Bulletin 1 (30) 2019

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Issue 1 (30) 2019

I. Vechtomov E. Ì. Binary relations and homomorphisms of Booleans

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The works deals with Binary relations between arbitrary sets A and B investigated in terms of corresponding complete V-homomorphisms from the Boolean B(A) to the Boolean B(B). The author proposes two duality theorems: for the category of all sets and binary relations between them considered as morphisms, and also for the category of all binary relations and their 2-morphisms.

Keywords: binary relation, Boolean, complete V-homomorphism, duality of categories.

References

  1. Kon P. Universalnaya algebra (Universal algebra), M.: Mir, 1968, 352 p.
  2. Arkhangelskiy A. V. Kantorovskaya teoriya mnozhestv (Cantor theory of sets), M.: Izd-vo MGU, 1988, 112 p.
  3. Birkgo G., Barti T. Sovremennaya prikladnaya algebra (Modern applied algebra), M.: Mir, 1976, 400 p.
  4. Vechtomov Ye. M. Binarnyye otnosheniya (Binary relations), Matematika v obrazovanii, 2007, v. 3, pp. 41-51.
  5. Vechtomov Ye. M. O binarnykh otnosheniyakh dlya matematikov i informatikov (On binary relations for mathematicians and computer scientists), Vestnik Vyatskogo gosudarstvennogo gumanitarnogo universiteta, 2012, 1 (3), pp. 51-58.
  6. Vechtomov Ye. M. Matematika: osnovnyye matematicheskiye struktury: uchebnoye posobiye dlya akademicheskogo bakalavriata (Mathematics: Basic Mathematical Structures: A Manual for Academic Baccalaureate), 2-ye izd, M.: Yurayt, 2018, 296 p.
  7. Kuk D., Beyz G. Kompyuternaya matematika (Computer Mathematics), M.: Nauka, 1990, 384 p.
  8. Maltsev A. I. Algebraicheskiye sistemy (Algebraic Systems), M.: Nauka, 1970, 392 p.
  9. Tsalenko M. SH. Modelirovaniye semantiki v bazakh dannykh (Simulation of semantics in databases), M.: Nauka, 1989, 288 p.
  10. Shreyder YU. A. Ravenstvo. Skhodstvo. Poryadok (Equality. Similarity. Order), M.: Nauka, 1971, 256 p.
  11. Grettser G. Obshchaya teoriya reshetok (The general theory of lattices), M.: Mir, 1982, 456 p.
  12. Plotkin B. I. Universalnaya algebra, algebraicheskaya logika i bazy dannykh (Universal algebra, algebraic logic and databases), M.: Nauka, 1991, 448 p.

For citation: Vechtomov E. Ì. Binary relations and homomorphisms of Booleans, Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2019, 1 (30), pp. 3-15.

II. Pimenov R. R. Lineup markup as an introduction to group theory

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The article reveals the relationship between the ruler markup, the group of permutations of the three elements, involutive transformations and linear fractional functions. Threefold symmetry is shown, without which the ruler marking would be impossible. Examples and tasks useful for teaching mathematics at school and university are given.

Keywords: lineup markup, symmetry, group theory, involution, education.

References

  1. Pimenov R. R. Troystvennaya simmetriya Fraktal’nogo kaleydoskopa (Triple symmetry of a fractal kaleidoscope), Mat. Pros., Ser. 3, 20, MCCME, Moscow, 2016, pp. 57-110.
  2. Pimenov R. R. K logicheskim i naglyadno-geometricheskim svojstvam orientacii 1 (About logic and visual-geometric properties of orientation 1), Matematichesky vestnik pedvuzov i yniversitetov Volgo- Viatskogo regiona: periodichesky mejvuzovsky sbornik nauchno-metodicheskyh rabot, Kirov: Naucn. izd-vo ViatGU, 2016, v. 18, pp. 99-114.
  3. Koganov L. Dvoynoye otnosheniye kak prostoye (Cross-ratio as ane ratio), Problems of theoretical cybernetics, abstracts of 14 inter. conferences, Penza May 23-28, M, ed. MSU, 2005, pp. 1-4.
  4. Pimenov R. R. Esteticheskaya geometriya ili teoriya simmetriy (Aesthetic geometry or theory of symmetries), SPb, School league, 2014, 288 p.
  5. Bachmann F. Postroyeniye geometrii na osnove ponyatiya simmetrii (Aufbau der Geometrie aus dem Spiegelungsbegriff), Moscow, Nauka, 1969, 380 p.

For citation: Pimenov R. R. Lineup markup as an introduction to group theory, Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2019, 1 (30), pp. 16-26.

III. Mingaleva A. E., Nekipelov S. V., Petrova O. V., Sivkov D. V., Sivkov V. N.   Apparatus distortion investigations in NEXAFS C1s-spectra on the example of fullerite C60

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The paper presents the results of the comparison of transmission and TEY methods in determining the absorption cross section in the NEXAFS C1s-spectra of the fullerite C60, as well as of «the thickness effect» modeling in the absorption cross section spectral dependences of the C60-films in the NEXAFS C1s-spectra. The calculations were performed with using absorption cross section spectra obtained in TEY mode as the true (undistorted) data. The modelling results are in good agreement with the experiment.

Keywords: absorption cross section, NEXAFS, fullerite, «thickness effect»,synchrotron radiation.

References

  1. St¨ohr J. NEXAFS Spectroscopy. Berlin: Springer Verlag, 1992. 403 p.
  2. Parratt L. G., Hempstead C. F., Jossem E. L. «Thickness Effect» in Absorption Spectra near Absorption Edges, Phys. Rev., 1957, V. 105, 1228 p.
  3. Sivkov V. N., Vinogradov A. S. Sila ostsillyatorov πg ― rezonansa formy v NK ― spektre pogloshcheniya molekuly azota (The oscillator strength of the πg― resonance form in the NK ― absorption spectrum of the nitrogen molecule), Opt. and spectrum, 2002, T. 93, 3, pp. 431-434.
  4. Sivkov V. N., Vinogradov A. S., Nekipelov S. V., Sivkov D. V.,Sluggish D. V., Molodtsov S. L. Sily ostsillyatorov dlya rezonansov formy v NK-spektre pogloshcheniya NaNO3, izmerennyye s ispol’zovaniyem sinkhrotronnogo izlucheniya (Oscillator strengths for form resonances in the NK absorption spectrum of NaNO3, measured usingsynchrotron radiation), Opt. and spectrum, 2006, T. 101, 5, pp. 782-788.
  5. Fedoseenko S. I., Vyalikh D. V., Iossifov I. E., Follath R.,Gorovikov S. A., P¨uttner R., Schmidt J.-S., Molodtsov S. L., Adamchuk V. K., Gudat W., Kaindl G. Commissioning results and performance of the high-resolution Russian-German Beamline at BESSY II, Nucl. Instr.and Meth. A., 2003, V. 505, pp. 718-728.
  6. Kummer K., Sivkov V. N., Vyalikh D. V., Maslyuk V. V., Bluher A., Nekipelov S. N., Bredow T., Mertig I., Molodtsov S. L. Oscillator strength of the peptide bond π∗ resonances at all relevant X-ray absorption edges, Phys. Rev., 2009, V. 80, pp. 155433-8 (2).
  7. Sivkov V. N., Obedkov A. M., Petrova O. V., Nekipelov S. V., Kremlin K. V., Kaverin B. S. , Semenov N. M., Gusev S. A. Rentgenovskiye i sinkhrotronnyye issledovaniya geterogennykh sistemna osnove mnogostennykh uglerodnykh nanotrubok (X-ray and synchrotron studies of heterogeneous systems based on multi-walled carbon nanotubes), Solid State Physics, 2015, 57, pp. 187-191.
  8. Gudat W., Kunz C. Close Similari between Photoelectric Yield and Photoabsorption Spectra in the Soft-X-Ray Range, Phys. Rev. Letters, 1972, V. 29, pp. 169-172.
  9. Petrova O. V. Raspredeleniye sil ostsillyatorov v ul’tramyagkikh rentgenovskikh spektrakh uglerodnykh nanostrukturirovannykh materialov i biopolimerov: dis. na soiskaniye uchonoy stepeni kand. fiz.- mat. nauk: 01.04.07 (Distribution of oscillator strengths in ultra-soft x-ray spectra of carbon nanostructured materials and biopolymers: dis. for the degree of Candidate Phys.-Mat. Sciences: 01.04.07), Mosk. state University, Moscow, 2018, 150 p.
  10. Maxwell A. J., Br¨uhwiler P. A., Arvanitis D., Hasselstr¨om J.,M¨artensson N. Carbon 1s near-edge-absorption fine structure in graphite, Chem. Phys. Lett., 1996, V. 260, pp. 71-77.
  11. Batson P. E. Carbon 1s near-edge-absorption fine structure in graphite, Phys. Rev., 1993, B. 48, pp. 2608-2610.

For citation: Mingaleva A. E., Nekipelov S. V., Petrova O. V., Sivkov D. V., Sivkov V. N. Apparatus distortion investigations in NEXAFS C1s-spectra on the example of fullerite C60, Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2019, 1 (30), pp. 27-39.

IV. Mikhailov A. V., Tarasov V. N. The stability of the reinforced arches under the boundary conditions of the hinged support

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The paper solves problem of stability of elastic systems in the presence of one-sided constraints on displacement. The stability problems of circular arches under uniform pressure were previously discussed in the works of E. L. Nikolai, A. N. Dinnik and other authors. This paper discusses the stability problems of circular arches, supported by inextensible threads thatdo not withstand the compressive forces under the boundary conditions of hinged support. Both ends of the thread are attached to the axis of the arch, so that the distance between the points of attachment as a result of the deformation cannot increase. This problem is reduced to finding and studying the bifurcation points of solutions of a certain nonlinear programming problem.

Keywords: arch, stability, support by threads, hinged edge, spline, variational problem, one-sided constraints.

References

  1. Nikolai E. L. Trudy po mekhanike (Works on mechanics), M.: Izd. tekhniko-tekhnicheskoy literatury, 1955, 584 p.
  2. Dinnik A. N. Ustoychivost’ arok (The stability of the arches), M.: Gostekhizdat, 1946, 128 p.
  3. Zav’yalov Y. S., Kvasov B. I., Miroshnichenko V. L. Metody splayn-funktsiy (Methods of spline functions), M.: Nauka. Glavnaya redaktsiya fiziko-matematicheskoy literatury, 1980, pp. 96-101.
  4. Tarasov V. N. Metody optimisatsii konstruktivno-nelineinnykh zadach mekaniki uprugikh system (Optimization methods in the study of structurally non-linear problems of the mechanics of elastic systems), Syktyvkar, 2013, 238 p.
  5. Sukharev A. G. Global’nyy ekstremum i metody ego otyskaniya (Global extremum and methods for finding it), Matematicheskiye metody v issledovanii operatsiy, M.: Izd. MGU, 1983, 193 с.
  6. Tarasov V. N. Ob ustoychivosti uprugikh sistem pri odnostoronnikh ogranicheniyakh na peremeshcheniya (On the stability of elastic systems with one-sided constraints on displacements), Trudy instituta matematiki i mekhaniki. Rossiyskaya akademya nauk. Ural’skoye otdeleniye, Tom 11, № 1, 2005, pp. 177-188.
  7. Feodos’yev 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: Mikhailov A. V., Tarasov V. N. The stability of the reinforced arches under the boundary conditions of the hinged support, Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2019, 1 (30), pp. 40-52.

V. Cheredov V. N. Percolation-nanoclusters model of the crystallization front

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A new nanocluster model of a first-order liquid-solid phase transition is proposed based on the model of oscillating bonds and the percolation lattice of bonds and assemblies. The nanocluster structure at the water crystallization front, the conditions of its formation, and its relation to the percolation threshold of the liquid structure are studied. The relationship between the parameters of nanoclusters and the ratio of the thermodynamic and percolation characteristics of the structure of intermolecular fluid bonds has been revealed. Within the framework of the constructed model, the dynamics of the water structure and its phase transitions is studied. Quantitative characteristics of liquid phase nanoclusters at the water crystallization front are studied.

Keywords: intermolecular bonds, phase transitions, nanoclusters, percolation threshold, model of oscillating bonds.

References

  1. Cheredov V. N., Kuratova L. A. Dinamika setki mezhmolekulyarnyh svjazej i fazovyje perehody v kondensirovannyh sredah (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. Kaplan I. G. Mezhmolekuljarnye vzaimodejstvija. Fizicheskaja interpretacija, komp’juternye raschjoty i model’nye potencialy (Intermolecular interactions. Physical interpretation, computer calculations and model potentials), M.: BINOM, Laboratorija znanij, 2012, 400 p.
  3. Cheredov V. N. Statika i dinamika defektov v sinteticheskih kristallah fljuorita (Statics and dynamics of defects in synthetic fluorite crystals), SPb: Nauka, 1993, 112 p.
  4. Landau L. D., Lifshic E. M. Statisticheskaja fizika. Ch.1 (Statistical physics. Part 1), M.: Fizmatlit, 2010, 616 p.
  5. Enohovich A. S. Spravochnik po fizike i tehnike (Reference book on physics and techniques), M.: Prosveshhenie, 1989, 224 p.
  6. Zacepina G. N. Fizicheskie svojstva i struktura vody (Physical properties and structure of water), M.: MGU, 1998, 184 p.
  7. Jejzenberg D., Kaucman V. Struktura i svojstva vody (Structure and properties of water), M.: Direkt-media, 2012, 284 p.
  8. Dorsey N. E. Properties of ordinary Watter-Suvstance, New York: Reinhold Publishing Corporation, 1940, 673 p.
  9. Giauque W. F., Stout J. W. The entropy of water and the third law of thermodynamics. The heat capacity of ice from 15 to 273◦K, Journal of the American Chemical Society, 1936, V. 58, pp. 1144-1150.
  10. McDougall D. P., Giauque W. F. The production of temperatures below 1◦ A. The heat capacities of water, gadolinium nitrobenzene sulfonate heptahydrate and gadolinium anthraquinone sulfonate, Journal of the American Chemical Society, 1936, V. 58, pp. 1032-1037.
  11. Kirrilin V. A., Sychev V. V., Shendlin A. E. Tekhnicheskaya termodinamika (Technical thermodynamics), Ìoscow: Izdatelstvo MEI, 2008, 486 p.
  12. Tarasevich Ju. Ju. Perkoljacija: teorija, prilozhenija, algoritmy (Percolation: theory, applications, algorithms), M.: Librokom, 2012. 116 p.
  13. Jefros A. L. Fizika i geometrija besporjadka (Physics and geometry of disorder), M.: Nauka, 1982, 176 p.
  14. Stanley H. E. A polychromatic correlated-site percolation problem with possible relevance to the unusual behaviour of supercooled H2O and D2O (A polychromatic correlated-site percolation problem with possible relevance to the unusual behaviour of supercooled H2O and D2O), Journal of Physics A: Mathematical and General, 1979, V. 12, 12, pp. L329-L337.
  15. Stanley H. E., Teixeira J. J. Interpretation of the unusual behavior of H2O and D2O at low temperatures: Tests of a percolation model (Interpretation of the unusual behavior of H2O and D2O at low temperatures: Tests of a percolation model), The Journal of Chemical Physics, 1980, V. 73, 7, pp. 3404-3422.
  16. Stanley H. E., Teixeira J., Geiger A., Blumberg R. L. Interpretation of the unusual behavior of H2O and D2O at low temperature: Are concepts of percolation relevant to the «puzzle of liquid wate»? (Interpretation of the unusual behavior of H2O and D2O at low temperature: Are concepts of percolation relevant to the«puzzle of liquid water»?), Physica A: Statistical Mechanics and its Applications, 1981, V. 106, 1-2, pp. 260-277.

For citation: Cheredov V. N. Percolation-nanoclusters model of the crystallization front, Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2019, 1 (30), pp. 53-66.

VI. Belykh E. A. Car number plate segmentation based on averaged models

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This paper is devoted to the problem of dividing an image with a car number into images of individual characters, as well as recognizing these characters. The paper proposes a method for solving this problem by constructing an averaged image.

Keywords: symbol recognition, car plate, computer vision, image segmentation.

References

  1. Malygin E. S. Ustoychivaya k shumam segmentatsiya avtomobil’nykh nomerov v nizkom razreshenii: bakalavrskaya rabota (Low-noise, noise-free segmentation: bachelor’s work), St. Petersburg State the university. St. Petersburg, 2015, 26 p.
  2. Bolotova Yu. A., Spitsyn V. G., Rudometkin M. N. Raspoznavaniye avtomobil’nykh nomerov na osnove metoda svyaznykh komponent i iyerarkhicheskoy vremennoy seti (Recognition of license plates based on the method of connected components and hierarchical temporary network), Computer Optics, 2015, T. 39, 2, pp. 275-280.
  3. Serikov A. S. Segmentatsiya i raspoznavaniye avtomobil’nykh registratsionnykh nomerov (Segmentation and recognition of car registration numbers), Youth and modern information technologies: collectionof works XIV International Scientific and Practical Conference of Students, Postgraduates and young scientists, 2016, Tomsk: TPU publishing house, 2016, T. 2, pp. 219-220.
  4. Viola P., Jones M. Rapid Object Detection using a Boosted Cascade of Simple Features, 2013 IEEE Conference on Computer Vision and Pattern Recognition, 2001, Vol. 1, pp. 511-518.
  5. Belykh E. A. Optimizatsiya algoritmov raspoznavaniya avtomobil’nykh nomerov dlya raboty s videopotokom: vypusknaya kvalifikatsionnaya rabota (Optimization of license plate recognition algorithms for work with video stream: final qualifying work), Syktyvkar Pitirim Sorokin State University, Syktyvkar, 2017, 64 p.

For citation: Belykh E. A. Car number plate segmentation based on averaged models, Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2019, 1 (30), pp. 67-76.

VII. Odyniec W. P. About Physicists Who Came to the URSS in the 1930s

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The article presents a slice of the development of physical science in the USSR in the 30s of twentieth century against the background of history of interaction with foreign physicists who came to the country.

Keywords: quantum and nuclear physics, low temperature physics, relativity theory, astrophysics, rigid body theory, statistical nuclear theory, A. Ioffe, V. Weisskopf, A. Weissberg, K. Weiselberg, F. G. Houtermans, M. Ruhemann, L. Tisza, G. Placzek, F. Lange, V. S. Spinel, V. A. Maslov, P. A. M. Dirac, L. Landau, A. Leipunsky, I. Obreimov, L. V. Schubnikow, V. Fomin, N. Rozen, B. Podolsky, V. A. Fock, I. Kurchatov.

References

  1. Ioffe A. F. Vstrechi s fizikami. Moi vospominaniya o zarubezhnykh fizikakh (Encounters with Physicists. Remeniscences of Foreign Physiciasts), Leningrad: Nauka, 1983, 262 p.
  2. Odyniec W. P. Immigratsiya v SSSR v dovoyennyy period: Profili matematikov (Immigration to the USSR in the pre-war period: Profiles of the mathematicians) / W.P. Odyniec, Syktyvkar: Syktyvkar State University named after Pitirim Sorokin, 2019, 124 p.
  3. Tolok V. T., Kozak V. S., Vlasov V. V. Fizika i Khar’kov (Physics and Kharkov), V.T. Tolok, Kharkov: Timchenko, 2009, 408 p.
  4. Fraenkel V. Ya. Professor Fridrikh Khoutermans: Raboty, zhizn’, sud’ba (Professor Friedrich Houtermans: Work, Life, Fate), St. Petersburg: PIYaPh RAN Press, 1997, 200 p.
  5. Khramov Yu. A. Fiziki: Biograficheskiy spravochnik (Biographical Handbook), eds. A.I. Akhiezer; 2nd ed.,enlarged and corrected, Moscow: Nauka, 1983, 400 p.
  6. Ranyuk Yu. Laboratoriya 1. Yaderna fizika v Ukraini (Laboratory No 1. Nuclear Physics in Ukraine), Yu. Ranyuk.-Kharkov: Añta, 2006, 590 p.
  7. Oleynikov P. V. German Scientists in the Soviet Atomic Project, The Nonproliferation Review/ Summer 2000, No 2, pp. 1-30.
  8. Khroniki. Uspekhi fizicheskikh nauk (Chronicles. Uspekhi Fizicheskich Nauk), Vol. XIV, 1934, pp. 516-520.
  9. Walther A. The second Union Conference on the atomic nucleus, Moscow: Physikalische Zeitschrift der Sowjetunion, Vol. 12, No. 5, 1937, pp. 610-622.

For citation: Odyniec W. P. About Physicists Who Came to the URSS in the 1930s, Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2019, 1 (30), pp. 77-91 .

VIII. Sotnikova O. A. Teaching logical and mathematical analysis based on higher algebra material to future math teachers

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The ability to perform logical and mathematical analysis of math instruction material is considered to be one of the key methodological competences for a math teacher. Traditionally, the issue has been dealt with in the course of methods-of-teaching subjects. However, the author follows the principle of vocational and pedagogic focus of education and substantiates the reasonability and feasibility of dealing with the issue when studying higher algebra. The article provides the list of activities in order to perform logical and mathematical analysis of algebra instruction material.

Keywords: Math teacher training at university, logical and mathematical analysis, methodological skills.

References

  1. Lyachenko E. I., Zobkova K. V., Kirichenko T. F. I dr. Labaratornye i prakticheskie raboty po metodike prepodavania matematike: Uchebnoe posobie dla studentov fiz.-mat. spez. ped. Insnitutov(Laboratory and practical work on the methods of teaching mathematics: A manual for students of Phys.-Mat. specialist. ped. institutions), Pod ped. E.I. Lyachenko, Ì.: Prosvyachenie, 1988, 223 p.
  2. Mordkovich A. G. O professionalno-pedagogicheskoy napravlennosti matematicheskoy podgotovki budushix uchiteley (On the professional and pedagogical orientation of the mathematical preparation of future teachers), Matematika v shkole, 1984, ― 6, pp. 42-45.
  3. Gorskiy D. P. Opredelenie (Definition), Ì.: Ìysl, 1974, 310 p.
  4. Kondakov N. I. Logicheskiy slovar (Logical dictionary), Ì.: Nauka, 1971, 637 p.
  5. Boltyanskiy V.G. Kak ustroena teorema? (How does the theorem work?), Matematika v shkole, 1973, ― 1, pp. 41-50.
  6. Gradshteyn I. S. Pryamaya i obratnaya teoremy (Direct and inverse theorems), Ì.: Nauka, 1973, 128 p.

For citation: Sotnikova O. A. Teaching logical and mathematical analysis based on higher algebra material to future math teachers, Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2019, 1 (30), pp. 92-112.

IX. Odyniec W. P. About the problems of mathematical training of physists

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Teaching physics relies heavily on the mathematical apparatus. Unfortunately, curricula in physics and mathematics are not always consistent. Therefore, in the process of giving lectures in physics, sections of mathematics that have not yet been studied have to either be offered to students to study on their own, or set forth directly in lectures in physics . The first option is actual for elite universities only, while the another is fraught with loss of generality in such disciplines as, for example, quantum logic. The development of new elective courses in physics (for example, in the framework of the magistracy) may require new supplementary courses in mathematics. It is noted that often due to the lack of required mathematics teachers and the reduction of study hours in physics and mathematics, it is not so easy to ensure the learning process. In our opinion,the following mathematical courses in signal processing theory, data compression, latticeanalysis, would be helpful such as: 1) wavelet analysis initiated by S. Mallat, (U.S.A.) and Y. Meyer (France) [3; 4]; 2) the theory of summation of divergent series [1]; 3) theory of fractal [2]. (The subject of the article was discussed at the round table of 15th International Conference «Physics in the system of modern education» (PSME-19) (3-6 June 2019, St. Petersburg)).

Keywords: quantum logic,wavelet analysis, summation of divergent series, fractal theory.

References

  1. Novikov I. Ya., Protasov V. Yu., Skopina M. A. Teoriya vspleskov (The wavelet theory), Moscow: Fizsmatlit, 2006, 616 p.
  2. Odyniec W. P. Ob istorii nekotorykh matematicheskikh metodov, ispol’zuyemykh pri prinyatii upravlencheskikh resheniy: uchebnoye posobiye (On the history of some mathematical methods used in the makingof managerial decisions), Syktyvkar: Pitirim Sorokin University Press, 2015, 108 p.
  3. Cook R. G. Beskonechnyye matritsy i prostranstva posledovatel’nostey (Infinite matrices and sequence spaces), London: MacMillan and Co.,1950, 360 p.
  4. Mandelbrot B. Fraktal’naya geometriya prirody (Fractals: form, chance and dimension), San Francisco: W.H. Freeman and Co., 1977, 365 p.

For citation: Odyniec W. P. About the problems of mathematical training of physists, Bulletin of Syktyvkar University. Series 1: Mathematics. Mechanics. Informatics, 2019, 1 (30), pp. 113-115.

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