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Ukrainian Mathematical Journal

, Volume 46, Issue 4, pp 331–342 | Cite as

On D. Ya. Petrina's works in contemporary mathematical physics

  • V. I. Gerasimenko
  • P. V. Malyshev
  • A. L. Rebenko
Article
  • 28 Downloads

Abstract

This is a brief survey of the works of Prof. D. Ya. Petrina in various branches of contemporary mathematical physics.

Keywords

Mathematical Physic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    D. Ya. Petrina, O. S. Parasyuk, and P. N. Tatsunyak, “Chellen-Lehmann theorem for quantum field theory in a space with indefinite metric,”Ukr. Mat. Zh.,10, No. 3, 344–346 (1958).Google Scholar
  2. 2.
    D. Ya. Petrina, “Dispersion relations for inelastic scattering in nonrelativistic approximation,”Ukr. Mat. Zh.,11, No. 3, 267–274 (1959).Google Scholar
  3. 3.
    D. Ya. Petrina, “A solution of the inverse diffraction problem,”Ukr. Mat. Zh.,12, No. 4, 476–479 (1960).Google Scholar
  4. 4.
    D. Ya. Petrina, “On the impossibility of a nonlocal field theory with a positive spectrum of the energy-momentum operator,”Ukr. Mat. Zh.,13, No. 4, 109–111 (1961).Google Scholar
  5. 5.
    D. Ya. Petrina, “Analytic properties of partial waves of the scattering amplitude in perturbation theory,”Dokl. Akad Nauk SSSR,144, No. 4, 755–758 (1962).Google Scholar
  6. 6.
    D. Ya. Petrina, “Analytic properties of the scattering amplitude for a potential on the first “nonphysical” sheet,”Zh. Éksp. Teor. Fiz.,44, Issue 1, 151–156 (1963).Google Scholar
  7. 7.
    D. Ya. Petrina, “Analytic properties of the contributions of Feynman diagrams,”Dokl. Akad Nauk SSSR,149, No. 4, 808–810 (1963).Google Scholar
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    D. Ya. Petrina, “Complex singular points of the contributions of Feynman diagrams and the continuity theorem,”Ukr. Mat. Zh.,16, No. 1, 31–40 (1964).Google Scholar
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    D. Ya. Petrina, “On the principle of maximal analyticity with respect to the complex orbital momentum,”Ukr. Mat. Zh.,16, No. 4, 502–512 (1964).Google Scholar
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    D. Ya. Petrina, “Mandelstam representation and the continuity theorem,”Zh. Éksp. Teor. Fiz.,46, Issue 2, 544–554 (1964).Google Scholar
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    D. Ya. Petrina, “A proof of the Mandelstam representation for a ladder diagram of the sixth order,”Zh. Éksp. Teor. Fiz.,47, Issue 8, 524–529 (1964).Google Scholar
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    D. Ya. Petrina, “Analytic properties of a class of functions defined by integrals over a manifold. I,”Ukr. Mat. Zh.,17, No. 5, 54–66 (1965).Google Scholar
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    D. Ya. Petrina, “Analytic properties of a class of functions defined by integrals over a manifold. II,”Ukr. Mat. Zh.,17, No. 6, 60–66 (1965).Google Scholar
  14. 14.
    D. Ya. Petrina, “On a holomorphic extension of the contributions of Feynman diagrams,”Dokl. Akad Nauk SSSR,168, No. 2, 308–309 (1966).Google Scholar
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    D. Ya. Petrina, “On the completeness of amplitudes of perturbation theory in the space of amplitudes,”Ukr. Mat. Zh.,19, No. 3, 62–78 (1967).Google Scholar
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    D. Ya. Petrina, “On the completeness of amplitudes of perturbation theory in the space of amplitudes,”Dokl. Akad Nauk SSSR,173, No. 2, 295–297 (1967).Google Scholar
  17. 17.
    D. Ya. Petrina, “On the summation of contributions of Feynman diagrams: the existence theorem,”Izv. Akad Nauk SSSR, Ser. Mat.,32, No. 5, 1052–1074 (1968).Google Scholar
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    D. Ya. Petrina,Mathematical Problems in the Theory of Scattering Matrices, Doctor's Degree Thesis (Physics and Mathematics), Kiev (1968).Google Scholar
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  20. 20.
    D. Ya. Petrina and S. S. Ivanov, “On equations appearing as a result of summation of perturbation series for scattering matrices,”Dokl. Akad Nauk SSSR,188, No. 4, 776–779 (1969).Google Scholar
  21. 21.
    D. Ya. Petrina, N. N. Bogolyubov, and B. I. Khatset, “Mathematical description of equilibrium states of classical systems on the basis of the formalism of canonical ensemble,”Teor. Mat. Fit.,1, No. 2, 251–274 (1969).Google Scholar
  22. 22.
    D. Ya. Petrina, “On Hamiltonians of quantum statistics and on the model Hamiltonian in the theory of superconductivity,”Teor. Mat. Fiz.,4, No. 3, 394–411 (1970).Google Scholar
  23. 23.
    D. Ya. Petrina and V. I. Skrypnyk, “Kirkwood-Salsburg equations for coefficient functions of the scattering matrix,”Teor. Mat. Fiz.,8, No. 3, 369–380 (1971).Google Scholar
  24. 24.
    D. Ya. Petrina and V. P. Yatsyshyn, “On the model Hamiltonian in the theory of superconductivity,”Teor. Mat. Fiz.,10, No. 2, 283–299 (1972).Google Scholar
  25. 25.
    D. Ya. Petrina, “On solutions of the Bogolyubov kinetic equations. Quantum statistics,”Teor. Mat. Fiz.,13, No. 3, 391–405 (1972).Google Scholar
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    D. Ya. Petrina and A. K. Vidybida, “Cauchy problem for the Bogolyubov kinetic equations,”Tr. Mat. Inst. Akad. Nauk SSSR,136, 370–378 (1975).Google Scholar
  27. 27.
    D. Ya. Petrina, S. S. Ivanov, and A. L. Rebenko, “On equations for the coefficient functions of theS-matrix in quantum field theory,”Teor. Mat. Fiz.,19, No. 1, 37–46 (1974).Google Scholar
  28. 28.
    D. Ya. Petrina, S. S. Ivanov, and A. L. Rebenko, “S-matrix in constructive field theory,”Teor. Mat. Fiz.,23, No. 2, 160–177 (1975).Google Scholar
  29. 29.
    D. Ya. Petrina, S. S. Ivanov, and A. L. Rebenko, “S-matrix in constructive field theory,”Fiz. Element. Chastic Atom. Yadra,7, Issue 3, 647–686 (1976).Google Scholar
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    D. Ya. Petrina and A. K. Vidybida, “Cauchy problem for the Bogolyubov equations,”Dokl. Akad Nauk SSSR,228, No. 3, 573–575 (1976).Google Scholar
  31. 31.
    D. Ya. Petrina and V. Z. Énol'skii,On Oscillations of One-Dimensional Systems [in Russian], Preprint, Institute of Theoretical Physics, Ukrainian Academy of Sciences, Kiev (1976).Google Scholar
  32. 32.
    D. Ya. Petrina and V. Z. Énol'skii, “On oscillations of one-dimensional systems,”Dokl. Akad Nauk Ukr. SSR, Ser. A., No. 8, 756–760 (1976).Google Scholar
  33. 33.
    D. Ya. Petrina and N. N. Bogolyubov (Jr.), “On a class of model systems that admit a reduction of the degree of the Hamiltonian in the thermodynamic limit. I,”Teor. Mat. Fiz.,33, No. 2, 231–245 (1977).Google Scholar
  34. 34.
    D. Ya. Petrina and N. N. Bogolyubov (Jr.), “On a class of model systems that admit a reduction of the degree of the Hamiltonian in the thermodynamic limit. II,”Teor. Mat. Fiz.,37, No. 2, 246–257 (1978).Google Scholar
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    D. Ya. Petrina, S. S. Ivanov, and A. L. Rebenko,Equations for the Coefficient Functions of the Scattering Matrix [in Russian], Nauka, Moscow (1979).Google Scholar
  36. 36.
    D. Ya. Petrina, “Mathematical description of the evolution of infinite systems of classical statistical physics. Locally perturbed one-dimensional systems,”Teor. Mat. Fiz.,38, No. 2, 230–250 (1979).Google Scholar
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    D. Ya. Petrina and V. I. Gerasimenko, “Statistical mechanics of quantum-classical systems. Nonequilibrium systems,”Teor. Mat. Fiz.,42, No. 1, 88–100 (1980).Google Scholar
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    D. Ya. Petrina and A. L. Rebenko, “Projection-iterative method for the solution of equations of quantum field theory and its connection with renormalization theory. Equations of quantum field theory and ill-posed problems of mathematical physics,”Teor. Mat. Fiz.,42, No. 2, 167–183 (1980).Google Scholar
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    D. Ya. Petrina, V. I. Gerasimenko, A. I. Pilyavskii, and P. V. Malyshev, “On the process of inverse osmosis as a boundary-value problem in domains with complicated structure,”Dokl Akad Nauk Ukr. SSR, Ser. A., No. 9, 75–78 (1980).Google Scholar
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    D. Ya. Petrina and A. I. Pilyavskii,Potential of the Electrostatic Field of Charged Particles and a Dynamical Membrane [in Russian], Preprint 80-141P, Institute of Theoretical Physics, Ukrainian Academy of Sciences, Kiev (1980).Google Scholar
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    D. Ya. Petrina and A. I. Pilyavskii, “Potential of the electrostatic field of a system of charged particles and a dynamical membrane,”Dokl. Akad Nauk Ukr. SSR, Ser. A., No. 7, 57–60 (1981).Google Scholar
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    D. Ya. Petrina, “Solution of a classical problem of electrostatics and the subtraction procedure,”Dokl. Akad Nauk SSSR,270, No. 1, 78–81 (1983).Google Scholar
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    D. Ya. Petrina and V. I. Gerasimenko, “Mathematical description of the evolution of states of infinite systems of classical statistical mechanics,”Usp. Mat. Nauk.,38, No. 5, 5–61 (1983).Google Scholar
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    D. Ya. Petrina, “Solution of a classical problem of electrostatics by using the subtraction procedure,”Zh. Vych. Mat. Mat. Fiz.,24, No. 5, 709–721 (1984).Google Scholar
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    D. Ya. Petrina,Quantum Field Theory [in Russian], Vyshcha Shkola, Kiev (1984).Google Scholar
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    D. Ya. Petrina and E. D. Belokolos, “On the relation between the methods of approximating Hamiltonian and finite-zone integration,”Dokl. Akad Nauk SSSR,275, No. 3, 580–582 (1984).Google Scholar
  47. 47.
    D. Ya. Petrina and E. D. Belokolos, “On the relation between the methods of approximating Hamiltonian and finite-zone integration,”Teor. Mat. Fiz.,58, No. 1, 61–71 (1984).Google Scholar
  48. 48.
    D. Ya. Petrina, “Distribution functions of systems of charged particles in spatially-inhomogeneous media,”Teor. Mat. Fiz.,60, No. 1, 104–116 (1985).Google Scholar
  49. 49.
    D. Ya. Petrina and V. I. Gerasimenko, “Evolution of states of infinite systems of classical statistical mechanics,”Sov. Sci. Rev., Ser. C,5, 1–51 (1985).Google Scholar
  50. 50.
    D. Ya. Petrina and A. I. Pilyavskii, “Problems of electrostatics in spatially-inhomogenecus media and the subtraction procedure,”Fiz. Mnogochastich. Sist., Issue7, 82–96 (1985).Google Scholar
  51. 51.
    D. Ya. Petrina and V. I. Gerasimenko, “Thermodynamic limit for nonequilibrium states of a three-dimensional system of hard spheres,”Teor. Mat. Fiz.,64, No. 1, 130–149 (1985).Google Scholar
  52. 52.
    D. Ya. Petrina and V. I. Gerasimenko, “Thermodynamic limit for nonequilibrium states of a three-dimensional system of hard spheres,”Dokl. Akad Nauk SSSR,282, No. 1, 130–136 (1985).Google Scholar
  53. 53.
    D. Ya. Petrina, V. I. Gerasimenko, and P. V. Malyshev,Mathematical Foundations of Classical Statistical Mechanics [in Russian], Naukova Dumka, Kiev (1985).Google Scholar
  54. 54.
    D. Ya. Petrina, Yu. A. Mitropolskii, and V. G. Baryakhtar, “Creative contribution of Academician N. N. Bogolyubov to the development of mathematics, nonlinear mechanics, and theoretical physics,”Vestn. Akad. Nauk Ukr. SSR, No. 11, 9–21 (1985).Google Scholar
  55. 55.
    D. Ya. Petrina and V. I. Gerasimenko, “Thermodynamic and Boltzmann-Grad limits for nonequilibrium states of a system of hard spheres,” in:Problems of Modern Statistical Physics [in Russian], Naukova Dumka, Kiev (1985), pp. 228–337.Google Scholar
  56. 56.
    D. Ya. Petrina, V. I. Gerasimenko, and P. V. Malyshev, “Thermodynamic limit for solutions of the Bogolyubov equations,”Sov. Sci. Rev., Ser. C,7, 280–336 (1987).Google Scholar
  57. 57.
    D. Ya. Petrina, “Mathematical problems of the description of the evolution of states of infinite systems of statistical mechanics,” in:Proceedings of the Internat. Workshop “Selected Problems of Statistical Physics,” Vol. 2, World Scientific, Singapore (1987), p. 332.Google Scholar
  58. 58.
    D. Ya. Petrina and V. I. Gerasimenko, “Boltzmann-Grad limit for the states of an infinite system of hard spheres,”Dokl. Akad Nauk SSSR,297, No. 2, 336–340 (1987).Google Scholar
  59. 59.
    D. Ya. Petrina, A. N. Bogolyubov, and T. M. Urbanskii,Nikolai Mitrofanovich Krylov [in Russian], Naukova Dumka, Kiev (1987), pp. 157–169.Google Scholar
  60. 60.
    D. Ya. Petrina and P. V. Malyshev, “Thermodynamic limit for nonequilibrium distributions functions of three-dimensional classical systems of interacting particles,”Dokl. Akad Nauk SSSR,301, No. 3, 585–589 (1988).Google Scholar
  61. 61.
    D. Ya. Petrina and A. V. Mishchenko, “Exact solutions a class of Boltzmann equations,”Dokl. Akad Nauk SSSR,298, No. 2, 338–342 (1988).Google Scholar
  62. 62.
    D. Ya. Petrina and A. V. Mishchenko, “Linearization and exact solutions a class of Boltzmann equations,”Teor. Mat. Fiz.,77, No. 1, 135–153 (1988).Google Scholar
  63. 63.
    D. Ya. Petrina and V. I. Gerasimenko, “Boltzmann-Grad limit for equilibrium states,”Dokl. Akad Nauk Ukr. SSR, Ser. A., No. 12, 17–19 (1988).Google Scholar
  64. 64.
    D. Ya. Petrina and V. I. Gerasimenko, “On the limiting Boltzmann-Grad theorem,”Dokl. Akad Nauk Ukr. SSR, Ser. A., No. 11, 12–16 (1989).Google Scholar
  65. 65.
    D. Ya. Petrina, V. I. Gerasimenko, and P. V. Malyshev,Mathematical Foundations of Classical Statistical Mechanics, Gordon and Breach, London (1989).Google Scholar
  66. 66.
    D. Ya. Petrina and V. I. Gerasimenko, “Existence of the Boltzmann-Grad limit for an infinite system of hard spheres,”Teor. Mat. Fiz.,83, No. 3, 92–114 (1990).Google Scholar
  67. 67.
    D. Ya. Petrina and V. I. Gerasimenko, “Mathematical problems of a statistical system of hard spheres,”Usp. Mat. Nauk,45, No. 3, 135–182 (1990).Google Scholar
  68. 68.
    D. Ya. Petrina,Exactly Solvable Models of Quantum Statistical Mechanics, Preprint No. 18, Dipartimento di Matematica Politecnico di Torino, Torino (1992).Google Scholar
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    D. Ya. Petrina,Mathematical Foundations of Quantum Statistical Mechanics, Kluwer, Dordrecht (1995).Google Scholar
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    D. Ya. Petrina,Mathematical Foundations of Quantum Statistical Mechanics [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1995).Google Scholar
  71. 71.
    D. Ya. Petrina, V. I. Gerasimenko, and V. Z. Enol'skii, “Equations of motion for a class of quantum-classical systems,”Dokl. Akad Nauk SSSR,315, No. 1, 75–80 (1990).Google Scholar

Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • V. I. Gerasimenko
    • 1
  • P. V. Malyshev
    • 1
  • A. L. Rebenko
    • 1
  1. 1.Institute of MathematicsUkrainian Academy of SciencesKiev

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