Publications by Dmitri Vassiliev

Monograph

The asymptotic distribution of eigenvalues of partial differential operators, American Mathematical Society, 1997 (hardcover), 1998 (softcover), series Translations of Mathematical Monographs, vol. 155. Jointly with Yu.Safarov. The book can be ordered from the AMS Bookstore. Here is the list of typos.

Papers

1975

1. On the zeros of determinants of Wronski type, Transactions of the 20 Scientific Conference (1974) of the Moscow Institute of Physics and Technology, series Aerophysics and applied mathematics, part 2, p. 116-121 (in Russian).

1976

2. On the dependence of eigenvalues on the length of the segment under the clamping condition at one of the ends, Transactions of the 21 Scientific Conference (1975) of the Moscow Institute of Physics and Technology, series Aerophysics and applied mathematics, p. 170-174 (in Russian).

1977

3. On the zeros of the bending deflection function in the theory of free vibrations of shells of revolution, Soviet Mathematics Doklady 18, p. 1470-1473.

1978

4. On the eigenfunctions of a thin elastic shell of revolution under the condition of internal resonance, Transactions of the 23 Scientific Conference (1977) of the Moscow Institute of Physics and Technology, series Aerophysics and applied mathematics, p. 133-136 (in Russian).

1979

5. A formula for the frequency distribution function of a shell of revolution immersed in a liquid, Soviet Physics Doklady 24, p. 720-722.

1980

6. Vibrations of thin elastic shells interacting with a liquid, Uspekhi Mat. Nauk 35, no. 2, p. 255-256 (in Russian). Jointly with A.G.Aslanyan and V.B.Lidskii.

7. Frequencies of free vibrations of a shell interacting with a liquid, Transactions of the 12 USSR Conference on the Theory of Shells and Plates (Yerevan, 1980), vol. 2, p. 86-92; Yerevan, publishing house of the Yerevan State University (in Russian). Jointly with A.G.Aslanyan and V.B.Lidskii.

8. Asymptotics of the distribution function of the spectrum of pseudodifferential operators with parameters, Functional Analysis and its Applications 14, p. 217-219.

1981

9. The distribution of eigenfrequencies of a thin elastic shell interacting with a liquid, Uspekhi Mat. Nauk 36, no. 4, p. 216 (in Russian). Jointly with A.G.Aslanyan and V.B.Lidskii.

10. Frequencies of free vibrations of a thin shell interacting with a liquid, Functional Analysis and its Applications 15, p. 157-164. Jointly with A.G.Aslanyan and V.B.Lidskii.

11. Asymptotic estimates of complex vibration frequencies of a shell in a liquid, preprint no. 186 of the Institute for Problems in Mechanics of the USSR Academy of Sciences, 68 p. (in Russian). Jointly with I.V.Simonov.

1982

12. Free vibrations of a shell interacting with a liquid, Transactions of the 2 USSR Symposium on the Physics of Acoustohydrodynamic Phenomena and Optoacoustics (Suzdal, 1979), p. 264-266; Moscow, Nauka publishing house (in Russian). Jointly with A.G.Aslanyan and V.B.Lidskii.

13. Vibrations of thin elastic shells contacting liquid, Uspekhi Mat. Nauk 37, no. 5, p. 224-225 (in Russian). Jointly with V.B.Lidskii.

1983

14.Vibrations and radiation of a shell of revolution acted upon by an annular load, Mechanics of Solids 18, no. 4, p. 183-191. Jointly with A.L.Gol'denveizer.

15. Binomial asymptotics of the spectrum of a boundary value problem, Functional Analysis and its Applications 17, p. 309-311.

16. Asymptotic analysis of vibrations of stiffened shells of revolution contacting liquid, Transactions of the 13 USSR Conference on the Theory of Plates and Shells (Tallinn, 1983), vol. 1, p. 169-174; Tallinn, publishing house of the Tallinn Polytechnic Institute (in Russian).

1984

17. Two-term asymptotics of the spectrum of a boundary value problem under an interior reflection of general type, Uspekhi Mat. Nauk 39, no. 4, p. 131 (in Russian).

18. Two-term asymptotics of the spectrum of a boundary value problem under an interior reflection of general form, Functional Analysis and its Applications 18, p. 267-277.

19. A refinement of the Courant asymptotic formula for the frequency distribution function of a plate, Transactions of the USSR Seminar on Theoretical and Numerical Methods for Plates and Shells (Tbilisi, 1984), vol. 2, p. 90-96; Tbilisi, publishing house of the Tbilisi State University (in Russian).

20. Asymptotics of the spectrum of pseudodifferential operators with small parameters, Math. USSR Sbornik 49, no. 1, p. 61-72.

1985

21. Two-term asymptotics of the spectrum of a boundary value problem with a piecewise smooth boundary, Uspekhi Mat. Nauk 40, no. 5, p. 233 (in Russian).

1986

22. Forced vibrations of a thin elastic shell interacting with a liquid, in Mathematical Methods of the Mechanics of Solids (series Strength and viscoelastoplasticity), p. 30-36; Moscow, Nauka publishing house (in Russian). Jointly with V.B.Lidskii.

23. Asymptotics of the spectrum of a differential operator on a surface of revolution, Izvestia of the Azerbaijani Academy of Sciences, series Physical-technical and mathematical sciences, 7, no. 6, p. 23-28 (in Russian). Jointly with L.G.Khalilova.

24. On the spectrum of the shell theory operator and the problem Au=Lambda Bu , in 11 USSR School on Operator Theory in Functional Spaces (Chelyabinsk, 1986), part 2, p. 22; Chelyabinsk, publishing house of the Chelyabinsk Polytechnic Institute (in Russian).

25. Two-term asymptotics of the spectrum of a boundary value problem in the case of a piecewise smooth boundary, Soviet Mathematics Doklady 33, no. 1, p. 227-230.

26. Forced vibrations of a thin elastic shell interacting with a liquid, Uspekhi Mat. Nauk 41, no. 4, p. 157-158 (in Russian). Jointly with V.B.Lidskii.

27. Quasiresonances in the problem of forced vibrations of a thin elastic shell interacting with a liquid, Functional Analysis and its Applications 20, p. 267-276. Jointly with V.B.Lidskii.

1987

28. Asymptotics of the spectrum of a boundary value problem, Transactions of the Moscow Mathematical Society, p. 173-245.

29. Hamiltonian systems with reflection, Uspekhi Mat. Nauk 42, no. 4, p. 139-140 (in Russian).

30. Vibrations of a shell filled with a viscous compressible liquid, Transactions of the 14 USSR Conference on the Theory of Plates and Shells (Kutaisi, 1987), vol. 1, p. 279-284; Tbilisi, publishing house of the Tbilisi State University (in Russian). Jointly with M.R.Levitin and V.B.Lidskii.

1988

31. Forced vibrations of a shell filled with a viscous compressible fluid, Uspekhi Mat. Nauk 43, no. 4, p. 164 (in Russian). Jointly with M.R.Levitin and V.B.Lidskii.

1989

32. Forced vibrations of a thin elastic shell filled with a viscous compressible fluid, Soviet Physics Doklady 34, no. 3, p. 215-217. Jointly with M.R.Levitin and V.B.Lidskii.

33. Branching Hamiltonian billiards, Soviet Mathematics Doklady 38, no. 1, p. 64-68. Jointly with Yu.G.Safarov.

34. High-frequency radiation of a thin elastic shell, Mechanics of Solids 24, no. 5, p. 163-174. Jointly with A.L.Gol'denveizer and E.V.Nolde.

35. Vibrations of a shell containing fluid, International Conference on Engineering Aero-Hydroelasticity (Prague, 1989), vol. 2, p. 185-190. Jointly with A.Gol'denveizer, J.Kaplunov and V.Lidskii.

36. Equations of magnetoelasticity of thin shells. Statement of the problem and error estimates, preprint no. 400 of the Institute for Problems in Mechanics of the USSR Academy of Sciences, 44 p. (in Russian). Jointly with A.L.Gol'denveizer, J.D.Kaplunov, M.R.Levitin, D.G.Markushevich, E.V.Nolde and A.L.Radovinskii.

37. Equations of magnetoelasticity of superconducting shells, Transactions of the 4 Symposium on Theoretical Problems of Magnetoelasticity (Yerevan, 1989), p. 49-52; Yerevan, publishing house of the Yerevan State University (in Russian).

1990

38. Two-term asymptotics of the spectrum of natural frequencies of a thin elastic shell, Soviet Mathematics Doklady 41, no. 1, p. 108-112.

39. On the problems of magnetoelasticity of thin superconducting shells, Transactions of the 15 USSR Conference on the Theory of Plates and Shells (Kazan, 1990), vol. 1, p. 265-271; Kazan, publishing house of the Kazan State University (in Russian). Jointly with M.R.Levitin.

40. The asymptotic distribution of eigenvalues of differential operators, Lectures at the 14 School on Operator Theory in Functional Spaces (Novgorod, 1989), part 2, 114 pages; Novgorod, publishing house of the Novgorod Pedagogical Institute (in Russian). Jointly with Yu.G.Safarov.

41. Tambour fractal: exemple d'une formule asymptotique a deux termes pour la ``fonction de comptage'', Comptes Rendus Acad. Sc. Paris, 1990, vol. 311, p. 867-872. Jointly with J.Fleckinger.

1991

42. Distribution of free vibration frequencies in two- and three-dimensional elastic bodies, in Mechanics of Deformable Solids (edited by N.Kh.Arutiunian, I.F.Obraztzov and V.Z.Parton), p. 227-242; Hemisphere Publishing Corporation. Jointly with A.L.Gol'denveizer.

43. One can hear the dimension of a connected fractal in  R^2 , in Petkov & Lazarov: Integral Equations and Inverse Problems, p. 270-273; Longman Academic, Scientific & Technical (1991).

44. A class of nonlinear variational problems appearing in the theory of magnetoelasticity of thin superconducting shells, Nonlinearity 4, p. 821-833. Jointly with A.Laptev and M.R.Levitin.

45. Forced vibrations of a shell immersed in a viscous compressible fluid, Functional Analysis and its Applications 25, p. 309-311. Jointly with M.R.Levitin and V.B.Lidskii.

1992

46. Remarks on vibration of thin elastic shells and their numerical computation, Comptes Rendus Acad. Sc. Paris, 1992, vol. 314, Ser. 2, p. 445-452. Jointly with E.Sanchez-Palencia.

47. The asymptotic distribution of eigenvalues of differential operators, translation into English of the revised version of reference 40, in ``Spectral Theory of Operators'' (ed. S.Gindikin), American Mathematical Society Translations, ser. 2, vol. 150 (1992), p. 55-111. Jointly with Yu.G.Safarov.

48. The distribution of eigenvalues of partial differential operators, Équations aux Dérivées Partielles, Séminaire 1991-1992, Exposé noXVII 17 Mars 1992, École Polytechnique, p. XVII-1 - XVII-17.

49. Characteristic properties of distributions associated with the wave group, Journees Équations aux Dérivées Partielles, Saint-Jean-De-Monts, 1992, p. XV-1 - XV-14.

50. On global representation of Lagrangian distributions and solutions of hyperbolic equations, preprint no. LiTH-MAT-R-92-25 of Linköping University (Sweden), 1992; 54 pages. Jointly with A.Laptev and Yu.Safarov.

51. Global representation of Lagrangian distributions, in ``Operator Calculus and Spectral Theory'' (ed. M.Demuth, B.Gramsch and B.-W.Schulze), series Operator Theory: Advances and Applications, vol. 57, Birkhäuser Verlag, 1992, p. 197-213. Jointly with A.Laptev and Yu.Safarov.

52. Vibration du Tambour Fractal, Matapli (Bulletin SMAI -- Société de Mathématiques Appliquées et Industrielles), no. 32 Octobre 1992, p. 29-36. Jointly with J.Fleckinger.

1993

53. Book review (Pseudodifferential operators on manifolds with singularities by B.-W.Schulze), Bulletin of the London Mathematical Society, 1993, no. 113, vol. 25 part 2, p. 203-204.

54. An example of a two-term asymptotics for the ``counting function'' of a fractal drum, Transactions of the American Mathematical Society, 1993, vol. 337, no. 1, p. 99-116. Jointly with J.Fleckinger.

1994

55. Existence theorems for trapped modes, Journal of Fluid Mechanics, 1994, vol. 261, p. 21-31. Jointly with D.V.Evans and M.Levitin.

56. Mathematics, Soviet style. Book review (Cubature formulas and modern analysis: an introduction by S.L.Sobolev), The Times Higher Education Supplement, no. 1143, 30 September 1994, p. 25.

57. On global representation of Lagrangian distributions and solutions of hyperbolic equations, Communications on Pure and Applied Mathematics, 1994, vol. 47, no. 11, p. 1411-1456. Jointly with A.Laptev and Yu.Safarov.

58. Formula for the frequency counting function of a thin shell of revolution for the case of a simple turning point, Functional Analysis and its Applications, 1994, vol. 28, no. 4, p. 282-284. Jointly with A.G.Aslanyan and V.B.Lidskii.

1995

59. Construction of the wave group for higher order elliptic boundary value problems, Équations aux Dérivées Partielles, Séminaire 1994-1995, Exposé noİX 22 Novembre 1994, École Polytechnique, p. IX-1 - IX-14.

60. Book review (Degenerate Elliptic Equations by S.Levendorskii), Bulletin of the London Mathematical Society, 1995, no. 125, vol. 27 part 2, p. 198-199.

61. Book review (Introduction to the general theory of singular perturbations by S.A.Lomov), Bulletin of the London Mathematical Society, 1995, no. 125, vol. 27 part 2, p. 201-202.

62. Some examples of two-term spectral asymptotics for sets with fractal boundary, series Operator Theory: Advances and Applications, vol. 78 (ed. M.Demuth and B.-W.Schulze), Birkhäuser Verlag, 1995, p. 227-233. Jointly with M.Levitin.

63. Vibrations of a shell submerged in fluid, in ``Spectral Analysis of Complex Structures'' (ed. E. Sanchez-Palencia), Hermann, 1995, p. 159-171.

64. Heat equation on the triadic von Koch snowflake: asymptotic and numerical analysis, Proceedings of the London Mathematical Society, 1995, vol. 71, p. 372-396. Jointly with J.Fleckinger and M.Levitin.

65. The symbol of a Lagrangian distribution, Journees Équations aux Dérivées Partielles, Saint-Jean-De-Monts, 1995, p. V.1 - V.12.

66. Heat content of the triadic von Koch snowflake, International Journal of Applied Science and Computations 1995, vol. 2, , no. 2, p. 289-305. Jointly with J.Fleckinger and M.Levitin.

1996

67. Spectral asymptotics, renewal theorem, and the Berry conjecture for a class of fractals, Proceedings of the London Mathematical Society, 1996, vol. 72, p. 188-214. Jointly with M.Levitin.

68. Book review (Fourier Integral Operators by J.J.Duistermaat, V.W.Guillemin and L.Hörmander, and Microlocal Analysis for Differential Operators - An Introduction by A.Grigis and J.Sjöstrand), Bulletin of the London Mathematical Society, 1996, no. 131, vol. 28 part 2, p. 208-212.

69. Vibrations of shells contacting fluid: asymptotic analysis, in ``Acoustic Interaction with Submerged Elastic Structures -- Part I'' (ed. A.Guran, J.Ripoche and F.Ziegler), World Scientific, 1996, p. 310-332. Jointly with M.Levitin.

1997

70. On spectral pollution in the finite element approximation of thin elastic ``membrane'' shells, Numerische Mathematik, 1997, vol. 75, p. 473-500. Jointly with J.Rappaz, J. Sanchez Hubert, and E. Sanchez Palencia.

1998

71. Edge resonance in an elastic semi-strip, Quarterly Journal of Mechanics and Applied Mathematics, 1998, vol. 51 part 1, p. 1-13. Jointly with I.Roitberg and T.Weidl.

1999

72. Book review (Elliptic boundary value problems in domains with point singularities by V.A.Kozlov, V.G.Maz'ya and J.Rossmann), Bulletin of the London Mathematical Society, 1999, no. 150, vol. 31 part 3, p. 370-371.

73. The Dirac equation without spinors, in ``Rostock Conference on Functional Analysis, Partial Differential Equations and Applications'' (ed. J.Rossmann, P.Takac and G.Wildenhain), series Operator Theory: Advances and Applications, vol. 110, Birkhäuser Verlag, 1999, p. 133-152. Jointly with D.M.Elton. Available as preprint math-ph/9808010.

74. Victor Borisovich Lidskii (on his 75th birthday), Russian Mathematical Surveys, 1999, vol. 54, no. 5, p. 1077-1085. Jointly with M.S.Agranovich, V.I.Arnol'd, I.M.Gel'fand, A.L.Gol'denveizer, A.G.Kostyuchenko, L.D.Kudryavtsev and M.I.Vishik.

2000

75. Trapped modes in a waveguide with a long obstacle, Journal of Fluid Mechanics, 2000, vol. 403, p. 251-261. Jointly with N.S.A.Khallaf and L.Parnovski.

76. Edge resonance in an elastic semi-infinite cylinder, Applicable Analysis, 2000, vol. 74, no. 3-4, p. 479-495. Jointly with A.Holst.

77. Complex resonances in acoustic waveguides, Quarterly Journal of Mechanics and Applied Mathematics, 2000, vol. 53 part 3, p. 429-447. Jointly with A.Aslanyan and L.Parnovski.

78. Book review (Spectral theory and geometry, ed. E.B.Davies and Yu.Safarov), Bulletin of the London Mathematical Society, 2000, vol. 32 issue 6, p. 751-752.

2001

79. A metric-affine field model for the neutrino, in ``Noncommutative structures in mathematics and physics'' (ed. S.Duplij and J.Wess), NATO Science Series II: Mathematics, Physics and Chemistry, vol. 22, Kluwer Academic Publishers, 2001, p. 427-439.

80. Torsion waves in metric-affine field theory, Classical and Quantum Gravity, 2001, vol. 18, p. 2317-2329. Jointly with A.D.King. Available as preprint gr-qc/0012046.

2002

81. Pseudoinstantons in metric-affine field theory, General Relativity and Gravitation, 2002, vol. 34, p. 1239-1265. Available as preprint gr-qc/0108028.

2003

82. Pseudoinstantons in metric-affine field theory, in ``Quark confinement and the hadron spectrum V'' (ed. N.Brambilla and G.M.Prosperi), World Scientific, 2003, p. 273-275.

2004

83. The pseudoinstanton - a Lorentzian analogue of the instanton, Proceedings of Institute of Mathematics of NAS of Ukraine, 2004, vol. 50 part 2, p. 981-985.

84. Quadratic non-Riemannian gravity, Journal of Nonlinear Mathematical Physics, 2004, vol. 11 Supplement, p. 204-216.

2005

85. Quadratic metric-affine gravity, Annalen der Physik, 2005, vol. 14, no. 4, p. 231-252. The paper is a heavily revised version of preprint gr-qc/0304028.

86. PP-waves with torsion and metric-affine gravity, Classical and Quantum Gravity, 2005, vol. 22, p. 3961-3975. Jointly with V.Pasic. Available as preprint gr-qc/0505157.

2007

87. Teleparallel model for the neutrino, Physical Review D, 2007, vol. 75, article number 025006 (6 pages). The paper is a heavily revised version of preprint gr-qc/0604011.

88. A teleparallel representation of the Weyl Lagrangian, International Journal of Geometric Methods in Modern Physics, 2007, vol. 4, no. 2, p. 325-332.

2008

89. A teleparallel representation of the Weyl Lagrangian, in ``Proceedings of the Eleventh Marcel Grossmann Meeting on General Relativity'' (in 3 volumes, editors H.Kleinert and R.T.Jantzen, series editor R.Ruffini), World Scientific, 2008, part B, p. 1245-1247. Available as preprint gr-qc/0702020.

2009

90. Dirac equation as a special case of Cosserat elasticity, in ``Analysis, Partial Differential Equations and Applications - The Vladimir Maz'ya Anniversary Volume'' (ed. A.Cialdea, F.Lanzara and P.E.Ricci), series Operator Theory: Advances and Applications, vol. 193, Birkhäuser Verlag, 2009, p. 15-29. Jointly with J.Burnett and O.Chervova. Available as preprint arXiv:0812.3948.

91. Weyl's Lagrangian in teleparallel form, Journal of Mathematical Physics, 2009, vol. 50, article number 102501 (17 pages). Jointly with J.Burnett. Available as preprint arXiv:0901.1070.

92. Massless Dirac equation as a special case of Cosserat elasticity, Journal of the Association of Arab Universities for Basic and Applied Sciences, 2009, vol. 7, p. 25-42. (Proceedings of the International Conference on Recent Trends in Mathematical Sciences, Bahrain, 10-12 November 2008.) Jointly with O.Chervova. Available as preprint arXiv:0902.1268.

2010

93. The stationary Weyl equation and Cosserat elasticity, Journal of Physics A: Mathematical and Theoretical, 2010, vol. 43, article number 335203 (14 pages). Jointly with O.Chervova. The paper is a heavily revised version of preprint arXiv:1001.4726.

94. Victor Borisovich Lidskii (1924-2008), in ``Operator Theory and Its Applications: In Memory of V. B. Lidskii (1924-2008)'' (ed. M.Levitin and D.Vassiliev), series American Mathematical Society Translations - Series 2, vol. 231, American Mathematical Society, 2010, p. 1-5. Jointly with M.Levitin. Available as preprint arXiv:1008.2669.

2011

95. Rotational elasticity, Quarterly Journal of Mechanics and Applied Mathematics, 2011, vol. 64, p. 415-439. Jointly with C.G.Boehmer and R.J.Downes. Click here for free online access. The paper is a heavily revised version of preprint arXiv:1008.3833.

2012

96. Modelling the neutrino in terms of Cosserat elasticity, in ``Proceedings of the Twelfth Marcel Grossmann Meeting on General Relativity'' (in 3 volumes, editors Thibault Damour and Robert Jantzen, series editor Remo Ruffini), World Scientific, 2012, part C, p. 2398-2400. Jointly with O.Chervova. Available as preprint arXiv:1003.0574.

97. Modelling the electron with Cosserat elasticity, Mathematika, 2012, vol. 58, p. 349-370. Jointly with J.Burnett. The paper is a heavily revised version of preprint arXiv:1007.3481.

2013

98. The spectral function of a first order elliptic system, Journal of Spectral Theory, 2013, vol. 3, p. 317-360. Jointly with O.Chervova and R.J.Downes. Available as preprint arXiv:1208.6015.

99. Spectral asymmetry of the massless Dirac operator on a 3-torus, Journal of Mathematical Physics, 2013, vol. 54, article number 111503 (18 pages). Jointly with R.J.Downes and M.Levitin. Available as preprint arXiv:1306.5689.

2014

100. Spectral theoretic characterization of the massless Dirac operator, Journal of the London Mathematical Society, 2014, vol. 89, p. 301-320. Jointly with O.Chervova and R.J.Downes. Available as preprint arXiv:1209.3510.

101. Spectral theoretic characterization of the massless Dirac action. Jointly with R.J.Downes. Preprint arXiv:1401.1951.

102. Analysis as a source of geometry: a non-geometric representation of the Dirac equation. Jointly with Yan-Long Fang. Preprint arXiv:1401.3160.

103. Analysis of first order systems of partial differential equations. Jointly with Yan-Long Fang. Preprint arXiv:1403.2663.



The background on this page uses an illustration from [64].

Updated 12 March 2014