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Bruno Alexander Dmitrievich

Statistics Math-Net.Ru
Total publications: 155
Scientific articles: 153
Presentations: 9

Number of views:
This page:3408
Abstract pages:16166
Full texts:6584
References:971
Professor
Doctor of physico-mathematical sciences (1969)
Speciality: 01.01.02 (Differential equations, dynamical systems, and optimal control)
Phone: +7 (499) 250 78 84
Fax: +7 (499) 972 07 37
E-mail:
Website: http://brunoa.name
Keywords: number theory, complex analysis, differential equations, algebraic equations, analytical mechanics, stability of motion, celestial mechanics, hydrodynamics.
UDC: 511.36, 513, 514.172, 517, 517.52, 517.9, 517.91, 517.913, 517.925, 517.93, 519, 521.1, 521.41, 529.7, 531.31, 531.38
MSC: 11, 30, 34, 35, 40, 41, 52, 70, 76, 85, 34C20

Subject:

(I) The new calculus "Power Geometry" was created for nonlinear equations and systems of equations of any type (algebraic, ordinary differential and partial differential). It gives the general algorithms for:

  1. the isolation of their first approximations by means of the Newton polyhedrons and their analogous;
  2. simplification of the first approximations by means of the power and logarithmic transformations;
  3. finding self-similar solutions to quasihomogeneous systems (to which belong all first approximations);
  4. finding asymptotic forms of their solutions and
  5. the computation of the asymptotic expansions of their solutions.
It allows to study any singularities (including the singular perturbations) in the mentioned equations and systems. For the autonomous ODE system in a neighborhood of the stationary solution (and also near the periodic solution or the invariant torus), there were proven: (a) the existence of the formal invertible change of coordinates transforming the system to the resonant normal form, (b) which can be reduced to a system of lower order (equal to the multiplicity of the resonance) by means of the power transformation. (c) There were found the conditions $\omega$ on eigenvalues and A on the normal form that are necessary and sufficient for the analyticity of the normalizing transformation. (d) If the condition A is violated, there are sets ${\cal A}$ (if small divisors are absent) and ${\cal B}$ (if they are present) on which the normalizing transformation is analytic. The sets are computed via the normal form, they contain all invariant tori found by means of the KAM theory and allow to simplify the study of bifurcations of the periodic solutions and of the invariant tori. (e) The further simplifications of the resonant normal form were considered. In particular, for systems with the one-fold resonance, there was given the polynomial normal form, all coefficients of which are the formal invariants. (f) Similar results were found for the resonant Hamiltonian normal form of the Hamiltonian system. In particular, the theory of the Hamiltonian normal form for the linear Hamiltonian systems with constant or periodic coefficients was finished. (g) It was shown that the normal form is very convenient for the study of stability. In particular, it was shown that the proof of the stability of the stationary point in the Hamiltonian system with two degrees of freedom, given by V. I. Arnol'd in 1963, contains the wrong statement. (h) The Power Geometry and the normal forms were applied in problems of Mechanics (in particular, all power expansions of motions of the rigid body were calculated for the generic case with $y_0=z_0=0$ and a lot of the new integrable cases was found), of Celestial Mechanics (the families of periodic solutions in the planar restricted three-body problem and in the Beletsky equation, describing the planar motions of a satellite around its masscenter, were studied) and of Hydrodynamics (on a needle the boundary layer was given and the surface waves on the water were studied). (i) For the ordinary differential equation of any order I proposed an algorithm of computing asymptotic expansions of its solutions near a singularity. I have find new types of such expansions: power-logarithmic, complicated, exotic and power-periodic. I obtained conditions of their convergency. All that was made for solutions, for which order of derivative differs from the order of the solution by $-1$ as well as for solution, for which that difference is arbitrary. Finally, by these methods we calculated all asymptotic expansions of solutions of all six Painlev\'e equations. (k) For algebraic equations of $n$ variables, I proposed new methods of computation of approximate values of roots for $n=1$ and of approximate uniformizations its solutions, i. e. algebraic curves and surfaces, for $n>1$. These methods are based on the Hadamar open polygon and polyhedron. I also developed an algorithm of computation of asymptotic expansions of its solutions near singularity (including infinity).

(II) In Number Theory it was shown that the continued fractions of the cubic irrationalities have the same structure as the continued fractions for the almost all numbers. There were attempts to find the multidimensional generalizations of the continued fractions, based on the Klein polyhedra. In particular, the quality of the matrix algorithms of Euler, Jacobi, Poincare, Brun, Parusnikov and Bruno was compared. It appears that the Poincare algorithm is the worst. For the multidimensional generalization of the continued fraction, I proposed a modular polyhedron instead of the Klein polyhedron (that name was given by me instead of the name «Arnold polyhedron»). Preimages of vertices of the modular polyhedron give the best Diophantine approximations. The modular polyhedron can be computed by means of a standard program for computing convex hulls. It gives a solution of the problem, which majority of main mathematicians of XIX century tried to solve. In the algebraic case, using the modular polyhedron it is possible to find all fundamental units of some rings. Using them it is possible to compute all periods of the generalized continued fraction and to compute all solutions to Diophantine equations of some class. This approach gives also simultaneous Diophantine approximations.

Biography

Graduated from Faculty of Mathematics and Mechanics of the M. V. Lomonosov Moscow State University (MSU) in 1962 (department of differential equations). Ph.D. thesis was defended in 1966. D.Sci. thesis was defended in 1969. Professor since 2007. The list of my publications contains more than 380 titles.

In 1956 and 1957 I received the 3rd and the 1st prizes at the Moscow mathematical olympiades for pupiles. In 1960 and 1961 I received the 2nd prizes at the competition of the students works in the Faculty of Mechanics and Mathematics. Since 1965 I am a member of the Moscow Mathematical Soc., since 1993 of the American Math. Soc. and since 1996 I am a member of the Academy of Nonlinear Sciences. My biographical data were published in Who's Who in the World, Marquis, 12th ed., 1995, p. 178; 16th ed., 1999, p. 222. Outstanding People of the 20th Century, Intern. Biogr. Centre, Cambridge, 1st ed., 1999, p. XXXIV–XXXV, 82. Five Hundred Leaders of Influence, ABI, 8th ed., 1999, p. 44; 2000 Outstanding Scholars of the 20th Century, IBC, 2000, p. 46–47; 2000 Outstanding Intellectuals of the 20th Century, IBC, 2000, p. 44; The First Five Hundred in the New Millennium, IBC, 2000, p. 52–53.

   
Main publications:
  • Brjuno A. D. Analytical form of differential equation // Transaction of Moscow Mathematical Society, 1971, 25, 131–288; 1972, 26, 199–239.
  • Bruno A. D. Local Methods in Nonlinear Differential Equations. Berlin: Springer-Verlag, 1989.
  • Bruno A. D. The Restricted 3-Body Problem. Berlin: Walter de Gruyter, 1994.
  • Bruno A. D. Power Geometry in Algebraic and Differential Equations. Amsterdam: Elsevier Science, 2000.
  • Bruno A. D., Shadrina T. V. Axisymmetric boundary layer on a needle // Transactions of Moscow Math. Soc. 68 (2007) 201--259

http://www.mathnet.ru/eng/person9101
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https://mathscinet.ams.org/mathscinet/MRAuthorID/198723
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http://www.scopus.com/authid/detail.url?authorId=7102246723
https://www.researchgate.net/profile/A_Bruno

Publications in Math-Net.Ru
2018
1. A. D. Bruno, “Normal form of the periodic Hamiltonian system with $n$ degrees of freedom”, Keldysh Institute preprints, 2018, 223  mathnet
2. A. D. Bruno, “Complicated and exotic expansions of solutions to the Painlevé equations”, Keldysh Institute preprints, 2018, 118  mathnet
3. A. D. Bruno, “Expansion of ODE solutions into transseries”, Keldysh Institute preprints, 2018, 117  mathnet
4. A. D. Bruno, “Asymptotic solution of some nonlinear problems”, Keldysh Institute preprints, 2018, 035  mathnet
5. A. D. Bruno, “Power geometry and expansions of solutions to the Painlevé equations”, Keldysh Institute preprints, 2018, 021  mathnet
6. A. D. Bruno, “On complicated expansions of solutions to ODES”, Zh. Vychisl. Mat. Mat. Fiz., 58:3 (2018),  346–364  mathnet  elib; Comput. Math. Math. Phys., 58:3 (2018), 328–347  isi  scopus
2017
7. A. D. Bruno, “Complicated and exotic expansions of solutions to the fifth Painlevé equation”, Keldysh Institute preprints, 2017, 107  mathnet
8. A. D. Bruno, “Calculation of exotic expansions of solutions to the third Painlevé equation”, Keldysh Institute preprints, 2017, 096  mathnet
9. A. D. Bruno, “Calculation of complicated asymptotic expansions of solutions to the Painlevé equations”, Keldysh Institute preprints, 2017, 055  mathnet
10. A. D. Bruno, “Calculation of fundamental units of number rings by means of the generalized continued fraction”, Keldysh Institute preprints, 2017, 046  mathnet
11. A. D. Bruno, “Solving the polynomial equations by algorithms of power geometry”, Keldysh Institute preprints, 2017, 034  mathnet
2016
12. A. D. Bruno, “From Diophantine approximations to Diophantine equations”, Chebyshevskii Sb., 17:3 (2016),  38–52  mathnet  elib
13. A. D. Bruno, “On solution of an algebraic equation”, Keldysh Institute preprints, 2016, 070  mathnet
14. A. D. Bruno, “From Diophantine approximations to Diophantine equations”, Keldysh Institute preprints, 2016, 001  mathnet
2015
15. A. D. Bruno, “Universal generalization of the continued fraction algorithm”, Chebyshevskii Sb., 16:2 (2015),  35–65  mathnet  elib
2014
16. A. B. Batkhin, A. D. Bruno, “On investigation of the certain real algebraic surface”, Keldysh Institute preprints, 2014, 083  mathnet
2013
17. A. D. Bruno, I. V. Goryuchkina, “Convergence of power expansions of solutions to an ODE”, Keldysh Institute preprints, 2013, 094  mathnet
18. A. D. Bruno, “Power Geometry and elliptic expansions of solutions to the Painlevé equations”, Keldysh Institute preprints, 2013, 088  mathnet
19. A. D. Bruno, “Asymptotic solving nonlinear equations and idempotent mathematics”, Keldysh Institute preprints, 2013, 056  mathnet
2012
20. A. D. Bruno, A. V. Parusnikova, “Expansions and asymptotic forms of solutions to the fifth Painlevé equation near infinity”, Keldysh Institute preprints, 2012, 061  mathnet
21. A. D. Bruno, “Power-elliptic expansions of solutions to an ordinary differential equation”, Zh. Vychisl. Mat. Mat. Fiz., 52:12 (2012),  2206–2218  mathnet  mathscinet  zmath  elib; Comput. Math. Math. Phys., 52:12 (2012), 1650–1661  isi  elib  scopus
2011
22. A. D. Bruno, A. V. Parusnikova, “Periodic and Elliptic Asymptotic Forms of Solutions to the Fifth Painlev'e Equation”, Keldysh Institute preprints, 2011, 061  mathnet
23. A. D. Bruno, “Power-elliptic expansions of solutions to an ODE”, Keldysh Institute preprints, 2011, 060  mathnet
24. A. D. Bruno, “Power-exponential expansions of solutions to an ODE”, Keldysh Institute preprints, 2011, 054  mathnet
25. A. B. Batkhin, A. D. Bruno, V. P. Varin, “Sets of stability of multiparameter Hamiltonian systems”, Keldysh Institute preprints, 2011, 042  mathnet
26. A. D. Bruno, “Exponential expansions of solutions to an ODE”, Keldysh Institute preprints, 2011, 036  mathnet
27. A. D. Bruno, A. V. Parusnikova, “Expansions of solutions to the fifth Painlevé equation near its nonsingular point”, Keldysh Institute preprints, 2011, 018  mathnet
28. A. D. Bruno, “On complicated expansions of solutions to ODE”, Keldysh Institute preprints, 2011, 015  mathnet
29. A. D. Bruno, A. B. Batkhin, “Resolution of an algebraic singularity by Power Geometry algorithms”, Keldysh Institute preprints, 2011, 010  mathnet
2010
30. A. D. Bruno, “Structure of the best diophantine approximations and multidimensional generalizations of the continued fraction”, Chebyshevskii Sb., 11:1 (2010),  68–73  mathnet  mathscinet
31. A. D. Bruno, A. V. Parusnikova, “Local Expansions of Solutions of the Fifth Painlevé Equation”, Keldysh Institute preprints, 2010, 072  mathnet
32. A. B. Aranson, A. D. Bruno, “Power expansions of the shifted solutions to the N. Kowalewski system”, Keldysh Institute preprints, 2010, 048  mathnet
33. A. D. Bruno, A. V. Parusnikova, “Asymptotic Expansions of Solutions to the Fifth Painlevé equation”, Keldysh Institute preprints, 2010, 039  mathnet
34. A. D. Bruno, A. B. Batkhin, V. P. Varin, “Computation of the Sets of Stability in Multiparameter Problems”, Keldysh Institute preprints, 2010, 023  mathnet
35. A. D. Bruno, “The structure of the multidimensional Diophantine approximations”, Keldysh Institute preprints, 2010, 011  mathnet
36. A. D. Bruno, A. V. Gridnev, “Nonpower expansions of solutions to the third Painlevé equation”, Keldysh Institute preprints, 2010, 010  mathnet
37. A. D. Bruno, A. B. Batkhin, V. P. Varin, “The stability set of a gyroscopic problems”, Keldysh Institute preprints, 2010, 004  mathnet
38. A. D. Bruno, “Sets of stability of multiparameter problems”, Keldysh Institute preprints, 2010, 003  mathnet
2009
39. A. D. Bruno, V. F. Edneral, “Algorithmic analysis of local integrability”, Dokl. Akad. Nauk, 424:3 (2009),  299–303  mathnet  mathscinet; Dokl. Math., 79:1 (2009), 48–52  isi  scopus
40. A. D. Bruno, I. V. Goryuchkina, “Non-formal solutions to ODE”, Keldysh Institute preprints, 2009, 061  mathnet
41. A. D. Bruno, I. V. Goryuchkina, “Elliptic asymptotic forms of solutions to the Painlev'e equations”, Keldysh Institute preprints, 2009, 006  mathnet
42. A. D. Bruno, V. F. Edneral, “On integrability of a planar system of ODEs near a degenerate stationary point”, Zap. Nauchn. Sem. POMI, 373 (2009),  34–47  mathnet; J. Math. Sci. (N. Y.), 168:3 (2010), 326–333  scopus
2008
43. A. D. Bruno, I. V. Goryuchkina, “All expansions of solutions to the sixth Painlevé equation near its nonsingular point”, Keldysh Institute preprints, 2008, 075  mathnet
44. A. D. Bruno, V. I. Parusnikov, “Two-sided generalization of the continued fraction”, Keldysh Institute preprints, 2008, 058  mathnet
45. A. D. Bruno, V. P. Varin, “The families c and i of periodic solutions of the restricted problem for $\mu=5\cdot10^{-5}$”, Keldysh Institute preprints, 2008, 022  mathnet
2007
46. A. D. Bruno, V. F. Edneral, “On integrability of the Euler–Poisson equations”, Fundam. Prikl. Mat., 13:1 (2007),  45–59  mathnet  mathscinet  zmath; J. Math. Sci., 152:4 (2008), 479–489  scopus
47. A. D. Bruno, I. V. Goryuchkina, “All asymptotic expansions of solutions to the equation P6 are obtained from base ones”, Keldysh Institute preprints, 2007, 077  mathnet
48. A. D. Bruno, I. V. Goryuchkina, “All asymptotic expansions of solutions to the equation P6 in the case $a\cdot b=0$”, Keldysh Institute preprints, 2007, 070  mathnet
49. A. D. Bruno, I. V. Goryuchkina, “All base asymptotic expansions of solutions to the equation P6 in the case $a\cdot b\ne0$”, Keldysh Institute preprints, 2007, 062  mathnet
50. A. D. Bruno, I. V. Goryuchkina, “Methods are used for researching of asymptotic expansions of solutions to the equation P6”, Keldysh Institute preprints, 2007, 061  mathnet
51. A. D. Bruno, I. V. Goryuchkina, “Review of all asymptotic expansions of solutions to the equation P6”, Keldysh Institute preprints, 2007, 060  mathnet
52. A. D. Bruno, V. F. Edneral, “Analysis of the local integrability by methods of normal form and power geometry”, Keldysh Institute preprints, 2007, 053  mathnet
53. A. D. Bruno, V. P. Varin, “Family $c$ of periodic solutions of the restricted problem”, Keldysh Institute preprints, 2007, 051  mathnet
54. A. D. Bruno, V. P. Varin, “Complicated families of periodic solutions of the restricted problem”, Keldysh Institute preprints, 2007, 035  mathnet
55. A. D. Bruno, V. P. Varin, “Periodic solutions of the restricted three-body problem for small $\mu$”, Keldysh Institute preprints, 2007, 034  mathnet
56. A. D. Bruno, “Power Geometry as a new mathematics”, Keldysh Institute preprints, 2007, 028  mathnet
57. A. D. Bruno, I. V. Goryuchkina, “All asymptotic expansions of solutions to the sixth Painlevé equation”, Keldysh Institute preprints, 2007, 019  mathnet
58. A. D. Bruno, V. F. Edneral, “Computation of normal forms of the Euler–Poisson equations”, Keldysh Institute preprints, 2007, 001  mathnet
2006
59. A. D. Bruno, “Complicated expansions of solutions to an ODE system”, Keldysh Institute preprints, 2006, 081  mathnet
60. A. D. Bruno, “Exotic expansions of solutions to an ordinary differential equation”, Keldysh Institute preprints, 2006, 066  mathnet
61. A. D. Bruno, V. Yu. Petrovich, “Desingularizations of the restricted three-body problem”, Keldysh Institute preprints, 2006, 053  mathnet
62. A. D. Bruno, V. P. Varin, “The generating family $i$ of periodic solutions of the restricted problem”, Keldysh Institute preprints, 2006, 036  mathnet
63. A. D. Bruno, “On movable singular points of solutions to the ordinary differential equations”, Keldysh Institute preprints, 2006, 026  mathnet
64. A. D. Bruno, I. V. Goryuchkina, “Expansions of solutions to the sixth Painlevé equation near singular points $x=0$ и $x=\infty$”, Keldysh Institute preprints, 2006, 013  mathnet
65. A. D. Bruno, I. V. Goryuchkina, “Expansions of solutions to the sixth painleve equation in cases $a=0$ and $b=0$”, Keldysh Institute preprints, 2006, 002  mathnet
2005
66. A. D. Bruno, “Theory of normal forms of the Euler-Poisson equations”, Keldysh Institute preprints, 2005, 100  mathnet
67. A. D. Bruno, “Properties of the modulus polyhedron”, Keldysh Institute preprints, 2005, 072  mathnet
68. A. D. Bruno, I. N. Gashenenko, “Simple finite solutions to the N. Kowalewski equations”, Keldysh Institute preprints, 2005, 068  mathnet
69. A. D. Bruno, V. P. Varin, “The family $h$ of periodic solutions of the restricted problem for small $\mu$”, Keldysh Institute preprints, 2005, 067  mathnet
70. A. D. Bruno, “Normal Forms and Integrability of the Euler–Poisson Equations”, Keldysh Institute preprints, 2005, 066  mathnet
71. A. D. Bruno, I. N. Gashenenko, “Last expansions of modified motions of a rigid body”, Keldysh Institute preprints, 2005, 065  mathnet
72. A. D. Bruno, V. P. Varin, “The family $h$ of periodic solutions of the restricted problem for big $\mu$”, Keldysh Institute preprints, 2005, 064  mathnet
73. A. D. Bruno, T. V. Shadrina, “On the viscous incompressible fluid flow around a plate”, Keldysh Institute preprints, 2005, 054  mathnet
74. A. D. Bruno, V. I. Parusnikov, “New generalizations of the continued fraction”, Keldysh Institute preprints, 2005, 052  mathnet
75. A. D. Bruno, V. P. Varin, “The family $h$ of periodic solutions of the restricted problem for small $\mu$”, Keldysh Institute preprints, 2005, 048  mathnet
76. A. D. Bruno, V. I. Parusnikov, “Further generalization of the continued fraction”, Keldysh Institute preprints, 2005, 040  mathnet
77. A. D. Bruno, “Complicated expansions of solutions to an ordinary differential equation”, Keldysh Institute preprints, 2005, 036  mathnet
78. A. D. Bruno, N. A. Kudryashov, “Power expansions of solutions to an analogy to the first Painlevé equation”, Keldysh Institute preprints, 2005, 017  mathnet
79. A. D. Bruno, V. P. Varin, “On families of periodic solutions to the restricted three-body problem”, Keldysh Institute preprints, 2005, 010  mathnet
80. A. D. Bruno, I. V. Goryuchkina, “Power expansions of solutions to the sixth Painlevé equation near a regular point”, Keldysh Institute preprints, 2005, 004  mathnet
2004
81. A. D. Bruno, V. Yu. Petrovich, “Singularities of solutions to the first Painlevé equation”, Keldysh Institute preprints, 2004, 075  mathnet
82. A. D. Bruno, “Algorithm of the generalizationued continued fraction”, Keldysh Institute preprints, 2004, 045  mathnet
83. A. D. Bruno, T. V. Shadrina, “The compressible heat conductive boundary layer on a needle”, Keldysh Institute preprints, 2004, 037  mathnet
84. A. D. Bruno, T. V. Shadrina, “About incompressible boundary layer on a needle”, Keldysh Institute preprints, 2004, 036  mathnet
85. A. D. Bruno, T. V. Shadrina, “Methods of a study of the boundary layer on a needle”, Keldysh Institute preprints, 2004, 035  mathnet
86. A. D. Bruno, “On generalisations of the continued fraction”, Keldysh Institute preprints, 2004, 010  mathnet
87. A. D. Bruno, “Asymptotic behaviour and expansions of solutions of an ordinary differential equation”, Uspekhi Mat. Nauk, 59:3(357) (2004),  31–80  mathnet  mathscinet  zmath  elib; Russian Math. Surveys, 59:3 (2004), 429–480  isi  scopus
2003
88. A. D. Bruno, V. I. Parusnikov, “Polyhedra of absolute values for triples of linear forms”, Keldysh Institute preprints, 2003, 093  mathnet
89. A. D. Bruno, “The сorrect generalization of the continued fraction”, Keldysh Institute preprints, 2003, 086  mathnet
90. A. D. Bruno, T. V. Shadrina, “Axisymmetric boundary layer on a needle”, Keldysh Institute preprints, 2003, 064  mathnet
91. A. D. Bruno, “Expansions of solutions to an ODE system”, Keldysh Institute preprints, 2003, 059  mathnet
92. A. D. Bruno, “Asymptotically сlose slutions to an ODE system”, Keldysh Institute preprints, 2003, 058  mathnet
93. A. D. Bruno, A. V. Gridnev, “Power and exponential expansions of solutions to the third Painlevé equation”, Keldysh Institute preprints, 2003, 051  mathnet
94. A. D. Bruno, E. S. Karulina, “Power expansions of solutions to the fifth Painlevé equation”, Keldysh Institute preprints, 2003, 050  mathnet
95. A. D. Bruno, I. V. Chukhareva, “Power expansions of solutions to the sixth Painlevé equation”, Keldysh Institute preprints, 2003, 049  mathnet
96. A. D. Bruno, Yu. V. Zavgorodnyaya, “Power series and nonpower asymptotics of solutions to the second Painlevé equation”, Keldysh Institute preprints, 2003, 048  mathnet
97. A. D. Bruno, “Asymptotically close solutions to an ordinary differential equation”, Keldysh Institute preprints, 2003, 031  mathnet
98. A. D. Bruno, “The asymptotical solution of nonlinear equations by means of Power Geometry”, Keldysh Institute preprints, 2003, 028  mathnet
99. A. D. Bruno, “Asymptotics and expansions of solutions to an ordinary differential equation”, Keldysh Institute preprints, 2003, 009  mathnet
2002
100. A. D. Bruno, V. P. Varin, “Classes of families of generalized periodic solutions to the Beletsky equation”, Keldysh Institute preprints, 2002, 064  mathnet
101. A. D. Bruno, “Analysis of the Euler-Poisson equations by methods of Power Geometry”, Keldysh Institute preprints, 2002, 041  mathnet
102. A. D. Bruno, “Asymptotics of solutions to the ordinary differential equations”, Keldysh Institute preprints, 2002, 040  mathnet
103. A. D. Bruno, V. V. Lunev, “Properties of expansions of modified motions of a rigid body”, Keldysh Institute preprints, 2002, 023  mathnet
2001
104. A. D. Bruno, V. V. Lunev, “Asymptotical expansions of modified motions of a rigid body”, Keldysh Institute preprints, 2001, 090  mathnet
105. A. D. Bruno, V. V. Lunev, “Local expansions of modified motions of a rigid body”, Keldysh Institute preprints, 2001, 073  mathnet
106. A. D. Bruno, V. V. Lunev, “The modified system of equations described motions of a rigid body”, Keldysh Institute preprints, 2001, 049  mathnet
2000
107. A. D. Bruno, “Power Expansions of Solutions of a System of Algebraic and Differential Equations”, Keldysh Institute preprints, 2000, 068  mathnet
108. A. D. Bruno, “Power Expansions of Solutions of One Algebraic or Differential Equation”, Keldysh Institute preprints, 2000, 063  mathnet
109. A. D. Bruno, “Families of Periodic Solutions of the Beletskii Equation”, Keldysh Institute preprints, 2000, 051  mathnet
110. A. D. Bruno, V. Yu. Petrovich, “Normal Forms of the ODE System”, Keldysh Institute preprints, 2000, 018  mathnet
111. A. D. Bruno, “Self-similar solutions and power geometry”, Uspekhi Mat. Nauk, 55:1(331) (2000),  3–44  mathnet  mathscinet  zmath  elib; Russian Math. Surveys, 55:1 (2000), 1–42  isi  elib  scopus
1999
112. A. D. Bruno, “A new generalization of the continued fraction”, Keldysh Institute preprints, 1999, 082  mathnet
113. A. D. Bruno, “On Complexity of Problems of Power Geometry”, Keldysh Institute preprints, 1999, 059  mathnet
114. A. D. Bruno, “Finding Self-Similar Solutions by Means of Power Geometry”, Keldysh Institute preprints, 1999, 057  mathnet
1997
115. A. D. Bruno, V. J. Petrovich, “Computation of periodic oscillations of a satellite”, Matem. Mod., 9:6 (1997),  82–94  mathnet  mathscinet  zmath
116. A. D. Bruno, V. I. Parusnikov, “Comparison of various generalizations of continued fractions”, Mat. Zametki, 61:3 (1997),  339–348  mathnet  mathscinet  zmath; Math. Notes, 61:3 (1997), 278–286  isi
1996
117. A. D. Bruno, “Zero-Multiple and Retrograde Periodic Solutions of the Restricted Three-Body Problem”, Keldysh Institute preprints, 1996, 093  mathnet
1995
118. A. D. Bruno, V. P. Varin, “The Second Limit Problem for the Equation of Oscillations of a Satellite”, Keldysh Institute preprints, 1995, 128  mathnet
119. A. D. Bruno, V. P. Varin, “The First Limit Problem for the Equation of Oscillations of a Satellite”, Keldysh Institute preprints, 1995, 124  mathnet
120. A. D. Bruno, A. Soleev, “The Hamiltonian Truncations of a Hamiltonian System”, Keldysh Institute preprints, 1995, 055  mathnet
121. A. D. Bruno, A. Soleev, “Homoclinic Solutions of an Invertible ODE System”, Keldysh Institute preprints, 1995, 054  mathnet
122. A. D. Bruno, “The Newton Polyhedron in the Nonlinear Analysis”, Keldysh Institute preprints, 1995, 048  mathnet
123. A. D. Bruno, A. Soleev, “Local Analysis of a Singularity of an Invertible ODE System. Complicated Cases”, Keldysh Institute preprints, 1995, 047  mathnet
124. A. D. Bruno, M. M. Vasiliev, “Newton Polyhedra and the Asymptotic Analysis of the Viscous Fluid Flow Around Flat Plate”, Keldysh Institute preprints, 1995, 044  mathnet
125. A. D. Bruno, A. Soleev, “Local Analysis of a Singularity of an Invertible ODE System. Simple Cases”, Keldysh Institute preprints, 1995, 040  mathnet
126. A. D. Bruno, S. Yu. Sadov, “Formal integral of a divergence-free system”, Mat. Zametki, 57:6 (1995),  803–813  mathnet  mathscinet  zmath; Math. Notes, 57:6 (1995), 565–572  isi
127. A. D. Bruno, A. Soleev, “Local analysis of singularities of an invertible system of ordinary differential equations”, Uspekhi Mat. Nauk, 50:6(306) (1995),  169–170  mathnet  mathscinet  zmath; Russian Math. Surveys, 50:6 (1995), 1258–1259  isi
1994
128. A. D. Bruno, V. I. Parusnikov, “Klein polyhedrals for two cubic Davenport forms”, Mat. Zametki, 56:4 (1994),  9–27  mathnet  mathscinet  zmath; Math. Notes, 56:4 (1994), 994–1007  isi
1991
129. A. D. Bruno, A. Soleev, “Local uniformization of the branches of a space curve, and Newton polyhedra”, Algebra i Analiz, 3:1 (1991),  67–101  mathnet  mathscinet  zmath; St. Petersburg Math. J., 3:1 (1992), 53–82
1990
130. A. D. Bruno, “The normal form of a system, close to a Hamiltonian system”, Mat. Zametki, 48:5 (1990),  35–46  mathnet  mathscinet  zmath; Math. Notes, 48:5 (1990), 1100–1108  isi
131. A. D. Bruno, “System, similar to a normal form”, Mat. Zametki, 48:3 (1990),  20–31  mathnet  mathscinet  zmath; Math. Notes, 48:3 (1990), 885–893  isi
1989
132. A. D. Bruno, “Normalization of a Hamiltonian system near an invariant cycle or torus”, Uspekhi Mat. Nauk, 44:2(266) (1989),  49–78  mathnet  mathscinet  zmath; Russian Math. Surveys, 44:2 (1989), 53–89  isi
1988
133. A. D. Bruno, “The normal form of a Hamiltonian system”, Uspekhi Mat. Nauk, 43:1(259) (1988),  23–56  mathnet  mathscinet  zmath; Russian Math. Surveys, 43:1 (1988), 25–66  isi
1986
134. A. D. Bruno, “Stability in a Hamiltonian system”, Mat. Zametki, 40:3 (1986),  385–392  mathnet  mathscinet  zmath; Math. Notes, 40:3 (1986), 726–730  isi
1983
135. A. D. Bryuno, “Noncanonical invariants of Hamiltonian systems”, Mat. Zametki, 33:3 (1983),  333–344  mathnet  mathscinet  zmath; Math. Notes, 33:3 (1983), 167–174
1982
136. A. D. Bruno, “Divergence of a real normalizing transformation”, Mat. Zametki, 31:3 (1982),  403–410  mathnet  mathscinet  zmath; Math. Notes, 31:3 (1982), 207–211  isi
1977
137. A. D. Bruno, “Properties of certain functions of celestial mechanics”, Mat. Zametki, 22:1 (1977),  109–116  mathnet  mathscinet  zmath; Math. Notes, 22:1 (1977), 550–554
1976
138. A. D. Bruno, “Normal form and averaging methods”, Dokl. Akad. Nauk SSSR, 230:2 (1976),  257–260  mathnet  mathscinet  zmath
1975
139. A. D. Bruno, “Integral analytic sets”, Dokl. Akad. Nauk SSSR, 220:6 (1975),  1255–1258  mathnet  mathscinet  zmath
140. A. D. Bruno, “Normal form of real differential equations”, Mat. Zametki, 18:2 (1975),  227–241  mathnet  mathscinet  zmath; Math. Notes, 18:2 (1975), 722–731
1974
141. A. D. Bruno, “Analytic integral manifolds”, Dokl. Akad. Nauk SSSR, 216:2 (1974),  253–256  mathnet  mathscinet  zmath
142. A. D. Bruno, “Normal form of differential equations with a small parameter”, Mat. Zametki, 16:3 (1974),  407–414  mathnet  mathscinet  zmath; Math. Notes, 16:3 (1974), 832–836
1973
143. A. D. Bruno, “Local invariants of differential equations”, Mat. Zametki, 14:4 (1973),  499–507  mathnet  mathscinet  zmath; Math. Notes, 14:4 (1973), 844–848
1972
144. A. D. Bruno, “Analytic form of differential equations”, Tr. Mosk. Mat. Obs., 26 (1972),  199–239  mathnet  zmath
1971
145. A. D. Bruno, “Analytic form of differential equations. I, II”, Tr. Mosk. Mat. Obs., 25 (1971),  119–262  mathnet  mathscinet  zmath
1970
146. A. D. Bruno, “Instability in a Hamiltonian system and the distribution of asteroids”, Mat. Sb. (N.S.), 83(125):2(10) (1970),  273–312  mathnet  mathscinet  zmath; Math. USSR-Sb., 12:2 (1970), 271–312
1969
147. A. D. Bruno, “An analytic form of differential equations”, Mat. Zametki, 6:6 (1969),  771–778  mathnet  mathscinet; Math. Notes, 6:6 (1969), 927–931
1967
148. A. D. Bruno, “The divergence of transformations to normal form of differential equations”, Dokl. Akad. Nauk SSSR, 174:5 (1967),  1003–1006  mathnet  mathscinet  zmath
149. A. D. Bryuno, “Formal stability of Hamiltonian systems”, Mat. Zametki, 1:3 (1967),  325–330  mathnet  mathscinet  zmath; Math. Notes, 1:3 (1967), 216–219
1965
150. A. D. Bruno, “On convergence of transforms of differential equations to the normal form”, Dokl. Akad. Nauk SSSR, 165:5 (1965),  987–989  mathnet  mathscinet  zmath
151. A. D. Bruno, “Power asymptotics of solutions of non-linear systems”, Izv. Akad. Nauk SSSR Ser. Mat., 29:2 (1965),  329–364  mathnet  mathscinet  zmath
1964
152. A. D. Bruno, “The normal form of differential equations”, Dokl. Akad. Nauk SSSR, 157:6 (1964),  1276–1279  mathnet  mathscinet  zmath
153. A. D. Bruno, “The expansion of algebraic numbers into continued fractions”, Zh. Vychisl. Mat. Mat. Fiz., 4:2 (1964),  211–221  mathnet  mathscinet  zmath; U.S.S.R. Comput. Math. Math. Phys., 4:2 (1964), 1–15
1962
154. A. D. Bruno, “Asymptotic behavior of solutions of systems of differential equations”, Dokl. Akad. Nauk SSSR, 143:4 (1962),  763–766  mathnet  mathscinet  zmath

2016
155. A. I. Aptekarev, A. B. Batkhin, A. D. Bruno, “Vladimir Igorevich Parusnikov”, Chebyshevskii Sb., 17:1 (2016),  286–298  mathnet  elib
1975
156. É. Dzhusti, M. I. Vishik, A. V. Fursikov, A. S. Schwarz, O. I. Bogoyavlenskii, B. M. Levitan, V. V. Kucherenko, A. G. Kushnirenko, M. V. Fedoryuk, M. A. Shubin, A. D. Bruno, “Sessions of the Petrovskii Seminar on differential equations and mathematical problems of physics”, Uspekhi Mat. Nauk, 30:2(182) (1975),  261–269  mathnet  mathscinet

Presentations in Math-Net.Ru
1. Solving a polynomial equation
A. D. Bruno
XV International Conference «Algebra, Number Theory and Discrete Geometry: modern problems and applications», dedicated to the centenary of the birth of the Doctor of Physical and Mathematical Sciences, Professor of the Moscow State University Korobov Nikolai Mikhailovich
May 29, 2018 10:40
2. Разложение решений ОДУ в трансряды
A. D. Bruno
Seminar on analytic theory of differential equations
April 25, 2018 14:30
3. Calculation of complex asympotics of solutions of Painleve equations
A. D. Bruno
Seminar on analytic theory of differential equations
April 12, 2017 14:30   
4. Solving the Polynomial Equations by Algorithms of Power Geometry
A. D. Bruno
Seminar on Complex Analysis (Gonchar Seminar)
February 20, 2017 17:00
5. From Diophantine approximations to fundamental units of algebraic fields
A. D. Bruno
Conference to the Memory of Anatoly Alekseevitch Karatsuba on Number theory and Applications
January 30, 2016 14:30
6. Asymptotic solutions to algebraic equation
A. D. Bruno, A. B. Batkhin
Seminar of the Department of Algebra
May 10, 2011 15:00
7. Power geometry as new mathematics
A. D. Bruno
Meetings of the St. Petersburg Mathematical Society
April 19, 2005
8. Степенная геометрия как новая математика
A. D. Bruno
Meetings of the Moscow Mathematical Society
April 5, 2005
9. A new generalization of the continued fraction
A. D. Bruno
Meetings of the St. Petersburg Mathematical Society
April 20, 2004

Organisations
 
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