Grigory L. Litvinov
Institutional Addresses:
INDEPENDENT UNIVERSITY OF MOSCOW (IMU)
and the RUSSIAN-FRENCH LABORATORY "J.-V. PONCELET"
Bol'shoi Vlasievskii per., 11
Moscow 119002, Russia
Phone: (7-495) 241 4086
E-mail: islc@dol.ru
glitvinov@mail.ru
Home Address:
Nagornaya, 27 - 4 - 72
Moscow 117186, Russia
Phone: (7-495) 127 3804
Personal Data:
Birthday: May 16, 1944
Birthplace: Russia, Moscow
Citizenship: Russia
Education and degrees:
1966-1969 Moscow State University,
Ph.D. in Mathematics.
1961-1966 Moscow State University, B.S. and
M.S. in Mathematics
EXPERIENCE IN HIGH EDUCATION
INSTITUTION RANK DATES
Independent University Research fellow 2002 -
of Moscow and lecturer Present
International University Visiting professor Sep. 2005 -
Bremen of mathematics June 2006
ICSL Mathematical Professor of 1992-2002
College mathematics
Moscow State University Part-time Lecturer 1998 and
1971-1972
Moscow State Correspondence Part-time Senior 1973-1974
Pedagogical Institute Lecturer
Engineering Academy Assistant 1969-1973
Moscow Professor
RESEARCH AND INDUSTRIAL EXPERIENCE
INSTITUTION RANK DATES
Russian-French Laboratory Researcher 2006-
"J.-V. Poncelet" Present
Independent University Research fellow 2002-
of Moscow Present
Institute for New Deputy Director 1990-2002
Technologies, of the Center
Center for Optimization
and Mathematical Modeling
Soviet-Bulgarian Chief Expert 1988-1990
Computer Joint Venture
"Lingvotehnika"
All-Union Patent Library Head of Software 1987-1988
Department
All-Union Research Institute Head of Software 1977-1987
of Documents and Archives Department
All-Union State Project Manager 1973-1977
Institute of Statistics
Technology
MEMBERSHIP IN HONORARY OR PROFESSIONAL SOCIETIES
Moscow Mathematical Society
American Mathematical Society
SERVICE TO MATHEMATICAL COMMUNITY
Member of the Editorial Council of the "Russian Journal
of Mathematical Physics".
AWARDS AND HONORS
Soros & Russian Acad. Nat. Sci. Award and Diploma for 1993
2000 Science/Interperiodica Award and Diploma for the best
publication in 1999 from Russian Acad. Sci. and Pleiades Publ.
INVITED LECTURER at
Dates
Independent University of Moscow September 2006
May 2007
Lecture course title: "The Maslov
dequantization and tropical mathematics
International University Bremen September 2005
- June 2006
Lecture courses:
- "Numerical analysis",
- "Analysys-2 and ODE",
- "Linear algebra, probability and
statistics",
- "Integration and manifolds".
Voronezh Winter mathematical School - 2004 January -
February
Lecture course title: "The Maslov 2004
dequantization and idempotent (tropical)
mathematics"
The Fields Institute for Research in June 2002
Mathematical Sciences (Toronto, Canada)
Lecture course title: "Smooth and
Nonsmooth Interval Analysis"
Voronezh Winter Mathematical School-2000 January -
(Voronezh, Russia) February
2000
Lecture course title: "Idempotent
Mathematics and Applications"
Moscow State University February -
- May
Lecture course title: "Idempotent 1999
Mathematics"
Istituto per le Applicazioni del Calcolo November,
"Mauro Picone" (Rome, Italy) December
1995
Lecture course title: "Idempotent Analysis
and computer calculations"
Oslo University (Oslo, Norway) March, April
1995
Lecture course title: "The theory of
non-unitary representations of Lie
groups and the orbit method"
Belorussian State University (Minsk, Belorussia) May 1993
Lecture course title: "Approximations
and Errors"
All-Union Winter Mathematical School
(Voronezh, Russia). 1990
Lecture course title: "Rational
Approximations"
Moscow State University 1971-1972
Lecture course title: "Representations
of Groups and Topological Group Algebras"
COLLOQUIUM AND CONFERENCE LECTURES
Dates
Workshop on Convex Analysis (Moscow State University, Feb 2 - Feb 5
Moscow, Russia; invited speaker) 2007
International Conference "OPERATOR ALGEBRAS AND Jan 29 - Feb 3
TOPOLOGY" (Moscow State University, Moscow, Russia; 2007
invited speaker)
International Conference "INTERVAL-06" (St. Petersburg July 2006
State University, Peterhof, Russia; invited speaker)
International University Bremen (Bremen, Germany) Oct. 2005
International Conference "Methods of Logic in July 2005
Mathematics" (Euler International Mathematical
Institute, St. Petersburg, Russia; invited speaker)
International Conference " Harmonic Analysis on April 25-29
Homogeneous Spaces, Representations of Lie Groups 2005
and Quantization" (Tambov State University, Tambov,
Russia; invited speaker)
International Conference "Topology, Analysis and Feb. 14-18
Applications in Mathematical Physics" (Moscow State 2005
University, Moscow, Russia; invited speaker)
Max Planck Institute for Mathematics
(Bonn, Germany) July 2004
Bonn University Institute for Applied Mathematics
(Bonn, Germany) July 2004
International Conference "Representation theory, June 8-13
Dynamical Systems, and Asimptotic Combinatorics" 2004
(Euler International Mathematical Institute,
St. Petersburg, Russia; invited speaker)
Charles University in Prague (Czech Republic) April 2004
International Conference "Geometry and Combinatorics
in Physics" (Independent University of Moscow, March 2-7
Moscow, Russia; invited speaker) 2004
Moscow State University (Math. Dept.,Moscow) March 1
2004
Inernational 25th Linz Seminar on Fuzzy Set Feb. 3-7
Theory "Mathematics of Fuzzy Systems" (Linz 2004
University, Linz, Austria; invited main speaker)
International Conference "Kantorovich Memorial. Jan. 8-13
Mathematics and economics: Old problems and 2004
new approaches" (Euler International Mathematical
Institute, St. Petersburg, Russia; invited speaker)
International Workshop "Amoebas and tropical Oct. 22-27
geometry" (American Institute of Mathematics, 2003
Palo Alto, California, USA, invited speaker)
University of California in San Diego, USA Oct. 2003
International Workshop on Max-algebra (IWMA; June 30 -
University of Birmingham, Birmingham, UK; July 3, 2003
invited speaker)
Steklov Mathematical Institute of the Russian March 29
Academy of Sciences (S.- Petersburg division) 2003
Institute for Theoretical and Experimental March 17
Physics (ITEP, Moscow) 2003
International Workshop "Idempotent Mathematics Feb. 3-10
and Mathematical Physics" (Erwin Schroedinger 2003
International Institute for Mathematical Physics,
Vienna, Austria; organizer and invited speaker)
The 2002 Twente Conference on Lie Groups Dec. 16-18
(Twente, the Netherlands; invited speaker) 2002
International Workshop "Representation Theory Dec. 9-13
of Lie groups, Harmonic Analysis on Homogeneous 2002
Spaces and Quantization" (Lorentz Center,
Leiden, the Netherlands)
Lund University (Lund, Sweden) November-
December 2002
Independent University of Moscow (Globus Oct. 10, 2002
Semin., Moscow, Russia)
Centre Universitaire de Luxembourg (Luxembourg) September-
October 2002
Universite de Metz (Metz, France) September-
October 2002
and Feb 3, 1995
International Conference on Harmonic Analysis Sep. 9-12
(Luxembourg-Metz; invited speaker) 2002
University of Toronto (Toronto, Canada) June 2002
January and
February 1996
University of Salerno (Salerno, Italy) Dec. 2001
University of Tunis (Tunis, Tunisia) Oct. 2001
Journees de Physique-Mathematique "Analyse Oct. 10-11
Harmonique et Quantification" (Faculte des 2001
Sciences de Sfax, Sfax, Tunisia; invited
speaker)
International Workshop "Topology, Analysis, Aug. 29-31
and Related Topics" (Moscow State University, 2001
Moscow, Russia; invited speaker)
European summer school "Asymptotic July 16-18
combinatorics with applications to 2001
mathematical physics" (NATO Advanced Study
Institute and the Euler International
Mathematical Institute, St. Petersburg;
invited speaker)
St. Petersburg Mathematical Society Meeting May 22
Title: "Idempotent Mathematics 2001
and Mathematical Physics"
Istituto per le Applicazioni del Calcolo May 2001
"Mauro Picone" (Rome, Italy)
Univ. of Rome "La Sapienza" (Rome, Italy) May 2001
European School in Group Theory Aug. 14-26
(SDU-Odense University, Denmark; invited speaker) 2000
11th IFAC International Workshop "Control July 3-6
Applications of Optimization" (St. Petersburg, 2000
Russia; invited speaker)
6th International IMACS Conference on June 25-28
Applications of Computer Algebra (St. 2000
Petersburg, Russia; invited speaker)
Erwin Schroedinger International Institute Apr. 2000
for Mathematical Physics (Vienna, Austria) October and
November 1998
May 1996
International Conference on Rational June 6-11,1999
Approximation (Antwerp, Belgium;
invited speaker)
International Meeting "Braude College Days May 1999
of Differential Equations and Applied Analysis
(Karmiel, Israel; invited speaker)
Technion (Haifa, Israel) May 16, 1999
Univ. of Haifa (Haifa, Israel) May 11, 1999
Univ. Bar-Ilan (Tel-Aviv, Israel) May 9, 1999
Mathematical Institute, Leiden University April 1999
(Leiden, The Netherlands) April 1997
May 1994
Univ. de Valenciennes (Valenciennes, France) February 1999
November 1997
Univ. de Lille (Lille, France) February 1999
November 1997
International Colloquium "Idempotent Analysis" January 1999
(Caen, France; invited speaker)
Sophus Lie Workshop (Kazan, Russia; Sept. 7-17,
invited speaker) 1998
International Conference Dedicated to the Aug. 31-
90th Anniversary of L.S.Pontryagin Sept. 6, 1998
(Moscow, Russia; invited speaker)
International Conference "Geometric August 1998
Analysis'98" (Kiel, Germany; invited speaker)
Lobachevsky Meeting, Kazan University December 1997
(Kazan, Russia; invited speaker)
Univ. Paul Sabatier (Toulouse, France) November 1997
S.G.Krein Conference on Functional Analysis October 1997
and Mathematical Physics (Voronezh, Russia;
invited speaker)
International Workshop on Idempotent Analysis April 4, 1997
(Nottingham, England; invited speaker)
International Conference "Interval-96" October 1996
(Wuerzburg, Germany; invited speaker)
International Workshop "Harmonic Analysis August 1996
on Homogeneous Spaces" (Tambov, Russia;
invited speaker)
Moscow State University (Moscow, Russia; April 1996
Faculty Seminar)
University of Miami (Coral Gables, USA) March 22, 1996
Pennsylvania State University (University March 19, 1996 and
Park, USA) Nov. 13, 1990
University of California (Riverside, USA) March 12, 1996
Northeastern University (Boston, USA) March 4 and 8, 1996
University of Michigan (Ann Arbor, USA) Feb. 29, 1996 and
Oct. 19, 1990
University of Missouri-St.Louis (St.Louis, USA) Feb. 26, 1996
International Conference on Harmonic Analysis December 1995
(Delhi, India; invited speaker,
Sci. Committee member)
Fifth Workshop on Representation Theory of August 1995
Lie Groups and its Applications
(Cordoba, Argentina; invited speaker)
Univ. Pierre et Marie Curie (Paris, France) Feb. 10, 1995
Ecole Polytechnique (Palaiseau, France) Jan. 31, 1995
LAAS - CNRS seminar, Toulouse, France Jan. 26,27
1995
Workshop on Models of Parallel Computations December 1994
I.A.C.- C.N.R., Rome, Italy (Invited speaker)
Hewlett-Packard's Basic Research Institute in October 1994
Mathematical Sciences (BRIMS) (Bristol, England)
(Idempotency workshop, invited speaker)
The University of York October 1994
The University of Nottingham October 1994
Classical and Quantum Geometry of Homogeneous August 1994
Spaces, International Conference, Moscow
(Invited speaker)
Math. Inst., Oslo University (Oslo, Norway) June 1994 and
May 1993
American Mathematical Soc. Summer Meeting August 1993
(Seattle, USA, invited speaker)
International Lie-Lobachevsky Oct. 26-30,
Colloquium (Tartu, Estonia, invited 1992
speaker)
International Conf.on Interval and Sept. 22-26,
Stochastic Methods in Sci.and Eng. 1992
(Interval-92), Moscow
University of Edinburgh, Scotland May 21, 1992
(Analysis Seminar)
King's College (University of London) May 14, 1992
(Modern Analysis Seminar)
School of Mathematical Sciences May 13, 1992
University of North London (Special Seminar)
Purdue University Oct. 17, 1990
School of Science at Indianapolis
Department of Math.Sci at IUPUI (Colloquium)
Massachusetts Institute of Technology Oct. 13, 1990
(D.Vogan's Seminar on Lie Groups)
Moscow Mathematical Society Meeting
Title: "Rational Approximations and Error March 5, 1991
Auto-correction"
Title: "Representations, Ideals, Characters" 1978
Leningrad State University Feb. 27, 1991
Leningrad Mathematical Society Meeting Feb. 26, 1991
All-Union Winter Mathematical School (Voronezh) 1976-1989, 2000, 2004
All-Union School on Operator Theory 1977, 1979, 1989
AND SO ON
RESEARCH INTERESTS:
Representation Theory and Lie groups
Functional Analysis
Hypergroups
Rational Approximations. Theory and Applications
Computer Technology (Algorithms and special hardware and software)
Idempotent Mathematics
Mathematical Modeling and Optimization
Numerical Analysis and Interval Analysis
PUBLICATIONS OF G.L. LITVINOV:
More than 185 publications and books in Russian and in English, a list
of selected publications is included. A full list of publications
is available here.
List of selected publications
SELECTED PUBLICATIONS IN RUSSIAN:
1. Ordinary Differential Equations - Lecture Notes for
Students of the Engineering Academy. - In: Textbook on Calculus. -
Moscow, Eng. Acad. Publ., part II, 1974, p. 173-207; part III,
1975, p. 116-188.
2. Editor and one of the authors of the book
"Mathematical Algorithms and Programs for Small Computers" -
"Finances and Statistics" Publ., Moscow, 1981, 112 pp.
3. Approximation properties of locally convex spaces
and the trace problem. - In: Theor.Funktsii, Funktsional.
Anal. i Prilozhen, vol.39, 1983, p.73-87.
4. Approximate Construction of Rational Approximations
and an Effect of Error Auto-correction. - In: Mathematics and
Modeling - USSR Academy of Sciences Publ., Puschino, 1990,
p.99-141.
SELECTED PUBLICATIONS IN ENGLISH:
5. On Completely Irreducible Representations of Complex
and Real Nilpotent Lie Groups. - Functional Anal.Appl.Vol.3,
# 4, 1970, p.332-334.
6. Group Algebras of Analytic Functionals and their
Representations. - Soviet Math.Dokl., Vol.11, # 1, 1970,
p.191-193.
7. The Laplace Transform on Lie Groups. - Functional
Anal.Appl. Vol.6, # 2, 1972, p. 76, 77.
8. Characters and Simple Representations. - Soviet
Math. Dokl., Vol.15, # 1, 1974, p. 179-183.
9. On Traces of Operators and Characters of Irreducible
Representations in Locally Convex Spaces. - Soviet Math.
Dokl., Vol.15, # 1, 1974, p. 308-312.
10. (with G.B.Shpiz) Primary Decompositions of Finite-
Dimensional Representations of Lie Algebras and Lie Groups.
- Functional Anal.Appl., Vol.12, # 2, 1978, p. 149-151.
11. Traces of Linear Operators in Locally Convex
Spaces. - Functional Anal.Appl. Vol.13, # 1, 1979, p. 60-62.
12. (with L.I.Vainerman). The Plansherel Formula and
the Inversion Formula for Generalized Translation Operators.
- Soviet Math.Dokl., Vol. 23, # 1, 1981, p. 333-337.
13. (with V.I.Lomonosov). Density Theorems in Locally
Convex Spaces and Irreducible Representations. - Soviet
Math. Dokl., Vol. 23, # 2, 1981, p.372-375.
14. (with A.P.Kryukov and A.Ya.Rodionov). Construction
of Rational Approximations by Means of REDUCE. - In:
Proceedings of the 1986 Symposium on Symbolic and Algebraic
Computation. Symsac'86 (July 21-23, 1986, Waterloo,
Ontario), Univ.of Waterloo, 1986, p. 31-33.
15. Hypergroups and Hypergroup Algebras. - Journal of
Soviet Math., Vol. 38, 1987, p. 1734-1761.
16. Representations of Groups in Locally Convex Spaces
and Topological Group Algebras. - Selecta Mathematica
Sovietica, Vol. 7, # 2, 1988, p. 101-182.
17. On Conditions under which a Representation is
Determined by its Character, up to Equivalence. - Selecta
Mathematica Sovietica, Vol. 8, # 1, 1989, p. 73-97.
18. On the Traces of Linear Operators in Locally Convex
Spaces. - Selecta Mathematica Sovietica, Vol.8, # 3, 1989,
p. 203-212.
19. (with V.I.Lomonosov). Density Theorems in Locally
Convex Spaces and Applications. - Selecta Mathematica
Sovietica, Vol.8, # 4, 1989, p. 323-339.
20. (with B.M.Levitan). Generalized Displacement
Operators. - Encyclopaedia of Mathematics, Vol.4. - Kluwer
Academic Publ., 1989, p. 224-228.
21. Nuclear Operator. - Encyclopaedia of Mathematics,
Vol.6. - Kluwer Academic Publ., 1990, p. 487-490.
22. Nuclear Space. - Encyclopaedia of Mathematics,
Vol.6. - Kluwer Academic Publ., 1990, p. 490-493.
23. General Concepts of Quantum Mechanics. - In:
F.A.Berezin a.o., The Schroedinger Equation - Kluwer
Academic Publ., Dordrecht/Boston/London. 1991, p. 1-49 +
p. 386-415 + p. 462-465.
24. Generalized Translation Operators and their
Representations. - Selecta Mathematica Sovietica, vol.10, #
4, 1991, p. 307-337.
25. Double Topological Algebras and Topological Hopf
Algebras. - Selecta Mathematica Sovietica, vol.10, # 4,
1991, p. 339-343.
26. Approximation Properties of Locally Convex Spaces
and the Problem of Uniqueness of the Trace of a Linear
Operator. - Selecta Mathematica Sovietica, vol.11, # 1,
1992, p. 25-40.
27. (with G.B.Shpiz). Primary Decompositions of Finite
Dimensional Representations of Lie Algebras and Lie Groups.
- Selecta Mathematica Sovietica, vol.11, # 1, 1992, p. 41-54.
28. Error Auto-Correction in Rational Approximation -
Proc. Intern. Conf. on Interval and Stochastic Methods in
Science and Engineering (Interval-92), Moscow, 1992, vol.2,
p.59-63; Interval Computations, 1992, # 4(6), p. 14-18.
29. Approximate Construction of Rational Approximations
and an Effect of Error Auto-correction. Applications -
Russian Journal of Mathematical Physics, vol. 1, # 3, 1994,
p. 313-352. See also E-print arXiv:math.NA/0101042
30. (with V.P. Maslov). The Correspondence Principle for
Idempotent Calculus and Some Computer Applications; Preprint
IHES, Bures-sur-Yvette, 1995; the same in: Idempotency/
J. Gunawardena (Editor), Cambridge University Press, Cambridge,
1998 (ISBN 0-521-55344-X), p. 420-443. see also E-print arXiv:
math.GM/0101021
31. (with V.P. Maslov). Idempotent Mathematics: Correspondence
principle and applications// Russian Mathematical Surveys, v.51,
# 6, 1996, p. 1210-1211.
32. (with V.N. Tulovsky). Generalized Fourier Transform,
Maslov's Canonical Operator and Hypergroups. - International
Sophus Lie Center, Moscow, 1997.
33. Five articles on Idempotent Mathematics for the
Encyclopaedia of Mathematics, - Kluwer Acad.Publ., Suppl. vol. 1,
1997, p. 312-317.
34. (with B. Komrakov, I.Krasil'shchik, and A.Sossinsky, Editors).
Lie Groups and Lie Algebras, their Representations, Generalizations,
and Applications. - Kluwer Acad.Publ., Dordrecht et al., 1998.
35. (with K. A. Ross et al., Editors). Harmonic Analysis and
Hypergroups ("Trends in Mathematics" series). - Birkhauser, 1998.
36. (with V.P. Maslov and G.B. Shpiz). Linear Functionals on
Idempotent Spaces: an Algebraic Approach// Doklady Mathematics,
v.58, # 3, 1998, p. 389-391. See E-print arXiv:math.FA/0012268
.
37. (with V.P. Maslov and A.N. Sobolevskii) Idempotent
Mathematics and Interval Analysis. - The Erwin Schroedinger
International Institute for Mathematical Physics, Vienna,
1998. See also E-print 1998 (http://www.esi.ac.at).
38. (with V.P. Maslov and G.B. Shpiz) Tensor Products
of Idempotent Semimodules. An Algebraic Approach // Mathematical
Notes, v. 65, # 4, 1999, p. 479-489. See also E-print arXiv:
math.FA/0101153
39. (with E.V. Maslova) Universal Numerical Algorithms
and Their Software Implementation // Programming and Computer
Software, v. 26, # 5, 2000, p. 275-280. See also E-print arXiv:
math.SC/0102114
40. (with A.N. Sobolevskii) Exact Interval Solutions of
the Discrete Bellman Equation and Polynomial Complexity of
Problems in Interval Idempotent Linear Algebra // Doklady
Mathematics, v. 62, # 2, 2000, p. 199-201. See also E-print
arXiv:math.LA/0101041
41. (with V.P. Maslov and A.Ya.Rodionov) A Unifying
Approach to Software and Hardware Design for Scientific
Calculations and Idempotent Mathematics. - International Sophus
Lie Centre, Moscow 2000. See also E-print arXiv:math.SC/0101069
.
42. Numerical Mathematics Should Be Taught Differently. -
In: The 9th International Congress on Mathematical Education,
ICME-9. Topics Study Group 10. Trends in Mathematics and
Mathematical Sciences. Reflections on Mathematics Education.
August 1-2, 2000, Tokyo/Makuhari, p. 18-21.
43. (with A.N. Sobolevskii) Idempotent Interval Analysis
and Optimization Problems // Reliable Computing, v. 7, # 5, 2001,
p. 353-377. See also E-print arXiv:math.SC/0101080
44. (with V.P. Maslov and G.B. Shpiz) Idempotent Functional
Analysis. An Algebraic Approach // Mathematical Notes, v. 69,
# 5, 2001, p. 696-729. See also E-print arXiv:math.FA/0009128
.
45. (with V.P. Maslov and G.B. Shpiz) Nondigital
Implementations of the Arithmetic of Real Numbers by Means of
Quantum Computer Media // Mathematical Notes, v. 70, # 1, 2001,
p. 53-60. See also E-print arXiv:quant-ph/9904025
46. (with V.P. Maslov and G.B. Shpiz) Idempotent (Asymptotic)
Analysis and the Representation Theory. - in: V.A. Malyshev and
A.M. Vershik (eds.), Asymptotic Combinatorics with Applications
to Mathematical Physics. - Kluwer Acad. Publ., Dordrecht et al.,
2002, p. 267-278. See also E-print arXiv:math.RT/0206025
.
47. (with G.B. Shpiz) Nuclear Semimodules and Kernel Theorems
in Idempotent Analysis: An Algebraic Approach//Doklady Mathematics,
v. 66, #2, 2002, p.197-199. See also E-print arXiv:math.FA/0202026
.
48. Error Autocorrection in Rational Approximation and Interval
Estimates//Central European Journal of Mathematics, V.1, #1, 2003.
E-print. See also E-print arXiv:math.NA/0207183
49. Dequantization of Mathematics, idempotent semirings and
fuzzy sets. -- In: LINZ 2004. 25th Linz Seminar on Fuzzy Set Theory
"Mathematics of Fuzzy Systems". Linz, Austria, February 3 - 7, 2004/
E.P. Klements, E. Pap, Editors, Johannes Kepler Univ., Linz, 2004,
p. 113-117.
50. (with V.P. Maslov, eds.) Idempotent Mathematics and
Mathematical Physics, AMS "Contemporary Mathematics", 2005.
51. The Maslov Dequantization,Idempotent and Tropical
Mathematics: A Very Brief Introduction -- in [50], p. 1-17.
See also E-print arXiv:math.GM/0501038
52. (with G.B. Shpiz) The dequantization transform and
generalized Newton polytopes. - In [50], 181-186. See also
E-print arXiv:
math-ph/0412090
53. (with G.B. Shpiz) Kernel theorems and nuclearity in
idempotent mathematics. An algebraic approach, Journal of
Mathematical Sciences, v. 141, #4, 2007, p. 1417-1428. See also E-print arXiv:
math.FA/0609033
54. The Maslov dequantization, idempotent and tropical
mathematics: A brief introduction, Journal of Mathematical
Sciences, v. 140, #3, 2007, p. 426-444. See also E-print arXiv:math.GM/
0507014
55. (with G.B. Shpiz) Engel type theorems for
representations of groups in idempotent spaces. - In preparation.
56. Irreducible Representations of Complex and Real
Nilpotent Lie Groups in Locally Convex Spaces. - In preparation.
57. Lie Hypergroups and Their Representations. -
In preparation.
BRIEF SUMMARY OF RESULTS
1. Representation Theory
My papers [5-11, 13, 16-19, 27, 46, 56] are devoted to
the theory of representations of Lie groups and locally
compact groups (generally speaking, representations that are
infinite-dimensional and not unitary). The description of
the group representations reduced to the description of the
group algebras, different group algebras corresponding to
different classes of representations. The group algebras
which I have studied are topological, and the study of the
representations of these algebras is connected with the
study of their closed ideals, right, left, and two-sided
[6,16]. Completely irreducible representations of complex
nilpotent Lie groups with holomorphic matrix elements were
studied. The description of such representations reduced to
the description of representation of a special group
algebra, the algebra of analytic functionals. Completely
irreducible (non-unitary) representations of real nilpotent
Lie groups were constructed by using the results obtained
[5]. A classification of all completely irreducible
representations of nilpotent Lie groups is obtained [5,16,56].
So the well-known A.A. Kirillov's problem is solved.
Some results about spectral analysis of arbitrary
representations of metabelian groups were presented [16].
The Laplace transform on Lie groups was studied [7, 16].
Using V.I.Lomonosov's techniques I have obtained a
solution of M.A.Naimark's problem about the complete
irreducibility of some representations of the Heisenberg
group [19, theorem 6].
Several approaches to the theory of characters for
infinite-dimensional representations of topological groups
and algebras were presented in [8, 9, 13, 17, 19]. Some
sufficient conditions were given under which a
representation is determined by its character up to some
natural equivalence relation.
In [10, 27] primary decompositions of finite
dimensional representations of Lie algebras and Lie groups
(that are similar to decompositions of linear operators with
respect to root subspaces) were studied. In particular, we
have described all the Lie algebras for which each complex
representation admits such a decomposition.
In [46, 55] representations of groups in idempotent
linear spaces are examined in the framework of Idempotent
Mathematics.
2. Functional Analysis
My papers [3, 9, 11, 13, 17, 18, 21, 26] are closely
related with classical problems of S.Banach and A.
Grothendieck (the basis problem, the approximation problem,
the trace problem, the invariant subspace problem) for the
case of arbitrary locally convex spaces.
In particular, the following results give a solution of
the A.Grothendieck's trace problem ("probleme de
biunivocite").
Theorem 1. Any nuclear operator in a locally convex
space E has a well-defined trace in the sense of
Grothendieck if E has the approximation property.
Theorem 2. There exists a quasi-complete locally convex
space with the approximation property in which one can not
define the trace for all Fredholm operators.
Theorem 3. Any Fredholm operator in a locally convex
space E has a well-defined trace in the sense of
Grothendieck if E has the bounded approximation property.
Theorem 4. If E is a locally convex space which has a
Schauder basis, then every nuclear operator (or Fredholm
operator) on E has a well-defined matrix trace which does
not depend on the choice of the basis, and coincides with
the trace in the sense of Grothendieck.
3. Hypergroups Theory
My papers [12, 15 , 20, 24, 25, 32, 57] are devoted to
the theory of hypergroups. Essential fragments of this
theory became familiar as the theory of the J.Delsarte-B.M.
Levitan generalized translation operators, the theory of the
Ju.M.Berezansky-S.G.Krein hypercomplex systems with
continuous basis, the theory of convolution algebras, etc.
Roughly speaking, a hypergroup is a topological space or
manifold with a supplementary structure, which permits one
to construct a Banach or topological algebra of the type of
a group algebra - a hypergroup algebra. Thus, in the theory
of hypergroups, just as in the theory of supergroups, the
object of generalization is not so much a group as a group
algebra (coalgebra). For some classes of hypergroups results
on a connection with infinitesimal objects in the spirit of
the Lie theory are found, see [15]. The ideas and methods of
the theory of group representations carries over to the case
of hypergroups, while it is convenient to treat
representations of hypergroups as representations of the
corresponding hypergroup algebras. In particular, in [12] an
abstract Plansherel theorem is stated and an inversion
formula is given for a broad class of hypergroups that
includes all locally compact groups. The theory of Lie
hypergroups, their Lie hyperalgebras and representations is
developed in [15, 24, 32].
Relations to Maslov's canonical operator and Integral
Geometry are discussed [32].
4. Rational Approximations. Theory and Applications
Methods of rational approximations of real-valued
functions were investigated and implemented. Some
paradoxical effects occurring in these approximations were
discovered and explained, see [2, 4, 14, 28, 29]. The nature
of one of them is the following. If "good" method of
constructing of rational approximations is used (for
example, Pade approximations, linear and non-linear Pade-
Chebyshev approximations etc.) then large errors in
calculated coefficients have no significant influence on the
overall precision of approximation constructed. Errors of
numerator and denominator cancel each other. The reason of
this cancellation is that the errors of the coefficients are
not completely independent, but form coefficients of another
approximant to the same function. Relations to interval
analysis are discussed [28, 48].
Efficient algorithms for calculation of functions were
designed and implemented for concrete small computers (see,
e.g., [2, 29]).
5. Computer Technologies
New computational schemes and algorithms for decimal
computers were designed, see [2, 4, 29]. Several devices for
fast calculation of functions were invented. I was a manager
of several large software projects. I didn't include the
corresponding papers to the list of publications. Some
important applications of V.P. Maslov's Idempotent Calculus
to hardware and software design are discussed in [30, 31, 33,
37, 39-43]. Quantum computing is discussed in [45].
6. Idempotent Mathematics
Idempotent Mathematics can be treated as a result of a
dequantization of the traditional Mathematics as the Planck
constant tends to zero taking pure imaginary values, see
[30, 31, 44, 46, 51, 54]. For example, the field of real
numbers can be treated as a quantum object whereas idempotent
semirings can be examined as "classical" or "semiclassical"
objects (a semiring is called idempotent if the semiring
addition is idempotent, i.e. x + x = x). Idempotent Mathema-
tics (in a modern sense) was founded by V.P. Maslov
and his collaborators.
In the spirit of N.Bohr's correspondence principle there
is a (heuristic) correspondence between important, useful,
and interesting constructions and results over fields and
similar results over idempotent semirings [30, 31]. For
example, the superposition principle in Quantum Mechanics
(the linearity of the Schroedinger equation) corresponds to
a linearity of the Hamilton-Jacobi and Bellman equations
over idempotent semirings. A systematic application of this
correspondence principle leads to a variety of results
including such exotic applications as a methodology to
construct computer devices (processors) for numerical
calculations and to get the corresponding patents, as well
as a methodology for software design see [30, 31, 39, 40,
41, 42, 43].
In the framework of Idempotent Mathematics a new version
of Functional Analysis is developed from idempotent variants
of basic theorems (e.g. of the Hahn-Banach type) to the
theory of tensor products, nuclear operators and nuclear
spaces in the spirit of A. Grothendieck as well as basic
concepts and results of the theory of representation of
groups in idempotent linear spaces [36, 38, 44, 46, 47, 51
-- 55].
7. Mathematical Modeling and Optimization
A new method of constructing analytical models using
rational and spline approximations was designed. A number of
models were worked out for concrete users. Some of my papers
(which I didn't include to the list) are devoted to
statistical models and methods.
Many algorithms, software and hardware facilities for
effective solving of a large number of problems in
optimization and mathematical modeling were constructed
and examined [2, 4, 14, 30, 37, 39 - 43, 48]. An essential
aspect of this activity is developing a system of algorithms,
utilities and programs based on a new mathematical calculus
which is called Idempotent Mathematics (Idempotent Analysis
or Idempotent Calculus), see [30, 31, 33, 36-44].
8. Numerical Analysis and Interval Analysis
Nonstandard basic numerical algorithms and their
implementations for decimal computers were described in the
book [2] and in [29]. For example, all elementary functions
can be calculated (for ordinal or double precision) using
a Jacobi fraction with a fixed number of parameters.
Several approximate construction methods for rational
approximations to functions and their computer realizations
were developed and an effect of error autocorrection was
discovered and formulated. It was shown that this effect
allows us to control calculation errors by changing the
construction procedure depending on the form of the
corresponding approximant [4, 29]. Construction methods for
linear and nonlinear Pade-Chebyshev rational approximations
involving a computer algebra system were also described [4,
14, 29].
A general form of the error autocorrection effect was
examined and it was shown that in the case of this effect
all the standard methods of "smooth" interval analysis
(including the so-called a posteriori interval analysis in
the sense of Yu. Matiyasevich) lead to nonreasonable (very
pessimistic) estimates.
In the framework of Idempotent Mathematics a version of
"nonsmooth" interval analysis was developed especially for
optimization problems [40, 43]. In this case all estimates
are exact for algorithms presented in terms of Idempotent
Mathematics. Interval solutions to a linear matrix equation
over idempotent semirings (the so-called discrete Bellman
equation) were constructed and it was proved that the
computational complexity of the problem is polynomial; in
the traditional interval analysis, standard problems of
linear algebra are NP-hard.
G. L. Litvinov
February 2007
PUBLICATIONS IN RUSSIAN (1-107)
(for publications in English see below)
1. On completely irreducible representations of complex and real
nilpotent Lie groups // Funktsional. Analiz i ego Prilozhen., v. 3, # 4,
1969, p. 87-88.
2. Group algebras of analytic functionals and their representations //
Doklady AN SSSR, v. 190, # 4, 1970, p. 769-771.
3. On continuous representations of metabelian Lie groups // Uspehi
Matemat. Nauk, v. 26, # 4, 1971, p. 237-238.
4. The Laplace transform on Lie groups // Funktsional. Analiz i ego
Prilozhen., v. 6, # 1, 1972, p. 83-84.
5. Representations of groups in locally convex spaces and topological
group algebras // Trudy Semin. Vektor and Tenzor Anal., v. 16, 1972,
p. 267-349.
6. General concepts of Quantum Mechanics. - In: F.A.Berezin et al.,
Lectures on Quantum Mechanics, Moscow State Univ., Dep. Math. Mech. Publ.,
1972, p. 5-44.
7. (with M.A.Shubin) Operators in Hilbert space. Spectral theorems.
Ibid., p. 222-262.
8. Characters and simple representations // Doklady AN SSSR, v. 214,
# 3, 1974, p. 29-32.
9. On traces of operators and characters of irreducible representations
in locally convex spaces // Ibid., v. 214, # 5, 1974, p. 1027-1030.
10. On conditions under which a representation is determined by its
character, up to equivalence // Trudy Semin. Vektor. And Tenzor. Anal., v. 17,
1974, p. 325-349.
11. (with B.G.Martynov) Textbook on Calculus, part 2. - Moscow, Eng.
Acad. Publ., 1974. - 220 pp.
12. (with M.S.Onuhov) Textbook on Calculus, part 3, Ibid., 1975. - 222 pp.
13. (with N.A.Berezina) Mathematical Statistics Algorithms Collection. -
All-Union State Institute for Statistics Technology Publ., 1975. - 19 pp.
14. (with G.A.Denisevich) Statistical data processing with
minicomputers of the M5000 type // Vestnik Statistiki, 1976, # 9, p. 49 - 52.
15. (with G.A.Denisevich) Evaluation of small computers possibilities
for data processing by means of mathematical methods. - In: Appendix to
Proceedings of the Association of M5000 computers users, Dnepropetrovsk,
1976, p. 22-28.
16. (with M.V.Dunskaya) Abstract Data Types, - Moscow, All-Union State
Institute for Statistics Technology Publ., 1977. - 40 pp.
17. (with M.V.Dunskaya) PVK M5000 (M5010) System Software Package for
Statistical Data Analysis, - Moscow, All-Union State Institute for Statistics
Technology Publ., 1977. - 52 pp.
18. Mathematical Statistics Software for ES EVM Computers, - Moscow,
All-Union State Institute for Statistics Technology Publ., 1977. - 51 pp.
19. (with E.A.Dmitrieva et al.) Standard Mathematical Programs.
Software for PVK M5000. - Moscow, All-Union State Institute for
Statistics Technology Publ., 1978. - 137 pp.
20. On generalized translation operators and their representations //
Trudy Semin. Vector. Tenzor. Anal., v. 18, 1978, p. 345-371.
21. Double topological algebras and topological Hopf algebras // Ibid.,
p. 372-375.
22. (with G.B.Shpiz) Primary decompositions of finite dimensional
representations of Lie algebras and Lie groups // Funktsional. Analiz i
ego Prilozhen., v. 12, # 2, 1978, p. 86-87.
23. (with A.P.Kurantov et al.) Computer-aided System for Science and
Technics Information based on the USSR State Archive Fund. Technical project.
State registration number GR 77018384. - Moscow, VNIIDAD, 1978. - 502 pp.
24. On traces of linear operators in locally convex spaces // Funktsional.
Analiz i ego Prilozhen., v. 13, # 1, 1979, p. 73-74.
25. (with G.B.Shpiz) Primary Decompositions of finite dimensional
representations of Lie algebras and Lie groups // Trudy Semin. Vector. Tenzor.
Anal., v. 19, 1979, p. 250-262.
26. On traces of linear operators in locally convex spaces, Ibid.,
v. 19, 1979, p. 263-272.
27. (with A.P.Kurantov et al.) Computer-aided System for Science
and Technics Information based on the USSR State Archive Fund. Working
project. State registration number GR 77018384. - Moscow, VNIIDAD,
1979. - 1555 pp.
28. (with G.P.Galadzheva and E.A.Kondratyeva) Software for the
computer-aided system ASNTI GAF SSSR // Express-information. Archives,
1979, # 4 (Moscow, VNIIDAD Publ.), p. 13-17.
29. (with G.P.Galadzheva, V.G.Zinchenko) Hardware for the
computer-aided system ASNTI GAF SSSR // Ibid., p. 25-31.
30. (with G.P.Galadzheva et al.) Interactive information system. -
Proceedings of the All-Union conference "Dialogue-79", Serpukhov, 1979,
p. 213-215.
31. Statistical computer data processing. - In: Creation problems
for the Computer-Aided Science and Technics Information System based on
the USSR State Archive Fund. - Moscow, VNIIDAD Publ., 1981, p. 118-163.
32. (with M.V.Dunskaya) Abstract data types and their applications
prospects. - Ibid., p. 164-208.
33. (with V.I.Lomonosov) Density theorems in locally convex spaces and
their applications// Trudy Semin. Vektor. Tenzor. Anal., v. 20, 1981, p.210-227.
34. (with B.M.Levitan) Generalized translation operators. - In:
Mathematical Encyclopedia, v. 3, - Moscow, Soviet Encyclopedia Publ., 1981,
p. 1111-1116.
35. (with O.V.Golosov et al.) Technical Specifications for the ASNTI
GAF SSSR system development. State registration number GR 77018384. -
Moscow, VNIIDAD, 1981. - 22 pp.
36. (with R.I.Tolstyh, L.I.Shadurskaya, E.A.Dmitrieva) Mathematical
Algorithms and Programs for Small Computers. - Moscow, Finances and Statistics
Publ., 1981. - 112pp.
37. (with R.M.Borisyuk) Program implementation of elementary functions
calculation algorithms for the "Mir" type computers. - In [36], Appendix,
p. 103-109.
38. Approximate construction of rational approximations. In: XV Voronezh
Winter Mathematical School. - Voronezh State University, 1981, p. 62-63.
39. Organization of elementary functions computer calculation. - Ibid.,
p.64.
40. Boolean algebras and automata. - In: L.A.Basova et al., Lectures
and Problems on Mathematics. - Moscow, Education Publ., 1981, p. 52-70.
41. (with L.I.Vainerman) The Plancherel formula and the inversion
formula for generalized translation operators // Doklady AN SSSR, v. 257,
# 4, 1981, p. 792-795.
42. (with V.I.Lomonosov) Density theorems in locally convex spaces and
irreducible representations // Doklady AN SSSR, v. 257, # 4, 1981, p. 826-830.
43. (with O.V.Golosov et al.) Basic System Analysis for the ASNTI GAF
SSSR System. - Moscow, VNIIDAD, 1981. - 479 pp.
44. (with N.A.Berezina et al.) Development of the ASNTI GAF SSSR System.
- In: Proc. of XII sci. seminar "GASNTI System Investigations", v. II,
1981, p. 60-62.
45. (with A.P.Kurantov et al.) Creation and exploitation start of
the ASNTI GAF SSSR system. Final Report. State registration number GR
7708384. - Moscow, VNIIDAD, 1981. - 48 pp.
46. (with G.P.Galadzheva et al.) Hardware and software for the ASNTI
GAF SSSR system. - In: High Efficiency Systems of Information Retrieval
and Data Base Management. - Proc. Of All-Union Symposium, - Kishinev, 1982,
p. 43-56.
47. (with N.A.Berezina et al.) Problems and construction principles
for the ASNTI GAF SSSR system. - In: Proc. of the All-Union Conference
"Problems of Information Support for Fundamental and Applied Scientific
Investigations after XXVI KPSS Congress Solutions. (Zvenigorod, 4-6 May,
1982)", v. 2. - Moscow, USSR Sci. Acad. Publ., 1982, p.123-125.
48. (with I.A.Andreeva et al.) Dialogue tools development for the ASNTI
GAF SSSR system. In: Proceedings of the All-Union conference "Man-Machine
Dialogue", v. 1 - Leningrad, LIAP Publ., 1982, p.24-26.
49. Approximation properties of locally convex spaces and the trace
problem // Teor. Funktsii, Funktsional. Anal. i Prilozhen., v. 39, 1983,
p. 73-87.
50. General concepts of quantum mechanics. - In: F.A.Berezin et al.,
The Schroedinger Equation. - Moscow, Moscow State Univ. Publ., 1983, p. 7-45.
51. (with M.A.Shubin) Spectral theory of operators in Hilbert spaces. -
In [49], Appendix 1, p. 294-366.
52. (with V.N.Friedman) Approximate construction of rational
approximations // Comptes Rendus de l'Academie Bulgare des Sciences,
v. 36, # 1, 1983, p. 49-52.
53. Software for the ASNTI GAF SSSR System. - In: Applications
of System Analysis Methods to the ASNTI GAF SSSR system. - Moscow, VNIIDAD
Publ., 1983, p. 132-150.
54. (with I.A.Andreeva et al.) The publication subsystem of the ASNTI
GAF SSSR system. - In: Proc. of the 4-d All-Union Conference "Problems of
Computer-Aided Support for Processing of Scientific and Technical
Information", - Moscow, VIMI Publ., 1983, p. 35-37.
55. (with I.A.Andreeva et al.) The ASNTI GAF SSSR System Development.
Technical project. State registration number GR 77018384. - Moscow,
VNIIDAD, 1983. - 500 pp.
56. (with I.A.Andreeva et al.) The ASNTI GAF SSSR System Development.
Working project. State registration number GR 77018384. - Moscow, VNIIDAD,
1984. - 980 pp.
57. (with I.A.Andreeva et al.) Dialogue preparation of publications
by means of the ASNTI GAF SSSR system. - In: Proc. Of the 3-d All-Union
‘onference "Man-Computer Dialogue", Serpukhov, 1984, p. 259.
58. Calculation system for decimal computers. - In: Personal Dialogue
Systems Based on Microcomputers (Personal Computers). Dialogue - 84 -
Micro. Proc. All-Union Conference. Leningrad, 19-22 November 1984. -
Leningrad, "Nauka Publ.", 1984, p. 86-88.
59. (with V.A.Sidorova) Problems of sample method applications for mass
documentation analysis. - Moscow, VNIIDAD, 1984. - 31 pp.
60. (with I.A.Andreeva) Extended time standards for research tasks in
VNIIDAD, - Moscow, VNIIDAD, 1985. - 55 pp.
61. (with I.A.Andreeva et al.) PADE - Rational Approximants Calculation.
Description and Program. - Pushchino, Research Computer Centre, Program
Fund, 1985. - 33 pp.
62. Fredholm kernel. - In: Mathematical Encyclopedia, v. 5. - Moscow,
"Soviet Encyclopedia Publ.", 1985, p. 660.
63. Nuclear C*-algebra.- Ibid., p. 1029.
64. Nuclear bilinear form.- Ibid., p. 1029-1030.
65. Nuclear norm.- Ibid., p. 1031-1032.
66. Nuclear space.- Ibid., p. 1033-1037.
67. Nuclear operator.- Ibid., p. 1037-1041.
68. Kernel of an integral operator. - Ibid., p. 1042.
69. Kernel of a linear operator. - Ibid., p. 1043.
70. Problem oriented small computers and information arrays processing.
- In: Computer-Aided Document Information Processing. - Moscow, VNIIDAD,
1985, p. 1-22.
71. Singularities of small computers software. - Ibid., p. 23-46.
72. (with I.A.Andreeva et al.) Formation and processing of information
arrays in the publication subsystem of the ASNTI GAF SSSR system. - In:
Proc. Semin. Computer-Aided Data Processing. - Moscow, VIMI Publ., 1985,
p. 38-40.
73. (with I.A.Andreeva) The publication subsystem of the ASNTI GAF
system // Express-information. Archives; 1985, # 3 (35), p.1-12.
74. Hypergroups and hypergroup algebras. - In: Modern Problems of
Mathematics. Recent Achievements. v. 25. - Moscow, VINITI, 1985, p. 57-106.
75. (with A.P.Krukov, A.Ya.Rodionov) Application of the REDUCE
system to rational approximants construction. - In: "Analytical
Computer Calculations and their Applications in Theoretical Physics".
Proc. Int. Conf. on Computer Algebra and its Applications in Theoretical
Physics. Dubna, 17-20 September, 1985. - OIYaI Publ., Dubna, 1985, p. 322-326.
76. (with O.V.Golosov et al.) Development of the ASNTI GAF SSSR system.
Sci. report. State registration number GR 01822040055. - Moscow,
VNIIDAD, 1985. - 78 pp.
77. (with I.A.Andreeva et al.) Standards for numbers of employers
in state archive institutions. - Moscow, VNIIDAD, 1985. - 11 pp.
78. (with I.A.Andreeva et al.) Standards for numbers of employers
in central republican archive administrations. - Moscow, VNIIDAD, 1985. - 4 pp.
79. (with I.A.Andreeva et al.) PADE - rational approximants calculation.
- In: Information Catalogue of Program Tools in Program Fund of the USSR
Sci. Acad. Research Computer Centre. - Pushchino, NIVTs AN SSSR Publ.,
1985, p. 10-11; 1987, p. 11-12.
80. (with I.A.Andreeva) SPLINE-PADE Program System and its Applications
to Standards Development. - Moscow, VNIIDAD, 1985. - 19 pp.
81. (with Ya.I.Fet) A device for function approximations. - Soviet
Patent # 1488838 with August 7, 1986 priority. - Moscow 1986, - 12 pp.;
"Discoveries. Inventions". Official Bull., 1989, # 23, p. 234.
82. (with V.A.Komarov et al.) Computer-Aided Library Information
System for VPTB (ASODI VPTB). Technical Report, State registration number
GR 01.87.0006435. - Moscow, USSR State Inventions and Discoveries
Committee, All-Union Patent Techn. Library, 1987. - 70 pp.
83. (with V.M.Zonov) Actual problems of artificial intellect methods for
computer input and transformation systems. - In: Proc. Semin. "Problems of
Linguistic Technics Development". - Moscow, "Lingvotekhnika Publ.", 1989,
p. 40-45.
84. Calculation organization method for decimal computers. - Ibid.,
p. 46-53.
85. Approximate construction of rational approximations and an effect
of error autocorrection. - In: Mathematics and Modelling (A.D.Bazykin, Yu.G.
Zarkhin, Eds). - Pushchino, Sci. Research Centre of the USSR Sci. Acad.,
1990, p. 99-141.
86. (with V.P. Maslov) Idempotent Mathematics: correspondence principle
and its computer applications // Uspekhi Matem. Nauk (Russian Matematicle
Surveys), v. 51, # 6, 1996, p. 209-210.
87. (with V.P. Maslov and G.B. Shpiz) Linear functionals on idempotent
spaces: algebraic approach // Doklady Akademii Nauk (Doklady Mathematics),
v. 363, # 3, 1998, p. 298-300.
88. (with V.P. Maslov and G.B. Shpiz) Tensor products of idempotent
semimodules. An algebraic approach // Matem. Zametki (Math. Notes), v. 65,
# 4, 1999, p. 572-585.
89. (with E.V. Maslova) Universal numerical algorithms and their
software implementations // Programming, 2000, # 5, p. 53-62.
90. (with E.V. Maslova) Universal algorithms and arbitrary precision
calculations // Computational Technologies, v. 5, # 2, 2000, p. 68-70.
91. (with V.P. Maslov) Idempotent Mathematics. - In: Modern Analysis
and its Applications. Voronezh Winter Mathematical School, Voronezh: Voronezh
State University, 2000, p. 20-22.
92. (with I.A. Andreeva) Spaces of functions on smooth spaces, Ibid.,
p. 110-111.
93. Universal algorithms and Idempotent Mathematics // Computer Tools
in Educations, 2000#6, (St. Petersburg), p. 12-18.
94. (with A.N. Sobolevskii) Exact interval solutions of the discrete
Bellman equation and polynomial complexity of problems in interval idempotent
linear algebra // Doklady Akademii Nauk (Doklady Math.), v. 374, # 3, 2000,
p. 304-306.
95. (with V.P. Maslov and A.N. Sobolevskii) Idempotent Mathematics and
interval analysis // Computational Technologies, v. 6, # 6, 2001, p. 47-70.
96. (with A.Ya. Rodionov and A.V. Churkin) Approximate rational arithmetic
with controlled round off errors // Computational Technologies, v. 6, # 5,
2001, p. 87-94.
97. (with V.P. Maslov and G.B. Shpiz) Nondigital implementation of the
arithmetic of real numbers by means of quantum computer media // Matem. Zametki
(Math. Notes), v. 70, # 1, 2001, p. 59-67.
98. (with V.P. Maslov and G.B. Shpiz) Idempotent functional analysis:
an algebraic approach// Matem. Zametki (Math. Notes), v. 69, # 5, 2001,
p. 758-797.
99. (with G.B. Shpiz) Nuclear Semimodules and Kernel Theorems in Idempotent
Analysis: An Algebraic Approach// Doklady Akademii Nauk, Vol. 386, # 3, 2002,
p. 300-303.
100. The Maslov dequantization and idempotent (tropical) mathematics. --
In : Voronezh Winter Mathematical School--2004, Voronezh State University,
Voronezh, 2004, p. 71-72.
101. (with G.B. Shpiz) Dequantization and Newton polytopes. -- Ibid.,
p.114-115.
102. (with G.B. Shpiz) Dequantization of mathematics and representations
of groups//Vestnik Tambovskogo Universiteta, ser. Estestvoznanie i
tekhnicheskie nauki, Tambov, v. 10, #4, 2005, p.390-411.
103. The Maslov dequantization, idempotent and tropical mathematics:
A brief introduction//Zapiski nauchnyh seminarov POMI, v. 326, 2005,
p.145-182. E-print arXiv:math.GM/0507014
104. (with A.N. Sobolevskii) Positive interval algorithms and idempotent
mathematics. -- In: Vserossijskoe (s mezhdunarodnym uchastiem) soveschanie
po interval'nomu analizu i ego prilozhenoyam "INTERVAL-06, July 1-4, 2006,
Peterhof, Russia. Extended abstracts. St. Petersburg, 2006, p. 72-74.
105. (with G.B. Shpiz) Interval solutions of generalized algebraic
equations in idempotent mathematics. -- Ibid., p.75-77.
106. Kernel theorems and nuclearity in idempotent mathematics. An algebraic
approach//Zapiski nauchnyh seminarov POMI, v. 331, 2006, p. 60-83.
107. Dequantization of mathematics. -- In: Globus: general-mathematical
seminar, Vol. 4. MCCME Publ., Moscow, in press.
PUBLICATIONS IN ENGLISH (108-186)
(for publications in Russian see above)
1 (108). On completely irreducible representations of complex and real
nilpotent Lie groups// Functional Anal. Appl. v. 3, # 4, 1970, p. 332-334.
2 (109). Group algebras of analytic functionals and their representations
// Soviet Math. Dokl., v. 11, # 1, 1970, p. 191-193.
3 (110). The Laplace transform on Lie groups // Functional Anal. Appl.,
v. 6, # 2, 1972, p. 76,77.
4 (111). Characters and simple representations // Soviet Math. Dokl.,
v. 15, # 1, 1974, p. 179-183.
5 (112). On Traces of operators and characters of irreducible
representations in locally convex spaces // Soviet Math. Dokl., v. 15, # 1,
1974, p. 308-312.
6 (113). (with G.B.Shpiz) Primary decompositions of finite dimensional
representations of Lie algebras and Lie groups // Functional Anal. Appl.,
v. 12, # 2, 1978, p. 149-151.
7 (114). Traces of linear operators in locally convex spaces //
Functional Anal. Appl. v. 13, # 1, 1979, p. 60-62.
8 (115). (with L.I.Vainerman) The Plansherel formula and the inversion
formula for generalized translation operators // Soviet Math. Dokl., v. 23,
# 1, 1981, p. 333-337.
9 (116). (with V.I.Lomonosov) Density theorems in locally convex
spaces and irreducible representations // Soviet Math. Dokl., v. 23, # 2,
1981, p. 372-375.
10 (117). (with A.P.Kryukov and A.Ya.Rodionov) Construction of rational
approximations by means of REDUCE. - In: Proceedings of the 1986 Symposium
on Symbolic and Algebraic Computation. Symsac'86 (July 21-23, 1986,
Waterloo, Ontario), Univ. of Waterloo, 1986, p. 31-33.
11 (118). Hypergroups and hypergroup algebras // Journal of Soviet
Math., v. 38, 1987, p. 1734-1761.
12 (119). Representations of groups in locally convex spaces and
topological group algebras // Selecta Mathematica Sovietica, v. 7, # 2, 1988,
p. 101-182.
13 (120). On conditions under which a representation is determined
by its character, up to equivalence // Selecta Mathematica Sovietica,
v. 8, # 1, 1989, p. 73-97.
14 (121). On the traces of linear operators in locally convex spaces //
Selecta Mathematica Sovietica, v. 8, # 3, 1989, p. 203-212.
15 (122). (with V.I.Lomonosov) Density theorems in locally convex
spaces and applications // Selecta Mathematica Sovietica, v. 8, # 4, 1989,
p. 323-339.
16 (123). Fredholm kernel. - Encyclopaedia of Mathematics, v. 4 - Kluwer
Academic Publ., 1989, p. 103.
17 (124). (with B.M.Levitan) Generalized displacement operators.
- Encyclopaedia of Mathematics, v.4. - Kluwer Academic Publ., 1989, p. 224-228.
18 (125). Kernel of a linear operator. - Encyclopaedia of Mathematics,
v. 5. - Kluwer Academic Publ., 1990, p. 263.
19 (126). Kernel of an integral operator. - Encyclopaedia of Mathematics,
v.5. - Kluwer Academic Publ., 1990, p. 265.
20 (127). Nuclear bilinear form. - Encyclopaedia of Mathematics,
v. 6. - Kluwer Academic Publ., 1990, p. 485.
21 (128). Nuclear C*-algebra. - Encyclopaedia of Mathematics, v. 6. -
Kluwer Academic Publ., 1990, p. 485-486.
22 (129). Nuclear Norm. - Encyclopaedia of Mathematics, v. 6. - Kluwer
Academic Publ., 1990, p. 486-487.
23 (130). Nuclear operator. - Encyclopaedia of Mathematics,v. 6. -
Kluwer Academic Publ., 1990, p. 487-490.
24 (131). Nuclear space. - Encyclopaedia of Mathematics, v. 6. - Kluwer
Academic Publ., 1990, p. 490-493.
25 (132). General concepts of Quantum Mechanics. - In: F.A.Berezin
et al., The Schroedinger Equation - Kluwer Academic Publ., Dordrecht/Boston/
London, 1991, p. 1-49.
26 (133). (with M.A.Shubin) Spectral theory of operators in Hilbert
space. - In: F.A. Berezin et al., The Schroedinger Equation. - Kluwer
Acad. Publ., Dordrecht/Boston/London, 1991, p. 386-465.
27 (134). Generalized translation operators and their representations //
Selecta Mathematica Sovietica, v. 10, # 4, 1991, p. 307-337.
28 (135). Double topological algebras and topological Hopf algebras //
Selecta Mathematica Sovietica, v. 10, # 4, 1991, p. 339-343.
29 (136). Approximation properties of locally convex spaces and the
problem of uniqueness of the trace of a linear operator // Selecta Mathematica
Sovietica, v. 11, # 1, 1992, p. 25-40.
30 (137). (with G.B.Shpiz) Primary decompositions of finite dimensional
representations of Lie algebras and Lie groups // Selecta Mathematica
Sovietica, v. 11, # 1, 1992, p. 41-54.
31 (138). Error auto-correction in rational approximation. - In: Proc.
Intern. Conf. on Interval and Stochastic Methods in Science and Engineering
(Interval-92), Moscow, 1992, v. 2, p.59-63.
32 (139). Error auto-correction in rational approximation // Interval
Computations, # 4(6), 1992, p. 14-18.
33 (140). On Lie hypergroups. - In: Abstracts Intern. Lie-Lobachevsky
Colloq. Tartu, October 26-30, 1992.
34 (141). Error auto-correction in rational approximation. - In: Intern.
Congress on Computer Systems and Applied Mathematics. CSAM'93. Abstracts.
St. Petersburg, July 19-23, 1993. - Polylog Publ., St. Petersburg, 1993,
p. 64-65.
35 (142). And the effect of error autocorrection. - In: A. Bazykin and
Yu. Zarkhin (Eds), Mathematics and Modelling. - Nova Science Publ., New York,
1993, p. 143-198.
36 (143). Approximate construction of rational approximations and an
effect of error autocorrection // Russian Journal of Mathematical Physics,
v. 1, # 3, 1994, p. 313-352 (and preprint: Inst. Math., University of Oslo,
# 8, May 1993 - 56 pp.) E-print arXiv:math.NA/0101042
37 (144). (with V.P. Maslov and A.Ya. Rodionov) Unifying Approach to
Software and Hardware Design for Scientific Calculations. International Sophus
Lie Centre, Moscow, 1995.
See also an updated e-version: E-print arXiv:quant-ph/9904024 v. 2
38 (145). (with V.P. Maslov) Correspondence principle for Idempotent
Calculus and some computer applications. - Preprint IHES, Bures-sur-Yvette,
1995; the same in: Idempotency/J. Gunawardena (Ed.), Cambridge University
Press, Cambridge, 1998, p. 420-443; E-print arXiv:math.GM/0101021
39 (146). (with V.P. Maslov) Idempotent Mathematics: correspondence
principle and applications//Russian Mathematical Surveys, v. 51, # 6, 1996,
p. 1210-1211.
40 (147). Interval computations and the effect of error auto-correction.
- In: Interval'96. International Conference on Interval Methods and Computer
Aided Proofs in Science and Engineering, Sep. 30 - Okt. 2, 1996, Wuerzburg,
Germany. Extended Abstracts. Wuerzburg, 1996, p. 76.
41 (148). (with V.P. Maslov and A.Ya. Rodionov) Idempotent mathematics
and scientific calculations, Ibid., p. 77-78.
42 (149). Idempotent Algorithm. - Encyclopaedia of Mathematics,
Supplemente Volume 1. - Kluwer Acad. Publ., 1997, p.312-313.
43 (150). Idempotent Analysis. - Ibid, p. 313-315.
44 (151). Idempotent Correspondence Principle. - Ibid, p. 315-316.
45 (152). Idempotent semiring. - Ibid, p. 316.
46 (153). Idempotent superposition principle. - Ibid, p. 317.
47 (154). (with V.M. Tulovsky) Generalized Fourier Transform, Maslov's
Canonical Operator and Hypergroups. - International Sophus Lie Centre,
Moscow, 1997.
48 (155). (with B. Komrakov, I. Krasil'shchik, and A. Sossinsky, Editors)
Lie Groups and Lie Algebras, Their Representations, Generalizations, and
Applications. - Kluwer Acad. Publ., Dordrecht et al., 1998.
49 (156). (with K.A. Ross et al., editors) Harmonic Analysis and
Hypergroups ("Trends in Mathematics" series) - Birkhauser, 1998.
50 (157). (with V.P. Maslov and G.B. Shpiz) Linear functionals on
idempotent spaces: an algebraic approach // Doklady Mathematics, v. 58, # 3,
1998, p. 389-391. E-print arXiv:math.FA/0012268
51 (158). (with V.P. Maslov and A.N. Sobolevskii) Idempotent Mathematics
and Interval Analysis. - The Erwin Schroedinger International Institute for
Mathematical Physics, Vienna, 1998, . E-print arXiv:math.NA/9911126
(see also http://www.esi.ac.at).
52 (159). (with V.P. Maslov and G.B. Shpiz) Idempotent Mathematics - In:
International Conference Dedicated to the 90th Anniversary of L.S. Pontryagin.
Abstracts. Algebra, Geometry and Topology. August, 31 - September, 6, 1998,
Moscow. Steklov Mathematical Institute and Moscow State (Lomonosov) University,
Moscow, 1998, p. 38-39.
53 (160). (with V.P. Maslov and G.B. Shpiz) Tensor products of idempotent
semimodules. An algebraic approach // Mathematical Notes, v. 65, # 4, 1999,
p. 479-489. E-print arXiv:math.FA/0101153
54 (161). (with E.V. Maslova) Universal numerical algorithms and their
software implementation // Programming and Computer Software, v. 26, # 5, 2000,
p. 275-280. E-print arXiv:math.SC/0102114
55 (162). (with A.N. Sobolevskii) Universal algorithms, idempotent
mathematics, and interval solutions of optimization problems. - In: 6th
International IMACS Conference on Applications of Computer Algebra, June 25-28,
2000, St. Petersburg, Russia. Abstracts. - Euler Institute and IMACS, St.
Petersburg, 2000, p. 84-85.
56 (163). (with V.P. Maslov) Idempotent Mathematics and Optimization. -
In: 11th IFAC International Workshop "Control Applications of Optimization",
CAO'2000, July 3-6, 2000, St. Petersburg, Russia. Abstracts. St. Petersburg,
2000, p. 144-145.
57 (164). (with A.N. Sobolevskii) Idempotent interval analysis, matrix
algebra, and optimization on graphs, Ibid, p. 242-243.
58 (165). (with A.N. Sobolevskii) Exact interval solutions of the discrete
Bellman equation and polynomial complexity of problems in interval idempotent
linear algebra // Doklady Mathematics, v. 62, # 2, 2000, p. 199-201.
E-print arXiv:math.LA/0101041
59 (166). (with V.P. Maslov and A.Ya.Rodionov) A unifying approach
to software and hardware design for scientific calculations and idempotent
mathematics. - International Sophus Lie Centre, Moscow 2000.
E-print arXiv:math.SC/0101069
60 (167). (with A.Ya. Rodionov and A.V. Churkin) Approximate rational
arithmetics and arbitrary precision calculations. - International Sophus Lie
Centre, Moscow, 2000. E-print arXiv:math.NA/0101152
61 (168). Numerical mathematics should be taught differently. - In:
The 9th International Congress on Mathematical Education, ICME-9. Topics Study
Group 10. Trends in Mathematics and Mathematical Sciences. Reflections on
Mathematics Education. August 1-2, 2000, Tokyo/ Makuhari, p. 18-21.
62 (169). (with A.N. Sobolevskii) Idempotent interval analysis and
optimization problems // Reliable Computing, v. 7, # 5, 2001, p. 353-377.
E-print arXiv:math.SC/0101080
63 (170). (with V.P. Maslov) Idempotent Mathematics and Mathematical
Physics. - In: International Workshop "Topology, Analysis and Related Topics.
Moscow State University, August 29-31, 2001". Moscow, 2001, p. 13.
64 (171). (with V.P. Maslov and G.B. Shpiz) Idempotent functional analysis.
An algebraic approach // Mathematical Notes, v. 69, # 5, 2001, p. 696-729.
E-print arXiv:math.FA/0009128
65 (172). (with V.P. Maslov and G.B. Shpiz) Nondigital implementations of
the arithmetic of real numbers by means of quantum computer media //
Mathematical Notes, v. 70, # 1, 2001, p. 53-60. E-print arXiv:quant-ph/9904025
66 (173). (with V.P. Maslov and G.B. Shpiz) Idempotent (Asymptotic)
Analysis and the Representation Theory. - In: V.A. Malyshev and A.M. Vershik
(eds.), Asymptotic Combinatorics with Applications to Mathematical Physics.
Kluwer Academic Publishers, Dordrecht et al, 2002, p. 267-278.
E-print arXiv:math.RT/0206025
67 (174). (with G.B. Shpiz) Nuclear Semimodules and Kernel Theorems in
Idempotent Analysis: An Algebraic Approach/Doklady Mathematics, Vol. 66, # 2,
2002, p. 197-199. E-print arXiv:math.FA/0202026
68 (175). Error Autocorrection in Rational Approximation and Interval
Estimates//Central European Journal of Mathematics, Vol. 1, # 1, 2003, p. 36-60.
See also math.NA/0207183
69 (176) (with M.I. Graev) Generalized Fourier Transforms and Hypergroups,
generated by Problems of Integral Geometry. - In: Intern. Conf. KOLMOGOROV and
CONTEMPORARY MATHEMATICS, Moscow, June 16-21, 2003. Abstracts. Moscow State
University and Russian Acad. Sci., Moscow, 2003, p. 172-173.
70. (177) Dequantization of Mathematics, idempotent semirings and fuzzy
sets -- In: LINZ 2004. 25th Linz Seminar on Fuzzy Set Theory "Matematics of
Fuzzy Systems". Linz, Austria, February 3-7, 2004. Abstracts/E.P. Klements,
E. Pap, Editors, Johannes Kepler Univ., Linz, 2004, p. 113-117.
71. (178) The Maslov dequantization, idempotent and tropical mathematics.
-- In: Intern. Conf. "Topology, Analysis and Applications to Mathematical
Physics. Moscow, 14-18 February, 2005. Abstracts", Moscow State University,
Moscow, 2005, p.43-44.
72. (179) (with G.B. Shpiz) Representation of groups in idempotent spaces:
an algebraic approach. -- In: Int. Conf. "Representation Theory, Dynamical
Systems, and Asymptotic Combinatorics. St. Ptersburg, Russia, June 8-13, 2004".
Abstracts of talks, St. Ptersburg, Euler Intern. Math. Inst., 2004, p. 15-16.
73. (180) (with V.P. Maslov, eds.) Idempotent Mathematics and Mathematical
Physics. Contemporary Mathematics, Vol. 377. Amer. Math. Soc., Providence,
Rhode Island, 2005. - 370 pp.
74. (181) The Maslov dequantization, idempotent and tropical mathematics:
a very brief introduction. -- Ibid., p.1-17. E-print arXiv:math.GM/0501038
75. (182) (with G.B. Shpiz) The dequantization transform and generalized
Newton polytopes. -- Ibid., p. 181-186. E-print arXiv:math-ph/0412090
76. (183) (with G.B. Shpiz) Dequantization of mathematics and
representations of groups. -- In: Intern. Conf. "Harmonic Analysis on
Homogeneous Spaces, Representations of Lie groups and Quantization.
April 25-29", 2005. Abstracts. Tambov State University, Tambov, 2005, p. 15.
77. (184) (with V.P. Maslov and G.B. Shpiz) Dequantization, logic, and
fuzzy sets. -- In: Intern. Conf. "Methods of Logic in Mathematics. July 18-14,
2005". Program. Abstracts. Euler Intern. Math. Inst., St. Petersburg, 2005,
p.27-28.
78. (185) (with G.B. Shpiz) Kernel theorems and nuclearity in idempotent
mathematics. An algebraic approach, Journal of Mathematical Sciences, v. 141,
#4, 2007, p. 1417-1428. E-print arXiv:math.FA/0609033
79. (186) The Maslov dequantization, idempotent and tropical mathematics:
A brief introduction, Journal of Mathematical Sciences, v. 140, #3, 2007,
p.426-444. E-print arXiv:math.GM/0507014
80. (187) (with G.B. Shpiz) The Maslov dequantization, convex geometry and
linear operators. -- In : International Conference "OPERATOR ALGEBRAS AND TOPOLOGY",
Moscow, January 29 - February 03, 2007. Abstracts. Moscow, Moscow State University,
2007, p. 21-22.
Last modified: Sat Mar 31 23:22:24 MSK 2007