- General Information
- Experience and Awards
- Lectures and Talks
- Research Interests
- Selected Publications
- Summary of Results
- Full List of Publications

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.

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