M. BURGIN

 

MAIN SCIENTIFIC ACHIEVEMENTS

Books are in bold

 

What is the most important for science are not books or other publications,

but discoveries and, even more, theories.

 

 

 

 

A. Originating and Development (Important Contributions)

 

1. Theory of Non-Diophantine Arithmetics  

          cf. 1. Non-Diophantine Arithmetics or is it Possible that 2+2 is not Equal to 4? Ukrainian Academy of Information Sciences, Kiev, 1997   (in Russian, English summary)

               2. Diophantine and Non-Diophantine Aritmetics: Operations with Numbers in Science and Everyday Life, LANL, Preprint Mathematics GM/0108149, 2001, 27 p.     (electronic edition:   http://arXiv.org)

               3. Infinite in Finite or Metaphysics and Dialectics of Scientific Abstractions,    Philosophical and Sociological Thought, 1992, No. 8, pp.21-32    (in Russian and Ukrainian)

                                                                                                            and others

 

2.  Theory of Fundamental Triads

               a) mathematics  (Theory of Named Sets):

cf   1. Theory of Named Sets as a Foundational Basis for Mathematics, in “Structures in Mathematical Theories”, San Sebastian, 1990, pp. 417-420

2. Operations with Named Sets,   Ordered Sets and Lattices, Saratov, 1986, pp.3-12   (in Russian)

3. Unified Foundations of Mathematics, Preprint Mathematics LO/0403186, 2004, 39 p.

and others

 

               b) science:

cf.  1. Fundamental Structures of Knowledge and Information: Achieving an Absolute, Ukrainian Academy of Information Sciences, Kiev, 1997   (in Russian)

2. What is the Surrounding World Built of,   Philosophical and Sociological Thought, 1991, No. 8, pp.54-67   (in Russian )

3. On the way to the "Absolute": Triad is the most fundamental structure in human society,    Visnik of the Academy of Sciences of Ukraine, 1993, No. 5, pp. 29-35    (in Ukrainian)
                                                                                                        and others

 

             c) philosophy:

cf. 1. On the Nature and Essence of MathematicsUkrainian Academy of Information Sciences, Kiev, 1998 (in Russian)

2. Triad as a fundamental structure in human culture, Studia Culturologia, 1993, v.2, pp.51-63

3. Named Sets as Basic Tool in Epistemology,   Epistemologia, 1995, XVIII, pp.87-110

and others

 

3. General Theory of Properties

cf. 1. Abstract Theory of Properties and Sociological Scaling, in “Expert Evaluation in Sociological Studies,”  Kiev, 1990, pp. 243-264                               (in Russian) 

2. Named Sets, General Theory of Properties, and LogicInstitute of Philophy, Kiev, 1989     (in Russian)

3. Quantifiers in the Theory of Properties,  in “Non-standard Semantics in Non-classical Logics”, Moscow, 1986, pp.99-107    (in Russian)

and others

 

4. Theory of Logical Varieties

cf. 1. Logical Tools for Program Integration and Interoperability, in Proceedings of the 8th IASTED International Conference on Software Engineering and Applications, MIT, Cambridge, 2004, pp. 743-748  *

2. Logical Varieties and Covarieties, in Methodological and Theoretical Problems of Mathematics and Information and Computer Sciences, Kiev, 1997, pp.18-34    (in Russian)

3. Logical Tools for Inconsistent Knowledge Systems, Information: Theories & Applications, 1995, v. 3, No. 10, pp.13-18

                                                                                                       and others

 

5. The Structure-Nominative Approach in Methodology of Science (in collaboration)

cf. 1. Introduction to the Modern Exact Methodology of Science, Moscow, ISF, 1994  (in Russian)

2. Axiological Aspects of Scientific TheoriesKiev, Naukova Dumka, 1991, 183 p. (in Russian)

3. The World of Theories and the Power of MindKiev, Naukova Dumka, 1992, 231 p. (in Russian)

                                                                                                                            and others

 

6. Theory of Hypermeasures and Hyperintegration  

cf. 1. Hyperfunctionals and Generalized Distributions, in “Stochastic Processes and Functional Analysis” (Eds. Krinik, A.C. and Swift, R.J.; A Dekker Series of Lecture Notes in Pure and Applied Mathematics, v.238) 2004, pp. 81 - 119

2. Hypermeasures and Hyperintegration, Doklady of Academy of  Sciences of Ukraine, 1990, No. 6, pp.10-13     (in Russian and Ukrainian)

3. Integral Calculus for Extrafunctions,  Doklady of Academy of  Sciences of Ukraine, 1995, No. 11, pp.14-17
                                                                                                          and others

 

7. Neoclassical Analysis

cf. 1. Uncertainty and Imprecision in Analytical Context: Fuzzy Limits and Fuzzy Derivatives, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, v. 9, No. 5, 2001, pp. 563-685

2. Theory of Fuzzy Limits, Fuzzy Sets and Systems, 2000, v. 115, No. 3, pp. 433-443 *

3. Neoclassical Analysis: Fuzzy Continuity and Convergence, Fuzzy Sets and Systems, 1995, v. 75, No. 2, pp.291-299
                                                                                                        and others

 

8. General Theory of Information

 cf. 1. Entertainment and Information, Los Angeles, UCMA, 2001, 76 p.

2.   Information Theory: A Multifaceted Model of Information, Entropy, v. 5, No. 2, 2003, pp. 146-160

3.   Information Algebras, Control Systems and Machines, 1997, No. 6, pp.5-16     (in Russian)

                                                                                                                               and others

 

9. System Theory of Time

cf.  1. The Problem of Time Scales in Computer Visualization, in “Computational Science”, Lecture Notes in Computer Science, v. 2074, part II, 2001, pp.728-737   (in collaboration)

2.   Time as a Factor of Science Development, Science and Science of Science, 1997, No. 1/2, pp.45-59

3.   Elements of the System Theory of Time, LANL, Preprint in Physics 0207055, 2002, 21 p.

                                                                                                                                   and others

 

10. Theory of Intellectual Activity 

cf.  1. Intellectual Components of Creativity,    Aerospace Academy of Ukraine, Kiyv, 1998  (in Ukrainian)

2.   Intellectual Activity as a Psychological Base of New Ideas Creation, DANI, 1996, No. 1, pp. 14-15  (in Ukrainian)

3.   Static and Dynamic Approach to Person's Intelligence, Ukrainian Psychology: Modern Potential, Kiev, 1996, pp.154-159 (in Ukrainian)
                                                                                                       and others

 

11. Theory of Multinumbers and Multicardinal Numbers

cf. 1.  Finite and Infinite, in On the Nature and Essence of Mathematics, Appendix, 1998, pp.97-108   (in Russian)

2. Algebraic Structures of Multicardinal Numbers,    Problems of Group Theory and Homological Algebra, Yaroslavl, 1992, pp.3-20   (in Russian)

3. Upper Multicardinals, Abstracts presented to the American Mathematical Society, 1986, v.7, No. 4

                                                                                                       and others

12. Theory of Hypernumbers and Extrafunctions  *    

cf. 1. Theory of Hypernumbers and Extrafunctions: Functional Spaces and Differentiation, Discrete Dynamics in Nature and Society, v. 7, No. 3, 2002, pp. 201-212

2. Differential Calculus for Extrafunctions,  Doklady of Academy of  Sciences of Ukraine, 1993, No. 11, pp.7-11

3. On the Hahn-Banach theorem for hyperfunctionals, Doklady of the Ukrainian Academy of Sciences, 1991, No. 7, pp. 9-14     (in Russian, in Ukrainian)

and others
 

 

 

 

B. Originating

1. Mathematical Theory of Technology 

cf. 1. Mathematical Models for Simulating Technological Processes,  in “Proceedings of the Business and Industry Simulation Symposium,” Society for Modeling and Simulation International, San Diego, California, 2002, pp. 165-168             

2. A Technological Approach to the System Science-Industry-Consumption, Science and Science of Science, 1997, No. 3/4, pp. 73-88        (in Russian)

3. Mathematical Theory of Technology,   Methodological Problems of Mathematics and  Information Sciences, Kiev, 1997, pp. 91-100   (in Russian)

                                                                                                           and others

 

2. Scalable/Discontinuous Topology

cf. 1. Discontinuity Structures in Topological Spaces, International Journal of Pure and Applied Mathematics, 2004, v. 16, No. 4, pp. 485-513

2. Continuity and Connectedness in Discontinuous Topology, University of California, Los Angeles, Mathematics Report Series, MRS Report 01-06, 2001, 28 p.

3. Extended Fixed Point Theorem, Methodological Problems of Mathematics and  Information Sciences, Kiev, 1997, pp.52-60  (in  Russian)

          and others

3. Quantum Theory of Knowledge Systems  

                cf.  1. Data, Information, and Knowledge, Information, v. 7, No.1, 2004, pp. 47-57

2. Knowledge and Data in Computer Systems, in Proceedings of the ISCA 17th International Conference “Computers and their Applications”, International Society for Computers and their Applications, San Francisco, California, 2002, pp. 307-310

3. Knowledge Discovery, Information Retrieval, and Data Mining, in Proceedings of the ISCA 17th International Conference “Computers and their Applications”, International Society for Computers and their Applications, San Francisco, California, 2002, pp. 55-58                           (in collaboration)

                                                                                                     and others

4. Theory of Grid Automata

cf.  1. From Neural networks to Grid Automata, in Proceedings of the IASTED International Conference ”Modeling and Simulation”, Palm Springs, California, 2003, pp. 307-312

2. Cluster Computers and Grid Automata, in Proceedings of the ISCA 17th International Conference “Computers and their Applications”, International Society for Computers and their Applications, Honolulu, Hawaii, 2003, pp. 106-109     

 

 

5. Axiomatic Theory of Algorithms  

cf.  1. Axiomatic Theory of Algorithms: Computability and Decidability in Algorithmic Classes, Preprint in Mathematics LO/0409142, 2004, 27 p.

2. Algorithms and algorithmic problems, Programming, 1985, No. 4, pp. 3-14   (Programming and Computer Software, 1985)          (translated from Russian)

3. Complexity measures in the axiomatic theory of algorithms, in “Methods of design of applied intellectual program systems”, Kiev, 1992, pp. 60-67    (in Russian)

 

 

 

 

C. Development (Important Contributions)

1. Theory of Superrecursive Algorithms  

cf.  1. Super-recursive Algorithms, Springer, New York/ Heidelberg/ Berlin, 2005, 304 p. *

2. Nonlinear Phenomena in Spaces of Algorithms, International Journal of Computer Mathematics, v. 80, No. 12, 2003, pp. 1449-1476

3. How We Know What Technology Can Do, Communications of the ACM, v. 44, No. 11, 2001, pp. 82-88 *

                                                                                                     and others

 

2. Theory of Complexity of Algorithms and Computations

cf. 1. Dual complexity measures and expert system functioning, Problem solving in intellectual computer media, Kiev, 1991, pp. 118-126     (in Russian)

2. Complexity Measures on Systems of Parallel Algorithms,    Programming, 1984, No. 1, pp.17-28

3. Generalized Kolmogorov Complexity and Duality in the Theory of Computational Complexity,    Notices of the Academy of Sciences of the USSR, 1982, v.264, No. 2,   pp. 19-23

                                                                                                     and others

 

3. Theory of Linear Algebras and W-algebras

           cf. 1. Schreier Varieties of Linear Algebras,    Soviet Math. Sbornik, 1974, v. 93, No. 4, pp. 555-573

                 2. Kurosh Varieties of Linear W-algebras,    Problems of Theory of Groups and Homological Algebra, Yaroslavl, 1981, pp.17-27  (in Russian)

                3. Linear  W-algebras,   Russian Mathematical Surveys, 1975, v. 30, No. 4, pp. 61-106              (translated from Russian, in collaboration)
                                                                                                      and others
 

4. Theory of Knowledge Megasystems (Scientific Theories) 

cf. 1. Model Part of a Scientific Theory, Epistemologia, 1992, XV, No. 1, pp. 41-64     (in collaboration)

2. Methodological models of scientific knowledge, Methodological Consciousness in Modern Science, Kiev, 1989, pp. 199-308       (in Russian, in collaboration)

3. New dimensions of scientific theory, Visnik of the Academy of Sciences of Ukraine, 1990, No. 10, pp. 26-30     (in Ukrainian, in collaboration)

                                                                                                     and others