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,
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”,
2. Operations with Named Sets, Ordered Sets and Lattices,
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,
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
and others
cf. 1. On
the Nature and Essence of Mathematics,
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,”
2. Named Sets, General Theory of
Properties, and Logic,
3. Quantifiers in the Theory of Properties, in
“Non-standard Semantics in Non-classical Logics”,
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,
2. Logical Varieties and Covarieties, in Methodological and
Theoretical Problems of Mathematics and Information and Computer Sciences,
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,
2. Axiological Aspects of Scientific
Theories,
3. The World of Theories and the
Power of Mind,
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
3. Integral Calculus for Extrafunctions, Doklady of Academy
of Sciences of
and others
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,
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
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,
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
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,
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,
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,
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,
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”,
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,
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”,
C. Development (Important Contributions)
1. Theory of Superrecursive Algorithms
cf. 1. Super-recursive
Algorithms, Springer,
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,
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
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,
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,
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