**M.S.BURGIN**

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

3. *Non-classical Models of Natural
Numbers*, Russian Mathematical Surveys, 1977, v.32, No. 6, pp.
209-210 (in Russian)

and others

**2. Theory of
Fundamental Triads**

**
a) mathematics (Theory of Named Sets):**

cf 1. ** Theory of Named Sets,** Nova Science Publishers, New York, 2011

2. *Unified Foundations of Mathematics*, Preprint Mathematics
LO/0403186, 2004, 39 p. (electronic
edition:
http://arXiv.org)

3. *Algebraic Structures of Multicardinal Numbers*, Problems of group
theory and homological algebra, Yaroslavl, 1992, pp. 3‑20 (in Russian)

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, N8, 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 Mathematics**, 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,” * *(in
Russian)*
*

2. *Abstract Theory of Properties*, Non-classical
Logics, Moscow, 1985, pp.109-118 (in Russian)

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

and others

**4. Theory of Logical Varieties **

cf. 1. *Consistent structuring of inconsistent knowledge*, Journal of Intelligent Information
Systems, v. 45, No. 1, 2015, pp. 5-28
(in collaboration)

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

3. *Logical Methods in Artificial Intelligence Systems*, Vestnik of
the VOIVT, 1991, N2, pp.66-78 (in Russian)

and others

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

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

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. *Integration in Bundles with a
Hyperspace Base*:* Definite Integration*,
Integration: Mathematical Theory and Applications, v. 3, No. 1, 2012, pp. 1-54

2. *Hypermeasures in General Spaces*, International Journal of Pure and
Applied Mathematics, 2005, v. 24, No. 3, pp. 299-323

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

and others

**7. Neoclassical Analysis**

cf. 1. ** Neoclassical Analysis: Calculus
closer to the Real World,** Nova Science Publishers, New York, 2008

2. *Fuzzification of the Theory of Continuous Functions*,
Fuzzy Sets and Systems, 1994, v. 62, No. 1, pp.71-81 (in
collaboration)

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. ** Theory of Information: Fundamentality, Diversity and Unification**, World Scientific, New
York/London/Singapore, 2010

2. *Information* *Operators in Categorical
Information Spaces*, Information, v. 1, No.1, 2010, pp. 119 - 152

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

and others

**9. 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
17^{th} International Conference “*Computers
and their Applications*”, International Society for Computers and their
Applications, San Francisco, California, 2002, pp. 307-310

3. *The Phenomenon of Knowledge*, Philosophical and Sociological
Thought, 1995, No. 3-4, pp. 41-63 (in
Russian and Ukrainian)

and others

**10. Theory of Intellectual Activity **

cf. 1. *Intellectual Components of
Creativity*,

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 Intellect*, Ukrainian Psychology:
Modern Potential, Kiev, 1996, pp.154-159 (in Ukrainian)

and others

**11. Theory of Multinumbers and Multicardinal Numbers**

cf. 1. *Nonuniform Operations on Named
Sets*, 5^{th} Annual International Conference on Statistics,
Mathematics and Related Fields, 2006 Conference Proceedings, Honolulu, Hawaii,
2006, pp. 245-271

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

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

and others

**12. Theory of Hypernumbers and Extrafunctions** ** **

cf. 1. *Hypernumbers and Extrafunctions***: Extending the
Classical Calculus**, Springer, New York, 2012

2. *Semitopological Vector Spaces and
Hyperseminorms*, Theory and Applications of Mathematics and Computer Science, v. 3,
No. 2, 2013, pp. 1 - 35

3.* 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

and others

**B.
Originating:**

**1. Mathematical Theory of Technology **

cf. 1. *Robustness of Information Systems and Technologies*, in “Proceedings of the 8^{th}
WSEAS International Conference on Data Networks, Communications, Computers”
(DNCOCO’09), Baltimore, Maryland, USA, November, 2009, pp. 67 - 72

2. *Mathematical Theory of
Information Technology*,
in “Proceedings of the 8^{th} WSEAS International Conference on Data
Networks, Communications, Computers” (DNCOCO’09), Baltimore, Maryland, USA,
November, 2009, pp. 42 - 47

3. *Levels of System Functioning Description*: *From Algorithm to Program to Technology, *in “Proceedings of the Business and
Industry Simulation Symposium,” Society for Modeling and Simulation
International, Orlando, Florida, 2003, pp. 3-7

and others

**2. Scalable and Discontinuous Topology**

cf. 1. *Scalable Topological
Spaces*, 5^{th} Annual International Conference on Statistics,
Mathematics and Related Fields, 2006 Conference Proceedings, Honolulu, Hawaii,
January, 2006, pp. 1865-1896

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

3. *Extended Fixed Point Theorem*, Methodological
Problems of Mathematics and Information Sciences,

and others

**3. Mathematical Schema Theory **

cf. 1.* Mathematical Schema Theory for
Network Design*, in
Proceedings of the ISCA 25^{th} International Conference “Computers and
their Applications” (CATA-2010), ISCA, Honolulu, Hawaii, 2010, pp. 157-162

2.
*From Craft to Engineering*: *Software Development and Schema Theory*, in Proceedings of the 2009 WRI World Congress Computer Science and
Information Engineering (CSIE 2000),** **WRI, Los Angeles, California, 2009
(CD edition, 5 p.)

3. *Operational and Program Schemas*, in Proceedings of the 15^{th}
International Conference on Software Engineering and Data Engineering
(SEDE-2006), ISCA, Los Angeles, California, July 6-8, 2006, pp. 74-78

and others

**4.
Hyperprobability Theory **(in collaboration)

cf. 1. *Hyperexpectations of random variables without expectations*,
Integration: Mathematical Theory and Applications, v. 3, No. 3, 2012, pp.
245-267

2. *Probabilities and Hyperprobabilities*,* *8^{th}* *Annual International
Conference on Statistics, Mathematics and Related Fields, 2009 Conference
Proceedings, Honolulu, Hawaii, January, 2009, pp. 351-367

3. *Introduction to Conditional Hyperprobabilities*, Integration:
Mathematical Theory and Applications, v. 2, No. 3, 2011, pp. 285 - 304

and others

**5. System Theory of Time**

cf. 1.* Elements of the System Theory of Time*, LANL, Preprint in Physics
0207055, 2002, 21 p. (electronic
edition: http://arXiv.org)

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

3. *The system Approach to the Theory of Time*, Philosophical and
Sociological Thought, 1992, No. 8, pp.160-163 (in
Russian)

and others

**C. Development (Important Contributions):**

**1. Theory of Superrecursive Algorithms **

cf. 1. *Topological
Algorithms*, in Proceedings of the ISCA 16^{th} International
Conference “Computers and their Applications”, ISCA,

2. *Super-recursive Algorithms as a Tool for High Performance
Computing, *Proceedings of the High
Performance Computing Symposium , San
Diego, 1999, pp. 224-228

3. 2.* Universal Limit Turing Machines*, Notices of the

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 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, 93, No. 4, pp.
555-573

2. *Kurosh Varieties of Linear *W*-algebras*, Problems
of Theory of Groups and Homological Algebra,

3. *Permutational products of linear *W*-algebras*,* *Soviet Math. Izvestiya, 1970, v. 4, No. 3, pp. 977-999 (translated from Russian)

and others

**4. Theory of Negative Probability **

cf. 1. *Extended Probabilities*: *Mathematical
Foundations*, Preprint in Physics,** **math-ph/0912.4767,
2009, 18 p. (electronic edition: http://arXiv.org)

2. *Interpretations of Negative Probabilities*, Preprint in Quantum
Physics,** **quant-ph/1008.1287, 2010,
17 p. (electronic edition: http://arXiv.org)

3. *Negative Probabilities in Modeling Random Financial Processes*,
Integration: Mathematical Theory and Applications, v. 2, No. 3, 2011, pp. 305 -
322 (in collaboration)

and others

**5. Algorithmic Information Theory **

cf. 1.** Theory of Information: Fundamentality, Diversity and Unification**, World Scientific, New
York/London/Singapore, 2010

2. *Algorithmic Complexity of Computational Problems*, International
Journal of Computing & Information Technology, 2010, v. 2, No. 1, pp.
149-187

3. *Evolutionary Information Theory*, Information, v. 4, No.2, 2013, pp.
224 – 268

and others