**MAIN SCIENTIFIC ACHIEVEMENTS**

Mark Burgin

**Books and booklets are in
bold letters**

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

*but **discoveries** and, even more, **theories**.*

**A.
Origination and Advanced Development (Important Contributions):**

**1. Theory of Fundamental Triads**

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

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

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

3. Burgin, M. *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.* *Burgin, M. ** Fundamental Structures of
Knowledge and Information: Achieving an Absolute**,

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

3. Burgin, M. *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. Burgin, M. ** On
the Nature and Essence of Mathematics**, Kiev, 1998 (in Russian)

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

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

and others

** c) technology:**

cf. 1. Burgin, M. *Structural Organization of Temporal Databases*,
in Proc. 17^{th} Int. Conf. on Software Eng. and Data Eng. (SEDE-2008),
Los Angeles, California, 2008, pp. 68-73

2. Burgin, M. *Knowledge in intellectual systems*,* *Conference on Intelligent Management
Systems, Varna, 1989, pp. 281‑286

3. Nocedal, A.S., Gerrikagoitia Arrien, J.K.
and Burgin, M. *A mathematical model for managing XML data*, International Journal of Metadata,
Semantics and Ontologies (IJMSO), v. 6, No. 1, 2011, pp. 56 - 73

and others

**2. Theory of Non-Diophantine Arithmetics
**

cf. 1. Burgin, M. ** 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. Burgin, M. *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. Burgin, M. *Non-classical Models of
Natural Numbers*, Russian Mathematical Surveys, 1977, v.32, No. 6, pp. 209-210
(in Russian)

and others

**3. General Theory of Properties**

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

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

3. Burgin, M. *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. Burgin, M. *Logical Tools for
Program Integration and Interoperability, *in Proceedings of the 8^{th}
IASTED International Conference on Software Engineering and Applications, MIT,
Cambridge, 2004, pp. 743-748

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

3. Burgin, M. *Logical Methods in
Artificial Intelligence Systems*, Vestnik of the VOIVT, 1991, No. 2,
pp. 66-78 (in Russian)

and others

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

cf. 1.* *Burgin, M. and Kuznetsov, V.I.* Introduction
to the Modern Exact Methodology of Science*, Moscow, 1994 (in Russian)

2. Burgin, M. and Kuznetsov, V.I.* Axiological Aspects of Scientific Theories*, Kiev,
1991 (in Russian)

3. Burgin, M. and Kuznetsov, V.I.* The World of Theories and the Power of Mind*, Kiev,
1992 (in Russian)

and others

**6. Theory of Hypermeasures and
Hyperintegration **

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

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

3. Burgin, M. *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. Burgin, M. ** Neoclassical Analysis: Calculus
closer to the Real World,** Nova Science Publishers, New York, 2008

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

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

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

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

and others

**9. Quantum Theory of Knowledge **

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

2. Burgin, M. *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. Burgin, M. *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. Burgin, M. ** Intellectual Components of
Creativity, **International Academy "Man in Aerospace
Systems",

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

3. Burgin, M. *Static and Dynamic Approach to Person's
Intellect*, Ukrainian Psychology: Modern Potential,

and others

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

cf. 1. Burgin, M. *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. Burgin, M. *Algebraic Structures of
Multicardinal Numbers*, Problems of Group Theory and
Homological Algebra, Yaroslavl, 1992, pp.3-20 (in Russian)

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

and others

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

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

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

3.* *Burgin, M. *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.
Origination and Development:**

**1. Mathematical Theory of Technology **

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

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

and others

**3. Mathematical Schema Theory **

cf. 1.* *Burgin, M. *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. Burgin,
M. *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. Burgin, M. *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 **

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

2. Burgin, M.
and Krinik, A.C. *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. Burgin, M.
and Krinik, A.C. *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.* *Burgin, M. *Elements of the
System Theory of Time*, LANL, Preprint in Physics 0207055, 2002, 21 p. (electronic edition: http://arXiv.org)

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

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

and others

**6.
Mathematical Theory of Oracles**

cf. 1.* *Burgin, M. Inaccessible Information and the Mathematical
Theory of Oracles, in *Information Studies
and the Quest for Transdisciplinarity*: *Unity
through Diversity*, World Scientific, New York/London/Singapore, 2017, 59 -
114

2. Burgin, M. *On
the power of oracles in the context of hierarchical intelligence*, Journal of* *Artificial Intelligence Research & Advances, v. 3, No. 2, 2016, pp. 6 - 17

3. Mikkilineni,
R., Morana, G. and Burgin, M. *Oracles in Software Networks*: *A New*
*Scientific and Technological Approach to
Designing Self-Managing Distributed Computing Processes*, Proceedings of the
2015 European Conference on Software* *Architecture
Workshops, Dubrovnik/Cavtat, Croatia, September 7-11, 2015, ACM, 2015,
pp. 11:1-11:8

and others

**7. General Theory of Structures**

cf. 1.** **Burgin,
M.

2. Burgin, M. *Structure Integration*, Integration:
Mathematical Theory and Applications, v. 2, No. 2, 2010, pp. 167-212

3. Burgin, M. *Structures in mathematics and beyond*, 8^{th}
Annual International Conference on Statistics, Mathematics and Related Fields, 2009
Conference Proceedings, Honolulu, Hawaii, 2009, pp. 449-469

and others

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

**1. Theory of Superrecursive Algorithms **

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

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

3. Burgin, M. *Universal Limit Turing Machines*,
Notices of the

and others

**2. Theory of Complexity of Algorithms and
Computations **

cf. 1. Burgin, M. *Dual complexity
measures and expert system functioning*, Problem solving in intellectual
computer media,

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

3. Burgin, M. *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

**4. Theory of Negative Probability **

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

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

3. Burgin, M. *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.** **Burgin,
M.

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

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

and others