Alexander GORBAN


Professor, Chair in Applied Mathematics,
Director of the Centre for Mathematical Modelling,
Department of Mathematics,
University of Leicester,
University Road,
Leicester LE1 7RH,
United Kingdom


Chief Scientists (now on leave)
Institute of Computational Modelling,
Russian Academy of Sciences,
Siberian Branch, Krasnoyarsk, Russia


Full Professor in Modeling & Simulation
Russian State Certificate of Professorship, 1993

Research interests:

Dynamics of systems of physical, chemical and biological kinetics;

Data Mining;


Human adaptation to hard living conditions;


Citation information (05 February 2014)

  • Web of Science citation index 1124, h-index 17 (24.03.2014), WoS Researcher ID: D-7310-2011:

·      Citations from Web of Science to all publications citation index (CItot) 1967, h-index 21 (including citations of books, 2013-04-12), calculated by the project “Who is who in Russian Science” using the Web of Science data, option "Cited Reference Search", )

·      Google Scholar system citation index 4942 h-index 31 (24.03.2014):


Academic degrees

·          1980, PhD in Physics & Math (Differential Equations & Math.Phys), Kuibyshev, USSR. Thesis: "Slow Relaxations and Bifurcations of Omega-Limit Sets of Dynamical Systems";

·          1990, Doctor of Physics & Math (Biophysics), (Advanced doctoral degree, Dr. Sc., analogue of Dr Habilit.), Institute of Biophysics, Russian Academy of Sciences, Krasnoyarsk, USSR. Thesis: "Extremal Principles and a priori Estimations in Biological and Formal Kinetics".

·          1993, Full Professor in Modelling & Simulation: Russian State Certificate of Professorship.




2004-present, University of Leicester:

·          2006-present, Director of the Centre for Mathematical Modelling;

·          2004-present, Chair in Applied Mathematics;

2003-2004, Swiss Federal Institute of technology (ETH), Zurich, Switzerland):

·          Senior Researcher, 2003-2004;

1983-2005, Institute of Computational Modelling, Russian Academy of Sciences, Siberian Branch, Krasnoyarsk, Russia (now on leave):

·          Deputy Director and Head of the Computer Sciences Department, 1995 – 2005;

·          Head of the Nonequilibrium Systems Laboratory, 1989 – 2006;

·          Senior researcher, 1983-1989;

·          Junior researcher, 1978-1983;

1977-1978, Institute of Catalysis, USSR Academy of Sciences, Siberian Branch, Novosibirsk, Russia:

·          Engineer, 1977-1978;

1978, Institute of Theoretical & Applied Mechanics, USSR Academy of Sciences, Siberian Branch, Novosibirsk, Russia:

·          Engineer, 1978;

1977, Tomsk Polytechnic University, Laboratory of Kinetics, Tomsk, Russia:

·          Junior researcher, 1977;

1976, Omsk State University, Laboratory of Kinetics, Omsk, Russia:

Junior researcher, 1976;

1973-1976, Omsk Railway Engineering Institute, Research Division, Omsk, Russia:

·          Engineer, 1973-1976.


Part-time teaching

Krasnoyarsk State Technical University, Krasnoyarsk, Russia: (last years was on leave):

·          Head of Neurocomputers Department, 1993-2006; Professor, 1993-2006;

Krasnoyask State Technological University, Krasnoyarsk, Russia::

·          Professor, Department of Automatization and Robots, 1993-2003;

Krasnoyarsk State University, Krasnoyarsk, Russia:

·          Professor, Psychological Faculty, 1998-2001;

·          Associate professor, Higher Mathematics Chair, 1981-1989.



  • Isaac Newton Institute for Mathematical Sciences (Cambridge, UK), 2010;
  • Institut des Hautes Etudes Scientiques (IHES, Bures-sur-Yvette, France), 2000, 2001, 2002, 2003, 2009.
  • Courant Mathematics Institute (New York, USA), 2000, 2007, 2008;
  • Clay Mathematics Institute (Cambridge, USA), 2000 (Clay Scholar);
  • ETH, Zurich, Switzerland: 1999, 2000, 2002.


Expert positions:

·         Vice-Chairman of Scientific Council at Krasnoyarsk State Technical University (direction: Software and tools for mathematical modelling) (1999-2006);

·         Head of Workgroup on Neurocomputing, Ministry of Science and Technology Russian Federation (1998-2000);

·         Vice-Chairman of Expert Council Krasnoyarsk Regional Science Foundation (1993-1996);

·         Chairman of the Analytic Games Committee, Krasnoyarsk (1989-1994);

·         Member of Jury of USSR National competition in mathematics for students of technical universities (1986-1990).


Membership of Associations

·         Member of SIAM;

·         Member of LMS;

·         Associated Member of ASME (American Society of Mechanical Engineers) (1997);


Some scientific achievements:

·   Kinetics.  A. Gorban has developed a family of methods for model reduction and coarse-graining: method of invariant manifold, method of natural projector, relaxation methods. He has solved problems in gas kinetics, polymer dynamics, chemical reaction kinetics and biological kinetics.  For this series of work AG received the I. Prigogine prize and medal, he has been Clay Scholar (Cambridge, USA, 2000).

·   Solution of Hilbert problem. Gorban with his student Karlin have recently received the recognition of the scientific community for solving an important part of the Hilbert 6th problem about the irreducibility of continuum mechanics to physical kinetics that remained unsolved almost for 100 years.

·   Stable Lattice Boltzmann methods. New family of numerical methods is developed. They are based on the ideas of lattice Boltzmann models (LBM) in combination with methods of invariant manifold and specific entropic stabilisers. Standard test examples demonstrate that the new methods erase spurious oscillations without blurring of shocks, and do not affect smooth solutions.  

·   Bioinformatics. The methods of genome analysis based on frequency dictionaries are elaborated and applied to various biological problems (genome redundancy, mosaic structure of genome, genetic species signature, etc.). For example, existence of a universal 7-cluster structure in all available bacterial genomes is proved.

·   Data mining and rules extraction. A general neural networks based technology of extraction of explicit knowledge from data was developed. This technology was implemented in a series of software libraries and allowed us to create dozens of knowledge-based expert systems in medical and technical diagnosis, ecology and other fields. For example new tools were developed for differential diagnosis of allergic and pseudoallergic reactions, for anticipation of myocardial infarction complications, and  for evaluation of the accumulated radiation dose based on parameters of human blood. 

·   Revealing and visualization of hidden structure of complex systems. A system of methods is developed to reveal the hidden intelligible models in complex systems: complex datasets and complex reaction networks. First of all, this is revealing of hidden geometry. New special tools have been proposed and elaborated, the grammars of elementary transformations which allow us to create the intelligible models of complex systems by the chains of simple steps and dominant systems that represent the complex networks by the simple networks, which dynamics can be studied analytically. Several biological and medical centres now use these methods and algorithms, for example, Institute Curie (France).

·   MicroRNAs kinetic signatures. MicroRNAs are key regulators of all important biological processes, including development, differentiation and cancer. Although remarkable progress has been made in deciphering the mechanisms used by miRNAs to regulate translation, many contradictory findings have been published that stimulate active debate in this field. A. Gorban with co-authors have developed computational tools for discriminating among different possible individual mechanisms of miRNA action based on translation kinetics data that can be experimentally measured (kinetic signatures). They have found sensitive parameters of the translation process for various conditions.

·   The crises anticipation. A. Gorban invented a new method to measure the stress caused by environmental factors. In particular, this is a possibility to measure the health of the groups of healthy people. This method is based on a universal effect discovered by A. Gorban in his study of human adaptation. This effect is supported by hundreds of experiments and observations and extended to systems of different nature. Now, this method is used for monitoring of Far North populations, for analysis of crises in national financial systems and in companies. It becomes a part of the modern approach to crises anticipation.



Organizer of:

·          International Research Workshop: Coping with Complexity:  Model Reduction and Data Analysis, Ambleside, Lake District, UK August 31 – September 4, 2009. 

·          International Research workshop: Mathematics of Model Reduction, University of Leicester, Leicester, UK August 28 – August 30, 2007,

·          International Research Workshop: Lattice Boltzmann at all-scales: from turbulence to DNA translocation, 15 November 2006, University of Leicester, Leicester, UK,

·          International Research Workshop: “Principal manifolds for data cartography and dimension reduction” August 24-26, 2006, Leicester University, Leicester, UK

·          International Research Workshop:Geometry of Genome: Unravelling of Structures Hidden in Genomic Sequences,” Leicester University, Leicester, UK, 22/09/2005 – 24/09/2005;

·          International Research Workshop: “Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena,” Leicester University, Leicester, UK August 24-26 2005;

·          International Research Workshop: "Invariance and Model Reduction for Multiscale Phenomena," Zurich, August, 2003;

·          USA-NIS Neurocomputing Opportunities Workshop, Washington, DC, July 1999, (Sponsored by the National Science Foundation of the USA and Applied Computational Intelligence Lab, TTU) (Co-Chair);

·          Russian annual National Conference “Neurionformatics” (1998-present);

·          Russian annual National Workshops “Neuroinformatics and Application,” Krasnoyarsk, 1992- present;

·          Russian annual National Workshops “Modeling of Nonequilibrium Systems,” Krasnoyarsk, 1999- present.

·          Russian National Conference “Problems of Regional Informatization”, Krasnoyarsk, 1998-2003.

·          Soviet Union National competition in Neuroinformatics and Neurocomputers for students and young scientists, 1991.


Grants and awards:

·         Lattice Boltzmann methods for analysis of permeability in geophysics, Weatherford Contract Research,  07.2013  06.2015;

·         Data Mining for Geological Information, NERC (Natural Environment Research Council), 07.2013  10.2013;

·         Data Mining for Lymphoma Differential Diagnosis, European Regional Development Fund, 2012;

·         Mathematical Modelling of Adaptation and Decision-Making in  Neural Systems, The Royal Society, UK: International Joint UK-Japan Project;

·         Modularity, Abstraction and Robustness of Network Models in Molecular Biology, Alliance : Franco-British Research Partnership Programme;

·         EPSRC and LMS grants for the International Workshop “Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena,” Leicester, UK August 24-26 2005;

·         Prigogine Award and Medal (2003);

·         Clay Scholar, (Clay Mathematics Institute, Cambridge, USA, 2000);

·         Russian Federal Grant of the “Integration” program, 4 times (1998-2003);

·         Grant of Russian Federal subprogram “New Information Processing Technology” (1999);

·         Russian Federal Fellowship for outstanding scientists, twice (6 years);

·         Grant of Russian Foundation of Basic Research (1996-1998);

·         1994-1996 American Mathematical Society Fellowship.


Scientific advisor of 29 PhD thesis and 5 DrSc, including:

·         D. Packwood, Non-Equilibrium Dynamics of Discrete Time Boltzmann Systems, (PhD, Applied Mathematics, University of Leicester, UK, 2012).

·         Jianxia Zhang, Nonequilibrium Entropic Filters for Lattice Boltzmann Methods and Shock Tube Case Studies (PhD, Applied Mathematics, University of Leicester, UK, 2012).

·         M. Shahzad, Slow Invariant Manifold and its approximations in kinetics of catalytic reactions, (PhD, Applied Mathematics, University of Leicester, UK, 2011).

·         H.A. Wahab, Quasichemical Models of Multicomponent Nonlinear Diffusion (PhD, Applied Mathematics, University of Leicester, UK, 2011).

·         O. Radulescu, Mathematical models of complexity in molecular biology and mechanics of complex fluids, (Dr. Habilitation, Applied Mathematics, Université de Rennes 1, France, 2006)

·         E.M. Mirkes, The structure and functioning of ideal neurocomputer (DrSc=“doctor nauk”, Computer Science, Krasnoyarsk State Technical University, Russia, 2002);

·         E.V. Smirnova, Measurement and modeling of adaptation (DrSc=“doctor nauk”, Modeling in Biophysics, Institute of Biophysics, Russian Academy of Sciences, Krasnoyarsk, Russia,  2001);

·         A.Yu. Zinovyev, Method of Elastic Maps for Data Visualization: Algorithms, Software and Applications in Bioinformatics (PhD=“kandidat nauk”, Computer Science, Krasnoyarsk State Technical University, Russia, 2001);

·         V.G. Tzaregorodtzev, Algorithms, technology and software for knowledge extraction using trainable neural networks (PhD=“kandidat nauk, Computer Science, Krasnoyarsk State Technical University, Russia, 2000);

·         A.A. Pitenko, Neural networks for geoinformatics (PhD=“kandidat nauk, Computer Science, Krasnoyarsk State Technical University, Russia, 2000);

·         A.A. Rossiev, Neural network modeling of data with gaps (PhD=“kandidat nauk, Computer Science, Krasnoyarsk State Technical University, Russia, 2000);

·         M.Yu. Senashova, Accuracy estimation for neural networks (PhD=“kandidat nauk, Computer Science, Krasnoyarsk State Technical University, Russia, 1999);

·         M.A. Dorrer, Psychological intuition of neural networks (PhD=“kandidat nauk, Computer Science, Krasnoyarsk State Technical University, Russia, 1999);

·         D.A. Rossiev, Neural networks based expert systems for medical  diagnostics (DrSc=“doctor nauk”, Biophysics, Institute of Biophysics, Russian Academy of Sciences, Krasnoyarsk, Russia,  1997);

·         I.V. Karlin, Method of invariant manifold in physical kinetics, (PhD, Physics, Krasnoyarsk AMSE Centre, Russia-France, 1991);

·         V.I. Verbitsky, Simultaneously dissipative operators and global stability (PhD=“kandidat nauk”, Mathematical Analysis, Ural University, Yekaterinburg, Russia, 1989);

·         M.G. Sadovskii, Optimization in space distributions of populations, (PhD=“kandidat nauk”, Biophysics, Institute of Biophysics, Russian Academy of Sciences, Krasnoyarsk, Russia, 1989);

·         V.A. Okhonin, Kinetic equations for population dynamics (PhD=“kandidat nauk”, Biophysics, Moscow University, Russia, 1986).



Selected Publications


Monographs (in reverse chronological order):


  1. A.N. Gorban and D. Roose (Eds.), Coping with Complexity: Model Reduction and Data Analysis, Lecture Notes in Computational Science and Engineering, 75, Springer: Heidelberg – Dordrecht - London -New York, 2011, pp. 311-344.
  2. A. Gorban, B. Kegl, D. Wunsch, A. Zinovyev  (Eds.), Principal Manifolds for Data Visualisation and Dimension Reduction, LNCSE 58, Springer, Berlin – Heidelberg – New York, 2007.
  3. Model Reduction and Coarse--Graining Approaches for Multiscale Phenomena, Ed. by Alexander N. Gorban, Nikolaos  Kazantzis, Ioannis G. Kevrekidis, Hans Christian Öttinger, Constantinos Theodoropoulos, Springer, Berlin-Heidelberg-New York, 2006.
  4. A.N. Gorban, B.M. Kaganovich, S.P. Filippov, A.V. Keiko, V.A. Shamansky, I.A. Shirkalin, Thermodynamic Equilibria and Extrema: Analysis of Attainability Regions and Partial Equilibria, Springer, Berlin-Heidelberg-New York, 2006.
  5. Invariant Manifolds for Physical and Chemical Kinetics, Lect. Notes Phys. 660, Springer, Berlin, Heidelberg, 2005 (498 pages). [Review in Bull. London Math. Soc. 38 (2006)] [Preface-Contents-Introduction(pdf)][Reviews(htm)] (With I.V. Karlin);
  6. Singularities of transition processes in dynamical systems: Qualitative theory of critical delays, Electron. J. Diff. Eqns., Monograph 05, 2004, (55 pages). Online:
  7. Neuroinformatics, Novosibirsk, Nauka Publ., 1998, 258 p. (With W.L.Dunin-Barkovskii, D.A.Rossiev, S.A.Terehov).
  8. Methods of neuroinformatics, Krasnoyarsk State University Press, 1998, 205 p. (A.N.Gorban ed.)
  9. Neural networks on PC, Novosibirsk, Nauka Publ., 1996, 276 p. (With D.A.Rossiev). [In Russian, Russian title “Neironnye seti na personal’nom komp’yutere”]
  10. New methods for solving the Boltzmann Equations, AMSE Press, Tassin, France, 1993, ISBN: 2-909214-51-6, 166 p. (With I.V.Karlin).

11.  Kinetic Models of Catalytic Reactions (Comprehensive Chemical Kinetics, V.32, ed. by R.G. Compton), Elsevier, Amsterdam, 1991, 396p. (With G.S.Yablonskii, V.I.Bykov and V.I.Elokhin). (Review on this book: Journal of American Chemical Society (JAChS), V.114, n 13, 1992; sections “Reviews on the book”, W. Henry Weinberg, review on the book "Comprehensive Chemical Kinetics", Volume 32, Kinetic Models of Catalytic Reactions, Elsevier, 1991).

  1. Training of Neural Networks, Moscow, USSR-USA JV "ParaGraph", 1990, 160 p. [In Russian, Russian title “Obuchenie neironnyh setei”]
  2. Demon of Darwin. The Idea of Optimality and Natural Selection, Nauka Pub., Moscow, 1988, 208p. (With R.G.Khlebopros).
  3. Essays on Chemical relaxation, Novosibirsk, Nauka Publ., 1986, 316 p. (With V.I.Bykov, G.S.Yablonskii);
  4. Equilibrium Encircling. Equations of Chemical Kinetics and their Thermodynamic Analysis, Novosibirsk, Nauka Publ., 1984, 256 p. [In Russian, Russian title “Obkhod Ravnovesiya”]
  5. Kinetic models of heterogeneous catalytic reactions, Novosibirsk, Nauka Publ., 1983, 256 p. (With V.I.Bykov, G.S.Yablonskii).



Articles (in reverse chronological order):


1.           E M Mirkes, I Alexandrakis, K Slater, R Tuli and A N Gorban, Computational diagnosis of canine lymphoma, J. Phys.: Conf. Ser. 490 012135 (2014)

2.           A A Akinduko and A N Gorban, Multiscale principal component analysis, J. Phys.: Conf. Ser. 490 012081 (2014)

3.           Y Shi, A N Gorban and T Y Yang, Is it possible to predict long-term success with k-NN? Case study of four market indices (FTSE100, DAX, HANGSENG, NASDAQ), J. Phys.: Conf. Ser. 490 012082 (2014)

4.           Spahn, F., Vieira Neto, E., Guimarães, A.H.F., Gorban, A.N., Brilliantov, N.V. A statistical model of aggregate fragmentation, New Journal of Physics 16, Article number 013031, 2014.

5.           K.I. Sofeikov, I. Romanenko, I. Tyukin, A.N. Gorban. Scene Analysis Assisting for AWB Using Binary Decision Trees and Average Image Metrics. In Proceedings of IEEE Conference on Consumer Electronics, 10-13 January, Las-Vegas, USA, 2014, pp. 488-491.

6.           A.N. Gorban, I. Karlin, Hilbert's 6th Problem: exact and approximate hydrodynamic manifolds for kinetic equations, Bulletin of the American Mathematical Society, 51(2), 2014, 186-246. PII: S 0273-0979(2013)01439-3

7.           A.N. Gorban, I. Tyukin, E. Steur, and H. Nijmeijer, Lyapunov-like conditions of forward invariance and boundedness for a class of unstable systems, SIAM J. Control Optim., Vol. 51, No. 3, 2013, pp. 2306-2334.

8.           E.M. Mirkes, A. Zinovyev, and A.N. Gorban, Geometrical Complexity of Data Approximators, in I. Rojas, G. Joya, and J. Cabestany (Eds.): IWANN 2013, Part I, Advances in Computation Intelligence, Springer LNCS 7902, pp. 500–509, 2013.

9.           I.Y. Tyukin, A.N. Gorban, Explicit Reduced-Order Integral Formulations of State and Parameter Estimation Problems for a Class of Nonlinear Systems. In Proceedings of the 52-th IEEE International Conference on Decision and Control (10-13 December, 2013, Florence, Italy), IEEE, 4284-4289.

10.       A.N. Gorban, G.S. Yablonsky Grasping Complexity, Computers & Mathematics with Applications, Volume 65, Issue 10, May 2013, 1421-1426. arXiv:1303.3855 [cs.GL]

11.       A.N. Gorban, Maxallent: Maximizers of all entropies and uncertainty of uncertainty, Computers & Mathematics with Applications, Volume 65, Issue 10, May 2013, 1438-1456. arXiv:1212.5142 []

12.       R.A. Brownlee, J. Levesley, D. Packwood, A.N. Gorban, Add-ons for Lattice Boltzmann Methods: Regularization, Filtering and Limiters, Progress in Computational Physics, 2013, vol. 3, 31-52. arXiv:1110.0270 [physics.comp-ph]

13.       A.N. Gorban, Thermodynamic Tree: The Space of Admissible Paths, SIAM J. Applied Dynamical Systems, Vol. 12, No. 1 (2013), pp. 246-278. DOI: 10.1137/120866919 arXiv e-print

14.       Zinovyev, N. Morozova, A.N. Gorban, and A. Harel-Belan, Mathematical Modeling of microRNA-Mediated Mechanisms of Translation Repression, in U. Schmitz et al. (eds.), MicroRNA Cancer Regulation: Advanced Concepts, Bioinformatics and Systems Biology Tools, Advances in Experimental Medicine and Biology Vol. 774, Springer, 2013, pp. 189-224.

15.       A.N. Gorban, E.M. Mirkes, G.S. Yablonsky, Thermodynamics in the limit of irreversible reactions, Physica A 392 (2013) 1318–1335.

16.       A.N. Gorban, Local equivalence of reversible and general Markov kinetics, Physica A 392 (2013) 1111–1121.

17.       A.N. Gorban and D. Packwood, Allowed and forbidden regimes of entropy balance in lattice Boltzmann collisions, Physical Review E 86, 025701(R) (2012).

18.       N. Morozova, A. Zinovyev, N. Nonne, L.-L. Pritchard, A.N. Gorban, and A. Harel-Bellan,

19.       Kinetic signatures of microRNA modes of action, RNA, Vol. 18, No. 9 (2012) 1635-1655, doi:10.1261/rna.032284.112

20.       O. Radulescu, A. N. Gorban, A. Zinovyev, V. Noel, Reduction of dynamical biochemical reaction networks in computational biology, Frontiers in Genetics (Bioinformatics and Computational Biology). July2012, Volume3, Article 131. (e-print arXiv:1205.2851 [q-bio.MN])

21.       A.N. Gorban, G.S.Yablonsky, Extended detailed balance for systems with irreversible reactions, Chemical Engineering Science 66 (2011) 5388–5399.

22.       A.N. Gorban, H.P. Sargsyan and H.A. Wahab, Quasichemical Models of Multicomponent Nonlinear Diffusion, Mathematical Modelling of Natural Phenomena, Volume 6 / Issue 05, (2011), 184262.

23.       A.Gorban and S. Petrovskii, Collective dynamics: when one plus one does not make two, Mathematical Medicine and Biology (2011) 28, 85−88.

24.       A.N. Gorban and M. Shahzad, The Michaelis-Menten-Stueckelberg Theorem. Entropy 2011, 13, 966-1019.

25.       G. S. Yablonsky, A. N. Gorban, D. Constales, V. V. Galvita and G. B. Marin, Reciprocal relations between kinetic curves, EPL, 93 (2011) 20004.

26.       A.N. Gorban, Self-simplification in Darwin’s Systems, In: Coping with Complexity: Model Reduction and Data Analysis, A.N. Gorban and D. Roose (eds.), Lecture Notes in Computational Science and Engineering, 75, Springer: Heidelberg – Dordrecht - London -New York, 2011, pp. 311-344.

27.       D.J. Packwood, J. Levesley, and A.N. Gorban, Time step expansions and the invariant manifold approach to lattice Boltzmann models, In: Coping with Complexity: Model Reduction and Data Analysis, A.N. Gorban and D. Roose (eds.), Lecture Notes in Computational Science and Engineering, 75, Springer: Heidelberg – Dordrecht - London -New York, 2011, pp. 169-206.

28.       A.N. Gorban, Kinetic path summation, multi-sheeted extension of master equation, and evaluation of ergodicity coefficient, Physica A 390 (2011) 1009–1025.

29.       A.N. Gorban, L.I. PokidyshevaE,V. Smirnova, T.A. Tyukina, Law of the Minimum Paradoxes, Bull Math Biol 73(9) (2011), 2013-2044.

30.       A.N. Gorban, E.V. Smirnova, T.A. Tyukina, Correlations, risk and crisis: From physiology to finance, Physica A, Vol. 389, Issue 16, 2010, 3193-3217.

31.       A.N. Gorban, A. Zinovyev, Principal manifolds and graphs in practice: from molecular biology to dynamical systems, International Journal of Neural Systems, Vol. 20, No. 3 (2010) 219–232.

32.       E. Chiavazzo, I.V. Karlin, A.N. Gorban, K. Boulouchos, Coupling of the model reduction technique with the lattice Boltzmann method, Combustion and Flame 157 (2010) 1833–1849

33.       Gorban A.N., Gorban P.A., Judge G. Entropy: The Markov Ordering Approach. Entropy. 2010; 12(5):1145-1193.

34.       AN. Gorban and V. M. Cheresiz, Slow Relaxations and Bifurcations of the Limit Sets of Dynamical Systems. I. Bifurcations of Limit Sets, Journal of Applied and Industrial Mathematics, 2010, Vol. 4, No. 1, pp. 54–64.

35.       A.N. Gorban and V. M. Cheresiz, Slow Relaxations and Bifurcations of the Limit Sets of Dynamical Systems. II. Slow Relaxations of a Family of Semiflows, Journal of Applied and Industrial Mathematics, 2010, Vol. 4, No. 2, pp. 182–190

36.       E. Chiavazzo, I.V. Karlin, and A.N. Gorban, The Role of Thermodynamics in Model Reduction when Using Invariant Grids, Commun. Comput. Phys., Vol. 8, No. 4 (2010), pp. 701-734.

37.       Andrei Zinovyev, Nadya Morozova, Nora Nonne, Emmanuel Barillot, Annick Harel-Bellan, Alexander N Gorban, Dynamical modeling of microRNA action on the protein translation process,  BMC Systems Biology 2010, 4:13 (24 February 2010).

38.       A.N. Gorban, O. Radulescu, A. Y. Zinovyev, Asymptotology of chemical reaction networks, Chemical Engineering Science 65 (2010) 2310–2324.

39.       A.N. Gorban, E.V. Smirnova, T.A. Tyukina, General Laws of Adaptation to Environmental Factors: from Ecological Stress to Financial Crisis. Math. Model. Nat. Phenom. Vol. 4, No. 6, 2009, pp. 1-53.

40.       A.N. Gorban, A. Y. Zinovyev, Principal Graphs and Manifolds, Chapter 2 in: Handbook of Research on Machine Learning Applications and Trends: Algorithms, Methods, and Techniques, Emilio Soria Olivas et al. (eds), IGI Global, Hershey, PA, USA, 2009, pp. 28-59.

41.       E. Chiavazzo, I. V. Karlin, A. N. Gorban and K Boulouchos, Combustion simulation via lattice Boltzmann and reduced chemical kinetics, J. Stat. Mech. (2009) P06013,

42.       Ovidiu Radulescu, Alexander N Gorban, Andrei Zinovyev, and Alain Lilienbaum Robust simplifications of multiscale biochemical networks, BMC Systems Biology 2008, 2:86 doi:10.1186/1752-0509-2-86.

43.       A.N. Gorban and O. Radulescu, Dynamic and Static Limitation in Multiscale Reaction Networks, Revisited, Advances in Chemical Engineering 34 (2008), 103-173.

44.       A.N. Gorban, Selection Theorem for Systems with Inheritance, Math. Model. Nat. Phenom., Vol. 2, No. 4, 2007, pp. 1-45.

45.       R. A. Brownlee, A. N. Gorban, and J. Levesley, Nonequilibrium entropy limiters in lattice Boltzmann methods, Physica A: Statistical Mechanics and its Applications Volume 387, Issues 2-3, 15 January 2008, Pages 385-406 .

46.       A.N. Gorban and O. Radulescu, Dynamical robustness of biological networks with hierarchical distribution of time scales, IET Syst. Biol., 2007, 1, (4), pp. 238–246.

47.       R. A. Brownlee, A. N. Gorban, and J. Levesley, Stability and stabilization of the lattice Boltzmann method, Phys. Rev. E 75, 036711 (2007) (17 pages)

48.       A.N. Gorban and  A.Y. Zinovyev The Mystery of Two Straight Lines in Bacterial Genome Statistics, Bulletin of Mathematical Biology (2007)

49.       E. Chiavazzo, A.N. Gorban, and I.V. Karlin, Comparison of Invariant Manifolds for Model Reduction in Chemical Kinetics, Commun. Comput. Phys. Vol. 2, No. 5 (2007), pp. 964-992

50.       A.N. Gorban, N.R. Sumner, and A.Y. Zinovyev, Topological grammars for data approximation, Applied Mathematics Letters Volume 20, Issue 4  (2007),  382-386

51.       A.N. Gorban, Order–disorder separation: Geometric revision, Physica A Volume 374, Issue 1 , 15 January 2007, Pages 85-102

52.       A.N. Gorban and O. Radulescu, Dynamical robustness of biological networks with hierarchical distribution of time scales, IET Syst. Biol., 2007, 1, (4), pp. 238–246

53.       R.A. Brownlee, A.N. Gorban, and J. Levesley, Stabilization of the lattice Boltzmann method using the Ehrenfests' coarse-graining idea, Phys. Rev. E 74, 037703 (2006)

54.       A.Gorban, I. Karlin, A. Zinovyev, Invariant Grids: Method of Complexity Reduction in Reaction Networks, Complexus, V. 2, 110–127.

55.       A.N. Gorban, I.V. Karlin, Quasi-Equilibrium Closure Hierarchies for the Boltzmann Equation, Physica A 360 (2006) 325–364

56.       A.Gorban, A. Zinovyev, Elastic Principal Graphs and Manifolds and their Practical Applications, Computing 75, 359–379 (2005),

57.       A.N. Gorban, I.V. Karlin, Invariance correction to Grad's equations: Where to go beyond approximations? Continuum Mechanics and Thermodynamics, 17(4) (2005), 311–335,

58.       A.N. Gorban,  T.G.Popova, A.Yu. Zinovyev, Codon usage trajectories and 7-cluster structure of 143 complete bacterial genomic sequences  •Physica A: Statistical and Theoretical Physics, 353C (2005), 365-387.

59.       A.N. Gorban,  T.G.Popova, A.Yu. Zinovyev, Four basic symmetry types in the universal 7-cluster structure of microbial genomic sequences, In Silico Biology, 5 (2005), 0039.

60.       A.N. Gorban, P.A.Gorban, and I. V. Karlin, Legendre Integrators, Post-Processing and Quasiequilibrium, J. Non-Newtonian Fluid Mech. 120 (2004) 149-167.

61.       A.N. Gorban, I.V. Karlin, A.Yu. Zinovyev, Constructive methods of invariant manifolds for kinetic problems, Physics Reports, V. 396, N 4-6 (2004), p. 197-403.

62.       A.N. Gorban, I.V. Karlin, A.Yu. Zinovyev, Invariant grids for reaction kinetics, Physica A, 333 (2004), 106--154.

63.       A.N. Gorban, I.V. Karlin, Uniqueness of thermodynamic projector and kinetic basis of molecular individualism, Physica A, 336, 3-4 (2004), 391-432.

64.       A.N. Gorban, I.V. Karlin, Methods of nonlinear kinetics, in: Encyclopedia of Life Support Systems, Encyclopedia of Mathematical Sciences,  EOLSS Publishers, Oxford, 2004.

65.       A.N. Gorban, T. G. Popova, and A. Yu. Zinovyev: Self-organizing approach for automated gene identification. Open Sys. Information Dyn. 10 (2003) 1-13.

66.       A.N. Gorban and I. V. Karlin, Family of additive entropy functions out of thermodynamic limit, Phys. Rev. E. 2003, V.67, 016104, E-print: http:,

67.       A.N. Gorban, I. V. Karlin and H. C. Ottinger, The additive generalization of the Boltzmann entropy. Phys. Rev. E. (2003), V. 67. E-print: http:,

68.       A.N. Gorban, I. V. Karlin, Method of invariant manifold for chemical kinetics. Chem. Eng. Sci. 58 (2003) 4751-4768.

69.       I.V. Karlin, L. L. Tatarinova, A. N. Gorban, and H. C. Öttinger, Irreversibility in the short memory approximation, Physica A 327 (2003) 399-424.

70.       A.Gorban, A. Zinovyev, T. Popova. Seven clusters in genomic triplet distributions. In Silico Biology. V.3 (2003), 471-482.

71.       A.N. Gorban, T.G Popova, M.G Sadovsky, Classification of nucleotide sequences over their frequency dictionaries reveals a relation between the structure of sequences and taxonomy of their bearers, Zh Obshch Biol 64 (1), 65-77. 2003

72.       A.Gorban', Braverman M., Silantyev V. Modified Kirchhoff flow with a partially penetrable obstacle and its application to the efficiency of free flow  turbines. Math. Comput. Modelling 35 (2002), No. 13, 1371-1375.

73.       Gorban', Silantyev V. Riabouchinsky Flow with Partially Penetrable Obstacle. Math. Comput. Modelling 35 (2002), no. 13, 1365-1370.

74.       I.V. Karlin, M. Grmela, and A.N. Gorban: Duality in nonextensive statistical mechanics, Phys. Rev. E 65 (2002) 036128.

75.       A.N. Gorban and I. V. Karlin, Reconstruction lemma and fluctuation-dissipation theorem, Revista Mexicana de Fisica, 2002. V. 48 Suplemento 1,  PP. 238-242.

76.       A.N. Gorban and I. V. Karlin, Geometry of irreversibility, in: Recent Developments in Mathematical and Experimental Physics, Volume C:  Hydrodynamics and Dynamical Systems, Ed. F. Uribe (Kluwer, Dordrecht, 2002), pp. 19-43.

77.       A.N. Gorban and I. V. Karlin, Macroscopic dynamics through coarse-graining: A solvable example, Phys. Rev. E. V 65. 026116(1-5) (2002).

78.       I.V. Karlin and A.N. Gorban, Hydrodynamics from Grad's equations: What can we learn from exact solutions? Ann. Phys. (Leipzig) 10-11 (2002),  pp. 783-833. E-print: http:,

79.       A.N. Gorban, Zinov'ev A.Y., Pitenko A.A., Data vizualization. The method of elastic maps, Neirocompjutery, 2002, 4, 19-30.

80.       A.N. Gorban, A.A Rossiev, Iterative modeling of data with gaps via submanifolds of small dimension, Neirocompjutery, 2002, 4, 40-44.

81.       A.Gorban, Rossiev A., Makarenko N., Kuandykov Y., Dergachev V. Recovering data gaps through neural network methods. International Journal of  Geomagnetism and Aeronomy, 2002, Vol. 3, No. 2, December 2002.

82.       A.N. Gorban, V.T. Manchuk, A.V.Perfil’eva, E.V.Smirnova, E.P. Cheusova, The mechanism of increasing the correlation between physiological parameters for high adaptation tension, Siberian Ecological Journal, 2001, No 5, 651-655.

83.       A.N. Gorban, Gorlov A.M., Silantyev V.M. Limits of the turbin efficiency for free fluid flow,  ASME Journal of Energy Resourses Technology, Dec.  2001, V. 123, Iss. 4, pp. 311-317.

84.       A.N. Gorban, Pitenko A.A., Zinov'ev A.Y., Wunsch D.C. Vizualization of any data uzing elastic map method ,  Smart Engineering System Design.  2001, V.11, p. 363-368.

85.       A.N. Gorban, Popova T.G., Sadovsky M.G., Wunsch D.C. Information content of the frequency dictionaries, reconstruction, transformation and  classification of dictionaries and genetic texts. Smart Engineering System Design, 2001, V.11, p. 657-663.

86.       A.N.Gorban, I.V.Karlin, P.Ilg and H.C.Ottinger Corrections and enhancements of quasi-equilibrium states, J. Non-Newtonian Fluid Mech., 2001,  V.96(1-2), PP. 203-219.

87.       A.N. Gorban, Karlin I.V., Ottinger H.C., Tatarinova L.L. Ehrenfest's argument extended to a formalism of nonequilibrium thermodynaics, Phys. Rev. E. 2001, V. 63. 066124.

88.       A.N. Gorban, Gorbunova K.O., Wunsch D.C. Liquid Brain: The Proof of Algorithmic Universality of Quasichemical Model of Fine-Grained Parallelism, Neural Network World, 2001, No. 4. P P. 391-412.

89.       A.N. Gorban, Zinovyev A. Yu. Method of Elastic Maps and its Applications in Data Visualization and Data Modeling. International Journal of Computing Anticipatory Systems, CHAOS. 2001. V. 12. PP. 353-369.

90.       V.A. Dergachev, Gorban A.N., Rossiev A.A., Karimova L.M., Kuandykov E., Makarenko N.G., Steier. The filling of gaps in geophysical time series by artificial neural networks, Radiocarbon, 2001, V. 43, No. 2, PP. 343-348.

91.       A.N.Gorban, V.P.Torchilin, M.V.Malyutov, M. Lu Modeling polymer brushes protective action ,  Simulation in Industry' 2000. Proceedings of 12-th  European Simulation Symposium ESS'2000. September 28-30, 2000, Hamburg, Germany. A publication of the Society of Computer Simulation  International. Printed in Delft, The Netherlands, 2000. PP. 651-655.

92.       A.N.Gorban, Neuroinformatics: What are us, where are we going, how to measure our way? Informacionnye technologii, 2000, 4. - С. 10-14.

93.       A.N. Gorban, K. O. Gorbunova, Liquid Brain: Kinetic Model of Structureless Parallelism, Internation Journal of Computing Anticipatory Systems, CHAOS, V. 6, 2000, P.117-126.

94.       A.N. Gorban, I.V. Karlin, V.B. Zmievskii and S.V. Dymova, Reduced description in reaction kinetics, Physica A, 2000. V. 275, No. 3-4, PP. 361-379.

95.       A.N Gorban, The generalized Stone approximation theorem for arbitrary algebras of continuous functions, Dokl Akad Nauk, 365 (5), 586-588, 1999

96.       A.N. Gorban, A.A Rossiev, Neural network iterative method of principal curves for data with gaps,  J Comput Sys Sc Int, 38 (5): 825-830, 1999.

97.       A.N. Gorban, I.V.Karlin and  V.B.Zmievskii, Two-step approximation of space-independent relaxation, Transp.Theory Stat.Phys., 1999. V. 28(3), PP. 271-296.

98.       A.N. Gorban, Approximation of Continuous Functions of Several Variables by an Arbitrary Nonlinear Continuous Function of One Variable, Linear Functions, and  Their Superpositions. Appl. Math. Lett., 1998. V. 11, No. 3, pp. 45-49.

99.       S.E. Gilev, A.N. Gorban, The completeness theorem for semigroups of continuous functions, Dokl Akad Nauk, 362 (6): 733-734, 1998

100.   N.N.Bugaenko, A. N. Gorban, M.G.Sadovskii, Maximum entropy method in analysis of genetic text and measurement of its information content ,  Open systems and information dynamics. #5, 1998. - pp.265-278. 

101.   A.N. Gorban, Neuroinformatics and applications, Otkrytye sistemy (Open Systems), 1998, No. 4-5. pp. 36-41.

102.   A.N. Gorban, I.V. Karlin, Sroedinger operator in a overfull set ,  Europhys. Lett., 1998, V. 42, No.2, pp. 113-117.

103.   I.V. Karlin, A. N. Gorban, S. Succi, V.  Boffi, Maximum Entropy Principle for Lattice Kinetic Equation ,  Physical Review Letters, 1998, V. 81, No. 1, pp. 6-9.

104.   A.N. Gorban, Yeugenii M. Mirkes and Donald Wunsch, High Order Orthogonal Tensor Networks: Information Capacity and Reliability, Proc. IEEE/INNS International Conference on Neural Networks, Houston, IEEE, 1997, pp. 1311-1314.

105.   A.N. Gorban, Masha Yu. Senashova and Donald Wunsch, Back-Propagation of Accuracy, Proc. IEEE/INNS International Conference on Neural Networks, Houston, IEEE, 1997, pp. 1998-2001.

106.   N.N. Bugaenko, A. N. Gorban, M.G.Sadovskii, Information content of nucleotid sequences and their fragments. Biofizika. 1997. V. 42, Iss. 5, pp. 1047-1053.

107.   V.I. Bykov, A.N. Gorban, S.V. Dymova, Method of invariant manifolds for the reduction of kinetic description, ACH-Models Chem 134 (1): 83-95 1997

108.   A.N. Gorban, I.V.Karlin, Scattering rates versus moments: Alternative Grad equations, Phys. Rev. E, 1996, 54(4), R3109.

109.   A.N. Gorban, I.V.Karlin, Short-Wave Limit of Hydrodynamics: A Soluble Example, Phys. Rev. Lett., 1996, V. 77, N. 2, P. 282-285.

110.   N.N. Bugaenko, A.N. Gorban, M.G. Sadovskii, Information content in nucleotide sequences, Mol Biol, 30 (3): 313-320, 1996.

111.   A.N. Gorban, T.G. Popova, M.G. Sadovskii, Human virus genes are less redundant than human genes, Genetika, 32 (2), 289-294, 1996.

112.   A.N. Gorban, I.V.Karlin, V.B.Zmievskii, T.F.Nonnenmacher, Relaxational trajectories: global approximations, Physica A, 1996, V.231, No.4, p.648-672.

113.   A.N. Gorban, D.N.Golub, Multi-Particle Networks for Associative Memory, Proc. of the World Congress on Neural Networks, Sept. 15-18, 1996, San Diego, CA, Lawrence  Erlbaum Associates, 1996, pp. 772-775.

114.   S.E. Gilev, A. N. Gorban, On Completeness of the Class of Functions Computable by Neural Networks, Proc. of the World Congress on Neural Networks, Sept. 15-18, 1996,  San Diego, CA, Lawrence Erlbaum Associates, 1996, pp. 984-991.

115.   A.N. Gorban, D.A. Rossiyev, E.V. Butakova, S.E. Gilev, S.E. Golovenkin, S.A. Dogadin, D.A. Kochenov, E.V. Maslennikova, G.V. Matyushin, Y.E. Mirkes, B.V. Nazarov, Medical and Physiological Applications of MultiNeuron Neural Simulator. Proceedings of the 1995 World Congress On Neural Networks, A Volume in the INNS Series of Texts, Monographs, and Proceedings, Vol. 1, 1995.

116.   M.G. Dorrer, A.N. Gorban, A.G. Kopytov, V.I. Zenkin, Psychological Intuition of Neural Networks. Proceedings of the 1995 World Congress On Neural Networks, A Volume in the INNS Series of Texts, Monographs, and Proceedings, Vol. 1, 1995.

117.   A.N. Gorban, C. Waxman, Neural Networks for Political Forecast. Proceedings of the 1995 World Congress On Neural Networks, A Volume in the INNS Series of Texts, Monographs, and Proceedings, Vol. 1, 1995.

118.   A.N. Gorban, T.G. Popova, M.G. Sadovskii, Redundancy of genetic texts, Mol Biol, 28 (2), 206-213, 1994.

119.   A.N. Gorban, T.G. Popova, M.G. Sadovskii, Correlation approach to comparing nucleotide-sequences, Zh Obshch Biol, 55 (4-5), 420-430, 1994.

120.   A.N. Gorban, I.V. Karlin, General approach to constructing models of the Boltzmann equation, Physica A, 206 (1994), 401-420.

121.   A.N. Gorban, I.V. Karlin, Method of invariant manifolds and regularization of acoustic spectra, Transport Theory and Stat. Phys., 23, 559-632, 1994.

122.   A.N. Gorban, E.M. Mirkes, T.G. Popova, M.G. Sadovskii, A new approach to the investigations of statistical properties of genetic texts, Biofizika 38 (5), 762-767, 1993.

123.   A.N. Gorban, E.M. Mirkes, T.G. Popova, M.G. Sadovskii, The comparative redundancy of genes of various organisms and viruses, Genetika 29 (9), 1413-1419, 1993.

124.   A.N. Gorban, I.V.Karlin, Structure and Approximations of the Chapman-Enskog Expansion for Linearized Grad Equations, Transport Theory and Stat.Phys, V.21, No 1&2,  P.101-117, 1992.

125.   V.I. Verbitskii, A.N. Gorban, Jointly dissipative operators and their applications, Siberian Math J, 33 (1), 19-23, 1992.

126.   A.N. Gorban, E.M. Mirkes, A.P. Svitin, Method of multiplet covering and its application for the prediction of atom and molecular-properties, Zh Fiz Khim, 66 (6): 1504-1510, 1992.

127.   V.I. Bykov, V.I. Verbitskii, A.N. Gorban, Evaluation of cauchy-problem solution with inaccurately given initial data and the right part, Izv Vuz Mat, (12), 5-8, 1991.

128.   A.N. Gorban, V.I.Verbitsky, Simultaneously Dissipative Operators and Quasi-Thermodynamicity of the Chemical Reactions Systems, Advances in Modelling and Simulation, 1992, V.26,  N1, p.13-21.

129.   N.N. Bugaenko, A. N. Gorban, I.V.Karlin  Universal Expansion of the Triplet Distribution Function, Teoreticheskaya i Matematicheskaya Fisica, V.88, No.3, P.430-441(1991).

130.   A.N. Gorban, I.V.Karlin, Approximations of the Chapman-Enskog Expansion, Zh.Exp.Teor.Fis., V.100, No.4(10), P.1153-1161(1991); Sov. Phys. JETP, V.73(4),  P.637-641.(1991).

131.   S.Ye. Gilev, A. N. Gorban and E.M. Mirkes, Small Experts and Internal Conflicts in Learnable Neural Networks, Doklady Acad. Nauk SSSR, V.320, No.1, (1991) P.220-223.

132.   A.N. Gorban, E.M. Mirkes, A.N. Bocharov, V.I. Bykov,  Thermodynamic consistency of kinetic data, Combust Explosion & Shock, 25 (5), 593-600, 1989.

133.   V.I. Verbitskii, A.N. Gorban, G.S. Utiubaev, Y.I. Shokin, Moores effect in interval spaces, Dokl Akad Nauk SSSR, 304 (1), 17-21 1989.

134.   A.N. Gorban, M.G. Sadovskii, Optimal strategies of spatial-distribution - Olli effect, Zh Obshch Biol 50 (1), 16-21, 1989.

135.   A.N. Gorban, K.R.Sedov and  E.V.Smirnova, Correlation Adaptometry as a Method for Measuring the Health, Vestnik Acad. Medic. Nauk SSSR, No.5, P.69-75(1989).

136.   V.I.Bykov, A. N. Gorban, A Model of Autooscillations in Association Reactions, Chem.Eng.Sci., V.42, No.5, P.1249-1251(1987).

137.   A.N. Gorban, M.G.Sadovskii, Evolutionary Mechanisms of Creation of Cellular Clusters in Flowrate Cultivators, Biotechnology and Biotechnics, No.5, P.34-36(1987).

138.   V.I.Bykov, A. N. Gorban, G.S.Yablonskii. Thermodynamic Function Analogue for Reactions Proceeding Without Interactions of Various Substances, Chem.Eng.Sci., V.41, No.11, P.2739-2745 (1986).

139.   V.I. Bykov, S.E. Gilev, A.N. Gorban, G.S. Yablonskii, Imitation modeling of the diffusion on the surface of a catalyst, Dokl Akad Nauk SSSR, 283 (5): 1217-1220 1985.

140.   V.I. Bykov, A.N. Gorban, Simplest model of self-oscillations in association reactions, React Kinet Catal Lett, 27 (1): 153-155 1985

141.   V.I. Bykov, A.N. Gorban, T.P. Pushkareva, Autooscillation model in reactions of the association, Zh Fiz Khim, 59 (2): 486-488, 1985.

142.   A.N. Gorban, V.I. Bykov, G.S. Yablonskii, Description of non-isothermal reactions using equations of nonideal chemical-kinetics, Kinet Catal, 24 (5), 1055-1063, 1983.

143.   V.I. Bykov, A.N. Gorban, L.P. Kamenshchikov, G.S. Yablonskii, Inhomogeneous stationary states in reaction of carbon-monoxide oxidation on platinum, Kinet Catal, 24 (3), 520-524, 1983

144.   V.I. Bykov, A.N. Gorban, Quasithermodynamic characteristic of reactions without the reaction of different substances, Zh Fiz Khim, 57 (12), 2942-2948, 1983.

145.   V.I. Bykov, A.N. Gorban, G.S. Yablonskii, Description of non-isothermal reactions in terms of Marcelin-De-Donder kinetics and its generalizations, React Kinet Catal Lett, 20 (3-4), 261-265, 1982.

146.   S.E. Gilev, A.N. Gorban, V.I. Bykov, G.S. Yablonskii, Simulative modeling of processes on a catalyst surface, Dokl Akad Nauk SSSR, 262 (6), 1413-1416, 1982.

147.   V.I. Elokhin, G.S. Yablonskii, A.N. Gorban, V.M. Ceresiz, Dynamics of chemical-reactions and non-physical steady-states, React Kinet Catal Lett, 15 (2), 245-250, 1980.

148.   A.N. Gorban, G.S. Yablonskii, On one unused possibility in the kinetic experiment design, Dokl Akad Nauk SSSR, 250 (5): 1171-1174, 1980.

149.   A.N. Gorban, V.I. Bykov, G.S. Yablonskii, The Path to Equilibrium, Intern. Chem. Eng. V.22, No.2, P.386-375(1982).

150.   A.N. Gorban, V.M.Ceresiz, Slow Relaxations of Dynamical Systems and Bifurcations of Omega-Limit Sets, Soviet Math. Dokl., V.24, P.645-649(1981).

151.   A.N. Gorban, V.I. Bykov, G.S. Yablonskii, Macroscopic Clusters Induced by Diffusion in Catalytic Oxidation Reactions, Chem. Eng. Sci., 1980. V. 35, N. 11. P. 2351-2352. .

152.   A.N. Gorban, V.I.Bykov, V.I.Dimitrov. Marcelin-De Donder Kinetics Near Equilibrium, React. Kinet. Catal. Lett., V.12, No.1, P.19-23(1979).

153.   A.N. Gorban, Priori evaluation of the region of linearity for kinetic-equations, React Kinet Catal Lett, 10 (2), 149-152, 1979

154.   A.N. Gorban, Invariant Sets for Kinetic Equations, React. Kinet. Catal. Lett., 1979, V.10, P.187-190.

155.   A.N. Gorban, Sets of Removable Singularities and Continuous Mappings, Siberian Math. Journ., V.19, P.1388-1391(1978).

156.   A.N. Gorban, V.B. Melamed, Certain properties of Fredholm analytic sets in Banach-spaces, Siberian Math J, 17 (3), 523-526, 1976.



Past Achievements and Future Research


Summary of completed research and its scientific and technological impact


Research Conducted During the Past 15 years

A collection of methods for construction of slow invariant manifolds has been developed, in particular the analogue of Kolmogorov-Arnold-Moser methods for dissipative systems. The nonperturbative deviation of physically consistent hydrodynamics from the Boltzmann equation and from reversible dynamics, for Knudsen numbers  near one, was obtained.


The theory of simultaneously dissipative operators and tools for global stability analysis were developed. An explicitly solvable mathematical model for estimating the maximum efficiency of turbines in a free (non-ducted) fluid was obtained. This result can be used for hydropower turbines where construction of dams is impossible or undesirable.


A family of fast training algorithms for neural networks and generalized technology of extraction of explicit knowledge from data was developed. These algorithms are now in use in medical expert systems and in anti-terrorism security systems in Russia (the system "Voron").


The geometric seven-cluster structure of the genome was discovered.


Summary of current research project


The Geometry of Irreversibility. A new general geometrical framework of nonequilibrium thermo-dynamics will be developed. Our approach is based on constructive methods of invariant manifolds elaborated during the past two decades. The new methods allow us to solve the problem of macro-kinetics even when there are no autonomous equations of macro-kinetics. These methods will be elaborated together with computational algorithms. Each step of these algorithms should be physically consistent. The notion of the invariant film of non-equilibrium states, and the method of its approximate construction transform the problem of nonequilibrium kinetics into a series of problems of equilibrium statistical physics. The main specific problem for application of developed methods will be the problem of dynamic memory appearance in macromolecular complexes. Such memory effects may be important for chromatin dynamics and its role in functional nuclear organization. Spatio-temporal organization of chromatin will be studied.


Results and Projects (1971-2004)


1. The beginning (1971-1975)

Two scientific contacts determined my scientific work during 1971-1975: Prof. V.P. Mikheev (technical sciences) and Prof. V.B. Melamed (functional analysis). With Prof. Mikheev we created models of contact net and contact devices and developed new stations for technical diagnosis. Perhaps the main results of our collaboration are: stations for technical diagnosis that were in use on the USSR railways, new methods for modeling of  the dynamics of contact net and contact devices, and applied software for implementation of these methods.


Prof. Melamed was from the Voronezh mathematical school. We introduced the notion of a Fredholm analytic subset of Banach space as a subset that admits a local representation by a set of zeros of an analytic mapping whose differential is Fredholm. The maximum modulus principle and an analogue of the Remmert-Stein theorem were proved.


2. Chemical kinetics and topological dynamics (1975-1980)

Global constraints for the dynamics of systems follow from the well-known local thermodynamic constraints.  If a chemical system is one-dimensional (the number of different substances is equal to the number of independent balances plus one), then the equilibrium encircling is impossible. Such systems move monotonically to their equilibrium states, if external conditions are equilibrium.  If the dimension of the system is greater than one, then this monotonicity may be broken.  The effects of “equilibrium encircling” appear. The exact thermodynamic boundaries for the maximal amplitudes of these effects were found. The search for boundaries of the equilibrium encircling is based on the analysis of Lyapunov function trees in the balance polyhedra. The constructive theory of trees of convex functions in convex polyhedra was developed. These results were summarized in the book Equilibrium Encircling: Equations of Chemical Kinetics and their Thermodynamic Analysis (Novosibirsk, Nauka Publ., 1984).


A systematic analysis of singularities of transition processes in general dynamical systems was also undertaken.  Dynamical systems depending on parameter were studied and a system of relaxation times was constructed. Each relaxation time depends on three variables: initial conditions, parameters of the system and accuracy e of relaxation. The singularities of the relaxation times as functions of initial data and parameters under fixed e were studied, leading to a classification of different bifurcations (explosions). The relationship between the singularities of relaxation times and bifurcations of limit sets was investigated. The peculiarities of transition processes under perturbations were studied. It was shown that perturbations simplify the situation: the interrelations between the singularities of relaxation times and other peculiarities of dynamics for general dynamical system under small perturbations are the same as for smooth two-dimensional structurally stable systems. These results were summarized in my PhD thesis (1980).


3. Biological kinetics and functional analysis (1980-1990)

Does the dynamics of distributed systems which models biological evolution always lead to a discrete distribution? (In the biological context this question can be reformulated as follows: is natural selection really effective if the initial diversity is sufficiently rich?)  In order to answer this question, a theory of special dynamical systems in the space of Radon measures on compact space was developed.  These are systems with a specific conservation law: the conservation of support of measures. There are characterization theorems for omega-limit points, and different theorems about efficiency of natural selection. The qualitative picture of these results was summarized in the book: Demon of Darwin. The Idea of Optimality and Natural Selection, A.N. Gorban, R.G. Khlebopros (Nauka Pub. Moscow, 1988, 208 pp). A short review of these results was given in the talk “Optimality, adaptation and natural selection - the mathematical way to separate sense from nonsense”, available on-line at .


This abstract theory has found very practical application. My former PhD student, E. V. Smirnova (now Professor Smirnova) discovered that the approximate dimension of the cloud of physiological data of a group precisely characterizes the level of adaptation of this group to the living conditions: when the group members exhaust their adaptation resource then the dimension usually decreases.  It decreases usually, but not always.  Sometimes the dimension goes another way. We explained the effect, and, on the other hand, predicted the exclusions. The results were confirmed by thousands of experiments with different populations and groups: from human to plants and fungi. Now the developed concept of correlation adaptometry is in use for monitoring needs in Siberia and Far North.


4. Neural networks (1985-now)

In 1985 I stated the problem of effective parallelism as a main problem for our group for the next decade. In 1986 V. Okhonin (former PhD student) published a new algorithm for training neural networks (for synchronized and non-synchronized networks, for discrete and continuous time, for systems with delays in time, and for many other cases).  The central idea was the flexible use of duality (it is a rather usual step in optimization methods). (At the same time, Rumelhart D.E., Hinton G.E., Williams R.J. published a particular case of this algorithm that became famous under the name “back propagation of errors”.) For several years we tried to make the training algorithms faster, and network skills more stable. During an interval of fifteen years (1987-2002) we developed a generalized technology of extraction of explicit knowledge from data.  This technology was implemented in a series of software libraries and allowed us to create dozens of knowledge-based expert systems in medical and technical diagnosis, ecology and other fields.


On the base of this approach, the Russian Close Corporation "Applied Radiophysics - Security Systems" developed neural network-based security systems (1997 – 2003). This Russian system "Voron" was the laureate of the international exhibition "Frontier-2000" (see, (in Russian).


The results were summarized in several monographs, 16 PhD theses were submitted, and 3 scientists prepared Doctor of Science degrees. The developed software is in widespread use in the former USSR, and our lab in Krasnoyarsk now serves as the Russian National Center for Neuroinformatics and Neurocomputing.


5. Physical Kinetics and Invariant Manifolds (1977-present)

The concept of the slow invariant manifold is recognized as the central idea underpinning a transition from micro to macro and model reduction in kinetic theories. We developed constructive methods of invariant manifolds for model reduction in physical and chemical kinetics. The physical problem of a reduced description is studied in the most general form as a problem of constructing the slow invariant manifold. A collection of methods to derive analytically and to compute numerically the slow invariant manifold is elaborated. Among them, iteration methods based on incomplete linearization, relaxation methods and the method of invariant grids have been developed. The systematic use of thermodynamic structures and of the quasi-chemical representation allows us to construct approximations which are consistent with physical restrictions at each step.


There are many examples of applications: nonperturbative derivation of physically consistent hydrodynamics from the Boltzmann equation and from reversible dynamics, for Knudsen numbers Kn near one; construction of the moment equations for nonequilibrium media and their dynamical correction in order to gain more accuracy in the description of highly nonequilibrium flows; the kinetic theory of phonons; model reduction in chemical kinetics; derivation and numerical implementation of constitutive equations for polymeric fluids. A review of this direction of work is now published in Physics Reports.


A new approach to the lattice Boltzmann method is developed. Beginning from thermodynamic considerations, the LBM can be recognised as a discrete dynamical system generated by entropic involution and free-flight and the stability analysis is more natural. We solve the stability problem of the LBM on the basis of this thermodynamic point of view. The main instability mechanisms are identified. The simplest and most effective receipt for stabilisation adds no artificial dissipation, preserves the second-order accuracy of the method, and prescribes coupled steps: to start from a local equilibrium, then, after free-flight, perform the overrelaxation collision, and after a second free-flight step go to new local equilibrium. Two other prescriptions (“salvation rules”) add some artificial dissipation locally and prevent the system from loss of positivity and local blow-up.


6. Bioinformatics and Geometry of Genome (1990-now)

Is it possible to study the genetic text on the same way as A. Kolmogorov studied poetry? Is there a footprint of biological sense in statistical features of the genome? This question needs to be carefully solved. The result may be positive or negative.  Nevertheless, we should study this problem.  We have investigated a numbe of questions in this direction.


Some positive results have been obtained and published during the past fourteen years. In particular, the clear seven-cluster structure of genome was identified. We studied cluster structure of several genomes in the space of olygomer frequencies. The result: many complete genomic sequences were analyzed, using visualization of tables of triplet counts in a sliding window. The distribution of 64-dimensional vectors of triplet frequencies displays a well-detectable cluster structure. The structure was found to consist of seven clusters, corresponding to protein-coding information in three possible phases in one of the two complementary strands and in the non-coding regions. Awareness of the existence of this structure allows development of methods for the segmentation of sequences into regions with the same coding phase and non-coding regions. This method may be completely unsupervised.