Abstract
The phenomenological and mathematical definitions of a class of dynamic systems with an entropy operator are formulated. Dynamic systems with an entropy operator are classified and the main theoretical results pertaining to the properties of entropy operators and these dynamic systems are studied within this classification. By way of examples, restoration of monochromatic images and modeling of labor market are examined.
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Boltzmann, L., On the Link between the Second Beginning of Mechanical Calory Theory and Probability Theory in Theorems of Thermal Equilibrium, 1877, in Izbr. tr. Ser. Klassiki nauki (Selected Papers in Classical Works of Science), Moscow: Nauka, 1984, pp. 190–236.
Landau, L.D. and Lifshits, E.M., Statisticheskaya fizika (Statistical Physics), Moscow: Nauka, 1964.
Meerkov, S.M., Mathematical Theory of Behavior—Individual and Collective Behavior of Retardable Elements, Math. Biosci., 1979, vol. 43, pp. 41–106.
Haken, H., Synergetics. An Introduction, Berlin: Springer-Verlag, 1978. Translated under the title Sinergetika, Moscow: Mir, 1980.
Loskutov, A.M. and Mikhailov, A.S., Vvedenie v sinergetiky (Introduction to Synergetics), Moscow: Nauka, 1990.
Chernavskii, D.S., Sinergetika i informatsiya (Synergetics and Information), Moscow: URSS, 2004.
Sinergetika i problemy teorii upravleniya (Synergetics and Control Problems), Kolesnikov, A.A., Ed., Moscow: Fizmatlit, 2004.
Kapitsa, S.P., Kurdyumov, S.P., and Malinetskii, G.G., Sinergetika i prognozy budushchego (Sinergetics and Prediction of the Future), Moscow: URSS, 2003.
Prigogine, I. and Stengers, I., Order Out of Chaos, London: Heinemann, 1984.
Weidlich, W. and Haag, G., Interregional Migration: Dynamic Theory and Comparative Analysis, Berlin: Springer-Verlag, 1988.
Weidlich, W., Sotsiodinamika. Sistemnyi podkhod k modelirovanie v sotsial’nykh naukakh (Sociodynamics: A System Approach to Modeling in Social Sciences), Moscow: URSS, 2004.
Nicolis, G. and Prigogine, I., Self-Organization in Non-Equilibrium Systems, New York: Wiley, 1977. Translated under the title Samoorganizatsiya v neravnovesnykh sistemakh Moscow: Mir, 1979.
Ebling, V., Engel, A., and Faisteil, R., Fizika protsessov evolyutsii (Physics of Evolution Processes), Moscow: URSS, 2001.
Rozonoer, L.I., Resource Exchange and Distribution: A Generalized Thermodynamic Approach. I–III, Avtom. Telemekh., 1973, no. 5, pp. 115–132; no. 6, pp. 65–79; no. 8, pp. 82–103.
Wilson, A.G., Entropy in Urban and Regional Modeling, London: Pion, 1970. Translated under the title Entropiinye metody modelirovaniya slozhnykh sistem, Moscow: Nauka, 1978.
Popkov, Yu.S., Teoriya makrosistem, Moscow: URSS, 1999. Translated under the title Macrosystem Theory and Its Applications, London: Springer, 1995.
Tsirlin, A.M., Optimal’nye protsessy v neobratimoi termodinamike i ekonomike (Optimal Processes in Irreversible Thermodynamics and Economics), Moscow: Fizmatlit, 2002.
Sieniutcz, S. and Farkas, H., Variational and Extremum Principles in Macroscopic Systems, Amsterdam: Elsevier, 2005.
Leontovich, M.A., Main Equation of the Kinetic Theory of Gases from the Viewpoint of the Theory of Random Processes, Zh. Eksp. Teor. Fiz., 1935, vol. 5, nos. 3, 4, pp. 211–231.
Stratonovich, R.L., Nelineinaya neravnovesnaya termodinamika (Nonlinear Nonequilibrium Thermodynamics), Moscow: Nauka, 1985.
Helbing, D., A Mathematical Model for the Behavior of Individuals in Social Field, J. Math. Soc., 1994, vol. 19(3), pp. 189–219.
Klimontovich, Yu.L., Statisticheskaya teoriya otkrytykh sistem (Statistical Theory of Open Systems), Moscow: Yanus, 1995.
Wilson, A.G., Catastrophe Theory and Bifurcations. Applications to Urban and Regional Systems, London: Croom Helm, 1981.
Popkov, Y.S. and Ryazantsev, A.N., Spatio-functional Models of Demographic Processes, Preprint of the United Nations Fund for Population Activity, 1980.
De Groot, S.T. and Mazur, P., Non-Equilibrium Thermodynamics, Amsterdam: North-Holland, 1962. Translated under the title Neravnovesnaya termodinamika, Moscow: Mir, 1964.
Popkov, Yu.S., Local Stationary Models of Nonequilibrium Dynamics of Self-Reproduction Macrosystems, Doklady Akad. Nauk SSSR, 1988, vol. 303, no. 3, pp. 14–16.
Popkov, Yu.S., A New Class of Dynamic Macrosystem Models with Self-Reproduction, Environment & Planning A, 1989, no. 21, pp. 739–751.
Shvetsov, V.I., Stationary States and Bifurcations in Regional Dynamics Models, in Dinamika Neodnorodnykh Sistem (Dynamics of Nonhomogeneous Systems), Moscow: VNIISI, 1989, pp. 34–45.
Popkov, Yu.S. and Shvetsov, V.I., The Local Equilibrium Principle in Regional Dynamic Models, Mat. Modelir., 1990, vol. 2, no. 5, pp. 40–59.
Popkov, Yu.S., Dynamic Models of Macrosystems with Self-Reproduction and Their Application to Analysis of Regional Systems, Ann. Regional Sci., 1993, no. 27, pp. 165–174.
Kitaev, O.V., Macrosystem Models of Population Dynamics, in Dinamika Neodnorodnykh Sistem (Dynamics of Nonhomogeneous Systems), Moscow: VNIISI, 1997, pp. 56–70.
Popkov, Yu.S., Shvetsov, V.I., and Weidlich, W., Settlement Formation Models with Entropy Operator, Ann. Regional Sci., 1998, vol. 32, pp. 267–294.
Volterra, V., Thèorie mathèmatique de la lutte pour la vie, Paris: Gauthier-Villars, 1931. Translated under the title Matematicheskaya teoriya bor’by za sushchestvovanie, Moscow: Nauka, 1976.
Eigen, M. and Schuster, P., The Hypercycles: A Principle of Natural Selforganization, Berlin: Springer-Verlag, 1979. Translated under the title Gipertsikl: printsipy samoorganizatsii makromolekul, Moscow: Mir, 1982.
Popkov, Yu.S., Macrodynamics of a Class of Nonlinear Markovian Processes (Comparison of Models of Geographical Systems), Geographical Syst., 1996, vol. 3, pp. 59–72.
Popkov, Yu.S., Macrosystems and GRID Technologies: Modeling of Dynamic Stochastic Networks, Avtom. Telemekh., 2003, no. 12, pp. 143–163.
Popkov, Yu.S., Positive Dynamic Systems with the Entropy Operator, Avtom. Telemekh., 2003, no. 3, pp. 104–113.
Popkov, Yu.S. and Rublev, M.V., Estimation of the Local Lipschitz Constant of the B q-Entropy Operator, Avtom. Telemekh., 2005, no. 7, pp. 54–65.
Gantmakher, F.R., Teoriya matrits, Moscow: Nauka, 1967. Translated under the title The Theory of Matrices, New York: Chelsea, 1959.
Byrne, C.L., Iterative Image Reconstruction Algorithms based on Cross-Entropy Minimization, IEEE Trans. Img. Proc., 1993, vol. 2, no. 1, pp. 444–466.
Popkov, Yu.S., A Variational Principle of Image Reconstruction from Projections, Avtom. Telemekh., 1997, no. 12, pp. 131–139.
Popkov, Yu.S. and Rublev, M.V., Dynamic Procedures of Image Reconstruction from Projections (Computer Tomography), Avtom. Telemekh., 2006, no. 2, pp. 70–79.
Vissen, L., Popkov, A.Yu., Popkov, E.Yu, and Popkov, Yu.S., A Labor Market Entropy-Operator Model, Ekonom. Mat. Metody, 2004, vol. 40, no. 2, pp. 99–112.
Tsypkin, Ya.Z. and Popkov, Yu.S., Teoriya nelineinykh impul’snykh sistem (Theory of Nonlinear Pulse Systems), Moscow: Nauka, 1973.
Antipin, S.A., Continuous and Iterative Projection Operators and Projection Types, in Vopr. Kibernetiki. Vychisl. Voprocy Analyza Bol’shikh System, Moscow: Akad. Nauk SSSR, 1989.
Bobylev, N.A. and Popkov, A.Yu., Forced Oscillations in Systems with Arg min Type Operators, Avtom. Telemekh., 2002, no. 11, pp. 13–23.
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Original Russian Text © Yu.S. Popkov, 2006, published in Avtomatika i Telemekhanika, 2006, No. 6, pp. 75–105.
This paper was recommended for publication by B.T. Polyak, a member of the Editorial Board