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Basic Results for the Behaviour of Discrete Iterations

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Disordered Systems and Biological Organization

Part of the book series: NATO ASI Series ((NATO ASI F,volume 20))

Abstract

There has been an improved effort for about 20 years in studying the dynamical behaviour of discrete, iterative systems. The reason for this is probably that different (but conceptually similar) discrete models are presently of interest in various domains of science, such as: Physics (spin glass problems, see (10), (12), (14), for example), Chemistry (diffusion reactions (9)), Biomathematics (neural networks, genetic nets, (10), (11), (14), (18), (20)), Computer Science (pattern recognition, associative memories (7), (10), cellular automata (1), (2), (9), (19), (21), (22), (23), cellular arrays for systolic computation in V.L.S.I. systems (13), (15), (17), (19) and so on: see especially (4), (5), (6)).

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References

  1. A.W. BURKS: Essays on cellular Automata., University of Illinois Press (1970).

    MATH  Google Scholar 

  2. E.F. CODD: Cellular Automata., Academic Press (1968).

    MATH  Google Scholar 

  3. M. COSNARD, E. GOLES: Dynamique d’un automate à mémoire modélisant le fonctionnement d’un neurone., C.R.A.S., t. 299, I N° 10, (1984). (see also the papers by M. Cosnard and E. Goles in the present volume)

    Google Scholar 

  4. M. COSNARD, J. DEMONGEOT, A. LEBRETON, Editors: Rythms in biology and other fields of Applications. Springer Verlag, Lecture notes in Mathematics n°49 (1983).

    Google Scholar 

  5. J. DELLA DORA, J. DEMONGEOT, B. LACOLLE, Editors: Numerical methods in study of critical phenomena., Springer Verlag (1981).

    MATH  Google Scholar 

  6. J. DEMONGEOT, E. GOLES, M. TCHUENTE, Editors: Dynamic behaviour of automata networks, Academic Press (1984).

    Google Scholar 

  7. F. FOGELMAN: Contributions à une théorie du calcul sur réseaux. Thesis, Grenoble 1985. (see also the paper by F. Fogelman in the present volume.)

    Google Scholar 

  8. E. GOLES: Comportement dynamique de réseaux d’automates. Thesis, Grenoble 1985. (see also the paper by E. Goles in the present volume.)

    Google Scholar 

  9. J.M. GREENBERG, B.D. HASSARD, S.P. HASTINGS: Pattern formation and periodic structures in systems modelled by reaction diffusion equations. Bull. Am. Math. Soc. 84. 6, p. 1296–1327 (1978).

    Article  MATH  MathSciNet  Google Scholar 

  10. J. J. HOPFIELD: Neural networks and physical systems with emergent collective computational abilities. Proc. Nat. Acad. Sc. U.S.A. (79), p. 2554–2558 (1982).

    Article  MathSciNet  Google Scholar 

  11. S. KAUFFMAN: Behaviour of randomly constructed genetic nets. in Towards a theoretical biology, Vol3, Edinburgh University Press, p. 18–46 (1970).

    Google Scholar 

  12. S. KIRKPATRICK: Models of disordered materials. In Ill condensed matter, Les Houches, North Holland (1979).

    Google Scholar 

  13. C.A. MEAD, M.A. CONWAY: Introduction to V.L.S.I systems Addison Wesley, (1980).

    Google Scholar 

  14. P. PERETTO: Collective properties of neural networks; A statistical physics approach. (to appear in Biological Cybernetics.)

    Google Scholar 

  15. P. QUINTON: The systematic design of systolic arrays. (to appear).

    Google Scholar 

  16. F. ROBERT: Discrete iterations Springer Verlag (to appear).

    Google Scholar 

  17. Y. ROBERT: Thesis, Grenoble (to appear).

    Google Scholar 

  18. R. SHINGAI: Maximum period of 2-dimensional uniform neural networks Inf and Control (11), 324–341, (1979).

    Google Scholar 

  19. M. TCHUENTE: Contribution à l’étude des méthodes de calcul pour des systèmes de type coopératif, Thesis, Grenoble (1982). (see also the paper by M. Tchuente in the present volume.)

    Google Scholar 

  20. R. THOMAS: Kinetic Logic, Lecture Notes in biomathematics, Vol 29, Springer Verlag (1979).

    Google Scholar 

  21. J. VON NEUMANN: Theory of self reproducing automata., A.W. Burks Editor; University of Illinois Press (1966).

    Google Scholar 

  22. S. WOLFRAM (Editor): Cellular automata. Los Alamos Science (1984).

    MATH  Google Scholar 

  23. S. WOLFRAM: Statistical mechanics of cellular automata. Rev. Mod. phys. 55, n° 3, 601–642 (1983).

    Article  MATH  MathSciNet  Google Scholar 

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© 1986 Springer-Verlag Berlin Heidelberg

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Robert, F. (1986). Basic Results for the Behaviour of Discrete Iterations. In: Bienenstock, E., Soulié, F.F., Weisbuch, G. (eds) Disordered Systems and Biological Organization. NATO ASI Series, vol 20. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-82657-3_3

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  • DOI: https://doi.org/10.1007/978-3-642-82657-3_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-82659-7

  • Online ISBN: 978-3-642-82657-3

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