E. Goles - Böcker

"Et moi, ..., si j'avait Sll comment en revenir. One sennce mathematics has rendered the human race. It has put common sense back je n'y serais point alle.' Jules Verne whe", it belongs, on the topmost shelf next to the dusty canister labelled 'discarded non- The series is divergent; therefore we may be smse'. able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non- linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics ...'; 'One service logic has rendered com- puter science ...'; 'One service category theory has rendered mathematics ...'. All arguably true. And all statements obtainable this way form part of the raison d'!ltre of this series.

One service mathematics has rendered the ~Et moi, ...si j'avait su comment en revenir, je human race. It has put common sense back n'y serais point aile.' where it belongs, on the topmost shelf next to Jules Verne the dusty canister labelled 'discarded nonsense'. Eric T. Bell The series is divergent; therefore we may be able to do something with it. O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and nonlineari- ties abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sci- ences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One ser- vice topology has rendered mathematical physics ...'; 'One service logic has rendered computer science ...'; 'One service category theory has rendered mathematics ...'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

This book contains the lectures given at the NATO ASI 910820 "Cellular Automata and Cooperative Systems" Meeting which was held at the Centre de Physique des Houches, France, from June 22 to July 2, 1992. This workshop brought together mathematical physicists, theoretical physicists and mathe- maticians working in fields related to local interacting systems, cellular and probabilistic automata, statistical physics, and complexity theory, as well as applications of these fields. We would like to thank our sponsors and supporters whose interest and help was essential for the success of the meeting: the NATO Scientific Affairs Division, the DRET (Direction des Recherches, Etudes et Techniques), the Ministere des Affaires Etrangeres, the National Science Foundation. We would also like to thank all the secretaries who helped us during the preparation of the meeting, in particular Maryse Cohen-Solal (CPT, Marseille) and Janice Nowinski (Courant Institute, New York). We are grateful for the fine work of Mrs. Gladys Cavallone in preparing this volume.

This volume contains the lectures given at the Third School on Statistical Physics and Cooperative Systems, Santiago, Chile, in December 1992. All lectures are related to recent interdisciplinary trends in statistical physics, including nonlinear phenomena, dynamical systems, ergodic theory, cellular automata, symbolic dynamics, large deviations theory and neural networks. Each contribution is devoted to one or more of the previous subjects. In most cases they are structured as surveys presenting an original point of view and new results at the same time. This volume should be of interest to researchers and graduate students interested in statistical physics and theoretical computer science.

This text contains the lectures given at the Fourth School on Statistical Physics and Cooperative Systems, held in Santiago, Chile, December 12-16, 1994. This school brings together scientists working on subjects related to recent trends in complex systems. Subjects dealt with include dynamical systems, ergodic theory, cellular automata, symbolic and arithmetic dynamics, spatial systems, large deviation theory and neural networks.

This text contains the courses given at the Fifth School on Complex Systems held at Santiago, Chile, from 9th to 13th December, 1996, by researchers working on areas related to recent trends in complex systems, including dynamical systems, cellular automata, symbolic dynamics, spatial systems, statistical physics and thermodynamics. Scientists in these subjects come from several areas: pure and applied mathematics, physics, biology, computer science and electrical engineering. Each contribution is devoted to one of the above subjects. In most cases they are structured as surveys, presenting an original point of view about the topic as well as showing mostly new results.

This volume contains the courses given at the Sixth Summer School on Complex Systems held at the Faculty of Physical and Mathematical Sciences, University of Chile at Santiago, Chile, 14-18 December 1998. The contributions, which in some cases have been structured as surveys, treat recoding Sturmian sequences on a subshift of finite type chaos from order; Lyapunov exponents and synchronization of cellular automata; dynamical systems and biological regulations; cellular automata and artificial life; Kolmogorov complexity; and cutoff for Markov chains. This book should be of interest to graduate students and researchers whose work involves mathematical modelling and industrial mathematics, statistical physics, thermodynamics, algorithms and computational theory, statistics and probability, and discrete mathematics.

This volume contains the lectures given at the Third School on Statistical Physics and Cooperative Systems, Santiago, Chile, in December 1992. All lectures are related to recent interdisciplinary trends in statistical physics: nonlinear phenomena, dynamical systems, ergodic theory, cellular automata, symbolic dynamics, large deviations theory and numeral networks. Each contribution is devoted to one or more of the previous subjects. In most cases they are structured as surveys presenting an original point of view and new results at the same time. For researchers and graduate students interested in statistical physics and theoretical computer science.

This book contains the lectures given at the Fourth School on Statistical Physics and Cooperative Systems, held in Santiago, Chile, December 12-16, 1994. This School brings together scientists working on subjects related to recent trends in complex systems. Subjects dealt with include dynamical systems, ergodic theory, cellular automata, symbolic and arithmetic dynamics, spatial systems, large deviation theory and neural networks. Audience: This volume will appeal to scientists working in the areas of pure and applied mathematics, nonlinear physics, biology, computer science, electrical engineering and artificial intelligence.

This book contains the courses given at the Fifth School on Complex Systems held at Santiago, Chile, from 9th to 13th December, 1996, by researchers working on areas related to recent trends in Complex Systems, including dynamical systems, cellular automata, symbolic dynamics, spatial systems, statistical physics and thermo-dynamics. Scientists working in these subjects come from several areas: pure and applied mathematics, physics, biology, computer science and electrical engineering. Each contribution is devoted to one of the above subjects. In most cases they are structured as surveys, presenting an original point of view about the topic as well as showing mostly new results.

This volume contains the courses given at the Sixth Summer School on Complex Systems held at Facultad de Ciencias Fisicas y Maternaticas, Universidad de Chile at Santiago, Chile, from 14th to 18th December 1998. This school was addressed to graduate students and researchers working on areas related with recent trends in Complex Systems, including dynamical systems, cellular automata, complexity and cutoff in Markov chains. Each contribution is devoted to one of these subjects. In some cases they are structured as surveys, presenting at the same time an original point of view and showing mostly new results. The paper of Pierre Arnoux investigates the relation between low complex systems and chaotic systems, showing that they can be put into relation by some re­ normalization operations. The case of quasi-crystals is fully studied, in particular the Sturmian quasi-crystals. The paper of Franco Bagnoli and Raul Rechtman establishes relations be­ tween Lyapunov exponents and synchronization processes in cellular automata. The principal goal is to associate tools, usually used in physical problems, to an important problem in cellularautomata and computer science, the synchronization problem. The paper of Jacques Demongeot and colleagues gives a presentation of at­ tractors of dynamical systems appearing in biological situations. For instance, the relation between positive or negative loops and regulation systems.

This book contains the lectures given at the NATO ASI 910820 "Cellular Automata and Cooperative Systems" Meeting which was held at the Centre de Physique des Houches, France, from June 22 to July 2, 1992. This workshop brought together mathematical physicists, theoretical physicists and mathe- maticians working in fields related to local interacting systems, cellular and probabilistic automata, statistical physics, and complexity theory, as well as applications of these fields. We would like to thank our sponsors and supporters whose interest and help was essential for the success of the meeting: the NATO Scientific Affairs Division, the DRET (Direction des Recherches, Etudes et Techniques), the Ministere des Affaires Etrangeres, the National Science Foundation. We would also like to thank all the secretaries who helped us during the preparation of the meeting, in particular Maryse Cohen-Solal (CPT, Marseille) and Janice Nowinski (Courant Institute, New York). We are grateful for the fine work of Mrs. Gladys Cavallone in preparing this volume.

One service mathematics has rendered the ~Et moi, ...si j'avait su comment en revenir, je human race. It has put common sense back n'y serais point aile.' where it belongs, on the topmost shelf next to Jules Verne the dusty canister labelled 'discarded nonsense'. Eric T. Bell The series is divergent; therefore we may be able to do something with it. O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and nonlineari- ties abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sci- ences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One ser- vice topology has rendered mathematical physics ...'; 'One service logic has rendered computer science ...'; 'One service category theory has rendered mathematics ...'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

"Et moi, ..., si j'avait Sll comment en revenir. One sennce mathematics has rendered the human race. It has put common sense back je n'y serais point alle.' Jules Verne whe", it belongs, on the topmost shelf next to the dusty canister labelled 'discarded non- The series is divergent; therefore we may be smse'. able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non- linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics ...'; 'One service logic has rendered com- puter science ...'; 'One service category theory has rendered mathematics ...'. All arguably true. And all statements obtainable this way form part of the raison d'!ltre of this series.