This book is devoted to the phenomenon of synchronization and its application for determining the values of Lyapunov exponents. In recent years, the idea of synchronization has become an object of great interest in many areas of science, e.g., biology, communication or laser physics. Over the last decade, new types of synchronization have been identified and some interesting new ideas concerning the synchronization have also appeared.
This book presents the complete synchronization problem rather than just results from the research. The problem is demonstrated in relation to a kind of coupling applied between dynamical systems, whereby a unique classification of possible couplings is introduced. Another novel feature is the connection presented between synchronization and the problem of determining the Lyapunov exponents, especially for non-differentiable systems. A detailed proposal of such an estimation method and examples of its application are included.
Theory:
Classification of Couplings
Determination of Complete Synchronization Thresholds
Synchronizability of Coupled Oscillators
Readership: Mathematicians, physicists, engineers and other researchers interested in synchronization and stability of dynamical systems.
224pp Pub. date: Jun 2009
ISBN: 978-981-283-766-0
This book aims to provide mathematical analyses of nonlinear differential equations, which have proved pivotal to understanding many phenomena in physics, chemistry and biology. Topics of focus are autocatalysis and dynamics of molecular evolution, relaxation oscillations, deterministic chaos, reaction diffusion driven chemical pattern formation, solitons and neuron dynamics. Included is a discussion of processes from the viewpoints of reversibility, reflected by conservative classical mechanics, and irreversibility introduced by the dissipative role of diffusion. Each chapter presents the subject matter from the point of one or a few key equations, whose properties and consequences are amplified by approximate analytic solutions that are developed to support graphical display of exact computer solutions
Readership: Advanced undergraduates, graduate students and researchers in physics, chemistry, biology or bioinformatics who are interested in mathematical modeling.
250pp Pub. date: Jul 2009
ISBN: 978-981-4271-59-2
The monograph, as its first main goal, aims to study the overconvergence
phenomenon of important classes of Bernstein-type operators of one or several
complex variables, that is, to extend their quantitative convergence properties
to larger sets in the complex plane rather than the real intervals. The
operators studied are of the following types: Bernstein, Bernstein-Faber,
Bernstein?Butzer, q?Bernstein, Bernstein?Stancu, Bernstein?Kantorovich,
Favard-Szasz-Mirakjan, Baskakov and Balazs?Szabados.
The second main objective is to provide a study of the approximation and
geometric properties of several types of complex convolutions: the de la
Vallee Poussin, Fejer, Riesz-Zygmund, Jackson, Rogosinski, Picard, Poisson-Cauchy,
Gauss-Weierstrass, q-Picard, q-Gauss-Weierstrass, Post-Widder, rotation-invariant,
Sikkema and nonlinear. Several applications to partial differential equations
(PDEs) are also presented.
Many of the open problems encountered in the studies are proposed at the end of each chapter. For further research, the monograph suggests and advocates similar studies for other complex Bernstein-type operators, and for other linear and nonlinear convolutions.
Researchers and graduate students in the field of complex approximation of functions and its applications, mathematical analysis and numerical analysis.
352pp Pub. date: Aug 2009
ISBN: 978-981-4282-42-0
This book was originally written in 1969 by Berkeley mathematician John Rhodes. It is the founding work in what is now called algebraic engineering, an emerging field created by using the unifying scheme of finite state machine models and their complexity to tie together many fields: finite group theory, semigroup theory, automata and sequential machine theory, finite phase space physics, metabolic and evolutionary biology, epistemology, mathematical theory of psychoanalysis, philosophy, and game theory. The author thus introduced a completely original algebraic approach to complexity and the understanding of finite systems. The unpublished manuscript, often referred to as “The Wild Book”, became an underground classic, continually requested in manuscript form, and read by many leading researchers in mathematics, complex systems, artificial intelligence, and systems biology. Yet it has never been available in print until now.
This first published edition has been edited and updated by Chrystopher Nehaniv for the 21st century. Its novel and rigorous development of the mathematical theory of complexity via algebraic automata theory reveals deep and unexpected connections between algebra (semigroups) and areas of science and engineering. Co-founded by John Rhodes and Kenneth Krohn in 1962, algebraic automata theory has grown into a vibrant area of research, including the complexity of automata, and semigroups and machines from an algebraic viewpoint, and which also touches on infinite groups, and other areas of algebra. This book sets the stage for the application of algebraic automata theory to areas outside mathematics.
The material and references have been brought up to date by the editor as much as possible, yet the book retains its distinct character and the bold yet rigorous style of the author. Included are treatments of topics such as models of time as algebra via semigroup theory; evolution-complexity relations applicable to both ontogeny and evolution; an approach to classification of biological reactions and pathways; the relationships among coordinate systems, symmetry, and conservation principles in physics; discussion of “punctuated equilibrium” (prior to Stephen Jay Gould); games; and applications to psychology, psychoanalysis, epistemology, and the purpose of life.
The approach and contents will be of interest to a variety of researchers and students in algebra as well as to the diverse, growing areas of applications of algebra in science and engineering. Moreover, many parts of the book will be intelligible to non-mathematicians, including students and experts from diverse backgrounds.
Prologue: Birth, Death, Time, Space, Existence, Understanding, Science and Religion
Introduction
What is Finite Group Theory?
A Generalization of Finite Group Theory to Finite Semigroups
A Reformulation of Physics
Automata Models and the Complexity of Finite State Machines
Applications: Introduction
Part I Analysis and Classification of Biochemical Reactions
Part II Complexity of Evolved Organisms, Appendix
Part III The Lagrangian of Life: A. The Laws of Growing and Evolving Organisms
Part III The Lagrangian of Life: B. Complexity, Emotion, Neurosis and
Schizophrenia
Part IV Complexity of Games
Students and researchers interested in understanding complexity in biology (evolution, genetics, metabolism, biochemistry), physics, mathematics, philosophy, mathematical psychology and psychoanalysis, artificial intelligence, automata theory (and its foundations in semigroup and group theory), game theory, and computational sciences.
300pp Pub. date: Sep 2009
ISBN: 978-981-283-696-0
ISBN: 978-981-283-697-7(pbk)
Series on Knots and Everything - Vol. 23
This book is about “diamond”, a logic of paradox. In diamond, a statement can be true yet false; an “imaginary” state, midway between being and non-being. Diamond's imaginary values solve many logical paradoxes unsolvable in two-valued boolean logic. In this volume, paradoxes by Russell, Cantor, Berry and Zeno are all resolved. This book has three sections: Paradox Logic, which covers the classic paradoxes of mathematical logic, shows how they can be resolved in this new system; The Second Paradox, which relates diamond to Boolean logic and the Spencer-Brown “modulator”; and Metamathematical Dilemma, which relates diamond to Godelian meta-mathematics and dilemma games.
Readership: Mathematicians and computer scientists.
300pp Pub. date: Dec 2009
ISBN: 978-981-4287-13-5