P-adic deterministic and random dynamicsP-adic确定性及随机动态
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作者： Andrei Yu. Khrennikov，Marcus Nilsson著
出 版 社： 化学工业出版社
出版时间： 2004-12-1字数：版次：页数： 267印刷时间： 2004/12/01开本： 16开印次：纸张： 胶版纸I S B N ： 9781402026591包装： 精装内容简介
This is the first monograph in the theory of p-adic (and more general non-Archimedean) dynamical systems. The theory of such systems is a new intensively developing discipline on the boundary between the theory of dynamical systems, theoretical physics, number theory, algebraic geometry and non-Archimedean analysis. Investigations on p-adic dynamical systems are motivated by physical applications (p-adic string theory, p-adic quantum mechanics and field theory, spin glasses) as well as natural inclination of mathematicians to generalize any theory as much as possible (e.g., to consider dynamics not only in the fields of real and complex numbers, but also in the fields of p-adic numbers). The main part of the book is devoted to discrete dynamical systems: cyclic behavior (especially when p goes to infinity), ergodicity, fuzzy cycles, dynamics in algebraic extensions, conjugate maps, small denominators. There are also studied p-adic random dynamical system, especially Markovian behavior (depending on p). In 1997 one of the authors proposed to apply p-adic dynamical systems for modeling of cognitive processes. In applications to cognitive science the crucial role is played not by the algebraic structure of fields of p-adic numbers, but by their tree-like hierarchical structures. In this book there is presented a model of probabilistic thinking on p-adic mental space based on ultrametric diffusion. There are also studied p-adic neural network and their applications to cognitive sciences: learning algorithms, memory recalling. Finally, there are considered wavelets on general ultrametric spaces, developed corresponding calculus of pseudo-differential operators and considered cognitive applications.
Audience: This book will be of interest to mathematicians working in the theory of dynamical systems, number theory, algebraic geometry, non-Archimedean analysis as well as general functional analysis, theory of pseudo-differential operators; physicists working in string theory, quantum mechanics, field theory, spin glasses; psychologists and other scientists working in cognitive sciences and even mathematically oriented philosophers.
目录
Dedication
Foreword
Acknowledgments
1. ON APPLICATIONS OF P-ADIC ANALYSIS
2. P-AD1C NUMBERS AND P-ADIC ANALYSIS
1 Ultrametric spaces
2 Non-archimedean fields
3 The field of p-adic numbers
4 Tree-like structure of the p-adic numbers
5 Extensions of the field of p-adic numbers
6 Analysis in complete non-Archimedean fields
7 Analytic functions
8 Hensel's lemma
9 Roots of unity
10 Some facts from number theory
3. P-ADIC DYNAMICAL SYSTEMS
1 Periodic points and their character
2 Monomial dynamical systems
4. PERTURBATION OF MONOMIAL SYSTEMS
1 Existence of Fixed Points of a Perturbated System
2 CycLes of Perturbed Systems
5. DYNAMICAL SYSTEMS IN FINITE EXTENSIONS OF Qp
1 Some examples on behaviour of polynomial dynamical systems in finite extensions.
2 Polynomial dynamical systems over local fields
6. CONJUGATE MAPS
1 Introduction
2 Attracting fixed points
3 Repelling fixed points
4 Small denominators
5 Neutrally stable fixed points in Cv
7. P-ADIC ERGODICITY
1 Minimality.
2 Unique ergodicity.
8. P-ADIC NEURAL NETWORKS
1 Hierarchical synaptic potentials
2 Multidimensional case
3 inimization algorithm of learning
4 Parametric dynamical networks
5 p-adic model for memory retrieval
9. DYNAMICS IN ULTRA-PSEUDOMETRIC SPACES
1 Extension of the p-adic mental model: associations and ideas
2 Dynamics in pseudometric spaces of sets
3 Existence of attractors
4 Thinking with constant sharpness of associations
5 Thinking with increasing sharpness of associations
6 Strong triangle inequality for Hausdorff's pseudometric
10. RANDOM DYNAMICS
1 Introduction to the theory of random dynamical systems
2 Random dynamics for monomial maps
3 Definition of Markovian dynamics
4 Conditions for Markovian dynamics
5 Concluding remarks
……
11. DYNAMICS OF PROBABILITYDISTRIBUTIONS ON THE
12. ULTRAMETRIC WAVELETS AND THEIR APPLICATIONS
References
Index
