P ADIC DETERMINISTIC AND RANDOM DYNAMICS MATHEMATICS AND ITS APPLICATIONS

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Author : Andrei Y. Khrennikov
ISBN : 9781402026607
Genre : Science
File Size : 29.94 MB
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This book provides an overview of the theory of p-adic (and more general non-Archimedean) dynamical systems. The main part of the book is devoted to discrete dynamical systems. It presents a model of probabilistic thinking on p-adic mental space based on ultrametric diffusion. Coverage also details p-adic neural networks and their applications to cognitive sciences: learning algorithms, memory recalling.

This book provides an introduction to the relatively new discipline of arithmetic dynamics. Whereas classical discrete dynamics is the study of iteration of self-maps of the complex plane or real line, arithmetic dynamics is the study of the number-theoretic properties of rational and algebraic points under repeated application of a polynomial or rational function. A principal theme of arithmetic dynamics is that many of the fundamental problems in the theory of Diophantine equations have dynamical analogs.This graduate-level text provides an entry for students into an active field of research and serves as a standard reference for researchers.

Author : Vladimir Anashin
ISBN : 9783110203004
Genre : Mathematics
File Size : 34.93 MB
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This monograph presents recent developments of the theory of algebraic dynamical systems and their applications to computer sciences, cryptography, cognitive sciences, psychology, image analysis, and numerical simulations. The most important mathematical results presented in this book are in the fields of ergodicity, p-adic numbers, and noncommutative groups. For students and researchers working on the theory of dynamical systems, algebra, number theory, measure theory, computer sciences, cryptography, and image analysis.

The subject of this conference was recent developments in p-adic mathematical physics and related areas. The field of p-Adic mathematical physics was conceived in 1987 as a result of attempts to find non-Archimedean approaches to space-time at the Planck scale as well as to strings. Since then, many applications of p-adic numbers and adeles in physics and related sciences have emerged. Some of them are p-adic and adelic string theory, p-adic and adelic quantum mechanics and quantum field theory, ultrametricity of spin glasses, biological and hierarchical systems, p-adic dynamical systems, p-adic probability theory, p-adic models of cognitive processes and cryptography, as well as p-adic and adelic cosmology.

Author : Andrei Khrennikov
ISBN : 1402018681
Genre : Computers
File Size : 51.99 MB
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This book develops a new physical/mathematical model for the functioning of the human brain, based, not on the modern Newton-Einstein view of physical reality, but on 'information reality'. The work is devoted to the physical-mathematical modeling of (conscious) cognitive phenomena. The most important distinguishing feature of the theory presented here is a new model of mental space, the so-called p-adic hierarchic tree space, and the development of mental analogs of classical and quantum mechanics. Mental processes and more general information processes are handled as a kind of new physical processes. In particular, the procedure of information quantization and an information analog of Bohmian mechanics are developed. Here, mind is a singularity in the mental pilot wave. Applications to neurophysiology, localization of mental function and brain ablations, and psychology (in particular, Freud's psychoanalysis) are considered.Audience: This book will be of interest to researchers working on physical, mathematical, cognitive, neurophysical, psychological and philosophical aspects of human consciousness.

Author : Bernard M. Dwork
ISBN : 0691036810
Genre : Mathematics
File Size : 36.28 MB
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Written for advanced undergraduate and first-year graduate students, this book aims to introduce students to a serious level of p-adic analysis with important implications for number theory. The main object is the study of G-series, that is, power series y=aij=0 Ajxj with coefficients in an algebraic number field K. These series satisfy a linear differential equation Ly=0 with LIK(x) [d/dx] and have non-zero radii of convergence for each imbedding of K into the complex numbers. They have the further property that the common denominators of the first s coefficients go to infinity geometrically with the index s. After presenting a review of valuation theory and elementary p-adic analysis together with an application to the congruence zeta function, this book offers a detailed study of the p-adic properties of formal power series solutions of linear differential equations. In particular, the p-adic radii of convergence and the p-adic growth of coefficients are studied. Recent work of Christol, Bombieri, André, and Dwork is treated and augmented. The book concludes with Chudnovsky's theorem: the analytic continuation of a G -series is again a G -series. This book will be indispensable for those wishing to study the work of Bombieri and André on global relations and for the study of the arithmetic properties of solutions of ordinary differential equations.

Author : Giuseppe Toscani
ISBN : UOM:39015019488066
Genre : Mathematics
File Size : 42.1 MB
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Contents: A.M. Anile: Modeling intense relativistic electron beams.- N. Bellomo, M. Lachowicz: On the asymptotic theory of the Boltzmann and Enskog equations. A rigorous H-theorem for the Enskog equation.- F. Brezzi et al.: On some numerical problems in semiconductor device simulation.- D.G. Cacuci, V. Protopopescu: Canonical propagators for nonlinear systems: theory and sample applications.- R.E. Caflisch: Singularity formation for vortex sheets and hyperbolic equations.- H. Cornille: Exact exponential type solutions to the discrete Boltzmann models.- P. Degond et al.: Semiconductor modelling via the Boltzmann equation.- G. Frosali: Functional-analytic techniques in the study of time-dependent electron swarms in weakly ionized gases.- G.P. Galdi, M. Padula: Further results in the nonlinear stability of the Bénard problem.- F. Golse: Particle transport in nonhomogeneous media.- K.R. Rajagopal: Some recent results on swirling flows of Newtonian and non-Newtonian fluids.- Y. Sone et al.: Evaporation and condensation of a rarefield gas between its two parallel planes.- G. Spiga: Rigorous solution to the extended kinetic equations for homogeneous gas mixtures.

Author : Erich Novak
ISBN : STANFORD:36105032438017
Genre : Mathematics
File Size : 39.91 MB
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In these notes different deterministic and stochastic error bounds of numerical analysis are investigated. For many computational problems we have only partial information (such as n function values) and consequently they can only be solved with uncertainty in the answer. Optimal methods and optimal error bounds are sought if only the type of information is indicated. First, worst case error bounds and their relation to the theory of n-widths are considered; special problems such approximation, optimization, and integration for different function classes are studied and adaptive and nonadaptive methods are compared. Deterministic (worst case) error bounds are often unrealistic and should be complemented by different average error bounds. The error of Monte Carlo methods and the average error of deterministic methods are discussed as are the conceptual difficulties of different average errors. An appendix deals with the existence and uniqueness of optimal methods. This book is an introduction to the area and also a research monograph containing new results. It is addressd to a general mathematical audience as well as specialists in the areas of numerical analysis and approximation theory (especially optimal recovery and information-based complexity).