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Dynamics and Randomness II

Author: Alejandro Maass

Publisher: Springer Science & Business Media

Published: 2004-05-31

Total Pages: 244

ISBN-13: 9781402019906

This book contains the lectures given at the Second Conference on Dynamics and Randomness held at the Centro de Modelamiento Matemático of the Universidad de Chile, from December 9-13, 2003. This meeting brought together mathematicians, theoretical physicists, theoretical computer scientists, and graduate students interested in fields related to probability theory, ergodic theory, symbolic and topological dynamics. The courses were on: -Some Aspects of Random Fragmentations in Continuous Times; -Metastability of Ageing in Stochastic Dynamics; -Algebraic Systems of Generating Functions and Return Probabilities for Random Walks; -Recurrent Measures and Measure Rigidity; -Stochastic Particle Approximations for Two-Dimensional Navier Stokes Equations; and -Random and Universal Metric Spaces. The intended audience for this book is Ph.D. students on Probability and Ergodic Theory as well as researchers in these areas. The particular interest of this book is the broad areas of problems that it covers. We have chosen six main topics and asked six experts to give an introductory course on the subject touching the latest advances on each problem.

Random Graph Dynamics

Author: Rick Durrett

Publisher: Cambridge University Press

Published: 2010-05-31

Total Pages: 203

ISBN-13: 1139460889

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The theory of random graphs began in the late 1950s in several papers by Erdos and Renyi. In the late twentieth century, the notion of six degrees of separation, meaning that any two people on the planet can be connected by a short chain of people who know each other, inspired Strogatz and Watts to define the small world random graph in which each site is connected to k close neighbors, but also has long-range connections. At a similar time, it was observed in human social and sexual networks and on the Internet that the number of neighbors of an individual or computer has a power law distribution. This inspired Barabasi and Albert to define the preferential attachment model, which has these properties. These two papers have led to an explosion of research. The purpose of this book is to use a wide variety of mathematical argument to obtain insights into the properties of these graphs. A unique feature is the interest in the dynamics of process taking place on the graph in addition to their geometric properties, such as connectedness and diameter.

P-adic Deterministic and Random Dynamics

Author: Andrei Y. Khrennikov

Publisher: Springer Science & Business Media

Published: 2004-10-18

Total Pages: 296

ISBN-13: 9781402026591

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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.

Small Worlds

Author: Duncan J. Watts

Publisher: Princeton University Press

Published: 2018-06-05

Total Pages: 279

ISBN-13: 0691188335

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Everyone knows the small-world phenomenon: soon after meeting a stranger, we are surprised to discover that we have a mutual friend, or we are connected through a short chain of acquaintances. In his book, Duncan Watts uses this intriguing phenomenon--colloquially called "six degrees of separation"--as a prelude to a more general exploration: under what conditions can a small world arise in any kind of network? The networks of this story are everywhere: the brain is a network of neurons; organisations are people networks; the global economy is a network of national economies, which are networks of markets, which are in turn networks of interacting producers and consumers. Food webs, ecosystems, and the Internet can all be represented as networks, as can strategies for solving a problem, topics in a conversation, and even words in a language. Many of these networks, the author claims, will turn out to be small worlds. How do such networks matter? Simply put, local actions can have global consequences, and the relationship between local and global dynamics depends critically on the network's structure. Watts illustrates the subtleties of this relationship using a variety of simple models---the spread of infectious disease through a structured population; the evolution of cooperation in game theory; the computational capacity of cellular automata; and the sychronisation of coupled phase-oscillators. Watts's novel approach is relevant to many problems that deal with network connectivity and complex systems' behaviour in general: How do diseases (or rumours) spread through social networks? How does cooperation evolve in large groups? How do cascading failures propagate through large power grids, or financial systems? What is the most efficient architecture for an organisation, or for a communications network? This fascinating exploration will be fruitful in a remarkable variety of fields, including physics and mathematics, as well as sociology, economics, and biology.

Random Operators

Author: Michael Aizenman

Publisher: American Mathematical Soc.

Published: 2015-12-11

Total Pages: 343

ISBN-13: 1470419130

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This book provides an introduction to the mathematical theory of disorder effects on quantum spectra and dynamics. Topics covered range from the basic theory of spectra and dynamics of self-adjoint operators through Anderson localization--presented here via the fractional moment method, up to recent results on resonant delocalization. The subject's multifaceted presentation is organized into seventeen chapters, each focused on either a specific mathematical topic or on a demonstration of the theory's relevance to physics, e.g., its implications for the quantum Hall effect. The mathematical chapters include general relations of quantum spectra and dynamics, ergodicity and its implications, methods for establishing spectral and dynamical localization regimes, applications and properties of the Green function, its relation to the eigenfunction correlator, fractional moments of Herglotz-Pick functions, the phase diagram for tree graph operators, resonant delocalization, the spectral statistics conjecture, and related results. The text incorporates notes from courses that were presented at the authors' respective institutions and attended by graduate students and postdoctoral researchers.

Topological Dynamics of Random Dynamical Systems

Author: Nguyen Dinh Cong

Publisher: Oxford University Press

Published: 1997

Total Pages: 216

ISBN-13: 9780198501572

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This book is the first systematic treatment of the theory of topological dynamics of random dynamical systems. A relatively new field, the theory of random dynamical systems unites and develops the classical deterministic theory of dynamical systems and probability theory, finding numerous applications in disciplines ranging from physics and biology to engineering, finance and economics. This book presents in detail the solutions to the most fundamental problems of topological dynamics: linearization of nonlinear smooth systems, classification, and structural stability of linear hyperbolic systems. Employing the tools and methods of algebraic ergodic theory, the theory presented in the book has surprisingly beautiful results showing the richness of random dynamical systems as well as giving a gentle generalization of the classical deterministic theory.

Dynamic Random Walks

Author: Nadine Guillotin-Plantard

Publisher: Elsevier

Published: 2006-02-08

Total Pages: 279

ISBN-13: 0080462847

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The aim of this book is to report on the progress realized in probability theory in the field of dynamic random walks and to present applications in computer science, mathematical physics and finance. Each chapter contains didactical material as well as more advanced technical sections. Few appendices will help refreshing memories (if necessary!).· New probabilistic model, new results in probability theory· Original applications in computer science· Applications in mathematical physics· Applications in finance