The Self Avoiding Walk
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The Self Avoiding Walk
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Author : Neal Madras
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-11-27
The Self Avoiding Walk written by Neal Madras and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-11-27 with Mathematics categories.
A self-avoiding walk is a path on a lattice that does not visit the same site more than once. In spite of this simple definition, many of the most basic questions about this model are difficult to resolve in a mathematically rigorous fashion. In particular, we do not know much about how far an n step self-avoiding walk typically travels from its starting point, or even how many such walks there are. These and other important questions about the self-avoiding walk remain unsolved in the rigorous mathematical sense, although the physics and chemistry communities have reached consensus on the answers by a variety of nonrigorous methods, including computer simulations. But there has been progress among mathematicians as well, much of it in the last decade, and the primary goal of this book is to give an account of the current state of the art as far as rigorous results are concerned. A second goal of this book is to discuss some of the applications of the self-avoiding walk in physics and chemistry, and to describe some of the nonrigorous methods used in those fields. The model originated in chem istry several decades ago as a model for long-chain polymer molecules. Since then it has become an important model in statistical physics, as it exhibits critical behaviour analogous to that occurring in the Ising model and related systems such as percolation.
The Self Avoiding Walk
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Author : Springer
language : en
Publisher:
Release Date : 2012-11-01
The Self Avoiding Walk written by Springer and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-11-01 with categories.
The Self Avoiding Walk
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Author : Neal Noah Madras
language : en
Publisher:
Release Date : 1993
The Self Avoiding Walk written by Neal Noah Madras and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1993 with Medical categories.
The Lace Expansion And Its Applications
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Author : Gordon Slade
language : en
Publisher: Springer Science & Business Media
Release Date : 2006-05-17
The Lace Expansion And Its Applications written by Gordon Slade and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-05-17 with Mathematics categories.
The lace expansion is a powerful and flexible method for understanding the critical scaling of several models of interest in probability, statistical mechanics, and combinatorics, above their upper critical dimensions. These models include the self-avoiding walk, lattice trees and lattice animals, percolation, oriented percolation, and the contact process. This volume provides a unified and extensive overview of the lace expansion and its applications to these models.
Protein Folding And Self Avoiding Walks Polyhedral Studies And Solutions
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Author : Agnes Dittel
language : en
Publisher: Logos Verlag Berlin GmbH
Release Date : 2008
Protein Folding And Self Avoiding Walks Polyhedral Studies And Solutions written by Agnes Dittel and has been published by Logos Verlag Berlin GmbH this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008 with Mathematics categories.
The protein folding problem refers to the correlation of a protein's amino acid sequence and its native three-dimensional structure which is essential for functionality. It still constitutes one of the major challenges in computational biology. One commonly studied model for the protein folding problem is the HP lattice model in which proteins are considered in a fairly abstract representation. However, the HP model proteins exhibit significant parallels to proteins occurring in nature. The solution of the HP lattice mode as a combinatorial optimization problem has been proven to be NP-complete, and there have already been developed various different approaches for efficient algorithms. We study an integer programming formulation of the problem. Starting with an analysis of this model, where we concentrate on symmetry issues, we show how the model can be consolidated by exploiting symmetry properties of the underlying lattice. The main focus lies in the development of specific components of a branch-and-cut framework for the computation of solutions for the HP model by means of integer programming methods. In order to understand the structure of the model, we perform a series of polyhedral studies from which we derive two main classes of cutting planes. Furthermore, we exploit the knowledge of folding principles which are also valid for HP model proteins for the development of related branching strategies. For the solution of a special class of instances, we present an implementation of a genetic algorithm for the generation of primal feasible start solutions. Finally, we document the performance of the methods developed for each of the four topics (model consolidation, primal method, branching strategy and cutting planes) within the branch-and-cut procedure. We present computational results for different types of lattices, where we both consider known benchmark instances from literature and random instances.
Polygons Polyominoes And Polycubes
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Author : A. J. Guttmann
language : en
Publisher: Springer
Release Date : 2009-03-30
Polygons Polyominoes And Polycubes written by A. J. Guttmann and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-03-30 with Science categories.
The problem of counting the number of self-avoiding polygons on a square grid, - therbytheirperimeterortheirenclosedarea,is aproblemthatis soeasytostate that, at ?rst sight, it seems surprising that it hasn’t been solved. It is however perhaps the simplest member of a large class of such problems that have resisted all attempts at their exact solution. These are all problems that are easy to state and look as if they should be solvable. They include percolation, in its various forms, the Ising model of ferromagnetism, polyomino enumeration, Potts models and many others. These models are of intrinsic interest to mathematicians and mathematical physicists, but can also be applied to many other areas, including economics, the social sciences, the biological sciences and even to traf?c models. It is the widespread applicab- ity of these models to interesting phenomena that makes them so deserving of our attention. Here however we restrict our attention to the mathematical aspects. Here we are concerned with collecting together most of what is known about polygons, and the closely related problems of polyominoes. We describe what is known, taking care to distinguish between what has been proved, and what is c- tainlytrue,but has notbeenproved. Theearlierchaptersfocusonwhatis knownand on why the problems have not been solved, culminating in a proof of unsolvability, in a certain sense. The next chapters describe a range of numerical and theoretical methods and tools for extracting as much information about the problem as possible, in some cases permittingexactconjecturesto be made.
Mathematical Constants
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Author : Steven R. Finch
language : en
Publisher: Cambridge University Press
Release Date : 2003-08-18
Mathematical Constants written by Steven R. Finch and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003-08-18 with Mathematics categories.
Steven Finch provides 136 essays, each devoted to a mathematical constant or a class of constants, from the well known to the highly exotic. This book is helpful both to readers seeking information about a specific constant, and to readers who desire a panoramic view of all constants coming from a particular field, for example, combinatorial enumeration or geometric optimization. Unsolved problems appear virtually everywhere as well. This work represents an outstanding scholarly attempt to bring together all significant mathematical constants in one place.
The Statistical Mechanics Of Interacting Walks Polygons Animals And Vesicles
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Author : E. J. Janse van Rensburg
language : en
Publisher: OUP Oxford
Release Date : 2015-05-14
The Statistical Mechanics Of Interacting Walks Polygons Animals And Vesicles written by E. J. Janse van Rensburg and has been published by OUP Oxford this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-05-14 with Mathematics categories.
The self-avoiding walk is a classical model in statistical mechanics, probability theory and mathematical physics. It is also a simple model of polymer entropy which is useful in modelling phase behaviour in polymers. This monograph provides an authoritative examination of interacting self-avoiding walks, presenting aspects of the thermodynamic limit, phase behaviour, scaling and critical exponents for lattice polygons, lattice animals and surfaces. It also includes a comprehensive account of constructive methods in models of adsorbing, collapsing, and pulled walks, animals and networks, and for models of walks in confined geometries. Additional topics include scaling, knotting in lattice polygons, generating function methods for directed models of walks and polygons, and an introduction to the Edwards model. This essential second edition includes recent breakthroughs in the field, as well as maintaining the older but still relevant topics. New chapters include an expanded presentation of directed models, an exploration of methods and results for the hexagonal lattice, and a chapter devoted to the Monte Carlo methods.
Soft Matter
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Author : Wim van Saarloos
language : en
Publisher: Princeton University Press
Release Date : 2024-03-26
Soft Matter written by Wim van Saarloos and has been published by Princeton University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-03-26 with Science categories.
"Soft matter science is an interdisciplinary field at the interface of physics, biology, chemistry, engineering, and materials science. It encompasses colloids, polymers, and liquid crystals as well as rapidly emerging topics such as metamaterials, memory formation and learning in matter, bioactive systems, and artificial life. This textbook introduces key phenomena and concepts in soft matter from a modern perspective, marrying established knowledge with the latest developments and applications. The presentation integrates statistical mechanics, dynamical systems, and hydrodynamic approaches, emphasizing conservation laws and broken symmetries as guiding principles while paying attention to computational and machine learning advances. The book features introductory chapters on fluid mechanics, elasticity, and stochastic phenomena and also covers advanced topics such as pattern formation and active matter. it discusses technological applications as well as relevant phenomena in the life sciences and offers perspectives on emerging research directions"--
Probability And Phase Transition
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Author : G.R. Grimmett
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-04-17
Probability And Phase Transition written by G.R. Grimmett and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-04-17 with Science categories.
This volume describes the current state of knowledge of random spatial processes, particularly those arising in physics. The emphasis is on survey articles which describe areas of current interest to probabilists and physicists working on the probability theory of phase transition. Special attention is given to topics deserving further research. The principal contributions by leading researchers concern the mathematical theory of random walk, interacting particle systems, percolation, Ising and Potts models, spin glasses, cellular automata, quantum spin systems, and metastability. The level of presentation and review is particularly suitable for postgraduate and postdoctoral workers in mathematics and physics, and for advanced specialists in the probability theory of spatial disorder and phase transition.