Mathematics Of Aperiodic Order
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The Mathematics Of Long Range Aperiodic Order
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Author : R.V. Moody
language : en
Publisher: Springer
Release Date : 2010-12-15
The Mathematics Of Long Range Aperiodic Order written by R.V. Moody and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-12-15 with Mathematics categories.
THEOREM: Rotational symmetries of order greater than six, and also five-fold rotational symmetry, are impossible for a periodic pattern in the plane or in three-dimensional space. The discovery of quasicrystals shattered this fundamental 'law', not by showing it to be logically false but by showing that periodicity was not synonymous with long-range order, if by 'long-range order' we mean whatever order is necessary for a crystal to produce a diffraction pat tern with sharp bright spots. It suggested that we may not know what 'long-range order' means, nor what a 'crystal' is, nor how 'symmetry' should be defined. Since 1984, solid state science has been under going a veritable K uhnian revolution. -M. SENECHAL, Quasicrystals and Geometry Between total order and total disorder He the vast majority of physical structures and processes that we see around us in the natural world. On the whole our mathematics is well developed for describing the totally ordered or totally disordered worlds. But in reality the two are rarely separated and the mathematical tools required to investigate these in-between states in depth are in their infancy.
Mathematics Of Aperiodic Order
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Author : Johannes Kellendonk
language : en
Publisher: Birkhäuser
Release Date : 2015-06-05
Mathematics Of Aperiodic Order written by Johannes Kellendonk and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-06-05 with Mathematics categories.
What is order that is not based on simple repetition, that is, periodicity? How must atoms be arranged in a material so that it diffracts like a quasicrystal? How can we describe aperiodically ordered systems mathematically? Originally triggered by the – later Nobel prize-winning – discovery of quasicrystals, the investigation of aperiodic order has since become a well-established and rapidly evolving field of mathematical research with close ties to a surprising variety of branches of mathematics and physics. This book offers an overview of the state of the art in the field of aperiodic order, presented in carefully selected authoritative surveys. It is intended for non-experts with a general background in mathematics, theoretical physics or computer science, and offers a highly accessible source of first-hand information for all those interested in this rich and exciting field. Topics covered include the mathematical theory of diffraction, the dynamical systems of tilings or Delone sets, their cohomology and non-commutative geometry, the Pisot substitution conjecture, aperiodic Schrödinger operators, and connections to arithmetic number theory.
Aperiodic Order Volume 1 A Mathematical Invitation
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Author : Michael Baake
language : en
Publisher: Cambridge University Press
Release Date : 2013-08-22
Aperiodic Order Volume 1 A Mathematical Invitation written by Michael Baake 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 2013-08-22 with Mathematics categories.
Quasicrystals are non-periodic solids that were discovered in 1982 by Dan Shechtman, Nobel Prize Laureate in Chemistry 2011. The underlying mathematics, known as the theory of aperiodic order, is the subject of this comprehensive multi-volume series. This first volume provides a graduate-level introduction to the many facets of this relatively new area of mathematics. Special attention is given to methods from algebra, discrete geometry and harmonic analysis, while the main focus is on topics motivated by physics and crystallography. In particular, the authors provide a systematic exposition of the mathematical theory of kinematic diffraction. Numerous illustrations and worked-out examples help the reader to bridge the gap between theory and application. The authors also point to more advanced topics to show how the theory interacts with other areas of pure and applied mathematics.
Aperiodic Order
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Author : Michael Baake
language : en
Publisher: Cambridge University Press
Release Date : 2013-08-22
Aperiodic Order written by Michael Baake 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 2013-08-22 with Mathematics categories.
A comprehensive introductory monograph on the theory of aperiodic order, with numerous illustrations and examples.
Aperiodic Order
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Author : Michael Baake
language : en
Publisher: Cambridge University Press
Release Date : 2013
Aperiodic Order written by Michael Baake 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 2013 with Mathematics categories.
The second volume in a series exploring the mathematics of aperiodic order. Covers various aspects of crystallography.
Aperiodic Order Volume 2 Crystallography And Almost Periodicity
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Author : Michael Baake
language : en
Publisher: Cambridge University Press
Release Date : 2017-11-02
Aperiodic Order Volume 2 Crystallography And Almost Periodicity written by Michael Baake 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 2017-11-02 with Mathematics categories.
Quasicrystals are non-periodic solids that were discovered in 1982 by Dan Shechtman, Nobel Prize Laureate in Chemistry 2011. The mathematics that underlies this discovery or that proceeded from it, known as the theory of Aperiodic Order, is the subject of this comprehensive multi-volume series. This second volume begins to develop the theory in more depth. A collection of leading experts, among them Robert V. Moody, cover various aspects of crystallography, generalising appropriately from the classical case to the setting of aperiodically ordered structures. A strong focus is placed upon almost periodicity, a central concept of crystallography that captures the coherent repetition of local motifs or patterns, and its close links to Fourier analysis. The book opens with a foreword by Jeffrey C. Lagarias on the wider mathematical perspective and closes with an epilogue on the emergence of quasicrystals, written by Peter Kramer, one of the founders of the field.
Directions In Mathematical Quasicrystals
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Author : Michael Baake
language : en
Publisher: American Mathematical Soc.
Release Date : 2000
Directions In Mathematical Quasicrystals written by Michael Baake and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000 with Crystallography, Mathematical categories.
This volume includes twelve solicited articles which survey the current state of knowledge and some of the open questions on the mathematics of aperiodic order. A number of the articles deal with the sophisticated mathematical ideas that are being developed from physical motivations. Many prominent mathematical aspects of the subject are presented, including the geometry of aperiodic point sets and their diffractive properties, self-affine tilings, the role of $C*$-algebras in tiling theory, and the interconnections between symmetry and aperiodic point sets. Also discussed are the question of pure point diffraction of general model sets, the arithmetic of shelling icosahedral quasicrystals, and the study of self-similar measures on model sets. From the physical perspective, articles reflect approaches to the mathematics of quasicrystal growth and the Wulff shape, recent results on the spectral nature of aperiodic Schrödinger operators with implications to transport theory, the characterization of spectra through gap-labelling, and the mathematics of planar dimer models. A selective bibliography with comments is also provided to assist the reader in getting an overview of the field. The book will serve as a comprehensive guide and an inspiration to those interested in learning more about this intriguing subject.
Introduction To The Mathematics Of Quasicrystals
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Author : Marko V. Jaric
language : en
Publisher: Elsevier
Release Date : 2012-12-02
Introduction To The Mathematics Of Quasicrystals written by Marko V. Jaric and has been published by Elsevier this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-02 with Science categories.
Introduction to the Mathematics of Quasicrystals provides a pedagogical introduction to mathematical concepts and results necessary for a quantitative description or analysis of quasicrystals. This book is organized into five chapters that cover the three mathematical areas most relevant to quasicrystals, namely, the theory of almost periodic functions, the theory of aperiodic tilings, and group theory. Chapter 1 describes the aspects of the theory of tiling in two- and three-dimensional space that are important for understanding some of the ways in which “classical mathematical crystallography is being generalized; this process is to include possible models for aperiodic crystals. Chapter 2 examines the non-local nature of assembly “mistakes that might have significance to the quasicrystals growth. This chapter also describes how closely a physical quasicrystal might be able to approximate a three-dimensional version of tilings. Chapter 3 discusses the theoretical background and concepts of group theory of icosahedral quasicrystals. Chapter 4 presents the local properties of the three-dimensional Penrose tilings and their global construction is described through the projection method. This chapter emphasizes the relationship between quasiperiodic sets of points and quasiperiodic tiling. Chapter 5 explores the analysis of defects in quasicrystals and their kinetics, as well as some properties of the perfect system. This book is of great value to physicists, crystallographers, metallurgists, and beginners in the field of quasicrystals.
Topology Of Tiling Spaces
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Author : Lorenzo Adlai Sadun
language : en
Publisher: American Mathematical Soc.
Release Date : 2008
Topology Of Tiling Spaces written by Lorenzo Adlai Sadun and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008 with Mathematics categories.
"This book is an introduction to the topology of tiling spaces, with a target audience of graduate students who wish to learn about the interface of topology with aperiodic order. It isn't a comprehensive and cross-referenced tome about everything having to do with tilings, which would be too big, too hard to read, and far too hard to write! Rather, it is a review of the explosion of recent work on tiling spaces as inverse limits, on the cohomology of tiling spaces, on substitution tilings and the role of rotations, and on tilings that do not have finite local complexity. Powerful computational techniques have been developed, as have new ways of thinking about tiling spaces." "The text contains a generous supply of examples and exercises."--BOOK JACKET.
Aperiodic Crystals
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Author : Siegbert Schmid
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-04-19
Aperiodic Crystals written by Siegbert Schmid 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 2013-04-19 with Technology & Engineering categories.
Aperiodic Crystals collects 37 selected papers from the scientific contributions presented at Aperiodic 2012 - the Seventh International Conference on Aperiodic Crystalsheld held in Cairns, Australia, 2-7 of September 2012. The volume discusses state-of-the-art discoveries, new trends and applications of aperiodic crystals - including incommensurately modulated crystals, composite crystals, and quasicrystals - from a wide range of different perspectives. Starting with a general historical introduction to aperiodic crystals, the book proceeds to examine the complex mathematics of aperiodic long-range order, as well as the theoretical approaches aimed at understanding some of the unique properties and mechanisms underlying the existence of aperiodic crystals. The book then explores in detail such topics as complex metallic alloys, modulated structures, quasicrystals and their approximants, dynamics, disorder and defects in quasicrystals. It concludes with an analysis of quasicrystal surfaces and their properties. By describing the latest research and the progress made on the structure determination of aperiodic crystals and the influence of this unique structure on their physical properties, this book represents a valuable resource to mathematicians, crystallographers, physicists, chemists, materials and surface scientists, and even architects and artists, interested in the fascinating nature of aperiodic crystals.