Ordered Equilibrium Structures of Patchy Particle Systems
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Collection Volume I
Collection volume I PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information. PDF generated at: Thu, 29 Jul 2010 21:47:23 UTC Contents Articles Abstraction 1 Analogy 6 Bricolage 15 Categorization 19 Computational creativity 21 Data mining 30 Deskilling 41 Digital morphogenesis 42 Heuristic 44 Hidden curriculum 49 Information continuum 53 Knowhow 53 Knowledge representation and reasoning 55 Lateral thinking 60 Linnaean taxonomy 62 List of uniform tilings 67 Machine learning 71 Mathematical morphology 76 Mental model 83 Montessori sensorial materials 88 Packing problem 93 Prior knowledge for pattern recognition 100 Quasi-empirical method 102 Semantic similarity 103 Serendipity 104 Similarity (geometry) 113 Simulacrum 117 Squaring the square 120 Structural information theory 123 Task analysis 126 Techne 128 Tessellation 129 Totem 137 Trial and error 140 Unknown unknown 143 References Article Sources and Contributors 146 Image Sources, Licenses and Contributors 149 Article Licenses License 151 Abstraction 1 Abstraction Abstraction is a conceptual process by which higher, more abstract concepts are derived from the usage and classification of literal, "real," or "concrete" concepts. An "abstraction" (noun) is a concept that acts as super-categorical noun for all subordinate concepts, and connects any related concepts as a group, field, or category. Abstractions may be formed by reducing the information content of a concept or an observable phenomenon, typically to retain only information which is relevant for a particular purpose. For example, abstracting a leather soccer ball to the more general idea of a ball retains only the information on general ball attributes and behavior, eliminating the characteristics of that particular ball. -
Tilings and Tessellations
August 24 { September 4, 2015. Isfahan, IRAN CIMPA Research School Tilings and Tessellations Isfahan University of Technology Isfahan Mathematics House Foreword These are the lectures notes of the CIMPA school Tilings and Tessellations that takes place from august 24 to september 4 in the Technological University of Isfahan and the Mathematics House of Isfahan, Iran. This school is supported by the Centre International de Math´ematiquesPures et Appliqu´ees(CIMPA), the french Agence National de la Recherche (project QuasiCool, ANR-12-JS02-011-01), the universities of the contributors (listed below), the International Mathematical Union (IMU) and the iranian Institute for research in Fundamental Sciences (IPM). List of the contributors: • Fr´ed´erique Bassino, Univ. Paris 13, France • Florent Becker, Univ. Orl´eans,France • Nicolas Bedaride´ , Univ. Aix-Marseille, France • Olivier Bodini, Univ. Paris 13, France • C´edric Boutillier, Univ. Paris 6, France • B´eatrice De Tilliere` , Univ. Paris 6, France • Thomas Fernique, CNRS & Univ. Paris 13, France • Pierre Guillon, CNRS & Univ. Aix-Marseille, France • Amir Hashemi, Isfahan Univrsity of Technology, Iran • Eric´ Remila´ , Univ. Saint-Etienne,´ France • Guillaume Theyssier, Univ. de Savoie, France iii Contents 1 Tileability 1 1.1 Tiling with dominoes . .1 1.2 Tiling with bars . .2 1.3 Tiling with rectangles . .2 1.4 Gr¨obnerbases . .5 1.5 Buchberger's algorithm . .7 1.6 Gr¨obnerbases and tilings . .9 2 Random Tilings 11 2.1 Dimer model and random tilings . 11 2.2 Kasteleyn/Temperley and Fisher's theorem . 15 2.3 Explicit expression for natural probability measures on dimers . 16 2.4 Computation of the number of tilings : a few examples . -
The Computation of Overlap Coincidence in Taylor–Socolar Substitution Tiling
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Osaka City University Repository Akiyama, S. and Lee, J.-Y. Osaka J. Math. 51 (2014), 597–607 THE COMPUTATION OF OVERLAP COINCIDENCE IN TAYLOR–SOCOLAR SUBSTITUTION TILING SHIGEKI AKIYAMA and JEONG-YUP LEE £ (Received August 27, 2012, revised November 15, 2012) Abstract Recently Taylor and Socolar introduced an aperiodic mono-tile. The associated tiling can be viewed as a substitution tiling. We use the substitution rule for this tiling and apply the algorithm of [1] to check overlap coincidence. It turns out that the tiling has overlap coincidence. So the tiling dynamics has pure point spectrum and we can conclude that this tiling has a quasicrystalline structure. 1. Introduction Aperiodic tiles are the set of prototiles which tile the space with their isomorphic images by Euclidean motions (composition of translations, rotations and reflections) but only in non-periodic way. There have been many examples of aperiodic tiles and study on them [2, 4, 7, 12, 14, 25, 26, 29, 30]. Two of well-known examples of aperiodic tiles with simple prototiles up to Euclidean motions are Penrose tiles and Ammann tiles which are uncovered in the mid ’70s. These sets consist of two prototiles and it has been the smallest number of prototiles which form aperiodic tiles until recently. Since then, people have been interested in finding a single prototile for an aperiodic tile. This problem is coined as ‘Mono-tile’ problem or ‘Einstein’ problem (one stone in German). It had taken quite some time before Taylor and Socolar announced in 2010 the existence of an aperiodic mono-tile. -
Crystallography of Quasicrystals Crystallography of Quasicrystals
Steurer Steurer Walter Steurer Soi a Deloudi • Deloudi SPRINGER SERIES IN MATERIALS SCIENCE 126 1 Crystallography Crystallography of Quasicrystals of Quasicrystals Concepts, Methods and Structures 123 Springer Series in materials science 126 Springer Series in materials science Editors: R. Hull R. M. Osgood, Jr. J. Parisi H. Warlimont The Springer Series in Materials Science covers the complete spectrum of materials physics, including fundamental principles, physical properties, materials theory and design. Recognizing the increasing importance of materials science in future device technologies, the book titles in this series reflect the state-of-the-art in understanding and controlling the structure and properties of all important classes of materials. Please view available titles in Springer Series in Materials Science on series homepage http://www.springer.com/series/856 Walter Steurer Sofia Deloudi Crystallography of Quasicrystals Concepts, Methods and Structures With177 Figures ABC Professor Dr. Walter Steurer Dr. Sofia Deloudi ETH Zurich,¨ Department of Materials, Laboratory of Crystallography Wolfgang-Pauli-Str. 10, 8093 Zurich,¨ Switzerland E-mail: [email protected], [email protected] Series Editors: ProfessorRobertHull Professor Jürgen Parisi University of Virginia Universitat¨ Oldenburg, Fachbereich Physik Dept. of Materials Science and Engineering Abt. Energie- und Halbleiterforschung Thornton Hall Carl-von-Ossietzky-Straße 9–11 Charlottesville, VA22903-2442, USA 26129 Oldenburg, Germany ProfessorR.M.Osgood,Jr. Professor Hans Warlimont Microelectronics Science Laboratory DSL Dresden Material-Innovation GmbH Department of Electrical Engineering Pirnaer Landstr. 176 Columbia University 01257 Dresden, Germany Seeley W. Mudd Building New York, NY 10027, USA Springer Series in Materials Science ISSN 0933-033X ISBN 978-3-642-01898-5 e-ISBN 978-3-642-01899-2 DOI 10.1007/978-3-642-01899-2 SpringerHeidelbergDordrechtLondonNewYork Library of Congress Control Number: 2009929706 c Springer-Verlag Berlin Heidelberg 2009 This work is subject to copyright. -
Arxiv:1602.06372V2 [Math.MG] 17 Jul 2017 Following Two Problems [5]
CONVEX POLYGONS FOR APERIODIC TILING TERUHISA SUGIMOTO Abstract. If all tiles in a tiling are congruent, the tiling is called monohedral. Tiling by convex polygons is called edge-to-edge if any two convex polygons are either disjoint or share one vertex or one entire edge in common. In this paper, we prove that a convex polygon that can generate an edge-to-edge monohedral tiling must be able to generate a periodic tiling. 1. Introduction A tiling of the plane is an exact covering of the plane by a collection of sets without gaps or overlaps (except for the boundaries of the sets). More precisely, a collection of sets (the \tiles") is a tiling of the plane if their union is the entire plane, but the interiors of different tiles are disjoint. The tiles are frequently polygons and are all congruent to one another or at least congruent to one of a small number of prototiles [4,5]. If all tiles in the tiling are of the same size and shape (i.e., congruent), the tiling is called monohedral [4]. Therefore, a prototile of a monohedral tiling by polygons is single polygon in the tiling, and we call the polygon the polygonal tile [4, 13{18]. A tiling of the plane is periodic if the tiling can be translated onto itself in two nonparallel directions. More precisely, a tiling is periodic if it coincides with its translation by a nonzero vector. A set of prototiles is called aperiodic if congruent copies of the prototiles admit infinitely many tilings of the plane, none of which are periodic. -
Symbolic Dynamics and Tilings of Rd
Proceedings of Symposia in Applied Mathematics Symbolic Dynamics and Tilings of Rd E. Arthur Robinson, Jr. Abstract. Aperiodic tilings of Euclidean space can profitably be studied from the point of view of dynamical systems theory. This study takes place via a kind of dynamical system called a tiling dynamical system. 1. Introduction In this chapter we study tilings of Euclidean space from the point of view of dy- namical systems theory, and in particular, symbolic dynamics. Our goal is to show that these two subjects share many common themes and that they can make useful contributions to each other. The tilings we study are tilings of Rd by translations of a finite number of basic tile types called “prototiles”. A good general reference on tilings is [GS87]. The link between tilings and dynamics will be established using a kind of dynamical system called a tiling dynamical system, first described by Dan Rudolph [Rud88], [Rud89]. The parts of the theory we concentrate on here are the parts most closely related to symbolic dynamics. Interestingly, these also tend to be the parts related to the theory of quasicrystals. A quasicrystal is a solid which, like a crystal, has a regular enough atomic struc- ture to produce sharp spots in its X-ray diffraction patterns, but unlike a crystal, has an aperiodic atomic structure. Because of this aperiodicity, quasicrystals can have “symmetries” forbidden to ordinary crystals, and these can be observed in their X-ray diffraction patterns. The first quasicrystals were discovered in 1984 by physicists at NIST (see [SBGC84]) who observed a diffraction pattern with 5-fold rotational symmetry. -
Tensor Network and Neural Network Methods in Physical Systems
Tensor network and neural network methods in physical systems Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Peiyuan Teng Graduate Program in Physics The Ohio State University 2018 Dissertation Committee: Dr. Yuan-Ming Lu, Advisor Dr. Ciriyam Jayaprakash Dr. Jay Gupta Dr. Comert Kural c Copyright by Peiyuan Teng 2018 Abstract In this dissertation, new ideas and methods from tensor network theory and neu- ral network theory are discussed. Firstly, common computational methods, such as the exact diagonalization method, the Density Matrix Renormalization Group ap- proach, and the tensor network theory are reviewed. Following this direction, a way of generalizing the tensor renormalization group (TRG) to all spatial dimensions is proposed. Mathematically, the connection between patterns of tensor renormalization group and the concept of truncation sequence in polytope geometry is discovered. A theoretical contraction framework is proposed. Furthermore, the canonical polyadic decomposition is introduced to tensor network theory. A numerical verification of this method on the 3-D Ising model is carried out. Secondly, this dissertation includes an efficient way of calculating the geomet- ric measure of entanglement using tensor decomposition methods. The connection between these two concepts is explored using the tensor representation of the wave- function. Numerical examples are benchmarked and compared. Furthermore, highly entangled qubit states are searched for to show the applicability of this method. Finally, machine learning approaches are reviewed. Machine learning methods are applied to quantum mechanics. The radial basis function network in a discrete basis is used as the variational wavefunction for the ground state of a quantum system. -
Combinatorics of Tilings
Combinatorics of tilings V. Berthé LIAFA-CNRS-Paris-France [email protected] http ://www.liafa.jussieu.fr/~berthe Substitutive tilings and fractal geometry-Guangzhou-July 2010 Tilings Definition Covering of the ambiant space without overlaps using some tiles Tile of Rd a compact set which is the closure of its interior Tiles are also often assumed to be homeomorphic to the ball Tilings of Rd A tiling is a set of tiles such that T d R = T 2T T[ where distinct tiles have non-intersecting interiors each compact set intersects a finite number of tiles in T Labeling Rigid motion Euclidean isometry preserving orientation (composition of translations and rotations) One might distinguish between congruent tiles by labeling them Two tiles are equivalent if they differ by a rigid motion and carry the same label A prototile set is a finite set of inequivalent tiles P Each tile of the tiling is equivalent to a prototile of T P We will work here mainly with translations Bibliography R. Robinson Symbolic Dynamics and Tilings of Rd , AMS Proceedings of Symposia in Applied Mathematics. N. Priebe Frank A primer on substitution tilings of the Euclidean plane – Expo. Math. 26 (2008) 295-326. B. Solomyak Dynamics of self-similar tilings, Ergodic Theory and Dynam. Sys. 17 (1997), 695-738 C. Goodman-Strauss http ://comp.uark.edu/ strauss/ The tiling ListServe Casey Mann [email protected] Tilings Encyclopedia http ://tilings.math.uni-bielefeld.de/E. Harriss, D. Frettlöh R. M. Robinson Undecidability and nonperiodicity for tilings of the plane, Inventiones -
Exploring the Einstein Problem
Exploring The Einstein Problem Thesis submitted at the University of Leicester in partial fulllment of the requirements for the degree of Master of Mathematics by Zoe C. Allwood Department of Mathematics University of Leicester May 2014 Contents Declaration iii Abstract iv Introduction 1 1 Describing the Tile 4 1.1 Lines and Flags . 4 1.2 Lines and Coloured Diagonals . 5 1.3 Other Representations . 6 2 Proof of Aperiodicity 7 3 The Substitution Tiling 12 3.1 Simplifying the Tile Set . 13 3.2 The Substitution . 17 3.3 The Substitution on the Simplied Tile Sets . 19 3.4 Amalgamation . 22 3.5 Proof of Aperiodicity . 22 4 A Better Einstein 23 4.1 Unconnected Tile . 23 4.2 3D Tile . 24 4.2.1 Construction . 24 4.2.2 Issues . 26 i 4.3 Schmitt, Conway and Danzer Tile . 26 4.4 Issues . 27 4.4.1 Height . 27 4.4.2 Tile Orientation . 27 4.4.3 Aperiodic Layers . 28 4.5 Danzer Implementation Attempt . 28 4.5.1 Rotation of 2π .............................. 28 3 4.5.2 Irrational Rotation . 30 4.6 Aperiodic Stack . 31 4.6.1 Thue Morse exploration . 32 4.6.2 General explanation . 33 4.6.3 Two dimensional exploration . 33 5 Conclusions 34 5.1 Socolar and Taylor Tile . 34 5.2 Einsteins and Dimension . 35 Bibliography 36 ii Declaration All sentences or passages quoted in this project dissertation from other people's work have been specically acknowledged by clear cross referencing to author, work and page(s). I understand that failure to do this amounts to plagiarism and will be considered grounds for failure in this module and the degree examination as a whole. -
Emmer M. Imagine Math.. Between Culture and Mathematics (Springer
To Mario Geymonat Michele Emmer Editor Imagine Math Between Culture and Mathematics Michele Emmer Department of Mathematics Sapienza University of Rome ISBN 978-88-470-2426-7 ISBN 978-88-470-2427-4 (eBook) DOI 10.1007/978-88–470-2427-4 Library of Congress Control Number: 2011944690 Springer Milan Dordrecht Heidelberg London New York c Springer-Verlag Italia 2012 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broa- dcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the Italian Copyright Law in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the Italian Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Translation: Kim Williams for the contributions by E. Fabbri, E. Gamba, M. Costamagna, S. Caran- dini, C. Viviani and M. Emmer (“Visual Harmonies: an Exhibition on Art and Math” and “The Fantastic World of Tor Bled-Nam”) Cover-Design: deblik, Berlin Cover-Image: Lucio Saffaro, “Il piano di Orfeo” (particular). Courtesy of Fondazione Lucio Saf- faro, Bologna; Springer thanks Gisella Vismara Typesetting with LaTeX: CompoMat S.r.l., Configni (RI) Printing and Binding: GECA Industrie Grafiche, Cesano Boscone (MI) Printed in Italy Springer-Verlag Italia S.r.l., Via Decembrio 28, I-20137 Milano Springer fa parte di Springer Science + Business Media (www.springer.com) Contents Introduction ..................................................