Sphere Packings

Author : J.H. Conway,N.J.A. Sloane
Publisher : Springer Science & Business Media
Page : 724 pages
File Size : 52,9 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9781475722499

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Sphere Packings, Lattices and Groups by J.H. Conway,N.J.A. Sloane Pdf

The second edition of this timely, definitive, and popular book continues to pursue the question: what is the most efficient way to pack a large number of equal spheres in n-dimensional Euclidean space? The authors also continue to examine related problems such as the kissing number problem, the covering problem, the quantizing problem, and the classification of lattices and quadratic forms. Like the first edition, the second edition describes the applications of these questions to other areas of mathematics and science such as number theory, coding theory, group theory, analog-to-digital conversion and data compression, n-dimensional crystallography, and dual theory and superstring theory in physics. Results as of 1992 have been added to the text, and the extensive bibliography - itself a contribution to the field - is supplemented with approximately 450 new entries.

Author : John Conway,Neil J. A. Sloane
Publisher : Springer Science & Business Media
Page : 778 pages
File Size : 45,7 Mb
Release : 2013-06-29
Category : Mathematics
ISBN : 9781475765687

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Sphere Packings, Lattices and Groups by John Conway,Neil J. A. Sloane Pdf

The third edition of this definitive and popular book continues to pursue the question: what is the most efficient way to pack a large number of equal spheres in n-dimensional Euclidean space? The authors also examine such related issues as the kissing number problem, the covering problem, the quantizing problem, and the classification of lattices and quadratic forms. There is also a description of the applications of these questions to other areas of mathematics and science such as number theory, coding theory, group theory, analogue-to-digital conversion and data compression, n-dimensional crystallography, dual theory and superstring theory in physics. New and of special interest is a report on some recent developments in the field, and an updated and enlarged supplementary bibliography with over 800 items.

Author : Chuanming Zong
Publisher : Springer Science & Business Media
Page : 242 pages
File Size : 53,6 Mb
Release : 2008-01-20
Category : Mathematics
ISBN : 9780387227801

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Sphere Packings by Chuanming Zong Pdf

Sphere packings is one of the most fascinating and challenging subjects in mathematics. In the course of centuries, many exciting results have been obtained, ingenious methods created, related challenging problems proposed, and many surprising connections with other subjects found. This book gives a full account of this fascinating subject, especially its local aspects, discrete aspects, and its proof methods. The book includes both classical and contemporary results and provides a full treatment of the subject.

Author : Thomas Callister Hales
Publisher : Cambridge University Press
Page : 286 pages
File Size : 40,9 Mb
Release : 2012-09-06
Category : Mathematics
ISBN : 9780521617703

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Dense Sphere Packings by Thomas Callister Hales Pdf

The definitive account of the recent computer solution of the oldest problem in discrete geometry.

Author : John H. Conway,Neil J.A. Sloane
Publisher : Springer Science & Business Media
Page : 690 pages
File Size : 40,8 Mb
Release : 2013-04-17
Category : Mathematics
ISBN : 9781475720167

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Sphere Packings, Lattices and Groups by John H. Conway,Neil J.A. Sloane Pdf

The main themes. This book is mainly concerned with the problem of packing spheres in Euclidean space of dimensions 1,2,3,4,5, . . . . Given a large number of equal spheres, what is the most efficient (or densest) way to pack them together? We also study several closely related problems: the kissing number problem, which asks how many spheres can be arranged so that they all touch one central sphere of the same size; the covering problem, which asks for the least dense way to cover n-dimensional space with equal overlapping spheres; and the quantizing problem, important for applications to analog-to-digital conversion (or data compression), which asks how to place points in space so that the average second moment of their Voronoi cells is as small as possible. Attacks on these problems usually arrange the spheres so their centers form a lattice. Lattices are described by quadratic forms, and we study the classification of quadratic forms. Most of the book is devoted to these five problems. The miraculous enters: the E 8 and Leech lattices. When we investigate those problems, some fantastic things happen! There are two sphere packings, one in eight dimensions, the E 8 lattice, and one in twenty-four dimensions, the Leech lattice A , which are unexpectedly good and very 24 symmetrical packings, and have a number of remarkable and mysterious properties, not all of which are completely understood even today.

Author : Thomas M. Thompson
Publisher : American Mathematical Soc.
Page : 228 pages
File Size : 44,8 Mb
Release : 1983-12-31
Category : Electronic
ISBN : 9781470454609

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From Error-Correcting Codes Through Sphere Packings to Simple Groups by Thomas M. Thompson Pdf

This book traces a remarkable path of mathematical connections through seemingly disparate topics. Frustrations with a 1940's electro-mechanical computer at a premier research laboratory begin this story. Subsequent mathematical methods of encoding messages to ensure correctness when transmitted over noisy channels lead to discoveries of extremely efficient lattice packings of equal-radius balls, especially in 24-dimensional space. In turn, this highly symmetric lattice, with each point neighboring exactly 196,560 other points, suggested the possible presence of new simple groups as groups of symmetries. Indeed, new groups were found and are now part of the "Enormous Theorem"—the classification of all simple groups whose entire proof runs some 10,000+ pages—and these connections, along with the fascinating history and the proof of the simplicity of one of those "sporatic" simple groups, are presented at an undergraduate mathematical level.

Author : Thomas M. Thompson
Publisher : Unknown
Page : 252 pages
File Size : 44,5 Mb
Release : 1983
Category : Error-correcting codes (Information theory)
ISBN : 0883850001

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From Error-correcting Codes Through Sphere Packings to Simple Groups by Thomas M. Thompson Pdf

Author : Samuel L. P. Ferguson
Publisher : Unknown
Page : 208 pages
File Size : 51,5 Mb
Release : 1997
Category : Sphere packings
ISBN : UOM:39015041239339

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Sphere Packings, V. by Samuel L. P. Ferguson Pdf

Author : Károly Bezdek
Publisher : Springer Science & Business Media
Page : 186 pages
File Size : 53,7 Mb
Release : 2013-08-04
Category : Mathematics
ISBN : 9781461481188

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Lectures on Sphere Arrangements – the Discrete Geometric Side by Károly Bezdek Pdf

This monograph gives a short introduction to the relevant modern parts of discrete geometry, in addition to leading the reader to the frontiers of geometric research on sphere arrangements. The readership is aimed at advanced undergraduate and early graduate students, as well as interested researchers. It contains more than 40 open research problems ideal for graduate students and researchers in mathematics and computer science. Additionally, this book may be considered ideal for a one-semester advanced undergraduate or graduate level course. The core part of this book is based on three lectures given by the author at the Fields Institute during the thematic program on “Discrete Geometry and Applications” and contains four core topics. The first two topics surround active areas that have been outstanding from the birth of discrete geometry, namely dense sphere packings and tilings. Sphere packings and tilings have a very strong connection to number theory, coding, groups, and mathematical programming. Extending the tradition of studying packings of spheres, is the investigation of the monotonicity of volume under contractions of arbitrary arrangements of spheres. The third major topic of this book can be found under the sections on ball-polyhedra that study the possibility of extending the theory of convex polytopes to the family of intersections of congruent balls. This section of the text is connected in many ways to the above-mentioned major topics and it is also connected to some other important research areas as the one on coverings by planks (with close ties to geometric analysis). This fourth core topic is discussed under covering balls by cylinders.

Author : Jeffrey C. Lagarias
Publisher : Springer Science & Business Media
Page : 456 pages
File Size : 48,8 Mb
Release : 2011-11-09
Category : Mathematics
ISBN : 9781461411291

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The Kepler Conjecture by Jeffrey C. Lagarias Pdf

The Kepler conjecture, one of geometry's oldest unsolved problems, was formulated in 1611 by Johannes Kepler and mentioned by Hilbert in his famous 1900 problem list. The Kepler conjecture states that the densest packing of three-dimensional Euclidean space by equal spheres is attained by the “cannonball" packing. In a landmark result, this was proved by Thomas C. Hales and Samuel P. Ferguson, using an analytic argument completed with extensive use of computers. This book centers around six papers, presenting the detailed proof of the Kepler conjecture given by Hales and Ferguson, published in 2006 in a special issue of Discrete & Computational Geometry. Further supporting material is also presented: a follow-up paper of Hales et al (2010) revising the proof, and describing progress towards a formal proof of the Kepler conjecture. For historical reasons, this book also includes two early papers of Hales that indicate his original approach to the conjecture. The editor's two introductory chapters situate the conjecture in a broader historical and mathematical context. These chapters provide a valuable perspective and are a key feature of this work.

Author : Douglas H. Everett
Publisher : Royal Society of Chemistry
Page : 236 pages
File Size : 51,7 Mb
Release : 1990-05-31
Category : Reference
ISBN : 9780851865188

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Colloid Science by Douglas H. Everett Pdf

Specialist Periodical Reports provide systematic and detailed review coverage of progress in the major areas of chemical research. Written by experts in their specialist fields the series creates a unique service for the active research chemist, supplying regular critical in-depth accounts of progress in particular areas of chemistry. For over 80 years the Royal Society of Chemistry and its predecessor, the Chemical Society, have been publishing reports charting developments in chemistry, which originally took the form of Annual Reports. However, by 1967 the whole spectrum of chemistry could no longer be contained within one volume and the series Specialist Periodical Reports was born. The Annual Reports themselves still existed but were divided into two, and subsequently three, volumes covering Inorganic, Organic and Physical Chemistry. For more general coverage of the highlights in chemistry they remain a 'must'. Since that time the SPR series has altered according to the fluctuating degree of activity in various fields of chemistry. Some titles have remained unchanged, while others have altered their emphasis along with their titles; some have been combined under a new name whereas others have had to be discontinued.

Author : Thomas Miller Thompson
Publisher : Unknown
Page : 536 pages
File Size : 40,5 Mb
Release : 1979
Category : Algebra
ISBN : UCAL:X14358

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From Error-correcting Codes Through Sphere Packings to Simple Groups by Thomas Miller Thompson Pdf

Author : René Weller
Publisher : Springer Science & Business Media
Page : 240 pages
File Size : 42,7 Mb
Release : 2013-07-12
Category : Computers
ISBN : 9783319010205

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New Geometric Data Structures for Collision Detection and Haptics by René Weller Pdf

Starting with novel algorithms for optimally updating bounding volume hierarchies of objects undergoing arbitrary deformations, the author presents a new data structure that allows, for the first time, the computation of the penetration volume. The penetration volume is related to the water displacement of the overlapping region, and thus corresponds to a physically motivated and continuous force. The practicability of the approaches used is shown by realizing new applications in the field of robotics and haptics, including a user study that evaluates the influence of the degrees of freedom in complex haptic interactions. New Geometric Data Structures for Collision Detection and Haptics closes by proposing an open source benchmarking suite that evaluates both the performance and the quality of the collision response in order to guarantee a fair comparison of different collision detection algorithms. Required in the fields of computer graphics, physically-based simulations, computer animations, robotics and haptics, collision detection is a fundamental problem that arises every time we interact with virtual objects. Some of the open challenges associated with collision detection include the handling of deformable objects, the stable computation of physically-plausible contact information, and the extremely high frequencies that are required for haptic rendering. New Geometric Data Structures for Collision Detection and Haptics presents new solutions to all of these challenges, and will prove to be a valuable resource for researchers and practitioners of collision detection in the haptics, robotics and computer graphics and animation domains.

Author : H. Konietzky
Publisher : CRC Press
Page : 342 pages
File Size : 48,6 Mb
Release : 2002-01-01
Category : Technology & Engineering
ISBN : 9058095320

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Numerical Modeling in Micromechanics via Particle Methods by H. Konietzky Pdf

Particle methods have seen increasing use in several engineering and scientific fields, both because of their unique modelling capabilities and the availability of the necessary computational power. This title focuses on their theory and application.

Author : Dan Romik
Publisher : Walter de Gruyter GmbH & Co KG
Page : 308 pages
File Size : 49,5 Mb
Release : 2023-08-21
Category : Mathematics
ISBN : 9783110796810

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Topics in Complex Analysis by Dan Romik Pdf

This graduate-level mathematics textbook provides an in-depth and readable exposition of selected topics in complex analysis. The material spans both the standard theory at a level suitable for a first-graduate class on the subject and several advanced topics delving deeper into the subject and applying the theory in different directions. The focus is on beautiful applications of complex analysis to geometry and number theory. The text is accompanied by beautiful figures illustrating many of the concepts and proofs. Among the topics covered are asymptotic analysis; conformal mapping and the Riemann mapping theory; the Euler gamma function, the Riemann zeta function, and a proof of the prime number theorem; elliptic functions, and modular forms. The final chapter gives the first detailed account in textbook format of the recent solution to the sphere packing problem in dimension 8, published by Maryna Viazovska in 2016 -- a groundbreaking proof for which Viazovska was awarded the Fields Medal in 2022. The book is suitable for self-study by graduate students or advanced undergraduates with an interest in complex analysis and its applications, or for use as a textbook for graduate mathematics classes, with enough material for 2-3 semester-long classes. Researchers in complex analysis, analytic number theory, modular forms, and the theory of sphere packing, will also find much to enjoy in the text, including new material not found in standard textbooks.