UID:
almahu_9947362922402882
Format:
XXVII, 665 p.
,
online resource.
ISBN:
9781475720167
Series Statement:
Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics, 290
Content:
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.
Note:
1 Sphere Packings and Kissing Numbers -- 2 Coverings, Lattices and Quantizers -- 3 Codes, Designs and Groups -- 4 Certain Important Lattices and Their Properties -- 5 Sphere Packing and Error-Correcting Codes -- 6 Laminated Lattices -- 7 Further Connections Between Codes and Lattices -- 8 Algebraic Constructions for Lattices -- 9 Bounds for Codes and Sphere Packings -- 10 Three Lectures on Exceptional Groups -- 11 The Golay Codes and the Mathieu Groups -- 12 A Characterization of the Leech Lattice -- 13 Bounds on Kissing Numbers -- 14 Uniqueness of Certain Spherical Codes -- 15 On the Classification of Integral Quadratic Forms -- 16 Enumeration of Unimodular Lattices -- 17 The 24-Dimensional Odd Unimodular Lattices -- 18 Even Unimodular 24-Dimensional Lattices -- 19 Enumeration of Extremal Self-Dual Lattices -- 20 Finding the Closest Lattice Point -- 21 Voronoi Cells of Lattices and Quantization Errors -- 22 A Bound for the Covering Radius of the Leech Lattice -- 23 The Covering Radius of the Leech Lattice -- 24 Twenty-Three Constructions for the Leech Lattice -- 25 The Cellular Structure of the Leech Lattice -- 26 Lorentzian Forms for the Leech Lattice -- 27 The Automorphism Group of the 26-Dimensional Even Unimodular Lorentzian Lattice -- 28 Leech Roots and Vinberg Groups -- 29 The Monster Group and its 196884-Dimensional Space -- 30 A Monster Lie Algebra?.
In:
Springer eBooks
Additional Edition:
Printed edition: ISBN 9781475720181
Language:
English
DOI:
10.1007/978-1-4757-2016-7
URL:
http://dx.doi.org/10.1007/978-1-4757-2016-7
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