Perfect Lattices in Euclidean Spaces

1. Auflage. Softcover version of original hardcover Auflage 2003
ISBN: 978-3-642-07921-4
Verlag: Springer

Buch, Englisch, Band 327, 526 Seiten, Paperback, Format (B × H): 155 mm x 235 mm, Gewicht: 826 g

Reihe: Grundlehren der mathematischen Wissenschaften

Lattices are discrete subgroups of maximal rank in a Euclidean space. To each such geometrical object, we can attach a canonical sphere packing which, assuming some regularity, has a density. The question of estimating the highest possible density of a sphere packing in a given dimension is a fascinating and difficult problem: the answer is known only up to dimension 3.

This book thus discusses a beautiful and central problem in mathematics, which involves geometry, number theory, coding theory and group theory, centering on the study of extreme lattices, i.e. those on which the density attains a local maximum, and on the so-called perfection property.

Written by a leader in the field, it is closely related to, though disjoint in content from, the classic book by J.H. Conway and N.J.A. Sloane, Sphere Packings, Lattices and Groups, published in the same series as vol. 290.

Every chapter except the first and the last contains numerous exercises. For simplicity those chapters involving heavy computational methods contain only few exercises. It includes appendices on Semi-Simple Algebras and Quaternions and Strongly Perfect Lattices.

Zielgruppe

Research

Autoren/Hrsg.

Martinet, Jacques

Fachgebiete

• Mathematik | Informatik Mathematik Geometrie Euklidische Geometrie
• Mathematik | Informatik EDV | Informatik Daten / Datenbanken Informationstheorie, Kodierungstheorie
• Mathematik | Informatik Mathematik Algebra Zahlentheorie
• Interdisziplinäres Wissenschaften Wissenschaften: Forschung und Information Informationstheorie, Kodierungstheorie

Inhaltsverzeichnis

1 General Properties of Lattices.- 2 Geometric Inequalities.- 3 Perfection and Eutaxy.- 4 Root Lattices.- 5 Lattices Related to Root Lattices.- 6 Low-Dimensional Perfect Lattices.- 7 The Voronoi Algorithm.- 8 Hermitian Lattices.- 9 The Configurations of Minimal Vectors.- 10 Extremal Properties of Families of Lattices.- 11 Group Actions.- 12 Cross-Sections.- 13 Extensions of the Voronoi Algorithm.- 14 Numerical Data.- 15 Appendix 1: Semi-Simple Algebras and Quaternions.- 16 Appendix 2: Strongly Perfect Lattices.- References.- List of Symbols.