Perfect Lattices in Euclidean Spaces / by Jacques Martinet.

Format:

Book

Author/Creator:

Martinet, Jacques, Author.

Series:

Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics, 2196-9701 ; 327

Language:

English

Subjects (All):

Geometry.

Number theory.

Discrete mathematics.

Number Theory.

Discrete Mathematics.

Local Subjects:

Geometry.

Number Theory.

Discrete Mathematics.

Physical Description:

1 online resource (XXII, 526 p.)

Edition:

1st ed. 2003.

Place of Publication:

Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 2003.

Language Note:

English

Summary:

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.

Contents:

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.

Notes:

Bibliographic Level Mode of Issuance: Monograph

Includes bibliographical references and index.

ISBN:

3-662-05167-2