Perfect lattices in Euclidean spaces:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English French |
| Published: |
Berlin ; Heidelberg ; New York ; Hong Kong ; London ; Milan ; Paris ; Tokyo
Springer
2003
|
| Series: | Die Grundlehren der mathematischen Wissenschaften
327 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXI, 523 Seiten |
| ISBN: | 3540442367 |
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| 245 | 1 | 0 | |a Perfect lattices in Euclidean spaces |c Jacques Martinet |
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Record in the Search Index
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| adam_text | Contents
1 General Properties of Lattices 1
1.1 Lattices in Real Vector Spaces 1
1.2 Lattices in Euclidean Spaces 4
1.3 Duality 7
1.4 Automorphism Groups 12
1.5 Bilinear and Quadratic Forms 17
1.6 Quadratic Forms 18
1.7 The Dictionary Relating Lattices and Quadratic Forms 20
1.8 Packings 24
1.9 More on Integral Lattices 26
1.10 Tensor Product and Exterior Powers 30
1.11 Notes on Chapter 1 33
2 Geometric Inequalities 37
2.1 The Hadamard Inequality 37
2.2 The Hermite Inequality 39
2.3 The Mordell Inequality 41
2.4 Mahler s Compactness Theorem 43
2.5 Lattice Constants 45
2.6 Extreme Lattices for an Open Star Body 47
2.7 The Lattice Constant for a Convex Star Body 51
2.8 Generalizations of the Hermite Invariant 55
2.9 The HKZ Reduction 59
2.10 Exercises for Chapter 2 62
2.11 Notes on Chapter 2 63
3 Perfection and Eutaxy 67
3.1 Symmetric Endomorphisms 67
3.2 Linear Forms on Spaces of Endomorphisms 73
3.3 Linear Inequalities 77
3.4 A Characterization of Extreme Lattices 78
3.5 Perfect Configurations 81
3.6 Eutactic Configurations and Extreme Lattices 84
3.7 The Lamination Process 92
XVIII Contents
3.8 Dual Extreme Lattices 94
3.9 Exercises for Chapter 3 100
3.10 Notes on Chapter 3 105
4 Root Lattices 109
4.1 The Z Lattice 110
4.2 The A« Lattice 110
4.3 The Dn Lattice 112
4.4 The ©+ Packing and the Eg Lattice 114
4.5 The Lattices E? and Ee 117
4.6 Graphs and Inclusions Between Root Lattices 120
4.7 Perfection and Eutaxy 124
4.8 Some Other Constructions for Root Lattices 125
4.9 Residual Quadratic Forms 129
4.10 Root Systems 131
4.11 Exercises for Chapter 4 138
4.12 Notes on Chapter 4 145
5 Lattices Related to Root Lattices 147
5.1 The Coxeter Barnes Lattices A£ 147
5.2 The Coxeter Lattices A£ 153
5.3 Barnes s Lattices Pn 157
5.4 Craig s Difference Lattices 163
5.5 Lattices Related to the Dn Lattice 171
5.6 Unimodular Lattices 174
5.7 Around the Leech Lattice 177
5.8 Exercises for Chapter 5 182
5.9 Notes on Chapter 5 187
6 Low Dimensional Perfect Lattices 189
6.1 A Combinatorial Characterization of the A« Lattices 190
6.2 Perfect Lattices up to Dimension 4 194
6.3 Dual Extreme Lattices up to Dimension 4 196
6.4 Perfect Lattices in Dimension 5 200
6.5 Perfect Lattices in Dimensions 6 and 7 208
6.6 Some Indications About 8 Dimensional Perfect Lattices 212
6.7 Exercises for Chapter 6 219
6.8 Notes on Chapter 6 223
7 The Voronoi Algorithm 227
7.1 Voronoi Domains 227
7.2 Contiguity 234
7.3 Finiteness Results 237
7.4 The Voronoi Graphs 238
7.5 Lattices Contiguous to An 241
Contents XIX
7.6 The Voronoi Algorithm in Dimension 4 242
7.7 The Facets of B)n and the 5 Dimensional Perfect Lattices .... 244
7.8 Determination of the Contiguous Form 253
7.9 Perfect Forms in Dimensions 6 and 7 254
7.10 Exercises for Chapter 7 258
7.11 Notes on Chapter 7 260
8 Hermitian Lattices 263
8.1 Complex and Quaternionic Structures 263
8.2 Hurwitz Lattices: Enlargements of Dn 268
8.3 Hurwitz Lattices: Around Dimension 16 274
8.4 Eisenstein Lattices: A Construction of Barnes 280
8.5 Eisenstein Lattices: The Coxeter Todd Lattice 284
8.6 A General Construction of Hermitian Lattices 292
8.7 Quadratic Hermitian Structures 298
8.8 Beyond Dimension 24 306
8.9 Exercises for Chapter 8 310
8.10 Notes on Chapter 8 316
9 The Configurations of Minimal Vectors 321
9.1 Minimal Equivalent Lattices 321
9.2 Classes of Dimension n 3 329
9.3 Classification in Dimension 4 333
9.4 Weakly Eutactic Lattices in a Minimal Class 339
9.5 The Classification of Eutactic Lattices 343
9.6 Perfect Pairs of Lattices 348
9.7 Complements 354
9.8 Exercises for Chapter 9 357
9.9 Notes on Chapter 9 361
10 Extremal Properties of Families of Lattices 363
10.1 Some Elementary Results on Lie Groups 364
10.2 Perfection and Eutaxy 366
10.3 Extremality 368
10.4 Minimal Classes 373
10.5 Dual Extreme Lattices 374
10.6 The Rankin Invariants 376
10.7 Exercises for Chapter 10 379
10.8 Notes on Chapter 10 380
11 Group Actions 383
11.1 Rational and Integral Representations 383
11.2 G Lattices 385
11.3 G extreme Lattices 388
11.4 Cyclotomic Lattices 393
XX Contents
11.5 Isodual Lattices, Modular Lattices, and Normal Lattices 400
11.6 Normal Lattices 403
11.7 Extreme Symmetric and Symplectic Lattices 407
11.8 Isodual Lattices: Examples and Classification Results 414
11.9 Rationality and Finiteness Questions 417
11.10 Exercises for Chapter 11 421
11.11 Notes on Chapter 11 425
12 Cross Sections 427
12.1 Embedding a Lattice in a Larger One 427
12.2 X Rays of Lattices 430
12.3 Lattices with a Fixed Cross Section 433
12.4 A Characterization of Relatively Extreme Lattices 437
12.5 Patchwork Lattices 438
12.6 Exercises for Chapter 12 439
12.7 Notes on Chapter 12 440
13 Extensions of the Voronoi Algorithm 443
13.1 Contiguity Relative to a Space of Symmetric Matrices 444
13.2 The Voronoi Algorithm Relative to a Space of Symmetric
Matrices 448
13.3 Perfect G Lattices 451
13.4 Two Dimensional Centralizers 454
13.5 Cyclotomic Lattices 458
13.6 Lattices with a Fixed Section and Patchwork Lattices 460
13.7 Examples 463
13.8 Exercises for Chapter 13 464
13.9 Notes on Chapter 13 465
14 Numerical Data 467
14.1 Low Dimensional Perfect Lattices 467
14.2 Root Lattices 468
14.3 Eutactic Lattices up to Dimension 4 469
14.4 The Hermite Constant 472
14.5 Invariants Related to Duality 474
14.6 The Kissing Number 476
15 Appendix 1: Semi Simple Algebras and Quaternions 479
15.1 Semi Simple Algebras 479
15.2 Quaternion Algebras 481
15.3 Algebraic Lattices over Dedekind Domains 482
15.4 Arithmetic in Separable Algebras 484
15.5 Number Fields 485
15.6 Quaternions Again 486
15.7 Ideal Class Set 487
Contents XXI
16 Appendix 2: Strongly Perfect Lattices 489
16.1 Spherical Designs 489
16.2 Strong Perfection 491
16.3 An Infinite Series 492
16.4 Modular Lattices 493
16.5 Group Theory 494
16.6 Designs on Grassmannian Varieties 495
References 497
List of Symbols 511
Index 517
|
| any_adam_object | 1 |
| author | Martinet, Jacques 1939- |
| author_GND | (DE-588)111235039X |
| author_facet | Martinet, Jacques 1939- |
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| ctrlnum | (OCoLC)440911341 (DE-599)BVBBV014878793 |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV014878793 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T19:08:19Z |
| institution | BVB |
| isbn | 3540442367 |
| language | English French |
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| physical | XXI, 523 Seiten |
| publishDate | 2003 |
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| publisher | Springer |
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| series | Die Grundlehren der mathematischen Wissenschaften |
| series2 | Die Grundlehren der mathematischen Wissenschaften |
| spelling | Martinet, Jacques 1939- (DE-588)111235039X aut Les réseaux parfaits des espaces euclidiens Perfect lattices in Euclidean spaces Jacques Martinet Berlin ; Heidelberg ; New York ; Hong Kong ; London ; Milan ; Paris ; Tokyo Springer 2003 XXI, 523 Seiten txt rdacontent n rdamedia nc rdacarrier Die Grundlehren der mathematischen Wissenschaften 327 Euklidischer Raum (DE-588)4309127-1 gnd rswk-swf Gitter Mathematik (DE-588)4157375-4 gnd rswk-swf Gitter Mathematik (DE-588)4157375-4 s Euklidischer Raum (DE-588)4309127-1 s DE-604 Die Grundlehren der mathematischen Wissenschaften 327 (DE-604)BV000000395 327 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010061592&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Martinet, Jacques 1939- Perfect lattices in Euclidean spaces Die Grundlehren der mathematischen Wissenschaften Euklidischer Raum (DE-588)4309127-1 gnd Gitter Mathematik (DE-588)4157375-4 gnd |
| subject_GND | (DE-588)4309127-1 (DE-588)4157375-4 |
| title | Perfect lattices in Euclidean spaces |
| title_alt | Les réseaux parfaits des espaces euclidiens |
| title_auth | Perfect lattices in Euclidean spaces |
| title_exact_search | Perfect lattices in Euclidean spaces |
| title_full | Perfect lattices in Euclidean spaces Jacques Martinet |
| title_fullStr | Perfect lattices in Euclidean spaces Jacques Martinet |
| title_full_unstemmed | Perfect lattices in Euclidean spaces Jacques Martinet |
| title_short | Perfect lattices in Euclidean spaces |
| title_sort | perfect lattices in euclidean spaces |
| topic | Euklidischer Raum (DE-588)4309127-1 gnd Gitter Mathematik (DE-588)4157375-4 gnd |
| topic_facet | Euklidischer Raum Gitter Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010061592&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000395 |
| work_keys_str_mv | AT martinetjacques lesreseauxparfaitsdesespaceseuclidiens AT martinetjacques perfectlatticesineuclideanspaces |