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Đề tài " A proof of the Kepler conjecture "

This project would not have been possible without the generous support of many people. I would particularly like to thank Kerri Smith, Sam Ferguson, Sean McLaughlin, Jeff Lagarias, Gabor Fejes T´oth, Robert MacPherson, and the referees for their support of this project. A more comprehensive list of those who contributed to this project in various ways appears in [Hal06b]. | Annals of Mathematics A proof of the Kepler conjecture By Thomas C. Hales Annals of Mathematics 162 2005 1065 1185 A proof of the Kepler conjecture By Thomas C. Hales To the memory of László Fejes Toth Contents Preface 1. The top-level structure of the proof 1.1. Statement of theorems 1.2. Basic concepts in the proof 1.3. Logical skeleton of the proof 1.4. Proofs of the central claims 2. Construction of the Q-system 2.1. Description of the Q-system 2.2. Geometric considerations 2.3. Incidence relations 2.4. Overlap of simplices 3. V -cells 3.1. V -cells 3.2. Orientation 3.3. Interaction of V-cells with the Q-system 4. Decomposition stars 4.1. Indexing sets 4.2. Cells attached to decomposition stars 4.3. Colored spaces 5. Scoring Ferguson Hales 5.1. Definitions 5.2. Negligibility 5.3. Fcc-compatibility 5.4. Scores of standard clusters 6. Local optimality 6.1. Results 6.2. Rogers simplices 6.3. Bounds on simplices 6.4. Breaking clusters into pieces 6.5. Proofs This research was supported by a grant from the NSF over the period 1995-1998. 1066 THOMAS C. HALES 7. Tame graphs 7.1. Basic definitions 7.2. Weight assignments 7.3. Plane graph properties 8. Classification of tame plane graphs 8.1. Statement of the theorems 8.2. Basic definitions 8.3. A finite state machine 8.4. Pruning strategies 9. Contravening graphs 9.1. A review of earlier results 9.2. Contravening plane graphs defined 10. Contravention is tame 10.1. First properties 10.2. Computer calculations and their consequences 10.3. Linear programs 10.4. A non-contravening 4-circuit 10.5. Possible 4-circuits 11. Weight assignments 11.1. Admissibility 11.2. Proof that tri v 2 11.3. Bounds when tri v e 3 4 11.4. Weight assignments for aggregates 12. Linear program estimates 12.1. Relaxation 12.2. The linear programs 12.3. Basic linear programs 12.4. Error analysis 13. Elimination of aggregates 13.1. Triangle and quad branching 13.2. A pentagonal hull with n 8 13.3. n 8 hexagonal hull 13.4. n 7 pentagonal hull 13.5.