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Issue 1-2: L'ENSEIGNEMENT MATHÉMATIQUE
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Front matter
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Table of Contents
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1
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Front matter
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2
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Article
CARACTÉRISATION GÉOMÉTRIQUE DES SOLUTIONS DE MINIMAX POUR L'ÉQUATION DE HAMILTON-JACOBI
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3
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Abstract
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3
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Chapter
Introduction
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3
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Chapter
1. MINIMAX D'UNE FONCTION QUADRATIQUE À L'INFINI
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5
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Chapter
1.1 Préliminaires
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5
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Chapter
1.2 Points critiques incidents, liés et libres
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8
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Chapter
1.3 Le niveau critique de minimax
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10
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Chapter
2. La solution de minimax
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13
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Chapter
2.1 Rappels de géométrie symplectique
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13
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Chapter
2.2 La solution géométrique de (PC)
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17
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Chapter
2.3 La solution de minimax
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20
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Chapter
3. Caractérisation géométrique de la solution de minimax
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22
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Chapter
3.1 Notations
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22
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Chapter
3.2 DÉCOMPOSITIONS ADMISSIBLES (D'APRÈS CHEKANOV ET PUSHKAR)
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23
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Chapter
3.3 Caractérisation géométrique du minimax
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25
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Chapter
3.4 Triangles évanescents
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29
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Bibliography
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33
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Article
LECTURES ON QUASI-INVARIANTS OF COXETER GROUPS AND THE CHEREDNIK ALGEBRA
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35
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Chapter
Introduction
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35
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Chapter
1. Lecture 1
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36
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Chapter
1.1 Définition of quasi-invariants
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36
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Chapter
1.2 Elementary properties of $Q_m$
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37
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Chapter
1.3 The variety $X_m$ and its bijective normalization
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38
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Chapter
1.4 FURTHER PROPERTIES OF $X_m$
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39
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Chapter
1.5 The Poincaré séries of $Q_m$
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40
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Chapter
1.6 The ring of differential operators on $X_m$
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43
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Chapter
2. Lecture 2
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43
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Chapter
2.1 Hamiltonian mechanics and integrable Systems
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43
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Chapter
2.2 The classical Calogero-Moser System
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44
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Chapter
2.3 The quantum Calogero-Moser System
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45
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Chapter
2.4 The algebra of differential-reflection operators
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46
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Chapter
2.5 Dunkl operators and symmetric quantum integrals
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48
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Chapter
2.6 Additional integrals for integer valued c
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50
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Chapter
2.7 An example
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52
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Chapter
3. Lecture 3
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53
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Chapter
3.1 Shift operator and construction of the Baker-Akhiezer function
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53
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Chapter
3.2 Berest's formula for $L_q$
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54
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Chapter
3.3 Differential operators on $X_m$
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57
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Chapter
3.4 The Cherednik algebra
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58
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Chapter
3.5 The spherical subalgebra
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59
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Chapter
3.6 Category O
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60
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Chapter
3.7 Generic c
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61
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Chapter
3.8 The Levasseur-Stafford theorem and its generalization
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62
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Chapter
3.9 The action of the Cherednik algebra to quasi-invariants
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62
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Chapter
3.10 Proof of Theorem 1.8
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63
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Chapter
3.11 Proof of Theorem 1.15
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63
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Bibliography
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64
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Article
SOME REMARKS ON NONCONNECTED COMPACT LIE GROUPS
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67
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Abstract
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67
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Chapter
1. Introduction
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67
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Chapter
2. Compact Lie groups :a review
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70
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Chapter
3. Compact Lie groups and extensions
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72
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Chapter
4. Proof of the Main Theorem and examples
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75
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Chapter
5. Splitting of the extension ASSOCIATED TO A NONCONNECTED COMPACT LIE GROUP
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80
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Bibliography
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83
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Article
ATIYAH'S $L^2$-INDEX THEOREM
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85
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Chapter
1. Introduction
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85
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Chapter
2. REVIEW OF THE $L^2$ -INDEX THEOREM
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85
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Chapter
3. Hilbert modules
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88
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Chapter
4. On K-homology
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89
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Chapter
5. Algebraic proof of Atiyah's $L2$-index theorem
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91
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Bibliography
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92
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Article
UNE PREUVE DU THÉORÈME DE LIOUVILLE EN GÉOMÉTRIE CONFORME DANS LE CAS ANALYTIQUE
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95
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Chapter
1. Introduction
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Chapter
2. Invariance conforme des géodésiques isotropes
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96
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Chapter
3. Une application: le théorème de Liouville dans le cas analytique
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98
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Bibliography
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100
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Article
IDEAL SOLUTIONS OF THE TARRY-ESCOTT PROBLEM OF DEGREES FOUR AND FIVE AND RELATED DIOPHANTINE SYSTEMS
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101
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Abstract
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101
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Chapter
1. Introduction
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101
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Chapter
2. Ideal non-symmetric solutions of the Tarry-Escott problem of degree four
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102
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Chapter
3. Ideal non-symmetric solutions of the Tarry-Escott problem of degree five
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105
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Bibliography
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108
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Article
ADDITIVE NUMBER THEORY SHEDS EXTRA LIGHT ON THE HOPF-STIEFEL o FUNCTION
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109
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Abstract
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109
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Chapter
1. Introduction
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109
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Chapter
2. Proof of Theorem 4
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112
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Chapter
2.1 The lower bound
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112
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Chapter
2.2 The upper bound
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113
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Chapter
3. From Theorem 3 to Theorem 1
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114
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Bibliography
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115
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Article
NOTE ON THE HOPF-STIEFEL FUNCTION
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117
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Chapter
Introduction
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117
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Chapter
1. Deriving Theorem 1 from Theorem 2
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118
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Chapter
2. Proof of Theorem 2
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120
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Bibliography
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122
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Article
TILE HOMOTOPY GROUPS
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123
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Abstract
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123
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Chapter
1. Introduction
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123
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Chapter
2. Tiling and integer programming
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126
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Chapter
3. Boundary words
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131
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Chapter
4. Tile homotopy groups
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136
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Chapter
5. Strategy for working with tile path groups
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140
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Chapter
6. Criteria for $\pi(\Tau)$ to be abelian
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148
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Appendix
7. Appendix: further examples
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151
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Bibliography
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154
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Article
SYMPLECTIC LOOK AT SURFACES OF REVOLUTION
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157
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Chapter
1. Introduction
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157
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Chapter
2. Abstract surfaces of revolution
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158
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Chapter
3. Metrics of specified curvature
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166
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Bibliography
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172
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Article
QUADRICS, ORTHOGONAL ACTIONS AND INVOLUTIONS IN COMPLEX PROJECTIVE SPACES
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173
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Chapter
0. Introduction
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173
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Chapter
1. ON THE TOPOLOGY OF A QUADRIC IN $P_C^n$
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177
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Chapter
2. ON THE GEOMETRY OF $P_C^n$
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181
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Chapter
3. $P_C^2$ AND THE 4-SPHERE $S^4$
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188
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Chapter
4. SOME APPLICATIONS AND REMARKS
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195
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Bibliography
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201
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Rubric
COMMISSION INTERNATIONALE DE L'ENSEIGNEMENT MATHÉMATIQUE (THE INTERNATIONAL COMMISSION ON MATHEMATICAL INSTRUCTION)
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Chapter
DISCUSSION DOCUMENT FOR THE FOURTEENTH ICMI STUDY APPLICATIONS AND MODELLING IN MATHEMATICS EDUCATION
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205
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Chapter
1. Rationale for the Study
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205
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Chapter
2. Framework for the Study
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207
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Chapter
2.1 Concepts and notions
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207
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Chapter
2.2 Structure of the topic applications and modelling in mathematics education
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208
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Chapter
3. Examples of important issues
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209
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Chapter
3.1 Epistemology
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209
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Chapter
3.2 Application problems
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210
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Chapter
3.3 Modelling abilities and competencies
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210
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Chapter
3.4 Beliefs, attitudes, and emotions
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210
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Chapter
3.5 Curriculum and goals
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211
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Chapter
3.6 Modelling pedagogy
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212
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Chapter
3.7 SUSTAINED IMPLEMENTATION
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212
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Chapter
3.8 Assessment and evaluation
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213
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Chapter
3.9 Technological impacts
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213
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Chapter
4. Call for contributions to the Study
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214
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Rubric
BULLETIN BIBLIOGRAPHIQUE
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1
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Chapter
Généralités
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1
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Chapter
Histoire
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5
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Chapter
Logique et fondements
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6
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Chapter
Théorie des ensembles
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7
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Chapter
Analyse combinatoire
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7
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Chapter
Ordre, treillis
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8
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Chapter
Théorie des nombres
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8
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Chapter
Corps et polynômes
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11
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Chapter
Géométrie algébrique
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12
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Chapter
Anneaux et algèbres
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12
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Chapter
Théorie des groupes et généralisations
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13
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Chapter
Mesure et intégration
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14
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Chapter
Fonctions d'une variable complexe
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14
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Chapter
Equations différentielles ordinaires
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15
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Chapter
Systèmes dynamiques et théorie ergodique
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15
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Chapter
Analyse de Fourier, analyse harmonique abstraite
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16
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Chapter
Analyse fonctionnelle
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16
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Chapter
Théorie des opérateurs
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18
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Chapter
Calcul des variations
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18
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Chapter
Géométrie
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19
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Chapter
Topologie générale
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20
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Chapter
Topologie algébrique
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21
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Chapter
Topologie des variétés, analyse globale et analyse des variétés
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21
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Chapter
Probabilités et processus stochastiques
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23
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Chapter
Statistique
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24
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Chapter
Analyse numérique
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25
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Chapter
Informatique
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25
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Chapter
Mécanique des solides, élasticité et plasticité
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26
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Chapter
Optique, électromagnétique
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26
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Chapter
Economie, recherche opérationnelle, jeux
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26
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Chapter
Systèmes, contrôle optimal
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27
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Back matter
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28
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Back matter
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Back matter
Endseiten
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Back matter
Endseiten
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Back matter
Endseiten
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Heft 3-4: L'ENSEIGNEMENT MATHÉMATIQUE
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Front matter
Titelseiten
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Table of Contents
Inhaltsverzeichnis
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215
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Front matter
Titelseiten
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216
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Article
ON THE ENTROPY OF HOLOMORPHIC MAPS
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217
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Chapter
§1. Notation and definitions
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218
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Chapter
§2. ESTIMATES OF DENSITY
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219
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Chapter
§3. KÄHLER MANIFOLDS
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221
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Chapter
§4. Real algebraic maps
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224
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Chapter
§5. Quasiconformal maps
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227
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Chapter
§6. Mean curvature
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229
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Appendix
Appendix : Examples of holomorphic endomorphisms
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230
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Bibliography
Bibliographie
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231
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Chapter
Note des Éditeurs
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232
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Article
NOTES SUR L'ARTICLE DE M. GROMOV
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232
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Bibliography
Bibliographie
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234
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Article
ENDOMORPHISMES DES VARIÉTÉS HOMOGÈNES
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237
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Chapter
1. Introduction
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237
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Chapter
2. Endomorphismes des variétés complexes compactes
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239
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Chapter
2.1 Dimension de Kodaira
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240
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Chapter
2.2 Action d'un endomorphisme et ramification
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241
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Chapter
2.3 Fibration d'Albanese
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242
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Chapter
2.4 Petite dimension
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243
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Chapter
2.5 Une question proche
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243
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Chapter
3. Variétés homogènes kählériennes
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244
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Chapter
3.1 Tores
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244
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Chapter
3.2 Variétés de drapeaux
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244
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Chapter
3.3 Cas général
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247
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Chapter
4. Invariance de la fibration de Tits
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248
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Chapter
4.1 La fibration de Tits
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248
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Chapter
4.2 Première application
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249
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Chapter
5. Quelques exemples
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249
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Chapter
6. Existence de facteurs inversibles.
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252
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Chapter
6.1 Structure de la fibration de Tits
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253
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Chapter
6.2 Un théorème de Jörg Winkelmann
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254
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Chapter
6.3 Endomorphismes agissant par automorphismes dans les fibres
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255
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Chapter
6.4 Application
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258
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Chapter
7. Endomorphismes irréductibles
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258
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Bibliography
Bibliographie
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261
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Article
HYPERBOLICITY OF MAPPING-TORUS GROUPS AND SPACES
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263
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Abstract
Kurzfassung
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263
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Chapter
Introduction
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263
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Chapter
1. An illustration
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268
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Chapter
2. Mapping-telescopes and forest-stacks
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271
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Chapter
3. Metrics
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272
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Chapter
3.1 Horizontal and vertical metrics
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272
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Chapter
3.2 Telescopic metric
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275
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Chapter
4. Main theorem
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276
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Chapter
5. PRELIMINARY WORK
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278
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Chapter
5.1 About dilatation in cancellations
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280
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Chapter
5.2 Straight telescopic paths
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282
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Chapter
6. About straight quasi geodesics
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283
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Chapter
7. Substitution of quasi geodesics
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285
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Chapter
8. Approximation of straight quasi geodesics in fine position
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288
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Chapter
9. PUTTING PATHS IN FINE POSITION
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289
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Chapter
10. Straight quasi geodesic bigons are thin
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291
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Chapter
11. Geodesic triangles are thin
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293
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Chapter
12. Back to mapping-telescopes
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296
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Chapter
12.1 Statement of the theorem
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296
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Chapter
12.2 Proof of Theorem 12.4
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298
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Chapter
13. About mapping-torus groups
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298
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Chapter
13.1 Relationships with mapping-telescopes
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299
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Chapter
13.2 Free group endomorphisms and forest-maps
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301
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Chapter
13.3 Proof of Theorem 13.2
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303
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Bibliography
Bibliographie
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304
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Article
THE BASIC GERBE OVER A COMPACT SIMPLE LIE GROUP
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307
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Abstract
Kurzfassung
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307
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Chapter
1. Introduction
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307
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Chapter
2. Gerbes with connections
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309
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Chapter
2.1 Chatterjee-Hitchin gerbes
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309
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Chapter
2.2 Bundle gerbes
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310
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Chapter
2.3 Simplicial gerbes
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311
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Chapter
2.4 Equivariant bundle gerbes
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315
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Chapter
3. Gerbes from principal bundles
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316
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Chapter
4. Gluing data
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319
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Chapter
5. The basic gerbe over a compact simple Lie group
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323
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Chapter
5.1 Notation
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323
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Chapter
5.2 The basic 3-form on G
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324
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Chapter
5.3 The special unitary group
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326
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Chapter
6. Pre-quantization of conjugacy classes
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329
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Appendix
Appendix A. Proof of Lemma 4.4
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331
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Bibliography
Bibliographie
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332
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Article
ANALYSE DE FOURIER DES FRACTIONS CONTINUES À QUOTIENTS RESTREINTS
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335
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Abstract
Kurzfassung
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335
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Chapter
1. Introduction
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335
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Chapter
2. Les ensembles F(A)
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337
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Chapter
3. Dimension de Hausdorff
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340
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Chapter
4. Une mesure spéciale
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341
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Chapter
5. Intégrales oscillantes
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344
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Chapter
6. Estimation de la transformée de Fourier
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345
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Chapter
7. Une question de Montgomery
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352
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Chapter
8. COMMENTAIRES ET QUESTIONS
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354
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Bibliography
Bibliographie
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355
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Article
ON THE CLASSIFICATION OF RATIONAL KNOTS
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357
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Abstract
Kurzfassung
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357
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Chapter
1. Introduction
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357
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Chapter
2. Rational tangles and their invariant fractions
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362
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Chapter
3. The classification of unoriented rational knots
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373
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Chapter
3.1 The cuts
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375
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Chapter
3.2 The flypes
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384
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Chapter
4. Rational knots and their mirror images
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392
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Chapter
5. On connectivity
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393
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Chapter
6. The oriented case
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395
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Chapter
7. Strongly invertible links
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405
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Bibliography
Bibliographie
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407
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Rubric
BULLETIN BIBLIOGRAPHIQUE
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29
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Chapter
Généralités
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29
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Chapter
Histoire
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35
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Chapter
Logique et fondements
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36
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Chapter
Analyse combinatoire
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36
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Chapter
Théorie des nombres
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37
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Chapter
Corps et polynômes
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38
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Chapter
Géométrie algébrique
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38
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Chapter
Anneaux et algèbres
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40
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Chapter
K-théorie
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41
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Chapter
Théorie des groupes et généralisations
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41
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Chapter
Groupes topologiques ; groupes et algèbres de Lie
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42
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Chapter
Fonctions d'une variable complexe
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42
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Chapter
Fonctions de plusieurs variables complexes
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43
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Chapter
Équations différentielles ordinaires
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44
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Chapter
Équations aux dérivées partielles
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44
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Chapter
Systèmes dynamiques et théorie ergodique
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44
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Chapter
Approximations et développements en série
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45
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Chapter
Analyse de Fourier, analyse harmonique abstraite
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46
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Chapter
Analyse fonctionnelle
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46
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Chapter
Théorie des opérateurs
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47
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Chapter
Calcul des variations et contrôle optimal
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47
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Chapter
Géométrie
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48
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Chapter
Géométrie différentielle
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48
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Chapter
Topologie algébrique
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49
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Chapter
Topologie des variétés, analyse globale et analyse des variétés
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50
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Chapter
Probabilités et processus stochastiques
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51
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Chapter
Statistique
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51
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Chapter
Analyse numérique
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52
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Chapter
Informatique
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53
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Chapter
Mécanique des fluides, acoustique
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53
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Chapter
Biologie et sciences du comportement
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54
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Chapter
Information, communication, circuits
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54
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Back matter
Endseiten
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Front matter
Titelseiten
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Index
Register
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Back matter
Endseiten
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Back matter
Endseiten
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Back matter
Endseiten
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