ETH Zurich
E-Periodica
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L'Enseignement Mathématique

Volume 43 (1997)
Rubric
Issue 1-2: L'ENSEIGNEMENT MATHÉMATIQUE
Front matter
Table of Contents 1
Front matter 2
Article: PARAMETRIZED PLANE CURVES, MINKOWSKI CAUSTICS, MINKOWSKI VERTICES AND CONSERVATIVE LINE FIELDS 3
Chapter: 0. Introduction 3
Chapter: 1. Finsler metric from the contact geometrical viewpoint 4
Chapter: 2. Minkowski geometry associated with a parametrized curve 7
Chapter: 3. Osculating indicatrices and Minkowski caustic 10
Chapter: 4. Minkowski vertices and Chebyshev Systems 16
Chapter: 5. Conservative transverse line fields 20
Bibliography 25
Article: EVEN NON-SPIN MANIFOLDS, SPINc STRUCTURES, AND DUALITY 27
Abstract 27
Rubric 28
Bibliography 32
Article: BOTT-CHERN FORMS AND ARITHMETIC INTERSECTIONS 33
Abstract 33
Chapter: 1. Introduction 33
Chapter: 2. Invariant and symmetric functions 35
Chapter: 3. Hermitian differential geometry 36
Chapter: 4. Calculating Bott-Chern Forms 39
Chapter: 5. $0 \rightarrow \bar{S} \rightarrow \bar{E} \rightarrow \bar{Q} \rightarrow 0$ WITH $\bar{E}$ FLAT 42
Chapter: 6. Calculations when $\bar{E}$ is projectively flat 44
Chapter: 7. Arithmetic intersection theory 48
Chapter: 8. Arakelov Chow rings of grassmannians 51
Bibliography 53
Article: ATKIN-LEHNER EIGENFORMS AND STRONGLY MODULAR LATTICES 55
Abstract 55
Chapter: Introduction 55
Chapter: 1. Standard involutions on lattices 56
Chapter: 2. ATKIN-LEHNER ACTION ON THETA FUNCTIONS 59
Chapter: 3. Some use of modular forms 61
Bibliography 65
Article: THE NORMALISER ACTION AND STRONGLY MODULAR LATTICES 67
Abstract 67
Chapter: 1. Introduction 67
Chapter: 2. PRELIMINAIRES AND NOTATION 68
Chapter: 3. Similarities Normalise 69
Chapter: 4. Obtaining Elements of N 70
Chapter: 5. Proof of the Theorem 73
Bibliography 75
Article: SUR LA LOI DE RÉCIPROCITÉ DE KATO POUR LES ANNEAUX LOCAUX DE DIMENSION 2 77
Chapter: 1. ÉNONCÉS 77
Chapter: 2. Préparation 79
Chapter: 3. Toujours préparation, mais à la Weierstrass 82
Chapter: 4. RÉDUCTIONS 84
Chapter: 5. Conclusion 87
Bibliography 90
Article: LA SOMMATION DE RAMANUJAN 93
Abstract 93
Chapter: 1. Introduction 93
Chapter: 2. DÉVELOPPEMENTS D'EULER-MACLAURIN FORMELS 94
Chapter: 3. Sommation de Ramanujan et transformation de Laplace-Borel 96
Chapter: 3.1. Sommation de Ramanujan 96
Chapter: 3.2. Liens avec la sommation de Cauchy 102
Chapter: 4. Propriétés de la sommation 104
Chapter: 4.1. Linéarité 104
Chapter: 4.2. Translation 104
Chapter: 4.3. Dérivation 105
Chapter: 4.4. Sommation par parties 105
Chapter: 4.5. SÉPARATION DES TERMES PAIRS ET IMPAIRS 108
Chapter: 4.6. Utilisations de développements en série entière 109
Chapter: 4.7. DÉPENDANCE ANALYTIQUE PAR RAPPORT À UN PARAMÈTRE 112
Chapter: 5. Exemples d'utilisation 113
Chapter: 5.1. DÉVELOPPEMENT EN SÉRIE DE LA FONCTION Ψ 113
Chapter: 5.2. Calcul de $\sum_{n \geq 1}^R n^{2q} \ln(n)$ 114
Chapter: 5.3. Une solution de l'équation de la chaleur 116
Chapter: 6. Interpolation de Newton et sommation de Ramanujan 117
Appendix: 7. Appendice: Transformation de Laplace-Borel 121
Chapter: 7.1. Notations 121
Chapter: 7.2. Transformation de Borel 121
Chapter: 7.3. Transformation de Laplace 124
Chapter: 7.4. Le cas intégrable 128
Chapter: 7.5. Quelques propriétés 131
Bibliography 131
Article: THE LOCAL LINEARIZATION PROBLEM FOR SMOOTH SL(n) -ACTIONS 133
Abstract 133
Abstract 133
Chapter: 1. Introduction 133
Chapter: 2. Background and Motivation 136
Chapter: 3. Preparatory results 142
Chapter: 4. SL(n,R)-ACTIONS ON $R^n$ FOR n ≥ 3 149
Chapter: 5. The adjoint representation of SL(2,R) 151
Chapter: 6. SL(2,R) -actions on $R^2$ 153
Chapter: 7. Examples of $C^0$-actions of SL(2,R) on $R^m$ 157
Chapter: 8. A $C^\infty$ -action of SL(2,R) which is not linearizable 159
Chapter: 9. A $C^\infty$ -ACTION OF SL(3,R) WHICH IS NOT LINEARIZABLE 161
Chapter: 10. LINEARIZABILITY OF SL(n,Z)-ACTIONS 164
Bibliography 168
Article: POLYGON SPACES AND GRASSMANNIANS 173
Abstract 173
Chapter: 1. Introduction 173
Chapter: 2. The polygon spaces 175
Chapter: 3. Quaternions, Grassmannians and structures on the full polygon spaces 178
Chapter: 4. POLYGONS WITH GIVEN SIDES – KÄHLER STRUCTURES 183
Chapter: 5. The Gel'fand-Cetlin action 187
Chapter: 6. Toric manifold structures on $mP_+^3(\alpha)(a)$ for m = 4,5,6 191
Chapter: 7. Remarks and open problems 196
Bibliography 197
Rubric: COMMISSION INTERNATIONALE DE L'ENSEIGNEMENT MATHÉMATIQUE (THE INTERNATIONAL COMMISSION ON MATHEMATICAL INSTRUCTION) 199
Chapter: THE ROLE OF THE HISTORY OF MATHEMATICS IN THE TEACHING AND LEARNING OF MATHEMATICS DISCUSSION DOCUMENT FOR AN ICMI STUDY (1997-2000) 199
Chapter: Some research questions 199
Chapter: Call for contributions 202
Rubric: BULLETIN BIBLIOGRAPHIQUE 1
Back matter 44
Back matter
Back matter
Issue 3-4: L'ENSEIGNEMENT MATHÉMATIQUE
Front matter
Table of Contents 205
Front matter 206
Article: AN ALGORITHM FOR CELLULAR MAPS OF CLOSED SURFACES 207
Abstract 207
Chapter: 1. Introduction 207
Chapter: 2. DIAGRAMS OF CELLULAR SURFACE MAPS 212
Chapter: 3. The algorithm 222
Chapter: 4. HOMOMORPHISMS OF SURFACE GROUPS 237
Chapter: 5. A WORKED EXAMPLE 247
Bibliography 251
Article: QUATERNARY CUBIC FORMS AND PROJECTIVE ALGEBRAIC THREEFOLDS 253
Chapter: Introduction 253
Chapter: I. Quaternary Cubic Forms 255
Chapter: 1. Normal Forms for Quaternary Cubic Forms 256
Chapter: 2. The Invariant Theory of Quaternary Cubic Forms 260
Chapter: II. Cubic Forms of Projective Threefolds 262
Chapter: 1. Preliminaries 262
Chapter: 2. A Projective Threefold with a Nodal Cubic as Cup Form 263
Chapter: 3. Quaternary Cubic Forms that are Cup Forms of Projective Algebraic Manifolds 264
Chapter: 4. Real Cubic Forms which are not Cup Forms of Projective Algebraic Manifolds 268
Bibliography 269
Article: 4-MANIFOLDS, GROUP INVARIANTS, AND $l_2$-BETTI NUMBERS 271
Chapter: 1. A BASIC CONSTRUCTION 271
Chapter: 2. The Hausmann-Weinberger invariant 272
Chapter: 3. The (λ+σ) -invariant 273
Chapter: 4. Deus ex machina: $l_2$-cohomology 275
Chapter: 5. The vanishing of q(G) 278
Bibliography 279
Article: THEOREM OF INGHAM IMPLYING THAT DIRICHLET'S L-FUNCTIONS HAVE NO ZEROS WITH REAL PART ONE 281
Chapter: §1. Introduction 281
Chapter: §2. Proof of Ingham's Theorem 283
Bibliography 284
Article: SUR LES TRANSFORMATIONS DE CREMONA DE BIDEGRÉ (3,3) 285
Abstract 285
Chapter: Introduction 285
Chapter: 1. Le résultat principal 286
Chapter: 2. Exemples 287
Chapter: 3. Preuve du théorème 289
Chapter: 4. Un corollaire et plus d'exemples 293
Chapter: 5. Comparaison avec les résultats classiques 295
Bibliography 297
Article: NONMINIMAL RATIONAL CURVES ON K3 SURFACES 299
Chapter: Introduction 299
Chapter: 1. About finiteness 303
Chapter: 2. About existence 304
Chapter: 3. Rational curves on quartics in $P^3$ 310
Chapter: 4. Rational curves on K3 surfaces in $P^4$ 313
Bibliography 316
Article: ON CYCLOTOMIC POLYNOMIALS, POWER RESIDUES, AND RECIPROCITY LAWS 319
Abstract 319
Chapter: 1. Introduction 319
Chapter: 2. Statement of results 320
Chapter: 3. Background 322
Chapter: 4. The odd case 325
Chapter: 5. Homogeneous polynomials 328
Chapter: 6. WHAT ABOUT q? 329
Chapter: 7. The even case 330
Chapter: 8. The second proof 333
Bibliography 335
Article: AMENABILITY AND GROWTH OF ONE-RELATOR GROUPS 337
Abstract 337
Chapter: 0. Introduction 337
Chapter: 1. An algorithm for checking amenability 339
Chapter: 2. Two-generated one-relator groups 342
Chapter: 3. Uniformly exponential growth and growth of graded algebras 346
Chapter: 4. More on uniformly exponential growth of one-relator groups 349
Bibliography 352
Article: ÉQUISINGULARITÉ DANS LES PINCEAUX DE GERMES DE COURBES PLANES ET $C^0$-SUFFISANCE 355
Chapter: §1. Introduction 355
Chapter: §2. ÉLIMINATION DE L'INDÉTERMINATION LOCALE D'UNE FONCTION MÉROMORPHE À DEUX VARIABLES 357
Chapter: §3. Valeurs spéciales d'un pinceau de germes de courbes planes 360
Chapter: §4. L'OUVERT D'ÉQUISINGULARITÉ D'UN PINCEAU LOCAL 363
Chapter: §5. Bonnes composantes dicritiques 369
Chapter: §6. Étude d'un cas particulier 371
Chapter: §7. Le calcul du degré de $C^0$ -suffisance 372
Chapter: §8. Un petit historique de la $C^0$-suffisance 377
Bibliography 379
Rubric: COMMISSION INTERNATIONALE DE L'ENSEIGNEMENT MATHÉMATIQUE (THE INTERNATIONAL COMMISSION ON MATHEMATICAL INSTRUCTION) 381
Chapter: ICMI STUDY ON THE TEACHING AND LEARNING OF MATHEMATICS AT UNIVERSITY LEVEL DISCUSSION DOCUMENT 381
Chapter: 1. WHY A STUDY ON THE TEACHING AND LEARNING OF MATHEMATICS AT UNIVERSITY LEVEL ? 382
Chapter: 2. THEMES AND ISSUES PERTAINING TO RESEARCH ON THE TEACHING AND LEARNING OF MATHEMATICS AT UNIVERSITY LEVEL 384
Chapter: 3. THEMES AND ISSUES PERTAINING TO PRACTICE 385
Chapter: 4. THEMES AND ISSUES RELATING TO POLICY 387
Chapter: 5. Call for reactions 388
Rubric: BULLETIN BIBLIOGRAPHIQUE 45
Back matter
Index
Back matter