ETH Zurich
E-Periodica
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L'Enseignement Mathématique

Band 36 (1990)
Heft 1-2: L'ENSEIGNEMENT MATHÉMATIQUE
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Artikel: STATE MODELS FOR LINK POLYNOMIALS 1
Kapitel: I. Introduction 1
Kapitel: II. Skein polynomials 2
Kapitel: III. A SKEIN MODEL FOR THE HOMFLY POLYNOMIAL 9
Kapitel: IV. A Skein Model for the Kauffman polynomial 12
Kapitel: V. Graph polynomials 14
Kapitel: VI. The Conway Polynomial 18
Kapitel: VII. Yang-Baxter Models 21
Kapitel: VIII. Applications and questions 27
Kapitel: IX. Relations with mathematical physics 28
Appendix: Appendix on state model formalism 31
Bibliography 33
Article: GAUSS SUMS AND THEIR PRIME FACTORIZATION 39
Chapter: Introduction 39
Chapter: 1. Gauss sums and some of their properties 39
Chapter: 2. The prime factorization of p in Q(pm) 42
Chapter: 3. The prime factorization of the Gauss sum: statement of the result 43
Chapter: 4. A USEFUL LEMMA 44
Chapter: 5. The prime factorization of the Gauss sum: proof of the result 46
Chapter: 6. Annihilators of the ideal class group of a cyclotomic field 47
Bibliography 51
Article: EXCEPTIONAL POLYNOMIALS AND THE REDUCIBILITY OF SUBSTITUTION POLYNOMIALS 53
Chapter: 1. Introduction 53
Chapter: 2. The semi-factorable families 55
Chapter: 3. Substitution polynomials with a quadratic factor 58
Chapter: 4. Substitution polynomials with a cubic factor 63
Bibliography 65
Article: THE POMPEIU PROBLEM REVISITED 67
Abstract 67
Chapter: 1. Introduction 67
Chapter: 2. Spectral analysis of radial functions 70
Chapter: 3. POMPEIU PROBLEM FOR THE M(2) ACTION ON $R^2$ 73
Chapter: 4. A LONG-STANDING CONJECTURE ! 77
Chapter: 5. Pompeiu property in non-compact symmetric spaces 78
Chapter: 6. Symmetric spaces of the compact type 82
Chapter: 7. Pompeiu property for two-sided translations on groups 85
Chapter: 8. CONCLUDING REMARKS 88
Bibliography 89
Article: LINK SIGNATURE, GOERITZ MATRICES AND POLYNOMIAL INVARIANTS 93
Abstract: Abstract 93
Chapter: 1. P-SKEINS AND SIGNATURE 93
Chapter: 1.1. PRELIMINARIES 93
Chapter: 1.2. Signature and oriented skeins 95
Chapter: 2. Goeritz matrices and the F-polynomial 100
Chapter: 2.1. The Goeritz matrix and graph of a link 100
Chapter: 2.2. Kauffman's polynomial and the Goeritz matrix 104
Bibliography 113
Article: TOPOLOGICAL SERIES OF ISOLATED PLANE CURVE SINGULARITIES 115
Abstract 115
Chapter: 1. Introduction 115
Chapter: 2. SPLICING AND SERIES 117
Chapter: 3. The definition of topological series 121
Chapter: 4. The spectrum of a plane curve singularity 125
Chapter: 5. Invariants in the case that f has only transversal $A_1$ singularities 130
Chapter: 6. Equations 132
Appendix: Appendix 136
Bibliography 140
Article: VALUES OF QUADRATIC FORMS AT INTEGRAL POINTS: AN ELEMENTARY APPROACH 143
Appendix: Appendix Trajectories of unipotent flows and minimal sets 162
Bibliography 173
Article: ON THE INVERSIVE DIFFERENTIAL GEOMETRY OF PLANE CURVES 175
Chapter: §1. Introduction 175
Chapter: §2. THE INFINITESIMAL COXETER INVARIANT 178
Chapter: §3. The four vertex theorem in $R^2$ 181
Chapter: §4. A GENERALIZATION OF THE INVARIANCE OF ω 181
Chapter: §5. A GENERALIZED FOUR VERTEX THEOREM 184
Chapter: §6. Normal form and inversive curvature 184
Chapter: §7. The canonical map g:γ→G 187
Chapter: §8. Relation with Cartan's moving frames 188
Chapter: §9. LOXODROMES 189
Chapter: §10. The complex of geometric forms on a curve in $R^2$ 192
Bibliography 196
Rubric: COMMISSION INTERNATIONALE DE L'ENSEIGNEMENT MATHÉMATIQUE (THE INTERNATIONAL COMMISSION ON MATHEMATICAL INSTRUCTION) 197
Chapter: ASSESSMENT IN MATHEMATICS EDUCATION AND ITS EFFECTS 197
Chapter: DISCUSSION DOCUMENT 197
Chapter: Background and outline of the study 197
Chapter: Call for papers 205
Obituary: GEORGES DE RHAM 1903-1990 207
Bibliography 213
Article: CONTACT GEOMETRY AND WAVE PROPAGATION 215
Chapter: §1. Basic definitions 215
Chapter: §2. Characteristics 222
Chapter: §3. Submanifolds 235
Chapter: §4. Legendre fibrations and singularities 246
Chapter: §5. Legendre varieties and the obstacle problem 252
Bibliography 265
Article: YANGIANS AND R-MATRICES 267
Chapter: 0. Introduction 267
Chapter: 1. Yangians 271
Chapter: 2. Highest weight representations 276
Chapter: 3. A COMBINATORIAL INTERLUDE 280
Chapter: 4. Classification 283
Chapter: 5. R-MATRICES AND INTERTWINING OPERATORS 294
Chapter: 6. CONCLUDING REMARKS 299
Bibliography 302
Article: EXTERIOR ALGEBRAS AND THE QUADRATIC RECIPROCITY LAW 303
Abstract 303
Rubric 304
Bibliography 307
Article: THE HADAMARD-CARTAN THEOREM IN LOCALLY CONVEX METRIC SPACES 309
Chapter: 1. Introduction 309
Chapter: 2. Conjugate points 311
Chapter: 3. Proof of Theorem 1 315
Chapter: 4. COHN-VOSSEN'S THEOREM AND SPACES WITHOUT CONJUGATE POINTS 317
Bibliography 320
Article: THE DISTANCE BETWEEN IDEALS IN THE ORDERS OF A REAL QUADRATIC FIELD 321
Chapter: 1. Introduction 321
Chapter: 2. Basic definitions 322
Chapter: 3. The homomorphism θ 330
Chapter: 4. Reduced ideals 335
Chapter: 5. Lagrange's reduction procedure 337
Chapter: 6. Periods of reduced cycles 343
Chapter: 7. COMPARISON OF DISTANCES BETWEEN CORRESPONDING IDEALS IN DIFFERENT ORDERS 350
Chapter: 8. GAUSS'S REDUCTION PROCESS 352
Bibliography 357
Article: THE ROLE OF GAMES AND PUZZLES IN THE POPULARIZATION OF MATHEMATICS 359
Abstract 359
Chapter: popularization of mathematics 359
Chapter: Games and mathematics 362
Chapter: Games and puzzles as a means for popularization 364
Bibliography 368
Article: CATÉGORIES DÉRIVÉES ET DUALITÉ, TRAVAUX DE J.-L. VERDIER 369
Chapter: 1. Catégories dérivées 369
Chapter: 2. Dualité 378
Bibliography 388
Bibliography 389
Bibliography 390
Chapter: Note sur la bibliographie 391
Article: MANIN'S PROOF OF THE MORDELL CONJECTURE OVER FUNCTION FIELDS 393
Chapter: I. The Theorem of the Kernel 393
Chapter: 0. Review of connections and hypercohomology 393
Chapter: 1. Extensions of connections 395
Chapter: 2. The Gauss-Manin connection 398
Chapter: 3. Sections of a family and extensions of connections 399
Chapter: 4. Abelian schemes 402
Chapter: 5. The algebraic proof 403
Chapter: 6. The analytic proof 409
Chapter: II. PICARD-FUCHS EQUATIONS 412
Chapter: 1. PICARD-FUCHS DIFFERENTIAL EQUATIONS 412
Chapter: 2. PICARD-FUCHS COMPUTATIONS 415
Chapter: III. MORDELL'S CONJECTURE 417
Chapter: 1. Sets of bounded height 417
Chapter: 2. Lang-Siegel towers 418
Chapter: 3. COROLLARIES OF THE THEOREM OF THE KERNEL 419
Chapter: 4. Proof of Mordell's conjecture 420
Appendix: Appendix: Chai's proof of the Theorem of the Kernel 424
Bibliography 427
Rubric
Rubric: BULLETIN BIBLIOGRAPHIQUE 1
Rubric: BULLETIN BIBLIOGRAPHIQUE 27
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