Mathematics and the 21st century: proceedings of the international conference, Cairo, Egypt, 15-20 January 2000

Organizing Committee......Page 6
Preface......Page 8
Contents......Page 10
1000 YEARS OF MATHEMATICS......Page 14
Trend 1: From the Linear Model to the Dynamic Model of Research......Page 16
Trend 2: From Theory + Experiment to Theory + Experiment + Computation......Page 17
Trend 3: From Disciplinary to Interdisciplinary Research......Page 19
Trend 4: Complementing Reductionism with the Study of Complex Systems......Page 21
Trend 5: Globalization and the Diffusion of Knowledge......Page 22
Conclusion......Page 24
Arabic Mathematics and Rewriting the History of Mathematics......Page 26
Rational Diophantine analysis......Page 27
Integer Diophantine Analysis......Page 29
The Tradition continues......Page 30
Numerical analysis and interpolation methods......Page 32
The Paradigm Shift in Mathematics Education: A Scenario for Change......Page 40
A Half Century of Swirl Progress:......Page 41
1. A Chinese Perspective (Er-Sheng 1998)......Page 43
II. A view From South Africa: Out-Comes Based Education (OBE) (Volmik 1998)......Page 44
III. U.S.A. Standards 2000 (NCTM 1998)......Page 45
V. An Australian Curriculum and Standards Framework (CSF). (Board of studies 1995).......Page 48
A Scenario For Change In Mathematics Education (Case study: Egypt).......Page 50
REFERENCES......Page 52
I. FOUNDATION......Page 54
II. SOME DEVELOPMENTS RESULTS AND PROBLEMS......Page 59
REFERENCES......Page 63
I. Moduli of compact Riemann Surfaces......Page 66
II. Moduli of vector bundles on a compact Riemann surfaces......Page 67
III. Gauge Theory......Page 68
IV. Construction of moduli spaces......Page 69
IV. Difference Geometric Interpretation of Stability......Page 70
2. The 3264 conics of Chasles......Page 72
5. Cohomology......Page 73
6. Homogeneous spaces......Page 74
7. Spherical varieties......Page 75
8. Strings and Moduli spaces......Page 76
9. Quantum cohomology......Page 78
10. Equivariant Theories......Page 79
Optical Solitons: Twenty-Seven Years of the Last Millennium and Three More Years of the New?......Page 82
1. INTRODUCTION: A NEW TECHNOLOGY AND A SHORT SURVEY OF SOLITON THEORY AND ITS MATHEMATICS......Page 83
2. MAXWELL-BLOCH HIERARCHY OF EQUATIONS AND THE SOLITONS OF SELF-INDUCED TRANSPARENCY (SIT)......Page 86
3. COMPLETE HAMILTONIAN INTEGRABILITY OF THE SOLITON SYSTEMS......Page 100
4. BOSE-EINSTEIN CONDENSATION (BEC)......Page 111
5. QUANTUM INFORMATION......Page 116
6. FINAL COMMENTS AND CONCLUSIONS......Page 122
REFERENCES......Page 125
1 Introduction......Page 136
2 The general resulting of non-smooth systems......Page 141
3 Lyapunov exponents......Page 144
References......Page 150
2. Structure theorems......Page 154
3. Closure operations on classes of rings......Page 156
4. Ring constructions......Page 158
5. Radical theory in other categories......Page 159
6. Cardinality condition......Page 162
References......Page 163
Introduction and results......Page 166
References......Page 170
1 Introduction......Page 172
2 Totally permutable products......Page 174
3 Mutually permutable products......Page 178
References......Page 179
Asymptotic Behaviour of Solutions of Evolution Equations......Page 182
REFERENCES......Page 185
1. Introduction......Page 186
2. Basic Equations of Water Waves......Page 187
3. The Korteweg-de Vries (KdV) Equation Near Resonant Speed......Page 188
4. The Nonlinear Schrodinger (NLS) Equation Near Resonant Conditions......Page 189
5. The Nonlinear Schrodinger Equation and Evolution of Wave Packets......Page 191
6. Higher-order Nonlinear Schrodinger Equations......Page 192
7. The Davey-Stewartson (DS) Equations in Water of Finite Depth......Page 196
Acknowledgement......Page 199
References......Page 200
1 THE FREE CONVECTION PROBLEM......Page 202
2 LAYER-RESOLVING METHOD FOR BLASIUS\' PROBLEM......Page 205
3 CONVERGENCE OF THE METHOD......Page 206
4 CONCLUSION......Page 210
REFERENCES......Page 211
1 Introduction......Page 212
2 Preliminaries......Page 214
4 Hadamard-Hayman convexity......Page 216
5 Extensions......Page 219
6 Gap- Series......Page 220
References......Page 221
INTRODUCTION.......Page 224
1. INVERTIBILITY PRESERVING MAPS AND JORDAN ISOMORPHISMS......Page 225
2. COMMUTATIVITY PRESERVING MAPS AND LIE ISOMORPHISMS......Page 227
3. RANK ONE PRESERVING MAPS......Page 230
REFERENCES......Page 232
Prediction: Advances and New Research......Page 236
1. INTRODUCTION......Page 237
2. PREDICTIVE DENSITY OF THE Rjth ORDER STATISTIC IN SAMPLE J......Page 240
3. A GENERAL CLASS OF POPULATION DISTRIBUTIONS AND A GENERAL CLASS OF PRIORS......Page 245
4. APPLICATIONS......Page 247
REFERENCES......Page 253
1 INTRODUCTION......Page 260
2 RECURRENCE RELATIONS FOR ORDER STATISTICS FROM TWO RELATED DISTRIBUTIONS......Page 262
3 BLUE\'S OF u AND o......Page 264
4 MLE\'S OF u AND o......Page 266
5 PIVOTAL QUANTITIES AND INFERENCE......Page 267
6 NUMERICAL ILLUSTRATION......Page 269
7 RESULTS FOR GENERAL CENSORED SAMPLES......Page 271
References......Page 273
0. Introduction......Page 284
1. Resource......Page 285
2. Generalized Sedyakin\'s model......Page 286
3. Additive accumulation of damages model......Page 290
4. Proportional hazards model......Page 293
5. Generalized proportional hazards models......Page 298
6. The main classes of GPH models.......Page 303
7. Parametrization of the function r in AAD and GPH models......Page 307
9. Changing shape and scale models......Page 310
10. Generalizations......Page 311
REFERENCES......Page 313
The Vibrations of a Drum with Fractal Boundary......Page 318
1 INTRODUCTION......Page 319
3 VIBRATIONS OF A DRUM; WEYL\'S ESTIMATE......Page 320
4 CAN ONE HEAR THE SHAPE OF A DRUM?......Page 323
5 DOMAINS WITH FRACTAL BOUNDARIES......Page 324
6 HEAT EQUATION ON THE TRIADIC VON KOCH SNOWFLAKE.......Page 330
REFERENCES......Page 332
1. Introduction.......Page 336
2. Even and odd coherent states.......Page 337
3. The Binomial states.......Page 341
4. The phased generalized binomial state.......Page 346
5 Generalized geometric state.......Page 355
6 Even geometric states.......Page 365
References......Page 367
1 Introduction:......Page 370
2 Derivation of the two-body equation:......Page 372
3 Solutions:......Page 378
4 References......Page 384
1 Introduction......Page 386
2 d-spaces......Page 388
3 A simple example......Page 391
4 Schwarzschild space-time......Page 392
5 The closed Friedmann Universe......Page 394
6 The origin of charge......Page 395
References......Page 398
1 Introduction......Page 400
2 Structure of Light Cone in Godel Universe......Page 402
3 The Inner Geometry of Light Cone......Page 403
REFERENCES......Page 406
List of Participants......Page 408