Contents -- Organizing Committee -- Preface -- Millennium Lecture - Cairo 15 January 2000 -- 1000 YEARS OF MATHEMATICS -- Trends for Science and Mathematics in the 21st Century -- Trend 1: From the Linear Model to the Dynamic Model of Research -- Trend 2: From Theory + Experiment to Theory + Experiment + Computation -- Trend 3: From Disciplinary to Interdisciplinary Research -- Trend 4: Complementing Reductionism with the Study of Complex Systems -- Trend 5: Globalization and the Diffusion of Knowledge -- Conclusion -- Arabic Mathematics and Rewriting the History of Mathematics -- Rational Diophantine analysis -- Integer Diophantine Analysis -- The Tradition continues -- Numerical analysis and interpolation methods -- The Paradigm Shift in Mathematics Education: A Scenario for Change -- A Half Century of Swirl Progress: -- 1. A Chinese Perspective (Er-Sheng 1998) -- II. A view From South Africa: Out-Comes Based Education (OBE) (Volmik 1998) -- III. U.S.A. Standards 2000 (NCTM 1998) -- IV. The Swedish "ADM"-Project (Bjork and Brolin 1998). -- V. An Australian Curriculum and Standards Framework (CSF). (Board of studies 1995). -- A Scenario For Change In Mathematics Education (Case study: Egypt). -- REFERENCES -- Einstein's Theory of Spacetime and Gravity -- INTRODUCTION -- I. FOUNDATION -- II. SOME DEVELOPMENTS RESULTS AND PROBLEMS -- REFERENCES -- Moduli Problems in Geometry -- I. Moduli of compact Riemann Surfaces -- II. Moduli of vector bundles on a compact Riemann surfaces -- III. Gauge Theory -- IV. Construction of moduli spaces -- V. Study of moduli spaces -- IV. Difference Geometric Interpretation of Stability -- Enumerative Geometry from the Greeks to Strings -- 1. The beginning -- 2. The 3264 conics of Chasles -- 3. Schubert Calculus -- 4. Hilbert's 15th Problem -- 5. Cohomology -- 6. Homogeneous spaces.

7. Spherical varieties -- 8. Strings and Moduli spaces -- 9. Quantum cohomology -- 10. Equivariant Theories -- Optical Solitons: Twenty-Seven Years of the Last Millennium and Three More Years of the New? -- 1. INTRODUCTION: A NEW TECHNOLOGY AND A SHORT SURVEY OF SOLITON THEORY AND ITS MATHEMATICS -- 2. MAXWELL-BLOCH HIERARCHY OF EQUATIONS AND THE SOLITONS OF SELF-INDUCED TRANSPARENCY (SIT) -- 3. COMPLETE HAMILTONIAN INTEGRABILITY OF THE SOLITON SYSTEMS -- 4. BOSE-EINSTEIN CONDENSATION (BEC) -- 5. QUANTUM INFORMATION -- 6. FINAL COMMENTS AND CONCLUSIONS -- REFERENCES -- Concepts for Non-smooth Dynamical Systems -- 1 Introduction -- 2 The general resulting of non-smooth systems -- 3 Lyapunov exponents -- References -- Radical Theory: Developments and Trends -- 1. Introduction -- 2. Structure theorems -- 3. Closure operations on classes of rings -- 4. Ring constructions -- 5. Radical theory in other categories -- 6. Cardinality condition -- References -- On Minimal Subgroups of Finite Groups -- Introduction and results -- References -- Totally and Mutually Permutable Products of Finite Groups -- 1 Introduction -- 2 Totally permutable products -- 3 Mutually permutable products -- References -- Asymptotic Behaviour of Solutions of Evolution Equations -- REFERENCES -- On Nonlinear Evolution Equations with Applications -- 1. Introduction -- 2. Basic Equations of Water Waves -- 3. The Korteweg-de Vries (KdV) Equation Near Resonant Speed -- 4. The Nonlinear Schrodinger (NLS) Equation Near Resonant Conditions -- 5. The Nonlinear Schrodinger Equation and Evolution of Wave Packets -- 6. Higher-order Nonlinear Schrodinger Equations -- 7. The Davey-Stewartson (DS) Equations in Water of Finite Depth -- Acknowledgement -- References.

A Robust Layer-Resolving Numerical Method for a Free Convection Problem -- 1 THE FREE CONVECTION PROBLEM -- 2 LAYER-RESOLVING METHOD FOR BLASIUS' PROBLEM -- 3 CONVERGENCE OF THE METHOD -- 4 CONCLUSION -- ACKNOWLEDGMENTS -- REFERENCES -- Growth Value-Distribution and Zero-Free Regions of Entire Functions and Sections -- 1 Introduction -- 2 Preliminaries -- 3 Growth of entire functions -- 4 Hadamard-Hayman convexity -- 5 Extensions -- 6 Gap- Series -- References -- Three Linear Preserver Problems -- INTRODUCTION. -- 1. INVERTIBILITY PRESERVING MAPS AND JORDAN ISOMORPHISMS -- 2. COMMUTATIVITY PRESERVING MAPS AND LIE ISOMORPHISMS -- 3. RANK ONE PRESERVING MAPS -- REFERENCES -- Prediction: Advances and New Research -- 1. INTRODUCTION -- 2. PREDICTIVE DENSITY OF THE Rjth ORDER STATISTIC IN SAMPLE J -- 3. A GENERAL CLASS OF POPULATION DISTRIBUTIONS AND A GENERAL CLASS OF PRIORS -- 4. APPLICATIONS -- 5. CONCLUDING REMARKS -- REFERENCES -- Inference on Parameters of the Laplace Distribution Based on Type-II Censored Samples Using Edgeworth Approximation -- 1 INTRODUCTION -- 2 RECURRENCE RELATIONS FOR ORDER STATISTICS FROM TWO RELATED DISTRIBUTIONS -- 3 BLUE'S OF u AND o -- 4 MLE'S OF u AND o -- 5 PIVOTAL QUANTITIES AND INFERENCE -- 6 NUMERICAL ILLUSTRATION -- 7 RESULTS FOR GENERAL CENSORED SAMPLES -- Acknowledgments -- References -- Mathematical Models in the Theory of Accelerated Experiments -- 0. Introduction -- 1. Resource -- 2. Generalized Sedyakin's model -- 3. Additive accumulation of damages model -- 4. Proportional hazards model -- 5. Generalized proportional hazards models -- 6. The main classes of GPH models. -- 7. Parametrization of the function r in AAD and GPH models -- 9. Changing shape and scale models -- 10. Generalizations -- 11. The heredity hypothesis -- REFERENCES.

The Vibrations of a Drum with Fractal Boundary -- 1 INTRODUCTION -- 2 AN EXAMPLE: VIBRATION OF A CORD -- 3 VIBRATIONS OF A DRUM -- WEYL'S ESTIMATE -- 4 CAN ONE HEAR THE SHAPE OF A DRUM? -- 5 DOMAINS WITH FRACTAL BOUNDARIES -- 6 HEAT EQUATION ON THE TRIADIC VON KOCH SNOWFLAKE. -- REFERENCES -- Intermediate States: Some Nonclassical Properties -- 1. Introduction. -- 2. Even and odd coherent states. -- 3. The Binomial states. -- 4. The phased generalized binomial state. -- 5 Generalized geometric state. -- 6 Even geometric states. -- 7 Conclusion. -- References -- On the Relativistic Two-Body Equation -- 1 Introduction: -- 2 Derivation of the two-body equation: -- 3 Solutions: -- 4 References -- Singularities in General Relativity and the Origin of Charge -- 1 Introduction -- 2 d-spaces -- 3 A simple example -- 4 Schwarzschild space-time -- 5 The closed Friedmann Universe -- 6 The origin of charge -- References -- The Inner Geometry of Light Cone in Godel Universe -- 1 Introduction -- 2 Structure of Light Cone in Godel Universe -- 3 The Inner Geometry of Light Cone -- REFERENCES -- List of Participants.