Mathematics and the 21st Century

The following sections are included:

• Organizing Committee

• Preface

• Contents

The following sections are included:

• 1000 YEARS OF MATHEMATICS

The following sections are included:

• Trend 1: From the Linear Model to the Dynamic Model of Research

• Trend 2: From Theory + Experiment, to Theory + Experiment + Computation

• The ozone hole

• Kepler's sphere packing conjecture

• Theoretical computer science

• Trend 3: From Disciplinary to Interdisciplinary Research

• The life sciences

• Infectious diseases

• Trend 4: Complementing Reductionism with the Study of Complex Systems

• Trend 5: Globalization and the Diffusion of Knowledge

• Some challenges

• The culture of "pure" mathematics

• Conclusion

When one is speaking about mathematics in Arabic from 9th century on, it is difficult to avoid understatement; it is difficult to do full justice to one's subject. Even an uncommmonly armed and industrious mind could scarcely produce the content of a large number of lost works, and a huge number of mathematical arabic manuscripts, neither yet edited nor studied. In spite of this situation, the results obtained in the last decades show easily that, if we take away the contribution of arab mathematicians, we will be unable to understand classical mathematics, I mean mathematics until the end of the 17th century. The old story about mathematics beginning in Greece and renewed in Renaissance Europe, with arab mathematicians as agents of transmission of greek legacy, does not fit if we are aware to get the facts right…

With the advent of the third millennium, the mathematics education institutions seem to be in a "tempestuous" zone. While there is a great progress in mathematics as a discipline and as a recognized effective tool for the advancement in science and technology to the extent that high technology is considered as a mathematical technology (David, 1984), there is a distress and dissatisfaction with mathematics education in terms of its content, pedagogies and delivery system at all levels. In general there are poor outcomes in spite of the rich mandated objectives…

The following sections are included:

• Introduction

• Foundation

• Some Developments, Results and Problems

• References

The aim of this expository talk is to explain some aspects of moduli problems both classical and modern. Moduli problems occur naturally in several areas of Mathematics, like Algebraic Geometry, Differential Geometry, Number theory, Partial Differential Equations and in Theoretical Physics. Indeed techniques from all these areas are used to study moduli problems…

The following sections are included:

• The beginning

• The 3264 conies of Chasles

• Schubert Calculus

• Hilbert's 15^th Problem

• Cohomology

• Homogeneous spaces

• Spherical varieties

• Strings and Moduli spaces

• Witten Conjecture Proved by Kontsevich

• Quantum cohomology

• Equivariant Theories

I give a short survey of soliton theory since the time of its creation in the late sixties and early seventies of the last century: mainly I here focus on the discovery of solitons in the dynamics of nonlinear optics - particularly reference optical solitons of the so-called Maxwell-Bloch equations in one time and one space (1+1) dimensions. These M-B equations include the sine-Gordon (s-G) and nonlinear Schrödinger (NLS) equations as final products under a succession of slowly varying envelope and phase approximations. These SVEPA's preserve the exact integrability of each family of nonlinear partial differential equations in 1+1 dimensions, and each family can be quantised and exactly solved as completely integrable quantum systems. Both the quantum s-G and the quantum NLS in 1+1 dimensions could prove to be natural candidates for applications in the newly developing science and technology of 'quantum information' including 'quantum computing': the attractive case of the quantum NLS is already a centre of interest in optical fibres. Neither the quantum nor the classical NLS equations are integrable in two and three space dimensions (i.e. in 2+1 and 3+1 dimensions). But, as a part of the theme of the paper illustrating the interplay of abstract mathematics, mathematical physics, and experimental physics and technologies, I show that the weakly coupled quantum NLS equations in 3+1 dimensions in particular find direct application to the theory of Bose condensates observed in 1995 in metal vapours held in magnetic traps at temperatures of nano-degrees Kelvin. Here approximated functional integral methods give a good description of these Bose condensates, but so far only for the repulsive case: predictions for the correlation functions in this repulsive case are in good agreement with recent experiments on magnetically trapped ⁸⁷Rb vapour held in very elongated traps in the temperature range 250-450 nano-Kelvin. Although these experiments rather confirm that these quantum condensates in 3+1 dimensions are in exact quantum coherent states asymptotically, nevertheless they must also confirm that these states are states of 'quantum chaos'. This is because the semiclassical equations in 3+1 dimensions are not integrable and so represent very large dimensional Hamiltonian chaos. Even so the quantum coherent states in 3+1 dimensions may herald still another new technology based on an atom, rather than a photon, laser.

We present some concepts within the area of dynamical systems which have been extended to non-smooth differential equations. These include the definition of Lyapunov exponents, extension of Conley-index or KAM-theory, an adaption of the Melnikov-technique for the detection of chaos and an approach to generalize Hopf bifurcation.

The following sections are included:

• Introduction

• Structure theorems

• Closure operations on classes of rings

• Ring constructions

• Radical theory in other categories

• Cardinality condition

• References

Many authors have investigated the structure of a finite group G under the assumption that all subgroups of G of prime order are well-situated in G. The aim of this talk is to introduce these investigations.

The following sections are included:

• Introduction

• Totally permutable products

• Mutually permutable products

• References

In this talk we study asymptotic behavior of solutions of abstract equations of the form (*) Bɸ(t) = Aɸ(t) + ψ(t) for t ∈ J, where A is the generator of a C₀–semigroup of operators on a Banach space X, ψ : J → X is continuous, Bɸ = ɸ′ and J ∈ {ℝ⁺, ℝ}. However, our treatment applies to more general operators A, B and to more general groups or semigrops J. The following are the main tools in our investigations. 1- Harmonic analysis: we itroduced the notion of a spectrum of a function ɸ with respect to a class and the resonance set where θ_B is the characteristic function of B. It contains the Beurling specrtra of all solutions of the homogenuous equation Bɸ = A ο ɸ. 2-Ergodicity. The notion of ergodicity is developed to include unbounded functions. Using ergodicity we developed a criteria for a function to be an element of a class . 3- Extension of the known classes to Mean classes. This enabled us to study not only uniformly continuous bounded solutions but also measurable bounded solutions. These tools are used by the speaker for the study of the abstract Cauchy problem and developed jointly with A. Pryde (Monash University , Australia) and Hans Günzler (Kiel University, Germany) to include more general equations.

This paper is concerned with the most general water wave equations in (3 + 1) dimensions. Special attention is given to the derivation of the (1 + 1)-dimensional forced Korteweg-de Vries (fKdV) equation and the (1 + 1)-dimensional forced nonlinear Schrödinger (fNLS) equation near resonant conditions. Using the multiple scale technique, the (1 + 1)-dimensional nonlinear Schrödinger (NLS) equation for the amplitude function A₀ of wake packets is derived. Section 6 deals with the fourth-order nonlinear Schrödinger equation for the amplitude of the wave potential leading to a major improvement in agreement with Longuet-Higgins analysis (1978a, b). A stability analysis is made based on the simplified version of the Dysthe's (1979) NLS equation. This is followed by Hogan's analysis of the fourth-order evolution equation for capillary-gravity waves in deep water. The final section deals with the Davey-Stewartson equations and the Kadomtsev and Petviashvili equation in water of finite depth.

We consider free convection near a semi-infinite vertical flat plate. This problem is singularly perturbed with perturbation parameter Gr, the Grashof number. Our aim is to find numerical approximations of the solution in a bounded domain, which does not include the leading edge of the plate, for arbitrary values of Gr ≥ 1. Thus, we need to determine values of the velocity components and temperature with errors that are Gr–independent. We use the Blasius approach to reformulate the problem in terms of two coupled non-linear ordinary differential equations on a semi– infinite interval. A novel iterative numerical method for the solution of the transformed problem is described and numerical approximations are obtained for the Blasius solution functions, their derivatives and the corresponding physical velocities and temperature. The numerical method is Gr–uniform in the sense that error bounds of the form C_pN^-p, where C_p and p are independent of the Gr, are valid for the interpolated numerical solutions. The numerical approximations are therefore of controllable accuracy.

The growth of a second derivative of the logarithm of the maximum modulus of an entire function provides information about the location of zeros of the function and its sections. We present a survey of work on this topic along with some recent sharp results and open questions.

Linear preserver problems are questions about characterising linear maps on spaces of matrices or spaces of operators (or more generally on rings or algebras) that preserve certain properties. We present an exposition of three such problems on preserving invertibility or commutativity or rank one.

Prediction is reviewed and the most recent advances in the area are presented.

An objective of this paper is to study the Bayesian multisample prediction and give a concise form for the predictive density function of the observable in sample j based on the informative sample(s).

Applications are shown to a general class of population distributions which specializes to a wide spectrum of life testing distribution models. The uncertainty about the true value of the parameter(s) is measured by a general class of prior density functions.

By deriving two recurrence relations which express the single and double moments of order statistics from a symmetric distribution in terms of the corresponding quantities from its folded distribution, Govindarajulu (1963, 1966) determined means, variances and covariances of Laplace order statistics (using the results on exponential order statistics) for sample sizes up to 20. He also tabulated the BLUE's (Best Linear Unbiased Estimators) of the location and scale parameter of the Laplace distribution based on complete and symmetrically Type-II censored samples. In this paper, we first establish similar relations for the computation of triple and quadruple moments. We then use these results to develop Edgeworth approximations for some pivotal quantities which will enable one to develop inference for the location and scale parameters. Next, we show that this method provides close approximations to percentage points of the pivotal quantities determined by Monte Carlo simulations. Finally, we present an example to illustrate the method of inference developed in this paper.

The following sections are included:

• Introduction

• Resource

• Generalized Sedyakin's model

• Additive accumulation of damages model

• Proportional hazards model

• Generalized proportional hazards models

• The main classes of GPH models.

• Parametrization of the function r in AAD and GPH models

• Changing shape and scale models

• Generalizations

• The heredity hypothesis

• References

Let Ω be a bounded domain in ℝ^N. We study the eigenvalues of the Dirichlet Laplacian defined on the domain Ω. There exists a countable sequence of eigenvalues. Their asymptotics are related to the geometry of the domain. We recall the results established during the previous century concerning the link between the geometry of the domain and the asymptotics of the eigenvalues; we try to answer M.Kac's question "Can one hear the shape of a drum?" especially in the case of domains with fractal boundary.

In this article we consider in some detail some new classes of states. These states are intermediate states either between the pure number (Fock) states, and the (non-pure) chaotic state (thermal state), such as geometric state, or between the coherent state and number state such as binomial state. We extend our discussion to include some other states such as even (odd) coherent states, even (odd) binomial states, phased generalized binomial state … etc. In our study of these states we pay attention to a discussion of the nonclassical properties, besides the statistical properties, for example correlation functions, squeezing, and quasiprobability distribution functions (P-representation, W-Wigner, and Q-function). Furthermore we consider the field distribution and the photon number distribution, as well as the phase properties. Finally, some schemes for the production of these states are presented.

Recently, an exact two body covariant wave equation has been derived from the field theory of coupled Maxwell Dirac equations (Barut & Komy, 1985). It involves only one common invariant center - of - mass time τ. It takes full account of Spain and recoil corrections of both particles. The equation that also includes self-energy effects in a non-linear fashion, is a16-component spinor equation for two Dirac particles.

Working directly form this equation, energy eigen values and eigen functions are calculated to order α² and α⁴, where α is the fine structure constant. The eigen values agree with those determined previously by perturbative techniques, inclding relativistic, recoil and spin corrections, for all the energy levels of orthopositronium. The eigen value problems that arise are of Sturm-Liouville type, but involve two or four coupled, second - order differential equations in the radial variable r, with up to four singular points. In this approach approximated eigen values are determined directly, and the corresponding approximate eigen functions are obtained (for the first time) in simple closed form.

d-spaces are a simple and very useful tool for the description of singularities in General Relativity. In the first part of this paper, we recall the basic definitions of the theory of d-spaces. Then a short review of the results is presented, which were obtained with this theory. It is shown that there are situations, where pointlike particles can pass through singularities. This is the case e.g. for the classical Big Bangs of a series of closed Friedmann universes. In Schwarzschild's solution, the problem of causality violation near the White Source can be solved, although some mild form of causality violation remains. Finally, the first example of a "wormhole" without exotic matter is presented. This means that an electric field proportional to 1/r² is generated only by topology without any electric charge.

The following sections are included:

• Introduction

• Structure of Light Cone in Godel Universe

• The Inner Geometry of Light Cone

• References

The following sections are included:

• List of Participants