Summer School/Summer Seminar in Applied Analysis

Summer School / Summer Seminar in Applied Analysis City University of Hong Kong July 3-13, 2001 Abstracts On q-Di®erence Equations Moemen H. Abu-Risha Department of Mathematics Cairo University, Eygpt E-mail: moemenha@math-sci.cairo.eun.eg We give an investigation of some properties of solutions of linear q¡di®erence equations de¯ned on an interval [a; 1): The cases a = 0 and a > 0 are essentially di®erent. The study of the latter case is necessary for the derivation of asymptotic formulae and oscillation criteria for solutions near 1: In the case a > 0; some q¡di®erence equations share with the delay di®erential equations the property that a set of initial conditions at the point a is insu±cient to guarantee the uniqueness of the solution. A set of initial functions de¯ned on an interval left to a is needed. q-Sturm-Liouville Problems and Applications Mahmoud Annaby Department of Mathematics Cairo University, Eygpt E-mail: annaby@math.la.asu.edu In this talk we discuss some problems concerning q¡di®erence equations. An existence and uniqueness theorem based on successive approximations will be introduced. The n th order linear equation is also considered and the construction of a fundamental system of solutions is developed. The general q¡type Wronskian as well as a Liouville type formula will be given. The self adjoint Sturm-Liouville problem is de¯ned and its properties are given. A detailed example will be presented with an application in sampling theory. The work presented in this talk is taken from some projects with di®erent colleagues. Richard Askey Department of Mathematics University of Wisconsin-Madison, USA E-mail: askey@math.wisc.edu 1 First talk| A Maximum Problem and the Integrals it leads to Over 100 years ago, T.J. Stieltjes formulated and solved a beautiful maximum problem about the location of charges on an interval when there are ¯xed charges at the end of the interval and there is a repulsive force between the particles. The solution leads to a classical di®erential equation, and to the problem of ¯nding the discriminant of a set of polynomials. Later work by A. Selberg allows one to complete this solution in a di®erent way, by evaluating a multidimensional beta type integral. This integral was extended by K. Aomoto, whose evaluation of it is simple enough to give in a lecture. Second talk| More Gamma and Beta Integrals and Orthogonal Polynomials The classical integrals of Euler and Cauchy for gamma and beta functions have now been extended in a number of ways. Some of these extensions will be given, and the connections with orthogonal polynomials described. These include polynomials on the real line and also polynomials on the unit circle. On Value Distribution Theory, Complex Oscillation of Di®erential Equations and Bessel Polynomials Yik Man Chiang Department of Mathematics Hong Kong University of Science and Technology, Hong Kong E-mail: machiang@ust.hk In this talk, we establishes a connection between the oscillation theory of certain ordinary di®erential equations with periodic coe±cients (under the description of Value Distribution theory of meromorphic functions in the complex plane) and certain special functions. In particular, we solve a problem of ¯nding the entire solutions that admit few zeros of certain di®erential equations with periodic coe±cients. Our results not only give new proofs of the results of Bank and Laine (J. Reine Angew. Math. 344 (1983), 1-21) for the representations of those solutions with few zeros, we even obtain a fairly complete description of the representations for those solutions with many zeros, in terms of special functions for the ¯rst time. The most interesting feature of this work is perhaps the discovery of the seemingly unrelated Bessel polynomials actually play a major role in the description of the solutions mentioned above. This is a joint work with Mourad E. H. Ismail The Stieltjes-Wigert and q-Laguerre Moment Problems Jacob Stordal Christiansen Department of Mathematics University of Copenhagen, Denmark E-mail: stordal@math.ku.dk 2 The moment problems associated with the Stieltjes-Wigert polynomials and the qLaguerre polynomials are examples of indeterminate moment problems. After a brief presentation of the well-known solutions we shall study a transformation of the set of solutions, which turns out to ¯x all the classical solutions but not the N-extremal solutions. Based on generating functions, expressions for the four entire functions from the Nevanlinna parametrization will be given. The connection between the two moment problems will be discussed; we show how to come from q-Laguerre solutions to Stieltjes-Wigert solutions by letting the parameter alpha tend to in¯nity, and we explain how to lift a Stieltjes-Wigert solution to a q-Laguerre solution at the level of Pick functions. Path Integrals in Financial Mathematics Omar E. Foda Department of Mathematics & Statistics University of Melbourne, Australia E-mail: foda@ms.unimelb.edu.au I would like to outline an application of a basic tool of mathematical physics, namely path or functional integrals, to a problem in industrial mathematics, namely pricing ¯nancial derivatives. On the one hand, ¯nancial derivatives are common contracts that are frequently used, in large quantities, by almost all industries and ¯nancial institutions. On the other hand, the accurate pricing of these contracts is a very technical problem that requires non-trivial mathematics. Ito calculus and martingales are currently the tools of choice used in pricing derivatives. I would like to argue that path integrals o®er an approach to pricing derivatives, that is natural, intuitive, technically less demanding than, and at least as powerful as the currently used methods. More importantly, path integrals could apply to cases where the linearity assumptions that are necessary for Ito calculus to apply are no longer valid. This is not an original proposal, and I would like to review the currently, very limited, known applications of path integrals to derivative pricing theory. Furthermore, if time allows, I would like to say something about a certain class of derivatives, namely Arithmatic average Asian options, whose valuations require non-trivial use of special functions. Dirichlet Series in Approximation Theory and Applications in Signal Analysis Brigitte Forster Zentrum Mathematik Technische UniversitÄat MÄ unchen, Germany E-mail: forsterb@ma.tum.de 3 Dirichlet series on convex polygons D are a generalization of the classical Fourier series. In the talk we give new direct and inverse theorems with k-th moduli of smoothness for the approximation of functions f 2 AC(D) with Dirichlet series X f (z) = ·f (¸n )e¸n z ; z 2 D: n2Z In the proofs the connection between the Dirichlet coe±cients ·f (¸n ) and the Fourier coe±cients of certain functions in C(T) is used. Furthermore we show an application of Dirichlet series for signal analysis. Ä chenig Karlheinz Gro University of Connecticut, USA and Department of Mathematics, University of Vienna, Austria E-mail: karlheinz.groechenig@univie.ac.at First talk| Weyl-Heisenberg-Frames: Theory and Applications in Signal Analysis signal analysis and in data transmission series expansions of the form f (t) = P Both in2¼iblt c e g(t ¡ ak), so-called Gabor expansions, are used. Usually the coe±cients k;l2Z kl ckl are determined by means of a dual window °. For most applications it is essential to construct pairs of windows (g; °) that are both well localized simultaneously in time and frequency. We will present new results on the problem of window design and discuss applications to signal compression and the design of pulse shapes in OFDM and BFDM systems. Second talk| Pseudodi®erential Operators and Modulation Spaces Modulation space norms are natural measures to quantify the time-frequency concentration of functions. We present these spaces of functions and distributions as natural symbol classes for proving boundedness results and trace class results for pseudodi®erential operators and give some extensions of the theorem of Calderµon and Vaillancourt. The Inversion Problem for Low Energy Probes: Nonlinear Problems in Di®use Tomography Änbaum F. Alberto Gru Department of Mathematics University of California at Berkeley, USA E-mail: grunbaum@math.berkeley.edu A description of the material in the course It is important to realize that all the existing medical imaging devices available at many hospitals (like CT scanners, MRI , PET,...) and which have trully revolutionized the practice of diagnostic medicine, depend on the ultrafast solution of very large systems 4 of linear equations. In all cases the mathematics that underlies these algorithms is based on some kind of linear theory, with the inversion of the Radon transform being a good example. Some new inroads in terms of interventionist radiology are becoming available in the form of "brain attack teams". It is amazing that a math-physics theory that essentially ignores scattering has produced such remarkably useful images. Optical, or di®use tomography, refers to the use of extremely low energy probes, like an infrared laser, to obtain images of highly scattering media. In this very young ¯eld one strives for the imaging of both attenuation and scattering characteristics of the medium being probed. This represents an emerging area for the application of new and old mathematical tools to important problems in medical imaging. It is our contention that this poses important mathematical challenges dealing with the solution of large systems of nonlinear equations. In this set of lectures we discuss an extremely simpli¯ed model, almost a caricature, of the real physical problem. The point of this model is that it captures in many ways the essential features of more realistic ones and it allows us to discuss all relevant issues. In particular I want to stress that in this case we will be able to give a completely rigorous discussion of all issues involving nonuniqueness, explicit inversion formulas, etc. A variety of mathematical models of the underlying physics are reviewed in [A1] including the ones considered in [G1], [GP1], [GP2], [GP3], [G4], where we have discussed both the \direct", as well as the more interesting \inverse problem" for a discrete model. The time evolution of the system is governed by a Markov chain with discrete state space and discrete time. The one step transition probability matrix is denoted by P . Some of the states are \incoming", some are \outgoing", and the rest are \hidden" states. The ¯rst two types correspond to sources and detectors respectively, and the hidden ones represent the state of a photon while it moves in the interior of the object. The simplest direct problem consists of going from P to the \input-output" matrix, namely a matrix whose rows and columns are labelled by the incoming and outgoing states respectively and whose entries are the probabilities of eventually making it from any input to any output state. The inverse problem consists of recovering as much of P as possible from this \input-output" matrix. For any source-detector pair one can use, given the present state of technology, not only the total photon count as described above, but also \time-of-°ight" information consisting of some information about the distribution of the arrival time for photons for any source-detector pair. This extra information was not used in [G1], [GP1], [GP2], [GP3], in large part because this kind of measurement was not available at the time. In other words, we will consider here a more complex direct problem and its corresponding inversion. In these notes we manage to reduce the nonlinear inversion problem in question to the solution of a ¯nite number of linear problems. I think that one of many mathematical challenges raised by these new developments is to understand the underlying conceptual picture that makes this possible. The immediate payo® of this reduction to an equivalent ¯nite number of linear systems is that we can produce explicit solutions. Explicit results that use only photon count had been given earlier, but those that use time of °ight information are given here for the ¯rst time. Some of the discussion here is adapted from [G4] and [GM]. Some results in the three dimensional case are given in [G3]. The main motivation for this line of work is, as mentioned above, recent work in di®use or optical tomography, i.e., the inverse problem resulting form the use of very low energy sources like an infrared laser to obtain images of diagnostic value. Presently this is used 5 in a neonatal clinic to measure oxigen content in the brain of premature babies as well as in the case of repeated mammographies. With the development of highly speci¯c markers that respond well in the optical or infrared region there are many potential applications of this emerging area, [S]. For a very nice and up-to-date discussion of work in this area one can see [A1], [A2]. It is also very worthwhile looking into [D], [NW]. We state now the main result to be given in these notes: for a small but very general model we will see that the zeroth and ¯rst moments of the time of °ight distribution determine the matrix P up to the choice of an arbitrary diagonal matrix D. Explicit formulas will be provided too. It will turn out that no extra information about time of °ight will resolve the underlying nonuniqueness given by the matrix D. The only way to make further progress in this inverse problem, and try to get a unique solution, is to use some extra physical information in the form of restrictions on the unknowns of the problem. In particular any restriction that would determine the arbitrary diagonal matrix D would give a unique solution. An example of possible physical conditions like \microscopic reversibility" was used in [GP2]. The case of \isotropic scattering" was considered in [G2] and [GZ]. The approach pursued here is that of analyzing the structure of the mathematical equations resulting from a certain interconnection network among di®erent states. At a later stage, and depending on the relevant physical situation some extra information can be thrown in. For systems larger in size that the one discussed here one should use some of the ideas in [P1] and [P2]. A1 Arridge, S., Optical tomography in medical imaging, Inverse Problems 15 (1999), R41{R93. A2 Arridge, S., and Hebden, J. C., Optical imaging in medicine: II. Modelling and reconstruction, Phys. Med. Biol. 42 (1997), 841{853. D Dorn, O., A transport-backtransport method for optical tomography, Inverse Problems 14 (1998), 1107{1130. G1 GrÄ unbaum, F. A., Tomography with di®usion, in \Inverse Problems in Action", P. C. Sabatier (ed.), Springer-Verlag, Berlin, pp. 16{21. , Di®use tomography: The isotropic case, Inverse Problems 8 (1992), G2 409{419. G3 , Di®use tomography: Venturing into dimension three. , Di®use Tomography: Using time-of-°ight information in a two-dimensional G4 model GM GrÄ unbaum, F. A., and Matusevich,L.F., Explicit inversion formulas for a model of di®use tomography, submitted for publication April 2001. GP1 GrÄ unbaum, F. A., and Patch, S., The use of Grassmann identities for inversion of a general model in di®use tomography, Proceedings of the Lapland Conference on Inverse Problems, Saariselka, Finland, June 1992. GP2 , Simpli¯cation of a general model in di®use tomography, in \Inverse Problems in Scattering and Imaging", M. A. Fiddy (ed.), Proc. SPIE 176, 744{754. 6 GP3 , How many parameters can one solve for in di®use tomography?, Proceedings of the IMA Workshop on Inverse Problems in Waves and Scattering, March 1995. GZ GrÄ unbaum, F. A., and Zubelli, J., Di®use tomography: Computational aspects of the isotropic case, Inverse Problems 8 (1992), 421{433. NW Natterer, F., and Wubbeling, F., Mathematical methods in image reconstruction, SIAM (2001). P1 Patch, S., Recursive recovery of a family of Markov transition probabilities from boundary value data, J. Math. Phys. 36(7) (July 1995), 3395{3412. P2 , A recursive algorithm for di®use planar tomography, Chapter 20 in \Discrete Tomography: Foundations, Algorithms, and Applications", G. Herman and A. Kuba (eds.), Birkhauser, Boston, 1999. S Schotland, J. . Private communication. Orthogonal Polynomials Mourad E. H. Ismail Department of Mathematics, University of South Florida, USA and Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong, Hong Kong E-mail: ismail@math.usf.edu or ismail@math.cityu.edu.hk This is introductory material to the theory of classical orthogonal polynomials including the Askey-Wilson polynomials. A Riemann-Hilbert Method Alexander R. Its Department of Mathematics Indiana University Purdue University Indianapolis, USA E-mail: itsa@math.iupui.edu The Riemann-Hilbert approach of the theory of integrable systems will be presented. Applications of the method to the asymptotic problems arising in the theory of orthogonal polynomials and random matrices will be discussed in detail. The speci¯c topics of the lectures include: 1. The setting and the elements of general theory of Riemann-Hilbert factorization problems. Relations to the inverse scattering problem for di®erential operators and to the inverse monodromy problem for Fuchsian systems. Integrable nonlinear equations of the KdV and Painlev¶e types. 7 2. Integrable Fredholm operators and the Riemann-Hilbert approach to random matices, exactly solvable statistical mechanics models, and random permutations. 3. Isomonodromy and the Deift-Zhou nonlinear steepest descent asymptotic methods for oscillatory Riemann-Hilbert problems. Connection formulae for Painlev¶e equations. Asymptotics of the distribution and correlation functions of the random matrix theory and the theory of integrable statistical mechanics . 4. The Riemann-Hilbert approach to orthogonal polynomials. Semiclassical asymptotics of orthogonal polynomials and universality in the matrix model. The lectures will be based on the material presented in the books, 1. P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach, Courant Lecture Notes in Mathematics, New York University, 1999 2. A.R. Its, V.Yu. Novokshenov, The Isomonodromic Deformation Method in the Theory of Painleve Equations, Lect. Notes in Math. 1191, 313, Springer Verlag, Heidelberg (1986). and in the original papers 1. P. Bleher, A. Its, Semiclassical Asymptotics of Orthogonal Polynomials, RiemannHilbert Problem, and Universality in the Matrix Model, Ann. of Math. 150 (1999), 185 - 266. 2. P. A. Deift, Integrable Operators, in Di®erential operators and spectral theory: M. Sh. Birman's 70th anniversary collection, V. Buslaev, M. Solomyak, D. Yafaev, eds., American mathematical Society Translations, ser. 2, v. 189, Providence, RI: AMS, 1999. 3. P. A. Deift and X. Zhou, A Steepest Descent Method for Oscillatory RiemannHilbert Problems. Asymptotics for the MKdV equation, Ann. of Math. 137 (1995), 295-368. 4. P. A. Deift, A. R. Its, X. Zhou, A Riemann-Hilbert Approach to Asymptotic Problems Arising in the Theory of Random Matrix Models, and Also in the Theory of Integrable Statistical Mechanics, Ann. of Math. 146 (1997), 149-235 . 5. A.S. Fokas, A.R. Its and A.V. Kitaev, The Isomonodromy Approach to Matrix Models in 2D Quantum Gravity, Commun. Math. Phys., 395-430 (1992). 6. A. R. Its, Connection Formulae for the Painleve Transcendents, in the book: The Stokes Phenomenon and Hilbert's 16th Problem, B. L. J. Braaksma, G. K. Immink, and M. van der Put, eds., World Scienti¯c, Singapore, pp. 139-165 (1996). 7. A. R. Its, A. S. Fokas, and A. A. Kapaev, On the Asymptotic Analysis of the Painleve Equations via the Isomonodromy Method, Nonlinearity 7 (1994), 1291-1325. 8. A.R. Its, A.G. Izergin, V.E. Korepin, N.A. Slavnov, Di®erential Equations for Quantum Correlation Functions, J. Mod. Phys. B, 1003, (1990); 8 (Mostly, Deift's book and papers [1], [2], [4], [6] will be used.) Only a standard background in complex analysis and in the basic functional analysis is needed in order to follow the lectures. All the necessary speci¯c facts concerning general Riemann-Hilbert theory, the theory of integrable systems, and the theory of orthogonal polynomials will be explained. The relevant general references are: [1]. M. J. Ablowitz, H. Segur, Solitons and inverse scattering transform, SIAM Stud. Appl. Math., SIAM, Philadelphia, 1981 [2]. R. Beals, P. A. Deift, and C. Tomei, Direct and inverse scattering on the line. Mathematical Surveys and Monographs, 28, AMS Providence, Rhode Island, 1988 [3]. K. Clancey and I. Gohberg, Factorization of matrix functions and singular integral operators. Operator Theory. 3, BirkhÄauser Verlag Basel, 1981 [4]. L. D. Faddeev, L. A. Takhtajan, Hamiltonian methods in the theory of solitons, Berlin-Heidelberg: Springer-Verlag, 1987 [5]. G. Litvinchuk and T. Spitkovskii, Factorization of measurable matrix functions. BirkhÄauser Verlag, Basel, 51, 1987 [6]. M. L. Mehta, Random Matrices, 2nd ed., Academic Press Inc., Boston, 1991. [7]. N. I. Muskhelishvili, Singular Integral Equations. Nordho® N. Y. Groningen , 1953 [9]. V.E. Zakharov, S. V. Manakov, S. P. Novikov, L. P. Pitaevskii, The theory of solitons. The method of inverse scattering, Moscow, Nauka, 1980 [10]. X. Zhou, Riemann-Hilbert problem and inverse scattering, SIAM J. Math. Anal. 20, 966{986, 1989. Lax Pair and the Riemann-Hilbert Method for Solving Scattering Problems in Geophysics Elizabeth Its Department of Mathematical Sciences Indiana University Purdue University Indianapolis, USA New Riemann-Hilbert method was suggested recently by A. Fokas for studying boundary problems for linear and integrable nonlinear PDEs. We extend this approach for solution of the vector elastodynamic equation in inhomogeneous geomaterials. Scattering of Rayleigh wave in an elastic quarter space is considered. The suitable Lax pair formulation of the elastodynamic equation is obtained. The integral representations for the solutions of the Lax pair equations are found. This reduces the problem to the analysis of certain Riemann-HIlbert problems. The results of this analysis, which uses the apparatus of the Jacobian elliptic functions will be reported at the meeting. 9 Darboux Transformation and Orthogonal Polynomials Kil H. Kwon and Gang J. Yoon Division of Applied Mathematics KAIST, Taejon 305-701, Korea E-mail: fkhkwon, ykjg@jacobi.kaist.ac.kr For quasi-de¯nite moment functionals ¾ and ¿ , we ¯nd the relations between the corresponding orthogonal polynomials when ¾ = (x ¡ c)¿ . Let fPn (x)g1 n=0 be a classical OPS relative to ¾, then we show that an OPS fQn (x)g1 relative to ¿ is obtained n=0 by means of a Darboux transformation of an in¯nite matrix induced by the three term 1 coe±cients of fPn (x)g1 n=0 if and only if fQn (x)gn=0 satis¯es a Darboux transformation ~ = L1 L2 of the operator L = L1 L2 which has fPn (x)g1 L n=0 as a family of eigenfunctions. In this case, there are two polynomials r(x) and s(x) such that r(x)¾ = s(x)¿ and r(x)s(x) = ®r (x) for some integer r ¸ 0 and some polynomial ®(x) of deg(®) · 2: And we ¯nd a necessary condition in order that fQn (x)g1 n=0 relative to ¿ = ¼(x)¾ for some polynomial ¼(x) satis¯es a di®erential equation of ¯nite order. These result explain the reason that Dirac delta functions and its derivatives appear at the zero of ®(x) in the form of the moment functional relative to which the polynomials obtained by Darboux processes are orthogonal. Polynomial Bases on the Sphere ¶ndez Noem¶³ La¶³n Ferna Institute of Mathematics Medical University of LÄ ubeck, Germany E-mail: fernande@math.mu-luebeck.de Topic of many areas of investigation, like meteorology or crystalography, is the reconstruction of a continuous signal on the sphere by scattered data. A classical approximation method is Polynomial Interpolation: Let Vn denote the set of polynomials of degree · n on S2 . It is known that Vn = n M Harmk (S2 ); k=0 so that dim Vn = n P dim Harmk ( S2 ) = k=0 n P 2k + 1 = (n + 1)2 k=0 (Harmk ( S2 ) stands fot the space of homogeneous harmonic polynomials of degree k). The so-called Spherical Harmonics form an orthonormal basis of the space Vn . As these functions are not localized around any point, we try to ¯nd better localized bases: For this purpouse let » 2 S2 . We de¯ne the scaling function '(»; ¢) by '(»; ¢) = n X 2k + 1 4¼ k=0 10 Pk (»¢); where Pk ist the Legendre polynomial of degree k. Using the Addition Theorem for the spaces Harmk (S2 ) it is easy to show, that the scaling function '(»; ¢) is a reproducing kernel for Vn and that it is localized around the point ». Our problem reduces therfore to the localization of constellations of (n + 1)2 points on the sphere, which yield well-conditioned bases for our space Vn . Applied Harmonic Analysis Rupert Lasser Institut fr Biomathematik und Biometrie GSF - Forschungszentrum fr Umwelt und Gesundheit, Germany E-mail: lasser@gsf.de The ¯rst three chapters are an introduction into the theory of discrete hypergroups and their harmonic analysis. The fourth chapter gives concrete examples of polynomial hypergroups, whereas chapter ¯ve and six deal with applications in form of approximation results respectively in form of time series analysis. The course in \Applied Harmonic Analysis" starts with chapters 4, 5 and 6. The students then should be more or less familiar with the ¯rst three chapters. Contents 1. Discrete hypergroups 2. Discrete commutative hypergroups 3. Harmonic analysis 4. Polynomial Hypergroups 5. Orthogonal polynomial expansions 6. Time series homogeneous with respect to polynomial hypergroups Orthogonal Polynomials of Discrete Variables and Lie Algebras of Matrices of Complex Size Dimitry Leites Department of Mathematics University of Stockholm, Sweden E-mail: mleites@matematik.su.se Classical orthogonal polynomials like sines, cosines and more sophisticated Chebychev and Hahn's polynomials, as well as their q-analogs (whatever this is), have so numerous applications that even schoolchildren know about some of them. These applications make the study of the properties of these polynomials vital. In the talk which REQUIRES NO 11 PRELIMINARY KNOWLEDGE, except how to di®erentiate a polynomial and what the trace of the matrix is, I will show a uniform way to introduce all the classical orthogonal polynomials. In particular, I will show how to get Chebychev and Hahn's polynomials in terms of the traces of nxn matrices. Particularly interesting is the case when n is a complex number! This method allows one to obtain multidimensional orthogonal polynomials in a most natural way. (Observe also that generalizations of Lie algebra of matrices of complex size provide us with a natural generalization of Korteveg-deVries hierarchy and appear as symmetry algebras in unconventional Supergravity Models of the Grand Uni¯cation Theory which allow particles of spin greater than 2.) Integration of the Soliton Hierarchy with Self-consistent Sources Runliang Liny , Wen-Xiu Maz and Yunbo Zeng] y] Department of Mathematical Sciences, Tsinghua University, P. R. China z Department of Mathematics, City University of Hong Kong E-mail: y rlin@math.tsinghua.edu.cn, z mawx@cityu.edu.hk ] yzeng@mail.tsinghua.edu.cn The evolution of the eigenfunctions in the Lax representation of the soliton hierarchy with self-consistent sources possesses singularity. By proposing a method to treat the singularity to determine the evolution of scattering data, the KdV hierarchy with selfconsistent sources, the AKNS hierarchy with self-consistent sources, and the Kaup-Newell hierarchy with self-consistent sources are integrated by the inverse scattering method. The soliton solutions of these equations are obtained. It is shown that the insertion of a source may cause the variation of the speed of soliton. This approach can be applied to solve all other (1+1)-dimensional soliton hierarchies. Stabilization and Controllability for the Transmission Wave Equation Weijiu Liu Department of Mathematics and Statistics Dalhousie University, Canada E-mail: weiliu@mscs.dal.ca In this talk we address the problem of control of the transmission wave equation. In particular, we consider the case where, due to total internal re°cetion of waves at the interface, the system may not be controlled from exterior boundaries. We show that such a system can be controlled by introducing both boundary control along the exterior boundary and distributed control near the transmission boundary and give a physical explanation why the additional control near the transmission boundary might be needed for some domains. 12 A New Chaotic Attractor and its Dynamical Behaviors Äz and Suochun Zhang] Guanrong Cheny , Jinhu Lu y Department of Electronic Engineering, City University of Hong Kong, Hong Kong z] Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, P. R. China E-mail: y gchen@ee.cityu.edu.hk, z] mathljh@china.com In 1963, Lorenz found the ¯rst chaotic attractor in a three-dimensional autonomous system [Sparrow, 1982]. In 1999, Chen found another chaotic attractor, also in a simple three-dimensional autonomous system, which is derived from the Lorenz system and has similarly simple structure but displays even more sophisticated dynamical behaviors [Chen & Ueta, 1999]. More interestingly, it has been found that the Lorenz and Chen systems are classi¯ed as two opposite classes of chaotic systems via a crucial condition set by Vanȩçek & Çelikovsk¶ y: The Lorenz system satis¯es a12 a21 > 0 while Chen system satis¯es a12 a21 < 0. Very recently, LÄ u et al. found a new chaotic system in [LÄ u et al., 2001], which therefore bridges the gap between the Lorenz and Chen systems, and represents the transition from one to the other. This paper is devoted to a more detailed analysis of this new chaotic attractor. Some basic properties, routes to chaos, bifurcations, topological structure, and periodic windows of the new system are studied either analytically or numerically. Also, the transition between the Lorenz attractor and Chen attractor through the new system is explored. References [1]. Lorenz, E. [1963] \Deterministic nonperiodic °ow", J. Atmos. Science, 20, 130-141. [2]. Chen, G. & Ueta, T. [1999] \Yet another chaotic attractor," Int. J. of Bifurcation and Chaos 9, 1465-1466. [3]. LÄ u, J., Chen, G. & Zhang, S. [2001] \A new chaotic attractor coined," Int. J. of Bifurcation and Chaos, to appear. [4]. LÄ u, J., Chen, G. & Zhang, S. [2001] \Dynamical analysis of a new chaotic attractor," Int. J. of Bifurcation and Chaos, to appear. Formal Solutions of Nonlinear Totally Characteristic PDE Zhuangchu Luo College of Mathematics Wuhan University, P. R. China E-mail: pde-g@whu.edu.cn In this paper, we calculate the formal Gevrey index of the formal solution of a class of nonlinear ¯rst order totally characteristic type partial di®erential equations with irregular 13 singularity in the space variable. We also prove that our index is the best possible one in a generic case. And we also consider the cases of higher order case. In this case, the formal solution belongs to a formal Gevrey class, and we also prove that our index is the best possible one in a class of PDE. On Singular Integral Operators in Orlicz Spaces Peide Liuy and Tao Meiz College of Mathematics Wuhan University, P. R. China E-mail: z m tao@163.net Let ©1 and ©2 be two nonnegative convex functions with ©1 (0) = ©2 (0) = 0; ª1 and ª2 be the complementary functions of ©1 and ©2 , respectively. In this paper, we establish a kind of (©1 ; ©2 ) type inequalities for Hardy's average operators. Using these inequalities, we prove the equivalence of the following statements for maximal singular integral operators T and maximal Hilbert operator H: Rt Rt (i) 0 Ã2s(s) ds · cÃ1 (ct) and 0 '1s(s) ds · c'2 (ct); 8t > 0; (ii) kT f k©1 · ckfk©2 , for all f 2 L©2 (Rn ; dx); (iii) Hf 2 L©1 (R+ ; dx) for all f 2 L©2 (R+ ; dx). where '1 ; '2 and Ã1 ; Ã2 denote the left continuous derivatives of ©1 ; ©2 and ª1 ; ª2 , respectively. Forced Oscillation of nth -Order Functional Di®erential Equations Chunhua Ou Department of Mathematics City University of Hong Kong, Hong Kong E-mail: 50002591@plink.cityu.edu.hk In this presentation we invesitigate the oscillation of forced functional di®erential equation x(n) (t) + a(t)f (x(q(t))) = e(t); when the forcing term e(t) is not required to be the nth derivative of an oscillatory function. Some new oscillatory criteria are given. We also apply our technique to the forced neutral di®erential equation of the form (x(t) + cx(t ¡ ¿ ))(n) + a(t)x(t) + b(t)x(t ¡ ¿ ) = e(t) + c(t)f1 (x(t)) + d(t)f2 (x(t ¡ ¿ )) 14 where xf1 (x) > 0 and xf2 (x) > 0 for x 6= 0; n ¸ 1; ¿; ± are nonnegative constants, a(t) > 0; b(t) > 0; c(t) > 0; d(t) > 0; which includes the special case f1 (x) = jxj¸ sgnx, f2 (x) = jxjµ sgnx, ¸ 6= 1 and µ 6= 1: Time-frequency-localization for Periodic Functions and Bases Ä rgen Prestin Ju Institute of Mathematics Medical University of LÄ ubeck, Germany E-mail: prestin@math.mu-luebeck.de It is the aim of these talks to present some ideas of time-frequency-localization and multiscale analysis for trigonometric polynomial spaces. Particularly we will focus on uncertainty principles for periodic functions. In a ¯rst part we study the connection between the Heisenberg principle for functions on the real line and a much more recent uncertainty principle for periodic functions stated by Breitenberger in 1983. It is proven that under certain smoothness conditions, the periodic uncertainty product of a periodized function converges to the real-line uncertainty product of the original function. These results are used to ¯nd asymptotically optimal sequences for the periodic uncertainty principle, based either on Theta functions or trigonometric polynomials obtained by sampling B-splines. As the second part, properties of certain polynomials which minimize the time localization measure of this uncertainty principle are investigated. In particular, it is shown how they can be used to form bases. This approach is closely related to the construction of periodic wavelets. It turns out that there is another uncertainty inbetween stability and localization. These results are joint work with E. Quak (SINTEF Oslo) and K. Selig, H. Rauhut (TU Munich). Feedback Stabilization of Rotating Timoshenko Beam with Adaptive Gain De-Xing Fengy and Dong-Hua Shiz y Department of Mathematical Sciences, Tsinghua University, P. R. China z Institute of Systems Science, Academy of Mathematics & Systems Science, Chinese Academy of Sciences, P. R. China E-mail: y dxfeng@iss03.iss.ac.cn, z dshi@math.tsinghua.edu.cn The problem of boundary feedback stabilization of rotating Timoshenko beam, arising from control of °exible robot arms, is studied in this paper. First, under gain adaptive direct strain feedback controls, a counterexample is given to show that the corresponding closed loop system is not asymptotically stable, which is contrary to traditional conjecture. 15 The counterexample given in this paper also exempli¯es an interesting result: a certain two second-order linear partial di®erential equations with ¯ve homogeneous boundary conditions have nontrivial solutions. Then, with an additional boundary feedback control, the related energy of the closed loop system is proved to be strongly stable, or more precisely, the con¯guration of the beam can be exponentially stabilized with some suitable nonlinear boundary feedback controls with adaptive gain. Di®erence Equations and Discriminants for Discrete Orthogonal Plamen Simeonov Department of Computer and Mathematical Sciences University of Houston-Downtown, USA E-mail: simeonov@math.usf.edu We prove that any set of polynomials orthogonal with respect to a discrete measure supported on equidistant points contained in a half line satisfy a second order di®erence equation. We also give a discrete analogue of the discriminant and give a general formula for the discrete discriminant of a discrete orthogonal polynomial. As an application we give explicit evaluations of the discrete discriminants of the Meixner and the Hahn polynomials. This is based on joint work with M. E. H. Ismail, and I Nikolova. Spectral Theory of Schrodinger Operators: An Introduction Barry Simon Department of Mathematics California Institute of Technology, USA E-mail: bsimon@bigfoot.com We'll discuss properties of spectral measures of continuum and discrete Schrodinger Operators, Borel transform methods, genericity of singular continuous spectrum, rank one perturbations and an introduction to Inverse Spectral Theory. Lecture 1: Introduction. We discuss continuum and discrete Schrodinger operators, spectral measures, Borel transforms and their relation to spectral measures, some examples. Lecture 2: Spectral measures and boundary behavior of the Borel transform Lecture 3: Baire genericity of sc measures, results on generic ac spectrum. Lecture 4: Rank One perturbations Lecture 5: An Introduction to Inverse Spectral Theory 16 Nonnegative Linearization of Orthogonal Polynomials and Group Representations Ryszard Szwarc Institute of Mathematics University of WrocÃlaw, Poland E-mail: szwarc@math.uni.wroc.pl The harmonic analysis on the unit circle essentially depends on the multiplicative property of the characters, or the product formula for the cosines 2 cos mµ cos nµ = cos(m ¡ n)µ + cos(m + n)µ: Studying harmonic analysis of compact matrix groups SO(n) (the unit circle SO(1) is the only commutative instance among them) leads to another class of special functions called Gegenbauer or ultraspherical polynomials. These functions show up naturally as matrix coe±cients of irreducible unitary representations. We will exhibit that these functions satisfy a certain nonnegative linearization formula, a counterpart of that for the cosines. It turns out that other orthogonal polynomial systems, not necessarily associated with groups, also can admit nonnegative linearization. We will give general condition when it may occur. Also we will show that basing on the nonnegative linearization one can build a convolution structure, similar to that of `1 (N); and even obtain some pointwise estimates for the polynomials. Multiple Othogonal Plynomials: Theory and Applications Walter Van Assche Department of Mathematics Katholieke Universiteit Leuven, Belgium E-mail: walter@wis.kuleuven.ac.be The concept of orthogonal polynomials is very well known and has many applications, such as least squares approximation, rational (Pad) approximation, Gauss quadrature, Lanczos method for computing eigenvalues, the Toda lattice, birth and death processes, just to name a few. Multiple orthogonal polynomials also go back at least a century and ¯rst came up in rational approximation (Hermite-Pad) of several functions simultaneousl. The orthogonality relations are now using several measures (or weight functions). We will de¯ne what we mean by multiple orthogonal polynomials (not to be confused with multivariate orthogonal polynomials: we are still using only one variable). We will present many examples and indicate how some properties of the usual orthogonal polynomials are generalized (e.g., Rodrigues formula, higher order recurrence relation, higher order di®erential equation). We will also point out some applications, such as simultaneous Gauss quadrature, certain higher order non-linear dynamical systems extending the Toda lattice, spectral theory of certain non-symmetric matrices, and especially several application from number theory, involving irrationality and transcendence of famous mathematical constants. 17 Some useful material (all available from my homepage) Multiple orthogonal polynomials, irrationality and transcendence (http://spwww.cc.kuleuven.ac.be/ pgaee03/missouri.ps) Nonsymmetric linear di®erence equations for multiple orthogonal polynomials (http://spwww.cc.kuleuven.ac.be/ pgaee03/sideIII.ps) Riemann-Hilbert problems for multiple orthogonal polynomials (http://spwww.cc.kuleuven.ac.be/ pgaee03/multRHP.ps) Some classical multiple orthogonal polynomials (http://spwww.cc.kuleuven.ac.be/ pgaee03/clasmult.ps) Multiple orthogonal polynomials for classical weights (http://spwww.cc.kuleuven.ac.be/ pgaee03/clasmops.ps) Asymptotics Roderick Wong Department of Mathematics City University of Hong Kong, Hong Kong E-mail: mawong@cityu.edu.hk In the ¯rst two lectures, I will present various classical methods in asymtotic analysis and illustrate them with some speci¯c examples from Orthogonal Polynomials and Special functions. In the third lecture, I will give a brief introduction to what is now known as Exponential Asymptotics. This is a new area in asymptotics, and started only 12 years ago. In the ¯nal lecture, I will talk about recent work on turning-point theory for secondorder linear di®erence equations, which promises to open up a new and interesting area of research with applications to special functions and orthogonal polynomials. The L2 Boundedness of High Order Generalized Riesz Transform Associated With Nondivergence Operators Ming Xu Department of Mathematics Zhongshan University, P. R. China E-mail: st96xmin@163.net In this paper, we consider the L2 boundedness of high order generalized Riesz transform associated with nondivergence di®erential operators. We solve the problem of the L2 boundedness of a generalized Riesz transform under conditions that the BMO norm of its coe±cients is small. The perturbation problem of quadratic function of nondivergence operators is also considered. In the paper, we use wavelets methods. 18 Orthogonal Polynomials on the Unit Ball of IRn Yuan Xu Department of Mathematics University of Oregon, USA E-mail: yuan@math.uoregon.edu Although the study of orthogonal polynomials of several variables can be traced back to the work of Hermite, the subject lags far behind that of orthogonal polynomials of one variable. Even for classical orthogonal polynomials on the regular regions (for example, cube, ball, simplex, sphere, and the entire IRd ), new properties are still being discovered and several questions are resolved only recently. In this talk, we use orthogonal polynomials on the unit ball as an example to explain questions and results in the subject, and discuss some recent development. Starting with the weight function W (x) = (1 ¡ kxk2 )¹¡1=2 , we present several orthogonal bases, reproducing kernels, di®erential equations, all given in explicit formulae. We will discuss questions about orthogonal expansion, norm of orthogonal projection operator, and summability. We will also discuss various extensions (to more general weight functions and to closely related weight functions on simplex and sphere) and open questions. A Singularly Perturbed Two-point Boundary Value Problem Heping Yang Department of Mathematics City University of Hong Kong, Hong Kong E-mail: mayhp@math.cityu.edu.hk We consider the singularly perturbed two-point boundary-value problem ( 00 "y + a(x)y 0 + b(x)y = 0; x 2 [x¡ ; x+ ]; x¡ < 0 < x+ ; y(x¡ ) = A; y(x+ ) = B; where A and B are two prescribed constants, and 0 < " ¿ 1 is a small positive parameter. We assume that the coe±cients a(x) and b(x) are su±ciently smooth real-valued functions. It is also assumed that, as x ! 0, a(x) » ®x and b(x) » ¯, where ® 6= 0 and ¯ are two constants. By making use of the method of successive approximation, we obtain asymptotic solutions which hold uniformly for x 2 [x¡ ; x+ ]. Some Orthogonal Polynomials Related to Elliptic Functions Mourad E. H. Ismaily , Galliano Valentz and Gang J. Yoon] Department of Mathematics, University of South Florida, USA and Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong, Hong Kong z Laboratoire de Physique Th¶eorique et des Hautes Energies, Universit¶e Paris 7, France y 19 ] Division of Applied Mathematics, KAIST, Korea E-mail: y ismail@math.usf.edu, z valent@lpthe.jussieu.fr ] ykj@jacobi.kaist.ac.kr We characterize the orthogonal polynomials in a class of polynomials de¯ned through their generating functions. This led to three new systems of orthogonal polynomials whose generating functions and orthogonality relations involve elliptic functions. The Hamburger moment problems associated with these polynomials are indeterminate. We give in¯nite families of weight functions in each case. The di®erent polynomials treated in this work are also polynomials in a parameter and as functions of this parameter they are orthogonal with respect to unique measures, which we ¯nd explicitly. Through a quadratic transformation we ¯nd a new exactly solvable birth and death process with quartic birth and death rates. The Boltzmann Equation and its History Shih-Hsien Yu Department of Mathematics City University of Hong Kong, Hong Kong E-mail: mashyu@cityu.edu.hk We will give the development history of Boltzmann equation from the very beginnig to the most recent. In particular, we are interested in its relation to °uid dynamics. How to Sum the Divergent Basic Hypergeometric Series 2 '0 (a; b; ¡; q; x) Changgui Zhang Department of Mathematics La Rochelle, France E-mail: czhang@univ-lr.fr The power series 2 '0 (a; b; ¡; q; x) de¯ned by X (a; q)n (b; q)n ' (a; b; ¡; q; x) = q¡n(n¡1)=2 (¡x)n 2 0 (q; q) n n¸0 (a; b 2 C; 0 < jqj < 1) diverge. This is a formal solution of the following q-di®erence equation: (1 ¡ abqx)y(q 2 x) ¡ (1 ¡ (a + b)qx)y(qx) ¡ qxy(x) = 0: In my talk, I de¯ne and then compare some analytical solutions that admit 2 '0 (a; b; ¡; q; x) as asymptotic expansion near x = 0. For each of these solutions I give its Stokes multipliers (due to divergence of 2 '0 (a; b; ¡; q; x)). Here is such a solution: X 2 '1 (a; b; 0; q; ¡q n ¸) : x 7! n ¸=x) µ(q n2Z 20 Here, ¸ is a ¯xed \generic" P value of C¤ (or, of the elliptic curve C¤ =q Z ), µ(t) is the theta function given by t 7! n2Z q n(n¡1)=2 tn , and one uses the analytical continuation on C n (¡1; ¡1) of 2 '1 (a; b; 0; q; ¡x). The Covering Number in Learning Theory Ding-Xuan Zhou Department of Mathematics City University of Hong Kong, Hong Kong E-mail: mazhou@math.cityu.edu.hk Learning Theory studies learning objects from random samples. The main question is: How many samples do we need to ensure an error bound with certain con¯dence? To answer this question, the covering numbers or entropy numbers play an essential role, as shown by Vapnik, Poggio, Cucker-Smale, and many others. For kernel machine learning such as the Support Vector Machine, a Reproducing Kernel Hilbert Space associated with a Mercer kernel K is often used and the covering number of a ball BR of such a space (as a subset IK (BR ) of C(X)) is needed. We show how to estimate the covering number N(IK (BR ); ´). Our estimates are based on the regularity of the kernel function K. For convolution type kernels K(x; t) = k(x ¡t) on [0; 1]n , we provide estimates depending on the decay of the Fourier transform of k. In particular, when k^ decays exponentially, we have ¶n+1 µ R : ln N(IK (BR ); ´) · const ln ´ This provides estimates for the covering number involving many important kernels used in Learning Theory. 21