Abstract. Some significant quantities in mathematics and physics are most naturally expressed as the Fredholm determinant of an integral operator, most notably many of the distribution functions in random matrix theory. Though their numerical values are of interest, there is no systematic numerical treatment of Fredholm determinants to be found in the literature. Instead, the few numerical evaluations that are available rely on eigenfunction expansions of the operator, if expressible in terms of special functions, or on alternative, numerically more straightforwardly accessible analytic expressions, e.g., in terms of Painlevé transcendents, that have masterfully been derived in some cases. In this paper we close the gap in the literature by studying projection methods and, above all, a simple, easily implementable, general method for the numerical evaluation of Fredholm determinants that is derived from the classical Nyström method for the solution of Fredholm equations of the second kind. Using Gauss–Legendre or Clenshaw– Curtis as the underlying quadrature rule, we prove that the approximation error essentially behaves like the quadrature error for the sections of the kernel. In particular, we get exponential convergence for analytic kernels, which are typical in random matrix theory. The application of the method to the distribution functions of the Gaussian unitary ensemble (GUE), in the bulk and the edge scaling limit, is discussed in detail. After extending the method to systems of integral operators, we evaluate the twopoint correlation functions of the more recently studied Airy and Airy 1 processes. Key words. Fredholm determinant, Nyström’s method, projection method, trace class operators, random

Asymptotic and numerical methods are used to study several classes of singularly perturbed boundary value problems for which the underlying homogeneous operators have exponentially small eigenvalues. Examples considered include the familiar boundary layer resonance problems and some extensions, and certain linearized equations associated with metastable internal layer motion. For the boundary layer resonance problems, a systematic projection method, motivated by the work of De Groen [SIAM J. Math. Anal. 11, (1980), pp. 1-22], is used to analytically calculate high order asymptotic solutions. This method justifies and extends some previous results obtained from the variational method of Grasman and Matkowsky [SIAM J. Appl. Math. 32, (1977), pp. 588-597]. A numerical approach, based on an integral equation formulation, is used to accurately compute boundary layer resonance solutions and their associated exponentially small eigenvalues. For various examples, the numerical results are show...

This paper studies the behavior of a large body of neurons in the continuum limit. A mathematical characterization of such systems is obtained by approximating the inverse input-output nonlinearity of a cell (or an assembly of cells) by three adjustable linearized sections. The associative spatio-temporal patterns for storage in the neural system are obtained by using approaches analogous to solving space-time field equations in physics. A noise-reducing equation is also derived from this neural model. In addition, conditions that make a noisy pattern retrievable are identified. Based on these analyses, a visual cortex model is proposed and an exact characterization of the patterns that are storable in this cortex is obtained. Furthermore, we show that this model achieves pattern association that is invariant to scaling, translation, rotation and mirror-reflection. 1 Please contact hjchang@pine.ece.utexas.edu or ghosh@pine.ece.utexas.edu for any questions or comments regarding this p...

. The height at which an unloaded column will buckle under its own weight is the fourth root of the least eigenvalue of a certain Sturm--Liouville operator. We show that the operator associated with the column proposed by Keller and Niordson (The Tallest Column, J. Math. Mech., 16 (1966), pp. 433--446) does not possess a discrete spectrum. This calls into question their formal use of perturbation theory and so we consider a class of designs that permits a tapered summit yet still guarantees a discrete spectrum. Within this class we prove that the least eigenvalue increases when one replaces a design with its decreasing rearrangement. This leads to a very simple proof of the existence of a tallest column. Key words. polynomials, SI model AMS subject classifications. 33H40, 35C01 1. Introduction. Euler [4],[5], posed and solved the problem of buckling of prismatic columns under self--weight. He found that a column, clamped at its base and free at its summit, could be built to a heigh...

Abstract: Reduced modelling techniques, based on a proper orthogonal decomposition (POD) method, are applied to an investigation of the incompressible Navier–Stokes equations with inputs. A circular cylinder in uniform � ow with and without inputs is studied. Reduced dynamic models are created by POD and by extended POD (EPOD) approaches for the forced � ow which is statistically non-stationary. A direct control action is applied to the � ow at particular points and this investigation provides insights into the applications of the proposed approaches coupled with a full solver.

Theoretical and computational aspects of scattering from rough surfaces: one-dimensional perfectly reflecting surfaces

The purpose of this thesis is to investigate the scattering of a train of small amplitude harmonic surface waves on water by undulating one-dimensional bed topography. The computational efficiency of an integral equation procedure that has been used to solve the mild-slope equation, an approximation to wave scattering, is improved by using a new choice of trial function. The coefficients of the scattered waves given by the mild-slope equation satisfy a set of relations. These coefficients are also shown to satisfy the set of relations when they are given by any approximation to the solution of the mild-slope equation. A new approximation to wave scattering is derived that includes both progressive and decaying wave mode terms and its accuracy is tested. In particular, this approximation is compared with older approximations that only contain progressive wave mode terms such as the mild-slope approximation. The results given by the new approximation are shown to agree much more closely with known test results over steep topography, where decaying wave modes are significant. During this analysis, a

In this article, we investigate the problem of simultaneously steering an uncountable family of finite dimensional time-varying linear systems. We call this class of control problems Ensemble Control, a notion coming from the study of spin dynamics in Nuclear Magnetic Resonance (NMR) spectroscopy and imaging (MRI). This subject involves controlling a continuum of parameterized dynamical systems with the same open-loop control input. From a viewpoint of mathematical control theory, this class of problems is challenging because it requires steering a continuum of dynamical systems between points of interest in an infinite dimensional state space by use of the same control function. The existence of such a control raises fundamental questions of ensemble controllability. We derive the necessary and sufficient controllability conditions and an accompanying analytical optimal control law for ensemble control of time-varying linear systems. We show that ensemble controllability is in connection with singular values of the operator characterizing the system dynamics. In addition, we study the problem of optimal ensemble control of harmonic oscillators to demonstrate our main results. We show that the optimal solutions are pertinent to the study of time-frequency limited signals and prolate spheroidal wave functions. A systematic study of ensemble control systems has immediate applications to systems with parameter uncertainties as well as to broad areas of quantum control systems as arising in coherent spectroscopy and quantum information processing. The new mathematical structures appearing in such problems are an excellent motivation for new developments in control theory.

During the last few years, interest has arisen in using light-front Tamm-Dancoff field theory to describe relativistic bound states for theories such as QCD. Unfortunately, difficult renormalization problems stand in the way. We introduce a general, non-perturbative approach to renormalization that is well suited for the ultraviolet and, presumably, the infrared divergences found in these systems. We reexpress the renormalization problem in terms of a set of coupled inhomogeneous integral equations, the “counterterm equation.” The solution of this equation provides a kernel for the Tamm-Dancoff integral equations which generates states that are independent of any cutoffs. We also introduce a Rayleigh-Ritz approach to numerical solution of the counterterm equation. Using our approach to renormalization, we examine several ultraviolet divergent models. Finally, we use the Rayleigh-Ritz approach to First attempted in the 1950’s by Tamm and Dancoff [1, 2], the idea of describing