Past

The trapezoidal rule for numerical integration is remarkably accurate when

the integrand under consideration is smooth and periodic. In this

situation it is superior to more sophisticated methods like Simpson's rule

and even the Gauss-Legendre rule. In the first part of the talk we

discuss this phenomenon and give a few elementary examples. In the second

part of the talk we discuss the application of this idea to the numerical

evaluation of contour integrals in the complex plane.

Demonstrations involving numerical differentiation, the computation

of special functions, and the inversion of the Laplace transform will be

presented.

Marstrand's Theorem is a one of the classic results of Geometric Measure Theory, amongst other things it says that fractal measures do not have density. All methods of proof have used symmetry properties of Euclidean space in an essential way. We will present an elementary history of the subject and state a version of Marstrand's theorem which holds for spaces whose unit ball is a polytope.

We present a large deviation result for the behaviour of the

end-point of a diffusion under the influence of a strong drift. The rate

function can be explicitely determined for both attracting and repelling

drift. It transpires that this problem cannot be solved using

Freidlin-Wentzel theory alone. We present the main ideas of a proof which

is based on the Girsanov-Formula and Tauberian theorems of exponential type.

Let {X_n} be a sequence of bounded, real-valued random variables.

Assume that the partial-sums processes {S_n}, where S_n=X_1+...+X_n,

satisfies the large deviation principle with a convex rate-function, I().

Given an observation of the process {X_n}, how would you estimate I()? This

talk will introduce an estimator that was proposed to tackle a problem in

telecommunications and discuss it's properties. In particular, recent

results regarding the large deviations of estimating I() will be presented.

The significance of these results for the problem which originally motivated

the estimator, estimating the tails of queue-length distributions, will be

demonstrated. Open problems will be mentioned and a tenuous link to Oxford's

Mathematical Institute revealed.

Plane Couette flow - the flow between two infinite parallel plates moving in opposite directions -

undergoes a discontinuous transition from laminar flow to turbulence as the Reynolds number is

increased. Due to its simplicity, this flow has long served as one of the canonical examples for understanding shear turbulence and the subcritical transition process typical of channel and pipe flows. Only recently was it discovered in very large aspect ratio experiments that this flow also exhibits remarkable pattern formation near transition. Steady, spatially periodic patterns of distinct regions of turbulent and laminar flow emerges spontaneously from uniform turbulence as the Reynolds number is decreased. The length scale of these patterns is more than an order of magnitude larger than the plate separation. It now appears that turbulent-laminar patterns are inevitable intermediate states on the route from turbulent to laminar flow in many shear flows. I will explain how we have overcome the difficulty of simulating these large scale patterns and show results from studies of three types of patterns: periodic, localized, and intermittent.

http://links.jstor.org/sici?sici=0036-8075%2819940415%293%3A264%3A5157%…

http://links.jstor.org/sici?sici=0036-8075%2819940204%293%3A263%3A5147%…

While the classification of crystals made up by just one atom per cell is well-known and understood (Bravais lattices), that for more complex structures is not. We present a geometric way classifying these crystals and an arithmetic one, the latter introduced in solid mechanics only recently. The two approaches are then compared. Our main result states that they are actually equivalent; this way a geometric interpretation of the arithmetic criterion in given. These results are useful for the kinematic description of solid-solid phase transitions. Finally we will reformulate the arithmetic point of view in terms of group cohomology, giving an intrinsic view and showing interesting features.

Under the nondegenerate condition as in the diffusion case, we show

that the linear stochastic jump diffusion process projected on the

unite sphere has an uni que invariant probabolity measure. The

Lyapunov exponentcan be represented as an integral over the

sphere. These results were extended to the degenerated and Levy jump

cases.

Anderson localisation is an important phenomenon describing a

transition between insulation and conductivity. The problem is to analyse

the spectrum of a Schroedinger operator with a random potential in the

Euclidean space or on a lattice. We say that the system exhibits

(exponential) localisation if with probability one the spectrum is pure

point and the corresponding eigen-functions decay exponentially fast.

So far in the literature one considered a single-particle model where the

potential at different sites is IID or has a controlled decay of

correlations. The present talk aims at $N$-particle systems (bosons or

fermions) where the potential sums over different sites, and the traditional

approach needs serious modifications. The main result is that if the

`randomness' is strong enough, the $N$-particle system exhibits

localisation.

The proof exploits the muli-scale analysis scheme going back to Froehlich,

Martinelli, Scoppola and Spencer and refined by von Drefus and Klein. No

preliminary knowledge of the related material will be assumed from the

audience, apart from basic facts.

This is a joint work with V Chulaevsky (University of Reims, France)

Sparse grids yield numerical solutions to PDEs with a

significantly reduced number of degrees of freedom. The relative

benefit increases with the dimensionality of the problem, which makes

multi-factor models for financial derivatives computationally tractable.

An outline of a convergence proof for the so called combination

technique will be given for a finite difference discretisation of the

heat equation, for which sharp error bounds can be shown.

Numerical examples demonstrate that by an adaptive (heuristic)

choice of the subspaces European and American options with up to thirty

(and most likely many more) independent variables can be priced with

high accuracy.

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