Past

Consider a spectrally one-sided Levy process X and reflect it at

its past infimum I. Call this process Y. We determine the law of the

first crossing time of Y of a positive level a in terms of its

'scale' functions. Next we study the exponential decay of the

transition probabilities of Y killed upon leaving [0,a]. Restricting

ourselves to the case where X has absolutely continuous transition

probabilities, we also find the quasi-stationary distribution of

this killed process. We construct then the process Y confined in

[0,a] and prove some properties of this process.

A reveiw of results about spectral analysis of generators of

some stochastic lattice models (a stochastic planar rotators model, a

stochastic Blume-Capel model etc.) will be presented. Then I'll discuss new

results by R.A. Minlos, Yu.G. Kondratiev and E.A. Zhizhina concerning spectral

analysis of the generator of stochastic continuous particle system. The

construction of one-particle subspaces of the generators and the spectral

analysis of the generator restricted on these subspaces will be the focus of

the talk.

Following the framework of Formaggia and Perotto (Numer.

Math. 2001 and 2003), anisotropic a posteriori error estimates have been

proposed for various elliptic and parabolic problems. The error in the

energy norm is bounded above by an error indicator involving the matrix

of the error gradient, the constant being independent of the mesh aspect

ratio. The matrix of the error gradient is approached using

Zienkiewicz-Zhu error estimator. Numerical experiments show that the

error indicator is sharp. An adaptive finite element algorithm which

aims at producing successive triangulations with high aspect ratio is

proposed. Numerical results will be presented on various problems such

as diffusion-convection, Stokes problem, dendritic growth.

Recently Friesecke, James and Muller established the following

quantitative version of the rigidity of SO(n) the group of special orthogonal

matrices. Let U be a bounded Lipschitz domain. Then there exists a constant

C(U) such that for any mapping v in the L2-Sobelev space the L^2-distance of

the gradient controlls the distance of v a a single roation.

This interesting inequality is fundamental in several problems concerning

dimension reduction in nonlinear elasticity.

In this talk, we will present a joint work with Muller and Zhong where we

investigate an analagous quantitative estimate where we replace SO(n) by an

arbitrary smooth, compact and SO(n) invariant subset of the conformal

matrices E. The main novelty is that exact solutions to the differential

inclusion Df(x) in E a.e.x in U are not necessarily affine mappings.

http://www.maths.ox.ac.uk/notices/events/abstracts/stochastic-analysis/…

Stability is central to the study of numerical algorithms for solving
partial differential equations. But stability can be subtle and elusive. In
fact, for a number of important classes of PDE problems, no one has yet
succeeded in devising stable numerical methods. In developing our
understanding of stability and instability, a wide range of mathematical
ideas--with origins as diverse as functional analysis,differential geometry,
and algebraic topology--have been enlisted and developed. The talk will
explore the concept of stability of discretizations to PDE, its significance,
and recent advances in its understanding.

While the average settling velocity of particles in a suspension has been successfully predicted, we are still unsuccessful with the r.m.s velocity, with theories suggesting a divergence with the size of

the container and experiments finding no such dependence. A possible resolution involves stratification originating from the spreading of the front between the clear liquid above and the suspension below. One theory describes the spreading front by a nonlinear diffusion equation

$\frac{\partial \phi}{\partial t} = D \frac{\partial }{\partial z}(\phi^{4/5}(\frac{\partial \phi}{\partial z})^{2/5})$.

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Experiments and computer simulations find differently.

Using a technique explored in unpublished work of Ball and Mizel I shall

show that already in 2 and 3 dimensions there are vectorfields which are

singular minimizers of integral functionals whose integrand is strictly

polyconvex and depends on the gradient of the map only. The analysis behind

these results gives rise to an interesting question about the relationship

between the regularity of a polyconvex function and that of its possible

convex representatives. I shall indicate why this question is interesting in

the context of the regularity results above and I shall answer it in certain

cases.

Consider a graph with n vertices placed randomly in the unit

square, each connected by an edge to its nearest neighbour in a

south-westerly direction. For many graphs of this type, the centred

total length is asymptotically normal for n large, but in the

present case the limit distribution is not normal, being defined in

terms of fixed-point distributions of a type seen more commonly in

the analysis of algorithms. We discuss related results. This is

joint work with Andrew Wade.

We consider a system of interacting Fisher-Wright diffusions

which arise in population genetics as the diffusion limit of a spatial

particle model in which frequencies of genetic types are changing due to

migration and reproduction.

For both models the historical processes are constructed,

which record the family structure and the paths of descent through space.

For any fixed time, particle representations for the

historical process of a collection of Moran models with increasing particle

intensity and of the limiting interacting Fisher-Wright diffusions are

provided on one and the same probability space by means of Donnelly and

Kurtz's look-down construction.

It will be discussed how this can be used to obtain new

results on the long term behaviour. In particular, we give representations for

the equilibrium historical processes. Based on the latter the behaviour of

large finite systems in comparison with the infinite system is described on

the level of the historical processes.

The talk is based on joint work with Andreas Greven and Vlada

Limic.

We consider the problem of recovering the position of a scattering surface

from measurements of the scattered field on a finite line above the surface.

A point source algorithm is proposed, based on earlier work by Potthast,

which reconstructs, in the first instance, the scattered field in the whole

region above the scattering surface. This information is used in a second

stage to locate the scatterer. We summarise the theoretical results that can

be obtained (error bounds on the reconstructed field as a function of the

noise level in the original measurements). For the case of a point source of

the incident field we present numerical experiments for both a steady source

(time harmonic excitation) and a pulse source typical of an antenna in

ground penetrating radar applications.

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This is joint work with Claire Lines (Brunel University).

After a review of the symmetries of supergravity theories and Kac-Moody

algebras, we explain show that M theory is likely to possess a very large

Kac-Moody symmetry of rank eleven, denoted by E_{11}. We also provide

evidence that even pure gravity and the closed bosonic string possess

analogous Kac-Moody symmetries.

Finally, we explain how the central charges of the maximal supergravity

theories arise naturally in E_{11}.