Past

Condition numbers are a central concept in numerical analysis.

They provide a natural parameter for studying the behavior of

algorithms, as well as sensitivity and geometric properties of a problem.

The condition number of a problem instance is usually a measure

of the distance to the set of ill-posed instances. For instance, the

classical Eckart and Young identity characterizes the condition

number of a square matrix as the reciprocal of its relative distance

to singularity.

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We present concepts of conditioning for optimization problems and

for more general variational problems. We show that the Eckart and

Young identity has natural extension to much wider contexts. We also

discuss conditioning under the presence of block-structure, such as

that determined by a sparsity pattern. The latter has interesting

connections with the mu-number in robust control and with the sign-real

spectral radius.

I present a calculation of the moduli Kahler potential for M-theory

on a G_2 manifold in a large radius approximation. The result is used to

analyze moduli dynamics and moduli stabilization in the context of the

associated four-dimensional effective theory.

Take any region omega and let function u defined inside omega be the

distance from the boundary, u solves the iconal equation \lt|Du\rt|=1 with

boundary condition zero. Functional u is also conjectured (in some cases

proved) to be the "limiting minimiser" of various functionals that

arise models of blistering and micro magnetics. The precise formulation of

these problems involves the notion of gamma convergence. The Aviles Giga

functional is a natural "second order" generalisation of the Cahn

Hilliard model which was one of the early success of the theory of gamma

convergence. These problems turn out to be surprisingly rich with connections

to a number of areas of pdes. We will survey some of the more elementary

results, describe in detail of one main problems in field and state some

partial results.

We construct and study different surface measures on the space of

paths in a compact Riemannian manifold embedded into the Euclidean

space. The idea of the constructions is to force a Brownian particle

in the ambient space to stay in a small neighbourhood of the manifold

and then to pass to the limit. Finally, we compare these surface

measures with the Wiener measure on the space of paths in the

manifold.

I discuss gauge theories for commutative but non-associative algebras

related to the SO(2k+1) covariant finite dimensional fuzzy 2k-sphere

algebras. A consequence of non-associativity is that gauge fields and

gauge parameters have to be generalized to be functions of coordinates as

well as derivatives. The usual gauge fields depending on coordinates only

are recovered after a partial gauge fixing.The deformation parameter for

these commutative but non-associative algebras is a scalar of the rotation

group. This suggests interesting string-inspired algebraic deformations of

spacetime which preserve Lorentz-invariance.

The talk will be based on hepth/0310153

The seminar will focus on mathematical modelling of physics phenomena,

with applications in e.g. mass and heat transfer, fluid flow, and

electromagnetic wave propagation. Simultaneous solutions of several

physics phenomena described by PDEs - multiphysics - will also be

presented and discussed.

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All models will be realised through the use of the MATLAB based finite

element package FEMLAB. FEMLAB is a multiphysics modelling environment

with built-in PDE solvers for linear, non-linear, time dependent and

eigenvalue problems. For ease-of-use, it comprises ready-to-use applications

for various physics phenomena, and tailored applications for Structural

Mechanics, Electromagnetics, and Chemical Engineering. But in addition,

FEMLAB facilitates straightforward implementation of arbitrary coupled

non-linear PDEs, which brings about a great deal of flexibility in problem

definition. Please see http://ww.uk.comsol.com for more info.

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FEMLAB is developed by the COMSOL Group, a Swedish headquartered spin-off

from the Royal Institute of Technology (KTH) in Stockholm, with offices

around the world. Its UK office is situated in The Oxford Science Park.

We will discuss the mixing rate of the standard random walk on the giant

component of the random graph G(n,p). We tie down the mixing rate precisely

for all values of p greater than (1+c)/n for any positive constant c. We need

to develop a new bound on the mixing time of general Markov chains, inspired

by and extending work of Kannan and Lovasz. This is joint work with Nick

Fountoulakis.

We consider an investment model that can operate in two different

modes. The transition from one mode to the other one is immediate and forms a

sequence of costly decisions made by the investment's management. Each of the

two modes is associated with a rate of payoff that is a function of a state

process which can be an economic indicator such as the price of a given

comodity. We model the state process by a general one-dimensional

diffusion. The objective of the problem is to determine the switching

strategy that maximises a long-term average criterion in a pathwise

sense. Our analysis results in analytic solutions that can easily be

computed, and exhibit qualitatively different optimal behaviours.

The process industries are one of the UK's major sectors and include

petrochemicals, pharmaceuticals, water, energy and the food industry,

amongst others. The design of a processing plant is a difficult task. This

is due to both the need to cater for multiple criteria (such as economics,

environmental and safety) and the use highly complex nonlinear models to

describe the behaviour of individual unit operations in the process. Early

in the design stages, an engineer may wish to use automated design tools to

generate conceptual plant designs which have potentially positive attributes

with respect to the main criteria. Such automated tools typically rely on

optimization for solving large mixed integer nonlinear programming models.

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This talk presents an overview of some of the work done in the Computer

Aided Process Engineering group at UCL. Primary emphasis will be given to

recent developments in hybrid optimization methods, including the use of

graphical interfaces based on problem specific visualization techniques to

allow the engineer to interact with embedded optimization procedures. Case

studies from petrochemical and water industries will be presented to

demonstrate the complexities involved and illustrate the potential benefits

of hybrid approaches.

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