Past

The Brownian snake (with lifetime given by a normalized

Brownian excursion) arises as a natural limit when studying random trees. This

may be used in both directions, i.e. to obtain asymptotic results for random

trees in terms of the Brownian snake, or, conversely, to deduce properties of

the Brownian snake from asymptotic properties of random trees. The arguments

are based on Aldous' theory of the continuum random tree.

I will discuss two such situations:

1. The Wiener index of random trees converges, after

suitable scaling, to the integral (=mean position) of the head of the Brownian

snake. This enables us to calculate the moments of this integral.

2. A branching random walk on a random tree converges, after

suitable scaling, to the Brownian snake, provided the distribution of the

increments does not have too large tails. For i.i.d increments Y with mean 0,

a necessary and sufficient condition is that the tails are o(y^{-4}); in

particular, a finite fourth moment is enough, but weaker moment conditions are

not.

The celebrated Levy-Khintchine formula provides us an explicit

structure of Levy processes on $R^d$. In this talk I shall present a

structure result for quasi-regular semi-Dirichlet forms, i.e., for

those semi-Dirichlet forms which are associated with right processes

on general state spaces. The result is regarded as an extension of

Levy-Khintchine formula in semi-Dirichlet forms setting. It can also

be regarded as an extension of Beurling-Deny formula which is up to

now available only for symmetric Dirichlet forms.

Consider approximating a set of discretely defined values $f_{1}, \ldots , f_{m}$ say at $x=x_{1}, x_{2}, \ldots, x_{m}$, with a chosen approximating form. Given prior knowledge that noise is present and that some might be outliers, a standard least squares approach based on $l_{2}$ norm of the error $\epsilon$ may well provide poor estimates. We instead consider a least squares approach based on a modified measure of the form $\tilde{\epsilon} = \epsilon (1+c^{2}\epsilon^{2})^{-\frac{1}{2}}$, where $c$ is a constant to be fixed.

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The choice of the constant $c$ in this estimator has a significant effect on the performance of the estimator both in terms of its algorithmic convergence to a solution and its ability to cope effectively with outliers. Given a prior estimate of the likely standard deviation of the noise in the data, we wish to determine a value of $c$ such that the estimator behaves like a robust estimator when outliers are present but like a least squares estimator otherwise.

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We describe approaches to determining suitable values of $c$ and illustrate their effectiveness on approximation with polynomial and radial basis functions. We also describe algorithms for computing the estimates based on an iteratively weighted linear least squares scheme.

In Clarendon Lab

In Corpus

We consider matrix groups defined in terms of scalar products. Examples of interest include the groups of

• complex orthogonal,
• real, complex, and conjugate symplectic,
• real perplectic,
• real and complex pseudo-orthogonal,
• pseudo-unitary

matrices. We

• Construct a variety of transformations belonging to these groups that imitate the actions of Givens rotations, Householder reflectors, and Gauss transformations.
• Describe applications for these structured transformations, including to generating random matrices in the groups.
• Show how to exploit group structure when computing the polar decomposition, the matrix sign function and the matrix square root on these matrix groups.

This talk is based on recent joint work with N. Mackey, D. S. Mackey, and N. J. Higham.

Let gamma be a closed knotted curve in R^3 such that the tubular

neighborhood U_r (gamma) with given radius r>0 does not intersect

itself. The length minimizing curve gamma_0 within a prescribed knot class is

called ideal knot. We use a special representation of curves and tools from

nonsmooth analysis to derive a characterization of ideal knots. Analogous

methods can be used for the treatment of self contact of elastic rods.

Weakly self-avoiding walk (WSAW) is obtained by giving a penalty for every

self-intersection to the simple random walk path. The Edwards model (EM) is

obtained by giving a penalty proportional to the square integral of the local

times to the Brownian motion path. Both measures significantly reduce the

amount of time the motion spends in self-intersections.

The above models serve as caricature models for polymers, and we will give

an introduction polymers and probabilistic polymer models. We study the WSAW

and EM in dimension one.

We prove that as the self-repellence penalty tends to zero, the large

deviation rate function of the weakly self-avoiding walk converges to the rate

function of the Edwards model. This shows that the speeds of one-dimensional

weakly self-avoiding walk (if it exists) converges to the speed of the Edwards

model. The results generalize results earlier proved only for nearest-neighbor

simple random walks via an entirely different, and significantly more

complicated, method. The proof only uses weak convergence together with

properties of the Edwards model, avoiding the rather heavy functional analysis

that was used previously.

The method of proof is quite flexible, and also applies to various related

settings, such as the strictly self-avoiding case with diverging variance.

This result proves a conjecture by Aldous from 1986. This is joint work with

Frank den Hollander and Wolfgang Koenig.

We give a construction of Brownian motion in a Weyl chamber, by a

multidimensional generalisation of Pitman's theorem relating one

dimensional Brownian motion with the three dimensional Bessel

process. There are connections representation theory, especially to

Littelmann path model.