Past

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.