Past

An important class of nonlinear parabolic equations is the class of quasi-linear equations, i.e., equations with a leading second-order (in space) linear part (e.g., the Laplacian) and a nonlinear part which depends on the first-order spatial derivatives of the unknown function. This class contains the Navier-Stokes system of fluid dynamics, as well as "viscous" versions (or "regularized") of the Hamilton-Jacobi equation, nonlinear hyperbolic conservation laws and more. The talk will present various recent results concerning existence/uniqueness (and nonexistence/nonuniqueness) of global solutions. In addition, a new class of "Bernstein-type" estimates of derivatives will be presented. These estimates are independent of the viscosity parameter and thus lead to results concerning the "zero-viscosity" limit.

The estimation of heat kernels has been of much interest in various settings. Often, the spaces considered have some kind of uniformity in the volume growth. Recent results have shown that this is not the case for certain random fractal sets. I will present heat kernel bounds for spaces admitting a suitable resistance form, when the volume growth is not uniform, which are motivated by these examples.

We study diploid branching particle models and its behaviour when rapid

stirring, i.e. rapid exchange of particles between neighbouring spatial

sites, is added to the interaction. The particle models differ from the

``usual'' models in that they all involve two types of particles, male

and female, and branching can only occur when both types of particles

are present. We establish the existence of nontrivial stationary

distributions for various models when birth rates are sufficiently large.

Most people acquire their

We develop a nonlinear delay-differential equation for the human cardiovascular control system, and use it to explore blood pressure and heart rate variability under short-term baroreflex control. The model incorporates an intrinsically stable heart rate in the absence of nervous control, and features baroreflex influence on both heart rate and peripheral resistance. Analytical simplications of the model allow a general investigation of the r\^{o}les played by gain and delay, and the effects of ageing. View diagram:  Download PDF

Abstract 1 Another Orthogonal Matrix

A householder reflection and a suitable product of Givens rotations are two well known examples of an orthogonal matrix with given first column. We present another way to produce such a matrix and apply it to produce a "fast Givens" method to compute the R factor of A, A = QR. This approach avoids the danger of under/overflow.
(joint work with Eric Barszcz)

Abstract 2 An application of Pfaff's Theorem (on skew-symmetric matrices)

There are no constraints on the eigenvalues of a product of two real symmetric matrices but what about the product of two real skew-symmetric matrices?
(joint work with A Dubrulle)

We aim at presenting some aspects of stochastic calculus via regularization

in relation with integrator processes which are generally not semimartingales.

Significant examples of those processes are Dirichlet processes, Lyons-Zheng

processes and fractional (resp. bifractional) Brownian motion. A Dirichlet

process X is the sum of a local martingale M and a zero quadratic variation

process A. We will put the emphasis on a generalization of Dirichlet processes.

A weak Dirichlet process is the sum of local martingale M and a process A such

that [A,N] = 0 where N is any martingale with respect to an underlying

filtration. Obviously a Dirichlet process is a weak Dirichlet process. We will

illustrate partly the following application fields.

Analysis of stochastic integrals related to fluidodynamical models considered

for instance by A. Chorin, F. Flandoli and coauthors...

Stochastic differential equations with distributional drift and related

stochastic control theory.

The talk will partially cover joint works with M. Errami, F. Flandoli, F.

Gozzi, G. Trutnau.