We present a second kind integral equation (SKIE) formulation

for calculating the electromagnetic modes of optical waveguides,

where the unknowns are only on material interfaces. The resulting numerical

algorithm can handle optical waveguides with a large number of inclusions of 

arbitrary irregular cross section. It is capable of finding the bound, leaky, and 

complex modes for optical fibers and waveguides including photonic crystal 

fibers (PCF), dielectric fibers and waveguides. Most importantly, the formulation 

is well conditioned even in the case of nonsmooth geometries. Our method is highly 

accurate and thus can be used to calculate the propagation loss of the electromagnetic 

modes accurately, which provides the photonics industry a reliable tool for the design 

of more compact and efficient photonic devices. We illustrate and validate the 

performance of our method through extensive numerical studies and by comparison 

with semi-analytical results and previously published results.

In the talk we consider inequalities of Hardy type for functions in Triebel-Lizorkin spaces. In particular, we discuss these inequalities for functions defined on domains whose boundary has the small Aikawa dimension  (the case of a 'thin' boundary). We also show the validity of Hardy inequalities on open sets under a combined fatness and visibility condition on the boundary (the case of a 'fat' set). In addition, we would like to give a short exposition of various fatness conditions related to the theory, and apply Hardy inequalities in connection to the boundedness of extension operators for Triebel-Lizorkin spaces.

In this talk, we consider kinetic equations containing random

terms. The kinetic models contain a small parameter and it is well

known that, after scaling, when this parameter goes to zero the limit

problem is a diffusion equation in the PDE sense, ie a parabolic equation

of second order. A smooth noise is added, accounting for external perturbation.

It scales also with the small parameter. It is expected that the limit

equation is then a stochastic parabolic equation where the noise is in

Stratonovitch form.

Our aim is to justify in this way several SPDEs commonly used.

We first treat linear equations with multiplicative noise. Then show how

to extend the methods to nonlinear equations or to the more physical

case of a random forcing term.

The results have been obtained jointly with S. De Moor and J. Vovelle.

Abstract:  For the second  phase transition line $\lambda=e^{\beta}$ of  almost Mathieu operator, we prove that for dense $\alpha$, the operator has purely singular continuous spectrum for every phases, and  for dense $\alpha$, the operator has pure point spectrum for almost every phases. This is joint work with Artur Avila and Svetlana Jitomirskaya.