It is well known that piecewise smooth signals are approximately sparse in a wavelet basis. However, other sparse representations are possible, such as the discrete gradient basis. It turns out that signals drawn from a random piecewise constant model have sparser representations in the discrete gradient basis than in Haar wavelets (with high probability). I will talk about this result and its implications, and also show some numerical experiments in which the use of the gradient basis improves compressive signal reconstruction.

The Banach-Tarski paradox is a celebrated result showing that, using the axiom of choice, it is possible to deconstruct a ball into finitely many pieces that may be rearranged to build two copies of that ball. In this seminar we will sketch the proof of the paradox trying to emphasize the key ideas.
 

Topics:

1) Marine Acoustics;

2) Air and water quality discharge and emission modelling;

3) Geospatial mapping, remote sensing and ecosystem services.

The dynamic flow of liquid crystals is described by the Ericksen-Leslie system. The Ericksen-Leslie system is a system of  the Navier-Stokes equations coupled with the gradient flow for the Oseen-Frank model,   which generalizes the heat flow for harmonic maps  into the $2$-sphere.   In this talk, we will outline a proof of global existence of solutions of the Ericksen-Leslie system for a general Oseen-Frank  model in 2D.