In this study, we investigate how can we concentrate energy in a point by focusing shock waves. One open problem is what happens when the shock waves are not circularly symmetric. Here we show a numerical simulation of a non-circular convergent shock wave on a gas with gamma=1.4. The second picture shows an advanced stage of the focusing at a scale 8 times smaller than the first picture. We found that under the right conditions, the shock waves form a triangular pattern that is stable.

Here we do a similar study, but instead of shock waves in a gas, we have a flow of gas into a porous medium. We found that there is a family of selfsimilar solutions describing the focusing. Contour lines of the selfsimilar solutions of the second kind: convergent currents in two dimensions. From left to right: 3-fold symmetry and m=1.3, 4-fold symmetry and m=1.2, 5-fold symmetry and m=1.1.

Zipper solutions describe 2D singularities on the solutions of the porous medium equation. They are selfsimilar travelling waves. Similar solutions also exist for flows driven by capillarity. These solutions are useful counterexamples of the regularity of the interfaces in special cases (here the interface is a half line).

VERY VISCOUS FLOWS
Spreading of a blob of viscous fluid on a flat surface: boundary elements method simulation for the Stokes flow. The insert shows the streamlines near the 'nose' of the viscous current.

STUDIES ON WETTING TRANSITION
Micro-interferogram of droplets of nonane near the wetting transition on a C8 Silane monolayer, at a temperature of 33.1C.

Sketch of the microscope that I designed and built with Jiang and Bruce for obtaining the previous picture

Here we explore a powerful method to denoise color images. Synthetic picture showing texture in color. The image is corrupted in order to destroy the texture and then the image is partially restored with a pseudo-local filter. SOURCE CODE.

Photograph (Piedra Movediza Del Tandil) restored with the pseudo-local filter. The original noisy image is on the left and it is restored on the right preserving the texture of the stone. Processing time: 3.5 sec in a PC-600 Mhz.

Synthetic 64x64x64 tomography of a hollow cube with texture (top). The color RGB components are proportional to the three components of the local unit normal of the surface. The data is corrupted with noise in order to ruin the texture (center). The image is restored with a pseudo-local filter (bottom).

Triangulated Moebius srip with a color pattern (top). Noise is added to both the color pattern and the shape (center). The shape is partially restored with a combination PDE's and a pseudo-local filter (bottom).

SKELETONS
Affine invariant skeleton (inner points) of a shape with noise (outer curve, left). The same figure was skewed out, in order to show the affine invariance of the skeleton (right). Straight lines in the skeleton may be associated with skew-symmetry.

GEODESIC CURVATURE FLOW
Evolution of a curve on a surface. The velocity of the curve is equal to the geodesic curvature. The curve converges to a geodesic in this example.