Filtering the Tau method with Frobenius-Pad\'e Approximants

In this work, we use rational approximation to improve the accuracy of spectral solutions of differential equations. When working in the vicinity of solutions with singularities, spectral methods may fail their propagated spectral rate of convergence and even they may fail their convergence at all. We describe a Pad\'e approximation based method to improve the approximation in the Tau method solution of ordinary differential equations. This process is suitable to build rational approximations to solutions of differential problems when their exact solutions have singularities close to their domain

Testing the Influence of the Gravitational Redshift on the Casimir Effect in Space

We show that the Casimir effect should be influenced by variations of the gravitational potential. This could be tested with a satellite in a highly elliptic orbit. Still significant technology development is required to achieve a relative accuracy of $<5 \times 10^{-10}$ necessary to detect the effect around Earth. That may be obtained in the future based on recent concepts for giant casimir forces. Such a space mission could evaluate both the laws of gravitation and quantum mechanics and their interaction. A dedicated mission, Gravity-Probe C, is proposed

On the Space Time of a Galaxy

We present an exact solution of the averaged Einstein's field equations in the presence of two real scalar fields and a component of dust with spherical symmetry. We suggest that the space-time found provides the characteristics required by a galactic model that could explain the supermassive central object and the dark matter halo at once, since one of the fields constitutes a central oscillaton surrounded by the dust and the other scalar field distributes far from the coordinate center and can be interpreted as a halo. We show the behavior of the rotation curves all along the background. Thus, the solution could be a first approximation of a ``long exposition photograph'' of a galaxy.Comment: 8 pages REVTeX, 11 eps figure