Equilibrium and eigenfunctions estimates in the semi-classical regime

We establish eigenfunctions estimates, in the semi-classical regime, for critical energy levels associated to an isolated singularity. For Schr\"odinger operators, the asymptotic repartition of eigenvectors is the same as in the regular case, excepted in dimension 1 where a concentration at the critical point occurs. This principle extends to pseudo-differential operators and the limit measure is the Liouville measure as long as the singularity remains integrable.Comment: 13 pages, 1 figure, perhaps to be revise

On distinct distances in homogeneous sets in the Euclidean space

A homogeneous set of $n$ points in the $d$-dimensional Euclidean space determines at least $\Omega(n^{2d/(d^2+1)} / \log^{c(d)} n)$ distinct distances for a constant $c(d)>0$. In three-space, we slightly improve our general bound and show that a homogeneous set of $n$ points determines at least $\Omega(n^{.6091})$ distinct distances

Numerical simulation of prominence oscillations

We present numerical simulations, obtained with the Versatile Advection Code, of the oscillations of an inverse polarity prominence. The internal prominence equilibrium, the surrounding corona and the inert photosphere are well represented. Gravity and thermodynamics are not taken into account, but it is argued that these are not crucial. The oscillations can be understood in terms of a solid body moving through a plasma. The mass of this solid body is determined by the magnetic field topology, not by the prominence mass proper. The model also allows us to study the effect of the ambient coronal plasma on the motion of the prominence body. Horizontal oscillations are damped through the emission of slow waves while vertical oscillations are damped through the emission of fast waves.Comment: 12 pages, 14 figures, accepted by Astronomy and Astrophysic

Satellite galaxy velocity dispersions in the SDSS and modified gravity models

The Sloan Digital Sky Survey (SDSS) provides data on several hundred thousand galaxies. Precise location of these galaxies in the sky, along with information about their luminosities and line-of-sight (Doppler) velocities allows one to construct a three-dimensional map of their location and estimate their line-of-sight velocity dispersion. This information, in principle, allows one to test dynamical gravity models, specifically models of satellite galaxy velocity dispersions near massive hosts. A key difficulty is the separation of true satellites from interlopers. We sidestep this problem by not attempting to derive satellite galaxy velocity dispersions from the data, but instead incorporate an interloper background into the mathematical models and compare the result to the actual data. We find that due to the presence of interlopers, it is not possible to exclude several gravitational theories on the basis of the SDSS data.Comment: 4 pages, 2 figures. Last section updated with an improved approach to compare models. Main conclusion unchange