The analytic torsion of the finite metric cone over a compact manifold

We give an explicit formula for the $L^2$ analytic torsion of the finite metric cone over an oriented compact connected Riemannian manifold. We provide an interpretation of the different factors appearing in this formula. We prove that the analytic torsion of the cone is the finite part of the limit obtained collapsing one of the boundaries, of the ratio of the analytic torsion of the frustum to a regularising factor. We show that the regularising factor comes from the set of the non square integrable eigenfunctions of the Laplace Beltrami operator on the cone.

Spitzer Observations of the Lambda Orionis cluster I: the frequency of young debris disks at 5 Myr

We present IRAC/MIPS Spitzer observations of intermediate-mass stars in the 5 Myr old Lambda Orionis cluster. In a representative sample of stars earlier than F5 (29 stars), we find a population of 9 stars with a varying degree of moderate 24um excess comparable to those produced by debris disks in older stellar groups. As expected in debris disks systems, those stars do not exhibit emission lines in their optical spectra. We also include in our study the star HD 245185, a known Herbig Ae object which displays excesses in all Spitzer bands and shows emission lines in its spectrum. We compare the disk population in the Lambda Orionis cluster with the disk census in other stellar groups studied using similar methods to detect and characterize their disks and spanning a range of ages from 3 Myr to 10 Myr. We find that for stellar groups of 5 Myr or older the observed disk frequency in intermediate mass stars (with spectral types from late B to early F) is higher than in low mass stars (with spectral types K and M). This is in contradiction with the observed trend for primordial disks evolution, in which stars with higher stellar masses dissipate their primordial disks faster. At 3 Myr the observed disk frequency in intermediate mass stars is still lower than for low mass stars indicating that second generation dusty disks start to dominate the disk population at 5 Myr for intermediate mass stars. This result agrees with recent models of evolution of solids in the region of the disk where icy objects form (>30 AU), which suggest that at 5-10 Myr collisions start to produce large amount of dust during the transition from runaway to oligarchic growth (reaching sizes of ~500 km) and then dust production peaks at 10-30 Myr, when objects reach their maximum sizes (>1000 km)

The analytic torsion of a cone over a sphere

We compute the analytic torsion of a cone over a sphere of dimension 1, 2, and 3, and we conjecture a general formula for the cone over an odd dimensional sphere.