Milky Way dust extinction measured with QSOs

We investigate reddening by Milky Way dust in the low-extinction regime of $E_{B-V}<0.15$. Using over 50,000 QSOs at $0.5<z<2.5$ from the SDSS DR7 QSO Catalogue we probe the residual SDSS colours after dereddening and correcting for the known spectroscopic redshifts. We find that the extinction vector of Schlafly & Finkbeiner (2011) is a better fit to the data than that used by Schlegel et al. (1998, SFD). There is evidence for a non-linearity in the SFD reddening map, which is similarly present in the V1.2 map of the Planck Collaboration. This non-linearity is similarly seen when galaxies or stars are used as probes of the SFD map.Comment: 7 pages, 5 figures, MNRAS, accepte

The affine preservers of non-singular matrices

When K is an arbitrary field, we study the affine automorphisms of M_n(K) that stabilize GL_n(K). Using a theorem of Dieudonn\'e on maximal affine subspaces of singular matrices, this is easily reduced to the known case of linear preservers when n>2 or #K>2. We include a short new proof of the more general Flanders' theorem for affine subspaces of M_{p,q}(K) with bounded rank. We also find that the group of affine transformations of M_2(F_2) that stabilize GL_2(F_2) does not consist solely of linear maps. Using the theory of quadratic forms over F_2, we construct explicit isomorphisms between it, the symplectic group Sp_4(F_2) and the symmetric group S_6.Comment: 13 pages, very minor corrections from the first versio