Symplectic matrices with predetermined left eigenvalues

We prove that given four arbitrary quaternion numbers of norm 1 there always exists a $2\times 2$ symplectic matrix for which those numbers are left eigenvalues. The proof is constructive. An application to the LS category of Lie groups is given.Comment: 7 page

Model of the polarized foreground diffuse Galactic emissions from 33 to 353 GHz

We present 3D models of the Galactic magnetic field including regular and turbulent components, and of the distribution of matter in the Galaxy including relativistic electrons and dust grains. By integrating along the line of sight, we construct maps of the polarized Galactic synchrotron and thermal dust emissions for each of these models. We perform a likelihood analysis to compare the maps of the Ka, Q, V and W bands of the Wilkinson Microwave Anisotropy Probe (Wmap) and the 353 GHz Archeops data to the models obtained by varying the pitch angle of the regular magnetic field, the relative amplitude of the turbulent magnetic field and the extrapolation spectral indices of the synchrotron and thermal dust emissions. The best-fit parameters obtained for the different frequency bands are very similar and globally the data seem to favor a negligible isotropic turbulent magnetic field component at large angular scales (an anisotropic line-of-sight ordered component can not be studied using these data). From this study, we conclude that we are able to propose a consistent model of the polarized diffuse Galac- tic synchrotron and thermal dust emissions in the frequency range from 33 to 353 GHz, where most of the CMB studies are performed and where we expect a mixture of these two main foreground emissions. This model can be very helpful to estimate the contamination by foregrounds of the polarized CMB anisotropies, for experiments like the Planck satellite.Comment: 22 pages, 4 figure

Lusternik-Schnirelmann category of simplicial complexes and finite spaces

In this paper we establish a natural definition of Lusternik-Schnirelmann category for simplicial complexes via the well known notion of contiguity. This category has the property of being homotopy invariant under strong equivalences, and only depends on the simplicial structure rather than its geometric realization. In a similar way to the classical case, we also develop a notion of geometric category for simplicial complexes. We prove that the maximum value over the homotopy class of a given complex is attained in the core of the complex. Finally, by means of well known relations between simplicial complexes and posets, specific new results for the topological notion of category are obtained in the setting of finite topological spaces.Comment: 18 pages, 10 figures, this is a new version with some minor changes and a new exampl