We study some issues concerning the existence of weak solutions for two Boltzmann like equations : a Modified Boltzmann model and the Boltzmann Dirac model. The analysis of the collision operators rests on suitable decompositions. These are also provided for the Generalised Boltzmann operator. This study is performed without assuming Grad's angular cutoff hypothesis on the cross sections.

We investigate the limit of the periodic compressible Navier-Stokes equations, when the Mach number goes to zero, and the density goes to a constant. We prove long time existence results for smooth solutions of the weakly compressible Navier-Stokes equations, for small Mach numbers, under a smallness condition on the initial velocity field, which depends only on the viscosity (that smallness condition is imposed only on the compressible part of the velocity in the bidimensional case). We also prove the convergence of the solutions of the compressible equations to the incompressible equations, once the fast waves, which satisfy a fully parabolic equation, have been removed.

We give direct proofs on the perfect block diagonalization and on the transformation to Arnold-Petkov's normal form of matrices of pseudo-differential operators in the ultradifferentiable classes.

It is shown that the Determinant line bundle generates the Picard group of the infinite Grassmannian and that it is defined by the Pliicker equations. An approach to its automorphism group is also offered.

We give a new elementary proof of Igusa’s theorem on the structure of Siegel modular forms of genus 2. The key point of the proof is the estimation of the dimension of Jacobi forms appearing in the Fourier-Jacobi development of Siegel modular forms. This proves not only Igusa’s theorem, but also gives the canonical lifting from Jacobi forms to Siegel modular forms of genus 2.