A simple approximation of the ground state eigenfunction for both
the quantum-mechanical anharmonic oscillator and the double-well
potential given by V = m2x2 + gx4 at arbitrary g≥0 for
m2≥0 and m2 < 0 , respectively, is proposed. The approximation models perturbation theory results at small distances and WKB asymptotics at large distances. It is shown that this approximation if taken as variational trial function defines the energies of low-lying states with relative accuracy 10 ^{- 8} - 10 ^{- 10} even in semiclassical domain. It is also shown that this approximation if taken as zero approximation
for solving the Schroedinger equation leads to an extremely fast convergent perturbation theory. A generalization to multidimensional anharmonic oscillators is briefly discussed.

I examine a codimension-two brane scenario in which all the observed Standard Model particles reside on a brane but the Higgs is an elementary extra-dimensional scalar in the bulk. I first present how the brane couplings between the Higgs field and the standard model particles ought to be renormalized for the low-energy theory to be regularization independent. I then show that these renormalized couplings yield two novel effects. First, they cause the Higgs vev to depend only logarithmically on the mass term of the bulk potential, thus providing a new mechanism for tackling the hierarchy problem. Second, the Higgs brane couplings cause the lowest mass KK mode to localize near the brane without any need for geometrical effects like warping. I will then finish by exploring some preliminary implications such models have for the Higgs signature at the LHC, both in the case where the extra dimensions arise at the TeV scale, and in ADD models having Large Extra Dimensions.

Following an introduction to nonequilibrium universality classes,
which occur in basic reaction-diffusion particle models I discuss
some results obtained by cluster mean-field approximations and
simulations of such models. These findings are unexpected by perturbative renormalization group (RG) methods, but are in agreement with non-perturbative RG. This suggests that diffusion may become relevant in case of competing reactions and nontrivial phase structures can emerge even in very simple system.