We study the entropy of the set traced by an n-step simple symmetric random walk on ℤd. We show that for d≥3, the entropy is of order n. For d=2, the entropy is of order n/log2n. These values are essentially governed by the size of the boundary of the trace.
We study the textures of smectic-A liquid crystals consisting in curved, but stricdy equidistant lamellae. Assuming translational symmetry, we can generate them from a single curve. The free energy is a non-trivial functional of it. We learn how to derive the equilibrium equation for this curve, when the texture is confined between two parallel plates, which exert a weak anchoring on the orientation of the lamellae, but do not interfere direcdy with their position. Finally, we describe an instability...
A kinetic collision operator of Landau type for Fermi-Dirac particles is considered. Equilibrium states are rigorously determined under minimal assumptions on the distribution function of the particles. The particular structure of the considered operator (strong non-linearity and degeneracy) requires a special investigation compared to the classical Boltzmann or Landau operator.
We answer some questions raised by Gantert, Löwe and Steif (Ann. Inst. Henri Poincaré Probab. Stat.41(2005) 767–780) concerning “signed” voter models on locally finite graphs. These are voter model like processes with the difference that the edges are considered to be either positive or negative. If an edge between a site and a site is negative (respectively positive) the site will contribute towards the flip rate of if and only if the two current spin values are equal (respectively opposed)....
We study the boundedness of the Hausdorff measure of the singular set of any solution for a semi-linear elliptic equation in general dimensional Euclidean space . In our previous paper, we have clarified the structures of the nodal set and singular set of a solution for the semi-linear elliptic equation. In particular, we showed that the singular set is -rectifiable. In this paper, we shall show that under some additive smoothness assumptions, the -dimensional Hausdorff measure of singular set...