Weil Representation of a Generalized Linear Group over a Ring of Truncated Polynomials over a Finite Field Endowed with a Second Class Involution

We construct a complex linear Weil representation $\rho$ of the generalized special linear group $G={\rm SL}_*^{1}(2,A_n)$ ($A_n=K[x]/\langle x^n\rangle$, $K$ the quadratic extension of the finite field $k$ of $q$ elements, $q$ odd), where $A_n$ is endowed with a second class involution. After the construction of a specific data, the representation is defined on the generators of a Bruhat presentation of $G$, via linear operators satisfying the relations of the presentation. The structure of a unitary group $U$ associated to $G$ is described. Using this group we obtain a first decomposition of $\rho$

Fermion localization on thick branes

We consider chiral fermion confinement in scalar thick branes, which are known to localize gravity, coupled through a Yukawa term. The conditions for the confinement and their behavior in the thin-wall limit are found for various different BPS branes, including double walls and branes interpolating between different AdS_5 spacetimes. We show that only one massless chiral mode is localized in all these walls, whenever the wall thickness is keep finite. We also show that, independently of wall's thickness, chiral fermionic modes cannot be localized in dS_4 walls embedded in a M_5 spacetime. Finally, massive fermions in double wall spacetimes are also investigated. We find that, besides the massless chiral mode localization, these double walls support quasi-localized massive modes of both chiralities.Comment: 8 pages, 3 figure

The asymmetric multitype contact process

In the multitype contact process, vertices of a graph can be empty or occupied by a type 1 or a type 2 individual; an individual of type $i$ dies with rate 1 and sends a descendant to a neighboring empty site with rate $\lambda_i$. We study this process on $\Z^d$ with $\lambda_1 > \lambda_2$ and $\lambda_1$ larger than the critical value of the (one-type) contact process. We prove that, if there is at least one type 1 individual in the initial configuration, then type 1 has a positive probability of never going extinct. Conditionally on this event, type 1 takes over a ball of radius growing linearly in time. We also completely characterize the set of stationary distributions of the process and prove that the process started from any initial configuration converges to a convex combination of distributions in this set