Smooth planar rr-splines of degree 2r2r

In \cite{as}, Alfeld and Schumaker give a formula for the dimension of the space of piecewise polynomial functions (splines) of degree $d$ and smoothness $r$ on a generic triangulation of a planar simplicial complex $\Delta$ (for $d \ge 3r+1$) and any triangulation (for $d\geq 3r+2$). In \cite{ss}, it was conjectured that the Alfeld-Schumaker formula actually holds for all $d \ge 2r+1$. In this note, we show that this is the best result possible; in particular, there exists a simplicial complex $\Delta$ such that for any $r$, the dimension of the spline space in degree $d=2r$ is not given by the formula of \cite{as}. The proof relies on the explicit computation of the nonvanishing of the first local cohomology module described in \cite{ss2}.Comment: 6 pages, 1 figur

Exponential stabilization without geometric control

We present examples of exponential stabilization for the damped wave equation on a compact manifold in situations where the geometric control condition is not satisfied. This follows from a dynamical argument involving a topological pressure on a suitable uncontrolled set

Toric surface codes and Minkowski sums

Toric codes are evaluation codes obtained from an integral convex polytope $P \subset \R^n$ and finite field \F_q. They are, in a sense, a natural extension of Reed-Solomon codes, and have been studied recently by J. Hansen and D. Joyner. In this paper, we obtain upper and lower bounds on the minimum distance of a toric code constructed from a polygon $P \subset \R^2$ by examining Minkowski sum decompositions of subpolygons of $P$. Our results give a simple and unifying explanation of bounds of Hansen and empirical results of Joyner; they also apply to previously unknown cases.Comment: 15 pages, 7 figures; This version contains some minor editorial revisions -- to appear SIAM Journal on Discrete Mathematic

Weyl laws for partially open quantum maps

We study a toy model for "partially open" wave-mechanical system, like for instance a dielectric micro-cavity, in the semiclassical limit where ray dynamics is applicable. Our model is a quantized map on the 2-dimensional torus, with an additional damping at each time step, resulting in a subunitary propagator, or "damped quantum map". We obtain analogues of Weyl's laws for such maps in the semiclassical limit, and draw some more precise estimates when the classical dynamic is chaotic.Comment: 35 pages, 5 figures. Corrected typos. Some proofs clarifie