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

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

Resonances near the real axis for manifolds with hyperbolic trapped sets

For manifolds Euclidian at infinity and compact perturbations of the Laplacian, we show that under assumptions involving hyperbolicity of the classical flow on the trapped set and its period spectrum, there are strips below the real axis where the resonance counting function grows sub-linearly. We also provide an inverse result, showing that the knowledge of the scattering poles can give some information about the Hausdorff dimension of the trapped set when the classical flow satisfies the Axiom-A condition

The Weak Lefschetz Property and powers of linear forms in K[x,y,z]

We show that an Artinian quotient of K[x, y, z] by an ideal I generated by powers of linear forms has the Weak Lefschetz property. If the syzygy bundle of I is semistable this follows from results of Brenner-Kaid; our proof works without this hypothesis, which typically does not hold.Comment: 5 pages, to appear in PAM

Reasoning About Liquids via Closed-Loop Simulation

Simulators are powerful tools for reasoning about a robot's interactions with its environment. However, when simulations diverge from reality, that reasoning becomes less useful. In this paper, we show how to close the loop between liquid simulation and real-time perception. We use observations of liquids to correct errors when tracking the liquid's state in a simulator. Our results show that closed-loop simulation is an effective way to prevent large divergence between the simulated and real liquid states. As a direct consequence of this, our method can enable reasoning about liquids that would otherwise be infeasible due to large divergences, such as reasoning about occluded liquid.Comment: Robotics: Science & Systems (RSS), July 12-16, 2017. Cambridge, MA, US