11,376 research outputs found
Smooth planar -splines of degree
In \cite{as}, Alfeld and Schumaker give a formula for the dimension of the
space of piecewise polynomial functions (splines) of degree and smoothness
on a generic triangulation of a planar simplicial complex (for ) and any triangulation (for ). In \cite{ss}, it was
conjectured that the Alfeld-Schumaker formula actually holds for all . In this note, we show that this is the best result possible; in
particular, there exists a simplicial complex such that for any ,
the dimension of the spline space in degree 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
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