1,888,207 research outputs found
Cone fields and topological sampling in manifolds with bounded curvature
Often noisy point clouds are given as an approximation of a particular
compact set of interest. A finite point cloud is a compact set. This paper
proves a reconstruction theorem which gives a sufficient condition, as a bound
on the Hausdorff distance between two compact sets, for when certain offsets of
these two sets are homotopic in terms of the absence of {\mu}-critical points
in an annular region. Since an offset of a set deformation retracts to the set
itself provided that there are no critical points of the distance function
nearby, we can use this theorem to show when the offset of a point cloud is
homotopy equivalent to the set it is sampled from. The ambient space can be any
Riemannian manifold but we focus on ambient manifolds which have nowhere
negative curvature. In the process, we prove stability theorems for
{\mu}-critical points when the ambient space is a manifold.Comment: 20 pages, 3 figure
Homotopy properties of endpoint maps and a theorem of Serre in subriemannian geometry
We discuss homotopy properties of endpoint maps for affine control systems.
We prove that these maps are Hurewicz fibrations with respect to some
topology on the space of trajectories, for a certain . We study critical
points of geometric costs for these affine control systems, proving that if the
base manifold is compact then the number of their critical points is infinite
(we use Lusternik-Schnirelmann category combined with the Hurewicz property).
In the special case where the control system is subriemannian this result can
be read as the corresponding version of Serre's theorem, on the existence of
infinitely many geodesics between two points on a compact riemannian manifold.
In the subriemannian case we show that the Hurewicz property holds for all
and the horizontal-loop space with the topology has the
homotopy type of a CW-complex (as long as the endpoint map has at least one
regular value); in particular the inclusion of the horizontal-loop space in the
ordinary one is a homotopy equivalence
A unifying model for several two-dimensional phase transitions
A relatively simple and physically transparent model based on quantum
percolation and dephasing is employed to construct a global phase diagram which
encodes and unifies the critical physics of the quantum Hall, "two-dimensional
metal-insulator", classical percolation and, to some extent,
superconductor-insulator transitions. Using real space renormalization group
techniques, crossover functions between critical points are calculated. The
critical behavior around each fixed point is analyzed and some experimentally
relevant puzzles are addressed.Comment: 4 pages, including 3 figure
Conformal invariants measuring the best constants for Gagliardo-Nirenberg-Sobolev inequalities
We introduce a family of conformal invariants associated to a smooth metric
measure space which generalize the relationship between the Yamabe constant and
the best constant for the Sobolev inequality to the best constants for
Gagliardo-Nirenberg-Sobolev inequalities . These invariants are constructed via a minimization
procedure for the weighted scalar curvature functional in the conformal class
of a smooth metric measure space. We then describe critical points which are
also critical points for variations in the metric or the measure. When the
measure is assumed to take a special form --- for example, as the volume
element of an Einstein metric --- we use this description to show that
minimizers of our invariants are only critical for certain values of and
. In particular, on Euclidean space our result states that either
or , giving a new characterization of the GNS inequalities whose
sharp constants were computed by Del Pino and Dolbeault.Comment: 20 page
Computational Method for Phase Space Transport with Applications to Lobe Dynamics and Rate of Escape
Lobe dynamics and escape from a potential well are general frameworks
introduced to study phase space transport in chaotic dynamical systems. While
the former approach studies how regions of phase space are transported by
reducing the flow to a two-dimensional map, the latter approach studies the
phase space structures that lead to critical events by crossing periodic orbit
around saddles. Both of these frameworks require computation with curves
represented by millions of points-computing intersection points between these
curves and area bounded by the segments of these curves-for quantifying the
transport and escape rate. We present a theory for computing these intersection
points and the area bounded between the segments of these curves based on a
classification of the intersection points using equivalence class. We also
present an alternate theory for curves with nontransverse intersections and a
method to increase the density of points on the curves for locating the
intersection points accurately.The numerical implementation of the theory
presented herein is available as an open source software called Lober. We used
this package to demonstrate the application of the theory to lobe dynamics that
arises in fluid mechanics, and rate of escape from a potential well that arises
in ship dynamics.Comment: 33 pages, 17 figure
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