172 research outputs found
Cut locus and heat kernel at Grushin points of 2 dimensional almost Riemannian metrics
This article deals with 2d almost Riemannian structures, which are
generalized Riemannian structures on manifolds of dimension 2. Such
sub-Riemannian structures can be locally defined by a pair of vector fields
(X,Y), playing the role of orthonormal frame, that may become colinear on some
subset. We denote D = span(X,Y). After a short introduction, I first give a
description of the local cut and conjugate loci at a Grushin point q (where Dq
has dimension 1 and Dq = TqM) that makes appear that the cut locus may have an
angle at q. In a second time I describe the local cut and conjugate loci at a
Riemannian point x in a neighborhood of a Grushin point q. Finally, applying
results of [6], I give the asymptotics in small time of the heat kernel
p_t(x,y) for y in the same neighborhood of q
Existence of planar curves minimizing length and curvature
In this paper we consider the problem of reconstructing a curve that is
partially hidden or corrupted by minimizing the functional , depending both on length and curvature . We fix
starting and ending points as well as initial and final directions.
For this functional we discuss the problem of existence of minimizers on
various functional spaces. We find non-existence of minimizers in cases in
which initial and final directions are considered with orientation. In this
case, minimizing sequences of trajectories can converge to curves with angles.
We instead prove existence of minimizers for the "time-reparameterized"
functional \int \| \dot\gamma(t) \|\sqrt{1+K_\ga^2} dt for all boundary
conditions if initial and final directions are considered regardless to
orientation. In this case, minimizers can present cusps (at most two) but not
angles
Normal forms and invariants for 2-dimensional almost-Riemannian structures
Two-dimensional almost-Riemannian structures are generalized Riemannian
structures on surfaces for which a local orthonormal frame is given by a Lie
bracket generating pair of vector fields that can become collinear.
Generically, there are three types of points: Riemannian points where the two
vector fields are linearly independent, Grushin points where the two vector
fields are collinear but their Lie bracket is not, and tangency points where
the two vector fields and their Lie bracket are collinear and the missing
direction is obtained with one more bracket. In this paper we consider the
problem of finding normal forms and functional invariants at each type of
point. We also require that functional invariants are "complete" in the sense
that they permit to recognize locally isometric structures. The problem happens
to be equivalent to the one of finding a smooth canonical parameterized curve
passing through the point and being transversal to the distribution. For
Riemannian points such that the gradient of the Gaussian curvature is
different from zero, we use the level set of as support of the
parameterized curve. For Riemannian points such that the gradient of the
curvature vanishes (and under additional generic conditions), we use a curve
which is found by looking for crests and valleys of the curvature. For Grushin
points we use the set where the vector fields are parallel. Tangency points are
the most complicated to deal with. The cut locus from the tangency point is not
a good candidate as canonical parameterized curve since it is known to be
non-smooth. Thus, we analyse the cut locus from the singular set and we prove
that it is not smooth either. A good candidate appears to be a curve which is
found by looking for crests and valleys of the Gaussian curvature. We prove
that the support of such a curve is uniquely determined and has a canonical
parametrization
Stability of Planar Nonlinear Switched Systems
We consider the time-dependent nonlinear system , where , and are two
% smooth vector fields, globally asymptotically stable at the origin
and is an arbitrary measurable function. Analysing the
topology of the set where and are parallel, we give some sufficient and
some necessary conditions for global asymptotic stability, uniform with respect
to . Such conditions can be verified without any integration or
construction of a Lyapunov function, and they are robust under small
perturbations of the vector fields
Local properties of almost-Riemannian structures in dimension 3
A 3D almost-Riemannian manifold is a generalized Riemannian manifold defined
locally by 3 vector fields that play the role of an orthonormal frame, but
could become collinear on some set \Zz called the singular set. Under the
Hormander condition, a 3D almost-Riemannian structure still has a metric space
structure, whose topology is compatible with the original topology of the
manifold. Almost-Riemannian manifolds were deeply studied in dimension 2. In
this paper we start the study of the 3D case which appear to be reacher with
respect to the 2D case, due to the presence of abnormal extremals which define
a field of directions on the singular set. We study the type of singularities
of the metric that could appear generically, we construct local normal forms
and we study abnormal extremals. We then study the nilpotent approximation and
the structure of the corresponding small spheres. We finally give some
preliminary results about heat diffusion on such manifolds
The sphere and the cut locus at a tangency point in two-dimensional almost-Riemannian geometry
We study the tangential case in 2-dimensional almost-Riemannian geometry. We
analyse the connection with the Martinet case in sub-Riemannian geometry. We
compute estimations of the exponential map which allow us to describe the
conjugate locus and the cut locus at a tangency point. We prove that this last
one generically accumulates at the tangency point as an asymmetric cusp whose
branches are separated by the singular set
Lipschitz classification of almost-Riemannian distances on compact oriented surfaces
International audienceTwo-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. We consider the Carnot--Caratheodory distance canonically associated with an almost-Riemannian structure and study the problem of Lipschitz equivalence between two such distances on the same compact oriented surface. We analyse the generic case, allowing in particular for the presence of tangency points, i.e., points where two generators of the distribution and their Lie bracket are linearly dependent. The main result of the paper provides a characterization of the Lipschitz equivalence class of an almost-Riemannian distance in terms of a labelled graph associated with it
Search for the standard model Higgs boson in the H to ZZ to 2l 2nu channel in pp collisions at sqrt(s) = 7 TeV
A search for the standard model Higgs boson in the H to ZZ to 2l 2nu decay
channel, where l = e or mu, in pp collisions at a center-of-mass energy of 7
TeV is presented. The data were collected at the LHC, with the CMS detector,
and correspond to an integrated luminosity of 4.6 inverse femtobarns. No
significant excess is observed above the background expectation, and upper
limits are set on the Higgs boson production cross section. The presence of the
standard model Higgs boson with a mass in the 270-440 GeV range is excluded at
95% confidence level.Comment: Submitted to JHE
Search for New Physics with Jets and Missing Transverse Momentum in pp collisions at sqrt(s) = 7 TeV
A search for new physics is presented based on an event signature of at least
three jets accompanied by large missing transverse momentum, using a data
sample corresponding to an integrated luminosity of 36 inverse picobarns
collected in proton--proton collisions at sqrt(s)=7 TeV with the CMS detector
at the LHC. No excess of events is observed above the expected standard model
backgrounds, which are all estimated from the data. Exclusion limits are
presented for the constrained minimal supersymmetric extension of the standard
model. Cross section limits are also presented using simplified models with new
particles decaying to an undetected particle and one or two jets
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