9,796 research outputs found
Conformal scattering on the Schwarzschild metric
We show that existing decay results for scalar fields on the Schwarzschild
metric are sufficient to obtain a conformal scattering theory. Then we
re-interpret this as an analytic scattering theory defined in terms of wave
operators, with an explicit comparison dynamics associated with the principal
null geodesic congruences. The case of the Kerr metric is also discussed.Comment: 36 pages, 6 figures. From the first version, recent references have
been added and the discussion has been modified to take the new references
into account. To appear in Annales de l'Institut Fourie
On Lars H\"ormander's remark on the characteristic Cauchy problem
We extend the results of a work by L. H\"ormander in 1990 concerning the
resolution of the characteristic Cauchy problem for second order wave equations
with regular first order potentials. The geometrical background of this work
was a spatially compact spacetime with smooth metric. The initial data surface
was spacelike or null at each point and merely Lipschitz. We lower the
regularity hypotheses on the metric and potential and obtain similar results.
The Cauchy problem for a spacelike initial data surface is solved for a
Lipschitz metric and coefficients of the first order potential that are
, with the same finite energy solution space as in the
smooth case. We also solve the fully characteristic Cauchy problem with very
slightly more regular metric and potential, namely a metric and a
potential with continuous first order terms and locally coefficients
for the terms of order 0.Comment: 21 pages Typing errors corrected in the estimates for the last
theorem, results extended from those of the previous versio
The characteristic Cauchy problem for Dirac fields on curved backgrounds
On arbitrary spacetimes, we study the characteristic Cauchy problem for Dirac
fields on a light-cone. We prove the existence and uniqueness of solutions in
the future of the light-cone inside a geodesically convex neighbourhood of the
vertex. This is done for data in and we give an explicit definition of
the space of data on the light-cone producing a solution in . The method
is based on energy estimates following L. H\"ormander (J.F.A. 1990). The data
for the characteristic Cauchy problem are only a half of the field, the other
half is recovered from the characteristic data by integration of the
constraints, consisting of the restriction of the Dirac equation to the cone. A
precise analysis of the dynamics of light rays near the vertex of the cone is
done in order to understand the integrability of the constraints; for this, the
Geroch-Held-Penrose formalism is used.Comment: 39 pages. An error in lemma 3.1 in the first version has been
corrected. Moreover, the treatment of the constraints (restriction of the
equations to the null cone) has been considerably extended and is now given
in full details. To appear in Journal of Hyperbolic Differential Equation
Regularity at space-like and null infinity
We extend Penrose's peeling model for the asymptotic behaviour of solutions
to the scalar wave equation at null infinity on asymptotically flat
backgrounds, which is well understood for flat space-time, to Schwarzschild and
the asymptotically simple space-times of Corvino-Schoen/Chrusciel-Delay. We
combine conformal techniques and vector field methods: a naive adaptation of
the ``Morawetz vector field'' to a conformal rescaling of the Schwarzschild
metric yields a complete scattering theory on Corvino-Schoen/Chrusciel-Delay
space-times. A good classification of solutions that peel arises from the use
of a null vector field that is transverse to null infinity to raise the
regularity in the estimates. We obtain a new characterization of solutions
admitting a peeling at a given order that is valid for both Schwarzschild and
Minkowski space-times. On flat space-time, this allows large classes of
solutions than the characterizations used since Penrose's work. Our results
establish the validity of the peeling model at all orders for the scalar wave
equation on the Schwarzschild metric and on the corresponding
Corvino-Schoen/Chrusciel-Delay space-times
On the semiclassical Laplacian with magnetic field having self-intersecting zero set
This paper is devoted to the spectral analysis of the Neumann realization of
the 2D magnetic Laplacian with semiclassical parameter h > 0 in the case when
the magnetic field vanishes along a smooth curve which crosses itself inside a
bounded domain. We investigate the behavior of its eigenpairs in the limit h
0. We show that each crossing point acts as a potential well,
generating a new decay scale of h 3/2 for the lowest eigenvalues, as well as
exponential concentration for eigenvectors around the set of crossing points.
These properties are consequences of the nature of associated model problems in
R 2 for which the zero set of the magnetic field is the union of two straight
lines. In this paper we also analyze the spectrum of model problems when the
angle between the two straight lines tends to 0
Critical reflexivity in financial markets: a Hawkes process analysis
We model the arrival of mid-price changes in the E-Mini S&P futures contract
as a self-exciting Hawkes process. Using several estimation methods, we find
that the Hawkes kernel is power-law with a decay exponent close to -1.15 at
short times, less than approximately 10^3 seconds, and crosses over to a second
power-law regime with a larger decay exponent of approximately -1.45 for longer
times scales in the range [10^3, 10^6] seconds. More importantly, we find that
the Hawkes kernel integrates to unity independently of the analysed period,
from 1998 to 2011. This suggests that markets are and have always been close to
criticality, challenging a recent study which indicates that reflexivity
(endogeneity) has increased in recent years as a result of increased automation
of trading. However, we note that the scale over which market events are
correlated has decreased steadily over time with the emergence of higher
frequency trading.Comment: 9 pages, 6 figures. Some clarification and correction made to section
II, minor alterations elsewher
Background subtraction based on Local Shape
We present a novel approach to background subtraction that is based on the
local shape of small image regions. In our approach, an image region centered
on a pixel is mod-eled using the local self-similarity descriptor. We aim at
obtaining a reliable change detection based on local shape change in an image
when foreground objects are moving. The method first builds a background model
and compares the local self-similarities between the background model and the
subsequent frames to distinguish background and foreground objects.
Post-processing is then used to refine the boundaries of moving objects.
Results show that this approach is promising as the foregrounds obtained are
com-plete, although they often include shadows.Comment: 4 pages, 5 figures, 3 tabl
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