9,796 research outputs found

    Conformal scattering on the Schwarzschild metric

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    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

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    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 Lloc∞L^\infty_\mathrm{loc}, 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 C1{\cal C}^1 metric and a potential with continuous first order terms and locally L∞L^\infty 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

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    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 L2L^2 and we give an explicit definition of the space of data on the light-cone producing a solution in H1H^1. 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

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    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

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    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 →\rightarrow 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

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    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

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    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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