Null Cones in Schwarzschild Geometry

Light cones of Schwarzschild geometry are studied in connection to the Null Surface Formulation and gravitational lensing. The paper studies the light cone cut function's singularity structure, gives exact gravitational lensing equations, and shows that the "pseudo-Minkowski" coordinates are well defined within the model considered.Comment: 31 pages, 5 figure

Mathematics of Gravitational Lensing: Multiple Imaging and Magnification

The mathematical theory of gravitational lensing has revealed many generic and global properties. Beginning with multiple imaging, we review Morse-theoretic image counting formulas and lower bound results, and complex-algebraic upper bounds in the case of single and multiple lens planes. We discuss recent advances in the mathematics of stochastic lensing, discussing a general formula for the global expected number of minimum lensed images as well as asymptotic formulas for the probability densities of the microlensing random time delay functions, random lensing maps, and random shear, and an asymptotic expression for the global expected number of micro-minima. Multiple imaging in optical geometry and a spacetime setting are treated. We review global magnification relation results for model-dependent scenarios and cover recent developments on universal local magnification relations for higher order caustics.Comment: 25 pages, 4 figures. Invited review submitted for special issue of General Relativity and Gravitatio

Large time wellposdness to the 3-D Capillary-Gravity Waves in the long wave regime

In the regime of weakly transverse long waves, given long-wave initial data, we prove that the nondimensionalized water wave system in an infinite strip under influence of gravity and surface tension on the upper free interface has a unique solution on [0,{T}/\eps] for some \eps independent of constant $T.$ We shall prove in the subsequent paper \cite{MZZ2} that on the same time interval, these solutions can be accurately approximated by sums of solutions of two decoupled Kadomtsev-Petviashvili (KP) equations.Comment: Split the original paper(The long wave approximation to the 3-D capillary-gravity waves) into two parts, this is the first on

Photoemission studies of Ga1−x_{1-x}Mnx_{x}As: Mn-concentration dependent properties

Using angle-resolved photoemission, we have investigated the development of the electronic structure and the Fermi level pinnning in Ga$_{1-x}$Mn$_{x}$As with Mn concentrations in the range 1--6%. We find that the Mn-induced changes in the valence-band spectra depend strongly on the Mn concentration, suggesting that the interaction between the Mn ions is more complex than assumed in earlier studies. The relative position of the Fermi level is also found to be concentration-dependent. In particular we find that for concentrations around 3.5--5% it is located very close to the valence-band maximum, which is in the range where metallic conductivity has been reported in earlier studies. For concentration outside this range, larger as well as smaller, the Fermi level is found to be pinned at about 0.15 eV higher energy.Comment: REVTeX style; 7 pages, 3 figure

Fuchsian convex bodies: basics of Brunn--Minkowski theory

The hyperbolic space \H^d can be defined as a pseudo-sphere in the $(d+1)$ Minkowski space-time. In this paper, a Fuchsian group $\Gamma$ is a group of linear isometries of the Minkowski space such that \H^d/\Gamma is a compact manifold. We introduce Fuchsian convex bodies, which are closed convex sets in Minkowski space, globally invariant for the action of a Fuchsian group. A volume can be associated to each Fuchsian convex body, and, if the group is fixed, Minkowski addition behaves well. Then Fuchsian convex bodies can be studied in the same manner as convex bodies of Euclidean space in the classical Brunn--Minkowski theory. For example, support functions can be defined, as functions on a compact hyperbolic manifold instead of the sphere. The main result is the convexity of the associated volume (it is log concave in the classical setting). This implies analogs of Alexandrov--Fenchel and Brunn--Minkowski inequalities. Here the inequalities are reversed

Global well-posedness of the 3-D full water wave problem

We consider the problem of global in time existence and uniqueness of solutions of the 3-D infinite depth full water wave problem. We show that the nature of the nonlinearity of the water wave equation is essentially of cubic and higher orders. For any initial interface that is sufficiently small in its steepness and velocity, we show that there exists a unique smooth solution of the full water wave problem for all time, and the solution decays at the rate $1/t$.Comment: 60 page

Dark soliton states of Bose-Einstein condensates in anisotropic traps

Dark soliton states of Bose-Einstein condensates in harmonic traps are studied both analytically and computationally by the direct solution of the Gross-Pitaevskii equation in three dimensions. The ground and self-consistent excited states are found numerically by relaxation in imaginary time. The energy of a stationary soliton in a harmonic trap is shown to be independent of density and geometry for large numbers of atoms. Large amplitude field modulation at a frequency resonant with the energy of a dark soliton is found to give rise to a state with multiple vortices. The Bogoliubov excitation spectrum of the soliton state contains complex frequencies, which disappear for sufficiently small numbers of atoms or large transverse confinement. The relationship between these complex modes and the snake instability is investigated numerically by propagation in real time.Comment: 11 pages, 8 embedded figures (two in color

Large time existence for 3D water-waves and asymptotics

We rigorously justify in 3D the main asymptotic models used in coastal oceanography, including: shallow-water equations, Boussinesq systems, Kadomtsev-Petviashvili (KP) approximation, Green-Naghdi equations, Serre approximation and full-dispersion model. We first introduce a ``variable'' nondimensionalized version of the water-waves equations which vary from shallow to deep water, and which involves four dimensionless parameters. Using a nonlocal energy adapted to the equations, we can prove a well-posedness theorem, uniformly with respect to all the parameters. Its validity ranges therefore from shallow to deep-water, from small to large surface and bottom variations, and from fully to weakly transverse waves. The physical regimes corresponding to the aforementioned models can therefore be studied as particular cases; it turns out that the existence time and the energy bounds given by the theorem are always those needed to justify the asymptotic models. We can therefore derive and justify them in a systematic way.Comment: Revised version of arXiv:math.AP/0702015 (notations simplified and remarks added) To appear in Inventione

Brain iron accumulation in Wilson disease: a post-mortem 7 Tesla MRI - histopathological study

Aims: In Wilson disease (WD), T2/T2*-weighted (T2*w) MRI frequently shows hypointensity in the basal ganglia that is suggestive of paramagnetic deposits. It is currently unknown whether this hypointensity is related to copper or iron deposition. We examined the neuropathological correlate of this MRI pattern, particularly in relation to iron and copper concentrations. Methods: Brain slices from nine WD and six control cases were investigated using a 7T-MRI system. High resolution T2*w images were acquired and R2* parametric maps were reconstructed using a multi-gradient recalled echo sequence. R2* was measured in the globus pallidus (GP) and the putamen. Corresponding histopathological sections containing the lentiform nucleus were examined using Turnbull iron staining, and double staining combining Turnbull with immunohistochemistry for macrophages or astrocytes. Quantitative densitometry of the iron staining as well as copper and iron concentrations were measured in the GP and putamen and correlated to R2* values. Results: T2*w hypointensity in the GP and/or putamen was apparent in WD cases and R2* values correlated with quantitative densitometry of iron staining. In WD, iron and copper concentrations were increased in the putamen compared to controls. R2* was correlated with the iron concentration in the GP and putamen whereas no correlation was observed for the copper concentration. Patients with more pronounced pathological severity in the putamen displayed increased iron concentration, which correlated with an elevated number of iron-containing macrophages. Conclusions: T2/T2*w hypointensity observed in vivo in the basal ganglia of WD patients is related to iron rather than copper deposits