Rigidity in topology C^0 of the Poisson bracket for Tonelli Hamiltonians

We prove the following rigidity result for the Tonelli Hamiltonians. Let T * M be the cotangent bundle of a closed manifold M endowed with its usual symplectic form. Let (F\_n) be a sequence of Tonelli Hamiltonians that C^0 converges on the compact subsets to a Tonelli Hamiltonian F. Let (G\_n) be a sequence of Hamiltonians that that C^0 converges on the compact subsets to a Hamiltonian G. We assume that the sequence of the Poisson brackets ({F\_n , G\_n }) C^0-converges on the compact subsets to a C^1 function H. Then H = {F, G}

Pseudographs and Lax-Oleinik semi-group: a geometric and dynamical interpretation

Let H be a Tonelli Hamiltonian defined on the cotangent bundle of a compact and connected manifold and let u be a semi-concave function defined on M. If E (u) is the set of all the super-differentials of u and (\phi t) the Hamiltonian flow of H, we prove that for t > 0 small enough, \phi-t (E (u)) is an exact Lagrangian Lipschitz graph. This provides a geometric interpretation/explanation of a regularization tool that was introduced by P.~Bernard to prove the existence of C 1,1 subsolutions

Lower and upper bounds for the Lyapunov exponents of twisting dynamics: a relationship between the exponents and the angle of the Oseledet's splitting

We consider locally minimizing measures for the conservative twist maps of the $d$-dimensional annulus or for the Tonelli Hamiltonian flows defined on a cotangent bundle $T^*M$. For weakly hyperbolic such measures (i.e. measures with no zero Lyapunov exponents), we prove that the mean distance/angle between the stable and the unstable Oseledet's bundles gives an upper bound of the sum of the positive Lyapunov exponents and a lower bound of the smallest positive Lyapunov exponent. Some more precise results are proved too

Towards adiabatic waveforms for inspiral into Kerr black holes: I. A new model of the source for the time domain perturbation equation

We revisit the problem of the emission of gravitational waves from a test mass orbiting and thus perturbing a Kerr black hole. The source term of the Teukolsky perturbation equation contains a Dirac delta function which represents a point particle. We present a technique to effectively model the delta function and its derivatives using as few as four points on a numerical grid. The source term is then incorporated into a code that evolves the Teukolsky equation in the time domain as a (2+1) dimensional PDE. The waveforms and energy fluxes are extracted far from the black hole. Our comparisons with earlier work show an order of magnitude gain in performance (speed) and numerical errors less than 1% for a large fraction of parameter space. As a first application of this code, we analyze the effect of finite extraction radius on the energy fluxes. This paper is the first in a series whose goal is to develop adiabatic waveforms describing the inspiral of a small compact body into a massive Kerr black hole.Comment: 21 pages, 6 figures, accepted by PRD. This version removes the appendix; that content will be subsumed into future wor

Increase of the Number of Detectable Gravitational Waves Signals due to Gravitational Lensing

This article deals with the gravitational lensing (GL) of gravitational waves (GW). We compute the increase in the number of detected GW events due to GL. First, we check that geometrical optics is valid for the GW frequency range on which Earth-based detectors are sensitive, and that this is also partially true for what concerns the future space-based interferometer LISA. To infer this result, both the diffraction parameter and a cut-off frequency are computed. Then, the variation in the number of GW signals is estimated in the general case, and applied to some lens models: point mass lens and singular isothermal sphere (SIS profile). An estimation of the magnification factor has also been done for the softened isothermal sphere and for the King profile. The results appear to be strongly model-dependent, but in all cases the increase in the number of detected GW signals is negligible. The use of time delays among images is also investigated.Comment: Accepted for publication in General Relativity and Gravitatio

The Role of Calcium in Osteoporosis

Calcium requirements may vary throughout the lifespan. During the growth years and up to age 25 to 30, it is important to maximize dietary intake of calcium to maintain positive calcium balance and achieve peak bone mass, thereby possibly decreasing the risk of fracture when bone is subsequently lost. Calcium intake need not be greater than 800 mg/day during the relatively short period of time between the end of bone building and the onset of bone loss (30 to 40 years). Starting at age 40 to 50, both men and women lose bone slowly, but women lose bone more rapidly around the menopause and for about 10 years after. Intestinal calcium absorption and the ability to adapt to low calcium diets are impaired in many postmenopausal women and elderly persons owing to a suspected functional or absolute decrease in the ability of the kidney to produce 1,25(OH)2D2. The bones then become more and more a source of calcium to maintain critical extracellular fluid calcium levels. Excessive dietary intake of protein and fiber may induce significant negative calcium balance and thus increase dietary calcium requirements. Generally, the strongest risk factors for osteoporosis are uncontrollable (e.g., sex, age, and race) or less controllable (e.g., disease and medications). However, several factors such as diet, physical activity, cigarette smoking, and alcohol use are lifestyle related and can be modified to help reduce the risk of osteoporosis

BEC-BCS crossover in an optical lattice

We present the microscopic theory for the BEC-BCS crossover of an atomic Fermi gas in an optical lattice, showing that the Feshbach resonance underlying the crossover in principle induces strong multiband effects. Nevertheless, the BEC-BCS crossover itself can be described by a single-band model since it occurs at magnetic fields that are relatively far away from the Feshbach resonance. A criterion is proposed for the latter, which is obeyed by most known Feshbach resonances in ultracold atomic gases.Comment: 4 pages, 3 figure

Fluorescence from a few electrons

Systems containing few Fermions (e.g., electrons) are of great current interest. Fluorescence occurs when electrons drop from one level to another without changing spin. Only electron gases in a state of equilibrium are considered. When the system may exchange electrons with a large reservoir, the electron-gas fluorescence is easily obtained from the well-known Fermi-Dirac distribution. But this is not so when the number of electrons in the system is prevented from varying, as is the case for isolated systems and for systems that are in thermal contact with electrical insulators such as diamond. Our accurate expressions rest on the assumption that single-electron energy levels are evenly spaced, and that energy coupling and spin coupling between electrons are small. These assumptions are shown to be realistic for many systems. Fluorescence from short, nearly isolated, quantum wires is predicted to drop abruptly in the visible, a result not predicted by the Fermi-Dirac distribution. Our exact formulas are based on restricted and unrestricted partitions of integers. The method is considerably simpler than the ones proposed earlier, which are based on second quantization and contour integration.Comment: 10 pages, 3 figures, RevTe