Weak KAM for commuting Hamiltonians

For two commuting Tonelli Hamiltonians, we recover the commutation of the Lax-Oleinik semi-groups, a result of Barles and Tourin ([BT01]), using a direct geometrical method (Stoke's theorem). We also obtain a "generalization" of a theorem of Maderna ([Mad02]). More precisely, we prove that if the phase space is the cotangent of a compact manifold then the weak KAM solutions (or viscosity solutions of the critical stationary Hamilton-Jacobi equation) for G and for H are the same. As a corrolary we obtain the equality of the Aubry sets, of the Peierls barrier and of flat parts of Mather's $\alpha$ functions. This is also related to works of Sorrentino ([Sor09]) and Bernard ([Ber07b]).Comment: 23 pages, accepted for publication in NonLinearity (january 29th 2010). Minor corrections, fifth part added on Mather's $\alpha$ function (or effective Hamiltonian

A new kind of Lax-Oleinik type operator with parameters for time-periodic positive definite Lagrangian systems

In this paper we introduce a new kind of Lax-Oleinik type operator with parameters associated with positive definite Lagrangian systems for both the time-periodic case and the time-independent case. On one hand, the new family of Lax-Oleinik type operators with an arbitrary $u\in C(M,\mathbb{R}^1)$ as initial condition converges to a backward weak KAM solution in the time-periodic case, while it was shown by Fathi and Mather that there is no such convergence of the Lax-Oleinik semigroup. On the other hand, the new family of Lax-Oleinik type operators with an arbitrary $u\in C(M,\mathbb{R}^1)$ as initial condition converges to a backward weak KAM solution faster than the Lax-Oleinik semigroup in the time-independent case.Comment: We give a new definition of Lax-Oleinik type operator; add some reference

Duality relations in the auxiliary field method

The eigenenergies $\epsilon^{(N)}(m;\{n_i,l_i\})$ of a system of $N$ identical particles with a mass $m$ are functions of the various radial quantum numbers $n_i$ and orbital quantum numbers $l_i$. Approximations $E^{(N)}(m;Q)$ of these eigenenergies, depending on a principal quantum number $Q(\{n_i,l_i\})$, can be obtained in the framework of the auxiliary field method. We demonstrate the existence of numerous exact duality relations linking quantities $E^{(N)}(m;Q)$ and $E^{(p)}(m';Q')$ for various forms of the potentials (independent of $m$ and $N$) and for both nonrelativistic and semirelativistic kinematics. As the approximations computed with the auxiliary field method can be very close to the exact results, we show with several examples that these duality relations still hold, with sometimes a good accuracy, for the exact eigenenergies $\epsilon^{(N)}(m;\{n_i,l_i\})$

Optimizing ISOCAM data processing using spatial redundancy

We present new data processing techniques that allow to correct the main instrumental effects that degrade the images obtained by ISOCAM, the camera on board the Infrared Space Observatory (ISO). Our techniques take advantage of the fact that a position on the sky has been observed by several pixels at different times. We use this information (1) to correct the long term variation of the detector response, (2) to correct memory effects after glitches and point sources, and (3) to refine the deglitching process. Our new method allows the detection of faint extended emission with contrast smaller than 1% of the zodiacal background. The data reduction corrects instrumental effects to the point where the noise in the final map is dominated by the readout and the photon noises. All raster ISOCAM observations can benefit from the data processing described here. These techniques could also be applied to other raster type observations (e.g. ISOPHOT or IRAC on SIRTF).Comment: 13 pages, 10 figures, to be published in Astronomy and Astrophysics Supplement Serie

On the number of Mather measures of Lagrangian systems

In 1996, Ricardo Ricardo Ma\~n\'e discovered that Mather measures are in fact the minimizers of a "universal" infinite dimensional linear programming problem. This fundamental result has many applications, one of the most important is to the estimates of the generic number of Mather measures. Ma\~n\'e obtained the first estimation of that sort by using finite dimensional approximations. Recently, we were able with Gonzalo Contreras to use this method of finite dimensional approximation in order to solve a conjecture of John Mather concerning the generic number of Mather measures for families of Lagrangian systems. In the present paper we obtain finer results in that direction by applying directly some classical tools of convex analysis to the infinite dimensional problem. We use a notion of countably rectifiable sets of finite codimension in Banach (and Frechet) spaces which may deserve independent interest

Collodial particles at a range of fluid-fluid particles

The study of solid particles residing at fluid-fluid interfaces has become an established area in surface and colloid science recently experiencing a renaissance since around 2000. Particles at interfaces arise in many industrial products and processes like anti-foam formulations, crude oil emulsions, aerated foodstuffs and flotation. Although they act in many ways like traditional surfactant molecules, they offer distinct advantages also and the area is now multi-disciplinary involving research in the fundamental science and potential applications. In this Feature Article, a flavour of some of this interest is given based on recent work from our own group and includes the behaviour of particles at oil-water, air-water, oil-oil, air-oil and water-water interfaces. The materials capable of being prepared by assembling various kinds of particles at fluid interfaces include particle-stabilised emulsions, particle-stabilised aqueous and oil foams, dry liquids, liquid marbles and powdered emulsions

Heavy-Light Semileptonic Decays in Staggered Chiral Perturbation Theory

We calculate the form factors for the semileptonic decays of heavy-light pseudoscalar mesons in partially quenched staggered chiral perturbation theory (\schpt), working to leading order in $1/m_Q$, where $m_Q$ is the heavy quark mass. We take the light meson in the final state to be a pseudoscalar corresponding to the exact chiral symmetry of staggered quarks. The treatment assumes the validity of the standard prescription for representing the staggered ``fourth root trick'' within \schpt by insertions of factors of 1/4 for each sea quark loop. Our calculation is based on an existing partially quenched continuum chiral perturbation theory calculation with degenerate sea quarks by Becirevic, Prelovsek and Zupan, which we generalize to the staggered (and non-degenerate) case. As a by-product, we obtain the continuum partially quenched results with non-degenerate sea quarks. We analyze the effects of non-leading chiral terms, and find a relation among the coefficients governing the analytic valence mass dependence at this order. Our results are useful in analyzing lattice computations of form factors $B\to\pi$ and $D\to K$ when the light quarks are simulated with the staggered action.Comment: 53 pages, 8 figures, v2: Minor correction to the section on finite volume effects, and typos fixed. Version to be published in Phys. Rev.

Low Dirac Eigenmodes and the Topological and Chiral Structure of the QCD Vacuum

Several lattice calculations which probe the chiral and topological structure of QCD are discussed. The results focus attention on the low-lying eigenmodes of the Dirac operator in typical gauge field configurations.Comment: Talk presented at the DPF2000 Conferenc

Extinction and dust/gas ratio in LMC molecular clouds

Aims. The goal of this paper is to measure the dust content and distribution in the Large Magellanic Cloud (LMC) by comparing extinction maps produced in the near-infrared wavelengths and the spatial distribution of the neutral and molecular gas, as traced by Hi and CO observations. Methods. In order to derive an extinction map of the LMC, we have developed a new method to measure the color excess of dark clouds, using the 2MASS all-sky survey. Classical methods to measure the color excess (including the NICE method) tend to underestimate the true color excess if the clouds are significantly contaminated by unreddened foreground stars, as is the case in the LMC. We propose a new method that uses the color of the X percentile reddest stars and which is robust against such contamination. Using this method, it is possible to infer the positions of dark clouds with respect to the star distribution by comparing the observed color excess as a function of the percentile used and that predicted by a model. Results. On the basis of the resulting extinction map, we perform a correlation analysis for a set of dark molecular clouds. Assuming similar infrared absorption properties for the dust in the neutral and molecular phases, we derive the absorption-to-column density ratio AV/NH and the CO-to-H2 conversion factor X_(CO). We show that AV/NH increases from the outskirts of the LMC towards the 30 Dor star-forming region. This can be explained either by a systematic increase of the dust abundance, or by the presence of an additional gas component not traced by Hi or CO, but strongly correlated to the Hi distribution. If dust abundance is allowed to vary, the derived X_(CO) factors for the selected regions are several times lower than those derived from a virial analysis of the CO data. This could indicate that molecular clouds in the LMC are not gravitationally bound, or that they are bounded by substantial external pressure. However, the X_(CO) values derived from absorption can be reconciled with the virial results assuming a constant value for the dust abundance and the existence of an additional, unseen gas component. These results are in agreement with those derived for the LMC from diffuse far-infrared emission