Numerical range for random matrices

We analyze the numerical range of high-dimensional random matrices, obtaining limit results and corresponding quantitative estimates in the non-limit case. For a large class of random matrices their numerical range is shown to converge to a disc. In particular, numerical range of complex Ginibre matrix almost surely converges to the disk of radius $\sqrt{2}$. Since the spectrum of non-hermitian random matrices from the Ginibre ensemble lives asymptotically in a neighborhood of the unit disk, it follows that the outer belt of width $\sqrt{2}-1$ containing no eigenvalues can be seen as a quantification the non-normality of the complex Ginibre random matrix. We also show that the numerical range of upper triangular Gaussian matrices converges to the same disk of radius $\sqrt{2}$, while all eigenvalues are equal to zero and we prove that the operator norm of such matrices converges to $\sqrt{2e}$.Comment: 23 pages, 4 figure

Energy cascade and the four-fifths law in superfluid turbulence

The 4/5-law of turbulence, which characterizes the energy cascade from large to small-sized eddies at high Reynolds numbers in classical fluids, is verified experimentally in a superfluid 4He wind tunnel, operated down to 1.56 K and up to R_lambda ~ 1640. The result is corroborated by high-resolution simulations of Landau-Tisza's two-fluid model down to 1.15 K, corresponding to a residual normal fluid concentration below 3 % but with a lower Reynolds number of order R_lambda ~ 100. Although the K\'arm\'an-Howarth equation (including a viscous term) is not valid \emph{a priori} in a superfluid, it is found that it provides an empirical description of the deviation from the ideal 4/5-law at small scales and allows us to identify an effective viscosity for the superfluid, whose value matches the kinematic viscosity of the normal fluid regardless of its concentration.Comment: 6 pages, 7 figure

Archeops: an instrument for present and future cosmology

Archeops is a balloon-borne instrument dedicated to measure the cosmic microwave background (CMB) temperature anisotropies. It has, in the millimetre domain (from 143 to 545 GHz), a high angular resolution (about 10 arcminutes) in order to constrain high l multipoles, as well as a large sky coverage fraction (30%) in order to minimize the cosmic variance. It has linked, before WMAP, Cobe large angular scales to the first acoustic peak region. From its results, inflation motivated cosmologies are reinforced with a flat Universe (Omega_tot=1 within 3%). The dark energy density and the baryonic density are in very good agreement with other independent estimations based on supernovae measurements and big bang nucleosynthesis. Important results on galactic dust emission polarization and their implications for Planck are also addressed.Comment: 4 pages, 2 figures, to appear in Proceedings of the Multiwavelength Cosmology Conference, June 2003, Mykonos Island, Greec

The ratio of pattern speeds in double-barred galaxies

We have obtained two-dimensional velocity fields in the ionized gas of a set of 8 double-barred galaxies, at high spatial and spectral resolution, using their H$\alpha$ emission fields measured with a scanning Fabry-Perot spectrometer. Using the technique by which phase reversals in the non-circular motion indicate a radius of corotation, taking advantage of the high angular and velocity resolution we have obtained the corotation radii and the pattern speeds of both the major bar and the small central bar in each of the galaxies; there are few such measurements in the literature. Our results show that the inner bar rotates more rapidly than the outer bar by a factor between 3.3 and 3.6.Comment: 5 pages, 1 figure, 1 tabl

Rare mutations limit of a steady state dispersal evolution model

International audienceThe evolution of dispersal is a classical question in evolutionary ecology, which has been widely studied with several mathematical models. The main question is to define the fittest dispersal rate for a population in a bounded domain, and, more recently, for traveling waves in the full space. In the present study, we reformulate the problem in the context of adaptive evolution. We consider a population structured by space and a genetic trait acting directly on the dispersal (diffusion) rate under the effect of rare mutations on the genetic trait. We show that, as in simpler models, in the limit of vanishing mutations, the population concentrates on a single trait associated to the lowest dispersal rate. We also explain how to compute the evolution speed towards this evolutionary stable distribution. The mathematical interest stems from the asymptotic analysis which requires a completely different treatment of the different variables. For the space variable, the ellipticity leads to the use the maximum principle and Sobolev-type regularity results. For the trait variable, the concentration to a Dirac mass requires a different treatment. This is based on the WKB method and viscosity solutions leading to an effective Hamiltonian (effective fitness of the population) and a constrained Hamilton-Jacobi equation

Semi-classical analysis of real atomic spectra beyond Gutzwiller's approximation

Real atomic systems, like the hydrogen atom in a magnetic field or the helium atom, whose classical dynamics are chaotic, generally present both discrete and continuous symmetries. In this letter, we explain how these properties must be taken into account in order to obtain the proper (i.e. symmetry projected) $\hbar$ expansion of semiclassical expressions like the Gutzwiller trace formula. In the case of the hydrogen atom in a magnetic field, we shed light on the excellent agreement between present theory and exact quantum results.Comment: 4 pages, 1 figure, final versio

Comment on "Turbulent heat transport near critical points: Non-Boussinesq effects" (cond-mat/0601398)

In a recent preprint (cond-mat/0601398), D. Funfschilling and G. Ahlers describe a new effect, that they interpret as non-Boussinesq, in a convection cell working with ethane, near its critical point. They argue that such an effect could have spoiled the Chavanne {\it et al.} (Phys. Rev. Lett. {\bf 79} 3648, 1997) results, and not the Niemela {\it et al.} (Nature, {\bf 404}, 837, 2000) ones, which would explain the differences between these two experiments. We show that:-i)Restricting the Chavanne's data to situations as far from the critical point than the Niemela's one, the same discrepancy remains.-ii)The helium data of Chavanne show no indication of the effect observed by D. Funfschilling and G. Ahlers.Comment: comment on cond-mat/060139