C2 piecewise cubic quasi-interpolants on a 6-direction mesh

We study two kinds of quasi-interpolants (abbr. QI) in the space of C2 piecewise cubics in the plane, or in a rectangular domain, endowed with the highly symmetric triangulation generated by a uniform 6-direction mesh. It has been proved recently that this space is generated by the integer translates of two multi-box splines. One kind of QIs is of differential type and the other of discrete type. As those QIs are exact on the space of cubic polynomials, their approximation order is 4 for sufficiently smooth functions. In addition, they exhibit nice superconvergent properties at some specific points. Moreover, the infinite norms of the discrete QIs being small, they give excellent approximations of a smooth function and of its first order partial derivatives. The approximation properties of the QIs are illustrated by numerical examples

Simple linear compactifications of odd orthogonal groups

We classify the simple linear compactifications of SO(2r+1), namely those compactifications with a unique closed orbit which are obtained by taking the closure of the SO(2r+1)xSO(2r+1)-orbit of the identity in a projective space P(End(V)), where V is a finite dimensional rational SO(2r+1)-module.Comment: v2: several simplifications, final version. To appear in J. Algebr

Collisional dissipation of Alfvén waves in a partially ionised solar chromosphere

Certain regions of the solar atmosphere are at sufficiently low temperatures to be only partially ionised. The lower chromosphere contains neutral atoms, the existence of which greatly increases the efficiency of the damping of waves due to collisional friction momentum transfer. More specifically the Cowling conductivity can be up to 12 orders of magnitude smaller than the Spitzer value, so that the main damping mechanism in this region is due to the collisions between neutrals and positive ions (Khodachenko et al. 2004, A&A, 422, 1073). Using values for the gas density and temperature as functions of height taken from the VAL C model of the quiet Sun (Vernazza et al. 1981, ApJS, 45, 635), an estimate is made for the dependance of the Cowling conductivity on height and strength of magnetic field. Using both analytic and numerical approaches the passage of Alfvén waves over a wide spectrum through this partially ionised region is investigated. Estimates of the efficiency of this region in the damping of Alfvén waves are made and compared for both approaches. We find that Alfvén waves with frequencies above 0.6 Hz are completely damped and frequencies below 0.01 Hz unaffected

Stopping Light on a Defect

Gap solitons are localized nonlinear coherent states which have been shown both theoretically and experimentally to propagate in periodic structures. Although theory allows for their propagation at any speed $v$, $0\le v\le c$, they have been observed in experiments at speeds of approximately 50% of $c$. It is of scientific and technological interest to trap gap solitons. We first introduce an explicit multiparameter family of periodic structures with localized defects, which support linear defect modes. These linear defect modes are shown to persist into the nonlinear regime, as {\it nonlinear defect modes}. Using mathematical analysis and numerical simulations we then investigate the capture of an incident gap soliton by these defects. The mechanism of capture of a gap soliton is resonant transfer of its energy to nonlinear defect modes. We introduce a useful bifurcation diagram from which information on the parameter regimes of gap soliton capture, reflection and transmission can be obtained by simple conservation of energy and resonant energy transfer principles.Comment: 45 pages, Submitted to Journal of the Optical Society

Magnetohydrodynamics of the Weakly Ionized Solar Photosphere

We investigate the importance of ambipolar diffusion and Hall currents for high-resolution comprehensive ('realistic') photospheric simulations. To do so we extended the radiative magnetohydrodynamics code \emph{MURaM} to use the generalized Ohm's law under the assumption of local thermodynamic equilibrium. We present test cases comparing analytical solutions with numerical simulations for validation of the code. Furthermore, we carried out a number of numerical experiments to investigate the impact of these neutral-ion effects in the photosphere. We find that, at the spatial resolutions currently used (5-20 km per grid point), the Hall currents and ambipolar diffusion begin to become significant -- with flows of 100 m/s in sunspot light bridges, and changes of a few percent in the thermodynamic structure of quiet-Sun magnetic features. The magnitude of the effects is expected to increase rapidly as smaller-scale variations are resolved by the simulations.Comment: accepted Ap

Box spline prewavelets of small support

The purpose of this paper is the construction of bi- and trivariate prewavelets from box-spline spaces, \ie\ piecewise polynomials of fixed degree on a uniform mesh. They have especially small support and form Riesz bases of the wavelet spaces, so they are stable. In particular, the supports achieved are smaller than those of the prewavelets due to Riemenschneider and Shen in a recent, similar constructio

Moving gap solitons in periodic potentials

We address existence of moving gap solitons (traveling localized solutions) in the Gross-Pitaevskii equation with a small periodic potential. Moving gap solitons are approximated by the explicit localized solutions of the coupled-mode system. We show however that exponentially decaying traveling solutions of the Gross-Pitaevskii equation do not generally exist in the presence of a periodic potential due to bounded oscillatory tails ahead and behind the moving solitary waves. The oscillatory tails are not accounted in the coupled-mode formalism and are estimated by using techniques of spatial dynamics and local center-stable manifold reductions. Existence of bounded traveling solutions of the Gross--Pitaevskii equation with a single bump surrounded by oscillatory tails on a finite large interval of the spatial scale is proven by using these technique. We also show generality of oscillatory tails in other nonlinear equations with a periodic potential.Comment: 22 pages, 2 figure

Optical correlation techniques for the investigation of colloidal systems

This review aims to provide a simple introduction to the application of optical correlation methods in colloidal science. In particular, I plan to show that full appraisal of the intimate relation between light scattering and microscopy allows designing novel powerful investigation techniques that combine their powers. An extended version of this paper will appear in "ColloidalFoundations of Nanoscience", edited by D. Berti and G. Palazzo, Elsevier (ISBN 978-0-444-59541-6). I am very grateful to the publisher for having granted me the permission to post this preprint on arXiv.Comment: 19 pages, 5 figure

Phase Reversal Diffraction in incoherent light

Phase reversal occurs in the propagation of an electromagnetic wave in a negatively refracting medium or a phase-conjugate interface. Here we report the experimental observation of phase reversal diffraction without the above devices. Our experimental results and theoretical analysis demonstrate that phase reversal diffraction can be formed through the first-order field correlation of chaotic light. The experimental realization is similar to phase reversal behavior in negatively refracting media.Comment: 8 pages, 5 figure

Surface-Enhanced Plasmon Splitting in a Liquid-Crystal-Coated Gold Nanoparticle

We show that, when a gold nanoparticle is coated by a thin layer of nematic liquid crystal, the deformation produced by the nanoparticle surface can enhance the splitting of the nanoparticle surface plasmon. We consider three plausible liquid crystal director configurations in zero electric field: boojum pair (north-south pole configuration), baseball (tetrahedral), and homogeneous. From a calculation using the Discrete Dipole Approximation, we find that the surface plasmon splitting is largest for the boojum pair, intermediate for the homogeneous, and smallest for the baseball configuration. The boojum pair results are in good agreement with experiment. We conclude that the nanoparticle surface has a strong effect on the director orientation, but, surprisingly, that this deformation can actually enhance the surface plasmon splitting.Comment: 5 pages, 3 figures To be published in PR