On the Reduction of Some Dispersionless Integrable Systems

We study the invariance, reduction, exact solutions and conservation laws of the dispersionless Kadomtsev-Petviashivili and the heavenly equation. The existence of nontrivial conservation laws lead to repeated reductions paving the way for determining exact solutions. A variety of such solutions exist dependent on the subalgebra of Lie point symmetries that is chosen in the reduction procedure.     

The exterior magnetic field for the multilayer ellipsoidal model of the brain

The magnetic induction field in the exterior of an ellipsoidallyinhomogeneous, four-conducting-layer model of the human headis obtained analytically up to its quadrupole approximation.The interior ellipsoidal core represents the homogeneous brainwhile each one of the shells represents the cerebrospinal fluid,the skull and the scalp, all characterized by different conductivities.The inhomogeneities of these four domains, together with theanisotropy imposed by the use of the ellipsoidal geometry, providethe most realistic physical and geometrical model of the brainfor which an analytic solution of the biomagnetic forward problemis possible. It is shown that in contrast to the spherical model,where shells of different conductivity are magnetically invisible,the magnetic induction field in ellipsoidal geometry is stronglydependent on the conductivity supports. The fact that sphericalshells of different conductivity are invisible has enhancedthe common belief that the biomagnetic forward solution doesnot depend on the conductivity profiles. As we demonstrate inthe present work, this is not true. Hence, the proposed multilayeredellipsoidal model provides a qualitative improvement of therealistic interpretation of magnetoencephalography (MEG) measurements.We show that the presence of the shells of different conductivitycan be incorporated in the form of the dipole vector for thehomogeneous model. Numerical investigations show that the effectsof shell inhomogeneities are almost as sound as the level ofMEG measurements themselves. The degenerate cases, where eitherthe differences of the conductivities within the shells disappear,or the ellipsoidal geometry is reduced to the spherical one,are also considered.     

Structure formation and phase behaviour in ferrofluid monolayers: theory and computer simulations

In order to investigate the peculiarities brought by the quasi-2D geometry into the aggregation processes in ferrofluids, a combination of density functional theory (DFT), and molecular dynamics (MD) simulations is presented. The microstructure formation and the phase behaviour of monodisperse ferrofluid monolayers is studied. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the inheritance of the strongly π-extending property

In this article, we focus on modules with the property that every projection invariant submodule is essential in a fully invariant direct summand. In contrast to π-extending condition, it is shown that the former property is inherited by direct summands and Morita invariant. An application of our results yields that the endomorphism ring of a free module enjoys the property. Moreover, we characterize generalized triangular matrix rings with the aforementioned property and apply to somewhat special cases.     

Hermite‐Hadamard‐type inequalities for interval‐valued coordinated convex functions involving generalized fractional integrals

In this paper, we define interval‐valued left‐sided and right‐sided generalized fractional double integrals. We establish inequalities of Hermite‐Hadamard like for coordinated interval‐valued convex functions by applying our newly defined integrals.     

A posteriori error estimation with finite element methods of lines for one-dimensional parabolic systems

Summary Consider the solution of one-dimensional linear initial-boundary value problems by a finite element method of lines using a piecewiseP ^th -degree polynomial basis. A posteriori estimates of the discretization error are obtained as the solutions of either local parabolic or local elliptic finite element problems using piecewise polynomial corrections of degreep+1 that vanish at element ends. Error estimates computed in this manner are shown to converge in energy under mesh refinement to the exact finite element discretization error. Computational results indicate that the error estimates are robust over a wide range of mesh spacings and polynomial degrees and are, furthermore, applicable in situations that are not supported by the analysis.This research was partially supported by the U.S. Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Number AFOSR 90-0194; by the U.S. Army Research Office under Contract Number DAAL03-91-G-0215; and by the National Science Foundation under Institutional Infrastructure Grant Number CDA-8805910     

Mixed formulations for a class of variational inequalitiesFormulations mixtes pour une classe d'inéquations variationnelles

This Note is an attempt to extend the mixed finite element method to a class of variational inequalities including the problems of Signorini and of unilateral contact in elasticity with or without friction. Existence and uniqueness for the continuous and the discrete problems as well as error estimates are established in a general abstract framework. As a result, the mixed approximation of the Signorini problem is proved to converge with an error bound in h^3/4. To cite this article: L. Slimane et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 87–92     

Structural and phase characteristics of the diamond/matrix interfacial zone in high-resistant diamond composites

The structural-phase state of the contact zone and the factors that influence on the strength of diamond retention in the diamond carbide composites were determined. Composites were obtained by the new hybrid technology that eliminates the reheating of the metalized coating. The elaborated technology combines the thermal diffusion metallization of a diamond and the sintering by the scheme of self-dosed impregnation in a one-stage technological cycle. By the methods of electron microscopy, X-ray diffraction analysis, and Raman spectroscopy the structural and phase characteristics of the interphase boundary were investigated. The improvement of chemical and mechanical adhesion between the diamond and carbide matrix was obtained. It was shown that the specific productivity of the samples with a metalized diamond component is 39% higher than those without metallization.     

Angular Multiplexing Gratings Characterization in Dichromated Gelatin and Thoughts on Diffractive Optical Memory

We were mainly concerned here with multiple exposure gratings that allow interactions and coupling between gratings. We started the study of interactions and coupling with five gratings recorded in dichromated gelatin (DCG). Having determined experimentally the characteristic parameters of these five gratings, we used a method developed by Oxford University to synthesize the modulation profile. Entering these parameters of each grating into our computing programs (based on the alternative theory of diffraction in modulated media) and by varying the amplitude modulation, the latter was deduced by adjusting the numerical curve to the experimental one. We then extended the study to nine multiplexed gratings. This study allowed us to know the range of the amplitude modulation such that a certain number of angular multiplexing can be done in DCG. Several undesirable effects which we had experimentally observed appeared then during the reconstruction of any one of the multiplexed images, such as cross-talk effect. We noted the good concordance between experimental and theoretical results. We finally illustrated cross-talk effect in a diffractive memory where, if the amplitude modulation was first calculated, the number of multiplexing for an angular range could then be determined and the angular separation achieved to avoid cross-talk.