Bronze roman mirrors: the secret of brightness

The surface microchemical structure of high tin leaded bronze Roman mirrors has been studied by means of scanning electron microscopy combined with energy dispersive spectrometry (SEM-EDS), X-ray diffraction (XRD) and optical microscopy (OM) techniques. The results allowed understanding of the origin of their high chemical stability and silvery-lustrous appearance. Indeed, some areas of the selected Roman mirrors are still characterised by a highly reflective and silver coloured surface even though they have been buried in the soil for about 2000 years. The micro-chemical results obtained from these areas have revealed that the mirror surface was tin enriched via inverse-segregation phenomenon by tailoring the cooling parameters. Furthermore, the presence of tin could be likely enhanced via cycles of oxidation and selective copper corrosion processes, thus resulting in a tin surface enrichment as a semi-transparent amorphous-like tin oxide (SnO₂) film, as well as a copper depletion at the outer surfaces.     

Zinn und Antimon in Legierungen

Ohne Zusammenfassung     

On energy densities reached in heavy-ion collisions at the CERN SPS

We present a few estimates of energy densities reached in heavy-ion collisions at the CERN SPS. The estimates are based on data and models of proton-nucleus and nucleus-nucleus interactions. In all of these estimates the maximum energy density in central Pb+Pb interactions is larger than the critical energy density GeV/fm³ following from lattice gauge theory computations. In estimates which we consider as realistic the maximum energy density is about . In this way our analysis gives some support to claims that deconfined matter has been produced at the CERN SPS. Any definite statement requires a deeper understanding of formation times of partons and hadrons in nuclear collisions. We also compare our results with implicit energy estimates contained in earlier models of anomalous suppression in nuclear collisions. Received: 3 February 2003 / Revised version: 5 March 2003 / Published online: 5 May 2003     

Shape Optimization for Semi-Linear Elliptic Equations Based on an Embedding Domain Method

We study a class of shape optimization problems for semi-linear elliptic equations with Dirichlet boundary conditions in smooth domains in ℝ². A part of the boundary of the domain is variable as the graph of a smooth function. The problem is equivalently reformulated on a fixed domain. Continuity of the solution to the state equation with respect to domain variations is shown. This is used to obtain differentiability in the general case, and moreover a useful formula for the gradient of the cost functional in the case where the principal part of the differential operator is the Laplacian. Online publication 23 January 2004.     

Stable products of spheres in the non-linear coupling of oscillators or quasi-periodic motions

For generic families of vector fields or transformations, normally hyperbolic invariant products of spheres appear near partially elliptic rest points. To cite this article: M. Kammerer-Colin de Verdière, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Teaching proof in the context of physics

This paper describes an application of statics to geometrical proofs in the classroom. The aim of the study was to find out whether the use of concepts and arguments from statics can help students understand and produce proofs of geometrical theorems. The two theorems studied were (1) that the medians in a triangle meet at a single point which is the centre of gravity of the triangle, and (2) the Varignon theorem, that the lines joining the midpoints of successive sides of a quadrilateral form a parallelogram. The classroom experiment showed that most students were successful in using arguments from statics in their proofs, and that they gained a better understanding of the theorems. These findings lend support to the claim that the introduction of statics helps students produce proofs and grasp their meaning.     

General covariance of the non-abelian DBI-action: checks and balances

We perform three tests on our proposal to implement diffeomorphism invariance in the non-abelian D0-brane DBI action as a base-point independence constraint between matrix Riemann normal coordinate systems. First, we show that T-duality along an isometry correctly interchanges the potential and kinetic terms in the action. Second, we show that the method to impose base-point independence using an auxiliary dN²-dimensional nonlinear sigma model also works for metrics which are curved along the brane, provided a physical gauge choice is made at the end. Third, we show that without alteration this method is applicable to higher order in velocities. Testing specifically to order 4, we elucidate the range of validity of the symmetrized trace approximation to the non-abelian DBI action.     

Large deviation for the empirical eigenvalue density of truncated Haar unitary matrices

Let U_m be an m×m Haar unitary matrix and U_[m,n] be its n×n truncation. In this paper the large deviation is proven for the empirical eigenvalue density of U_[m,n] as m/n→λ and n→∞. The rate function and the limit distribution are given explicitly. U_[m,n] is the random matrix model of quq, where u is a Haar unitary in a finite von Neumann algebra, q is a certain projection and they are free. The limit distribution coincides with the Brown measure of the operator quq.     

A finite element-spherical harmonics radiation transport model for photon migration in turbid media

In this paper, we solve the steady-state form of the Boltzmann transport equation in homogeneous and heterogeneous tissue-like media with a finite element-spherical harmonics (FE-P_N) radiation transport method. We compare FE-transport and diffusion solutions in terms of the ratio of absorption to reduced scattering coefficient, (μ_a/μ_s′) and the anisotropy factor g. Two different scattering phase function formulas are employed to model anisotropic scattering in the slab media with high g-value. Influence of void-like heterogeneities, and of their boundaries with the surrounding medium on the transport of photons are also examined.