The Empty Hexagon Theorem

Let P be a finite set of points in general position in the plane. Let C(P) be the convex hull of P and let Cⁱ_P be the ith convex layer of P. A minimal convex set S of P is a convex subset of P such that every convex set of P ∩ C(S) different from S has cardinality strictly less than |S|. Our main theorem states that P contains an empty convex hexagon if C¹_P is minimal and C⁴_P is not empty. Combined with the Erdos-Szekeres theorem, this result implies that every set P with sufficiently many points contains an empty convex hexagon, giving an affirmative answer to a question posed by Erdos in 1977.     

Variable space search for graph coloring

Let G=(V,E) be a graph with vertex set V and edge set E. The k-coloring problem is to assign a color (a number chosen in {1,…,k}) to each vertex of G so that no edge has both endpoints with the same color. We propose a new local search methodology, called Variable Space Search, which we apply to the k-coloring problem. The main idea is to consider several search spaces, with various neighborhoods and objective functions, and to move from one to another when the search is blocked at a local optimum in a given search space. The k-coloring problem is thus solved by combining different formulations of the problem which are not equivalent, in the sense that some constraints are possibly relaxed in one search space and always satisfied in another. We show that the proposed algorithm improves on every local search used independently (i.e., with a unique search space), and is competitive with the currently best coloring methods, which are complex hybrid evolutionary algorithms.     

Asymptotic estimates for white noise distributions

Let θ be a Young function. Using properties of the Laplace and Legendre transforms, it is shown that white noise measures in the dual of a test function space of θ-exponential growth satisfy an exponential decay property with rate θ. An application to stochastic differential equations is given. To cite this article: H. Ouerdiane, N. Privault, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

The role of the researcher’s epistemology in mathematics education: an essay on the case of proof

Is there a shared meaning of “mathematical proof” among researchers in mathematics education? Almost all researchers may agree on a formal definition of mathematical proof. But beyond this minimal agreement, what is the state of our field? After three decades of activity in this area, being familiar with the most influential pieces of work, I realize that the sharing of keywords hides important differences in the understanding. These differences could be obstacles to scientific progress in this area, if they are not made explicit and addressed as such. In this essay I take a sample of research projects which have impacted the teaching and learning of mathematical proof, in order to describe where the gaps are. Then I suggest a possible scientific programme which aspires to strengthen the research practice in this domain. Eventually, I make the additional claim that this programme could hold for other areas of research in mathematics education.     

Tensor Gauge Fields in Arbitrary Representations of GL( D, $${\mathbb{R})}$$ : II. Quadratic Actions

Quadratic, second-order, non-local actions for tensor gauge fields transforming in arbitrary irreducible representations of the general linear group in D-dimensional Minkowski space are explicitly written in a compact form by making use of Levi–Civita tensors. The field equations derived from these actions ensure the propagation of the correct massless physical degrees of freedom and are shown to be equivalent to non-Lagrangian local field equations proposed previously. Moreover, these actions allow a frame-like reformulation à la MacDowell–Mansouri, without any trace constraint in the tangent indices. Chargé de Recherches FNRS, Belgium     

Cation mobility upon adsorption of methanol in NaY faujasite type zeolite: A molecular dynamics study compared to dielectric relaxation experiments

Molecular Dynamics simulations have been carried out to address the question of cation migration upon adsorption of methanol in NaY Faujasite system as a function of the loading. It has been shown that at low and intermediate loadings, SII cations can migrate toward the center of the supercage due strong interaction with the adsorbates, followed by hopping of SI' cations from the sodalite cage into the supercage to fill vacant SII sites. SI cations mainly remain trapped in their initial sites whatever the loading. At higher loading, only limited motion is observed for SII cations due to steric effects induced by the adsorbates within the supercage. These simulated results are in good agreement with those extracted by Complex Impedance Spectroscopy measurements, which provided the evolution of the number of extraframework cations in the different crystallographic sites as a function of the treatment temperature.     

A laser-wire system for the international linear collider

A new laser-wire has been installed in the extraction line of the ATF at KEK. It aims at demonstrating that laser-wires can be used to measure micrometre scale beam size. In parallel, studies have been made to specify a laser suitable for the ILC laser-wires.      

Internal derivatization of oligonucleotides with selenium for X-ray crystallography using MAD.

We have developed a route for the synthesis of 2'-selenium uridine analogues and oligonucleotides containing selenium labels, and have demonstrated for the first time a new strategy to covalently derivatize nucleotides with selenium for phase and structure determination in X-ray crystallography.     

Study of a nanocomposite based on a conducting polymer: polyaniline

A simple way to obtain a conducting nanocomposite is described, and the conducting particles are characterized. Core-shell particles [polystyrene-polyaniline (PANI)] have been obtained by the dispersion process from three types of polystyrene latexes: a no-cross-linked core stabilized by a nonylphenolethoxylate (NP40) and two cross-linked cores stabilized by NP40 and a mixture NP40/Surfamid (a surfactant bearing an amide group). The surface of these particles has been extensively characterized by X-ray photoelectron spectroscopy (XPS), atomic force microscopy, and scanning electron microscopy. A maximum coverage of 94% was obtained for the high PANI content as revealed by XPS analysis. A better coverage was obtained for the cross-linked polystyrene latex stabilized by the Surfamid. The amide group of this surfactant allows the H-bonding formation with the PANI backbone and, thus, improves the conductivity. It was shown that a uniform coverage of the core particles was not required to ensure a good conductivity.