Pullback attractors for non-autonomous 2D-Navier–Stokes equations in some unbounded domains

In this Note we first introduce the concept of pullback asymptotic compactness. Next, we establish a result ensuring the existence of a pullback attractor for a non-autonomous dynamical system under the general assumptions of pullback asymptotic compactness and the existence of a pullback absorbing family of sets. Finally, we prove the existence of a pullback attractor for a non-autonomous 2D Navier–Stokes model in an unbounded domain, a case in which the theory of uniform attractors does not work since the non-autonomous term is quite general. To cite this article: T. Caraballo et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

The critical probability for random Voronoi percolation in the plane is 1/2

We study percolation in the following random environment: let Z be a Poisson process of constant intensity on ℝ², and form the Voronoi tessellation of ℝ² with respect to Z. Colour each Voronoi cell black with probability p, independently of the other cells. We show that the critical probability is 1/2. More precisely, if p>1/2 then the union of the black cells contains an infinite component with probability 1, while if p<1/2 then the distribution of the size of the component of black cells containing a given point decays exponentially. These results are analogous to Kesten's results for bond percolation in ℤ². The result corresponding to Harris' Theorem for bond percolation in ℤ² is known: Zvavitch noted that one of the many proofs of this result can easily be adapted to the random Voronoi setting. For Kesten's results, none of the existing proofs seems to adapt. The methods used here also give a new and very simple proof of Kesten's Theorem for ℤ²; we hope they will be applicable in other contexts as well. Research supported in part by NSF grant ITR 0225610 and DARPA grant F33615-01-C-1900 Research partially undertaken during a visit to the Forschungsinstitut für Mathematik, ETH Zürich, Switzerland     

Use of the Apparent Content Curves for the spectrophotometric identification of substances: identification of amphetamines

The possibility of identification of substances which have similar spectral behaviour by means of Apparent Content Curves has been studied. This study is carried out with absorption, excitation and emission spectra of several amphetamines of widespread pharmaceutical use. Results obtained show that amphetamine, phenylpropanolamine, pseudoephedrine, phenylephrine, epinephrine, dopamine and methoxyamphetamine can be identified with a probability of >95%.     

Large number hypothesis

Dirac's large number hypothesis (LNH) and the Whitrow-Randall-Sciama relation are related to some cosmological models. The LNH and Whitrow-Randall-Sciama relations are neither equivalent nor consistent in general relativity, but they may both be valid in the Brans-Dicke theory and in another theory considered in this paper.     

Multiplicities of interpoint distances in finite planar sets

What is the maximum number of unit distances between the vertices of a convex n-gon in the plane? We review known partial results for this and other open questions on multiple occurrences of the same interpoint distance in finite planar subsets. Some new results are proved for small n. Challenging conjectures, both old and new, are highlighted.     

On the quiver-theoretical quantum Yang–Baxter equation

Quivers over a fixed base set form a monoidal category with tensor product given by pullback. The quantum Yang–Baxter equation, or more properly the braid equation, is investigated in this setting. A solution of the braid equation in this category is called a “solution” for short. Results of Etingof–Schedler–Soloviev, Lu–Yan–Zhu and Takeuchi on the set-theoretical quantum Yang–Baxter equation are generalized to the context of quivers, with groupoids playing the role of groups. The notion of “braided groupoid” is introduced. Braided groupoids are solutions and are characterized in terms of bijective 1-cocycles. The structure groupoid of a non-degenerate solution is defined; it is shown that it is a braided groupoid. The reduced structure groupoid of a non-degenerate solution is also defined. Non-degenerate solutions are classified in terms of representations of matched pairs of groupoids. By linearization we construct star-triangular face models and realize them as modules over quasitriangular quantum groupoids introduced in papers by M. Aguiar, S. Natale and the author.     

A comparison theorem for a piecewise Lipschitz continuous Hamiltonian and application to Shape-from-Shading problems

Summary The reconstruction from a shaded image of a Lambertian and not self-shadowing surface illuminated by a single distant pointwise light source may be written as a first-order Hamilton-Jacobi equation.In this paper, we continue the investigation begun in E. Rouy and A. Tourin into the uniqueness of the solution of this equation; the approach is based on the viscosity solutions theory and the dynamic programming principle.More precisely, we concentrate here on the uniqueness of the viscosity solution of this equation in case the measured luminous intensity reflected by the surface is discontinuous along a smooth curve. We prove a general comparison result for a piecewise Lipschitz continuous Hamiltonian and illustrate it by numerical experiments.