Approximation Algorithms for Steiner and Directed Multicuts

In this paper we consider theSteiner multicutproblem. This is a generalization of the minimum multicut problem where instead of separating nodepairs, the goal is to find a minimum weight set of edges that separates all givensetsof nodes. A set is considered separated if it is not contained in a single connected component. We show anO(log³(kt)) approximation algorithm for the Steiner multicut problem, wherekis the number of sets andtis the maximum cardinality of a set. This improves theO(t log k) bound that easily follows from the previously known multicut results. We also consider an extension of multicuts to directed case, namely the problem of finding a minimum-weight set of edges whose removal ensures that none of the strongly connected components includes one of the prespecifiedknode pairs. In this paper we describe anO(log² k) approximation algorithm for this directed multicut problem. Ifk ? n, this represents an improvement over theO(log n log log n) approximation algorithm that is implied by the technique of Seymour.     

Elliptic Interface Problems in Axisymmetric Domains. Part I: Singular Functions of Non - Tensorial Type

We study the regularity of solutions of interface problems for the Poisson equation in axisynunetric domains. The Fourier decomposition of the 3D-problem into a sequence of 2D-variational equations end uniform (with respect to the sequence parameter) a prior; estimates of their solutions are derived. Some non-tensorial singular functions describing the behaviour of the solution near interface edges are given and the smoothness of the stress intensity distribution as well as the tangential regularity are characterized in tenns of Sobolev spaces. In a forthcoming part II of this paper, the results will be applied to error estimates of the so-called Fourier-finite-element method for solving approximately elliptic interface problems in 3D.     

Tail behavior of random products and stochastic exponentials

In this paper we study the distributional tail behavior of the solution to a linear stochastic differential equation driven by infinite variance α$\alpha$-stable Lévy motion. We show that the solution is regularly varying with index α$\alpha$. An important step in the proof is the study of a Poisson number of products of independent random variables with regularly varying tail. The study of these products merits its own interest because it involves interesting saddle-point approximation techniques.     

Singularities of the quad curl problem

We consider the quad curl problem in smooth and non smooth domains of the space. We first give an augmented variational formulation equivalent to the one from [25] if the datum is divergence free. We describe the singularities of the variational space which correspond to the ones of the Maxwell system with perfectly conducting boundary conditions. The edge and corner singularities of the solution of the corresponding boundary value problem with smooth data are also characterized. We finally obtain some regularity results of the variational solution.     

Méthode particulaire anisotropeAnisotropic particle method

A particle method has been used to simulate the vorticity transport in a two-dimensional flow of an incompressible inviscid fluid. In this method, not only the location and the circulation of the particle are used but also the moments of the internal vorticity. The transport equation for these moments has been derived from the vorticity transport equation. The method has been compared to the usual particle method as well as to Teng's elliptic vortex model. The test case is that of the evolution of two circular patches of vorticity already used by Teng. To cite this article: A. Beaudoin et al., C. R. Mecanique 330 (2002) 51–56     

Singular Solutions to Conformal Hessian Equations

The authors show that for any e ∈]0,1[,there exists an analytic outside zero solution to a uniformly elliptic conformal Hessian equation in a ball B C R5 which belongs to C1'ε(B) \ C1'ε+(B).     

Congruence preserving functions on free monoids

A function on an algebra is congruence preserving if for any congruence, it maps congruent elements to congruent elements. We show that on a free monoid generated by at least three letters, a function from the free monoid into itself is congruence preserving if and only if it is of the form \({x \mapsto w_{0}xw_{1} \cdots w_{n-1}xw_n }\) for some finite sequence of words \({w_0,\ldots ,w_n}\). We generalize this result to functions of arbitrary arity. This shows that a free monoid with at least three generators is a (noncommutative) affine complete algebra. As far as we know, it is the first (nontrivial) case of a noncommutative affine complete algebra.     

Vanishing evaluations of simple functors

The classification of simple biset functors is known, but the evaluation of a simple biset functor at a finite group G$G$ may be zero. We investigate various situations where this happens, as well as cases where this does not occur. We also prove a closed formula for such an evaluation under some restrictive conditions on  G$G$.     

A proof of Culter’s theorem on the existence of periodic orbits in polygonal outer billiards

We prove a recent theorem by C. Culter every polygonal outer billiard in the affine plane has a periodic trajectory.      

Z-scan measurement of the nonlinear refractive index of graphene

Under strong laser illumination, few-layer graphene exhibits both a transmittance increase due to saturable absorption and a nonlinear phase shift. Here, we unambiguously distinguish these two nonlinear optical effects and identify both real and imaginary parts of the complex nonlinear refractive index of graphene. We show that graphene possesses a giant nonlinear refractive index n(2)?10(-7) cm(2) W(-1), almost 9 orders of magnitude larger than bulk dielectrics. We find that the nonlinear refractive index decreases with increasing excitation flux but slower than the absorption. This suggests that graphene may be a very promising nonlinear medium, paving the way for graphene-based nonlinear photonics.