Publications

2013

• La conjecture de Syracuse -visualisation bidimensionnelle
  
  • Colonna Jean-François
  , 2013. The Syracuse conjecture -bidimensional display- (La conjecture de Syracuse -visualisation bidimensionnelle-)
• La conjecture de Syracuse -visualisation monodimensionnelle
  
  • Colonna Jean-François
  , 2013. The Syracuse conjecture -monodimensional display- (La conjecture de Syracuse -visualisation monodimensionnelle-)
• La conjecture de Syracuse -visualisation monodimensionnelle
  
  • Colonna Jean-François
  , 2013. The Syracuse conjecture -monodimensional display- (La conjecture de Syracuse -visualisation monodimensionnelle-)
• La conjecture de Syracuse -visualisation monodimensionnelle
  
  • Colonna Jean-François
  , 2013. The Syracuse conjecture -monodimensional display- (La conjecture de Syracuse -visualisation monodimensionnelle-)
• La conjecture de Syracuse -visualisation monodimensionnelle
  
  • Colonna Jean-François
  , 2013. The Syracuse conjecture -monodimensional display- (La conjecture de Syracuse -visualisation monodimensionnelle-)
• La conjecture de Syracuse -visualisation monodimensionnelle
  
  • Colonna Jean-Francois
  , 2013. The Syracuse conjecture -monodimensional display- (La conjecture de Syracuse -visualisation monodimensionnelle-)
• La conjecture de Syracuse -visualisation monodimensionnelle
  
  • Colonna Jean-Francois
  , 2013. The Syracuse conjecture -monodimensional display- (La conjecture de Syracuse -visualisation monodimensionnelle-)
• Les deux premières itérations de la construction de la courbe de von Koch
  
  • Colonna Jean-Francois
  , 2013. The first two iterations of the construction of the von Koch curve (Les deux premières itérations de la construction de la courbe de von Koch)
• Une surface intermédiaire entre une 'double sphère' et un cylindre
  
  • Colonna Jean-Francois
  , 2013. A surface between a 'double sphere' and a cylinder (Une surface intermédiaire entre une 'double sphère' et un cylindre)
• Un cylindre défini à l'aide de trois champs bidimensionnels
  
  • Colonna Jean-Francois
  , 2013. A cylinder defined by means of three bidimensional fields (Un cylindre défini à l'aide de trois champs bidimensionnels)
• Une surface intermédiaire entre une 'double sphère' et un cylindre
  
  • Colonna Jean-Francois
  , 2013. A surface between a 'double sphere' and a cylinder (Une surface intermédiaire entre une 'double sphère' et un cylindre)
• Une surface intermédiaire entre une 'double sphère' et un cylindre
  
  • Colonna Jean-Francois
  , 2013. A surface between a 'double sphere' and a cylinder (Une surface intermédiaire entre une 'double sphère' et un cylindre)
• La conjecture de Goldbach
  
  • Colonna Jean-Francois
  , 2013. The Goldbach conjecture (La conjecture de Goldbach)
• L’opérateur de Laplace-Beltrami en Géométrie presque-Riemannienne
  
  • Boscain Ugo
  • Laurent Camille
  Annales de l'Institut Fourier, Association des Annales de l'Institut Fourier, 2013, 63 (5), pp.1739 - 1770. Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. Generically, the singular set is an embedded one dimensional manifold and there are three type of points: Riemannian points where the two vector fields are linearly independent, Grushin points where the two vector fields are collinear but their Lie bracket is not and tangency points where the two vector fields and their Lie bracket are collinear and the missing direction is obtained with one more bracket. Generically tangency points are isolated. In this paper we study the Laplace-Beltrami operator on such a structure. In the case of a compact orientable surface without tangency points, we prove that the Laplace-Beltrami operator is essentially self-adjoint and has discrete spectrum. As a consequence a quantum particle in such a structure cannot cross the singular set and the heat cannot flow through the singularity. This is an interesting phenomenon since when approaching the singular set (i.e. where the vector fields become collinear), all Riemannian quantities explode, but geodesics are still well defined and can cross the singular set without singularities. This phenomenon appears also in sub-Riemannian structure which are not equiregular i.e. in which the grow vector depends on the point. We show this fact by analyzing the Martinet case. (10.5802/aif.2813)
  DOI : 10.5802/aif.2813
• Un 'double sphere' défini à l'aide de trois champs bidimensionnels
  
  • Colonna Jean-Francois
  , 2013. A 'double sphere' defined by means of three bidimensional fields (Un 'double sphere' défini à l'aide de trois champs bidimensionnels)
• La conjecture de Goldbach
  
  • Colonna Jean-Francois
  , 2013. The Goldbach conjecture (La conjecture de Goldbach)
• Agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb') pour seize éclairages différents -section tridimensionnelle
  
  • Colonna Jean-François
  , 2013. Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') for sixteen different lightings -tridimensional cross-section- (Agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb') pour seize éclairages différents -section tridimensionnelle-)
• An Optimal Affine Invariant Smooth Minimization Algorithm
  
  • d'Aspremont Alexandre
  • Guzmán Cristóbal
  • Jaggi Martin
  , 2013. We formulate an affine invariant implementation of the algorithm in Nesterov (1983). We show that the complexity bound is then proportional to an affine invariant regularity constant defined with respect to the Minkowski gauge of the feasible set. We also detail matching lower bounds when the feasible set is an ℓp ball. In this setting, our bounds on iteration complexity for the algorithm in Nesterov (1983) are thus optimal in terms of target precision, smoothness and problem dimension.
• Statistics of animal movement
  
  • Berthelot Geoffroy C.B.
  • Bansaye Vincent
  • Calenge C.
  , 2013.
• A numerical method for kinetic equations with discontinuous equations : application to mathematical modeling of cell dynamics
  
  • Aymard Benjamin
  • Clément Frédérique
  • Coquel Frédéric
  • Postel Marie
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2013, 35 (6), pp.27 pages. Abstract: In this work, we propose a numerical method to handle discontinuous fluxes arising in transport-like equations. More precisely, we study hyperbolic PDEs with flux transmission conditions at interfaces between subdomains where coefficients are discontinuous. A dedicated finite volume scheme with a limited high order enhancement is adapted to treat the discontinuities arising at interfaces. The validation of the method is done on 1D and 2D toy problems for which exact solutions are available, allowing us to do a thorough convergence study. We then apply the method to a biological model focusing on complex cell dynamics, that initially motivated this study, and illustrates the full potentialities of the scheme. (10.1137/120904238)
  DOI : 10.1137/120904238
• A conformal mapping method in inverse obstacle scattering
  
  • Haddar Houssem
  • Kress Rainer
  Complex Variables and Elliptic Equations, Taylor & Francis, 2013. Akduman, Haddar and Kress [1, 5, 11] have employed a conformal map- ping technique for the inverse problem to reconstruct a perfectly conducting inclusion in a homogeneous background medium from Cauchy data for elec- trostatic imaging, that is, for solving an inverse boundary value problem for the Laplace equation. We propose an extension of this approach to inverse obstacle scattering for time-harmonic waves, that is, to the solution of an inverse boundary value problem for the Helmholtz equation. The main idea is to use the conformal mapping algorithm in an iterative procedure to ob- tain Cauchy data for a Laplace problem from the given Cauchy data for the Helmholtz problem. We present the foundations of the method together with a convergence result and exhibit the feasibility of the method via numerical examples.
• An improved time domain linear sampling method for Robin and Neumann obstacles
  
  • Haddar Houssem
  • Lechleiter Armin
  • Marmorat Simon
  Applicable Analysis, Taylor & Francis, 2013, pp.1-22. We consider inverse obstacle scattering problems for the wave equation with Robin or Neu- mann boundary conditions. The problem of reconstructing the geometry of such obstacles from measurements of scattered waves in the time domain is tackled using a time domain linear sampling method. This imaging technique yields a picture of the scatterer by solving a linear operator equation involving the measured data for many right-hand sides given by singular so- lutions to the wave equation. We analyze this algorithm for causal and smooth impulse shapes, we discuss the effect of different choices of the singular solutions used in the algorithm, and finally we propose a fast FFT-based implementation. (10.1080/00036811.2013.772583)
  DOI : 10.1080/00036811.2013.772583
• Diffraction of Bloch Wave Packets for Maxwell's Equations
  
  • Allaire Grégoire
  • Palombaro Mariapia
  • Rauch Jeffrey
  Communications in Contemporary Mathematics, World Scientific Publishing, 2013, 15 (6), pp.1350040. We study, for times of order 1/h, solutions of Maxwell's equations in an O(h^2) modulation of an h-periodic medium. The solutions are of slowly varying amplitude type built on Bloch plane waves with wavelength of order h. We construct accurate approximate solutions of three scale WKB type. The leading profile is both transported at the group velocity and dispersed by a Schrödinger equation given by the quadratic approximation of the Bloch dispersion relation. A weak ray average hypothesis guarantees stability. Compared to earlier work on scalar wave equations, the generator is no longer elliptic. Coercivity holds only on the complement of an infinite dimensional kernel. The system structure requires many innovations. (10.1142/S0219199713500405)
  DOI : 10.1142/S0219199713500405
• Adaptation and migration of a population between patches
  
  • Mirrahimi Sepideh
  Discrete and Continuous Dynamical Systems - Series B, American Institute of Mathematical Sciences, 2013, 18, pp.753-768. A Hamilton-Jacobi formulation has been established previously for phenotypically structured population models where the solution concentrates as Dirac masses in the limit of small diffusion. Is it possible to extend this approach to spatial models? Are the limiting solutions still in the form of sums of Dirac masses? Does the presence of several habitats lead to polymorphic situations? We study the stationary solutions of a structured population model, while the population is structured by continuous phenotypical traits and discrete positions in space. The growth term varies from one habitable zone to another, for instance because of a change in the temperature. The individuals can migrate from one zone to another with a constant rate. The mathematical modeling of this problem, considering mutations between phenotypical traits and competitive interaction of individuals within each zone via a single resource, leads to a system of coupled parabolic integro-differential equations. We study the asymptotic behavior of the stationary solutions to this model in the limit of small mutations. The limit, which is a sum of Dirac masses, can be described with the help of an effective Hamiltonian. The presence of migration can modify the dominant traits and lead to polymorphic situations. (10.3934/dcdsb.2013.18.753)
  DOI : 10.3934/dcdsb.2013.18.753
• The Moutard transformation and two-dimensional multi-point delta-type potentials
  
  • Novikov Roman
  • Taimanov Iskander
  Russian Mathematical Surveys, Turpion, 2013, 68 (5), pp.957–959. In the framework of the Moutard transformation formalism we find multi-point delta-type potentials for two-dimensional Schrodinger operators and their isospectral deformations on the zero energy level. In particular, these potentials are "reflectionless" in the sense of the Faddeev generalized "scattering" data.