Publications

2012

• Visualisation tridimensionnelle artistique de la conjecture de Goldbach (vue aérienne)
  
  • Colonna Jean-François
  , 2012. Artistic tridimensional visualization of the Goldbach conjecture (bird's-eye view) (Visualisation tridimensionnelle artistique de la conjecture de Goldbach (vue aérienne))
• La conjecture de Goldbach
  
  • Colonna Jean-François
  , 2012. The Goldbach conjecture (La conjecture de Goldbach)
• Visualisation tridimensionnelle artistique de la conjecture de Goldbach
  
  • Colonna Jean-François
  , 2012. Artistic tridimensional visualization of the Goldbach conjecture (Visualisation tridimensionnelle artistique de la conjecture de Goldbach)
• Mathematical analysis of light propagation in optical fibers with randomly varying birefringence
  
  • Gazeau Maxime
  , 2012. The study of light propagation in monomode optical fibers requires to take care of various complex phenomena such as the polarization mode dispersion (PMD) and the Kerr effect. It has been proved that the slowly varying envelope of the electric field is well described by a coupled non linear schrödinger equation with random coefficients called the Manakov PMD equation. The particularity of this equation is the presence of various length scales whose ratio is given by a small parameter. The first part of this thesis is concerned with the theoretical study of the asymptotic dynamic of the solution of the Manakov PMD equation as this parameter goes to zero. Generalizing the theory of the Diffusion Approximation in the infinite dimensional setting, we were able to prove that the asymptotic dynamic is given by a stochastic partial differential equation driven by three Brownian motions. In a second part, we propose a Crank Nicolson scheme for this equation and we prove that the order of convergence is 1/2. The discretization of the noise term is taken implicit so that the scheme is conservative and stable. Finally the last part is concerned with numerical simulations of the PMD and propagation and collision of Manakov solitons. The above scheme is implemented and we propose a variance reduction method valid in the context of stochastic partial differential equations.
• Necessary conditions involving Lie brackets for impulsive optimal control problems; the commutative case
  
  • Aronna Maria Soledad
  • Rampazzo Franco
  , 2012. In this article we study control problems with systems that are governed by ordinary differential equations whose vector fields depend linearly in the time derivatives of some components of the control. The remaining components are considered as classical controls. This kind of system is called 'impulsive system'. We assume that the vector fields multiplying the derivatives of each component of the control are commutative. We derive new necessary conditions in terms of the adjoint state and the Lie brackets of the data functions.
• A remark on Lipschitz stability for inverse problems
  
  • Bourgeois Laurent
  , 2012. An abstract Lipschitz stability estimate is proved for a class of inverse problems. It is then applied to the inverse Robin problem for the Laplace equation and to the inverse medium problem for the Helmholtz equation.
• La conjecture de Goldbach
  
  • Colonna Jean-François
  , 2012. The Goldbach conjecture (La conjecture de Goldbach)
• Structure fractale tridimensionnelle
  
  • Colonna Jean-François
  , 2012. Tridimensional fractal structure (Structure fractale tridimensionnelle)
• Absence of solitons with sufficient algebraic localization for the Novikov-Veselov equation at nonzero energy
  
  • Kazeykina Anna
  , 2012. We show that the Novikov--Veselov equation (an analog of KdV in dimension 2 + 1) at positive and negative energies does not have solitons with the space localization stronger than O( | x |^{ -3 } ) as | x | \to \infty.
• Structure fractale tridimensionnelle
  
  • Colonna Jean-François
  , 2012. Tridimensional fractal structure (Structure fractale tridimensionnelle)
• Anaglyphe d'un polyèdre fractal
  
  • Colonna Jean-François
  , 2012. Anaglyph of a fractal polyhedron (Anaglyphe d'un polyèdre fractal)
• Un ensemble de 4x3 stéréogrammes d'un polyèdre fractal
  
  • Colonna Jean-François
  , 2012. A set of 4x3 stereograms of a fractal polyhedron (Un ensemble de 4x3 stéréogrammes d'un polyèdre fractal)
• On the Spherical Hausdorff Measure in Step 2 Corank 2 sub-Riemannian Geometry
  
  • Boscain Ugo
  • Gauthier Jean-Paul
  , 2012. In this paper, we consider generic corank 2 sub-Riemannian structures, and we show that the Spherical Hausdorf measure is always a C^1-smooth volume, which is in fact generically C^2- smooth out of a stratified subset of codimension 7. In particular, for rank 4, it is generically C^2 . This is the continuation of a previous work by the auhors.
• Anaglyphe de l'ensemble de Julia dans le corps des quaternions calculé pour A=(0,1,0,0) -ou 'la danseuse d'Yr'- -coupe tridimensionnelle
  
  • Colonna Jean-François
  , 2012. Anaglyph of a the quaternionic Julia set computed with A=(0,1,0,0) -tridimensional cross-section- (Anaglyphe de l'ensemble de Julia dans le corps des quaternions calculé pour A=(0,1,0,0) -ou 'la danseuse d'Yr'- -coupe tridimensionnelle-)
• Semi-Lagrangian discontinuous Galerkin schemes for some first and second order partial differential equations
  
  • Bokanowski Olivier
  • Simarmata Giorevinus
  Journal of Scientific Computing, Springer Verlag, 2012, pp.DOI 10.1007/s10915-012-9648-x. Explicit, unconditionally stable, high order schemes for the approximation of some first and second order linear, time-dependent partial differential equations (PDEs) are proposed. The schemes are based on a weak formulation of a semi-Lagrangian scheme using discontinuous Galerkin elements. It follows the ideas of the recent works of Crouseilles, Mehrenberger and Vecil (2010) and of Qiu and Shu (2011), for first order equations, based on exact integration, quadrature rules, and splitting techniques. In particular we obtain high order schemes, unconditionally stable and convergent, in the case of linear second order PDEs with constant coefficients. In the case of non-constant coefficients, we construct "almost" unconditionally stable second order schemes and give precise convergence results. The schemes are tested on several academic examples, including the Black and Scholes PDE in finance.
• Anaglyphe de l'attracteur de Lorenz
  
  • Colonna Jean-François
  , 2012. Anaglyph of the Lorenz attractor (Anaglyphe de l'attracteur de Lorenz)
• Anaglyphe de la courbe de Hilbert tridimensionnelle -itération 4
  
  • Colonna Jean-François
  , 2012. Anaglyph of the tridimensional Hilbert Curve -iteration 4- (Anaglyphe de la courbe de Hilbert tridimensionnelle -itération 4-)
• Un ensemble de 4x3 stéréogrammes de la courbe de Hilbert tridimensionnelle -itération 4
  
  • Colonna Jean-François
  , 2012. A set of 4x3 stereograms of the tridimensional Hilbert Curve -iteration 4- (Un ensemble de 4x3 stéréogrammes de la courbe de Hilbert tridimensionnelle -itération 4-)
• Anaglyphe de la courbe de Hilbert tridimensionnelle -itération 3
  
  • Colonna Jean-François
  , 2012. Anaglyph of the tridimensional Hilbert Curve -iteration 3- (Anaglyphe de la courbe de Hilbert tridimensionnelle -itération 3-)
• Un ensemble de 4x3 stéréogrammes de la courbe de Hilbert tridimensionnelle -itération 3
  
  • Colonna Jean-François
  , 2012. A set of 4x3 stereograms of the tridimensional Hilbert Curve -iteration 3- (Un ensemble de 4x3 stéréogrammes de la courbe de Hilbert tridimensionnelle -itération 3-)
• MODEL UNCERTAINTY IN FINANCE AND SECOND ORDER BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS
  
  • Zhou Chao
  , 2012. The main objective of this PhD thesis is to study some financial mathematics problems in an incomplete market with model uncertainty. In recent years, the theory of second order backward stochastic differential equations (2BSDEs for short) has been developed by Soner, Touzi and Zhang on this topic. In this thesis, we adopt their point of view. This thesis contains of four key parts related to 2BSDEs. In the first part, we generalize the 2BSDEs theory initially introduced in the case of Lipschitz continuous generators to quadratic growth generators. This new class of 2BSDEs will then allow us to consider the robust utility maximization problem in non-dominated models. In the second part, we study this problem for exponential utility, power utility and logarithmic utility. In each case, we give a characterization of the value function and an optimal investment strategy via the solution to a 2BSDE. In the third part, we provide an existence and uniqueness result for second order reflected BSDEs with lower obstacles and Lipschitz generators, and then we apply this result to study the problem of American contingent claims pricing with uncertain volatility. In the fourth part, we define a notion of 2BSDEs with jumps, for which we prove the existence and uniqueness of solutions in appropriate spaces. We can interpret these equations as standard BSDEs with jumps, under both volatility and jump measure uncertainty. As an application of these results, we shall study a robust exponential utility maximization problem under model uncertainty, where the uncertainty affects both the volatility process and the jump measure.
• Anaglyphe d'un ensemble de Mandelbrot dans l'ensemble des pseudo-quaternions (un 'MandelBulb') -'la ronde des enfants' ou 'la conscience émergeant des Mathématiques'
  
  • Colonna Jean-François
  , 2012. Anaglyph of a pseudo-quaternionic Mandelbrot set (a 'MandelBulb') -'the children round' or 'the consciousness emerging from Mathematics'- (Anaglyphe d'un ensemble de Mandelbrot dans l'ensemble des pseudo-quaternions (un 'MandelBulb') -'la ronde des enfants' ou 'la conscience émergeant des Mathématiques'-)
• Anaglyphe d'une structure fractale filamenteuse tridimensionnelle
  
  • Colonna Jean-François
  , 2012. Anaglyph of a tridimensional filamentous fractal structure (Anaglyphe d'une structure fractale filamenteuse tridimensionnelle)
• Anaglyphe d'un ruban de Möbius fractal
  
  • Colonna Jean-François
  , 2012. Anaglyph of a fractal Möbius strip (Anaglyphe d'un ruban de Möbius fractal)
• Anaglyphe d'une structure fractale tridimensionnelle -la 'mousse' de l'espace-temps ?
  
  • Colonna Jean-François
  , 2012. Anaglyph of a tridimensional fractal structure -the space-time foam ?- (Anaglyphe d'une structure fractale tridimensionnelle -la 'mousse' de l'espace-temps ?-)