Publications

2012

• 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'une interpolation tridimensionnelle entre une structure fractale et un réseau cubique
  
  • Colonna Jean-François
  , 2012. Anaglyph of a tridimensional interpolation between a fractal structure and a cubic mesh (Anaglyphe d'une interpolation tridimensionnelle entre une structure fractale et un réseau cubique)
• Anaglyphe d'un agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb')
  
  • Colonna Jean-François
  , 2012. Anaglyph of a close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') (Anaglyphe d'un agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb'))
• Global stability for the multi-channel Gel'fand-Calderón inverse problem in two dimensions
  
  • Santacesaria Matteo
  Bulletin des Sciences Mathématiques, Elsevier, 2012, 136 (7), pp.731-744. We prove a global logarithmic stability estimate for the multi-channel Gel'fand-Calderón inverse problem on a two-dimensional bounded domain, i.e. the inverse boundary value problem for the equation $-\Delta \psi + v\, \psi = 0$ on $D$, where $v$ is a smooth matrix-valued potential defined on a bounded planar domain $D$. (10.1016/j.bulsci.2012.02.004)
  DOI : 10.1016/j.bulsci.2012.02.004
• Anaglyphe d'une structure fractale tridimensionnelle
  
  • Colonna Jean-François
  , 2012. Anaglyph of a tridimensional fractal structure (Anaglyphe d'une structure fractale tridimensionnelle)
• Local stability and robustness of sparse dictionary learning in the presence of noise
  
  • Jenatton Rodolphe
  • Gribonval Rémi
  • Bach Francis
  , 2012, pp.41. A popular approach within the signal processing and machine learning communities consists in modelling signals as sparse linear combinations of atoms selected from a learned dictionary. While this paradigm has led to numerous empirical successes in various fields ranging from image to audio processing, there have only been a few theoretical arguments supporting these evidences. In particular, sparse coding, or sparse dictionary learning, relies on a non-convex procedure whose local minima have not been fully analyzed yet. In this paper, we consider a probabilistic model of sparse signals, and show that, with high probability, sparse coding admits a local minimum around the reference dictionary generating the signals. Our study takes into account the case of over-complete dictionaries and noisy signals, thus extending previous work limited to noiseless settings and/or under-complete dictionaries. The analysis we conduct is non-asymptotic and makes it possible to understand how the key quantities of the problem, such as the coherence or the level of noise, can scale with respect to the dimension of the signals, the number of atoms, the sparsity and the number of observations.
• Anaglyphe d'un agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-quaternions (un 'Mandelbulb')
  
  • Colonna Jean-François
  , 2012. Anaglyph of a close-up on a pseudo-quaternionic Mandelbrot set (a 'Mandelbulb') (Anaglyphe d'un agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-quaternions (un 'Mandelbulb'))
• Anaglyphe d'noeud '3-trèfle' torique fractal
  
  • Colonna Jean-François
  , 2012. Anaglyph of a fractal 3-foil torus knot (Anaglyphe d'noeud '3-trèfle' torique fractal)
• Anaglyphe d'une interpolation tridimensionnelle entre une structure fractale et un réseau cubique
  
  • Colonna Jean-François
  , 2012. Anaglyph of a tridimensional interpolation between a fractal structure and a cubic mesh (Anaglyphe d'une interpolation tridimensionnelle entre une structure fractale et un réseau cubique)
• Modified Suliciu relaxation system and exact resolution of isolated shock waves
  
  • Chalons Christophe
  • Coquel Frédéric
  , 2012. We present a new Approximate Riemann Solver (ARS) for the gas dynamics equations in Lagrangian coordinates and with general non linear pressure laws. The design of this new ARS relies on a generalized Suliciu pressure relaxation approach. It gives by construction the exact solutions for isolated entropic shocks and we prove that it is Lipschitz-continuous and satisfies an entropy inequality. Finally, the ARS is used to develop either a classical entropy conservative Godunov-type method, or a Glimm-type (random sampling based Godunov-type) method able to generate infinitely sharp discrete shock profiles. Numerical experiments are proposed to prove the validity of these approaches.
• Asymptotic expansions for interior penalty solutions of control constrained linear-quadratic problems
  
  • Alvarez Felipe
  • Bolte Jérôme
  • Bonnans J. Frederic
  • Silva Francisco
  Mathematical Programming, Series A, Springer, 2012, 135 (1-2), pp.473-507. We consider a quadratic optimal control problem governed by a nonautonomous affine differential equation subject to nonnegativity control constraints. For a general class of interior penalty functions, we show how to compute the principal term of the pointwise expansion of the state and the adjoint state. Our main argument relies on the following fact: If the control of the initial problem satisfies strict complementarity conditions for its Hamiltonian except for a finite number of times, the estimates for the penalized optimal control problem can be derived from the estimations of a related stationary problem. Our results provide several types of efficiency measures of the penalization technique: error estimations of the control for $L^s$ norms ($s$ in $[1,+\infty]$), error estimations of the state and the adjoint state in Sobolev spaces $W^{1,s}$ ($s$ in $[1,+\infty)$) and also error estimates for the value function. For the $L^1$ norm and the logarithmic penalty, the optimal results are given. In this case we indeed establish that the penalized control and the value function errors are of order $O(\eps|\log\eps|)$.
• Anaglyphe d'une bouteille de Klein fractale
  
  • Colonna Jean-François
  , 2012. Anaglyph of a fractal Klein bottle (Anaglyphe d'une bouteille de Klein fractale)
• Anaglyphe d'noeud '5-trèfle' torique fractal
  
  • Colonna Jean-François
  , 2012. Anaglyph of a fractal 5-foil torus knot (Anaglyphe d'noeud '5-trèfle' torique fractal)
• Anaglyphe d'un tore fractal
  
  • Colonna Jean-François
  , 2012. Anaglyph of a fractal torus (Anaglyphe d'un tore fractal)
• 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)
• A universality result for the global fluctuations of the eigenvectors of Wigner matrices
  
  • Benaych-Georges Florent
  Random Matrices: Theory and Applications, World Scientific, 2012, 01 (04), pp.23. Let $U_n=[u_{i,j}]$ be the eigenvectors matrix of a Wigner matrix. We prove that under some moments conditions, the bivariate random process indexed by $[0,1]^2$ with value at $(s,t)$ equal to the sum, over $1\le i \le ns$ and $1\le j \le nt$, of $|u_{i,j}|^2 - 1/n$, converges in distribution to the bivariate Brownian bridge. This result has already been proved for GOE and GUE matrices. It is conjectured here that the necessary and sufficient condition, for the result to be true for a general Wigner matrix, is the matching of the moments of orders $1$, $2$ and $4$ of the entries of the Wigner with the ones of a GOE or GUE matrix. Surprisingly, the third moment of the entries of the Wigner matrix has no influence on the limit distribution. (10.1142/S2010326312500116)
  DOI : 10.1142/S2010326312500116
• Enhanced controllability of low Reynolds number swimmers in the presence of a wall
  
  • Alouges François
  • Giraldi Laetitia
  , 2012. Swimming, i.e., being able to advance in the absence of external forces by performing cyclic shape changes, is particularly demanding at low Reynolds numbers which is the regime of interest for micro-organisms and micro-robots. We focus on self-propelled stokesian robots composed of assemblies of balls and we prove that the presence of a wall has an effect on their motility. More precisely, we demonstrate that a controllable swimmer remains controllable in a half space whereas the reachable set of a non fully controllable one is affected by the presence of a wall.
• Variation artistique sur des noeuds '3/5/7-trèfle' toriques fractals
  
  • Colonna Jean-François
  , 2012. Artistic variation on fractal 3/5/7-foil torus knots (Variation artistique sur des noeuds '3/5/7-trèfle' toriques fractals)
• Visualisation tridimensionnelle de la conjecture ABC
  
  • Colonna Jean-François
  , 2012. Tridimensional visualization of the ABC conjecture (Visualisation tridimensionnelle de la conjecture ABC)
• La conjecture ABC
  
  • Colonna Jean-François
  , 2012. The ABC conjecture (La conjecture ABC)
• Variation artistique sur des noeuds '3/5/7-trèfle' toriques fractals
  
  • Colonna Jean-François
  , 2012. Artistic variation on fractal 3/5/7-foil torus knots (Variation artistique sur des noeuds '3/5/7-trèfle' toriques fractals)
• Un ensemble de 4x3 stéréogrammes d'un noeud '5-trèfle' torique fractal
  
  • Colonna Jean-François
  , 2012. A set of 4x3 stereograms of a fractal 5-foil torus knot (Un ensemble de 4x3 stéréogrammes d'un noeud '5-trèfle' torique fractal)
• Un ensemble de 4x3 stéréogrammes d'un noeud '7-trèfle' torique fractal
  
  • Colonna Jean-François
  , 2012. A set of 4x3 stereograms of a fractal 7-foil torus knot (Un ensemble de 4x3 stéréogrammes d'un noeud '7-trèfle' torique fractal)
• Un ensemble de 4x3 stéréogrammes d'un noeud '3-trèfle' torique fractal
  
  • Colonna Jean-François
  , 2012. A set of 4x3 stereograms of a fractal 3-foil torus knot (Un ensemble de 4x3 stéréogrammes d'un noeud '3-trèfle' torique fractal)