Publications

2012

• Transformée en ondelettes d'un champ fractal bidimensionnel
  
  • Colonna Jean-François
  , 2012. Wavelet transform of a bidimensional fractal field (Transformée en ondelettes d'un champ fractal bidimensionnel)
• Transformée en ondelettes d'un champ fractal bidimensionnel
  
  • Colonna Jean-François
  , 2012. Wavelet transform of a bidimensional fractal field (Transformée en ondelettes d'un champ fractal bidimensionnel)
• Transformée en ondelettes d'un champ fractal bidimensionnel (vue aérienne)
  
  • Colonna Jean-François
  , 2012. Wavelet transform of a bidimensional fractal field (bird's-eye view) (Transformée en ondelettes d'un champ fractal bidimensionnel (vue aérienne))
• Visualisation tridimensionnelle de la conjecture de Goldbach
  
  • Colonna Jean-François
  , 2012. Tridimenional display of the Goldbach conjecture (Visualisation tridimensionnelle de la conjecture de Goldbach)
• Visualisation tridimensionnelle de la conjecture de Goldbach
  
  • Colonna Jean-François
  , 2012. Tridimenional display of the Goldbach conjecture (Visualisation tridimensionnelle de la conjecture de Goldbach)
• Physiologically structured cell population dynamic models with applications to combined drug delivery optimisation in oncology
  
  • Clairambault Jean
  • Fercoq Olivier
  , 2012. Optimising drug delivery in the general circulation targeted towards cancer cell pop- ulations, but inevitably reaching also proliferating healthy cell populations imposes to design optimised drug infusion algorithms in a dynamic way, i.e., controlling the growth of both populations simultaneously by the action of the drugs in use, wanted for cancer cells, and unwanted for toxic side effects on healthy cells. Towards this goal, we design models and methods, with optional representation of circadian clock control on proliferation in both populations, according to three axes [15, 16]: a) representing the oncologist's main weapons, drugs, and their fates in the organism by molecular-based pharmacokinetic-pharmacodynamic equations; b) representing the cell populations under attack by drugs, and their proliferation dynamics, including in the models molecular and functional targets for the drugs at stake, by physiologically structured equations; c) using numerical algorithms, optimising drug delivery under different constraints at the whole organism level, representing impacts of multiple drugs with different targets on cell populations. In the present study, two molecular pharmacological ODE models, one for oxali- platin, and one for 5-Fluorouracil, have been designed, using law of mass action and enzyme kinetics, to describe the fate of these two cytotoxic drugs in the organism. An age-structured PDE cell population model has been designed with drug control. targets to represent the effects of oxaliplatin and 5-Fluorouracil on the cell cycle in proliferating cell populations. The models for proliferating cell population dynam- ics involve possible physiological fixed (i.e., out of reach of therapeutic influence) circadian clock control, and varying drug control to be optimised, connected with pharmacological models. Concentrations of drugs, represented by outputs of ODEs, are assumed to be homogeneous in the cell populations under attack by cytotoxic drugs. The possi- bility to describe the effects of other drugs, cytostatic (including in this category anti-angiogenic drugs, considered as acting on the G1 phase, choking its entries and slowing it down), is also presented, but not put in pharmacokinetic equations and actual simulations in this study, that is focused on the combination of 5-FU and oxaliplatin, a classic therapeutic association in the treatment of colorectal cancer. We then set conditions to numerically solve drug delivery optimisation problems (maximisation of cancer cell kill under the constraint of preserving healthy cells over a tolerability threshold) by considering a trade-off between therapeutic and toxic effects. The observed effects on proliferation are growth exponents, i.e., first eigenvalues of the linear PDE systems, in the two populations, healthy and cancer. The solutions to an optimisation problem taking into account circadian clock control are presented as best delivery time schedules for the two drugs used in combined treatments, to be implemented in programmable delivery pumps in the clinic.
• Small time heat kernel asymptotics at the cut locus on surfaces of revolution
  
  • Barilari Davide
  • Jendrej Jacek
  , 2012. In this paper we investigate the small time heat kernel asymptotics on the cut locus on the class of two-spheres of revolution, which is the simplest class of 2-dimensional Riemannian manifolds different from the sphere with non trivial cut-conjugate locus. We determine the degeneracy of the exponential map near a cut-conjugate point and present the consequences of this result to the small time heat kernel asymptotics at this point. These results give a first example where the minimal degeneration of the asymptotic expansion at the cut locus is attained. (10.1016/j.anihpc.2013.03.003)
  DOI : 10.1016/j.anihpc.2013.03.003
• L'ensemble de Julia dans le corps des quaternions calculé pour A=(0,1,0,0) -coupe tridimensionnelle
  
  • Colonna Jean-François
  , 2012. The quaternionic Julia set computed with A=(0,1,0,0) -tridimensional cross-section- (L'ensemble de Julia dans le corps des quaternions calculé pour A=(0,1,0,0) -coupe tridimensionnelle-)
• L'ensemble de Julia dans le corps des quaternions calculé pour A=(0,1,0,0) -coupe tridimensionnelle
  
  • Colonna Jean-François
  , 2012. The quaternionic Julia set computed with A=(0,1,0,0) -tridimensional cross-section- (L'ensemble de Julia dans le corps des quaternions calculé pour A=(0,1,0,0) -coupe tridimensionnelle-)
• Un 'mélange' entre une structure fractale tridimensionnelle et un agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb') -coupe tridimensionnelle
  
  • Colonna Jean-François
  , 2012. A 'mixing' between a tridimensional fractal structure and a close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') -tridimensional cross-section- (Un 'mélange' entre une structure fractale tridimensionnelle et un agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb') -coupe tridimensionnelle-)
• Time minimal trajectories for two-level quantum systems with two bounded controls
  
  • Boscain Ugo
  • Grönberg Fredrik
  • Long Ruixing
  • Rabitz Herschel
  , 2012. In this paper we consider the minimum time population transfer problem for a two level quantum system driven by two external fields with bounded amplitude. The controls are modeled as real functions and we do not use the Rotating Wave Approximation. After projection on the Bloch sphere, we tackle the time-optimal control problem with techniques of optimal synthesis on 2-D manifolds. Based on the Pontryagin Maximum Principle, we characterize a restricted set of candidate optimal trajectories. Properties on this set, crucial for complete optimal synthesis, are illustrated by numerical simulations. Furthermore, when the two controls have the same bound and this bound is small with respect to the difference of the two energy levels, we get a complete optimal synthesis up to a small neighborhood of the antipodal point of the starting point.
• Un ensemble de 4x3 stéréogrammes d'une structure fractale tridimensionnelle
  
  • Colonna Jean-François
  , 2012. A set of 4x3 stereograms of a tridimensional fractal structure (Un ensemble de 4x3 stéréogrammes d'une structure fractale tridimensionnelle)
• L'ensemble de Julia dans le corps des quaternions calculé pour A=(0,1,0,0) -coupe tridimensionnelle
  
  • Colonna Jean-François
  , 2012. The quaternionic Julia set computed with A=(0,1,0,0) -tridimensional cross-section- (L'ensemble de Julia dans le corps des quaternions calculé pour A=(0,1,0,0) -coupe tridimensionnelle-)
• L'ensemble de Julia dans le corps des quaternions calculé pour A=(-0.5815147625160462,+0.6358885017421603,0,0) -coupe tridimensionnelle
  
  • Colonna Jean-François
  , 2012. The quaternionic Julia set computed with A=(-0.5815147625160462,+0.6358885017421603,0,0) -tridimensional cross-section- (L'ensemble de Julia dans le corps des quaternions calculé pour A=(-0.5815147625160462,+0.6358885017421603,0,0) -coupe tridimensionnelle-)
• Anaglyphe d'une structure fractale tridimensionnelle
  
  • Colonna Jean-François
  , 2012. Anaglyph of a tridimensional fractal structure (Anaglyphe d'une structure fractale tridimensionnelle)
• Comparison between classes of state-quadratic Lyapunov functions for discrete-time linear polytopic and switched systems
  
  • Mason Paolo
  • Sigalotti Mario
  • Daafouz Jamal
  Systems and Control Letters, Elsevier, 2012, 61 (11), pp.1062-1068. The paper deals with the stability properties of linear discrete-time switched systems with polytopic sets of modes. The most classical way of studying the uniform asymptotic stability of such a system is to check for the existence of a quadratic Lyapunov function. It is known from the literature that letting the Lyapunov function depend on the time-varying switching parameter improves the chance that a quadratic Lyapunov function exists. Our objective is to compare different notions of quadratic stability. The contribution of this paper is twofold. In the first part we consider switching systems satisfying a certain non-degeneracy assumption and we prove that, for such systems, no gain in the stability analysis is obtained if we allow the Lyapunov function to depend explicitly also on time. In the second part we consider the case where the non-degeneracy assumption is violated. We prove that in this case allowing the Lyapunov function to depend on time is less conservative. We also show that new LMI conditions can be used in order to characterize the existence of a time-dependent quadratic Lyapunov function. Moreover in the paper we discuss the case where the variation of the switching parameter is bounded by a prescribed constant between two subsequent times. (10.1016/j.sysconle.2012.07.010)
  DOI : 10.1016/j.sysconle.2012.07.010
• L'ensemble de Julia dans le corps des quaternions calculé pour A=(0,1,0,0) -coupe tridimensionnelle
  
  • Colonna Jean-François
  , 2012. The quaternionic Julia set computed with A=(0,1,0,0) -tridimensional cross-section- (L'ensemble de Julia dans le corps des quaternions calculé pour A=(0,1,0,0) -coupe tridimensionnelle-)
• Phase field method for mean curvature flow with boundary constraints
  
  • Bretin Elie
  • Perrier Valérie
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2012, 46 (6), pp.1509-1526. This paper is concerned with the numerical approximation of mean curvature flow $t \to \Omega(t)$ satisfying an additional inclusion-exclusion constraint $\Omega_1 \subset \Omega(t) \subset \Omega_2$. Classical phase field model to approximate these evolving interfaces consists in solving the Allen-Cahn equation with Dirichlet boundary conditions. In this work, we introduce a new phase field model, which can be viewed as an Allen Cahn equation with a penalized double well potential. We first justify this method by a $\Gamma$-convergence result and then show some numerical comparisons of these two different models. (10.1051/m2an/2012014)
  DOI : 10.1051/m2an/2012014
• Structure fractale tridimensionnelle
  
  • Colonna Jean-François
  , 2012. Tridimensional fractal structure (Structure fractale tridimensionnelle)
• Elastic limit of square lattices with three point interactions
  
  • Meunier Nicolas
  • Pantz Olivier
  • Raoult Annie
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2012, 22 (11), pp.10.1142/S0218202512500327. We derive the equivalent energy of a square lattice that either deforms into the three-dimensional Euclidean space or remains planar. Interactions are not restricted to pairs of points and take into account changes of angles. Under some relationships between the local energies associated with the four vertices of an elementary square, we show that the limit energy can be obtained by mere quasiconvexification of the elementary cell energy and that the limit process does not involve any relaxation at the atomic scale. In this case, it can be said that the Cauchy-Born rule holds true. Our results apply to classical models of mechanical trusses that include torques between adjacent bars and to atomic models. (10.1142/S0218202512500327)
  DOI : 10.1142/S0218202512500327
• L'ensemble des champs bidimensionnels définissant les trois coordonnées d'une interpolation entre un rectangle et un cube arrondi
  
  • Colonna Jean-François
  , 2012. The set of bidimensional fields defining the three coordinates of an interpolation between a rectangle and a rounded cube (L'ensemble des champs bidimensionnels définissant les trois coordonnées d'une interpolation entre un rectangle et un cube arrondi)
• Intégration du problème des N-corps (N=3) montrant deux planètes avec des conditions initiales symétriques sur des trajectoires elliptiques
  
  • Colonna Jean-François
  , 2012. N-body problem integration (N=3) displaying two planets with symmetrical initial conditions on elliptic trajectories (Intégration du problème des N-corps (N=3) montrant deux planètes avec des conditions initiales symétriques sur des trajectoires elliptiques)
• Intégration du problème des N-corps (N=3) montrant deux planètes avec des conditions initiales symétriques sur des trajectoires circulaires
  
  • Colonna Jean-François
  , 2012. N-body problem integration (N=3) displaying two planets with symmetrical initial conditions on circular trajectories (Intégration du problème des N-corps (N=3) montrant deux planètes avec des conditions initiales symétriques sur des trajectoires circulaires)
• An improved time domain linear sampling method for Robin and Neumann obstacles
  
  • Haddar Houssem
  • Lechleiter Armin
  • Marmorat Simon
  , 2012, pp.32. We consider inverse obstacle scattering problems for the wave equation with Robin or Neumann 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 solutions 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.
• 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)