Institut Élie Cartan de Lorraine

About: Institut Élie Cartan de Lorraine is a facility organization based out in Vandœuvre-lès-Nancy, France. It is known for research contribution in the topics: Boundary value problem & Stochastic differential equation. The organization has 345 authors who have published 1084 publications receiving 15512 citations. The organization is also known as: Institut Élie-Cartan de Nancy.

Initial-boundary value problem for the heat equation—A stochastic algorithm

Abstract: The initial-boundary value problem for the heat equation is solved by using an algorithm based on a random walk on heat balls. Even if it represents a sophisticated generalization of the Walk on Spheres (WOS) algorithm introduced to solve the Dirichlet problem for Laplace’s equation, its implementation is rather easy. The construction of this algorithm can be considered as a natural consequence of previous works the authors completed on the hitting time approximation for Bessel processes and Brownian motion [Ann. Appl. Probab. 23 (2013) 2259–2289, Math. Comput. Simulation 135 (2017) 28–38, Bernoulli 23 (2017) 3744–3771]. A similar procedure was introduced previously in the paper [Random Processes for Classical Equations of Mathematical Physics (1989) Kluwer Academic]. The definition of the random walk is based on a particular mean value formula for the heat equation. We present here a probabilistic view of this formula. The aim of the paper is to prove convergence results for this algorithm and to illustrate them by numerical examples. These examples permit to emphasize the efficiency and accuracy of the algorithm.

Average-case Analysis of the Chip Problem

Abstract: In the system level, adaptive fault diagnosis problem we must determine which components (chips) in a system are defective, assuming the majority of them are good. Chips are tested as follows: Take two chips, say x and y, and have x report whether y is good or bad. If x is good, the answer is correct, but if x is bad, the answer is unreliable and can be either wrong with probability α or right with probability 1 − α. The key to identifying all defective chips is to identify a single good chip which can then be used to diagnose the other chips; the chip problem is to identify a single good chip. In [1] we have shown that the chip problem is closely related to a modified majority problem in the worst case and have used this fact to obtain upper and lower bounds on algorithms for the chip problem. In this paper, we show the limits of this relationship by showing that algorithms for the chip problem can violate lower bounds on average performance for the modified majority problem and we give an algorithm for the “biased chip” problem (a chip is bad with probability p) whose average performance is better than the average cost of the best algorithm for the biased majority problem.

Recherche statistique de biomarqueurs du cancer et de l'allergie à l'arachide

Abstract: La premiere partie de la these traite de la recherche de biomarqueurs du cancer. Lors de la transcription, il apparait que certains nucleotides peuvent etre remplaces par un autre nucleotide. On s'interesse alors a la comparaison des probabilites de survenue de ces infidelites de transcription dans des ARNm cancereux et dans des ARNm sains. Pour cela, une procedure de tests multiples menee sur les positions des sequences de reference de 17 genes est realisee via les EST (Expressed Sequence Tag). On constate alors que ces erreurs de transcription sont majoritairement plus frequentes dans les tissus cancereux que dans les tissus sains. Ce phenomene conduirait ainsi a la production de proteines dites aberrantes, dont la mesure permettrait par la suite de detecter les patients atteints de formes precoces de cancer. La deuxieme partie de la these s'attache a l'etude de l'allergie a l'arachide. Afin de diagnostiquer l'allergie a l'arachide et de mesurer la severite des symptomes, un TPO (Test de Provocation Orale) est realise en clinique. Le protocole consiste a faire ingerer des doses croissantes d'arachide au patient jusqu'a l'apparition de symptomes objectifs. Le TPO pouvant se reveler dangereux pour le patient, des analyses discriminantes de l'allergie a l'arachide, du score du TPO, du score du premier accident et de la dose reactogene sont menees a partir d'un echantillon de 243 patients, recrutes dans deux centres differents, et sur lesquels sont mesures 6 dosages immunologiques et 30 tests cutanes. Les facteurs issus d'une Analyse Factorielle Multiple sont egalement utilises comme predicteurs. De plus, un algorithme regroupant simultanement en classes des intervalles comprenant les doses reactogenes et selectionnant des variables explicatives est propose, afin de mettre ensuite en competition des regles de classement. La principale conclusion de cette etude est que les mesures de certains anticorps peuvent apporter de l'information sur l'allergie a l'arachide et sa severite, en particulier ceux diriges contre rAra-h1, rAra-h2 et rAra-h3.

Transformation de Poisson sur un arbre localement fini

Abstract: Dans cet article on etudie en premier lieu la resolvante (le noyau de Green) d'un operateur agissant sur un arbre localement fini. Ce noyau est suppose invariant par un groupe G d'automorphismes de l'arbre. On donne l'expression generique de cette resolvante et on etablit des simplifications sous differentes hypotheses sur G. En second lieu on introduit la transformation de Poisson qui associe a une mesure additive finie sur l'espace Ω des bouts de l'arbre une fonction propre de 1' operateur. On montre que la bijectivite de cette transformation se deduit de la non nullite de certains determinants et on montre celle-ci pour des cas assez generaux.

The Hardy-Schrödinger operator on the Poincaré Ball: compactness and multiplicity

Abstract: Let $\Omega$ be a compact smooth domain containing zero in the Poincar\'e ball model of the Hyperbolic space $\mathbb{B}_n$ ($n \geq 3$) and let $-\Delta_{\mathbb{B}_n}$ be the Laplace-Beltrami operator on $\mathbb{B}_n$, associated with the metric $g_{\mathbb{B}_n}= \frac{4}{(1-|x|^{2})^2}g_{_{\hbox{Eucl}}}$. We consider issues of non-existence, existence, and multiplicity of variational solutions for the borderline Dirichlet problem, \begin{eqnarray} (E)\left\{ \begin{array}{lll} -\Delta_{\mathbb{B}_n}u-\gamma{V_2}u -\lambda u&=V_{2^*(s)}|u|^{2^*(s)-2}u &\hbox{ in }\Omega \\ \hfill u &=0 & \hbox{ on } \partial \Omega, \end{array} \right. \end{eqnarray} where $0\leq \gamma \leq \frac{(n-2)^2}{4}$, $0 0$. The latter result also holds true for $n\geq 3$ and $\gamma > \frac{(n-2)^2}{4}-1$ provided the domain has a positive ``hyperbolic mass". On the other hand, the same analysis yields that if $\gamma > \frac{(n-2)^2}{4}-1$ and the mass is non vanishing, then there is a surprising stability of regimes where no variational positive solution exists. As for higher energy solutions to (E), we show that there are infinitely many of them provided $n\geq 5$, $0\leq \gamma \frac{n-2}{n-4} \left(\frac{n(n-4)}{4}-\gamma \right)$.