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.

A Milstein-type scheme without Levy area terms for SDEs driven by fractional Brownian motion

Abstract: In this article, we study the numerical approximation of stochastic differential equations driven by a multidimensional fractional Brownian motion (fBm) with Hurst parameter greater than 1/3. We introduce an implementable scheme for these equations, which is based on a second order Taylor expansion, where the usual Levy area terms are replaced by products of increments of the driving fBm. The convergence of our scheme is shown by means of a combination of rough paths techniques and error bounds for the discretisation of the Levy area terms.

The first eigenvalue of Dirac and Laplace operators on surfaces

Abstract: Let $(M,g,\sigma)$ be a compact Riemmannian surface equipped with a spin structure $\sigma$. For any metric $\tilde{g}$ on $M$, we denote by $\mu_1(\tilde{g})$ (resp. $\lambda_1(\tilde{g})$) the first positive eigenvalue of the Laplacian (resp. the Dirac operator) with respect to the metric $\tilde{g}$. In this paper, we show that $$\inf \frac{\lambda_1(\tilde{g})^2 }{\mu_1(\tilde{g})} \leqslant \frac{1}{2}.$$ where the infimum is taken over the metrics $\tilde{g}$ conformal to $g$. This answer a question asked by Agricola, Ammann and Friedrich

Fonctions arithmétiques et formes binaires irréductibles de degré $3$

Abstract: Dans cet article, nous obtenons des estimations de l'ordre moyen $\sum_{\substack{|n_1|\leq x,|n_2|\leq x}}h(n_1^3+2n_2^3)$ pour des fonctions arithmetiques $h$ soumises a certaines conditions. On donne en particulier une formule asymptotique du nombre d'entiers $y$-friables de la forme $n^3_1+2n_2^2$ ou $(n_1,n_2)$ parcourent $[1,x]^2$, uniforme dans la region $\exp\left(\frac{\log x}{(\log\log x)^{1/2-\varepsilon}}\right)\leq y\leq x$. La methode utilisee s'applique egalement a des fonctions multiplicatives oscillantes comme la fonction $\mu$ de Mobius : il s'ensuit une nouvelle preuve de la conjecture de Chowla pour la forme $X_1^3+2X_2^3$, recemment demontree par Helfgott dans le cas plus general des formes binaires cubiques irreductibles.

Simplicial Differential Calculus, Divided Differences, and Construction of Weil Functors

Abstract: We define a simplicial differential calculus by generalizing divided differences from the case of curves to the case of general maps, defined on general topological vector spaces, or even on modules over a topological ring K. This calculus has the advantage that the number of evaluation points growths linearly with the degree, and not exponentially as in the classical, ``cubic'' approach. In particular, it is better adapted to the case of positive characteristic, where it permits to define Weil functors corresponding to scalar extension from K to truncated polynomial rings K[X]/(X^{k+1}).

Surgery and the spinorial tau-invariant

Abstract: We associate to a compact spin manifold M a real-valued invariant \tau(M) by taking the supremum over all conformal classes over the infimum inside each conformal class of the first positive Dirac eigenvalue, normalized to volume 1. This invariant is a spinorial analogue of Schoen's $\sigma$-constant, also known as the smooth Yamabe number. We prove that if N is obtained from M by surgery of codimension at least 2 , then $\tau(N) \geq \min\{\tau(M),\Lambda_n\}$ with $\Lambda_n>0$. Various topological conclusions can be drawn, in particular that \tau is a spin-bordism invariant below $\Lambda_n$. Below $\Lambda_n$, the values of $\tau$ cannot accumulate from above when varied over all manifolds of a fixed dimension.