We study the performance of empirical risk minimization (ERM), with respect to the quadratic risk, in the context of convex aggregation, in which one wants to construct a procedure whose risk is as close as possible to the best function in the convex hull of an arbitrary finite class F . We show that ERM performed in the convex hull of F is an optimal aggregation procedure for the convex aggreg...

A normed space $mathfrak{X}$ is said to have the fixed point property, if for each nonexpansive mapping $T : E longrightarrow E $ on a nonempty bounded closed convex subset $ E $ of $ mathfrak{X} $ has a fixed point. In this paper, we first show that if $ X $ is a locally compact Hausdorff space then the following are equivalent: (i) $X$ is infinite set, (ii) $C_0(X)$ is infinite dimensional, (...

RÉSUMÉ. De nombreuses approches ont proposé de pré-calculer des cubes de données afin de répondre efficacement aux requêtes OLAP. La notion de cube de données a été déclinée de différentes manières : cubes icebergs, cubes intervallaires ou encore cubes différentiels. Dans cet article, nous introduisons le concept de cube convexe qui permet de capturer tous les tuples d’un cube de données satisf...

A space $Y$ is called an {em extension} of a space $X$, if $Y$ contains $X$ as a dense subspace. Two extensions of $X$ are said to be {em equivalent}, if there is a homeomorphism between them which fixes $X$ point-wise. For two (equivalence classes of) extensions $Y$ and $Y'$ of $X$ let $Yleq Y'$, if there is a continuous function of $Y'$ into $Y$ which fixes $X$ point-wise. An extension $Y$ ...

Suppose that $A$ is a semi-simple and commutative Banach algebra. In this paper we try to characterize the character space of the Banach algebra $C_{rm{BSE}}(Delta(A))$ consisting of all  BSE-functions on $Delta(A)$ where $Delta(A)$ denotes the character space of $A$. Indeed, in the case that $A=C_0(X)$ where $X$ is a non-empty locally compact Hausdroff space, we give a complete characterizatio...

We consider a family of elliptic equations introduced in the context of traffic congestion. They have the form ∇·(∇F(∇u)) = f , where F is a convex function which vanishes inside some convex set and is elliptic outside. Under some natural assumptions on F and f , we prove that the function ∇F(∇u) is continuous in any dimension, extending a previous result valid only in dimension 2 [14]. Résumé....

In this note we establish the existence and uniqueness of solutions for optimal control problems for the 2D Navier-Stokes equations in a 2D-channel. Our approach is based on infinite-dimensional optimization ; the cost functional is shown to be strictly convex. Generalization to other control problems as well as a gradient algorithm are presented. Existence et unicité du contrôle optimal des éq...