A preorder consists of linearly ordered equivalence classes called blocks, and an alignment is a sequence of cycles on n labelled elements. We investigate the block structure of a random preorder chosen uniformly at random among all preorders on n elements, and also the distribution of cycles in a random alignment chosen uniformly at random among all alignments on n elements, as n →∞.

In the context of structural optimization by the level-set method, we propose an extension of the velocity of the underlying Hamilton-Jacobi equation. The gradient method is endowed with an Hilbertian structure based on the H Sobolev space. Numerical results for compliance minimization and mechanism design shows a strong improvement of the rate of convergence of the level-set method. Another im...

Following Davie’s example of a Banach space failing the approximation property ([D]), we show how to construct a Banach space E which is asymptotically Hilbertian and fails the approximation property. Moreover, the space E is shown to be a subspace of a space with an unconditional basis which is “almost” a weak Hilbert space and which can be written as the direct sum of two subspaces all of who...

This paper is devoted to weaken “classical” assumptions and give new arguments to prove existence of sweeping process (associated to the proximal normal cone of sets). Mainly we define the concept of a “directional prox-regularity” and give assumptions about a Banach space to insure the existence of such sweeping process (which permit to generalize the existing results requiring a Hilbertian st...

There is a strong analogy between proving theorems in mathematics and writing programs in computer science. This paper is devoted to an analysis, from the perspective of this analogy, of proof in mathematics. We will argue that while the Hilbertian notion of proof has few chances to change, future proofs will be of various types, will play different roles, and their truth will be checked differ...

In this paper we study shorted operators relative to two different subspaces, for bounded operators on infinite dimensional Hilbert spaces. We define two notions of “complementability” in the sense of Ando for operators, and study the properties of the shorted operators when they can be defined. We use these facts in order to define and study the notions of parallel sum and subtraction, in this...