Les solutions élements finis des équations de Navier–Stokes périodiques en dimension trois sont « appropriées »

It is shown that the limits of Faedo–Galerkin approximations of the Navier–Stokes equations in the three-dimensional torus are suitable weak solutions to the Navier–Stokes equations provided they are constructed using finite-dimensional spaces having a discrete commutator property and satisfying a proper inf–sup condition. Low order mixed finite element spaces appear to be acceptable for this purpose. This question was open since the notion of suitable solution was introduced. To cite this article: J.-L. Guermond, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Regular subsets of valued fields and Bhargava’s v-orderings

We generalize notions and results obtained by Amice for regular compact subsets S of a local field K and extended by Bhargava to general compact subsets of K. Considering any ultrametric valued field K and subsets S that are regular in a generalized sense (but not necessarily compact), we show that they still have strong properties such as having v-orderings ${\{a_n\}_{n\geq0}}$ which satisfy a generalized Legendre formula, which are very well ordered and well distributed sequences in the sense of Helsmoortel and which remain v-orderings when a finite number of the initial terms of the sequence are deleted.     

A threshold property for intersections in a finite set

Let F be any family of subsets of a finite set E and let n be an integer, n<|F|. Under what condition does the knowledge of cardinals of m-intersections in F, for all m≤n, univocally determine the cardinal of any intersection in F, and what is the minimal condition? We give a complete answer to that. For any n, this determination property is satisfied by n if and only if |E|<2ⁿ, without further condition on F.     

About Polynomials Whose Divided Differences are Integer-Valued on Prime Numbers

We first globalize some recent results about polynomials whose divided differences are integer-valued on subsets of discrete valuation domains. This allows us to describe explicit computations, in particular for polynomials which are integer-valued on the prime numbers. Then, we specialize other recent results about integer-valued polynomials on triangular matrices to matrices whose coefficients are in some subsets, and in particular, when the coefficients are primes.     

Distribution functions of linear combinations of lattice polynomials from the uniform distribution

We give the distribution functions, the expected values, and the moments of linear combinations of lattice polynomials from the uniform distribution. Linear combinations of lattice polynomials, which include weighted sums, linear combinations of order statistics, and lattice polynomials, are actually those continuous functions that reduce to linear functions on each simplex of the standard triangulation of the unit cube. They are mainly used in aggregation theory, combinatorial optimization, and game theory, where they are known as discrete Choquet integrals and Lovász extensions.     

Axiomatizations of quasi-Lovász extensions of pseudo-Boolean functions

We introduce the concept of quasi-Lovász extension as being a mapping \({f\colon I^n\to\mathbb{R}}\) defined on a nonempty real interval I containing the origin and which can be factorized as f(x ₁, . . . , x _n ) =  L(φ(x ₁), . . . , φ(x _n )), where L is the Lovász extension of a pseudo-Boolean function \({\psi\colon \{0, 1\}^n \to \mathbb{R}}\) (i.e., the function \({L\colon \mathbb{R}^n \to \mathbb{R}}\) whose restriction to each simplex of the standard triangulation of [0, 1] ⁿ is the unique affine function which agrees with ψ at the vertices of this simplex) and \({\varphi\colon I \to \mathbb{R}}\) is a nondecreasing function vanishing at the origin. These functions appear naturally within the scope of decision making under uncertainty since they subsume overall preference functionals associated with discrete Choquet integrals whose variables are transformed by a given utility function. To axiomatize the class of quasi-Lovász extensions, we propose generalizations of properties used to characterize Lovász extensions, including a comonotonic version of modularity and a natural relaxation of homogeneity. A variant of the latter property enables us to axiomatize also the class of symmetric quasi-Lovász extensions, which are compositions of symmetric Lovász extensions with 1-place nondecreasing odd functions.     

Topological Entropy and Secondary Folding

A convenient measure of a map or flow’s chaotic action is the topological entropy. In many cases, the entropy has a homological origin: it is forced by the topology of the space. For example, in simple toral maps, the topological entropy is exactly equal to the growth induced by the map on the fundamental group of the torus. However, in many situations the numerically computed topological entropy is greater than the bound implied by this action. We associate this gap between the bound and the true entropy with ‘secondary folding’: material lines undergo folding which is not homologically forced. We examine this phenomenon both for physical rod stirring devices and toral linked twist maps, and show rigorously that for the latter secondary folds occur.     

Starch Nanocrystal Fillers in an Acrylic Polymer Matrix

Aqueous suspensions of nanocrystals can be obtained by acid hydrolysis of native waxy maize starch granules. The disruption of waxy maize starch granules by 2.2N HCl hydrolysis has been followed by scanning and transmission electron microscopy and laser granulometry. The mechanical properties of composite materials made of poly(styrene-co-butyl acrylate) filled with starch nanocrystals were characterized by dynamic mechanical analysis. These nanocrystals appeared to be an interesting reinforcing phase in a thermoplastic matrix.     

Blends of a new thermoplastic in a thermoset epoxy matrix

Properties of thermoplastic modified epoxy network have been studied. The particularity of this work is the use of new thermoplastic epoxies whose structure is close to the final matrix. Blends of thermoset epoxy (Diglycidyl Ether of Bisphenol A/4-4′ methylenebis [3-chloro 2,6-diethylaniline]) with a thermoplastic content from 5 to 40%w have been realised. Initial miscibility in the thermoset precursors shows an UCST behaviour with a maximal value near 130°C for a thermoplastic content of 10%. Due to the presence of tertiary amine and pendant hydroxyl groups on the thermoplastic backbone, epoxy amine reactions are faster than for the neat system but the thermoplastic seems not to have reacted with the thermoset network. The final blends are transparent but toughening increase is rather low.