The range of the derivative of a differentiable bumpL'image de la dérivée d'une fonction bosse différentiable

We study the range of the derivative of a Frechet differentiable bump. X is an infinite dimensional separable C^p-smooth Banach space. We first prove that any connected open subset of $\text{X}{}^1$ containing 0 is the range of the derivative of a C^p-bump. Next, analytic subsets of $\text{X}{}^1$ which satisfy a natural linkage condition are the range of the derivative of a C¹-bump. We find analogues of these results in finite dimensions. We finally show that $\text{f′(}\text{R}{}^2\text{)}$ is the closure of its interior, if f is a C²-bump on $\text{R}{}^2$. To cite this article: T. Gaspari, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 189–194.     

Sufficiency and invariance

Suppose that a statistical decision problem is invariant under a group of transformations g?G. T (X) is equivariant if there exists $\text{g}{}^1\text{? G}{}^1$ such that $\text{T(g(X)) = g}{}^1\text{(T((X))}$. We show that the minimal sufficient statistic is equivalent and that if T(X) is an equivariant sufficient statistics and d(X) is invariant under G, then $\text{d}{}^1\text{(T) = Ed(X)∥T}$ is invariant under $\text{G}{}^1$.     

Strong law for the bootstrap

Let X₁, X₂, X₃, … be i.i.d. r.v. with E|X₁| < ∞, E X₁ = μ. Given a realization X = (X₁,X₂,…) and integers n and m, construct Y_n,i, i = 1, 2, …, m as i.i.d. r.v. with conditional distribution $\text{P}{}^1\text{(Y}{}_{\mathrm{n,i}}\text{= X}{}_{\mathrm{j}}\text{) =}\text{1}\text{n}$ for 1 ? j ? n. ($\text{P}{}^1$ denotes conditional distribution given X). Conditions relating the growth rate of m with n and the moments of X₁ are given to ensure the almost sure convergence of $\text{(}\text{1}\text{m}\text{)Σ}{}^{\mathrm{m}}{}_{\mathrm{i=1}}\text{Y}{}_{\mathrm{n,i}}$ toμ. This equation is of some relevance in the theory of Bootstrap as developed by Efron (1979) and Bickel and Freedman (1981).     

Sur la compatibilité des extensions ponctuelles d'un matroïde

A natural sufficient condition for a finite family of single element extensions of a matroid to be compatible is given. Characterizations of all the finite extensions N of a matroid M(E) are given for which the rank function satisfies

$\text{ρ}{}_{\mathrm{N}}\text{(X)=}\text{Min}\text{Z?E}\text{{ρ}{}_{\mathrm{M}}\text{(Z)+|X?}\text{Z}{}^{\mathrm{N}}\text{|}}$

or equivalently the closure operator satisfies $\text{X}{}^{\mathrm{N}}\text{=}\text{X}{}^{\mathrm{N}}\text{? E}{}^{\mathrm{N}}\text{? X}$. The single element extensions and the principal extensions are examples of such matroids. The notion of a sheaf of flats of M. Las Vergnas is used in the proof of a new necessary and sufficient condition for two single element extensions of a matroid to be compatible. An initial announcement of part of these results appeared in R. Cordovil (C. R. Acad. Sci. Paris. A284 (1977), 1249–1252).     

On convergence rates and the expectation of sums of random variables

Let X_i be iidrv's and S_n=X₁+X₂+…+X_n. When EX²₁<+∞, by the law of the iterated logarithm $\text{(S}{}_{\mathrm{n}}\text{?α}{}_{\mathrm{n}}\text{)}\text{(n log n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{→0 a.s.}$ for some constants α_n. Thus the r.v. $\text{Y=sup}{}_{\mathrm{n?1}}\text{[|S}{}_{\mathrm{n}}\text{?α}{}_{\mathrm{n}}\text{|?(δn log n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{]}{}^{\mathrm{+}}$ is a.s.finite when δ>0. We prove a rate of convergence theorem related to the classical results of Baum and Katz, and apply it to show, without the prior assumption EX²₁<+∞ that EY^h<+∞ if and only if $\text{E}\text{|X}{}_1\text{|}{}^{\mathrm{2+h}}\text{[log|X}{}_1\text{|]}{}^{-1}\text{<+∞}$ for 0<h<1 and δ> $\text{h}\text{E}\text{(X}{}_1\text{?}\text{E}\text{X}{}_1\text{)}{}^2$, whereas $\text{E}\text{Y}{}^{\mathrm{h}}\text{=+∞}$ whenever h>0 and $\text{0<δ<h}\text{E}\text{(X}{}_1\text{?}\text{E}\text{X}{}_1\text{)}{}^2$.     

The minimal generating space of a polynomial

Given a polynomial $\text{P(X}{}_1\text{,…,X}{}_{\mathrm{N}}\text{)∈}\text{R}\text{[}\text{X}\text{]}$, we calculate a subspace G_p of the linear space 〈X〉 generated by the indeterminates which is minimal with respect to the property $\text{P∈}\text{R}\text{[}\text{G}{}_{\mathrm{p}}\text{]}$ (the algebra generated by G_p, and prove its uniqueness. Furthermore, we use this result to characterize the pairs (P,Q) of polynomials P(X₁,…,X_n) and Q(X₁,…,X_n) for which there exists an isomorphism T:〈X〉 →〈X〉 that “separates P from Q,” i.e., such that for some k(1<k<n) we can write P and Q as $\text{P}{}^1\text{(Y}{}_1\text{,…,Y}{}_{\mathrm{k}}\text{)}$ and $\text{Q}{}^1\text{(Y}{}_{\mathrm{<mtext>k+</mtext><mtext>1</mtext>}}\text{,…,Y}{}_{\mathrm{n}}\text{)}$ respectively, where $\text{Y}\text{=T}\text{X}$.     

On rigid displacements and their relation to the infinitesimal rigid displacement lemma in shell theory

Let ω be an open connected subset of $\text{R}{}^2$ and let $\text{θ}$ be an immersion from ω into $\text{R}{}^3$. It is established that the set formed by all rigid displacements of the surface $\text{θ}\text{(ω)}$ is a submanifold of dimension 6 and of class $\text{C}{}^{\mathrm{\infty }}$ of the space $\text{H}{}^1\text{(ω)}$. It is shown that the infinitesimal rigid displacements of the same surface $\text{θ}\text{(ω)}$ span the tangent space at the origin to this submanifold. To cite this article: P.G. Ciarlet, C. Mardare, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Non-explosion en temps grand et stabilité de solutions globales des équations de Navier–StokesNon-blowup at large times and stability for global solutions to the Navier–Stokes equations

Suppose there exists a global solution u to the incompressible Navier–Stokes equations, such that $\text{u∈}\text{C}{}_{\mathrm{t}}\text{(}\text{H}\text{?}{}^{\mathrm{1/2}}\text{)}$. We prove that its $\text{H}\text{?}{}^{\mathrm{1/2}}$ norm goes to 0 at infinity. We next use this fact to control the $\text{L}{}^2{}_{\mathrm{t}}\text{(}\text{H}\text{?}{}^{\mathrm{3/2}}\text{)}$ norm of u, and finally we prove that such a solution is stable. To cite this article: I. Gallagher et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 289–292.     

On the distance distribution of homogeneous codes

A code C is called homogeneous if the distance of code words are equally distributed among the coordinates. It is shown that if C is a nontrivia homogeneous code the distance distribution of the punctured code $\text{C}{}^1$ and that of C determine each other, and the minimum distances $\text{m(C)}\text{and}\text{m(C}{}^1\text{)}$ are related by $\text{m(C)=m(C}{}^1\text{)+1}$. If C is a linear code of dimension at least 2, moreover m(C)=m(C′) where C′ is a suitable hyperlane of $\text{C}{}^1$.     

Algebraic braided model of the affine line and difference calculus on a topological spaceModèle algébrique tressé de la droite affine et calcul aux différences sur un espace topologique

In a previous Note [1], we suggested a quantum model of the unit interval [0,1], using convergent power series, parametrized by a variable q (a remarkable example is the quantum exponential, defined by Euler). In the present Note, we suggest a simpler model based on functions $\text{f=f(x):}\text{Z}\text{→k}$ (with an arbitrary commutative ring k) which are constant when x?+∞ or x??∞ and their “differentials” considered as functions x?f(x+1)?f(x) (difference calculus). Thanks to this new “differential calculus over the integers”, we can associate to any simplicial set or topological space X a braided differential graded algebra $\text{D}{}^1\text{(X)}$ which is similar in spirit to the algebra $\text{W}{}^1\text{(X)}$ introduced in [1]. We notice that the p-homotopy type of X can be read from the braiding of $\text{D}{}^1\text{(X)}$. In particular, if $\text{k=}\text{Z}$, we recover in a purely algebraic way the integral cohomology, Steenrod operations, homotopy groups from this braiding. To cite this article: M. Karoubi, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 121–126.     

Banach spaces whose duals contain complemented subspaces isomorphic to C[0, 1]

We give several characterizations of those Banach spaces X such that the dual $\text{X}{}^1$ contains a complemented subspace isomorphic to $\text{C[0, 1]}{}^1$. We investigate operators on separable $\text{L}$_∞ spaces whose adjoints have nonseparable ranges and apply our results to obtain a structure theorem for $\text{L}$_∞ spaces whose duals are not isomorphic to l₁(Γ).     

On the Łojasiewicz numbers

Let f be a holomorphic function of two complex variables with an isolated critical point at $\text{0∈}\text{C}{}^2$. We give some necessary conditions for a rational number to be the smallest θ>0 in the ?ojasiewicz inequality |gradf(z)|?C|z|^θ for z near $\text{0∈}\text{C}{}^2$. To cite this article: E. Garc??a Barroso, A. P?oski, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Sur l'équation divu=fOn the equation divu=f

The main result is the following. Let $\text{Ω}$ be a bounded Lipschitz domain in $\text{R}{}^{\mathrm{d}}$, d?2. Then for every $\text{f∈}\text{L}{}^{\mathrm{d}}\text{(Ω)}$ with ∫f=0, there exists a solution $\text{u∈}\text{C}{}^0\text{(}\text{Ω}\text{)∩}\text{W}{}^{\mathrm{1,d}}\text{(Ω)}$ of the equation divu=f in $\text{Ω}$, satisfying in addition u=0 on $\text{?Ω}$ and the estimate

where C depends only on $\text{Ω}$. However one cannot choose u depending linearly on f. To cite this article: J. Bourgain, H. Brezis, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 973–976.     

Projections on spaces of invariant functions

Let (X, ∑, μ) be a measure space and S be a semigroup of measure-preserving transformations T:X → X. In case μ(X) < ∞, Aribaud [1] proved the existence of a positive contractive projection P of L¹(μ) such that for every $\text{? ? L}{}^1\text{(μ), Pf}$ belongs to the closure $\text{C}{}_1\text{(?)}\text{in}\text{L}{}^1\text{(μ)}$ of the convex hull $\text{C(?)}$ of the set {$\text{? ○ T:T ? S}$}. In this paper we extend this result in three directions: we consider infinite measure spaces, vector-valued functions, and L^p spaces with 1 ? p < ∞, and prove that P is in fact the conditional expectation with respect to the σ-algebra Λ of sets of ∑ which are invariant with respect to all T?S.     

Sur l'analyticité des applications CR lisses à valeurs dans un ensemble algébrique réelOn the analyticity of smooth CR maps into a real algebraic set

Let f:M→M′ be a $\text{C}{}^{\mathrm{\infty }}$-smooth CR mapping between a generic real analytic submanifold $\text{M?}\text{C}{}^{\mathrm{n}}$ and a real algebraic subset $\text{M′?}\text{C}{}^{\mathrm{n\prime }}$. We prove that if M is minimal at a point p and if M′ does not contain complex curves, then f is real-analytic at p. To cite this article: B. Coupet et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 953–956.     

A transfer spectral sequence for fixed point free involutions with an application to stunted real projective spaces

We show that if X is a finite CW-complex admitting a fixed point free involution then there is a singly graded spectral sequence with $\text{E}{}^1{}_1\text{? H}{}^1\text{(X;}\text{Z}{}_2\text{)}$ and $\text{E}{}^1\text{∞ = 0}$. As an application we prove that for any n > 0 there is a natural number k(n) such that if n > k(n) and X is a homotopy $\text{R}\text{P}{}^{\mathrm{n+k}}\text{R}\text{P}{}^{\mathrm{n}}$, then X will not admit a fixed point free involution.     

Extensions of factorial states of C1-algebras

Let A be a $\text{C}{}^1$-algebra, B be a $\text{C}{}^1$-subalgebra of A, and φ be a factorial state of B. Sometimes, φ may be extended to a factorial state of A by a tensor product method of Sakai (“$\text{C}{}^1$-algebras and $\text{W}{}^1$-algebras, Springer-Verlag, Berlin/Heidelberg/ New York 1971”). Sometimes, there is a weak expectation of A into $\text{π}{}_{\mathrm{\phi }}\text{(B)}$, and then factorial extensions may be found by a method of Sakai and Tsui (Yokohama Math. J.29 (1981), 157–160). These two methods are shown to have the same effect, and the factorial extensions produced by them are analysed.     

Duality theorems for linear systems and convex systems

We show that, if (F →^uX) is a linear system, $\text{Ω ? X}$ a convex target set and $\text{h: X →}\text{R}\text{?}$ a convex functional, then, under suitable assumptions, the computation of inf $\text{h({y ? F ¦ u(y) ? Ω}}$) can be reduced to the computation of the infimum of h on certain strips or hyperplanes in F, determined by elements of $\text{u}{}^1\text{(X}{}^1\text{)}$, or of the infima on F of Lagrangians, involving elements of $\text{u}{}^1\text{(X}{}^1\text{)}$. Also, we prove similar results for a convex system (F →^uX) and the convex cone Ω of all non-positive elements in X.