Sur quelques résultats de convergence dans l'inférence d'une classe de processus ponctuelsConvergence results for point processes estimators

Our aim is to generalize some results obtained for a Poisson point process in [7], to a general point process. Those results are in field of complete convergence of two like Parzen–Rosenblatt estimates of density of mean measure function and regression curves. Those estimates are defined from the superposition of n i.i.d. point processes as:

$\text{f}{}_{\mathrm{n}}\text{(x)=}\text{1}\text{nh}\text{∑}\text{i=1}\text{m}\text{K}\text{x?X}{}_{\mathrm{i}}{}^{\mathrm{(n)}}\text{h(n)}\text{and}\text{Ψ}{}_{\mathrm{n}}\text{(x)=}\text{∑}{}_{\mathrm{i=1}}{}^{\mathrm{m}}\text{Y}{}_{\mathrm{i}}\text{K}\text{x?X}{}_{\mathrm{i}}{}^{\mathrm{(n)}}\text{h(n)}\text{∑}{}_{\mathrm{i=1}}{}^{\mathrm{m}}\text{K}\text{x?X}{}_{\mathrm{i}}{}^{\mathrm{(n)}}\text{h(n)}\text{,}$

where m is the number of seem generics points of the superposition. We give some sufficient conditions for the convergence of those kernel-like estimators. To cite this article: A. Diakhaby, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 597–602.     

Répartition des densités des sous-suites d'une suite d'entiers

Let $\text{A}$ denote a strictly increasing sequence of integers; for any integer n, define A(n) to be the number of positive elements of $\text{A}$ not exceeding n. The upper and lower asymptotic densities of $\text{A}$ are defined by

We describe the set of pairs (d$\text{B}$, $\text{d}$$\text{B}$), where $\text{B}$ runs over all subsequences of $\text{A}$, as being a closed convex region of the plane. The converse statement is also proved.     

Sharp Sobolev type inequalities for higher fractional derivativesInégalités optimales de type Sobolev pour les dérivées fractionelles d'ordre supérieur

On $\text{R}{}^{\mathrm{n}}$, n?1 and n≠2, we prove the existence of a sharp constant for Sobolev inequalities with higher fractional derivatives. Let s be a positive real number. For n>2s and $\text{q=}\text{2n}\text{n?2s}$ any function $\text{f∈}\text{H}{}^{\mathrm{s}}\text{(}\text{R}{}^{\mathrm{n}}\text{)}$ satisfies

$\text{6f6}{}^2{}_{\mathrm{q}}\text{?S}{}_{\mathrm{n,s}}\text{(?Δ)}{}^{\mathrm{s/2}}\text{f}{}^2{}_2\text{,}$

where the operator (?Δ)^s in Fourier spaces is defined by $\text{(?Δ)}{}^{\mathrm{s}}\text{f}\text{(k):=(2π|k|)}{}^{\mathrm{2s}}\text{f}\text{(k)}$. To cite this article: A. Cotsiolis, N.C. Tavoularis, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 801–804.     

<Emphasis Type="Italic">F</Emphasis>-isocristaux surconvergents et surcohérence différentielle

Résumé Soient un anneau de valuation discrète complet d’inégales caractéristiques, de corps résiduel parfait k, un -schéma formel propre et lisse, T un diviseur de la fibre spéciale P de , U l’ouvert de P complémentaire de T, Y un sous-k-schéma fermé lisse de U. Nous prouvons que la catégorie des F-isocristaux surconvergents sur Y est équivalente à celle des F-isocristaux surcohérents sur Y (voir [Car, 6.2.1 et 6.4.3.a)]). Plus généralement, nous établissons par recollement une telle équivalence pour tout k-schéma séparé lisse Y. Nous vérifions de plus que les F-complexes de -modules à cohomologie bornée et -surcohérente se dévissent en F-isocristaux surconvergents.     

Parties semi-algébriques d'une variété algébrique <Emphasis Type="Italic">p</Emphasis>-adique

Let k be a field complete with respect to an ultrametric absolute value, let A be a finitely generated k-algebra and let X be its spectrum. We denote by X^an the analytification à la Berkovich of the algebraic variety X. We say that a subset of X^an is semi-algebraic if it can be defined by a boolean combination of inequalities |f||g| where f and g are in A, where is a symbol belonging to {<,>,,} and where is a positive real number. In this text we show that the image of any semi-algebraic subset under an algebraic map between affine varieties is semi-algebraic, and that a semi-algebraic subset has only finitely many connected components, each of which is semi-algebraic. In fact our results concern not only algebraic varieties, but also analytic families of algebraic varieties which are parametrized by an affinoid space.     

Sous-groupes discrets de <Emphasis Type="Italic">PU</Emphasis>(2,1) engendrés par <Emphasis Type="Italic">n</Emphasis> réflexions complexes et Déformation

Let PU(2,1) be the group of holomorphic isometries in the hyperbolic complex plane and let G _n be a sub-group of PU(2,1) which is generated by n complex reflections with respect to complex lines in . Under certain conditions, we prove that G _n is discrete. We construct representations ρ of the fundamental group Γ _g of the compact surface Σ _g of genus g, into PU(2,1), we prove they are discrete, faithful and we compute the dimension their deformation space.      

Propriétés de l'intégrale de Cauchy Harish-Chandra pour certaines paires duales d'algèbres de LieProperties of the Cauchy Harish-Chandra integral for some dual pairs of Lie algebras

We consider a symplectic group Sp and an reductive and irreductible dual pair (G,G′) in Sp in the sense of R. Howe. Let $\text{g}$ (resp. $\text{g}\text{′}$) be the Lie algebra of G (resp. G′). T. Przebinda has defined a map $\text{Chc}$, called the Cauchy Harish-Chandra integral from the space of smooth compactly supported functions of $\text{g}$ to the space of functions defined on the open set $\text{g}{}^{\mathrm{<mtext>\prime </mtext><mspace\; sp="0.2"\; width="2px"\; linebreak="nobreak"\; is="true"></mspace><mtext>reg</mtext>}}$ of semisimple regular elements of $\text{g}\text{′}$. We prove that these functions are invariant integrals if G and G′ are linear groups and they behave locally like invariant integrals if G and G′ are unitary groups of same rank. In this last case, we obtain the jump relations up to a multiplicative constant which only depends on the dual pair. To cite this article: F. Bernon, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 945–948.     

Représentations <Emphasis Type="Italic">p</Emphasis>-adiques de GL<Subscript>2</Subscript>(L) et Catégories Dérivées

We construct some locally ℚ _p -analytic representations of GL₂(L), L a finite extension of ℚ _p , associated to some p-adic representations of the absolute Galois group of L. We prove that the space of morphisms from these representations to the de Rham complex of Drinfel’d’s upper half space has a structure of rank 2 admissible filtered (φ, N)-module. Finally, we prove that this filtered module is associated, via Fontaine’s theory, to the initial Galois representation.     

Existence and uniqueness results for nonlinear elliptic problems with a lower order term and measure datumExistence et unicité de solutions renormalisées d'équations elliptiques non linéaires avec des termes d'ordre inférieur et données mesures

In this Note we consider a class of noncoercive nonlinear problems whose prototype is

$\text{?△}{}_{\mathrm{p}}\text{u+b(x)|?u|}{}^{\mathrm{\lambda }}\text{=μ}\text{in}\text{Ω,}\text{u=0}\text{on}\text{?Ω,}$

where $\text{Ω}$ is a bounded open subset of $\text{R}{}^{\mathrm{N}}$ (N?2), △_p is the so called p-Laplace operator (1<p<N) or a variant of it, μ is a Radon measure with bounded variation on $\text{Ω}$ or a function in $\text{L}{}^1\text{(Ω)}$, λ?0 and b belongs to the Lorentz space $\text{L}{}^{\mathrm{N,1}}\text{(Ω)}$ or to the Lebesgue space $\text{L}{}^{\mathrm{\infty }}\text{(Ω)}$. We prove existence and uniqueness of renormalized solutions. To cite this article: M.F. Betta et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 757–762.     

Sous-groupes canoniques et cycles évanescents <Emphasis Type="Italic">p</Emphasis>-adiques pour les variétés abéliennes

Sans résumé     

Précis of <Emphasis Type="Italic">Self-Expression</Emphasis> (Oxford, 2007)

I give a brief overview of the major contentions and methodologies of my book, Self-Expression.     

Éclosions combinatoires appliquées à l'inversion multidimensionnelle des séries formelles

Using general methods from the theory of combinatorial species, in the sense of A. Joyal (Adv. in Math.42 (1981), 1–82), symmetric powers of suitably chosen differential operators are interpreted combinatorially in terms of “éclosions” (bloomings) of certain kinds of points, called “bourgeons” (buds), into certain kinds of structures, called “gerbes” (bundles). This gives rise to a combinatorial setting and simple proof of a general multidimensional power series reversion formula of the Lie-Gröbner type (14., 15.). Some related functional equations are also treated and an adaptation of the results to the reversion of cycle index (indicatrix) series, in the sense of Pólya-Joyal (Joyal, loc. cit.), is given.     

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.     

Cramér type moderate deviations for trimmed <Emphasis Type="Italic">L</Emphasis>-statistics

We establish Cramér type moderate deviation results for heavy trimmed L-statistics; we obtain our results under a very mild smoothness condition on the inversion F ^?1 (F is the underlying distribution function of i.i.d. observations) near two points, where trimming occurs, we assume also some smoothness of weights of the L-statistic. Our results complement previous work on Cramér type large deviations for trimmed L-statistics [8] and [5].     

A solution to the Bézout equation in <Emphasis Type="Italic">A</Emphasis>(<Emphasis Type="Italic">K</Emphasis>) without Gelfand theory

Let \({M\subseteq \mathbb{C}}\) be compact and \({K\subseteq M}\) closed, and let A(K, M) be the uniform algebra of all functions continuous on M and holomorphic in the interior K° of K. We present a constructive proof of Arens’ classical result that for \({(f_{1},\ldots,f_{n})\in A(K,M)^{n}}\) the Bézout equation \({\sum_{j=1}^{n} a_{j}f_{j}=1}\) has a solution in A(K, M) if and only if the functions f _j have no common zero in K. We shall also consider matrix-valued Bézout equations.     

On the convergence of Fejér means of Walsh-Fourier series in the space <Emphasis Type="Italic">H</Emphasis> <Subscript> <Emphasis Type="Italic">p</Emphasis> </Subscript>

The main aim of this paper is to find necessary and sufficient conditions for a modulus of continuity of a martingale F ∈ H_p, for which the Fejér means of Walsh-Fourier series converge in H_p-norm, when 0 < p ≤ 1/2.     

On the equation Δ<Emphasis Type="Italic">u</Emphasis> + <Emphasis Type="Italic">q</Emphasis>(<Emphasis Type="Italic">x</Emphasis>)<Emphasis Type="Italic">u</Emphasis> = 0

Sufficient conditions for the blow-up of nontrivial generalized solutions of the interior Dirichlet problem with homogeneous boundary condition for the homogeneous elliptic-type equation Δu + q(x)u = 0, where either q(x) ≠ const or q(x) = const= λ > 0, are obtained. A priori upper bounds (Theorem 4 and Remark 6) for the exact constants in the well-known Sobolev and Steklov inequalities are established.     

Potential spaces and traces of Lévy processes on <Emphasis Type="Italic">h</Emphasis>-sets

Potential spaces and Dirichlet forms associated with Lévy processes subordinate to Brownian motion in ℝ ⁿ with generator f(−Δ) are investigated. Estimates for the related Rieszand Bessel-type kernels of order s are derived which include the classical case f(r) = r ^α/2 with 0 < α < 2 corresponding to α-stable Lévy processes. For general (tame) Bernstein functions f potential representations of the trace spaces, the trace Dirichlet forms, and the trace processes on fractal h-sets are derived. Here we suppose the trace condition ∫₀¹ r ⁻⁽ⁿ⁺¹⁾ f(r ⁻²)⁻¹ h(r) dr < ∞ on f and the gauge function h. Dedicated to the 80th birthday of Klaus Krickeberg     

The smooth structure set of <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> × <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>

We calculate \({\mathcal{S}^{{\it Diff}}(S^p \times S^q)}\), the smooth structure set of S ^p × S ^q , for p, q ≥ 2 and p + q ≥ 5. As a consequence we show that in general \({\mathcal{S}^{Diff}(S^{4j-1}\times S^{4k})}\) cannot admit a group structure such that the smooth surgery exact sequence is a long exact sequence of groups. We also show that the image of the forgetful map \({\mathcal{S}^{Diff}(S^{4j}\times S^{4k}) \rightarrow \mathcal{S}^{Top}(S^{4j}\times S^{4k})}\) is not in general a subgroup of the topological structure set.