Asymptotics with sharp remainder estimates are recovered for number $\mathbf{N}\left(\tau \right)$ of eigenvalues of the generalized Maxwell problem and for related Laplacians which are similar to Neumann Laplacian. We consider domains with ultra-thin cusps (with $exp(-|x{|}^{m+1}$) width ; $m\>0$) and recover eigenvalue asymptotics with sharp remainder estimates.

RésuméNous considérons les applications exponentielles pour les groupes $C^{\infty }(M,GL(N,ℂ))$ où M est une variété lisse compacte. Nous montrons que l’application $P:C^{\infty }(M,gl(N,ℂ))\to C^{\infty }(M,GL(N,ℂ))$ définie par $P\left(f\right)=Exp\left(f_1\right)·...·Exp\left(f_k\right)$ pour $f_i\in g_i$, $g=g_1\oplus ...\oplus g_k$ est (sous certaines conditions sur la décomposition de g) une bijection locale lisse (d’un voisinage de zéro sur un voisinage de l’unité). Nous montrons aussi que pour M = S¹ l’application Q définie par $Q\left(f\right)\left(t\right)={\prod }_{j=-\infty }^{\infty }Exp\left(A_j\left(f\right)e^{ijt}\right)$ est une bijection locale lisse. TABLE DES MATIÈRESIntroduction......................................................................................................................................................................5Chapitre...

It is shown the existence of an uncountable infinity of asymptotic structures (i.e. equivalence's classes of quasi-isometric riemannian metrics) on the non compact manifold ${ℝ}^n$.

It is shown the existence of an uncountable infinity of asymptotic structures (i.e. equivalence's classes of quasi-isometric riemannian metrics) on the non compact manifold ${ℝ}^n$.

En s’appuyant sur un théorème des fonctions implicites de Hamilton, nous montrons la persistance d’une courbe invariante indifférente pour une dynamique holomorphe fibrée de classe $C^{\infty }$. Une condition diophantienne sur la paire de nombres de rotation est demandée. On montre également que cette condition est optimale.

Let p be a prime, and let ℱ be the category of functors from the finite ${}_p$-vector spaces to all ${}_p$-vector spaces. The object Id of ℱ is the inclusion functor. Let F and G be two objects in ℱ. If F and G satisfy suitable conditions, the main result of this paper allows one to compute $Ext{*}_{ℱ}(Id,G\circ F)$ from the knowledge of $Ext{*}_{ℱ}(Id,F)$ and $Ext{*}_{ℱ}(Id,G)$.

Notre étude porte sur une catégorie de structures de Poisson singulières holomorphes au voisinage de $0\in {ℂ}^n$ et admettant une forme normale formelle polynomiale i.e. un nombre fini d’invariants formels. Les séries normalisantes sont divergentes en général. On montre l’existence de transformations normalisantes holomorphes sur des domaines sectoriels de la forme $a\<\arg x^R\<b$, où $x^R$ est un monôme associé au problème. Il suit une classification analytique.

We give a description of Kähler manifolds $M$ equipped with an integrable subbundle $ℱ$ of $TM$ of rank $n-1$ ($n=\dim M$) under the assumption that the line bundle $\mathrm{D}é\mathrm{t}ℱ$ is numerically trivial. This is a sort of foliated version of Bogomolov’s theorem concerning Kähler manifolds with trivial canonical class.

We prove in this paper that the types of system $ℱ$ inhabited uniquely by λ-terms (the -types) have a positive quantifier. We give also consequences of this result and some examples.

On montre, pour une classe particulière de groupes non-unimodulaires $G=N⋌ℝ$, où $N$ est un groupe de Lie stratifié et où l’action de $ℝ$ est définie par les dilatations naturelles de $N$, et pour les sous-laplaciens invariants à gauche correspondants $\Delta$, que toute fonction $m\in H^{2+ϵ}\left(ℝ\right)$ possédant un support compact dans ${ℝ}_{+}$ définit un opérateur $m\left(\Delta \right)$ borné sur les espaces de Lebesgue $L^p(G,d^{r}g)$ associés à la mesure de Haar invariante à droite sur $G$, $1\le p\le \infty$.

By getting uniformly asymptotic estimates for the Poisson kernel on Heisenberg groups ${ℍ}_{2n+1}$, we prove that there exists a constant A > 0, independent of n ∈ ℕ*, such that for all $f\in L¹\left({ℍ}_{2n+1}\right)$, we have ${\left|\right|Mf\left|\right|}_{L^{1,\infty }}\le An\left|\right|f\left|\right|₁$, where M denotes the centered Hardy-Littlewood maximal function defined by the Carnot-Carathéodory distance or by the Korányi norm.

On montre que si $\Gamma$ est un pseudogroupe de transformations locales holomorphes de ${ℂ}^n$ en zéro contenant deux éléments “en position générale” et proches de l’identité, alors : 1) L’action de $\Gamma$ sur le fibré des jets d’ordre infini sur un petit voisinage épointé $ℬ$ de $0$ est minimale (c’est-à-dire que si $z_0,z_1\in ℬ$ et si $\phi :z_0\to z_1$ est un germe de biholomorphisme alors il existe une suite ${\gamma }_n\in \Gamma$ qui converge vers $\phi$ uniformément au voisinage de $z_0$). 2) $\Gamma$ ne préserve aucune structure géométrique au voisinage de $0$ (c’est...

Let p be a prime number ≥ 11 and c be a square-free integer ≥ 3. In this paper, we study the diophantine equation $x^p-y^p=cz²$ in the case where p belongs to 11,13,17. More precisely, we prove that for those primes, there is no integer solution (x,y,z) to this equation satisfying gcd(x,y,z) = 1 and xyz ≠ 0 if the integer c has the following property: if ℓ is a prime number dividing c then ℓ ≢ 1 mod p. To obtain this result, we consider the hyperelliptic curves $C_p:y²={\Phi }_p\left(x\right)$ and $D_p:py²={\Phi }_p\left(x\right)$, where ${\Phi }_p$ is the pth cyclotomic...

Taking an odd increasing Lipschitz-continuous function with polynomial growth β, an odd Lipschitz-continuous and bounded function ϕ satisfying sgn(x)ϕ(x) ≥ 0 and a parameter a ∈ [1/2,1], we consider the (nonlinear) stochastic differential system ⎧$X_t=X₀+B_t+a{\int }_0^t\varphi *v_s\left(X_s\right)ds-(1-a){\int }_0^t\beta *u_s\left(X_s\right)ds$, (E)⎨ ⎩$Y_t=Y₀+B{̃}_t+(1-a){\int }_0^t\varphi *u_s\left(Y_s\right)ds-a{\int }_0^t\beta *v_s\left(Y_s\right)ds$, $\mathbb{P}\left(X_t\in dx\right)=u_t\left(dx\right)$ and $\mathbb{P}\left(Y_t\in dx\right)=v_t\left(dx\right)$, where $\beta *u_t\left(x\right)={\int }_{ℝ}\beta (x-y)u_t\left(dy\right)$, ${\left(B_t\right)}_{t\ge 0}$ and ${\left(B{̃}_t\right)}_{t\ge 0}$ are independent Brownian motions. We show that (E) admits a stationary probability measure, and, under some additional conditions, that $(X_t,Y_t)$ converges in distribution to this invariant measure. Moreover we...

We prove existence (uniqueness is easy) of a weak solution to a boundary value problem for an equation like ${(v-1)}_t^{+}=v_{xx}+F{\left(v\right)}_x$ where the function $F:ℝ\to ℝ$ is only supposed to be locally lipschitz continuous. In order to replace the lack of compactness in on , we use nonlinear semigroup theory.

Let Ω(n) and ω(n) denote the number of distinct prime factors of the positive integer n, counted respectively with and without multiplicity. Let $d_k\left(n\right)$ denote the Piltz function (which counts the number of ways of writing n as a product of k factors). We obtain a precise estimate of the sum ${\sum }_{n\le x,\Omega \left(n\right)-\omega \left(n\right)=q}f\left(n\right)$ for a class of multiplicative functions f, including in particular $f\left(n\right)=d_k\left(n\right)$, unconditionally if 1 ≤ k ≤ 3, and under some reasonable assumptions if k ≥ 4. The result also applies to f(n) = φ(n)/n (where φ is...

Let k be a commutative field. For any a,b∈ k, we denote by $J_{a,b}\left(k\right)$ the deformation of the 2-dimensional Weyl algebra over k associated with the Jordanian Hecke symmetry with parameters a and b. We prove that: (i) any $J_{a,b}\left(k\right)$ can be embedded in the usual Weyl algebra A₂(k), and (ii) $J_{a,b}\left(k\right)$ is isomorphic to A₂(k) if and only if a = b.

We denote by the unit circle and by the unit disc of ℂ. Let s be a non-negative real and ω a weight such that $\omega \left(n\right)={(1+n)}^s$ (n ≥ 0) and the sequence ${(\omega (-n)/{(1+n)}^s)}_{n\ge 0}$ is non-decreasing. We define the Banach algebra $A_{\omega }\left(\right)={f\in \left(\right):\left|\right|f\left|\right|}_{\omega }={\sum }_{n=-\infty }^{+\infty }\left|f̂\left(n\right)\right|\omega \left(n\right)<+\infty$. If I is a closed ideal of $A_{\omega }\left(\right)$, we set $h⁰\left(I\right)=z\in :f\left(z\right)=0\left(f\in I\right)$. We describe all closed ideals I of $A_{\omega }\left(\right)$ such that h⁰(I) is at most countable. A similar result is obtained for closed ideals of the algebra $A{⁺}_s\left(\right)=f\in A_{\omega }\left(\right):f̂\left(n\right)=0(n<0)$ without inner factor. Then we use this description to establish a link between operators with countable spectrum and interpolating...

On a metric measure space (X,ϱ,μ), consider the weight functions $w_{\alpha }\left(x\right)=\varrho {(x,z₀)}^{-\alpha ₀}$ if ϱ(x,z₀) < 1, $w_{\alpha }\left(x\right)=\varrho {(x,z₀)}^{-\alpha ₁}$ if ϱ(x,z₀) ≥ 1, $w_{\beta }\left(x\right)=\varrho {(x,z₀)}^{-\beta ₀}$ if ϱ(x,z₀) < 1, $w_{\beta }\left(x\right)=\varrho {(x,z₀)}^{-\beta ₁}$ if ϱ(x,z₀) ≥ 1, where z₀ is a given point of X, and let ${\kappa }_a:X\times X\to ℝ₊$ be an operator kernel satisfying ${\kappa }_a(x,y)\le c\varrho {(x,y)}^{a-d}$ for all x,y ∈ X such that ϱ(x,y) < 1, ${\kappa }_a(x,y)\le c\varrho {(x,y)}^{a-D}$ for all x,y ∈ X such that ϱ(x,y)≥ 1, where 0 < a < min(d,D), and d and D are respectively the local and global volume growth rate of the space X. We determine conditions on a, α₀, α₁, β₀, β₁ ∈ ℝ for the Hardy-Littlewood-Sobolev...

En utilisant la méthode du double quantique, nous construisons une $R$-matrice universelle pour la quantification de la superalgèbre de Lie $D(2,1,x)$. Nous utilisons ce résultat pour construire un invariant d’entrelacs et nous montrons qu’il est égal à une spécialisation du polynôme de Dubrovnik introduit par Kauffman.