In this Note we first establish a result on the structure of the set of fixed points of a multi-valued contraction with convex values. As a consequence of this result, we then obtain the following theorem: Let $(U,\parallel \cdot {\parallel }_U)$, $(V,\parallel \cdot {\parallel }_V)$ be two real Banach spaces and let $\mathrm{\Phi }$ be a continuous linear operator from $U$ onto $V$. Put: $\alpha =sup\{inf\{{\parallel u\parallel }_U:u\in {\mathrm{\Phi }}^{-1}(v)\}:v\in V,{\parallel v\parallel }_V\le 1\}$. Then, for every $v\in V$ and every lipschitzian operator $\mathrm{\Psi }:U\to V$, with Lipschitz constant $L$ such that , the set $\{u\in U:\mathrm{\Phi }(u)+\mathrm{\Psi }(u)=v\}$ is non-empty and arc wise connected.

Nello spazio proiettivo $P^{2k-2}$, consideriamo il numero $v_k$ dei sottospazi $(k-2)$-dimensionali, $k$-secanti una curva C, di grado $n$ e genere $g$. Nel 1889, una formula per $v_k$ è stata data da Castelnuovo [2]; in [1] si trova una dimostrazione moderna. In questo lavoro, mi propongo di costruire la funzione generatrice della serie ${\sum }_{k\ge 0}{v}_k{t}^k\in Z[[t]]$, senza usare i risultati di Castelnuovo.

Si dimostra resistenza e l'unicità della soluzione del problema $Au=f$, $u\in H_0^1(\mathrm{\Omega })$ nel caso in cui $\mathrm{\Omega }$ è un aperto di ${ℝ}^n$ non limitato, $A$ è un operatore variazionale ellittico del secondo ordine a coefficienti misurabili e limitati e $f$ appartiene a $H^{-1}(\mathrm{\Omega })$.

Si dimostra resistenza e l'unicità della soluzione del problema $Au=f$, $u\in H_0^1(\mathrm{\Omega })$ nel caso in cui $\mathrm{\Omega }$ è un aperto di ${ℝ}^n$ non limitato, $A$ è un operatore variazionale ellittico del secondo ordine a coefficienti misurabili e limitati e $f$ appartiene a $H^{-1}(\mathrm{\Omega })$.

Si discute l'esistenza di soluzioni ${\mathrm{\Gamma }}^{(d)}$ su insiemi aperti $\mathrm{\Omega }\subset {ℝ}^n$ per equazioni differenziali iperbolico-ipoellittiche. Si dà una caratterizzazione geometrica quasi completa per aperti $\mathrm{\Omega }\subset {ℝ}^2$.

Si discute l'esistenza di soluzioni ${\mathrm{\Gamma }}^{(d)}$ su insiemi aperti $\mathrm{\Omega }\subset {ℝ}^n$ per equazioni differenziali iperbolico-ipoellittiche. Si dà una caratterizzazione geometrica quasi completa per aperti $\mathrm{\Omega }\subset {ℝ}^2$.

Soit $(D,\varphi )$ un isocristal de $\nu \in {\left({\mathbb{Q}}^d\right)}_{+}$. On associe à une filtration ${ℱ}^{•}$ de $D$ son $\mu \left({ℱ}^{•}\right)\in {\left({\mathbb{Z}}^d\right)}_{+}$. Si ${ℱ}^{•}$ est (. $(D,\varphi ,{ℱ}^{•})$ est faiblement admissible en tant qu’isocristal filtré), alors $\mu \left({ℱ}^{•}\right)\ge \nu$. Réciproquement, on démontre qu’étant donné $\mu \in {\left({\mathbb{Z}}^d\right)}_{+}$ avec $\mu \ge \nu$, il existe une filtration admissible ${ℱ}^{•}$ de $D$ avec $\mu =\mu \left({ℱ}^{•}\right)$. On en déduit, à l’aide d’un théorème de Laffaille, l’existence d’un réseau $M$ dans $D$ de type $\mu$. On donne aussi une variante pour un groupe quasi-déployé quelconque.

Siano $A$, $B$ sottoinsiemi convessi, chiusi e limitati di uno spazio normato $X$, con le frontiere $frA$, $frB$. Dimostriamo che $h(A,B)=h(frA,frB)$, dove $h$ è la metrica di Hausdorff tra sottoinsiemi chiusi di $X$. Studiamo inoltre la continuità e la semicontinuità superiore ed inferiore di una multifunzione di tipo «frontiera».

Siano $A$, $B$ sottoinsiemi convessi, chiusi e limitati di uno spazio normato $X$, con le frontiere $frA$, $frB$. Dimostriamo che $h(A,B)=h(frA,frB)$, dove $h$ è la metrica di Hausdorff tra sottoinsiemi chiusi di $X$. Studiamo inoltre la continuità e la semicontinuità superiore ed inferiore di una multifunzione di tipo «frontiera».

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 $Ext{*}_{ℱ}(Id,G)$.

Nous prouvons l’hyper-réflexivité du shift bilatéral $S_{\omega }$ sur ${\ell }_{\omega }^2\left(\mathbb{Z}\right)$, lorsque le poids vérifie $\omega \left(n\right)=1$ for $n\ge 0$ et ${lim}_{n\to -\infty }\omega \left(n\right)=+\infty$.

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...

Let [0;a₁(x),a₂(x),…] be the regular continued fraction expansion of an irrational x ∈ [0,1]. We prove mainly that, for α > 0, β ≥ 0 and for almost all x ∈ [0,1], $li{m}_{n\to \infty }(aⁿ₁\left(x\right)+\dots +aⁿₙ\left(x\right))/nlogn=\alpha /log2$ if α < 1 and β ≥ 0, $li{m}_{n\to \infty }(aⁿ₁\left(x\right)+\dots +aⁿₙ\left(x\right))/nlogn=1/log2$ if α = 1 and β < 1, and, if α > 1 or α = 1 and β >1, $lim in{f}_{n\to \infty }(aⁿ₁\left(x\right)+\dots +aⁿₙ\left(x\right))/nlogn=1/log2$, $lim su{p}_{n\to \infty }(aⁿ₁\left(x\right)+\dots +aⁿₙ\left(x\right))/nlogn=\infty$, where $a{ⁿ}_i\left(x\right)=a_i\left(x\right)$ if $a_i\left(x\right)\le n^{\alpha }lo{g}^{\beta }n$ and $a{ⁿ}_i\left(x\right)=0$ otherwise, for all i ∈ 1,…,n.

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...

Generalizing the classical BMO spaces defined on the unit circle with vector or scalar values, we define the spaces $BM{O}_{{\psi }_q}\left(\right)$ and $BM{O}_{{\psi }_q}(,B)$, where ${\psi }_q\left(x\right)=e^{x^q}-1$ for x ≥ 0 and q ∈ [1,∞[, and where B is a Banach space. Note that $BM{O}_{{\psi }_1}\left(\right)=BMO\left(\right)$ and $BM{O}_{{\psi }_1}(,B)=BMO(,B)$ by the John-Nirenberg theorem. Firstly, we study a generalization of the classical Paley inequality and improve the Blasco-Pełczyński theorem in the vector case. Secondly, we compute the idempotent multipliers of $BM{O}_{{\psi }_q}\left(\right)$. Pisier conjectured that the supports of idempotent multipliers of...

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...

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...

Soit $F_0$ un difféomorphisme d’une surface possédant deux fers à cheval $\Lambda ,{\Lambda }^{\text{'}}$ tels que $W^s\Lambda$ et $W^u{\Lambda }^{\text{'}}$ aient en un point $q$ une tangence quadratique isolée. Nous montrons que, si la somme des dimensions transverses de $W^s\Lambda$ et $W^u{\Lambda }^{\text{'}}$ est strictement plus grande que 1, les difféomorphismes voisins de $F_0$ tels que $W^s\Lambda$ et $W^u{\Lambda }^{\text{'}}$ soient stablement tangents au voisinage de $q$ forment une partie de densité inférieure strictement positive en $F_0$.

On considère une extension finie $k$ de ${\mathbb{Q}}_p$, avec $p$ un nombre premier, $H$ un sous-groupe d’indice fini de $k^{*}$ et le groupe $\mathrm{SL}(n,k)$. Nous montrons que $\mathrm{SL}(n,k)$ admet un sous-groupe ${\mathbb{Q}}_p$-Zariski-dense dont toutes les matrices ont leur spectre inclus dans $H$ si et seulement si soit $-1$ est dans le sous-groupe $H$, soit $n$ n’est pas congru à 2 modulo 4.