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...

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 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 $Ext{*}_{ℱ}(Id,G)$.

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...

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 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 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.

Soient $X$ une variété algébrique complexe, lisse, irréductible, $E$ et $F$ deux espaces vectoriels complexes de dimension finie et $\mu$ un morphisme de $X$ dans l’espace Lin$(E,F)$ des applications linéaires de $E$ dans $F$. Pour $x\in X$, on note $E\left(x\right)$ et $x·E$ le noyau et l’image de $\mu \left(x\right)$, ${\overline{\mu }}_x$ le morphisme de $X$ dans Lin$\left(E\right(x),F/(x·E\left)\right)$ qui associe à $y$ l’application linéaire $v\mapsto \mu \left(y\right)\left(v\right)+x·E$. Soit i${}_{\mu }$ la dimension minimale de $E\left(x\right)$. On dit que $\mu$ asi i${}_{{\overline{\mu }}_x}$ est inférieur à i${}_{\mu }$. Soient $F^{*}$ le dual de $F$, S$\left(F\right)$ l’algèbre symétrique de $F$, ${ℐ}_{\mu }$ l’idéal de ${𝒪}_X{\otimes }_{ℂ}\mathrm{S}\left(F\right)$ engendré par les fonctions...

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.

Soit $q$ dans $\mathbb{Z}$ tel que $\left|q\right|\ge 2$. Dans cette note, nous démontrons que si une fonction entière $f$ a une croissance assez lente et si $f(q^n+i{q}^m)\in \mathbb{Z}\left[i\right]$ pour $n,m\in \mathbb{N}$, alors $f$ est un polynôme.