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 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...

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 $Ext{*}_{ℱ}(Id,G)$.

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...

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...

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.

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...

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.

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...

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 give in this paper a purely syntactical definition of input and output types of system $ℱ$. We define the syntactical data types as input and output types. We show that any type with positive quantifiers is a syntactical data type and that an input type is an output type. We give some restrictions on the -elimination rule in order to prove that an output type is an input type.

A powerfull method has been developped in [2] for the study of $L^1$-stability of travelling waves in conservation laws or more generally in equations which display $L^1$-contractivity, maximum principle and mass conservation. We recall shortly the general procedure. We also show that it partly applies to the waves of a model of radiating gas. These waves have first been studied by Kawashima and Nishibata [5,6] in a different framework. Therefore, shock fronts for this model are stable under...

Let G be a compact metric infinite abelian group and let X be a Banach space. We study the following question: if the dual X* of X does not have the Radon-Nikodym property, is $L^p(G,X*)$ complemented in $L^q(G,X)*$, 1 < p ≤ ∞, 1/p + 1/q = 1, or, if p = 1, in the subspace of C(G,X)* consisting of the measures that are absolutely continuous with respect to the Haar measure? We show that the answer is negative if X is separable and does not contain ℓ¹, and if 1 ≤ p < ∞. If p = 1, this answers a question...

Consider a where is a neighborhood of 0 in , is a identified to ker , being the 1-form: $\omega =dz-\frac{y^2}{2}dx$, and is a on which can be taken in the : $g=a\left(q\right)d{x}^2+c\left(q\right)d{y}^2$, , , $G_{{|}_{x=y=0}}=0$. In a previous article we analyze : ; we describe the , the and the . The objectif of this article is to provide a geometric and computational framework to analyze the general case. This frame is obtained by analysing three of the flat case which clarify the role of the three parameters $\alpha ,\beta ,\gamma$ in the where:...

Dans ce papier nous prouvons que si une solution de KdV est suffisamment décroissante à l'infini (c'est-à-dire comme e${}^{-x^{\alpha }}$ où $\alpha >9/4$) et si la donnée de Cauchy est nulle pour assez grand alors la solution est nulle. Ce résultat est la conséquence d'une inégalité de Carleman adaptée à la décroissance de la solution à l'infini.