We show that any quasi-arithmetic mean $A_{\phi }$ and any non-quasi-arithmetic mean M (reasonably regular) are inconsistent in the sense that the only solutions f of both equations $f\left(M(x,y)\right)=A_{\phi }(f\left(x\right),f\left(y\right))$ and $f\left(A_{\phi }(x,y)\right)=M(f\left(x\right),f\left(y\right))$ are the constant ones.

We present an example of an o-minimal structure which does not admit $C^{\infty }$ cellular decomposition. To this end, we construct a function $H$ whose germ at the origin admits a $C^k$ representative for each integer $k$, but no $C^{\infty }$ representative. A number theoretic condition on the coefficients of the Taylor series of $H$ then insures the quasianalyticity of some differential algebras ${𝒜}_n\left(H\right)$ induced by $H$. The o-minimality of the structure generated by $H$ is deduced from this quasianalyticity property.

A Haar wavelet operational matrix is applied to fractional integration, which has not been undertaken before. The Haar wavelet approximating method is used to reduce the fractional Volterra and Abel integral equations to a system of algebraic equations. A global error bound is estimated and some numerical examples with smooth, nonsmooth, and singular solutions are considered to demonstrate the validity and applicability of the developed method.

Se exponen las estimaciones numéricas preliminares de las singularidades de una ecuación diferencial fraccionaria no lineal. Dicha ecuación aparece en el estudio de las ondas viajeras asociadas a una ecuación de ondas que es una interpolación entre la ecuación de ondas clásica y la ecuación de Benjamin-Ono.

En dimension 1 on analyse la fonction irrégulière $r\left(x\right)={\sum }_{n=1}^{\infty }{n}^{-p}sin\left(n^{p}x\right)$ (p entier ≥ 2) en un point $x_0$ de dérivabilité (π est un tel point) et on démontre que le terme d’erreur est un chirp de classe (1 + 1/(2p-2), 1/(p-1), (p-1)/p). La fonction r(x) est dans l’espace 2-microlocal $C_{x_0}^{s,s^{\text{'}}}$ si et seulement si s+s’ ≤ 1 - 1/p et ps+s’≤ p - 1/2. En dimension 2, on obtient en (π,π) l’existence d’un plan tangent pour la surface $z={\sum }_{m,n=1}^{\infty }{(m^2+n^2)}^{-\gamma }sin(m^{2}x+n^{2}y)$ dès que γ>1.

Soit $X$ un espace de Banach de dual topologique $X^{\text{'}}$. $𝒞\left(X\right)$ (resp. $𝒞\left(X^{\text{'}}\right)$) désigne l’ensemble des parties non vides convexes fermées de $X$ (resp. $w^{*}$-fermées de $X^{\text{'}}$) muni de la topologie de la convergence uniforme sur les bornés des fonctions distances. Cette topologie se réduit à celle de la métrique de Hausdorff sur les convexes fermés bornés [16] et admet en général une représentation en terme de cette dernière [11]. De plus, la métrique qui lui est associée s’est révélée très adéquate pour l’étude quantitative...

Let X be a Banach space and X' its continuous dual. C(X) (resp. C(X')) denotes the set of nonempty convex closed subsets of X (resp. ω*-closed subsets of X') endowed with the topology of uniform convergence of distance functions on bounded sets. This topology reduces to the Hausdorff metric topology on the closed and bounded convex sets [16] and in general has a Hausdorff-like presentation [11]. Moreover, this topology is well suited for estimations and constructive approximations [6-9]. We...