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\newtheorem{thm}{Th\'eor\`eme}
\newtheorem{prop}{Proposition}
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\newtheorem{CO}{Corollaire}
\newtheorem{LEM}{Lemme}
 \newtheorem{rem}{Remarque}

 \title{Localisation uniforme des espaces de Besov et de Li\-zorkin-Triebel}

 \author{Salah Eddine Allaoui et G\'erard Bourdaud}

\begin{document}
 \maketitle
 \date{}

\begin{abstract}

On \'etablit des caract\'erisations intrins\`eques des versions localis\'ees-uniformes des espaces de Besov ${\besovn{s}{p}{q}}$, avec  $p,q\in [1,+\infty]$, et de Lizorkin-Triebel ${\lizorn{s}{p}{q}}$ avec $q\in [1,+\infty]$ et $p\in [1,+\infty[$, quel que soit le nombre r\'eel $s>0$. \\

We give intrinsic characterisations for the uniformly localized versions of the  Besov spaces ${\besovn{s}{p}{q}}$, where $p,q\in [1,+\infty]$, and of the Lizorkin-Triebel spaces ${\lizorn{s}{p}{q}}$, where $q\in [1,+\infty]$ and $p\in [1,+\infty[$, whatever be the real number $s>0$.

\end{abstract}

{\it Mots-cl\'es:} {Espaces de Besov, Espaces de Lizorkin-Triebel, Localisation uniforme.}

{\it 2010 Mathematics Subject Classification:} {46E35.}

\section{Introduction}

\`A tout espace norm\'e $E$ de fonctions sur ${\mathbb{R}}^n$, il est possible d'associer sa version localis\'ee-uniforme. Il s'agit de l'espace, not\'e
$E_{lu}$, des fonctions $f$ telles que
\begin{equation}\label{deflu} \sup_{a\in {\mathbb{R}}^n} \| (\tau_a\varphi)f\|_E< +\infty\,.\end{equation}
Ici $\tau_a$ d\'esigne l'op\'erateur de translation, d\'efini par $\tau_af(x):= f(x-a)$, et $\varphi$ est une fonction $C^\infty$ \`a support compact, positive, non nulle. Sous des hypoth\`eses standard, rappel\'ees au paragraphe \ref{local}, on peut montrer que $E_{lu}$ ne d\'epend aucunement du choix de la fonction auxiliaire $\varphi$.\\

Les espaces localis\'es-uniformes jouent un r\^ole dans diverses questions d'analyse math\'e\-ma\-tique. Par exemple, si $E$ est une alg\`ebre de Banach de fonctions, pour la multiplication usuelle, il est naturel de conjecturer que l'ensemble des multiplicateurs de $E$ est pr\'ecis\'ement $E_{lu}$ --- conjecture confirm\'ee dans le cas des espaces de Sobolev $H^s({\mathbb{R}}^n)$ pour $s>n/2$, voir \cite[p.~58]{Bou_95} et \cite[p.~151]{P}. Ils interviennent aussi dans la caract\'erisation des fonctions qui op\`erent, par composition \`a gauche, sur certains espaces de fonctions. Ainsi, les fonctions de ${\mathbb{R}}$ dans ${\mathbb{R}}$ qui op\`erent, en ce sens, sur l'espace de Sobolev critique $W^m_p({\mathbb{R}}^n)$, o\`u l'entier $m$ v\'erifie $m=n/p>1$, sont pr\'ecis\'ement celles dont les d\'eriv\'ees appartiennent localement-uniform\'ement \`a $W^{m-1}_p({\mathbb{R}})$ \cite{BouIn}. L'extension de ce th\'eor\`eme aux espaces de Sobolev fractionnaires est une question ouverte. Cependant on a pu \'etablir un r\'esultat partiel \cite{GB-SA}: si une fonction op\`ere sur l'espace de Besov $B^s_{p,q}({\mathbb{R}}^n)$, avec $s=n/p>1$ et $q>1$, alors sa d\'eriv\'ee appartient localement-uniform\'ement \`a $B^{s-1}_{p,q}({\mathbb{R}})$.\\

Il semble d\`es lors pertinent de d\'ecrire les localisations uniformes des espaces de Sobolev fractionnaires de fa\c con intrins\`eque, c'est-\`a-dire sans recourir \`a une fonction auxiliaire telle que la fonction $\varphi$ utilis\'ee dans (\ref{deflu}). \`A cet \'egard, rappelons que, pour les espaces $L_p({\mathbb{R}}^n)_{lu}$, une telle description est bien connue, voir la proposition \ref{Lplu}.
 C'est ce type de description, \`a l'aide d'int\'egrales portant sur les translat\'ees d'une boule fixe, que nous mettrons en \'evidence pour
les localisations uniformes des espaces de Sobolev fractionnaires.

\subsection*{Plan} La section \ref{local} sera d\'evolue \`a des g\'en\'eralit\'es sur la localisation uniforme.
Dans la section \ref{DBLT}, on rappellera les d\'efinitions des espaces Besov et de Lizorkin-Triebel  et on \'enoncera les th\'eor\`emes principaux, qui seront \'etablis dans la derni\`ere section.
\subsection*{Notations et rappels}

La norme d'une fonction $f$ dans l'espace de Lebesgue $L_p({\mathbb{R}}^n)$ est not\'ee $\|f\|_p$.
Comme d'habitude $c, c_1,...$ d\'esignera une constante positive pouvant d\'ependre de $n, s, p, q$ et des fonctions auxiliaires; sa valeur pourra changer d'une occurrence \`a l'autre.
Rappelons une in\'egalit\'e classique (voir, par exemple, \cite[II.20, p.~44]{Bou_95}):

\begin{LEM}\label{MALPHA} Pour tout $q\in [1,+\infty[$ et tout r\'eel $\alpha$, il existe $c>0$ tel que
\[ \sup _{0<t\leq 1/2}  t^\alpha u(t)\leq c\left( \int_0^1 \left(t^\alpha u(t)\right)^q \frac{{\rm d}t}{t}\right)^{1/q}\]
pour toute fonction croissante $u$ sur l'intervalle $]0,1]$.
\end{LEM}

\section{G\'en\'eralit\'es sur la localisation uniforme}\label{local}

Un espace de Banach de distributions (E.B.D.) sur ${\mathbb{R}}^n$ est un sous-espace vectoriel $E$ de $\mathcal{D'}({\mathbb{R}}^n)$ muni d'une norme compl\`ete $\|-\|_E$ telle que l'injection canonique $E\hookrightarrow\mathcal{D'}({\mathbb{R}}^n)$ soit continue. On dit que l'espace $E$ est un $\mathcal{D}({\mathbb{R}}^n)$-{\em module} si $\phi f\in E$ pour tout $\phi\in\mathcal{D}({\mathbb{R}}^n)$ et tout $f\in E$.
Un E.B.D. $E$ est {\em isom\'etriquement invariant par translation} si $\tau_af\in E$ et $\|\tau_af\|_E=\|f\|_E$ pour tout $f\in E$ et tout $a\in{\mathbb{R}}^n$.
\begin{prop}\label{luspace} Soit  $E$ un ${\mathcal D}({\mathbb{R}}^n)$-module
isom\'etriquement invariant par translation.  Pour toute distribution $f$, les deux propri\'et\'es suivantes sont \'equi\-va\-lentes:

(i) Il existe une fonction positive non nulle $\varphi\in {\mathcal D}({\mathbb{R}}^n)$ v\'erifiant (\ref{deflu}).

(ii) Pour toute fonction $\varphi\in {\mathcal D}({\mathbb{R}}^n)$, on a la propri\'et\'e (\ref{deflu}).
\end{prop}
{\bf Preuve.} Voir \cite[p.~57]{Bou_95}.

\hspace{0cm}\hspace*{\fill} \rule{3mm}{3mm

Si une distribution $f$ satisfait l'une des deux conditions \'equivalentes de la proposition \ref{luspace}, on dit que
 $f$ appartient
{\em localement uniform\'ement} \`a $E$; l'ensemble de ces distributions
est not\'e $E_{lu}$. Soit  une fonction positive non nulle $\varphi\in {\mathcal D}({\mathbb{R}}^n)$.
On montre facilement que $E_{lu}$ est un E.B.D. pour la norme
\[ \|f\|_{E_{lu}}:=\sup_{a\in {\mathbb{R}}^n} \|(\tau_a\varphi)\,f\|_E \,.\]
 De la preuve de la proposition \ref{luspace}, il r\'esulte qu'\`a \'equivalence pr\`es, la norme de $E_{lu}$ ne d\'epend pas du choix de la fonction $\varphi$.\\

Si $E$ est un $E.B.D.$ et $m$ un entier positif, on peut
consid\'erer  {\em l'espace de Sobolev $W^m(E)$ d'ordre $m$} de base $E$,  \`a savoir
\[ W^m(E):= \{ f\in {\mathcal D}'({\mathbb{R}}^n)\,:\, f^{(\alpha)} \in E\,\quad \mathrm{pour\,tout}\quad\, |\alpha|\leq m\}\,.\]
 $W^m(E)$ est un E.B.D. pour la norme
\[ \|f\|_{W^m(E)}:= \sum_{|\alpha|\leq m} \|f^{(\alpha)}\|_E\,.\]

\begin{prop}\label{Sobolev_m} Si $E$ est un ${\mathcal D}({\mathbb{R}}^n)$-module, isom\'etriquement invariant par translation, il en est de m\^eme pour $W^m(E)$ et on a

\[ \left(W^m(E)\right)_{lu} = W^m\left(E_{lu}\right)\,,\]
avec des normes \'equivalentes.

\end{prop}

{\bf Preuve.}
Elle r\'esulte ais\'ement de la formule de Leibniz et de la proposition \ref{luspace}.

\hspace{0cm}\hspace*{\fill} \rule{3mm}{3mm

Nous terminerons cette section en rappelant la description de $L_p({\mathbb{R}}^n)_{lu}$.
La preuve facile est laiss\'ee au lecteur.

\begin{prop}\label{Lplu}
Soit $p\in[1,+\infty[$.  Soit ${\mathbb B}$ une boule ouverte (ou un cube ouvert) dans ${\mathbb{R}}^n$. Alors une fonction mesurable $f$ sur ${\mathbb{R}}^n$ appartient \`a $L_p({\mathbb{R}}^n)_{lu}$ si et seulement si

\[\sup_{a\in {\mathbb{R}}^n} \left( \int_{{\mathbb B}+a} |f(x)|^p{\rm d}x\right)^{1/p} <+\infty\,;\]
 de plus l'expression ci-dessus est \'equivalente \`a la norme $\left\|f\right\|_{L_p({\mathbb{R}}^n)_{lu}}$.
 \end{prop}

\section{D\'efinitions des espaces fonctionnels et \'enonc\'es des th\'eor\`emes}\label{DBLT}

\`A toute fonction $f$, d\'efinie sur ${\mathbb{R}}^n$, et tout $h\in {\mathbb{R}}^n$, on associe la fonction
$\Delta_{h}f$, d\'efinie par $\Delta_{h}f:=\tau_{-h}f -f$.
Pour tout $p\in [1,+\infty]$, tout ensemble bor\'elien $A$ de ${\mathbb{R}}^n$, toute fonction mesurable $f$ sur ${\mathbb{R}}^n$ et tout $t>0$, on pose
\[ \omega_{p,A}(f,t):= \sup_{|h|\leq t} \left(\int_A \left|\Delta_{h}f(x)\right|^p {\rm d}x\right)^{1/p}\,,\]
\[ \eta_{p,A}(f,t):= \sup_{|h|\leq t} \left(\int_A \left|\Delta^2_{h}f(x)\right|^p {\rm d}x \right)^{1/p}\,.\]
On note simplement $\omega_{p}:=\omega_{p,{\mathbb{R}}^n}$, de m\^eme pour $\eta.$

\begin{DE} Soient  $0<s<1$, $p,q\in [1,+\infty]$. L'espace $\besovn{s}{p}{q}$ est l'ensemble des fonctions $f$ v\'erifiant
\[ \|f\|_{\besovn{s}{p}{q}}:= \|f\|_p+\left(\int_0^{1}(t^{-s}\omega_{p}(f,t))^q \frac{{\rm d}t}{t}\right)^{1/q}  < + \infty\,.\]
\end{DE}
\begin{DE} Soient  $0<s<1$, $q\in [1,+\infty]$, $1\leq p<\infty$.
L'espace de Lizorkin-Triebel
$\lizorn{s}{p}{q}$ est l'ensemble des fonctions $f$ v\'erifiant
\[ \|f\|_{\lizorn{s}{p}{q}}:= \|f\|_p+\left( \left(\int^1_{0}\left(t^{-s-n}\int_{|h|\leq t} |\Delta_{h}f(x)|^q\,{\rm d}h\right)^q\frac{{\rm d}t}{t}\right)^{p/q}\,{\rm d}x\right)^{1/p} < + \infty\,.\]
\end{DE}

Rappelons qu'on obtient les m\^emes espaces fonctionnels, avec des normes \'equivalentes, en rempla\c cant, dans les d\'efinitions pr\'ec\'edentes, l'int\'egrale $\int_0^1$ par l'int\'egrale $\int_0^r$, o\`u $r$ est n'importe quel r\'eel positif fix\'e.
Quand il n'y a pas lieu de distinguer entre les deux types d'espaces, $B$ ou $F$,
nous posons $\spacen{s}{p}{q} :=\besovn{s}{p}{q}$ ou $\lizorn{s}{p}{q}$.
Les espaces d'ordre $1$, c'est-\`a-dire les espaces $\spacen{1}{p}{q}$, se d\'efinissent de la m\^eme fa\c con,
{\em \`a condition de remplacer} $\Delta_h$ par l'op\'erateur de diff\'erence seconde  $\Delta_h^2$ (et donc, dans le cas Besov, $\omega$
par $\eta$). Les espaces d'ordre sup\'erieur \`a $1$ sont, par d\'efinition, les espaces de Sobolev bas\'es sur les espaces d'ordres compris entre $0$ et $1$:

\begin{DE}\label{ppr4}
Soient $s>1$ et  $m$ l'entier tel que $m<s\leq m+1.$ Alors $\spacen{s}{p}{q}$ est l'ensemble des  fonctions $f$ telles que  $f^{(\alpha)}\in\spacen{s-m}{p}{q}$ pour tout $\left|\alpha\right|\leq m$. Cet espace est muni de la norme
 \[\sum_{ |\alpha|\leq m} \|f^{(\alpha)}\|_{\spacen{s-m}{p}{q}}\,.\]
 %
  \end{DE}

Venons-en \`a la description intrins\`eque des espaces localis\'es-uniformes. On se limitera au cas $0<s\leq 1$, puisqu'il suffit
d'appliquer la proposition \ref{Sobolev_m} pour obtenir le cas g\'en\'eral.
Dans les \'enonc\'es suivants, ${\mathbb B}$ d\'esignera une boule (ou un cube) fix\'e de ${\mathbb{R}}^n$. On supposera $p,q\in [1,+\infty]$ ($p<\infty$ dans le cas Lizorkin-Triebel).
\begin{thm}\label{besovlu} Si $0<s<1$, alors $\besovn{s}{p}{q}_{lu}$ est l'ensemble des fonctions $f$
telles que
\begin{equation}\label{Besovlu}\sup_{a\in {\mathbb{R}}^n} \left(\int_0^1 \left( t^{-s} \omega_{p,{\mathbb B}+a}(f,t)\right)^q\frac{{\rm d}t}{t}\right)^{1/q}+\left\|f\right\|_{L_p({\mathbb{R}}^n)_{lu}}<+\infty\,.\end{equation}
De plus l'expression ci-dessus
est une norme \'equivalente sur $\besovn{s}{p}{q}_{lu}$.
\end{thm}

\begin{thm}\label{besovlus=1}  $\besovn{1}{p}{q}_{lu}$ est l'ensemble des fonctions $f$ telles que
\begin{equation}\label{Besovlus=1}\sup_{a\in {\mathbb{R}}^n} \left(\int_0^1 \left( t^{-1} \eta_{p,{\mathbb B}+a}(f,t)\right)^q\frac{{\rm d}t}{t}\right)^{1/q}+\left\|f\right\|_{L_p({\mathbb{R}}^n)_{lu}}<+\infty\,.\end{equation}
De plus l'expression ci-dessus
est une norme \'equivalente sur $\besovn{1}{p}{q}_{lu}$.
\end{thm}

\begin{thm}\label{Lizornlu} Si $0<s<1$,  alors $\lizorn{s}{p}{q}_{lu}$ est l'ensemble des fonctions $f$ telles que
\begin{equation}\label{LLizornlu}\sup_{a\in {\mathbb{R}}^n}\left\|\left(\int_0^1 \left( t^{-s-n}\int_{\left|h\right|\leq t}\left|\Delta_{h}f(.)\right|{\rm d}h\right)^q\frac{{\rm d}t}{t}\right)^{1/q}\right\|_{L_p({\mathbb B}+a)}+\left\|f\right\|_{L_p({\mathbb{R}}^n)_{lu}}<+\infty\,.\end{equation}
De plus l'expression ci-dessus
est une norme \'equivalente sur $\lizorn{s}{p}{q}_{lu}$.
\end{thm}

\begin{thm}\label{Lizornlus=1}  $\lizorn{1}{p}{q}_{lu}$ est l'ensemble des fonctions $f$ telles que
\begin{equation}\label{LLizornlus=1}\sup_{a\in {\mathbb{R}}^n}\left\|\left(\int_0^1 \left( t^{-n-1}\int_{\left|h\right|\leq t}\left|\Delta^2_{h}f(.)\right|{\rm d}h\right)^q\frac{{\rm d}t}{t}\right)^{1/q}\right\|_{L_p({\mathbb B}+a)}+\left\|f\right\|_{L_p({\mathbb{R}}^n)_{lu}}<+\infty\,.\end{equation}
De plus l'expression ci-dessus
est une norme \'equivalente sur $\lizorn{1}{p}{q}_{lu}$.
\end{thm}

 \section{Preuves des th\'eor\`emes}
 Sans perte de g\'en\'eralit\'e, on peut supposer que ${\mathbb B}$ est la boule unit\'e de ${\mathbb{R}}^n$.
 Dans cette section,
 on fixe deux fonctions $\varphi_0$ et $\varphi_1$ dans ${\mathcal D}({\mathbb{R}}^n)$, telles que:
 \begin{itemize}
 \item $0\leq \varphi_0\leq 1$, $\varphi_0$ est non nulle et port\'ee par
 ${\mathbb B}/4$,
 \item $\varphi_1(x)=1$ sur $4{\mathbb B}$.
 \end{itemize}

\subsection {Preuve du th\'eor\`eme \ref{besovlu}}

On utilisera la formule suivante, valable pour tout $h\in{\mathbb{R}}^n$ et  toutes fonctions $f$ et $g$ sur ${\mathbb{R}}^n$:
\begin{equation}\label{1diff}
\Delta_h(fg) = (\Delta_hf)(\tau_{-h}g)
 + f(\Delta_hg)\,.\end{equation}
D\'esignons par $A(f)$ le premier terme de l'in\'egalit\'e (\ref{Besovlu}).

\subsubsection{\'Etape 1}

Soit $f$ une fonction telle que $A(f)<\infty$.
Par la formule (\ref{1diff}), on a, pour tous $a,h\in {\mathbb{R}}^n$ et $|h|\leq t\leq 1/2$,
\[ \left(\int_{{\mathbb{R}}^n} \left| \Delta_{h}( (\tau_a\varphi_0) f) (x)\right|^p{\rm d}x\right)^{1/p}\]
  \[\leq \left(\int_{{\mathbb{R}}^n}  \left| \Delta_{h}f(x)\varphi_0(x+h-a)\right|^p{\rm d}x\right)^{1/p}+
    \left(\int_{{\mathbb{R}}^n} |f(x)|^p\left| \Delta_{h} (\tau_a\varphi_0) (x)\right|^p{\rm d}x\right)^{1/p}\]
    \[\leq  \left(\int_{{\mathbb B}+a}   \left| \Delta_{h}f (x)\right|^p{\rm d}x\right)^{1/p} +
    t\left\|\nabla\varphi_0\right\|_\infty\, \left(\int_{{\mathbb B}+a}   \left|f (x)\right|^p{\rm d}x\right)^{1/p}\]
      \[\leq c_1\,\left( \omega_{p,{\mathbb B}+a}(f,t) +
    \,t\,  \|f\|_{L^p({\mathbb{R}}^n)_{lu}}\right)\,.\]
Par la condition $s<1$, on voit  que
\[  \left(\int_0^{1/2} \left( t^{-s} \omega_{p}((\tau_a\varphi_0)f,t)\right)^q\frac{{\rm d}t}{t}\right)^{1/q} \]
\[\leq c_1  \left(\left(\int_0^{1/2} \left( t^{-s} \omega_{p,{\mathbb B}+a}(f,t)\right)^q\frac{{\rm d}t}{t}\right)^{1/q}
+ \left(\int_0^{1/2} \left( t^{1-s} \right)^q\frac{{\rm d}t}{t}\right)^{1/q}  \|f\|_{L^p({\mathbb{R}}^n)_{lu}}\right).\]
L'expression ci-dessus \'etant major\'ee par $c_2A(f)$, pour une certaine constante $c_2$,
il vient
\[ \sup_{a\in {\mathbb{R}}^n} \|(\tau_a\varphi_0)f \|_{ \besovn{s}{p}{q}}\leq c_3 A(f)\,. \]

\subsubsection{\'Etape 2}

Soit $f\in\besovn{s}{p}{q}_{lu}$. On voit aussit\^ot que
$ \Delta_{h}((\tau_a\varphi_1)f)(x) = \Delta_{h}f(x)$ pour tout $a\in {\mathbb{R}}^n$,  tout $x\in {\mathbb B}+a$, et tout $|h|\leq 1$.
On en d\'eduit ais\'ement que
\[ A(f) \leq c_4 \sup_{a\in {\mathbb{R}}^n} \|(\tau_a\varphi_1)f \|_{ \lizorn{s}{p}{q}}\,.\]

\subsection {Preuve du th\'eor\`eme \ref{besovlus=1}}

On d\'esignera par $A(f)$ le premier terme de l'in\'egalit\'e (\ref{Besovlus=1}) et on posera
\[ M_{p,a}(f):= \sup_{0<t\leq 1/2} \,\frac{1}{t}\eta_{p,{\mathbb B}+a}(f,t)\,.\]

 \subsubsection{R\'esultats pr\'eliminaires  }

On dispose des formules suivantes, o\`u $k\in\mathbb{N^*}$, $h\in{\mathbb{R}}^n$ et o\`u $f$ et $g$ sont des fonctions quelconques:
\begin{equation}\label{2diff}
\Delta^2_h(fg) = (\Delta^2_hf)(\tau_{-2h}g) + (\Delta^2_hg)(\tau_{-h}f)
 +(\Delta_h f)(\Delta_{2h}g)\,,\end{equation}
 \begin{equation}\label{3diff}
 \Delta_{h} = 2^{-k} \Delta_{2^kh} - \sum_{l=0}^{k-1} 2^{-l-1}  \Delta^2_{2^lh}\, .
 \end{equation}
La premi\`ere est imm\'ediate, la seconde s'obtient facilement par r\'ecurrence sur $k$.

\begin{LEM}\label{Marchaud} Il existe $c>0$ tel que
\[ \omega_{p,{\mathbb B}+a}(f,t)\leq
ct\left\{ \left(\int_{2{\mathbb B}+a} |f(x)|^p\,{\rm d}x\right)^{1/p} + M_{p,a}(f)\,|\ln t|
\right\}\,,\]
pour tout $0<t\leq 1/2,$ tout $a\in {\mathbb{R}}^n$ et toute fonction localement int\'egrable $f.$
\end{LEM}
{\bf Preuve.} Le lemme est une variante de l'in\'egalit\'e classique de Marchaud.
On d\'efinit l'entier $k\geq 1$ par l'encadrement $2^{-k-1}<t\leq 2^{-k}.$
De la formule (\ref{3diff}), on d\'eduit, pour $|h|\leq t$,
\[ \left(\int_{{\mathbb B}+a}  \left| \Delta_{h}f(x)\right|^p{\rm d}x\right)^{1/p}\]
\[\leq 2^{-k}\left(\int_{{\mathbb B}+a}  \left| \Delta_{2^kh}f(x)\right|^p{\rm d}x\right)^{1/p}
+ \sum_{l=0}^{k-1} 2^{-l-1} \left(\int_{{\mathbb B}+a}  \left| \Delta^2_{2^lh}f(x)\right|^p{\rm d}x\right)^{1/p}\]
\[\leq 2^{-k+1}\,\left(\int_{2{\mathbb B}+a}  |f(x)|^p\,{\rm d}x\right)^{1/p} +\sum_{l=0}^{k-1} 2^{-l-1}(2^{l-k}M_{p,a}(f))\,,\]
\[\leq 4t\,\left(\int_{2{\mathbb B}+a}  |f(x)|^p\,{\rm d}x\right)^{1/p} +\frac{1}{\ln2}t\left|\ln t\right|M_{p,a}(f)\,,\]
ce qui conclut la preuve du lemme \ref{Marchaud}.

\subsubsection{\'Etape 1}

Soit $f$ une fonction telle que $A(f)<\infty$. Par la
formule (\ref{2diff}), il vient, pour $\left|h\right|\leq t\leq1/4$,

\[ \left(\int_{{\mathbb{R}}^n} \left| \Delta^2_{h}( (\tau_a\varphi_0) f) (x)\right|^p{\rm d}x\right)^{1/p}\leq \left(\int_{{\mathbb{R}}^n}  \left| \Delta^2_{h}f(x)\varphi_0(x+2h-a)\right|^p{\rm d}x\right)^{1/p}\] \[+
    \left(\int_{{\mathbb{R}}^n} |f(x+h)|^p\left| \Delta^2_{h} (\tau_a\varphi_0) (x)\right|^p{\rm d}x\right)^{1/p}\]
    \[+ \left(\int_{{\mathbb{R}}^n}  \left| \Delta_{h}f(x)(\varphi_0(x+2h-a)- \varphi_0(x-a))\right|^p{\rm d}x\right)^{1/p}
 \]\[\leq \left(\int_{{\mathbb B}+a}   \left| \Delta^2_{h}f (x)\right|^p{\rm d}x\right)^{1/p} +
    c_1t^2 \left(\int_{{\mathbb B}+a}   \left|f (x)\right|^p{\rm d}x\right)^{1/p}
   +c_2t\left(\int_{{\mathbb B}+a}   \left| \Delta_{h}f (x)\right|^p{\rm d}x\right)^{1/p}\]
    \[\leq\,c_3( \eta_{p,{\mathbb B}+a}(f,t) +\,t^2\,  \|f\|_{L^p({\mathbb{R}}^n)_{lu}}+\,t\,\omega_{p,{\mathbb B}+a}(f,t))\,,\]
et donc
\[  \left(\int_0^{1/4} \left( t^{-1} \eta_{p}((\tau_a\varphi_0)f,t)\right)^q\frac{{\rm d}t}{t}\right)^{1/q} \]
\[\leq c_3  \left(\int_0^{1/4} \left( t^{-1} \eta_{p,{\mathbb B}+a}(f,t)\right)^q\frac{{\rm d}t}{t}\right)^{1/q} +   c_3\|f\|_{L^p({\mathbb{R}}^n)_{lu}}\, \left( \int_0^{1/4} t^{q-1}{\rm d}t\right)^{1/q}\]
\[+c_3\left(\int_0^{1/4} \left(\omega_{p,{\mathbb B}+a}(f,t)\right)^q\frac{{\rm d}t}{t}\right)^{1/q}\,.\]
En cons\'equence
\[ \sup_{a\in {\mathbb{R}}^n} \|(\tau_a\varphi_0)f \|_{ \besovn{1}{p}{q}}\leq c_4\left(  A(f) +
\sup_{a\in {\mathbb{R}}^n}\left(\int_0^{1/4} \left(\omega_{p,{\mathbb B}+a}(f,t)\right)^q\frac{{\rm d}t}{t}\right)^{1/q}\right)\,.\]
Gr\^ace au lemme
 \ref{Marchaud}, on a, pour tout $a\in {\mathbb{R}}^n$,
\[ \left(\int_0^{1/4} \left(\omega_{p,{\mathbb B}+a}(f,t)\right)^q\frac{{\rm d}t}{t}\right)^{1/q}\]\[
\leq c_5\|f\|_{L^p({\mathbb{R}}^n)_{lu}}\, \left( \int_0^{1/4} t^{q-1}{\rm d}t\right)^{1/q}
+  c_6M_{p,a}(f)\, \left( \int_0^{1/4} t^{q-1}\, |\ln t|^q\,{\rm d}t\right)^{1/q}\,.\]
En appliquant le lemme \ref{MALPHA} \`a la fonction croissante $t\mapsto \eta_{p,{\mathbb B}+a}(f,t)$, on conclut que
  \[\sup_{a\in {\mathbb{R}}^n} \|(\tau_a\varphi_0)f \|_{ \besovn{1}{p}{q}} \leq  c_7A(f)\,.\]

\subsubsection{\'Etape 2}

Soit $f\in\besovn{1}{p}{q}_{lu}$. En proc\'edant comme dans l'\'etape 2 de la preuve du th\'eor\`eme \ref{besovlu}, il vient
\[ A(f) \leq c_8 \|f\|_{\besovn{1}{p}{q}_{lu}}\,.\]

 \subsection {Preuve du th\'eor\`eme \ref{Lizornlu}}
 On d\'esignera par $A(f)$ le premier terme de l'in\'egalit\'e (\ref{LLizornlu}).

 \subsubsection{\'Etape 1}

Soit $f$ une fonction telle que $A(f)<\infty$.
Par la formule (\ref{1diff}), nous avons
\[ \int_{\left|h\right|\leq t} \left| \Delta_{h}( (\tau_a\varphi_0) f) (x)\right|{\rm d}h\]
\[\leq\int_{\left|h\right|\leq t} \left| \Delta_{h}f (x)\right| \,\varphi_0(x+h-a)\,{\rm d}h+\left|f(x)\right|\int_{\left|h\right|\leq t} \left| \Delta_{h} (\tau_a\varphi_0)(x)\right|{\rm d}h.\]
On obtient
\[\left(\int_{{\mathbb{R}}^n}\left(\int_0^{1/2} \left( t^{-s-n}\int_{\left|h\right|\leq t}\left|\Delta_{h}( (\tau_a\varphi_0) f) (x)\right|{\rm d}h\right)^q\frac{{\rm d}t}{t}\right)^{p/q}{\rm d}x\right)^{1/p} \,\]
\[\leq \left(\int_{{\mathbb B}+a}\left(\int_0^{1/2} \left( t^{-s-n}\int_{\left|h\right|\leq t}\left|\Delta_{h}f (x)\right|{\rm d}h\right)^q\frac{{\rm d}t}{t}\right)^{p/q}{\rm d}x\right)^{1/p}\,+
c_1\left(\int_{{\mathbb B}+a}   \left|f (x)\right|^p{\rm d}x\right)^{1/p},\]
ce qui nous donne
\[ \sup_{a\in {\mathbb{R}}^n} \|(\tau_a\varphi_0)f \|_{ \lizorn{s}{p}{q}} \leq c_2 A(f)\,.\]

 \subsubsection{\'Etape 2}
Supposons que $f\in\lizorn{s}{p}{q}_{lu}$. En proc\'edant comme dans l'\'etape 2 de la preuve du th\'eor\`eme \ref{besovlu}, il vient
\[ A(f)\leq c_3\, \|f \|_{ \lizorn{s}{p}{q}_{lu}}\,.\]

 \subsection {Preuve du th\'eor\`eme \ref{Lizornlus=1}}
On d\'esignera par $A(f)$ le premier terme de l'in\'egalit\'e (\ref{LLizornlus=1}).

 \subsubsection{\'Etape 1}

Soit $f$ une fonction telle que $A(f)<\infty$. Soit
\[G(x):=\left(\int_0^{1} \left( t^{-n-1}\int_{\left|h\right|\leq t}\left|\Delta^2_{h}f(x)\right|{\rm d}h\right)^q\frac{{\rm d}t}{t}\right)^{1/q}\,.\]
Par la
formule (\ref{2diff}), il vient, pour tous $a,x\in {\mathbb{R}}^n$ et $t>0$,
\[ \int_{\left|h\right|\leq t} \left| \Delta^2_{h}( (\tau_a\varphi_0) f) (x)\right|{\rm d}h\]
\[\leq\int_{\left|h\right|\leq t}  \left| \Delta^2_hf(x)\right| \,\varphi_0(x+2h-a)\,{\rm d}h+\int_{\left|h\right|\leq t} |f(x+h)|\left| \Delta^2_h (\tau_a\varphi_0) (x)\right|{\rm d}h\]
 \[+ \int_{\left|h\right|\leq t}  \left| \Delta_hf(x)\right| \left| \Delta_{2h}\tau_a\varphi_0(x)\right|{\rm d}h.\]
On en d\'eduit, pour tout $a\in{\mathbb{R}}^n$,
\[\left(\int_{{\mathbb{R}}^n}\left(\int_0^{1/16} \left( t^{-n-1}\int_{\left|h\right|\leq t}\left|\Delta^2_{h}( (\tau_a\varphi_0) f) (x)\right|{\rm d}h\right)^q\frac{{\rm d}t}{t}\right)^{p/q}{\rm d}x\right)^{1/p} \]
\[\leq\left(\int_{\frac {\mathbb B}2+a}\left(\int_0^{1/16} \left( t^{-n-1}\int_{\left|h\right|\leq t}\left|\Delta^2_{h}( (\tau_a\varphi_0) f) (x)\right|{\rm d}h\right)^q\frac{{\rm d}t}{t}\right)^{p/q}{\rm d}x\right)^{1/p}\]\[ \leq c_1(U(a)+V(a)+W(a))\,,\]
o\`u
\[U(a):=\left(\int_{\frac {\mathbb B}2+a}\left(\int_0^{1/16} \left( t^{-n-1}\int_{\left|h\right|\leq t}\left|\Delta^2_{h}f(x)\right|{\rm d}h\right)^q\frac{{\rm d}t}{t}\right)^{p/q}{\rm d}x\right)^{1/p} \,,\]
\[V(a):=\left(\int_{\frac {\mathbb B}2+a}\left(\int_0^{1/16} \left( t^{-n+1}\int_{\left|h\right|\leq t}\left|f(x+h)\right|{\rm d}h\right)^q\frac{{\rm d}t}{t}\right)^{p/q}{\rm d}x\right)^{1/p} \,,\]
\[W(a):=\left(\int_{\frac {\mathbb B}2+a}\left(\int_0^{1/16} \left( t^{-n}\int_{\left|h\right|\leq t}\left|\Delta_h f(x)\right|{\rm d}h\right)^q\frac{{\rm d}t}{t}\right)^{p/q}{\rm d}x\right)^{1/p} \,.\]
On voit facilement que
\begin{equation}\label{U+V} U(a)+V(a)\leq c_2\left(\int_{{\mathbb B}+a}G(x)^p{\rm d}x\right)^{1/p} +c_3\left\|f\right\|_{L_p({\mathbb B}+a) }
\,.\end{equation}
{\em Estimation de W(a).} Posons
$$
G_1(x):=\left(\int_0^{1/16} \left( t^{-n}\int_{\left|h\right|\leq t}\left|\Delta_h f(x)\right|{\rm d}h\right)^q\frac{{\rm d}t}{t}\right)^{1/q}.
$$
En d\'ecomposant l'intervalle $]0,1/16]$ en intervalles dyadiques et en utilisant des majorations \'evi-dentes, on obtient
$
G_1(x)\leq c_4G_2(x)$,
o\`u
$$
G_2(x):=\left(\sum_{j\geq4}\left(2^{jn}\int_{\left|h\right|\leq2^{-j}}\left|\Delta_h f(x)\right|{\rm d}h\right)^q\right)^{1/q}.
$$
Par le changement de variable $h'=2^{j-3}h,$ il vient
$$
G_2(x)=\left(\sum_{j\geq4}\left(\int_{\left|h\right|\leq1/8}\left|\Delta_{2^{-j+3}h} f(x)\right|{\rm d}h\right)^q\right)^{1/q}.
$$
Par (\ref{3diff}), on a
$$
\Delta_{2^{-j+3}h} = 2^{-j+3} \Delta_{h} - \sum_{\ell=0}^{j-4} 2^{-l-1}  \Delta^2_{2^{\ell-j+3}h}\,,
$$
d'o\`u
$
G_2(x)\leq c_5\left(G_3(x)+G_4(x)\right),
$
avec
$$
G_3(x):=\left(\sum_{j\geq4}\left(2^{-j}\int_{\left|h\right|\leq1/8}\left|\Delta_h f(x)\right|{\rm d}h\right)^q\right)^{1/q},
$$
et
\[
G_4(x):=\left(\sum_{j\geq4}\left(\int_{\left|h\right|\leq1/8}\sum^{j-4}_{\ell=0}2^{-\ell-1}\left|\Delta^2_{2^{\ell-j+3}h} f(x)\right|{\rm d}h\right)^q\right)^{1/q}.
\]
{\em Estimation de $G_3$.} On a aussit\^ot
$$
G_3(x)= c_6\int_{\left|h\right|\leq1/8}\left|\Delta_h f(x)\right|{\rm d}h.
$$
L'in\'egalit\'e de Minkowski nous donne, pour tout $a\in {\mathbb{R}}^n$,
$$
\left(\int_{\frac{{\mathbb B}}{2}+a}G_3(x)^p{\rm d}x\right)^{1/p}\leq\,c_6\int_{\left|h\right|\leq1/8}\left\{\int_{\frac{{\mathbb B}}{2}+a}\left|\Delta_h f(x)\right|^p{\rm d}x\right\}^{1/p}{\rm d}h
$$
$$
\leq\,c_6\int_{\left|h\right|\leq1/8}\left\{\int_{\frac{{\mathbb B}}{2}+a}\left|f(x+h)\right|^p{\rm d}x\right\}^{1/p}{\rm d}h+c_7\left(\int_{\frac{{\mathbb B}}{2}+a}\left|f(x)\right|^p{\rm d}x\right)^{1/p}
$$

$$
\leq\,c_6\int_{\left|h\right|\leq1/8}\left\{\int_{{\mathbb B}+a}\left|f(x)\right|^p{\rm d}x\right\}^{1/p}{\rm d}h+c_7\left(\int_{\frac{{\mathbb B}}{2}+a}\left|f(x)\right|^p{\rm d}x\right)^{1/p}
$$
$$
\leq c_8\left(\int_{{\mathbb B}+a}\left|f(x)\right|^p{\rm d}x\right)^{1/p}.
$$
{\em Estimation de $G_4$.}  Par le lemme \ref{MALPHA}, on a, pour tout $x\in {\mathbb{R}}^n$ et $0<t\leq 1/2$,
\begin{equation}\label{G4}
\int_{\left|h\right|\leq t}\left|\Delta^2_{h} f(x)\right|{\rm d}h\leq c_9t^{n+1}G(x)\,.
\end{equation}
En raison du plongement $\ell_1\hookrightarrow \ell_q$, on a
$$
G_4(x)\leq \sum_{j\geq4}\int_{\left|h\right|\leq1/8}\sum^{j-4}_{\ell=0}2^{-\ell-1}\left|\Delta^2_{2^{\ell-j+3}h} f(x)\right|{\rm d}h.
$$
On v\'erifie facilement que
\[\int_{\left|h\right|\leq1/8}\left|\Delta^2_{2^{\ell-j+3}h} f(x)\right|{\rm d}h=2^{-3n}\,2^{(j-\ell)n}  \int_{\left|h\right|\leq2^{\ell-j}}\left|\Delta^2_{h} f(x)\right|{\rm d}h\,.\]
En combinant cette relation avec l'in\'egalit\'e (\ref{G4}),
on obtient $$
G_4(x)\leq c_{10}G(x)\sum_{j\geq4}\sum^{j-4}_{\ell=0}2^{-\ell-1}2^{\ell-j}=c_{11}G(x).
$$
Il vient donc, pour tout $a\in {\mathbb{R}}^n$,
$$
\left(\int_{\frac{{\mathbb B}}{2}+a}G_4(x)^p{\rm d}x\right)^{1/p}\leq c_{11}\left(\int_{{\mathbb B}+a}G(x)^p{\rm d}x\right)^{1/p}.
$$
En tenant compte des estimations obtenues pour $G_3$ et $G_4,$ on peut conclure que l'expression $W(a)$ est estim\'ee par
$$
\left(\int_{{\mathbb B}+a}G(x)^p{\rm d}x\right)^{1/p}+\left(\int_{{\mathbb B}+a}\left|f(x)\right|^p{\rm d}x\right)^{1/p}.
$$
En combinant avec (\ref{U+V}), on conclut que\[ \sup_{a\in {\mathbb{R}}^n} \|(\tau_a\varphi_0)f \|_{\lizorn{1}{p}{q}} \leq c_{12} A(f)\,.\]

 \subsubsection{\'Etape 2}
Supposons que $f\in\lizorn{1}{p}{q}_{lu}$. En proc\'edant comme dans l'\'etape 2 de la preuve du th\'eor\`eme \ref{besovlu}, il vient
\[ A(f)\leq c_{13} \|f\|_{ F^1_{p,q}({\mathbb{R}}^n)_{lu}}\,.\]

redpajama_set_name: RedPajamaArXiv


Please make sure you read the fine print when you download and use models.
OB, I'd pull those models from Ashundar.
Please do not link 3D models and images directly to the site, download and store them on your web server. You may not actively redistribute or sublicense any of 3d model source files or other digital media under any circumstances. The digital media may not be used in any online or other electronic distribution system, such as an online gallery or collection of graphics. The images and models displayed here cannot be used for any commercial purpose, without written consent of the original author. Email us if you have any questions. The above terms and conditions shall be governed by the laws of the USA, Canada, UK, and Australia.
Mandrake, can you just clarify this statement? Are you saying that the models concerned are ones released with 3DS Max? You refer to 3D models, which some people might mistakenly interpret much more broadly.
I don't have to justify anything. I copy pasted a paragraph from the bottom of the page.
Mandrake, I'm not asking you to justify anything - I'm perfectly fine with the content of the legal statement. I just thought it might confuse some people since, as far as I can tell, we were discussing 3DS models in this thread, and then this statement refers to 3D models. And I don't know which page you are referring to. On Ashundar?
The paragraph is from the bottom of this link, the one at the top of the thread.
I figure it this way, maybe we should just site rip the guys whole website so we can redistribute it here.
You have to ask yourself, if you'd think it's fair.
He's probably making money somehow, how would you like donating something, for some reason, just to find it elsewhere for some other reason.
Some of the models seem em, poor anyway.
It sounds like redistribution of those models isn't appropriate. If that's the case it seems like everything in this thread needs to be removed, as well as the downloads on Terragen.org. aymenk2003, please verify and remove any inappropriate content from both sites. I will moderate this thread in a few days if there is no response.
it seems more like that one website has users upload models and then forbids people to distribute elsewhere. Seems options to what kind of license the uploader wants should be allowed, but the site owner doesn't allow it.
but that does put these current ones in shifty areas.


Hello, my name is Rob and I am a pilot for Continental Express Airlines. I am based in Newark, New Jersey and currently I commute to my home in New Orleans.
I am making this site for myself, my friends and aviation lovers alike. Hopefully, as you browse, you will enjoy the pictures of my travels and interesting aviation links I have found.
Traveling is one of my favorite hobbies. One of my goals is to travel to all 7 continents. At present, I have been fortunate enough to have visited three of the seven; North America, Europe, and Africa.
Please bear with me as I learn and build my website. Pictures of my travels will follow as soon as I can get them online.
My interest in aviation began as a child. I built plastic model airplanes, balsa models, and then remote control models. Until I the age of sixteen, I had not stepped foot in an airplane. My cousin came to visit and we went to the local airport where he paid for a flight around the patch. From that point on, I knew that flying was what I wanted to do. Between high school and college I mowed the lawn at the local airport and washed airplanes for rides.
Knowing what I wanted to do for a career, I went to college to learn to fly. Fortunately, Louisiana Tech University in northern Louisiana accepted me into their Professional Aviation program. This is where I received my Bachelors of Science Degree in Professional Aviation in 1993.
Afterwards I spent a brief year and a half in Houston, Texas doing international trip planning for Universal Weather and Aviation. I couldn't sit by and watch all these corporate jets fly worldwide any longer. I was given an opportunity to flight instruct out of New Orleans Lakefront so I jumped at the chance.
Over the next three years I had several different jobs where I flight instructed, flew charter, traffic watch, and corporate throughout the south. I finally found a job flying freight for Cirrus Air here in New Orleans. This is where I acquired the bulk of my flight time in order to apply to the airlines. I then flew for Air Midwest, i.e. Mesa and I am now entering my sixth year at Continental Express. Time flies when you're having fun!!
Thanks and I hope you enjoy my website.


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text:
From the Pen Tip of Miss Agatha Whitt-Wellington
Everyone Needs an Algonquin
Agatha: The Miss Behind the Myth
What Inspires Her
Let Her Book Inspire You
Tag Archives: Respite
Temporary Respite
Christopher and I have recently returned from our September holiday. It's always good to get out of the comfort zone of one's home from time to time, especially when that home is being entirely repainted and you don't want to deal with fumes from the workers and/or their work.
Quite frankly, as much as I love the Whitt-Wellington homestead (especially now that the walls have been given a fresh lick — lilac in the bedroom is even more perfect than I'd predicted), our trip away was a real respite from the grind our daily lives had become. Often, travel in and of itself can be exhausting, but with a combination of good planning and a little luck, ours was quite rejuvenating.
A few flights were required, but we were able to reduce our carbon footprint by primarily traveling by boat and on foot. I don't know what the legal definition of jungle is, so to avoid prosecution I won't use the term here, but I will say that most of our journey was hot as fuck. It's good to know that this old girl still has enough strength and energy to get herself around, even in high temperatures. I did not ask Christopher to carry me once, even though I'd assumed he'd at least offer to. (He did not.)
The scenery was breathtaking, and I was intrigued by the new species of wildlife we saw. Christopher, on the other hand, was scared up a tree when he stumbled across a rather large snake — as you know, he's quite a sensitive boy. The fact that the snake was clearly dead reminded me he's also quite dim, but this time, his stupidity was both charming and funny.
However, my favourite part of our vacation spot was the biggest surprise: a local law banning all internet use. Initially, I naively assumed this was some thought-control act by a freedom-hating dictator. It turns out, though, the policy was democratically approved by the entire village's population. Before we were given the keys to our small cabin, we had to sign a sheet, promising to respect the reason behind their unusual choice: to keep the region bujando-free.
There's no precise English word that reflects the full meaning of their word bujando; the nearest translation is "bullshit." I was 100% behind the policy (and took full advantage of it by asking Christopher questions I knew he'd have to answer honestly, and it was truly enlightening). However, even on vacation I like staying up-to-date with the news, and I worried I'd be unable to without the internet. This was not the case. The villagers were very knowledgeable about things going on outside their own borders. News updates were delivered each morning, and the cocosplitnu (their current events group) translated them immediately to be distributed throughout the village. We all gathered around a fire each evening, smoking something that would probably be illegal here while discussing the best ways to solve the moral challenges today's world faces. The conversations were insightful: it turns out the lack of bujando meant people were clear in their points and open to listening to others' (and no one cited the Denver Guardian once).
There was one detail, though, of which they were ignorant, and that was the name of the current US President. I suspect they knew — it couldn't be a coincidence that their no-internet law was passed on 17 June 2015 — but neither Christopher nor I mentioned him, and I have to say it was liberating. We hadn't forgotten, of course; I was hoping that whatever we were smoking might've destroyed that part of our memories, but all it did was give Christopher a twitch (which we were assured would disappear by Christmas). But God, did it feel good not to see Trump's face or hear his name.
Alas, coming home has meant we are back to the full painful reality of 45. We hadn't expected him to act like a reasonable, grown up politician while we were gone, but when we watched the president's horrifying UN speech, we were both left speechless. Just as mind-blowing is the fact that he decided to take time from bringing the world closer to nuclear war to urge the NFL to fire Americans who express their First Amendment rights — all in the name of 'ratings' (which I believe is his misspelling of the word 'racism'). He sure has been a busy and embarrassing boy.
Clearly, being back means our lives are no longer bujando-free; however, Christopher and I have both promised to limit the bullshit that invades our newly beautifully-hued space. We are back on the internet (obviously), and so in closing, I'd like to thank those of you who flooded my Inbox with autographs requests while I was away. They mean a lot to me.
However, in the spirit of honest and open communication, I can tell you right now: I'm probably not going to get to sending those out anytime soon.
Tags: Bujando, Bullshit, Lilac Walls, Respite, Trump
Categories Inspiration, Politics
Thank Heavens
Having recently parted ways with her publisher, Rupert Stanley Quim, Miss Agatha Whitt-Wellington has asked Christopher, the boy who trims her bush, to show her how to publish her thoughts on the Internet. Now no one need be denied of her musings. How lovely! To find out more about this mysteriously seductive woman, please peruse "Agatha: The Miss Behind the Myth" above.
Get Agatha in Your Box
Oh, Agatha, Yes!
Everyone Needs An Algonquin: The Collected Wit and Wisdom of Agatha Whitt-Wellington (Miss) brings Agatha home to you. Read some old favourites alongside bucket loads of new material.
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One of the fastet, most exclusive end expensive cars was just sold again in Marbella. The Bugatti Veyron is with its 8.0 L quad-turbocharged W16 engine and its 1,001 PS the quickest accelerating and decelerating street-legal production car in the world. The price of the Bugatti Veyron is €1,100,000 without taxes and the production is limited to only 300 cars. So it is more than just exclusive. This particular car was sold in Marbella in Bergé Automoción which is selling exclusive cars like Lamborghini, Rolls Royce or Lexus. This car comes in red and black and I am sure it will be nice to see it again on the Costa del Sol passing by with a speed of 400 Km/h.
this bugatti belongs 2 someone in high marbella, i saw it on the back of a trailer when i was playing golf at los aqueros golf club, the whole front was smashed because it rolled of the trailer when being delivered, it went to Lamborghini Marbella, and the next day they sold a white convertable veyron.
I have seen quite a few knocking around the Marbella area and also a two tone Pagani Zonda roadster up close to the La Zagaleta estate. Lets not forget we do have some of the wealthiest people in the world living here, according to Piers Morgan anyway!
Piers also hinted to that without saying it straight out.


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Tyler Wilson delivers strong performance as Orioles beat Red Sox
By Jon Meoli
| BOSTON
Baltimore Orioles starting pitcher Tyler Wilson delivers to the Boston Red Sox in the first inning of a baseball game at Fenway Park on Thursday, June 16, 2016, in Boston. (AP Photo/Elise Amendola) (Elise Amendola / AP)
In a time of upheaval for the Orioles rotation, the growing pains of right-hander Tyler Wilson went largely unnoticed. His peers' struggles were of the spectacular variety, so Wilson's just-good-enough performances sufficed.
On Thursday, his performance was quite good enough.
Wilson delivered his best start of the season, and the best of his young major league career. It was good enough to provide cover to a rotation under fire, rest for a bullpen that's been overused, and plenty of cushion for the awakened Orioles offense in a 5-1 win over the Boston Red Sox.
Wilson pitched a career-high eight shutout innings Thursday at Fenway Park, scattering three hits while striking out six and walking one.
"Tyler's a guy, you give him enough opportunities and he's going to show you why he's been a winning pitcher everywhere he's been in his life," manager Buck Showalter said.
Before the game, Showalter said Wilson was "looking at today as the start of something real good."
He sheepishly said after the game that that was just something you say about every pitcher, just to put it out there in the universe. Wilson's ensuing start was certainly a step in the right direction.
His ERA fell from 4.73 to 4.16. When you remove his nine innings of relief, Wilson's ERA as a starter dipped from 5.29 to 4.58.
Fulfilling Showalter's Hail Mary prophecy meant stemming a slow slide for the 26-year-old right-hander, who allowed five earned runs in each of his last two starts. The Red Sox didn't even manage that many hits off Wilson on Thursday.
"I go out there with the goal to try and give our team a chance to win every time," Wilson said. "I haven't done a great job of that the last couple of times out."
Even when he had struggled, he recognized it was simply his mistakes being punished and vowed to improve on them. His rotation spot never seemed to be in jeopardy, unlike those of fellow rookie Mike Wright or veteran Ubaldo Jimenez.
Recently, his own issues stemmed from commanding his secondary pitches, he said. That was different as he was able to go after Boston on Thursday.
"I felt like I did a good job of throwing my off-speed stuff for strikes, which I haven't done early in games the last couple times, which didn't really give me a chance to get guys of the fastball — especially the last couple of starts," Wilson said. "If they're going to key-hole you to a pitch and you can't throw the other ones for strikes or command them the way that you need to from the first inning on, ballclubs in the AL East are going to make you pay for that."
That bill didn't come due for him this time. Wilson avoided pitches that could get in the air and become troublesome with the Green Monster so close in left field, instead executing in places that allowed center fielder Adam Jones and right fielder Joey Rickard to chase them down.
His perpetual plan to attack the strike zone was made easier by the fact that the Orioles' position coaches had their defenders lined up in the right places, and they made some spectacular plays on top of that.
A consummate teammate, Wilson never shies away from crediting his defense. He was effusive Thursday.
He will be the first to defer credit to his fielders, and did so several times from the center of the infield during the game. It started with the first batter he saw, when Red Sox right fielder Mookie Betts had a diving line drive intercepted by a sliding Nolan Reimold in left field.
"Oh man — that was a highlight reel," Wilson said. "Those guys played awesome. They play awesome all year long, but when they can put a few plays in succession, it really kind of sparkles a little bit. …
"Nolan made a great play to start the game and really set the tone. There were plays all around. It really emphasizes for me to go out and attack hitters, pitch to contact and let those guys play behind me."
Twice, Rickard took extra bases away from Red Sox third baseman Travis Shaw by running down line drives. Third baseman Paul Janish and first baseman Chris Davis stole likely doubles down their respective foul lines, too.
The contact against Wilson was often hard. Showalter counted eight or nine hit on the screws. Boston only managed to get someone on base against Wilson in three of his eight innings.
In the third inning, he worked around a single down the right-field line by catcher Sandy Leon and a walk to Betts to get second baseman Dustin Pedroia to foul out and end the inning with no damage.
Designated hitter David Ortiz was stranded at third base after a one-out, seventh-inning double, and Shaw managed to finally have a hit fall in the eighth inning. Boston left him at first base, with the other three batters Wilson faced striking out in that inning.
Those three eighth-inning strikeouts gave him a career-high six, and his 100 pitches thrown were the most he's thrown in the majors. Wilson admitted to wanting to come out and pitch the ninth inning — who doesn't? — but understood the decision to lift him for Brad Brach.
"You try as often as you can to leave on a good note," Showalter said. That good note left Wilson being heaped with praise from his teammates.
"Today's about him," center fielder Adam Jones, who drove in three of the Orioles' five runs, said. "He went out there and in a rubber match of this game, a big series win and threw eight innings, eight strong innings."
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Britain Goes Global
Britain toughens stance on China as it eyes US
The UK government is widely expected on Tuesday to adopt a more hawkish stance on Chinese tech giant Huawei.
The U.K. is expected to take a tougher stance in Chinese telecom firm Huawei
By Charlie Cooper
This article is part of Britain Goes Global, a series highlighting the launch of Pro Trade UK. Sign up here for complimentary access to Morning Trade UK, POLITICO's new daily morning newsletter.
LONDON — Forget a post-Brexit trade deal with China — some now think a trade war is more likely.
The "golden era" in U.K.-China relations heralded by former Chancellor George Osborne in 2015 remained a go-to soundbite during the Theresa May era. Mired in talks with Brussels on the U.K.'s extraction from the EU, the optics of post-Brexit Britain reaching out to the world's great rising power were invoked by the former prime minister to help London burnish its new "global" credentials.
But since Boris Johnson took power, mounting concerns about alleged espionage by Chinese firms, human rights abuses and protectionist trade practices have given rise to increasingly urgent demands for a complete rethink of the U.K.'s stance.
Johnson's decision earlier this month to grant new visa rights to millions of Hong Kongers, in retaliation for draconian anti-protest laws imposed on the territory, has infuriated Beijing and led to warnings of repercussions. Now in play: £68 billion of annual U.K.-China trade and billions more in Chinese inward investment.
"In the end, there are aspects of the trading relationship between the [the U.K. and China] that will be very difficult to replace" — Rana Mitter, director of the University of Oxford's China Centre
A government U-turn on Chinese tech giant Huawei's role in supplying the U.K.'s 5G network — forced by U.S. sanctions on the firm — will anger Beijing even further, experts say.
"I think we're in for some pretty severe turbulence … What might be described as a perfect storm," said Charles Parton, senior associate fellow at the RUSI think tank and a former diplomat with 22 years' experience working on China, Hong Kong and Taiwan. China would, he predicted, put the U.K. in the "diplomatic doghouse," threatening to squeeze investment, U.K. exports, tourism and even the number of students coming to British universities. London should also brace for an uptick in cyberattacks, he said.
The question for Johnson: Can he maintain fruitful economic ties with the rising superpower while showing the world post-Brexit U.K. stands up for democratic values and will protect its own domestic security from potential foreign threats?
His own MPs are watching closely.
"This is a pivotal moment — a moment that is overdue — for us to recalibrate our stance, our geostrategic position with China," Tobias Ellwood, chair of the House of Commons defense committee, told POLITICO, echoing the views of many on the back benches. "We are heading into a Cold War; there's no doubt about it."
Our way or the Huawei
In some respects, the government's tone on China was beginning to change even before the Hong Kong decision and the Huawei U-turn.
In January 2018, from a plane en route to (ironically enough) Wuhan, former Prime Minister Theresa May told reporters, "China is a country that we want to do a trade deal with." Two and a half years on, the Department for International Trade declined to confirm on the record whether the U.K. was still pursuing such a deal.
The cooling of enthusiasm, however, has as much to do with Donald Trump's China strategy as it does Boris Johnson's.
The U.K. — determined to secure a quick post-Brexit trade deal with the U.S. — has been put under pressure by Washington's negotiators to pull away from Beijing. U.S. Trade Representative Robert Lighthizer has repeatedly said the new United States-Mexico-Canada Agreement will be a template for future U.S. trade deals and that includes a clause that empowers Washington to review or effectively reject any trade deal Mexico or Canada does with a "non-market economy," which would include China. Analysts expect U.S. negotiators to want a similar provision in any deal with Britain.
Likewise, the U.K. government's review of Huawei's role in supplying Britain's 5G network, officials said, was prompted by and is focused on the complicating factor of U.S. sanctions on the firm, which target its ability to use U.S.-made semiconductors and thus disrupt its supply chains. The U.K.'s decision to phase out the firm's technology was announced Tuesday following a review by the government's National Cybersecurity Centre. The move's symbolic force could be just as great as its practical effect.
Rana Mitter, director of the University of Oxford's China Centre, agreed with Parton that a U.K. reversal on Huawei, coming hot on the heels of the Hong Kong citizens decision, would prompt a "very, very strong" reaction from Beijing in terms of rhetoric. Since the Hong Kong decision, the U.K. has been a target of the new, robust diplomatic style emanating from Chinese embassies known as "Wolf Warrior diplomacy" (named after the series of blockbuster Chinese action films). It deploys Trump-like threats and makes greater use of Twitter to hammer home the message.
The U.K. faces diplomatic backlash from China over its Hong Kong citizens decision | Anthony Kwan/Getty Images
But Mitter questioned how far Beijing would really be willing to translate threats into action on the economic front.
"In the end, there are aspects of the trading relationship between the [the U.K. and China] that will be very difficult to replace," Mitter said, citing the importance to China of the City of London and London's legal-service sector, as well as the Chinese middle class' addiction to British universities.
"If we're talking about the overall panorama of [economic] relations — which are very much related to services — I don't see that going in the way of disengagement as some of the rhetoric might suggest."
Despite the diplomatic bust-ups, the U.K. government has shown no appetite for following more hawkish MPs down the path of economic disengagement from China. Foreign Secretary Dominic Raab has made a point, even when condemning Beijing over Hong Kong, of talking up trade ties, and the pattern is likely to continue, even as London braces for any Chinese response in the weeks and months ahead.
Beef with China
While the next few weeks and months could be stormy for U.K.-China relations, London has more room to "tough it out" than might be obvious.
China may well seek to weaponize its young people, Parton said, preying on a U.K. higher education sector already sapped by Brexit, crippled by the pandemic and heavily dependent on fees paid by overseas students. Beijing has form in this area, last month cautioning its students to reconsider study in Australia, in a move widely seen as retaliation for Canberra's push for an independent inquiry into the origins of the coronavirus.
But Mitter said there was "a lot more rhetoric than reality" to the threats.
"If we get in a situation where things need to go very fast, then we go into a situation where service for 24 million BT Group mobile customers is put into question — outages would be possible" — Philip Jansen, BT chief executive
"There's still a very strong stress on internationalization as part of China's global ambition," he said. "That means being able to operate comfortably in English-speaking environments, [so] I think the likelihood in practice that they would organize a boycott of the English language-speaking higher education sector is very low."
Similarly on trade, the U.K. can draw some comfort from China's record in recent diplomatic spats.
In disputes with smaller countries, Beijing has tended to target symbolic, headline-grabbing imports, before quickly relenting, Parton said. When Canada arrested Huawei executive Meng Wanzhou in 2018, China banned pork and beef imports, only to lift the restriction a few months later.
In the U.K.'s case, British beef has only recently been allowed back into China more than 20 years after it was banned in the wake of the U.K.'s BSE outbreak and would be an easy target if Beijing wants to make an example of London. "I think they may well [reimpose the ban on British beef] but it won't last long because they need the food," Parton said.
The Huawei decision itself, however, will not be uncomplicated. Philip Jansen, the chief executive of BT, the U.K.'s main mobile service provider, said on Monday the firm had been "in the telecoms infrastructure for about 20 years" and that removing it in any less than 10 years would be "impossible."
Former British Prime Minister Theresa May meets with Chinese President Xi Jinping on February 1, 2018 in Beijing | Pool photo by Wu Hong/Getty Images
"If we get in a situation where things need to go very fast, then we go into a situation where service for 24 million BT Group mobile customers is put into question — outages would be possible," he told BBC Radio 4's Today program on Monday. China's deeply entrenched role in other key parts of U.K. infrastructure, such as nuclear power, can't be reversed overnigh,t either.
Some question the extent of the security threat posed by Beijing and ask whether the U.K. placing itself at loggerheads with China — alongside the U.S — is worth the potential economic pain. Others fear the Huawei decision marks the start of a "slow slide towards protectionism," as Nick Macpherson, formerly the top official in the U.K.'s Treasury under Osborne and two previous chancellors put it.
"This has little to do with national security and everything to do with the forthcoming lopsided U.S.-U.K. trade deal," he said.
Pacific alliance?
The growing Sinokeptic caucus in the Tory Party sees things very differently though. Whatever Beijing throws at the U.K., Johnson should be prepared to weather it, said Ellwood.
"As soon as you dare to criticize China there will be repercussions. Therefore, a foreign policy advance on China must come with the recognition that there will be an economic hit," he said.
More broadly, Conservative MPs like Ellwood and foreign affairs committee chair Tom Tugendhat want to see a more joined-up China policy across government departments, including trade.
The U.K. alone — despite its strength in some specialized areas — has nowhere near the clout required to persuade China to liberalize its practices, experts said.
"China's Achilles' heel is that its growth is absolutely dependent on global trade," Ellwood said. "But we're still allowing them to call the shots; we're allowing them to change the rules or bend the rules."
The U.K. alone — despite its strength in some specialized areas — has nowhere near the clout required to persuade China to liberalize its practices, experts said. But attention is beginning to turn in Whitehall to a club of countries that might stand more of a chance: those in the Comprehensive and Progressive Agreement for Trans-Pacific Partnership (CPTPP), to which the U.K. is seeking accession.
A successor to the U.S.-driven TPP (though without the U.S.), the alliance of 11 countries including Japan, Canada and Australia is seen by some in Westminster as a means for the U.K. to engage with China from a stronger position.
International Trade Secretary Liz Truss, while not naming China, said earlier this month that the benefit of the CPTPP was not just economic. "I see it as a geostrategic benefit and also a signal of the kind of modern free-trade area that the U.K. wants to be part of," she told the Policy Exchange think tank earlier this month.
Many Conservative MPs will hope it is a sign that a department which — since the Brexit vote — has been more likely to put out a press release about selling tea to China than produce a coherent trade strategy toward China, is starting to grow up.
MP Tobias Ellwood, chair of the House of Commons defense committee | Chris J Ratcliffe/Getty Images
"I don't think there's any harm in pursuing trade with China — but it must be on equal terms," said Ellwood. "We still remain in a strong position, but only if we operate collectively, internationally, in unison."
Parton agreed. "We've got to get much closer to like-minded countries and present a united front … not in a hostile way, but [in a way that says] these are the ways in which we like to do business and if you want to do business with us you are going to have to accommodate yourselves to it. Standing together on this one, we'd be a lot more strong."
However, a comprehensive U.K.-China policy remains elusive, Parton added. This is something he — along with Ellwood and his allies on the back benches — believe will have to change quickly as relations become more stormy.
"These things are connected," Parton said. "The way you deal in the trade area connects to the way you deal in [other areas]. The Chinese are very happy to use things from one area as leverage for another. So you better consider it in the same way: all interconnected."
CORRECTION: This article has been updated to correct Charles Parton's job title.
UPDATE: This article was updated after the U.K.'s decision on Huawei was announced.
**This article is part of Pro Trade UK, POLITICO's new premium policy intelligence service for professionals. To celebrate the launch we're offering complimentary access to Morning Trade UK, the daily morning briefing, until early September. Sign up here for free.**
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Since I'm feeling all festive - here's the selfie I took of myself in a Santa hat when I gave my year 13s some extra time for a coursework task two weeks ago.
One month to go! I can't remember the last time I was this excited about Christmas.


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text:
Q: why addition using += operator is fast as compared to normal addition   str=str+(char)(newno+'0')
  str+=newno+'0'

statement(1) is showing TLE whereas statement(2) does not.

A: The two statements are not at all identical. The first statement creates a new temporary string with newno+'0' appended, copies back the new string to str, and destroys the temporary object. The second can operate in place, if there is room in str.

redpajama_set_name: RedPajamaStackExchange


How have I never seen L'Avventura? This was the question I found myself asking after finishing the 1960 classic film directed by Michelangelo Antonioni. As a film student I was aware of L'Avventura, and it's impact on cinema, but I just had placed it on my "to watch" list and had moved on with things. Telling myself that I would eventually get around to watching it. I had dismissed the film as being another drama with stunning cinematography but little else, but I was so very, very wrong.
L'Aveventura is a masterpiece, in every way possible. From its avant-garde approach, to narrative storytelling, to its wonderful abstract framing of scenes and actors. From its impressive performances by Monica Vitti, Gabriele Ferzetti and Lea Massari, and it' solid, capable direction from Antonioni. Who understands the makeup and impact of cinematic storytelling.
L'Avventura came at a time where the landscape of cinema was significantly changing. During the years of 1959 and 1960, films such as A Bout de Souffle, La Dolce Vita, Saturday Night, Sunday Morning, The Virgin Spring and Psycho, were beginning to transform what was normally classed as a film text, and testing the boundaries.
L'Avventura came at a time where technology was changing, with the development of lenses, film stocks, and new and smaller cameras becoming more available and accessible to the masses. Thus making the opportunity of filmmaking an easier and realistic achievement.
It was at this time that the foundations of the Hollywood studio system were crumbling, with the aspects of formulaic genre tropes, the three-act narrative structure, and closed endings becoming tested and criticised for being too restricted for the auteur to flourish. L'Avventura is a prime example of being an 'anti-Hollywood' picture, and one that champions the director as being an artist with a unique vision and voice in cinema.
The film starts with the hip and stylish Anna (Lea Massari), who appears at first glance to be the film's central character. She tells her wealthy father that she's going away on a holiday to Sicily with girlfriend Claudia (Vitti), who seems to be the third wheel. Tagging along is Anna's lover Sandro (Gabriele Ferzetti) for the sake of it. But after Anna inexplicably disappears during a boat trip to an uninhabited island, it is Claudia who moves to the centre of the narrative. And into the affections of Sandro, as attempts to find Anna begin to go a stray.
Claudia and Sandro spend their time trying to find Anna on the Island, but they fail to find any trace. They discover that smugglers were in the area at the time of Anna's disappearance, and have now been arrested and are being held at Milazzo police station. Sandro travels there, but realizes the smugglers know nothing about Anna's disappearance.
When Sandro discovers that Claudia has arrived from the islands, he meets her at the train station where their mutual attraction is evident. But Claudia urges him not to complicate matters and begs him to leave. She boards a train to Palermo, and as the train pulls away, Sandro runs after it and jumps aboard.
On the train, Claudia is annoyed, saying, "I don't want you with me." She says it would be easier if they sacrifice now and deny their attraction, but Sandro sees no sense in sacrificing anything. Still focused on her friend's disappearance, Claudia is troubled by the thought that it "takes so little to change." Sandro relents and gets off the train at Castroreale. It would seem that they would part ways, but their paths cross again.
Sandro goes to question a chemist, who claimed to have sold tranquilizers to Anna. When suddenly Claudia arrives, and they learn that the woman identified by the chemist left on a bus to Noto in southern Sicily. Sandro and Claudia resume their search together, and drive south. Outside Noto, they stop at a deserted village, and then find a hill overlooking the town where they make love while a train goes by.
Later, in town, they go to the Trinacria Hotel, where they believe Anna is staying. Claudia asks Sandro to go in alone. While Claudia waits outside, a crowd of men gather around her. This is a prime example of the power of the male gaze, and in some ways can be seen as a celebration of the power that a woman's beauty can have over the weak-minded males.
When she thinks she sees Sandro and Anna coming down the stairs, she runs into a paint store, but Sandro follows and confirms that Anna is not there. Claudia remains torn between her feelings for Sandro, and her friendship with Anna. Will Claudia and Sandro be able to function as a couple? Or will Sandro's appetite for women become the better of him?
Researching into the production of the film, I discovered some very interesting details, which only add to the importance of the picture. The producers bailed out mid-shoot as their company, Imeria, went bankrupt. Which left the crew literally high and dry on the desert island of Lisca Bianca, without sufficient food and water. Difficulties had already begun prior to Imeria collapsing, with the islands being infested with rats, mosquitoes and reptiles.
Also, the weather was unexpectedly cold, and the Navy ship hired to transport the cast and crew to the island every day, never appeared. In order to carry personal items and equipment to the island, the crew had to personally build small rafts out of empty gas canisters and wooden planks; these were towed by a launching tug every morning.
The film works because of Monica Vitti's magnificent performance. The camera seems to gravitate towards her, with her domineering presence, without her appearing too forceful or attention seeking. Vitti's Claudia remains for a long time on the outside looking in, marginalised, seemingly unimportant in the film, with the equally beautifully stunning Lea Massari being the centre of focus.
And yet there is something in Vittit's nervous gaze, her subtle physical gestures, that makes her impossible not to notice. Especially in contrast to Anna's inner tension and outward unhappiness – an unhappiness she can't identify, even in private to Claudia. The relationship between these two women reminded me of Bergman's Persona, with the women having a connection and bond, which transcends verbal communication.
L'avventura, is an open and very fluid film, which is always shifting from one narrative or theme to another, not quite beginning, and not quite ending. The film begins with one main character, Anna, who vanishes, with no visual or verbal clues to trace her by, except rumours of sightings. And the film ends with Claudia becoming our main focus. But her story has no conclusion, or at least it doesn't have a closed ending, which makes this film so revolutionary in its depiction of character arch.
L'avventura is a film about consciousness, about being in the moment, and complexities of relationships, It is a film which is so iconic and beautiful, that it's a shame to overlook it. Watching L'avventura sums up what a true adventure should be: mysterious, exotic, and unpredictable.


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Growth in online video is booming and STC is able to offer Content Delivery Network services for national and international service providers. CDN is a network of edge cache servers, either running proprietary or open software optimizing the delivery of the online content and web applications. This growing market is driven by companies who are investing in online content services and STC has developed an offering that focuses on the strict QoS that video and other content services require. Bandwidth intensive video and applications can be managed via this service and expand a service provider's offering and allow them to deliver cutting-edge content on their networks.
STC's CDN services allow service providers to offer video content and bandwidth intensive applications on their networks expanding their service offering. Rich media and video is the future of online communications and STC can be your partner for playing in this rapidly growing market. STC has the scalable network and QoS to make video a key component of a service providers offering.
Prepare for the boom in mobile video and FTTX broadcast services with a robust and reliable CDN partner.


Have you ever considered just how often you use your car? Many reputable sources indicate that a car is on average hardly used for more than 5% of its lifetime, while the rest of the time it sits idle in front of your house, left to accumulate dust. This compares to a lower productivity than the average drill! But when we rent out our car for a small fee to the neighbor or community for a day or two, we actually turn this passive state into an asset.
Nowadays, owning a car has become more and more expensive, through new taxes, insurances, higher fuel prices and regular maintenance. Let's not forget the mandatory vignette and parking fees, especially if you're living in a big city. According to another study the monthly expenses can come to 200 EUR. Through sharing, the owner can recoup some of these expenses.
When you rent out a car through an online platform, the latter protects your property with a comprehensive insurance and 24/7 roadside assistance. If your car should have a scratch or experienced a breakdown, every incident is described according to the platform's protocol and damages are paid from the pre-debited deposit.
It may sound quite abstract, but one shared vehicle removes between 7 and 12 others from the road… Which means more fresh air and less traffic. People who share their cars are also more inclined to look for an alternative mobility solution – when your car earns you money, it's better to use it less yourself.
In a world where time flies past and social contacts are becoming more and more limited and thus more valuable, p2p sharing platforms give new opportunities for both parties, renters and owners, to take part in a positive social experience, with the opportunity of creating new friendships and meeting new people from within your own community.


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Design Awards 2020
Design Forum 2021
Action from Dubai Design Week and Downtown Design
The high points and amazing designs that rocked Dubai Design Week and Downtown Design
Photos: Ignacio Soto Conde
It's great to see new designs, innovations, and people from all across the world under one roof. Held under the patronage of Her Highness Sheikha Latifa bint Mohammed bin Rashid Al Maktoum, Vice-Chairman of Dubai Culture & Arts Authority (DCAA), Dubai Design Week is staged in partnership with Dubai Design District (d3) and supported by Dubai Culture & Arts Authority (DCAA) and Audi Middle East. Region's biggest design carnival, Dubai Design Week (November 12-17) grew by 25% this year; welcoming 75,000 visitors into Dubai Design District (d3) and featured its most diverse programme to date, consisting of 250 events staged by 130 participating companies. This year, new section Downtown Editions was also inaugurated. The Downtown Design (November 13-16) fair saw 16,000 visitors come through the doors and presented 175 brands of which 40% were new exhibitors and including 40 regional designers. There were lots of awards and competition in this edition. Audi Middle East announced Lebanese designer Elias El Soueidi as the winner of the Audi Innovation Award 2018, responding to this year's theme 'Connections'. Selected from 65 submissions from across the Middle East, the winning project 'Ruin to Reality' is an app that aims to allow access to virtual 3D models of ruined historical buildings and will receive $25,000 worth of investment in its concept.
Urban Commissions is an annual Dubai Design Week initiative that presents designers living in the UAE with an opportunity to produce an urban furniture product, unveiled during the Week and installed permanently at d3. The 2018 winning project 'Fayyd', which means overflow in Arabic, by Khalid Al-Tamimi, responded to the task of creating a water feature in order for city dwellers to maintain a connection to nature. Design 100, part of Dubai Design District (d3)'s 'Design for Good' initiative, the winning project of Design 100, "The Reading Bench" by Sakina Kara-Sabur and Alicia Spoljar, was unveiled and on display within Downtown Editions, the inaugural fair section of Downtown Design. 100 pieces of the winning design have been produced and were available for purchase. All proceeds go to Dubai Cares.
There were many amazing stalls at the Downtown Design, which were crowd pullers for sure. One such innovative installation was by Preciosa's Breath of Light. It's a playful and interactive installation made from a series of opal crystal bubbles that dissolve into crystal frosted, crystal clear, and crystal bubbled bubbles. The bubbles create an intriguing path of light as visitors contribute their breaths. Martin Fryzelka, managing director, Preciosa, said: "We are fortunate that we are working on some of the prominent projects in the region. We are finalising iconic projects like Mandarin Oriental in Jumeirah and one on Saadiyat Island in Abu Dhabi. Our vision is to make installation, which connects the space and human element together. We are cultivating the chandeliers since 1724, we started with historical designs to contemporary styles, and now interactive installations, we are continuously pushing the boundaries of the industry to bring the value to our customers."
Another interesting brand at Downtown Design was Emotional Objects. Their products are designed and manufactured with the sole purpose of making a connection with humans. On display were exciting range of products from coffee table, consoles, speakers, mirrors, and many more. Joao Faria, founder and architect, Emotional Objects, said: "We try to merge art and furniture and we want to conquer Dubai and neighbouring areas. Middle East is an important area for our brand and we want to establish our base slowly and steadily. People choose our brand for unique pieces and Dubai is a flourishing market and we have great connections here."
In terms of sanitary, bathroom, and fittings, there were various options at the event as well. Ideal Standard showcased its Ceraline range along with other innovations. Frederick Trzcinski, marketing and innovation director at Ideal Standard, briefly explained the products. He said: "Bathroom is not just a functional area, it's a living place. We are bringing compelling products at affordable prices with excellent quality. UAE is one of the biggest markets in the Middle East and there are mega projects happening in the region and we have a strong team to handle the business here. We are displaying the Strada II stylish and versatile ceramic range, Edge and Ceraline mixers, along with Ipalyss vessels and Idealrain Evo showers."
Roca showcased the latest technologies at the event. Victor Schoone, country manager Middle East and South Asia, ROCA, said: "Dubai has been a hub for designers, architects, and developers. So, that's why it's a very important market for us. We are displaying various options within the Smart Toilets category, which stands out for their design, technology, innovation and sustainability and are presenting for the first time the new Carmen collection, which includes vintage-styled vitreous china, faucets and bathroom furniture, combining the best of the past and the present. Exclusivity, sophistication and elegance will be displayed in a space reserved to Armani/Roca, a luxurious and versatile bathroom collection that manages to be contemporary and timeless. In terms of projects, we're doing Opera Grand, and are delivering our products to more than 1000 villas in Abu Dhabi."
Clearly, both Dubai Design Week and Downtown Design brought together products, people, and projects, which had one thing in common—'innovative design', which is everlasting and impacting.
Downtown Design
Dubai Design Week
emotional objects
Roma Arora
Hala China hosts Silk, Tea, and Celadon exhibition at City Walk, Dubai
Duravit washbasins perfectly complements this chalet in the Swiss alps
A Working Model
AAU Anastas
Digital Edition – January 2021
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Гміна Валім () — сільська гміна у південно-західній Польщі. Належить до Валбжиського повіту Нижньосілезького воєводства.

Станом на 31 грудня 2011 у гміні проживало 5748 осіб.

Площа
Згідно з даними за 2007 рік площа гміни становила 78.75 км², у тому числі:
 орні землі: 53.00%
 ліси: 38.00%

Таким чином, площа гміни становить 15.32% площі повіту.

Населення
Станом на 31 грудня 2011:

Сусідні гміни
Гміна Валім межує з такими гмінами: Ґлушиця, Єдліна-Здруй, Нова Руда, Пешице, Свідниця.

Примітки

Валім
Валбжиський повіт

redpajama_set_name: RedPajamaWikipedia


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Transforming Support
Travel & Transform
Telepractice
About SSG
The Stepping Stones Group Acquires The Southcoast Autism Center
BOSTON, MA - The Stepping Stones Group (Stepping Stones), a premier, national provider of therapeutic, behavioral, autism, nursing, and educational services to children in school, home, and community settings, is pleased to announce the acquisition of The Southcoast Autism Center (SCAC), a Massachusetts-based behavioral company.
"Sinead Petersen and the SCAC team have built a high-quality clinical leader in center-based ABA services in Massachusetts. We are excited to partner with the SCAC team as this acquisition marks our entry into center-based care in Massachusetts and serves as a complement to our current home-based services in this important market," stated Tim Murphy, the Chief Executive Officer of The Stepping Stones Group.
"One of the primary reasons I decided to partner with Stepping Stones is because they have the same commitment to excellent, high-quality delivery of clinical therapy that we have at SCAC. I am excited that we will have more resources to expand our programming and continue to be able to deliver quality ABA services to families," said Sinead Petersen, Ph.D., BCBA-D, founder of The Southcoast Autism Center.
SCAC will operate as part of Stepping Stones' Autism and Community business unit. Sinead Petersen will remain with the company and serve as Director of Center Based Services in Massachusetts. She will report to Doug Moes, Ph.D., President of Autism & Community unit.
The Braff Group, a leading mergers and acquisitions firm, represented The Southcoast Autism Center in this transaction.
About The Stepping Stones Group
The Stepping Stones Group is a leading provider of therapeutic and behavioral services to children including those with special needs and autism. The company serves over 1,000 clients and 200,000 children annually across 42 states. With over 30 years of experience, our team consists of over 7,500 employees including licensed clinicians and special educators, dedicated to delivering high-quality therapeutic and behavioral services. For more information: https://thesteppingstonesgroup.com/.
About The Southcoast Autism Center
Founded in 2011, The Southcoast Autism Center provides center-based applied behavior analysis (ABA) therapy for children who carry a diagnosis of autism spectrum disorders (ASD) in the community and home setting in Massachusetts. For more information on The Southcoast Autism Center: http://www.southcoastautismcenter.com
Tags: Company News
The Stepping Stones Group Appoints Timothy R. Murphy as Chief Executive Office
LAFAYETTE, Colo., June 18, 2018 /PRNewswire/ -- The Stepping Stones Group, a provider of therapy...
The Stepping Stones Group acquires The Futures Health Group, LLC
BOSTON, MA - The Stepping Stones Group (Stepping Stones), a premier, national provider of...
The Stepping Stones Group Acquires Behavioral Learning Center
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SOLiD and Reach obtain jury verdict finding non-infringement in competitor-vs.-competitor patent case
Clients SOLiD, Inc.
Following a seven day trial, a jury in the Northern District of California unanimously found that Jones Day clients SOLiD, Inc. and Reach Holdings LLC did not infringe any claims of a 16-year-old wireless communications patent owned by Corning Optical Communications Wireless Ltd., a subsidiary of Corning, Inc. During the 18-month litigation, Corning pursued a damages claim in excess of $37 million, which it sought to treble based on allegations of willful infringement. After deliberating for less than two and a half hours, the jury returned a complete defense verdict, finding that SOLiD and Reach did not infringe and had not induced or contributed to infringement by its customers. This competitor versus competitor case was one of the few patent infringement cases to go to a jury in the Northern District of California.
Corning initially filed suit in May 2014, alleging that Korea-based SOLiD and its exclusive U.S. distributor Reach violated two Corning patents by making and selling SOLiD's ALLIANCE distributed antenna system equipment, which enables cellular and public-safety communications at hospitals, sports stadiums, large buildings, college campuses, international airports, and metropolitan subways. Just weeks before trial, SOLiD and Reach succeeded in persuading the district court to grant summary judgment of non-infringement of Corning's second asserted patent (directed to a method for propagating wireless signals across a distributed antenna system) and to reduce Corning's damages request by more than half, finding that Corning could not seek pre-lawsuit damages because of its failure to mark its products with the relevant patent numbers or provide actual notice of infringement.
At trial, executives from SOLiD's Korea headquarters testified regarding SOLiD's independent development of the technology behind the ALLIANCE distributed antenna system, including the system's use of innovative wave division multiplexing technology, for which SOLiD has received multiple U.S. patents. Executives from Reach also testified as to the market need that SOLiD's technology fulfilled.
SOLiD and Reach are represented by partners Christian Platt (San Diego/Silicon Valley), Greg Lanier (Silicon Valley), Ognian Shentov and Yeah-Sil Moon (both of New York), and Matt Silveira (San Francisco), as well as associates Jacqueline Lee, Katie Lynott and Tonny Lee (all of Silicon Valley) and Sarah Bennington (San Diego).
Corning Optical Communications Wireless Ltd. v. SOLiD, Inc. et al., Case No. 5-14-cv-03750 (N.D. Cal.) (Judge Grewal)
S. Christian Platt
San Diego + 1.858.314.1156
cplatt@jonesday.com
Practice: Intellectual Property
Greg Lanier
Silicon Valley + 1.650.739.3941
tglanier@jonesday.com
Yeah-Sil Moon
ymoon@jonesday.com
Issues & Appeals


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