From 84295a275db55cdbf5a40b2021a3029eff78919a Mon Sep 17 00:00:00 2001
From: =?utf8?q?Jean-Fran=C3=A7ois=20Couchot?=
 <couchot@couchot.iut-bm.univ-fcomte.fr>
Date: Wed, 14 Oct 2015 13:50:08 +0200
Subject: [PATCH 01/16] =?utf8?q?ahmad=20->=20d=C3=A9codage?=
MIME-Version: 1.0
Content-Type: text/plain; charset=utf8
Content-Transfer-Encoding: 8bit

---
 ahmad.tex | 89 ++++++++++++++++++++++++++++++++++++++++++++++++-------
 1 file changed, 78 insertions(+), 11 deletions(-)

diff --git a/ahmad.tex b/ahmad.tex
index 6aed620..b94f1b9 100644
--- a/ahmad.tex
+++ b/ahmad.tex
@@ -1,4 +1,4 @@
-En étudiant le watermarking,
+En étudiant les schémas de watermarking,
 nous avons constaté que très peu de travaux ciblaient les documents PDF
 qui représentent cependant une part non anecdotique des données
 échangées en ligne.
@@ -13,7 +13,7 @@ le positionnements des caractères. D'autres éléments de postionnement
 sont intégrés dans~\cite{WT08}.
 Une attaque qui remodifierait  aléatoirement de manière faible ces positions
  détruirait la marque dans les deux cas.
-La quantification (au sens du traitemetn du signal) est une réponse
+La quantification (au sens du traitement du signal) est une réponse
 à ces attaques: des positions modifiées de manière mal intentionnée  
 peuvent grâce cette démarche être rapprochées (abstraites) en des positions
 préétablies et conserver ainsi leur information et donc la marque.
@@ -22,25 +22,24 @@ STDM~\cite{CW01} est une instance de ces schémas de marquage.
 Ce chapitre présente une application de STDM au marquage de documents PDFs.
 \JFC{annonce du plan}
 
-\section{Rappels sur STDM}
-
-\section{Spread Transform Dither Modulation}
+\section{Rappels sur la Spread Transform Dither Modulation}
 \label{sec:STDM}
 Les paramètres de ce schéma sont
 \begin{itemize}
-\item le facteur de quantification $\Delta$ est un réel positif; plus $\Delta$
+\item le facteur de quantification $\Delta$ qui est un réel positif; plus $\Delta$
 est grand, plus la distortion peut être importante;
 \item le niveau d'indécision  $d_0$ qui est un réel dans
 $[-\dfrac{\Delta}{2},\dfrac{\Delta}{2}]$; plus ce nombre a une valeur absolue
 élevée, plus les erreurs peuvent être corrigées;
-On définit $d_1$ par 
+on définit $d_1$ par 
 $$d_1 = \begin{cases} 
   d_0 + \Delta/2, & \textrm{ si }~~d_0<0 \\  
   d_0 - \Delta/2, & \textrm{ sinon } 
 \end{cases}
 $$
-\item un nombre $L$ d'éléments dans lequel chaque bit est embarqué;
-\item un vecteur $p$ de projection de taille $L$; 
+\item un nombre $L$ d'éléments dans lequel chaque bit de la marque 
+  est embarqué;
+\item un vecteur $p$ de projection de taille $L$. 
 
 \end{itemize}
 
@@ -55,7 +54,7 @@ x' = f(x,m) = x+ ((\lfloor(\frac{(x^T p) -d_m}{\Delta})\rfloor\Delta +d_m )~ - x
 Avec les mêmes paramètres $\Delta$, $d_0$ , $L$ et $p$ le message 
 $\hat{m}$ extrait de 
 $x'$ de taille $L$ est défini par:
-\begin{equation}
+\begin{equation}\label{eq:stdm:ext}
 \hat{m} = arg \min_{ m \in \{0, 1\}} \mid x'^T p - f(x,m) \mid
 \end{equation}
 
@@ -63,9 +62,77 @@ Les auteurs de~\cite{CW01} ont montré que la variance de l'erreur
 est égale à $\Delta^2/12L$ 
 lorsque chacun des $L$ éléments de $x$ suit une ditribution uniforme 
 $U(\Delta)$. 
-
+Tous les éléments sont en place pour embarquer une marque 
+dans un fichier PDF selon le schéma STDM.
 
 \section{Application au marquage de documents PDF}
 
+On détaille successivement comment insérer une marque dans un document PDF, 
+puis comment l'extraire.
+
+\subsection{Insertion de la marque}
+
+On cherche à ajouter à un document PDF une marque $m$ de $k$ bits 
+déjà codée (cryptée, correction d'erreurs incluse). 
+L'insertion de celle-ci dans le document s'effectue 
+en quatre étapes.
+
+On considère comme fixés les paramètres  
+$\Delta$,  $d_0$  et la manière de construire le vecteur $p$ pour une taille 
+$L$. 
+
+
+\begin{enumerate}
+\item Le vecteur hôte $x$ de taille $N$ 
+  est constitué de l'abscice (flottante) 
+  de chaque caractère rencontré dans le document PDF. 
+  La dimension $L$ est calculée comme la partie entière de $N/k$.
+
+\item Un générateur pseudo aléatoire (initialisé par une clef) 
+construit $k$ ensembles $M_1$, \ldots, $M_k$ 
+de taille $L$ mutuellement disjoints dans $[1,N]$. Ainsi 
+$\bigcup_{1\le i \le k} M_i \subseteq [N]$. 
+
+
+\item Pour chacun des ensembles $M_i$, $ 1 \le i \le k$, 
+  de l'étape précédente,  le vecteur $\dot{x} = (x_{j_1}, \ldots ,x_{j_L})$,
+  est construit où $\{j_1, \ldots, j_L\} = M_i$.
+  Le vecteur $\dot{x'} = f(\dot{x},m_i)$ est
+  construit selon l'équation~(\ref{eq:stdm}).
+  Dans $x$, chacun des $x_{j_1}, \ldots, x_{j_L}$ est remplacé par 
+  $\dot{x'}_{j_1}, \ldots, \dot{x'}_{j_L}$.
+
+\item L'abscisse de chaque caractère est ainsi redéfini 
+  selon le nouveau vecteur de positions ${x'}$. 
+\end{enumerate}
+
+Voyons comment extraire une marque d'une document PDF.
+
+\subsection{Exctraction de la marque}
+
+On considère comme connue la taille de la marque: c'est $k$ bits.
+Les paramètres $\Delta$,  $d_0$ et la manière de construire 
+$p$ en fonction de $L$ sont les mêmes qu'à l'étape précédente d'insertion de 
+marque.
+
+\begin{enumerate}
+\item on récupère le vecteur $x'$ (de taille $N$ lui aussi) des abscices des
+  caractères du document PDF comme dans la phase d'insertion. 
+  la valeur de $L$ est définie comme précédement.
+
+\item le même générateur pseudo aléatoire (initialisé avec la même clef) 
+construit les $k$ mêmes ensembles $M_1$, \ldots, $M_k$ 
+de taille $L$ mutuellement disjoints dans $[1,N]$. 
+
+\item Pour chacun des ensembles $M_i$, $ 1 \le i \le k$, 
+  de l'étape précédente,  le vecteur $\dot{x'} = (x'_{j_1}, \ldots, x'_{j_L})$,
+  est construit où $\{j_1, \ldots, j_L\} = M_i$.
+  Le bit $\hat{m}_i$  est défini selon l'équation~(\ref{eq:stdm:ext})
+  en remplaçant $x'$ par $\dot{x'}$ .
+\end{enumerate}
 
+\section{Choix des paramètres}
+Le schéma de marquage est paramétré par $\Delta$,  $d_0$ et la manière de construire le vecteur $p$ pour une taille $L$. 
+Les travaux réalisés se sont focalisés sur l'influence du paramètre 
+$\Delta$ dans l'algorithme.
 
-- 
2.39.5


From e760413d8a2e7bc9385c7fd3e94e629b4edee6e7 Mon Sep 17 00:00:00 2001
From: couchot <jf.couchot@gmail.com>
Date: Wed, 21 Oct 2015 22:48:36 +0200
Subject: [PATCH 02/16] fin travail bittar

---
 ahmad.tex | 111 +++++++++++++++++++++++++++++++++++++++++++++++-------
 main.tex  |   4 ++
 2 files changed, 101 insertions(+), 14 deletions(-)

diff --git a/ahmad.tex b/ahmad.tex
index b94f1b9..1b92dbc 100644
--- a/ahmad.tex
+++ b/ahmad.tex
@@ -9,9 +9,9 @@ ajoutent des caractères invisibles dans le document.
 En supprimant ces espaces ou caractères invisibles, la marque s'enlève
 facilement.
 Dans~\cite{PD2008}, les auteurs modifient de manière imperceptible
-le positionnements des caractères. D'autres éléments de postionnement
+le positionnements des caractères. D'autres éléments de positionnement
 sont intégrés dans~\cite{WT08}.
-Une attaque qui remodifierait  aléatoirement de manière faible ces positions
+Une attaque qui modifierait  aléatoirement de manière faible ces positions
  détruirait la marque dans les deux cas.
 La quantification (au sens du traitement du signal) est une réponse
 à ces attaques: des positions modifiées de manière mal intentionnée  
@@ -27,7 +27,7 @@ Ce chapitre présente une application de STDM au marquage de documents PDFs.
 Les paramètres de ce schéma sont
 \begin{itemize}
 \item le facteur de quantification $\Delta$ qui est un réel positif; plus $\Delta$
-est grand, plus la distortion peut être importante;
+est grand, plus la distorsion peut être importante;
 \item le niveau d'indécision  $d_0$ qui est un réel dans
 $[-\dfrac{\Delta}{2},\dfrac{\Delta}{2}]$; plus ce nombre a une valeur absolue
 élevée, plus les erreurs peuvent être corrigées;
@@ -54,13 +54,14 @@ x' = f(x,m) = x+ ((\lfloor(\frac{(x^T p) -d_m}{\Delta})\rfloor\Delta +d_m )~ - x
 Avec les mêmes paramètres $\Delta$, $d_0$ , $L$ et $p$ le message 
 $\hat{m}$ extrait de 
 $x'$ de taille $L$ est défini par:
-\begin{equation}\label{eq:stdm:ext}
+\begin{equation}
 \hat{m} = arg \min_{ m \in \{0, 1\}} \mid x'^T p - f(x,m) \mid
+\label{eq:stdm:ext}
 \end{equation}
 
 Les auteurs de~\cite{CW01} ont montré que la variance de l'erreur 
-est égale à $\Delta^2/12L$ 
-lorsque chacun des $L$ éléments de $x$ suit une ditribution uniforme 
+est égale à $D_s = \Delta^2/12L$ 
+lorsque chacun des $L$ éléments de $x$ suit une distribution uniforme 
 $U(\Delta)$. 
 Tous les éléments sont en place pour embarquer une marque 
 dans un fichier PDF selon le schéma STDM.
@@ -78,13 +79,13 @@ L'insertion de celle-ci dans le document s'effectue
 en quatre étapes.
 
 On considère comme fixés les paramètres  
-$\Delta$,  $d_0$  et la manière de construire le vecteur $p$ pour une taille 
-$L$. 
+$\Delta$,  $d_0$ , $L$ et la manière de construire le vecteur $p$ 
+pour ce $L$ donné. 
 
 
 \begin{enumerate}
 \item Le vecteur hôte $x$ de taille $N$ 
-  est constitué de l'abscice (flottante) 
+  est constitué de l'abscisse (flottante) 
   de chaque caractère rencontré dans le document PDF. 
   La dimension $L$ est calculée comme la partie entière de $N/k$.
 
@@ -108,7 +109,7 @@ $\bigcup_{1\le i \le k} M_i \subseteq [N]$.
 
 Voyons comment extraire une marque d'une document PDF.
 
-\subsection{Exctraction de la marque}
+\subsection{Extraction de la marque}
 
 On considère comme connue la taille de la marque: c'est $k$ bits.
 Les paramètres $\Delta$,  $d_0$ et la manière de construire 
@@ -116,9 +117,9 @@ $p$ en fonction de $L$ sont les mêmes qu'à l'étape précédente d'insertion d
 marque.
 
 \begin{enumerate}
-\item on récupère le vecteur $x'$ (de taille $N$ lui aussi) des abscices des
+\item on récupère le vecteur $x'$ (de taille $N$ lui aussi) des abscisse des
   caractères du document PDF comme dans la phase d'insertion. 
-  la valeur de $L$ est définie comme précédement.
+  la valeur de $L$ est définie comme précédemment.
 
 \item le même générateur pseudo aléatoire (initialisé avec la même clef) 
 construit les $k$ mêmes ensembles $M_1$, \ldots, $M_k$ 
@@ -131,8 +132,90 @@ de taille $L$ mutuellement disjoints dans $[1,N]$.
   en remplaçant $x'$ par $\dot{x'}$ .
 \end{enumerate}
 
-\section{Choix des paramètres}
+\section{Expérimentations }
 Le schéma de marquage est paramétré par $\Delta$,  $d_0$ et la manière de construire le vecteur $p$ pour une taille $L$. 
 Les travaux réalisés se sont focalisés sur l'influence du paramètre 
-$\Delta$ dans l'algorithme.
+$D_S = \frac{\Delta^2}{12L}$ dans l'algorithme en satisfaisant 
+les deux contraintes antagonistes
+de fournir une marque suffisamment robuste
+et suffisamment transparente.
+On cherche deux réels $a$ et $b$   tels que
+$a$ et $b$ correspondent respectivement 
+au seuil maximum pour être transparent et 
+au seuil minimum pour être robuste. 
+Les études de perceptibilité doivent permettre de déterminer $a$ tandis 
+que celles sur la robustesse devront fixer le seuil $b$.
+Finalement, les contraintes précédentes seront  satisfaites si et seulement si  
+$a > b$ et $D_s \in [b,a]$.
+
+Concernant la transparence, 
+les expériences présentées dans l'article~\cite{BDCC16} ont consisté en 
+choisir un texte d'un nombre fixe de caractères $n$
+dans lequel doit être embarqué une marque de taille fixe $k$.
+En faisant varier la valeur de $\Delta$, nous avons remarqué que la 
+valeur $a= 0,01335$ est le seuil au delà duquel il est visuellement 
+possible de remarquer une différence entre le document original 
+et le document marqué.
+
+Il nous reste à détailler les expériences d'étude de robustesse de la démarche.
+Comme dans l'évaluation de la transparence, il s'est agit de faire 
+varier le paramètre  $\Delta$.
+Pour chacune de ces valeurs, le document a été altéré selon 
+un flou gaussien (de paramètre 0,1 et 0,25)  
+et une attaque de type poivre et sel (de paramètre 0,1 et 0,25 aussi).
+Le rapport entre le nombre de bits erronés par rapport au nombre total 
+de bits (nommé BER ci-après) après l'extraction du message est alors calculé. 
+Le facteur de quantification a été choisi entre 0.1 et 10. 
+L'expérience a été répétée 500 fois et les moyennes sont représentées 
+à la figure~\ref{fig:pdf:atq:ber}.
+Sur cette figure, on constate que pour peu que la quantification $\Delta$
+soit supérieure à 1,  le taux d'erreur est inférieur à 12,5\%. Ce taux peut 
+être corrigé par un code correcteur usuel.
+Avec les paramètres de l'expérimentation, cela revient à considérer un seuil 
+$b=0,00214$. 
+Ces expériences ont ainsi pu valider l'existence de seuils de distorsion
+permettant d'avoir une méthode à la fois robuste et transparente.
+
+
+
+\begin{figure}[ht]
+\begin{center}
+\begin{tikzpicture}
+
+        \begin{axis}[%
+            axis x line=bottom,
+            axis y line=left,
+            xlabel=$\Delta$,
+            ylabel=$BER~(\%)$,
+width=0.66\textwidth,
+            legend pos=north east]
+            \addplot[mark=none, dashed, red,thick] coordinates {(0.1, 13.8742) (0.5, 12.8721) (1, 8.4680) (1.1, 7.3940) (1.2, 6.5020) (1.3, 5.7960) (1.4, 4.9580) (1.5, 4.1180) (1.6, 3.8080) (1.7, 3.2580) (1.8, 2.8320) (1.9, 2.5000) (2, 2.2100) (2.1, 2.0420) (2.2, 1.8120) (2.3, 1.6080) (2.4, 1.4040) (2.5, 1.3860) (3, 1.1100) (5, 1) (10, 1)};
+
+ \addplot[mark=none, dotted, green,thick] coordinates {(0.1, 10.3501) (0.5, 7.1) (1, 4.7420) (1.1, 4.0580) (1.2, 3.3620) (1.3, 2.8260) (1.4, 2.3900) (1.5, 2.1220) (1.6, 1.9260) (1.7, 1.6540) (1.8, 1.4460) (1.9, 1.3680) (2, 1.3400) (2.1, 1.2460) (2.2, 1.1420) (2.3, 1.0920) (2.4, 1.0600) (2.5, 1.0460) (3, 1.0100) (5, 1) (10, 1)};
+
+ \addplot[mark=none, dashdotted, blue,thick] coordinates {(0.1, 15.3222) (0.5, 13) (1, 11.1560) (1.1, 10.2920) (1.2, 9.8520) (1.3, 8.7860) (1.4, 8.3960) (1.5, 7.3480) (1.6, 7.0880) (1.7, 6.0940) (1.8, 5.2100) (1.9, 4.8860) (2, 4.5940) (2.1, 4.0140) (2.2, 3.6060) (2.3, 3.3520) (2.4, 2.9300) (2.5, 2.6140) (3, 1.7000) (5, 1.0140) (10, 1)};
+
+ \addplot[mark=none, dash pattern=on 10pt off 2pt on 5pt off 6pt, black,thick] coordinates {(0.1, 13) (0.5, 10.7) (1, 9.3340) (1.1, 8.7580) (1.2, 7.7080) (1.3, 6.7580) (1.4, 5.9260) (1.5, 5.4320) (1.6, 4.7260) (1.7, 4.3020) (1.8, 3.6200) (1.9, 3.1380) (2, 2.9920) (2.1, 2.5780) (2.2, 2.4340) (2.3, 2.1240) (2.4, 1.8760) (2.5, 1.7386) (3, 1.2880) (5, 1) (10, 1)};
+
+            \legend{$Gaussian (0.1)$,$Salt\&pepper (0.1)$,$Gaussian (0.25)$,$Salt\&pepper (0.25)$};
+        \end{axis}
+    \end{tikzpicture}
+\\
+\end{center}
+\caption{Représentation du BER pour des attaques de type flou gaussien et
+poivre et sel}\label{fig:pdf:atq:ber}
+\end{figure}
+
+
+\section{Conclusion}\label{pdf:s:conclusion}
+Ce travail a présenté une démarche outillée
+basée sur la Spread Transform Dither Modulation 
+permettant d'embarquer une marque dans un document PDF.
+Les éléments modifiés sont les abscisses des caractères présents 
+dans le document.
+
+Deux des propriétés essentielles des algorithmes de marquage ont été étudiées:
+la transparence et la robustesse. La notion d'intervalle de distorsion 
+acceptable a été définie et calculée sur un exemple jouet. 
+
 
diff --git a/main.tex b/main.tex
index e81e5e3..cacbb94 100644
--- a/main.tex
+++ b/main.tex
@@ -13,6 +13,10 @@
 \usepackage{dsfont}
 \usepackage{graphicx}
 \usepackage{listings}
+\usepackage{tikz}
+\usepackage{pgfplots}
+\usepgfplotslibrary{groupplots}
+
 %\usepackage[font=footnotesize]{subfig}
 \usepackage[utf8]{inputenc}
 \usepackage{thmtools, thm-restate}
-- 
2.39.5


From 73637978d1eddcf2e4938ff290939b708d05e185 Mon Sep 17 00:00:00 2001
From: couchot <couchot@localhost.localdomain>
Date: Mon, 22 Aug 2016 17:52:38 +0200
Subject: [PATCH 03/16] ajout d'image et reprise

---
 demandeInscription/Training_wow_01.pdf | Bin 0 -> 6020 bytes
 demandeInscription/Training_wow_04.pdf | Bin 0 -> 5254 bytes
 demandeInscription/synthese.tex        | 264 +++++++++++++++++--------
 3 files changed, 186 insertions(+), 78 deletions(-)
 create mode 100644 demandeInscription/Training_wow_01.pdf
 create mode 100644 demandeInscription/Training_wow_04.pdf

diff --git a/demandeInscription/Training_wow_01.pdf b/demandeInscription/Training_wow_01.pdf
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literal 0
HcmV?d00001

diff --git a/demandeInscription/synthese.tex b/demandeInscription/synthese.tex
index 791646a..fce2b58 100755
--- a/demandeInscription/synthese.tex
+++ b/demandeInscription/synthese.tex
@@ -271,8 +271,9 @@ est une suite directe des travaux de thèse.
 Le second concerne les systèmes dynamiques discrets 
 et leurs applications (Sec.~\ref{sub:sdd}), thématique 
 pour laquelle j'ai été recruté dans l'équipe AND du département
-DISC. Enfin la section~\ref{sub:gen} présente comment je me suis investi dans 
-le  domaine de la bio-informatique à l'aide de compétences connexes.
+DISC. Enfin la section~\ref{sec:appl}
+présente comment j'ai appliqué ces compétences théoriques et pratiques
+en bio-informatique  et  les réseaux de capteurs.
 Ces travaux ont été valorisés par des publications dont les références sont données à la section~\ref{sec:publi} et synthétisées à la figure~\ref{fig:bilan}
 en fin de document.
 
@@ -398,14 +399,9 @@ qui permettent d'obtenir des graphes d'itérations fortement connexes.
 Ce théorème a aussi été prouvé dans~\cite{bcgr11:ip}
 et des instanciations effectives 
 ont été produites. 
-Une troisième méthode~\cite{chgw+14:onp} s'appuie sur les codes 
+Une troisième méthode~\cite{chgw+14:oip,chgw+14:onp} s'appuie sur les codes 
 de Gray, ou de manière équivalente sur les cycles hamiltoniens du graphe des
 itérations.
-% : un cycle qui visite chaque n{\oe}ud exactement une fois est un  
-% \emph{cycle hamiltonien}.
-%La démarche consiste à enlever du graphe un de ses cycles hamiltoniens dont 
-%la démarche de génération est un problème connu. 
-
 Ces méthodes ont permis d'étendre à l'infini la classe des fonctions 
 dont les itérations sont chaotiques.
 
@@ -495,19 +491,22 @@ des batteries de tests telles que Die-Hard, NIST, TestU01.
 Plus récemment, nous avons entrepris de trouver des bornes du temps d'arrêt
 d'obtention d'une distribution uniforme d'un générateur 
 construit en enlevant un chemin hamiltonien équilibré dans un $n$-cube. Le travail 
-en collaboration avec Pr. P.-C. Heam du DISC
-est en cours de soumission~\cite{cgh16:onp}
+en collaboration avec Pr. P.-C. Heam, Pr. C. Guyeux du DISC et Pr. S. Contassot
+du LORIA est en cours de soumission~\cite{cgh16:onp}
 dans un  journal international.
 
-Enfin j'ai été sollicité pour encadrer une thèse sur l'implantation 
+Enfin je co-encadre une thèse sur l'implantation 
 de générateurs de nombre pseudo-aléatoires à bases d'itérations 
 chaotiques sur des circuits logiques 
 programmables. J'ai commencé ce travail en encadrant une étude exhaustive 
 de toutes les instances d'implantations de cette classe.
 Ce travail complet théorique et pratique est terminé aujourd'hui et 
 est en cours de soumission~\cite{bgco16:onp} dans un journal international.
-
-
+18 PRNGs de 4 familles (xorshift, LFSR, TGFSR, and LCG) ont été 
+implantés sur  FPGA et ont pu être ainsi comparés pratiquement sur 
+la même plate-forme. Une boite de post-processing implantant les itérations
+chaotiques a été déployée et a permis d'améliorer les résultats statistiques.
+Ceci a été valorisé par la publication~\cite{BCG16:ip}.
 
 
 
@@ -547,13 +546,15 @@ d'intégrer ceci à du marquage de document PDF puis de
 présenter ce problème comme un problème d'optimisation. 
 Grâce à une telle présentation nous avons pu trouver les paramètre optimaux
 des méthodes QIM assurant à la fois robustesse et indétectabilité.
-Le travail est en cours de soumission dans un  journal international~\cite{bdcc15:onp}.
+Ce travail a été accepté dans un  journal international~\cite{bdcc15:onp}.
 
 Lorsque l'objectif visé est l'indétectabilité, on parle de \emph{stéganographie}.
 Ce domaine a été adressé en critiquant notamment les scénarios usuels d'évaluation
 des algorithmes de stéganographie. J'ai proposé un cadre complémentaire permettant 
-d'évaluer ces schémas de masquage~\cite{fccg15:ip}.
-Ceci se réalise en co-encadrant le doctorat de Y. Fadil.
+d'évaluer ces schémas de masquage~\cite{fccg15:ip,accfg15:ip}.
+Ceci se réalise en co-encadrant le doctorat de Y. Fadil. Nous avons de plus dans celui-ci
+proposé d'exprimer les fonctions de distorsion comme des méthodes de calcul de gradient 
+ou de matrice Hessienne. Ceci a donné lieu à la publication~\cite{ccfg16:ip}.  
 
 J'ai de plus participé à l'élaboration de l'algorithme STABYLO~\cite{ccg15:ij}
 qui est un schéma de
@@ -601,7 +602,39 @@ fournie et complexité.
   \caption{Embarquement d'un message dans une image selon STABYLO}\label{fig:STABYLO}
 \end{figure}
 
-  
+Enfin, les méthodes d'apprentissage ont largement progressé cette dernière décennie, 
+particulièrement grâce aux implantations efficaces de deep-learning. Dans un travail en collaboration
+avec Pr R. Couturier, Pr. C. Guyeux et Dr. M. Salomon, nous avons voulu étudier la 
+faisabilité d'utiliser du deep-learning comme outil de stéganalyse.
+Ceci a donné lieu à l'encadrement d'un stage de M2, D Fache, et 
+un rapport technique~\cite{CCGS16:ir}. Lorsque la clef d'embarquement est 
+la même pour toutes les images, nous avons exhibé une architecture de réseau 
+deep learning qui améliore de manière significative tous les résultats connus en 
+stéganalyse. La figure~\ref{fig:deep} montre que la détection est valide à 95\% 
+(respectivement à 80\%) lorsque le taux d'embarquement est de 0,4 bit par pixel (resp. 0.1)
+tandis que les meilleurs résultats connus sont de l'ordre de 85\% (resp. 70\%).
+Cependant, ceci ne s'applique pas aux cas où les clefs sont distinctes.
+
+
+\begin{figure}[h]
+\centering
+\subfigure[][WOW, $\alpha=0.4$~bpp]{%
+\label{fig:accuracy_WOW_0.4}%
+%\label{fig:ex3-a}%
+\includegraphics[scale=0.39]{Training_wow_04.pdf}
+}%
+\hspace{4pt}%
+\subfigure[][WOW, $\alpha=0.1$~bpp]{%
+\label{fig:accuracy_WOW_0.1}%
+\includegraphics[scale=0.39]{Training_wow_01.pdf}
+}
+\caption{Stéganalyse de WOW avec différents ratios d'embarquement}\label{fig:deep}
+\end{figure}
+
+
+
+
+
 
 \subsubsection{Les fonctions de hachage}\label{sub:hash}
 Une fonction qui calcule une empreinte servant à identifier rapidement toute 
@@ -627,9 +660,9 @@ Nous avons de plus vérifié statistiquement le critère d'avalanche.
 Ces résultats ont été valorisés dans les publications~\cite{bcg11:ip,bcg12:ij}.
 
 
+\subsection{Applications à d'autres sujets}\label{sec:appl}
 
-
-\subsection{Application à la génomique}\label{sub:gen}
+\subsubsection{Génomique}\label{sub:gen}
 
 Ayant acquis des compétences sur certaines structures de mathématiques 
 discrètes (particulièrement théorie des graphes, 
@@ -652,13 +685,27 @@ petits cardinaux. J'ai contribué notamment
 d'une adaptation discrète la méthode d'essaims particulaires~\cite{aagp+15:ip}.
 D'autres travaux (dont la première étape est cette construction de 
 core-génome) ont été ensuite réalisés, principalement dans le but de 
-construire des arbres phylogénétiques de qualité~\cite{agcs+15:ip,aangc+15:ip}.
-Une synthèse de ces arbres a été réalisée et est présentée à l'adresse~\url{http://meso.univ-fcomte.fr/peg/phylo}.
+construire des arbres phylogénétiques de qualité~\cite{agcs+15:ip,aangc+15:ip,aagc+15:ip}.
+Une synthèse de ces arbres a été réalisée et est présentée à 
+l'adresse~\url{http://meso.univ-fcomte.fr/peg/phylo}.
 
 
 Tous ces travaux ont été réalisés en collaboration avec M. Salomon et en co-encadrant 
-le doctorat de B. Alkindy.
+le doctorat de B. Alkindy, soutenu en décembre 2015.
+
+\subsubsection{Réseaux de capteurs}\label{sub:optim}
 
+J'ai abordé avec Dr. A. Mostefaoui la thématique des réseaux de capteurs multimédia:
+ce contexte scientifique 
+a la particularité d'avoir des données volumineuses et qui requièrent un traitement
+coûteux en énergie. L'enjeu consiste a optimiser la dépense énergétique en fonction de la qualité 
+attendue. 
+Dans le cadre d'un projet soutenu par la région de Franche-Comté, \og capteurs multimédias collaboratifs: une approche
+intégrée de la sécurité et de la robustesse\fg{}, nous encadrons la doctorante 
+N. Khernane sur ce sujet depuis novembre 2015.
+Nous avons proposé un nouvel algorithme complètement distribué permettant d'optimiser 
+la durée de vie d'un tel réseau de capteurs en fonction d'une qualité visuelle attendue.
+Ce premier travail a été valorisé par une publication en conférence internationale~\cite{kcm16:ip}.
 
 
 
@@ -808,18 +855,20 @@ la logique informatique.
 Suite à mon recrutement dans l'équipe AND, mes recherches se sont réorientées
 autour des SDDs et donc de l'analyse numérique plus généralement.
 Ce défi de changement de thématique a été attaqué dès mon recrutement.
-En plus des 5 journaux dans cette nouvelle thématique, 
-j'ai publié après ma thèse notamment dans
-les conférences reconnues suivantes: 
-\cite{couchot07cade}, de rang A, 
-\cite{bcgr11:ip}, de rang A  et 
-\cite{chgw+14:oip}, de rang B.
+En plus des 6 journaux dans cette nouvelle thématique, 
+j'ai publié sur ces thématiques notamment dans
+les conférences reconnues suivantes:
+\begin{itemize}
+\item  \cite{bcgr11:ip} de rang A et
+\item \cite{BCG16:ip,ccfg16:ip,kcm16:ip, chgw+14:oip} de rang B.
+\end{itemize}
+
 
 
 \subsection{Appels à projet}
 
 
-En 2014 (les dates a verifier), j'ai participé 
+En 2014, j'ai participé 
 au projet Jeune chercheur de l'UFC porté par
 Christophe Guyeux dont le thème était 
 \og la sécurisation numérique par chaos\fg{}.
@@ -829,7 +878,7 @@ Franche-Comté en 2015: j'ai participé à l'élaboration du
 projet de \og capteurs multimédias collaboratifs: une approche
 intégrée de la sécurité et de la robustesse\fg{} dont le porteur 
 est M. A. Mostefaoui. Nous avons obtenu le financement d'une bourse doctorale 
-pour la période d'oct. 2015--oct. 2018. Ce doctorat sera co-encadré par 
+pour la période d'oct. 2015--oct. 2018. Ce doctorat est co-encadré par 
 M. A. Mostefaoui et moi-même.
 
 J'ai répondu à de nombreux autres appels à projets sans succès: 
@@ -839,7 +888,7 @@ A FRAMEwork for predicting the behavior of Dicrete Dynamic Systems\fg{}
 avec l'I3S et le LORIA. 
 \item participant à un projet ANR blanc en 2010: 
 \og Unification des Systèmes Dynamiques Discrets \fg{}
-avec l'I3S, le LORIA et le LIF de MArseille.
+avec l'I3S, le LORIA et le LIF de Marseille.
 \item porteur d'un projet PEPS 2011 avec l'équipe MIE du département Energie de  FEMTO-ST.
 \item participant à un projet PHC CAI YUANPEI 2014: \og Theoretical study, design, and circuit implementation of high dimensional integer domain chaotic systems \fg{} avec l'Université Technlogique de Guangdong.
 \item porteur d'un projet PHC Cedre 2015: \og surveillance vidéo sécurisée et robuste par réseau de capteurs\fg{} avec l'Université Antonine.
@@ -862,14 +911,15 @@ la publication~\cite{bcgr11:ip}.
 \item avec R. Darazy de l'Université Antonine au Liban. 
   J'ai co-encadré trois stages de Master 2 recherche avec
   des membres de cette université.
-  J'ai un article en cours de soumission\cite{bdcc15:onp} résumant un travail en collaboration 
+  J'ai un article de journal accepté~\cite{bdcc15:onp} résumant un travail en collaboration 
   avec un ancien étudiant et un enseignant chercheur de cette université.
 
 \item avec Pr. S. Contassot-Vivier du LORIA (Nancy)- UMR CNRS 7503 
 sur la thématique des SDDs, de leur convergence dans les itérations 
 mixtes et de la générations de matrices de Markov doublement
 stochastiques. Cela s'est traduit notamment avec 
-le rapport~\cite{BCVC10:ir}.
+le rapport~\cite{BCVC10:ir} et l'article de journal~\cite{cgh16:onp}, en cours de soumission
+en journal international.
 
 \item avec Pr. Y. Bailly et J.-C. Roy de l'équipe 
   MIE du département \'Energie de FEMTO-ST sur la thématique 
@@ -909,22 +959,37 @@ Le comité de programme était composé de 18 chercheurs internationaux
 Il s'est  déroulé sur 1/2 journée et 5 chercheurs ont présenté leurs travaux.
 Le taux d'acceptation des articles a été de 41\%. 
 
+
+J'ai été responsable de la publication des actes dans les conferences 
+{\em 19th IEEE International Conference on Computational Science and Engineering (CSE 2016)},
+{\em 15th International Symposium on Distributed Computing and Applications to Business, Engineering and Science (DCABES 2016)}
+et
+{\em 14th IEEE/IFIP International Conference on Embedded and Ubiquitous Computing (EUC 2016)}.
+
+
+
 J'ai fait partie des comités de programme des conférences \og int. conf. on Evolving Internet \fg{}, (2013, 2014) 
 et IHTIAP 2012.
 
 J'ai rédigé des rapports de relecture pour les journaux internationaux 
 suivants: 
 \begin{itemize}
-\item The computer Journal (2); 
-\item Springer Multimedia Tool and application (1);
-\item Springer Annals in Telecommunication (1);
-\item International Journal of Bioinformatics Research (1);
+\item Springer Annals of Telecommunications 2013 (2);
+\item The computer Journal 2014 (1); 
+\item Springer Multimedia Tools and Applications 2014 (1);
+\item International Journal of Bioinformatics Research 2015 (1);
+\item International Journal of Computer Science and Application 2015 (1);
+\item MDPI Symmetry 2016 (3);
+\item HIndawi Journal of Electrical and Computer Engineering 2016 (1).
 \end{itemize}
 et pour les conférences suivantes:
 \begin{itemize}
 \item MAJECSTIC (2005 (2), 2007 (2));
 \item B 2007 (1), A-BZ 2008 (1);
 \item Evolving Internet \fg{}, (2012 (2), 2013 (2), 2014 (1));
+\item 2015 IEEE/SICE International Symposium on System Integration (1);
+\item 2016 IEEE International Conference on Computational Science and Engineering (CSE 2016)  (4);
+
 \end{itemize}
 
 Membre du GDR codes et cryptographie, j'ai 
@@ -1201,27 +1266,15 @@ Au DISC à FEMTO-ST&
 \addtocounter{\@listctr}{6}
 \makeatother
 
-\bibitem{bcg12:ij}
-Jacques Bahi, Jean-Fran\c{c}ois Couchot, and Christophe Guyeux.
-\newblock Quality analysis of a chaotic proven keyed hash function.
-\newblock {\em International Journal On Advances in Internet Technology},
-  5(1):26--33, 2012.
 
 
-\bibitem{bcg11:ij}
-Jacques Bahi, Jean-Fran\c{c}ois Couchot, and Christophe Guyeux.
-\newblock Steganography: a class of secure and robust algorithms.
-\newblock {\em The Computer Journal}, 55(6):653--666, 2012.
-
-\bibitem{bcgs12:ij}
-Jacques Bahi, Jean-Fran\c{c}ois Couchot, Christophe Guyeux, and Michel Salomon.
-\newblock Neural networks and chaos: Construction, evaluation of chaotic
-  networks, and prediction of chaos with multilayer feedforward network.
-\newblock {\em Chaos, An Interdisciplinary Journal of Nonlinear Science},
-  22(1):013122--1 -- 013122--9, March 2012.
-\newblock 9 pages.
-
 
+\bibitem{bdcc15:onp}
+Ahmad Bitar, Rony Darazi, Jean-François Couchot, and Raphaël Couturier.
+\newblock Blind digital watermarking in pdf documents using spread transform
+  dither modulation.
+\newblock{\em Multimedia Tools and Applications}, 
+1573-7721:1--19, November 2015.
 
 \bibitem{ccg15:ij}
 Jean-Fran\c{c}ois Couchot, Rapha\"el Couturier, and Christophe Guyeux.
@@ -1230,13 +1283,40 @@ Jean-Fran\c{c}ois Couchot, Rapha\"el Couturier, and Christophe Guyeux.
 \newblock {\em Annals of Telecommunications}, 2015.
 \newblock Available online. Paper version to appear.
 
-
 \bibitem{cds13:ij}
 Jean-Fran\c{c}ois Couchot, Karine Deschinkel, and Michel Salomon.
 \newblock Active {MEMS}-based flow control using artificial neural network.
 \newblock {\em Mechatronics}, 23(7):898--905, October 2013.
 \newblock Available online. Paper version to appear.
 
+
+\bibitem{bcgs12:ij}
+Jacques Bahi, Jean-Fran\c{c}ois Couchot, Christophe Guyeux, and Michel Salomon.
+\newblock Neural networks and chaos: Construction, evaluation of chaotic
+  networks, and prediction of chaos with multilayer feedforward network.
+\newblock {\em Chaos, An Interdisciplinary Journal of Nonlinear Science},
+  22(1):013122--1 -- 013122--9, March 2012.
+\newblock 9 pages.
+
+\bibitem{bcg11:ij}
+Jacques Bahi, Jean-Fran\c{c}ois Couchot, and Christophe Guyeux.
+\newblock Steganography: a class of secure and robust algorithms.
+\newblock {\em The Computer Journal}, 55(6):653--666, 2012.
+
+\bibitem{bcg12:ij}
+Jacques Bahi, Jean-Fran\c{c}ois Couchot, and Christophe Guyeux.
+\newblock Quality analysis of a chaotic proven keyed hash function.
+\newblock {\em International Journal On Advances in Internet Technology},
+  5(1):26--33, 2012.
+
+
+
+
+
+
+
+
+
 \end{thebibliography}
 
 
@@ -1248,24 +1328,16 @@ Jean-Fran\c{c}ois Couchot, Karine Deschinkel, and Michel Salomon.
 \begin{thebibliography}{CHG{\etalchar{+}}14b}
 
 \makeatletter
-\addtocounter{\@listctr}{11}
+\addtocounter{\@listctr}{12}
 \makeatother
 
 
-\bibitem{bdcc15:onp}
-Ahmad Bitar, Rony Darazi, Jean-François Couchot, and Raphaël Couturier.
-\newblock Blind digital watermarking in pdf documents using spread transform
-  dither modulation.
-\newblock in submission to the Multimedia Tools and Applications journal,
-  December 2014.
 
 \bibitem{cgh16:onp}
-Jean-François Couchot, Christophe Guyeux, and Pierre-Cyrille H\'eam.
+Jean-François Couchot, Sylvain Contassot-Vivier, Christophe Guyeux, and Pierre-Cyrille H\'eam.
 \newblock Random walk in a n-cube without hamiltonian cycle to chaotic
   pseudorandom number generation: Theoretical and practical considerations.
-\newblock in submission to the RAIRO - Theoretical Informatics and Applications
-  special issue to the 15-th Mons Theoretical Computer Science Days, March
-  2015.
+\newblock in submission.
 
 \bibitem{bgco16:onp}
 Mohammed Bakiri, Christophe Guyeux, Jean-Fran\c{c}cois Couchot, and
@@ -1290,6 +1362,13 @@ Mohammed Bakiri, Christophe Guyeux, Jean-Fran\c{c}cois Couchot, and
 \addtocounter{\@listctr}{14}
 \makeatother
 
+
+\bibitem{BCG16:ip}
+Bakiri Mohammed, Jean-François Couchot, and Christophe Guyeux.
+\newblock  FPGA Implementation of F2-Linear Pseudorandom Number Generators based on Zynq MPSoC: A Chaotic Iterations Post Processing Case Study
+\newblock In {\em Secrypt 2016, 13th Int. Conf. on Security and Cryptography},  pages ***--***,
+  Lisboa, Portugal, July 2016.
+
 \bibitem{aangc+15:ip}
 Bassam Alkindy, Huda Al'Nayyef, Christophe Guyeux, Jean-Fran\c{c}ois Couchot,
   Michel Salomon, and Jacques Bahi.
@@ -1322,9 +1401,9 @@ Bassam Alkindy, Christophe Guyeux, Jean-Fran\c{c}ois Couchot, Michel Salomon,
 \newblock Hybrid genetic algorithm and lasso test approach for inferring well
   supported phylogenetic trees based on subsets of chloroplastic core genes.
 \newblock In {\em AlCoB 2015, 2nd Int. Conf. on Algorithms for Computational
-  Biology}, volume~* of {\em LNCS/LNBI}, pages ***--***, Mexico City, Mexico,
+  Biology}, volume~9199 of {\em LNCS}, pages 83--96, Mexico City, Mexico,
   August 2015. Springer.
-\newblock To appear in the LNCS/LNBI series.
+
 
 \bibitem{aagp+15:ip}
 Reem Alsrraj, Bassam Alkindy, Christophe Guyeux, Laurent Philippe, and
@@ -1336,6 +1415,15 @@ Reem Alsrraj, Bassam Alkindy, Christophe Guyeux, Laurent Philippe, and
   September 2015.
 \newblock To appear.
 
+\bibitem{aagc+15:ip}
+B. AlKindy, B. Al{-}Nuaimi, C. Guyeux, J.-F. Couchot, M. Salomon, R. Alsrraj, and Laurent Philippe,
+\newblock Binary Particle Swarm Optimization Versus Hybrid Genetic Algorithm
+               for Inferring Well Supported Phylogenetic Trees.
+\newblock{\em Computational Intelligence Methods for Bioinformatics and Biostatistics
+               - 12th International Meeting, {CIBB} 2015,
+             Revised Selected Papers}, pages 165--179, September 2015.
+
+
 
 \bibitem{bcfg12a:ip}
 Jacques Bahi, Jean-Fran\c{c}ois Couchot, Nicolas Friot, and Christophe Guyeux.
@@ -1441,15 +1529,31 @@ Jean-Fran\c{c}ois Couchot and S.~Lescuyer.
 \newblock In {\em CADE'07, 21st Int. Conf. on Automated Deduction}, volume 4603
   of {\em LNCS}, pages 263--278, Bremen, Germany, July 2007. Springer.
 
-
 \bibitem{fccg15:ip}
 Yousra~Ahmed Fadil, Jean-Fran\c{c}ois Couchot, Rapha\"el Couturier, and
   Christophe Guyeux.
 \newblock Steganalyzer performances in operational contexts.
 \newblock In {\em IIH-MSP 2015, 11th Int. Conf. on Intelligent Information
-  Hiding and Multimedia Signal Processing}, pages ***--***, Adelaide,
+  Hiding and Multimedia Signal Processing}, pages 429--432, Adelaide,
   Australia, September 2015.
-\newblock To appear.
+
+\bibitem{accfg15:ip}
+B. Al Bouna and J. F. Couchot and R. Couturier and Y. A. Fadil and C. Guyeux.
+\newblock Performance Study of Steganalysis Techniques.
+\newblock In {\em Applied Research in Computer Science and Engineering (ICAR), 2015 International Conference on}, pages 1--7, Lebanon, October 2015.
+
+
+\bibitem{ccfg16:ip}
+Jean-François Couchot, Raphaël Couturier, Yousra Ahmed Fadil and Christophe Guyeux.
+\newblock A Second Order Derivatives based Approach for Steganography.
+\newblock In {\em Secrypt 2016, 13th Int. Conf. on Security and Cryptography},  pages ***--***,
+  Lisboa, Portugal, July 2016.
+
+\bibitem{kcm16:ip}
+Nesrine Khernane, J.-F. Couchot, A. Mostefaoui.
+\newblock Maximizing Network Lifetime in Wireless Video Sensor Networks Under Quality Constraints.
+\newblock In {\em The 19th ACM International Conference on Modeling, Analysis
+and Simulation of Wireless and Mobile Systems, MSWiM'16} Malta,  to appear, November, 2016.
 
 \end{thebibliography}
 
@@ -1472,17 +1576,21 @@ J'ai été invité:
 \begin{thebibliography}{CHG{\etalchar{+}}14b}
 
 \makeatletter
-\addtocounter{\@listctr}{34}
+\addtocounter{\@listctr}{39}
 \makeatother
 
-\bibitem{acgs13:onp}
-Bassam Alkindy, Jean-Fran\c{c}ois Couchot, Christophe Guyeux, and Michel
-  Salomon.
-\newblock Finding the core-genes of chloroplast species.
-\newblock Journ\'ees SeqBio 2013, Montpellier, November 2013.
+\bibitem{CCGS16:ir}
+J.-F. Couchot,  R. Couturier, C. Guyeux, and M. Salomon
+\newblock Steganalysis via a Convolutional Neural Network using Large Convolution Filters for Embedding Process with Same Stego Key.
+\newblock July, 2016.
 
+\bibitem{acgs13:onp}
+ Bassam Alkindy, Jean-François Couchot, Christophe Guyeux, and Michel Salomon. 
+\newblock Finding the core-genes of Chloroplast Species. 
+\newblock  Journées SeqBio 2013, Montpellier, November 2013. 
 
 
+\newblock Finding the core-genes of chloroplast species.
 
 \bibitem{BCVC10:ir}
 J.~M. Bahi, S.~Contassot-Vivier, and J.-F. Couchot.
-- 
2.39.5


From 52b5aa442f61f15f582a3412a4a283118e812859 Mon Sep 17 00:00:00 2001
From: couchot <couchot@localhost.localdomain>
Date: Tue, 23 Aug 2016 22:10:46 +0200
Subject: [PATCH 04/16] =?utf8?q?apr=C3=A8s=20relecture=20sylvaine?=
MIME-Version: 1.0
Content-Type: text/plain; charset=utf8
Content-Transfer-Encoding: 8bit

---
 demandeInscription/synthese.tex | 291 ++++++++++++++++++--------------
 1 file changed, 166 insertions(+), 125 deletions(-)

diff --git a/demandeInscription/synthese.tex b/demandeInscription/synthese.tex
index fce2b58..5553fd9 100755
--- a/demandeInscription/synthese.tex
+++ b/demandeInscription/synthese.tex
@@ -27,7 +27,7 @@
 \input{symboles.sty}
 
 
-\geometry{hmargin=1cm, vmargin=1.5cm }
+\geometry{hmargin=1.5cm, vmargin=1.5cm }
 
 
 \newcommand{\JFC}[1]{\begin{color}{green}\textit{#1}\end{color}}
@@ -273,7 +273,7 @@ et leurs applications (Sec.~\ref{sub:sdd}), thématique
 pour laquelle j'ai été recruté dans l'équipe AND du département
 DISC. Enfin la section~\ref{sec:appl}
 présente comment j'ai appliqué ces compétences théoriques et pratiques
-en bio-informatique  et  les réseaux de capteurs.
+en bio-informatique  et  dans les réseaux de capteurs.
 Ces travaux ont été valorisés par des publications dont les références sont données à la section~\ref{sec:publi} et synthétisées à la figure~\ref{fig:bilan}
 en fin de document.
 
@@ -540,7 +540,7 @@ D'autre méthodes de watermarking ont été investies,
 particulièrement celles basées sur la Quantization Index Modulation (QIM), méthodes 
 étant supposées comme les plus robustes. Mes principales contributions 
 sur ce travail --en collaboration avec Dr R. Darazy de l'Université Antonine au Liban
-et en co-encadrant le stage de M2 de Ahmad Bittar--,
+et en co-encadrant le stage de M2 de Ahmad Bitar--,
 ont été 
 d'intégrer ceci à du marquage de document PDF puis de 
 présenter ce problème comme un problème d'optimisation. 
@@ -560,22 +560,22 @@ J'ai de plus participé à l'élaboration de l'algorithme STABYLO~\cite{ccg15:ij
 qui est un schéma de
 stéganographie basé sur l'enfouissement de l'information dans les contours 
 présents dans une image et illustré à la figure~\ref{fig:STABYLO}.
-Ce travail est en collaboration avec Pr. R. Couturier.
+Ce travail est en collaboration avec Pr C. Guyeux et Pr. R. Couturier.
 Mes contributions ont principalement été la formalisation de l'algorithme et 
 son étude de complexité. Grâce a l'optimisation de cette dernière,
 nous avons  pu montrer 
 que cet algorithme présente un excellent compromis entre sécurité
-fournie et complexité.
+fournie et complexité algorithmique.
 
 
 
 \begin{figure}[h]
   \begin{center}
     \subfigure[Support]{
-      \includegraphics[scale=0.20]{lena512}
+      \includegraphics[scale=0.17]{lena512}
     }
     \subfigure[Message]{
-      \begin{minipage}[b]{0.28\linewidth}
+      \begin{minipage}[b]{0.33\linewidth}
         \begin{flushleft}
           \begin{scriptsize}
             The skies they were ashen and sober;\linebreak
@@ -592,10 +592,10 @@ fournie et complexité.
       \end{minipage}
     }
     \subfigure[Contours]{
-      \includegraphics[scale=0.20]{edge7}
+      \includegraphics[scale=0.17]{edge7}
     }
     \subfigure[Image avec msg.]{
-      \includegraphics[scale=0.20]{lena7}
+      \includegraphics[scale=0.17]{lena7}
     }
 
   \end{center}
@@ -605,11 +605,11 @@ fournie et complexité.
 Enfin, les méthodes d'apprentissage ont largement progressé cette dernière décennie, 
 particulièrement grâce aux implantations efficaces de deep-learning. Dans un travail en collaboration
 avec Pr R. Couturier, Pr. C. Guyeux et Dr. M. Salomon, nous avons voulu étudier la 
-faisabilité d'utiliser du deep-learning comme outil de stéganalyse.
-Ceci a donné lieu à l'encadrement d'un stage de M2, D Fache, et 
+faisabilité d'utiliser du deep-learning à base de convolutions comme outil de stéganalyse.
+Ceci a donné lieu à l'encadrement d'un stagiaire en M2, D. Fache, et 
 un rapport technique~\cite{CCGS16:ir}. Lorsque la clef d'embarquement est 
 la même pour toutes les images, nous avons exhibé une architecture de réseau 
-deep learning qui améliore de manière significative tous les résultats connus en 
+deep learning à base de convolutions qui améliore de manière significative tous les résultats connus en 
 stéganalyse. La figure~\ref{fig:deep} montre que la détection est valide à 95\% 
 (respectivement à 80\%) lorsque le taux d'embarquement est de 0,4 bit par pixel (resp. 0.1)
 tandis que les meilleurs résultats connus sont de l'ordre de 85\% (resp. 70\%).
@@ -680,17 +680,17 @@ de tous les génomes. Ceci a donné lieu aux
 publications~\cite{acgs13:onp,akgcs+14:oip,acgm+14:ij}.
 
 L'approche précédente souffrait de n'engendrer que des core-génomes de (trop)
-petits cardinaux. J'ai contribué notamment 
+petit cardinal. J'ai contribué notamment 
 à l'amélioration de la méthode en proposant une étape d'optimisation issue 
 d'une adaptation discrète la méthode d'essaims particulaires~\cite{aagp+15:ip}.
 D'autres travaux (dont la première étape est cette construction de 
 core-génome) ont été ensuite réalisés, principalement dans le but de 
 construire des arbres phylogénétiques de qualité~\cite{agcs+15:ip,aangc+15:ip,aagc+15:ip}.
-Une synthèse de ces arbres a été réalisée et est présentée à 
-l'adresse~\url{http://meso.univ-fcomte.fr/peg/phylo}.
+%Une synthèse de ces arbres a été réalisée et est présentée à 
+%l'adresse~\url{http://meso.univ-fcomte.fr/peg/phylo}.
 
 
-Tous ces travaux ont été réalisés en collaboration avec M. Salomon et en co-encadrant 
+Tous ces travaux ont été réalisés en collaboration avec Dr. M. Salomon et en co-encadrant 
 le doctorat de B. Alkindy, soutenu en décembre 2015.
 
 \subsubsection{Réseaux de capteurs}\label{sub:optim}
@@ -698,7 +698,7 @@ le doctorat de B. Alkindy, soutenu en décembre 2015.
 J'ai abordé avec Dr. A. Mostefaoui la thématique des réseaux de capteurs multimédia:
 ce contexte scientifique 
 a la particularité d'avoir des données volumineuses et qui requièrent un traitement
-coûteux en énergie. L'enjeu consiste a optimiser la dépense énergétique en fonction de la qualité 
+coûteux en énergie. L'enjeu consiste à optimiser la dépense énergétique en fonction de la qualité 
 attendue. 
 Dans le cadre d'un projet soutenu par la région de Franche-Comté, \og capteurs multimédias collaboratifs: une approche
 intégrée de la sécurité et de la robustesse\fg{}, nous encadrons la doctorante 
@@ -716,32 +716,6 @@ autour de la thématique des systèmes dynamiques discrets.
 
 
 
-\subsection{Les codes de Gray}
-L'utilisation des codes de Gray dans une démarche d'apprentissage 
-(d'écoulement d'air ou  de fonctions chaotiques) ne s'est pas révélée comme 
-concluante. Dans chacun des cas, la distance de Hamming entre deux 
-configurations voisines peut être très petite tandis que le chemin (dans le 
-cycle  hamiltonien) qui les relie peut être long et ce même 
-pour des codes équilibrés.
-Je propose de travailler sur ce problème discret en mesurant la qualité 
-du code de Gray à l'aide d'une fonction basée sur la longueur des chemins
-(du cycle hamiltonien) entre les configurations voisines.
-Je pense ainsi réduire ce problème à un problème d'optimisation et dégager 
-une démarche de génération, comme je l'ai fait en bio-informatique.
- 
-Jusqu'à présent, la production de codes de Gray équilibrés pour la génération 
-de nombres pseudo-aléatoires bute sur des problèmes d'explosion combinatoire:
-les seuls algorithmes connus répondant à ce problème nécessitent a priori
-plus de $10^{36}$ évaluations pour $n=8$. 
-Il n'est ainsi pas raisonnable de mettre en  
-pratique ce genre d'approche lorsque chacune de ces évaluations prend 1s.
-On peut peut-être
-se contenter de codes ``presque'' équilibrés , à défaut de pouvoir 
-trouver ceux qui seront équilibrés.
-Je propose d'investiguer  
-dans cette thématique en exploitant des approches itératives permettant 
-d'obtenir des optimums locaux et trouver ainsi des codes presque équilibrés.
-
 
  
 \subsection{Génération de nombres pseudo-aléatoires}
@@ -762,31 +736,15 @@ Les propriété établies notamment sur les temps d'arrêt devraient être conse
 Il restera à le prouver.
 
 
-Jusqu'à présent, une seule expérimentation d'implantation de nos générateurs 
-sur des dispositifs physiques comme les FPGAs a été réalisée. Celle-ci 
-s'est faite automatiquement à l'aide de l'outil Matlab. Si le code engendré
-sur le circuit est bien une implantation fidèle à la spécification,
-il n'en est pas pour autant efficace: le nombre de bits générés par surface
-est plutôt faible. Nous allons exploiter les meilleures démarches mises en 
-exergue lors de la rédaction d'un état de l'art exhaustif sur les PRNGs 
-implantés sur FPGA pour produire du code optimisé. 
-Je prévois de réaliser ceci dans la thèse de M. Bakiri, en cours.
+Dans le cadre du doctorat de M. Bakiri, nous avons implanté sur  FPGA 
+18 PRNGs de 4 familles (xorshift, LFSR, TGFSR, and LCG) ainsi que 
+le post-processing implantant les itérations
+chaotiques. Nous pensons étudier d'abord pratiquement l'intérêt de telles itérations
+en termes de surface requise lors du déploiement et du nombre d'opérateurs nécessaires.
+Les améliorations escomptées dans le paragraphe précédent seront réinvesties dans cette 
+partie et celle-ci viendront nourrir le travail précédent. 
+
 
-Pour générer une fonction dont la matrice de Markov est doublement
-stochastique, nous avons proposé principalement deux méthodes 
-(génération puis test, suppression de chemin hamiltonien dans un $n$-cube).  
-Ces deux méthodes ne passent pas à l'échelle, même pour des $n$ de petite taille.
-Je pense attaquer ce problème algébriquement et en programmation logique avec 
-contraintes. Dans le premier cas, on peut remarquer qu'un matrice
-composée de $1$ uniquement 
-en $(i,i+1)$ est une réponse triviale au problème. Je pense continuer l'étude 
-de ce genre de matrices et proposer une méthode plus générale de génération.
-Je prévois de réaliser ce travail avec M. S. Contassot, Pr. à l'Université de Lorraine.
-Le département DISC et l'équipe VESONTIO 
-a de fortes compétences en programmation logique avec 
-contraintes. J'ai déjà démontré que ce problème peut être soluble par cette
-approche, sans avoir pour autant réussi à le faire.
-Je prévois des collaborations avec l'équipe VESONTIO du DISC sur ce sujet.
 
 
 Enfin, marcher dans une partie d'un $n$-cube est le modèle théorique que 
@@ -800,6 +758,40 @@ Pour l'instant, nous n'avons pas réussi à obtenir des bornes
 du temps d'arrêt. Je propose d'investiguer aussi dans cette direction.
 
 
+\subsection{Les chemin Hamiltonien/ les codes de Gray}
+
+Pour générer une fonction dont la matrice de Markov est doublement
+stochastique
+--condition nécessaire pour fournir une sortie uniformément distribuée--, nous avons proposé principalement la méthode de
+suppression de chemin hamiltonien dans un $n$-cube. 
+
+Un chemin hamiltonien pouvant être vu comme un code de Gray, nous sommes intéressés à la 
+génération de ce genre de codes en privilégiant les codes de Gray équilibrés, 
+c'est-à-dire ceux dont chaque bit est nié le même nombre de fois.
+Cependant la méthode de Robinson et Cohn (permettant d'obtenir des codes de Gray globalement ``presque'' 
+équilibrés) et leurs extensions sont indéterministes dans le sens où 
+elles montrent l'existence d'une suite particulière sans donner sa construction.
+Un premier verrou scientifique que je propose de lever consiste à proposer une solution 
+algébrique (et donc un algorithme) donnant une solution à ce problème. 
+
+Il est apparu récemment des algorithmes permettant d'obtenir des codes de Gray 
+localement équilibrés, c.-à-d. où la longueur du plus grand nombre d'étapes entre 
+deux changements d'un même bit est aussi petite que possible.
+Dans tous les cas, aucun des ces codes n'est globalement équilibré ni même presque équilibré.
+Cette double propriété serait cependant très intéressante aussi bien théoriquement que pratiquement
+pour nos générateurs.
+Un second verrou consistera à adapter ces algorithmes pour proposer des codes possédant les 
+deux propriétés d'équilibrage.
+Ce travail consistera d'abord à vérifier l'existence de codes pour des instances de petite 
+taille, sans que cela soit trivial (par exemple pour $n=6$) et ce en travaillant en collaboration 
+avec Pr S. Contassot, spécialiste en SDD et en calcul distribué.
+ 
+
+
+
+
+
+
 
 \subsection{Masquage d'information}
 
@@ -836,8 +828,20 @@ la classification. Ces méthodes ont déjà été
 appliquées avec succès lorsqu'elles sont combinées avec des méthodes 
 d'apprentissage, par exemple dans de la reconnaissance faciale.
 Je propose d'étudier cette piste dans ce domaine. 
-Ceci se réalisera notamment au travers du doctorat de Y. Fadil.
+%Ceci se réalisera notamment au travers du doctorat de Y. Fadil.
 
+De plus les résultats obtenus en stéganalyse à l'aide de deep learning à base de convolutions
+sont très prometteurs  lorsque la clef qui a servi à l'embarquement est constante. 
+Malheureusement, lorsque la clef varie, nous n'avons pas réussi à généraliser ces avancées.
+Les démarches les plus efficaces demeurent 
+celles obtenues par des approches classiques à base de caractéristiques statistiques (features) 
+d'images.
+Cependant, en étudiant plus finement les features, on constate que nombreuses sont celles qui sont aussi
+basées sur des produits de convolution.
+Je propose d'étudier exhaustivement ces features pour d'abord traduire
+en deep-learning celles qui sont des convolutions directes. Il restera ensuite 
+à adapter l'outil de deep learning aux caractéristiques restantes ce qui est un autre challenge 
+scientifique. 
 
 
 
@@ -911,8 +915,8 @@ la publication~\cite{bcgr11:ip}.
 \item avec R. Darazy de l'Université Antonine au Liban. 
   J'ai co-encadré trois stages de Master 2 recherche avec
   des membres de cette université.
-  J'ai un article de journal accepté~\cite{bdcc15:onp} résumant un travail en collaboration 
-  avec un ancien étudiant et un enseignant chercheur de cette université.
+  J'ai un article de journal~\cite{bdcc15:onp} résumant un travail en collaboration
+  avec R. Darazy.
 
 \item avec Pr. S. Contassot-Vivier du LORIA (Nancy)- UMR CNRS 7503 
 sur la thématique des SDDs, de leur convergence dans les itérations 
@@ -937,6 +941,9 @@ Nous avons collaboré sur deux publications~\cite{bcgw11:ip,chgw+14:oip}.
 \subsection{Invitations}
 J'ai été invité:
 \begin{itemize}
+\item à la soutenance de Thèse de M. J.-L. Joly sur {\em 
+Contributions à la génération aléatoire pour des classes d'automates finis} en tant qu'invité
+le 23 mars 2016 à l'Université de Franche-Comté;
 \item au workshop \og Théorie des réseaux booléens et ses applications en
   biologie\fg{}  à Nice en novembre 2014 (20 participants);
 \item à la journée \og Advances in Mobile Technologies Day\ fg{} 
@@ -980,7 +987,7 @@ suivants:
 \item International Journal of Bioinformatics Research 2015 (1);
 \item International Journal of Computer Science and Application 2015 (1);
 \item MDPI Symmetry 2016 (3);
-\item HIndawi Journal of Electrical and Computer Engineering 2016 (1).
+\item Hindawi Journal of Electrical and Computer Engineering 2016 (1).
 \end{itemize}
 et pour les conférences suivantes:
 \begin{itemize}
@@ -1004,52 +1011,69 @@ où j'ai présenté \og Steganography: secure and robust algorithms \fg{} et en
 \section{Encadrement et co-encadrement d'étudiants} 
 
 \subsection{Thèse d'université}
-
-Je co-encadre:
+J'ai co-encadré:
 \begin{itemize}
 \item
 (à hauteur de 40\%) avec Michel Salomon (40\%) et  
 Jacques M. Bahi  (20\%) le doctorat de Bassam Alkindy
 sur les combinaisons d'approches pour la prédiction de l'évolution génomique.	
-Le doctorat a commencé en septembre 2012 et devrait être soutenu en 
-novembre 2015.	
+Le doctorat a commencé en septembre 2012 et a été soutenu le 17 décembre 2015.
 Ce travail de thèse a donné lieu à 6 publications dans des conférences 
-internationales.
-\item  (à hauteur de 50\%) avec Christophe Guyeux (50\%)
+internationales~\cite{aangc+15:ip,acgm+14:ij,akgcs+14:oip,agcs+15:ip,aagp+15:ip, aagc+15:ip} et une communication nationale~\cite{acgs13:onp}.
+\end{itemize}
+
+Je co-encadre:
+\begin{itemize}
+\item  (à hauteur de 50\%) avec Pr. C. Guyeux (50\%)
 le doctorat de  Youssra Fadil
 sur
 l'étude du comportement des outils d'intelligence artificielle
 face à des dynamiques complexes.	
 Le doctorat a commencé en décembre 2013 et devrait être soutenu en 
 décembre 2016.	
-Ce travail de thèse a donné lieu à 1 publication dans une conférence
-internationale.
+Ce travail de thèse a donné lieu à 3 publications dans des conférences
+internationales~\cite{fccg15:ip,accfg15:ip,accfg15:ip}.
 		
-\item  (à hauteur de 50\%) avec Christophe Guyeux (50\%)
+\item  (à hauteur de 50\%) avec Pr. C. Guyeux (50\%)
 le doctorat de  Mohamed Bakiri
 sur
 l'implémentation matérielle de générateurs de nombres pseudo-aléatoires basés sur les itérations chaotiques.
 Le doctorat a commencé en septembre 2014 et devrait être soutenu en 
 août 2017.	
 Ce travail de thèse a donné lieu à 1 publication dans un journal international 
-en cours de soumission~\cite{bgco16:onp}.
+en cours de soumission~\cite{bgco16:onp} et une conférence internationale~\cite{bgco16:onp}.
+
+\item  (à hauteur de 50\%) avec  Dr. A. Mostefaoui (50\%)
+le doctorat de  Nesrine Khernane
+sur
+une approche
+intégrée de la sécurité et de la robustesse dans les capteurs multimédias collaboratifs. 
+Le doctorat a commencé en novembre 2015 et devrait être soutenu en 
+novembre 2018.	
+Ce travail de thèse a donné lieu à 1 publication dans une conférence internationale~\cite{fccg15:ip}.
 \end{itemize}
 
 
 \subsection{Master 2 recherche}
 
+J'encadre:
+
+\begin{itemize}
+\item le stage de recherche de Master 2 de D. Fache
+sur une étude théorique et pratique du Deep Learning, application à la stéganalyse.
+Le stage a commencé en mars 2016 et devrait être soutenu en septembre 2016. 
+\end{itemize}
 
 J'ai co-encadré:
 \begin{itemize}
 \item  (à hauteur de 34\%) avec Raphael Couturier (33\%) et 
 Rony Darazi -- Université Antonine au Liban--(33\%)
-le stage de recherche de Master 2 de  Ahmad Bittar
+le stage de recherche de Master 2 de  Ahmad Bitar
 sur
 la 
 Steganography in PDF documents based on Unprintable Control ASCII Codes.
 Le stage a commencé le 01 mai 2013 et a été soutenu le 30 septembre 2013. 
-Ce travail de Master a donné lieu à 1 publication dans un journal international 
-en cours de soumission~\cite{bdcc15:onp} (en révision mineure).
+Ce travail de Master a donné lieu à 1 publication dans un journal international~\cite{bdcc15:onp}.
 \item  (à hauteur de 80\%) avec   
 Talar Atechian -- Université Antonine au Liban--(20\%)
 le stage de recherche de Master 2 de Hussein Nasser
@@ -1065,10 +1089,11 @@ Talar Atechian -- Université Antonine au Liban--(20\%)
 le stage de recherche de Master 2 de Khaled Daher 
 sur la désynchronisation du calcul d'optimisation de la durée de vie 
 d'un réseau de capteurs.
-Le stage a commencé le 01 avril 2015 et sera soutenu le 31 août 2015. 
+Le stage a commencé le 01 avril 2015 et a été soutenu le 31 août 2015. 
 \end{itemize}
 
 
+
 %\newpage
 \section{Participation à des tâches d'intérêt collectif}
 
@@ -1123,13 +1148,13 @@ de l'IUT BM.
 J'ai monté avec succès deux projets BQF au sein de l'UFC autour
 de la programmation mobile en 2009 et 2014. 
 Ces projets BQF ont permis l'achat de smartphones et ordinateurs 
-portables pour les projets d'étudiants du département d'info de l'IUTBM. 
+portables pour les projets d'étudiants du département d'informatique de l'IUTBM. 
 
 
 \subsection{\`A l'extérieur de l'université}
 
 Je suis régulièrement membre de jury des épreuves TIPE, épreuves communes
-à 90 concours d'ingénieurs (sessions 2012 et  2015) et fournis  
+à 90 concours d'ingénieurs (sessions 2012, 2015 et 2016) et fournis  
 à chaque session des sujets scientifiques pour l'épreuve.
 
 
@@ -1190,8 +1215,8 @@ En post-doctorat &
 %journaux
 &
 % conf inter
-\cite{aangc+15:ip,aagp+15:ip,acgm+14:ij,
-  akgcs+14:oip,agcs+15:ip}
+\cite{BCG16:ip,aangc+15:ip,acgm+14:ij,
+  akgcs+14:oip,agcs+15:ip,aagp+15:ip}
 
 &
 % conf nat
@@ -1206,7 +1231,7 @@ En post-doctorat &
 
 Au DISC à FEMTO-ST&
 %journaux
-\cite{bcg12:ij,bcg11:ij,bcgs12:ij,ccg15:ij,cds13:ij}
+\cite{bcg12:ij,bcg11:ij,bcgs12:ij,cds13:ij,ccg15:ij,bdcc15:onp}
 
 
 
@@ -1216,7 +1241,7 @@ Au DISC à FEMTO-ST&
 
 &
 % conf inter
-\cite{bcfg+13:ip,bcfg12a:ip,bcfg12b:ip,bcg11:ip,bcg11b:ip}
+\cite{aagc+15:ip,bcfg12a:ip,bcfg12b:ip,bcfg+13:ip,bcg11:ip,bcg11b:ip}
 
 &
 % conf nat
@@ -1235,7 +1260,7 @@ Au DISC à FEMTO-ST&
 
 &
 % conf inter
-\cite{bcgr11:ip,bcgw11:ip,cds12:ip,chgw+14:oip,fccg15:ip}
+\cite{bcgr11:ip,bcgw11:ip,cds12:ip,chgw+14:oip,fccg15:ip,accfg15:ip}
 
 &
 % conf nat
@@ -1244,6 +1269,21 @@ Au DISC à FEMTO-ST&
 &
 % divers
 
+\\ %\hline  
+
+&
+%journaux
+
+&
+% conf inter
+\cite{ccfg16:ip,kcm16:ip}
+
+&
+% conf nat
+
+
+&
+% divers
 
 
 %%%%%%%%%%%%%
@@ -1477,21 +1517,6 @@ Jacques Bahi, Jean-Fran\c{c}ois Couchot, Christophe Guyeux, and Qianxue Wang.
 \newblock In {\em INTERNET 2011, the 3-rd Int. Conf. on Evolving Internet},
   pages 72--77, Luxembourg, Luxembourg, June 2011.
 
-\bibitem{BCDG07}
-Fabrice Bouquet, Jean-Fran\c{c}ois Couchot, Fr\'ed\'eric Dadeau, and Alain
-  Giorgetti.
-\newblock Instantiation of parameterized data structures for model-based
-  testing.
-\newblock In Jacques Julliand and Olga Kouchnarenko, editors, {\em B'2007, the
-  7th Int. B Conference}, volume 4355 of {\em LNCS}, pages 96--110, Besancon,
-  France, January 2007. Springer.
-
-\bibitem{CouchotD07IFM}
-Jean-Fran\c{c}ois Couchot and Fr\'ed\'eric Dadeau.
-\newblock Guiding the correction of parameterized specifications.
-\newblock In {\em IFM'07, 6th Int. Conf. on Integrated Formal Methods}, volume
-  4591 of {\em LNCS}, pages 176--194, Oxford, UK, July 2007. Springer.
-
 
 \bibitem{cds12:ip}
 Jean-Fran\c{c}ois Couchot, Karine Deschinkel, and Michel Salomon.
@@ -1502,12 +1527,6 @@ Jean-Fran\c{c}ois Couchot, Karine Deschinkel, and Michel Salomon.
   MEMS}, pages 1--6, Besan\c{c}on, France, April 2012. IEEE CPS.
 
 
-\bibitem{cgs09:ip}
-Jean-Fran\c{c}ois Couchot, Alain Giorgetti, and Nicolas Stouls.
-\newblock {G}raph {B}ased {R}eduction of {P}rogram {V}erification {C}onditions.
-\newblock In Hassen Sa\"{i}di and N.~Shankar, editors, {\em AFM'09, {A}utomated
-  {F}ormal {M}ethods (colocated with {CAV}'09)}, pages 40--47, Grenoble,
-  France, 2009. ACM Press.
 
 \bibitem{chgw+14:oip}
 Jean-Fran\c{c}ois Couchot, Pierre-Cyrille H\'eam, Christophe Guyeux, Qianxue
@@ -1517,18 +1536,6 @@ Jean-Fran\c{c}ois Couchot, Pierre-Cyrille H\'eam, Christophe Guyeux, Qianxue
   pages 469--475, Vienna, Austria, August 2014.
 
 
-\bibitem{couchot07FTP}
-Jean-Fran\c{c}ois Couchot and T.~Hubert.
-\newblock A graph-based strategy for the selection of hypotheses.
-\newblock In {\em FTP'07, Int. Workshop on First-Order Theorem Proving},
-  Liverpool, UK, September 2007.
-
-\bibitem{couchot07cade}
-Jean-Fran\c{c}ois Couchot and S.~Lescuyer.
-\newblock Handling polymorphism in automated deduction.
-\newblock In {\em CADE'07, 21st Int. Conf. on Automated Deduction}, volume 4603
-  of {\em LNCS}, pages 263--278, Bremen, Germany, July 2007. Springer.
-
 \bibitem{fccg15:ip}
 Yousra~Ahmed Fadil, Jean-Fran\c{c}ois Couchot, Rapha\"el Couturier, and
   Christophe Guyeux.
@@ -1555,6 +1562,40 @@ Nesrine Khernane, J.-F. Couchot, A. Mostefaoui.
 \newblock In {\em The 19th ACM International Conference on Modeling, Analysis
 and Simulation of Wireless and Mobile Systems, MSWiM'16} Malta,  to appear, November, 2016.
 
+\bibitem{BCDG07}
+Fabrice Bouquet, Jean-Fran\c{c}ois Couchot, Fr\'ed\'eric Dadeau, and Alain
+  Giorgetti.
+\newblock Instantiation of parameterized data structures for model-based
+  testing.
+\newblock In Jacques Julliand and Olga Kouchnarenko, editors, {\em B'2007, the
+  7th Int. B Conference}, volume 4355 of {\em LNCS}, pages 96--110, Besancon,
+  France, January 2007. Springer.
+
+\bibitem{CouchotD07IFM}
+Jean-Fran\c{c}ois Couchot and Fr\'ed\'eric Dadeau.
+\newblock Guiding the correction of parameterized specifications.
+\newblock In {\em IFM'07, 6th Int. Conf. on Integrated Formal Methods}, volume
+  4591 of {\em LNCS}, pages 176--194, Oxford, UK, July 2007. Springer.
+
+\bibitem{cgs09:ip}
+Jean-Fran\c{c}ois Couchot, Alain Giorgetti, and Nicolas Stouls.
+\newblock {G}raph {B}ased {R}eduction of {P}rogram {V}erification {C}onditions.
+\newblock In Hassen Sa\"{i}di and N.~Shankar, editors, {\em AFM'09, {A}utomated
+  {F}ormal {M}ethods (colocated with {CAV}'09)}, pages 40--47, Grenoble,
+  France, 2009. ACM Press.
+
+\bibitem{couchot07FTP}
+Jean-Fran\c{c}ois Couchot and T.~Hubert.
+\newblock A graph-based strategy for the selection of hypotheses.
+\newblock In {\em FTP'07, Int. Workshop on First-Order Theorem Proving},
+  Liverpool, UK, September 2007.
+
+\bibitem{couchot07cade}
+Jean-Fran\c{c}ois Couchot and S.~Lescuyer.
+\newblock Handling polymorphism in automated deduction.
+\newblock In {\em CADE'07, 21st Int. Conf. on Automated Deduction}, volume 4603
+  of {\em LNCS}, pages 263--278, Bremen, Germany, July 2007. Springer.
+
 \end{thebibliography}
 
 \subsection{Communications orales invitées}
-- 
2.39.5


From d419382ccd1a95aa44a16d1e7cfff6fd4aba0d05 Mon Sep 17 00:00:00 2001
From: couchot <couchot@localhost.localdomain>
Date: Wed, 24 Aug 2016 15:26:53 +0200
Subject: [PATCH 05/16] ajout qques fichiers

---
 .../AvisHDR_JFCouchot_JB2015.pdf              | Bin 0 -> 795136 bytes
 .../AvisHDR_JFCouchot_OK_2015.docx            | Bin 0 -> 592276 bytes
 demandeInscription/DiplomeDoctoratCouchot.pdf | Bin 0 -> 506687 bytes
 demandeInscription/L_FMTO_OK_2015-08-30.pdf   | Bin 0 -> 5393484 bytes
 .../LettreAuPresident150831.pdf               | Bin 0 -> 27118 bytes
 demandeInscription/avec_jef_hdr.docx          | Bin 0 -> 590595 bytes
 images/h3prng.eps                             | 621 ++++++++++++++++++
 7 files changed, 621 insertions(+)
 create mode 100644 demandeInscription/AvisHDR_JFCouchot_JB2015.pdf
 create mode 100644 demandeInscription/AvisHDR_JFCouchot_OK_2015.docx
 create mode 100644 demandeInscription/DiplomeDoctoratCouchot.pdf
 create mode 100644 demandeInscription/L_FMTO_OK_2015-08-30.pdf
 create mode 100644 demandeInscription/LettreAuPresident150831.pdf
 create mode 100644 demandeInscription/avec_jef_hdr.docx
 create mode 100644 images/h3prng.eps

diff --git a/demandeInscription/AvisHDR_JFCouchot_JB2015.pdf b/demandeInscription/AvisHDR_JFCouchot_JB2015.pdf
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2.39.5


From 78cd1e6fef0e2664357da1584c958f06b0820a88 Mon Sep 17 00:00:00 2001
From: couchot <couchot@localhost.localdomain>
Date: Fri, 26 Aug 2016 15:58:51 +0200
Subject: [PATCH 06/16]  fin prng

---
 14Secrypt.tex                   | 170 ++++++++++++++++++++++++--------
 demandeInscription/synthese.tex |   2 +-
 main.tex                        |   5 +
 3 files changed, 134 insertions(+), 43 deletions(-)

diff --git a/14Secrypt.tex b/14Secrypt.tex
index 597acbb..f1fae6f 100644
--- a/14Secrypt.tex
+++ b/14Secrypt.tex
@@ -380,43 +380,65 @@ vérifiant $\sum_{i=1}^nNT_n(i) = 2^n$.
 \section{Génération de codes de Gray équilibrés par induction}
 \label{sec:induction}
 
-Dans  leur  article de  2004~\cite{ZanSup04},  Zanten  et  Suparta proposent  un
-algorithme inductif  pour générer  des codes  de Gray équilibrés  de $n$  bits à
-partir   de  codes   de  $n-2$   bits.   Cependant,   leur  méthode   n'est  pas
-constructive. En effet, elle effectue  des manipulations sur un partitionnement du
-code de Gray  initial de $n-2$ bits pour  obtenir un code de Gray  sur $n$ bits,
-mais le  résultat n'est pas  systématiquement équilibré. Il est  donc nécessaire
-d'évaluer les résultats obtenus à  partir de tous les partitionnements réalisables
-en suivant les  contraintes spécifiées.  Or, le nombre  de possibilités augmente
-exponentiellement (voir~\cite{Mons14} pour  l'évaluation détaillée), ce qui rend
-déraisonnable    tout   parcours    exhaustif.    Une    amélioration   proposée
-dans~\cite{Mons14} permet  de réduire le nombre  de partitionnements considérés,
-mais l'ordre  de grandeur  reste similaire. On  constate donc clairement  ici la
-nécessité de trouver  des algorithmes de génération de  codes de Gray équilibrés
-plus  efficaces.  Ce  problème  représente  une des  voies  que nous  souhaitons
-explorer dans la suite de nos travaux.
-
-Le   tableau~\ref{table:nbFunc}  donne  le   nombre  de   fonctions  différentes
-compatibles avec les codes de  Gray équilibrés générés par l'approche précédente
-selon le nombre  de bits. Il donne  donc la taille de la  classe des générateurs
-pouvant être produits.  Les  cas 7 et 8 ne sont que  des bornes minimales basées
-sur des sous-ensembles des partitionnements possibles.
+De nombreuses approches ont été developpées pour résoudre le problème de construire
+un code de Gray dans un $\mathsf{N}$-cube~\cite{Robinson:1981:CS,DBLP:journals/combinatorics/BhatS96,ZanSup04}, 
+selon les propriétés que doit vérifier ce code.
+
+Dans les travaux~\cite{Robinson:1981:CS}, les auteurs 
+proposent une approche inductive de construction de code de Gray équilibrés 
+(on passe du $\mathsf{N}-2$ à $\mathsf{N}$)
+pour peu que l'utilisateur fournisse une sous-séquence possédant certaines 
+propriétés à chaque pas inductif.
+Ce travail a été renforcé dans ~\cite{DBLP:journals/combinatorics/BhatS96}
+où les auteurs donnent une manière explicite de construire une telle sous-séquence.
+Enfin, les autheurs de~\cite{ZanSup04} présentent une extension de l'algorithme de
+\emph{Robinson-Cohn}. La présentation rigoureuse de cette extension leur permet 
+principalement de prouver que si $\mathsf{N}$ est une puissance de 2, 
+le code de Gray équilibré engendré par l'extension est toujours totalement équilibré et 
+que $S_{\mathsf{N}}$ est la séquence de transition d'un code de Gray de $\mathsf{N}$ bits 
+si  $S_{\mathsf{N}-2}$ l'est aussi.. 
+Cependant les auteurs ne prouvent pas que leur approche fournit systématiquement  
+un code de Gray (totalement) équilibré. 
+Cette section montre que ceci est vrai en rappelant tout d'abord
+l'extension de l'algorithme de \emph{Robinson-Cohn} pour un 
+code de Gray avec $\mathsf{N}-2$ bits.
 
-\begin{table}[ht]
-  \begin{center}
-    \begin{tabular}{|l|c|c|c|c|c|}
-      \hline
-      $n$              & 4 & 5 & 6    & 7      & 8      \\
-      \hline
-      nb. de fonctions & 1 & 2 & 1332 & $>$ 2300 & $>$ 4500 \\
-      \hline
-    \end{tabular}
-  \end{center}
-\caption{Nombre de codes de Gray équilibrés selon le nombre de bits.}\label{table:nbFunc}
-\end{table}
+\begin{enumerate}
+\item \label{item:nondet}Soit $l$ un entier positif pair. Trouver des sous-sequences 
+$u_1, u_2, \dots , u_{l-2}, v$ (possiblement vides) de $S_{\mathsf{N}-2}$ 
+telles que $S_{\mathsf{N}-2}$ est la concaténation de  
+$$
+s_{i_1}, u_0, s_{i_2}, u_1, s_{i_3}, u_2, \dots , s_{i_l-1}, u_{l-2}, s_{i_l}, v
+$$
+où $i_1 = 1$, $i_2 = 2$, et $u_0 = \emptyset$ (la séquence vide).
+\item\label{item:u'} Remplacer dans $S_{\mathsf{N}-2}$ les sequences $u_0, u_1, u_2, \ldots, u_{l-2}$ 
+  par 
+  $\mathsf{N} - 1,  u'(u_1,\mathsf{N} - 1, \mathsf{N}) , u'(u_2,\mathsf{N}, \mathsf{N} - 1), u'(u_3,\mathsf{N} - 1,\mathsf{N}), \dots, u'(u_{l-2},\mathsf{N}, \mathsf{N} - 1)$
+  respectivement, où $u'(u,x,y)$ est la séquence $u,x,u^R,y,u$ telle que 
+  $u^R$ est $u$, mais dans l'ordre inverse. La séquence obtenue est ensuite notée $U$.
+\item\label{item:VW} Contruire les séquences $V=v^R,\mathsf{N},v$, $W=\mathsf{N}-1,S_{\mathsf{N}-2},\mathsf{N}$. Soit  alors $W'$ définie commé étant égale à $W$ sauf pour les 
+deux premiers éléments qui ont été intervertis.
+\item La séquence de transition  $S_{\mathsf{N}}$ est la concatenation $U^R, V, W'$.
+\end{enumerate} 
+
+L'étape~(\ref{item:nondet}) n'est pas constructive: il n'est pas précisé
+comment sélectionner des sous-séquences qui assurent que le code obtenu est équilibré.
+La théoreme suivante montre que c'est possible et sa preuve explique comment le faire. 
+
+
+\begin{theorem}\label{prop:balanced}
+Soit $\mathsf{N}$ dans $\Nats^*$, et $a_{\mathsf{N}}$ défini par 
+$a_{\mathsf{N}}= 2 \left\lfloor \dfrac{2^{\mathsf{N}}}{2\mathsf{N}} \right\rfloor$. 
+il existe une séquence $l$ dans l'étape~(\ref{item:nondet}) de l'extension
+de l'algorithme de \emph{Robinson-Cohn} extension telle que 
+le nombres de transitions $\textit{TC}_{\mathsf{N}}(i)$ 
+sont tous $a_{\mathsf{N}}$ ou $a_{\mathsf{N}}+2$ 
+pour chaque  $i$, $1 \le i \le \mathsf{N}$.
+\end{theorem}
 
+La preuve de ce théorème est donnée en annexes~\ref{anx:generateur}.
 
-Ces fonctions étant générée, on s'intéresse à étudier à quelle vitesse 
+Ces fonctions étant générées, on s'intéresse à étudier à quelle vitesse 
 un générateur les embarquant converge vers la distribution uniforme.
 C'est l'objectif de la section suivante. 
 
@@ -541,19 +563,22 @@ Si $\tau$ est un temps d'arrêt fort, alors $d(t)\leq \max_{X\in\Bool^{\mathsf{N
 \P_X(\tau > t)$.
 \end{theorem}
 
+
+Soit alors $\ov{h} : \Bool^{\mathsf{N}} \rightarrow \Bool^{\mathsf{N}}$ la fonction 
+telle que pour $X \in \Bool^{\mathsf{N}} $, 
+$(X,\ov{h}(X)) \in E$ et $X\oplus\ov{h}(X)=0^{{\mathsf{N}}-h(X)}10^{h(X)-1}$. 
+La fonction $\ov{h}$ est dite  {\it anti-involutive} si pour tout $X\in \Bool^{\mathsf{N}}$,
+$\ov{h}(\ov{h}(X))\neq X$. 
+
+
 \begin{theorem} \label{prop:stop}
-If $\ov{h}$ is bijective et telle que if for every $X\in \Bool^{\mathsf{N}}$,
+Si $\ov{h}$ is bijective et anti involutive 
 $\ov{h}(\ov{h}(X))\neq X$, alors
 $E[\ts]\leq 8{\mathsf{N}}^2+ 4{\mathsf{N}}\ln ({\mathsf{N}}+1)$. 
 \end{theorem}
 
-Sans entrer dans les détails de la preuve, on remarque tout d'abord 
-que le calcul 
-de cette borne n'intègre pas le fait qu'on préfère enlever des 
-chemins hamiltoniens équilibrés. 
-En intégrant cette contrainte, la borne supérieure pourrait être réduite.
-
-On remarque ensuite que la chaîne de Markov proposée ne suit pas exactement
+Les détails de la preuve sont donnés en annexes~\ref{anx:generateur}.
+On remarque tout d'abord que la chaîne de Markov proposée ne suit pas exactement
 l'algorithme~\ref{CI Algorithm}. En effet dans la section présente, 
 la probabilité de rester dans une configuration donnée 
 est fixée à $frac{1}{2}+\frac{1}{2n}$.
@@ -563,6 +588,67 @@ a été introduite pour simplifier le calcul de la borne sup
 du temps d'arrêt.   
 
 
+Sans entrer dans les détails de la preuve, on remarque aussi
+que le calcul  de cette borne impose uniquement que 
+pour chaque n{\oe}ud du $\mathsf{N}$-cube 
+un arc entrant et un arc sortant sont supprimés.
+Le fait qu'on enlève un cycle  hamiltonien et que ce dernier 
+soit équilibré n'est pas pris en compte.
+En intégrant cette contrainte, la borne supérieure pourrait être réduite.
+
+En effet, le temps de mixage est en $\Theta(N\ln N)$ lors d'une
+marche aléatoire classique paresseuse dans le $\mathsf{N}$-cube.
+On peut ainsi conjecturer que cet ordre de grandeur reste le même 
+dans le contexte du $\mathsf{N}$-cube privé d'un chemin hamiltonien.
+
+On peut évaluer ceci pratiquement: pour une fonction
+$f: \Bool^{\mathsf{N}} \rightarrow \Bool^{\mathsf{N}}$ et une graine initiale
+$x^0$, le code donné à l'algorithme algorithm~\ref{algo:stop} retourne le 
+nombre d'itérations suffisant tel que tous les éléments $\ell\in \llbracket 1,{\mathsf{N}} \rrbracket$ sont équitables. Il permet de déduire une approximation de $E[\ts]$
+en l'instanciant un grand nombre de fois: pour chaque nombre $\mathsf{N}$, 
+$ 3 \le \mathsf{N} \le 16$, 10 fonctionss ont été générées comme dans 
+ce chapitre. Pour chacune d'elle, le calcul d'une approximation de $E[\ts]$
+est exécuté 10000 fois avec une graine aléatoire.La Figure~\ref{fig:stopping:moy}
+résume ces resultats. Dans celle-ci, un cercle  représente une approximation de 
+$E[\ts]$ pour un  $\mathsf{N}$ donné tandis que la courbe est une représentation de 
+la fonction $x \mapsto 2x\ln(2x+8)$. 
+On  cosntate que l'approximation de $E[\ts]$ est largement inférieure 
+à la borne quadratique donnée au thérème~\ref{prop:stop} et que la conjecture 
+donnée au paragraphe précédent est sensée.
+
+
+\begin{algorithm}[ht]
+%\begin{scriptsize}
+\KwIn{a function $f$, an initial configuration $x^0$ ($\mathsf{N}$ bits)}
+\KwOut{a number of iterations $\textit{nbit}$}
+
+$\textit{nbit} \leftarrow 0$\;
+$x\leftarrow x^0$\;
+$\textit{fair}\leftarrow\emptyset$\;
+\While{$\left\vert{\textit{fair}}\right\vert < \mathsf{N} $}
+{
+        $ s \leftarrow \textit{Random}(\mathsf{N})$ \;
+        $\textit{image} \leftarrow f(x) $\;
+        \If{$\textit{Random}(1) \neq 0$ and $x[s] \neq \textit{image}[s]$}{
+            $\textit{fair} \leftarrow \textit{fair} \cup \{s\}$\;
+            $x[s] \leftarrow \textit{image}[s]$\;
+          }
+        $\textit{nbit} \leftarrow \textit{nbit}+1$\;
+}
+\Return{$\textit{nbit}$}\;
+%\end{scriptsize}
+\caption{Pseudo Code of stoping time calculus }
+\label{algo:stop}
+\end{algorithm}
+
+
+\begin{figure}
+\centering
+\includegraphics[width=0.49\textwidth]{images/complexityET}
+\caption{Average Stopping Time Approximation}\label{fig:stopping:moy}
+\end{figure}
+
+
 
 
 \section{Et les itérations généralisées?}
diff --git a/demandeInscription/synthese.tex b/demandeInscription/synthese.tex
index 5553fd9..03c44f7 100755
--- a/demandeInscription/synthese.tex
+++ b/demandeInscription/synthese.tex
@@ -1377,7 +1377,7 @@ Jacques Bahi, Jean-Fran\c{c}ois Couchot, and Christophe Guyeux.
 Jean-François Couchot, Sylvain Contassot-Vivier, Christophe Guyeux, and Pierre-Cyrille H\'eam.
 \newblock Random walk in a n-cube without hamiltonian cycle to chaotic
   pseudorandom number generation: Theoretical and practical considerations.
-\newblock in submission.
+\newblock  in submission to Elsevier Chaos, Solitons \& Fractal, August 2016.
 
 \bibitem{bgco16:onp}
 Mohammed Bakiri, Christophe Guyeux, Jean-Fran\c{c}cois Couchot, and
diff --git a/main.tex b/main.tex
index cacbb94..0a4591e 100644
--- a/main.tex
+++ b/main.tex
@@ -244,6 +244,10 @@ On montre qu'on a des résultats similaires.
 \chapter{Une démarches plus classique de dissimulation: STABYLO}
  \input{stabylo}
 
+\chapter{Une démarches plus classique de dissimulation: STABYLO}
+ \input{stabylo}
+
+
 
 \part{Conclusion et Perspectives}
 
@@ -332,6 +336,7 @@ du chapitre 8}
 
 \chapter{Preuves sur les générateurs de nombres pseudo-aléatoires}\label{anx:generateur}
 \input{annexePreuveDistribution}
+\input{annexePreuveGrayEquilibre}
 \input{annexePreuveStopping}
 
 \chapter{Preuves sur le marquage de média}\label{anx:marquage}
-- 
2.39.5


From 322d0e3286e2bc9cae4178aa7488b5631c57a8ba Mon Sep 17 00:00:00 2001
From: couchot <couchot@localhost.localdomain>
Date: Fri, 26 Aug 2016 16:59:12 +0200
Subject: [PATCH 07/16] secrypt 16

---
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 images/38_dp_diff-eps-converted-to.pdf        |  Bin 0 -> 26728 bytes
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 images/38_poly-eps-converted-to.pdf           |  Bin 0 -> 76131 bytes
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 ...03\251_de_Franche-Comt\303\251_(logo).svg" |  636 +++
 images/complexity-eps-converted-to.pdf        |  Bin 0 -> 26292 bytes
 images/complexityET-eps-converted-to.pdf      |  Bin 0 -> 15091 bytes
 images/complexityET.eps                       | 1324 +++++
 images/dec-eps-converted-to.pdf               |  Bin 0 -> 9053 bytes
 images/emb-eps-converted-to.pdf               |  Bin 0 -> 10598 bytes
 images/h3prng-eps-converted-to.pdf            |  Bin 0 -> 5394 bytes
 main.tex                                      |   22 +-
 stegoyousra.tex                               |  954 ++++
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840.982,161.957 840.984,161.957 840.986,161.955 C 840.986,161.952 840.984,161.951 840.982,161.949 L 840.987,161.944 C 840.979,161.939 840.986,161.934 840.98,161.927 C 840.991,161.92 840.975,161.919 840.979,161.913 C 840.965,161.905 840.982,161.898 840.982,161.893 L 840.959,161.883 C 840.972,161.879 840.968,161.859 840.98,161.853 C 840.975,161.848 840.982,161.837 840.972,161.836 L 840.977,161.82 C 840.975,161.808 840.966,161.804 840.962,161.792 C 840.968,161.789 840.962,161.782 840.972,161.784 C 840.973,161.778 840.975,161.77 840.969,161.765 C 840.982,161.753 840.952,161.748 840.968,161.734 C 840.98,161.71 840.938,161.72 840.946,161.699 C 840.948,161.698 840.953,161.695 840.949,161.693 C 840.965,161.696 840.961,161.675 840.968,161.67 C 840.965,161.665 840.969,161.658 840.972,161.651 C 840.961,161.644 840.98,161.636 840.972,161.628 C 840.968,161.631 840.966,161.634 840.966,161.636 C 840.959,161.639 840.941,161.626 840.935,161.636 C 840.933,161.634 840.931,161.631 840.928,161.626 C 840.938,161.621 840.944,161.615 840.955,161.618 C 840.959,161.615 840.973,161.613 840.965,161.605 C 840.973,161.601 840.961,161.597 840.965,161.589 C 840.952,161.578 840.977,161.572 840.959,161.562 C 840.953,161.551 840.962,161.537 840.955,161.532 C 840.958,161.521 840.946,161.517 840.948,161.505 C 840.934,161.501 840.949,161.487 840.942,161.48 L 840.939,161.48 C 840.942,161.478 840.942,161.473 840.939,161.47 C 840.92,161.463 840.946,161.442 840.934,161.428 L 840.939,161.423 C 840.938,161.421 840.935,161.419 840.934,161.418 L 840.939,161.413 C 840.933,161.41 840.939,161.403 840.935,161.396 C 840.931,161.391 840.938,161.378 840.925,161.376 C 840.927,161.366 840.934,161.361 840.933,161.351 C 840.916,161.344 840.933,161.327 840.931,161.315 C 840.916,161.308 840.928,161.295 840.92,161.286 C 840.928,161.282 840.92,161.276 840.927,161.271 C 840.921,161.263 840.928,161.255 840.928,161.247 C 840.927,161.243 840.925,161.247 840.921,161.245 L 840.921,161.242 L 840.924,161.242 C 840.921,161.234 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840.959,160.807 C 840.975,160.789 840.939,160.776 840.953,160.756 C 840.962,160.758 840.962,160.751 840.966,160.745 C 840.965,160.742 840.959,160.743 840.958,160.739 C 840.958,160.729 840.966,160.727 840.969,160.721 C 840.987,160.721 840.986,160.695 841.001,160.688 C 841.006,160.66 841.018,160.634 841.006,160.609 C 841.01,160.606 841.018,160.602 841.017,160.594 L 841.025,160.591 C 841.018,160.581 841.014,160.575 841.006,160.571 L 841.005,160.571 L 841.005,160.568 C 840.991,160.563 841.004,160.546 840.994,160.538 C 841.001,160.538 841.006,160.538 841.011,160.535 C 841.017,160.532 841.011,160.519 841.007,160.522 C 841.005,160.512 841.02,160.511 841.013,160.504 C 841.02,160.504 841.025,160.491 841.034,160.489 C 841.041,160.488 841.034,160.476 841.043,160.476 C 841.041,160.447 841.077,160.474 841.091,160.465 C 841.105,160.471 841.106,160.455 841.115,160.45 C 841.132,160.459 841.152,160.442 841.162,160.458 L 841.165,160.454 C 841.172,160.455 841.188,160.465 841.198,160.453 C 841.203,160.454 841.209,160.455 841.209,160.462 C 841.217,160.468 841.222,160.459 841.229,160.459 C 841.242,160.466 841.247,160.466 841.255,160.473 C 841.276,160.483 841.295,160.492 841.317,160.496 C 841.313,160.5 841.322,160.506 841.329,160.506 C 841.332,160.496 841.347,160.502 841.347,160.494 C 841.35,160.491 841.353,160.498 841.356,160.496 L 841.355,160.502 C 841.356,160.506 841.353,160.514 841.359,160.514 C 841.366,160.512 841.363,160.511 841.366,160.506 C 841.376,160.512 841.388,160.498 841.398,160.514 C 841.418,160.528 841.453,160.514 841.468,160.54 C 841.478,160.546 841.487,160.553 841.498,160.55 C 841.507,160.545 841.518,160.532 841.526,160.532 C 841.532,160.535 841.544,160.532 841.544,160.542 C 841.549,160.545 841.556,160.55 841.563,160.546 C 841.563,160.53 841.577,160.535 841.584,160.525 C 841.597,160.523 841.587,160.538 841.598,160.537 L 841.603,160.522 C 841.605,160.522 841.607,160.525 841.611,160.525 C 841.613,160.533 841.615,160.54 841.611,160.548 C 841.617,160.571 841.641,160.568 841.656,160.58 L 841.659,160.577 L 841.659,160.583 L 841.666,160.583 C 841.67,160.581 841.679,160.571 841.685,160.574 C 841.678,160.564 841.691,160.561 841.685,160.555 C 841.695,160.556 841.727,160.555 841.715,160.579 C 841.722,160.587 841.732,160.587 841.742,160.585 C 841.743,160.575 841.753,160.583 841.76,160.583 C 841.774,160.594 841.786,160.583 841.797,160.575 C 841.802,160.571 841.81,160.574 841.811,160.579 C 841.81,160.585 841.813,160.599 841.811,160.602 C 841.815,160.605 841.815,160.607 841.819,160.607 C 841.826,160.609 841.834,160.609 841.835,160.602 L 841.837,160.606 L 841.847,160.597 C 841.849,160.602 841.856,160.597 841.856,160.602 C 841.872,160.606 841.866,160.626 841.882,160.623 C 841.889,160.62 841.903,160.625 841.908,160.615 C 841.92,160.613 841.933,160.606 841.945,160.606 C 841.949,160.611 841.96,160.62 841.968,160.613 C 841.975,160.613 841.986,160.609 841.988,160.62 L 841.991,160.616 L 841.992,160.618 L 841.99,160.62 L 841.995,160.623 C 841.999,160.621 842.016,160.623 842.024,160.618 L 842.021,160.615 C 842.033,160.62 842.039,160.611 842.044,160.602 C 842.053,160.621 842.072,160.623 842.09,160.626 C 842.092,160.623 842.106,160.626 842.112,160.618 C 842.116,160.628 842.12,160.615 842.126,160.618 C 842.14,160.609 842.159,160.623 842.177,160.623 C 842.177,160.606 842.192,160.625 842.192,160.609 C 842.205,160.616 842.205,160.597 842.217,160.602 C 842.229,160.602 842.221,160.615 842.228,160.62 C 842.234,160.616 842.237,160.623 842.244,160.621 C 842.258,160.62 842.28,160.625 842.291,160.618 C 842.295,160.613 842.306,160.621 842.303,160.611 C 842.316,160.613 842.314,160.597 842.321,160.594 C 842.327,160.594 842.331,160.597 842.331,160.605 C 842.34,160.611 842.355,160.606 842.361,160.609 C 842.372,160.606 842.375,160.609 842.388,160.609 C 842.403,160.597 842.418,160.621 842.438,160.613 C 842.445,160.606 842.451,160.615 842.455,160.611 C 842.455,160.616 842.461,160.62 842.466,160.623 C 842.473,160.625 842.487,160.626 842.494,160.62 C 842.497,160.626 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843.186,160.641 843.209,160.641 C 843.215,160.643 843.212,160.632 843.218,160.639 C 843.229,160.628 843.244,160.634 843.26,160.637 C 843.27,160.632 843.284,160.632 843.288,160.618 L 843.294,160.62 C 843.302,160.616 843.301,160.628 843.308,160.626 C 843.317,160.62 843.335,160.628 843.349,160.623 C 843.359,160.623 843.376,160.625 843.387,160.626 C 843.392,160.597 843.423,160.628 843.44,160.62 C 843.453,160.625 843.461,160.605 843.474,160.615 C 843.487,160.609 843.48,160.615 843.492,160.609 C 843.501,160.613 843.518,160.611 843.517,160.625 C 843.53,160.632 843.547,160.625 843.557,160.625 C 843.56,160.611 843.572,160.623 843.577,160.623 C 843.585,160.644 843.605,160.621 843.616,160.636 C 843.616,160.651 843.638,160.643 843.644,160.652 C 843.668,160.646 843.668,160.677 843.692,160.675 C 843.698,160.669 843.704,160.664 843.712,160.66 C 843.712,160.661 843.717,160.666 843.718,160.663 C 843.719,160.67 843.734,160.671 843.732,160.682 C 843.764,160.692 843.791,160.705 843.827,160.708 C 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C 844.268,161.338 844.256,161.355 844.256,161.378 C 844.249,161.405 844.264,161.431 844.25,161.46 C 844.248,161.47 844.251,161.478 844.249,161.488 C 844.249,161.494 844.254,161.499 844.258,161.494 C 844.271,161.507 844.248,161.52 844.248,161.537 C 844.245,161.545 844.261,161.554 844.248,161.558 C 844.237,161.57 844.257,161.572 844.267,161.575 C 844.27,161.584 844.258,161.589 844.256,161.592 C 844.251,161.601 844.244,161.613 844.251,161.621 C 844.251,161.632 844.248,161.644 844.256,161.654 C 844.25,161.667 844.275,161.663 844.268,161.678 C 844.261,161.673 844.258,161.684 844.254,161.686 C 844.256,161.699 844.249,161.714 844.25,161.73 C 844.257,161.73 844.251,161.737 844.256,161.739 C 844.262,161.747 844.267,161.753 844.27,161.76 L 844.289,161.782 C 844.283,161.786 844.278,161.789 844.278,161.796 C 844.272,161.799 844.268,161.802 844.27,161.808 C 844.289,161.822 844.258,161.834 844.261,161.848 C 844.254,161.86 844.272,161.859 844.27,161.872 C 844.28,161.885 844.268,161.9 844.268,161.915 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844.085,162.126 844.085,162.11 C 844.083,162.11 844.081,162.105 844.079,162.105 C 844.081,162.102 844.081,162.099 844.081,162.095 C 844.068,162.095 844.079,162.081 844.065,162.081 C 844.053,162.062 844.067,162.036 844.071,162.017 C 844.055,162.014 844.071,161.997 844.061,161.989 C 844.071,161.986 844.058,161.979 844.065,161.975 C 844.051,161.97 844.061,161.955 844.052,161.951 C 844.052,161.941 844.061,161.927 844.051,161.924 C 844.048,161.905 844.067,161.888 844.045,161.874 C 844.052,161.868 844.058,161.859 844.057,161.851 C 844.051,161.848 844.04,161.848 844.032,161.841 C 844.031,161.828 844.047,161.81 844.032,161.802 C 844.038,161.795 844.051,161.789 844.045,161.779 C 844.038,161.779 844.038,161.77 844.037,161.765 C 844.038,161.76 844.051,161.763 844.052,161.755 L 844.045,161.748 C 844.045,161.733 844.061,161.715 844.045,161.706 C 844.048,161.698 844.034,161.696 844.04,161.687 C 844.044,161.68 844.052,161.673 844.047,161.663 C 844.037,161.659 844.045,161.648 844.034,161.646 C 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842.379,160.811 842.375,160.813 842.372,160.811 L 842.369,160.815 C 842.374,160.815 842.369,160.816 842.369,160.82 C 842.362,160.815 842.362,160.83 842.354,160.823 C 842.341,160.818 842.327,160.827 842.314,160.811 C 842.31,160.818 842.306,160.808 842.3,160.811 C 842.291,160.808 842.284,160.804 842.278,160.801 C 842.273,160.807 842.264,160.803 842.261,160.81 C 842.243,160.807 842.227,160.8 842.21,160.815 C 842.198,160.813 842.196,160.799 842.183,160.801 C 842.188,160.81 842.171,160.808 842.169,160.811 C 842.151,160.813 842.136,160.801 842.12,160.811 C 842.106,160.807 842.09,160.807 842.078,160.811 C 842.076,160.82 842.069,160.823 842.064,160.826 C 842.058,160.826 842.051,160.823 842.05,160.818 C 842.056,160.815 842.058,160.804 842.05,160.801 C 842.04,160.8 842.031,160.801 842.02,160.807 C 842.008,160.807 841.995,160.8 841.99,160.791 C 841.986,160.791 841.986,160.795 841.981,160.796 C 841.979,160.789 841.974,160.795 841.971,160.796 C 841.967,160.807 841.953,160.796 841.949,160.804 C 841.947,160.796 841.944,160.807 841.94,160.803 C 841.923,160.791 841.911,160.815 841.894,160.803 C 841.879,160.813 841.854,160.807 841.835,160.807 C 841.819,160.796 841.808,160.82 841.791,160.803 C 841.775,160.801 841.764,160.781 841.743,160.784 C 841.725,160.8 841.703,160.768 841.684,160.786 C 841.678,160.786 841.671,160.784 841.671,160.791 C 841.653,160.777 841.637,160.803 841.618,160.786 C 841.613,160.786 841.613,160.781 841.611,160.777 C 841.587,160.772 841.561,160.756 841.537,160.765 C 841.527,160.763 841.519,160.761 841.511,160.763 C 841.5,160.795 841.471,160.758 841.449,160.768 C 841.446,160.765 841.44,160.758 841.433,160.755 C 841.429,160.753 841.428,160.761 841.426,160.755 C 841.429,160.742 841.407,160.747 841.4,160.739 C 841.391,160.734 841.383,160.742 841.374,160.742 L 841.369,160.735 C 841.362,160.747 841.35,160.734 841.342,160.732 L 841.34,160.729 C 841.333,160.727 841.332,160.729 841.322,160.727 C 841.31,160.729 841.304,160.705 841.297,160.721 C 841.287,160.721 841.287,160.73 841.273,160.729 C 841.262,160.732 841.251,160.743 841.24,160.745 C 841.232,160.736 841.228,160.732 841.215,160.732 C 841.218,160.739 841.207,160.743 841.207,160.747 C 841.201,160.747 841.201,160.758 841.191,160.758 C 841.186,160.763 841.186,160.776 841.174,160.773 C 841.172,160.778 841.176,160.784 841.17,160.789 C 841.179,160.791 841.176,160.803 841.184,160.807 C 841.177,160.81 841.174,160.822 841.163,160.815 L 841.162,160.816 C 841.157,160.829 841.174,160.83 841.163,160.843 C 841.155,160.841 841.152,160.849 841.152,160.856 C 841.158,160.877 841.129,160.865 841.143,160.886 C 841.136,160.897 841.129,160.91 841.125,160.924 L 841.129,160.933 C 841.12,160.934 841.132,160.942 841.125,160.944 L 841.129,160.962 C 841.124,160.965 841.118,160.974 841.12,160.984 C 841.132,160.989 841.129,161.007 841.138,161.012 C 841.129,161.028 841.163,161.051 841.129,161.055 C 841.129,161.063 841.122,161.067 841.125,161.074 L 841.132,161.082 C 841.125,161.089 841.13,161.097 841.124,161.102 C 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841.15,161.476 C 841.142,161.48 841.142,161.492 841.142,161.505 C 841.143,161.511 841.152,161.521 841.149,161.53 C 841.157,161.542 841.152,161.557 841.158,161.566 C 841.157,161.572 841.146,161.57 841.149,161.578 C 841.165,161.585 841.146,161.601 841.157,161.608 C 841.155,161.631 841.165,161.636 841.157,161.657 C 841.158,161.661 841.163,161.667 841.169,161.668 C 841.177,161.684 841.159,161.699 841.169,161.712 C 841.172,161.716 841.162,161.722 841.165,161.73 L 841.162,161.733 C 841.166,161.748 841.186,161.76 841.174,161.779 C 841.19,161.779 841.174,161.789 841.176,161.795 C 841.179,161.805 841.166,161.822 841.179,161.833 C 841.17,161.851 841.191,161.851 841.201,161.857 C 841.201,161.866 841.204,161.878 841.201,161.883 C 841.202,161.889 841.19,161.893 841.191,161.903 C 841.195,161.902 841.195,161.905 841.195,161.906 C 841.191,161.917 841.191,161.927 841.191,161.941 C 841.198,161.949 841.207,161.954 841.207,161.965 C 841.212,161.97 841.21,161.981 841.204,161.989 C 841.203,161.999 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841.152,162.587 C 841.169,162.598 841.146,162.616 841.163,162.623 C 841.163,162.634 841.157,162.647 841.166,162.655 C 841.165,162.66 841.169,162.662 841.172,162.665 C 841.17,162.667 841.169,162.668 841.166,162.667 C 841.169,162.68 841.162,162.695 841.172,162.704 C 841.165,162.714 841.155,162.72 841.155,162.732 C 841.143,162.729 841.158,162.745 841.146,162.742 C 841.143,162.751 841.152,162.755 841.15,162.766 L 841.152,162.768 C 841.15,162.775 841.152,162.789 841.146,162.797 C 841.138,162.798 841.149,162.805 841.144,162.812 C 841.142,162.818 841.135,162.827 841.139,162.836 C 841.135,162.84 841.139,162.844 841.143,162.846 C 841.142,162.848 841.139,162.848 841.138,162.848 C 841.146,162.852 841.136,162.863 841.146,162.867 C 841.146,162.871 841.138,162.872 841.143,162.877 C 841.138,162.881 841.143,162.889 841.132,162.889 C 841.129,162.907 841.113,162.926 841.128,162.945 C 841.124,162.947 841.125,162.953 841.125,162.957 C 841.129,162.96 841.143,162.96 841.139,162.969 C 841.142,162.974 841.136,162.976 841.132,162.976 C 841.125,162.987 841.12,163.003 841.129,163.011 C 841.125,163.015 841.13,163.025 841.135,163.025 C 841.129,163.036 841.138,163.053 841.128,163.061 C 841.124,163.072 841.111,163.084 841.125,163.095 C 841.128,163.105 841.115,163.113 841.117,163.12 C 841.1,163.122 841.115,163.141 841.106,163.15 C 841.111,163.159 841.104,163.174 841.11,163.185 C 841.106,163.19 841.113,163.195 841.118,163.199 C 841.117,163.209 841.111,163.213 841.111,163.225 C 841.12,163.227 841.125,163.231 841.128,163.241 C 841.125,163.247 841.128,163.265 841.113,163.263 L 841.105,163.271 C 841.107,163.276 841.106,163.284 841.113,163.289 C 841.111,163.299 841.11,163.307 841.1,163.31 L 841.1,163.315 C 841.1,163.315 841.1,163.311 841.105,163.313 C 841.1,163.318 841.107,163.325 841.113,163.326 C 841.113,163.339 841.106,163.349 841.1,163.359 C 841.11,163.363 841.113,163.375 841.11,163.387 C 841.1,163.399 841.1,163.416 841.096,163.431 C 841.093,163.444 841.079,163.463 841.094,163.47 C 841.093,163.475 841.096,163.478 841.1,163.481 C 841.096,163.487 841.09,163.487 841.087,163.494 C 841.09,163.504 841.094,163.507 841.1,163.512 L 841.094,163.527 C 841.1,163.542 841.084,163.556 841.1,163.572 C 841.11,163.567 841.106,163.58 841.111,163.584 C 841.096,163.593 841.105,163.607 841.107,163.619 L 841.113,163.623 C 841.106,163.628 841.104,163.634 841.093,163.632 C 841.091,163.634 841.1,163.639 841.105,163.639 C 841.11,163.646 841.105,163.653 841.107,163.66 C 841.11,163.665 841.111,163.676 841.115,163.683 C 841.117,163.683 841.113,163.696 841.12,163.695 C 841.117,163.714 841.124,163.737 841.12,163.76 C 841.125,163.768 841.122,163.777 841.125,163.786 C 841.12,163.789 841.12,163.794 841.124,163.797 C 841.118,163.798 841.12,163.801 841.122,163.805 C 841.118,163.812 841.124,163.823 841.115,163.831 C 841.118,163.841 841.11,163.853 841.115,163.864 C 841.115,163.867 841.111,163.867 841.11,163.869 C 841.118,163.874 841.106,163.879 841.113,163.884 C 841.111,163.89 841.107,163.898 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843.085,161.44 843.086,161.44 843.09,161.438 C 843.097,161.44 843.105,161.447 843.111,161.444 C 843.112,161.445 843.116,161.449 843.118,161.445 L 843.119,161.444 C 843.129,161.452 843.139,161.452 843.153,161.457 C 843.162,161.445 843.178,161.454 843.19,161.457 C 843.197,161.463 843.198,161.476 843.205,161.483 C 843.221,161.466 843.235,161.483 843.249,161.488 C 843.254,161.498 843.244,161.505 843.252,161.511 C 843.268,161.514 843.277,161.528 843.29,161.534 C 843.302,161.545 843.288,161.557 843.302,161.562 C 843.288,161.577 843.32,161.578 843.304,161.594 C 843.301,161.608 843.321,161.597 843.32,161.61 C 843.329,161.618 843.335,161.631 843.329,161.646 C 843.333,161.651 843.324,161.658 843.328,161.661 C 843.333,161.659 843.346,161.654 843.35,161.661 C 843.354,161.672 843.349,161.686 843.365,161.684 C 843.375,161.695 843.354,161.715 843.376,161.722 C 843.385,161.737 843.353,161.753 843.38,161.763 L 843.38,161.765 C 843.385,161.763 843.385,161.768 843.387,161.77 C 843.387,161.778 843.381,161.787 843.389,161.789 C 843.392,161.799 843.387,161.81 843.385,161.817 C 843.397,161.833 843.381,161.851 843.383,161.863 C 843.394,161.875 843.37,161.871 843.377,161.883 L 843.388,161.893 C 843.373,161.898 843.367,161.915 843.361,161.929 C 843.37,161.941 843.38,161.957 843.37,161.973 C 843.388,161.983 843.367,162.001 843.369,162.017 C 843.361,162.014 843.359,162.024 843.354,162.03 L 843.356,162.041 C 843.37,162.036 843.357,162.052 843.361,162.059 L 843.359,162.059 C 843.37,162.06 843.362,162.076 843.365,162.084 C 843.35,162.097 843.361,162.114 843.35,162.128 C 843.35,162.133 843.349,162.138 843.342,162.141 C 843.335,162.141 843.333,162.153 843.329,162.162 C 843.328,162.166 843.328,162.172 843.333,162.175 L 843.335,162.176 C 843.32,162.172 843.326,162.193 843.315,162.198 C 843.311,162.201 843.321,162.216 843.308,162.214 C 843.297,162.229 843.317,162.221 843.313,162.235 C 843.315,162.237 843.317,162.243 843.315,162.247 C 843.291,162.271 843.301,162.307 843.282,162.335 C 843.265,162.335 843.272,162.359 843.257,162.361 C 843.254,162.371 843.249,162.38 843.252,162.39 C 843.254,162.388 843.26,162.39 843.257,162.394 C 843.26,162.413 843.235,162.411 843.241,162.43 L 843.228,162.442 C 843.235,162.447 843.231,162.456 843.228,162.464 C 843.218,162.47 843.209,162.46 843.203,162.472 C 843.195,162.47 843.19,162.478 843.183,162.48 C 843.18,162.486 843.179,162.491 843.179,162.499 C 843.195,162.501 843.179,162.515 843.184,162.521 C 843.178,162.525 843.184,162.532 843.179,162.537 C 843.188,162.547 843.171,162.546 843.169,162.551 C 843.162,162.547 843.164,162.558 843.157,162.558 C 843.149,162.563 843.139,162.553 843.142,162.567 C 843.152,162.575 843.138,162.585 843.139,162.597 C 843.135,162.594 843.127,162.603 843.12,162.602 C 843.118,162.616 843.111,162.625 843.097,162.631 C 843.085,162.632 843.07,162.634 843.06,162.646 C 843.07,162.655 843.063,162.665 843.058,162.675 C 843.046,162.687 843.032,162.664 843.024,162.683 C 843.014,162.68 843.019,162.692 843.012,162.695 L 843.007,162.691 C 843.001,162.694 842.992,162.695 842.986,162.694 C 842.987,162.698 842.982,162.702 842.979,162.699 C 842.973,162.699 842.972,162.698 842.971,162.702 C 842.969,162.692 842.955,162.698 842.959,162.687 C 842.946,162.685 842.935,162.714 842.928,162.687 C 842.915,162.695 842.907,162.672 842.892,162.683 C 842.882,162.684 842.883,162.695 842.874,162.691 C 842.875,162.677 842.859,162.683 842.849,162.677 L 842.848,162.691 C 842.844,162.692 842.839,162.692 842.837,162.687 C 842.831,162.692 842.817,162.681 842.807,162.685 C 842.808,162.694 842.8,162.704 842.796,162.714 C 842.787,162.704 842.787,162.724 842.778,162.717 C 842.775,162.72 842.775,162.726 842.77,162.727 L 842.767,162.726 C 842.769,162.729 842.765,162.73 842.761,162.732 C 842.755,162.729 842.748,162.729 842.745,162.72 C 842.734,162.72 842.721,162.704 842.714,162.717 C 842.702,162.714 842.708,162.73 842.697,162.732 C 842.679,162.736 842.682,162.719 842.67,162.714 C 842.659,162.711 842.65,162.717 842.644,162.726 L 842.644,162.727 C 842.63,162.732 842.625,162.722 842.617,162.714 C 842.611,162.719 842.61,162.704 842.604,162.714 L 842.6,162.711 C 842.592,162.722 842.579,162.707 842.57,162.714 L 842.568,162.711 C 842.568,162.714 842.566,162.714 842.565,162.716 L 842.557,162.708 L 842.556,162.71 L 842.552,162.704 C 842.544,162.699 842.531,162.702 842.524,162.694 C 842.53,162.687 842.53,162.675 842.524,162.67 C 842.514,162.668 842.51,162.662 842.506,162.655 C 842.488,162.644 842.465,162.647 842.452,162.631 C 842.443,162.623 842.445,162.606 842.433,162.598 C 842.429,162.581 842.441,162.565 842.445,162.55 C 842.445,162.543 842.433,162.537 842.44,162.529 C 842.433,162.516 842.421,162.53 842.413,162.525 C 842.409,162.508 842.389,162.496 842.402,162.478 C 842.398,162.48 842.393,162.473 842.392,162.47 L 842.393,162.468 L 842.388,162.464 C 842.395,162.456 842.395,162.453 842.4,162.447 C 842.398,162.432 842.366,162.421 842.381,162.406 C 842.374,162.4 842.382,162.382 842.374,162.38 C 842.369,162.373 842.372,162.366 842.375,162.359 L 842.385,162.357 C 842.385,162.351 842.379,162.349 842.38,162.343 C 842.375,162.335 842.355,162.341 842.362,162.326 L 842.359,162.322 C 842.368,162.322 842.362,162.315 842.368,162.311 C 842.365,162.302 842.365,162.301 842.369,162.294 C 842.375,162.294 842.374,162.289 842.375,162.285 C 842.361,162.274 842.374,162.262 842.366,162.248 C 842.366,162.229 842.392,162.233 842.385,162.211 C 842.388,162.204 842.388,162.193 842.398,162.191 C 842.411,162.199 842.411,162.185 842.421,162.185 C 842.42,162.178 842.428,162.176 842.431,162.178 C 842.429,162.162 842.452,162.159 842.447,162.143 C 842.461,162.134 842.465,162.123 842.48,162.114 C 842.49,162.121 842.494,162.105 842.504,162.107 C 842.504,162.092 842.521,162.097 842.524,162.084 C 842.534,162.09 842.53,162.072 842.54,162.076 C 842.545,162.067 842.554,162.067 842.565,162.066 C 842.572,162.046 842.594,162.076 842.603,162.056 C 842.607,162.052 842.618,162.052 842.625,162.045 L 842.63,162.052 C 842.633,162.046 842.64,162.046 842.648,162.045 L 842.651,162.049 L 842.651,162.043 C 842.653,162.041 842.663,162.046 842.659,162.05 C 842.658,162.049 842.658,162.046 842.659,162.046 C 842.666,162.046 842.664,162.058 842.673,162.056 C 842.682,162.069 842.697,162.083 842.714,162.083 C 842.722,162.087 842.724,162.095 842.728,162.099 C 842.73,162.1 842.731,162.102 842.734,162.105 C 842.731,162.107 842.73,162.107 842.727,162.107 L 842.725,162.105 C 842.71,162.11 842.727,162.119 842.728,162.126 L 842.743,162.138 C 842.737,162.145 842.727,162.149 842.728,162.159 L 842.734,162.162 C 842.734,162.17 842.734,162.183 842.724,162.185 C 842.722,162.202 842.699,162.199 842.692,162.213 C 842.679,162.209 842.683,162.229 842.672,162.221 C 842.672,162.229 842.659,162.229 842.652,162.229 L 842.651,162.226 C 842.648,162.24 842.63,162.224 842.625,162.233 C 842.622,162.226 842.618,162.235 842.613,162.237 C 842.611,162.237 842.61,162.235 842.61,162.235 C 842.611,162.24 842.605,162.242 842.603,162.242 C 842.592,162.24 842.585,162.247 842.577,162.247 L 842.563,162.254 C 842.568,162.258 842.57,162.262 842.566,162.266 C 842.556,162.268 842.559,162.274 842.554,162.281 C 842.561,162.286 842.559,162.297 842.57,162.301 L 842.568,162.302 C 842.579,162.305 842.57,162.324 842.577,162.33 L 842.577,162.357 L 842.579,162.354 C 842.581,162.362 842.583,162.371 842.596,162.375 C 842.59,162.385 842.605,162.39 842.607,162.401 C 842.61,162.401 842.611,162.404 842.613,162.406 C 842.615,162.401 842.62,162.4 842.625,162.403 C 842.631,162.408 842.622,162.421 842.63,162.423 C 842.636,162.432 842.623,162.445 842.64,162.447 C 842.644,162.447 842.648,162.444 842.651,162.449 C 842.653,162.438 842.666,162.44 842.672,162.434 C 842.678,162.435 842.679,162.423 842.69,162.426 C 842.705,162.411 842.727,162.44 842.745,162.426 C 842.755,162.434 842.761,162.421 842.77,162.418 C 842.767,162.416 842.76,162.404 842.773,162.403 C 842.783,162.406 842.785,162.394 842.797,162.395 C 842.8,162.385 842.81,162.394 842.81,162.385 C 842.826,162.377 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842.896,161.962 842.895,161.944 842.882,161.936 C 842.879,161.915 842.86,161.898 842.856,161.875 L 842.859,161.874 C 842.852,161.868 842.837,161.868 842.835,161.874 C 842.817,161.878 842.81,161.867 842.791,161.863 C 842.776,161.857 842.763,161.851 842.758,161.836 C 842.756,161.834 842.752,161.834 842.752,161.83 C 842.763,161.834 842.76,161.82 842.763,161.817 C 842.755,161.81 842.741,161.811 842.731,161.81 C 842.725,161.822 842.714,161.814 842.708,161.828 C 842.696,161.824 842.686,161.845 842.679,161.828 C 842.67,161.834 842.662,161.83 842.651,161.825 C 842.639,161.828 842.633,161.829 842.63,161.84 C 842.622,161.841 842.617,161.851 842.607,161.843 C 842.592,161.843 842.586,161.848 842.576,161.853 C 842.568,161.857 842.556,161.846 842.556,161.859 C 842.544,161.853 842.547,161.871 842.537,161.867 C 842.524,161.879 842.507,161.866 842.492,161.866 C 842.484,161.867 842.479,161.855 842.468,161.859 C 842.469,161.871 842.452,161.868 842.448,161.878 C 842.445,161.872 842.438,161.874 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844.217,155.15 844.232,155.158 844.237,155.168 L 844.229,155.177 C 844.238,155.189 844.238,155.197 844.245,155.211 C 844.248,155.223 844.232,155.225 844.229,155.235 C 844.235,155.239 844.237,155.25 844.245,155.247 C 844.249,155.252 844.265,155.252 844.261,155.264 C 844.249,155.262 844.254,155.282 844.243,155.278 C 844.232,155.283 844.235,155.292 844.231,155.301 C 844.232,155.304 844.232,155.308 844.238,155.308 C 844.232,155.325 844.243,155.33 844.249,155.342 C 844.243,155.345 844.243,155.351 844.238,155.354 C 844.231,155.367 844.241,155.378 844.249,155.385 C 844.255,155.397 844.239,155.408 844.249,155.42 C 844.245,155.434 844.258,155.448 844.249,155.463 C 844.257,155.481 844.245,155.506 844.264,155.516 C 844.264,155.525 844.272,155.525 844.271,155.537 C 844.268,155.546 844.261,155.544 844.249,155.546 C 844.249,155.553 844.249,155.562 844.257,155.565 L 844.261,155.564 C 844.261,155.565 844.257,155.57 844.261,155.574 C 844.268,155.574 844.269,155.583 844.276,155.579 C 844.284,155.589 844.275,155.596 844.281,155.606 C 844.295,155.61 844.284,155.622 844.287,155.629 C 844.275,155.644 844.281,155.667 844.283,155.682 C 844.292,155.691 844.29,155.706 844.302,155.714 C 844.303,155.72 844.298,155.724 844.298,155.729 C 844.307,155.734 844.298,155.741 844.313,155.744 C 844.327,155.755 844.314,155.761 844.307,155.77 C 844.322,155.773 844.309,155.791 844.318,155.799 C 844.317,155.805 844.31,155.807 844.305,155.807 C 844.307,155.815 844.302,155.831 844.307,155.838 C 844.305,155.845 844.303,155.857 844.305,155.864 C 844.314,155.862 844.31,155.871 844.314,155.874 C 844.29,155.886 844.317,155.902 844.318,155.917 C 844.318,155.929 844.302,155.926 844.305,155.938 C 844.317,155.941 844.305,155.952 844.307,155.959 C 844.302,155.976 844.328,155.971 844.328,155.987 C 844.328,155.997 844.322,156.007 844.315,156.01 L 844.321,156.014 C 844.321,156.021 844.321,156.031 844.313,156.035 C 844.292,156.04 844.313,156.062 844.305,156.078 C 844.314,156.078 844.307,156.093 844.317,156.095 C 844.313,156.097 844.318,156.109 844.307,156.107 C 844.302,156.114 844.303,156.123 844.307,156.13 C 844.302,156.147 844.298,156.166 844.309,156.185 C 844.329,156.197 844.317,156.213 844.313,156.228 C 844.307,156.24 844.327,156.242 844.318,156.255 C 844.297,156.276 844.321,156.296 844.315,156.321 C 844.317,156.332 844.315,156.349 844.31,156.361 C 844.31,156.371 844.314,156.381 844.321,156.387 C 844.323,156.397 844.31,156.39 844.313,156.4 C 844.321,156.403 844.313,156.415 844.322,156.415 C 844.321,156.421 844.322,156.437 844.309,156.439 C 844.307,156.445 844.307,156.451 844.313,156.454 C 844.298,156.465 844.314,156.48 844.303,156.493 C 844.302,156.514 844.31,156.534 844.327,156.546 C 844.317,156.549 844.317,156.56 844.313,156.569 C 844.31,156.586 844.328,156.591 844.331,156.609 L 844.313,156.616 C 844.314,156.622 844.305,156.629 844.314,156.632 C 844.315,156.641 844.322,156.653 844.323,156.662 C 844.321,156.667 844.327,156.677 844.317,156.677 C 844.327,156.683 844.314,156.694 844.315,156.702 C 844.315,156.707 844.314,156.712 844.309,156.713 C 844.313,156.719 844.302,156.724 844.309,156.729 C 844.298,156.746 844.322,156.755 844.321,156.773 C 844.31,156.79 844.309,156.808 844.314,156.827 C 844.315,156.835 844.321,156.84 844.317,156.845 C 844.317,156.851 844.322,156.851 844.327,156.852 z M 844.094,156.344 L 844.097,156.342 L 844.084,156.332 C 844.087,156.328 844.092,156.325 844.091,156.32 C 844.084,156.317 844.086,156.308 844.087,156.301 C 844.092,156.299 844.09,156.292 844.09,156.289 C 844.087,156.287 844.086,156.285 844.084,156.283 C 844.094,156.273 844.086,156.247 844.101,156.246 C 844.112,156.234 844.092,156.227 844.106,156.216 L 844.11,156.216 C 844.109,156.213 844.101,156.207 844.101,156.2 C 844.092,156.199 844.101,156.19 844.098,156.185 C 844.092,156.183 844.087,156.178 844.084,156.173 C 844.087,156.157 844.087,156.137 844.101,156.13 C 844.11,156.126 844.105,156.116 844.106,156.112 C 844.092,156.106 844.109,156.086 844.099,156.078 C 844.101,156.071 844.101,156.062 844.106,156.055 C 844.101,156.048 844.098,156.047 844.099,156.036 L 844.109,156.035 C 844.105,156.027 844.105,156.02 844.097,156.016 C 844.101,156.014 844.11,156.009 844.105,156.002 C 844.101,155.998 844.101,155.997 844.101,155.993 C 844.098,155.986 844.091,155.976 844.099,155.968 C 844.109,155.964 844.099,155.955 844.101,155.952 C 844.09,155.952 844.099,155.938 844.098,155.933 C 844.101,155.925 844.099,155.914 844.098,155.907 C 844.105,155.896 844.083,155.902 844.087,155.89 C 844.078,155.887 844.083,155.879 844.084,155.874 C 844.094,155.876 844.091,155.865 844.094,155.861 C 844.092,155.857 844.094,155.85 844.094,155.843 C 844.087,155.838 844.092,155.834 844.09,155.826 L 844.092,155.824 C 844.087,155.821 844.092,155.812 844.091,155.807 C 844.097,155.8 844.097,155.789 844.098,155.782 C 844.087,155.778 844.094,155.765 844.087,155.757 C 844.098,155.753 844.086,155.744 844.09,155.736 C 844.078,155.724 844.092,155.712 844.087,155.696 C 844.084,155.684 844.074,155.676 844.071,155.664 L 844.077,155.627 C 844.067,155.624 844.076,155.617 844.074,155.61 C 844.077,155.603 844.067,155.608 844.067,155.603 C 844.067,155.598 844.077,155.598 844.071,155.594 C 844.077,155.591 844.068,155.586 844.068,155.583 L 844.065,155.584 C 844.058,155.575 844.068,155.565 844.059,155.558 L 844.061,155.557 C 844.041,155.554 844.059,155.534 844.057,155.527 C 844.053,155.51 844.072,155.498 844.058,155.482 C 844.074,155.459 844.034,155.457 844.052,155.432 C 844.05,155.416 844.01,155.419 844.035,155.402 C 844.032,155.397 844.035,155.387 844.027,155.383 C 844.022,155.375 844.035,155.363 844.024,155.354 C 844.026,155.347 844.034,155.349 844.034,155.34 C 844.02,155.323 844.048,155.302 844.035,155.285 C 844.035,155.275 844.027,155.264 844.017,155.26 C 844.017,155.25 844.024,155.249 844.015,155.242 C 844.011,155.226 844.027,155.215 844.02,155.202 L 844.022,155.2 C 844.007,155.197 844.024,155.178 844.013,155.172 L 844.02,155.166 C 844.007,155.164 844.024,155.153 844.011,155.15 C 844.006,155.147 844.011,155.139 844.002,155.139 C 844.005,155.126 844.01,155.115 844.001,155.107 C 844.001,155.095 844.007,155.079 844.006,155.067 L 844.002,155.07 C 843.985,155.054 844.017,155.045 844.006,155.029 C 844,155.033 843.991,155.028 843.985,155.025 C 843.975,155.013 844.001,154.996 843.984,154.986 C 843.985,154.975 844.006,154.972 843.994,154.959 C 843.989,154.957 843.987,154.952 843.989,154.946 C 844,154.946 844.001,154.933 843.998,154.926 C 844.001,154.907 844.005,154.89 843.991,154.876 C 844,154.862 843.998,154.846 843.989,154.835 C 843.981,154.832 843.985,154.817 843.979,154.813 L 843.979,154.803 L 843.974,154.803 C 843.965,154.787 843.994,154.787 843.979,154.775 L 843.969,154.773 C 843.972,154.766 843.981,154.762 843.987,154.76 C 843.984,154.755 843.985,154.743 843.978,154.743 C 843.979,154.737 843.974,154.733 843.969,154.73 C 843.975,154.727 843.98,154.722 843.981,154.717 C 843.979,154.717 843.972,154.713 843.975,154.708 C 843.98,154.7 843.984,154.691 843.98,154.682 C 843.987,154.669 843.993,154.649 843.984,154.635 C 843.991,154.617 843.994,154.596 843.985,154.579 C 843.987,154.575 843.997,154.575 843.991,154.57 C 843.974,154.568 844.001,154.548 843.979,154.546 C 843.979,154.537 843.969,154.532 843.975,154.521 C 843.98,154.521 843.987,154.52 843.985,154.513 C 843.972,154.507 843.994,154.489 843.979,154.482 C 843.985,154.482 843.979,154.477 843.979,154.473 C 843.979,154.458 843.993,154.449 843.989,154.439 C 843.984,154.428 844,154.414 843.985,154.406 L 843.985,154.402 L 843.979,154.404 C 843.975,154.398 843.974,154.39 843.974,154.382 C 843.978,154.373 843.979,154.375 843.989,154.369 C 843.989,154.366 843.991,154.364 843.994,154.366 C 844,154.358 843.989,154.349 843.991,154.342 C 843.98,154.341 843.987,154.33 843.987,154.325 C 843.991,154.325 843.993,154.319 843.989,154.316 C 843.989,154.304 844.002,154.287 843.98,154.285 C 843.981,154.269 843.964,154.258 843.969,154.242 C 843.975,154.24 843.981,154.238 843.981,154.231 C 843.985,154.221 843.975,154.214 843.968,154.213 C 843.975,154.202 843.964,154.197 843.961,154.19 L 843.958,154.19 C 843.958,154.174 843.939,154.185 843.932,154.176 C 843.929,154.166 843.922,154.157 843.913,154.148 C 843.913,154.143 843.91,154.14 843.908,154.137 C 843.892,154.14 843.892,154.117 843.875,154.126 C 843.874,154.122 843.86,154.113 843.855,154.124 C 843.848,154.122 843.83,154.121 843.834,154.107 C 843.823,154.111 843.829,154.097 843.82,154.098 C 843.809,154.103 843.801,154.107 843.79,154.107 C 843.783,154.105 843.793,154.098 843.788,154.097 C 843.782,154.095 843.775,154.097 843.771,154.1 C 843.763,154.083 843.741,154.093 843.732,154.076 C 843.723,154.071 843.718,154.076 843.712,154.083 C 843.716,154.088 843.706,154.084 843.704,154.086 C 843.682,154.088 843.699,154.065 843.678,154.068 C 843.673,154.065 843.664,154.065 843.658,154.062 C 843.642,154.052 843.63,154.079 843.619,154.053 C 843.6,154.059 843.603,154.034 843.587,154.034 C 843.589,154.024 843.574,154.022 843.567,154.021 C 843.554,154.033 843.533,154.014 843.519,154.024 C 843.507,154.026 843.498,154.016 843.487,154.026 C 843.479,154.017 843.472,154.003 843.462,153.998 L 843.466,153.995 C 843.445,153.979 843.425,153.998 843.403,154 C 843.399,154.003 843.389,154.002 843.388,153.997 C 843.38,153.983 843.355,153.987 843.341,153.983 C 843.329,153.985 843.322,153.991 843.31,153.991 C 843.313,154.006 843.295,153.991 843.295,154.006 C 843.269,154.003 843.24,153.979 843.216,154.006 C 843.212,154.006 843.206,154 843.203,154.006 L 843.203,154.008 C 843.193,154.014 843.183,154.002 843.177,154.016 C 843.162,154.007 843.147,154.024 843.135,154.014 L 843.119,154.019 C 843.111,154.008 843.104,153.998 843.104,153.987 C 843.091,154.006 843.059,153.985 843.05,154.01 C 843.039,154.008 843.025,154 843.015,154.01 C 843.006,154.008 843.008,154.002 842.999,154.008 C 842.984,154.006 842.967,153.995 842.951,154.006 C 842.945,154.002 842.942,154.01 842.939,154.01 C 842.927,154.01 842.913,154.008 842.902,154.016 C 842.901,154.026 842.887,154.014 842.887,154.026 L 842.85,154.019 L 842.846,154.026 C 842.841,154.022 842.828,154.024 842.823,154.019 C 842.817,154.024 842.815,154.007 842.809,154.017 L 842.803,154.01 C 842.796,154.014 842.787,154.014 842.777,154.019 C 842.771,154.017 842.767,154.01 842.76,154.014 C 842.748,154.007 842.741,154.019 842.73,154.019 C 842.722,154.027 842.708,154.021 842.697,154.019 C 842.69,154.026 842.69,154.037 842.679,154.041 C 842.67,154.034 842.67,154.014 842.653,154.021 C 842.648,154.022 842.645,154.003 842.638,154.01 C 842.636,154.007 842.633,154.008 842.631,154.01 C 842.616,154.006 842.604,153.998 842.59,154.007 C 842.585,153.998 842.581,154.01 842.577,154.006 C 842.554,153.998 842.54,154.012 842.519,154.01 C 842.494,154.002 842.475,154.008 842.451,154.006 C 842.435,154.012 842.419,154 842.406,154.01 C 842.396,154.007 842.393,154.014 842.386,154.016 C 842.368,154.006 842.35,154.017 842.331,154.014 C 842.324,154.021 842.306,154.01 842.302,154.027 C 842.291,154.034 842.28,154.022 842.27,154.031 C 842.257,154.017 842.231,154.016 842.211,154.008 C 842.199,154.026 842.179,154.002 842.165,154.016 C 842.161,154.016 842.157,154.017 842.153,154.022 C 842.146,154.022 842.135,154.017 842.132,154.01 C 842.126,154.007 842.118,154.008 842.117,154.014 L 842.115,154.021 L 842.089,154.016 C 842.072,154.012 842.057,154.026 842.043,154.016 C 842.037,154.012 842.037,154.019 842.033,154.021 L 842.017,154.017 L 842.01,154.034 C 842.003,154.033 841.992,154.034 841.99,154.027 L 841.995,154.022 C 841.987,154.019 841.995,154.008 841.984,154.008 C 841.972,154.006 841.968,154.017 841.958,154.014 C 841.953,154.019 841.946,154.014 841.94,154.016 C 841.939,154.006 841.928,154.002 841.918,154.003 C 841.913,154.007 841.906,154.007 841.905,154.014 C 841.892,154.017 841.876,154.006 841.866,154.014 C 841.85,154.012 841.842,154.022 841.826,154.016 C 841.813,154.031 841.793,154.008 841.78,154.022 C 841.77,154.021 841.761,154.016 841.751,154.021 C 841.75,154.026 841.745,154.024 841.742,154.024 C 841.738,154.021 841.738,154.017 841.735,154.016 C 841.731,154.026 841.727,154.012 841.719,154.014 C 841.71,154.003 841.692,153.998 841.678,153.998 L 841.675,154.006 C 841.656,154.008 841.638,153.987 841.626,154 C 841.616,154.006 841.6,153.991 841.596,154.003 C 841.579,153.997 841.57,154.017 841.557,154.002 L 841.553,154.003 C 841.559,153.993 841.544,153.993 841.54,153.987 C 841.534,153.987 841.528,153.987 841.524,153.985 C 841.514,153.987 841.493,153.971 841.48,153.983 C 841.473,153.976 841.466,153.987 841.466,153.977 C 841.442,153.965 841.442,154.007 841.422,153.989 C 841.411,153.985 841.396,153.981 841.386,153.987 C 841.383,153.977 841.378,153.979 841.373,153.974 L 841.37,153.974 C 841.363,153.955 841.341,153.96 841.326,153.957 C 841.307,153.958 841.302,153.955 841.284,153.953 C 841.274,153.938 841.262,153.951 841.25,153.941 C 841.248,153.936 841.244,153.935 841.24,153.936 C 841.235,153.938 841.225,153.938 841.225,153.947 C 841.204,153.941 841.184,153.985 841.166,153.953 C 841.163,153.951 841.159,153.948 841.155,153.95 C 841.149,153.958 841.144,153.965 841.139,153.972 C 841.129,153.972 841.13,153.991 841.118,153.991 C 841.113,153.998 841.107,154.008 841.113,154.019 C 841.12,154.024 841.12,154.03 841.115,154.034 C 841.111,154.04 841.106,154.037 841.1,154.034 C 841.105,154.04 841.098,154.052 841.105,154.057 C 841.094,154.065 841.096,154.086 841.082,154.091 C 841.079,154.114 841.059,154.136 841.07,154.161 C 841.065,154.171 841.072,154.185 841.062,154.195 C 841.062,154.207 841.073,154.216 841.072,154.228 C 841.082,154.235 841.073,154.252 841.087,154.259 L 841.091,154.266 C 841.085,154.275 841.069,154.272 841.066,154.285 C 841.069,154.292 841.066,154.297 841.073,154.3 C 841.074,154.313 841.065,154.321 841.063,154.331 C 841.07,154.345 841.065,154.361 841.072,154.373 C 841.073,154.379 841.066,154.39 841.077,154.392 C 841.07,154.399 841.078,154.404 841.069,154.406 C 841.069,154.407 841.066,154.425 841.073,154.43 C 841.082,154.439 841.066,154.447 841.069,154.455 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841.046,146.8 C 841.052,146.803 841.05,146.796 841.053,146.794 C 841.059,146.792 841.06,146.794 841.062,146.8 C 841.065,146.792 841.072,146.8 841.078,146.794 C 841.087,146.792 841.102,146.794 841.107,146.798 C 841.111,146.807 841.118,146.792 841.125,146.8 C 841.139,146.78 841.143,146.805 841.158,146.807 C 841.172,146.792 841.19,146.812 841.204,146.813 C 841.217,146.828 841.238,146.831 841.255,146.834 C 841.255,146.839 841.262,146.849 841.27,146.846 C 841.276,146.836 841.284,146.839 841.294,146.834 C 841.301,146.832 841.296,146.841 841.296,146.844 C 841.299,146.846 841.295,146.855 841.301,146.855 C 841.304,146.849 841.324,146.844 841.333,146.846 C 841.347,146.865 841.374,146.865 841.395,146.862 C 841.406,146.877 841.414,146.888 841.432,146.891 C 841.441,146.885 841.458,146.88 841.46,146.868 C 841.472,146.873 841.487,146.87 841.491,146.884 C 841.507,146.891 841.501,146.868 841.518,146.868 C 841.525,146.864 841.537,146.868 841.539,146.875 C 841.541,146.87 841.543,146.862 841.55,146.862 C 841.556,146.862 841.551,146.87 841.556,146.873 C 841.543,146.891 841.567,146.895 841.572,146.907 C 841.591,146.911 841.607,146.924 841.628,146.911 C 841.629,146.907 841.628,146.903 841.623,146.901 C 841.63,146.899 841.625,146.895 841.628,146.893 C 841.636,146.893 841.637,146.885 841.648,146.891 C 841.643,146.895 841.662,146.898 841.652,146.907 C 841.655,146.916 841.666,146.924 841.675,146.924 C 841.689,146.908 841.708,146.927 841.722,146.922 C 841.728,146.919 841.734,146.916 841.734,146.912 C 841.764,146.888 841.742,146.932 841.758,146.943 C 841.761,146.945 841.763,146.94 841.763,146.94 C 841.782,146.95 841.788,146.922 841.804,146.94 C 841.802,146.948 841.815,146.951 841.819,146.955 C 841.83,146.953 841.846,146.959 841.859,146.95 C 841.867,146.945 841.872,146.935 841.882,146.94 C 841.898,146.959 841.919,146.938 841.936,146.955 C 841.946,146.961 841.958,146.946 841.971,146.951 C 841.975,146.948 841.979,146.945 841.982,146.94 C 841.986,146.938 841.988,146.938 841.991,146.941 C 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842.586,146.946 842.591,146.941 842.591,146.938 C 842.607,146.929 842.622,146.935 842.639,146.94 C 842.64,146.948 842.64,146.955 842.648,146.96 C 842.665,146.968 842.683,146.966 842.702,146.966 C 842.701,146.953 842.714,146.96 842.716,146.95 C 842.743,146.955 842.762,146.968 842.791,146.968 C 842.794,146.957 842.801,146.95 842.814,146.95 C 842.818,146.957 842.831,146.957 842.828,146.966 C 842.848,146.974 842.876,146.974 842.894,146.959 C 842.887,146.95 842.885,146.948 842.888,146.938 C 842.894,146.932 842.895,146.922 842.903,146.924 L 842.905,146.932 C 842.903,146.94 842.896,146.94 842.894,146.945 C 842.907,146.946 842.898,146.963 842.913,146.961 C 842.921,146.963 842.921,146.957 842.928,146.955 C 842.934,146.957 842.939,146.96 842.941,146.966 C 842.952,146.966 842.966,146.968 842.977,146.96 C 842.987,146.968 842.997,146.948 843.002,146.96 C 843.007,146.953 843.007,146.963 843.012,146.961 C 843.026,146.951 843.043,146.978 843.056,146.96 C 843.072,146.974 843.074,146.946 843.093,146.957 C 843.115,146.943 843.129,146.966 843.149,146.966 C 843.166,146.946 843.186,146.959 843.206,146.959 C 843.214,146.955 843.221,146.953 843.221,146.945 C 843.235,146.938 843.249,146.957 843.264,146.945 C 843.284,146.955 843.306,146.94 843.326,146.95 C 843.328,146.946 843.33,146.938 843.334,146.934 C 843.353,146.938 843.369,146.945 843.388,146.943 C 843.39,146.935 843.401,146.943 843.405,146.932 L 843.409,146.938 C 843.422,146.932 843.434,146.932 843.446,146.932 C 843.454,146.934 843.452,146.941 843.456,146.946 C 843.465,146.946 843.479,146.955 843.484,146.945 L 843.487,146.948 C 843.492,146.945 843.501,146.945 843.505,146.94 C 843.514,146.935 843.52,146.946 843.525,146.953 C 843.532,146.955 843.54,146.953 843.544,146.948 C 843.555,146.946 843.547,146.961 843.56,146.963 C 843.572,146.963 843.578,146.975 843.591,146.975 C 843.598,146.972 843.605,146.98 843.607,146.986 C 843.619,146.984 843.639,147.002 843.643,146.984 C 843.649,146.984 843.653,146.98 843.658,146.984 C 843.662,146.989 843.672,146.991 843.67,147.001 L 843.709,147.012 C 843.723,147.021 843.751,147.022 843.77,147.03 C 843.764,147.047 843.794,147.036 843.794,147.037 C 843.796,147.033 843.809,147.041 843.809,147.033 L 843.815,147.037 C 843.822,147.041 843.834,147.044 843.84,147.037 C 843.851,147.036 843.86,147.044 843.868,147.053 C 843.875,147.04 843.892,147.052 843.903,147.053 C 843.909,147.056 843.914,147.044 843.922,147.048 C 843.928,147.035 843.945,147.041 843.955,147.047 C 843.958,147.041 843.966,147.043 843.971,147.043 C 843.978,147.047 843.975,147.056 843.985,147.064 C 843.997,147.07 844.006,147.07 844.022,147.067 L 844.022,147.072 C 844.027,147.078 844.031,147.087 844.038,147.082 C 844.047,147.087 844.059,147.092 844.055,147.101 C 844.059,147.103 844.068,147.105 844.073,147.101 L 844.084,147.115 C 844.101,147.122 844.118,147.103 844.136,147.115 C 844.152,147.145 844.19,147.116 844.209,147.14 L 844.212,147.136 C 844.219,147.142 844.226,147.139 844.235,147.14 C 844.256,147.124 844.262,147.155 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C 844.201,147.699 844.201,147.708 844.201,147.719 C 844.208,147.746 844.196,147.78 844.201,147.81 L 844.205,147.81 C 844.212,147.811 844.212,147.819 844.211,147.825 C 844.197,147.831 844.202,147.845 844.196,147.855 C 844.201,147.858 844.205,147.869 844.197,147.872 C 844.189,147.888 844.209,147.888 844.216,147.894 C 844.208,147.906 844.197,147.917 844.196,147.932 C 844.209,147.937 844.193,147.95 844.203,147.958 C 844.203,147.969 844.205,147.982 844.218,147.986 C 844.222,147.997 844.209,147.992 844.205,147.995 C 844.203,148.005 844.202,148.021 844.199,148.033 C 844.209,148.04 844.199,148.048 844.203,148.057 C 844.216,148.067 844.228,148.082 844.238,148.099 C 844.226,148.1 844.23,148.114 844.219,148.12 L 844.228,148.134 L 844.218,148.148 C 844.212,148.159 844.209,148.174 844.223,148.179 C 844.226,148.202 844.223,148.219 844.224,148.241 C 844.234,148.238 844.23,148.255 844.238,148.248 C 844.242,148.259 844.23,148.263 844.224,148.269 C 844.219,148.281 844.224,148.294 844.237,148.3 C 844.222,148.312 844.242,148.325 844.244,148.336 C 844.25,148.35 844.237,148.361 844.23,148.37 C 844.235,148.376 844.245,148.381 844.254,148.381 C 844.262,148.39 844.26,148.403 844.254,148.411 C 844.249,148.412 844.242,148.409 844.238,148.415 C 844.23,148.427 844.234,148.438 844.238,148.45 C 844.231,148.465 844.261,148.482 844.238,148.491 C 844.23,148.499 844.244,148.504 844.241,148.515 C 844.27,148.523 844.235,148.543 844.245,148.563 C 844.257,148.574 844.245,148.59 844.249,148.603 C 844.257,148.619 844.245,148.638 844.261,148.651 C 844.264,148.658 844.271,148.665 844.268,148.676 C 844.267,148.686 844.256,148.679 844.249,148.684 C 844.248,148.697 844.264,148.696 844.264,148.71 C 844.27,148.717 844.28,148.711 844.28,148.726 C 844.275,148.729 844.282,148.735 844.28,148.741 C 844.301,148.763 844.264,148.782 844.282,148.806 C 844.275,148.818 844.296,148.832 844.285,148.842 C 844.29,148.849 844.305,148.846 844.302,148.86 C 844.292,148.872 844.32,148.883 844.317,148.897 C 844.294,148.905 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844.311,149.533 C 844.311,149.543 844.317,149.551 844.323,149.554 C 844.323,149.572 844.305,149.567 844.305,149.583 C 844.308,149.6 844.309,149.616 844.301,149.633 C 844.298,149.652 844.311,149.673 844.327,149.682 L 844.309,149.701 C 844.301,149.723 844.327,149.729 844.328,149.746 C 844.327,149.754 844.318,149.752 844.313,149.752 C 844.313,149.759 844.303,149.764 844.308,149.772 C 844.317,149.783 844.32,149.793 844.321,149.808 C 844.309,149.814 844.32,149.821 844.318,149.829 C 844.305,149.832 844.321,149.848 844.308,149.85 C 844.305,149.855 844.301,149.86 844.308,149.865 C 844.297,149.882 844.315,149.896 844.318,149.907 C 844.313,149.924 844.298,149.94 844.313,149.958 C 844.311,149.967 844.313,149.974 844.315,149.985 C 844.318,149.988 844.321,149.99 844.327,149.99 L 844.323,149.997 C 844.328,150.007 844.339,150.01 844.342,150.016 z M 844.097,149.321 C 844.092,149.318 844.085,149.315 844.081,149.312 C 844.092,149.295 844.084,149.274 844.101,149.266 C 844.107,149.262 844.105,149.255 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843.699,147.207 C 843.684,147.196 843.662,147.186 843.644,147.197 C 843.624,147.209 843.633,147.173 843.612,147.186 C 843.602,147.177 843.592,147.162 843.583,147.155 C 843.577,147.15 843.568,147.154 843.564,147.155 C 843.544,147.146 843.522,147.16 843.504,147.15 C 843.498,147.154 843.492,147.157 843.488,147.155 C 843.482,147.151 843.488,147.142 843.479,147.142 C 843.468,147.135 843.469,147.115 843.453,147.119 C 843.438,147.117 843.426,147.135 843.409,147.13 C 843.406,147.135 843.399,147.131 843.394,147.131 C 843.388,147.108 843.361,147.119 843.343,147.113 C 843.324,147.115 843.309,147.126 843.291,147.135 C 843.276,147.127 843.256,147.13 843.241,147.122 C 843.241,147.135 843.224,147.124 843.223,147.135 C 843.217,147.13 843.211,147.135 843.205,147.135 C 843.201,147.142 843.183,147.135 843.178,147.145 C 843.169,147.143 843.156,147.142 843.146,147.146 L 843.143,147.142 C 843.136,147.143 843.124,147.146 843.118,147.145 C 843.118,147.14 843.115,147.134 843.11,147.134 C 843.111,147.127 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841.99,147.146 L 841.999,147.145 C 842.001,147.139 841.986,147.142 841.993,147.131 C 841.982,147.124 841.978,147.14 841.965,147.136 L 841.962,147.142 C 841.955,147.139 841.938,147.142 841.938,147.127 C 841.923,147.119 841.914,147.131 841.903,147.136 C 841.879,147.126 841.855,147.143 841.834,147.139 C 841.819,147.151 841.797,147.134 841.781,147.143 C 841.773,147.142 841.758,147.136 841.749,147.145 C 841.745,147.13 841.729,147.146 841.723,147.135 C 841.723,147.136 841.723,147.136 841.723,147.136 C 841.719,147.124 841.703,147.126 841.692,147.121 C 841.685,147.121 841.679,147.119 841.675,147.126 C 841.657,147.126 841.643,147.108 841.628,147.122 C 841.617,147.122 841.604,147.116 841.597,147.124 C 841.584,147.117 841.574,147.14 841.563,147.126 C 841.561,147.117 841.55,147.113 841.544,147.108 C 841.543,147.105 841.541,147.109 841.539,147.111 C 841.521,147.103 841.5,147.096 841.48,147.101 C 841.473,147.094 841.46,147.101 841.453,147.1 C 841.451,147.113 841.439,147.119 841.426,147.111 C 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844.249,140.753 844.219,140.776 844.234,140.799 C 844.25,140.806 844.23,140.828 844.241,140.843 C 844.238,140.848 844.238,140.854 844.23,140.854 C 844.228,140.87 844.215,140.885 844.224,140.901 C 844.219,140.91 844.222,140.93 844.215,140.941 L 844.228,140.968 L 844.219,141.003 C 844.23,141.005 844.218,141.014 844.224,141.02 C 844.228,141.039 844.205,141.034 844.202,141.05 C 844.203,141.052 844.202,141.06 844.202,141.065 C 844.215,141.081 844.197,141.099 844.209,141.113 C 844.211,141.124 844.199,141.137 844.212,141.143 C 844.218,141.151 844.209,141.155 844.212,141.164 C 844.208,141.17 844.209,141.178 844.209,141.185 C 844.203,141.195 844.209,141.216 844.192,141.218 C 844.189,141.228 844.186,141.242 844.193,141.249 C 844.183,141.262 844.193,141.276 844.199,141.286 C 844.196,141.301 844.205,141.319 844.211,141.334 C 844.228,141.346 844.196,141.356 844.201,141.373 C 844.211,141.388 844.197,141.402 844.208,141.414 C 844.201,141.433 844.199,141.456 844.193,141.48 C 844.193,141.485 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841.395,143.705 841.387,143.697 841.383,143.708 C 841.378,143.701 841.368,143.708 841.36,143.704 C 841.355,143.705 841.352,143.709 841.347,143.708 C 841.346,143.709 841.34,143.711 841.342,143.715 C 841.336,143.713 841.323,143.716 841.315,143.711 C 841.311,143.713 841.308,143.72 841.303,143.716 C 841.291,143.711 841.283,143.72 841.271,143.72 C 841.266,143.734 841.247,143.716 841.238,143.729 C 841.217,143.724 841.204,143.725 841.183,143.73 L 841.184,143.734 C 841.181,143.729 841.171,143.735 841.165,143.739 C 841.146,143.741 841.128,143.742 841.107,143.741 C 841.102,143.745 841.084,143.735 841.08,143.746 C 841.079,143.746 841.079,143.745 841.078,143.745 C 841.072,143.75 841.06,143.741 841.06,143.751 L 841.06,143.75 C 841.057,143.742 841.053,143.75 841.05,143.75 C 841.038,143.745 841.03,143.755 841.018,143.749 C 841.016,143.738 841.007,143.755 841.006,143.742 C 840.998,143.738 840.981,143.742 840.969,143.749 C 840.958,143.739 840.949,143.75 840.941,143.75 C 840.92,143.724 840.889,143.732 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840.919,143.024 C 840.92,143.006 840.903,142.993 840.908,142.976 C 840.909,142.976 840.911,142.976 840.911,142.974 C 840.911,142.971 840.911,142.964 840.908,142.964 C 840.914,142.953 840.909,142.937 840.918,142.921 C 840.908,142.911 840.914,142.896 840.919,142.884 C 840.905,142.878 840.919,142.86 840.911,142.848 C 840.911,142.827 840.934,142.795 840.908,142.777 C 840.914,142.754 840.9,142.74 840.907,142.72 C 840.9,142.708 840.909,142.693 840.911,142.677 C 840.905,142.675 840.898,142.668 840.896,142.663 C 840.907,142.652 840.896,142.637 840.896,142.626 C 840.883,142.595 840.918,142.567 840.901,142.533 C 840.907,142.522 840.905,142.506 840.914,142.493 C 840.914,142.486 840.914,142.48 840.907,142.476 C 840.914,142.455 840.901,142.437 840.9,142.418 C 840.908,142.406 840.9,142.397 840.9,142.388 C 840.889,142.376 840.9,142.362 840.9,142.351 C 840.896,142.344 840.893,142.335 840.893,142.33 C 840.88,142.311 840.908,142.291 840.892,142.274 C 840.878,142.267 840.889,142.253 840.896,142.248 C 840.893,142.227 840.898,142.208 840.903,142.191 C 840.892,142.174 840.911,142.155 840.901,142.136 C 840.909,142.112 840.879,142.103 840.887,142.077 C 840.883,142.073 840.893,142.071 840.887,142.067 C 840.887,142.057 840.892,142.05 840.896,142.043 C 840.892,142.04 840.896,142.037 840.896,142.03 C 840.879,142.014 840.919,142.005 840.898,141.988 C 840.905,141.979 840.909,141.973 840.918,141.968 C 840.914,141.963 840.918,141.958 840.914,141.952 C 840.894,141.939 840.909,141.92 840.905,141.902 C 840.903,141.888 840.929,141.871 840.914,141.854 C 840.924,141.836 840.907,141.818 840.905,141.801 C 840.92,141.786 840.907,141.756 840.929,141.748 C 840.931,141.736 840.926,141.725 840.92,141.718 C 840.918,141.704 840.914,141.696 840.909,141.683 C 840.914,141.673 840.911,141.66 840.92,141.655 C 840.924,141.643 840.92,141.626 840.908,141.621 C 840.908,141.618 840.905,141.611 840.911,141.609 C 840.907,141.604 840.911,141.594 840.901,141.594 C 840.9,141.574 840.918,141.566 840.929,141.559 C 840.929,141.553 840.934,141.546 840.927,141.538 C 840.952,141.523 840.926,141.501 840.922,141.481 C 840.892,141.466 840.927,141.434 840.909,141.414 C 840.909,141.401 840.919,141.397 840.927,141.388 C 840.941,141.367 840.911,141.362 840.909,141.346 L 840.922,141.346 C 840.931,141.345 840.927,141.334 840.933,141.327 C 840.931,141.32 840.924,141.32 840.924,141.314 C 840.901,141.307 840.922,141.283 840.908,141.268 C 840.905,141.253 840.922,141.254 840.927,141.248 C 840.926,141.238 840.935,141.238 840.934,141.227 C 840.931,141.217 840.941,141.205 840.929,141.195 C 840.939,141.175 840.922,141.162 840.919,141.145 C 840.903,141.143 840.908,141.13 840.901,141.122 C 840.894,141.116 840.9,141.103 840.908,141.1 C 840.9,141.087 840.918,141.072 840.907,141.059 C 840.914,141.055 840.905,141.05 840.907,141.045 C 840.896,141.035 840.909,141.019 840.907,141.005 C 840.903,140.996 840.893,140.999 840.887,140.993 C 840.883,140.984 840.88,140.972 840.883,140.963 C 840.903,140.957 840.893,140.937 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840.815,140.268 C 840.821,140.247 840.829,140.219 840.834,140.201 L 840.835,140.201 C 840.845,140.186 840.827,140.176 840.823,140.164 C 840.829,140.143 840.839,140.117 840.819,140.102 C 840.816,140.086 840.812,140.066 840.829,140.059 C 840.832,140.05 840.829,140.041 840.826,140.035 C 840.787,140.029 840.827,139.997 840.823,139.973 C 840.839,139.968 840.834,139.945 840.846,139.936 C 840.834,139.927 840.842,139.907 840.834,139.895 C 840.835,139.88 840.848,139.867 840.839,139.849 C 840.837,139.829 840.841,139.81 840.823,139.793 C 840.822,139.78 840.803,139.766 840.813,139.754 C 840.834,139.745 840.812,139.73 840.822,139.717 C 840.807,139.707 840.821,139.69 840.819,139.678 C 840.808,139.675 840.819,139.662 840.807,139.661 C 840.803,139.651 840.789,139.642 840.789,139.63 C 840.79,139.625 840.797,139.621 840.8,139.616 C 840.801,139.608 840.801,139.595 840.796,139.589 C 840.797,139.582 840.797,139.566 840.81,139.568 C 840.821,139.559 840.807,139.553 840.804,139.547 C 840.793,139.542 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842.743,137.164 842.748,137.172 C 842.762,137.166 842.775,137.181 842.789,137.171 C 842.794,137.174 842.804,137.168 842.804,137.179 C 842.821,137.172 842.839,137.189 842.852,137.178 C 842.86,137.18 842.87,137.168 842.876,137.164 C 842.876,137.163 842.875,137.157 842.879,137.153 C 842.895,137.153 842.911,137.166 842.925,137.158 C 842.945,137.164 842.974,137.153 842.988,137.175 C 842.992,137.172 842.993,137.175 842.997,137.179 C 843.005,137.174 843.019,137.181 843.026,137.174 C 843.033,137.179 843.033,137.172 843.038,137.18 C 843.044,137.168 843.045,137.189 843.05,137.179 C 843.063,137.18 843.077,137.18 843.09,137.179 C 843.105,137.166 843.12,137.179 843.138,137.179 C 843.143,137.174 843.149,137.175 843.157,137.175 C 843.179,137.192 843.21,137.172 843.231,137.18 L 843.234,137.179 C 843.234,137.186 843.245,137.189 843.25,137.192 C 843.256,137.205 843.27,137.206 843.283,137.207 L 843.306,137.196 C 843.311,137.198 843.317,137.19 843.323,137.194 C 843.336,137.198 843.336,137.179 843.35,137.185 C 843.36,137.19 843.36,137.194 843.366,137.201 C 843.369,137.215 843.386,137.217 843.397,137.22 C 843.401,137.206 843.418,137.217 843.423,137.207 C 843.428,137.211 843.438,137.206 843.439,137.215 C 843.442,137.21 843.454,137.217 843.456,137.206 C 843.468,137.207 843.482,137.206 843.491,137.217 C 843.492,137.215 843.494,137.217 843.498,137.217 C 843.501,137.212 843.504,137.212 843.505,137.21 C 843.525,137.212 843.54,137.211 843.561,137.21 C 843.566,137.203 843.566,137.189 843.581,137.19 C 843.587,137.203 843.605,137.205 843.617,137.21 C 843.637,137.22 843.653,137.194 843.673,137.201 C 843.678,137.186 843.691,137.2 843.698,137.201 C 843.703,137.212 843.718,137.215 843.73,137.217 C 843.737,137.207 843.757,137.215 843.758,137.196 C 843.764,137.2 843.769,137.192 843.772,137.189 C 843.781,137.189 843.784,137.184 843.794,137.185 C 843.824,137.19 843.854,137.207 843.886,137.2 L 843.892,137.194 C 843.902,137.196 843.915,137.189 843.928,137.19 C 843.952,137.194 843.967,137.207 843.989,137.215 C 843.995,137.205 844,137.217 844.006,137.207 C 844.008,137.198 844.026,137.201 844.033,137.198 C 844.055,137.189 844.085,137.207 844.101,137.189 C 844.136,137.198 844.167,137.203 844.202,137.211 C 844.212,137.205 844.238,137.207 844.254,137.217 C 844.249,137.232 844.267,137.237 844.267,137.251 C 844.256,137.268 844.272,137.283 844.264,137.299 C 844.272,137.319 844.261,137.341 844.26,137.361 C 844.272,137.375 844.264,137.383 844.267,137.398 C 844.256,137.417 844.271,137.434 844.268,137.453 C 844.272,137.474 844.275,137.499 844.285,137.516 C 844.284,137.528 844.278,137.542 844.285,137.553 C 844.287,137.564 844.275,137.568 844.275,137.578 C 844.278,137.587 844.275,137.599 844.272,137.61 C 844.284,137.63 844.268,137.646 844.267,137.667 C 844.276,137.686 844.261,137.709 844.28,137.723 C 844.276,137.73 844.284,137.744 844.282,137.749 C 844.285,137.752 844.282,137.762 844.287,137.764 C 844.298,137.785 844.29,137.807 844.285,137.832 C 844.296,137.843 844.278,137.869 844.29,137.885 C 844.285,137.893 844.292,137.905 844.29,137.911 C 844.301,137.922 844.29,137.937 844.301,137.949 z M 844.079,139.838 L 844.078,139.837 C 844.084,139.833 844.078,139.829 844.076,139.827 C 844.067,139.822 844.057,139.823 844.052,139.818 C 844.078,139.81 844.064,139.788 844.073,139.77 C 844.067,139.756 844.047,139.746 844.053,139.73 C 844.059,139.728 844.065,139.719 844.063,139.714 C 844.057,139.7 844.076,139.684 844.059,139.671 C 844.06,139.662 844.048,139.661 844.047,139.652 C 844.041,139.636 844.053,139.625 844.052,139.609 L 844.037,139.579 C 844.045,139.563 844.012,139.548 844.033,139.538 C 844.038,139.535 844.037,139.53 844.037,139.522 C 844.027,139.517 844.037,139.507 844.026,139.502 C 844.019,139.476 844.037,139.45 844.026,139.424 C 844.021,139.412 844.034,139.4 844.024,139.389 C 844.01,139.386 844.021,139.369 844.012,139.362 C 844.012,139.355 844.021,139.35 844.018,139.344 C 844.019,139.329 844.01,139.314 844.021,139.3 C 844.014,139.288 844.015,139.278 844.01,139.263 C 844.006,139.246 844.006,139.231 844.012,139.211 C 844.019,139.211 844.019,139.201 844.015,139.196 C 844.008,139.195 844.006,139.188 844.001,139.18 C 844.014,139.175 843.997,139.162 844.001,139.15 C 843.993,139.143 844.006,139.128 844.004,139.117 C 843.995,139.103 844.008,139.086 844.018,139.075 C 844.027,139.051 844.001,139.03 844.004,139.006 C 843.993,139.002 844.004,138.996 844.001,138.991 C 844.01,138.987 844,138.978 844,138.973 C 844.01,138.955 844.021,138.936 844.012,138.915 C 844.019,138.9 844.015,138.886 844.006,138.878 C 844.001,138.871 844,138.861 843.999,138.854 C 843.993,138.837 843.988,138.815 843.995,138.797 C 844,138.786 843.997,138.77 843.995,138.755 C 843.986,138.753 843.971,138.749 843.979,138.737 C 843.985,138.732 843.999,138.732 843.997,138.723 C 844.006,138.713 843.988,138.7 844.001,138.691 C 844.006,138.689 844.01,138.685 844.008,138.679 C 843.999,138.666 844.014,138.652 844.012,138.639 C 844,138.633 844.012,138.615 844,138.61 C 843.989,138.605 843.993,138.594 843.988,138.587 C 843.989,138.582 843.992,138.579 843.997,138.582 C 844.001,138.574 844.01,138.567 844.012,138.557 C 844.006,138.549 844.012,138.54 844.004,138.537 C 844.008,138.53 844.018,138.522 844.015,138.512 C 844.01,138.512 844.006,138.503 844.012,138.5 C 844.026,138.491 844.03,138.47 844.027,138.455 C 844.019,138.455 844.027,138.449 844.024,138.444 C 844.019,138.432 844.01,138.417 843.995,138.413 C 843.992,138.39 844.026,138.382 844.015,138.355 C 844.008,138.35 844.012,138.34 844.012,138.329 C 844.024,138.322 844.041,138.32 844.041,138.303 C 844.044,138.28 844.041,138.267 844.027,138.253 C 844.027,138.241 844.041,138.241 844.038,138.229 C 844.034,138.216 844.014,138.207 844.022,138.193 C 844.03,138.188 844.045,138.188 844.045,138.175 C 844.03,138.169 844.031,138.155 844.026,138.144 C 844.037,138.148 844.044,138.138 844.047,138.131 C 844.053,138.118 844.04,138.108 844.037,138.097 C 844.031,138.097 844.027,138.091 844.031,138.087 C 844.037,138.079 844.048,138.092 844.052,138.081 C 844.065,138.079 844.067,138.062 844.068,138.051 C 844.068,138.035 844.048,138.039 844.038,138.03 C 844.038,138.018 844.044,138.017 844.052,138.009 C 844.055,138.009 844.059,138.013 844.06,138.008 C 844.059,137.994 844.077,137.978 844.063,137.967 C 844.06,137.957 844.068,137.952 844.076,137.951 C 844.078,137.938 844.089,137.926 844.084,137.915 C 844.077,137.901 844.063,137.896 844.053,137.887 C 844.063,137.882 844.084,137.882 844.079,137.866 C 844.065,137.853 844.092,137.842 844.078,137.832 C 844.084,137.821 844.068,137.825 844.068,137.816 C 844.055,137.807 844.081,137.796 844.065,137.787 C 844.052,137.791 844.051,137.779 844.045,137.771 C 844.04,137.755 844.06,137.74 844.057,137.724 C 844.06,137.711 844.055,137.698 844.047,137.69 C 844.052,137.68 844.041,137.671 844.044,137.662 C 844.045,137.652 844.048,137.64 844.044,137.631 C 844.052,137.609 844.048,137.58 844.048,137.558 C 844.041,137.554 844.04,137.547 844.037,137.542 C 844.038,137.538 844.041,137.534 844.038,137.531 C 844.047,137.521 844.055,137.512 844.055,137.499 C 844.044,137.481 844.06,137.469 844.048,137.451 C 844.051,137.44 844.053,137.434 844.041,137.433 C 844.026,137.428 844.015,137.435 844.004,137.443 L 843.989,137.431 C 843.975,137.435 843.959,137.429 843.945,137.438 C 843.926,137.428 843.902,137.433 843.888,137.413 C 843.873,137.412 843.857,137.4 843.843,137.412 C 843.828,137.417 843.811,137.424 843.794,137.422 C 843.782,137.431 843.767,137.416 843.758,137.428 C 843.741,137.417 843.739,137.438 843.722,137.433 C 843.703,137.44 843.684,137.428 843.666,137.424 C 843.658,137.417 843.644,137.417 843.635,137.42 C 843.624,137.424 843.614,137.438 843.603,137.428 C 843.591,137.416 843.577,137.407 843.56,137.408 C 843.555,137.408 843.555,137.412 843.551,137.413 C 843.537,137.412 843.52,137.4 843.508,137.41 C 843.501,137.41 843.504,137.422 843.494,137.417 L 843.484,137.429 C 843.491,137.434 843.482,137.438 843.484,137.443 C 843.456,137.445 843.436,137.424 843.408,137.431 C 843.406,137.433 843.401,137.425 843.399,137.431 C 843.381,137.404 843.353,137.439 843.34,137.407 C 843.328,137.4 843.311,137.399 843.301,137.404 C 843.29,137.404 843.275,137.399 843.261,137.404 C 843.25,137.399 843.249,137.391 843.241,137.386 L 843.236,137.387 C 843.235,137.386 843.234,137.383 843.234,137.382 C 843.218,137.387 843.202,137.377 843.186,137.387 C 843.175,137.383 843.163,137.394 843.152,137.398 C 843.136,137.386 843.115,137.404 843.105,137.386 C 843.098,137.386 843.09,137.381 843.086,137.387 C 843.086,137.386 843.085,137.386 843.085,137.383 C 843.074,137.39 843.07,137.369 843.06,137.381 C 843.056,137.382 843.044,137.378 843.045,137.371 C 843.031,137.369 843.015,137.351 843.005,137.36 C 842.99,137.355 842.974,137.36 842.962,137.368 C 842.954,137.372 842.941,137.368 842.939,137.378 L 842.888,137.369 C 842.876,137.377 842.862,137.369 842.849,137.368 C 842.836,137.36 842.818,137.365 842.803,137.368 C 842.796,137.371 842.785,137.375 842.778,137.377 C 842.76,137.381 842.744,137.36 842.727,137.375 C 842.697,137.382 842.692,137.344 842.665,137.356 C 842.651,137.364 842.637,137.35 842.624,137.356 C 842.596,137.346 842.573,137.36 842.547,137.351 L 842.546,137.355 C 842.507,137.341 842.475,137.375 842.44,137.346 C 842.425,137.354 842.409,137.364 842.392,137.364 C 842.38,137.361 842.368,137.371 842.357,137.368 C 842.35,137.376 842.339,137.368 842.333,137.381 C 842.321,137.378 842.306,137.371 842.295,137.381 C 842.261,137.359 842.228,137.386 842.197,137.39 C 842.188,137.391 842.191,137.407 842.182,137.4 C 842.178,137.41 842.173,137.4 842.165,137.403 C 842.165,137.397 842.159,137.394 842.153,137.393 C 842.142,137.408 842.119,137.387 842.109,137.398 C 842.097,137.397 842.081,137.397 842.076,137.387 C 842.069,137.383 842.059,137.383 842.051,137.386 C 842.041,137.382 842.034,137.393 842.023,137.393 L 842.02,137.397 C 842.008,137.393 841.993,137.394 841.985,137.382 C 841.975,137.381 841.967,137.382 841.961,137.391 C 841.936,137.383 841.919,137.376 841.899,137.397 C 841.901,137.404 841.888,137.404 841.885,137.412 C 841.879,137.41 841.874,137.412 841.872,137.417 C 841.855,137.397 841.822,137.394 841.8,137.403 C 841.795,137.413 841.783,137.4 841.782,137.413 C 841.77,137.407 841.758,137.428 841.748,137.412 C 841.722,137.399 841.701,137.378 841.671,137.378 C 841.668,137.382 841.655,137.381 841.649,137.387 C 841.637,137.383 841.622,137.382 841.607,137.378 C 841.604,137.378 841.6,137.375 841.596,137.376 C 841.589,137.383 841.587,137.397 841.577,137.398 C 841.565,137.399 841.551,137.394 841.541,137.387 C 841.516,137.39 841.494,137.369 841.468,137.381 L 841.433,137.376 C 841.429,137.383 841.419,137.377 841.411,137.382 C 841.401,137.394 841.385,137.39 841.374,137.397 C 841.35,137.381 841.34,137.403 841.315,137.4 C 841.317,137.413 841.302,137.403 841.302,137.413 C 841.29,137.417 841.281,137.428 841.271,137.417 C 841.26,137.41 841.231,137.417 841.228,137.397 C 841.212,137.391 841.197,137.387 841.181,137.386 C 841.178,137.39 841.171,137.393 841.168,137.39 C 841.149,137.398 841.131,137.386 841.113,137.397 C 841.102,137.394 841.09,137.397 841.079,137.399 C 841.078,137.403 841.074,137.408 841.078,137.412 L 841.079,137.413 C 841.08,137.425 841.06,137.424 841.069,137.438 C 841.078,137.443 841.065,137.449 841.067,137.455 L 841.064,137.455 C 841.065,137.474 841.059,137.493 841.057,137.514 C 841.067,137.52 841.056,137.526 841.06,137.534 C 841.067,137.534 841.067,137.548 841.072,137.543 C 841.072,137.552 841.086,137.566 841.072,137.572 C 841.065,137.592 841.043,137.597 841.049,137.624 C 841.056,137.631 841.072,137.626 841.072,137.64 C 841.086,137.644 841.07,137.65 841.074,137.656 C 841.065,137.656 841.067,137.666 841.064,137.667 C 841.064,137.676 841.072,137.682 841.067,137.69 C 841.062,137.692 841.049,137.688 841.049,137.697 L 841.052,137.697 C 841.045,137.701 841.057,137.708 841.052,137.714 C 841.065,137.718 841.052,137.727 841.059,137.734 C 841.069,137.75 841.087,137.764 841.072,137.785 C 841.067,137.79 841.067,137.8 841.069,137.807 C 841.078,137.812 841.079,137.823 841.09,137.827 C 841.093,137.839 841.07,137.847 841.086,137.86 C 841.093,137.868 841.104,137.869 841.094,137.88 C 841.09,137.88 841.086,137.884 841.086,137.889 C 841.094,137.896 841.08,137.899 841.09,137.908 C 841.08,137.911 841.091,137.917 841.083,137.92 C 841.08,137.922 841.086,137.929 841.079,137.93 L 841.083,137.935 C 841.079,137.937 841.079,137.941 841.08,137.943 C 841.084,137.948 841.086,137.959 841.083,137.967 L 841.078,137.969 C 841.08,137.98 841.079,137.993 841.074,138.004 C 841.064,138.001 841.065,138.013 841.057,138.013 C 841.056,138.014 841.059,138.017 841.057,138.018 C 841.057,138.02 841.053,138.02 841.052,138.02 C 841.062,138.029 841.041,138.052 841.059,138.056 C 841.062,138.065 841.07,138.074 841.072,138.082 C 841.072,138.106 841.07,138.114 841.074,138.138 C 841.076,138.138 841.076,138.139 841.078,138.139 C 841.07,138.149 841.078,138.175 841.06,138.18 C 841.057,138.184 841.064,138.188 841.057,138.19 C 841.062,138.195 841.064,138.2 841.064,138.206 C 841.065,138.205 841.067,138.207 841.067,138.207 L 841.064,138.214 C 841.069,138.216 841.062,138.222 841.062,138.222 C 841.06,138.235 841.069,138.246 841.069,138.258 L 841.072,138.258 C 841.07,138.269 841.083,138.279 841.074,138.285 C 841.078,138.295 841.069,138.309 841.074,138.32 C 841.072,138.325 841.083,138.336 841.08,138.346 C 841.087,138.361 841.065,138.36 841.064,138.372 C 841.072,138.386 841.045,138.407 841.07,138.408 C 841.078,138.408 841.072,138.419 841.08,138.42 C 841.078,138.434 841.113,138.432 841.093,138.449 C 841.084,138.451 841.086,138.465 841.072,138.461 C 841.074,138.473 841.062,138.474 841.057,138.478 C 841.053,138.478 841.052,138.473 841.049,138.476 C 841.045,138.485 841.049,138.491 841.053,138.495 C 841.05,138.5 841.053,138.503 841.052,138.511 C 841.053,138.512 841.056,138.511 841.057,138.512 C 841.049,138.522 841.07,138.534 841.059,138.543 C 841.052,138.546 841.045,138.556 841.049,138.563 C 841.049,138.561 841.052,138.563 841.053,138.563 C 841.046,138.567 841.057,138.574 841.056,138.579 C 841.062,138.587 841.053,138.6 841.064,138.605 L 841.06,138.609 C 841.065,138.61 841.07,138.618 841.078,138.618 C 841.084,138.633 841.064,138.633 841.057,138.641 C 841.05,138.648 841.052,138.66 841.043,138.666 L 841.046,138.671 C 841.045,138.674 841.041,138.671 841.039,138.671 C 841.046,138.679 841.038,138.687 841.037,138.693 C 841.041,138.697 841.037,138.7 841.037,138.704 C 841.056,138.718 841.032,138.74 841.043,138.753 C 841.046,138.761 841.034,138.774 841.043,138.781 L 841.031,138.792 C 841.038,138.796 841.027,138.799 841.031,138.802 C 841.039,138.812 841.038,138.829 841.053,138.829 C 841.052,138.837 841.045,138.841 841.049,138.849 C 841.038,138.858 841.037,138.871 841.045,138.884 C 841.052,138.894 841.056,138.904 841.046,138.912 C 841.05,138.916 841.052,138.921 841.052,138.926 C 841.046,138.934 841.037,138.932 841.032,138.941 C 841.032,138.947 841.043,138.955 841.037,138.961 C 841.041,138.967 841.031,138.971 841.032,138.977 C 841.024,138.973 841.025,138.986 841.021,138.987 C 841.032,138.991 841.041,138.998 841.039,139.009 C 841.049,139.017 841.062,139.015 841.064,139.027 C 841.074,139.03 841.069,139.044 841.067,139.05 C 841.06,139.058 841.046,139.046 841.041,139.06 C 841.039,139.067 841.056,139.077 841.043,139.087 C 841.032,139.086 841.032,139.101 841.027,139.108 C 841.025,139.118 841.038,139.118 841.041,139.127 C 841.038,139.128 841.034,139.13 841.037,139.137 C 841.045,139.157 841.024,139.179 841.027,139.201 C 841.024,139.205 841.024,139.21 841.023,139.215 L 841.03,139.221 C 841.031,139.237 841.016,139.257 841.032,139.272 C 841.031,139.276 841.037,139.284 841.041,139.284 C 841.038,139.294 841.043,139.302 841.049,139.313 C 841.057,139.315 841.05,139.321 841.052,139.328 C 841.045,139.331 841.046,139.34 841.046,139.346 C 841.03,139.355 841.024,139.381 841.025,139.398 C 841.045,139.407 841.032,139.425 841.031,139.442 C 841.024,139.445 841.031,139.452 841.03,139.458 C 841.038,139.461 841.032,139.472 841.034,139.48 C 841.025,139.486 841.016,139.495 841.021,139.506 C 841.023,139.513 841.017,139.527 841.025,139.541 C 841.03,139.551 841.03,139.56 841.025,139.573 C 841.023,139.59 841.013,139.609 841.021,139.626 C 841.012,139.631 841.025,139.64 841.027,139.647 C 841.034,139.651 841.034,139.665 841.039,139.671 L 841.037,139.684 C 841.024,139.677 841.027,139.69 841.021,139.693 L 841.023,139.694 C 841.016,139.698 841.016,139.703 841.016,139.709 C 841.023,139.718 841.012,139.727 841.012,139.735 C 841.006,139.746 841.004,139.754 841.006,139.77 C 841.016,139.77 841.006,139.78 841.012,139.785 C 841.024,139.788 841.031,139.804 841.018,139.811 C 841.023,139.815 841.018,139.815 841.017,139.821 L 841.018,139.833 L 841.023,139.833 C 841.025,139.84 841.017,139.854 841.03,139.855 C 841.023,139.859 841.012,139.868 841.016,139.877 C 841.013,139.881 841.025,139.884 841.018,139.889 C 841.018,139.895 841.027,139.905 841.021,139.91 C 841.021,139.92 841.021,139.932 841.021,139.942 C 841.031,139.941 841.024,139.952 841.031,139.958 C 841.025,139.962 841.032,139.976 841.023,139.979 L 841.023,139.988 L 841.018,139.988 C 841.024,139.998 841.034,140.008 841.031,140.024 C 841.037,140.028 841.027,140.045 841.037,140.05 C 841.037,140.061 841.016,140.072 841.025,140.087 C 841.017,140.088 841.031,140.096 841.023,140.098 C 841.025,140.107 841.03,140.113 841.039,140.118 C 841.039,140.14 841.024,140.16 841.027,140.18 C 841.034,140.186 841.03,140.2 841.037,140.206 C 841.024,140.227 841.039,140.25 841.037,140.274 C 841.041,140.277 841.038,140.281 841.037,140.283 C 841.038,140.289 841.03,140.302 841.039,140.306 C 841.034,140.316 841.052,140.314 841.049,140.324 C 841.05,140.324 841.052,140.32 841.056,140.323 C 841.059,140.33 841.053,140.336 841.049,140.341 L 841.043,140.34 C 841.039,140.358 841.027,140.372 841.027,140.391 C 841.038,140.402 841.031,140.414 841.03,140.428 C 841.031,140.441 841.037,140.455 841.038,140.465 C 841.03,140.471 841.039,140.476 841.038,140.481 C 841.045,140.487 841.032,140.505 841.046,140.507 C 841.045,140.515 841.041,140.522 841.049,140.528 C 841.049,140.537 841.046,140.543 841.046,140.552 C 841.059,140.559 841.046,140.57 841.059,140.579 C 841.057,140.583 841.057,140.589 841.052,140.59 C 841.05,140.59 841.046,140.587 841.045,140.59 C 841.034,140.608 841.037,140.631 841.046,140.646 C 841.05,140.66 841.041,140.673 841.038,140.688 C 841.037,140.703 841.046,140.708 841.052,140.716 C 841.059,140.714 841.062,140.72 841.06,140.729 C 841.057,140.735 841.052,140.741 841.057,140.75 L 841.059,140.75 C 841.059,140.754 841.06,140.766 841.065,140.77 L 841.06,140.775 C 841.069,140.78 841.056,140.786 841.059,140.789 C 841.052,140.789 841.052,140.797 841.052,140.802 C 841.06,140.807 841.05,140.819 841.064,140.819 C 841.072,140.829 841.064,140.844 841.074,140.853 C 841.072,140.859 841.069,140.868 841.072,140.875 C 841.079,140.882 841.09,140.888 841.087,140.9 C 841.08,140.907 841.09,140.916 841.094,140.927 C 841.087,140.947 841.097,140.959 841.093,140.978 C 841.107,140.989 841.087,141.003 841.104,141.01 C 841.098,141.015 841.1,141.023 841.1,141.029 C 841.111,141.039 841.097,141.052 841.107,141.061 C 841.105,141.068 841.117,141.079 841.11,141.086 C 841.113,141.092 841.104,141.108 841.113,141.112 C 841.124,141.133 841.091,141.147 841.113,141.164 C 841.111,141.169 841.115,141.17 841.118,141.171 C 841.113,141.183 841.135,141.186 841.12,141.2 C 841.124,141.206 841.117,141.211 841.12,141.218 C 841.153,141.222 841.115,141.253 841.135,141.268 C 841.139,141.28 841.111,141.28 841.128,141.295 C 841.12,141.305 841.135,141.305 841.131,141.315 C 841.12,141.315 841.124,141.327 841.117,141.335 C 841.115,141.34 841.12,141.34 841.125,141.341 C 841.118,141.352 841.132,141.362 841.125,141.371 C 841.12,141.386 841.136,141.394 841.126,141.413 C 841.117,141.419 841.113,141.424 841.111,141.437 C 841.113,141.442 841.115,141.451 841.125,141.45 C 841.118,141.463 841.138,141.471 841.125,141.483 C 841.105,141.497 841.111,141.528 841.1,141.548 C 841.098,141.56 841.11,141.564 841.118,141.568 C 841.129,141.566 841.124,141.578 841.129,141.584 C 841.132,141.6 841.098,141.586 841.104,141.614 C 841.11,141.629 841.104,141.651 841.098,141.669 L 841.11,141.704 C 841.12,141.713 841.105,141.731 841.118,141.744 C 841.126,141.748 841.124,141.753 841.124,141.76 C 841.11,141.756 841.115,141.771 841.105,141.771 C 841.117,141.775 841.097,141.778 841.106,141.786 C 841.093,141.805 841.129,141.815 841.11,141.828 C 841.118,141.839 841.097,141.842 841.107,141.848 C 841.106,141.848 841.105,141.852 841.107,141.853 C 841.105,141.865 841.106,141.877 841.104,141.889 C 841.104,141.893 841.111,141.894 841.105,141.9 C 841.104,141.915 841.097,141.937 841.1,141.953 C 841.093,141.957 841.098,141.963 841.093,141.967 C 841.104,141.972 841.086,141.973 841.094,141.979 C 841.083,141.991 841.105,142.01 841.093,142.024 C 841.087,142.037 841.105,142.043 841.09,142.05 C 841.083,142.057 841.086,142.073 841.083,142.081 C 841.083,142.09 841.08,142.107 841.097,142.111 L 841.098,142.108 C 841.094,142.116 841.097,142.124 841.093,142.129 C 841.083,142.136 841.08,142.15 841.079,142.164 C 841.084,142.173 841.08,142.186 841.087,142.196 C 841.086,142.204 841.08,142.217 841.087,142.226 L 841.083,142.231 L 841.086,142.237 C 841.076,142.24 841.09,142.244 841.084,142.25 C 841.097,142.25 841.09,142.264 841.098,142.269 C 841.098,142.278 841.08,142.278 841.093,142.287 C 841.087,142.296 841.09,142.304 841.083,142.31 C 841.084,142.311 841.084,142.315 841.083,142.319 C 841.083,142.33 841.074,142.345 841.084,142.352 C 841.083,142.357 841.083,142.367 841.09,142.366 C 841.086,142.378 841.091,142.392 841.093,142.406 C 841.105,142.404 841.102,142.415 841.107,142.42 C 841.111,142.429 841.117,142.446 841.1,142.445 L 841.094,142.454 C 841.1,142.458 841.093,142.462 841.098,142.466 C 841.102,142.475 841.102,142.485 841.098,142.496 C 841.111,142.493 841.1,142.508 841.111,142.511 C 841.111,142.515 841.105,142.518 841.11,142.52 C 841.106,142.522 841.102,142.523 841.104,142.528 L 841.106,142.53 C 841.1,142.528 841.1,142.538 841.094,142.542 C 841.091,142.549 841.102,142.555 841.098,142.563 C 841.098,142.568 841.104,142.568 841.106,142.569 C 841.107,142.58 841.1,142.594 841.107,142.6 C 841.105,142.606 841.11,142.61 841.115,142.611 C 841.111,142.63 841.093,142.642 841.094,142.663 C 841.102,142.671 841.097,142.682 841.105,142.69 C 841.105,142.698 841.1,142.702 841.094,142.704 L 841.098,142.704 C 841.097,142.706 841.087,142.709 841.093,142.717 C 841.091,142.719 841.09,142.719 841.09,142.72 C 841.117,142.733 841.087,142.756 841.1,142.777 C 841.094,142.785 841.086,142.791 841.087,142.801 C 841.093,142.818 841.097,142.833 841.098,142.85 C 841.106,142.854 841.097,142.866 841.107,142.869 C 841.104,142.888 841.098,142.911 841.087,142.928 C 841.087,142.931 841.09,142.933 841.087,142.936 C 841.083,142.942 841.104,142.955 841.09,142.964 L 841.084,142.967 C 841.091,142.974 841.094,142.982 841.105,142.988 C 841.105,142.993 841.102,143 841.105,143.006 C 841.1,143.019 841.08,143.02 841.087,143.036 C 841.097,143.042 841.087,143.054 841.087,143.062 C 841.083,143.068 841.086,143.077 841.09,143.082 C 841.09,143.098 841.074,143.116 841.087,143.13 C 841.09,143.138 841.079,143.149 841.09,143.154 L 841.084,143.155 C 841.094,143.162 841.084,143.166 841.083,143.174 C 841.08,143.176 841.087,143.181 841.08,143.185 C 841.08,143.195 841.094,143.199 841.093,143.211 C 841.093,143.21 841.093,143.211 841.094,143.211 C 841.079,143.211 841.084,143.23 841.079,143.242 C 841.078,143.251 841.091,143.248 841.083,143.256 C 841.087,143.261 841.086,143.264 841.09,143.272 C 841.08,143.274 841.09,143.276 841.087,143.283 C 841.084,143.289 841.098,143.303 841.084,143.304 C 841.091,143.306 841.084,143.316 841.087,143.321 C 841.08,143.325 841.086,143.332 841.078,143.332 L 841.08,143.34 L 841.078,143.34 C 841.086,143.345 841.076,143.355 841.083,143.359 C 841.078,143.361 841.083,143.366 841.084,143.368 C 841.08,143.382 841.07,143.394 841.083,143.407 C 841.083,143.413 841.084,143.422 841.087,143.428 C 841.084,143.435 841.08,143.442 841.084,143.45 C 841.09,143.456 841.093,143.461 841.1,143.462 C 841.098,143.47 841.094,143.48 841.094,143.485 C 841.104,143.492 841.106,143.497 841.107,143.507 C 841.106,143.519 841.098,143.531 841.102,143.542 L 841.098,143.548 C 841.107,143.553 841.124,143.554 841.135,143.554 C 841.131,143.553 841.135,143.552 841.135,143.549 C 841.145,143.554 841.165,143.558 841.181,143.552 C 841.19,143.552 841.195,143.542 841.197,143.537 C 841.217,143.54 841.245,143.531 841.271,143.533 C 841.299,143.542 841.315,143.528 841.343,143.542 C 841.355,143.545 841.365,143.54 841.373,143.542 C 841.392,143.538 841.411,143.563 841.429,143.545 C 841.444,143.542 841.458,143.556 841.468,143.548 C 841.473,143.544 841.485,143.556 841.487,143.544 C 841.492,143.548 841.499,143.542 841.506,143.542 L 841.507,143.538 C 841.516,143.54 841.532,143.542 841.544,143.544 L 841.55,143.538 C 841.552,143.544 841.559,143.538 841.565,143.538 C 841.58,143.524 841.596,143.549 841.617,143.542 C 841.625,143.531 841.639,143.545 841.652,143.544 C 841.673,143.535 841.692,143.553 841.711,143.542 C 841.719,143.54 841.732,143.552 841.74,143.54 C 841.749,143.54 841.768,143.549 841.782,143.538 C 841.801,143.544 841.81,143.519 841.833,143.528 L 841.834,143.527 C 841.84,143.533 841.842,143.52 841.847,143.528 C 841.865,143.535 841.888,143.531 841.905,143.542 C 841.911,143.54 841.916,143.542 841.922,143.537 C 841.929,143.538 841.938,143.537 841.944,143.533 L 841.947,143.538 C 841.958,143.532 841.978,143.544 841.993,143.537 C 841.998,143.538 842.005,143.54 842.012,143.538 C 842.008,143.524 842.024,143.538 842.026,143.527 C 842.03,143.537 842.037,143.52 842.037,143.532 C 842.064,143.535 842.091,143.532 842.116,143.535 C 842.12,143.528 842.133,143.535 842.133,143.524 L 842.173,143.531 C 842.176,143.516 842.179,143.535 842.186,143.528 C 842.202,143.552 842.228,143.544 842.254,143.553 C 842.258,143.544 842.269,143.553 842.275,143.542 C 842.288,143.538 842.306,143.556 842.31,143.542 C 842.317,143.52 842.336,143.548 842.344,143.531 C 842.35,143.531 842.357,143.531 842.357,143.524 C 842.362,143.527 842.374,143.524 842.38,143.528 L 842.383,143.533 C 842.386,143.524 842.392,143.535 842.398,143.533 C 842.398,143.535 842.4,143.535 842.399,143.538 C 842.405,143.531 842.416,143.542 842.414,143.528 C 842.432,143.524 842.447,143.544 842.465,143.532 C 842.469,143.535 842.477,143.532 842.477,143.527 C 842.485,143.527 842.5,143.531 842.507,143.527 C 842.513,143.531 842.519,143.532 842.524,143.532 C 842.536,143.524 842.548,143.537 842.561,143.537 C 842.568,143.537 842.576,143.532 842.58,143.531 C 842.586,143.524 842.594,143.512 842.606,143.516 C 842.612,143.507 842.622,143.511 842.63,143.51 C 842.636,143.511 842.633,143.52 842.639,143.519 C 842.633,143.528 842.651,143.527 842.655,143.532 C 842.659,143.524 842.669,143.535 842.669,143.527 C 842.666,143.537 842.681,143.535 842.686,143.537 L 842.696,143.537 L 842.696,143.535 L 842.703,143.54 L 842.706,143.537 C 842.706,143.542 842.714,143.54 842.716,143.54 C 842.727,143.542 842.738,143.556 842.747,143.545 C 842.762,143.545 842.777,143.542 842.794,143.544 L 842.797,143.542 C 842.803,143.545 842.807,143.538 842.811,143.542 C 842.821,143.544 842.826,143.533 842.831,143.528 C 842.842,143.528 842.852,143.527 842.863,143.533 L 842.867,143.528 C 842.869,143.533 842.873,143.533 842.876,143.532 C 842.898,143.538 842.921,143.532 842.934,143.54 L 842.939,143.537 C 842.945,143.538 842.934,143.544 842.942,143.545 L 842.947,143.538 C 842.954,143.544 842.96,143.535 842.967,143.533 L 842.99,143.542 C 842.99,143.545 842.99,143.548 842.993,143.548 L 842.997,143.544 C 843.005,143.544 843.018,143.548 843.026,143.542 C 843.045,143.54 843.067,143.556 843.089,143.548 C 843.115,143.556 843.136,143.542 843.163,143.553 L 843.166,143.548 C 843.176,143.56 843.198,143.56 843.215,143.568 C 843.218,143.563 843.228,143.572 843.234,143.564 C 843.235,143.574 843.242,143.559 843.245,143.57 C 843.263,143.58 843.272,143.553 843.291,143.568 C 843.306,143.577 843.315,143.563 843.324,143.559 C 843.324,143.56 843.326,143.56 843.326,143.563 C 843.35,143.559 843.386,143.58 843.401,143.559 C 843.422,143.564 843.439,143.542 843.456,143.54 C 843.46,143.54 843.458,143.533 843.461,143.537 C 843.464,143.542 843.466,143.538 843.468,143.537 C 843.469,143.548 843.482,143.549 843.488,143.549 C 843.498,143.545 843.499,143.549 843.505,143.548 C 843.532,143.553 843.56,143.56 843.589,143.564 C 843.594,143.558 843.602,143.564 843.607,143.563 C 843.612,143.554 843.618,143.545 843.629,143.54 C 843.663,143.544 843.695,143.56 843.73,143.563 L 843.734,143.558 C 843.737,143.565 843.747,143.556 843.751,143.558 C 843.757,143.549 843.767,143.554 843.774,143.556 C 843.785,143.549 843.8,143.54 843.808,143.556 C 843.814,143.553 843.822,143.56 843.824,143.554 C 843.836,143.559 843.855,143.574 843.87,143.564 C 843.888,143.57 843.906,143.572 843.926,143.575 L 843.933,143.57 C 843.951,143.563 843.969,143.568 843.988,143.575 C 843.997,143.56 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840.924,131.127 C 840.939,131.126 840.934,131.107 840.939,131.094 C 840.938,131.078 840.924,131.081 840.918,131.065 C 840.921,131.053 840.894,131.057 840.896,131.041 C 840.905,131.036 840.905,131.026 840.907,131.02 C 840.901,131.015 840.907,131.003 840.907,130.998 C 840.891,130.986 840.913,130.972 840.901,130.959 C 840.899,130.946 840.899,130.929 840.899,130.918 C 840.889,130.906 840.899,130.89 840.892,130.88 C 840.896,130.872 840.882,130.875 840.883,130.866 C 840.889,130.854 840.887,130.842 840.883,130.829 C 840.889,130.819 840.892,130.808 840.892,130.796 C 840.892,130.781 840.876,130.783 840.869,130.776 C 840.865,130.767 840.879,130.757 840.865,130.756 C 840.868,130.752 840.862,130.742 840.872,130.74 C 840.873,130.732 840.873,130.719 840.866,130.715 L 840.883,130.693 C 840.879,130.69 840.883,130.686 840.883,130.683 C 840.869,130.679 840.879,130.673 840.876,130.662 C 840.873,130.652 840.865,130.652 840.859,130.643 C 840.865,130.636 840.861,130.622 840.873,130.622 C 840.887,130.612 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841.136,127.347 L 841.139,127.346 C 841.143,127.346 841.138,127.351 841.143,127.353 C 841.156,127.356 841.17,127.365 841.184,127.357 C 841.21,127.366 841.225,127.351 841.251,127.36 C 841.256,127.353 841.273,127.363 841.275,127.353 C 841.288,127.356 841.301,127.343 841.31,127.341 C 841.322,127.347 841.336,127.33 841.346,127.346 C 841.35,127.339 841.355,127.346 841.362,127.346 C 841.369,127.363 841.391,127.357 841.406,127.363 C 841.412,127.359 841.407,127.357 841.414,127.36 C 841.429,127.354 841.446,127.373 841.459,127.36 C 841.473,127.359 841.488,127.346 841.495,127.33 C 841.53,127.336 841.567,127.342 841.597,127.359 C 841.603,127.359 841.611,127.359 841.618,127.357 C 841.622,127.354 841.632,127.354 841.638,127.353 C 841.638,127.343 841.653,127.354 841.653,127.342 C 841.671,127.349 841.684,127.325 841.701,127.334 C 841.711,127.319 841.73,127.335 841.745,127.335 C 841.749,127.336 841.757,127.336 841.756,127.33 C 841.764,127.332 841.776,127.334 841.782,127.342 L 841.787,127.336 C 841.791,127.347 841.795,127.341 841.8,127.351 C 841.815,127.356 841.83,127.356 841.843,127.351 C 841.846,127.351 841.846,127.346 841.843,127.346 L 841.847,127.346 C 841.856,127.343 841.857,127.335 841.86,127.328 C 841.866,127.319 841.875,127.321 841.882,127.325 C 841.882,127.339 841.901,127.335 841.912,127.341 C 841.92,127.328 841.936,127.343 841.947,127.332 C 841.972,127.336 841.998,127.341 842.025,127.347 L 842.027,127.342 L 842.031,127.347 C 842.037,127.342 842.046,127.334 842.058,127.339 C 842.072,127.325 842.091,127.346 842.102,127.334 C 842.117,127.351 842.14,127.351 842.164,127.356 C 842.165,127.346 842.173,127.354 842.181,127.353 C 842.191,127.354 842.196,127.347 842.199,127.341 C 842.206,127.339 842.211,127.342 842.211,127.351 C 842.221,127.356 842.228,127.346 842.237,127.347 C 842.243,127.339 842.257,127.347 842.258,127.335 C 842.275,127.335 842.27,127.311 842.288,127.311 C 842.302,127.309 842.293,127.326 842.307,127.325 C 842.309,127.321 842.32,127.319 842.316,127.311 C 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842.807,127.321 L 842.808,127.326 C 842.811,127.33 842.815,127.325 842.818,127.325 C 842.839,127.335 842.855,127.325 842.873,127.319 C 842.882,127.316 842.874,127.308 842.874,127.303 C 842.889,127.295 842.907,127.316 842.921,127.303 C 842.935,127.301 842.949,127.315 842.964,127.303 C 842.966,127.309 842.971,127.303 842.975,127.308 C 842.972,127.312 842.979,127.311 842.98,127.312 C 842.987,127.321 842.999,127.323 843.012,127.325 C 843.025,127.311 843.041,127.334 843.063,127.326 C 843.072,127.316 843.084,127.332 843.094,127.319 C 843.115,127.321 843.136,127.33 843.153,127.319 C 843.157,127.319 843.164,127.312 843.164,127.319 L 843.162,127.321 C 843.172,127.335 843.186,127.323 843.196,127.326 C 843.215,127.312 843.232,127.325 843.249,127.332 C 843.25,127.342 843.263,127.343 843.27,127.349 C 843.276,127.349 843.284,127.351 843.291,127.347 C 843.291,127.335 843.306,127.349 843.308,127.336 C 843.321,127.339 843.333,127.335 843.348,127.326 C 843.359,127.325 843.357,127.339 843.365,127.341 C 843.365,127.353 843.383,127.353 843.386,127.363 L 843.389,127.357 L 843.394,127.365 L 843.403,127.359 C 843.409,127.346 843.422,127.359 843.432,127.353 C 843.436,127.36 843.449,127.354 843.456,127.351 C 843.471,127.342 843.487,127.356 843.495,127.359 C 843.508,127.341 843.528,127.363 843.544,127.351 C 843.564,127.365 843.559,127.323 843.581,127.332 C 843.599,127.346 843.625,127.359 843.65,127.346 C 843.652,127.332 843.67,127.347 843.675,127.336 C 843.684,127.335 843.698,127.339 843.703,127.346 C 843.706,127.342 843.706,127.349 843.709,127.351 C 843.716,127.354 843.727,127.36 843.734,127.351 C 843.751,127.357 843.754,127.335 843.767,127.332 C 843.782,127.316 843.804,127.328 843.822,127.325 C 843.827,127.343 843.847,127.332 843.856,127.335 C 843.862,127.334 843.877,127.342 843.881,127.33 C 843.882,127.332 843.882,127.334 843.882,127.335 C 843.887,127.33 843.899,127.334 843.899,127.326 C 843.908,127.325 843.92,127.334 843.929,127.325 C 843.938,127.326 843.942,127.326 843.948,127.334 L 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844.291,128.216 C 844.287,128.231 844.296,128.238 844.296,128.253 C 844.307,128.258 844.296,128.272 844.303,128.28 C 844.297,128.288 844.289,128.288 844.282,128.29 C 844.298,128.316 844.271,128.343 844.282,128.368 C 844.291,128.377 844.275,128.383 844.276,128.392 C 844.258,128.392 844.265,128.414 844.251,128.421 C 844.257,128.424 844.254,128.429 844.254,128.43 C 844.261,128.443 844.261,128.459 844.255,128.473 C 844.265,128.487 844.25,128.501 844.264,128.51 C 844.255,128.522 844.269,128.527 844.265,128.538 C 844.262,128.552 844.261,128.568 844.261,128.579 C 844.258,128.587 844.249,128.587 844.244,128.589 C 844.242,128.595 844.234,128.596 844.239,128.601 C 844.23,128.619 844.238,128.631 844.239,128.648 C 844.25,128.649 844.238,128.66 844.248,128.663 C 844.25,128.672 844.254,128.683 844.254,128.693 C 844.264,128.703 844.261,128.719 844.255,128.731 C 844.269,128.74 844.248,128.754 844.255,128.766 C 844.255,128.779 844.239,128.797 844.254,128.806 C 844.248,128.808 844.251,128.817 844.251,128.825 C 844.251,128.838 844.235,128.833 844.235,128.845 C 844.244,128.855 844.235,128.874 844.23,128.886 C 844.228,128.89 844.219,128.893 844.224,128.897 C 844.219,128.901 844.234,128.907 844.224,128.912 C 844.23,128.913 844.223,128.925 844.23,128.922 C 844.234,128.925 844.231,128.92 844.234,128.918 C 844.231,128.928 844.239,128.92 844.244,128.921 C 844.258,128.933 844.239,128.955 844.239,128.969 C 844.234,128.976 844.235,128.981 844.234,128.988 C 844.223,128.987 844.234,128.997 844.228,128.999 C 844.228,129.009 844.235,129.014 844.244,129.021 L 844.228,129.032 C 844.223,129.04 844.234,129.045 844.234,129.056 C 844.244,129.061 844.244,129.066 844.244,129.076 C 844.242,129.087 844.255,129.088 844.255,129.097 C 844.264,129.104 844.268,129.117 844.28,129.119 C 844.275,129.129 844.257,129.125 844.255,129.138 C 844.255,129.143 844.254,129.15 844.258,129.156 C 844.255,129.166 844.245,129.169 844.235,129.175 C 844.244,129.185 844.228,129.19 844.238,129.197 L 844.235,129.201 C 844.238,129.21 844.248,129.222 844.238,129.233 C 844.235,129.243 844.249,129.245 844.254,129.25 C 844.255,129.267 844.276,129.261 844.269,129.28 C 844.276,129.288 844.268,129.298 844.276,129.306 C 844.28,129.32 844.262,129.322 844.258,129.333 C 844.249,129.332 844.254,129.342 844.249,129.346 C 844.255,129.351 844.264,129.355 844.272,129.356 C 844.276,129.361 844.265,129.376 844.28,129.376 C 844.269,129.394 844.287,129.393 844.285,129.408 C 844.28,129.416 844.268,129.415 844.258,129.42 C 844.257,129.43 844.269,129.434 844.275,129.439 C 844.275,129.446 844.269,129.457 844.261,129.459 C 844.262,129.463 844.254,129.468 844.261,129.473 C 844.255,129.477 844.262,129.486 844.261,129.491 C 844.271,129.489 844.262,129.503 844.272,129.505 C 844.268,129.519 844.271,129.534 844.255,129.543 C 844.254,129.548 844.251,129.556 844.258,129.562 C 844.254,129.578 844.257,129.594 844.258,129.606 C 844.271,129.609 844.255,129.626 844.271,129.626 C 844.262,129.639 844.275,129.653 844.28,129.667 C 844.276,129.674 844.271,129.678 844.275,129.686 L 844.269,129.686 C 844.276,129.692 844.265,129.705 844.28,129.705 C 844.276,129.722 844.289,129.736 844.297,129.75 C 844.314,129.767 844.282,129.772 844.289,129.792 C 844.296,129.797 844.296,129.805 844.3,129.812 C 844.297,129.83 844.313,129.842 844.3,129.857 C 844.285,129.879 844.315,129.877 844.317,129.893 C 844.32,129.907 844.307,129.904 844.302,129.911 L 844.298,129.911 C 844.283,129.923 844.309,129.931 844.297,129.945 C 844.303,129.954 844.297,129.968 844.303,129.979 C 844.3,129.995 844.315,129.994 844.309,130.01 C 844.314,130.016 844.314,130.029 844.321,130.037 C 844.314,130.044 844.324,130.06 844.313,130.068 C 844.313,130.073 844.31,130.082 844.317,130.086 C 844.331,130.109 844.297,130.113 844.307,130.137 C 844.313,130.141 844.313,130.149 844.32,130.145 C 844.32,130.154 844.328,130.168 844.32,130.176 C 844.305,130.178 844.31,130.197 844.307,130.206 L 844.322,130.227 C 844.32,130.244 844.3,130.249 844.297,130.266 C 844.297,130.272 844.297,130.279 844.297,130.286 C 844.309,130.287 844.294,130.302 844.307,130.301 C 844.309,130.313 844.327,130.313 844.331,130.32 C 844.331,130.327 844.329,130.332 844.328,130.341 C 844.321,130.34 844.315,130.348 844.31,130.342 L 844.307,130.349 C 844.305,130.355 844.302,130.369 844.31,130.372 C 844.309,130.377 844.309,130.387 844.305,130.396 C 844.309,130.402 844.32,130.398 844.321,130.406 C 844.32,130.413 844.317,130.424 844.31,130.429 C 844.321,130.443 844.291,130.453 844.29,130.46 C 844.289,130.465 844.282,130.473 844.291,130.477 L 844.289,130.481 C 844.297,130.486 844.291,130.496 844.302,130.498 C 844.302,130.504 844.302,130.512 844.291,130.512 C 844.294,130.517 844.285,130.52 844.29,130.525 C 844.303,130.524 844.29,130.539 844.302,130.538 C 844.31,130.539 844.321,130.543 844.321,130.553 C 844.321,130.567 844.31,130.56 844.302,130.56 C 844.297,130.569 844.29,130.58 844.29,130.591 C 844.294,130.597 844.302,130.597 844.307,130.6 C 844.313,130.605 844.315,130.614 844.31,130.621 C 844.285,130.622 844.309,130.663 844.282,130.669 C 844.28,130.683 844.283,130.693 844.282,130.709 C 844.305,130.717 844.283,130.745 844.296,130.757 C 844.291,130.757 844.29,130.756 844.29,130.757 C 844.289,130.761 844.29,130.762 844.291,130.766 L 844.29,130.767 C 844.283,130.785 844.302,130.792 844.31,130.804 C 844.309,130.811 844.297,130.807 844.291,130.808 C 844.291,130.814 844.283,130.818 844.289,130.825 C 844.287,130.844 844.268,130.842 844.269,130.86 C 844.278,130.869 844.269,130.881 844.272,130.895 C 844.28,130.897 844.271,130.902 844.272,130.906 C 844.28,130.912 844.262,130.911 844.271,130.918 C 844.262,130.932 844.287,130.937 844.283,130.95 C 844.272,130.96 844.282,130.983 844.268,130.993 C 844.272,130.998 844.258,130.998 844.265,131.005 L 844.261,131.011 C 844.269,131.03 844.251,131.055 844.254,131.078 C 844.262,131.085 844.264,131.097 844.271,131.108 C 844.264,131.112 844.265,131.118 844.265,131.122 C 844.258,131.13 844.262,131.146 844.268,131.156 C 844.262,131.175 844.261,131.169 844.244,131.177 C 844.228,131.19 844.255,131.201 844.245,131.213 C 844.255,131.22 844.242,131.235 844.251,131.243 L 844.249,131.27 C 844.258,131.27 844.25,131.279 844.255,131.282 C 844.245,131.306 844.257,131.327 844.245,131.349 C 844.242,131.349 844.235,131.351 844.235,131.357 C 844.234,131.364 844.226,131.374 844.237,131.377 L 844.234,131.405 C 844.237,131.406 844.238,131.408 844.238,131.414 C 844.245,131.416 844.245,131.42 844.245,131.429 C 844.235,131.444 844.255,131.454 844.254,131.468 C 844.276,131.482 844.237,131.486 844.245,131.507 C 844.242,131.513 844.254,131.519 844.249,131.524 C 844.254,131.542 844.248,131.56 844.249,131.579 C 844.248,131.586 844.239,131.591 844.248,131.599 C 844.248,131.605 844.242,131.615 844.244,131.623 C 844.248,131.626 844.251,131.63 844.249,131.636 C 844.242,131.645 844.255,131.657 844.244,131.661 C 844.239,131.665 844.237,131.674 844.239,131.679 C 844.245,131.688 844.242,131.699 844.248,131.708 C 844.244,131.71 844.244,131.713 844.245,131.716 C 844.239,131.737 844.251,131.756 844.25,131.776 C 844.249,131.778 844.245,131.787 844.25,131.788 C 844.248,131.792 844.257,131.799 844.25,131.801 C 844.25,131.809 844.248,131.817 844.249,131.826 C 844.255,131.832 844.251,131.84 844.251,131.851 L 844.257,131.851 C 844.251,131.852 844.257,131.855 844.257,131.859 C 844.257,131.868 844.25,131.87 844.245,131.874 C 844.248,131.874 844.248,131.876 844.249,131.876 C 844.25,131.886 844.249,131.897 844.244,131.907 C 844.239,131.916 844.248,131.924 844.245,131.93 C 844.254,131.942 844.238,131.949 844.239,131.959 C 844.25,131.972 844.245,131.992 844.238,132.008 C 844.239,132.015 844.239,132.025 844.248,132.028 L 844.249,132.027 C 844.249,132.033 844.248,132.042 844.245,132.052 C 844.245,132.059 844.237,132.072 844.248,132.076 C 844.251,132.089 844.242,132.095 844.239,132.108 C 844.242,132.124 844.25,132.134 844.239,132.149 C 844.249,132.156 844.248,132.168 844.25,132.176 C 844.244,132.187 844.248,132.199 844.239,132.208 C 844.244,132.215 844.235,132.224 844.239,132.231 C 844.257,132.234 844.254,132.246 844.251,132.258 C 844.265,132.266 844.257,132.286 844.269,132.294 C 844.269,132.3 844.264,132.304 844.258,132.305 C 844.249,132.32 844.268,132.327 844.265,132.341 C 844.254,132.341 844.264,132.356 844.261,132.361 C 844.258,132.369 844.269,132.377 844.264,132.386 C 844.264,132.387 844.268,132.39 844.269,132.39 C 844.268,132.402 844.287,132.396 844.282,132.41 C 844.29,132.418 844.275,132.432 844.278,132.442 L 844.272,132.446 C 844.282,132.452 844.269,132.456 844.278,132.463 L 844.282,132.462 C 844.287,132.465 844.282,132.472 844.29,132.474 C 844.296,132.484 844.298,132.494 844.291,132.507 C 844.28,132.504 844.282,132.514 844.276,132.519 C 844.282,132.533 844.285,132.542 844.276,132.56 C 844.289,132.586 844.261,132.613 844.276,132.642 C 844.271,132.652 844.269,132.667 844.269,132.674 C 844.28,132.681 844.282,132.686 844.28,132.699 C 844.283,132.713 844.278,132.718 844.276,132.732 C 844.261,132.74 844.275,132.76 844.275,132.773 L 844.276,132.771 C 844.275,132.769 844.272,132.775 844.276,132.778 C 844.271,132.792 844.287,132.8 844.278,132.813 C 844.254,132.821 844.282,132.828 844.272,132.846 C 844.285,132.854 844.262,132.877 844.28,132.881 C 844.269,132.897 844.28,132.91 844.29,132.922 C 844.283,132.933 844.3,132.93 844.294,132.939 C 844.307,132.945 844.297,132.956 844.302,132.968 L 844.29,132.979 C 844.296,132.98 844.285,132.993 844.297,132.994 C 844.294,133.002 844.313,133.004 844.302,133.011 C 844.321,133.025 844.298,133.044 844.305,133.061 C 844.317,133.071 844.297,133.089 844.31,133.098 C 844.313,133.105 844.309,133.111 844.315,133.119 C 844.313,133.124 844.31,133.132 844.309,133.138 C 844.29,133.144 844.309,133.16 844.302,133.173 C 844.317,133.185 844.302,133.198 844.302,133.212 C 844.314,133.217 844.305,133.225 844.305,133.234 C 844.297,133.239 844.3,133.243 844.297,133.248 C 844.302,133.256 844.296,133.266 844.294,133.275 C 844.294,133.284 844.283,133.296 844.296,133.302 C 844.29,133.317 844.297,133.333 844.303,133.343 C 844.303,133.356 844.296,133.353 844.291,133.362 C 844.291,133.375 844.296,133.389 844.3,133.404 C 844.297,133.419 844.303,133.429 844.302,133.445 L 844.305,133.445 L 844.302,133.447 C 844.31,133.466 844.291,133.48 844.298,133.503 C 844.282,133.519 844.307,133.527 844.313,133.538 L 844.31,133.54 C 844.297,133.559 844.317,133.577 844.317,133.598 C 844.313,133.61 844.328,133.62 844.327,133.631 C 844.327,133.637 844.321,133.647 844.322,133.654 C 844.337,133.655 844.324,133.675 844.334,133.683 C 844.334,133.694 844.32,133.689 844.321,133.699 z M 843.357,133.598 C 843.354,133.606 843.361,133.615 843.362,133.623 L 843.367,133.631 C 843.365,133.631 843.366,133.634 843.366,133.637 C 843.376,133.641 843.377,133.652 843.388,133.655 C 843.383,133.667 843.401,133.66 843.399,133.672 C 843.406,133.678 843.419,133.681 843.422,133.689 C 843.431,133.694 843.441,133.699 843.449,133.704 C 843.454,133.704 843.458,133.71 843.464,133.71 C 843.478,133.704 843.493,133.704 843.506,133.694 C 843.512,133.679 843.551,133.689 843.535,133.663 C 843.544,133.642 843.537,133.621 843.535,133.6 C 843.526,133.589 843.54,133.575 843.53,133.565 C 843.535,133.559 843.53,133.551 843.532,133.546 L 843.526,133.545 C 843.532,133.538 843.525,133.532 843.522,133.528 C 843.514,133.524 843.504,133.517 843.495,133.517 C 843.488,133.502 843.466,133.506 843.452,133.498 C 843.439,133.512 843.422,133.493 843.409,133.503 C 843.403,133.502 843.395,133.506 843.389,133.509 C 843.394,133.524 843.375,133.511 843.375,133.524 C 843.362,133.527 843.361,133.545 843.356,133.551 C 843.359,133.565 843.34,133.586 843.357,133.598 z M 841.106,131.036 L 841.111,131.038 C 841.107,131.038 841.11,131.043 841.11,131.046 C 841.12,131.043 841.12,131.056 841.125,131.061 L 841.124,131.084 L 841.129,131.082 C 841.128,131.085 841.125,131.087 841.129,131.09 C 841.13,131.097 841.136,131.088 841.138,131.094 C 841.136,131.108 841.124,131.127 841.136,131.143 C 841.132,131.16 841.142,131.169 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842.648,127.512 C 842.626,127.496 842.596,127.513 842.572,127.504 C 842.566,127.518 842.559,127.537 842.559,127.551 C 842.572,127.564 842.551,127.581 842.566,127.594 C 842.57,127.605 842.576,127.613 842.572,127.622 C 842.564,127.63 842.576,127.636 842.572,127.641 C 842.57,127.657 842.581,127.664 842.586,127.677 C 842.59,127.679 842.593,127.684 842.593,127.689 C 842.607,127.701 842.612,127.715 842.631,127.72 C 842.633,127.733 842.651,127.73 842.649,127.745 C 842.663,127.757 842.685,127.754 842.701,127.754 C 842.708,127.75 842.721,127.759 842.724,127.746 C 842.737,127.746 842.737,127.741 842.751,127.736 C 842.776,127.73 842.761,127.698 842.776,127.684 C 842.778,127.668 842.782,127.647 842.797,127.643 C 842.807,127.63 842.822,127.627 842.834,127.616 C 842.846,127.623 842.86,127.606 842.865,127.62 L 842.868,127.615 C 842.87,127.618 842.868,127.623 842.874,127.622 C 842.876,127.612 842.876,127.63 842.883,127.626 C 842.896,127.63 842.902,127.641 842.912,127.647 C 842.92,127.664 842.913,127.689 842.918,127.699 C 842.901,127.707 842.909,127.733 842.896,127.745 C 842.896,127.746 842.893,127.748 842.892,127.746 C 842.887,127.757 842.87,127.754 842.87,127.767 C 842.859,127.771 842.853,127.784 842.844,127.789 C 842.828,127.782 842.835,127.805 842.821,127.798 L 842.818,127.803 C 842.807,127.798 842.797,127.812 842.787,127.817 C 842.782,127.834 842.767,127.815 842.76,127.829 C 842.745,127.824 842.741,127.837 842.725,127.833 C 842.721,127.837 842.715,127.829 842.711,127.837 C 842.704,127.837 842.695,127.828 842.69,127.833 C 842.685,127.821 842.678,127.831 842.675,127.822 C 842.669,127.813 842.652,127.824 842.649,127.817 C 842.644,127.824 842.642,127.812 842.639,127.813 C 842.617,127.817 842.597,127.798 842.579,127.788 C 842.572,127.782 842.56,127.779 842.552,127.772 C 842.552,127.763 842.54,127.759 842.534,127.751 C 842.527,127.748 842.524,127.734 842.513,127.733 C 842.515,127.723 842.502,127.728 842.504,127.719 C 842.482,127.707 842.492,127.679 842.471,127.664 C 842.462,127.66 842.46,127.647 842.456,127.641 C 842.447,127.636 842.451,127.622 842.443,127.615 C 842.44,127.596 842.436,127.579 842.438,127.56 C 842.423,127.554 842.438,127.537 842.428,127.53 C 842.438,127.532 842.429,127.52 842.434,127.518 C 842.425,127.509 842.443,127.508 842.433,127.501 C 842.433,127.502 842.423,127.502 842.428,127.508 L 842.425,127.512 C 842.421,127.508 842.419,127.509 842.415,127.512 L 842.411,127.516 C 842.413,127.514 842.411,127.514 842.408,127.513 C 842.402,127.514 842.386,127.512 842.382,127.52 C 842.379,127.516 842.37,127.518 842.363,127.518 L 842.362,127.523 C 842.36,127.512 842.353,127.527 842.347,127.523 C 842.343,127.533 842.331,127.525 842.33,127.533 L 842.289,127.529 C 842.291,127.532 842.288,127.537 842.284,127.537 C 842.288,127.543 842.283,127.557 842.291,127.56 C 842.295,127.574 842.283,127.584 842.293,127.594 C 842.286,127.597 842.295,127.605 842.286,127.606 L 842.288,127.613 L 842.284,127.615 C 842.286,127.626 842.28,127.636 842.284,127.647 C 842.288,127.665 842.275,127.689 842.281,127.707 C 842.283,127.719 842.273,127.728 842.275,127.736 L 842.269,127.742 L 842.273,127.745 C 842.263,127.748 842.275,127.76 842.262,127.76 L 842.263,127.767 L 842.26,127.765 C 842.255,127.771 842.254,127.781 842.249,127.786 C 842.241,127.788 842.237,127.801 842.235,127.806 C 842.231,127.805 842.224,127.806 842.224,127.813 C 842.221,127.815 842.206,127.81 842.21,127.822 C 842.199,127.812 842.202,127.834 842.189,127.829 C 842.188,127.833 842.188,127.834 842.185,127.837 C 842.168,127.833 842.151,127.85 842.136,127.854 C 842.124,127.859 842.112,127.854 842.099,127.863 L 842.051,127.855 L 842.047,127.859 C 842.033,127.837 841.998,127.846 841.986,127.822 C 841.973,127.819 841.96,127.817 841.958,127.805 C 841.946,127.802 841.946,127.789 841.933,127.789 C 841.927,127.789 841.925,127.791 841.923,127.797 L 841.92,127.793 C 841.905,127.803 841.916,127.817 841.902,127.829 C 841.892,127.837 841.892,127.848 841.876,127.854 C 841.863,127.848 841.843,127.85 841.835,127.834 C 841.821,127.831 841.813,127.822 841.801,127.813 C 841.761,127.797 841.709,127.803 841.689,127.76 C 841.697,127.748 841.705,127.738 841.721,127.736 C 841.734,127.742 841.745,127.734 841.757,127.728 C 841.757,127.716 841.778,127.715 841.768,127.699 C 841.771,127.689 841.77,127.668 841.774,127.654 C 841.771,127.647 841.775,127.641 841.771,127.632 C 841.771,127.63 841.771,127.626 841.768,127.622 C 841.774,127.608 841.77,127.589 841.768,127.575 C 841.751,127.571 841.734,127.57 841.721,127.557 C 841.714,127.543 841.695,127.552 841.689,127.537 C 841.668,127.533 841.65,127.551 841.629,127.543 C 841.622,127.546 841.619,127.541 841.613,127.544 C 841.609,127.552 841.611,127.564 841.616,127.57 C 841.609,127.591 841.605,127.612 841.597,127.632 C 841.592,127.649 841.591,127.676 841.574,127.688 C 841.57,127.705 841.553,127.706 841.552,127.723 C 841.539,127.726 841.544,127.745 841.53,127.742 L 841.521,127.759 C 841.521,127.757 841.518,127.755 841.519,127.754 C 841.492,127.754 841.521,127.779 841.5,127.782 C 841.488,127.789 841.478,127.802 841.464,127.803 C 841.453,127.809 841.446,127.817 841.432,127.819 C 841.421,127.821 841.414,127.831 841.403,127.828 C 841.385,127.837 841.359,127.831 841.347,127.85 C 841.332,127.85 841.311,127.834 841.296,127.846 C 841.29,127.844 841.281,127.844 841.277,127.85 C 841.273,127.851 841.27,127.844 841.266,127.848 L 841.258,127.843 L 841.255,127.844 C 841.255,127.837 841.249,127.843 841.245,127.843 C 841.228,127.843 841.207,127.834 841.192,127.825 L 841.19,127.829 C 841.19,127.815 841.17,127.819 841.17,127.809 C 841.156,127.797 841.135,127.801 841.125,127.786 C 841.111,127.782 841.111,127.765 841.098,127.761 C 841.103,127.76 841.098,127.757 841.096,127.755 C 841.083,127.755 841.094,127.741 841.083,127.739 C 841.09,127.733 841.076,127.723 841.073,127.715 L 841.069,127.716 C 841.064,127.707 841.077,127.695 841.062,127.693 C 841.07,127.683 841.05,127.688 841.056,127.677 C 841.05,127.671 841.058,127.664 841.049,127.658 L 841.036,127.662 L 841.041,127.667 L 841.036,127.668 C 841.041,127.671 841.036,127.687 841.046,127.687 C 841.05,127.693 841.032,127.701 841.046,127.71 C 841.045,127.71 841.045,127.714 841.045,127.715 C 841.046,127.716 841.05,127.715 841.051,127.719 L 841.045,127.72 C 841.049,127.73 841.041,127.748 841.056,127.751 C 841.053,127.755 841.058,127.757 841.059,127.76 C 841.062,127.775 841.045,127.781 841.053,127.793 C 841.059,127.802 841.08,127.794 841.077,127.812 C 841.077,127.824 841.062,127.828 841.069,127.842 C 841.08,127.844 841.069,127.848 841.069,127.855 L 841.056,127.855 L 841.056,127.863 L 841.053,127.863 L 841.053,127.869 L 841.056,127.869 C 841.058,127.875 841.051,127.889 841.062,127.894 C 841.058,127.904 841.076,127.898 841.069,127.909 L 841.084,127.927 C 841.08,127.934 841.084,127.948 841.072,127.948 C 841.07,127.952 841.069,127.955 841.069,127.958 C 841.072,127.96 841.072,127.962 841.073,127.964 C 841.077,127.978 841.059,127.993 841.072,128.006 C 841.066,128.01 841.072,128.014 841.069,128.02 L 841.077,128.025 C 841.073,128.03 841.077,128.043 841.066,128.048 C 841.076,128.057 841.056,128.056 841.066,128.062 L 841.062,128.066 C 841.066,128.072 841.064,128.082 841.059,128.088 C 841.062,128.099 841.053,128.112 841.053,128.123 L 841.058,128.13 C 841.054,128.135 841.054,128.145 841.054,128.154 C 841.056,128.161 841.062,128.172 841.058,128.185 C 841.064,128.189 841.064,128.192 841.059,128.196 L 841.058,128.217 C 841.07,128.214 841.064,128.231 841.072,128.234 C 841.066,128.238 841.07,128.244 841.072,128.249 C 841.079,128.257 841.07,128.262 841.08,128.263 C 841.084,128.272 841.072,128.283 841.084,128.296 C 841.085,128.306 841.076,128.316 841.084,128.325 C 841.073,128.335 841.065,128.349 841.069,128.363 C 841.084,128.372 841.065,128.382 841.073,128.391 L 841.069,128.398 C 841.07,128.395 841.077,128.401 841.073,128.404 C 841.072,128.414 841.08,128.428 841.087,128.437 C 841.085,128.447 841.073,128.468 841.087,128.475 C 841.077,128.476 841.087,128.482 841.084,128.487 C 841.094,128.487 841.096,128.473 841.11,128.476 C 841.11,128.465 841.125,128.472 841.129,128.464 C 841.148,128.464 841.163,128.443 841.182,128.445 C 841.192,128.437 841.204,128.449 841.214,128.437 C 841.225,128.444 841.243,128.443 841.26,128.444 L 841.263,128.441 C 841.267,128.441 841.267,128.447 841.273,128.443 C 841.301,128.451 841.328,128.447 841.352,128.465 C 841.352,128.478 841.373,128.473 841.374,128.487 C 841.383,128.494 841.399,128.491 841.4,128.506 C 841.414,128.508 841.411,128.522 841.424,128.525 C 841.422,128.549 841.448,128.549 841.455,128.57 C 841.467,128.573 841.459,128.586 841.469,128.587 C 841.466,128.596 841.466,128.608 841.471,128.614 C 841.467,128.626 841.477,128.634 841.477,128.645 C 841.478,128.646 841.481,128.646 841.484,128.652 C 841.48,128.665 841.492,128.686 841.473,128.691 C 841.469,128.705 841.453,128.694 841.446,128.7 C 841.432,128.69 841.426,128.675 841.421,128.662 C 841.412,128.649 841.407,128.634 841.395,128.628 C 841.395,128.614 841.378,128.617 841.369,128.609 C 841.362,128.586 841.333,128.589 841.322,128.573 C 841.301,128.573 841.284,128.573 841.263,128.568 L 841.26,128.573 C 841.243,128.564 841.231,128.58 841.214,128.573 C 841.21,128.58 841.197,128.573 841.197,128.584 C 841.19,128.589 841.176,128.589 841.169,128.6 C 841.163,128.603 841.156,128.607 841.156,128.614 C 841.15,128.611 841.15,128.622 841.144,128.626 C 841.146,128.632 841.138,128.628 841.136,128.632 C 841.125,128.648 841.132,128.667 841.11,128.675 C 841.113,128.683 841.106,128.684 841.103,128.684 C 841.115,128.705 841.077,128.706 841.098,128.725 C 841.098,128.745 841.08,128.767 841.093,128.785 L 841.09,128.8 C 841.105,128.806 841.122,128.812 841.142,128.812 L 841.143,128.808 C 841.146,128.814 841.162,128.806 841.17,128.814 C 841.184,128.798 841.204,128.82 841.222,128.81 C 841.242,128.817 841.256,128.806 841.277,128.81 L 841.287,128.806 C 841.295,128.812 841.314,128.806 841.332,128.814 L 841.335,128.808 C 841.337,128.814 841.34,128.814 841.343,128.81 C 841.35,128.814 841.359,128.815 841.366,128.814 C 841.37,128.806 841.385,128.806 841.394,128.812 C 841.411,128.82 841.428,128.817 841.441,128.817 C 841.457,128.817 841.46,128.831 841.469,128.838 C 841.473,128.846 841.469,128.859 841.473,128.866 C 841.473,128.888 841.455,128.908 841.457,128.922 C 841.459,128.949 841.437,128.951 841.433,128.978 C 841.419,128.981 841.414,128.994 841.406,129.001 C 841.398,129.002 841.398,129.016 841.388,129.02 C 841.391,129.021 841.392,129.025 841.388,129.026 C 841.378,129.022 841.378,129.038 841.367,129.035 C 841.36,129.045 841.34,129.044 841.34,129.061 C 841.324,129.059 841.311,129.074 841.295,129.073 C 841.295,129.085 841.29,129.071 841.284,129.078 C 841.28,129.078 841.274,129.079 841.273,129.083 C 841.255,129.085 841.242,129.087 841.222,129.087 L 841.218,129.091 C 841.188,129.078 841.152,129.082 841.129,129.059 C 841.115,129.051 841.094,129.053 841.09,129.036 C 841.08,129.04 841.079,129.022 841.072,129.029 C 841.076,129.021 841.064,129.021 841.059,129.016 C 841.058,129.017 841.054,129.017 841.053,129.016 C 841.054,129.009 841.05,129.005 841.045,129.002 L 841.041,129.002 L 841.041,128.992 L 841.027,128.992 L 841.027,128.997 C 841.028,128.997 841.031,128.995 841.032,128.995 C 841.036,128.999 841.028,128.999 841.031,129.002 C 841.039,129.012 841.014,129.025 841.036,129.029 C 841.036,129.035 841.032,129.044 841.036,129.049 C 841.034,129.051 841.032,129.051 841.032,129.052 C 841.049,129.056 841.027,129.073 841.041,129.076 C 841.043,129.082 841.043,129.088 841.041,129.094 C 841.043,129.097 841.051,129.1 841.046,129.108 C 841.054,129.118 841.046,129.129 841.049,129.14 C 841.043,129.146 841.034,129.146 841.034,129.152 C 841.046,129.154 841.054,129.166 841.056,129.18 L 841.062,129.181 C 841.056,129.195 841.069,129.216 841.053,129.228 C 841.059,129.231 841.069,129.239 841.064,129.247 C 841.051,129.25 841.053,129.267 841.043,129.273 L 841.05,129.284 L 841.051,129.283 C 841.064,129.289 841.049,129.305 841.058,129.311 C 841.054,129.32 841.056,129.33 841.051,129.337 C 841.039,129.367 841.051,129.393 841.045,129.424 C 841.049,129.439 841.066,129.45 841.062,129.47 C 841.07,129.47 841.066,129.482 841.072,129.488 C 841.066,129.491 841.072,129.498 841.065,129.498 C 841.072,129.505 841.062,129.511 841.062,129.516 C 841.045,129.522 841.053,129.544 841.043,129.553 C 841.045,129.56 841.051,129.569 841.056,129.571 C 841.059,129.586 841.051,129.6 841.049,129.613 C 841.054,129.622 841.054,129.631 841.059,129.641 C 841.056,129.646 841.054,129.653 841.046,129.653 C 841.046,129.661 841.039,129.672 841.043,129.683 C 841.053,129.692 841.054,129.672 841.062,129.683 C 841.083,129.689 841.099,129.674 841.12,129.684 C 841.124,129.686 841.128,129.683 841.128,129.679 C 841.148,129.674 841.144,129.698 841.165,129.699 C 841.179,129.713 841.156,129.718 841.156,129.728 C 841.144,129.724 841.139,129.736 841.128,129.736 C 841.118,129.74 841.11,129.749 841.106,129.757 C 841.098,129.758 841.098,129.771 841.09,129.775 C 841.091,129.779 841.091,129.782 841.085,129.782 C 841.085,129.792 841.091,129.797 841.084,129.803 C 841.09,129.807 841.079,129.812 841.084,129.817 C 841.08,129.832 841.072,129.842 841.083,129.855 C 841.072,129.875 841.091,129.889 841.083,129.904 L 841.085,129.907 L 841.084,129.938 C 841.08,129.942 841.08,129.948 841.079,129.954 C 841.08,129.954 841.083,129.955 841.084,129.957 L 841.079,129.96 C 841.085,129.969 841.076,129.981 841.084,129.989 C 841.083,129.999 841.076,130.016 841.085,130.027 C 841.079,130.04 841.08,130.054 841.077,130.071 C 841.087,130.073 841.079,130.082 841.083,130.087 C 841.083,130.086 841.084,130.087 841.085,130.087 C 841.085,130.091 841.083,130.093 841.087,130.097 C 841.085,130.102 841.083,130.116 841.091,130.12 C 841.084,130.134 841.094,130.147 841.099,130.16 C 841.091,130.168 841.107,130.17 841.105,130.176 L 841.12,130.187 C 841.115,130.19 841.122,130.192 841.124,130.196 L 841.129,130.196 C 841.122,130.204 841.135,130.207 841.139,130.207 C 841.144,130.196 841.162,130.21 841.169,130.196 C 841.177,130.203 841.181,130.19 841.19,130.187 C 841.195,130.178 841.207,130.175 841.217,130.172 C 841.24,130.174 841.269,130.176 841.281,130.194 C 841.296,130.196 841.284,130.21 841.297,130.213 C 841.293,130.222 841.301,130.219 841.303,130.227 C 841.303,130.232 841.301,130.24 841.307,130.246 C 841.301,130.249 841.311,130.254 841.304,130.259 C 841.307,130.263 841.311,130.265 841.317,130.265 C 841.326,130.246 841.355,130.275 841.359,130.249 C 841.367,130.246 841.378,130.241 841.385,130.237 C 841.388,130.218 841.407,130.197 841.394,130.178 C 841.399,130.166 841.401,130.151 841.394,130.138 C 841.398,130.117 841.403,130.097 841.4,130.081 C 841.403,130.077 841.4,130.07 841.407,130.068 C 841.4,130.051 841.401,130.031 841.399,130.014 C 841.407,129.995 841.388,129.983 841.4,129.964 C 841.4,129.955 841.407,129.945 841.406,129.933 C 841.401,129.917 841.411,129.899 841.406,129.885 C 841.406,129.877 841.408,129.867 841.4,129.864 C 841.4,129.853 841.403,129.842 841.398,129.833 C 841.407,129.828 841.394,129.823 841.401,129.819 C 841.403,129.798 841.412,129.778 841.422,129.766 L 841.422,129.749 C 841.428,129.744 841.435,129.734 841.441,129.731 C 841.448,129.724 841.444,129.716 841.455,129.713 C 841.46,129.699 841.473,129.698 841.484,129.693 C 841.484,129.679 841.498,129.693 841.5,129.68 C 841.514,129.684 841.518,129.668 841.53,129.676 C 841.53,129.658 841.545,129.672 841.552,129.664 C 841.567,129.667 841.577,129.658 841.592,129.658 C 841.611,129.672 841.625,129.657 841.644,129.667 C 841.649,129.665 841.649,129.676 841.652,129.668 C 841.663,129.684 841.697,129.68 841.701,129.699 C 841.714,129.71 841.723,129.72 841.741,129.722 C 841.751,129.724 841.754,129.741 841.768,129.736 C 841.77,129.718 841.774,129.695 841.783,129.678 L 841.782,129.678 C 841.786,129.676 841.791,129.674 841.795,129.668 C 841.829,129.665 841.853,129.684 841.882,129.692 C 841.885,129.701 841.891,129.686 841.894,129.698 C 841.898,129.695 841.899,129.695 841.902,129.698 C 841.918,129.693 841.933,129.716 841.946,129.699 C 841.953,129.697 841.961,129.697 841.968,129.704 C 841.968,129.709 841.965,129.722 841.975,129.726 C 841.977,129.728 841.975,129.734 841.98,129.736 C 841.98,129.747 841.98,129.762 841.967,129.762 C 841.957,129.767 841.942,129.767 841.933,129.775 C 841.927,129.781 841.927,129.786 841.918,129.784 C 841.916,129.798 841.906,129.817 841.916,129.83 C 841.909,129.847 841.906,129.867 841.902,129.89 C 841.912,129.898 841.899,129.911 841.912,129.917 C 841.909,129.933 841.901,129.944 841.908,129.959 C 841.905,129.969 841.906,129.986 841.901,130 L 841.908,130.007 C 841.908,130.02 841.899,130.034 841.902,130.046 C 841.905,130.055 841.901,130.062 841.906,130.073 C 841.912,130.081 841.905,130.091 841.909,130.097 L 841.906,130.104 C 841.908,130.104 841.909,130.107 841.909,130.108 C 841.908,130.114 841.906,130.121 841.906,130.132 C 841.899,130.138 841.914,130.145 841.905,130.151 C 841.909,130.16 841.901,130.172 841.912,130.176 C 841.898,130.18 841.916,130.187 841.906,130.192 C 841.912,130.203 841.894,130.218 841.908,130.227 C 841.902,130.241 841.916,130.253 841.906,130.267 C 841.895,130.282 841.872,130.276 841.866,130.298 C 841.856,130.313 841.841,130.294 841.827,130.302 C 841.819,130.31 841.804,130.298 841.794,130.308 L 841.749,130.301 L 841.745,130.305 C 841.741,130.299 841.727,130.301 841.716,130.302 C 841.708,130.305 841.695,130.305 841.689,130.299 C 841.679,130.29 841.675,130.28 841.682,130.267 C 841.685,130.265 841.695,130.265 841.695,130.258 C 841.709,130.265 841.715,130.249 841.727,130.253 C 841.741,130.246 841.756,130.241 841.763,130.227 L 841.775,130.227 C 841.768,130.204 841.781,130.187 841.774,130.161 C 841.776,130.147 841.786,130.126 841.774,130.109 C 841.774,130.104 841.775,130.095 841.77,130.092 C 841.771,130.074 841.786,130.055 841.774,130.042 C 841.776,130.028 841.778,130.014 841.778,129.999 C 841.776,129.986 841.781,129.973 841.775,129.963 C 841.781,129.954 841.781,129.934 841.775,129.923 C 841.776,129.915 841.781,129.904 841.776,129.896 L 841.774,129.892 C 841.775,129.89 841.778,129.889 841.775,129.885 C 841.776,129.875 841.781,129.859 841.774,129.847 C 841.768,129.847 841.767,129.84 841.767,129.834 C 841.747,129.832 841.743,129.813 841.727,129.81 C 841.723,129.798 841.704,129.8 841.698,129.789 L 841.692,129.793 C 841.673,129.788 841.646,129.786 841.629,129.778 C 841.618,129.779 841.611,129.781 841.598,129.782 C 841.591,129.792 841.584,129.782 841.574,129.789 C 841.574,129.8 841.561,129.797 841.559,129.807 C 841.544,129.813 841.539,129.83 841.53,129.843 C 841.532,129.849 841.532,129.85 841.53,129.855 C 841.532,129.871 841.523,129.886 841.536,129.896 L 841.53,129.899 C 841.536,129.911 841.526,129.92 841.526,129.933 C 841.527,129.938 841.532,129.944 841.532,129.951 C 841.532,129.955 841.545,129.957 841.536,129.963 C 841.536,129.976 841.527,129.997 841.533,130.009 C 841.533,130.017 841.53,130.028 841.536,130.037 C 841.523,130.039 841.54,130.051 841.527,130.052 C 841.537,130.058 841.521,130.06 841.527,130.064 C 841.526,130.077 841.526,130.087 841.53,130.097 L 841.526,130.099 C 841.532,130.108 841.521,130.12 841.527,130.128 C 841.533,130.137 841.523,130.151 841.533,130.156 C 841.53,130.16 841.532,130.162 841.533,130.166 C 841.523,130.18 841.533,130.201 841.527,130.211 C 841.524,130.223 841.537,130.237 841.53,130.249 C 841.526,130.269 841.518,130.296 841.497,130.301 C 841.493,130.302 841.495,130.305 841.493,130.306 L 841.447,130.301 C 841.441,130.311 841.437,130.298 841.429,130.302 C 841.426,130.311 841.412,130.306 841.411,130.313 C 841.399,130.313 841.378,130.302 841.362,130.31 C 841.347,130.308 841.321,130.308 841.315,130.293 C 841.31,130.287 841.31,130.274 841.297,130.279 L 841.296,130.286 C 841.287,130.279 841.288,130.298 841.277,130.298 C 841.255,130.317 841.228,130.317 841.202,130.328 C 841.191,130.327 841.185,130.328 841.181,130.32 C 841.162,130.318 841.15,130.306 841.136,130.299 C 841.13,130.298 841.124,130.288 841.115,130.286 C 841.115,130.279 841.106,130.275 841.098,130.274 C 841.084,130.279 841.09,130.298 841.077,130.306 C 841.065,130.301 841.056,130.313 841.045,130.311 C 841.046,130.32 841.046,130.327 841.041,130.335 L 841.036,130.337 C 841.036,130.341 841.045,130.346 841.036,130.348 L 841.039,130.352 C 841.039,130.363 841.024,130.379 841.037,130.383 C 841.032,130.394 841.05,130.4 841.043,130.41 C 841.051,130.42 841.045,130.434 841.051,130.446 C 841.041,130.451 841.056,130.455 841.049,130.462 C 841.058,130.473 841.05,130.486 841.058,130.498 C 841.056,130.503 841.056,130.508 841.056,130.514 C 841.066,130.518 841.059,130.534 841.058,130.547 C 841.065,130.557 841.059,130.569 841.066,130.58 C 841.064,130.59 841.064,130.595 841.062,130.601 C 841.077,130.612 841.056,130.63 841.07,130.641 C 841.069,130.648 841.079,130.661 841.073,130.67 C 841.085,130.682 841.072,130.693 841.084,130.704 C 841.08,130.714 841.079,130.727 841.085,130.74 C 841.087,130.742 841.093,130.742 841.093,130.745 C 841.093,130.749 841.091,130.753 841.087,130.756 C 841.073,130.752 841.077,130.77 841.072,130.776 C 841.065,130.788 841.08,130.796 841.077,130.807 C 841.091,130.818 841.072,130.839 841.073,130.856 L 841.079,130.86 L 841.073,130.862 C 841.08,130.869 841.083,130.883 841.094,130.885 C 841.091,130.891 841.105,130.891 841.099,130.897 C 841.091,130.893 841.096,130.908 841.09,130.909 C 841.11,130.911 841.128,130.908 841.15,130.912 C 841.159,130.909 841.176,130.912 841.188,130.915 C 841.202,130.924 841.225,130.922 841.234,130.935 L 841.238,130.935 L 841.238,130.933 C 841.24,130.941 841.263,130.943 841.269,130.954 C 841.29,130.96 841.303,130.974 841.315,130.989 C 841.332,130.993 841.326,131.003 841.335,131.015 C 841.341,131.02 841.346,131.033 841.356,131.035 L 841.355,131.036 C 841.369,131.055 841.383,131.065 841.385,131.088 C 841.387,131.092 841.391,131.088 841.391,131.092 C 841.383,131.104 841.401,131.113 841.394,131.122 C 841.401,131.133 841.392,131.14 841.403,131.146 C 841.399,131.15 841.407,131.156 841.403,131.161 C 841.418,131.168 841.406,131.182 841.415,131.188 C 841.414,131.194 841.408,131.208 841.419,131.212 C 841.414,131.218 841.415,131.232 841.421,131.243 C 841.418,131.261 841.408,131.281 841.408,131.302 L 841.414,131.308 C 841.411,131.327 841.4,131.347 841.398,131.364 C 841.398,131.372 841.392,131.378 841.388,131.385 C 841.388,131.395 841.388,131.398 841.379,131.405 C 841.383,131.411 841.373,131.408 841.374,131.415 C 841.374,131.434 841.352,131.433 841.353,131.45 C 841.343,131.454 841.34,131.468 841.336,131.478 C 841.329,131.478 841.326,131.484 841.322,131.488 C 841.315,131.501 841.297,131.501 841.293,131.519 C 841.284,131.519 841.277,131.525 841.27,131.532 C 841.26,131.532 841.251,131.544 841.24,131.544 C 841.232,131.553 841.218,131.546 841.211,131.555 C 841.202,131.55 841.192,131.562 841.182,131.56 C 841.182,131.565 841.176,131.565 841.171,131.565 C 841.166,131.574 841.166,131.587 841.156,131.594 L 841.159,131.6 L 841.156,131.6 C 841.152,131.605 841.163,131.606 841.158,131.615 C 841.171,131.615 841.159,131.627 841.17,131.631 C 841.177,131.658 841.142,131.665 841.155,131.697 C 841.152,131.701 841.152,131.708 841.144,131.713 C 841.142,131.72 841.15,131.719 841.148,131.726 C 841.152,131.74 841.181,131.73 841.171,131.752 C 841.17,131.768 841.152,131.756 841.152,131.77 L 841.155,131.774 C 841.148,131.786 841.152,131.802 841.152,131.812 C 841.158,131.826 841.132,131.837 841.152,131.854 C 841.144,131.864 841.169,131.862 841.156,131.872 L 841.165,131.878 C 841.155,131.88 841.169,131.888 841.158,131.889 C 841.159,131.901 841.166,131.904 841.166,131.913 C 841.177,131.928 841.152,131.932 841.152,131.938 C 841.148,131.949 841.159,131.953 841.152,131.962 C 841.144,131.975 841.166,131.972 841.159,131.988 L 841.165,131.988 C 841.162,131.988 841.156,131.993 841.162,131.999 L 841.156,132.003 C 841.156,132.008 841.148,132.02 841.155,132.027 C 841.15,132.037 841.15,132.056 841.155,132.067 C 841.152,132.079 841.148,132.094 841.146,132.108 C 841.146,132.116 841.144,132.126 841.146,132.135 C 841.138,132.137 841.148,132.142 841.142,132.145 C 841.146,132.152 841.135,132.159 841.142,132.165 C 841.152,132.176 841.177,132.172 841.184,132.185 C 841.198,132.194 841.191,132.218 841.211,132.218 C 841.218,132.225 841.222,132.238 841.228,132.246 C 841.229,132.257 841.222,132.262 841.218,132.273 C 841.215,132.276 841.207,132.271 841.207,132.28 C 841.198,132.279 841.192,132.276 841.185,132.27 C 841.174,132.271 841.152,132.269 841.143,132.276 C 841.152,132.292 841.129,132.31 841.143,132.328 C 841.146,132.334 841.135,132.334 841.142,132.341 C 841.136,132.357 841.128,132.371 841.138,132.387 C 841.139,132.398 841.124,132.413 841.139,132.421 C 841.142,132.425 841.148,132.436 841.152,132.442 C 841.143,132.438 841.144,132.45 841.139,132.454 C 841.136,132.471 841.129,132.49 841.135,132.507 C 841.125,132.523 841.142,132.529 841.139,132.545 C 841.148,132.55 841.136,132.566 841.15,132.57 C 841.146,132.572 841.146,132.573 841.148,132.58 C 841.159,132.577 841.155,132.588 841.162,132.594 C 841.166,132.605 841.155,132.613 841.155,132.622 C 841.15,132.622 841.146,132.62 841.144,132.626 C 841.158,132.628 841.144,132.647 841.155,132.652 C 841.146,132.657 841.159,132.662 841.152,132.668 C 841.165,132.668 841.156,132.68 841.165,132.682 C 841.156,132.694 841.144,132.714 841.152,132.73 C 841.152,132.735 841.156,132.742 841.162,132.745 C 841.163,132.755 841.15,132.765 841.162,132.773 C 841.163,132.778 841.165,132.782 841.17,132.783 C 841.163,132.798 841.152,132.809 841.15,132.828 C 841.152,132.836 841.15,132.849 841.162,132.852 C 841.158,132.859 841.162,132.873 841.15,132.875 C 841.148,132.878 841.142,132.881 841.144,132.889 L 841.148,132.889 C 841.146,132.902 841.166,132.897 841.156,132.914 C 841.146,132.931 841.163,132.943 841.15,132.958 C 841.144,132.963 841.143,132.964 841.144,132.973 C 841.152,132.979 841.143,132.993 841.152,132.999 C 841.152,133.002 841.152,133.006 841.152,133.011 L 841.156,133.01 C 841.158,133.01 841.156,133.011 841.156,133.014 C 841.156,133.015 841.152,133.014 841.152,133.015 C 841.159,133.016 841.152,133.03 841.163,133.032 C 841.165,133.041 841.156,133.053 841.162,133.061 C 841.165,133.064 841.155,133.064 841.162,133.071 C 841.152,133.078 841.146,133.097 841.144,133.109 C 841.163,133.12 841.125,133.148 841.156,133.151 L 841.165,133.161 C 841.162,133.169 841.169,133.175 841.162,133.185 C 841.155,133.189 841.148,133.194 841.146,133.202 C 841.152,133.213 841.155,133.229 841.146,133.239 C 841.152,133.252 841.152,133.264 841.146,133.272 C 841.144,133.281 841.143,133.29 841.148,133.295 C 841.152,133.298 841.146,133.304 841.15,133.309 C 841.143,133.312 841.15,133.318 841.148,133.322 L 841.152,133.327 C 841.142,133.33 841.158,133.339 841.144,133.341 C 841.146,133.343 841.15,133.346 841.146,133.348 C 841.143,133.35 841.148,133.358 841.144,133.362 C 841.156,133.362 841.148,133.374 841.158,133.379 C 841.15,133.382 841.146,133.391 841.144,133.399 C 841.152,133.412 841.165,133.405 841.17,133.418 C 841.182,133.423 841.199,133.423 841.207,133.436 C 841.221,133.448 841.238,133.455 841.25,133.471 C 841.255,133.472 841.262,133.478 841.262,133.484 C 841.277,133.479 841.269,133.5 841.281,133.503 C 841.287,133.517 841.303,133.52 841.301,133.536 C 841.304,133.546 841.311,133.554 841.314,133.565 C 841.32,133.573 841.326,133.585 841.328,133.594 C 841.329,133.592 841.332,133.592 841.333,133.595 C 841.328,133.6 841.329,133.611 841.332,133.62 C 841.337,133.626 841.329,133.637 841.326,133.646 C 841.32,133.642 841.311,133.647 841.307,133.648 C 841.296,133.641 841.283,133.633 841.28,133.62 C 841.274,133.612 841.275,133.603 841.266,133.6 C 841.266,133.589 841.25,133.586 841.25,133.577 C 841.231,133.576 841.231,133.551 841.211,133.554 L 841.203,133.549 L 841.202,133.55 C 841.185,133.534 841.169,133.531 841.148,133.53 C 841.152,133.546 841.135,133.559 841.146,133.573 C 841.148,133.577 841.146,133.585 841.152,133.586 C 841.152,133.598 841.152,133.606 841.148,133.62 C 841.171,133.625 841.148,133.655 841.171,133.66 L 841.171,133.669 C 841.176,133.669 841.179,133.671 841.181,133.672 C 841.176,133.675 841.179,133.679 841.179,133.683 C 841.174,133.686 841.176,133.695 841.169,133.695 C 841.171,133.699 841.169,133.702 841.165,133.704 L 841.17,133.71 C 841.159,133.712 841.17,133.718 841.169,133.721 C 841.17,133.72 841.174,133.72 841.176,133.721 C 841.176,133.727 841.179,133.727 841.182,133.73 C 841.191,133.733 841.191,133.72 841.197,133.727 C 841.199,133.723 841.21,133.728 841.214,133.721 C 841.221,133.735 841.232,133.72 841.243,133.725 C 841.243,133.714 841.26,133.723 841.256,133.714 C 841.263,133.714 841.263,133.708 841.269,133.704 C 841.284,133.718 841.301,133.695 841.317,133.704 C 841.32,133.702 841.321,133.7 841.322,133.699 C 841.332,133.702 841.343,133.7 841.353,133.706 C 841.359,133.706 841.369,133.709 841.373,133.704 L 841.374,133.709 C 841.391,133.702 841.408,133.712 841.426,133.712 C 841.426,133.71 841.429,133.709 841.429,133.709 C 841.44,133.712 841.46,133.71 841.473,133.72 C 841.485,133.723 841.481,133.716 841.49,133.71 C 841.488,133.706 841.488,133.698 841.492,133.691 L 841.487,133.685 L 841.492,133.685 C 841.484,133.671 841.497,133.654 841.492,133.638 C 841.49,133.626 841.497,133.607 841.488,133.596 C 841.49,133.589 841.493,133.579 841.485,133.576 C 841.488,133.565 841.481,133.549 841.487,133.543 C 841.49,133.532 841.492,133.517 841.485,133.509 C 841.481,133.497 841.473,133.484 841.46,133.479 C 841.459,133.471 841.447,133.47 841.437,133.47 L 841.435,133.472 C 841.429,133.474 841.433,133.462 841.426,133.467 L 841.426,133.47 C 841.424,133.459 841.398,133.459 841.407,133.445 C 841.407,133.432 841.421,133.436 841.426,133.427 C 841.441,133.435 841.455,133.426 841.473,133.429 C 841.492,133.426 841.512,133.427 841.533,133.431 C 841.543,133.427 841.55,133.432 841.557,133.426 C 841.573,133.426 841.591,133.435 841.605,133.426 C 841.625,133.432 841.644,133.42 841.663,133.429 C 841.673,133.419 841.684,133.431 841.695,133.424 C 841.697,133.423 841.698,133.42 841.701,133.42 C 841.695,133.435 841.718,133.431 841.705,133.445 C 841.698,133.455 841.685,133.455 841.679,133.47 C 841.668,133.466 841.668,133.482 841.657,133.478 C 841.657,133.486 841.644,133.482 841.649,133.492 L 841.646,133.49 C 841.645,133.496 841.633,133.502 841.638,133.507 C 841.625,133.506 841.626,133.528 841.622,133.54 L 841.623,133.543 C 841.622,133.543 841.619,133.546 841.622,133.549 C 841.622,133.55 841.625,133.549 841.626,133.55 C 841.616,133.554 841.63,133.562 841.622,133.568 C 841.625,133.573 841.616,133.581 841.626,133.585 C 841.622,133.585 841.623,133.589 841.625,133.592 C 841.613,133.615 841.632,133.633 841.623,133.659 C 841.63,133.66 841.626,133.664 841.625,133.668 L 841.625,133.69 C 841.626,133.689 841.626,133.69 841.629,133.691 C 841.625,133.698 841.625,133.699 841.629,133.704 C 841.646,133.694 841.661,133.714 841.679,133.706 C 841.691,133.7 841.704,133.714 841.711,133.704 C 841.719,133.708 841.727,133.708 841.737,133.709 C 841.745,133.706 841.743,133.704 841.751,133.704 L 841.757,133.709 C 841.77,133.708 841.791,133.714 841.802,133.708 C 841.808,133.709 841.816,133.709 841.82,133.706 C 841.823,133.698 841.829,133.712 841.831,133.704 C 841.843,133.712 841.849,133.7 841.861,133.704 C 841.872,133.69 841.86,133.677 841.863,133.664 C 841.865,133.655 841.869,133.646 841.863,133.637 C 841.866,133.634 841.865,133.629 841.863,133.626 C 841.865,133.617 841.869,133.602 841.861,133.595 C 841.872,133.573 841.86,133.555 841.865,133.534 C 841.853,133.528 841.866,133.513 841.854,133.507 C 841.865,133.497 841.843,133.498 841.849,133.488 C 841.826,133.478 841.802,133.471 841.776,133.467 C 841.775,133.459 841.767,133.453 841.761,133.451 C 841.761,133.444 841.76,133.426 841.775,133.429 C 841.783,133.419 841.797,133.431 841.809,133.42 C 841.821,133.423 841.837,133.431 841.846,133.423 C 841.857,133.424 841.872,133.423 841.882,133.42 L 841.886,133.424 C 841.901,133.42 841.916,133.435 841.931,133.424 C 841.94,133.423 841.953,133.432 841.96,133.42 C 841.979,133.435 841.995,133.412 842.016,133.426 C 842.037,133.429 842.069,133.418 842.073,133.444 C 842.076,133.444 842.078,133.441 842.078,133.445 C 842.069,133.448 842.076,133.455 842.067,133.459 C 842.06,133.462 842.065,133.478 842.053,133.471 C 842.05,133.48 842.028,133.474 842.027,133.49 C 842.003,133.507 842.006,133.532 842.001,133.561 C 842.008,133.571 841.998,133.589 842.001,133.602 C 841.994,133.607 842.008,133.615 841.999,133.62 C 842.005,133.625 841.994,133.629 841.998,133.637 C 841.994,133.652 842.006,133.66 841.998,133.675 C 842.012,133.679 841.99,133.706 842.016,133.7 L 842.019,133.695 L 842.021,133.702 C 842.046,133.698 842.078,133.709 842.099,133.689 C 842.105,133.699 842.11,133.683 842.113,133.695 C 842.125,133.683 842.164,133.712 842.151,133.685 C 842.152,133.655 842.169,133.621 842.173,133.59 C 842.176,133.583 842.185,133.577 842.181,133.571 C 842.196,133.546 842.211,133.52 842.224,133.498 L 842.227,133.498 L 842.221,133.496 C 842.234,133.498 842.231,133.479 842.243,133.482 C 842.257,133.478 842.258,133.459 842.276,133.462 C 842.286,133.445 842.309,133.445 842.326,133.436 C 842.336,133.429 842.344,133.427 842.359,133.426 C 842.368,133.419 842.386,133.418 842.399,133.419 C 842.414,133.416 842.433,133.436 842.445,133.42 C 842.479,133.432 842.519,133.426 842.541,133.457 C 842.559,133.453 842.563,133.474 842.576,133.478 C 842.577,133.482 842.583,133.493 842.586,133.497 C 842.599,133.503 842.604,133.523 842.62,133.53 C 842.623,133.536 842.625,133.549 842.633,133.554 C 842.64,133.561 842.637,133.573 842.648,133.577 C 842.642,133.592 842.653,133.6 842.652,133.616 C 842.658,133.621 842.651,133.631 842.659,133.638 C 842.663,133.651 842.653,133.659 842.652,133.669 C 842.644,133.669 842.644,133.675 842.639,133.678 C 842.626,133.672 842.607,133.675 842.605,133.659 C 842.596,133.657 842.607,133.644 842.596,133.646 C 842.603,133.634 842.583,133.633 842.586,133.623 C 842.579,133.62 842.581,133.611 842.576,133.602 C 842.577,133.589 842.552,133.589 842.554,133.577 C 842.53,133.569 842.521,133.551 842.495,133.549 C 842.479,133.549 842.456,133.532 842.44,133.543 C 842.428,133.538 842.413,133.549 842.4,133.554 C 842.398,133.549 842.386,133.551 842.381,133.55 L 842.38,133.555 C 842.379,133.551 842.374,133.555 842.37,133.555 L 842.368,133.564 C 842.366,133.561 842.36,133.562 842.359,133.564 C 842.357,133.565 842.359,133.569 842.353,133.571 C 842.352,133.568 842.348,133.571 842.347,133.571 C 842.347,133.575 842.344,133.577 842.341,133.579 C 842.34,133.577 842.336,133.575 842.334,133.577 C 842.326,133.586 842.317,133.594 842.307,133.6 C 842.302,133.602 842.307,133.603 842.307,133.607 C 842.298,133.598 842.296,133.615 842.295,133.617 C 842.293,133.646 842.281,133.671 842.27,133.691 C 842.267,133.691 842.278,133.698 842.27,133.7 C 842.281,133.706 842.296,133.712 842.309,133.706 L 842.313,133.709 L 842.317,133.706 C 842.321,133.714 842.328,133.706 842.334,133.709 C 842.334,133.694 842.348,133.708 842.357,133.702 C 842.355,133.7 842.357,133.699 842.359,133.699 L 842.362,133.704 C 842.369,133.695 842.375,133.709 842.382,133.7 C 842.382,133.694 842.389,133.694 842.39,133.689 C 842.398,133.694 842.411,133.694 842.419,133.689 C 842.423,133.679 842.428,133.686 842.434,133.683 C 842.436,133.672 842.441,133.686 842.447,133.683 L 842.445,133.686 C 842.455,133.689 842.466,133.699 842.479,133.695 C 842.487,133.689 842.493,133.678 842.504,133.686 C 842.508,133.689 842.521,133.691 842.53,133.691 C 842.54,133.691 842.551,133.675 842.56,133.683 C 842.576,133.679 842.592,133.694 842.607,133.683 C 842.615,133.686 842.617,133.695 842.623,133.689 C 842.627,133.694 842.633,133.686 842.639,133.69 C 842.644,133.689 842.644,133.679 842.649,133.685 L 842.662,133.675 L 842.658,133.671 L 842.662,133.668 C 842.664,133.672 842.673,133.669 842.679,133.669 L 842.682,133.664 C 842.683,133.671 842.689,133.663 842.693,133.664 C 842.701,133.669 842.716,133.669 842.711,133.681 C 842.731,133.679 842.745,133.698 842.764,133.689 C 842.767,133.689 842.764,133.694 842.768,133.691 C 842.77,133.691 842.773,133.69 842.774,133.689 C 842.78,133.694 842.793,133.691 842.797,133.7 C 842.807,133.702 842.818,133.699 842.827,133.7 C 842.83,133.694 842.835,133.695 842.841,133.694 C 842.844,133.683 842.837,133.678 842.839,133.668 L 842.841,133.664 C 842.848,133.642 842.837,133.623 842.839,133.6 C 842.85,133.577 842.835,133.55 842.844,133.524 C 842.835,133.519 842.844,133.506 842.834,133.5 C 842.83,133.48 842.811,133.474 842.801,133.464 C 842.787,133.453 842.776,133.47 842.763,133.459 C 842.768,133.453 842.758,133.453 842.758,133.448 C 842.758,133.436 842.76,133.429 842.764,133.42 C 842.787,133.405 842.798,133.423 842.826,133.42 C 842.839,133.409 842.859,133.429 842.87,133.416 C 842.902,133.418 842.931,133.409 842.966,133.416 C 842.975,133.412 842.99,133.42 842.999,133.412 C 843.023,133.419 843.045,133.405 843.064,133.429 L 843.067,133.441 C 843.053,133.448 843.053,133.464 843.034,133.457 C 843.025,133.464 843.012,133.455 843.007,133.47 L 843.007,133.474 C 843.004,133.471 842.998,133.472 842.997,133.478 C 842.997,133.493 842.973,133.498 842.98,133.517 C 842.986,133.536 842.966,133.564 842.975,133.585 C 842.979,133.615 842.975,133.634 842.968,133.663 C 842.971,133.66 842.972,133.66 842.973,133.664 C 842.964,133.668 842.98,133.675 842.972,133.679 C 842.987,133.685 843.008,133.694 843.023,133.69 C 843.038,133.677 843.056,133.699 843.073,133.695 L 843.076,133.69 L 843.077,133.694 L 843.116,133.694 C 843.115,133.691 843.116,133.691 843.118,133.69 C 843.119,133.691 843.119,133.695 843.12,133.695 C 843.135,133.69 843.149,133.7 843.162,133.7 C 843.176,133.69 843.191,133.702 843.203,133.698 C 843.206,133.702 843.211,133.694 843.215,133.698 C 843.244,133.695 843.27,133.72 843.301,133.71 C 843.304,133.712 843.306,133.716 843.31,133.712 C 843.301,133.704 843.29,133.7 843.283,133.689 C 843.274,133.686 843.27,133.677 843.261,133.675 C 843.249,133.648 843.228,133.634 843.224,133.607 C 843.212,133.603 843.222,133.59 843.212,133.589 C 843.206,133.565 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840.966,123.602 C 840.975,123.598 840.962,123.592 840.972,123.588 C 840.969,123.578 840.972,123.567 840.978,123.558 C 840.975,123.555 840.975,123.55 840.972,123.552 C 840.974,123.547 840.965,123.541 840.972,123.537 C 840.965,123.524 840.966,123.512 840.972,123.504 C 840.981,123.501 840.969,123.49 840.98,123.489 C 840.969,123.471 840.984,123.449 840.986,123.431 C 840.978,123.421 840.997,123.401 840.981,123.395 C 840.982,123.387 840.978,123.38 840.978,123.374 L 840.984,123.373 C 840.981,123.365 840.995,123.36 840.982,123.356 C 840.984,123.35 840.98,123.346 840.984,123.344 C 840.99,123.328 840.988,123.309 840.982,123.296 L 840.98,123.296 C 840.982,123.292 840.981,123.29 840.978,123.287 C 840.975,123.277 840.982,123.274 840.978,123.265 C 840.982,123.253 840.98,123.234 840.984,123.22 C 840.978,123.208 840.982,123.195 840.984,123.182 C 840.969,123.149 840.988,123.118 840.99,123.086 C 840.978,123.076 840.972,123.058 840.975,123.043 C 840.958,123.027 840.982,123.011 840.969,122.996 C 840.965,122.982 840.982,122.985 840.982,122.972 C 840.975,122.97 840.969,122.967 840.965,122.962 C 840.972,122.951 840.966,122.939 840.962,122.927 C 840.952,122.914 840.966,122.896 840.966,122.883 C 840.981,122.861 840.961,122.837 840.972,122.817 C 840.981,122.807 840.98,122.794 840.984,122.782 C 840.966,122.78 840.986,122.761 840.974,122.757 C 840.975,122.749 840.975,122.743 840.972,122.738 C 840.972,122.726 840.96,122.711 840.969,122.704 C 840.969,122.693 840.961,122.683 840.962,122.676 C 840.966,122.665 840.966,122.65 840.966,122.642 C 840.947,122.624 840.966,122.6 840.961,122.583 C 840.961,122.577 840.961,122.564 840.954,122.561 C 840.966,122.54 840.965,122.515 840.972,122.49 C 840.961,122.473 840.975,122.461 840.974,122.441 C 840.969,122.433 840.969,122.423 840.966,122.415 C 840.965,122.406 840.96,122.397 840.955,122.387 C 840.949,122.365 840.974,122.338 840.96,122.317 C 840.96,122.302 840.978,122.308 840.972,122.295 C 840.961,122.287 840.974,122.276 840.974,122.27 L 840.978,122.271 C 840.986,122.271 840.984,122.262 840.988,122.26 C 840.986,122.252 840.982,122.243 840.978,122.24 C 840.972,122.231 840.972,122.224 840.975,122.217 C 840.966,122.2 840.975,122.183 840.984,122.168 C 840.984,122.167 840.982,122.165 840.982,122.162 C 840.988,122.158 840.984,122.147 840.984,122.141 C 840.984,122.134 840.981,122.131 840.984,122.127 C 840.975,122.12 840.975,122.117 840.978,122.108 C 840.978,122.104 840.975,122.099 840.972,122.1 L 840.972,122.096 C 840.965,122.09 840.96,122.094 840.952,122.094 C 840.949,122.076 840.975,122.09 840.982,122.077 C 840.981,122.076 840.981,122.076 840.98,122.076 C 840.981,122.068 840.975,122.055 840.988,122.054 C 841.003,122.049 840.997,122.032 840.999,122.021 C 840.988,122.017 840.995,122.008 840.984,122.005 L 840.988,122.001 C 840.974,121.984 840.988,121.959 840.993,121.937 C 840.995,121.924 840.972,121.929 840.978,121.913 C 840.99,121.898 840.969,121.894 840.969,121.883 C 840.969,121.863 840.993,121.867 840.999,121.854 C 840.995,121.837 841.013,121.819 841.003,121.804 C 841.004,121.792 840.993,121.791 840.995,121.781 C 840.972,121.778 840.984,121.757 840.978,121.744 C 840.986,121.743 840.984,121.732 840.988,121.728 C 840.982,121.726 840.986,121.717 840.982,121.715 C 840.982,121.706 840.978,121.7 840.98,121.695 C 840.99,121.696 840.986,121.682 840.995,121.686 C 841,121.682 841.003,121.677 841.004,121.674 C 841.003,121.665 840.997,121.659 840.99,121.656 C 840.965,121.639 841,121.642 841,121.628 C 841.008,121.616 840.988,121.612 840.984,121.601 C 840.982,121.597 840.986,121.591 840.981,121.586 C 840.991,121.573 840.969,121.556 840.986,121.547 C 841.004,121.547 841,121.526 841.006,121.518 C 841.007,121.495 841,121.483 841,121.464 C 840.99,121.458 840.999,121.436 840.98,121.439 C 840.966,121.429 840.981,121.419 840.969,121.412 C 840.96,121.4 840.984,121.404 840.98,121.393 C 840.969,121.377 840.981,121.359 840.98,121.343 C 840.966,121.337 840.981,121.323 840.974,121.317 C 840.975,121.314 840.978,121.31 840.974,121.307 C 840.982,121.294 840.961,121.297 840.958,121.291 C 840.957,121.279 840.947,121.27 840.955,121.258 C 840.966,121.252 840.966,121.235 840.972,121.223 C 840.969,121.214 840.966,121.205 840.955,121.203 C 840.957,121.186 840.942,121.179 840.931,121.173 C 840.933,121.164 840.931,121.155 840.938,121.151 L 840.935,121.148 C 840.942,121.142 840.934,121.132 840.941,121.124 C 840.928,121.115 840.942,121.097 840.935,121.086 C 840.933,121.07 840.934,121.055 840.934,121.043 C 840.922,121.029 840.935,121.016 840.928,121.002 C 840.905,120.995 840.931,120.972 840.918,120.964 C 840.914,120.958 840.92,120.949 840.926,120.944 C 840.925,120.93 840.931,120.919 840.922,120.908 C 840.896,120.904 840.914,120.879 840.902,120.868 C 840.914,120.862 840.903,120.85 840.905,120.84 C 840.909,120.835 840.909,120.82 840.922,120.819 C 840.922,120.804 840.903,120.794 840.909,120.778 C 840.879,120.767 840.914,120.747 840.914,120.73 C 840.903,120.724 840.918,120.713 840.907,120.708 C 840.905,120.692 840.909,120.676 840.905,120.665 C 840.889,120.656 840.907,120.639 840.903,120.626 C 840.896,120.612 840.878,120.602 840.886,120.582 C 840.881,120.56 840.887,120.535 840.898,120.516 C 840.902,120.504 840.909,120.493 840.905,120.474 C 840.9,120.473 840.896,120.469 840.898,120.465 C 840.883,120.451 840.903,120.432 840.9,120.415 C 840.893,120.409 840.898,120.401 840.889,120.397 C 840.883,120.386 840.881,120.371 840.889,120.359 C 840.893,120.351 840.903,120.345 840.9,120.337 C 840.893,120.334 840.896,120.323 840.887,120.327 C 840.879,120.322 840.874,120.313 840.881,120.303 C 840.887,120.277 840.9,120.262 840.909,120.24 C 840.926,120.228 840.896,120.215 840.907,120.199 C 840.9,120.181 840.919,120.153 840.907,120.133 C 840.909,120.125 840.9,120.112 840.905,120.1 C 840.893,120.086 840.883,120.069 840.875,120.056 C 840.874,120.04 840.894,120.051 840.893,120.037 C 840.883,120.023 840.896,120.007 840.883,119.996 C 840.883,119.986 840.889,119.981 840.889,119.972 C 840.883,119.969 840.883,119.96 840.878,119.955 C 840.874,119.949 840.868,119.941 840.859,119.934 C 840.863,119.925 840.85,119.923 840.862,119.919 C 840.883,119.902 840.853,119.885 840.868,119.864 C 840.875,119.864 840.883,119.862 840.883,119.853 C 840.881,119.851 840.879,119.846 840.875,119.85 L 840.879,119.846 C 840.878,119.84 840.872,119.841 840.868,119.839 L 840.869,119.831 C 840.85,119.819 840.869,119.801 840.872,119.784 C 840.878,119.764 840.863,119.756 840.874,119.737 C 840.887,119.717 840.856,119.721 840.853,119.71 C 840.852,119.702 840.859,119.704 840.862,119.701 C 840.879,119.684 840.872,119.648 840.868,119.63 C 840.881,119.619 840.883,119.599 840.881,119.586 C 840.872,119.577 840.869,119.565 840.862,119.557 C 840.859,119.548 840.852,119.54 840.859,119.534 C 840.875,119.519 840.845,119.505 840.863,119.493 C 840.848,119.489 840.859,119.47 840.859,119.464 C 840.869,119.463 840.856,119.452 840.868,119.455 L 840.863,119.446 C 840.863,119.436 840.872,119.426 840.872,119.42 C 840.883,119.41 840.863,119.391 840.869,119.376 C 840.868,119.364 840.886,119.359 840.879,119.353 C 840.881,119.353 840.883,119.353 840.883,119.35 C 840.878,119.343 840.887,119.332 840.874,119.332 C 840.867,119.325 840.868,119.318 840.869,119.31 C 840.902,119.293 840.855,119.293 840.859,119.276 C 840.867,119.262 840.848,119.255 840.848,119.244 C 840.85,119.241 840.853,119.239 840.85,119.236 C 840.863,119.229 840.863,119.211 840.875,119.205 C 840.883,119.197 840.872,119.194 840.874,119.184 C 840.869,119.175 840.872,119.163 840.863,119.158 C 840.862,119.151 840.863,119.137 840.868,119.13 C 840.902,119.116 840.867,119.09 840.875,119.068 C 840.85,119.052 840.883,119.039 840.878,119.018 C 840.869,119.016 840.872,119.008 840.874,119.004 C 840.893,118.991 840.883,118.965 840.883,118.949 C 840.874,118.946 840.883,118.934 840.881,118.927 C 840.874,118.929 840.868,118.924 840.863,118.92 C 840.859,118.907 840.879,118.907 840.889,118.902 C 840.896,118.897 840.889,118.886 840.902,118.886 C 840.902,118.876 840.902,118.864 840.893,118.86 C 840.894,118.847 840.889,118.833 840.9,118.825 C 840.893,118.821 840.894,118.809 840.887,118.806 C 840.896,118.794 840.878,118.775 840.896,118.765 C 840.889,118.761 840.879,118.754 840.883,118.745 L 840.887,118.745 C 840.907,118.737 840.898,118.72 840.902,118.705 C 840.911,118.692 840.907,118.68 840.914,118.666 C 840.919,118.654 840.908,118.645 840.9,118.643 C 840.905,118.635 840.893,118.634 840.894,118.626 C 840.893,118.604 840.919,118.604 840.931,118.596 C 840.934,118.588 840.929,118.582 840.933,118.575 C 840.919,118.561 840.929,118.547 840.931,118.534 C 840.928,118.53 840.926,118.527 840.925,118.526 C 840.902,118.52 840.896,118.495 840.886,118.475 C 840.892,118.474 840.886,118.472 840.886,118.469 C 840.896,118.472 840.894,118.456 840.905,118.461 C 840.905,118.457 840.905,118.456 840.903,118.453 C 840.908,118.444 840.922,118.433 840.918,118.417 L 840.911,118.417 C 840.918,118.411 840.907,118.407 840.902,118.404 L 840.914,118.372 C 840.919,118.363 840.908,118.347 840.92,118.339 C 840.929,118.334 840.926,118.324 840.928,118.316 C 840.919,118.306 840.908,118.291 840.908,118.277 C 840.903,118.26 840.929,118.243 840.909,118.231 C 840.898,118.229 840.907,118.215 840.896,118.214 C 840.903,118.195 840.903,118.178 840.911,118.163 C 840.909,118.158 840.909,118.156 840.918,118.153 L 840.909,118.148 C 840.908,118.137 840.902,118.13 840.893,118.125 C 840.896,118.119 840.889,118.111 840.9,118.108 C 840.902,118.102 840.903,118.096 840.896,118.094 C 840.903,118.074 840.909,118.058 840.907,118.04 C 840.911,118.03 840.9,118.005 840.909,117.995 C 840.905,117.987 840.907,117.979 840.896,117.975 C 840.896,117.96 840.875,117.965 840.875,117.947 C 840.886,117.925 840.875,117.925 840.879,117.899 C 840.881,117.898 840.886,117.898 840.883,117.892 L 840.889,117.894 C 840.892,117.891 840.889,117.886 840.886,117.884 C 840.887,117.873 840.889,117.873 840.896,117.865 C 840.905,117.853 840.898,117.838 840.889,117.822 L 840.896,117.821 C 840.896,117.815 840.9,117.811 840.894,117.806 C 840.905,117.789 840.872,117.78 840.886,117.76 C 840.887,117.754 840.894,117.751 840.892,117.745 C 840.896,117.742 840.892,117.732 840.902,117.733 C 840.894,117.723 840.903,117.706 840.902,117.692 C 840.889,117.681 840.909,117.671 840.9,117.656 C 840.919,117.645 840.883,117.633 840.905,117.623 C 840.887,117.617 840.903,117.597 840.902,117.587 C 840.903,117.582 840.894,117.576 840.902,117.572 C 840.894,117.557 840.909,117.539 840.9,117.527 C 840.903,117.51 840.907,117.494 840.918,117.479 L 840.938,117.48 C 840.942,117.472 840.952,117.483 840.955,117.473 C 840.966,117.458 840.995,117.462 841.01,117.464 C 841.023,117.477 841.047,117.486 841.065,117.479 C 841.079,117.48 841.093,117.469 841.106,117.461 C 841.118,117.461 841.132,117.472 841.139,117.458 L 841.145,117.464 C 841.149,117.458 841.15,117.462 841.155,117.461 C 841.168,117.452 841.188,117.461 841.201,117.466 C 841.214,117.48 841.24,117.486 841.264,117.491 C 841.28,117.48 841.297,117.496 841.314,117.486 C 841.334,117.491 841.352,117.486 841.372,117.477 C 841.388,117.449 841.412,117.478 841.435,117.477 C 841.444,117.474 841.439,117.488 841.447,117.483 C 841.446,117.488 841.457,117.492 841.46,117.492 C 841.48,117.488 841.499,117.494 841.519,117.496 C 841.528,117.492 841.541,117.492 841.551,117.486 C 841.563,117.48 841.554,117.472 841.563,117.464 C 841.573,117.462 841.589,117.472 841.603,117.464 C 841.606,117.474 841.618,117.474 841.628,117.477 C 841.636,117.479 841.655,117.473 841.651,117.48 C 841.656,117.486 841.665,117.491 841.67,117.492 C 841.677,117.491 841.688,117.494 841.695,117.488 C 841.712,117.494 841.722,117.474 841.737,117.477 C 841.744,117.472 841.76,117.461 841.775,117.466 C 841.788,117.457 841.804,117.466 841.817,117.472 C 841.82,117.468 841.831,117.473 841.833,117.466 C 841.84,117.466 841.844,117.469 841.85,117.474 C 841.856,117.474 841.85,117.466 841.854,117.472 C 841.852,117.474 841.856,117.474 841.859,117.477 C 841.863,117.469 841.868,117.479 841.874,117.484 C 841.885,117.484 841.9,117.496 841.907,117.484 L 841.915,117.486 C 841.919,117.483 841.919,117.479 841.92,117.477 C 841.931,117.478 841.931,117.466 841.934,117.461 C 841.94,117.461 841.951,117.451 841.952,117.462 C 841.968,117.478 841.993,117.469 842.007,117.474 C 842.017,117.469 842.034,117.478 842.044,117.469 C 842.064,117.473 842.081,117.48 842.102,117.483 C 842.109,117.484 842.109,117.473 842.117,117.477 C 842.128,117.473 842.145,117.472 842.159,117.474 C 842.171,117.466 842.186,117.48 842.198,117.484 L 842.203,117.48 L 842.205,117.488 L 842.21,117.484 L 842.214,117.491 C 842.23,117.491 842.257,117.498 842.273,117.479 C 842.278,117.479 842.284,117.483 842.284,117.491 L 842.288,117.486 C 842.289,117.484 842.291,117.488 842.294,117.491 C 842.304,117.491 842.308,117.478 842.32,117.48 C 842.33,117.477 842.349,117.472 842.35,117.453 C 842.359,117.451 842.369,117.457 842.376,117.462 C 842.386,117.466 842.383,117.451 842.394,117.453 C 842.401,117.453 842.412,117.457 842.412,117.464 C 842.413,117.468 842.418,117.462 842.418,117.464 C 842.432,117.462 842.447,117.469 842.465,117.472 C 842.473,117.464 842.485,117.466 842.495,117.469 C 842.502,117.468 842.513,117.464 842.513,117.455 C 842.538,117.434 842.545,117.473 842.567,117.469 C 842.568,117.479 842.579,117.469 842.579,117.479 C 842.591,117.477 842.609,117.478 842.609,117.461 C 842.62,117.462 842.63,117.468 842.64,117.472 C 842.65,117.469 842.666,117.479 842.672,117.472 C 842.682,117.472 842.688,117.457 842.697,117.466 C 842.703,117.483 842.727,117.478 842.74,117.479 C 842.741,117.483 842.743,117.478 842.744,117.477 L 842.749,117.48 C 842.755,117.477 842.774,117.492 842.775,117.477 C 842.784,117.479 842.782,117.464 842.791,117.469 C 842.806,117.455 842.833,117.474 842.849,117.466 C 842.872,117.469 842.889,117.474 842.909,117.477 C 842.913,117.468 842.926,117.478 842.926,117.468 C 842.939,117.474 842.935,117.455 842.947,117.461 C 842.94,117.452 842.948,117.449 842.955,117.446 C 842.962,117.451 842.965,117.447 842.972,117.453 C 842.974,117.455 842.977,117.455 842.978,117.455 C 842.988,117.449 843.004,117.458 843.017,117.455 C 843.023,117.447 843.026,117.462 843.031,117.453 C 843.04,117.455 843.05,117.458 843.052,117.466 C 843.059,117.473 843.067,117.473 843.076,117.474 C 843.092,117.464 843.111,117.479 843.129,117.477 C 843.137,117.468 843.143,117.478 843.151,117.474 C 843.164,117.462 843.18,117.477 843.199,117.474 C 843.205,117.468 843.215,117.478 843.221,117.468 C 843.225,117.472 843.234,117.464 843.234,117.474 C 843.25,117.484 843.265,117.474 843.282,117.472 C 843.302,117.473 843.315,117.488 843.33,117.498 C 843.334,117.494 843.334,117.498 843.337,117.499 C 843.343,117.492 843.346,117.504 843.356,117.5 C 843.365,117.494 843.382,117.486 843.397,117.491 C 843.401,117.48 843.412,117.486 843.416,117.479 C 843.422,117.483 843.426,117.488 843.431,117.492 C 843.431,117.504 843.446,117.505 843.449,117.508 L 843.453,117.514 C 843.464,117.523 843.464,117.505 843.474,117.51 C 843.484,117.503 843.499,117.508 843.512,117.51 C 843.517,117.499 843.531,117.51 843.534,117.5 C 843.541,117.5 843.555,117.503 843.555,117.513 C 843.564,117.518 843.572,117.505 843.581,117.503 C 843.584,117.505 843.591,117.5 843.594,117.508 C 843.602,117.504 843.609,117.509 843.614,117.509 C 843.63,117.51 843.629,117.496 843.637,117.486 C 843.649,117.486 843.659,117.498 843.672,117.5 C 843.692,117.519 843.715,117.5 843.736,117.492 C 843.741,117.498 843.75,117.492 843.757,117.491 C 843.76,117.503 843.776,117.5 843.784,117.51 C 843.791,117.509 843.796,117.509 843.805,117.509 C 843.808,117.5 843.826,117.509 843.823,117.496 C 843.834,117.492 843.846,117.477 843.862,117.483 C 843.874,117.479 843.893,117.483 843.902,117.494 C 843.914,117.473 843.922,117.503 843.94,117.496 C 843.945,117.492 843.947,117.488 843.952,117.491 L 843.954,117.484 C 843.959,117.483 843.961,117.492 843.969,117.484 C 843.985,117.469 844.006,117.491 844.026,117.486 C 844.04,117.488 844.03,117.504 844.045,117.505 C 844.059,117.513 844.07,117.504 844.084,117.496 C 844.097,117.498 844.114,117.488 844.127,117.492 C 844.137,117.486 844.152,117.499 844.159,117.486 L 844.162,117.488 C 844.171,117.48 844.183,117.483 844.193,117.486 C 844.205,117.488 844.222,117.494 844.231,117.498 C 844.264,117.514 844.289,117.494 844.321,117.51 C 844.321,117.514 844.321,117.523 844.327,117.523 L 844.324,117.527 C 844.348,117.538 844.323,117.567 844.334,117.587 C 844.333,117.592 844.342,117.597 844.333,117.6 C 844.339,117.615 844.328,117.633 844.327,117.652 L 844.333,117.654 L 844.33,117.656 C 844.348,117.679 844.32,117.698 844.334,117.723 C 844.334,117.737 844.337,117.747 844.337,117.764 C 844.344,117.772 844.346,117.784 844.346,117.795 C 844.359,117.806 844.349,117.82 844.348,117.834 C 844.36,117.847 844.339,117.863 844.342,117.878 C 844.342,117.887 844.346,117.898 844.341,117.903 L 844.344,117.907 C 844.341,117.913 844.346,117.921 844.341,117.93 C 844.327,117.951 844.349,117.97 844.337,117.99 C 844.348,117.992 844.333,118.005 844.346,118.006 C 844.346,118.022 844.353,118.03 844.35,118.047 C 844.364,118.048 844.35,118.056 844.356,118.064 C 844.368,118.07 844.353,118.084 844.364,118.091 z M 844.15,120.227 L 844.149,120.222 C 844.155,120.218 844.152,120.215 844.153,120.209 C 844.147,120.201 844.159,120.193 844.149,120.189 L 844.149,120.174 C 844.143,120.166 844.122,120.17 844.127,120.156 C 844.137,120.152 844.144,120.146 844.145,120.136 C 844.155,120.113 844.089,120.12 844.133,120.1 C 844.133,120.087 844.136,120.078 844.138,120.065 C 844.131,120.053 844.118,120.047 844.118,120.032 C 844.123,120.022 844.136,120.013 844.127,120.005 C 844.131,119.992 844.133,119.977 844.126,119.966 C 844.13,119.956 844.118,119.956 844.112,119.951 C 844.109,119.928 844.13,119.895 844.103,119.878 C 844.107,119.867 844.101,119.853 844.092,119.845 C 844.086,119.831 844.105,119.841 844.103,119.827 C 844.1,119.819 844.1,119.809 844.093,119.799 C 844.092,119.778 844.09,119.757 844.099,119.736 C 844.089,119.729 844.096,119.717 844.089,119.71 C 844.09,119.704 844.1,119.696 844.092,119.69 C 844.093,119.684 844.09,119.681 844.086,119.677 C 844.072,119.658 844.093,119.638 844.081,119.617 C 844.084,119.603 844.092,119.591 844.081,119.578 C 844.076,119.565 844.076,119.548 844.073,119.531 C 844.072,119.517 844.086,119.504 844.084,119.49 C 844.059,119.483 844.081,119.467 844.065,119.455 C 844.07,119.448 844.065,119.437 844.072,119.43 C 844.061,119.407 844.073,119.38 844.085,119.364 L 844.084,119.356 C 844.085,119.356 844.086,119.355 844.086,119.355 C 844.079,119.349 844.089,119.346 844.084,119.335 C 844.077,119.329 844.072,119.318 844.072,119.306 C 844.055,119.296 844.085,119.277 844.065,119.267 C 844.076,119.251 844.089,119.235 844.078,119.215 C 844.09,119.193 844.078,119.172 844.065,119.156 C 844.073,119.137 844.055,119.135 844.061,119.117 C 844.051,119.105 844.068,119.091 844.064,119.075 C 844.052,119.068 844.073,119.044 844.051,119.044 C 844.045,119.043 844.04,119.038 844.04,119.033 C 844.04,119.026 844.05,119.026 844.053,119.022 C 844.055,119.023 844.059,119.023 844.061,119.023 C 844.061,119.017 844.065,119.008 844.063,119 L 844.059,118.993 C 844.065,118.987 844.076,118.985 844.073,118.973 C 844.072,118.97 844.077,118.96 844.068,118.959 C 844.076,118.944 844.079,118.933 844.072,118.919 C 844.077,118.909 844.065,118.907 844.063,118.899 C 844.052,118.898 844.059,118.889 844.055,118.882 C 844.063,118.873 844.077,118.867 844.077,118.852 C 844.084,118.841 844.064,118.826 844.079,118.816 C 844.072,118.799 844.09,118.786 844.09,118.769 C 844.096,118.758 844.093,118.741 844.089,118.728 C 844.078,118.724 844.078,118.711 844.065,118.71 C 844.063,118.705 844.053,118.696 844.063,118.692 L 844.078,118.679 C 844.084,118.672 844.086,118.665 844.084,118.658 C 844.077,118.65 844.086,118.639 844.077,118.631 C 844.073,118.617 844.092,118.623 844.093,118.616 L 844.1,118.616 C 844.103,118.592 844.119,118.571 844.099,118.552 C 844.099,118.535 844.109,118.518 844.09,118.505 L 844.085,118.494 C 844.093,118.487 844.096,118.473 844.107,118.475 C 844.109,118.473 844.109,118.468 844.109,118.464 C 844.097,118.462 844.096,118.448 844.092,118.438 C 844.099,118.433 844.111,118.438 844.112,118.426 L 844.112,118.404 C 844.105,118.4 844.099,118.388 844.093,118.38 L 844.093,118.379 C 844.103,118.374 844.118,118.381 844.125,118.37 C 844.133,118.363 844.138,118.35 844.133,118.339 C 844.13,118.323 844.109,118.334 844.105,118.32 C 844.109,118.313 844.112,118.307 844.118,118.302 C 844.119,118.303 844.122,118.303 844.125,118.302 C 844.123,118.286 844.144,118.27 844.127,118.258 C 844.133,118.249 844.14,118.243 844.145,118.236 L 844.144,118.235 C 844.15,118.229 844.147,118.224 844.152,118.216 C 844.15,118.215 844.147,118.212 844.15,118.209 C 844.138,118.202 844.133,118.187 844.119,118.18 C 844.125,118.172 844.14,118.176 844.145,118.166 C 844.143,118.152 844.143,118.141 844.15,118.129 C 844.14,118.129 844.145,118.114 844.137,118.114 C 844.127,118.102 844.149,118.087 844.131,118.083 C 844.118,118.084 844.117,118.073 844.111,118.065 C 844.101,118.047 844.136,118.036 844.123,118.016 C 844.127,118.002 844.119,117.992 844.112,117.986 C 844.117,117.965 844.107,117.947 844.114,117.93 C 844.101,117.914 844.118,117.891 844.118,117.869 C 844.119,117.856 844.103,117.851 844.107,117.836 C 844.111,117.822 844.109,117.806 844.112,117.794 C 844.118,117.791 844.112,117.784 844.112,117.779 C 844.105,117.777 844.109,117.76 844.109,117.754 C 844.118,117.751 844.117,117.737 844.117,117.732 C 844.093,117.721 844.073,117.745 844.055,117.727 L 844.051,117.731 C 844.044,117.731 844.028,117.723 844.021,117.732 C 843.997,117.727 843.967,117.723 843.948,117.711 C 843.941,117.711 843.928,117.707 843.92,117.706 C 843.915,117.706 843.906,117.702 843.908,117.71 C 843.906,117.711 843.905,117.714 843.905,117.711 C 843.899,117.71 843.889,117.715 843.885,117.717 C 843.879,117.716 843.869,117.715 843.863,117.721 C 843.839,117.717 843.81,117.72 843.788,117.731 C 843.77,117.737 843.76,117.72 843.744,117.723 C 843.73,117.715 843.715,117.711 843.701,117.717 C 843.698,117.727 843.683,117.72 843.683,117.732 C 843.665,117.728 843.653,117.701 843.637,117.71 C 843.625,117.706 843.613,117.705 843.606,117.711 C 843.603,117.707 843.593,117.705 843.586,117.707 C 843.577,117.698 843.566,117.715 843.559,117.717 C 843.551,117.727 843.553,117.742 843.537,117.742 C 843.531,117.739 843.51,117.743 843.508,117.727 L 843.493,117.727 C 843.494,117.728 843.493,117.728 843.491,117.731 L 843.487,117.725 C 843.465,117.731 843.442,117.715 843.418,117.716 C 843.408,117.714 843.408,117.696 843.394,117.7 L 843.387,117.702 C 843.386,117.692 843.381,117.705 843.376,117.7 C 843.365,117.698 843.356,117.705 843.343,117.702 C 843.335,117.696 843.315,117.701 843.315,117.688 C 843.297,117.675 843.271,117.676 843.255,117.681 L 843.25,117.685 C 843.247,117.68 843.24,117.685 843.234,117.683 C 843.231,117.696 843.21,117.688 843.204,117.689 C 843.197,117.689 843.189,117.692 843.179,117.689 C 843.158,117.68 843.139,117.676 843.115,117.671 C 843.104,117.654 843.081,117.652 843.063,117.653 C 843.058,117.654 843.045,117.655 843.038,117.659 C 843.031,117.656 843.027,117.662 843.026,117.667 C 843.018,117.663 843.008,117.667 843.002,117.671 C 842.99,117.67 842.974,117.656 842.965,117.67 C 842.965,117.669 842.962,117.667 842.961,117.665 C 842.935,117.68 842.921,117.648 842.896,117.659 C 842.88,117.654 842.867,117.665 842.853,117.671 C 842.843,117.665 842.835,117.679 842.828,117.671 C 842.822,117.675 842.817,117.669 842.811,117.669 L 842.81,117.665 C 842.8,117.667 842.784,117.663 842.774,117.669 C 842.765,117.652 842.744,117.649 842.725,117.649 C 842.706,117.662 842.691,117.639 842.67,117.652 C 842.658,117.649 842.64,117.643 842.629,117.649 C 842.626,117.645 842.624,117.644 842.623,117.648 C 842.581,117.635 842.546,117.663 842.506,117.639 C 842.494,117.643 842.491,117.656 842.472,117.654 C 842.466,117.662 842.46,117.653 842.455,117.655 C 842.441,117.662 842.423,117.656 842.408,117.667 C 842.401,117.675 842.389,117.667 842.38,117.674 C 842.372,117.67 842.355,117.669 842.34,117.665 C 842.324,117.659 842.322,117.67 842.307,117.669 C 842.303,117.679 842.287,117.667 842.28,117.676 C 842.271,117.675 842.269,117.685 842.26,117.683 C 842.261,117.696 842.245,117.692 842.242,117.698 C 842.228,117.701 842.227,117.67 842.212,117.689 C 842.204,117.692 842.191,117.68 842.185,117.692 C 842.183,117.688 842.179,117.688 842.176,117.689 C 842.152,117.689 842.128,117.665 842.106,117.681 C 842.078,117.701 842.054,117.659 842.026,117.68 C 842.016,117.679 841.997,117.667 841.99,117.679 C 841.981,117.683 841.964,117.685 841.961,117.701 C 841.958,117.7 841.951,117.702 841.947,117.707 C 841.944,117.705 841.94,117.706 841.933,117.702 C 841.936,117.694 841.923,117.694 841.919,117.692 C 841.914,117.689 841.914,117.696 841.912,117.698 C 841.908,117.685 841.901,117.705 841.9,117.692 C 841.893,117.696 841.874,117.683 841.874,117.698 C 841.862,117.698 841.848,117.701 841.837,117.707 C 841.822,117.706 841.8,117.702 841.791,117.688 C 841.778,117.679 841.762,117.685 841.758,117.669 C 841.742,117.671 841.73,117.671 841.717,117.676 C 841.705,117.674 841.685,117.676 841.675,117.669 C 841.668,117.665 841.668,117.676 841.664,117.669 C 841.655,117.674 841.662,117.681 841.656,117.688 C 841.644,117.688 841.633,117.698 841.622,117.688 C 841.607,117.671 841.582,117.676 841.565,117.669 C 841.557,117.676 841.539,117.667 841.525,117.674 L 841.521,117.671 L 841.518,117.674 C 841.512,117.671 841.5,117.667 841.492,117.67 C 841.491,117.68 841.478,117.671 841.477,117.681 C 841.472,117.681 841.467,117.683 841.464,117.689 C 841.447,117.681 841.428,117.683 841.412,117.688 L 841.409,117.692 C 841.403,117.69 841.399,117.692 841.392,117.69 C 841.394,117.702 841.376,117.694 841.378,117.707 C 841.369,117.7 841.369,117.715 841.363,117.714 L 841.31,117.7 C 841.303,117.698 841.299,117.692 841.299,117.688 C 841.28,117.679 841.257,117.676 841.236,117.681 C 841.228,117.689 841.21,117.679 841.202,117.689 C 841.197,117.683 841.184,117.685 841.174,117.688 C 841.171,117.694 841.145,117.68 841.15,117.7 C 841.152,117.715 841.136,117.725 841.139,117.742 C 841.126,117.766 841.142,117.789 841.129,117.811 C 841.128,117.82 841.146,117.819 841.132,117.821 C 841.138,117.831 841.138,117.836 841.149,117.842 C 841.15,117.851 841.149,117.863 841.144,117.872 C 841.126,117.873 841.132,117.894 841.124,117.903 L 841.125,117.904 C 841.116,117.909 841.126,117.914 841.125,117.921 C 841.131,117.925 841.142,117.925 841.149,117.93 C 841.146,117.939 841.146,117.951 841.136,117.955 C 841.139,117.96 841.136,117.966 841.139,117.974 C 841.138,117.987 841.118,117.975 841.124,117.991 C 841.125,118.002 841.126,118.011 841.128,118.022 C 841.136,118.036 841.158,118.053 841.149,118.073 C 841.139,118.07 841.139,118.078 841.138,118.083 C 841.139,118.094 841.142,118.11 841.158,118.114 C 841.158,118.129 841.144,118.145 841.161,118.153 C 841.172,118.15 841.161,118.162 841.168,118.166 C 841.161,118.182 841.155,118.205 841.149,118.227 C 841.168,118.24 841.138,118.262 841.153,118.28 C 841.153,118.282 841.146,118.284 841.152,118.289 C 841.144,118.289 841.142,118.302 841.131,118.298 C 841.125,118.309 841.125,118.329 841.118,118.342 C 841.12,118.343 841.124,118.339 841.125,118.342 C 841.125,118.342 841.126,118.343 841.126,118.345 L 841.125,118.347 C 841.138,118.35 841.142,118.368 841.146,118.38 C 841.145,118.394 841.135,118.407 841.145,118.417 C 841.146,118.426 841.139,118.438 841.146,118.448 C 841.145,118.462 841.126,118.468 841.131,118.484 C 841.142,118.495 841.128,118.51 841.132,118.524 C 841.132,118.527 841.132,118.544 841.142,118.546 C 841.138,118.556 841.152,118.561 841.145,118.572 C 841.135,118.588 841.149,118.606 841.145,118.622 C 841.161,118.624 841.142,118.638 841.153,118.641 C 841.162,118.651 841.144,118.65 841.139,118.658 C 841.132,118.655 841.135,118.665 841.132,118.668 C 841.142,118.678 841.129,118.687 841.132,118.697 C 841.136,118.706 841.153,118.701 841.149,118.714 C 841.152,118.716 841.153,118.714 841.155,118.718 C 841.149,118.726 841.161,118.727 841.164,118.731 C 841.162,118.74 841.158,118.757 841.145,118.754 L 841.145,118.757 C 841.138,118.754 841.132,118.777 841.118,118.765 L 841.12,118.769 C 841.118,118.769 841.118,118.771 841.116,118.771 C 841.126,118.775 841.115,118.789 841.126,118.789 C 841.128,118.799 841.118,118.813 841.129,118.821 C 841.131,118.837 841.116,118.845 841.118,118.86 L 841.125,118.86 C 841.12,118.864 841.129,118.871 841.128,118.877 C 841.135,118.883 841.15,118.883 841.164,118.886 C 841.169,118.886 841.169,118.878 841.174,118.886 C 841.188,118.881 841.204,118.889 841.217,118.881 C 841.228,118.883 841.24,118.889 841.25,118.881 C 841.262,118.883 841.276,118.889 841.288,118.882 C 841.311,118.89 841.332,118.878 841.356,118.886 L 841.36,118.881 C 841.363,118.886 841.366,118.886 841.368,118.882 C 841.378,118.886 841.392,118.887 841.402,118.882 C 841.412,118.883 841.428,118.889 841.439,118.882 C 841.464,118.89 841.48,118.878 841.504,118.886 C 841.518,118.877 841.539,118.894 841.552,118.882 C 841.573,118.893 841.594,118.878 841.618,118.886 L 841.619,118.882 C 841.624,118.89 841.632,118.878 841.639,118.883 C 841.645,118.887 841.659,118.889 841.664,118.881 L 841.666,118.883 C 841.679,118.881 841.692,118.89 841.705,118.882 C 841.717,118.883 841.729,118.889 841.741,118.883 C 841.751,118.882 841.765,118.893 841.775,118.882 C 841.786,118.882 841.801,118.89 841.811,118.882 C 841.814,118.883 841.817,118.887 841.822,118.883 C 841.841,118.893 841.856,118.881 841.879,118.886 C 841.895,118.877 841.914,118.893 841.931,118.883 C 841.941,118.882 841.953,118.893 841.962,118.883 C 841.987,118.889 842.007,118.878 842.034,118.883 C 842.044,118.881 842.061,118.893 842.074,118.882 C 842.099,118.889 842.124,118.881 842.15,118.886 C 842.176,118.889 842.192,118.878 842.221,118.886 C 842.234,118.877 842.253,118.893 842.264,118.883 C 842.287,118.893 842.307,118.877 842.329,118.886 C 842.342,118.877 842.361,118.893 842.376,118.883 C 842.392,118.886 842.402,118.886 842.416,118.883 C 842.425,118.882 842.438,118.893 842.445,118.882 C 842.453,118.889 842.456,118.882 842.465,118.883 C 842.484,118.889 842.495,118.878 842.514,118.886 L 842.517,118.881 C 842.52,118.887 842.521,118.881 842.527,118.882 C 842.541,118.887 842.558,118.886 842.571,118.883 L 842.585,118.883 L 842.585,118.881 C 842.587,118.883 842.594,118.882 842.598,118.881 C 842.609,118.883 842.62,118.886 842.629,118.881 C 842.64,118.883 842.656,118.887 842.666,118.881 C 842.688,118.893 842.708,118.873 842.729,118.883 C 842.741,118.876 842.765,118.893 842.77,118.868 C 842.789,118.862 842.791,118.841 842.811,118.836 C 842.81,118.833 842.807,118.831 842.811,118.829 C 842.814,118.84 842.822,118.829 842.829,118.829 L 842.833,118.825 C 842.835,118.836 842.841,118.821 842.847,118.829 C 842.882,118.823 842.919,118.829 842.948,118.847 C 842.943,118.858 842.959,118.855 842.959,118.863 C 842.972,118.868 842.972,118.883 842.988,118.887 C 842.992,118.894 842.997,118.902 843,118.909 C 843.004,118.919 842.999,118.933 843.005,118.94 C 843.002,118.949 843,118.955 843.004,118.965 C 843.008,118.981 843.027,118.982 843.031,118.999 C 843.04,118.993 843.035,119.004 843.04,119.007 C 843.052,119.02 843.07,119.034 843.081,119.048 C 843.102,119.047 843.089,119.066 843.105,119.07 C 843.119,119.074 843.119,119.094 843.135,119.095 C 843.131,119.103 843.143,119.103 843.145,119.107 C 843.149,119.105 843.151,119.111 843.151,119.114 C 843.158,119.121 843.166,119.117 843.166,119.13 C 843.178,119.134 843.178,119.147 843.191,119.151 C 843.201,119.158 843.205,119.175 843.223,119.178 C 843.219,119.178 843.218,119.183 843.221,119.184 C 843.234,119.184 843.231,119.199 843.245,119.202 C 843.243,119.211 843.252,119.21 843.256,119.218 C 843.263,119.215 843.26,119.223 843.261,119.224 C 843.275,119.236 843.283,119.246 843.298,119.259 C 843.298,119.261 843.298,119.262 843.298,119.265 C 843.309,119.271 843.32,119.28 843.327,119.291 C 843.342,119.292 843.34,119.31 843.354,119.31 C 843.356,119.322 843.366,119.332 843.376,119.335 C 843.379,119.342 843.382,119.347 843.387,119.35 C 843.394,119.349 843.395,119.354 843.395,119.356 C 843.423,119.376 843.436,119.41 843.469,119.42 C 843.494,119.415 843.531,119.417 843.553,119.436 C 843.553,119.447 843.57,119.443 843.568,119.456 C 843.578,119.459 843.583,119.473 843.593,119.474 C 843.598,119.483 843.605,119.49 843.61,119.5 C 843.619,119.513 843.606,119.529 843.614,119.543 C 843.613,119.548 843.61,119.553 843.61,119.565 C 843.611,119.568 843.611,119.572 843.617,119.572 L 843.611,119.573 C 843.617,119.582 843.613,119.591 843.61,119.599 C 843.606,119.612 843.594,119.614 843.591,119.626 C 843.581,119.621 843.578,119.639 843.57,119.64 L 843.55,119.667 C 843.541,119.669 843.532,119.673 843.522,119.677 C 843.488,119.677 843.449,119.687 843.416,119.67 C 843.408,119.661 843.394,119.649 843.386,119.643 C 843.375,119.626 843.363,119.621 843.354,119.604 C 843.342,119.591 843.354,119.568 843.347,119.552 C 843.349,119.548 843.349,119.543 843.349,119.538 C 843.342,119.528 843.33,119.519 843.324,119.508 C 843.304,119.489 843.281,119.467 843.263,119.443 C 843.249,119.441 843.249,119.426 843.237,119.421 C 843.237,119.422 843.235,119.422 843.234,119.422 C 843.235,119.411 843.219,119.411 843.218,119.401 C 843.202,119.385 843.189,119.371 843.175,119.356 C 843.156,119.342 843.146,119.328 843.129,119.314 C 843.119,119.302 843.112,119.292 843.099,119.286 C 843.093,119.271 843.076,119.265 843.066,119.249 C 843.059,119.245 843.052,119.231 843.043,119.229 C 843.038,119.22 843.028,119.209 843.02,119.205 C 843.011,119.189 842.997,119.184 842.985,119.167 C 842.974,119.166 842.977,119.152 842.965,119.151 C 842.956,119.141 842.951,119.131 842.94,119.125 C 842.929,119.114 842.921,119.103 842.911,119.099 C 842.907,119.094 842.905,119.09 842.898,119.088 C 842.885,119.096 842.867,119.079 842.853,119.091 C 842.849,119.086 842.844,119.09 842.839,119.09 L 842.803,119.079 C 842.781,119.062 842.77,119.027 842.736,119.039 C 842.711,119.034 842.692,119.038 842.666,119.038 C 842.655,119.044 842.637,119.028 842.625,119.042 L 842.62,119.038 C 842.615,119.042 842.6,119.037 842.594,119.039 C 842.591,119.034 842.587,119.039 842.584,119.042 C 842.576,119.042 842.564,119.037 842.554,119.039 C 842.551,119.033 842.547,119.039 842.545,119.042 C 842.53,119.031 842.513,119.043 842.501,119.038 C 842.494,119.038 842.48,119.034 842.475,119.042 C 842.472,119.038 842.469,119.039 842.465,119.039 C 842.455,119.039 842.443,119.034 842.434,119.042 C 842.432,119.038 842.428,119.037 842.427,119.039 C 842.421,119.039 842.408,119.033 842.405,119.042 C 842.402,119.031 842.394,119.044 842.386,119.039 C 842.38,119.039 842.366,119.033 842.362,119.042 C 842.353,119.039 842.336,119.034 842.327,119.042 C 842.3,119.033 842.284,119.044 842.26,119.038 C 842.245,119.047 842.228,119.033 842.214,119.042 C 842.204,119.039 842.191,119.039 842.179,119.038 C 842.173,119.044 842.153,119.034 842.143,119.042 C 842.136,119.043 842.12,119.038 842.111,119.038 C 842.1,119.047 842.085,119.033 842.072,119.042 C 842.071,119.034 842.066,119.043 842.061,119.042 C 842.033,119.039 842.01,119.037 841.982,119.042 C 841.978,119.039 841.971,119.038 841.964,119.039 L 841.962,119.044 L 841.958,119.039 C 841.947,119.043 841.939,119.038 841.926,119.039 C 841.914,119.048 841.895,119.031 841.885,119.043 C 841.872,119.044 841.859,119.034 841.85,119.044 C 841.85,119.039 841.844,119.038 841.842,119.042 C 841.833,119.043 841.82,119.033 841.811,119.043 C 841.788,119.033 841.77,119.051 841.745,119.042 C 841.732,119.044 841.717,119.037 841.701,119.044 C 841.689,119.047 841.673,119.034 841.658,119.044 C 841.651,119.044 841.638,119.037 841.63,119.044 C 841.628,119.038 841.624,119.043 841.618,119.043 C 841.607,119.043 841.596,119.037 841.589,119.044 C 841.584,119.037 841.58,119.047 841.574,119.042 C 841.567,119.039 841.554,119.038 841.55,119.043 C 841.539,119.047 841.523,119.034 841.517,119.044 L 841.517,119.042 C 841.491,119.039 841.46,119.043 841.434,119.042 C 841.421,119.044 841.402,119.038 841.392,119.042 C 841.382,119.043 841.366,119.033 841.356,119.043 C 841.354,119.042 841.352,119.038 841.35,119.042 L 841.348,119.043 C 841.342,119.039 841.335,119.039 841.328,119.042 L 841.328,119.043 C 841.324,119.037 841.317,119.047 841.313,119.042 C 841.302,119.037 841.287,119.047 841.28,119.042 C 841.271,119.043 841.26,119.038 841.25,119.042 C 841.236,119.047 841.221,119.034 841.208,119.044 C 841.204,119.039 841.201,119.039 841.197,119.042 C 841.179,119.028 841.165,119.052 841.15,119.042 C 841.144,119.043 841.126,119.043 841.112,119.043 C 841.118,119.054 841.11,119.066 841.109,119.079 C 841.089,119.079 841.109,119.094 841.104,119.101 C 841.115,119.101 841.105,119.111 841.112,119.117 L 841.115,119.114 C 841.115,119.122 841.126,119.117 841.125,119.126 C 841.125,119.131 841.115,119.134 841.12,119.137 C 841.115,119.14 841.109,119.152 841.109,119.162 C 841.11,119.172 841.131,119.189 841.118,119.204 C 841.118,119.21 841.126,119.218 841.118,119.225 L 841.104,119.228 C 841.105,119.231 841.109,119.234 841.104,119.235 C 841.11,119.246 841.1,119.257 841.102,119.267 C 841.098,119.275 841.09,119.271 841.089,119.277 C 841.094,119.284 841.116,119.286 841.11,119.302 C 841.116,119.3 841.118,119.308 841.124,119.306 C 841.125,119.316 841.138,119.316 841.139,119.327 C 841.144,119.334 841.139,119.349 841.128,119.347 C 841.128,119.349 841.125,119.349 841.125,119.349 C 841.115,119.343 841.116,119.356 841.112,119.356 C 841.118,119.362 841.116,119.354 841.124,119.354 C 841.135,119.355 841.15,119.364 841.161,119.354 C 841.165,119.359 841.165,119.359 841.171,119.355 C 841.181,119.356 841.194,119.362 841.202,119.355 C 841.204,119.355 841.208,119.359 841.21,119.355 C 841.214,119.356 841.23,119.364 841.231,119.354 C 841.244,119.354 841.262,119.36 841.276,119.355 C 841.283,119.355 841.302,119.362 841.303,119.354 L 841.303,119.356 C 841.316,119.35 841.332,119.364 841.343,119.354 C 841.346,119.362 841.35,119.353 841.354,119.355 C 841.363,119.356 841.373,119.359 841.382,119.354 L 841.412,119.355 L 841.412,119.354 C 841.418,119.362 841.421,119.347 841.431,119.355 C 841.457,119.355 841.467,119.354 841.491,119.356 L 841.492,119.353 C 841.495,119.36 841.499,119.353 841.504,119.355 C 841.514,119.355 841.526,119.36 841.536,119.355 L 841.539,119.356 L 841.563,119.355 C 841.574,119.35 841.594,119.36 841.606,119.354 C 841.61,119.359 841.612,119.353 841.616,119.354 C 841.633,119.359 841.649,119.354 841.668,119.356 L 841.677,119.353 C 841.677,119.359 841.682,119.354 841.684,119.354 C 841.698,119.355 841.705,119.359 841.718,119.353 C 841.728,119.356 841.742,119.359 841.755,119.353 L 841.786,119.355 L 841.786,119.354 L 841.793,119.355 C 841.791,119.354 841.794,119.354 841.794,119.353 C 841.807,119.354 841.823,119.362 841.837,119.354 C 841.844,119.355 841.855,119.359 841.863,119.355 C 841.863,119.354 841.866,119.354 841.866,119.353 L 841.87,119.355 C 841.888,119.356 841.905,119.353 841.918,119.355 L 841.934,119.355 C 841.934,119.354 841.936,119.354 841.936,119.353 C 841.94,119.359 841.941,119.35 841.947,119.354 C 841.971,119.36 841.987,119.35 842.01,119.356 L 842.017,119.353 C 842.03,119.354 842.044,119.36 842.057,119.355 C 842.061,119.356 842.059,119.368 842.064,119.368 C 842.069,119.377 842.054,119.395 842.066,119.407 C 842.057,119.422 842.069,119.441 842.064,119.456 C 842.074,119.477 842.057,119.497 842.067,119.517 C 842.069,119.53 842.054,119.543 842.066,119.553 C 842.066,119.565 842.06,119.581 842.067,119.588 C 842.067,119.597 842.059,119.599 842.052,119.603 C 842.038,119.604 842.027,119.597 842.019,119.607 C 842.008,119.603 841.993,119.599 841.982,119.607 C 841.971,119.607 841.966,119.601 841.952,119.603 L 841.947,119.608 C 841.946,119.607 841.951,119.603 841.946,119.603 C 841.934,119.609 841.919,119.594 841.908,119.607 C 841.895,119.604 841.881,119.599 841.868,119.607 C 841.846,119.597 841.829,119.609 841.809,119.604 C 841.794,119.612 841.778,119.601 841.767,119.608 C 841.757,119.609 841.74,119.599 841.729,119.609 C 841.705,119.599 841.685,119.612 841.662,119.604 L 841.658,119.609 L 841.656,119.607 C 841.643,119.607 841.629,119.601 841.617,119.609 C 841.594,119.601 841.573,119.612 841.552,119.604 L 841.549,119.612 C 841.544,119.603 841.541,119.609 841.536,119.608 C 841.526,119.607 841.513,119.603 841.504,119.609 C 841.503,119.604 841.499,119.609 841.497,119.609 C 841.477,119.599 841.458,119.613 841.437,119.607 L 841.435,119.612 C 841.432,119.609 841.435,119.607 841.432,119.607 C 841.425,119.612 841.414,119.604 841.403,119.607 C 841.392,119.612 841.372,119.601 841.356,119.612 C 841.346,119.609 841.334,119.604 841.324,119.612 C 841.303,119.604 841.283,119.612 841.26,119.608 C 841.255,119.609 841.251,119.614 841.25,119.609 C 841.245,119.614 841.242,119.607 841.238,119.609 C 841.231,119.609 841.219,119.603 841.216,119.612 C 841.204,119.609 841.195,119.609 841.184,119.608 C 841.172,119.619 841.155,119.601 841.144,119.613 C 841.138,119.607 841.129,119.612 841.125,119.612 L 841.118,119.628 L 841.116,119.627 C 841.126,119.648 841.094,119.645 841.1,119.669 C 841.098,119.675 841.09,119.681 841.098,119.687 C 841.097,119.696 841.11,119.7 841.109,119.71 C 841.104,119.723 841.098,119.736 841.1,119.749 C 841.104,119.749 841.1,119.753 841.104,119.757 C 841.11,119.758 841.102,119.764 841.106,119.77 C 841.105,119.778 841.09,119.778 841.09,119.79 C 841.09,119.799 841.093,119.81 841.09,119.819 C 841.089,119.825 841.104,119.824 841.097,119.831 L 841.1,119.84 C 841.094,119.841 841.102,119.846 841.1,119.849 C 841.1,119.853 841.097,119.862 841.098,119.867 C 841.098,119.867 841.105,119.864 841.105,119.868 C 841.118,119.871 841.139,119.871 841.152,119.868 C 841.157,119.868 841.171,119.873 841.171,119.864 C 841.179,119.864 841.198,119.873 841.214,119.872 C 841.224,119.864 841.245,119.874 841.253,119.868 C 841.266,119.864 841.283,119.874 841.294,119.867 C 841.295,119.874 841.301,119.867 841.304,119.868 C 841.321,119.873 841.339,119.864 841.356,119.871 C 841.36,119.871 841.368,119.873 841.372,119.868 C 841.383,119.868 841.406,119.871 841.414,119.868 C 841.425,119.871 841.435,119.873 841.444,119.868 C 841.457,119.868 841.468,119.874 841.477,119.867 C 841.48,119.871 841.49,119.867 841.497,119.871 C 841.504,119.873 841.499,119.88 841.504,119.886 C 841.492,119.907 841.51,119.928 841.5,119.949 C 841.51,119.961 841.494,119.98 841.504,119.995 C 841.506,120.005 841.495,120.015 841.506,120.023 C 841.504,120.038 841.5,120.047 841.499,120.063 C 841.491,120.063 841.482,120.071 841.477,120.065 C 841.467,120.069 841.451,120.061 841.44,120.069 C 841.431,120.067 841.411,120.059 841.402,120.067 C 841.401,120.065 841.399,120.064 841.398,120.063 L 841.394,120.065 C 841.382,120.065 841.365,120.057 841.352,120.065 C 841.329,120.056 841.307,120.065 841.283,120.063 L 841.283,120.065 C 841.28,120.065 841.277,120.057 841.273,120.064 C 841.261,120.065 841.247,120.056 841.24,120.065 C 841.236,120.057 841.231,120.067 841.228,120.063 C 841.204,120.061 841.188,120.064 841.165,120.061 L 841.161,120.065 L 841.157,120.063 C 841.139,120.057 841.124,120.069 841.106,120.063 C 841.102,120.061 841.098,120.064 841.1,120.067 C 841.1,120.059 841.091,120.069 841.086,120.064 C 841.079,120.064 841.083,120.07 841.083,120.075 C 841.084,120.08 841.093,120.08 841.097,120.084 C 841.098,120.105 841.078,120.118 841.094,120.133 C 841.091,120.14 841.094,120.146 841.1,120.146 C 841.093,120.15 841.1,120.161 841.089,120.158 C 841.086,120.162 841.084,120.167 841.086,120.17 L 841.091,120.17 C 841.089,120.175 841.093,120.184 841.091,120.192 C 841.093,120.192 841.094,120.192 841.097,120.193 C 841.089,120.197 841.098,120.206 841.091,120.212 C 841.091,120.219 841.09,120.222 841.091,120.234 C 841.093,120.234 841.094,120.231 841.097,120.235 L 841.094,120.239 C 841.093,120.241 841.1,120.244 841.104,120.244 C 841.102,120.254 841.1,120.265 841.094,120.276 L 841.091,120.277 C 841.089,120.291 841.106,120.291 841.102,120.303 L 841.104,120.307 C 841.1,120.31 841.104,120.313 841.104,120.314 C 841.1,120.322 841.105,120.327 841.104,120.334 L 841.106,120.334 C 841.106,120.342 841.112,120.363 841.094,120.363 C 841.09,120.369 841.104,120.374 841.097,120.38 C 841.097,120.386 841.093,120.391 841.094,120.399 C 841.1,120.405 841.115,120.402 841.11,120.415 C 841.11,120.432 841.093,120.447 841.102,120.463 L 841.104,120.465 C 841.097,120.469 841.102,120.48 841.104,120.49 C 841.112,120.503 841.094,120.524 841.106,120.535 C 841.105,120.542 841.1,120.554 841.109,120.557 L 841.105,120.56 C 841.106,120.569 841.105,120.581 841.105,120.59 C 841.106,120.595 841.11,120.595 841.106,120.599 C 841.116,120.607 841.112,120.618 841.126,120.619 C 841.126,120.625 841.12,120.634 841.115,120.637 L 841.118,120.638 L 841.112,120.638 C 841.109,120.656 841.09,120.671 841.104,120.692 C 841.104,120.708 841.091,120.73 841.109,120.743 C 841.097,120.758 841.118,120.773 841.109,120.789 C 841.116,120.796 841.112,120.809 841.118,120.816 C 841.12,120.826 841.112,120.841 841.124,120.847 C 841.118,120.859 841.131,120.868 841.129,120.881 C 841.126,120.884 841.12,120.881 841.116,120.881 C 841.112,120.892 841.104,120.9 841.109,120.914 L 841.112,120.914 C 841.105,120.918 841.115,120.925 841.112,120.93 C 841.128,120.944 841.1,120.96 841.118,120.977 C 841.11,120.976 841.11,120.983 841.109,120.986 C 841.115,120.995 841.116,121.008 841.129,121.008 C 841.126,121.012 841.129,121.016 841.132,121.017 C 841.142,121.022 841.126,121.022 841.131,121.029 C 841.128,121.033 841.124,121.033 841.125,121.039 C 841.126,121.038 841.128,121.034 841.129,121.038 C 841.124,121.043 841.135,121.054 841.135,121.063 C 841.132,121.074 841.126,121.082 841.125,121.094 C 841.131,121.1 841.118,121.106 841.129,121.111 C 841.129,121.116 841.135,121.116 841.138,121.117 C 841.144,121.132 841.144,121.148 841.139,121.164 C 841.144,121.173 841.158,121.168 841.155,121.18 C 841.168,121.191 841.146,121.21 841.164,121.218 C 841.162,121.223 841.168,121.223 841.171,121.229 C 841.169,121.232 841.161,121.24 841.165,121.246 C 841.161,121.25 841.165,121.258 841.164,121.262 C 841.165,121.265 841.168,121.266 841.169,121.269 C 841.162,121.27 841.171,121.275 841.168,121.278 C 841.162,121.292 841.179,121.298 841.169,121.313 L 841.169,121.319 C 841.171,121.317 841.172,121.318 841.174,121.318 L 841.174,121.319 L 841.171,121.323 C 841.178,121.328 841.169,121.337 841.174,121.344 C 841.181,121.349 841.171,121.36 841.181,121.359 C 841.184,121.36 841.183,121.363 841.181,121.366 C 841.183,121.38 841.178,121.393 841.184,121.402 C 841.179,121.406 841.184,121.41 841.188,121.412 C 841.19,121.421 841.184,121.427 841.181,121.433 L 841.179,121.432 C 841.174,121.437 841.181,121.447 841.181,121.457 C 841.195,121.458 841.178,121.47 841.191,121.47 L 841.197,121.485 C 841.191,121.49 841.186,121.499 841.19,121.508 C 841.208,121.508 841.202,121.528 841.202,121.538 L 841.201,121.538 C 841.198,121.541 841.202,121.542 841.202,121.545 C 841.191,121.547 841.209,121.552 841.202,121.564 C 841.204,121.561 841.208,121.564 841.209,121.565 C 841.201,121.572 841.195,121.578 841.195,121.588 C 841.204,121.588 841.198,121.598 841.202,121.603 C 841.195,121.616 841.186,121.63 841.197,121.642 C 841.201,121.656 841.188,121.67 841.198,121.682 C 841.197,121.687 841.202,121.688 841.204,121.692 C 841.195,121.695 841.208,121.709 841.195,121.705 C 841.19,121.712 841.184,121.722 841.184,121.735 C 841.181,121.743 841.197,121.736 841.191,121.744 C 841.202,121.752 841.195,121.766 841.202,121.771 C 841.19,121.774 841.191,121.789 841.184,121.796 L 841.186,121.798 C 841.174,121.8 841.191,121.81 841.179,121.814 C 841.186,121.829 841.165,121.84 841.174,121.854 C 841.181,121.857 841.186,121.863 841.195,121.868 C 841.202,121.87 841.194,121.882 841.202,121.883 C 841.194,121.884 841.191,121.895 841.181,121.892 C 841.176,121.899 841.174,121.915 841.179,121.926 C 841.176,121.94 841.174,121.956 841.169,121.971 C 841.178,121.981 841.181,121.995 841.19,122.008 C 841.181,122.013 841.184,122.019 841.186,122.028 L 841.19,122.03 L 841.195,122.053 C 841.184,122.05 841.188,122.064 841.178,122.061 L 841.176,122.073 L 841.179,122.076 C 841.179,122.077 841.172,122.08 841.178,122.084 L 841.179,122.086 C 841.174,122.088 841.179,122.094 841.178,122.099 C 841.186,122.1 841.181,122.112 841.186,122.12 C 841.181,122.126 841.172,122.141 841.178,122.15 L 841.179,122.153 L 841.174,122.184 L 841.178,122.184 L 841.174,122.188 C 841.183,122.196 841.174,122.206 841.176,122.217 C 841.171,122.225 841.169,122.239 841.169,122.251 C 841.161,122.252 841.168,122.262 841.165,122.267 C 841.169,122.27 841.165,122.271 841.165,122.272 L 841.162,122.272 L 841.164,122.282 L 841.161,122.283 C 841.169,122.287 841.158,122.301 841.171,122.301 L 841.164,122.305 C 841.174,122.312 841.157,122.313 841.165,122.32 C 841.162,122.328 841.169,122.341 841.158,122.348 C 841.161,122.365 841.142,122.392 841.164,122.399 C 841.162,122.405 841.165,122.407 841.169,122.411 C 841.164,122.412 841.164,122.416 841.165,122.42 C 841.153,122.426 841.153,122.445 841.15,122.461 C 841.157,122.467 841.149,122.484 841.155,122.494 L 841.158,122.495 L 841.155,122.501 C 841.158,122.513 841.149,122.526 841.155,122.536 C 841.152,122.541 841.157,122.55 841.161,122.555 L 841.171,122.559 C 841.176,122.573 841.153,122.567 841.158,122.581 L 841.164,122.587 C 841.162,122.593 841.153,122.59 841.158,122.597 C 841.15,122.597 841.153,122.607 841.15,122.612 C 841.152,122.612 841.153,122.612 841.155,122.616 C 841.145,122.619 841.158,122.625 841.15,122.63 C 841.15,122.635 841.149,122.643 841.15,122.65 C 841.162,122.651 841.15,122.665 841.161,122.668 C 841.158,122.68 841.158,122.692 841.165,122.704 L 841.179,122.711 C 841.178,122.717 841.176,122.726 841.183,122.731 C 841.178,122.738 841.164,122.745 841.164,122.753 C 841.172,122.754 841.164,122.77 841.174,122.77 C 841.161,122.77 841.181,122.779 841.169,122.784 C 841.164,122.796 841.183,122.799 841.181,122.811 C 841.169,122.821 841.169,122.84 841.165,122.848 C 841.165,122.847 841.169,122.848 841.171,122.848 C 841.158,122.852 841.176,122.858 841.174,122.866 C 841.184,122.876 841.164,122.89 841.179,122.901 C 841.176,122.908 841.191,122.913 841.181,122.918 C 841.184,122.927 841.169,122.924 841.171,122.934 C 841.171,122.945 841.157,122.961 841.171,122.97 C 841.164,122.977 841.178,122.979 841.176,122.989 C 841.171,122.993 841.165,122.996 841.164,123.001 L 841.164,123.007 L 841.161,123.007 L 841.161,123.016 C 841.169,123.016 841.164,123.027 841.176,123.025 C 841.184,123.016 841.201,123.027 841.21,123.025 L 841.216,123.019 L 841.217,123.022 C 841.242,123.028 841.257,123.012 841.28,123.023 C 841.282,123.022 841.284,123.022 841.284,123.017 C 841.295,123.023 841.314,123.023 841.328,123.023 C 841.336,123.016 841.352,123.027 841.356,123.017 C 841.357,123.019 841.356,123.022 841.36,123.022 C 841.373,123.016 841.388,123.028 841.403,123.019 L 841.434,123.022 L 841.434,123.019 C 841.447,123.016 841.464,123.028 841.477,123.016 C 841.484,123.022 841.497,123.025 841.506,123.019 C 841.508,123.023 841.514,123.022 841.518,123.019 C 841.526,123.022 841.541,123.023 841.549,123.017 C 841.557,123.019 841.573,123.022 841.584,123.025 C 841.598,123.019 841.606,123.002 841.622,122.993 C 841.625,122.985 841.638,122.981 841.641,122.973 C 841.649,122.972 841.649,122.961 841.658,122.958 C 841.668,122.949 841.682,122.94 841.689,122.927 C 841.691,122.92 841.696,122.929 841.698,122.924 C 841.697,122.915 841.714,122.918 841.708,122.91 L 841.714,122.913 C 841.715,122.901 841.73,122.901 841.734,122.887 C 841.741,122.889 841.74,122.878 841.75,122.88 C 841.755,122.866 841.768,122.866 841.774,122.853 C 841.773,122.853 841.773,122.852 841.773,122.849 C 841.777,122.845 841.778,122.853 841.782,122.848 C 841.786,122.837 841.793,122.835 841.801,122.83 C 841.807,122.813 841.829,122.813 841.836,122.796 C 841.846,122.795 841.85,122.782 841.859,122.78 C 841.868,122.774 841.874,122.758 841.886,122.754 L 841.92,122.724 C 841.919,122.722 841.919,122.72 841.918,122.72 C 841.919,122.717 841.92,122.718 841.922,122.72 C 841.936,122.711 841.94,122.699 841.947,122.683 L 841.953,122.651 C 841.952,122.65 841.947,122.65 841.946,122.649 C 841.947,122.634 841.959,122.621 841.947,122.608 C 841.951,122.597 841.958,122.581 841.951,122.567 C 841.955,122.564 841.951,122.561 841.951,122.559 C 841.955,122.54 841.952,122.524 841.953,122.505 C 841.953,122.502 841.951,122.499 841.947,122.498 C 841.955,122.485 841.958,122.467 841.952,122.452 C 841.953,122.442 841.952,122.437 841.953,122.428 C 841.955,122.422 841.946,122.418 841.958,122.415 C 841.952,122.412 841.94,122.41 841.934,122.415 L 841.933,122.418 L 841.927,122.411 C 841.914,122.418 841.898,122.401 841.883,122.415 C 841.874,122.411 841.866,122.411 841.855,122.41 L 841.854,122.415 C 841.852,122.405 841.844,122.416 841.84,122.412 C 841.833,122.411 841.822,122.406 841.814,122.41 C 841.814,122.411 841.814,122.412 841.811,122.415 L 841.81,122.41 C 841.803,122.418 841.797,122.405 841.789,122.407 C 841.757,122.412 841.744,122.38 841.722,122.372 C 841.725,122.364 841.712,122.362 841.71,122.354 L 841.708,122.355 C 841.714,122.344 841.696,122.349 841.696,122.341 C 841.695,122.343 841.692,122.343 841.691,122.343 C 841.691,122.338 841.695,122.335 841.689,122.332 C 841.685,122.334 841.679,122.332 841.679,122.326 C 841.679,122.318 841.666,122.32 841.665,122.312 L 841.649,122.301 C 841.652,122.301 841.651,122.296 841.651,122.295 C 841.641,122.289 841.638,122.276 841.625,122.272 C 841.625,122.261 841.607,122.267 841.61,122.255 C 841.599,122.251 841.599,122.236 841.587,122.236 C 841.585,122.23 841.58,122.226 841.574,122.221 L 841.572,122.224 C 841.572,122.209 841.549,122.202 841.558,122.188 C 841.557,122.183 841.559,122.178 841.554,122.177 C 841.554,122.165 841.558,122.152 841.558,122.142 L 841.554,122.134 C 841.559,122.131 841.552,122.125 841.557,122.12 C 841.563,122.108 841.551,122.1 841.557,122.09 C 841.557,122.084 841.561,122.073 841.554,122.068 C 841.559,122.063 841.557,122.059 841.554,122.055 C 841.552,122.046 841.563,122.033 841.554,122.027 C 841.557,122.016 841.559,122.001 841.554,121.987 C 841.558,121.962 841.552,121.94 841.557,121.914 C 841.552,121.903 841.559,121.888 841.552,121.876 C 841.559,121.85 841.552,121.826 841.557,121.804 C 841.558,121.781 841.552,121.758 841.558,121.735 C 841.549,121.722 841.561,121.702 841.554,121.688 C 841.561,121.667 841.549,121.647 841.557,121.625 C 841.552,121.612 841.559,121.592 841.552,121.578 C 841.559,121.555 841.549,121.538 841.557,121.518 C 841.546,121.504 841.561,121.495 841.554,121.483 L 841.55,121.48 L 841.554,121.476 C 841.551,121.463 841.558,121.439 841.552,121.428 C 841.559,121.411 841.549,121.396 841.554,121.377 C 841.544,121.358 841.563,121.34 841.552,121.319 C 841.561,121.306 841.546,121.288 841.554,121.275 C 841.552,121.258 841.551,121.245 841.554,121.229 C 841.554,121.218 841.551,121.205 841.554,121.193 L 841.55,121.189 C 841.554,121.188 841.551,121.183 841.55,121.179 C 841.558,121.158 841.549,121.139 841.552,121.117 C 841.544,121.104 841.561,121.086 841.55,121.073 C 841.558,121.044 841.546,121.033 841.554,121.007 C 841.546,120.995 841.558,120.976 841.551,120.964 C 841.558,120.937 841.546,120.92 841.552,120.894 C 841.551,120.894 841.551,120.893 841.55,120.893 C 841.551,120.889 841.554,120.885 841.55,120.884 C 841.552,120.859 841.552,120.837 841.55,120.814 C 841.55,120.804 841.557,120.793 841.55,120.782 C 841.551,120.773 841.552,120.758 841.551,120.747 C 841.55,120.737 841.554,120.724 841.55,120.716 C 841.552,120.708 841.55,120.699 841.55,120.692 C 841.554,120.684 841.546,120.671 841.55,120.661 C 841.551,120.653 841.552,120.641 841.549,120.633 C 841.558,120.613 841.541,120.595 841.551,120.576 C 841.551,120.572 841.551,120.566 841.549,120.564 C 841.55,120.556 841.55,120.546 841.551,120.538 C 841.554,120.53 841.546,120.526 841.55,120.52 C 841.546,120.513 841.558,120.499 841.546,120.494 L 841.55,120.493 C 841.546,120.48 841.554,120.467 841.549,120.454 C 841.559,120.421 841.541,120.385 841.55,120.354 C 841.554,120.351 841.549,120.348 841.549,120.345 C 841.549,120.334 841.55,120.323 841.55,120.314 C 841.544,120.303 841.551,120.291 841.551,120.277 L 841.546,120.274 C 841.552,120.274 841.544,120.267 841.549,120.262 C 841.549,120.254 841.557,120.241 841.546,120.235 C 841.549,120.231 841.552,120.234 841.551,120.228 L 841.55,120.227 C 841.566,120.215 841.57,120.2 841.587,120.192 C 841.591,120.181 841.604,120.178 841.606,120.169 C 841.616,120.173 841.612,120.157 841.622,120.158 C 841.619,120.151 841.632,120.146 841.632,120.138 C 841.648,120.142 841.641,120.113 841.658,120.12 C 841.668,120.112 841.677,120.1 841.688,120.09 C 841.697,120.087 841.695,120.069 841.708,120.07 C 841.714,120.063 841.722,120.063 841.723,120.052 C 841.73,120.047 841.74,120.044 841.74,120.034 C 841.748,120.034 841.75,120.03 841.753,120.023 L 841.751,120.02 C 841.762,120.018 841.768,120.011 841.77,120.005 C 841.778,120.005 841.788,119.991 841.794,119.981 C 841.809,119.981 841.807,119.964 841.817,119.961 C 841.823,119.947 841.844,119.941 841.854,119.925 C 841.862,119.916 841.874,119.905 841.885,119.894 C 841.888,119.886 841.903,119.882 841.907,119.871 C 841.911,119.873 841.915,119.868 841.918,119.864 C 841.936,119.872 841.955,119.859 841.978,119.864 C 841.979,119.864 841.982,119.859 841.985,119.864 C 841.992,119.859 842.006,119.867 842.01,119.859 C 842.013,119.862 842.016,119.862 842.019,119.859 C 842.023,119.857 842.019,119.851 842.024,119.851 C 842.03,119.855 842.034,119.85 842.037,119.845 C 842.043,119.845 842.06,119.855 842.066,119.845 L 842.067,119.849 C 842.083,119.845 842.1,119.857 842.113,119.846 C 842.123,119.859 842.136,119.864 842.145,119.874 C 842.146,119.872 842.149,119.874 842.15,119.877 L 842.152,119.885 L 842.175,119.903 C 842.168,119.911 842.185,119.911 842.178,119.919 C 842.182,119.925 842.169,119.938 842.179,119.945 C 842.173,119.947 842.179,119.951 842.179,119.954 C 842.179,119.961 842.173,119.977 842.182,119.985 C 842.178,119.991 842.179,120.005 842.169,120.005 C 842.168,120.023 842.138,120.028 842.153,120.047 L 842.149,120.051 C 842.156,120.065 842.142,120.082 842.15,120.096 C 842.146,120.104 842.149,120.113 842.149,120.122 C 842.149,120.125 842.153,120.122 842.153,120.129 C 842.152,120.129 842.15,120.129 842.149,120.129 C 842.152,120.138 842.145,120.158 842.15,120.169 L 842.149,120.169 C 842.149,120.181 842.145,120.197 842.15,120.208 C 842.15,120.213 842.149,120.219 842.149,120.227 L 842.152,120.227 L 842.149,120.23 C 842.153,120.23 842.156,120.244 842.161,120.236 C 842.168,120.258 842.188,120.267 842.201,120.287 C 842.205,120.287 842.21,120.288 842.21,120.295 C 842.219,120.291 842.209,120.299 842.217,120.301 C 842.23,120.301 842.227,120.317 842.24,120.317 C 842.236,120.327 842.253,120.327 842.254,120.332 C 842.257,120.332 842.257,120.334 842.256,120.337 C 842.268,120.351 842.287,120.363 842.298,120.374 C 842.3,120.386 842.316,120.385 842.322,120.397 C 842.331,120.399 842.331,120.412 842.342,120.415 C 842.34,120.427 842.356,120.425 842.357,120.432 L 842.359,120.428 C 842.356,120.44 842.374,120.437 842.374,120.448 C 842.386,120.454 842.385,120.467 842.401,120.473 C 842.405,120.475 842.409,120.486 842.418,120.486 L 842.414,120.488 C 842.425,120.493 842.432,120.504 842.441,120.509 C 842.442,120.521 842.458,120.519 842.46,120.531 C 842.485,120.548 842.51,120.531 842.536,120.542 C 842.547,120.535 842.565,120.543 842.58,120.541 L 842.584,120.538 C 842.585,120.541 842.587,120.541 842.591,120.539 C 842.599,120.541 842.613,120.546 842.62,120.538 C 842.632,120.538 842.643,120.546 842.653,120.538 C 842.663,120.538 842.681,120.548 842.691,120.538 C 842.704,120.539 842.721,120.546 842.734,120.538 C 842.743,120.539 842.755,120.543 842.762,120.535 L 842.765,120.539 C 842.777,120.539 842.791,120.542 842.803,120.538 C 842.807,120.541 842.809,120.539 842.811,120.538 C 842.831,120.542 842.849,120.539 842.872,120.539 C 842.876,120.534 842.895,120.542 842.907,120.539 C 842.923,120.531 842.945,120.548 842.965,120.539 C 842.972,120.542 842.982,120.535 842.99,120.539 C 842.994,120.534 843.002,120.542 843.008,120.541 L 843.02,120.539 L 843.02,120.538 C 843.026,120.542 843.033,120.535 843.038,120.539 C 843.045,120.538 843.058,120.546 843.065,120.538 C 843.085,120.543 843.104,120.531 843.127,120.541 C 843.145,120.53 843.159,120.55 843.175,120.535 L 843.204,120.539 C 843.229,120.542 843.252,120.531 843.281,120.539 C 843.291,120.531 843.304,120.542 843.315,120.534 C 843.327,120.531 843.34,120.543 843.346,120.531 C 843.368,120.542 843.388,120.529 843.409,120.535 L 843.418,120.531 C 843.426,120.535 843.449,120.531 843.455,120.548 C 843.468,120.551 843.467,120.564 843.48,120.569 C 843.48,120.582 843.501,120.577 843.501,120.59 C 843.515,120.59 843.513,120.609 843.527,120.612 C 843.527,120.622 843.546,120.618 843.545,120.63 C 843.56,120.634 843.561,120.649 843.574,120.654 C 843.572,120.664 843.586,120.659 843.586,120.666 C 843.613,120.682 843.62,120.726 843.614,120.753 C 843.617,120.753 843.617,120.755 843.619,120.755 C 843.61,120.775 843.623,120.801 843.602,120.816 C 843.599,120.821 843.589,120.823 843.586,120.834 L 843.586,120.837 C 843.577,120.835 843.576,120.85 843.564,120.85 C 843.559,120.868 843.543,120.873 843.522,120.881 C 843.501,120.879 843.493,120.884 843.468,120.881 C 843.455,120.885 843.431,120.881 843.415,120.872 C 843.402,120.85 843.381,120.837 843.363,120.816 C 843.365,120.811 843.359,120.807 843.354,120.807 C 843.359,120.8 843.349,120.793 843.346,120.785 C 843.32,120.773 843.296,120.737 843.261,120.747 L 843.26,120.749 C 843.257,120.737 843.25,120.753 843.245,120.748 C 843.235,120.747 843.225,120.743 843.218,120.749 C 843.215,120.744 843.214,120.748 843.21,120.749 C 843.199,120.747 843.189,120.747 843.179,120.749 C 843.169,120.749 843.156,120.743 843.146,120.752 L 843.143,120.747 C 843.131,120.752 843.116,120.743 843.104,120.749 C 843.093,120.748 843.077,120.743 843.067,120.752 C 843.05,120.742 843.027,120.755 843.008,120.749 C 843.008,120.752 843.007,120.753 843.005,120.755 C 843.005,120.752 843.004,120.752 843.004,120.749 C 842.993,120.753 842.978,120.744 842.967,120.752 C 842.947,120.743 842.923,120.757 842.9,120.749 C 842.889,120.757 842.874,120.744 842.866,120.755 C 842.862,120.753 842.866,120.749 842.862,120.752 C 842.855,120.758 842.848,120.748 842.839,120.749 C 842.831,120.748 842.822,120.749 842.817,120.753 C 842.793,120.747 842.773,120.755 842.749,120.749 L 842.744,120.755 C 842.734,120.749 842.717,120.749 842.704,120.753 C 842.697,120.752 842.682,120.752 842.675,120.755 C 842.67,120.749 842.669,120.753 842.666,120.753 C 842.658,120.749 842.645,120.752 842.64,120.752 C 842.632,120.755 842.615,120.747 842.604,120.755 C 842.6,120.752 842.599,120.752 842.597,120.755 C 842.572,120.744 842.561,120.761 842.536,120.753 L 842.533,120.755 L 842.53,120.752 L 842.527,120.755 C 842.504,120.747 842.484,120.758 842.461,120.752 C 842.445,120.758 842.431,120.744 842.413,120.755 C 842.401,120.752 842.385,120.748 842.372,120.755 C 842.361,120.752 842.343,120.749 842.335,120.752 C 842.33,120.743 842.316,120.742 842.313,120.732 C 842.298,120.73 842.298,120.712 842.284,120.71 C 842.283,120.697 842.268,120.699 842.264,120.69 C 842.247,120.687 842.25,120.666 842.234,120.666 C 842.23,120.654 842.212,120.654 842.209,120.641 C 842.191,120.625 842.171,120.609 842.152,120.595 C 842.156,120.583 842.138,120.587 842.136,120.578 C 842.13,120.581 842.12,120.572 842.116,120.573 C 842.1,120.581 842.085,120.562 842.069,120.576 C 842.059,120.572 842.041,120.569 842.032,120.573 L 842,120.569 L 842,120.576 L 841.994,120.569 C 841.985,120.578 841.964,120.564 841.951,120.573 C 841.945,120.573 841.931,120.569 841.927,120.576 C 841.923,120.573 841.914,120.573 841.911,120.576 L 841.908,120.581 C 841.905,120.578 841.901,120.578 841.898,120.581 C 841.893,120.598 841.867,120.602 841.867,120.619 C 841.863,120.622 841.862,120.614 841.859,120.619 C 841.85,120.63 841.848,120.641 841.833,120.645 C 841.822,120.654 841.819,120.665 841.809,120.669 C 841.803,120.676 841.801,120.686 841.791,120.687 C 841.791,120.702 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841.675,121.205 L 841.673,121.205 C 841.684,121.214 841.665,121.217 841.675,121.223 C 841.675,121.232 841.665,121.245 841.677,121.254 C 841.677,121.265 841.666,121.278 841.677,121.286 C 841.673,121.286 841.673,121.288 841.675,121.291 C 841.675,121.3 841.671,121.317 841.677,121.325 C 841.675,121.329 841.675,121.343 841.673,121.35 L 841.677,121.354 C 841.67,121.355 841.675,121.359 841.677,121.363 C 841.677,121.373 841.668,121.385 841.677,121.393 C 841.675,121.406 841.67,121.421 841.679,121.433 C 841.675,121.443 841.67,121.46 841.677,121.469 C 841.673,121.484 841.67,121.5 841.677,121.514 C 841.665,121.526 841.688,121.541 841.673,121.555 C 841.673,121.564 841.673,121.574 841.679,121.582 C 841.67,121.594 841.679,121.612 841.677,121.621 C 841.675,121.629 841.673,121.642 841.677,121.65 C 841.677,121.66 841.67,121.682 841.679,121.688 C 841.673,121.696 841.673,121.706 841.673,121.715 C 841.682,121.721 841.67,121.723 841.679,121.73 C 841.673,121.736 841.675,121.747 841.673,121.754 L 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841.736,122.243 841.748,122.248 C 841.744,122.255 841.758,122.252 841.76,122.26 L 841.765,122.26 C 841.767,122.271 841.786,122.271 841.791,122.287 C 841.801,122.283 841.795,122.298 841.809,122.301 C 841.819,122.301 841.834,122.309 841.841,122.301 C 841.842,122.302 841.846,122.305 841.85,122.301 C 841.862,122.301 841.878,122.312 841.885,122.301 C 841.885,122.309 841.888,122.301 841.893,122.302 C 841.901,122.301 841.914,122.312 841.92,122.301 C 841.929,122.298 841.946,122.313 841.952,122.298 C 841.952,122.309 841.964,122.296 841.968,122.304 C 841.978,122.298 841.988,122.297 841.993,122.288 L 841.998,122.287 C 841.998,122.286 841.997,122.286 841.997,122.283 C 841.998,122.278 842,122.282 842.003,122.283 C 842.017,122.256 842.056,122.255 842.081,122.26 C 842.093,122.248 842.113,122.265 842.13,122.262 C 842.146,122.265 842.146,122.281 842.161,122.286 C 842.162,122.297 842.183,122.297 842.176,122.313 C 842.178,122.312 842.179,122.309 842.182,122.309 L 842.183,122.312 C 842.182,122.32 842.178,122.324 842.182,122.335 L 842.182,122.341 L 842.185,122.341 C 842.182,122.348 842.185,122.355 842.182,122.368 L 842.186,122.372 L 842.182,122.372 C 842.189,122.384 842.179,122.395 842.176,122.403 C 842.169,122.406 842.169,122.415 842.164,122.423 C 842.169,122.432 842.15,122.45 842.165,122.454 C 842.164,122.457 842.162,122.457 842.161,122.457 C 842.164,122.471 842.158,122.477 842.161,122.493 C 842.164,122.506 842.153,122.52 842.162,122.536 C 842.161,122.545 842.156,122.564 842.162,122.573 C 842.162,122.582 842.156,122.603 842.162,122.613 C 842.161,122.621 842.156,122.635 842.162,122.643 C 842.158,122.649 842.165,122.655 842.161,122.662 L 842.161,122.674 C 842.162,122.674 842.165,122.67 842.165,122.674 C 842.165,122.676 842.164,122.68 842.162,122.677 L 842.161,122.677 C 842.169,122.68 842.159,122.693 842.169,122.692 C 842.173,122.705 842.185,122.712 842.183,122.724 C 842.183,122.732 842.176,122.743 842.185,122.749 C 842.176,122.773 842.197,122.795 842.171,122.807 C 842.165,122.82 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841.791,123.095 C 841.775,123.094 841.781,123.116 841.767,123.112 C 841.767,123.122 841.748,123.121 841.75,123.134 C 841.74,123.129 841.741,123.143 841.734,123.145 C 841.737,123.147 841.734,123.147 841.734,123.151 L 841.729,123.147 C 841.728,123.161 841.708,123.155 841.708,123.172 L 841.698,123.172 C 841.705,123.182 841.688,123.178 841.692,123.187 C 841.685,123.187 841.675,123.199 841.668,123.205 C 841.656,123.216 841.639,123.2 841.628,123.21 C 841.616,123.209 841.599,123.203 841.589,123.21 C 841.574,123.208 841.559,123.203 841.549,123.21 C 841.532,123.208 841.517,123.203 841.504,123.209 C 841.494,123.205 841.478,123.203 841.468,123.208 C 841.458,123.21 841.442,123.199 841.435,123.209 C 841.426,123.208 841.403,123.199 841.394,123.21 C 841.387,123.208 841.376,123.205 841.368,123.205 L 841.365,123.21 C 841.363,123.209 841.365,123.208 841.36,123.205 C 841.346,123.21 841.329,123.199 841.314,123.209 C 841.291,123.204 841.273,123.21 841.251,123.205 C 841.238,123.213 841.217,123.199 841.204,123.208 C 841.195,123.204 841.186,123.208 841.176,123.205 L 841.176,123.208 C 841.171,123.208 841.169,123.205 841.165,123.21 C 841.164,123.225 841.15,123.225 841.158,123.24 L 841.164,123.244 L 841.157,123.252 C 841.165,123.251 841.157,123.274 841.171,123.278 C 841.178,123.276 841.172,123.284 841.176,123.29 C 841.174,123.291 841.172,123.292 841.174,123.296 C 841.176,123.297 841.179,123.296 841.181,123.299 C 841.179,123.309 841.165,123.307 841.165,123.317 C 841.162,123.317 841.161,123.317 841.161,123.321 L 841.164,123.324 L 841.157,123.33 C 841.158,123.33 841.162,123.332 841.164,123.334 C 841.157,123.338 841.164,123.343 841.161,123.346 C 841.165,123.35 841.174,123.354 841.181,123.354 L 841.186,123.346 L 841.19,123.353 L 841.195,123.349 C 841.197,123.353 841.201,123.353 841.204,123.35 C 841.21,123.354 841.228,123.357 841.228,123.349 C 841.23,123.353 841.234,123.35 841.236,123.35 L 841.264,123.353 C 841.273,123.353 841.287,123.35 841.299,123.354 C 841.303,123.344 841.31,123.357 841.316,123.35 C 841.324,123.353 841.336,123.357 841.342,123.35 C 841.354,123.349 841.373,123.36 841.378,123.35 L 841.378,123.354 C 841.388,123.35 841.403,123.356 841.416,123.354 L 841.418,123.35 C 841.424,123.35 841.434,123.353 841.444,123.356 C 841.464,123.346 841.486,123.36 841.508,123.353 C 841.523,123.365 841.546,123.346 841.563,123.356 C 841.577,123.343 841.598,123.366 841.61,123.35 C 841.61,123.354 841.613,123.354 841.617,123.353 C 841.628,123.354 841.639,123.36 841.645,123.35 C 841.651,123.36 841.652,123.353 841.659,123.353 C 841.666,123.354 841.679,123.361 841.688,123.35 C 841.705,123.364 841.723,123.346 841.741,123.356 L 841.757,123.354 L 841.757,123.353 C 841.778,123.361 841.801,123.344 841.822,123.356 C 841.846,123.354 841.874,123.353 841.898,123.356 C 841.918,123.346 841.934,123.364 841.951,123.353 C 841.964,123.353 841.979,123.361 841.992,123.353 C 842.001,123.36 842.02,123.353 842.027,123.35 C 842.032,123.343 842.043,123.339 842.047,123.328 C 842.066,123.324 842.066,123.303 842.083,123.297 C 842.087,123.287 842.097,123.283 842.102,123.274 C 842.111,123.271 842.113,123.261 842.123,123.257 C 842.123,123.246 842.142,123.248 842.14,123.234 C 842.152,123.229 842.153,123.216 842.165,123.21 C 842.168,123.198 842.185,123.203 842.185,123.187 C 842.196,123.183 842.204,123.173 842.209,123.165 C 842.216,123.163 842.219,123.157 842.223,123.153 C 842.219,123.142 842.234,123.149 842.235,123.141 C 842.235,123.106 842.243,123.069 842.238,123.035 C 842.24,123.028 842.238,123.022 842.242,123.011 C 842.238,122.982 842.238,122.955 842.242,122.929 C 842.235,122.927 842.243,122.919 842.238,122.915 C 842.242,122.889 842.236,122.868 842.243,122.841 C 842.235,122.831 842.245,122.816 842.238,122.805 C 842.24,122.795 842.242,122.778 842.238,122.766 C 842.238,122.753 842.247,122.739 842.24,122.727 C 842.245,122.707 842.236,122.687 842.242,122.668 C 842.236,122.664 842.247,122.659 842.238,122.655 C 842.235,122.643 842.25,122.627 842.238,122.616 C 842.238,122.603 842.247,122.589 842.24,122.577 C 842.242,122.564 842.236,122.555 842.243,122.545 C 842.24,122.544 842.24,122.547 842.238,122.545 L 842.242,122.544 C 842.24,122.53 842.243,122.515 842.238,122.506 C 842.236,122.49 842.249,122.472 842.24,122.457 C 842.249,122.433 842.236,122.415 842.243,122.397 C 842.242,122.397 842.24,122.397 842.238,122.399 L 842.236,122.397 L 842.243,122.392 C 842.235,122.385 842.243,122.375 842.243,122.368 C 842.242,122.349 842.24,122.339 842.243,122.322 C 842.24,122.316 842.242,122.302 842.243,122.289 C 842.231,122.289 842.242,122.272 842.228,122.275 C 842.228,122.266 842.209,122.266 842.214,122.256 C 842.196,122.248 842.188,122.221 842.165,122.218 C 842.164,122.209 842.15,122.206 842.143,122.204 C 842.116,122.194 842.099,122.205 842.067,122.199 C 842.057,122.204 842.034,122.196 842.02,122.2 C 842.001,122.198 841.987,122.204 841.966,122.199 C 841.952,122.204 841.944,122.202 841.931,122.198 C 841.915,122.184 841.908,122.168 841.889,122.165 C 841.888,122.156 841.878,122.152 841.874,122.143 C 841.87,122.141 841.862,122.141 841.863,122.133 C 841.854,122.123 841.837,122.117 841.833,122.108 C 841.82,122.104 841.822,122.085 841.807,122.085 C 841.797,122.076 841.791,122.068 841.793,122.053 C 841.784,122.043 841.795,122.027 841.788,122.019 C 841.788,122.007 841.795,121.991 841.788,121.978 C 841.789,121.974 841.791,121.966 841.789,121.959 C 841.789,121.949 841.788,121.926 841.788,121.91 C 841.784,121.902 841.794,121.888 841.786,121.879 C 841.791,121.876 841.786,121.872 841.786,121.868 C 841.793,121.845 841.788,121.824 841.786,121.8 C 841.793,121.774 841.784,121.752 841.788,121.726 C 841.791,121.705 841.781,121.687 841.789,121.665 C 841.778,121.65 841.794,121.628 841.786,121.611 C 841.789,121.598 841.789,121.583 841.786,121.573 C 841.788,121.55 841.788,121.53 841.788,121.506 C 841.788,121.494 841.788,121.48 841.786,121.466 C 841.788,121.46 841.789,121.442 841.784,121.433 C 841.8,121.411 841.768,121.373 841.801,121.363 C 841.807,121.345 841.817,121.349 841.823,121.331 C 841.833,121.335 841.831,121.321 841.837,121.321 C 841.85,121.319 841.844,121.302 841.859,121.3 C 841.86,121.288 841.874,121.291 841.878,121.278 C 841.888,121.278 841.889,121.261 841.9,121.26 C 841.903,121.249 841.918,121.244 841.922,121.232 C 841.936,121.223 841.944,121.208 841.958,121.198 C 841.964,121.196 841.959,121.186 841.966,121.186 C 841.961,121.173 841.972,121.158 841.962,121.143 C 841.968,121.13 841.964,121.106 841.982,121.104 C 841.99,121.094 841.998,121.085 842.007,121.073 C 842.012,121.068 842.019,121.063 842.026,121.058 C 842.048,121.048 842.069,121.06 842.09,121.052 C 842.1,121.059 842.117,121.055 842.12,121.068 C 842.128,121.075 842.136,121.078 842.143,121.086 C 842.146,121.091 842.158,121.089 842.152,121.097 C 842.164,121.101 842.171,121.121 842.185,121.109 L 842.189,121.115 C 842.198,121.109 842.216,121.12 842.227,121.109 C 842.234,121.106 842.249,121.121 842.256,121.109 C 842.262,121.109 842.28,121.115 842.289,121.111 C 842.298,121.106 842.317,121.115 842.327,121.106 C 842.349,121.115 842.369,121.101 842.392,121.111 C 842.406,121.101 842.421,121.116 842.439,121.106 C 842.445,121.105 842.456,121.113 842.465,121.109 L 842.466,121.104 C 842.48,121.111 842.504,121.101 842.51,121.121 C 842.519,121.127 842.521,121.135 842.53,121.139 L 842.533,121.146 C 842.536,121.143 842.537,121.143 842.537,121.143 C 842.543,121.155 842.547,121.164 842.559,121.172 C 842.567,121.169 842.559,121.178 842.568,121.179 C 842.579,121.189 842.592,121.203 842.604,121.21 C 842.609,121.222 842.615,121.226 842.623,121.232 C 842.625,121.232 842.629,121.239 842.626,121.241 L 842.643,121.252 C 842.646,121.258 842.659,121.265 842.656,121.27 C 842.67,121.287 842.69,121.3 842.704,121.314 C 842.708,121.319 842.717,121.317 842.714,121.325 C 842.725,121.337 842.74,121.346 842.751,121.358 C 842.751,121.373 842.769,121.37 842.77,121.381 C 842.777,121.39 842.789,121.396 842.795,121.406 C 842.797,121.415 842.815,121.41 842.811,121.421 C 842.822,121.429 842.835,121.441 842.844,121.447 C 842.847,121.443 842.849,121.442 842.853,121.446 C 842.863,121.447 842.879,121.449 842.889,121.443 C 842.901,121.449 842.921,121.447 842.933,121.458 L 842.936,121.457 C 842.931,121.467 842.947,121.466 842.954,121.47 C 842.951,121.484 842.967,121.483 842.969,121.495 C 842.98,121.495 842.981,121.508 842.982,121.518 C 842.981,121.524 842.978,121.538 842.987,121.542 C 842.978,121.561 842.992,121.577 842.981,121.594 C 842.977,121.601 842.982,121.612 842.971,121.612 C 842.954,121.628 842.926,121.647 842.936,121.677 C 842.933,121.679 842.933,121.682 842.936,121.686 C 842.936,121.695 842.927,121.709 842.936,121.715 C 842.926,121.717 842.941,121.726 842.935,121.732 C 842.933,121.742 842.935,121.748 842.936,121.754 C 842.927,121.77 842.947,121.785 842.936,121.802 C 842.943,121.806 842.933,121.81 842.941,121.816 C 842.939,121.822 842.939,121.835 842.939,121.843 C 842.94,121.845 842.941,121.845 842.943,121.846 C 842.943,121.848 842.941,121.85 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842.401,121.21 C 842.379,121.203 842.365,121.213 842.342,121.208 C 842.336,121.21 842.335,121.204 842.331,121.21 C 842.323,121.208 842.308,121.208 842.3,121.213 C 842.288,121.21 842.271,121.205 842.26,121.213 C 842.249,121.209 842.236,121.209 842.227,121.213 C 842.219,121.214 842.212,121.209 842.203,121.213 L 842.201,121.217 C 842.189,121.218 842.178,121.208 842.169,121.218 C 842.153,121.225 842.145,121.239 842.132,121.25 C 842.124,121.26 842.111,121.265 842.102,121.27 C 842.094,121.269 842.079,121.262 842.072,121.271 C 842.061,121.271 842.047,121.261 842.039,121.273 C 842.024,121.266 842.016,121.286 842.007,121.293 C 841.993,121.298 841.99,121.314 841.979,121.319 C 841.981,121.325 841.974,121.325 841.972,121.325 C 841.968,121.344 841.946,121.346 841.94,121.363 C 841.936,121.366 841.926,121.368 841.933,121.373 C 841.926,121.363 841.929,121.385 841.918,121.381 C 841.914,121.397 841.895,121.4 841.892,121.416 C 841.895,121.419 841.892,121.421 841.889,121.425 C 841.892,121.433 841.881,121.446 841.892,121.449 L 841.888,121.455 C 841.895,121.466 841.879,121.484 841.892,121.498 C 841.892,121.5 841.885,121.503 841.889,121.506 C 841.879,121.525 841.895,121.545 841.886,121.564 L 841.892,121.568 C 841.886,121.572 841.892,121.583 841.886,121.591 L 841.893,121.599 C 841.879,121.601 841.895,121.608 841.889,121.616 C 841.889,121.624 841.881,121.634 841.893,121.638 C 841.885,121.64 841.895,121.65 841.889,121.656 C 841.892,121.664 841.883,121.671 841.892,121.677 C 841.893,121.688 841.885,121.706 841.892,121.718 C 841.892,121.727 841.886,121.738 841.893,121.748 C 841.885,121.771 841.895,121.788 841.889,121.814 C 841.9,121.829 841.883,121.843 841.895,121.86 L 841.892,121.862 C 841.895,121.867 841.888,121.872 841.889,121.879 L 841.893,121.895 C 841.895,121.907 841.885,121.924 841.893,121.936 C 841.889,121.944 841.889,121.953 841.889,121.962 C 841.895,121.978 841.881,121.999 841.898,122.013 C 841.903,122.013 841.9,122.023 841.907,122.019 C 841.912,122.032 841.92,122.038 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diff --git a/images/complexity-eps-converted-to.pdf b/images/complexity-eps-converted-to.pdf
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diff --git a/main.tex b/main.tex
index 0a4591e..217f8f4 100644
--- a/main.tex
+++ b/main.tex
@@ -22,6 +22,8 @@
 \usepackage{thmtools, thm-restate}
 \usepackage{multirow}
 \usepackage{algorithm2e}
+\usepackage{mathtools}
+
 %\declaretheorem{theorem}
 
 %%--------------------
@@ -135,7 +137,21 @@
 \def \P {\mathbb{P}}
 \def \ov {\overline}
 \def \ts {\tau_{\rm stop}}
-
+\def\rl{{^{.}}}
+
+\DeclarePairedDelimiter\abs{\lvert}{\rvert}%
+\DeclarePairedDelimiter\norm{\lVert}{\rVert}%
+
+% Swap the definition of \abs* and \norm*, so that \abs
+% and \norm resizes the size of the brackets, and the 
+% starred version does not.
+\makeatletter
+\let\oldabs\abs
+\def\abs{\@ifstar{\oldabs}{\oldabs*}}
+%
+\let\oldnorm\norm
+\def\norm{\@ifstar{\oldnorm}{\oldnorm*}}
+\makeatother
 
 \newtheorem{theorem}{Théorème}
 \newtheorem{lemma}{Lemme}
@@ -244,8 +260,8 @@ On montre qu'on a des résultats similaires.
 \chapter{Une démarches plus classique de dissimulation: STABYLO}
  \input{stabylo}
 
-\chapter{Une démarches plus classique de dissimulation: STABYLO}
- \input{stabylo}
+\chapter{Schéma de stéganographie: les dérivées du second ordre}
+ \input{stegoyousra}
 
 
 
diff --git a/stegoyousra.tex b/stegoyousra.tex
new file mode 100644
index 0000000..dc17a7a
--- /dev/null
+++ b/stegoyousra.tex
@@ -0,0 +1,954 @@
+La plupart des schémas de stéganographie sont conçus de sorte  à minimiser une 
+fonction de distortion. Dans les exemples du chapitre précédent, 
+ces fonctions de distortion sont construites dans l'obejctif de préserver 
+les caractéristiques de l'images. 
+On comprend aisément que dans des régions uniformes ou sur des bords clairement définis,
+une modification même mineure de l'image est facilement détectable.
+Au contraire dans les textures, le bruit ou les régions chaotiques 
+sont  difficiles à modéliser. Les caractéristiques des images 
+dont ces zones ont été modifiées sont ainsi similaires à celles
+des images initiales.
+
+Ces régions sont caractérisées par des courbes de niveau très perturbées.
+Ce chapitre présente une nouvelle fonction de distortion pour la stéganographie 
+qui est basée sur les dérivées du second ordre, l'outil mathématique usuel 
+pour les courbes de niveau.
+
+Pour peu qu'on sache définir une fonction $P$ 
+qui associe à chaque pixel $(x,y)$ sa valeur $P(x,y)$,
+les pixels tels que les dérivées secondes de $P$ ont des valeurs élevées 
+sont des bon candidats pour contenir un bit du message.
+Cependant, une telle fonction $P$ n'est connue que de manière discrète,
+\textit{i.e.}, en un nombre fini de points. 
+Les dérivées premières et secondes ne peuvent donc pas être évaluées mathématiquement.
+Au mieux, on peut construire une fonction qui approxime ces $P$ sur cet ensemble 
+de pixels. Ordonner alors les pixels selon la matrice hessienne 
+(\textit{i.e.}, la matrice des dérivées secondes) n'est pas trivial puisque celle-ci 
+contient de nombreuses valeurs pour un seul pixel donné. 
+
+On verra dans ce chapitre comment des approximations des dérivées 
+premières et secondes pour des images numriques (Section~\ref{sec:gradient}) on peu être
+obtenues.
+Deux propositions de dérivées secondes sont ensuite 
+données et prouvées (Section~\ref{sec:second} et Section~\ref{sec:poly}).
+Une adaptation d'une fonction de distortion existante est étudiée
+en Section~\ref{sec:distortion} et des expériences sont présentées 
+en Section~\ref{sec:experiments}.
+
+
+
+\section{Derivatives in an Image}\label{sec:gradient}
+
+This section first recalls links between level curves, gradient, and
+Hessian matrix (Section~\ref{sub:general}).
+It next analyses them using kernels from signal theory 
+(Section~\ref{sub:class:1} and Section~\ref{sub:class:2}).
+
+
+\subsection{Hessian Matrix}\label{sub:general}
+Let us consider that an image can be seen as a numerical function $P$ that 
+associates a value $P(x,y)$ to each pixel of coordinates $(x,y)$.
+The variations of this function in $(x_0,y_0)$ 
+can be evaluated thanks to its gradient
+$\nabla{P}$, which is the vector whose two components 
+are the partial derivatives in $x$ and in $y$ of $P$: 
+
+\[\nabla{P}(x_0,y_0) = \left(\frac{\partial P}{\partial x}(x_0,y_0),\frac{\partial P}{\partial y}(x_0,y_0)\right).
+\]
+
+In the context of two variables, the gradient vector
+points to the direction where the function has the highest increase.
+Pixels with close values thus follow level curve that is orthogonal 
+to the one of highest increase.
+
+The variations of the gradient vector are expressed in the 
+Hessian matrix $H$ of second-order partial derivatives of $P$.
+
+\[
+H = \begin{bmatrix} 
+\dfrac{\partial^2 P}{\partial x^2} & 
+\dfrac{\partial^2 P}{\partial x \partial y} \\
+\dfrac{\partial^2 P}{\partial y \partial x} & 
+\dfrac{\partial^2 P}{\partial y^2} \\
+\end{bmatrix}. 
+\]
+
+
+
+
+In one pixel $(x_0,y_0)$, the larger the absolute values of this matrix are,
+the more the gradient is varying around $(x_0,y_0)$.
+We are then left to evaluate such an Hessian matrix.
+
+This task is not as easy as it appears since natural images are not defined 
+with differentiable functions from $\R^2$ to $\R$.
+Following subsections provide various approaches to compute these 
+Hessian matrices.
+
+\subsection{Classical Gradient Image Approaches}\label{sub:class:1}
+In the context of image values, the most used approaches to evaluate gradient vectors are the well-known ``Sobel'', ``Prewitt'', ``Central Difference'', and ``Intermediate Difference'' ones.
+
+
+\begin{table}[ht]
+\caption{Kernels of usual image gradient operators\label{table:kernels:usual}
+}
+\centering
+\scriptsize
+\begin{tabular}{|c|c|c|}
+    \hline
+    Name&   Sobel & Prewitt \\
+    \hline
+    Kernel & $\textit{Ks}= \begin{bmatrix} -1 & 0 & +1 \\ -2 & 0 & +2 \\ -1 & 0 & +1 \end{bmatrix} $ &
+    $\textit{Kp}= \begin{bmatrix} -1 & 0 & +1 \\ -1 & 0 & +1 \\ -1 & 0 & +1 \end{bmatrix} $\\
+    \hline\hline
+    Name & Central & Intermediate \\
+            & Difference &Difference \\ 
+    \hline
+    Kernel & $\textit{Kc}= \begin{bmatrix} 0&0&0 \\ -\dfrac{1}{2} & 0 & +\dfrac{1}{2} \\ 0&0&0 \end{bmatrix} $ &
+    $\textit{Ki}= \begin{bmatrix} 0&0&0 \\ 0 & -1 & 1 \\ 0&0&0 \end{bmatrix} $ \\
+    \hline
+\end{tabular}
+\end{table}
+
+
+Each of these approaches applies a convolution product $*$ between a  kernel $K$ (recalled in Table~\ref{table:kernels:usual}) and 
+ a $3\times 3$ window of pixel values $A$. The result 
+ $A * K$ is an evaluation of the horizontal gradient, 
+\textit{i.e.}, $\dfrac{\partial P}{\partial x}$
+expressed as a matrix in $\R$.
+Let $K\rl$ be the result of a $\pi/2$ rotation applied on $K$. 
+The vertical gradient $\dfrac{\partial P}{\partial y}$ is similarly obtained by computing $A * K\rl$, which is again expressed as a matrix in $\R$.
+
+The two elements of the first line 
+of the Hessian matrix are the result
+of applying the horizontal gradient calculus 
+first on  $\dfrac{\partial P}{\partial x}$ and next 
+on  $\dfrac{\partial P}{\partial y}$. 
+Let us study these Hessian matrices in the next section.
+
+
+\subsection{Hessian Matrices induced by Gradient Image Approaches}\label{sub:class:2}
+
+First of all, it is well known that 
+$\dfrac{\partial^2 P}{\partial x \partial y} $ is equal to 
+$\dfrac{\partial^2 P}{\partial y \partial x}$ if  
+the approach that computes the gradient and the
+one which evaluates the Hessian matrix are the same.
+For instance, in the Sobel approach, 
+it is easy to verify that the calculus of 
+$\dfrac{\partial^2 P}{\partial x \partial y}$ 
+and of 
+$\dfrac{\partial^2 P}{\partial y \partial x}$ 
+are both the result 
+of a convolution product with the Kernel 
+$\textit{Ks}''_{xy}$ given in Table~\ref{table:hessian:usual}.
+This one summarizes kernels
+$K_{x^2}''$ and 
+$K_{xy}''$  
+that allow to respectively compute  
+$\dfrac{\partial^2 P}{\partial x^2}$ and
+$\dfrac{\partial^2 P}{\partial x \partial y}$ with a convolution product
+for each of the usual image gradient operator.
+
+\begin{table}[ht]
+\caption{Kernels of second order gradient operators\label{table:hessian:usual}
+}\centering
+\tiny
+\begin{tabular}{|c|c|}
+    \hline
+    Sobel & Prewitt \\
+    \hline
+    $
+    \textit{Ks}_{x^2}''= 
+    \begin{bmatrix} 
+    1 &  0  & -2  & 0 & 1 \\ 
+    4 &  0  & -8  & 0 & 4 \\ 
+    6 &  0  & -12 & 0 & 6 \\
+    4 &  0  & -8  & 0 & 4 \\ 
+    1 &  0  & -2  & 0 & 1 
+    \end{bmatrix} 
+    $
+    & 
+    $
+    \textit{Kp}_{x^2}''= 
+    \begin{bmatrix} 
+    1 &  0  & -2  & 0 & 1 \\ 
+    2 &  0  & -4  & 0 & 2 \\ 
+    3 &  0  & -6 & 0 & 3 \\
+    2 &  0  & -4  & 0 & 2 \\ 
+    1 &  0  & -2  & 0 & 1 
+    \end{bmatrix} 
+    $
+    \\
+    \hline
+    $
+    \textit{Ks}_{xy}''=
+    \begin{bmatrix} 
+    -1 & -2 & 0 & 2 & 1 \\ 
+    -2 & -4 & 0 & 4 & 2 \\ 
+    0  & 0  & 0 & 0 & 0 \\
+    2 & 4 & 0 & -4 & -2 \\ 
+    1 & 2 & 0 & -2 & -1 
+    \end{bmatrix} 
+    $
+    &
+    $
+    \textit{Kp}_{xy}''=
+    \begin{bmatrix} 
+    -1 & -1 & 0 & 1 & 1 \\ 
+    -1 & -1 & 0 & 1 & 1 \\ 
+    0  & 0  & 0 & 0 & 0 \\
+    1 & 1 & 0 & -1 & -1 \\ 
+    1 & 1 & 0 & -1 & -1 
+    \end{bmatrix} 
+    $ \\
+    
+    \hline\hline
+    Central & Intermediate \\
+    Difference &Difference \\ 
+    \hline
+    $
+    \textit{Kc}_{x^2}''= 
+    \begin{bmatrix} 
+    0 &  0  & 0  & 0 & 0 \\ 
+    0 &  0  & 0  & 0 & 0 \\ 
+    \dfrac{1}{4} &  0  & -\dfrac{1}{2} & 0 & \dfrac{1}{4} \\
+    0 &  0  & 0  & 0 & 0 \\ 
+    0 &  0  & 0  & 0 & 0  
+    \end{bmatrix} 
+    $
+    & 
+    $
+    \textit{Ki}_{x^2}''= 
+    \begin{bmatrix} 
+    0 &  0  & 0  & 0 & 0 \\ 
+    0 &  0  & 0  & 0 & 0 \\ 
+    0 &  0  & 1  & -2 & 1 \\
+    0 &  0  & 0  & 0 & 0 \\ 
+    0 &  0  & 0  & 0 & 0  
+    \end{bmatrix} 
+    $
+    \\
+    \hline
+    $
+    \textit{Kc}_{xy}''= 
+    \begin{bmatrix} 
+     -\dfrac{1}{4}  & 0 & \dfrac{1}{4}\\ 
+     0  & 0 & 0 \\ 
+     \dfrac{1}{4} & 0 &  -\dfrac{1}{4} 
+    \end{bmatrix} 
+    $
+    & 
+    $
+    \textit{Ki}_{xy}''= 
+    \begin{bmatrix} 
+     0  & -1 & 1 \\ 
+     0  & 1 & -1 \\ 
+     0  &  0 & 0 
+    \end{bmatrix} 
+    $\\
+    \hline
+\end{tabular}
+
+\end{table}
+
+The Sobel kernel $\textit{Ks}_{x^2}''$ allows to detect whether the central 
+pixel belongs to a ``vertical'' edge, even if this one is noisy, by considering its 
+vertical neighbours. The introduction of these vertical neighbours in this kernel is meaningful 
+in the context of finding edges, but not very accurate when the objective is to 
+precisely find the level curves of the image. 
+Moreover, all the pixels that are in the  second and the fourth column in this kernel
+are ignored.
+The Prewitt Kernel has similar drawbacks in this context.
+
+
+The Central Difference kernel $\textit{Kc}_{x^2}''$ is not influenced by 
+the vertical neighbours of the central pixel and is thus more accurate here.
+However, the kernel $\textit{Kc}_{xy}''$ again looses the values of the pixels that 
+are vertically and diagonally aligned with the central one.
+
+Finally, the Intermediate Difference kernel $\textit{Ki}_{x^2}''$ shifts 
+to the left the value of horizontal variations of $\dfrac{\partial P}{\partial x}$:
+the central pixel $(0,0)$ exactly receives the value
+$\dfrac{P(0,2)-P(0,1)}{1} - \dfrac{P(0,1)-P(0,0)}{1}$,
+which is an approximation of 
+$\dfrac{\partial P}{\partial x}(0,1)$ and not of 
+$\dfrac{\partial P}{\partial x}(0,0)$.
+Furthermore the Intermediate Difference kernel $\textit{Ki}_{xy}''$ only deals 
+with pixels in the upper right corner, loosing all the other information.
+
+Due to these drawbacks, we are then left to produce another approach to find the level curves with strong accuracy.
+
+\section{Second Order Kernels for Accurate Level Curves}\label{sec:second}
+
+This step aims at finding 
+accurate level curve variations in an image. 
+We do not restrict the kernel to have a fixed size (\textit{e.g.}, $3\times3$ or $5 \times 5$ as in the 
+aforementioned schemes).
+This step is thus defined with kernels of size 
+$(2n+1)\times (2n+1)$, $n \in \{1,2,\dots,N\}$, where 
+$N$ is a parameter of the approach.
+
+The horizontal gradient variations are thus captured thanks to $(2n+1)\times (2n+1)$ square kernels
+\begin{small}
+\[
+\arraycolsep=1.4pt
+\def\arraystretch{1.4}
+\def\arraystretch{1.4}
+    \textit{Ky}_{x^2}''=
+    \left(
+    \begin{array}{ccccccccc}
+         0           & & & & \dots& & & & 0  \\
+         \vdots      & & & & & & & & \vdots \\
+         0           & & & & \dots & & & & 0  \\
+         \dfrac{1}{2n}& 0 & \dots & 0  & -\dfrac{2}{2n} & 0  & \dots & 0& \dfrac{1}{2n} \\
+         0           & & & & \dots& & & & 0  \\
+        \vdots      & & & & & & & & \vdots \\
+         0           & & & & \dots & & & & 0  
+    \end{array}
+    \right)
+\]
+\end{small}
+
+When the convolution product is applied on a $(2n+1)\times(2n+1)$ window,
+the result is 
+$\dfrac{1}{2}\left(\dfrac{P(0,n)-P(0,0)}{n} - \dfrac{P(0,0)-P(0,-n)}{n}\right)$, which is indeed 
+the variation between the gradient around the central pixel. 
+This proves that this calculus is a correct approximation of 
+$\dfrac{\partial^2 P}{\partial x^2}$.
+
+When $n$ is 1, this kernel is a centered version of the horizontal Intermediate Difference kernel $\textit{Ki}_{x^2}''$ modulo a multiplication by $1/2$. When $n$ is 2, this kernel is equal to $\textit{Kc}_{x^2}''$.
+
+The vertical gradient variations are again obtained by applying 
+a $\pi/2$ rotation to each horizontal kernel $\textit{Ky}_{x^2}''$.
+
+The diagonal gradient variations are obtained thanks to the $(2n+1)\times (2n+1)$ square kernels 
+$\textit{Ky}_{xy}''$ defined by
+
+\begin{small}
+\[
+\arraycolsep=1.4pt
+\def\arraystretch{1.4}
+\textit{Ky}_{xy}'' = \dfrac{1}{4}
+\left(
+    \begin{array}{ccccccccc}
+     \frac{1}{n^2}& \dots & \frac{1}{2n} & \frac{1}{n} 
+     & 0 &
+     -\frac{1}{n}&-\frac{1}{2n} & \dots &  -\frac{1}{n^2} 
+     \\
+     \vdots & 0     &    &
+     & \dots &
+      &  &  0 & \vdots 
+     \\
+     \frac{1}{2n} & 0 &        &
+     & \dots &
+        &  & 0& -\frac{1}{2n} 
+     \\
+     \frac{1}{n} & 0 &    &    
+     & \dots &
+      &  & 0 &  -\frac{1}{n}
+     \\
+     0      &      & & & \dots& & & & 0  \\
+     -\frac{1}{n} & 0 &        &
+     & \dots &
+        &  &0 & \frac{1}{n}
+     \\
+     -\frac{1}{2n} & 0 &       &
+     & \dots &
+      &  & 0 &  \frac{1}{2n} 
+     \\
+      \vdots & 0 &    &    
+     & \dots &
+        &  & 0& \vdots 
+     \\
+     -\frac{1}{n^2}&  \dots & -\frac{1}{2n} & -\frac{1}{n} 
+     & 0 &
+     \frac{1}{n}& \frac{1}{2n} & \dots  & \frac{1}{n^2}
+    \end{array}
+    \right).
+\]
+\end{small}
+
+When $n$ is 1, $\textit{Ky}_{xy}''$ is equal to the kernel
+$\textit{Kc}_{xy}''$, and %.
+%
+%When $n$ is 1, 
+the average vertical variations of the horizontal variations are 
+\[
+\begin{array}{l}
+\dfrac{1}{4}
+\left[ 
+\left((P(0,1)-P(0,0))-(P(1,1)-P(1,0))\right)+ \right.\\
+\quad \left((P(-1,1)-P(-1,0))-(P(0,1)-P(0,0))\right)+\\
+\quad \left((P(0,0)-P(0,-1))-(P(1,0)-P(1,-1))\right)+\\
+\quad \left. \left((P(-1,0)-P(-1,-1))-(P(0,0)-P(0,-1))\right)
+\right]  \\
+=\dfrac{1}{4}
+\left[ P(1,-1) -P(1,1) - P(-1,-1) + P(-1,1)\right].
+\end{array}
+\]
+which is $\textit{Ky}_{xy}''$.
+
+Let us now consider any number $n$, $1 \le n \le N$.
+Let us first investigate the vertical variations related to 
+the horizontal vector $\vec{P_{0,0}P_{0,1}}$ 
+(respectively  $\vec{P_{0,-1}P_{0,0}}$)
+of length 1 that starts from (resp. that points to) $(0,0)$. 
+As with the case $n=1$, there are 2 new vectors of 
+length 1, namely 
+$\vec{P_{n,0}P_{n,1}}$ and $\vec{P_{-n,0}P_{-n,1}}$ 
+(resp. 
+$\vec{P_{n,-1}P_{n,0}}$, and $\vec{P_{-n,-1}P_{-n,0}}$)
+that are vertically aligned with $\vec{P_{0,0}P_{0,1}}$ 
+(resp. with $\vec{P_{0,-1}P_{0,0}}$).
+
+The vertical variation is now equal to $n$. Following the case where $n$ is 1 to compute the average variation, 
+the coefficients of the first and last line around the central 
+vertical line are thus from left to right:
+$\dfrac{1}{4n}$,
+$\dfrac{-1}{4n}$,
+$\dfrac{-1}{4n}$, and
+$\dfrac{1}{4n}$.
+
+Cases are similar with vectors $\vec{P_{0,0}P_{0,1}}$, \ldots  
+$\vec{P_{0,0}P_{0,n}}$ which respectively lead to coefficients 
+$-\dfrac{1}{4 \times 2n}$, \ldots, 
+$-\dfrac{1}{4 \times n.n}$, and the proof is omitted.
+Finally, let us consider the vector $\vec{P_{0,0}P_{0,1}}$
+and its vertical variations when $\delta y$ is $n-1$.
+As in the case where $n=1$, we thus obtain the coefficients 
+$\dfrac{1}{4 \times (n-1)n}$ and 
+$-\dfrac{1}{4 \times (n-1)n}$ 
+ (resp. $-\dfrac{1}{4 \times (n-1)n}$ and 
+$\dfrac{1}{4 \times (n-1)n}$)
+in the second line (resp. in the 
+penultimate line) since the vector has length $n$
+and $\delta y$ is $n-1$.
+Coefficient in the other lines are similarly obtained and the proof is thus omitted.
+
+We are then left to compute an approximation of the partial second order derivatives 
+$\dfrac{\partial^2 P}{\partial x^2}$,  $\dfrac{\partial^2 P}{\partial y^2}$, and $\dfrac{\partial^2 P}{\partial x \partial y}$
+with the kernels, 
+$\textit{Ky}_{x^2}''$, $\textit{Ky}_{y^2}''$, and $\textit{Ky}_{xy}''$ respectively.
+However, the size of each of these kernels is varying from $3\times3$ to $(2N+1)\times (2N+1)$.
+Let us explain the approach on the former partial derivative.
+The other can be immediately deduced.
+
+Since the objective is to detect large variations, the second order derivative is approximated as 
+the maximum of the approximations. More formally, 
+let $n$, $1 \le n \le N$, be an integer number and 
+$\dfrac{\partial^2 P}{\partial x^2}_n$ be the result of applying the Kernel 
+$\textit{Ky}_{x^2}''$ of size $(2n+1)\times (2n+1)$. The derivative 
+$\dfrac{\partial^2 P}{\partial x^2}$ is defined by
+
+\begin{equation}
+\dfrac{\partial^2 P}{\partial x^2}
+= \max \left\{ 
+\abs{\dfrac{\partial^2 P}{\partial x^2}_1}, \dots, \abs{\dfrac{\partial^2 P}{\partial x^2}_N}
+\right \}. 
+\label{eq:d2p_dx2}
+\end{equation}
+
+The same iterative approach is applied to compute approximations of
+$\dfrac{\partial^2 P}{\partial y \partial x}$ 
+and of 
+$\dfrac{\partial^2 P}{\partial y^2}$.
+Next section studies the suitability of approximating second order derivatives
+when considering an image as a polynomial.
+
+
+\section{Polynomial Interpolation of Images for Hessian Matrix Computation}\label{sec:poly}
+Let $P(x,y)$ be the discrete value of the pixel $(x,y)$ in the image.
+Let $n$, $1 \le n \le N$, be an integer such that the objective is to find a polynomial interpolation 
+on the $(2n+1)\times(2n+1)$ window where the central pixel has index $(0,0)$.  
+There exists an unique polynomial $L : \R\times \R \to \R$ of degree $(2n+1)\times(2n+1)$ defined  
+such that $L(x,y)=P(x,y)$ for each pixel $(x,y)$ in this window.
+Such a polynomial is defined by 
+\begin{equation}
+\begin{array}{l}
+L(x,y) =  
+\sum_{i=-n}^{n}
+\sum_{j=-n}^{n} \\
+\quad P(i,j)
+\left( 
+\prod_{\stackrel{-n\leq j'\leq n}{j'\neq j}}
+\frac{x-j'}{i-j'}
+\right)
+\left( 
+\prod_{\stackrel{-n\leq i'\leq n}{i'\neq i}}
+\frac{x-i'}{i-i'}
+\right)
+\end{array}
+\end{equation}
+
+It is not hard to prove that the first order horizontal derivative of the polynomial $L(x,y)$ 
+is
+\begin{equation}
+\begin{array}{l}
+\dfrac{\partial L}{\partial x} =  
+\sum_{i=-n}^{n}
+\sum_{j=-n}^{n} 
+P(i,j)
+\left( 
+\prod_{\stackrel{-n\leq j'\leq n}{j'\neq j}}
+\frac{y-j'}{j-j'}
+\right)\\
+\quad
+\left( 
+\sum_{\stackrel{-n\leq i'\leq n}{i'\neq i}}
+\frac{1}{i-i'}
+\prod_{\stackrel{-n\leq i''\leq n}{i''\neq i,i'}}
+\frac{x-i''}{i-i''}
+\right)
+\end{array}
+\end{equation}
+\noindent and thus to deduce that the
+second order ones are
+
+\begin{equation}
+\begin{array}{l}
+\dfrac{\partial^2 L}{\partial x^2} =  
+\sum_{i=-n}^{n}
+\sum_{j=-n}^{n} 
+P(i,j)
+\left( 
+\prod_{\stackrel{-n\leq j'\leq n}{j'\neq j}}
+\frac{y-j'}{j-j'}
+\right)\\
+\quad
+\left( 
+\sum_{\stackrel{-n\leq i'\leq n}{i'\neq i}}
+\frac{1}{i-i'}
+\sum_{\stackrel{-n\leq i''\leq n}{i''\neq i,i'}}
+\frac{1}{i-i''}
+\prod_{\stackrel{-n\leq i'''\leq n}{i'''\neq i,i',i''}}
+\frac{x-i'''}{i-i'''}
+\right)
+\end{array}
+\label{eq:deriv:poly:x2}
+\end{equation}
+
+\begin{equation}
+\begin{array}{l}
+\dfrac{\partial^2 L}{\partial y \partial x} =  
+\sum_{i=-n}^{n}
+P(i,j) \\
+\quad
+\left( 
+\sum_{\stackrel{-n\leq j'\leq n}{j'\neq j}}
+\frac{1}{j-j'}
+\prod_{\stackrel{-n\leq j''\leq n}{j''\neq j, j'}}
+\frac{y-j''}{j-j''}
+\right)\\
+\quad
+\left( 
+\sum_{\stackrel{-n\leq i'\leq n}{i'\neq i}}
+\frac{1}{i-i'}
+\prod_{\stackrel{-n\leq i''\leq n}{i''\neq i, i'}}
+\frac{x-i''}{i-i''}
+\right)
+\end{array}
+\label{eq:deriv:poly:yx}
+\end{equation}
+
+These second order derivatives  are computed for each moving 
+window and are associated to the central pixel, \textit{i.e.}, to the pixel $(0,0)$ inside this one.
+
+Let us first simplify $\dfrac{\partial^2 L}{\partial x^2}$ when $(x,y)=(0,0)$
+defined in Equation~(\ref{eq:deriv:poly:x2}). If $j$ is not null, the index $j'$
+is going to be null and the product  
+$\left( 
+\prod_{\stackrel{-n\leq j'\leq n}{j'\neq j}}
+\frac{-j'}{j-j'}
+\right)$ is null too. 
+In this equation, we thus only consider $j=0$.
+It is obvious that the product indexed with $j'$ is thus equal to 1.
+This equation can thus be simplified in:
+
+\begin{equation}
+\begin{array}{l}
+\dfrac{\partial^2 L}{\partial x^2} =  
+\sum_{i=-n}^{n}
+P(i,0)\\
+\quad
+\left( 
+\sum_{\stackrel{-n\leq i'\leq n}{i'\neq i}}
+\frac{1}{i-i'}
+\sum_{\stackrel{-n\leq i''\leq n}{i''\neq i,i'}}
+\frac{1}{i-i''}
+\prod_{\stackrel{-n\leq i'''\leq n}{i'''\neq i,i',i''}}
+\frac{i'''}{i'''-i}
+\right)
+\end{array}
+\label{eq:deriv:poly:x2:simpl}
+\end{equation}
+
+
+and then in:
+
+\begin{equation}
+\begin{array}{l}
+\dfrac{\partial^2 L}{\partial x^2} =  
+\sum_{i=-n}^{n}
+P(i,0)\\
+\quad
+\left( 
+\sum_{\stackrel{-n\leq i' < i'' \le n}{i',i''\neq i}}
+\frac{2}{(i-i')(i-i'')}
+\prod_{\stackrel{-n\leq i'''\leq n}{i'''\neq i,i',i''}}
+\frac{i'''}{i'''-i}
+\right).
+\end{array}
+\label{eq:deriv:poly:x2:simpl:2}
+\end{equation}
+
+From this equation, the kernel allowing to evaluate horizontal 
+second order derivatives 
+can be computed for any $n$.
+It is further denoted as $Ko''_{x^2}$. 
+Instances of such matrix when $n=2$, $3$, and $4$
+are given in Table~\ref{table:sod:hori:poly}.
+
+\begin{table}[ht]
+    \caption{Kernels $Ko''_{x^2}$ 
+    for second order horizontal derivatives induced 
+    by polynomial interpolation}
+    \centering
+    \scriptsize
+\def\arraystretch{1.4}
+    \begin{tabular}{|c|c|}
+         \hline
+         $n$ & $Ko''_{x^2}$ \\
+         \hline
+         $2$ & $\left[\dfrac{-1}{12}, \dfrac{4}{3} , \dfrac{-5}{2}, \dfrac{4}{3} \dfrac{-1}{12}\right]$ \\
+         \hline
+         $3$ & $\left[\dfrac{1}{90}, \dfrac{-3}{20}, \dfrac{3}{2}, \dfrac{-49}{18}, \dfrac{3}{2}, \dfrac{-3}{20}, \dfrac{1}{90}\right]$ \\
+         \hline
+         $4$ & $\left[\dfrac{-1}{560}, \dfrac{8}{315}, \dfrac{-1}{5}, \dfrac{8}{5}, \dfrac{-205}{72}, \dfrac{8}{5}, \dfrac{-1}{5}, \dfrac{8}{315}, \dfrac{-1}{560}\right]$\\
+         \hline
+         %$5$ & $\left[\dfrac{1}{3150}, \dfrac{-5}{1008}, \dfrac{5}{126}, %\dfrac{-5}{21}, \dfrac{5}{3}, \dfrac{-5269}{1800}, \dfrac{5}{3}, %\dfrac{-5}{21}, \dfrac{5}{126}, \dfrac{-5}{1008}, \dfrac{1}{3150}\right]$\\
+         %\hline
+    \end{tabular}
+
+    \label{table:sod:hori:poly}
+\end{table}
+
+
+\begin{table}[ht]
+\caption{Kernels for second order diagonal derivatives induced 
+    by polynomial interpolation \label{table:sod:diag:poly}
+}
+\centering
+\scriptsize
+\def\arraystretch{1.5}
+\begin{tabular}{|c|c|}
+    \hline
+    $n$ & $Ko''_{xy}$\\
+    \hline
+    2 & $
+\begin{bmatrix} 
+\dfrac{1}{4} & 0 &  \dfrac{-1}{4}\\
+ 0 & 0 &0\\
+\dfrac{-1}{4} & 0 &  \dfrac{1}{4}\\
+\end{bmatrix}
+$
+ \\   
+ \hline
+   3 &  
+   $\begin{bmatrix} 
+\dfrac{1}{144} & \dfrac{-1}{18} & 0 & \dfrac{1}{18} & \dfrac{-1}{144}\\
+\dfrac{-1}{18} & \dfrac{4}{9} & 0 & \dfrac{-4}{9} & \dfrac{1}{18}\\
+0 & 0 & 0 & 0 &0\\
+\dfrac{1}{18} & \dfrac{-4}{9} & 0 & \dfrac{4}{9} & \dfrac{-1}{18}\\
+\dfrac{-1}{144} & \dfrac{1}{18} & 0 & \dfrac{-1}{18} & \dfrac{1}{144}
+\end{bmatrix}
+$\\
+  \hline
+\end{tabular}
+\end{table}
+
+
+From Equation~(\ref{eq:deriv:poly:yx}), kernels allowing to evaluate diagonal 
+second order derivatives (\textit{i.e.}, 
+$\dfrac{\partial^2 L}{\partial y \partial x}$)  are computed.
+They are denoted as $Ko''_{xy}$.
+Table~\ref{table:sod:diag:poly} gives two examples of them when $n=1$ and $n=2$.
+Notice that for $n=1$, the kernel $Ko''_{xy}$ is equal to $Kc''_{xy}$. 
+
+
+
+
+
+
+\section{Distortion Cost}\label{sec:distortion}
+
+The distortion function has to associate to each pixel $(i,j)$ 
+the cost $\rho_{ij}$ of its modification
+by $\pm 1$.
+
+The objective is to map a small value to a pixel when all its second order derivatives 
+are high and a large value otherwise. 
+%\YF{The WOW  use the  wavelet relative distortion (UNIWARD) as a function to force the embedding operation to the noisy areas and avert the smooth areas.}
+In WOW and UNIWARD  the distortion function is based on the H\"older norm with
+\[
+\rho_{ij}^w = 
+\left(
+\abs{\xi_{ij}^h}^{p} +
+\abs{\xi_{ij}^v}^{p} +
+\abs{\xi_{ij}^d}^{p} 
+\right)^{-\frac{1}{p}}
+\]
+where $p$ is a negative number and 
+$\xi_{ij}^h$ (resp. $\xi_{ij}^v$ and $\xi_{ij}^d$)
+represents the horizontal (resp. vertical and diagonal) suitability.
+A small suitability in one direction means an inaccurate position to embed a message.
+
+We propose here to adapt such a distortion cost as follows:
+\[
+\rho_{ij} = 
+\left(
+\abs{\dfrac{\partial^2 P}{\partial x^2}(i,j)} +
+\abs{\dfrac{\partial^2 P}{\partial y^2}(i,j)} +
+\abs{\dfrac{\partial^2 P}{\partial y \partial x}(i,j)}
+\right)^{-\frac{1}{p}}
+\]
+It is not hard to check that such a function has large 
+value when at least one of its derivatives is null. Otherwise,
+the larger the derivatives are, the smaller the returned value
+is.
+
+
+
+
+%Practically, it is computed as a double convolution product with Daubechies Wavelet Kernels. 
+
+%It is not hard to see that this approach 
+%does not consider the range of the values. For %instance, if one direction only has large suitability
+%absolute values, all the elements $\abs{\xi_{ij}}^{p}$ are so small, and 
+%suitable pixels cannot be distinguished from inaccurate ones. The distortion function has to 
+%provide a way to discriminate pixels even when the distribution has small deviation
+
+%Because of each pixel value ranges in $[0,255]$, each first order derivative belongs to
+%$[-255,255]$ and thus each second order derivative has a real value in 
+%$[-255,255]$.  
+%Due to the definition of the approximation of $\dfrac{\partial^2 P}{\partial x^2}$, 
+%this one is a positive or null real number lower than 255. 
+
+%The higher the second order derivatives are, 
+%the 
+
+%To enlarge the discrimination power, 
+%data are firstly linearly mapped to $[0,1]$. 
+%The tuple 
+%$(\dfrac{\partial^2 P}{\partial x^2} , \dfrac{\partial^2 P}{\partial y^2}, \dfrac{\partial^2 %P}{\partial x \partial y})$ belongs to $[0,1]\times[0,1]\times[0,1] = [0,1]^3$.
+
+%Basically, for each dimension the idea is to select a subset of pixels of size $k$ and next to retain those which have been selected three times, \textit{i.e.}, those with large values 
+%in any element of 
+%$(\dfrac{\partial^2 P}{\partial x^2} , \dfrac{\partial^2 P}{\partial y^2}, \dfrac{\partial^2 %P}{\partial x \partial y})$.
+%Since the dimension is $3$, the size $k$ can be defined by
+%\[
+%k = W \times H \times \alpha ^{\dfrac{1}{3}}
+%\]
+%where $W$ and $H$ respectively stand for the width and the length of the image (expressed in pixels)
+%and $\alpha$ is the payload. For instance if $\alpha$ is 0.1, 
+%about $0.46\%$ of the image pixels with higher values in $\dfrac{\partial^2 P}{\partial x^2}$ should be selected, and similarly with $\dfrac{\partial^2 P}{\partial y^2}$ and $\dfrac{\partial^2  P}{\partial x \partial y}$.
+
+
+% Dire comment on les ordonne, les autres et le sens
+
+\begin{figure*}
+    \centering
+    \begin{tabular}{|c|c|c|}
+    \hline
+    Scheme & Stego. content & Changes with cover \\
+    \hline
+    $Ky$ based approach &\includegraphics[scale=0.20]{images/38_dp}&
+    \includegraphics[scale=0.20]{images/38_dp_diff} \\
+    \hline
+    $Ko$ based approach & \includegraphics[scale=0.20]{images/38_poly} &
+    \includegraphics[scale=0.20]{images/38_poly_diff} \\
+    \hline
+    \end{tabular}
+    \caption{Embedding changes instance with payload  $\alpha = 0.4$}
+    \label{fig:oneimage}
+\end{figure*}
+
+
+\section{Experiments}\label{sec:experiments}
+
+First of all, the whole steganographic approach code is available online\footnote{\url{https://github.com/stego-content/SOS}}.
+
+Figure~\ref{fig:oneimage} presents the results of embedding data in a cover image
+from the BOSS contest database~\cite{Boss10} with respect to the two second order derivative schemes presented in this work. 
+The $Ky$ based approach (resp. the $Ko$ based one)
+corresponds to the scheme detailed in Section~\ref{sec:second} 
+(resp. in Section~\ref{sec:poly}).
+The payload $\alpha$ is set to 0.4 and kernels are computed with $N=4$.    
+The central column outputs the embedding result whereas the right one displays 
+differences between the cover image and the stego one.
+It can be observed that pixels in smooth area (the sky, the external access steps) and pixels in clean edges (the columns, 
+the step borders) are not modified by the approach.
+On the contrary, an unpredictable area (a monument for example) concentrates pixel changes.
+
+
+\subsection{Choice of parameters}
+
+The two methods proposed in Section~\ref{sec:second} and  
+in Section~\ref{sec:poly} are based on kernels of size up to 
+$(2N+1)\times(2N+1)$. This section aims at finding the 
+value of the $N$ parameter that maximizes the security level.
+For each approach, we have built 1,000 stego images with 
+$N=2$, $4$, $6$, $8$, $10$, $12$, and $14$ where the covers belong 
+to the BOSS contest database. 
+This set contains 10,000 grayscale $512\times 512$ images in a RAW format. 
+The security of the approach has been evaluated thanks to the Ensemble Classifier~\cite{DBLP:journals/tifs/KodovskyFH12} based steganalyser,
+which is considered as a state of the art steganalyser tool.
+This steganalysis process embeds the rich model (SRM) features~\cite{DBLP:journals/tifs/FridrichK12}
+of size 34,671. 
+For a payload $\alpha$, either equal to $0.1$ or to $0.4$, average testing errors (expressed in percentages) have been studied and are summarized in
+Table~\ref{table:choice:parameter}.
+
+\begin{table}[ht]
+\caption{Average Testing Errors with respect to the the Kernel Size}
+\begin{scriptsize}
+\centering
+\setlength{\tabcolsep}{3pt} 
+\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
+\cline{2-9}
+\multicolumn{1}{c|}{} & \multirow{2}{*}{$\alpha$} & \multicolumn{7}{c|}{$N$} \\
+\cline{3-9}
+\multicolumn{1}{c|}{}&  & $2$ & $4$&  $6$&  $8$& $10$& $12$ & $14$ \\
+\hline{}
+Average testing  
+& \textit{0.1} & 39& 40.2&  39.7&  39.8& 40.1& $39.9$&  $39.8$  \\
+\cline{2-9}
+error for Kernel $K_y$ & \textit{0.4}& 15& 18.8& 19.1&  19.0& 18.6& 18.7 & 18.7 \\
+\hline
+Average testing & \textit{0.1} & 35.2 & 36.6&  36.7&  36.6& 37.1& 37.2 & 37.2 \\
+\cline{2-9}
+error for Kernel $K_o$ & \textit{0.4}  & 5.2 & 6.8& 7.5 & 7.9 & 8.1 & 8.2 & 7.6 \\
+\hline
+\end{tabular}
+\label{table:choice:parameter}
+\end{scriptsize}
+\end{table}
+
+Thanks to these experiments, we observe that the size $N=4$ (respectively $N=12$) obtains sufficiently  large average testing errors for the $Ky$ based approach 
+(resp. for the $Ko$ based one). In what follows, these values are retained for these two methods.
+
+\subsection{Security Evaluation}
+As in the previous section, the BOSS contest database has been retained.
+To achieve a complete comparison with other steganographic tools,
+the whole database of 10,000 images has been used.
+Ensemble Classifier with SRM features is again used to evaluate the security of the approach.
+
+We have chosen 4 different payloads, 0.1, 0.2, 0.3, and 0.4, as in many steganographic evaluations.
+Three values are systematically given for each experiment:
+the area under the ROC curve (AUC),
+the average testing error (ATE),
+and the OOB error (OOB).
+
+All the results are summarized in Table~\ref{table:experiments:summary}.
+%and  in Figure~\ref{fig1}, Figure~\ref{fig2}, and Figure~\ref{fig3}.
+Let us analyse these experimental results. 
+The security approach is often lower than those observed with state of the art tools:
+for instance with payload $\alpha=0.1$, the most secure approach is WOW 
+with an average testing error equal to 0.43 whereas our approach reaches 0.38.
+However these results are  promising and for two reasons.
+First, our approaches give more resistance towards Ensemble Classifier (contrary to HUGO)
+for large payloads.
+Secondly, without any optimisation, our approach is not so far from state of the art steganographic tools.
+Finally, we explain the lack of security of the $Ko$ based approach with large payloads as follows:
+second order derivatives are indeed directly extracted from polynomial interpolation.
+This easy construction however induces large variations between the polynomial $L$ and 
+the pixel function $P$.
+
+\begin{table}
+\caption{Summary of experiments}\label{table:experiments:summary}
+\begin{small}
+\centering
+\begin{tabular}{|l|l|l|l|l|}
+\hline
+
+ & Payload & AUC & ATE   &  OOB \\ \hline
+{WOW}   
+            & 0.1 & 0.6501 & 0.4304 & 0.3974\\
+            & 0.2 & 0.7583 & 0.3613 & 0.3169\\
+            & 0.3 & 0.8355 & 0.2982 & 0.2488\\
+            & 0.4 & 0.8876 & 0.2449 & 0.1978\\ 
+                                       \hline
+
+{SUNIWARD}  & 0.1 & 0.6542 & 0.4212 & 0.3972\\
+            & 0.2 & 0.7607 & 0.3493 & 0.3170\\
+            & 0.3 & 0.8390 & 0.2863 & 0.2511\\
+            & 0.4 & 0.8916 & 0.2319 & 0.1977\\ 
+ \hline
+ {MVG}      & 0.1 & 0.6340 & 0.4310 &0.4124 \\
+            & 0.2 & 0.7271 & 0.3726 &0.3399 \\
+            & 0.3 & 0.7962 & 0.3185& 0.2858\\
+            & 0.4 & 0.8486& 0.2719& 0.2353 \\ 
+ \hline
+  {HUGO}    & 0.1 & 0.6967 & 0.3982 & 0.3626 \\
+            & 0.2 & 0.8012 & 0.3197 & 0.2847 \\
+            & 0.3 & 0.8720 & 0.2557 & 0.2212 \\
+            & 0.4 & 0.9517 & 0.1472 & 0.1230 \\ 
+ \hline
+{$Ky$ based approach} 
+            & 0.1 & 0.7378 & 0.3768 & 0.3306 \\
+            & 0.2 & 0.8568 & 0.2839 & 0.2408 \\
+            & 0.3 & 0.9176 & 0.2156 & 0.1710 \\
+            & 0.4 & 0.9473 & 0.1638 & 0.1324\\
+ \hline
+{$Ko$ based approach} 
+            & 0.1 & 0.6831 & 0.3696  & 0.3450 \\
+            & 0.2 & 0.8524 & 0.1302  & 0.2408 \\
+            & 0.3 & 0.9132 & 0.1023  & 0.1045 \\
+            & 0.4 & 0.9890 & 0.0880  & 0.0570 \\
+                              \hline 
+                              
+\end{tabular}
+\end{small}
+\end{table}
+
+
+%\begin{figure}[ht]
+%\centering
+%\includegraphics[width=9cm]{22} 
+%\caption{Average Testing Error(ATE) } 
+%\label{fig1}
+%\end{figure}
+
+%\begin{figure}[ht]
+%\centering
+%\includegraphics[width=9cm]{11} 
+%\caption{Out of Bag($E_{\textit{OOB}}$)} 
+%\label{fig2}
+%\end{figure}
+
+%\begin{figure}[ht]
+%\centering
+%\includegraphics[width=9cm]{33} 
+%\caption{Area Under Curve(AUC)} %\label{fig3}
+%\end{figure}
+
+
+\section{Conclusion}
+
+The first contribution of this paper is to propose of a distortion
+function which is based on second order derivatives. These
+partial derivatives allow to accurately compute 
+the level curves and thus to look favorably on pixels
+without clean level curves. 
+Two approaches to build these derivatives have been proposed.
+The first one is based on revisiting kernels usually embedded 
+in edge detection algorithms. 
+The second one is based on the polynomial approximation
+of the bitmap image.
+These two methods have been completely implemented.
+The first experiments have shown that the security level 
+is slightly inferior the one of the most stringent approaches. These first promising results encourage us to deeply investigate this research direction. 
+
+Future works aiming at improving the security of this proposal are planned as follows. The authors want first to focus on other approaches to provide second order derivatives with larger discrimination power.
+Then, the objective will be to deeply investigate whether the H\"older norm is optimal when the objective is to avoid null second order derivatives, and to give priority to the largest second order values.
+
+
-- 
2.39.5


From f4997f36924772119b653509ff5ae4e8aef4200a Mon Sep 17 00:00:00 2001
From: couchot <couchot@localhost.localdomain>
Date: Sat, 27 Aug 2016 12:18:56 +0200
Subject: [PATCH 08/16] =?utf8?q?avanc=C3=A9es=20ds=20YsousraStego?=
MIME-Version: 1.0
Content-Type: text/plain; charset=utf8
Content-Transfer-Encoding: 8bit

---
 stegoyousra.tex | 117 ++++++++++++++++++++++--------------------------
 1 file changed, 54 insertions(+), 63 deletions(-)

diff --git a/stegoyousra.tex b/stegoyousra.tex
index dc17a7a..efd326d 100644
--- a/stegoyousra.tex
+++ b/stegoyousra.tex
@@ -37,33 +37,26 @@ en Section~\ref{sec:experiments}.
 
 
 
-\section{Derivatives in an Image}\label{sec:gradient}
+\section{Des dérivées dans une image}\label{sec:gradient}
 
-This section first recalls links between level curves, gradient, and
-Hessian matrix (Section~\ref{sub:general}).
-It next analyses them using kernels from signal theory 
-(Section~\ref{sub:class:1} and Section~\ref{sub:class:2}).
+Cette section rappelle d'abord les liens entre lignes de niveau, gradient et 
+matrice hessienne puis analyse ensuite leur construction à l'aide
+de noyaux de la théorie du signal.
 
 
-\subsection{Hessian Matrix}\label{sub:general}
-Let us consider that an image can be seen as a numerical function $P$ that 
-associates a value $P(x,y)$ to each pixel of coordinates $(x,y)$.
-The variations of this function in $(x_0,y_0)$ 
-can be evaluated thanks to its gradient
-$\nabla{P}$, which is the vector whose two components 
-are the partial derivatives in $x$ and in $y$ of $P$: 
-
-\[\nabla{P}(x_0,y_0) = \left(\frac{\partial P}{\partial x}(x_0,y_0),\frac{\partial P}{\partial y}(x_0,y_0)\right).
-\]
-
-In the context of two variables, the gradient vector
-points to the direction where the function has the highest increase.
-Pixels with close values thus follow level curve that is orthogonal 
-to the one of highest increase.
-
-The variations of the gradient vector are expressed in the 
-Hessian matrix $H$ of second-order partial derivatives of $P$.
+\subsection{Matrice hessienne}\label{sub:general}
+On considère qu'une image peut être assimilée à une fonction de $\R^+\times \R^+$
+dans $\R^+$ telle que la valeur $P(x,y)$ est associée à chaque  pixel de coordonnées $(x,y)$.
+Les variations d'une telle fonction en $(x_0,y_0)$ peuvent être évaluées 
+grace au gradient 
+$\nabla{P}(x_0,y_0) = \left(\frac{\partial P}{\partial x}(x_0,y_0),\frac{\partial P}{\partial y}(x_0,y_0)\right).
+$
+Le vecteur gradient pointe dans la direction où la fonction a le plus fort acroissement.
+Des pixels ayant des valeurs voisines sont sur des lignes de niveaux qui sont orthogonales
+à ce vecteur.
 
+Les variations du vecteur gradient s'expriment usuellement à l'aide de la matrice
+hessienne $H$ des dérivées partielles de second ordre de $P$.
 \[
 H = \begin{bmatrix} 
 \dfrac{\partial^2 P}{\partial x^2} & 
@@ -73,67 +66,65 @@ H = \begin{bmatrix}
 \end{bmatrix}. 
 \]
 
-
-
-
-In one pixel $(x_0,y_0)$, the larger the absolute values of this matrix are,
-the more the gradient is varying around $(x_0,y_0)$.
-We are then left to evaluate such an Hessian matrix.
-
-This task is not as easy as it appears since natural images are not defined 
-with differentiable functions from $\R^2$ to $\R$.
-Following subsections provide various approaches to compute these 
-Hessian matrices.
-
-\subsection{Classical Gradient Image Approaches}\label{sub:class:1}
-In the context of image values, the most used approaches to evaluate gradient vectors are the well-known ``Sobel'', ``Prewitt'', ``Central Difference'', and ``Intermediate Difference'' ones.
+En un pixel $(x_0,y_0)$, plus les valeurs de cette matrice sont éloignées de zéro,
+plus le gradient varie en ce point. Evaluer ce type de matrice est ainsi primordial
+en stéganographie. Cependant cette tâche n'est pas aussi triviale qu'elle n'y 
+paraît puisque les images naturelles ne sont pas  définies à l'aide 
+de fonction différentiables de $\R^+\times \R^+$
+dans $\R^+$. La suite montre comment obtenir des approximations de telles matrices. 
+
+\subsection{Approches classiques pour évaluer le gradient dans des images}\label{sub:class:1}
+Dans ce contexte, les approches les plus utilisées pour évaluer un gradient 
+sont ``Sobel'', ``Prewitt'', ``Différence centrale'' et `` Difference intermediaire''.
+Chacune de ces  approches applique un produit de convolution $*$ entre un noyau $K$
+(rappelé dans le tableau~\ref{table:kernels:usual}) et une fenêtre $A$ de taille 
+$3\times 3$. Le résultat 
+ $A * K$ est une approximation du gradient horizontal
+\textit{i.e.}, $\dfrac{\partial P}{\partial x}$.
+Soit $K\rl$ le résultat de la rotation  d'un angle $\pi/2$ sur $K$.
+La composante verticale du gradient,  $\dfrac{\partial P}{\partial y}$ est obtenue 
+de manière similaire en évaluant $A * K\rl$. Lorsqu'on applique ceci sur toute 
+la matrice image, on obtient peu ou prou une matrice de même taille pour chacune des 
+dérivées partielles. 
+
+Les deux éléments de la première ligne (respectivement de la seconde ligne) 
+de la matrice hessienne
+sont le résultat du calcul du gradient sur la matrice $\dfrac{\partial P}{\partial x}$
+(resp. sur la matrice  $\dfrac{\partial P}{\partial y}$).
 
 
 \begin{table}[ht]
-\caption{Kernels of usual image gradient operators\label{table:kernels:usual}
+\caption{Noyaux usuels pour évaluer des gradients d'images\label{table:kernels:usual}
 }
 \centering
 \scriptsize
 \begin{tabular}{|c|c|c|}
     \hline
-    Name&   Sobel & Prewitt \\
+    Nom&   Sobel & Prewitt \\
     \hline
-    Kernel & $\textit{Ks}= \begin{bmatrix} -1 & 0 & +1 \\ -2 & 0 & +2 \\ -1 & 0 & +1 \end{bmatrix} $ &
+    Noyau & $\textit{Ks}= \begin{bmatrix} -1 & 0 & +1 \\ -2 & 0 & +2 \\ -1 & 0 & +1 \end{bmatrix} $ &
     $\textit{Kp}= \begin{bmatrix} -1 & 0 & +1 \\ -1 & 0 & +1 \\ -1 & 0 & +1 \end{bmatrix} $\\
     \hline\hline
-    Name & Central & Intermediate \\
-            & Difference &Difference \\ 
+    Nom & Difference  & Différence  \\
+            & centrale & Intermediaire \\ 
     \hline
-    Kernel & $\textit{Kc}= \begin{bmatrix} 0&0&0 \\ -\dfrac{1}{2} & 0 & +\dfrac{1}{2} \\ 0&0&0 \end{bmatrix} $ &
+    Noyau & $\textit{Kc}= \begin{bmatrix} 0&0&0 \\ -\dfrac{1}{2} & 0 & +\dfrac{1}{2} \\ 0&0&0 \end{bmatrix} $ &
     $\textit{Ki}= \begin{bmatrix} 0&0&0 \\ 0 & -1 & 1 \\ 0&0&0 \end{bmatrix} $ \\
     \hline
 \end{tabular}
 \end{table}
 
 
-Each of these approaches applies a convolution product $*$ between a  kernel $K$ (recalled in Table~\ref{table:kernels:usual}) and 
- a $3\times 3$ window of pixel values $A$. The result 
- $A * K$ is an evaluation of the horizontal gradient, 
-\textit{i.e.}, $\dfrac{\partial P}{\partial x}$
-expressed as a matrix in $\R$.
-Let $K\rl$ be the result of a $\pi/2$ rotation applied on $K$. 
-The vertical gradient $\dfrac{\partial P}{\partial y}$ is similarly obtained by computing $A * K\rl$, which is again expressed as a matrix in $\R$.
-
-The two elements of the first line 
-of the Hessian matrix are the result
-of applying the horizontal gradient calculus 
-first on  $\dfrac{\partial P}{\partial x}$ and next 
-on  $\dfrac{\partial P}{\partial y}$. 
-Let us study these Hessian matrices in the next section.
 
 
-\subsection{Hessian Matrices induced by Gradient Image Approaches}\label{sub:class:2}
+\subsection{Matrices hessiennes induites  par des approches 
+de gradient d'images}\label{sub:class:2}
+Il est connu que  
+$\dfrac{\partial^2 P}{\partial x \partial y} $ est égal à 
+$\dfrac{\partial^2 P}{\partial y \partial x}$ si 
+les méthode qui calculent le grandient et le gradient du gradient (la matrice hessienne)
+sont les mêmes.
 
-First of all, it is well known that 
-$\dfrac{\partial^2 P}{\partial x \partial y} $ is equal to 
-$\dfrac{\partial^2 P}{\partial y \partial x}$ if  
-the approach that computes the gradient and the
-one which evaluates the Hessian matrix are the same.
 For instance, in the Sobel approach, 
 it is easy to verify that the calculus of 
 $\dfrac{\partial^2 P}{\partial x \partial y}$ 
-- 
2.39.5


From 02fd942d6a30fe7197b732c94450ca466b7a49f5 Mon Sep 17 00:00:00 2001
From: couchot <couchot@localhost.localdomain>
Date: Mon, 29 Aug 2016 20:54:07 +0200
Subject: [PATCH 09/16] -> prng inclus

---
 11FCT.tex                           |   1 -
 14Secrypt.tex                       |  51 ++-
 chaosANN.tex                        |  35 +-
 demandeInscription/avis_jef_hdr.pdf | Bin 438585 -> 438682 bytes
 main.tex                            |  49 ++-
 modelchecking.tex                   |  29 +-
 sdd.tex                             |  10 -
 stegoyousra.tex                     | 635 ++++++++++------------------
 8 files changed, 329 insertions(+), 481 deletions(-)

diff --git a/11FCT.tex b/11FCT.tex
index 0e9c64f..45d91c9 100644
--- a/11FCT.tex
+++ b/11FCT.tex
@@ -90,4 +90,3 @@ Deux fonctions sont équivalentes si leurs \textsc{giu} sont isomorphes
   \end{center}
 \caption{Exemple de graphe d'interactions vérifiant le théorème~\ref{th:Adrien}} \label{fig:Adrien:G}
 \end{figure}
-
diff --git a/14Secrypt.tex b/14Secrypt.tex
index f1fae6f..bc59ba1 100644
--- a/14Secrypt.tex
+++ b/14Secrypt.tex
@@ -1,16 +1,29 @@
 On  a vu  dans  le chapitre précédent  que  pour avoir
 un  générateur à  sortie
 uniforme, il est nécessaire que  la matrice d'adjacence du graphe d'itération du
-système  soit doublement stochastique.   Nous présentons  dans cette  partie une
-méthode permettant de générer de telles matrices.
-
-Les approches théoriques basées sur la programmation logique par contraintes sur
-domaines  finis ne  sont pas  envisageables en  pratique dès  que la  taille des
-matrices considérées devient suffisamment grande.
-
+système  soit doublement stochastique.   Nous présentons  dans cette  partie
+des méthodes effectives permettant de générer de telles matrices.
+La première est basée sur la programmation logique par contraintes
+(Section~\ref{sec:plc}).
+Cependant celle-ci souffre de ne pas passer à l'échelle et ne fournit pas 
+une solution en un temps raisonnable dès que la fonction à engendrer 
+porte sur un grand nombre de bits.
 Une approche plus pragmatique consiste  à supprimer un cycle hamiltonien dans le
-graphe d'itérations, ce qui revient à supprimer en chaque n{\oe}ud de ce graphe une
-arête sortante et une arête entrante.
+graphe d'itérations $\textsc{giu}(\neg)$ (section~\ref{sec:hamiltonian}). 
+Pour obtenir plus rapidement une distribution uniforme, l'idéal serait
+de supprimer un cycle hamiltonien qui nierait autant de fois chaque bit. 
+Cette forme de cycle est dit équilibré. La section~\ref{sub:gray} établit le
+lien avec les codes de Gray équilibrés, étudiés dans la litérature. 
+La section suivante présente une démarche de génération automatique de code de Gray équilibré (section~\ref{sec:induction}).
+La vitesse avec laquelle l'algorithme de PRNG converge en interne vers 
+une distribution unifiorme est étduiée théoriquement et pratiquement à la 
+section~\ref{sec:mixing}.
+L'extension du travail aux itérations généralisées est présenté à la 
+section~\ref{sec:prng:gray:general}.
+Finalement, des instances de PRNGS engendrés selon les méthodes détaillées dans 
+ce chapitre sont présentés en section~\ref{sec:prng;gray:tests}.
+Les sections~\ref{sec:plc} à~\ref{sub:gray} ont été publiées 
+à~\ref{chgw+14:oip}.
 
 
 % This aim of this section is to show 
@@ -442,7 +455,7 @@ Ces fonctions étant générées, on s'intéresse à étudier à quelle vitesse
 un générateur les embarquant converge vers la distribution uniforme.
 C'est l'objectif de la section suivante. 
 
-\section{Quantifier l'écart par rapport à la distribution uniforme} 
+\section{Quantifier l'écart par rapport à la distribution uniforme}\label{sec:mixing} 
 On considère ici une fonction construite comme à la section précédente.
 On s'intéresse ici à étudier de manière théorique les 
 itérations définies à l'équation~(\ref{eq:asyn}) pour une 
@@ -651,7 +664,7 @@ $\textit{fair}\leftarrow\emptyset$\;
 
 
 
-\section{Et les itérations généralisées?}
+\section{Et les itérations généralisées?}\label{sec:prng:gray:general}
 Le chaptire précédent a présenté un algorithme de 
 PRNG construit à partir d'itérations unaires. 
 On pourrait penser que cet algorithme est peu efficace puisqu'il 
@@ -924,7 +937,7 @@ $$
 
 
 
-\section{Tests statistiques}
+\section{Tests statistiques}\label{sec:prng;gray:tests}
 
 La qualité des séquences aléatoires produites par 
 le générateur des itérations unaires ainsi que 
@@ -1072,4 +1085,16 @@ Complexité linaire& 0.005 (0.98)& 0.534 (0.99)& 0.085 (0.97)& 0.996 (1.0)\\ \hl
 \end{table}
 
 
-%
+\section{Conclusion}
+Ce chaptitre a montré comment construire un PRNG chaotique, notamment à partir 
+de codes de Gray équilibrés. Une méthode completement automatique de
+construction de ce type de codes a été présentée étendant les méthodes 
+existantes. 
+Dans le cas des itérations unaires, 
+l'algorithme qui en découle a un temps de mélange qui a 
+une borne sup quadratique de convergence vers la distribution uniforme. 
+Pratiquement,  ce temps de mélange se rapproche de $N\ln N$.
+Les expérimentations au travers de la batterie de test de NIST ont montré
+que toutes les propriétés statistiques sont obtenues pour
+ $\mathsf{N} = 4, 5, 6, 7, 8$.
+
diff --git a/chaosANN.tex b/chaosANN.tex
index b814c12..4ea34b9 100644
--- a/chaosANN.tex
+++ b/chaosANN.tex
@@ -1,5 +1,3 @@
-% \section{Introduction}
-% \label{S1}
 
 Les réseaux de neurones chaotiques ont été étudiés à de maintes reprises 
 par le passé en raison notamment de leurs applications potentielles:
@@ -28,8 +26,6 @@ fonctions chaotiques.
 
 
 
-
-
 Ces réseaux de neurones partagent le fait qu'ils sont qualifiés de 
 ``chaotiques'' sous prétexte qu'ils embarquent une fonction de ce type 
 et ce sans aucune preuve rigoureuse. Ce chapitre caractérise la 
@@ -37,23 +33,13 @@ classe des réseaux de neurones MLP chaotiques. Il
 s'intéresse ensuite à l'étude de prévisibilité de systèmes dynamiques
 discrets chaotiques par cette famille de MLP.
 
-\JFC{revoir plan}
-
-The remainder of this research  work is organized as follows. The next
-section is devoted to the basics of Devaney's chaos.  Section~\ref{S2}
-formally  describes  how  to  build  a neural  network  that  operates
-chaotically.  Section~\ref{S3} is devoted to the dual case of checking
-whether  an existing neural  network is  chaotic or  not.  Topological
-properties of chaotic neural networks are discussed in Sect.~\ref{S4}.
-The  Section~\ref{section:translation}  shows  how to  translate  such
-iterations  into  an Artificial  Neural  Network  (ANN),  in order  to
-evaluate the  capability for this  latter to learn  chaotic behaviors.
-This  ability  is  studied in  Sect.~\ref{section:experiments},  where
-various ANNs try to learn two  sets of data: the first one is obtained
-by chaotic iterations while the  second one results from a non-chaotic
-system.  Prediction success rates are  given and discussed for the two
-sets.  The paper ends with a conclusion section where our contribution
-is summed up and intended future work is exposed.
+La section~\ref{S2} définit la construction d'un réseau de neurones 
+chaotique selon Devanay. La section~\ref{S3} présente l'approche duale 
+de vérification si un réseau de neurones est chaotique ou non.
+La section~\ref{sec:ann:approx} s'intéresse à étudier pratiquement
+si un réseau de 
+neurones peut approximer des itération unaires chaotiques. Ces itérations
+étant obtenues à partir de fonction générées à l'aide du chapitre précédent.
 
 
 \section{Un réseau de neurones chaotique au sens de Devaney}
@@ -159,7 +145,7 @@ $\textsc{giu}(f)$ est fortement connexe.
 
 
 \section{Un réseau de neurones peut-il approximer
-des itération unaires chaotiques?}
+des itération unaires chaotiques?}\label{sec:ann:approx}
 
 Cette section s'intéresse à étudier le comportement d'un réseau de neurones 
 face à des itérations unaires chaotiques, comme définies à 
@@ -424,6 +410,11 @@ Nous avons présenté ensuite comment vérifier si un réseau de neurones
 Nous avons enfin montré en pratique qu'il est difficile pour un 
 réseau de neurones d'apprendre le comportement global d'itérations
 chaotiques.
+Comme il est difficile (voir impossible) d'apprendre le comportement 
+de telles fonction, il paraît naturelle de savoir si celles ci peuvent être 
+utilisées pour générer des nombres pseudo aléatoires.
+
+
 % \appendix{}
 
 % \begin{Def} \label{def2}
diff --git a/demandeInscription/avis_jef_hdr.pdf b/demandeInscription/avis_jef_hdr.pdf
index d706ae772cc960a418738b4b6fe30b712d499d3f..81569ce8619983c3a18c2798c4d0abff7b47c96b 100644
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diff --git a/main.tex b/main.tex
index 217f8f4..c6f86cd 100644
--- a/main.tex
+++ b/main.tex
@@ -175,19 +175,28 @@ Blabla blabla.
 \part{Réseaux Discrets}
 
 \chapter{Iterations discrètes de réseaux booléens}
-\JFC{chapeau à refaire}
-\section{Formalisation}
+
+Ce chapitre formalise tout d'abord ce qu'est 
+un réseau booléen (section~\ref{sec:sdd:formalisation}. On y revoit 
+les différents modes opératoires, leur représentation à l'aide de 
+graphes et les résultats connus de convergence).
+Ce chapitre montre ensuite à la section~\ref{sec:sdd:mixage}
+comment combiner ces modes pour converger aussi 
+souvent sans, mais plus rapidement. Cette dernière section 
+a fait l'objet du rapport~\cite{BCVC10:ir}.
+
+\section{Formalisation}\label{sec:sdd:formalisation}
 \input{sdd}
 
-\section{Combinaisons synchrones et asynchrones}
+\section{Combinaisons synchrones et asynchrones}\label{sec:sdd:mixage}
 \input{mixage}
 
 \section{Conclusion}
-\JFC{Conclusion à refaire}
 
 Introduire de l'asynchronisme peut permettre de réduire le temps 
 d'exécution global, mais peut aussi introduire de la divergence. 
-Dans ce chapitre, nous avons exposé comment construire un mode combinant les
+Dans ce chapitre, après avoir introduit les bases sur les réseaux bouléens,
+nous avons exposé comment construire un mode combinant les
 avantage du synchronisme en terme de convergence avec les avantages 
 de l'asynchronisme en terme de vitesse de convergence.
 
@@ -209,12 +218,18 @@ au chaos}
   discrets chaotiques]{Caracterisation des systèmes 
   discrets chaotiques pour les schémas unaires et généralisés}\label{chap:carachaos}
 
-La première section  rappelle ce que sont les systèmes dynamiques chaotiques.
-Dire que cette caractérisation dépend du type de stratégie : unaire (TIPE), 
-généralisée (TSI).  Pour chacune d'elle, 
-on introduit une distance différente.
-
-On montre qu'on a des résultats similaires.
+La suite de ce document se focalise sur des systèmes dynamiques discrets qui ne 
+convergent pas. Parmi ceux-ci se trouvent ceux qui sont \og chaotiques\fg{}.
+La première section  de ce chapitre rappelle ce que sont les systèmes 
+dynamiques chaotiques et leur caractéristiques. Celles-ci dépendent 
+tout d'abord de la stratégie itérée. La section~\ref{sec:TIPE12} 
+se focalise sur le schéma unaire tandis que la section~\ref{sec:chaos:TSI}
+considère le mode généralisé. Pour chacun de ces modes, 
+une distance est définie. Finalement, la section~\ref{sec:11FCT}
+exhibe des conditions suffisantes premettant d'engendrer 
+des fonctions chaotiques seon le mode unaire.
+Les sections~\ref{sec:TIPE12} et~\ref{sec:11FCT} ont été publiées 
+dans~\cite{bcgr11:ip}.
 
 \section{Systèmes dynamiques chaotiques selon Devaney}
 \label{subsec:Devaney}
@@ -223,13 +238,23 @@ On montre qu'on a des résultats similaires.
 \section{Schéma unaire}\label{sec:TIPE12}
 \input{12TIPE}
 
-\section{Schéma généralisé}
+\section{Schéma généralisé}\label{sec:chaos:TSI}
 \input{15TSI}
 
 
 \section{Générer des fonctions chaotiques}\label{sec:11FCT}
 \input{11FCT} 
 
+\section{Conclusion}
+Ce chapitre a montré que les itérations unaires sont chaotiques si
+et seulement si le graphe $\textsc{giu}(f)$ est fortement connexe et 
+que les itérations généralisées sont chaotiques si
+et seulement si le graphe $\textsc{gig}(f)$ est aussi fortement connexe.
+On dispose ainsi à priori d'une collection infinie de fonctions chaotiques.
+Le chapitre suivant s'intéresse à essayer de prédire le comportement 
+de telles fonctions. 
+
+
 \chapter{Prédiction des systèmes chaotiques}
 \input{chaosANN}
 
diff --git a/modelchecking.tex b/modelchecking.tex
index 1919416..2a7107f 100644
--- a/modelchecking.tex
+++ b/modelchecking.tex
@@ -1,6 +1,4 @@
 
-
-
 L'étude de convergence de systèmes dynamiques discrets est simple à vérifier 
 pratiquement pour le mode synchrone. Lorsqu'on introduit des stratégies 
 pseudo périodiques pour les modes unaires et généralisées, le problème 
@@ -43,7 +41,7 @@ Les théorèmes de correction et de complétude de la démarche
 sont ensuite donnés à la (\Sec{sec:spin:proof}). 
 Des données pratiques comme la complexité et des synthèses d'expérimentation
 sont ensuite fournies (\Sec{sec:spin:practical}). 
-
+Ce chapitre a fait l'objet du rapport~\cite{Cou10:ir}.
 
 
 
@@ -782,7 +780,7 @@ pour établir un verdict.
           $\bot$ & 374 & 7.7s&
           $\bot$ & 370 & 0.51s \\
           \hline %\cline{2-13}
-          AC2D 
+          \cite{RC07}
           &$\bot$ & 2.5 & 0.001s  % RC07_async_mixed.spin
           &$\bot$ & 2.5 & 0.01s   % RC07_async_mixed_all.spin
           &$\bot$ & 2.5 & 0.01s   % RC07_async.spin
@@ -816,7 +814,7 @@ du graphe complet.
 
 
 L'exemple  \textit{RE} a été prouvé comme universellement convergent.
-\JFC{statuer sur AC2D}
+%\JFC{statuer sur AC2D}
 Comme la convergence n'est déjà pas établie pour les itérations synchrones
 de~\cite{RC07}, il en est donc 
 de même pour les itérations asynchrones.
@@ -826,16 +824,13 @@ $\{1,2\};\{1,2\};\{1\};\{1,2\};\{1,2\}$ et débute avec $x=(0,0)$.
 En raison de la dépendance forte entre les éléments
 de~\cite{BM99}, 
 $\delta_0$ est réduit à 1. Cela aboutit cependant à $2^{100}$ 
-configurations dans le mode des itérations asynchrones.
-
-\JFC{Quid de ceci?}
-La convergence des itérations asynchrones de l'exemple~\cite{BCVC10:ir} n'est pas établie 
-lorsque pour $\delta_0$ vaut 1. Il ne peut donc y avoir convergence universelle.
+configurations dans le mode des itérations asynchrones, montrant les limites de
+l'approche.
 
 \begin{figure}
 \centering
 \includegraphics[scale=0.6]{images/RC07ce.eps}   
-\caption{Contre exemple de convergence pour~\ref{fig:RC07CE}} \label{fig:RC07CE}
+\caption{Contre exemple de convergence pour~~\cite{RC07}} \label{fig:RC07CE}
 \end{figure}
 
 
@@ -868,7 +863,17 @@ lorsque pour $\delta_0$ vaut 1. Il ne peut donc y avoir convergence universelle.
 \section{Conclusion}
 \label{sec:spin:concl}
 
-
+L'idée principale de ce chapitre est que l'on peut, 
+pour des réseaux bouléens à délais bornés de  petite taille, obtenir 
+une preuve de la convergence ou de sa divergence et ce
+de manière automatique.  
+L'idée principale est de traduire le réseau en PROMELA et de laisser 
+le model checker établir la preuve.
+Toute l'approche a été prouvée: le verdict rendu par a donc valeur de vérité.
+L'approche a cependant ses limites et ne peut donc pas être 
+apliquée qu'à des modèles simplifiés de programmes.
+La suite de ce travail consiste à se focaliser sur les systèmes qui ne 
+convergent pas.
 
 
 
diff --git a/sdd.tex b/sdd.tex
index 355e33c..69e09b6 100644
--- a/sdd.tex
+++ b/sdd.tex
@@ -1,17 +1,7 @@
 
-\JFC{Chapeau chapitre à faire}
 
 
 
-% Cette section énonce quelques notions suffisantes 
-% à la compréhension de ce document.
-% Elle commence par formaliser ce que sont les systèmes dynamiques booléens 
-% (section \ref{sub:sdd}) 
-% et montre comment en extraire leur graphe d'itérations (section~\ref{sub:grIter})
-% et d'interactions (section~\ref{sub:sdd:inter}). 
-% Elle se termine en définissant une distance sur l'espace 
-% $\llbracket 1;n\rrbracket^{\Nats}\times \Bool^n$ (section~\ref{sub:metric}).
-
 
 
 
diff --git a/stegoyousra.tex b/stegoyousra.tex
index efd326d..cfda20f 100644
--- a/stegoyousra.tex
+++ b/stegoyousra.tex
@@ -1,6 +1,6 @@
 La plupart des schémas de stéganographie sont conçus de sorte  à minimiser une 
-fonction de distortion. Dans les exemples du chapitre précédent, 
-ces fonctions de distortion sont construites dans l'obejctif de préserver 
+fonction de distorsion. Dans les exemples du chapitre précédent, 
+ces fonctions de distorsion sont construites dans l'objectif de préserver 
 les caractéristiques de l'images. 
 On comprend aisément que dans des régions uniformes ou sur des bords clairement définis,
 une modification même mineure de l'image est facilement détectable.
@@ -10,7 +10,7 @@ dont ces zones ont été modifiées sont ainsi similaires à celles
 des images initiales.
 
 Ces régions sont caractérisées par des courbes de niveau très perturbées.
-Ce chapitre présente une nouvelle fonction de distortion pour la stéganographie 
+Ce chapitre présente une nouvelle fonction de distorsion pour la stéganographie 
 qui est basée sur les dérivées du second ordre, l'outil mathématique usuel 
 pour les courbes de niveau.
 
@@ -27,11 +27,11 @@ de pixels. Ordonner alors les pixels selon la matrice hessienne
 contient de nombreuses valeurs pour un seul pixel donné. 
 
 On verra dans ce chapitre comment des approximations des dérivées 
-premières et secondes pour des images numriques (Section~\ref{sec:gradient}) on peu être
+premières et secondes pour des images numériques (Section~\ref{sec:gradient}) on peu être
 obtenues.
 Deux propositions de dérivées secondes sont ensuite 
 données et prouvées (Section~\ref{sec:second} et Section~\ref{sec:poly}).
-Une adaptation d'une fonction de distortion existante est étudiée
+Une adaptation d'une fonction de distorsion existante est étudiée
 en Section~\ref{sec:distortion} et des expériences sont présentées 
 en Section~\ref{sec:experiments}.
 
@@ -48,10 +48,10 @@ de noyaux de la théorie du signal.
 On considère qu'une image peut être assimilée à une fonction de $\R^+\times \R^+$
 dans $\R^+$ telle que la valeur $P(x,y)$ est associée à chaque  pixel de coordonnées $(x,y)$.
 Les variations d'une telle fonction en $(x_0,y_0)$ peuvent être évaluées 
-grace au gradient 
+grâce au gradient 
 $\nabla{P}(x_0,y_0) = \left(\frac{\partial P}{\partial x}(x_0,y_0),\frac{\partial P}{\partial y}(x_0,y_0)\right).
 $
-Le vecteur gradient pointe dans la direction où la fonction a le plus fort acroissement.
+Le vecteur gradient pointe dans la direction où la fonction a le plus fort accroissement.
 Des pixels ayant des valeurs voisines sont sur des lignes de niveaux qui sont orthogonales
 à ce vecteur.
 
@@ -67,7 +67,7 @@ H = \begin{bmatrix}
 \]
 
 En un pixel $(x_0,y_0)$, plus les valeurs de cette matrice sont éloignées de zéro,
-plus le gradient varie en ce point. Evaluer ce type de matrice est ainsi primordial
+plus le gradient varie en ce point. Évaluer ce type de matrice est ainsi primordial
 en stéganographie. Cependant cette tâche n'est pas aussi triviale qu'elle n'y 
 paraît puisque les images naturelles ne sont pas  définies à l'aide 
 de fonction différentiables de $\R^+\times \R^+$
@@ -75,7 +75,7 @@ dans $\R^+$. La suite montre comment obtenir des approximations de telles matric
 
 \subsection{Approches classiques pour évaluer le gradient dans des images}\label{sub:class:1}
 Dans ce contexte, les approches les plus utilisées pour évaluer un gradient 
-sont ``Sobel'', ``Prewitt'', ``Différence centrale'' et `` Difference intermediaire''.
+sont ``Sobel'', ``Prewitt'', ``Différence centrale'' et `` Différence intermédiaire''.
 Chacune de ces  approches applique un produit de convolution $*$ entre un noyau $K$
 (rappelé dans le tableau~\ref{table:kernels:usual}) et une fenêtre $A$ de taille 
 $3\times 3$. Le résultat 
@@ -97,7 +97,6 @@ sont le résultat du calcul du gradient sur la matrice $\dfrac{\partial P}{\part
 \caption{Noyaux usuels pour évaluer des gradients d'images\label{table:kernels:usual}
 }
 \centering
-\scriptsize
 \begin{tabular}{|c|c|c|}
     \hline
     Nom&   Sobel & Prewitt \\
@@ -105,8 +104,8 @@ sont le résultat du calcul du gradient sur la matrice $\dfrac{\partial P}{\part
     Noyau & $\textit{Ks}= \begin{bmatrix} -1 & 0 & +1 \\ -2 & 0 & +2 \\ -1 & 0 & +1 \end{bmatrix} $ &
     $\textit{Kp}= \begin{bmatrix} -1 & 0 & +1 \\ -1 & 0 & +1 \\ -1 & 0 & +1 \end{bmatrix} $\\
     \hline\hline
-    Nom & Difference  & Différence  \\
-            & centrale & Intermediaire \\ 
+    Nom & Différence  & Différence  \\
+            & centrale & Intermédiaire \\ 
     \hline
     Noyau & $\textit{Kc}= \begin{bmatrix} 0&0&0 \\ -\dfrac{1}{2} & 0 & +\dfrac{1}{2} \\ 0&0&0 \end{bmatrix} $ &
     $\textit{Ki}= \begin{bmatrix} 0&0&0 \\ 0 & -1 & 1 \\ 0&0&0 \end{bmatrix} $ \\
@@ -122,29 +121,21 @@ de gradient d'images}\label{sub:class:2}
 Il est connu que  
 $\dfrac{\partial^2 P}{\partial x \partial y} $ est égal à 
 $\dfrac{\partial^2 P}{\partial y \partial x}$ si 
-les méthode qui calculent le grandient et le gradient du gradient (la matrice hessienne)
+les méthode qui calculent le gradient et le gradient du gradient (la matrice hessienne)
 sont les mêmes.
-
-For instance, in the Sobel approach, 
-it is easy to verify that the calculus of 
-$\dfrac{\partial^2 P}{\partial x \partial y}$ 
-and of 
-$\dfrac{\partial^2 P}{\partial y \partial x}$ 
-are both the result 
-of a convolution product with the Kernel 
-$\textit{Ks}''_{xy}$ given in Table~\ref{table:hessian:usual}.
-This one summarizes kernels
-$K_{x^2}''$ and 
+Le tableau~\ref{table:hessian:usual} résume les les noyaux 
+$K_{x^2}''$ et
 $K_{xy}''$  
-that allow to respectively compute  
-$\dfrac{\partial^2 P}{\partial x^2}$ and
-$\dfrac{\partial^2 P}{\partial x \partial y}$ with a convolution product
-for each of the usual image gradient operator.
+qui permettent de calculer respectivement 
+$\dfrac{\partial^2 P}{\partial x^2}$ et 
+$\dfrac{\partial^2 P}{\partial x \partial y}$ comme un produit de convolution 
+pour chacun des opérateurs de gradient rappelés à la section précédente.
+
 
 \begin{table}[ht]
-\caption{Kernels of second order gradient operators\label{table:hessian:usual}
+\caption{Noyaux usuels pour évaluer des gradients de second ordre d'images
+\label{table:hessian:usual}
 }\centering
-\tiny
 \begin{tabular}{|c|c|}
     \hline
     Sobel & Prewitt \\
@@ -195,8 +186,8 @@ for each of the usual image gradient operator.
     $ \\
     
     \hline\hline
-    Central & Intermediate \\
-    Difference &Difference \\ 
+    Différence & Différence  \\
+    centrale  &intermédiaire \\ 
     \hline
     $
     \textit{Kc}_{x^2}''= 
@@ -243,45 +234,34 @@ for each of the usual image gradient operator.
 
 \end{table}
 
-The Sobel kernel $\textit{Ks}_{x^2}''$ allows to detect whether the central 
-pixel belongs to a ``vertical'' edge, even if this one is noisy, by considering its 
-vertical neighbours. The introduction of these vertical neighbours in this kernel is meaningful 
-in the context of finding edges, but not very accurate when the objective is to 
-precisely find the level curves of the image. 
-Moreover, all the pixels that are in the  second and the fourth column in this kernel
-are ignored.
-The Prewitt Kernel has similar drawbacks in this context.
-
-
-The Central Difference kernel $\textit{Kc}_{x^2}''$ is not influenced by 
-the vertical neighbours of the central pixel and is thus more accurate here.
-However, the kernel $\textit{Kc}_{xy}''$ again looses the values of the pixels that 
-are vertically and diagonally aligned with the central one.
-
-Finally, the Intermediate Difference kernel $\textit{Ki}_{x^2}''$ shifts 
-to the left the value of horizontal variations of $\dfrac{\partial P}{\partial x}$:
-the central pixel $(0,0)$ exactly receives the value
+Le noyau $\textit{Ks}_{x^2}''$ permet de détecter si le le pixel central 
+pixel central appartient à une bord ``vertical'', même si celui contient du bruit,
+en considérant ces voisins verticaux. Ces derniers sont vraiment 
+pertinents dans un objectif de détecter les bords. Cependant, leur lien avec 
+les lignes de niveau n'est pas direct. De plus tous les pixels qui sont dans la 
+deuxième et la quatrième colonne de ce noyau sont ignorés.
+Le noyau de Prewitt a des propriétés similaires.
+Le noyau de différence centrale $\textit{Kc}_{x^2}''$ n'est pas influencé par les 
+voisins verticaux du pixel central et peu paraître plus adapté ici.
+Cependant, le noyau $\textit{Kc}_{xy}''$ perd aussi les valeurs des pixels 
+qui sont alignés verticalement et diagonalement avec le pixel central.
+Enfin, le noyau de différence intermédiaire  $\textit{Ki}_{x^2}''$ décale
+à gauche la valeur des variations horizontales de $\dfrac{\partial P}{\partial x}$:
+Le pixel central  $(0,0)$ reçoit  exactement la valeur 
 $\dfrac{P(0,2)-P(0,1)}{1} - \dfrac{P(0,1)-P(0,0)}{1}$,
-which is an approximation of 
-$\dfrac{\partial P}{\partial x}(0,1)$ and not of 
+qui est une approximation de 
+$\dfrac{\partial P}{\partial x}(0,1)$ et non de 
 $\dfrac{\partial P}{\partial x}(0,0)$.
-Furthermore the Intermediate Difference kernel $\textit{Ki}_{xy}''$ only deals 
-with pixels in the upper right corner, loosing all the other information.
-
-Due to these drawbacks, we are then left to produce another approach to find the level curves with strong accuracy.
-
-\section{Second Order Kernels for Accurate Level Curves}\label{sec:second}
-
-This step aims at finding 
-accurate level curve variations in an image. 
-We do not restrict the kernel to have a fixed size (\textit{e.g.}, $3\times3$ or $5 \times 5$ as in the 
-aforementioned schemes).
-This step is thus defined with kernels of size 
-$(2n+1)\times (2n+1)$, $n \in \{1,2,\dots,N\}$, where 
-$N$ is a parameter of the approach.
-
-The horizontal gradient variations are thus captured thanks to $(2n+1)\times (2n+1)$ square kernels
-\begin{small}
+De plus, le noyau de différence intermédiaire $\textit{Ki}_{xy}''$ ne concerne 
+que les pixels du coin supérieur droit, en perdant toutes les autres informations.
+La section suivante propose une autre approche pour calculer les lignes de niveau avec une précision accrue.
+
+\section{Des noyaux pour des lignes de niveau}\label{sec:second}
+On ne restreint pas aux noyaux de taille fixe (comme  $3\times3$ or $5 \times 5$ 
+dans les schémas précédents). Au contraire, on considère des noyaux de taille variable  
+$(2n+1)\times (2n+1)$, $n \in \{1,2,\dots,N\}$, où
+$N$ est un paramètre de l'approche.
+Les variations horizontales du gradient sont extraites grâce au noyau de taille $(2n+1)\times (2n+1)$: 
 \[
 \arraycolsep=1.4pt
 \def\arraystretch{1.4}
@@ -299,24 +279,22 @@ The horizontal gradient variations are thus captured thanks to $(2n+1)\times (2n
     \end{array}
     \right)
 \]
-\end{small}
 
-When the convolution product is applied on a $(2n+1)\times(2n+1)$ window,
-the result is 
-$\dfrac{1}{2}\left(\dfrac{P(0,n)-P(0,0)}{n} - \dfrac{P(0,0)-P(0,-n)}{n}\right)$, which is indeed 
-the variation between the gradient around the central pixel. 
-This proves that this calculus is a correct approximation of 
+Lorsque le produit de convolution est appliqué sur une fenêtre $(2n+1)\times(2n+1)$,
+le résultat est 
+$\dfrac{1}{2}\left(\dfrac{P(0,n)-P(0,0)}{n} - \dfrac{P(0,0)-P(0,-n)}{n}\right)$, 
+qui représente en effet les variation horizontales de la partie horizontale 
+du gradient autour du pixel central. On obtient donc bien une approximation de 
 $\dfrac{\partial^2 P}{\partial x^2}$.
+Lorsque $n$ vaut 1, ce noyau est une version centrée du noyau horizontal de différence 
+intermédiaires. $\textit{Ki}_{x^2}''$ à un facteur  $1/2$ près).
+Lorsque $n$ vaut 2, on retrouve $\textit{Kc}_{x^2}''$.
 
-When $n$ is 1, this kernel is a centered version of the horizontal Intermediate Difference kernel $\textit{Ki}_{x^2}''$ modulo a multiplication by $1/2$. When $n$ is 2, this kernel is equal to $\textit{Kc}_{x^2}''$.
+Les variations verticales du gradient sont aussi obtenus en faisant subir 
+à $\textit{Ky}_{x^2}''$ une rotation d'angle  $\pi/2$.
+Les variations diagonales sont obtenues à l'aide du gradient 
+$\textit{Ky}_{xy}''$ défini par:
 
-The vertical gradient variations are again obtained by applying 
-a $\pi/2$ rotation to each horizontal kernel $\textit{Ky}_{x^2}''$.
-
-The diagonal gradient variations are obtained thanks to the $(2n+1)\times (2n+1)$ square kernels 
-$\textit{Ky}_{xy}''$ defined by
-
-\begin{small}
 \[
 \arraycolsep=1.4pt
 \def\arraystretch{1.4}
@@ -358,79 +336,24 @@ $\textit{Ky}_{xy}''$ defined by
     \end{array}
     \right).
 \]
-\end{small}
 
-When $n$ is 1, $\textit{Ky}_{xy}''$ is equal to the kernel
-$\textit{Kc}_{xy}''$, and %.
-%
-%When $n$ is 1, 
-the average vertical variations of the horizontal variations are 
-\[
-\begin{array}{l}
-\dfrac{1}{4}
-\left[ 
-\left((P(0,1)-P(0,0))-(P(1,1)-P(1,0))\right)+ \right.\\
-\quad \left((P(-1,1)-P(-1,0))-(P(0,1)-P(0,0))\right)+\\
-\quad \left((P(0,0)-P(0,-1))-(P(1,0)-P(1,-1))\right)+\\
-\quad \left. \left((P(-1,0)-P(-1,-1))-(P(0,0)-P(0,-1))\right)
-\right]  \\
-=\dfrac{1}{4}
-\left[ P(1,-1) -P(1,1) - P(-1,-1) + P(-1,1)\right].
-\end{array}
-\]
-which is $\textit{Ky}_{xy}''$.
-
-Let us now consider any number $n$, $1 \le n \le N$.
-Let us first investigate the vertical variations related to 
-the horizontal vector $\vec{P_{0,0}P_{0,1}}$ 
-(respectively  $\vec{P_{0,-1}P_{0,0}}$)
-of length 1 that starts from (resp. that points to) $(0,0)$. 
-As with the case $n=1$, there are 2 new vectors of 
-length 1, namely 
-$\vec{P_{n,0}P_{n,1}}$ and $\vec{P_{-n,0}P_{-n,1}}$ 
-(resp. 
-$\vec{P_{n,-1}P_{n,0}}$, and $\vec{P_{-n,-1}P_{-n,0}}$)
-that are vertically aligned with $\vec{P_{0,0}P_{0,1}}$ 
-(resp. with $\vec{P_{0,-1}P_{0,0}}$).
-
-The vertical variation is now equal to $n$. Following the case where $n$ is 1 to compute the average variation, 
-the coefficients of the first and last line around the central 
-vertical line are thus from left to right:
-$\dfrac{1}{4n}$,
-$\dfrac{-1}{4n}$,
-$\dfrac{-1}{4n}$, and
-$\dfrac{1}{4n}$.
-
-Cases are similar with vectors $\vec{P_{0,0}P_{0,1}}$, \ldots  
-$\vec{P_{0,0}P_{0,n}}$ which respectively lead to coefficients 
-$-\dfrac{1}{4 \times 2n}$, \ldots, 
-$-\dfrac{1}{4 \times n.n}$, and the proof is omitted.
-Finally, let us consider the vector $\vec{P_{0,0}P_{0,1}}$
-and its vertical variations when $\delta y$ is $n-1$.
-As in the case where $n=1$, we thus obtain the coefficients 
-$\dfrac{1}{4 \times (n-1)n}$ and 
-$-\dfrac{1}{4 \times (n-1)n}$ 
- (resp. $-\dfrac{1}{4 \times (n-1)n}$ and 
-$\dfrac{1}{4 \times (n-1)n}$)
-in the second line (resp. in the 
-penultimate line) since the vector has length $n$
-and $\delta y$ is $n-1$.
-Coefficient in the other lines are similarly obtained and the proof is thus omitted.
-
-We are then left to compute an approximation of the partial second order derivatives 
-$\dfrac{\partial^2 P}{\partial x^2}$,  $\dfrac{\partial^2 P}{\partial y^2}$, and $\dfrac{\partial^2 P}{\partial x \partial y}$
-with the kernels, 
-$\textit{Ky}_{x^2}''$, $\textit{Ky}_{y^2}''$, and $\textit{Ky}_{xy}''$ respectively.
-However, the size of each of these kernels is varying from $3\times3$ to $(2N+1)\times (2N+1)$.
-Let us explain the approach on the former partial derivative.
-The other can be immediately deduced.
-
-Since the objective is to detect large variations, the second order derivative is approximated as 
-the maximum of the approximations. More formally, 
-let $n$, $1 \le n \le N$, be an integer number and 
-$\dfrac{\partial^2 P}{\partial x^2}_n$ be the result of applying the Kernel 
-$\textit{Ky}_{x^2}''$ of size $(2n+1)\times (2n+1)$. The derivative 
-$\dfrac{\partial^2 P}{\partial x^2}$ is defined by
+
+
+En effet, lorsque  $n$ vaut 1, $\textit{Ky}_{xy}''$ se retrouve en calculant la moyenne
+des variations horizontales de la composante verticale du gradient calculé à l'aide de 
+$\textit{Ky}_{y}'$. Pour cette valeur de $n$, on a
+$\textit{Ky}_{xy}'' =   \textit{Kc}_{xy}''$. 
+Pour chaque nombre  $n$, $1 <  n \le N$, $\textit{Ky}_{xy}''$ se retrouve de la même
+manière, c'est-à-dire en effectuant des moyennes de variations. 
+Une preuve de la construction se trouve dans l'article~\cite{ccfg16:ip}.
+
+
+
+L'objectif est de détecter les grandes variations des dérivées premières. 
+Ainsi les dérivées secondes seront approximées comme les maximums des 
+matrices hessiennes  obtenues lorsque $n$ varie entre  $1$ et $ N$.
+
+La dérivée partielle $\dfrac{\partial^2 P}{\partial x^2}$ est définie par 
 
 \begin{equation}
 \dfrac{\partial^2 P}{\partial x^2}
@@ -439,28 +362,30 @@ $\dfrac{\partial^2 P}{\partial x^2}$ is defined by
 \right \}. 
 \label{eq:d2p_dx2}
 \end{equation}
-
-The same iterative approach is applied to compute approximations of
+où $\dfrac{\partial^2 P}{\partial x^2}_n$ est le résultat de l'application 
+du noyau $\textit{Ky}_{x^2}''$ de taille $(2n+1)\times (2n+1)$.
+La même approche itérative est appliquée pour construire les 
+approximations de 
 $\dfrac{\partial^2 P}{\partial y \partial x}$ 
-and of 
+et de 
 $\dfrac{\partial^2 P}{\partial y^2}$.
-Next section studies the suitability of approximating second order derivatives
-when considering an image as a polynomial.
 
+La section suivante étudie la pertinence d'interpoler une image par un polynome 
+lorsqu'on cherche a obtenir ces dérivées secondes.
 
-\section{Polynomial Interpolation of Images for Hessian Matrix Computation}\label{sec:poly}
-Let $P(x,y)$ be the discrete value of the pixel $(x,y)$ in the image.
-Let $n$, $1 \le n \le N$, be an integer such that the objective is to find a polynomial interpolation 
-on the $(2n+1)\times(2n+1)$ window where the central pixel has index $(0,0)$.  
-There exists an unique polynomial $L : \R\times \R \to \R$ of degree $(2n+1)\times(2n+1)$ defined  
-such that $L(x,y)=P(x,y)$ for each pixel $(x,y)$ in this window.
-Such a polynomial is defined by 
+
+\section{Interpolation polynomiale pour le calcul de la matrice hessienne}\label{sec:poly}
+Soit $P(x,y)$ la  valeur du pixel $(x,y)$ et soit  $n$, $1 \le n \le N$, 
+tel que l'objectif est de trouver un polynome d'interpolation dans la fenêtre de taille  
+$(2n+1)\times(2n+1)$ dont le pixel central a pour indice $(0,0)$.  
+Il existe un unique polynôme $L : \R\times \R \to \R$ 
+de degré $(2n+1)\times(2n+1)$ tel que  $L(x,y)=P(x,y)$ pour chaque pixel  
+$(x,y)$ de cette fenêtre et ce polynome est défini par  
 \begin{equation}
 \begin{array}{l}
 L(x,y) =  
 \sum_{i=-n}^{n}
-\sum_{j=-n}^{n} \\
-\quad P(i,j)
+\sum_{j=-n}^{n}P(i,j)
 \left( 
 \prod_{\stackrel{-n\leq j'\leq n}{j'\neq j}}
 \frac{x-j'}{i-j'}
@@ -471,9 +396,7 @@ L(x,y) =
 \right)
 \end{array}
 \end{equation}
-
-It is not hard to prove that the first order horizontal derivative of the polynomial $L(x,y)$ 
-is
+On peut facilement prouver que les dérivées partielles  de $L$ selon $x$ est 
 \begin{equation}
 \begin{array}{l}
 \dfrac{\partial L}{\partial x} =  
@@ -483,8 +406,7 @@ P(i,j)
 \left( 
 \prod_{\stackrel{-n\leq j'\leq n}{j'\neq j}}
 \frac{y-j'}{j-j'}
-\right)\\
-\quad
+\right)
 \left( 
 \sum_{\stackrel{-n\leq i'\leq n}{i'\neq i}}
 \frac{1}{i-i'}
@@ -493,20 +415,17 @@ P(i,j)
 \right)
 \end{array}
 \end{equation}
-\noindent and thus to deduce that the
-second order ones are
-
-\begin{equation}
-\begin{array}{l}
-\dfrac{\partial^2 L}{\partial x^2} =  
+\noindent et ainsi en déduire que les dérivées partielles de second ordre 
+sont 
+\begin{eqnarray}
+\dfrac{\partial^2 L}{\partial x^2} &=&  
 \sum_{i=-n}^{n}
 \sum_{j=-n}^{n} 
 P(i,j)
 \left( 
 \prod_{\stackrel{-n\leq j'\leq n}{j'\neq j}}
 \frac{y-j'}{j-j'}
-\right)\\
-\quad
+\right) 
 \left( 
 \sum_{\stackrel{-n\leq i'\leq n}{i'\neq i}}
 \frac{1}{i-i'}
@@ -515,74 +434,33 @@ P(i,j)
 \prod_{\stackrel{-n\leq i'''\leq n}{i'''\neq i,i',i''}}
 \frac{x-i'''}{i-i'''}
 \right)
-\end{array}
-\label{eq:deriv:poly:x2}
-\end{equation}
-
-\begin{equation}
-\begin{array}{l}
-\dfrac{\partial^2 L}{\partial y \partial x} =  
+\label{eq:deriv:poly:x2} \\
+\dfrac{\partial^2 L}{\partial y \partial x} &= & 
 \sum_{i=-n}^{n}
-P(i,j) \\
-\quad
+P(i,j) 
 \left( 
 \sum_{\stackrel{-n\leq j'\leq n}{j'\neq j}}
 \frac{1}{j-j'}
 \prod_{\stackrel{-n\leq j''\leq n}{j''\neq j, j'}}
 \frac{y-j''}{j-j''}
-\right)\\
-\quad
+\right)
 \left( 
 \sum_{\stackrel{-n\leq i'\leq n}{i'\neq i}}
 \frac{1}{i-i'}
 \prod_{\stackrel{-n\leq i''\leq n}{i''\neq i, i'}}
 \frac{x-i''}{i-i''}
 \right)
-\end{array}
 \label{eq:deriv:poly:yx}
-\end{equation}
-
-These second order derivatives  are computed for each moving 
-window and are associated to the central pixel, \textit{i.e.}, to the pixel $(0,0)$ inside this one.
-
-Let us first simplify $\dfrac{\partial^2 L}{\partial x^2}$ when $(x,y)=(0,0)$
-defined in Equation~(\ref{eq:deriv:poly:x2}). If $j$ is not null, the index $j'$
-is going to be null and the product  
-$\left( 
-\prod_{\stackrel{-n\leq j'\leq n}{j'\neq j}}
-\frac{-j'}{j-j'}
-\right)$ is null too. 
-In this equation, we thus only consider $j=0$.
-It is obvious that the product indexed with $j'$ is thus equal to 1.
-This equation can thus be simplified in:
-
-\begin{equation}
-\begin{array}{l}
-\dfrac{\partial^2 L}{\partial x^2} =  
-\sum_{i=-n}^{n}
-P(i,0)\\
-\quad
-\left( 
-\sum_{\stackrel{-n\leq i'\leq n}{i'\neq i}}
-\frac{1}{i-i'}
-\sum_{\stackrel{-n\leq i''\leq n}{i''\neq i,i'}}
-\frac{1}{i-i''}
-\prod_{\stackrel{-n\leq i'''\leq n}{i'''\neq i,i',i''}}
-\frac{i'''}{i'''-i}
-\right)
-\end{array}
-\label{eq:deriv:poly:x2:simpl}
-\end{equation}
-
-
-and then in:
+\end{eqnarray}
+Ces dérivées secondes sont calculées pour chaque pixel central, \textit{i.e.} le pixel dont l'indice est  $(0,0)$ dans la fenêtre.
+En considérant cette particularisation, l'équation~(\ref{eq:deriv:poly:x2}) peut 
+se simplifie en 
 
 \begin{equation}
 \begin{array}{l}
 \dfrac{\partial^2 L}{\partial x^2} =  
 \sum_{i=-n}^{n}
-P(i,0)\\
-\quad
+P(i,0)
 \left( 
 \sum_{\stackrel{-n\leq i' < i'' \le n}{i',i''\neq i}}
 \frac{2}{(i-i')(i-i'')}
@@ -593,19 +471,9 @@ P(i,0)\\
 \label{eq:deriv:poly:x2:simpl:2}
 \end{equation}
 
-From this equation, the kernel allowing to evaluate horizontal 
-second order derivatives 
-can be computed for any $n$.
-It is further denoted as $Ko''_{x^2}$. 
-Instances of such matrix when $n=2$, $3$, and $4$
-are given in Table~\ref{table:sod:hori:poly}.
-
 \begin{table}[ht]
-    \caption{Kernels $Ko''_{x^2}$ 
-    for second order horizontal derivatives induced 
-    by polynomial interpolation}
+    \caption{Noyaux $Ko''_{x^2}$ pour  calculer des dérivées de second ordre à partir d'interpolation polynomiale}
     \centering
-    \scriptsize
 \def\arraystretch{1.4}
     \begin{tabular}{|c|c|}
          \hline
@@ -626,11 +494,10 @@ are given in Table~\ref{table:sod:hori:poly}.
 
 
 \begin{table}[ht]
-\caption{Kernels for second order diagonal derivatives induced 
-    by polynomial interpolation \label{table:sod:diag:poly}
+\caption{ Noyaux pour les dérivéees secondes en $x$ et $y$ lors de l'interpolation polynomiale\label{table:sod:diag:poly}
 }
 \centering
-\scriptsize
+%\scriptsize
 \def\arraystretch{1.5}
 \begin{tabular}{|c|c|}
     \hline
@@ -658,29 +525,26 @@ $\\
 \end{tabular}
 \end{table}
 
+Cette dérivée partielle peut s'écrire comme un produit de convolution avec un noyau
+noté $Ko''_{x^2}$. Des instances de tels noyaux, pour $n=2$, $3$ et  $4$
+sont données au tableau~\ref{table:sod:hori:poly}. De manière similaire,  
+le tableau~\ref{table:sod:diag:poly} donne deux exemples pour $n=1$ et $n=2$ 
+de noyaux $Ko''_{xy}$ permettant de calculer directement les dérivées 
+de second ordre selon $x$ et $y$ en $(0,0)$.
+On remarque que pour $n=1$, le noyau est égal à $Kc''_{xy}$. 
 
-From Equation~(\ref{eq:deriv:poly:yx}), kernels allowing to evaluate diagonal 
-second order derivatives (\textit{i.e.}, 
-$\dfrac{\partial^2 L}{\partial y \partial x}$)  are computed.
-They are denoted as $Ko''_{xy}$.
-Table~\ref{table:sod:diag:poly} gives two examples of them when $n=1$ and $n=2$.
-Notice that for $n=1$, the kernel $Ko''_{xy}$ is equal to $Kc''_{xy}$. 
 
 
 
 
 
 
-\section{Distortion Cost}\label{sec:distortion}
-
-The distortion function has to associate to each pixel $(i,j)$ 
-the cost $\rho_{ij}$ of its modification
-by $\pm 1$.
-
-The objective is to map a small value to a pixel when all its second order derivatives 
-are high and a large value otherwise. 
-%\YF{The WOW  use the  wavelet relative distortion (UNIWARD) as a function to force the embedding operation to the noisy areas and avert the smooth areas.}
-In WOW and UNIWARD  the distortion function is based on the H\"older norm with
+\section{Fonction de distorsion}\label{sec:distortion}
+Une fonction de distorsion associe à chaque pixel  $(i,j)$ 
+le coût $\rho_{ij}$ du modification  par $\pm 1$. L'objectif est d'associer une 
+valeur faible aux pixels dont toutes les dérivées secondes sont éloignées de 0 
+et une valeur rédhibitoire sinon.
+Dans WOW comme dans  UNIWARD la fonction de distorsion est définie par 
 \[
 \rho_{ij}^w = 
 \left(
@@ -689,119 +553,79 @@ In WOW and UNIWARD  the distortion function is based on the H\"older norm with
 \abs{\xi_{ij}^d}^{p} 
 \right)^{-\frac{1}{p}}
 \]
-where $p$ is a negative number and 
-$\xi_{ij}^h$ (resp. $\xi_{ij}^v$ and $\xi_{ij}^d$)
-represents the horizontal (resp. vertical and diagonal) suitability.
-A small suitability in one direction means an inaccurate position to embed a message.
-
-We propose here to adapt such a distortion cost as follows:
+où $p$ est un nombre négatif et  
+$\xi_{ij}^h$ (resp. $\xi_{ij}^v$ et $\xi_{ij}^d$)
+représentent la pertinence horizontale (resp. verticale et diagonale) de modification.
+Une faible pertinence dans une direction signifie que l'embarquement 
+dans ce pixel est inapproprié.
+La fonction de distorsion que l'on a retenu est une particularisation ($p=-1$) 
+de cette dernière:
 \[
 \rho_{ij} = 
 \left(
-\abs{\dfrac{\partial^2 P}{\partial x^2}(i,j)} +
-\abs{\dfrac{\partial^2 P}{\partial y^2}(i,j)} +
-\abs{\dfrac{\partial^2 P}{\partial y \partial x}(i,j)}
-\right)^{-\frac{1}{p}}
+\abs{\dfrac{\partial^2 P}{\partial x^2}(i,j)}^{-1} +
+\abs{\dfrac{\partial^2 P}{\partial y^2}(i,j)}^{-1} +
+\abs{\dfrac{\partial^2 P}{\partial y \partial x}(i,j)}^{-1}
+\right)
 \]
-It is not hard to check that such a function has large 
-value when at least one of its derivatives is null. Otherwise,
-the larger the derivatives are, the smaller the returned value
-is.
-
-
-
-
-%Practically, it is computed as a double convolution product with Daubechies Wavelet Kernels. 
-
-%It is not hard to see that this approach 
-%does not consider the range of the values. For %instance, if one direction only has large suitability
-%absolute values, all the elements $\abs{\xi_{ij}}^{p}$ are so small, and 
-%suitable pixels cannot be distinguished from inaccurate ones. The distortion function has to 
-%provide a way to discriminate pixels even when the distribution has small deviation
 
-%Because of each pixel value ranges in $[0,255]$, each first order derivative belongs to
-%$[-255,255]$ and thus each second order derivative has a real value in 
-%$[-255,255]$.  
-%Due to the definition of the approximation of $\dfrac{\partial^2 P}{\partial x^2}$, 
-%this one is a positive or null real number lower than 255. 
 
-%The higher the second order derivatives are, 
-%the 
+\section{Experimentations}\label{sec:experiments}
 
-%To enlarge the discrimination power, 
-%data are firstly linearly mapped to $[0,1]$. 
-%The tuple 
-%$(\dfrac{\partial^2 P}{\partial x^2} , \dfrac{\partial^2 P}{\partial y^2}, \dfrac{\partial^2 %P}{\partial x \partial y})$ belongs to $[0,1]\times[0,1]\times[0,1] = [0,1]^3$.
+Tout d'abord, l'ensemble du code est accessible en ligne\footnote{\url{https://github.com/stego-content/SOS}}.
+La Figure~\ref{fig:oneimage} représente les résultats d'embarquement de données dans 
+l'image 38 du challenge BOSS~\cite{Boss10} en suivant les deux schémas basés 
+sur les dérivées secondes présentés dans ce chapitre.
+Le taux d'embarquement $\alpha$ est fixé à 0.4 bits par pixel et les noyaux sont 
+construits avec $N=4$. On remarque bien que les pixels dans les zones uniformes 
+et les pixels dans les bords bien définis ne sont pas modifiés par l'approche tandis 
+qu'au contraire les zones peu prévisibles (le monument par exemple) 
+concentrent les changements.
 
-%Basically, for each dimension the idea is to select a subset of pixels of size $k$ and next to retain those which have been selected three times, \textit{i.e.}, those with large values 
-%in any element of 
-%$(\dfrac{\partial^2 P}{\partial x^2} , \dfrac{\partial^2 P}{\partial y^2}, \dfrac{\partial^2 %P}{\partial x \partial y})$.
-%Since the dimension is $3$, the size $k$ can be defined by
-%\[
-%k = W \times H \times \alpha ^{\dfrac{1}{3}}
-%\]
-%where $W$ and $H$ respectively stand for the width and the length of the image (expressed in pixels)
-%and $\alpha$ is the payload. For instance if $\alpha$ is 0.1, 
-%about $0.46\%$ of the image pixels with higher values in $\dfrac{\partial^2 P}{\partial x^2}$ should be selected, and similarly with $\dfrac{\partial^2 P}{\partial y^2}$ and $\dfrac{\partial^2  P}{\partial x \partial y}$.
 
 
-% Dire comment on les ordonne, les autres et le sens
 
-\begin{figure*}
+\begin{figure}[ht]
     \centering
     \begin{tabular}{|c|c|c|}
     \hline
-    Scheme & Stego. content & Changes with cover \\
+    Schéma & Image. Stego. & Différence avec le support \\
     \hline
-    $Ky$ based approach &\includegraphics[scale=0.20]{images/38_dp}&
+    Approche à base de $Ky$  &\includegraphics[scale=0.20]{images/38_dp}&
     \includegraphics[scale=0.20]{images/38_dp_diff} \\
     \hline
-    $Ko$ based approach & \includegraphics[scale=0.20]{images/38_poly} &
+    Approche à base de  $Ko$  & \includegraphics[scale=0.20]{images/38_poly} &
     \includegraphics[scale=0.20]{images/38_poly_diff} \\
     \hline
     \end{tabular}
-    \caption{Embedding changes instance with payload  $\alpha = 0.4$}
+    \caption{Exemple de changements dus à un embarquement avec  $\alpha = 0.4$}
     \label{fig:oneimage}
-\end{figure*}
-
-
-\section{Experiments}\label{sec:experiments}
-
-First of all, the whole steganographic approach code is available online\footnote{\url{https://github.com/stego-content/SOS}}.
-
-Figure~\ref{fig:oneimage} presents the results of embedding data in a cover image
-from the BOSS contest database~\cite{Boss10} with respect to the two second order derivative schemes presented in this work. 
-The $Ky$ based approach (resp. the $Ko$ based one)
-corresponds to the scheme detailed in Section~\ref{sec:second} 
-(resp. in Section~\ref{sec:poly}).
-The payload $\alpha$ is set to 0.4 and kernels are computed with $N=4$.    
-The central column outputs the embedding result whereas the right one displays 
-differences between the cover image and the stego one.
-It can be observed that pixels in smooth area (the sky, the external access steps) and pixels in clean edges (the columns, 
-the step borders) are not modified by the approach.
-On the contrary, an unpredictable area (a monument for example) concentrates pixel changes.
-
-
-\subsection{Choice of parameters}
-
-The two methods proposed in Section~\ref{sec:second} and  
-in Section~\ref{sec:poly} are based on kernels of size up to 
-$(2N+1)\times(2N+1)$. This section aims at finding the 
-value of the $N$ parameter that maximizes the security level.
-For each approach, we have built 1,000 stego images with 
-$N=2$, $4$, $6$, $8$, $10$, $12$, and $14$ where the covers belong 
-to the BOSS contest database. 
-This set contains 10,000 grayscale $512\times 512$ images in a RAW format. 
-The security of the approach has been evaluated thanks to the Ensemble Classifier~\cite{DBLP:journals/tifs/KodovskyFH12} based steganalyser,
-which is considered as a state of the art steganalyser tool.
-This steganalysis process embeds the rich model (SRM) features~\cite{DBLP:journals/tifs/FridrichK12}
-of size 34,671. 
-For a payload $\alpha$, either equal to $0.1$ or to $0.4$, average testing errors (expressed in percentages) have been studied and are summarized in
-Table~\ref{table:choice:parameter}.
+\end{figure}
+
+
+
+
+
+\subsection{Choix des paramètres}
+
+Les deux méthodes présentées ici dépendent de noyaux dont la taille va jusqu'à  
+$(2N+1)\times(2N+1)$. Cette section montre comment évaluer $N$ pour maximiser 
+le niveau de sécurité.
+Pour chaque approche, 1,000 images stegos avec  
+$N=2$, $4$, $6$, $8$, $10$, $12$ et $14$ et dont les supports appartiennent 
+à l'ensemble des 10000 images du challenge BOSS. 
+LA sécurité de l'approche a été évaluée avec le stéganalyseur 
+Ensemble Classifier~\cite{DBLP:journals/tifs/KodovskyFH12}.
+Pour un taux d'embarquement   $\alpha$ égal soit à  $0.1$ ou soit à  $0.4$, 
+l'erreur moyenne de test (exprimée en pourcentage) a été calculée. 
+Le tableau~\ref{table:choice:parameter} synthétise les résultats.
+On observe que la taille $N=4$ (respectivement $N=12$) 
+permet d'obtenir des erreurs suffisamment élevées pour l'approche basée sur $Ky$  
+(resp. pour celle basée sur  $Ko$). 
+Ces deux valeurs de paramètre sont retenues par la suite.
 
 \begin{table}[ht]
-\caption{Average Testing Errors with respect to the the Kernel Size}
-\begin{scriptsize}
+\caption{Erreur moyenne de test en fonction de la taille du noyau}
 \centering
 \setlength{\tabcolsep}{3pt} 
 \begin{tabular}{|c|c|c|c|c|c|c|c|c|}
@@ -810,58 +634,48 @@ Table~\ref{table:choice:parameter}.
 \cline{3-9}
 \multicolumn{1}{c|}{}&  & $2$ & $4$&  $6$&  $8$& $10$& $12$ & $14$ \\
 \hline{}
-Average testing  
+Erreur moyenne    
 & \textit{0.1} & 39& 40.2&  39.7&  39.8& 40.1& $39.9$&  $39.8$  \\
 \cline{2-9}
-error for Kernel $K_y$ & \textit{0.4}& 15& 18.8& 19.1&  19.0& 18.6& 18.7 & 18.7 \\
+de test pour le noyau $K_y$ & \textit{0.4}& 15& 18.8& 19.1&  19.0& 18.6& 18.7 & 18.7 \\
 \hline
-Average testing & \textit{0.1} & 35.2 & 36.6&  36.7&  36.6& 37.1& 37.2 & 37.2 \\
+Erreur moyenne    & \textit{0.1} & 35.2 & 36.6&  36.7&  36.6& 37.1& 37.2 & 37.2 \\
 \cline{2-9}
-error for Kernel $K_o$ & \textit{0.4}  & 5.2 & 6.8& 7.5 & 7.9 & 8.1 & 8.2 & 7.6 \\
+de test pour le noyau $K_o$ & \textit{0.4}  & 5.2 & 6.8& 7.5 & 7.9 & 8.1 & 8.2 & 7.6 \\
 \hline
 \end{tabular}
 \label{table:choice:parameter}
-\end{scriptsize}
+
 \end{table}
 
-Thanks to these experiments, we observe that the size $N=4$ (respectively $N=12$) obtains sufficiently  large average testing errors for the $Ky$ based approach 
-(resp. for the $Ko$ based one). In what follows, these values are retained for these two methods.
-
-\subsection{Security Evaluation}
-As in the previous section, the BOSS contest database has been retained.
-To achieve a complete comparison with other steganographic tools,
-the whole database of 10,000 images has been used.
-Ensemble Classifier with SRM features is again used to evaluate the security of the approach.
-
-We have chosen 4 different payloads, 0.1, 0.2, 0.3, and 0.4, as in many steganographic evaluations.
-Three values are systematically given for each experiment:
-the area under the ROC curve (AUC),
-the average testing error (ATE),
-and the OOB error (OOB).
-
-All the results are summarized in Table~\ref{table:experiments:summary}.
-%and  in Figure~\ref{fig1}, Figure~\ref{fig2}, and Figure~\ref{fig3}.
-Let us analyse these experimental results. 
-The security approach is often lower than those observed with state of the art tools:
-for instance with payload $\alpha=0.1$, the most secure approach is WOW 
-with an average testing error equal to 0.43 whereas our approach reaches 0.38.
-However these results are  promising and for two reasons.
-First, our approaches give more resistance towards Ensemble Classifier (contrary to HUGO)
-for large payloads.
-Secondly, without any optimisation, our approach is not so far from state of the art steganographic tools.
-Finally, we explain the lack of security of the $Ko$ based approach with large payloads as follows:
-second order derivatives are indeed directly extracted from polynomial interpolation.
-This easy construction however induces large variations between the polynomial $L$ and 
-the pixel function $P$.
+
+\subsection{Évaluation de la sécurité}
+Comme dans ce qui précède, la base du challenge BOSS a été retenue.
+Ici c'est cependant l'ensemble des 10000 images qui a été utilisé pour évaluer 
+la sécurité.
+C'est aussi les caractéristiques SRM et Ensemble Classifier qui ont été utilisées 
+pour évaluer la sécurité de l'approche..
+Quatre taux d'embarquement  0.1, 0.2, 0.3 et  0.4
+ont été retenus. Pour chaque expérience, 
+l'aire sous la courbe de ROC (AUC),
+l'erreur moyenne de test (ATE), 
+l'erreur OOB (OOB) sont données et tous les résultats sont synthétisés 
+dans le tableau~\ref{table:experiments:summary}.
+Même si la sécurité est souvent plus faible que celle observée 
+pour les outils les plus récents, 
+les résultats concernant $K_y$ sont encourageants car ils ne sont pas éloignés de 
+ceux de l'état de l'art sans aucune optimisation.
+Enfin la faible sécurité de $K_o$ s'explique par le fait que le polynôme interpole 
+exactement l'image en tous les points de la fenêtre, mais il ne tient pas forcément 
+compte des variations dans celle-ci. Les dérivées secondes sont certes faciles 
+à exprimer, mais elles ne représentent pas nécessairement fidèlement celles de l'image.
 
 \begin{table}
-\caption{Summary of experiments}\label{table:experiments:summary}
-\begin{small}
+\caption{Évaluation de la sécurité}\label{table:experiments:summary}
 \centering
 \begin{tabular}{|l|l|l|l|l|}
 \hline
-
- & Payload & AUC & ATE   &  OOB \\ \hline
+ & Taux & AUC & ATE   &  OOB \\ \hline
 {WOW}   
             & 0.1 & 0.6501 & 0.4304 & 0.3974\\
             & 0.2 & 0.7583 & 0.3613 & 0.3169\\
@@ -884,13 +698,13 @@ the pixel function $P$.
             & 0.3 & 0.8720 & 0.2557 & 0.2212 \\
             & 0.4 & 0.9517 & 0.1472 & 0.1230 \\ 
  \hline
-{$Ky$ based approach} 
+{Approche à base de $Ky$} 
             & 0.1 & 0.7378 & 0.3768 & 0.3306 \\
             & 0.2 & 0.8568 & 0.2839 & 0.2408 \\
             & 0.3 & 0.9176 & 0.2156 & 0.1710 \\
             & 0.4 & 0.9473 & 0.1638 & 0.1324\\
  \hline
-{$Ko$ based approach} 
+{Approche à base de $Ko$} 
             & 0.1 & 0.6831 & 0.3696  & 0.3450 \\
             & 0.2 & 0.8524 & 0.1302  & 0.2408 \\
             & 0.3 & 0.9132 & 0.1023  & 0.1045 \\
@@ -898,7 +712,6 @@ the pixel function $P$.
                               \hline 
                               
 \end{tabular}
-\end{small}
 \end{table}
 
 
@@ -923,23 +736,23 @@ the pixel function $P$.
 %\end{figure}
 
 
-\section{Conclusion}
-
-The first contribution of this paper is to propose of a distortion
-function which is based on second order derivatives. These
-partial derivatives allow to accurately compute 
-the level curves and thus to look favorably on pixels
-without clean level curves. 
-Two approaches to build these derivatives have been proposed.
-The first one is based on revisiting kernels usually embedded 
-in edge detection algorithms. 
-The second one is based on the polynomial approximation
-of the bitmap image.
-These two methods have been completely implemented.
-The first experiments have shown that the security level 
-is slightly inferior the one of the most stringent approaches. These first promising results encourage us to deeply investigate this research direction. 
-
-Future works aiming at improving the security of this proposal are planned as follows. The authors want first to focus on other approaches to provide second order derivatives with larger discrimination power.
-Then, the objective will be to deeply investigate whether the H\"older norm is optimal when the objective is to avoid null second order derivatives, and to give priority to the largest second order values.
+% \section{Conclusion}
+
+% The first contribution of this paper is to propose of a distortion
+% function which is based on second order derivatives. These
+% partial derivatives allow to accurately compute 
+% the level curves and thus to look favorably on pixels
+% without clean level curves. 
+% Two approaches to build these derivatives have been proposed.
+% The first one is based on revisiting kernels usually embedded 
+% in edge detection algorithms. 
+% The second one is based on the polynomial approximation
+% of the bitmap image.
+% These two methods have been completely implemented.
+% The first experiments have shown that the security level 
+% is slightly inferior the one of the most stringent approaches. These first promising results encourage us to deeply investigate this research direction. 
+
+% Future works aiming at improving the security of this proposal are planned as follows. The authors want first to focus on other approaches to provide second order derivatives with larger discrimination power.
+% Then, the objective will be to deeply investigate whether the H\"older norm is optimal when the objective is to avoid null second order derivatives, and to give priority to the largest second order values.
 
 
-- 
2.39.5


From a8bd04acf5ce24f5fe4686f33781d6127c7f1f49 Mon Sep 17 00:00:00 2001
From: couchot <couchot@localhost.localdomain>
Date: Mon, 29 Aug 2016 22:28:49 +0200
Subject: [PATCH 10/16] -> jsutqu'a watermarking

---
 main.tex   |  1 -
 oxford.tex | 45 ++++++++++++++++++++++++++++++++-------------
 2 files changed, 32 insertions(+), 14 deletions(-)

diff --git a/main.tex b/main.tex
index c6f86cd..da87f22 100644
--- a/main.tex
+++ b/main.tex
@@ -275,7 +275,6 @@ de telles fonctions.
 
 
 \chapter{Des embarquement préservant le chaos}\label{chap:watermarking} 
-% OXFORD
 \input{oxford}
 
 \chapter{Une démarche de  marquage de PDF}
diff --git a/oxford.tex b/oxford.tex
index 6fbda16..0be2951 100644
--- a/oxford.tex
+++ b/oxford.tex
@@ -1,14 +1,27 @@
-\JFC{Dire que c'est une synthèse du chapitre 22 de la thèse de Tof}
-\JFC{Dire qu'on est d'abord binaire, puisqu'on étend ceci à un message
-récupérable}
+La propriété de régularité des fonctions chaotiques est à l'origine du marquage de documents numériques: de tout média, même tronqué, on peut réextraire la 
+marque. 
+Dans ce chapitre, le processus d'embarquement d'un message dans 
+un média est formalisé en section~\ref{sec:watermarking:formulation}.
+La sécurité des approches de watermarking est étudiée selon deux approches:
+l'approche probabiliste~\ref{sec:watermarking:security:probas} 
+et l'approche chaotique~\ref{sec:watermarking:security:chaos}.
+Une proposition d'embarquement dans le domaine fréquentiel est abordée
+en section~\ref{sec:watermarking:frequentiel}.
 
+On remarque cependant que l'algorithme formalisé dans ces sections ne permet
+d'embarquer \textit{in fine} qu'un bit qui est vrai si l'image est marquée 
+et faux dans le cas contraire. 
+Il ne permet pas d'extraire le contenu du message initial à partir de
+l'image marquée. La section~\ref{sec:watermarking:extension}
+propose une solution à ce problème.
 
-This section has focused on security with regards to probabilistic behaviors. 
-Next section studies it in the perspective of topological ones.
+Les trois premières sections de ce chapitre sont une reformulation 
+du chapitre 22 de~\cite{guyeux10}. Elles ont été publiées à~\cite{bcg11:ij}.
+L'extension a quant à elle été publiée dans~\cite{bcfg+13:ip}.
 
 
 
-\section{Processus de marquage binaire}
+\section{Processus de marquage binaire}\label{sec:watermarking:formulation}
 
 Par la suite, le message numérique qu'on cherche à embarquer est 
 noté $y$ et le support dans lequel se fait l'insertion est noté $x$. 
@@ -315,7 +328,7 @@ La preuve de double-stochasiticité de la matrice associée
 On dispose ainsi d'un nouvel algorithme de marquage $\epsilon$-stego-secure et 
 chaos-sécure.
 
-\section{Applications aux domaines fréquentiels}
+\section{Applications aux domaines fréquentiels}\label{sec:watermarking:frequentiel}
 Le schéma d'algorithme présenté dans ce chapitre a été appliqué au marquage d'images 
 dans les coefficients DCT et les DWT.
 
@@ -469,7 +482,7 @@ L'approche est résistante à:
 \end{itemize}
 
 
-\section{Embarquons d'avantage qu'1 bit}
+\section{Embarquons d'avantage qu'1 bit}\label{sec:watermarking:extension}
 L'algorithme présenté dans les sections précédentes
 ne  permet de savoir, \textit{in fine}, 
 que si l'image est marquée ou pas. Cet algorithme ne permet pas
@@ -609,14 +622,20 @@ et produit un réel dans $[0;1]$. Si elle est inférieure à un certain
 seuil (à définir), le média $y'$ est declaré 
 comme marqué et le message doit pouvoir être extrait.
 
-\section{Etude de robustesse} 
+\subsection{Etude de robustesse}\label{sec:watermarking:robustesse} 
 La méthode d'expérimentation de robustesse appliquée à la section précédente 
 pourrait être réappliquée ici et nous pourrions obtenir, grâce aux courbes de 
 ROC une valeur seuil pour déterminer si une marque est présente ou pas.
-
-Nous n'avons cependant pas poussé la démarche plus loin que de l'embarquement 
+Dans~\cite{bcfg+13:ip}, nous n'avons cependant pas poussé
+la démarche plus loin que de l'embarquement 
 dans les bits de poids faible en spatial et l'on sait que ceci est 
-particulièrement peu robuste. Il reste ainsi à combiner cette approche avec 
-une sélection appropriés des bits à modifier pour qu'elle devienne intéressante.
+particulièrement peu robuste. 
+
 
+\section{Conclusion}
+Grace à la formalisation du processus de watermarking par itérations discrètes, nous avons pu dans ce chapitre montrer que le processus possédait les propriétés
+attendues, à savoir stego-sécurité, chaos sécurité et une robustesse relative.
+Pour étendre le champ applicatif, nous avons proposé un second algorithme
+permettant de particulariser la marque à embarquer et donc à extraire.
+Le chapitre suivant s'intéresse au marquage, mais dans un autre domaine que celui des images.
 
-- 
2.39.5


From 9e9f22c16917d3bf287f5e1f0df739200c392594 Mon Sep 17 00:00:00 2001
From: couchot <couchot@localhost.localdomain>
Date: Wed, 31 Aug 2016 08:46:41 +0200
Subject: [PATCH 11/16] preuve promela:debut de traduction

---
 ahmad.tex                       |  12 ++-
 annexePreuveMixage.tex          |  14 ++-
 annexePromelaProof.tex          | 149 +++++++++++++++++---------------
 demandeInscription/synthese.tex |   5 +-
 main.tex                        |   8 +-
 mixage.tex                      |  13 ++-
 modelchecking.tex               |  43 +++++----
 sdd.tex                         |  29 +++----
 stabylo.tex                     |   2 +-
 stegoyousra.tex                 |  34 +++-----
 10 files changed, 153 insertions(+), 156 deletions(-)

diff --git a/ahmad.tex b/ahmad.tex
index 1b92dbc..05d9177 100644
--- a/ahmad.tex
+++ b/ahmad.tex
@@ -20,7 +20,13 @@ préétablies et conserver ainsi leur information et donc la marque.
 STDM~\cite{CW01} est une instance de ces schémas de marquage.
 
 Ce chapitre présente une application de STDM au marquage de documents PDFs.
-\JFC{annonce du plan}
+La première section fournit quelques rappels sur la STDM. Le schéma basé sur 
+cette approche est présenté à la section~\ref{sec:stdm:schema}. 
+Finalement, la démarche expérimentale permettant de trouver un compromis entre 
+robustesse et qualité visuelle est présentée à la section~\ref{sec:stdm:exp}
+Ce travail a été publié dans~\cite{BDCC16}.
+
+
 
 \section{Rappels sur la Spread Transform Dither Modulation}
 \label{sec:STDM}
@@ -66,7 +72,7 @@ $U(\Delta)$.
 Tous les éléments sont en place pour embarquer une marque 
 dans un fichier PDF selon le schéma STDM.
 
-\section{Application au marquage de documents PDF}
+\section{Application au marquage de documents PDF}\label{sec:stdm:schema}
 
 On détaille successivement comment insérer une marque dans un document PDF, 
 puis comment l'extraire.
@@ -132,7 +138,7 @@ de taille $L$ mutuellement disjoints dans $[1,N]$.
   en remplaçant $x'$ par $\dot{x'}$ .
 \end{enumerate}
 
-\section{Expérimentations }
+\section{Expérimentations}\label{sec:stdm:exp}
 Le schéma de marquage est paramétré par $\Delta$,  $d_0$ et la manière de construire le vecteur $p$ pour une taille $L$. 
 Les travaux réalisés se sont focalisés sur l'influence du paramètre 
 $D_S = \frac{\Delta^2}{12L}$ dans l'algorithme en satisfaisant 
diff --git a/annexePreuveMixage.tex b/annexePreuveMixage.tex
index d4087d4..12c3223 100644
--- a/annexePreuveMixage.tex
+++ b/annexePreuveMixage.tex
@@ -3,11 +3,9 @@ $\preceq$  entre les  classes d'équivalences.
 Formellement, \class{p}
 $\preceq$   \class{q}   
 s'il existe un chemin de longueur   $\alpha$
-($0<\alpha<|\mathcal{K}|$)  entre un élément le la classe
+($0\le\alpha<|\mathcal{K}|$)  entre un élément le la classe
 \class{p} vers un élément  de
-\class{q}.  
-On remarque que si la  \class{p}$\preceq$\class{q}, 
-il n'est alors pas possible que  \class{q}$\preceq$\class{p}.
+\class{q}. 
 
 \begin{lemma}
   Il existe un processus de renommage qui effecte un nouvel identifiant aux
@@ -17,7 +15,7 @@ il n'est alors pas possible que  \class{q}$\preceq$\class{p}.
   \begin{Proof}
     
     Tout d'abord,  soit  \class{p_1},   \ldots,  \class{p_l}  des   classes 
-    contenant respectivement $n_1$,\ldots,  $n_l$  éléments respectively
+    contenant respectivement les éléments $n_1$,\ldots,  $n_l$
     qui ne dépendent d'aucune autre classe.  
     Les éléments de  \class{p_1} sont renommés par $1$, \ldots,  $n_1$,
     les elements  de \class{p_i},   $2  \le   i  \le   l$  sont renommés par 
@@ -72,7 +70,7 @@ On peut remarquer que ce processus de renommage est inspiré des \emph{graphes
 %   Processes numbers are already compliant with the order $\preceq$.
 % \end{xpl}
 
-\begin{Proof}[of Theorem~\ref{th:cvg}]
+\begin{Proof}[du théorème~\ref{th:cvg}]
 
   Le reste de la preuve est fait par induction sur le numéro de classe. 
   Considérons la première  classe \class{b_1} de $n_1$ éléments 
@@ -95,10 +93,10 @@ On peut remarquer que ce processus de renommage est inspiré des \emph{graphes
   pour chaque 
   $p_{k+1} \in$ \class{b_{k+1}} et $p_j \in$ \class{b_j}, $1 \le j \le k$.
 
-  Il nous reste donc des itérations synchronous entre les 
+  Il ne reste donc que des itérations synchrones entre les 
   elements  of \class{b_{k+1}} en démarant dans des configurations
   où tous les éléments de  \class{b_j}, $1 \le j
-  \le k$, on des valeurs constantes.  
+  \le k$, ont  des valeurs constantes.  
   D'après les hypothèses du théorème, cela converge.
 \end{Proof}
 
diff --git a/annexePromelaProof.tex b/annexePromelaProof.tex
index 70981a9..881383c 100644
--- a/annexePromelaProof.tex
+++ b/annexePromelaProof.tex
@@ -1,10 +1,10 @@
-\JFC{Voir section~\ref{sec:spin:proof}}
+%\JFC{Voir section~\ref{sec:spin:proof}}
 
 Cette section donne les preuves des deux théorèmes de correction et complétude 
 du chapitre~\ref{chap:promela}.
 
 
-\begin{lemma}[Strategy Equivalence]\label{lemma:strategy}
+\begin{lemma}[Stratégie équivalente]\label{lemma:strategy}
   Soit $\phi$  un système dynamique discret de stratégie  $(S^t)^{t \in  \Nats}$ 
   et $\psi$  sa traduction en promela. 
   Il existe une exécution de $\psi$ sous hypothèse d'équité faible telle 
@@ -20,7 +20,7 @@ du chapitre~\ref{chap:promela}.
 \end{Proof}
 
 Dans ce qui suit, soit     $Xd^t_{ji}$     la valeur de 
-\verb+Xd[+$j$\verb+].v[+$i$\verb+]+  après le   $t^{\text{th}}$ appel 
+\verb+Xd[+$j$\verb+].v[+$i$\verb+]+  après le   $t^{\text{ème}}$ appel 
 à la fonction 
 \verb+fetch_values+. 
 De plus, soit $Y^k_{ij}$  l'élément à l'indice $k$ 
@@ -62,11 +62,11 @@ permet de modéliser l'équation  \Equ{eq:async}.
 La  function   $M_{ij}^{t+1}$  est  obtenue à l'aide de mises à jour successives
 de  $M_{ij}^{t}$  au travers des deux   functions   \verb+fetch_values+  and
 \verb+diffuse_values+.   Par abus,   soit  $M_{ij}^{t+1/2}$  
-la valeur de  $M_{ij}^{t}$ après la première fonctions pendant l'itération
+la valeur de  $M_{ij}^{t}$ après la première fonction pendant l'itération
  $t$.
 
 Dans ce qui suit, on  considère les éléments  $i$ et  $j$
-dans  $\llbracket  n  \rrbracket$. 
+dans  $[  \mathsf{N}  ]$. 
 A l'itération   $t$,  $t   \geq  1$,   soit
 $(Y^0_{ij},a^0_{ij},c^0_{ij})$ la valeur de  $M_{ij}^t(0)$ en entrant 
 dans la fonction 
@@ -97,7 +97,7 @@ est  exécutée et $M_{ij}^{t+1} = M_{ij}^{t+1/2}$.
   Pour chaque  sequence $(S^t)^{t  \in \Nats}$,\linebreak $(D^t)^{t \in \Nats}$, 
   pour chaque fonction $F$,
   il existe une exécution SPIN  telle que pour toute itération $t$, $t
-  \ge  1$, et pour chaque  $i$ et $j$ in  $\llbracket n \rrbracket$  
+  \ge  1$, et pour chaque  $i$ et $j$ dans  $[ \mathsf{N} ]$  
   on a la propriété suivante:
    
 \noindent Si le domaine de $M_{ij}^t$ n'est pas vide, alors
@@ -105,7 +105,7 @@ est  exécutée et $M_{ij}^{t+1} = M_{ij}^{t+1/2}$.
   \left\{
     \begin{array}{rcl}
       M_{ij}^1(0) & = & \left(X_i^{D_{ji}^{0}}, 0,0 \right) \\
-      \textrm{sit $t \geq 2$ alors }M_{ij}^t(0) & = &
+      \textrm{si $t \geq 2$ alors }M_{ij}^t(0) & = &
       \left(X_i^{D_{ji}^{c}},D_{ji}^{c},c \right) \textrm{, }
       c = \min\{l | D_{ji}^l > D_{ji}^{t-2} \}
     \end{array}
@@ -117,11 +117,11 @@ est  exécutée et $M_{ij}^{t+1} = M_{ij}^{t+1/2}$.
   \forall t'\, .\,   1 \le t' \le t  \Rightarrow   Xd^{t'}_{ji} = X^{D^{t'-1}_{ji}}_i
   \label{eq:correct_retrieve}
 \end{equation}
-\noindent Enfin, pour chaque $k\in S^t$, la valeurde 
+\noindent Enfin, pour chaque $k\in S^t$, la valeur de 
 la variable  \verb+Xp[k]+  en sortant du processus 
 \verb+update_elems+  est  égale à
 $X_k^{t}$          \textit{i.e.},          $F_{k}\left(         X_1^{D_{k\,1}^{t-1}},\ldots,
-  X_{n}^{D_{k\,{n}}^{t-1}}\right)$ à la fin de la  $t^{\text{th}}$ itération.
+  X_{\mathsf{N}}^{D_{k\,{\mathsf{N}}}^{t-1}}\right)$ à la fin de la  $t^{\text{th}}$ itération.
 \end{lemma}
 \begin{Proof}
 La preuve est faite par induction sur le nombre d'itérations.
@@ -130,20 +130,20 @@ La preuve est faite par induction sur le nombre d'itérations.
 Pour le premier item, par definition de $M_{ij}^t$, on a $M_{ij}^1(0) = \left(
   \verb+Xp[i]+, 0,0 \right)$ qui est égal à  $\left(X_i^{D_{ji}^{0}},
   0,0 \right)$.
-Ensuite, lepremier appel à la  fonction \verb+fetch_value+ 
+Ensuite, le premier appel à la  fonction \verb+fetch_value+ 
 soit affecte à la tête de \verb+channels[i].sent[j]+  à   \verb+Xd[j].v[i]+ soit ne modifie par 
 \verb+Xd[j].v[i]+. 
 Grâce au processus \verb+init+ process, 
 les deux cas sont égaux à 
 \verb+Xp[i]+,  \textit{i.e.}, $X_i^0$.  L'equation (\ref{eq:correct_retrieve})  est ainsi établie.
 
-Pour le dernier item, soit $k$, $0  \le  k \le  n-1$. 
+Pour le dernier item, soit $k$, $0  \le  k \le  \mathsf{N}-1$. 
 A la fin de la première exécution du processus \verb+update_elems+,
-la valur   de
+la valeur   de
 \verb+Xp[k]+       est       $F(\verb+Xd[+k\verb+].v[0]+,       \ldots,
-\verb+Xd[+k\verb+].v[+n-1\verb+]+)$.  
+\verb+Xd[+k\verb+].v[+\mathsf{N}-1\verb+]+)$.  
 Ainsi par définition de  $Xd$, ceci est égal à 
-$F(Xd^1_{k\,0}, \ldots,Xd^1_{k\,n-1})$.  Grâce à l'équation \Equ{eq:correct_retrieve},
+$F(Xd^1_{k\,0}, \ldots,Xd^1_{k\,\mathsf{N}-1})$.  Grâce à l'équation \Equ{eq:correct_retrieve},
 on peut conclure la preuve.
 
 
@@ -157,79 +157,84 @@ n'est pas vide, par hypothèse d'induction $M_{ij}^{l}(0)$  est
 $\left(X_i^{D_{ji}^{c}}, D_{ji}^{c},c
 \right)$ où $c$ est $\min\{k | D_{ji}^k > D_{ji}^{l-2} \}$.
 
-A l'itération $l$, si  $l < c + 1$ alors  \verb+skip+ statement is executed in
-the   \verb+fetch_values+  function.   Thus,   $M_{ij}^{l+1}(0)$  is   equal  to
-$M_{ij}^{l}(0)$.  Since $c > l-1$  then $D_{ji}^c > D_{ji}^{l-1}$ and hence, $c$
-is $\min\{k  | D_{ji}^k  > D_{ji}^{l-1} \}$.  Obviously, this implies  also that
-$D_{ji}^c > D_{ji}^{l-2}$ and $c=\min\{k | D_{ji}^k > D_{ji}^{l-2} \}$.
-
-We now consider that at iteration $l$, $l$ is $c + 1$.  In other words, $M_{ij}$
-is modified depending on the domain $\dom(M^l_{ij})$ of $M^l_{ij}$:
+A l'itération $l$, si  $l < c + 1$ alors l'instruction 
+ \verb+skip+ est exécutée dans la fonction \verb+fetch_values+. 
+ Ainsi,   $M_{ij}^{l+1}(0)$  est égal à
+$M_{ij}^{l}(0)$.  Puisque $c > l-1$, alors $D_{ji}^c > D_{ji}^{l-1}$ et donc, $c$
+est $\min\{k  | D_{ji}^k  > D_{ji}^{l-1} \}$.
+Cela implique que 
+$D_{ji}^c > D_{ji}^{l-2}$ et $c=\min\{k | D_{ji}^k > D_{ji}^{l-2} \}$.
+
+On considère maintenant qu'à l'itération $l$, celui-ci vaut $c + 1$. 
+Dit autrement, $M_{ij}$ est modifié en fonction du domaine $\dom(M^l_{ij})$ de
+ $M^l_{ij}$:
 \begin{itemize}
-\item  if $\dom(M_{ij}^{l})=\{0\}$  and $\forall  k\,  . \,  k\ge l  \Rightarrow
-  D^{k}_{ji} \neq l$  is established then $\dom(M_{ij}^{l+1})$ is  empty and the
-  first item of the lemma  is established; 
-\item if $\dom(M_{ij}^{l})=\{0\}$ and $\exists  k\, . \, k\ge l \land D^{k}_{ji}
-  = l$  is established then $M_{ij}^{l+1}(0)$  is $(\verb+Xp[i]+,l,c_{ij})$ that
-  is  added  in  the  \verb+diffuse_values+ function  s.t.\linebreak  $c_{ij}  =
-  \min\{k  \mid  D^{k}_{ji}  = l  \}  $.   Let  us  prove  that we  can  express
-  $M_{ij}^{l+1}(0)$  as  $\left(X_i^{D_{ji}^{c'}},D_{ji}^{c'},c' \right)$  where
-  $c'$ is  $\min\{k |  D_{ji}^k > D_{ji}^{l-1}  \}$.  First,  it is not  hard to
-  establish that  $D_{ji}^{c_{ij}}= l \geq  D_{ji}^{l} > D_{ji}^{l-1}$  and thus
-  $c_{ij}  \geq   c'$.   Next,  since   $\dom(M_{ij}^{l})=\{0\}$,  then  between
-  iterations $D_{ji}^{c}+1$ and $l-1$, the \texttt{diffuse\_values} function has
-  not updated $M_{ij}$.  Formally we have
+\item  si $\dom(M_{ij}^{l})=\{0\}$  et $\forall  k\,  . \,  k\ge l  \Rightarrow
+  D^{k}_{ji} \neq l$ sont vraies, alors  $\dom(M_{ij}^{l+1})$ est vide et le premier
+  item du lemme est vérifié;
+\item si $\dom(M_{ij}^{l})=\{0\}$ et $\exists  k\, . \, k\ge l \land D^{k}_{ji}
+  = l$  sont vraies, alors $M_{ij}^{l+1}(0)$  vaut  $(\verb+Xp[i]+,l,c_{ij})$ qui est ajouté 
+  dans  la fonction \verb+diffuse_values+ de sorte que   $c_{ij}  =
+  \min\{k  \mid  D^{k}_{ji}  = l  \}  $.   
+  Prouvons qu'on peut exprimer 
+  $M_{ij}^{l+1}(0)$  comme  $\left(X_i^{D_{ji}^{c'}},D_{ji}^{c'},c' \right)$  où
+  $c'$ vaut  $\min\{k |  D_{ji}^k > D_{ji}^{l-1}  \}$.  
+  Tout d'abord, il n'est pas difficile de prouver que 
+  $D_{ji}^{c_{ij}}= l \geq  D_{ji}^{l} > D_{ji}^{l-1}$  et que 
+  $c_{ij}  \geq   c'$.   
+  Ensuite, comme   $\dom(M_{ij}^{l})=\{0\}$,  alors, entre les 
+  itérations $D_{ji}^{c}+1$ et $l-1$, la fonction  \texttt{diffuse\_values} n'a pas mis à jour
+   $M_{ij}$. On a ainsi la propriété
 $$
 \forall t,k  \, .\, D_{ji}^c <  t < l \land  k \geq t  \Rightarrow D_{ji}^k \neq
 t.$$
 
-Particularly, $D_{ji}^{c'} \not  \in \{D_{ji}^{c}+1,\ldots,l-1\}$.  We can apply
-the     third    item     of    the     induction    hypothesis     to    deduce
-$\verb+Xp[i]+=X_i^{D_{ji}^{c'}}$ and we can conclude.
-
-\item  if   $\{0,1\}  \subseteq  \dom(M_{ij}^{l})$   then  $M_{ij}^{l+1}(0)$  is
-  $M_{ij}^{l}(1)$.   Let  $M_{ij}^{l}(1)=  \left(\verb+Xp[i]+, a_{ij}  ,  c_{ij}
-  \right)$.   By  construction $a_{ij}$  is  $\min\{t'  |  t' >  D_{ji}^c  \land
-  (\exists k \, .\, k \geq t' \land D_{ji}^k = t')\}$ and $c_{ij}$ is $\min\{k |
-  D_{ji}^k = a_{ij}\}$.  Let us show  $c_{ij}$ is equal to $\min\{k | D_{ji}^k >
-  D_{ji}^{l-1} \}$ further  referred as $c'$.  First we  have $D_{ji}^{c_{ij}} =
-  a_{ij} >  D_{ji}^c$. Since $c$  by definition is  greater or equal to  $l-1$ ,
-  then $D_{ji}^{c_{ij}}>  D_{ji}^{l-1}$ and then $c_{ij} \geq  c'$.  Next, since
-  $c$ is  $l-1$, $c'$ is $\min\{k |  D_{ji}^k > D_{ji}^{c} \}$  and then $a_{ij}
-  \leq  D_{ji}^{c'}$. Thus,  $c_{ij} \leq  c'$  and we  can conclude  as in  the
-  previous part.
+En particulier, on a   $D_{ji}^{c'} \not  \in \{D_{ji}^{c}+1,\ldots,l-1\}$.  
+On peut donc appliquer le troisième item de l'hypothèse d'induction pour déduire
+$\verb+Xp[i]+=X_i^{D_{ji}^{c'}}$ et on peut conclure.
+
+\item  Si   $\{0,1\}  \subseteq  \dom(M_{ij}^{l})$, alors  $M_{ij}^{l+1}(0)$ vaut
+  $M_{ij}^{l}(1)$.   Soit  $M_{ij}^{l}(1)=  \left(\verb+Xp[i]+, a_{ij}  ,  c_{ij}
+  \right)$.   Par construction,  $a_{ij}$  vaut $\min\{t'  |  t' >  D_{ji}^c  \land
+  (\exists k \, .\, k \geq t' \land D_{ji}^k = t')\}$  et  $c_{ij}$ est $\min\{k |
+  D_{ji}^k = a_{ij}\}$.  Montrons que   $c_{ij}$ est égal à  $\min\{k | D_{ji}^k >
+  D_{ji}^{l-1} \}$, noté plus tard $c'$.  On a tout d'abord $D_{ji}^{c_{ij}} =
+  a_{ij} >  D_{ji}^c$. Puisque $c$  par définition est supérieur ou égal à  $l-1$,
+  alors $D_{ji}^{c_{ij}}>  D_{ji}^{l-1}$ et donc $c_{ij} \geq  c'$.  Ensuite, puisque
+  $c=l-1$, $c'$ vaut  $\min\{k |  D_{ji}^k > D_{ji}^{c} \}$  et donc $a_{ij}
+  \leq  D_{ji}^{c'}$. Ainsi,  $c_{ij} \leq  c'$  et on peut conclure comme dans la partie précédente.
 \end{itemize}
 
 
-The case where  the domain $\dom(M^l_{ij})$ is empty but  the formula $\exists k
-\, .\, k \geq  l \land D_{ji}^k = l$ is established  is equivalent to the second
-case given above and then is omitted.
+Le cas où le domaine $\dom(M^l_{ij})$ est vide mais où la formule  $\exists k
+\, .\, k \geq  l \land D_{ji}^k = l$ est vraie est équivalent au second
+cas ci-dessus et n'est pas présenté.
 
 
-Secondly, let us focus on the formula~(\ref{eq:correct_retrieve}).  At iteration
-$l+1$, let $c'$ be defined as $\min\{k | D_{ji}^k > D_{ji}^{l-1} \}$.  Two cases
-have to be  considered depending on whether $D_{ji}^{l}$  and $D_{ji}^{l-1}$ are
-equal or not.
+Concentrons nous sur la formule~(\ref{eq:correct_retrieve}).  A l'itération 
+$l+1$, soit $c'$ défini par  $c'=\min\{k | D_{ji}^k > D_{ji}^{l-1} \}$.  Deux cas peuvent 
+apparaître selon que $D_{ji}^{l}$ et  $D_{ji}^{l-1}$ sont égaux ou non.
 \begin{itemize}
-\item If  $D_{ji}^{l} = D_{ji}^{l-1}$, since $D_{ji}^{c'}  > D_{ji}^{l-1}$, then
-  $D_{ji}^{c'} > D_{ji}^{l}$ and then $c'$  is distinct from $l$. Thus, the SPIN
-  execution detailed  above does not  modify $Xd_{ji}^{l+1}$.  It is  obvious to
-  establish   that   $Xd_{ji}^{l+1}  =   Xd_{ji}^{l}   =  X_i^{D_{ji}^{l-1}}   =
+\item Si  $D_{ji}^{l} = D_{ji}^{l-1}$, puisque $D_{ji}^{c'}  > D_{ji}^{l-1}$, alors
+  $D_{ji}^{c'} > D_{ji}^{l}$ et donc $c'$  est différent de  $l$. L'exécution de SPIN
+  ne modifie pas $Xd_{ji}^{l+1}$.  On a ainsi  $Xd_{ji}^{l+1}  =   Xd_{ji}^{l}   =  X_i^{D_{ji}^{l-1}}   =
   X_i^{D_{ji}^{l}}$.
-\item Otherwise $D_{ji}^{l}$ is greater than $D_{ji}^{l-1}$ and $c$ is thus $l$.
-  According     to     \Equ{eq:Mij0}     we     have    proved,     we     have
-  $M_{ij}^{l+1}(0)=(X_i^{D_{ji}^{l}},D_{ji}^{l},l)$.   Then  the SPIN  execution
-  detailed above  assigns $X_i^{D_{ji}^{l}}$ to $Xd_{ji}^{l+1}$,  which ends the
-  proof of (\ref{eq:correct_retrieve}).
+\item Sinon, $D_{ji}^{l}$ et plus grand que $D_{ji}^{l-1}$ et $c$ est donc égal à $l$.
+  Selon l'équation \Equ{eq:Mij0},    on a 
+  $M_{ij}^{l+1}(0)=(X_i^{D_{ji}^{l}},D_{ji}^{l},l)$.  Ainsi l'exécution SPIN 
+  affecte  $X_i^{D_{ji}^{l}}$ à $Xd_{ji}^{l+1}$,  ce qui termine la preuve 
+(\ref{eq:correct_retrieve}).
 \end{itemize}
 
-We are left to prove the induction of  the third part of the lemma.  Let $k$, $k
+Il reste à prouver la partie inductive de la troisième partie du lemme.
+Soit $k$, $k
 \in S^{l+1}$. % and $\verb+k'+ = k-1$.
-At the  end of the first  execution of the \verb+update_elems+  process, we have
+A l'issue de la première exécutions 
+du processus \verb+update_elems+, on a 
 $\verb+Xp[+k\verb+]+=                                   F(\verb+Xd[+k\verb+][0]+,
-\ldots,\verb+Xd[+k\verb+][+n\verb+-1]+)+$.  By  definition of $Xd$,  it is equal
-to      $F(Xd^{l+1}_{k\,0},      \ldots,Xd^{l+1}_{k\,n-1})$.      Thanks      to
-\Equ{eq:correct_retrieve} we have proved, we can conclude the proof.
+\ldots,\verb+Xd[+k\verb+][+n\verb+-1]+)+$.  
+Par définition $Xd=F(Xd^{l+1}_{k\,0},      \ldots,Xd^{l+1}_{k\,n-1})$.      
+Grace à~\Equ{eq:correct_retrieve} déjà prouvée, on peut conclure la preuve.
 \end{Proof}
 
 
@@ -284,7 +289,7 @@ to      $F(Xd^{l+1}_{k\,0},      \ldots,Xd^{l+1}_{k\,n-1})$.      Thanks      to
 %   performed under weak  fairness property; we then detail  what are continuously
 %   enabled:
 % \begin{itemize}
-% \item if the strategy is not  defined as periodic, elements $0$, \ldots, $n$ are
+% \item if the strategy is not  defined as periodic, elements $0$, \ldots, $\mathsf{N}$ are
 %   infinitely often updated leading to pseudo-periodic strategy;
 % \item  instructions  that  write  or read  into  \verb+channels[j].sent[i]+  are
 %   continuously enabled leading to convenient available dates $D_{ji}$.
diff --git a/demandeInscription/synthese.tex b/demandeInscription/synthese.tex
index 03c44f7..58ad6b7 100755
--- a/demandeInscription/synthese.tex
+++ b/demandeInscription/synthese.tex
@@ -1557,10 +1557,9 @@ Jean-François Couchot, Raphaël Couturier, Yousra Ahmed Fadil and Christophe Gu
   Lisboa, Portugal, July 2016.
 
 \bibitem{kcm16:ip}
-Nesrine Khernane, J.-F. Couchot, A. Mostefaoui.
+N. Khernane, J.-F. Couchot, A. Mostefaoui.
 \newblock Maximizing Network Lifetime in Wireless Video Sensor Networks Under Quality Constraints.
-\newblock In {\em The 19th ACM International Conference on Modeling, Analysis
-and Simulation of Wireless and Mobile Systems, MSWiM'16} Malta,  to appear, November, 2016.
+\newblock In {\em MOBIWAC 2016: The 14th ACM* International Symposium on Mobility Management and Wireless Access} Malta,  to appear, November, 2016.
 
 \bibitem{BCDG07}
 Fabrice Bouquet, Jean-Fran\c{c}ois Couchot, Fr\'ed\'eric Dadeau, and Alain
diff --git a/main.tex b/main.tex
index da87f22..33a3ba1 100644
--- a/main.tex
+++ b/main.tex
@@ -172,7 +172,7 @@ Blabla blabla.
 
 \mainmatter
 
-\part{Réseaux Discrets}
+\part{Réseaux discrets}
 
 \chapter{Iterations discrètes de réseaux booléens}
 
@@ -182,8 +182,8 @@ les différents modes opératoires, leur représentation à l'aide de
 graphes et les résultats connus de convergence).
 Ce chapitre montre ensuite à la section~\ref{sec:sdd:mixage}
 comment combiner ces modes pour converger aussi 
-souvent sans, mais plus rapidement. Cette dernière section 
-a fait l'objet du rapport~\cite{BCVC10:ir}.
+souvent, mais plus rapidement vers un point fixe. Les deux 
+dernières sections ont fait l'objet du rapport~\cite{BCVC10:ir}.
 
 \section{Formalisation}\label{sec:sdd:formalisation}
 \input{sdd}
@@ -339,7 +339,7 @@ du chapitre 8}
 
 \appendix
 
-\chapter{Preuves sur les SDD}
+\chapter{Preuves sur les réseaux discrets}
 
 \section{Convergence du mode mixe}\label{anx:mix}
 \input{annexePreuveMixage}
diff --git a/mixage.tex b/mixage.tex
index e4a283c..4ee1171 100644
--- a/mixage.tex
+++ b/mixage.tex
@@ -21,9 +21,6 @@ s^{t}=24 \textrm{ si  $t$ est pair et  } s^{t}=15 \textrm{ sinon }
 \end{equation}
 \noindent active successivement les deux premiers éléments (24 est 11000) 
 et les quatre derniers éléments (15  est  01111).  
-On dit que la stratégie est
-\emph{pseudo-periodique}  si tous les éléments sont activés infiniment 
-souvent.
 % , it is sufficient to establish  that the set $\{t \mid t \in \mathbb{N}
 % \land \textit{bin}(s^t)[i]  = 1\}$  is infinite for  any $i$,  $1 \le i  \le n$,
 % where
@@ -54,7 +51,7 @@ souvent.
 Dans le mode asynchrone, a chaque itération $t$, chaque composant peut
 mettre à jour son état en 
 fonction des dernières valeurs qu'il connaît des autre composants.
-Obtenir ou non les valeurs les plus à jours dépend du temps de calcul et 
+Obtenir ou non les valeurs les plus à jour dépend du temps de calcul et 
 du temps d'acheminement de celles-ci. On parle de latence, de délai.
 
 Formalisons le mode les itérations asynchrone.
@@ -190,7 +187,7 @@ Les itérations asynchrones sont conservées entre les groupes.
   La  \emph{relation de synchronisation} $\eqNode$ est
   définie sur l'ensemble des n{\oe}uds par:
   $i \eqNode j$ si $i$ et $j$  appartiennent à la même composante fortement
-  connexe (CFC) dans $\Gamma(F)$.
+  connexe (CFC) dans $\Gamma(f)$.
 \end{Def}
 
 On peut facilement démontrer que la relation de synchronisation est une 
@@ -214,7 +211,7 @@ Ainsi $p_1$ et $p_2$ sont distinguables même s'ils appartiennent à la même
 classe.
 Pour gommer cette distinction, on définit le mode suivant:
 \begin{Def}[Itérations mixes avec delais uniformes]
-  Le mode mixe a des \emph{délais uniformes}si pour chaque 
+  Le mode mixe a des \emph{délais uniformes} si pour chaque 
   $t=0,1,\ldots$ et pour chaque paire de  classes  $(\class{p}, \class{q})$,
   il existe une constante $d^t_{pq}$  telle que la propriété suivante est 
   établie:
@@ -420,7 +417,7 @@ ascendants pour converger. On a dans ce cas:
   mode synchrone.
 \end{xpl}
 
-\subsection{Le mode généralisé asynchrone}
+\subsection{Le mode unaire asynchrone}
 \label{sec:evalasync}
 En terme de durée de convergence, ce mode peut être vu comme un 
 cas particulier du mode mixe où toutes les classes sont des singletons.
@@ -442,7 +439,7 @@ et apparaît lorsque chaque élément dépend des autres et que les calculs
 ne recouvrent nullement les communications.
 
 \begin{xpl}
-  La \Fig{fig:evalasync} présente un exemple d'exécution du mode généralisé 
+  La \Fig{fig:evalasync} présente un exemple d'exécution du mode unaire
   asynchrone.
   Certaines communications issues de l'élément $4$ n'ont pas été représentées
   pour des raisons de clarté.
diff --git a/modelchecking.tex b/modelchecking.tex
index 2a7107f..b01a526 100644
--- a/modelchecking.tex
+++ b/modelchecking.tex
@@ -19,18 +19,19 @@ automatique pour construire cette fonction.
 Un outil qui construirait automatiquement toutes 
 les transitons serait le bienvenu. 
 Pour peu qu'on établisse la preuve de correction et de complétude de la 
-démarche, la convergence du réseau discret ne repose alors que sur le verdict 
+démarche, la convergence du réseau discret ne reposerait
+ alors que sur le verdict 
 donné par l'outil. 
 Cependant, même pour des réseaux discrets à peu d'éléments, 
 le nombre de configurations induites explose rapidement.
 Les \emph{Model-Checkers}~\cite{Hol03,nusmv02,Blast07,MCErlang07,Bogor03}  
-sont des classes d'outils qui adressent le problème de vérifier automatiquement
-qu'un modèle vérifie une propriété donnée. Pour traiter le problème d'explosion 
+sont des classes d'outils qui adressent le problème de détecter automatiquement
+si un modèle vérifie une propriété donnée. Pour traiter le problème d'explosion 
 combinatoire, ces outils appliquent des méthodes d'ordre partiel, d'abstraction,
 de quotientage selon une relation d'équivalence.
 
 Ce chapitre montre comment nous simulons 
-des réseaux discrets selon toutes les sortes d'itérations pour établir 
+des réseaux discrets selon pour établir 
 formellement leur convergence (ou pas).
 Nous débutons par un exemple et faisons quelques rappels sur 
 le langage PROMELA qui est le langage du model-checker 
@@ -38,8 +39,8 @@ SPIN~\cite{Hol03} (\Sec{sec:spin:promela}).
 Nous présentons ensuite la démarche de traduction 
 de réseaux discrets dans PROMELA (\Sec{sec:spin:translation}).   
 Les théorèmes de correction et de complétude de la démarche
-sont ensuite donnés à la (\Sec{sec:spin:proof}). 
-Des données pratiques comme la complexité et des synthèses d'expérimentation
+sont ensuite donnés à la \Sec{sec:spin:proof}. 
+Des données pratiques comme la complexité et des synthèses d'expérimentations
 sont ensuite fournies (\Sec{sec:spin:practical}). 
 Ce chapitre a fait l'objet du rapport~\cite{Cou10:ir}.
 
@@ -88,11 +89,12 @@ Ce chapitre a fait l'objet du rapport~\cite{Cou10:ir}.
 
 
 
-On peut facilement vérifier que toutes les itérations synchrones initialisées 
+On peut facilement vérifier que toutes les itérations parallèles 
+synchrones initialisées 
 avec $x^0 \neq 7$ soit $(111)$ 
 convergent vers $2$ soit $(010)$; celles initialisées avec 
 $x^0=7$ restent en 7.
-Pour les  mode unaires ou généralisés  avec  une 
+Pour les  modes unaires ou généralisés  avec  une 
 stratégie pseudo périodique, on a des comportements qui dépendent 
 de la configuration initiale:
 \begin{itemize}
@@ -153,7 +155,7 @@ déclarations de variables qui servent dans l'exemple de ce chapitre.
 Il définit:
 \begin{itemize}
 \item les constantes \verb+N+ et \verb+d_0+ qui précisent respectivement le nombre
- $n$ d'éléments et le délais maximum $\delta_0$;
+ ${\mathsf{N}}$ d'éléments et le délais maximum $\delta_0$;
 \item les deux tableaux  (\verb+X+ et \verb+Xp+) de \verb+N+ variables booléennes; 
 les cellules \verb+X[i]+ et \verb+Xp[i]+ sont associées à la variables $x_{i+1}$
 d'un système dynamique discret;
@@ -208,8 +210,8 @@ réception de messages. Pour un canal
 L'instruction de réception consomme la valeur en tête du canal \verb+ch+
 et l'affecte à la variable \verb+m+ (pour peu que  \verb+ch+ soit initialisé et non vide).
 De manière similaire, l'instruction d'envoi ajoute la valeur de \verb+m+ à la queue du canal
-\verb+ch+ (pour peu que celui-ci soit initialisé et non rempli).
-Dans les cas problématiques, canal non initialisé et vide pour une réception ou bien rempli pour un envoi,
+\verb+ch+ (pour peu que celui-ci soit initialisé et pas plein).
+Dans les cas problématiques, canal non initialisé et vide pour une réception ou plein pour un envoi,
 le processus est bloqué jusqu'à ce que les  conditions soient  remplies.
 
 La structures de contrôle   \verb+if+   (resp.   \verb+do+)   définit un choix non déterministe 
@@ -218,9 +220,9 @@ si plus d'une des conditions est établie, l'ensemble des instructions correspon
 sera choisi aléatoirement puis exécuté.
 
 Dans le process \verb+init+ détaillé à la {\sc Figure}~\ref{fig:spin:init}, 
-une boucle de taille $N$ initialise aléatoirement la variable  globale de type tableau \verb+Xp+.
+une boucle de taille ${\mathsf{N}}$ initialise aléatoirement la variable  globale de type tableau \verb+Xp+.
 Ceci permet par la suite de vérifier si les itérations sont  convergentes pour n'importe  
-quelle configuration initiale $x^{(0)}$.
+quelle configuration initiale $x^{0}$.
 
 
 
@@ -343,7 +345,7 @@ les éléments possiblement mis à jour à l'itération $t$.
 
 Basiquement, le process est une boucle qui est débloquée lorsque la valeur du sémaphore
 \verb+sync_mutex+ est 1. Dans ce cas, les éléments à modifier sont choisis
-aléatoirement  (grâce à  $n$ choix successifs) et sont mémorisés dans le tableau
+aléatoirement  (grâce à  ${\mathsf{N}}$ choix successifs) et sont mémorisés dans le tableau
 \verb+mods+,  dont la taille est  \verb+ar_len+.   
 Dans la séquence d'exécution, le choix d'un élément mis à jour est directement
 suivi par des mises à jour: ceci est réalisé grâce à la modification de la valeur du sémaphore
@@ -588,7 +590,7 @@ délai.
 \subsection{Propriété de convergence universelle}
 Il reste à formaliser dans le model checker SPIN le fait que les 
 itérations d'un système 
-dynamique à $n$ éléments est universellement convergent.
+dynamique à ${\mathsf{N}}$ éléments est universellement convergent.
 
 Rappelons tout d'abord que les variables \verb+X+  et \verb+Xp+ 
 contiennent respectivement la valeur de $x$ avant et après la mise à jour. 
@@ -650,7 +652,7 @@ Cette section donne tout d'abord quelques mesures de complexité de l'approche
 puis présente ensuite les expérimentations issues de ce travail.
 
 \begin{theorem}[Nombre d'états ]
-  Soit  $\phi$  un modèle de système dynamique discret à  $n$ éléments, $m$ 
+  Soit  $\phi$  un modèle de système dynamique discret à  ${\mathsf{N}}$ éléments, $m$ 
   arcs dans le graphe d'incidence
   et $\psi$ sa traduction en PROMELA.  Le nombre de configurations 
   de l'exécution en SPIN de $\psi$ est bornée par $2^{m(\delta_0+1)+n(n+2)}$.
@@ -667,7 +669,7 @@ puis présente ensuite les expérimentations issues de ce travail.
   Puisque le nombre d'arêtes du graphe d'incidence  est $m$,  
   il y a  $m$ canaux non constants,  ce qui génère approximativement $2^{m(\delta_0+1)}$ états. 
   Le nombre de configurations est donc borné par $2^{m(\delta_0+1)+n(n+2)}$.
-  On remarque que cette borne est traitable par SPIN pour des valeurs raisonnables de  $n$, 
+  On remarque que cette borne est traitable par SPIN pour des valeurs raisonnables de  ${\mathsf{N}}$, 
   $m$ et $\delta_0$.
   %\JFC{Donner un ordre de grandeur de cet ordre de grandeur}
   
@@ -804,10 +806,7 @@ pour établir un verdict.
 
 L'exemple \textit{RE} est l'exemple de ce chapitre,
 \cite{RC07} concerne un réseau composé de deux gènes
-à valeur dans $\{0,1,2\}$,
-AC2D est un automate cellulaire  avec 9 elements prenant des
-valeurs booléennes en fonction de 
-de 4 voisins et
+à valeur dans $\{0,1,2\}$ et
 \cite{BM99} consiste en  10 process
 qui modifient leurs valeurs booléennes dans un graphe d'adjacence proche 
 du graphe complet.
@@ -818,7 +817,7 @@ L'exemple  \textit{RE} a été prouvé comme universellement convergent.
 Comme la convergence n'est déjà pas établie pour les itérations synchrones
 de~\cite{RC07}, il en est donc 
 de même pour les itérations asynchrones.
-La {\sc Figure}~\ref{fig:RC07CE} donne une trace de la sortie de SPIN de menant à la violation 
+La {\sc Figure}~\ref{fig:RC07CE} donne une trace de la sortie de SPIN  menant à la violation 
 de la convergence. Celle-ci correspond à une stratégie périodique qui répète
 $\{1,2\};\{1,2\};\{1\};\{1,2\};\{1,2\}$ et débute avec $x=(0,0)$.
 En raison de la dépendance forte entre les éléments
diff --git a/sdd.tex b/sdd.tex
index 69e09b6..c41efa9 100644
--- a/sdd.tex
+++ b/sdd.tex
@@ -30,7 +30,7 @@ et $\overline{x}^i$ pour $\overline{x}^{\{i\}}$ pour $i \in [{\mathsf{N}}]$
 Pour tout $x$ et  $y$ dans  $\Bool^{\mathsf{N}}$, l'ensemble 
 $\Delta(x, y)$,  contient les $i \in [{\mathsf{N}}]$
 tels que $x_i \neq y_i$.
-Soit enfin $f : \Bool^n \rightarrow \Bool^{\mathsf{N}}$. Son $i^{\textrm{ème}}$ composant
+Soit enfin $f : \Bool^{\mathsf{N}} \rightarrow \Bool^{\mathsf{N}}$. Son $i^{\textrm{ème}}$ composant
 est nommé $f_i$ qui est une fonction de $\Bool^{\mathsf{N}}$ dans $\Bool$.
 Pour chaque 
 $x$ dans $\Bool^{\mathsf{N}}$, l'ensemble 
@@ -95,8 +95,8 @@ Pour $x=(0,1,0)$ les assertions suivantes se déduisent directement du tableau:
 \end{xpl}
 
 \subsection{Réseau booléen}
-Soit $n$ un entier naturel représentant le nombre 
-d'éléments étudiés (gènes, protéines,\ldots).
+Soit ${\mathsf{N}}$ un entier naturel représentant le nombre 
+d'éléments étudiés.
 Un réseau booléen  est 
 défini à partir d'une fonction booléenne:
 \[
@@ -118,7 +118,7 @@ schémas suivants :
   défini  par une  suite 
   $S  = \left(s^t\right)^{t \in  \mathds{N}}$ qui est  une séquence
   d'indices dans $[{\mathsf{N}}]$. Cette suite est appelée \emph{stratégie unaire}. 
-  Il est basé sur la relation définie pour tout $i \in [{\mathsf{N}}]$ par
+  Ce mode est défini pour tout $i \in [{\mathsf{N}}]$ par
   
 \begin{equation}
   x^{t+1}_i=
@@ -274,7 +274,7 @@ On a la proposition suivante:
 
 
 \begin{theorem}\label{Prop:attracteur}
-Le point $x$ est un point fixe si et seulement si 
+La configuration $x$ est un point fixe si et seulement si 
 $\{x\}$ est un attracteur du graphe d'itération (synchrone, unaire, généralisé).
 En d'autres termes, les attracteurs non cycliques de celui-ci 
 sont les points fixes de $f$.
@@ -331,7 +331,7 @@ $f'_{ij}(x)=0$. Ainsi $B(f)_{ij}$ est nulle. La réciproque est aussi vraie.
 
 En outre, les interactions peuvent se  représenter à l'aide d'un 
 graphe $\Gamma(f)$ orienté et signé défini ainsi: 
-l'ensemble des sommet %s est 
+l'ensemble des sommets est 
 $[{\mathsf{N}}]$ et il existe un arc de $j$ à $i$ de signe
  $s\in\{-1,1\}$, noté $(j,s,i)$, si $f_{ij}(x)=s$ pour au moins
 un $x\in\Bool^{\mathsf{N}}$. 
@@ -527,22 +527,21 @@ Parmi toutes les stratégies unaires de
 $[{\mathsf{N}}]^{\Nats}$, on qualifie de:
 \begin{itemize}
 \item \emph{périodiques} celles 
-qui sont constituées par une répétition indéfinie 
+qui sont constituées par une répétition infinie 
 d'une même séquence $S$ complète relativement à $[{\mathsf{N}}]$.
 En particulier toute séquence périodique est complète.
 \item \emph{pseudo-périodiques} celles 
-qui sont constituées par une succession indéfinie de séquences 
+qui sont constituées par une succession infinie de séquences 
 (de longueur éventuellement variable non supposée bornée) complètes.
 Autrement dit dans chaque stratégie pseudo-périodique, chaque indice de
-$1$ à ${\mathsf{N}}$ revient indéfiniment.
+$1$ à ${\mathsf{N}}$ revient infiniment.
 \end{itemize}
 
 
-François Robert~\cite{Rob95}
-a énoncé en 1995 le théorème suivant de convergence 
+On a le théorème suivant de convergence 
 dans le mode des itérations unaires.
 
-\begin{theorem}\label{Th:conv:GIU}
+\begin{theorem}[~\cite{Rob95}]\label{Th:conv:GIU}
 Si le graphe $\Gamma(f)$ n'a pas de cycle et si la stratégie unaire est
 pseudo-périodique, alors tout chemin de $\textsc{giu}(f)$ atteint 
 l'unique point fixe $\zeta$ en au plus ${\mathsf{N}}$ pseudo-périodes.
@@ -551,11 +550,11 @@ l'unique point fixe $\zeta$ en au plus ${\mathsf{N}}$ pseudo-périodes.
 Le qualificatif \emph{pseudo-périodique} peut aisément 
 s'étendre aux stratégies généralisées comme suit.
 Lorsqu'une stratégie généralisée est constituée d'une 
-succession indéfinie de séquences de parties de $[{\mathsf{N}}]$ 
+succession infinie de séquences de parties de $[{\mathsf{N}}]$ 
 dont l'union est $[{\mathsf{N}}]$, cette stratégie est {pseudo-périodique}.
-J. Bahi~\cite{Bah00} a démontré le théorème suivant:
+On a le théorème suivant.
 
-\begin{theorem}\label{Th:Bahi}
+\begin{theorem}[~\cite{Bah00}]\label{Th:Bahi}
 Si le graphe $\Gamma(f)$ n'a pas de cycle et si la stratégie généralisée
 est pseudo-périodique alors
 tout chemin de $\textsc{gig}(f)$ (et donc de $\textsc{giu}(f)$) 
diff --git a/stabylo.tex b/stabylo.tex
index dda07a4..8801599 100644
--- a/stabylo.tex
+++ b/stabylo.tex
@@ -27,7 +27,7 @@ A partir de ces cartes de distortions, chacun de ces algorithmes selectionne
 les pixels dont les modifications induisent la distortion la plus faible 
 possible. Ceci revient à définir une fonction de signification $u$.
 La complexité du schéma de stéganographie est peu ou prou celle du calcul
-de cette carte, et elle est élevée (cf partie~\ref{XXXXXXXX}) dans le cas
+de cette carte, et elle est élevée  dans le cas
 de ces algorithmes.
 Nous avons proposé un algorithme~\cite{ccg15:ij}
 de complexité beaucoup plus faible 
diff --git a/stegoyousra.tex b/stegoyousra.tex
index cfda20f..7f2add3 100644
--- a/stegoyousra.tex
+++ b/stegoyousra.tex
@@ -34,6 +34,7 @@ données et prouvées (Section~\ref{sec:second} et Section~\ref{sec:poly}).
 Une adaptation d'une fonction de distorsion existante est étudiée
 en Section~\ref{sec:distortion} et des expériences sont présentées 
 en Section~\ref{sec:experiments}.
+Ce chapitre a été publié dans~\cite{ccfg16:ip}.
 
 
 
@@ -736,23 +737,16 @@ compte des variations dans celle-ci. Les dérivées secondes sont certes faciles
 %\end{figure}
 
 
-% \section{Conclusion}
-
-% The first contribution of this paper is to propose of a distortion
-% function which is based on second order derivatives. These
-% partial derivatives allow to accurately compute 
-% the level curves and thus to look favorably on pixels
-% without clean level curves. 
-% Two approaches to build these derivatives have been proposed.
-% The first one is based on revisiting kernels usually embedded 
-% in edge detection algorithms. 
-% The second one is based on the polynomial approximation
-% of the bitmap image.
-% These two methods have been completely implemented.
-% The first experiments have shown that the security level 
-% is slightly inferior the one of the most stringent approaches. These first promising results encourage us to deeply investigate this research direction. 
-
-% Future works aiming at improving the security of this proposal are planned as follows. The authors want first to focus on other approaches to provide second order derivatives with larger discrimination power.
-% Then, the objective will be to deeply investigate whether the H\"older norm is optimal when the objective is to avoid null second order derivatives, and to give priority to the largest second order values.
-
-
+\section{Conclusion}
+La principale contribution de ce chapitre est de proposer des 
+fonctions de distorsion basées sur des approximations de dérivées 
+secondes, l'idée sous-jacente étant qu'une zone où les lignes de niveau 
+ne sont pas clairement définies est peu prédictible.
+Deux approches d'approximation ont été présentées.
+La première  basée 
+sur un produit de convolution, exploite des noyaux déjà intégrés dans des
+algorithmes de détection de bords.
+La seconde s'appuie sur une interpolation polynomiale de l'image.
+Ces deux méthodes onté été complètement implantées et leur sécurité 
+face à des stéganalyseurs a été étudiée. Les résultats encouragent 
+à poursuivre dans cette direction.
\ No newline at end of file
-- 
2.39.5


From 416d383eafc79d519cc2910697507e81bdc0d3c7 Mon Sep 17 00:00:00 2001
From: couchot <couchot@localhost.localdomain>
Date: Wed, 31 Aug 2016 11:46:10 +0200
Subject: [PATCH 12/16] preuve promela traduite

---
 annexePromelaProof.tex | 34 +++++++++++++++++++---------------
 1 file changed, 19 insertions(+), 15 deletions(-)

diff --git a/annexePromelaProof.tex b/annexePromelaProof.tex
index 881383c..3675094 100644
--- a/annexePromelaProof.tex
+++ b/annexePromelaProof.tex
@@ -239,15 +239,17 @@ Grace à~\Equ{eq:correct_retrieve} déjà prouvée, on peut conclure la preuve.
 
 
 \begin{lemma}
-  Bounding the size of channels  to $\textit{max} = \delta_0$ is sufficient when
-  simulating a DDN where delays are bounded by $\delta_0$.
+  Borner la taille du cannal à $\textit{max} = \delta_0$ est suffisant 
+  lorsqu'on simule un système dynamique dont les délais sont bornés par  
+  $\delta_0$.
 \end{lemma}
 
 \begin{Proof}
-  For  any $i$,  $j$, at  each  iteration $t+1$,  thanks to  bounded delays  (by
-  $\delta_0$),  element $i$  has to  know at  worst $\delta_0$  values  that are
-  $X_j^{t}$, \ldots, $X_j^{t-\delta_0+1}$.  They  can be stored into any channel
-  of size $\delta_0$.
+  Pour chaque $i$ et $j$, à chaque itération $t+1$, comme les délais sont bornés par 
+  $\delta_0$,  l'élément $i$  doit connaître au plus $\delta_0$ 
+  valeurs qui sont
+  $X_j^{t}$, \ldots, $X_j^{t-\delta_0+1}$. Elles peuvent être mémorisées dans n'importe quel cannal
+  de taille $\delta_0$.
 \end{Proof}
 
 
@@ -259,12 +261,12 @@ Grace à~\Equ{eq:correct_retrieve} déjà prouvée, on peut conclure la preuve.
 %   to  convenient  available  dates  $D_{ji}$.   It is  then  easy  to  construct
 %   corresponding iterations of the DDN that are convergent.
 % \ANNOT{quel sens donnes-tu a \emph{convenient} ici ?}
-
-  Let us show the contraposition of the theorem.  The previous lemmas have shown
-  that for any  sequence of iterations of the DDN, there  exists an execution of
-  the PROMELA  model that  simulates them.   If some iterations  of the  DDN are
-  divergent, then  they prevent  the PROMELA model  from stabilizing,  \textit{i.e.},  not
-  verifying the LTL property (\ref{eq:ltl:conv}).
+  Montrons la conraposée du théorème.
+  Le lemme  précédent a montré que pour chaque séquence d'itérations du système dynamique discret,
+  Il existe une exécution du modèle PROMELA qui la simule.
+  Si des itérations du système dynamique discret sont divergentes, leur exécution vont empêcher 
+  le modèle PROMELA de se stabiliser,  \textit{i.e.} 
+  ce dernier ne verifiera pas la propriété LTL (\ref{eq:ltl:conv}).
 \end{Proof}
 
 
@@ -281,9 +283,11 @@ Grace à~\Equ{eq:correct_retrieve} déjà prouvée, on peut conclure la preuve.
 \promelacomplete*
 
 \begin{Proof}
-  For models $\psi$  that do not verify the  LTL property (\ref{eq:ltl:conv}) it
-  is easy  to construct corresponding iterations  of the DDN,  whose strategy is
-  pseudo-periodic since weak fairness property is taken into account.
+  Pour chaque  modele  $\psi$ 
+  qui ne vérifie pas la propriété  LTL  (\ref{eq:ltl:conv}), 
+  il est immédiat de construire les itérations correpondantes du
+  système dynamique, dont la stratégie est pseudo périodique en raison 
+  de la propriété d'équité faible..
 
 %   i.e. iterations that  are divergent.   Executions are
 %   performed under weak  fairness property; we then detail  what are continuously
-- 
2.39.5


From d69000ebda300fc836232f34cebb88ddfce4ac98 Mon Sep 17 00:00:00 2001
From: couchot <couchot@localhost.localdomain>
Date: Thu, 1 Sep 2016 11:33:43 +0200
Subject: [PATCH 13/16] chapitre chaos repris

---
 11FCT.tex                        |  7 ++++---
 12TIPE.tex                       | 35 ++++++++++++++++----------------
 15TSI.tex                        | 32 ++++++++++++++++++++---------
 annexesccg.tex                   |  6 ++++--
 caracgeneralise.tex              | 19 +++++------------
 demandeInscription/president.tex |  3 ++-
 demandeInscription/synthese.tex  |  4 +---
 main.tex                         | 24 +++++++++++-----------
 preuveDistanceGeneralisee.tex    |  3 ++-
 9 files changed, 70 insertions(+), 63 deletions(-)

diff --git a/11FCT.tex b/11FCT.tex
index 45d91c9..498af15 100644
--- a/11FCT.tex
+++ b/11FCT.tex
@@ -55,7 +55,8 @@ $f_j$ et qui permet de n'engendrer que des fonctions
 $f$ dont le  graphe d'itérations
  $\textsc{giu}(f)$  est fortement connexe.
 
-\begin{theorem}\label{th:Adrien}
+\begin{restatable}{theorem}{thAdrien}
+\label{th:Adrien}
 Soit $f$ une fonction de $\Bool^{\mathsf{N}}$ vers lui-même telle que:
 \begin{enumerate}
 \item 
@@ -68,11 +69,11 @@ chaque sommet de $\Gamma(f)$ est accessible depuis un sommet qui possède
 une boucle négative.
 \end{enumerate}
 Alors, $\textsc{giu}(f)$ est fortement connexe.
-\end{theorem}
+\end{restatable}
 
 La preuve de ce théorème est donnée en annexe~\ref{anx:sccg}.
 
-Illustrons ce théorème par un exemple. On considère par le graphe d'interactions 
+Illustrons ce théorème par un exemple. On considère le graphe d'interactions 
 $\Gamma(f)$ donné en figure~\ref{fig:Adrien:G}. 
 Il vérifie le théorème~\ref{th:Adrien}: 
 toutes les fonctions $f$ possédant un tel graphe d'interactions
diff --git a/12TIPE.tex b/12TIPE.tex
index 7ebcfde..afd6077 100644
--- a/12TIPE.tex
+++ b/12TIPE.tex
@@ -8,11 +8,11 @@ $\Bool^{{\mathsf{N}}}$ dans lui-même.
 
 Dans le schéma unaire, à la  $t^{\textrm{ème}}$ itération, 
 seul le  $s_{t}^{\textrm{ème}}$ 
-composant (entre 1 et $n$) est mis à jour.
+composant (entre 1 et $\mathsf{N}$) est mis à jour.
 Pour une stratégie $s = \left(s_t\right)_{t \in \mathds{N}}$ 
 (\textit{i.e.}, une séquence d'indices
-de $\llbracket 1;\mathsf{N} \rrbracket$), on peut définir
-la fonction $F_{f_u}: \Bool^{\mathsf{N}} \times \llbracket1;\mathsf{N}\rrbracket$
+de $[\mathsf{N}]$), on peut définir
+la fonction $F_{f_u}: \Bool^{\mathsf{N}} \times [\mathsf{N}]$
 vers $\Bool^\mathsf{N}$ par 
 \[
 F_{f_u}(x,i)=(x_1,\dots,x_{i-1},f_i(x),x_{i+1},\dots,x_\mathsf{N}).
@@ -20,7 +20,7 @@ F_{f_u}(x,i)=(x_1,\dots,x_{i-1},f_i(x),x_{i+1},\dots,x_\mathsf{N}).
 
 Dans le schéma des itérations unaires pour une configuration initiale
 $x^0\in\Bool^\mathsf{N}$ et une stratégie $s\in
-\llbracket1;\mathsf{N}\rrbracket^\Nats$, les configurations $x^t$
+[\mathsf{N}]^\Nats$, les configurations $x^t$
 sont définies par la récurrence
 \begin{equation}\label{eq:asyn}
 x^{t+1}=F_{f_u}(x^t,s_t).
@@ -29,7 +29,7 @@ x^{t+1}=F_{f_u}(x^t,s_t).
 
 On peut alors construire l'espace 
 $\mathcal{X}_u =
-\Bool^{{\mathsf{N}}} \times \llbracket1;{\mathsf{N}}\rrbracket^{\Nats}$ 
+\Bool^{{\mathsf{N}}} \times [{\mathsf{N}}]^{\Nats}$ 
 et la fonction d'itération $G_{f_u}$ définie  de 
 $\mathcal{X}_u$ 
 dans lui-même par 
@@ -39,8 +39,8 @@ G_{f_u}(x,s)=(F_{f_u}(x,s_0),\sigma(s)).
 \end{equation}
 
 Dans cette définition, la fonction 
-$\sigma: \llbracket1;{\mathsf{N}}\rrbracket^{\Nats} \longrightarrow
- \llbracket1;{\mathsf{N}}\rrbracket^{\Nats} 
+$\sigma: {\mathsf{N}}]^{\Nats} \longrightarrow
+ [{\mathsf{N}}]^{\Nats} 
 $
 décale
 la stratégie fournie en argument d'un élément vers la gauche en supprimant 
@@ -59,7 +59,7 @@ La section suivante introduit une métrique sur $\mathcal{X}_u$.
 Sur $\mathcal{X}_u$, 
 on définit la distance $d$ entre les points $X=(x,s)$ et
 $X'=(x',s')$ de $\mathcal{X}_u$ par 
-\[
+\begin{equation}
 d(X,X')= d_H(x,x')+d_S(s,s'),~\textrm{où}~
 \left\{
 \begin{array}{l}
@@ -67,7 +67,8 @@ d(X,X')= d_H(x,x')+d_S(s,s'),~\textrm{où}~
 \displaystyle{d_S(s,s')=\frac{9}{n}\sum_{t\in\Nats}\frac{|s_t-s'_t|}{10^{t+1}}}.
 \end{array}
 \right.\,.
-\]
+\end{equation}
+
 On note que dans le calcul de $d_H(x,x')$-- 
 appelée distance de Hamming entre $x$ et $x'$-- 
 les termes $x_i$ et $x'_i$ sont considérés comme des entiers naturels 
@@ -92,8 +93,8 @@ apporte une réponse à cette question.
 chaotiques $G_{f_u}$ sur $\mathcal{X}_u$}
 
 
-On peut tout d'abord démontrer que pour toute fonction booléenne $f$, 
-$G_{f_u}$ est continue sur $\mathcal{X}_u$ (cf annexe~\ref{anx:cont}).   
+% On peut tout d'abord démontrer que pour toute fonction booléenne $f$, 
+% $G_{f_u}$ est continue sur $\mathcal{X}_u$ (cf annexe~\ref{anx:cont}).   
 
 Pour caractériser les fonctions rendant chaotiques dans $\mathcal{X}_u$ les itérations de $G_{f_u}$ 
 on se focalise donc que sur la régularité et sur la transitivité de $G_{f_u}$.
@@ -103,16 +104,16 @@ $\mathcal{R}$ des fonctions régulières
 et $\mathcal{C}$ des fonctions chaotiques définis respectivement ci-dessous:
 \begin{itemize}
 \item   $\mathcal{T}   =    \left\{f   :   \mathds{B}^n   \to
-\mathds{B}^n \big/ G_{f_u} \textrm{ est transitive} \right\}$,
+\mathds{B}^n \textrm{ t. q. } G_{f_u} \textrm{ est transitive} \right\}$,
 \item   $\mathcal{R}   =    \left\{f   :   \mathds{B}^n   \to
-\mathds{B}^n \big/ G_{f_u} \textrm{ est régulière} \right\}$,
+\mathds{B}^n \textrm{ t. q. } G_{f_u} \textrm{ est régulière} \right\}$,
 \item   $\mathcal{C}   =    \left\{f   :   \mathds{B}^n   \to
-\mathds{B}^n  \big/  G_{f_u}  \textrm{  est chaotique} \right\}$.
+\mathds{B}^n  \textrm{ t. q. }  G_{f_u}  \textrm{  est chaotique} \right\}$.
 \end{itemize}
 
 
-On énonce les théorèmes successifs suivants.
-Leur preuve est donnée en annexe~\ref{anx:chaos:unaire}.
+On énonce les théorèmes successifs suivants dont les preuves sont données 
+dans~\cite{guyeux10}.
 
 \begin{theorem} $G_{f_u}$  est transitive si et seulement si 
  $\textsc{giu}(f)$ est fortement connexe.
@@ -127,7 +128,7 @@ On peut conclure  que $\mathcal{C} = \mathcal{R} \cap \mathcal{T}
 
 \begin{theorem}%[Characterization of $\mathcal{C}$]
 \label{Th:CaracIC}  
-Soit $f:\Bool^n\to\Bool^n$. La fonction $G_{f_u}$ est chaotique  
+Soit $f:\Bool^{\mathsf{N}}\to\Bool^{\mathsf{N}}$. La fonction $G_{f_u}$ est chaotique  
 si et seulement si $\textsc{giu}(f)$ est fortement connexe.
 \end{theorem}
 
diff --git a/15TSI.tex b/15TSI.tex
index f2fe17b..138036f 100644
--- a/15TSI.tex
+++ b/15TSI.tex
@@ -7,7 +7,7 @@ On reprend ici le même plan que dans la section précédente.
 
 Dans le schéma généralisé, à la  $t^{\textrm{ème}}$ itération, 
 c'est l'ensemble 
-des $s_{t}^{\textrm{ème}}$ éléments (inclus dans $[n]$) qui 
+des $s_{t}^{\textrm{ème}}$ éléments (inclus dans $[{\mathsf{N}}]$) qui 
 sont  mis à jour (cf. équation~(\ref{eq:schema:generalise})).
 On redéfinit la fonction la fonction
   $F_{f_g}:  \Bool^{\mathsf{N}} \times \mathcal{P}(\{1, \ldots, \mathsf{N}\}) 
@@ -36,7 +36,7 @@ configurations $x^t$ sont définies par la récurrence
   \] 
   où la fonction $\sigma$ est définie comme à la section précédente.
   A nouveau, les itérations généralisées 
-  de $f$ induites par $x^0$ et la  stratégie $S$.
+  de $f$ induites par $x^0$ et la  stratégie $S$ 
   décrivent la même orbite que les 
   itérations parallèles de $G_{f_g}$ depuis un point initial
   $X^0=(x^0,S)$ 
@@ -62,7 +62,8 @@ On considère l'espace $\mathcal{X}_g=\mathcal{P}(\{1, \ldots, {\mathsf{N}}\})^{
 \Bool^{\mathsf{N}}$ et
 on définit la distance $d$ entre les points $X=(S,x)$ et
 $X'=(S',x')$ de $\mathcal{X}_g$ par 
-\[
+
+\begin{equation}
 d(X,X')= d_H(x,x')+d_S(S,S'),~\textrm{où}~
 \left\{
 \begin{array}{l}
@@ -70,7 +71,8 @@ d(X,X')= d_H(x,x')+d_S(S,S'),~\textrm{où}~
 \displaystyle{d_S(S,S')=\frac{9}{{\mathsf{N}}}\sum_{t\in\Nats}\frac{|S_t \Delta S'_t|}{10^{t+1}}}.
 \end{array}
 \right.\,.
-\]
+\label{eq:distance:Xg}
+\end{equation}
 
 La fonction $d$ est une somme de deux fonctions.
 La fonction $d_H$ est la distance de Hamming; il est aussi établi que la 
@@ -79,7 +81,7 @@ Ainsi, pour montrer que $d$ est aussi une distance, il suffit
 de montrer que $d_S$ en une aussi, ce qui est fait en annexe~\ref{anx:distance:generalise}.
 
 La section suivante caractérise les fonctions $f$ qui sont  
-chaotiques pour le schéma généralisées.
+chaotiques pour le schéma généralisé.
 
 \subsection{Caractérisation des fonctions rendant 
 chaotiques $G_{f_g}$ sur $\mathcal{X}_g$}
@@ -88,13 +90,23 @@ en les adaptant à  $G_{f_g}$.
 On a les théorèmes suivants dont les preuves sont données en 
 annexe~\ref{anx:chaos:generalise}.
 
-\begin{theorem} $G_{f_g}$  est transitive si et seulement si 
+
+
+\begin{restatable}{theorem}{caractransitivegeneralise}
+\label{Theo:carac:transitive:gen}
+$G_{f_g}$  est transitive si et seulement si 
  $\textsc{gig}(f)$ est fortement connexe.
-\end{theorem}
+\end{restatable}
 
-\begin{theorem}
-\label{Prop: T est dans R:g} $\mathcal{T} \subset \mathcal{R}$.
-\end{theorem}
+
+
+\begin{restatable}{theorem}{caracsubgeneralise}
+\label{Prop: T est dans R:g}
+ $\mathcal{T} \subset \mathcal{R}$.
+\end{restatable}
+
+On peut conclure  que $\mathcal{C} = \mathcal{R} \cap \mathcal{T}
+= \mathcal{T}$. On a alors la  caractérisation suivante:
 
 
 \begin{theorem}%[Characterization of $\mathcal{C}$]
diff --git a/annexesccg.tex b/annexesccg.tex
index 9d32b22..257f86c 100644
--- a/annexesccg.tex
+++ b/annexesccg.tex
@@ -59,8 +59,10 @@ $\textsc{giu}(f)^\alpha$ a un  arc de $h(x)$ vers $h(y)$.
 \end{Proof}
 
 
+On peut alors prouver le théorème:
+\thAdrien*
+
 \begin{Proof}
-du Théorème~\ref{th:Adrien}.
 La preuve se fait par induction sur ${\mathsf{N}}$. 
 Soit $f$ une fonction de $\Bool^{\mathsf{N}}$ dans lui-même et qui vérifie les hypothèses 
 du théorème.
@@ -106,7 +108,7 @@ D'après la seconde hypothèse,
 ${\mathsf{N}}$ n'a pas de boucle, \emph{i.e.}, la valeur de $f_{\mathsf{N}}(x)$
 ne dépend pas de la valeur de $x_{\mathsf{N}}$. 
 D'après la troisième hypothèse, 
-il existe $i\in \llbracket 1;{\mathsf{N}} \rrbracket$ tel que $G(f)$ a un arc de 
+il existe $i\in [{\mathsf{N}}]$ tel que $G(f)$ a un arc de 
 $i$ vers ${\mathsf{N}}$.
 Ainsi, il existe $x \in \Bool^{\mathsf{N}}$ tel que $f_{{\mathsf{N}}i}(x) \neq 0$ et donc 
 %$n$ n'est donc pas de degré zéro dans $G(f)$, \emph{i.e.} 
diff --git a/caracgeneralise.tex b/caracgeneralise.tex
index 54d99af..3014a6b 100644
--- a/caracgeneralise.tex
+++ b/caracgeneralise.tex
@@ -1,9 +1,8 @@
 
-Commençons par caractériser l'ensemble $\mathcal{T}$ des fonctions transitives:
+Commençons par caractériser l'ensemble $\mathcal{T}$ des fonctions transitives dans le cas 
+des itérations généralisées.
 
-\begin{theorem} $G_{f_g}$  est transitive si et seulement si 
- $\textsc{gig}(f)$ est fortement connexe.
-\end{theorem}
+\caractransitivegeneralise*
 
 \begin{Proof} 
 
@@ -55,9 +54,7 @@ par contraposée, on a la démonstration souhaitée.
 
 Prouvons à présent le théorème suivant: 
 
-\begin{theorem}
-\label{Prop: T est dans R:gp} $\mathcal{T} \subset \mathcal{R}$.
-\end{theorem}
+\caracsubgeneralise*
 
 
 \begin{Proof}  
@@ -88,10 +85,4 @@ choix de  $t_1$, on a  $d((x,S),(x,\tilde S))<\epsilon$.
 \end{Proof}
 
 On peut conclure  que $\mathcal{C} = \mathcal{R} \cap \mathcal{T}
-= \mathcal{T}$. On a alors la  caractérisation suivante:
-
-\begin{theorem}%[Characterization of $\mathcal{C}$]
-\label{Th:CaracIC:gp}  
-Soit $f:\Bool^{\mathsf{N}}\to\Bool^{\mathsf{N}}$. La fonction $G_{f_g}$ est chaotique  
-si et seulement si $\textsc{gig}(f)$ est fortement connexe.
-\end{theorem}
+= \mathcal{T}$. 
\ No newline at end of file
diff --git a/demandeInscription/president.tex b/demandeInscription/president.tex
index 4dd1c77..c01b381 100644
--- a/demandeInscription/president.tex
+++ b/demandeInscription/president.tex
@@ -41,7 +41,8 @@ Par le présent courrier, je vous sollicite pour une inscription à
 l'Habilitation à Diriger des Recheches (HDR).
 Je vous informe que je n'ai effectué aucune demande inscription dans une 
 autre université cette année universitaire et que cette requête
-constitue ma première demande.
+constitue ma seconde demande. Ma première demande d'inscription
+a été formulé le 31 aout 2015. 
 
 
 
diff --git a/demandeInscription/synthese.tex b/demandeInscription/synthese.tex
index 58ad6b7..4a31101 100755
--- a/demandeInscription/synthese.tex
+++ b/demandeInscription/synthese.tex
@@ -148,9 +148,7 @@ Je suis membre de l'équipe Algorithmique Numérique Distribuée (AND) du
 Département d'Informatique des Systèmes Complexes (DISC)
 du laboratoire FEMTO-ST. 
 Je relève de l'école doctorale 37 Sciences Pour l'Ingénieur et Microtechniques (SPIM) de l'UFC.
-Mon directeur de recherche pour cette HDR est Pr. J. {\sc Bahi} 
-du département DISC. Son avis, ainsi que celui du directeur de l'équipe (Pr. R. {\sc Couturier}, du directeur de l'école doctorale (PR. P. {\sc Lutz}) 
-et du directeur du département (Pr. O. {\sc Kouchnarenko}) sont donnés en annexes. 
+Les avis du directeur de l'équipe AND (Pr. R. {\sc Couturier} et du directeur du département DISC (Pr. O. {\sc Kouchnarenko}) sont donnés en annexes. 
 
 
 % \subsection{Avis du directeur de l'équipe}\label{sec:avis:directeur:equipe}
diff --git a/main.tex b/main.tex
index 33a3ba1..628e4f9 100644
--- a/main.tex
+++ b/main.tex
@@ -221,15 +221,16 @@ au chaos}
 La suite de ce document se focalise sur des systèmes dynamiques discrets qui ne 
 convergent pas. Parmi ceux-ci se trouvent ceux qui sont \og chaotiques\fg{}.
 La première section  de ce chapitre rappelle ce que sont les systèmes 
-dynamiques chaotiques et leur caractéristiques. Celles-ci dépendent 
-tout d'abord de la stratégie itérée. La section~\ref{sec:TIPE12} 
-se focalise sur le schéma unaire tandis que la section~\ref{sec:chaos:TSI}
-considère le mode généralisé. Pour chacun de ces modes, 
-une distance est définie. Finalement, la section~\ref{sec:11FCT}
+dynamiques chaotiques et leur caractéristiques.
+La section~\ref{sec:TIPE12}, qui est une reformulation de~\cite{guyeux10},
+se focalise sur le schéma unaire. Elle est rappelée pour avoir un document se 
+suffisant à lui-même.
+La section~\ref{sec:chaos:TSI} étend ceci au mode généralisé. Pour chacun de ces modes, 
+une métrique est définie. Finalement, la section~\ref{sec:11FCT}
 exhibe des conditions suffisantes premettant d'engendrer 
 des fonctions chaotiques seon le mode unaire.
 Les sections~\ref{sec:TIPE12} et~\ref{sec:11FCT} ont été publiées 
-dans~\cite{bcgr11:ip}.
+dans~\cite{bcg11:ij,bcgr11:ip}.
 
 \section{Systèmes dynamiques chaotiques selon Devaney}
 \label{subsec:Devaney}
@@ -354,13 +355,12 @@ du chapitre 8}
 \chapter{Preuves sur les systèmes chaotiques}
 
 
-\section{Continuité de $G_f$ dans $(\mathcal{X}_u,d)$}\label{anx:cont}
-\input{annexecontinuite.tex}
+%\section{Continuité de $G_f$ dans $(\mathcal{X}_u,d)$}\label{anx:cont}
+%\input{annexecontinuite.tex}
 
 
-\section{Caractérisation des fonctions $f$ rendant chaotique $G_{f_u}$ dans $(\mathcal{X}_u,d)$}\label{anx:chaos:unaire}
-\input{caracunaire.tex}
-
+%\section{Caractérisation des fonctions $f$ rendant chaotique $G_{f_u}$ dans $(\mathcal{X}_u,d)$}\label{anx:chaos:unaire}
+%\input{caracunaire.tex}
 
 \section{Preuve que $d$ est une distance sur $\mathcal{X}_g$}\label{anx:distance:generalise}
 \input{preuveDistanceGeneralisee}
@@ -370,7 +370,7 @@ du chapitre 8}
 \input{caracgeneralise.tex}
 
 
-\section{Théorème~\ref{th:Adrien}}\label{anx:sccg}
+\section{Conditions suffisantes pour un $\textsc{giu}(f)$ fortement connexe \label{anx:sccg}}
 \input{annexesccg}
 
 
diff --git a/preuveDistanceGeneralisee.tex b/preuveDistanceGeneralisee.tex
index 9b7f3b6..c8a6c5c 100644
--- a/preuveDistanceGeneralisee.tex
+++ b/preuveDistanceGeneralisee.tex
@@ -3,7 +3,8 @@ on définit
 \[
 \displaystyle{d_S(S,S')=\frac{9}{{\mathsf{N}}}\sum_{t\in\Nats}\frac{|S_t \Delta S'_t|}{10^{t+1}}}.
 \]
-Montrons que $d_S$ est une distance sur $\mathcal{P}(\{1, \ldots, {\mathsf{N}}\})$.
+Montrons que $d_S$ est une distance sur $\mathcal{P}(\{1, \ldots, {\mathsf{N}}\})$ et ainsi 
+que $d$ définie à l'equation~(\ref{eq:distance:Xg}) est une distance.
 
 
 
-- 
2.39.5


From 75aa438e61284f634375e2c1e62c79f2af12678f Mon Sep 17 00:00:00 2001
From: couchot <couchot@localhost.localdomain>
Date: Fri, 2 Sep 2016 17:56:54 +0200
Subject: [PATCH 14/16] correction prng chaos chap 5

---
 12TIPE.tex     |   6 +-
 14Secrypt.tex  |   2 +-
 15RairoGen.tex | 142 +++++++++++++++++++++++++----------------------
 chaosANN.tex   | 147 ++++++++++++++++++++++++++-----------------------
 4 files changed, 157 insertions(+), 140 deletions(-)

diff --git a/12TIPE.tex b/12TIPE.tex
index afd6077..7c04d49 100644
--- a/12TIPE.tex
+++ b/12TIPE.tex
@@ -14,9 +14,11 @@ Pour une stratégie $s = \left(s_t\right)_{t \in \mathds{N}}$
 de $[\mathsf{N}]$), on peut définir
 la fonction $F_{f_u}: \Bool^{\mathsf{N}} \times [\mathsf{N}]$
 vers $\Bool^\mathsf{N}$ par 
-\[
+
+\begin{equation}
 F_{f_u}(x,i)=(x_1,\dots,x_{i-1},f_i(x),x_{i+1},\dots,x_\mathsf{N}).
-\]
+\label{eq:iterations:unaires}
+\end{equation}
 
 Dans le schéma des itérations unaires pour une configuration initiale
 $x^0\in\Bool^\mathsf{N}$ et une stratégie $s\in
diff --git a/14Secrypt.tex b/14Secrypt.tex
index bc59ba1..10a4f85 100644
--- a/14Secrypt.tex
+++ b/14Secrypt.tex
@@ -23,7 +23,7 @@ section~\ref{sec:prng:gray:general}.
 Finalement, des instances de PRNGS engendrés selon les méthodes détaillées dans 
 ce chapitre sont présentés en section~\ref{sec:prng;gray:tests}.
 Les sections~\ref{sec:plc} à~\ref{sub:gray} ont été publiées 
-à~\ref{chgw+14:oip}.
+à~\cite{chgw+14:oip}.
 
 
 % This aim of this section is to show 
diff --git a/15RairoGen.tex b/15RairoGen.tex
index a950d85..06f84b1 100644
--- a/15RairoGen.tex
+++ b/15RairoGen.tex
@@ -5,15 +5,17 @@ a de \og bonnes\fg{} propriétés chaotiques,
 le mot $x^b$ devrait  \og sembler ne plus dépendre\fg{} de $x^0$.
 On peut penser à exploiter une de ces fonctions $G_f$ 
 comme un générateur aléatoire. 
-Enfin, un bon générateur aléatoire se doit de 
-fournir  des nombres selon une distribution uniforme 
+
+Ce chapitre  présente une application directe
+de la théorie développée ci-avant
+à la génération de nombres pseudo aléatoires. 
+
+
 La suite de ce document donnera
 une condition nécessaire est suffisante pour que
 cette propriété soit satisfaite.
 
 
-Cette section présente une application directe de la théorie développée ci-avant
-à la génération de nombres pseudo aléatoires. 
 On présente tout d'abord le générateur
 basé sur des fonctions chaotiques (section~\ref{sub:prng:algo}), 
 puis comment intégrer la contrainte de distribution uniforme
@@ -33,16 +35,15 @@ L'approche est évaluée dans la dernière section.
 \begin{algorithm}[h]
 %\begin{scriptsize}
 \KwIn{une fonction $f$, un nombre d'itérations $b$, 
-une configuration initiale $x^0$ ($n$ bits)}
-\KwOut{une configuration $x$ ($n$ bits)}
+une configuration initiale $x^0$ (${\mathsf{N}}$ bits)}
+\KwOut{une configuration $x$ (${\mathsf{N}}$ bits)}
 $x\leftarrow x^0$\;
-$k\leftarrow b $\;
 %$k\leftarrow b + \textit{XORshift}(b+1)$\;
-\For{$i=1,\dots,k$}
+\For{$i=1,\dots,b$}
 {
-$s\leftarrow{\textit{Random}(n)}$\;
+$s\leftarrow{\textit{Random}({\mathsf{N}})}$\;
 %$s\leftarrow{\textit{XORshift}(n)}$\;
-$x\leftarrow{F_{f_u}(s,x)}$\;
+$x\leftarrow{F_{f_u}(x,s)}$\;
 }
 return $x$\;
 %\end{scriptsize}
@@ -56,11 +57,11 @@ aléatoires comme dans l'algorithme~\ref{CI Algorithm} donné ci-dessous.
 
 
 Celui-ci prend en entrée: une fonction $f$;
-un entier $b$, qui assure que le nombre d'itérations
-est compris entre $b+1 $ et  $2b+1$ (et donc supérieur à $b$) 
+un entier $b$, qui est le nombre d'itérations à effectuer entre deux sorties 
 et une configuration initiale $x^0$.
 Il retourne une nouvelle configuration $x$ en appliquant 
-la fonction $F_{f_u}$ vue au chapitre~\ref{chap:carachaos} et correspondant 
+la fonction $F_{f_u}$ (équation~\ref{eq:iterations:unaires}
+vue au chapitre~\ref{chap:carachaos}) et correspondant 
 à des itérations unaires.
 En interne, il exploite un algorithme de génération
 de nombres pseudo aléatoires donné en paramètre.
@@ -105,10 +106,13 @@ Notre approche vise a donner des propriétés de chaos a ce générateur embarqu
 Nous avons vu au chapitre~\ref{chap:carachaos} que 
 $G_{f_u}$ est chaotique dans l'espace $\mathcal{X}_u$ 
 si et seulement le graphe d'itérations $\textsc{giu}(f)$ 
-doit être fortement connexe.
+est fortement connexe.
 Pour $b=1$, l'algorithme itère la fonction $F_{f_u}$.
-Regardons comment l'uniformité de la distribution a
-contraint la fonction.
+
+Pour simuler au mieux l'aléa, un bon générateur de nombre pseudo-aléatoires
+se doit de fournir  des nombres selon une distribution uniforme.
+Regardons comment l'uniformité de la distribution 
+contraint la fonction $f$ à itérer.
 
 \subsection{Un générateur à sortie uniformément distribuée}\label{sub:prng:unif}
 
@@ -121,7 +125,7 @@ Enfin, une matrice stochastique de taille $n \times n$ est régulière
 si  la propriété suivante est établie:
 $$\exists k \in \mathds{N}^\ast, \forall i,j \in \llbracket 1; n \rrbracket, M_{ij}^k>0.$$
 
-On énonce enfin le théorème suivant liant les 
+On énonce le théorème classique suivant liant les 
 vecteur de probabilités 
 et les chaînes de Markov.
 
@@ -316,10 +320,12 @@ On énonce directement le théorème suivant dont la preuve est donnée en annex
 
 \subsection{Quelques exemples}
 
-On reprend le graphe d'interactions $\Gamma(f)$ donné en figure~\ref{fig:G} à la section~\ref{sec:11FCT}.
-On a vu qu'il y avait  520 fonctions $f$ non isomorphes de graphe d'interactions  $\Gamma(f)$, 
-dont seulement 16 d'entre elles possèdent une matrice doublement stochastique.
+On considère le graphe d'interactions $\Gamma(f)$ donné
+en figure~\ref{fig:G}. C'est le même qui a été présenté
+à la section~\ref{sec:11FCT}.
+On a vu qu'il y avait  520 fonctions $f$ non isomorphes de graphe d'interactions  $\Gamma(f)$. 
 
+Seulement 16 d'entre elles possèdent une matrice doublement stochastique.
 La figure~\ref{fig:listfonction} explicite ces 16 fonctions en 
 définissant les images des éléments de la liste
 0, 1, 2,\ldots, 14, 15 en respectant  l'ordre.
@@ -343,11 +349,11 @@ ce vecteur au vecteur $\pi=(\frac{1}{2^n},\ldots,\frac{1}{2^n})$
 Ainsi, on a 
 \begin{equation}
 b = \max\limits_{i \in \llbracket 1, 2^n \rrbracket} 
-\{
-\min \{
+\left\{
+\min \left\{
  t \mid t \in \Nats, \vectornorm{e_i M_f^t - \pi} < 10^{-4}
-\}
-\}. 
+\right\}
+\right\}. 
 \label{eq:mt:ex}
 \end{equation}
 
@@ -444,7 +450,7 @@ Ceci est difficilement compatible avec la volonté d'avoir une sortie uniformém
 se rapprocher de cette distribution nécessite en effet un nombre plus élevé
 d'itérations $b$ entre chaque sortie. Par exemple, dans l'exemple précédent, il est nécessaire 
 d'itérer au moins 42 fois entre chaque sortie pour suivre une loi uniforme à $10^{-4}$ près.
-Montrer les sous-séquences de suites chaotiques ainsi générées  demeurent chaotiques
+Montrer que les sous-séquences de suites chaotiques ainsi générées  demeurent chaotiques
 est l'objectif de la section suivante.
 
 
@@ -464,15 +470,14 @@ Intuitivement, c'est le nombre d'itérations qu'il est autorisé de faire.
 On ordonne les $\mathsf{p}$ éléments de $\mathcal{P}$ comme suit: 
 $\mathcal{P} = \{ p_1, p_2, \hdots, p_\mathsf{p}\}$
 et $p_1< p_2< \hdots < p_\mathsf{p}$.
+
 Dans l'algorithme~\ref{CI Algorithm}, 
 $\mathsf{p}$ vaut 1 et  $p_1=b$. 
-
-
 Cet  algorithme peut être vu comme $b$ compostions de la fonction $F_{f_u}$.
 Ceci peut cependant se généraliser à $p_i$, $p_i \in \mathcal{P}$,
 compositions fonctionnelles de $F_{f_u}$.
 Ainsi, pour chaque $p_i \in \mathcal{P}$, on construit la fonction
-$F_{{f_u},p_i} :  \mathds{B}^\mathsf{N} \times \llbracket 1, \mathsf{N} \rrbracket^{p_i}
+$F_{{f_u},p_i} :  \mathds{B}^\mathsf{N} \times [\mathsf{N}]^{p_i}
 \rightarrow \mathds{B}^\mathsf{N}$ définie par
 
 $$
@@ -484,11 +489,11 @@ $$
 on construit l'espace 
  $\mathcal{X}_{\mathsf{N},\mathcal{P}}=  \mathds{B}^\mathsf{N} \times \mathds{S}_{\mathsf{N},\mathcal{P}}$, où
 $\mathds{S}_{\mathsf{N},\mathcal{P}}=
-\llbracket 1, \mathsf{N} \rrbracket^{\Nats}\times 
+[\mathsf{N}]^{\Nats}\times 
 \mathcal{P}^{\Nats}$. 
 Chaque élément de l'espace est une paire où le premier élément est 
 un $\mathsf{N}$-uplet de  $\Bool^{\mathsf{N}}$ (comme dans $\mathcal{X}_u$).
-Le second élément est aussi une paire $((u^k)_{k \in \Nats},(v^k)_{k \in \Nats})$ de suites infinies.
+Le second élément est une paire $((u^k)_{k \in \Nats},(v^k)_{k \in \Nats})$ de suites infinies.
 La suite $(v^k)_{k \in \Nats}$ définit combien d'itérations sont exécutées au temps $k$ entre deux sorties.
 La séquence $(u^k)_{k \in \Nats}$ définit quel élément est modifié (toujours au temps $k$).
 
@@ -529,7 +534,7 @@ Soit  $x=(e,s)$ et $\check{x}=(\check{e},\check{s})$ dans
 $\mathcal{X}_{\mathsf{N},\mathcal{P}} = \mathds{B}^\mathsf{N} \times \mathds{S}_{\mathsf{N},\mathcal{P}} $,
 où $s=(u,v)$ et $\check{s}=(\check{u},\check{v})$ sont dans $ \mathds{S}_{\mathsf{N},\mathcal{P}} = 
 \mathcal{S}_{\llbracket 1, \mathsf{N} \rrbracket} \times \mathcal{S}_\mathcal{P}$. 
-\begin{itemize}
+\begin{enumerate}
 \item $e$ et $\check{e}$ sont des entiers appartenant à $\llbracket 0, 2^{\mathsf{N}-1} \rrbracket$. 
 La distance de Hamming $d_{\mathds{B}^\mathsf{N}}$ sur entre les 
 décompositions binaires de $e$ et de $\check{e}$ (\textit{i.e.}, le 
@@ -544,7 +549,7 @@ $\check{u}^{\check{v}^0}, \check{u}^{\check{v}^0+1}, \hdots, \check{u}^{\check{v
 Plus précisément, soit 
 $p = \lfloor \log_{10}{(\max{\mathcal{P}})}\rfloor +1$ et 
 $n = \lfloor \log_{10}{(\mathsf{N})}\rfloor +1$.
-\begin{itemize}
+\begin{enumerate}
 \item Les $p$ premiers éléments de $d(x,\check{x})$ sont $|v^0-\check{v}^0|$ 
   écrits en base 10 et sur $p$ indices;
 \item les $n\times \max{(\mathcal{P})}$ éléments suivants servent 
@@ -552,7 +557,7 @@ $n = \lfloor \log_{10}{(\mathsf{N})}\rfloor +1$.
   $\check{u}^0, \check{u}^1, \hdots, \check{u}^{\check{v}^0-1}$. 
   Les $n$ premiers éléments sont $|u^0-\check{u}^0|$. Il sont suivis de 
 $|u^1-\check{u}^1|$ écrits à l'aide de $n$ éléments, etc.
-\begin{itemize}
+\begin{enumerate}
 \item Si
 $v^0=\check{v}^0$,
 alors le processus se continue jusqu'à $|u^{v^0-1}-\check{u}^{\check{v}^0-1}|$ et la 
@@ -563,10 +568,10 @@ $p+n\times \max{(\mathcal{P})}$ éléments.
 éléments sont $|u^0-\check{u}^0|$, ..., $|u^{v^0-1}-\check{u}^{v^0-1}|$,
 $\check{u}^{v^0}$ (sur $n$ éléments), ..., $\check{u}^{\check{v}^0-1}$ (sur $n$ éléments), suivi par des 0, si besoin.
 \item Le cas $v^0>\check{v}^0$ est similaire, et donc omis
-\end{itemize}
+\end{enumerate}
 \item Les $p$ suivants sont $|v^1-\check{v}^1|$, etc.
-\end{itemize}
-\end{itemize}
+\end{enumerate}
+\end{enumerate}
 
 
 La fonction $d$ peut se formaliser comme suit:
@@ -606,8 +611,10 @@ $\check{s}=\left\{
 \check{v}=2,1,...
 \end{array}
 \right.$.
-
-Ainsi $d_{\mathds{S}_{\mathsf{N},\mathcal{P}}}(s,\check{s}) = 0.010004000000000000000000011005 ...$
+Ainsi 
+\[
+d_{\mathds{S}_{\mathsf{N},\mathcal{P}}}(s,\check{s}) = 
+0.01~0004000000000000000000~01~1005 \dots\]
 En effet, les $p=2$ premiers éléments sont  01, c'est-à-dire 
 $|v^0-\check{v}^0|=1$, 
 et on utilise $p$ éléments pour représenter cette différence
@@ -621,35 +628,35 @@ la chaîne obtenue a $n\times \max{(\mathcal{P})}=22$ éléments, soit:
 De manière similaire, les $\check{v}^0=2$ premiers
 termes de $\check{u}$ sont représentés par 
 0604000000000000000000.
-LA valeur absolue de leur différence est égale à 
+La valeur absolue de leur différence est égale à 
 0004000000000000000000.
 Ces éléments sont concaténés avec 01. On peut construire alors le reste de 
 la séquence.
 \end{xpl}
 
 
-\begin{xpl}
-On considère à présent que  $\mathsf{N}=9$, que $\mathcal{P}=\{2,7\}$ et que
-$$s=\left\{
-\begin{array}{l}
-u=\underline{6,7,} ~ \underline{4,2,} ...\\
-v=2,2,...
-\end{array}
-\right.$$
-avec
-$$\check{s}=\left\{
-\begin{array}{l}
-\check{u}=\underline{4, 9, 6, 3, 6, 6, 7,} ~ \underline{9, 8}, ...\\
-\check{v}=7,2,...
-\end{array}
-\right.
-$$
+% \begin{xpl}
+% On considère à présent que  $\mathsf{N}=9$, que $\mathcal{P}=\{2,7\}$ et que
+% $$s=\left\{
+% \begin{array}{l}
+% u=\underline{6,7,} ~ \underline{4,2,} ...\\
+% v=2,2,...
+% \end{array}
+% \right.$$
+% avec
+% $$\check{s}=\left\{
+% \begin{array}{l}
+% \check{u}=\underline{4, 9, 6, 3, 6, 6, 7,} ~ \underline{9, 8}, ...\\
+% \check{v}=7,2,...
+% \end{array}
+% \right.
+% $$
 
-Ainsi $d_{\mathds{S}_{\mathsf{N},\mathcal{P}}}(s,\check{s}) = 0.5173633305600000...$, 
-puisque 
-$|v^0-\check{v}^0|=5$, $|4963667-6700000| = 1736333$, $|v^1-\check{v}^1|=0$,
-et $|9800000-4200000| = 5600000$.
-\end{xpl}
+% Ainsi $d_{\mathds{S}_{\mathsf{N},\mathcal{P}}}(s,\check{s}) = 0.5173633305600000...$, 
+% puisque 
+% $|v^0-\check{v}^0|=5$, $|4963667-6700000| = 1736333$, $|v^1-\check{v}^1|=0$,
+% et $|9800000-4200000| = 5600000$.
+% \end{xpl}
 
 
 
@@ -668,8 +675,9 @@ définit le graphe orienté $\textsc{giu}_{\mathcal{P}}(f)$ de la manière suiva
 %\item Each vertex has $\displaystyle{\sum_{i=1}^\mathsf{p} \mathsf{N}^{p_i}}$ arrows, namely all the $p_1, p_2, \hdots, p_\mathsf{p}$ tuples 
 %  having their elements in $\llbracket 1, \mathsf{N} \rrbracket $.
 \item il y a un arc libellé $u_0, \hdots, u_{p_i-1}$, $i \in \llbracket 1, \mathsf{p} \rrbracket$ entre les n{\oe}uds $x$ et $y$ si et seulement si $p_i$ est un élément de 
-$\mathcal{P}$ (\textit{i.e.}, on peut itérer $p_i$ fois), 
-chaque $u_k$ de la suite appartient à $[\mathsf{N}]$ et 
+$\mathcal{P}$ (\textit{i.e.}, on peut itérer $p_i$ fois), et pour chaque 
+$k$, $0 \le k \le p_i-1$, on a 
+ $u_k$ qui apaprtient à  $[\mathsf{N}]$ et 
 $y=F_{f_u,p_i} (x, (u_0, \hdots, u_{p_i-1})) $.
 \end{itemize}
 Il n'est pas difficile de constater que $\textsc{giu}_{\{1\}}(f)$ est $\textsc{giu}(f)$.
@@ -683,7 +691,7 @@ Il n'est pas difficile de constater que $\textsc{giu}_{\{1\}}(f)$ est $\textsc{g
     \subfigure[$\textsc{giu}_{\{2\}}(h)$]{
       \begin{minipage}{0.30\textwidth}
         \begin{center}
-          \includegraphics[height=4cm]{images/h2prng}
+          \includegraphics[scale=0.5]{images/h2prng}
         \end{center}
       \end{minipage}
       \label{fig:h2prng}
@@ -691,7 +699,7 @@ Il n'est pas difficile de constater que $\textsc{giu}_{\{1\}}(f)$ est $\textsc{g
     \subfigure[$\textsc{giu}_{\{3\}}(h)$]{
       \begin{minipage}{0.40\textwidth}
         \begin{center}
-          \includegraphics[height=4cm]{images/h3prng}
+          \includegraphics[scale=0.5]{images/h3prng}
         \end{center}
       \end{minipage}
       \label{fig:h3prng}
@@ -699,7 +707,7 @@ Il n'est pas difficile de constater que $\textsc{giu}_{\{1\}}(f)$ est $\textsc{g
     \subfigure[$\textsc{giu}_{\{2,3\}}(h)$]{
       \begin{minipage}{0.40\textwidth}
         \begin{center}
-          \includegraphics[height=4cm]{images/h23prng}
+          \includegraphics[scale=0.5]{images/h23prng}
         \end{center}
       \end{minipage}
       \label{fig:h23prng}
@@ -732,7 +740,7 @@ Le dernier donnerait le comportement d'un générateur qui s'autoriserait
 
 \subsection{le PRNG de l'algorithme~\ref{CI Algorithm} est chaotique sur $\mathcal{X}_{\mathsf{N},\mathcal{P}}$}
 
-Le théorème suivant, similaire à celui dans $\mathcal{X}_u$ et dans $\mathcal{X}_g$
+Le théorème suivant, similaire à ceux dans $\mathcal{X}_u$ et dans $\mathcal{X}_g$
 est prouvé en annexes~\ref{anx:generateur}.
 
 \begin{theorem}
@@ -751,7 +759,7 @@ On alors corollaire suivant
 \end{corollary}
 \begin{proof}
   Dans cet algorithme, $\mathcal{P}$ est le singleton $\{b\}$.
-  Que $b$ soit pair ou impair,  $\textsc{giu}_{\mathcal{b}}(f)$
+  Que $b$ soit pair ou impair,  $\textsc{giu}_{\{b\}}(f)$
   n'est pas fortement connexe.
 \end{proof}
 
diff --git a/chaosANN.tex b/chaosANN.tex
index 4ea34b9..89ac939 100644
--- a/chaosANN.tex
+++ b/chaosANN.tex
@@ -1,6 +1,6 @@
 
 Les réseaux de neurones chaotiques ont été étudiés à de maintes reprises 
-par le passé en raison notamment de leurs applications potentielles:
+en raison de leurs applications potentielles:
 %les mémoires associatives~\cite{Crook2007267}  
 les  composants utiles à la  sécurité comme les fonctions de 
 hachage~\cite{Xiao10},
@@ -20,7 +20,7 @@ universels~\cite{Cybenko89,DBLP:journals/nn/HornikSW89}.
 Ils permettent, dans certains cas, de simuler des comportements 
 physiques chaotiques comme le  circuit de Chua~\cite{dalkiran10}.  
 Parfois~\cite{springerlink:10.1007/s00521-010-0432-2},
-la fonction de transfert de cette famille de réseau celle 
+la fonction de transfert de cette famille de réseau et celle 
 d'initialisation sont toutes les deux définies à l'aide de 
 fonctions chaotiques. 
 
@@ -28,7 +28,8 @@ fonctions chaotiques.
 
 Ces réseaux de neurones partagent le fait qu'ils sont qualifiés de 
 ``chaotiques'' sous prétexte qu'ils embarquent une fonction de ce type 
-et ce sans aucune preuve rigoureuse. Ce chapitre caractérise la 
+et ce sans aucune preuve rigoureuse ne soit fournie.
+Ce chapitre caractérise la 
 classe des réseaux de neurones MLP chaotiques. Il  
 s'intéresse ensuite à l'étude de prévisibilité de systèmes dynamiques
 discrets chaotiques par cette famille de MLP.
@@ -39,21 +40,22 @@ de vérification si un réseau de neurones est chaotique ou non.
 La section~\ref{sec:ann:approx} s'intéresse à étudier pratiquement
 si un réseau de 
 neurones peut approximer des itération unaires chaotiques. Ces itérations
-étant obtenues à partir de fonction générées à l'aide du chapitre précédent.
-
+étant obtenues à partir de fonctions issues de la démarche détaillée dans 
+le chapitre précédent.
+Ce travail a été publié dans~\cite{bcgs12:ij}.
 
 \section{Un réseau de neurones chaotique au sens de Devaney}
 \label{S2}
 
 On considère une fonction 
-$f:\Bool^n\to\Bool^n$ telle que  $\textsc{giu}(f)$ est fortement connexe.
+$f:\Bool^{\mathsf{N}}\to\Bool^{\mathsf{N}}$ telle que  $\textsc{giu}(f)$ est fortement connexe.
 Ainsi $G_{f_u}$ est chaotique d'après le théorème~\ref{Th:CaracIC}.
 
 On considère ici le schéma unaire défini par l'équation (\ref{eq:asyn}).
 On construit un Perceptron multi-couches associé à la fonction  
 $F_{f_u}$. 
 Plus précisément, pour chaque entrée 
- $(x,s)   \in  \mathds{B}^n \times [n]$,
+ $(x,s)   \in  \mathds{B}^{\mathsf{N}} \times [{\mathsf{N}}]$,
 la couche de sortie doit générer $F_{f_u}(x,s)$.   
 On peut ainsi lier la couche de sortie avec celle d'entrée pour représenter 
 les dépendance entre deux itérations successives.
@@ -62,18 +64,18 @@ On obtient une réseau de neurones dont le comportement est le suivant
 
 \begin{itemize}
 \item   Le réseau est initialisé avec le vecteur d'entrée
-  $\left(x^0,S^0\right)  \mathds{B}^n \times [n]$      
+  $\left(x^0,S^0\right)  \mathds{B}^{\mathsf{N}} \times [{\mathsf{N}}]$      
   et calcule le  vecteur de sortie 
   $x^1=F_{f_u}\left(x^0,S^0\right)$. 
-  Ce vecteur est publié comme une sortie et est aussi retournée sur la couche d'entrée 
+  Ce vecteur est publié comme une sortie et est aussi retourné sur la couche d'entrée 
   à travers les liens de retours.
 \item Lorsque le réseau est  activé à la $t^{th}$  itération, l'état du 
-  système $x^t  \in \mathds{B}^n$ reçu depuis la couche de sortie ainsi que le 
+  système $x^t  \in \mathds{B}^{\mathsf{N}}$ reçu depuis la couche de sortie ainsi que le 
   premier terme de la  séquence $(S^t)^{t  \in \Nats}$
-  (\textit{i.e.},  $S^0  \in [n]$)  servent à construire le nouveau vecteur de sortie.
+  (\textit{i.e.},  $S^0  \in [{\mathsf{N}}]$)  servent à construire le nouveau vecteur de sortie.
   Ce nouveau vecteur, qui représente le nouvel état du système dynamique, satisfait:
   \begin{equation}
-    x^{t+1}=F_{f_u}(x^t,S^0) \in \mathds{B}^n \enspace .
+    x^{t+1}=F_{f_u}(x^t,S^0) \in \mathds{B}^{\mathsf{N}} \enspace .
   \end{equation}
 \end{itemize}
 
@@ -85,7 +87,7 @@ On obtient une réseau de neurones dont le comportement est le suivant
 \end{figure}
 
 Le comportement de ce réseau de neurones est tel que lorsque l'état 
-initial est composé de $x^0~\in~\mathds{B}^n$ et d'une séquence
+initial est composé de $x^0~\in~\mathds{B}^{\mathsf{N}}$ et d'une séquence
  $(S^t)^{t  \in \Nats}$, alors la séquence  contenant les vecteurs successifs 
 publiés $\left(x^t\right)^{t \in \mathds{N}^{\ast}}$ est exactement celle produite 
 par les itérations unaires décrites à la section~\ref{sec:TIPE12}.
@@ -102,41 +104,44 @@ réseau de neurones de type Perceptron multi-couches
 dont on cherche à savoir s'il est chaotique (parce qu'il a par exemple été 
 déclaré comme tel) au sens de Devaney. 
 On considère de plus que sa topologie est la suivante:
-l'entrée est constituée de  $n$ bits et un entier, la sortie est constituée de $n$ bits
+l'entrée est constituée de  ${\mathsf{N}}$ bits et un entier, 
+la sortie est constituée de ${\mathsf{N}}$ bits
 et chaque sortie est liée à une entrée par une boucle.
 
 \begin{itemize}
-\item Le réseau est initialisé avec  $n$~bits
-   $\left(x^0_1,\dots,x^0_n\right)$ et une valeur entière $S^0 \in [n]$.
+\item Le réseau est initialisé avec  ${\mathsf{N}}$~bits
+   $\left(x^0_1,\dots,x^0_{\mathsf{N}}\right)$ et une valeur entière $S^0 \in [{\mathsf{N}}]$.
 \item  A l'itération~$t$,     le vecteur 
-  $\left(x^t_1,\dots,x^t_n\right)$  permet de construire les $n$~bits 
-  servant de sortie $\left(x^{t+1}_1,\dots,x^{t+1}_n\right)$.
+  $\left(x^t_1,\dots,x^t_{\mathsf{N}}\right)$  permet de construire les ${\mathsf{N}}$~bits 
+  servant de sortie $\left(x^{t+1}_1,\dots,x^{t+1}_{\mathsf{N}}\right)$.
 \end{itemize}
 
 Le comportement de ce type de réseau de neurones peut être prouvé comme 
-étant chaotique en suivant la démarche énoncée maintenant.
-On nomme tout d'abord $F:    \mathds{B}^n  \times [n] \rightarrow
-\mathds{B}^n$     la      fonction qui associe  
+étant chaotique en suivant la démarche suivante.
+On nomme tout d'abord $F:    \mathds{B}^{\mathsf{N}}  \times [{\mathsf{N}}] \rightarrow
+\mathds{B}^{\mathsf{N}}$     la      fonction qui associe  
 au vecteur 
-$\left(\left(x_1,\dots,x_n\right),s\right)    \in   \mathds{B}^n \times[n]$ 
+$\left(\left(x_1,\dots,x_{\mathsf{N}}\right),s\right)    \in   \mathds{B}^{\mathsf{N}} \times[{\mathsf{N}}]$ 
 le vecteur 
-$\left(y_1,\dots,y_n\right)       \in       \mathds{B}^n$, où
-$\left(y_1,\dots,y_n\right)$  sont les sorties du réseau neuronal
+$\left(y_1,\dots,y_{\mathsf{N}}\right)       \in       \mathds{B}^{\mathsf{N}}$, où
+$\left(y_1,\dots,y_{\mathsf{N}}\right)$  sont les sorties du réseau neuronal
 après l'initialisation de la couche d'entrée avec 
-$\left(s,\left(x_1,\dots, x_n\right)\right)$.  Ensuite, on définie $f:
-\mathds{B}^n       \rightarrow      \mathds{B}^n$  telle que 
-$f\left(x_1,x_2,\dots,x_n\right)$ est égal à 
+$\left(s,\left(x_1,\dots, x_{\mathsf{N}}\right)\right)$.  Ensuite, on définie $f:
+\mathds{B}^{\mathsf{N}}       \rightarrow      \mathds{B}^{\mathsf{N}}$  telle que 
+$f\left(x_1,x_2,\dots,x_{\mathsf{N}}\right)$ est égal à 
 \begin{equation}
-\left(F\left(\left(x_1,x_2,\dots,x_n\right),1\right),\dots,
-  F\left(\left(x_1,x_2,\dots,x_n\right)\right),n\right) \enspace .
+\left(
+  F\left(\left(x_1,x_2,\dots,x_{\mathsf{N}}\right),1\right),\dots,
+  F\left(\left(x_1,x_2,\dots,x_{\mathsf{N}}\right),{\mathsf{N}}\right) 
+\right).
 \end{equation}
-Ainsi pour chaque $j$, $1 \le j \le n$, on a 
-$f_j\left(x_1,x_2,\dots,x_n\right) = 
-F\left(\left(x_1,x_2,\dots,x_n\right),j\right)$.
+Ainsi pour chaque $j$, $j \in [{\mathsf{N}}]$, on a 
+$f_j\left(x_1,x_2,\dots,x_{\mathsf{N}}\right) = 
+F\left(\left(x_1,x_2,\dots,x_{\mathsf{N}}\right),j\right)$.
 Si ce réseau de neurones est initialisé avec 
-$\left(x_1^0,\dots,x_n^0\right)$   et $S   \in    [n]^{\mathds{N}}$, 
+$\left(x_1^0,\dots,x_{\mathsf{N}}^0\right)$   et $S   \in    [{\mathsf{N}}]^{\mathds{N}}$, 
 il produit exactement les même sorties que les itérations de  $F_{f_u}$ avec une 
-condition initiale $\left((x_1^0,\dots,  x_n^0),S\right)  \in  \mathds{B}^n \times [n]^{\mathds{N}}$.
+condition initiale $\left((x_1^0,\dots,  x_{\mathsf{N}}^0),S\right)  \in  \mathds{B}^{\mathsf{N}} \times [{\mathsf{N}}]^{\mathds{N}}$.
 Les itérations de $F_{f_u}$ 
 sont donc un modèle formel de cette classe de réseau de neurones.
 Pour vérifier si un de ces représentants est chaotique, il suffit ainsi 
@@ -144,7 +149,7 @@ de vérifier si le graphe d'itérations
 $\textsc{giu}(f)$ est fortement connexe.
 
 
-\section{Un réseau de neurones peut-il approximer
+\section[Approximation des itérations unaires chaotiques par RN]{Un réseau de neurones peut-il approximer
 des itération unaires chaotiques?}\label{sec:ann:approx}
 
 Cette section s'intéresse à étudier le comportement d'un réseau de neurones 
@@ -152,7 +157,7 @@ face à des itérations unaires chaotiques, comme définies à
 la section~\ref{sec:TIPE12}.
 Plus précisément, on considère dans cette partie une fonction  dont le graphe 
 des itérations unaires est fortement connexe et une séquence dans 
-$[n]^{\mathds{N}}$. On cherche à construire un réseau de neurones
+$[{\mathsf{N}}]^{\mathds{N}}$. On cherche à construire un réseau de neurones
 qui approximerait les itérations de la fonction $G_{f_u}$ comme définie 
 à l'équation~(\ref{eq:sch:unaire}).
 
@@ -191,11 +196,11 @@ mémoriser des configurations. On en considère deux principalement.
 Dans le premier cas, on considère une entrée booléenne par élément
 tandis que dans le second cas, les configurations  sont mémorisées comme 
 des entiers naturels. Dans ce dernier cas, une approche naïve pourrait 
-consister à attribuer à chaque configuration de $\Bool^n$ 
+consister à attribuer à chaque configuration de $\Bool^{\mathsf{N}}$ 
 l'entier naturel naturel correspondant.
 Cependant, une telle représentation rapproche 
 arbitrairement des configurations diamétralement
-opposées dans le $n$-cube comme  une puissance de
+opposées dans le ${\mathsf{N}}$-cube comme  une puissance de
 deux et la configuration immédiatement précédente: 10000 serait modélisée 
 par 16 et  et 01111 par 15 alors que leur distance de Hamming est 15.
 De manière similaire, ce codage éloigne des configurations qui sont 
@@ -207,13 +212,13 @@ Concentrons nous sur la traduction de la stratégie.
 Il n'est naturellement pas possible de traduire une stratégie 
 infinie quelconque à l'aide d'un nombre fini d'éléments.
 On se restreint donc à des stratégies de taille 
-$l \in \llbracket 2,k\rrbracket$, où $k$ est un paramètre défini
+$l$, $2 \le l \le k$, où $k$ est un paramètre défini
 initialement. 
 Chaque stratégie est mémorisée comme un entier naturel exprimé en base 
-$n+1$: à chaque itération, soit aucun élément n'est modifié, soit un 
+${\mathsf{N}}+1$: à chaque itération, soit aucun élément n'est modifié, soit un 
 élément l'est. 
-Enfin, on donne une dernière entrée: $m  \in \llbracket
-1,l-1\rrbracket$, qui est le nombre d'itérations successives que l'on applique 
+Enfin, on donne une dernière entrée: $m  \in [l-1]
+$, qui est le nombre d'itérations successives que l'on applique 
 en commençant à $x$. 
 Les sorties (stratégies et configurations) sont mémorisées 
 selon les mêmes règles.
@@ -221,33 +226,33 @@ selon les mêmes règles.
 Concentrons nous sur la complexité du problème.
 Chaque entrée, de l'entrée-sortie de l'outil est un triplet 
 composé d'une configuration $x$, d'un extrait  $S$ de la stratégie à 
-itérer de taille $l \in \llbracket 2, k\rrbracket$ et d'un nombre $m \in  \llbracket 1, l-1\rrbracket$ d'itérations à exécuter.
-Il y a  $2^n$  configurations $x$ et  $n^l$ stratégies de
+itérer de taille $l$, $2 \le l \le  k$ et d'un nombre $m \in [l-1]$ d'itérations à exécuter.
+Il y a  $2^{\mathsf{N}}$  configurations $x$ et  ${\mathsf{N}}^l$ stratégies de
 taille $l$. 
 De plus, pour une  configuration donnée, il y a 
-$\omega = 1 \times n^2 +  2 \times n^3 + \ldots+ (k-1) \times n^k$
+$\omega = 1 \times {\mathsf{N}}^2 +  2 \times {\mathsf{N}}^3 + \ldots+ (k-1) \times {\mathsf{N}}^k$
 manières d'écrire le couple $(m,S)$. Il n'est pas difficile d'établir que 
 \begin{equation}
-\displaystyle{(n-1) \times \omega = (k-1)\times n^{k+1} - \sum_{i=2}^k n^i} \nonumber
+\displaystyle{({\mathsf{N}}-1) \times \omega = (k-1)\times {\mathsf{N}}^{k+1} - \sum_{i=2}^k {\mathsf{N}}^i} \nonumber
 \end{equation}
 donc
 \begin{equation}
 \omega =
-\dfrac{(k-1)\times n^{k+1}}{n-1} - \dfrac{n^{k+1}-n^2}{(n-1)^2} \enspace . \nonumber
+\dfrac{(k-1)\times {\mathsf{N}}^{k+1}}{{\mathsf{N}}-1} - \dfrac{{\mathsf{N}}^{k+1}-{\mathsf{N}}^2}{({\mathsf{N}}-1)^2} \enspace . \nonumber
 \end{equation}
 \noindent
 Ainsi le nombre de paire d'entrée-sortie pour les réseaux de neurones considérés
 est 
 $$
-2^n \times \left(\dfrac{(k-1)\times n^{k+1}}{n-1} - \dfrac{n^{k+1}-n^2}{(n-1)^2}\right) \enspace .
+2^{\mathsf{N}} \times \left(\dfrac{(k-1)\times {\mathsf{N}}^{k+1}}{{\mathsf{N}}-1} - \dfrac{{\mathsf{N}}^{k+1}-{\mathsf{N}}^2}{({\mathsf{N}}-1)^2}\right) \enspace .
 $$
 Par exemple, pour $4$   éléments binaires et une stratégie d'au plus 
 $3$~termes on obtient 2304 couples d'entrée-sortie.
 
 \subsection{Expérimentations}
-\label{section:experiments}
-On se focalise dans cette section sur l'entraînement d'un Perceptron 
-multi-couches pour apprendre des itérations chaotiques. Ce type de réseau
+\label{section:ann:experiments}
+On se focalise dans cette section sur l'entraînement d'un MLP
+pour apprendre des itérations chaotiques. Ce type de réseau
 ayant déjà été évalué avec succès dans la prédiction de 
 séries chaotiques temporelles. En effet, les auteurs de~\cite{dalkiran10} 
 ont montré qu'un MLP pouvait apprendre la dynamique du circuit de Chua.
@@ -274,39 +279,40 @@ section précédente.
 \hline
 \hline
 \multicolumn{2}{|c||}{Neurones cachés} & \multicolumn{3}{c|}{10 neurones} \\
-\cline{3-5} 
+\hline
 \multicolumn{2}{|c||}{Epochs} & 125 & 250 & 500 \\ 
 \hline
-\multirow{6}{*}{\rotatebox{90}{Chaotique $g$ }}&Entrée~(1) & 90.92\% & 91.75\% & 91.82\% \\ 
-& Entrée~(2) & 69.32\% & 78.46\% & 82.15\% \\
-& Entrée~(3) & 68.47\% & 78.49\% & 82.22\% \\
-& Entrée~(4) & 91.53\% & 92.37\% & 93.4\% \\
+\multirow{6}{*}{\rotatebox{90}{Chaotique $g$ }}&Sortie~(1) & 90.92\% & 91.75\% & 91.82\% \\ 
+& Sortie~(2) & 69.32\% & 78.46\% & 82.15\% \\
+& Sortie~(3) & 68.47\% & 78.49\% & 82.22\% \\
+& Sortie~(4) & 91.53\% & 92.37\% & 93.4\% \\
 & Config. & 36.10\% & 51.35\% & 56.85\% \\
 & Stratégie~(5) & 1.91\% & 3.38\% & 2.43\% \\
 \hline
-\multirow{6}{*}{\rotatebox{90}{Non-chaotic $f$}}&Entrée~(1) & 97.64\% & 98.10\% & 98.20\% \\
-& Entrée~(2) & 95.15\% & 95.39\% & 95.46\% \\
-& Entrée~(3) & 100\% & 100\% & 100\% \\
-& Entrée~(4) & 97.47\% & 97.90\% & 97.99\% \\
+\multirow{6}{*}{\rotatebox{90}{Non-chaotic $f$}}&Sortie~(1) & 97.64\% & 98.10\% & 98.20\% \\
+& Sortie~(2) & 95.15\% & 95.39\% & 95.46\% \\
+& Sortie~(3) & 100\% & 100\% & 100\% \\
+& Sortie~(4) & 97.47\% & 97.90\% & 97.99\% \\
 & Config. & 90.52\% & 91.59\% & 91.73\% \\
 & Stratégie~(5) & 3.41\% & 3.40\% & 3.47\% \\
 \hline
 \hline
 \multicolumn{2}{|c||}{Neurones cachés} & \multicolumn{3}{c|}{25 neurones} \\
-\cline{3-5} \\%& \multicolumn{3}{|c|}{40 neurons} \\
+%\cline{3-5} \\%& \multicolumn{3}{|c|}{40 neurons} \\
+\hline
 \multicolumn{2}{|c||}{Epochs} & 125 & 250 & 500 \\ %& 125 & 250 & 500 \\ 
 \hline
-\multirow{6}{*}{\rotatebox{90}{Chaotique $g$}}&Entrée~(1) & 91.65\% & 92.69\% & 93.93\% \\ %& 91.94\% & 92.89\% & 94.00\% \\
-& Entrée~(2) & 72.06\% & 88.46\% & 90.5\% \\ %& 74.97\% & 89.83\% & 91.14\% \\
-& Entrée~(3) & 79.19\% & 89.83\% & 91.59\% \\ %& 76.69\% & 89.58\% & 91.84\% \\
-& Entrée~(4) & 91.61\% & 92.34\% & 93.47\% \\% & 82.77\% & 92.93\% & 93.48\% \\
+\multirow{6}{*}{\rotatebox{90}{Chaotique $g$}}&Sortie~(1) & 91.65\% & 92.69\% & 93.93\% \\ %& 91.94\% & 92.89\% & 94.00\% \\
+& Sortie~(2) & 72.06\% & 88.46\% & 90.5\% \\ %& 74.97\% & 89.83\% & 91.14\% \\
+& Sortie~(3) & 79.19\% & 89.83\% & 91.59\% \\ %& 76.69\% & 89.58\% & 91.84\% \\
+& Sortie~(4) & 91.61\% & 92.34\% & 93.47\% \\% & 82.77\% & 92.93\% & 93.48\% \\
 & Config. & 48.82\% & 67.80\% & 70.97\% \\%& 49.46\% & 68.94\% & 71.11\% \\
 & Stratégie~(5) & 2.62\% & 3.43\% & 3.78\% \\% & 3.10\% & 3.10\% & 3.03\% \\
 \hline
-\multirow{6}{*}{\rotatebox{90}{Non-chaotique $f$}}&Entrée~(1) & 97.87\% & 97.99\% & 98.03\% \\ %& 98.16\% \\
-& Entrée~(2) & 95.46\% & 95.84\% & 96.75\% \\ % & 97.4\% \\
-& Entrée~(3) & 100\% & 100\% & 100\% \\%& 100\% \\
-& Entrée~(4) & 97.77\% & 97.82\% & 98.06\% \\%& 98.31\% \\
+\multirow{6}{*}{\rotatebox{90}{Non-chaotique $f$}}&Sortie~(1) & 97.87\% & 97.99\% & 98.03\% \\ %& 98.16\% \\
+& Sortie~(2) & 95.46\% & 95.84\% & 96.75\% \\ % & 97.4\% \\
+& Sortie~(3) & 100\% & 100\% & 100\% \\%& 100\% \\
+& Sortie~(4) & 97.77\% & 97.82\% & 98.06\% \\%& 98.31\% \\
 & Config. & 91.36\% & 91.99\% & 93.03\% \\%& 93.98\% \\
 & Stratégie~(5) & 3.37\% & 3.44\% & 3.29\% \\%& 3.23\% \\
 \hline
@@ -412,7 +418,8 @@ réseau de neurones d'apprendre le comportement global d'itérations
 chaotiques.
 Comme il est difficile (voir impossible) d'apprendre le comportement 
 de telles fonction, il paraît naturelle de savoir si celles ci peuvent être 
-utilisées pour générer des nombres pseudo aléatoires.
+utilisées pour générer des nombres pseudo aléatoires, ce que propose la partie
+suivante.
 
 
 % \appendix{}
-- 
2.39.5


From d81b15b2024adaf639e9d4a85934a5b5722c1bf1 Mon Sep 17 00:00:00 2001
From: couchot <couchot@localhost.localdomain>
Date: Mon, 5 Sep 2016 17:14:37 +0200
Subject: [PATCH 15/16] jusqu'au chpitre 6

---
 14Secrypt.tex                | 134 +++++++++++++++++------------------
 15RairoGen.tex               |  77 ++++++++++----------
 annexePreuveDistribution.tex |  50 +++++--------
 annexePreuveMixage.tex       |   8 +--
 annexePromelaProof.tex       |  20 +++---
 annexesccg.tex               |  12 ++--
 caracgeneralise.tex          |   8 +--
 caracunaire.tex              |   8 +--
 main.tex                     |   3 +-
 modelchecking.tex            |   4 +-
 10 files changed, 154 insertions(+), 170 deletions(-)

diff --git a/14Secrypt.tex b/14Secrypt.tex
index 10a4f85..76e635b 100644
--- a/14Secrypt.tex
+++ b/14Secrypt.tex
@@ -13,12 +13,12 @@ graphe d'itérations $\textsc{giu}(\neg)$ (section~\ref{sec:hamiltonian}).
 Pour obtenir plus rapidement une distribution uniforme, l'idéal serait
 de supprimer un cycle hamiltonien qui nierait autant de fois chaque bit. 
 Cette forme de cycle est dit équilibré. La section~\ref{sub:gray} établit le
-lien avec les codes de Gray équilibrés, étudiés dans la litérature. 
+lien avec les codes de Gray équilibrés, étudiés dans la littérature. 
 La section suivante présente une démarche de génération automatique de code de Gray équilibré (section~\ref{sec:induction}).
 La vitesse avec laquelle l'algorithme de PRNG converge en interne vers 
-une distribution unifiorme est étduiée théoriquement et pratiquement à la 
+une distribution uniforme est étudiée théoriquement et pratiquement à la 
 section~\ref{sec:mixing}.
-L'extension du travail aux itérations généralisées est présenté à la 
+L'extension du travail aux itérations généralisées est présentée à la 
 section~\ref{sec:prng:gray:general}.
 Finalement, des instances de PRNGS engendrés selon les méthodes détaillées dans 
 ce chapitre sont présentés en section~\ref{sec:prng;gray:tests}.
@@ -58,7 +58,7 @@ la matrice est stochastique à droite;
 \item Toutes les éléments de la somme $\sum_{1\le k\le 2^{\mathsf{N}}}M^k$ sont strictement positif, \textit{i.e.}, le graphe $\textsc{giu}(f)$ est fortement connexe;
 \end{enumerate}
 Ce problème s'exprime sur des domaines finis entiers avec des opérateurs  
-arithmétiques simples (sommes et produits). il pourrait théoriquement être 
+arithmétiques simples (sommes et produits). Il pourrait théoriquement être 
 traité par des démarches de programmation logique par contrainte
 sur des domaines finis (comme en PROLOG).
 L'algorithme donné en Figure~\ref{fig:prolog}
@@ -70,7 +70,7 @@ ici pour  $\mathsf{N} = 2$. Dans ce code,
 valent True si et seulement si $R$ 
 est le produit matriciel  (ou la somme matricielle) 
 entre  $X$ and $Y$ respectivement. 
-il n'est pas difficile d'adapter ce code à n'importe quelle valeur 
+Il n'est pas difficile d'adapter ce code à n'importe quelle valeur 
 entière naturelle  $\mathsf{N}$.  
 
 \begin{figure}[ht]
@@ -100,15 +100,15 @@ bistoc(X):-
 \end{figure}
 
 Enfin, on définit la relation $\mathcal{R}$, qui est établie pour les deux 
-fonctions  $f$ et $g$ si leur graphes 
-respectifs  $\textsf{giu}(f)$ et $\textsf{giu}(g)$ 
+fonctions  $f$ et $g$ si leurs graphes 
+respectifs  $\textsc{giu}(f)$ et $\textsc{giu}(g)$ 
 sont isomorphes.
 C'est évidemment une relation d'équivalence.
 
 
 
 %\subsection{Analyse de l'approche}\label{sub:prng:ana}
-Exécutée sur un ordinateur personnelle, PROLOG trouve 
+Exécutée sur un ordinateur personnel, PROLOG trouve 
 en moins d'une seconde les
 49 solutions pour  $n=2$, 
 dont 2 seulement ne sont pas équivalentes, 
@@ -122,7 +122,7 @@ en s'appuyant sur l'efficience de l'algorithme de backtrack natif de PROLOG.
 
 Cependant, pour des valeurs de $n$ petites, nous avons 
 comparé les fonctions non équivalentes selon leur proportion
-à engendrer des temps de mélange petits (cf. équation~\ref{eq:mt:ex}).
+à engendrer des temps de mélange petits (cf. équation~(\ref{eq:mt:ex})).
 
 
 
@@ -169,7 +169,7 @@ Cependant, le graphe $\textsc{giu}(f^*)$
 (donné à la Figure~\ref{fig:iteration:f*})
 est le $3$-cube dans lequel le cycle 
 $000,100,101,001,011,111,110,010,000$ 
-a été enlevé. Dans cette figure, le le graphe $\textsc{giu}(f)$ est
+a été enlevé. Dans cette figure, le  graphe $\textsc{giu}(f)$ est
 en continu tandis que le cycle est en pointillés.
 Ce cycle qui visite chaque n{\oe}ud exactement une fois est un  
 \emph{cycle hamiltonien}.
@@ -278,12 +278,12 @@ connexité du graphe d'itérations.
   La suppression d'un cycle hamiltonien dans une matrice de Markov $M$, issue du
   $n$-cube, produit une matrice doublement stochastique.
 \end{theorem}
-\begin{Proof}
+\begin{proof}
 
 Un cycle hamiltonien passe par chaque n{\oe}ud une et une seule fois.
 Pour chaque n{\oe}ud $v$ dans le $n$-cube $C_1$,
 une arête entrante $(o,v)$ et une arête sortante $(v,e)$ 
-est ainsi enlevée.
+sont ainsi enlevées.
 Considérons un autre  $n$-cube $C_2$ auquel on ajoute les arêtes 
 pour le rendre complet. La matrice de Markov $M$ correspondante est
 remplie de $\frac{1}{2^n}$ et est doublement stochastique.
@@ -302,7 +302,7 @@ $2^{n-1}$ arêtes menant à $v$ qui sont enlevées.
 Dans $M$ les $2^{n-1}$ coefficients correspondants sont mis à 0 et 
 $M_{vv}$ vaut alors $\frac{2^{n-1} +1}{2}$.
 $M$ est donc doublement stochastique.
-\end{Proof}
+\end{proof}
 
 
 
@@ -312,7 +312,7 @@ $M$ est donc doublement stochastique.
 \end{theorem}
 
 
-\begin{Proof}
+\begin{proof}
 On considère les deux $n$-cubes $C_1$ et $C_2$ définis 
 dans la preuve du théorème~\ref{th:supprCH}.
 Dans $C_1$ on considère le cycle inverse $r$ du cycle
@@ -325,15 +325,15 @@ Tous les n{\oe}uds de $C_1$ dans lequel $c$ a été enlevé sont accessibles
 depuis n'importe quel n{\oe}ud. Le graphe des itérations $\textsf{giu}$ qui
 étend le précédent graphe est ainsi fortement connexe. 
 
-\end{Proof}
+\end{proof}
 
 
 
 %Les preuves, relativement directes, sont  laissées en exercices au lecteur.  
-La génération de  cycles hamiltoniens dans le
-$n$-cube,  ce qui  revient à  trouver des  \emph{codes de  Gray  cycliques}.  On
-rappelle que  les codes de  Gray sont des  séquences de mots binaires  de taille
-fixe ($n$),  dont les éléments successifs ne  différent que par un  seul bit. Un
+Générer un  cycle hamiltonien dans le
+$n$-cube revient à  trouver un  \emph{code de  Gray  cyclique}.  On
+rappelle qu'un code de  Gray est une  séquence de mots binaires  de taille
+fixe ($\mathsf{N}$),  dont les éléments successifs ne  différent que par un  seul bit. Un
 code  de  Gray est  \emph{cyclique}  si  le premier  élément  et  le dernier  ne
 différent que par un seul bit.
 
@@ -344,37 +344,35 @@ La borne  inférieure du  nombre de codes  de Gray  ($\left(\frac{n*\log2}{e \lo
     \log   n}\times(1  -  o(1))\right)^{2^n}$),   donnée  dans~\cite{Feder2009NTB},
 indique  une explosion combinatoire  pour notre  recherche.  Afin  de contourner
 cette  difficulté,  nous  nous   restreignons  aux  codes  induisant  un  graphe
-d'itérations $\textsf{giu}(f)$  \emph{uniforme}.  Cette uniformité se  traduit par des
+d'itérations $\textsc{giu}(f)$  \emph{uniforme}.  Cette uniformité se  traduit par des
 nombres d'arcs  équilibrés entre les  \emph{dimensions} du graphe,  la dimension
 $i$  correspondant aux  seules variations  du  bit $i$  (parmi les  $n$ bits  au
 total).   Cette  approche  revient  à  chercher  des  codes  de  Gray  cycliques
 \emph{équilibrés}.
 
-Un code de Gray équilibré peut être défini de la façon suivante :
-
-\begin{Def}[Code de Gray cyclique équilibré]\label{def:grayequ}
-  Soit $L = w_1,  w_2, \dots, w_{2^n}$ la séquence d'un code  de Gray cyclique à
-  $n$ bits.  Soit $S = s_1,  s_2, \dots, s_{2^n}$ la séquence des transitions où
-  $s_i$, $1  \le i \le 2^n$  est l'indice du seul  bit qui varie  entre les mots
-  $w_i$ et  $w_{i+1}$. Enfin, soit  $\textit{NT}_n : \{1,\dots,  n\} \rightarrow
-  \{0, \ldots, 2^n\}$  la fonction qui au paramètre  $i$ associe le \emph{nombre
-    de transitions} présentes dans la séquence $L$ pour le bit $i$, c'est-à-dire
-  le nombre d'occurrences de $i$ dans $S$.
+On formalise un code de Gray équilibré comme suit.
+Soit $L = w_1,  w_2, \dots, w_{2^n}$ la séquence d'un code  de Gray cyclique à
+$n$ bits.  Soit $S = s_1,  s_2, \dots, s_{2^n}$ la séquence des transitions où
+$s_i$, $1  \le i \le 2^n$  est l'indice du seul  bit qui varie  entre les mots
+$w_i$ et  $w_{i+1}$. Enfin, soit  $\textit{TC}_n : \{1,\dots,  n\} \rightarrow
+\{0, \ldots, 2^n\}$  la fonction qui au paramètre  $i$ associe le \emph{nombre
+  de transitions} présentes dans la séquence $L$ pour le bit $i$, c'est-à-dire
+le nombre d'occurrences de $i$ dans $S$.
   
-  Le      code       $L$      est      \textbf{équilibré}       si      $\forall
-  i,j\in\{1,...,n\},~|\textit{NT}_n(i)  -  \textit{NT}_n(j)|  \le  2$.   Il  est
-  \textbf{totalement             équilibré}             si             $\forall
-  i\in\{1,...,n\},~\textit{NT}_n(i)=\frac{2^n}{n}$.
-\end{Def}
+Le      code       $L$      est      \textbf{équilibré}       si      $\forall
+i,j\in\{1,...,n\},~|\textit{TC}_n(i)  -  \textit{TC}_n(j)|  \le  2$.   Il  est
+\textbf{totalement             équilibré}             si             $\forall
+i\in\{1,...,n\},~\textit{TC}_n(i)=\frac{2^n}{n}$.
+
 
 On peut  donc déjà déduire  qu'il ne peut  exister des codes de  Gray totalement
 équilibrés que  pour les  systèmes ayant un  nombre d'éléments $n=2^k,  k>0$. De
-plus,  comme  dans tout  code  de  Gray  cyclique, $\textit{NT}_n(i)$  est  pair
+plus,  comme  dans tout  code  de  Gray  cyclique, $\textit{TC}_n(i)$  est  pair
 $\forall  i\in\{1,...,n\}$,  alors  les  systèmes  ayant  un  nombre  d'éléments
 différent  de $2^k$,  ne peuvent  avoir  que des  codes de  Gray équilibrés  avec
-$\textit{NT}_n(i)=\lfloor\frac{2^n}{n}\rfloor$                                 ou
+$\textit{TC}_n(i)=\lfloor\frac{2^n}{n}\rfloor$                                 ou
 $\textit{NT}_n(i)=\lceil\frac{2^n}{n}\rceil,    \forall    i\in\{1,...,n\}$   et
-vérifiant $\sum_{i=1}^nNT_n(i) = 2^n$.
+vérifiant $\sum_{i=1}^n\textit{TC}_n(i) = 2^n$.
 
 \begin{xpl}
   Soit  $L^*=000,100,101,001,011,111,110,010$ le code  de Gray  correspondant au
@@ -393,7 +391,7 @@ vérifiant $\sum_{i=1}^nNT_n(i) = 2^n$.
 \section{Génération de codes de Gray équilibrés par induction}
 \label{sec:induction}
 
-De nombreuses approches ont été developpées pour résoudre le problème de construire
+De nombreuses approches ont été développées pour résoudre le problème de construire
 un code de Gray dans un $\mathsf{N}$-cube~\cite{Robinson:1981:CS,DBLP:journals/combinatorics/BhatS96,ZanSup04}, 
 selon les propriétés que doit vérifier ce code.
 
@@ -404,7 +402,7 @@ pour peu que l'utilisateur fournisse une sous-séquence possédant certaines
 propriétés à chaque pas inductif.
 Ce travail a été renforcé dans ~\cite{DBLP:journals/combinatorics/BhatS96}
 où les auteurs donnent une manière explicite de construire une telle sous-séquence.
-Enfin, les autheurs de~\cite{ZanSup04} présentent une extension de l'algorithme de
+Enfin, les auteurs de~\cite{ZanSup04} présentent une extension de l'algorithme de
 \emph{Robinson-Cohn}. La présentation rigoureuse de cette extension leur permet 
 principalement de prouver que si $\mathsf{N}$ est une puissance de 2, 
 le code de Gray équilibré engendré par l'extension est toujours totalement équilibré et 
@@ -414,43 +412,43 @@ Cependant les auteurs ne prouvent pas que leur approche fournit systématiquemen
 un code de Gray (totalement) équilibré. 
 Cette section montre que ceci est vrai en rappelant tout d'abord
 l'extension de l'algorithme de \emph{Robinson-Cohn} pour un 
-code de Gray avec $\mathsf{N}-2$ bits.
+code de Gray avec $\mathsf{N}-2$ bits
+défini à partir de la séquence $S_{\mathsf{N}-2}$.
 
 \begin{enumerate}
-\item \label{item:nondet}Soit $l$ un entier positif pair. Trouver des sous-sequences 
+\item \label{item:nondet}Soit $l$ un entier positif pair. Trouver des sous-séquences 
 $u_1, u_2, \dots , u_{l-2}, v$ (possiblement vides) de $S_{\mathsf{N}-2}$ 
 telles que $S_{\mathsf{N}-2}$ est la concaténation de  
 $$
 s_{i_1}, u_0, s_{i_2}, u_1, s_{i_3}, u_2, \dots , s_{i_l-1}, u_{l-2}, s_{i_l}, v
 $$
 où $i_1 = 1$, $i_2 = 2$, et $u_0 = \emptyset$ (la séquence vide).
-\item\label{item:u'} Remplacer dans $S_{\mathsf{N}-2}$ les sequences $u_0, u_1, u_2, \ldots, u_{l-2}$ 
+\item\label{item:u'} Remplacer dans $S_{\mathsf{N}-2}$ les séquences $u_0, u_1, u_2, \ldots, u_{l-2}$ 
   par 
   $\mathsf{N} - 1,  u'(u_1,\mathsf{N} - 1, \mathsf{N}) , u'(u_2,\mathsf{N}, \mathsf{N} - 1), u'(u_3,\mathsf{N} - 1,\mathsf{N}), \dots, u'(u_{l-2},\mathsf{N}, \mathsf{N} - 1)$
   respectivement, où $u'(u,x,y)$ est la séquence $u,x,u^R,y,u$ telle que 
   $u^R$ est $u$, mais dans l'ordre inverse. La séquence obtenue est ensuite notée $U$.
-\item\label{item:VW} Contruire les séquences $V=v^R,\mathsf{N},v$, $W=\mathsf{N}-1,S_{\mathsf{N}-2},\mathsf{N}$. Soit  alors $W'$ définie commé étant égale à $W$ sauf pour les 
+\item\label{item:VW} Construire les séquences $V=v^R,\mathsf{N},v$, $W=\mathsf{N}-1,S_{\mathsf{N}-2},\mathsf{N}$. Soit  alors $W'$ définie comme étant égale à $W$ sauf pour les 
 deux premiers éléments qui ont été intervertis.
-\item La séquence de transition  $S_{\mathsf{N}}$ est la concatenation $U^R, V, W'$.
+\item La séquence de transition  $S_{\mathsf{N}}$ est la concaténation $U^R, V, W'$.
 \end{enumerate} 
 
 L'étape~(\ref{item:nondet}) n'est pas constructive: il n'est pas précisé
 comment sélectionner des sous-séquences qui assurent que le code obtenu est équilibré.
-La théoreme suivante montre que c'est possible et sa preuve explique comment le faire. 
+La théorème suivante montre que c'est possible et sa preuve,
+donnée en annexes~\ref{anx:generateur}, explique comment le faire. 
 
 
 \begin{theorem}\label{prop:balanced}
 Soit $\mathsf{N}$ dans $\Nats^*$, et $a_{\mathsf{N}}$ défini par 
 $a_{\mathsf{N}}= 2 \left\lfloor \dfrac{2^{\mathsf{N}}}{2\mathsf{N}} \right\rfloor$. 
 il existe une séquence $l$ dans l'étape~(\ref{item:nondet}) de l'extension
-de l'algorithme de \emph{Robinson-Cohn} extension telle que 
+de l'algorithme de \emph{Robinson-Cohn}  telle que 
 le nombres de transitions $\textit{TC}_{\mathsf{N}}(i)$ 
 sont tous $a_{\mathsf{N}}$ ou $a_{\mathsf{N}}+2$ 
 pour chaque  $i$, $1 \le i \le \mathsf{N}$.
 \end{theorem}
 
-La preuve de ce théorème est donnée en annexes~\ref{anx:generateur}.
-
 Ces fonctions étant générées, on s'intéresse à étudier à quelle vitesse 
 un générateur les embarquant converge vers la distribution uniforme.
 C'est l'objectif de la section suivante. 
@@ -585,7 +583,7 @@ $\ov{h}(\ov{h}(X))\neq X$.
 
 
 \begin{theorem} \label{prop:stop}
-Si $\ov{h}$ is bijective et anti involutive 
+Si $\ov{h}$ est bijective et anti involutive 
 $\ov{h}(\ov{h}(X))\neq X$, alors
 $E[\ts]\leq 8{\mathsf{N}}^2+ 4{\mathsf{N}}\ln ({\mathsf{N}}+1)$. 
 \end{theorem}
@@ -594,8 +592,8 @@ Les détails de la preuve sont donnés en annexes~\ref{anx:generateur}.
 On remarque tout d'abord que la chaîne de Markov proposée ne suit pas exactement
 l'algorithme~\ref{CI Algorithm}. En effet dans la section présente, 
 la probabilité de rester dans une configuration donnée 
-est fixée à $frac{1}{2}+\frac{1}{2n}$.
-Dans l'algorithme initial, celle-ci est de ${1}{n}$.
+est fixée à $\frac{1}{2}+\frac{1}{2n}$.
+Dans l'algorithme initial, celle-ci est de $\frac{1}{n}$.
 Cette version, qui reste davantage sur place que l'algorithme original,
 a été introduite pour simplifier le calcul de la borne sup 
 du temps d'arrêt.   
@@ -616,17 +614,17 @@ dans le contexte du $\mathsf{N}$-cube privé d'un chemin hamiltonien.
 
 On peut évaluer ceci pratiquement: pour une fonction
 $f: \Bool^{\mathsf{N}} \rightarrow \Bool^{\mathsf{N}}$ et une graine initiale
-$x^0$, le code donné à l'algorithme algorithm~\ref{algo:stop} retourne le 
+$x^0$, le code donné à l'algorithme  ~\ref{algo:stop} retourne le 
 nombre d'itérations suffisant tel que tous les éléments $\ell\in \llbracket 1,{\mathsf{N}} \rrbracket$ sont équitables. Il permet de déduire une approximation de $E[\ts]$
 en l'instanciant un grand nombre de fois: pour chaque nombre $\mathsf{N}$, 
-$ 3 \le \mathsf{N} \le 16$, 10 fonctionss ont été générées comme dans 
+$ 3 \le \mathsf{N} \le 16$, 10 fonctions ont été générées comme dans 
 ce chapitre. Pour chacune d'elle, le calcul d'une approximation de $E[\ts]$
-est exécuté 10000 fois avec une graine aléatoire.La Figure~\ref{fig:stopping:moy}
-résume ces resultats. Dans celle-ci, un cercle  représente une approximation de 
+est exécuté 10000 fois avec une graine aléatoire. La Figure~\ref{fig:stopping:moy}
+résume ces résultats. Dans celle-ci, un cercle  représente une approximation de 
 $E[\ts]$ pour un  $\mathsf{N}$ donné tandis que la courbe est une représentation de 
 la fonction $x \mapsto 2x\ln(2x+8)$. 
-On  cosntate que l'approximation de $E[\ts]$ est largement inférieure 
-à la borne quadratique donnée au thérème~\ref{prop:stop} et que la conjecture 
+On  constate que l'approximation de $E[\ts]$ est largement inférieure 
+à la borne quadratique donnée au théorème~\ref{prop:stop} et que la conjecture 
 donnée au paragraphe précédent est sensée.
 
 
@@ -650,7 +648,7 @@ $\textit{fair}\leftarrow\emptyset$\;
 }
 \Return{$\textit{nbit}$}\;
 %\end{scriptsize}
-\caption{Pseudo Code of stoping time calculus }
+\caption{Pseudo Code pour évaluer le temps d'arrêt}
 \label{algo:stop}
 \end{algorithm}
 
@@ -658,14 +656,14 @@ $\textit{fair}\leftarrow\emptyset$\;
 \begin{figure}
 \centering
 \includegraphics[width=0.49\textwidth]{images/complexityET}
-\caption{Average Stopping Time Approximation}\label{fig:stopping:moy}
+\caption{Interpolation du temps d'arrêt}\label{fig:stopping:moy}
 \end{figure}
 
 
 
 
 \section{Et les itérations généralisées?}\label{sec:prng:gray:general}
-Le chaptire précédent a présenté un algorithme de 
+Le chapitre précédent a présenté un algorithme de 
 PRNG construit à partir d'itérations unaires. 
 On pourrait penser que cet algorithme est peu efficace puisqu'il 
 dispose d'une fonction $f$ de $\Bool^n$ dans lui même mais il ne modifie à 
@@ -698,7 +696,7 @@ la ligne $s\leftarrow{\textit{Set}(\textit{Random}(2^n))}$ est différente.
 Dans celle-ci la fonction  $\textit{Set}   :    \{1,\ldots,2^n\}   \rightarrow
 \mathcal{P}(\{1,\ldots   n\})$   retourne  l'ensemble   dont   la   fonction
 caractéristique  serait  représentée par  le  nombre  donné  en argument.
-Par exemple, pour $n=3$, l'ensemble $\textit{Set}(6)$ vaudraitt $\{3,2\}$.
+Par exemple, pour $n=3$, l'ensemble $\textit{Set}(6)$ vaudrait $\{3,2\}$.
 On remarque aussi que l'argument de la fonction  $\textit{Random}$
 passe de $n$ à $2^n$.
 
@@ -874,7 +872,7 @@ du mode, une fonction peut être optimale pour un mode et  ne pas l'être pour l
 
 Un second  résultat est  que ce nouvel  algorithme réduit grandement  le nombre
 d'itérations  suffisant pour  obtenir une  faible  déviation par  rapport à  une
-distribution uniforme.  On constate de  plus que ce nombre décroit avec
+distribution uniforme.  On constate de  plus que ce nombre décroît avec
 le nombre d'éléments alors qu'il augmente dans l'approche initiale où 
 l'on marche.
 
@@ -912,7 +910,7 @@ donc $b'*\ln(n)/(n*\ln(2))$ appels pour 1 bit généré en moyenne.
 Le tableau~\ref{table:marchevssaute} donne des instances de 
 ces valeurs pour $n \in\{4,5,6,7,8\}$ et les fonctions  
 données au tableau~\ref{table:functions}.
-On constate que le nombre d'appels par bit généré décroit avec $n$ dans le 
+On constate que le nombre d'appels par bit généré décroît avec $n$ dans le 
 cas des itérations généralisées et est toujours plus faible
 que celui des itérations unaires.
 
@@ -959,13 +957,13 @@ permet de générer la stratégie aléatoire.
 Les tableau~\ref{fig:TEST:generalise} donnent
 une vision synthétique de ces expérimentations. 
 Nous avons évalué les fonctions préfixées par 
-$f$ (respecitvement $g$) avec les générateurs issus des itérations 
+$f$ (respectivement $g$) avec les générateurs issus des itérations 
 généralisées (resp. unaires).
 Quelle que soit la méthode utilisée, on constate que chacun des 
 générateurs passe 
-avec succes le test de NIST. 
+avec succès le test de NIST. 
 
-Interpréter ces resultats en concluant que ces générateurs sont 
+Interpréter ces résultats en concluant que ces générateurs sont 
 tous équivalents serait erroné: la meilleur des 
 méthodes basées sur le mode des itérations
 généralisées (pour $n=8$ par exemple) 
@@ -1087,7 +1085,7 @@ Complexité linaire& 0.005 (0.98)& 0.534 (0.99)& 0.085 (0.97)& 0.996 (1.0)\\ \hl
 
 \section{Conclusion}
 Ce chaptitre a montré comment construire un PRNG chaotique, notamment à partir 
-de codes de Gray équilibrés. Une méthode completement automatique de
+de codes de Gray équilibrés. Une méthode complètement automatique de
 construction de ce type de codes a été présentée étendant les méthodes 
 existantes. 
 Dans le cas des itérations unaires, 
diff --git a/15RairoGen.tex b/15RairoGen.tex
index 06f84b1..d905e51 100644
--- a/15RairoGen.tex
+++ b/15RairoGen.tex
@@ -6,25 +6,17 @@ le mot $x^b$ devrait  \og sembler ne plus dépendre\fg{} de $x^0$.
 On peut penser à exploiter une de ces fonctions $G_f$ 
 comme un générateur aléatoire. 
 
-Ce chapitre  présente une application directe
+Ce chapitre  présente donc une application directe
 de la théorie développée ci-avant
 à la génération de nombres pseudo aléatoires. 
+La section~\ref{sub:prng:algo} 
+présente tout d'abord l'algorithme de PRNG. La contrainte de  
+distribution uniforme de la sortie est discutée dans cette section.
+La chaoticité du générateur est ensuite étudiée en 
+section~\ref{prng:unaire:chaos}.
+La section~\ref{sub:prng:algo}  a été publiéeà~\cite{bcgw11:ip,bcgr11:ip}.
 
 
-La suite de ce document donnera
-une condition nécessaire est suffisante pour que
-cette propriété soit satisfaite.
-
-
-On présente tout d'abord le générateur
-basé sur des fonctions chaotiques (section~\ref{sub:prng:algo}), 
-puis comment intégrer la contrainte de distribution uniforme
-de la sortie 
-dans le choix de la fonction à itérer (section~\ref{sub:prng:unif}). 
-L'approche est évaluée dans la dernière section.
-\JFC{plan à revoir}
- 
-
 \section{ Nombres pseudo aléatoires construits par itérations unaires}\label{sub:prng:algo}
 
 
@@ -304,7 +296,7 @@ ait une distribution suffisamment proche de la distribution uniforme.
 
 On énonce directement le théorème suivant dont la preuve est donnée en annexes~\ref{anx:generateur}.
 
-\begin{theorem}\label{thm:prng:u}
+\begin{restatable}[Uniformité de la sortie de l'algorithme~\ref{CI Algorithm}]{theorem}{PrngCIUniforme}\label{thm:prng:u}
   Soit $f: \Bool^{n} \rightarrow \Bool^{n}$, $\textsc{giu}(f)$ son 
   graphe d'itérations , $\check{M}$ sa matrice d'adjacence
   et $M$ une matrice  $2^n\times 2^n$  
@@ -315,7 +307,7 @@ On énonce directement le théorème suivant dont la preuve est donnée en annex
   l'algorithme~\ref{CI Algorithm} suit une loi qui 
   tend vers la distribution uniforme si 
   et seulement si  $M$ est une matrice doublement stochastique.
-\end{theorem}
+\end{restatable}
 
 
 \subsection{Quelques exemples}
@@ -454,7 +446,7 @@ Montrer que les sous-séquences de suites chaotiques ainsi générées  demeuren
 est l'objectif de la section suivante.
 
 
-\section{Un PRNG basé sur des itérations unaires qui est chaotique }
+\section{Un PRNG basé sur des itérations unaires qui est chaotique }\label{prng:unaire:chaos}
 
 Cette section présente un espace métrique adapté au générateur de nombres pseudo-aléatoires 
 présenté à l'algorithme~\ref{CI Algorithm} et prouve ensuite que la fonction qu'il représente 
@@ -661,9 +653,11 @@ la séquence.
 
 
 On a la proposition suivante, qui est démontrée en annexes~\ref{anx:generateur}.
-\begin{lemma}
+
+
+\begin{restatable}[Une distance dans $\mathcal{X}_{\mathsf{N},\mathcal{P}}$]{theorem}{distancedsxnp}
 $d$ est une distance sur $\mathcal{X}_{\mathsf{N},\mathcal{P}}$.
-\end{lemma}
+\end{restatable}
 
 
 \subsection{Le graphe $\textsc{giu}_{\mathcal{P}}(f)$ étendant  $\textsc{giu}(f)$}
@@ -743,24 +737,33 @@ Le dernier donnerait le comportement d'un générateur qui s'autoriserait
 Le théorème suivant, similaire à ceux dans $\mathcal{X}_u$ et dans $\mathcal{X}_g$
 est prouvé en annexes~\ref{anx:generateur}.
 
-\begin{theorem}
+\begin{restatable}[Conditions pour la choticité de $G_{f_u,\mathcal{P}}$]{theorem}{thmchoticitgfp}
 La fonction $G_{f_u,\mathcal{P}}$ est chaotique sur 
  $(\mathcal{X}_{\mathsf{N},\mathcal{P}},d)$ si et seulement si 
-graphe d'itération $\textsc{giu}_{\mathcal{P}}(f)$ 
+le graphe d'itération $\textsc{giu}_{\mathcal{P}}(f)$ 
 est fortement connexe.
-\end{theorem}
-On alors corollaire suivant 
-
-\begin{corollary}
-  Le générateur de nombre pseudo aléatoire détaillé 
-  à l'algorithme~\ref{CI Algorithm}
-  n'est pas chaotique 
-  sur $(\mathcal{X}_{\mathsf{N},\{b\}},d)$ pour la fonction négation.
-\end{corollary}
-\begin{proof}
-  Dans cet algorithme, $\mathcal{P}$ est le singleton $\{b\}$.
-  Que $b$ soit pair ou impair,  $\textsc{giu}_{\{b\}}(f)$
-  n'est pas fortement connexe.
-\end{proof}
-
+\end{restatable}
+% On alors corollaire suivant 
+
+% \begin{corollary}
+%   Le générateur de nombre pseudo aléatoire détaillé 
+%   à l'algorithme~\ref{CI Algorithm}
+%   n'est pas chaotique 
+%   sur $(\mathcal{X}_{\mathsf{N},\{b\}},d)$ pour la fonction négation.
+% \end{corollary}
+% \begin{proof}
+%   Dans cet algorithme, $\mathcal{P}$ est le singleton $\{b\}$.
+%   Que $b$ soit pair ou impair,  $\textsc{giu}_{\{b\}}(f)$
+%   n'est pas fortement connexe.
+% \end{proof}
+
+
+\section{Conclusion}
+Ce chapitre a proposé un algorithme permettant de construire un 
+PRNG chaotique à partir d'un PRNG existant. Pour ce faire, il est nécessaire 
+et suffisant que la fonction $f$ qui est itérée un nombre $b$ de fois 
+possède un $\textsc{giu}_{\{b\}}(f)$ fortement connexe et que sa matrice de Markov assosiée soit doublement stochastique.
+Le chapitre suivant montre comment construire une telle fonction.
 
+ 
+ 
diff --git a/annexePreuveDistribution.tex b/annexePreuveDistribution.tex
index f53e3b9..ea5f4f1 100644
--- a/annexePreuveDistribution.tex
+++ b/annexePreuveDistribution.tex
@@ -1,4 +1,5 @@
- 
+
+\section{Chaînes de Markov associées à $\textsc{giu}(f)$}  
 Considérons le lemme technique suivant:
 \begin{lemma}\label{lem:stoc}
 Soit $f: \Bool^{n} \rightarrow \Bool^{n}$, $\textsc{giu}(f)$ son graphe d'itérations, $\check{M}$ la matrice d'adjacence de $\textsc{giu}(f)$, et  $M$  la matrice 
@@ -8,7 +9,7 @@ Alors $M$ est une matrice stochastique régulière si et seulement si
 $\textsc{giu}(f)$ est fortement connexe.
 \end{lemma}
 
-\begin{Proof}  
+\begin{proof}  
 On remarque tout d'abord que $M$ 
 est une matrice stochastique par construction.
 Supposons $M$ régulière. 
@@ -28,22 +29,11 @@ on peut conclure que, si
 $k$ est le plus petit multiple commun de $\{k_{ij}  \big/ i,j  \in \llbracket 1,  2^n \rrbracket  \}$ alors
 $\forall i,j  \in \llbracket  1, 2^n \rrbracket,  \check{M}_{ij}^{k}>0$. 
 Ainsi, $\check{M}$ et donc $M$ sont régulières.
-\end{Proof}
-
-Ces résultats permettent formuler et de prouver le théorème suivant:
-
-\begin{theorem}
-  Soit $f: \Bool^{n} \rightarrow \Bool^{n}$, $\textsc{giu}(f)$ son 
-  graphe d'itérations , $\check{M}$ sa matrice d'adjacence
-  et $M$ une matrice  $2^n\times 2^n$  définie comme dans le lemme précédent.
-  Si $\textsc{giu}(f)$ est fortement connexe, alors 
-  la sortie du générateur de nombres pseudo aléatoires détaillé par 
-  l'algorithme~\ref{CI Algorithm} suit une loi qui 
-  tend vers la distribution uniforme si 
-  et seulement si  $M$ est une matrice doublement stochastique.
-\end{theorem}
-
-\begin{Proof}
+\end{proof}
+
+Ces résultats permettent formuler et de prouver le théorème annoncé.
+\PrngCIUniforme*
+\begin{proof}
   $M$ est une matrice stochastique régulière (Lemme~\ref{lem:stoc}) 
   qui a un unique vecteur de probabilités stationnaire
   (Théorème \ref{th}).
@@ -53,14 +43,13 @@ Ces résultats permettent formuler et de prouver le théorème suivant:
   la somme des valeurs de chaque colonne de $M$  est 1, 
   \textit{i.e.} si et seulement si 
   $M$ est  doublement  stochastique.
-\end{Proof}
-
+\end{proof}
 
+\section{Chaoticité de la fonction $G_{f_u,\mathcal{P}}$ dans
+$(\mathcal{X}_{\mathsf{N},\mathcal{P}},d)$}
 
-Montrons que
-\begin{lemma}
-$d$ est une distance sur $\mathcal{X}_{\mathsf{N},\mathcal{P}}$.
-\end{lemma}
+Montrons le théorème
+\distancedsxnp*
 
 
 \begin{proof}
@@ -172,15 +161,8 @@ $$a_0^0, ...,a_{k_2}^{|a_{k_2}|}...),(v^0, ..., v^{k_1}, |a_0|, ..., |a_{k_2}|,v
 est un point périodique dans le voisinage $\mathcal{B}(x,\varepsilon)$ de $x$.
 \end{proof}
 
-$G_{f_u,\mathcal{P}}$ étant topologiquement transitive and regulière, 
-on peut conclure le théorème:
-
-
-\begin{theorem}
-La fonction $G_{f_u,\mathcal{P}}$ est chaotique sur 
- $(\mathcal{X}_{\mathsf{N},\mathcal{P}},d)$ si et seulement si 
-graphe d'itération $\textsc{giu}_{\mathcal{P}}(f)$ 
-est fortement connexe.
-\end{theorem}
+$G_{f_u,\mathcal{P}}$ étant topologiquement transitive and régulière, 
+on peut démontrer le théorème:
+\thmchoticitgfp*
 
 
diff --git a/annexePreuveMixage.tex b/annexePreuveMixage.tex
index 12c3223..8132c68 100644
--- a/annexePreuveMixage.tex
+++ b/annexePreuveMixage.tex
@@ -12,7 +12,7 @@ s'il existe un chemin de longueur   $\alpha$
   élément  $i\in$ \class{p}  et $j  \in$ \class{q}  tel que 
   $i  \le  j$ si et seulement si
   \class{p} $\preceq$ \class{q}.
-  \begin{Proof}
+  \begin{proof}
     
     Tout d'abord,  soit  \class{p_1},   \ldots,  \class{p_l}  des   classes 
     contenant respectivement les éléments $n_1$,\ldots,  $n_l$
@@ -59,7 +59,7 @@ s'il existe un chemin de longueur   $\alpha$
     \class{p}  $\preceq$ \class{q'}  et pour chaque  $i$, $k$  tels que $i  \in$
     \class{p} et  $k \in$ \class{q'}, $i \le k$
     et le résultat est établi.
-  \end{Proof}
+  \end{proof}
 \end{lemma}
 
 On peut remarquer que ce processus de renommage est inspiré des \emph{graphes 
@@ -70,7 +70,7 @@ On peut remarquer que ce processus de renommage est inspiré des \emph{graphes
 %   Processes numbers are already compliant with the order $\preceq$.
 % \end{xpl}
 
-\begin{Proof}[du théorème~\ref{th:cvg}]
+\begin{proof}[du théorème~\ref{th:cvg}]
 
   Le reste de la preuve est fait par induction sur le numéro de classe. 
   Considérons la première  classe \class{b_1} de $n_1$ éléments 
@@ -98,7 +98,7 @@ On peut remarquer que ce processus de renommage est inspiré des \emph{graphes
   où tous les éléments de  \class{b_j}, $1 \le j
   \le k$, ont  des valeurs constantes.  
   D'après les hypothèses du théorème, cela converge.
-\end{Proof}
+\end{proof}
 
 
 
diff --git a/annexePromelaProof.tex b/annexePromelaProof.tex
index 3675094..6b3adaa 100644
--- a/annexePromelaProof.tex
+++ b/annexePromelaProof.tex
@@ -11,13 +11,13 @@ du chapitre~\ref{chap:promela}.
   le le scheduler met à jour les  elements of $S^t$
   donnés par \verb+update_elems+ à l'iteration $t$.
 \end{lemma}
-\begin{Proof}
+\begin{proof}
   La preuve est directe pour $t=0$.
   Supposons qu'elle est établie jusqu'en $t$ vallant un certain $t_0$. 
   On considère des stratégies pseudo périodiques.
   Grâce à l'hypothèse d'équité faible, \verb+update_elems+ modifie 
   les éléments de $S^t$ à l'iteration $t$.
-\end{Proof}
+\end{proof}
 
 Dans ce qui suit, soit     $Xd^t_{ji}$     la valeur de 
 \verb+Xd[+$j$\verb+].v[+$i$\verb+]+  après le   $t^{\text{ème}}$ appel 
@@ -123,7 +123,7 @@ la variable  \verb+Xp[k]+  en sortant du processus
 $X_k^{t}$          \textit{i.e.},          $F_{k}\left(         X_1^{D_{k\,1}^{t-1}},\ldots,
   X_{\mathsf{N}}^{D_{k\,{\mathsf{N}}}^{t-1}}\right)$ à la fin de la  $t^{\text{th}}$ itération.
 \end{lemma}
-\begin{Proof}
+\begin{proof}
 La preuve est faite par induction sur le nombre d'itérations.
 
 \paragraph{Situation initiale:}
@@ -235,7 +235,7 @@ $\verb+Xp[+k\verb+]+=                                   F(\verb+Xd[+k\verb+][0]+
 \ldots,\verb+Xd[+k\verb+][+n\verb+-1]+)+$.  
 Par définition $Xd=F(Xd^{l+1}_{k\,0},      \ldots,Xd^{l+1}_{k\,n-1})$.      
 Grace à~\Equ{eq:correct_retrieve} déjà prouvée, on peut conclure la preuve.
-\end{Proof}
+\end{proof}
 
 
 \begin{lemma}
@@ -244,17 +244,17 @@ Grace à~\Equ{eq:correct_retrieve} déjà prouvée, on peut conclure la preuve.
   $\delta_0$.
 \end{lemma}
 
-\begin{Proof}
+\begin{proof}
   Pour chaque $i$ et $j$, à chaque itération $t+1$, comme les délais sont bornés par 
   $\delta_0$,  l'élément $i$  doit connaître au plus $\delta_0$ 
   valeurs qui sont
   $X_j^{t}$, \ldots, $X_j^{t-\delta_0+1}$. Elles peuvent être mémorisées dans n'importe quel cannal
   de taille $\delta_0$.
-\end{Proof}
+\end{proof}
 
 
 \promelasound*
-\begin{Proof}
+\begin{proof}
 %   For  the  case  where  the  strategy  is  finite,  one  notice  that  property
 %   verification  is achieved  under  weak fairness  property.  Instructions  that
 %   write or read into \verb+channels[j].sent[i]+ are continuously enabled leading
@@ -267,7 +267,7 @@ Grace à~\Equ{eq:correct_retrieve} déjà prouvée, on peut conclure la preuve.
   Si des itérations du système dynamique discret sont divergentes, leur exécution vont empêcher 
   le modèle PROMELA de se stabiliser,  \textit{i.e.} 
   ce dernier ne verifiera pas la propriété LTL (\ref{eq:ltl:conv}).
-\end{Proof}
+\end{proof}
 
 
 % \begin{Corol}[Soundness wrt universall convergence property]\label{Theo:sound}
@@ -282,7 +282,7 @@ Grace à~\Equ{eq:correct_retrieve} déjà prouvée, on peut conclure la preuve.
 
 \promelacomplete*
 
-\begin{Proof}
+\begin{proof}
   Pour chaque  modele  $\psi$ 
   qui ne vérifie pas la propriété  LTL  (\ref{eq:ltl:conv}), 
   il est immédiat de construire les itérations correpondantes du
@@ -299,5 +299,5 @@ Grace à~\Equ{eq:correct_retrieve} déjà prouvée, on peut conclure la preuve.
 %   continuously enabled leading to convenient available dates $D_{ji}$.
 % \end{itemize}
 % The simulated DDN does not stabilize and its iterations are divergent.
- \end{Proof}
+ \end{proof}
 
diff --git a/annexesccg.tex b/annexesccg.tex
index 257f86c..bbfc12d 100644
--- a/annexesccg.tex
+++ b/annexesccg.tex
@@ -23,7 +23,7 @@ arcs dont ${\mathsf{N}}$ est soit l'extrémité, soit l'origine (et dans ce dern
 cas, les arcs sont des boucles sur ${\mathsf{N}}$).
 \end{lemma}
 
-\begin{Proof}
+\begin{proof}
 Supposons que $G(f^{\alpha})$ possède un arc de  $j$ vers $i$ de signe 
 $s$. Par définition, il existe un sommet $x\in\Bool^{{\mathsf{N}}-1}$ tel que
 $f^{\alpha}_{ij}(x)=s$, et puisque 
@@ -42,13 +42,13 @@ Ainsi $f_{ij}(x,\alpha)$ est égal à $s$.
 On a donc aussi
 $f^{\alpha}_{ij}(x)=s$. Ainsi $G(f^\alpha)$ possède un arc 
 arc de $j$ vers $i$ de signe $s$.
-\end{Proof}
+\end{proof}
 
 \begin{lemma}\label{lemma:iso}
 Les graphes $\textsc{giu}(f^\alpha)$ et $\textsc{giu}(f)^\alpha$ sont isomorphes.
 \end{lemma}
 
-\begin{Proof}
+\begin{proof}
 Soit $h$ la bijection de $\Bool^{{\mathsf{N}}-1}$ vers
 $\Bool^{{\mathsf{N}}-1}\times \{\alpha\}$ définie par $h(x)=(x,\alpha)$ pour chaque
 $x\in\Bool^{{\mathsf{N}}-1}$.
@@ -56,13 +56,13 @@ On voit facilement que $h$ permet de définir un isomorphisme
 entre $\textsc{giu}(f^\alpha)$ et $\textsc{giu}(f)^\alpha$: 
 $\textsc{giu}(f^\alpha)$ possède un arc de $x$ vers $y$ si et seulement si 
 $\textsc{giu}(f)^\alpha$ a un  arc de $h(x)$ vers $h(y)$.
-\end{Proof}
+\end{proof}
 
 
 On peut alors prouver le théorème:
 \thAdrien*
 
-\begin{Proof}
+\begin{proof}
 La preuve se fait par induction sur ${\mathsf{N}}$. 
 Soit $f$ une fonction de $\Bool^{\mathsf{N}}$ dans lui-même et qui vérifie les hypothèses 
 du théorème.
@@ -122,7 +122,7 @@ Puisque la valeur de $f_{\mathsf{N}}(x)$
 on a $f_{\mathsf{N}}(x')=f_{\mathsf{N}}(x)=1\neq x'_{\mathsf{N}}$ (resp. $f_{\mathsf{N}}(y')=f_{\mathsf{N}}(y)=0\neq
 y'_{\mathsf{N}}$). 
 Ainsi la  condition ($*$) est établie, et le théorème est prouvé.
-\end{Proof}
+\end{proof}
 
 
 
diff --git a/caracgeneralise.tex b/caracgeneralise.tex
index 3014a6b..9ebc5d1 100644
--- a/caracgeneralise.tex
+++ b/caracgeneralise.tex
@@ -4,7 +4,7 @@ des itérations généralisées.
 
 \caractransitivegeneralise*
 
-\begin{Proof} 
+\begin{proof} 
 
 $\Longleftarrow$ Supposons que $\textsc{gig}(f)$ soit fortement connexe.
 Soient $(x,S)$ et $(x',S')$ deux points de $\mathcal{X}_g$ et  $\varepsilon >0$. 
@@ -49,7 +49,7 @@ Pour tout entier naturel $t$, on a
 $G_{f_g}^t(x'',S'') \neq (x',S')$.
 Ainsi $G_{f_g}$ n'est pas transitive et 
 par contraposée, on a la démonstration souhaitée.
-\end{Proof}
+\end{proof}
 
 
 Prouvons à présent le théorème suivant: 
@@ -57,7 +57,7 @@ Prouvons à présent le théorème suivant:
 \caracsubgeneralise*
 
 
-\begin{Proof}  
+\begin{proof}  
 Soit $f:\Bool^{\mathsf{N}}\to\Bool^{\mathsf{N}}$ telle que  $G_{f_g}$ est transitive (\textit{i.e.}
 $f$ appartient à $\mathcal{T}$). 
 Soit $(x,S) \in\mathcal{X}_g$ et $\varepsilon >0$. Pour 
@@ -82,7 +82,7 @@ Il est évident que  $(x,\tilde S)$ s'obtient à partir de  $(x,\tilde S)$ aprè
 $t_1+t_2$ itérations parallèles de $G_{f_g}$. Ainsi $(x,\tilde S)$ est un point 
 périodique. Puisque $\tilde s_t$ est égal à $s_t$ pour $t<t_1$, d'après le 
 choix de  $t_1$, on a  $d((x,S),(x,\tilde S))<\epsilon$.
-\end{Proof}
+\end{proof}
 
 On peut conclure  que $\mathcal{C} = \mathcal{R} \cap \mathcal{T}
 = \mathcal{T}$. 
\ No newline at end of file
diff --git a/caracunaire.tex b/caracunaire.tex
index 1111939..35b28b5 100644
--- a/caracunaire.tex
+++ b/caracunaire.tex
@@ -8,7 +8,7 @@ On prouve les théorèmes suivants
 
 
 
-\begin{Proof} 
+\begin{proof} 
 
 $\Longleftarrow$ Supposons que $\textsc{giu}(f)$ soit fortement connexe.
 Soient $(x,S)$ et $(x',S')$ deux points de $\mathcal{X}_u$ et  $\varepsilon >0$. 
@@ -53,7 +53,7 @@ Pour tout entier naturel $t$, on a
 $G_{f_u}^t(x'',S'') \neq (x',S')$.
 Ainsi $G_{f_u}$ n'est pas transitive et 
 par contraposée, on a la démonstration souhaitée.
-\end{Proof}
+\end{proof}
 
 
 Prouvons à présent le théorème suivant: 
@@ -63,7 +63,7 @@ Prouvons à présent le théorème suivant:
 \end{theorem}
 
 
-\begin{Proof}  
+\begin{proof}  
 Soit $f:\Bool^{\mathsf{N}}\to\Bool^{\mathsf{N}}$ telle que  $G_{f_u}$ est transitive (\textit{i.e.}
 $f$ appartient à $\mathcal{T}$). 
 Soit $(x,S) \in\mathcal{X}_u$ et $\varepsilon >0$. Pour 
@@ -88,7 +88,7 @@ Il est évident que  $(x,\tilde S)$ s'obtient à partir de  $(x,\tilde S)$ aprè
 $t_1+t_2$ itérations parallèles de $G_{f_u}$. Ainsi $(x,\tilde S)$ est un point 
 périodique. Puisque $\tilde s_t$ est égal à $s_t$ pour $t<t_1$, d'après le 
 choix de  $t_1$, on a  $d((x,S),(x,\tilde S))<\epsilon$.
-\end{Proof}
+\end{proof}
 
 On peut conclure  que $\mathcal{C} = \mathcal{R} \cap \mathcal{T}
 = \mathcal{T}$. On a alors la  caractérisation suivante:
diff --git a/main.tex b/main.tex
index 628e4f9..e2161f1 100644
--- a/main.tex
+++ b/main.tex
@@ -157,7 +157,7 @@
 \newtheorem{lemma}{Lemme}
 \newtheorem{corollary}{Corollaire}
 \newtheorem*{xpl}{Exemple}
-\newtheorem*{Proof}{Preuve}
+
 \newtheorem{Def}{Définition}
 
 \begin{document}
@@ -376,6 +376,7 @@ du chapitre 8}
 
 \chapter{Preuves sur les générateurs de nombres pseudo-aléatoires}\label{anx:generateur}
 \input{annexePreuveDistribution}
+\section{Codes de Gray équilibrés par induction}
 \input{annexePreuveGrayEquilibre}
 \input{annexePreuveStopping}
 
diff --git a/modelchecking.tex b/modelchecking.tex
index b01a526..d9e626e 100644
--- a/modelchecking.tex
+++ b/modelchecking.tex
@@ -657,7 +657,7 @@ puis présente ensuite les expérimentations issues de ce travail.
   et $\psi$ sa traduction en PROMELA.  Le nombre de configurations 
   de l'exécution en SPIN de $\psi$ est bornée par $2^{m(\delta_0+1)+n(n+2)}$.
 \end{theorem}
-\begin{Proof}
+\begin{proof}
   Une configuration est une évaluation des variables globales.
   Leur nombre ne dépend que de celles qui ne sont pas constantes.
 
@@ -673,7 +673,7 @@ puis présente ensuite les expérimentations issues de ce travail.
   $m$ et $\delta_0$.
   %\JFC{Donner un ordre de grandeur de cet ordre de grandeur}
   
-\end{Proof}
+\end{proof}
 
 La méthode détaillée ici a pu être appliquée sur l'exemple
 pour prouver formellement sa convergence universelle.
-- 
2.39.5


From 749714e242c186d9017fa85964d6c67edf1bf4d1 Mon Sep 17 00:00:00 2001
From: couchot <couchot@localhost.localdomain>
Date: Wed, 7 Sep 2016 22:04:52 +0200
Subject: [PATCH 16/16] retouche preuve gray

---
 14Secrypt.tex | 20 ++++++++++----------
 main.tex      |  5 ++++-
 2 files changed, 14 insertions(+), 11 deletions(-)

diff --git a/14Secrypt.tex b/14Secrypt.tex
index 76e635b..69fb6d8 100644
--- a/14Secrypt.tex
+++ b/14Secrypt.tex
@@ -340,7 +340,7 @@ différent que par un seul bit.
 \section{Lien avec les codes de Gray cycliques (totalement) équilibrés}
 \label{sub:gray}
 
-La borne  inférieure du  nombre de codes  de Gray  ($\left(\frac{n*\log2}{e \log
+Un minorant du  nombre de codes  de Gray  ($\left(\frac{n*\log2}{e \log
     \log   n}\times(1  -  o(1))\right)^{2^n}$),   donnée  dans~\cite{Feder2009NTB},
 indique  une explosion combinatoire  pour notre  recherche.  Afin  de contourner
 cette  difficulté,  nous  nous   restreignons  aux  codes  induisant  un  graphe
@@ -438,16 +438,16 @@ comment sélectionner des sous-séquences qui assurent que le code obtenu est é
 La théorème suivante montre que c'est possible et sa preuve,
 donnée en annexes~\ref{anx:generateur}, explique comment le faire. 
 
-
-\begin{theorem}\label{prop:balanced}
+\begin{restatable}[Existence d'un code de Gray équilibré]{theorem}{theograyequilibre}
+\label{prop:balanced}
 Soit $\mathsf{N}$ dans $\Nats^*$, et $a_{\mathsf{N}}$ défini par 
 $a_{\mathsf{N}}= 2 \left\lfloor \dfrac{2^{\mathsf{N}}}{2\mathsf{N}} \right\rfloor$. 
 il existe une séquence $l$ dans l'étape~(\ref{item:nondet}) de l'extension
-de l'algorithme de \emph{Robinson-Cohn}  telle que 
+de l'algorithme de \emph{Robinson-Cohn} extension telle que 
 le nombres de transitions $\textit{TC}_{\mathsf{N}}(i)$ 
 sont tous $a_{\mathsf{N}}$ ou $a_{\mathsf{N}}+2$ 
 pour chaque  $i$, $1 \le i \le \mathsf{N}$.
-\end{theorem}
+\end{restatable}
 
 Ces fonctions étant générées, on s'intéresse à étudier à quelle vitesse 
 un générateur les embarquant converge vers la distribution uniforme.
@@ -595,17 +595,17 @@ la probabilité de rester dans une configuration donnée
 est fixée à $\frac{1}{2}+\frac{1}{2n}$.
 Dans l'algorithme initial, celle-ci est de $\frac{1}{n}$.
 Cette version, qui reste davantage sur place que l'algorithme original,
-a été introduite pour simplifier le calcul de la borne sup 
+a été introduite pour simplifier le calcul d'un majorant 
 du temps d'arrêt.   
 
 
 Sans entrer dans les détails de la preuve, on remarque aussi
-que le calcul  de cette borne impose uniquement que 
+que le calcul  de ce majorant impose uniquement que 
 pour chaque n{\oe}ud du $\mathsf{N}$-cube 
 un arc entrant et un arc sortant sont supprimés.
 Le fait qu'on enlève un cycle  hamiltonien et que ce dernier 
 soit équilibré n'est pas pris en compte.
-En intégrant cette contrainte, la borne supérieure pourrait être réduite.
+En intégrant cette contrainte, ce majorant  pourrait être réduite.
 
 En effet, le temps de mixage est en $\Theta(N\ln N)$ lors d'une
 marche aléatoire classique paresseuse dans le $\mathsf{N}$-cube.
@@ -624,7 +624,7 @@ résume ces résultats. Dans celle-ci, un cercle  représente une approximation
 $E[\ts]$ pour un  $\mathsf{N}$ donné tandis que la courbe est une représentation de 
 la fonction $x \mapsto 2x\ln(2x+8)$. 
 On  constate que l'approximation de $E[\ts]$ est largement inférieure 
-à la borne quadratique donnée au théorème~\ref{prop:stop} et que la conjecture 
+à le majorant quadratique donné au théorème~\ref{prop:stop} et que la conjecture 
 donnée au paragraphe précédent est sensée.
 
 
@@ -1090,7 +1090,7 @@ construction de ce type de codes a été présentée étendant les méthodes
 existantes. 
 Dans le cas des itérations unaires, 
 l'algorithme qui en découle a un temps de mélange qui a 
-une borne sup quadratique de convergence vers la distribution uniforme. 
+un majorant quadratique de convergence vers la distribution uniforme. 
 Pratiquement,  ce temps de mélange se rapproche de $N\ln N$.
 Les expérimentations au travers de la batterie de test de NIST ont montré
 que toutes les propriétés statistiques sont obtenues pour
diff --git a/main.tex b/main.tex
index e2161f1..1c00650 100644
--- a/main.tex
+++ b/main.tex
@@ -376,8 +376,11 @@ du chapitre 8}
 
 \chapter{Preuves sur les générateurs de nombres pseudo-aléatoires}\label{anx:generateur}
 \input{annexePreuveDistribution}
+
 \section{Codes de Gray équilibrés par induction}
 \input{annexePreuveGrayEquilibre}
+
+\section{Majoration du temps d'arrêt}
 \input{annexePreuveStopping}
 
 \chapter{Preuves sur le marquage de média}\label{anx:marquage}
@@ -391,7 +394,7 @@ du chapitre 8}
 \input{annexePreuveMarquageCorrectioncompletude}
 \backmatter
 
-\section{Complexité d'Algorithmes de stéganographie}
+\section{Complexités d'algorithmes de stéganographie}
 \label{anx:preuve:cplxt}
 \input{annexePreuvesComplexiteStego}
 
-- 
2.39.5