From aa282765b88f449ee03206e4995360fcced37f56 Mon Sep 17 00:00:00 2001
From: guyeux <guyeux@gmail.com>
Date: Fri, 1 Jun 2012 12:16:29 +0200
Subject: [PATCH 1/1] fdskl

---
 review_prng.tex | 109 ++++++++++++++++++++++--------------------------
 1 file changed, 50 insertions(+), 59 deletions(-)

diff --git a/review_prng.tex b/review_prng.tex
index a5d1388..158f90b 100644
--- a/review_prng.tex
+++ b/review_prng.tex
@@ -3,7 +3,7 @@
 \usepackage[standard]{ntheorem}
 \usepackage[english]{babel}
 
-
+\usepackage{stmaryrd}
 \usepackage{amsmath}
 \usepackage{color}
 \usepackage{dsfont}
@@ -93,70 +93,61 @@ $$\left\{
 where $f^n:\mathcal{X}^{n+1}\rightarrow \mathcal{X}$.
 \end{definition}
 
+Some particular cases of these iterative systems are well documented,
+namely the serial, parallel, or chaotic modes.
+In the serial mode, each component is updated one by one, whereas the
+parallel mode consists in updating all the components at each iteration,
+leading to an usual discrete dynamical system.
+These modes are compiliant with the definition above, 
 
+On appelle \emph{itérations parallèles} \index{itérations!parallèles} de $f$ sur 
+$\mathcal{X}$ toute itération synchrone $\Sigma = (\mathcal{X}, \mathcal{F})$ 
+telle que la suite $\mathcal{F}$ est constante égale à $f: \X \to \X$.
+On parle encore d'itérations parallèles $\left(\mathcal{X},f\right)$.
 
 
 
-%%  \end{block}
-%%\uncover<2->{
-%%Différents modes d'itérations: séries, parallèles, chaotiques, asynchrones...
-%%}
-%%}
-
-
-
-
-
-
-
-
-
-
-%\subsection*{Cas des Itérations chaotiques}
-%\frame{
-%  \frametitle{Les « itérations chaotiques »}
-%    \begin{block}{Définition (Itérations chaotiques)}
-% Soient $f: \mathds{B}^\mathsf{N} \longrightarrow \mathds{B}^\mathsf{N}$ et $S \subset \mathcal{P} \left(\llbracket1,\mathsf{N}\rrbracket\right)^\mathds{N}$. Les \emph{itérations chaotiques} sont:
-%$$\left\{
-%\begin{array}{l}
-%x^0 \in \mathds{B}^\mathsf{N} \\
-%\forall n \in \mathds{N}^*, \forall i \in \llbracket 1; \mathsf{N} \rrbracket,  x^{n}_i = \left\{
-%\begin{array}{ll}
-%x^{n-1}_{i} & \textrm{ si } i \notin S^n\\
-%f(x^{n-1})_{i} & \textrm{ si } i \in S^n
-%\end{array}
-%\right.
-%\end{array}
-%\right.$$
-%\end{block}
-%%\uncover<2->{
-%%Itérations chaotiques et théorie du chaos: a priori, rien à voir.
-%%}
-%%\uncover<3->{Y a-t-il un lien ?}\uncover<4->{ Pour quoi faire ?}
-%}
-
-
-
-
-
-%\frame{
-%  \frametitle{Non-convergence des IC}
-%   \begin{alertblock}{Théorème (Condition nécessaire de non-convergence)}
-%    % Soit $f : \mathds{B}^\mathsf{N} \to \mathds{B}^\mathsf{N}$ et $S \in \mathcal{S}$. 
-%    Si les itérations chaotiques $\left(f,(x^0,S)\right)$ sont non convergentes, alors:
-%\begin{itemize}
-%\item soit $f$ n'est pas contractante,
-%\item soit $S$ n'est pas pseudo-périodique (complète).
-%\end{itemize}
-%   \end{alertblock}
-%   \uncover<2->{
-%   Quelle quantité de désordre ?
-%   }
-%}
-
-
+Finally, iterative systems in chaotic mode, simply called chaotic iterations,
+ are defined as follows.
+ 
 
+\begin{definition}
+Let $f: \mathds{B}^\mathsf{N} \longrightarrow \mathds{B}^\mathsf{N}$ and 
+$S \subset \mathcal{P} \left(\llbracket1,\mathsf{N}\rrbracket\right)^\mathds{N}$. 
+\emph{Chaotic iterations} are defined by:
+$$\left\{
+\begin{array}{l}
+x^0 \in \mathds{B}^\mathsf{N} \\
+\forall n \in \mathds{N}^*, \forall i \in \llbracket 1; \mathsf{N} \rrbracket,  x^{n}_i = \left\{
+\begin{array}{ll}
+x^{n-1}_{i} & \textrm{ if } i \notin S^n\\
+f(x^{n-1})_{i} & \textrm{ if } i \in S^n
+\end{array}
+\right.
+\end{array}
+\right.$$
+\end{definition}
 
+\emph{A priori}, there is no relation between these chaotic iterations
+and the mathematical theory of chaos recalled in the previous section.
+On our side, we have regarded whether these chaotic iterations can 
+behave chaotically, as it is defined for instance by Devaney, and if so, 
+in which applicative context this behavior can be profitable.
+This questionning has led to a first necessary condition of non convergence.
+
+\begin{proposition}
+Let $f : \mathds{B}^\mathsf{N} \to \mathds{B}^\mathsf{N}$ and 
+$S \in \llbracket 1, \mathsf{N} \rrbracket^{\mathds{N}}$. 
+If the chaotic iterations $\left(f,(x^0,S)\right)$ are not convergent, then:
+\begin{itemize}
+\item either $f$ is not a contraction,
+\item or $S$ is not pseudo-periodic.
+\end{itemize}
+\end{proposition}
+
+The quantity of disorder generated by such chaotic iterations, when satisfying
+the proposition above, has then been measured. To do so, chaotic iterations
+have first been rewriten as simple discrete dynamical systems, as follows.
 
 
 
-- 
2.39.5