From 020defdbb2ac938563eba1071c78520973093e4b Mon Sep 17 00:00:00 2001
From: couchot <jf.couchot@gmail.com>
Date: Wed, 22 Jul 2015 08:47:19 +0200
Subject: [PATCH] =?utf8?q?referencesd=C3=A9finiesplusieurs=20fois?=
MIME-Version: 1.0
Content-Type: text/plain; charset=utf8
Content-Transfer-Encoding: 8bit
---
12TIPE.tex | 2 +-
14Secrypt.tex | 7 +-
15TSI.tex | 6 +-
caracgeneralise.tex | 4 +-
caracunaire.tex | 2 +-
images/iter_f0d.eps | 902 ++++++++++++++++++++++++++++++++++++++++++++
images/iter_f0d.pdf | Bin 0 -> 5280 bytes
7 files changed, 913 insertions(+), 10 deletions(-)
create mode 100644 images/iter_f0d.eps
create mode 100644 images/iter_f0d.pdf
diff --git a/12TIPE.tex b/12TIPE.tex
index 4cbc26d..85efe78 100644
--- a/12TIPE.tex
+++ b/12TIPE.tex
@@ -119,7 +119,7 @@ Leur preuve est donnée en annexe~\ref{anx:chaos:unaire}.
\end{theorem}
\begin{theorem}
-\label{Prop: T est dans R} $\mathcal{T} \subset \mathcal{R}$.
+\label{Prop: T est dans R:u} $\mathcal{T} \subset \mathcal{R}$.
\end{theorem}
On peut conclure que $\mathcal{C} = \mathcal{R} \cap \mathcal{T}
diff --git a/14Secrypt.tex b/14Secrypt.tex
index b7cb0d3..d4f76f4 100644
--- a/14Secrypt.tex
+++ b/14Secrypt.tex
@@ -83,7 +83,7 @@ bistoc(X):-
allpositive(S4).
\end{lstlisting}
\end{scriptsize}
-\caption{Prolog Problem to Find DSSC Matrix when $n=2$}\label{fig:prolog}
+\caption{Code PROLOG permettant de trouver toutes les matrices DSSC pour $n=2$}\label{fig:prolog}
\end{figure}
Enfin, on définit la relation $\mathcal{R}$, qui est établie pourles deux
@@ -156,7 +156,8 @@ 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.
+a été enlevé. Dans cette figure, le 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}.
La matrice de Markov correspondante est donnée Ã
@@ -173,7 +174,7 @@ On s'intéresse par la suite à la génération de ce genre de cycles.
\label{fig:iteration:f*}]{
\begin{minipage}{0.55\linewidth}
\centering
- \includegraphics[width=\columnwidth]{images/iter_f0c}%
+ \includegraphics[width=\columnwidth]{images/iter_f0d}%
\end{minipage}
}%
\subfigure[Matrice de Markov associée à $\textsc{giu}(f^*)$
diff --git a/15TSI.tex b/15TSI.tex
index 7d1d842..76e0203 100644
--- a/15TSI.tex
+++ b/15TSI.tex
@@ -26,7 +26,7 @@ $x^0\in\Bool^{\mathsf{N}}$ et une stratégie $S = \left(s_t\right)_{t \in \math
\in \mathcal{P}(\{1, \ldots, {\mathsf{N}}\})^{\Nats}$,
les
configurations $x^t$ sont définies par la récurrence
-\begin{equation}\label{eq:asyn}
+\begin{equation}\label{eq:asyn:g}
x^{t+1}=F_{f_g}(s_t,x^t).
\end{equation}
Soit alors $G_{f_g}$ une fonction de $\Bool^{\mathsf{N}} \times \mathcal{P}(\{1, \ldots, {\mathsf{N}}\})^{\Nats}$
@@ -93,12 +93,12 @@ annexe~\ref{anx:chaos:generalise}.
\end{theorem}
\begin{theorem}
-\label{Prop: T est dans R} $\mathcal{T} \subset \mathcal{R}$.
+\label{Prop: T est dans R:g} $\mathcal{T} \subset \mathcal{R}$.
\end{theorem}
\begin{theorem}%[Characterization of $\mathcal{C}$]
-\label{Th:CaracIC}
+\label{Th:CaracIC:g}
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}
diff --git a/caracgeneralise.tex b/caracgeneralise.tex
index 0427db8..54d99af 100644
--- a/caracgeneralise.tex
+++ b/caracgeneralise.tex
@@ -56,7 +56,7 @@ par contraposée, on a la démonstration souhaitée.
Prouvons à présent le théorème suivant:
\begin{theorem}
-\label{Prop: T est dans R} $\mathcal{T} \subset \mathcal{R}$.
+\label{Prop: T est dans R:gp} $\mathcal{T} \subset \mathcal{R}$.
\end{theorem}
@@ -91,7 +91,7 @@ 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}
+\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}
diff --git a/caracunaire.tex b/caracunaire.tex
index c74aaf0..1111939 100644
--- a/caracunaire.tex
+++ b/caracunaire.tex
@@ -94,7 +94,7 @@ 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}
+\label{Th:CaracIC:up}
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}
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