referencesdéfiniesplusieurs fois

diff --git a/12TIPE.tex b/12TIPE.tex
index 4cbc26dbf3263c619e9ab6faa50caccece609b51..85efe7887f31637058b7eba14c7e397640342c05 100644 (file)
--- a/12TIPE.tex
+++ b/12TIPE.tex
@@ -119,7 +119,7 @@ Leur preuve est donnée en annexe~\ref{anx:chaos:unaire}.
 \end{theorem}
 
 \begin{theorem}
 \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}
 \end{theorem}
 
 On peut conclure  que $\mathcal{C} = \mathcal{R} \cap \mathcal{T}

diff --git a/14Secrypt.tex b/14Secrypt.tex
index b7cb0d398c9e99eeb53a93f531134b820d910661..d4f76f43b71f1fa00568c314a0fcfef58446aedc 100644 (file)
--- a/14Secrypt.tex
+++ b/14Secrypt.tex
@@ -83,7 +83,7 @@ bistoc(X):-
   allpositive(S4).
 \end{lstlisting}
 \end{scriptsize}
   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 
 \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 
 $000,100,101,001,011,111,110,010,000$ 
 (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 à 
 Ce cycle qui visite chaque n{\oe}ud exactement une fois est un  
 \emph{cycle hamiltonien}.
 La matrice de Markov correspondante est donnée à 


@@ -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
     \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 $\textsc{giu}(f^*)$
       \end{minipage}
     }%
     \subfigure[Matrice de Markov associée à $\textsc{giu}(f^*)$

diff --git a/15TSI.tex b/15TSI.tex
index 7d1d8429d9b21e6015aad11847303cc23aac668e..76e0203ebf681fa2c274c0ceaf26a6fd60e24aa1 100644 (file)
--- 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
 \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 $\Bool^{\mathsf{N}}  \times  \mathcal{P}(\{1, \ldots, {\mathsf{N}}\})^{\Nats}$ 
     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}
 \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 $\mathcal{C}$]
 \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}
 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 0427db87648bac22bf28aab93b5943f6f6cb4475..54d99afba1bd4b8b0cb69b8de08c487633980134 100644 (file)
--- 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}
 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}
 
 
 \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 $\mathcal{C}$]
 = \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}
 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 c74aaf063e463c156d6342829e7e9ceca306731c..1111939ac3e8596062ee0388e2c33fb40936a37f 100644 (file)
--- 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 $\mathcal{C}$]
 = \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}
 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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