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diff --git a/prng_gpu.tex b/prng_gpu.tex
index 25214fdd48735df7cd383019f240dc838a8f3b8e..32055e7c41eadc7c7c62f8534bbf6b752d48f894 100644 (file)
--- a/prng_gpu.tex
+++ b/prng_gpu.tex
@@ -192,12 +192,11 @@ The remainder of this paper  is organized as follows. In Section~\ref{section:re
   and on an iteration process called ``chaotic
 iterations'' on which the post-treatment is based. 
 The proposed PRNG and its proof of chaos are given in  Section~\ref{sec:pseudorandom}.
   and on an iteration process called ``chaotic
 iterations'' on which the post-treatment is based. 
 The proposed PRNG and its proof of chaos are given in  Section~\ref{sec:pseudorandom}.

-
-Section~\ref{The generation of pseudorandom sequence} illustrates the statistical
-improvement related to the chaotic iteration based post-treatment, for
-our previously released PRNGs and a new efficient 
+Section~\ref{sec:efficient PRNG} %{The generation of pseudorandom sequence} %illustrates the statistical
+%improvement related to the chaotic iteration based post-treatment, for
+%our previously released PRNGs and
+ contains a new efficient 

 implementation on CPU.
 implementation on CPU.

-

  Section~\ref{sec:efficient PRNG
   gpu}   describes and evaluates theoretically  the  GPU   implementation. 
 Such generators are experimented in 
  Section~\ref{sec:efficient PRNG
   gpu}   describes and evaluates theoretically  the  GPU   implementation. 
 Such generators are experimented in 


@@ -726,15 +725,7 @@ investigated in Annex~\ref{A-deuxième def}, leading to the following result.
 
  \begin{theorem}
  \label{t:chaos des general}
 
  \begin{theorem}
  \label{t:chaos des general}

-  The general chaotic iterations defined by
- \begin{equation}
-   x_i^n=\left\{
- \begin{array}{ll}
-   x_i^{n-1} &  \text{ if  } i \notin \mathcal{S}^n \\
-   \left(f(x^{n-1})\right)_{S^n} & \text{ if }i \in \mathcal{S}^n.
- \end{array}\right.
- \label{general CIs}
- \end{equation}
+  The general chaotic iterations defined in Equation~\ref{eq:generalIC}

 satisfy
  the Devaney's property of chaos.
  \end{theorem}
 satisfy
  the Devaney's property of chaos.
  \end{theorem}


@@ -998,10 +989,10 @@ satisfy
 %%RAF : mis en supplementary
 
 
 %%RAF : mis en supplementary
 
 

-\section{Statistical Improvements Using Chaotic Iterations}
-\label{The generation of pseudorandom sequence}
-The content is this section is given in Section~\ref{A-The generation of pseudorandom sequence} of the annex document.
-
+%\section{Statistical Improvements Using Chaotic Iterations}
+%\label{The generation of pseudorandom sequence}
+%The content is this section is given in Section~\ref{A-The generation of pseudorandom sequence} of the annex document.
+The reasons to desire chaos to achieve randomness are given in Annex~\ref{A-The generation of pseudorandom sequence}.

 
 %% \label{The generation of pseudorandom sequence}
 
 
 %% \label{The generation of pseudorandom sequence}
 


@@ -1325,7 +1316,7 @@ The content is this section is given in Section~\ref{A-The generation of pseudor
 %% raise ambiguity.
 
 
 %% raise ambiguity.
 
 

-\subsection{First Efficient Implementation of a PRNG based on Chaotic Iterations}
+\section{First Efficient Implementation of a PRNG based on Chaotic Iterations}

 \label{sec:efficient PRNG}
 %
 %Based on the proof presented in the previous section, it is now possible to 
 \label{sec:efficient PRNG}
 %
 %Based on the proof presented in the previous section, it is now possible to