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diff --git a/prng_gpu.tex b/prng_gpu.tex
index 11a1d56bbd4b098c2abc1326113723497a16f7b5..25214fdd48735df7cd383019f240dc838a8f3b8e 100644 (file)
--- a/prng_gpu.tex
+++ b/prng_gpu.tex
@@ -14,6 +14,7 @@
 \usepackage{algorithmic}
 \usepackage{slashbox}
 \usepackage{ctable}
+\usepackage{cite}
 \usepackage{tabularx}
 \usepackage{multirow}
 % Pour mathds : les ensembles IR, IN, etc.
@@ -54,8 +55,8 @@ Guyeux, and Pierre-Cyrille Héam\thanks{Authors in alphabetic order}}
 \IEEEcompsoctitleabstractindextext{
 \begin{abstract}
 In this paper we present a new pseudorandom number generator (PRNG) on
-graphics processing units  (GPU). This PRNG is based  on the so-called chaotic iterations.  It
-is firstly proven  to be chaotic according to the Devaney's  formulation. We thus propose  an efficient
+graphics processing units  (GPU). This PRNG is based  on the so-called chaotic iterations and
+it is thus chaotic according to the Devaney's  formulation. We propose  an efficient
 implementation  for  GPU that successfully passes the   {\it BigCrush} tests, deemed to be the  hardest
 battery of tests in TestU01.  Experiments show that this PRNG can generate
 about 20 billion of random numbers  per second on Tesla C1060 and NVidia GTX280
@@ -718,17 +719,33 @@ in what follows).
 However, proofs
 of chaos obtained in~\cite{bg10:ij} have been established
 only for chaotic iterations of the form presented in Definition 
-\ref{Def:chaotic iterations}. The question is now to determine whether the
+\ref{Def:chaotic iterations}. The question to determine whether the
 use of more general chaotic iterations to generate pseudorandom numbers 
-faster, does not deflate their topological chaos properties.
+faster, does not deflate their topological chaos properties, has been
+investigated in Annex~\ref{A-deuxième def}, leading to the following result.
+
+ \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}
+satisfy
+ the Devaney's property of chaos.
+ \end{theorem}
 
 
 %%RAF proof en supplementary, j'ai mis le theorem.
 % A vérifier
 
- \subsection{Proofs of Chaos of the General Formulation of the Chaotic Iterations}
-\label{deuxième def}
-The proof is given in Section~\ref{A-deuxième def} of the annex document.
+% \subsection{Proofs of Chaos of the General Formulation of the Chaotic Iterations}
+%\label{deuxième def}
+%The proof is given in Section~\ref{A-deuxième def} of the annex document.
 %% \label{deuxième def}
 %% Let us consider the discrete dynamical systems in chaotic iterations having 
 %% the general form: $\forall    n\in     \mathds{N}^{\ast     }$, $  \forall     i\in