X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/prng_gpu.git/blobdiff_plain/a17d0a6c41e0afab24e5a0c8c806533db3ad753a..893deb2477f17914ca2f0420009697c0925d5cbe:/prng_gpu.tex?ds=inline

diff --git a/prng_gpu.tex b/prng_gpu.tex
index 11a1d56..32055e7 100644
--- 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
@@ -191,12 +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}.
-
-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.
-
  Section~\ref{sec:efficient PRNG
   gpu}   describes and evaluates theoretically  the  GPU   implementation. 
 Such generators are experimented in 
@@ -718,17 +718,25 @@ 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 in Equation~\ref{eq:generalIC}
+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
@@ -981,10 +989,10 @@ The proof is given in Section~\ref{A-deuxième def} of the annex document.
 %%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}
 
@@ -1308,7 +1316,7 @@ The content is this section is given in Section~\ref{A-The generation of pseudor
 %% 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