Merge branch 'master' of ssh://info.iut-bm.univ-fcomte.fr/prng_gpu

[prng_gpu.git] / prng_gpu.tex

diff --git a/prng_gpu.tex b/prng_gpu.tex
index 20e256606089108345f7624f6f22121113fd72ec..55fc756560f07eab5d8a70a102c4798e8551a695 100644 (file)
--- a/prng_gpu.tex
+++ b/prng_gpu.tex
@@ -224,7 +224,10 @@ We can finally remark that, to the best of our knowledge, no GPU implementation
 \label{section:BASIC RECALLS}
 
 This section is devoted to basic definitions and terminologies in the fields of
 \label{section:BASIC RECALLS}
 
 This section is devoted to basic definitions and terminologies in the fields of

-topological chaos and chaotic iterations.
+topological chaos and chaotic iterations. We assume the reader is familiar
+with basic notions on topology (see for instance~\cite{Devaney}).
+
+

 \subsection{Devaney's Chaotic Dynamical Systems}
 
 In the sequel $S^{n}$ denotes the $n^{th}$ term of a sequence $S$ and $V_{i}$
 \subsection{Devaney's Chaotic Dynamical Systems}
 
 In the sequel $S^{n}$ denotes the $n^{th}$ term of a sequence $S$ and $V_{i}$


@@ -237,7 +240,7 @@ Consider a topological space $(\mathcal{X},\tau)$ and a continuous function $f :
 \mathcal{X} \rightarrow \mathcal{X}$.
 
 \begin{definition}
 \mathcal{X} \rightarrow \mathcal{X}$.
 
 \begin{definition}

-$f$ is said to be \emph{topologically transitive} if, for any pair of open sets
+The function $f$ is said to be \emph{topologically transitive} if, for any pair of open sets

 $U,V \subset \mathcal{X}$, there exists $k>0$ such that $f^k(U) \cap V \neq
 \varnothing$.
 \end{definition}
 $U,V \subset \mathcal{X}$, there exists $k>0$ such that $f^k(U) \cap V \neq
 \varnothing$.
 \end{definition}


@@ -256,7 +259,7 @@ necessarily the same period).
 
 
 \begin{definition}[Devaney's formulation of chaos~\cite{Devaney}]
 
 
 \begin{definition}[Devaney's formulation of chaos~\cite{Devaney}]

-$f$ is said to be \emph{chaotic} on $(\mathcal{X},\tau)$ if $f$ is regular and
+The function $f$ is said to be \emph{chaotic} on $(\mathcal{X},\tau)$ if $f$ is regular and

 topologically transitive.
 \end{definition}
 
 topologically transitive.
 \end{definition}
 


@@ -264,12 +267,12 @@ The chaos property is strongly linked to the notion of ``sensitivity'', defined
 on a metric space $(\mathcal{X},d)$ by:
 
 \begin{definition}
 on a metric space $(\mathcal{X},d)$ by:
 
 \begin{definition}

-\label{sensitivity} $f$ has \emph{sensitive dependence on initial conditions}
+\label{sensitivity} The function $f$ has \emph{sensitive dependence on initial conditions}

 if there exists $\delta >0$ such that, for any $x\in \mathcal{X}$ and any
 neighborhood $V$ of $x$, there exist $y\in V$ and $n > 0$ such that
 $d\left(f^{n}(x), f^{n}(y)\right) >\delta $.
 
 if there exists $\delta >0$ such that, for any $x\in \mathcal{X}$ and any
 neighborhood $V$ of $x$, there exist $y\in V$ and $n > 0$ such that
 $d\left(f^{n}(x), f^{n}(y)\right) >\delta $.
 

-$\delta$ is called the \emph{constant of sensitivity} of $f$.
+The constant $\delta$ is called the \emph{constant of sensitivity} of $f$.

 \end{definition}
 
 Indeed, Banks \emph{et al.} have proven in~\cite{Banks92} that when $f$ is
 \end{definition}
 
 Indeed, Banks \emph{et al.} have proven in~\cite{Banks92} that when $f$ is


@@ -807,7 +810,11 @@ where $(s^0,s^1, \hdots)$ is the strategy of $Y$, satisfies the properties
 claimed in the lemma.
 \end{proof}
 
 claimed in the lemma.
 \end{proof}
 

+<<<<<<< HEAD
+We can now prove the Theorem~\ref{t:chaos des general}.
+=======

 We can now prove Theorem~\ref{t:chaos des general}...
 We can now prove Theorem~\ref{t:chaos des general}...

+>>>>>>> e55d237aba022a66cc2d7650d295b29169878f45

 
 \begin{proof}[Theorem~\ref{t:chaos des general}]
 Firstly, strong transitivity implies transitivity.
 
 \begin{proof}[Theorem~\ref{t:chaos des general}]
 Firstly, strong transitivity implies transitivity.


@@ -1232,8 +1239,10 @@ $y\bigoplus_{i=1}^{i=j} w_i^\prime=y\bigoplus_{i=1}^{i=j} w_i$. It follows,
 by a direct induction, that $w_i=w_i^\prime$. Furthermore, since $\mathbb{B}^{kN}$
 is finite, each $\varphi_y$ is bijective. Therefore, and using (\ref{PCH-1}),
 one has
 by a direct induction, that $w_i=w_i^\prime$. Furthermore, since $\mathbb{B}^{kN}$
 is finite, each $\varphi_y$ is bijective. Therefore, and using (\ref{PCH-1}),
 one has

+$\mathrm{Pr}[D^\prime(U_{kN})=1]=\mathrm{Pr}[D(\varphi_y(U_{kN}))=1]$ and,
+therefore, 

 \begin{equation}\label{PCH-2}
 \begin{equation}\label{PCH-2}

-\mathrm{Pr}[D^\prime(U_{kN})=1]=\mathrm{Pr}[D(\varphi_y(U_{kN}))=1]=\mathrm{Pr}[D(U_{kN})=1].
+\mathrm{Pr}[D^\prime(U_{kN})=1]=\mathrm{Pr}[D(U_{kN})=1].

 \end{equation}
 
 Now, using (\ref{PCH-1}) again, one has  for every $x$,
 \end{equation}
 
 Now, using (\ref{PCH-1}) again, one has  for every $x$,