Corrections en anglais

diff --git a/prng_gpu.tex b/prng_gpu.tex
index 307f55d4e0fb8626f134f8ba717465e7c94dd20e..73bc958598d278d6393d5df27a22f1fbfae59d7e 100644 (file)
--- a/prng_gpu.tex
+++ b/prng_gpu.tex
@@ -251,13 +251,13 @@ In other words, $\mathcal{X}$ is approximately equal to $\big[ 0, 2^\mathsf{N} \
 \end{figure}
 
 
 \end{figure}
 
 

-We have written a Python program to represent the chaotic iterations with the vectorial negation on the real line $\mathds{R}$. Various representations of these CIs are given in Figures \ref{fig:ICs}, \ref{fig:ICs2} and \ref{fig:ICs3}. It can be remarked that the function $g$ is \alert{affine par morceaux}: it is linear on each interval having the form $\left[ \dfrac{n}{10}, \dfrac{n+1}{10}\right[$, $n \in \llbracket 0;2^{10}\times 10 \rrbracket$ \alert{and its line has a pent equal to 10}. Let us justify these claims:
+We have written a Python program to represent the chaotic iterations with the vectorial negation on the real line $\mathds{R}$. Various representations of these CIs are given in Figures \ref{fig:ICs}, \ref{fig:ICs2} and \ref{fig:ICs3}. It can be remarked that the function $g$ is a piecewise linear function: it is linear on each interval having the form $\left[ \dfrac{n}{10}, \dfrac{n+1}{10}\right[$, $n \in \llbracket 0;2^{10}\times 10 \rrbracket$ and its slope is equal to 10. Let us justify these claims:

 
 \begin{proposition}
 \label{Prop:derivabilite des ICs}
 
 \begin{proposition}
 \label{Prop:derivabilite des ICs}

-Chaotic iterations $g$ defined on $\mathds{R}$ are \alert{infiniment dérivables} on $\big[ 0, 2^{10} \big[$, except on the 10241 points in $I$ defined by $\left\{ \dfrac{n}{10} ~\big/~ n \in \llbracket 0;2^{10}\times 10\rrbracket \right\}$.
+Chaotic iterations $g$ defined on $\mathds{R}$ have derivatives of all orders on $\big[ 0, 2^{10} \big[$, except on the 10241 points in $I$ defined by $\left\{ \dfrac{n}{10} ~\big/~ n \in \llbracket 0;2^{10}\times 10\rrbracket \right\}$.

 
 

-Furthermore, on each interval of the form $\left[ \dfrac{n}{10}, \dfrac{n+1}{10}\right[$, with $n \in \llbracket 0;2^{10}\times 10 \rrbracket$, the function $g$ is \emph{affine}. \alert{It is a line of pent equal to 10}: $\forall x \notin I, g'(x)=10$.
+Furthermore, on each interval of the form $\left[ \dfrac{n}{10}, \dfrac{n+1}{10}\right[$, with $n \in \llbracket 0;2^{10}\times 10 \rrbracket$, $g$ is a linear function, having a slope equal to 10: $\forall x \notin I, g'(x)=10$.

 \end{proposition}
 
 
 \end{proposition}
 
 


@@ -271,7 +271,7 @@ To sum up, the action of $g$ on the points of $I$ is as follows: first, make a m
 \end{proof}
 
 \begin{remark}
 \end{proof}
 
 \begin{remark}

-Finally, chaotic iterations are elements of the large family of functions that are \alert{chaotiques linéaires par morceaux}, like the tent map, the \alert{doublement de l'angle}, \emph{etc.}
+Finally, chaotic iterations are elements of the large family of functions that are both chaotic and piecewise linear (like the tent map).

 \end{remark}
 
 
 \end{remark}
 
 


@@ -308,7 +308,7 @@ If $D(x^n,x) \to 0$, then $D_e(x^n,x) = 0$ at least for $n$ larger than a given
 Additionally, $D_s(x^n, x) \to 0$, then $\forall k \in \mathds{N}^*, \exists N_k \in \mathds{N}, n \geqslant N_k \Rightarrow D_s(x^n,x) \leqslant 10^{-k}$. This means that for all $k$, an index $N_k$ can be found such that, $\forall n \geqslant N_k$, all the $x^n$ have the same $k$ firsts digits, which are the digits of $x$. We can deduce the convergence $\Delta(x^n,x) \to 0$, and thus the result.
 \end{proof}
 
 Additionally, $D_s(x^n, x) \to 0$, then $\forall k \in \mathds{N}^*, \exists N_k \in \mathds{N}, n \geqslant N_k \Rightarrow D_s(x^n,x) \leqslant 10^{-k}$. This means that for all $k$, an index $N_k$ can be found such that, $\forall n \geqslant N_k$, all the $x^n$ have the same $k$ firsts digits, which are the digits of $x$. We can deduce the convergence $\Delta(x^n,x) \to 0$, and thus the result.
 \end{proof}
 

-The conclusion of these propositions is that the proposed metric is more \alert{précise} than the Euclidian distance, that is:
+The conclusion of these propositions is that the proposed metric is more precise than the Euclidian distance, that is:

 
 \begin{corollary}
 $D$ is finer than the Euclidian distance $\Delta$.
 
 \begin{corollary}
 $D$ is finer than the Euclidian distance $\Delta$.


@@ -330,22 +330,22 @@ This corollary can be reformulated as follows:
 
 \subsubsection{Chaos according to Devaney}
 
 
 \subsubsection{Chaos according to Devaney}
 

-We have recalled previously that the chaotic iterations $\left(\Go, \mathcal{X}_d\right)$ are chaotic according to the formulation of Devaney. We can deduce that they are chaotic on $\mathds{R}$ too, when considering the \alert{topology of order}, because:
+We have recalled previously that the chaotic iterations $\left(\Go, \mathcal{X}_d\right)$ are chaotic according to the formulation of Devaney. We can deduce that they are chaotic on $\mathds{R}$ too, when considering the order topology, because:

 \begin{itemize}
 \item $\left(\Go, \mathcal{X}_d\right)$ and $\left(g, \big[ 0, 2^{10} \big[_D\right)$ are semiconjugate by $\varphi$,
 \item Then $\left(g, \big[ 0, 2^{10} \big[_D\right)$ is a system chaotic according to Devaney, because the semiconjugacy preserve this character.
 \begin{itemize}
 \item $\left(\Go, \mathcal{X}_d\right)$ and $\left(g, \big[ 0, 2^{10} \big[_D\right)$ are semiconjugate by $\varphi$,
 \item Then $\left(g, \big[ 0, 2^{10} \big[_D\right)$ is a system chaotic according to Devaney, because the semiconjugacy preserve this character.

-\item But the topology generated by $D$ is finer than the topology generated by the Euclidian distance $\Delta$ -- which is the \alert{topology of order}.
-\item According to Theorem \ref{Th:chaos et finesse}, we can deduce that the chaotic iterations $g$ are indeed chaotic, as defined by Devaney, for the \alert{topology of order} on $\mathds{R}$.
+\item But the topology generated by $D$ is finer than the topology generated by the Euclidian distance $\Delta$ -- which is the order topology.
+\item According to Theorem \ref{Th:chaos et finesse}, we can deduce that the chaotic iterations $g$ are indeed chaotic, as defined by Devaney, for the order topology on $\mathds{R}$.

 \end{itemize}
 
 This result can be formulated as follows.
 
 \begin{theorem}
 \label{th:IC et topologie de l'ordre}
 \end{itemize}
 
 This result can be formulated as follows.
 
 \begin{theorem}
 \label{th:IC et topologie de l'ordre}

-The chaotic iterations $g$ on $\mathds{R}$ are chaotic according to the Devaney's formulation, when $\mathds{R}$ has his usual topology, which is the \alert{topology of order}.
+The chaotic iterations $g$ on $\mathds{R}$ are chaotic according to the Devaney's formulation, when $\mathds{R}$ has his usual topology, which is the order topology.

 \end{theorem}
 
 \end{theorem}
 

-Indeed this result is \alert{weaker} than the theorem establishing the chaos for the finer topology $d$. However the Theorem \ref{th:IC et topologie de l'ordre} still remains important. Indeed, we have studied in our previous works a set different from the usual set of study ($\mathcal{X}$ instead of $\mathds{R}$), in order to be as close as possible from the computer: the properties of disorder proved theoretically will then be preserved when computing. However, we could wonder whether this change does not lead to a disorder of a lower quality. In other words, have we replaced a situation of a good disorder lost when computing, to another situation of a disorder preserved but of bad quality. Theorem \ref{th:IC et topologie de l'ordre} prove exactly the contrary.
+Indeed this result is weaker than the theorem establishing the chaos for the finer topology $d$. However the Theorem \ref{th:IC et topologie de l'ordre} still remains important. Indeed, we have studied in our previous works a set different from the usual set of study ($\mathcal{X}$ instead of $\mathds{R}$), in order to be as close as possible from the computer: the properties of disorder proved theoretically will then be preserved when computing. However, we could wonder whether this change does not lead to a disorder of a lower quality. In other words, have we replaced a situation of a good disorder lost when computing, to another situation of a disorder preserved but of bad quality. Theorem \ref{th:IC et topologie de l'ordre} prove exactly the contrary.