Ajout figure 3-cube

[16dcc.git] / prng.tex

diff --git a/prng.tex b/prng.tex
index 8925e51326f8ebea7013e5ab1fc7cc5186c11753..2d1d7f28a3f5ccb9931caa4b8eafec913fd219f4 100644 (file)
--- a/prng.tex
+++ b/prng.tex
@@ -6,7 +6,7 @@ a PRNG \textit{Random},
 an integer $b$ that corresponds to an iteration number (\textit{i.e.}, the length of the walk), and 
 an initial configuration $x^0$. 
 Starting from $x^0$, the algorithm repeats $b$ times 
 an integer $b$ that corresponds to an iteration number (\textit{i.e.}, the length of the walk), and 
 an initial configuration $x^0$. 
 Starting from $x^0$, the algorithm repeats $b$ times 

-a random choice of which edge to follow, and traverses this edge 
+a random choice of which edge to follow, and crosses this edge 

 provided it is allowed to do so, \textit{i.e.}, 
 when $\textit{Random}(1)$ is not null. 
 The final configuration is thus outputted.
 provided it is allowed to do so, \textit{i.e.}, 
 when $\textit{Random}(1)$ is not null. 
 The final configuration is thus outputted.


@@ -46,7 +46,7 @@ Sect.~\ref{sec:hypercube}.
 
 
 Notice that the chaos property of $G_f$ given in Sect.\ref{sec:proofOfChaos}
 
 
 Notice that the chaos property of $G_f$ given in Sect.\ref{sec:proofOfChaos}

-only requires that the graph $\Gamma_{\{b\}}(f)$ is strongly connected.
+only requires the graph $\Gamma_{\{b\}}(f)$ to be  strongly connected.

 Since the $\chi_{\textit{16HamG}}$ algorithm 
 only adds probability constraints on existing edges, 
 it preserves this property. 
 Since the $\chi_{\textit{16HamG}}$ algorithm 
 only adds probability constraints on existing edges, 
 it preserves this property. 


@@ -61,7 +61,7 @@ whose Markov Matrix (issued from Eq.~(\ref{eq:Markov:rairo}))
 has the smallest practical mixing time.
 Such functions are 
 given in Table~\ref{table:nc}.
 has the smallest practical mixing time.
 Such functions are 
 given in Table~\ref{table:nc}.

-In this table, let us consider for instance 
+In this table, let us consider, for instance, 

 the function $\textcircled{a}$ from $\Bool^4$ to $\Bool^4$
 defined by the following images : 
 $[13, 10, 9, 14, 3, 11, 1, 12, 15, 4, 7, 5, 2, 6, 0, 8]$.
 the function $\textcircled{a}$ from $\Bool^4$ to $\Bool^4$
 defined by the following images : 
 $[13, 10, 9, 14, 3, 11, 1, 12, 15, 4, 7, 5, 2, 6, 0, 8]$.


@@ -71,7 +71,7 @@ the  second  list (namely~14).
 
 In this table the column that is labeled with $b$ %(respectively by $E[\tau]$)
 gives the practical mixing time 
 
 In this table the column that is labeled with $b$ %(respectively by $E[\tau]$)
 gives the practical mixing time 

-where the deviation to the standard distribution is lesser than $10^{-6}$.
+where the deviation to the standard distribution is inferior than $10^{-6}$.

 %(resp. the theoretical upper bound of stopping time as described in Sect.~\ref{sec:hypercube}).
 
 
 %(resp. the theoretical upper bound of stopping time as described in Sect.~\ref{sec:hypercube}).
 
 


@@ -172,7 +172,7 @@ $\textcircled{e}$&151, 149, 19, 210, 144, 152, 141, 206, 13, 12, 171, 10, 201, 1
 \end{tabular}
 \end{scriptsize}
 \end{center}
 \end{tabular}
 \end{scriptsize}
 \end{center}

-\caption{Functions with DSCC Matrix and smallest MT\label{table:nc}}
+\caption{Functions with DSCC Matrix and smallest MT}\label{table:nc}

 \end{table*}
 
 
 \end{table*}