quelques corrections après remarques de Sylvain

diff --git a/chaos.tex b/chaos.tex
index dafc635458f21fa5851632657325f79351d1e89c..acf42e33893ed388a70ccc519d5e14cbcf944629 100644 (file)
--- a/chaos.tex
+++ b/chaos.tex
@@ -510,7 +510,8 @@ $$\left\{(e, ((u^0, \dots, u^{v^{k_1-1}},U^0, U^1, \dots),(v^0, \dots, v^{k_1},V
 $$\left.\forall i,j \in \mathds{N}, U^i \in \llbracket 1, \mathsf{N} \rrbracket, V^j \in \mathcal{P}\right\}
 \subset \mathcal{B}(x,\varepsilon),$$
 and $y=G_f^{k_1}(e,(u,v))$. $\Gamma_{\mathcal{P}}(f)$ being strongly connected,
 $$\left.\forall i,j \in \mathds{N}, U^i \in \llbracket 1, \mathsf{N} \rrbracket, V^j \in \mathcal{P}\right\}
 \subset \mathcal{B}(x,\varepsilon),$$
 and $y=G_f^{k_1}(e,(u,v))$. $\Gamma_{\mathcal{P}}(f)$ being strongly connected,

-there is at least a path from the Boolean state $y_1$ of $y$ and $e$ \ANNOT{Phrase pas claire : "from \dots " mais pas de "to \dots"}.
+there is at least a path from the Boolean state $y_1$ of $y$ to $e$.
+%\ANNOT{Phrase pas claire : "from \dots " mais pas de "to \dots"}.

 Denote by $a_0, \hdots, a_{k_2}$ the edges of such a path.
 Then the point:\linebreak
 $(e,((u^0, \dots, u^{v^{k_1-1}},a_0^0, \dots, a_0^{|a_0|}, a_1^0, \dots, a_1^{|a_1|},\dots, 
 Denote by $a_0, \hdots, a_{k_2}$ the edges of such a path.
 Then the point:\linebreak
 $(e,((u^0, \dots, u^{v^{k_1-1}},a_0^0, \dots, a_0^{|a_0|}, a_1^0, \dots, a_1^{|a_1|},\dots, 

diff --git a/generating.tex b/generating.tex
index d512a9803b59c27f0a95a6a6bff958b887a8f30d..0837fdaec746214beea9f8734d672881c96634cc 100644 (file)
--- a/generating.tex
+++ b/generating.tex
@@ -51,7 +51,7 @@ The question which remains to solve is:
 The answer is indeed positive. We furthermore have the following strongest 
 result.
 \begin{thrm}
 The answer is indeed positive. We furthermore have the following strongest 
 result.
 \begin{thrm}

-There exist $b \in \Nats$ such that $\Gamma_{\{b\}}(f)$ is complete.
+There exists $b \in \Nats$ such that $\Gamma_{\{b\}}(f)$ is complete.

 \end{thrm}
 \begin{proof}
 There is an arc $(x,y)$ in the 
 \end{thrm}
 \begin{proof}
 There is an arc $(x,y)$ in the 


@@ -65,13 +65,12 @@ This section ends with the idea of removing a Hamiltonian cycle in the
 $\mathsf{N}$-cube. 
 In such a context, the Hamiltonian cycle is equivalent to a Gray code.
 Many approaches have been proposed a way to build such codes, for instance 
 $\mathsf{N}$-cube. 
 In such a context, the Hamiltonian cycle is equivalent to a Gray code.
 Many approaches have been proposed a way to build such codes, for instance 

-the Reflected Binary Code. In this one, one of the bits is switched 
-exactly $2^{\mathsf{N}-}$ \ANNOT{formule incomplète : $2^{\mathsf{N}-1}$ ??} for a $\mathsf{N}$-length cycle. 
-
-%%%%%%%%%%%%%%%%%%%%%
-
-The function that is built 
-from the \ANNOT{Phrase non terminée}
+the Reflected Binary Code. In this one and 
+for a $\mathsf{N}$-length cycle, one of the bits is exactly switched 
+$2^{\mathsf{N}-1}$ times whereas the others bits are modified at most 
+$\left\lfloor \dfrac{2^{\mathsf{N-1}}}{\mathsf{N}-1} \right\rfloor$ times.
+It is clear that the function that is built from such a code would
+not provide a uniform output.  

 
 The next section presents how to build balanced Hamiltonian cycles in the 
 $\mathsf{N}$-cube with the objective to embed them into the 
 
 The next section presents how to build balanced Hamiltonian cycles in the 
 $\mathsf{N}$-cube with the objective to embed them into the 

diff --git a/main.bbl b/main.bbl
index 76558380e60c4783fb12a7ec477ec635f9a0f794..9cc3880bb6335c22eac70cd3d5917a1302613faf 100644 (file)
--- a/main.bbl
+++ b/main.bbl
@@ -138,4 +138,10 @@ D.~A. Levin, Y.~Peres, and E.~L. Wilmer, \emph{{Markov chains and mixing
 M.~Mitzenmacher and E.~Upfal, \emph{Probability and Computing}.\hskip 1em plus
   0.5em minus 0.4em\relax Cambridge University Press, 2005.
 
 M.~Mitzenmacher and E.~Upfal, \emph{Probability and Computing}.\hskip 1em plus
   0.5em minus 0.4em\relax Cambridge University Press, 2005.
 

+\bibitem{matsumoto1998mersenne}
+M.~Matsumoto and T.~Nishimura, ``Mersenne twister: a 623-dimensionally
+  equidistributed uniform pseudo-random number generator,'' \emph{ACM
+  Transactions on Modeling and Computer Simulation (TOMACS)}, vol.~8, no.~1,
+  pp. 3--30, 1998.
+

 \end{thebibliography}
 \end{thebibliography}

diff --git a/main.pdf b/main.pdf
index 14bb15cadca1029381d2cc292530c9f20adb8fe2..d8fbf55fe871eee191fa92fb1d69eb1f6564b775 100644 (file)
Binary files a/main.pdf and b/main.pdf differ

diff --git a/review.txt b/review.txt
index 8ec57afd3e24a90f3fadd0244acb0195145a9b96..ad465f7ca46b54cae9a20fc35595f794acc2b58e 100644 (file)
--- a/review.txt
+++ b/review.txt
@@ -1,4 +1,3 @@
-jfjucobo16

 
 Review 1
 
 
 Review 1
 


@@ -7,7 +6,8 @@ number generators (PRNG) introduced in a previous work by the same authors.
 These PRNGs are based on iterating continuous functions on a discrete domain.
        The paper first recalls Devaney’s definition of chaos and presents the proof of
 the main results. Next, the authors study the stopping time, i.e. the time until
 These PRNGs are based on iterating continuous functions on a discrete domain.
        The paper first recalls Devaney’s definition of chaos and presents the proof of
 the main results. Next, the authors study the stopping time, i.e. the time until

-a uniform distribution is reached. Finally, they evaluate the PRNG against the
+a uniform distribut
+ion is reached. Finally, they evaluate the PRNG against the

 NIST suite.
 Review 1
 
 NIST suite.
 Review 1