From: couchot <couchot@localhost.localdomain>
Date: Wed, 24 Aug 2016 12:36:35 +0000 (+0200)
Subject: quelques corrections après remarques de Sylvain
X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/16dcc.git/commitdiff_plain/c9d6a79d174af327059607a3e0a1f23ab7921a49?ds=sidebyside

quelques corrections après remarques de Sylvain
---

diff --git a/chaos.tex b/chaos.tex
index dafc635..acf42e3 100644
--- 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,
-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, 
diff --git a/generating.tex b/generating.tex
index d512a98..0837fda 100644
--- 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}
-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 
@@ -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 
-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 
diff --git a/main.bbl b/main.bbl
index 7655838..9cc3880 100644
--- 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.
 
+\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}
diff --git a/main.pdf b/main.pdf
index 14bb15c..d8fbf55 100644
Binary files a/main.pdf and b/main.pdf differ
diff --git a/review.txt b/review.txt
index 8ec57af..ad465f7 100644
--- a/review.txt
+++ b/review.txt
@@ -1,4 +1,3 @@
-jfjucobo16
 
 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
-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