Merge branch 'master' of ssh://bilbo.iut-bm.univ-fcomte.fr/16dcc

diff --git a/main.pdf b/main.pdf
index 58e266f7178b4897eccea184fe3b48b5984a83e4..ff7928d3826af47791ba6a0a1c9bfd8a8f9d1716 100644 (file)
Binary files a/main.pdf and b/main.pdf differ

diff --git a/main.tex b/main.tex
index f49eef653f907d5b3093d9fddf98eb8a1167eda9..62257511298d9bf93aaa0130d7ac4f235c281aa3 100644 (file)
--- a/main.tex
+++ b/main.tex
@@ -551,15 +551,15 @@ Moreover, it is not clear that embedding such kind of functions into a PRNG
 allows to get a chaotic output, which could be required for simulating 
 some chaotic behaviours.
 
-In a previous work, some of the authors have proposed the idea of walking into a 
-$\mathsf{N}$-cube where a balanced Hamiltonian cycle have been removed 
-as the basis of a chaotic PRNG.
-In this article, all the difficult issues observed in the previous work have been tackled.
-The chaotic behavior of the whole PRNG is proven.
-The construction of the balanced Hamiltonian cycle is  theoretically and practically solved.
-A upper bound of the length of the walk to obtain a uniform distribution is calculated.
-Finally practical experiments show that the generators successfully pass 
-the classical statistical tests.
+In a previous work, some of the authors have proposed the idea of walking
+into a $\mathsf{N}$-cube where a balanced Hamiltonian cycle have been
+removed as the basis of a chaotic PRNG. In this article, all the difficult
+issues observed in the previous work have been tackled. The chaotic behavior
+of the whole PRNG is proven. The construction of the balanced Hamiltonian
+cycle is theoretically and practically solved. An upper bound of the
+expected length of the walk to obtain a uniform distribution is calculated.
+Finally practical experiments show that the generators successfully pass the
+classical statistical tests.
 
 
 \end{abstract}

diff --git a/stopping.tex b/stopping.tex
index 284fc49e57b604847b7ebed6b8be603ece7f5b87..539d653dd3fa66c209cb9b7d11a49c05b32d6021 100644 (file)
--- a/stopping.tex
+++ b/stopping.tex
@@ -60,12 +60,14 @@ $$\tv{\pi-\mu}=\frac{1}{2}\sum_{X\in\Bool^{\mathsf{N}}}|\pi(X)-\mu(X)|.$$ Moreov
 $\nu$ is a distribution on $\Bool^{\mathsf{N}}$, one has
 $$\tv{\pi-\mu}\leq \tv{\pi-\nu}+\tv{\nu-\mu}$$
 
-Let $P$ be the matrix of a Markov chain on $\Bool^{\mathsf{N}}$. $P(X,\cdot)$ is the
-distribution induced by the $X$-th row of $P$. If the Markov chain induced by
-$P$ has a stationary distribution $\pi$, then we define
+Let $P$ be the matrix of a Markov chain on $\Bool^{\mathsf{N}}$. For any
+$X\in \Bool^{\mathsf{N}}$, let $P(X,\cdot)$ be the distribution induced by the
+${\rm bin}(X)$-th row of $P$, where ${\rm bin}(X)$ is the integer whose
+binary encoding is $X$. If the Markov chain induced by $P$ has a stationary
+distribution $\pi$, then we define
 $$d(t)=\max_{X\in\Bool^{\mathsf{N}}}\tv{P^t(X,\cdot)-\pi}.$$
 
-\ANNOT{incohérence de notation $X$ : entier ou dans $B^N$ ?}
+%\ANNOT{incohérence de notation $X$ : entier ou dans $B^N$ ?}
 and
 
 $$t_{\rm mix}(\varepsilon)=\min\{t \mid d(t)\leq \varepsilon\}.$$