$$\left.\forall i,j \in \mathds{N}, U^i \in \llbracket 1, \mathsf{N} \rrbracket, V^j \in \mathcal{P}\right\}
\subset \mathcal{B}(x,\varepsilon),$$
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,
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
$\mathsf{N}$-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
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}
-jfjucobo16
Review 1
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
