From c9d6a79d174af327059607a3e0a1f23ab7921a49 Mon Sep 17 00:00:00 2001 From: couchot Date: Wed, 24 Aug 2016 14:36:35 +0200 Subject: [PATCH 1/1] =?utf8?q?quelques=20corrections=20apr=C3=A8s=20remarq?= =?utf8?q?ues=20de=20Sylvain?= MIME-Version: 1.0 Content-Type: text/plain; charset=utf8 Content-Transfer-Encoding: 8bit --- chaos.tex | 3 ++- generating.tex | 15 +++++++-------- main.bbl | 6 ++++++ main.pdf | Bin 400526 -> 378245 bytes review.txt | 4 ++-- 5 files changed, 17 insertions(+), 11 deletions(-) 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 14bb15cadca1029381d2cc292530c9f20adb8fe2..d8fbf55fe871eee191fa92fb1d69eb1f6564b775 100644 GIT binary patch delta 124082 zcmV(=K-s^Jxfq3u7qA|e0yZ;~AeSwF+_shP`4!qejlOEQ#GQ1S?w6!a>a62rYdh_J zX&y!rB~3D#QRZUV`|I}{9FPPs;E;2%>$H6lNel%72j_kc4)U|xv;6GqPri8d$$x&u z1Uu7NX+&}M>gG%qS*hij(RpS}dG_l1?8nhJ-@X3Ji*unypFckz3zd()`--lA&(D8) z^$qnPv$9~q_F%GH>ob-yE$IJ0Z(nOa#51MX+1PvS6~1;Rvb-#?U1|6Er>>?Tu=X==lp82o-XDqYP_s&;5A`J6+PdcvwT#|p(mzB@Z$WI+UcFuX0~2s)be7n zo?caU_&I=(Wu?+Il=aM659)M!U 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