From: Jean-François Couchot <couchot@couchot.iut-bm.univ-fcomte.fr>
Date: Thu, 7 Apr 2016 18:23:02 +0000 (+0200)
Subject: khkhj
X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/16dcc.git/commitdiff_plain/1b5eed47c5d9fada5fd208ab9a007f7d4355102e
khkhj
---
diff --git a/hamilton.tex b/hamilton.tex
index 694bbcf..f4dc64a 100644
--- a/hamilton.tex
+++ b/hamilton.tex
@@ -133,24 +133,62 @@ the code is neither balanced nor totally balanced.
\begin{thrm}\label{prop:balanced}
Let $\mathsf{N}$ in $\Nats$, and $a_{\mathsf{N}}$ be defined by
$a_{\mathsf{N}}= 2 \lfloor \dfrac{2^{\mathsf{N}}}{2\mathsf{N}} \rfloor$.
-Then, there exists a sequence $l$ in
+There exists then a sequence $l$ in
step~(\ref{item:nondet}) of the \emph{Robinson-Cohn extension} algorithm
such that all the transition counts $\textit{TC}_{\mathsf{N}}(i)$
are $a_{\mathsf{N}}$ or $a_{\mathsf{N}}+2$
for any $i$, $1 \le i \le \mathsf{N}$.
+\end{thrm}
-\end{thrm}
-
-
+The proof is done by induction on $\mathsf{N}$. Let us imadialty verify
+that it is established for both odd and even smallest values, \textit{i.e.}
+$3$ and $4$.
+For the initial case where $\mathsf{N}=3$, \textit{i.e.} $\mathsf{N-2}=1$ we successively have: $S_1=1,1$, $l=2$, $u_0 = \emptyset$, and $v=\emptyset$.
+Thus again the algorithm successively produces
+$U= 1,2,1$, $V = 3$, $W= 2, 1, 1,3$, and $W' = 1,2,1,3$.
+Finally, $S_3$ is $1,2,1,3,1,2,1,3$ which obviously verifies the theorem.
+ For the initial case where $\mathsf{N}=4$, \textit{i.e.} $\mathsf{N-2}=2$
+we successively have: $S_1=1,2,1,2$, $l=4$,
+$u_0,u_1,u_2 = \emptyset,\emptyset,\emptyset$, and $v=\emptyset$.
+Thus again the algorithm successively produces
+$U= 1,3,2,3,4,1,4,3,2$, $V = 4$, $W= 3, 1, 2, 1,2, 4$, and $W' = 1, 3, 2, 1,2, 4 $.
+Finally, $S_4$ is
+$
+2,3,4,1,4,3,2,3,1,4,1,3,2,1,2,4
+$
+such that $\textit{TC}_4(i) = 4$ and the theorem is established for
+odd and even initial values.
-\begin{proof}
+For the inductive case, let us first define some variables.
+Let $c_{\mathsf{N}}$ (resp. $d_{\mathsf{N}}$) be the number of elements
+whose transistion count is exactly $a_{\mathsf{N}}$ (resp $a_{\mathsf{N}} +2$).
+These two variables are defined by the system
+
+$$
+\left\{
+\begin{array}{lcl}
+c_{\mathsf{N}} + d_{\mathsf{N}} & = & \mathsf{N} \\
+c_{\mathsf{N}}a_{\mathsf{N}} + d_{\mathsf{N}}(a_{\mathsf{N}}+2) & = & 2^{\mathsf{N}}
+\end{array}
+\right.
+\qquad
+\Leftrightarrow
+\qquad
+\left\{
+\begin{array}{lcl}
+d_{\mathsf{N}} & = & \dfrac{2^{\mathsf{N} -n.a_{\mathsf{N}}}{2} \\
+c_{\mathsf{N}} = \mathsf{N} - d_{\mathsf{N}}
+\end{array}
+\right.
+$$
+Since $a_{\mathsf{N}$ is even, $d_{\mathsf{N}}$ is defined.
+Both $c_{\mathsf{N}}$ and $d_{\mathsf{N}}$ are obviously positves.
-\end{proof}
\subsection{Toward a local uniform distribution of switches}
diff --git a/main.aux b/main.aux
index 6f13901..af7621c 100644
--- a/main.aux
+++ b/main.aux
@@ -38,13 +38,13 @@
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.2}{A Metric Space for PRNG Iterations}}{5}}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.3}{A metric on $\mathcal {X}_{\mathsf {N},\mathcal {P}}$}}{6}}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.4}{$\Gamma _{\mathcal {P}}(f)$ as an extension of $\Gamma (f)$}}{8}}
+\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.5}{Proofs of chaos}}{8}}
\newlabel{graphe1}{{2a}{9}}
\newlabel{sub@graphe1}{{a}{9}}
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\newlabel{fig:itg}{{2}{9}}
-\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.5}{Proofs of chaos}}{9}}
\newlabel{prop:trans}{{3.8}{9}}
\citation{bcgr11:ip}
\citation{DBLP:conf/secrypt/CouchotHGWB14}
@@ -69,51 +69,4 @@
\citation{ZanSup04}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{5.1}{Analysis of the Robinson-Cohn extension algorithm}}{12}}
\newlabel{item:nondet}{{1}{12}}
-\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{5.2}{Balanced Codes}}{13}}
-\newlabel{prop:balanced}{{5.1}{13}}
-\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{5.3}{Toward a local uniform distribution of switches}}{13}}
-\@writefile{toc}{\contentsline {section}{\tocsection {}{6}{Stopping Time}}{13}}
-\newlabel{sec:hypercube}{{6}{13}}
-\citation{LevinPeresWilmer2006}
-\newlabel{eq:Markov:rairo}{{3}{15}}
-\newlabel{lm:h}{{6.2}{15}}
-\citation{proba}
-\newlabel{prop:stop}{{6.4}{16}}
-\newlabel{prop:lambda}{{6.5}{16}}
-\newlabel{lm:stopprime}{{6.6}{17}}
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-\newlabel{sec:prng}{{7}{17}}
-\@writefile{loa}{\contentsline {algocf}{\numberline {2}{\ignorespaces Pseudo Code of the $\chi _{\textit {15Rairo}}$ PRNG\relax }}{18}}
-\newlabel{CI Algorithm:2}{{2}{18}}
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-\newlabel{table:nc}{{1}{19}}
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-\bibstyle{alpha}
-\bibdata{biblio}
-\bibcite{Banks92}{BBCS92}
-\bibcite{bcgr11:ip}{BCGR11}
-\bibcite{Nist10}{BR10}
-\bibcite{DBLP:journals/combinatorics/BhatS96}{BS96}
-\bibcite{Bykov2016}{Byk16}
-\bibcite{chgw14oip}{CHG{$^{+}$}14a}
-\@writefile{lot}{\contentsline {table}{\numberline {2}{\ignorespaces NIST SP 800-22 test results ($\mathbb {P}_T$)\relax }}{20}}
-\newlabel{The passing rate}{{2}{20}}
-\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{20}}
-\bibcite{DBLP:conf/secrypt/CouchotHGWB14}{CHG{$^{+}$}14b}
-\bibcite{5376454}{CMZ09}
-\bibcite{Devaney}{Dev89}
-\bibcite{guyeuxTaiwan10}{GWB10}
-\bibcite{LevinPeresWilmer2006}{LPW06}
-\bibcite{LEcuyerS07}{LS07}
-\bibcite{Marsaglia1996}{Mar96}
-\bibcite{proba}{MU05}
-\bibcite{Robinson:1981:CS}{RC81}
-\bibcite{915385}{SK01}
-\bibcite{915396}{SPK01}
-\bibcite{ZanSup04}{SZ04}
-\bibcite{wbg10ip}{WBGF10}
-\newlabel{tocindent-1}{0pt}
-\newlabel{tocindent0}{13.28564pt}
-\newlabel{tocindent1}{18.39551pt}
-\newlabel{tocindent2}{26.57129pt}
-\newlabel{tocindent3}{0pt}
+\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{5.2}{Balanced Codes}}{12}}
diff --git a/main.blg b/main.blg
index b2d7619..01af0c2 100644
--- a/main.blg
+++ b/main.blg
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The top-level auxiliary file: main.aux
The style file: alpha.bst
Database file #1: biblio.bib
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You've used 19 entries,
2543 wiz_defined-function locations,
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index 749adc1..8af4a0a 100644
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-### bottom level
+Runaway argument?
+{2^{\mathsf {N} -n.a_{\mathsf {N}}}{2} \\ c_{\mathsf {N}} = \mathsf {\ETC.
+! Paragraph ended before \@genfrac was complete.
+<to be read again>
+ \par
+l.189
+
+? r
+OK, entering \nonstopmode...
+! Missing $ inserted.
+<inserted text>
+ $
+l.190 Since $a_
+ {\mathsf{N}$ is even, $d_{\mathsf{N}}$ is defined.
+I've inserted a begin-math/end-math symbol since I think
+you left one out. Proceed, with fingers crossed.
+
+! Missing } inserted.
+<inserted text>
+ }
+l.190 Since $a_{\mathsf{N}$
+ is even, $d_{\mathsf{N}}$ is defined.
+I've inserted something that you may have forgotten.
+(See the <inserted text> above.)
+With luck, this will get me unwedged. But if you
+really didn't forget anything, try typing `2' now; then
+my insertion and my current dilemma will both disappear.
+
+
+! LaTeX Error: Something's wrong--perhaps a missing \item.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type H <return> for immediate help.
+ ...
+
+l.194 \subsection
+ {Toward a local uniform distribution of switches}
+Try typing <return> to proceed.
+If that doesn't work, type X <return> to quit.
+
+
+! LaTeX Error: Something's wrong--perhaps a missing \item.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type H <return> for immediate help.
+ ...
+
+l.194 \subsection
+ {Toward a local uniform distribution of switches}
+Try typing <return> to proceed.
+If that doesn't work, type X <return> to quit.
+
+! Missing } inserted.
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+ }
+l.194 ...a local uniform distribution of switches}
+
+I've inserted something that you may have forgotten.
+(See the <inserted text> above.)
+With luck, this will get me unwedged. But if you
+really didn't forget anything, try typing `2' now; then
+my insertion and my current dilemma will both disappear.
+
+! Missing \cr inserted.
+<inserted text>
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+l.194 ...a local uniform distribution of switches}
+
+I'm guessing that you meant to end an alignment here.
+
+! Missing $ inserted.
+<inserted text>
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+l.194 ...a local uniform distribution of switches}
+
+I've inserted a begin-math/end-math symbol since I think
+you left one out. Proceed, with fingers crossed.
+
+)
+! Interruption.
+<to be read again>
+ \vskip
+l.124 \section{Stopping Time}
+ \label{sec:hypercube}
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