khkhj

diff --git a/hamilton.tex b/hamilton.tex
index 694bbcf82c301ab14b9a96a88f0552fdf2affb7f..f4dc64acc1ad5528c25446919d4b670dd891423f 100644 (file)
--- 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 6f139013412f5a308b95fe8583437a513839220f..af7621c94ce02019acd4fb6eb0bcd38334980c87 100644 (file)
--- 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}}
 \newlabel{graphe2}{{2b}{9}}
 \newlabel{sub@graphe2}{{b}{9}}
 \@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Iterating $f_0:(x_1,x_2) \DOTSB \mapstochar \rightarrow (\overline  {x_1}, \overline  {x_2})$\relax }}{9}}
 \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}}
-\@writefile{toc}{\contentsline {section}{\tocsection {}{7}{Experiments}}{17}}
-\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}}
-\@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces Functions with DSCC Matrix and smallest MT\relax }}{19}}
-\newlabel{table:nc}{{1}{19}}
-\@writefile{toc}{\contentsline {section}{\tocsection {}{8}{Conclusion}}{19}}
-\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 b2d76191b07b9db23757cf048f8cca48cfe96e4c..01af0c23f3f0c87d18c9c9ddf80f4668a5a28ca0 100644 (file)
--- a/main.blg
+++ b/main.blg
@@ -3,7 +3,6 @@ Capacity: max_strings=35307, hash_size=35307, hash_prime=30011
 The top-level auxiliary file: main.aux
 The style file: alpha.bst
 Database file #1: biblio.bib
-Warning--I didn't find a database entry for ""
 You've used 19 entries,
             2543 wiz_defined-function locations,
             695 strings with 8037 characters,
@@ -45,4 +44,3 @@ warning$ -- 0
 while$ -- 80
 width$ -- 21
 write$ -- 259
-(There was 1 warning)

diff --git a/main.log b/main.log
index 749adc158e82154a132d3a564b662c0175fa12a6..8af4a0aa65bce0cb0678cc1663d91e40997945b9 100644 (file)
--- a/main.log
+++ b/main.log
@@ -1,11 +1,11 @@
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 Document Class: ita 1999/03/01 v1.1 EDP-Sciences
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+
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+
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+
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+
+
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+
+)
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- [3]) (./chaos.tex [4 <./images/iter_f0c.pdf>]
+
+
+LaTeX Warning: Citation `DBLP:conf/secrypt/CouchotHGWB14' on page 3 undefined o
+n input line 66.
+
+) [3] (./chaos.tex
+
+LaTeX Warning: Citation `bcgr11:ip' on page 4 undefined on input line 2.
+
+
+LaTeX Warning: Citation `bcgr11:ip' on page 4 undefined on input line 21.
+
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+
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+LaTeX Warning: Citation `Banks92' on page 5 undefined on input line 79.
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@@ -1301,178 +1376,246 @@ Underfull \hbox (badness 10000) in paragraph at lines 472--477
 ; :::; a[]; :::; a[];$
  []
 
-[9 <./graphe1.pdf> <./graphe2.pdf>]) (./generating.tex [10]) (./hamilton.tex
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+ []
 
-### simple group (level 1) entered at line 10 ({)
-### 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.
+
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+                                                  
+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.
+
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+l.194 ...a local uniform distribution of switches}
+                                                  
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+l.194 ...a local uniform distribution of switches}
+                                                  
+I've inserted a begin-math/end-math symbol since I think
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+
+)
+! Interruption.
+<to be read again> 
+                   \vskip 
+l.124 \section{Stopping Time}
+                             \label{sec:hypercube}
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