\batchmode
\documentstyle[epsf]{article}
\makeatletter























\def\stack#1#2{{#1\atop #2}}

\def\SGMPlim#1{\def  \tempcs{#1}\ifx \tempcs\empty
		\let\SGMPdolim=\displaylimits
	\else \if #1c
		\let\SGMPdolim=\limits
	\else \if #1r
		\let\SGMPdolim=\nolimits
	\else
		\let\SGMPdolim=\relax
	\fi \fi \fi
}

\def\Rad#1{\begingroup
	\def  \RadTempCs{{#1}}\let\RdxTempCs=\empty
}

\def\Rdx#1{\def  \RdxTempCs{#1}}

\def\DoRad{\relax
	\ifx \RdxTempCs\empty
		\sqrt\RadTempCs
	\else
		\root \RdxTempCs \of \RadTempCs
	\fi
	\endgroup
}

\def\MthAcnt#1#2{#2{#1}}

\makeatother
\newenvironment{tex2html_wrap}{}{}
\newwrite\lthtmlwrite
\def\lthtmltypeout#1{{\let\protect\string\immediate\write\lthtmlwrite{#1}}}%
\newbox\sizebox
\begin{document}
\pagestyle{empty}
\stepcounter{section}
\stepcounter{section}
{\newpage
\clearpage
\samepage \(
n=b=2\)
}

{\newpage
\clearpage
\samepage \(
b^{n}\)
}

\stepcounter{subsection}
\stepcounter{subsection}
{\newpage
\clearpage
\samepage \(
L{\ifmmode<\else$<$\fi}N\)
}

{\newpage
\clearpage
\samepage \(
N=4\)
}

{\newpage
\clearpage
\samepage \begin{equation}\label{x:lattice size}\vcenter{\halign{\strut\hfil#\hfil&#\hfil\cr 
$\displaystyle{N_{i}={\left( {N\atop i}
\right) }}$\cr 
}}\end{equation}
}

{\newpage
\clearpage
\samepage \begin{figure}\epsfysize=3.5in
\hspace*{\fill}
\epsffile{fig1.ps}
\hspace*{\fill}


\end{figure}
}

{\newpage
\clearpage
\samepage \(
L=2\)
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\{$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2380: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2382: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \(
N=nb\)
}

{\newpage
\clearpage
\samepage \(
L=n\)
}

{\newpage
\clearpage
\samepage \(
n{\left( {b\atop 2}\right) }
\)
}

{\newpage
\clearpage
\samepage \(
b{\ifmmode\geq\else$\geq$\fi}2\)
}

{\newpage
\clearpage
\samepage \(
L{\ifmmode\leq\else$\leq$\fi}N/2\)
}

{\newpage
\clearpage
\samepage \(
n=2\)
}

{\newpage
\clearpage
\samepage \(
v_{1}\)
}

{\newpage
\clearpage
\samepage \(
v_{2}\)
}

{\newpage
\clearpage
\samepage \(
b=2\)
}

{\newpage
\clearpage
\samepage \(
v_{1}\not=v_{2}\)
}

{\newpage
\clearpage
\samepage \(
N=nb=4\)
}

{\newpage
\clearpage
\samepage \(
v_{1}=1,\hskip 0.212em v_{1}=2,\hskip 0.212em 
v_{2}=1,\hskip 0.212em \hskip 0.212em v_{2}
=2\)
}

{\newpage
\clearpage
\samepage \(
1,2,3,4\)
}

{\newpage
\clearpage
\samepage \(
{\left\lbrace v_{1}=1,\hskip 0.167em v_{2}=1
\right\rbrace }\)
}

{\newpage
\clearpage
\samepage \(
{\left\lbrace v_{1}=2,\hskip 0.167em v_{2}=2
\right\rbrace }\)
}

{\newpage
\clearpage
\samepage \(
{\left\lbrace {\rm 1},{\rm 3}\right\rbrace }\hskip 0.212em 
\hbox{\rm and}\hskip 0.212em {\left\lbrace {\rm 2},{\rm 4}
\right\rbrace }\)
}

{\newpage
\clearpage
\samepage \(
{\left\lbrace v_{1}=1,\hskip 0.167em v_{1}=2
\right\rbrace }\)
}

{\newpage
\clearpage
\samepage \(
{\left\lbrace v_{2}=1,\hskip 0.167em v_{2}=2
\right\rbrace }\)
}

{\newpage
\clearpage
\samepage \(
{\left\lbrace {\rm 1},{\rm 2}\right\rbrace }\hskip 0.212em 
\hbox{\rm and}\hskip 0.212em {\left\lbrace {\rm 3},{\rm 4}
\right\rbrace }\)
}

{\newpage
\clearpage
\samepage \(
{\left\lbrace {\rm 2},{\rm 3}\right\rbrace }\hskip 0.212em 
\hbox{\rm and}\hskip 0.212em {\left\lbrace {\rm 1},{\rm 4}
\right\rbrace }\)
}

{\newpage
\clearpage
\samepage \(
{\left\lbrace v_{1}=2,\hskip 0.167em v_{2}=1
\right\rbrace }\)
}

{\newpage
\clearpage
\samepage \(
{\left\lbrace v_{1}=1,\hskip 0.167em v_{2}=2
\right\rbrace }\)
}

\stepcounter{section}
\stepcounter{subsection}
{\newpage
\clearpage
\samepage \(
|\alpha \rangle \)
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\alpha$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2518: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \(
N=2^{n}\)
}

{\newpage
\clearpage
\samepage \(
s_{1}=0,\ldots ,s_{N}=2^{n}-1\)
}

{\newpage
\clearpage
\samepage \(
|s_{i}\rangle \)
}

{\newpage
\clearpage
\samepage \(
i-1\)
}

{\newpage
\clearpage
\samepage \(
|{\left\lbrace v_{1}=1,v_{2}=1\right\rbrace }
,\hskip 0.167em \hbox{\rm soln}=\hbox{\rm False}\rangle \)
}

{\newpage
\clearpage
\samepage \(
|s_{1}\rangle ,\ldots ,|s_{N}\rangle 
\)
}

{\newpage
\clearpage
\samepage \(
|s\rangle =\sum \psi _{i}|
s_{i}\rangle \)
}

{\newpage
\clearpage
\samepage \(
\psi _{i}\)
}

{\newpage
\clearpage
\samepage \(
|s\rangle \)
}

{\newpage
\clearpage
\samepage \(
{\left| \psi _{i}\right| }^{2}
\)
}

{\newpage
\clearpage
\samepage \begin{equation}\label{x:normalization}\vcenter{\halign{\strut\hfil#\hfil&#\hfil\cr 
$\displaystyle{\sum _{i}{\left| \psi _{i}\right| }^{2}=1}$\cr 
}}\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${}^{th}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2522: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \(
\psi \equiv {\left( \psi _{1},\ldots ,\psi _{N}\right) }\)
}

{\newpage
\clearpage
\samepage \(
\phi \cdot \psi =\sum _{i=1}^{N}
\phi ^{*}_{i}\psi _{i}\)
}

{\newpage
\clearpage
\samepage \(
\phi ^{*}_{i}\)
}

{\newpage
\clearpage
\samepage \(
\phi _{i}\)
}

{\newpage
\clearpage
\samepage \(
\phi ^{{\ifmmode\dagger\else$\dagger$\fi}}\psi \)
}

{\newpage
\clearpage
\samepage \(
\psi \)
}

{\newpage
\clearpage
\samepage \(
\phi ^{{\ifmmode\dagger\else$\dagger$\fi}}\)
}

{\newpage
\clearpage
\samepage \(
\phi \)
}

{\newpage
\clearpage
\samepage \(
\psi ^{{\ifmmode\dagger\else$\dagger$\fi}}\psi =1\)
}

{\newpage
\clearpage
\samepage \(
f{\left( s_{i}\right) }\)
}

{\newpage
\clearpage
\samepage \(
f{\left( |s\rangle \right) }=\sum 
\psi _{i}|f{\left( s_{i}\right) }
\rangle \)
}

{\newpage
\clearpage
\samepage \(
f{\left( s_{1}\right) }=f{\left( 
s_{2}\right) }=s_{3}\)
}

{\newpage
\clearpage
\samepage \(
|s\rangle ={{1}\over{\begingroup
\def  \RadTempCs{{2}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup }}
{\left( |s_{1}\rangle +|s_{2}\rangle 
\right) }\)
}

{\newpage
\clearpage
\samepage \(
f{\left( |s\rangle \right) }={{1}
\over{\begingroup
\def  \RadTempCs{{2}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup }}{\left( |f{\left( 
s_{1}\right) }\rangle +|f{\left( 
s_{2}\right) }\rangle \right) }
\)
}

{\newpage
\clearpage
\samepage \(
\begingroup
\def  \RadTempCs{{2}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup |s_{3}\rangle \)
}

{\newpage
\clearpage
\samepage \(
|s\rangle =\sum \psi _{k}|
s_{k}\rangle \)
}

{\newpage
\clearpage
\samepage \(
|s^{{\ifmmode\prime\else${}^\prime$\fi}}\rangle =U|s\rangle =\sum 
\psi ^{'}_{j}|s_{j}\rangle 
\)
}

{\newpage
\clearpage
\samepage \(
\psi ^{{\ifmmode\prime\else${}^\prime$\fi}}_{j}=\sum _{k}
U_{jk}\psi _{k}\)
}

{\newpage
\clearpage
\samepage \(
\psi ^{{\ifmmode\prime\else${}^\prime$\fi}}=U\psi \)
}

{\newpage
\clearpage
\samepage \(
{\left( \psi ^{{\ifmmode\prime\else${}^\prime$\fi}}\right) }^{{\ifmmode\dagger\else$\dagger$\fi}}\psi ^{{\ifmmode\prime\else${}^\prime$\fi}}=1\)
}

{\newpage
\clearpage
\samepage \(
U^{{\ifmmode\dagger\else$\dagger$\fi}}\)
}

{\newpage
\clearpage
\samepage \(
{\left( U^{{\ifmmode\dagger\else$\dagger$\fi}}\right) }_{j
k}={\left( U_{kj}\right) }^{*}\)
}

{\newpage
\clearpage
\samepage \begin{equation}\label{x:U condition}\vcenter{\halign{\strut\hfil#\hfil&#\hfil\cr 
$\displaystyle{1={\left( U\psi \right) }^{{\ifmmode\dagger\else$\dagger$\fi}
}{\left( U\psi \right) }=\psi ^{{\ifmmode\dagger\else$\dagger$\fi}}{\left( U^{{\ifmmode\dagger\else$\dagger$\fi}}U\right) }
\psi }$\cr 
}}\end{equation}
}

{\newpage
\clearpage
\samepage \(
A\equiv U^{{\ifmmode\dagger\else$\dagger$\fi}}U\)
}

{\newpage
\clearpage
\samepage \(
\psi =\hat {{\bf e}}_{j}={\left( 
\ldots ,0,1,0,\ldots \right) }\)
}

{\newpage
\clearpage
\samepage \(
|s_{j}\rangle \)
}

{\newpage
\clearpage
\samepage \(
\psi _{j}=1\)
}

{\newpage
\clearpage
\samepage \(
\psi ^{{\ifmmode\dagger\else$\dagger$\fi}}A\psi =A_{jj}\)
}

{\newpage
\clearpage
\samepage \(
U^{{\ifmmode\dagger\else$\dagger$\fi}}U\)
}

{\newpage
\clearpage
\samepage \(
\psi ={{1}\over{\begingroup
\def  \RadTempCs{{2}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup }}{\left( 
\hat {{\bf e}}_{j}+\hat {{\bf e
}}_{k}\right) }\)
}

{\newpage
\clearpage
\samepage \(
j\not=k\)
}

{\newpage
\clearpage
\samepage \begin{equation}\vcenter{\halign{\strut\hfil#\hfil&#\hfil\cr 
$\displaystyle{\psi ^{{\ifmmode\dagger\else$\dagger$\fi}}A\psi }$\hfilneg&$\displaystyle{{}={{1}\over{2}}{\left( {\bf \hat {e}
}_{j}+{\bf \hat {e}}_{k}
\right) }A{\left( {\bf \hat {e}
}_{j}+{\bf \hat {e}}_{k}
\right) }}$\cr 
$\displaystyle{}$\hfilneg&$\displaystyle{{}={{1}\over{2}}{\left[ 
A_{jj}+A_{kk}+A_{jk}+A_{kj}\right] }
}$\cr 
}}\end{equation}
}

{\newpage
\clearpage
\samepage \(
A_{jk}=-A_{kj}\)
}

{\newpage
\clearpage
\samepage \(
\psi ={{1}\over{\begingroup
\def  \RadTempCs{{2}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup }}{\left( 
\hat {{\bf e}}_{j}+i\hat {{\bf 
e}}_{k}\right) }\)
}

{\newpage
\clearpage
\samepage \(
A_{jk}=A_{kj}\)
}

{\newpage
\clearpage
\samepage \(
U^{{\ifmmode\dagger\else$\dagger$\fi}}U=I\)
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \(
|0\rangle \)
}

{\newpage
\clearpage
\samepage \(
|1\rangle \)
}

{\newpage
\clearpage
\samepage \(
\hbox{\rm NOT}{\left( |0\rangle \right) }
=|1\rangle \)
}

{\newpage
\clearpage
\samepage \(
\hbox{\rm NOT}{\left( |1\rangle \right) }
=|0\rangle \)
}

{\newpage
\clearpage
\samepage \begin{equation}\vcenter{\halign{\strut\hfil#\hfil&#\hfil\cr 
$\displaystyle{\hbox{\rm NOT}{\left( {\psi _{0}
\atop \psi _{1}}\right) }
\equiv \hbox{\rm NOT}{\left( \psi _{0}|
0\rangle +\psi _{1}|1\rangle \right) }
=\psi _{0}|1\rangle +\psi _{1}|
0\rangle \equiv {\left( {\psi _{1}\atop \psi _{0}}\right) }}$\cr 
}}\end{equation}
}

{\newpage
\clearpage
\samepage \(
{\left( 
\matrix{0&1\cr 
1&0\cr 
}\right) }\)
}

{\newpage
\clearpage
\samepage \begin{equation}\vcenter{\halign{\strut\hfil#\hfil&#\hfil\cr 
$\displaystyle{U{\left( \theta \right) }={\left( 
\matrix{\cos \theta &-\sin \theta 
\cr 
\sin \theta &\cos \theta 
\cr 
}\right) }}$\cr 
}}\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\label{x:create}\vcenter{\halign{\strut\hfil#\hfil&#\hfil\cr 
$\displaystyle{U{\left( {{\pi }\over{4}}\right) }
{\left( {1\atop 0}\right) }
\equiv U{\left( {{\pi }\over{4}}\right) }
|0\rangle ={{1}\over{\begingroup
\def  \RadTempCs{{2}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup }}
{\left( |0\rangle +|1\rangle \right) }
\equiv {{1}\over{\begingroup
\def  \RadTempCs{{2}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup }}{\left( 
{1\atop 1}\right) }}$\cr 
}}\end{equation}
}

{\newpage
\clearpage
\samepage \(
R_{0}\)
}

{\newpage
\clearpage
\samepage \(
3/4\)
}

{\newpage
\clearpage
\samepage \(
R_{1}\)
}

{\newpage
\clearpage
\samepage \(
1/4\)
}

{\newpage
\clearpage
\samepage \(
{{3}\over{4}}p+{{1}\over{4}}{\left( 
1-p\right) }\)
}

{\newpage
\clearpage
\samepage \begin{equation}\vcenter{\halign{\strut\hfil#\hfil&#\hfil\cr 
$\displaystyle{P_{classical}={{1}\over{4}}+{{p}\over{2}}}$\cr 
}}\end{equation}
}

{\newpage
\clearpage
\samepage \(
p=0\)
}

{\newpage
\clearpage
\samepage \(
\theta ={{\pi }\over{3}}\)
}

{\newpage
\clearpage
\samepage \begin{equation}\vcenter{\halign{\strut\hfil#\hfil&#\hfil\cr 
$\displaystyle{U{\left( {{\pi }\over{3}}\right) }
{\left( {1\atop 0}\right) }
={{1}\over{2}}{\left( {\begingroup
\def  \RadTempCs{{
3}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup \atop 1}\right) }}$\cr 
$\displaystyle{U{\left( {{\pi }\over{3}}\right) }
{\left( {0\atop 1}\right) }
={{1}\over{2}}{\left( {-1\atop \begingroup
\def  \RadTempCs{{3}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup }\right) }
}$\cr 
}}\end{equation}
}

{\newpage
\clearpage
\samepage \(
{\left( {\cos \phi \atop \sin 
\phi }\right) }\)
}

{\newpage
\clearpage
\samepage \(
p=\def  \tempcs{r}\ifx \tempcs\empty
\let\SGMPdolim=\displaylimits
\else \if rc
\let\SGMPdolim=\limits
\else \if rr
\let\SGMPdolim=\nolimits
\else
\let\SGMPdolim=\relax
\fi \fi \fi\cos \SGMPdolim ^{2}\phi 
\)
}

{\newpage
\clearpage
\samepage \(
U{\left( {{\pi }\over{3}}\right) }
{\left( {\cos \phi \atop \sin 
\phi }\right) }\)
}

{\newpage
\clearpage
\samepage \begin{equation}\vcenter{\halign{\strut\hfil#\hfil&#\hfil\cr 
$\displaystyle{P_{quantum}}$\hfilneg&$\displaystyle{{}={{1}\over{4}}
+{{\def  \tempcs{r}\ifx \tempcs\empty
\let\SGMPdolim=\displaylimits
\else \if rc
\let\SGMPdolim=\limits
\else \if rr
\let\SGMPdolim=\nolimits
\else
\let\SGMPdolim=\relax
\fi \fi \fi\cos \SGMPdolim ^{2}
\phi }\over{2}}-{{\begingroup
\def  \RadTempCs{{3}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup }
\over{4}}\sin {\left( 2\phi \right) }
}$\cr 
$\displaystyle{}$\hfilneg&$\displaystyle{{}=P_{classical}-{{\begingroup
\def  \RadTempCs{{
3}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup }\over{4}}\sin {\left( 2\phi 
\right) }}$\cr 
}}\end{equation}
}

{\newpage
\clearpage
\samepage \(
\phi ={{\pi }\over{3}}\)
}

{\newpage
\clearpage
\samepage \(
\hbox{\rm SET}{\left( |0\rangle \right) }
=|1\rangle \)
}

{\newpage
\clearpage
\samepage \(
\hbox{\rm SET}{\left( |1\rangle \right) }
=|1\rangle \)
}

{\newpage
\clearpage
\samepage \(
\hbox{\rm SET}{\left( {{1}\over{\begingroup
\def  \RadTempCs{{2}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup 
}}{\left( |0\rangle +|1\rangle \right) }
\right) }={{1}\over{\begingroup
\def  \RadTempCs{{2}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup 
}}{\left( \hbox{\rm SET}{\left( |0\rangle 
\right) }+\hbox{\rm SET}{\left( |1\rangle 
\right) }\right) }\)
}

{\newpage
\clearpage
\samepage \(
{{1}\over{\begingroup
\def  \RadTempCs{{2}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup }}2|1\rangle 
=\begingroup
\def  \RadTempCs{{2}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup |1\rangle \)
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \(
2^{n}\)
}

{\newpage
\clearpage
\samepage \(
f{\left( s\right) }\)
}

{\newpage
\clearpage
\samepage \(
s_{1},\ldots ,s_{N}\)
}

{\newpage
\clearpage
\samepage \(
{{1}\over{\begingroup
\def  \RadTempCs{{N}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup }}{\left( 
|s_{1}\rangle +\ldots +|s_{N}\rangle 
\right) }\)
}

{\newpage
\clearpage
\samepage \(
{{1}\over{\begingroup
\def  \RadTempCs{{N}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup }}\sum 
|s_{i},\hbox{\rm soln}=f{\left( s_{i}\right) }\rangle \)
}

{\newpage
\clearpage
\samepage \(
|s_{i},\hbox{\rm soln}=f{\left( s_{i}
\right) }\rangle \)
}

{\newpage
\clearpage
\samepage \(
s_{i}\)
}

{\newpage
\clearpage
\samepage \(
1/N\)
}

{\newpage
\clearpage
\samepage \(
O{\left( N\right) }\)
}

{\newpage
\clearpage
\samepage \(
i+1\)
}

{\newpage
\clearpage
\samepage \(
n-1\)
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \(
N=6\)
}

{\newpage
\clearpage
\samepage \(
|{\left\lbrace 1,2\right\rbrace }\rangle \)
}

{\newpage
\clearpage
\samepage \(
|{\left\lbrace 1,2,3\right\rbrace }\rangle +\ldots 
+|{\left\lbrace 1,2,6\right\rbrace }\rangle \)
}

{\newpage
\clearpage
\samepage \(
j!\)
}

{\newpage
\clearpage
\samepage \(
L!\)
}

{\newpage
\clearpage
\samepage \(
k=2\)
}

{\newpage
\clearpage
\samepage \(
r_{j}\)
}

{\newpage
\clearpage
\samepage \(
r_{k}=k!\)
}

{\newpage
\clearpage
\samepage \(
j{\ifmmode>\else$>$\fi}k\)
}

{\newpage
\clearpage
\samepage \begin{equation}\vcenter{\halign{\strut\hfil#\hfil&#\hfil\cr 
$\displaystyle{\psi {\left( s\right) }=\sum 
_{m=1}^{j}\psi {\left( s_{m}
\right) }e^{i\phi _{m}}}$\cr 
}}\end{equation}
}

{\newpage
\clearpage
\samepage \(
s_{m}\)
}

{\newpage
\clearpage
\samepage \(
j-1\)
}

{\newpage
\clearpage
\samepage \(
\phi _{m}\)
}

{\newpage
\clearpage
\samepage \(
\psi {\left( s_{m}\right) }\)
}

{\newpage
\clearpage
\samepage \(
r_{j-1}\)
}

{\newpage
\clearpage
\samepage \(
\theta _{m}\)
}

{\newpage
\clearpage
\samepage \begin{equation}\vcenter{\halign{\strut\hfil#\hfil&#\hfil\cr 
$\displaystyle{r_{j}=r_{j-1}{\left\langle \sum 
_{m=1}^{j}e^{i\theta _{m}}
\right\rangle }=r_{j-1}\begingroup
\def  \RadTempCs{{j}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup }$\cr 
}}\end{equation}
}

{\newpage
\clearpage
\samepage \(
\begingroup
\def  \RadTempCs{{j}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup \)
}

{\newpage
\clearpage
\samepage \(
r_{j}=r_{k}\begingroup
\def  \RadTempCs{{j!/k!}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup \)
}

{\newpage
\clearpage
\samepage \(
r_{j}=\begingroup
\def  \RadTempCs{{j!k!}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup \)
}

{\newpage
\clearpage
\samepage \(
L!^{2}\)
}

{\newpage
\clearpage
\samepage \(
{\left( N_{L}-1\right) }r^{2}_{L}\sim N_{L}L!k!\)
}

{\newpage
\clearpage
\samepage \(
N_{L}\)
}

{\newpage
\clearpage
\samepage \(
L=N/b\)
}

{\newpage
\clearpage
\samepage \begin{equation}\vcenter{\halign{\strut\hfil#\hfil&#\hfil\cr 
$\displaystyle{\ln {\left( {{P_{soln}}\over{P_{nogood}}}\right) }=\ln 
{\left( {{L!}\over{N_{L}k!}}\right) }
\sim {{N}\over{b}}\ln N+O{\left( 
N\right) }}$\cr 
}}\end{equation}
}

{\newpage
\clearpage
\samepage \(
N=3\)
}

{\newpage
\clearpage
\samepage \(
|{\left\lbrace 1\right\rbrace }\rangle ,|{\left\lbrace 
2\right\rbrace }\rangle ,|{\left\lbrace 3\right\rbrace }
\rangle \)
}

{\newpage
\clearpage
\samepage \(
|{\left\lbrace 1,2\right\rbrace }\rangle ,|{\left\lbrace 
1,3\right\rbrace }\rangle ,|{\left\lbrace 2,3\right\rbrace }
\rangle \)
}

{\newpage
\clearpage
\samepage \begin{equation}\label{x:M[1,3]}\vcenter{\halign{\strut\hfil#\hfil&#\hfil\cr 
$\displaystyle{M={\left( 
\matrix{1&1&0\cr 
1&0&1\cr 
0&1&1\cr 
}\right) }}$\cr 
}}\end{equation}
}

{\newpage
\clearpage
\samepage \(
|{\left\lbrace 1\right\rbrace }\rangle \)
}

{\newpage
\clearpage
\samepage \(
|{\left\lbrace 1,3\right\rbrace }\rangle \)
}

{\newpage
\clearpage
\samepage \(
|{\left\lbrace 2,3\right\rbrace }\rangle \)
}

{\newpage
\clearpage
\samepage \(
\begingroup
\def  \RadTempCs{{2}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup \)
}

{\newpage
\clearpage
\samepage \(
N-i\)
}

{\newpage
\clearpage
\samepage \(
i{\ifmmode<\else$<$\fi}N/2\)
}

\stepcounter{subsection}
\stepcounter{subsubsection}
{\newpage
\clearpage
\samepage \(
k{\ifmmode\leq\else$\leq$\fi}m\)
}

{\newpage
\clearpage
\samepage \(
M=A^{{\ifmmode\dagger\else$\dagger$\fi}}\Lambda B\)
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\Lambda$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2838: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \(
A^{{\ifmmode\dagger\else$\dagger$\fi}}\)
}

{\newpage
\clearpage
\samepage \(
U=A^{{\ifmmode\dagger\else$\dagger$\fi}}B\)
}

{\newpage
\clearpage
\samepage \(
{\left| U-M\right| }^{2}\equiv \sum _{rs}{\left| U_{rs}-M_{rs}\right| }^{2}\)
}

{\newpage
\clearpage
\samepage \(
k{\ifmmode>\else$>$\fi}m\)
}

{\newpage
\clearpage
\samepage \(
N_{i}{\ifmmode\leq\else$\leq$\fi}N_{i+1}\)
}

{\newpage
\clearpage
\samepage \(
L{\ifmmode\leq\else$\leq$\fi}{\left\lceil N/2\right\rceil }\)
}

{\newpage
\clearpage
\samepage \begin{equation}\vcenter{\halign{\strut\hfil#\hfil&#\hfil\cr 
$\displaystyle{A^{{\ifmmode\dagger\else$\dagger$\fi}}\Lambda B={\left( 
\matrix{{{1}\over{\begingroup
\def  \RadTempCs{{3}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup }}
&-{{1}\over{\begingroup
\def  \RadTempCs{{2}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup }}
&{{1}\over{\begingroup
\def  \RadTempCs{{6}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup }}
\cr 
{{1}\over{\begingroup
\def  \RadTempCs{{3}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup }}
&{{1}\over{\begingroup
\def  \RadTempCs{{2}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup }}
&{{1}\over{\begingroup
\def  \RadTempCs{{6}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup }}
\cr 
{{1}\over{\begingroup
\def  \RadTempCs{{3}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup }}
&0&-\begingroup
\def  \RadTempCs{{{{2}\over{3}}
}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup \cr 
}\right) }{\left( 
\matrix{2&0&0\cr 
0&1&0\cr 
0&0&1\cr 
}\right) }{\left( 
\matrix{{{1}\over{\begingroup
\def  \RadTempCs{{3}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup }}
&{{1}\over{\begingroup
\def  \RadTempCs{{3}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup }}
&{{1}\over{\begingroup
\def  \RadTempCs{{3}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup }}
\cr 
0&-{{1}\over{\begingroup
\def  \RadTempCs{{2}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup 
}}&{{1}\over{\begingroup
\def  \RadTempCs{{2}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup 
}}\cr 
\begingroup
\def  \RadTempCs{{{{2}\over{3}}}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup 
&-{{1}\over{\begingroup
\def  \RadTempCs{{6}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup }}
&-{{1}\over{\begingroup
\def  \RadTempCs{{6}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup }}
\cr 
}\right) }}$\cr 
}}\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\label{x:U[1,3]}\vcenter{\halign{\strut\hfil#\hfil&#\hfil\cr 
$\displaystyle{U=A^{{\ifmmode\dagger\else$\dagger$\fi}}B={{1}\over{3}}{\left( 
\matrix{2&2&-1\cr 
2&-1&2\cr 
-1&2&2\cr 
}\right) }}$\cr 
}}\end{equation}
}

{\newpage
\clearpage
\samepage \(
U_{r\beta }=a_{{\left| r\cap \beta \right| }
}\)
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\beta$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2856: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \(
{\left| r\cap \beta \right| }\)
}

{\newpage
\clearpage
\samepage \(
\beta \subset r\)
}

{\newpage
\clearpage
\samepage \begin{equation}\label{x:coef1}\vcenter{\halign{\strut\hfil#\hfil&#\hfil\cr 
$\displaystyle{1={\left( U^{{\ifmmode\dagger\else$\dagger$\fi}}U\right) }_{\alpha \alpha }}$\hfilneg&$\displaystyle{{}=\sum _{k=0}
^{i}n_{k}a^{2}_{k}}$\cr 
}}\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\label{x:n(k)}\vcenter{\halign{\strut\hfil#\hfil&#\hfil\cr 
$\displaystyle{n_{k}={\left( {i\atop k}
\right) }{\left( {N-i\atop i+
1-k}\right) }}$\cr 
}}\end{equation}
}

{\newpage
\clearpage
\samepage \(
{\left| \alpha \cap \beta \right| }=p{\ifmmode<\else$<$\fi}i
\)
}

{\newpage
\clearpage
\samepage \begin{equation}\label{x:coef2}\vcenter{\halign{\strut\hfil#\hfil&#\hfil\cr 
$\displaystyle{0={\left( U^{{\ifmmode\dagger\else$\dagger$\fi}}U\right) }_{\alpha \beta }}$\hfilneg&$\displaystyle{{}=\sum _{j,k=0
}^{i}n^{{\left( p\right) }
}_{jk}a_{j}a_{k}}$\cr 
}}\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\vcenter{\halign{\strut\hfil#\hfil&#\hfil\cr 
$\displaystyle{n^{{\left( p\right) }}_{jk}
=\sum _{x}{\left( {i-p\atop k-x}\right) }{\left( {p\atop x}\right) }{\left( {i-p\atop j-x}\right) }{\left( 
{N-2i+p\atop i+1-j-k+x}\right) }
}$\cr 
}}\end{equation}
}

{\newpage
\clearpage
\samepage \(
{\left| r\cap \alpha \right| }=k{\ifmmode\leq\else$\leq$\fi}i\)
}

{\newpage
\clearpage
\samepage \(
{\left| r\cap \beta \right| }=j{\ifmmode\leq\else$\leq$\fi}i\)
}

{\newpage
\clearpage
\samepage \(
a_{0},\ldots ,a_{i}\)
}

{\newpage
\clearpage
\samepage \(
a^{{\left( i\right) }}_{k}\)
}

{\newpage
\clearpage
\samepage \(
a_{k}\)
}

{\newpage
\clearpage
\samepage \(
i=1\)
}

{\newpage
\clearpage
\samepage \(
1=a^{2}_{0}+2a^{2}_{1}\)
}

{\newpage
\clearpage
\samepage \(
0=2a_{0}a_{1}+a^{2}_{1}\)
}

{\newpage
\clearpage
\samepage \(
a_{0}=-{{1}\over{3}},\hskip 0.212em 
a_{1}={{2}\over{3}}\)
}

{\newpage
\clearpage
\samepage \(
a^{{\left( i\right) }}_{i}=
{{1}\over{\begingroup
\def  \RadTempCs{{n_{i}}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup }}
={{1}\over{\begingroup
\def  \RadTempCs{{N-i}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup }}\)
}

{\newpage
\clearpage
\samepage \(
b^{{\left( i\right) }}_{k}\equiv 
{\left( -1\right) }^{k}a^{{\left( 
i\right) }}_{i-k}\begingroup
\def  \RadTempCs{{n_{i-k
}}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup \)
}

{\newpage
\clearpage
\samepage \(
n_{i-k}\)
}

{\newpage
\clearpage
\samepage \(
N=10\)
}

{\newpage
\clearpage
\samepage \(
b^{{\left( i\right) }}_{0}\)
}

{\newpage
\clearpage
\samepage \begin{figure}\epsfysize=2in
\hspace*{\fill}
\epsffile{fig2.eps}
\hspace*{\fill}


\end{figure}
}

\stepcounter{subsubsection}
{\newpage
\clearpage
\samepage \(
a^{{\left( i\right) }}_{i-1}
\)
}

\stepcounter{subsubsection}
{\newpage
\clearpage
\samepage \(
K=2\)
}

{\newpage
\clearpage
\samepage \(
N^{2}\)
}

{\newpage
\clearpage
\samepage \(
L-1\)
}

{\newpage
\clearpage
\samepage \(
\psi ^{{\left( j\right) }}_{\alpha 
}\)
}

{\newpage
\clearpage
\samepage \begin{equation}\label{x:map}\vcenter{\halign{\strut\hfil#\hfil&#\hfil\cr 
$\displaystyle{\psi ^{{\left( j+1\right) }}_{r}=\sum _{\alpha }U_{r\alpha 
}\rho _{\alpha }\psi ^{{\left( j
\right) }}_{\alpha }=\sum 
_{k}a^{{\left( j\right) }}_{k}\sum _{{\left| r\cap \alpha \right| }
=k}\rho _{\alpha }\psi ^{{\left( 
j\right) }}_{\alpha }}$\cr 
}}\end{equation}
}

{\newpage
\clearpage
\samepage \(
\rho _{\alpha }\)
}

{\newpage
\clearpage
\samepage \(
\rho _{\alpha }=1\)
}

{\newpage
\clearpage
\samepage \(
\rho _{\alpha }=-1\)
}

{\newpage
\clearpage
\samepage \(
\rho _{\alpha }=e^{i\theta }\)
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\theta$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2926: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \(
[0,2\pi )\)
}

{\newpage
\clearpage
\samepage \begin{equation}\label{x:p(soln)}\vcenter{\halign{\strut\hfil#\hfil&#\hfil\cr 
$\displaystyle{p_{soln}=\sum _{s}{\left| 
\psi ^{{\left( L\right) }}_{s
}\right| }^{2}}$\cr 
}}\end{equation}
}

{\newpage
\clearpage
\samepage \(
2^{N}\)
}

{\newpage
\clearpage
\samepage \(
p_{soln}\)
}

{\newpage
\clearpage
\samepage \(
1/p_{soln}\)
}

{\newpage
\clearpage
\samepage \(
L-K{\ifmmode<\else$<$\fi}N/2\)
}

{\newpage
\clearpage
\samepage \(
O{\left( N^{K}\right) }\)
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \(
K=0\)
}

{\newpage
\clearpage
\samepage \(
|\emptyset \rangle \)
}

{\newpage
\clearpage
\samepage \(
{{1}\over{\begingroup
\def  \RadTempCs{{3}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup }}{\left( 
|{\left\lbrace 1\right\rbrace }\rangle +|{\left\lbrace 
2\right\rbrace }\rangle +|{\left\lbrace 3\right\rbrace }
\rangle \right) }\)
}

{\newpage
\clearpage
\samepage \(
{{1}\over{\begingroup
\def  \RadTempCs{{3}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup }}{\left( 
|{\left\lbrace 1\right\rbrace }\rangle +|{\left\lbrace 
2\right\rbrace }\rangle +e^{i\theta }|{\left\lbrace 
3\right\rbrace }\rangle \right) }\)
}

{\newpage
\clearpage
\samepage \begin{equation}\vcenter{\halign{\strut\hfil#\hfil&#\hfil\cr 
$\displaystyle{{{1}\over{3\begingroup
\def  \RadTempCs{{3}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup }}{\left( 
{\left( 4-e^{i\theta }\right) }|
{\left\lbrace 1,2\right\rbrace }\rangle +{\left( 1
+2e^{i\theta }\right) }|{\left\lbrace 
1,3\right\rbrace }\rangle +{\left( 1+2e^{i\theta 
}\right) }|{\left\lbrace 2,3\right\rbrace }
\rangle \right) }}$\cr 
}}\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\vcenter{\halign{\strut\hfil#\hfil&#\hfil\cr 
$\displaystyle{p_{soln}={\left| {{1}\over{3\begingroup
\def  \RadTempCs{{
3}}\let\RdxTempCs=\empty\relax
\ifx \RdxTempCs\empty
\sqrt\RadTempCs
\else
\root \RdxTempCs \of \RadTempCs
\fi
\endgroup }}{\left( 4-e^{i\theta }\right) }
\right| }^{2}={{1}\over{27}}
{\left( 17-8\cos \theta \right) }}$\cr 
}}\end{equation}
}

{\newpage
\clearpage
\samepage \(
p_{soln}={{17}\over{27}}=0.63\)
}

{\newpage
\clearpage
\samepage \(
\cos \theta \)
}

{\newpage
\clearpage
\samepage \(
\theta =\pi \)
}

{\newpage
\clearpage
\samepage \(
p_{soln}={{25}\over{27}}=0.93\)
}

\stepcounter{section}
{\newpage
\clearpage
\samepage \(
10^{-11}\)
}

{\newpage
\clearpage
\samepage \(
N=14\)
}

{\newpage
\clearpage
\samepage \(
L=N/2\)
}

\stepcounter{subsection}
\stepcounter{subsubsection}
{\newpage
\clearpage
\samepage \(
L+1\)
}

{\newpage
\clearpage
\samepage \begin{figure}\epsfysize=2in
\hspace*{\fill}
\epsffile{fig3.eps}
\hspace*{\fill}


\end{figure}
}

{\newpage
\clearpage
\samepage \begin{equation}\vcenter{\halign{\strut\hfil#\hfil&#\hfil\cr 
$\displaystyle{\rho _{L}={{{\left( {{\left( 
{N\atop 2}\right) }-{\left( 
{L\atop 2}\right) }\atop m}\right) }}\over{{\left( 
{{\left( {N\atop 2}\right) }
\atop m}\right) }}}}$\cr 
}}\end{equation}
}

{\newpage
\clearpage
\samepage \(
{\left( {N\atop 2}\right) }
\)
}

{\newpage
\clearpage
\samepage \(
N_{soln}=N_{L}\rho _{L}\)
}

{\newpage
\clearpage
\samepage \(
m=\beta N\)
}

{\newpage
\clearpage
\samepage \begin{equation}\vcenter{\halign{\strut\hfil#\hfil&#\hfil\cr 
$\displaystyle{\ln N_{soln}\sim N{\left( h{\left( 
{{1}\over{b}}\right) }+\beta \ln 
{\left( 1-{{1}\over{b^{2}}}\right) }
\right) }}$\cr 
}}\end{equation}
}

{\newpage
\clearpage
\samepage \(
h{\left( x\right) }\equiv -x\ln x-{\left( 
1-x\right) }\ln {\left( 1-x\right) }
\)
}

{\newpage
\clearpage
\samepage \begin{equation}\vcenter{\halign{\strut\hfil#\hfil&#\hfil\cr 
$\displaystyle{\beta _{crit}={{h{\left( 1/b\right) }
}\over{-\ln {\left( 1-1/b^{2}\right) }
}}}$\cr 
}}\end{equation}
}

{\newpage
\clearpage
\samepage \(
\beta _{crit}=2.41\)
}

{\newpage
\clearpage
\samepage \begin{equation}\vcenter{\halign{\strut\hfil#\hfil&#\hfil\cr 
$\displaystyle{\beta _{poly}={{b^{2}-1}\over{2
b}}\ln(b-1)}$\cr 
}}\end{equation}
}

{\newpage
\clearpage
\samepage \(
\beta _{poly}=0\)
}

{\newpage
\clearpage
\samepage \(
\beta {\ifmmode>\else$>$\fi}0\)
}

\stepcounter{subsubsection}
{\newpage
\clearpage
\samepage \(
T=1/p_{soln}\)
}

{\newpage
\clearpage
\samepage \begin{figure}\epsfysize=2in
\hspace*{\fill}
\epsffile{fig4.eps}
\hspace*{\fill}


\end{figure}
}

{\newpage
\clearpage
\samepage \begin{figure}\epsfysize=2in
\hspace*{\fill}
\epsffile{fig5.eps}
\hspace*{\fill}


\end{figure}
}

{\newpage
\clearpage
\samepage \begin{figure}\epsfysize=2in
\hspace*{\fill}
\epsffile{fig6.eps}
\hspace*{\fill}


\end{figure}
}

\stepcounter{subsubsection}
{\newpage
\clearpage
\samepage \begin{figure}\epsfysize=1.64in
\hspace*{\fill}
\epsffile{fig7.eps}
\hspace*{\fill}


\end{figure}
}

{\newpage
\clearpage
\samepage \begin{figure}\epsfysize=1.64in
\hspace*{\fill}
\epsffile{fig8.eps}
\hspace*{\fill}


\end{figure}
}

{\newpage
\clearpage
\samepage \(
\beta =2\)
}

{\newpage
\clearpage
\samepage \begin{figure}\epsfysize=2in
\hspace*{\fill}
\epsffile{fig9.eps}
\hspace*{\fill}


\end{figure}
}

{\newpage
\clearpage
\samepage \begin{figure}\epsfysize=2in
\hspace*{\fill}
\epsffile{fig10.eps}
\hspace*{\fill}


\end{figure}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \(
N=2n\)
}

{\newpage
\clearpage
\samepage \(
n=5\)
}

{\newpage
\clearpage
\samepage \(
c=4.2n\)
}

{\newpage
\clearpage
\samepage \(
c/n\)
}

{\newpage
\clearpage
\samepage \(
n=3\)
}

\stepcounter{subsubsection}
{\newpage
\clearpage
\samepage \begin{figure}\epsfysize=2in
\hspace*{\fill}
\epsffile{fig11.eps}
\hspace*{\fill}


\end{figure}
}

\stepcounter{subsubsection}
{\newpage
\clearpage
\samepage \(
c/n=2\)
}

{\newpage
\clearpage
\samepage \begin{figure}\epsfysize=1.64in
\hspace*{\fill}
\epsffile{fig12.eps}
\hspace*{\fill}


\end{figure}
}

{\newpage
\clearpage
\samepage \begin{figure}\epsfysize=2in
\hspace*{\fill}
\epsffile{fig13.eps}
\hspace*{\fill}


\end{figure}
}

{\newpage
\clearpage
\samepage \(
c=0\)
}

{\newpage
\clearpage
\samepage \(
\sum _{s}{\left| \psi ^{{\left( 
i\right) }}_{s}\right| }^{2}\)
}

{\newpage
\clearpage
\samepage \begin{figure}\epsfysize=2in
\hspace*{\fill}
\epsffile{fig14.eps}
\hspace*{\fill}


\end{figure}
}

{\newpage
\clearpage
\samepage \begin{figure}\epsfysize=2in
\hspace*{\fill}
\epsffile{fig15.eps}
\hspace*{\fill}


\end{figure}
}

\stepcounter{section}
\stepcounter{section}

\end{document}
