\batchmode

\documentstyle[a4,jair,itmath,theapa,11pt]{article}

\let\providecommand\newcommand
\RequirePackage{ifthen}


              %% \set{1, 2, 3} produces {1, 2, 3}
              %% \seq{S} produces S*
  %% \tuple{1, 2, 3} produces <1, 2, 3>
            %% produces ->
               %% function overriding (a symbol from VDM)
                 %% produces upside down A
                 %% produces upside down E
                  %% produces ~
                %% produces ^
             %% produces big ^
                   %% produces v
                %% produces big v
       %% produces =>
\let\implies=\Rightarrow
       %% produces <=>
\let\biimplies=\Leftrightarrow

                  %% powerset
                %% set union
                %% set intersection
             %% big set union
             %% big set intersection

                  %% natural numbers
                 %% reals
\newtheorem{defn}{Definition}\newtheorem{exam}{Example}\newtheorem{thm}{Theorem}\newtheorem{lem}{Lemma}\newtheorem{cor}{Corollary}\newtheorem{fct}{Fact}\newtheorem{fact}{Fact}\newtheorem{question}{Question}\newtheorem{problem}{Problem}\newtheorem{claim}{Claim}%
\newenvironment{proof}{

\vspace*{0.5ex}\noindent\rm {\em Proof.} }%%
{{\hfill\tiny$\Box$}

\vspace*{1ex}}\jairheading{22}{2004}{XX--YY}{07/04}{12/04}
\ShortHeadings{Extremal Behaviour in Multiagent Contract Negotiation}{Dunne}






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\lthtmlinlinemathA{tex2html_wrap_inline4207}%
$-\pi_i$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4209}%
$\pi_i>0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4217}%
$\sigma_u(Q)>\sigma_u(P)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4219}%
$\Phi$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3673}%
\begin{displaymath}
\langle P^{(1)},\ P^{(2)}\ ,\ldots,\ P^{(t-1)},\ P^{(t)}\rangle 
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4231}%
$1\leq i < t$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4233}%
$\langle P^{(i)},P^{(i+1)}\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4237}%
$\Phi(P^{(i)},P^{(i+1)})$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4247}%
$\Psi$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4253}%
$\Psi(\delta)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4259}%
$\langle P^{(1)},\ldots,P^{(t)}\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4263}%
$\Phi(P^{(t)},Q)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4267}%
$\Phi(P,Q)=\top$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4279}%
$\ \ \ $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4287}%
$n=2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4305}%
$\Psi(P,Q)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4309}%
$\langle{\mathcal A}_n,{\mathcal R}_m,{\mathcal U}\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4323}%
$g(m)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4339}%
$L^{{\rm opt}}(\delta,\langle{\mathcal A},{\mathcal R},{\mathcal U}\rangle ,\Phi)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4345}%
$L^{\max}(\langle{\mathcal A},{\mathcal R},{\mathcal U}\rangle ,\Phi,\Psi)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3674}%
\begin{displaymath}
L^{\max}(\langle{\mathcal A},{\mathcal R},{\mathcal U}\rangle ,\Phi,\Psi)\ =\ 
\max_{\mbox{$\Psi$-deals $\delta$}}\ L^{{\rm opt}}(\delta,\langle{\mathcal A},{\mathcal R},{\mathcal U}\rangle ,\Phi)
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4351}%
$\rho^{\max}(n,m,\Phi,\Psi)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3675}%
\begin{displaymath}
\rho^{\max}(n,m,\Phi,\Psi)\ =\
\max_{{\mathcal U}=\langle u_1,u_2,\ldots,u_n\rangle }\ \ 
L^{\max}(\langle{\mathcal A}_n,{\mathcal R}_m,{\mathcal U}\rangle ,\Phi,\Psi)
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4359}%
$L^{{\rm opt}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4361}%
$L^{\max}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4363}%
$\rho^{\max}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4415}%
$L^{\max}(\langle{\mathcal A},{\mathcal R},{\mathcal U}\rangle ,\Phi,\Psi)\leq K$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4423}%
$K$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4435}%
$\langle u\mbox{ : }2^{\mathcal R}\ \rightarrow\ {\bf Q}\rangle ^n$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4437}%
$\langle\Phi,\Psi\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4451}%
$\langle n,m,\Phi,\Psi\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4469}%
$\rho_{\Phi,\Psi}^{\max}(n,m)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4471}%
$\rho^{\max}(n.m,\Phi,\Psi)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4481}%
$\langle n,m\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3676}%
\begin{displaymath}
\rho^{\max}_{{\rm mono}}(n,m,\Phi,\Psi)\ =\
\max_{{\mathcal U}=\langle u_1,u_2,\ldots,u_n\rangle \mbox{ : each $u_i$\  is monotone}}\ \ 
L^{\max}(\langle{\mathcal A}_n,{\mathcal R}_m,{\mathcal U}\rangle ,\Phi,\Psi)
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4489}%
$\rho^{\max}_{{\rm mono}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4497}%
$\langle{\mathcal A},{\mathcal R}_m,{\mathcal U}\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4517}%
$M(k+1)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4519}%
$M(k-1)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4531}%
$\Delta_m=\langle P^{(1)},~P^{(2)}~,\ldots,~P^{(t)}\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4533}%
$\langle P^{(1)},P^{(t)}\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4535}%
$\Phi'$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4539}%
$\Delta_m$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4545}%
$\langle P^{(i)},P^{(i+1}\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4549}%
$P^{(i)}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4551}%
$P^{(i+j)}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4555}%
$j\geq 2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4557}%
$\langle P^{(i)},P^{(i+j)}\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4563}%
$P^{(i+1)}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4573}%
${\mathcal U}_n=\langle u_1,\ldots,u_n\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4579}%
$\Phi''$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4595}%
$\langle P^{(i)},Q\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4625}%
${\mathcal H}_m$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4631}%
$2^m$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4635}%
$\alpha=a_1a_2\ldots a_m$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3677}%
\begin{displaymath}
\{\ \langle\alpha,\beta\rangle \mbox{ : $\alpha$\  and $\beta$\  differ in {\em exactly one} bit position}\}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4647}%
$S^{\alpha}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4649}%
${\mathcal R}_m$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4651}%
$r_i\in S^{\alpha}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4653}%
$a_i=1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4659}%
$\beta(S)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4663}%
$\beta(S)=b_1b_2\ldots b_m$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4665}%
$b_i=1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4667}%
$r_i\in S$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4669}%
$\alpha$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4671}%
$|\alpha|$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4673}%
$1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4681}%
$\beta$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4685}%
$\alpha\beta$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4687}%
$2m$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4691}%
${\mathcal T}_m$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4699}%
$S^{\beta}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4707}%
$\overline{\alpha}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4709}%
$0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4723}%
${\mathcal R}_m\setminus S^{\alpha}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4731}%
$\alpha\subset\beta$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4733}%
$S^{\alpha}\subset S^{\beta}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4749}%
$\langle\alpha,\overline{\alpha}\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4751}%
$\langle A_1,A_2\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4755}%
${\mathcal G}_m$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4759}%
$\beta(P)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4761}%
$\beta(Q)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4767}%
$\langle P,{\mathcal R}\setminus P\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4769}%
$\langle Q,{\mathcal R}\setminus Q\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4779}%
$\Phi(P,Q)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4789}%
$m\geq 7$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3678}%
\begin{displaymath}
\rho^{\max}(2,m,\Phi,\Psi)\ \geq \left({\frac{77}{256}}\right)\ 2^m\ -\ 2
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4791}%
${\mathcal C}=\langle\alpha_1, \alpha_2,\ldots,\alpha_t\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3679}%
\begin{displaymath}
\forall\ 1\leq i<j\leq t\ (j\geq i+2)\ \Rightarrow\ (\mbox{$\alpha_i$\  and 
$\alpha_j$\  differ in at least 2 positions) }\ \ \ \mbox{ (SC)}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3680}%
\begin{displaymath}
\emptyset,\{r_1\},\ \{r_1,r_3\},\{r_1,r_2,r_3\},\{r_2,r_3\},\{r_2,r_3,r_4\},\{r_2,r_4\},
\{r_1,r_2,r_4\}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4801}%
$\langle 0000,1000,1010,1110,0110,0111,0101,1101\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4803}%
${\mathcal C}^{(m)}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4807}%
$\Delta_m=\langle P^{(1)},P^{(2)},\ldots,P^{(t)}\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4809}%
$P^{(i)}=\langle\alpha_i,\overline{\alpha_i}\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4811}%
$u_1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4813}%
$u_2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4815}%
$\gamma\subseteq{\mathcal R}_m$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3681}%
\begin{displaymath}
u_1(\gamma)+u_2(\overline{\gamma})=
\left\{
{
\begin{array}{lcl}
k&\mbox{ if }&\gamma=\alpha_k\\
0&\mbox{ if }&\gamma\not\in\{\alpha_1,\alpha_2,\ldots,\alpha_t\}
\end{array}
}
\right.
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3682}%
\begin{displaymath}
\sigma_u(P^{(i+1)})=i+1 > i=\sigma_u(P^{(i)})
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4827}%
$1\leq i\leq t$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4829}%
$i+2\leq j\leq t$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4831}%
$\langle P^{(i)},P^{(j)}\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4843}%
$\langle 0000,1000,1001,1101\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4847}%
$\langle 1000,1001\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4851}%
$\rho^{\max}(2,m,\Phi,\Psi)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4855}%
$(77/256)2^m-2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4857}%
$\Box$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3683}%
\begin{displaymath}
\begin{array}{lcl}
{\mathcal C}^{(4)}&\mbox{ $=$\  }&\langle 0000,1000,1010,1110,0110,0111,0101,1101\rangle \\
&\mbox{ $=$\  }&\langle\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5,\alpha_6,\alpha_7,\alpha_8\rangle 
\end{array}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4863}%
$\langle\{a_1,a_2\},\{r_1,r_2,r_3,r_4\},\langle u_1,u_2\rangle \rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4865}%
$\sigma_u(\langle\alpha_1,\overline{\alpha}_1\rangle )=1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4867}%
$\sigma_u(\langle\alpha_8,\overline{\alpha}_8\rangle )=8$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4869}%
${\mathcal C}^{(4)}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4873}%
$\langle\langle\alpha_1,\overline{\alpha}_1\rangle ,~\langle\alpha_8,\overline{\alpha}_8\rangle \rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4877}%
${\mathcal R}\setminus S$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4879}%
$u_1(S)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4881}%
$u_2({\mathcal R}\setminus S)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4895}%
$0000$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4897}%
$1111$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4905}%
$\alpha_1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4907}%
$1000$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4909}%
$0111$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4915}%
$2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4917}%
$\alpha_2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4919}%
$0001$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4921}%
$1110$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4929}%
$1001$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4931}%
$0110$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4939}%
$0010$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4941}%
$1101$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4949}%
$1010$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4951}%
$0101$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4957}%
$3$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4959}%
$\alpha_3$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4961}%
$0011$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4963}%
$1100$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4971}%
$1011$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4973}%
$0100$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5005}%
$4$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5009}%
$7$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5011}%
$\alpha_7$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5021}%
$8$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5023}%
$\alpha_8$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5033}%
$5$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5035}%
$\alpha_5$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5047}%
$\alpha_4$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5057}%
$6$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5059}%
$\alpha_6$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5079}%
$u_i(Q_i)\geq u_i(P_i)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5083}%
$u_j(Q_j)>u_j(P_j)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5087}%
$\min_{i\in{\mathcal A}^\delta} u_i(Q_i) >\min_{i\in{\mathcal A}^\delta} u_i(P_i)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5093}%
$u_i(P_i)+u_j(P_j)=u_i(Q_i)+u_j(Q_j)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5095}%
$|u_i(Q_i)-u_j(Q_j)|<|u_i(P_i)-u_j(P_j)|$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5097}%
$|\ldots|$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5101}%
$\Phi(\delta)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5133}%
$\langle u_1,u_2\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5137}%
$u_2(\gamma)=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5139}%
$\gamma\subseteq{\mathcal R}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3685}%
\begin{displaymath}
u_1(\gamma)=\left\{
{
\begin{array}{lcl}
k&\mbox{ if }&\gamma=\alpha_k\\
0&\mbox{ if }&\gamma\not\in\{\alpha_1,\ldots,\alpha_t\}
\end{array}
}
\right.
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5155}%
$u_2(\gamma)=u_1(\overline{\gamma})$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3687}%
\begin{displaymath}
u_1(\gamma)=\left\{
{
\begin{array}{lcl}
k&\mbox{ if }&\gamma=\alpha_k\\
0&\mbox{ if }&\gamma\not\in\{\alpha_1,\ldots,\alpha_t\}
\end{array}
}
\right.\ \ \ ;\ \ \ 
u_2(\gamma)=\left\{
{
\begin{array}{lcl}
2^m-k&\mbox{ if }&\overline{\gamma}=\alpha_k\\
2^m&\mbox{ if }&\overline{\gamma}\not\in\{\alpha_1,\ldots,\alpha_t\}
\end{array}
}
\right.
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5171}%
$u_1(\alpha_i)+u_2(\overline{\alpha_i})=2^m$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5175}%
$|u_2(\overline{\alpha_{i+1}})-u_1(\alpha_{i+1})|=2^m-2i-2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5177}%
$|u_2(\overline{\alpha_i})-u_1(\alpha_i)|=2^m-2i$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3688}%
\begin{displaymath}
|u_2(Q_2)-u_1(Q_1)| = 2^m\ >\ |u_2(P_2)-u_1(P_1)|
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5187}%
$n\geq2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3689}%
\begin{displaymath}
\rho^{\max}(n,m,\Phi,\Psi)\ \geq\ \left({\frac{77}{256}}\right)2^m\ -\ 2
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5195}%
$\overline{\alpha_1}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5199}%
$\emptyset$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5201}%
$3\leq j\leq n$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5205}%
$u_j$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5207}%
$u_j(\emptyset)=1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5209}%
$u_j(S)=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5211}%
$S\not=\emptyset$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5219}%
$u_j(\emptyset)=2^m$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5225}%
$u_j(S)=2^m$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5245}%
$\langle P^{(t)},Q\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5251}%
$P^{(t)}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5257}%
$\sigma_e(P)=\min_{1\leq i\leq n} u_i(P_i)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5267}%
$t$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5271}%
$\sigma_e(P^{(t)})=t$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5279}%
$\rho_{{\rm mono}}^{\max}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5285}%
$m\geq 14$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3690}%
\begin{displaymath}
\rho^{\max}_{{\rm mono}}(2,m,\Phi,\Psi)
~\geq~
\left\{
{
\begin{array}{lcl}
\left({\frac{77}{128}}\right)\ 2^{m/2}~-~3&\mbox{ if }&\mbox{$m$\  is even}\\
&&\\
\left({\frac{77}{128}}\right)\ 2^{(m-1)/2}~-~3&\mbox{ if }&\mbox{$m$\  is odd}
\end{array}
}
\right.
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5295}%
$m=2s$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5297}%
$s\geq 7$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3691}%
\begin{displaymath}
\Delta_s~=~\langle\alpha_1,\alpha_2,\ldots,\alpha_t\rangle 
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5299}%
${\mathcal H}_s$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5301}%
$\alpha_i$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5303}%
${\mathcal R}_s$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5305}%
$s$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5307}%
$double(\Delta_s)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5309}%
${\mathcal H}_{2s}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3692}%
\begin{displaymath}
\begin{array}{lcl}
double(\Delta_s)&~=~&\langle~
\alpha_1\overline{\alpha_1},~\alpha_2\overline{\alpha_2}~,\ldots,~
\alpha_i\overline{\alpha_i},~\alpha_{i+1}\overline{\alpha_{i+1}}~,\ldots,~
\alpha_t\overline{\alpha_t}~\rangle \\
&~=~&\langle\beta_1,\beta_3,\ldots,\beta_{2i-1},\beta_{2i+1},\ldots,\beta_{2t-1}\rangle 
\end{array}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5321}%
$\beta_{2i-1}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5323}%
$\overline{\beta_{2i-1}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5325}%
$\beta_{2i+1}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5327}%
$\beta_{2i}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3693}%
\begin{displaymath}
\beta_{2i}~=~
\left\{
{
\begin{array}{lcl}
\alpha_{i+1}\overline{\alpha_i}&\mbox{ if }&\alpha_i\subset\alpha_{i+1}\\
\alpha_i\overline{\alpha_{i+1}}&\mbox{ if }&\alpha_i\supset\alpha_{i+1}
\end{array}
}
\right.
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5333}%
$ext(\Delta_s)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3694}%
\begin{displaymath}
ext(\Delta_s)~=~
\langle\beta_1,\beta_2,\beta_3~,\ldots,~\beta_{2(t-1)},\beta_{2t-1}\rangle 
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5337}%
$\Delta_s$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5345}%
$j$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5347}%
$|\beta_j|=s$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5351}%
$|\beta_j|=s+1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5357}%
$\langle\beta_i,\overline{\beta_i}\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5363}%
$P^{(1)}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5365}%
$P^{(2t-1)}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5369}%
$j\geq i+2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5387}%
$\langle P^{(1)},P^{(2t-1)}\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5391}%
$S\subseteq{\mathcal R}_{2s}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3695}%
\begin{displaymath}
u_2(S)~=~
\left\{
{
\begin{array}{lcl}
0&\mbox{ if }&|S|\leq s-2\\
2t+1&\mbox{ if }&|S|=s-1\\
2t+2&\mbox{ if }&|S|\geq s
\end{array}
}
\right.
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5401}%
$\beta_i$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5405}%
$|\beta_i|$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5409}%
$s+1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5411}%
$i$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5433}%
$\gamma$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5437}%
$\beta_{2j-1}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5439}%
$\beta_{2k-1}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5441}%
$|\gamma|=s+1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3696}%
\begin{displaymath}
\beta_{2i-1}\subset\gamma~~;~~\beta_{2j-1}\subset\gamma~~;~~\beta_{2k-1}\subset\gamma
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5443}%
$\beta_{2i-1}=\alpha_i\overline{\alpha_i}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5449}%
$\{\alpha_i,\alpha_j,\alpha_k\}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5453}%
$\alpha_k$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5459}%
$s-2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5461}%
$|\beta_{2i-1}\cap\beta_{2k-1}|\leq s-2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5465}%
$\beta_{2k-1}=\alpha_k\overline{\alpha_k}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5471}%
$\overline{\alpha_i}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5473}%
$\overline{\alpha_k}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5475}%
$|\beta_{2i-1}\setminus\beta_{2k-1}|\geq2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5481}%
$|\beta_{2k-1}\setminus\beta_{2i-1}|\geq2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3697}%
\begin{displaymath}
\beta_{2i-1}\cap\beta_{2k-1}~\cup~\beta_{2i-1}\setminus\beta_{2k-1}~\cup
~\beta_{2k-1}\setminus\beta_{2i-1}\subseteq\gamma
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3698}%
\begin{displaymath}
|~\beta_{2i-1}\cap\beta_{2k-1}~\cup~\beta_{2i-1}\setminus\beta_{2k-1}~\cup
~\beta_{2k-1}\setminus\beta_{2i-1}~|~~\geq~~s+2
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5503}%
$\gamma=\beta_{2i-1}\cup\beta_{2i+1}=\beta_{2i}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5509}%
${\mathcal R}_{2s}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5513}%
$\{\gamma~:~\gamma=\beta_{2i}\}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3699}%
\begin{displaymath}
\{~\gamma~:~\beta_{2i-1}\subset\gamma\mbox{, }\gamma\not=\beta_{2i}\mbox{ and }i<t\}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3700}%
\begin{displaymath}
\{~\gamma~:~\mbox{$\gamma$\  is neither $Good$\  nor a $Digression$}\}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5521}%
$Good$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5527}%
$Digressions$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5541}%
$Inaccessible$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5553}%
$\beta_{2t-1}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3701}%
\begin{displaymath}
u_1(\gamma)~=~
\left\{
{
\begin{array}{lcl}
2i-1&\mbox{ if }&\gamma=\beta_{2i-1}\\
2i+1&\mbox{ if }&\gamma=\beta_{2i}\\
2i&\mbox{ if }&\mbox{$|\gamma|=s+1$\  and $\gamma$\  is a $Digression$\  from $\beta_{2i-1}$}\\
0&\mbox{ if }& |\gamma|\leq s-1\\
0&\mbox{ if }& |\gamma|=s\mbox{ and }\gamma\not\in ext(\Delta_s)\\
2t-1&\mbox{ if }&\mbox{$\gamma$\  is $Inaccessible$\  or $|\gamma|\geq s+2$}
\end{array}
}
\right.
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5579}%
$\langle P^{(1)},\ldots,P^{(2t-1)}\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3702}%
\begin{displaymath}
\forall~1\leq j < 2t-1~~u_1(\beta_j)+u_2(\overline{\beta_j})~<~
u_1(\beta_{j+1})+u_2(\overline{\beta_{j+1}})
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3703}%
\begin{displaymath}
\begin{array}{lcl}
u_1(\beta_{2i-1})+u_2(\overline{\beta_{2i-1}})&~=~&2(t+i)+1\\
&~<~&u_1(\beta_{2i})+u_2(\overline{\beta_{2i}})\\
&~=~&2(t+i)+2\\
&~<~&u_1(\beta_{2i+1})+u_2(\overline{\beta_{2i+1}})\\
&~=~&2(t+i)+3
\end{array}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5601}%
$P^{(j)}=\langle\beta_j,\overline{\beta_j}\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5605}%
$P^{(j+1)}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5607}%
$P^{(j-1)}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5609}%
$\langle P^{(j)},Q\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5613}%
$j=2i-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5615}%
$\sigma_u(P^{(2i-1)})=2(t+i)+1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5619}%
$|Q_1|\in\{s-1,s+1\}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5621}%
$u_1(Q_1)=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5623}%
$u_2(Q_2)=2t+2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5625}%
$\sigma_u(Q)=2t+2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5629}%
$u_1(Q_1)=2i$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5631}%
$Q_1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5633}%
$Digression$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5637}%
$u_2(Q_2)=2t+1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5639}%
$\sigma_u(Q)=2(t+i)+1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5647}%
$j=2i$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5649}%
$\sigma_u(P^{(2i)})=2(t+i)+2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5657}%
$|Q_1|\in\{s,s+2\}$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5677}%
$u_1(Q_1)=2t-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5679}%
$u_2(Q_2)=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5681}%
$|Q_2|=s-2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5683}%
$P^{(j)}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5687}%
$\langle P^{(j)},P^{(j+1)}\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5697}%
$S\subset T$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5699}%
$u_1(S)\leq u_1(T)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5701}%
$|S|= s$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5707}%
$|T|=s+1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5709}%
$|S|<s$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5715}%
$u_1(S)=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5717}%
$|S|\geq s+1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5721}%
$|T|\geq s+2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5723}%
$u_1(T)=2t-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5739}%
$T=\beta_{2i}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5749}%
$u_1(\beta_{2i})=2i+1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5751}%
$\max\{u_1(\beta_{2i-1}),u_1(\beta_{2i+1})\}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5765}%
$S=\beta_{2i-1}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5767}%
$u_1(T)=2i$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5769}%
$u_1(S)=2i-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5779}%
$m=2s+1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5783}%
$r_m$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5789}%
$s=3$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5791}%
$\Delta_3~=~\langle 000,001,101,111,110\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5793}%
$t=5$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5795}%
$double(\Delta_3)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3704}%
\begin{displaymath}
\langle 000111,001110,101010,111000,110001\rangle 
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5799}%
$ext(\Delta_3)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3705}%
\begin{displaymath}
\begin{array}{cl}
&\langle 000111,001111,001110,101110,101010,111010,111000,111001,110001\rangle \\
\mbox{ $=$\  }&\langle\beta_1,\beta_2,\beta_3,\beta_4,\beta_5,\beta_6,\beta_7,\beta_8,\beta_9\rangle 
\end{array}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5803}%
$s+1=4$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3706}%
\begin{displaymath}
\begin{array}{lcl}
Good&\mbox{ $=$\  }&\{001111,~101110,~111010,~111001\}\\
Digression&\mbox{ $=$\  }&\{010111,100111,101011,011110,111100\}\\
Inaccessible&\mbox{ $=$\  }&\{011011,011101,101101,110110,110011,110101\}
\end{array}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5811}%
$\{110011,110101\}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5815}%
$\beta_9=110001$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5817}%
$\beta_9$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5819}%
$S\subseteq{\mathcal R}=\{r_1,r_2,r_3,r_4,r_5,r_6\}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3707}%
\begin{displaymath}
u_2(S)~~=~~\left\{
{
\begin{array}{lcl}
0&\mbox{ {\rm if }}&|S|\leq 1\\
11&\mbox{ {\rm if }}&|S|=2\\
12&\mbox{ {\rm if }}&|S|\geq 3
\end{array}
}
\right.
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3708}%
\begin{displaymath}
u_1(S)~=~
\left\{
{
\begin{array}{lcl}
0&\mbox{ {\rm if} }&|S|\leq 2\\
0&\mbox{ {\rm if} }&|S|=3\mbox{ {\rm and} }S\not\in\{000111,001110,101010,111000,110001\}\\
1&\mbox{ {\rm if} }&S=000111~~(\beta_1)\\
2&\mbox{ {\rm if} }&S=010111\mbox{ ({\rm digression from $\beta_1$})}\\
2&\mbox{ {\rm if} }&S=100111\mbox{ ({\rm digression from $\beta_1$})}\\
3&\mbox{ {\rm if} }&S=001111~~(\beta_2)\\
3&\mbox{ {\rm if} }&S=001110~~(\beta_3)\\
4&\mbox{ {\rm if} }&S=011110\mbox{ ({\rm digression from $\beta_3$})}\\
5&\mbox{ {\rm if} }&S=101110~~(\beta_4)\\
5&\mbox{ {\rm if} }&S=101010~~(\beta_5)\\
6&\mbox{ {\rm if} }&S=101011\mbox{ ({\rm digression from $\beta_5$})}\\
7&\mbox{ {\rm if} }&S=111010~~(\beta_6)\\
7&\mbox{ {\rm if} }&S=111000~~(\beta_7)\\
8&\mbox{ {\rm if} }&S=111100\mbox{ ({\rm digression from $\beta_7$})}\\
9&\mbox{ {\rm if} }&S=111001~~(\beta_8)\\
9&\mbox{ {\rm if} }&S=110001~~(\beta_9)\\
9&\mbox{ {\rm if} }&|S|\geq5\mbox{ {\rm or }}S\in\{011011,011101,101101,110110,110011,110101\}
\end{array}
}
\right.
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5871}%
$ext(\Delta_{s})$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3710}%
\begin{displaymath}
\langle u_1(\gamma),u_2(\overline{\gamma})\rangle ~=~
\left\{
{
\begin{array}{lcl}
\langle i,i\rangle &\mbox{ if }&\gamma=\beta_{2i-1}\\
\langle i+1,i\rangle &\mbox{ if }&\gamma=\beta_{2i}\\
\langle i,i-1\rangle &\mbox{ if }&\mbox{$|\gamma|=s+1$\  and $\gamma$\  is a $Digression$\  from $\beta_{2i-1}$}\\
\langle 0,2t-1\rangle &\mbox{ if }& |\gamma|\leq s-1\\
\langle 0,2t-1\rangle &\mbox{ if }& |\gamma|=s\mbox{ and }{\gamma}\not\in ext(\Delta_s)\\
\langle 2t-1,0\rangle &\mbox{ if }&\mbox{$\gamma$\  is $Inaccessible$\  or $|\gamma|\geq s+2$}
\end{array}
}
\right.
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3711}%
\begin{displaymath}
\begin{array}{lcl}
\langle u_1(\beta_{2i-1}),u_2(\overline{\beta_{2i-1}})\rangle &\mbox{ $=$\  }&\langle i,i\rangle \\
\langle u_1(\beta_{2i}),u_2(\overline{\beta_{2i}})\rangle &\mbox{ $=$\  }&\langle i+1,i\rangle \\
\langle u_1(\beta_{2i+1}),u_2(\overline{\beta_{2i+1}})\rangle &\mbox{ $=$\  }&\langle i+1,i+1\rangle 
\end{array}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5907}%
$Q=\langle\gamma,{\overline{\gamma}}\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5919}%
$Q\not\in\{P^{(j-1)},P^{(j+1)}\}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5923}%
$\langle u_1(\beta_j),u_2(\overline{\beta_j})\rangle =\langle i,i\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5925}%
$|\gamma|\in\{s-1,s+1\}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5939}%
$\langle u_1(\beta_j),u_2(\overline{\beta_j})\rangle =\langle i+1,i\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5941}%
$|\gamma|\in\{s,s+2\}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5943}%
$\gamma\not\in ext(\Delta_s)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5967}%
$|S|\leq s-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5987}%
$S=\beta_{2i-1},$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5999}%
$u_1(S)=i$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6001}%
$T\in\{\beta_{2i},\beta_{2i-2}\}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6003}%
$u_1(T)\geq i = u_1(S)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6011}%
$u_1(T)=i=u_1(S)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6027}%
$|T|\geq s+1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6029}%
${\mathcal R}_{2s}\setminus T$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6033}%
$u_2(T)=2t-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6035}%
$|S|\leq s-2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6037}%
${\mathcal R}_{2s}\setminus S$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6041}%
$u_2(S)=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6043}%
$T=\overline{\beta_{2i-1}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6045}%
$u_2(T)=i$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6047}%
$|S|=s-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6055}%
$\{\beta_{2i},\beta_{2i-2}\}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6061}%
$u_2(S)=i-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6065}%
$\beta_{2i-2}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6067}%
$u_2(S)=i-1~<~u_2(T)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6073}%
$u_2(S)=i=u_2(T)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3712}%
\begin{displaymath}
\begin{array}{lcl}
\langle u_1(\gamma),u_2(\overline{\gamma})\rangle ~=~
\left\{
{
\begin{array}{lcl}
\langle 2i-1,2i\rangle &\mbox{ if }&\gamma=\beta_{2i-1}\\
\langle 2i+1,2i\rangle &\mbox{ if }&\gamma=\beta_{2i}\\
\langle 2i,2i-1\rangle &\mbox{ if }&\mbox{$|\gamma|=s+1$\  and $\gamma$\  is a $Digression$\  from $\beta_{2i-1}$}\\
\langle 0,2t-1\rangle &\mbox{ if }& |\gamma|\leq s-1\\
\langle 0,2t-1\rangle &\mbox{ if }& |\gamma|=s\mbox{ and }{\gamma}\not\in ext(\Delta_s)\\
\langle 2t-1,0\rangle &\mbox{ if }&\mbox{$\gamma$\  is $Inaccessible$\  or $|\gamma|\geq s+2$}
\end{array}
}
\right.
\end{array}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3713}%
\begin{displaymath}
\begin{array}{lcl}
\min\{u_1(\beta_{2i-1}),u_2(\overline{\beta_{2i-1}})\}&\mbox{ $=$\  }&2i-1\\
\min\{u_1(\beta_{2i}),u_2(\overline{\beta_{2i}})\}&\mbox{ $=$\  }&2i\\
\min\{u_1(\beta_{2i+1}),u_2(\overline{\beta_{2i+1}})\}&\mbox{ $=$\  }&2i+1
\end{array}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6105}%
$Q=\langle\gamma,\overline{\gamma}\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6121}%
$P^{(j)}=\langle\beta_{2i-1},\overline{\beta_{2i-1}}\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6123}%
$\min\{u_1(\beta_{2i-1}),u_2(\overline{\beta_{2i-1}})\}=2i-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6127}%
$\min\{u_1(\gamma),u_2(\overline{\gamma})\}=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6129}%
$\min\{u_1(\gamma),u_2(\overline{\gamma})\}=2i-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6137}%
$P^{(j)}=\langle\beta_{2i},\overline{\beta_{2i}}\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6139}%
$\min\{u_1(\beta_{2i}),u_2(\overline{\beta_{2i}})\}=2i$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6143}%
$|\gamma|=s$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6145}%
$\min\{u_1(\gamma),u_2(\overline{\gamma})\}\leq 2i-1$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6147}%
$\gamma=\beta_{2i-1}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6149}%
$|\gamma|=s+2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3715}%
\begin{displaymath}
\langle u_1(S),u_2({\mathcal R}\setminus S)\rangle ~=~
\left\{
{
\begin{array}{lcl}
\langle 0,9\rangle &\mbox{ {\rm if} }&|S|\leq 2\\
\langle 0,9\rangle &\mbox{ {\rm if} }&|S|=3\mbox{ {\rm and} }S\not\in\{000111,001110,101010,111000,110001\}\\
\langle 1,1\rangle &\mbox{ {\rm if} }&S=000111~~(\beta_1)\\
\langle 1,0\rangle &\mbox{ {\rm if} }&S=010111\mbox{ {\rm digression from $\beta_1$}}\\
\langle 1,0\rangle &\mbox{ {\rm if} }&S=100111\mbox{ {\rm digression from $\beta_1$}}\\
\langle 2,1\rangle &\mbox{ {\rm if} }&S=001111~~(\beta_2)\\
\langle 2,2\rangle &\mbox{ {\rm if} }&S=001110~~(\beta_3)\\
\langle 2,1\rangle &\mbox{ {\rm if} }&S=011110\mbox{ {\rm digression from $\beta_3$}}\\
\langle 3,2\rangle &\mbox{ {\rm if} }&S=101110~~(\beta_4)\\
\langle 3,3\rangle &\mbox{ {\rm if} }&S=101010~~(\beta_5)\\
\langle 3,2\rangle &\mbox{ {\rm if} }&S=101011\mbox{ {\rm digression from $\beta_5$}}\\
\langle 4,3\rangle &\mbox{ {\rm if} }&S=111010~~(\beta_6)\\
\langle 4,4\rangle &\mbox{ {\rm if} }&S=111000~~(\beta_7)\\
\langle 4,3\rangle &\mbox{ {\rm if} }&S=111100\mbox{ {\rm digression from $\beta_7$}}\\
\langle 5,4\rangle &\mbox{ {\rm if} }&S=111001~~(\beta_8)\\
\langle 5,5\rangle &\mbox{ {\rm if} }&S=110001~~(\beta_9)\\
\langle 9,0\rangle &\mbox{ {\rm if} }&|S|\geq5\mbox{ {\rm or }}S\in\{011011,011101,101101,110110,110011,110101\}
\end{array}
}
\right.
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6205}%
$C$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6207}%
$M$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6227}%
$\Phi_k(P,Q)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6233}%
$k\geq 3$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6235}%
$n\geq k$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6239}%
$m\geq\mbox{{\scriptsize$\left({\begin{array}{c}k\\2\end{array}}\right)$}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3717}%
\begin{displaymath}
\begin{array}{lcclr}
L^{{\rm opt}}(\delta,\langle{\mathcal A},{\mathcal R},{\mathcal U}\rangle ,\Phi_k)&\mbox{ $=$\  }&1&\mbox{ }&(a)\\
L^{{\rm opt}}(\delta,\langle{\mathcal A},{\mathcal R},{\mathcal U}\rangle ,\Phi_{k-1})&\mbox{ $\geq$\  }&
2^{\lfloor{2m/k(k-1)}\rfloor} -1&\mbox{ }&(b)\\
L^{{\rm opt}}(\delta,\langle{\mathcal A},{\mathcal R},{\mathcal U}\rangle ,\Phi_{k-2})&&\mbox{ {\rm is undefined}}&
\mbox{ }&(c)
\end{array}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6255}%
$L^{{\rm opt}}(\delta,\langle{\mathcal A},{\mathcal R},{\mathcal U}\rangle ,\Phi_{k-1})$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6257}%
$\rho^{\max}(n,m,\Phi_{k-1},\Psi)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6277}%
$M(k-2)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6295}%
$\left({\begin{array}{c}k\\2\end{array}}\right)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6301}%
$m~=~s\mbox{{\scriptsize$\left({\begin{array}{c}k\\2\end{array}}\right)$}}+q$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6303}%
$s\geq 1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6307}%
$1\leq q<\mbox{{\scriptsize$\left({\begin{array}{c}k\\2\end{array}}\right)$}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6309}%
$m-q$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6311}%
$q$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6319}%
$\lfloor{\ldots}\rfloor$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6325}%
$k=4$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6327}%
$m=18=3\times6$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3718}%
\begin{displaymath}
\Delta\ =\ \langle P^{(1)},P^{(2)},\ldots,P^{(t)}\rangle 
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6341}%
$d$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6345}%
$1\leq d\leq t$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6349}%
${\mathcal A}=\{A_0,A_1,\ldots,A_{k-1}\}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6353}%
$m=s\mbox{{\scriptsize$\left({\begin{array}{c}k\\2\end{array}}\right)$}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6365}%
$0\leq i<j\leq k-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6367}%
${\mathcal R}^{i,j}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6369}%
$\{r_{1}^{\{i,j\}},~r_{2}^{\{i,j\}}~,\ldots,~r_{s}^{\{i,j\}}\}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6373}%
${\mathcal R}^{\{i,j\}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6389}%
$\langle P^{(d)},P^{(d+1)}\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6391}%
$P^{(d)}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6393}%
$\{i,j\}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6395}%
$h\not\in\{i,j\}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6397}%
$P_{h}^{(d)}\cap{\mathcal R}^{\{i,j\}}=\emptyset$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6399}%
$\delta=\langle P^{(d)},P^{(d+1)}\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6405}%
$\{A_i,A_j\}\subseteq{\mathcal A}^\delta$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6417}%
$P^{(d)}_{i}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6419}%
$P^{(d+1)}_{j}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6423}%
$P^{(d)}_{j}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6425}%
$P^{(d+1)}_{i}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6433}%
$\left({\begin{array}{c}k-1\\2\end{array}}\right)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6439}%
$\{A_i,A_j\}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6443}%
${\mathcal A}^{\delta}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6447}%
$s\geq 2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6453}%
$2^s$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6455}%
$s=2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6457}%
$\langle 00,01,11,10,00\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6459}%
${\mathcal H}_2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6461}%
$\langle\alpha_1,\alpha_2,\ldots,\alpha_p,\alpha_1\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6463}%
$p=2^s$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6467}%
$\langle 0\alpha_1,1\alpha_1,1\alpha_p,1\alpha_{p-1},\ldots,1\alpha_2,0\alpha_2\ldots,0\alpha_p,0\alpha_1\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6469}%
${\mathcal H}_{s+1}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3719}%
\begin{displaymath}
{\mathcal S}^{(v)}=\underline{v}^{(0)},\underline{v}^{(1)},\ldots,
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\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3720}%
\begin{displaymath}
{\mathcal S}^{(w)}=\underline{w}^{(0)},\underline{w}^{(1)},\ldots,\underline{w}^{(i)},\ldots,
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\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6473}%
$\underline{w}^{(i)}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6475}%
$\underline{v}^{(i)}$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6479}%
$\underline{v}=v_1v_2\ldots v_s$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6483}%
$r_{k}^{\{i,j\}}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6485}%
$v_k=1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6491}%
$P_{i}^{(d)}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6501}%
$P_{i}^{(d)}\cap{\mathcal R}^{\{i,j\}}$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6503}%
$i\not\in{\mathcal A}^{\delta}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6505}%
$j\not\in{\mathcal A}^{\delta}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6507}%
$P_{i}^{(d+1)}\cap{\mathcal R}^{\{i,j\}}=P_{i}^{(d)}\cap{\mathcal R}^{\{i,j\}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6509}%
$P_{j}^{(d+1)}\cap{\mathcal R}^{\{i,j\}}=P_{j}^{(d)}\cap{\mathcal R}^{\{i,j\}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6511}%
$\{i,j\}\subseteq{\mathcal A}^{\delta}$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6515}%
$P_{j}^{(d)}\cap{\mathcal R}^{\{i,j\}}$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6521}%
$\langle\underline{v}^{(h)},\underline{w}^{(h)}\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6523}%
$P_{i}^{(d+1)}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6525}%
$P_{j}^{(d+1)}$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6529}%
$P_{i}^{(d+1)}\cap{\mathcal R}^{\{i,j\}}$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6531}%
$P_{j}^{(d+1)}\cap{\mathcal R}^{\{i,j\}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6535}%
$\underline{v}^{(h+1)}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6537}%
$\underline{w}^{(h+1)}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6539}%
$\underline{v}^{(h)}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6541}%
$\underline{w}^{(h)}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6549}%
$\underline{v}^{(0)}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6551}%
$\underline{w}^{(0)}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6555}%
$\underline{v}^{(2^s-1)}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6557}%
$\underline{w}^{(2^s-1)}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6573}%
$p\geq k-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6575}%
$M(p)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6585}%
$P^{(d+1)}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6597}%
$\langle P^{(1)},Q\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3721}%
\begin{displaymath}
{\mathcal R}^{\{i,j\}}=\{r_{1}^{\{i,j\}},r_{2}^{\{i,j\}},\ldots,r_{s}^{\{i,j\}}\}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6615}%
${\mathcal A}=\{A_0,A_1,A_2,A_3\}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3722}%
\begin{displaymath}
\begin{array}{lcl}
{\mathcal R}^{\{0,1\}}&=&\{r_{1}^{\{0,1\}},r_{2}^{\{0,1\}},r_{3}^{\{0,1\}}\}\\
{\mathcal R}^{\{0,2\}}&=&\{r_{1}^{\{0,2\}},r_{2}^{\{0,2\}},r_{3}^{\{0,2\}}\}\\
{\mathcal R}^{\{0,3\}}&=&\{r_{1}^{\{0,3\}},r_{2}^{\{0,3\}},r_{3}^{\{0,3\}}\}\\
{\mathcal R}^{\{1,2\}}&=&\{r_{1}^{\{1,2\}},r_{2}^{\{1,2\}},r_{3}^{\{1,2\}}\}\\
{\mathcal R}^{\{1,3\}}&=&\{r_{1}^{\{1,3\}},r_{2}^{\{1,3\}},r_{3}^{\{1,3\}}\}\\
{\mathcal R}^{\{2,3\}}&=&\{r_{1}^{\{2,3\}},r_{2}^{\{2,3\}},r_{3}^{\{2,3\}}\}
\end{array}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6621}%
$\underline{v}^{(0)}=111\ldots11$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6623}%
$\underline{w}^{(0)}=000\ldots00$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3725}%
\begin{displaymath}
\begin{array}{llcl}
\mbox{a. }&{\mathcal S}^{(v)}&=&\langle 111,110,010,011,001,000,100,101\rangle \\
\mbox{b. }&{\mathcal S}^{(w)}&=&\langle 000,001,101,100,110,111,011,010\rangle 
\end{array}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6635}%
$k\times k$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6637}%
$B=[b_{i,j}]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6639}%
$0\leq i,j\leq k-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3727}%
\begin{displaymath}
b_{i,j}\ =\ \left\{
{
\begin{array}{lcl}
\bot&\mbox{ if }&i=j\\
\neg b_{j,i}&\mbox{ if }&i>j\\
\neg b_{i,j-1}&\mbox{ if }&i<j
\end{array}
}
\right.
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6641}%
$1\leq i\leq k$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3728}%
\begin{displaymath}
P_{i}^{(1)}\ =\ \bigcup_{j=0}^{i-1}\{\ R^{\{j,i\}}\mbox{ : }b_{i,j}=\top\}\ \cup
                \bigcup_{j=i+1}^{k-1}\{\ R^{\{i,j\}}\mbox{ : }b_{i,j}=\top\}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3729}%
\begin{displaymath}
B\ =\ 
\left[
{
\begin{array}{cccc}
\bot&\top&\bot&\top\\
\bot&\bot&\top&\bot\\
\top&\bot&\bot&\top\\
\bot&\top&\bot&\bot
\end{array}
}
\right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3730}%
\begin{displaymath}
\begin{array}{lclclcl}
P_0^{(1)}&=&{\mathcal R}^{\{0,1\}}\cup{\mathcal R}^{\{0,3\}}&\mbox{ $=$\  }&\langle 111,000,111\rangle &
\mbox{ $\subseteq$\  }&{\mathcal R}^{\{0,1\}}\cup{\mathcal R}^{\{0,2\}}\cup{\mathcal R}^{\{0,3\}}\\
P_1^{(1)}&=&{\mathcal R}^{\{1,2\}}&\mbox{ $=$\  }&\langle 000,111,000\rangle &
\mbox{ $\subseteq$\  }&{\mathcal R}^{\{0,1\}}\cup{\mathcal R}^{\{1,2\}}\cup{\mathcal R}^{\{1,3\}}\\
P_2^{(1)}&=&{\mathcal R}^{\{0,2\}}\cup{\mathcal R}^{\{2,3\}}&\mbox{ $=$\  }&\langle 111,000,111\rangle &
\mbox{ $\subseteq$\  }&{\mathcal R}^{\{0,2\}}\cup{\mathcal R}^{\{1,2\}}\cup{\mathcal R}^{\{2,3\}}\\
P_3^{(1)}&=&{\mathcal R}^{\{1,3\}}&\mbox{ $=$\  }&\langle 000,111,000\rangle &
\mbox{ $\subseteq$\  }&{\mathcal R}^{\{0,3\}}\cup{\mathcal R}^{\{1,3\}}\cup{\mathcal R}^{\{2,3\}}
\end{array}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6659}%
$P_{i}^{(1)}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3731}%
\begin{displaymath}
Q^1,Q^2,\ldots,Q^z
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6673}%
$Q^{x,p}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6677}%
$x$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6679}%
$0\leq p\leq k-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6683}%
$Q^{0,k-1}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6685}%
$Q^{x,0}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6689}%
$Q^{x-1,k-1}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6691}%
$x\geq1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6697}%
$Q^{x,p-1}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6699}%
$0<p\leq k-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6707}%
$\chi(i,j,P)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6709}%
$\underline{u}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6713}%
$P_i\cap{\mathcal R}^{\{i,j\}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6721}%
$S^{(v)}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6723}%
${\mathcal R}^{\{i,j\}}\subseteq P_{i}^{(1)}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6725}%
$S^{(w)}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6727}%
${\mathcal R}^{\{i,j\}}\subseteq P_{j}^{(1)}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6729}%
$P=Q^{x,p}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6731}%
$\chi(i,j,x,p)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6733}%
$\chi(i,j,x,p)=\chi(j,i,x,p)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6735}%
$P^{(1)}=Q^{0,k-1}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6737}%
$\chi(i,j,0,k-1)=\chi(j,i,0,k-1)=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6741}%
$Q^{1,0}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6749}%
$p>0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6751}%
$Q^{x+1,0}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6753}%
$Q^{x,k-1}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6755}%
$Q^{y,q}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6759}%
$\delta=\langle Q^{x,p},Q^{y,q}\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6763}%
${\mathcal A}^{\delta}={\mathcal A}\setminus \{A_q\}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6781}%
$\chi(i,j,y,q)=\chi(i,j,x,p)+1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6789}%
$\chi(j,i,y,q)=\chi(j,i,x,p)+1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6805}%
$P^{(d)}=Q^{x,p}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3732}%
\begin{displaymath}
\chi(i,j,x,p)~=~
\left\{
{
\begin{array}{lcl}
0&\mbox{ if }&x=0\mbox{ and }p=k-1\\
1~+~\chi(i,j,x-1,k-1)&\mbox{ if }&\mbox{$x\geq 1$, $p=0$, and $p\not\in\{i,j\}$}\\
\chi(i,j,x-1,k-1)&\mbox{ if }&\mbox{$x\geq 1$, $p=0$, and $p\in\{i,j\}$}\\
1~+~\chi(i,j,x,p-1)&\mbox{ if }&\mbox{$1\leq p\leq k-1$, and $p\not\in\{i,j\}$}\\
\chi(i,j,x,p-1)&\mbox{ if }&\mbox{$1\leq p\leq k-1$, and $p\in\{i,j\}$}
\end{array}
}
\right.
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3733}%
\begin{displaymath}
\begin{array}{lcl}
\mbox{{
\large$\underline{v}^{(\chi(i,j,x,p))}$}}&\mbox{ {\rm if} }&{\mathcal R}^{\{i,j\}}\subseteq P_{i}^{(1)}\\
\mbox{{
\large$\underline{w}^{(\chi(i,j,x,p))}$}}&\mbox{ {\rm if} }&{\mathcal R}^{\{i,j\}}\subseteq P_{j}^{(1)}
\end{array}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3734}%
\begin{displaymath}
\begin{array}{c}
\begin{array}{c|ll|ccc|ccc|ccc|ccc|c}
& & & &A_0& & &A_1& & &A_2& & &A_3& &\\
& & &i~j&i~j&i~j&i~j&i~j&i~j&i~j&i~j&i~j&i~j&i~j&i~j&\mbox{{\small${\mathcal A}^{\langle P^{(d-1)},P^{(d)}\rangle }$}}\\
d&x&p&0~1&0~2&0~3&1~0&1~2&1~3&2~0&2~1&2~3&3~0&3~1&3~2&\\\hline
1&0&3&111&000&111&000&111&000&111&000&111&000&111&000&\mbox{--}\\\hline
2&1&0&111&000&111&000&110&001&111&001&110&000&110&001&\mbox{{\small$\{A_1,A_2,A_3\}$}}\\
3&1&1&111&001&110&000&110&001&110&001&010&001&110&101&\mbox{{\small$\{A_0,A_2,A_3\}$}}\\
4&1&2&110&001&010&001&110&101&110&001&010&101&010&101&\mbox{{\small$\{A_0,A_1,A_3\}$}}\\
5&1&3&010&101&010&101&010&101&010&101&010&101&010&101&\mbox{{\small$\{A_0,A_1,A_2\}$}}\\\hline
6&2&0&010&101&011&101&011&100&010&100&011&101&011&100&\mbox{{\small$\{A_1,A_2,A_3\}$}}\\
7&2&1&010&100&001&101&011&100&011&100&001&100&011&110&\mbox{{\small$\{A_0,A_2,A_3\}$}}\\
8&2&2&011&100&001&100&011&110&011&100&001&110&001&110&\mbox{{\small$\{A_0,A_1,A_3\}$}}\\
9&2&3&001&110&001&110&001&110&001&110&001&110&001&110&\mbox{{\small$\{A_0,A_1,A_2\}$}}\\\hline
\vdots&\vdots&\vdots& &\vdots& & &\vdots& & &\vdots& & &\vdots& &\vdots
\end{array}\\
\mbox{{\bf Subsets of ${\mathcal R}^{\{i,j\}}$\  held by $A_i$\  in $Q^{x,p}$\  ($k=4$, $s=3$)}}
\end{array}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3735}%
\begin{displaymath}
\begin{array}{c}
\begin{array}{c|ll|ccc|ccc|ccc|ccc|c}
& & & &A_0& & &A_1& & &A_2& & &A_3& &\\
& & &i~j&i~j&i~j&i~j&i~j&i~j&i~j&i~j&i~j&i~j&i~j&i~j&{\mathcal A}^{\langle P^{(d-1)},P^{(d)}\rangle }\\
d&x&p&0~1&0~2&0~3&1~0&1~2&1~3&2~0&2~1&2~3&3~0&3~1&3~2&\\\hline
1&0&3&0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0&\mbox{--}\\\hline
2&1&0&0 &0 &0 &0 &1 &1 &0 &1 &1 &0 &1 &1&\{A_1,A_2,A_3\}\\
3&1&1&0 &1 &1 &0 &1 &1 &1 &1 &2 &1 &1 &2&\{A_0,A_2,A_3\}\\
4&1&2&1 &1 &2 &1 &1 &2 &1 &1 &2 &2 &2 &2&\{A_0,A_1,A_3\}\\
5&1&3&2 &2 &2 &2 &2 &2 &2 &2 &2 &2 &2 &2&\{A_0,A_1,A_2\}\\\hline
6&2&0&2 &2 &2 &2 &3 &3 &2 &3 &3 &2 &3 &3&\{A_1,A_2,A_3\}\\
7&2&1&2 &3 &3 &2 &3 &3 &3 &3 &4 &3 &3 &4&\{A_0,A_2,A_3\}\\
8&2&2&3 &3 &4 &3 &3 &4 &3 &3 &4 &4 &4 &4&\{A_0,A_1,A_3\}\\
9&2&3&4 &4 &4 &4 &4 &4 &4 &4 &4 &4 &4 &4&\{A_0,A_1,A_2\}\\\hline
\vdots&\vdots&\vdots& &\vdots& & &\vdots& & &\vdots& & &\vdots& &\vdots
\end{array}\\
\mbox{{\bf Cube Positions $\chi(i,j,x,p)$\  ($k=4$, $s=3$)}}
\end{array}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6885}%
$P^{(d+r)}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6887}%
$r\geq 2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6901}%
$Q^{x,p}_i=Q^{y,q}_i$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6903}%
$\langle P^{(d)},P^{(d+r)}\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6907}%
$r>1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6911}%
$j\not=i$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6913}%
$P_{i}^{(d+r)}\cap{\mathcal R}^{\{i,j\}}=P_{i}^{(d)}\cap{\mathcal R}^{\{i,j\}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6917}%
$P_{i}^{(d+r)}=P_{i}^{(d)}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6919}%
$d=i=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6921}%
$P_{0}^{(1)}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6923}%
$Q^{x}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6925}%
$\chi(0,j,x-1,k-1)=p$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6927}%
$\chi(0,j,x,k-1)=p+k-2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6931}%
$\chi(0,j,x-1,k-1)=\chi(0,j,x,0)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6933}%
$\chi(0,j,x,j)=\chi(0,j,x,j-1)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6935}%
$P_{0}^{(1)}=Q_{0}^{0,k-1}\not=Q_{0}^{x,p}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6939}%
$0<p<k-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3736}%
\begin{displaymath}
\chi(0,1,x,p)=\chi(0,2,x,p)=\ldots=\chi(0,k-1,x,p)
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6941}%
$p=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6943}%
$p=k-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6947}%
$qk$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6951}%
$q(k-2)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3737}%
\begin{displaymath}
\min\{\mbox{ $r>1$\  : $P_{0}^{(1)}=P_{0}^{(1+r)}$\  }\}\ =\ 
\min\{\mbox{ $qk$\  : $q(k-2)$\  is a multiple of $2^s$}\}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6967}%
$q=2^s$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6969}%
$r=k2^s$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6973}%
$z2^l+2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6975}%
$z$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6981}%
$l\geq s$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6983}%
$2^{s-l}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6985}%
$l\leq s$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6987}%
$r=k$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6989}%
$r=z2^s+2^{s-l+1}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6995}%
$k=1\times2^1+2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6997}%
$r=2^3+2^{3-1+1}=16$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6999}%
$P_{0}^{(1)}=P_{0}^{(17)}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3738}%
\begin{displaymath}
r\geq\left\{
{
\begin{array}{lcl}
k2^s&\mbox{ if }&\mbox{$k$\  is odd}\\
k&\mbox{ if }&\mbox{$k=z2^l+2$, $z$\  is odd, and $l\geq s$}\\
2^s&\mbox{ if }&\mbox{$k=z2^l+2$, $z$\  is odd and $l\leq s$}
\end{array}
}
\right.
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7015}%
$r\geq 2^s$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7017}%
$k\geq2^s$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3740}%
\begin{displaymath}
2^{m/\mbox{{\scriptsize$\left({\begin{array}{c}k\\2\end{array}}\right)$}}}\ -1\ =\ 
2^{\frac{2m}{k(k-1)}}~-~1
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7041}%
${\mathcal U}=\langle u_0,\ldots,u_{k-1}\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3741}%
\begin{displaymath}
\Delta=\langle Q^{0,k-1},Q^{1,0},Q^{1,1},\ldots,Q^{1,k-1},\ldots,Q^{x,p},\ldots,Q^{r,k-1}\rangle 
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7055}%
$rk\geq2^s$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7065}%
$Q_{i}^{x,p}\cap{\mathcal R}^{\{i,j\}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7067}%
${\mathcal S}^{(v)}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7069}%
${\mathcal R}^{\{i,j\}}\subseteq Q_{i}^{0,k}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7071}%
${\mathcal S}^{(w)}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7073}%
${\mathcal R}^{\{i,j\}}\subseteq Q_{j}^{0,k-1}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7081}%
$\forall~x,p\ \chi(i,j,x,p)=\chi(j,i,x,p)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7089}%
$\chi(i,j,y,q)\leq \chi(i,j,x,p)+1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7091}%
$q\not\in\{i,j\}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7093}%
$\forall~i',j'$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7095}%
$0\leq i',j'\leq k-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7097}%
$\chi(i,j,x,k-1)=\chi(i',j',x,k-1)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7105}%
$Q^{x-1,0}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7111}%
$S\subseteq{\mathcal R}_m$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3742}%
\begin{displaymath}
u_i(S)=
\left\{
{
\begin{array}{ll}
\sum_{j\not=i}~\chi(i,j,x,p)&\mbox{ if }S=Q_{i}^{x,p}\mbox{ for some $0\leq x\leq r$, $0\leq p\leq k-1$}\\
-2^{km}&\mbox{ otherwise}
\end{array}
}
\right.
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7121}%
$\langle Q^{0,k-1},Q^{r,k-1}\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7129}%
$|{\mathcal A}^\delta|=k-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7133}%
${\mathcal A}^\delta={\mathcal A}\setminus \{A_q\}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7135}%
$Q_{i}^{y,q}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7145}%
$\sigma_u(Q^{y,q})>\sigma_u(Q^{x,p})$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7147}%
$\langle Q^{x,p},Q^{y,q}\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7155}%
$\delta=\langle Q^{x,p},P\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7165}%
$P_i\subseteq\cup_{j\not=i}{\mathcal R}^{\{i,j\}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7169}%
$\langle(x_0,p_0),\ldots,(x_{k-1},p_{k-1})\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7171}%
$P_i=Q_{i}^{x_{i},p_{i}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7175}%
$u_i(P_i)=-2^{km}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7177}%
$\sigma_u(P)<0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7185}%
$P\not=Q^{y,q}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7187}%
$Q_{i}^{{x_i},p_{i}}\not=Q_{i}^{y,q}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7189}%
$Q_{i}^{{x_i},p_{i}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7197}%
$P_i=Q_{i}^{{x_i},p_{i}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7205}%
$(y_i,p_i)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7209}%
$Q_{i}^{x,p}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7261}%
$\Phi_k(\delta)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7293}%
${\mathcal U}=\langle u_1,u_2,\ldots,u_k\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3745}%
\begin{displaymath}
\begin{array}{lcl}
\Delta_m&\mbox{ $=$\  }&\langle P^{(1)},P^{(2)},\ldots,P^{(r)},\ldots,P^{(t)}\rangle \\
        &\mbox{ $=$\  }&\langle Q^{0,k-1},Q^{1,0},\ldots,Q^{x,p},\ldots,Q^{z,r}\rangle 
\end{array}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7307}%
$t\geq 2^{\mbox{{\scriptsize$\lfloor{2m/k(k-1)}$}}}-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3746}%
\begin{displaymath}
u_i(S)=\left\{
{
\begin{array}{lcl}
-1&\mbox{ if }&S\not=Q_{i}^{x,p}\mbox{ for any $Q^{x,p}\in\Delta_m$}\\
xk^2+k-i&\mbox{ if }&S=Q_{i}^{x,k-1}\\
(x-1)k^2+k+p&\mbox{ if }&\mbox{$S=Q_{0}^{x,p}$, $p<k-1$\  and $i=0$}\\
(x-1)k^2+k-i+p+1&\mbox{ if }&\mbox{$S=Q_{i}^{x,p}$, $p<i-1$\  and $i\not=0$.}\\
xk^2+1&\mbox{ if }&\mbox{$S=Q_{i}^{x,i-1}=Q_{i}^{x,i}$\  and $i\not=0$.}\\
xk^2+1+p-i&\mbox{ if }&\mbox{$S=Q_{i}^{x,p}$, $p>i$\  and $i\not=0$}
\end{array}
}
\right.
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7359}%
$\langle Q^{0,k-1},Q^{z,r}\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3747}%
\begin{displaymath}
\min_{0\leq i\leq k-1}~~\{u_i(Q_{i}^{z,r})\}~>~1~=~\min_{0\leq i\leq k-1}~~\{u_{i}(Q_{i}^{0,k-1})\}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3748}%
\begin{displaymath}
\min_{0\leq i\leq k-1}~~\{u_i(Q_{i}^{x,p})\}~\not=~u_q(Q_{q}^{x,p})
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7367}%
$A_q\not\in{\mathcal A}^\delta$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7369}%
$u_i(Q_{i}^{y,q})>u_i(Q_{i}^{x,p})$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7371}%
$q=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7375}%
$y=x+1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7377}%
$q>0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7379}%
$p=q-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7381}%
$u_0(Q_{0}^{x,k-1})=xk^2+k$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3749}%
\begin{displaymath}
\min\{u_i(Q_{i}^{x,k-1})\}=xk^2+1=u_{k-1}(Q_{k-1}^{x,k-1})
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7383}%
$\langle Q^{x,k-1},Q^{x+1,0}\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7387}%
$u_q(Q_{q}^{x,q-1})=xk^2+1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3750}%
\begin{displaymath}
\begin{array}{lcl}
\min\{u_i(Q_{i}^{x,q-1})\}&\mbox{ $\leq$\  }&u_0(Q_{0}^{x,q-1})\\
&\mbox{ $=$\  }&(x-1)k^2+k+q-1\\
&\mbox{ $<$\  }&xk^2-(k^2-2k+1)\\
&\mbox{ $=$\  }&xk^2-(k-1)^2\\
&\mbox{ $<$\  }&xk^2+1\\
&\mbox{ $=$\  }&u_{q}(Q_{q}^{x,q-1})
\end{array}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7401}%
$\langle Q^{x,q-1},Q^{x,q}\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7415}%
$\rho^{\max}_{{\rm mono}}(n,m,\Phi_k,\Psi)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7427}%
$2s$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7429}%
$\alpha\subseteq{\mathcal R}_s$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7431}%
$\overline{\alpha}\subseteq{\mathcal T}_s$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7433}%
$multi(\Delta)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7451}%
$\{i,j\}\subseteq{\mathcal A}^\delta$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7457}%
${\mathcal T}^{\{i,j\}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3751}%
\begin{displaymath}
\alpha^{(d)}_{i,0}~\alpha^{(d)}_{i,1}~\cdots~\alpha^{(d)}_{i,i-1}~\alpha^{(d)}_{i,i+1}~\cdots~
\alpha^{(d)}_{i,k-1}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7463}%
$\alpha_{i,j}^{(d)}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7467}%
$R^{\{i,j\}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3752}%
\begin{displaymath}
multi(\Delta)~=~\langle C^{(1)},C^{(2)},\ldots,C^{(t)}\rangle 
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7477}%
${\mathcal R}_m\cup{\mathcal T}_m$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7479}%
$C^{(d)}_{i}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3753}%
\begin{displaymath}
\beta^{(d)}_{i,0}~\beta^{(d)}_{i,1}~\cdots~\beta^{(d)}_{i,i-1}~\beta^{(d)}_{i,i+1}~\cdots~
\beta^{(d)}_{i,k-1}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7481}%
$\beta_{i,j}^{(d)}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7483}%
${\mathcal R}^{\{i,j\}}\cup{\mathcal T}^{\{i,j\}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3754}%
\begin{displaymath}
\beta^{(d)}_{i,j}~~=~~\alpha_{i,j}^{(d)}~\overline{\alpha_{i,j}^{(d)}}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7491}%
$\overline{\alpha_{i,j}^{(d)}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7495}%
$C^{(d)}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7501}%
$|C^{(d)}_{i}|=(k-1)s$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7505}%
$(k-1)s$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7507}%
$S\subseteq{\mathcal R}_m\cup{\mathcal T}_m$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7509}%
$|S|<(k-1)s$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7511}%
$u_i(S)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7513}%
$|S|>(k-1)s$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7519}%
$\langle C^{(d)},C^{(d+1)}\rangle $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7543}%
$m\geq\mbox{{\scriptsize$2\left({\begin{array}{c}k\\2\end{array}}\right)$}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7547}%
$u\in{\mathcal U}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7571}%
$n^m - (n-1)m$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7581}%
$|{\mathcal A}|=2$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7601}%
$S\subseteq{\mathcal R}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7603}%
$u(S)=\sum_{r\in S} u(r)$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7609}%
$0-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7617}%
$\rho^{\max}_{{\rm add}}(n,m,\Phi,\Psi)$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7619}%
$\rho^{\max}_{{\rm0-1}}(n,m,\Phi,\Psi)$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7625}%
$\Phi_1(P,Q)$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7631}%
$\Phi_2(P,Q)$%
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{\newpage\clearpage
\lthtmldisplayA{displaymath3756}%
\begin{displaymath}
\begin{array}{lclcr}
\rho^{\max}_{{\rm add}}(n,m,\Phi_1,\Psi)&\mbox{ $=$\  }&m&\mbox{ $~$\  }&\shortcite[Theorem~9]{Endriss-et-al:2004}\\
\rho^{\max}_{{\rm0-1}}(n,m,\Phi_2,\Psi)&\mbox{ $=$\  }&m&\mbox{ $~$\  }&\shortcite[Theorem~11]{Endriss-et-al:2004}
\end{array}
\end{displaymath}%
\lthtmldisplayZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7657}%
$\Omega(2^m)$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7659}%
$\Omega(2^{m/2})$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7685}%
$\Omega(2^{2m/k^2})$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7699}%
$m^p$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7717}%
$O(m)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7721}%
$\langle X,Y,Z\rangle $%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7723}%
$u_1(X)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7725}%
$Y\cup Z$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7729}%
$A_3$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7761}%
$(\gamma(m),O)$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7765}%
$\gamma(m)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7769}%
$\gamma(m)=0$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7771}%
$(0,O)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7773}%
$\gamma(m)=m$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7775}%
$(m,O)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7777}%
$\gamma(m)=o(m)$%
\lthtmlinlinemathZ
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\end{document}
