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%\principaladviser{Gregory J. E. Rawlins, Ph.D.}
\firstreader{Christopher T. Haynes, Ph.D.}
\secondreader{Douglas R. Hofstadter, Ph.D.}
\externalreader{Esther Thelen, Ph.D.}
\submitdate{August 4, 1993}
\submitmonth{August 1993}
\copyrightyear{1993}
\title{GENETIC ALGORITHMS AS A COMPUTATIONAL TOOL FOR DESIGN}
\author{Sushil J. Louis}
\setcounter{secnumdepth}{5}















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\newtheorem {Pf}{Proof}

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\samepage \begin{figure}[htp]
\centerline{
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	\psfig{figure=figures/scales.ps,height=3.0in,width=4.0in}
}
}
 
\label{scales}
\end{figure}
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{\newpage
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\samepage \begin{figure}[htp]
\centerline{
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}
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\label{sieve}
\end{figure}
}

{\newpage
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\samepage \begin{equation}\label{st}
        m(h,t+1) \geq \frac{m(h,t) f(h)}{\overline{f}_t} \left[1 - P_c
\frac{\delta(h)}{l - 1} - O(h) P_m \right]
\end{equation}
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{\label{3pga}} 
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\samepage \begin{tabular}{|llllllll|} \hline {\label{aptable}}
	& \multicolumn{2}{c}{\sf Length}& \multicolumn{2}{c} {\sf Width}
	& \multicolumn{2}{c} {\sf Area}	&  \\  
Room	& Min	& Max	& Min	& Max	& Min	& Max	& Proportion \\  \hline
Living	& 8	& 20	& 8	& 20	& 120	& 300	& 1.5 	\\ 
Kitchen	& 6 	& 18	& 6	& 18	& 50	& 120	& any	\\ 
Bath	& 5.5	& 5.5	& 8.5	& 8.5	&	&	& 	\\ 
Hall	& 5.5	& 5.5	& 3.5	& 6	& 19	& 72	& any	\\ 
Bed1	& 10	& 17	& 10	& 17	& 100	& 180	& 1.5	\\ 
Bed2	& 9	& 20	& 9 	& 20	& 100	& 180	& 1.5	\\ 
Bed3	& 8 	& 18	& 8	& 18	& 100	& 180	& 1.5	\\  \hline
\end{tabular}
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{\newpage
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\samepage \begin{figure}[hpt]
\centerline{
        \psfig{figure=./figures/apt.ps,height=3.0in,width=4.0in}}

\label{apt}
\end{figure}
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{\newpage
\clearpage
\samepage \begin{tabular}{|lll|} \hline
{\sf Room} 	& {\sf Length/Width}	& {\sf Label} \\  \hline
Living		& length		& $X_1$ \\ 
Living		& width			& $X_2$ \\ 
Kitchen		& length		& $X_3$ \\ 
Kitchen		& width			& $X_4$ \\ 
Bed1		& length		& $X_5$ \\ 
Bed2		& length		& $X_6$ \\ 
Bed2		& width			& $X_7$ \\ 
Bed3 		& width			& $X_8$ \\  \hline
\end{tabular}
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{\newpage
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\samepage \begin{displaymath}\begin{array}{r}
	F = X_1 X_2 + 2 X_3 X_4 + 93.5 \;\mbox{(cost of bath)} \; + \\ 
		5.5 (X_2 - 8.5) + X_4 X_5 + X_6 X_7 + \\ 
		 X_6 X_8
\end{array}
\end{displaymath}
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{\newpage
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\samepage \begin{displaymath}\begin{array}{c}
\mbox{Living}
\end{array}
\;
\;
\begin{array}{lcl}
  X_1, X_2 & \leq & 20 \\ 
  X_1, X_2 & \geq & 8 \\ 
  X_1 X_2  & \leq & 300 \\ 
  X_1 X_2  & \geq & 120 \\ 
  X_1/X_2  & \leq & 1.5 \\ 
\end{array}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\begin{array}{c}
\mbox{Kitchen}
\end{array}
\;
\;
\begin{array}{lcl}
  X_3, X_4 & \leq & 18 \\ 
  X_1, X_2 & \geq & 6\\ 
  X_1 X_2  & \leq & 120 \\ 
  X_1 X_2  & \geq & 50 \\ 
\end{array}
\end{displaymath}
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{\newpage
\clearpage
\samepage \begin{displaymath}\begin{array}{c}
\mbox{Hall}
\end{array}
\;
\;
\begin{array}{lcl}
  5.5 (X_2 - 8.5)  & \leq & 72 \\ 
  5.5 (X_2 - 8.5)  & \geq & 19 \\ 
\end{array}
\end{displaymath}
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{\newpage
\clearpage
\samepage \begin{displaymath}\begin{array}{c}
\mbox{Bed1}
\end{array}
\;
\;
\begin{array}{lcl}
  X_5, X_4 & \leq & 17 \\ 
  X_5, X_4 & \geq & 10 \\ 
  X_5 X_4  & \leq & 180 \\ 
  X_5 X_4  & \geq & 100 \\ 
  X_5/X_4  & \leq & 1.5 \\ 
\end{array}
\end{displaymath}
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{\newpage
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\samepage \begin{displaymath}\begin{array}{c}
\mbox{Bed2}
\end{array}
\;
\;
\begin{array}{lcl}
  X_6, X_7 & \leq & 20 \\ 
  X_6, X_7 & \geq & 9\\ 
  X_6 X_7  & \leq & 180 \\ 
  X_6 X_7  & \geq & 100 \\ 
  X_6/X_7  & \leq & 1.5 \\ 
\end{array}
\end{displaymath}
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{\newpage
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\samepage \begin{displaymath}\begin{array}{c}
\mbox{Bed3}
\end{array}
\;
\;
\begin{array}{lcl}
  X_8, X_6 & \leq & 18 \\ 
  X_8, X_6 & \geq & 8\\ 
  X_8 X_6  & \leq & 180 \\ 
  X_8 X_6  & \geq & 100 \\ 
  X_6/X_8  & \leq & 1.5 \\ 
\end{array}
\end{displaymath}
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{\newpage
\clearpage
\samepage \begin{displaymath}\begin{array}{c}
\mbox{Doorways in hall}
\end{array}
\;
\;
\begin{array}{lcr}
 X_7 - 8.5 & \geq & 3 \\ 
 X_8 - X_4 & \geq & 3 
\end{array}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\begin{array}{c}
\mbox{Wall alignments}
\end{array}
\;
\;
\begin{array}{ccccc}
 X_1 + 5.5 & = & X_3 + X_5 & & \\ 
 8.5 + (X_2 - 8.5) + X_4 & = & X_7 + X_8 & = & X_2 + X_4 \\ 
\end{array}
\end{displaymath}
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{\newpage
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\samepage \begin{displaymath}X_1 = \mbox{Min}(X_1) + \frac{100.0 \times 12.0}{y_1} 
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{\newpage
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\samepage \begin{displaymath}\begin{array}{r}
	F = X_1 X_2 + 2 X_3 X_4 + 93.5 \;\mbox{(cost of bath)} \; + \\ 
		5.5 (X_2 - 8.5) + X_4 X_5 + X_6 X_7 + \\ 
		X_6 X_8 + \mbox{penalties}
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{\newpage
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{\newpage
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\samepage \begin{figure}[hpt]
\centerline{ 	
\fbox{
{\psfig{figure=figures/platsch.ps,height=3.0in,width=4.0in}}} 	
}

\label{plateau}
\end{figure}
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\samepage \begin{figure}[hpb]
\centerline{ 	
\fbox{
	{\psfig{figure=figures/plat.ps,height=3.0in,width=4.0in}}} 	
}

\label{liner}
\end{figure}
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{\newpage
\clearpage
\samepage \begin{equation}\label{func}
\begin{array}{ll}
 	\mbox{if} \; \;\\  
	& (x_1 + x_2) \geq (l - p) \\ 
	\mbox{then} & \\ 
 	& \mbox{fitness}(\mbox{string}) = (x_1 x_2)^2 \\ 
	\mbox{else} & \\ 
 	& \mbox{fitness}(\mbox{string}) = \;\mid 2 x_2 - x_1 \mid + 
					\mid 2 x_1 - x_2\mid \\ 
\end{array}
\end{equation}
}

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\samepage \begin{displaymath}(p_m)^{l/2} (1 - p_m)^{l/2}
\end{displaymath}
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{\newpage
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\samepage \setbox\sizebox=\hbox{$(d_1, d_2, \ldots, d_n)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2433: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
\clearpage
\samepage \begin{displaymath}\forall_i(c_i \geq d_i) \; \mbox{AND} \; \exists_i (c_i > d_i)
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{figure}[hp]
\centerline{ 	
\fbox{
	{\psfig{figure=figures/pareto.ps,height=3.2in,width=4.0in}}} 	
}

\label{popt}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{equation}\label{p1}
\begin{array}{ll}
 	\mbox{if} \; \;\\  & (x_1 + x_2) \geq (l - p) \\ 
	\mbox{then} & \\ 
 	& c_1(\mbox{string}) = x_1 x_2 \\ 
	\mbox{else} & \\ 
 	& c_1(\mbox{string}) =\; \mid 2 x_1 - x_2 \mid\\ 
\end{array}
\end{equation}
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{\newpage
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{\newpage
\clearpage
\samepage \begin{equation}\label{p2}
\begin{array}{ll}
 	\mbox{if} \; \;\\  & (x_1 + x_2) \geq (l - p) \\ 
	\mbox{then} & \\ 
 	& c_2(\mbox{string}) = x_1 x_2 \\ 
	\mbox{else} & \\ 
 	& c_2(\mbox{string}) =\; \mid 2 x_2 - x_1 \mid\\ 
\end{array}
\end{equation}
}

\stepcounter{section}
{\newpage
\clearpage
\samepage \begin{figure}[htp]
\centerline{
{\psfig{figure=figures/l30-b2.ps,height=3.0in,width=4.0in}}}

\label{graph30b2}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{figure}[htb]
\centerline{
{\psfig{figure=figures/l30-b4.ps,height=3.0in,width=4.0in}}}

\label{graph30b4}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{table}\begin{center}
\begin{tabular}{|c|c c c c| } \hline & & & &\\  
Length	& Width	& PGA	& CGA & PSHC \\  
& & & & \\  \hline
 & & & & \\  
$20$ 	& 1	& 100 \% 	& 55 \% & 45 \% \\  
& & & & \\  
$30$	& 2	& 100 \% 	& 27 \% 	& 18 \%\\  
& & & &\\  
$40$	& 3	& 73 \%		& 0 \%		& 0 \% \\  
& & & &\\  
$50$	& 4	& 45 \%		& 0\%		& 0\% \\  
& & & & \\  \hline
\end{tabular}
\end{center}

\end{table}
}

{\newpage
\clearpage
\samepage \begin{table}\begin{center}
\begin{tabular}{|c|c c c c| } \hline & & & &\\  
Length	& Width	& PGA	& CGA & PSHC \\  
& & & & \\  \hline & & & & \\  
$20$ 	& 2	& 100 \% 	& 91 \% & 100 \% \\  
& & & &\\  
$30$	& 4	& 100 \% 	& 82 \% 	& 55 \% \\  
& & & &\\  
$40$	& 6	& 100 \%	& 64 \%		& 0 \% \\  
& & & &\\  
$50$	& 8	& 82 \%		& 0\%		& 0\% \\  
& & & &\\  \hline
\end{tabular}
\end{center}

\end{table}
}

{\newpage
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}

{\newpage
\clearpage
\samepage \begin{table}\begin{center}
\begin{tabular}{|c|c l l|}  \hline & & & \\  
Length	& Width	& PGA vs CGA	& CGA vs PSHC \\  
& & Significant? & Significant?  \\  \hline
 & & &  \\  
$20$ 	& 2	& No 	& No \% \\  
& & &  \\  
$30$	& 4	& Yes ($75\%$) 	& Yes ($75\%$)     \\  
& & & \\  
$40$	& 6	& Yes ($95\%$)	& Yes ($95\%$) \\  
& & & \\  
$50$	& 8	& Yes ($99\%$)	& No \\  
& & &  \\  \hline
\end{tabular}
\end{center}

\end{table}
}

\stepcounter{section}
{\newpage
\clearpage
\samepage \begin{figure}[hp]
\centerline{ 	
\fbox{
	{\psfig{figure=figures/pl2.ps,height=3.0in,width=4.0in}}} 	
}

\label{pl2.ps}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{figure}[hp]
\centerline{ 	
\fbox{
	{\psfig{figure=figures/shc.ps,height=3.5in,width=4.0in}}} 	
}

\label{shc_plot}
\end{figure}
}

{\newpage
\clearpage
\samepage \( \frac{1}{2l} \)
}

\stepcounter{section}
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{\newpage
\clearpage
\samepage \begin{figure}\centerline{
\mbox{\psfig{figure=figures/inv.ps,height=1.0in,width=3.0in}}}

{\label{inv}} 
\end{figure}
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{\newpage
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\fbox{\psfig{figure=figures/ax.ps,height=2.0in,width=5.0in}}}
 
{\label{axover}} 
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\samepage \begin{figure}\centerline{
\begin{tabular}{|l|r|} \hline
	{\bf Case}  & {\bf Rule} \\  \hline
	Both good & $MF_{gg}$ \\  \hline
	Both bad & $MF_{bb}$ \\  \hline
	Both average & $MF_{aa}$\\  \hline
	One good, one bad & $MF_{gb}$ \\  \hline
	One good, one average & $MF_{ga}$ \\  \hline
	One average, one bad & $MF_{ab}$ \\  \hline
\end{tabular}
}
 
{\label{nine2}}
\end{figure}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$MF_{gg}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2759: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{figure}\centerline{
\fbox{\psfig{figure=figures/mf6.ps,height=2.0in,width=5.0in}}}

{\label{mf6a}} 
\end{figure}
}

\stepcounter{section}
{\newpage
\clearpage
\samepage \begin{figure}\centerline{
	\fbox{\psfig{figure=figures/mp12d.ps,height=2.5in,width=4.0in}}}

{\label{map12d}} 
\end{figure}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$S_{i,j}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2833: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$S_{i,j-1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2835: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$S_{i+1,j-1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2837: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$S_{i-1,j-1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2839: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{figure}\centerline{
	\fbox{\psfig{figure=figures/gate_in.ps,height=2.5in,width=4.0in}}}

{\label{gate_input}} 
\end{figure}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$S_{i,0}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2841: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$2^n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2845: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$3*2^4$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2865: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{figure}\centerline{
\mbox{\psfig{figure=figures/2b_gaterow_max.ps,height=3.2in,width=3.2in}}}

{\label{2add_max}} 
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{figure}\centerline{
\mbox{\psfig{figure=figures/2b_gaterow_avg.ps,height=3.2in,width=3.2in}}}

{\label{2add_avg}} 
\end{figure}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$S_{33}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2869: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{figure}\centerline{
\mbox{\psfig{figure=figures/4bit_dga_addercirc.ps,height=2.5in,width=4.0in}}}
 
{\label{2bdga_adder}} 
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{figure}\centerline{
\mbox{\psfig{figure=figures/4bit_cga_addercirc.ps,height=2.5in,width=4.0in}}}

{\label{2bcga_adder}} 
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{figure}[hp]
\centerline{
\mbox{\psfig{figure=figures/par_bbh_map.ps,height=5.0in,width=5.5in}}}
 
{\label{Parity_checker}}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{figure}[htp]
\centerline{
\mbox{\psfig{figure=figures/4bit_rest_max.ps,height=3.2in,width=3.2in}}}

{\label{4bit_mx}} 
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{figure}[hbp]
\centerline{
\mbox{\psfig{figure=figures/4bit_rest_avg.ps,height=3.2in,width=3.2in}}}

{\label{4bit_avg}} 
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{figure}[htp]
\centerline{
\mbox{\psfig{figure=figures/new/5bitpar_max.ps,height=3.2in,width=3.2in}}}

{\label{5bit_mx}} 
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{figure}[hbp]
\centerline{
\mbox{\psfig{figure=figures/new/5bitpar.ps,height=3.2in,width=3.2in}}}

{\label{5bit_avg}} 
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{figure}[htp]
\centerline{
\mbox{\psfig{figure=figures/new/6bitpar_max.ps,height=3.2in,width=3.2in}}}

{\label{6bitmax}} 
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{figure}[hbp]
\centerline{
\mbox{\psfig{figure=figures/new/6bitpar.ps,height=3.2in,width=3.2in}}}

{\label{6bitavg}} 
\end{figure}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$95\%$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2879: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{figure}[thp]
\center{
\begin{minipage}[t]{3.5in}
\mbox{\psfig{figure=figures/new/5bitpar_max.ps,height=3.2in,width=3.2in}}
\end{minipage}
\begin{minipage}[t]{3.5in}
\mbox{\psfig{figure=figures/new/6bitpar_scl_max.ps,height=3.2in,width=3.2in}}
\end{minipage}
}
 
{\label{6bitpar_scl_max}} 
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{figure}[htp]
\centerline{
\mbox{\psfig{figure=figures/5b_par_3cmp_max.ps,height=3.2in,width=3.2in}}}
 
{\label{5bit_sel_max}} 
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{figure}[hbp]
\centerline{
\mbox{\psfig{figure=figures/5b_par_3cmp_avg.ps,height=3.2in,width=3.2in}}}
 
{\label{5bit_sel_avg}} 
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{figure}[htp]
\centerline{
\mbox{\psfig{figure=figures/4bit_dga_parcircuit.ps,height=2.5in,width=4.0in}}}
 
{\label{4dga}} 
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{figure}[hbp]
\centerline{
\mbox{\psfig{figure=figures/4bit_cga_parcircuit.ps,height=2.5in,width=4.0in}}}
 
{\label{4cga}} 
\end{figure}
}

\stepcounter{section}
\stepcounter{chapter}
\stepcounter{section}
\stepcounter{section}
\stepcounter{section}
{\newpage
\clearpage
\samepage \begin{figure}[h]
\centerline{
{\psfig{figure=figures/sys.ps,height=4in,width=5in}}}

\label{system}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{figure}[th]
\centerline{
\fbox{
	{\psfig{figure=figures/ctree.ps,height=3.0in,width=5in}}}
}

\label{simpletree}
\end{figure}
}

\stepcounter{section}
\stepcounter{section}
\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{figure}[hbp]
\centerline{
\fbox{\psfig{figure=figures/tindex.ps,height=3.0in,width=5in}}}
 
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{figure}[htp]
\centerline{
\fbox{\psfig{figure=figures/tzoom.ps,height=2.5in,width=5in}}}
 
\end{figure}
}

\stepcounter{subsection}
\stepcounter{section}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$2^{80}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2981: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{figure}[hpt]
\centerline{
\mbox{\psfig{figure=figures/cbrpar.ps,height=3.0in,width=4.5in}}}

\label{cbrpar}
\end{figure}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$50^{th}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3015: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{section}
{\newpage
\clearpage
\samepage \begin{table}[hpt]
\centerline{
\begin{tabular}{|l|l|}
 		           \hline 
$f(x) = c$		& c is a constant (control) 	\\ 
$f(x) = ones(x)$ 	& number of ones in genotype\\ 
$f(x) = x$		& signed, genotype length 10	\\ 
$f(x) = x^{2}$		& signed, g-length 10		\\ 	
$f(x) = x^{3}$ 		& signed, g-length 10		\\ 
$f(x,y) = x^{y}$	& signed, g-length 20		\\ 
$f(x) = 2^{x}$		& all positive, g-length 10		\\ 
$f(x) = log(x)$ 	& all positive, g-length 10		\\ 
$f(x,y) = {x^{2}-y^{2}}$& all positive, g-length 20 	\\  \hline
$f(x_i)=\sum_i^3 x_{i}^2$& DeJong F1, $(-5.12..5.12)$\\ 
$f(x_i)= 100(x_{1}^2 - x_{2})^2$ & DeJong F2 $(-2.048..2.048)$\\ 
\hspace{1.5cm}$+ \sum_i^2(1 - x_{i})^2$   &    \\ 
$f(x_i)= \sum_i^5 \mbox{integer}(x_i)$ & DeJong F3 $(-5.12..5.12)$\\  \hline
\multicolumn{2}{|c|}{A pair of deceptive
functions.} \\  \hline 
\end{tabular}}

\label{functions}
\end{table}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$10\%$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3061: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{section}
\stepcounter{chapter}
\stepcounter{section}
\stepcounter{section}
{\newpage
\clearpage
\samepage \begin{displaymath}\frac{N(N - 1)}{2}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$h_0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3075: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}h_0 = \frac{l}{2}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$s_0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3077: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}s_0 = \frac{\sqrt{l}}{2} 
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\epsilon > 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3079: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{section}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\ldots$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3091: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{Tm}Traditional crossover operators do not change the average hamming
distance of a given population.
\end{Tm}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\{a,b\}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3099: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$h_{i,t}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3107: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}h_t = \sum_{i = 1}^{l} h_{i,t}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$h_{t+1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3111: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}h_{t+1} = \sum_{i = 1}^{l} h_{i,t+1}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}h_{i,t} = h_{i,t+1}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}h_{t+1} = h_t
\end{displaymath}
}

\stepcounter{section}
{\newpage
\clearpage
\samepage \begin{displaymath}f(x_i) = \mbox{constant} \; 
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$p_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3129: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}\left(
        \begin{array}{c}
                N \\ 
                k
        \end{array}
\right)
p_i^k
(1 - p_i)^{N-k}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \(\Phi(p,t) \)
}

{\newpage
\clearpage
\samepage \( 0 < p < 1 \)
}

{\newpage
\clearpage
\samepage \begin{displaymath}\Phi(p,t) = \frac{6p_0(1 - p_0)}{N} \; \left( 1 - \frac{2}{N} \right)^t
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}{\cal P}(t) = 1 - \Phi(p,t)
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}{\cal P}(t) = 1 - \frac{6p_0(1 - p_0)}{N} \; \left( 1 -
\frac{2}{N} \right)^t 
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{equation}\label{calptl}
        {\cal P}(t,l) = \left[1 - \frac{6p_0(1 - p_0)}{N} \; \left( 1 -
\frac{2}{N} \right)^t \right]^l
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$92\%$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3157: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$73\%$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3163: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{section}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$l_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3173: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}\mbox{Deceptive}(x_i) = \sum_{i = 1}^{n}
		\left\{ \begin{array}{ll}
			x_i & \mbox{if } \; x_i \neq 0 \\ 
			2^{l_i+2} &\mbox{if } \;  x_i = 0 \\ 
			\end{array}
		\right \}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\mbox{Dec1:} \;  \mbox{Deceptive}(x_1^{2},\; x_2^{8}) 
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\mbox{Dec2:} \; \mbox{Deceptive}(x_1^{2},\; x_2^{8},\; x_3^{5}, \; x_4^{5})
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{figure}[tph]
\centerline{
        \psfig{figure=./figures/dec1_m1_r1.ps,height=3.0in,width=5.0in}}

\end{figure}
}

{\newpage
\clearpage
\samepage \begin{figure}[hpt]
\centerline{
\psfig{figure=figures/dec1_l.ps,width=4.0in,height=3.4in}
}

\label{dec1T}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{figure}[hpt]
\centerline{
\psfig{figure=figures/dec2_l.ps,width=4.0in,height=3.4in}
}

\label{dec2T}
\end{figure}
}

\stepcounter{section}
\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{equation}\label{hm1}
        h_{t+1} = {\cal F}(h_t)
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$h_t$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3229: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\cal F$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3233: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\label{hamming_sum}
        h_t = \sum_{i = 0}^{l - 1} h_{i,t}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$m(a_i,t)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3255: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$m(b_i,t)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3257: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\label{Iham}
        h_{i,t} = \frac{m(a_i,t) m(b_i,t)}{ N (N - 1) / 2}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\label{schema_theorem}
        m(s,t+1) = \frac{m(s,t) f(s)}{\overline{f}_t} 
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{displaymath}h_{i,t+1} = \frac{m(a_i,t+1)  m(b_i,t+1)}{N (N - 1) / 2}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$h_{i,t+1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3283: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}h_{i,t+1} = \frac{m(a_i,t) f(a_i)}{\overline{f}_t} 
                    \frac{m(b_i,t) f(b_i)}{\overline{f}_t}
                    \frac{1}{N (N - 1) / 2}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{equation}\label{ham_rec}
        h_{i,t+1} = h_{i,t} \frac{f(a_i) f(b_i)}{\overline{f}^{2}_t}        
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\label{recurrence_solution}
        h_{i,t} = \left[f(a_i) f(b_i)\right]^t 
                  \frac{h_{i,0}}{{\displaystyle\prod_{k =
                  0}^{t}\overline{f}^{2}_k}} 
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\overline{f}_{i,t} = \frac{f(a_i) m(a_i,t) + f(b_i) m(b_i,t)}{N}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\overline{f}_{i,t}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3291: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\label{fitness_recurrence}
        \overline{f}_{t} = \frac{f(a_i) m(a_i,t) + f(b_i) m(b_i,t)}{N}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\overline{f}_{t} = \frac{1}{\overline{f}_{t-1}}
                                \frac{f(a_i)^2 m(a_i,t-1) + f(b_i)^2
                                m(b_i,t-1)}{N} 
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\overline{f}_{t-1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3295: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}\overline{f}_{t} \overline{f}_{t-1} = 
        \frac{ m(a_i,t-1) f(a_i)^2}{N} + \frac{m(b_i,t-1) f(b_i)^2}{N}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}m(a_i,t-1) = \frac{m(a_i,t-2) f(a_i)}{\overline{f}_{t-2}} 
 & \mbox{And} &
         m(b_i,t-1) = \frac{m(b_i,t-2) f(b_i)}{\overline{f}_{t-2}} 
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\overline{f}_{t} \overline{f}_{t-1} =
        \frac{m(a_i,t-2) f(a_i)^3}{N \overline{f}_{t-2}} + 
        \frac{m(b_i,t-2) f(b_i)^3}{N \overline{f}_{t-2}}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\overline{f}_{t-2}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3297: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}\overline{f}_{t} \overline{f}_{t-1} \overline{f}_{t-2} =
        \frac{m(a_i,t-2) f(a_i)^3}{N} +
        \frac{m(b_i,t-2) f(b_i)^3}{N}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\vdots$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3299: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\label{fit_prod}
        \prod_{k=0}^{t}\overline{f}_k = 
        \frac{m(a_i,0) f(a_i)^{t+1}}{N} + 
        \frac{m(b_i,0) f(b_i)^{t+1}}{N}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{displaymath}h_{i,t} = \left[f(a_i) f(b_i)\right]^t 
                \frac{h_{i,0}}{\left[\frac{m(a_i,0) f(a_i)^{t+1}}{N} + 
               \frac{m(b_i,0) f(b_i)^{t+1}}{N}\right]^2} 
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}m(a_i,0) = m(b_i,0) = N/2
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}h_{i,t} = \left[f(a_i) f(b_i)\right]^t 
                  \frac{4 h_{i,0} }{\left[ f(a_i)^{t+1} + f(b_i)^{t+1}\right]^2
                  }
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}h_{i,t} = \frac{4  \left[f(a_i) f(b_i)\right]^t}{\left[ f(a_i)^{t+1} +
        f(b_i)^{t+1}\right]^2} h_{i,0}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$h_{i,0}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3303: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}h_{i,t} = \frac{2  \left[f(a_i) f(b_i)\right]^t}{\left[ f(a_i)^{t+1} +
        f(b_i)^{t+1}\right]^2}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}h_t = \sum_{i = 0}^{l - 1} h_{i,t} = 2
        \sum_{i = 0}^{l - 1} \frac{\left[f(a_i) f(b_i)\right]^t}
        {\left[ f(a_i)^{t+1} + 
        f(b_i)^{t+1}\right]^2} 
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{equation}\label{heq}
        h_t =  2
        \sum_{i = 0}^{l - 1} \frac{\left[f(a_i) f(b_i)\right]^t}
        {\left[ f(a_i)^{t+1} + 
        f(b_i)^{t+1}\right]^2} 
\end{equation}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{displaymath}\begin{array}{c}
        m(a_i,t+1) = m(a_i,t)\left[1 - p_m\right] + m(b_i,t)p_m \\ 
        m(b_i,t+1) = m(b_i,t)\left[1 - p_m\right] + m(a_i,t)p_m
\end{array}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{equation}\label{mschemat}
\begin{array}{c}
        m(a_i,t+1) = \displaystyle{\frac{f(a_i) m(a_i,t)\left[1 - p_m\right] +
        f(b_i)m(b_i,t)p_m}{\overline{f}_t}   }  \\ 
        \mbox{\vspace{1cm}} \\ 
        m(b_i,t+1) = \displaystyle{\frac{f(b_i) m(b_i,t)\left[1 - p_m\right] +
        f(a_i)m(a_i,t)p_m}{\overline{f}_t}   }
\end{array}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\label{mrec_ham}
        h_{i,t} = \frac{m(a_i,t)  m(b_i,t)}{N (N - 1) / 2}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$m(a_i,0)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3355: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$m(b_i,t) = N - m(a_i,t)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3357: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}m(a_i,t) = \frac{
f(a_i) m(a_i,t-1) (1 - p_m) + f(b_i)p_m \left[N - m(a_i,t-1)\right]}
{\overline{f}_{t-1}}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}m(a_i,t) = 
\frac{m(a_i,t-1) \left[ f(a_i)(1-p_m) - f(b_i)p_m\right]}{\overline{f}_{t-1}}+ 
\frac{f(b_i)p_m N}{\overline{f}_{t-1}}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}m(a_i,t) = \frac{c\: m(a_i,t-1)}{\overline{f}_{t-1}} +
\frac{d}{\overline{f}_{t-1}} 
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\begin{array}{l}
c =  f(a_i)(1-p_m) - f(b_i)p_m \\ 
\mbox{\vspace{1cm}} \\ 
d = f(b_i)p_m N
\end{array}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}m(a_i,t) = \frac{c^t
m(a_i,0)}{\displaystyle{\prod_{k=0}^{t-1}\overline{f}_k}}\: +\: 
           \frac{d}{\displaystyle{\prod_{k=0}^{t-2}\overline{f}_k }}
           \sum_{i = 0}^{t-2} c^i \prod_{k=i+1}^{t-2} \overline{f}_k
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{equation}\label{masol}
m(a_i,t) = \frac{c^t
m(a_i,0)}{\displaystyle{\prod_{k=0}^{t-1}\overline{f}_k}}\: +\: 
           \frac{d
           \:\overline{f}_{t-1}}{\displaystyle{\prod_{k=0}^{t-1}\overline{f}_k 
           }} \sum_{i = 0}^{t-2} c^i \prod_{k=i+1}^{t-2} \overline{f}_k
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\begin{array}{l}
u =  f(b_i)(1-p_m) - f(a_i)p_m \\ 
\mbox{\vspace{1cm}} \\ 
w = f(a_i)p_m N
\end{array}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{equation}\label{mbsol}
m(b_i,t) = \frac{u^t
m(b_i,0)}{\displaystyle{\prod_{k=0}^{t-1}\overline{f}_k}}\: +\: 
           \frac{w \:
           \overline{f}_{t-1}}{\displaystyle{\prod_{k=0}^{t-1}\overline{f}_k 
           }} \sum_{i = 0}^{t-2} u^i \prod_{k=i+1}^{t-2} \overline{f}_k
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\label{mheq}
h_{i,t} = \frac{ \displaystyle{
\left[ c^t m(a_i,0) + d \: \overline{f}_{t-1} \sum_{i = 0}^{t-2} c^i
\prod_{k=i+1}^{t-2} 
\overline{f}_k \right] 
\left[ u^t m(b_i,0) + w \: \overline{f}_{t-1} \sum_{i = 0}^{t-2} u^i
\prod_{k=i+1}^{t-2} 
\overline{f}_k \right]  }
}{\frac{N (N-1)}{2} \displaystyle{\prod_{k=0}^{t-1}\overline{f}_k^2}}
\end{equation}
}

\stepcounter{section}
{\newpage
\clearpage
\samepage \begin{displaymath}f(a_i) =   \frac{m(a_i,t+1) \overline{f}_t}{m(a_i,t)}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$a_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3371: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\label{m2sch}
\begin{array}{c}
   \overline{f}_t m(a_i,t+1) = 
\displaystyle{f(a_i) m(a_i,t)\left[1 - p_m\right] + f(b_i)m(b_i,t)p_m}
        \mbox{\vspace{1cm}} \\ 
    \overline{f}_t m(b_i,t+1) = 
\displaystyle{f(b_i) m(b_i,t)\left[1 - p_m\right] + f(a_i)m(a_i,t)p_m}
\end{array}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$f(b_i)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3373: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}f(b_i) = \frac{\overline{f}_t m(b_i,t+1) - f(a_i) m(a_i,t)p_m}
		 {m(b_i,t)(1 - p_m)}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$f(a_i)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3375: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}\overline{f}_t m(a_i, t+1) = 
	f(a_i) m(a_i, t) (1 - p_m) + 
	\left[m(b_i,t) p_m\right] 
	\left[\frac{\overline{f}_t m(b_i,t+1) - f(a_i) m(a_i,t)p_m}
		 {m(b_i,t)(1 - p_m)}\right]
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\overline{f}_t m(a_i, t+1) = 
	f(a_i) m(a_i, t) (1 - p_m) + 
	\frac{\overline{f}_t m(b_i,t+1) p_m}{(1 - p_m)} - 
	\frac{f(a_i) m(a_i,t)p_m^2}{(1 - p_m)}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$(1 - p_m)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3377: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}\overline{f}_t m(a_i, t+1) (1 - p_m) - \overline{f}_t m(b_i,t+1) p_m =
	f(a_i) m(a_i, t) (1 - p_m)^2 - f(a_i) m(a_i,t)p_m^2
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{equation}\label{fai}
	f(a_i) = \frac{
	\overline{f}_t m(a_i, t+1) (1 - p_m) - \overline{f}_t m(b_i,t+1) p_m}
	{m(a_i, t) (1 - 2 p_m)}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\label{fbi}
	f(b_i) = \frac{
	\overline{f}_t m(b_i, t+1) (1 - p_m) - \overline{f}_t m(a_i,t+1) p_m}
	{m(b_i, t) (1 - 2 p_m)}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\label{mfit}
	\overline{f}_{t} = \frac{f(a_i)m(a_i,t) + f(b_i) m(b_i,t)}{N}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$b_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3397: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$ O(\log{N})$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3399: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{section}
\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{displaymath}h_{t+1} = a h_t
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{equation}\label{simpleh}
	h_t = a^t h_0 
\end{equation}
}

{\newpage
\clearpage
\samepage \(f(x_1, x_2) = 10, \)
}

{\newpage
\clearpage
\samepage \(50\)
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$F1 \ldots F5$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3417: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \( f(X) = |X| \; \; \)
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\lceil \log{30} \rceil = 5$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3435: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{table}% latex2html id marker 1465
\begin{center}
\begin{tabular}{|l|l l | c | c c |} \hline  Function & \multicolumn{2}{c|}{\em Observed} & \multicolumn{3}{c|}{\em
Predicted} \\  \cline{2-6}
	& $h_0$ &  $h_{50}$  & $h_{50}$ eqn~\ref{simpleh}  & $h_{50}$ lower &
$h_{50}$ upper \\  \hline
Flat    & $25$   & $4.6 \;(2.4)$  & $5.1 \;(2.2)$  & $4.3\; (1.3)$ & $9.3\;
(1.1)$ \\ 
F1	& $15$   & $2.1 \;(1.1)$  & $2.2\; (1.2)$  & $2.1\; (0.8)$ & $4.6\;
(1.3)$\\  
F2	& $12$   & $2.1 \;(0.9)$  & $2.5\; (1.5)$  & $1.7\; (1.0)$ & $4.3\;
(1.1)$ \\ 
F3	& $25$   & $3.5 \;(1.7)$  & $3.4\; (1.5)$  & $3.2\; (1.2)$ & $7.9\;
(2.0)$ \\ 
F4	& $120$  & $18.6\; (8.0)$ & $25.3 \;(11.5)$ & $17.4\; (6.5)$ &$42.2 \;
(5.7)$ \\ 
F5	& $17$   & $2.9\; (1.7)$  & $4.2\; (2.3)$  & $2.8\; (1.1)$ & $5.7\;
(1.0)$ \\ 
OneMaxI & $25$   & $3.7 \;(2.3)$  & $4.7 \;(3.1)$  & $4.2 \;(1.0)$ & $8.0\;
 (2.1)$ \\ 
OneMaxII & $100$  & $15.8\; (8.1)$ & $16.0\; (9.7)$ & $12.8 \; (5.4)$ &
$29.8\; (4.8)$ \\  \hline
\end{tabular}
\end{center}
 
\end{table}
}

{\newpage
\clearpage
\samepage \begin{equation}\label{lin_ham}
	h_{t+1} = a h_t + b
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{displaymath}h_t = a^t h_0 + b\frac{1 - a^t}{1 - a}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\epsilon = h_t = h_{t+1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3547: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\epsilon$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3549: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}\epsilon = \frac{b}{1 - a}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{table}\begin{center}
\begin{tabular}{|l|l c | l l | l l |} \hline  Function  & \multicolumn{2}{c|}{\em Observed} & \multicolumn{4}{c|}{\em
Predicted} \\  \cline{2-7} 
 	& $h_0$ & $h_{500}$ & $h$ from $25$ & $h$ from $45$ & $h$ lower & $h$
upper \\  \hline  
Flat    & $25$ & $11.2\; (1.9)$ & $14.4\; (6.0)^*$ &$11.4\; (1.5)$ &
$7.7 \;(0.9)$ & $11.2\; (1.3)$ \\  
F1 & $15$ & $5.8 \;(1.1)$ & $9.8 \;(10.2)$ & $6.4 \;(1.3)$ & $4.1\;
(0.5)$ & $6.1 (1.3)$\\  
F2 & $12$ & $4.3 \;(1.2)$ & $7.3 \;(2.4)$ & $5.7 \;(1.4)$ & $3.5\;
(0.9)$ & $5.4\; (1.0)$ \\ 
F3 & $25$ & $8.9 \;(1.7)$ & $8.9\;(3.7)^*$ & $9.1\;(2.1)$ &
$6.3\;(1.1)$ & $9.5 (1.3)$ \\ 
F4	& $120$& $48.9\; (4.7)$ & $64.3\;(14.0)^*$  &$49.3\;(12.2)^*$ &
$32.3 \;(3.9)$ & $51.4\;(4.7)$\\ 
F5 & $17$ & $6.8 \;(1.3)$ & $11.3\;(4.4)^*$ & $7.2\;(2.3)$ &
$4.8\;(0.7)$ & $6.8\;(1.3)$  \\  
One Max & $25$ & $9.2\; (1.7)$  & $10.3\; (4.0)^*$ &$9.0\;(2.2)$ & $7.3
\;(1.3)$ & $11.3 (1.7)$\\  
One Max & $100$& $45.7\; (5.3)$ & $51.5\; (11.4)$&$40.3\; (13.6)$ &
$25.7 \;(4.6)$ & $38.7 (4.5)$\\  \hline
\end{tabular}
\end{center}

\end{table}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{table}\begin{center}
\begin{tabular}{|l|l l | l l | l l|} \hline  
{\footnotesize Function}  & \multicolumn{2}{c|}{\em {\footnotesize Observed}} &
\multicolumn{4}{c|}{\em {\footnotesize Predicted}} \\  \cline{2-7} 

  & ${\scriptstyle F_0}$ & ${\scriptstyle F_{500}}$  & ${\scriptstyle F/10}$ &
${\scriptstyle F/20}$ &${\scriptstyle F}$ {\footnotesize lower} &
${\scriptstyle F}$ {\footnotesize upper} \\  \hline 
{\footnotesize F1}	& ${\scriptstyle 53.6}$   & ${\scriptstyle 74.5\;(2.8)}$   &
${\scriptstyle 72.9\;(14.6)}$  & ${\scriptstyle 72.8\;(5.2)}$  &
${\scriptstyle 71.7\;(4.3)}$  & ${\scriptstyle 72.8\;(4.5)}$\\  
{\footnotesize F2}	& ${\scriptstyle 3462.6}$ & ${\scriptstyle 3931.1\;(69.1)}$  &
${\scriptstyle 3856.2\;(71.4)}$ & ${\scriptstyle 3872.6\;(65.7)}$ &
${\scriptstyle 4031.0\;(77.2)}$ & ${\scriptstyle 4098.9\;(109.0)}$ \\ 
{\footnotesize F3}	& ${\scriptstyle 32.4}$   & ${\scriptstyle 49.3\;(2.9)}$   &
${\scriptstyle 43.0\;(3.3)}$  & ${\scriptstyle 51.0\;(14.4)}$   &
${\scriptstyle 40.8\;(2.3)}$ & ${\scriptstyle 41.9\;(2.5)}$ \\  
{\footnotesize F4}	& ${\scriptstyle 970.0}$  & ${\scriptstyle 1055.4\;(40.8)}$  &
${\scriptstyle 1023.0\;(31.0)}$ & ${\scriptstyle 1020.1\;(57.2)}$&
${\scriptstyle 1040.1\;(27.5)}$ & ${\scriptstyle 1055.5\;(27.8)}$  \\  
{\footnotesize F5}	& ${\scriptstyle 3.99651}$ & ${\scriptstyle 3.99796\;(0.0)}$  &
${\scriptstyle 3.9972\;(0.0)}$ & ${\scriptstyle 3.99679\;(0.0)}$ &
${\scriptstyle 4.2\;(0.0)}$ & ${\scriptstyle 4.3\;(0.0)}$ \\ 
{\footnotesize One Max} & ${\scriptstyle 24.6}$   & ${\scriptstyle 33.5\;(2.3)}$   &
${\scriptstyle 36.0\;(12.2)}$   & ${\scriptstyle 30.6\;(11.2)}$ & 
${\scriptstyle 29.8\;(1.2)}$ & ${\scriptstyle 30.3\;(1.3)}$ \\ 
{\footnotesize One Max} & ${\scriptstyle 99.9}$   & ${\scriptstyle 110.9\;(5.4)}$
&${\scriptstyle 124.8\;(59.8)}$ & ${\scriptstyle 109.8\;(5.9)}$ &
${\scriptstyle 109.5\;(3.2)}$ & ${\scriptstyle 111.5\;(2.7)}$  \\  \hline 
\end{tabular} 

\end{center}
\end{table}
}

{\newpage
\clearpage
\samepage \begin{table}\begin{center}
\begin{tabular}{|l|l c| l l | c c|} \hline  {\footnotesize Function}  & \multicolumn{2}{c|}{\em {\footnotesize Observed}} & \multicolumn{4}{c|}{\em
{\footnotesize Predicted}} \\  \cline{2-7} 
        & ${\scriptstyle  F_0}$ & ${\scriptstyle  F_{500}}$ & ${\scriptstyle
F/20}$ & ${\scriptstyle  F/45}$  & ${\scriptstyle  F}$ {\footnotesize lower} & ${\scriptstyle  F}$ {\footnotesize upper} \\ 
\hline  
{\footnotesize F1}	& ${\scriptstyle  53.6}$   & ${\scriptstyle  75.0\;(1.8)}$   &
${\scriptstyle  71.2\;(3.8)}$   & ${\scriptstyle  73.4\;(2.6)}$
& ${\scriptstyle  70.8\;(4.2)}$ & ${\scriptstyle  72.4\;(4.5)}$ \\ 
{\footnotesize F2}	& ${\scriptstyle  3462.6}$ & ${\scriptstyle  3919.4\;(66.1)}$  &
${\scriptstyle  3842.6\;(43.3)}$ &
${\scriptstyle  3866.8\;(39.9)}$  & ${\scriptstyle  3959.4\;(115.0)}$ &
${\scriptstyle  4029.0\;(142.0)}$ \\  
{\footnotesize F3}	& ${\scriptstyle  32.4}$   & ${\scriptstyle  49.6\;(1.9)}$   &
${\scriptstyle  55.3\;(21.6)}$   & ${\scriptstyle  46.9\;(2.6)}$
& ${\scriptstyle  40.4 (2.9)}$ & ${\scriptstyle  41.5\;(2.9)}$ \\  
{\footnotesize F4}	& ${\scriptstyle  970.0}$  & ${\scriptstyle  1058.6\;(24.5)}$  &
${\scriptstyle  1243.3\;(615.8)}$ &
${\scriptstyle  1077.0\;(68.6) }$  & ${\scriptstyle  1031.1\;(22.4)}$ &
${\scriptstyle  1052.7\;(23.5)}$ \\  
{\footnotesize F5}	& ${\scriptstyle  3.996}$    & ${\scriptstyle  3.997\;(0.00)}$   &
${\scriptstyle  3.996\;(0.00)}$    & ${\scriptstyle  3.997\;(0.00)}$
& ${\scriptstyle  4.20\;(0.04)}$ & ${\scriptstyle  4.25\;(0.06)}$ \\  
{\footnotesize One Max} & ${\scriptstyle  24.6}$   & ${\scriptstyle  35.2\;(1.6)}$   &
${\scriptstyle  31.0\;(4.5)}$   & ${\scriptstyle  34.8\;(5.0)}$
& ${\scriptstyle  28.8\;(1.1)}$ &${\scriptstyle  29.3\;(1.2)}$ \\ 
{\footnotesize One Max} & ${\scriptstyle  99.9}$   & ${\scriptstyle  110.7\;(3.6)}$   &
${\scriptstyle  107.4\;(3.6)}$  & ${\scriptstyle  108.8\;(5.2)}$
& ${\scriptstyle  108.9\;(2.9)}$ & ${\scriptstyle  111.2\;(3.0)}$ \\  \hline
\end{tabular}
\end{center}

\end{table}
}

\stepcounter{section}
\stepcounter{chapter}
\stepcounter{section}
\stepcounter{section}
{\newpage
\clearpage
\samepage \begin{figure}[hp]
\centerline{ 	
{\psfig{figure=figures/pl2.ps,height=3.0in,width=4.0in}}} 	
 
\label{easyspace}
\end{figure}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$F_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3913: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$F_j$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3921: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{figure}[hp]
\centerline{ 	
{\psfig{figure=figures/saga.ps,height=3.0in,width=3.0in}}} 	
 
\label{savsga}
\end{figure}
}

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{\newpage
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\fbox{\psfig{figure=figures/mf6.ps,height=2.0in,width=5.0in}}}

{\label{mf6}} 
\end{figure}
}

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{\newpage
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\fbox{\psfig{figure=figures/mf1.ps,height=2.0in,width=5.0in}}}

{\label{mf1}} 
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\fbox{\psfig{figure=figures/mf21.ps,height=2.0in,width=5.0in}}}

{\label{mf21}} 
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\fbox{\psfig{figure=figures/mf22.ps,height=2.0in,width=5.0in}}}

{\label{mf22}} 
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\fbox{\psfig{figure=figures/mf4.ps,height=2.0in,width=5.0in}}}

{\label{mf4}} 
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{\newpage
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\samepage \begin{figure}\centerline{
	\fbox{\psfig{figure=figures/mf51.ps,height=2.0in,width=5.0in}}}

{\label{mf51}}
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}

{\newpage
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\samepage \begin{figure}\centerline{
	\fbox{\psfig{figure=figures/mf52.ps,height=2.0in,width=5.0in}}}

{\label{mf52}} 
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{\newpage
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\samepage \begin{figure}\centerline{
	\fbox{\psfig{figure=figures/mf3.ps,height=2.0in,width=5.0in}}}

{\label{mf3}} 
\end{figure}
}

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{\newpage
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\samepage \begin{figure}[hp]
\centerline{
\fbox{\psfig{figure=figures/5tree.ps,height=3.0in,width=5in}}}

\end{figure}
}

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\end{document}
