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{\newpage
\clearpage
\samepage \begin{eqnarray*}i=1 \ldots N & \hspace{1em} & \mbox{label cities} \\ 
\eta_{ip} = 1 & \hspace{1em} & \mbox{if traveller is at city $i$ 
at time step} \\ 
& & p = 1 \ldots N \\ 
d_{ij} = & \hspace{1em} & \mbox{distance from city $i$ to city $j$}.
\end{eqnarray*}
}

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\samepage \begin{tabular}% latex2html id marker 114
{l}
Maximum Likelihood \\ 
$\chi^2$ \\ 
Maximum Entropy \\ 
Combinational Optimization \\ 
$\alpha$-$\beta$ Pruning \\ 
Branch and Bound \\ 
Branch and Cut \\ 
Linear Programming \\ 
Nonlinear Programming \\ 
Integer Programming \\ 
Heuristic \\ 
Greedy Algorithm \\ 
Conjugate Gradient \\ 
Simulated Annealing \\ 
Simulated Tempering \\ 
Neural Networks \\ 
Deterministic Annealing \\ 
Genetic Algorithms
\end{tabular}
}

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\samepage \begin{eqnarray*}V & = & 0 \qquad \mbox{doesn't vote} \\ 
& = & 1 \qquad \mbox{votes for Clinton} \\ 
& = & 2 \qquad \mbox{votes for Bush} \\ 
& = & 3 \qquad \mbox{votes for Perot, etc.} 
\end{eqnarray*}
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\samepage \begin{eqnarray*}p\ \ \overline{p}\ \ & \hspace{1em} & \rightarrow N\ \mbox{particles} \\ 
\ \ \rightarrow\ \ \leftarrow & \hspace{1em} & N = 2 \rightarrow 1000? 
\end{eqnarray*}
}

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\samepage \begin{displaymath}% latex2html id marker 1265
P_i \left( x_i \right) = c\, \exp \left\lbrace - \frac{1}{2\sigma_0^2}
\left( x_i - \alpha_0 \right)^2 \right\rbrace
\end{displaymath}
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\samepage \begin{eqnarray*}% latex2html id marker 236
L \left( x_1 \ldots x_N \right) & = & \prod_{i=1}^N c\, \exp 
\left\lbrace - \frac{1}{2\sigma_0^2} \left( x_i - \alpha_0 \right)^2
\right\rbrace \\ 
& = & \tilde{c}\, \exp \left\lbrace - \frac{1}{2\sigma_0^2} \sum_i \left(
x_i - \alpha_0 \right)^2 \right\rbrace 
\end{eqnarray*}
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L \left( x_1 \ldots x_N \right) = {\rm Pr} \left\lbrace x_1 \ldots x_n
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\end{displaymath}
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P \left( \alpha | x_i\,, T \right) = P \left( x_i | \alpha\,, T \right)
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\samepage \begin{displaymath}% latex2html id marker 1369
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\samepage \begin{displaymath}% latex2html id marker 1383
L \left( x_1 \ldots x_N \right) = c\, \exp \left\lbrace - \frac{1}{2\sigma^2} 
\sum_{i=1}^N \left( x_i - \alpha \right)^2 \right\rbrace
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\samepage \begin{eqnarray*}% latex2html id marker 271
L (\alpha ) & = & c\, \exp \left\lbrace - \frac{N}{2\sigma^2} \left( \alpha
- \frac{1}{N} \sum_{i=1}^n x_i \right)^2 \right. \\ 
& - & \frac{1}{2\sigma^2} \left\lbrack \left. \sum_i\, x_i^2 - \frac{1}{N} 
\left( \sum_i\, x_i \right)^2 \right\rbrack \right\rbrace 
\end{eqnarray*}
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\samepage \begin{displaymath}% latex2html id marker 1421
L(\alpha ) = \exp \left\lbrace {\rm const.} - \frac{1}{2\Delta\alpha^2}
\left( \alpha - \alpha^* \right)^2 \right\rbrace
\end{displaymath}
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{\newpage
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\samepage \begin{displaymath}% latex2html id marker 1422
- \ln L = - \sum_{i=1}^N \ln P \left( \underline{x}_i,\,
\underline{\alpha} \right)
\end{displaymath}
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\samepage \begin{eqnarray*}{dL\over d\tau} & = & 0 \qquad \mbox{or} \\ 
\frac{1}{\tau^{*2}} \sum_{i=1}^N t_i - \frac{N}{\tau^*} & = & 0 \\ 
\underline{\rm or} \qquad \tau^* & = & \frac{1}{N} \sum_{i=1}^N t_i 
\end{eqnarray*}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{eqnarray*}\sigma^2 & = & \frac{1}{N} \left\lbrace \left\langle t_i^2 \right\rangle
- \left\langle t_i \right\rangle^2 \right\rbrace \\ 
& = & \frac{1}{N} \left\lbrace \frac{1}{\tau_0} \int t^2
e^{- \frac{t}{\tau_0}} dt - \tau_0^2 \right\rbrace \\ 
\mbox{or}\ \ \sigma & = & \frac{\tau_0}{\sqrt{N}} 
\end{eqnarray*}
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\samepage \begin{eqnarray*}% latex2html id marker 366
\bullet\ \ L_1 = & \prod^{N_1}_{i=1} p\left( x_i\,, \alpha \right) & \quad 
\mbox{is likelihood of} \\ 
&& \quad \mbox{experiment \#1} \\ 
&& \\ 
\bullet\ \ L_2 = & \prod^{N_2}_{i=1} q\left( y_i\,, \alpha \right) & \quad 
\mbox{is likelihood of} \\ 
&& \quad \mbox{experiment \#2} 
\end{eqnarray*}
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{\newpage
\clearpage
\samepage \begin{eqnarray*}\left\langle \chi^2 \right\rangle & = \phantom{2}N & \\ 
\sigma_{\chi^2}^2\ \ \ & = 2N & (**) 
\end{eqnarray*}
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\sigma (\theta ) = a_0 + a_1 \cos \theta + a_2 \cos^2 \theta + a_3
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\end{displaymath}
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{\newpage
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\samepage \begin{eqnarray*}x_i & = & N_i \ \ \mbox{(up to constant)} \\ 
\mbox{error} & \mbox{in} & x_i \ \ \mbox{is}\ \  \sqrt{N_i}
\end{eqnarray*}
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\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{eqnarray*}\mbox{minimize:} & & \sum_{i=1}^N \frac{\left( x_i - \xi_i 
\right)^2}{\sigma_i^2} \qquad - \left( \chi^2 \right) \\ 
x_i & = & \ \mbox{const.}\ N_i \\ 
\sigma_i & = &\ \mbox{const.}\ \sqrt{N_i} \qquad \sigma_i = 
\frac{x_i}{\sqrt{N_i}} 
\end{eqnarray*}
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\samepage \begin{eqnarray*}\sum_{i=1}^N \frac{\left( x_i - \xi_i \right)}{\sigma_i^2}\,
\frac{\partial\xi_i}{\partial a_m} & = & 0\ \ \ \ \  \hbox{or} \\ 
\sum_l A_{ml}\ a^*_l & = & b_m 
\end{eqnarray*}
}

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\samepage \begin{eqnarray*}\chi^2 & = & \sum_{i=1}^N \left( \frac{e_i - \xi_i}{\sigma_i} \right)^2 \\ 
& = & \sum_{i=1}^N \hbox{res}_{\,i}^{\,2} \ \ \ \ \ \ \leftarrow\ \mbox{scaled
residual} 
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}\xi \left( \underline{x} \right) = \xi \left( \underline{x}_0 \right)
& + & \left( \underline{x} - \underline{x}_0 \right)^T
\frac{\partial\xi}{\partial\underline{x}} \\ 
& + & \mbox{Higher order in} \left( \underline{x} - \underline{x}_0
\right) 
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}\chi^2 \left( \underline{x} \right) = \chi^2 \left( \underline{x}_0
\right) & + & \frac{1}{2} \left( \underline{x} - \underline{x}_0
\right)^T A \left( \underline{x} - \underline{x}_0 \right) \\ 
& - & \left( \underline{x} - \underline{x}_0 \right)^T \underline{b} +
\mbox{corrections} 
\end{eqnarray*}
}

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\samepage \begin{eqnarray*}\xi_i & = & \xi_i \left( \underline{x}_0 \right) + \sum_{d=1}^m t_{id}
\left( x - x_0 \right)_d \\ 
\mbox{where}\ \ \ t_{id} & = & \frac{\partial\xi_i}{\partial x_d} \\ 
A_{dd'} & = & \sum_i t_{id}\, t_{id'} 
\end{eqnarray*}
}

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\samepage \begin{eqnarray*}\sum_{j=1}^r a_{ij} x_j - x_{r+1} & = & b_1 \\ 
a_{1r+1} & = & -1 \\ 
c_{r+1} & = & 0 
\end{eqnarray*}
}

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\samepage \begin{displaymath}% latex2html id marker 1923
-\ln L = - \sum_{i=1}^N \ln p \left( \underline{x}_i ,
\underline{\alpha} \right)
\end{displaymath}
}

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$\underline{\alpha}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1937: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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\samepage \begin{displaymath}% latex2html id marker 1924
\ln p \left( \underline{x}_i , \underline{\alpha} \right)
\end{displaymath}
}

{\newpage
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\samepage \begin{displaymath}% latex2html id marker 1925
\frac{\partial}{\partial\underline{\alpha}} \ln p \left( \underline{x}_i , 
\underline{\alpha} \right)
\end{displaymath}
}

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$-Ln L \left( \underline{\alpha}
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}

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\samepage \begin{displaymath}% latex2html id marker 1953
\underline{\underline{M}} \underline{\alpha} = \underline{b}
\hspace{7em} (**)
\end{displaymath}
}

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\samepage \begin{displaymath}% latex2html id marker 1954
\mbox{``value,''}\ \ \frac{\partial}{\partial\alpha_k}\ \ \mbox{``value''} 
\ \ \ \ \  \mbox{NOT} \ \ 
\frac{\partial^2}{\partial\alpha_k\,\partial\alpha_j}\ \ \mbox{``value''}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}& = & \ln p \left( \underline{x}_i \right) \mbox{for Likelihood method} \\ 
& = & \xi_i \left( \underline{x}_i \right) \mbox{for}\ \chi^2
\mbox{method} \\ 
\end{eqnarray*}
}

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\samepage \begin{eqnarray*}\eta_l\ \hbox{(event)} & = & 1 \ \ \mbox{event\ lies\ in}\ l\,\mbox{th\
bin} \\ 
& = & 0 \ \ \mbox{event\ lies outside}\ l\,\mbox{th\ bin} 
\end{eqnarray*}
}

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\samepage \begin{eqnarray*}\langle \eta_l \rangle & = & \left( 1 - \mu_l \right) \cdot {\rm O} +
\mu_l \cdot 1 = \mu_l \\ 
\langle \eta_l^2 \rangle & = & \left( 1 - \mu_l \right) \cdot {\rm O}^2
+ \mu_l \cdot 1^2 = \mu_l \\ 
\langle \eta_l^2 \rangle - \langle \eta_l \rangle^2 & = & \mu_l -
\mu_l^2 \hspace{8em} \mbox{(Fulle)} \\ 
& = & \mu_l \ \ \ \mbox{if}\ \ \ \mu_l << 1 \hspace{3em} \mbox{(Approxe)}
\end{eqnarray*}
}

{\newpage
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\samepage \begin{eqnarray*}N_l & = & \sum_{i=1}^{\rm \#\ events} \eta_l \ \ \mbox{(event}\ i) \\ 
& = & \mbox{\#\ events in}\ l \mbox{th\ bin}
\end{eqnarray*}
}

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\samepage \begin{eqnarray*}N_l & = & 250 \\ 
\sqrt{N_l} & \approx & 16 
\end{eqnarray*}
}

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\samepage \begin{eqnarray*}i & = & 1 \ldots d \\ 
d & = & 1 \\ 
f_1 & = & \left( p_1 + p_2 \right)^2 \\ 
\mbox{or}\ \ \ f_1 & = & \mbox{choice\ for\ President} 
\end{eqnarray*}
}

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\samepage \begin{displaymath}% latex2html id marker 2115
f(x,y) = A \exp \left\lbrack - \frac{(x-\alpha )^2}{2 \gamma^2} -
\frac{(y-\beta )^2}{2 \delta^2} \right\rbrack
\end{displaymath}
}

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\samepage \begin{eqnarray*}N_{\rm expected} & \propto & {\displaystyle \int_{\rm bin}} f(x, y) dx dy \\ 
& \sim & f \left( x_{\rm central} , y_{\rm central} \right) dx dy \\ 
\\ 
\mbox{with error} & & \sqrt{N_{\rm expected}} 
\end{eqnarray*}
}

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\end{document}
