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% !!! IMAGES START HERE !!!

\stepcounter{section}
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%
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%
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%
\providecommand{\fv}{\svector{f}}%


%
\providecommand{\dv}{\svector{d}}%


%
\providecommand{\pv}{\svector{p}}%


%
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%
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%
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%
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%
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%
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$\svector{d}^m$%
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$\svector{p}^m$%
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$m$%
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$\svector{x}^{m,n}$%
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$\svector{f}^n$%
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$\stensor{W}^n$%
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{\newpage\clearpage
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$\displaystyle x_i^{m,n}$%
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$\textstyle =$%
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$\displaystyle \sum_j W_{ij}^n f_j^n ( \svector{x}^{m,n-1} ),$%
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$\displaystyle \mbox{Find parameters }~ \svector{p}^m$%
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$\textstyle \approx$%
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$\displaystyle \svector{x}^{m,N}$%
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$\displaystyle \mbox{from data}~ \svector{d}^m$%
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$\displaystyle \svector{x}^{m,N}(\svector{d}^m ,\stensor{W}^1,\ldots,\stensor{W}^N) = \svector{x}^{m,N}.$%
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{\newpage\clearpage
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$F$%
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{\newpage\clearpage
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$\displaystyle \min_{\{W_{ij}^n\}}
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$\delta F$%
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$\displaystyle \delta F(\stensor{W}^1,\ldots,\stensor{W}^N)$%
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$\displaystyle \mbox{and}~ \delta x_i^{m,n}$%
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{\newpage\clearpage
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$\displaystyle \sum_{j} \, \delta W_{ij}^n \cdot f_j^n (\svector{x}^{m,n-1})  +
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{\newpage\clearpage
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$\displaystyle \mbox{or} ~
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay684}%
$\displaystyle \delta {\stensor{W}}^n \cdot \svector{f}^n ( \svector{x}^{m,n-1} )
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{\newpage\clearpage
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{\newpage\clearpage
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$\displaystyle \delta x_k^{m,N}$%
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$\displaystyle \sum_{n,i,j} G_{kij}^{m,n} \delta W_{ij}^n$%
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$\displaystyle \mbox{or} ~
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$\displaystyle \sum_n \stensor{G}^{m,n} : \delta {\stensor{W}}^n .$%
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{\newpage\clearpage
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$\displaystyle \mbox{or} ~
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{\newpage\clearpage
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$\displaystyle \sum_m \delta {\svector{x}}^{m,n} [ \svector{f}^n (\svector{x}^{m,n-1} ) ]^{*}  ~~
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$\displaystyle \mbox{where} ~ \delta \svector{x}^{m,N}$%
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{\newpage\clearpage
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{\newpage\clearpage
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$\displaystyle \mbox{and} ~ \delta x_k^{m,n-1}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay707}%
$\displaystyle \sum_j
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{\newpage\clearpage
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$\displaystyle \mbox{or} ~
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay710}%
$\displaystyle [ \stensor{\nabla} \svector{f}^n ( \svector{x}^{m,n-1} )]^{*}
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{\newpage\clearpage
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$\displaystyle \sum_{m,k} G_{kij}^{m,n} \delta x_k^{m,N} ,$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay718}%
$\displaystyle ( \stensor{G}^{m,n} )^{*} \cdot \delta {\svector{x}}^{m,N} .$%
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\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline419}%
$\Delta W_{ij}^n 
= \sum_{m,k} G_{kij}^{m,n} [ p_k^m - x_k^{m,N} ( \svector{d}^m , \stensor{W}^1,\ldots,\stensor{W}^N) ],$%
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{\newpage\clearpage
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$\Delta \stensor{W}^n = ( \stensor{G}^{m,n} )^{*} \cdot ( \svector{p}^m - \svector{x}^{m,N} )$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline425}%
$\stensor{W}^n + \epsilon \Delta \stensor{W}^n$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline427}%
$\epsilon$%
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{\newpage\clearpage
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$\Delta \stensor{W}^n $%
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{\newpage\clearpage
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$\alpha$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline435}%
$\sum_{m,k} \,  [ p_k^m - x_k^{m,N}(\svector{d}^m , 
\stensor{W}^1 + \alpha \Delta \stensor{W}^1 , \ldots ,
\stensor{W}^N + \alpha \Delta \stensor{W}^N )]^2$%
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{\newpage\clearpage
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$\stensor{W}^n + \alpha \Delta \stensor{W}^n$%
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{\newpage\clearpage
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$\Delta \svector{x}^{m,N} = \sum_n \stensor{G}^{m,n} : \Delta \stensor{W}^n$%
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{\newpage\clearpage
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$\sum_{m,k} \, [ p_k^m - \alpha \Delta x_k^{m,N} ]^2$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline451}%
$\alpha = (\sum_{m,k} \, p_k^m \Delta x_k^{m,N} ) / 
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{\newpage\clearpage
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$\{\stensor{w}^1,\ldots,\stensor{w}^N\}$%
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{\newpage\clearpage
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$\sum_{m,k} \, [ p_k^m - x_k^{m,N} ( \svector{d}^m , \stensor{W}^1,\ldots,\stensor{W}^N) 
	- \sum_{n,i,j} G_{kij}^{m,n} w_{ij}^n ]^2$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline465}%
$\stensor{W}^n + \stensor{w}^n$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline467}%
$\stensor{w}^n$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}


\end{document}
