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% !!! IMAGES START HERE !!!

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$ v ({\svector x}) $%
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$ \{ v_i\}$%
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$ n$%
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$\displaystyle =$%
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$\displaystyle \sum_{i=1}^nw_i\cdot v({\svector x}_i),$%
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{\newpage\clearpage
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$\displaystyle \sum_{i=1}^nw_iv_i, ~$%
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{\newpage\clearpage
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$\displaystyle ~
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$\displaystyle \equiv$%
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{\newpage\clearpage
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{\newpage\clearpage
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$ {\svector w}$%
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{\newpage\clearpage
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$\displaystyle E (v_i) \equiv 0, ~$%
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$ \sigma_v^2$%
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{\newpage\clearpage
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{\newpage\clearpage
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$ \stensor C$%
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{\newpage\clearpage
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$\displaystyle C_{ij}$%
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$\displaystyle E (v_i v_j ), ~$%
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{\newpage\clearpage
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$\displaystyle \stensor C$%
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$\displaystyle E ({\svector v}{\svector v}^\intercal ).$%
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$ \svector c$%
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$\displaystyle c_i$%
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{\newpage\clearpage
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$ \sigma_E^2$%
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{\newpage\clearpage
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$\displaystyle \sigma_E^2$%
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{\newpage\clearpage
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{\newpage\clearpage
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$\displaystyle E(v_0v_0)  - 2 \sum_{i=1}^nE ( v_iv_0) w_i+ \sum_{i=1}^n\sum_{j=1}^nE (v_iv_j) w_iw_j$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1027}%
$\displaystyle \sigma_v^2- 2 \sum_{i=1}^nc_iw_i+ \sum_{i=1}^n\sum_{j=1}^nC_{ij}w_iw_j.$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1034}%
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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$\displaystyle \sigma_v^2-  \svector c^\intercal \cdot {\svector w}.$%
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\stepcounter{section}
{\newpage\clearpage
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$\displaystyle C_{ij}\equiv
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$\displaystyle ~$%
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{\newpage\clearpage
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$\displaystyle \|{\svector x}_i- {\svector x}_j\|^{-2}$%
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{\newpage\clearpage
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{\newpage\clearpage
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$\displaystyle \approx$%
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$\displaystyle \infty$%
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{\newpage\clearpage
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$\displaystyle \stensor I$%
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{\newpage\clearpage
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$ \Delta x$%
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{\newpage\clearpage
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$\displaystyle \min(\Delta x^{-2},  \|{\svector x}_i- {\svector x}_j\| ^{-2} ) .$%
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$ x$%
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$ v_2$%
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$ v_1$%
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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$ {{\svector w}^\intercal}= [ 0.8 , 0.2 ]$%
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{\newpage\clearpage
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$ \Delta x= 0.5$%
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{\newpage\clearpage
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{\newpage\clearpage
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$ {{\svector w}^\intercal}= [ 0.82 , 0.18 ]$%
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{\newpage\clearpage
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{\newpage\clearpage
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$ {{\svector w}^\intercal}= [ 1 ,0 ]$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1169}%
$\displaystyle \gamma_v ({\svector x}_i, {\svector x}_j)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1173}%
$\displaystyle \frac{1}{2} E \{ [v ({\svector x}_i) - v({\svector x}_j) ]^2 \} = \sigma_v^2- C_{ij}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1176}%
$\displaystyle \gamma_v ({\svector x}_i- {\svector x}_j) =  \sigma_v^2-  C( {\svector x}_i - {\svector x}_j )$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1178}%
$ {\svector h}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1181}%
$\displaystyle {\svector h}_{ij}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1185}%
$\displaystyle {\svector x}_i- {\svector x}_j;$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1187}%
$\displaystyle \gamma_v ({\svector h}_{ij})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1191}%
$\displaystyle \sigma_v^2-  C( {\svector h}_{ij}).$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1194}%
$\displaystyle \gamma ({\svector h}) = C ({\svector 0} ) - C ({\svector h}).$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1197}%
$\displaystyle C( {\svector \infty})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1202}%
$\displaystyle \Rightarrow  ~~ \gamma( {\svector \infty})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1206}%
$\displaystyle C({\svector 0})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1208}%
$\displaystyle \Rightarrow  ~~ C ({\svector h})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1212}%
$\displaystyle \gamma ({\svector \infty}) - \gamma({\svector h}).$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1215}%
$\displaystyle \gamma_{\exp} ({\svector h})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1219}%
$\displaystyle 1 - \exp( - 3 \| {\svector h}\| / a),  ~~$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1221}%
$\displaystyle C_{\exp} ({\svector h})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1225}%
$\displaystyle \exp( - 3 \| {\svector h}\| / a).$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1227}%
$ a$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1232}%
$ {\svector m}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1234}%
$ {\svector d}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1236}%
$ {\stensor F}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1238}%
$ {\svector n}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1241}%
$\displaystyle {\svector d}= {\stensor F}{\svector m}+ {\svector n}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1244}%
$\displaystyle S ({\svector m}) = ({\svector d}- {\stensor F}{\svector m})^{\intercal} {\stensor C}_n^{-1}({\svector d}- {\stensor F}{\svector m}) + {\svector m}^{\intercal} {\stensor C}_m^{-1}{\svector m}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1246}%
$ {\stensor C}_m$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1250}%
$ {\stensor C}_n$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1252}%
$ {\stensor C}_d$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1255}%
$\displaystyle {\stensor C}_d= {\stensor F}{\stensor C}_m{\stensor F}^\intercal + {\stensor C}_n.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1259}%
$ \hat{{\svector m}}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1262}%
$\displaystyle \frac{\partial S}{\partial {\svector m}^{\intercal}} (\hat{{\svector m}})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1266}%
$\displaystyle - {\stensor F}^\intercal {\stensor C}_n^{-1}( {\svector d}- {\stensor F}\hat{{\svector m}}) + {\stensor C}_m^{-1}\hat{{\svector m}}= 0 ;$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1268}%
$\displaystyle \Rightarrow ~~ \hat{{\svector m}}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1272}%
$\displaystyle ( {\stensor F}^\intercal {\stensor C}_n^{-1}{\stensor F}+ {\stensor C}_m^{-1}) ^ {-1} {\stensor F}^\intercal {\stensor C}_n^{-1}{\svector d};$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1278}%
$\displaystyle {\stensor C}_m{\stensor F}^\intercal ( {\stensor F}{\stensor C}_m{\stensor F}^\intercal + {\stensor C}_n)^{-1} {\svector d}~=~  {\stensor C}_m{\stensor F}^\intercal {\stensor C}_d^{-1}{\svector d}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1281}%
$\displaystyle ( {\stensor F}^\intercal {\stensor C}_n^{-1}{\stensor F}+ {\stensor C}_m^{-1}) ^ {-1} {\stensor F}^\intercal {\stensor C}_n^{-1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1285}%
$\displaystyle {\stensor C}_m{\stensor F}^\intercal ( {\stensor F}{\stensor C}_m{\stensor F}^\intercal + {\stensor C}_n)^{-1} ;$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1287}%
$\displaystyle \Rightarrow ~~ {\stensor F}^\intercal {\stensor C}_n^{-1}( {\stensor F}{\stensor C}_m{\stensor F}^\intercal + {\stensor C}_n)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1291}%
$\displaystyle ( {\stensor F}^\intercal {\stensor C}_n^{-1}{\stensor F}+ {\stensor C}_m^{-1}) {\stensor C}_m{\stensor F}^\intercal$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1295}%
$\displaystyle {\stensor F}^\intercal {\stensor C}_n^{-1}{\stensor F}{\stensor C}_m{\stensor F}^\intercal + {\stensor F}^\intercal .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1309}%
$ {\stensor F}^\intercal $%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1315}%
$ {\stensor F}{\stensor F}^\intercal = \stensor I$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1317}%
$ {\stensor C}_n= \epsilon \stensor I; \epsilon \rightarrow 0$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1319}%
$ \hat{{\svector m}}\rightarrow {\stensor F}^\intercal {\svector d}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1324}%
$\displaystyle ~~
{\stensor C}_n$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1328}%
$\displaystyle 0  ;$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1330}%
$\displaystyle \Rightarrow ~~ {\stensor C}_d$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1334}%
$\displaystyle {\stensor F}{\stensor C}_m{\stensor F}^\intercal$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1337}%
$\displaystyle \hat{{\svector m}}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1341}%
$\displaystyle ({\stensor C}_m{\stensor F}^\intercal ) ( {\stensor F}{\stensor C}_m{\stensor F}^\intercal )^{-1} {\svector d}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1345}%
$\displaystyle ({\stensor F}{\stensor C}_m)^{\intercal} {\stensor C}_d^{-1}{\svector d}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1349}%
$\displaystyle \stensor W^{\intercal}{\svector d};$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1351}%
$\displaystyle ~~ \stensor W$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1355}%
$\displaystyle {\stensor C}_d^{-1}({\stensor F}{\stensor C}_m)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}


\end{document}
