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\begin{document}

\boldmath
\LARGE \bf

%-----------------------------------------------------------
% Page 1

%\begin{landscape}


\begin{center}
\centerline{\resizebox{11.5in}{!}{\includegraphics*{MIT3.jpg}}}
\vskip-6.8in


{\huge\bf \cyr Rolyata na polinomite na Shur\\
\medskip
v triizmernoto razpoznavane na tseli}
\bigskip

\end{center}

\bigskip
\bigskip

\hrule
\bigskip
\bigskip
\bigskip
\bigskip
\bigskip
\bigskip

{\LARGE \bf
\begin{center}
{\cyr Plamen Koev }\\
{ {\cyr Fakultet po Matematika \\
Masachuzet\-ski Tehnologichen Institut}
}


\vspace*{.5in}

%\vspace*{1in}


\vspace*{1in}

{\rm \large \bf \cyr Pioneri na B\cyrhrdsn lgarskata Matematika, 8 Yuli 2006g.}
\end{center}
}



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%\mathversion{bold}
{%\sffamily


% ---------------------- TANKS SLIDE -----------------------------

\slide{\cyr Tsel: Avtomatichno razpoznavane na tseli}

{\large\bf
\begin{center}
\begin{tabular}{ccc}
\resizebox{3in}{!}{\includegraphics*{Tank-Blazer.jpg}} &
\resizebox{3in}{!}{\includegraphics*{Tank-hmmwv-2.jpg}} &
\resizebox{3in}{!}{\includegraphics*{Tank-M1-A1_Abrams2.jpg}} \\
Blazer  & HMMWV & M1 A1 Abrams \\
\\
\resizebox{3in}{!}{\includegraphics*{Tank-Leopard.jpg}}     &
\resizebox{3in}{!}{\includegraphics*{Tank-T62-2.jpg}}    &
\resizebox{3in}{!}{\includegraphics*{TankBritishChallenger2.jpg}}\\
 Leopard & T62 & Challenger
\end{tabular}
\end{center}
}


% ---------------------- TANKS SLIDE -----------------------------


\slide{\cyr Tsel: Avtomatichno razpoznavane na tseli}

{\large
\begin{center}
\begin{tabular}{ccc}
\resizebox{3in}{!}{\includegraphics*{Tank-Blazer.jpg}} &
\resizebox{3in}{!}{\includegraphics*{Tank-hmmwv-2.jpg}} &
\resizebox{3in}{!}{\includegraphics*{Tank-M1-A1_Abrams2.jpg}} \\
Blazer  & HMMWV & M1 A1 Abrams \\
\\
\resizebox{3in}{!}{\includegraphics*{Tank-Leopard.jpg}}     &
\resizebox{3in}{!}{\includegraphics*{Tank-T62-2.jpg}}    &
\resizebox{3in}{!}{\includegraphics*{TankBritishChallenger2.jpg}}\\
 Leopard & T62 & Challenger
\end{tabular}

\smallskip

\resizebox{!}{1.8in}{\includegraphics*{Tank-LeopardBorden1.jpg}}
\end{center}
}



% ----------------------------------------------------------------

\slide{\cyr Do sega: Dvuizmerno, po kartinka}

\begin{center}

\begin{tabular}{ll}

\begin{tabular}{l}
{\cyr Izgledi}:
\\ \phantom{X}
\\ \phantom{X}
\\ \phantom{X}
\end{tabular}

&

\resizebox{1.8in}{!}{\includegraphics*{Tank-M1-A1_Abrams.jpg}}
\resizebox{1.8in}{!}{\includegraphics*{Tank-M1A1Abrams1.jpg}}
\resizebox{1.8in}{!}{\includegraphics*{Tank-M1A1Abrams2.jpg}}
\resizebox{1.8in}{!}{\includegraphics*{Tank-US-M1A1.jpg}}
\begin{tabular}{c}
{\Huge\tr{$\cdots$}}
\\
\phantom{X}
\\
\phantom{X}
\end{tabular}
\\
\begin{tabular}{l}
\hskip0.4in \tr{$\times$}\\
{\cyr Razmeri}:
\\ \phantom{X}
\\ \phantom{X}
\\ \phantom{X}
\end{tabular}
&


\resizebox{0.3in}{!}{\includegraphics*{Tank-M1-A1_Abrams.jpg}}
\resizebox{0.5in}{!}{\includegraphics*{Tank-M1-A1_Abrams.jpg}}
\resizebox{0.9in}{!}{\includegraphics*{Tank-M1-A1_Abrams.jpg}}
\resizebox{1.3in}{!}{\includegraphics*{Tank-M1-A1_Abrams.jpg}}
\resizebox{1.75in}{!}{\includegraphics*{Tank-M1-A1_Abrams.jpg}}
\resizebox{2.2in}{!}{\includegraphics*{Tank-M1-A1_Abrams.jpg}}
\begin{tabular}{c}
{\Huge\tr{$\cdots$}}
\\
\phantom{X}
\\
\phantom{X}
\end{tabular}

\\


\begin{tabular}{l}
\hskip0.4in \tr{$\times$}\\
{\cyr Vidove}:
\\ \phantom{X}
\\ \phantom{X}
\\ \phantom{X}
\end{tabular}
&

\resizebox{1.8in}{!}{\includegraphics*{Tank-M1-A1_Abrams.jpg}}
\resizebox{1.8in}{!}{\includegraphics*{Tank-Blazer.jpg}}
\resizebox{1.8in}{!}{\includegraphics*{Tank-T62-2.jpg}}
\resizebox{1.8in}{!}{\includegraphics*{TankBritishChallenger2.jpg}}
\begin{tabular}{c}
{\Huge\tr{$\cdots$}}
\\
\phantom{X}
\\
\phantom{X}
\\
\phantom{X}
\end{tabular}
\end{tabular}

\hrule

\end{center}

\tr{\cyr Neefektivno}, {\cyr no istinskiyat problem e ...}


% ----------------------------------------------------------------
\slide{\cyr Istinskiyat problem v dve izmereniya}


\centerline{\resizebox{8.5in}{!}{\includegraphics*{TankOnFire.jpg}}}

%-----------------------------------------------------------------
\slide{\cyr  V tri izmereniya ...}

\begin{center}

\resizebox{3.2in}{!}{\includegraphics*{Tank1.jpg}}

\bigskip

\hskip4.5in
\resizebox{0.1in}{0.8in}{\includegraphics*{red_arrow.jpg}}
\hskip0.1in
\begin{tabular}{l}
{\cyr \tr{S}intetichen \tr{A}perturen \tr{R}adar}
\\
\phantom{X}
\\
\phantom{X}
\end{tabular}

\vskip-0.3in
\resizebox{3.4in}{!}{\includegraphics*{Tank4.jpg}}
\end{center}

\begin{itemize}
\item {\cyr Raboti v m\cyrhrdsn gla, push\cyrhrdsn k, prez oblatsi}
\item {\cyr Vr\cyrhrdsn wa matritsa }$n\times 3,\;${\cyr  no
kak se razpoznavat matritsi?}

\item {\cyr Lesna za izvezhdane formula, no trudna za smyatane}
\end{itemize}



\IGNORE{
%-----------------------------------------------------------
\slide{\cyr Algoritmite imat znachenie}


\begin{itemize}

\item {\cyr ... zawoto kompyutrite smyatat  $\sim 10^9$ op/sek.}

\item {\cyr Naprimer: Kak da smetnem}
$
f(x)=x^{1024}?
$
\vskip1in
\begin{itemize}
\item $1023:\;\;\; \;\;\;\underbrace{x.x\ldots x}_{1024}$
\item $10: \;\;\;\;\;\;\; (((x^2\underbrace{)^2)\ldots)}_{10}\mbox{}^2$
\item $O(1): \;\; e^{1024\,\ln x}$



\end{itemize}
\end{itemize}
}


%-----------------------------------------------------------------
\slide{\cyr Matematicheskata zadacha}

\begin{itemize}

\item {\cyr Katalog}: $X_1, X_2, \ldots, X_m$ ($n\times 3$)

\item {\cyr Nablyudenie:} $ X= (X_i+E)Q, \quad\quad e_{ij}\sim N(0,\sigma^2)$

\item \tb{\cyr V\cyrhrdsn pros: $i=?$}
$$
L(i|X)\sim \sum_{k=0}^\infty c_k \sum_{\kappa\vdash k} \tr{s_\kappa(x_1,x_2,x_3)}
$$


\item $s_\kappa$ -- {\cyr polinom na Shur};
$\quad\quad\{x_1,x_2,x_3\}= {\rm eig}(X^TX)$

\item {\cyr Bavno s\-hodyaw red} $\Rightarrow$ {$\infty=100,150,200$}

\item {\cyr Broya ednochleni v $s_\kappa(x_1,x_2,x_3)$ e $3^{|\kappa|}$}
$\Rightarrow$ {\cyr godini ($\sim 10^9$/sek.)}

\item {\cyr Smyatame $s_\kappa$ kato $\chi$ na ${\rm GL}_3(\mathbb{C})$ s cena:
$\mathcal{O}(3)$}

($\mathcal{O}= \#\{\mu|\mu<\kappa\}\ll 3^{|\kappa|}$)

\item {\cyr Sega rabotim nad nov algorit\cyrhrdsn m}:
$\sqrt{\#\{\mu|\mu<\kappa\}}$



\end{itemize}



%-----------------------------------------------------------

\newpage
\phantom{X}
\centerline{\resizebox{15.5in}{!}{\includegraphics*{Google3.jpg}}}

\begin{center}

\vskip-2.2in

{\Huge\cyr Line\u inata algebra na Internet t\cyrhrdsn rsachkite}

\end{center}

%-----------------------------------------------------------

\newpage
\phantom{X}
\centerline{\resizebox{15.5in}{!}{\includegraphics*{Google3.jpg}}}

\begin{center}

\vskip-2.2in

{\Huge\cyr Line\u inata algebra na Internet t\cyrhrdsn rsachkite}

\vskip1in

\begin{itemize}

\item {\cyr Chast 1: Namirane na hilyadi stranici s\cyrhrdsn d\cyrhrdsn rzhawi t\cyrhrdsn senoto}

\item {\cyr Chast 2: Sortirane na namerenoto}

\bigskip
\bigskip

\item {\cyr Primer: SU Kliment Ohridski}
\end{itemize}

\end{center}



% ----------------------------------------------------------

\slide{{\cyr Namerihme $242,000$ stranici. Koya da e} No.\ 1?}


\centerline{\resizebox{8.5in}{!}{\includegraphics*{SU2.jpg}}}

\begin{itemize}


\item {\cyr Otgovor: Na\u i-posewavanata ot sluchaen ,,G-n Brauz\cyrhrdsn r''}

\end{itemize}

\IGNORE{

% ------------------------------------------------------------
\slide{\cyr Gospodin Brauz\cyrhrdsn r:}

\vskip1in
\centerline{\includegraphics{MrBrowser4.jpg}}

% ------------------------------------------------------------
\slide{\cyr Kak go namerihme?}

\centerline{\resizebox{9in}{!}{\includegraphics{MrBrowser2.jpg}}}

% ------------------------------------------------------------
\slide{\cyr Eto mu i stranicata ... }

\centerline{\resizebox{9in}{!}{\includegraphics{MrBrowser.jpg}}}

% ------------------------------------------------------------

}

\slide{\cyr Edin prost model}

\centerline{\resizebox{9in}{!}{\includegraphics{Graph2.jpg}}}

% ------------------------------------------------------------

\slide{\cyr Edin prost model}

\centerline{\resizebox{9in}{!}{\includegraphics{Graph4.jpg}}}

\vskip1in


\centerline{$p_1,q_1=$ {\cyr veroyatnost G-n Brauz\cyrhrdsn r e na stranitsa 1, 2}}
\bigskip
\centerline{$p_1+q_1=1$}

% ------------------------------------------------------------

\slide{\cyr Kak se promenyat veroyatnostite sled 1 st\cyrhrdsn pka}

\centerline{\resizebox{9in}{!}{\includegraphics{Graph4.jpg}}}

\vskip1in


\centerline{$p_1,q_1=$ {\cyr veroyatnost G-n Brauz\cyrhrdsn r e na stranitsa 1, 2}}

\bigskip
\centerline{$p_1+q_1=1$}

\bigskip

\tr{\cyr Sled 1 st\cyrhrdsn pka:}
\bigskip

\centerline{$p_2,q_2=$ {\cyr veroyatnost G-n Brauz\cyrhrdsn r e na stranitsa 1, 2}}
\bigskip
\centerline{$p_2+q_2=1$}

% ------------------------------------------------------------

\slide{\cyr Sled 1 st\cyrhrdsn pka}

\centerline{\resizebox{9in}{!}{\includegraphics{Graph4.jpg}}}

\vskip0.5in
$
\begin{array}{ccc}
p_2&=& \tfrac{1}{4}p_1 + \tfrac{2}{3}q_1\\
\\
q_2&=& \tfrac{3}{4}p_1 + \tfrac{1}{3}q_1
\end{array}
$

% ------------------------------------------------------------

\slide{\cyr Sled 2 st\cyrhrdsn pki}

\centerline{\resizebox{9in}{!}{\includegraphics{Graph4.jpg}}}

\vskip0.5in
$
\begin{array}{ccc}
p_2&=& \tfrac{1}{4}p_1 + \tfrac{2}{3}q_1\\
\\
q_2&=& \tfrac{3}{4}p_1 + \tfrac{1}{3}q_1
\end{array}
$
\hskip0.7in
$
\begin{array}{ccc}
p_3&=& \tfrac{1}{4}p_2 + \tfrac{2}{3}q_2\\
\\
q_3&=& \tfrac{3}{4}p_2 + \tfrac{1}{3}q_2
\end{array}
$

% ------------------------------------------------------------

\slide{\cyr Sled $n$ st\cyrhrdsn pki}

\centerline{\resizebox{9in}{!}{\includegraphics{Graph4.jpg}}}

\vskip0.5in
$
\begin{array}{ccc}
p_2&=& \tfrac{1}{4}p_1 + \tfrac{2}{3}q_1\\
\\
q_2&=& \tfrac{3}{4}p_1 + \tfrac{1}{3}q_1
\end{array}
$
\hskip0.7in
$
\begin{array}{ccc}
p_3&=& \tfrac{1}{4}p_2 + \tfrac{2}{3}q_2\\
\\
q_3&=& \tfrac{3}{4}p_2 + \tfrac{1}{3}q_2
\end{array}
$
\hskip0.7in
$
\begin{array}{ccc}
p_n&=& \tfrac{1}{4}p_{n-1} + \tfrac{2}{3}q_{n-1}\\
\\
q_n&=& \tfrac{3}{4}p_{n-1} + \tfrac{1}{3}q_{n-1}
\end{array}
$

% ------------------------------------------------------------

\slide{\cyr Sled $n$ st\cyrhrdsn pki}

\centerline{\resizebox{9in}{!}{\includegraphics{Graph4.jpg}}}

\vskip0.5in
$
\begin{array}{ccc}
p_2&=& \tfrac{1}{4}p_1 + \tfrac{2}{3}q_1\\
\\
q_2&=& \tfrac{3}{4}p_1 + \tfrac{1}{3}q_1
\end{array}
$
\hskip0.7in
$
\begin{array}{ccc}
p_3&=& \tfrac{1}{4}p_2 + \tfrac{2}{3}q_2\\
\\
q_3&=& \tfrac{3}{4}p_2 + \tfrac{1}{3}q_2
\end{array}
$
\hskip0.7in
$
\begin{array}{ccc}
p_n&=& \tfrac{1}{4}p_{n-1} + \tfrac{2}{3}q_{n-1}\\
\\
q_n&=& \tfrac{3}{4}p_{n-1} + \tfrac{1}{3}q_{n-1}
\end{array}
$

\bigskip
\bigskip

{\cyr Neka:}
$
p_n, q_n\stackrel{n\rightarrow\infty}{\longrightarrow} p,q.
$

% ------------------------------------------------------------

\slide{\cyr Sled $n$ st\cyrhrdsn pki}

\centerline{\resizebox{9in}{!}{\includegraphics{Graph4.jpg}}}

\vskip0.5in
$
\begin{array}{ccc}
p_2&=& \tfrac{1}{4}p_1 + \tfrac{2}{3}q_1\\
\\
q_2&=& \tfrac{3}{4}p_1 + \tfrac{1}{3}q_1
\end{array}
$
\hskip0.7in
$
\begin{array}{ccc}
p_3&=& \tfrac{1}{4}p_2 + \tfrac{2}{3}q_2\\
\\
q_3&=& \tfrac{3}{4}p_2 + \tfrac{1}{3}q_2
\end{array}
$
\hskip0.7in
$
\begin{array}{ccc}
p_n&=& \tfrac{1}{4}p_{n-1} + \tfrac{2}{3}q_{n-1}\\
\\
q_n&=& \tfrac{3}{4}p_{n-1} + \tfrac{1}{3}q_{n-1}
\end{array}
$

\bigskip
\bigskip

{\cyr Neka:}
$
p_n, q_n\stackrel{n\rightarrow\infty}{\longrightarrow} p,q.
$

\bigskip
\bigskip
$
\begin{array}{ccc}
p&=& \tfrac{1}{4}p + \tfrac{2}{3}q\\
\\
q&=& \tfrac{3}{4}p + \tfrac{1}{3}q
\end{array}
$
\hskip0.6in
$
\begin{array}{ccc}
\tfrac{3}{4}p &=&  \tfrac{2}{3}q\\
\\
 \tfrac{2}{3}q&=& \tfrac{3}{4}p
\end{array}
$\hskip0.7in

% ------------------------------------------------------------

\slide{\cyr Sled $n$ st\cyrhrdsn pki}

\centerline{\resizebox{9in}{!}{\includegraphics{Graph4.jpg}}}

\vskip0.5in
$
\begin{array}{ccc}
p_2&=& \tfrac{1}{4}p_1 + \tfrac{2}{3}q_1\\
\\
q_2&=& \tfrac{3}{4}p_1 + \tfrac{1}{3}q_1
\end{array}
$
\hskip0.7in
$
\begin{array}{ccc}
p_3&=& \tfrac{1}{4}p_2 + \tfrac{2}{3}q_2\\
\\
q_3&=& \tfrac{3}{4}p_2 + \tfrac{1}{3}q_2
\end{array}
$
\hskip0.7in
$
\begin{array}{ccc}
p_n&=& \tfrac{1}{4}p_{n-1} + \tfrac{2}{3}q_{n-1}\\
\\
q_n&=& \tfrac{3}{4}p_{n-1} + \tfrac{1}{3}q_{n-1}
\end{array}
$

\bigskip
\bigskip

{\cyr Neka:}
$
p_n, q_n\stackrel{n\rightarrow\infty}{\longrightarrow} p,q.
$

\bigskip
\bigskip
$
\begin{array}{ccc}
p&=& \tfrac{1}{4}p + \tfrac{2}{3}q\\
\\
q&=& \tfrac{3}{4}p + \tfrac{1}{3}q
\end{array}
$
\hskip0.6in
$
\begin{array}{ccc}
\tfrac{3}{4}p &=&  \tfrac{2}{3}q\\
\\
 \tfrac{2}{3}q&=& \tfrac{3}{4}p
\end{array}
$\hskip0.5in
$p+q=1 \quad\quad\Rightarrow\quad\quad
\begin{array}{ccc}
p &=&0.47\\
\\
q&=&0.53
\end{array}
$



% ----------------------------------------------------------
\slide{\cyr Sega model s 5 stranitsi}

%\centerline{\resizebox{!}{5.5in}{
\centerline{\includegraphics*{TheWeb7.jpg}}
%}}

% ----------------------------------------------------------
\slide{\cyr Sega model s 5 stranitsi}

%\centerline{\resizebox{!}{5.5in}{
\centerline{\includegraphics*{TheWeb6.jpg}}
%}}


% ----------------------------------------------------------

\slide{\cyr $p,q,r,s,t = $ veroyatnosti G-n Brauz\cyrhrdsn r e na $1,2,3,4,5$}


\noindent\hskip5.8in\resizebox{!}{2in}{ \includegraphics*{TheWeb6.jpg}}

\vskip-2in
\noindent$\begin{array}{cccccccccccccc}
p_2&=& &&\frac{1}{3}q_1\\
\\
q_2&=& \frac{1}{3}p_1 & & & +&\frac{1}{2}r_1    \\
\\
r_2&=& \frac{1}{3}p_1& +&\frac{1}{3}q_1&&&&&+&\frac{1}{2}t_1   \\
\\
s_2&=& \frac{1}{3}p_1&&&+ &\frac{1}{2}r_1&&&+&\frac{1}{2}t_1   \\
\\
t_2&=& &&\frac{1}{3}q_1& &&+& 1s_1\\
\end{array}
$

% ----------------------------------------------------------

\slide{\cyr $p,q,r,s,t = $ veroyatnosti G-n Brauz\cyrhrdsn r e na $1,2,3,4,5$}


\noindent\hskip5.8in\resizebox{!}{2in}{ \includegraphics*{TheWeb6.jpg}}

\vskip-2in
\noindent$\begin{array}{cccccccccccccc}
p\phantom{_2}&=& &&\frac{1}{3}q\phantom{_1}\\
\\
q\phantom{_2}&=& \frac{1}{3}p\phantom{_1} & & & +&\frac{1}{2}r\phantom{_1}    \\
\\
r\phantom{_2}&=& \frac{1}{3}p\phantom{_1}& +&\frac{1}{3}q\phantom{_1}&&&&&+&\frac{1}{2}t\phantom{_1}   \\
\\
s\phantom{_2}&=& \frac{1}{3}p\phantom{_1}&&&+ &\frac{1}{2}r\phantom{_1}&&&+&\frac{1}{2}t\phantom{_1}   \\
\\
t\phantom{_2}&=& &&\frac{1}{3}q\phantom{_1}&&&+& 1s\phantom{_1}\\
\\
\\
1&=&p&+&q&+&r&+&s&+&t
\end{array}
$

% ----------------------------------------------------------


\slide{\cyr $p,q,r,s,t = $ veroyatnosti G-n Brauz\cyrhrdsn r e na $1,2,3,4,5$}


\noindent\hskip5.8in\resizebox{!}{2in}{ \includegraphics*{TheWeb6.jpg}}

\vskip-2in
\noindent$\begin{array}{cccccccccccccc}
p\phantom{_2}&=& &&\frac{1}{3}q\phantom{_1}\\
\\
q\phantom{_2}&=& \frac{1}{3}p\phantom{_1} & & & +&\frac{1}{2}r\phantom{_1}    \\
\\
r\phantom{_2}&=& \frac{1}{3}p\phantom{_1}& +&\frac{1}{3}q\phantom{_1}&&&&&+&\frac{1}{2}t\phantom{_1}   \\
\\
s\phantom{_2}&=& \frac{1}{3}p\phantom{_1}&&&+ &\frac{1}{2}r\phantom{_1}&&&+&\frac{1}{2}t\phantom{_1}   \\
\\
t\phantom{_2}&=& &&\frac{1}{3}q\phantom{_1}&&&+& 1s\phantom{_1}\\
\\
\\
1&=&p&+&q&+&r&+&s&+&t
\end{array}
$


\bigskip
\bigskip


\centerline{\cyr Reshenie $(p,q,r,s,t)=(0.04,0.12,0.22,0.29,0.33)$}

\bigskip

\centerline{{\cyr Tova e t.nar.} \tr{PageRank} {\cyr vektor.}}


% ----------------------------------------------------------
\slide{\cyr Zawo stranitsa nomer 5?}


%\centerline{\resizebox{!}{5.5in}{
\centerline{\includegraphics*{TheWeb7.jpg}}
%}}


% ----------------------------------------------------------
\slide{\cyr Eksperiment --- 15-ti Yanuari 2006}
\centerline{\resizebox{9.5in}{!}{
\centerline{\includegraphics*{plamen-1-15-06-2.jpg}}
}}

% ----------------------------------------------------------
\slide{\cyr Eksperiment --- 29-ti Yanuari 2006}
\centerline{\resizebox{9.5in}{!}{
\centerline{\includegraphics*{plamen1-29-06.jpg}}
}}

% ----------------------------------------------------------
\slide{\cyr Eksperiment --- 8-mi Fevruari 2006}
\centerline{\resizebox{9in}{!}{
\centerline{\includegraphics*{plamen-2-08-06.jpg}}
}}



}

\end{document}

