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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\title{2���v����ɂ�������݃t�B�{�i�b�`�o�ΐ�
}%\vspace{10mm}}
\subtitle{(Alternately Fibonacci Duality in Quadratic Programming)}
\author{
��B��w���_����\hspace{2zw}��{�@���� (Seiichi ~IWAMOTO)\\[0mm]
\hspace*{-2mm}�H�c������w\hspace{4zw}�ؑ��@�� \hspace{4mm}(Yutaka ~KIMURA)
%��B��w���_����\hspace{2zw}��{�@���� (Seiichi ~Iwamoto)\\[2mm]
%\hspace*{-0.5mm}�H�c������w\hspace{4zw}�ؑ��@�� \hspace{4mm}(Yutaka ~Kimura)
%��B��w���_����\hspace{2zw}��{�@���� \\[2mm]
%\hspace*{-2.5mm}�H�c������w\hspace{14.5mm}�ؑ��@��
}
\vspace{15mm}
\institute{}
\date{{\footnotesize 
OR�w�������u�m���œK�����f���Ƃ��̉��p�v������E��2��\\[1mm]
June~~11,~2011\\[1mm]
(��t��w����t�L�����p�X���w��1����)}}
\begin{document}
\bf
%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Contents}
\begin{itemize}
\item \textcolor<2,8>{red}{���i�������}
\item \textcolor<3>{red}{\color<8>{gray} ���݃t�B�{�i�b�`�i4 : 6�^�j}
\item \textcolor<4>{red}{\color<8>{gray} ���݃t�B�{�i�b�`����o�ΐ�}
\item \textcolor<5>{red}{\color<8>{gray} ���݃t�B�{�i�b�`�i6 : 4�^�j}
\item \textcolor<6>{red}{\color<8>{gray} ���݃t�B�{�i�b�`�E�V�t�g�o�ΐ�}
\item \textcolor<7>{red}{\color<8>{gray} ���݃_�E���B���`�E�R�[�h}
\end{itemize}
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%%%%%%%%%%%%%%%%%%%%%
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\frametitle{2�i�K2�������}
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%�}1:~~�ł���ߕ����i����݁j
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\[
1^{2} + 4^{2} + 3^{2} + 1^{2} \dql  27 
\]
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%%%%%%%%%%%%%%%%%%%%%
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\[
3^{2} + 2^{2} + 1^{2} + 1^{2} \dql 15 \text{\RED{�i�œK�����j}}
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%%%%%%%%%%%%%%%%%%%%%
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\[
3^{2} + 10^{2} + 6^{2} + 4^{2} +1^{2} + 3^{2} \dql 171
\]
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%%%%%%%%%%%%%%%%%%%%%
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\[
7^{2} + 6^{2} + 3^{2} + 3^{2} +2^{2} + 1^{2} \dql 108 \,\text{\BLUE{�i���X�����j}}
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\[
8^{2} + 5^{2} + 3^{2} + 2^{2} +1^{2} + 1^{2} \dql 104 \,\text{\RED{�i�œK�����j}}
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%%%%%%%%%%%%%%%%%%%%%
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\[
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\uncover<5->{
\[
8^{2} + (-5)^{2} + (-3)^{2} + 2^{2} +1^{2} + (-1)^{2} \dql 104 \,\text{\RED{�i�œK�����j}}
\]
}
\end{frame}










%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	���݃t�B�{�i�b�`�@�S�F�U
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Primal Problem $({\rm P_{c4}})$}
����(primal problem)�Ƃ��Ď��̍ŏ��������l����. 
\begin{eqnarray*}
& &{\rm minimize} \hspace{3mm} 
\dsum_{n=0}^{3}\left[(x_{n} + x_{n+1})^{2} + x_{n+1}^{2} \right] \\ 
({\rm P_{c4}})& &{\rm subject~~ to}
\hspace{2ex} {\rm (i)}~~ -\infty < x_{n} < \infty ~~~~n= 1,2,3,4 \\
& &\hspace*{21mm} {\rm (ii)}~~x_{0} = c.
\end{eqnarray*}

\pause 

����$({\rm P_{c4}})$�̍ŏ���$\hat{x}$�ƍŏ��l$m_{4}$��,
\[
\begin{array}{lcl}
\hat{x} & = &
(x_{0}, ~\hat{x}_{1}, ~\hat{x}_{2}, ~\hat{x}_{3}, ~\hat{x}_{4})  \\[3mm]
 & = & \dfrac{c}{\,\,34 \,\,}
\left(
\begin{array}{ccccc}
      34 \,, & -13 \,, & 5 \,, &  -2 \,, & 1 
\end{array}
\right) \\[6mm]
m_{4} & = & 
\dfrac{\,21 \,}{\,\,34 \,\,}c^{2}
\end{array}
\]
%\[
%\begin{array}{lcl}
%\hat{x} & = &
%(x_{0}, ~\hat{x}_{1}, ~\hat{x}_{2}, ~\hat{x}_{3}, ~\hat{x}_{4})  \\[3mm]
% & = & \dfrac{c}{\,\,\framebox(19,12){34}\,\,}
%\left(
%\begin{array}{ccccc}
%      \framebox(19,12){$34$}\,, & \framebox(20,12){$-13$}\,, & \framebox(19,12)%{$5$}\,, &  \framebox(19,12){$-2$}\,, & \framebox(19,12){1} 
%\end{array}
%\right) \\[6mm]
%m_{4} & = & 
%\dfrac{\,\framebox(19,12){21}\,}{\,\,\framebox(19,12){34}\,\,}c^{2}
%\end{array}
%\]
�ł���. 
\end{frame}

%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Dual Problem $({\rm D_{c4}})$}
����$({\rm P_{c4}})$�ɑ΂���o�Ζ��(dual problem)$({\rm D_{c4}})$�́A
���ŕ\�����. 
\begin{eqnarray*}
\hspace*{-3mm}& &{\rm Maximize} \hspace{3mm} 2c\mu_{0} - \mu_{0}^{2} - \dsum_{n=0}^{2}\left[(\mu_{n} + \mu_{n+1})^{2} + \mu_{n+1}^{2} \right] 
- \mu_{3}^{2} \\[-2,56mm] 
\hspace*{-3mm}({\rm D_{c4}})& & \\[-2,56mm] 
\hspace*{-3mm}& &{\rm subject~\, to}\hspace{2ex} {\rm (i)}~~~ -\infty < \mu_{n} < \infty ~~~~n= 0,1,2,3.
\end{eqnarray*}

\pause 

�o�Ζ��$({\rm D_{c4}})$�̍ő��$\mu^{*}$�ƍő�l$M_{4}$��,
\[
\begin{array}{lcl}
\mu^{*} & = &
(\mu^{*}_{0},~ \mu^{*}_{1},~ \mu^{*}_{2},~ \mu^{*}_{3}) \\[3mm]
 & = & 
\dfrac{c}{\,\,34 \,\,}
\left(
\begin{array}{cccc}
       21 \,, & -8 \,, &  3 \,, & -1 
\end{array}
\right) \\[6mm]
M_{4} & = & 
\dfrac{\,21 \,}{\,\,34 \,\,}c^{2}
\end{array}
\]
%\[
%\begin{array}{lcl}
%\mu^{*} & = &
%(\mu^{*}_{0},~ \mu^{*}_{1},~ \mu^{*}_{2},~ \mu^{*}_{3}) \\[3mm]
% & = & 
%\dfrac{c}{\,\,\dashbox(19,12){34}\,\,}
%\left(
%\begin{array}{cccc}
%       \dashbox(19,12){$21$}\,, & \dashbox(19,12){$-8$}\,, &  \dashbox(19,12){$%3$}\,, & \dashbox(19,12){$-1$} 
%\end{array}
%\right) \\[6mm]
%M_{4} & = & 
%\dfrac{\,\dashbox(19,12){21}\,}{\,\,\dashbox(19,12){34}\,\,}c^{2}
%\end{array}
%\]
�ł���. 
\end{frame}


%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{}
\[
\mbox{����}({\rm P_{c4}})
\left\{
\begin{array}{lcl}
\hat{x} & = &
(x_{0}, ~\hat{x}_{1}, ~\hat{x}_{2}, ~\hat{x}_{3}, ~\hat{x}_{4})  \\[3mm]
 & = & \dfrac{c}{\,\,\framebox(19,12){34}\,\,}
\left(
\begin{array}{ccccc}
\framebox(19,12){$34$}\,, & \framebox(19,12){$-13$}\,, & \framebox(19,12){$5$}\,, &  \framebox(19,12){$-2$}\,, & \framebox(19,12){$1$} 
\end{array}
\right), \\[6mm]
m_{4} & = & 
\dfrac{\,\framebox(19,12){$21$}\,}{\,\,\framebox(19,12){$34$}\,\,}c^{2}.
\end{array}
\right.
\]
\[
\hspace*{-8mm}
\mbox{�o�Ζ��}({\rm D_{c4}})
\left\{
\begin{array}{lcl}
\mu^{*} & = &
(\mu^{*}_{0},~ \mu^{*}_{1},~ \mu^{*}_{2},~ \mu^{*}_{3}) \\[3mm]
 & = & 
\dfrac{c}{\,\,\dashbox(19,12){$34$}\,\,}
\left(
\begin{array}{cccc}
       \dashbox(19,12){$21$}\,, & \dashbox(19,12){$-8$}\,, &  \dashbox(19,12){$3$}\,, & \dashbox(19,12){$-1$} 
\end{array}
\right), \\[6mm]
M_{4} & = & 
\dfrac{\,\dashbox(19,12){21}\,}{\,\,\dashbox(19,12){34}\,\,}c^{2}.
\end{array}
\right.
\]
\pause
����$({\rm P_{c4}})$�̍ŏ���$\hat{x}$�ƁA
�o�Ζ��$({\rm D_{c4}})$�̍ő��$\mu^{*}$�y�сA
�ŏ��l$m_4$�ƍő�l$M_4$�̊Ԃɂ�
\RED{���݃t�B�{�i�b�`����o�ΐ�}�ƌĂ΂��֌W�����藧���Ă���. 
\end{frame}



%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Fibonacci sequence}
\textbf{�t�B�{�i�b�`����} 
%$ \{ F_{n} \} $ 
�͎���2�K���`�����������i3���ԑQ�����j 
\[
 F_{n+2} - F_{n+1} - F_{n} = 0,~~~~~F_{1} = 1, ~F_{0} = 0 \label{eqn:gd1}
\]
�̉��Œ�܂�. 

\begin{table}[htbp]
%
\begin{center}
% \begin{tabular}{c|cccccccccccccccccc} \hline
\begin{tabular}{c|ccccccccccccc} \hline
$ n $     &  $ 0 $  &  $ 1 $  &  $ 2  $ &  $ 3 $ &  $ 4 $ &  $ 5 $  & $ 6 $ & $ 7 $ & $ 8 $ & $ 9 $ & $ 10 $ & $ 11 $ & $ 12 $ \\ \hline
%& $ 13 $ & $ 14 $ & $ 15 $ & $ 16 $   \\ \hline 
$ F_{n} $ &  $ 0 $  &  $ 1 $  &  $ 1 $ &  $ 2 $ &  $3 $ &  $ 5  $  & $ 8 $ & $ 13 $  & $ 21 $ & $ 34 $ & $ 55 $ & $ 89 $ & $ 144 $ \\ \hline
%& $ 233 $ & $ 377 $ & $ 610 $ & $ 987 $   \\ \hline
\end{tabular}
\end{center}
%\caption{stop reward} \label{tab:h1}
\end{table}
\vspace{-5mm}
\begin{center}
Table 1:~~Fibonacci sequence $ \{F_{n}\} $
%�\ 1 ~~�t�B�{�i�b�`���� $ \{F_{n}\} $
\end{center}
\end{frame}






%%%%%%%%%%%%%%%%%
%\begin{frame}
\begin{frame}[allowframebreaks]
\frametitle{���݃t�B�{�i�b�`����o�ΐ�\\ 
(Alternately Fibonacci Complementary Duality)}

����$({\rm P_{c4}})$�Ƒo�Ζ��$({\rm D_{c4}})$�̊Ԃɂ�
����3�̊֌W�����藧��. 
�����\RED{���݃t�B�{�i�b�`����o�ΐ�}�Ƃ���. 
\vspace{3mm}

\begin{enumerate}
\item \BLUE{(�o�ΐ�)}\\[2mm]
����$({\rm P_{c4}})$�̍ŏ��l$m_4$�Ƒo�Ζ��$({\rm D_{c4}})$��
�ő�l$M_4$��������. ���Ȃ킿�A
\[
\begin{array}{lcl}
m_4 & = & \dfrac{\,21\,}{\,\,34\,\,}c^{2} \dql \dfrac{\,F_{8}\,}{F_{9}} c^{2}, \\[5mm]
M_4 & = & \dfrac{\,21\,}{\,\,34\,\,}c^{2} \dql \dfrac{\,F_{8}\,}{F_{9}} c^{2}, 
\end{array}
\]
�ł���A$m_{4} = M_{4}$ �����藧��. 
\vspace{1mm}

\item \BLUE{(2�i�K�t�B�{�i�b�`)}\\[2mm]
����̍ŏ���$\hat{x}$��
�o�Ζ��̍ő��$\mu^{*}$
�͋���\RED{���}�t�B�{�i�b�`����̌�����2�i���тł���. 
\[
\begin{array}{lcl}
\hat{x} & = &
(x_{0}, ~\hat{x}_{1}, ~\hat{x}_{2}, ~\hat{x}_{3}, ~\hat{x}_{4})  \\[3mm]
 & = & \dfrac{c}{\,\,34\,\,}
\left(
\begin{array}{ccccc}
\MagRedframebox{19}{12}{$34$}\,, & \MagBlueframebox{19}{12}{$-13$}\,, 
& \MagRedframebox{19}{12}{$5$}\,, 
&  \MagBlueframebox{19}{12}{$-2$}\,, & \MagRedframebox{19}{12}{$1$} 
\end{array}
\right) \\[4mm] 
 & = & \dfrac{c}{\,\,F_{9} \,\,}
\left(
\begin{array}{ccccc}
\MagRedframebox{19}{12}{$F_{9}$}\,, & \MagBlueframebox{19}{12}{$-F_{7}$}\,, 
& \MagRedframebox{19}{12}{$F_{5}$}\,, &  \MagBlueframebox{19}{12}{$-F_{3}$}\,, 
& \MagRedframebox{19}{12}{$F_{1}$} 
\end{array}
\right),
\end{array}
\]
%
%\[
%\begin{array}{lcl}
%\hat{x} & = &
%(x_{0}, ~\hat{x}_{1}, ~\hat{x}_{2}, ~\hat{x}_{3}, ~\hat{x}_{4})  \\[3mm]
% & = & \dfrac{c}{\,\,\framebox(19,12){34}\,\,}
%\left(
%\begin{array}{ccccc}
%\framebox(19,12){$34$}\,, & \framebox(20,12){$-13$}\,, & \framebox(19,12){$5$}\%,, &  \framebox(19,12){$-2$}\,, & \framebox(19,12){$1$} 
%\end{array}
%\right), \\[4mm] 
% & = & \dfrac{c}{\,\,\framebox(19,12){$F_{9}$}\,\,}
%\left(
%\begin{array}{ccccc}
%\framebox(19,12){$F_{9}$}\,, & \framebox(20,12){\MAGENTA{$-F_{7}$}}\,, 
%& \framebox(19,12){$F_{5}$}\,, &  \framebox(19,12){\MAGENTA{$-F_{3}$}}\,, & \fr%amebox(19,12){$F_{1}$} 
%\end{array}
%\right).
%\end{array}
%\]
\[
\hspace*{-15mm}
\begin{array}{lcl}
\mu^{*} & = &
(\mu^{*}_{0},~ \mu^{*}_{1},~ \mu^{*}_{2},~ \mu^{*}_{3}) \\[3mm]
 & = & 
\dfrac{c}{\,\,34 \,\,}
\left(
\begin{array}{cccc}
       \CyanRedframebox{19}{12}{$21$}\,, & \CyanBlueframebox{19}{12}{$-8$}\,, 
&  \CyanRedframebox{19}{12}{$3$}\,, & \CyanBlueframebox{19}{12}{$-1$} 
\end{array}
\right) \\[4mm]
 & = & 
\dfrac{c}{\,\,F_{9} \,\,}
\left(
\begin{array}{cccc}
       \CyanRedframebox{19}{12}{$F_{8}$}\,, 
& \CyanBlueframebox{19}{12}{$-F_{6}$}\,, 
&  \CyanRedframebox{19}{12}{$F_{4}$}\,, 
& \CyanBlueframebox{19}{12}{$-F_{2}$} 
\end{array}
\right). 
\end{array}
\]
%\[
%\hspace*{-15mm}
%\begin{array}{lcl}
%\mu^{*} & = &
%(\mu^{*}_{0},~ \mu^{*}_{1},~ \mu^{*}_{2},~ \mu^{*}_{3}) \\[3mm]
% & = & 
%\dfrac{c}{\,\,\dashbox(19,12){$34$}\,\,}
%\left(
%\begin{array}{cccc}
%       \dashbox(19,12){$21$}\,, & \dashbox(19,12){$-8$}\,, &  \dashbox(19,12){$%3$}\,, & \dashbox(19,12){$-1$} 
%\end{array}
%\right) \\[4mm]
% & = & 
%\dfrac{c}{\,\,\dashbox(19,12){$34$}\,\,}
%\left(
%\begin{array}{cccc}
%       \dashbox(19,12){$F_{8}$}\,, & \dashbox(19,12){\MAGENTA{$-F_{6}$}}\,, &  %\dashbox(19,12){$F_{4}$}\,, & \dashbox(19,12){\MAGENTA{$-F_{2}$}} 
%\end{array}
%\right)
%\end{array}
%\]

\item \BLUE{(����t�B�{�i�b�`)}\\[2mm]
����̍ŏ���$\hat{x}$��
�o�Ζ��̍ő��$\mu^{*}$�����݂ɕ҂ނƁA\\
\RED{2�A���}�t�B�{�i�b�`����̌������i1�i���сj�ł���. 
���Ȃ킿�A�萔 $ \dfrac{c}{\,F_{9}\,} $ �𖳎�����΁A


\[
\hspace*{-6.3mm}
\begin{array}{ccccccccccc}
( & \framebox(19,12){$x_{0}$}, & 
& \framebox(19,12){$\hat{x}_{1}$}, 
& & \framebox(19,12){$\hat{x}_{2}$}, & 
& \framebox(19,12){$\hat{x}_{3}$}, & 
& ~\framebox(19,12){$\hat{x}_{4}$} & ) \\[1.56mm]
( & & \dashbox(19,12){$\mu_{0}^{*}$}, & 
& \dashbox(19,12){$\mu_{1}^{*}$}, & 
& \dashbox(19,12){$\mu_{2}^{*}$}, & 
& \dashbox(19,12){$\mu_{3}^{*}$} & & )\\[1.56mm]
\end{array}
\]



$$ 
\framebox(24,15){$x_{0}$}~~\,~\dashbox(24,15){$\mu^{*}_{0}$}~~\,~\framebox(24,15){$\hat{x}_{1}$}~~\,~\dashbox(24,15){$\mu^{*}_{1}$}~~\,~\framebox(24,15){$\hat{x}_{2}$}~~\,~\dashbox(24,15){$\mu^{*}_{2}$}~~\,~\framebox(24,15){$\hat{x}_{3}$}~~\,~\dashbox(24,15){$\mu^{*}_{3}$}~~\,~\framebox(24,15){$\hat{x}_{4}$}
$$
$$ 
\MagRedframebox{24}{15}{$F_{9}$}~~\,~
\CyanRedframebox{24}{15}{$F_{8}$}~~
\,~\MagBlueframebox{24}{15}{$-F_{7}$}~~
\,~\CyanBlueframebox{24}{15}{$-F_{6}$}~~\,~
\MagRedframebox{24}{15}{$F_{5}$}~~\,~
\CyanRedframebox{24}{15}{$F_{4}$}~~\,~
\MagBlueframebox{24}{15}{$-F_{3}$}~~
\,~\CyanBlueframebox{24}{15}{$-F_{2}$}~~\,~
\MagRedframebox{24}{15}{$F_{1}$}
$$
%$$ 
%\framebox(24,15){$x_{0}$}~~\,~\dashbox(24,15){$\mu^{*}_{0}$}~~\,~\framebox(24,1%5){$\hat{x}_{1}$}~~\,~\dashbox(24,15){$\mu^{*}_{1}$}~~\,~\framebox(24,15){$\hat%{x}_{2}$}~~\,~\dashbox(24,15){$\mu^{*}_{2}$}~~\,~\framebox(24,15){$\hat{x}_{3}$%}~~\,~\dashbox(24,15){$\mu^{*}_{3}$}~~\,~\framebox(24,15){$\hat{x}_{4}$}
%$$
%$$ 
%\framebox(24,15){\BLACK{$F_{9}$}}~~\,~\dashbox(24,15){$F_{8}$}~~
%\,~\framebox(24,15){$-F_{7}$}~~
%\,~\dashbox(24,15){$-F_{6}$}~~\,~\framebox(24,15){$F_{5}$}~~
%\,~\dashbox(24,15){$F_{4}$}~~\,~\framebox(24,15){\MAGENTA{$-F_{3}$}}~~
%\,~\dashbox(24,15){\MAGENTA{$-F_{2}$}}~~\,~\framebox(24,15){$F_{1}$}
%$$
�ƂȂ�. 

\end{enumerate}
\end{frame}


%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{���݃t�B�{�i�b�`���� \\
(Alternately Fibonacci Condition)}
����$({\rm P_{c4}})$�y�сA�o�Ζ��$({\rm D_{c4}})$�ɂ����āA
���ꂼ�ꎟ��4�̓�������Ȃ�n${\rm (AF)_{P_{c}}}$�y��${\rm (AF)_{D_{c}}}$���A
���ꂼ����$({\rm P_{c4}})$�A$({\rm D_{c4}})$�ɑ΂���
\RED{���݃t�B�{�i�b�`����}�Ƃ���. 
\[
\textcolor<2>{red}{
({\rm AF})_{\rm P_{c}}
\left\{
\begin{array}{ll}
\dfrac{\,c + x_{1}\,}{F_{8}} \dql \dfrac{\,x_{1}\,}{-F_{7}}, &
\dfrac{\,x_{1} + x_{2}\,}{-F_{6}} \dql \dfrac{\,x_{2}\,}{F_{5}}, \\[5mm]
\dfrac{\,x_{2} + x_{3}\,}{F_{4}} \dql \dfrac{\,x_{3}\,}{-F_{3}}, &
\dfrac{\,x_{3} + x_{4}\,}{-F_{2}}  \dql \dfrac{\,x_{4}\,}{F_{1}}.
\end{array}
\right.
}
\]
\vspace{2mm}
\[
\textcolor<2>{gray}{
\hspace{3.6mm}({\rm AF})_{\rm D_{c}}
\left\{
\begin{array}{l}
\dfrac{\,c - \mu_{0}\,}{F_{7}}  
\dql \dfrac{\,\mu_{0} + \mu_{1}\,}{F_{7}} \dql 
\dfrac{\,\mu_{1}\,}{\,-F_{6}\,}, \\[5mm]
\dfrac{\,\mu_{1} + \mu_{2}\,}{\,-F_{5}\,} \dql 
\dfrac{\,\mu_{2}\,}{F_{4}}, \ \ 
\dfrac{\,\mu_{2} + \mu_{3}\,}{F_{3}}  \dql \dfrac{\,\mu_{3}\,}{\,-F_{2}\,}.
\end{array}
\right.
}
\]

\end{frame}



%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{${\rm (AF)_{P_{c}}}$�̓��l����}
����$({\rm P_{c4}})$�̖ړI�֐��� 
$ x = (x_{1}, x_{2}, x_{3}, x_{4}) $ ��4�ϐ��֐� $ f(x) $ �ŕ\��. 
\[
f(x) : \dql \dsum_{n=0}^{3}\left[(x_{n} + x_{n+1})^{2} + x_{n+1}^{2} \right].
\]
\begin{eqnarray*}
%& & \mbox{���݃t�B�{�i�b�`����}{\rm (AF)_{P}} \\[3mm]
{\rm (AF)_{P_{c}}}
& \Longleftrightarrow & 
\left\{
\begin{array}{rcl}
(c + x_{1}) + x_{1} + (x_{1} + x_{2}) & \dql &  0 \\[0.56mm]
(x_{1} + x_{2}) + x_{2} + (x_{2} + x_{3}) & \dql & 0 \\[-2.46mm]
 & &  \\[-2.46mm]
(x_{2} + x_{3}) + x_{3} + (x_{3} + x_{4}) & \dql &  0 \\[0.56mm]
(x_{3} + x_{4}) + x_{4} & \dql & 0 
\end{array}
\right. \\[2mm]
 & \Longleftrightarrow &  
\dfrac{\partial f}{\partial x_{n}} \dql 0 ~~~~~n = 1, 2, 3, 4. \\
\end{eqnarray*}

�ړI�֐�$f(x)$��$x$��2���̓ʊ֐��ł��邱�Ƃ���A
���$({\rm P_{c4}})$�̍ŏ���
$\hat{x} = (x_{0}, ~\hat{x}_{1}, ~\hat{x}_{2}, ~\hat{x}_{3}, ~\hat{x}_{4})$
�́A�œK����1�K�����𖞂���. 
\end{frame}


%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{���$({\rm P_{c4}})$�̌��݃t�B�{�i�b�`����}
\begin{eqnarray*}
({\rm AF})_{\rm P_{c}}
& \stackrel{\mbox{\itshape\scriptsize def}}{\Longleftrightarrow} &
\left\{
\begin{array}{ll}
\dfrac{\,c + x_{1}\,}{F_{8}} \dql \dfrac{\,x_{1}\,}{-F_{7}}, &
\dfrac{\,x_{1} + x_{2}\,}{-F_{6}} \dql \dfrac{\,x_{2}\,}{F_{5}}, \\[5mm]
\dfrac{\,x_{2} + x_{3}\,}{F_{4}} \dql \dfrac{\,x_{3}\,}{-F_{3}}, &
\dfrac{\,x_{3} + x_{4}\,}{-F_{2}}  \dql \dfrac{\,x_{4}\,}{F_{1}}, 
\end{array}
\right. \\[2mm]
\uncover<2->{
& \Longleftrightarrow &
\left\{
\begin{array}{ll}
\dfrac{\,c - (- x_{1})\,}{F_{8}} \dql \dfrac{\,-x_{1}\,}{F_{7}}, &
\dfrac{\,-x_{1} - x_{2}\,}{F_{6}} \dql \dfrac{\,x_{2}\,}{F_{5}}, \\[5mm]
\dfrac{\,x_{2} - (- x_{3}) \,}{F_{4}} \dql \dfrac{\,-x_{3}\,}{F_{3}}, &
\dfrac{\,-x_{3} - x_{4}\,}{F_{2}}  \dql \dfrac{\,x_{4}\,}{F_{1}}, 
\end{array}
\right. } \\[2mm]
\uncover<3>{
& \Longleftrightarrow &
\left\{
\begin{array}{ll}
\dfrac{\,c - (- x_{1})\,}{21} \dql \dfrac{\,-x_{1}\,}{13}, &
\dfrac{\,-x_{1} - x_{2}\,}{8} \dql \dfrac{\,x_{2}\,}{5}, \\[5mm]
\dfrac{\,x_{2} - (- x_{3}) \,}{3} \dql \dfrac{\,-x_{3}\,}{2}, &
\dfrac{\,-x_{3} - x_{4}\,}{1}  \dql \dfrac{\,x_{4}\,}{1}.
\end{array}
\right. } \\
\end{eqnarray*}
\end{frame}



%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{���$({\rm P_{c4}})$�̌��݃t�B�{�i�b�`�����ɂ��œK���̉�@}
$x_{0} = c \,~(\, > 0 \,) $ \\
\[
\begin{array}{ll}
\textcolor<2>{red}{\dfrac{\,c - (- x_{1})\,}{21} 
\dql \dfrac{\,-x_{1}\,}{13}}, &
\textcolor<3>{red}{\dfrac{\,-x_{1} - x_{2}\,}{8} 
\dql \dfrac{\,x_{2}\,}{5}}, \\[5mm]
\textcolor<4>{red}{\dfrac{\,x_{2} - (- x_{3}) \,}{3} 
\dql \dfrac{\,-x_{3}\,}{2}}, &
\textcolor<5>{red}{\dfrac{\,-x_{3} - x_{4}\,}{1}  \dql \dfrac{\,x_{4}\,}{1}}.
\end{array}
\]
%\vspace{3mm}
%
%\uncover<2>{
%$
%\MAGENTA{\dfrac{\,c - (- x_{1})\,}{21} \dql \dfrac{\,-x_{1}\,}{13}}
%$
%}
%\uncover<3>{\hspace*{-39mm}
%$
%\MAGENTA{\dfrac{\,-x_{1} - x_{2}\,}{8} \dql \dfrac{\,x_{2}\,}{5}}
%$
%}
%\uncover<4>{\hspace*{-34mm}
%$
%\MAGENTA{\dfrac{\,x_{2} - (- x_{3}) \,}{3} \dql \dfrac{\,-x_{3}\,}{2}}
%$
%}
%\uncover<5>{\hspace*{-41mm}
%$
%\MAGENTA{\dfrac{\,-x_{3} - x_{4}\,}{1}  \dql \dfrac{\,x_{4}\,}{1}}
%$
%}

\begin{center}
\unitlength=1mm
\begin{picture}(111,10)(0,0)
\uncover<1->{
\allinethickness{0.8mm}
\put(0,0){\line(1,0){111}}	%
\put(0,-1){\line(0,1){2}}	%
\put(-2,3){\makebox(0,0){\large $0$}}	%
\put(111,-1){\line(0,1){2}}	%
\put(111,3){\makebox(0,0){\large $c$}}	%
}
%% -x_1
\uncover<2->{
\allinethickness{0.3mm}			
\put(43,-1){\line(0,1){2}}	
\put(43,3){\makebox(0,0){\large $-x_1$}}
}
\uncover<2>{
\bezier{200}(0,0)(22,-4)(43,0)	
\put(22,-6){\makebox(0,0){\large $13$}}
\put(22,-11){\makebox(0,0){\large $F_{7}$}}
\bezier{200}(43,0)(76,-4)(111,0)	
\put(76,-6){\makebox(0,0){\large $21$}}
\put(76,-11){\makebox(0,0){\large $F_{8}$}}
}
%% x_2
\uncover<3->{
\allinethickness{0.3mm}			
\put(17,-1){\line(0,1){2}}	
\put(17,3){\makebox(0,0){\large $x_2$}}
}
\uncover<3>{
\bezier{100}(0,0)(9,-3)(17,0)	
\put(8,-6){\makebox(0,0){\large $5$}}
\put(8,-11){\makebox(0,0){\large $F_{5}$}}
\bezier{100}(17,0)(29,-3)(43,0)	
\put(29,-6){\makebox(0,0){\large $8$}}
\put(29,-11){\makebox(0,0){\large $F_{6}$}}
}
%% -x_3
\uncover<4->{
\allinethickness{0.3mm}			
\put(7,-1){\line(0,1){2}}	
\put(9,3){\makebox(0,0){\large $-x_3$}}
}
\uncover<4>{
\bezier{50}(0,0)(4,-2)(7,0)	
\put(3,-6){\makebox(0,0){\large $2$}}
\put(3,-11){\makebox(0,0){\large $F_{3}$}}
\bezier{50}(7,0)(12,-2)(17,0)	
\put(12,-6){\makebox(0,0){\large $3$}}
\put(12,-11){\makebox(0,0){\large $F_{4}$}}
}
%% x_4
\uncover<5->{
\allinethickness{0.3mm}			
\put(4,-1){\line(0,1){2}}	
\put(3,3){\makebox(0,0){\large $x_4$}}
\put(2,-4){\makebox(0,0){\large $1$}}
\put(1,-9){\makebox(0,0){\large $F_{1}$}}
\put(5,-4){\makebox(0,0){\large $1$}}
\put(6,-9){\makebox(0,0){\large $F_{2}$}}
}
\end{picture}
\end{center}

\end{frame}


%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{}
\begin{center}
\unitlength=1mm
\begin{picture}(111,10)(0,0)
\allinethickness{0.8mm}
\put(0,0){\line(1,0){111}}	%
\put(0,-1){\line(0,1){2}}	%
\put(-2,3){\makebox(0,0){\large $0$}}	%
\put(111,-1){\line(0,1){2}}	%
\put(111,3){\makebox(0,0){\large $c$}}	%
%% -x_1
\allinethickness{0.3mm}	
\uncover<2->{
\put(43,-1){\line(0,1){2}}	
\put(43,6){\makebox(0,0){\large $-\hat{x}_1$}}
\put(76,-6){\makebox(0,0){\large $F_{8}$}}
\bezier{300}(43,0)(77,-4)(111,0)
}
\uncover<2>{
\bezier{200}(0,0)(22,-4)(43,0)	
\put(22,-6){\makebox(0,0){\large $F_{7}$}}
}
%% x_2
\uncover<3->{
\put(17,-1){\line(0,1){2}}	
\put(17,6){\makebox(0,0){\large $\hat{x}_2$}}
\put(29,-6){\makebox(0,0){\large $F_{6}$}}
\bezier{100}(17,0)(30,-4)(43,0)	
}
\uncover<3>{
\bezier{100}(0,0)(9,-4)(17,0)	
\put(9,-6){\makebox(0,0){\large $F_{5}$}}
}
%% -x_3
\uncover<4->{
\put(8,-1){\line(0,1){2}}	
\put(9,6){\makebox(0,0){\large $-\hat{x}_3$}}
\put(12,-6){\makebox(0,0){\large $F_{4}$}}
\bezier{50}(8,0)(13,-2)(17,0)
}
\uncover<4>{
\bezier{50}(0,0)(4,-2)(8,0)	
\put(3,-6){\makebox(0,0){\large $F_{3}$}}
}
%% x_4
\uncover<5->{
\put(4,-1){\line(0,1){2}}
\bezier{50}(4,0)(6,-2)(8,0)	
\put(2,6){\makebox(0,0){\large $\hat{x}_4$}}
\bezier{50}(0,0)(2,-2)(4,0)	
\put(1,-6){\makebox(0,0){\large $F_{1}$}}
\put(6,-6){\makebox(0,0){\large $F_{2}$}}
}
\end{picture}
\end{center}
\vspace{5mm}

\[
\hspace*{-4mm}
\mbox{$({\rm P_{c4}})$�̍œK�� \ }
\left\{
\begin{array}{rclcl}
\uncover<2->{- \hat{x}_1 & = &  \dfrac{F_{7}}{F_{7}+F_{8}} \ c }
& \uncover<6->{= & \dfrac{F_7}{F_9} \, c \,,} \\[4mm]
\uncover<3->{\hat{x}_2 & = & \dfrac{F_{5}}{F_{5}+F_{6}} \, (- \hat{x}_{1})} 
& \uncover<6->{= & \dfrac{F_5}{F_7} \ndot 
\dfrac{F_{7}}{F_{9}} \, c \,,} \\[4mm]
\uncover<4->{- \hat{x}_3 & = & \dfrac{F_{3}}{F_{3}+F_{4}} \, \hat{x}_{2}} 
& \uncover<6->{= & \dfrac{F_3}{F_5} \ndot \dfrac{F_5}{F_7} \ndot 
\dfrac{F_{7}}{F_{9}} \, c \,,} \\[4mm]
\uncover<5->{\hat{x}_4 & = & \dfrac{F_{1}}{F_{1}+F_{2}} \, (- \hat{x}_{3})} 
& \uncover<6->{= & \dfrac{F_1}{F_3} \ndot \dfrac{F_3}{F_5} \ndot 
\dfrac{F_5}{F_7} \ndot \dfrac{F_{7}}{F_{9}} \, c \,.}
\end{array}
\right.
%\mbox{�ŋ��߂���. }
\]

\end{frame}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
\begin{comment}
%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{}
\begin{center}
\unitlength=1mm
\begin{picture}(111,10)(0,0)
\allinethickness{0.8mm}
\put(0,0){\line(1,0){111}}	%
\put(0,-1){\line(0,1){2}}	%
\put(-2,3){\makebox(0,0){\large $0$}}	%
\put(111,-1){\line(0,1){2}}	%
\put(111,3){\makebox(0,0){\large $c$}}	%
%% -x_1
\allinethickness{0.3mm}			
\put(43,-1){\line(0,1){2}}	
\put(43,6){\makebox(0,0){\large $-\hat{x}_1$}}
\put(76,-4){\makebox(0,0){\large $21$}}
\bezier{300}(43,0)(77,-3)(111,0)	
%% x_2
\put(17,-1){\line(0,1){2}}	
\put(17,3){\makebox(0,0){\large $\hat{x}_2$}}
\put(29,-4){\makebox(0,0){\large $8$}}
\bezier{100}(17,0)(30,-3)(43,0)	
%% -x_3
\put(8,-1){\line(0,1){2}}	
\put(9,6){\makebox(0,0){\large $-\hat{x}_3$}}
\put(12,-4){\makebox(0,0){\large $3$}}
\bezier{50}(8,0)(13,-2)(17,0)	
%% x_4
\put(4,-1){\line(0,1){2}}
\bezier{50}(4,0)(6,-2)(8,0)	
\put(4,3){\makebox(0,0){\large $\hat{x}_4$}}
\bezier{50}(0,0)(2,-2)(4,0)	
\put(2,-4){\makebox(0,0){\large $1$}}
\put(5,-4){\makebox(0,0){\large $1$}}
\end{picture}
\end{center}
\vspace{5mm}

\[
\mbox{$({\rm P_{4}})$�̍œK�� \ }
\left\{
\begin{array}{rclcl}
- \hat{x}_1 & = &  \dfrac{13}{34} \, c & \uncover<2->{=} 
	& \uncover<2->{\dfrac{F_7}{F_9} \, c} \\[4mm]
\hat{x}_2 & = & \dfrac{5}{13}(- \hat{x}_{1}) 
	& \uncover<2->{=} & \uncover<2->{\dfrac{F_5}{F_7}(- \hat{x}_1)} \\[4mm]
- \hat{x}_3 & = & \dfrac{2}{5} \, \hat{x}_{2} 
& \uncover<2->{=} & \uncover<2->{\dfrac{F_3}{F_5} \, \hat{x}_{2}} \\[4mm]
\hat{x}_4 & = & \dfrac{1}{2}(- \hat{x}_{3}) 
	& \uncover<2->{=} & \uncover<2->{\dfrac{F_1}{F_3}(- \hat{x}_{3})}
\end{array}
\right.
%\mbox{�ŋ��߂���. }
\]

\end{frame}
%%%%%%%%%%%%%%%
\end{comment}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{}
���Ȃ킿�A����$({\rm P_{c4}})$�̍ŏ���$\hat{x}$�́A
\[
\begin{array}{lcl}
\hat{x} & = &
(x_{0}, ~\hat{x}_{1}, ~\hat{x}_{2}, ~\hat{x}_{3}, ~\hat{x}_{4})  \\[3mm]
% & = & \dfrac{c}{\,\,\framebox(19,12){34}\,\,}
%\left(
%\begin{array}{ccccc}
%      \framebox(19,12){$34$}\,, & \framebox(20,12){$-13$}\,, & \framebox(19,12)%{$5$}\,, &  \framebox(19,12){$-2$}\,, & \framebox(19,12){1} 
%\end{array}
%\right) \\[4mm]
 & = & \dfrac{c}{\,\,F_{9} \,\,}
\left(
\begin{array}{ccccc}
F_{9} \,, & -F_{7} \,, & F_{5} \,, &  -F_{3} \,, & F_{1}
\end{array}
\right) 
\end{array}
\]
�ƂȂ�. 

\begin{center}
\unitlength=1mm
\begin{picture}(111,10)(0,0)
\allinethickness{0.8mm}
\put(0,0){\line(1,0){111}}	%
\put(0,-1){\line(0,1){2}}	%
\put(0,-3){\makebox(0,0){\large $-c$}}	%
\put(111,-1){\line(0,1){2}}	%
\put(111,-3){\makebox(0,0){\large $c$}}	%
\put(53,-1){\line(0,1){2}}	%
\put(53,-3){\makebox(0,0){\large $0$}}	%
%% x_0
\allinethickness{0.3mm}			
\put(111,5){\makebox(0,0){\large \RED{\small $x_0$}}}	%
\put(111,12){\makebox(0,0){\large \MAGENTA{\small $\dfrac{F_{9}}{F_{9}}c$}}}
%% -x_1
\put(76,-1){\line(0,1){2}}	
\put(76,-5){\makebox(0,0){\large \GRAY{\small $-x_1$}}}
\put(76,-12){\makebox(0,0){\large \GRAY{\small $\dfrac{F_{7}}{F_{9}}c$}}}
%% x_1
\put(30,-1){\line(0,1){2}}	
\put(30,5){\makebox(0,0){\large \RED{\small $x_1$}}}
\put(30,12){\makebox(0,0){\large \MAGENTA{\small $-\dfrac{F_{7}}{F_{9}}c$}}}
%% x_2
\put(63,-1){\line(0,1){2}}	
\put(63,5){\makebox(0,0){\large \RED{\small $x_2$}}}
\put(65,12){\makebox(0,0){\large \MAGENTA{\small $\dfrac{F_{5}}{F_{9}}c$}}}
%% -x_3
\put(57,-1){\line(0,1){2}}	
\put(57,-5){\makebox(0,0){\large \GRAY{\small $-x_3$}}}
\put(57,-12){\makebox(0,0){\large \GRAY{\small $\dfrac{F_{3}}{F_{9}}c$}}}
%% x_3
\put(49,-1){\line(0,1){2}}	
\put(49,5){\makebox(0,0){\large \RED{\small $x_3$}}}
\put(45,12){\makebox(0,0){\large \MAGENTA{\small $-\dfrac{F_{3}}{F_{9}}c$}}}
%% x_4
\put(55,-1){\line(0,1){2}}	
\put(55,5){\makebox(0,0){\large \RED{\small $x_4$}}}
\put(56,12){\makebox(0,0){\large \MAGENTA{\small $\dfrac{F_{1}}{F_{9}}c$}}}
\end{picture}
\end{center}
\vspace{17mm}

\pause

���̗�A$(x_{0}, \hat{x}_{1}, \hat{x}_{2}, \hat{x}_{3}, \hat{x}_{4})$
��\RED{���݃t�B�{�i�b�`�o�H}�Ƃ���. 
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{comment}
%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{}
���Ȃ킿�A����$({\rm P_{4}})$�̍ŏ���$\hat{x}$�́A
\[
\begin{array}{lcl}
\hat{x} & = &
(x_{0}, ~\hat{x}_{1}, ~\hat{x}_{2}, ~\hat{x}_{3}, ~\hat{x}_{4})  \\[3mm]
% & = & \dfrac{c}{\,\,\framebox(19,12){34}\,\,}
%\left(
%\begin{array}{ccccc}
%      \framebox(19,12){$34$}\,, & \framebox(20,12){$-13$}\,, & \framebox(19,12)%{$5$}\,, &  \framebox(19,12){$-2$}\,, & \framebox(19,12){1} 
%\end{array}
%\right) \\[4mm]
 & = & \dfrac{c}{\,\,\framebox(19,12){$F_{9}$}\,\,}
\left(
\begin{array}{ccccc}
      \framebox(19,12){$F_{9}$}\,, & \framebox(20,12){$-F_{7}$}\,, & \framebox(19,12){$F_{5}$}\,, &  \framebox(19,12){$-F_{3}$}\,, & \framebox(19,12){$F_{1}$} 
\end{array}
\right) 
\end{array}
\]
�ƂȂ�. 

\begin{center}
\unitlength=1mm
\begin{picture}(111,10)(0,0)
\allinethickness{0.8mm}
\put(0,0){\line(1,0){111}}	%
\put(0,-1){\line(0,1){2}}	%
\put(0,-3){\makebox(0,0){\large $-c$}}	%
\put(111,-1){\line(0,1){2}}	%
\put(111,-3){\makebox(0,0){\large $c$}}	%
\put(43,-1){\line(0,1){2}}	%
\put(43,-3){\makebox(0,0){\large $0$}}	%
%% x_0
\allinethickness{0.3mm}			
\put(111,3){\makebox(0,0){\large \RED{\small $x_0$}}}	%
\put(111,12){\makebox(0,0){\large \MAGENTA{\small $\dfrac{F_{9}}{F_{9}}c$}}}
%% -x_1
\put(69,-1){\line(0,1){2}}	
\put(69,-4){\makebox(0,0){\large \GRAY{\small $-x_1$}}}
\put(69,-11){\makebox(0,0){\large \GRAY{\small $\dfrac{F_{7}}{F_{9}}c$}}}
%% x_1
\put(17,-1){\line(0,1){2}}	
\put(17,3){\makebox(0,0){\large \RED{\small $x_1$}}}
\put(16,12){\makebox(0,0){\large \MAGENTA{\small $-\dfrac{F_{7}}{F_{9}}c$}}}
%% x_2
\put(53,-1){\line(0,1){2}}	
\put(53,3){\makebox(0,0){\large \RED{\small $x_2$}}}
\put(55,12){\makebox(0,0){\large \MAGENTA{\small $\dfrac{F_{5}}{F_{9}}c$}}}
%% -x_3
\put(47,-1){\line(0,1){2}}	
\put(47,-4){\makebox(0,0){\large \GRAY{\small $-x_3$}}}
\put(48,-11){\makebox(0,0){\large \GRAY{\small $\dfrac{F_{3}}{F_{9}}c$}}}
%% x_3
\put(39,-1){\line(0,1){2}}	
\put(39,3){\makebox(0,0){\large \RED{\small $x_3$}}}
\put(35,12){\makebox(0,0){\large \MAGENTA{\small $-\dfrac{F_{3}}{F_{9}}c$}}}
%% x_4
\put(45,-1){\line(0,1){2}}	
\put(45,3){\makebox(0,0){\large \RED{\small $x_4$}}}
\put(46,12){\makebox(0,0){\large \MAGENTA{\small $\dfrac{F_{1}}{F_{9}}c$}}}
\end{picture}
\end{center}

\end{frame}
%%%%%%%%%%%%%%%%%
\end{comment}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%
\begin{frame}
\frametitle{���$({\rm P_{c4}})$�̍œK�l�̓��o}
����$({\rm P_{c4}})$�̖ړI�֐��́A
\[
f(x) : \dql \dsum_{n=0}^{3}\left[(x_{n} + x_{n+1})^{2} + x_{n+1}^{2} \right]
\]
�ł���A�ŏ���$\hat{x}$�́A
\[
\begin{array}{lcl}
\hat{x} & = &
\dfrac{c}{\,\,F_{9}\,\,}
\left(
\begin{array}{ccccc}
F_{9} \,, & -F_{7} \,, & F_{5} \,, & -F_{3} \,, & F_{1} 
\end{array}
\right) 
\end{array}
\]
�Ȃ̂ŁA
\pause
\begin{eqnarray*}
f(\hat{x}) \eql 
\dfrac{c^2}{F_{9}^2} \Biggl\{ \left[(F_{9} - F_{7})^{2} 
+ (-F_{7})^{2} \right] + 
\left[(-F_{7} + F_{5})^{2} + F_{5}^{2} \right] \\[1.16mm]
 & & \ \ \ \ \ \ \ \ \ \ + \left[(F_{5} - F_{3})^{2} + (-F_{3})^{2} \right] 
	+ \left[(-F_{3} + F_{1})^{2} + F_{1}^{2} \right] \Biggr\} \\[1.56mm]
\end{eqnarray*}
�ƂȂ�B


\end{frame}



%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Lucas Formula}
\begin{block}{\textbf{�����J�̌����iLucas formula�j}}
\begin{eqnarray*}
\dsum_{k=1}^{n} F_{k}^{2} \eql F_{n}F_{n+1}
\end{eqnarray*}
\end{block}
\vspace{3mm}

\begin{table}[htbp]
%
\begin{center}
% \begin{tabular}{c|cccccccccccccccccc} \hline
\begin{tabular}{c|ccccccccccccc} \hline
$ n $     &  $ 0 $  &  $ 1 $  &  $ 2  $ &  $ 3 $ &  $ 4 $ &  $ 5 $  & $ 6 $ & $ 7 $ & $ 8 $ & $ 9 $ & $ 10 $ & $ 11 $ & $ 12 $ \\ \hline
%& $ 13 $ & $ 14 $ & $ 15 $ & $ 16 $   \\ \hline 
$ F_{n} $ &  $ 0 $  &  $ 1 $  &  $ 1 $ &  $ 2 $ &  $3 $ &  $ 5  $  & $ 8 $ & $ 13 $  & $ 21 $ & $ 34 $ & $ 55 $ & $ 89 $ & $ 144 $ \\ \hline
%& $ 233 $ & $ 377 $ & $ 610 $ & $ 987 $   \\ \hline
\end{tabular}
\end{center}
%\caption{stop reward} \label{tab:h1}
\end{table}
\vspace{-5mm}
\begin{center}
Table 1:~~Fibonacci sequence $ \{F_{n}\} $
%�\ 1 ~~�t�B�{�i�b�`���� $ \{F_{n}\} $
\end{center}

\pause
�Ⴆ�΁A$n \dql 5$�ł́A
\begin{eqnarray*}
& & F_{1}^{2} + F_{2}^{2} + F_{3}^{2} + F_{4} ^{2} + F_{5}^{2} \\[0.56mm]
& = & 1^{2} + 1^{2} + 2^{2} + 3^{2} + 5^{2} \\[0.56mm]
& = & 40 \\[0.56mm]
& = & F_{5} F_{6}
\end{eqnarray*}

\end{frame}






%%%%%%%%%%%%%%
\begin{frame}
\frametitle{}
%\frametitle{Lucas Formula}
%\begin{block}{\textbf{�����J�̌����iLucas formula�j}}
%\begin{eqnarray*}
%\dsum_{k=1}^{n} F_{k}^{2} \eql F_{n}F_{n+1}
%\end{eqnarray*}
%\end{block}
%
%\pause

\begin{eqnarray*}
f(\hat{x}) \eql 
\dfrac{c^2}{F_{9}^2} \Biggl\{ \left[(F_{9} - F_{7})^{2} 
+ (-F_{7})^{2} \right] + 
\left[(-F_{7} + F_{5})^{2} + F_{5}^{2} \right] \\[1.16mm]
 & & \ \ \ \ \ \ \ \ \ \ + \left[(F_{5} - F_{3})^{2} + (-F_{3})^{2} \right] 
	+ \left[(-F_{3} + F_{1})^{2} + F_{1}^{2} \right] \Biggr\} \\[1.56mm]
\end{eqnarray*}
���A
\pause

\begin{eqnarray*}
F_{9}^{2}\dfrac{\,f(\hat{x})\,}{c^{2}} \eql \left[(F_{9} - F_{7})^{2} 
+ (-F_{7})^{2} \right] + \left[(-F_{7} + F_{5})^{2} 
+ F_{5}^{2} \right] \\[1.16mm]
 & & \ \ \ \ \ \ \ \ \ \ + \left[(F_{5} - F_{3})^{2} + (-F_{3})^{2} \right] 
	+ \left[(-F_{3} + F_{1})^{2} + F_{1}^{2} \right] \\[1.56mm]
%
\eql  F_{8}^{2} + F_{7}^{2} + F_{6}^{2} + F_{5}^{2} + F_{4}^{2} 
+ F_{3}^{2} + F_{2}^{2} + F_{1}^{2} \\[1.56mm]
%
\eql  F_{8} \ndot F_{9}.
\end{eqnarray*}

���Ȃ킿�A�ŏ��l�� \ 
%\begin{eqnarray*}
$
f(\hat{x}) \dql \dfrac{F_{8}}{\,F_{9}\,}c^{2} = \dfrac{21}{\,34\,} \, c^{2}
$
%\end{eqnarray*}
\ �ł���. 

\end{frame}


%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{���݃t�B�{�i�b�`���� \\
(Alternately Fibonacci Condition)}
����$({\rm P_{c4}})$�y�сA�o�Ζ��$({\rm D_{c4}})$�ɂ����āA
���ꂼ�ꎟ��4�̓�������Ȃ�n${\rm (AF)_{P_{c}}}$�y��${\rm (AF)_{D_{c}}}$���A
���ꂼ����$({\rm P_{c4}})$�A$({\rm D_{c4}})$�ɑ΂���
\RED{���݃t�B�{�i�b�`����}�Ƃ���. 
\[
\textcolor<2>{gray}{
({\rm AF})_{\rm P_{c}}
\left\{
\begin{array}{ll}
\dfrac{\,c + x_{1}\,}{F_{8}} \dql \dfrac{\,x_{1}\,}{-F_{7}}, &
\dfrac{\,x_{1} + x_{2}\,}{-F_{6}} \dql \dfrac{\,x_{2}\,}{F_{5}}, \\[5mm]
\dfrac{\,x_{2} + x_{3}\,}{F_{4}} \dql \dfrac{\,x_{3}\,}{-F_{3}}, &
\dfrac{\,x_{3} + x_{4}\,}{-F_{2}}  \dql \dfrac{\,x_{4}\,}{F_{1}}.
\end{array}
\right.
}
\]
\vspace{2mm}
\[
\textcolor<2>{red}{
\hspace{3.6mm}({\rm AF})_{\rm D_{c}}
\left\{
\begin{array}{l}
\dfrac{\,c - \mu_{0}\,}{F_{7}}  
\dql \dfrac{\,\mu_{0} + \mu_{1}\,}{F_{7}} \dql 
\dfrac{\,\mu_{1}\,}{\,-F_{6}\,}, \\[5mm]
\dfrac{\,\mu_{1} + \mu_{2}\,}{\,-F_{5}\,} \dql 
\dfrac{\,\mu_{2}\,}{F_{4}}, \ \ 
\dfrac{\,\mu_{2} + \mu_{3}\,}{F_{3}}  \dql \dfrac{\,\mu_{3}\,}{\,-F_{2}\,}.
\end{array}
\right.
}
\]

\end{frame}





%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{${\rm (AF)_{D_{c}}}$�̓��l����}
�o�Ζ��$({\rm D_{c4}})$�̖ړI�֐��� 
$ \mu = (\mu_{0}, \mu_{1}, \mu_{2}, \mu_{3}) $ ��4�ϐ��֐� $ g(\mu) $ �ŕ\��. 
\[
g(\mu) : \dql 
2c\mu_{0} - \mu_{0}^{2} - \dsum_{n=0}^{2}\left[(\mu_{n} + \mu_{n+1})^{2} + \mu_{n+1}^{2} \right] - \mu_{3}^{2}.
\]


\begin{eqnarray*}
%& & \mbox{���݃t�B�{�i�b�`����}{\rm (AF)_{D}} \\[3mm]
{\rm (AF)_{D_{c}}}
& \Longleftrightarrow & 
\left\{
\begin{array}{rcl}
-(c - \mu_{0}) + (\mu_{0} + \mu_{1}) & \dql & 0 \\[0.56mm]
(\mu_{0} + \mu_{1}) + \mu_{1} + (\mu_{1} + \mu_{2}) & \dql & 0  \\[-2.45mm]
& &  \\[-2.45mm]
(\mu_{1} + \mu_{2}) + \mu_{2} + (\mu_{2} + \mu_{3}) & \dql & 0  \\[0.56mm]
(\mu_{2} + \mu_{3}) + 2\mu_{3}  & \dql & 0
\end{array}
\right. \\[2mm]
 & \Longleftrightarrow &  
\dfrac{\partial g}{\partial \mu_{n}} \dql 0 ~~~~~n = 0, 1, 2, 3. \\
\end{eqnarray*}

�ړI�֐�$g(\mu)$��$\mu$��2���̉��֐��ł��邱�Ƃ���A
���$({\rm D_{c4}})$�̍ő��
$\mu^{*} = (\mu_{0}^{*}, ~\mu_{1}^{*}, ~\mu_{2}^{*}, ~\mu_{3}^{*})$
�́A�œK����1�K�����𖞂����B
\end{frame}



%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{���$({\rm D_{c4}})$�̌��݃t�B�{�i�b�`����}
\begin{eqnarray*}
({\rm AF})_{\rm D_{c}}
& \stackrel{\mbox{\itshape\scriptsize def}}{\Longleftrightarrow} &
\left\{
\begin{array}{l}
\dfrac{\,c - \mu_{0}\,}{F_{7}}  
\dql \dfrac{\,\mu_{0} + \mu_{1}\,}{F_{7}} \dql 
\dfrac{\,\mu_{1}\,}{\,-F_{6}\,}, \\[5mm]
\dfrac{\,\mu_{1} + \mu_{2}\,}{\,-F_{5}\,} \dql 
\dfrac{\,\mu_{2}\,}{F_{4}}, \ \ 
\dfrac{\,\mu_{2} + \mu_{3}\,}{F_{3}}  \dql \dfrac{\,\mu_{3}\,}{\,-F_{2}\,},
\end{array}
\right. \\[2mm]
\uncover<2->{
& \Longleftrightarrow &
\left\{
\begin{array}{l}
\dfrac{\,c - \mu_{0}\,}{F_{7}}  
\dql \dfrac{\,\mu_{0} - (- \mu_{1}) \,}{F_{7}} \dql 
\dfrac{\, - \mu_{1}\,}{\,F_{6}\,}, \\[5mm]
\dfrac{\,- \mu_{1} - \mu_{2}\,}{\,F_{5}\,} \dql 
\dfrac{\,\mu_{2}\,}{F_{4}}, \ \ 
\dfrac{\,\mu_{2} - (- \mu_{3} )\,}{F_{3}}  
\dql \dfrac{\,- \mu_{3}\,}{\,F_{2}\,},
\end{array}
\right. } \\[2mm]
\uncover<3>{
& \Longleftrightarrow &
\left\{
\begin{array}{l}
\dfrac{\,c - \mu_{0}\,}{13}  
\dql \dfrac{\,\mu_{0} - (- \mu_{1}) \,}{13} \dql 
\dfrac{\, - \mu_{1}\,}{\,8\,}, \\[5mm]
\dfrac{\,- \mu_{1} - \mu_{2}\,}{\,5\,} \dql 
\dfrac{\,\mu_{2}\,}{3}, \ \ 
\dfrac{\,\mu_{2} - (- \mu_{3} )\,}{2}  
\dql \dfrac{\,- \mu_{3}\,}{\,1\,}.
\end{array}
\right. } \\
\end{eqnarray*}
\end{frame}



%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{���$({\rm D_{c4}})$�̌��݃t�B�{�i�b�`�����ɂ��œK���̉�@}
$x_{0} = c \,~(\, > 0 \,) $ \\
\[
\begin{array}{l}
\textcolor<2>{red}{\dfrac{\,c - \mu_{0}\,}{13}  
\dql \dfrac{\,\mu_{0} - (- \mu_{1}) \,}{13} \dql 
\dfrac{\, - \mu_{1}\,}{\,8\,}}, \\[5mm]
\textcolor<3>{red}{\dfrac{\,- \mu_{1} - \mu_{2}\,}{\,5\,} \dql 
\dfrac{\,\mu_{2}\,}{3}}, \ \ 
\textcolor<4>{red}{\dfrac{\,\mu_{2} - (- \mu_{3} )\,}{2}  
\dql \dfrac{\,- \mu_{3}\,}{\,1\,}}.
\end{array}
\]

\begin{center}
\unitlength=1mm
\begin{picture}(111,10)(0,0)
\uncover<1->{
\allinethickness{0.8mm}
\put(0,0){\line(1,0){111}}	%
\put(0,-1){\line(0,1){2}}	%
\put(-2,3){\makebox(0,0){\large $0$}}	%
\put(111,-1){\line(0,1){2}}	%
\put(111,3){\makebox(0,0){\large $c$}}	%
}
%% -\mu_{1}, \mu_{0}
\uncover<2->{
\allinethickness{0.3mm}
\put(27,-1){\line(0,1){2}}	
\put(27,3){\makebox(0,0){\large $- \mu_{1}^{*}$}}		
\put(69,-1){\line(0,1){2}}	
\put(69,3){\makebox(0,0){\large $\mu_{0}^{*}$}}
}
\uncover<2>{
\bezier{200}(0,0)(13,-4)(27,0)	
\put(13,-6){\makebox(0,0){\large $8$}}
\put(13,-11){\makebox(0,0){\large $F_{6}$}}
\bezier{200}(27,0)(48,-4)(69,0)	
\put(48,-6){\makebox(0,0){\large $13$}}
\put(48,-11){\makebox(0,0){\large $F_{7}$}}
\bezier{200}(69,0)(90,-4)(111,0)	
\put(90,-6){\makebox(0,0){\large $13$}}
\put(90,-11){\makebox(0,0){\large $F_{7}$}}
}
%% \mu_{2}
\uncover<3->{
\allinethickness{0.3mm}			
\put(12,-1){\line(0,1){2}}	
\put(12,3){\makebox(0,0){\large $\mu_{2}^{*}$}}
}
\uncover<3>{
\bezier{100}(0,0)(6,-3)(12,0)	
\put(6,-6){\makebox(0,0){\large $3$}}
\put(6,-11){\makebox(0,0){\large $F_{4}$}}
\bezier{100}(12,0)(18,-3)(27,0)	
\put(18,-6){\makebox(0,0){\large $5$}}
\put(18,-11){\makebox(0,0){\large $F_{5}$}}
}
%% -\mu_{3}
\uncover<4->{
\allinethickness{0.3mm}			
\put(3,-1){\line(0,1){2}}	
\put(3,6){\makebox(0,0){\large $-\mu_{3}^{*}$}}
}
\uncover<4>{
\bezier{50}(0,0)(2,-2)(4,0)	
\put(2,-6){\makebox(0,0){\large $1$}}
\put(2,-11){\makebox(0,0){\large $F_{2}$}}
\bezier{50}(4,0)(8,-2)(12,0)	
\put(8,-6){\makebox(0,0){\large $2$}}
\put(8,-11){\makebox(0,0){\large $F_{3}$}}
}
\end{picture}
\end{center}

\end{frame}




%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{}
\begin{center}
\unitlength=1mm
\begin{picture}(111,10)(0,0)
\allinethickness{0.8mm}
\put(0,0){\line(1,0){111}}	%
\put(0,-1){\line(0,1){2}}	%
\put(-3,6){\makebox(0,0){\large $0$}}	%
\put(111,-1){\line(0,1){2}}	%
\put(111,6){\makebox(0,0){\large $c$}}	%
%% -\mu_{1}, \mu_{0}
\uncover<2->{
\allinethickness{0.3mm}
\put(27,-1){\line(0,1){2}}	
\put(27,6){\makebox(0,0){\large $- \mu_{1}^{*}$}}		
\put(69,-1){\line(0,1){2}}	
\put(69,6){\makebox(0,0){\large $\mu_{0}^{*}$}}
\bezier{200}(27,0)(48,-4)(69,0)	
\put(48,-6){\makebox(0,0){\large $F_{7}$}}
\bezier{200}(69,0)(90,-4)(111,0)	
\put(90,-6){\makebox(0,0){\large $F_{7}$}}
}
\uncover<2>{
\bezier{200}(0,0)(13,-4)(27,0)	
\put(13,-6){\makebox(0,0){\large $F_{6}$}}
}
%% \mu_{2}
\uncover<3->{
\allinethickness{0.3mm}			
\put(12,-1){\line(0,1){2}}	
\put(13,6){\makebox(0,0){\large $\mu_{2}^{*}$}}
\bezier{100}(12,0)(18,-3)(27,0)	
\put(18,-6){\makebox(0,0){\large $F_{5}$}}
}
\uncover<3>{
\bezier{100}(0,0)(6,-3)(12,0)	
\put(6,-6){\makebox(0,0){\large $F_{4}$}}
}
%% -\mu_{3}
\uncover<4->{
\allinethickness{0.3mm}			
\put(3,-1){\line(0,1){2}}	
\put(5,6){\makebox(0,0){\large $-\mu_{3}^{*}$}}
\bezier{50}(0,0)(2,-2)(4,0)	
\put(2,-6){\makebox(0,0){\large $F_{2}$}}
\bezier{50}(4,0)(8,-2)(12,0)	
\put(8,-6){\makebox(0,0){\large $F_{3}$}}}
\end{picture}
\end{center}
\vspace{5mm}

\[
\mbox{$({\rm D_{c4}})$�̍œK�� \ }
\left\{
\begin{array}{rclcl}
\uncover<2->{\mu_{0}^{*} & = &  \dfrac{F_{6} + F_{7}}{F_{6}+F_{7}+F_{7}} \ c }
& \uncover<2->{= & \dfrac{F_8}{F_9} \, c \,,} \\[4mm]
\uncover<2->{-\mu_{1}^{*} & = & \dfrac{F_{6}}{F_{6}+F_{7}+F_{7}} \ c}
& \uncover<2->{= & \dfrac{F_6}{F_9} \, c \,,} \\[4mm]
\uncover<3->{\mu_{2}^{*} & = & \dfrac{F_{4}}{F_{4}+F_{5}} \,(-\mu_{1}^{*})} 
& \uncover<5->{= & \dfrac{F_4}{F_6} \ndot \dfrac{F_6}{F_9} \, c \,,} \\[4mm]
\uncover<4->{-\mu_{3}^{*} & = & \dfrac{F_{2}}{F_{2}+F_{3}} \, \mu_{2}^{*}} 
& \uncover<5->{= & \dfrac{F_2}{F_4} \ndot \dfrac{F_4}{F_6} 
\ndot \dfrac{F_6}{F_9} \, c \,.}
\end{array}
\right.
%\mbox{�ŋ��߂���. }
\]

\end{frame}


%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{}
���Ȃ킿�A�o�Ζ��$({\rm D_{c4}})$�̍ő��$\mu^{*}$�́A
\[
\begin{array}{lcl}
\mu^{*} 
& = &
(\mu^{*}_{0},~ \mu^{*}_{1},~ \mu^{*}_{2},~ \mu^{*}_{3}) \\[3mm]
 & = & 
\dfrac{c}{\,\,F_{9}\,\,}
\left(
\begin{array}{cccc}
       F_{8}\,, & -F_{6}\,, &  F_{4}\,, & -F_{2} 
\end{array}
\right) \\
\end{array}
\]
�ƂȂ�. 


\begin{center}
\unitlength=1mm
\begin{picture}(111,10)(0,0)
\allinethickness{0.8mm}
\put(0,0){\line(1,0){111}}	%
\put(0,-1){\line(0,1){2}}	%
\put(0,-3){\makebox(0,0){\large $-c$}}	%
\put(111,-1){\line(0,1){2}}	%
\put(111,-3){\makebox(0,0){\large $c$}}	%
\put(53,-1){\line(0,1){2}}	%
\put(53,-3){\makebox(0,0){\large $0$}}	%
\allinethickness{0.3mm}			
\put(67,-1){\line(0,1){2}}	%
\put(67,-8){\makebox(0,0){\large $-\mu_{1}^{*}$}}	%
\put(39,-1){\line(0,1){2}}	%
\put(39,6){\makebox(0,0){\large $\RED{\mu_{1}^{*}}$}}	%
\put(39,15){\makebox(0,0){\large \MAGENTA{\small $-\dfrac{F_{6}}{F_{9}}c$}}}
\put(89,-1){\line(0,1){2}}	%
\put(89,6){\makebox(0,0){\large $\RED{\mu_{0}^{*}}$}}	%
\put(89,15){\makebox(0,0){\large \MAGENTA{\small $\dfrac{F_{8}}{F_{9}}c$}}}
\put(59,-1){\line(0,1){2}}	%
\put(59,6){\makebox(0,0){\large $\RED{\mu_{2}^{*}}$}}	%
\put(61,15){\makebox(0,0){\large \MAGENTA{\small $\dfrac{F_{4}}{F_{9}}c$}}}
\put(55,-1){\line(0,1){2}}	%
\put(55,-8){\makebox(0,0){\large $-\mu_{3}^{*}$}}	%
\put(51,-1){\line(0,1){2}}	%
\put(51,6){\makebox(0,0){\large $\RED{\mu_{3}^{*}}$}}	%
\put(51,15){\makebox(0,0){\large \MAGENTA{\small $-\dfrac{F_{2}}{F_{9}}c$}}}
\end{picture}
\end{center}
\vspace{10mm}

\pause

�� $(\mu_{0}^{*}, \mu_{1}^{*}, \mu_{2}^{*}, \mu_{3}^{*} )$
�͌��݃t�B�{�i�b�`�o�H���\�����Ă���. 

\end{frame}



%%%%%%%%%%%%%
\begin{frame}
\frametitle{���$({\rm D_{c4}})$�̍œK�l�̓��o}
�o�Ζ��$({\rm D_{c4}})$�̖ړI�֐��́A
\[
g(\mu) : \dql 
2c\mu_{0} - \mu_{0}^{2} - \dsum_{n=0}^{2}\left[(\mu_{n} + \mu_{n+1})^{2} + \mu_{n+1}^{2} \right] - \mu_{3}^{2}
\]
�ł���A�ő��$\mu^{*}$�́A
\[
\begin{array}{lcl}
\mu^{*} 
%& = &
%(\mu^{*}_{0},~ \mu^{*}_{1},~ \mu^{*}_{2},~ \mu^{*}_{3}) \\[3mm]
 & = & 
\dfrac{c}{\,\,F_{9}\,\,}
\left(
\begin{array}{cccc}
       F_{8}\,, & -F_{6}\,, &  F_{4}\,, & -F_{2} 
\end{array}
\right) \\
\end{array}
\]
�Ȃ̂ŁA
\pause
\begin{eqnarray*}
g(\mu^{*}) 
\eql  \dfrac{c^2}{F_{9}^{2}} \Biggl\{ 2F_{8}F_{9} - F_{8}^{2} - \left[(F_{8} - F_{6})^{2} + (-F_{6})^{2} \right] \\[1.56mm]
& & \ \ \ \ \ \ - \left[ (-F_{6} + F_{4})^{2} + F_{4}^{2}  \right]  \\[1.56mm]
& & \ \ \ \ \ \ - \left[ (F_{4} - F_{2})^{2} + (-F_{2})^{2} \right] 
- (-F_{2})^{2} \Biggr\}.
\end{eqnarray*}

\end{frame}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{}
�����J�̌������A
\begin{eqnarray*}
g(\mu^{*})F_{9}^{2}/c^{2} 
\eql  2F_{8}F_{9} - F_{8}^{2} 
- \left[(F_{8} - F_{6})^{2} + (-F_{6})^{2} \right] \\[1.56mm]
& & \ \ \ \ \ \ - \left[ (-F_{6} + F_{4})^{2} + F_{4}^{2}  \right]  \\[1.56mm]
& & \ \ \ \ \ \ - \left[ (F_{4} - F_{2})^{2} + (-F_{2})^{2} \right] - (-F_{2})^{2} \\[2.56mm]
%
\eql  2F_{8}F_{9} - \left[F_{8}^{2} + F_{7}^{2} + F_{6}^{2} + F_{5}^{2} + F_{4}^{2} + F_{3}^{2} + F_{2}^{2} + F_{1}^{2}\right] \\[2.56mm]
%
\eql  2F_{8}F_{9} - F_{8}F_{9} \\[1.56mm]
%
\eql  F_{8}F_{9} 
\end{eqnarray*}
�ƂȂ�A
���Ȃ킿�A�ő�l�� \ 
%\begin{eqnarray*}
$
g(\mu^{*}) \dql \dfrac{F_{8}}{\,F_{9}\,}c^{2} = \dfrac{21}{\,34\,} \, c^{2}
$\ �ł���. 
%\end{eqnarray*} 



\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{comment}
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{���݃_�E���B���`�E�R�[�h}
�f��u�_�E���B���`�E�R�[�h�v�i2006�N�j�ł͖`���̃V�[����10���̈Ïؔԍ�
{\large
\[
1~~~1~~~2~~~3~~~5~~~8~~~1~~~3~~~2~~~1
\]
}
������Ă���. 

\pause 

����10�̐����� 8�̐�������Ȃ��
$$ 
\RedBlackframebox{24}{15}{$1$}~~\,~\RedBlackframebox{24}{15}{$1$}~~\,
~\RedBlackframebox{24}{15}{$2$}~~\,~\RedBlackframebox{24}{15}{$3$}~~\,
~\RedBlackframebox{24}{15}{$5$}~~\,~\RedBlackframebox{24}{15}{$8$}~~\,
~\RedBlackframebox{24}{15}{$13$}~~\,~\RedBlackframebox{24}{15}{$21$} 
$$
���Ȃ킿�A�t�B�{�i�b�`����̑�1�� 
$ F_{1} = 1 $ �����8�� $ F_{8} = 21 $ �܂ł̐���ɑ��Ȃ�Ȃ�. 

\begin{table}[htbp]
%
\begin{center}
% \begin{tabular}{c|cccccccccccccccccc} \hline
\begin{tabular}{c|ccccccccccccc} \hline
$ n $     &  $ 0 $  &  $ 1 $  &  $ 2  $ &  $ 3 $ &  $ 4 $ &  $ 5 $  & $ 6 $ & $ 7 $ & $ 8 $ & $ 9 $ & $ 10 $ & $ 11 $ & $ 12 $ \\ \hline
%& $ 13 $ & $ 14 $ & $ 15 $ & $ 16 $   \\ \hline 
$ F_{n} $ &  $ 0 $  &  $ 1 $  &  $ 1 $ &  $ 2 $ &  $3 $ &  $ 5  $  & $ 8 $ & $ 13 $  & $ 21 $ & $ 34 $ & $ 55 $ & $ 89 $ & $ 144 $ \\ \hline
%& $ 233 $ & $ 377 $ & $ 610 $ & $ 987 $   \\ \hline
\end{tabular}
\end{center}
%\caption{stop reward} \label{tab:h1}
\end{table}
\vspace{-5mm}
\begin{center}
Table 1:~~Fibonacci sequence $ \{F_{n}\} $
%�\ 1 ~~�t�B�{�i�b�`���� $ \{F_{n}\} $
\end{center}

\end{frame}



%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{���]���}
���� $({\rm P_{4}})$ ����ёo�Ζ�� $({\rm D_{4}})$ �̕ϐ��𔽓]�����āA���]���
%
\begin{eqnarray*}
& &{\rm minimize} \hspace{3mm} \dsum_{n=1}^{4}\left[ x_{n}^{2} + (x_{n} + x_{n+1})^{2} \right] \\ 
({\rm RP_{4}})& &{\rm subject~~ to}\hspace{2ex} {\rm (i)}~~ -\infty < x_{n} < \infty ~~~~n= 1,2,3,4 \\
& &\hspace*{21.2mm} {\rm (ii)}~~x_{5} = c
\end{eqnarray*}
%
�����
%
\begin{eqnarray*}
\hspace*{-7mm}& &{\rm Maximize} \hspace{3mm} - \mu_{1}^{2} - \dsum_{n=1}^{3}\left[\mu_{n}^{2} + (\mu_{n} + \mu_{n+1})^{2}  \right] 
- \mu_{4}^{2} + 2c\mu_{4}  \\[-2,56mm] 
\hspace*{-7mm}({\rm RD}_{4})& & \\[-2,56mm] 
\hspace*{-7mm}& &{\rm subject~\, to}\hspace{2ex} {\rm (i)}~~~ -\infty < \mu_{n} < \infty ~~~~n= 1,2,3,4
\end{eqnarray*}
%
���l����. 
%�����]���̍œK���̊Ԃɂ͍��x��\textbf{�O������}
%���݃t�B�{�i�b�`����o�ΐ������藧��. 


\end{frame}





%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{}
���$({\rm RP_{4}})$�̍ŏ���$\tilde{x}$�ƍŏ��l$m^{\prime}_{4}$�́A
\[
\hspace{10mm}\begin{array}{lcl}
\tilde{x} & \dql & (\tilde{x}_{1}, ~\tilde{x}_{2}, ~\tilde{x}_{3}, 
~\tilde{x}_{4}, ~x_{5}) \\[3mm]
 & \dql & \dfrac{c}{\,\,34 \,\,}
\left(
\begin{array}{ccccc}
      1 \,, & -2 \,, & 5 \,, &  -13 \,, & 34 
\end{array}
\right) \\[6mm]
m_{4}^{\prime} & = & 
\dfrac{\,21 \,}{\,\,34 \,\,}c^{2}
\end{array}
\]
�ł���A
���$({\rm RD_{4}})$�̍ő��$\mu^{\star}$�ƍő�l$M^{\prime}_{4}$�́A
\[
\begin{array}{lcl}
\mu^{\star} & \dql & 
( \mu^{\star}_{1},~ \mu^{\star}_{2},~ \mu^{\star}_{3},
~ \mu^{\star}_{4}) \\[3mm]
 & \dql & \dfrac{c}{\,\,34 \,\,}
\left(
\begin{array}{cccc}
      -1 \,, & 3 \,, & -8 \,, &  21 
\end{array}
\right) \\[6mm]
M_{4}^{\prime} & = & 
\dfrac{\,21 \,}{\,\,34 \,\,}c^{2}
\end{array}
\]
�ƂȂ�. 
\end{frame}



%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{}
�����]���̍œK���̊Ԃɂ͍��x��\RED{�O������}���݃t�B�{�i�b�`����o�ΐ���
���藧��. 
%���Ȃ킿�A
%���]���� $ ({\rm RP}_{4}) $ �̍ŏ����Ɣ��]�o�Ζ�� $ ({\rm RD}_{4}) $ �̍ő�%���̊Ԃɂ͎��̎O�ʈ�̂̊֌W�����藧���Ă���.  
\begin{itemize}
	\item \BLUE{(�o�ΐ�)}\\[2mm]
	�ŏ��l�ƍő�l���������F
	\[ 
	m_{4}^{\prime} \dql M_{4}^{\prime} \dql \dfrac{21}{\,34\,} \,c^{2}.
	\]
	\item \BLUE{(2�i�K�t�B�{�i�b�`)}\\[2mm]
	�ŏ��� $ \tilde{x} $
%$ (\tilde{x}_{1}, ~\tilde{x}_{2}, ~\tilde{x}_{3}, ~\tilde{x}_{4}) $ 
�ƍő�_ $\mu^{\star}$
%$( \mu^{\star}_{1},~ \mu^{\star}_{2},~ \mu^{\star}_{3},~ \mu^{\star}_{4}) $ 
�͋��Ɍ��t�B�{����̑O����2�i���тł���. 
\[
\begin{array}{lcl}
\tilde{x} & = &
(\tilde{x}_{1}, ~\tilde{x}_{2}, ~\tilde{x}_{3}, 
~\tilde{x}_{4}, ~x_{5})  \\[3mm]
 & = & \dfrac{c}{\,\,34\,\,}
\left(
\begin{array}{ccccc}
\MagRedframebox{19}{12}{$1$}\,, & \MagBlueframebox{19}{12}{$-2$}\,, 
& \MagRedframebox{19}{12}{$5$}\,, 
&  \MagBlueframebox{19}{12}{$-13$}\,, & \MagRedframebox{19}{12}{$34$} 
\end{array}
\right) \\[4mm] 
 & = & \dfrac{c}{\,\,F_{9} \,\,}
\left(
\begin{array}{ccccc}
\MagRedframebox{19}{12}{$F_{1}$}\,, & \MagBlueframebox{19}{12}{$-F_{3}$}\,, 
& \MagRedframebox{19}{12}{$F_{5}$}\,, &  \MagBlueframebox{19}{12}{$-F_{7}$}\,, 
& \MagRedframebox{19}{12}{$F_{9}$} 
\end{array}
\right),
\end{array}
\]
\end{itemize}
\end{frame}






%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{}
\[
\hspace*{-15mm}
\begin{array}{lcl}
\mu^{\star} & = &
(\mu^{\star}_{1},~ \mu^{\star}_{2},~ \mu^{\star}_{3},~ \mu^{\star}_{4}) \\[3mm]
 & = & 
\dfrac{c}{\,\,34 \,\,}
\left(
\begin{array}{cccc}
       \CyanBlueframebox{19}{12}{$-1$}\,, & \CyanRedframebox{19}{12}{$3$}\,, 
&  \CyanBlueframebox{19}{12}{$-8$}\,, & \CyanRedframebox{19}{12}{$21$} 
\end{array}
\right) \\[4mm]
 & = & 
\dfrac{c}{\,\,F_{9} \,\,}
\left(
\begin{array}{cccc}
       \CyanBlueframebox{19}{12}{$-F_{2}$}\,, 
& \CyanRedframebox{19}{12}{$F_{4}$}\,, 
&  \CyanBlueframebox{19}{12}{$-F_{6}$}\,, 
& \CyanRedframebox{19}{12}{$F_{8}$} 
\end{array}
\right). 
\end{array}
\]

\pause

\begin{itemize}
\item \BLUE{(����t�B�{�i�b�`)}\\[2mm]
�ŏ���$\tilde{x}$�ƍő��$\mu^{\star}$�����݂ɕ҂ނƁA
$$ 
\framebox(24,15){$\tilde{x}_{1}$}~~\,~\dashbox(24,15){$\mu^{\star}_{1}$}~~\,~\framebox(24,15){$\tilde{x}_{2}$}~~\,~\dashbox(24,15){$\mu^{\star}_{2}$}~~\,~\framebox(24,15){$\tilde{x}_{3}$}~~\,~\dashbox(24,15){$\mu^{\star}_{3}$}~~\,~\framebox(24,15){$\tilde{x}_{4}$}~~\,~\dashbox(24,15){$\mu^{\star}_{4}$}~~\,~\framebox(24,15){$x_{5}$}
$$
$$ 
\MagRedframebox{24}{15}{$1$}~~\,~
\CyanBlueframebox{24}{15}{$-1$}~~
\,~\MagBlueframebox{24}{15}{$-2$}~~
\,~\CyanRedframebox{24}{15}{$3$}~~\,~
\MagRedframebox{24}{15}{$5$}~~\,~
\CyanBlueframebox{24}{15}{$-8$}~~\,~
\MagBlueframebox{24}{15}{$-13$}~~
\,~\CyanRedframebox{24}{15}{$21$}~~\,~
\MagRedframebox{24}{15}{$34$}
$$
�ɂȂ�. 
\end{itemize}

\pause

\RED{���Ȃ킿�A����́u���݃_�E���B���`�E�R�[�h�v���\�����Ă���. }
\end{frame}


\end{comment}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	���݃t�B�{�i�b�`�@�U�F�S
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%
\begin{comment}
%%%%%%%%%%%%%
\begin{frame}
\frametitle{\large Mathematical Programming Problems : 
$({\rm P_{4}})$ and $({\rm D_{4}})$}
����(primal problem)�Ƃ��Ď��̍ŏ��������l����D
\begin{eqnarray*}
& &{\rm minimize} \hspace{3mm} 
\dsum_{n=0}^{3}\left[(x_{n} + x_{n+1})^{2} + x_{n+1}^{2} \right] \\ 
({\rm P_{4}})& &{\rm subject~~ to}
\hspace{2ex} {\rm (i)}~~ -\infty < x_{n} < \infty ~~~~n= 1,2,3,4 \\
& &\hspace*{20.2mm} {\rm (ii)}~~x_{0} = c.
\end{eqnarray*}

\pause

���̎���$({\rm P_{4}})$�ɑ΂���o�Ζ��(dual problem)$({\rm D_{4}})$�́A
���ŕ\�����D
\begin{eqnarray*}
& &{\rm Maximize} \hspace{3mm} 2c\mu_{0} - \mu_{0}^{2} - \dsum_{n=0}^{2}\left[(\mu_{n} + \mu_{n+1})^{2} + \mu_{n+1}^{2} \right] - \mu_{3}^{2} \\[-2,56mm] 
({\rm D}_{4})& & \\[-2,56mm] 
& &{\rm subject~\, to}\hspace{2ex} {\rm (i)}~~~ -\infty < \mu_{n} < \infty ~~~~n= 0,1,2,3.
\end{eqnarray*}
\end{frame}
%%%%%%%%%%%%%
\end{comment}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Primal Problem $({\rm P_{s8}})$}
����(primal problem)�Ƃ��Ď��̍ŏ��������l����B
\begin{eqnarray*}
& &{\rm minimize} \hspace{3mm} \dsum_{n=0}^{7}\left[F_{8-n}(x_{n} + x_{n+1})^{2} + \dfrac{F_{7-n}^{2}}{F_{8-n}} x_{n+1}^{2} \right] 
+ \dfrac{F_{0}F_{1}}{F_{2}} x_{8}^{2} \\ 
({\rm P_{s8}})& &{\rm subject~~ to}\hspace{2ex} 
{\rm (i)}~~ -\infty < x_{n} < \infty ~~~~n= 1, 2, \cdots, 8 \\
& &\hspace*{20.5mm} {\rm (ii)}~~x_{0} = c.
%& &{\rm minimize} \hspace{3mm} 
%\dsum_{n=0}^{3}\left[(x_{n} + x_{n+1})^{2} + x_{n+1}^{2} \right] \\ 
%({\rm P_{4}})& &{\rm subject~~ to}
%\hspace{2ex} {\rm (i)}~~ -\infty < x_{n} < \infty ~~~~n= 1,2,3,4 \\
%& &\hspace*{20.2mm} {\rm (ii)}~~x_{0} = c.
\end{eqnarray*}

\pause 

����$({\rm P_{s8}})$�̍ŏ���$\hat{x}$�ƍŏ��l$m_{8}$�́A
\[
\begin{array}{lcl}
\hat{x} & = &
(\hat{x}_{1}, ~\hat{x}_{2}, ~\cdots, ~\hat{x}_{8})  \\[3mm]
 & = & \dfrac{c}{\,\,34 \,\,}
\left(
\begin{array}{cccccccc}
      -21 \,, & 13 \,, & -8 \,, & 5 \,, &  -3 \,, & 2 \,, & -1 \,, & 1
\end{array}
\right) \\[6mm]
m_{8} & = & 
\dfrac{\,13 \cdot 21 \,}{\,\,34 \,\,}c^{2}
\end{array}
\]
%\[
%\begin{array}{lcl}
%\hat{x} & = &
%(x_{0}, ~\hat{x}_{1}, ~\hat{x}_{2}, ~\hat{x}_{3}, ~\hat{x}_{4})  \\[3mm]
% & = & \dfrac{c}{\,\,\framebox(19,12){34}\,\,}
%\left(
%\begin{array}{ccccc}
%      \framebox(19,12){$34$}\,, & \framebox(20,12){$-13$}\,, & \framebox(19,12)%{$5$}\,, &  \framebox(19,12){$-2$}\,, & \framebox(19,12){1} 
%\end{array}
%\right) \\[6mm]
%m_{4} & = & 
%\dfrac{\,\framebox(19,12){21}\,}{\,\,\framebox(19,12){34}\,\,}c^{2}
%\end{array}
%\]
�ł���B
\end{frame}

%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Dual Problem $({\rm D_{s8}})$}
����$({\rm P_{s8}})$�ɑ΂���o�Ζ��(dual problem)$({\rm D_{s8}})$�́A
���ŕ\�����B
\begin{eqnarray*}
& \hspace*{-5mm}&{\rm Maximize} \hspace{3mm} 
2cF_{8}\mu_{0} - \dsum_{n=0}^{6}F_{8-n}
\left[\mu_{n}^{2} + \left( \dfrac{F_{8-n}}{F_{7-n}}\mu_{n} + \mu_{n+1} 
\right)^{2} \right] - F_{1}\mu_{7}^{2} \\[-2,56mm] 
\hspace*{-30mm}({\rm D_{s8}})& \hspace*{-7mm}& \\[-2,56mm] 
& \hspace*{-5mm}&{\rm subject~\, to}\hspace{2ex} {\rm (i)}~~~ -\infty < \mu_{n} < \infty ~~~~n= 0,1,\cdots, 6 \\
& \hspace*{-5mm}&\hspace*{21mm} {\rm (ii)}~~~\mu_{7} = 0.
%\hspace*{-3mm}& &{\rm Maximize} \hspace{3mm} 2c\mu_{0} - \mu_{0}^{2} - \dsum_{n%=0}^{2}\left[(\mu_{n} + \mu_{n+1})^{2} + \mu_{n+1}^{2} \right] 
%- \mu_{3}^{2} \\[-2,56mm] 
%\hspace*{-3mm}({\rm D}_{4})& & \\[-2,56mm] 
%\hspace*{-3mm}& &{\rm subject~\, to}\hspace{2ex} {\rm (i)}~~~ -\infty < \mu_{n}% < \infty ~~~~n= 0,1,2,3.
\end{eqnarray*}

\pause 

�o�Ζ��$({\rm D_{s8}})$�̍ő��$\mu^{*}$�ƍő�l$M_{8}$�́A
\[
\begin{array}{lcl}
\mu^{*} & = &
(\mu^{*}_{0},~ \mu^{*}_{1},~ \cdots,~ \mu^{*}_{6},~ \mu_{7}) \\[3mm]
 & = & 
\dfrac{c}{\,\,34 \,\,}
\left(
\begin{array}{cccccccc}
       13 \,, & -8 \,, &  5 \,, & -3 \,, & 2 \,, & -1 \,, & 1 \,, & 0 
\end{array}
\right) \\[6mm]
M_{8} & = & 
\dfrac{\,13 \cdot 21 \,}{\,\,34 \,\,}c^{2}
\end{array}
\]
%\[
%\begin{array}{lcl}
%\mu^{*} & = &
%(\mu^{*}_{0},~ \mu^{*}_{1},~ \mu^{*}_{2},~ \mu^{*}_{3}) \\[3mm]
% & = & 
%\dfrac{c}{\,\,\dashbox(19,12){34}\,\,}
%\left(
%\begin{array}{cccc}
%       \dashbox(19,12){$21$}\,, & \dashbox(19,12){$-8$}\,, &  \dashbox(19,12){$%3$}\,, & \dashbox(19,12){$-1$} 
%\end{array}
%\right) \\[6mm]
%M_{4} & = & 
%\dfrac{\,\dashbox(19,12){21}\,}{\,\,\dashbox(19,12){34}\,\,}c^{2}
%\end{array}
%\]
�ł���B
\end{frame}


%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{}
\hspace*{-7mm}
$({\rm P_{s8}})$�ɂ����āA
\vspace{-2mm}
\[
\hspace*{-7mm}
%({\rm P_{8}})
%\left\{
\begin{array}{lcl}
\hat{x} & = &
(\hat{x}_{1}, ~\hat{x}_{2}, ~\hat{x}_{3}, ~\hat{x}_{4}, ~\hat{x}_{5}, 
~\hat{x}_{6}, ~\hat{x}_{7}, ~\hat{x}_{8})  \\[3mm]
 & = & \dfrac{c}{\,\,\framebox(19,12){34}\,\,}
\left(
\begin{array}{cccccccc}
\framebox(19,12){$-21$}, & \framebox(19,12){$13$}, & \framebox(19,12){$-8$}, &  \framebox(19,12){$5$}, & \framebox(19,12){$-3$}, & 
\framebox(19,12){$2$}, & \framebox(19,12){$-1$}, & \framebox(19,12){$1$}
\end{array}
\right) \\[6mm]
m_{4} & = & 
\dfrac{\,\framebox(19,12){$13$} \cdot \framebox(19,12){$21$}\,}{\,\,\framebox(19,12){$34$}\,\,}c^{2}.
\end{array}
%\right.
\]
\hspace*{-7mm}
$({\rm D_{s8}})$�ɂ����āA
\vspace{-2mm}
\[
\hspace*{-7mm}
%\mbox{�o�Ζ��}({\rm D_{4}})
%\left\{
\begin{array}{lcl}
\mu^{*} & = &
(\mu^{*}_{0},~ \mu^{*}_{1},~ \mu^{*}_{2},~ \mu^{*}_{3},~ \mu^{*}_{4},~ \mu^{*}_{5},~ \mu^{*}_{6},~ \mu_{7}) \\[3mm]
 & = & 
\dfrac{c}{\,\,\framebox(19,12){$34$}\,\,}
\left(
\begin{array}{cccccccc}
       \framebox(19,12){$13$}, & \framebox(19,12){$-8$}, &  
\framebox(19,12){$5$}, & \framebox(19,12){$-3$}, & \framebox(19,12){$2$}, 
& \framebox(19,12){$-1$}, &  \framebox(19,12){$1$}, & \dashbox(19,12){$0$}
\end{array}
\right) \\[6mm]
M_{8} & = & 
\dfrac{\,\framebox(19,12){$13$} \cdot \framebox(19,12){21}\,}{\,\,\framebox(19,12){34}\,\,}c^{2}.
\end{array}
%\right.
\]
\pause
\hspace*{-7mm}
\begin{minipage}{120mm}
����$({\rm P_{s8}})$�̍ŏ���$\hat{x}$�ƁA
�o�Ζ��$({\rm D_{s8}})$�̍ő��$\mu^{*}$�y�сA
�ŏ��l$m_8$�ƍő�l$M_8$�̊Ԃɂ�
\RED{���݃t�B�{�i�b�`�E�V�t�g�o�ΐ�}�ƌĂ΂��֌W�����藧���Ă���B
\end{minipage}
\end{frame}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
\begin{comment}
%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Fibonacci sequence}
\textbf{�t�B�{�i�b�`����} 
%$ \{ F_{n} \} $ 
�͎���2�K���`�����������i3���ԑQ�����j 
\[
 F_{n+2} - F_{n+1} - F_{n} = 0,~~~~~F_{1} = 1, ~F_{0} = 0 \label{eqn:gd1}
\]
�̉��Œ�܂�B

\begin{table}[htbp]
%
\begin{center}
% \begin{tabular}{c|cccccccccccccccccc} \hline
\begin{tabular}{c|ccccccccccccc} \hline
$ n $     &  $ 0 $  &  $ 1 $  &  $ 2  $ &  $ 3 $ &  $ 4 $ &  $ 5 $  & $ 6 $ & $ 7 $ & $ 8 $ & $ 9 $ & $ 10 $ & $ 11 $ & $ 12 $ \\ \hline
%& $ 13 $ & $ 14 $ & $ 15 $ & $ 16 $   \\ \hline 
$ F_{n} $ &  $ 0 $  &  $ 1 $  &  $ 1 $ &  $ 2 $ &  $3 $ &  $ 5  $  & $ 8 $ & $ 13 $  & $ 21 $ & $ 34 $ & $ 55 $ & $ 89 $ & $ 144 $ \\ \hline
%& $ 233 $ & $ 377 $ & $ 610 $ & $ 987 $   \\ \hline
\end{tabular}
\end{center}
%\caption{stop reward} \label{tab:h1}
\end{table}
\vspace{-5mm}
\begin{center}
Table 1:~~Fibonacci sequence $ \{F_{n}\} $
%�\ 1 ~~�t�B�{�i�b�`���� $ \{F_{n}\} $
\end{center}
\end{frame}

\end{comment}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%
%\begin{frame}
\begin{frame}[allowframebreaks]
\frametitle{���݃t�B�{�i�b�`�E�V�t�g�o�ΐ�\\ 
(Alternately Fibonacci Shift Duality)}

����$({\rm P_{s8}})$�Ƒo�Ζ��$({\rm D_{s8}})$�̊Ԃɂ�
����3�̊֌W�����藧�B
�����\RED{���݃t�B�{�i�b�`�E�V�t�g�o�ΐ�}�Ƃ����B
\vspace{3mm}

\begin{enumerate}
\item \BLUE{(�o�ΐ�)}\\[2mm]
����$({\rm P_{s8}})$�̍ŏ��l$m_8$�Ƒo�Ζ��$({\rm D_{s8}})$��
�ő�l$M_8$���������B���Ȃ킿�A
\[
\begin{array}{lclcl}
m_8 & = & \dfrac{\,13 \cdot 21\,}{\,\,34\,\,}c^{2}
& = & \dfrac{\,F_{7}F_{8}\,}{F_{9}}c^{2}, \\[5mm]
M_8 & = & \dfrac{\,13 \cdot 21\,}{\,\,34\,\,}c^{2}
& = & \dfrac{\,F_{7}F_{8}\,}{F_{9}}c^{2}, 
\end{array}
\]
�ł���A$m_{8} = M_{8}$ �����藧�B
\vspace{1mm}

\item \BLUE{(���݃t�B�{�i�b�`)}\\[2mm]
����̍ŏ���$\hat{x}$��
�o�Ζ��̍ő��$\mu^{*}$
�͋���\RED{���}�t�B�{�i�b�`����̌������ł���B
\[
\hspace*{-5mm}
\begin{array}{lcl}
\hat{x} & = &
%(\hat{x}_{1}, ~\hat{x}_{2}, ~\hat{x}_{3}, ~\hat{x}_{4}, ~\hat{x}_{5}, 
%~\hat{x}_{6}, ~\hat{x}_{7}, ~\hat{x}_{8})  \\[3mm]
(\hat{x}_{1}, ~\hat{x}_{2}, ~\cdots, ~\hat{x}_{8})  \\[3mm]
 & = & \dfrac{c}{\,\,34\,\,}
\left(
\begin{array}{cccccccc}
\BLUE{-21}, & \RED{13}, & \BLUE{-8}, &  \RED{5}, & \BLUE{-3}, & 
\RED{2}, & \BLUE{-1}, & \RED{1}
%%\begin{array}{ccccc}
%%\MagRedframebox{19}{12}{$34$}, & \MagBlueframebox{19}{12}{$-13$}, 
%%& \MagRedframebox{19}{12}{$5$}, 
%%&  \MagBlueframebox{19}{12}{$-2$}, & \MagRedframebox{19}{12}{$1$} 
\end{array}
\right) \\[4mm] 
 & = & \dfrac{c}{\,\,F_{9} \,\,}
\left(
\begin{array}{cccccccc}
\BLUE{-F_{8}}, & \RED{F_{7}}, & \BLUE{-F_{6}}, & \RED{F_{5}}, & 
\BLUE{-F_{4}}, & \RED{F_{3}}, & \BLUE{-F_{2}}, & \RED{F_{1}}
%\MagRedframebox{19}{12}{$F_{9}$}\,, & \MagBlueframebox{19}{12}{$-F_{7}$}\,, 
%& \MagRedframebox{19}{12}{$F_{5}$}\,, &  
%\MagBlueframebox{19}{12}{$-F_{3}$}\,, 
%& \MagRedframebox{19}{12}{$F_{1}$} 
\end{array}
\right),
\end{array}
\]
%
%\[
%\begin{array}{lcl}
%\hat{x} & = &
%(x_{0}, ~\hat{x}_{1}, ~\hat{x}_{2}, ~\hat{x}_{3}, ~\hat{x}_{4})  \\[3mm]
% & = & \dfrac{c}{\,\,\framebox(19,12){34}\,\,}
%\left(
%\begin{array}{ccccc}
%\framebox(19,12){$34$}\,, & \framebox(20,12){$-13$}\,, & \framebox(19,12){$5$}\%,, &  \framebox(19,12){$-2$}\,, & \framebox(19,12){$1$} 
%\end{array}
%\right), \\[4mm] 
% & = & \dfrac{c}{\,\,\framebox(19,12){$F_{9}$}\,\,}
%\left(
%\begin{array}{ccccc}
%\framebox(19,12){$F_{9}$}\,, & \framebox(20,12){\MAGENTA{$-F_{7}$}}\,, 
%& \framebox(19,12){$F_{5}$}\,, &  \framebox(19,12){\MAGENTA{$-F_{3}$}}\,, & \fr%amebox(19,12){$F_{1}$} 
%\end{array}
%\right).
%\end{array}
%\]
\[
\hspace*{-5mm}
\begin{array}{lcl}
\mu^{*} & = &
(\mu^{*}_{0},~ \mu^{*}_{1},~ \cdots,~ \mu^{*}_{6}, ~ \mu_{7}) \\[3mm]
 & = & 
\dfrac{c}{\,\,34 \,\,}
\left(
\begin{array}{cccccccc}
\RED{13}, & \BLUE{-8}, &  \RED{5}, & \BLUE{-3}, & 
\RED{2}, & \BLUE{-1}, & \RED{1}, & \BLUE{0}
%       \CyanRedframebox{19}{12}{$21$}\,, & 
%\CyanBlueframebox{19}{12}{$-8$}\,, 
%&  \CyanRedframebox{19}{12}{$3$}\,, & \CyanBlueframebox{19}{12}{$-1$} 
\end{array}
\right) \\[4mm]
 & = & 
\dfrac{c}{\,\,F_{9} \,\,}
\left(
\begin{array}{cccccccc}
\RED{F_{7}}, & \BLUE{-F_{6}}, & \RED{F_{5}}, & 
\BLUE{-F_{4}}, & \RED{F_{3}}, & \BLUE{-F_{2}}, & \RED{F_{1}}, 
& \BLUE{-F_{0}}
%       \CyanRedframebox{19}{12}{$F_{8}$}\,, 
%& \CyanBlueframebox{19}{12}{$-F_{6}$}\,, 
%&  \CyanRedframebox{19}{12}{$F_{4}$}\,, 
%& \CyanBlueframebox{19}{12}{$-F_{2}$} 
\end{array}
\right). 
\end{array}
\]

\item \BLUE{(�V�t�g)}\\[2mm]
����̍ŏ���$\hat{x}$��
�o�Ζ��̍ő��$\mu^{*}$�͂��݂���1�i\RED{�V�t�g}
���������̂ł���B
\[
\hspace*{-6.3mm}
\begin{array}{ccccccccccc}
( & \hat{x}_{1}, &~~\hat{x}_{2}, &~~\hat{x}_{3}, &~~\hat{x}_{4}, &
~~\hat{x}_{5}, &~~\hat{x}_{6}, &~~\hat{x}_{7}, &~~\hat{x}_{8} & ) & \\[1.56mm]
( & \BLUE{-F_{8}}, & \RED{F_{7}}, & \BLUE{-F_{6}}, & \RED{F_{5}}, &
\BLUE{-F_{4}}, & \RED{F_{3}}, & \BLUE{-F_{2}}, & \RED{F_{1}} & ) & \\[7.56mm]
 & ( &~~\mu^{*}_{0}, &~~\mu^{*}_{1}, &~~\mu^{*}_{2}, &
~~\mu^{*}_{3}, &~~\mu^{*}_{4}, &~~\mu^{*}_{5}, &~~\mu^{*}_{6}, 
& ~~\mu_{7} & ) \\[1.56mm]
& ( & \RED{F_{7}}, & \BLUE{-F_{6}}, & \RED{F_{5}}, &
\BLUE{-F_{4}}, & \RED{F_{3}}, & \BLUE{-F_{2}}, & \RED{F_{1}}, 
& \MagBlueframebox{19}{12}{$-F_{0}$} & ) \\
\end{array}
\]

\end{enumerate}
\end{frame}


%%%%%%%%%%%%%%%%%%
%\begin{frame}
\begin{frame}[allowframebreaks]
\frametitle{���݃t�B�{�i�b�`���� \\
(Alternately Fibonacci Condition)}
����$({\rm P_{s8}})$�ɂ����āA
���̓�������Ȃ�n${\rm (AF)_{P_{s}}}$���A
���$({\rm P_{s8}})$�ɑ΂���
\RED{���݃t�B�{�i�b�`����}�Ƃ����B
\[
\textcolor<2>{red}{
({\rm AF})_{\rm P_{s}}
\left\{
\begin{array}{ll}
\dfrac{\,c + x_{1}\,}{F_{7}} \dql \dfrac{\,x_{1}\,}{-F_{8}}, &
\dfrac{\,x_{1} + x_{2}\,}{-F_{6}} \dql \dfrac{\,x_{2}\,}{F_{7}}, \\[5mm]
\dfrac{\,x_{2} + x_{3}\,}{F_{5}} \dql \dfrac{\,x_{3}\,}{-F_{6}}, &
\dfrac{\,x_{3} + x_{4}\,}{-F_{4}}  \dql \dfrac{\,x_{4}\,}{F_{5}}, \\[5mm]
\dfrac{\,x_{4} + x_{5}\,}{F_{3}} \dql \dfrac{\,x_{5}\,}{-F_{4}}, &
\dfrac{\,x_{5} + x_{6}\,}{-F_{2}} \dql \dfrac{\,x_{6}\,}{F_{3}}, \\[5mm]
\dfrac{\,x_{6} + x_{7}\,}{F_{1}} \dql \dfrac{\,x_{7}\,}{-F_{2}}, &
x_{7} \dql -x_{8}.
\end{array}
\right.
}
\]
\vspace{5mm}
\newpage

�܂��A
���$({\rm D_{s8}})$�ɑ΂�����݃t�B�{�i�b�`������
���̓�������Ȃ�n${\rm (AF)_{D_{s}}}$�ŗ^������B
\[
\textcolor<2>{gray}{
\hspace{3.6mm}({\rm AF})_{\rm D_{s}}
\left\{
\begin{array}{l}
\dfrac{\,c - \mu_{0}\,}{F_{8}}  
\dql \dfrac{\,\dfrac{\,F_{8}\,}{F_{7}}\mu_{0} + \mu_{1}\,}{F_{7}} \dql 
\dfrac{\,\mu_{1}\,}{\,-F_{6}\,}, \\[5mm]
\dfrac{\,\dfrac{\,F_{7}\,}{F_{6}}\mu_{1} + \mu_{2}\,}{\,-F_{6}\,} \dql 
\dfrac{\,\mu_{2}\,}{F_{5}}, \ \ 
\dfrac{\,\dfrac{\,F_{6}\,}{F_{5}}\mu_{2} + \mu_{3}\,}{F_{5}}  
\dql \dfrac{\,\mu_{3}\,}{\,-F_{4}\,}, \\[5mm]
\dfrac{\,\dfrac{\,F_{5}\,}{F_{4}}\mu_{3} + \mu_{4}\,}{\,-F_{4}\,} \dql 
\dfrac{\,\mu_{4}\,}{F_{3}}, \ \ 
\dfrac{\,\dfrac{\,F_{4}\,}{F_{3}}\mu_{4} + \mu_{5}\,}{F_{3}}  
\dql \dfrac{\,\mu_{5}\,}{\,-F_{2}\,}, \\[5mm]
\dfrac{\,\dfrac{\,F_{3}\,}{F_{2}}\mu_{5} + \mu_{6}\,}{\,-F_{2}\,} \dql 
\dfrac{\,\mu_{6}\,}{F_{1}}, \ \ \mu_{7} \dql 0.
\end{array}
\right.
}
\]

\end{frame}



%%%%%%%%%%%%%%%%%%%
%\begin{frame}
\begin{frame}[allowframebreaks]
\frametitle{${\rm (AF)_{P_{s}}}$�̓��l����}
����$({\rm P_{s8}})$�̖ړI�֐��� $ x = (x_{1}, x_{2}, \cdots, x_{8}) $ 
��8�ϐ��֐� $ f(x) $ �ŕ\���B
\[
f(x) : \dql 
\dsum_{n=0}^{7}\left[F_{8-n}(x_{n} + x_{n+1})^{2} + \dfrac{F_{7-n}^{2}}{F_{8-n}} x_{n+1}^{2} \right] 
+ \dfrac{F_{0}F_{1}}{F_{2}} x_{8}^{2}
\]

\begin{eqnarray*}
%& & \mbox{���݃t�B�{�i�b�`����}{\rm (AF)_{P_{s}}} \\[3mm]
{\rm (AF)_{P_{s}}}
& \Longleftrightarrow & 
\left\{
\begin{array}{rcl}
F_{8}(x_{0} + x_{1}) + \dfrac{\,F_{7}^{2}\,}{F_{8}}x_{1} 
+ F_{7}(x_{1} + x_{2}) & \dql &  0 \\[2.56mm]
F_{7}(x_{1} + x_{2}) + \dfrac{\,F_{6}^{2}\,}{F_{7}}x_{2} 
+ F_{6}(x_{2} + x_{3}) & \dql & 0 \\[1.56mm]
\vdots \hspace{23mm} & & \\[0.56mm]
%F_{6}(x_{2} + x_{3}) + \dfrac{\,F_{5}^{2}\,}{F_{6}}x_{3} 
%+ F_{5}(x_{3} + x_{4}) & \dql &  0 \\[0.56mm]
%F_{5}(x_{3} + x_{4}) + \dfrac{\,F_{4}^{2}\,}{F_{5}}x_{4} 
%+ F_{4}(x_{4} + x_{5}) & \dql & 0 \\[0.56mm]
%F_{4}(x_{4} + x_{5}) + \dfrac{\,F_{3}^{2}\,}{F_{4}}x_{5} 
%+ F_{3}(x_{5} + x_{6}) & \dql & 0 \\[0.56mm]
F_{3}(x_{5} + x_{6}) + \dfrac{\,F_{2}^{2}\,}{F_{3}}x_{6} 
+ F_{2}(x_{6} + x_{7}) & \dql & 0 \\[2.56mm]
F_{2}(x_{6} + x_{7}) + \dfrac{\,F_{1}^{2}\,}{F_{2}}x_{7} 
+ F_{1}(x_{7} + x_{8}) & \dql & 0 \\[1.56mm]
F_{1}(x_{7} + x_{8}) + \dfrac{\,F_{0}^{2}\,}{F_{1}}x_{8} & \dql & 0 
\end{array}
\right. \\[2mm]
 & \Longleftrightarrow &  
\dfrac{\partial f}{\partial x_{n}} \dql 0 ~~~~~n = 1, 2, \cdots, 8. \\
\end{eqnarray*}
�ړI�֐�$f(x)$��$x$��2���̓ʊ֐��ł��邱�Ƃ���A
���$({\rm P_{s8}})$�̍ŏ���
$\hat{x} = (\hat{x}_{1}, ~\hat{x}_{2}, ~\cdots, ~\hat{x}_{8})$
�́A�œK����1�K�����𖞂����B

\end{frame}


%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink]
\frametitle{���݃t�B�{�i�b�`����${\rm (AF)_{P_{s}}}$}
%\frametitle{����$({\rm P_{8}})$�̌��݃t�B�{�i�b�`����}
\begin{eqnarray*}
({\rm AF})_{\rm P_{s}}
& \stackrel{\mbox{\itshape\scriptsize def}}{\Longleftrightarrow} &
\left\{
\begin{array}{ll}
\dfrac{\,c + x_{1}\,}{F_{7}} \dql \dfrac{\,x_{1}\,}{-F_{8}}, &
\dfrac{\,x_{1} + x_{2}\,}{-F_{6}} \dql \dfrac{\,x_{2}\,}{F_{7}}, \\[5mm]
\dfrac{\,x_{2} + x_{3}\,}{F_{5}} \dql \dfrac{\,x_{3}\,}{-F_{6}}, &
\dfrac{\,x_{3} + x_{4}\,}{-F_{4}}  \dql \dfrac{\,x_{4}\,}{F_{5}}, \\[5mm]
\dfrac{\,x_{4} + x_{5}\,}{F_{3}} \dql \dfrac{\,x_{5}\,}{-F_{4}}, &
\dfrac{\,x_{5} + x_{6}\,}{-F_{2}} \dql \dfrac{\,x_{6}\,}{F_{3}}, \\[5mm]
\dfrac{\,x_{6} + x_{7}\,}{F_{1}} \dql \dfrac{\,x_{7}\,}{-F_{2}}, &
x_{7} \dql -x_{8}.
\end{array}
\right. \\[2mm]
\uncover<2->{
& \Longleftrightarrow &
\left\{
\begin{array}{ll}
\dfrac{\,c - (-x_{1})\,}{13} \dql \dfrac{\,-x_{1}\,}{21}, &
\dfrac{\,-x_{1} - x_{2}\,}{8} \dql \dfrac{\,x_{2}\,}{13}, \\[5mm]
\dfrac{\,x_{2} - (-x_{3})\,}{5} \dql \dfrac{\,-x_{3}\,}{8}, &
\dfrac{\,-x_{3} - x_{4}\,}{3}  \dql \dfrac{\,x_{4}\,}{5}, \\[5mm]
\dfrac{\,x_{4} - (-x_{5})\,}{2} \dql \dfrac{\,-x_{5}\,}{3}, &
\dfrac{\,-x_{5} - x_{6}\,}{1} \dql \dfrac{\,x_{6}\,}{2}, \\[5mm]
\dfrac{\,x_{6} - (-x_{7})\,}{1} \dql \dfrac{\,-x_{7}\,}{1}, &
-x_{7} \dql x_{8}.
\end{array}
\right. } \\
\end{eqnarray*}
\end{frame}



%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{����$({\rm P_{s8}})$�̌��݃t�B�{�i�b�`����}
$x_{0} = c \,~(\, > 0 \,) $ \\
\[
\begin{array}{ll}
\textcolor<2>{red}{
\dfrac{\,c - (-x_{1})\,}{13} \dql \dfrac{\,-x_{1}\,}{21}}, &
\textcolor<3>{red}{
\dfrac{\,-x_{1} - x_{2}\,}{8} \dql \dfrac{\,x_{2}\,}{13}}, \\[5mm]
\textcolor<4>{red}{
\dfrac{\,x_{2} - (-x_{3})\,}{5} \dql \dfrac{\,-x_{3}\,}{8}}, &
\textcolor<5>{red}{
\dfrac{\,-x_{3} - x_{4}\,}{3}  \dql \dfrac{\,x_{4}\,}{5}}, \\[5mm]
\textcolor<6>{red}{
\dfrac{\,x_{4} - (-x_{5})\,}{2} \dql \dfrac{\,-x_{5}\,}{3}}, &
\textcolor<7>{red}{
\dfrac{\,-x_{5} - x_{6}\,}{1} \dql \dfrac{\,x_{6}\,}{2}}, \\[5mm]
\textcolor<8>{red}{
\dfrac{\,x_{6} - (-x_{7})\,}{1} \dql \dfrac{\,-x_{7}\,}{1}}, &
\textcolor<8>{red}{
-x_{7} \dql x_{8}}.
\end{array}
\]
\begin{center}
\unitlength=1mm
\begin{picture}(111,10)(0,0)
\uncover<1->{
\allinethickness{0.8mm}
\put(0,0){\line(1,0){111}}	%
\put(0,-1){\line(0,1){2}}	%
\put(-2,3){\makebox(0,0){\large $0$}}	%
\put(111,-1){\line(0,1){2}}	%
\put(111,3){\makebox(0,0){\large $c$}}	%
}
%% -x_1
\uncover<2->{
\allinethickness{0.3mm}			
\put(69,-1){\line(0,1){2}}	
\put(69,3){\makebox(0,0){\large $-x_1$}}
}
\uncover<2>{
\bezier{300}(0,0)(34,-4)(69,0)	
\put(34,-6){\makebox(0,0){\large $21$}}
\put(34,-11){\makebox(0,0){\large $F_{8}$}}
\bezier{200}(69,0)(90,-4)(111,0)	
\put(90,-6){\makebox(0,0){\large $13$}}
\put(90,-11){\makebox(0,0){\large $F_{7}$}}
}
%% x_2
\uncover<3->{
\allinethickness{0.3mm}			
\put(43,-1){\line(0,1){2}}	
\put(43,3){\makebox(0,0){\large $x_2$}}
}
\uncover<3>{
\bezier{200}(0,0)(22,-3)(43,0)	
\put(22,-6){\makebox(0,0){\large $13$}}
\put(22,-11){\makebox(0,0){\large $F_{7}$}}
\bezier{100}(43,0)(56,-3)(69,0)	
\put(56,-6){\makebox(0,0){\large $8$}}
\put(56,-11){\makebox(0,0){\large $F_{6}$}}
}
%% -x_3
\uncover<4->{
\allinethickness{0.3mm}			
\put(26,-1){\line(0,1){2}}	
\put(26,3){\makebox(0,0){\large $-x_3$}}
}
\uncover<4>{
\bezier{100}(0,0)(13,-3)(26,0)	
\put(13,-6){\makebox(0,0){\large $8$}}
\put(13,-11){\makebox(0,0){\large $F_{6}$}}
\bezier{100}(26,0)(35,-3)(43,0)	
\put(35,-6){\makebox(0,0){\large $5$}}
\put(35,-11){\makebox(0,0){\large $F_{5}$}}
}
%% x_4
\uncover<5->{
\allinethickness{0.3mm}
\put(17,-1){\line(0,1){2}}	
\put(17,3){\makebox(0,0){\large $x_4$}}
}
\uncover<5>{
\bezier{100}(0,0)(9,-3)(17,0)	
\put(9,-6){\makebox(0,0){\large $5$}}
\put(9,-11){\makebox(0,0){\large $F_{5}$}}
\bezier{100}(17,0)(22,-3)(26,0)	
\put(22,-6){\makebox(0,0){\large $3$}}
\put(22,-11){\makebox(0,0){\large $F_{4}$}}
}
%% -x_5
\uncover<6->{
\allinethickness{0.3mm}
\put(10,-1){\line(0,1){2}}	
\put(10,3){\makebox(0,0){\large $-x_5$}}
}
\uncover<6>{
\bezier{100}(0,0)(5,-3)(10,0)	
\put(5,-6){\makebox(0,0){\large $3$}}
\put(5,-11){\makebox(0,0){\large $F_{4}$}}
\bezier{100}(10,0)(14,-3)(17,0)	
\put(14,-6){\makebox(0,0){\large $2$}}
\put(14,-11){\makebox(0,0){\large $F_{3}$}}
}
%% x_6
\uncover<7->{
\allinethickness{0.3mm}
\put(6,-1){\line(0,1){2}}	
\put(7,-4){\makebox(0,0){\large $x_6$}}
}
\uncover<7>{
\bezier{100}(0,0)(3,-3)(6,0)	
\put(3,-6){\makebox(0,0){\large $2$}}
\put(3,-11){\makebox(0,0){\large $F_{3}$}}
\bezier{100}(6,0)(8,-3)(10,0)	
\put(8,-6){\makebox(0,0){\large $1$}}
\put(8,-11){\makebox(0,0){\large $F_{2}$}}
}
%% -x_7, x_8
\uncover<8->{
\allinethickness{0.3mm}
\put(3,-1){\line(0,1){2}}	
\put(3,3){\makebox(0,0){\large $-x_7$}}
\put(3,6){\makebox(0,0){\large $x_8$}}
}
\uncover<8>{
\bezier{100}(0,0)(2,-3)(3,0)	
\put(2,-6){\makebox(0,0){\large $1$}}
\put(1,-11){\makebox(0,0){\large $F_{2}$}}
\bezier{100}(3,0)(5,-3)(6,0)	
\put(6,-6){\makebox(0,0){\large $1$}}
\put(7,-11){\makebox(0,0){\large $F_{1}$}}
}
\end{picture}
\end{center}

\end{frame}


%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{}
\begin{center}
\unitlength=1mm
\begin{picture}(111,10)(0,0)
\allinethickness{0.8mm}
\put(0,0){\line(1,0){111}}	%
\put(0,-1){\line(0,1){2}}	%
\put(-2,3){\makebox(0,0){\large $0$}}	%
\put(111,-1){\line(0,1){2}}	%
\put(111,3){\makebox(0,0){\large $c$}}	%
%% -x_1
\allinethickness{0.3mm}	
\uncover<2->{
\put(69,-1){\line(0,1){2}}	
\put(69,6){\makebox(0,0){\large $-\hat{x}_1$}}
\put(90,-6){\makebox(0,0){\large $F_{7}$}}
\bezier{300}(69,0)(90,-4)(111,0)
}
\uncover<2>{
\bezier{300}(0,0)(34,-4)(69,0)	
\put(34,-6){\makebox(0,0){\large $F_{8}$}}
}
%% x_2
\uncover<3->{
\put(43,-1){\line(0,1){2}}	
\put(43,6){\makebox(0,0){\large $\hat{x}_2$}}
\put(56,-6){\makebox(0,0){\large $F_{6}$}}
\bezier{100}(43,0)(56,-4)(69,0)	
}
\uncover<3>{
\bezier{200}(0,0)(22,-4)(43,0)	
\put(22,-6){\makebox(0,0){\large $F_{7}$}}
}
%% -x_3
\uncover<4->{
\put(26,-1){\line(0,1){2}}	
\put(26,6){\makebox(0,0){\large $-\hat{x}_3$}}
\put(35,-6){\makebox(0,0){\large $F_{5}$}}
\bezier{100}(26,0)(35,-3)(43,0)
}
\uncover<4>{
\bezier{200}(0,0)(13,-3)(26,0)	
\put(13,-6){\makebox(0,0){\large $F_{6}$}}
}
%% x_4
\uncover<5->{
\put(17,-1){\line(0,1){2}}	
\put(17,6){\makebox(0,0){\large $\hat{x}_4$}}
\put(22,-6){\makebox(0,0){\large $F_{4}$}}
\bezier{100}(17,0)(22,-3)(26,0)
}
\uncover<5>{
\bezier{100}(0,0)(9,-3)(17,0)	
\put(9,-6){\makebox(0,0){\large $F_{5}$}}
}
%% -x_5, x_6, -x_7, x_8
\uncover<6->{
\put(10,-1){\line(0,1){2}}	
\put(10,6){\makebox(0,0){\large $-\hat{x}_5$}}
\put(6,-1){\line(0,1){2}}	
\put(7,3){\makebox(0,0){\large $\hat{x}_6$}}
\put(3,-1){\line(0,1){2}}	
\put(1,6){\makebox(0,0){\large $-\hat{x}_7$}}
\put(3,11){\makebox(0,0){\large $\hat{x}_8$}}
\bezier{100}(10,0)(14,-3)(17,0)	
\put(14,-6){\makebox(0,0){\large $F_{3}$}}
\bezier{100}(6,0)(8,-3)(10,0)	
\put(9,-6){\makebox(0,0){\large $F_{2}$}}
\bezier{100}(0,0)(2,-3)(3,0)	
\put(-1,-6){\makebox(0,0){\large $F_{2}$}}
\bezier{100}(3,0)(5,-3)(6,0)	
\put(5,-6){\makebox(0,0){\large $F_{1}$}}
}
\end{picture}
\end{center}
\vspace{5mm}

\[
\hspace*{-4mm}
\mbox{�œK�� \ }
\left\{
\begin{array}{rclcl}
\uncover<2->{- \hat{x}_1 & = &  \dfrac{F_{8}}{F_{8}+F_{7}} \ c }
& \uncover<8->{= & \dfrac{F_8}{F_9} \, c \,,} \\[4mm]
\uncover<3->{\hat{x}_2 & = & \dfrac{F_{7}}{F_{7}+F_{6}} \, (- \hat{x}_{1})} 
& \uncover<8->{= & \dfrac{F_7}{F_8} \ndot 
\dfrac{F_{8}}{F_{9}} \, c \,,} \\[4mm]
\uncover<4->{- \hat{x}_3 & = & \dfrac{F_{6}}{F_{6}+F_{5}} \, \hat{x}_{2}} 
& \uncover<8->{= & \dfrac{F_6}{F_7} \ndot \dfrac{F_7}{F_8} \ndot 
\dfrac{F_{8}}{F_{9}} \, c \,,} \\[4mm]
\uncover<5->{\hat{x}_4 & = & \dfrac{F_{5}}{F_{5}+F_{4}} \, (- \hat{x}_{3})} 
& \uncover<8->{= & \dfrac{F_5}{F_6} \ndot \dfrac{F_6}{F_7} \ndot 
\dfrac{F_7}{F_8} \ndot \dfrac{F_{8}}{F_{9}} \, c \,,} \\[4mm]
\uncover<6->{\vdots} & & \uncover<6->{\vdots} & & \\[2mm]
\uncover<7->{-\hat{x}_7 = \hat{x}_8 
& = & \dfrac{F_{2}}{F_{2}+F_{1}} \, \hat{x}_{6}} 
& \uncover<8->{= & \dfrac{F_2}{F_3} \ndot \dfrac{F_3}{F_4} \ndot \cdots \ndot
\dfrac{F_7}{F_8} \ndot \dfrac{F_{8}}{F_{9}} \, c \,.}
\end{array}
\right.
%\mbox{�ŋ��߂���. }
\]

\end{frame}




%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{}
���Ȃ킿�A����$({\rm P_{s8}})$�̍ŏ���$\hat{x}$�́A
\[
\begin{array}{lcl}
\hat{x} & = &
(\hat{x}_{1}, ~\hat{x}_{2}, ~\cdots, ~\hat{x}_{8})  \\[3mm]
 & = & \dfrac{c}{\,\,F_{9} \,\,}
\left(
\begin{array}{cccccccc}
-F_{8} \,, & F_{7} \,, & -F_{6} \,, & F_{5} \,, & -F_{4} \,, & F_{3} 
\,, & -F_{2} \,, & F_{1}
\end{array}
\right) 
\end{array}
\]
�ƂȂ�B

\begin{center}
\unitlength=1mm
\begin{picture}(111,10)(0,0)
\allinethickness{0.8mm}
\put(0,0){\line(1,0){111}}	%
\put(0,-1){\line(0,1){2}}	%
\put(0,-3){\makebox(0,0){\large $-c$}}	%
\put(111,-1){\line(0,1){2}}	%
\put(111,-3){\makebox(0,0){\large $c$}}	%
\put(53,-1){\line(0,1){2}}	%
\put(53,-3){\makebox(0,0){\large $0$}}	%
%% -x_1
\put(89,-1){\line(0,1){2}}	
\put(89,-5){\makebox(0,0){\large \GRAY{\small $-\hat{x}_1$}}}
\put(89,-12){\makebox(0,0){\large \GRAY{\small $\dfrac{F_{8}}{F_{9}}c$}}}
%% x_1
\put(17,-1){\line(0,1){2}}	
\put(17,5){\makebox(0,0){\large \RED{\small $\hat{x}_1$}}}
\put(17,12){\makebox(0,0){\large \MAGENTA{\small $-\dfrac{F_{8}}{F_{9}}c$}}}
%% x_2
\put(76,-1){\line(0,1){2}}	
\put(76,5){\makebox(0,0){\large \RED{\small $\hat{x}_2$}}}
\put(76,12){\makebox(0,0){\large \MAGENTA{\small $\dfrac{F_{7}}{F_{9}}c$}}}
%% -x_3
\put(68,-1){\line(0,1){2}}	
\put(68,-5){\makebox(0,0){\large \GRAY{\small $-\hat{x}_3$}}}
\put(68,-12){\makebox(0,0){\large \GRAY{\small $\dfrac{F_{6}}{F_{9}}c$}}}
%% x_3
\put(38,-1){\line(0,1){2}}	
\put(38,5){\makebox(0,0){\large \RED{\small $\hat{x}_3$}}}
\put(38,12){\makebox(0,0){\large \MAGENTA{\small $-\dfrac{F_{6}}{F_{9}}c$}}}
%% x_4
\put(63,-1){\line(0,1){2}}	
\put(63,5){\makebox(0,0){\large \RED{\small $\hat{x}_4$}}}
\put(63,12){\makebox(0,0){\large \MAGENTA{\small $\dfrac{F_{5}}{F_{9}}c$}}}
%% x_5
\put(48,-1){\line(0,1){2}}	
\put(48,5){\makebox(0,0){\large \RED{\small $\hat{x}_5$}}}
\put(48,12){\makebox(0,0){\large \MAGENTA{\small $-\dfrac{F_{4}}{F_{9}}c$}}}
%% x_6
\put(56,-1){\line(0,1){2}}	
\put(56,5){\makebox(0,0){\large \RED{\small $\hat{x}_6$}}}
\put(56,12){\makebox(0,0){\large \MAGENTA{\small $\dfrac{F_{3}}{F_{9}}c$}}}
%% x_7
%\put(52,-1){\line(0,1){2}}	
%\put(52,5){\makebox(0,0){\large \RED{\small $\hat{x}_7$}}}
%\put(52,12){\makebox(0,0){\large \MAGENTA{\small $\dfrac-{F_{2}}{F_{9}}c$}}}
%% x_8
%\put(54,-1){\line(0,1){2}}	
%\put(54,5){\makebox(0,0){\large \RED{\small $\hat{x}_6$}}}
%\put(54,12){\makebox(0,0){\large \MAGENTA{\small $\dfrac{F_{1}}{F_{9}}c$}}}
\end{picture}
\end{center}
\vspace{17mm}

%\pause

���̗�A$(\hat{x}_{1}, \hat{x}_{2}, \cdots, \hat{x}_{8})$
��\RED{���݃t�B�{�i�b�`�o�H}���\�����Ă���B
\end{frame}




%%%%%%%%%%%%%
\begin{frame}
\frametitle{����$({\rm P_{s8}})$�̍ŏ��l}
����$({\rm P_{s8}})$�̖ړI�֐��́A
\[
f(x) : \dql 
\dsum_{n=0}^{7}\left[F_{8-n}(x_{n} + x_{n+1})^{2} + \dfrac{F_{7-n}^{2}}{F_{8-n}} x_{n+1}^{2} \right] 
+ \dfrac{F_{0}F_{1}}{F_{2}} x_{8}^{2}
\]
�ł���A�ŏ���$\hat{x}$�́A
\[
\begin{array}{lcl}
\hat{x} & = &
(\hat{x}_{1}, ~\hat{x}_{2}, ~\cdots, ~\hat{x}_{8})  \\[3mm]
 & = & \dfrac{c}{\,\,F_{9} \,\,}
\left(
\begin{array}{cccccccc}
-F_{8} \,, & F_{7} \,, & -F_{6} \,, & F_{5} \,, & -F_{4} \,, & F_{3} 
\,, & -F_{2} \,, & F_{1}
\end{array}
\right) 
\end{array}
\]
�Ȃ̂ŁA
\pause
\begin{eqnarray*}
f(\hat{x}) \eql 
\dfrac{c^2}{F_{9}^2} \Biggl\{ (F_{8}F_{7}^{2} + F_{7}^{2}F_{8}) + 
(F_{7}F_{6}^{2} + F_{6}^{2}F_{7}) + \cdots \\[1.16mm]
 & & \ \ \ \ \ \ \ \ \ \ \cdots + (F_{2}F_{1}^{2} + F_{1}^{2}F_{2}) 
	+ (F_{1}F_{0}^{2} + F_{0}^{2}F_{1}) \Biggr\} 
\end{eqnarray*}
�ƂȂ�B


\end{frame}




%%%%%%%%%%%%%%
\begin{frame}
%\frametitle{Lucas Formula}
\frametitle{�������a}
\begin{block}{\textbf{���i�������a�j}}
�t�B�{�i�b�`��$\{F_{k}\}$�ɂ����āA�������藧�B
\begin{eqnarray*}
2 \dsum_{k=1}^{n} F_{k}^{2}F_{k+1} \eql F_{n}F_{n+1}F_{n+2}. 
\end{eqnarray*}
\end{block}
\begin{table}[htbp]
%
\begin{center}
% \begin{tabular}{c|cccccccccccccccccc} \hline
\begin{tabular}{c|ccccccccccccc} \hline
$ n $     &  $ 0 $  &  $ 1 $  &  $ 2  $ &  $ 3 $ &  $ 4 $ &  $ 5 $  & $ 6 $ & $ 7 $ & $ 8 $ & $ 9 $ & $ 10 $ & $ 11 $ & $ 12 $ \\ \hline
%& $ 13 $ & $ 14 $ & $ 15 $ & $ 16 $   \\ \hline 
$ F_{n} $ &  $ 0 $  &  $ 1 $  &  $ 1 $ &  $ 2 $ &  $3 $ &  $ 5  $  & $ 8 $ & $ 13 $  & $ 21 $ & $ 34 $ & $ 55 $ & $ 89 $ & $ 144 $ \\ \hline
%& $ 233 $ & $ 377 $ & $ 610 $ & $ 987 $   \\ \hline
\end{tabular}
\end{center}
%\caption{stop reward} \label{tab:h1}
\end{table}
\vspace{-3mm}
\begin{center}
Table 1:~~Fibonacci sequence $ \{F_{n}\} $
%�\ 1 ~~�t�B�{�i�b�`���� $ \{F_{n}\} $
\end{center}

\pause
\vspace{-3mm}
�Ⴆ�΁A$n \dql 4$�ł́A
\begin{eqnarray*}
& & 2 \times \left(F_{1}^{2} F_{2} + F_{2}^{2} F_{3} +  F_{3}^{2} F_{4} 
+ F_{4}^{2} F_{5} \right)\\[0.56mm]
& = &  2 \times \left(1^{2} \cdot 1 + 1^{2} \cdot 2 + 2^{2} \cdot 3 
+ 3^{2} \cdot 5 \right) \\[0.56mm]
& = & 120 \\[0.56mm]
& = & F_{4}F_{5}F_{6}
\end{eqnarray*}

\end{frame}


%%%%%%%%%%%%%%
\begin{frame}
\frametitle{}
����āA
\begin{eqnarray*}
f(\hat{x}) \eql 
\dfrac{c^2}{F_{9}^2} \Biggl\{ (F_{8}F_{7}^{2} + F_{7}^{2}F_{8}) + 
(F_{7}F_{6}^{2} + F_{6}^{2}F_{7}) + \cdots \\[1.16mm]
 & & \ \ \ \ \ \ \ \ \ \ \cdots + (F_{2}F_{1}^{2} + F_{1}^{2}F_{2}) 
	+ (F_{1}F_{0}^{2} + F_{0}^{2}F_{1}) \Biggr\} 
\end{eqnarray*}
���A
\begin{eqnarray*}
F_{9}^{2}\dfrac{\,f(\hat{x})\,}{c^{2}} \eql 
(F_{8}F_{7}^{2} + F_{7}^{2}F_{8}) + 
(F_{7}F_{6}^{2} + F_{6}^{2}F_{7}) + \cdots \\[1.16mm]
 & & \ \ \ \ \ \ \ \ \ \ \cdots + (F_{2}F_{1}^{2} + F_{1}^{2}F_{2}) 
	+ (F_{1}F_{0}^{2} + F_{0}^{2}F_{1}) \\[1.56mm]
%
\eql 2 (F_{8}F_{7}^{2} + F_{7}F_{6}^{2} + \cdots + F_{2}F_{1}^{2} 
+ F_{1}F_{0}^{2}) \\[1.56mm]
%
\eql  F_{7} \ndot F_{8} \ndot F_{9}.
\end{eqnarray*}

���Ȃ킿�A�ŏ��l�� \ 
%\begin{eqnarray*}
$
f(\hat{x}) \dql \dfrac{F_{7} F_{8}}{\,F_{9}\,}c^{2} 
= \dfrac{13 \cdot 21}{\,34\,} \, c^{2}
$
%\end{eqnarray*}
\ �ł���B 

\end{frame}






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{comment}
%%%%%%%%%%%%%%
\begin{frame}
%\frametitle{Lucas Formula}
\frametitle{�������a}
%\begin{block}{\textbf{�����J�̌����iLucas formula�j}}
%\begin{eqnarray*}
%\dsum_{k=1}^{n} F_{k}^{2} \eql F_{n}F_{n+1}
%\end{eqnarray*}
%\end{block}
\begin{block}{\textbf{���i�������a�j}}
�t�B�{�i�b�`��$\{F_{k}\}$�ɂ����āA�������藧�B
\begin{eqnarray*}
2 \dsum_{k=1}^{n} F_{k}^{2}F_{k+1} \eql F_{n}F_{n+1}F_{n+2}. 
\end{eqnarray*}
\end{block}

\pause

\begin{eqnarray*}
F_{9}^{2}\dfrac{\,f(\hat{x})\,}{c^{2}} \eql 
(F_{8}F_{7}^{2} + F_{7}^{2}F_{8}) + 
(F_{7}F_{6}^{2} + F_{6}^{2}F_{7}) + \cdots \\[1.16mm]
 & & \ \ \ \ \ \ \ \ \ \ \cdots + (F_{2}F_{1}^{2} + F_{1}^{2}F_{2}) 
	+ (F_{1}F_{0}^{2} + F_{0}^{2}F_{1}) \\[1.56mm]
%
\eql 2 (F_{8}F_{7}^{2} + F_{7}F_{6}^{2} + \cdots + F_{2}F_{1}^{2} 
+ F_{1}F_{0}^{2}) \\[1.56mm]
%
\eql  F_{7} \ndot F_{8} \ndot F_{9}.
\end{eqnarray*}

���Ȃ킿�A�ŏ��l�� \ 
%\begin{eqnarray*}
$
f(\hat{x}) \dql \dfrac{F_{7} F_{8}}{\,F_{9}\,}c^{2} 
= \dfrac{13 \cdot 21}{\,34\,} \, c^{2}
$
%\end{eqnarray*}
\ �ł���B 

\end{frame}



%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Lucas Formula}
\begin{block}{\textbf{�����J�̌����iLucas formula�j}}
\begin{eqnarray*}
\dsum_{k=1}^{n} F_{k}^{2} \eql F_{n}F_{n+1}
\end{eqnarray*}
\end{block}
\vspace{3mm}

\begin{table}[htbp]
%
\begin{center}
% \begin{tabular}{c|cccccccccccccccccc} \hline
\begin{tabular}{c|ccccccccccccc} \hline
$ n $     &  $ 0 $  &  $ 1 $  &  $ 2  $ &  $ 3 $ &  $ 4 $ &  $ 5 $  & $ 6 $ & $ 7 $ & $ 8 $ & $ 9 $ & $ 10 $ & $ 11 $ & $ 12 $ \\ \hline
%& $ 13 $ & $ 14 $ & $ 15 $ & $ 16 $   \\ \hline 
$ F_{n} $ &  $ 0 $  &  $ 1 $  &  $ 1 $ &  $ 2 $ &  $3 $ &  $ 5  $  & $ 8 $ & $ 13 $  & $ 21 $ & $ 34 $ & $ 55 $ & $ 89 $ & $ 144 $ \\ \hline
%& $ 233 $ & $ 377 $ & $ 610 $ & $ 987 $   \\ \hline
\end{tabular}
\end{center}
%\caption{stop reward} \label{tab:h1}
\end{table}
\vspace{-5mm}
\begin{center}
Table 1:~~Fibonacci sequence $ \{F_{n}\} $
%�\ 1 ~~�t�B�{�i�b�`���� $ \{F_{n}\} $
\end{center}
\end{frame}

\end{comment}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{���݃t�B�{�i�b�`���� \\
(Alternately Fibonacci Condition)}
���$({\rm D_{s8}})$�ɑ΂�����݃t�B�{�i�b�`������
���̓�������Ȃ�n${\rm (AF)_{D_{s}}}$�ŗ^������B
\[
%\textcolor<2>{gray}{
\hspace{3.6mm}({\rm AF})_{\rm D_{s}}
\left\{
\begin{array}{l}
\dfrac{\,c - \mu_{0}\,}{F_{8}}  
\dql \dfrac{\,\dfrac{\,F_{8}\,}{F_{7}}\mu_{0} + \mu_{1}\,}{F_{7}} \dql 
\dfrac{\,\mu_{1}\,}{\,-F_{6}\,}, \\[5mm]
\dfrac{\,\dfrac{\,F_{7}\,}{F_{6}}\mu_{1} + \mu_{2}\,}{\,-F_{6}\,} \dql 
\dfrac{\,\mu_{2}\,}{F_{5}}, \ \ 
\dfrac{\,\dfrac{\,F_{6}\,}{F_{5}}\mu_{2} + \mu_{3}\,}{F_{5}}  
\dql \dfrac{\,\mu_{3}\,}{\,-F_{4}\,}, \\[5mm]
\dfrac{\,\dfrac{\,F_{5}\,}{F_{4}}\mu_{3} + \mu_{4}\,}{\,-F_{4}\,} \dql 
\dfrac{\,\mu_{4}\,}{F_{3}}, \ \ 
\dfrac{\,\dfrac{\,F_{4}\,}{F_{3}}\mu_{4} + \mu_{5}\,}{F_{3}}  
\dql \dfrac{\,\mu_{5}\,}{\,-F_{2}\,}, \\[5mm]
\dfrac{\,\dfrac{\,F_{3}\,}{F_{2}}\mu_{5} + \mu_{6}\,}{\,-F_{2}\,} \dql 
\dfrac{\,\mu_{6}\,}{F_{1}}, \ \ \mu_{7} \dql 0.
\end{array}
\right.
%}
\]
\end{frame}





%%%%%%%%%%%%%%%%%%%
%\begin{frame}[shrink]
\begin{frame}[allowframebreaks]
\frametitle{${\rm (AF)_{D_{s}}}$�̓��l����}
�o�Ζ��$({\rm D_{s8}})$�̖ړI�֐��� 
$ \mu = (\mu_{0}, \mu_{1}, \cdots, \mu_{7}) $ ��8�ϐ��֐� $ g(\mu) $ �ŕ\���B
\[
g(\mu) : \dql 
2cF_{8}\mu_{0} - \dsum_{n=0}^{6}F_{8-n}
\left[\mu_{n}^{2} + \left( \dfrac{F_{8-n}}{F_{7-n}}\mu_{n} + \mu_{n+1} 
\right)^{2} \right] - F_{1}\mu_{7}^{2}. 
\]


\begin{eqnarray*}
%& & \mbox{���݃t�B�{�i�b�`����}{\rm (AF)_{D_{s}}} \\[3mm]
\hspace*{-5mm}& & {\rm (AF)_{D_{s}}} \\[2.56mm]
\hspace*{-5mm}& \Longleftrightarrow & 
\left\{
\begin{array}{rcl}
c - \mu_{0} + \dfrac{\,F_{8}\,}{F_{7}}\left(\dfrac{\,F_{8}\,}{F_{7}}\mu_{0} 
+ \mu_{1}\right) & \dql & 0 \\[3.56mm]
-\dfrac{\,F_{8}\,}{F_{7}}\left(\dfrac{\,F_{8}\,}{F_{7}}\mu_{0} 
+ \mu_{1}\right) 
- \mu_{1} - \dfrac{\,F_{7}\,}{F_{6}}\left(\dfrac{\,F_{7}\,}{F_{6}}\mu_{1} 
+ \mu_{2}\right) & \dql & 0  \\[2.56mm]
\vdots \hspace{53mm} & & \\[-0.45mm]
-\dfrac{\,F_{4}\,}{F_{3}}\left(\dfrac{\,F_{4}\,}{F_{3}}\mu_{4} 
+ \mu_{5}\right) 
- \mu_{5} - \dfrac{\,F_{3}\,}{F_{2}}\left(\dfrac{\,F_{3}\,}{F_{2}}\mu_{5} 
+ \mu_{6}\right) & \dql & 0  \\[3.56mm]
-\dfrac{\,F_{3}\,}{F_{2}}\left(\dfrac{\,F_{3}\,}{F_{2}}\mu_{5} 
+ \mu_{6}\right) 
- \mu_{6} -\dfrac{\,F_{2}\,}{F_{1}}\left(\dfrac{\,F_{2}\,}{F_{1}}\mu_{6} 
+ \mu_{7}\right)  & \dql & 0 \\[2.56mm]
\mu_{7} & \dql & 0
\end{array}
\right. \\[2mm]
\hspace*{-5mm}& \Longleftrightarrow &  
\dfrac{\partial g}{\partial \mu_{n}} \dql 0 ~~~~~n = 0, 1, \cdots, 6, \ \ 
\mu_{7} = 0. 
\end{eqnarray*}

�ړI�֐�$g(\mu)$��$\mu$��2���̉��֐��ł��邱�Ƃ���A
���$({\rm D_{s8}})$�̍ő��
$\mu^{*} = (\mu_{0}^{*}, ~\mu_{1}^{*}, ~\cdots, ~\mu_{6}^{*})$
�́A�œK����1�K�����𖞂����B
\end{frame}

%%%%%%%%%%%%%%%%%%%%
\begin{frame}[shrink]
\begin{eqnarray*}
({\rm AF})_{\rm D_{s}}
& \stackrel{\mbox{\itshape\scriptsize def}}{\Longleftrightarrow} &
\left\{
\begin{array}{ll}
\dfrac{\,c - \mu_{0}\,}{F_{8}} \dql \dfrac{\,\mu_{0}\,}{F_{7}}, &
\dfrac{\,\mu_{0} + \mu_{1}\,}{F_{5}} \dql \dfrac{\,\mu_{1}\,}{-F_{6}}, \\[5mm]
\dfrac{\,\mu_{1} + \mu_{2}\,}{-F_{4}} \dql \dfrac{\,\mu_{2}\,}{F_{5}}, &
\dfrac{\,\mu_{2} + \mu_{3}\,}{F_{3}}  \dql \dfrac{\,\mu_{3}\,}{-F_{4}}, \\[5mm]
\dfrac{\,\mu_{3} + \mu_{4}\,}{-F_{2}} \dql \dfrac{\,\mu_{4}\,}{F_{3}}, &
\dfrac{\,\mu_{4} + \mu_{5}\,}{F_{1}} \dql \dfrac{\,\mu_{5}\,}{-F_{2}}, \\[5mm]
\mu_{5} + \mu_{6} \dql -\dfrac{\,F_{0}\,}{F_{1}}\mu_{6}, &
\mu_{7} = 0.
\end{array}
\right. \\[2mm]
\uncover<2->{
& \Longleftrightarrow &
\left\{
\begin{array}{ll}
\dfrac{\,c - \mu_{0}\,}{21} \dql \dfrac{\,\mu_{0}\,}{13}, &
\dfrac{\,\mu_{0} - (-\mu_{1})\,}{5} \dql 
\dfrac{\,-\mu_{1}\,}{8}, \\[5mm]
\dfrac{\,-\mu_{1} - \mu_{2}\,}{3} \dql \dfrac{\,\mu_{2}\,}{5}, &
\dfrac{\,\mu_{2} - (-\mu_{3})\,}{2}  \dql \dfrac{\,-\mu_{3}\,}{3}, \\[5mm]
\dfrac{\,-\mu_{3} - \mu_{4}\,}{1} \dql \dfrac{\,\mu_{4}\,}{2}, &
\dfrac{\,\mu_{4} - (-\mu_{5})\,}{1} \dql \dfrac{\,-\mu_{5}\,}{1}, \\[5mm]
\mu_{5} + \mu_{6} \dql 0, &
\mu_{7} = 0.
\end{array}
\right. } \\
\end{eqnarray*}
\end{frame}



%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{�o�Ζ��$({\rm D_{s8}})$�̌��݃t�B�{�i�b�`����}
$x_{0} = c \,~(\, > 0 \,) $ \\
\[
\begin{array}{ll}
\textcolor<2>{red}{
\dfrac{\,c - \mu_{0}\,}{21} \dql \dfrac{\,\mu_{0}\,}{13}}, &
\textcolor<3>{red}{
\dfrac{\,\mu_{0} - (-\mu_{1})\,}{5} \dql 
\dfrac{\,-\mu_{1}\,}{8}}, \\[5mm]
\textcolor<4>{red}{
\dfrac{\,-\mu_{1} - \mu_{2}\,}{3} \dql \dfrac{\,\mu_{2}\,}{5}}, &
\textcolor<5>{red}{
\dfrac{\,\mu_{2} - (-\mu_{3})\,}{2}  \dql \dfrac{\,-\mu_{3}\,}{3}}, \\[5mm]
\textcolor<6>{red}{
\dfrac{\,-\mu_{3} - \mu_{4}\,}{1} \dql \dfrac{\,\mu_{4}\,}{2}}, &
\textcolor<7>{red}{
\dfrac{\,\mu_{4} - (-\mu_{5})\,}{1} \dql \dfrac{\,-\mu_{5}\,}{1}}, \\[5mm]
\textcolor<8>{red}{
\mu_{5} + \mu_{6} \dql 0}, &
\mu_{7} = 0.
\end{array}
\]

\begin{center}
\unitlength=1mm
\begin{picture}(111,10)(0,0)
\uncover<1->{
\allinethickness{0.8mm}
\put(0,0){\line(1,0){111}}	%
\put(0,-1){\line(0,1){2}}	%
\put(-2,3){\makebox(0,0){\large $0$}}	%
\put(111,-1){\line(0,1){2}}	%
\put(111,3){\makebox(0,0){\large $c$}}	%
}
%% \mu_{0}
\uncover<2->{
\allinethickness{0.3mm}
\put(46,-1){\line(0,1){2}}	
\put(46,3){\makebox(0,0){\large $\mu_{0}^{*}$}}
}
\uncover<2>{
\bezier{200}(0,0)(23,-4)(46,0)	
\put(23,-6){\makebox(0,0){\large $13$}}
\put(23,-11){\makebox(0,0){\large $F_{7}$}}
\bezier{300}(46,0)(79,-4)(111,0)	
\put(79,-6){\makebox(0,0){\large $21$}}
\put(79,-11){\makebox(0,0){\large $F_{8}$}}
}
%% -\mu_{1}
\uncover<3->{
\allinethickness{0.3mm}			
\put(30,-1){\line(0,1){2}}	
\put(30,3){\makebox(0,0){\large $-\mu_{1}^{*}$}}
}
\uncover<3>{
\bezier{200}(0,0)(15,-3)(30,0)	
\put(15,-6){\makebox(0,0){\large $8$}}
\put(15,-11){\makebox(0,0){\large $F_{6}$}}
\bezier{100}(30,0)(36,-3)(46,0)	
\put(36,-6){\makebox(0,0){\large $5$}}
\put(36,-11){\makebox(0,0){\large $F_{5}$}}
}
%% \mu_{2}
\uncover<4->{
\allinethickness{0.3mm}			
\put(20,-1){\line(0,1){2}}	
\put(20,6){\makebox(0,0){\large $\mu_{2}^{*}$}}
}
\uncover<4>{
\bezier{100}(0,0)(10,-3)(20,0)	
\put(10,-6){\makebox(0,0){\large $5$}}
\put(10,-11){\makebox(0,0){\large $F_{5}$}}
\bezier{100}(20,0)(25,-3)(30,0)	
\put(20,-6){\makebox(0,0){\large $3$}}
\put(20,-11){\makebox(0,0){\large $F_{4}$}}
}
%% -\mu_{3}
\uncover<5->{
\allinethickness{0.3mm}			
\put(12,-1){\line(0,1){2}}	
\put(12,3){\makebox(0,0){\large $-\mu_{3}^{*}$}}
}
\uncover<5>{
\bezier{100}(0,0)(6,-3)(12,0)	
\put(6,-6){\makebox(0,0){\large $3$}}
\put(6,-11){\makebox(0,0){\large $F_{4}$}}
\bezier{100}(12,0)(16,-3)(20,0)	
\put(16,-6){\makebox(0,0){\large $2$}}
\put(16,-11){\makebox(0,0){\large $F_{3}$}}
}
%% \mu_{4}
\uncover<6->{
\allinethickness{0.3mm}			
\put(8,-1){\line(0,1){2}}	
\put(8,6){\makebox(0,0){\large $\mu_{4}^{*}$}}
}
\uncover<6>{
\bezier{100}(0,0)(4,-3)(8,0)	
\put(4,-6){\makebox(0,0){\large $2$}}
\put(4,-11){\makebox(0,0){\large $F_{3}$}}
\bezier{100}(8,0)(10,-3)(12,0)	
\put(10,-6){\makebox(0,0){\large $1$}}
\put(10,-11){\makebox(0,0){\large $F_{2}$}}
}
%% -\mu_{5}, \mu_{6}
\uncover<7->{
\allinethickness{0.3mm}			
\put(4,-1){\line(0,1){2}}	
\put(4,3){\makebox(0,0){\large $-\mu_{5}^{*}$}}
}
\uncover<7>{
\bezier{50}(0,0)(2,-3)(4,0)	
\put(2,-6){\makebox(0,0){\large $1$}}
\put(2,-11){\makebox(0,0){\large $F_{2}$}}
\bezier{50}(4,0)(6,-3)(8,0)	
\put(6,-6){\makebox(0,0){\large $1$}}
\put(6,-11){\makebox(0,0){\large $F_{1}$}}
}
\uncover<8->{
\put(4,10){\makebox(0,0){\large $\mu_{6}^{*}$}}
}
\end{picture}
\end{center}
\end{frame}



%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{}
\begin{center}
\unitlength=1mm
\begin{picture}(111,10)(0,0)
\uncover<1->{
\allinethickness{0.8mm}
\put(0,0){\line(1,0){111}}	%
\put(0,-1){\line(0,1){2}}	%
\put(-2,3){\makebox(0,0){\large $0$}}	%
\put(111,-1){\line(0,1){2}}	%
\put(111,3){\makebox(0,0){\large $c$}}	%
}
%% \mu_{0}
\uncover<2->{
\allinethickness{0.3mm}
\put(46,-1){\line(0,1){2}}	
\put(46,3){\makebox(0,0){\large $\mu_{0}^{*}$}}
\bezier{300}(46,0)(79,-4)(111,0)	
\put(79,-6){\makebox(0,0){\large $F_{8}$}}
}
\uncover<2>{
\bezier{200}(0,0)(23,-4)(46,0)	
\put(23,-6){\makebox(0,0){\large $F_{7}$}}
}
%% -\mu_{1}
\uncover<3->{
\allinethickness{0.3mm}			
\put(30,-1){\line(0,1){2}}	
\put(30,3){\makebox(0,0){\large $-\mu_{1}^{*}$}}
\bezier{100}(30,0)(36,-3)(46,0)	
\put(36,-6){\makebox(0,0){\large $F_{5}$}}
}
\uncover<3>{
\bezier{100}(0,0)(15,-3)(30,0)	
\put(15,-6){\makebox(0,0){\large $F_{6}$}}
}
%% \mu_{2}
\uncover<4->{
\allinethickness{0.3mm}			
\put(20,-1){\line(0,1){2}}	
\put(20,6){\makebox(0,0){\large $\mu_{2}^{*}$}}
\bezier{100}(20,0)(25,-3)(30,0)	
\put(25,-6){\makebox(0,0){\large $F_{4}$}}
}
\uncover<4>{
\bezier{100}(0,0)(10,-3)(20,0)	
\put(10,-6){\makebox(0,0){\large $F_{5}$}}
}
%% -\mu_{3}
\uncover<5->{
\allinethickness{0.3mm}			
\put(12,-1){\line(0,1){2}}	
\put(12,3){\makebox(0,0){\large $-\mu_{3}^{*}$}}
\bezier{100}(12,0)(16,-3)(20,0)	
\put(16,-6){\makebox(0,0){\large $F_{3}$}}
}
\uncover<5>{
\bezier{100}(0,0)(6,-3)(12,0)	
\put(6,-6){\makebox(0,0){\large $F_{4}$}}
}
%% \mu_{4}
\uncover<6->{
\allinethickness{0.3mm}			
\put(8,-1){\line(0,1){2}}	
\put(8,6){\makebox(0,0){\large $\mu_{4}^{*}$}}
\bezier{100}(8,0)(10,-3)(12,0)	
\put(10,-6){\makebox(0,0){\large $F_{2}$}}
}
\uncover<6>{
\bezier{100}(0,0)(4,-3)(8,0)	
\put(4,-6){\makebox(0,0){\large $F_{3}$}}
}
%% -\mu_{5}, \mu_{6}
\uncover<7->{
\allinethickness{0.3mm}			
\put(4,-1){\line(0,1){2}}	
\put(4,3){\makebox(0,0){\large $-\mu_{5}^{*}$}}
\bezier{50}(0,0)(2,-3)(4,0)	
\put(2,-6){\makebox(0,0){\large $F_{2}$}}
\bezier{50}(4,0)(6,-3)(8,0)	
\put(6,-6){\makebox(0,0){\large $F_{1}$}}
}
\uncover<7->{
\put(4,10){\makebox(0,0){\large $\mu_{6}^{*}$}}
}
\end{picture}
\end{center}
\vspace{5mm}

\[
\hspace*{-4mm}
\mbox{�œK�� \ }
\left\{
\begin{array}{rclcl}
\uncover<2->{\mu_{0}^{*} & = &  \dfrac{F_{7}}{F_{7}+F_{8}} \ c }
& \uncover<8->{= & \dfrac{F_7}{F_9} \, c \,,} \\[4mm]
\uncover<3->{-\mu_{1}^{*} & = & \dfrac{F_{6}}{F_{6}+F_{5}} \, \mu_{0}^{*}} 
& \uncover<8->{= & \dfrac{F_6}{F_7} \ndot 
\dfrac{F_{7}}{F_{9}} \, c \,,} \\[4mm]
\uncover<4->{\mu_{2}^{*} & = & \dfrac{F_{5}}{F_{5}+F_{4}} \, (-\mu_{1}^{*})} 
& \uncover<8->{= & \dfrac{F_5}{F_6} \ndot \dfrac{F_6}{F_7} \ndot 
\dfrac{F_{7}}{F_{9}} \, c \,,} \\[4mm]
\uncover<5->{-\mu_{3}^{*} & = & \dfrac{F_{4}}{F_{4}+F_{3}} \, \mu_{2}^{*}} 
& \uncover<8->{= & \dfrac{F_4}{F_5} \ndot \dfrac{F_5}{F_6} \ndot 
\dfrac{F_6}{F_7} \ndot \dfrac{F_{7}}{F_{9}} \, c \,,} \\[4mm]
\uncover<6->{\vdots} & & \hspace{10mm} \uncover<6->{\vdots} & & 
\hspace{10mm}\uncover<8->{\vdots}\\[2mm]
\uncover<7->{\mu_{6}^{*} = -\mu_{5}^{*} 
& = & \dfrac{F_{2}}{F_{2}+F_{1}} \, \mu_{4}^{*}} 
& \uncover<8->{= & \dfrac{F_2}{F_3} \ndot \dfrac{F_3}{F_4} \ndot \cdots \ndot
\dfrac{F_{7}}{F_{9}} \, c \,.}
\end{array}
\right.
%\mbox{�ŋ��߂���. }
\]

\end{frame}



%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{}
���Ȃ킿�A�o�Ζ��$({\rm D_{s8}})$�̍ő��$\mu^{*}$�́A
\[
\begin{array}{lcl}
\mu^{*} 
& = &
(\mu^{*}_{0},~ \mu^{*}_{1},~ \cdots,~ \mu^{*}_{6},~ \mu_{7}) \\[3mm]
 & = & 
\dfrac{c}{\,\,F_{9}\,\,}
\left(
\begin{array}{cccccccc}
       F_{7}\,, & -F_{6}\,, &  F_{5}\,, & -F_{4}\,, & F_{3}\,, 
& -F_{2}\,, & F_{1}\,, & -F_{0}
\end{array}
\right) \\
\end{array}
\]
�ƂȂ�. 


\begin{center}
\unitlength=1mm
\begin{picture}(111,10)(0,0)
\allinethickness{0.8mm}
\put(0,0){\line(1,0){111}}	%
\put(0,-1){\line(0,1){2}}	%
\put(0,-3){\makebox(0,0){\large $-c$}}	%
\put(111,-1){\line(0,1){2}}	%
\put(111,-3){\makebox(0,0){\large $c$}}	%
\put(53,-1){\line(0,1){2}}	%
\put(53,-3){\makebox(0,0){\large $0$}}	%
\allinethickness{0.3mm}	
\put(79,-1){\line(0,1){2}}	%
\put(79,6){\makebox(0,0){\large $\RED{\mu_{0}^{*}}$}}	%
\put(79,15){\makebox(0,0){\large \MAGENTA{\small $\dfrac{F_{7}}{F_{9}}c$}}}
\put(69,-1){\line(0,1){2}}	%
\put(69,-8){\makebox(0,0){\large $-\mu_{1}^{*}$}}	%
\put(37,-1){\line(0,1){2}}	%
\put(37,6){\makebox(0,0){\large $\RED{\mu_{1}^{*}}$}}	%
\put(37,15){\makebox(0,0){\large \MAGENTA{\small $-\dfrac{F_{6}}{F_{9}}c$}}}
\put(63,-1){\line(0,1){2}}	%
\put(63,6){\makebox(0,0){\large $\RED{\mu_{2}^{*}}$}}	%
\put(63,15){\makebox(0,0){\large \MAGENTA{\small $\dfrac{F_{5}}{F_{9}}c$}}}
\put(59,-1){\line(0,1){2}}	%
\put(59,-8){\makebox(0,0){\large $-\mu_{3}^{*}$}}	%
\put(47,-1){\line(0,1){2}}	%
\put(47,6){\makebox(0,0){\large $\RED{\mu_{3}^{*}}$}}	%
\put(47,15){\makebox(0,0){\large \MAGENTA{\small $-\dfrac{F_{4}}{F_{9}}c$}}}
\end{picture}
\end{center}
\vspace{10mm}

�� $(\mu_{0}^{*}, \mu_{1}^{*}, \cdots, \mu_{6}^{*}, \mu_{7})$
�͌��݃t�B�{�i�b�`�o�H���\�����Ă���B

\end{frame}



%%%%%%%%%%%%%
\begin{frame}
\frametitle{�o�Ζ��$({\rm D_{s8}})$�̍ő�l}
�o�Ζ��$({\rm D_{s8}})$�̖ړI�֐��́A
\[
g(\mu) : \dql 
2cF_{8}\mu_{0} - \dsum_{n=0}^{6}F_{8-n}
\left[\mu_{n}^{2} + \left( \dfrac{F_{8-n}}{F_{7-n}}\mu_{n} + \mu_{n+1} 
\right)^{2} \right] - F_{1}\mu_{7}^{2}. 
\]
�ł���A�ő��$\mu^{*}$�́A
\[
\begin{array}{lcl}
\mu^{*} 
%& = &
%(\mu^{*}_{0},~ \mu^{*}_{1},~ \cdots,~ \mu^{*}_{6},~\mu_{7}) \\[3mm]
 & = & 
\dfrac{c}{\,\,F_{9}\,\,}
\left(
\begin{array}{cccccccc}
       F_{7}\,, & -F_{6}\,, &  F_{5}\,, & -F_{4}\,, & F_{3}\,, & -F_{2}\,, 
& F_{1}\,, & -F_{0}
\end{array}
\right) \\
\end{array}
\]
�Ȃ̂ŁA
\pause
\begin{eqnarray*}
g(\mu^{*}) 
\eql  \dfrac{c^2}{F_{9}^{2}} \Biggl\{ 
2F_{7}F_{8}F_{9}-F_{8}\left[F_{7}^{2} + (F_{8}-F_{6})^{2} \right] \\[1.56mm]
& & \ \ \ \ \ \ - F_{7} \left[F_{6}^{2} + (F_{7} - F_{5})^{2} \right] - \cdots \\[1.56mm]
& & \ \ \ \ \ \ \cdots -F_{2} \left[F_{1}^{2} + (F_{2}-F_{0})^{2} \right]
- F_{1}F_{2}F_{0}^{2}
\Biggr\}
\end{eqnarray*}

\end{frame}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{}
���i�������a�j���A
\begin{eqnarray*}
g(\mu^{*})F_{9}^{2}/c^{2} 
\eql  2F_{7}F_{8}F_{9} 
-F_{8}\left[F_{7}^{2} + (F_{8}-F_{6})^{2} \right] \\[1.56mm]
& & \ \ \ \ \ \ - F_{7} \left[F_{6}^{2} + (F_{7} - F_{5})^{2} \right] - \cdots \\[1.56mm]
& & \ \ \ \ \ \ \cdots -F_{2} \left[F_{1}^{2} + (F_{2}-F_{0})^{2} \right]
- F_{1}F_{2}F_{0}^{2} \\[1.56mm]
\eql 2F_{7}F_{8}F_{9}-2(F_{8}F_{7}^{2} + \cdots + F_{2}F_{1}^{2}) \\[1.56mm]
\eql F_{7}F_{8}F_{9} 
\end{eqnarray*}
�ƂȂ�A
���Ȃ킿�A�ő�l�� \ 
%\begin{eqnarray*}
$
g(\mu^{*}) \dql \dfrac{F_{7} F_{8}}{\,F_{9}\,}c^{2} 
= \dfrac{13 \cdot 21}{\,34\,} \, c^{2}
$\ �ł���B
%\end{eqnarray*} 


\end{frame}




%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{�_�E���B���`�E�R�[�h}
�f��u�_�E���B���`�E�R�[�h�v�i2006�N�j�ł͖`���̃V�[����10���̈Ïؔԍ�
{\large
\[
1~~~1~~~2~~~3~~~5~~~8~~~1~~~3~~~2~~~1
\]
}
������Ă���B

\pause 

����10�̐����� 8�̐�������Ȃ��
$$ 
\RedBlackframebox{24}{15}{$1$}~~\,~\RedBlackframebox{24}{15}{$1$}~~\,
~\RedBlackframebox{24}{15}{$2$}~~\,~\RedBlackframebox{24}{15}{$3$}~~\,
~\RedBlackframebox{24}{15}{$5$}~~\,~\RedBlackframebox{24}{15}{$8$}~~\,
~\RedBlackframebox{24}{15}{$13$}~~\,~\RedBlackframebox{24}{15}{$21$} 
$$
���Ȃ킿�A�t�B�{�i�b�`����̑�1�� 
$ F_{1} = 1 $ �����8�� $ F_{8} = 21 $ �܂ł̐���ɑ��Ȃ�Ȃ��B

\begin{table}[htbp]
%
\begin{center}
% \begin{tabular}{c|cccccccccccccccccc} \hline
\begin{tabular}{c|ccccccccccccc} \hline
$ n $     &  $ 0 $  &  $ 1 $  &  $ 2  $ &  $ 3 $ &  $ 4 $ &  $ 5 $  & $ 6 $ & $ 7 $ & $ 8 $ & $ 9 $ & $ 10 $ & $ 11 $ & $ 12 $ \\ \hline
%& $ 13 $ & $ 14 $ & $ 15 $ & $ 16 $   \\ \hline 
$ F_{n} $ &  $ 0 $  &  $ 1 $  &  $ 1 $ &  $ 2 $ &  $3 $ &  $ 5  $  & $ 8 $ & $ 13 $  & $ 21 $ & $ 34 $ & $ 55 $ & $ 89 $ & $ 144 $ \\ \hline
%& $ 233 $ & $ 377 $ & $ 610 $ & $ 987 $   \\ \hline
\end{tabular}
\end{center}
%\caption{stop reward} \label{tab:h1}
\end{table}
\vspace{-5mm}
\begin{center}
Table 1:~~Fibonacci sequence $ \{F_{n}\} $
%�\ 1 ~~�t�B�{�i�b�`���� $ \{F_{n}\} $
\end{center}

\end{frame}



%%%%%%%%%%%%%%%%%
\begin{frame}
%\frametitle{4 : 6 �^���]���}
\frametitle{���݃_�E���B���`�E�R�[�h46}
���� $({\rm P_{c4}})$ ����ёo�Ζ�� $({\rm D_{c4}})$ �̕ϐ��𔽓]�����āA���]���
%
\begin{eqnarray*}
& &{\rm minimize} \hspace{3mm} \dsum_{n=1}^{4}\left[ x_{n}^{2} + (x_{n} + x_{n+1})^{2} \right] \\ 
({\rm RP_{c4}})& &{\rm subject~~ to}\hspace{2ex} 
{\rm (i)}~~ -\infty < x_{n} < \infty ~~~~n= 1,2,3,4 \\
& &\hspace*{21.2mm} {\rm (ii)}~~x_{5} = c
\end{eqnarray*}
%
�����
%
\begin{eqnarray*}
\hspace*{-7mm}& &{\rm Maximize} \hspace{3mm} - \mu_{1}^{2} - \dsum_{n=1}^{3}\left[\mu_{n}^{2} + (\mu_{n} + \mu_{n+1})^{2}  \right] 
- \mu_{4}^{2} + 2c\mu_{4}  \\[-2,56mm] 
\hspace*{-7mm}({\rm RD_{c4}})& & \\[-2,56mm] 
\hspace*{-7mm}& &{\rm subject~\, to}\hspace{2ex} {\rm (i)}~~~ -\infty < \mu_{n} < \infty ~~~~n= 1,2,3,4
\end{eqnarray*}
%
���l����. 
%�����]���̍œK���̊Ԃɂ͍��x��\textbf{�O������}
%���݃t�B�{�i�b�`����o�ΐ������藧��. 


\end{frame}





%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{}
���$({\rm RP_{c4}})$�̍ŏ���$\tilde{x}$�ƍŏ��l$m^{\prime}_{4}$�́A
\[
\hspace{10mm}\begin{array}{lcl}
\tilde{x} & \dql & (\tilde{x}_{1}, ~\tilde{x}_{2}, ~\tilde{x}_{3}, 
~\tilde{x}_{4}, ~x_{5}) \\[3mm]
 & \dql & \dfrac{c}{\,\,34 \,\,}
\left(
\begin{array}{ccccc}
      1 \,, & -2 \,, & 5 \,, &  -13 \,, & 34 
\end{array}
\right) \\[6mm]
m_{4}^{\prime} & = & 
\dfrac{\,21 \,}{\,\,34 \,\,}c^{2}
\end{array}
\]
�ł���A
���$({\rm RD_{c4}})$�̍ő��$\mu^{\star}$�ƍő�l$M^{\prime}_{4}$�́A
\[
\begin{array}{lcl}
\mu^{\star} & \dql & 
( \mu^{\star}_{1},~ \mu^{\star}_{2},~ \mu^{\star}_{3},
~ \mu^{\star}_{4}) \\[3mm]
 & \dql & \dfrac{c}{\,\,34 \,\,}
\left(
\begin{array}{cccc}
      -1 \,, & 3 \,, & -8 \,, &  21 
\end{array}
\right) \\[6mm]
M_{4}^{\prime} & = & 
\dfrac{\,21 \,}{\,\,34 \,\,}c^{2}
\end{array}
\]
�ƂȂ�. 
\end{frame}



%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{}
�����]���̍œK���̊Ԃɂ͍��x��\RED{�O������}���݃t�B�{�i�b�`����o�ΐ���
���藧��. 
%���Ȃ킿�A
%���]���� $ ({\rm RP}_{4}) $ �̍ŏ����Ɣ��]�o�Ζ�� $ ({\rm RD}_{4}) $ �̍ő�%���̊Ԃɂ͎��̎O�ʈ�̂̊֌W�����藧���Ă���.  
\begin{itemize}
	\item \BLUE{(�o�ΐ�)}\\[2mm]
	�ŏ��l�ƍő�l���������F
	\[ 
	m_{4}^{\prime} \dql M_{4}^{\prime} \dql \dfrac{21}{\,34\,} \,c^{2}.
	\]
	\item \BLUE{(2�i�K�t�B�{�i�b�`)}\\[2mm]
	�ŏ��� $ \tilde{x} $
%$ (\tilde{x}_{1}, ~\tilde{x}_{2}, ~\tilde{x}_{3}, ~\tilde{x}_{4}) $ 
�ƍő�_ $\mu^{\star}$
%$( \mu^{\star}_{1},~ \mu^{\star}_{2},~ \mu^{\star}_{3},~ \mu^{\star}_{4}) $ 
�͋��Ɍ��t�B�{����̑O����2�i���тł���. 
\[
\begin{array}{lcl}
\tilde{x} & = &
(\tilde{x}_{1}, ~\tilde{x}_{2}, ~\tilde{x}_{3}, 
~\tilde{x}_{4}, ~x_{5})  \\[3mm]
 & = & \dfrac{c}{\,\,34\,\,}
\left(
\begin{array}{ccccc}
\MagRedframebox{19}{12}{$1$}\,, & \MagBlueframebox{19}{12}{$-2$}\,, 
& \MagRedframebox{19}{12}{$5$}\,, 
&  \MagBlueframebox{19}{12}{$-13$}\,, & \MagRedframebox{19}{12}{$34$} 
\end{array}
\right) \\[4mm] 
 & = & \dfrac{c}{\,\,F_{9} \,\,}
\left(
\begin{array}{ccccc}
\MagRedframebox{19}{12}{$F_{1}$}\,, & \MagBlueframebox{19}{12}{$-F_{3}$}\,, 
& \MagRedframebox{19}{12}{$F_{5}$}\,, &  \MagBlueframebox{19}{12}{$-F_{7}$}\,, 
& \MagRedframebox{19}{12}{$F_{9}$} 
\end{array}
\right),
\end{array}
\]
\end{itemize}
\end{frame}






%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{}
\[
\hspace*{-15mm}
\begin{array}{lcl}
\mu^{\star} & = &
(\mu^{\star}_{1},~ \mu^{\star}_{2},~ \mu^{\star}_{3},~ \mu^{\star}_{4}) \\[3mm]
 & = & 
\dfrac{c}{\,\,34 \,\,}
\left(
\begin{array}{cccc}
       \CyanBlueframebox{19}{12}{$-1$}\,, & \CyanRedframebox{19}{12}{$3$}\,, 
&  \CyanBlueframebox{19}{12}{$-8$}\,, & \CyanRedframebox{19}{12}{$21$} 
\end{array}
\right) \\[4mm]
 & = & 
\dfrac{c}{\,\,F_{9} \,\,}
\left(
\begin{array}{cccc}
       \CyanBlueframebox{19}{12}{$-F_{2}$}\,, 
& \CyanRedframebox{19}{12}{$F_{4}$}\,, 
&  \CyanBlueframebox{19}{12}{$-F_{6}$}\,, 
& \CyanRedframebox{19}{12}{$F_{8}$} 
\end{array}
\right). 
\end{array}
\]

\pause

\begin{itemize}
\item \BLUE{(����t�B�{�i�b�`)}\\[2mm]
�ŏ���$\tilde{x}$�ƍő��$\mu^{\star}$�����݂ɕ҂ނƁA
$$ 
\framebox(24,15){$\tilde{x}_{1}$}~~\,~\dashbox(24,15){$\mu^{\star}_{1}$}~~\,~\framebox(24,15){$\tilde{x}_{2}$}~~\,~\dashbox(24,15){$\mu^{\star}_{2}$}~~\,~\framebox(24,15){$\tilde{x}_{3}$}~~\,~\dashbox(24,15){$\mu^{\star}_{3}$}~~\,~\framebox(24,15){$\tilde{x}_{4}$}~~\,~\dashbox(24,15){$\mu^{\star}_{4}$}~~\,~\framebox(24,15){$x_{5}$}
$$
$$ 
\MagRedframebox{24}{15}{$1$}~~\,~
\CyanBlueframebox{24}{15}{$-1$}~~
\,~\MagBlueframebox{24}{15}{$-2$}~~
\,~\CyanRedframebox{24}{15}{$3$}~~\,~
\MagRedframebox{24}{15}{$5$}~~\,~
\CyanBlueframebox{24}{15}{$-8$}~~\,~
\MagBlueframebox{24}{15}{$-13$}~~
\,~\CyanRedframebox{24}{15}{$21$}~~\,~
\MagRedframebox{24}{15}{$34$}
$$
�ɂȂ�. 
\end{itemize}

\pause

���Ȃ킿�A�����\RED{�u���݃_�E���B���`�E�R�[�h�v}���\������B
\end{frame}



%%%%%%%%%%%%%%%%%
\begin{frame}
%\frametitle{6 : 4�^���]���}
\frametitle{���݃_�E���B���`�E�R�[�h64}
���� $({\rm P_{s8}})$ ����ёo�Ζ�� $({\rm D_{s8}})$ �̕ϐ��𔽓]�����āA
���]���
\begin{eqnarray*}
& \hspace*{-5mm} &{\rm minimize} \hspace{3mm} \dfrac{\,F_{-1}F_{0}\,}{\,F_{1}\,}x_{1}^{2} + \dsum_{k=1}^{8}\left[ \dfrac{\,F_{k-1}^{2}\,}{\,F_{k}\,}x_{k}^{2} + F_{k}(x_{k} + x_{k+1})^{2} \right ] \\
%\\[-2.26mm]
\hspace*{-30mm} ({\rm RP_{s8}})& \hspace*{-5mm} & 
{\rm subject~~ to}\hspace{2ex} 
{\rm (i)}~~ -\infty < x_{n} < \infty ~~~~n= 1, 2, \cdots, 8 \\
& \hspace*{-5mm} &\hspace*{21mm} {\rm (ii)}~~x_{9} = c
%({\rm RP_{8}})& & \hspace{16mm}  \\[-2.26mm]
%& & \hspace{16mm}  \\[-1.26mm]
%({\rm RP_{8}}) & &{\rm subject~~ to}\hspace{3ex} {\rm (i)}~~ x \in R^{8} \\
%& & \hspace{24.5mm}{\rm (ii)}~~ x_{9} = c.
%& &{\rm subject~~ to}\hspace{3ex} 
%{\rm (i)}~~ (x_{1}, \cdots, x_{8} ) \in R^{8},
%~~{\rm (ii)}~~ x_{9} = c.
\end{eqnarray*}
%
�����
%
\begin{eqnarray*}
& \hspace*{-5mm} &{\rm Maximize} \hspace{3mm} -F_{1}F_{2} \mu_{1}^{2} 
- \dsum_{k=1}^{7} \Bigl[\dfrac{\,F_{k+1}\,}{\,F_{k}^{2}\,}
\left(F_{k} \mu_{k} +  F_{k+1} \mu_{k+1}\right)^{2} \\[-1.56mm]
& \hspace*{-5mm} & \hspace*{60mm}+  F_{k+1} 
\mu_{k+1}^{2} \Bigr] + 2F_{8}c \mu_{8} \\
\hspace*{-30mm} ({\rm RD_{s8}})& \hspace*{-5mm} &{\rm subject~~ to}\hspace{2ex} 
{\rm (i)}~~ -\infty < \mu_{n} < \infty ~~~~n= 2, 3, \cdots, 8\\
& \hspace*{-5mm} & \hspace{21mm}{\rm (ii)}~~\mu_{1} = 0
%\\[-2.26mm] 
%({\rm RD_{8}})& &\hspace{23mm} \\[-2.26mm] 
%%& &\hspace{23mm} \\[-2.26mm] 
%%
%& &{\rm subject~~ to}\hspace{2ex} 
%{\rm (i)}~~ (\mu_{1}, \cdots, \mu_{8} ) \in R^{8},~~~ 
%{\rm (ii)}~~\mu_{1} = 0. 
%
\end{eqnarray*}
���l����B
%���l����ƁA�����]���̍œK���̊Ԃɂ͍��x��\textbf{�O������}
%���݃t�B�{�i�b�`�E�V�t�g�o�ΐ������藧�B
\end{frame}





%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{}
����$({\rm RP_{s8}})$�̍ŏ���$\tilde{x}$�ƍŏ��l$m^{\prime}_{8}$�́A
\[
\begin{array}{lcl}
\tilde{x} & \dql & (\tilde{x}_{1}, ~\tilde{x}_{2}, ~\cdots, 
~\tilde{x}_{8}) \\[3mm]
 & \dql & \dfrac{c}{\,\,34 \,\,}
\left(
\begin{array}{cccccccc}
      1 \,, & -1 \,, & 2 \,, & -3 \,, & 
5 \,, & -8 \,, & 13 \,, & -21
%
%\begin{array}{cccccccc}
%      \MagRedframebox{19}{12}{1}, & \MagBlueframebox{19}{12}{-1}, & 
%\MagRedframebox{19}{12}{2}, & \MagBlueframebox{19}{12}{-3}, & 
%\MagRedframebox{19}{12}{5}, & \MagBlueframebox{19}{12}{-8}, & 
%\MagRedframebox{19}{12}{13}, & \MagBlueframebox{19}{12}{-21}
\end{array}
\right) \\[6mm]
m_{8}^{\prime} & = & 
\dfrac{\,13 \cdot 21 \,}{\,\,34 \,\,}c^{2}
\end{array}
\]
�ł���A
�o�Ζ��$({\rm RD_{s8}})$�̍ő��$\mu^{\star}$�ƍő�l$M^{\prime}_{8}$�́A
\[
\begin{array}{lcl}
\mu^{\star} & \dql & 
( \mu_{1},~ \mu^{\star}_{2},~ \cdots,
~ \mu^{\star}_{8}) \\[3mm]
 & \dql & \dfrac{c}{\,\,34 \,\,}
\left(
\begin{array}{cccccccc}
      0 \,, & 1 \,, & -1 \,, &  2 \,, &  -3 \,, & 5 \,, & -8 \,, & 13
\end{array}
\right) \\[6mm]
M_{8}^{\prime} & = & 
\dfrac{\,13 \cdot 21 \,}{\,\,34 \,\,}c^{2}
\end{array}
\]
�ƂȂ�B
\end{frame}



%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{}
�����]���̍œK���̊Ԃɂ͍��x��\RED{�O������}���݃t�B�{�i�b�`�E
�V�t�g�o�ΐ������藧�B
%���Ȃ킿�A
%���]���� $ ({\rm RP}_{4}) $ �̍ŏ����Ɣ��]�o�Ζ�� $ ({\rm RD}_{4}) $ ��
%�ő���̊Ԃɂ͎��̎O�ʈ�̂̊֌W�����藧���Ă���.  
\begin{itemize}
	\item \BLUE{(�o�ΐ�)}\\[2mm]
	�ŏ��l�ƍő�l���������F
	\[ 
	m_{8}^{\prime} \dql M_{8}^{\prime} \dql 
	\dfrac{13 \cdot 21}{\,34\,} \,c^{2}.
	\]
	\item \BLUE{(���݃t�B�{�i�b�`)}\\[2mm]
	�ŏ��� $ \tilde{x} $
%$ (\tilde{x}_{1}, ~\tilde{x}_{2}, ~\tilde{x}_{3}, ~\tilde{x}_{4}) $ 
�ƍő�_ $\mu^{\star}$
%$( \mu^{\star}_{1},~ \mu^{\star}_{2},~ \mu^{\star}_{3},~ \mu^{\star}_{4}) $ 
�͋��Ɍ��t�B�{�i�b�`����̑O�����ł���B
\[
\hspace*{-15mm}\begin{array}{lcl}
\tilde{x} & \dql & (\tilde{x}_{1}, ~\tilde{x}_{2}, ~\cdots, 
~\tilde{x}_{8}) \\[3mm]
 & \dql & \dfrac{c}{\,\,34 \,\,}
\left(
\begin{array}{cccccccc}
      \MagRedframebox{19}{12}{$1$}, & \MagBlueframebox{19}{12}{$-1$}, & 
\MagRedframebox{19}{12}{$2$}, & \MagBlueframebox{19}{12}{$-3$}, & 
\MagRedframebox{19}{12}{$5$}, & \MagBlueframebox{19}{12}{$-8$}, & 
\MagRedframebox{19}{12}{$13$}, & \MagBlueframebox{19}{12}{$-21$}
\end{array}
\right) \\[6mm]
 & \dql & \dfrac{c}{\,\,F_{9} \,\,}
\left(
\begin{array}{cccccccc}
      \MagRedframebox{19}{12}{$F_{1}$}, & \MagBlueframebox{19}{12}{$-F_{2}$}, 
& \MagRedframebox{19}{12}{$F_{3}$}, & \MagBlueframebox{19}{12}{$-F_{4}$}, & 
\MagRedframebox{19}{12}{$F_{5}$}, & \MagBlueframebox{19}{12}{$-F_{6}$}, & 
\MagRedframebox{19}{12}{$F_{7}$}, & \MagBlueframebox{19}{12}{$-F_{8}$}
\end{array}
\right) 
\end{array}
\]
\end{itemize}
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{}
\[
\hspace*{-5mm}\begin{array}{lcl}
\mu^{\star} & \dql & 
(\mu_{1}, ~\mu^{\star}_{2}, ~\cdots, ~\mu^{\star}_{8}) \\[3mm]
 & \dql & \dfrac{c}{\,\,34 \,\,} 
\left(
\begin{array}{cccccccc}
      \CyanBlueframebox{19}{12}{$0$}, & \MagRedframebox{19}{12}{$1$}, & 
\MagBlueframebox{19}{12}{$-1$}, & \MagRedframebox{19}{12}{$2$}, & 
\MagBlueframebox{19}{12}{$-3$}, & 
\MagRedframebox{19}{12}{$5$}, & \MagBlueframebox{19}{12}{$-8$}, & 
\MagRedframebox{19}{12}{$13$}
\end{array}
\right) \\[6mm]
 & \dql & \dfrac{c}{\,\,F_{9} \,\,} 
\left(
\begin{array}{cccccccc}
      \CyanBlueframebox{19}{12}{$-F_{0}$}, & 
\MagRedframebox{19}{12}{$F_{1}$}, & 
\MagBlueframebox{19}{12}{$-F_{2}$}, & 
\MagRedframebox{19}{12}{$F_{3}$}, & 
\MagBlueframebox{19}{12}{$-F_{4}$}, & 
\MagRedframebox{19}{12}{$F_{5}$}, & 
\MagBlueframebox{19}{12}{$-F_{6}$}, & 
\MagRedframebox{19}{12}{$F_{7}$}
\end{array}
\right) 
\end{array}
\]

\pause

\begin{itemize}
\item \BLUE{(�V�t�g)}\\[2mm]
�ŏ���$\tilde{x}$�ƍő��$\mu^{\star}$�͂��݂�1�i�V�t�g�����Ă���B
\[
\hspace*{-15mm}\begin{array}{lcl}
\tilde{x} & \dql & \dfrac{c}{\,\,34 \,\,}
\left(
\begin{array}{cccccccc}
      \MagRedframebox{19}{12}{$1$}, & \MagBlueframebox{19}{12}{$-1$}, & 
\MagRedframebox{19}{12}{$2$}, & \MagBlueframebox{19}{12}{$-3$}, & 
\MagRedframebox{19}{12}{$5$}, & \MagBlueframebox{19}{12}{$-8$}, & 
\MagRedframebox{19}{12}{$13$}, & \MagBlueframebox{19}{12}{$-21$}
\end{array}
\right) \\[6mm]

\mu^{\star} & \dql & \dfrac{c}{\,\,34 \,\,}
\left(
\begin{array}{cccccccc}
\CyanBlueframebox{19}{12}{$0$}, &
      \MagRedframebox{19}{12}{$1$}, & \MagBlueframebox{19}{12}{$-1$}, & 
\MagRedframebox{19}{12}{$2$}, & \MagBlueframebox{19}{12}{$-3$}, & 
\MagRedframebox{19}{12}{$5$}, & \MagBlueframebox{19}{12}{$-8$}, & 
\MagRedframebox{19}{12}{$13$}
\end{array}
\right) 
\end{array}
\]
�ɂȂ�B
\end{itemize}

\pause

$\tilde{x}  \dql  (\tilde{x}_{1}, ~\tilde{x}_{2}, ~\cdots, 
~\tilde{x}_{8})$�́A
\RED{�u���݃_�E���B���`�E�R�[�h�v }���\������B

\end{frame}




%%%%%%%%%%%%%%%
\begin{comment}



%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{}
\[
%\hspace*{-15mm}\begin{array}{lcl}
%\mu^{\star} & \dql & (\mu_{1}, ~\mu^{\star}_{2}, ~\cdots, 
%~\mu^{\star}_{8}) \\[3mm]
% & \dql & \dfrac{c}{\,\,34 \,\,} \\
%\left(
%\begin{array}{cccccccc}
%      \MagBlueframebox{19}{12}{0}, & \MagRedframebox{19}{12}{1}, & \MagBluefram%ebox{19}{12}{-1}, & \MagRedframebox{19}{12}{2}, & 
%\MagBlueframebox{19}{12}{-3}, & 
%\MagRedframebox{19}{12}{5}, & \MagBlueframebox{19}{12}{-8}, & 
%\MagRedframebox{19}{12}{13}
%\end{array}
%\right) \\[6mm]
% & \dql & \dfrac{c}{\,\,F_{9} \,\,}
%\left(
%\begin{array}{cccccccc}
%\CyanBlueframebox{19}{12}{-F_{0}}, &
%      \MagRedframebox{19}{12}{$F_{1}$}, & \MagBlueframebox{19}{12}{$-F_{2}$}, 
%& \MagRedframebox{19}{12}{$F_{3}$}, & \MagBlueframebox{19}{12}{$-F_{4}$}, & 
%\MagRedframebox{19}{12}{$F_{5}$}, & \MagBlueframebox{19}{12}{$-F_{6}$}, & 
%\MagRedframebox{19}{12}{$F_{7}$}
%\end{array}
%\right) 
%\end{array}
\]



\pause

\begin{itemize}
\item \BLUE{(�V�t�g)}\\[2mm]
�ŏ���$\tilde{x}$�ƍő��$\mu^{\star}$�͂��݂�1�i�V�t�g�����Ă���B
\[
\hspace*{-15mm}\begin{array}{lcl}
\tilde{x} & \dql & \dfrac{c}{\,\,34 \,\,}
\left(
\begin{array}{cccccccc}
      \MagRedframebox{19}{12}{1}, & \MagBlueframebox{19}{12}{-1}, & 
\MagRedframebox{19}{12}{2}, & \MagBlueframebox{19}{12}{-3}, & 
\MagRedframebox{19}{12}{5}, & \MagBlueframebox{19}{12}{-8}, & 
\MagRedframebox{19}{12}{13}, & \MagBlueframebox{19}{12}{-21}
\end{array}
\right) \\[6mm]

\mu^{\star} & \dql & \dfrac{c}{\,\,34 \,\,}
\left(
\begin{array}{cccccccc}
\CyanBlueframebox{19}{12}{0}, &
      \MagRedframebox{19}{12}{1}, & \MagBlueframebox{19}{12}{-1}, & 
\MagRedframebox{19}{12}{2}, & \MagBlueframebox{19}{12}{-3}, & 
\MagRedframebox{19}{12}{5}, & \MagBlueframebox{19}{12}{-8}, & 
\MagRedframebox{19}{12}{13}
\end{array}
\right) 
\end{array}
\]
�ɂȂ�. 
\end{itemize}

\pause


\RED{���Ȃ킿�A����́u���݃_�E���B���`�E�R�[�h�v���\�����Ă���. }
\end{frame}


\end{comment}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%










%%%%%%%%%%%%%%%%%%%%
%\begin{frame}
\begin{frame}[allowframebreaks]
\frametitle{References}

{\footnotesize 
\noindent
[1]\ R.E.~Bellman, \textit{Introduction to Matrix Analysis}, 
McGraw-Hill, New York, NY, 1970 
(Second Edition is a SIAM edition 1997). \\[.7zw]


\noindent
[2]\ ��{����C���I�v��_�C���o�ŉ�C1987. \\[.7zw]

\noindent
[3]\ S.~Iwamoto, Cross dual on the Golden optimum solutions, 
�u�o�ς̐�����́v, ���吔�����u���^1443, 2005�N7��, pp. 27--43.\\[.7zw]

\noindent
[4]\ S.~Iwamoto, The Golden trinity --- optimility, inequality, identity ---, 
�u�o�ς̐�����́v, ���吔�����u���^, 2006�N5��, pp. 1--14.\\[.7zw]

\noindent
[5]\ S.~Iwamoto, 
The Golden optimum solution in quadratic programming, 
Ed. W. Takahashi and T. Tanaka, Nonlinear Analysis and Convex Analysis (NACA05), Yokohama Publishers, Yokohama, 2007, pp. 199--205.\\[.7zw]


\noindent
[6]\ ��{ ����C�����œK�����ӏ܂���\�\ �o�ϐ��w�ւ̃v�������[�h �iV�j�\�\, 
�o�ϊw�����E�ʍ� ��12��(���o�ϊw��), ����18(2006)�N4��, pp.39--43�D\\[.7zw]


\noindent
[7]\ ��{ ����C�œK���u�_�E���B���`�E�R�[�h�v �\�\ �o�ϐ��w�ւ̃v�������[�h�iVI�j�\�\, �o�ϊw�����E�ʍ� ��13��(���o�ϊw��). 
����19(2007)�N4��, pp.45--52�D\\[.7zw]

\noindent
[8]\ ��{ ����C
�_�E���B���`�E�R�[�h�͍œK���H�C
�����o�ϊw�����Z���^�[���C��37���C����21(2009)�N9���Cpp.1--9. \\[.7zw]

\noindent
[9]\ ��{ ����C�g�� �m���C�A�� �M�V�C
�_�E���B���`�E�R�[�h, 
�o�ϊw����(���o�ϊw��), ��76��(2009�N10��) 2�3��, pp.1--21. \\[.7zw]

\noindent
[10]\ ��{ ����C�g�� �m���C�A�� �M�V�C
�_�E���B���`�E�R�[�h64, 
�o�ϊw����(���o�ϊw��), ��77�� 1��, 2010�N, pp.1--25. \\[.7zw]

\noindent
[11]\ ��{ ����C�ؑ� ���C
���݃_�E���B���`�E�R�[�h, 
�o�ϊw����(���o�ϊw��), ��76�� 4��, 2010�N, pp.1--18. \\[.7zw]

\noindent
[12]\ S.~Iwamoto and Y.~Kimura, 
The Alternately Fibonacci Complementary Duality in 
Quadratic Optimization Problem, 
Journal of Nonlinear Analysis and Optimization, Vol.2, No.1, 2011, pp.93--103.
\\[.7zw]

\noindent
[13]\ ��{ ����C�ؑ� ���C
���݃_�E���B���`�E�R�[�h64, 
�o�ϊw����(���o�ϊw��), ��78�� 1��, 2011�N, 
pp.1--26 (�f�ڗ\��). \\[.7zw]

\noindent
[14]\ S.~Iwamoto and A.~Kira, 
The Fibonacci complementary duality in quadratic programming, 
Ed. W. Takahashi and T. Tanaka, Proceedings of the International Conference on Nonlinear Analysis and Convex Analysis (NACA2007 Taiwan), in press. \\[.7zw]

\newpage

\noindent
[15]\ S. Iwamoto and M. Yasuda, \lq\lq Dynamic programming creates the Golden Ratio, too," \textit{Proc. of the Sixth Intl Conference on Optimization: Techniques and Applications (ICOTA 2004)}, Ballarat, Australia, December, 2004.\\[.7zw]

\noindent
[16]\ S.~Iwamoto and M.~Yasuda, 
Golden optimal path in discrete-time dynamic optimization processes, 
Ed. S.~Elaydi, K.~Nishimura, M.~Shishikura and N.~Tose,  
Advanced Studies in Pure Mathematics 53, June 2009, 
Advances in Discrete Dynamic Systems, pp.77--86. 
Proceedings of The International Conference on Differential Equations and Applications (ICDEA2006), Kyoto University, Kyoto, Japan, July, 2006. \\[.7zw]


\noindent
[17]\ A. Kira and S. Iwamoto, 
Golden complementary dual in quadratic optimization, 
Modeling Decisions for Artificial Intelligence, Proceedings of the Fifth International Confernece (MDAI 2008), Sabadell (Barcelona), Catalonia, Spain, October 30-31, 2008, 
Eds. V. Torra and Y. Narukawa, Springer-Verlag Lecture Notes in Artificial Intelligence, Vol.5285, 2008, pp.191--202. 
}

\end{frame}
\end{document}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{comment}


%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{�s���͂ɂ����$({\rm P_{4}})$�̍œK���̓��o}
$f: \R \rightarrow \R$ �Ƃ���B

\begin{block}{\textbf{���~1}}
$ a > 0 $ �̂Ƃ��C$2$���� 
\[
f(x) = ax^{2} + 2bx + c
\]
�́A
$ x = -\dfrac{\,b\,}{\,a\,} = -a^{-1}b $ �̂Ƃ��A
�ŏ��l 
\begin{eqnarray*}
m  \eql c - \dfrac{\,b^{2}\,}{\,a\,} \\[1.56mm]
 \eql c -  ba^{-1}b
\end{eqnarray*}
�����B
%�������A���萔 $b, \, c $ �͗^�����Ă���Ƃ���B
\end{block}


\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{}
$A = (a_{ij})$ �� $n \times n$ ���Ώ̍s��Ƃ���B
\vspace{5mm}

�C�ӂ�$x \in R^{n} ( x \neq \theta)$�ɑ΂��āA
%$ (0, ~0, ~\ldots, ~0) $�łȂ��C�ӂ̃x�N�g�� $ x = (x_{1}, ~x_{2}, ~\ldots, ~x_%{n}) $�ɑ΂���
$2$���`��
\[
(x, A x) = \dsum_{i=1}^{n}\dsum_{j=1}^{n}a_{ij}x_{i}x_{j}  \ > 0
\]
�Ȃ�΁A$ A $ �� \RED{����l}(positive definite)
�ł���Ƃ����A�L���@\RED{$ A > 0 $}�ŕ\���B
�������A�L�� $(~, ~)$�͓��ς�\���B

�܂����̂Ƃ��A�t�s��$ A^{-1} $ �����݂��āA$ A^{-1} > 0 $ �ł���[1]�B
\vspace{10mm}

{\footnotesize 
[1] R.E.~Bellman, \textit{Introduction to Matrix Analysis}, McGraw-Hill, New York, NY, 1970 (Second Edition is a SIAM edition 1997).
}

\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{}
$f: \R^n \rightarrow \R$ �Ƃ���B

\begin{block}{\textbf{�藝~1}}
�s��$A$��$n \times n$�̎��Ώ̍s��A
$b \in \R^n$, $c \in \R$�Ƃ���B
\vspace{2mm}

$ A > 0 $�̂Ƃ��A�׃N�g��$2$���� 
\[
f(x) = (x, A x) + 2(b, x) + c
\]
�́A $ x = -A^{-1}b $ �̂Ƃ��A�ŏ��l
\[
m \dql c - (b, A^{-1}b)
\]
�����B
%�������A���x�N�g�� $ b = (b_{1}, ~b_{2}, ~\ldots, ~b_{n}) $ ����ю��萔 $ c $ %�͗^�����Ă���Ƃ���B
\end{block}

\end{frame}



%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{}
�ړI�֐�$f(x)$ �̓x�N�g�� $ x = (x_{1}, ~x_{2}, ~x_{3}, ~x_{4}) $ ��$2$����  
\begin{eqnarray*}
f(x) = (x, B x) + 2(b, x) + c^{2}
\end{eqnarray*}
�ɂȂ�B�����ɁA
\[
b \dql (c, 0, 0, 0)^{T}
\]�ŁA4 $\stimes$ 4 �s�� 
$$
B = \left(
\begin{array}{cccc}
 3 &  1  & 0 &  0     \\[1.26mm]
 1 &  3 & 1 & 0     \\[1.26mm]
    0 & 1  & 3 & 1      \\[1.26mm]
     0 & 0 & 1 & 2
\end{array}
\right) 
$$
�ł���A�s��$B$�͐���l�ƂȂ�B

\end{frame}




%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{}
���̋t�s���
$$ B^{-1} = \dfrac{1}{\,34\,}\left(
\begin{array}{rrrr}
    13 & -5  & 2 & -1 \\[1.26mm]
   -5 &  15  & -6  & 3   \\[1.26mm]
2 &  -6 &  16  & -8   \\[1.26mm]
-1 &  3 &  -8  & 21  
\end{array}
\right) 
$$
�ɂȂ�B���������Ē藝~1���A
��� $ ({\rm P}_{4}) $ �̍ŏ��� $ \hat{x} $ �́A
%\begin{equation*}
\begin{eqnarray*}
\hat{x} \eql -B^{-1}b = \dfrac{1}{\,34\,}\left(
\begin{array}{rrrr}
    -13 & 5  & -2 & 1 \\[1.26mm]
   5 &  -15  & 6  & -3   \\[1.26mm]
-2 &  6 &  -16  & 8   \\[1.26mm]
1 &  -3 &  8  & -21  
\end{array}
\right)
\left(
\begin{array}{c}
c \\[1.26mm]
0  \\[1.26mm]
0 \\[1.26mm]
0  
\end{array}
\right) \\[1.56mm]
\eql 
\dfrac{c}{\,34\,}
\left(
\begin{array}{r}
-13 \\[1.26mm]
5  \\[1.26mm]
-2 \\[1.26mm]
1  
\end{array}
\right).
\end{eqnarray*}
%\end{equation*}
\end{frame}


%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{}
���Ȃ킿�A
\[
\hat{x} \dql 
\dfrac{c}{\,34\,}
\left(
\begin{array}{r}
-13 \\[1.26mm]
5  \\[1.26mm]
-2 \\[1.26mm]
1  
\end{array}
\right) 
\dql
\dfrac{c}{\,F_{9}\,}
\left(
\begin{array}{r}
-F_{7} \\[1.26mm]
F_{5}  \\[1.26mm]
-F_{3} \\[1.26mm]
F_{1}  
\end{array}
\right) 
\]
�̂Ƃ��A�ŏ��l 
\begin{eqnarray*}
m_{4} \eql c^{2} - (b, B^{-1}b) \\[1.56mm]
\eql c^{2} - \dfrac{\,13\,}{\,34\,}\,c^{2} \\[1.56mm]
\eql \dfrac{\,21\,}{\,34\,}\,c^{2} \\[1.56mm]
\eql \dfrac{\,F_{8}\,}{F_{9}}\, c^{2}
\end{eqnarray*}
�����B

\end{frame}




%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{�s���͂ɂ����$({\rm D_{4}})$�̍œK���̓��o}
�ړI�֐�$g(\mu)$ �̓x�N�g�� 
$ \mu = (\mu_{0}, ~\mu_{1}, ~\mu_{2}, ~\mu_{3}) $ ��$2$����  
\begin{eqnarray*}
g(\mu) = 2(b, \mu) - (\mu, C \mu) 
\end{eqnarray*}
�ɂȂ�B�����ɁA
\[
b \dql (c, 0, 0, 0)^{T}
\]�ŁA4 $\stimes$ 4 �s�� 
\[
C \dql \left(
\begin{array}{cccc}
  2 & 1 & 0 & 0      \\[1.26mm]
 1 & 3 & 1 & 0     \\[1.26mm]
 0 & 1 & 3 & 1  \\[1.26mm]
  0 & 0 & 1 & 3
\end{array}
\right)
\]
�ł���A�s��$C$�͐���l�ƂȂ�B

\end{frame}



%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{}
���̋t�s��� 
\begin{eqnarray*}
C^{-1} \dql  \dfrac{1}{\,34\,} \left(
\begin{array}{rrrr}
 21 & -8 & 3 & -1      \\[1.26mm]
 -8 & 16 & -6 & 2     \\[1.26mm]
 3 & -6 & 15 & -5     \\[1.26mm]
 -1 & 2 & -5 & 13  
\end{array}
\right)
\end{eqnarray*}
�ł���B���������āA$ ({\rm D}_{4}) $ �͓_ 
%\begin{eqnarray*}
\[
\mu^{*} = C^{-1}b = 
\dfrac{c}{\,34\,}
\left(
\begin{array}{r}
21 \\[1.26mm]
-8 \\[1.26mm]
3 \\[1.26mm]
-1  
\end{array}
\right) 
=
\dfrac{c}{\,F_{9}\,}
\left(
\begin{array}{r}
F_{8} \\[1.26mm]
-F_{6}  \\[1.26mm]
F_{4} \\[1.26mm]
-F_{2}  
\end{array}
\right) 
\]
%\end{eqnarray*}
�ōő�l
\[
M_{4} = (b, C^{-1}b) = \dfrac{21}{\,34\,}c^{2}=\dfrac{\,F_{8}\,}{F_{9}}\,c^{2}
\]
�����B
\end{frame}







\end{comment}
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