# LaTeX2HTML 2002 (1.62)
# Associate images original text with physical files.


$key = q/{displaymath}lambdax_{i}=sum^k_{j=-k}a_{i,i+j}x_{i+j}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img41.png"
 ALT="\begin{displaymath}
\lambda x_{i} = \sum^k_{j = -k} a_{i,i+j} x_{i + j}
\end{displaymath}">|; 

$key = q/{displaymath}alpha=left(lambda+frac{2k}{m}right)frac{sqrt{Mm}}{k}qquadqquadbeta=left(lambda+frac{2k}{M}right)frac{sqrt{Mm}}{k}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="377" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img303.png"
 ALT="\begin{displaymath}
\alpha = \left(\lambda + \frac{2k}{m}\right) \frac{\sqrt{Mm...
...a = \left(\lambda + \frac{2k}{M} \right)
\frac{\sqrt{Mm}}{k}
\end{displaymath}">|; 

$key = q/{displaymath}T_{i+1}=left[array{{ccc}frac{lambda-a_{0}}{a_{1}}&-1&1&0array{right]+:epsilonleft[array{{ccc}1&-1&0&0array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="269" HEIGHT="66" BORDER="0"
 SRC="|."$dir".q|img274.png"
 ALT="\begin{displaymath}
T_{i + 1} = \left[ \begin{array}{ccc}
\frac{\lambda-a_{0}}...
...{array}{ccc}
1 &amp; -1 \\\\
&amp; \\\\
0 &amp; 0
\end{array} \right]
\end{displaymath}">|; 

$key = q/includegraphics[width=3.7in]{f17p79.eps};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="425" HEIGHT="586" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img351.png"
 ALT="\includegraphics[width=3.7in]{f17p79.eps}">|; 

$key = q/{displaymath}widetilde{W}=frac{W}{mbox{cosh}theta}eqno{(mbox{IV.18b})}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="319" HEIGHT="39" BORDER="0"
 SRC="|."$dir".q|img431.png"
 ALT="\begin{displaymath}
\widetilde{W} = \frac{W}{\mbox{cosh}  \theta}
\eqno{(\mbox{IV.18b})}
\end{displaymath}">|; 

$key = q/{displaymath}mu=mbox{cosh}:varphipmmbox{senh}:varphi{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="27" BORDER="0"
 SRC="|."$dir".q|img159.png"
 ALT="\begin{displaymath}
\mu = \mbox{cosh} \: \varphi \pm  \mbox{senh} \: \varphi
\end{displaymath}">|; 

$key = q/xi=0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img333.png"
 ALT="$\xi = 0$">|; 

$key = q/includegraphics[width=4.7in]{f10p62.eps};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="541" HEIGHT="123" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img295.png"
 ALT="\includegraphics[width=4.7in]{f10p62.eps}">|; 

$key = q/T^{N-i-1}QT^{i-1};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img286.png"
 ALT="$T^{N-i-1}  Q  T^{i-1}$">|; 

$key = q/{displaymath}Z_{N}=T^{N}Z_{0}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="27" BORDER="0"
 SRC="|."$dir".q|img151.png"
 ALT="\begin{displaymath}
Z_{N} = T^{N} Z_{0}
\end{displaymath}">|; 

$key = q/{displaymath}frac{c}{2}=mbox{cosh}:varphi{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="35" BORDER="0"
 SRC="|."$dir".q|img156.png"
 ALT="\begin{displaymath}
\frac{c}{2} = \mbox{cosh} \: \varphi
\end{displaymath}">|; 

$key = q/{displaymath}K=k(1+epsilon){displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img270.png"
 ALT="\begin{displaymath}
K = k (1 + \epsilon)
\end{displaymath}">|; 

$key = q/{displaymath}mbox{cosh}^2:varphi-1=mbox{senh}^2:varphi{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="27" BORDER="0"
 SRC="|."$dir".q|img158.png"
 ALT="\begin{displaymath}
\mbox{cosh}^2 \: \varphi - 1 = \mbox{senh}^2 \: \varphi
\end{displaymath}">|; 

$key = q/{displaymath}frac{mbox{senh}(N+1)varphi}{frac{mbox{senh}(N-L)varphimbox{senh}(L+1)varphi}{mbox{senh}varphi}}=-epsilon{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img249.png"
 ALT="\begin{displaymath}
\frac{\mbox{senh}(N+1)\varphi}{\frac{\mbox{senh}(N-L)\varphi \
\mbox{senh}(L+1)\varphi}{\mbox{senh} \varphi}} = - \epsilon
\end{displaymath}">|; 

$key = q/Tr(U)=0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img395.png"
 ALT="$Tr (U) = 0$">|; 

$key = q/{displaymath}widetilde{W}=-frac{1}{w_{2}}(w_{1}:J+w_{2}:K+w_{3}:L){displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="225" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img500.png"
 ALT="\begin{displaymath}
\widetilde{W} = - \frac{1}{w_{2}}(w_{1}\:J + w_{2}\:K + w_{3} \:L)
\end{displaymath}">|; 

$key = q/{displaymath}widehat{U}widehat{V}=mbox{cosh}thetaI+mbox{senh}thetawidehat{W}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="25" BORDER="0"
 SRC="|."$dir".q|img470.png"
 ALT="\begin{displaymath}
\widehat{U}  \widehat{V} = \mbox{cosh}  \theta  I + \mbox{senh}
 \theta  \widehat{W}
\end{displaymath}">|; 

$key = q/(x_{1},x_{2});MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img488.png"
 ALT="$(x_{1}, x_{2})$">|; 

$key = q/2K;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img50.png"
 ALT="$2K$">|; 

$key = q/T_1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img62.png"
 ALT="$T_1$">|; 

$key = q/{displaymath}tabular{{|c|c|c|c|c|}hlinebullet&mbox{I}&mbox{J}&mbox{K}&mbox{L}hlix{L}&mbox{K}&mbox{-J}&mbox{-I}hlinetabular{eqno{(4.11)}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="360" HEIGHT="106" BORDER="0"
 SRC="|."$dir".q|img405.png"
 ALT="\begin{displaymath}
\begin{tabular}{\vert c\vert c\vert c\vert c\vert c\vert} \...
... \mbox{-J} &amp; \mbox{-I} \ \hline
\end{tabular} \eqno{(4.11)}
\end{displaymath}">|; 

$key = q/{displaymath}[widehat{X},Nwidehat{W}]=0{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img476.png"
 ALT="\begin{displaymath}[\widehat{X}, N  \widehat{W}]= 0
\end{displaymath}">|; 

$key = q/{displaymath}mu^{2}-cmu+1=0{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="27" BORDER="0"
 SRC="|."$dir".q|img154.png"
 ALT="\begin{displaymath}
\mu^{2} - c \mu + 1 = 0
\end{displaymath}">|; 

$key = q/{displaymath}a=-frac{k}{m}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="39" BORDER="0"
 SRC="|."$dir".q|img210.png"
 ALT="\begin{displaymath}
a = -\frac{k}{m}
\end{displaymath}">|; 

$key = q/{displaymath}left[array{{ccc}a&bc&darray{right]=mbox{cosh}varphiI+mbox{senh}varphiwidehat{U}eqno{(mbox{IV.10'})}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="376" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img403.png"
 ALT="\begin{displaymath}
\left[ \begin{array}{ccc}
a &amp; b \\\\
c &amp; d
\end{array} ...
...i I + \mbox{senh} \varphi \widehat{U}
\eqno{(\mbox{IV.10'})}
\end{displaymath}">|; 

$key = q/{displaymath}v=-mu_pm{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img174.png"
 ALT="\begin{displaymath}
v = -\mu_\pm
\end{displaymath}">|; 

$key = q/{eqnarraystar}Z_{N}&=&T_{N}Z_{N-1}Z_{N-1}&=&T_{N-1}Z_{N-2}&vdots&Z_{1}&=&T_{1}Z_{0}{eqnarraystar};MSF=1.6;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="108" BORDER="0"
 SRC="|."$dir".q|img111.png"
 ALT="\begin{eqnarray*}
Z_{N} &amp; = &amp; T_{N} Z_{N - 1} \\\\
Z_{N - 1} &amp; = &amp; T_{N - 1}    Z_{N - 2} \\\\
&amp; \vdots &amp; \\\\
Z_{1} &amp; = &amp; T_{1} Z_{0}
\end{eqnarray*}">|; 

$key = q/{displaymath}left[array{{ccc}0y_{N}array{right]=left[array{{ccc}tau^{(N)}_{11}&t-2)}_{22}array{right]left[array{{ccc}y_{1}0array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="270" HEIGHT="73" BORDER="0"
 SRC="|."$dir".q|img123.png"
 ALT="\begin{displaymath}
\left [ \begin{array}{ccc}
0 \\\\
\\\\
y_{N}
\end{arra...
...[ \begin{array}{ccc}
y_{1} \\\\
\\\\
0
\end{array} \right]
\end{displaymath}">|; 

$key = q/displaystylefrac{k}{m};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img150.png"
 ALT="$\displaystyle \frac{k}{m}$">|; 

$key = q/{displaymath}-v=mu_pm{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img173.png"
 ALT="\begin{displaymath}
-v = \mu_\pm
\end{displaymath}">|; 

$key = q/x_{N+k};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img14.png"
 ALT="$x_{N + k}$">|; 

$key = q/2K-1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img51.png"
 ALT="$2K - 1$">|; 

$key = q/{displaymath}Mddot{X}=KX{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="24" BORDER="0"
 SRC="|."$dir".q|img26.png"
 ALT="\begin{displaymath}
M \ddot{X} = K X
\end{displaymath}">|; 

$key = q/{displaymath}M=left[array{{cccc}m_1&0&dots&00&m_{2}&dots&0cdot&cdot&cdot&cdotcdo&cdot&cdotcdot&cdot&cdot&cdot0&cdots&&m_{N}array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="121" BORDER="0"
 SRC="|."$dir".q|img23.png"
 ALT="\begin{displaymath}
M = \left[ \begin{array}{cccc}
m_1 &amp; 0 &amp; \dots &amp; 0 \\\\
0 ...
... \cdot &amp; \cdot \\\\
0 &amp; \cdots &amp; &amp; m_{N}
\end{array} \right]
\end{displaymath}">|; 

$key = q/x_{N+k-1}cdots;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img15.png"
 ALT="$x_{N + k - 1} \cdots$">|; 

$key = q/{displaymath}(widehat{a},widehat{b})=a_{1}b_{1}+a_{2}b_{2}-a_{3}b_{3}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img416.png"
 ALT="\begin{displaymath}
(\widehat{a}, \widehat{b}) = a_{1} b_{1} + a_{2} b_{2} - a_{3}
b_{3}
\end{displaymath}">|; 

$key = q/{displaymath}widetilde{U}=frac{U}{mbox{cosh}varphi}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="42" BORDER="0"
 SRC="|."$dir".q|img439.png"
 ALT="\begin{displaymath}
\widetilde{U} = \frac{U}{\mbox{cosh}  \varphi}
\end{displaymath}">|; 

$key = q/i=1,2,...,N;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img42.png"
 ALT="$i = 1, 2, ..., N$">|; 

$key = q/{displaymath}Z_{0}=left[array{{cccc}x_{k}x_{k-1}vdotsx_{1}x_{0}vdotsx_{-k+1}array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="161" BORDER="0"
 SRC="|."$dir".q|img67.png"
 ALT="\begin{displaymath}
Z_{0} = \left[ \begin{array}{cccc}
x_{k} \\\\
x_{k - 1} \\\\...
...} \\\\
x_{0} \\\\
\vdots \\\\
x_{-k + 1}
\end{array} \right]
\end{displaymath}">|; 

$key = q/{displaymath}m_{i}=m_{0}+i_{epsilon}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="27" BORDER="0"
 SRC="|."$dir".q|img352.png"
 ALT="\begin{displaymath}
m_{i} = m_{0} + i_{\epsilon}
\end{displaymath}">|; 

$key = q/rho^2;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img302.png"
 ALT="$\rho^2$">|; 

$key = q/cosh;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img427.png"
 ALT="$cosh$">|; 

$key = q/M^{-1};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img27.png"
 ALT="$M^{-1}$">|; 

$key = q/i+k;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img56.png"
 ALT="$i + k$">|; 

$key = q/1,2,...,N;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img200.png"
 ALT="$1, 2, ...,N$">|; 

$key = q/Mrightarrowinfty;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img254.png"
 ALT="$M \rightarrow \infty$">|; 

$key = q/{displaymath}widetilde{W}=mbox{tanh}thetawidehat{W}eqno{(mbox{IV.18a})}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="329" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img429.png"
 ALT="\begin{displaymath}
\widetilde{W} = \mbox{tanh}  \theta  \widehat{W}
\eqno{(\mbox{IV.18a})}
\end{displaymath}">|; 

$key = q/{displaymath}tau_{11}=mbox{cosh}theta+mbox{senh}thetaw_{2}=0{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="202" HEIGHT="27" BORDER="0"
 SRC="|."$dir".q|img498.png"
 ALT="\begin{displaymath}
\tau_{11} = \mbox{cosh}  \theta + \mbox{senh}  \theta  w_{2} =
0
\end{displaymath}">|; 

$key = q/includegraphics[width=6in]{portada.eps};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="581" HEIGHT="733" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2.png"
 ALT="\includegraphics[width=6in]{portada.eps}">|; 

$key = q/{displaymath}left[array{{cc}frac{M}{k}lambda+2&-11&0array{right]left[array{{cc}fght]left[array{{cc}frac{m}{k}lambda+2&-11&0array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="375" HEIGHT="64" BORDER="0"
 SRC="|."$dir".q|img261.png"
 ALT="\begin{displaymath}
\left[ \begin{array}{cc}
\frac{M}{k}\lambda + 2 &amp; -1 \ 
...
...c{m}{k}\lambda + 2 &amp; -1 \\\\
\\\\
1 &amp; 0
\end{array} \right]
\end{displaymath}">|; 

$key = q/{displaymath}quad=frac{1}{1+epsilon}left[array{{ccc}frac{lambda-a_{0}}{a_{1}}&-1ilon^{2}}{1+epsilon}left[array{{ccc}0&0&0&0array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="652" HEIGHT="69" BORDER="0"
 SRC="|."$dir".q|img276.png"
 ALT="\begin{displaymath}
\quad = \frac{1}{1+\epsilon} \left[ \begin{array}{ccc}
\fr...
...n{array}{ccc}
0 &amp; 0 \\\\
&amp; \\\\
0 &amp; 0
\end{array} \right]
\end{displaymath}">|; 

$key = q/includegraphics[width=3.7in]{f15p76.eps};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="425" HEIGHT="744" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img349.png"
 ALT="\includegraphics[width=3.7in]{f15p76.eps}">|; 

$key = q/{displaymath}left[array{{ccc}1&00&0array{right]qquadleft[array{{ccc}0&01&0array{10&0array{right]qquadleft[array{{ccc}0&00&1array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="354" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img406.png"
 ALT="\begin{displaymath}
\left[\begin{array}{ccc}
1 &amp; 0 \\\\
0 &amp; 0
\end{array} \ri...
...eft[\begin{array}{ccc}
0 &amp; 0 \\\\
0 &amp; 1
\end{array} \right]
\end{displaymath}">|; 

$key = q/{displaymath}mu=frac{a+d}{2}pmsqrt{left(frac{a+d}{2}right)^{2}-1}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img367.png"
 ALT="\begin{displaymath}
\mu = \frac{a + d}{2} \pm \sqrt{\left(\frac{a + d}{2}\right)^{2} -
1}
\end{displaymath}">|; 

$key = q/{displaymath}varphi=frac{ellpii}{N+1}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img191.png"
 ALT="\begin{displaymath}
\varphi = \frac{\ell \pi  i}{N + 1}
\end{displaymath}">|; 

$key = q/{displaymath}w_1x_1+w_2x_2-w_3x_3=0{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="27" BORDER="0"
 SRC="|."$dir".q|img477.png"
 ALT="\begin{displaymath}
w_1 x_1 + w_2 x_2 - w_3 x_3 = 0
\end{displaymath}">|; 

$key = q/{displaymath}left[array{{cccc}(lambda+frac{2k}{m})frac{sqrt{Mm}}{k}&-1&1&0array{right]qquadmbox{part'{i}culaspares}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="315" HEIGHT="67" BORDER="0"
 SRC="|."$dir".q|img298.png"
 ALT="\begin{displaymath}
\left[ \begin{array}{cccc}
(\lambda + \frac{2k}{m}) \frac{...
... 0
\end{array} \right] \qquad \mbox{part\'{\i}culas  pares}
\end{displaymath}">|; 

$key = q/includegraphics[width=3.7in]{f06p52.eps};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="425" HEIGHT="741" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img267.png"
 ALT="\includegraphics[width=3.7in]{f06p52.eps}">|; 

$key = q/{displaymath}alphabeta=2(xi+rho)cdot2(xi+frac{1}{rho})=4left[xi^{2}+xi(rho+frac{1}{rho})+1right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="335" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img316.png"
 ALT="\begin{displaymath}
\alpha \beta = 2(\xi + \rho) \cdot 2(\xi + \frac{1}{\rho}) =
4\left[\xi^{2} + \xi(\rho + \frac{1}{\rho}) + 1\right]
\end{displaymath}">|; 

$key = q/{displaymath}mbox{senh}(N+1)varphi-frac{2(a-a_{0})}{a_{1}}mbox{senh}Nvarphi+frac{(a-a_{0})^{2}}{a^{2}_{1}}mbox{senh}(N-1)varphi=0{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="450" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img227.png"
 ALT="\begin{displaymath}
\mbox{senh} (N + 1)\varphi - \frac{2(a - a_{0})}{a_{1}}
\m...
...{(a - a_{0})^{2}}{a^{2}_{1}}
\mbox{senh} (N - 1) \varphi = 0
\end{displaymath}">|; 

$key = q/{displaymath}qquad=frac{e^{pm2varphi}-e^{varphi}e^{pmvarphi}-e^{-varphi}e^{pmvarphi}+ad}{b^{2}}=frac{ad-1}{b^{2}}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="292" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img379.png"
 ALT="\begin{displaymath}
\qquad = \frac{e^{\pm 2\varphi} - e^{\varphi} e^{\pm\varphi...
...{-\varphi} e^{\pm\varphi} + ad}{b^{2}} = \frac{ad - 1}{b^{2}}
\end{displaymath}">|; 

$key = q/{displaymath}nu_{pm}+upsilon_{pm}=frac{2e^{pmvarphi}-e^{varphi}-e^{-varphi}}{b}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="197" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img376.png"
 ALT="\begin{displaymath}
\nu_{\pm} + \upsilon_{\pm} = \frac{2e^{\pm \varphi} - e^{\varphi}
- e^{-\varphi}}{b}
\end{displaymath}">|; 

$key = q/{displaymath}P_pm=frac{arrowvertu_pm><v_pmarrowvert}{<v_pmarrowvertu_pm>}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="44" BORDER="0"
 SRC="|."$dir".q|img176.png"
 ALT="\begin{displaymath}
P_\pm = \frac{\arrowvert u_\pm &gt;&lt; v_\pm \arrowvert}{&lt; v_\pm
\arrowvert u_\pm &gt;}
\end{displaymath}">|; 

$key = q/{displaymath}frac{mbox{senh}(N+1)varphi}{mbox{senh}:varphi}=0{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img187.png"
 ALT="\begin{displaymath}
\frac{\mbox{senh}(N + 1) \varphi}{\mbox{senh} \: \varphi }= 0
\end{displaymath}">|; 

$key = q/includegraphics[width=5in]{f01p04.eps};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="575" HEIGHT="168" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4.png"
 ALT="\includegraphics[width=5in]{f01p04.eps}">|; 

$key = q/y_{i-1};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img108.png"
 ALT="$y_{i - 1}$">|; 

$key = q/epsilonrightarrowinfty;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img289.png"
 ALT="$\epsilon
\rightarrow \infty$">|; 

$key = q/{displaymath}mbox{cosh}theta=mbox{cosh}varphimbox{cosh}psi+mbox{senh}varphimbox{senh}psi(widehat{U},widehat{V}){displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="335" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img426.png"
 ALT="\begin{displaymath}
\mbox{cosh}  \theta = \mbox{cosh}  \varphi  \mbox{cosh} ...
... \varphi  \mbox{senh}  \psi  (\widehat{U}, \
\widehat{V})
\end{displaymath}">|; 

$key = q/x_{ell};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img10.png"
 ALT="$x_{\ell}$">|; 

$key = q/varphi<0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img163.png"
 ALT="$\varphi &lt; 0$">|; 

$key = q/{displaymath}arrowvertT_{i}arrowvert=frac{a_{i,i-1}}{a_{i,i+1}}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="40" BORDER="0"
 SRC="|."$dir".q|img360.png"
 ALT="\begin{displaymath}\arrowvert T_{i}
\arrowvert = \frac{a_{i, i-1}}{a_{i, i+1}}
\end{displaymath}">|; 

$key = q/{displaymath}frac{+epsilon}{mbox{senh}^{2}varphi}left[array{{cc}mbox{senh}(N-L)vi&-mbox{senh}(N-L-1)varphimbox{senh}Lvarphiarray{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="537" HEIGHT="64" BORDER="0"
 SRC="|."$dir".q|img247.png"
 ALT="\begin{displaymath}
\frac{+ \epsilon}{\mbox{senh}^{2}  \varphi} \left[
\begin...
...L-1)  \varphi  \mbox{senh} L  \varphi
\end{array} \right]
\end{displaymath}">|; 

$key = q/M^{-1slash2}KM^{-1slash2};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img34.png"
 ALT="$M^{-1/2}K   M^{-1/2}$">|; 

$key = q/tau^{(i)^{-1}};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img133.png"
 ALT="$\tau^{(i)^{-1}}$">|; 

$key = q/includegraphics[width=5in]{f04p47.eps};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="575" HEIGHT="337" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img253.png"
 ALT="\includegraphics[width=5in]{f04p47.eps}">|; 

$key = q/{displaymath}TZ=muZ{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="27" BORDER="0"
 SRC="|."$dir".q|img76.png"
 ALT="\begin{displaymath}
T Z = \mu Z
\end{displaymath}">|; 

$key = q/2k+1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img21.png"
 ALT="$2k+1$">|; 

$key = q/{displaymath}tauleft[array{{ccc}frac{lambda-a_{0}}{a_{1}}&-11&0array{right]^{N-Lay{{ccc}frac{lambda-a_{0}}{a_{1}}&-11&0array{right]^{L}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="395" HEIGHT="70" BORDER="0"
 SRC="|."$dir".q|img239.png"
 ALT="\begin{displaymath}
\tau \left[ \begin{array}{ccc}
\frac{\lambda - a_{0}}{a_{1...
... a_{0}}{a_{1}} &amp; -1 \\\\
\\\\
1 &amp; 0
\end{array} \right]^{L}
\end{displaymath}">|; 

$key = q/includegraphics[width=3.2in]{f26p114.eps};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="368" HEIGHT="354" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img491.png"
 ALT="\includegraphics[width=3.2in]{f26p114.eps}">|; 

$key = q/{displaymath}T=sum^{2k}_{i=1}lambda_{i}frac{arrowverti><iarrowvert}{<iarrowverti>}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img80.png"
 ALT="\begin{displaymath}
T = \sum^{2k}_{i = 1}  \lambda_{i}  \frac{\arrowvert i
&gt;&lt; i \arrowvert}{&lt; i \arrowvert i &gt;}
\end{displaymath}">|; 

$key = q/mu>0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img341.png"
 ALT="$\mu &gt; 0$">|; 

$key = q/M^{-1slash2};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img39.png"
 ALT="$M^{-1/2}$">|; 

$key = q/{displaymath}T=frac{e^{varphi}}{2mbox{senh}varphi}left[array{{ccc}e^{varphi}&-11left[array{{ccc}-e^{-varphi}&1-1&e^{varphi}array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="369" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img183.png"
 ALT="\begin{displaymath}
T = \frac{e^{\varphi}}{2 \mbox{senh} \varphi} \left[
\begi...
...
-e^{-\varphi} &amp; 1 \\\\
-1 &amp; e^{\varphi}
\end{array} \right]
\end{displaymath}">|; 

$key = q/alphabeta;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img315.png"
 ALT="$\alpha \beta$">|; 

$key = q/{displaymath}arrowvertT_{i}arrowvert=1{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img361.png"
 ALT="\begin{displaymath}
\arrowvert T_{i} \arrowvert = 1
\end{displaymath}">|; 

$key = q/{displaymath}(widehat{U},widehat{U})=1{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img451.png"
 ALT="\begin{displaymath}
(\widehat{U},  \widehat{U}) = 1
\end{displaymath}">|; 

$key = q/{displaymath}(widehat{U}timeswidehat{V},widehat{U}timeswidehat{V})=mbox{cosh}^{2}theta-1=mbox{senh}^{2}theta{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="280" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img464.png"
 ALT="\begin{displaymath}
(\widehat{U} \times \widehat{V}, \widehat{U} \times
\wideh...
...) = \mbox{cosh}^{2}  \theta - 1
= \mbox{senh}^{2}  \theta
\end{displaymath}">|; 

$key = q/{displaymath}AY=lambdaY{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="24" BORDER="0"
 SRC="|."$dir".q|img99.png"
 ALT="\begin{displaymath}
A Y = \lambda Y
\end{displaymath}">|; 

$key = q/{displaymath}frac{mbox{senh}(N+1)varphi}{mbox{senh}varphi}+frac{epsilon}{mbox{se^{2}varphi}mbox{senh}(N-L)varphimbox{senh}(L+1)varphi=0{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="406" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img248.png"
 ALT="\begin{displaymath}
\frac{\mbox{senh} (N+1)\varphi}{\mbox{senh}  \varphi} +
\...
...hi} \
\mbox{senh}(N-L)\varphi  \mbox{senh}(L+1) \varphi = 0
\end{displaymath}">|; 

$key = q/{eqnarraystar}<v_pmarrowvertu_pm>&=&left[array{{ccc}v_pm&1array{right]left[array{{cc}1u_pmarray{right]&=&v_pm+u_pm{eqnarraystar};MSF=1.6;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="243" HEIGHT="64" BORDER="0"
 SRC="|."$dir".q|img178.png"
 ALT="\begin{eqnarray*}
&lt; v_\pm \arrowvert u_\pm &gt; &amp;=&amp; \left [
\begin{array}{ccc}
v...
... 1 \\\\
u_\pm
\end{array} \right]
\\\\
&amp; = &amp; v_\pm +  u_\pm
\end{eqnarray*}">|; 

$key = q/includegraphics[width=3.7in]{f21p85.eps};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="425" HEIGHT="722" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img357.png"
 ALT="\includegraphics[width=3.7in]{f21p85.eps}">|; 

$key = q/{displaymath}widehat{U}widehat{V}=(widehat{U},widehat{V})I+widehat{U}timeswidehat{V}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img469.png"
 ALT="\begin{displaymath}
\widehat{U} \widehat{V} = (\widehat{U}, \widehat{V}) I +
\widehat{U} \times \widehat{V}
\end{displaymath}">|; 

$key = q/{displaymath}W_{2}=-1{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="27" BORDER="0"
 SRC="|."$dir".q|img502.png"
 ALT="\begin{displaymath}
W_{2} = -1
\end{displaymath}">|; 

$key = q/a_{11};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img94.png"
 ALT="$a_{11}$">|; 

$key = q/T^{-1};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img215.png"
 ALT="$T^{-1}$">|; 

$key = q/widetilde{W};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img492.png"
 ALT="$\widetilde{W}$">|; 

$key = q/T_{i};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img55.png"
 ALT="$T_{i}$">|; 

$key = q/x_{0};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img17.png"
 ALT="$x_{0}$">|; 

$key = q/{displaymath}ddot{X}=M^{-1}KX{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="24" BORDER="0"
 SRC="|."$dir".q|img28.png"
 ALT="\begin{displaymath}
\ddot{X} = M^{-1}K X
\end{displaymath}">|; 

$key = q/{displaymath}T_{i+1}=left[array{{cc}frac{lambda-a_{i+1,i+1}}{a_{i+1,i+2}}&-frac{a_{i+1,i}}{a_{i+1,i+2}}1&0array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="232" HEIGHT="69" BORDER="0"
 SRC="|."$dir".q|img114.png"
 ALT="\begin{displaymath}
T_{i + 1} = \left[ \begin{array}{cc}
\frac{\lambda - a_{i ...
..., i}}{a_{i + 1, i + 2}} \\\\
\\\\
1 &amp; 0
\end{array} \right]
\end{displaymath}">|; 

$key = q/frac{lambda-a_{0}}{2a_{1}}<-1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img166.png"
 ALT="$\frac{\lambda - a_{0}}{2a_{1}} &lt; -1$">|; 

$key = q/{displaymath}left[array{{ccc}frac{lambda-a_{0}}{a_{1}}+epsilon&-(1+epsilon)&1&0array{right]qquadmbox{para}i+1{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="270" HEIGHT="66" BORDER="0"
 SRC="|."$dir".q|img272.png"
 ALT="\begin{displaymath}
\left[ \begin{array}{ccc}
\frac{\lambda-a_{0}}{a_{1}} + \...
...
&amp; \\\\
1 &amp; 0
\end{array} \right] \qquad \mbox{para } i + 1
\end{displaymath}">|; 

$key = q/{displaymath}[4(xi^{2}+2xiP+1)][4xi^{2}+8xiP+4-4]=0{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="281" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img326.png"
 ALT="\begin{displaymath}[4(\xi^{2} + 2 \xi P + 1)][4 \xi^{2} + 8 \xi P + 4 - 4] = 0
\end{displaymath}">|; 

$key = q/epsilonrightarrow-1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img291.png"
 ALT="$\epsilon \rightarrow -1$">|; 

$key = q/alpha;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img312.png"
 ALT="$\alpha$">|; 

$key = q/tau_{ij}=tau_{ij}(lambda)i,j=1,2,dots,k;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="223" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img71.png"
 ALT="$\tau_{i j} = \tau_{i j} (\lambda)     \
 i, j = 1,2, \dots, k$">|; 

$key = q/{displaymath}(N+1)varphi=ellpiiell=0,pm1,pm2,...,pm(N+1){displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="328" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img190.png"
 ALT="\begin{displaymath}
(N + 1) \varphi = \ell \pi  i       \ell = 0, \pm 1, \pm 2,
..., \pm (N + 1)
\end{displaymath}">|; 

$key = q/{displaymath}k_{ii}=-k_{ii-1}-k_{ii+1}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img87.png"
 ALT="\begin{displaymath}
k_{i i} = -k_{i i -1} -k_{i i + 1}
\end{displaymath}">|; 

$key = q/{displaymath}AX=lambdaX{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="24" BORDER="0"
 SRC="|."$dir".q|img40.png"
 ALT="\begin{displaymath}
A X = \lambda X
\end{displaymath}">|; 

$key = q/{displaymath}tau=T^{N-i-1}left[frac{1}{1+epsilon}(T^{2}+epsilonQ)right]T^{i-1}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="244" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img281.png"
 ALT="\begin{displaymath}
\tau = T^{N-i-1} \left[\frac{1}{1+\epsilon}(T^{2} + \epsilon
Q)\right]
T^{i-1}
\end{displaymath}">|; 

$key = q/widehat{V}widehat{V};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img454.png"
 ALT="$\widehat{V}   \widehat{V}$">|; 

$key = q/{displaymath}mbox{senh}varphineqinfty{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="27" BORDER="0"
 SRC="|."$dir".q|img188.png"
 ALT="\begin{displaymath}
\mbox{senh}\varphi \neq \infty
\end{displaymath}">|; 

$key = q/includegraphics[width=5in]{f03p43.eps};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="575" HEIGHT="104" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img238.png"
 ALT="\includegraphics[width=5in]{f03p43.eps}">|; 

$key = q/varphi;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img79.png"
 ALT="$\varphi$">|; 

$key = q/{displaymath}T^{N}=frac{1}{mbox{senh}varphi}left[array{{ccc}mbox{senh}(N+1)varphphimbox{senh}Nvarphi&-mbox{senh}(N-1)varphiarray{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="350" HEIGHT="64" BORDER="0"
 SRC="|."$dir".q|img186.png"
 ALT="\begin{displaymath}
T^{N} = \frac{1}{\mbox{senh } \varphi} \left[
\begin{array...
... \varphi &amp; - \mbox{senh} ( N - 1)\varphi
\end{array} \right]
\end{displaymath}">|; 

$key = q/{displaymath}(mu+1)[(mu+1)-alphabeta]+alphabeta=0{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="215" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img310.png"
 ALT="\begin{displaymath}
(\mu +1)[(\mu + 1) - \alpha \beta] + \alpha \beta = 0
\end{displaymath}">|; 

$key = q/{displaymath}xi+2P=0eqno{(mbox{III.46b})}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="314" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img331.png"
 ALT="\begin{displaymath}
\xi + 2P = 0 \eqno{(\mbox{III.46b})}
\end{displaymath}">|; 

$key = q/{displaymath}mbox{senh}varphi=sqrt{left(frac{a+d}{2}right)^{2}-ad+bc}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="224" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img398.png"
 ALT="\begin{displaymath}
\mbox{senh}  \varphi = \sqrt{\left(\frac{a + d}{2}\right)^{2} -
ad + bc}
\end{displaymath}">|; 

$key = q/{displaymath}left[array{{c}m_{1}ddot{x}_{1}m_{2}ddot{x}_{2}vdotsm_{N}ddot{x}_{N}ay{right]left[array{{c}x_{1}x_{2}vdotsx_{N}array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="335" HEIGHT="94" BORDER="0"
 SRC="|."$dir".q|img89.png"
 ALT="\begin{displaymath}
\left [ \begin{array}{c}
m_{1} \ddot{x}_{1} \\\\
m_{2} \d...
...
x_{1} \\\\
x_{2} \\\\
\vdots \\\\
x_{N}
\end{array} \right]
\end{displaymath}">|; 

$key = q/{displaymath}left[array{{cc}alphabeta-1-mu&-alpha&beta&-1-muarray{right]=0{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="203" HEIGHT="64" BORDER="0"
 SRC="|."$dir".q|img309.png"
 ALT="\begin{displaymath}
\left[ \begin{array}{cc}
\alpha \beta - 1 -\mu &amp; -\alpha \\\\
&amp; \\\\
\beta &amp; -1 -\mu
\end{array} \right] = 0
\end{displaymath}">|; 

$key = q/{eqnarraystar}e^{varphiwidehat{U}}e^{psiwidehat{V}}&=&e^{thetawidehat{W}}&=&mboxthetaI+mbox{cosh}thetambox{tanh}thetawidehat{W}{eqnarraystar};MSF=1.6;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="306" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img425.png"
 ALT="\begin{eqnarray*}
e^{\varphi  \widehat{U}}  e^{\psi  \widehat{V}} &amp;=&amp; e^{\th...
... I + \mbox{cosh}  \theta  \mbox{tanh}
 \theta  \widehat{W}
\end{eqnarray*}">|; 

$key = q/{displaymath}(widehat{U},widehat{W})=u_1w_1+u_2w_2-u_3w_3{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="212" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img473.png"
 ALT="\begin{displaymath}
(\widehat{U}, \widehat{W}) = u_1 w_1 + u_2 w_2 - u_3 w_3
\end{displaymath}">|; 

$key = q/lambda_i;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img82.png"
 ALT="$\lambda_i$">|; 

$key = q/M^{-1}K;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img29.png"
 ALT="$M^{-1} K$">|; 

$key = q/{displaymath}frac{arrowverti><iarrowvert}{<iarrowverti>}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="44" BORDER="0"
 SRC="|."$dir".q|img81.png"
 ALT="\begin{displaymath}
\frac{\arrowvert i &gt;&lt; i \arrowvert}{&lt; i \arrowvert i&gt;}
\end{displaymath}">|; 

$key = q/T^{N}_{11};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img284.png"
 ALT="$T^{N}_{11}$">|; 

$key = q/{displaymath}widetilde{W}=frac{mbox{cosh}varphimbox{cosh}psi}{mbox{cosh}varphimbi}(widetilde{U}+widetilde{V}+widetilde{U}+widetilde{V}){displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="379" HEIGHT="42" BORDER="0"
 SRC="|."$dir".q|img433.png"
 ALT="\begin{displaymath}
\widetilde{W} = \frac{\mbox{cosh}  \varphi  \mbox{cosh} \...
...idetilde{U} + \widetilde{V} +
\widetilde{U} + \widetilde{V})
\end{displaymath}">|; 

$key = q/M=m;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img256.png"
 ALT="$M = m$">|; 

$key = q/{displaymath}xi=pmfrac{mu-1}{2sqrt{mu}}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="44" BORDER="0"
 SRC="|."$dir".q|img335.png"
 ALT="\begin{displaymath}
\xi = \pm \frac{\mu - 1}{2\sqrt{\mu}}
\end{displaymath}">|; 

$key = q/{displaymath}y_{i}=y_{i}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="27" BORDER="0"
 SRC="|."$dir".q|img104.png"
 ALT="\begin{displaymath}
y_{i} = y_{i}
\end{displaymath}">|; 

$key = q/includegraphics[width=3.7in]{f23p88.eps};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="425" HEIGHT="741" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img359.png"
 ALT="\includegraphics[width=3.7in]{f23p88.eps}">|; 

$key = q/{displaymath}left[array{{ccc}frac{(lambda-a_{i+2,i+2})(lambda-a_{i+1,i+1})(lambd(lambda-a_{i+1,i+1})}{a_{i,i+1}a_{i+1,i+2}}array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="495" HEIGHT="121" BORDER="0"
 SRC="|."$dir".q|img121.png"
 ALT="\begin{displaymath}
\left[ \begin{array}{ccc}
\frac{(\lambda - a_{i + 2, i + ...
...)}{a_{i, i + 1}  \
 a_{i + 1, i + 2}}
\end{array} \right]
\end{displaymath}">|; 

$key = q/{displaymath}C_j=-frac{a_{ii+j}}{a_{i,i+k}}jneq0eqno{(mbox{I.11c})}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="363" HEIGHT="40" BORDER="0"
 SRC="|."$dir".q|img47.png"
 ALT="\begin{displaymath}
C_j = -\frac{a_{i i + j}} {a_{i,i + k}}         j \neq 0
\eqno{(\mbox{I.11c})}
\end{displaymath}">|; 

$key = q/{displaymath}mu^{2}-(a+d)mu+1=0{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img366.png"
 ALT="\begin{displaymath}
\mu^{2} - (a + d)\mu + 1 = 0
\end{displaymath}">|; 

$key = q/{eqnarraystar}tau&=&left[array{{ccc}1&frac{-(a-a_{0})}{a_{1}}0&1array{right]leftsenh}Nvarphi&-mbox{senh}(N-1)varphiarray{right]{eqnarraystar};MSF=1.6;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="634" HEIGHT="142" BORDER="0"
 SRC="|."$dir".q|img232.png"
 ALT="\begin{eqnarray*}
\tau &amp; = &amp; \left[ \begin{array}{ccc}
1 &amp; \frac{-(a - a_{0})}...
... N \varphi &amp; -\mbox{senh} (N - 1) \varphi
\end{array} \right]
\end{eqnarray*}">|; 

$key = q/z;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img266.png"
 ALT="$z$">|; 

$key = q/x_{2};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img484.png"
 ALT="$x_{2}$">|; 

$key = q/T_{i+2};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img120.png"
 ALT="$T_{i+2}$">|; 

$key = q/ell;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img11.png"
 ALT="$\ell$">|; 

$key = q/{displaymath}left[array{{c}y_{j+1}y_{j}array{right]=T^{j}left[array{{cccc}y_{1}0array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img201.png"
 ALT="\begin{displaymath}
\left [ \begin{array}{c}
y_{j + 1} \\\\
y_{j}
\end{array...
...\left[ \begin{array}{cccc}
y_{1} \\\\
0
\end{array} \right]
\end{displaymath}">|; 

$key = q/{displaymath}array{{ccc}lambdax_{2i}&=&frac{k}{sqrt{Mm}}x_{2i-1}-frac{2k}{m}x_{22i}-frac{2k}{M}x_{2i+1}+frac{k}{sqrt{Mm}}x_{2i+2}array{{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="317" HEIGHT="71" BORDER="0"
 SRC="|."$dir".q|img297.png"
 ALT="\begin{displaymath}
\begin{array}{ccc}
\lambda x_{2i} &amp; = &amp; \frac{k}{\sqrt{Mm}...
...k}{M}
x_{2i+1} + \frac{k}{\sqrt{Mm}} x_{2i + 2}
\end{array}
\end{displaymath}">|; 

$key = q/includegraphics[width=4in,angle=-90]{f12p71.eps};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="462" HEIGHT="2" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img342.png"
 ALT="\includegraphics[width=4in,angle=-90]{f12p71.eps}">|; 

$key = q/Mrightarrow0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img257.png"
 ALT="$M \rightarrow 0$">|; 

$key = q/T_D;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img308.png"
 ALT="$T_D$">|; 

$key = q/{displaymath}widehat{V}timeswidehat{V}=widehat{0}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="25" BORDER="0"
 SRC="|."$dir".q|img458.png"
 ALT="\begin{displaymath}
\widehat{V} \times \widehat{V} = \widehat{0}
\end{displaymath}">|; 

$key = q/a;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img250.png"
 ALT="$a$">|; 

$key = q/{displaymath}P_{pm}=frac{left[array{{ccc}upsilon_{pm}&1nu_{pm}upsilon_{pm}&nu_{pm}array{right]}{nu_{pm}+upsilon_{pm}}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="66" BORDER="0"
 SRC="|."$dir".q|img373.png"
 ALT="\begin{displaymath}
P_{\pm} = \frac{\left[ \begin{array}{ccc}
\upsilon_{\pm} &amp;...
...&amp; \nu_{\pm}
\end{array} \right]}{\nu_{\pm} + \upsilon_{\pm}}
\end{displaymath}">|; 

$key = q/{displaymath}L_{n}T=frac{varphi}{2mbox{senh}varphi}left[array{{ccc}a-d&2b2c&d-aarray{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="236" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img389.png"
 ALT="\begin{displaymath}
L_{n}  T = \frac{\varphi}{2\mbox{senh}  \varphi} \left[
...
...{array}{ccc}
a - d &amp; 2b \\\\
2c &amp; d - a
\end{array} \right]
\end{displaymath}">|; 

$key = q/S_{2}^{2}=S_{1}^{2}=I;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img443.png"
 ALT="$S_{2}^{2} = S_{1}^{2} = I$">|; 

$key = q/{displaymath}P=frac{mu^{2}-(4xi^{2}+2)mu+1}{8ximu}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="44" BORDER="0"
 SRC="|."$dir".q|img332.png"
 ALT="\begin{displaymath}
P = \frac{\mu^{2} - (4 \xi^{2} + 2) \mu + 1}{8 \xi \mu}
\end{displaymath}">|; 

$key = q/k+1leqileqN-k;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img12.png"
 ALT="$k + 1 \leq i \leq N - k$">|; 

$key = q/displaystylea_{0};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img146.png"
 ALT="$\displaystyle a_{0}$">|; 

$key = q/{displaymath}lambda=a_{0}+2a_{1}mbox{cos}frac{ellpi}{N+1}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img194.png"
 ALT="\begin{displaymath}
\lambda = a_{0} + 2 a_{1} \mbox{cos}\frac{\ell  \pi}{N + 1}
\end{displaymath}">|; 

$key = q/epsilon=0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img290.png"
 ALT="$\epsilon = 0$">|; 

$key = q/T;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img75.png"
 ALT="$T$">|; 

$key = q/T^{-1}T_{1};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img218.png"
 ALT="$T^{-1} T_{1}$">|; 

$key = q/{displaymath}widetilde{W}:=:mbox{tanh}:theta:widehat{W}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="24" BORDER="0"
 SRC="|."$dir".q|img493.png"
 ALT="\begin{displaymath}
\widetilde{W} \: = \: \mbox{tanh} \: \theta \: \widehat{W}
\end{displaymath}">|; 

$key = q/{displaymath}frac{lambda-a_{0}}{2a_{1}}=mbox{cos}frac{ellpi}{N+1}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="42" BORDER="0"
 SRC="|."$dir".q|img193.png"
 ALT="\begin{displaymath}
\frac{\lambda - a_{0}}{2 a_{1}} = \mbox{cos}\frac{\ell  \pi}{N +
1}
\end{displaymath}">|; 

$key = q/mbox{tanh}varphi;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img438.png"
 ALT="$\mbox{tanh}  \varphi$">|; 

$key = q/{displaymath}widehat{U}widehat{V}=I(widehat{U},widehat{V})+widehat{U}timeswidehat{V}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img423.png"
 ALT="\begin{displaymath}
\widehat{U}  \widehat{V} = I (\widehat{U}, \widehat{V}) +
\widehat{U} \times \widehat{V}
\end{displaymath}">|; 

$key = q/{displaymath}Z_{N}=left[array{{cccc}x_{N+k}x_{N+k-1}vdotsx_{N+1}x_{N}vdotsx_{N-k+1}array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="161" BORDER="0"
 SRC="|."$dir".q|img66.png"
 ALT="\begin{displaymath}
Z_{N} = \left[ \begin{array}{cccc}
x_{N + k} \\\\
x_{N + k...
...
x_{N} \\\\
\vdots \\\\
x_{N - k + 1}
\end{array} \right]
\end{displaymath}">|; 

$key = q/{eqnarraystar}(widehat{U}timeswidehat{V},widehat{U}timeswidehat{V})&=&([(u_{2}v_3}v_{1}-u_{1}v_{3})K+(u_{2}v_{1}-u_{1}v_{2})L]){eqnarraystar};MSF=1.6;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="421" HEIGHT="76" BORDER="0"
 SRC="|."$dir".q|img460.png"
 ALT="\begin{eqnarray*}
(\widehat{U} \times \widehat{V}, \widehat{U} \times \widehat{...
... v_{1} - u_{1} v_{3})  K + (u_{2} v_{1} - u_{1}
v_{2})  L])
\end{eqnarray*}">|; 

$key = q/B;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img411.png"
 ALT="$B$">|; 

$key = q/M^{-1slash2};K;M^{-1slash2};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img91.png"
 ALT="$M^{-1/2} \;K \;
M^{-1/2}$">|; 

$key = q/cos;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img195.png"
 ALT="$cos$">|; 

$key = q/{displaymath}R=S_{3}:S_{1}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="27" BORDER="0"
 SRC="|."$dir".q|img448.png"
 ALT="\begin{displaymath}
R = S_{3} \: S_{1}
\end{displaymath}">|; 

$key = q/{displaymath}x_3=frac{w_1x_1+w_2x_2}{w_{3}}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="40" BORDER="0"
 SRC="|."$dir".q|img480.png"
 ALT="\begin{displaymath}
x_3 = \frac{w_1 x_1 + w_2 x_2}{w_{3}}
\end{displaymath}">|; 

$key = q/{displaymath}m_{i}x_{i}=sum^{K}_{j=1}k_{ij}(x_{i+j}+x_{i-j}-2x_{i}){displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img6.png"
 ALT="\begin{displaymath}
m_{i} x_{i} = \sum^{K}_{j = 1} k_{ij}(x_{i+j} + x_{i-j} - 2x_{i})
\end{displaymath}">|; 

$key = q/{displaymath}(mu+1)^{2}-alphabeta(mu+1)+alphabeta=0{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="213" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img311.png"
 ALT="\begin{displaymath}
(\mu + 1)^{2} - \alpha \beta (\mu + 1) + \alpha \beta = 0
\end{displaymath}">|; 

$key = q/e^{thetawidehat{W}};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img472.png"
 ALT="$e^{\theta \widehat{W}}$">|; 

$key = q/k_{ij};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7.png"
 ALT="$k_{ij}$">|; 

$key = q/{displaymath}P_{+}=frac{left[array{{ccc}e^{varphi}&-11&-e^{-varphi}array{right]}{2mbox{senh}varphi}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="65" BORDER="0"
 SRC="|."$dir".q|img180.png"
 ALT="\begin{displaymath}
P _{+} = \frac{ \left[ \begin{array}{ccc}
e^{\varphi} &amp; ...
... -e^{-\varphi}
\end{array} \right]}{2 \mbox{senh}  \varphi}
\end{displaymath}">|; 

$key = q/6;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img347.png"
 ALT="$6$">|; 

$key = q/{displaymath}xi(xi+2P)=0{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img329.png"
 ALT="\begin{displaymath}
\xi (\xi + 2 P) = 0
\end{displaymath}">|; 

$key = q/{displaymath}mu_pm=e^{pmvarphi}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img160.png"
 ALT="\begin{displaymath}
\mu_\pm = e^{\pm \varphi}
\end{displaymath}">|; 

$key = q/{displaymath}=frac{1}{2mbox{senh}varphi}left[array{{ccc}e^{(N+1)varphi}-e^{-(N+1varphi}&-(e^{(N-1)varphi}-e^{-(N-1)varphi})array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="419" HEIGHT="65" BORDER="0"
 SRC="|."$dir".q|img185.png"
 ALT="\begin{displaymath}
= \frac{1}{2 \mbox{senh} \varphi} \left[
\begin{array}{ccc...
...^{(N - 1)\varphi}
-e^{-(N - 1)\varphi})
\end{array} \right]
\end{displaymath}">|; 

$key = q/{displaymath}mu=frac{c}{2}pmsqrt{left(frac{c}{2}right)^{2}-1}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img155.png"
 ALT="\begin{displaymath}
\mu = \frac{c}{2} \pm \sqrt{\left(\frac{c}{2}\right)^{2} -1}
\end{displaymath}">|; 

$key = q/{displaymath}mu=frac{alphabeta-2}{2}pmfrac{sqrt{alphabeta(alphabeta-4)}}{2}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="200" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img323.png"
 ALT="\begin{displaymath}
\mu = \frac{\alpha \beta - 2}{2} \pm \frac{\sqrt{\alpha
\beta(\alpha \beta - 4)}}{2}
\end{displaymath}">|; 

$key = q/{displaymath}nu_{pm}upsilon_{pm}=frac{ad-ad+bc}{b^{2}}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="39" BORDER="0"
 SRC="|."$dir".q|img380.png"
 ALT="\begin{displaymath}
\nu_{\pm} \upsilon_{\pm} = \frac{ad - ad + bc}{b^{2}}
\end{displaymath}">|; 

$key = q/A;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img37.png"
 ALT="$A$">|; 

$key = q/includegraphics[width=4.8in]{f11p69.eps};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="552" HEIGHT="398" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img336.png"
 ALT="\includegraphics[width=4.8in]{f11p69.eps}">|; 

$key = q/T^{2};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img278.png"
 ALT="$T^{2}$">|; 

$key = q/{displaymath}widehat{U}timeswidehat{V}=mbox{senh}:theta:widehat{W}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="25" BORDER="0"
 SRC="|."$dir".q|img465.png"
 ALT="\begin{displaymath}
\widehat{U} \times \widehat{V} = \mbox{senh} \: \theta \:
\widehat{W}
\end{displaymath}">|; 

$key = q/{displaymath}N+1leqellleqN+k{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="26" BORDER="0"
 SRC="|."$dir".q|img45.png"
 ALT="\begin{displaymath}
N + 1 \leq \ell \leq N + k
\end{displaymath}">|; 

$key = q/{displaymath}left[array{{c}m_{1}ddot{x}_{1}m_{2}ddot{x}_{2}vdotsm_{k}ddot{x}_{k}eft[array{{c}x_{1}x_{2}vdotsx_{k}vdotsx_{N}array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="572" HEIGHT="202" BORDER="0"
 SRC="|."$dir".q|img20.png"
 ALT="\begin{displaymath}
\left[ \begin{array}{c}
m_{1} \ddot{x}_{1} \\\\
m_{2} \dd...
..._{k} \\\\
\\\\
\vdots \\\\
\\\\
x_{N}
\end{array} \right]
\end{displaymath}">|; 

$key = q/{displaymath}ddot{X}=lambdaX{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="24" BORDER="0"
 SRC="|."$dir".q|img31.png"
 ALT="\begin{displaymath}
\ddot{X} = \lambda X
\end{displaymath}">|; 

$key = q/{displaymath}leftarrowvertarray{{cc}a&bc&darray{rightarrowvert=ad-bc=1{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img363.png"
 ALT="\begin{displaymath}
\left \arrowvert \begin{array}{cc}
a &amp; b \\\\
c &amp; d
\end{array} \right \arrowvert = ad - bc = 1
\end{displaymath}">|; 

$key = q/Z;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img84.png"
 ALT="$Z$">|; 

$key = q/{displaymath}mu=e^{varphi}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="27" BORDER="0"
 SRC="|."$dir".q|img78.png"
 ALT="\begin{displaymath}
\mu = e^{\varphi}
\end{displaymath}">|; 

$key = q/{displaymath}left[array{{ccc}c-mu&-11&-muarray{right]=0{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img153.png"
 ALT="\begin{displaymath}
\left[ \begin{array}{ccc}
c - \mu &amp; -1 \\\\
1 &amp; -\mu
\end{array} \right] = 0
\end{displaymath}">|; 

$key = q/{displaymath}P_{+}=frac{left[array{{ccc}-e^{varphi}&1-1&e^{-varphi}array{right]}{-e^{varphi}+e^{-varphi}}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="63" BORDER="0"
 SRC="|."$dir".q|img179.png"
 ALT="\begin{displaymath}
P _{+} = \frac{ \left[ \begin{array}{ccc}
-e^{\varphi} &amp; ...
...{-\varphi}
\end{array} \right]}{-e^{\varphi} + e^{-\varphi}}
\end{displaymath}">|; 

$key = q/{displaymath}left[array{{ccc}frac{lambda-a_0}{a_1}&-1&1&0array{right]^{L}=frac{1hi&mbox{senh}Lvarphi&-mbox{senh}(L-1)varphiarray{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="442" HEIGHT="70" BORDER="0"
 SRC="|."$dir".q|img245.png"
 ALT="\begin{displaymath}
\left[ \begin{array}{ccc}
\frac{\lambda - a_0}{a_1} &amp; -1 ...
... \varphi &amp; -\mbox{senh}  (L - 1)\varphi
\end{array} \right]
\end{displaymath}">|; 

$key = q/{displaymath}left[array{{ccc}a&bc&darray{right]=e^{varphiwidehat{U}}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img402.png"
 ALT="\begin{displaymath}
\left[ \begin{array}{ccc}
a &amp; b \\\\
c &amp; d
\end{array} \right] = e^{\varphi \widehat{U}}
\end{displaymath}">|; 

$key = q/W;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img430.png"
 ALT="$W$">|; 

$key = q/{displaymath}a'_{11}=-a_{12}qquada'_{NN}=-a_{N,N-1}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="240" HEIGHT="29" BORDER="0"
 SRC="|."$dir".q|img97.png"
 ALT="\begin{displaymath}
a'_{11} = -a_{12}  \qquad   a'_{NN} = -a_{N,N-1}
\end{displaymath}">|; 

$key = q/m;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img233.png"
 ALT="$m$">|; 

$key = q/{displaymath}u_pm=frac{1}{mu_pm}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="42" BORDER="0"
 SRC="|."$dir".q|img170.png"
 ALT="\begin{displaymath}
u_\pm = \frac{1}{\mu_\pm}
\end{displaymath}">|; 

$key = q/{displaymath}xi=0eqno{(mbox{III.46a})}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="294" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img330.png"
 ALT="\begin{displaymath}
\xi = 0 \eqno{(\mbox{III.46a})}
\end{displaymath}">|; 

$key = q/{displaymath}array{{ccc}A&=&a_{0}I+widehat{a}B&=&b_{0}I+widehat{b}array{{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img412.png"
 ALT="\begin{displaymath}
\begin{array}{ccc}
A &amp; = &amp; a_{0} I + \widehat{a} \\\\
B &amp; = &amp; b_{0} I + \widehat{b}
\end{array}
\end{displaymath}">|; 

$key = q/{displaymath}(U,U)=u_{1}^{2}+u_{2}^{2}-u_{3}^{2}=1{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img441.png"
 ALT="\begin{displaymath}
(U,  U) = u_{1}^{2} + u_{2}^{2} - u_{3}^{2} = 1
\end{displaymath}">|; 

$key = q/(P,xi);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img337.png"
 ALT="$(P, \xi)$">|; 

$key = q/{displaymath}mu=mu(xi,P){displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img324.png"
 ALT="\begin{displaymath}
\mu = \mu (\xi, P)
\end{displaymath}">|; 

$key = q/{displaymath}1=mu_pmu{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img169.png"
 ALT="\begin{displaymath}
1 = \mu_\pm  u
\end{displaymath}">|; 

$key = q/includegraphics[width=3.2in]{f25p113.eps};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="368" HEIGHT="292" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img490.png"
 ALT="\includegraphics[width=3.2in]{f25p113.eps}">|; 

$key = q/{displaymath}=left[array{{ccc}frac{(lambda-a_{i+1,i+1})(lambda-a_{ii})}{a_{i+1,iii}}{a_{i,i+1}}&-frac{a_{i,i-1}}{a_{i,i+1}}array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="385" HEIGHT="74" BORDER="0"
 SRC="|."$dir".q|img119.png"
 ALT="\begin{displaymath}
= \left[ \begin{array}{ccc}
\frac{(\lambda - a_{i + 1, i ...
...}} &amp; -\frac{a_{i, i
- 1}}{a_{i, i + 1}}
\end{array} \right]
\end{displaymath}">|; 

$key = q/mu;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img77.png"
 ALT="$\mu$">|; 

$key = q/{displaymath}ddot{Y}=(M^{-1slash2}KM^{-1slash2})Y{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img33.png"
 ALT="\begin{displaymath}
\ddot{Y} = (M^{-1/2}K   M^{-1/2}) Y
\end{displaymath}">|; 

$key = q/epsilon;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img287.png"
 ALT="$\epsilon$">|; 

$key = q/{displaymath}P_{-}=frac{left[array{{ccc}-e^{-varphi}&1-1&e^{varphi}array{right]}{2mbox{senh}varphi}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="65" BORDER="0"
 SRC="|."$dir".q|img182.png"
 ALT="\begin{displaymath}
P _{-} = \frac{ \left[ \begin{array}{ccc}
-e^{-\varphi} &amp;...
... &amp; e^{\varphi}
\end{array} \right]}{2 \mbox{senh}  \varphi}
\end{displaymath}">|; 

$key = q/varphi=0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img198.png"
 ALT="$\varphi = 0$">|; 

$key = q/{displaymath}mu^{2}-4muxi^{2}-2mu+1=0{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="27" BORDER="0"
 SRC="|."$dir".q|img334.png"
 ALT="\begin{displaymath}
\mu^{2} - 4 \mu \xi^{2} - 2 \mu + 1 = 0
\end{displaymath}">|; 

$key = q/{displaymath}(widehat{V},widehat{V})=1{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img452.png"
 ALT="\begin{displaymath}
(\widehat{V},  \widehat{V}) = 1
\end{displaymath}">|; 

$key = q/P;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img318.png"
 ALT="$P$">|; 

$key = q/{displaymath}L_{n}left[array{{ccc}a&bc&darray{right]=frac{varphi}{mbox{senh}varp{(b+c)}{2}J+frac{(a-d)}{2}K+frac{(b-c)}{2}Larray{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="371" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img393.png"
 ALT="\begin{displaymath}
L_{n} \left[\begin{array}{ccc}
a &amp; b \\\\
c &amp; d
\end{arra...
...frac{(a - d)}{2} K + \frac{(b - c)}{2} L
\end{array} \right]
\end{displaymath}">|; 

$key = q/{eqnarraystar}(widehat{U}timeswidehat{V},widehat{U}timeswidehat{V})&=&(u_{2}v_{3^{2}&=&(u_{1}v_{1}+u_{2}v_{2}-u_{3}v_{3})^{2}-1{eqnarraystar};MSF=1.6;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="540" HEIGHT="195" BORDER="0"
 SRC="|."$dir".q|img461.png"
 ALT="\begin{eqnarray*}
(\widehat{U} \times \widehat{V},   \widehat{U} \times
\wid...
... \\\\
&amp; = &amp; (u_{1} v_{1} + u_{2} v_{2} - u_{3}  v_{3})^{2} - 1
\end{eqnarray*}">|; 

$key = q/{displaymath}P=frac{1}{2}left(rho+frac{1}{rho}right){displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img317.png"
 ALT="\begin{displaymath}
P = \frac{1}{2}\left(\rho + \frac{1}{\rho}\right)
\end{displaymath}">|; 

$key = q/{displaymath}widehat{U}=u_{1}J+u_{2}K+u_{3}L{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="27" BORDER="0"
 SRC="|."$dir".q|img440.png"
 ALT="\begin{displaymath}
\widehat{U} = u_{1}  J + u_{2}  K + u_{3}  L
\end{displaymath}">|; 

$key = q/{displaymath}widetilde{U}=mbox{tanh}varphiwidehat{U}qquadqquadwidetilde{V}=mbox{tanh}psiwidehat{V}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="261" HEIGHT="27" BORDER="0"
 SRC="|."$dir".q|img432.png"
 ALT="\begin{displaymath}
\widetilde{U} = \mbox{tanh}  \varphi  \widehat{U} \qquad \qquad
\widetilde{V} = \mbox{tanh}  \psi  \widehat{V}
\end{displaymath}">|; 

$key = q/a_1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img211.png"
 ALT="$a_1$">|; 

$key = q/xi=-2P;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img340.png"
 ALT="$\xi = -2 P$">|; 

$key = q/x_{-k-1};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img19.png"
 ALT="$x_{-k-1}$">|; 

$key = q/L;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img236.png"
 ALT="$L$">|; 

$key = q/theta;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img468.png"
 ALT="$\theta$">|; 

$key = q/{displaymath}frac{mbox{senh}(N+1)varphi-frac{2(a-a_{0})}{a_{1}}mbox{senh}Nvarphi}}{a^{2}_{1}}mbox{senh}(N-1)varphi}{mbox{senh}varphi}=0{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="418" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img225.png"
 ALT="\begin{displaymath}
\frac{\mbox{senh}(N + 1)\varphi - \frac{2(a - a_{0})}{a_{1}...
...2}_{1}}\mbox{senh} (N - 1) \varphi}{\mbox{senh}
\varphi} = 0
\end{displaymath}">|; 

$key = q/{displaymath}L_{n}T=frac{varphi}{2mbox{senh}varphi}left[array{{ccc}mu_{+}-d&bc&mvarphi}left[array{{ccc}mu_{-}-d&bc&mu_{-}-aarray{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="480" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img386.png"
 ALT="\begin{displaymath}
L_{n} T = \frac{\varphi}{2\mbox{senh}  \varphi} \left[
\b...
...c}
\mu_{-} - d &amp; b \\\\
c &amp; \mu_{-} - a
\end{array} \right]
\end{displaymath}">|; 

$key = q/y_{i};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img107.png"
 ALT="$y_{i}$">|; 

$key = q/{displaymath}T_{D}=left[array{{cc}alphabeta-1&-alpha&beta&-1array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="64" BORDER="0"
 SRC="|."$dir".q|img307.png"
 ALT="\begin{displaymath}
T_{D} = \left[ \begin{array}{cc}
\alpha \beta - 1 &amp; -\alpha \\\\
&amp; \\\\
\beta &amp; -1
\end{array} \right]
\end{displaymath}">|; 

$key = q/{displaymath}(widehat{U},widehat{W})=[widehat{U},Nwidehat{W}]=[u_1,u_2,u_3]left[&00&0&-1array{right]left[array{{c}w_1w_2w_3array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="384" HEIGHT="64" BORDER="0"
 SRC="|."$dir".q|img474.png"
 ALT="\begin{displaymath}
(\widehat{U}, \widehat{W}) = [\widehat{U}, N \widehat{W}] =...
...egin{array}{c}
w_1 \\\\
w_2 \\\\
w_3
\end{array} \right]
\end{displaymath}">|; 

$key = q/{displaymath}widehat{U}=frac{U}{mbox{senh}varphi}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="42" BORDER="0"
 SRC="|."$dir".q|img401.png"
 ALT="\begin{displaymath}
\widehat{U} = \frac{U}{\mbox{senh} \varphi}
\end{displaymath}">|; 

$key = q/Y;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img98.png"
 ALT="$Y$">|; 

$key = q/{displaymath}P_{-}=frac{left[array{{ccc}-e^{-varphi}&1-1&e^{varphi}array{right]}{-e^{-varphi}+e^{varphi}}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="63" BORDER="0"
 SRC="|."$dir".q|img181.png"
 ALT="\begin{displaymath}
P _{-} = \frac{ \left[ \begin{array}{ccc}
-e^{-\varphi} &amp;...
...^{\varphi}
\end{array} \right]}{-e^{-\varphi} + e^{\varphi}}
\end{displaymath}">|; 

$key = q/{displaymath}T_{i+1}T_{i}=frac{1}{1+epsilon}left[array{{ccc}frac{lambda-a_{0}}{a{{ccc}frac{2(lambda-a_{0})}{a_{1}}-2&-1&1&0array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="427" HEIGHT="68" BORDER="0"
 SRC="|."$dir".q|img277.png"
 ALT="\begin{displaymath}
T_{i+1}  T_{i} = \frac{1}{1+\epsilon} \left[ \begin{array}...
..._{0})}{a_{1}}-2 &amp; -1 \\\\
&amp; \\\\
1 &amp; 0
\end{array} \right]
\end{displaymath}">|; 

$key = q/{displaymath}Z_{i}=T_{i}Z_{i-1}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img54.png"
 ALT="\begin{displaymath}
Z_{i} = T_{i}  Z_{i - 1}
\end{displaymath}">|; 

$key = q/{displaymath}T_{i}=frac{1}{1+epsilon}left[array{{ccc}frac{lambda-a_{0}}{a_{1}}&-arepsilon}{1+epsilon}left[array{{ccc}1&01&0array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="305" HEIGHT="66" BORDER="0"
 SRC="|."$dir".q|img273.png"
 ALT="\begin{displaymath}
T_{i} = \frac{1}{1+\epsilon} \left[ \begin{array}{ccc}
\fr...
...ft[ \begin{array}{ccc}
1 &amp; 0 \\\\
1 &amp; 0
\end{array} \right]
\end{displaymath}">|; 

$key = q/{displaymath}tau=frac{1}{1+epsilon}T^{N}+frac{epsilon}{1+epsilon}T^{N-i-1}QT^{i-1}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="262" HEIGHT="40" BORDER="0"
 SRC="|."$dir".q|img282.png"
 ALT="\begin{displaymath}
\tau = \frac{1}{1+\epsilon}  T^{N} + \frac{\epsilon}{1+\epsilon}
 T^{N-i-1} Q  T^{i-1}
\end{displaymath}">|; 

$key = q/{displaymath}frac{mbox{senh}(N+1)varphi}{frac{lambdambox{senh}(N-L)varphimbox{se}{mbox{senh}varphi}}=-frac{M-m}{k}eqno{(mbox{III.33}')}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="408" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img251.png"
 ALT="\begin{displaymath}
\frac{\mbox{senh}(N+1)\varphi}{\frac{\lambda \
\mbox{senh}...
...{senh} \
\varphi}} = - \frac{M-m}{k} \eqno{(\mbox{III.33}')}
\end{displaymath}">|; 

$key = q/{displaymath}mbox{senh}(N+1)varphi-frac{(a-a_{0})}{a_{1}}mbox{senh}Nvarphi=0{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="266" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img230.png"
 ALT="\begin{displaymath}
\mbox{senh} (N + 1) \varphi - \frac{(a - a_{0})}{a_{1}} \
\mbox{senh} N \varphi = 0
\end{displaymath}">|; 

$key = q/{displaymath}R=R_{2}:R_{1}=S_{3}:S_{2}:S_{2}:S_{1}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="27" BORDER="0"
 SRC="|."$dir".q|img446.png"
 ALT="\begin{displaymath}
R = R_{2} \: R_{1} = S_{3} \: S_{2} \: S_{2} \: S_{1}
\end{displaymath}">|; 

$key = q/includegraphics[width=5in]{f02p16.eps};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="575" HEIGHT="83" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img85.png"
 ALT="\includegraphics[width=5in]{f02p16.eps}">|; 

$key = q/frac{lambda-a_{0}}{2a_{1}}geq1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img161.png"
 ALT="$\frac{\lambda - a_{0}}{2  a_{1}} \geq 1$">|; 

$key = q/Q;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img279.png"
 ALT="$Q$">|; 

$key = q/{displaymath}mbox{tanh}theta=-frac{1}{w_{2}}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img499.png"
 ALT="\begin{displaymath}
\mbox{tanh}  \theta = -\frac{1}{w_{2}}
\end{displaymath}">|; 

$key = q/{displaymath}Y=M^{1slash2}X{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="24" BORDER="0"
 SRC="|."$dir".q|img32.png"
 ALT="\begin{displaymath}
Y = M^{1/2} X
\end{displaymath}">|; 

$key = q/{displaymath}-(k+1)leqellleq0{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img43.png"
 ALT="\begin{displaymath}
-(k + 1) \leq \ell \leq 0 \end{displaymath}">|; 

$key = q/{eqnarraystar}T^{N}T^{-1}T_{1}&=&frac{1}{mbox{senh}varphi}left[array{{cc}mbox{se}(N-1)varphi&-mbox{senh}(N-1)varphiarray{right]{eqnarraystar};MSF=1.6;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="535" HEIGHT="138" BORDER="0"
 SRC="|."$dir".q|img223.png"
 ALT="\begin{eqnarray*}
T^{N} T^{-1} T_{1} &amp; = &amp; \frac{1}{\mbox{senh} \varphi} \left ...
... -
1)\varphi &amp; -\mbox{senh}(N - 1)\varphi
\end{array} \right]
\end{eqnarray*}">|; 

$key = q/{displaymath}alphabeta(alphabeta-4)=0{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img325.png"
 ALT="\begin{displaymath}
\alpha \beta (\alpha \beta - 4) = 0
\end{displaymath}">|; 

$key = q/a_{N};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img143.png"
 ALT="$a_{N}$">|; 

$key = q/{displaymath}u_pm=e^{mpvarphi}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img171.png"
 ALT="\begin{displaymath}
u_\pm = e^{\mp \varphi}
\end{displaymath}">|; 

$key = q/Q_{11};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img285.png"
 ALT="$Q_{11}$">|; 

$key = q/{displaymath}L_{n}T=frac{varphi}{2mbox{senh}varphi}left[array{{ccc}frac{a-d}{2}&bc&frac{d-a}{2}array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="222" HEIGHT="65" BORDER="0"
 SRC="|."$dir".q|img390.png"
 ALT="\begin{displaymath}
L_{n}  T = \frac{\varphi}{2\mbox{senh}  \varphi} \left[
...
...d}{2} &amp; b \\\\
\\\\
c &amp; \frac{d - a}{2}
\end{array} \right]
\end{displaymath}">|; 

$key = q/{displaymath}left[array{{ccc}y_{i+1}y_{i}array{right]=left[array{{ccc}tau^{(i)}_-2)}_{22}array{right]left[array{{ccc}y_{1}0array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="266" HEIGHT="73" BORDER="0"
 SRC="|."$dir".q|img127.png"
 ALT="\begin{displaymath}
\left [ \begin{array}{ccc}
y_{i + 1} \\\\
\\\\
y_{i}
\e...
...[ \begin{array}{ccc}
y_{1} \\\\
\\\\
0
\end{array} \right]
\end{displaymath}">|; 

$key = q/{displaymath}x_{i+k}=sum^{k-1}_{j=-k}C_{j}X_{i+j}eqno{(mbox{I.11b})}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="346" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img46.png"
 ALT="\begin{displaymath}
x_{i + k} = \sum^{k - 1}_{j = -k} C_{j} X_{i + j}
\eqno{(\mbox{I.11b})}
\end{displaymath}">|; 

$key = q/{displaymath}AB=(a_{0}I+a_{1}J+a_{2}K+a_{3}L)(b_{0}I+b_{1}J+b_{2}K+b_{3}L){displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="393" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img409.png"
 ALT="\begin{displaymath}
AB = (a_{0} I + a_{1} J + a_{2} K + a_{3} L)(b_{0} I + b_{1} J +
b_{2} K + b_{3} L)
\end{displaymath}">|; 

$key = q/1slash2.5;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img343.png"
 ALT="$1/2.5$">|; 

$key = q/mbox{cosh}varphi;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img396.png"
 ALT="$\mbox{cosh}  \varphi$">|; 

$key = q/displaystylearray{{ccc}y_{i+1}<col_mark>=<col_mark>tau^{(i)}_{11}y_{1}<row_mark>mark><row_mark>y_{i}<col_mark>=<col_mark>tau^{(i-1)}_{21}y_{1}array{;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="79" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img128.png"
 ALT="$\displaystyle \begin{array}{ccc}
y_{i + 1} &amp; = &amp; \tau^{(i)}_{11}   y_{1} \\\\
&amp; &amp; \\\\
y_{i} &amp; = &amp; \tau^{(i - 1)}_{21}   y_{1}
\end{array}$">|; 

$key = q/x_{1},x_{2},dots,x_{k};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img72.png"
 ALT="$x_{1}, x_{2}, \dots , x_{k}$">|; 

$key = q/beta;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img313.png"
 ALT="$\beta$">|; 

$key = q/(2K+1);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img36.png"
 ALT="$(2K + 1)$">|; 

$key = q/{displaymath}x^{2}_{3}=frac{w^{2}_{1}x^{2}_{1}+w^{2}_{2}x^{2}_{2}+2w_{1}w_{2}x_{1}x_{2}}{w^{2}_{3}}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="227" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img481.png"
 ALT="\begin{displaymath}
x^{2}_{3} = \frac {w^{2}_{1} x^{2}_{1} + w^{2}_{2} x^{2}_{2} +
2 w_{1} w_{2} x_{1} x_{2}}{w^{2}_{3}}
\end{displaymath}">|; 

$key = q/{displaymath}C_{0}=frac{lambda-a_{ii}}{a_{i,i+k}}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="44" BORDER="0"
 SRC="|."$dir".q|img48.png"
 ALT="\begin{displaymath}
C_{0} = \frac{\lambda - a_{i i}}{a_{i,i + k}}
\end{displaymath}">|; 

$key = q/xi;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img301.png"
 ALT="$\xi$">|; 

$key = q/{displaymath}array{{ccc}widehat{U}widehat{U}&=&Iwidehat{V}widehat{V}&=&Iarray{{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img459.png"
 ALT="\begin{displaymath}
\begin{array}{ccc}
\widehat{U}  \widehat{U} &amp; = &amp; I \\\\
\widehat{V}  \widehat{V} &amp; = &amp; I
\end{array}
\end{displaymath}">|; 

$key = q/{displaymath}T_{i+1}:T_{i}=frac{1}{1+epsilon}(T^{2}+epsilonQ){displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="40" BORDER="0"
 SRC="|."$dir".q|img280.png"
 ALT="\begin{displaymath}
T_{i+1} \: T_{i} = \frac{1}{1 + \epsilon} (T^{2} + \epsilon Q)
\end{displaymath}">|; 

$key = q/y_i;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img135.png"
 ALT="$y_i$">|; 

$key = q/{displaymath}mbox{senh}varphi=sqrt{left(frac{a+d}{2}right)^{2}-1}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img397.png"
 ALT="\begin{displaymath}
\mbox{senh}  \varphi = \sqrt{\left(\frac{a + d}{2}\right)^{2} -
1}
\end{displaymath}">|; 

$key = q/';MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="9" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img252.png"
 ALT="$'$">|; 

$key = q/{displaymath}K=left[array{{ccc}1&00&-1array{right]L=left[array{{ccc}0&1-1&0array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="258" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img392.png"
 ALT="\begin{displaymath}
K = \left[\begin{array}{ccc}
1 &amp; 0 \\\\
0 &amp; -1
\end{array...
...ft[\begin{array}{ccc}
0 &amp; 1 \\\\
-1 &amp; 0
\end{array} \right]
\end{displaymath}">|; 

$key = q/x_{N+1};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img16.png"
 ALT="$x_{N + 1}$">|; 

$key = q/{displaymath}widehat{V}=v_{1}J+v_{2}K+v_{3}L{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="27" BORDER="0"
 SRC="|."$dir".q|img450.png"
 ALT="\begin{displaymath}
\widehat{V} = v_{1}  J + v_{2}  K + v_{3}  L
\end{displaymath}">|; 

$key = q/{displaymath}left[array{{ccc}v&1array{right]left[array{{cccc}c&-11&0array{right]=mu_pmleft[array{{ccc}v&1array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="241" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img172.png"
 ALT="\begin{displaymath}
\left [ \begin{array}{ccc}
v &amp; 1
\end{array} \right]
\l...
... \mu_\pm \left[ \begin{array}{ccc} v &amp; 1
\end{array} \right]
\end{displaymath}">|; 

$key = q/{displaymath}left[array{{ccc}frac{frac{lambda-a_{0}}{a_{1}}+epsilon}{1+epsilon}&-frac{1}{1+epsilon}&1&0array{right]qquadmbox{parala}i{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="229" HEIGHT="74" BORDER="0"
 SRC="|."$dir".q|img271.png"
 ALT="\begin{displaymath}
\left[ \begin{array}{ccc}
\frac{\frac{\lambda-a_{0}}{a_{1...
...
&amp; \\\\
1 &amp; 0
\end{array} \right] \qquad \mbox{para la } i
\end{displaymath}">|; 

$key = q/{displaymath}ddot{Y}=M^{-1slash2};K;M^{-1slash2};Y{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="24" BORDER="0"
 SRC="|."$dir".q|img90.png"
 ALT="\begin{displaymath}
\ddot{Y} = M^{-1/2} \; K \; M^{-1/2} \; Y
\end{displaymath}">|; 

$key = q/x_{k},x_{k-1},dots,x_{1};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img74.png"
 ALT="$x_{k}, x_{k-1}, \dots, x_{1}$">|; 

$key = q/includegraphics[width=3.8in]{f09p61.eps};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="437" HEIGHT="742" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img294.png"
 ALT="\includegraphics[width=3.8in]{f09p61.eps}">|; 

$key = q/{displaymath}tau=frac{1}{mbox{senh}}varphileft[array{{cc}mbox{senh}(N+1)varphi&-phimbox{senh}Nvarphi&-mbox{senh}(N+1)varphiarray{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="346" HEIGHT="64" BORDER="0"
 SRC="|."$dir".q|img246.png"
 ALT="\begin{displaymath}
\tau = \frac{1}{\mbox{senh}}  \varphi \left[ \begin{array}...
...varphi &amp; -\mbox{senh}  (N + 1)
\varphi
\end{array} \right]
\end{displaymath}">|; 

$key = q/{displaymath}ddot{Y}=lambdaY{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="24" BORDER="0"
 SRC="|."$dir".q|img100.png"
 ALT="\begin{displaymath}
\ddot{Y} = \lambda Y
\end{displaymath}">|; 

$key = q/textstyle=;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img147.png"
 ALT="$\textstyle =$">|; 

$key = q/{displaymath}y_{1}=mbox{sen}(ell:pislashN+1){displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img207.png"
 ALT="\begin{displaymath}
y_{1} = \mbox{sen} (\ell \: \pi / N + 1)
\end{displaymath}">|; 

$key = q/{eqnarraystar}e^{psiwidehat{V}}e^{psiwidehat{V}}&=&[mbox{cosh}(I+mbox{tanh}varphx{tanh}varphimbox{tanh}psiwidehat{U}widehat{V}]{eqnarraystar};MSF=1.6;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="549" HEIGHT="80" BORDER="0"
 SRC="|."$dir".q|img420.png"
 ALT="\begin{eqnarray*}
e^{\psi  \widehat{V}} e^{\psi  \widehat{V}} &amp; = &amp;[\mbox{cos...
...h}  \varphi  \mbox{tanh}  \psi \
\widehat{U}  \widehat{V}]
\end{eqnarray*}">|; 

$key = q/k_{i}=kslash2;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img260.png"
 ALT="$k_{i}=
k/2$">|; 

$key = q/-1leqfrac{lambda-a_{0}}{2a_{1}}<1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img164.png"
 ALT="$-1 \leq \frac{\lambda - a_{0}}{2 a_{1}} &lt; 1$">|; 

$key = q/X;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img25.png"
 ALT="$X$">|; 

$key = q/includegraphics[width=4.2in]{f24p106.eps};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="483" HEIGHT="519" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img449.png"
 ALT="\includegraphics[width=4.2in]{f24p106.eps}">|; 

$key = q/{displaymath}T^{j}=frac{1}{mbox{senh}varphi}left[array{{ccc}mbox{senh}(j+1)varphphimbox{senh}jvarphi&-mbox{senh}(j-1)varphiarray{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="323" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img203.png"
 ALT="\begin{displaymath}
T^{j} = \frac{1}{\mbox{senh} \varphi} \left[
\begin{array}...
... \varphi &amp; - \mbox{senh} (j - 1)\varphi
\end{array} \right]
\end{displaymath}">|; 

$key = q/{displaymath}P_{pm}=pmfrac{1}{2mbox{senh}varphi}left[array{{ccc}mu_{pm}-d&bc&mu_{pm}-aarray{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="257" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img383.png"
 ALT="\begin{displaymath}
P_{\pm} = \pm \frac{1}{2\mbox{senh}  \varphi} \left[
\beg...
...
\mu_{\pm} - d &amp; b \\\\
c &amp; \mu_{\pm}- a
\end{array} \right]
\end{displaymath}">|; 

$key = q/{displaymath}widehat{U}timeswidehat{U}=(u_{2}u_{3}-u_{3}u_{2}):J+(u_{3}u_{1}-u_{1}u_{3}):K+(u_{2}u_{1}-u_{1}u_{2}):L=widehat{0}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="492" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img457.png"
 ALT="\begin{displaymath}
\widehat{U} \times \widehat{U} = (u_{2} u_{3} - u_{3} u_{2}...
...}) \: K  +  (u_{2} u_{1} -
u_{1} u_{2}) \: L = \widehat{0}
\end{displaymath}">|; 

$key = q/mbox{senh}theta;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img467.png"
 ALT="$\mbox{senh} \
\theta$">|; 

$key = q/{displaymath}T^{N}=frac{e^{Nvarphi}}{2mbox{senh}varphi}left[array{{ccc}e^{varphileft[array{{ccc}-e^{-varphi}&1-1&e^{varphi}array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="381" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img184.png"
 ALT="\begin{displaymath}
T^{N} = \frac{e^{N \varphi}}{2 \mbox{senh} \varphi} \left[
...
...
-e^{-\varphi} &amp; 1 \\\\
-1 &amp; e^{\varphi}
\end{array} \right]
\end{displaymath}">|; 

$key = q/{displaymath}T^{N}=sum^{2k}_{i=1}lambda^{N}_{i}frac{arrowverti><iarrowvert}{<iarrowverti>}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img83.png"
 ALT="\begin{displaymath}
T^{N} = \sum^{2k}_{i = 1}  \lambda^{N}_{i}   \frac{\arrowvert i
&gt;&lt; i \arrowvert}{&lt; i \arrowvert i&gt;}
\end{displaymath}">|; 

$key = q/widehat{a};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img414.png"
 ALT="$\widehat{a}$">|; 

$key = q/{displaymath}widehat{U}widehat{V}=e^{thetawidehat{W}}eqno{(mbox{IV.26'})}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="315" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img471.png"
 ALT="\begin{displaymath}
\widehat{U}  \widehat{V} = e^{\theta \widehat{W}}
\eqno{(\mbox{IV.26'})}
\end{displaymath}">|; 

$key = q/{displaymath}tau=frac{1}{mbox{senh}varphi}left[array{{ccc}mbox{senh}(N+1)varphi-enh}(N-1)varphi&qquad-mbox{senh}(N-1)varphiarray{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="492" HEIGHT="73" BORDER="0"
 SRC="|."$dir".q|img229.png"
 ALT="\begin{displaymath}
\tau = \frac{1}{\mbox{senh} \varphi} \left[ \begin{array}{c...
...hi &amp; \qquad -\mbox{senh} (N - 1) \varphi
\end{array} \right]
\end{displaymath}">|; 

$key = q/Z_{N};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img64.png"
 ALT="$Z_{N}$">|; 

$key = q/{displaymath};;;left[array{{cccc}(lambda+frac{2k}{m})frac{sqrt{Mm}}{k}&-1&1&0array{right]qquadmbox{part'{i}culasimpares}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="333" HEIGHT="67" BORDER="0"
 SRC="|."$dir".q|img299.png"
 ALT="\begin{displaymath}
\; \; \; \left[ \begin{array}{cccc}
(\lambda + \frac{2k}{m...
...
\end{array} \right] \qquad \mbox{part\'{\i}culas  impares}
\end{displaymath}">|; 

$key = q/{displaymath}x^{2}_{1}+x^{2}_{2}-frac{w^{2}_{1}x^{2}_{1}+w^{2}_{2}x^{2}_{2}+2w_{1}w_{2}x_{1}x_{2}}{w^{2}_{3}}=1{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="290" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img482.png"
 ALT="\begin{displaymath}
x^{2}_{1} + x^{2}_{2} - \frac{w^{2}_{1} x^{2}_{1} + w^{2}_{2}
x^{2}_{2} + 2 w_{1} w_{2} x_{1} x_{2}}{w^{2}_{3}} = 1
\end{displaymath}">|; 

$key = q/varphi>0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img162.png"
 ALT="$\varphi &gt; 0$">|; 

$key = q/{displaymath}nu_{pm}+upsilon_{pm}=pmfrac{2mbox{senh}varphi}{b}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="39" BORDER="0"
 SRC="|."$dir".q|img377.png"
 ALT="\begin{displaymath}
\nu_{\pm} + \upsilon_{\pm} = \pm \frac{2 \mbox{senh} \varphi}{b}
\end{displaymath}">|; 

$key = q/{displaymath}y_{j}=frac{mbox{senh}jvarphi}{mbox{senh}varphi}:y_{1}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="42" BORDER="0"
 SRC="|."$dir".q|img204.png"
 ALT="\begin{displaymath}
y_{j} = \frac{\mbox{senh}  j  \varphi}{\mbox{senh}  \varphi} \:
 y_{1}
\end{displaymath}">|; 

$key = q/N-L-1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img237.png"
 ALT="$N-L-1$">|; 

$key = q/{displaymath}T=left[array{{cc}a&bc&darray{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img362.png"
 ALT="\begin{displaymath}
T = \left[ \begin{array}{cc}
a &amp; b \\\\
c &amp; d
\end{array} \right]
\end{displaymath}">|; 

$key = q/{displaymath}widetilde{U}=mbox{tanh}varphiwidehat{U}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="27" BORDER="0"
 SRC="|."$dir".q|img436.png"
 ALT="\begin{displaymath}
\widetilde{U} = \mbox{tanh}  \varphi  \widehat{U}
\end{displaymath}">|; 

$key = q/{displaymath}u_1v_1+u_2v_2-u_3v_3=mbox{cosh}theta{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="197" HEIGHT="27" BORDER="0"
 SRC="|."$dir".q|img462.png"
 ALT="\begin{displaymath}
u_1 v_1 + u_2 v_2 - u_3 v_3 = \mbox{cosh}  \theta
\end{displaymath}">|; 

$key = q/{displaymath}leftvertarray{{cccc}tau_{11}&tau_{12}&dots&tau_{1k}tau_{21}&tau_{22dots&&vdotstau_{k1}&tau_{k2}&&tau_{kk}array{rightvert=0{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="94" BORDER="0"
 SRC="|."$dir".q|img73.png"
 ALT="\begin{displaymath}
\left \vert \begin{array}{cccc}
\tau_{11} &amp; \tau_{12} &amp; \...
..._{k1} &amp; \tau_{k2} &amp; &amp; \tau_{kk}
\end{array} \right \vert = 0
\end{displaymath}">|; 

$key = q/widehat{V};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img422.png"
 ALT="$\widehat{V}$">|; 

$key = q/{displaymath}mbox{tanh}thetawidehat{W}=frac{mbox{cosh}varphimbox{cosh}psi}{mbox{ox{tanh}varphimbox{tanh}psi(widehat{U}timeswidehat{V})]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="531" HEIGHT="39" BORDER="0"
 SRC="|."$dir".q|img428.png"
 ALT="\begin{displaymath}\mbox{tanh}  \theta  \widehat{W} =
\frac{\mbox{cosh}  \va...
...phi  \mbox{tanh}  \psi \
(\widehat{U} \times \widehat{V})]
\end{displaymath}">|; 

$key = q/T^{j};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img202.png"
 ALT="$T^{j}$">|; 

$key = q/varphi=ibeta;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img165.png"
 ALT="$\varphi = i \beta$">|; 

$key = q/{displaymath}x_{i+j}=x_{i+j}-k+1leqjleqk-1{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="265" HEIGHT="29" BORDER="0"
 SRC="|."$dir".q|img52.png"
 ALT="\begin{displaymath}
x_{i + j} = x_{i + j}         -k + 1 \leq j \leq k - 1
\end{displaymath}">|; 

$key = q/{displaymath}m_{i}ddot{x}_{i}=k_{ii-1}x_{i-1}+k_{ii}x_{i}+k_{ii+1}x_{i+1}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="254" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img88.png"
 ALT="\begin{displaymath}
m_{i} \ddot{x}_{i} = k_{i i -1} x_{i-1} + k_{i i} x_{i} + k_{i i +
1} x_{i + 1}
\end{displaymath}">|; 

$key = q/N-1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img255.png"
 ALT="$N - 1$">|; 

$key = q/Krightarrow0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img268.png"
 ALT="$K\rightarrow 0$">|; 

$key = q/2.5;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img344.png"
 ALT="$2.5$">|; 

$key = q/{displaymath}mu_{+}+mu_{-}=a+d{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img388.png"
 ALT="\begin{displaymath}
\mu_{+} + \mu_{-} = a + d
\end{displaymath}">|; 

$key = q/{displaymath}Z_N=T_NT^{N-2}T_1Z_0{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="27" BORDER="0"
 SRC="|."$dir".q|img212.png"
 ALT="\begin{displaymath}
Z_N = T_N  T^{N - 2}  T_1  Z_0
\end{displaymath}">|; 

$key = q/i+1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img109.png"
 ALT="$i+1$">|; 

$key = q/{displaymath}Z_{N}=left[array{{cccc}00vdots0x_{N}vdotsx_{N-k+1}array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="161" BORDER="0"
 SRC="|."$dir".q|img68.png"
 ALT="\begin{displaymath}
Z_{N} = \left[ \begin{array}{cccc}
0 \\\\
0 \\\\
\vdots \\\\...
...\
x_{N} \\\\
\vdots \\\\
x_{N - k + 1}
\end{array} \right]
\end{displaymath}">|; 

$key = q/tau_{11}=0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img497.png"
 ALT="$\tau_{11} = 0$">|; 

$key = q/{displaymath}m_{i}=mqquadqquadi=1,2,...,N{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="210" HEIGHT="27" BORDER="0"
 SRC="|."$dir".q|img138.png"
 ALT="\begin{displaymath}
m_{i} = m \qquad \qquad i = 1, 2, ..., N
\end{displaymath}">|; 

$key = q/{displaymath}mu=frac{alphabeta-2}{2}pmfrac{sqrt{(alphabeta-2)^{2}-4}}{2}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="215" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img322.png"
 ALT="\begin{displaymath}
\mu = \frac{\alpha \beta - 2}{2} \pm \frac{\sqrt{(\alpha \beta -
2)^{2} - 4}}{2}
\end{displaymath}">|; 

$key = q/{eqnarraystar}AB&=&(a_{0}b_{0}+a_{1}b_{1}+a_{2}b_{2}-a_{3}b_{3})I+(a_{0}b_{1}+a_+(a_{0}b_{3}+a_{3}b_{0}+a_{2}b_{1}-a_{1}b_{2})L{eqnarraystar};MSF=1.6;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="500" HEIGHT="76" BORDER="0"
 SRC="|."$dir".q|img410.png"
 ALT="\begin{eqnarray*}
AB &amp; = &amp; (a_{0} b_{0} + a_{1} b_{1} + a_{2} b_{2} - a_{3} b_{...
... K +
(a_{0} b_{3} + a_{3} b_{0} + a_{2} b_{1} - a_{1} b_{2}) L
\end{eqnarray*}">|; 

$key = q/{displaymath}widehat{V}widehat{V}=(widehat{V},widehat{V})I+widehat{V}timeswidehat{V}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img456.png"
 ALT="\begin{displaymath}
\widehat{V}  \widehat{V} = (\widehat{V},  \widehat{V})  I +
\widehat{V} \times \widehat{V}
\end{displaymath}">|; 

$key = q/{displaymath}left[array{{ccc}c&-11&0array{right]left[array{{cccc}1uarray{right]=mu_pmleft[array{{ccc}1uarray{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="200" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img168.png"
 ALT="\begin{displaymath}
\left [ \begin{array}{ccc}
c &amp; -1 \ 1 &amp; 0
\end{array} \...
..._\pm \left[ \begin{array}{ccc}
1 \\\\
u
\end{array} \right]
\end{displaymath}">|; 

$key = q/{displaymath}Tleft[array{{ccc}frac{lambda-a_{0}}{a_{1}}&-11&0array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img145.png"
 ALT="\begin{displaymath}
T \left[ \begin{array}{ccc}
\frac{\lambda - a_{0}}{a_{1}} &amp; -1 \\\\
1 &amp; 0
\end{array} \right]
\end{displaymath}">|; 

$key = q/{displaymath}T_{i+2}=left[array{{cc}frac{lambda-a_{i+2,i+2}}{a_{i+2,i+3}}&-frac{a_{i+2,i+1}}{a_{i+2,i+3}}1&0array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="232" HEIGHT="69" BORDER="0"
 SRC="|."$dir".q|img115.png"
 ALT="\begin{displaymath}
T_{i + 2} = \left[ \begin{array}{cc}
\frac{\lambda - a_{i ...
...+ 1}}{a_{i + 2, i + 3}} \\\\
\\\\
1 &amp; 0
\end{array} \right]
\end{displaymath}">|; 

$key = q/{displaymath}left[array{{c}x_{i+k}x_{i+k-1}vdotsx_{i-k+1}array{right]=left[arrayeft[array{{c}x_{i+k-1}x_{i+k-2}vdotsx_{i-k}array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="408" HEIGHT="94" BORDER="0"
 SRC="|."$dir".q|img53.png"
 ALT="\begin{displaymath}
\left [ \begin{array}{c}
x_{i + k} \\\\
x_{i + k - 1} \ ...
...
x_{i + k -2} \\\\
\vdots \\\\
x_{i - k}
\end{array} \right]
\end{displaymath}">|; 

$key = q/widehat{b};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img415.png"
 ALT="$\widehat{b}$">|; 

$key = q/{displaymath}left[array{{ccc}frac{lambda+2a}{a}&-11&0array{right]=left[array{{ccarray{right]+left[array{{cc}varepsilon&00&0array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="315" HEIGHT="66" BORDER="0"
 SRC="|."$dir".q|img240.png"
 ALT="\begin{displaymath}
\left[ \begin{array}{ccc}
\frac{\lambda + 2a}{a} &amp; -1 \ ...
...y}{cc}
\varepsilon &amp; 0 \\\\
\\\\
0 &amp; 0
\end{array} \right]
\end{displaymath}">|; 

$key = q/{displaymath}frac{K_{ij}}{sqrt{m_{i}m_{j}}}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img35.png"
 ALT="\begin{displaymath}
\frac{K_{i j}}{\sqrt{m_{i} m_{j}}}
\end{displaymath}">|; 

$key = q/{displaymath}left[array{{cc}0&1-1&frac{lambda-a_{0}}{a_{1}}array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="66" BORDER="0"
 SRC="|."$dir".q|img216.png"
 ALT="\begin{displaymath}
\left [ \begin{array}{cc}
0 &amp; 1 \\\\
\\\\
-1 &amp; \frac{\lambda - a_{0}}{a_{1}}
\end{array} \right]
\end{displaymath}">|; 

$key = q/{displaymath}(xi^{2}+2Pxi+1)(xi^{2}+2Pxi)=0{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="205" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img327.png"
 ALT="\begin{displaymath}
(\xi^{2} + 2 P \xi + 1)(\xi^{2} + 2 P \xi) = 0
\end{displaymath}">|; 

$key = q/{displaymath}Z_{0}=tau^{(i)^{-1}}Z_{i}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="27" BORDER="0"
 SRC="|."$dir".q|img132.png"
 ALT="\begin{displaymath}
Z_{0} = \tau^{(i)^{-1}} Z_{i}
\end{displaymath}">|; 

$key = q/{displaymath}R=S_{2}S_{1}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="27" BORDER="0"
 SRC="|."$dir".q|img442.png"
 ALT="\begin{displaymath}
R = S_{2}  S_{1}
\end{displaymath}">|; 

$key = q/{displaymath}frac{T_{11}^{N}}{Q_{11}}=-epsilon{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img283.png"
 ALT="\begin{displaymath}
\frac{T_{11}^{N}}{Q_{11}} = -\epsilon
\end{displaymath}">|; 

$key = q/{displaymath}frac{a+d}{2}=mbox{cosh}varphi{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="39" BORDER="0"
 SRC="|."$dir".q|img368.png"
 ALT="\begin{displaymath}
\frac{a + d}{2} = \mbox{cosh}  \varphi
\end{displaymath}">|; 

$key = q/{eqnarraystar}T^{-1}T_{1}&=&left[array{{cc}0&1-1&frac{lambda-a_{0}}{a_{1}}array{array{{cc}1&0frac{a-a_{0}}{a_{1}}&1array{right]{eqnarraystar};MSF=1.6;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="412" HEIGHT="66" BORDER="0"
 SRC="|."$dir".q|img220.png"
 ALT="\begin{eqnarray*}
T^{-1} T_{1} &amp; = &amp; \left [ \begin{array}{cc}
0 &amp; 1 \\\\
\ ...
...&amp; 0 \\\\
\\\\
\frac{a - a_{0}}{a_{1}} &amp; 1
\end{array} \right]
\end{eqnarray*}">|; 

$key = q/{displaymath}left[array{{cccc}frac{-2k}{M}&frac{k}{sqrt{Mm}}&0&&frac{k}{sqrt{Mm}ac{-2k}{m}&frac{k}{sqrt{Mm}}&&0&ddots&ddotsarray{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="121" BORDER="0"
 SRC="|."$dir".q|img296.png"
 ALT="\begin{displaymath}
\left[ \begin{array}{cccc}
\frac{-2k}{M} &amp; \frac{k}{\sqrt{...
...{Mm}} \\\\
&amp; &amp; \\\\
0 &amp; \ddots &amp; \ddots
\end{array} \right]
\end{displaymath}">|; 

$key = q/U;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img400.png"
 ALT="$U$">|; 

$key = q/mbox{cosh}theta;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img463.png"
 ALT="$\mbox{cosh}  \theta$">|; 

$key = q/{displaymath}leftarrowvertarray{{cc}a-mu&bc&d-muarray{rightarrowvert=0{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img364.png"
 ALT="\begin{displaymath}
\left \arrowvert \begin{array}{cc}
a - \mu &amp; b \\\\
c &amp; d - \mu
\end{array} \right \arrowvert = 0
\end{displaymath}">|; 

$key = q/{displaymath}T_{i+1}T_{i}=left(left[array{{ccc}frac{lambda-a_{0}}{a_{1}}&-1&1&0aon}{1+epsilon}left[array{{ccc}1&0&1&0array{right]right){displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="624" HEIGHT="66" BORDER="0"
 SRC="|."$dir".q|img275.png"
 ALT="\begin{displaymath}
T_{i + 1}   T_{i} = \left(\left[ \begin{array}{ccc}
\fra...
...}{ccc}
1 &amp; 0 \\\\
&amp; \\\\
1 &amp; 0
\end{array} \right]\right)
\end{displaymath}">|; 

$key = q/{eqnarraystar}mbox{cosh}varphi&=&mbox{cosh}frac{ellpii}{N+1}&=&mbox{cos}frac{ellpi}{N+1}{eqnarraystar};MSF=1.6;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="81" BORDER="0"
 SRC="|."$dir".q|img192.png"
 ALT="\begin{eqnarray*}
\mbox{cosh} \varphi &amp; = &amp; \mbox{cosh} \frac{\ell \pi  i}{N + 1} \\\\
&amp; = &amp; \mbox{cos} \frac{\ell  \pi}{N + 1}
\end{eqnarray*}">|; 

$key = q/widehat{U}widehat{U};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img453.png"
 ALT="$\widehat{U}   \widehat{U}$">|; 

$key = q/2;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img418.png"
 ALT="$2$">|; 

$key = q/{displaymath}T_{i+1}T_{i}=left[array{{ccc}frac{lambda-a_{i+1,i+1}}{a_{i+1,i+2}}&}{a_{i,i+1}}&-frac{a_{i,i-1}}{a_{i,i+1}}1&0array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="390" HEIGHT="69" BORDER="0"
 SRC="|."$dir".q|img118.png"
 ALT="\begin{displaymath}
T_{i + 1} T_{i} = \left [ \begin{array}{ccc}
\frac{\lambda...
...- 1}}{a_{i, i + 1}} \\\\
\\\\
1 &amp; 0 \\\\
\end{array} \right]
\end{displaymath}">|; 

$key = q/I;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img413.png"
 ALT="$I$">|; 

$key = q/{eqnarraystar}e^{varphiwidehat{U}}e^{psiwidehat{V}}&=&I[mbox{cos}hvarphimbox{cosvarphimbox{tanh}psi(widehat{U}timeswidehat{V})]{eqnarraystar};MSF=1.6;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="580" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img424.png"
 ALT="\begin{eqnarray*}
e^{\varphi  \widehat{U}}  e^{\psi  \widehat{V}} &amp;=&amp;
I[\m...
...arphi  \mbox{tanh}  \psi \
(\widehat{U} \times \widehat{V})]
\end{eqnarray*}">|; 

$key = q/{displaymath}left[array{{ccc}frac{lambda-a_0}{a_1}&-1&1&0array{right]^{N-L-1}=frsenh}(N-L-1)varphi&-mbox{senh}(N-L-2)varphiarray{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="552" HEIGHT="70" BORDER="0"
 SRC="|."$dir".q|img244.png"
 ALT="\begin{displaymath}
\left[ \begin{array}{ccc}
\frac{\lambda - a_0}{a_1} &amp; -1 ...
...hi &amp; -\mbox{senh}  (N - L - 2)
\varphi
\end{array} \right]
\end{displaymath}">|; 

$key = q/kslash2;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img263.png"
 ALT="$k/2$">|; 

$key = q/a_{NN};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img95.png"
 ALT="$a_{NN}$">|; 

$key = q/{displaymath}M^{-1}KX=lambdaX{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="24" BORDER="0"
 SRC="|."$dir".q|img30.png"
 ALT="\begin{displaymath}
M^{-1} K X = \lambda X
\end{displaymath}">|; 

$key = q/{displaymath}xi^{2}+2Pxi+1=0{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="27" BORDER="0"
 SRC="|."$dir".q|img328.png"
 ALT="\begin{displaymath}
\xi^{2} + 2 P \xi + 1 = 0
\end{displaymath}">|; 

$key = q/{displaymath}I=left[array{{ccc}1&00&1array{right]J=left[array{{ccc}0&11&0array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="227" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img391.png"
 ALT="\begin{displaymath}
I = \left[\begin{array}{ccc}
1 &amp; 0 \\\\
0 &amp; 1
\end{array}...
...eft[\begin{array}{ccc}
0 &amp; 1 \\\\
1 &amp; 0
\end{array} \right]
\end{displaymath}">|; 

$key = q/includegraphics[width=4in,angle=-90]{f16p78.eps};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="462" HEIGHT="2" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img350.png"
 ALT="\includegraphics[width=4in,angle=-90]{f16p78.eps}">|; 

$key = q/{displaymath}y_{i+1}=frac{lambda-a_{ii}}{a_{i,i+1}}y_{i};-;frac{a_{i,i-1}}{a_{i,i+1}}y_{i-1}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="222" HEIGHT="44" BORDER="0"
 SRC="|."$dir".q|img103.png"
 ALT="\begin{displaymath}
y_{i + 1} = \frac{\lambda - a_{i i}}{a_{i, i + 1}}  y_{i} \; - \;
\frac{a_{i, i -1}}{a_{i, i + 1}}  y_{i - 1}
\end{displaymath}">|; 

$key = q/ax_3;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img479.png"
 ALT="$a  x_3$">|; 

$key = q/y_{1}=0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img129.png"
 ALT="$y_{1} = 0$">|; 

$key = q/M;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img22.png"
 ALT="$M$">|; 

$key = q/ell=N+1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img197.png"
 ALT="$\ell = N + 1$">|; 

$key = q/K;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img24.png"
 ALT="$K$">|; 

$key = q/includegraphics[width=3.7in]{f19p82.eps};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="425" HEIGHT="732" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img354.png"
 ALT="\includegraphics[width=3.7in]{f19p82.eps}">|; 

$key = q/T^{N}T^{-1}T_{1};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img222.png"
 ALT="$T^{N} T^{-1}
T_{1}$">|; 

$key = q/{displaymath}A=frac{k}{m}left[array{{cccccc}a_{1}&1&&&1&-2&1&0&&1&-2&1&0&ddots&ddots&ddots&1&&&1&a_{N}array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="245" HEIGHT="113" BORDER="0"
 SRC="|."$dir".q|img141.png"
 ALT="\begin{displaymath}
A = \frac{k}{m} \left[ \begin{array}{cccccc}
a_{1} &amp; 1 &amp; &amp;...
...ddots &amp; \ddots &amp; 1 \\\\
&amp; &amp; &amp; 1 &amp; a_{N}
\end{array} \right]
\end{displaymath}">|; 

$key = q/mslashk;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img259.png"
 ALT="$m/k$">|; 

$key = q/ell=0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img196.png"
 ALT="$\ell = 0$">|; 

$key = q/i;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="10" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5.png"
 ALT="$i$">|; 

$key = q/I,J,K;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img404.png"
 ALT="$I, J, K$">|; 

$key = q/Krightarrowinfty;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img269.png"
 ALT="$K\rightarrow
\infty$">|; 

$key = q/{displaymath}widehat{W}=w_{1}:J+w_{2}:K+w_{3}:L{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="27" BORDER="0"
 SRC="|."$dir".q|img495.png"
 ALT="\begin{displaymath}
\widehat{W} = w_{1} \: J + w_{2} \: K + w_{3} \: L
\end{displaymath}">|; 

$key = q/includegraphics[width=3.7in]{f13p73.eps};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="425" HEIGHT="711" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img345.png"
 ALT="\includegraphics[width=3.7in]{f13p73.eps}">|; 

$key = q/{displaymath}left[array{{ccc}frac{lambda+2a}{a}&-11&0array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="64" BORDER="0"
 SRC="|."$dir".q|img235.png"
 ALT="\begin{displaymath}
\left[ \begin{array}{ccc}
\frac{\lambda + 2a}{a} &amp; -1 \\\\
\\\\
1 &amp; 0
\end{array} \right]
\end{displaymath}">|; 

$key = q/includegraphics[width=4.4in]{f07p57.eps};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="506" HEIGHT="324" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img288.png"
 ALT="\includegraphics[width=4.4in]{f07p57.eps}">|; 

$key = q/{displaymath}U=frac{(b+c)}{2}J+frac{(a-d)}{2}K+frac{(b-c)}{2}L=left[array{{ccc}frac{a-d}{2}&bc&frac{d-a}{2}array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="399" HEIGHT="65" BORDER="0"
 SRC="|."$dir".q|img394.png"
 ALT="\begin{displaymath}
U = \frac{(b + c)}{2}  J + \frac{(a - d)}{2}  K + \frac{(...
...d}{2} &amp; b \\\\
\\\\
c &amp; \frac{d - a}{2}
\end{array} \right]
\end{displaymath}">|; 

$key = q/widehat{U};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img421.png"
 ALT="$\widehat{U}$">|; 

$key = q/M^{-1slash2}K;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img38.png"
 ALT="$M^{-1/2} K$">|; 

$key = q/x;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img265.png"
 ALT="$x$">|; 

$key = q/T_{i+1};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img117.png"
 ALT="$T_{i+1}$">|; 

$key = q/3;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img346.png"
 ALT="$3$">|; 

$key = q/{displaymath}R_{1}=S_{2}:S_{1}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="27" BORDER="0"
 SRC="|."$dir".q|img444.png"
 ALT="\begin{displaymath}
R_{1} = S_{2} \: S_{1}
\end{displaymath}">|; 

$key = q/{displaymath}[widehat{U},Nwidehat{V}]=mbox{cosh}theta{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img487.png"
 ALT="\begin{displaymath}[\widehat{U}, N  \widehat{V}]= \mbox{cosh}  \theta
\end{displaymath}">|; 

$key = q/{displaymath}nu_{pm}upsilon_{pm}=frac{c}{b}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="35" BORDER="0"
 SRC="|."$dir".q|img381.png"
 ALT="\begin{displaymath}
\nu_{\pm} \upsilon_{\pm} = \frac{c}{b}
\end{displaymath}">|; 

$key = q/Z_{n};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img122.png"
 ALT="$Z_{n}$">|; 

$key = q/j;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img8.png"
 ALT="$j$">|; 

$key = q/{displaymath}y_{j}=frac{mbox{sen}:jell:pislashN+1}{mbox{sen}ell:pi:slashN+1}:y_{1}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="44" BORDER="0"
 SRC="|."$dir".q|img206.png"
 ALT="\begin{displaymath}
y_{j} = \frac{\mbox{sen}\: j \ell \: \pi  / N + 1}{\mbox{sen
}\ell \: \pi \: / N + 1} \:y_{1}
\end{displaymath}">|; 

$key = q/{displaymath}P=1{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="24" BORDER="0"
 SRC="|."$dir".q|img338.png"
 ALT="\begin{displaymath}
P = 1
\end{displaymath}">|; 

$key = q/{displaymath}mu^{2}+(2-4xi^{2}-8Pxi-4)mu+1=0{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="240" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img320.png"
 ALT="\begin{displaymath}
\mu^{2} + (2 - 4 \xi^{2} - 8 P \xi - 4) \mu + 1 = 0
\end{displaymath}">|; 

$key = q/{displaymath}tau=T^{N}T^{-1}T_{1}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="27" BORDER="0"
 SRC="|."$dir".q|img228.png"
 ALT="\begin{displaymath}
\tau = T^{N} T^{-1} T_{1}
\end{displaymath}">|; 

$key = q/{displaymath}lambda;y_{i}=a_{i,i-1}y_{i-1};+;a_{ii}y_{i};+;a_{i,i+1}y_{i+1}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="284" HEIGHT="29" BORDER="0"
 SRC="|."$dir".q|img101.png"
 ALT="\begin{displaymath}
\lambda \; y_{i} = a_{i, i -1}  y_{i - 1} \; + \; a_{i i} \
y_{i} \; + \; a_{i, i + 1}  y_{i + 1}
\end{displaymath}">|; 

$key = q/{displaymath}widetilde{W}=frac{widetilde{U}+widetilde{V}+widetilde{U}timeswidetilde{V}}{1+(widetilde{U},widetilde{V})}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img434.png"
 ALT="\begin{displaymath}
\widetilde{W} = \frac{\widetilde{U} + \widetilde{V} +
\wid...
...U} \times \widetilde{V}}{1 + (\widetilde{U},
\widetilde{V})}
\end{displaymath}">|; 

$key = q/{displaymath}y_{j}=frac{mbox{senh}(jellpiislashN+1)}{mbox{senh}(ellpiislashN+1)}:y_{1}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="44" BORDER="0"
 SRC="|."$dir".q|img205.png"
 ALT="\begin{displaymath}
y_{j} = \frac{\mbox{senh} (j \ell  \pi  i / N + 1)}
{\mbox{senh} (\ell  \pi  i / N + 1)} \:y_{1}
\end{displaymath}">|; 

$key = q/{displaymath}y_{0}=y_{N+1}=0{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img110.png"
 ALT="\begin{displaymath}
y_{0} = y_{N + 1} = 0
\end{displaymath}">|; 

$key = q/lambda;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1.png"
 ALT="$\lambda $">|; 

$key = q/rho;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img304.png"
 ALT="$\rho$">|; 

$key = q/{displaymath}left[array{{cc}a&bc&darray{right]left[array{{cc}1nuarray{right]=mu_pmleft[array{{c}1nuarray{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img369.png"
 ALT="\begin{displaymath}
\left[ \begin{array}{cc}
a &amp; b \\\\
c &amp; d
\end{array} \r...
...pm
\left[ \begin{array}{c}
1 \\\\
\nu
\end{array} \right]
\end{displaymath}">|; 

$key = q/{displaymath}left[array{{cc}tau_{11}&tau_{12}tau_{21}&tau_{22}array{right]=mbox{cosh}thetaI+mbox{senh}thetawidehat{W}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="250" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img494.png"
 ALT="\begin{displaymath}
\left[ \begin{array}{cc}
\tau_{11} &amp; \tau_{12} \\\\
\tau_{...
...\mbox{cosh}  \theta  I + \mbox{senh}  \theta  \widehat{W}
\end{displaymath}">|; 

$key = q/{displaymath}x^{2}_{1}(w^{2}_{3}+w^{2}_{1})+x^{2}_{2}(w^{2}_{3}-w^{2}_{2})-2w_{1}w_{2}x_{1}x_{2}-w^{2}_{3}=0{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="356" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img483.png"
 ALT="\begin{displaymath}
x^{2}_{1} (w^{2}_{3} + w^{2}_{1}) + x^{2}_{2} (w^{2}_{3} -
w^{2}_{2}) - 2 w_{1} w_{2} x_{1} x_{2} - w^{2}_{3} = 0
\end{displaymath}">|; 

$key = q/epsilonrightarrow0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img293.png"
 ALT="$\epsilon\rightarrow 0$">|; 

$key = q/displaystylefrac{-2k}{m};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img148.png"
 ALT="$\displaystyle \frac{-2k}{m}$">|; 

$key = q/T_i;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img116.png"
 ALT="$T_i$">|; 

$key = q/{displaymath}mu^{2}-(a+d)mu+(ad-bc)=0{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="205" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img365.png"
 ALT="\begin{displaymath}
\mu^{2} - (a + d) \mu + (ad - bc) = 0
\end{displaymath}">|; 

$key = q/k;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img13.png"
 ALT="$k$">|; 

$key = q/tau;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img61.png"
 ALT="$\tau$">|; 

$key = q/{displaymath}nu_{pm}+upsilon_{pm}=frac{mu_{pm}-a}{b}+frac{mu_{pm}-d}{b}=frac{2mu_{pm}-(a+d)}{b}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="322" HEIGHT="40" BORDER="0"
 SRC="|."$dir".q|img374.png"
 ALT="\begin{displaymath}
\nu_{\pm} + \upsilon_{\pm} = \frac{\mu_{\pm} - a}{b} +
\frac{\mu_{\pm} - d}{b} = \frac{2\mu_{\pm} - (a + d)}{b}
\end{displaymath}">|; 

$key = q/{displaymath}mbox{senh}(N+1)varphi=0{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img189.png"
 ALT="\begin{displaymath}
\mbox{senh} (N + 1) \varphi = 0
\end{displaymath}">|; 

$key = q/{displaymath}L_{n}T=frac{varphi}{2mbox{senh}varphi}left[array{{ccc}mu_{+}+mu_{-}-2d&2b2c&mu_{+}+mu_{-}-2aarray{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="350" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img387.png"
 ALT="\begin{displaymath}
L_{n} T = \frac{\varphi}{2\mbox{senh}  \varphi} \left[
\b...
...2d &amp; 2b \\\\
2c &amp; \mu_{+} + \mu_{-} - 2a
\end{array} \right]
\end{displaymath}">|; 

$key = q/{displaymath}upsilon_pm=frac{mu_pm-alpha}{b}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="37" BORDER="0"
 SRC="|."$dir".q|img372.png"
 ALT="\begin{displaymath}
\upsilon_\pm = \frac{\mu_\pm - \alpha}{b}
\end{displaymath}">|; 

$key = q/{displaymath}left[array{{cc}upsilon&1array{right]left[array{{ccccccc}a&bc&darray{right]=mu_pmleft[array{{cc}upsilon&1array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="231" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img371.png"
 ALT="\begin{displaymath}
\left[ \begin{array}{cc}
\upsilon &amp; 1
\end{array} \right...
...pm \left[ \begin{array}{cc} \upsilon &amp; 1
\end{array} \right]
\end{displaymath}">|; 

$key = q/x_{-1}cdots;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img18.png"
 ALT="$x_{-1} \cdots$">|; 

$key = q/{displaymath}A=a_{0}I+a_{1}J+a_{2}K+a_{3}L{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="198" HEIGHT="27" BORDER="0"
 SRC="|."$dir".q|img407.png"
 ALT="\begin{displaymath}
A = a_{0} I + a_{1} J + a_{2} K + a_{3} L
\end{displaymath}">|; 

$key = q/a_{1};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img142.png"
 ALT="$a_{1}$">|; 

$key = q/{displaymath}Z_{0}=left[array{{cccc}x_{k}x_{k-1}vdotsx_{1}0vdots0array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="161" BORDER="0"
 SRC="|."$dir".q|img69.png"
 ALT="\begin{displaymath}
Z_{0} = \left[ \begin{array}{cccc}
x_{k} \\\\
x_{k - 1} \\\\...
...ots \\\\
x_{1} \\\\
0 \\\\
\vdots \\\\
0
\end{array} \right]
\end{displaymath}">|; 

$key = q/{displaymath}N=left[array{{cccc}1&0&00&1&00&0&-1array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="64" BORDER="0"
 SRC="|."$dir".q|img475.png"
 ALT="\begin{displaymath}
N = \left[ \begin{array}{cccc}
1 &amp; 0 &amp; 0 \\\\
0 &amp; 1 &amp; 0 \\\\
0 &amp; 0 &amp; -1
\end{array} \right]
\end{displaymath}">|; 

$key = q/varphi=pii;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img199.png"
 ALT="$\varphi = \pi  i$">|; 

$key = q/includegraphics[width=4in,angle=-90]{f18p81.eps};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="462" HEIGHT="2" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img353.png"
 ALT="\includegraphics[width=4in,angle=-90]{f18p81.eps}">|; 

$key = q/mu<0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img339.png"
 ALT="$\mu &lt; 0$">|; 

$key = q/LnT;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img385.png"
 ALT="$LnT$">|; 

$key = q/1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img57.png"
 ALT="$1$">|; 

$key = q/T_{N}T^{-1};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img219.png"
 ALT="$ T_{N} T^{-1}$">|; 

$key = q/{displaymath}array{{ccccc}a_{1}&=&a_{N}&=&-2mbox{extremosfijos}a_{1}&=&a_{N}&=&-olibre}a_{1}&=&-2a_{N}&=&-1mbox{yelotroconfinado}array{{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="333" HEIGHT="178" BORDER="0"
 SRC="|."$dir".q|img144.png"
 ALT="\begin{displaymath}
\begin{array}{ccccc}
a_{1} &amp; = &amp; a_{N} &amp; = &amp; -2      ...
...} &amp; = &amp; -1      \mbox{y el otro
confinado}
\end{array}
\end{displaymath}">|; 

$key = q/y_{1}neq0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img125.png"
 ALT="$y_{1}
\neq 0$">|; 

$key = q/{displaymath}T=frac{mu_{+}}{2mbox{senh}varphi}left[array{{ccc}mu_{+}-d&bc&mu_{+}varphi}left[array{{ccc}mu_{-}-d&bc&mu_{-}-aarray{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="468" HEIGHT="64" BORDER="0"
 SRC="|."$dir".q|img384.png"
 ALT="\begin{displaymath}
T = \frac{\mu_{+}}{2\mbox{senh}  \varphi} \left[
\begin{a...
...mu_{-} - d &amp; b \\\\
\\\\
c &amp; \mu_{-}- a
\end{array} \right]
\end{displaymath}">|; 

$key = q/displaystyleZ_{N}=T_{N}T_{N-1}dots;T_{1}Z_{0}=tauZ_{0};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="218" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img60.png"
 ALT="$\displaystyle Z_{N} = T_{N} T_{N-1} \dots \; T_{1} Z_{0} = \tau Z_{0}$">|; 

$key = q/tau_{11};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img113.png"
 ALT="$\tau_{11}$">|; 

$key = q/includegraphics[width=4in,angle=-90]{f22p87.eps};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="462" HEIGHT="2" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img358.png"
 ALT="\includegraphics[width=4in,angle=-90]{f22p87.eps}">|; 

$key = q/{displaymath}x_{ell}=0eqno{(mbox{I.11a})}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="298" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img44.png"
 ALT="\begin{displaymath}
x_{\ell} = 0 \eqno{(\mbox{I.11a})} \end{displaymath}">|; 

$key = q/{displaymath}tau=T_{N}T^{-1}T^{N}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="27" BORDER="0"
 SRC="|."$dir".q|img231.png"
 ALT="\begin{displaymath}
\tau = T_{N} T^{-1} T^{N}
\end{displaymath}">|; 

$key = q/{displaymath}nu_{pm}upsilon_{pm}=frac{(mu_{pm}-a)(mu_{pm}-d)}{b^{2}}=frac{mu_{pm}^{2}-(a+d)mu_{pm}+ad}{b^{2}}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="353" HEIGHT="42" BORDER="0"
 SRC="|."$dir".q|img378.png"
 ALT="\begin{displaymath}
\nu_{\pm} \upsilon_{\pm} = \frac{(\mu_{\pm} - a)(\mu_{\pm} ...
...b^{2}} = \frac{\mu_{\pm}^{2} - (a + d) \mu_{\pm} + ad}{b^{2}}
\end{displaymath}">|; 

$key = q/{displaymath}alpha=2(xi+rho)qquadqquadbeta=2(xi+frac{1}{rho}){displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="241" HEIGHT="42" BORDER="0"
 SRC="|."$dir".q|img305.png"
 ALT="\begin{displaymath}
\alpha = 2(\xi + \rho) \qquad \qquad \beta = 2(\xi +
\frac{1}{\rho})
\end{displaymath}">|; 

$key = q/includegraphics[width=4in,angle=-90]{f20p84.eps};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="462" HEIGHT="2" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img356.png"
 ALT="\includegraphics[width=4in,angle=-90]{f20p84.eps}">|; 

$key = q/{displaymath}mbox{senh}varphi=sqrt{left(frac{b+c}{2}right)^{2}+left(frac{a-d}{2}right)^{2}-left(frac{b-c}{2}right)^{2}}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="329" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img399.png"
 ALT="\begin{displaymath}
\mbox{senh}  \varphi = \sqrt{\left(\frac{b + c}{2}\right)^...
...rac{a - d}{2}\right)^{2} - \left(\frac{b -
c}{2}\right)^{2}}
\end{displaymath}">|; 

$key = q/{displaymath}R_{2}=S_{3}:S_{2}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="27" BORDER="0"
 SRC="|."$dir".q|img445.png"
 ALT="\begin{displaymath}
R_{2} = S_{3} \: S_{2}
\end{displaymath}">|; 

$key = q/{displaymath}frac{lambda-a_{0}}{2a_{1}}=mbox{cosh}varphi{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="42" BORDER="0"
 SRC="|."$dir".q|img243.png"
 ALT="\begin{displaymath}
\frac{\lambda - a_{0}}{2a_{1}} = \mbox{cosh}  \varphi
\end{displaymath}">|; 

$key = q/{displaymath}widehat{U}widehat{U}=(widehat{U},widehat{U})I+widehat{U}timeswidehat{U}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img455.png"
 ALT="\begin{displaymath}
\widehat{U}  \widehat{U} = (\widehat{U},  \widehat{U})  I +
\widehat{U} \times \widehat{U}
\end{displaymath}">|; 

$key = q/{displaymath}a=frac{k}{M}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="39" BORDER="0"
 SRC="|."$dir".q|img234.png"
 ALT="\begin{displaymath}
a = \frac{k}{M}
\end{displaymath}">|; 

$key = q/{displaymath}xi=frac{lambdasqrt{Mm}}{2k}rho=frac{sqrt{M}}{sqrt{m}}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="191" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img300.png"
 ALT="\begin{displaymath}
\xi = \frac{\lambda \sqrt{Mm}}{2k}         \rho =
\frac{\sqrt{M}}{\sqrt{m}}
\end{displaymath}">|; 

$key = q/{displaymath}T_{i}=left[array{{cc}frac{lambda-a_{ii}}{a_{i,i+1}}&-frac{a_{i,i-1}}{a_{i,i+1}}1&0array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img106.png"
 ALT="\begin{displaymath}
T_{i} = \left[ \begin{array}{cc}
\frac{\lambda - a_{i i}}{...
...ac{a_{i, i -1}}{a_{i, i + 1}} \\\\
1 &amp; 0
\end{array} \right]
\end{displaymath}">|; 

$key = q/{displaymath}T_{D}=left[array{{cc}alpha&-1&1&0array{right]left[array{{cc}beta&-1&1&0array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="207" HEIGHT="64" BORDER="0"
 SRC="|."$dir".q|img306.png"
 ALT="\begin{displaymath}
T_{D} = \left[ \begin{array}{cc}
\alpha &amp; -1 \\\\
&amp; \\\\
...
...ray}{cc}
\beta &amp; -1 \\\\
&amp; \\\\
1 &amp; 0
\end{array} \right]
\end{displaymath}">|; 

$key = q/{eqnarraystar}arrowvertu_pm><v_pmarrowvert&=&left[array{{c}1u_pmarray{right]leftleft[array{{ccc}v_pm&1v_pmu_pm&u_pmarray{right]{eqnarraystar};MSF=1.6;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="248" HEIGHT="89" BORDER="0"
 SRC="|."$dir".q|img177.png"
 ALT="\begin{eqnarray*}
\arrowvert u_\pm &gt;&lt; v_\pm \arrowvert &amp;=&amp; \left [
\begin{arra...
...ccc}
v_\pm &amp; 1 \\\\
v_\pm  u_\pm &amp; u_\pm
\end{array} \right]
\end{eqnarray*}">|; 

$key = q/{displaymath}mu^{2}-(4xi^{2}+8Pxi+2)mu+1=0eqno{(mbox{III.42'})}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="382" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img321.png"
 ALT="\begin{displaymath}
\mu^{2} - (4 \xi^{2} + 8 P \xi + 2)\mu + 1 = 0
\eqno{(\mbox{III.42'})}
\end{displaymath}">|; 

$key = q/J,K,L;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img435.png"
 ALT="$J,K,L$">|; 

$key = q/{eqnarraystar}tau&=&left[array{{cc}1&-frac{a-a_{0}}{a_{1}}0&1array{right]left[ar}(N-1)varphi&-mbox{senh}(N-1)varphiarray{right]{eqnarraystar};MSF=1.6;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="629" HEIGHT="190" BORDER="0"
 SRC="|."$dir".q|img224.png"
 ALT="\begin{eqnarray*}
\tau &amp; = &amp; \left [\begin{array}{cc}
1 &amp; - \frac{a - a_{0}}{a...
...
1) \varphi &amp; -\mbox{senh} (N - 1)\varphi
\end{array} \right]
\end{eqnarray*}">|; 

$key = q/{displaymath}P_{pm}=pmfrac{b}{2mbox{senh}varphi}left[array{{ccc}frac{mu_{pm}-d}{b}&1frac{c}{b}&frac{mu_{pm}-a}{b}array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="240" HEIGHT="68" BORDER="0"
 SRC="|."$dir".q|img382.png"
 ALT="\begin{displaymath}
P_{\pm} = \pm \frac{b}{2\mbox{senh}  \varphi} \left[
\beg...
...\
\frac{c}{b} &amp; \frac{\mu_{\pm}- a}{b}
\end{array} \right]
\end{displaymath}">|; 

$key = q/{displaymath}S_{2}:S_{2}=S_{2}^{2}=I{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img447.png"
 ALT="\begin{displaymath}
S_{2} \: S_{2} = S_{2}^{2} = I
\end{displaymath}">|; 

$key = q/{displaymath}tau=left[array{{cc}frac{lambda-a_0}{a_1}&-1&1&0array{right]^{N}+var[array{{cc}frac{lambda-a_0}{a_1}&-1&1&0array{right]^{L}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="524" HEIGHT="70" BORDER="0"
 SRC="|."$dir".q|img242.png"
 ALT="\begin{displaymath}
\tau = \left[ \begin{array}{cc}
\frac{\lambda - a_0}{a_1} ...
...- a_0}{a_1} &amp; -1 \\\\
&amp; \\\\
1 &amp; 0
\end{array} \right]^{L}
\end{displaymath}">|; 

$key = q/varphineqinfty;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img226.png"
 ALT="$\varphi \neq \infty$">|; 

$key = q/N;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3.png"
 ALT="$N$">|; 

$key = q/{displaymath}x^{2}_{1}(w^{2}_{1}+w^{2}_{2})-x^{2}_{3}(w^{2}_{2}-w^{2}_{3})-2w_{1}w_{3}x_{1}x_{3}-w^{2}_{2}=0{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="356" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img485.png"
 ALT="\begin{displaymath}
x^{2}_{1} (w^{2}_{1}+w^{2}_{2}) - x^{2}_{3} (w^{2}_{2} -
w^{2}_{3}) - 2 w_{1} w_{3} x_{1}x_{3} - w^{2}_{2} = 0
\end{displaymath}">|; 

$key = q/{displaymath}left[array{{cc}tau_{11}&tau_{12}tau_{21}&tau_{22}array{right]=mbox{cosh}:theta:I+(w_{1}:J+w_{2}:K+w_{3}L):mbox{senh}:theta{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="380" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img496.png"
 ALT="\begin{displaymath}
\left[ \begin{array}{cc}
\tau_{11} &amp; \tau_{12} \\\\
\tau_{...
... (w_{1} \:J + w_{2}\: K + w_{3}  L)\:
\mbox{senh} \: \theta
\end{displaymath}">|; 

$key = q/{displaymath}mu^{2}+(2-alphabeta)mu+1=0{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img314.png"
 ALT="\begin{displaymath}
\mu^{2} + (2 - \alpha \beta)\mu + 1 = 0
\end{displaymath}">|; 

$key = q/{displaymath}left[array{{ccc}y_{1}0array{right]=left[array{{ccc}tau^{(i)^{-1}}_{array{{ccc}y_{i+1}y_{i}array{right]eqno{(mbox{II.24'})}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="424" HEIGHT="74" BORDER="0"
 SRC="|."$dir".q|img134.png"
 ALT="\begin{displaymath}
\left [ \begin{array}{ccc}
y_{1} \\\\
\\\\
0
\end{array...
...} \\\\
\\\\
y_{i}
\end{array} \right]\eqno{(\mbox{II.24'})}
\end{displaymath}">|; 

$key = q/{displaymath}Z_{i}=T_{i}T_{i-1}dotsT_{1}Z_{0}=tau^{(i)}Z_{0}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="202" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img126.png"
 ALT="\begin{displaymath}
Z_{i} = T_{i} T_{i - 1} \dots T_{1} Z_{0} = \tau^{(i)} Z_{0}
\end{displaymath}">|; 

$key = q/x_2;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img489.png"
 ALT="$x_2$">|; 

$key = q/{displaymath}frac{k_{ij}}{sqrt{m_{i}m_{j}}}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img92.png"
 ALT="\begin{displaymath}
\frac{k_{i j}}{\sqrt{m_{i}  m_{j}}}
\end{displaymath}">|; 

$key = q/includegraphics[width=4in,angle=-90]{f05p50.eps};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="462" HEIGHT="2" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img264.png"
 ALT="\includegraphics[width=4in,angle=-90]{f05p50.eps}">|; 

$key = q/T^{N};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img217.png"
 ALT="$T^{N}$">|; 

$key = q/{displaymath}v_pm=-e^{pm:varphi}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img175.png"
 ALT="\begin{displaymath}
v_\pm = -e^{\pm \: \varphi}
\end{displaymath}">|; 

$key = q/{displaymath}k_{ij}=left{array{{ccc}k&qquad&ineqj-2k&qquad&i=jarray{right.{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img140.png"
 ALT="\begin{displaymath}
k_{i j} = \left\{
\begin{array}{ccc}
k &amp; \qquad &amp; i \neq j \\\\
-2k &amp; \qquad &amp; i = j
\end{array} \right.
\end{displaymath}">|; 

$key = q/{displaymath}frac{m_{i}}{m_i+1}=epsilon{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="37" BORDER="0"
 SRC="|."$dir".q|img355.png"
 ALT="\begin{displaymath}
\frac{m_{i}}{m_i + 1} = \epsilon
\end{displaymath}">|; 

$key = q/{eqnarraystar}e^{varphiwidehat{u}}&=&mbox{cosh}varphi(I+mbox{tanh}varphiwidehat{t{V}}&=&mbox{cosh}psi(I+mbox{tan}psiwidehat{V}){eqnarraystar};MSF=1.6;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="57" BORDER="0"
 SRC="|."$dir".q|img419.png"
 ALT="\begin{eqnarray*}
e^{\varphi  \widehat{u}} &amp; = &amp; \mbox{cosh}  \varphi  ( I +...
...} &amp; = &amp; \mbox{cosh}  \psi  (I + \mbox{tan}
\psi \widehat{V})
\end{eqnarray*}">|; 

$key = q/varphi=alpha+ipi;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img167.png"
 ALT="$\varphi = \alpha + i \pi$">|; 

$key = q/{displaymath}c=frac{lambda-a_{0}}{a_{1}}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="42" BORDER="0"
 SRC="|."$dir".q|img152.png"
 ALT="\begin{displaymath}
c = \frac{\lambda - a_{0}}{a_{1}}
\end{displaymath}">|; 

$key = q/{displaymath}a_{11}=-a_{12}-a_{10}a_{NN}=-a_{N,N-1}-a_{N,N+1}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="314" HEIGHT="29" BORDER="0"
 SRC="|."$dir".q|img96.png"
 ALT="\begin{displaymath}
a_{11} = -a_{12} -a_{10}   a_{NN} = -a_{N,N-1} -a_{N,N+1}
\end{displaymath}">|; 

$key = q/widetilde{U};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img437.png"
 ALT="$\widetilde{U}$">|; 

$key = q/widehat{W};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img466.png"
 ALT="$\widehat{W}$">|; 

$key = q/y_{i+1};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img102.png"
 ALT="$y_{i + 1}$">|; 

$key = q/N+K;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img58.png"
 ALT="$N + K$">|; 

$key = q/{displaymath}array{{ccc}tau_{11}x_{k}+tau_{12}x_{k-1}+dots+tau_{1k}x_{1}&=&0tau_{k-1}x_{k}+tau_{k2}x_{k-1}+dots+tau_{kk}x_{1}&=&0array{{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="270" HEIGHT="94" BORDER="0"
 SRC="|."$dir".q|img70.png"
 ALT="\begin{displaymath}
\begin{array}{ccc}
\tau_{11}x_{k} + \tau_{12}x_{k-1} + \do...
...{k2} x_{k-1} + \dots + \tau_{k k} x_{1} &amp; =
&amp; 0
\end{array}
\end{displaymath}">|; 

$key = q/{displaymath}Z_{N}=T_{N}T_{N-1}dotsT_{1}Z_{0}=tauZ_{0}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="208" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img112.png"
 ALT="\begin{displaymath}
Z_{N} = T_{N} T_{N-1} \dots T_{1} Z_{0} = \tau Z_{0}
\end{displaymath}">|; 

$key = q/displaystylea_{1};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img149.png"
 ALT="$\displaystyle a_{1}$">|; 

$key = q/{displaymath}T_{1}=T_{N}=left[array{{ccc}frac{lambda-a}{a_1}&-11&0array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img209.png"
 ALT="\begin{displaymath}
T_{1} = T_{N} = \left[ \begin{array}{ccc}
\frac{\lambda - a}{a_1} &amp; -1 \\\\
1 &amp; 0
\end{array} \right]
\end{displaymath}">|; 

$key = q/x_{i+k};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img49.png"
 ALT="$x_{i + k}$">|; 

$key = q/{eqnarraystar}Z_{N}&=&T_{N}Z_{N-1}Z_{N-1}&=&T_{N-1}Z_{N-2}vdotsZ_{1}&=&T_{1}Z_{0}{eqnarraystar};MSF=1.6;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="108" BORDER="0"
 SRC="|."$dir".q|img59.png"
 ALT="\begin{eqnarray*}
Z_{N} &amp; = &amp; T_{N} Z_{N-1} \\\\
Z_{N - 1} &amp; = &amp; T_{N - 1} Z_{N-2} \\\\
\vdots \\\\
Z_{1} &amp; = &amp; T_{1} Z_{0}
\end{eqnarray*}">|; 

$key = q/{displaymath}y_{j}=mbox{sen}(jellpislashN+1){displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="29" BORDER="0"
 SRC="|."$dir".q|img208.png"
 ALT="\begin{displaymath}
y_{j} = \mbox{sen} ( j  \ell  \pi / N + 1)
\end{displaymath}">|; 

$key = q/Z_{0};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img65.png"
 ALT="$Z_{0}$">|; 

$key = q/tau^{(N)}_{11}y_{1}=0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img124.png"
 ALT="$\tau^{(N)}_{11}  y_{1} = 0$">|; 

$key = q/{displaymath}B=b_{0}I+b_{1}J+b_{2}K+b_{3}L{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="27" BORDER="0"
 SRC="|."$dir".q|img408.png"
 ALT="\begin{displaymath}
B = b_{0} I + b_{1} J + b_{2} K + b_{3} L
\end{displaymath}">|; 

$key = q/tau_{11}^{N}(lambda);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img137.png"
 ALT="$\tau_{11} ^{N}(\lambda) $">|; 

$key = q/{displaymath}x^{2}_{1}+x^{2}_{2}-x^{2}_{3}=1{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img478.png"
 ALT="\begin{displaymath}
x^{2}_{1} + x^{2}_{2} - x^{2}_{3} = 1
\end{displaymath}">|; 

$key = q/{displaymath}tau_{11}^{N}(lambda)=0{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img136.png"
 ALT="\begin{displaymath}
\tau_{11} ^{N}(\lambda) = 0
\end{displaymath}">|; 

$key = q/J,K;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img503.png"
 ALT="$J,  K$">|; 

$key = q/{displaymath}arrowvertT_{i}arrowvert=frac{a_{i,i-1}}{a_{i,i+1}}neq0mbox{paratoda}i{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="40" BORDER="0"
 SRC="|."$dir".q|img131.png"
 ALT="\begin{displaymath}
\arrowvert T_{i} \arrowvert = \frac{a_{i, i - 1}}{a_{i, i + 1}}
\neq 0        \mbox{para toda}  i
\end{displaymath}">|; 

$key = q/{displaymath}T=left[array{{cc}frac{lambda-a_{0}}{a_{1}}&-11&0array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="66" BORDER="0"
 SRC="|."$dir".q|img214.png"
 ALT="\begin{displaymath}
T = \left [ \begin{array}{cc}
\frac{\lambda - a_{0}}{a_{1}} &amp; -1 \\\\
\\\\
1 &amp; 0
\end{array} \right]
\end{displaymath}">|; 

$key = q/{displaymath}left[array{{cc}frac{m}{k}lambda+frac{1}{2}+1&-frac{1}{2}1&0array{ri]left[array{{cc}frac{2m}{k}lambda+2+1&-21&0array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="320" HEIGHT="64" BORDER="0"
 SRC="|."$dir".q|img262.png"
 ALT="\begin{displaymath}
\left[ \begin{array}{cc}
\frac{m}{k}  \lambda+ \frac{1}...
...}{k}\lambda + 2 +1 &amp; -2 \\\\
\\\\
1 &amp; 0
\end{array} \right]
\end{displaymath}">|; 

$key = q/{displaymath}widetilde{W}=-frac{w_{1}}{w_{2}}:J-1-frac{w_{3}}{w_{2}}:L{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="37" BORDER="0"
 SRC="|."$dir".q|img501.png"
 ALT="\begin{displaymath}
\widetilde{W} = - \frac{w_{1}}{w_{2}}\:J - 1 -
\frac{w_{3}}{w_{2}}\:L
\end{displaymath}">|; 

$key = q/{displaymath}varepsilon=lambda(frac{1}{a}-frac{1}{a_{1}}){displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img241.png"
 ALT="\begin{displaymath}
\varepsilon = \lambda(\frac{1}{a} - \frac{1}{a_{1}})
\end{displaymath}">|; 

$key = q/{displaymath}left[array{{c}y_{i+1}y_{i}array{right]=left[array{{cccc}frac{lambda}}1&0array{right]left[array{{c}y_{i}y_{i-1}array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="277" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img105.png"
 ALT="\begin{displaymath}
\left [ \begin{array}{c}
y_{i + 1} \\\\
y_{i}
\end{array...
...[ \begin{array}{c}
y_{i} \\\\
y_{i - 1}
\end{array} \right]
\end{displaymath}">|; 

$key = q/{displaymath}a+d=e^{varphi}+e^{-varphi}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="25" BORDER="0"
 SRC="|."$dir".q|img375.png"
 ALT="\begin{displaymath}
a + d = e^{\varphi} + e^{-\varphi}
\end{displaymath}">|; 

$key = q/includegraphics[width=4in,angle=-90]{f08p59.eps};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="462" HEIGHT="2" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img292.png"
 ALT="\includegraphics[width=4in,angle=-90]{f08p59.eps}">|; 

$key = q/M=0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img258.png"
 ALT="$M = 0$">|; 

$key = q/{eqnarraystar}T_{N}T^{-1}&=&left[array{{cc}frac{lambda-a}{a_{1}}&-11&0array{righrray{{cc}1&-frac{a-a_{0}}{a_{1}}0&1array{right]{eqnarraystar};MSF=1.6;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="430" HEIGHT="66" BORDER="0"
 SRC="|."$dir".q|img221.png"
 ALT="\begin{eqnarray*}
T_{N} T^{-1} &amp; = &amp; \left [ \begin{array}{cc}
\frac{\lambda -...
... -\frac{a - a_{0}}{a_{1}} \\\\
\\\\
0 &amp; 1
\end{array} \right]
\end{eqnarray*}">|; 

$key = q/{displaymath}mu=mbox{cosh}:varphipmsqrt{mbox{cosh}^2:varphi-1}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="189" HEIGHT="36" BORDER="0"
 SRC="|."$dir".q|img157.png"
 ALT="\begin{displaymath}
\mu = \mbox{cosh} \: \varphi \pm \sqrt{\mbox{cosh}^2 \: \varphi
-1}
\end{displaymath}">|; 

$key = q/{displaymath}P=frac{1}{2}left(frac{sqrt{M}}{sqrt{m}}+frac{sqrt{m}}{sqrt{M}}right)=frac{1}{2}frac{M+m}{sqrt{m}}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="253" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img319.png"
 ALT="\begin{displaymath}
P = \frac{1}{2} \left(\frac{\sqrt{M}}{\sqrt{m}} +
\frac{\sqrt{m}}{\sqrt{M}}\right) = \frac{1}{2}  \frac{M +
m}{\sqrt{m}}
\end{displaymath}">|; 

$key = q/x_{3};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img486.png"
 ALT="$x_{3}$">|; 

$key = q/m_{i};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img9.png"
 ALT="$m_{i}$">|; 

$key = q/{displaymath}frac{d^{2}}{dt^{2}}left[array{{c}y_{1}y_{2}vdotsy_{N}array{right]=lay{right]left[array{{c}y_{1}y_{2}vdotsy_{N}array{right]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="341" HEIGHT="94" BORDER="0"
 SRC="|."$dir".q|img93.png"
 ALT="\begin{displaymath}
\frac{d^{2}}{d t^{2}} \left [ \begin{array}{c}
y_{1} \\\\
...
...
y_{1} \\\\
y_{2} \\\\
\vdots \\\\
y_{N}
\end{array} \right]
\end{displaymath}">|; 

$key = q/{displaymath}tau=T^{N}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="24" BORDER="0"
 SRC="|."$dir".q|img63.png"
 ALT="\begin{displaymath}
\tau = T^{N}
\end{displaymath}">|; 

$key = q/y_{i+1}=y_{i}=0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img130.png"
 ALT="$y_{i + 1} = y_{i}
= 0$">|; 

$key = q/{displaymath}array{{ccl}Z_{N}&=&T_{n}:T^{-1}:T:T^{N-2}:T:T^{-1}:T_{1}:T_{0}&&&=&-1}:T^{N}:T^{-1}:T_{1}:Z_{0}array{eqno{(mbox{III.22}')}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="406" HEIGHT="64" BORDER="0"
 SRC="|."$dir".q|img213.png"
 ALT="\begin{displaymath}
\begin{array}{ccl}
Z_{N} &amp; = &amp; T_{n} \: T^{-1} \: T \: T^{...
...T^{-1} \: T_{1} \: Z_{0}
\end{array} \eqno{(\mbox{III.22}')}
\end{displaymath}">|; 

$key = q/{displaymath}nu_pm=frac{mu_pm-a}{b}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="37" BORDER="0"
 SRC="|."$dir".q|img370.png"
 ALT="\begin{displaymath}
\nu_\pm = \frac{\mu_\pm - a}{b}
\end{displaymath}">|; 

$key = q/{displaymath}m_{i}ddot{x}_{i}=-k_{ii-1}(x_{i}-x_{i-1})-k_{ii+1}(x_{i}-x_{i+1}){displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="307" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img86.png"
 ALT="\begin{displaymath}
m_{i} \ddot{x}_{i} = -k_{i i -1} (x_{i} - x_{i-1}) -k_{i i + 1}
(x_{i} - x_{i + 1})
\end{displaymath}">|; 

$key = q/includegraphics[width=4in,angle=-90]{f14p75.eps};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="462" HEIGHT="2" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img348.png"
 ALT="\includegraphics[width=4in,angle=-90]{f14p75.eps}">|; 

$key = q/{displaymath}widehat{a}timeswidehat{b}:=:(a_{2}b_{3}-a_{3}b_{2})J+(a_{3}b_{1}-a_{1}b_{3})K+(a_{2}b_{1}-a_{1}b_{2})L{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="410" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img417.png"
 ALT="\begin{displaymath}
\widehat{a} \times \widehat{b} \: = \: (a_{2}b_{3} - a_{3}b...
...(a_{3} b_{1} - a_{1} b_{3}) K + (a_{2} b_{1} - a_{1} b_{2}) L
\end{displaymath}">|; 

1;

