# LaTeX2HTML 99.1 release (March 30, 1999)
# Associate images original text with physical files.


$key = q/K^3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img2.gif"
 ALT="$ K^3$">|; 

$key = q/[f];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img15.gif"
 ALT="$ [f]$">|; 

$key = q/displaystylef(x,y,z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img5.gif"
 ALT="$\displaystyle f(x,y,z)$">|; 

$key = q/mathcal{Q};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img4.gif"
 ALT="$ \mathcal{Q}$">|; 

$key = q/displaystyle=;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="img6.gif"
 ALT="$\displaystyle =$">|; 

$key = q/f;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img23.gif"
 ALT="$ f$">|; 

$key = q/hspace{1cm}^tXmathcal{A}hspace{1mm}X=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img19.gif"
 ALT="$ \hspace{1cm}
^tX\mathcal{A}\hspace{1mm}X =0 $">|; 

$key = q/i,j=0,1,2,3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="img12.gif"
 ALT="$ i,j = 0,1,2,3$">|; 

$key = q/,A^3(K),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img1.gif"
 ALT="$ \,A^3(K)\,$">|; 

$key = q/{displaymath}mathcal{A}_0=left(array{{ccc}a_{11}<col_mark>a_{12}<col_mark>a_{13}ark>a_{32}<col_mark>a_{33}<row_mark>array{right){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="196" HEIGHT="90" BORDER="0"
 SRC="img21.gif"
 ALT="\begin{displaymath}
\mathcal{A}_0 = \left(
\begin{array}{ccc}
a_{11} &amp; a_{12}...
..._{23}\\\\
a_{31} &amp; a_{32} &amp; a_{33}\\\\
\end{array}
\right)
\end{displaymath}">|; 

$key = q/displaystylea_{11}x^2+a_{22}y^2+a_{33}z^2+2a_{12}xy+2a_{13}xz+;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="344" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="img7.gif"
 ALT="$\displaystyle a_{11}x^2+a_{22}y^2+a_{33}z^2+2a_{12}xy+2a_{13}xz+$">|; 

$key = q/,mathcal{A},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="img16.gif"
 ALT="$ \,\mathcal{A}\,$">|; 

$key = q/a_{ij}inK;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img10.gif"
 ALT="$ a_{ij}\in
K$">|; 

$key = q/x,y,z,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="img9.gif"
 ALT="$ x, y, z,$">|; 

$key = q/q;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="img22.gif"
 ALT="$ q$">|; 

$key = q/i>j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="img13.gif"
 ALT="$ i &gt; j$">|; 

$key = q/{displaymath}mathcal{A}=left(array{{cccc}a_{00}<col_mark>a_{01}<col_mark>a_{02}<ark>a_{32}<col_mark>a_{33}<row_mark>array{right){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="228" HEIGHT="118" BORDER="0"
 SRC="img14.gif"
 ALT="\begin{displaymath}\mathcal{A} =
\left(
\begin{array}{cccc}
a_{00} &amp; a_{01} &amp;...
...
a_{30} &amp; a_{31} &amp; a_{32} &amp; a_{33}\\\\
\end{array}
\right) \end{displaymath}">|; 

$key = q/a_{ij}=a_{ji};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="img11.gif"
 ALT="$ a_{ij} = a_{ji}$">|; 

$key = q/E^3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img3.gif"
 ALT="$ E^3$">|; 

$key = q/displaystyleleft(array{{cccc}1<col_mark>x<col_mark>y<col_mark>z<row_mark>array{r{c}1<row_mark>x<row_mark>y<row_mark>z<row_mark>array{right)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="246" HEIGHT="128" ALIGN="MIDDLE" BORDER="0"
 SRC="img18.gif"
 ALT="$\displaystyle \left(
 \begin{array}{cccc}
 1 &amp; x &amp; y &amp; z\\\\  
 \end{array}
 \rig...
...al{A} \left(
 \begin{array}{c}
 1 \\\\  x \\\\  y \\\\  z\\\\  
 \end{array}
 \right)=0$">|; 

$key = q/displaystyle+a_{23}yz+2a_{01}x+2a_{02}y+2a_{03}z+a_{00};=;0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="355" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="img8.gif"
 ALT="$\displaystyle +a_{23}yz+2a_{01}x+2a_{02}y+2a_{03}z+a_{00} \; = \; 0$">|; 

$key = q/{displaymath}X=left(array{{c}1<row_mark>x<row_mark>y<row_mark>z<row_mark>array{right){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="118" BORDER="0"
 SRC="img17.gif"
 ALT="\begin{displaymath}X = \left(
\begin{array}{c}
1 \\\\  x \\\\  y \\\\  z\\\\
\end{array}
\right) \end{displaymath}">|; 

$key = q/,mathcal{A}_0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img20.gif"
 ALT="$ \,\mathcal{A}_0\,$">|; 

1;

