# LaTeX2HTML 2022 (Released January 1, 2022)
# Associate images original text with physical files.


$key = q//;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 16.09ex; vertical-align: -6.92ex; " SRC="|."$dir".q|img20.png"
 ALT="$\displaystyle {\frac{{\pm
\left\vert\begin{array}{ccc}
x_1-x_2 &amp; y_1-y_2 &amp; z_1-...
...^2
+ \left\vert\begin{array}{cc}m_2&amp;n_2\\\\
n_2&amp;l_2\end{array}\right\vert^2}}}}$">|; 

$key = q/displaystyleacute{{a}};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.68ex; vertical-align: -0.12ex; " SRC="|."$dir".q|img21.png"
 ALT="$\displaystyle \acute{{a}}$">|; 

$key = q/displaystylebar{{a}};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.47ex; vertical-align: -0.12ex; " SRC="|."$dir".q|img25.png"
 ALT="$\displaystyle \bar{{a}}$">|; 

$key = q/displaystylebreve{{a}};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.68ex; vertical-align: -0.12ex; " SRC="|."$dir".q|img26.png"
 ALT="$\displaystyle \breve{{a}}$">|; 

$key = q/displaystylecheck{{a}};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.54ex; vertical-align: -0.12ex; " SRC="|."$dir".q|img27.png"
 ALT="$\displaystyle \check{{a}}$">|; 

$key = q/displaystyleddot{{a}};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.68ex; vertical-align: -0.12ex; " SRC="|."$dir".q|img23.png"
 ALT="$\displaystyle \ddot{{a}}$">|; 

$key = q/displaystyledot{{a}};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.68ex; vertical-align: -0.12ex; " SRC="|."$dir".q|img30.png"
 ALT="$\displaystyle \dot{{a}}$">|; 

$key = q/displaystylegrave{{a}};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.68ex; vertical-align: -0.12ex; " SRC="|."$dir".q|img22.png"
 ALT="$\displaystyle \grave{{a}}$">|; 

$key = q/displaystylehat{{a}};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.68ex; vertical-align: -0.12ex; " SRC="|."$dir".q|img28.png"
 ALT="$\displaystyle \hat{{a}}$">|; 

$key = q/displaystyleint^{t}_{0};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 5.80ex; vertical-align: -2.23ex; " SRC="|."$dir".q|img4.png"
 ALT="$\displaystyle \int^{t}_{0}$">|; 

$key = q/displaystyleleft(vphantom{sum_{k=1}^{m-1}lfloor(mslashk)slashlceilmslashkrceilrfloor}right.;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 6.99ex; vertical-align: -2.98ex; " SRC="|."$dir".q|img14.png"
 ALT="$\displaystyle \left(\vphantom{\sum_{k=1}^{m-1}
\lfloor(m/k)/\lceil m/k\rceil
\rfloor }\right.$">|; 

$key = q/displaystyleleft(vphantom{sum_{kgeq0}a_{jk}z^k}right.;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 8.46ex; vertical-align: -3.72ex; " SRC="|."$dir".q|img6.png"
 ALT="$\displaystyle \left(\vphantom{\sum_{k\geq 0}a_{jk} z^k}\right.$">|; 

$key = q/displaystyleleft(vphantom{sum_{{k_0,k_1,ldotsgeq0}atop{k_0+k_1+ldots=n}}a{_0k_0}a_{1k_1}ldots}right.;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 9.86ex; vertical-align: -4.47ex; " SRC="|."$dir".q|img9.png"
 ALT="$\displaystyle \left(\vphantom{ \sum_{{k_0,k_1,\ldots\geq 0}
\atop{k_0+k_1+\ldots=n} }
a{_0k_0}a_{1k_1}\ldots }\right.$">|; 

$key = q/displaystyleleft.vphantom{left(sum_{k=1}^{m-1}lfloor(mslashk)slashlceilmslashkrceilrfloorright)^{-1}}rightrfloor;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 8.39ex; vertical-align: -3.72ex; " SRC="|."$dir".q|img17.png"
 ALT="$\displaystyle \left.\vphantom{ \left(\sum_{k=1}^{m-1}
\lfloor(m/k)/\lceil m/k\rceil
\rfloor \right)^{-1}
}\right\rfloor$">|; 

$key = q/displaystyleleft.vphantom{sum_{k=1}^{m-1}lfloor(mslashk)slashlceilmslashkrceilrfloor}right)^{{-1}}_{};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 7.62ex; vertical-align: -3.02ex; " SRC="|."$dir".q|img16.png"
 ALT="$\displaystyle \left.\vphantom{\sum_{k=1}^{m-1}
\lfloor(m/k)/\lceil m/k\rceil
\rfloor }\right)^{{-1}}_{}$">|; 

$key = q/displaystyleleft.vphantom{sum_{kgeq0}a_{jk}z^k}right);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 8.46ex; vertical-align: -3.72ex; " SRC="|."$dir".q|img8.png"
 ALT="$\displaystyle \left.\vphantom{\sum_{k\geq 0}a_{jk} z^k}\right)$">|; 

$key = q/displaystyleleft.vphantom{sum_{{k_0,k_1,ldotsgeq0}atop{k_0+k_1+ldots=n}}a{_0k_0}a_{1k_1}ldots}right);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 9.86ex; vertical-align: -4.47ex; " SRC="|."$dir".q|img11.png"
 ALT="$\displaystyle \left.\vphantom{ \sum_{{k_0,k_1,\ldots\geq 0}
\atop{k_0+k_1+\ldots=n} }
a{_0k_0}a_{1k_1}\ldots }\right)$">|; 

$key = q/displaystyleleftlfloorvphantom{left(sum_{k=1}^{m-1}lfloor(mslashk)slashlceilmslashkrceilrfloorright)^{-1}}right.;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 8.39ex; vertical-align: -3.72ex; " SRC="|."$dir".q|img13.png"
 ALT="$\displaystyle \left\lfloor\vphantom{ \left(\sum_{k=1}^{m-1}
\lfloor(m/k)/\lceil m/k\rceil
\rfloor \right)^{-1}
}\right.$">|; 

$key = q/displaystyleprod_{{jgeq0}}^{};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 5.32ex; vertical-align: -3.21ex; " SRC="|."$dir".q|img5.png"
 ALT="$\displaystyle \prod_{{j\geq 0}}^{}$">|; 

$key = q/displaystylesum_{{k=1}}^{{m-1}};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 6.85ex; vertical-align: -2.98ex; " SRC="|."$dir".q|img15.png"
 ALT="$\displaystyle \sum_{{k=1}}^{{m-1}}$">|; 

$key = q/displaystylesum_{{kgeq0}}^{};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 5.39ex; vertical-align: -3.28ex; " SRC="|."$dir".q|img7.png"
 ALT="$\displaystyle \sum_{{k\geq 0}}^{}$">|; 

$key = q/displaystylesum_{{m=2}}^{{n}};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 6.43ex; vertical-align: -2.90ex; " SRC="|."$dir".q|img12.png"
 ALT="$\displaystyle \sum_{{m=2}}^{{n}}$">|; 

$key = q/displaystylesum_{{{k_0,k_1,ldotsgeq0}atop{k_0+k_1+ldots=n}}}^{};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 6.57ex; vertical-align: -4.47ex; " SRC="|."$dir".q|img10.png"
 ALT="$\displaystyle \sum_{{{k_0,k_1,\ldots\geq 0}
\atop{k_0+k_1+\ldots=n} }}^{}$">|; 

$key = q/displaystyletilde{{a}};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.68ex; vertical-align: -0.12ex; " SRC="|."$dir".q|img24.png"
 ALT="$\displaystyle \tilde{{a}}$">|; 

$key = q/displaystyleunderbrace{{<comment_mark>2overbrace{mathstruta,ldots,a}^{ka's},overbrace{mathstrutb,ldots,b}^{lb's}}}_{{k+1mathrm{elements}}}^{},;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 9.16ex; vertical-align: -4.08ex; " SRC="|."$dir".q|img18.png"
 ALT="$\displaystyle \underbrace{{%
\overbrace{\mathstrut a,\ldots,a}^{k a's},
\overbrace{\mathstrut b,\ldots,b}^{l b's}}}_{{k+1 \mathrm{elements}}}^{} $">|; 

$key = q/displaystylevec{{a}},;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.75ex; vertical-align: -0.12ex; " SRC="|."$dir".q|img29.png"
 ALT="$\displaystyle \vec{{a}} $">|; 

$key = q/displaystyle{frac{{1}}{{sqrt{2pi}}}};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 5.25ex; vertical-align: -2.28ex; " SRC="|."$dir".q|img3.png"
 ALT="$\displaystyle {\frac{{1}}{{\sqrt{2\pi}}}}$">|; 

$key = q/mathcal{ABCDE};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.82ex; vertical-align: -0.12ex; " SRC="|."$dir".q|img2.png"
 ALT="$\mathcal {ABCDE}$">|; 

$key = q/simeq;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.05ex; vertical-align: -0.02ex; " SRC="|."$dir".q|img1.png"
 ALT="$\simeq$">|; 

1;

