# LaTeX2HTML 2022 (Released January 1, 2022)
# Associate images original text with physical files.
$key = q//;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 7.98ex; vertical-align: 150.40ex; " SRC="|."$dir".q|img33.png"
 ALT="$\displaystyle \left.\vphantom{ \begin{array}{ccc}
x_{11} &amp; x_{12} &amp; \ldots \\\\
x_{21} &amp; x_{22} &amp; \ldots \\\\
\vdots &amp; \vdots &amp; \ddots
\end{array} }\right)$">|; 
$key = q/displaystyleBigl(;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 5.56ex; vertical-align: 156.86ex; " SRC="|."$dir".q|img26.png"
 ALT="$\displaystyle \Bigl($">|; 
$key = q/displaystyleBigr)^{{2}}_{};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 3.90ex; vertical-align: 158.41ex; " SRC="|."$dir".q|img27.png"
 ALT="$\displaystyle \Bigr)^{{2}}_{}$">|; 
$key = q/displaystylebf{X;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 4.79ex; vertical-align: 158.89ex; " SRC="|."$dir".q|img30.png"
 ALT="$\displaystyle \bf X$">|; 
$key = q/displaystylechoosek;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.43ex; vertical-align: 161.32ex; " SRC="|."$dir".q|img17.png"
 ALT="$\displaystyle \choosek$">|; 
$key = q/displaystyleint;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 4.34ex; vertical-align: 157.63ex; " SRC="|."$dir".q|img28.png"
 ALT="$\displaystyle \int$">|; 
$key = q/displaystyleint_{{0}}^{{frac{pi}{2}}};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 6.13ex; vertical-align: 155.54ex; " SRC="|."$dir".q|img20.png"
 ALT="$\displaystyle \int_{{0}}^{{\frac{\pi}{2}}}$">|; 
$key = q/displaystyleint_{{0}}^{{x}};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 9.01ex; vertical-align: 152.41ex; " SRC="|."$dir".q|img34.png"
 ALT="$\displaystyle \int_{{0}}^{{x}}$">|; 
$key = q/displaystyleint_{{D}}^{};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 4.73ex; vertical-align: 157.07ex; " SRC="|."$dir".q|img29.png"
 ALT="$\displaystyle \int_{{D}}^{}$">|; 
$key = q/displaystyleintlimits_{{-infty}}^{{+infty}};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 5.49ex; vertical-align: 154.71ex; " SRC="|."$dir".q|img21.png"
 ALT="$\displaystyle \int\limits_{{-\infty}}^{{+\infty}}$">|; 
$key = q/displaystyleleft(vphantom{frac{1}{1-x^{2}}}right.;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.43ex; vertical-align: 159.35ex; " SRC="|."$dir".q|img23.png"
 ALT="$\displaystyle \left(\vphantom{ \frac{1}{ 1-x^{2} }
}\right.$">|; 
$key = q/displaystyleleft.vphantom{frac{1}{1-x^{2}}}right)^{3}_{};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 4.79ex; vertical-align: 156.88ex; " SRC="|."$dir".q|img25.png"
 ALT="$\displaystyle \left.\vphantom{ \frac{1}{ 1-x^{2} }
}\right)^{3}_{}$">|; 
$key = q/displaystylelim_{{xto0}}^{};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 4.85ex; vertical-align: 156.53ex; " SRC="|."$dir".q|img12.png"
 ALT="$\displaystyle \lim_{{x \to 0}}^{}$">|; 
$key = q/displaystylesqrt{{a^{2}+b^{2}}};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 179.22ex; vertical-align: -0.38ex; " SRC="|."$dir".q|img6.png"
 ALT="$\displaystyle \sqrt{{ a^{2}+b^{2} }}$">|; 
$key = q/displaystylesum_{{i=1}}^{{n}};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 4.73ex; vertical-align: 156.22ex; " SRC="|."$dir".q|img19.png"
 ALT="$\displaystyle \sum_{{i=1}}^{{n}}$">|; 
$key = q/displaystyle{frac{{1}}{{1-x^{2}}}};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 5.11ex; vertical-align: 156.85ex; " SRC="|."$dir".q|img24.png"
 ALT="$\displaystyle {\frac{{1}}{{ 1-x^{2} }}}$">|; 
$key = q/displaystyle{frac{{sinx}}{{x}}};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 3.13ex; vertical-align: 158.32ex; " SRC="|."$dir".q|img13.png"
 ALT="$\displaystyle {\frac{{\sin x}}{{x}}}$">|; 
$key = q/displaystyle{frac{{x^{2}}}{{2!}}};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 5.24ex; vertical-align: 156.91ex; " SRC="|."$dir".q|img38.png"
 ALT="$\displaystyle {\frac{{x^{2}}}{{2!}}}$">|; 
$key = q/displaystyle{frac{{x^{2}}}{{k+1}}};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.87ex; vertical-align: 159.08ex; " SRC="|."$dir".q|img15.png"
 ALT="$\displaystyle {\frac{{ x^{2} }}{{ k+1 }}}$">|; 
$key = q/displaystyle{frac{{x^{3}}}{{3!}}};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 4.98ex; vertical-align: 157.17ex; " SRC="|."$dir".q|img35.png"
 ALT="$\displaystyle {\frac{{x^{3}}}{{3!}}}$">|; 
$key = q/displaystyle{frac{{x^{4}}}{{4!}}};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 5.17ex; vertical-align: 156.98ex; " SRC="|."$dir".q|img39.png"
 ALT="$\displaystyle {\frac{{x^{4}}}{{4!}}}$">|; 
$key = q/displaystyle{frac{{x^{5}}}{{5!}}};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 5.24ex; vertical-align: 156.91ex; " SRC="|."$dir".q|img36.png"
 ALT="$\displaystyle {\frac{{x^{5}}}{{5!}}}$">|; 
$key = q/displaystyle{frac{{x^{6}}}{{6!}}};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 5.17ex; vertical-align: 156.98ex; " SRC="|."$dir".q|img40.png"
 ALT="$\displaystyle {\frac{{x^{6}}}{{6!}}}$">|; 
$key = q/displaystyle{frac{{x^{7}}}{{7!}}};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 5.24ex; vertical-align: 156.91ex; " SRC="|."$dir".q|img37.png"
 ALT="$\displaystyle {\frac{{x^{7}}}{{7!}}}$">|; 
$key = q/displaystyle{xatopy+2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 179.22ex; vertical-align: -2.15ex; " SRC="|."$dir".q|img18.png"
 ALT="$\displaystyle {x\atop y+2}$">|; 
$key = q/overline{{m+n}};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.36ex; vertical-align: 161.06ex; " SRC="|."$dir".q|img10.png"
 ALT="$\overline{{m+n}}$">|; 
$key = q/scriptstyle{frac{{2}}{{k+1}}};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 5.37ex; vertical-align: 157.54ex; " SRC="|."$dir".q|img16.png"
 ALT="$\scriptstyle {\frac{{2}}{{k+1}}}$">|; 
$key = q/sqrt[3]{{2}};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 3.90ex; vertical-align: 159.47ex; " SRC="|."$dir".q|img9.png"
 ALT="$\sqrt[3]{{2}}$">|; 
$key = q/sqrt{{x^{2}+sqrt{y}}};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.36ex; vertical-align: 159.93ex; " SRC="|."$dir".q|img8.png"
 ALT="$\sqrt{{ x^{2}+\sqrt{y} }}$">|; 
$key = q/sqrt{{x}};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.62ex; vertical-align: 160.50ex; " SRC="|."$dir".q|img7.png"
 ALT="$\sqrt{{x}}$">|; 
$key = q/underbrace{{a+b+cdots+z}}_{{26}}^{},;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 1.92ex; vertical-align: 158.32ex; " SRC="|."$dir".q|img11.png"
 ALT="$\underbrace{{ a+b+\cdots+z }}_{{26}}^{} $">|; 
$key = q/updownarrow;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 7.60ex; vertical-align: 155.31ex; " SRC="|."$dir".q|img22.png"
 ALT="$\updownarrow$">|; 
$key = q/{center}vbox{input{lksym}}{center};LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
  STYLE="height: 5.24ex; vertical-align: 158.51ex; " SRC="|."$dir".q|img41.png"
 ALT="\begin{center}\vbox{\input{lksym}
}\end{center}">|; 
$key = q/{figure}vspace{6cm}{{{figure};LFS=11;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 286.76ex; " SRC="|."$dir".q|img5.png"
 ALT="\begin{figure}\vspace{6cm}
\end{figure}">|; 
$key = q/{frac{{1}}{{2}}};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 4.85ex; vertical-align: 158.09ex; " SRC="|."$dir".q|img14.png"
 ALT="${\frac{{1}}{{2}}}$">|;