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$key = q/{_inline}$tinI${_inline}/;
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$key = q/{_inline}$(4+4+1)3=3${_inline}/;
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$key = q/{_inline}$f(t)=a_t+t${_inline}/;
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$key = q/{_inline}$f(t)=a_t+t=f(u)=a_u+u${_inline}/;
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$key = q/{_inline}$df(t)=a_t${_inline}/;
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$key = q/{_inline}$iin[n]${_inline}/;
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$key = q/{_inline}$B(f)=b${_inline}/;
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$key = q/{_inline}$ttot+3${_inline}/;
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$key = q/{_inline}$pi(i)${_inline}/;
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$key = q/{_inline}$t+a_tbmodn${_inline}/;
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$key = q/{displaymath}a_0a_1cdotsa_n-1{displaymath}/;
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$key = q/{_inline}$f(t)equivpi_f(t)bmodn${_inline}/;
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$key = q/{displaymath}B(f)=sum_t=0^n-1g(t).{displaymath}/;
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$key = q/{_inline}$df(t)${_inline}/;
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$key = q/{displaymath}lim_|I|rightarrowinfty~sum_xinIdf(x)over|I|{displaymath}/;
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$key = q/{_inline}$pih${_inline}/;
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$key = q/{_inline}$hatpi=36201754${_inline}/;
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$key = q/{displaymath}de_n(k)=natopwithdelimslangleranglek.{displaymath}/;
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$key = q/{_inline}$f(t)=t+2cdotT(t)${_inline}/;
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$key = q/{_inline}$N(b,n)${_inline}/;
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$key = q/{displaymath}natopwithdelimslangleranglek{displaymath}/;
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$key = q/{_inline}$i=1,,2,,5,,6${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img185.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img185.gif"></A>'; 
$key = q/{_inline}$df(1)=5${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img67.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img67.gif"></A>'; 
$key = q/{displaymath}N_;SPM_lt;(b,n)=sum_k=0^n-1de_n(k)~pmatrixn+b-k-1crncr=sum_k=0^n-1natopwithdelimslangleranglekn+b-k-1choosen.{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img199.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple199.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img199.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img199.gif"></A>'; 
$key = q/{_inline}$gcolon[n]toZ${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img115.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img115.gif"></A>'; 
$key = q/{_inline}$f(t)get${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img20.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img20.gif"></A>'; 
$key = q/{_inline}$G(t)=g(t)-d_pi(t)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img133.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img133.gif"></A>'; 
$key = q/{_inline}$g(t)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img86.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img86.gif"></A>'; 
$key = q/{_inline}$f(t')=u${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img105.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img105.gif"></A>'; 
$key = q/{displaymath}sequivtbmodn~~Longrightarrow~~f(s)equivf(t)bmodn.{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img70.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple70.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img70.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img70.gif"></A>'; 
$key = q/{_inline}$f(t)=t+a_{t}equivubmodn${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img103.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img103.gif"></A>'; 
$key = q/{displaymath}N_;SPM_lt;(b,n)=sum_k=0^n-1natopwithdelimslangleranglek~pmatrixb+kcrncr.{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img200.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple200.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img200.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img200.gif"></A>'; 
$key = q/{_inline}$f(t)=pi_f(t)+ncdotg(t)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img87.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img87.gif"></A>'; 
$key = q/{_inline}$(b+1)^n-b^n${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img203.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img203.gif"></A>'; 
$key = q/{_inline}$T(t)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img57.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img57.gif"></A>'; 
$key = q/{_inline}$a_t+tbmodn${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img82.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img82.gif"></A>'; 
$key = q/{_inline}$f(t)=t+df(t)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img72.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img72.gif"></A>'; 
$key = q/{displaymath}f(t)=pi(t)+ncdotd_pi(t)+ncdotG(t).{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img134.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple134.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img134.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img134.gif"></A>'; 
$key = q/{_inline}$ep${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img50.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img50.gif"></A>'; 
$key = q/{_inline}$f(x)=x+4${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img17.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img17.gif"></A>'; 
$key = q/{_inline}${0,1,ldots,n-1}${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img75.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img75.gif"></A>'; 
$key = q/{_inline}$f(t)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img97.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img97.gif"></A>'; 
$key = q/{_inline}$tequivubmodn${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img96.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img96.gif"></A>'; 
$key = q/{_inline}$B(f)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img40.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img40.gif"></A>'; 
$key = q/{_inline}$df(t)=3${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img23.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img23.gif"></A>'; 
$key = q/{_inline}$pi(t);SPM_lt;t${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img131.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img131.gif"></A>'; 
$key = q/{_inline}$t+df(t)=f(t)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img33.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img33.gif"></A>'; 
$key = q/{_inline}$piinS_n${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img130.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img130.gif"></A>'; 
$key = q/{_inline}$[n]${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img122.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img122.gif"></A>'; 
$key = q/{_inline}$f:bfZrightarrowZ${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img61.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img61.gif"></A>'; 
$key = q/{_inline}${4n+a:ninbfZ}${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img6.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img6.gif"></A>'; 
$key = q/{_inline}$f(t)=t${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img30.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img30.gif"></A>'; 
$key = q/{_inline}$df(t)=0${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img31.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img31.gif"></A>'; 
$key = q/{_inline}$0leqi;SPM_lt;n-1${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img153.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img153.gif"></A>'; 
$key = q/{_inline}$a_t=df(t)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img84.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img84.gif"></A>'; 
$key = q/{_inline}$pi(i);SPM_gt;pi(i+1)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img152.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img152.gif"></A>'; 
$key = q/{_inline}$a_0a_1cdotsa_n-1${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img80.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img80.gif"></A>'; 
$key = q/{displaymath}f(x)~=~casesy;SPM_amp;iftheballthrownattime$x$isnextthrownattime$y$crx;SPM_amp;ifthereisnothrowattime$x$.cr{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img10.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple10.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img10.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img10.gif"></A>'; 
$key = q/{_inline}$pi(t)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img121.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img121.gif"></A>'; 
$key = q/{displaymath}f(t)equivpi_f(t)bmodn,qquad1letlen.{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img79.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple79.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img79.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img79.gif"></A>'; 
$key = q/{_inline}$inZ${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img21.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img21.gif"></A>'; 
$key = q/{displaymath}d_pi(t)~=~cases1;SPM_amp;if$t$isadropfor$pi$cr0;SPM_amp;if$t$isnotadropfor$pi$.cr{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img132.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple132.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img132.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img132.gif"></A>'; 
$key = q/{_inline}$pi~=~pi(0)pi(1)ldotspi(n-1)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img187.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img187.gif"></A>'; 
$key = q/{_inline}$N_;SPM_lt;(b,n)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img198.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img198.gif"></A>'; 
$key = q/{_inline}$pi_finS_n${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img78.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img78.gif"></A>'; 
$key = q/{_inline}$a_t=a_u${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img98.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img98.gif"></A>'; 
$key = q/{_inline}$pi'~=~pi(n-1)pi(n-2)ldotspi(0)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img188.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img188.gif"></A>'; 
$key = q/{_inline}$3^rmrd${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img209.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img209.gif"></A>'; 

1;

