# LaTeX2HTML 95 (Thu Jan 19 1995)
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$key = q/{_inline}$x_N${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img313.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img313.gif"></A>'; 
$key = q/{_inline}$lambda${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img202.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img202.gif"></A>'; 
$key = q/{_inline}$k_1:=k(q^p)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img261.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img261.gif"></A>'; 
$key = q/{_inline}$log_23=1.58cdots;SPM_lt;2${_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}$3sqrt7${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img100.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img100.gif"></A>'; 
$key = q/{_inline}$q=e^-pisqrtr${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img255.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img255.gif"></A>'; 
$key = q/{_inline}$x:=(x_0,x_1,x_2,cdots,x_n-1)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img102.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img102.gif"></A>'; 
$key = q/{displaymath}s_n+1:=frac25(z+xz+1)^2s_n,{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img15.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple15.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img15.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img15.gif"></A>'; 
$key = q/{_inline}$lambda(t)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img207.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img207.gif"></A>'; 
$key = q/{displaymath}C(x,y)=F^-1[F(x)F(y)].{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img112.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple112.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img112.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img112.gif"></A>'; 
$key = q/{displaymath}z:=left[frac12xleft(y+sqrty^2-4x^3right)right]^15.{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img17.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple17.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img17.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img17.gif"></A>'; 
$key = q/{_inline}$k_i+1${_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/{displaymath}arctanx=int^x_0fracdt1+t^2=x-fracx^33+fracx^55-cdotsquad|x|leq1{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img71.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple71.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img71.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img71.gif"></A>'; 
$key = q/{_inline}$pi${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img323.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img323.gif"></A>'; 
$key = q/{displaymath}3^10_71;SPM_lt;pi;SPM_lt;3^1_7.{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img68.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple68.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img68.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img68.gif"></A>'; 
$key = q/{displaymath}F[C(x,y)]=F(x)F(y){displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img111.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple111.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img111.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img111.gif"></A>'; 
$key = q/{_inline}$O(n(logn)(loglogn))${_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>'; 
$key = q/{_inline}$o(n)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img137.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img137.gif"></A>'; 
$key = q/{_inline}$p(p+1)(p-1)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img223.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img223.gif"></A>'; 
$key = q/{_inline}$10^n${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img84.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img84.gif"></A>'; 
$key = q/{_inline}$Kpi${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img199.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img199.gif"></A>'; 
$key = q/{displaymath}fracTheta_2(q^p)Theta_3(q^p)=v^2quadmboxandquadfracTheta_2(q)Theta_3(q)=u^2{displaymath}/;
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$key = q/{_inline}$lambda_5${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img238.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img238.gif"></A>'; 
$key = q/{_inline}$W_5${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img217.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img217.gif"></A>'; 
$key = q/{_inline}$2^27${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img74.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img74.gif"></A>'; 
$key = q/{_inline}$O(nM(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>'; 
$key = q/{_inline}$O(M(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>'; 
$key = q/{_inline}$K(k)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img270.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img270.gif"></A>'; 
$key = q/{_inline}$k(q)${_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>'; 
$key = q/{_inline}$b_0:=3${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img66.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img66.gif"></A>'; 
$key = q/{displaymath}eta(q):=q^frac112prod^infty_n=1(1-q^2n).{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img245.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple245.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img245.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img245.gif"></A>'; 
$key = q/{_inline}$O(n^2)${_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>'; 
$key = q/{_inline}$C(x,y)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img106.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img106.gif"></A>'; 
$key = q/{_inline}$W_58${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img304.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img304.gif"></A>'; 
$key = q/{_inline}$times${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img77.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img77.gif"></A>'; 
$key = q/{_inline}$e^pi${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img41.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img41.gif"></A>'; 
$key = q/{_inline}$rrightarrowp^2r${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img273.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img273.gif"></A>'; 
$key = q/{_inline}$lambda_i${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img228.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img228.gif"></A>'; 
$key = q/{_inline}$y:=(y_0,y_1,y_2,cdots,y_n-1)${_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>'; 
$key = q/{_inline}$d_n(N)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img312.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img312.gif"></A>'; 
$key = q/{displaymath}k:=k(q)quadmboxandquadq:=e^-pisqrtr.{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img251.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple251.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img251.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img251.gif"></A>'; 
$key = q/{_inline}$p:=2,3,4,5,${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img274.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img274.gif"></A>'; 
$key = q/{displaymath}frac1pi=fracsqrt89801sum^infty_n=0frac(4n)!(n!)^4frac[1103+26390n]396^4n.{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img5.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple5.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img5.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img5.gif"></A>'; 
$key = q/{_inline}$k_0:=k(q)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img260.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img260.gif"></A>'; 
$key = q/{_inline}$O((logn)M(n))${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img128.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img128.gif"></A>'; 
$key = q/{_inline}$pi+log2+sqrt2log3${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img43.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img43.gif"></A>'; 
$key = q/{_inline}${mboximenskipt;SPM_gt;0}${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img176.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img176.gif"></A>'; 
$key = q/{_inline}$2^n${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img247.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img247.gif"></A>'; 
$key = q/{_inline}$p:=4${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img279.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img279.gif"></A>'; 
$key = q/{_inline}$e^-2piiN${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img116.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img116.gif"></A>'; 
$key = q/{displaymath}fracpi4=4arctanleft(frac15right)-arctanleft(frac1239right).{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img72.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple72.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img72.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img72.gif"></A>'; 
$key = q/{_inline}$318${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img60.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img60.gif"></A>'; 
$key = q/{_inline}$2^k${_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>'; 
$key = q/{_inline}$lambda_p${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img241.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img241.gif"></A>'; 
$key = q/{_inline}$exp(lambda^-1(g(z)))${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img211.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img211.gif"></A>'; 
$key = q/{_inline}$6cdot2^n${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img62.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img62.gif"></A>'; 
$key = q/{displaymath}lambda_i:=lambda(z^iq^1p)quadi;SPM_lt;p{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img185.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple185.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img185.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img185.gif"></A>'; 
$key = q/{displaymath}y_n+1:=frac1-(1-y^4_n)^141+(1-y^4_n)^14{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img8.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple8.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img8.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img8.gif"></A>'; 
$key = q/{displaymath}0;SPM_lt;leftvertbeta-fracpqrightvert;SPM_lt;frac1q^n.{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img37.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple37.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img37.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img37.gif"></A>'; 
$key = q/{_inline}$g_58${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img306.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img306.gif"></A>'; 
$key = q/{_inline}$Theta_2${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img151.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img151.gif"></A>'; 
$key = q/{_inline}$O(M(n)logn)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img118.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img118.gif"></A>'; 
$key = q/{_inline}$sqrty${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img91.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img91.gif"></A>'; 
$key = q/{_inline}$a_0:=2sqrt3${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img65.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img65.gif"></A>'; 
$key = q/{_inline}$p:=5${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img276.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img276.gif"></A>'; 
$key = q/{displaymath}q:=e^ipit,quadmboxthenquadtaurightarrowtau+2quadmboxandquadtaurightarrowfractau(2tau+1){displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img225.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple225.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img225.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img225.gif"></A>'; 
$key = q/{_inline}$b_n${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img64.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img64.gif"></A>'; 
$key = q/{_inline}$2+epsilon${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img39.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img39.gif"></A>'; 
$key = q/{_inline}$2^60${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img70.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img70.gif"></A>'; 
$key = q/{displaymath}frac1pi=sum^infty_n=0left(arrayc2nnarrayright)^3frac42n+52^12n+4.{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img316.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple316.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img316.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img316.gif"></A>'; 
$key = q/{_inline}$exp${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img210.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img210.gif"></A>'; 
$key = q/{_inline}$dKdk${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img291.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img291.gif"></A>'; 
$key = q/{displaymath}z_k=C_k(x,y)=sum^N-1_j=0x_jy_k-j,{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img107.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple107.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img107.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img107.gif"></A>'; 
$key = q/{displaymath}q:=e^ipit.{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img158.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple158.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img158.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img158.gif"></A>'; 
$key = q/{_inline}$Gamma${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img164.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img164.gif"></A>'; 
$key = q/{displaymath}x_k+1=2x_k-x^2_ky{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img88.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple88.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img88.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img88.gif"></A>'; 
$key = q/{displaymath}k^prime:=sqrt1-k^2{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img140.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple140.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img140.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img140.gif"></A>'; 
$key = q/{_inline}$pi+e${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img45.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img45.gif"></A>'; 
$key = q/{_inline}$1y${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img89.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img89.gif"></A>'; 
$key = q/{_inline}$infty${_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>'; 
$key = q/{_inline}$z:=(z_0,z_1,z_2,cdots,z_n-1)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img104.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img104.gif"></A>'; 
$key = q/{_inline}$16cdot2^12n${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img318.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img318.gif"></A>'; 
$key = q/{displaymath}w(t):=fracat+bct+d,{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img172.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple172.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img172.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img172.gif"></A>'; 
$key = q/{displaymath}x:=frac5Theta_3(q^5)Theta_3(q)-1quadandquady:=(x-1)^2+7.{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img214.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple214.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img214.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img214.gif"></A>'; 
$key = q/{_inline}$_3F_2${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img293.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img293.gif"></A>'; 
$key = q/{_inline}$bfQ(v)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img220.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img220.gif"></A>'; 
$key = q/{_inline}$beta${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img36.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img36.gif"></A>'; 
$key = q/{displaymath}O(n^log_23).{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img85.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple85.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img85.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img85.gif"></A>'; 
$key = q/{_inline}$E(k)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img271.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img271.gif"></A>'; 
$key = q/{_inline}$v:=lambda^18${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img192.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img192.gif"></A>'; 
$key = q/{displaymath}K^prime(k):=K(k^prime)quadmboxandquadE^prime(k):=E(k^prime).{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img143.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple143.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img143.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img143.gif"></A>'; 
$key = q/{_inline}$logpi${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img47.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img47.gif"></A>'; 
$key = q/{_inline}$k_i${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img204.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img204.gif"></A>'; 
$key = q/{_inline}$dot${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img264.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img264.gif"></A>'; 
$key = q/{_inline}$M_p:=M_p(k_0,k_1)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img262.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img262.gif"></A>'; 
$key = q/{_inline}$10^9${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img48.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img48.gif"></A>'; 
$key = q/{_inline}$(0,-1),(0,0),(i,infty)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img180.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img180.gif"></A>'; 
$key = q/{_inline}$5m2^m${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img114.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img114.gif"></A>'; 
$key = q/{_inline}$N:=58${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img300.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img300.gif"></A>'; 
$key = q/{displaymath}x_N:=frac4k_N(k^prime_N)^2(1+k^2_N)^2:=left(fracg^12_N+g_N^-122right)^1.{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img286.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple286.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img286.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img286.gif"></A>'; 
$key = q/{_inline}$qrightarrowq^p${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img272.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img272.gif"></A>'; 
$key = q/{_inline}$pgeq5${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img224.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img224.gif"></A>'; 
$key = q/{_inline}$F_G${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img166.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img166.gif"></A>'; 
$key = q/{_inline}$a_n${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img63.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img63.gif"></A>'; 
$key = q/{_inline}$bfQ(lambda)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img230.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img230.gif"></A>'; 
$key = q/{displaymath}f(g(z))=f(z)quadforallginG.{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img160.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple160.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img160.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img160.gif"></A>'; 
$key = q/{_inline}$1pi${_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>'; 
$key = q/{_inline}$sqrt2${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img99.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img99.gif"></A>'; 
$key = q/{_inline}$lambda^18,:Omega_5${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img218.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img218.gif"></A>'; 
$key = q/{displaymath}Theta_3(t,q):=sum^infty_n=-inftyq^n^2e^piintenskipquad(mboximenskipt;SPM_gt;0){displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img150.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple150.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img150.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img150.gif"></A>'; 
$key = q/{_inline}$(t=0)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img149.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img149.gif"></A>'; 
$key = q/{displaymath}k_N:=k(e^-pisqrtN),quadg^12_N=(k^prime_N)^2(2k_N).{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img288.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple288.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img288.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img288.gif"></A>'; 
$key = q/{_inline}$F^-1(x)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img109.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img109.gif"></A>'; 
$key = q/{_inline}$14^n${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img124.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img124.gif"></A>'; 
$key = q/{displaymath}lambda(w(t)):=lambda(t){displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img177.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple177.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img177.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img177.gif"></A>'; 
$key = q/{displaymath}x_k+1=fracx_k(3-yx^2_k)2{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img93.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple93.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img93.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img93.gif"></A>'; 
$key = q/{_inline}$_2F_1${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img292.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img292.gif"></A>'; 
$key = q/{displaymath}frac2K(k_0)pi=M^-1_p(k_0,k_1)M_p^-1(k_1,k_2)M_p^-1(k_2,k_3)cdots.{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img201.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple201.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img201.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img201.gif"></A>'; 
$key = q/{_inline}$zeta(3)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img127.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img127.gif"></A>'; 
$key = q/{_inline}$lambda^-1${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img208.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img208.gif"></A>'; 
$key = q/{_inline}$gamma${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img126.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img126.gif"></A>'; 
$key = q/{_inline}$W_p${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img234.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img234.gif"></A>'; 
$key = q/{_inline}$k(q^p)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img197.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img197.gif"></A>'; 
$key = q/{_inline}$1sqrty${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img92.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img92.gif"></A>'; 
$key = q/{_inline}$O(logn)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img119.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img119.gif"></A>'; 
$key = q/{_inline}$e^pi=(-1)^-i${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img42.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img42.gif"></A>'; 
$key = q/{displaymath}frac4k(k^prime)^2(1+k^2)^2=left(fracg^12+g^-122right)^-1{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img295.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple295.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img295.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img295.gif"></A>'; 
$key = q/{_inline}$u:=x^18${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img191.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img191.gif"></A>'; 
$key = q/{_inline}$2^6n${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img317.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img317.gif"></A>'; 
$key = q/{displaymath}k_1=frac2sqrtk1+k.{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img280.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple280.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img280.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img280.gif"></A>'; 
$key = q/{_inline}$k_i:=k(q^p^i)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img200.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img200.gif"></A>'; 
$key = q/{_inline}$k_N${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img301.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img301.gif"></A>'; 
$key = q/{displaymath}x:=5s_n-1quady:=(x-1)^2+7{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img16.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple16.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img16.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img16.gif"></A>'; 
$key = q/{_inline}$0;SPM_lt;k;SPM_lt;1)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img145.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img145.gif"></A>'; 
$key = q/{_inline}$pie${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img46.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img46.gif"></A>'; 
$key = q/{_inline}$x_k${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img95.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img95.gif"></A>'; 
$key = q/{_inline}$c^overlinen${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img289.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img289.gif"></A>'; 
$key = q/{_inline}$A_5${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img221.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img221.gif"></A>'; 
$key = q/{displaymath}frac1pi=fracsqrt89801sum^infty_n=0frac(4n)!(n!)^4frac[1103+26390n]396^4n{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img282.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple282.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img282.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img282.gif"></A>'; 
$key = q/{_inline}$K^prime${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img141.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img141.gif"></A>'; 
$key = q/{_inline}$M_p${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img265.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img265.gif"></A>'; 
$key = q/{_inline}$s_n${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img29.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img29.gif"></A>'; 
$key = q/{_inline}$epsilon;SPM_gt;0${_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>'; 
$key = q/{_inline}$y_0:=sqrt2-1${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img7.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img7.gif"></A>'; 
$key = q/{_inline}$E^prime${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img142.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img142.gif"></A>'; 
$key = q/{displaymath}27090964785531389931563200281035226311929052227303times2^923^195^2011^253.{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img248.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple248.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img248.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img248.gif"></A>'; 
$key = q/{_inline}$lambda_0${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img239.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img239.gif"></A>'; 
$key = q/{_inline}$c^overlinen:=c(c+1)(c+2)cdots(c+n-1).${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img290.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img290.gif"></A>'; 
$key = q/{displaymath}g^2_58=left(fracsqrt29+52right){displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img298.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple298.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img298.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img298.gif"></A>'; 
$key = q/{_inline}$O((logn)^2+epsilonn)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img101.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img101.gif"></A>'; 
$key = q/{_inline}$N=2^m${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img113.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img113.gif"></A>'; 
$key = q/{_inline}$O(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>'; 
$key = q/{_inline}$Omega_5${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img219.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img219.gif"></A>'; 
$key = q/{_inline}$K(k_1)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img268.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img268.gif"></A>'; 
$key = q/{_inline}$(N=58)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img283.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img283.gif"></A>'; 
$key = q/{_inline}$(p=11)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img246.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img246.gif"></A>'; 
$key = q/{_inline}$O(c^p^i)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img206.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img206.gif"></A>'; 
$key = q/{_inline}$0.10100100001cdots${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img134.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img134.gif"></A>'; 
$key = q/{displaymath}lambda_p:=lambda(q^p).{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img186.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple186.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img186.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img186.gif"></A>'; 
$key = q/{_inline}$s_0:=5(sqrt5-2)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img13.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img13.gif"></A>'; 
$key = q/{displaymath}w(t)=fracat+bct+d,{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img163.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple163.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img163.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img163.gif"></A>'; 
$key = q/{_inline}$O((logn)^2M(n))${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img129.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img129.gif"></A>'; 
$key = q/{displaymath}x_k+1=frac12left(x_k+fracyx_kright){displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img90.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple90.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img90.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img90.gif"></A>'; 
$key = q/{_inline}$O(m(n))${_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>'; 
$key = q/{_inline}$bfQ(lambda,lambda_0)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img231.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img231.gif"></A>'; 
$key = q/{_inline}$F(x)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img108.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img108.gif"></A>'; 
$key = q/{_inline}$2^2m+3${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img115.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img115.gif"></A>'; 
$key = q/{_inline}$s:=1M_5(k_0,k_1)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img277.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img277.gif"></A>'; 
$key = q/{_inline}$31381${_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>'; 
$key = q/{_inline}$M(n)${_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>'; 
$key = q/{_inline}$p(p-1)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img237.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img237.gif"></A>'; 
$key = q/{displaymath}r_i:=frac12xleft(ypmsqrty^2-4x^3right){displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img213.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple213.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img213.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img213.gif"></A>'; 
$key = q/{displaymath}a_n+1:=frac2a_nb_na_n+b_nquadmboxandquadb_n+1:=sqrta_n+1b_n.{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img67.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple67.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img67.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img67.gif"></A>'; 
$key = q/{_inline}$k_0${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img266.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img266.gif"></A>'; 
$key = q/{displaymath}(a+b10^n)(c+d10^n)=ac-[(a-b)(c-d)-ac-bd]10^n+bd10^2n,{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img83.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple83.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img83.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img83.gif"></A>'; 

1;

