# LaTeX2HTML 95 (Thu Jan 19 1995)
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$key = q/{_inline}$Q_infty:bZ_2rightarrowbZ_2${_inline}/;
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$key = q/{_inline}$+In${_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}$0leqileqk-1${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img114.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img114.gif"></A>'; 
$key = q/{_inline}$lambda_k(n)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img117.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img117.gif"></A>'; 
$key = q/{_inline}$leqx${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img147.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img147.gif"></A>'; 
$key = q/{_inline}$barQ_k${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img125.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img125.gif"></A>'; 
$key = q/{displaymath}displaystylefrac125leqleft{3choose2^kxiright}leqdisplaystylefrac2425~;~~k=1,2,3,ldots,{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img691.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple691.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img691.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img691.gif"></A>'; 
$key = q/{_inline}$|0|_2=0${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img463.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img463.gif"></A>'; 
$key = q/{_inline}$bfv=bv_k(n)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img93.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img93.gif"></A>'; 
$key = q/{_inline}$n_1=(3n_0+1)8${_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>'; 
$key = q/{_inline}$(x_0(n),x_1(n),x_2(n),ldots)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img573.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img573.gif"></A>'; 
$key = q/{_inline}$sigma_infty(n)=40${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img303.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img303.gif"></A>'; 
$key = q/{displaymath}n_0;SPM_lt;T(n_0);SPM_lt;cdots;SPM_lt;T^(i)(n_0){displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img410.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple410.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img410.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img410.gif"></A>'; 
$key = q/{_inline}$n_0=1${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img415.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img415.gif"></A>'; 
$key = q/{_inline}$2^r_1-1k+n_1${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img324.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img324.gif"></A>'; 
$key = q/{_inline}$Q_In^(k)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img575.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img575.gif"></A>'; 
$key = q/{_inline}$n_0(bfv)${_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}|klog2-llog3|geqk^-c_0~.{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img266.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple266.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img266.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img266.gif"></A>'; 
$key = q/{_inline}$(a+b)b${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img421.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img421.gif"></A>'; 
$key = q/{displaymath}n_0(bfv)equiv(barQ_k)^-1left(sum_i=0^k-1v_i2^iright)~~(bmod~2^k){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>'; 
$key = q/{_inline}$U:bfNrightarrowbN${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img600.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img600.gif"></A>'; 
$key = q/{_inline}${T^(i)(n):i=0,1,2,ldots}${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img510.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img510.gif"></A>'; 
$key = q/{_inline}$q_13=190737${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img408.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img408.gif"></A>'; 
$key = q/{_inline}$sigma(n)${_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}displaystylefrac1xsum_n=1^Xsigma_infty(n)sim2left(lndisplaystylefrac43right)^-1lnx~.{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img347.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple347.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img347.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img347.gif"></A>'; 
$key = q/{_inline}$U_D={n:ninD${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img439.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img439.gif"></A>'; 
$key = q/{_inline}$sigma_infty(12)=sigma_infty(13)=7${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img299.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img299.gif"></A>'; 
$key = q/{_inline}$(m,d)=1${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img635.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img635.gif"></A>'; 
$key = q/{_inline}$n=n(bfv)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img359.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img359.gif"></A>'; 
$key = q/{_inline}$T^(k)(n)neq1${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img32.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img32.gif"></A>'; 
$key = q/{_inline}${a_i:0leqi;SPM_lt;infty}${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img452.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img452.gif"></A>'; 
$key = q/{displaymath}Q_k(n)=sum_i=0^k-1x_i(n)2^i{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>'; 
$key = q/{_inline}$delta_3,delta_4;SPM_gt;0${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img652.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img652.gif"></A>'; 
$key = q/{_inline}$kgeq1${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img617.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img617.gif"></A>'; 
$key = q/{_inline}$H(x)=-xlog_2x-(1-x)log_2(1-x)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img208.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img208.gif"></A>'; 
$key = q/{_inline}$sigma_infty(340)=sigma_infty(341)=10${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img302.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img302.gif"></A>'; 
$key = q/{_inline}$bv${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img167.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img167.gif"></A>'; 
$key = q/{displaymath}sum_bfvinC_2~mboxweight(bfv)=2^-k|C_2|=1-F(k)~,{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img224.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple224.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img224.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img224.gif"></A>'; 
$key = q/{_inline}$g(n)=2n+1${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img732.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img732.gif"></A>'; 
$key = q/{_inline}$0leqileqk-2${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img142.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img142.gif"></A>'; 
$key = q/{_inline}$F(k)rightarrow1${_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>'; 
$key = q/{_inline}$n;SPM_lt;2^32cong4.5times10^9${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img748.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img748.gif"></A>'; 
$key = q/{_inline}$Q_In^-1${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img551.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img551.gif"></A>'; 
$key = q/{_inline}$sigma_infty(n_1)equivsigma_infty(n_2)(bmod~2)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img314.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img314.gif"></A>'; 
$key = q/{displaymath}S_2leq#{bfv:bfv~mboxadmissible~and~length~(bfv)leqk}+c_3~,{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img291.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple291.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img291.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img291.gif"></A>'; 
$key = q/{_inline}$0leql;SPM_lt;k${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img594.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img594.gif"></A>'; 
$key = q/{_inline}$2^k${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img126.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img126.gif"></A>'; 
$key = q/{_inline}$n_0leqM(k)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img388.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img388.gif"></A>'; 
$key = q/{_inline}$Q_infty(-54)=-827${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img571.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img571.gif"></A>'; 
$key = q/{_inline}$Q_infty(bN^+)subseteqfrac13bZ${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img558.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img558.gif"></A>'; 
$key = q/{_inline}${beta_n}${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img546.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img546.gif"></A>'; 
$key = q/{_inline}$2^i${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img396.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img396.gif"></A>'; 
$key = q/{displaymath}C_1={bfv:bfv~mboxis~admissible~and~length~(bfv)leqk}{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img211.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple211.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img211.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img211.gif"></A>'; 
$key = q/{_inline}$b;SPM_gt;10^199${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img425.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img425.gif"></A>'; 
$key = q/{_inline}$n_1${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img322.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img322.gif"></A>'; 
$key = q/{tabular}|cc|cc|cc|cc|hline_inline$sigma_infty(n)$_inline;SPM_amp;freq.;SPM_amp;_inline$sigma_infty(n)$_inline;SPM_amp;freq.;SPM_amp;_inline$sigma_infty(n)$_inline;SPM_amp;freq.;SPM_amp;_inline$sigma_infty(n)$_inline;SPM_amp;freq.hline10;SPM_amp;1;SPM_amp;29;SPM_amp;1;SPM_amp;50;SPM_amp;3;SPM_amp;88;SPM_amp;515;SPM_amp;1;SPM_amp;31;SPM_amp;7;SPM_amp;53;SPM_amp;4;SPM_amp;91;SPM_amp;118;SPM_amp;6;SPM_amp;34;SPM_amp;6;SPM_amp;61;SPM_amp;4;SPM_amp;99;SPM_amp;223;SPM_amp;17;SPM_amp;42;SPM_amp;9;SPM_amp;72;SPM_amp;6;SPM_amp;107;SPM_amp;226;SPM_amp;6;SPM_amp;45;SPM_amp;3;SPM_amp;80;SPM_amp;16;SPM_amp;;SPM_amp;hline{tabular}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img339.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple339.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img339.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img339.gif"></A>'; 
$key = q/{_inline}$T^(r_1-1)(n_1)=T^(r_1-1)(n_2)=2${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img319.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img319.gif"></A>'; 
$key = q/{_inline}$g^(k)(2^m)=2^f(m)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img619.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img619.gif"></A>'; 
$key = q/{_inline}$D={T^(k)(n_0):kgeq1}${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img438.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img438.gif"></A>'; 
$key = q/{_inline}$32${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img740.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img740.gif"></A>'; 
$key = q/{_inline}$leqk}${_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>'; 
$key = q/{_inline}$(n,W(n),W^(2)(n),ldots)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img682.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img682.gif"></A>'; 
$key = q/{displaymath}S(bfv)={n_0(bfv)+2^ki:0leqi;SPM_lt;infty}~.{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img136.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple136.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img136.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img136.gif"></A>'; 
$key = q/{displaymath}(1-lambda_k(bfv))nleqrho_k(bfv)~,{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img367.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple367.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img367.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img367.gif"></A>'; 
$key = q/{_inline}$C_2${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img226.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img226.gif"></A>'; 
$key = q/{_inline}${x}=x-[x]${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img670.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img670.gif"></A>'; 
$key = q/{_inline}$0leqn_0(bfv);SPM_lt;2^k${_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>'; 
$key = q/{_inline}$sigma(n)=In${_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>'; 
$key = q/{_inline}$(bmod~2^k-1)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img511.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img511.gif"></A>'; 
$key = q/{_inline}$[n,n+1)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img671.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img671.gif"></A>'; 
$key = q/{_inline}$ngeq1${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img683.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img683.gif"></A>'; 
$key = q/{_inline}$bfv=(v_0,v_1,ldots,v_k-1)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img139.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img139.gif"></A>'; 
$key = q/{_inline}$bfv=(x_0,x_1,ldots,x_k)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img368.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img368.gif"></A>'; 
$key = q/{_inline}$Q_In${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img535.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img535.gif"></A>'; 
$key = q/{_inline}$F_q[x]${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img739.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img739.gif"></A>'; 
$key = q/{_inline}$c_3=#{n:omega(n)leqk_0${_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>'; 
$key = q/{_inline}$p_n${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img623.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img623.gif"></A>'; 
$key = q/{_inline}$S(bfv)~(bmod~2^k)${_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>'; 
$key = q/{_inline}$bN${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img613.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img613.gif"></A>'; 
$key = q/{_inline}$(32)^n${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img745.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img745.gif"></A>'; 
$key = q/{_inline}$F(k)${_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>'; 
$key = q/{_inline}$(n,T(n),T^(2)(n),T^(3)(n),ldots)${_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>'; 
$key = q/{tabular}|c|c|c|c|c|c|c|c|c|c|c|hline;SPM_amp;1000;SPM_amp;1010;SPM_amp;1020;SPM_amp;1030;SPM_amp;1040;SPM_amp;1050;SPM_amp;1060;SPM_amp;1070;SPM_amp;1080;SPM_amp;1090;SPM_amp;_inline$-1009$_inline;SPM_amp;_inline$-1019$_inline;SPM_amp;_inline$-1029$_inline;SPM_amp;_inline$-1039$_inline;SPM_amp;_inline$-1049$_inline;SPM_amp;_inline$-1059$_inline;SPM_amp;_inline$-1069$_inline;SPM_amp;_inline$-1079$_inline;SPM_amp;_inline$-1089$_inline;SPM_amp;_inline$-1099$_inlinehline0;SPM_amp;72;SPM_amp;42;SPM_amp;34;SPM_amp;80;SPM_amp;23;SPM_amp;23;SPM_amp;80;SPM_amp;18;SPM_amp;31;SPM_amp;311;SPM_amp;91;SPM_amp;42;SPM_amp;34;SPM_amp;26;SPM_amp;80;SPM_amp;61;SPM_amp;80;SPM_amp;107;SPM_amp;88;SPM_amp;312;SPM_amp;72;SPM_amp;72;SPM_amp;42;SPM_amp;80;SPM_amp;80;SPM_amp;53;SPM_amp;80;SPM_amp;23;SPM_amp;31;SPM_amp;233;SPM_amp;29;SPM_amp;72;SPM_amp;42;SPM_amp;99;SPM_amp;80;SPM_amp;53;SPM_amp;50;SPM_amp;18;SPM_amp;88;SPM_amp;234;SPM_amp;45;SPM_amp;26;SPM_amp;10;SPM_amp;80;SPM_amp;23;SPM_amp;53;SPM_amp;23;SPM_amp;18;SPM_amp;31;SPM_amp;235;SPM_amp;45;SPM_amp;26;SPM_amp;26;SPM_amp;80;SPM_amp;23;SPM_amp;107;SPM_amp;50;SPM_amp;18;SPM_amp;31;SPM_amp;506;SPM_amp;45;SPM_amp;34;SPM_amp;26;SPM_amp;80;SPM_amp;80;SPM_amp;23;SPM_amp;23;SPM_amp;23;SPM_amp;88;SPM_amp;617;SPM_amp;61;SPM_amp;99;SPM_amp;26;SPM_amp;80;SPM_amp;80;SPM_amp;53;SPM_amp;42;SPM_amp;23;SPM_amp;88;SPM_amp;888;SPM_amp;72;SPM_amp;34;SPM_amp;80;SPM_amp;42;SPM_amp;23;SPM_amp;23;SPM_amp;18;SPM_amp;34;SPM_amp;15;SPM_amp;619;SPM_amp;72;SPM_amp;42;SPM_amp;80;SPM_amp;42;SPM_amp;42;SPM_amp;23;SPM_amp;18;SPM_amp;34;SPM_amp;31;SPM_amp;23hline{tabular}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img336.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple336.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img336.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img336.gif"></A>'; 
$key = q/{_inline}$sigma_infty(n)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img345.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img345.gif"></A>'; 
$key = q/{_inline}$T^(k)(n)${_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>'; 
$key = q/{_inline}$n_0;SPM_gt;2^40${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img407.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img407.gif"></A>'; 
$key = q/{_inline}$T^(k)(n)=1${_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/{_inline}$7083leqnleq7099${_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>'; 
$key = q/{_inline}$ageq4${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img420.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img420.gif"></A>'; 
$key = q/{_inline}$g^(k)(2^m)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img621.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img621.gif"></A>'; 
$key = q/{_inline}$lambda_k(bfv),~rho_k(bfv)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img94.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img94.gif"></A>'; 
$key = q/{_inline}$xrightarrowIn${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img650.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img650.gif"></A>'; 
$key = q/{_inline}$f(2n)=n${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img726.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img726.gif"></A>'; 
$key = q/{displaymath}mboxweight~(bfv)=summboxweight(bfw)~,{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img219.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple219.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img219.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img219.gif"></A>'; 
$key = q/{_inline}$g(n)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img12.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img12.gif"></A>'; 
$key = q/{_inline}$(v_0+ldots+v_i)ln3;SPM_gt;(i+1)ln2${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img141.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img141.gif"></A>'; 
$key = q/{_inline}$frac12${_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>'; 
$key = q/{displaymath}d(I_k)=lim_xrightarrowinftydisplaystylefrac1x#{n:nleqx~mboxand~$ninI_k}${displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img194.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple194.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img194.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img194.gif"></A>'; 
$key = q/{_inline}$bfv=(v_0,ldots,v_k-1)in(bZ2bZ)^k${_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>'; 
$key = q/{_inline}$log_23${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img405.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img405.gif"></A>'; 
$key = q/{_inline}$g^(k)(2^m)=2^j${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img618.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img618.gif"></A>'; 
$key = q/{_inline}$1leqileqk-1${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img159.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img159.gif"></A>'; 
$key = q/{_inline}$n=n_0(bfv)${_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>'; 
$key = q/{_inline}$lambda_i(bfv);SPM_gt;1${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img158.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img158.gif"></A>'; 
$key = q/{_inline}$S_1=#{nleq2^k:omega(n)geqk+1}${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img287.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img287.gif"></A>'; 
$key = q/{_inline}$f(3n-2)=4n-3${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img738.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img738.gif"></A>'; 
$key = q/{displaymath}S_2leqc_52^(l-eta)k+c_3leq(2c_5+c_3)x^1-eta~.{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>'; 
$key = q/{displaymath}N^ast(k)=|{j:jleqk~mboxand~$T^(j)(n_0)equiv1~(bmod~2)}|$~,{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img436.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple436.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img436.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img436.gif"></A>'; 
$key = q/{_inline}$T(n)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img25.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img25.gif"></A>'; 
$key = q/{_inline}$kgeqk_0${_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>'; 
$key = q/{_inline}$k,lgeqc_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>'; 
$key = q/{_inline}$g_0^(k)(n)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img629.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img629.gif"></A>'; 
$key = q/{_inline}$bQ_2${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img563.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img563.gif"></A>'; 
$key = q/{_inline}$1~(bmod~3)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img734.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img734.gif"></A>'; 
$key = q/{_inline}$jcl${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img333.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img333.gif"></A>'; 
$key = q/{_inline}$a_k${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img466.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img466.gif"></A>'; 
$key = q/{_inline}$bfv=(x_0,ldots,x_k-1)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img357.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img357.gif"></A>'; 
$key = q/{_inline}$10^199${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img428.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img428.gif"></A>'; 
$key = q/{_inline}$c_1${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img282.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img282.gif"></A>'; 
$key = q/{_inline}$rho_k(n)${_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>'; 
$key = q/{_inline}$lambda_k(bfv);SPM_gt;1${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img182.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img182.gif"></A>'; 
$key = q/{_inline}$ngeqM(k)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img374.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img374.gif"></A>'; 
$key = q/{_inline}$lnn${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img346.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img346.gif"></A>'; 
$key = q/{_inline}${(32)^k:k=1,2,ldots}${_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>'; 
$key = q/{_inline}$(bmod~2^r_1-1)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img332.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img332.gif"></A>'; 
$key = q/{_inline}$2^k-1leqxleq2^k${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img285.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img285.gif"></A>'; 
$key = q/{_inline}$(v_0+ldots+v_k-1)ln3;SPM_lt;kln2${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img140.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img140.gif"></A>'; 
$key = q/{_inline}$bw${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img216.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img216.gif"></A>'; 
$key = q/{_inline}$U=U(m,d,R)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img633.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img633.gif"></A>'; 
$key = q/{_inline}$sigma_infty(2^r_1-1k+n_1)=sigma_infty(2^r_1-1k+n_2)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img328.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img328.gif"></A>'; 
$key = q/{_inline}$bfvinC${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img214.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img214.gif"></A>'; 
$key = q/{_inline}$nleq2^50${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img743.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img743.gif"></A>'; 
$key = q/{_inline}$nleqM(k)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img379.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img379.gif"></A>'; 
$key = q/{_inline}${(32)^kxi${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img687.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img687.gif"></A>'; 
$key = q/{_inline}$Q_In^(l)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img578.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img578.gif"></A>'; 
$key = q/{_inline}$n=1,2,3,ldots${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img672.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img672.gif"></A>'; 
$key = q/{_inline}$rho_k(n_0)neq0${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img254.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img254.gif"></A>'; 
$key = q/{_inline}$U^(N)(n);SPM_gt;nx^-delta_1}=O(x^1-delta_2)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img649.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img649.gif"></A>'; 
$key = q/{_inline}$x_n${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img702.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img702.gif"></A>'; 
$key = q/{_inline}$g(n)n${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img615.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img615.gif"></A>'; 
$key = q/{_inline}$^Z${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img708.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img708.gif"></A>'; 
$key = q/{displaymath}1-F(k)geq2^-(eta+epsilon)k{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img241.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple241.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img241.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img241.gif"></A>'; 
$key = q/{displaymath}left(displaystylefrac32right)^12left(displaystylefrac34right)^14left(displaystylefrac38right)^18ldots=displaystylefrac34;SPM_lt;1~.{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img69.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple69.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img69.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img69.gif"></A>'; 
$key = q/{_inline}$T(n)(bmod~2^k)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img73.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img73.gif"></A>'; 
$key = q/{_inline}$lambda_k(bfv);SPM_lt;1${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img157.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img157.gif"></A>'; 
$key = q/{_inline}$(bfv)leqk${_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>'; 
$key = q/{_inline}$n;SPM_lt;2^40approx1.2times10^12${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img58.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img58.gif"></A>'; 
$key = q/{_inline}$epsilon;SPM_gt;0${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img232.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img232.gif"></A>'; 
$key = q/{_inline}$bZ_2${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img527.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img527.gif"></A>'; 
$key = q/{_inline}$1leqileqd-1}${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img637.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img637.gif"></A>'; 
$key = q/{_inline}$2^-k${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img472.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img472.gif"></A>'; 
$key = q/{_inline}$p_kq_k${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img400.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img400.gif"></A>'; 
$key = q/{_inline}$f:N^+rightarrowN^+${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img735.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img735.gif"></A>'; 
$key = q/{_inline}$kgeq1}${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img441.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img441.gif"></A>'; 
$key = q/{displaymath}|2^k-3^l|geqdisplaystylefrac122^kk^-c_1~,{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img269.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple269.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img269.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img269.gif"></A>'; 
$key = q/{_inline}$ninS(bfv)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img138.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img138.gif"></A>'; 
$key = q/{_inline}$s(n)=+In${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img51.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img51.gif"></A>'; 
$key = q/{_inline}$x_n+1=3x+1${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img703.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img703.gif"></A>'; 
$key = q/{displaymath}s(27)=displaystylefracsup_kgeq0T^(k)(27)27=displaystylefrac461627approx171~.{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img52.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple52.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img52.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img52.gif"></A>'; 
$key = q/{_inline}$bF${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img607.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img607.gif"></A>'; 
$key = q/{displaymath}pi^ast(x)=|{n:nleqx~mboxand~sigma(n);SPM_lt;infty}|~.{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>'; 
$key = q/{_inline}$g_0(n)n${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img627.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img627.gif"></A>'; 
$key = q/{_inline}$logn${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img71.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img71.gif"></A>'; 
$key = q/{_inline}$2^r_1-1k+n_2${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img325.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img325.gif"></A>'; 
$key = q/{_inline}$m;SPM_lt;d^d(d-1)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img645.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img645.gif"></A>'; 
$key = q/{_inline}$ninbZ${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img680.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img680.gif"></A>'; 
$key = q/{displaymath}1-F(k)=lim_xrightarrowinftydisplaystylefrac1x#{n:nleqx~mboxand~sigma(n);SPM_gt;k}~,{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img227.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple227.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img227.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img227.gif"></A>'; 
$key = q/{_inline}$T^(0)(n)=n${_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>'; 
$key = q/{_inline}$beta_n${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img542.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img542.gif"></A>'; 
$key = q/{_inline}$leqk${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img385.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img385.gif"></A>'; 
$key = q/{_inline}$sigma_infty(52)=sigma_infty(53)=9${_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>'; 
$key = q/{_inline}$r_1-1${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img326.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img326.gif"></A>'; 
$key = q/{displaymath}sum_bfvinC~mboxweight(bfv)=sum_mboxlength(bfw)=k~mboxweight(bfw)=1~.{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img223.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple223.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img223.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img223.gif"></A>'; 
$key = q/{_inline}$f(2^p_n)=2^p_n+1${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img622.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img622.gif"></A>'; 
$key = q/{_inline}$f(2n+1)=3n+2${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img727.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img727.gif"></A>'; 
$key = q/{_inline}$n_1=(3n_0+1)4,~frac18${_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>'; 
$key = q/{_inline}$n_1=(3n_0+1)2,~frac14${_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>'; 
$key = q/{_inline}$T:bZ_2rightarrowbZ_2${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img483.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img483.gif"></A>'; 
$key = q/{_inline}$2^i+1${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img111.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img111.gif"></A>'; 
$key = q/{displaymath}S_k={n:n~mboxhas~stopping~time~$k$}{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img196.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple196.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img196.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img196.gif"></A>'; 
$key = q/{_inline}$(a,b,h)=1,1,1)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img419.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img419.gif"></A>'; 
$key = q/{_inline}$T^(k)(n)geqn${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img183.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img183.gif"></A>'; 
$key = q/{_inline}${T^(k)(n_0):0leqk;SPM_lt;infty}${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img434.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img434.gif"></A>'; 
$key = q/{_inline}$f_0${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img625.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img625.gif"></A>'; 
$key = q/{tabular}|clc|hline_inline$k$_inline;SPM_amp;_inline$barQ_k$_inline;SPM_amp;orderhline1;SPM_amp;identity;SPM_amp;12;SPM_amp;identity;SPM_amp;13;SPM_amp;(1,5);SPM_amp;24;SPM_amp;(1,5)(2,10)(9,13);SPM_amp;25;SPM_amp;(1,21)(2,10)(4,20)(5,17)(7,23)(9,29,25,13)(18,26);SPM_amp;46;SPM_amp;(1,21)(2,42)(3,35)(4.20)(5,17,37,49);SPM_amp;;SPM_amp;(7,23)(8,40)(9,29,25,13)(10,34);SPM_amp;;SPM_amp;(18,58,50,26)(19,51)(27,59)(33,53);SPM_amp;;SPM_amp;(36,52)(29,55)(41,61,57,45);SPM_amp;4hline{tabular}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img128.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple128.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img128.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img128.gif"></A>'; 
$key = q/{_inline}$1leqk;SPM_lt;infty}${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img688.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img688.gif"></A>'; 
$key = q/{_inline}$M=prod(2^l-1)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img567.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img567.gif"></A>'; 
$key = q/{_inline}$#{n:nleqx${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img654.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img654.gif"></A>'; 
$key = q/{_inline}$q_14=10590737${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img409.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img409.gif"></A>'; 
$key = q/{_inline}$jgeq4${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img406.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img406.gif"></A>'; 
$key = q/{_inline}$~~Box${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img236.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img236.gif"></A>'; 
$key = q/{_inline}$etaapprox.05004ldots${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img284.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img284.gif"></A>'; 
$key = q/{_inline}$lambda_k(n_0);SPM_lt;1${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img256.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img256.gif"></A>'; 
$key = q/{_inline}$Q_infty(bmod~2^m)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img591.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img591.gif"></A>'; 
$key = q/{_inline}$kgeq0${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img327.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img327.gif"></A>'; 
$key = q/{_inline}$Q_infty(bQ_2)=bQ_2${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img568.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img568.gif"></A>'; 
$key = q/{_inline}$n~(bmod~2^k)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img154.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img154.gif"></A>'; 
$key = q/{_inline}$S_k={n:n${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img76.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img76.gif"></A>'; 
$key = q/{_inline}$bv_k(n_0)=(x_0(n_0),ldots,x_k-1(n_0))${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img412.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img412.gif"></A>'; 
$key = q/{_inline}$pounds${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img18.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img18.gif"></A>'; 
$key = q/{_inline}$n2${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img3.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img3.gif"></A>'; 
$key = q/{_inline}$p_iq_j${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img404.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img404.gif"></A>'; 
$key = q/{_inline}$(v_0,ldots,v_i)${_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>'; 
$key = q/{_inline}$M(k)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img390.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img390.gif"></A>'; 
$key = q/{_inline}$H^-1(E)=E${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img489.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img489.gif"></A>'; 
$key = q/{_inline}$N=[logxlogd]${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img647.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img647.gif"></A>'; 
$key = q/{_inline}$T^(k)(n);SPM_gt;n${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img440.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img440.gif"></A>'; 
$key = q/{_inline}$m;SPM_gt;d^d(d-1)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img659.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img659.gif"></A>'; 
$key = q/{_inline}$sigma_infty(20)=sigma_infty(21)=6${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img298.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img298.gif"></A>'; 
$key = q/{_inline}$kleq21${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img725.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img725.gif"></A>'; 
$key = q/{_inline}$xi${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img667.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img667.gif"></A>'; 
$key = q/{_inline}$Q_infty(N^+)subseteqfrac13bZ-bZ${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img559.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img559.gif"></A>'; 
$key = q/{_inline}$R={r_i:r_iequivi~(bmod~d)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img636.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img636.gif"></A>'; 
$key = q/{displaymath}pi_rmtotal(x);SPM_gt;x^c_4{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img342.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple342.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img342.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img342.gif"></A>'; 
$key = q/{_inline}$epsilon${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img242.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img242.gif"></A>'; 
$key = q/{displaymath}C_2={bfv:bfv~mboxis~inflating~and~length~(bfv)=k}~.{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img212.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple212.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img212.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img212.gif"></A>'; 
$key = q/{_inline}$barQ_n${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img544.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img544.gif"></A>'; 
$key = q/{tabular}|c|c|c|c|hline_inline$n$_inline;SPM_amp;_inline$sigma(n)$_inline;SPM_amp;_inline$sigma_infty(n)$_inline;SPM_amp;_inline$s(n)$_inlinehline1;SPM_amp;_inline$In$_inline;SPM_amp;2;SPM_amp;27;SPM_amp;7;SPM_amp;11;SPM_amp;3.727;SPM_amp;59;SPM_amp;70;SPM_amp;171._inline$2^50-1$_inline;SPM_amp;143;SPM_amp;383;SPM_amp;_inline$6.37times10^8$_inline_inline$2^50$_inline;SPM_amp;1;SPM_amp;50;SPM_amp;1_inline$2^50+1$_inline;SPM_amp;2;SPM_amp;223;SPM_amp;1.50_inline$2^500-1$_inline;SPM_amp;1828;SPM_amp;4331;SPM_amp;_inline$1.11times10^88$_inline_inline$2^500+1$_inline;SPM_amp;2;SPM_amp;2204;SPM_amp;1.50hline{tabular}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img55.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple55.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img55.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img55.gif"></A>'; 
$key = q/{displaymath}pi_rmtotal(x)=|{n:nleqx~mboxand~$sigma_infty(n);SPM_lt;infty$}|~.{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img340.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple340.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img340.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img340.gif"></A>'; 
$key = q/{_inline}$sigma(n)neqomega(n)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img244.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img244.gif"></A>'; 
$key = q/{displaymath}s(n)=displaystylefracsup_kgeq0T^(k)(n)n~,{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img50.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple50.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img50.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img50.gif"></A>'; 
$key = q/{_inline}$lim_krightarrowinftyT^(k)(n)=In${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img34.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img34.gif"></A>'; 
$key = q/{_inline}$k,lgeq1${_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>'; 
$key = q/{_inline}$g^(k)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img11.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img11.gif"></A>'; 
$key = q/{_inline}$n;SPM_lt;2^40=1.2times10^12${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img695.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img695.gif"></A>'; 
$key = q/{_inline}$Q_infty(-827)=?15${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img572.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img572.gif"></A>'; 
$key = q/{_inline}$omega(n_0)leqk${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img257.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img257.gif"></A>'; 
$key = q/{_inline}$|C_2|${_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>'; 
$key = q/{_inline}$T^(k)(n);SPM_lt;n${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img37.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img37.gif"></A>'; 
$key = q/{_inline}$(32)^k${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img660.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img660.gif"></A>'; 
$key = q/{_inline}$omega(n)=sigma(n)=k${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img174.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img174.gif"></A>'; 
$key = q/{_inline}$f(n)=n3${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img730.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img730.gif"></A>'; 
$key = q/{_inline}$omega(n)=sigma(n)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img378.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img378.gif"></A>'; 
$key = q/{_inline}$(bmod~1)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img664.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img664.gif"></A>'; 
$key = q/{_inline}$1000leqnleq1099${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img307.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img307.gif"></A>'; 
$key = q/{_inline}$phi=log_23${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img398.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img398.gif"></A>'; 
$key = q/{_inline}$bG${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img639.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img639.gif"></A>'; 
$key = q/{_inline}$n_2${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img317.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img317.gif"></A>'; 
$key = q/{_inline}$krightarrowIn${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img81.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img81.gif"></A>'; 
$key = q/{_inline}$sigma_infty(n_1)=r_1geqsigma_infty(n_2)=r_2${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img315.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img315.gif"></A>'; 
$key = q/{_inline}$n_0${_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>'; 
$key = q/{_inline}$barQ_k:bZ2^kbZrightarrowbZ2^2bZ${_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>'; 
$key = q/{_inline}$omega(n_0)neqsigma(n_0)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img258.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img258.gif"></A>'; 
$key = q/{_inline}$sigma_infty(84)=sigma_infty(85)=8${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img300.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img300.gif"></A>'; 
$key = q/{_inline}$bQ_2subseteqbZ_2${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img564.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img564.gif"></A>'; 
$key = q/{_inline}$mu_2(bZ_2)=1${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img478.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img478.gif"></A>'; 
$key = q/{_inline}$ninU_D${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img443.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img443.gif"></A>'; 
$key = q/{displaymath}mboxweight~(bfv)=2^-~mboxlength~(bfv)~.{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img202.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple202.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img202.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img202.gif"></A>'; 
$key = q/{_inline}$omega(n)leqk${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img369.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img369.gif"></A>'; 
$key = q/{_inline}$epsilon_k;SPM_gt;0${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img163.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img163.gif"></A>'; 
$key = q/{_inline}$g_0:bfNrightarrowbN${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img626.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img626.gif"></A>'; 
$key = q/{_inline}$lambda_r_1-1(n_1)=lambda_r_1-1(n_2)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img321.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img321.gif"></A>'; 
$key = q/{displaymath}epsilon_k=1-lambda_k(bfv)=1-displaystylefrac3^v_0+ldots+v_k-12^k{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img166.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple166.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img166.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img166.gif"></A>'; 
$key = q/{_inline}$n_0=2^bh-1${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img418.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img418.gif"></A>'; 
$key = q/{_inline}$f(3n-1)=4n-1${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img737.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img737.gif"></A>'; 
$key = q/{_inline}$omega(n)${_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>'; 
$key = q/{_inline}$rho_i(bfv)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img392.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img392.gif"></A>'; 
$key = q/{_inline}$g:bfNrightarrowbN${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img611.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img611.gif"></A>'; 
$key = q/{_inline}$sigma(n_0)=In${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img387.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img387.gif"></A>'; 
$key = q/{_inline}$k~(bmod~d)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img602.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img602.gif"></A>'; 
$key = q/{_inline}$barQ_m${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img596.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img596.gif"></A>'; 
$key = q/{_inline}$beta${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img458.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img458.gif"></A>'; 
$key = q/{displaymath}epsilon_kgeqk^-c_1~.{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img270.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple270.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img270.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img270.gif"></A>'; 
$key = q/{_inline}$phi${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img399.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img399.gif"></A>'; 
$key = q/{_inline}$etaapprox.05004${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img445.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img445.gif"></A>'; 
$key = q/{_inline}$sigma_infty(n)=In${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img44.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img44.gif"></A>'; 
$key = q/{_inline}$(k(32)^n)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img709.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img709.gif"></A>'; 
$key = q/{_inline}$S(bfv)subseteqS(bfw)${_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>'; 
$key = q/{_inline}${(32)^kxi:1leqk;SPM_lt;infty}${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img666.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img666.gif"></A>'; 
$key = q/{_inline}$x_i(n)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img110.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img110.gif"></A>'; 
$key = q/{_inline}$S_k${_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>'; 
$key = q/{_inline}$f(3n)=2n${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img736.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img736.gif"></A>'; 
$key = q/{_inline}$ngeq2${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img245.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img245.gif"></A>'; 
$key = q/{_inline}$nequiv0~(bmod~3)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img731.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img731.gif"></A>'; 
$key = q/{displaymath}Q_infty(bZ)subseteqdisplaystylefrac1MbZ~.{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img566.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple566.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img566.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img566.gif"></A>'; 
$key = q/{_inline}$c_4${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img341.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img341.gif"></A>'; 
$key = q/{_inline}$H:bZ_2rightarrowbZ_2${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img487.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img487.gif"></A>'; 
$key = q/{_inline}$f^(k)(n)=1${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img742.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img742.gif"></A>'; 
$key = q/{_inline}$S(bfw)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img189.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img189.gif"></A>'; 
$key = q/{_inline}$g(k)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img741.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img741.gif"></A>'; 
$key = q/{_inline}$U^(N)(n);SPM_lt;nx^delta_3}=O(x^1-delta_4)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img655.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img655.gif"></A>'; 
$key = q/{_inline}$bN^+${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img557.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img557.gif"></A>'; 
$key = q/{_inline}$T:bZrightarrowbZ${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img482.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img482.gif"></A>'; 
$key = q/{_inline}$C=C_1cupC_2${_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>'; 
$key = q/{_inline}$[x]${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img146.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img146.gif"></A>'; 
$key = q/{_inline}$n;SPM_lt;2^50${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img59.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img59.gif"></A>'; 
$key = q/{_inline}$(bmod~2^k)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img684.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img684.gif"></A>'; 
$key = q/{_inline}$b;SPM_lt;10^199${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img426.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img426.gif"></A>'; 
$key = q/{_inline}$1leql;SPM_lt;k${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img579.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img579.gif"></A>'; 
$key = q/{_inline}$I_k${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img191.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img191.gif"></A>'; 
$key = q/{_inline}$phi=[1;1,1,2,2,3,1,5,2,23,2,2,1,1,55,1,4,3ldots]${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img401.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img401.gif"></A>'; 
$key = q/{displaymath}-1=1+1cdot2+1cdot2^2+cdots~.{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img481.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple481.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img481.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img481.gif"></A>'; 
$key = q/{_inline}$0leqlleqk${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img589.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img589.gif"></A>'; 
$key = q/{_inline}$rho_k${_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>'; 
$key = q/{_inline}$sigma(n);SPM_lt;In${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img375.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img375.gif"></A>'; 
$key = q/{_inline}$Q_infty(3)=-203${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img561.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img561.gif"></A>'; 
$key = q/{_inline}$S(bfv)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img184.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img184.gif"></A>'; 
$key = q/{_inline}${(32)^k:1leqk;SPM_lt;infty}${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img662.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img662.gif"></A>'; 
$key = q/{_inline}$c_0;SPM_gt;0${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img264.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img264.gif"></A>'; 
$key = q/{_inline}$S_2=#{nleq2^k:omega(n)leqk${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img288.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img288.gif"></A>'; 
$key = q/{_inline}$x_i${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img365.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img365.gif"></A>'; 
$key = q/{displaymath}F(k)=lim_xrightarrowinftydisplaystylefrac1x#{n:nleqx~~mboxand~$sigma(n)leqk}${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>'; 
$key = q/{_inline}$bZ${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img508.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img508.gif"></A>'; 
$key = q/{_inline}$bfw=(v_0,v_1,ldots,v_i)${_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>'; 
$key = q/{_inline}$(32)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img711.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img711.gif"></A>'; 
$key = q/{_inline}$mu_2${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img491.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img491.gif"></A>'; 
$key = q/{_inline}$c_1=2c_2+c_3+2c_5${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img296.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img296.gif"></A>'; 
$key = q/{_inline}$k(32)^n${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img704.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img704.gif"></A>'; 
$key = q/{_inline}$nequiv0${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img733.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img733.gif"></A>'; 
$key = q/{displaymath}|{n:nleqx~mboxand~sigma(n)=infty}|=|pi^ast(x)-x|leqS_1+S_2~,{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>'; 
$key = q/{_inline}$delta_1,delta_2;SPM_gt;0${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img646.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img646.gif"></A>'; 
$key = q/{_inline}$n(bfv)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img364.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img364.gif"></A>'; 
$key = q/{_inline}$Q_infty(-263)=-54${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img570.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img570.gif"></A>'; 
$key = q/{_inline}$x_n+1=x_n2${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img701.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img701.gif"></A>'; 
$key = q/{_inline}$Q_infty(10)=-263${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img569.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img569.gif"></A>'; 
$key = q/{_inline}$omega(n)neqsigma(n)}${_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>'; 
$key = q/{_inline}$omega(n_0)=ileqk${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img386.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img386.gif"></A>'; 
$key = q/{displaymath}T^(i)(n)geqdisplaystylefrac3^v_0+ldots+v_i-12^ingeqn~~mboxfor~~$1leqileqk-1$~,{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img161.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple161.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img161.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img161.gif"></A>'; 
$key = q/{_inline}$s(n)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img49.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img49.gif"></A>'; 
$key = q/{displaymath}S(bfv)={n:bfv=(x_0(n),ldots,x_k-1(n))}~.{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img131.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple131.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img131.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img131.gif"></A>'; 
$key = q/{_inline}$Q_k:bZrightarrowbZ2^kbZ${_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>'; 
$key = q/{_inline}$||_2${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img468.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img468.gif"></A>'; 
$key = q/{_inline}$(bmod~2)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img435.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img435.gif"></A>'; 
$key = q/{_inline}$Q_k(n)${_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>'; 
$key = q/{_inline}$lambda_k(n),~rho_k(n))${_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/{displaymath}T^(i)(n_0);SPM_gt;T^(i+1)(n_0);SPM_gt;cdots;SPM_gt;T^(k)(n_0)=n_0~,{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img411.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple411.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img411.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img411.gif"></A>'; 
$key = q/{_inline}$~~~Box${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img279.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img279.gif"></A>'; 
$key = q/{displaymath}d(I_k)=displaystylefrac12^k#{bfv:bfv~mboxis~admissible~and~of~length~$k$}~.{displaymath}/;
$cached_env_img{$key} ='<A HREF=/cgi-bin/organics/nmpform?mfname=img195.mpl><IMG ALIGN=RIGHT BORDER=0 SRC="../gif/maple/maple195.gif"> </A><A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img195.tex"  ><IMG BORDER=0 ALIGN=BOTTOM ALT=""" SRC="../gif/img195.gif"></A>'; 
$key = q/{_inline}$k,lgeq0${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img502.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img502.gif"></A>'; 
$key = q/{_inline}$Q_infty(1)=sum_i=0^infty2^2n=-13,~Q_infty(2)=-23${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img560.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img560.gif"></A>'; 
$key = q/{_inline}$k_0${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img259.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img259.gif"></A>'; 
$key = q/{_inline}$(bmod~d)${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img638.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img638.gif"></A>'; 
$key = q/{_inline}$T^(i)(n_0);SPM_gt;n_0${_inline}/;
$cached_env_img{$key} ='<A HREF="/cgi-bin/organics/evaleqn.pl?tfname=img251.tex"  ><IMG BORDER=0 ALIGN=MIDDLE ALT=""" SRC="../gif/img251.gif"></A>'; 
$key = q/{_inline}$lambda_k${_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>'; 

1;

