{VERSION 5 0 "IBM INTEL LINUX" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 307 "#This program calcu lates the values at and successive ratios of the points q0, q2, q01, a nd q12 on the F2 contraction of the Sierpinski Gasket with a singulari ty. q1 is equal to zero, and therefore unnecessary to program. To ru n this program, first run \"RunV2\" to calculate the alphas, betas, b' s and c's." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "j:=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"jG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "q[2]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "summing2[0]:=sum('(1/5^(j-n))*(b[ n]*beta[j-n]+3*c[n]*alpha[j-n+1])','n'=0..j):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "q[0]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "summi ng0[0]:=sum('(1/5^(j-n))*(b[n]*beta[j-n]-3*c[n]*alpha[j-n+1])','n'=0.. j):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 384 "#Here I am calculating the values of interm ediate points I will need for the calculation of values at our necessa ry points. Delta is the value of q01 and q12 on each successive level of P2, and epsilon is the value of q12 and -q01 on each successive le vel of P3 (since P3 is skew symmetric). These values will need to be s ummed with appropriate coefficients to get the values for P4." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 29 "delta[j]:=3*5^(-j-1)*beta[j]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "epsilon[j]:=5^(-j-1)*(3*alpha[j+1]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "#Here I'm setting initial values of q01 and q12." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 78 "summing01[0]:=sum('(1/5^(j-n))*(b[n]*delta[j-n]-c[n ]*epsilon[j-n])','n'=0..j):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "summing12[0]:=sum('(1/5^(j-n))*(b[n]*delta[j-n]+c[n]*epsilon[j-n]) ','n'=0..j):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 445 "#Here \+ we run the recursion relation which allows us to calculate the values \+ of the points q0, q2, q01, and q12 on Pj4. q1 is 0 on all levels, and q02 has it's own program. To find the formulas for these, I refer yo u to the \"Calculus on the Sierpinski Gasket\" paper by Professor Stri chartz, Jonathon Needleman, and Po-Lam Yung. You can calculate these u p to an arbitrary level, but it needs to be less than or equal to the \+ level set in \"RunV2\"." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " for j from 1 to 22 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "summing0[j ]:=sum('(1/5^(j-n))*(b[n]*beta[j-n]-3*c[n]*alpha[j-n+1])','n'=0..j);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "summing2[j]:=sum('(1/5^(j-n))*(b[ n]*beta[j-n]+3*c[n]*alpha[j-n+1])','n'=0..j);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "delta[j]:=3*5^(-j-1)*beta[j];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "epsilon[j]:=5^(-j-1)*(3*alpha[j+1]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "summing01[j]:=sum('(1/5^(j-n))*(b[n]*delta[j-n]- c[n]*epsilon[j-n])','n'=0..j);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "s umming12[j]:=sum('(1/5^(j-n))*(b[n]*delta[j-n]+c[n]*epsilon[j-n])','n' =0..j);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "#Now we evaluate the values \+ at point q0 for each successive level." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "for j from 0 to 22 do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "summing0[j]:=evalf(summing0[j],20);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)summin g0G6#\"\"!$\"\"$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)summing0G6# \"\"\"$!5LLLLLLLLLt!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)summing0 G6#\"\"#$!5@Vl()4Kaw)4#!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)summ ing0G6#\"\"$$!5cCf)4>2;BD*!#D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)s umming0G6#\"\"%$!5)y&G9bgp>#f\"!#F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >&%)summing0G6#\"\"&$!50#[`6,)*4FT\"!#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)summing0G6#\"\"'$!5Q-)Q@$355pu!#M" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)summing0G6#\"\"($!5\"Q\\y,)3N3!f#!#P" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%)summing0G6#\"\")$!5S%y=zz(*3$*G'!#T" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)summing0G6#\"\"*$!549esrx]Q@6!#W" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)summing0G6#\"#5$!5EY3v%y.:>_\"!# [" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)summing0G6#\"#6$!5W*y))G2h([= ;!#_" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)summing0G6#\"#7$!5#Q;XBFU( 3z8!#c" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)summing0G6#\"#8$!5Bjx#3E tLWj*!#h" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)summing0G6#\"#9$!5!ph@ !)ys`J^&!#l" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)summing0G6#\"#:$!5L KXSQ**HPwG!#p" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)summing0G6#\"#;$! 5L1^-@zX\"Q`'!#u" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)summing0G6#\"# <$!5n%)R$o%Rr01=!#x" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)summing0G6# \"#=$\"5^%3)[i!Q)G6S!#\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)summi ng0G6#\"#>$!5^0Gq'en)*RA\"!#%)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%) summing0G6#\"#?$\"5]?5Q\"fKS/g$!#))" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>&%)summing0G6#\"#@$!5#\\BB%\\r_bi5!#\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)summing0G6#\"#A$\"52GA8F_1(\\8$!#&*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "#Here are the point values for q2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "for j from 0 to 22 do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "summing2[j]:=evalf(summing2[j],20);" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)summing2G6#\"\"!$\"\"\"F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)summing2G6#\"\"\"$!5AAAAAAAAA7!#? " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)summing2G6#\"\"#$!5,'yj:M>r*[< !#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)summing2G6#\"\"$$!5!>!3\"Hm pJ^&Q!#B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)summing2G6#\"\"%$!5#R& 4)[6+xqJ$!#D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)summing2G6#\"\"&$! 5!Qa^\\f*Gdr9!#F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)summing2G6#\" \"'$!5df*Qn^xc,*Q!#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)summing2G6 #\"\"($!5iF[Y]P\"4]u'!#L" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)summin g2G6#\"\")$!5&Q3S5%)R/#*=)!#O" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)s umming2G6#\"\"*$!5\")oK&%)summing2G6#\"#5$!5K%\\_EQ^]T&\\!#U" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)summing2G6#\"#6$!5S(fr?*yrDME!#X" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)summing2G6#\"#7$!5PGU6r&%)summing2G6#\"#8$!5bjznm@6E?R!#_" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)summing2G6#\"#9$!50Dj9P(*Rl@6!#b" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)summing2G6#\"#:$!56+1\"yT#z*f#H!#f" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)summing2G6#\"#;$!5ri!)=#z%*pKK$!#j" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)summing2G6#\"#<$!5'>N#\\*4#3/$f%!# m" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)summing2G6#\"#=$\"5H1*on!Rph+ ^!#p" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)summing2G6#\"#>$!5Mm(GkK+u >y(!#s" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)summing2G6#\"#?$\"5#oHyD b:\\X9\"!#u" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)summing2G6#\"#@$!55 -*>5.7')))o\"!#x" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)summing2G6#\"# A$\"5;z!4K\"G3X\"\\#!#!)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "#Here is q01." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 23 "for j from 0 to 22 do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "summing01[j]:=evalf(summing01[j],20);" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*summing01G6#\"\"!$\"5+++++++++9!#> " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*summing01G6#\"\"\"$!566666666 \">(!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*summing01G6#\"\"#$!57/I *=yq'R9L!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*summing01G6#\"\"$$! 5(p&o!*)o3.l\"=!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*summing01G6# \"\"%$!5AsQ!o-W)yjI!#F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*summing0 1G6#\"\"&$!5f8-Pp&z=L9$!#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*summ ing01G6#\"\"'$!5jPa5#QU5y!Q!#M" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%* summing01G6#\"\"($!5%o6T7N7%fUW!#O" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >&%*summing01G6#\"\")$\"5fX#z8+%Hm$Q\"!#Q" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*summing01G6#\"\"*$!5W2EH9/V*p)\\!#T" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*summing01G6#\"#5$\"5@&)='*H#HC@y\"!#V" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*summing01G6#\"#6$!5U@#\\dQ$oGvj!#Y " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*summing01G6#\"#7$\"50(zZUN5k6G #!#[" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*summing01G6#\"#8$!58w2(z&= (4J;)!#^" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*summing01G6#\"#9$\"5R[ 3)G_vr7#H!#`" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*summing01G6#\"#:$! 5W/>P0Q4VX5!#b" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*summing01G6#\"#; $\"5vpog'=E#HTP!#e" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*summing01G6# \"#<$!5K7vD(y*H!*Q8!#g" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*summing0 1G6#\"#=$\"5xIuqim;c\"z%!#j" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*sum ming01G6#\"#>$!5$)fA1#*Gvw9&% *summing01G6#\"#?$\"5pl$R=UO!oOh!#o" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>&%*summing01G6#\"#@$!5)4Nzol$*[h>#!#q" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*summing01G6#\"#A$\"5lqM)3ih7%fy!#t" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "# \+ And finally, q12." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "for j \+ from 0 to 22 do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "summing12[j]:= evalf(summing12[j],20);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*summing12G6#\"\"!$\"\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> &%*summing12G6#\"\"\"$!5cbbbbbbbbz!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*summing12G6#\"\"#$!5,'yj:M>r*[j!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*summing12G6#\"\"$$!5-X2<#R'zr&%*summing12G6#\"\"%$!5e9b*4<-Q`a$!#E" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*summing12G6#\"\"&$!51!>*H$=N22:)!#H" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*summing12G6#\"\"'$!5vQ3pCFnR66!#J " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*summing12G6#\"\"($!5&[()4zEXv@ !**!#N" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*summing12G6#\"\")$!5-7E( >ygXs<'!#Q" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*summing12G6#\"\"*$!5 &%*summing12G6 #\"#5$!5f<*H,s>)oJ**!#X" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*summing 12G6#\"#6$!5cx1=BB&)[-D!#[" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*summ ing12G6#\"#7$!5F*4'z(4nq3S)!#_" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%* summing12G6#\"#8$\"50H*Hn6'G&*=F!#b" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>&%*summing12G6#\"#9$!5\\5Td%)QbV3d!#e" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*summing12G6#\"#:$\"5J(Rdtw)>*G<)!#h" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*summing12G6#\"#;$!5AMD)H'o&e(37!#j" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*summing12G6#\"#<$\"5Xv!oI!*HsGy\"!#m" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*summing12G6#\"#=$!57SH1m\"Hv,j#!#p" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*summing12G6#\"#>$\"5#*Qv-PA)4,)Q!# s" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*summing12G6#\"#?$!5(*>]ZO.E0C d!#v" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*summing12G6#\"#@$\"5'Hs!*= F?\"HW%)!#y" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*summing12G6#\"#A$!5 QfT.&>uEdC\"!#!)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 208 "#The numbers aren't very pretty, but they're not r eally what we're interested in anyway. What we really want to see is \+ the ratio of point values from one level to the next, which should be \+ known eigenvalues." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for j from 1 to 22 do" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "ratio0[j]:=summing0[j-1]/summing0[j ];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio0G6#\"\"\"$!+\"4444%!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio0G6#\"\"#$\"+Zw6%\\$!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio0G6#\"\"$$\"+MwOoA!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio0G6#\"\"%$\"+!RP5\"e!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio0G6#\"\"&$\"+U:0F6!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio0G6#\"\"'$\"+WcS\"*=!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio0G6#\"\"($\"+.)HP)G!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio0G6#\"\")$\"+aAB=T!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio0G6#\"\"*$\"+?&=&3c!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio0G6#\"#5$\"+)>]#ot!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio0G6#\"#6$\"+efJ.%*!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio0G6#\"#7$\"+#>$ft6!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio0G6#\"#8$\"+K^TJ9!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio0G6#\"#9$\"+Se`Z&%'ratio0G6#\"#:$\"+QLq;>!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio0G6#\"#;$\"+guG-W!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio0G6#\"#<$\"+rIs&%'ratio0G6#\"#=$!+VlV-X!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio0G6#\"#>$!+N**>xK!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio0G6#\"#?$!+50e*R$!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio0G6#\"#@$!+^MZ)Q$!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio0G6#\"#A$!+PFO*Q$!\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "#T he ratio for q0 is 5 times the first eigenvalue in the 6-series" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 21 "for j from 1 to 50 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "ratio2[j]:=summing2[j-1]/summing2[j];" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'rati o2G6#\"\"\"$!+$====)!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2 G6#\"\"#$\"+%HN#))p!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G 6#\"\"$$\"+n_tOX!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6# \"\"%$\"+yu?i6!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\" \"&$\"+$3.TD#!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"\" '$\"+)G6Gy$!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"\"($ \"+2'fuw&!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"\")$\" +4XYO#)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"\"*$\"+/ Pq@6!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"#5$\"+R+lt9 !\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"#6$\"+#>j1)=!\" '" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"#7$\"+%Q'=ZB!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"#8$\"+j-$G'G!\"'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"#9$\"+#or]\\$!\"'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"#:$\"+umSLQ!\"'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"#;$\"+@\\d/))!\"'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"#<$\"+ShWNs!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"#=$!+*3t[+*!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"#>$!+q)*Ral!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"#?$!+<5;*z'!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"#@$!+.p%px'!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"#A$!+wasyn!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"#B$!+0()fyn!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"#C$!+\"z1'yn!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"#D$!+Ejgyn!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"#E$!+`jgyn!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"#F$!+[jgyn!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"#G$!+]jgyn!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"#H$!+\\jgyn!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"#I$!+^jgyn!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"#J$!+\\jgyn!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"#K$!+_jgyn!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"#L$!+Zjgyn!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"#M$!+^jgyn!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"#N$!+[jgyn!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"#O$!+]jgyn!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"#P$!+]jgyn!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"#Q$!+_jgyn!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"#R$!+\\jgyn!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"#S$!+\\jgyn!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"#T$!+]jgyn!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"#U$!+\\jgyn!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"#V$!+\\jgyn!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"#W$!+]jgyn!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"#X$!+]jgyn!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"#Y$!+]jgyn!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"#Z$!+]jgyn!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"#[$!+]jgyn!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"#\\$!+]jgyn!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'ratio2G6#\"#]$!+\\jgyn!\"(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 58 "#The ratio for q2 is the first eigenvalue in the 6- series." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for j from 1 to 50 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "ratio01[j]:=summing01[j-1]/summing01[j];" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio01G6#\"\"\"$!+gz%o%>!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>&%(ratio01G6#\"\"#$\"+S\"f'p@!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>&%(ratio01G6#\"\"$$\"+\\GgC=!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>&%(ratio01G6#\"\"%$\"+bU%*Gf!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >&%(ratio01G6#\"\"&$\"+Kg)pu*!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> &%(ratio01G6#\"\"'$\"+0Y#\\D)!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> &%(ratio01G6#\"\"($\"+5=9r&)!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>& %(ratio01G6#\"\")$!++)[2@$!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%( ratio01G6#\"\"*$!+_GauF!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(rat io01G6#\"#5$!+)eU$)z#!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio 01G6#\"#6$!+7LO&z#!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio01G 6#\"#7$!+9:v%z#!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio01G6# \"#8$!+*GzWz#!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio01G6#\"# 9$!+#[oVz#!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio01G6#\"#:$! +QIK%z#!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio01G6#\"#;$!+(Q /Vz#!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio01G6#\"#<$!+DnH%z #!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio01G6#\"#=$!+vNH%z#! \"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio01G6#\"#>$!+zAH%z#!\"( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio01G6#\"#?$!+Y&%(ratio01G6#\"#@$!+E:H%z#!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio01G6#\"#A$!+P9H%z#!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio01G6#\"#B$!+)R\"H%z#!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio01G6#\"#C$!+%Q\"H%z#!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio01G6#\"#D$!+x8H%z#!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio01G6#\"#E$!+t8H%z#!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio01G6#\"#F$!+u8H%z#!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio01G6#\"#G$!+t8H%z#!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio01G6#\"#H$!+t8H%z#!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio01G6#\"#I$!+t8H%z#!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio01G6#\"#J$!+s8H%z#!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio01G6#\"#K$!+t8H%z#!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio01G6#\"#L$!+t8H%z#!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio01G6#\"#M$!+s8H%z#!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio01G6#\"#N$!+u8H%z#!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio01G6#\"#O$!+s8H%z#!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio01G6#\"#P$!+s8H%z#!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio01G6#\"#Q$!+u8H%z#!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio01G6#\"#R$!+t8H%z#!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio01G6#\"#S$!+t8H%z#!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio01G6#\"#T$!+t8H%z#!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio01G6#\"#U$!+s8H%z#!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio01G6#\"#V$!+t8H%z#!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio01G6#\"#W$!+t8H%z#!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio01G6#\"#X$!+s8H%z#!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio01G6#\"#Y$!+s8H%z#!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio01G6#\"#Z$!+t8H%z#!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio01G6#\"#[$!+t8H%z#!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio01G6#\"#\\$!+s8H%z#!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio01G6#\"#]$!+t8H%z#!\"(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 67 "#The ratio for q01 is 5 times the first eigenvalue \+ in the 5-series." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for j from 1 to 50 do" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "ratio12[j]:=summing12[j-1]/summing1 2[j];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"\"\"$!+SK)pD\"!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"\"#$\"+4k/`7!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"\"$$\"+/x[E#)!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"\"%$\"+!zjo<#!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"\"&$\"+#[I(\\V!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"\"'$\"+W9vLt!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"\"($\"+BjPA6!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"\")$\"+:#3Ig\"!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"\"*$\"+tU'f>#!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"#5$\"+:uMKG!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"#6$\"+uZsoR!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"#7$\"+UU%)yH!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"#8$!+w[u*3$!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"#9$!+>Q/jZ!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"#:$!+?of%)p!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"#;$!+;KRhn!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"#<$!+%=Q)zn!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"#=$!+I.`yn!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"#>$!+L/hyn!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"#?$!+dhgyn!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"#@$!+djgyn!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"#A$!+\\jgyn!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"#B$!+^jgyn!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"#C$!+\\jgyn!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"#D$!+[jgyn!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"#E$!+]jgyn!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"#F$!+]jgyn!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"#G$!+]jgyn!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"#H$!+^jgyn!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"#I$!+[jgyn!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"#J$!+^jgyn!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"#K$!+]jgyn!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"#L$!+\\jgyn!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"#M$!+^jgyn!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"#N$!+[jgyn!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"#O$!+\\jgyn!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"#P$!+^jgyn!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"#Q$!+]jgyn!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"#R$!+\\jgyn!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"#S$!+`jgyn!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"#T$!+Yjgyn!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"#U$!+]jgyn!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"#V$!+^jgyn!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"#W$!+]jgyn!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"#X$!+\\jgyn!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"#Y$!+\\jgyn!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"#Z$!+_jgyn!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"#[$!+]jgyn!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"#\\$!+]jgyn!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(ratio12G6#\"#]$!+\\jgyn!\"(" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 " #The ratio fo q12 is the first eigenvalue in the 6-series." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "#As you can see, these numbers were much prettier, a nd in fact are familiar values related to the Dirichlet Eigenvalues." }}}}{MARK "53 0 0" 58 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }