{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 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 
0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 
{CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 
0 0 0 0 0 -1 0 }}
{SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 315 "#These sigmas are the sums related to the normal der
ivative of the Dirichlet Eigenfunction.  bigsig is used to represent a
n uppercase Sigma, which is reserved.  First, we will run them for the
 2-series.  In case it is unclear, small-sigma at level zero is only t
he j=1 value, and at level one is only the j=2 value." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "read \"values_2.m\";" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "sigma:=sigma;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&s
igmaGF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "sigma[0]:=dn[1]^
2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&sigmaG6#\"\"!$\"+&>7V#H!\")
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for k from 1 to 9 do" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "sigma[k]:=sum('(dn[j])^2','j'=(2^(
k-1)+1)..2^k);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>&%&sigmaG6#\"\"\"$\"+tMK$)H!\"(" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>&%&sigmaG6#\"\"#$\"+[)QjW#!\"'" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>&%&sigmaG6#\"\"$$\"+s)HW.#!\"&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%&sigmaG6#\"\"%$\"+hcL&p\"!\"%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%&sigmaG6#\"\"&$\"+a'zFT\"!\"$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%&sigmaG6#\"\"'$\"+$H;t<\"!\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%&sigmaG6#\"\"($\"+(=p4\")*!\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%&sigmaG6#\"\")$\"+c!3e<)!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%&sigmaG6#\"\"*$\"+[T<8o\"\"!" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 21 "for k from 0 to 9 do " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "bigsig[k]:=sum('(dn[j])^2','j'=1..2^k)" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'bigs
igG6#\"\"!$\"+&>7V#H!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'bigsig
G6#\"\"\"$\"+#padF$!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'bigsigG
6#\"\"#$\"+<V\"Rx#!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'bigsigG6
#\"\"$$\"+/8#=J#!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'bigsigG6#
\"\"%$\"+\"z<l#>!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'bigsigG6#
\"\"&$\"+L9V0;!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'bigsigG6#\"
\"'$\"+O%fyL\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'bigsigG6#\"
\"($\"+iG)[6\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'bigsigG6#\"
\")$\"+=4p!H*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'bigsigG6#\"
\"*$\"+VKCUx\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for k \+
from 0 to 9 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "littlesig2growth[
k]:=(sigma[k])/((25/3)^k);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%1littlesig2growthG6#\"\"!$\"+&>7V#
H!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%1littlesig2growthG6#\"\"\"
$\"+o\"))*zN!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%1littlesig2grow
thG6#\"\"#$\"+TzsAN!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%1littles
ig2growthG6#\"\"$$\"+>[\\:N!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%
1littlesig2growthG6#\"\"%$\"+F![a^$!\")" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>&%1littlesig2growthG6#\"\"&$\"+pyW:N!\")" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%1littlesig2growthG6#\"\"'$\"+9wW:N!\")" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>&%1littlesig2growthG6#\"\"($\"+UwW:N!\")" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%1littlesig2growthG6#\"\")$\"+9yW:N!
\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%1littlesig2growthG6#\"\"*$\"
+5#[a^$!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for k from 0
 to 9 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "bigsig2growth[k]:=(bigs
ig[k])/((25/3)^k);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>&%.bigsig2growthG6#\"\"!$\"+&>7V#H!\")" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%.bigsig2growthG6#\"\"\"$\"+Ic!4$R!
\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%.bigsig2growthG6#\"\"#$\"+;m
V%*R!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%.bigsig2growthG6#\"\"$$
\"+8s#[*R!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%.bigsig2growthG6#
\"\"%$\"+\"HF[*R!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%.bigsig2gro
wthG6#\"\"&$\"+Vr#[*R!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%.bigsi
g2growthG6#\"\"'$\"+qo#[*R!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%.
bigsig2growthG6#\"\"($\"+lo#[*R!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>&%.bigsig2growthG6#\"\")$\"+Qq#[*R!\")" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>&%.bigsig2growthG6#\"\"*$\"+du#[*R!\")" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 33 "#Now we will work on the 5-series" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "read \"values_5.m\";" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "sigma:=sigma;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&sigmaGF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
19 "sigma[0]:=dn5[1]^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&sigmaG6
#\"\"!$\"+fQ1J<!\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for \+
k from 1 to 9 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "sigma[k]:=sum('
(dn5[j])^2','j'=(2^(k-1)+1)..2^k);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
3 "od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&sigmaG6#\"\"\"$\"+rZ8*>#
!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&sigmaG6#\"\"#$\"++o#zr$!\"
'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&sigmaG6#\"\"$$\"+[PQcI!\"&" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&sigmaG6#\"\"%$\"+b$GIa#!\"%" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&sigmaG6#\"\"&$\"+s%p\">@!\"$" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&sigmaG6#\"\"'$\"+9W(fw\"!\"#" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&sigmaG6#\"\"($\"+B`kr9!\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&sigmaG6#\"\")$\"+[7PE7\"\"!" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&sigmaG6#\"\"*$\"+Lg(>-\"\"\"\"" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for k from 0 to 9 do " }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "bigsig[k]:=sum('(dn5[j])^2','j'=1..
2^k)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%'bigsigG6#\"\"!$\"+fQ1J<!\"(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%'bigsigG6#\"\"\"$\"+I')>IR!\"(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%'bigsigG6#\"\"#$\"+jm%46%!\"'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%'bigsigG6#\"\"$$\"+:%yuY$!\"&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%'bigsigG6#\"\"%$\"+'>w(*)G!\"%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%'bigsigG6#\"\"&$\"+#4Z\"3C!\"$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%'bigsigG6#\"\"'$\"+A\"*y1?!\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%'bigsigG6#\"\"($\"+NUKs;!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%'bigsigG6#\"\")$\"+pOg$R\"\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%'bigsigG6#\"\"*$\"++kLh6\"\"\"" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 20 "for k from 0 to 9 do" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 43 "littlesig5growth[k]:=(sigma[k])/((25/3)^k);" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%
1littlesig5growthG6#\"\"!$\"+fQ1J<!\"(" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>&%1littlesig5growthG6#\"\"\"$\"+D<'*QE!\")" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%1littlesig5growthG6#\"\"#$\"+#f9QN&!\")" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>&%1littlesig5growthG6#\"\"$$\"+<6V\"G&!\")" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%1littlesig5growthG6#\"\"%$\"+(fBKF&
!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%1littlesig5growthG6#\"\"&$
\"+\"yrJF&!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%1littlesig5growth
G6#\"\"'$\"+X8<t_!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%1littlesig
5growthG6#\"\"($\"+m7<t_!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%1li
ttlesig5growthG6#\"\")$\"+\"*=<t_!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>&%1littlesig5growthG6#\"\"*$\"+b=<t_!\")" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 20 "for k from 0 to 9 do" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "bigsig5growth[k]:=(bigsig[k])/((25/3)^k);" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%.b
igsig5growthG6#\"\"!$\"+fQ1J<!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
&%.bigsig5growthG6#\"\"\"$\"+c$Qir%!\")" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>&%.bigsig5growthG6#\"\"#$\"+&>j(>f!\")" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%.bigsig5growthG6#\"\"$$\"+,F!=*f!\")" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>&%.bigsig5growthG6#\"\"%$\"+?*RA*f!\")" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>&%.bigsig5growthG6#\"\"&$\"+s0C#*f!\")" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%.bigsig5growthG6#\"\"'$\"+5-C#*f!\"
)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%.bigsig5growthG6#\"\"($\"+!4SA
*f!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%.bigsig5growthG6#\"\")$\"
+\"pSA*f!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%.bigsig5growthG6#\"
\"*$\"+R2C#*f!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "#As you can see, in each ca
se, the values of lowercase sigma and uppercase sigma are proportional
 to (25/3)^k where k is from the number of points being 2^k.  " }}}}
{MARK "18 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 
2 33 1 1 }