{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 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "with(plots): read \" values_2.m\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 317 "#Below I'm graphing the normal der ivative of the heat kernel over time for the 2-series. The graph take s a while to plot, so be patient. The graph is on a logarithmic scale , so as time goes on , you move down and to the left. You can change \+ the time scale as well, by changing the x-range within the plot functi on." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 59 "summ:=log(sum('(exp(-(El[j])*x))*((dn[j])^2) ','j'=1..256)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "plot([log(1/x),summ,x=10^(- 9)..1], title=\"Plot of function with norm at same point on j=256, 2-s eries\");" }}{PARA 13 "" 1 "" {GLPLOT2D 375 375 375 {PLOTDATA 2 "6&-%' CURVESG6$7gn7$$\"IToyFf@4B>;c6TYp$eEB2#!#Q$\"I_/iud9B7&Q\"o:_xIs'*>j?F *7$$\"Ih?vax]^hN02.^7C'H5.bqKH&p&*G@92N^2nlF07$$\"INqvKAy=-Q5Kf@ KH')*)4$>'F0$\"IE%z*H\\85!y\"p+OVamm&Qw-'F07$$\"I6-rCuH=lS(Q\\!=:=OqT0 fF0$\"Icju2_Q4(3%e[;5&**eU5%=cF07$$\"I?XlF%*H8S+A\"=gW8/0_**\\&F0$\"IN L%[))=&Rt!)>AiMuf[rwr]F07$$\"I/=R-]XCZF07$$\"Iclxqghf*4u^E!3O>#*\\!o![F0$\"I03;UY'f8_^'H?0o'[<3_@% F07$$\"IZ.u@qP>*e69ni&oK]H7>XF0$\"I.*o$)GZ!R8\\(e)H')oXNCzJQF07$$\"IRY :KK^\"f^,n))>s.G(yl8TF0$\"Ie1=FRt\"*\\8xnsixqi'fcL$F07$$\"Ig+$)Hs7Wv'f xjkXWc\"e(f#QF0$\"I!=G,NWdG)z(pEUHZ6;&e%3$F07$$\"IZ)HE437.IxutPRN2+yZY $F0$\"IgU[$H)p8K+B!p\\+&>/j-fGF07$$\"I._5#Q%=.(*))\\$R5?63z')**>$F0$\" I#zmJjS**3Np92T>'[U;J\"p#F07$$\"IJPYJZB.]!o]X#yqq5d9zFF0$\"IgGh$yu3!=N ()Qb:.%f*H_JBF07$$\"I;gh)>8T``'[*\\'*o?Fa\"\\1sb;8e_`M' 3r/(>F07$$\"I>(3v76m8\"z.!*R8-kMNYaAF0$\"I?&\\K[4A0yq4@suIcmW6h\"F07$$ \"IeB3:I_T9#*\\k\\q`8\\s^\"3#F0$\"I'Gf&[@TL-mjk4D]_\\J+y7F07$$\"I4&e0T hJfF`!3>z*=%*zr#H>F0$\"Izb(RVqJp,]'p.^_\"f1W0L*!#S7$$\"I4iz'y**QMplv4Uza1,7M(*[jOO?BzIw&F]q7$$\"IeVuBIzWE(>!Rgp/FG;Ot;F0 $\"ILSt\"4mpVbAeg$\\X7.`02AF]q7$$\"I4X]34xbPFjN:XP\\O`mj:F0$!Inh6`9,Hu P#\\U8%Hm)R_2X\"F]q7$$\"I!))pP[K)yF`#fQ)4hxxN5w9F0$!I?#)QV\"F07$$\" I)[z!e]R%p.0f?p\\FlHq)G7F0$!Ig/;%eI!G/'Rz?B)\\I4J5X:F07$$\"I%fm_KJQ:+ \"yN!*ob7/L6m6F0$!Ig*Q5Z^L/2T\"Qw6KR%Q5Q'=F07$$\"I'3_(eMJ&)G&**)e6Vl,R (yi4\"F0$!I#y%*\\G^>f)=n?xR2]gTyUAF07$$\"IZPTvtV\\UTccFj@)G+62/\"F0$!I &p.UKw!f9-'=1k'e$*eY#Qc#F07$$\"IAmQ&yxEq@Z![LU`+t8L(z*F]q$!I!4fZd2w^Ol Fg#))RFz5FPHF07$$\"I'pWejVwLwC_**[!z^.+&pG*F]q$!IjHwCj#yE6RL;o'RLRW$yE $F07$$\"Id/ab(H9oD$Go%4%er]4Tb()F]q$!I6Iwuv5I/Oj&*e?RzO[^IOF07$$\"IK1) [e))HD$zl=_Oguo4Cu#)F]q$!I7Iwu+<,SwIL^f'=;Mse(RF07$$\"IciRP!Gh89:pLkrc ;17dz(F]q$!I7IwC)))[z)3:+12>_nm@OVF07$$\"I2%**\\toiks>eb0zl*4CdvtF]q$! 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Since the graph was parametric, if you would lie to find a val ue at a certain spot on the x-axis, say x=a, you'll need to check the \+ value at exp(-a), since x-axis-location=log(1/time)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "intvalue:= x -> lo g(sum('(exp(-(El[j])*x))*((dn[j])^2)','j'=1..256));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%)intvalueGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$log G6#-%$sumG6$.*&-%$expG6#,$*&&%#ElG6#%\"jG\"\"\"9$F=!\"\"F=)&%#dnGF;\" \"#F=/.F<;F=\"$c#F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " evalf(intvalue(1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!I,jZKwXdhsz\" yH9G!\\a.W8!#Q" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "evalf(int value(exp(-13)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"Ie;\\/G\"e3a&3 ]Ek$*z]oC+:!#Q" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "evalf(int value(exp(-14)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"I)=azSo@c(z#>$ yExJ&[7(G;!#Q" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "evalf(intv alue(exp(-15)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"I'eTRM8T+HNeBX3 ig1k*f " 0 "" {MPLTEXT 1 0 26 "evalf(intvalu e(exp(-16)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"I.Hl6rPcl " 0 "" {MPLTEXT 1 0 26 "evalf(intvalue(e xp(-17)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"IXK*)3)=4S5.p=e`q;9'* \\*>!#Q" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "evalf(intvalue(e xp(-18)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"I'4#p75K!Q[(pUoADzE4T Q?!#Q" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "evalf(intvalue(exp (-19)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"I2')o " 0 "" {MPLTEXT 1 0 26 "evalf(intvalue(exp(- 20)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"I())e-l#eriM9#H)4\\kqbKh? !#Q" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "evalf(intvalue(exp(- 21)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"IL/g0>'z$*p$p9IH1smrij?!# Q" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "evalf(intvalue(10^(-9) ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"I_/iud9B7&Q\"o:_xIs'*>j?!#Q " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 56 "#Here is all the same information, but with j \+ up to 512." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "sumo:=log(sum('(exp(-(El[j])*x))*(( dn[j])^2)','j'=1..512)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "plot([log(1/x),sumo,x=10^(-9)..1], title=\"Plot of function with norm at same point on 2-series\");" }}{PARA 13 "" 1 "" {GLPLOT2D 375 375 375 {PLOTDATA 2 "6&-%'CURVESG6$7gn7$$\"IToyFf@4B>;c6TYp$eEB2#!#Q$\"Id* zLPDJElV;51Xv[Z#=oAF*7$$\"Ih?vax]^hN02.^7C'H5.bqKH&p&*G@92N^2nlF 07$$\"INqvKAy=-Q5Kf@KH')*)4$>'F0$\"IE%z*H\\85!y\"p+OVamm&Qw-'F07$$\"I6 -rCuH=lS(Q\\!=:=OqT0fF0$\"Icju2_Q4(3%e[;5&**eU5%=cF07$$\"I?XlF%*H8S+A \"=gW8/0_**\\&F0$\"INL%[))=&Rt!)>AiMuf[rwr]F07$$\"I/=R-]XCZF07$$\"Iclxqghf*4u^E!3O>#*\\!o![F0$\"I0 3;UY'f8_^'H?0o'[<3_@%F07$$\"IZ.u@qP>*e69ni&oK]H7>XF0$\"I.*o$)GZ!R8\\(e )H')oXNCzJQF07$$\"IRY:KK^\"f^,n))>s.G(yl8TF0$\"Ie1=FRt\"*\\8xnsixqi'fc L$F07$$\"Ig+$)Hs7Wv'fxjkXWc\"e(f#QF0$\"I!=G,NWdG)z(pEUHZ6;&e%3$F07$$\" IZ)HE437.IxutPRN2+yZY$F0$\"IgU[$H)p8K+B!p\\+&>/j-fGF07$$\"I._5#Q%=.(*) )\\$R5?63z')**>$F0$\"I#zmJjS**3Np92T>'[U;J\"p#F07$$\"IJPYJZB.]!o]X#yqq 5d9zFF0$\"IgGh$yu3!=N()Qb:.%f*H_JBF07$$\"I;gh)>8T``'[*\\'*o?Fa\"\\1sb;8e_`M'3r/(>F07$$\"I>(3v76m8\"z.!*R8-kMNYaAF0$\"I?&\\K[ 4A0yq4@suIcmW6h\"F07$$\"IeB3:I_T9#*\\k\\q`8\\s^\"3#F0$\"I'Gf&[@TL-mjk4 D]_\\J+y7F07$$\"I4&e0ThJfF`!3>z*=%*zr#H>F0$\"Izb(RVqJp,]'p.^_\"f1W0L*! #S7$$\"I4iz'y**QMplv4Uza1,7M(*[jOO?BzIw&F]q7$$\"Ie VuBIzWE(>!Rgp/FG;Ot;F0$\"ILSt\"4mpVbAeg$\\X7.`02AF]q7$$\"I4X]34xbPFjN: XP\\O`mj:F0$!Inh6`9,HuP#\\U8%Hm)R_2X\"F]q7$$\"I!))pP[K)yF`#fQ)4hxxN5w9 F0$!I?#)QV\"F07$$\"I)[z!e]R%p.0f?p\\FlHq)G7F0$!Ig/;%eI!G/'Rz?B)\\I4 J5X:F07$$\"I%fm_KJQ:+\"yN!*ob7/L6m6F0$!Ig*Q5Z^L/2T\"Qw6KR%Q5Q'=F07$$\" I'3_(eMJ&)G&**)e6Vl,R(yi4\"F0$!I#y%*\\G^>f)=n?xR2]gTyUAF07$$\"IZPTvtV \\UTccFj@)G+62/\"F0$!I&p.UKw!f9-'=1k'e$*eY#Qc#F07$$\"IAmQ&yxEq@Z![LU`+ t8L(z*F]q$!I!4fZd2w^OlFg#))RFz5FPHF07$$\"I'pWejVwLwC_**[!z^.+&pG*F]q$! 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