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\\HN\\MV=\\;::R@F=:::::::::\\;kjEr?T;eb>qcWmuJGtK\\N^XN ]DkK`QwdqvpMJEC_VEMWEWsEYsEWSEMCE[FZKwdldglPoOtdilQ;::::::::benO::jPSe ::S_f\\elO[d\\bXnqOkTPa<_[aVcK^`o^ke^o`h[kW[LV\\M:P;TKEnCIdaGfXquXoeJIt^cF_aFHObEYsEx\\hv^ G`bNvndXOpDUOQcexDY?::::::::ZhLQ;::U_g<:Be^OrdVONOBESsb[BFEtV;CaSBVGw< oc@]b>HKD^LsDMsCQLLtLR[B>[b>_jC:P;T;g:v=:::mZS:::::::::>=\\;iZQb?vnXx]AFpaxbBGcDOCGtH]vWkUXwv\\Ava_V^EGX= cE[SFv_hnbooZsNnKVof`OfOW_hlQse:::::::::W_h<::RefO::S_fodvkqGckHhbv_j>`jn_g>lWhg_X\\?Poe`oghOUOQcevOxD:::: ::::JQce:::pDU?:ZfLP;hDQ?O_dTdFO1:" }{TEXT -1 1 " " }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "load " } {TEXT 0 11 "fractalplot" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 263 9 "fractal.m" }{TEXT -1 38 " contain s the code for the procedures " }{TEXT 0 11 "fractalplot" }{TEXT -1 25 " used in this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 122 "It can \+ be read into a Maple session by a command similar to the one that foll ows, where the file path gives its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "read \"K:\\\\MapleMath/procdrs/fractal.m\";" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 51 "load ext ra colours and utility routines for colours" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 263 9 "colours.m" }{TEXT -1 87 " cont ains the code for the procedures giving the extra colours used in this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 122 "It can be read into a Ma ple session by a command similar to the one that follows, where the fi le path gives its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "read \"K:\\\\MapleMath/procdrs/colours.m\";" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 33 "Newton's method for the ze ros of " }{XPPEDIT 18 0 "z^6-1" "6#,&*$%\"zG\"\"'\"\"\"F'!\"\"" } {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 163 "Using the formulas for the factorisation of a \+ sum of two squares and then the formulas for the factorisation of a su m and difference of two cubes we can factorise " }{XPPEDIT 18 0 "z^6-1 " "6#,&*$%\"zG\"\"'\"\"\"F'!\"\"" }{TEXT -1 13 " as follows. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "z^6-1=(z^3+1)*(z^3-1)" "6#/,&*$%\"zG\"\"'\"\"\"F(!\"\"* &,&*$F&\"\"$F(F(F(F(,&*$F&F-F(F(F)F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(z+1)*(z^2-z+1)*(z-1)*(z^2+z+1) " "6#/%!G**,&%\"zG\"\"\"F(F(F(,(*$F'\"\"#F(F'!\"\"F(F(F(,&F'F(F(F,F(,( *$F'F+F(F'F(F(F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "The n " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "z^2-z+1=0" "6#/ ,(*$%\"zG\"\"#\"\"\"F&!\"\"F(F(\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 4 "when" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "z^2-z+1/4=-3/4" "6#/,(*$%\"zG\"\"#\"\"\"F&!\"\"*&F(F(\"\"%F)F(,$*&\" \"$F(F+F)F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is " } }{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "(z-1/2)^2=-3/4" "6# /*$,&%\"zG\"\"\"*&F'F'\"\"#!\"\"F*F),$*&\"\"$F'\"\"%F*F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "z-1/2=``" "6#/,&%\"zG\"\"\"*&F&F&\"\"#!\"\"F) %!G" }{TEXT 269 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(3)/2" "6#*& -%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 1 " " }{TEXT 270 1 "i" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "z=1/2" "6#/%\"zG*&\"\"\"F&\"\"#!\"\" " }{TEXT -1 1 " " }{TEXT 271 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqr t(3)/2" "6#*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 1 " " }{TEXT 272 1 "i" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 11 "Similarly, \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "z^2+z+1=0" "6#/,( *$%\"zG\"\"#\"\"\"F&F(F(F(\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "when " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " z = -1/2;" "6#/%\"zG,$*&\"\"\"F'\"\"#!\"\"F)" }{TEXT -1 1 " " }{TEXT 273 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(3)/2" "6#*&-%%sqrtG6#\" \"$\"\"\"\"\"#!\"\"" }{TEXT -1 1 " " }{TEXT 274 1 "i" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Hence the equation " }{XPPEDIT 18 0 "z^6-1=0" "6#/,&*$%\"zG\"\"'\"\"\"F(! \"\"\"\"!" }{TEXT -1 23 " has the six solutions " }{XPPEDIT 18 0 "z=`` " "6#/%\"zG%!G" }{TEXT 275 1 "+" }{TEXT -1 3 "1, " }{XPPEDIT 18 0 "1/2 " "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 1 " " }{TEXT 276 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(3)/2" "6#*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\" " }{TEXT -1 1 " " }{TEXT 277 1 "i" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "-1 /2" "6#,$*&\"\"\"F%\"\"#!\"\"F'" }{TEXT -1 1 " " }{TEXT 278 1 "+" } {TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(3)/2" "6#*&-%%sqrtG6#\"\"$\"\"\"\" \"#!\"\"" }{TEXT -1 1 " " }{TEXT 279 1 "i" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 48 "Alternatively, writing the equation in the form \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "z^6=exp(2*k*Pi*i) " "6#/*$%\"zG\"\"'-%$expG6#**\"\"#\"\"\"%\"kGF,%#PiGF,%\"iGF," }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 287 1 "k" } {TEXT -1 24 " is an integer, we have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "z=exp(k*Pi*i/3)" "6#/%\"zG-%$expG6#**%\"kG\"\"\"%# PiGF*%\"iGF*\"\"$!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "z=cos(k *Pi/3)+i*sin(k*Pi/3)" "6#/%\"zG,&-%$cosG6#*(%\"kG\"\"\"%#PiGF+\"\"$!\" \"F+*&%\"iGF+-%$sinG6#*(F*F+F,F+F-F.F+F+" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 7 "Taking " }{XPPEDIT 18 0 "k=0,1,2,3,4,5" "6(/%\"kG\" \"!\"\"\"\"\"#\"\"$\"\"%\"\"&" }{TEXT -1 30 " in turn gives the soluti ons: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "PIECEWISE([k = 0, cos(0)+i*sin(0) = 1],[k = 1, cos(Pi/3)+i*sin(P i/3) = 1/2+sqrt(3)/2*i],[k = 2, cos(2*Pi/3)+i*sin(2*Pi/3) = -1/2+sqrt( 3)/2*i],[k = 3, cos(Pi)+i*sin(Pi) = -1],[k = 4, cos(4*Pi/3)+i*sin(4*Pi /3) = -1/2-sqrt(3)/2*i],[k = 5, cos(5*Pi/3)+i*sin(5*Pi/3) = 1/2-sqrt(3 )/2*i]);" "6#-%*PIECEWISEG6(7$/%\"kG\"\"!/,&-%$cosG6#F)\"\"\"*&%\"iGF/ -%$sinG6#F)F/F/F/7$/F(F//,&-F-6#*&%#PiGF/\"\"$!\"\"F/*&F1F/-F36#*&FF/F/,&*&F/F/\"\"#F>F/*(-%%sqrtG6#F=F/FEF>F1F/F/7$/F(FE/,&-F-6#*(FE F/FF/*&F1F/-F36#*(FEF/FF/F/,&*&F/F/FEF>F>*(-FH6#F=F/FEF> F1F/F/7$/F(F=/,&-F-6#F7$/F(\"\"%/,&-F-6#*(F` oF/FF/*&F1F/-F36#*(F`oF/FF/F/,&*&F/F/FEF>F>*(-FH6#F=F/FE F>F1F/F>7$/F(\"\"&/,&-F-6#*(FapF/FF/*&F1F/-F36#*(FapF/FF /F/,&*&F/F/FEF>F/*(-FH6#F=F/FEF>F1F/F>" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "The six solutions a re spaced evenly around the unit circle in the complex plane with its \+ centre at the origin given by " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "z=exp(i*theta)" "6#/%\"zG-%$expG6#*&%\"iG\"\"\"%&thetaG F*" }{XPPEDIT 18 0 "`` = cos*theta+i*sin*theta;" "6#/%!G,&*&%$cosG\"\" \"%&thetaGF(F(*(%\"iGF(%$sinGF(F)F(F(" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "0<=theta" "6#1\"\"!%&thetaG" }{XPPEDIT 18 0 "``<=2*Pi" "6#1%!G*& \"\"#\"\"\"%#PiGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "seq(convert(exp(k*Pi*I/3),s incos),k=0..5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(\"\"\",&#F#\"\"#F#* &^#F%F#\"\"$F%F#,&#F#F&!\"\"F'F#F,,&#F#F&F,*&^##F,F&F#F)F%F#,&F%F#F/F# " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 347 265 265 {PLOTDATA 2 "6.-%'CURVESG6%7S7$$ !3/+++++++8!#<$\"3t++!4]`b](!#=7$$!3omm;*RFLC\"F*$\"3EO%pt3a$yrF-7$$!3 GL$3A(o,%>\"F*$\"3A\\A=F'fO*oF-7$$!3zmmTF?cQ6F*$\"3]lwVJ4\\tlF-7$$!3om m\">fRF3\"F*$\"3%ff[(\\**>^iF-7$$!3YL$32Z#=F5F*$\"3#p&p$3wS/$fF-7$$!3= nmTv!Rnv*F-$\"3SX;oKf0LcF-7$$!3-,+vQxRB#*F-$\"3BC$G$o68D`F-7$$!3KmmTl` \"=n)F-$\"3i!>Vl$\\n1]F-7$$!3j***\\P#>+A\")F-$\"3G!=Kz**R#*o%F-7$$!3AL LL=7XcvF-$\"3+jq&e9>FO%F-7$$!3Qmm;\\QJeqF-$\"3i%H?;S>^2%F-7$$!3G+++b\" Hv\\'F-$\"3S^QU2-N^PF-7$$!3%3++]#=WMfF-$\"3uL*3ee^iU$F-7$$!3%******\\% G!=R&F-$\"3!fDtX#)eH6$F-7$$!3?nmTlY.**[F-$\"3L[$)Q$)*e%GGF-7$$!3!QLL$Q c38VF-$\"3g7d![:h,\\#F-7$$!3uLLL`*4n\"QF-$\"3i!*pR>&yN?#F-7$$!3K++vGqI RKF-$\"3c&>%R&y9-(=F-7$$!3nMLL.w?GFF-$\"3C33\\%RJ^d\"F-7$$!3+***\\(G0X n@F-$\"3UR#ok9y8D\"F-7$$!3g)**\\i=xMj\"F-$\"3uH%oP$\\)3V*!#>7$$!3oKL$3 -Kj2\"F-$\"3UyL%3#e?9iFar7$$!3'zKL3P))pk&Far$\"3Q!fMmD!HgKFar7$$!3)=gm m\"HL$G\"!#?$\"3qq,R]gK4u!#@7$$\"3nQL$eM)*Rg&Far$!3tKO7a*paB$Far7$$\"3 )))**\\())4Sf5F-$!3V\")\\^hWX;hFar7$$\"3iML$e!fL)f\"F-$!3>^tkcm*zA*Far 7$$\"3a******4%3^:#F-$!3e%>%)3U_UC\"F-7$$\"3S++]#o,)*p#F-$!3?ozWG7te:F -7$$\"3H***\\7V?oA$F-$!3KH\\'\\k0I'=F-7$$\"3O++](*[)>\"QF-$!3?e\"Rr]]3 ?#F-7$$\"3tlmm'>\"yPVF-$!3'f(*f?9>W]#F-7$$\"3[,++v4=**[F-$!3S(ol_XV&GG F-7$$\"3qJLe9E*yS&F-$!3an^Pd#[A7$F-7$$\"3y,++0)[S'fF-$!3B%4&o$=XLW$F-7 $$\"3gkm\"zV[t['F-$!3%37uwOsau$F-7$$\"3(>+]iLZV.(F-$!3OwO$*GBGhSF-7$$ \"3ynmmT(Q\"pvF-$!3a8.(3HW+P%F-7$$\"3Y(**\\7y:!H\")F-$!35O'[/X*G$p%F-7 $$\"3MLLL8RCo')F-$!3F'3@x&Hh/]F-7$$\"3[NL$3c#o>#*F-$!3mRwN4i)HK&F-7$$ \"39nm\"z,blw*F-$!3CauspJsQcF-7$$\"36+++,u!p-\"F*$!3*>8N^k_)GfF-7$$\"3 emmT[<]%3\"F*$!3kKo/lPPhiF-7$$\"3ELLL>[,O6F*$!3G3*3(=YyelF-7$$\"35+]7( \\Q4>\"F*$!3Mh&[.i')e(oF-7$$\"3>+]([Y2NC\"F*$!3u;9A(p$RzrF-7$$\"3/++++ +++8F*$!3t++!4]`b](F--%'COLOURG6&%$RGBG$\"\"!F`[lF_[l$\"*++++\"!\")-%* THICKNESSG6#\"\"#-F$6%7S7$F(Fiz7$F/$!3EO%pt3a$yrF-7$F4$!3A\\A=F'fO*oF- 7$F9$!3]lwVJ4\\tlF-7$F>$!3%ff[(\\**>^iF-7$FC$!3#p&p$3wS/$fF-7$FH$!3SX; oKf0LcF-7$FM$!3BC$G$o68D`F-7$FR$!3i!>Vl$\\n1]F-7$FW$!3G!=Kz**R#*o%F-7$ Ffn$!3+jq&e9>FO%F-7$F[o$!3i%H?;S>^2%F-7$F`o$!3S^QU2-N^PF-7$Feo$!3uL*3e e^iU$F-7$Fjo$!3!fDtX#)eH6$F-7$F_p$!3L[$)Q$)*e%GGF-7$Fdp$!3g7d![:h,\\#F -7$Fip$!3i!*pR>&yN?#F-7$F^q$!3c&>%R&y9-(=F-7$Fcq$!3C33\\%RJ^d\"F-7$Fhq $!3UR#ok9y8D\"F-7$F]r$!3uH%oP$\\)3V*Far7$Fcr$!3UyL%3#e?9iFar7$Fhr$!3Q! fMmD!HgKFar7$F]s$!3qq,R]gK4uFbs7$Fds$\"3tKO7a*paB$Far7$Fis$\"3V\")\\^h WX;hFar7$F^t$\"3>^tkcm*zA*Far7$Fct$\"3e%>%)3U_UC\"F-7$Fht$\"3?ozWG7te: F-7$F]u$\"3KH\\'\\k0I'=F-7$Fbu$\"3?e\"Rr]]3?#F-7$Fgu$\"3'f(*f?9>W]#F-7 $F\\v$\"3S(ol_XV&GGF-7$Fav$\"3an^Pd#[A7$F-7$Ffv$\"3B%4&o$=XLW$F-7$F[w$ \"3%37uwOsau$F-7$F`w$\"3OwO$*GBGhSF-7$Few$\"3a8.(3HW+P%F-7$Fjw$\"35O'[ /X*G$p%F-7$F_x$\"3F'3@x&Hh/]F-7$Fdx$\"3mRwN4i)HK&F-7$Fix$\"3CauspJsQcF -7$F^y$\"3*>8N^k_)GfF-7$Fcy$\"3kKo/lPPhiF-7$Fhy$\"3G3*3(=YyelF-7$F]z$ \"3Mh&[.i')e(oF-7$Fbz$\"3u;9A(p$RzrF-7$FgzF+F[[lFd[l-F$6%7$7$F_[lF(7$F _[lFgzF[[lFd[l-F$6&7(7$$\"\"\"F`[lF_[l7$$\"3++++++++]F-$\"3a+++SSDg')F -7$$!3++++++++]F-Fhel7$$!\"\"F`[lF_[l7$F[fl$!3a+++SSDg')F-7$FfelFafl-% 'SYMBOLG6#%'CIRCLEG-%&COLORG6&F^[lFdelF`[l$Fg[lF_fl-%&STYLEG6#%&POINTG -F$6&Fael-Fefl6#%(DIAMONDGFhflF\\gl-F$6&Fael-Fefl6#%&CROSSGFhflF\\gl-% %TEXTG6%7$$\"#8F_fl$!#8!\"#Q%Real6\"-Fifl6&F^[l$FdelFbhlFghlFghl-F[hl6 %7$$FbhlF_flF^hlQ%ImagFdhlFehl-%*AXESTICKSG6$\"\"$F`il-%(SCALINGG6#%,C ONSTRAINEDG-%+AXESLABELSG6%%!GFhil-%%FONTG6#%(DEFAULTG-%%VIEWG6$;$Fahl F_flF^hlF\\jl" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7 " "Curve 8" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 347 "The compl ex plane may be divided into six regions as shown such that a starting value in the given region might be expected to converge via Newton's \+ method to the root which it contains. Again this is only a rough appro ximation to the observed phonomenon in which we obtain a fractal struc ture for the regions giving convergence to the six roots. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "The iterative for mula for Newton's method in this case is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 280 1 "z" }{TEXT -1 1 "'" }{XPPEDIT 18 0 "`` = z - (z^6-1)/(6*z^5)" "6#/%!G,&%\"zG\"\"\"*&,&*$F&\"\"'F'F'!\"\"F'*&F+F' *$F&\"\"&F'F,F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is , " }}{PARA 256 "" 0 "" {TEXT -1 3 " z'" }{XPPEDIT 18 0 "`` =(5*z^6+1) /(6*z^5)" "6#/%!G*&,&*&\"\"&\"\"\"*$%\"zG\"\"'F)F)F)F)F)*&F,F)*$F+F(F) !\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 281 1 "z" }{TEXT -1 1 "'" }{XPPEDIT 18 0 "``=(5*z+1/z^5) /6" "6#/%!G*&,&*&\"\"&\"\"\"%\"zGF)F)*&F)F)*$F*F(!\" \"F)F)\"\"'F-" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "Drawing Newton fracta ls for " }{XPPEDIT 18 0 "z^6-1" "6#,&*$%\"zG\"\"'\"\"\"F'!\"\"" } {TEXT -1 6 " .. 1 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 33 "The iterative formula to use is: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 282 1 "z" }{TEXT -1 1 "'" } {XPPEDIT 18 0 "``=(5*z+1/z^5) /6" "6#/%!G*&,&*&\"\"&\"\"\"%\"zGF)F)*&F )F)*$F*F(!\"\"F)F)\"\"'F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "In order to calculate " } {XPPEDIT 18 0 "z^5" "6#*$%\"zG\"\"&" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "z=x+y*I" "6#/%\"zG,&%\"xG\"\"\"*&%\"yGF'%\"IGF'F'" }{TEXT -1 87 " a s rapidly as possible we make use of expressions for the real and imag inary parts of " }{XPPEDIT 18 0 "z^5" "6#*$%\"zG\"\"&" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "z := x + y*I:\nevalc(Re(z^5));\nevalc(Im(z^5));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"&\"\"\"F(*(\"#5F()F&\"\"$F()%\"yG \"\"#F(!\"\"*(F'F(F&F()F.\"\"%F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,(*(\"\"&\"\"\")%\"xG\"\"%F&%\"yGF&F&*(\"#5F&)F(\"\"#F&)F*\"\"$F&!\"\" *$)F*F%F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure \+ " }{TEXT 0 11 "newton6_hue" }{TEXT -1 39 " below determines which of t he 6 zeros " }{TEXT 264 1 "+" }{TEXT -1 3 "1, " }{XPPEDIT 18 0 "1/2" " 6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 1 " " }{TEXT 265 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(3)/2" "6#*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"" } {TEXT -1 1 " " }{TEXT 266 1 "i" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "-1/2 " "6#,$*&\"\"\"F%\"\"#!\"\"F'" }{TEXT -1 1 " " }{TEXT 267 1 "+" } {TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(3)/2" "6#*&-%%sqrtG6#\"\"$\"\"\"\" \"#!\"\"" }{TEXT -1 1 " " }{TEXT 268 1 "i" }{TEXT -1 5 " of " } {XPPEDIT 18 0 "f(z) = z^6-1;" "6#/-%\"fG6#%\"zG,&*$F'\"\"'\"\"\"F+!\" \"" }{TEXT -1 29 " a particular starting value " }{XPPEDIT 18 0 "x+y*i " "6#,&%\"xG\"\"\"*&%\"yGF%%\"iGF%F%" }{TEXT -1 151 " eventually conve rges to when Newton's formula is applied repeatedly. It is assumed tha t once an iteration gives a value which is within a distance of " } {XPPEDIT 18 0 "sqrt(1/20);" "6#-%%sqrtG6#*&\"\"\"F'\"#?!\"\"" }{TEXT -1 140 " or about 0.2236 of one of the 6 roots, then convergence to th at root is guaranteed. This is a bit more stringent than is actually r equired." }}{PARA 0 "" 0 "" {TEXT -1 10 "The roots " }{XPPEDIT 18 0 "1 ,1/2+sqrt(3)/2" "6$\"\"\",&*&F#F#\"\"#!\"\"F#*&-%%sqrtG6#\"\"$F#F&F'F# " }{TEXT -1 1 " " }{TEXT 283 1 "i" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "-1 /2+sqrt(3)/2" "6#,&*&\"\"\"F%\"\"#!\"\"F'*&-%%sqrtG6#\"\"$F%F&F'F%" } {TEXT -1 1 " " }{TEXT 284 1 "i" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "-1,- 1/2-sqrt(3)/2" "6$,$\"\"\"!\"\",&*&F$F$\"\"#F%F%*&-%%sqrtG6#\"\"$F$F(F %F%" }{TEXT -1 1 " " }{TEXT 288 1 "i" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "1/2-sqrt(3)/2" "6#,&*&\"\"\"F%\"\"#!\"\"F%*&-%%sqrtG6#\"\"$F%F&F 'F'" }{TEXT -1 1 " " }{TEXT 289 1 "i" }{TEXT -1 93 " are associated wi th the colours green, purple, red, blue, yellow and magenta, respectiv ely. " }}{PARA 0 "" 0 "" {TEXT -1 100 "The return value is the HUE val ue corresponding to the colour associated with the root converged to. " }}{PARA 0 "" 0 "" {TEXT -1 175 "If the iterate is not in the vicinit y of any of the 6 roots after 500 iterations, the value 0.5 is returne d. This may happen if, for example, the starting value is close to 0. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "newton6_hue: impleme ntation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2132 "newton6_hue := proc(x,y)\n local xx,yy,xs,ys,xf,y f,a,b,c,d,e,sum,i,p,q,r,s,t,u,\n comp,notclosetoroot,dx,rootThree,yT imesRootThree,\n oneSixth,fiveSixths;\011\011\011\011\n \n xx : = x; \n yy := y;\n xs := xx^2;\n ys := yy^2;\n\011\011\011\011 \n notclosetoroot := true;\n rootThree := 1.73205080756887729;\n \+ oneSixth := .166666666666666667;\n fiveSixths := .83333333333333333 3;\n\011\011\011\n for i from 1 to 500 while notclosetoroot do\n\011 \011\011\011\011 # Apply Newton's method for calculating roots of z^6- 1=0.\n\011\011\011\011\011 xf := xs^2;\n\011\011\011\011\011 yf := ys^ 2;\n\011\011\011\011\011 c := ((ys-2*xs)*ys*5+xf)*xx;\011# real part o f z^5\n\011\011\011\011\011 d := ((xs-2*ys)*xs*5+yf)*yy; # imag part o f z^5\n\011\011\011\011\011\n\011\011\011\011\011 if abs(c)>abs(d) the n\n\011\011\011\011\011\011 e := d/c;\n\011\011\011\011\011\011 a \+ := 1/(c+d*e); # real part of 1/z^5\011\n\011\011\011\011\011\011 b : = e*a;\011 \011# minus imag part of 1/z^5\n\011\011\011\011\011\011els e\n\011\011\011\011\011\011 e := c/d;\n\011\011\011\011\011\011 b \+ := 1/(c*e+d);\011# minus imag part of 1/z^5\n\011\011\011\011\011\011 \+ a := e*b;\011 # real part of 1/z^5\n\011\011\011\011\011 end if; \011\n # real part of 5/6*z+1/(6*z^5)\011\011\011\011\011\011\011 \011\011\011\011\n\011\011\011\011\011 xx := fiveSixths*xx+oneSixth*a; \n # imag part of 5/6*z+1/(6*z^5)\n\011\011\011\011\011 yy := fiv eSixths*yy-oneSixth*b;\n\011\011\011\011\011\n\011\011\011\011\011 # H ow close are we to one of the roots?\n\011\011\011\011\011 xs := xx^2; \n ys := yy^2;\n\011\011\011\011\011 sum := xs+ys;\n\n\011\011 \011\011\011 comp := sum+.99;\n\011\011\011\011\011 dx := 2*xx;\n\011 \011\011\011\011 yTimesRootThree := yy*rootThree;\n\n\011\011\011\011 \011 # Are we close to the root -1?\n\011\011\011\011 \011p := evalb(- dx>comp);\n\011\011\011\011\011\n\011\011\011\011\011 # Are we close t o the root -1/2-sqrt(3)/2*I?\n\011\011\011\011\011 q := evalb(-xx-yTim esRootThree>comp);\n\011\011\011\011\011\n\011\011\011\011\011 # Are w e close to the root 1/2-sqrt(3)/2*I?\n\011\011\011\011\011 r := evalb( xx-yTimesRootThree>comp);\n\n\011\011\011\011\011 # Are we close to th e root 1?\n\011\011\011\011 s := evalb(dx>comp);\n\011\011\011\011 \011\n\011\011\011\011\011 # Are we close to the root 1/2+sqrt(3)/2*I? \n\011\011\011\011\011 t := evalb(xx+yTimesRootThree>comp);\n\n\011 \011\011\011\011 # Are we close to the troot -1/2+sqrt(3)/2*I?\n\011 \011\011\011 \011u := evalb(yTimesRootThree-xx>comp);\n\011\011\011 \011\011\n\011\011\011\011\011 # Set flag to end loop if sufficiently \+ close to a root.\n\011\011\011\011\011 notclosetoroot := not(p or q or r or s or t or u);\n end do;\n\n # Determime the value to return. \n if p then .6666666667 elif q then .1666666667 \n elif r then .8 333333333 elif s then .3333333333\n elif t then .75 \n elif u then 0 else .5416666667 end if;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "evalhf(newton6_hue(0.00 1,0.5231));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"3++++++++v!#=" }}} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 76 "The sequ ence of iterates from Newton's formula starting with 0.001 + 0.5231 " }{TEXT 285 1 "i" }{TEXT -1 14 " converges to " }{XPPEDIT 18 0 "1/2+sqr t(3)/2" "6#,&*&\"\"\"F%\"\"#!\"\"F%*&-%%sqrtG6#\"\"$F%F&F'F%" }{TEXT -1 1 " " }{TEXT 286 1 "i" }{TEXT -1 39 ", and the associated colour is purple. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 50 "In the following it is assumed that the procedure " } {TEXT 0 11 "newton6_hue" }{TEXT -1 18 " has been defined." }}{PARA 0 " " 0 "" {TEXT -1 52 "The fractal is drawn by using the special procedur e " }{TEXT 0 11 "fractalplot" }{TEXT -1 25 " from another worksheet. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "fractalplot(newton6_hue,-1.1..1.1,-1.1..1.1,grid=[216,216]);" }}{PARA 13 "" 1 "" {GLPLOT3D 936 639 639 {PLOTDATA 3 "6'-%%GRIDG6&;$!# 6!\"\"$\"#6F)F&X,6\"6\"F-[gl'!%\"!!#[_av\"cx\"cx0000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 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dKPyLEuQoTmoPNDQJ=B<>;j=r;b;nZ?B;AR?SBdYeIMDMUTGgcGgcGf`of`oRMlN[\\tQ@ nwpQytax;umwLnDyLDUSEUSE[O@i>P^iN`lN`mV`mV`l>`oO\\e@pjHpkHmSHmSHpb@kiV ]>fZ?Z@J>;:v:X:GZ?jGprR=MyhTMp nTT=g;f@rL_E[mUYseYwuYyuYyMytQyViF@pJ`lPplQxlQxLgf_aFWIwE_sFakSJSllxvo kHpSlFCfCGYmB;Eb;Gj=vZAr@YGO::::::rcvN:::::nNTD:: ::RcfN:::C_^<:VNHD:?_\\<=_[LJ;lSFALI`L\\xqZ\\N_XLGPj=XJ>B<:ZAj=n:PZ=Va PFnOxfgXcon`pbG:P=Xnv@TBeqxxqyxqyZxSYoeLaR^ev_iv_i>TsB]j=XSSHmSLMTLMTH mSxrKekQAuFacTPmTPmSxoWt@Eb;Gr;>;>oyOiVVavfaxnayrIyKYhmDerrHQtPAgeWgeW eEWIVHWGTauKWv]Y>R^DC_>:@sBGL1:" }{TEXT -1 1 " " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "Drawing Newton fractals for " } {XPPEDIT 18 0 "z^6-1;" "6#,&*$%\"zG\"\"'\"\"\"F'!\"\"" }{TEXT -1 5 " . . 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Th e procedure " }{TEXT 0 11 "fractalplot" }{TEXT -1 98 " can construct i mages in which the colour is determined by its RGB (red, green, blue) \+ components. " }}{PARA 0 "" 0 "" {TEXT -1 27 "For example, the procedur e " }{TEXT 0 11 "newton6_rgb" }{TEXT -1 29 " is similar to the procedu re " }{TEXT 0 11 "newton6_hue" }{TEXT -1 67 " of the last section in t hat it determines which of the 6 zeros of " }{XPPEDIT 18 0 "f(z) = z^6 -1;" "6#/-%\"fG6#%\"zG,&*$F'\"\"'\"\"\"F+!\"\"" }{TEXT -1 29 " a parti cular starting value " }{XPPEDIT 18 0 "x+y*i" "6#,&%\"xG\"\"\"*&%\"yGF %%\"iGF%F%" }{TEXT -1 69 " eventually converges to when Newton's formu la is applied repeatedly." }}{PARA 0 "" 0 "" {TEXT -1 21 "As before, t he roots " }{XPPEDIT 18 0 "1,1/2+sqrt(3)/2" "6$\"\"\",&*&F#F#\"\"#!\" \"F#*&-%%sqrtG6#\"\"$F#F&F'F#" }{TEXT -1 1 " " }{TEXT 292 1 "i" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "-1/2+sqrt(3)/2" "6#,&*&\"\"\"F%\"\"#! \"\"F'*&-%%sqrtG6#\"\"$F%F&F'F%" }{TEXT -1 1 " " }{TEXT 293 1 "i" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "-1,-1/2-sqrt(3)/2" "6$,$\"\"\"!\"\",& *&F$F$\"\"#F%F%*&-%%sqrtG6#\"\"$F$F(F%F%" }{TEXT -1 1 " " }{TEXT 294 1 "i" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "1/2-sqrt(3)/2" "6#,&*&\"\" \"F%\"\"#!\"\"F%*&-%%sqrtG6#\"\"$F%F&F'F'" }{TEXT -1 1 " " }{TEXT 295 1 "i" }{TEXT -1 277 " are associated with the colours green, purple, r ed, blue, yellow and magenta, respectively, but this time the shade of the colour is determined by the number of iterations of the Newton fo rmula that are required to bring the iterate within the specified dist ance of the root. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "ne wton6_rgb: implementation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2465 "newton6_rgb := proc(x,y)\n loca l xx,yy,xs,ys,xf,yf,a,b,c,d,e,sum,i,p,q,r,s,t,u,\n comp,notclosetoro ot,dx,rootThree,yTimesRootThree,h,ans,\n oneSixth,fiveSixths;\011 \011\011\011\n \n xx := x; \n yy := y;\n xs := xx^2;\n ys \+ := yy^2;\n\011\011\011\011\n notclosetoroot := true;\n rootThree : = 1.73205080756887729;\n oneSixth := .166666666666666667;\n fiveSi xths := .833333333333333333;\n\011\011\011\n for i from 1 to 500 whi le notclosetoroot do\n\011\011\011\011\011 # Apply Newton's method for calculating roots of z^6-1=0.\n\011\011\011\011\011 xf := xs^2;\n\011 \011\011\011\011 yf := ys^2;\n\011\011\011\011\011 c := ((ys-2*xs)*ys* 5+xf)*xx;\011# real part of z^5\n\011\011\011\011\011 d := ((xs-2*ys)* xs*5+yf)*yy; # imag part of z^5\n\011\011\011\011\011\n\011\011\011 \011\011 if abs(c)>abs(d) then\n\011\011\011\011\011\011 e := d/c;\n \011\011\011\011\011\011 a := 1/(c+d*e); # real part of 1/z^5\011\n \011\011\011\011\011\011 b := e*a;\011 \011# minus imag part of 1/z^ 5\n\011\011\011\011\011\011else\n\011\011\011\011\011\011 e := c/d; \n\011\011\011\011\011\011 b := 1/(c*e+d);\011# minus imag part of 1 /z^5\n\011\011\011\011\011\011 a := e*b;\011 # real part of 1/z^5\n \011\011\011\011\011 end if;\011\011\011\011\011\011\011\011\011\011 \011\011\n # real part of 5/6*z+1/(6*z^5)\011\011\011\011\011\011 \011\011\011\011\011\n\011\011\011\011\011 xx := fiveSixths*xx+oneSixt h*a;\n # imag part of 5/6*z+1/(6*z^5)\n\011\011\011\011\011 yy := fiveSixths*yy-oneSixth*b;\n\011\011\011\011\011\n\011\011\011\011\011 # How close are we to one of the roots?\n\011\011\011\011\011 xs := x x^2;\n ys := yy^2;\n\011\011\011\011\011 sum := xs+ys;\n\n\011 \011\011\011\011 comp := sum+.99;\n\011\011\011\011\011 dx := 2*xx;\n \011\011\011\011\011 yTimesRootThree := yy*rootThree;\n\n\011\011\011 \011\011 # Are we close to the root -1?\n\011\011\011\011 \011p := eva lb(-dx>comp);\n\011\011\011\011\011\n\011\011\011\011\011 # Are we clo se to the root -1/2-sqrt(3)/2*I?\n\011\011\011\011\011 q := evalb(-xx- yTimesRootThree>comp);\n\011\011\011\011\011\n\011\011\011\011\011 # A re we close to the root 1/2-sqrt(3)/2*I?\n\011\011\011\011\011 r := ev alb(xx-yTimesRootThree>comp);\n\n\011\011\011\011\011 # Are we close t o the root 1?\n\011\011\011\011 s := evalb(dx>comp);\n\011\011\011 \011\011\n\011\011\011\011\011 # Are we close to the root 1/2+sqrt(3)/ 2*I?\n\011\011\011\011\011 t := evalb(xx+yTimesRootThree>comp);\n\n \011\011\011\011\011 # Are we close to the troot -1/2+sqrt(3)/2*I?\n \011\011\011\011 \011u := evalb(yTimesRootThree-xx>comp);\n\011\011 \011\011\011\n\011\011\011\011\011 # Set flag to end loop if sufficien tly close to a root.\n\011\011\011\011\011 notclosetoroot := not(p or \+ q or r or s or t or u);\n end do;\n\n h := irem(i,25)^2;\n h := \+ 1+0.000004*(h-625)*h;\n\n # Determine the value to return.\n ans : = hfarray(1..3);\n if p then\n ans[1]:=0; ans[2]:=0; ans[3]:=h; \n elif q then\n ans[1]:=h; ans[2]:=h; ans[3]:=0; \n elif r \+ then\n ans[1]:=h; ans[2]:=0; ans[3]:=h; \n elif s then\n a ns[1]:=0; ans[2]:=h; ans[3]:=0; \n elif t then\n ans[1]:=h*0.6 ; ans[2]:=0; ans[3]:=h; \n elif u then\n ans[1]:=h; ans[2]:=0; \+ ans[3]:=0; \n else\n ans[1]:=.5; ans[2]:=.5; ans[3]:=.5\n end if;\n ans;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "evalhf(newton6_rgb(-.5,.8));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")/YX8-%'VECTORG6#7%$\"3b +++++k+**!#=$\"\"!F/F.%&ArrayG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 69 "The sequence of iterates from Newton's fo rmula starting with 0.3+0.4 " }{TEXT 291 1 "i" }{TEXT -1 14 " converge s to " }{XPPEDIT 18 0 "-1" "6#,$\"\"\"!\"\"" }{TEXT -1 50 " and the co rresponding colour is a shade of blue. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "In the following it is as sumed that the procedure " }{TEXT 0 11 "newton6_rgb" }{TEXT -1 18 " ha s been defined." }}{PARA 0 "" 0 "" {TEXT -1 49 "The fractal is drawn u sing the special procedure " }{TEXT 0 11 "fractalplot" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "fractalplot(newton6_rgb,-1.1..1.1,-1.1..1.1,grid=[216,216]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 70 "The picture is saved as a 2-dimensional Windows bitmap to save memory." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{OLE 1 284162 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j:[Z:F:y;;\\:FBNBVB^BfBnBvB>CFCNCVC^CfCnCvC>DIFINIVI^IfInIvI>J>ELq^H_mvJ::::::::gjR< :T><::HL\\ur?Aq]Aj;J:@:<:=j[vGUMrvC?MoJ::::::::JCNZ;^:vYxI>:<::::::j`J 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