{VERSION 6 0 "IBM INTEL LINUX" "6.0" } {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 "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 " " 0 1 0 0 255 1 0 0 0 0 0 0 1 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 "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 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 58 "#################### ######################################" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "# Details of the computations in section 4.3" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "# yields a configuration of \+ type 5,4,1 with a planar end" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "# ########################################################" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 11 "Digits:=20;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'D igitsG\"#?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "N1:=5;N2:=4;N 3:=1;c1:=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#N1G\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#N2G\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#N3G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c1G\"\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "# set the value of c2 to hav e a planar end at level 2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "c2:=5/4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c2G#\"\"&\"\"%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 41 "# compute the value of c3 -- do not touch" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "c3:=(N1*(N1-1)*c1^2+N2*(N2-1 )*c2^2-N1*N2*c1*c2)/(N2*c2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c3G #\"#6\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "# compute log arithmic growths" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "Q1:=-N1 *c1;Q2:=N1*c1-N2*c2;Q3:=N2*c2-N3*c3;Q3:=N3*c3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#Q1G!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#Q2G\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#Q3G#\"\"*\"\"%" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#Q3G#\"#6\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "# compute the configuration -- notations as in sectio n 4.3 of the paper" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "P1:=z ^5+sum(a[i]*z^i,i=0..4);P2:=z^4+sum(b[i]*z^i,i=0..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#P1G,.*$)%\"zG\"\"&\"\"\"F*&%\"aG6#\"\"!F**&&F,6 #F*F*F(F*F**&&F,6#\"\"#F*)F(F5F*F**&&F,6#\"\"$F*)F(F:F*F**&&F,6#\"\"%F *)F(F?F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#P2G,,*$)%\"zG\"\"%\" \"\"F*&%\"bG6#\"\"!F**&&F,6#F*F*F(F*F**&&F,6#\"\"#F*)F(F5F*F**&&F,6#\" \"$F*)F(F:F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "x:='x';b[ 1]:=0;b[2]:=x;b[3]:=2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xGF$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"bG6#\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"bG6#\"\"#%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"bG6#\"\"$\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "P 1p:=diff(P1,z);P1pp:=diff(P1p,z);P2p:=diff(P2,z);P2pp:=diff(P2p,z);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$P1pG,,*&\"\"&\"\"\")%\"zG\"\"%F(F( &%\"aG6#F(F(*(\"\"#F(&F-6#F0F(F*F(F(*(\"\"$F(&F-6#F4F()F*F0F(F(*(F+F(& F-6#F+F()F*F4F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%P1ppG,**&\"#? \"\"\")%\"zG\"\"$F(F(*&\"\"#F(&%\"aG6#F-F(F(*(\"\"'F(&F/6#F+F(F*F(F(*( \"#7F(&F/6#\"\"%F()F*F-F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$P2pG ,(*&\"\"%\"\"\")%\"zG\"\"$F(F(*(\"\"#F(%\"xGF(F*F(F(*&\"\"'F()F*F-F(F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%P2ppG,(*&\"#7\"\"\")%\"zG\"\"#F (F(*&F+F(%\"xGF(F(*&F'F(F*F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "# below is equation 2.5 of the paper" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "EQ:=collect(expand(c1^2*z*P1pp*P2+c2^2*z*P2pp*P1 -c1*c2*z*P1p*P2p-c2*c3*P1*P2p),z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# >%#EQG,0*&,&#\"\"&\"\")\"\"\"*&\"\"$F+&%\"aG6#\"\"%F+!\"\"F+)%\"zG\"\" (F+F+*&,(*(\"#:F+F1F2%\"xGF+F+*&#\"#jF*F+F.F+F2*&F1F+&F/6#F-F+F2F+)F4 \"\"'F+F+*&,(*&#\"#**F*F+F?F+F2*&F-F+&F/6#\"\"#F+F2*&#F5F1F+*&F.F+F:F+ F+F2F+)F4F)F+F+*&,(*&#\"#@F1F+*&F?F+F:F+F+F2*&#\"$.\"F*F+FIF+F2*&\"#?F +&%\"bG6#\"\"!F+F+F+)F4F1F+F+*&,**&#\"#vF*F+&F/6#F+F+F2*&#\"#FF1F+*&FI F+F:F+F+F2*&F)F+&F/FgnF+F+*(\"#7F+F.F+FenF+F+F+)F4F-F+F+*&,(*&#\"#DF1F +*&F:F+F_oF+F+F2*(FBF+F?F+FenF+F+*&#F9F*F+FfoF+F2F+)F4FKF+F+*&,&*(FKF+ FIF+FenF+F+*&#F9F1F+*&F:F+FfoF+F+F2F+F4F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "for j from 1 to 7 do eq[j]:=coeff(EQ,z,j):od:print (eq);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#K%&TABLEG6#7)/\"\"\",&*(\"\"#F(&%\"aG6#F+F(&%\"bG6#\"\" !F(F(*&#\"#:\"\"%F(*&%\"xGF(&F-F1F(F(!\"\"/F+,(*&#\"#DF6F(*&F8F(&F-6#F (F(F(F:*(\"\"'F(&F-6#\"\"$F(F/F(F(*&#F5\"\")F(F9F(F:/FG,**&#\"#vFJF(FA F(F:*&#\"#FF6F(*&F,F(F8F(F(F:*&\"\"&F(F9F(F(*(\"#7F(&F-6#F6F(F/F(F(/F6 ,(*&#\"#@F6F(*&FEF(F8F(F(F:*&#\"$.\"FJF(F,F(F:*&\"#?F(F/F(F(/FU,(*&#\" #**FJF(FEF(F:*&FGF(F,F(F:*&#\"\"(F6F(*&FXF(F8F(F(F:/FD,(*(F5F(F6F:F8F( F(*&#\"#jFJF(FXF(F:*&F6F(FEF(F:/Fgo,&#FUFJF(*&FGF(FXF(F:Q(pprint16\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "sol:=solve(\{eq[7],eq[6], eq[5],eq[4],eq[3],eq[2]\},\{a[0],a[1],a[2],a[3],a[4],b[0]\});" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%$solG<(/&%\"aG6#\"\"%#\"\"&\"#C/&F(6 #\"\"$,&*(\"#:\"\"\"\"#;!\"\"%\"xGF5F5#\"$0\"\"$c#F7/&F(6#F5,$*(\"(!3m >F7,**&\")k[zHF5F8F5F5\"(V8N)F7*&\"('HnpF5)F8F1F5F5*&\")?Z\"o#F5)F8\" \"#F5F7F5,&*&F6F5F8F5F5\"\"*F5F7F5/&F(6#\"\"!,$*(\"(%=P%*F7,**&\"*oz`$ \\F5F8F5F7\"*zcT@\"F5*&\"*wh%*o(F5FHF5F5*&\"*S%pA;F5FKF5F5F5FMF7F7/&% \"bGFR,(*&#\"#jF;F5*$FKF5F5F5*&#\"&8$\\\"&K%=F5F8F5F7#\"&z8(\"&Ob'F5/& F(6#FL,&*(\"%&f%F5\"%_6F7F8F5F7#\"%lM\"%[?F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "P:=numer(normal(expand(subs(sol,eq[1]))));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG,,*&\".?&R&y,\\\"\"\"\")%\"xG\" \"%F(F(*&\".!3QY8$=#F()F*\"\"$F(F(*&\".+)o!4U_\"F()F*\"\"#F(!\"\"*&\"- ?>6p^GF(F*F(F4\"-:LG-.=F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "# check that P has four simple roots, each of them gives a nondege nerate configuration" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "dis crim(P,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"co+++O2]!G\"RI!fjEM=e' Qmv@FP=@iBVpvqq1is8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "# co mpute roots of P" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "x:=[sol ve(evalf(P),x)][2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG$!5^.B6_= AVhO!#?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "# compute points of the configuration" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "p1 :=[solve(subs(sol,P1),z)];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#p1G7' ^$$!5(3)*3$=U\"3,U\"!#>$!5Yf&e'*4n:f$>!#?^$F'$\"5Yf&e'*4n:f$>F,^$$\"51 b(>!eA]#fK&F,$!5ZY50$zRPxk\"F)^$F1$\"5ZY50$zRPxk\"F)$\"5Sx1o^Yzpm:F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "p2:=[solve(subs(sol,P2),z )];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p2G7&$!5**4X%o0I8u(=!#>$!5T( [*fw/**fw7F(^$$\"5,()*>sm-m+x&!#?$!5Hn+Z=%>]bP(F.^$F,$\"5Hn+Z=%>]bP(F. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "p3:=[0];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p3G7#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "# plot the configuration" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 94 "# note that several points are equal when c2=1. To \+ see all 10 points, perturb the value of c2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots);" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warni ng, the name changecoords has been redefined\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7Z%(animateG%*animate3dG%-animatecurveG%&arrowG%-change coordsG%,complexplotG%.complexplot3dG%*conformalG%,conformal3dG%,conto urplotG%.contourplot3dG%*coordplotG%,coordplot3dG%-cylinderplotG%,dens ityplotG%(displayG%*display3dG%*fieldplotG%,fieldplot3dG%)gradplotG%+g radplot3dG%,graphplot3dG%-implicitplotG%/implicitplot3dG%(inequalG%,in teractiveG%-listcontplotG%/listcontplot3dG%0listdensityplotG%)listplot G%+listplot3dG%+loglogplotG%(logplotG%+matrixplotG%(odeplotG%'paretoG% ,plotcompareG%*pointplotG%,pointplot3dG%*polarplotG%,polygonplotG%.pol ygonplot3dG%4polyhedra_supportedG%.polyhedraplotG%'replotG%*rootlocusG %,semilogplotG%+setoptionsG%-setoptions3dG%+spacecurveG%1sparsematrixp lotG%+sphereplotG%)surfdataG%)textplotG%+textplot3dG%)tubeplotG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "plot1:=complexplot(p1,axes=n one,style=point,color=blue,scaling=constrained):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "plot2:=complexplot(p2,axes=none,style=point,c olor=red,scaling=constrained):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "plot3:=complexplot(p3,axes=none,style=point,color=green,scalin g=constrained):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "display( plot1,plot2,plot3);" }}{PARA 13 "" 1 "" {GLPLOT2D 529 529 529 {PLOTDATA 2 "6)-%'CURVESG6%7'7$$!5(3)*3$=U\"3,U\"!#>$!5Yf&e'*4n:f$>!#? 7$F($\"5Yf&e'*4n:f$>F-7$$\"51b(>!eA]#fK&F-$!5ZY50$zRPxk\"F*7$F2$\"5ZY5 0$zRPxk\"F*7$$\"5Sx1o^Yzpm:F*$\"\"!F=-%'COLOURG6&%$RGBGFsm-m+x&F-$!5Hn+Z=%>]bP(F-7$FS$\"5Hn+Z=%>]bP(F--F?6&FAFBF " 0 "" {MPLTEXT 1 0 0 "" }}}} {MARK "38 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }