{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 "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }1 0 0 0 8 4 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 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 287 "We begin with the functio nal equation for the series denoted $\\bar M$ in the paper. It counts \+ planar maps, weighted by their Potts polynomial (divided by q), by the number of edges (variable t), non-root vertices (variable w), degree \+ of the root vertex (x), degree of the root face (y).\n" }}{PARA 0 "" 0 "" {TEXT -1 35 "The equation is given in Section 3." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 157 "eqM:=1+x*y*w*t*((nu-1)*(y-1)+q*y)*M(x,y)*M(1,y) + \+ x*y*t*(x*nu-1)*M(x,y)*M(x,1)\n+x*y*w*t*(nu-1)*(x*M(x,y)-M(1,y))/(x-1)+ x*y*t*(y*M(x,y)-M(x,1))/(y-1)-M(x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqMG,.\"\"\"F&*0%\"xGF&%\"yGF&%\"wGF&%\"tGF&,&*&,&%#nuGF&F&! \"\"F&,&F)F&F&F0F&F&*&%\"qGF&F)F&F&F&-%\"MG6$F(F)F&-F56$F&F)F&F&*.F(F& F)F&F+F&,&*&F/F&F(F&F&F&F0F&F4F&-F56$F(F&F&F&*0F(F&F)F&F*F&F+F&F.F&,&* &F(F&F4F&F&F7F0F&,&F(F&F&F0F0F&*,F(F&F)F&F+F&,&*&F)F&F4F&F&F " 0 "" {MPLTEXT 1 0 175 "Mbser:=proc(n )\noption remember:\nif n=0 then 1 else\nfactor(series(subs(M(x,y)=Mbs er(n-1), M(x,1)=subs(y=1,Mbser(n-1)), M(1,y)=subs(x=1,Mbser(n-1)) ,eqM +M(x,y)),t,n+1));\nfi:end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "The series denoted M in the paper does not depend on x, and has an additi onal factor w" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "Mser:=proc (n)\nfactor(series(subs(x=1,w*Mbser(n)),t,n+1));end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "Mser(2);" }}{PARA 12 "" 1 "" {XPPMATH 20 " 6#++%\"tG%\"wG\"\"!*(F%\"\"\"%\"yGF(,**(%#nuGF(F%F(F)F(F(*(%\"qGF(F%F( F)F(F(*&F%F(F)F(!\"\"F,F(F(F(*(F%F(F)F(,D**\"\"#F()F,F4F()F%F4F()F)\" \"$F(F(*,\"\"%F(F,F(F.F(F6F(F7F(F(**F4F()F.F4F(F6F(F7F(F(**F:F(F,F(F6F (F7F(F0**F:F(F.F(F6F(F7F(F0**F8F(F5F(F%F()F)F4F(F(*,F8F(F,F(F.F(F%F(F@ F(F(*(F4F(F6F(F7F(F(*(F5F(F%F(F)F(F(**F8F(F,F(F%F(F@F(F0*&F5F(F%F(F(*& F5F(F)F(F(*(F,F(F.F(F%F(F(*(F.F(F%F(F)F(F(*$F5F(F(*&F,F(F%F(F0*&F%F(F) F(F0F(F4-%\"OG6#F(F8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Let us ch eck with the list of small maps given in the figure (Section 2.1)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "serfig:=w+t*(w^2*y^2*(q-1+n u)+w*y*nu)+\nt^2*(2*w^3*y^4*(q-1+nu)^2+w^2*y^2*(q-1+nu^2)+(y+3*y^3)*w^ 2*nu*(q-1+nu)+(y+y^2)*w*nu^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'s erfigG,(%\"wG\"\"\"*&%\"tGF',&*()F&\"\"#F')%\"yGF-F',(%#nuGF'%\"qGF'F' !\"\"F'F'*(F1F'F&F'F/F'F'F'F'*&)F)F-F',***F-F')F&\"\"$F')F/\"\"%F')F0F -F'F'*(F,F'F.F',(*$)F1F-F'F'F2F'F'F3F'F'**,&*&F:F')F/F:F'F'F/F'F'F,F'F 1F'F0F'F'*(,&*$F.F'F'F/F'F'F&F'FAF'F'F'F'" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 35 "normal(series(Mser(2)-serfig,t,3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+%%\"tG-%\"OG6#\"\"\"\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "The duality relation of Section 2.3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "n:=7: factor(series(w^2*q*subs(\{n u=1+q/(nu-1),w=1/w/q,t=t*w*(nu-1),y=1\},Mser(n))-subs(y=1,Mser(n)),t,n +1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+%%\"tG-%\"OG6#\"\"\"\"\")" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 36 "The most important polynomial D(t,x)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "DD:=(q*nu+beta^2)*x^2-q*(nu+1)*x+beta*t*( q-4)*(w*q+beta)+q;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DDG,**&,&*$)% %betaG\"\"#\"\"\"F,*&%#nuGF,%\"qGF,F,F,)%\"xGF+F,F,*(F/F,,&F.F,F,F,F,F 1F,!\"\"**F*F,%\"tGF,,&F/F,\"\"%F4F,,&*&F/F,%\"wGF,F,F*F,F,F,F/F," }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 161 "The diffe rential system: Uniqueness of the solution, and determination of the f irst few coefficients of each series\n(Main theorem, and then Sections 4.6 and 4.7)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Dx:=subs(nu =1+beta,DD);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DxG,**&,&*$)%%betaG \"\"#\"\"\"F,*&%\"qGF,,&F,F,F*F,F,F,F,)%\"xGF+F,F,*(F.F,,&F+F,F*F,F,F1 F,!\"\"**F*F,%\"tGF,,&F.F,\"\"%F4F,,&*&F.F,%\"wGF,F,F*F,F,F,F.F," }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "The system without initial conditi ons" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "Psol:=P4(t)*x^4+P3( t)*x^3+P2(t)*x^2+P1(t)*x+P0(t);\nQsol:=x^2*Q2(t)+Q1(t)*x+Q0(t);\nRsol: =R0(t)+R1(t)*x+x^2*R2(t);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Psol G,,*&-%#P4G6#%\"tG\"\"\")%\"xG\"\"%F+F+*&-%#P3GF)F+)F-\"\"$F+F+*&-%#P2 GF)F+)F-\"\"#F+F+*&-%#P1GF)F+F-F+F+-%#P0GF)F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%QsolG,(*&)%\"xG\"\"#\"\"\"-%#Q2G6#%\"tGF*F**&-%#Q1GF -F*F(F*F*-%#Q0GF-F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%RsolG,(-%#R0 G6#%\"tG\"\"\"*&-%#R1GF(F*%\"xGF*F**&)F.\"\"#F*-%#R2GF(F*F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "edplan:=2*Psol*Dx*diff(Qsol ,t)-Qsol*diff(Psol,t)*Dx-2*Qsol*Psol*diff(Dx,t) -(2*Psol*Dx*diff(Rsol, x)-Rsol*diff(Psol,x)*Dx-2*Rsol*Psol*diff(Dx,x)):" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 44 "This gives 9 equations for 11 unknown series" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "degree(edplan,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "factor(coeff(edplan,x,8));\nfactor(coeff(edplan,x,0));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&,(*$)%%betaG\"\"#\"\"\"F)*&F'F)%\"qG F)F)F+F)F),&*(F(F)-%#P4G6#%\"tGF)-%%diffG6$-%#Q2GF0F1F)F)*&F5F)-F36$F. F1F)!\"\"F)" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,V**\"\"#\"\"\"-%#P0G6# %\"tGF&-%%diffG6$-%#Q0GF)F*F&%\"qGF&F&*(F.F&-F,6$F'F*F&F0F&!\"\"**\"\" )F&F'F&F.F&)%%betaGF%F&F&**F%F&F'F&-%#R1GF)F&F0F&F4*(-%#R0GF)F&-%#P1GF )F&F0F&F&**\"\"%F&F=F&F'F&F0F&F4*0F%F&F'F&F+F&F8F&)F0F%F&F*F&%\"wGF&F& *0F6F&F'F&F+F&F8F&F0F&F*F&FEF&F4*.F.F&F2F&F8F&FDF&F*F&FEF&F4*0FBF&F.F& F2F&F8F&F0F&F*F&FEF&F&*0F%F&F'F&F:F&F8F&FDF&F*F&FEF&F4*0F6F&F'F&F:F&F8 F&F0F&F*F&FEF&F&*.F=F&F?F&F8F&FDF&F*F&FEF&F&*0FBF&F=F&F?F&F8F&F0F&F*F& FEF&F4*.F%F&F'F&F+F&F7F&F0F&F*F&F&*,F.F&F2F&F7F&F0F&F*F&F4*.F%F&F'F&F. F&F8F&FDF&FEF&F4*.F6F&F'F&F.F&F8F&F0F&FEF&F&*.F%F&F'F&F:F&F7F&F0F&F*F& F4*,F=F&F?F&F7F&F0F&F*F&F&*,F%F&F=F&F'F&F0F&F8F&F4*,F6F&F'F&F+F&F7F&F* F&F4*,FBF&F.F&F2F&F7F&F*F&F&*,F%F&F'F&F.F&F7F&F0F&F4*,F6F&F'F&F:F&F7F& F*F&F&*,FBF&F=F&F?F&F7F&F*F&F4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Now we take into account the prescribed values of P4 and R2" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "Psol:=x^4+P3(t)*x^3+P2(t)*x ^2+P1(t)*x+P0(t);\nQsol:=Q1(t)*x+x^2*Q2(t)+Q0(t);\nRsol:=R0(t)+R1(t)*x +x^2*(beta+2-w*(q+2*beta));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Ps olG,,*$)%\"xG\"\"%\"\"\"F**&-%#P3G6#%\"tGF*)F(\"\"$F*F**&-%#P2GF.F*)F( \"\"#F*F**&-%#P1GF.F*F(F*F*-%#P0GF.F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%QsolG,(*&)%\"xG\"\"#\"\"\"-%#Q2G6#%\"tGF*F**&-%#Q1GF-F*F(F*F*-%# Q0GF-F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%RsolG,(-%#R0G6#%\"tG\"\" \"*&-%#R1GF(F*%\"xGF*F**&)F.\"\"#F*,(%%betaGF*F1F**&%\"wGF*,&%\"qGF**& F1F*F3F*F*F*!\"\"F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "The diff erential system" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "edplan0 :=collect(numer(factor(eval(1/(Rsol)*diff(Rsol^2/Psol/Dx^2,x)-1/Qsol*d iff(Qsol^2/Psol/Dx^2,t)))),x,factor):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "edplan:=2*Psol*Dx*diff(Qsol,t)-Qsol*diff(Psol,t)*Dx- 2*Qsol*Psol*diff(Dx,t) -(2*Psol*Dx*diff(Rsol,x)-Rsol*diff(Psol,x)*Dx-2 *Rsol*Psol*diff(Dx,x)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 " factor(edplan/edplan0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "The system still contains 9 equati ons, for 9 unknown series" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "degree(edplan,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "for i from 0 to 8 do ED[i]:= factor(coeff(edplan,x,i)) od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "seq(nops(ED[i]),i=0..8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+\"#E \"#j\"#%*\"$4\"\"$3\"\"#w\"#]\"#@\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "The 8th equation is easily solved, with the initial condi tion Q2(0)=1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "ED[8];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#\"\"\"-%%diffG6$-%#Q2G6#%\"tG F-F&,(*$)%%betaGF%F&F&*&F1F&%\"qGF&F&F3F&F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "This simplifies the system: 8 unknown series and 8 dif ferential equations" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "for \+ i from 0 to 7 do ED[i]:= factor(eval(subs(Q2(t)=1,ED[i]))) od:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "seq(nops(ED[i]),i=0..7);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6*\"#E\"#j\"#%)\"#**\"#)*F$\"#O\"\"#" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "ED[7];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,(*$)%%betaG\"\"#\"\"\"F)*&F'F)%\"qGF)F)F+F)F),.*(\" \")F)F'F)%\"wGF)!\"\"*(\"\"%F)F+F)F/F)F0*&F(F)-%%diffG6$-%#Q1G6#%\"tGF :F)F)-F56$-%#P3GF9F:F0*&F2F)F'F)F)F.F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "We now want to determine coefficients" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "expand(x^2*(x-1)^2);expand(x*(x-1));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"%\"\"\"F(*&\"\"#F()F&\" \"$F(!\"\"*$)F&F*F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%\"xG\" \"#\"\"\"F(F&!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "Expansions \+ of the 8 unknown series, taking into account the values at t=0 prescri bed for P and R" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 344 "P3ser:= proc(N) -2+add(p3[n]*t^n,n=1..N) end:\nP2ser:=proc(N) 1+add(p2[n]*t^n, n=1..N) end:\nP1ser:=proc(N) add(p1[n]*t^n,n=1..N) end:\nP0ser:=proc(N ) add(p0[n]*t^n,n=1..N) end:\nQ1ser:=proc(N) -1+add(q1[n]*t^n,n=1..N) \+ end:\nQ0ser:=proc(N) add(q0[n]*t^n,n=1..N) end:\nR1ser:=proc(N) add(r1 [n]*t^n,n=0..N) end:\nR0ser:=proc(N) add(r0[n]*t^n,n=0..N) end:\n" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Recursive determination of the coe fficients (details in Section 4.6)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "unassign('p3','p2','p1','p0','q1','q0','r1','r0'); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "The n-th set of unknown coeffic ients" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "cC:=proc(n) \{p0[n ],p1[n],p2[n],p3[n],q0[n],q1[n],r0[n-1],r1[n-1]\}: end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "cC(2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<*&%#p0G6#\"\"#&%#p1GF&&%#p2GF&&%#p3GF&&%#q0GF&&%#q1GF&&%#r0G6# \"\"\"&%#r1GF4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "NB: the variabl e denoted by i in the paper is n here." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 116 "The coefficient of t^\{n-1\} in the equation ED[0] does \+ not involve the coefficients of cC(n). For instance, when n=1:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "n:=1: coeff(series(eval(sub s( P1(t)=P1ser(n), P0(t)=P0ser(n), \nQ1(t)=Q1ser(n), Q0(t)=Q0ser(n), R 1(t)=R1ser(n), R0(t)=R0ser(n), ED[0])),t,n),t,n-1);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Or when n=2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 182 "n:=2: coeff(serie s(eval(subs( P1(t)=P1ser(n), P0(t)=P0ser(n), \nQ1(t)=Q1ser(n), Q0(t)=Q 0ser(n), R1(t)=R1ser(n), R0(t)=R0ser(n), ED[0])),t,n),t,n-1);\nindets( %) intersect cC(n);cC(n-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*,\" \"#\"\"\"%%betaGF&%\"qGF&&%#p0G6#F&F&&%#r0G6#\"\"!F&!\"\"*(F(F&F)F&&%# q0GF+F&F&**\"\"%F&F(F&F)F&F,F&F0**F%F&F(F&F)F&&%#r1GF.F&F0*(F(F&&%#p1G F+F&F,F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<*&%#p0G6#\"\"\"&%#p1GF&&%#p2GF&&%#p3GF&&%#q0GF&&%#q1GF &&%#r0G6#\"\"!&%#r1GF4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 116 "This i s not linear in the unknowns of cC(2). But when n>2, [t^\{n-1\}]ED[0] \+ becomes linear in the unknowns of cC(n-1)." }}{PARA 0 "" 0 "" {TEXT -1 9 "For n=3: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 218 "n:=3: c ollect(coeff(series(eval(subs( P1(t)=P1ser(n), P0(t)=P0ser(n), \nQ1(t) =Q1ser(n), Q0(t)=Q0ser(n), R1(t)=R1ser(n), R0(t)=R0ser(n), ED[0])),t,n ),t,n-1),cC(n-1),distributed,factor);\nindets(%) intersect cC(n);cC(n- 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.**\"\"#\"\"\"%\"qGF&,(*&%%bet aGF&&%#r0G6#\"\"!F&F&*&F%F&F+F&F&&%#r1GF-F&F&&%#p0G6#F%F&!\"\"*(F+F&F' F&&%#p1GF4F&F&**F*F&,(*&&F36#F&F&&%#q0GF=F&F&*(F%F&F " 0 "" {MPLTEXT 1 0 25 "EDsum:=add(ED[j],j=0..7): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 207 "n:=1:collect(normal(co eff(series(eval(subs(P3(t)=P3ser(n), P2(t)=P2ser(n), P1(t)=P1ser(n), P 0(t)=P0ser(n), \nQ1(t)=Q1ser(n), Q0(t)=Q0ser(n), R1(t)=R1ser(n), R0(t) =R0ser(n), EDsum)),t,n),t,n-1)),cC(n),factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "For n=2, \+ this is non-linear in cC(1)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 249 "n:=2:collect(normal(coeff(series(eval(subs(P3(t)=P3ser(n), P2(t)= P2ser(n), P1(t)=P1ser(n), P0(t)=P0ser(n), \nQ1(t)=Q1ser(n), Q0(t)=Q0se r(n), R1(t)=R1ser(n), R0(t)=R0ser(n), EDsum)),t,n),t,n-1)),cC(n-1),dis tributed,factor);\nindets(%) intersect cC(n); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,J*(&%#q0G6#\"\"\"F(&%#p1GF'F()%%betaG\"\"#F(F(*,F-F(F, F(,&*&\"\"$F(F,F(F(%\"qGF(F(&%#r0G6#\"\"!F(&%#p2GF'F(F(**F,F(,&*&\"\"( F(F,F(F(*&F-F(F2F(F(F(F3F(&%#p3GF'F(F(*(F%F(F7F(F+F(F(*(F%F(F>F(F+F(F( *(&%#p0GF'F(&%#q1GF'F(F+F(F(*(F+F(F7F(FEF(F(*(F)F(FEF(F+F(F(*(F+F(F>F( FEF(F(*,F-F(F,F(,&F2F(*&F-F(F,F(F(F(&%#r1GF5F(F7F(F(**F,F(,&*&\"\"&F(F ,F(F(*&F-F(F2F(F(F(FMF(F>F(F(**F,F(,&*&F1F(F,F(F(*&F-F(F2F(F(F(FMF(F)F (F(*,F-F(F,F(,&F,F(F2F(F(,**(F-F(F,F(%\"wGF(F(*&F2F(FfnF(F(F,!\"\"F-Fh nF(F7F(Fhn**F,F(FUF(FZF(F>F(Fhn**F,F(,&F,F(*&F-F(F2F(F(F(FZF(F)F(Fhn*, F-F(F,F(F2F(FZF(FCF(Fhn*,F-F(F,F(FKF(F3F(FCF(F(**F,F(FPF(F3F(F)F(F(*,F -F(F,F(FYF(FCF(FMF(F(*(F%F(FCF(F+F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "But it is linear for n= 3 and beyond" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 249 "n:=3:colle ct(normal(coeff(series(eval(subs(P3(t)=P3ser(n), P2(t)=P2ser(n), P1(t) =P1ser(n), P0(t)=P0ser(n), \nQ1(t)=Q1ser(n), Q0(t)=Q0ser(n), R1(t)=R1s er(n), R0(t)=R0ser(n), EDsum)),t,n),t,n-1)),cC(n-1),distributed,factor );\nindets(%) intersect cC(n); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,4* *%%betaG\"\"\",&%\"qGF&\"\"%!\"\"F&,&*&F(F&%\"wGF&F&F%F&F&,hn**\"\")F& F%F&F-F&&%#p0G6#F&F&F&**\"\"'F&F%F&F-F&&%#p1GF3F&F&**F)F&F%F&F-F&&%#p2 GF3F&F&**\"\"#F&F%F&F-F&&%#p3GF3F&F&**F)F&F(F&F-F&F1F&F&**\"\"$F&F(F&F -F&F6F&F&**FFjoF&F***FAF&F^pF&,*F1F&F6F&F9F&F=F&F&&FJFjoF&F&**FAF&F^pF&Fi rF&&FMFjoF&F&*(F%F&,2*(F)F&F%F&F1F&F&*(FpF&F%F&F6F&F&*(F5F&F%F&F9F&F& *(F]rF&F%F&F=F&F&*(F " 0 "" {MPLTEXT 1 0 570 "n:=1: ens:=normal([coeff(series(subs(P3(t)=P3ser(n+1 ), P2(t)=P2ser(n+1), P1(t)=P1ser(n+1), P0(t)=P0ser(n+1), \nQ1(t)=Q1ser (n+1), Q0(t)=Q0ser(n+1), R1(t)=R1ser(n+1), R0(t)=R0ser(n+1), EDsum),t, n+1),t,n),\ncoeff(series(subs(P3(t)=P3ser(n+1), P2(t)=P2ser(n+1), P1(t )=P1ser(n+1), P0(t)=P0ser(n), \nQ1(t)=Q1ser(n+1), Q0(t)=Q0ser(n+1), R1 (t)=R1ser(n+1), R0(t)=R0ser(n+1), ED[0]),t,n+1),t,n),\nseq(coeff(serie s(eval(subs(P3(t)=P3ser(n), P2(t)=P2ser(n), P1(t)=P1ser(n), P0(t)=P0se r(n), \nQ1(t)=Q1ser(n), Q0(t)=Q0ser(n), R1(t)=R1ser(n), R0(t)=R0ser(n) , ED[j])),t,n),t,n-1),j=2..7)]); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#> %$ensG7*,^q*,\"\"(\"\"\"&%#p3G6#F)F))%%betaG\"\"#F)%\"qGF)%\"wGF)!\"\" *,\"\"'F)&%#p2GF,F)F-F)F0F)F1F)F2*,F/F)F.F)F0F)F*F)&%#r0G6#\"\"!F)F)*, F/F)F.F)F0F)F5F)&%#r1GF:F)F)*,F/F)F.F)F0F)F*F)F=F)F)*,F/F)F.F)F0F)F5F) F8F)F)*,F/F)F=F)&%#p1GF,F)F.F)F0F)F)*,F/F)F.F)&%#p0GF,F))F0F/F)F1F)F2* ,F/F)F.F)FBF)FGF)F1F)F2*,F/F)F5F)F.F)FGF)F1F)F2*,F/F)F*F)F.F)FGF)F1F)F 2*,\"\"%F)FEF)F-F)F0F)F1F)F2*,\"\"&F)FBF)F-F)F0F)F1F)F2*(\"\"$F)F*F))F .FPF)F)*(F/F)F5F)FQF)F)*&FBF)FQF)F)*(FLF)F5F)F-F)F)*(F/F)FBF)F-F)F)*(F 4F)F*F)F-F)F)**FLF)F-F)F5F)F=F)F)**F4F)F-F)F5F)F8F)F)**F(F)F-F)F*F)F8F )F)*(F-F)F5F)&%#q1GF,F)F)*(F-F)F*F)FenF)F)**FNF)F-F)F*F)F=F)F)**FLF)F. F)FBF)F0F)F)**FLF)F8F)FEF)F-F)F)**F/F)FBF)FQF)F1F)F2**FLF)F5F)FQF)F1F) F2**F4F)F*F)FQF)F1F)F2*(&%#q0GF,F)F5F)F-F)F)*(F_oF)F*F)F-F)F)*(FEF)Fen F)F-F)F)*(FBF)FenF)F-F)F)**FPF)F=F)FBF)F-F)F)*(F_oF)FBF)F-F)F)**F/F)FE F)F-F)F0F)F)**F/F)FBF)F-F)F0F)F)**FLF)F5F)F.F)F0F)F)**FLF)F*F)F.F)F0F) F)**F/F)F5F)F-F)F0F)F)**F/F)F*F)F-F)F0F)F)*,F/F)F8F)FEF)F0F)F.F)F)*,F/ F)FEF)F=F)F.F)F0F)F)*,F/F)F8F)FBF)F.F)F0F)F)**FLF)FEF)F0F)F.F)F)**F/F) FEF)F=F)F-F)F)**FNF)F8F)FBF)F-F)F)*(F_oF)FEF)F-F)F),,*,F/F)F.F)F0F)FEF )F8F)F2*(F0F)FEF)F_oF)F)**FLF)F0F)FEF)F8F)F2**F/F)F0F)FEF)F=F)F2*(F0F) FBF)F8F)F),.*(F.F)F0F)FEF)F2**FLF)F.F)F0F)F8F)F2*(FPF)F0F)FEF)F2*&F0F) FBF)F)*(F/F)F0F)F_oF)F)*(\"#9F)F0F)F8F)F2,L*&F5F)F0F)F)*(F/F)FenF)F0F) F)*&FEF)F-F)F)*(F4F)F8F)F-F)F)*(F.F)FBF)F0F)F2**F/F)F_oF)F.F)F0F)F2**F /F)F=F)F.F)F0F)F2**\"#;F)F8F)F0F)F.F)F)**F/F)FEF)F0F)F.F)F)*(\"#IF)F8F )F0F)F)*(F4F)F=F)F0F)F2*(\"\")F)F_oF)F0F)F2*(FPF)FEF)F0F)F)*(FPF)FBF)F 0F)F2*(F/F)F-F)F0F)F)*(F/F)FGF)F1F)F)**F/F)F.F)FGF)F1F)F)**FLF)F.F)F0F )F1F)F2*&FLF)F0F)F2*(F/F)F.F)F0F)F2*&FrF)F-F)F2,X*(FPF)F5F)F0F)F2*(F rF)FenF)F0F)F2*(F/F)F_oF)F-F)F)*&F*F)F0F)F)*&FBF)F-F)F)FdqF2*(FLF)F=F) F-F)F)*(FqF)F8F)F-F)F2**F/F)F.F)FBF)F0F)F)**F4F)F_oF)F.F)F0F)F)*(F5F) F.F)F0F)F2**F/F)FenF)F.F)F0F)F2**\"#5F)F=F)F.F)F0F)F)**\"#?F)F8F)F0F)F .F)F2*(FEF)F0F)F.F)F2*(\"#EF)F8F)F0F)F2*(\"#=F)F=F)F0F)F)*(\"#7F)F_oF) F0F)F)*&FEF)F0F)F2*(FPF)FBF)F0F)F)*(F4F)F-F)F0F)F2*(F/F)FGF)F1F)F2**F4 F)F.F)FGF)F1F)F2**FisF)F.F)F0F)F1F)F)*&FLF)F0F)F)*(F/F)F.F)F0F)F)*&\"# CF)F-F)F),hn*(FPF)F5F)F0F)F)*(FtF)FenF)F0F)F)*(FLF)F_oF)F-F)F2*(FPF)F *F)F0F)F2*&F5F)F-F)F)F_sF2*(FgsF)F=F)F-F)F2*(FrF)F8F)F-F)F)*(F/F)FenF )F-F)F)FfqF2**F4F)F_oF)F.F)F0F)F2**F/F)F5F)F.F)F0F)F)**F4F)FenF)F.F)F0 F)F)*(F*F)F.F)F0F)F2**FqF)F=F)F.F)F0F)F2**FrF)F8F)F0F)F.F)F)*(FrF)F 8F)F0F)F)*(F^tF)F=F)F0F)F2*(FrF)F_oF)F0F)F2*&FBF)F0F)F2*(FgsF)F-F)F0F )F)*(F4F)FGF)F1F)F2**FgsF)F-F)F0F)F1F)F2**F/F)F.F)FGF)F1F)F)**\"#OF)F. F)F0F)F1F)F2*&F/F)FQF)F)*&FtF)F0F)F)*(FLF)FQF)F1F)F2*(FqF)F.F)F0F)F) *&FisF)F-F)F2,T*(FtF)FQF)F1F)F)**\"#AF)F-F)F0F)F1F)F)**F4F)F.F)FGF)F1 F)F)*&F4F)FQF)F2*(FgsF)F-F)F0F)F2*&F-F)F5F)F2*&F-F)F*F)F)*(F/F)F-F)F_o F)F)*(FLF)F-F)FenF)F2*(F4F)F-F)F=F)F)**\"#GF)F.F)F0F)F1F)F)*(F.F)F0F)F 5F)F2**F/F)F.F)F0F)F*F)F)**F/F)F.F)F0F)F_oF)F)**F4F)F.F)F0F)FenF)F2**F 4F)F.F)F0F)F=F)F)*(FgsF)FGF)F1F)F)*&FLF)F-F)F2*(F\\tF)F.F)F0F)F2*&F0F) F5F)F2*(FPF)F0F)F*F)F)*(F/F)F0F)F_oF)F)*(FrF)F0F)FenF)F2*(F4F)F0F)F=F )F)*&FisF)F0F)F2,$*&,(*$F-F)F)*&F.F)F0F)F)F0F)F),.*(FrF)F.F)F1F)F)*(F LF)F0F)F1F)F)*&FLF)F.F)F2F*F)*&F/F)FenF)F2FrF2F)F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "indets(ens) minus \{beta,q,w\}; nops(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<*&%#p0G6#\"\"\"&%#p1GF&&%#p2GF&&% #p3GF&&%#q0GF&&%#q1GF&&%#r0G6#\"\"!&%#r1GF4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "The syste m is not linear. For example," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "coeff(ens[2],p1[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%#r0G 6#\"\"!\"\"\"%\"qGF(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "But there is still a unique solution" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "solve(ens,indets(ens) minus \{beta,q,w\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<*/&%#p0G6#\"\"\"!\"%/&%#p1GF',&*(\"\"#F(%\"qGF(%\"wGF( !\"\"\"\")F(/&%#p2GF',**(\"\"%F(%%betaGF(F1F(F2*(F9F(F0F(F1F(F(*&F/F(F :F(F2F9F2/&%#p3GF',&*(F/F(F0F(F1F(F2*&F/F(F:F(F(/&%#q0GF',(*&F0F(F1F(F (*&F/F(F:F(F(F9F(/&%#q1GF',**(F9F(F:F(F1F(F(*&F0F(F1F(F(F:F2F9F2/&%#r0 G6#\"\"!F//&%#r1GFR,(*&F0F(F1F(F(F:F2F9F2" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 10 "assign(%):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 " From now on (n>1) we have a linear system in the unknowns of cC(n)" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 542 "n:=1+n:ens:=normal([coeff( series(subs(P3(t)=P3ser(n), P2(t)=P2ser(n), P1(t)=P1ser(n), P0(t)=P0se r(n), \nQ1(t)=Q1ser(n), Q0(t)=Q0ser(n), R1(t)=R1ser(n), R0(t)=R0ser(n) , EDsum),t,n+1),t,n),\ncoeff(series(subs(P3(t)=P3ser(n), P2(t)=P2ser(n ), P1(t)=P1ser(n), P0(t)=P0ser(n), \nQ1(t)=Q1ser(n), Q0(t)=Q0ser(n), R 1(t)=R1ser(n), R0(t)=R0ser(n), ED[0]),t,n+1),t,n),\nseq(coeff(series(e val(subs(P3(t)=P3ser(n), P2(t)=P2ser(n), P1(t)=P1ser(n), P0(t)=P0ser(n ), \nQ1(t)=Q1ser(n), Q0(t)=Q0ser(n), R1(t)=R1ser(n), R0(t)=R0ser(n), E D[j])),t,n+1),t,n-1),j=2..7)]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "indets(ens) minus \{beta,q,w\}; nops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<*&%#p0G6#\"\"#&%#p1GF&&%#p2GF&&%#p3GF&&%#q0GF&&%#q1GF& &%#r0G6#\"\"\"&%#r1GF4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 311 "Mat:=Matrix(1..8,1..8):\nfo r j from 1 to 8 do \nMat[1,j]:=coeff(ens[j], p0[n]):Mat[2,j]:=coeff(en s[j], p1[n]): Mat[3,j]:=coeff(ens[j], p2[n]):\nMat[4,j]:=coeff(ens[j], p3[n]): Mat[5,j]:=coeff(ens[j], q0[n]): Mat[6,j]:=coeff(ens[j], q1[n] ): Mat[7,j]:=coeff(ens[j], r0[n-1]): Mat[8,j]:=coeff(ens[j], r1[n-1]): \nod: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "map(factor,Mat) :factor(Determinant(Mat));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*.\"&% Q;\"\"\")%%betaG\"\"(F&)%\"qG\"\"$F&%\"wGF&,&F+F&\"\"%!\"\"F&),(*$)F( \"\"#F&F&*&F(F&F+F&F&F+F&F5F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "factor(%/(256*n^6*q^3*w*beta^7*(q-4)*(beta^2+beta*q+q)^2));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "factor(solve(ens, indets(ens) minus \{w,beta,q\})):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "assign(%):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "The values in cC(1)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "n:=1: P0ser(n);P1ser(n);P2ser(n);P3ser(n);\nQ 0ser(n);Q1ser(n);\nR0ser(n-1); R1ser(n-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"%\"\"\"%\"tGF&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*(\"\"#\"\"\"%\"qGF'%\"wGF'!\"\"\"\")F'F'%\"tGF'" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$*&,**(\"\"%F$%%betaGF$%\"wG F$!\"\"*(F(F$%\"qGF$F*F$F$*&\"\"#F$F)F$F+F(F+F$%\"tGF$F$" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,&\"\"#!\"\"*&,&*(F$\"\"\"%\"qGF)%\"wGF)F%*&F$F) %%betaGF)F)F)%\"tGF)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,(*&%\"qG \"\"\"%\"wGF'F'*&\"\"#F'%%betaGF'F'\"\"%F'F'%\"tGF'" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,&\"\"\"!\"\"*&,**(\"\"%F$%%betaGF$%\"wGF$F$*&%\"qGF$ F+F$F$F*F%F)F%F$%\"tGF$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&%\"qG\"\"\"%\"wGF&F&%%betaG!\"\"\" \"%F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "The coefficients seem to be polynomials in q, nu and w" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "map(factor,P0ser(3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*& \"\"%\"\"\"%\"tGF&!\"\"*&,**&)%\"qG\"\"#F&)%\"wGF.F&F&*(\"#;F&%%betaGF &F0F&F&*(F%F&F-F&F0F&F(*&\"\")F&F3F&F&F&)F'F.F&F&*(F.F&,4*(F3F&F,F&)F0 \"\"$F&F(*&)F-F " 0 "" {MPLTEXT 1 0 102 "eqM1bis: =-2*(beta^2+beta*q+q)*diff(M1t(t),t)+1/2*(2*beta*w+q*w-beta-2)*P3(t)+R 1(t)+2+2*(1+beta)*w-2*w;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(eqM1bis G,.*(\"\"#\"\"\",(*$)%%betaGF'F(F(*&F,F(%\"qGF(F(F.F(F(-%%diffG6$-%$M1 tG6#%\"tGF5F(!\"\"*&#F(F'F(*&,**(F'F(F,F(%\"wGF(F(*&F.F(F " 0 "" {MPLTEXT 1 0 51 "unassign('p3','p2','p1','p0 ','q1','q0','r1','r0'); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 344 "P3ser:=proc(N) -2+add(p3[n]*t^n,n=1..N) end:\nP2ser:=proc(N) 1+ad d(p2[n]*t^n,n=1..N) end:\nP1ser:=proc(N) add(p1[n]*t^n,n=1..N) end:\nP 0ser:=proc(N) add(p0[n]*t^n,n=1..N) end:\nQ1ser:=proc(N) -1+add(q1[n]* t^n,n=1..N) end:\nQ0ser:=proc(N) add(q0[n]*t^n,n=1..N) end:\nR1ser:=pr oc(N) add(r1[n]*t^n,n=0..N) end:\nR0ser:=proc(N) add(r0[n]*t^n,n=0..N) end:\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "We drop the last equat ion and the last unknown of the system" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "Fr n>1, we have a linear system in the unknowns of cC(n) \+ minus r1[n-1]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "n:=1:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 542 "n:=n+1:ens:=normal([coeff(s eries(subs(P3(t)=P3ser(n), P2(t)=P2ser(n), P1(t)=P1ser(n), P0(t)=P0ser (n), \nQ1(t)=Q1ser(n), Q0(t)=Q0ser(n), R1(t)=R1ser(n), R0(t)=R0ser(n), EDsum),t,n+1),t,n),\ncoeff(series(subs(P3(t)=P3ser(n), P2(t)=P2ser(n) , P1(t)=P1ser(n), P0(t)=P0ser(n), \nQ1(t)=Q1ser(n), Q0(t)=Q0ser(n), R1 (t)=R1ser(n), R0(t)=R0ser(n), ED[0]),t,n+1),t,n),\nseq(coeff(series(ev al(subs(P3(t)=P3ser(n), P2(t)=P2ser(n), P1(t)=P1ser(n), P0(t)=P0ser(n) , \nQ1(t)=Q1ser(n), Q0(t)=Q0ser(n), R1(t)=R1ser(n), R0(t)=R0ser(n), ED [j])),t,n+1),t,n-1),j=2..6)]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "indets(ens) minus \{beta,q,w\}; nops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<2&%#p0G6#\"\"\"&F%6#\"\"#&%#p1GF&&F,F)&%#p2GF&&F/F)&%# p3GF&&F2F)&%#q0GF&&F5F)&%#q1GF&&F8F)&%#r0G6#\"\"!&F;F&&%#r1GF<&F@F&" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 278 "Mat:=Matrix(1..7,1..7):\nfor j from 1 to 7 do \nMat[ 1,j]:=coeff(ens[j], p0[n]):Mat[2,j]:=coeff(ens[j], p1[n]): Mat[3,j]:=c oeff(ens[j], p2[n]):\nMat[4,j]:=coeff(ens[j], p3[n]): Mat[5,j]:=coeff( ens[j], q0[n]): Mat[6,j]:=coeff(ens[j], q1[n]): Mat[7,j]:=coeff(ens[j] , r0[n-1]): \nod: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "map (factor,Mat):dete:=factor(Determinant(Mat));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%deteG,$*.\"#K\"\"\")%%betaG\"\"$F()%\"qG\"\"#F(,F*$) F*\"\")F(F(*(F+F()F*\"\"(F(F-F(F(*(F+F()F*\"\"'F(F,F(F(*&)F*\"\"&F()F- F+F(F(*&F8F(F4F(F(*(\"#CF(F7F(F-F(F(*(\"#MF(F:F(F,F(F(*(\"#@F()F*\"\"% F(F " 0 "" {MPLTEXT 1 0 349 "factor(dete/(2*n^4*beta^3*q^2*((2*n-1)*p0[1]-2* r0[0])*(4*beta*w+2*q*w-2*beta-(2*n-1)*(p0[1]+p1[1]+p2[1]+p3[1])-2*r0[0 ]-2*r1[0]-4)*(beta^8+3*beta^7*q+3*beta^6*q^2+beta^5*q^3+6*beta^7+24*be ta^6*q+34*beta^5*q^2+21*beta^4*q^3+6*beta^3*q^4+16*beta^5*q+42*beta^4* q^2+36*beta^3*q^3+18*beta^2*q^4+12*beta^3*q^2+11*beta^2*q^3+18*beta*q^ 4-6*beta*q^3+6*q^4)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Inserting the correct initial valu es (of cC(1))" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "factor(su bs(p0[1]=-4,p1[1]=8-2*w*q,p2[1]=4*w*(q-beta)-2*beta-4, p3[1]=2*beta-2* w*q, r0[0]=2, r1[0]=w*q-beta-4,dete));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*,\"%#>)\"\"\")%%betaG\"\"%F&)%\"qG\"\"#F&,F*$)F(\"\")F&F&*(\" \"$F&)F(\"\"(F&F+F&F&*(F2F&)F(\"\"'F&F*F&F&*&)F(\"\"&F&)F+F2F&F&*&F7F& F3F&F&*(\"#CF&F6F&F+F&F&*(\"#MF&F9F&F*F&F&*(\"#@F&F'F&F;F&F&*(F7F&)F(F 2F&)F+F)F&F&*(\"#;F&F9F&F+F&F&*(\"#UF&F'F&F*F&F&*(\"#OF&FDF&F;F&F&*(\" #=F&)F(F,F&FEF&F&*(\"#7F&FDF&F*F&F&*(\"#6F&FNF&F;F&F&*(FMF&F(F&FEF&F&* (F7F&F(F&F;F&!\"\"*&F7F&FEF&F&F&%\"wGF&FU" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "version in the paper" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 193 "factor(%/(-128*n^6*q^2*beta^4*w*(6*q^4*(1+beta)^3+be ta*q^3*(beta ^4+21*beta^3+36*beta^2+11*beta-6)\n+beta^3*q^2*(3*beta^3+ 34*beta^2+42*beta+12)+beta^5*q*(3*beta^2+24*beta+16)+beta^7*(beta+6))) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 58 "We modify the again the system, introducing the series M1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "eqM1 :=(12*beta^2+1 2*beta*q+12*q)*M1t(t)+2*Q0(t)+1/4*P3(t)^2+(-4*beta*w-2*q*w+2*beta+4)*t *P3(t)-P2(t)+(-12*beta*w-4*q*w-4)*t;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%%eqM1G,.*&,(*&\"#7\"\"\")%%betaG\"\"#F*F**(F)F*F,F*%\"qGF*F**&F)F* F/F*F*F*-%$M1tG6#%\"tGF*F**&F-F*-%#Q0GF3F*F**&#F*\"\"%F**$)-%#P3GF3F-F *F*F**(,**(F:F*F,F*%\"wGF*!\"\"*(F-F*F/F*FBF*FC*&F-F*F,F*F*F:F*F*F4F*F =F*F*-%#P2GF3FC*&,(*(F)F*F,F*FBF*FC*(F:F*F/F*FBF*FCF:FCF*F4F*F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "eqM1bis := -2*(beta^2+beta* q+q)*diff(M1t(t),t)+1/2*(2*beta*w+q*w-beta-2)*P3(t)+R1(t)+2+2*(1+beta) *w-2*w;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(eqM1bisG,.*(\"\"#\"\"\", (*$)%%betaGF'F(F(*&F,F(%\"qGF(F(F.F(F(-%%diffG6$-%$M1tG6#%\"tGF5F(!\" \"*&#F(F'F(*&,**(F'F(F,F(%\"wGF(F(*&F.F(F " 0 "" {MPLTEXT 1 0 44 "M1tser:=proc(n) add(t^i*m1t[i],i=2..n): end :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 249 "eqQ0R := (nu-1)*(q-4) *(q*w+nu-1)*Q0(t)+q*(nu+1)*R0(t)+(2*nu*q^2*t*w+2*nu^2*q*t-8*nu*q*t*w-2 *q^2*t*w-8*nu^2*t-4*nu*q*t+8*q*t*w+16*nu*t+2*q*t+2*q-8*t)*R1(t)-2*(q*w -2)*(nu*q^2*t*w+nu^2*q*t-4*nu*q*t*w-q^2*t*w-4*nu^2*t-2*nu*q*t+4*q*t*w+ 8*nu*t+q*t+q-4*t);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&eqQ0RG,***,&% #nuG\"\"\"F)!\"\"F),&%\"qGF)\"\"%F*F),(*&F,F)%\"wGF)F)F(F)F)F*F)-%#Q0G 6#%\"tGF)F)*(F,F),&F(F)F)F)F)-%#R0GF3F)F)*&,8*,\"\"#F)F(F))F,F " 0 "" {MPLTEXT 1 0 163 "eqQ1R := (nu-1)*(q-4)*(q*w+ nu-1)*Q1(t)+(-2*nu^2-2*nu*q+4*nu-2)*R0(t)-q*(nu+1)*R1(t)-2*(q-4)*(nu-1 )*(2*nu*w+q*w-nu-2*w-1)*(q*w+nu-1)*t+2*q*(nu*q*w-2*nu*w-nu+2*w-1);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%&eqQ1RG,,**,&%#nuG\"\"\"F)!\"\"F),&% \"qGF)\"\"%F*F),(*&F,F)%\"wGF)F)F(F)F)F*F)-%#Q1G6#%\"tGF)F)*&,**&\"\"# F))F(F8F)F**(F8F)F(F)F,F)F**&F-F)F(F)F)F8F*F)-%#R0GF3F)F)*(F,F),&F(F)F )F)F)-%#R1GF3F)F**.F8F)F+F)F'F),,*(F8F)F(F)F0F)F)*&F,F)F0F)F)F(F**&F8F )F0F)F*F)F*F)F.F)F4F)F**(F8F)F,F),,*(F(F)F,F)F0F)F)*(F8F)F(F)F0F)F*F(F **&F8F)F0F)F)F)F*F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 32 "For n>1 we have a linear system " }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 5 "n:=1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 959 "n:=n+1:ens:=normal(subs(nu=1+beta,[coeff(series(subs(P3(t)=P3se r(n), P2(t)=P2ser(n), P1(t)=P1ser(n), P0(t)=P0ser(n),Q1(t)=Q1ser(n), Q 0(t)=Q0ser(n), R1(t)=R1ser(n), R0(t)=R0ser(n), EDsum),t,n+1),t,n),\nse q(coeff(series(eval(subs(P3(t)=P3ser(n), P2(t)=P2ser(n), P1(t)=P1ser(n ), P0(t)=P0ser(n), \nQ1(t)=Q1ser(n), Q0(t)=Q0ser(n), R1(t)=R1ser(n), R 0(t)=R0ser(n), ED[j])),t,n+1),t,n-1),j=4..7),\ncoeff(series(subs(P3(t) =P3ser(n+1), P2(t)=P2ser(n+1), P1(t)=P1ser(n+1), P0(t)=P0ser(n+1), \nQ 1(t)=Q1ser(n+1), Q0(t)=Q0ser(n+1), R1(t)=R1ser(n+1), R0(t)=R0ser(n+1), M1t(t)=M1tser(n+2),eqM1bis),t,n+1),t,n),\ncoeff(series(subs(P3(t)=P3s er(n+1), P2(t)=P2ser(n+1), P1(t)=P1ser(n+1), P0(t)=P0ser(n+1), \nQ1(t) =Q1ser(n+1), Q0(t)=Q0ser(n+1), R1(t)=R1ser(n+1), R0(t)=R0ser(n+1), eqQ 0R),t,n+1),t,n),\ncoeff(series(subs(P3(t)=P3ser(n+1), P2(t)=P2ser(n+1) , P1(t)=P1ser(n+1), P0(t)=P0ser(n+1), \nQ1(t)=Q1ser(n+1), Q0(t)=Q0ser( n+1), R1(t)=R1ser(n+1), R0(t)=R0ser(n+1), eqQ1R),t,n+1),t,n)])):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "indets(ens) minus \{beta,q,w \}; nops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<5&%$m1tG6#\"\"$&%#p0G 6#\"\"\"&F)6#\"\"#&%#p1GF*&F0F-&%#p2GF*&F3F-&%#p3GF*&F6F-&%#q0GF*&F9F- &%#q1GF*&F" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 310 "M at:=Matrix(1..8,1..8):\nfor j from 1 to 8 do \nMat[1,j]:=coeff(ens[j], p0[n]):Mat[2,j]:=coeff(ens[j], p1[n]): Mat[3,j]:=coeff(ens[j], p2[n]) :\nMat[4,j]:=coeff(ens[j], p3[n]): Mat[5,j]:=coeff(ens[j], q0[n]): Mat [6,j]:=coeff(ens[j], q1[n]): Mat[7,j]:=coeff(ens[j], r0[n]):Mat[8,j]:= coeff(ens[j], r1[n]): \nod: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 156 "map(factor,Mat):factor(Determinant(Mat));\nfactor(su bs(p0[1]=-4,p1[1]=8-2*w*q,p2[1]=4*w*(q-beta)-2*beta-4, p3[1]=2*beta-2* w*q, r0[0]=2, r1[0]=w*q-beta-4,%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,$*,\"#K\"\"\")%%betaG\"\"%F&,&%\"qGF&F)!\"\"F&),(*$)F(\"\"#F&F&*&F(F &F+F&F&F+F&\"\"&F&,6*(F)F&F(F&%\"wGF&F&*(F1F&F+F&F6F&F&*&F1F&F(F&F,*& \"\"$F&&%#p0G6#F&F&F,*&F:F&&%#p1GF=F&F,*&F:F&&%#p2GF=F&F,*&F:F&&%#p3GF =F&F,*&F1F&&%#r0G6#\"\"!F&F,*&F1F&&%#r1GFJF&F,F)F,F&F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*,\"$7&\"\"\")%%betaG\"\"&F&,&%\"qGF&\"\"%!\"\"F &),(*$)F(\"\"#F&F&*&F(F&F+F&F&F+F&F)F&%\"wGF&F-" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 55 "factor(%/(-16*n^5*beta^5*(q-4)*w*(beta^2+beta* q+q)^5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Another system" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "factor(subs(nu=1+beta,eqQ0R+eqQ1R));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*&%%betaG\"\"\",\\o*,\"#;F%F$F%%\"qGF%%\"tGF%)%\"w G\"\"#F%F%*,F-F%-%#R1G6#F*F%)F)F-F%F*F%F,F%F%*,F-F%F/F%F$F%F)F%F*F%F%* ,\"\")F%F/F%F)F%F*F%F,F%!\"\"*,\"\"%F%)F$F-F%F)F%F*F%F,F%F6**F5F%F)F%F *F%F$F%F%**F8F%)F)\"\"$F%F*F%F+F%F6**F(F%F2F%F*F%F+F%F%**F(F%F9F%F*F%F ,F%F%*(-%#Q0GF1F%F2F%F,F%F%*(-%#Q1GF1F%F2F%F,F%F%*(FAF%F$F%F)F%F%**F8F %FAF%F)F%F,F%F6**F5F%F/F%F$F%F*F%F6*(FDF%F$F%F)F%F%**F8F%FDF%F)F%F,F%F 6*(F8F%F$F%FAF%F6*(F-F%-%#R0GF1F%F$F%F6*(F8F%F$F%FDF%F6**\"#KF%F)F%F*F %F,F%F6**F5F%F2F%F*F%F,F%F%*,F8F%F$F%F2F%F*F%F+F%F6*(F-F%F2F%F,F%F%*&F /F%F)F%F6*&FMF%F)F%F6**F-F%F9F%F)F%F*F%F%*&F-F%F)F%F6*(F8F%F)F%F,F%F6* (F5F%F9F%F*F%F6*(FQF%F$F%F*F%F6*,F-F%F$F%F2F%F*F%F,F%F6*,F5F%F$F%F)F%F *F%F,F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "n:=1:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 974 "n:=n+1: ens:=normal(subs(nu =1+beta,[coeff(series(subs(P3(t)=P3ser(n), P2(t)=P2ser(n), P1(t)=P1ser (n), P0(t)=P0ser(n),Q1(t)=Q1ser(n), Q0(t)=Q0ser(n), R1(t)=R1ser(n), R0 (t)=R0ser(n), ED[0]),t,n+1),t,n),\nseq(coeff(series(eval(subs(P3(t)=P3 ser(n), P2(t)=P2ser(n), P1(t)=P1ser(n), P0(t)=P0ser(n), \nQ1(t)=Q1ser( n), Q0(t)=Q0ser(n), R1(t)=R1ser(n), R0(t)=R0ser(n), ED[j])),t,n+1),t,n -1),j=4..7),\ncoeff(series(subs(P3(t)=P3ser(n+1), P2(t)=P2ser(n+1), P1 (t)=P1ser(n+1), P0(t)=P0ser(n+1), \nQ1(t)=Q1ser(n+1), Q0(t)=Q0ser(n+1) , R1(t)=R1ser(n+1), R0(t)=R0ser(n+1), M1t(t)=M1tser(n+2),eqM1bis),t,n+ 1),t,n),\ncoeff(series(subs(P3(t)=P3ser(n+1), P2(t)=P2ser(n+1), P1(t)= P1ser(n+1), P0(t)=P0ser(n+1), \nQ1(t)=Q1ser(n+1), Q0(t)=Q0ser(n+1), R1 (t)=R1ser(n+1), R0(t)=R0ser(n+1), eqQ0R),t,n+1),t,n),\ncoeff(series(su bs(P3(t)=P3ser(n+1), P2(t)=P2ser(n+1), P1(t)=P1ser(n+1), P0(t)=P0ser(n +1), \nQ1(t)=Q1ser(n+1), Q0(t)=Q0ser(n+1), R1(t)=R1ser(n+1), R0(t)=R0s er(n+1), (eqQ1R+eqQ0R)/beta),t,n+1),t,n)\n])):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 38 "indets(ens) minus \{beta,q,w\}; nops(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<5&%$m1tG6#\"\"$&%#p0G6#\"\"\"&F)6#\" \"#&%#p1GF*&F0F-&%#p2GF*&F3F-&%#p3GF*&F6F-&%#q0GF*&F9F-&%#q1GF*&F" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "nops(ens);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 310 "Mat:=Matrix(1..8,1..8):\nfor j from 1 to 8 do \nMat[ 1,j]:=coeff(ens[j], p0[n]):Mat[2,j]:=coeff(ens[j], p1[n]): Mat[3,j]:=c oeff(ens[j], p2[n]):\nMat[4,j]:=coeff(ens[j], p3[n]): Mat[5,j]:=coeff( ens[j], q0[n]): Mat[6,j]:=coeff(ens[j], q1[n]): Mat[7,j]:=coeff(ens[j] , r0[n]):Mat[8,j]:=coeff(ens[j], r1[n]): \nod: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "map(factor,Mat):factor(Determinant(Mat)); \nfactor(subs(p0[1]=-4,r0[0]=2, %));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#,$*.\"#K\"\"\")%\"qG\"\"#F&%\"wGF&,&F(F&\"\"%!\"\"F&),(*$)%%betaGF)F &F&*&F2F&F(F&F&F(F&\"\"&F&,&*&\"\"$F&&%#p0G6#F&F&F&*&F)F&&%#r0G6#\"\"! F&F-F&F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*,\"$7&\"\"\")%\"qG\"\"# F&%\"wGF&,&F(F&\"\"%!\"\"F&),(*$)%%betaGF)F&F&*&F2F&F(F&F&F(F&\"\"&F&F &" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "factor(%/(16*n^5*q^2*( q-4)*w*(beta^2+beta*q+q)^5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\" \"" }}}{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 0 "" }}}{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 "" {TEXT -1 56 "Below, an earlier way of ruling out singularities at q=4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Let's get rid of the fake singularity at q=4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "fact or(subs(q=4, nu=1+beta,Dx));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$),(* &%%betaG\"\"\"%\"xGF(F(*&\"\"#F(F)F(F(F+!\"\"F+F(" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 11 "solve(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,$*&\"\"#\"\"\",&F%F&%%betaGF&!\"\"F&F#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "An equation replacing ED2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "ED2bis:=factor(subs(nu=1+beta,x=2/(2+beta), Q2(t)=1,e dplan))/(q-4):indets(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<4%%betaG% \"qG%\"tG%\"wG-%#P0G6#F&-%#P1GF*-%#P2GF*-%#P3GF*-%#Q0GF*-%#Q1GF*-%#R0G F*-%#R1GF*-%%diffG6$F(F&-F:6$F+F&-F:6$F-F&-F:6$F/F&-F:6$F1F&-F:6$F3F& " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "unassign('p3','p2','p1' ,'p0','q1','q0','r1','r0'); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 744 "n:=1: ens:=normal(\{coeff(series(subs(P3(t)=P3ser(n+1), P2(t)=P 2ser(n+1), P1(t)=P1ser(n+1), P0(t)=P0ser(n+1), \nQ1(t)=Q1ser(n+1), Q0( t)=Q0ser(n+1), R1(t)=R1ser(n+1), R0(t)=R0ser(n+1), EDsum),t,n+1),t,n), \ncoeff(series(subs(P3(t)=P3ser(n+1), P2(t)=P2ser(n+1), P1(t)=P1ser(n+ 1), P0(t)=P0ser(n), \nQ1(t)=Q1ser(n+1), Q0(t)=Q0ser(n+1), R1(t)=R1ser( n+1), R0(t)=R0ser(n+1), ED[0]),t,n+1),t,n),\ncoeff(series(eval(subs(P3 (t)=P3ser(n), P2(t)=P2ser(n), P1(t)=P1ser(n), P0(t)=P0ser(n), \nQ1(t)= Q1ser(n), Q0(t)=Q0ser(n), R1(t)=R1ser(n), R0(t)=R0ser(n), ED2bis)),t,n ),t,n-1),\nseq(coeff(series(eval(subs(P3(t)=P3ser(n), P2(t)=P2ser(n), \+ P1(t)=P1ser(n), P0(t)=P0ser(n), \nQ1(t)=Q1ser(n), Q0(t)=Q0ser(n), R1(t )=R1ser(n), R0(t)=R0ser(n), ED[j])),t,n),t,n-1),j=3..7)\}); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$ensG<*,$*&,(*$)%%betaG\"\"#\"\"\"F-*&F+F- %\"qGF-F-F/F-F-,.*(\"\")F-F+F-%\"wGF-F-*(\"\"%F-F/F-F3F-F-*&F5F-F+F-! \"\"&%#p3G6#F-F-*&F,F-&%#q1GF:F-F7F2F7F-F7,$**F,F-F+\"\"$,Z*&F2F-)F+F@ F-F7*&F2F-F8F-F-*&F2F-&%#p2GF:F-F-*&F2F-&%#p1GF:F-F-*&F2F-&%#p0GF:F-F- *&\"#;F-&%#r1G6#\"\"!F-F7*&FOF-&%#r0GFRF-F7*(F2F-FPF-F+F-F-*(FOF-F*F-F 3F-F7*(F2F-&%#q0GF:F-F+F-F-*(F5F-FZF-F*F-F-*&FLF-FCF-F-*(\"#7F-FLF-F+F -F-*(\"\"'F-FLF-F*F-F-*(F2F-F " 0 "" {MPLTEXT 1 0 38 "indets(ens) minus \+ \{beta,q,w\}; nops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<*&%#p0G6#\" \"\"&%#p1GF&&%#p2GF&&%#p3GF&&%#q0GF&&%#q1GF&&%#r0G6#\"\"!&%#r1GF4" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "The system is not linear. For example," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "coeff(ens[7],p1[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&&%#r0G6#\"\"!\"\"\"%\"qGF(" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 36 "But there is still a unique solution" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "solve(ens,indets(ens) minus \{beta,q,w\}) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<*/&%#p0G6#\"\"\"!\"%/&%#p1GF',&* (\"\"#F(%\"qGF(%\"wGF(!\"\"\"\")F(/&%#p2GF',**(\"\"%F(%%betaGF(F1F(F2* (F9F(F0F(F1F(F(*&F/F(F:F(F2F9F2/&%#p3GF',&*(F/F(F0F(F1F(F2*&F/F(F:F(F( /&%#q0GF',(*&F0F(F1F(F(*&F/F(F:F(F(F9F(/&%#q1GF',**(F9F(F:F(F1F(F(*&F0 F(F1F(F(F:F2F9F2/&%#r0G6#\"\"!F//&%#r1GFR,(*&F0F(F1F(F(F:F2F9F2" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "assign(%):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "From now on (n>1) we have a linear system in th e unknowns of cC(n)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 718 "n:= 1+n:ens:=normal(\{coeff(series(subs(P3(t)=P3ser(n), P2(t)=P2ser(n), P1 (t)=P1ser(n), P0(t)=P0ser(n), \nQ1(t)=Q1ser(n), Q0(t)=Q0ser(n), R1(t)= R1ser(n), R0(t)=R0ser(n), EDsum),t,n+1),t,n),\ncoeff(series(subs(P3(t) =P3ser(n), P2(t)=P2ser(n), P1(t)=P1ser(n), P0(t)=P0ser(n), \nQ1(t)=Q1s er(n), Q0(t)=Q0ser(n), R1(t)=R1ser(n), R0(t)=R0ser(n), ED[0]),t,n+1),t ,n),\ncoeff(series(eval(subs(P3(t)=P3ser(n), P2(t)=P2ser(n), P1(t)=P1s er(n), P0(t)=P0ser(n), \nQ1(t)=Q1ser(n), Q0(t)=Q0ser(n), R1(t)=R1ser(n ), R0(t)=R0ser(n), ED2bis)),t,n+1),t,n-1),\nseq(coeff(series(eval(subs (P3(t)=P3ser(n), P2(t)=P2ser(n), P1(t)=P1ser(n), P0(t)=P0ser(n), \nQ1( t)=Q1ser(n), Q0(t)=Q0ser(n), R1(t)=R1ser(n), R0(t)=R0ser(n), ED[j])),t ,n+1),t,n-1),j=3..7)\}):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "indets(ens) minus \{beta,q,w\}; nops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<*&%#p0G6#\"\"&&%#p1GF&&%#p2GF&&%#p3GF&&%#q0GF&&%#q1GF& &%#r0G6#\"\"%&%#r1GF4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 311 "Mat:=Matrix(1..8,1..8):\nfo r j from 1 to 8 do \nMat[1,j]:=coeff(ens[j], p0[n]):Mat[2,j]:=coeff(en s[j], p1[n]): Mat[3,j]:=coeff(ens[j], p2[n]):\nMat[4,j]:=coeff(ens[j], p3[n]): Mat[5,j]:=coeff(ens[j], q0[n]): Mat[6,j]:=coeff(ens[j], q1[n] ): Mat[7,j]:=coeff(ens[j], r0[n-1]): Mat[8,j]:=coeff(ens[j], r1[n-1]): \nod: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "map(factor,Mat) :factor(Determinant(Mat));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*.\"(+ ++)\"\"\"%%betaG\"\")%\"qG\"\"$%\"wGF&,(*$)F'\"\"#F&F&*&F'F&F)F&F&F)F& F/,&F/F&F'F&!\"#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "fa ctor(%/(n^6*q^3*w*beta^8*(beta^2+beta*q+q)^2/(2+beta)^2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!$7&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "factor(solve(ens, indets(ens) minus \{w,beta,q\})):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "assign(%):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 126 "A toy example: Tutte's equation for uncoloured planar maps: \nhow to obtain DEs rather than a polynomial equation (S ection 3.1)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "eqMu:=-M+1+t* y^2*M^2+t*y/(y-1)*(y*M-M1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqMu G,*%\"MG!\"\"\"\"\"F(*(%\"tGF()%\"yG\"\"#F()F&F-F(F(**F*F(F,F(,&F,F(F( F'F',&*&F&F(F,F(F(%#M1GF'F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "Muser:=proc(n)\nif n=0 then 1 else\nnormal(series(subs(M=Muser( n-1),M1=subs(y=1,Muser(n-1)),eqMu+M),t,n+1)):fi: end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Muser(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+-%\"tG\"\"\"\"\"!,&*$)%\"yG\"\"#F%F%F*F%F%*&,**&F+F%)F *\"\"$F%F%*&F0F%F)F%F%*&F+F%F*F%F%F+F%F%F*F%F+*&,.*&\"\"&F%)F*F6F%F%*& \"#5F%)F*\"\"%F%F%*&F9F%F/F%F%*&\"#6F%F)F%F%*&\"\"*F%F*F%F%F@F%F%F*F%F 0-%\"OG6#F%F;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "subs(y=1,M user(3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+-%\"tG\"\"\"\"\"!\"\"#F% \"\"*F'\"#a\"\"$-%\"OG6#F%\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "The quadratic method" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "The l eft-hand side is the square of :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "eqdM:=collect(numer(diff(eqMu,M)),M,factor);;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqdMG,*%\"yG!\"\"\"\"\"F(*,\"\"#F(% \"tGF()F&F*F(%\"MGF(,&F&F(F(F'F(F(*&F+F(F,F(F(" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 79 "Delta:=subs(M1=M1(t),(y-1)^2*factor(discrim(eq Mu,M)));collect(Delta,M1,factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %&DeltaG,4**\"\"%\"\"\"-%#M1G6#%\"tGF()F,\"\"#F()%\"yGF'F(F(**F'F(F)F( F-F()F0\"\"$F(!\"\"*&F-F(F/F(F(*(F'F(F,F(F/F(F4*(\"\"'F(F,F(F2F(F(*(F. F(F,F()F0F.F(F4*$F:F(F(*&F.F(F0F(F4F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,2*,\"\"%\"\"\")%\"tG\"\"#F&)%\"yG\"\"$F&,&F+F&F&!\"\"F&-%#M1G6# F(F&F&*&F'F&)F+F%F&F&*(F%F&F(F&F3F&F.*(\"\"'F&F(F&F*F&F&*(F)F&F(F&)F+F )F&F.*$F8F&F&*&F)F&F+F&F.F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 " Checking the expression in the paper" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "eqpap:=eqdM^2-Delta;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&eqpapG,6*$),*%\"yG!\"\"\"\"\"F+*,\"\"#F+%\"tGF+)F)F-F+%\"MGF+,&F )F+F+F*F+F+*&F.F+F/F+F+F-F+F+**\"\"%F+-%#M1G6#F.F+)F.F-F+)F)F4F+F***F4 F+F5F+F8F+)F)\"\"$F+F+*&F8F+F9F+F**(F4F+F.F+F9F+F+*(\"\"'F+F.F+F;F+F** (F-F+F.F+F/F+F+*$F/F+F**&F-F+F)F+F+F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "Deltapap:=(y-1-t*y^2)^2-4*t*y^2*(y-1)^2+4*t^2*y^3*(y- 1)*M1(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)DeltapapG,(*$),(*&%\"t G\"\"\")%\"yG\"\"#F+!\"\"F-F+F+F/F.F+F+**\"\"%F+F,F+F*F+),&F-F+F+F/F.F +F/*,F1F+)F*F.F+)F-\"\"$F+F3F+-%#M1G6#F*F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "factor(Deltapap/Delta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "fa ctor(subs(M1(t)=M1,eqpap/eqMu));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$ **\"\"%\"\"\")%\"yG\"\"#F&%\"tGF&),&F(F&F&!\"\"F)F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "Yser:=proc(n)\nif n=0 then 1 else \nconvert(normal(series(subs(M=convert(Muser(n-1),polynom),y=Yser(n-1) ,eqdM+y),t,n+1)),polynom):fi: end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "Yser(4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&\"$!> \"\"\")%\"tG\"\"%F&F&*&\"#DF&)F(\"\"$F&F&*&F)F&)F(\"\"#F&F&F(F&F&F&" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "M1eq:=op(2,factor(discrim( Delta,y)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%M1eqG,,*(\"#F\"\"\") -%#M1G6#%\"tG\"\"#F()F-F.F(F(*(\"#=F(F*F(F-F(!\"\"F*F(*&\"#;F(F-F(F(F( F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "collect(%,M1,factor); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**(\"#F\"\"\")-%#M1G6#%\"tG\"\"#F &)F+F,F&F&*&,&*&\"#=F&F+F&!\"\"F&F&F&F(F&F&*&\"#;F&F+F&F&F&F2" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "factor([solve(%,M1(t))]);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#7$,$*(\"#a!\"\"%\"tG!\"#,(*&\"#=\"\" \"F(F-F-F-F'*$,$*$),&*&\"#7F-F(F-F-F-F'\"\"$F-F'#F-\"\"#F-F-F-,$*(F&F' ,(*&F,F-F(F-F'F-F-F.F-F-F(F)F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "map(series,%,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$+-%\"tG \"\"\"\"\"!\"\"#F&\"\"*F(\"#a\"\"$-%\"OG6#F&\"\"%+1F%#!\"\"\"#F!\"##F( F+F2F2F'F4F&!\"*F(!#aF+F,F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "M1sol:=op(1,%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&M1solG,$*( \"#a!\"\"%\"tG!\"#,(*&\"#=\"\"\"F)F.F.F.F(*$,$*$),&*&\"#7F.F)F.F.F.F( \"\"$F.F(#F.\"\"#F.F.F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Now, t o the differential system" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "The \+ reciprocal of Y" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "Zser:=pr oc(n) normal(series(1/Yser(n),t,n+1)): end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "Zser(4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+/%\"tG \"\"\"\"\"!!\"\"F%!\"$\"\"#!#=\"\"$!$N\"\"\"%-%\"OG6#F%\"\"&" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "Differential approach (does not re quire Maple... except maybe to compute M1)" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 102 "Psol:=y^2*P2(t)+y*P1(t)+P0(t);\nQsol:=y^2*Q2(t)+y* Q1(t)+Q0(t);\nRsol:=y^3*R3(t)+y^2*R2(t)+y*R1(t)+R0(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%PsolG,(*&)%\"yG\"\"#\"\"\"-%#P2G6#%\"tGF*F**&F( F*-%#P1GF-F*F*-%#P0GF-F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%QsolG,( *&)%\"yG\"\"#\"\"\"-%#Q2G6#%\"tGF*F**&F(F*-%#Q1GF-F*F*-%#Q0GF-F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%RsolG,**&)%\"yG\"\"$\"\"\"-%#R3G6#% \"tGF*F**&)F(\"\"#F*-%#R2GF-F*F**&F(F*-%#R1GF-F*F*-%#R0GF-F*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "The system: five DEs" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "collect(numer(factor(1/Qsol*diff(Qs ol^2/Psol,t)-1/Rsol*diff(Rsol^2/Psol,y))),y,factor);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,2*&,(*&-%#Q2G6#%\"tG\"\"\"-%%diffG6$-%#P2GF)F*F+!\" \"*(\"\"#F+-F-6$F'F*F+F/F+F+*(\"\"%F+-%#R3GF)F+F/F+F1F+)%\"yGF7F+F+*&, .*&F'F+-F-6$-%#P1GF)F*F+F1*&-%#Q1GF)F+F,F+F1*(F3F+-F-6$FDF*F+F/F+F+*(F 3F+-%#R2GF)F+F/F+F1*(F3F+F4F+FAF+F+*(\"\"&F+F8F+FAF+F1F+)F;\"\"$F+F+*& ,2*&F'F+-F-6$-%#P0GF)F*F+F1*&FDF+F?F+F1*&-%#Q0GF)F+F,F+F1*(F3F+-F-6$FZ F*F+F/F+F+*(F3F+FGF+FAF+F+*(FPF+FJF+FAF+F1*(F3F+F4F+FVF+F+*(\"\"'F+F8F +FVF+F1F+)F;F3F+F+*&,0*&FDF+FTF+F1*&FZF+F?F+F1*(F3F+-%#R0GF)F+F/F+F+*( F3F+FgnF+FAF+F+*&-%#R1GF)F+FAF+F1*(F3F+FGF+FVF+F+*(F7F+FJF+FVF+F1F+F;F +F+*&FZF+FTF+F1*&FAF+FdoF+F+*(F3F+FVF+FgnF+F+*(F3F+FVF+FhoF+F1" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "factor(coeff(normal(Delta*Qs ol^2-Psol*diff(Delta,y)^2),y,8));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#* (%\"tG\"\"\",(*(\"\"%F%-%#M1G6#F$F%F$F%F%F$F%F(!\"\"F%,***\"#kF%-%#P2G F+F%F)F%)F$\"\"#F%F,*(\"#;F%F0F%F2F%F,*$)-%#Q2GF+F3F%F%*(F/F%F0F%F$F%F %F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "-op(3,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,***\"#k\"\"\"-%#P2G6#%\"tGF&-%#M1GF)F&)F*\" \"#F&F&*(\"#;F&F'F&F-F&F&*$)-%#Q2GF)F.F&!\"\"*(F%F&F'F&F*F&F5" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{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 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "We will now write the invariant equation of Sec tion 4.1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "The first invariant ( denoted I(t,y) in the paper)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "I1:=y*t*q*M(y)+(y-1)/y+t*y/(y-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#I1G,(**%\"yG\"\"\"%\"tGF(%\"qGF(-%\"MG6#F'F(F(*&,&F'F(F(!\"\" F(F'F0F(*(F)F(F'F(F/F0F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Polyn omials N and D" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Nsubs:=be ta*(4-q)*(1/y-1)+(q+2*beta)*x-q;;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %&NsubsG,(*(%%betaG\"\"\",&\"\"%F(%\"qG!\"\"F(,&*&F(F(%\"yGF,F(F(F,F(F (*&,&F+F(*&\"\"#F(F'F(F(F(%\"xGF(F(F+F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "DD;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&,&*$)%%beta G\"\"#\"\"\"F**&%#nuGF*%\"qGF*F*F*)%\"xGF)F*F**(F-F*,&F,F*F*F*F*F/F*! \"\"**F(F*%\"tGF*,&F-F*\"\"%F2F*,&*&F-F*%\"wGF*F*F(F*F*F*F-F*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "The second invariant is J=DD^(m/2) T_m(usubs) with" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "usubs:= Nsubs/2/sqrt(DD);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&usubsG,$*(\"\" #!\"\",(*(%%betaG\"\"\",&\"\"%F,%\"qGF(F,,&*&F,F,%\"yGF(F,F,F(F,F,*&,& F/F,*&F'F,F+F,F,F,%\"xGF,F,F/F(F,,**&,&*$)F+F'F,F,*&%#nuGF,F/F,F,F,)F6 F'F,F,*(F/F,,&F=F,F,F,F,F6F,F(**F+F,%\"tGF,,&F/F,F.F(F,,&*&F/F,%\"wGF, F,F+F,F,F,F/F,#F(F'F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Chebyshev polynomials" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "T:=proc(m) subs(cos(x)=u,expand(cos (m*x))): end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "T(4);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&\"\")\"\"\")%\"uG\"\"%F&F&*&F%F&)F (\"\"#F&!\"\"F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 78 "When q=2 (m=4,k=1): invariant equ ation, values of the series C_r (Section 4.1)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "eval(subs(m=4, k=1, 2+2*cos(2*k*Pi/m)));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "T(4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&\"\")\"\" \")%\"uG\"\"%F&F&*&F%F&)F(\"\"#F&!\"\"F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "If we just consider proper colourings (bipartite maps)" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "subs(q=2,nu=0,beta=-1,w=1, Nsubs);\nsubs(q=2,nu=0,beta=-1,w=1,DD);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"#\"\"\"%\"yG!\"\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, **$)%\"xG\"\"#\"\"\"F(*&F'F(%\"tGF(F(*&F'F(F&F(!\"\"F'F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "The invariant equation " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "eqinv:=subs(u=usubs,x=I1,q=2,nu=0,beta=-1 ,w=1,m=4,DD^2*T(4) - add(C[r]*I1^r,r=0..4));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&eqinvG,.*&),.*$),(**\"\"#\"\"\"%\"yGF.%\"tGF.-%\"MG6 #F/F.F.*&,&F/F.F.!\"\"F.F/F6F.*(F0F.F/F.F5F6F.F-F.F.**\"\"%F.F/F.F0F.F 1F.F6*(F-F.F5F.F/F6F6**F-F.F0F.F/F.F5F6F6*&F-F.F0F.F.F-F.F-F.,(*(\"\") F.F/!\"%F(!\"#F.*(F?F.F/FAF(F6F6F.F.F.F.&%\"CG6#\"\"!F6*&&FD6#F.F.F+F. F6*&&FD6#F-F.F*F.F6*&&FD6#\"\"$F.)F+FPF.F6*&&FD6#F9F.)F+F9F.F6" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 126 "Expansion around y=1 (we want the first five coefficients, and Maple does not seem to expand things at \+ deterministic order...)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " factor(series(eqinv,y=1,1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+',&% \"yG\"\"\"F&!\"\",$*&)%\"tG\"\"%F&,&&%\"CG6#F,F&F&F'F&F'!\"%-%\"OG6#F& !\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "factor(series(subs( C[4]=1,eqinv),y=1,2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+',&%\"yG\" \"\"F&!\"\",$*&)%\"tG\"\"$F&,&&%\"CG6#F,F&\"\"%F&F&F'!\"$-%\"OG6#F&!\" #" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "factor(series(subs(C[4 ]=1,C[3]=-4,eqinv),y=1,3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+',&%\" yG\"\"\"F&!\"\"*&)%\"tG\"\"#F&,&*&\"\"%F&F*F&F&&%\"CG6#F+F'F&!\"#-%\"O G6#F&F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "factor(series(su bs(C[4]=1,C[3]=-4,C[2]=4*t,eqinv),y=1,4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+',&%\"yG\"\"\"F&!\"\"*&%\"tGF&,(&%\"CG6#F&F'*&\"\")F&F )F&F&F/F&F&F'-%\"OGF-\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "factor(series(subs(C[4]=1,C[3]=-4,C[2]=4*t,C[1]=8*(1+t),eqinv),y=1 ,5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+',&%\"yG\"\"\"F&!\"\",,*(\"# KF&-%\"MG6#F&F&)%\"tG\"\"#F&F&*&\"\"%F&F.F&F'*&\"#SF&F/F&F'&%\"CG6#\" \"!F'F2F'F8-%\"OGF-F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "fa ctor(subs(C[4]=1,C[3]=-4,C[2]=4*t,C[1]=8*(1+t),C[0]=32*M(1)*t^2-4*t^2- 40*t-4,eqinv));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"#K\"\"\"%\"yG !\"#%\"tGF&,2*()-%\"MG6#F'\"\"#F&F)F&)F'\"\"%F&!\"\"*(F,F&F)F&)F'F0F&F &*(-F.6#F&F&F)F&F5F&F&*(F5F&F)F&F-F&F3*&F-F&F5F&F&*$F5F&F3F-F3F&F&F&F3 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "collect(op(3,numer(%)), M,factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**(-%\"MG6#\"\"\"F(%\"t GF()%\"yG\"\"#F(F(*,F)F(F*F(,&F+F(F(!\"\"F(,&F+F(F(F(F()-F&6#F+F,F(F/* &,(*&F)F(F*F(F/*$F*F(F(F(F/F(F2F(F(*&F.F(F0F(F/" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 36 "collect((-%/(y^2-1)+M(y)),M,factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**,%\"tG\"\"\"%\"yG\"\"#,&F'F&F&!\"\"F*,&F' F&F&F&F*-%\"MG6#F&F&F*F&F&*()-F-6#F'F(F&F%F&)F'F(F&F&*,F%F&F'F(F)F*F+F *F1F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 48 "Computing C_m, C_\{m-1\}, . ..C_\{m-3\} (Section 4.1)" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "We ta ke the invariant equation, multiply it by (y-1)^m so that it becomes n on singular at y=1." }}{PARA 0 "" 0 "" {TEXT -1 144 "Then we expand it around y=1, and write that the first (four) coefficients are zero. Th is gives us the values of the 4 series Cm, ...c, C_\{m-3\}." }}{PARA 0 "" 0 "" {TEXT -1 208 "\nWe write Dt = D(t,I)*(y-1)^2 and It=(y-1)*I, again to avoid singularities at y=1. The following expression then co incides with the invariant equation (multiplied by (y-1)^m) up to the \+ coefficient of (y-1)^4." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 " m:='m': expr:=Dt(y)^(m/2)*Tm(G(y))-sum(C[m-r]*(y-1)^(r)*It(y)^(m-r),r= 0..4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%exprG,.*&)-%#DtG6#%\"yG,$ *&\"\"#!\"\"%\"mG\"\"\"F1F1-%#TmG6#-%\"GGF*F1F1*&&%\"CG6#F0F1)-%#ItGF* F0F1F/*(&F96#,&F0F1F1F/F1,&F+F1F1F/F1)F " 0 "" {MPLTEXT 1 0 121 "Ge:=subs(De=DD,x=I1,usubs):\nDte:=factor((y- 1)^2*subs(x=I1,DD)):\nIte:=factor((y-1)*I1):\nNte:=(y-1)*subs(De=DD,x= I1,Nsubs):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "We take out the fac tor delta^(m/2) from the series C_r" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "exprmod:=subs(seq(C[m-r]=(beta^2+nu*q)^(m/2)*Cmod[m-r ],r=0..4),expr);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(exprmodG,.*&)-% #DtG6#%\"yG,$*&\"\"#!\"\"%\"mG\"\"\"F1F1-%#TmG6#-%\"GGF*F1F1*(),&*$)%% betaGF.F1F1*&%#nuGF1%\"qGF1F1F,F1&%%CmodG6#F0F1)-%#ItGF*F0F1F/**F8F1&F A6#,&F0F1F1F/F1,&F+F1F1F/F1)FDFIF1F/**F8F1&FA6#,&F0F1F.F/F1)FJF.F1)FDF OF1F/**F8F1&FA6#,&F0F1\"\"$F/F1)FJFVF1)FDFUF1F/**F8F1&FA6#,&F0F1\"\"%F /F1)FJFgnF1)FDFfnF1F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "expansio ns around y=1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "Itser:=map (Z->simplify(Z,symbolic),series(Ite,y=1,1) assuming y>1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ItserG+',&%\"yG\"\"\"F(!\"\"%\"tG\"\"!-%\"OG 6#F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "Gser:=map(Z->simp lify(Z,symbolic),series(Ge,y=1,2) assuming y>1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%GserG+',&%\"yG\"\"\"F(!\"\",$*(\"\"#F),&%\"qGF(*&F,F (%%betaGF(F(F(,&*$)F0F,F(F(*&%#nuGF(F.F(F(#F)F,F(\"\"!-%\"OG6#F(F(" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "Ntser:=map(Z->factor(simpli fy(Z,symbolic)),series(Nte,y=1,2) assuming y>1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&NtserG+),&%\"yG\"\"\"F(!\"\"*&,&%\"qGF(*&\"\"#F(%%be taGF(F(F(%\"tGF(\"\"!,,*,F.F(-%\"MG6#F(F(F/F(F,F(F0F(F(*(F4F()F,F.F(F0 F(F(*(F.F(F/F(F0F(F(*&F0F(F,F(F(F,F)F(-%\"OGF6F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "Dtser:=map(Z->factor(simplify(Z,symbolic)),se ries(Dte,y=1,1) assuming y>1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&D tserG+',&%\"yG\"\"\"F(!\"\"*&,&*$)%%betaG\"\"#F(F(*&%#nuGF(%\"qGF(F(F( )%\"tGF/F(\"\"!-%\"OG6#F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "simplify(series(subs(G(y)=Gser,It(y)=Itser,Dt(y)=Dtser,exprmod/t^ m/(beta^2+nu*q)^(m/2)),y=1),symbolic);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+',&%\"yG\"\"\"F&!\"\",&-%#TmG6#,$*(\"\"#F',&%\"qGF&*&F.F&%%beta GF&F&F&,&*$)F2F.F&F&*&%#nuGF&F0F&F&#F'F.F&F&&%%CmodG6#%\"mGF'\"\"!-%\" OG6#F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "We'll denote by Ti th e i-th derivative of T_m, evaluated at gamma=(q+2*beta)/2/sqrt(beta^2+ nu*q)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "subs(Tm(1/2*(q+2*b eta)/(beta^2+nu*q)^(1/2))=T0,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+' ,&%\"yG\"\"\"F&!\"\",&%#T0GF&&%%CmodG6#%\"mGF'\"\"!-%\"OG6#F&F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "Cmodsol:=simplify(solve(coef f(subs(y=1+s,%),s,0),Cmod[m]),symbolic);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(CmodsolG%#T0G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "C msol:=(beta^2+nu*q)^(m/2)*Cmodsol;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%&CmsolG*&),&*$)%%betaG\"\"#\"\"\"F,*&%#nuGF,%\"qGF,F,,$*&F+!\"\"%\"m GF,F,F,%#T0GF," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "One step furthe r: determination of C\{m-1\}" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "Itser:=map(Z->simplify(Z,symbolic),series(Ite,y=1,2) assuming y> 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ItserG+),&%\"yG\"\"\"F(!\"\" %\"tG\"\"!,&*(F*F(%\"qGF(-%\"MG6#F(F(F(F*F(F(-%\"OGF1\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "Gser:=map(Z->simplify(Z,symbolic),s eries(subs(nu=1+beta,Ge),y=1,3) assuming y>1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%GserG+),&%\"yG\"\"\"F(!\"\",$*(\"\"#F),&%\"qGF(*&F,F (%%betaGF(F(F(,(*$)F0F,F(F(*&F0F(F.F(F(F.F(#F)F,F(\"\"!,$*.\"\"%F)F.F( F0F(,&F.F(F9F)F(%\"tGF)F1#!\"$F,F)F(-%\"OG6#F(F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "Dtser:=map(Z->factor(simplify(Z,symbolic)),se ries(subs(nu=1+beta,Dte),y=1,2) assuming y>1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&DtserG+),&%\"yG\"\"\"F(!\"\"*&,(*$)%%betaG\"\"#F(F(* &F.F(%\"qGF(F(F1F(F()%\"tGF/F(\"\"!*&F3F(,2*,F/F(-%\"MG6#F(F(F-F(F1F(F 3F(F(*,F/F(F8F(F.F()F1F/F(F3F(F(**F/F(F8F(F " 0 "" {MPLTEXT 1 0 234 "simplify(subs(D(Tm)( 1/2*(q+2*beta)/(beta^2+beta*q+q)^(1/2))=T1,Tm(1/2*(q+2*beta)/(beta^2+b eta*q+q)^(1/2))=T0,nu=beta+1,series(subs(G(y)=Gser,It(y)=Itser,Dt(y)=D tser,Cmod[m]=Cmodsol,exprmod/t^m/(beta^2+nu*q)^(1/2*m)),y=1,2)),symbol ic);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+',&%\"yG\"\"\"F&!\"\",$*&#F& \"\"%F&*(,(*$)%%betaG\"\"#F&F&*&F0F&%\"qGF&F&F3F&#!\"$F1%\"tGF',0*(%#T 1GF&F0F&)F3F1F&F&**F+F&F9F&F3F&F0F&F'*.F1F&F-#F&F1%\"mGF&%#T0GF&F3F&F0 F&F&*,F+F&F-F=F>F&F?F&F3F&F&**F+F&F-F=F/F&&%%CmodG6#,&F>F&F&F'F&F&*,F+ F&F-F=F0F&F3F&FBF&F&**F+F&F-F=F3F&FBF&F&F&F&F'F&-%\"OG6#F&F1" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "Cmod1sol:=collect(solve(coef f(subs(y=1+s,%),s,1),Cmod[m-1]),[T0,T1],factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)Cmod1solG,&*.\"\"#!\"\"%\"qG\"\"\",(*$)%%betaGF'F*F* *&F.F*F)F*F*F)F*F(%\"mGF*,&F'F*F.F*F*%#T0GF*F(*.\"\"%F(F)F*F.F*,&F)F*F 4F(F*F+#!\"$F'%#T1GF*F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 " Cmod1pap:=-q/4*delta^(-3/2)*(2*m*(nu+1)*T0*sqrt(delta)+beta*(q-4)*T1); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)Cmod1papG,$**\"\"%!\"\"%\"qG\" \"\"%&deltaG#!\"$\"\"#,&*,F.F*%\"mGF*,&%#nuGF*F*F*F*%#T0GF*F+#F*F.F**( %%betaGF*,&F)F*F'F(F*%#T1GF*F*F*F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "simplify(factor(expand(rationalize(factor(subs(beta= nu-1,delta=(nu^2+nu*q-2*nu+1),Cmod1sol/Cmod1pap))))),symbolic);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Cm1sol:=(beta^2+nu*q)^(m/2)*Cmod1pap;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'Cm1solG,$*,\"\"%!\"\"),&*$)%%betaG\"\"#\"\"\"F/ *&%#nuGF/%\"qGF/F/,$*&F.F(%\"mGF/F/F/F2F/%&deltaG#!\"$F.,&*,F.F/F5F/,& F1F/F/F/F/%#T0GF/F6#F/F.F/*(F-F/,&F2F/F'F(F/%#T1GF/F/F/F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "We use expressions of T2 and T3 in terms of T0 and T1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "checking the DE \+ for T_m" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "for mm from 1 to 6 do factor((1-u^2)*diff(T(mm),u$2)-u*diff(T(mm),u)+mm^2*T(mm));od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "m:='m':;edTm:=(1-x^2)*diff(Tm(x),x$2)-x*diff(Tm(x),x) +m^2*Tm(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%edTmG,(*&,&*$)%\"xG \"\"#\"\"\"!\"\"F,F,F,-%%diffG6$-%#TmG6#F*-%\"$G6$F*F+F,F,*&F*F,-F/6$F 1F*F,F-*&)%\"mGF+F,F1F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "gammae:=1/2*(2*beta+q)/(beta^2+nu*q)^(1/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'gammaeG,$*(\"\"#!\"\",&%\"qG\"\"\"*&F'F+%%betaGF+F+F +,&*$)F-F'F+F+*&%#nuGF+F*F+F+#F(F'F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "T2e:=subs((diff(Tm(x),x))=T1,Tm(x)=T0,x=gammae,op(2,i solate(edTm,(diff(Tm(x),$(x,2))))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$T2eG*&,&*&%#T0G\"\"\")%\"mG\"\"#F)!\"\"**F,F-,&%\"qGF)*&F,F)% %betaGF)F)F),&*$)F2F,F)F)*&%#nuGF)F0F)F)#F-F,%#T1GF)F)F),&*(\"\"%F-F/F ,F3F-F-F)F)F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "diff(edTm, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**(\"\"$\"\"\"%\"xGF&-%%diffG6$-%#TmG6#F'-%\"$G6$F'\"\"#F&!\"\"*&,&*$)F'F1F&F2F&F&F&-F)6$F+-F/6$F'F% F&F&-F)6$F+F'F2*&)%\"mGF1F&F;F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "isolate(%,(diff(Tm(x),$(x,3))));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%%diffG6$-%#TmG6#%\"xG-%\"$G6$F*\"\"$*&,(*&)%\"mG\" \"#\"\"\"-F%6$F'F*F5!\"\"*(F.F5F*F5-F%6$F'-F,6$F*F4F5F5F6F5F5,&*$)F*F4 F5F8F5F5F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "T3e:=factor(s ubs((diff(Tm(x),$(x,2)))=T2e,(diff(Tm(x),x))=T1,Tm(x)=T0,x=gammae,op (2,%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$T3eG,$*,\"\"%\"\"\",&*$)%%betaG\"\"#F(F(*&%#nuGF(%\"qGF(F(F(,4*,\"#7F(F)#F(F-%#T0GF(F,F()%\"m GF-F(F(*,\"\"'F(F)F4F5F(F6F(F0F(F(*,F'F(F6F(%#T1GF(F0F(F,F(!\"\"*,F'F( F6F(F;F(F0F(F/F(F(*(F6F(F;F()F0F-F(F<*(F3F(F;F(F+F(F<**\"\")F(F;F(F0F( F,F(F<**F'F(F;F(F/F(F0F(F<*(F-F(F;F(F?F(F " 0 "" {MPLTEXT 1 0 67 "I tser:=map(Z->simplify(Z,symbolic),series(Ite,y=1,3) assuming y>1):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "Gser:=map(Z->simplify(Z,sym bolic),series(Ge,y=1,4) assuming y>1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "Dtser:=map(Z->factor(simplify(Z,symbolic)),series(Dte ,y=1,3) assuming y>1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 331 " collect( simplify(subs(@@(D,2)(Tm)(1/2*(q+2*beta)/(beta^2+nu*q)^(1/2 ))=T2e,\nD(Tm)(1/2*(q+2*beta)/(beta^2+nu*q)^(1/2))=T1,Tm(1/2*(q+2*beta )/(beta^2+nu*q)^(1/2))=T0,\nnu=1+beta,\nseries(subs(G(y)=Gser,It(y)=It ser,Dt(y)=Dtser,Cmod[m]=Cmodsol,Cmod[m-1]=Cmod1sol,exprmod/t^m/(beta^2 +nu*q)^(1/2*m)),y=1,3)),symbolic),[Cmod[m-2]],factor);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+',&%\"yG\"\"\"F&!\"\",$*&#F&\"\")F&*(,dq*2\"\"%F& ,(*$)%%betaG\"\"#F&F&*&F3F&%\"qGF&F&F6F&#F&F4%#T0GF&)F3\"\"$F&%\"mGF&) F6F4F&%\"tGF&%\"wGF&F&*2F/F&F0F7F8F&F2F&F;F&)F6F:F&F=F&F>F&F&*2F/F&F0F 7F8F&F3F&F;F&F@F&F=F&F>F&F&*2\"#;F&F0F7F8F&F9F&F;F&F6F&F=F&F>F&F'*2FCF &F0F7F8F&F2F&F;F&FF&F'*2FCF&F0F7F8F&F3F&F;F&FF&F'*0F CF&F0F7F8F&F2F&F;F&F6F&F=F&F'*0F/F&F0F7F=F&F8F&F2F&F;F&FF&F&*.\"#CF&FUF&F9F&FF&F& *.FCF&FUF&F2F&FF&F&*.F+F&FUF&F3F&F@F&F=F&F>F&F&*.F/F&FUF&F2F& F@F&F=F&F>F&F&*.F/F&FUF&FJF&FF&F'*,F4F&FUF&FJF&FF&F'*.FinF&FUF&F9F&F@F&F=F&F>F&F'*.F4F&FUF&F2F&F^pF&F=F&F>F&F '*,F+F&FUF&FJF&F6F&F=F&F'*,FCF&FUF&F9F&F6F&F=F&F'*,FWF&FUF&F2F&F " 0 "" {MPLTEXT 1 0 76 "Cmod2sol:=collect(solve(coef f(subs(y=1+s,%),s,2),Cmod[m-2]),[T0,T1],factor);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)Cmod2solG,&*,\"\"%!\"\"%\"mG\"\"\",J*,\"\"#F*)%%beta G\"\"$F*)%\"qGF-F*%\"tGF*%\"wGF*F**,F-F*)F/F-F*)F2F0F*F3F*F4F*F***F-F* )F/F'F*F2F*F3F*F***F-F*F.F*F1F*F3F*F**,\"\")F*F.F*F2F*F3F*F4F*F(*,F**F'F*F.F*F 3F*F4F*F**,\"\"'F*F6F*F2F*F3F*F4F*F**,F-F*F/F*F1F*F3F*F4F*F**(F-F*F.F* F3F*F(**F-F*F6F*F2F*F3F*F(*,F'F*F/F*F2F*F3F*F4F*F***F-F*F1F*F3F*F4F*F* *(F'F*F6F*F3F*F(*(F/F*F)F*F2F*F(**FVF*F/F*F2F*F3F*F(*&F/F*F2F*F**(F-F* F)F*F2F*F(*(F'F*F2F*F3F*F(*&F-F*F2F*F*F*FL#!\"&F-%#T1GF*F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Comparing with the expression of the pape r" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "Cmod2pap:=1/8*delta^(- 5/2)*(2*m*a0pap*T0*sqrt(delta)+beta*q*(q-4)*a1pap*T1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)Cmod2papG,$*(\"\")!\"\"%&deltaG#!\"&\"\"#,&*,F, \"\"\"%\"mGF/%&a0papGF/%#T0GF/F)#F/F,F/*,%%betaGF/%\"qGF/,&F6F/\"\"%F( F/%&a1papGF/%#T1GF/F/F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "a0pap:=2*t*beta*delta*(q-4)*(beta+w*q)+q*(m-1)*(q*(1+nu^2)-2*beta^ 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&a0papG,&*.\"\"#\"\"\"%\"tGF( %%betaGF(%&deltaGF(,&%\"qGF(\"\"%!\"\"F(,&*&F-F(%\"wGF(F(F*F(F(F(*(F-F (,&%\"mGF(F(F/F(,&*&F-F(,&*$)%#nuGF'F(F(F(F(F(F(*&F'F()F*F'F(F/F(F(" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "a1pap:=2*t*delta*(nu+1-w*( 2*beta+q))+q*(m-1)*(nu+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&a1pap G,&**\"\"#\"\"\"%\"tGF(%&deltaGF(,(%#nuGF(F(F(*&%\"wGF(,&%\"qGF(*&F'F( %%betaGF(F(F(!\"\"F(F(*(F0F(,&%\"mGF(F(F3F(,&F,F(F(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "simplify(factor(subs(\nbeta=nu-1,de lta=(nu^2+nu*q-2*nu+1),nu=1+beta,Cmod2sol/Cmod2pap)),symbolic);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Cm2sol:=(beta^2+nu*q)^(m/2)*Cmod2sol;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'Cm2solG*&),&*$)%%betaG\"\"#\"\"\"F,*&%#nuGF,%\" qGF,F,,$*&F+!\"\"%\"mGF,F,F,,&*,\"\"%F2F3F,,J*,F+F,)F*\"\"$F,)F/F+F,% \"tGF,%\"wGF,F,*,F+F,F)F,)F/F:F,F**F6F,F9F,F " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Last step: determination of C_\{m- 3\}" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "Itser:=map(Z->simpli fy(Z,symbolic),series(Ite,y=1,4) assuming y>1):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 65 "Gser:=map(Z->simplify(Z,symbolic),series(Ge,y= 1,5) assuming y>1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "Dtse r:=map(Z->factor(simplify(Z,symbolic)),series(Dte,y=1,4) assuming y>1) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 409 "collect(simplify(subs(\n@@(D,3)(Tm)(1/2*(q+ 2*beta)/(beta^2+nu*q)^(1/2))=T3e,\n@@(D,2)(Tm)(1/2*(q+2*beta)/(beta^ 2+nu*q)^(1/2))=T2e,\nD(Tm)(1/2*(q+2*beta)/(beta^2+nu*q)^(1/2))=T1,Tm(1 /2*(q+2*beta)/(beta^2+nu*q)^(1/2))=T0,\nnu=1+beta,series(subs(G(y)=Gse r,It(y)=Itser,Dt(y)=Dtser,Cmod[m]=Cmodsol,Cmod[m-1]=Cmod1sol,Cmod[m-2] =Cmod2sol,exprmod/t^m/(nu^2+nu*q-2*nu+1)^(1/2*m)),y=1,4)),symbolic),[C mod[m-3]],factor);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+',&%\"yG\"\"\"F &!\"\",$*&#F&\"#CF&*(,d]l*.\"#7F&,(*$)%%betaG\"\"#F&F&*&F3F&%\"qGF&F&F 6F&#F&F4%#T0GF&F2F&)%\"mG\"\"$F&)F6F4F&F&*.\"\"'F&F0F7F8F&F3F&F9F&)F6F ;F&F'*.F>F&%#T1GF&)F3\"\"&F&F:F&F?F&%\"tGF&F'*.F>F&FAF&)F3\"\"%F&F:F&) F6FGF&FDF&F'**FAF&)F3F;F&)F:F4F&FHF&F'*.\"#=F&F0F7F8F&F2F&FKF&F?F&F&*. \"#sF&FAF&FJF&F:F&F?F&FDF&F&*.F+F&FAF&F2F&F:F&FHF&FDF&F'*.F/F&FAF&F3F& )F6FCF&FDF&%\"wGF&F'*.F/F&F0F7F8F&F2F&F:F&F?F&F'*.F/F&FAF&F3F&F:F&FHF& FDF&F'*.F>F&F0F7F8F&FJF&FKF&F?F&F&*.F>F&F0F7F8F&F2F&F9F&F?F&F'*.\"#OF& FAF&FBF&F:F&FF&FF&F0F7F8F&FJF&F9F&FF&FAF&FJF&FRF&FDF&FSF&F'*.FMF&FAF&FJF&F:F&FHF&FDF &F'*.FYF&FinF&F\\oF&FAF&FaoF&F?F&F&*.\"$3\"F&FinF&F\\oF&FAF&FJF&FHF&F' *.FYF&FinF&F\\oF&FAF&F2F&FRF&F&*.F/F&FinF&F\\oF&FAF&F3F&FRF&F&*.F+F&FD F&FAF&FFF&F:F&F?F&F&*.FrF&FinF&F\\oF&FAF&FBF&F?F&F'*.F+F&FinF&F\\oF&F AF&FFF&FHF&F&*.FfoF&FinF&F\\oF&FAF&FJF&F?F&F'*.FfoF&FinF&F\\oF&FAF&F2F &FHF&F'*.FfoF&FinF&F\\oF&FAF&FBF&FF&FAF&FBF&F:F&F?F&FDF&FSF&F&*0F^sF&F0F7F8F &F2F&FKF&FF&FAF&F2F&F:F&FRF&FDF&FSF&F&*0FfnF&F0F7F8F&FBF&FKF&F6F&FDF &F&*0F/F&F0F7F8F&FBF&F:F&FF&FAF &F3F&F:F&FRF&FDF&FSF&F&*0F/F&F0F7F8F&FFF&FKF&F?F&FDF&F'*0F>F&FAF&FFF&F :F&FHF&FDF&FSF&F&*0FfnF&F0F7FDF&F8F&FBF&F:F&F6F&F'*0F^sF&F0F7FDF&F8F&F FF&FKF&F6F&F&*0FfoF&F0F7FDF&F8F&FJF&FKF&FF&FAF&FBF&F?F &FDF&F&*,F>F&FAF&FFF&FHF&FDF&F&*,F;F&FAF&FJF&F:F&FHF&F&*,FGF&F0F7F8F&F 9F&F?F&F'*,FGF&FAF&FJF&FKF&FF&FAF&FFF &FF&FAF&F2F&FHF&F'**F>F&FAF&F3F&FHF&F'*.FfnF &FAF&F3F&FHF&FDF&FSF&F&*.FjrF&FAF&FJF&F?F&FDF&FSF&F&*.FYF&FAF&F2F&FHF& FDF&FSF&F&*,FfnF&FAF&F3F&F?F&FDF&F'*,FYF&FAF&F2F&F:F&F?F&F'F&F0#!\"(F4 FD!\"$F&F&F;-%\"OGF[oFG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 " Cmod3sol:=collect(solve(coeff(subs(y=1+s,%),s,3),Cmod[m-3]),[T0,T1,M(1 )],factor);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)Cmod3solG,&*.\"#7!\" \"%\"qG\"\"\"%\"mGF*,fq*.\"\"'F*)%%betaG\"\"&F*F+F*F)F*%\"tGF*%\"wGF*F **.F.F*)F0\"\"%F*F+F*)F)\"\"#F*F2F*F3F*F**,F.F*F/F*F+F*F)F*F2F*F(*,\"# CF*F/F*F+F*F2F*F3F*F(*,F.F*F5F*F+F*F7F*F2F*F(*.F;F*F5F*F+F*F)F*F2F*F3F *F(*,F.F*F5F*F7F*F2F*F3F*F**,F.F*)F0\"\"$F*)F)FAF*F2F*F3F*F**.F.F*)F0F 8F*F+F*FBF*F2F*F3F*F(**F;F*F/F*F+F*F2F*F***F.F*F/F*F)F*F2F*F**,F'F*F5F *F+F*F)F*F2F*F***F.F*F5F*F7F*F2F*F**,F;F*F5F*F)F*F2F*F3F*F(*(F@F*)F+F8 F*F7F*F(*,\"#=F*F@F*F+F*F7F*F2F*F(*,F'F*F@F*F7F*F2F*F3F*F(*.F;F*FDF*F+ F*F7F*F2F*F3F*F**,FMF*FDF*FBF*F2F*F3F*F**.F.F*F0F*F+F*FBF*F2F*F3F*F(*( F;F*F/F*F2F*F(**\"#[F*F5F*F+F*F2F*F***F'F*F5F*F)F*F2F*F(**FAF*F@F*FKF* F)F*F***FAF*F@F*F+F*F7F*F**,\"#sF*F@F*F+F*F)F*F2F*F***FMF*F@F*F7F*F2F* F**,FTF*F@F*F)F*F2F*F3F*F(**FAF*FDF*FKF*F7F*F(*,F'F*FDF*F+F*F7F*F2F*F( *,FYF*FDF*F7F*F2F*F3F*F(*.F;F*F0F*F+F*F7F*F2F*F3F*F**,F'F*F0F*FBF*F2F* F3F*F**(FTF*F5F*F2F*F(**\"\"*F*F@F*F+F*F)F*F(*(F8F*F@F*F7F*F(**FYF*F@F *F)F*F2F*F(**F.F*FDF*FKF*F)F*F***F]oF*FDF*F+F*F7F*F**,FTF*FDF*F+F*F)F* F2F*F***F'F*FDF*F7F*F2F*F***FAF*F0F*FKF*F7F*F(*,FTF*F0F*F7F*F2F*F3F*F( *(F.F*F@F*F)F*F***FMF*FDF*F+F*F)F*F(*(F.F*FDF*F7F*F(**FTF*FDF*F)F*F2F* F(**F]oF*F0F*F+F*F7F*F**(F8F*FKF*F7F*F(*(F'F*FDF*F)F*F**(F.F*F0F*F7F*F (*(F.F*F+F*F7F*F**&F6F*F7F*F(F*,(*$FDF*F**&F0F*F)F*F*F)F*!\"$%#T0GF*F* *&,&*&#F*F8F**.,&F)F*F6F(F*F2F8F)F*F0F*Fp#F(F8-%\"MG6#F*F*F*F**.F;F(F )F*F0F*FjpF*,\\p*.F.F*F5F*F+F*F)F*F2F*F3F*F(*.F.F*F@F*F+F*F7F*F2F*F3F* F(**F'F*F/F*F2F*F3F*F**,F.F*F5F*F+F*F)F*F2F*F**,\"#OF*F5F*F)F*F2F*F3F* F**,F.F*F@F*F+F*F7F*F2F*F**.F;F*F@F*F+F*F)F*F2F*F3F*F(*,\"#IF*F@F*F7F* F2F*F3F*F**.FfqF*FDF*F+F*F7F*F2F*F3F*F(*,F.F*FDF*FBF*F2F*F3F*F**.F.F*F 0F*F+F*FBF*F2F*F3F*F(**F'F*F5F*F+F*F2F*F(**F.F*F5F*F)F*F2F*F(**F.F*F@F *F7F*F2F*F(*,FTF*F@F*F)F*F2F*F3F*F**(FDF*FKF*F7F*F**,FMF*FDF*F+F*F7F*F 2F*F**,FYF*FDF*F7F*F2F*F3F*F**.F;F*F0F*F+F*F7F*F2F*F3F*F(*,FMF*F0F*FBF *F2F*F3F*F**,F.F*F+F*FBF*F2F*F3F*F(*(F'F*F5F*F2F*F**(FDF*FKF*F)F*F(**F AF*FDF*F+F*F7F*F(**FMF*FDF*F7F*F2F*F(**FAF*F0F*FKF*F7F*F**,F;F*F0F*F+F *F7F*F2F*F**,FfqF*F0F*F7F*F2F*F3F*F***F'F*FBF*F2F*F3F*F***FAF*FDF*F+F* F)F*F**(F8F*FDF*F7F*F***F]oF*F0F*F+F*F7F*F(**F;F*F0F*F7F*F2F*F(*(FAF*F KF*F7F*F***F'F*F+F*F7F*F2F*F**(F8F*FDF*F)F*F(*(F.F*F0F*F7F*F**(F]oF*F+ F*F7F*F(*(F'F*F7F*F2F*F(*&F.F*F7F*F*F*Fp#!\"(F8F(F*%#T1GF*F*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Comparing with the expression of t he paper" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "Cmod3pap:=q/24 *delta^(-7/2)*(2*m*b0pap*sqrt(delta)*T0+beta*(q-4)*b1pap*T1)+q*(q-4)*b eta/2*t^2*M(1)*delta^(-1/2)*T1;;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% )Cmod3papG,&**\"#C!\"\"%\"qG\"\"\"%&deltaG#!\"(\"\"#,&*,F.F*%\"mGF*%&b 0papGF*F+#F*F.%#T0GF*F***%%betaGF*,&F)F*\"\"%F(F*%&b1papGF*%#T1GF*F*F* F**&F3F**0F)F*F7F*F6F*%\"tGF.-%\"MG6#F*F*F+#F(F.F:F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "b0pap:=6*t*w*(q-4)*beta*delta*(m*( beta^2-q)+q*(nu+1))-6*(m-1)*(nu+1)*(q-4)*beta^2*t*delta-q*(m-1)*(m-2)* (nu+1)*(q*(nu^2-nu+1)-3*beta^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% &b0papG,(*0\"\"'\"\"\"%\"tGF(%\"wGF(,&%\"qGF(\"\"%!\"\"F(%%betaGF(%&de ltaGF(,&*&%\"mGF(,&*$)F/\"\"#F(F(F,F.F(F(*&F,F(,&%#nuGF(F(F(F(F(F(F(*0 F'F(,&F3F(F(F.F(F9F(F+F(F6F(F)F(F0F(F.*,F,F(F \+ " 0 "" {MPLTEXT 1 0 137 "b1pap:=6*t*w*delta*(q*(m-1)*(nu^2+2*nu+q-3)-d elta*(2*beta+q))-6*t*delta*(m-1)*(q*(1+nu^2)-2*beta^2)+q*(m-1)*(m-2)*( beta^2-q*(nu^2+nu+1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&b1papG,(* ,\"\"'\"\"\"%\"tGF(%\"wGF(%&deltaGF(,&*(%\"qGF(,&%\"mGF(F(!\"\"F(,**$) %#nuG\"\"#F(F(*&F6F(F5F(F(F.F(\"\"$F1F(F(*&F+F(,&F.F(*&F6F(%%betaGF(F( F(F1F(F(*,F'F(F)F(F+F(F/F(,&*&F.F(,&*$F4F(F(F(F(F(F(*&F6F()F " 0 "" {MPLTEXT 1 0 97 "simplify(factor(subs(\nbeta= nu-1,delta=(nu^2+nu*q-2*nu+1),nu=1+beta,Cmod3sol/Cmod3pap)),symbolic); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 37 "Cm3sol:=(beta^2+nu*q)^(m/2)*Cmod3sol:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 55 "When q=1 (m=3, k=1): speci al values of y (Section 4.2)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "eq:=numer(factor(sub s(q=1,x=I1,beta=nu-1,y=1+t*Z,q=1,Nsubs^2-DD)));\nfactor(subs(t=0,%/3/( nu-1)));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#eqG,$*(\"\"$\"\"\",&%#n uGF(F(!\"\"F(,\\o*(%\"tGF()%\"ZG\"\"#F(F*F(F+*()F0\"\"%F(F*F()F.F4F(F( *(F3F(F*F()F.F'F(F+*(F3F(F7F(%\"wGF(F(**F4F()F0F'F(F*F(F7F(F(**F1F(F;F (F*F()F.F1F(F+**F1F(F;F(F=F(F9F(F(**\"\"'F(F/F(F*F(F=F(F(*(F/F(F.F(F9F (F(**F4F(F0F(F*F(F.F(F(*(-%\"MG6#,&*&F0F(F.F(F(F(F(F()F0\"\"&F(F5F(F(* (FDF(F3F(F7F(F(*(FDF(F;F(F=F(F+*(FDF(F/F(F.F(F+*(F1F(F.F(F/F(F+**)FDF1 F()F0F@F(F*F()F.F@F(F(*,F4F(FPF(FIF(F*F()F.FJF(F(*,F1F(FDF(FIF(F*F(FTF (F(*,F@F(FPF(F3F(F*F(F5F(F(*,F1F(FDF(FIF(F*F(F5F(F+*,\"\")F(FDF(F3F(F* F(F5F(F(*,F4F(FPF(F;F(F*F(F7F(F(*,F4F(FDF(F3F(F*F(F7F(F+*,\"#7F(FDF(F; F(F*F(F7F(F(**FPF(F/F(F*F(F=F(F(*,F1F(FDF(F;F(F*F(F=F(F+*,FYF(FDF(F/F( F*F(F=F(F(*,F1F(FDF(F0F(F*F(F.F(F(*&F;F(F=F(F+*&F;F(F.F(F(F*F(F0F+F(F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%#nuG\"\"\"%\"ZG!\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "We now determine the first few ter ms of Y (or Z)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "unassign( 'z'):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Zser:=proc(n) add( z[i]*t^i,i=0..n): end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "n :=0: normal(series(subs(M(Z*t+1)=subs(y=Z*t+1,Mser(n)), Z=Zser(n), eq) ,t,n+1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+'%\"tG,$*(\"\"$\"\"\",&% #nuGF(F(!\"\"F(,&&%\"zG6#\"\"!F+F*F(F(F(F0-%\"OG6#F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "z[0]:=nu:" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 23 "From now on we can loop" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "n:=1+n: normal(series(subs(M(Z*t+1)=convert(subs(y=Z *t+1,Mser(n)),polynom), Z=Zser(n), q=1,eq),t,n+1));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#+'%\"tG,$*(\"\"$\"\"\",&%#nuGF(F(!\"\"F(,,*(\"#?F()F* \"\"%F()%\"wGF'F(F(*(\"##)F(F/F()F2\"\"#F(F(*(\"#uF(F/F(F2F(F(*&\"#9F( F/F(F(&%\"zG6#F'F+F(F(F'-%\"OG6#F(F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "z[n]:=factor(solve(coeff(%,t,n),z[n]));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%\"zG6#\"\"$,$*(\"\"#\"\"\")%#nuG\"\"%F+,**&\" #5F+)%\"wGF'F+F+*&\"#TF+)F3F*F+F+*&\"#PF+F3F+F+\"\"(F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "Zser(n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*%#nuG\"\"\"**\"\"#F%)F$F'F%,&%\"wGF%F%F%F%%\"tGF%F%*( )F$\"\"$F%,(*&\"\"'F%)F*F'F%F%*&\"#9F%F*F%F%\"\"&F%F%)F+F'F%F%**F'F%)F$\"\"%F%,**&\"#5F%)F*F.F%F%*&\"#TF%F2F%F%*&\"#PF%F*F%F%\"\"(F%F%)F+F.F %F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "The properties of the lemm a" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "normal(series(subs(M(y )=Mser(n), y=1+t*Zser(n), q=1,I1),t,2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+)%\"tG*&\"\"\"F&%#nuG!\"\"\"\"!,(F&F(%\"wGF(F'F&F&-%\"OG6#F&\" \"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "factor(series(subs(x =I1, M(y)=Mser(n), y=1+t*Zser(n), q=1,beta=nu-1,DD),t,1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+'%\"tG*&,&%#nuG\"\"\"F(!\"\"\"\"#F'!\"#\"\" !-%\"OG6#F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "I1;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,(**%\"yG\"\"\"%\"tGF&%\"qGF&-%\"MG6#F %F&F&*&,&F%F&F&!\"\"F&F%F.F&*(F'F&F%F&F-F.F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "normal(series(subs(y=1+t*Zser(n), diff(subs(M(y) =Mser(n), q=1,I1),y)),t,1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+'%\"t G,$*&\"\"\"F'*$)%#nuG\"\"#F'!\"\"F,F,-%\"OG6#F'\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 33 "Special values of Y (Section 4.2)" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Let us denote z0=[t^0]Z =[t] Y" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "m:='m': j:='j': Pj:=(q+2*beta-q*z0)^2-qj*(q*beta+q +beta^2-q*z0*(beta+2)+q*z0^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#P jG,&*$),(*&%\"qG\"\"\"%#z0GF+!\"\"*&\"\"#F+%%betaGF+F+F*F+F/F+F+*&%#qj GF+,,*&F*F+F0F+F+F*F+*$)F0F/F+F+*(F*F+F,F+,&F/F+F0F+F+F-*&F*F+)F,F/F+F +F+F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "collect(factor(%), z0,factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,2*(%\"qG\"\"\",&F%F&%# qjG!\"\"F&)%#z0G\"\"#F&F&*(F%F&,**&%%betaGF&F(F&F&*&\"\"%F&F0F&F)*&F,F &F%F&F)*&F,F&F(F&F&F&F+F&F&*&)F0F,F&F(F&F)*(F0F&F%F&F(F&F)*&F2F&F6F&F& *(F2F&F%F&F0F&F&*$)F%F,F&F&*&F%F&F(F&F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "e0:=factor(subs(z0=1+beta*v,%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e0G*&)%%betaG\"\"#\"\"\",.*&)%\"qGF(F))%\"vGF(F)F)*( F-F)%#qjGF)F.F)!\"\"*(F-F)F1F)F/F)F)*(\"\"%F)F-F)F/F)F2F1F2F5F)F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "e1:=collect(e0/beta^2,v,fact or);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e1G,**(%\"qG\"\"\",&F'F(%#q jG!\"\"F()%\"vG\"\"#F(F(*(F'F(,&F*F(\"\"%F+F(F-F(F(F*F+F1F(" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Solutions for q=4 cos(pi/m)^2" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "e2:=collect(factor(subs(q=4* cos(pi/m)^2,qj=4*cos(j*pi/m)^2,e1)),v,factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e2G,(*,\"#;\"\"\")-%$cosG6#*&%#piGF(%\"mG!\"\"\"\"#F (,&-F+6#*(%\"jGF(F.F(F/F0F(F*F0F(,&F3F(F*F(F()%\"vGF1F(F0*,F'F(F)F(,&F 3F(F(F0F(,&F3F(F(F(F(F9F(F(*(\"\"%F(F;F(F " 0 " " {MPLTEXT 1 0 92 "vjp:=sin(j*pi/m)/2/cos(pi/m)/sin((1+j)*pi/m);\nvjm: =sin(j*pi/m)/2/cos(pi/m)/sin((-1+j)*pi/m);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$vjpG,$*&#\"\"\"\"\"#F(*(-%$sinG6#*(%\"jGF(%#piGF(%\" mG!\"\"F(-%$cosG6#*&F0F(F1F2F2-F,6#*(,&F(F(F/F(F(F0F(F1F2F2F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$vjmG,$*&#\"\"\"\"\"#F(*(-%$sinG6#*( %\"jGF(%#piGF(%\"mG!\"\"F(-%$cosG6#*&F0F(F1F2F2-F,6#*(,&F(F2F/F(F(F0F( F1F2F2F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "factor(simpl ify(expand(numer(factor(subs(v=vjp,e2)))),trig));\nfactor(simplify(exp and(numer(factor(subs(v=vjm,e2)))),trig));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "factor(subs(z0=1+beta*v,diff (Pj,z0)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(%\"qG\"\"\"%%betaGF%,* *(\"\"#F%F$F%%\"vGF%F%*(F)F%%#qjGF%F*F%!\"\"F,F%\"\"%F-F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Properties of the values vjp" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "factor(combine(factor(expand(subs(j =j+1,vjp)-vjp)),trig));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&\"\" \"!\"\"-%$cosG6#,$*(\"\"#F&%#piGF&%\"mGF'F&F&F&,*F(F&-F)6#,$**F-F&,&F- F&%\"jGF&F&F.F&F/F'F&F'F&F&-F)6#,$**F-F&,&F&F&F6F&F&F.F&F/F'F&F'F'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Properties of the values vjm" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "factor(combine(factor(expa nd(subs(j=j+1,vjm)-vjm)),trig));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&\"\"\"!\"\"-%$cosG6#,$*(\"\"#F&%#piGF&%\"mGF'F&F&F&,*F(F'-F)6#,$** F-F&%\"jGF&F.F&F/F'F&F&F&F'-F)6#,$**F-F&,&F&F'F5F&F&F.F&F/F'F&F&F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 25 "The value of D at (t,I_j)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "factor(subs(z0=1+beta*v,(q*beta+q+beta^2-q*z0*(beta+2 )+q*z0^2)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%%betaG\"\"#\"\"\", (*&%\"qGF')%\"vGF&F'F'*&F*F'F,F'!\"\"F'F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "factor(combine(factor(expand(factor(subs(q=4*cos(p i/m)^2,v=vjp,%)))),sin));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(%%betaG \"\"#,&\"\"\"!\"\"-%$cosG6#,$*(F%F'%#piGF'%\"mGF(F'F'F',&F'F(-F*6#,$** F%F',&F'F'%\"jGF'F'F.F'F/F(F'F'F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "The last property" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "re s:=(subs(y=1+t*Z,diff(I1,y)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$r esG,.*(%\"tG\"\"\"%\"qGF(-%\"MG6#,&*&%\"ZGF(F'F(F(F(F(F(F(**F-F(F'F(F) F(-%%diffG6$F*F-F(F(*&F(F(F-!\"\"F(*(F'F(F/F(F-!\"#F5*&F(F(F/F5F(*(F'F 5F-F(F/F7F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 74 "The differential system for P-hat, Q-hat and R- hat (Section 4.4) NEW!!!" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "expr1:=(De(t,x)*diff(C(t,x),x)-m/2*C(t,x)*diff(De(t,x),x))^2/(C(t, x)^2-De(t,x)^m)/De(t,x)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&expr1 G*(,&*&-%#DeG6$%\"tG%\"xG\"\"\"-%%diffG6$-%\"CGF*F,F-F-*&#F-\"\"#F-*(% \"mGF-F1F--F/6$F(F,F-F-!\"\"F5,&*$)F1F5F-F-)F(F7F:F:F(!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "D1:=factor(diff(expr1,t));" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%#D1G,$*&#\"\"\"\"\"%F(**,&*(%\"mGF(- %\"CG6$%\"tG%\"xGF(-%%diffG6$-%#DeGF0F2F(!\"\"*(\"\"#F(F6F(-F46$F.F2F( F(F(,6**F)F()F6F:F()F6F-F(-F46%F.F1F2F(F(*.F:F(F6F(F3F(F@F(-F46$F.F1F( F-F(F8*.F:F(F6F(F.F(F@F(-F46%F6F1F2F(F-F(F8**F)F(F?F()F.F:F(FAF(F8*,F: F(F6F()F.\"\"$F(FGF(F-F(F(*,F.F(F3F(F@F(-F46$F6F1F()F-F:F(F(*,F)F(F?F( F;F(F.F(FDF(F(*.F:F(F6F(F;F(F@F(FOF(F-F(F8*,F:F(FLF(F3F(FOF(F-F(F8*.F: F(F.F(F3F(F@F(FOF(F-F(F(F(,&*$FJF(F(F@F8!\"#F6!\"$F(F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "expr2:=(De(t,x)*diff(C(t,x),t)-m/2* C(t,x)*diff(De(t,x),t))^2/(C(t,x)^2-De(t,x)^m)/De(t,x)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&expr2G*(,&*&-%#DeG6$%\"tG%\"xG\"\"\"-%%diffG 6$-%\"CGF*F+F-F-*&#F-\"\"#F-*(%\"mGF-F1F--F/6$F(F+F-F-!\"\"F5,&*$)F1F5 F-F-)F(F7F:F:F(!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "D2:= factor(diff(%,x)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "facto r(simplify(factor(D1/D2),symbolic));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#*&,&*(%\"mG\"\"\"-%\"CG6$%\"tG%\"xGF'-%%diffG6$-%#DeGF*F,F'!\"\"*(\" \"#F'F0F'-F.6$F(F,F'F'F',&*(F&F'F(F'-F.6$F0F+F'F2*(F4F'F0F'-F.6$F(F+F' F'F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 68 "Leading co efficients (in x) of P-hat, Q-hat and R-hat (Section 4.4) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "exprC:=sum(C[m-r]*x^(m-r),r=0..4); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&exprCG,,*&&%\"CG6#%\"mG\"\"\")% \"xGF*F+F+*&&F(6#,&F*F+F+!\"\"F+)F-F1F+F+*&&F(6#,&F*F+\"\"#F2F+)F-F7F+ F+*&&F(6#,&F*F+\"\"$F2F+)F-F=F+F+*&&F(6#,&F*F+\"\"%F2F+)F-FCF+F+" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "We want to extract the coefficient of x^(2m) in: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "e1:=expr C^2-DD^m;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e1G,&*$),,*&&%\"CG6#% \"mG\"\"\")%\"xGF-F.F.*&&F+6#,&F-F.F.!\"\"F.)F0F4F.F.*&&F+6#,&F-F.\"\" #F5F.)F0F:F.F.*&&F+6#,&F-F.\"\"$F5F.)F0F@F.F.*&&F+6#,&F-F.\"\"%F5F.)F0 FFF.F.F;F.F.),**&,&*$)%%betaGF;F.F.*&%#nuGF.%\"qGF.F.F.)F0F;F.F.*(FRF. ,&FQF.F.F.F.F0F.F5**FOF.%\"tGF.,&FRF.FGF5F.,&*&FRF.%\"wGF.F.FOF.F.F.FR F.F-F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "simplify(subs(x= 1/v,e1*v^(2*m)),symbolic): limit(%,v=0) assuming m>0;\nP4hatsol:=simpl ify(subs(C[m]=Cmsol,%),symbolic);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, &*$)&%\"CG6#%\"mG\"\"#\"\"\"F+),&*$)%%betaGF*F+F+*&%#nuGF+%\"qGF+F+F)! \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)P4hatsolG*&,&*$)%#T0G\"\"# \"\"\"F+F+!\"\"F+),&*$)%%betaGF*F+F+*&%#nuGF+%\"qGF+F+%\"mGF+" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Connection between T_m and its der ivative" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "seq(expand((u^2- 1)*diff(T(m),u)^2-m^2*(T(m)^2-1)),m=0..4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"!F#F#F#F#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "E valuated at u=gamma" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "gamma e;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#!\"\",&%\"qG\"\"\"*&F%F )%%betaGF)F)F),&*$)F+F%F)F)*&%#nuGF)F(F)F)#F&F%F)" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 37 "eT01:=(gammae^2-1)*T1^2-m^2*(T0^2-1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eT01G,&*&,&*(\"\"%!\"\",&%\"qG\"\" \"*&\"\"#F-%%betaGF-F-F/,&*$)F0F/F-F-*&%#nuGF-F,F-F-F*F-F-F*F-)%#T1GF/ F-F-*&)%\"mGF/F-,&*$)%#T0GF/F-F-F-F*F-F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "P4hatsol:=simplify(factor(subs(nu=beta+1,solve(numer (factor(resultant(eT01,P4hatsol-P4,T0))),P4))),symbolic);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)P4hatsolG,$*.\"\"%!\"\"),(*$)%%betaG\"\"#\" \"\"F/*&F-F/%\"qGF/F/F1F/,&%\"mGF/F/F(F/%#T1GF.F1F/,&F1F/F'F(F/F3!\"#F /" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "Now for Q-hat, we want to ex tract the coefficient of x^m in:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "e2:=DD*diff(exprC,x)-m/2*diff(DD,x)*exprC:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 166 "simplify(subs(x=1/v,e2*v^(m )),symbolic): limit(%,v=0) assuming m>1;\nQ2hatsol:=simplify(subs(C[m] =Cmsol,C[m-1]=Cm1sol,nu=beta+1,delta=((beta^2+beta*q+q)),%),symbolic); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&#\"\"\"\"\"#F&*(&%\"CG6#%\"mGF &F,F&%\"qGF&F&!\"\"*(&F*6#,&F,F&F&F.F&%#nuGF&F-F&F.*&F0F&)%%betaGF'F&F .*&#F&F'F&**F)F&F,F&F-F&F3F&F&F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% )Q2hatsolG,$*.\"\"%!\"\",&%\"qG\"\"\"F'F(F+%%betaGF+%#T1GF+),(*$)F,\" \"#F+F+*&F,F+F*F+F+F*F+,&*&F2F(%\"mGF+F+#F+F2F(F+F*F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "Finally for R-hat, we need to take deriv atives w.r.t. t. However, recall that C_m and C_\{m-1\} are independen t from t." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "exprCt:=subs(s eq(C[m-r]=C[m-r](t),r=2..10),exprC);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%'exprCtG,,*&&%\"CG6#%\"mG\"\"\")%\"xGF*F+F+*&&F(6#,&F*F+F+!\"\"F+) F-F1F+F+*&-&F(6#,&F*F+\"\"#F26#%\"tGF+)F-F8F+F+*&-&F(6#,&F*F+\"\"$F2F: F+)F-FAF+F+*&-&F(6#,&F*F+\"\"%F2F:F+)F-FHF+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "We want to extract the coefficient of x^m in:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "e3:=DD*diff(exprCt,t)-m/2*di ff(DD,t)*exprCt:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 197 "simpli fy(subs(x=1/v,e3*v^(m)),symbolic): limit(%,v=0) assuming m>2;\nR2hatso l:=factor(simplify(eval(subs(C[m]=Cmsol,C[m-1]=Cm1sol,C[m-2](t)=Cm2sol ,nu=beta+1,delta=((beta^2+beta*q+q)),%)),symbolic));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*(-%%diffG6$-&%\"CG6#,&%\"mG\"\"\"\"\"#!\"\"6#%\"t GF2F.%#nuGF.%\"qGF.F.**F/F.F-F.)%%betaGF/F.&F*6#F-F.F.*&#F.F/F.**F-F.F 6F.F8F.F4F.F.F0*&#F.F/F.*,F-F.F7F.F8F.)F4F/F.%\"wGF.F.F0*.F/F.F-F.F7F. F8F.F4F.FAF.F.*&F%F.F6F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)R2hat solG,$*0\"\"%!\"\"%\"qG\"\"\"),(*$)%%betaG\"\"#F*F**&F/F*F)F*F*F)F*,&* &F0F(%\"mGF*F*#F*F0F(F*%#T1GF*F/F*,&F)F*F'F(F*,**(F0F*F/F*%\"wGF*F**&F )F*F:F*F*F/F(F0F(F*F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Checking with the value given in the paper" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "R2hatpap:=q*beta*(q/4-1)*delta^(m/2-1/2)*(beta+2-w*(q +2*beta))*T1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)R2hatpapG*.%\"qG\" \"\"%%betaGF',&*&\"\"%!\"\"F&F'F'F'F,F')%&deltaG,&*&\"\"#F,%\"mGF'F'#F 'F1F,F',(F(F'F1F'*&%\"wGF',&F&F'*&F1F'F(F'F'F'F,F'%#T1GF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "factor(R2hatsol/R2hatpap);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&),(*$)%%betaG\"\"#\"\"\"F**&F(F*%\"q GF*F*F,F*,&*&F)!\"\"%\"mGF*F*#F*F)F/F*)%&deltaGF-F/" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 26 "The case t =0 (Section 4.4)" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "The dominant c oefficient in T" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "seq(coef f(T(m),u,m),m=1..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,\"\"\"\"\"#\" \"%\"\")\"#;\"#K\"#k\"$G\"\"$c#\"$7&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "We just check that some factors occur in this and that expressi on" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Dcal:=subs(t=0,DD);; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%DcalG,(*&,&*$)%%betaG\"\"#\"\" \"F,*&%#nuGF,%\"qGF,F,F,)%\"xGF+F,F,*(F/F,,&F.F,F,F,F,F1F,!\"\"F/F," } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Ncal:=subs(y=1,Nsubs);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%NcalG,&*&,&%\"qG\"\"\"*&\"\"#F)%%be taGF)F)F)%\"xGF)F)F(!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "The \+ factor (N^2-4D) contains (x-1)^2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "factor(subs(beta=nu-1,Ncal^2-4*Dcal));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#*(%\"qG\"\"\"),&%\"xGF%F%!\"\"\"\"#F%,&F$F%\"\"% F)F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "factor(subs(nu=1+be ta,Ncal^2-q*Dcal));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**%%betaG\"\" \",&%\"qGF&\"\"%!\"\"F&%\"xGF&,(*&F%F&F+F&F&*&F(F&F+F&F&F(F*F&F*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "Finally, for Q-hat we need to chec k that the folowing values are non-zero" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "factor(subs(x=0,Dcal));factor(subs(x=1,Dcal));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%#DbG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%#DbG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 63 "Expression of M(1) and M'(1) in terms of P, Q, R (Sectio n 4.5)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "DD;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&,&*$)%%betaG\"\"#\"\"\"F**&%#nuGF*%\"qGF*F*F*) %\"xGF)F*F**(F-F*,&F,F*F*F*F*F/F*!\"\"**F(F*%\"tGF*,&F-F*\"\"%F2F*,&*& F-F*%\"wGF*F*F(F*F*F*F-F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "Psol:=x^4+P3*x^3+P2*x^2+P1*x+P0; Qsol:=x^2+Q1*x+Q0;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%%PsolG,,*&%#P3G\"\"\")%\"xG\"\"$F(F(*$)F*\"\"% F(F(*&%#P2GF()F*\"\"#F(F(*&%#P1GF(F*F(F(%#P0GF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%QsolG,(*&%#Q1G\"\"\"%\"xGF(F(*$)F)\"\"#F(F(%#Q0GF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "exprC:=C[m]*x^m+C[m-1]*x^ (m-1)+C[m-2]*x^(m-2)+C[m-3]*x^(m-3)+C[m-4]*x^(m-4);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%&exprCG,,*&&%\"CG6#%\"mG\"\"\")%\"xGF*F+F+*&&F(6#,& F*F+F+!\"\"F+)F-F1F+F+*&&F(6#,&F*F+\"\"#F2F+)F-F7F+F+*&&F(6#,&F*F+\"\"$F2F+)F-F=F+F+*&&F(6#,&F*F+\"\"%F2F+)F-FCF+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "The term we want to expand at x=infinity" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "diffe:=(D(x)*diff(C(x),x)-m/2*diff (D(x),x)*C(x))^2*Psol-q*m^2*beta^2*(q/4-1)*Qsol^2*(C(x)^2-D(x)^m);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&diffeG,&*&),&*&-%\"DG6#%\"xG\"\"\"- %%diffG6$-%\"CGF,F-F.F.*&#F.\"\"#F.*(%\"mGF.-F06$F*F-F.F2F.F.!\"\"F6F. ,,*&%#P3GF.)F-\"\"$F.F.*$)F-\"\"%F.F.*&%#P2GF.)F-F6F.F.*&%#P1GF.F-F.F. %#P0GF.F.F.*.%\"qGF.)F8F6F.)%%betaGF6F.,&*&FCF;FKF.F.F.F;F.),(*&%#Q1GF .F-F.F.*$FFF.F.%#Q0GF.F6F.,&*$)F2F6F.F.)F*F8F;F.F;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "We recall the connection between the first two de rivatives of T_m taken at gamma" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "eT01:=map(factor,(1-gammae^2)*T1^2-m^2*(1-T0^2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eT01G,&*,\"\"%!\"\"%\"qG\"\"\",(*&F'F*%%beta GF*F*F)F**&F'F*%#nuGF*F(F*%#T1G\"\"#,&*$)F-F1F*F**&F/F*F)F*F*F(F(*()% \"mGF1F*,&%#T0GF*F*F(F*,&F:F*F*F*F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "We set v=1/x and ask for an expansion at v=0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "res:=simplify(v^(2*m+4)*expand(sub s(x=1/v,normal(eval(subs(C(x)=exprC,D(x)=DD,diffe/(nu^2+nu*q-2*nu+1)^( m))))))) assuming v>0:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "The coe fficient of x^(2m+4)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "fa ctor(subs(simplify(subs(C[m]=Cmsol,C[m-1]=Cm1sol,C[m-2]=Cm2sol,delta=b eta^2+q*nu,beta=nu-1,series(res,v,1))) assuming q>0 assuming nu>0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+'%\"vG,$*.\"#;!\"\"%\"qG\"\"\",&%#n uGF*F*F(\"\"#,&F)F*\"\"%F(F*,6**F/F*)%#T0GF-F*)%\"mGF-F*)F,F-F*F**,F/F *F2F*F4F*F,F*F)F*F***\"\")F*F2F*F4F*F,F*F(*(F/F*F2F*F4F*F**&)%#T1GF-F* )F)F-F*F(*(F/F*F4F*F6F*F(**F/F*F4F*F,F*F)F*F(*(F/F*F " 0 "" {MPLTEXT 1 0 50 "factor(subs(n u=1+beta,rem(coeff(%,v,0),eT01,T1)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "The coefficient of x^ (2m+3)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 202 "res1:=factor(sub s(nu=1+beta,map(Z->factor(rem(Z,eT01,T1)),\nfactor(subs(simplify(subs( C[m]=Cmsol,C[m-1]=Cm1sol,C[m-2]=Cm2sol,delta=beta^2+q*nu,beta=nu-1,ser ies(res,v,2))) assuming q>0 assuming nu>0)))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%res1G+'%\"vG,$*2\"\"%!\"\"%\"mG\"\"#%%betaGF,%\"qG\" \"\",&%#T0GF/F/F*F/,&F1F/F/F/F/,&F.F/F)F*F/,.**\"\")F/F-F/%\"tGF/%\"wG F/F/**F)F/F.F/F7F/F8F/F/*(F)F/F-F/F7F/F*%#P3GF/*&F,F/%#Q1GF/F**&F6F/F7 F/F*F/F/F/-%\"OG6#F/F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "e qP3Q1:=collect(op(8,(coeff(res1,v,1))),[Q1,P3,w],factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'eqP3Q1G,**&\"\"#\"\"\"%#Q1GF(!\"\"%#P3GF(** \"\"%F(,&%\"qGF(*&F'F(%%betaGF(F(F(%\"tGF(%\"wGF(F(*(F-F(F2F(,&F'F(F1F (F(F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "P3Q1:=isolate(eqP3 Q1,P3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%P3Q1G/%#P3G,(*&\"\"#\"\" \"%#Q1GF*F***\"\"%F*,&%\"qGF**&F)F*%%betaGF*F*F*%\"tGF*%\"wGF*!\"\"*(F -F*F2F*,&F)F*F1F*F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 156 "Let us \+ check this identity, knowing the first few coefficients of Q1 and P3 ( This requires to have executed the section where we compute these coef ficients)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "n:=3: normal(s eries(subs(P3=P3ser(n),Q1=Q1ser(n),eqP3Q1),t,n+1));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Q 1P3:=isolate(eqP3Q1,Q1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Q1P3G/% #Q1G,(*&\"\"#!\"\"%#P3G\"\"\"F,**F)F,,&%\"qGF,*&F)F,%%betaGF,F,F,%\"tG F,%\"wGF,F,*(F)F,F2F,,&F)F,F1F,F,F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "The coefficient of x^(2m+2) (note that we use the connection be tween Q1 and P3)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 221 "res2:= factor(subs(nu=1+beta,map(Z->factor(rem(Z,eT01,T1)),\nfactor(subs(simp lify(subs(C[m]=Cmsol,C[m-1]=Cm1sol,C[m-2]=Cm2sol,C[m-3]=Cm3sol,delta=b eta^2+q*nu,beta=nu-1,Q1P3,series(res,v,3))) assuming q>0 assuming nu>0 )))));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%res2G+'%\"vG,$*&#\"\"\"\" #;F**0)%\"mG\"\"#F*)%%betaGF/F*%\"qGF*,&%#T0GF*F*!\"\"F*,&F4F*F*F*F*,& F2F*\"\"%F5F*,<**\"#[F*-%\"MG6#F*F*F0F*)%\"tGF/F*F**,F;F*F\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "r es:=coeff(%,v,2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "eqM1:= collect(op(8,res)/4,[M,Q0,P3,P2,t],factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqM1G,.*(,(*&\"#7\"\"\")%%betaG\"\"#F*F**(F)F*F,F*% \"qGF*F**&F)F*F/F*F*F*)%\"tGF-F*-%\"MG6#F*F*F**&F-F*%#Q0GF*F**&\"\"%! \"\"%#P3GF-F**(,**(F9F*F,F*%\"wGF*F:*(F-F*F/F*F?F*F:*&F-F*F,F*F*F9F*F* F2F*F;F*F*%#P2GF:*&,(*(F)F*F,F*F?F*F:*(F9F*F/F*F?F*F:F9F:F*F2F*F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "M1sol:=factor(solve(eqM1,M(1 )));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&M1solG,$**\"#[!\"\",6*,\"#; \"\"\"%#P3GF,%%betaGF,%\"tGF,%\"wGF,F,*,\"\")F,F-F,%\"qGF,F/F,F0F,F,** F2F,F-F,F.F,F/F,F(**F'F,F.F,F/F,F0F,F,**F+F,F3F,F/F,F0F,F,*$)F-\"\"#F, F(*(F+F,F-F,F/F,F(*&\"\"%F,%#P2GF,F,*&F2F,%#Q0GF,F(*&F+F,F/F,F,F,,(*$) F.F9F,F,*&F.F,F3F,F,F3F,F(F/!\"#F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "pap:=12*t^2*(q*nu+beta^2)*M(1)+P3^2/4-P2+2*Q0+2*t*(1+ nu-w*(2*beta+q))*P3-4*t*(1+w*(3*beta+q));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$papG,.**\"#7\"\"\")%\"tG\"\"#F(,&*$)%%betaGF+F(F(*&% #nuGF(%\"qGF(F(F(-%\"MG6#F(F(F(*&\"\"%!\"\"%#P3GF+F(%#P2GF8*&F+F(%#Q0G F(F(**F+F(F*F(,(F1F(F(F(*&%\"wGF(,&F2F(*&F+F(F/F(F(F(F8F(F9F(F(*(F7F(F *F(,&F(F(*&F@F(,&*&\"\"$F(F/F(F(F2F(F(F(F(F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "factor(subs(beta=nu-1,eqM1-pap));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Th e other characterisation of M(1)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 " Qsol:=Q1(t)*x+x^2+Q0(t);Rsol:=R1(t)*x+x^2*(nu+1-w*(q +2*beta))+R0(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%QsolG,(*&-%#Q1G 6#%\"tG\"\"\"%\"xGF+F+*$)F,\"\"#F+F+-%#Q0GF)F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%RsolG,(*&-%#R1G6#%\"tG\"\"\"%\"xGF+F+*&)F,\"\"#F+,(% #nuGF+F+F+*&%\"wGF+,&%\"qGF+*&F/F+%%betaGF+F+F+!\"\"F+F+-%#R0GF)F+" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "diffe1:=(De(x,t)*diff(C(x, t),x)-m/2*diff(De(x,t),x)*C(x,t))*(Rsol)-(De(x,t)*diff(C(x,t),t)-m/2*d iff(De(x,t),t)*C(x,t))*(Qsol);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'d iffe1G,&*&,&*&-%#DeG6$%\"xG%\"tG\"\"\"-%%diffG6$-%\"CGF+F,F.F.*&#F.\" \"#F.*(%\"mGF.-F06$F)F,F.F2F.F.!\"\"F.,(*&-%#R1G6#F-F.F,F.F.*&)F,F6F., (%#nuGF.F.F.*&%\"wGF.,&%\"qGF.*&F6F.%%betaGF.F.F.F;F.F.-%#R0GF@F.F.F.* &,&*&F)F.-F06$F2F-F.F.*&#F.F6F.*(F8F.-F06$F)F-F.F2F.F.F;F.,(*&-%#Q1GF@ F.F,F.F.*$FBF.F.-%#Q0GF@F.F.F;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "exprC:=C[m]*x^m+C[m-1]*x^(m-1)+C[m-2](t)*x^(m-2)+C[m-3](t)*x^( m-3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&exprCG,**&&%\"CG6#%\"mG\" \"\")%\"xGF*F+F+*&&F(6#,&F*F+F+!\"\"F+)F-F1F+F+*&-&F(6#,&F*F+\"\"#F26# %\"tGF+)F-F8F+F+*&-&F(6#,&F*F+\"\"$F2F:F+)F-FAF+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Cm3solt:=subs(M(1)=M1(t),Cm3sol):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "res:=simplify(v^(m+1)*expan d(subs(x=1/v,normal(eval(subs(C(x,t)=exprC,De(x,t)=DD,diffe1/(nu^2+nu* q-2*nu+1)^(m/2))))))) assuming v>0:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "The coefficient of x^(m+1)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "factor(subs(simplify(subs(C[m]=Cmsol,C[m-1]=Cm1sol,C [m-2](t)=Cm2sol,C[m-3](t)=Cm3solt,delta=beta^2+q*nu,beta=nu-1,series(r es,v,2))) assuming q>0 assuming nu>0));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+'%\"vG,$*&#\"\"\"\"\"%F(*.%\"qGF(%#T1GF(,&F+F(F)!\"\"F(,&%#nuGF (F(F.F(,**$)F0\"\"#F(F(*&F0F(F+F(F(*&F4F(F0F(F.F(F(#F.F4,hn*(\"\")F()% \"wGF4F(%\"tGF(F.*(F)F(-%#M1G6#F=F(F=F(F.**F:F(F;F(F=F(F3F(F.**\"#;F(F ;F(F=F(F0F(F(**F4F(-%#Q1GFAF(F0F(F " 0 "" {MPLTEXT 1 0 26 "res4:=numer (coeff(%,v,0));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%res4G*,%\"qG\"\" \"%#T1GF',&F&F'\"\"%!\"\"F',&%#nuGF'F'F+F',hn*(\"\")F')%\"wG\"\"#F'%\" tGF'F+*(F*F'-%#M1G6#F4F'F4F'F+**F0F'F1F'F4F')F-F3F'F+**\"#;F'F1F'F4F'F -F'F'**F3F'-%#Q1GF8F'F-F'F2F'F'**F*F'F6F'F:F'F4F'F+**F0F'F6F'F-F'F4F'F '**F3F'-%%diffG6$F6F4F'F:F')F4F3F'F+**F*F'FCF'F-F'FFF'F'**F3F')F&F3F'F 1F'F4F'F+**F0F'F&F'F1F'F4F'F'*(F>F'F&F'F2F'F'**F0F'F:F'F4F'F2F'F'*,F*F 'F&F'F4F'F2F'F-F'F'*,F3F'F&F'FCF'F-F'FFF'F+*,F*F'F&F'F6F'F-F'F4F'F+*,F 0F'F&F'F1F'F4F'F-F'F+**F*F'F&F'F4F'F2F'F'*(F3F'F-F'F2F'F'*(F3F'F:F'F4F 'F+*(F*F'F-F'F4F'F+*&F>F'F-F'F+*(F3F'F>F'F2F'F+*(F3F'FFF'FCF'F+F>F+-%# R1GF8F'*&F3F'F2F'F+*&F3F'F4F'F+*(F0F'F2F'F4F'F+F3F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "eqMt1Q1:=collect(factor(eval(subs(M 1(t)=M1t(t)/t^2,op(5,res4)))),[diff(M1t(t),t),Q1,t],factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(eqMt1Q1G,0*&,**&\"\"#\"\"\")%#nuGF)F*!\" \"*(F)F*F,F*%\"qGF*F-*&\"\"%F*F,F*F*F)F-F*-%%diffG6$-%$M1tG6#%\"tGF8F* F**&,,*(F)F*F,F*%\"wGF*F**&F/F*F " 0 "" {MPLTEXT 1 0 30 "map(factor,subs(nu=1+beta,%) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0*(\"\"#\"\"\",(*$)%%betaGF%F&F &*&F*F&%\"qGF&F&F,F&F&-%%diffG6$-%$M1tG6#%\"tGF3F&!\"\"*&,**(F%F&F*F&% \"wGF&F&*&F,F&F8F&F&F*F4F%F4F&-%#Q1GF2F&F&*(F%F&)F6F%F&F3F&F4*(F%F&,&F &F&F*F&F&F8F&F&-%#R1GF2F&*&F%F&F8F&F4F%F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Using P3 instead of Q1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "collect(subs(Q1(t)=Q1,Q1P3,eqMt1Q1),[(diff(M1t(t),t)) ,P3,R1,t],factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0*&,**&\"\"#\" \"\")%#nuGF'F(!\"\"*(F'F(F*F(%\"qGF(F+*&\"\"%F(F*F(F(F'F+F(-%%diffG6$- %$M1tG6#%\"tGF6F(F(*&,,*&F*F(%\"wGF(F(*&#F(F'F(*&F-F(F:F(F(F(*&#F(F'F( F*F(F+F:F+#F(F'F+F(%#P3GF(F(-%#R1GF5F(*,F'F(,&*&F'F(F:F(F(F(F+F(,,*(F' F(F*F(F:F(F(*&F-F(F:F(F(F*F+*&F'F(F:F(F+F(F+F(,(%%betaGF(F*F+F(F(F(F6F (F(*(F'F(F*F(F:F(F(*&F'F(F:F(F+F'F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "eqMt1P3:=map(factor,subs(nu=1+beta,%));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%(eqMt1P3G,.*(\"\"#\"\"\",(*$)%%betaGF'F(F(*&F, F(%\"qGF(F(F.F(F(-%%diffG6$-%$M1tG6#%\"tGF5F(!\"\"*(F'F6,**(F'F(F,F(% \"wGF(F(*&F.F(F:F(F(F,F6F'F6F(%#P3GF(F(-%#R1GF4F(F'F(*(F'F(,&F(F(F,F(F (F:F(F(*&F'F(F:F(F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "eqpa p:=2*(beta^2+q*nu)*diff(M1t(t),t)+(1+nu-w*(2*beta+q))*P3/2-R1(t)-2-2*b eta*w;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&eqpapG,,*(\"\"#\"\"\",&*$)%%betaGF'F(F(*&%#nuGF(%\"qGF(F(F(-%%diffG6$-%$M1tG6#%\"tGF6F(F(*(F'! \"\",(F.F(F(F(*&%\"wGF(,&F/F(*&F'F(F,F(F(F(F8F(%#P3GF(F(-%#R1GF5F8F'F8 *(F'F(F;F(F,F(F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "factor( subs(beta=nu-1,eqpap/eqMt1P3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\" \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "Results of th e above section: three generic non-differential equations" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "eqP3Q1 := -2*Q1(t)+P3(t)+4*(q+2*bet a)*t*w-4*t*(2+beta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'eqP3Q1G,**& \"\"#\"\"\"-%#Q1G6#%\"tGF(!\"\"-%#P3GF+F(**\"\"%F(,&%\"qGF(*&F'F(%%bet aGF(F(F(F,F(%\"wGF(F(*(F1F(F,F(,&F'F(F5F(F(F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "eqM1 := (12*beta^2+12*beta*q+12*q)*M1t(t)+2*Q0( t)+1/4*P3(t)^2+(-4*beta*w-2*q*w+2*beta+4)*t*P3(t)-P2(t)+(-12*beta*w-4* q*w-4)*t;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqM1G,.*&,(*&\"#7\"\" \")%%betaG\"\"#F*F**(F)F*F,F*%\"qGF*F**&F)F*F/F*F*F*-%$M1tG6#%\"tGF*F* *&F-F*-%#Q0GF3F*F**&#F*\"\"%F**$)-%#P3GF3F-F*F*F**(,**(F:F*F,F*%\"wGF* !\"\"*(F-F*F/F*FBF*FC*&F-F*F,F*F*F:F*F*F4F*F=F*F*-%#P2GF3FC*&,(*(F)F*F ,F*FBF*FC*(F:F*F/F*FBF*FCF:FCF*F4F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "eqM1bis := -2*(beta^2+beta*q+q)*diff(M1t(t),t)+1/2*( 2*beta*w+q*w-beta-2)*P3(t)+R1(t)+2+2*(1+beta)*w-2*w;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(eqM1bisG,.*(\"\"#\"\"\",(*$)%%betaGF'F(F(*&F,F(% \"qGF(F(F.F(F(-%%diffG6$-%$M1tG6#%\"tGF5F(!\"\"*&#F(F'F(*&,**(F'F(F,F( %\"wGF(F(*&F.F(F " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 36 "Simpl ifying the system (Section 6.2)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "DD;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&,&*$)%%betaG\"\"#\" \"\"F**&%#nuGF*%\"qGF*F*F*)%\"xGF)F*F**(F-F*,&F,F*F*F*F*F/F*!\"\"**F(F *%\"tGF*,&F-F*\"\"%F2F*,&*&F-F*%\"wGF*F*F(F*F*F*F-F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Dx:=subs(beta=nu-1,DD);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DxG,**&,&*$),&%#nuG\"\"\"F,!\"\"\"\"#F,F,*&%\"qGF ,F+F,F,F,)%\"xGF.F,F,*(F0F,,&F+F,F,F,F,F2F,F-**F*F,%\"tGF,,&F0F,\"\"%F -F,,(*&F0F,%\"wGF,F,F+F,F,F-F,F,F0F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "7 unknown series" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "Psol:=x^4+P3(t)*x^3+P2(t)*x^2+P1(t)*x+P0(t);\nQsol:=Q1(t)*x+x^2+Q0(t) ;\nRsol:=R0(t)+R1(t)*x+x^2*(nu+1-w*(q+2*nu-2));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%PsolG,,*$)%\"xG\"\"%\"\"\"F**&-%#P3G6#%\"tGF*)F(\"\"$F*F**&-%#P2GF.F*)F(\"\"#F*F**&-%#P1GF.F*F(F*F*-%#P0GF.F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%QsolG,(*&-%#Q1G6#%\"tG\"\"\"%\"xGF+F+*$)F,\" \"#F+F+-%#Q0GF)F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%RsolG,(-%#R0G6 #%\"tG\"\"\"*&-%#R1GF(F*%\"xGF*F**&)F.\"\"#F*,(%#nuGF*F*F**&%\"wGF*,(% \"qGF**&F1F*F3F*F*F1!\"\"F*F9F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "edplan:=collect(numer(factor(eval(1/(Rsol)*diff(Rsol ^2/Psol/Dx^2,x)-1/Qsol*diff(Qsol^2/Psol/Dx^2,t)))),x,factor):" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "The system of 8 equations" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "degree(edplan,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "for i from 0 to 7 do ED[i]:=factor(coeff(edplan,x,i)) od:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "seq(nops(ED[i]),i=0..7);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6*\"#c\"$F\"\"$\"\"$x\"\"$s\"\"#&*\"#Z \"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "ED[7];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,**$)%#nuG\"\"#\"\"\"F**&F(F*%\"qGF*F**&F)F *F(F*!\"\"F*F*F*,0*(\"\")F*F(F*%\"wGF*F.*(\"\"%F*F2F*F,F*F.*&F)F*-%%di ffG6$-%#Q1G6#%\"tGF \+ " 0 "" {MPLTEXT 1 0 7 "eqP3Q1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**& \"\"#\"\"\"-%#Q1G6#%\"tGF&!\"\"-%#P3GF)F&**\"\"%F&,&%\"qGF&*&F%F&%%bet aGF&F&F&F*F&%\"wGF&F&*(F/F&F*F&,&F%F&F3F&F&F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "factor(eval(subs(isolate(eqP3Q1 ,Q1(t)),beta=nu- 1,ED[7])));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Playing with the roots of DD" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "Dx;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,** &,&*$),&%#nuG\"\"\"F*!\"\"\"\"#F*F**&%\"qGF*F)F*F*F*)%\"xGF,F*F**(F.F* ,&F)F*F*F*F*F0F*F+**F(F*%\"tGF*,&F.F*\"\"%F+F*,(*&F.F*%\"wGF*F*F)F*F*F +F*F*F.F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "The two roots of DD " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "deltasol:=factor([solve (Dx,x)]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)deltasolG7$,$*(\"\"#! \"\",**$)%#nuGF(\"\"\"F.*&F-F.%\"qGF.F.*&F(F.F-F.F)F.F.F),(*&F0F.F-F.F .F0F.*$,$*(,&F0F.\"\"%F)F.,&F-F.F.F)F.,:*,F8F.F,F.F0F.%\"tGF.%\"wGF.F. *,F8F.F-F.)F0F(F.F " 0 "" {MPLTEXT 1 0 39 "map(series,deltasol,t,1);map(factor,%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#7$+'%\"tG,$*(\"\"#!\"\",**$)%#nuGF(\"\"\"F.*&F-F .%\"qGF.F.*&F(F.F-F.F)F.F.F),(*&F0F.F-F.F.F0F.*$*(,&F0F.\"\"%F)F.),&F- F.F.F)F(F.F0F.#F.F(F.F.F.\"\"!-%\"OG6#F.F.+'F%,$*(F(F),(F3F)F4F.F0F)F. F*F)F)F;F " 0 "" {MPLTEXT 1 0 95 "e1:=subs(x=d1,Qsol*diff(DD,t)-Rsol*diff(DD,x));\ne2:= subs(x=d2,Qsol*diff(DD,t)-Rsol*diff(DD,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e1G,&**,(*&-%#Q1G6#%\"tG\"\"\"%#d1GF-F-*$)F.\"\"#F-F --%#Q0GF+F-F-%%betaGF-,&%\"qGF-\"\"%!\"\"F-,&*&F6F-%\"wGF-F-F4F-F-F-*& ,(-%#R0GF+F-*&-%#R1GF+F-F.F-F-*&F0F-,(%#nuGF-F-F-*&F;F-,(F6F-*&F1F-FEF -F-F1F8F-F8F-F-F-,&*(F1F-,&*$)F4F1F-F-*&FEF-F6F-F-F-F.F-F-*&F6F-,&FEF- F-F-F-F8F-F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e2G,&**,(*&-%#Q1G6# %\"tG\"\"\"%#d2GF-F-*$)F.\"\"#F-F--%#Q0GF+F-F-%%betaGF-,&%\"qGF-\"\"%! \"\"F-,&*&F6F-%\"wGF-F-F4F-F-F-*&,(-%#R0GF+F-*&-%#R1GF+F-F.F-F-*&F0F-, (%#nuGF-F-F-*&F;F-,(F6F-*&F1F-FEF-F-F1F8F-F8F-F-F-,&*(F1F-,&*$)F4F1F-F -*&FEF-F6F-F-F-F.F-F-*&F6F-,&FEF-F-F-F-F8F-F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "solens:=factor(subs(d1=deltasol[1],d2=deltasol[2 ],beta=nu-1,solve(\{e1,e2\},\{Q0(t),Q1(t)\})));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'solensG<$/-%#Q0G6#%\"tG,$**,jn*,\"\"#\"\"\"%#nuGF0)% \"qG\"\"$F0F*F0)%\"wGF/F0!\"\"*,\"\")F0F1F0)F3F/F0F*F0F5F0F0**F/F0F2F0 F*F0F5F0F0**F9F0F:F0F*F0F5F0F7*,F/F0)F1F/F0F:F0F*F0F6F0F7*,F9F0F>F0F3F 0F*F0F6F0F0*,F/F0-%#R1GF)F0F>F0F3F0F*F0F0*,F/F0FAF0F:F0F*F0F6F0F7*,\" \"%F0FAF0F1F0F3F0F*F0F7*,F9F0FAF0F3F0F*F0F6F0F0*.F/F0FAF0F1F0F:F0F*F0F 6F0F0*.F9F0FAF0F1F0F3F0F*F0F6F0F7**F9F0FAF0F>F0F*F0F7**\"#;F0FAF0F1F0F *F0F0**F/F0FAF0F3F0F*F0F0*(-%#R0GF)F0F1F0F3F0F0*(F/F0FAF0F3F0F0*&FNF0F 3F0F0*(F9F0FAF0F*F0F7*(F/F0F:F0F6F0F7*,F9F0F1F0F:F0F*F0F6F0F0*,\"#KF0F 1F0F3F0F*F0F6F0F7**FEF0F>F0F3F0F*F0F0**\"\"'F0F:F0F*F0F6F0F7**F9F0F1F0 F3F0F*F0F7**\"#CF0F3F0F*F0F6F0F0*(FKF0F>F0F*F0F7*(FVF0F1F0F*F0F0*(FEF0 F3F0F*F0F0*&FKF0F*F0F7*&FEF0F3F0F0F0,&F1F0F0F7F7,&F3F0FEF7F7,(*&F3F0F6 F0F0F1F0F0F7F7F7/-%#Q1GF)**,fo*,FKF0F>F0F3F0F*F0F5F0F7*,FVF0F1F0F3F0F* F0F5F0F0*,FEF0F>F0F:F0F*F0F5F0F0*,F/F0F1F0F2F0F*F0F5F0F0*,FKF0F1F0F:F0 F*F0F5F0F7**FKF0)F1F4F0F*F0F6F0F7**\"#[F0F>F0F*F0F6F0F0**FKF0F3F0F*F0F 5F0F7**F]pF0F1F0F*F0F6F0F7**F/F0F2F0F*F0F5F0F7**\"#7F0F:F0F*F0F5F0F0*( F9F0F[pF0F*F0F0**F/F0F[pF0F3F0F*F0F7*,FEF0F[pF0F3F0F*F0F6F0F0*,FbpF0F> F0F3F0F*F0F6F0F7*(FAF0F1F0F3F0F0**F/F0FNF0F1F0F3F0F0**F/F0F1F0F:F0F6F0 F7*(FKF0F6F0F*F0F0*(F/F0FNF0F>F0F0*(FEF0FNF0F1F0F7*&FAF0F3F0F0*,FEF0F1 F0F:F0F*F0F6F0F7*,\"#GF0F1F0F3F0F*F0F6F0F0**F/F0F>F0F3F0F*F0F0**FEF0F: F0F*F0F6F0F0**F/F0F1F0F3F0F*F0F0**\"#?F0F3F0F*F0F6F0F7*(F9F0F>F0F*F0F7 *(F9F0F1F0F*F0F7*(F/F0F3F0F*F0F7*(FEF0F6F0F3F0F7*(F/F0F3F0F1F0F0*&F/F0 FNF0F0*&F9F0F*F0F0*&F/F0F3F0F0**FEF0F1F0F3F0F6F0F0F0F\\oF7F]oF7F^oF7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "eqQ0R:=collect(numer(Q0(t )-op(2,op(1,solens))),[Q0,R0,R1],factor);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&eqQ0RG,***,&%#nuG\"\"\"F)!\"\"F),&%\"qGF)\"\"%F*F),( *&F,F)%\"wGF)F)F(F)F)F*F)-%#Q0G6#%\"tGF)F)*(F,F),&F(F)F)F)F)-%#R0GF3F) F)*&,8*,\"\"#F)F(F))F,F " 0 "" {MPLTEXT 1 0 134 "eqpap:=beta*(w*q+beta)*(q-4)*Q0(t)+q*(beta+2)*R0(t)+2*(beta*( q-4)*(w*q+beta)*t+q)*R1(t)-2*beta*(q-4)*(w*q-2)*(w*q+beta)*t-2*q*(w*q- 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&eqpapG,,**%%betaG\"\"\",&*&% \"qGF(%\"wGF(F(F'F(F(,&F+F(\"\"%!\"\"F(-%#Q0G6#%\"tGF(F(*(F+F(,&\"\"#F (F'F(F(-%#R0GF2F(F(*(F6F(,&**F'F(F3F(F-F(F)F(F(F+F(F(-%#R1GF2F(F(*.F6F (F'F(F-F(,&*&F+F(F,F(F(F6F/F(F)F(F3F(F/*(F6F(F+F(F?F(F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "factor(subs(nu=1+beta,eqpap-eqQ0R)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 68 "eqQ1R:=collect(numer(Q1(t)-op(2,op(2,solens))),[Q1, R0,R1,t],factor);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&eqQ1RG,,**,&%# nuG\"\"\"F)!\"\"F),&%\"qGF)\"\"%F*F),(*&F,F)%\"wGF)F)F(F)F)F*F)-%#Q1G6 #%\"tGF)F)*&,**&\"\"#F))F(F8F)F**(F8F)F(F)F,F)F**&F-F)F(F)F)F8F*F)-%#R 0GF3F)F)*(F,F),&F(F)F)F)F)-%#R1GF3F)F**.F8F)F+F)F'F)F.F),,*(F8F)F(F)F0 F)F)*&F,F)F0F)F)F(F**&F8F)F0F)F*F)F*F)F4F)F**(F8F)F,F),,*(F(F)F,F)F0F) F)*(F8F)F(F)F0F)F*F(F**&F8F)F0F)F)F)F*F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Checking with the formula in the paper" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "eqpap:=beta*(w*q+beta)*(q-4)*Q1(t)-2*(be ta^2+beta*q+q)*R0(t)-q*(beta+2)*R1(t)-2*beta*(q-4)*(2*beta*w+w*q-beta- 2)*(w*q+beta)*t+2*q*(beta*w*q-2*beta*w+w*q-beta-2);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%&eqpapG,,**%%betaG\"\"\",&*&%\"qGF(%\"wGF(F(F'F(F(, &F+F(\"\"%!\"\"F(-%#Q1G6#%\"tGF(F(*(\"\"#F(,(*$)F'F5F(F(*&F'F(F+F(F(F+ F(F(-%#R0GF2F(F/*(F+F(,&F5F(F'F(F(-%#R1GF2F(F/*.F5F(F'F(F-F(,**(F5F(F' F(F,F(F(*&F+F(F,F(F(F'F/F5F/F(F)F(F3F(F/*(F5F(F+F(,,*(F'F(F+F(F,F(F(*( F5F(F'F(F,F(F/*&F+F(F,F(F(F'F/F5F/F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "factor(subs(nu=1+beta,eqpap-eqQ1R));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "We no w use the five relations we have obtained to express five series in te rms of T1, T1', P0 and P3." }}{PARA 0 "" 0 "" {TEXT -1 42 "We are the n left with T1, P1, P3, and P0." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Q1qe:=isolate(eqP3Q1,Q1(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Q1qeG/-%#Q1G6#%\"tG,(*&#\"\"\"\"\"#F--%#P3GF(F-F-**F .F-,&%\"qGF-*&F.F-%%betaGF-F-F-F)F-%\"wGF-F-*(F.F-F)F-,&F.F-F5F-F-!\" \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "R1qe:=isolate(eqM1bis ,R1(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%R1qeG/-%#R1G6#%\"tG,,*( \"\"#\"\"\",(*$)%%betaGF,F-F-*&F1F-%\"qGF-F-F3F-F--%%diffG6$-%$M1tGF(F )F-F-*&#F-F,F-*&,**(F,F-F1F-%\"wGF-F-*&F3F-F>F-F-F1!\"\"F,F@F--%#P3GF( F-F-F@F,F@*(F,F-,&F-F-F1F-F-F>F-F@*&F,F-F>F-F-" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 99 "R0qe:=collect(isolate(factor(subs(Q1qe,R1qe,be ta=nu-1,eqQ1R)),R0(t)),[P3,(diff(M1t(t),t))],factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%R0qeG/-%#R0G6#%\"tG,(*&,&*(\"\"#!\"\"%\"wG\"\" \"%\"qGF1F1F1F/F1-%#P3GF(F1F1*(F2F1,&%#nuGF1F1F1F1-%%diffG6$-%$M1tGF(F )F1F/*&F0F1F2F1F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "Q0qe:= collect(isolate(factor(subs(R0qe,R1qe,beta=nu-1,eqQ0R)),Q0(t)),[(diff( M1t(t),t)),P3],factor);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%Q0qeG/-% #Q0G6#%\"tG,(*(,:*,\"\"%\"\"\")%#nuG\"\"#F/%\"qGF/F)F/%\"wGF/F/*,F.F/F 1F/)F3F2F/F)F/F4F/F/*(F.F/)F1\"\"$F/F)F/F/**F.F/F0F/F3F/F)F/F/*,\"\")F /F1F/F3F/F)F/F4F/!\"\"*(\"#7F/F0F/F)F/F=**F.F/F1F/F3F/F)F/F=**F.F/F3F/ F)F/F4F/F/*&F1F/F3F/F=*(F?F/F1F/F)F/F/F3F/*&F.F/F)F/F=F/,(*&F3F/F4F/F/ F1F/F/F=F=-%%diffG6$-%$M1tGF(F)F/F=*&#F/F2F/*(,6*,F.F/F1F/F3F/F)F/)F4F 2F/F/**F2F/F6F/F)F/FQF/F/**F.F/F0F/F)F/F4F/F/**F.F/F3F/F)F/FQF/F=*(F2F /F0F/F)F/F=**F " 0 "" {MPLTEXT 1 0 5 "eqM1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&,(*&\"#7\"\"\")%%betaG\"\"#F(F(*(F'F(F*F(%\"qGF(F(* &F'F(F-F(F(F(-%$M1tG6#%\"tGF(F(*&F+F(-%#Q0GF1F(F(*&#F(\"\"%F(*$)-%#P3G F1F+F(F(F(*(,**(F8F(F*F(%\"wGF(!\"\"*(F+F(F-F(F@F(FA*&F+F(F*F(F(F8F(F( F2F(F;F(F(-%#P2GF1FA*&,(*(F'F(F*F(F@F(FA*(F8F(F-F(F@F(FAF8FAF(F2F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "P2qe:=collect(subs(Q0qe, Q1qe,beta=nu-1,isolate(eqM1,P2(t))),[(diff(M1t(t),t)),Mt1(t),P3,w],fac tor);;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%P2qeG/-%#P2G6#%\"tG,***\" \"#\"\"\",&*,\"\"%F-%\"qGF-F)F-,**$)%#nuGF,F-F-*&F5F-F1F-F-*&F,F-F5F-! \"\"F-F-F-%\"wGF-F-*&,&F5F-F-F8F-,,*(F0F-F4F-F)F-F-**F0F-F5F-F1F-F)F-F -*(\"\")F-F5F-F)F-F8F1F8*&F0F-F)F-F-F-F-F-,(*&F1F-F9F-F-F5F-F-F8F8-%%d iffG6$-%$M1tGF(F)F-F8*&#F-F0F-*$)-%#P3GF(F,F-F-F-**F9F-F1F-FBF8FMF-F8* (F,F-,H*,\"\"'F-FGF-F4F-F1F-F9F-F-*,FSF-FGF-F5F-)F1F,F-F9F-F-*,F,F-F5F -F1F-F)F-)F9F,F-F8*(FSF-FGF-)F5\"\"$F-F-**FSF-FGF-F4F-F1F-F-*,\"#7F-FG F-F5F-F1F-F9F-F8**F,F-F4F-F)F-F9F-F8**F,F-F1F-F)F-FWF-F-*(\"#=F-FGF-F4 F-F8**FSF-FGF-F5F-F1F-F8**FSF-FGF-F1F-F9F-F-**F0F-F5F-F)F-F9F-F-**F,F- F1F-F)F-F9F-F8*(F[oF-FGF-F5F-F-*(F,F-F5F-F)F-F8*&F9F-F1F-F8*(F,F-F9F-F )F-F8*&FSF-FGF-F8*&F,F-F)F-F-F-FBF8F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Let us inject this in our diffrential system" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "Psoln:=subs(R1qe,Q1qe,R0qe,Q0qe,P2qe,Psol):\nQsoln:=subs(R1qe ,Q1qe,R0qe,Q0qe,P2qe,Qsol):\nRsoln:=subs(R1qe,Q1qe,R0qe,Q0qe,P2qe,Rsol ):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "indets(Psoln*Qsoln* Rsoln);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<-%%betaG%#nuG%\"qG%\"tG%\" wG%\"xG-%$M1tG6#F'-%#P0GF,-%#P1GF,-%#P3GF,-%%diffG6$F*F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "Dx;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&,&*$),&%#nuG\"\"\"F*!\"\"\"\"#F*F**&%\"qGF*F)F*F*F*)%\"xGF,F*F* *(F.F*,&F)F*F*F*F*F0F*F+**F(F*%\"tGF*,&F.F*\"\"%F+F*,(*&F.F*%\"wGF*F*F )F*F*F+F*F*F.F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "edplann :=collect(numer(factor(subs(beta=nu-1,eval(1/(Rsoln)*diff(Rsoln^2/Psol n/Dx^2,x)-1/Qsoln*diff(Qsoln^2/Psoln/Dx^2,t))))),x,factor):" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "It remains four equations" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "degree(edplann,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "for i from 0 to 3 do EDn[i]:=factor(coeff(edplann,x,i)) od:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "seq(nops(EDn[i]),i=0..3);s eq(indets(EDn[i]),i=0..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"$F# \"$V(F#" }}{PARA 12 "" 1 "" {XPPMATH 20 "6&<.%#nuG%\"qG%\"tG%\"wG-%$M1 tG6#F&-%#P0GF*-%#P1GF*-%#P3GF*-%%diffG6$F(F&-F26$F(-%\"$G6$F&\"\"#-F26 $F+F&-F26$F/F&<-F$F%F&F'F(F-F/F1F 4F?F<" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Let us look at this syst em" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "The 0-th equation contains \+ P1 but not dP1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "ed0:=col lect(op(3,EDn[0]),[diff(M1t(t),$(t,2)),(diff(P0(t),t)),(diff(M1t(t), t)),P0,P1,P3,M1t,diff(P0(t),t),diff(P3(t),t)],distributed,factor);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%$ed0G,8*(,:*,\"#;\"\"\")%#nuG\"\"#F* %\"qGF*%\"tGF*%\"wGF*!\"\"*,F)F*F,F*)F.F-F*F/F*F0F*F1*(F)F*)F,\"\"$F*F /F*F1**F)F*F+F*F.F*F/F*F1*,\"#KF*F,F*F.F*F/F*F0F*F**(\"#[F*F+F*F/F*F** *F)F*F,F*F.F*F/F*F***F)F*F.F*F/F*F0F*F1*(\"\"%F*F,F*F.F*F**(F;F*F,F*F/ F*F1*&F?F*F.F*F1*&F)F*F/F*F*F*-%%diffG6$-%$M1tG6#F/-%\"$G6$F/F-F*-%#P0 GFHF*F**(,:*,\"\")F*F+F*F.F*F/F*F0F*F**,FQF*F,F*F3F*F/F*F0F*F**(FQF*F5 F*F/F*F***FQF*F+F*F.F*F/F*F**,F)F*F,F*F.F*F/F*F0F*F1*(\"#CF*F+F*F/F*F1 **FQF*F,F*F.F*F/F*F1**FQF*F.F*F/F*F0F*F**(F-F*F,F*F.F*F1*(FWF*F,F*F/F* F**&F-F*F.F*F**&FQF*F/F*F1F*-FD6$FFF/F*-FD6$FLF/F*F***,&*&F.F*F0F*F*F- F1F*,(*&F.F*F0F*F*F,F*F*F1F*-%#P3GFHF*-%#P1GFHF*F**,F-F*F.F*F0F*F_oF*F coF*F***FQF*,(*&F,F*F0F*F*F0F1F*F*F*F_oF*FLF*F**,F-F*,,*(F-F*F,F*F0F*F **&F.F*F0F*F*F,F1*&F-F*F0F*F1F*F1F*F_oF*FaoF*FLF*F***F-F*,4*,F?F*F,F*F .F*F/F*F0F*F***F-F*F3F*F/F*F0F*F**(F?F*F+F*F/F*F***F-F*F,F*F.F*F/F*F** *F?F*F.F*F/F*F0F*F1*(FQF*F,F*F/F*F1*(F-F*F.F*F/F*F1F.F1*&F?F*F/F*F*F*F 0F*FjnF*F1*(,6*,F?F*F,F*F.F*F/F*)F0F-F*F1**F-F*F3F*F/F*F[qF*F1**F?F*F+ F*F/F*F0F*F1**F?F*F.F*F/F*F[qF*F**(F-F*F+F*F/F*F***FQF*F,F*F/F*F0F*F** *F?F*F.F*F/F*F0F*F**&F.F*F0F*F**(F?F*F/F*F0F*F1*&F-F*F/F*F1F*FaoF*FjnF *F**.F-F*F.F*,&F,F*F*F*F*F_oF*FcoF*FhnF*F1*(,6*,FQF*F,F*F.F*F/F*F[qF*F ***F?F*F3F*F/F*F[qF*F***FQF*F+F*F/F*F0F*F***FQF*F.F*F/F*F[qF*F1*(F?F*F +F*F/F*F1**F)F*F,F*F/F*F0F*F1**FQF*F.F*F/F*F0F*F1*(F-F*F.F*F0F*F1*(FQF *F/F*F0F*F**&F?F*F/F*F*F*FLF*-FD6$FaoF/F*F**,FQF*,**$F+F*F**&F,F*F.F*F **&F-F*F,F*F1F*F*F*F_oF*FLF*FhnF*F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "ed1:=collect(op( 3,EDn[1]),[diff(M1t(t),$(t,2)),(diff(P0(t),t)),(diff(M1t(t),t)),P0,P 1,P3,M1t,diff(P1(t),t),diff(P0(t),t),diff(P3(t),t)],distributed,factor );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$ed1G,P**\"#;\"\"\"%\"qGF(%\"w GF(,4*,\"\"#F(%#nuGF(F)F(%\"tGF()F*F-F(F(**F-F()F.F-F(F/F(F*F(F(**F-F( F)F(F/F(F0F(!\"\"**\"\"%F(F.F(F/F(F*F(F4**F-F(F)F(F/F(F*F(F(*(F-F(F.F( F/F(F(*&F)F(F*F(F(*(F-F(F/F(F*F(F(*&F-F(F/F(F4F(F4*(,:*,\"#KF(F2F(F)F( F/F(F*F(F4*,F?F(F.F()F)F-F(F/F(F*F(F4*(F?F()F.\"\"$F(F/F(F4**F?F(F2F(F )F(F/F(F4*,\"#kF(F.F(F)F(F/F(F*F(F(*(\"#'*F(F2F(F/F(F(**F?F(F.F(F)F(F/ F(F(**F?F(F)F(F/F(F*F(F4*(\"\")F(F.F(F)F(F(*(FIF(F.F(F/F(F4*&FMF(F)F(F 4*&F?F(F/F(F(F(-%%diffG6$-%$M1tG6#F/-%\"$G6$F/F-F(-%#P1GFVF(F(*(,:*,F' F(F2F(F)F(F/F(F*F(F(*,F'F(F.F(FAF(F/F(F*F(F(*(F'F(FCF(F/F(F(**F'F(F2F( F)F(F/F(F(*,F?F(F.F(F)F(F/F(F*F(F4*(\"#[F(F2F(F/F(F4**F'F(F.F(F)F(F/F( F4**F'F(F)F(F/F(F*F(F(*(F6F(F.F(F)F(F4*(F^oF(F.F(F/F(F(*&F6F(F)F(F(*&F 'F(F/F(F4F(-FR6$FTF/F(-FR6$FZF/F(F(*,F'F(F)F(,&F.F(F(F(F(,:*,F6F(F2F(F )F(F/F(F*F(F(*,F6F(F.F(FAF(F/F(F*F(F(*(F6F(FCF(F/F(F(**F6F(F2F(F)F(F/F (F(*,FMF(F.F(F)F(F/F(F*F(F4*(\"#7F(F2F(F/F(F4**F6F(F.F(F)F(F/F(F4**F6F (F)F(F/F(F*F(F(*&F.F(F)F(F4*(FbpF(F.F(F/F(F(F)F(*&F6F(F/F(F4F()FeoF-F( F(*(,&*&F)F(F*F(F(F-F4F(,(*&F)F(F*F(F(F.F(F(F4F()-%#P3GFVFDF(F(*,F-F(F )F(F*F(,(*&F)F(F*F(F4F.F(FDF(F()F_qF-F(F(*,F-F(,,*(F-F(F.F(F*F(F(*&F)F (F*F(F(F.F4*&F-F(F*F(F4F(F4F(F\\qF(F_qF(FZF(F(*&,@*,F'F(F.F(FAF(F/F()F *FDF(F4*,F'F(F2F(F)F(F/F(F0F(F4**F'F(FAF(F/F(F]rF(F(*,FGF(F.F(F)F(F/F( F0F(F(**F'F(FAF(F/F(F0F(F4**F?F(F2F(F/F(F*F(F(*,F'F(F.F(F)F(F/F(F*F(F4 *(F'F(FAF(F0F(F4**F^oF(F)F(F/F(F0F(F4**FGF(F.F(F/F(F*F(F4**F^oF(F)F(F/ 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6F(F0F(F4F(F)F(F(*&F,F(F0F(F-*(FJF(F,F(F4F(F(*(F6F(F0F(F)F(F-*(F=F(F4F (F)F(F(F0F(*&F3F(F4F(F-F(FPF(F(*,FaoF(,6*,F3F(F,F(F0F(F4F(FgoF(F(**F6F (F7F(F4F(FgoF(F(**F3F(F9F(F4F(F)F(F(**F3F(F0F(F4F(FgoF(F-*(F6F(F9F(F4F (F-**F=F(F,F(F4F(F)F(F-**F3F(F0F(F4F(F)F(F-*&F0F(F)F(F(*(F3F(F4F(F)F(F (*&F6F(F4F(F(F(FAF(FdsF(FXF(F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "ed3:=collect(op(3,EDn[ 3]),[diff(M1t(t),$(t,2)),(diff(P0(t),t)),(diff(M1t(t),t)),P0,P1,P3,M 1t,diff(P1(t),t),diff(P0(t),t),diff(P3(t),t)],distributed,factor);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%$ed3G,8**\"\")\"\"\"%\"qGF(%\"wGF(,( *&%#nuGF(F*F(F(F-!\"\"F*F.F(F(**\"\"%F(,(*&F)F()F*\"\"#F(F(*&F)F(F*F(F .F(F(F(,&F-F(F(F.F(-%#P3G6#%\"tGF(F(**F7F(-%%diffG6$F7F:F(F)F(F*F(F(*( ,:*,F0F()F-F4F(F)F(F:F(F*F(F(*,F0F(F-F()F)F4F(F:F(F*F(F(*(F0F()F-\"\"$ F(F:F(F(**F0F(FBF(F)F(F:F(F(*,F'F(F-F(F)F(F:F(F*F(F.*(\"#7F(FBF(F:F(F. **F0F(F-F(F)F(F:F(F.**F0F(F)F(F:F(F*F(F(*&F-F(F)F(F.*(FKF(F-F(F:F(F(F) F(*&F0F(F:F(F.F(-F=6$-%$M1tGF9F:F(FF(FFF(F:F(F*F(F.*,F_qF(FBF(F)F (F:F(F*F(F.*,FepF(F-F(FDF(F:F(F*F(F.*,F_qF(F-F(F)F(F:F(F3F(F(**FipF(FD F(F:F(F3F(F(*(FepF(FFF(F:F(F(**F'F(FBF(F)F(F*F(F.**\"$)GF(FBF(F:F(F*F( F(*,F_qF(F-F(F)F(F:F(F*F(F(**FepF(F)F(F:F(F3F(F.*(F'F(FBF(F)F(F(**FipF (F-F(F)F(F:F(F(**FfnF(F-F(F)F(F*F(F(**FeqF(F-F(F:F(F*F(F.*(F'F(FDF(F*F (F(**FepF(F)F(F:F(F*F(F.*(FepF(F-F(F:F(F.*(F'F(F)F(F*F(F.*(FepF(F:F(F* F(F(*&F'F(F)F(F.*&FipF(F:F(F(F(FQF(F(*,FipF(F^pF(F]oF(FgoF(FSF(F.*(F@F (FaoF(F7F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "systplan:=\{ed0,ed1,ed2,ed3\}:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "indets(systplan);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "Results of th is section: two new generic non-differential equations" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 249 "eqQ0R := (nu-1)*(q-4)*(q*w+nu-1)*Q 0(t)+q*(nu+1)*R0(t)+(2*nu*q^2*t*w+2*nu^2*q*t-8*nu*q*t*w-2*q^2*t*w-8*nu ^2*t-4*nu*q*t+8*q*t*w+16*nu*t+2*q*t+2*q-8*t)*R1(t)-2*(q*w-2)*(nu*q^2*t *w+nu^2*q*t-4*nu*q*t*w-q^2*t*w-4*nu^2*t-2*nu*q*t+4*q*t*w+8*nu*t+q*t+q- 4*t);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&eqQ0RG,***,&%#nuG\"\"\"F)! \"\"F),&%\"qGF)\"\"%F*F),(*&F,F)%\"wGF)F)F(F)F)F*F)-%#Q0G6#%\"tGF)F)*( F,F),&F(F)F)F)F)-%#R0GF3F)F)*&,8*,\"\"#F)F(F))F,F " 0 "" {MPLTEXT 1 0 163 "eqQ1R := (nu-1)*(q-4)*(q*w+nu-1)*Q1(t)+(-2*n u^2-2*nu*q+4*nu-2)*R0(t)-q*(nu+1)*R1(t)-2*(q-4)*(nu-1)*(2*nu*w+q*w-nu- 2*w-1)*(q*w+nu-1)*t+2*q*(nu*q*w-2*nu*w-nu+2*w-1);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&eqQ1RG,,**,&%#nuG\"\"\"F)!\"\"F),&%\"qGF)\"\"%F*F),( *&F,F)%\"wGF)F)F(F)F)F*F)-%#Q1G6#%\"tGF)F)*&,**&\"\"#F))F(F8F)F**(F8F) F(F)F,F)F**&F-F)F(F)F)F8F*F)-%#R0GF3F)F)*(F,F),&F(F)F)F)F)-%#R1GF3F)F* *.F8F)F+F)F'F),,*(F8F)F(F)F0F)F)*&F,F)F0F)F)F(F**&F8F)F0F)F*F)F*F)F.F) F4F)F**(F8F)F,F),,*(F(F)F,F)F0F)F)*(F8F)F(F)F0F)F*F(F**&F8F)F0F)F)F)F* F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 53 "q=4 : D is a square, inde pendent from t (Section 8.2)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Dq4:=factor(subs(q=4,beta=nu-1,DD));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Dq4G*$),(*&%#nuG\"\"\"%\"xGF*F*F+F*\"\"#!\"\"F,F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "The non-differential equations" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "subs(q=4,eqP3Q1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&\"\"#\"\"\"-%#Q1G6#%\"tGF&!\"\"-%#P3GF)F &**\"\"%F&,&F/F&*&F%F&%%betaGF&F&F&F*F&%\"wGF&F&*(F/F&F*F&,&F%F&F2F&F& F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "factor(subs(q=4,\{eqQ 0R,eqQ1R\}));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$,$*(\"\"#\"\"\",&%# nuGF'F'F'F',,*&-%#R0G6#%\"tGF'F)F'F'F,F'*&F&F'-%#R1GF.F'F'*&\"\")F'%\" wGF'!\"\"\"\"%F'F'F6,,*(F7F'F,F'F)F'F'*&F7F'F,F'F'*&F4F'F1F'F'*&\"#KF' F5F'F6\"#;F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "solve(%,R1( t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#/-%#R1G6#%\"tG,**&#\"\"\"\" \"#F,*&-%#R0GF'F,%#nuGF,F,!\"\"*&#F,F-F,F/F,F2*&\"\"%F,%\"wGF,F,F-F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "eqR:=map(factor,subs(q=4, eqQ0R/4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqRG,*\"\"%\"\"\"*&,& %#nuGF'F'F'F'-%#R0G6#%\"tGF'F'*&\"\"#F'-%#R1GF-F'F'*&\"\")F'%\"wGF'!\" \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Rsol:=(nu+1)*(1-2*w)* x^2+R1(t)*x+R0(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%RsolG,(*(,&%# nuG\"\"\"F)F)F),&F)F)*&\"\"#F)%\"wGF)!\"\"F))%\"xGF,F)F)*&-%#R1G6#%\"t GF)F0F)F)-%#R0GF4F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Thus R(t,x ) has a factor sqrt(D)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "f actor(subs(isolate(eqR,R1(t)),Rsol));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"#F&*&,(*&%#nuGF&%\"xGF&F&F,F&F'!\"\"F&,(*(\"\"%F&F,F &%\"wGF&F&-%#R0G6#%\"tGF&*&F'F&F,F&F-F&F&F-" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 58 "Another equation derived from the equations in Q_j and \+ R_j" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "factor(subs(isolate( eqQ1R,R1(t)),eqQ0R));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*,,&%\"qG\"\" \"\"\"%!\"\"F&,&%#nuGF&F&F(F&,bv**\"\"'F&)F%\"\"$F&%\"tGF&)%\"wG\"\"#F &F(**\"#kF&)F%F3F&)F0F3F&F1F&F&**\"$#>F&)F*F3F&F7F&F2F&F&**\"#7F&F6F&F 7F&F2F&F&**\"#;F&F6F&F0F&F1F&F&**F5F&F%F&F7F&F1F&F(**\"\")F&F*F&F%F&F7 F&F&**\"$G\"F&F*F&F7F&F2F&F(**\"#cF&F%F&F7F&F2F&F(**FAF&-%#Q1G6#F0F&)F *F/F&F0F&F(**F'F&-%#R0GFIF&FJF&F0F&F(*(-%#Q0GFIF&F:F&F%F&F&*(FOF&F6F&F 2F&F&**\"#CF&FGF&F:F&F0F&F&**F3F&FGF&F6F&F2F&F&**FF&F7F&F&*(F>F&)F*F'F&F7F&F(*(\"#KF&FJF&F7F& F&*(FinF&F*F&F7F&F(*(F3F&F6F&F2F&F&*(F'F&F%F&F7F&F(*(FinF&F7F&F2F&F&*& FOF&F%F&F(*(F3F&FGF&F%F&F(*(FAF&FGF&F0F&F&*&FLF&F%F&F(*(F'F&FLF&F0F&F& 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F1F&F&*,F-F&FGF&F:F&F%F&F0F&F(*,F'F&FGF&F6F&F0F&F2F&F&*,F'F&FLF&F:F&F% F&F0F&F(*,F-F&FGF&F*F&F%F&F0F&F&*,F>F&FGF&F%F&F0F&F2F&F(*,F'F&FLF&F*F& F%F&F0F&F&*,F'F&FLF&F%F&F0F&F2F&F(**FOF&F*F&F6F&F2F&F&**F'F&FbpF&F7F&F pF&F&**FSF&F.F&F7F&FpF&F(**F'F&FgnF&F%F&F7F&F&**FinF&FgnF&F7F&F2F&F& **FF&F%F&F0F&F2F&F(**\"#5F&F6F&F0F&F2F&F&**F'F&F:F&F%F&F0 F&F&F&F%F(,&F*F&F&F&F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "f actor(subs(q=4,numer(%)/(q-4)/(nu-1)/4));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,R**\"#K\"\"\"%#nuGF&%\"tGF&)%\"wG\"\"#F&!\"\"**\"\")F& )F'F+F&F(F&F*F&F,**\"\"%F&-%#R0G6#F(F&F(F&F*F&F,**F1F&-%#Q0GF4F&F'F&F* F&F&**\"#;F&F'F&F(F&F*F&F&*(F2F&)F'\"\"$F&F(F&F,*(F2F&F/F&F(F&F,*(F2F& F'F&F(F&F&*(F%F&F(F&F)F&F,*&F6F&F/F&F&*(F1F&F6F&F*F&F&*(F.F&-%#Q1GF4F& F*F&F&*(F+F&FCF&F'F&F&*(\"#CF&F(F&F*F&F&*&F2F&F(F&F&*&F2F&F'F&F&*(F1F& F/F&F(F&F&*&F+F&FCF&F,F6F,F2F,*,F1F&F2F&F/F&F(F&F*F&F,*,F.F&F2F&F'F&F( F&F*F&F,*&F.F&F*F&F&*&F1F&F(F&F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "collect(%,[R0,Q0,Q1,t],factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&,(*(),&%#nuG\"\"\"F*F*\"\"#F*,(F)F**&\"\"%F*%\"wGF* F*F*!\"\"F*%\"tGF*F0F)F*F*F0F*-%#R0G6#F1F*F**(F(F*F,F*-%#Q0GF4F*F**&,( *&F+F*F)F*F**&\"\")F*F/F*F*F+F0F*-%#Q1GF4F*F**,F.F*,&F*F0*&F+F*F/F*F*F *F(F*F,F*F1F*F0*&F " 0 "" {MPLTEXT 1 0 54 "eqQR0:=collect(subs(nu=1+beta,%),[R0,Q0,Q1,t],factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&eqQR0G,,*&,&*(),&\"\"#\"\"\"%%betaGF,F+F, ,&F-F,*&\"\"%F,%\"wGF,F,F,%\"tGF,!\"\"F-F,F,-%#R0G6#F2F,F,*(F*F,F.F,-% #Q0GF6F,F,*&,&*&F+F,F-F,F,*&\"\")F,F1F,F,F,-%#Q1GF6F,F,*,F0F,,&F,F3*&F +F,F1F,F,F,F*F,F.F,F2F,F3*&F>F,F1F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "Q0Q1R:=collect(isolate(%,Q0(t)),[R0,Q1,Q0],factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&Q0Q1RG/-%#Q0G6#%\"tG,(**,0*&)%%be taG\"\"$\"\"\"F)F1F1**\"\"%F1)F/\"\"#F1F)F1%\"wGF1F1*(F3F1F4F1F)F1F1** \"#;F1F/F1F)F1F6F1F1*(F3F1F/F1F)F1F1*(F9F1F)F1F6F1F1F/!\"\"F1,&F5F1F/F 1F<,&F/F1*&F3F1F6F1F1F<-%#R0GF(F1F1*(F5F1F=F<-%#Q1GF(F1F<**F3F1,0**F5F 1F4F1F)F1F6F1F1**\"\")F1F/F1F)F1)F6F5F1F1*&F4F1F)F1F<*(F9F1F)F1FJF1F1* (F5F1F/F1F)F1F<*(FIF1F)F1F6F1F<*&F5F1F6F1FF " 0 "" {MPLTEXT 1 0 23 "factor(subs(q=4,eqM1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,<*(\"#7\"\"\"-%$M1tG6#%\"tGF&)%%betaG\"\"#F &F&*(\"#[F&F'F&F,F&F&*&F/F&F'F&F&*&F-F&-%#Q0GF)F&F&*&#F&\"\"%F&*$)-%#P 3GF)F-F&F&F&*,F6F&F9F&F,F&F*F&%\"wGF&!\"\"**F-F&F9F&F,F&F*F&F&**\"\")F &F9F&F*F&F " 0 "" {MPLTEXT 1 0 26 "fa ctor(subs(q=4,eqM1bis));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,6*(\"\"# \"\"\"-%%diffG6$-%$M1tG6#%\"tGF-F&)%%betaGF%F&!\"\"*(\"\")F&F'F&F/F&F0 *&F2F&F'F&F0*(-%#P3GF,F&F/F&%\"wGF&F&*&#F&F%F&*&F5F&F/F&F&F0*(F%F&F5F& F7F&F&F5F0-%#R1GF,F&F%F&*(F%F&F/F&F7F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Dq4;\{solve(%,x)\};" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#*$),(*&%#nuG\"\"\"%\"xGF(F(F)F(\"\"#!\"\"F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#,$*&\"\"#\"\"\",&%#nuGF'F'F'!\"\"F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "A change of variable" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Xx:=x-2/(1+nu); xX:=X+2/(1+nu);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#XxG,&%\"xG\"\"\"*&\"\"#F',&%#nuGF'F'F'!\"\"F," }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xXG,&%\"XG\"\"\"*&\"\"#F',&%#nuGF'F 'F'!\"\"F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "Psol:=x^4+P3 (t)*x^3+P2(t)*x^2+P1(t)*x+P0(t);\nQsol:=x^2+Q1(t)*x+Q0(t);\nRsol:=(nu+ 1)*(1-2*w)*x^2+R1(t)*x+solve(eqR,R0(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%PsolG,,*$)%\"xG\"\"%\"\"\"F**&-%#P3G6#%\"tGF*)F(\"\"$F*F**&-% #P2GF.F*)F(\"\"#F*F**&-%#P1GF.F*F(F*F*-%#P0GF.F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%QsolG,(*$)%\"xG\"\"#\"\"\"F**&-%#Q1G6#%\"tGF*F(F*F*- %#Q0GF.F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%RsolG,(*(,&%#nuG\"\"\" F)F)F),&F)F)*&\"\"#F)%\"wGF)!\"\"F))%\"xGF,F)F)*&-%#R1G6#%\"tGF)F0F)F) *(F,F),(F,F)F2F)*&\"\"%F)F-F)F.F)F'F.F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Expressions of P and Q and R in terms of the new unknown \+ coefficients" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "PsolX:=X^4 +P3n(t)*X^3+P2n(t)*X^2+P1n(t)*X+P0n(t);\nQsolX:=X^2+Q1n(t)*X+Q0n(t);\n RsolX:=(nu+1)*(1-2*w)*X^2+X*R1n(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%&PsolXG,,*$)%\"XG\"\"%\"\"\"F**&-%$P3nG6#%\"tGF*)F(\"\"$F*F**&-%$P2 nGF.F*)F(\"\"#F*F**&-%$P1nGF.F*F(F*F*-%$P0nGF.F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&QsolXG,(*$)%\"XG\"\"#\"\"\"F**&-%$Q1nG6#%\"tGF*F(F*F *-%$Q0nGF.F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&RsolXG,&*(,&%#nuG\" \"\"F)F)F),&F)F)*&\"\"#F)%\"wGF)!\"\"F))%\"XGF,F)F)*&F0F)-%$R1nG6#%\"t GF)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Expressions of the new c oefficients in terms of the old ones..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 260 "P0nexpr:=coeff(subs(x=xX,Psol),X,0);\nP1nexpr:=coeff (subs(x=xX,Psol),X,1);\nP2nexpr:=coeff(subs(x=xX,Psol),X,2);\nP3nexpr: =coeff(subs(x=xX,Psol),X,3);\nQ0nexpr:=coeff(subs(x=xX,Qsol),X,0);\nQ1 nexpr:=coeff(subs(x=xX,Qsol),X,1);\nR1nexpr:=coeff(subs(x=xX,Rsol),X,1 );\n\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(P0nexprG,,*&\"#;\"\"\",&% #nuGF(F(F(!\"%F(*(\"\")F(-%#P3G6#%\"tGF(F)!\"$F(*(\"\"%F(-%#P2GF0F(F)! \"#F(*(\"\"#F(-%#P1GF0F(F)!\"\"F(-%#P0GF0F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(P1nexprG,**&\"#K\"\"\",&%#nuGF(F(F(!\"$F(*(\"#7F(-%# P3G6#%\"tGF(F)!\"#F(*(\"\"%F(-%#P2GF0F(F)!\"\"F(-%#P1GF0F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(P2nexprG,(*&\"#C\"\"\",&%#nuGF(F(F(!\"#F( *(\"\"'F(-%#P3G6#%\"tGF(F)!\"\"F(-%#P2GF0F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(P3nexprG,&*&\"\")\"\"\",&%#nuGF(F(F(!\"\"F(-%#P3G6#% \"tGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Q0nexprG,(*&\"\"%\"\"\",& %#nuGF(F(F(!\"#F(*(\"\"#F(-%#Q1G6#%\"tGF(F)!\"\"F(-%#Q0GF0F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Q1nexprG,&*&\"\"%\"\"\",&%#nuGF(F(F(!\"\" F(-%#Q1G6#%\"tGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(R1nexprG,(\"\" %\"\"\"*&\"\")F'%\"wGF'!\"\"-%#R1G6#%\"tGF'" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 14 "and vice-versa" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 297 "P0expr:=coeff(subs(X=Xx,PsolX),x,0);\nP1expr:=coeff(subs(X=Xx,P solX),x,1);\nP2expr:=coeff(subs(X=Xx,PsolX),x,2);\nP3expr:=coeff(subs( X=Xx,PsolX),x,3);\nQ0expr:=coeff(subs(X=Xx,QsolX),x,0);\nQ1expr:=coeff (subs(X=Xx,QsolX),x,1);\nR0expr:=coeff(subs(X=Xx,RsolX),x,0);\nR1expr: =coeff(subs(X=Xx,RsolX),x,1);\n\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %'P0exprG,,*&\"#;\"\"\",&%#nuGF(F(F(!\"%F(*(\"\")F(-%$P3nG6#%\"tGF(F)! \"$!\"\"*(\"\"%F(-%$P2nGF0F(F)!\"#F(*(\"\"#F(-%$P1nGF0F(F)F3F3-%$P0nGF 0F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'P1exprG,**&\"#K\"\"\",&%#nuG F(F(F(!\"$!\"\"*(\"#7F(-%$P3nG6#%\"tGF(F)!\"#F(*(\"\"%F(-%$P2nGF1F(F)F ,F,-%$P1nGF1F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'P2exprG,(*&\"#C\" \"\",&%#nuGF(F(F(!\"#F(*(\"\"'F(-%$P3nG6#%\"tGF(F)!\"\"F2-%$P2nGF0F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'P3exprG,&*&\"\")\"\"\",&%#nuGF(F( F(!\"\"F+-%$P3nG6#%\"tGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'Q0expr G,(*&\"\"%\"\"\",&%#nuGF(F(F(!\"#F(*(\"\"#F(-%$Q1nG6#%\"tGF(F)!\"\"F2- %$Q0nGF0F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'Q1exprG,&*&\"\"%\"\" \",&%#nuGF(F(F(!\"\"F+-%$Q1nG6#%\"tGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'R0exprG,&*(\"\"%\"\"\",&%#nuGF(F(F(!\"\",&F(F(*&\"\"#F(%\"wGF (F+F(F(*(F.F(F)F+-%$R1nG6#%\"tGF(F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%'R1exprG,(\"\"%!\"\"*&\"\")\"\"\"%\"wGF*F*-%$R1nG6#%\"tGF*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 126 "We now translate the four remaini ng non-differential equations in terms of the new unknown series (we h ave already used R0n=0)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 " Q1nP3n:=collect(isolate(subs(P3(t)=P3expr,Q1(t)=Q1expr,q=4,eqP3Q1),Q1n (t)),[P3n,t],factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'Q1nP3nG/-%$Q1nG6#%\"tG,&*&#\"\"\"\"\"#F--%$P3nGF(F-F-**F.F-,&F-!\"\"*&F.F-%\"wGF -F-F-,&F.F-%%betaGF-F-F)F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "Q0nR1n:=collect(isolate(subs(Q0(t)=Q0expr,R0(t)=R0expr,Q1(t)=Q1 expr,nu=1+beta,eqQR0),Q0n(t)),[Q1n,R0,t],factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'Q0nR1nG/-%$Q0nG6#%\"tG,&*(\"\"#\"\"\"-%$R1nGF(F-F)F- !\"\"*,F,F-%%betaGF-F.F-,&F,F-F2F-!\"#,&F2F-*&\"\"%F-%\"wGF-F-F0F-" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "collect(beta^3*t+4*beta^2*t *w+4*beta^2*t+16*beta*t*w+4*beta*t+16*t*w-beta,t,factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(),&\"\"#\"\"\"%%betaGF(F'F(,&F)F(*&\"\"%F(% \"wGF(F(F(%\"tGF(F(F)!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "eqM1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&,(*&\"#7\"\"\")%%betaG \"\"#F(F(*(F'F(F*F(%\"qGF(F(*&F'F(F-F(F(F(-%$M1tG6#%\"tGF(F(*&F+F(-%#Q 0GF1F(F(*&#F(\"\"%F(*$)-%#P3GF1F+F(F(F(*(,**(F8F(F*F(%\"wGF(!\"\"*(F+F (F-F(F@F(FA*&F+F(F*F(F(F8F(F(F2F(F;F(F(-%#P2GF1FA*&,(*(F'F(F*F(F@F(FA* (F8F(F-F(F@F(FAF8FAF(F2F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "He re, we use Q1nP3n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "eqP2n :=collect(isolate(subs(q=4,P3(t)=P3expr,P2(t)=P2expr,Q0(t)=Q0expr,Q1(t )=Q1expr,Q1nP3n,nu=1+beta,eqM1),P2n(t)),[P3n,M1t,Q0n,t],factor);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&eqP2nG/-%$P2nG6#%\"tG,,*&#\"\"\"\" \"%F-*$)-%$P3nGF(\"\"#F-F-F-*,F3F-,&F-!\"\"*&F3F-%\"wGF-F-F-,&F3F-%%be taGF-F-F)F-F1F-F6*(\"#7F-)F9F3F--%$M1tGF(F-F-*&F3F--%$Q0nGF(F-F-*&,&*( F " 0 "" {MPLTEXT 1 0 155 "eqR1M:=collect(isolate(subs(q=4,P3(t)=P3expr,P2(t)=P2expr,Q0(t)=Q 0expr,Q1(t)=Q1expr,R1(t)=R1expr,nu=1+beta,eqM1bis),R1n(t)),[diff(M1t(t ),t),P3n,t],factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&eqR1MG/-%$R 1nG6#%\"tG,**(\"\"#\"\"\"),&F,F-%%betaGF-F,F--%%diffG6$-%$M1tGF(F)F-F- *&#F-F,F-*(,&F-!\"\"*&F,F-%\"wGF-F-F-F/F--%$P3nGF(F-F-F:*(F,F-F0F-F " 0 "" {MPLTEXT 1 0 65 "P0n 0:=factor(subs(P3(t)=-2,P2(t)=1,P1(t)=0,P0(t)=0,q=4,P0nexpr));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%P0n0G,$*(\"\"%\"\"\",&%#nuGF(F(!\" \"\"\"#,&F*F(F(F(!\"%F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 " P1n0:=factor(subs(P3(t)=-2,P2(t)=1,P1(t)=0,q=4,P1nexpr));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%P1n0G,$**\"\"%\"\"\",&%#nuGF(F(!\"\"F(,&F*F( \"\"$F+F(,&F*F(F(F(!\"$F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "Q0n0:=factor(subs(Q0(t)=0,Q1(t)=-1,q=4,Q0nexpr));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Q0n0G,$*(\"\"#\"\"\",&%#nuGF(F(!\"\"F(,&F*F(F(F (!\"#F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Q1n0:=factor(sub s(Q1(t)=-1,q=4,Q1nexpr));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Q1n0G,$*&,&%#nuG\"\"\"\"\"$!\"\"F),&F(F)F)F)F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Now plug P and Q and R in the differential system" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "Psolnew:=subs(X=Xx,PsolX);\n Qsolnew:=subs(X=Xx,QsolX);\nRsolnew:=subs(X=Xx,RsolX);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(PsolnewG,,*$),&%\"xG\"\"\"*&\"\"#F*,&%#nuGF*F*F *!\"\"F/\"\"%F*F**&-%$P3nG6#%\"tGF*)F(\"\"$F*F**&-%$P2nGF4F*)F(F,F*F** &-%$P1nGF4F*F(F*F*-%$P0nGF4F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Qs olnewG,(*$),&%\"xG\"\"\"*&\"\"#F*,&%#nuGF*F*F*!\"\"F/F,F*F**&-%$Q1nG6# %\"tGF*F(F*F*-%$Q0nGF3F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Rsolnew G,&*(,&%#nuG\"\"\"F)F)F),&F)F)*&\"\"#F)%\"wGF)!\"\"F)),&%\"xGF)*&F,F)F 'F.F.F,F)F)*&F0F)-%$R1nG6#%\"tGF)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "edplan:=factor(eval(1/(Rsolnew)*diff(Rsolnew^2/Psoln ew/Dq4^2,x)-1/Qsolnew*diff(Qsolnew^2/Psolnew/Dq4^2,t))):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "nops(edplan);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "seq (nops(op(i,edplan)),i=1..nops(edplan));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"\"\"\"#\"$(fF$F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "edplan:=op(3,edplan):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "The \+ new system, with the variable X" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "edplanX:=collect (factor(subs(x=xX,nu=1+beta,edplan))/(beta+2)^5,X,factor);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(edplanXG,2*&,.*(\"\")\"\"\"%%betaGF*%\"wG F*!\"\"*&\"\"#F*-%%diffG6$-%$Q1nG6#%\"tGF6F*F*-F16$-%$P3nGF5F6F-*&\"\" %F*F+F*F**&\"#;F*F,F*F-F)F*F*)%\"XG\"\"&F*F**&,4**\"\"'F*F9F*F+F*F,F*F -*(F/F*F0F*F9F*F**&F7F*F3F*F-*(\"\"$F*F9F*F+F*F**(\"#7F*F9F*F,F*F-*&F/ F*-F16$-%$Q0nGF5F6F*F*-F16$-%$P2nGF5F6F-*&FEF*-%$R1nGF5F*F**&FEF*F9F*F *F*)F@F " 0 "" {MPLTEXT 1 0 18 "degree(edplanX,X);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "for i from 0 to 5 do ED[i]:=collect(coeff(edplanX,X, i),[diff(Q0n(t),t),diff(P0n(t),t),diff(Q1n(t),t),diff(P1n(t),t),diff(P 2n(t),t),diff(P3n(t),t),P1n,P0n,P2n,P3n],distributed,factor);od;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#EDG6#\"\"!,(*(\"\"#\"\"\"-%$P0nG6# %\"tGF+-%%diffG6$-%$Q0nGF.F/F+F+*(F*F+F,F+-%$R1nGF.F+F+*&-F16$F,F/F+F3 F+!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#EDG6#\"\"\",,*(\"\"#F'- %$P1nG6#%\"tGF'-%%diffG6$-%$Q0nGF-F.F'F'*(\"\"$F'F+F'-%$R1nGF-F'F'*(F* F'-%$P0nGF-F'-F06$-%$Q1nGF-F.F'F'*&-F06$F+F.F'F2F'!\"\"*&-F06$F9F.F'F= F'FB" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#EDG6#\"\"#,0*(F'\"\"\"-%$P 1nG6#%\"tGF*-%%diffG6$-%$Q1nGF-F.F*F**(,&F*!\"\"*&F'F*%\"wGF*F*F*,&F'F *%%betaGF*F*F+F*F6*(F'F*-F06$-%$Q0nGF-F.F*-%$P2nGF-F*F**&-F06$F@F.F*F> F*F6*&-F06$F+F.F*F2F*F6-F06$-%$P0nGF-F.F6*(\"\"%F*F@F*-%$R1nGF-F*F*" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#EDG6#\"\"$,0*(\"\"#\"\"\"-%%diffG 6$-%$Q1nG6#%\"tGF2F+-%$P2nGF1F+F+*(F*F+-F-6$-%$Q0nGF1F2F+-%$P3nGF1F+F+ *&-F-6$F:F2F+F8F+!\"\"*&-F-6$F3F2F+F/F+F?-F-6$-%$P1nGF1F2F?**F*F+,&F+F ?*&F*F+%\"wGF+F+F+,&F*F+%%betaGF+F+F3F+F?*(\"\"&F+-%$R1nGF1F+F:F+F+" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#EDG6#\"\"%,.*(\"\"#\"\"\"-%%diffG 6$-%$Q1nG6#%\"tGF2F+-%$P3nGF1F+F+*&F*F+-F-6$-%$Q0nGF1F2F+F+*&-F-6$F3F2 F+F/F+!\"\"-F-6$-%$P2nGF1F2F=**\"\"$F+,&F+F=*&F*F+%\"wGF+F+F+,&F*F+%%b etaGF+F+F3F+F=*&\"\"'F+-%$R1nGF1F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>&%#EDG6#\"\"&,(*&\"\"#\"\"\"-%%diffG6$-%$Q1nG6#%\"tGF2F+F+-F-6$-%$P 3nGF1F2!\"\"*(\"\"%F+,&F+F7*&F*F+%\"wGF+F+F+,&F*F+%%betaGF+F+F7" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "The last equation is solved via th e connection between P3n and Q1n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "eval(subs(Q1nP3n,ED[5]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "Let us no w work with ED[4], and first get rid of Q1n, using its expression in t erms of P3n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "collect(fact or(eval(subs(Q1nP3n,ED[4]))),[(diff(P3n(t),t)),P3n],factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&,&*&#\"\"\"\"\"#F(-%$P3nG6#%\"tGF(F(**F) F(,&F(!\"\"*&F)F(%\"wGF(F(F(,&F)F(%%betaGF(F(F-F(F0F(-%%diffG6$F*F-F(F (*(F/F(F3F(F*F(F(*&F)F(-F66$-%$Q0nGF,F-F(F(-F66$-%$P2nGF,F-F0*&\"\"'F( -%$R1nGF,F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Now we get rid o f P2n, using another of the generic equations" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "factor(eval(subs(eqP2n,%)));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,6**\"\"'\"\"\"-%$P3nG6#%\"tGF&%%betaGF&%\"wGF&F&*(\" #7F&-%%diffG6$-%$M1tGF)F*F&)F+\"\"#F&!\"\"*(\"\"$F&F'F&F+F&F6*(F.F&F'F &F,F&F&*(\"#[F&F/F&F+F&F6*(F.F&F+F&F,F&F&*&F%F&-%$R1nGF)F&F&*&F%F&F'F& F6*&F;F&F/F&F6F.F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "What remain s is the last generic equation, and thus ED[4] is solved." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "factor(subs(eqR1M,%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "L et us now work with ED[0], in which we get rid of R1n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "(subs(isolate(Q0nR1n,R1n(t)),ED[0]) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*(\"\"#\"\"\"-%$P0nG6#%\"tGF&- %%diffG6$-%$Q0nGF)F*F&F&**F%F&F'F&F.F&,&*&F%F&F*F&F&**F%F&%%betaGF&,&F %F&F4F&!\"#,&F4F&*&\"\"%F&%\"wGF&F&!\"\"F;F;F;*&-F,6$F'F*F&F.F&F;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "P0nQ0n:=dsolve(%,P0n(t));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'P0nQ0nG/-%$P0nG6#%\"tG*(%$_C1G\"\" \"-%$Q0nGF(\"\"#,&*&,.*$)%%betaG\"\"$F,F,*(\"\"%F,)F5F/F,%\"wGF,F,*&F8 F,F9F,F,*(\"#;F,F5F,F:F,F,*&F8F,F5F,F,*&F=F,F:F,F,F,F)F,F,F5!\"\"F@" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "factor(isolate(subs(P0n(t) =P0n0,Q0n(t)=Q0n0,t=0,%),_C1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$ _C1G,$%%betaG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "P0nQ0 n:=subs(%,P0nQ0n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'P0nQ0nG/-%$P0 nG6#%\"tG,$*(%%betaG\"\"\"-%$Q0nGF(\"\"#,&*&,.*$)F,\"\"$F-F-*(\"\"%F-) F,F0F-%\"wGF-F-*&F8F-F9F-F-*(\"#;F-F,F-F:F-F-*&F8F-F,F-F-*&F=F-F:F-F-F -F)F-F-F,!\"\"F@F@" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "facto r(beta^3+4*beta^2*w+4*beta^2+16*beta*w+4*beta+16*w);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&),&\"\"#\"\"\"%%betaGF'F&F',&F(F'*&\"\"%F'%\"wGF'F 'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 44 "Elimination of R1n between the first two EDs" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "ED01:=numer(factor(subs(isol ate(ED[0],R1n(t)),ED[1])));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ED01 G,,**\"\"#\"\"\"-%$P1nG6#%\"tGF(-%%diffG6$-%$Q0nGF+F,F(-%$P0nGF+F(!\" \"**\"\"$F(F)F(-F.6$F2F,F(F0F(F(*(\"\"%F()F2F'F(-F.6$-%$Q1nGF+F,F(F(** F'F(-F.6$F)F,F(F0F(F2F(F4**F'F(F7F(F>F(F2F(F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Q1nQP:=dsolve(ED01,Q1n(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&Q1nQPG/-%$Q1nG6#%\"tG,&*&#\"\"\"\"\"#F-*(-%$P0nGF(! \"\"-%$P1nGF(F--%$Q0nGF(F-F-F-*&F0F,%$_C1GF-F-" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 69 "isolate(subs(P1n(t)=P1n0,Q1n(t)=Q1n0,Q0n(t)=Q0 n0,P0n(t)=P0n0,%),_C1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$_C1G\"\" !" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Q1nQP:=subs(%,Q1nQP); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&Q1nQPG/-%$Q1nG6#%\"tG,$*&#\"\" \"\"\"#F-*(-%$P0nGF(!\"\"-%$P1nGF(F--%$Q0nGF(F-F-F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Now we express all our series in terms of P3n a nd M1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Q1nP3n;\neqR1M;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$Q1nG6#%\"tG,&*&#\"\"\"\"\"#F+-%$P 3nGF&F+F+**F,F+,&F+!\"\"*&F,F+%\"wGF+F+F+,&F,F+%%betaGF+F+F'F+F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$R1nG6#%\"tG,**(\"\"#\"\"\"),&F*F+% %betaGF+F*F+-%%diffG6$-%$M1tGF&F'F+F+*&#F+F*F+*(,&F+!\"\"*&F*F+%\"wGF+ F+F+F-F+-%$P3nGF&F+F+F8*(F*F+F.F+F:F+F8F*F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Q0nP3M:=subs(eqR1M,Q0nR1n);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'Q0nP3MG/-%$Q0nG6#%\"tG,&*(\"\"#\"\"\",**(F,F-),&F,F- %%betaGF-F,F--%%diffG6$-%$M1tGF(F)F-F-*&#F-F,F-*(,&F-!\"\"*&F,F-%\"wGF -F-F-F1F--%$P3nGF(F-F-F<*(F,F-F2F-F>F-FF-F-F " 0 "" {MPLTEXT 1 0 27 "P2nP3M:=subs(Q0nP3M,eqP2n);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#> %'P2nP3MG/-%$P2nG6#%\"tG,.*&#\"\"\"\"\"%F-*$)-%$P3nGF(\"\"#F-F-F-*,F3F -,&F-!\"\"*&F3F-%\"wGF-F-F-,&F3F-%%betaGF-F-F)F-F1F-F6*(\"#7F-)F9F3F-- %$M1tGF(F-F-*(F.F-,**(F3F-F=F--%%diffG6$F>F)F-F-*&#F-F3F-*(F5F-F9F-F1F -F-F6*(F3F-F:F-F8F-F6F3F6F-F)F-F6*,F.F-F:F-FAF-F9!\"#,&F:F-*&F.F-F8F-F -F6F-*&,&*(F " 0 "" {MPLTEXT 1 0 36 "P0nP3M:=factor(subs(Q0nP3M,P0nQ0n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'P0nP3MG/-%$P0nG6#%\"tG,$*,,0*&)%%betaG\"\"$\"\" \"F)F1F1**\"\"%F1)F/\"\"#F1F)F1%\"wGF1F1*(F3F1F4F1F)F1F1**\"#;F1F/F1F) F1F6F1F1*(F3F1F/F1F)F1F1*(F9F1F)F1F6F1F1F/!\"\"F1,4**F5F1-%$P3nGF(F1F/ F1F6F1F1*(F3F1-%%diffG6$-%$M1tGF(F)F1F4F1F<*&F?F1F/F1F<*(F3F1F?F1F6F1F 1*(F9F1FBF1F/F1F<*(F3F1F/F1F6F1F1*&F5F1F?F1F<*&F9F1FBF1F " 0 "" {MPLTEXT 1 0 57 "P1nP3M:=subs(Q1nP3n,P0nP3M,Q0nP3M,isolate(Q1nQP,P1n(t )));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'P1nP3MG/-%$P1nG6#%\"tG,$*2 \"\"#\"\"\",&*&#F-F,F--%$P3nGF(F-F-**F,F-,&F-!\"\"*&F,F-%\"wGF-F-F-,&F ,F-%%betaGF-F-F)F-F-F-,0*&)F9\"\"$F-F)F-F-**\"\"%F-)F9F,F-F)F-F7F-F-*( F?F-F@F-F)F-F-**\"#;F-F9F-F)F-F7F-F-*(F?F-F9F-F)F-F-*(FCF-F)F-F7F-F-F9 F5F-,4**F,F-F1F-F9F-F7F-F-*(F?F--%%diffG6$-%$M1tGF(F)F-F@F-F5*&F1F-F9F -F5*(F?F-F1F-F7F-F-*(FCF-FIF-F9F-F5*(F?F-F9F-F7F-F-*&F,F-F1F-F5*&FCF-F IF-F5F?F-F,F9F-,&F9F-*&F?F-F7F-F-!\"#F8!\"%,&*(F,F-,**(F,F-)F8F,F-FIF- F-*&#F-F,F-*(F4F-F8F-F1F-F-F5*(F,F-F9F-F7F-F5F,F5F-F)F-F5*,F,F-F9F-FZF -F8FVFTF5F-F5F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "We plug this in the two DEs we hav e not used yet (the other four DEs are zero)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "e dP3Ma:=collect(factor(numer(factor(eval(subs(P1nP3M,P0nP3M,P2nP3M,Q0nP 3M,Q1nP3n,eqR1M,ED[2]))))/2),[diff(M1t(t),$(t,2)),diff(M1t(t),t),dif f(P3n(t),t),M1t,P3n,t],factor);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%' edP3MaG,.*&,.*&,&**\"#;\"\"\"),&\"\"#F,%%betaGF,\"\"%F,,&F0F,*&F1F,%\" wGF,F,F,)%\"tGF/F,F,**F+F,F0F,)F.F/F,F6F,!\"\"F,-%%diffG6$-%$M1tG6#F6F 6F,F,*&,&**\"#[F,F-F,F2F,F6F,F9*(FCF,F0F,F8F,F,F,F=F,F,**F8F,F2F,F6F,) -%$P3nGF?F/F,F9*&,&*,F1F,,&F,F9*&F/F,F4F,F,F,)F.\"\"$F,F2F,F5F,F,*,\"# 7F,F0F,FLF,F.F,F6F,F9F,FGF,F,**F+F,F8F,,,*(F/F,)F0F/F,F4F,F,*(F1F,F0F, )F4F/F,F,*(F1F,F0F,F4F,F,F0F,*&\"\")F,F4F,F,F,F5F,F,**\"#KF,F0F,,&*&F0 F,F4F,F,F,F,F,F6F,F9F,-F;6$F=-%\"$G6$F6F/F,F,*,FZF,F-F,F2F,F6F,)F:F/F, F,*&,(*&,(**F8F,F2F,F6F,FGF,F,*,FZF,FLF,FNF,F2F,F5F,F9*,FQF,F0F,FLF,F. F,F6F,F,F,-F;6$FGF6F,F,*.F/F,FLF,FNF,F2F,F6F,FGF,F9**FZF,F8F,FSF,F6F,F 9F,F:F,F,*&,(*&,&*,FQF,FLF,FNF,F2F,F6F,F,**FQF,F0F,FLF,F.F,F9F,F=F,F,* *F4F,F8F,F6F,FGF,F9*,F1F,F4F,FNF,FLF,F5F,F9F,FgoF,F,*.F/F,F4F,FNF,FLF, F6F,FGF,F,*,FZF,F4F,F8F,FgnF,F6F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 349 "eqpap:=16*a^2*(p-b)*(t*M1p-3*M1)*M1pp-p*P3^2*M1pp+4* t*(2*w-1)*a*(p-3*b)*P3*M1pp\n+16*t*(t*a^2*(2*b^2*w+b*(2*w+1)^2+8*w)-2* b*(b*w+1))*M1pp+8*a^2*p*M1p^2\n-4*t*a*(2*w-1)*(2*p-3*b)*P3p*M1p+p*(P3p -2*(2*w-1)*a)*P3*M1p+12*a*(2*w-1)*(p-b)*P3p*M1\n-8*t*a^2*(4*b*w^2+2*w* (b^2+2*b+4)+b)*M1p-t*w*a^2*(P3+4*a*t*(2*w-1))*P3p\n+2*t*w*a^3*(2*w-1)* P3+8*t*w*a^2*(1+w*b);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&eqpapG,:*, \"#;\"\"\")%\"aG\"\"#F(,&%\"pGF(%\"bG!\"\"F(,&*&%$M1pGF(%\"tGF(F(*&\" \"$F(%#M1GF(F/F(%%M1ppGF(F(*(F-F()%#P3GF+F(F7F(F/*0\"\"%F(F3F(,&F(F/*& F+F(%\"wGF(F(F(F*F(,&F-F(*&F5F(F.F(F/F(F:F(F7F(F(**F'F(F3F(,&*(F3F(F)F (,(*(F+F()F.F+F(F?F(F(*&F.F(),&*&F+F(F?F(F(F(F(F+F(F(*&\"\")F(F?F(F(F( F(*(F+F(F.F(,&*&F.F(F?F(F(F(F(F(F/F(F7F(F(**FMF(F)F(F-F()F2F+F(F(*0F " 0 "" {MPLTEXT 1 0 224 "collect(factor(edP3Ma-subs(P3=P3n(t),P3p=diff(P3n(t) ,t), M1=M1t(t), M1p=diff(M1t(t),t), M1pp=diff(M1t(t),t$2), p=a^2*(b+4* w)*t,a=beta+2, b=beta,eqpap)),[diff(M1t(t),$(t,2)),diff(M1t(t),t),di ff(P3n(t),t),M1t,P3n,t],factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" \"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 183 "edP3Mb:=collect(fac tor(numer(factor(eval(subs(P1nP3M,P0nP3M,P2nP3M,Q0nP3M,Q1nP3n,eqR1M,ED [3]))))/2/(2*beta+4)),[diff(M1t(t),$(t,2)),diff(M1t(t),t),diff(P3n(t ),t),M1t,P3n,t],factor);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'edP3MbG ,**&,(**,&\"\"#\"\"\"%%betaGF+F+,&F,F+*&\"\"%F+%\"wGF+F+F+%\"tGF+-%$P3 nG6#F1F+!\"\"*,F/F+,&F+F5*&F*F+F0F+F+F+)F)F*F+F-F+)F1F*F+F+**\"\")F+F, F+F7F+F1F+F5F+-%%diffG6$-%$M1tGF4-%\"$G6$F1F*F+F+*&,&**F)F+F-F+F1F+-F> 6$F2F1F+F5*,\"\"'F+F7F+F9F+F-F+F1F+F5F+-F>6$F@F1F+F+*&,&**\"\"$F+F)F+F -F+F@F+F+**F*F+F0F+F)F+F1F+F5F+FHF+F+*,FKF+F7F+F9F+F-F+F@F+F+" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "eqpap:=(p*P3/a-4*t*(2*w-1)*( p-2*b))*M1pp+p*(P3p/a+12*w-6)*M1p-3*p/t*(P3p/a+4*w-2)*M1+2*t*w*a*P3p; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&eqpapG,**&,&*(%\"pG\"\"\"%#P3GF *%\"aG!\"\"F***\"\"%F*%\"tGF*,&F*F-*&\"\"#F*%\"wGF*F*F*,&F)F**&F3F*%\" bGF*F-F*F-F*%%M1ppGF*F**(F)F*,(*&%$P3pGF*F,F-F**&\"#7F*F4F*F*\"\"'F-F* %$M1pGF*F**,\"\"$F*F)F*F0F-,(F;F**&F/F*F4F*F*F3F-F*%#M1GF*F-*,F3F*F0F* F4F*F,F*F " 0 "" {MPLTEXT 1 0 224 "collect( factor(edP3Mb+subs(P3=P3n(t),P3p=diff(P3n(t),t), M1=M1t(t), M1p=diff(M 1t(t),t), M1pp=diff(M1t(t),t$2), p=a^2*(b+4*w)*t,a=beta+2, b=beta,eqpa p)),[diff(M1t(t),$(t,2)),diff(M1t(t),t),diff(P3n(t),t),M1t,P3n,t],fa ctor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "Check ing the equations (this requires to have determined the series P_i, Q_ i and R_i in the relevant section above)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "n:=10: normal(series(subs(P3n(t)=P3nexpr,P3(t)=P3ser (n),M1t(t)=subs(y=1,t^2*Mser(n+1)), q=4,beta=nu-1,edP3Ma),t,n+1));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#+%%\"tG-%\"OG6#\"\"\"\"#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "n:=10: normal(series(subs(P3n(t)=P 3nexpr,P3(t)=P3ser(n),M1t(t)=subs(y=1,t^2*Mser(n+1)), q=4,beta=nu-1,ed P3Mb),t,n+1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+%%\"tG-%\"OG6#\"\" \"\"#6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Finally we eliminate P3 n." }}{PARA 0 "" 0 "" {TEXT -1 39 "We first build an equation without \+ dP3n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "edP3Mabis:=numer(factor(subs(isolate(edP3Mb ,diff(P3n(t),t)),edP3Ma))): nops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#\"$o&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "We differentiate this \+ equation" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "edP3Mc:=diff(ed P3Mabis,t):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "We finally elimin ate P3n and dP3n in the (polynomial) system consisting of edP3Mabis, e dP3Mb and edP3Mc" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "edP3Mcb is:=numer(factor(subs(isolate(edP3Mb,diff(P3n(t),t)),edP3Mc))): nops(% );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%=@" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "indets(edP3Mabis);indets(edP3Mcbis);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#<)%%betaG%\"tG%\"wG-%$M1tG6#F%-%$P3nGF)-%%diffG6$F'F%-F-6$F'-%\"$G6$F%\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<*%%bet aG%\"tG%\"wG-%$M1tG6#F%-%$P3nGF)-%%diffG6$F'F%-F-6$F'-%\"$G6$F%\"\"#-F -6$F'-F26$F%\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "res:= numer(factor(resultant(subs(P3n(t)=P3n,edP3Mabis),subs(P3n(t)=P3n,edP3 Mcbis),P3n))): nops(res);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "seq(indets(op(i,res)),i=1.. nops(res));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'<\"<#%%betaG<$F%%\"wG<' F%%\"tGF'-%$M1tG6#F)-%%diffG6$F*F)<)F%F)F'F*F--F.6$F*-%\"$G6$F)\"\"#-F .6$F*-F46$F)\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "op(4,r es);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*(%%betaG\"\"\"%\"tGF&-%%dif fG6$-%$M1tG6#F'F'F&F&**\"\"%F&F(F&F'F&%\"wGF&F&*(\"\"$F&F+F&F%F&!\"\"* (\"#7F&F+F&F0F&F3*(\"\"#F&F0F&F'F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "M1tser:=proc(n)\nsubs(q=4,y=1,t^2*Mser(n)): end:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "n:=3: normal(series(subs(M1t (t)=M1tser(n+2),beta=nu-1,op(4,res)),t,n+1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+)%\"tG,$*&\"\"#\"\"\"%\"wGF(F(F(,(*&%#nuGF(F)F(!\"\"*& \"\"%F()F)F'F(F-F)F(F'-%\"OG6#F(F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "edM1t:=collect(op(5,res),[diff(M1t(t),$(t,3)),diff( M1t(t),$(t,2)),diff(M1t(t),t),M1t],factor):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "let us check" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "n:=10: normal(series(subs(M1t(t)=M1tser(n),beta=nu-1,edM1t),t,n+ 1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+%%\"tG-%\"OG6#\"\"\"\"#6" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "The degree" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 92 "subs(diff(M1t(t),$(t,3))=T*X,diff(M1t(t),$ (t,2))=Z*X,diff(M1t(t),t)=Y*X,M1t(t)=X,edM1t):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 12 "degree(%,X);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#\"#6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Initial terms" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "unassign('m'):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "M1tserf:=proc(n)\nadd(m[i]*t^i,i=2. .n): end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "n:=9: factor(s eries(subs(M1t(t)=M1tserf(n+3),beta=nu-1,edM1t),t,n+1));" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#+'%\"tG,$**\"#s\"\"\")%\"wG\"\"&F(,(*&%#nuGF(F*F (F(F*!\"\"F(F(F(,fp**\"\"%F()F.\"\"$F()F*F4F()&%\"mG6#\"\"#F4F(F(**\"# ;F()F.F:F()F*F2F(F6F(F(**F2F(F3F()F*F:F(F6F(F/**\"\"*F(F3F(F*F()F7F2F( F(*,F2F(F=F(F>F(F7F(&F86#F4F(F/**\"#7F(F=F(F5F(F6F(F/**\"#SF(F=F(F@F(F CF(F(**\"#KF(F.F(F>F(F6F(F(**FF(F7F(FEF(F/**\"#OF(F.F(F5F(F6F(F/*,FF(F6F(F(*(FF(F7F(FEF(F/*(\"#?F(F5F(F6F(F/**F " 0 "" {MPLTEXT 1 0 20 "series(M1tser(2),t);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#++%\"tG%\"wG\"\"#*&F%\"\"\",(*&%#nuGF( F%F(F(F+F(*&\"\"$F(F%F(F(F(F-*&F%F(,0*(F&F()F+F&F()F%F&F(F(*(\"\"&F(F1 F(F%F(F(*(\"#7F(F+F(F2F(F(*&F&F(F1F(F(*(F6F(F+F(F%F(F(*&\"#=F(F2F(F(*& F-F(F%F(F(F(\"\"%-%\"OG6#F(F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "m[2]:=w:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "n:=9: fact or(series(subs(M1t(t)=M1tserf(n+3),beta=nu-1,edM1t),t,n+1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+'%\"tG,$*,\"#s\"\"\")%\"wG\"\"(F(,(*&%#nuGF (F*F(F(F*!\"\"F(F(F(,**&F.F()F*\"\"#F(F(*&F.F(F*F(F(*&\"\"$F(F2F(F(&% \"mG6#F6F/F(,4*(\"\"%F()F.F3F(F2F(F(*(\"#;F(F.F()F*F6F(F(*&F=F(F*F(F(* *F " 0 " " {MPLTEXT 1 0 22 "m[3]:=w*(nu*w+nu+3*w):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "From now on we can loop" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "n:=n+1: factor(series(subs(M1t(t)=M1tserf(n+3),beta=n u-1,edM1t),t,n+1));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+'%\"tG,$*2\"%? D\"\"\")%\"wG\"\")F(,&*&\"\"#F(F*F(F(F(!\"\"F(,&*&F.F(F*F(F(F(F(F(,&%# nuGF(F(F(F(,&F3F(\"\"$F/F(,(*&F3F(F*F(F(F*F/F(F(F(,\\o*(\"#UF()F3\"\"& F()F*\"\"'F(F(*(\"$'QF(F;F()F*F " 0 "" {MPLTEXT 1 0 43 "m[n-6]:=factor(solve(coef f(%,t,n),m[n-6]));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>&%\"mG6#\"\"(*& %\"wG\"\"\",jn*(\"#UF*)%#nuG\"\"&F*)F)F0F*F**(\"$'QF*F.F*)F)\"\"%F*F** (\"$I'F*)F/F5F*F1F*F**(\"%I5F*F.F*)F)\"\"$F*F**(\"%+RF*F8F*F4F*F**(\"% !y$F*)F/FF*F8F*FCF*F**(\" &Sw\"F*FAF*F;F*F**(\"&+F$F*FKF*F4F*F**(\"&5q\"F*F/F*F1F*F**&F-F*F.F*F* *(F7F*F8F*F)F*F**(\"%+dF*FAF*FCF*F**(\"&qF#F*FKF*F;F*F**(\"&II$F*F/F*F 4F*F**&\"&1-\"F*F1F*F**(\"$g$F*FAF*F)F*F**(\"%!3%F*FKF*FCF*F**(\"&]U\" F*F/F*F;F*F**&\"&SG\"F*F4F*F**(\"$N\"F*FKF*F)F*F**(\"%I:F*F/F*FCF*F**& \"%&\\$F*F;F*F**(\"#IF*F/F*F)F*F**&\"$S#F*FCF*F**&F " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 38 "The self-dual Potts model (Section 10)" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "Ds:=factor(subs(nu=1+beta,q= beta^2,w=1/beta,DD));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DsG*&)%%be taG\"\"#\"\"\",0*(F(F)F&F)%\"tGF)F)*&F'F))%\"xGF(F)F)*&F'F)F/F)!\"\"*& F(F)F.F)F)*&\"\")F)F,F)F1*&F(F)F/F)F1F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "factor(diff(Ds,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*()%%betaG\"\"#\"\"\",&*&F&F'%\"xGF'F'F'!\"\"F',&F&F'F%F'F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "factor(subs(x=1-x,Ds)-Ds);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "A change of variable" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Xx:=x-1/2; xX:=X+1/2;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#XxG,&%\"xG\"\"\"#F'\"\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xXG,&%\"XG\"\"\"#F'\"\"#F'" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 32 "collect(subs(x=xX,Ds),X,factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*()%%betaG\"\"#\"\"\",&F'F(F&F(F()%\"XGF'F (F(**\"\"%!\"\"F&F',&F&F(F'F.F(,(*(\"\")F(F&F(%\"tGF(F(*&\"#;F(F3F(F(F (F.F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Note that R2 vanishes \+ for this specialization" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 " Psol:=x^4+P3(t)*x^3+P2(t)*x^2+P1(t)*x+P0(t);\nQsol:=x^2+Q1(t)*x+Q0(t); \nRsol:=R1(t)*x+R0(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%PsolG,,*$)%\"xG\"\"%\"\"\"F**&-%#P3G6#%\"tGF*)F(\"\"$F*F**&-%#P2GF.F*)F(\"\"#F* F**&-%#P1GF.F*F(F*F*-%#P0GF.F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Q solG,(*$)%\"xG\"\"#\"\"\"F**&-%#Q1G6#%\"tGF*F(F*F*-%#Q0GF.F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%RsolG,&*&-%#R1G6#%\"tG\"\"\"%\"xGF+F+-%#R 0GF)F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "subs(x=xX,x*(x-1) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"XG\"\"\"#F&\"\"#F&F&,&F%F &#F&F(!\"\"F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 126 "As explained in the paper, this implies that P and Q are even functions of X, while R is odd: so we only have 4 unknown series" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Expressions of P and Q and R in terms of the new unknown \+ series" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "PsolX:=X^4+P2n(t) *X^2+P0n(t);\nQsolX:=X^2+Q0n(t);\nRsolX:=X*R1n(t);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%&PsolXG,(*$)%\"XG\"\"%\"\"\"F**&-%$P2nG6#%\"tGF*)F( \"\"#F*F*-%$P0nGF.F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&QsolXG,&*$) %\"XG\"\"#\"\"\"F*-%$Q0nG6#%\"tGF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%&RsolXG*&%\"XG\"\"\"-%$R1nG6#%\"tGF'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "Expressions of the original coefficients in terms of the \+ new ones" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 297 "P0expr:=coeff( subs(X=Xx,PsolX),x,0);\nP1expr:=coeff(subs(X=Xx,PsolX),x,2);\nP2expr:= coeff(subs(X=Xx,PsolX),x,2);\nP3expr:=coeff(subs(X=Xx,PsolX),x,3);\nQ0 expr:=coeff(subs(X=Xx,QsolX),x,0);\nQ1expr:=coeff(subs(X=Xx,QsolX),x,1 );\nR0expr:=coeff(subs(X=Xx,RsolX),x,0);\nR1expr:=coeff(subs(X=Xx,Rsol X),x,1);\n\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'P0exprG,(#\"\"\"\"# ;F'*&#F'\"\"%F'-%$P2nG6#%\"tGF'F'-%$P0nGF.F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'P1exprG,&#\"\"$\"\"#\"\"\"-%$P2nG6#%\"tGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'P2exprG,&#\"\"$\"\"#\"\"\"-%$P2nG6#%\"tGF )" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'P3exprG!\"#" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%'Q0exprG,&#\"\"\"\"\"%F'-%$Q0nG6#%\"tGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'Q1exprG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'R0exprG,$*&#\"\"\"\"\"#F(-%$R1nG6#%\"tGF(!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'R1exprG-%$R1nG6#%\"tG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "We now translate the five non-differentia l equations in terms of the new unknown series" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "The first one holds automatically" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "factor(subs(P3(t)=P3expr,Q1(t)=Q1expr,q=b eta^2,w=1/beta,eqP3Q1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "The second one" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "Q0nR1n:=factor(isolate(subs(Q0(t)=Q0expr,R 0(t)=R0expr,R1(t)=R1expr,Q1(t)=Q1expr,nu=1+beta,q=beta^2,w=1/beta,eqQ0 R),Q0n(t)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'Q0nR1nG/-%$Q0nG6#% \"tG,$*&#\"\"\"\"\"%F-*(,(*(\"\")F-%%betaGF-F)F-F-*&\"#;F-F)F-F-F-!\" \"F-,(-%$R1nGF(F-F3F6\"\"#F-F-,&F:F-F3F-F6F-F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "The third one holds automatically" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "factor(subs(Q0(t)=Q0expr,R0(t)=R0expr,R1 (t)=R1expr,Q1(t)=Q1expr,nu=1+beta,q=beta^2,w=1/beta,eqQ1R));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "In the last two generic equations, we introduce the notation S= be ta Mt1 = beta t^2 M" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "eqM 1s:=collect(factor(subs(q=beta^2,w=1/beta,nu=1+beta,Q0(t)=Q0expr,P3(t) =P3expr,P2(t)=P2expr,M1t(t)=S(t)/beta,eqM1)),[S,Q0n,P2n,R0n],factor); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&eqM1sG,***\"#7\"\"\"%%betaGF(,& \"\"#F(F)F(F(-%\"SG6#%\"tGF(F(*&F+F(-%$Q0nGF.F(F(-%$P2nGF.!\"\"*(\"\"% F(,&F)F(F7F(F(F/F(F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "eq M1biss:=collect(eval(subs(q=beta^2,w=1/beta,nu=1+beta,R1(t)=R1expr,M1t (t)=S(t)/beta,eqM1bis)),[diff(S(t),t),R1n],factor);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%)eqM1bissG,(**\"\"#\"\"\"%%betaGF(,&F'F(F)F(F(-%%di ffG6$-%\"SG6#%\"tGF1F(!\"\"\"\"%F(-%$R1nGF0F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Now plug P and Q and R in the differential system" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 80 "Psolnew:=subs(X=Xx,PsolX);\nQsolnew:=subs(X=Xx,Qsol X);\nRsolnew:=subs(X=Xx,RsolX);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%( PsolnewG,(*$),&%\"xG\"\"\"#F*\"\"#!\"\"\"\"%F*F**&-%$P2nG6#%\"tGF*)F(F ,F*F*-%$P0nGF2F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(QsolnewG,&*$),& %\"xG\"\"\"#F*\"\"#!\"\"F,F*F*-%$Q0nG6#%\"tGF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(RsolnewG*&,&%\"xG\"\"\"#F(\"\"#!\"\"F(-%$R1nG6#%\"tG F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "edplan:=factor(subs( nu=1+beta,eval(1/(Rsolnew)*diff(Rsolnew^2/Psolnew/Ds^2,x)-1/Qsolnew*di ff(Qsolnew^2/Psolnew/Ds^2,t)))):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "nops(edplan);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\" &" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "seq(nops(op(i,edplan)) ,i=1..nops(edplan));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"\"\"$%>\" \"#F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "edplan:=op(2,edp lan):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "The new system, with the variable X" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "edplanX:=col lect(factor(subs(x=xX,edplan)),X,factor):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 18 "degree(edplanX,X);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "for i from 0 to 6 do ED[i]:=factor(coeff(edplanX,X,i)): od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "seq(nops(ED[i]),i=0..6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)\"\"$\"\"\"\"#AF$\"#CF$F#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "The equations corresponding to odd powers of X are trivia l, as they should" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "ED[1]; ED[3]; ED[5];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "So we are left with 4 equations . " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "We now plug in this system \+ the expressions of R1n, P2n, Q0n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Q0nR1n;eqM1s; eqM1biss;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$Q0nG6#%\"tG,$*&#\"\"\"\"\"%F+*(,(*(\"\")F+%%betaGF+F'F+F+*& \"#;F+F'F+F+F+!\"\"F+,(-%$R1nGF&F+F1F4\"\"#F+F+,&F8F+F1F+F4F+F4" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,***\"#7\"\"\"%%betaGF&,&\"\"#F&F'F&F& -%\"SG6#%\"tGF&F&*&F)F&-%$Q0nGF,F&F&-%$P2nGF,!\"\"*(\"\"%F&,&F'F&F5F&F &F-F&F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(**\"\"#\"\"\"%%betaGF&,&F %F&F'F&F&-%%diffG6$-%\"SG6#%\"tGF/F&!\"\"\"\"%F&-%$R1nGF.F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "factor(eval(subs(isolate(eqM 1s,P2n(t)),Q0nR1n,isolate(eqM1biss,R1n(t)),ED[6])));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "ed4:=collect(op(3,factor(eval(subs(isolate(eqM1s,P2n(t)),Q0nR1n,isola te(eqM1biss,R1n(t)),ED[4])))),[diff(S(t),$(t,2)),diff(S(t),t)],facto r);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$ed4G,0*&,&**\"\"#\"\"\")%%be taGF)F*),(*(\"\")F*F,F*%\"tGF*F**&\"#;F*F1F*F*F*!\"\"F)F*-%%diffG6$-% \"SG6#F1F1F*F**(F.F*F,F*,,*(\"#[F*F+F*F8F*F**(\"#'*F*F,F*F8F*F**(F0F*F ,F*F1F*F4*&F>F*F1F*F4F*F4F*F4F*-F66$F8-%\"$G6$F1F)F*F**,F0F*F+F*,&F)F* F,F*F*F.F*)F5F)F*F**,\"\"%F*F,F*,&F,F*\"\"'F*F*F.F*F5F*F4*(\"#kF*F,F*F 1F*F**&FLF*-F66$-%$P0nGF:F1F*F4*&\"$G\"F*F1F*F*F0F4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "as written in the paper" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 208 "eqpap:=-4*(diff(P0n(t),t))+(diff(S(t),$(t,2 )))*(2*beta^2*p^2*(diff(S(t),t))-p*beta*(48*alpha*beta*S(t)-8*beta*t-4 8*t-1))+8*beta^2*alpha*p*(diff(S(t),t))^2-4*beta*(beta+6)*p*(diff(S(t) ,t))+64*beta*t+128*t-8;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&eqpapG,0 *&\"\"%\"\"\"-%%diffG6$-%$P0nG6#%\"tGF/F(!\"\"*&-F*6$-%\"SGF.-%\"$G6$F /\"\"#F(,&**F9F()%%betaGF9F()%\"pGF9F(-F*6$F4F/F(F(*(F?F(F=F(,***\"#[F (%&alphaGF(F=F(F4F(F(*(\"\")F(F=F(F/F(F0*&FEF(F/F(F0F(F0F(F0F(F(*,FHF( F " 0 "" {MPLTEXT 1 0 57 "factor(ed4/subs(alpha=beta+2,p=(8*beta*t+16*t-1),eqpap));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "ed2:=c ollect(factor(eval(subs(isolate(eqM1s,P2n(t)),Q0nR1n,isolate(eqM1biss, R1n(t)),ED[2]))),[diff(S(t),$(t,2)),diff(S(t),t),(diff(P0n(t),t))],f actor);;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$ed2G,>*&,&*,\"\"#\"\"\" )%%betaGF)F*,&F,F*F)!\"\"F*),(*(\"\")F*F,F*%\"tGF*F**&\"#;F*F3F*F*F*F. \"\"$F*-%%diffG6$-%\"SG6#F3F3F*F**(F0F*F,F*,B**\"$%QF*F:F*)F,\"\"%F*F3 F*F***\"$o(F*F:F*)F,F6F*F3F*F**(\"#kF*FEF*)F3F)F*F.*(\"#[F*F:F*FEF*F.* *\"%O:F*F:F*F+F*F3F*F.*(F@F*F+F*FHF*F.**\"%sIF*F:F*F,F*F3F*F.*(\"$c#F* F,F*FHF*F**(\"$#>F*F,F*F:F*F**(F5F*-%$P0nGF " 0 "" {MPLTEXT 1 0 152 "ed0:=collect(op(3,fact or(eval(subs(isolate(eqM1s,P2n(t)),Q0nR1n,isolate(eqM1biss,R1n(t)),ED[ 0])))),[(diff(P0n(t),t)),P0n,(diff(S(t),$(t,2)))],factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ed0G,&*(,(*(\"\")\"\"\"%%betaGF*%\"tGF*F* *&\"#;F*F,F*F*F*!\"\"F*,&*(\"\"#F*F+F*-%%diffG6$-%\"SG6#F,F,F*F*F*F/F* -F46$-%$P0nGF8F,F*F/*&,***\"\"%F*F'F*F+F*-F46$F6-%\"$G6$F,F2F*F**(F.F* )F+F2F*F3F*F**(\"#KF*F+F*F3F*F*FIF/F*F;F*F*" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 23 "as written in the paper" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "eqpap:=4*(p*beta*(diff(S(t),$(t,2)))+4*alpha*beta* (diff(S(t),t))-8)*P0n(t)-p*(2*beta*diff(S(t),t)-1)*diff(P0n(t),t);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&eqpapG,&*(\"\"%\"\"\",(*(%\"pGF(%%b etaGF(-%%diffG6$-%\"SG6#%\"tG-%\"$G6$F3\"\"#F(F(**F'F(%&alphaGF(F,F(-F .6$F0F3F(F(\"\")!\"\"F(-%$P0nGF2F(F(*(F+F(,&*(F7F(F,F(F:F(F(F(F=F(-F.6 $F>F3F(F=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "factor(ed0/sub s(alpha=beta+2,p=(8*beta*t+16*t-1),eqpap));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "The three equations are linearly dependent" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "coeff(ed0,(di ff(P0n(t),t)));coeff(ed2,(diff(P0n(t),t)));coeff(ed4,(diff(P0n(t),t))) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,(*(\"\")\"\"\"%%betaGF(%\"tG F(F(*&\"#;F(F*F(F(F(!\"\"F(,&*(\"\"#F(F)F(-%%diffG6$-%\"SG6#F*F*F(F(F( F-F(F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*,\"\")\"\"\"%%betaGF&,&\" \"#F&F'F&F&,(*(F%F&F'F&%\"tGF&F&*&\"#;F&F,F&F&F&!\"\"F&-%%diffG6$-%\"S G6#F,F,F&F&*(F%F&F*F&F'F&F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"%" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 110 "Taking a vanishing linear combination (careful with th e signs, which may vary from one execution to the other)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "factor((ed2+4*(2+beta)*ed0)/ed4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%%betaG\"\"\"\"\"#!\"\"F&,(*(\" \")F&F%F&%\"tGF&F&*&\"#;F&F,F&F&F&F(F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "So we work with ed0 and ed4 (the smallest two). We elimin ate dP0n: this gives an expression of P0n" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 125 "collect(factor(eval(subs(isolate(ed4,(diff(P0n(t), t))),ed0))),[(diff(S(t),$(t,2))),(diff(S(t),t)),P0n],distributed,fac tor);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,6*&\"\"#\"\"\"),(*(\"\")F&%% betaGF&%\"tGF&F&*&\"#;F&F,F&F&F&!\"\"F%F&F&*,\"\"%F&)F+\"\"$F&,&F%F&F+ F&F&F'F&)-%%diffG6$-%\"SG6#F,F,F3F&F/*,F1F&)F+F%F&,&F+F&F1F&F&F'F&)F6F %F&F&*,F=F&F'F&,**(\"#CF&F=F&F9F&F&*(\"#[F&F+F&F9F&F&*&F.F&F,F&F/F&F/F &-F76$F9-%\"$G6$F,F%F&F6F&F&**F2F&)F(F3F&FGF&F?F&F/*,F.F&F4F&F+F&-%$P0 nGF;F&F6F&F&*,F1F&F(F&F+F&FOF&FGF&F&**F+F&,&F+F&\"#5F&F&F'F&F6F&F/*&#F &F1F&**F+F&F'F&,,*(FEF&F=F&F9F&F&*(\"#'*F&F+F&F9F&F&*(F*F&F+F&F,F&F/*& FEF&F,F&F/F&F/F&FGF&F&F/*&\"#KF&FOF&F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "We differentiate this expression, and compare it to the e xpression coming from ed4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "res:=numer(factor(diff(solve(%,P0n(t)),t)-solve(ed4,(diff(P0n(t),t ))))):nops(res);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "op(3,res);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*(\"\")\"\"\"%%betaGF&%\"tGF&F&*&\"#;F&F(F&F&F&!\"\" " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 108 "edS:=collect(op(4,res),[(diff(S(t),$(t,3))) ,(diff(S(t),$(t,2))),(diff(S(t),t)),M1t],distributed,factor);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%$edSG,B\"#K\"\"\"**\"#;F')%%betaG\" \"$F'),&\"\"#F'F+F'F,F')-%%diffG6$-%\"SG6#%\"t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ioF'F+F'F4F'F'*(F)F'F+F'F7F'F'F+F?*&F&F'F7F'F'FinF?F'FMF' F1F'F'*,F/F'F+F',6*(FRF'F4F')F+F8F'F'*(\"$)GF'F4F'F*F'F'*(F=F'F*F'F7F' F?*(\"$w&F'FGF'F4F'F'*(\"$7\"F'FGF'F7F'F?*(\"$%QF'F+F'F4F'F'*$FGF'F?*( \"$W&F'F+F'F7F'F?*&\"$/(F'F7F'F?\"#GF'F'F@F'F1F'F'*0F/F'F+F',&F+F'FinF 'F'F;F'FfnF'F\\oF'F1F'F'*&,(*&F&F'FGF'F?*&\"$g\"F'F+F'F?\"#kF?F'F1F'F' **FGF'FLF'FfnF')F@F,F'F'*,F/F'F;F'F+F',0*(F,F'F4F'F*F'F'*(FgoF'FGF'F4F 'F?*&FGF'F7F'F?*(\"#gF'F+F'F4F'F?*(F/F'F+F'F7F'F?*&FgoF'F7F'F'F'F'F'FM F'F'*&,0*(FTF'F4F'F*F'F?*(FipF'FGF'F4F'F?*(F&F'FGF'F7F'F'*(\"$?$F'F+F' F7F'F'*&F8F'F+F'F'*&\"$c#F'F7F'F'F)F?F'F@F'F'**F8F'F;F'FfnF'F\\oF'F?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Let us check" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "Sser:=proc(n)\nsubs(q=beta^2,nu=1+beta,w= 1/beta,y=1,t^2*beta*Mser(n)): end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "n:=9:normal(series(subs(S(t)=Sser(n+1),edS),t,n+1)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+%%\"tG-%\"OG6#\"\"\"\"#5" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "edformal:=collect(subs(diff (S(t),$(t,3))=T,diff(S(t),$(t,2))=Z,diff(S(t),t)=Y,S(t)=X,edS),[Z, T],factor);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)edformalG,,**)%%beta G\"\"#\"\"\"),(*(\"\")F*F(F*%\"tGF*F**&\"#;F*F/F*F*F*!\"\"F)F*,**(\"\" $F*%\"XGF*F'F*F**(\"\"'F*F6F*F(F*F**&F(F*F/F*F2*&\"\"%F*F/F*F2F*)%\"ZG F5F*F***F,F*F(F*,H**F1F*)%\"YGF)F*)F(F;F*F/F*F2**\"#CF*F6F*FBF*FCF*F** *\"#kF*FAF*)F(F5F*F/F*F2**\"#'*F*F6F*FBF*FHF*F**(F)F*FAF*FHF*F***FGF*F AF*F'F*F/F*F2**FJF*F6F*FBF*F'F*F**(F8F*F6F*FHF*F**(F;F*FAF*F'F*F***F1F *FBF*F'F*F/F*F**(\"#[F*F6F*F'F*F2*&FBF*F'F*F2**\"#KF*FBF*F(F*F/F*F**(F )F*F'F*F/F*F2*(\"$?\"F*F6F*F(F*F2*(F8F*FBF*F(F*F2*(F;F*F(F*F/F*F2*&FRF *F/F*F*F)F*F*)F=F)F*F***F)F*,(*&FBF*F'F*F**(F)F*FBF*F(F*F*F)F2F*,J**F1 F*FAF*FCF*F/F*F***FRF*F6F*FBF*FCF*F***FGF*FAF*FHF*F/F*F***\"$#>F*F6F*F BF*FHF*F**(F)F*FAF*FHF*F2**FGF*FAF*F'F*F/F*F***F.F*FBF*FHF*F/F*F2**Fo F*F6F*FBF*F'F*F**(FRF*F6F*FHF*F2*(F;F*FAF*F'F*F2**\"$g\"F*FBF*F'F*F/F* F2*(FJF*F6F*F'F*F2*&FBF*F'F*F2**\"$)GF*FBF*F(F*F/F*F2*(F.F*F'F*F/F*F** (\"#5F*FBF*F(F*F**(\"#!)F*F(F*F/F*F*F(F**&FGF*F/F*F*F;F2F*F=F*F2*.F)F* FhnF*,&*(F)F*FBF*F(F*F*F*F2F*F,F*F3F*%\"TGF*F2**F.F*FdpF*,(*&FBF*F(F*F **&F)F*FBF*F*F*F2F*)FhnF)F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 " The version in the paper" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 316 "eqpap:=2*(Y*a*b-2)*(2*Y*b-1)*p*(3*X*a*b-b*t-4*t)*T-b^2*p^2*(3*X*a *b-b*t-4*t)*Z^3\n-p*b*(-2*b^2*a*p*Y^2+b*(16*a*t-b-6)*Y+24*b^2*a^2*X*Y+ 6*a*b*(b-10)*X-2*a*b*t+48*t+2)*Z^2\n+2*(Y*b*a-2)*(2*b^2*a*p*Y^2-b*(8*b ^2*t+160*b*t+b+288*t-10)*Y+48*a^2*b^2*X*Y-48*a*b^2*X\n+8*b^2*t+80*b*t+ b+64*t-4)*Z-8*(2*Y*b-1)*(a*Y-1)*(a*b*Y-2)^2;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&eqpapG,,*.\"\"#\"\"\",&*(%\"YGF(%\"aGF(%\"bGF(F(F'! \"\"F(,&*(F'F(F+F(F-F(F(F(F.F(%\"pGF(,(**\"\"$F(%\"XGF(F,F(F-F(F(*&F-F (%\"tGF(F.*&\"\"%F(F7F(F.F(%\"TGF(F(**)F-F'F()F1F'F(F2F()%\"ZGF4F(F.** F1F(F-F(,0*,F'F(F " 0 "" {MPLTEXT 1 0 77 "collect(factor(edformal/subs(p=8*a*t-1,a=beta+2,b=bet a,eqpap)),[Z,T],factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "The degree" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 " subs(T=T*X,Z=Z*X,Y=Y*X,edformal): degree(%,X);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Initial c onditions" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "unassign(s):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Sserf:=proc(n)\nadd(s[i]* t^i,i=2..n): end: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "n:=0: factor(series(subs(S(t)=Sserf(n+4),edS),t,n+1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+'%\"tG,$*(\"\")\"\"\",&&%\"sG6#\"\"#F(F(!\"\"F(,&*&%%b etaGF(F*F(F(\"\"%F(F(F.\"\"!-%\"OG6#F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "s[2]:=1:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "From \+ now on we can loop" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "n:=n+ 1:factor(series(subs(S(t)=Sserf(n+4),edS),t,n+1));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#+'%\"tG,$*(\"%35\"\"\",&%%betaGF(\"\"%F(F(,0*&\"%W8F( )F*\"\"&F(F(*&\"%U!*F()F*F+F(F(*&\"&gT#F()F*\"\"$F(F(*&\"&q?$F()F*\"\" #F(F(*&\"&_6#F(F*F(F(&%\"sG6#\"\"(!\"\"\"%WbF(F(F(F0-%\"OG6#F(\"\"'" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "s[n+2]:=solve(coeff(%,t,n) ,s[n+2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"sG6#\"\"(,.*&\"%W8\" \"\")%%betaG\"\"&F+F+*&\"%U!*F+)F-\"\"%F+F+*&\"&gT#F+)F-\"\"$F+F+*&\"& q?$F+)F-\"\"#F+F+*&\"&_6#F+F-F+F+\"%WbF+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Sserf(n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%\"t G\"\"#\"\"\"F(*&,&*&F'F(%%betaGF(F(F'F(F()F&\"\"$F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{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 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{MARK "18" 0 }{VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }