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2 2 2 0 0 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 215 257 "In this Maple file we c ompute the Hamiltonian evolutions for the second element of the Painle v\351 1 hierarchy and we obtain the Lax matrices in the geometric gaug e in terms of the (q_i,p_i) Darboux coordinates. Formulas are compared with the theoretical ones." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 222 46 "N otation: The general deformation operator is " }{TEXT 222 56 "\\mathca l\{L\}=\\hbar (alpha7*\\partial_\{t_\{\\infty^\{(1)\},7\} +" }{TEXT 222 75 "alpha6*\\partial_\{t_\{\\infty^\{(1)\},6\}+... alpha1*\\partia l_\{t_\{\\infty^\{(1)\},1\}" }{TEXT 222 1 ")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart:\n" }{MPLTEXT 1 0 21 "with(LinearAlgebra): \n" }{MPLTEXT 1 0 11 "rinfty:=4:\n" }{MPLTEXT 1 0 13 "g:=rinfty-2:\n" }{MPLTEXT 1 0 21 "tinfty27:=-tinfty17:\n" }{MPLTEXT 1 0 21 "tinfty25:= -tinfty15:\n" }{MPLTEXT 1 0 21 "tinfty23:=-tinfty13:\n" }{MPLTEXT 1 0 21 "tinfty21:=-tinfty11:\n" }{MPLTEXT 1 0 21 "tinfty20:=-tinfty10:\n" }{MPLTEXT 1 0 20 "tinfty26:=tinfty16:\n" }{MPLTEXT 1 0 20 "tinfty24:=t infty14:\n" }{MPLTEXT 1 0 20 "tinfty22:=tinfty12:\n" }{MPLTEXT 1 0 13 "tinfty10:=0:\n" }{MPLTEXT 1 0 23 "Pinfty01 := -tinfty12;\n" }{MPLTEXT 1 0 23 "Pinfty11 := -tinfty14;\n" }{MPLTEXT 1 0 23 "Pinfty21 := -tinf ty16;\n" }{MPLTEXT 1 0 103 "Pinfty22 := -(1/2)*tinfty11*tinfty17+(1/2) *tinfty12*tinfty16-(1/2)*tinfty13*tinfty15+(1/4)*tinfty14^2;\n" } {MPLTEXT 1 0 79 "Pinfty32 := -(1/2)*tinfty13*tinfty17+(1/2)*tinfty14*t infty16-(1/4)*tinfty15^2;\n" }{MPLTEXT 1 0 55 "Pinfty42 := -(1/2)*tinf ty15*tinfty17+(1/4)*tinfty16^2;\n" }{MPLTEXT 1 0 31 "Pinfty52 := -(1/4 )*tinfty17^2;\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 42 "P1:=x-> Pinfty 01+Pinfty11*x+Pinfty21*x^2;\n" }{MPLTEXT 1 0 81 "P2:=x-> Pinfty02+Pinf ty12*x+Pinfty22*x^2+Pinfty32*x^3+Pinfty42*x^4+Pinfty52*x^5;\n" } {MPLTEXT 1 0 69 "tdP2:=unapply(Pinfty22*x^2+Pinfty32*x^3+Pinfty42*x^4+ Pinfty52*x^5,x);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 53 "dP1dlambda:=unapply(diff(P1(lambda),lambda),lambda):\n" } {MPLTEXT 1 0 52 "dP2dlambda:=unapply(diff(P2(lambda),lambda),lambda):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 56 "dtdP2dlambda:=unapply(diff(tdP 2(lambda),lambda),lambda):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 58 "c3:=-(7*alpha6*tinfty17-6*alpha7*tinfty16)/(42*tinft y17):\n" }{MPLTEXT 1 0 2 "c2" }{MPLTEXT 1 0 128 ":=-(35*alpha4*tinfty1 7^2-28*alpha5*tinfty16*tinfty17-20*alpha7*tinfty14*tinfty17+20*alpha7* tinfty15*tinfty16)/(140*tinfty17^2):\n" }{MPLTEXT 1 0 2 "c1" }{MPLTEXT 1 0 276 ":=-(105*alpha2*tinfty17^3-70*alpha3*tinfty16*tinfty17^2-42*a lpha5*tinfty14*tinfty17^2+42*alpha5*tinfty15*tinfty16*tinfty17-30*alph a7*tinfty12*tinfty17^2+30*alpha7*tinfty13*tinfty16*tinfty17+30*alpha7* tinfty14*tinfty15*tinfty17-30*alpha7*tinfty15^2*tinfty16)/(210*tinfty1 7^3):\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 33 "nuMoins1:=2*alpha7/(7* tinfty17):\n" }{MPLTEXT 1 0 64 "nu0:=(2*(7*alpha5*tinfty17-5*alpha7*ti nfty15))/(35*tinfty17^2):\n" }{MPLTEXT 1 0 119 "nu1:=-(2*(tinfty13*tin fty17-tinfty15^2))*alpha7/(7*tinfty17^3)-2*tinfty15*alpha5/(5*tinfty17 ^2)+2*alpha3/(3*tinfty17):\n" }{MPLTEXT 1 0 203 "nu2:=-(2*(tinfty11*ti nfty17^2-2*tinfty13*tinfty15*tinfty17+tinfty15^3))*alpha7/(7*tinfty17^ 4)-(2*(tinfty13*tinfty17-tinfty15^2))*alpha5/(5*tinfty17^3)-2*tinfty15 *alpha3/(3*tinfty17^2)+2*alpha1/tinfty17:\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 359 "mu1:=2*(-35*alpha3*q2*tinfty17^3+21*alpha5*q2*tinfty 15*tinfty17^2+15*alpha7*q2*tinfty13*tinfty17^2-15*alpha7*q2*tinfty15^2 *tinfty17+105*alpha1*tinfty17^3-35*alpha3*tinfty15*tinfty17^2-21*alpha 5*tinfty13*tinfty17^2+21*alpha5*tinfty15^2*tinfty17-15*alpha7*tinfty11 *tinfty17^2+30*alpha7*tinfty13*tinfty15*tinfty17-15*alpha7*tinfty15^3) /(105*tinfty17^4*(q1-q2)):\n" }{MPLTEXT 1 0 362 "mu2:=-1/(105*tinfty17 ^4*(q1-q2))* 2*(-35*alpha3*q1*tinfty17^3+21*alpha5*q1*tinfty15*tinfty1 7^2+15*alpha7*q1*tinfty13*tinfty17^2-15*alpha7*q1*tinfty15^2*tinfty17+ 105*alpha1*tinfty17^3-35*alpha3*tinfty15*tinfty17^2-21*alpha5*tinfty13 *tinfty17^2+21*alpha5*tinfty15^2*tinfty17-15*alpha7*tinfty11*tinfty17^ 2+30*alpha7*tinfty13*tinfty15*tinfty17-15*alpha7*tinfty15^3):" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 53 "dP1dlambda: =unapply(diff(P1(lambda),lambda),lambda):\n" }{MPLTEXT 1 0 53 "dP2dlam bda:=unapply(diff(P2(lambda),lambda),lambda):\n" }{MPLTEXT 1 0 18 "L:= Matrix(2,2,0):\n" }{MPLTEXT 1 0 11 "L[1,1]:=0:\n" }{MPLTEXT 1 0 11 "L[ 1,2]:=1:\n" }{MPLTEXT 1 0 8 "L[2,1]:=" }{MPLTEXT 1 0 48 "-tdP2(lambda) +C1*lambda+C0 - h*p1/(lambda-q1)-h*" }{MPLTEXT 1 0 14 "p2/(lambda-q2)" }{MPLTEXT 1 0 2 ":\n" }{MPLTEXT 1 0 9 "L[2,2]:= " }{MPLTEXT 1 0 26 "P 1(lambda) +h/(lambda-q1)" }{MPLTEXT 1 0 14 "+h/(lambda-q2)" }{MPLTEXT 1 0 3 " :\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 18 "A:=Matrix(2,2,0): \n" }{MPLTEXT 1 0 8 "A[1,1]:=" }{MPLTEXT 1 0 12 "c3*lambda^3+" } {MPLTEXT 1 0 60 "c2*lambda^2+c1*lambda+c0+ rho1/(lambda-q1)+ rho2/(lam bda-q2)" }{MPLTEXT 1 0 2 ":\n" }{MPLTEXT 1 0 8 "A[1,2]:=" }{MPLTEXT 1 0 53 "nuMoins1*lambda+nu0+ mu1/(lambda-q1)+ mu2/(lambda-q2)" }{MPLTEXT 1 0 2 ":\n" }{MPLTEXT 1 0 22 "A[2,1]:=AA21(lambda):\n" }{MPLTEXT 1 0 22 "A[2,2]:=AA22(lambda):\n" }{MPLTEXT 1 0 26 "dAdlambda:=Matrix(2,2,0 ):\n" }{MPLTEXT 1 0 88 "for i from 1 to 2 do for j from 1 to 2 do dAdl ambda[i,j]:=diff(A[i,j],lambda): od: od:\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 3 "L:\n" }{MPLTEXT 1 0 3 "A:\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 69 "Q2:=unapply( -p1*(lambda-q2)/(q1-q2)-p2*(lambda-q1)/( q2-q1),lambda);\n" }{MPLTEXT 1 0 18 "simplify(Q2(q1));\n" }{MPLTEXT 1 0 18 "simplify(Q2(q2));\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 18 "J:=M atrix(2,2,0):\n" }{MPLTEXT 1 0 11 "J[1,1]:=1:\n" }{MPLTEXT 1 0 11 "J[1 ,2]:=0:\n" }{MPLTEXT 1 0 8 "J[2,1]:=" }{MPLTEXT 1 0 10 "Q2(lambda)" } {MPLTEXT 1 0 12 "/(lambda-q1)" }{MPLTEXT 1 0 12 "/(lambda-q2)" } {MPLTEXT 1 0 2 ":\n" }{MPLTEXT 1 0 22 "J[2,2]:=1/(lambda-q1)/" } {MPLTEXT 1 0 11 "(lambda-q2)" }{MPLTEXT 1 0 2 ":\n" }{MPLTEXT 1 0 26 " dJdlambda:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 87 "for i from 1 to 2 do fo r j from 1 to 2 do dJdlambda[i,j]:=diff(J[i,j],lambda): od: od:\n" } {MPLTEXT 1 0 3 "J:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 79 "Lcheck:=s implify(Multiply(Multiply(J,L),J^(-1))+h*Multiply(dJdlambda,J^(-1))): \n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 62 "Multiply(Multiply(J,A),J^(-1))[1,1]-A[1,1]+Q2(lambda)*A[1,2]);" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty01G6\",$I)tinfty12GF$!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty11G6\",$I)tinfty14GF$!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty21G6\",$I)tinfty16GF$!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty22G6\",**&I)tinfty11GF$\"\"\"I) tinfty17GF$F(#!\"\"\"\"#*&I)tinfty12GF$F(I)tinfty16GF$F(#F(F,*&I)tinft y13GF$F(I)tinfty15GF$F(F**$I)tinfty14GF$F,#F(\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty32G6\",(*&I)tinfty13GF$\"\"\"I)tinfty17GF$F(#! \"\"\"\"#*&I)tinfty14GF$F(I)tinfty16GF$F(#F(F,*$I)tinfty15GF$F,#F+\"\" %" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty42G6\",&*&I)tinfty15GF$\" \"\"I)tinfty17GF$F(#!\"\"\"\"#*$I)tinfty16GF$F,#F(\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty52G6\",$*$I)tinfty17GF$\"\"##!\"\"\"\"%" } }{PARA 11 "" 1 "" {XPPMATH 20 ">I#P1G6\"f*6#I\"xGF$F$6$I)operatorGF$I& arrowGF$F$,(I)Pinfty01GF$\"\"\"*&I)Pinfty11GF$F-9$F-F-*&I)Pinfty21GF$F -F0\"\"#F-F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#P2G6\"f*6#I\"xGF$ F$6$I)operatorGF$I&arrowGF$F$,.I)Pinfty02GF$\"\"\"*&I)Pinfty12GF$F-9$F -F-*&I)Pinfty22GF$F-F0\"\"#F-*&I)Pinfty32GF$F-F0\"\"$F-*&I)Pinfty42GF$ F-F0\"\"%F-*&I)Pinfty52GF$F-F0\"\"&F-F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%tdP2G6\"f*6#I\"xGF$F$6$I)operatorGF$I&arrowGF$F$,**&9$ \"\"&I)tinfty17GF$\"\"##!\"\"\"\"%*&F-F3,&*&I)tinfty15GF$\"\"\"F/F8#F2 F0*$I)tinfty16GF$F0#F8F3F8F8*&F-\"\"$,(*&I)tinfty13GF$F8F/F8F9*&I)tinf ty14GF$F8F;F8#F8F0*$F7F0F1F8F8*&F-F0,**&I)tinfty11GF$F8F/F8F9*&I)tinft y12GF$F8F;F8FD*&FAF8F7F8F9*$FCF0FI#Q2G6\"f*6#I'lambdaGF$F$6$I)operatorGF$I&arrowGF$F$,&*( I#p1GF$\"\"\",&9$F.I#q2GF$!\"\"F.,&I#q1GF$F.F1F2F2F2*(I#p2GF$F.,&F0F.F 4F2F.,&F1F.F4F2F2F2F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$I#p1G6\"! \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$I#p2G6\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 222 67 "The co mpatibility equation reads \\mathcal\{L\}L=h\\partial_x A+[A,L]. " } {TEXT 222 107 "Because the first line of L is trivial (0,1) we may obt ain A[2,1] and A[2,2] to complete the knowledge of A" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "CurlyL:=h*dAdlambda+(Multiply(A,L)-Multip ly(L,A)):\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 22 "Entry11:=CurlyL[1, 1]:\n" }{MPLTEXT 1 0 22 "Entry12:=CurlyL[1,2]:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 53 "AA21:=unapply(solve(Entry11=0,AA21(lambda)),lambda) :\n" }{MPLTEXT 1 0 40 "AA21bis:=h*dAdlambda[1,1]+A[1,2]*L[2,1]:" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 32 "simplify(AA 21(lambda)-AA21bis);\n" }{MPLTEXT 1 0 53 "AA22:=unapply(solve(Entry12= 0,AA22(lambda)),lambda):\n" }{MPLTEXT 1 0 48 "AA22bis:=h*dAdlambda[1,2 ]+A[1,1]+A[1,2]*L[2,2]:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 32 "simp lify(AA22(lambda)-AA22bis);\n" }{MPLTEXT 1 0 19 "simplify(Entry11);\n" }{MPLTEXT 1 0 19 "simplify(Entry12);\n" }{MPLTEXT 1 0 50 "CurlyL:=h*d Adlambda+(Multiply(A,L)-Multiply(L,A)):" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 222 65 "Let us now compute the action of \\mathc al\{L\} on L[2,2] and L[2,1]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Entry22:=simplify(CurlyL[2,2]):\n" }{MPLTEXT 1 0 80 "Entry22Term LambdaMoinsq1Cube:=factor(residue(Entry22*(lambda-q1)^2,lambda=q1));\n " }{MPLTEXT 1 0 80 "Entry22TermLambdaMoinsq1Square:=factor(residue(Ent ry22*(lambda-q1),lambda=q1)):\n" }{MPLTEXT 1 0 62 "Entry22TermLambdaMo insq1:=factor(residue(Entry22,lambda=q1)):\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 80 "Entry22TermLambdaMoinsq2Cube:=factor(residue(Entry22* (lambda-q2)^2,lambda=q2));\n" }{MPLTEXT 1 0 80 "Entry22TermLambdaMoins q2Square:=factor(residue(Entry22*(lambda-q2),lambda=q2)):\n" }{MPLTEXT 1 0 62 "Entry22TermLambdaMoinsq2:=factor(residue(Entry22,lambda=q2)): \n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 77 "Entry22TermLambdaInfty4:=fa ctor(-residue(Entry22/lambda^5,lambda=infinity));\n" }{MPLTEXT 1 0 77 "Entry22TermLambdaInfty3:=factor(-residue(Entry22/lambda^4,lambda=infi nity));\n" }{MPLTEXT 1 0 77 "Entry22TermLambdaInfty2:=factor(-residue( Entry22/lambda^3,lambda=infinity));\n" }{MPLTEXT 1 0 77 "Entry22TermLa mbdaInfty1:=factor(-residue(Entry22/lambda^2,lambda=infinity));\n" } {MPLTEXT 1 0 75 "Entry22TermLambdaInfty0:=factor(-residue(Entry22/lamb da,lambda=infinity));\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 101 "simpl ify( Entry22-(Entry22TermLambdaMoinsq1Square/(lambda-q1)^2+Entry22Term LambdaMoinsq1/(lambda-q1)\n" }{MPLTEXT 1 0 1 "+" }{MPLTEXT 1 0 81 "Ent ry22TermLambdaMoinsq2Square/(lambda-q2)^2+Entry22TermLambdaMoinsq2/(la mbda-q2)" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 159 "+Entry22TermLambdaIn fty0+Entry22TermLambdaInfty1*lambda+Entry22TermLambdaInfty2*lambda^2+E ntry22TermLambdaInfty3*lambda^3+Entry22TermLambdaInfty4*lambda^4) );\n " }{MPLTEXT 1 0 7 "L[2,2];" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I=Entry22 TermLambdaMoinsq1CubeG6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I=Ent ry22TermLambdaMoinsq2CubeG6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I 8Entry22TermLambdaInfty4G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8 Entry22TermLambdaInfty3G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8E ntry22TermLambdaInfty2G6\",$*&I'alpha6GF$\"\"\"I\"hGF$F(!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry22TermLambdaInfty1G6\",$*&I'alph a4GF$\"\"\"I\"hGF$F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry22T ermLambdaInfty0G6\",$*&I'alpha2GF$\"\"\"I\"hGF$F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ",,*&I)tinfty1 6G6\"\"\"\"I'lambdaGF%\"\"#!\"\"*&I)tinfty14GF%F&F'F&F)I)tinfty12GF%F) *&I\"hGF%F&,&F'F&I#q1GF%F)F)F&*&F.F&,&F'F&I#q2GF%F)F)F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 222 37 "Because the deformation operator is " } {TEXT 222 131 "\\mathcal\{L\}=\\hbar (alpha7*\\partial_\{t_\{\\infty^ \{(1)\},7\} +alpha6*\\partial_\{t_\{\\infty^\{(1)\},6\}+... alpha1*\\p artial_\{t_\{\\infty^\{(1)\},1\}" }{TEXT 222 9 "), we get" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "L22OrderLambda3:=-residue(L[2,2]/la mbda^4,lambda=infinity);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 60 "L22Or derLambda2:=-residue(L[2,2]/lambda^3,lambda=infinity);\n" }{MPLTEXT 1 0 60 "L22OrderLambda1:=-residue(L[2,2]/lambda^2,lambda=infinity);\n" } {MPLTEXT 1 0 60 "L22OrderLambda0:=-residue(L[2,2]/lambda^1,lambda=infi nity);\n" }{MPLTEXT 1 0 325 "Equation1:=factor(simplify(h*(alpha7*diff (L22OrderLambda2,tinfty17)+alpha6*diff(L22OrderLambda2,tinfty16)+alpha 5*diff(L22OrderLambda2,tinfty15)+alpha4*diff(L22OrderLambda2,tinfty14) +alpha3*diff(L22OrderLambda2,tinfty13)+alpha2*diff(L22OrderLambda2,tin fty12)+alpha1*diff(L22OrderLambda2,tinfty11))- Entry22TermLambdaInfty2 ));\n" }{MPLTEXT 1 0 30 "Equation2:=factor(simplify(h*(" }{MPLTEXT 1 0 38 "alpha7*diff(L22OrderLambda1,tinfty17)+" }{MPLTEXT 1 0 37 "alpha6 *diff(L22OrderLambda1,tinfty16)" }{MPLTEXT 1 0 1 "+" }{MPLTEXT 1 0 38 "alpha5*diff(L22OrderLambda1,tinfty15)+" }{MPLTEXT 1 0 6 "alpha4" } {MPLTEXT 1 0 32 "*diff(L22OrderLambda1,tinfty14)+" }{MPLTEXT 1 0 6 "al pha3" }{MPLTEXT 1 0 32 "*diff(L22OrderLambda1,tinfty13)+" }{MPLTEXT 1 0 6 "alpha2" }{MPLTEXT 1 0 32 "*diff(L22OrderLambda1,tinfty12)+" } {MPLTEXT 1 0 6 "alpha1" }{MPLTEXT 1 0 61 "*diff(L22OrderLambda1,tinfty 11))- Entry22TermLambdaInfty1));\n" }{MPLTEXT 1 0 30 "Equation3:=facto r(simplify(h*(" }{MPLTEXT 1 0 38 "alpha7*diff(L22OrderLambda0,tinfty17 )+" }{MPLTEXT 1 0 38 "alpha6*diff(L22OrderLambda0,tinfty16)+" } {MPLTEXT 1 0 76 "alpha5*diff(L22OrderLambda0,tinfty15)+alpha4*diff(L22 OrderLambda0,tinfty14)+" }{MPLTEXT 1 0 6 "alpha3" }{MPLTEXT 1 0 32 "*d iff(L22OrderLambda0,tinfty13)+" }{MPLTEXT 1 0 6 "alpha2" }{MPLTEXT 1 0 32 "*diff(L22OrderLambda0,tinfty12)+" }{MPLTEXT 1 0 6 "alpha1" } {MPLTEXT 1 0 61 "*diff(L22OrderLambda0,tinfty11))- Entry22TermLambdaIn fty0));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0L22OrderLambda3G6\"\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0L22OrderLambda2G6\",$I)tinfty16GF $!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0L22OrderLambda1G6\",$I)tinf ty14GF$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0L22OrderLambda0G6\",$ I)tinfty12GF$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Equation1G6\"\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Equation2G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Equation3G6\"\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 215 146 "We have verified that the computations of c_\{\\infty, 1\}, c_\{\\infty,2\} and c_\{\\infty,3\} are correct. Let us now look \+ at the evolution of q_1 and q_2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "CurlyLq1:=factor(Entry22TermLambdaMoinsq1Square/h):\n" } {MPLTEXT 1 0 51 "CurlyLq2:=factor(Entry22TermLambdaMoinsq2Square/h):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 222 38 "Let us now look at \\mathcal\{ L\}[L[2,1]]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Entry21:=sim plify(CurlyL[2,1]):\n" }{MPLTEXT 1 0 80 "Entry21TermLambdaMoinsq1Cube: =factor(residue(Entry21*(lambda-q1)^2,lambda=q1));\n" }{MPLTEXT 1 0 80 "Entry21TermLambdaMoinsq1Square:=factor(residue(Entry21*(lambda-q1) ,lambda=q1)):\n" }{MPLTEXT 1 0 62 "Entry21TermLambdaMoinsq1:=factor(re sidue(Entry21,lambda=q1)):\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 80 "E ntry21TermLambdaMoinsq2Cube:=factor(residue(Entry21*(lambda-q2)^2,lamb da=q2));\n" }{MPLTEXT 1 0 80 "Entry21TermLambdaMoinsq2Square:=factor(r esidue(Entry21*(lambda-q2),lambda=q2)):\n" }{MPLTEXT 1 0 61 "Entry21Te rmLambdaMoinsq2:=factor(residue(Entry21,lambda=q2)):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 76 "Entry21TermLambdaInfty8:=f actor(-residue(Entry21/lambda^9,lambda=infinity));" }{MPLTEXT 1 0 1 " \n" }{MPLTEXT 1 0 76 "Entry21TermLambdaInfty7:=factor(-residue(Entry21 /lambda^8,lambda=infinity));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 77 "E ntry21TermLambdaInfty6:=factor(-residue(Entry21/lambda^7,lambda=infini ty));\n" }{MPLTEXT 1 0 77 "Entry21TermLambdaInfty5:=factor(-residue(En try21/lambda^6,lambda=infinity));\n" }{MPLTEXT 1 0 77 "Entry21TermLamb daInfty4:=factor(-residue(Entry21/lambda^5,lambda=infinity)):\n" } {MPLTEXT 1 0 77 "Entry21TermLambdaInfty3:=factor(-residue(Entry21/lamb da^4,lambda=infinity)):\n" }{MPLTEXT 1 0 77 "Entry21TermLambdaInfty2:= factor(-residue(Entry21/lambda^3,lambda=infinity)):\n" }{MPLTEXT 1 0 77 "Entry21TermLambdaInfty1:=factor(-residue(Entry21/lambda^2,lambda=i nfinity)):\n" }{MPLTEXT 1 0 75 "Entry21TermLambdaInfty0:=factor(-resid ue(Entry21/lambda,lambda=infinity)):\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 144 "simplify( Entry21-(Entry21TermLambdaMoinsq1Cube/(lambda-q1)^ 3+Entry21TermLambdaMoinsq1Square/(lambda-q1)^2+Entry21TermLambdaMoinsq 1/(lambda-q1)\n" }{MPLTEXT 1 0 1 "+" }{MPLTEXT 1 0 124 "Entry21TermLam bdaMoinsq2Cube/(lambda-q2)^3+Entry21TermLambdaMoinsq2Square/(lambda-q2 )^2+Entry21TermLambdaMoinsq2/(lambda-q2)" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 122 "+Entry21TermLambdaInfty0+Entry21TermLambdaInfty1*lam bda+Entry21TermLambdaInfty2*lambda^2+Entry21TermLambdaInfty3*lambda^3 \n" }{MPLTEXT 1 0 100 "+Entry21TermLambdaInfty4*lambda^4+Entry21TermLa mbdaInfty5*lambda^5+Entry21TermLambdaInfty6*lambda^6\n" }{MPLTEXT 1 0 1 "+" }{MPLTEXT 1 0 32 "Entry21TermLambdaInfty7*lambda^7" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 5 ") );\n" }{MPLTEXT 1 0 8 "L[2,1];\n" } {MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I=Entry21TermLamb daMoinsq1CubeG6\",$**,<**I'alpha3GF$\"\"\"I#p1GF$F*I#q2GF$F*I)tinfty17 GF$\"\"$!#q*,I'alpha5GF$F*F+F*F,F*I)tinfty15GF$F*F-\"\"#\"#U*,I'alpha7 GF$F*F+F*F,F*I)tinfty13GF$F*F-F3\"#I*,F6F*F+F*F,F*F2F3F-F*!#I*(I#q1GF$ F*I%rho1GF$F*F-\"\"%\"$0\"*(F,F*F=F*F-F>!$0\"*(I'alpha1GF$F*F+F*F-F.\" $5#**F)F*F+F*F2F*F-F3F/**F1F*F+F*F7F*F-F3!#U**F1F*F+F*F2F3F-F*F4**F6F* F+F*I)tinfty11GF$F*F-F3F:*,F6F*F+F*F7F*F2F*F-F*\"#g*(F6F*F+F*F2F.F:F*I \"hGF$F3,&FI=Entry21TermLambdaMoinsq2CubeG6\",$**,<**I'alpha3GF$\"\"\"I#p2GF$F*I #q1GF$F*I)tinfty17GF$\"\"$!#q*,I'alpha5GF$F*F+F*F,F*I)tinfty15GF$F*F- \"\"#\"#U*,I'alpha7GF$F*F+F*F,F*I)tinfty13GF$F*F-F3\"#I*,F6F*F+F*F,F*F 2F3F-F*!#I*(F,F*I%rho2GF$F*F-\"\"%!$0\"*(I#q2GF$F*FI8Entry21TermLambdaInfty8G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry21TermLambdaInfty7G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry21TermLambdaInfty6G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry21TermLambdaInfty5G6\",$*(I'alpha7GF$\"\"\"I)t infty17GF$F(I\"hGF$F(#F(\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ",2*&I)tinfty17G6\"\"\"#I'lambdaGF%\"\" &#\"\"\"\"\"%*&,&*&I)tinfty15GF%F*F$F*#!\"\"F&*$I)tinfty16GF%F&F)F*F'F +F1*&,(*&I)tinfty13GF%F*F$F*F0*&I)tinfty14GF%F*F3F*#F*F&*$F/F&#F1F+F*F '\"\"$F1*&,**&I)tinfty11GF%F*F$F*F0*&I)tinfty12GF%F*F3F*F:*&F7F*F/F*F0 *$F9F&F)F*F'F&F1*&I#C1GF%F*F'F*F*I#C0GF%F**(I\"hGF%F*I#p1GF%F*,&F'F*I# q1GF%F1F1F1*(FJF*I#p2GF%F*,&F'F*I#q2GF%F1F1F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 215 65 "We check that rho_j=-p_j*mu_j cancel the cubic pole a t lambda=q_j" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "rho1:=-p1*m u1:\n" }{MPLTEXT 1 0 15 "rho2:=-p2*mu2:\n" }{MPLTEXT 1 0 40 "simplify( Entry21TermLambdaMoinsq1Cube);\n" }{MPLTEXT 1 0 39 "simplify(Entry21Te rmLambdaMoinsq2Cube);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 215 66 "We check that the value of \\td\{P\}_2 are compatible with the system " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "L21OrderLambda5:=-resid ue(L[2,1]/lambda^6,lambda=infinity);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 60 "L21OrderLambda4:=-residue(L[2,1]/lambda^5,lambda=infinity);\n" }{MPLTEXT 1 0 60 "L21OrderLambda3:=-residue(L[2,1]/lambda^4,lambda=in finity);\n" }{MPLTEXT 1 0 60 "L21OrderLambda2:=-residue(L[2,1]/lambda^ 3,lambda=infinity);\n" }{MPLTEXT 1 0 60 "L21OrderLambda1:=-residue(L[2 ,1]/lambda^2,lambda=infinity);\n" }{MPLTEXT 1 0 60 "L21OrderLambda0:=- residue(L[2,1]/lambda^1,lambda=infinity);\n" }{MPLTEXT 1 0 264 "Equati on1:=factor(simplify(h*(alpha7*diff(L21OrderLambda5,tinfty17)+alpha6*d iff(L21OrderLambda5,tinfty16)+alpha5*diff(L21OrderLambda5,tinfty15)+al pha4*diff(L21OrderLambda5,tinfty14)+alpha3*diff(L21OrderLambda5,tinfty 13)+alpha2*diff(L21OrderLambda5,tinfty12)+alpha1" }{MPLTEXT 1 0 60 "*d iff(L21OrderLambda5,tinfty11))- Entry21TermLambdaInfty5));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 324 "Equation2:=factor(simplify(h*(alpha7*dif f(L21OrderLambda4,tinfty17)+alpha6*diff(L21OrderLambda4,tinfty16)+alph a5*diff(L21OrderLambda4,tinfty15)+alpha4*diff(L21OrderLambda4,tinfty14 )+alpha3*diff(L21OrderLambda4,tinfty13)+alpha2*diff(L21OrderLambda4,ti nfty12)+alpha1*diff(L21OrderLambda4,tinfty11))- Entry21TermLambdaInfty 4));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 30 "Equation3:=factor(simplif y(h*(" }{MPLTEXT 1 0 38 "alpha7*diff(L21OrderLambda3,tinfty17)+" } {MPLTEXT 1 0 38 "alpha6*diff(L21OrderLambda3,tinfty16)+" }{MPLTEXT 1 0 76 "alpha5*diff(L21OrderLambda3,tinfty15)+alpha4*diff(L21OrderLambda 3,tinfty14)+" }{MPLTEXT 1 0 6 "alpha3" }{MPLTEXT 1 0 32 "*diff(L21Orde rLambda3,tinfty13)+" }{MPLTEXT 1 0 6 "alpha2" }{MPLTEXT 1 0 32 "*diff( L21OrderLambda3,tinfty12)+" }{MPLTEXT 1 0 6 "alpha1" }{MPLTEXT 1 0 61 "*diff(L21OrderLambda3,tinfty11))- Entry21TermLambdaInfty3));\n" } {MPLTEXT 1 0 30 "Equation4:=factor(simplify(h*(" }{MPLTEXT 1 0 234 "al pha7*diff(L21OrderLambda2,tinfty17)+alpha6*diff(L21OrderLambda2,tinfty 16)+alpha5*diff(L21OrderLambda2,tinfty15)+alpha4*diff(L21OrderLambda2, tinfty14)+alpha3*diff(L21OrderLambda2,tinfty13)+alpha2*diff(L21OrderLa mbda2,tinfty12)+alpha1" }{MPLTEXT 1 0 60 "*diff(L21OrderLambda2,tinfty 11))- Entry21TermLambdaInfty2));" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0L 21OrderLambda5G6\",$*$I)tinfty17GF$\"\"##\"\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0L21OrderLambda4G6\",&*&I)tinfty15GF$\"\"\"I)tinfty1 7GF$F(#F(\"\"#*$I)tinfty16GF$F+#!\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0L21OrderLambda3G6\",(*&I)tinfty13GF$\"\"\"I)tinfty17GF $F(#F(\"\"#*&I)tinfty14GF$F(I)tinfty16GF$F(#!\"\"F+*$I)tinfty15GF$F+#F (\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0L21OrderLambda2G6\",**&I)ti nfty11GF$\"\"\"I)tinfty17GF$F(#F(\"\"#*&I)tinfty12GF$F(I)tinfty16GF$F( #!\"\"F+*&I)tinfty13GF$F(I)tinfty15GF$F(F**$I)tinfty14GF$F+#F0\"\"%" } }{PARA 11 "" 1 "" {XPPMATH 20 ">I0L21OrderLambda1G6\"I#C1GF$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0L21OrderLambda0G6\"I#C0GF$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Equation1G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Equation2G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Equation3G 6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Equation4G6\"\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "CurlyLp1Fonction:=unapply(-E ntry21TermLambdaMoinsq1/h,C0,C1):\n" }{MPLTEXT 1 0 61 "CurlyLp2Fonctio n:=unapply(-Entry21TermLambdaMoinsq2/h,C0,C1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "Equation5:=simplify(Entry21TermLambdaMoinsq1Squ are-(-p1*h*CurlyLq1)):\n" }{MPLTEXT 1 0 69 "Equation6:=simplify(Entry2 1TermLambdaMoinsq2Square-(-p2*h*CurlyLq2)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 661 "C0:=-(-q1^5*q2*tinfty17^2+q1*q2^5*tinfty17^2-2*q1 ^4*q2*tinfty15*tinfty17+q1^4*q2*tinfty16^2+2*q1*q2^4*tinfty15*tinfty17 -q1*q2^4*tinfty16^2-2*q1^3*q2*tinfty13*tinfty17+2*q1^3*q2*tinfty14*tin fty16-q1^3*q2*tinfty15^2+2*q1*q2^3*tinfty13*tinfty17-2*q1*q2^3*tinfty1 4*tinfty16+q1*q2^3*tinfty15^2+4*p1*q1^2*q2*tinfty16-4*p2*q1*q2^2*tinft y16-2*q1^2*q2*tinfty11*tinfty17+2*q1^2*q2*tinfty12*tinfty16-2*q1^2*q2* tinfty13*tinfty15+q1^2*q2*tinfty14^2+2*q1*q2^2*tinfty11*tinfty17-2*q1* q2^2*tinfty12*tinfty16+2*q1*q2^2*tinfty13*tinfty15-q1*q2^2*tinfty14^2+ 4*p1*q1*q2*tinfty14-4*p2*q1*q2*tinfty14+4*p1^2*q2+4*p1*q2*tinfty12-4*p 2^2*q1-4*p2*q1*tinfty12+4*h*p1-4*h*p2)/(4*(q1-q2)):\n" }{MPLTEXT 1 0 562 "C1:=1/4/(q1-q2)*(-q1^5*tinfty17^2+q2^5*tinfty17^2-2*q1^4*tinfty15 *tinfty17+q1^4*tinfty16^2+2*q2^4*tinfty15*tinfty17-q2^4*tinfty16^2-2*q 1^3*tinfty13*tinfty17+2*q1^3*tinfty14*tinfty16-q1^3*tinfty15^2+2*q2^3* tinfty13*tinfty17-2*q2^3*tinfty14*tinfty16+q2^3*tinfty15^2+4*p1*q1^2*t infty16-4*p2*q2^2*tinfty16-2*q1^2*tinfty11*tinfty17+2*q1^2*tinfty12*ti nfty16-2*q1^2*tinfty13*tinfty15+q1^2*tinfty14^2+2*q2^2*tinfty11*tinfty 17-2*q2^2*tinfty12*tinfty16+2*q2^2*tinfty13*tinfty15-q2^2*tinfty14^2+4 *p1*q1*tinfty14-4*p2*q2*tinfty14+4*p1^2+4*p1*tinfty12-4*p2^2-4*p2*tinf ty12):\n" }{MPLTEXT 1 0 21 "simplify(Equation5);\n" }{MPLTEXT 1 0 20 " simplify(Equation6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 215 69 "We check that C_0 and C_1 are in accordance with the theoretical ones " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Vinfty:=Matrix(g,g,0): \n" }{MPLTEXT 1 0 16 "Vinfty[1,1]:=1:\n" }{MPLTEXT 1 0 16 "Vinfty[1,2] :=1:\n" }{MPLTEXT 1 0 17 "Vinfty[2,1]:=q1:\n" }{MPLTEXT 1 0 17 "Vinfty [2,2]:=q2:\n" }{MPLTEXT 1 0 8 "Vinfty;\n" }{MPLTEXT 1 0 21 "RHSC:=Matr ix(2,2,0):\n" }{MPLTEXT 1 0 54 "RHSC[1,1]:=p1^2-P1(q1)*p1+tdP2(q1)+h*( p2-p1)/(q1-q2):\n" }{MPLTEXT 1 0 53 "RHSC[2,1]:=p2^2-P1(q2)*p2+tdP2(q2 )+h*(p1-p2)/(q2-q1):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 54 "CVectorTh eo:=Multiply((Transpose(Vinfty))^(-1),RHSC):\n" }{MPLTEXT 1 0 8 "C0The o:=" }{MPLTEXT 1 0 18 "CVectorTheo[1,1]:\n" }{MPLTEXT 1 0 26 "C1Theo:= CVectorTheo[2,1]:\n" }{MPLTEXT 1 0 21 "simplify(C0-C0Theo);\n" } {MPLTEXT 1 0 20 "simplify(C1-C1Theo);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6'-I%mrowGF$6 #-I'mtableGF$66-I$mtrGF$6'-I$mtdGF$6(-I#mnGF$6%Q\"1F'/%+foregroundGQ([ 0,0,0]F'/%,mathvariantGQ'normalF'/%)rowalignGQ!F'/%,columnalignGFC/%+g roupalignGFC/%(rowspanGF:/%+columnspanGF:F4FAFDFF-F26'-F56(-I#miGF$6&Q #q1F'/%'italicGQ%trueF'F;/F?Q'italicF'FAFDFFFHFJ-F56(-FQ6&Q#q2F'FTF;FW FAFDFFFHFJFAFDFF/%&alignGQ%axisF'/FBQ)baselineF'/FEQ'centerF'/FGQ'|frl eft|hrF'/%/alignmentscopeGFV/%,columnwidthGQ%autoF'/%&widthGFeo/%+rows pacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowlinesGQ%noneF'/%,colum nlinesGF`p/%&frameGF`p/%-framespacingGQ,0.4em~0.5exF'/%*equalrowsGQ&fa lseF'/%-equalcolumnsGFjp/%-displaystyleGFjp/%%sideGQ&rightF'/%0minlabe lspacingGF]pF;F>/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 215 43 "We now compute the evolution of p_1 and \+ p_2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "CurlyLp1:=factor(sim plify(CurlyLp1Fonction(C0,C1))):\n" }{MPLTEXT 1 0 52 "CurlyLp2:=factor (simplify(CurlyLp2Fonction(C0,C1))):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 215 83 "We check the evolutions of q_1,q_2,p_1,p_2 in accordance with \+ our general formulas:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "Cu rlyLq1Theo:=2*mu1*(p1-1/2*P1(q1))-h*nu0-h*nuMoins1*q1-h*(mu1+mu2)/(q1- q2):\n" }{MPLTEXT 1 0 76 "CurlyLq2Theo:=2*mu2*(p2-1/2*P1(q2))-h*nu0-h* nuMoins1*q2-h*(mu2+mu1)/(q2-q1):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 9 "CurlyLq1-" }{MPLTEXT 1 0 14 "CurlyLq1Th eo);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 23 "CurlyLq2-CurlyLq2Theo);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 145 "CurlyLp1Theo:=h*(mu2+mu1)*(p2-p1)/(q1-q2)^2+mu1*(p 1*dP1dlambda(q1)-dtdP2dlambda(q1)+1*C1*q1^0) +h*nuMoins1*p1+ h*(1*c1*q 1^0+2*c2*q1^1+3*c3*q1^2):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 145 "Cur lyLp2Theo:=h*(mu1+mu2)*(p1-p2)/(q2-q1)^2+mu2*(p2*dP1dlambda(q2)-dtdP2d lambda(q2)+1*C1*q2^0) +h*nuMoins1*p2+ h*(1*c1*q2^0+2*c2*q2^1+3*c3*q2^2 ):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 14 "series(factor(" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 31 "CurlyLp1-CurlyLp1Theo)),p1=0);\n" }{MPLTEXT 1 0 32 "simplify(CurlyLp2-CurlyLp2Theo);" }{MPLTEXT 1 0 1 " \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 215 50 "We check the expression of the general Hamiltonian" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Hamiltonian:=-h/2*(mu1+mu2)*(p1-p2)/(q1-q2)" }{MPLTEXT 1 0 70 "-h/2*(mu2+mu1)*(p2-p1)/(q2-q1)-h*(nu0*(p1+p2)+nuMoin s1*(q1*p1+q2*p2))\n" }{MPLTEXT 1 0 1 "+" }{MPLTEXT 1 0 10 " mu1*(p1^2" }{MPLTEXT 1 0 10 "-p1*P1(q1)" }{MPLTEXT 1 0 10 "+tdP2(q1))" }{MPLTEXT 1 0 1 "+" }{MPLTEXT 1 0 58 " mu2*(p2^2-p2*P1(q2)+tdP2(q2))-h*(c1*q1^1 +c2*q1^2+c3*q1^3+" }{MPLTEXT 1 0 25 "c1*q2^1+c2*q2^2+c3*q2^3):" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 44 "simplify(CurlyLp1-(-diff(Hamilto nian,q1)));\n" }{MPLTEXT 1 0 43 "simplify(CurlyLq1-(diff(Hamiltonian,p 1)));\n" }{MPLTEXT 1 0 44 "simplify(CurlyLp2-(-diff(Hamiltonian,q2))); \n" }{MPLTEXT 1 0 42 "simplify(CurlyLq2-(diff(Hamiltonian,p2)));" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "fact or(simplify(Hamiltonian-(nu1*C0+nu2*C1-h*nu0*(p1+p2)-h*nuMoins1*(q1*p1 +q2*p2)-h*c1*(q1+q2)-h*c2*(q1^2+q2^2)" }{MPLTEXT 1 0 17 "-h*c3*(q1^3+q 2^3)" }{MPLTEXT 1 0 5 " )));" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" } }}{EXCHG {PARA 0 "" 0 "" {TEXT 215 72 "We check the first gauge transf ormation to remove apparent singularities" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "L" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 7 "[1,1];\n" }{MPLTEXT 1 0 1 "L" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 7 "[1,2];\n " }{MPLTEXT 1 0 1 "L" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 7 "[2,2]: \n" }{MPLTEXT 1 0 61 "ProductLambdaMinusq:=unapply((lambda-q1)*(lambda -q2),lambda):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "L" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 21 "11Theo:=-Q2(lambda):\n" }{MPLTEXT 1 0 1 "L " }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 33 "12Theo:=(lambda-q1)*(lambd a-q2):\n" }{MPLTEXT 1 0 1 "L" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 30 "22Theo:=P1(lambda)+Q2(lambda):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "L" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 8 "21Theo:=" }{MPLTEXT 1 0 183 "h*diff(Q2(lambda)/ProductLambdaMinusq(lambda),lambda) +L[2,1] /ProductLambdaMinusq(lambda)-P1(lambda)*Q2(lambda)/ProductLambdaMinusq (lambda)-(Q2(lambda))^2/ProductLambdaMinusq(lambda):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 10 "simplify(L" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 7 "[1,1]-L" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 8 "11Theo);" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 10 "simplify(L" }{MPLTEXT 1 0 5 "che ck" }{MPLTEXT 1 0 7 "[1,2]-L" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 8 "12Theo);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 10 "simplify(L" } {MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 7 "[2,2]-L" }{MPLTEXT 1 0 5 "chec k" }{MPLTEXT 1 0 9 "22Theo);\n" }{MPLTEXT 1 0 10 "simplify(L" } {MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 7 "[2,1]-L" }{MPLTEXT 1 0 5 "chec k" }{MPLTEXT 1 0 8 "21Theo);" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,(*&,& I#p1G6\"\"\"\"I#p2GF'!\"\"F(I'lambdaGF'F(F(*&F&F(I#q2GF'F(F**&F)F(I#q1 GF'F(F(F(,&F/F(F-F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,&I'lambdaG6 \"!\"\"I#q1GF%\"\"\"F(,&F$F&I#q2GF%F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 215 44 "We now check the second gauge transforma tion" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "G:=Matrix(2,2,0):\n " }{MPLTEXT 1 0 11 "G[1,1]:=1:\n" }{MPLTEXT 1 0 11 "G[2,2]:=1:\n" } {MPLTEXT 1 0 22 "G[2,1]:=1/2*tinfty16:\n" }{MPLTEXT 1 0 3 "G;\n" } {MPLTEXT 1 0 26 "dGdlambda:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 87 "for i \+ from 1 to 2 do for j from 1 to 2 do dGdlambda[i,j]:=diff(G[i,j],lambda ): od: od:\n" }{MPLTEXT 1 0 24 "CurlyLG:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 42 "for i from 1 to 2 do for j from 1 to 2 do " }{MPLTEXT 1 0 7 "C urlyLG" }{MPLTEXT 1 0 8 "[i,j]:=\n" }{MPLTEXT 1 0 15 "h*(alpha7*diff(" }{MPLTEXT 1 0 6 "G[i,j]" }{MPLTEXT 1 0 23 ",tinfty17)+alpha6*diff(" } {MPLTEXT 1 0 6 "G[i,j]" }{MPLTEXT 1 0 23 ",tinfty16)+alpha5*diff(" } {MPLTEXT 1 0 6 "G[i,j]" }{MPLTEXT 1 0 23 ",tinfty15)+alpha4*diff(" } {MPLTEXT 1 0 6 "G[i,j]" }{MPLTEXT 1 0 23 ",tinfty14)+alpha3*diff(" } {MPLTEXT 1 0 6 "G[i,j]" }{MPLTEXT 1 0 23 ",tinfty13)+alpha2*diff(" } {MPLTEXT 1 0 6 "G[i,j]" }{MPLTEXT 1 0 23 ",tinfty12)+alpha1*diff(" } {MPLTEXT 1 0 6 "G[i,j]" }{MPLTEXT 1 0 11 ",tinfty11))" }{MPLTEXT 1 0 10 ": od: od:\n" }{MPLTEXT 1 0 11 "dGdlambda;\n" }{MPLTEXT 1 0 9 "Curl yLG;\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 38 "Ltilde:=simplify(Multip ly(Multiply(G,L" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 41 "),G^(-1))+h *Multiply(dGdlambda,G^(-1))):\n" }{MPLTEXT 1 0 1 "A" }{MPLTEXT 1 0 5 " tilde" }{MPLTEXT 1 0 32 ":=simplify(Multiply(Multiply(G,A" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 36 "),G^(-1))+Multiply(CurlyLG,G^(-1))):" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,Typese ttingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableGF$66-I$mtrGF$6'-I$mtdGF$6(-I #mnGF$6%Q\"1F'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'normalF'/%)row alignGQ!F'/%,columnalignGFC/%+groupalignGFC/%(rowspanGF:/%+columnspanG F:-F56(-F86%Q\"0F'F;F>FAFDFFFHFJFAFDFF-F26'-F56(-F,6%-I&mfracGF$6)F7-F 86%Q\"2F'F;F>/%.linethicknessGF:/%+denomalignGQ'centerF'/%)numalignGF[ o/%)bevelledGQ&falseF'F;-I#moGF$6-Q1⁢F'F>/%&fenceGF`o/% *separatorGF`o/%)stretchyGF`o/%*symmetricGF`o/%(largeopGF`o/%.movablel imitsGF`o/%'accentGF`o/%'lspaceGQ&0.0emF'/%'rspaceGFep-F,6#-I#miGF$6&Q )tinfty16F'/%'italicGQ%trueF'F;/F?Q'italicF'FAFDFFFHFJF4FAFDFF/%&align GQ%axisF'/FBQ)baselineF'/FEF[o/FGQ'|frleft|hrF'/%/alignmentscopeGF`q/% ,columnwidthGQ%autoF'/%&widthGF_r/%+rowspacingGQ&1.0exF'/%.columnspaci ngGQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGFjr/%&frameGFjr/%-frame spacingGQ,0.4em~0.5exF'/%*equalrowsGF`o/%-equalcolumnsGF`o/%-displayst yleGF`o/%%sideGQ&rightF'/%0minlabelspacingGFgrF;F>/%%openGQ\"[F'/%&clo seGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG 6\"I,TypesettingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableGF$66-I$mtrGF$6'-I $mtdGF$6(-I#mnGF$6%Q\"0F'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'nor malF'/%)rowalignGQ!F'/%,columnalignGFC/%+groupalignGFC/%(rowspanGQ\"1F '/%+columnspanGFJF4FAFDFFF1/%&alignGQ%axisF'/FBQ)baselineF'/FEQ&rightF '/FGQ'|frleft|hrF'/%/alignmentscopeGQ%trueF'/%,columnwidthGQ%autoF'/%& widthGFen/%+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowlinesG Q%noneF'/%,columnlinesGF`o/%&frameGF`o/%-framespacingGQ,0.4em~0.5exF'/ %*equalrowsGQ&falseF'/%-equalcolumnsGFjo/%-displaystyleGFjo/%%sideGFS/ %0minlabelspacingGF]oF;F>/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,TypesettingGI(_sysli bGF'6'-I%mrowGF$6#-I'mtableGF$66-I$mtrGF$6'-I$mtdGF$6(-I#mnGF$6%Q\"0F' /%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'normalF'/%)rowalignGQ!F'/%,c olumnalignGFC/%+groupalignGFC/%(rowspanGQ\"1F'/%+columnspanGFJF4FAFDFF -F26'-F56(-F,6%-I&mfracGF$6)-F86%FJF;F>-F86%Q\"2F'F;F>/%.linethickness GFJ/%+denomalignGQ'centerF'/%)numalignGFin/%)bevelledGQ&falseF'F;-I#mo GF$6-Q1⁢F'F>/%&fenceGF^o/%*separatorGF^o/%)stretchyGF^o /%*symmetricGF^o/%(largeopGF^o/%.movablelimitsGF^o/%'accentGF^o/%'lspa ceGQ&0.0emF'/%'rspaceGFcp-F,6%-I#miGF$6&Q)α6F'/%'italicGQ%trueF' F;/F?Q'italicF'F_o-Fip6&Q\"hF'F\\qF;F_qFAFDFFFHFKF4FAFDFF/%&alignGQ%ax isF'/FBQ)baselineF'/FEFin/FGQ'|frleft|hrF'/%/alignmentscopeGF^q/%,colu mnwidthGQ%autoF'/%&widthGF`r/%+rowspacingGQ&1.0exF'/%.columnspacingGQ& 0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGF[s/%&frameGF[s/%-framespaci ngGQ,0.4em~0.5exF'/%*equalrowsGF^o/%-equalcolumnsGF^o/%-displaystyleGF ^o/%%sideGQ&rightF'/%0minlabelspacingGFhrF;F>/%%openGQ\"[F'/%&closeGQ \"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Ltilde11Orderlambd a3:=" }{MPLTEXT 1 0 7 "factor(" }{MPLTEXT 1 0 49 "-residue(Ltilde[1,1] /lambda^4,lambda=infinity));\n" }{MPLTEXT 1 0 22 "Ltilde12Orderlambda3 :=" }{MPLTEXT 1 0 7 "factor(" }{MPLTEXT 1 0 49 "-residue(Ltilde[1,2]/l ambda^4,lambda=infinity));\n" }{MPLTEXT 1 0 78 "Ltilde21Orderlambda3:= factor(-residue(Ltilde[2,1]/lambda^4,lambda=infinity));\n" }{MPLTEXT 1 0 22 "Ltilde22Orderlambda3:=" }{MPLTEXT 1 0 7 "factor(" }{MPLTEXT 1 0 48 "-residue(Ltilde[2,2]/lambda^4,lambda=infinity));" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5Ltilde11Orderlambda3G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5Ltilde12Orderlambda3G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5Ltilde21Orderlambda3G6\",$*$I)tinfty17GF$\"\"##\"\"\" \"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5Ltilde22Orderlambda3G6\"\"\" !" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "Ltilde11Orderlambda2:= factor(-residue(Ltilde[1,1]/lambda^3,lambda=infinity));\n" }{MPLTEXT 1 0 78 "Ltilde12Orderlambda2:=factor(-residue(Ltilde[1,2]/lambda^3,lam bda=infinity));\n" }{MPLTEXT 1 0 78 "Ltilde21Orderlambda2:=factor(-res idue(Ltilde[2,1]/lambda^3,lambda=infinity));\n" }{MPLTEXT 1 0 77 "Ltil de22Orderlambda2:=factor(-residue(Ltilde[2,2]/lambda^3,lambda=infinity ));" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5Ltilde11Ord erlambda2G6\",$I)tinfty16GF$#!\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5Ltilde12Orderlambda2G6\"\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5Ltilde21Orderlambda2G6\",$*&I)tinfty17GF$\"\"\",(*&I#q1GF$F(F'F(F( *&I#q2GF$F(F'F(F(I)tinfty15GF$\"\"#F(#F(\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5Ltilde22Orderlambda2G6\",$I)tinfty16GF$#!\"\"\"\"#" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Multiply(G,J);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,TypesettingGI(_sy slibGF'6'-I%mrowGF$6#-I'mtableGF$66-I$mtrGF$6'-I$mtdGF$6(-I#mnGF$6%Q\" 1F'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'normalF'/%)rowalignGQ!F'/ %,columnalignGFC/%+groupalignGFC/%(rowspanGF:/%+columnspanGF:-F56(-F86 %Q\"0F'F;F>FAFDFFFHFJFAFDFF-F26'-F56(-F,6%-F,6%-I&mfracGF$6)F7-F86%Q\" 2F'F;F>/%.linethicknessGF:/%+denomalignGQ'centerF'/%)numalignGF]o/%)be velledGQ&falseF'F;-I#moGF$6-Q1⁢F'F>/%&fenceGFbo/%*separ atorGFbo/%)stretchyGFbo/%*symmetricGFbo/%(largeopGFbo/%.movablelimitsG Fbo/%'accentGFbo/%'lspaceGQ&0.0emF'/%'rspaceGFgp-F,6#-I#miGF$6&Q)tinft y16F'/%'italicGQ%trueF'F;/F?Q'italicF'-Fdo6.Q\"+F'F;F>FgoFioF[pF]pF_pF apFcp/FfpQ,0.2222222emF'/FipFiq-FZ6)-F,6#-F,6&-Fdo6.Q*&uminus0;F'F;F>F goFioF[pF]pF_pFapFcpFhqFjq-FZ6)-F,6%-F]q6&Q#p1F'F`qF;FcqFco-F#6%-F,6%- F]q6&Q)λF'/FaqFboF;F>-Fdo6.Q(−F'F;F>FgoFioF[pF]pF_pFapFcp FhqFjq-F]q6&Q#q2F'F`qF;FcqF;F>-F,6#-F,6%-F]q6&Q#q1F'F`qF;FcqFcsFfsFinF [oF^oF`oF;Fcs-FZ6)-F,6%-F]q6&Q#p2F'F`qF;FcqFco-F#6%-F,6%F_sFcsF]tF;F>- F,6#-F,6%FfsFcsF]tFinF[oF^oF`oF;-F,6%FgtFcoF[sFinF[oF^oF`oF;FAFDFFFHFJ -F56(-FZ6)F7F_uFinF[oF^oF`oF;FAFDFFFHFJFAFDFF/%&alignGQ%axisF'/FBQ)bas elineF'/FEF]o/FGQ'|frleft|hrF'/%/alignmentscopeGFbq/%,columnwidthGQ%au toF'/%&widthGFav/%+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)ro wlinesGQ%noneF'/%,columnlinesGF\\w/%&frameGF\\w/%-framespacingGQ,0.4em ~0.5exF'/%*equalrowsGFbo/%-equalcolumnsGFbo/%-displaystyleGFbo/%%sideG Q&rightF'/%0minlabelspacingGFivF;F>/%%openGQ\"[F'/%&closeGQ\"]F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "CurlyLJ:=Matrix(2,2,0):\n" } {MPLTEXT 1 0 17 "CurlyLJ[1,1]:=0:\n" }{MPLTEXT 1 0 17 "CurlyLJ[1,2]:=0 :\n" }{MPLTEXT 1 0 64 "CurlyLJ[2,2]:=diff(J[2,2],q1)*CurlyLq1+diff(J[2 ,2],p1)*CurlyLp1+" }{MPLTEXT 1 0 49 "diff(J[2,2],q2)*CurlyLq2+diff(J[2 ,2],p2)*CurlyLp2" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 210 "+h*(alpha7*d iff(J[2,2],tinfty17)+alpha6*diff(J[2,2],tinfty16)+alpha5*diff(J[2,2],t infty15)+alpha4*diff(J[2,2],tinfty14)+alpha3*diff(J[2,2],tinfty13)+alp ha2*diff(J[2,2],tinfty12)+alpha1*diff(J[2,2],tinfty11)) :\n" }{MPLTEXT 1 0 64 "CurlyLJ[2,1]:=diff(J[2,1],q1)*CurlyLq1+diff(J[2,1],p1)*CurlyL p1+" }{MPLTEXT 1 0 49 "diff(J[2,1],q2)*CurlyLq2+diff(J[2,1],p2)*CurlyL p2" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 209 "+h*(alpha7*diff(J[2,1],tin fty17)+alpha6*diff(J[2,1],tinfty16)+alpha5*diff(J[2,1],tinfty15)+alpha 4*diff(J[2,1],tinfty14)+alpha3*diff(J[2,1],tinfty13)+alpha2*diff(J[2,1 ],tinfty12)+alpha1*diff(J[2,1],tinfty11)):\n" }{MPLTEXT 1 0 9 "CurlyLJ :\n" }{MPLTEXT 1 0 74 "Acheck:=simplify(Multiply(Multiply(J,A),J^(-1)) +Multiply(CurlyLJ,J^(-1))):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 223 51 "Expression in canonical \+ coordinates after reduction" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "tinfty17:=2:\n" }{MPLTEXT 1 0 13 "tinfty15:=0:\n" }{MPLTEXT 1 0 13 "tinfty16:=0:\n" }{MPLTEXT 1 0 13 "tinfty14:=0:\n" }{MPLTEXT 1 0 13 "tinfty12:=0:\n" }{MPLTEXT 1 0 18 "tinfty13:=2*tau1:\n" }{MPLTEXT 1 0 17 "tinfty11:=2*tau2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "tdP2(lambda);\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 27 "C0bis:=-h* (p1-p2)/(q1-q2)+(" }{MPLTEXT 1 0 3 "q1*" }{MPLTEXT 1 0 84 "p2^2-q2*p1^ 2)/(q1-q2) +2*q1*q2*tau2+2*(q1+q2)*q1*q2*tau1+(q1+q2)*(q1^2+q2^2)*q1*q 2;\n" }{MPLTEXT 1 0 107 "C1bis:=(p1^2-p2^2)/(q1-q2) -2*(q1^2+q1*q2+q2^ 2)*tau1 -2*(q1+q2)*tau2 -q1^4-q2*q1^3-q1^2*q2^2-q1*q2^3-q2^4;\n" } {MPLTEXT 1 0 42 "factor(simplify(series(C0-C0bis,tau2=0)));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 42 "factor(simplify(series(C1-C1bis,tau2=0)) );" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*$I'lambdaG6\"\"\"&!\"\"*&F$\"\" $I%tau1GF%\"\"\"!\"#*&F$\"\"#I%tau2GF%F+F," }}{PARA 11 "" 1 "" {XPPMATH 20 ">I&C0bisG6\",,*(I\"hGF$\"\"\",&I#p1GF$F(I#p2GF$!\"\"F(,&I #q1GF$F(I#q2GF$F,F,F,*&,&*&F*\"\"#F/F(F,*&F+F3F.F(F(F(F-F,F(*(F.F(F/F( I%tau2GF$F(F3**,&F.F(F/F(F(F.F(F/F(I%tau1GF$F(F3**F8F(,&*$F.F3F(*$F/F3 F(F(F.F(F/F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I&C1bisG6\",2*&,&*$I# p1GF$\"\"#\"\"\"*$I#p2GF$F*!\"\"F+,&I#q1GF$F+I#q2GF$F.F.F+*&,(*$F0F*F+ *&F0F+F1F+F+*$F1F*F+F+I%tau1GF$F+!\"#*&,&F0F+F1F+F+I%tau2GF$F+F8*$F0\" \"%F.*&F1F+F0\"\"$F.*&F0F*F1F*F.*&F0F+F1F?F.*$F1F=F." }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "mu1fonction:=unapply(mu1,alp ha7," }{MPLTEXT 1 0 7 "alpha6," }{MPLTEXT 1 0 7 "alpha5," }{MPLTEXT 1 0 7 "alpha4," }{MPLTEXT 1 0 7 "alpha3," }{MPLTEXT 1 0 7 "alpha2," } {MPLTEXT 1 0 9 "alpha1);\n" }{MPLTEXT 1 0 76 "mu2fonction:=unapply(mu2 ,alpha7,alpha6,alpha5,alpha4,alpha3,alpha2,alpha1);\n" }{MPLTEXT 1 0 18 "CurlyLq1fonction:=" }{MPLTEXT 1 0 8 "unapply(" }{MPLTEXT 1 0 8 "Cu rlyLq1" }{MPLTEXT 1 0 52 ",alpha7,alpha6,alpha5,alpha4,alpha3,alpha2,a lpha1):\n" }{MPLTEXT 1 0 18 "CurlyLp1fonction:=" }{MPLTEXT 1 0 68 "una pply(CurlyLp1,alpha7,alpha6,alpha5,alpha4,alpha3,alpha2,alpha1):\n" } {MPLTEXT 1 0 18 "CurlyLq2fonction:=" }{MPLTEXT 1 0 68 "unapply(CurlyLq 2,alpha7,alpha6,alpha5,alpha4,alpha3,alpha2,alpha1):\n" }{MPLTEXT 1 0 85 "CurlyLp2fonction:=unapply(CurlyLp2,alpha7,alpha6,alpha5,alpha4,alp ha3,alpha2,alpha1):" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,mu1fonctionG6 \"f*6)I'alpha7GF$I'alpha6GF$I'alpha5GF$I'alpha4GF$I'alpha3GF$I'alpha2G F$I'alpha1GF$F$6$I)operatorGF$I&arrowGF$F$,$*&,,*(9$\"\"\"I#q2GF$F6I%t au1GF$F6\"$?\"*&9(F6F7F6!$!G*&9&F6F8F6!$o\"*&F5F6I%tau2GF$F6!$?\"9*\"$ S)F6,&I#q1GF$F6F7!\"\"FG#F6FDF$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "> I,mu2fonctionG6\"f*6)I'alpha7GF$I'alpha6GF$I'alpha5GF$I'alpha4GF$I'alp ha3GF$I'alpha2GF$I'alpha1GF$F$6$I)operatorGF$I&arrowGF$F$,$*&,&I#q1GF$ \"\"\"I#q2GF$!\"\"F7,,*(9$F5F4F5I%tau1GF$F5\"$?\"*&9(F5F4F5!$!G*&9&F5F ;F5!$o\"*&F:F5I%tau2GF$F5!$?\"9*\"$S)F5#F7FGF$F$F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "mu1tau1:=mu1fonction(0,0,0,0,2,0,0);\n" } {MPLTEXT 1 0 36 "mu1tau2:=mu1fonction(0,0,0,0,0,0,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(mu1tau1G6\",$*&I#q2GF$\"\"\",&I#q1GF$F(F'!\"\"F+#! \"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(mu1tau2G6\",$*$,&I#q1GF$ \"\"\"I#q2GF$!\"\"F+\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "mu2tau1:=mu2fonction(0,0,0,0,2,0,0);\n" }{MPLTEXT 1 0 36 "mu2tau2: =mu2fonction(0,0,0,0,0,0,2);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(mu2tau1G6\",$*&,&I#q1GF$\"\"\"I#q2GF$!\"\"F+F(F)#\"\"# \"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(mu2tau2G6\",$*$,&I#q1GF$\"\" \"I#q2GF$!\"\"F+!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Cur lyLq1tau1:=CurlyLq1fonction(0,0,0,0,2,0,0):\n" }{MPLTEXT 1 0 51 "Curly Lq1tau1bis:=-2*h/3/(q1-q2)-4/3*q2*p1/(q1-q2);\n" }{MPLTEXT 1 0 52 "sim plify(series(CurlyLq1tau1-CurlyLq1tau1bis,h=0));\n" }{MPLTEXT 1 0 47 " CurlyLp1tau1:=CurlyLp1fonction(0,0,0,0,2,0,0):\n" }{MPLTEXT 1 0 154 "C urlyLp1tau1bis:=-2/3*(q2*p1^2-q2*p2^2)/(q1-q2)^2-2*h/3*(p1-p2)/(q1-q2) ^2-2/3*(q2^4+2*q2^3*q1+3*q2^2*q1^2+4*q2*q1^3) -4/3*q2*(q2+2*q1)*tau1- 4/3*q2*tau2;\n" }{MPLTEXT 1 0 52 "simplify(series(CurlyLp1tau1-CurlyLp 1tau1bis,p2=0));" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "> I0CurlyLq1tau1bisG6\",&*&I\"hGF$\"\"\",&I#q1GF$F(I#q2GF$!\"\"F,#!\"#\" \"$*(F)F,I#p1GF$F(F+F(#!\"%F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0CurlyLp1tau1bisG6\",2*&,&*&I#p1GF$ \"\"#I#q2GF$\"\"\"F,*&I#p2GF$F*F+F,!\"\"F,,&I#q1GF$F,F+F/!\"##F2\"\"$* (I\"hGF$F,,&F)F,F.F/F,F0F2F3*&F+F,F1F4#!\")F4*&F1F*F+F*F2*&F1F,F+F4#! \"%F4*$F+\"\"%F3*(F+F,,&F+F,F1F*F,I%tau1GF$F,F=*&F+F,I%tau2GF$F,F=" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "CurlyLq2" }{MPLTEXT 1 0 6 "tau1:=" }{MPLTEXT 1 0 16 "C urlyLq2fonction" }{MPLTEXT 1 0 16 "(0,0,0,0,2,0,0):" }{MPLTEXT 1 0 1 " \n" }{MPLTEXT 1 0 50 "CurlyLq2tau1bis:=2*h/3/(q1-q2)+4/3*q1*p2/(q1-q2) ;\n" }{MPLTEXT 1 0 16 "simplify(series(" }{MPLTEXT 1 0 8 "CurlyLq2" } {MPLTEXT 1 0 5 "tau1-" }{MPLTEXT 1 0 22 "CurlyLq2tau1bis,h=0));" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 47 "CurlyLp2tau1:=CurlyLp2fonction(0 ,0,0,0,2,0,0):\n" }{MPLTEXT 1 0 17 "CurlyLp2tau1bis:=" }{MPLTEXT 1 0 32 "2/3*(q1*p1^2-q1*p2^2)/(q1-q2)^2+" }{MPLTEXT 1 0 103 "2*h/3*(p1-p2) /(q1-q2)^2-2/3*(q1^4+2*q1^3*q2+3*q1^2*q2^2+4*q1*q2^3) -4/3*q1*(q1+2*q 2)*tau1-4/3*q1*tau2;" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 16 "simplify( series(" }{MPLTEXT 1 0 36 "CurlyLp2tau1-CurlyLp2tau1bis,p2=0));" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I0CurlyLq2tau1bisG6\",&*&I\"hGF$\"\"\", &I#q1GF$F(I#q2GF$!\"\"F,#\"\"#\"\"$*(F)F,I#p2GF$F(F*F(#\"\"%F/" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "> I0CurlyLp2tau1bisG6\",2*&,&*&I#p1GF$\"\"#I#q1GF$\"\"\"F,*&I#p2GF$F*F+F ,!\"\"F,,&F+F,I#q2GF$F/!\"##F*\"\"$*(I\"hGF$F,,&F)F,F.F/F,F0F2F3*$F+\" \"%#F2F4*&F1F,F+F4#!\"%F4*&F+F*F1F*F2*&F+F,F1F4#!\")F4*(F+F,,&F+F,F1F* F,I%tau1GF$F,F<*&F+F,I%tau2GF$F,F<" }}{PARA 11 "" 1 "" {XPPMATH 20 "\" \"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "CurlyLq1tau2:=CurlyL q1fonction(0,0,0,0,0,0,2):\n" }{MPLTEXT 1 0 31 "CurlyLq1tau2bis:=4*p1/ (q1-q2);\n" }{MPLTEXT 1 0 52 "simplify(series(CurlyLq1tau2-CurlyLq1tau 2bis,h=0));\n" }{MPLTEXT 1 0 47 "CurlyLp1tau2:=CurlyLp1fonction(0,0,0, 0,0,0,2):\n" }{MPLTEXT 1 0 104 "CurlyLp1tau2bis:=2*(p1^2-p2^2)/(q1-q2) ^2+4*(2*q1+q2)*tau1+4*tau2+2*(4*q1^3+3*q1^2*q2+2*q1*q2^2+1*q2^3);\n" } {MPLTEXT 1 0 54 "simplify(series(CurlyLp1tau2-CurlyLp1tau2bis,tau1=0)) ;" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0CurlyLq1tau2b isG6\",$*&I#p1GF$\"\"\",&I#q1GF$F(I#q2GF$!\"\"F,\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0CurlyLp1tau 2bisG6\",0*&,&*$I#p1GF$\"\"#\"\"\"*$I#p2GF$F*!\"\"F+,&I#q1GF$F+I#q2GF$ F.!\"#F**&,&F1F+F0F*F+I%tau1GF$F+\"\"%I%tau2GF$F6*$F0\"\"$\"\")*&F0F*F 1F+\"\"'*&F0F+F1F*F6*$F1F9F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "CurlyLq2tau2:=CurlyLq2fonc tion(0,0,0,0,0,0,2):\n" }{MPLTEXT 1 0 32 "CurlyLq2tau2bis:=-4*p2/(q1-q 2);\n" }{MPLTEXT 1 0 52 "simplify(series(CurlyLq2tau2-CurlyLq2tau2bis, h=0));\n" }{MPLTEXT 1 0 47 "CurlyLp2tau2:=CurlyLp2fonction(0,0,0,0,0,0 ,2):\n" }{MPLTEXT 1 0 105 "CurlyLp2tau2bis:=-2*(p1^2-p2^2)/(q1-q2)^2+4 *(2*q2+q1)*tau1+4*tau2+2*(4*q2^3+3*q2^2*q1+2*q2*q1^2+1*q1^3);\n" } {MPLTEXT 1 0 54 "simplify(series(CurlyLp2tau2-CurlyLp2tau2bis,tau1=0)) ;" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0CurlyLq2tau2b isG6\",$*&I#p2GF$\"\"\",&I#q1GF$F(I#q2GF$!\"\"F,!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0CurlyLp2tau2 bisG6\",0*&,&*$I#p1GF$\"\"#\"\"\"*$I#p2GF$F*!\"\"F+,&I#q1GF$F+I#q2GF$F .!\"#F2*&,&F0F+F1F*F+I%tau1GF$F+\"\"%I%tau2GF$F6*$F0\"\"$F**&F0F*F1F+F 6*&F0F+F1F*\"\"'*$F1F9\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "HamiltonianFonction:=unappl y(simplify(Hamiltonian)," }{MPLTEXT 1 0 7 "alpha7," }{MPLTEXT 1 0 35 " alpha6,alpha5,alpha4,alpha3,alpha2," }{MPLTEXT 1 0 9 "alpha1):\n" } {MPLTEXT 1 0 4 "H1:=" }{MPLTEXT 1 0 20 "HamiltonianFonction(" } {MPLTEXT 1 0 15 "0,0,0,0,2,0,0):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 10 "H1bis:=2*(" }{MPLTEXT 1 0 8 "q1*p2^2-" }{MPLTEXT 1 0 110 "q2*p1^2) /3/(q1-q2)-2*h/3*(p1-p2)/(q1-q2)+2/3*(q1+q2)*(q1^2+q2^2)*q1*q2+4/3*(q1 +q2)*q1*q2*tau1+4/3*q1*q2*tau2;\n" }{MPLTEXT 1 0 34 "simplify(series(H 1-H1bis,tau2=0));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 4 "H2:=" }{MPLTEXT 1 0 36 "HamiltonianFonction(0,0,0,0,0,0,2):\n" }{MPLTEXT 1 0 110 "H2bis:=2*(p1^2-p2^2)/(q1-q2)-2*(q1^4+q1^3*q2+q1^2* q2^2+q1*q2^3+q2^4)-4*(q1^2+q1*q2+q2^2)*tau1-4*tau2*(q1+q2);\n" } {MPLTEXT 1 0 32 "simplify(series(H2-H2bis,p1=0));" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I&H1bisG6\",,*&,&*&I#p1GF$\"\"#I#q2GF$\"\"\"!\"\"*&I#p2G F$F*I#q1GF$F,F,F,,&F0F,F+F-F-#F*\"\"$*(I\"hGF$F,,&F)F,F/F-F,F1F-#!\"#F 3**,&F0F,F+F,F,,&*$F0F*F,*$F+F*F,F,F0F,F+F,F2**F:F,F0F,F+F,I%tau1GF$F, #\"\"%F3*(F0F,F+F,I%tau2GF$F,F@" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 ">I&H2bisG6\",2*&,&*$I#p1GF$\"\"#\"\" \"*$I#p2GF$F*!\"\"F+,&I#q1GF$F+I#q2GF$F.F.F**$F0\"\"%!\"#*&F1F+F0\"\"$ F4*&F0F*F1F*F4*&F0F+F1F6F4*$F1F3F4*&,(*$F0F*F+*&F0F+F1F+F+*$F1F*F+F+I% tau1GF$F+!\"%*&,&F0F+F1F+F+I%tau2GF$F+F@" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "simplify(Ltild e);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,T ypesettingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableGF$66-I$mtrGF$6'-I$mtdGF $6(-I&mfracGF$6)-F,6#-F,6'-F,6%-F#6%-F,6%-I#miGF$6&Q#p1F'/%'italicGQ%t rueF'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'italicF'-I#moGF$6.Q(&mi nus;F'FK/FOQ'normalF'/%&fenceGQ&falseF'/%*separatorGFY/%)stretchyGFY/% *symmetricGFY/%(largeopGFY/%.movablelimitsGFY/%'accentGFY/%'lspaceGQ,0 .2222222emF'/%'rspaceGFbo-FE6&Q#p2F'FHFKFNFKFU-FR6-Q1⁢F 'FUFWFZFfnFhnFjnF\\oF^o/FaoQ&0.0emF'/FdoF\\p-FE6&Q)λF'/FIFYFKFU FQ-F,6%FDFho-FE6&Q#q2F'FHFKFN-FR6.Q\"+F'FKFUFWFZFfnFhnFjnF\\oF^oF`oFco -F,6%FeoFho-FE6&Q#q1F'FHFKFN-F,6#-F,6%F\\qFQFdp/%.linethicknessGQ\"1F' /%+denomalignGQ'centerF'/%)numalignGFhq/%)bevelledGFYFK/%)rowalignGQ!F '/%,columnalignGF_r/%+groupalignGF_r/%(rowspanGFeq/%+columnspanGFeq-F5 6(-F,6%-F#6%-F,6&-FR6.Q*&uminus0;F'FKFUFWFZFfnFhnFjnF\\oF^oF`oFcoF^pFg pF\\qFKFUFho-F#6%-F,6&F`sF^pFgpFdpFKFUF]rF`rFbrFdrFfrF]rF`rFbr-F26'-F5 6(-F86)-F,6#-F,65-F,6#-I%msupGF$6%F\\q-I#mnGF$6%Q\"5F'FKFU/%1superscri ptshiftGQ\"0F'Fgp-F,6%-F#6%-F,6%F^pFQFdpFKFUFho-Fdt6%F\\q-Fgt6%Q\"4F'F KFUFjtFgp-F,6%-F#6%-F,6'-F,6#-Fdt6%F^p-Fgt6%Q\"2F'FKFUFjtFQ-F,6%F^pFho FdpFgp-F,6%FbvFho-FE6&Q'τ1F'FHFKFNFKFUFho-Fdt6%F\\q-Fgt6%Q\"3F'FKF UFjtFgp-F,6%-F#6%-F,6)-F,6%-F#6%-F,6&F`sF^vFQFgvFKFUFhoFdpFgp-F,6#-Fdt 6%F^pF^wFjtFgp-F,6'FbvFhoF^pFhoFivFgp-F,6%FbvFho-FE6&Q'τ2F'FHFKFNF KFUFho-Fdt6%F\\qFbvFjtFQ-F,6'-F#6%-F,6--F,6#-Fdt6%FdpF^wFjtFgp-F,6%-Fd t6%FdpFbvFjtFhoF^pFgp-F,6%-F#6%-F,6%F^vFgpFgvFKFUFhoFdpFgp-F,6%FbvFhoF _xFgp-F,6'FeuFhoF^pFhoFivFgp-F,6%FeuFhoFexFKFUFhoFdpFhoF\\qFgp-F,6#-Fd t6%FdpFftFjtFgp-F,6%F^pFho-Fdt6%FdpFeuFjtFgp-F,6%FjyFhoFbyFgp-F,6%-F#6 %-F,6'F]xFgpFaxFgpFcxFKFUFhoFfyFQ-F,6#-Fdt6%F@FbvFjt-F,6#-Fdt6%-F#6%Fa qFKFUFbvFjtFcqFfqFiqF[rFKF]rF`rFbrFdrFfr-F56(-F86)-F,6#-F,6'-F,6%-F#6% -F,6%FeoFQFDFKFUFhoF^pFgpFbpFQFjpF_qFcqFfqFiqF[rFKF]rF`rFbrFdrFfrF]rF` rFbr/%&alignGQ%axisF'/F^rQ)baselineF'/FarFhq/FcrQ'|frleft|hrF'/%/align mentscopeGFJ/%,columnwidthGQ%autoF'/%&widthGFh]l/%+rowspacingGQ&1.0exF '/%.columnspacingGQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGFc^l/%&f rameGFc^l/%-framespacingGQ,0.4em~0.5exF'/%*equalrowsGFY/%-equalcolumns GFY/%-displaystyleGFY/%%sideGQ&rightF'/%0minlabelspacingGF`^lFKFU/%%op enGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Ltilde21:=simplify(Ltilde[2,1]):\n" }{MPLTEXT 1 0 71 "Ltilde21bis:=la mbda^3+(q1+q2)*lambda^2+(q1^2+q1*q2+q2^2+2*tau1)*lambda\n" }{MPLTEXT 1 0 50 "+q1^3+q1^2*q2+q1*q2^2+q2^3+ 2*(q1+q2)*tau1+2*tau2\n" }{MPLTEXT 1 0 23 "-(p1-p2)^2/(q1-q2)^2 ;\n" }{MPLTEXT 1 0 16 "simplify(series(" }{MPLTEXT 1 0 8 "Ltilde21" }{MPLTEXT 1 0 1 "-" }{MPLTEXT 1 0 11 "Ltil de21bis" }{MPLTEXT 1 0 8 ",p1=0));" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,Ltilde21bisG6\",6*$I'lambdaGF$\"\"$\"\"\"*&,&I#q 1GF$F)I#q2GF$F)F)F'\"\"#F)*&,**$F,F.F)*&F,F)F-F)F)*$F-F.F)I%tau1GF$F.F )F'F)F)*$F,F(F)*&F,F.F-F)F)*&F,F)F-F.F)*$F-F(F)*&F+F)F4F)F.I%tau2GF$F. *&,&I#p1GF$F)I#p2GF$!\"\"F.,&F,F)F-F?!\"#F?" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "c0:=0: \n" }{MPLTEXT 1 0 39 "Acheck11:=unapply(simplify(Acheck[1,1])" } {MPLTEXT 1 0 8 ",alpha7," }{MPLTEXT 1 0 35 "alpha6,alpha5,alpha4,alpha 3,alpha2," }{MPLTEXT 1 0 8 "alpha1):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 91 "Acheck12:=unapply(simplify(Acheck[1,2]),alpha7,alpha6,alpha5,a lpha4,alpha3,alpha2,alpha1):\n" }{MPLTEXT 1 0 91 "Acheck21:=unapply(si mplify(Acheck[2,1]),alpha7,alpha6,alpha5,alpha4,alpha3,alpha2,alpha1): \n" }{MPLTEXT 1 0 90 "Acheck22:=unapply(simplify(Acheck[2,2]),alpha7,a lpha6,alpha5,alpha4,alpha3,alpha2,alpha1):" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 1 "A" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 21 "111:=fact or(Acheck11(" }{MPLTEXT 1 0 17 "0,0,0,0,2,0,0));\n" }{MPLTEXT 1 0 1 "A " }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 5 "112:=" }{MPLTEXT 1 0 16 "fa ctor(Acheck12(" }{MPLTEXT 1 0 17 "0,0,0,0,2,0,0));\n" }{MPLTEXT 1 0 1 "A" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 5 "121:=" }{MPLTEXT 1 0 16 " factor(Acheck21(" }{MPLTEXT 1 0 17 "0,0,0,0,2,0,0));\n" }{MPLTEXT 1 0 1 "A" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 5 "122:=" }{MPLTEXT 1 0 16 "factor(Acheck22(" }{MPLTEXT 1 0 17 "0,0,0,0,2,0,0));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Acheck111G6\",$*&,&I#p1GF$\"\"\"I#p2GF$!\"\"F ),&I#q1GF$F)I#q2GF$F+F+#\"\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I *Acheck112G6\",(I'lambdaGF$#\"\"#\"\"$I#q1GF$#!\"#F)I#q2GF$F+" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I*Acheck121G6\",.*$I'lambdaGF$\"\"##F( \"\"$*&I#q1GF$\"\"\"F'F-F)*&I#q2GF$F-F'F-F)*$F,F(F)*$F/F(F)I%tau1GF$# \"\"%F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Acheck122G6\",$*&,&I#p1GF$ \"\"\"I#p2GF$!\"\"F),&I#q1GF$F)I#q2GF$F+F+#!\"#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "A" }{MPLTEXT 1 0 43 "check111:=factor(Ache ck11(0,0,0,0,0,0,2));\n" }{MPLTEXT 1 0 44 "Acheck112:=factor(Acheck12( 0,0,0,0,0,0,2));\n" }{MPLTEXT 1 0 44 "Acheck121:=factor(Acheck21(0,0,0 ,0,0,0,2));\n" }{MPLTEXT 1 0 27 "Acheck122:=factor(Acheck22(" } {MPLTEXT 1 0 16 "0,0,0,0,0,0,2));" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Acheck111G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Acheck112G6\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Acheck12 1G6\",(I#q1GF$\"\"%I'lambdaGF$\"\"#I#q2GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Acheck122G6\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }