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}{CSTYLE "Prompt" -1 1 "Courier" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Bold" -1 5 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "Times" 1 12 0 128 128 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "Caption Text" -1 224 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 217 240 "In this Maple file, we \+ compute the evolution equations for the Painlev\351 2 equations using \+ the compatibility equation of the Lax system. We also obtain the expre ssion of the Lax matrices in the geometric gauge without apparent sing ularities." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 225 56 "Lax matrices in t he oper gauge from previous Maple files" }}{EXCHG {PARA 0 "" 0 "" {TEXT 226 102 "Summary of previous files: We have the expression for \+ some coefficients of the Lax matrix L and of A." }}{PARA 0 "" 0 "" {TEXT 226 1 "\n" }{TEXT 226 244 "The operator is \\hbar (alpha13\\part ial_\{t_\{\\infty^\{(1)\},3\} +alpha23\\partial_\{t_\{\\infty^\{(2)\}, 3\}+alpha12\\partial_\{t_\{\\infty^\{(1)\},2\} +alpha22\\partial_\{t_ \{\\infty^\{(2)\},2\}+alpha11\\partial_\{t_\{\\infty^\{(1)\},1\} +alph a21\\partial_\{t_\{\\infty^\{(2)\},1\})) " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart:\n" }{MPLTEXT 1 0 21 "with(LinearAlgebra):\n" }{MPLTEXT 1 0 9 "Coherence" }{MPLTEXT 1 0 30 "Equation1:=tinfty10+tin fty20;\n" }{MPLTEXT 1 0 21 "tinfty20:=-tinfty10:\n" }{MPLTEXT 1 0 31 " Pinfty42 := tinfty13*tinfty23;\n" }{MPLTEXT 1 0 49 "Pinfty32 := tinfty 12*tinfty23+tinfty13*tinfty22;\n" }{MPLTEXT 1 0 67 "Pinfty22 := tinfty 12*tinfty22+tinfty13*tinfty21+tinfty11*tinfty23;\n" }{MPLTEXT 1 0 85 " Pinfty12 := tinfty20*tinfty13+tinfty12*tinfty21+tinfty10*tinfty23+tinf ty11*tinfty22;\n" }{MPLTEXT 1 0 32 "Pinfty01 := -tinfty11-tinfty21;\n" }{MPLTEXT 1 0 32 "Pinfty11 := -tinfty12-tinfty22;\n" }{MPLTEXT 1 0 32 "Pinfty21 := -tinfty13-tinfty23;\n" }{MPLTEXT 1 0 42 "P1:=x-> Pinft y01+Pinfty11*x+Pinfty21*x^2:\n" }{MPLTEXT 1 0 68 "P2:=x-> Pinfty02+Pin fty12*x+Pinfty22*x^2+Pinfty32*x^3+Pinfty42*x^4:\n" }{MPLTEXT 1 0 33 "t dP2:=unapply(P2(x)-Pinfty02,x):\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 53 "dP1dlambda:=unapply(diff(P1(lambda),lambda),lambda):\n" }{MPLTEXT 1 0 53 "dP2dlambda:=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 69 "L[2,1]:=-P2(lambda)+ Pinfty02 +C -h*lambda*tinfty13 - p*h/(lambda-q):\n" }{MPLTEXT 1 0 37 " L[2,2]:= P1(lambda) +h/(lambda-q) :\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 18 "A:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 120 "A[1,1]:=1/3*(alpha13* tinfty23-alpha23*tinfty13)/(tinfty13-tinfty23)*lambda^3+c2*lambda^2+c1 *lambda +c0+ rho/(lambda-q):\n" }{MPLTEXT 1 0 74 "A[1,2]:=(alpha13-alp ha23)/3/(tinfty13-tinfty23)*lambda+nu+ mu/(lambda-q):\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 "f or i from 1 to 2 do for j from 1 to 2 do dAdlambda[i,j]:=diff(A[i,j],l ambda): 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 24 "Q2:=unapp ly(-p,lambda):\n" }{MPLTEXT 1 0 18 "J:=Matrix(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 31 " J[2,1]:=Q2(lambda)/(lambda-q):\n" }{MPLTEXT 1 0 22 "J[2,2]:=1/(lambda- q):\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 26 "dJdlambda:=Matrix(2,2,0) :\n" }{MPLTEXT 1 0 87 "for i from 1 to 2 do for j from 1 to 2 do dJdla mbda[i,j]:=diff(J[i,j],lambda): od: od:\n" }{MPLTEXT 1 0 3 "J:\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 19 "LJ:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 12 "LJ[1,1]:=0:\n" }{MPLTEXT 1 0 12 "LJ[1,2]:=0:\n" }{MPLTEXT 1 0 63 "LJ[2,2]:=diff(J[2,2],q)*Lq+diff(J[2,2],p)*Lp+h*diff(J[2,2],t):\n" }{MPLTEXT 1 0 63 "LJ[2,1]:=diff(J[2,1],q)*Lq+diff(J[2,1],p)*Lp+h*diff( J[2,1],t):\n" }{MPLTEXT 1 0 4 "LJ:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 79 "checkL:=simplify(Multiply(Multiply(J,L),J^(-1))+h*Multiply(dJd lambda,J^(-1))):\n" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 65 "A:=simpl ify(Multiply(Multiply(J,A),J^(-1))+Multiply(LJ,J^(-1))):\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I3Coh erenceEquation1G6\",&I)tinfty10GF$\"\"\"I)tinfty20GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty42G6\"*&I)tinfty13GF$\"\"\"I)tinfty23GF$F' " }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty32G6\",&*&I)tinfty12GF$\"\" 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F,6%-Fbp6&Q\"pF'FepF;FhpFjpFav-F,6#-F,6%F[sF_s-Fbp6&Q\"qF'FepF;Fhp/%.l inethicknessGFJ/%+denomalignGQ'centerF'/%)numalignGFgw/%)bevelledGFgnF ;FAFDFFFHFK-F56(-F,6,FWF]rF_sFhrFcq-F,6%-F#6%-F,6&FWFerF_sF`rF;F>FjpF[ sFcq-F,6%-F#6%-F,6&FWF`qF_sFhqF;F>FjpF^tFcq-Fev6)-F,6#FavF\\wFcwFewFhw FjwF;FAFDFFFHFKFAFDFF/%&alignGQ%axisF'/FBQ)baselineF'/FEFgw/FGQ'|frlef t|hrF'/%/alignmentscopeGFgp/%,columnwidthGQ%autoF'/%&widthGF\\z/%+rows pacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowlinesGQ%noneF'/%,colum nlinesGFgz/%&frameGFgz/%-framespacingGQ,0.4em~0.5exF'/%*equalrowsGFgn/ %-equalcolumnsGFgn/%-displaystyleGFgn/%%sideGQ&rightF'/%0minlabelspaci ngGFdzF;F>/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6'-I%mrowG F$6#-I'mtableGF$66-I$mtrGF$6'-I$mtdGF$6(-F,6+-F,6%-I&mfracGF$6)-I#mnGF $6%Q\"1F'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'normalF'-F?6%Q\"3F' FBFE/%.linethicknessGFA/%+denomalignGQ'centerF'/%)numalignGFO/%)bevell 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FbuFduF[uF^uF`u-F26'-F56(-F,6%-Fap6&Q%AA21F'FdpFBFgp-FV6-Q0&ApplyFunct ion;F'FEFYFenFgnFinF[oF]oF_oFaoFdo-F#6%-F,6#F]rFBFEF[uF^uF`uFbuFdu-F56 (-F,6%-Fap6&Q%AA22F'FdpFBFgpFgwFjwF[uF^uF`uFbuFduF[uF^uF`u/%&alignGQ%a xisF'/F\\uQ)baselineF'/F_uFO/FauQ'|frleft|hrF'/%/alignmentscopeGFfp/%, columnwidthGQ%autoF'/%&widthGFay/%+rowspacingGQ&1.0exF'/%.columnspacin gGQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGF\\z/%&frameGF\\z/%-fram espacingGQ,0.4em~0.5exF'/%*equalrowsGFT/%-equalcolumnsGFT/%-displaysty leGFT/%%sideGQ&rightF'/%0minlabelspacingGFiyFBFE/%%openGQ\"[F'/%&close GQ\"]F'" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 225 73 "Solving the compati bility equations to obtain the Hamiltonian evolutions." }}{EXCHG {PARA 0 "" 0 "" {TEXT 226 69 "The compatibility equation is \\mathcal \{L\}L=h\\partial_\\lambda A+[A,L]\n" }{TEXT 226 111 "Since the first \+ line of L is trivial, we may easily obtain A[2,1] et A[2,2] to obtain \+ the full expression for A" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "LL:=h*dAdlambda+(Multiply(A,L)-Multiply(L,A)):\n" }{MPLTEXT 1 0 18 "Entry11:=LL[1,1]:\n" }{MPLTEXT 1 0 18 "Entry12:=LL[1,2]:\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 32 "simplify(AA21(lambda)-AA21bis); \n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 53 "AA22:=unapply(solve(Entry12 =0,AA22(lambda)),lambda):\n" }{MPLTEXT 1 0 47 "AA22bis:=h*dAdlambda[1, 2]+A[1,1]+A[1,2]*L[2,2]:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 32 "simpl ify(AA22(lambda)-AA22bis);\n" }{MPLTEXT 1 0 19 "simplify(Entry11);\n" }{MPLTEXT 1 0 19 "simplify(Entry12);\n" }{MPLTEXT 1 0 47 "LL:=h*dAdlam bda+(Multiply(A,L)-Multiply(L,A)):\n" }{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 226 95 "We now compute the acti on of \\mathcal\{L\} on L[2,2] et L[2,1] to obtain the evolution equa tions" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 26 "Evolution of entry L_\{ 2,2\}" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Entry22:=simplify( LL[2,2]):\n" }{MPLTEXT 1 0 77 "Entry22TermLambdaMinusqCube:=factor(res idue(Entry22*(lambda-q)^2,lambda=q));\n" }{MPLTEXT 1 0 18 "Entry22Term LambdaM" }{MPLTEXT 1 0 4 "inus" }{MPLTEXT 1 0 55 "qSquare:=factor(resi due(Entry22*(lambda-q),lambda=q));\n" }{MPLTEXT 1 0 18 "Entry22TermLam bdaM" }{MPLTEXT 1 0 4 "inus" }{MPLTEXT 1 0 38 "q:=factor(residue(Entry 22,lambda=q));\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 77 "Entry22TermLa mbdaInfty4:=factor(-residue(Entry22/lambda^5,lambda=infinity));\n" } {MPLTEXT 1 0 77 "Entry22TermLambdaInfty3:=factor(-residue(Entry22/lamb da^4,lambda=infinity));\n" }{MPLTEXT 1 0 77 "Entry22TermLambdaInfty2:= factor(-residue(Entry22/lambda^3,lambda=infinity));\n" }{MPLTEXT 1 0 77 "Entry22TermLambdaInfty1:=factor(-residue(Entry22/lambda^2,lambda=i nfinity));\n" }{MPLTEXT 1 0 75 "Entry22TermLambdaInfty0:=factor(-resid ue(Entry22/lambda,lambda=infinity));\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 37 "simplify( Entry22-(Entry22TermLambdaM" }{MPLTEXT 1 0 4 "inus" }{MPLTEXT 1 0 1 "q" }{MPLTEXT 1 0 6 "Square" }{MPLTEXT 1 0 32 "/(lamb da-q)^2+Entry22TermLambdaM" }{MPLTEXT 1 0 4 "inus" }{MPLTEXT 1 0 13 "q /(lambda-q)\n" }{MPLTEXT 1 0 159 "+Entry22TermLambdaInfty0+Entry22Term LambdaInfty1*lambda+Entry22TermLambdaInfty2*lambda^2+Entry22TermLambda Infty3*lambda^3+Entry22TermLambdaInfty4*lambda^4) );\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">II>Entry22TermLambdaMinusqSquareG6\",$*(I\"hGF$\"\"\", B*(I#muGF$F(I\"qGF$\"\"#I)tinfty13GF$F-!\"$*(F+F(F,F-I)tinfty23GF$F-\" \"$**F+F(F,F(I)tinfty12GF$F(F.F(F/**F+F(F,F(F4F(F1F(F2**F+F(F,F(F.F(I) tinfty22GF$F(F/**F+F(F,F(F7F(F1F(F2*(I(alpha13GF$F(F'F(F,F(F(*(I(alpha 23GF$F(F'F(F,F(!\"\"*(F'F(I#nuGF$F(F.F(F2*(F'F(F?F(F1F(F/*(F+F(I)tinft y11GF$F(F.F(F/*(F+F(FBF(F1F(F2*(F+F(F.F(I)tinfty21GF$F(F/*(F+F(FEF(F1F (F2*&I$rhoGF$F(F.F(\"\"'*&FHF(F1F(!\"'F(,&F.F(F1F=F=#F=F2" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry22TermLambdaMinusqG6\"\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 ">I8Entry22TermLambdaInfty4G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry22TermLambdaInfty3G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry22TermLambdaInfty2G6\",$*&,&I(alpha23GF$\"\" \"I(alpha13GF$F)F)I\"hGF$F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8E ntry22TermLambdaInfty1G6\",$*(I\"hGF$\"\"\",2*&I#nuGF$F(I)tinfty13GF$ \"\"#\"\"$*&F+F(I)tinfty23GF$F-!\"$*&I(alpha13GF$F(I)tinfty12GF$F(F(*& F3F(I)tinfty22GF$F(F(*&I(alpha23GF$F(F4F(!\"\"*&F8F(F6F(F9*&I#c2GF$F(F ,F(!\"'*&FI8Entry22TermLambdaInfty0G6\",$*(I\"hGF$\"\"\",:*&I#muGF $F(I)tinfty13GF$\"\"#\"\"$*&F+F(I)tinfty23GF$F-!\"$*(I#nuGF$F(I)tinfty 12GF$F(F,F(F.*(F3F(F4F(F0F(F1*(F3F(F,F(I)tinfty22GF$F(F.*(F3F(F7F(F0F( F1*&I(alpha13GF$F(I)tinfty11GF$F(F(*&F:F(I)tinfty21GF$F(F(*&I(alpha23G F$F(F;F(!\"\"*&F?F(F=F(F@*&I#c1GF$F(F,F(!\"'*&FCF(F0F(\"\"'F(,&F,F(F0F @F@#F@F." }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 226 9 "We have: " }{XPPEDIT 2 0 "Typesetting:-mrow(Typesett ing:-msub(Typesetting:-mi(\"L\", italic = \"true\", mathvariant = \"it alic\"), Typesetting:-mrow(Typesetting:-mn(\"2\", mathvariant = \"norm al\"), Typesetting:-mo(\",\", mathvariant = \"normal\", fence = \+ \"false\", separator = \"true\", stretchy = \"false\", symmetric = \"f alse\", largeop = \"false\", movablelimits = \"false\", accent = \"fal se\", lspace = \"0.0em\", rspace = \"0.3333333em\"), Typesetting:-mn( \"2\", mathvariant = \"normal\")), subscriptshift = \"0\"), Typesettin g:-mo(\"≔\", mathvariant = \"normal\", fence = \"false\", separ ator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \+ \"0.2777778em\", rspace = \"0.2777778em\"), Typesetting:-mo(\"&uminus0 ;\", mathvariant = \"normal\", fence = \"false\", separator = \"false \", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", \+ movablelimits = \"false\", accent = \"false\", lspace = \"0.2222222em \", rspace = \"0.2222222em\"), Typesetting:-mi(\"tinfty11\", italic = \+ \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"−\", math variant = \"normal\", fence = \"false\", separator = \"false\", stretc hy = \"false\", symmetric = \"false\", largeop = \"false\", movablelim its = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspace \+ = \"0.2222222em\"), Typesetting:-mi(\"tinfty21\", italic = \"true\", m athvariant = \"italic\"), Typesetting:-mo(\"+\", mathvariant = \" normal\", fence = \"false\", separator = \"false\", stretchy = \"false \", symmetric = \"false\", largeop = \"false\", movablelimits = \"fals e\", accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.222222 2em\"), Typesetting:-mrow(Typesetting:-mfenced(Typesetting:-mrow(Types etting:-mo(\"&uminus0;\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", l argeop = \"false\", movablelimits = \"false\", accent = \"false\", lsp ace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mi(\"ti nfty12\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-m o(\"−\", mathvariant = \"normal\", fence = \"false\", separator \+ = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \" false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2 222222em\", rspace = \"0.2222222em\"), Typesetting:-mi(\"tinfty22\", i talic = \"true\", mathvariant = \"italic\")), mathvariant = \"normal\" ), Typesetting:-mo(\"⁢\", mathvariant = \"normal\", fen ce = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi( \"λ\", italic = \"false\", mathvariant = \"normal\")), Typesett ing:-mo(\"+\", mathvariant = \"normal\", fence = \"false\", separ ator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \+ \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mrow(Typesett ing:-mfenced(Typesetting:-mrow(Typesetting:-mo(\"&uminus0;\", mathvari ant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits \+ = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspace = \" 0.2222222em\"), Typesetting:-mi(\"tinfty13\", italic = \"true\", mathv ariant = \"italic\"), Typesetting:-mo(\"−\", mathvariant = \"nor mal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\" , accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em \"), Typesetting:-mi(\"tinfty23\", italic = \"true\", mathvariant = \" italic\")), mathvariant = \"normal\"), Typesetting:-mo(\"&InvisibleTim es;\", mathvariant = \"normal\", fence = \"false\", separator = \"fals e\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rs pace = \"0.0em\"), Typesetting:-msup(Typesetting:-mi(\"λ\", ita lic = \"false\", mathvariant = \"normal\"), Typesetting:-mn(\"2\", mat hvariant = \"normal\"), superscriptshift = \"0\")), Typesetting:-mo(\" +\", mathvariant = \"normal\", fence = \"false\", separator = \"f alse\", stretchy = \"false\", symmetric = \"false\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", lspace = \"0.222222 2em\", rspace = \"0.2222222em\"), Typesetting:-mfrac(Typesetting:-mi( \"h\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mrow (Typesetting:-mi(\"λ\", italic = \"false\", mathvariant = \"nor mal\"), Typesetting:-mo(\"−\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \+ \"false\", largeop = \"false\", movablelimits = \"false\", accent = \" false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesett ing:-mi(\"q\", italic = \"true\", mathvariant = \"italic\")), linethic kness = \"1\", denomalign = \"center\", numalign = \"center\", bevelle d = \"false\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibG F'6.-I%msubGF$6%-I#miGF$6%Q\"LF'/%'italicGQ%trueF'/%,mathvariantGQ'ita licF'-F#6%-I#mnGF$6$Q\"2F'/F6Q'normalF'-I#moGF$6-Q(,F'F>/%&fence GQ&falseF'/%*separatorGF4/%)stretchyGFF/%*symmetricGFF/%(largeopGFF/%. movablelimitsGFF/%'accentGFF/%'lspaceGQ&0.0emF'/%'rspaceGQ,0.3333333em F'F:/%/subscriptshiftGQ\"0F'-FA6-Q)≔F'F>FD/FHFFFIFKFMFOFQ/FTQ,0 .2777778emF'/FWF[o-FA6-Q*&uminus0;F'F>FDFinFIFKFMFOFQ/FTQ,0.2222222emF '/FWFao-F/6%Q)tinfty11F'F2F5-FA6-Q(−F'F>FDFinFIFKFMFOFQF`oFbo-F/ 6%Q)tinfty21F'F2F5-FA6-Q'+F'F>FDFinFIFKFMFOFQF`oFbo-F#6%-I(mfence dGF$6$-F#6&F]o-F/6%Q)tinfty12F'F2F5Ffo-F/6%Q)tinfty22F'F2F5F>-FA6-Q1&I nvisibleTimes;F'F>FDFinFIFKFMFOFQFS/FWFU-F/6%Q)λF'/F3FFF>F\\p-F #6%-Fbp6$-F#6&F]o-F/6%Q)tinfty13F'F2F5Ffo-F/6%Q)tinfty23F'F2F5F>F\\q-I %msupGF$6%F`qF:/%1superscriptshiftGFenF\\p-I&mfracGF$6(-F/6%Q\"hF'F2F5 -F#6%F`qFfo-F/6%Q\"qF'F2F5/%.linethicknessGQ\"1F'/%+denomalignGQ'cente rF'/%)numalignGFes/%)bevelledGFF" }{TEXT 226 1 "\n" }{TEXT 226 299 "Si nce the operator is \\hbar (alpha13\\partial_\{t_\{\\infty^\{(1)\},3 \} +alpha23\\partial_\{t_\{\\infty^\{(2)\},3\}+alpha12*\\partial_\{t_ \{\\infty^\{(1)\},2\} +alpha22*\\partial_\{t_\{\\infty^\{(2)\},2\}+alp ha11*\\partial_\{t_\{\\infty^\{(1)\},1\} +alpha21*\\partial_\{t_\{\\in fty^\{(2)\},1\}) we can deduce the action of \\mathcal\{L\} on q" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "L22OrderLambda2:=-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 "L22Ord erLambda0:=-residue(L[2,2]/lambda^1,lambda=infinity):\n" }{MPLTEXT 1 0 274 "simplify(h*(alpha13*diff(L22OrderLambda2,tinfty13)+alpha23*diff (L22OrderLambda2,tinfty23)+alpha12*diff(L22OrderLambda2,tinfty12)+alph a22*diff(L22OrderLambda2,tinfty22)+alpha11*diff(L22OrderLambda2,tinfty 11)+alpha21*diff(L22OrderLambda2,tinfty21))- Entry22TermLambdaInfty2); \n" }{MPLTEXT 1 0 88 "Equation1:=factor(simplify(h*(alpha13*diff(L22Or derLambda1,tinfty13)+alpha23*diff(L22Ord" }{MPLTEXT 1 0 2 "er" } {MPLTEXT 1 0 37 "Lambda1,tinfty23)+alpha12*diff(L22Ord" }{MPLTEXT 1 0 2 "er" }{MPLTEXT 1 0 37 "Lambda1,tinfty12)+alpha22*diff(L22Ord" } {MPLTEXT 1 0 2 "er" }{MPLTEXT 1 0 37 "Lambda1,tinfty22)+alpha11*diff(L 22Ord" }{MPLTEXT 1 0 2 "er" }{MPLTEXT 1 0 37 "Lambda1,tinfty11)+alpha2 1*diff(L22Ord" }{MPLTEXT 1 0 2 "er" }{MPLTEXT 1 0 47 "Lambda1,tinfty21 ))- Entry22TermLambdaInfty1));\n" }{MPLTEXT 1 0 49 "Equation2:=factor( simplify(h*(alpha13*diff(L22Ord" }{MPLTEXT 1 0 2 "er" }{MPLTEXT 1 0 37 "Lambda0,tinfty13)+alpha23*diff(L22Ord" }{MPLTEXT 1 0 2 "er" } {MPLTEXT 1 0 37 "Lambda0,tinfty23)+alpha12*diff(L22Ord" }{MPLTEXT 1 0 2 "er" }{MPLTEXT 1 0 37 "Lambda0,tinfty12)+alpha22*diff(L22Ord" } {MPLTEXT 1 0 2 "er" }{MPLTEXT 1 0 37 "Lambda0,tinfty22)+alpha11*diff(L 22Ord" }{MPLTEXT 1 0 2 "er" }{MPLTEXT 1 0 37 "Lambda0,tinfty11)+alpha2 1*diff(L22Ord" }{MPLTEXT 1 0 2 "er" }{MPLTEXT 1 0 47 "Lambda0,tinfty21 ))- Entry22TermLambdaInfty0));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\" !" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Equation1G6\",$*(I\"hGF$\"\"\",: *&I#nuGF$F(I)tinfty13GF$\"\"#!\"'*&F+F(I)tinfty23GF$F-\"\"'*&I(alpha12 GF$F(F,F(\"\"$*&F3F(F0F(!\"$*&I(alpha13GF$F(I)tinfty12GF$F(!\"#*&F8F(I )tinfty22GF$F(F:*&I(alpha22GF$F(F,F(F4*&F>F(F0F(F6*&I(alpha23GF$F(F9F( F-*&FAF(FI*Equation2G6\",$*(I\"hGF$\"\"\",B*&I #muGF$F(I)tinfty13GF$\"\"#!\"$*&F+F(I)tinfty23GF$F-\"\"$*(I#nuGF$F(I)t infty12GF$F(F,F(F.*(F3F(F4F(F0F(F1*(F3F(F,F(I)tinfty22GF$F(F.*(F3F(F7F (F0F(F1*&I(alpha11GF$F(F,F(F1*&F:F(F0F(F.*&I(alpha13GF$F(I)tinfty11GF$ F(!\"\"*&F=F(I)tinfty21GF$F(F?*&I(alpha21GF$F(F,F(F1*&FCF(F0F(F.*&I(al pha23GF$F(F>F(F(*&FFF(FAF(F(*&I#c1GF$F(F,F(\"\"'*&FIF(F0F(!\"'F(,&F,F( F0F?F?#F?F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Lq:=factor(" }{MPLTEXT 1 0 22 "Entry22TermLambdaMinus" }{MPLTEXT 1 0 7 "qSquare" } {MPLTEXT 1 0 5 "/h):\n" }{MPLTEXT 1 0 73 "Lqbis:=-mu*P1(q)+2*rho-h*nu- h*(alpha13-alpha23)/3/(tinfty13-tinfty23)*q;\n" }{MPLTEXT 1 0 39 "fact or(simplify(series(Lq-Lqbis,q=0)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "> I&LqbisG6\",**&I#muGF$\"\"\",*I)tinfty11GF$!\"\"I)tinfty21GF$F+*&,&I)t infty12GF$F+I)tinfty22GF$F+F(I\"qGF$F(F(*&,&I)tinfty13GF$F+I)tinfty23G F$F+F(F1\"\"#F(F(F+I$rhoGF$F6*&I\"hGF$F(I#nuGF$F(F+**F9F(,&I(alpha13GF $F(I(alpha23GF$F+F(,&F4F(F5F+F+F1F(#F+\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "+%I\"qG6\",$I$rhoGF$!\"%\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 226 34 "Let us look at \\mathcal\{L\}[L[2,1]]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Entry21:=simplify(LL[2,1]):\n" }{MPLTEXT 1 0 77 "Entry21TermLambdaMinusqCube:=factor(residue(Entry21*(lambda-q) ^2,lambda=q));\n" }{MPLTEXT 1 0 77 "Entry21TermLambdaMinusqSquare:=fac tor(residue(Entry21*(lambda-q),lambda=q));\n" }{MPLTEXT 1 0 60 "Entry2 1TermLambdaMinusq:=factor(residue(Entry21,lambda=q));\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 77 "Entry21TermLambdaInfty6:=factor(-residue(En try21/lambda^7,lambda=infinity));\n" }{MPLTEXT 1 0 77 "Entry21TermLamb daInfty5:=factor(-residue(Entry21/lambda^6,lambda=infinity));\n" } {MPLTEXT 1 0 77 "Entry21TermLambdaInfty4:=factor(-residue(Entry21/lamb da^5,lambda=infinity));\n" }{MPLTEXT 1 0 77 "Entry21TermLambdaInfty3:= factor(-residue(Entry21/lambda^4,lambda=infinity));\n" }{MPLTEXT 1 0 77 "Entry21TermLambdaInfty2:=factor(-residue(Entry21/lambda^3,lambda=i nfinity));\n" }{MPLTEXT 1 0 77 "Entry21TermLambdaInfty1:=factor(-resid ue(Entry21/lambda^2,lambda=infinity));\n" }{MPLTEXT 1 0 75 "Entry21Ter mLambdaInfty0:=factor(-residue(Entry21/lambda,lambda=infinity));\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 78 "simplify( Entry21-(Entry21TermLa mbdaMinusqCube/(lambda-q)^3+Entry21TermLambdaM" }{MPLTEXT 1 0 4 "inus" }{MPLTEXT 1 0 1 "q" }{MPLTEXT 1 0 6 "Square" }{MPLTEXT 1 0 32 "/(lamb da-q)^2+Entry21TermLambdaM" }{MPLTEXT 1 0 4 "inus" }{MPLTEXT 1 0 13 "q /(lambda-q)\n" }{MPLTEXT 1 0 122 "+Entry21TermLambdaInfty0+Entry21Term LambdaInfty1*lambda+Entry21TermLambdaInfty2*lambda^2+Entry21TermLambda Infty3*lambda^3\n" }{MPLTEXT 1 0 100 "+Entry21TermLambdaInfty4*lambda^ 4+Entry21TermLambdaInfty5*lambda^5+Entry21TermLambdaInfty6*lambda^6\n" }{MPLTEXT 1 0 5 ") );\n" }{MPLTEXT 1 0 8 "L[2,1];\n" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">II>Entry21TermLambdaMinusqSquareG6\",$*( I\"hGF$\"\"\",fo**I#muGF$F(I\"qGF$\"\"%I)tinfty13GF$\"\"#I)tinfty23GF$ F(!\"'**F+F(F,F-F.F(F0F/\"\"'*,F+F(F,\"\"$I)tinfty12GF$F(F.F(F0F(F1**F +F(F,F5F6F(F0F/F3**F+F(F,F5F.F/I)tinfty22GF$F(F1*,F+F(F,F5F.F(F9F(F0F( F3*,F+F(F,F/I)tinfty11GF$F(F.F(F0F(F1**F+F(F,F/FI8Ent ry21TermLambdaMinusqG6\",$*(I\"hGF$\"\"\",bo**I#muGF$F(I\"qGF$\"\"$I)t infty13GF$\"\"#I)tinfty23GF$F(!#7**F+F(F,F-F.F(F0F/\"#7*,F+F(F,F/I)tin fty12GF$F(F.F(F0F(!\"***F+F(F,F/F5F(F0F/\"\"***F+F(F,F/F.F/I)tinfty22G F$F(F6*,F+F(F,F/F.F(F:F(F0F(F8**I(alpha13GF$F(F'F(F,F/F0F(F-**I(alpha2 3GF$F(F'F(F,F/F.F(!\"$*,F+F(F,F(I)tinfty11GF$F(F.F(F0F(!\"'**F+F(F,F(F BF(F0F/\"\"'*,F+F(F,F(F5F(F.F(F:F(FC*,F+F(F,F(F5F(F:F(F0F(FE**F+F(F,F( F.F/I)tinfty21GF$F(FC*,F+F(F,F(F.F(FIF(F0F(FE**I#c2GF$F(F'F(F,F(F.F(FE **FLF(F'F(F,F(F0F(FC*(F'F(F+F(F.F/F@**F'F(F+F(F.F(F0F(F-*(F+F(I)tinfty 10GF$F(F.F/F-**F+F(FQF(F.F(F0F(FC*(F+F(FQF(F0F/F-**F+F(FBF(F.F(F:F(F@* *F+F(FBF(F:F(F0F(F-**F+F(F5F(F.F(FIF(F@**F+F(F5F(FIF(F0F(F-*(F,F(I$rho GF$F(F.F/FE*(F,F(FYF(F0F/FC*(F=F(F'F(I\"pGF$F(F(*(F?F(F'F(FfnF(!\"\"*( I#c1GF$F(F'F(F.F(F-*(FjnF(F'F(F0F(F@*(FYF(F5F(F.F(F-*(FYF(F5F(F0F(F@*( FYF(F.F(F:F(F-*(FYF(F:F(F0F(F@F(,&F.F(F0FhnFhn#FhnF-" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry21TermLambdaInfty6G6\"\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 ">I8Entry21TermLambdaInfty5G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry21TermLambdaInfty4G6\",$*&,&*&I(alpha13GF$\"\"\" I)tinfty23GF$F*F**&I(alpha23GF$F*I)tinfty13GF$F*F*F*I\"hGF$F*!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry21TermLambdaInfty3G6\",$*(I\"hGF $\"\"\",6*(I#nuGF$F(I)tinfty13GF$\"\"#I)tinfty23GF$F(\"#7*(F+F(F,F(F.F -!#7*(I(alpha13GF$F(I)tinfty12GF$F(F.F(F-*(F3F(F,F(I)tinfty22GF$F(\"\" &*(F3F(F6F(F.F(!\"$*(I(alpha23GF$F(F4F(F,F(\"\"$*(F;F(F4F(F.F(!\"&*(F; F(F,F(F6F(!\"#*&I#c2GF$F(F,F-!\"'*&FBF(F.F-\"\"'F(,&F,F(F.!\"\"FG#FGF< " }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry21TermLambdaInfty2G6\",$*(I \"hGF$\"\"\",J*(I#muGF$F(I)tinfty13GF$\"\"#I)tinfty23GF$F(\"\"'*(F+F(F ,F(F.F-!\"'**I#nuGF$F(I)tinfty12GF$F(F,F(F.F(\"\"**(F3F(F4F(F.F-!\"**( F3F(F,F-I)tinfty22GF$F(F5**F3F(F,F(F9F(F.F(F7*(I(alpha13GF$F(I)tinfty1 1GF$F(F.F(F(*(FI8Entry21TermLambdaInfty1G6\",$*(I\"hGF $\"\"\",Z**I#muGF$F(I)tinfty12GF$F(I)tinfty13GF$F(I)tinfty23GF$F(F(*(F +F(F,F(F.\"\"#!\"\"*(F+F(F-F0I)tinfty22GF$F(F(**F+F(F-F(F3F(F.F(F1**I# nuGF$F(I)tinfty11GF$F(F-F(F.F(F0*(F6F(F7F(F.F0!\"#**F6F(F,F(F-F(F3F(F0 **F6F(F,F(F3F(F.F(F9*(F6F(F-F0I)tinfty21GF$F(F0**F6F(F-F(F=F(F.F(F9*(I (alpha13GF$F(F'F(F-F(F(*(F@F(F'F(F.F(F1*(F@F(I)tinfty10GF$F(F-F(F1*(F@ F(FCF(F.F(F(*(F@F(F7F(F3F(F(*(F@F(F,F(F=F(F(*(I(alpha23GF$F(FCF(F-F(F( *(FHF(FCF(F.F(F1*(FHF(F7F(F3F(F1*(FHF(F,F(F=F(F1*(I#c1GF$F(F,F(F-F(F1* (FMF(F,F(F.F(F(*(FMF(F-F(F3F(F1*(FMF(F3F(F.F(F(*(I#c2GF$F(F7F(F-F(F9*( FRF(F7F(F.F(F0*(FRF(F-F(F=F(F9*(FRF(F=F(F.F(F0F(,&F-F(F.F1F1F1" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry21TermLambdaInfty0G6\",$*(I\"hGF $\"\"\",T**I#muGF$F(I\"qGF$\"\"#I)tinfty13GF$F-I)tinfty23GF$F(\"\"'**F +F(F,F-F.F(F/F-!\"'*,F+F(F,F(I)tinfty12GF$F(F.F(F/F(\"\"$**F+F(F,F(F4F (F/F-!\"$**F+F(F,F(F.F-I)tinfty22GF$F(F5*,F+F(F,F(F.F(F9F(F/F(F7**I(al pha13GF$F(F'F(F,F(F/F(F7**I(alpha23GF$F(F'F(F,F(F.F(F5*(F'F(I#nuGF$F(F .F-F7**F'F(F@F(F.F(F/F(F5*(F@F(I)tinfty10GF$F(F.F-F5**F@F(FCF(F.F(F/F( F2*(F@F(FCF(F/F-F5**F@F(I)tinfty11GF$F(F.F(F9F(F7**F@F(FGF(F9F(F/F(F5* *F@F(F4F(F.F(I)tinfty21GF$F(F7**F@F(F4F(FJF(F/F(F5*(I#c1GF$F(FGF(F.F(F 5*(FMF(FGF(F/F(F7*(FMF(F.F(FJF(F5*(FMF(FJF(F/F(F7*&I$rhoGF$F(F.F-F7*&F RF(F/F-F5*&I\"CGF$F(FF(!\"#F(,&F.F(F/!\"\"FY#F(F5" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ", 0*&,**&I)tinfty10G6\"\"\"\"I)tinfty13GF'F(!\"\"*&F&F(I)tinfty23GF'F(F( *&I)tinfty11GF'F(I)tinfty22GF'F(F(*&I)tinfty12GF'F(I)tinfty21GF'F(F(F( I'lambdaGF'F(F**&,(*&F.F(F,F(F(*&F1F(F/F(F(*&F)F(F2F(F(F(F3\"\"#F**&,& *&F1F(F,F(F(*&F)F(F/F(F(F(F3\"\"$F**(F)F(F,F(F3\"\"%F*I\"CGF'F(*(I\"hG F'F(F3F(F)F(F**(I\"pGF'F(FCF(,&F3F(I\"qGF'F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "rho:=factor(solve(Entry21TermLambdaMinusqCube ,rho));\n" }{MPLTEXT 1 0 23 "simplify(rho-(-p*mu));\n" }{MPLTEXT 1 0 39 "simplify(Entry21TermLambdaMinusqCube);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$rhoG6\",$*&I\"pGF$\"\"\"I#muGF$F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {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=infinity):\n" }{MPLTEXT 1 0 60 "L21Ord erLambda2:=-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=infi nity):\n" }{MPLTEXT 1 0 274 "simplify(h*(alpha13*diff(L21OrderLambda4, tinfty13)+alpha23*diff(L21OrderLambda4,tinfty23)+alpha12*diff(L21Order Lambda4,tinfty12)+alpha22*diff(L21OrderLambda4,tinfty22)+alpha11*diff( L21OrderLambda4,tinfty11)+alpha21*diff(L21OrderLambda4,tinfty21))- Ent ry21TermLambdaInfty4);\n" }{MPLTEXT 1 0 293 "Equation3:=factor(simplif y(h*(alpha13*diff(L21OrderLambda3,tinfty13)+alpha23*diff(L21OrderLambd a3,tinfty23)+alpha12*diff(L21OrderLambda3,tinfty12)+alpha22*diff(L21Or derLambda3,tinfty22)+alpha11*diff(L21OrderLambda3,tinfty11)+alpha21*di ff(L21OrderLambda3,tinfty21))- Entry21TermLambdaInfty3));\n" }{MPLTEXT 1 0 293 "Equation4:=factor(simplify(h*(alpha13*diff(L21OrderLambda2,t infty13)+alpha23*diff(L21OrderLambda2,tinfty23)+alpha12*diff(L21OrderL ambda2,tinfty12)+alpha22*diff(L21OrderLambda2,tinfty22)+alpha11*diff(L 21OrderLambda2,tinfty11)+alpha21*diff(L21OrderLambda2,tinfty21))- Entr y21TermLambdaInfty2));\n" }{MPLTEXT 1 0 293 "Equation5:=factor(simplif y(h*(alpha13*diff(L21OrderLambda1,tinfty13)+alpha23*diff(L21OrderLambd a1,tinfty23)+alpha12*diff(L21OrderLambda1,tinfty12)+alpha22*diff(L21Or derLambda1,tinfty22)+alpha11*diff(L21OrderLambda1,tinfty11)+alpha21*di ff(L21OrderLambda1,tinfty21))- Entry21TermLambdaInfty1));\n" }{MPLTEXT 1 0 40 "Equation1:=factor(simplify(Equation1));\n" }{MPLTEXT 1 0 40 " Equation2:=factor(simplify(Equation2));\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Equation3G6\",$*(I\"hGF$\"\"\",:*(I#nuGF$F(I)t infty13GF$\"\"#I)tinfty23GF$F(!#7*(F+F(F,F(F.F-\"#7*(I(alpha12GF$F(F,F (F.F(\"\"$*&F3F(F.F-!\"$*(I(alpha13GF$F(I)tinfty12GF$F(F.F(!\"#*(F8F(F ,F(I)tinfty22GF$F(F:*&I(alpha22GF$F(F,F-F4*(F>F(F,F(F.F(F6*(I(alpha23G F$F(F9F(F.F(F-*(FAF(F,F(FI*Equation4G6\",$*( I\"hGF$\"\"\",V*(I#muGF$F(I)tinfty13GF$\"\"#I)tinfty23GF$F(!\"'*(F+F(F ,F(F.F-\"\"'**I#nuGF$F(I)tinfty12GF$F(F,F(F.F(!\"**(F3F(F4F(F.F-\"\"** (F3F(F,F-I)tinfty22GF$F(F5**F3F(F,F(F9F(F.F(F7*(I(alpha11GF$F(F,F(F.F( \"\"$*&FI*Equation5G6\",$*(I\"hGF$\"\"\",hn**I#muGF$F(I)tin fty12GF$F(I)tinfty13GF$F(I)tinfty23GF$F(!\"\"*(F+F(F,F(F.\"\"#F(*(F+F( F-F1I)tinfty22GF$F(F/**F+F(F-F(F3F(F.F(F(**I#nuGF$F(I)tinfty11GF$F(F-F (F.F(!\"#*(F6F(F7F(F.F1F1**F6F(F,F(F-F(F3F(F8**F6F(F,F(F3F(F.F(F1*(F6F (F-F1I)tinfty21GF$F(F8**F6F(F-F(F=F(F.F(F1*(I(alpha11GF$F(F-F(F3F(F(*( F@F(F3F(F.F(F/*(I(alpha12GF$F(F-F(F=F(F(*(FCF(F=F(F.F(F/*(I(alpha13GF$ F(F7F(F3F(F/*(FFF(F,F(F=F(F/*(I(alpha21GF$F(F,F(F-F(F(*(FIF(F,F(F.F(F/ *(I(alpha22GF$F(F7F(F-F(F(*(FLF(F7F(F.F(F/*(I(alpha23GF$F(F7F(F3F(F(*( FOF(F,F(F=F(F(*(I#c1GF$F(F,F(F-F(F(*(FRF(F,F(F.F(F/*(FRF(F-F(F3F(F(*(F RF(F3F(F.F(F/*(I#c2GF$F(F7F(F-F(F1*(FWF(F7F(F.F(F8*(FWF(F-F(F=F(F1*(FW F(F=F(F.F(F8F(,&F-F(F.F/F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Equat ion1G6\",$*(I\"hGF$\"\"\",:*&I#nuGF$F(I)tinfty13GF$\"\"#!\"'*&F+F(I)ti nfty23GF$F-\"\"'*&I(alpha12GF$F(F,F(\"\"$*&F3F(F0F(!\"$*&I(alpha13GF$F (I)tinfty12GF$F(!\"#*&F8F(I)tinfty22GF$F(F:*&I(alpha22GF$F(F,F(F4*&F>F (F0F(F6*&I(alpha23GF$F(F9F(F-*&FAF(FI*Equation 2G6\",$*(I\"hGF$\"\"\",B*&I#muGF$F(I)tinfty13GF$\"\"#!\"$*&F+F(I)tinft y23GF$F-\"\"$*(I#nuGF$F(I)tinfty12GF$F(F,F(F.*(F3F(F4F(F0F(F1*(F3F(F,F (I)tinfty22GF$F(F.*(F3F(F7F(F0F(F1*&I(alpha11GF$F(F,F(F1*&F:F(F0F(F.*& I(alpha13GF$F(I)tinfty11GF$F(!\"\"*&F=F(I)tinfty21GF$F(F?*&I(alpha21GF $F(F,F(F1*&FCF(F0F(F.*&I(alpha23GF$F(F>F(F(*&FFF(FAF(F(*&I#c1GF$F(F,F( \"\"'*&FIF(F0F(!\"'F(,&F,F(F0F?F?#F?F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "LpFunction:=unapply(-Entry21TermLambdaMinusq/h,C):\n" }{MPLTEXT 1 0 62 "Equation7:=simplify(Entry21TermLambdaMinusqSquare-( -p*h*Lq)):\n" }{MPLTEXT 1 0 26 "Csol:=solve(Equation7,C):\n" }{MPLTEXT 1 0 51 "Csolbis:=p^2- P1(q)*p+P2(q)-Pinfty02+h*q*tinfty13:\n" } {MPLTEXT 1 0 33 "factor(series(Csol-Csolbis,p=0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Lp:= factor(simplify(LpFunction(Csol))):\n" }{MPLTEXT 1 0 54 "Lpbis:=mu*(p* diff(P1(q),q) -diff(P2(q),q)-h*tinfty13)\n" }{MPLTEXT 1 0 107 "+h/3*(a lpha13-alpha23)/(tinfty13-tinfty23)*p+h*(alpha13*tinfty23-alpha23*tinf ty13)/(tinfty13-tinfty23)*q^2\n" }{MPLTEXT 1 0 16 "+2*h*c2*q+h*c1;\n" }{MPLTEXT 1 0 29 "factor(series(Lp-Lpbis,q=0));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 74 "Lqbis:=-mu*P1(q)+2*p*mu-h*nu-h* (alpha13-alpha23)/3/(tinfty13-tinfty23)*q:\n" }{MPLTEXT 1 0 38 "factor (simplify(series(Lq-Lqbis,q=0)))" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I&L pbisG6\",,*&I#muGF$\"\"\",4*&I\"pGF$F(,(I)tinfty12GF$!\"\"I)tinfty22GF $F.*&,&I)tinfty13GF$F.I)tinfty23GF$F.F(I\"qGF$F(\"\"#F(F(*&I)tinfty10G F$F(F2F(F(*&F7F(F3F(F.*&I)tinfty11GF$F(F/F(F.*&F-F(I)tinfty21GF$F(F.*& ,(*&F:F(F3F(F(*&F-F(F/F(F(*&F2F(F " 0 "" {MPLTEXT 1 0 186 "Hamiltonian:= mu*(p^2+tdP2(q)-p*P1(q)+h*tinfty13*q)- h*nu*p-h/3*(alpha13-alpha23)/(tinfty13-tinfty23)*p*q-h/3*(alpha13*tinf ty23-alpha23*tinfty13)/(tinfty13-tinfty23)*q^3-h*c2*q^2-h*c1*q:\n" } {MPLTEXT 1 0 37 "simplify(Lp-(-diff(Hamiltonian,q)));\n" }{MPLTEXT 1 0 35 "simplify(Lq-(diff(Hamiltonian,p)));" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 225 99 "Decomposition of the tangent space: shift of Darboux coordinates and non-trivial isomonodromic tim e" }}{EXCHG {PARA 0 "" 0 "" {TEXT 217 89 "From previous Maple sheet, w e have some expressions for the coefficients (c_1,c_2,mu,nu)." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 215 "nualter:= -1/6*(3*alpha12*t infty23+3*alpha22*tinfty13-3*alpha12*tinfty13+2*alpha23*tinfty22+2*alp ha13*tinfty12-2*alpha13*tinfty22-3*alpha22*tinfty23-2*alpha23*tinfty12 )/(-2*tinfty13*tinfty23+tinfty23^2+tinfty13^2):\n" }{MPLTEXT 1 0 817 " mualter := 1/6*(-2*tinfty13*alpha13*tinfty11+2*tinfty23*alpha23*tinfty 21-3*alpha12*tinfty22*tinfty23-3*alpha22*tinfty12*tinfty23+6*alpha11*t infty23^2-6*alpha21*tinfty13^2+3*alpha12*tinfty22*tinfty13-12*alpha11* tinfty23*tinfty13+3*alpha22*tinfty12*tinfty13+12*alpha21*tinfty13*tinf ty23+6*alpha11*tinfty13^2-6*alpha21*tinfty23^2+2*alpha13*tinfty11*tinf ty23-3*tinfty22*alpha22*tinfty13+3*tinfty22*alpha22*tinfty23+4*alpha23 *tinfty12*tinfty22-2*alpha23*tinfty13*tinfty21-2*alpha23*tinfty11*tinf ty23-4*alpha13*tinfty12*tinfty22+2*alpha13*tinfty13*tinfty21-2*alpha23 *tinfty22^2+2*alpha13*tinfty22^2-2*alpha23*tinfty12^2+2*alpha13*tinfty 12^2-3*tinfty12*alpha12*tinfty13+3*tinfty12*alpha12*tinfty23+2*tinfty1 3*alpha23*tinfty11-2*alpha13*tinfty21*tinfty23)/(3*tinfty13*tinfty23^2 -3*tinfty13^2*tinfty23+tinfty13^3-tinfty23^3):\n" }{MPLTEXT 1 0 274 "c 2alter := 1/6*(-3*alpha22*tinfty13^2+3*alpha22*tinfty13*tinfty23+2*alp ha13*tinfty13*tinfty22+3*alpha12*tinfty23*tinfty13-2*alpha23*tinfty13* tinfty22-2*alpha13*tinfty12*tinfty23-3*alpha12*tinfty23^2+2*alpha23*ti nfty12*tinfty23)/(-2*tinfty13*tinfty23+tinfty23^2+tinfty13^2):\n" } {MPLTEXT 1 0 1060 "c1alter:=factor(1/6*(-6*alpha21*tinfty13^3+6*alpha1 1*tinfty23^3-3*tinfty13*alpha12*tinfty22*tinfty23+3*tinfty13*alpha22*t infty12*tinfty23-2*tinfty13*alpha13*tinfty11*tinfty23+3*tinfty13*tinft y22*alpha22*tinfty23+2*tinfty13*alpha23*tinfty12*tinfty22+2*tinfty13*a lpha23*tinfty11*tinfty23-2*tinfty13*alpha13*tinfty12*tinfty22-3*tinfty 13*tinfty12*alpha12*tinfty23+2*tinfty23*alpha23*tinfty12*tinfty22+2*ti nfty23*alpha23*tinfty13*tinfty21-2*tinfty23*alpha13*tinfty12*tinfty22- 2*tinfty23*alpha13*tinfty13*tinfty21+2*alpha13*tinfty13^2*tinfty21+6*a lpha11*tinfty23*tinfty13^2+3*alpha12*tinfty22*tinfty13^2-6*alpha21*tin fty13*tinfty23^2+12*alpha21*tinfty13^2*tinfty23-3*tinfty22*alpha22*tin fty13^2-3*alpha22*tinfty12*tinfty23^2+2*alpha13*tinfty11*tinfty23^2-2* alpha23*tinfty13^2*tinfty21-2*alpha23*tinfty11*tinfty23^2+3*tinfty12*a lpha12*tinfty23^2-12*tinfty13*alpha11*tinfty23^2-2*tinfty13*alpha23*ti nfty22^2+2*tinfty13*alpha13*tinfty22^2-2*tinfty23*alpha23*tinfty12^2+2 *tinfty23*alpha13*tinfty12^2)/(3*tinfty13*tinfty23^2-3*tinfty13^2*tinf ty23+tinfty13^3-tinfty23^3)):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 226 82 "Expression of the Lax matrix in the geometric gaug e without apparent singularities" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "C:=Csol:\n" }{MPLTEXT 1 0 12 "Q2(lambda);\n" }{MPLTEXT 1 0 5 " check" }{MPLTEXT 1 0 21 "L11bis:=-Q2(lambda);\n" }{MPLTEXT 1 0 5 "chec k" }{MPLTEXT 1 0 1 "L" }{MPLTEXT 1 0 19 "12bis:=(lambda-q);\n" } {MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 1 "L" }{MPLTEXT 1 0 30 "22bis:=P1 (lambda)+Q2(lambda);\n" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 1 "L" } {MPLTEXT 1 0 95 "21bis:=h*diff(Q2(lambda)/(lambda-q),lambda)+L[2,1]/(l ambda-q)-P1(lambda)*Q2(lambda)/(lambda-q)\n" }{MPLTEXT 1 0 26 "-Q2(lam bda)^2/(lambda-q):\n" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 6 "che ckL" }{MPLTEXT 1 0 6 "[1,1]-" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 1 "L" }{MPLTEXT 1 0 8 "11bis);\n" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 1 "L" }{MPLTEXT 1 0 6 "[1,2]-" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 1 "L" }{MPLTEXT 1 0 8 "12bis);\n" } {MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 1 "L" }{MPLTEXT 1 0 6 "[2,2]-" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 9 "L22 bis);\n" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 1 "L" }{MPLTEXT 1 0 6 "[2,1]-" }{MPLTEXT 1 0 6 "checkL" }{MPLTEXT 1 0 8 "21bis);\n" }{MPLTEXT 1 0 17 "simplify(residue(" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 1 "L" }{MPLTEXT 1 0 18 "[2,1],lambda=q));\n" } {MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$I\"pG6\"!\"\"" } }{PARA 11 "" 1 "" {XPPMATH 20 ">I,checkL11bisG6\"I\"pGF$" }}{PARA 11 " " 1 "" {XPPMATH 20 ">I,checkL12bisG6\",&I'lambdaGF$\"\"\"I\"qGF$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,checkL22bisG6\",,*&,&I)tinfty13GF$ !\"\"I)tinfty23GF$F)\"\"\"I'lambdaGF$\"\"#F+*&,&I)tinfty12GF$F)I)tinft y22GF$F)F+F,F+F+I\"pGF$F)I)tinfty11GF$F)I)tinfty21GF$F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 217 91 "Expression of the evolution in the traceless setting a nd decomposition of the tangent space" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "p:=tdp+P1(q)/2:\n" }{MPLTEXT 1 0 38 "Ltdp:=simplify( \+ Lp-dP1dlambda(q)/2*Lq\n" }{MPLTEXT 1 0 187 "- 1/2*h*(alpha13*diff(P1(q ),tinfty13)+alpha23*diff(P1(q),tinfty23)+alpha12*diff(P1(q),tinfty12)+ alpha22*diff(P1(q),tinfty22)+alpha11*diff(P1(q),tinfty11)+alpha21*diff (P1(q),tinfty21)) ):\n" }{MPLTEXT 1 0 38 "Ltdpbis:=mu*(diff(P1(q)^2/4- P2(q),q))\n" }{MPLTEXT 1 0 47 "+h/3*(alpha13-alpha23)/(tinfty13-tinfty 23)*tdp\n" }{MPLTEXT 1 0 121 "+( 2*c2+ ((alpha12+alpha22)/2-nu*(tinfty 13+tinfty23))-1/3*(tinfty12+tinfty22)*(alpha13-alpha23)/(tinfty13-tinf ty23))*h*q\n" }{MPLTEXT 1 0 128 "+h*(c1-mu*tinfty13-1/6*(tinfty11+tinf ty21)*(alpha13-alpha23)/(tinfty13-tinfty23)+1/2*((alpha11+alpha21)-nu* (tinfty12+tinfty22)))" }{MPLTEXT 1 0 1 ":" }{MPLTEXT 1 0 2 " \n" } {MPLTEXT 1 0 33 "factor(series(Ltdp-Ltdpbis,q=0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "Qua ntiteq:=unapply( 2*c2+ ((alpha12+alpha22)/2-nu*(tinfty13+tinfty23))-1/ 3*(tinfty12+tinfty22)*(alpha13-alpha23)/(tinfty13-tinfty23), c2,nu);\n " }{MPLTEXT 1 0 126 "QuantiteConstant:=unapply(c1-mu*tinfty13-1/6*(tin fty11+tinfty21)*(alpha13-alpha23)/(tinfty13-tinfty23)+1/2*((alpha11+al pha21)-" }{MPLTEXT 1 0 34 "nu*(tinfty12+tinfty22)),c1,nu,mu);" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I*QuantiteqG6\"f*6$I#c2GF$I#nuGF$F$6$I) operatorGF$I&arrowGF$F$,,9$\"\"#I(alpha12GF$#\"\"\"F.I(alpha22GF$F0*&9 %F1,&I)tinfty13GF$F1I)tinfty23GF$F1F1!\"\"*(,&I)tinfty12GF$F1I)tinfty2 2GF$F1F1,&I(alpha13GF$F1I(alpha23GF$F8F1,&F6F1F7F8F8#F8\"\"$F$F$F$" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I1QuantiteConstantG6\"f*6%I#c1GF$I#nuGF $I#muGF$F$6$I)operatorGF$I&arrowGF$F$,.9$\"\"\"*&9&F/I)tinfty13GF$F/! \"\"*(,&I)tinfty21GF$F/I)tinfty11GF$F/F/,&I(alpha13GF$F/I(alpha23GF$F3 F/,&F2F/I)tinfty23GF$F3F3#F3\"\"'I(alpha11GF$#F/\"\"#I(alpha21GF$F@*&, &I)tinfty12GF$F/I)tinfty22GF$F/F/9%F/#F3FAF$F$F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "mu:=mualter:\n" }{MPLTEXT 1 0 13 "nu:=nualter :\n" }{MPLTEXT 1 0 13 "c1:=c1alter:\n" }{MPLTEXT 1 0 12 "c2:=c2alter:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "Lpfunction:=unapply(simp lify(Lp),alpha13,alpha23,alpha12,alpha22,alpha11,alpha21):\n" } {MPLTEXT 1 0 53 "Ltdpfunction:=unapply(simplify(Ltdp),alpha13,alpha23, " }{MPLTEXT 1 0 31 "alpha12,alpha22,alpha11,alpha21" }{MPLTEXT 1 0 3 " ):\n" }{MPLTEXT 1 0 49 "Lqfunction:=unapply(simplify(Lq),alpha13,alpha 23," }{MPLTEXT 1 0 31 "alpha12,alpha22,alpha11,alpha21" }{MPLTEXT 1 0 3 "):\n" }{MPLTEXT 1 0 44 "c1function:=unapply(c1alter,alpha13,alpha23 ," }{MPLTEXT 1 0 32 "alpha12,alpha22,alpha11,alpha21)" }{MPLTEXT 1 0 2 ":\n" }{MPLTEXT 1 0 22 "c2function:=unapply(c2" }{MPLTEXT 1 0 5 "alt er" }{MPLTEXT 1 0 17 ",alpha13,alpha23," }{MPLTEXT 1 0 31 "alpha12,alp ha22,alpha11,alpha21" }{MPLTEXT 1 0 3 "):\n" }{MPLTEXT 1 0 22 "nufunct ion:=unapply(nu" }{MPLTEXT 1 0 5 "alter" }{MPLTEXT 1 0 17 ",alpha13,al pha23," }{MPLTEXT 1 0 31 "alpha12,alpha22,alpha11,alpha21" }{MPLTEXT 1 0 3 "):\n" }{MPLTEXT 1 0 1 "m" }{MPLTEXT 1 0 21 "ufunction:=unapply( mu" }{MPLTEXT 1 0 5 "alter" }{MPLTEXT 1 0 17 ",alpha13,alpha23," } {MPLTEXT 1 0 31 "alpha12,alpha22,alpha11,alpha21" }{MPLTEXT 1 0 2 "):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "factor(Ltdpfunction(1,1, 0,0,0,0));\n" }{MPLTEXT 1 0 33 "factor(Lqfunction(1,1,0,0,0,0));\n" } {MPLTEXT 1 0 33 "factor(c1function(1,1,0,0,0,0));\n" }{MPLTEXT 1 0 33 "factor(c2function(1,1,0,0,0,0));\n" }{MPLTEXT 1 0 33 "factor(nufuncti on(1,1,0,0,0,0));\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 33 "factor(Lqfunction(0,0,1,1,0,0));\n" }{MPLTEXT 1 0 35 "factor(Ltdpfunction(0,0,1,1,0,0));\n" }{MPLTEXT 1 0 33 "factor(c1func tion(0,0,1,1,0,0));\n" }{MPLTEXT 1 0 33 "factor(c2function(0,0,1,1,0,0 ));\n" }{MPLTEXT 1 0 33 "factor(nufunction(0,0,1,1,0,0));\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 33 "factor(Lqfunction(0 ,0,0,0,1,1));\n" }{MPLTEXT 1 0 35 "factor(Ltdpfunction(0,0,0,0,1,1)); \n" }{MPLTEXT 1 0 33 "factor(c1function(0,0,0,0,1,1));\n" }{MPLTEXT 1 0 33 "factor(c2function(0,0,0,0,1,1));\n" }{MPLTEXT 1 0 33 "factor(nuf unction(0,0,0,0,1,1));\n" }{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 "\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "#!\"\"\"\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " \"\"!" }}{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 35 "simplify(mufunction(1,1,0,0,0,0));\n" }{MPLTEXT 1 0 35 "simplify(m ufunction(0,0,1,1,0,0));\n" }{MPLTEXT 1 0 35 "simplify(mufunction(0,0, 0,0,1,1));\n" }{MPLTEXT 1 0 67 "simplify(mufunction(0,0,2*tinfty13,2*t infty23,tinfty12,tinfty22));\n" }{MPLTEXT 1 0 85 "simplify(mufunction( 3*tinfty13,3*tinfty23,2*tinfty12,2*tinfty22,tinfty11,tinfty21));\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 35 "simplify(nufunction(1,1,0,0,0,0) );\n" }{MPLTEXT 1 0 35 "simplify(nufunction(0,0,1,1,0,0));\n" } {MPLTEXT 1 0 35 "simplify(nufunction(0,0,0,0,1,1));\n" }{MPLTEXT 1 0 67 "simplify(nufunction(0,0,2*tinfty13,2*tinfty23,tinfty12,tinfty22)); \n" }{MPLTEXT 1 0 85 "simplify(nufunction(3*tinfty13,3*tinfty23,2*tinf ty12,2*tinfty22,tinfty11,tinfty21));\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 49 "simplify(mufunction(0,0,0,0,tinfty13,tinfty23));\n" }{MPLTEXT 1 0 48 "simplify(nufunction(0,0,0,0,tinfty13,tinfty23));" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{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 67 "simplify(mufunction(0,0,2*tinfty13,2*tinfty23,ti nfty12,tinfty22));\n" }{MPLTEXT 1 0 67 "factor(Ltdpfunction(0,0,2*tinf ty13,2*tinfty23,tinfty12,tinfty22));\n" }{MPLTEXT 1 0 65 "factor(Lpfun ction(0,0,2*tinfty13,2*tinfty23,tinfty12,tinfty22));\n" }{MPLTEXT 1 0 72 "factor(Lqfunction(0,0,2*tinfty13,2*tinfty23,tinfty12,tinfty22) - ( -h));\n" }{MPLTEXT 1 0 65 "factor(c1function(0,0,2*tinfty13,2*tinfty23 ,tinfty12,tinfty22));\n" }{MPLTEXT 1 0 65 "factor(c2function(0,0,2*tin fty13,2*tinfty23,tinfty12,tinfty22));\n" }{MPLTEXT 1 0 64 "factor(nufu nction(0,0,2*tinfty13,2*tinfty23,tinfty12,tinfty22));" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{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 85 "simplify(mufunction(3*tinfty13,3*ti nfty23,2*tinfty12,2*tinfty22,tinfty11,tinfty21));\n" }{MPLTEXT 1 0 87 "factor(Lpfunction(3*tinfty13,3*tinfty23,2*tinfty12,2*tinfty22,tinfty1 1,tinfty21)-h*p);\n" }{MPLTEXT 1 0 92 "factor(Ltdpfunction(3*tinfty13, 3*tinfty23,2*tinfty12,2*tinfty22,tinfty11,tinfty21)- h*tdp);\n" } {MPLTEXT 1 0 91 "factor(Lqfunction(3*tinfty13,3*tinfty23,2*tinfty12,2* tinfty22,tinfty11,tinfty21) -(-h*q));\n" }{MPLTEXT 1 0 85 "simplify(c1 function(3*tinfty13,3*tinfty23,2*tinfty12,2*tinfty22,tinfty11,tinfty21 ));\n" }{MPLTEXT 1 0 85 "simplify(c2function(3*tinfty13,3*tinfty23,2*t infty12,2*tinfty22,tinfty11,tinfty21));\n" }{MPLTEXT 1 0 84 "simplify( nufunction(3*tinfty13,3*tinfty23,2*tinfty12,2*tinfty22,tinfty11,tinfty 21));" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{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 57 "factor(simplify(mu function(0,0,0,0,tinfty13,tinfty23)));\n" }{MPLTEXT 1 0 103 "factor(Lt dpfunction(0,0,0,0,tinfty13,tinfty23)- (diff((P1(q)^2/4-P2(q)),q) -h/2 *(tinfty13-tinfty23)));\n" }{MPLTEXT 1 0 57 "factor(Lqfunction(0,0,0,0 ,tinfty13,tinfty23)- (2*tdp) );\n" }{MPLTEXT 1 0 49 "simplify(c1functi on(0,0,0,0,tinfty13,tinfty23));\n" }{MPLTEXT 1 0 49 "simplify(c2functi on(0,0,0,0,tinfty13,tinfty23));\n" }{MPLTEXT 1 0 48 "simplify(nufuncti on(0,0,0,0,tinfty13,tinfty23));" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\" \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{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 33 "factor(Lqfunction(1,1,0,0,0, 0));\n" }{MPLTEXT 1 0 33 "factor(Lqfunction(0,0,1,1,0,0));\n" } {MPLTEXT 1 0 33 "factor(Lqfunction(0,0,0,0,1,1));\n" }{MPLTEXT 1 0 72 "factor(Lqfunction(0,0,2*tinfty13,2*tinfty23,tinfty12,tinfty22) - (-h) );\n" }{MPLTEXT 1 0 90 "factor(Lqfunction(3*tinfty13,3*tinfty23,2*tinf ty12,2*tinfty22,tinfty11,tinfty21) -(-h*q));" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{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 10 "pdsolve(\{\n" }{MPLTEXT 1 0 63 "diff(Q(t1,t2,t3,t4,t5 ,t6),t1)+diff(Q(t1,t2,t3,t4,t5,t6),t2)=0,\n" }{MPLTEXT 1 0 63 "diff(Q( t1,t2,t3,t4,t5,t6),t3)+diff(Q(t1,t2,t3,t4,t5,t6),t4)=0,\n" }{MPLTEXT 1 0 63 "diff(Q(t1,t2,t3,t4,t5,t6),t5)+diff(Q(t1,t2,t3,t4,t5,t6),t6)=0, \n" }{MPLTEXT 1 0 140 "2*t1*diff(Q(t1,t2,t3,t4,t5,t6),t3)+2*t2*diff(Q( t1,t2,t3,t4,t5,t6),t4)+t3*diff(Q(t1,t2,t3,t4,t5,t6),t5)+t4*diff(Q(t1,t 2,t3,t4,t5,t6),t6)=-1,\n" }{MPLTEXT 1 0 228 "3*t1*diff(Q(t1,t2,t3,t4,t 5,t6),t1)+3*t2*diff(Q(t1,t2,t3,t4,t5,t6),t2)+2*t3*diff(Q(t1,t2,t3,t4,t 5,t6),t3)+2*t4*diff(Q(t1,t2,t3,t4,t5,t6),t4)+t5*diff(Q(t1,t2,t3,t4,t5, t6),t5)+t6*diff(Q(t1,t2,t3,t4,t5,t6),t6)=-Q(t1,t2,t3,t4,t5,t6)\n" } {MPLTEXT 1 0 24 "\},Q(t1,t2,t3,t4,t5,t6));" }}{PARA 11 "" 1 "" {XPPMATH 20 "<#/-I\"QG6\"6(I#t1G6\"I#t2G6\"I#t3G6\"I#t4G6\"I#t5G6\"I#t 6G6\",$*&,&I#t1G6\"!\"\"I#t2G6\"\"\"\"#!\"%\"\"$,&*&,&I#t1G6\"\"\"\"I# t2G6\"!\"\"\"\"\"-I$_F1G6\"6#*&,(*&,&I#t5G6\"\"\"\"I#t6G6\"!\"\"\"\"\" I#t1G6\"\"\"\"\"\"\"*&I#t2G6\"\"\"\",&I#t5G6\"!\"\"I#t6G6\"\"\"\"\"\" \"\"\"\"*$,&I#t3G6\"\"\"\"I#t4G6\"!\"\"\"\"##!\"\"\"\"%\"\"\",&I#t1G6 \"!\"\"I#t2G6\"\"\"\"#!\"%\"\"$\"\"\"\"\"\"*&,&I#t3G6\"\"\"\"I#t4G6\"! \"\"\"\"\",&I#t1G6\"!\"\"I#t2G6\"\"\"\"#\"\"\"\"\"$#!\"\"\"\"#\"\"\"! \"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "factor(Ltdpfunction (1,1,0,0,0,0));\n" }{MPLTEXT 1 0 35 "factor(Ltdpfunction(0,0,1,1,0,0)) ;\n" }{MPLTEXT 1 0 35 "factor(Ltdpfunction(0,0,0,0,1,1));\n" }{MPLTEXT 1 0 67 "factor(Ltdpfunction(0,0,2*tinfty13,2*tinfty23,tinfty12,tinfty 22));\n" }{MPLTEXT 1 0 93 "factor(Ltdpfunction(3*tinfty13,3*tinfty23,2 *tinfty12,2*tinfty22,tinfty11,tinfty21) -(h*tdp));" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "pdsolve(\{\n" }{MPLTEXT 1 0 67 "diff(tdP(t1,t2,t3,t4, t5,t6),t1)+diff(tdP(t1,t2,t3,t4,t5,t6),t2)=0,\n" }{MPLTEXT 1 0 67 "dif f(tdP(t1,t2,t3,t4,t5,t6),t3)+diff(tdP(t1,t2,t3,t4,t5,t6),t4)=0,\n" } {MPLTEXT 1 0 67 "diff(tdP(t1,t2,t3,t4,t5,t6),t5)+diff(tdP(t1,t2,t3,t4, t5,t6),t6)=0,\n" }{MPLTEXT 1 0 147 "2*t1*diff(tdP(t1,t2,t3,t4,t5,t6),t 3)+2*t2*diff(tdP(t1,t2,t3,t4,t5,t6),t4)+t3*diff(tdP(t1,t2,t3,t4,t5,t6) ,t5)+t4*diff(tdP(t1,t2,t3,t4,t5,t6),t6)=0,\n" }{MPLTEXT 1 0 241 "3*t1* diff(tdP(t1,t2,t3,t4,t5,t6),t1)+3*t2*diff(tdP(t1,t2,t3,t4,t5,t6),t2)+2 *t3*diff(tdP(t1,t2,t3,t4,t5,t6),t3)+2*t4*diff(tdP(t1,t2,t3,t4,t5,t6),t 4)+t5*diff(tdP(t1,t2,t3,t4,t5,t6),t5)+t6*diff(tdP(t1,t2,t3,t4,t5,t6),t 6)=tdP(t1,t2,t3,t4,t5,t6)\n" }{MPLTEXT 1 0 26 "\},tdP(t1,t2,t3,t4,t5,t 6));" }}{PARA 11 "" 1 "" {XPPMATH 20 "<#/-I$tdPG6\"6(I#t1G6\"I#t2G6\"I #t3G6\"I#t4G6\"I#t5G6\"I#t6G6\"*&-I$_F1G6\"6#*&,(*&,&I#t5G6\"\"\"\"I#t 6G6\"!\"\"\"\"\"I#t1G6\"\"\"\"\"\"\"*&I#t2G6\"\"\"\",&I#t5G6\"!\"\"I#t 6G6\"\"\"\"\"\"\"\"\"\"*$,&I#t3G6\"\"\"\"I#t4G6\"!\"\"\"\"##!\"\"\"\"% \"\"\",&I#t1G6\"!\"\"I#t2G6\"\"\"\"#!\"%\"\"$\"\"\",&I#t1G6\"!\"\"I#t2 G6\"\"\"\"#\"\"\"\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 88 "This g ives the shift of the Darboux coordinates and the non-trivial isomonod romic times." }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 225 112 "Expression of the Lax matrices in the geometric gauge after the symplectic reductio n and the Painlev\351 2 equation" }}{EXCHG {PARA 0 "" 0 "" {TEXT 217 183 "Simplification of the formulas after the reduction and expression of the Lax matrices in the geometric gauge after reduction. In this c ase, we have \\check\{q\}=q and \\check\{p\}=p=\\td\{p\}." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "tinfty23:=-tinfty13:\n" }{MPLTEXT 1 0 21 "tinfty22:=-tinfty12:\n" }{MPLTEXT 1 0 21 "tinfty21:=-tinfty11: \n" }{MPLTEXT 1 0 21 "tinfty20:=-tinfty10:\n" }{MPLTEXT 1 0 17 "tinfty 11:=tau/2:\n" }{MPLTEXT 1 0 13 "tinfty13:=1:\n" }{MPLTEXT 1 0 13 "tinf ty12:=0:\n" }{MPLTEXT 1 0 7 "q:=tdq:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "c2:=c2alter;\n" }{MPLTEXT 1 0 13 "c1:=c1alter;\n" } {MPLTEXT 1 0 7 "c0:=0:\n" }{MPLTEXT 1 0 13 "nu:=nualter;\n" }{MPLTEXT 1 0 13 "mu:=mualter;\n" }{MPLTEXT 1 0 14 "alpha11:=1/2:\n" }{MPLTEXT 1 0 7 "alpha21" }{MPLTEXT 1 0 8 ":=-1/2:\n" }{MPLTEXT 1 0 12 "alpha13: =0:\n" }{MPLTEXT 1 0 12 "alpha23:=0:\n" }{MPLTEXT 1 0 7 "alpha12" } {MPLTEXT 1 0 5 ":=0:\n" }{MPLTEXT 1 0 7 "alpha22" }{MPLTEXT 1 0 5 ":=0 :\n" }{MPLTEXT 1 0 17 "checkL:=simplify(" }{MPLTEXT 1 0 5 "check" } {MPLTEXT 1 0 4 "L);\n" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 12 "A:=si mplify(" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 3 "A);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#c2G6\",&I(alpha12GF$#!\"\"\"\"%I(alpha22GF$F'" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I#c1G6\",&I(alpha11GF$#!\"\"\"\"#I(alph a21GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#nuG6\",&I(alpha12GF$#\"\" \"\"\"%I(alpha22GF$#!\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#muG6\" ,**&I(alpha13GF$\"\"\"I$tauGF$F(#!\"\"\"#7*&I(alpha23GF$F(F)F(#F(F,I(a lpha11GF$#F(\"\"#I(alpha21GF$#F+F2" }}{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#miGF$6&Q$tdpF'/%'italicGQ%trueF'/ %+foregroundGQ([0,0,0]F'/%,mathvariantGQ'italicF'/%)rowalignGQ!F'/%,co lumnalignGFF/%+groupalignGFF/%(rowspanGQ\"1F'/%+columnspanGFM-F56(-F,6 %-F86&Q)λF'/F/FBQ'normalF'-I#moGF$6.Q(−F'F>FY /%&fenceGFX/%*separatorGFX/%)stretchyGFX/%*symmetricGFX/%(largeopGFX/% .movablelimitsGFX/%'accentGFX/%'lspaceGQ,0.2222222emF'/%'rspaceGFio-F8 6&Q$tdqF'F;F>FAFDFGFIFKFNFDFGFI-F26'-F56(-F,6/-F,6#-I%msupGF$6%FT-I#mn GF$6%Q\"3F'F>FY/%1superscriptshiftGQ\"0F'-Ffn6.Q\"+F'F>FYFinF[oF]oF_oF aoFcoFeoFgoFjo-F,6%F\\p-Ffn6-Q1⁢F'FYFinF[oF]oF_oFaoFcoF eo/FhoQ&0.0emF'/F[pFjq-Fhp6%FT-F[q6%Q\"2F'F>FYF^qFaq-F,6%-F#6%-F,6%-F, 6#-Fhp6%F\\pF^rF^qFaq-F86&Q&τF'FWF>FYF>FYFfqFTFaq-F,6#-Fhp6%F\\pFj pF^qFaq-F,6%F[sFfqF\\pFen-F86&Q\"hF'F;F>FAFaq-F,6%F^rFfq-F86&Q)tinfty1 0F'F;F>FAFDFGFIFKFN-F56(-F,6$-Ffn6.Q*&uminus0;F'F>FYFinF[oF]oF_oFaoFco FeoFgoFjoF7FDFGFIFKFNFDFGFI/%&alignGQ%axisF'/FEQ)baselineF'/FHQ'center F'/FJQ'|frleft|hrF'/%/alignmentscopeGF=/%,columnwidthGQ%autoF'/%&width GF`u/%+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowlinesGQ%non eF'/%,columnlinesGF[v/%&frameGF[v/%-framespacingGQ,0.4em~0.5exF'/%*equ alrowsGFX/%-equalcolumnsGFX/%-displaystyleGFX/%%sideGQ&rightF'/%0minla belspacingGFhuF>FY/%%openGQ\"[F'/%&closeGQ\"]F'" }}{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\"0F'/%+fo regroundGQ([0,0,0]F'/%,mathvariantGQ'normalF'/%)rowalignGQ!F'/%,column alignGFC/%+groupalignGFC/%(rowspanGQ\"1F'/%+columnspanGFJ-F56(-I&mfrac GF$6)-F86%FJF;F>-F86%Q\"2F'F;F>/%.linethicknessGFJ/%+denomalignGQ'cent erF'/%)numalignGFen/%)bevelledGQ&falseF'F;FAFDFFFHFKFAFDFF-F26'-F56(-F ,6)-F,6%FO-I#moGF$6-Q1⁢F'F>/%&fenceGFjn/%*separatorGFjn /%)stretchyGFjn/%*symmetricGFjn/%(largeopGFjn/%.movablelimitsGFjn/%'ac centGFjn/%'lspaceGQ&0.0emF'/%'rspaceGFgp-F,6#-I%msupGF$6%-I#miGF$6&Q)& lambda;F'/%'italicGFjnF;F>FT/%1superscriptshiftGF:-Fdo6.Q\"+F'F;F>FgoF ioF[pF]pF_pFapFcp/FfpQ,0.2222222emF'/FipF[r-F,6%F_qFco-F`q6&Q$tdqF'/Fd qQ%trueF'F;/F?Q'italicF'Fgq-F,6%-FP6)-F86%Q\"3F'F;F>FTFWFYFfnFhnF;Fco- F,6#-F]q6%F_rFTFeqFgq-F,6%FOFco-F,6#-F`q6&Q&τF'FcqF;F>FAFDFFFHFKF4 FAFDFF/%&alignGQ%axisF'/FBQ)baselineF'/FEFen/FGQ'|frleft|hrF'/%/alignm entscopeGFcr/%,columnwidthGQ%autoF'/%&widthGFdt/%+rowspacingGQ&1.0exF' /%.columnspacingGQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGF_u/%&fra meGF_u/%-framespacingGQ,0.4em~0.5exF'/%*equalrowsGFjn/%-equalcolumnsGF jn/%-displaystyleGFjn/%%sideGQ&rightF'/%0minlabelspacingGF\\uF;F>/%%op enGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "G1:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 12 "G1[1,1]:=1:\n" }{MPLTEXT 1 0 12 "G1[2,2]:=1:\n" }{MPLTEXT 1 0 12 "G1[1,2]:=0:\n" }{MPLTEXT 1 0 23 " G1[2,1]:=g1*lambda+g0:\n" }{MPLTEXT 1 0 14 "g1:=tinfty13:\n" }{MPLTEXT 1 0 25 "g0:=tinfty13*q+tinfty12:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 27 "dG1dlambda:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 89 "for i from 1 to \+ 2 do for j from 1 to 2 do dG1dlambda[i,j]:=diff(G1[i,j],lambda): od: o d:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 24 "dG1dtau:=Matrix(2,2,0):\n " }{MPLTEXT 1 0 126 "for i from 1 to 2 do for j from 1 to 2 do dG1dtau [i,j]:=diff(G1[i,j],tau)+diff(G1[i,j],q)*dqdt+diff(G1[i,j],p)*dpdt : o d: od:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 12 " dqdt:=Lq/h:\n" }{MPLTEXT 1 0 1 "d" }{MPLTEXT 1 0 11 "pdt:=Lp/h:\n" } {MPLTEXT 1 0 13 "tdp:=checkp:\n" }{MPLTEXT 1 0 2 "td" }{MPLTEXT 1 0 3 "q:=" }{MPLTEXT 1 0 7 "checkq:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 11 "dcheckqdt:=" }{MPLTEXT 1 0 4 "dqdt" }{MPLTEXT 1 0 2 ":\n" }{MPLTEXT 1 0 9 "dcheckqdt" }{MPLTEXT 1 0 7 ":=dpdt:" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 85 "tdL:=simplify(Multiply(Multiply( G1,checkL),G1^(-1))+h*Multiply(dG1dlambda,G1^(-1))):\n" }{MPLTEXT 1 0 3 "tdA" }{MPLTEXT 1 0 32 ":=simplify(Multiply(Multiply(G1," }{MPLTEXT 1 0 6 "checkA" }{MPLTEXT 1 0 41 "),G1^(-1))+h*Multiply(dG1dtau,G1^(-1) )):\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 6 "tdL);\n" }{MPLTEXT 1 0 4 "tdA;" }}{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(-F,6'-F,6#-I%msupGF$6%-I#miGF$6&Q'che ckqF'/%'italicGQ%trueF'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'itali 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