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"In this Maple file, we \+ reduced the general formulas to the canonical set of trivial times in \+ order to obtain the reduced formulas for the Lax matrices and Hamilton ians for the second element of the Painlev\351 1 hierarchy. We check w ith the theoretical formulas." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 54 "Let us start by enumerating the symmetric polynomials." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart:\n" }{MPLTEXT 1 0 21 "with(L inearAlgebra):\n" }{MPLTEXT 1 0 17 "with(ListTools):\n" }{MPLTEXT 1 0 16 "with(combinat):\n" }{MPLTEXT 1 0 23 "with(PolynomialTools):\n" } {MPLTEXT 1 0 15 "with(Groebner):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 12 "chk:=proc()\n" }{MPLTEXT 1 0 26 "local VV,AA, pp,LL,K,N,KK:\n" }{MPLTEXT 1 0 31 "VV:=[seq(args[i],i=2..nargs)];\n" } {MPLTEXT 1 0 34 "AA:=[seq(sigma[i],i=1..nargs-1)];\n" }{MPLTEXT 1 0 53 "pp:=simplify(expand(mul(x_-args[i],i=2..nargs)),x_);\n" }{MPLTEXT 1 0 62 "LL := Reverse([seq((-1)^(r+nargs-1)*coeff(pp, x_, r), r = 0.. \n" }{MPLTEXT 1 0 15 "nargs-2)])-AA;\n" }{MPLTEXT 1 0 25 "K:=Basis(LL, tdeg(VV[])):\n" }{MPLTEXT 1 0 37 "N:=NormalForm(args[1],K,tdeg(VV[])); \n" }{MPLTEXT 1 0 26 "KK:=Basis(AA,tdeg(AA[]));\n" }{MPLTEXT 1 0 29 "N ormalForm(N,KK,tdeg(AA[]));\n" }{MPLTEXT 1 0 65 "if is(NormalForm(N,KK ,tdeg(AA[]))=0) then print(\"symmetric\")else\n" }{MPLTEXT 1 0 26 "pri nt(\"not symmetric\")fi;\n" }{MPLTEXT 1 0 10 "end proc:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 11 "es:=proc()\n" }{MPLTEXT 1 0 25 "local VV, AA, pp, LL, K;\n" }{MPLTEXT 1 0 64 "VV:=[seq(args[i],i=2..nargs)];AA: =[seq(sigma[i],i=1..nargs-1)];\n" }{MPLTEXT 1 0 53 "pp:=simplify(expan d(mul(x_-args[i],i=2..nargs)),x_);\n" }{MPLTEXT 1 0 62 "LL := Reverse( [seq((-1)^(r+nargs-1)*coeff(pp, x_, r), r = 0..\n" }{MPLTEXT 1 0 15 "n args-2)])-AA;\n" }{MPLTEXT 1 0 25 "K:=Basis(LL,tdeg(VV[]));\n" } {MPLTEXT 1 0 34 "NormalForm(args[1],K,tdeg(VV[]));\n" }{MPLTEXT 1 0 10 "end proc:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 41 "ss:=proc() loc al LL, LLL, t, LLLL, H, K;\n" }{MPLTEXT 1 0 31 "LL:=[seq(args[i],i=2.. nargs)];\n" }{MPLTEXT 1 0 41 "LLL:=[seq(map(x->x^r,LL),r=1..nargs-1)]; \n" }{MPLTEXT 1 0 27 "t:=seq(s[i],i=1..nargs-1);\n" }{MPLTEXT 1 0 48 " LLLL:=[seq(add(i,i in LLL[u]),u=1..nops(LLL))];\n" }{MPLTEXT 1 0 13 "H :=LLLL-[t];\n" }{MPLTEXT 1 0 25 "K:=Basis(H,grlex(LL[]));\n" }{MPLTEXT 1 0 35 "NormalForm(args[1],K,grlex(LL[]));\n" }{MPLTEXT 1 0 9 "end pr oc:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 22 "Eleme ntaryS:= proc(k)\n" }{MPLTEXT 1 0 19 "local aux,i,Coeff:\n" }{MPLTEXT 1 0 50 "aux:=0: for i from 1 to g do aux:=aux+q[i]^k od: \n" }{MPLTEXT 1 0 53 "Coeff:=unapply(es(aux,q[1],q[2]),sigma[1],sigma[2]):\n" } {MPLTEXT 1 0 26 "return(Coeff(Q[1],Q[2])):\n" }{MPLTEXT 1 0 10 "end pr oc:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 11 "rinfty:=5:\n" }{MPLTEXT 1 0 13 "g:=rinfty-3:\n" }{MPLTEXT 1 0 22 "S[0]:=ElementaryS(0);\n" } {MPLTEXT 1 0 22 "S[1]:=ElementaryS(1);\n" }{MPLTEXT 1 0 22 "S[2]:=Elem entaryS(2);\n" }{MPLTEXT 1 0 22 "S[3]:=ElementaryS(3);\n" }{MPLTEXT 1 0 22 "S[4]:=ElementaryS(4);\n" }{MPLTEXT 1 0 22 "S[5]:=ElementaryS(5); \n" }{MPLTEXT 1 0 22 "S[6]:=ElementaryS(6);\n" }{MPLTEXT 1 0 22 "S[7]: =ElementaryS(7);\n" }{MPLTEXT 1 0 21 "S[8]:=ElementaryS(8);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I\"SG6\"6#\"\"!\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I\"SG6\"6#\"\"\"&I\"QGF%F&" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I\"SG6\"6#\"\"#,&*$&I\"QGF%6#\"\"\"F'F-&F+F&!\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 ">&I\"SG6\"6#\"\"$,&*$&I\"QGF%6#\"\"\"F'F -*&F*F-&F+6#\"\"#F-!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I\"SG6\"6# \"\"%,(*$&I\"QGF%6#\"\"\"F'F-*&F*\"\"#&F+6#F/F-!\"%*$F0F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I\"SG6\"6#\"\"&,(*$&I\"QGF%6#\"\"\"F'F-*&F* \"\"$&F+6#\"\"#F-!\"&*&F*F-F0F2F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I \"SG6\"6#\"\"',**$&I\"QGF%6#\"\"\"F'F-*&F*\"\"%&F+6#\"\"#F-!\"'*&F*F2F 0F2\"\"**$F0\"\"$!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I\"SG6\"6#\" \"(,**$&I\"QGF%6#\"\"\"F'F-*&F*\"\"&&F+6#\"\"#F-!\"(*&F*\"\"$F0F2\"#9* &F*F-F0F5F3" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I\"SG6\"6#\"\"),,*$&I\" QGF%6#\"\"\"F'F-*&F*\"\"'&F+6#\"\"#F-!\")*&F*\"\"%F0F2\"#?*&F*F2F0\"\" $!#;*$F0F5F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "res:=-lambd a^(2*rinfty-5):\n" }{MPLTEXT 1 0 67 "for k from (rinfty-2) to (2*rinft y-7) do aux:=2*tau[2*rinfty-k-6]:\n" }{MPLTEXT 1 0 89 "for m from (k-r infty+6) to (rinfty-3) do aux:=aux+tau[rinfty-m-2]*tau[rinfty-k+m-5]: \+ od:\n" }{MPLTEXT 1 0 27 "res:=res-aux*lambda^k: od:\n" }{MPLTEXT 1 0 23 "aux2:=2*tau[rinfty-3]:\n" }{MPLTEXT 1 0 71 "for m from 3 to (rinft y-3) do aux2:=aux2+tau[rinfty-m-2]*tau[m-2]: od:\n" }{MPLTEXT 1 0 33 " res:=res-aux2*lambda^(rinfty-3):\n" }{MPLTEXT 1 0 5 "res;\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 69 "tdP2:=unapply(-lambda^5-2*tau[1]*lambda^ 3-2*tau[2]*lambda^2,lambda);\n" }{MPLTEXT 1 0 102 "for k from rinfty-3 to 2*rinfty-5 do P2[k]:=-residue(tdP2(lambda)/lambda^(k+1), lambda=in finity): od:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 7 "P2[2];\n" } {MPLTEXT 1 0 7 "P2[3];\n" }{MPLTEXT 1 0 7 "P2[4];\n" }{MPLTEXT 1 0 6 " P2[5];" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 10 "q1:=q[1]:\n" }{MPLTEXT 1 0 9 "q2:=q[2]:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 10 "p1:=p[1]:\n" }{MPLTEXT 1 0 10 "p2:=p[2]:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 22 " Elementaryh:= proc(k)\n" }{MPLTEXT 1 0 19 "local aux,i,Coeff:\n" } {MPLTEXT 1 0 55 "aux:=1: for i from 1 to g do aux:=aux/(1-t*q[i]): od: \n" }{MPLTEXT 1 0 74 "Coeff:=unapply(es(residue(aux/t^(k+1),t=0),q[1] ,q[2]),sigma[1],sigma[2]):\n" }{MPLTEXT 1 0 26 "return(Coeff(Q[1],Q[2] )):\n" }{MPLTEXT 1 0 10 "end proc:\n" }{MPLTEXT 1 0 32 "h[0]:=simplify (Elementaryh(0));\n" }{MPLTEXT 1 0 6 "h[1]:=" }{MPLTEXT 1 0 9 "simplif y(" }{MPLTEXT 1 0 17 "Elementaryh(1));\n" }{MPLTEXT 1 0 6 "h[2]:=" } {MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 17 "Elementaryh(2));\n" } {MPLTEXT 1 0 6 "h[3]:=" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 17 " Elementaryh(3));\n" }{MPLTEXT 1 0 6 "h[4]:=" }{MPLTEXT 1 0 9 "simplify (" }{MPLTEXT 1 0 17 "Elementaryh(4));\n" }{MPLTEXT 1 0 6 "h[5]:=" } {MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 17 "Elementaryh(5));\n" } {MPLTEXT 1 0 6 "h[6]:=" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 17 " Elementaryh(6));\n" }{MPLTEXT 1 0 6 "h[7]:=" }{MPLTEXT 1 0 9 "simplify (" }{MPLTEXT 1 0 17 "Elementaryh(7));\n" }{MPLTEXT 1 0 6 "h[8]:=" } {MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 17 "Elementaryh(8));\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 9 "Q[0]:=1:\n" }{MPLTEXT 1 0 17 "Q[1 ]:=q[1]+q[2]:\n" }{MPLTEXT 1 0 17 "Q[2]:=q[1]*q[2]:\n" }{MPLTEXT 1 0 26 "SymMatrix:=Matrix(g,g,0):\n" }{MPLTEXT 1 0 83 "for i from 1 to g d o for j from 1 to g do SymMatrix[i,j]:=diff(Q[j],q[i]): od: od:\n" } {MPLTEXT 1 0 11 "SymMatrix;\n" }{MPLTEXT 1 0 24 "Vectorp:=Matrix(g,1,0 ):\n" }{MPLTEXT 1 0 45 "for i from 1 to g do Vectorp[i,1]:=p[i]: od:\n " }{MPLTEXT 1 0 9 "Vectorp:\n" }{MPLTEXT 1 0 43 "VectorP:=Multiply(Sym Matrix^(-1),Vectorp);\n" }{MPLTEXT 1 0 44 "for i from 1 to g do P[i]:= VectorP[i,1]: od:" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*$I'lambdaG6\"\" \"&!\"\"*&F$\"\"$&I$tauGF%6#\"\"\"F-!\"#*&F$\"\"#&F+6#F0F-F." }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%tdP2G6\"f*6#I'lambdaGF$F$6$I)operatorGF$I&a rrowGF$F$,(*$9$\"\"&!\"\"*&F-\"\"$&I$tauGF$6#\"\"\"F5!\"#*&F-\"\"#&F36 #F8F5F6F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$&I$tauG6\"6#\"\"#!\"# " }}{PARA 11 "" 1 "" {XPPMATH 20 ",$&I$tauG6\"6#\"\"\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 ">&I\"hG6\"6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I\"hG6\"6#\"\"\"&I\"QGF%F&" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I\"hG6\"6#\"\"#,&*$&I\"QGF%6#\"\"\"F'F-&F+F&!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 ">&I\"hG6\"6#\"\"$*&&I\"QGF%6#\"\"\"F,,&* $F)\"\"#F,&F*6#F/!\"#F," }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I\"hG6\"6# \"\"%,(*$&I\"QGF%6#\"\"\"F'F-*&F*\"\"#&F+6#F/F-!\"$*$F0F/F-" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I\"hG6\"6#\"\"&,(*$&I\"QGF%6#\"\"\"F'F-*&F* \"\"$&F+6#\"\"#F-!\"%*&F*F-F0F2F/" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I \"hG6\"6#\"\"',**$&I\"QGF%6#\"\"\"F'F-*&F*\"\"%&F+6#\"\"#F-!\"&*&F*F2F 0F2F'*$F0\"\"$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I\"hG6\"6#\"\"( ,**$&I\"QGF%6#\"\"\"F'F-*&F*\"\"&&F+6#\"\"#F-!\"'*&F*\"\"$F0F2\"#5*&F* F-F0F5!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I\"hG6\"6#\"\"),,*$&I\"Q GF%6#\"\"\"F'F-*&F*\"\"'&F+6#\"\"#F-!\"(*&F*\"\"%F0F2\"#:*&F*F2F0\"\"$ !#5*$F0F5F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenam eG6\"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'n ormalF'/%)rowalignGQ!F'/%,columnalignGFC/%+groupalignGFC/%(rowspanGF:/ %+columnspanGF:-F56(-I%msubGF$6%-I#miGF$6&Q\"qF'/%'italicGQ%trueF'F;/F ?Q'italicF'-F,6#-F86%Q\"2F'F;F>/%/subscriptshiftGQ\"0F'FAFDFFFHFJFAFDF F-F26'F4-F56(-FO6%FQ-F,6#F7FinFAFDFFFHFJFAFDFF/%&alignGQ%axisF'/FBQ)ba selineF'/FEQ'centerF'/FGQ'|frleft|hrF'/%/alignmentscopeGFW/%,columnwid thGQ%autoF'/%&widthGFap/%+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8em F'/%)rowlinesGQ%noneF'/%,columnlinesGF\\q/%&frameGF\\q/%-framespacingG Q,0.4em~0.5exF'/%*equalrowsGQ&falseF'/%-equalcolumnsGFfq/%-displaystyl eGFfq/%%sideGQ&rightF'/%0minlabelspacingGFipF;F>/%%openGQ\"[F'/%&close GQ\"]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(-F,6%-I&mfracGF$6)-F,6%-I%msubGF$6%-I#miGF$6&Q\"qF'/%'italicGQ %trueF'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'italicF'-F,6#-I#mnGF$ 6%Q\"1F'FH/FLQ'normalF'/%/subscriptshiftGQ\"0F'-I#moGF$6-Q1&InvisibleT imes;F'FT/%&fenceGQ&falseF'/%*separatorGFin/%)stretchyGFin/%*symmetric GFin/%(largeopGFin/%.movablelimitsGFin/%'accentGFin/%'lspaceGQ&0.0emF' /%'rspaceGFho-F?6%-FB6&Q\"pF'FEFHFKFNFV-F,6#-F,6%F>-FZ6.Q(−F'FHF TFgnFjnF\\oF^oF`oFboFdo/FgoQ,0.2222222emF'/FjoFhp-F?6%FA-F,6#-FQ6%Q\"2 F'FHFTFV/%.linethicknessGFS/%+denomalignGQ'centerF'/%)numalignGFeq/%)b evelledGFinFHFdp-F:6)-F,6%FjpFY-F?6%F]pF\\qFVF`pFaqFcqFfqFhqFH/%)rowal ignGQ!F'/%,columnalignGFbr/%+groupalignGFbr/%(rowspanGFS/%+columnspanG FSF`rFcrFer-F26&-F56(-F,6&-FZ6.Q*&uminus0;F'FHFTFgnFjnF\\oF^oF`oFboFdo FgpFip-F:6)-F,6#F[pF`pFaqFcqFfqFhqFH-FZ6.Q\"+F'FHFTFgnFjnF\\oF^oF`oFbo FdoFgpFip-F:6)-F,6#F^rF`pFaqFcqFfqFhqFHF`rFcrFerFgrFirF`rFcrFer/%&alig nGQ%axisF'/FarQ)baselineF'/FdrFeq/FfrQ'|frleft|hrF'/%/alignmentscopeGF G/%,columnwidthGQ%autoF'/%&widthGF[u/%+rowspacingGQ&1.0exF'/%.columnsp acingGQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGFfu/%&frameGFfu/%-fr amespacingGQ,0.4em~0.5exF'/%*equalrowsGFin/%-equalcolumnsGFin/%-displa ystyleGFin/%%sideGQ&rightF'/%0minlabelspacingGFcuFHFT/%%openGQ\"[F'/%& closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "V:=Matrix(2 ,2,0):\n" }{MPLTEXT 1 0 11 "V[1,1]:=1:\n" }{MPLTEXT 1 0 12 "V[1,2]:=q1 :\n" }{MPLTEXT 1 0 11 "V[2,1]:=1:\n" }{MPLTEXT 1 0 12 "V[2,2]:=q2:\n" }{MPLTEXT 1 0 3 "V;\n" }{MPLTEXT 1 0 24 "HVector:=Matrix(2,1,0):\n" } {MPLTEXT 1 0 18 "HVector[1,1]:=H0:\n" }{MPLTEXT 1 0 18 "HVector[2,1]:= H1:\n" }{MPLTEXT 1 0 9 "HVector;\n" }{MPLTEXT 1 0 21 "RHSH:=Matrix(2,1 ,0):\n" }{MPLTEXT 1 0 44 "RHSH[1,1]:=p1^2+tdP2(q1)+h*(p2-p1)/(q1-q2): \n" }{MPLTEXT 1 0 44 "RHSH[2,1]:=p2^2+tdP2(q2)+h*(p1-p2)/(q2-q1):\n" } {MPLTEXT 1 0 6 "RHSH;\n" }{MPLTEXT 1 0 9 "HVector:=" }{MPLTEXT 1 0 32 "simplify(Multiply(V^(-1),RHSH));" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,TypesettingGI(_syslibG F'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'/%,col umnalignGFC/%+groupalignGFC/%(rowspanGF:/%+columnspanGF:-F56(-I%msubGF $6%-I#miGF$6&Q\"qF'/%'italicGQ%trueF'F;/F?Q'italicF'-F,6#F7/%/subscrip tshiftGQ\"0F'FAFDFFFHFJFAFDFF-F26'F4-F56(-FO6%FQ-F,6#-F86%Q\"2F'F;F>Ff nFAFDFFFHFJFAFDFF/%&alignGQ%axisF'/FBQ)baselineF'/FEQ'centerF'/FGQ'|fr left|hrF'/%/alignmentscopeGFW/%,columnwidthGQ%autoF'/%&widthGFap/%+row spacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowlinesGQ%noneF'/%,colu mnlinesGF\\q/%&frameGF\\q/%-framespacingGQ,0.4em~0.5exF'/%*equalrowsGQ &falseF'/%-equalcolumnsGFfq/%-displaystyleGFfq/%%sideGQ&rightF'/%0minl abelspacingGFipF;F>/%%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#miGF$6&Q#H0F'/%'it alicGQ%trueF'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'italicF'/%)rowa lignGQ!F'/%,columnalignGFF/%+groupalignGFF/%(rowspanGQ\"1F'/%+columnsp anGFMFDFGFI-F26&-F56(-F86&Q#H1F'F;F>FAFDFGFIFKFNFDFGFI/%&alignGQ%axisF '/FEQ)baselineF'/FHQ'centerF'/FJQ'|frleft|hrF'/%/alignmentscopeGF=/%,c olumnwidthGQ%autoF'/%&widthGF^o/%+rowspacingGQ&1.0exF'/%.columnspacing GQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGFio/%&frameGFio/%-framesp acingGQ,0.4em~0.5exF'/%*equalrowsGQ&falseF'/%-equalcolumnsGFcp/%-displ aystyleGFcp/%%sideGQ&rightF'/%0minlabelspacingGFfoF>/FBQ'normalF'/%%op enGQ\"[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(-F,6+-F,6#-I(msubsupGF$6'-I#miGF$6&Q\"pF'/%'it alicGQ%trueF'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'italicF'-F,6#-I #mnGF$6%Q\"1F'FE/FIQ'normalF'-FN6%Q\"2F'FEFQ/%1superscriptshiftGQ\"0F' /%/subscriptshiftGFX-I#moGF$6.Q(−F'FEFQ/%&fenceGQ&falseF'/%*sepa ratorGF[o/%)stretchyGF[o/%*symmetricGF[o/%(largeopGF[o/%.movablelimits GF[o/%'accentGF[o/%'lspaceGQ,0.2222222emF'/%'rspaceGFjo-F,6#-F<6'-F?6& Q\"qF'FBFEFHFK-FN6%Q\"5F'FEFQFVFYFen-F,6'FS-Ffn6-Q1⁢F'F QFinF\\oF^oF`oFboFdoFfo/FioQ&0.0emF'/F\\pF]q-F<6'FapFK-FN6%Q\"3F'FEFQF VFYFip-I%msubGF$6%-F?6&Q&τF'/FCF[oFEFQFKFYFen-F,6'FSFip-F<6'FapFKF SFVFYFip-Feq6%Fgq-F,6#FSFY-Ffn6.Q\"+F'FEFQFinF\\oF^oF`oFboFdoFfoFhoF[p -I&mfracGF$6)-F,6%-F?6&Q\"hF'FBFEFHFip-F#6%-F,6%-Feq6%F>FarFYFen-Feq6% F>FKFYFEFQ-F,6#-F,6%-Feq6%FapFKFYFen-Feq6%FapFarFY/%.linethicknessGFP/ %+denomalignGQ'centerF'/%)numalignGFbt/%)bevelledGF[oFE/%)rowalignGQ!F '/%,columnalignGFit/%+groupalignGFit/%(rowspanGFP/%+columnspanGFPFgtFj tF\\u-F26&-F56(-F,6+-F,6#-F<6'F>FarFSFVFYFen-F,6#-F<6'FapFarFdpFVFYFen -F,6'FSFip-F<6'FapFarFaqFVFYFipFdqFen-F,6'FSFip-F<6'FapFarFSFVFYFipF_r Fcr-Fgr6)-F,6%F[sFip-F#6%-F,6%FdsFenFbsFEFQ-F,6#-F,6%F\\tFenFjsF^tF`tF ctFetFEFgtFjtF\\uF^uF`uFgtFjtF\\u/%&alignGQ%axisF'/FhtQ)baselineF'/F[u Fbt/F]uQ'|frleft|hrF'/%/alignmentscopeGFD/%,columnwidthGQ%autoF'/%&wid thGF`x/%+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowlinesGQ%n oneF'/%,columnlinesGF[y/%&frameGF[y/%-framespacingGQ,0.4em~0.5exF'/%*e qualrowsGF[o/%-equalcolumnsGF[o/%-displaystyleGF[o/%%sideGQ&rightF'/%0 minlabelspacingGFhxFEFQ/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,TypesettingGI(_syslibG F'6'-I%mrowGF$6#-I'mtableGF$66-I$mtrGF$6&-I$mtdGF$6(-I&mfracGF$6)-F,6# -F,6--F,6%-I#miGF$6&Q\"hF'/%'italicGQ%trueF'/%+foregroundGQ([0,0,0]F'/ %,mathvariantGQ'italicF'-I#moGF$6-Q1⁢F'/FKQ'normalF'/%& fenceGQ&falseF'/%*separatorGFU/%)stretchyGFU/%*symmetricGFU/%(largeopG FU/%.movablelimitsGFU/%'accentGFU/%'lspaceGQ&0.0emF'/%'rspaceGF^o-F#6% -F,6%-I%msubGF$6%-FA6&Q\"pF'FDFGFJ-F,6#-I#mnGF$6%Q\"2F'FGFQ/%/subscrip tshiftGQ\"0F'-FN6.Q(−F'FGFQFSFVFXFZFfnFhnFjn/F]oQ,0.2222222emF'/ F`oFhp-Ffo6%Fho-F,6#-F^p6%Q\"1F'FGFQFapFGFQ-FN6.Q\"+F'FGFQFSFVFXFZFfnF hnFjnFgpFip-F,6%-I(msubsupGF$6'-FA6&Q\"qF'FDFGFJF\\q-F^p6%Q\"5F'FGFQ/% 1superscriptshiftGFcpFapFM-Ffo6%FiqF[pFapFaq-F,6)F]pFM-Fgq6'FiqF\\q-F^ p6%Q\"3F'FGFQF_rFapFM-Ffo6%-FA6&Q&τF'/FEFUFGFQF\\qFapFMFarFaq-F,6) F]pFM-Fgq6'FiqF\\qF]pF_rFapFM-Ffo6%F\\sF[pFapFMFarFaq-F,6%-F#6%-F,6*-F N6.Q*&uminus0;F'FGFQFSFVFXFZFfnFhnFjnFgpFip-F,6#-Fgq6'FiqF[pF\\rF_rFap Fdp-F,6'F]pFM-Fgq6'FiqF[pFgrF_rFapFMFjrFdp-F,6'F]pFM-Fgq6'FiqF[pF]pF_r FapFMFdsFaq-F,6#-Fgq6'FhoF[pF]pF_rFapFGFQFM-Ffo6%FiqF\\qFapFdp-F,6%-Fg q6'FhoF\\qF]pF_rFapFMFar-F,6#-F,6%F_uFdpFar/%.linethicknessGF`q/%+deno malignGQ'centerF'/%)numalignGF]v/%)bevelledGFUFG/%)rowalignGQ!F'/%,col umnalignGFdv/%+groupalignGFdv/%(rowspanGF`q/%+columnspanGF`qFbvFevFgv- F26&-F56(-F86)-F,6#-F,62F\\t-F,6#FfqFaqF_tFdp-F,6'F]pFMFerFMFjrFaqFctF dp-F,6'F]pFMFbsFMFdsFaqFgtFaq-F,6#FcuFdpF[uFeuFiuF[vF^vF`vFGFbvFevFgvF ivF[wFbvFevFgv/%&alignGQ%axisF'/FcvQ)baselineF'/FfvF]v/FhvQ'|frleft|hr F'/%/alignmentscopeGFF/%,columnwidthGQ%autoF'/%&widthGF[y/%+rowspacing GQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlines GFfy/%&frameGFfy/%-framespacingGQ,0.4em~0.5exF'/%*equalrowsGFU/%-equal columnsGFU/%-displaystyleGFU/%%sideGQ&rightF'/%0minlabelspacingGFcyFGF Q/%%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "H0:=-(p1-p2)*h/(q1-q2)+(p2^2*q1-p1^2*q2)/(q1-q2)+(q1+q2)" } {MPLTEXT 1 0 6 "*q1*q2" }{MPLTEXT 1 0 38 "*(q1^2+q2^2+2*tau[1])+2*q1*q 2*tau[2];\n" }{MPLTEXT 1 0 106 "H1:=(p1^2-p2^2)/(q1-q2) -2*tau[2]*(q1+ q2)-2*(q1^2+q1*q2+q2^2)*tau[1]-q1^4-q1^3*q2-q1^2*q2^2-q1*q2^3-q2^4;\n" }{MPLTEXT 1 0 7 "factor(" }{MPLTEXT 1 0 17 "HVector[1,1]-H0);" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 7 "factor(" }{MPLTEXT 1 0 18 "HVecto r[2,1]-H1);\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 31 "nu1tau1:=1/(2*ri nfty-2*1-5)*1;\n" }{MPLTEXT 1 0 12 "nu2tau1:=0;\n" }{MPLTEXT 1 0 12 "n u1tau2:=0;\n" }{MPLTEXT 1 0 29 "nu2tau2:=1/(2*rinfty-2*2-5)*1" } {MPLTEXT 1 0 2 ";\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 29 "nutau1Vect or:=Matrix(2,1,0):\n" }{MPLTEXT 1 0 19 "nutau1Vector[1,1]:=" }{MPLTEXT 1 0 9 "nu1tau1:\n" }{MPLTEXT 1 0 19 "nutau1Vector[2,1]:=" }{MPLTEXT 1 0 8 "nu2tau1:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 13 "nutau1Vector;" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 58 "mutau1Vec tor:=Multiply((LinearAlgebra[Transpose](V))^(-1)," }{MPLTEXT 1 0 13 "n utau1Vector)" }{MPLTEXT 1 0 2 ";\n" }{MPLTEXT 1 0 9 "mu1tau1:=" } {MPLTEXT 1 0 19 "mutau1Vector[1,1]:\n" }{MPLTEXT 1 0 9 "mu2tau1:=" } {MPLTEXT 1 0 19 "mutau1Vector[2,1]:\n" }{MPLTEXT 1 0 13 "mutau1Vector; " }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 29 "nutau2Ve ctor:=Matrix(2,1,0):\n" }{MPLTEXT 1 0 28 "nutau2Vector[1,1]:=nu1tau2: \n" }{MPLTEXT 1 0 28 "nutau2Vector[2,1]:=nu2tau2:\n" }{MPLTEXT 1 0 13 "nutau2Vector;" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 73 "mutau2Vector:=Multiply((LinearAlgebra[Transpose](V))^(-1),nutau2 Vector);\n" }{MPLTEXT 1 0 28 "mu1tau2:=mutau2Vector[1,1]:\n" }{MPLTEXT 1 0 28 "mu2tau2:=mutau2Vector[2,1]:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 33 "Hamtau1:= nu1tau1*H0+nu2tau1*H1;\n" }{MPLTEXT 1 0 33 "Hamtau2 := nu1tau2*H0+nu2tau2*H1;\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 68 "QQ :=unapply(-p1*(lambda-q2)/(q1-q2)-p2*(lambda-q1)/(q2-q1),lambda);\n" } {MPLTEXT 1 0 1 "\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 44 "J[2,1]:=QQ(lambda)/(lambda-q1)/(lambda-q2):\n" }{MPLTEXT 1 0 35 "J [2,2]:=1/(lambda-q1)/(lambda-q2):\n" }{MPLTEXT 1 0 3 "J;\n" }{MPLTEXT 1 0 26 "dJdlambda:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 42 "for i from 1 to 2 do for j from 1 to 2 do\n" }{MPLTEXT 1 0 45 "dJdlambda[i,j]:=diff(J [i,j],lambda): od: od:\n" }{MPLTEXT 1 0 10 "dJdlambda;" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\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 70 "L[2,1]:=-tdP2(lambda)+H1*lambda+H0-h*p1/(lambda-q1)-h *p2/(lambda-q2):\n" }{MPLTEXT 1 0 37 "L[2,2]:=h/(lambda-q1)+h/(lambda- q2):\n" }{MPLTEXT 1 0 2 "L;" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 40 "dq1dtau1:=factor(1/h*diff(Hamtau1,p1));\n" } {MPLTEXT 1 0 41 "dp1dtau1:=factor(-1/h*diff(Hamtau1,q1));\n" }{MPLTEXT 1 0 40 "dq2dtau1:=factor(1/h*diff(Hamtau1,p2));\n" }{MPLTEXT 1 0 41 " dp2dtau1:=factor(-1/h*diff(Hamtau1,q2));\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 40 "dq1dtau2:=factor(1/h*diff(Hamtau2,p1));\n" }{MPLTEXT 1 0 41 "dp1dtau2:=factor(-1/h*diff(Hamtau2,q1));\n" }{MPLTEXT 1 0 40 " dq2dtau2:=factor(1/h*diff(Hamtau2,p2));\n" }{MPLTEXT 1 0 41 "dp2dtau2: =factor(-1/h*diff(Hamtau2,q2));\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 22 "Atau1:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 60 "Atau1[1,1]:=-p1*mu1tau1 /(lambda-q1)-p2*mu2tau1/(lambda-q2):\n" }{MPLTEXT 1 0 54 "Atau1[1,2]:= mu1tau1/(lambda-q1)+mu2tau1/(lambda-q2):\n" }{MPLTEXT 1 0 57 "Atau1[2 ,1]:=h*diff(Atau1[1,1],lambda)+Atau1[1,2]*L[2,1]:\n" }{MPLTEXT 1 0 68 "Atau1[2,2]:=h*diff(Atau1[1,2],lambda)+Atau1[1,1]+Atau1[1,2]*L[2,2]:\n " }{MPLTEXT 1 0 7 "Atau1:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 22 "At au2:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 60 "Atau2[1,1]:=-p1*mu1tau2/(lamb da-q1)-p2*mu2tau2/(lambda-q2):\n" }{MPLTEXT 1 0 53 "Atau2[1,2]:=mu1tau 2/(lambda-q1)+mu2tau2/(lambda-q2):\n" }{MPLTEXT 1 0 57 "Atau2[2,1]:=h* diff(Atau2[1,1],lambda)+Atau2[1,2]*L[2,1]:\n" }{MPLTEXT 1 0 68 "Atau2[ 2,2]:=h*diff(Atau2[1,2],lambda)+Atau2[1,1]+Atau2[1,2]*L[2,2]:\n" } {MPLTEXT 1 0 7 "Atau2:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 24 "dJdtau1:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 42 "for i fr om 1 to 2 do for j from 1 to 2 do\n" }{MPLTEXT 1 0 141 "dJdtau1[i,j]:= diff(J[i,j],tau1)+diff(J[i,j],q1)*dq1dtau1+diff(J[i,j],p1)*dp1dtau1+di ff(J[i,j],q2)*dq2dtau1+diff(J[i,j],p2)*dp2dtau1: od: od:\n" }{MPLTEXT 1 0 9 "dJdtau1:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 24 "dJdtau2:=Mat rix(2,2,0):\n" }{MPLTEXT 1 0 42 "for i from 1 to 2 do for j from 1 to \+ 2 do\n" }{MPLTEXT 1 0 141 "dJdtau2[i,j]:=diff(J[i,j],tau2)+diff(J[i,j] ,q1)*dq1dtau2+diff(J[i,j],p1)*dp1dtau2+diff(J[i,j],q2)*dq2dtau2+diff(J [i,j],p2)*dp2dtau2: od: od:\n" }{MPLTEXT 1 0 9 "dJdtau2:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 78 "CheckL:=simplify(Multiply(Multiply(J,L),J ^(-1))+h*Multiply(dJdlambda,J^(-1)));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 85 "CheckAtau1:=simplify(Multiply(Multiply(J,Atau1),J^(-1))+h*Mult iply(dJdtau1,J^(-1)));\n" }{MPLTEXT 1 0 84 "CheckAtau2:=simplify(Multi ply(Multiply(J,Atau2),J^(-1))+h*Multiply(dJdtau2,J^(-1)));" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#H0G6\",**(,&&I\"pGF$6#\"\"\"F+&F)6#\"\"#!\"\" F+I\"hGF$F+,&&I\"qGF$F*F+&F3F-F/F/F/*&,&*&F(F.F4F+F/*&F,F.F2F+F+F+F1F/ F+**,&F2F+F4F+F+F2F+F4F+,(*$F2F.F+*$F4F.F+&I$tauGF$F*F.F+F+*(F2F+F4F+& F?F-F+F." }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#H1G6\",2*&,&*$&I\"pGF$6# \"\"\"\"\"#F,*$&F*6#F-F-!\"\"F,,&&I\"qGF$F+F,&F4F0F1F1F,*&&I$tauGF$F0F ,,&F3F,F5F,F,!\"#*&,(*$F3F-F,*&F3F,F5F,F,*$F5F-F,F,&F8F+F,F:*$F3\"\"%F 1*&F3\"\"$F5F,F1*&F3F-F5F-F1*&F3F,F5FDF1*$F5FBF1" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(nu1tau1G6\"#\"\"\"\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(nu2tau1G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(n u1tau2G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(nu2tau2G6\"\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,Types ettingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableGF$66-I$mtrGF$6&-I$mtdGF$6(- I&mfracGF$6)-I#mnGF$6%Q\"1F'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ' normalF'-F;6%Q\"3F'F>FA/%.linethicknessGF=/%+denomalignGQ'centerF'/%)n umalignGFK/%)bevelledGQ&falseF'F>/%)rowalignGQ!F'/%,columnalignGFS/%+g roupalignGFS/%(rowspanGF=/%+columnspanGF=FQFTFV-F26&-F56(-F;6%Q\"0F'F> FAFQFTFVFXFZFQFTFV/%&alignGQ%axisF'/FRQ)baselineF'/FUFK/FWQ'|frleft|hr F'/%/alignmentscopeGQ%trueF'/%,columnwidthGQ%autoF'/%&widthGFjo/%+rows pacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowlinesGQ%noneF'/%,colum nlinesGFep/%&frameGFep/%-framespacingGQ,0.4em~0.5exF'/%*equalrowsGFP/% -equalcolumnsGFP/%-displaystyleGFP/%%sideGQ&rightF'/%0minlabelspacingG FbpF>FA/%%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$6(-F,6$-I#moGF$6.Q*&uminus0;F'/%+for egroundGQ([0,0,0]F'/%,mathvariantGQ'normalF'/%&fenceGQ&falseF'/%*separ atorGFE/%)stretchyGFE/%*symmetricGFE/%(largeopGFE/%.movablelimitsGFE/% 'accentGFE/%'lspaceGQ,0.2222222emF'/%'rspaceGFT-F,6%-I&mfracGF$6)-I#mn GF$6%Q\"1F'F=F@-Fgn6%Q\"3F'F=F@/%.linethicknessGFin/%+denomalignGQ'cen terF'/%)numalignGFao/%)bevelledGFEF=-F:6-Q1⁢F'F@FCFFFHF JFLFNFP/FSQ&0.0emF'/FVFjo-FZ6)-F,6#-I%msubGF$6%-I#miGF$6&Q\"qF'/%'ital icGQ%trueF'F=/FAQ'italicF'-F,6#-Fgn6%Q\"2F'F=F@/%/subscriptshiftGQ\"0F '-F,6#-F,6%-Fap6%Fcp-F,6#FfnFaq-F:6.Q(−F'F=F@FCFFFHFJFLFNFPFRFUF `pF]oF_oFboFdoF=/%)rowalignGQ!F'/%,columnalignGFar/%+groupalignGFar/%( rowspanGFin/%+columnspanGFinF_rFbrFdr-F26&-F56(-F,6%FYFfo-FZ6)-F,6#Fhq FdqF]oF_oFboFdoF=F_rFbrFdrFfrFhrF_rFbrFdr/%&alignGQ%axisF'/F`rQ)baseli neF'/FcrFao/FerQ'|frleft|hrF'/%/alignmentscopeGFip/%,columnwidthGQ%aut oF'/%&widthGF`t/%+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)row linesGQ%noneF'/%,columnlinesGF[u/%&frameGF[u/%-framespacingGQ,0.4em~0. 5exF'/%*equalrowsGFE/%-equalcolumnsGFE/%-displaystyleGFE/%%sideGQ&righ tF'/%0minlabelspacingGFhtF=F@/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,TypesettingGI(_s yslibGF'6'-I%mrowGF$6#-I'mtableGF$66-I$mtrGF$6&-I$mtdGF$6(-F,6$-I#moGF $6.Q*&uminus0;F'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'normalF'/%&f enceGQ&falseF'/%*separatorGFE/%)stretchyGFE/%*symmetricGFE/%(largeopGF E/%.movablelimitsGFE/%'accentGFE/%'lspaceGQ,0.2222222emF'/%'rspaceGFT- F,6%-I&mfracGF$6)-I#mnGF$6%Q\"1F'F=F@-Fgn6%Q\"3F'F=F@/%.linethicknessG Fin/%+denomalignGQ'centerF'/%)numalignGFao/%)bevelledGFEF=-F:6-Q1&Invi sibleTimes;F'F@FCFFFHFJFLFNFP/FSQ&0.0emF'/FVFjo-FZ6)-F,6#-I%msubGF$6%- I#miGF$6&Q\"qF'/%'italicGQ%trueF'F=/FAQ'italicF'-F,6#-Fgn6%Q\"2F'F=F@/ %/subscriptshiftGQ\"0F'-F,6#-F,6%-Fap6%Fcp-F,6#FfnFaq-F:6.Q(−F'F =F@FCFFFHFJFLFNFPFRFUF`pF]oF_oFboFdoF=/%)rowalignGQ!F'/%,columnalignGF ar/%+groupalignGFar/%(rowspanGFin/%+columnspanGFinF_rFbrFdr-F26&-F56(- F,6%FYFfo-FZ6)-F,6#FhqFdqF]oF_oFboFdoF=F_rFbrFdrFfrFhrF_rFbrFdr/%&alig nGQ%axisF'/F`rQ)baselineF'/FcrFao/FerQ'|frleft|hrF'/%/alignmentscopeGF ip/%,columnwidthGQ%autoF'/%&widthGF`t/%+rowspacingGQ&1.0exF'/%.columns pacingGQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGF[u/%&frameGF[u/%-f ramespacingGQ,0.4em~0.5exF'/%*equalrowsGFE/%-equalcolumnsGFE/%-display styleGFE/%%sideGQ&rightF'/%0minlabelspacingGFhtF=F@/%%openGQ\"[F'/%&cl oseGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulename G6\"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'no rmalF'/%)rowalignGQ!F'/%,columnalignGFC/%+groupalignGFC/%(rowspanGQ\"1 F'/%+columnspanGFJFAFDFF-F26&-F56(-F86%FJF;F>FAFDFFFHFKFAFDFF/%&alignG Q%axisF'/FBQ)baselineF'/FEQ&rightF'/FGQ'|frleft|hrF'/%/alignmentscopeG Q%trueF'/%,columnwidthGQ%autoF'/%&widthGF[o/%+rowspacingGQ&1.0exF'/%.c olumnspacingGQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGFfo/%&frameGF fo/%-framespacingGQ,0.4em~0.5exF'/%*equalrowsGQ&falseF'/%-equalcolumns GF`p/%-displaystyleGF`p/%%sideGFY/%0minlabelspacingGFcoF;F>/%%openGQ\" [F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+mo dulenameG6\"I,TypesettingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableGF$66-I$m trGF$6&-I$mtdGF$6(-I&mfracGF$6)-I#mnGF$6%Q\"1F'/%+foregroundGQ([0,0,0] F'/%,mathvariantGQ'normalF'-F,6#-F,6%-I%msubGF$6%-I#miGF$6&Q\"qF'/%'it alicGQ%trueF'F>/FBQ'italicF'-F,6#F:/%/subscriptshiftGQ\"0F'-I#moGF$6.Q (−F'F>FA/%&fenceGQ&falseF'/%*separatorGFin/%)stretchyGFin/%*symm etricGFin/%(largeopGFin/%.movablelimitsGFin/%'accentGFin/%'lspaceGQ,0. 2222222emF'/%'rspaceGFho-FI6%FK-F,6#-F;6%Q\"2F'F>FAFV/%.linethicknessG F=/%+denomalignGQ'centerF'/%)numalignGFfp/%)bevelledGFinF>/%)rowalignG Q!F'/%,columnalignGF]q/%+groupalignGF]q/%(rowspanGF=/%+columnspanGF=F[ qF^qF`q-F26&-F56(-F,6$-FZ6.Q*&uminus0;F'F>FAFgnFjnF\\oF^oF`oFboFdoFfoF ioF7F[qF^qF`qFbqFdqF[qF^qF`q/%&alignGQ%axisF'/F\\qQ)baselineF'/F_qFfp/ FaqQ'|frleft|hrF'/%/alignmentscopeGFQ/%,columnwidthGQ%autoF'/%&widthGF [s/%+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowlinesGQ%noneF '/%,columnlinesGFfs/%&frameGFfs/%-framespacingGQ,0.4em~0.5exF'/%*equal rowsGFin/%-equalcolumnsGFin/%-displaystyleGFin/%%sideGQ&rightF'/%0minl abelspacingGFcsF>FA/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(Hamtau1G6\",**(,&&I\"pGF$6#\"\"\"F+&F)6#\"\"#!\"\"F+I \"hGF$F+,&&I\"qGF$F*F+&F3F-F/F/#F/\"\"$*&,&*&F(F.F4F+F/*&F,F.F2F+F+F+F 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">I)dp1dtau1G6\",$*(,<*&&I\"qGF$6#\"\"\"\"\"&&F*6#\"\"#F, \"\"%*&F)F1F.F0!\"&*$F.\"\"'F,*(F)\"\"$F.F,&I$tauGF$F+F,F1*(F)F0F.F0F8 F,!\"'*&F.F1F8F,F0*(F)F0F.F,&F9F/F,F0*(F)F,F.F0F>F,!\"%*&F.F7F>F,F0*&& I\"pGF$F+F0F.F,F,*&&FDF/F0F.F,!\"\"*&I\"hGF$F,FCF,F,*&FIF,FFF,FGF,FIFG ,&F)F,F.FG!\"##FGF7" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)dq2dtau1G6\",$ *(,&*&&I\"pGF$6#\"\"#\"\"\"&I\"qGF$6#F-F-F,I\"hGF$F-F-F1!\"\",&F.F-&F/ F+F2F2#F-\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)dp2dtau1G6\",$*(,<* $&I\"qGF$6#\"\"\"\"\"'!\"\"*&F)\"\"#&F*6#F0\"\"%\"\"&*&F)F,F1F4!\"%*&F )F3&I$tauGF$F+F,!\"#*(F)F0F1F0F8F,F-*(F)F,F1\"\"$F8F,F6*&F)F=&F9F2F,F: *(F)F0F1F,F?F,F3*(F)F,F1F0F?F,F:*&&I\"pGF$F+F0F)F,F,*&&FDF2F0F)F,F.*&I \"hGF$F,FCF,F,*&FHF,FFF,F.F,FHF.,&F)F,F1F.F:#F,F=" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)dq1dtau2G6\",$*(I\"hGF$!\"\",&&I\"qGF$6#\"\"\"F-&F+6# \"\"#F(F(&I\"pGF$F,F-F0" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)dp1dtau2G6 \"*(,8*$&I\"qGF$6#\"\"\"\"\"&\"\"%*&F(F-&F)6#\"\"#F+!\"&*$F/F,F+*&F(\" 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;F>FAFDFFFHFKFAFDFF-F26'-F56(-F,6'-I#miGF$6&Q)λF'/%'italicGQ&fa lseF'F;F>-I#moGF$6.Q\"+F'F;F>/%&fenceGFgn/%*separatorGFgn/%)stretchyGF gn/%*symmetricGFgn/%(largeopGFgn/%.movablelimitsGFgn/%'accentGFgn/%'ls paceGQ,0.2222222emF'/%'rspaceGF\\p-F,6%-F86%Q\"2F'F;F>-Fin6-Q1&Invisib leTimes;F'F>F\\oF^oF`oFboFdoFfoFho/F[pQ&0.0emF'/F^pFhp-I%msubGF$6%-FX6 &Q\"qF'/FfnQ%trueF'F;/F?Q'italicF'-F,6#FO/%/subscriptshiftGF:Fhn-F,6%F apFdp-F[q6%F]q-F,6#FapFfqFAFDFFFHFKF4FAFDFF/%&alignGQ%axisF'/FBQ)basel ineF'/FEQ'centerF'/FGQ'|frleft|hrF'/%/alignmentscopeGFaq/%,columnwidth GQ%autoF'/%&widthGF[s/%+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8emF' /%)rowlinesGQ%noneF'/%,columnlinesGFfs/%&frameGFfs/%-framespacingGQ,0. 4em~0.5exF'/%*equalrowsGFgn/%-equalcolumnsGFgn/%-displaystyleGFgn/%%si deGQ&rightF'/%0minlabelspacingGFcsF;F>/%%openGQ\"[F'/%&closeGQ\"]F'" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "CheckL21:=CheckL[2,1]:" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 134 "CheckL21bis:=lambda^3+(q1+q2)*l ambda^2+(q1^2+q1*q2+q2^2+2*tau[1])*lambda-(p1-p2)^2/(q1-q2)^2 + (q1^2+ q2^2+2*tau[1])*(q1+q2)+2*tau[2]:\n" }{MPLTEXT 1 0 40 "factor(series(Ch eckL21-CheckL21bis,p2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "tdL11theo:=0:\n" }{MPLTEXT 1 0 132 "for j from 0 to g-1 do aux:=0: for i from j+1 to g do aux:=au x+P[i]*Q[i-j-1]: od: tdL11theo:=tdL11theo-(-1)^(j-1)*aux*lambda^j: od: \n" }{MPLTEXT 1 0 20 "tdL11theo:=simplify(" }{MPLTEXT 1 0 10 "tdL11the o)" }{MPLTEXT 1 0 2 ";\n" }{MPLTEXT 1 0 19 "simplify(tdL11theo-" } {MPLTEXT 1 0 13 "CheckL[1,1]);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*tdL11theoG6\"*&,&*&&I\"pGF$6#\"\"\"F+,&I'lambdaGF$F+&I \"qGF$6#\"\"#!\"\"F+F+*&&F)F0F+,&F-F+&F/F*F2F+F2F+,&F6F+F.F2F2" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "tdL12theo:=0:\n" }{MPLTEXT 1 0 74 "for m from 0 to g \+ do tdL12theo:=tdL12theo+(-1)^(g-m)*Q[g-m]*lambda^m: od:\n" }{MPLTEXT 1 0 30 "tdL12theo:=simplify(tdL12theo)" }{MPLTEXT 1 0 1 ";" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 19 "simplify(tdL12theo-" }{MPLTEXT 1 0 13 "Ch eckL[1,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*tdL12theoG6\"*&,&I'lam bdaGF$\"\"\"&I\"qGF$6#F(!\"\"F(,&F'F(&F*6#\"\"#F,F(" }}{PARA 11 "" 1 " " {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Ter m1:=0:\n" }{MPLTEXT 1 0 107 "for i from 0 to rinfty-2 do for j from g+ i to 2*rinfty-5 do Term1:=Term1- P2[j]*h[j-g-i]*lambda^i: od: od:\n" } {MPLTEXT 1 0 7 "Term1:\n" }{MPLTEXT 1 0 10 "Term2:=0:\n" }{MPLTEXT 1 0 61 "for i from 0 to g-2 do for j1 from i+1 to g-1 do for j2 from " } {MPLTEXT 1 0 6 "g+i-j1" }{MPLTEXT 1 0 145 " to g-1 do for i1 from j1+1 to g do for i2 from j2+1 to g do Term2:=Term2- (-1)^(j1+j2)*P[i1]*Q[i 1-j1-1]*P[i2]*Q[i2-j2-1]*h[j1+j2-g-i]*lambda^i:\n" }{MPLTEXT 1 0 20 "o d: od: od: od: od:\n" }{MPLTEXT 1 0 7 "Term2:\n" }{MPLTEXT 1 0 34 "tdL 21theo:=simplify(Term1+Term2):\n" }{MPLTEXT 1 0 32 "simplify(tdL21theo -CheckL[2,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 56 "We have verified the \\td\{L\} formula. \+ Let us do A^\{tau1\}." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "nu[ 1]:=" }{MPLTEXT 1 0 7 "nu1tau1" }{MPLTEXT 1 0 2 ";\n" }{MPLTEXT 1 0 7 "nu[2]:=" }{MPLTEXT 1 0 7 "nu2tau1" }{MPLTEXT 1 0 2 ";\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 17 "nu[rinfty-2]:=0:\n" }{MPLTEXT 1 0 79 "for k from 1 to g do nu[rinfty-2]:=nu[rinfty-2]+(-1)^(g-k)*nu[k]*Q[g+1-k]: \+ od:\n" }{MPLTEXT 1 0 13 "nu[rinfty-2];" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I#nuG6\"6#\"\"\"#F'\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I#nuG6\"6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ", $*&&I\"qG6\"6#\"\"\"F(&F%6#\"\"#F(#!\"\"\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "tdA11theo:=0:\n" }{MPLTEXT 1 0 84 "for i from 0 \+ to g-2 do for m from 1 to g-1-i do for r from i+m+1 to g do tdA11theo: =" }{MPLTEXT 1 0 66 "tdA11theo-(-1)^(i+m-1)*nu[m]*P[r]*Q[r-i-m-1]*lamb da^i od: od: od:\n" }{MPLTEXT 1 0 31 "tdA11theo:=simplify(tdA11theo);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 7 "factor(" }{MPLTEXT 1 0 9 "simpl ify(" }{MPLTEXT 1 0 9 "tdA11theo" }{MPLTEXT 1 0 19 "-CheckAtau1[1,1])) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*tdA11theoG6\"*&,&&I\"pGF$6#\"\" \"F*&F(6#\"\"#!\"\"F*,&&I\"qGF$F)\"\"$&F1F,!\"$F." }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "tdA12 theo:=0:\n" }{MPLTEXT 1 0 113 "for j from 0 to g-1 do for m from 1 to \+ g-j do tdA12theo:=tdA12theo+(-1)^(g-j-m)*nu[m]*Q[g-j-m]*lambda^j: od: \+ od:\n" }{MPLTEXT 1 0 32 "tdA12theo:=simplify(tdA12theo);\n" }{MPLTEXT 1 0 44 "factor(simplify(tdA12theo-CheckAtau1[1,2]));" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*tdA12theoG6\",(&I\"qGF$6#\"\"\"# !\"\"\"\"$I'lambdaGF$#F)F,&F'6#\"\"#F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "tdA21theoTerm1:= 0:\n" }{MPLTEXT 1 0 111 "for i from 0 to g do lowerpoint:=max(g,g+i-1) : for j from lowerpoint to 2*rinfty-5 do for m from 1 to j-g-i do\n" } {MPLTEXT 1 0 66 " tdA21theoTerm1:=tdA21theoTerm1-nu[m]*h[j-g-m-i]*P2[j ]*lambda^i: \n" }{MPLTEXT 1 0 12 "od: od: od:\n" }{MPLTEXT 1 0 16 "tdA 21theoTerm1;\n" }{MPLTEXT 1 0 19 "tdA21theoTerm2:=0:\n" }{MPLTEXT 1 0 149 "for i from 0 to g do for j1 from 0 to g-1 do for j2 from 1 to g-1 do for m from 1 to j1+j2-g-i do for r1 from j1+1 to g do for r2 from \+ j2+1 to g do \n" }{MPLTEXT 1 0 109 "tdA21theoTerm2:=tdA21theoTerm2-(-1 )^(j1+j2)*nu[m]*h[j1+j2-g-i-m]*P[r1]*P[r2]*Q[r1-j1-1]*Q[r2-j2-1]*lambd a^i:\n" }{MPLTEXT 1 0 25 "od: od: od: od: od: od: \n" }{MPLTEXT 1 0 16 "tdA21theoTerm2;\n" }{MPLTEXT 1 0 51 "tdA21theo:=simplify(tdA21theo Term1+tdA21theoTerm2);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 44 "factor( simplify(tdA21theo-CheckAtau1[2,1]));" }}{PARA 11 "" 1 "" {XPPMATH 20 ",,&I$tauG6\"6#\"\"\"#\"\"#\"\"$*$,&&I\"qGF%F&F'&F.6#F)F'F)#F'F**&F-F' F/F'#!\"#F**&F,F'I'lambdaGF%F'F1*$F6F)F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*tdA21theoG6\",,*$I'lambd aGF$\"\"##\"\"\"\"\"$*&,&&I\"qGF$6#F*F*&F/6#F(F*F*F'F*F)*$F.F(F)*$F1F( F)&I$tauGF$F0#F(F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Term1Hamtheo:=0:\n" }{MPLTEXT 1 0 115 "for i from 1 to g do for k from i+1 to g do Term1Hamtheo:=Term1Ha mtheo-h*nu[i]*(-1)^i*(g-i)*P[k]*Q[k-1-i]: od: od:\n" }{MPLTEXT 1 0 69 "for i from 1 to g do for k from i+1 to g do for m from i+1 to k-1 do " }{MPLTEXT 1 0 76 "Term1Hamtheo:=Term1Hamtheo-h*nu[i]*(-1)^m*P[k]*Q[k -1-m]*S[m-i]: od: od: od:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 17 "Te rm2Hamtheo:=0:\n" }{MPLTEXT 1 0 109 "for i from 1 to g do for k1 from \+ 1 to g do for k2 from 1 to g do for r1 from max(0,i-k2) to min(k1-1,i- 1) do\n" }{MPLTEXT 1 0 14 "Term2Hamtheo:=" }{MPLTEXT 1 0 81 "Term2Hamt heo+nu[i]*P[k1]*P[k2]*(-1)^(i-1)*Q[k1-1-r1]*Q[k2-i+r1]: od: od: od: od :\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 154 "for i from 1 to g do for \+ k1 from 1 to g do for k2 from 1 to g do for r1 from 0 to k1-1 do for r 2 from 0 to k2-1 do for m from i to g do if r1+r2>=g then " }{MPLTEXT 1 0 14 "Term2Hamtheo:=" }{MPLTEXT 1 0 130 "Term2Hamtheo+nu[i]*P[k1]*P [k2]*(-1)^(r1+r2)*Q[k1-1-r1]*Q[k2-1-r2]*(-1)^(g-m)*Q[g-m]*h[r1+r2+m-i- g+1]: fi: od: od: od: od: od: od:\n" }{MPLTEXT 1 0 1 " " }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 17 "Term3Hamtheo:=0:\n" }{MPLTEXT 1 0 72 "for i from 1 to g do for r from g to 2*rinfty-5 do for m from i to g do " } {MPLTEXT 1 0 14 "Term3Hamtheo:=" }{MPLTEXT 1 0 69 "Term3Hamtheo+nu[i]* (-1)^(g-m)*P2[r]*Q[g-m]*h[r+m-i-g+1]: od: od: od:\n" }{MPLTEXT 1 0 1 " \n" }{MPLTEXT 1 0 19 "Hamilton:=simplify(" }{MPLTEXT 1 0 13 "Term1Hamt heo+" }{MPLTEXT 1 0 13 "Term2Hamtheo+" }{MPLTEXT 1 0 15 "Term3Hamtheo) :\n" }{MPLTEXT 1 0 18 "simplify(Hamilton-" }{MPLTEXT 1 0 10 "Hamtau1); \n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 17 "Term1H amtheo:=0:\n" }{MPLTEXT 1 0 115 "for i from 1 to g do for k from i+1 t o g do Term1Hamtheo:=Term1Hamtheo-h*nu[i]*(-1)^i*(g-i)*P[k]*Q[k-1-i]: \+ od: od:\n" }{MPLTEXT 1 0 145 "for i from 1 to g do for k from i+1 to g do for m from i+1 to k-1 do Term1Hamtheo:=Term1Hamtheo-h*nu[i]*(-1)^m *P[k]*Q[k-1-m]*S[m-i]: od: od: od:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 17 "Term2Hamtheo:=0:\n" }{MPLTEXT 1 0 153 "for i from 1 to g do fo r k1 from 1 to g do for k2 from 1 to g do for r1 from 0 to k1-1 do for r2 from 0 to k2-1 do if r1+r2<=g-1 then if r1+r2=i-1 then \n" } {MPLTEXT 1 0 79 "Term2Hamtheo:=Term2Hamtheo+nu[i]*P[k1]*P[k2]*(-1)^(r1 +r2)*Q[k1-1-r1]*Q[k2-1-r2]" }{MPLTEXT 1 0 29 ": fi: fi: od: od: od: od : od:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 298 "fo r i from 1 to g do for k1 from 1 to g do for k2 from 1 to g do for r1 \+ from 0 to k1-1 do for r2 from 0 to k2-1 do for m from i to g do if r1+ r2>=g then Term2Hamtheo:=Term2Hamtheo+nu[i]*P[k1]*P[k2]*(-1)^(r1+r2)* Q[k1-1-r1]*Q[k2-1-r2]*(-1)^(g-m)*Q[g-m]*h[r1+r2+m-i-g+1]: fi: od: od: \+ od: od: od: od:\n" }{MPLTEXT 1 0 2 " \n" }{MPLTEXT 1 0 17 "Term3Hamthe o:=0:\n" }{MPLTEXT 1 0 155 "for i from 1 to g do for r from g to 2*rin fty-5 do for m from i to g do Term3Hamtheo:=Term3Hamtheo+nu[i]*(-1)^(g -m)*P2[r]*Q[g-m]*h[r+m-i-g+1]: od: od: od:\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 60 "Hamilton:=simplify(Term1Hamtheo+Term2Hamtheo+Term3Ham theo):\n" }{MPLTEXT 1 0 27 "simplify(Hamilton-Hamtau1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "simplify(Hamtau1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,.*&I\"hG6\"\"\"\",&&I\"pGF&6#\"\"#F'&F*6#F' !\"\"F'F'*&&I\"qGF&F.\"\"&&F2F+F'F'*(F1\"\"$&I$tauGF&F.F'F4F'F,*(F1F,& F8F+F'F4F'F,*&,**$F4F3F/*&F4F6F7F'!\"#*&F4F,F:F'F?*$F)F,F'F'F1F'F'*&F- F,F4F'F/F',&F1F6F4!\"$F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "factor(P[1]^2+2*P[1]*P[2]*Q[1]+P[2]^2*(Q[1]^2-Q[2]));" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*&,&*&&I\"pG6\"6#\"\"\"\"\"#&I\"qGF(6#F+F*F**&&F'F .F+&F-F)F*!\"\"F*,&F1F*F,F2F2F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "simplify(Hamtau1-(h*P[2]/3 +1/3*(" }{MPLTEXT 1 0 46 "P[1]^2+2* P[1]*P[2]*Q[1]+P[2]^2*(Q[1]^2-Q[2])))" }{MPLTEXT 1 0 3 ");\n" } {MPLTEXT 1 0 24 "f1:=unapply(es(simplify(" }{MPLTEXT 1 0 16 "Hamtau1-h *P[2]/3" }{MPLTEXT 1 0 6 "-1/3*(" }{MPLTEXT 1 0 45 "P[1]^2+2*P[1]*P[2] *Q[1]+P[2]^2*(Q[1]^2-Q[2]))" }{MPLTEXT 1 0 1 ")" }{MPLTEXT 1 0 32 ",q[ 1],q[2]),sigma[1],sigma[2]):\n" }{MPLTEXT 1 0 11 "f1(Q1,Q2);\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 23 "simplify(Hamtau1- 1/3*(" } {MPLTEXT 1 0 7 "h*P[2]+" }{MPLTEXT 1 0 45 "P[1]^2+2*P[1]*P[2]*Q[1]+P[2 ]^2*(Q[1]^2-Q[2])+" }{MPLTEXT 1 0 61 "Q[1]^3*Q[2]-2*Q[1]*Q[2]^2+2*Q[1] *Q[2]*tau[1]+2*Q[2]*tau[2]));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 " \n" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*(,.*$&I\"qG6 \"6#\"\"\"\"\"$F**&F&\"\"#&F'6#F-F*F**&,&*$F.F-F*&I$tauGF(F)F-F*F&F*F* *$F.F+F**&F.F*F3F*F-&F4F/F-F*F&F*F.F*#F*F+" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**&I#Q1G6\"\"\"$I#Q2GF%\"\"\"#F(F&*&F$F(F'\"\"##!\"#F&*( F$F(F'F(&I$tauGF%6#F(F(#F+F&*&F'F(&F06#F+F(F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "simpli fy(" }{MPLTEXT 1 0 10 "CheckAtau1" }{MPLTEXT 1 0 20 "[1,1]-(-1/3*P[2]) );\n" }{MPLTEXT 1 0 45 "simplify(CheckAtau1[1,2]-1/3*(lambda-Q[1]));\n " }{MPLTEXT 1 0 25 "simplify(CheckAtau1[2,1]-" }{MPLTEXT 1 0 4 "1/3*" }{MPLTEXT 1 0 47 "(lambda^2+Q[1]*lambda+Q[1]^2-2*Q[2]+2*tau[1]));" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 212 63 "We have verified the \\td\{A\}^\{tau1\} formula. Let u s do A^\{tau2\}." }{TEXT 212 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "nu[1]:=nu1tau2;\n" }{MPLTEXT 1 0 16 "nu[2]:=nu2tau2;\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 17 "nu[rinfty-2]:=0:\n" }{MPLTEXT 1 0 79 "for k from 1 to g do nu[rinfty-2]:=nu[rinfty-2]+(-1)^(g-k)*nu[k] *Q[g+1-k]: od:\n" }{MPLTEXT 1 0 13 "nu[rinfty-2];" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I#nuG6\"6#\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I#nuG6\"6#\"\"#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&&I\"qG6\"6#\"\"\"F'&F$6#\"\"#F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "tdA11theo:=0:\n" }{MPLTEXT 1 0 150 "for i from \+ 0 to g-2 do for m from 1 to g-1-i do for r from i+m+1 to g do tdA11the o:=tdA11theo-(-1)^(i+m-1)*nu[m]*P[r]*Q[r-i-m-1]*lambda^i od: od: od:\n " }{MPLTEXT 1 0 32 "tdA11theo:=simplify(tdA11theo);\n" }{MPLTEXT 1 0 44 "factor(simplify(tdA11theo-CheckAtau2[1,1]));" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*tdA11theoG6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\" \"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "tdA12theo:=0:\n" } {MPLTEXT 1 0 113 "for j from 0 to g-1 do for m from 1 to g-j do tdA12t heo:=tdA12theo+(-1)^(g-j-m)*nu[m]*Q[g-j-m]*lambda^j: od: od:\n" } {MPLTEXT 1 0 32 "tdA12theo:=simplify(tdA12theo);\n" }{MPLTEXT 1 0 44 " factor(simplify(tdA12theo-CheckAtau2[1,2]));" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*tdA12theoG6\"\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " \"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "tdA21theoTerm1:=0: \n" }{MPLTEXT 1 0 111 "for i from 0 to g do lowerpoint:=max(g,g+i-1): \+ for j from lowerpoint to 2*rinfty-5 do for m from 1 to j-g-i do\n" } {MPLTEXT 1 0 66 " tdA21theoTerm1:=tdA21theoTerm1-nu[m]*h[j-g-m-i]*P2[j ]*lambda^i: \n" }{MPLTEXT 1 0 12 "od: od: od:\n" }{MPLTEXT 1 0 16 "tdA 21theoTerm1;\n" }{MPLTEXT 1 0 19 "tdA21theoTerm2:=0:\n" }{MPLTEXT 1 0 149 "for i from 0 to g do for j1 from 0 to g-1 do for j1 from 1 to g-1 do for m from 1 to j1+j2-g-i do for r1 from j1+1 to g do for r2 from \+ j2+1 to g do \n" }{MPLTEXT 1 0 109 "tdA21theoTerm2:=tdA21theoTerm2-(-1 )^(j1+j2)*nu[m]*h[j1+j2-g-i-m]*P[r1]*P[r2]*Q[r1-j1-1]*Q[r2-j2-1]*lambd a^i:\n" }{MPLTEXT 1 0 25 "od: od: od: od: od: od: \n" }{MPLTEXT 1 0 16 "tdA21theoTerm2;\n" }{MPLTEXT 1 0 52 "tdA21theo:=simplify(tdA21theo Term1+tdA21theoTerm2);\n" }{MPLTEXT 1 0 44 "factor(simplify(tdA21theo- CheckAtau2[2,1]));" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(I'lambdaG6\"\"\" \"&I\"qGF$6#F%\"\"#&F'6#F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" } }{PARA 11 "" 1 "" {XPPMATH 20 ">I*tdA21theoG6\",(I'lambdaGF$\"\"\"&I\" qGF$6#F'\"\"#&F)6#F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Term1Hamtheo:=0:\n" } {MPLTEXT 1 0 115 "for i from 1 to g do for k from i+1 to g do Term1Ham theo:=Term1Hamtheo-h*nu[i]*(-1)^i*(g-i)*P[k]*Q[k-1-i]: od: od:\n" } {MPLTEXT 1 0 145 "for i from 1 to g do for k from i+1 to g do for m fr om i+1 to k-1 do Term1Hamtheo:=Term1Hamtheo-h*nu[i]*(-1)^m*P[k]*Q[k-1- m]*S[m-i]: od: od: od:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 17 "Term2 Hamtheo:=0:\n" }{MPLTEXT 1 0 153 "for i from 1 to g do for k1 from 1 t o g do for k2 from 1 to g do for r1 from 0 to k1-1 do for r2 from 0 to k2-1 do if r1+r2<=g-1 then if r1+r2=i-1 then \n" }{MPLTEXT 1 0 45 "Te rm2Hamtheo:=Term2Hamtheo+nu[i]*P[k1]*P[k2]*" }{MPLTEXT 1 0 34 "(-1)^(r 1+r2)*Q[k1-1-r1]*Q[k2-1-r2]" }{MPLTEXT 1 0 30 ": fi: fi: od: od: od: o d: od:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 298 "for i from 1 to g do for k1 from 1 to g do for k2 from 1 to g do for r1 from 0 to k1-1 do \+ for r2 from 0 to k2-1 do for m from i to g do if r1+r2>=g then Term2H amtheo:=Term2Hamtheo+nu[i]*P[k1]*P[k2]*(-1)^(r1+r2)*Q[k1-1-r1]*Q[k2-1- r2]*(-1)^(g-m)*Q[g-m]*h[r1+r2+m-i-g+1]: fi: od: od: od: od: od: od:\n" }{MPLTEXT 1 0 2 " \n" }{MPLTEXT 1 0 17 "Term3Hamtheo:=0:\n" }{MPLTEXT 1 0 155 "for i from 1 to g do for r from g to 2*rinfty-5 do for m fro m i to g do Term3Hamtheo:=Term3Hamtheo+nu[i]*(-1)^(g-m)*P2[r]*Q[g-m]*h [r+m-i-g+1]: od: od: od:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 60 "Ham ilton:=simplify(Term1Hamtheo+Term2Hamtheo+Term3Hamtheo):\n" }{MPLTEXT 1 0 27 "simplify(Hamilton-Hamtau2);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Term1Hamtheo:=0:\n" }{MPLTEXT 1 0 115 "for i fr om 1 to g do for k from i+1 to g do Term1Hamtheo:=Term1Hamtheo-h*nu[i] *(-1)^i*(g-i)*P[k]*Q[k-1-i]: od: od:\n" }{MPLTEXT 1 0 145 "for i from \+ 1 to g do for k from i+1 to g do for m from i+1 to k-1 do Term1Hamtheo :=Term1Hamtheo-h*nu[i]*(-1)^m*P[k]*Q[k-1-m]*S[m-i]: od: od: od:\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 17 "Term2Hamtheo:=0:\n" }{MPLTEXT 1 0 109 "for i from 1 to g do for k1 from 1 to g do for k2 from 1 to g d o for r1 from max(0,i-k2) to min(k1-1,i-1) do\n" }{MPLTEXT 1 0 95 "Ter m2Hamtheo:=Term2Hamtheo+nu[i]*P[k1]*P[k2]*(-1)^(i-1)*Q[k1-1-r1]*Q[k2-i +r1]: od: od: od: od:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 298 "for i from 1 to g do for k1 from 1 to g do for k2 from 1 to g do for r1 fro m 0 to k1-1 do for r2 from 0 to k2-1 do for m from i to g do if r1+r2> =g then Term2Hamtheo:=Term2Hamtheo+nu[i]*P[k1]*P[k2]*(-1)^(r1+r2)*Q[k 1-1-r1]*Q[k2-1-r2]*(-1)^(g-m)*Q[g-m]*h[r1+r2+m-i-g+1]: fi: od: od: od: od: od: od:\n" }{MPLTEXT 1 0 2 " \n" }{MPLTEXT 1 0 17 "Term3Hamtheo:= 0:\n" }{MPLTEXT 1 0 155 "for i from 1 to g do for r from g to 2*rinfty -5 do for m from i to g do Term3Hamtheo:=Term3Hamtheo+nu[i]*(-1)^(g-m) *P2[r]*Q[g-m]*h[r+m-i-g+1]: od: od: od:\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 60 "Hamilton:=simplify(Term1Hamtheo+Term2Hamtheo+Term3Ham theo):\n" }{MPLTEXT 1 0 18 "simplify(Hamilton-" }{MPLTEXT 1 0 9 "Hamta u2);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "simplify(Hamtau2-" }{MPLTEXT 1 0 33 "factor(-2*P[1]*P[2]-P[2]^2*Q[1]))" }{MPLTEXT 1 0 1 ";" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 97 "f2:=unapply(es(simplify(Hamtau2- factor(-2*P[1]*P[2]-P[2]^2*Q[1])),q[1],q[2]),sigma[1],sigma[2]):\n" } {MPLTEXT 1 0 11 "f2(Q1,Q2);\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 118 "simplify(Hamtau2- (-2*P[1]*P[2]-P[2]^2*Q[1]-Q[1]^4+3*Q[1]^2*Q[2]-2*Q[ 1]^2*tau[1]-2*Q[1]*tau[2]-Q[2]^2+2*Q[2]*tau[1]));" }}{PARA 11 "" 1 "" {XPPMATH 20 ",,*$&I\"qG6\"6#\"\"\"\"\"%!\"\"*&F$\"\"$&F%6#\"\"#F(F**&, &*$F-F/F*&I$tauGF&F'!\"#F(F$F/F(*&,(*$F-F,F**&F-F(F3F(F5&F4F.F5F(F$F(F (*&,(F8F(F9F/F:F/F(F-F(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ",.*$I#Q1G6 \"\"\"%!\"\"*&F$\"\"#I#Q2GF%\"\"\"\"\"$*&F$F)&I$tauGF%6#F+F+!\"#*&F$F+ &F/6#F)F+F1*$F*F)F'*&F*F+F.F+F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"! " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "factor(P[1]);\n" } {MPLTEXT 1 0 14 "factor(P[2]);\n" }{MPLTEXT 1 0 14 "factor(Q[1]);\n" } {MPLTEXT 1 0 13 "factor(Q[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,&*& &I\"pG6\"6#\"\"\"F)&I\"qGF'F(F)F)*&&F&6#\"\"#F)&F+F.F)!\"\"F),&F*F)F0F 1F1" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*&,&&I\"pG6\"6#\"\"\"F)&F&6#\" \"#!\"\"F),&&I\"qGF'F(F)&F0F+F-F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 ", &&I\"qG6\"6#\"\"\"F'&F$6#\"\"#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&&I \"qG6\"6#\"\"\"F'&F$6#\"\"#F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "factor(CheckL[1,1]-(-P[2]*lambda)-Q[1]*P[2]-P[1]);\n" }{MPLTEXT 1 0 49 "factor(CheckL[1,2]-(lambda^2-Q[1]*lambda+Q[2]));\n" }{MPLTEXT 1 0 50 "factor(CheckL[2,2]-(P[2]*lambda)+Q[1]*P[2]+P[1]);\n" }{MPLTEXT 1 0 142 "factor(series(CheckL[2,1]-(lambda^3+Q[1]*lambda^2+(Q[1]^2-Q[ 2]+2*tau[1])*lambda -P[2]^2+Q[1]*(Q[1]^2-2*Q[2])+2*Q[1]*tau[1]+2*tau[2 ]),p[2]=0));" }}{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 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 }