{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "restart:\nwith(linal g):\nwith(LinearAlgebra):\nDigits:=30:" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and trace have been redefined and u nprotected\n" }}{PARA 7 "" 1 "" {TEXT -1 64 "Warning, the assigned nam e GramSchmidt now has a global binding\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "INITIAL CONDI TIONS and PARAMETERS" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "R:= 30:\nAM:=x^2:\nAR:=x^2:\nAU:= x^3:\nAl:=x:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 32 "AAM:=0:\nAAR:=0:\nAAU:= 0:\nAAl:=x:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 616 "for i from 0 to R do\n\nAAM:=AM:\nAAR:=AR:\nAAU:= AU :\n\nexp(sum('1/r*subs(x=x^r,AAM+AAU+AAl)',r=1..R))-1-(AAM+AAU+AAl):\n AR:=convert(series(%,x=0,R),'polynom'):\n\nexp(sum('1/r*subs(x=x^r,AAR +AAU+AAl)',r=1..R))-1-(AAR+AAU+AAl):\nAM:=convert(series(%,x=0,R),'pol ynom'):\n\nexp(sum('1/r*subs(x=x^r,AAM+AAR+AAU+AAl)',r=1..R))*sum('sub s(x=x^r,(AAM+AAR+AAU+AAl))',r=1..R)-(AAM+AAR+AAU+AAl)\n-1/2*2*((AAM+AA R+AAU+AAl)^2+subs(x=x^2,(AAM+AAR+AAU+AAl)))\n-2*exp(sum('1/r*subs(x=x^ r,AAM+AAR+AAU+AAl)',r=1..R))+\n2+2*(AAM+AAR+AAU+AAl)+2/2*((AAM+AAR+AAU +AAl)^2+subs(x=x^2,(AAM+AAR+AAU+AAl))):\n\nAU:=convert(series(%,x=0,R) ,'polynom'):\n\nend do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Asymptotic enumeration" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "TAM:=AM:\nTAU:=AU:\nTAR:=AM:\nTAl:=Al:\nC:=M:\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 290 "Eq1:=exp(M+U+x)*exp(sum('1/ r*subs(x=x^r,TAM+TAU+TAl)',r=2..R))-1-(M+U+x):\n\nEq3:=exp(C+M+U+x)*ex p(sum('1/r*subs(x=x^r,TAM+TAR+TAU+TAl)',r=2..R))*(C+M+U+x+sum('subs(x= x^r,(TAM+TAR+TAU+TAl))',r=2..R))-(M+C+U+x)\n-2*exp(C+M+U+x)*exp(sum('1 /r*subs(x=x^r,TAM+TAR+TAU+TAl)',r=2..R))+\n2+2*(M+C+U+x):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "\n\nMat:=Matrix([[diff(Eq1,M)-1, d iff(Eq1, U)],[diff(Eq3,M), diff(Eq3, U)-1]]): \nEq4:=Determinant(Mat): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "fsolve(\{Eq1=M,Eq3=U,Eq4=0\},\{x=0.03,M,U\});\n" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#<%/%\"MG$\"?;[%36DE5^#yFu\"HN\"!#I/%\" xG$\"?c5QKwPe@H')3%e*[?F(/%\"UG$\"?lmL\"R2qs)HVtGn@p!#J" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Dyssimmetry Theorem" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "x_30:=.204895840886292158377632381056:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 125 "rho:= x_30:\nU0:=.692167287343298727007391336665e- 1:\nM0:=.135291742778251102625110844816:\nR0:=.13529174277825110262511 0844816:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "x_30^(-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?W6[[C00Sp.W&G0)[!#H" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Let us look for the asymptotic expansion of each GF" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 91 "Sing_M:=M0+M1*X+M2*X^2+M3*X^3+M4*X^4+M5*X^5:\n\nSin g_U:=U0+U1*X+U2*X^2+U3*X^3+U4*X^4+U5*X^5:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "Eq_sing1:= subs(M=Sing_M, C=Sing_M,U=Sing_U,x=rho *(1-X^2),Eq1):\nEq_sing3:= subs(M=Sing_M, C=Sing_M,U=Sing_U,x=rho*(1-X ^2),Eq3):\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "First coefficient \+ of the singular expansion" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "First0:=coeftayl(Eq_s ing1, X=0,0);\n\nThird0:=coeftayl(Eq_sing3,X=0,0);\nfsolve(\{First0=M0 ,Third0=U0\},\{M0,U0\});" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'First0G $\">![%36DE5^#yFu\"HN\"!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'Third 0G$\"=nL\"R2qs)HVtGn@p!#H" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Thos e are the first coefficients" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "U0 := .6921672873432987270073913367e-1:\nM0 := .135291742778251 10262511084480:\nR0:= .13529174277825110262511084480:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Let us look the second coefficients" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "First1:=coeftayl(Eq_sing1, \+ X=0,1);\nThird1:=coeftayl(Eq_sing3,X=0,1);\nfsolve(\{First1=M1, Third1 =U1\},\{M1,U1\});\nevalf(subs(x=rho,M=M0,U=U0, Mat));\nDeterminant(%); \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'First1G,&*&$\">N/KfGjBCr%'Third1G,&*&$\">iWfNS&HVhLA>1&*e!#H\"\"\"%#M1GF*F**&$\">Jszn&z19njd(G#[%RIb%!#H$\">L/KfGjBCrcWfNS&HVhLA>1 &*eF.$!?Ax-A)H_$G>$)Q!pC0(!#I%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"#$)!#I" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "Here we have an indeterminate cond ition which links M1 and U1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "isolate(Third1=U1,U1):\nisolate(First1=M1,U1):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "U1:=rhs(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#U1G,$*&$\"?83Q1iEeK3NMD')e$)!#I\"\"\"%#M1GF*F*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "No information about this equation : only the relations between " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "First2:=coeftayl(Eq_sing1, X=0,2);\n\nThird2:=coeftayl(Eq_sing3, X=0,2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'First2G,**&$\"?]V?$fGjB CrB,?F)!\"\"*&F'F*%#U2GF*F**& $\"???)G\"*[1]\\$\\%H#=.E!#HF*)%#M1G\"\"#F*F*" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'Third2G,**&$\"?qY`%>$*H#G8PU.W#G%!#H\"\"\")%#M1G\"\" #F*F**&$\"?5B(z1&*eF1F*%# M2GF*F*$\"??(3H$\\Nau`3l`)e]\"F1!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Eqq1:=First2-M2=0;\nEqq2:=Third2-U2=0;" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#>%%Eqq1G/,**&$\"?]cz19njd(G#[%RIb%!#I\"\"\"%#M2G F+!\"\"$\"?v5I]\\M0O;sA>B,?F*F-*&$\"?]V?$fGjBCr%%Eqq2G/,**&$\"?qY`%>$*H#G8PU.W#G%!#H\"\"\")%#M1G\"\"# F+F+*&$\"?!pF?#)H_$G>$)Q!pC0(!#IF+%#U2GF+!\"\"*&$\"??Y%fNS&HVhLA>1&*eF 2F+%#M2GF+F+$\"??(3H$\\Nau`3l`)e]\"F2F4\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 " MM2:=isolate(Eqq1,M2);\nUU2:=isolate(Eqq1,U2);\nsubs(MM2,Eqq2);\nsubs( U2=0,%);\nfsolve(%,M1);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$MM2G/%#M 2G,($\"?aU\\Te-u^\"!#H\"\"\"%#U 2GF0F0*&$\"?>bX7E2i,v$RWgur&F/F0)%#M1G\"\"#F0F0" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$UU2G/%#U2G,(*&$\"?73Q1iEeK3NMD')e$)!#I\"\"\"%#M2GF,F ,$\"?lb,>\"o\"[7,'o7y;%yZ-Aw8B!#I$\"?TCD#>7y;%yZ-Aw8BF% " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "subs(UU2,Eqq2);\nsubs(M 2=0,%);\nfsolve(%,M1);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&$\"?L 1m*Rk%)\\BBzn=Hl(!#H\"\"\")%#M1G\"\"#F)F)*&$\"#=!#IF)%#M2GF)!\"\"$\"?F .3)*yoz.q2$f')p4%F0F2\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&$\" ?L1m*Rk%)\\BBzn=Hl(!#H\"\"\")%#M1G\"\"#F)F)$\"?F.3)*yoz.q2$f')p4%!#I! \"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$!?UCD#>7y;%yZ-Aw8B!#I$\" ?UCD#>7y;%yZ-Aw8BF%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "M1:= -.231376220247784167812192252442;\nU1;\nR1:=M1:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M1G$!?UCD#>7y;%yZ-Aw8B!#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!??)=>)fPf4*o(=?/M>!#I" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Eqq1;\nM2:=solve(Eqq1,M2);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&$\"?]cz19njd(G#[%RIb%!#I\"\"\"%#M2GF)!\"\"$\">;JRN $p'GAs3X%>wgF(F+*&$\"?]V?$fGjBCr%#M2G,&$\"?91Am(3/koMx'e`M8!#I!\"\"*&$\"?gWD*Q5 &)\\YEF-Nj>\"!#H\"\"\"%#U2GF.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Third coefficients" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "First3:=coeftayl(Eq_sing1, \+ X=0,3);\nThird3:=coeftayl(Eq_sing3, X=0,3);\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'First3G,**&$\"?]V?$fGjBCr%'Third3G,**&$\"?5B(z1&*eF)F*%#M3GF*F*$\"?UV/E2qXNpURzVoKF)F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "Eqqq1:=First3-M3=0;\nEqqq2:=Third3 -U3=0;\nMM3:=rhs(isolate(Eqqq1,M3)):\nsubs(M3=MM3,Eqqq2):\nsubs(U3=0,% );\nU2:=solve(%,U2);\nM2;\n\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&Eq qq1G/,**&$\"?]cz19njd(G#[%RIb%!#I\"\"\"%#M3GF+!\"\"*&$\"?]V?$fGjBCr%&Eqqq2G/,**&$\"?!pF ?#)H_$G>$)Q!pC0(!#I\"\"\"%#U3GF+!\"\"*&$\"?8Kv9)4lULlDu)yqB!#HF+%#U2GF +F-*&$\"?1&*eF*F+%#M3GF+F+$\"?UV/E2qXNpURzVoKF*F+\"\"!" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&$\"?%e\\alO*=ejv$QlUP'!#I\"\"\"*&$ \"?%>x%)>fNdCF(y*3nB%!#HF(%#U2GF(!\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#U2G$\"?Nf#\\Ret'eDdrA`/:!#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\">B**p'p_l0xp:y)Ql%!#I" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "U2;\nR2:=M2;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" ?Nf#\\Ret'eDdrA`/:!#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#R2G$\">B** p'p_l0xp:y)Ql%!#I" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Next coeffic ient" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "First4:=coeftayl(Eq_sing1, X=0,4);\nThird4: =coeftayl(Eq_sing3, X=0,4);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'Firs t4G,,*&$\"?]V?$fGjBCr%'Third4G,,*& $\"?3B(z1&*eF)F*%#M4GF*F**&$\"?Z^:N,kt3+AWYf(R\"!#HF* %#M3GF*F0$\"?]fC2+z&H8NonMS%z!#JF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "Eqqqq1:=First4-M4=0;\nEqqqq2:=Third4-U4=0;\nMM4:=rhs (isolate(Eqqqq1,M4)):\nsubs(M4=MM4,Eqqqq2);\nsubs(U4=0,%):\nUU3:=solve (%,U3):\nsubs(U3=UU3,Eqqq1):\nfsolve(%,M3);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'Eqqqq1G/,,*&$\"?]V?$fGjBCr%'Eqqqq2G/,,*&$\"?#pF?#)H_$G>$)Q! pC0(!#I\"\"\"%#U4GF+!\"\"*&$\"?@dxv1?oV+5@K(z)pF*F+%#U3GF+F-*&$\"?DY%f NS&HVhLA>1&*eF*F+%#M4GF+F+*&$\"?Z^:N,kt3+AWYf(R\"!#HF+%#M3GF+F-$\"?]fC 2+z&H8NonMS%z!#JF+\"\"!" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,**&$\"# \+ " 0 "" {MPLTEXT 1 0 57 "\nM3:=.628133238401507458454903259964e-1:\nR3: =M3:\nU3:=UU3:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "Fin_M:=su bs(M4=0,M5=0,Sing_M);\nFin_R:=subs(M4=0,M5=0,Sing_M);\nFin_U:=subs(U4= 0,U5=0,Sing_U);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&Fin_MG,*$\"?;[%3 6DE5^#yFu\"HN\"!#I\"\"\"*&$\"?UCD#>7y;%yZ-Aw8BF(F)%\"XGF)!\"\"*&$\">B* *p'p_l0xp:y)Ql%F(F))F-\"\"#F)F)*&$\"?k*fK!\\Xeu],%QK8G'!#JF))F-\"\"$F) F)" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&Fin_RG,*$\"?;[%36DE5^#yFu\"HN \"!#I\"\"\"*&$\"?UCD#>7y;%yZ-Aw8BF(F)%\"XGF)!\"\"*&$\">B**p'p_l0xp:y)Q l%F(F))F-\"\"#F)F)*&$\"?k*fK!\\Xeu],%QK8G'!#JF))F-\"\"$F)F)" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&Fin_UG,*$\"?lmL\"R2qs)HVtGn@p!#J\"\"\"*&$ \"??)=>)fPf4*o(=?/M>!#IF)%\"XGF)!\"\"*&$\"?Nf#\\Ret'eDdrA`/:F-F))F.\" \"#F)F)*&$\"?'4&ovIqX-n__w0=5F(F))F.\"\"$F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Rooting \+ and edge" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "E:=convert(ser ies(AR*AM+AR*AU+AR*Al+\nAM*AU+AM*Al+\nAU*Al+1/2*(AU^2+subs(x=x^2,AU)), x=0,R),'polynom');" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"EG,X*&\"\"# \"\"\")%\"xG\"\"$F(F(*&\"0u;i-.96&F()F*\"#EF(F(*&\"&k/*F()F*\"#6F(F(*& \"(UJm\"F()F*\"#8F(F(*&\"*c-pB'F()F*\"#_7F()F*\"#>F(F(* &\".Q6dC@I&F()F*\"#BF(F(*&\"0av5A**=6\"F()F*\"#DF(F(*&\"2[f09Dx5.&F()F *\"#HF(F(*&\"\"'F()F*\"\"%F(F(*&\"##)F()F*FMF(F(*&\"%%=&F()F*\"\"*F(F( *&\"%t7F()F*\"\")F(F(*&\"'adQF()F*\"#7F(F(*&\"&*[@F()F*\"#5F(F(*&\")WQ vJF()F*\"#:F(F(*&\"+K+$yy#F()F*\"#=F(F(*&\"(yxB(F()F*\"#9F(F(*&\"-mdcS eDF()F*\"#@F(F(*&\"*D[HS\"F()F*\"#;F(F(*&\"/Zga:xCCF()F*\"#CF(F(*&\"1u )y%Q[6bBF()F*\"#FF(F(*&\",Jf')*[cF()F*\"#?F(F(*&\".x4k5G;\"F()F*\"#AF( F(*&\"2#==beeW(3\"F()F*\"#GF(F(*&F`qF()F*\"\"&F(F(*&\"$?$F()F*\"\"(F(F (" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Rooting a vertex" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "AU-(exp(sum('1/r*(subs(x=x^r,AM+AR+AU+Al))',r=1..R))-1-(AM+AR+A U+x)-1/2*((AM+AR+AU+x)^2+subs(x=x^2,(AM+AR+AU+x)))):\nseries(%,x=0,R): \nAUdot:=convert(%,'polynom');\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>% &AUdotG,V*&\"/\"\\>3ya$z\"\"\")%\"xG\"#EF(F(*&\"&mX\"F()F*\"#6F(F(*&\" '#*REF()F*\"#8F(F(*&\")?ar(*F()F*\"#F()F*\"#>F(F(*&\"- iatbY#)F()F*\"#BF(F(*&\"/U`It9FR(oPF()F*\"#CF(F(*&\"0S+9,)faOF()F*\" #FF(F(*&\"+$p*G5))F()F*\"#?F(F(*&\"-r;r(*4=F()F*\"#AF(F(*&\"15/JDGv'o \"F()F*\"#GF(F(*&FJF()F*\"\"&F(F(*&\"#cF()F*\"\"(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "V:= x*(AU+AM+AR)+AR-1/2*((AM+AU+x)^2+subs(x=x^2,AM+A U+x))\n+AM-1/2*((AR+AU+x)^2+subs(x=x^2,AR+AU+x))+AUdot:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "series(V,x=0,30);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+en%\"xG\"\"%\"\"$\"#5F%\"#K\"\"&\"$3\"\"\"'\"$)R\" \"(\"%::\"\")\"%-g\"\"*\"&XV#F'\"'545\"#6\"'c]U\"#7\"(?]\"=\"#8\"(Hm$y \"#9\"))[dT$\"#:\"*fL3]\"\"#;\"*[41k'\"#<\"+\"*y9cH\"#=\",eG]IK\"\"#> \",a@Q(\\f\"#?\"--b=.(o#\"#@\".[VP#>=7\"#A\".WtVE?a&\"#B\"/hivQEHD\"#C \"0157LSw:\"\"#D\"0K\">?5b7`\"#E\"1-G9sF%RW#\"#F\"2iC)\\R\\#o7\"\"#G\" 2%Q`_l&yi?&\"#H-%\"OG6#\"\"\"\"#I" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Rooting at an oriented edge" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "DD:=AR*AM+AR*AU+AR*Al+\nAM*AR+AM*AU+AM*Al+\nAl*AR+Al*AM+Al*AU +\nAU*AR+AU*AM+AU*Al+ AU^2:\nseries(DD,x=0,20);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+G%\"xG\"\"%\"\"$\"#7F%\"#W\"\"&\"$j\"\"\"'\"$S'\"\"(\" %UD\"\")\"&o.\"\"\"*\"&jH%\"#5\"'G4=\"#6\"'_9xF'\"(%GEL\"#8\")W`Z9\"#9 \"))o2N'\"#:\"*7))e!G\"#;\"+70QZ7\"#<\"+!occd&\"#=\",_o*R/D\"#>-%\"OG6 #\"\"\"\"#?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Final counting formula" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "E+V-DD:\nconvert(series(%,x=0,R),'polynom'):\nFinal:=series(%,x= 0,30);\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&FinalG+Y%\"xG\"\"#\"\"$ \"\"%F)\"#5\"\"&\"#F\"\"'\"#y\"\"(\"$Y#\"\")\"$=)\"\"*\"%rGF*\"&Y/\"\" #6\"&e$R\"#7\"'y=:\"#8\"'j!*f\"#9\"(WOS#\"#:\"(s$*y*\"#;\")#pq.%\"#<\" *V7Ko\"\"#=\"*KW]3(\"#>\"+<-`2I\"#?\",O(>E'G\"\"#@\",b@z\"Qb\"#A\"-1i' =!*R#\"#B\".!zgM#\\/\"\"#C\"._M,6Td%\"#D\"/TJ,_<\"#H" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Singular expansions of E , V, DD and E+V-DD" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 234 "DD_s ing1:=AAAR*AAAM+AAAR*AAAU+AAAR*AAAl+\nAAAM*AAAR+AAAM*AAAU+AAAM*AAAl+\n AAAl*AAAR+AAAl*AAAM+AAAl*AAAU+\nAAAU*AAAR+AAAU*AAAM+AAAU*AAAl+ AAAU^2 :\nDD_Sing:=series(subs(AAAR=Fin_M, AAAM=Fin_M, AAAU=Fin_U, AAAl=rho*( 1-X^2),DD_sing1),X=0,4);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(DD_Sing G+-%\"XG$\"?MuKqzZ)4n]evP5=#!#I\"\"!$!?BsjrUskD-e5q)f*eF)\"\"\"$\"?= " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 206 "E_sing1:=AAAR*AAAM+AAAR*AAAU+AAAR*AAAl+\nAAAM*AAAU +AAAM*AAAl+\nAAAU*AAAl+1/2*(AAAU^2+subs(x=x^2,AU)):\nsubs(x=rho*(1-X^2 ),%):\nE_Sing:=series(subs(AAAR=Fin_M, AAAM=Fin_M, AAAU=Fin_U, AAAl=rh o*(1-X^2),%),X=0,4);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'E_SingG+-% \"XG$\"?^#\\^*)>3:y'o&ei44\"!#I\"\"!$!?6'=e8iBG6!H0N*z%HF)\"\"\"$\"?1) \\#)oq)e6C7%\\h)=@F)\"\"#$\"?gGIq6@,CPU>&oiJ&!#J\"\"$-%\"OG6#F-\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 289 "V_dot_sing1:=AAAU\n-((exp(AAAM+AAAR+AAAU+AAAl)* exp(sum('1/r*(subs(x=x^r,AM+AR+AU+Al))',r=2..R))\n-1-(AAAM+AAAR+AAAU+x )\n-1/2*((AAAM+AAAR+AAAU+AAAl)^2+subs(x=x^2,(AM+AR+AU+Al))))):\n\nsubs (x=rho*(1-X^2),%):\nV_dot_sing:=series(subs(AAAR=Fin_M, AAAM=Fin_M, AA AU=Fin_U, AAAl=rho*(1-X^2),%),X=0,4);\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+V_dot_singG+-%\"XG$\"?D!f#pG3H`QsSc[*\\\"!#J\"\"!$!=?qXAyL@BB K;qTX!#H\"\"\"$\"?d;&>a2!)GS%=u[VXMF)\"\"#$\"?C)*>w'))[ZVe7!Rj'4\"F)\" \"$-%\"OG6#F.\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 243 "V_si ng:= x*(AAAU+AAAM+AAAR)\n+AAAR-1/2*((AAAM+AAAU+x)^2+subs(x=x^2,AM+AU+x ))\n+AAAM-1/2*((AAAR+AAAU+x)^2+subs(x=x^2,AR+AU+x))\n+V_dot_sing:\nsub s(x=rho*(1-X^2),%):\nV_Sing:=series(subs(AAAR=Fin_M, AAAM=Fin_M, AAAU= Fin_U, AAAl=rho*(1-X^2),%),X=0,4);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# >%'V_SingG+-%\"XG$\"?C4&4ra_R)Q=#zpeV\"!#I\"\"!$!?5'=e8iBG6!H0N*z%HF) \"\"\"$\"?f7WDa(ye_`w#p(*[E!#J\"\"#$\"?`<5-@\"*RkzW'\\.QK#F)\"\"$-%\"O G6#F-\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "T_Sing:=serie s(V_Sing+E_Sing-DD_Sing,X=0,4);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%' T_SingG+-%\"XG$\">TFxNmfZ%**>?i%zX$!#I\"\"!$\"\"#F)\"\"\"$!?'ync*G/jZj cq$Q'f=F)F,$\"?m92&)4z*>f0Xkw@z\"F)\"\"$-%\"OG6#F-\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "C:=evalf(.1792176644505591997909850 71480/GAMMA(-3/2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG$\"?/<3gP 66*=2.YbMe(!#J" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "The symmetries, solving the equation" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Suma:=subs(x=x^2,AM+AR+AU);\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#> %%SumaG,Z*&\"+%f\"e%R\"\"\"\")%\"xG\"#MF(F(*&\"13E&=OfOD&F()F*\"#aF(F( *&\",`EUxz#F()F*\"#QF(F(*&\"-'eR\\F()F*\"#7F(F(*&\"#`F()F*\"#5F(F(*&\"&_<\"F() F*\"#=F(F(*&\"$Q(F()F*\"#9F(F(*&\"%9HF()F*FgnF(F(*&\"'[l')F()F*\"#CF(F (*&\"&I&[F()F*\"#?F(F(*&\")(y%3rF()F*\"#IF(F(*&\"'#e.#F()F*\"#AF(F(*& \"+W$y4B'F()F*\"#OF(F(*&\")aj@;F()F*\"#GF(F(*&\"-v>+$Hr&F()F*\"#UF(F(* &\"*7n'QJF()F*\"#KF(F(*&\"/A8dlH6aF()F*\"#[F(F(" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 315 "\nSU:=0;\nR:=20:\nfor i from 0 to R do\n(exp( sum('1/r*subs(x=x^r,SU+x)',r=1..R))-1-(SU+x)-1/2*((SU+x)^2+subs(x=x^2, SU+x)))+\n(exp((sum('1/r*(subs(x=x^(2*r),AR+AM+AU-SU))',r=1..R)))-1)*( exp(sum('1/r*subs(x=x^r,SU+x)',r=1..R))-1-1/2*((SU+x)^2+subs(x=x^2,SU+ x))):\n\nprint(i):\nSU:=convert(series(%,x=0,R),'polynom'):\n\nend do: " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#SUG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#8 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#=" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#>" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#\"#?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "SUT:=SU;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$SUTG,D*$)%\"xG\"\"$\"\"\"F**$)F(\"\"%F*F* *&F-F*)F(\"\"&F*F**&F)F*)F(\"\"'F*F**&\"#;F*)F(\"\"(F*F**&F5F*)F(\"\") F*F**&\"#sF*)F(\"\"*F*F**&\"#yF*)F(\"#5F*F**&\"$H$F*)F(\"#6F*F**&\"$6% F*)F(\"#7F*F**&\"%$e\"F*)F(\"#8F*F**&\"%d@F*)F(\"#9F*F**&\"%yxF*)F(\"# :F*F**&\"&)f6F*)F(F5F*F**&\"&n\"RF*)F(\"#F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "let us f ind the constant growth" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 256 "EQ:=SSU- (exp(SSU+x)*exp(sum('1/r*subs(x=x^r,SU+x)',r=2..R))-1-(SSU+x )-1/2*((SU+x)^2+subs(x=x^2,SUT+x)))+\n(exp((sum('1/r*(subs(x=x^(2*r),A R+AM+AU-SUT))',r=1..R)))-1)*(exp(SSU+x)\n*exp(sum('1/r*subs(x=x^r,SU+x )',r=2..R))-1-1/2*((SSU+x)^2+subs(x=x^2,SUT+x))):" }}}{PARA 12 "" 1 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "EQQ:=diff( EQ,SSU):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "fsolve(\{EQ=0,E QQ=0\},\{SSU,x=0.17\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"xG$\" ?fF^hGc#zL6SqHDx%!#I/%$SSUG$\"?>:+hht;H&*3i\"ouH&!#H" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "1/.477252970401133792562861512759;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?%)p0j#o#\\))G%>$[K&4#!#H" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "sqrt(.2048958408862921583776 32381056);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?)fiH\"f@C " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 " K:=convert(Final,'polynom');\nT_Sing;\nrho;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"KG,X*&\"\"#\"\"\")%\"xG\"\"$F(F(*&\"/F(F(*&\"-1i'=!*R#F()F*\"#BF(F(*&\"._M,6Td%F()F*\" #DF(F(*&\"1Ou>TJ,_E'G\"F()F*\"#@F(F(*&\"(s$*y*F()F*\"#;F(F(*&\".!zgM# \\/\"F()F*\"#CF(F(*&\"/GRmLz#)))F()F*FPF(F(*&\"+<-`2IF()F*\"#?F(F(*&\" ,b@z\"QbF()F*\"#AF(F(*&\"09[;93z$RF()F*\"#GF(F(*&F\\oF()F*\"\"&F(F(*& \"#yF()F*\"\"(F(F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+-%\"XG$\">TFxNm fZ%**>?i%zX$!#I\"\"!$\"\"#F'\"\"\"$!?'ync*G/jZjcq$Q'f=F'F*$\"?m92&)4z* >f0Xkw@z\"F'\"\"$-%\"OG6#F+\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"?c5QKwPe@H')3%e*[?!#I" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 " C:=exp(sum('subs(x=rho,1/r*subs(x=x^r,K))',r=2..30));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG$\"?uZyxi2Xw__\"Q\"3+5!#H" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "C*convert(series(exp(T_Sing),X=0,4),polynom);" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#,*$\"?A=&=5-^_^1!G&o_.\"!#H\"\"\"*&$\"?WOq.U?]II,cq`q ?!#fF'%\"XGF'F'*&$\"?T)fJI[[sl?&y]AD>!#IF')F,\"\"#F'!\"\"*&$\"?HE3ndh* *f\\owSQb=F0F')F,\"\"$F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "evalf(.185538407668495999615767082629/GAMMA(-3/2));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"?jtsmN[@%>&frF\"4&y!#J" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "102 1 0" 0 }{VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }{RTABLE_HANDLES 53231620 } {RTABLE M7R0 I5RTABLE_SAVE/53231620X,%)anythingG6"6"[gl!"%!!!#%"#"#$!>n&z19njd(G#[%RIb%!#H$" >cWfNS&HVhLA>1&*eF)$">L/KfGjBCr$)Q!pC0(!#IF& }