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0 2 2 -1 1 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 105 "- ---------------------------------------------------------------------- ---------------------------------\n" }{TEXT 217 119 "This session cont ains the calculations (generating functions) related to the bond-perco lation model on triangulations.\n" }{TEXT 217 104 "------------------- ---------------------------------------------------------------------- ---------------" }}{PARA 0 "" 0 "" {TEXT 217 151 "The computation boil s down to the enumeration of triangulations with a simple boundary acc ording to the number of edges incident to the outer vertices." }} {PARA 0 "" 0 "" {TEXT 217 104 "--------------------------------------- -----------------------------------------------------------------" }} {PARA 0 "" 0 "" {TEXT 217 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }{MPLTEXT 1 0 52 "with(gfun):with(algcurves):with(plots,implicitplot):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "mafactor:=t->map(factor,t):" }{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 217 52 "Let S(x,y,z) be the series of near-tritangulations " } {TEXT 217 109 "with a simple boudary (+ the bridge map) counted by out er degree (x), number of reef edges (y) and edges (z)." }{TEXT 217 0 " " }}{PARA 0 "" 0 "" {TEXT 217 56 "------------------------------------ --------------------" }}{PARA 0 "" 0 "" {TEXT 217 169 "Our first goal \+ is to find an algebraic equation for S(x,y,z). Before we do this, we n eed an easier result: an algebraic equation for the specialization F(x ,z):=S(x,1,z)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 221 141 "Preliminary calculation: an algebraic equation for the \+ series F(x,z) of triangulations with a simple boundary (without reef e dges parameter)." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 217 166 "Functional equation for the seri es F(x,z)=S(x,1,z) of near tritangulations with simple boudary (+ the \+ bridge map) counted by outer degree (x), and edges (z). F1=[x]F." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "eqF:=x^2*z+z/x*(F-x*F1)+z*F^ 2/x-F;\n" }{MPLTEXT 1 0 17 "eqF:=factor(%*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**&)I\"xG6\"\"\"#\"\"\"I\"zGF&F(F(*(F)F(F%!\"\",&*&I#F1G F&F(F%F(F+I\"FGF&F(F(F(*(F)F()F/F'F(F%F+F(F/F+" }}{PARA 11 "" 1 "" {XPPMATH 20 ",,*&)I\"xG6\"\"\"$\"\"\"I\"zGF&F(F(*&)I\"FGF&\"\"#F(F)F(F (*(I#F1GF&F(F%F(F)F(!\"\"*&F,F(F%F(F0*&F,F(F)F(F(" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 36 "coefF:= proc(n) local i,Coef,C,eq2;\n" } {MPLTEXT 1 0 19 "Coef:=array(0..n);\n" }{MPLTEXT 1 0 26 "for i from 1 \+ by 1 to n do\n" }{MPLTEXT 1 0 28 "C:=sum(Coef[k]*z^k,k=1..i);\n" } {MPLTEXT 1 0 38 "eq2:=subs(\{F=C,F1=coeff(C,x,1)\},eqF);\n" }{MPLTEXT 1 0 40 "Coef[i]:=solve(coeff(eq2,z,i),Coef[i]);\n" }{MPLTEXT 1 0 4 "od ;\n" }{MPLTEXT 1 0 30 "sum(Coef[k]*z^k,k=1..n); end:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 14 "FFF:=coefF(5);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",,*&)I\"xG6\"\"\"#\"\"\"I\"zGF&F(F(*&F%F()F)F'F(F(*&)F%\"\"$F()F)F .F(F(*(F.F(F$F()F)\"\"%F(F(**F'F(F%F(,&*$F-F(F(F'F(F()F)\"\"&F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 222 9 "solve eqF" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "eqdF 1:=diff(eqF,F);\n" }{MPLTEXT 1 0 19 "eqdF2:=diff(eqF,x);" }}{PARA 11 " " 1 "" {XPPMATH 20 ",(*(\"\"#\"\"\"I\"FG6\"F%I\"zGF'F%F%I\"xGF'!\"\"F( F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*(\"\"$\"\"\")I\"xG6\"\"\"#F%I\" zGF(F%F%*&I#F1GF(F%F*F%!\"\"I\"FGF(F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "r1:=factor(resultant(eqF,eqdF1,F))/z;\n" }{MPLTEXT 1 0 38 "r2:=factor(resultant(eqF,eqdF2,F))/z;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ",,*(\"\"%\"\"\")I\"xG6\"\"\"$F%)I\"zGF(\"\"#F%F%**F$F%I#F 1GF(F%F'F%F*F%!\"\"*$)F'F,F%F/*(F,F%F'F%F+F%F%*$F*F%F/" }}{PARA 11 "" 1 "" {XPPMATH 20 ",.*(\"\"*\"\"\")I\"xG6\"\"\"%F%)I\"zGF(\"\"#F%F%**\" \"'F%I#F1GF(F%)F'F,F%F*F%!\"\"*&)F/F,F%F*F%F%*&F,F%)F'\"\"$F%F1*(F6F%F 0F%F+F%F%*&F/F%F+F%F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "fa ctor(resultant(r1,r2,x));\n" }{MPLTEXT 1 0 18 "eqF1:=op(1,%/z^4);" }} {PARA 11 "" 1 "" {XPPMATH 20 "*&)I\"zG6\"\"\"%\"\"\"),0*(\"#kF')I#F1GF %\"\"$F')F$\"\"&F'F'*(\"#'*F')F-\"\"#F'F#F'!\"\"*&\"#FF'F/F'F5*(\"#IF' F-F')F$F.F'F'*&F3F'F$F'F'*$)F$F4F'F'F-F5F4F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ",0*(\"#k\"\"\")I#F1G6\"\"\"$F%)I\"zGF(\"\"&F%F%*(\"#'*F%) F'\"\"#F%)F+\"\"%F%!\"\"*&\"#FF%F*F%F3*(\"#IF%F'F%)F+F)F%F%*&F/F%F+F%F %*$)F+F0F%F%F'F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "factor(resultant(eqF,eqF1,F1 )):\n" }{MPLTEXT 1 0 14 "eqFalg:=%/z^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",dp*(\"#k\"\"\")I\"xG6\"\"\"*F%)I\"zGF(\"\"'F%!\"\"**\"$#>F%)I\"FG F(\"\"#F%)F'F,F%F*F%F-**F/F%)F1\"\"%F%)F'\"\"$F%F*F%F-**F/F%F1F%)F'\" \"(F%)F+\"\"&F%F%**F/F%F1F%F3F%F*F%F-*(F$F%)F1F,F%F*F%F-**\"$%QF%)F1F8 F%)F'F6F%F " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 168 "Functional equation for the series F(x,z)=S(x,1,z) of n ear-triangulations with a simple boundary (+ the bridge map) counted b y outer degree (x), and edges (z). F1=[x]F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "eqF \+ := x^3*z+F^2*z-F1*x*z-F*x+F*z;\n" }{MPLTEXT 1 0 64 "eqF1 := 64*F1^3*z^ 5-96*F1^2*z^4-27*z^5+30*F1*z^3+F1^2*z+z^2-F1;\n" }{MPLTEXT 1 0 565 "eq Falg := -64*x^9*z^6-192*F^2*x^6*z^6-192*F^4*x^3*z^6+192*F*x^7*z^5-192* F*x^6*z^6-64*F^6*z^6+384*F^3*x^4*z^5-384*F^3*x^3*z^6+96*x^7*z^5+192*F^ 5*x*z^5-192*F^5*z^6-192*F^2*x^5*z^4+576*F^2*x^4*z^5-192*F^2*x^3*z^6-19 2*F^4*x^2*z^4+480*F^4*x*z^5-192*F^4*z^6-192*F*x^5*z^4+192*F*x^4*z^5+64 *F^3*x^3*z^3-384*F^3*x^2*z^4+384*F^3*x*z^5-64*F^3*z^6-x^7*z^2-30*x^5*z ^4+27*x^3*z^6-2*F^2*x^4*z^2+96*F^2*x^3*z^3-222*F^2*x^2*z^4+96*F^2*x*z^ 5-F^4*x*z^2+2*F*x^5*z-2*F*x^4*z^2+30*F*x^3*z^3-30*F*x^2*z^4+2*F^3*x^2* z-2*F^3*x*z^2+x^5*z-x^3*z^3-F^2*x^3+3*F^2*x^2*z-F^2*x*z^2-F*x^3+F*x^2* z;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ",,*&)I\"xG6\"\"\"$\"\"\"I\"zGF&F (F(*&)I\"FGF&\"\"#F(F)F(F(*(I#F1GF&F(F%F(F)F(!\"\"*&F,F(F%F(F0*&F,F(F) F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ",0*(\"#k\"\"\")I#F1G6\"\"\"$F%)I \"zGF(\"\"&F%F%*(\"#'*F%)F'\"\"#F%)F+\"\"%F%!\"\"*&\"#FF%F*F%F3*(\"#IF %F'F%)F+F)F%F%*&F/F%F+F%F%*$)F+F0F%F%F'F3" }}{PARA 11 "" 1 "" {XPPMATH 20 ",dp*(\"#k\"\"\")I\"xG6\"\"\"*F%)I\"zGF(\"\"'F%!\"\"**\"$# >F%)I\"FGF(\"\"#F%)F'F,F%F*F%F-**F/F%)F1\"\"%F%)F'\"\"$F%F*F%F-**F/F%F 1F%)F'\"\"(F%)F+\"\"&F%F%**F/F%F1F%F3F%F*F%F-*(F$F%)F1F,F%F*F%F-**\"$% QF%)F1F8F%)F'F6F%F " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "#just checking\n" }{MPLTEXT 1 0 33 "algeqtoseries(eqFalg,z,F,5,true);" }}{PARA 11 "" 1 " " {XPPMATH 20 "7$+/I\"zG6\"!\"\"\"\"!*$)I\"xGF%\"\"#\"\"\"F,,$F*F&F+*$ )F*\"\"$F,F0,$*&F0F,F)F,F&\"\"%-I\"OG%*protectedG6#F,\"\"&+/F$F(F,F*F+ F.F0,$F2F,F3,$*(F+F,F*F,,&F.F,F+F,F,F,F8F4\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 221 56 "Computation of an algebraic \+ equation (denoted eqS) for S" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 204 "Let R(w) be the series of \+ triangulations with a simple boundary counted according to outer verti ces active (x) or inactive (w) and edges incident to black (y) and all edges (z) [see precise def in paper]\n" }{TEXT 217 1 "\n" }{TEXT 217 62 "We observe that S=R(0), F=subs(y=1,S) and F1=[x^1]subs(y=1,S)." }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 54 "We start f rom the functional equation for R.(denoting " }{TEXT 217 28 "Fw=w[x1]R (w; x; 1; z)=F(w,z)" }{TEXT 217 1 ")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "eqR:=-R+y*z*(x*(x+w)+(R-S)/w+S*R/x+(R-S)*Fw/w);\n" } {MPLTEXT 1 0 19 "eqR:=factor(w*x*%);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&I\"RG6\"!\"\"*(I\"yGF$\"\"\"I\"zGF$F(,**&I\"xGF$F (,&F,F(I\"wGF$F(F(F(*&,&F#F(I\"SGF$F%F(F.F%F(*(F1F(F#F(F,F%F(*(F0F(I#F wGF$F(F.F%F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ",2**)I\"wG6\"\"\"#\" \"\")I\"xGF&F'F(I\"yGF&F(I\"zGF&F(F(**F%F()F*\"\"$F(F+F(F,F(F(*,I#FwGF &F(I\"RGF&F(F*F(F+F(F,F(F(*,F1F(I\"SGF&F(F*F(F+F(F,F(!\"\"*,F2F(F4F(F% F(F+F(F,F(F(**F2F(F*F(F+F(F,F(F(**F4F(F*F(F+F(F,F(F5*(F2F(F%F(F*F(F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 222 48 "computa tion of first coefficients (needed below)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "n:=8;\n" }{MPLTEXT 1 0 36 "algeqtoseries(eqFalg,z,F,n+ 2,true);\n" }{MPLTEXT 1 0 9 "op(2,%):\n" }{MPLTEXT 1 0 37 "serFx:=sum( coeff(%,z,i)*z^i,i=0..n);\n" }{MPLTEXT 1 0 19 "serFw:=subs(x=w,%);" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 24 "serF1:=coeff(serFx,x,1);" } {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$+9I\"zG6\"!\"\"\"\"!*$)I\"xGF%\"\"#\"\"\"F,,$F*F& F+*$)F*\"\"$F,F0,$*&F0F,F)F,F&\"\"%,$*(F+F,F*F,,&F.F,F+F,F,F,\"\"&,$*& \"#5F,F/F,F&\"\"'*&F)F,,&*&F7F,F/F,F,\"#CF,F,\"\"(,$*&F*F,,&*&\"#NF,F/ F,F,\"#KF,F,F&\"\"),$*(F+F,F/F,,&*&F@F,F/F,F,\"#gF,F,F,\"\"*-I\"OG%*pr otectedG6#F,F:+9F$F(F,F*F+F.F0,$F2F,F3F4F7,$F9F,F;F " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "coefR:= proc(N) local i,Coef,C,eq2;\n" }{MPLTEXT 1 0 18 "Coef:=array(0..N);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 26 "for i f rom 0 by 1 to N do\n" }{MPLTEXT 1 0 28 "C:=sum(Coef[k]*z^k,k=0..i);\n" }{MPLTEXT 1 0 45 "eq2:=subs(\{Fw=serFw,R=C,S=subs(w=0,C)\},eqR);\n" } {MPLTEXT 1 0 40 "Coef[i]:=solve(coeff(eq2,z,i),Coef[i]);\n" }{MPLTEXT 1 0 4 "od;\n" }{MPLTEXT 1 0 30 "sum(Coef[k]*z^k,k=0..N); end:\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 4 "serR" }{MPLTEXT 1 0 11 ":=coefR(n );" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ",2**I\"xG6\"\" \"\"I\"yGF%F&,&F$F&I\"wGF%F&F&I\"zGF%F&F&*(F$F&)F*\"\"#F&)F'F-F&F&**F$ F&F.F&,(*(F)F&F$F&F'F&F&*&)F$F-F&F'F&F&*$)F)F-F&F&F&)F*\"\"$F&F&**F$F& F.F&,**&F)F&F.F&F&*(F7F&F$F&F.F&F&*&F)F&F'F&F&F)F&F&)F*\"\"%F&F&**F$F& F.F&,2**F-F&F)F&F3F&)F'F7F&F&*(F-F&)F$F7F&FBF&F&**F-F&F5F&F$F&F.F&F&*& )F)F7F&F'F&F&*$FGF&F&*&F-F&FBF&F&*$F.F&F&F'F&F&)F*\"\"&F&F&**F$F&F.F&, 2**FLF&F)F&F$F&)F'F>F&F&*(\"#5F&F3F&FPF&F&*(F-F&F5F&FBF&F&**F7F&F)F&F$ F&FBF&F&*(F-F&F5F&F.F&F&**F-F&F)F&F$F&F.F&F&*(F7F&F5F&F'F&F&*&F7F&F5F& F&F&)F*\"\"'F&F&**F$F&F.F&,D**FLF&F)F&FDF&)F'FLF&F&*(FLF&)F$F>F&FhnF&F &**FLF&F5F&F3F&FPF&F&**F7F&FGF&F$F&FBF&F&*&)F)F>F&F.F&F&**F-F&FGF&F$F& F.F&F&*(F7F&F)F&FhnF&F&*(\"#:F&F$F&FhnF&F&*(F-F&F^oF&F'F&F&*(F>F&F)F&F PF&F&*(FLF&F$F&FPF&F&*&F-F&F^oF&F&*(FLF&F)F&FBF&F&*(F>F&F$F&FBF&F&*(F> F&F)F&F.F&F&*(F>F&F)F&F'F&F&*&F>F&F)F&F&F&)F*\"\"(F&F&**F$F&F.F&,H**\" #@F&F)F&F3F&)F'FZF&F&*(\"#NF&FDF&FbpF&F&**\"#7F&F5F&F$F&FhnF&F&**\"\"* F&F)F&F3F&FhnF&F&*(F7F&FGF&FPF&F&**\"\")F&F5F&F$F&FPF&F&**FLF&F)F&F3F& FPF&F&*(FLF&FGF&FBF&F&**FhpF&F5F&F$F&FBF&F&*&F]pF&FbpF&F&*(F]pF&FGF&F. F&F&**FZF&F5F&F$F&F.F&F&*&FZF&FhnF&F&*(FRF&FGF&F'F&F&*&F]pF&FPF&F&*&FR F&FGF&F&*&F>F&FBF&F&*&F>F&F.F&F&*&F>F&F'F&F&F&)F*F[qF&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "subser:=A->collect(subs(R=serR,S=subs(w=0,serR),F1=co eff(subs(w=0,y=1,serR),x,1),A),z,factor);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"AG6\"F%6$I)operatorGF%I&arrowGF%F%-I(co llectG6$%*protectedGI(_syslibGF%6%-I%subsGF,6&/I\"RGF%I%serRGF%/I\"SGF %-F06$/I\"wGF%\"\"!F4/I#F1GF%-I&coeffGF,6%-F06%F9/I\"yGF%\"\"\"F4I\"xG F%FEF$I\"zGF%I'factorGF+F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 98 "We observe that Fw=F(w,z), and use the algebraic equatio n obtained for F to eliminate Fw from eqR." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 26 "eqFw:=subs(x=w,F=Fw,eqF);\n" }{MPLTEXT 1 0 32 "eqFa lgw:=subs(x=w,F=Fw,eqFalg);\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 42 " eqR2:=factor(resultant(eqR,eqFw,Fw))/w/z;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ",,*&)I\"wG6\"\"\"$\"\"\"I\"zGF&F(F(*(I#F1GF&F(F%F(F)F(!\" \"*&)I#FwGF&\"\"#F(F)F(F(*&F/F(F%F(F,*&F/F(F)F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ",dp*(\"#k\"\"\")I\"wG6\"\"\"*F%)I\"zGF(\"\"'F%!\"\"**\"$# >F%)I#FwGF(\"\"#F%)F'F,F%F*F%F-**F/F%)F1\"\"%F%)F'\"\"$F%F*F%F-**F/F%F 1F%)F'\"\"(F%)F+\"\"&F%F%**F/F%F1F%F3F%F*F%F-*(F$F%)F1F,F%F*F%F-**\"$% QF%)F1F8F%)F'F6F%FF(F ,F(F0F(F(*,F:F(F%F(F)F(F,F(F0F(F(*,F:F(F%F(F>F(F,F(F/F(F(**F:F(F)F(F,F (F/F(F(*,F=F(F2F(F>F(F,F(F0F(FC*,F=F(F%F(F)F(F,F(F0F(FC*,F=F(F%F(F>F(F ,F(F/F(FC**F=F(F)F(F,F(F/F(FC*.F9F(F=F(F%F(F*F(F,F(F0F(F(*,F9F(F=F(F*F (F,F(F/F(F(*.F:F(F@F(F%F(F*F(F,F(F0F(FC*,F:F(F@F(F*F(F,F(F/F(FC*.F.F(F :F(F2F(F>F(F-F(F0F(FC*.F.F(F:F(F%F(F)F(F-F(F0F(FC*0F.F(F9F(F=F(F%F(F*F (F-F(F0F(FC**F9F(F;F(F,F(F0F(F(*.F.F(F:F(F=F(F;F(F,F(F0F(FC**F@F(F;F(F ,F(F0F(F(**F9F(F%F(F;F(F-F(FC**F9F(F;F(F-F(F0F(FC*,F:F(F=F(F%F(F;F(F-F (F(*,F:F(F=F(F;F(F-F(F0F(F(*(F9F(F%F(F;F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 25 "We now wa nt to solve eqR2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "eqA:=" } {MPLTEXT 1 0 43 "subs(F1=t,R=r,S=s,collect(eqR2,R,factor));\n" } {MPLTEXT 1 0 1 "#" }{MPLTEXT 1 0 35 "coeff(eqA,r,2);collect(%,s,factor );" }{MPLTEXT 1 0 10 "latex(%);\n" }{MPLTEXT 1 0 1 "#" }{MPLTEXT 1 0 14 "coeff(eqA,r,1)" }{MPLTEXT 1 0 4 "/x/y" }{MPLTEXT 1 0 21 ";collect( %,s,factor);" }{MPLTEXT 1 0 9 "latex(%);" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 1 "#" }{MPLTEXT 1 0 47 "coeff(eqA,r,0)/(x^2*y^2*z);collec t(%,s,factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*&,6**)I\"wG6\"\"\"# \"\"\")I\"xGF(F)F*)I\"yGF(F)F*)I\"zGF(F)F*F***)I\"sGF(F)F*F'F*F-F*F/F* F***I\"tGF(F*F+F*F-F*F/F*!\"\"*,F3F*F'F*F,F*F-F*F0F*F***F3F*F,F*F-F*F/ F*F**.F)F*F3F*F'F*F,F*F.F*F0F*F6*(F+F*F-F*F0F*F**(F'F*F+F*F.F*F6*(F+F* F.F*F0F*F6*&F'F*F+F*F*F*)I\"rGF(F)F*F***F,F*F.F*,<*.F)F*F3F*F'F*F+F*F. F*F/F*F**.F)F*F3F*F5F*F,F*F.F*F/F*F***F&F*F+F*F.F*F0F*F***F'F*)F,\"\"$ F*F.F*F0F*F***F'F*F+F*F.F*F/F*F**(FFF*F.F*F/F*F***F2F*F'F*F.F*F0F*F6*( F2F*F.F*F/F*F6**F)F*F&F*F+F*F0F*F6**F)F*F'F*FFF*F0F*F6*,F)F*F3F*F,F*F. F*F0F*F6*(F3F*F'F*F,F*F**(F3F*F,F*F0F*F*F*F?F*F***F+F*F-F*F0F*,6*()F'F GF*F+F*F0F*F6**F)F*F&F*FFF*F0F*F6*(F'F*)F,\"\"%F*F0F*F6*(F2F*F&F*F0F*F 6*(F2F*F5F*F0F*F**(F3F*F&F*F,F*F**(F3F*F'F*F+F*F***F3F*F'F*F,F*F0F*F** (F3F*F+F*F0F*F**$F2F*F6F*F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 217 52 "We follow the strategy of Bousquet-Melou and Jeha nne" }{TEXT 217 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "eq1: =eqR2;\n" }{MPLTEXT 1 0 27 "eq2:=factor(diff(eqR2,R));\n" }{MPLTEXT 1 0 27 "eq3:=factor(-diff(eqR2,w));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 83 " #(eq1,eq2,eq3) is the system of equation that w e need to solve.\n" }{MPLTEXT 1 0 46 "res1:=factor(resultant(eq1,eq2,R ))/(y^2*x^2);\n" }{MPLTEXT 1 0 21 "eq41:=op(nops(%),%);\n" }{MPLTEXT 1 0 23 "eq42:=op(1,op(1,res1));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 16 "eq4:=eq41*eq42;\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 35 "res2:=fa ctor(resultant(eq2,eq3,R));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 20 "eq 5:=op(nops(%),%);\n" }{MPLTEXT 1 0 64 " #(eq4,eq5) is the system after eliminating R.\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ",^ o**)I\"wG6\"\"\"$\"\"\")I\"xGF&\"\"%F()I\"yGF&\"\"#F()I\"zGF&F.F(F(*,F .F()F%F.F()F*\"\"&F(F,F(F/F(F(**F%F()F*\"\"'F(F,F(F/F(F(*,)I\"RGF&F.F( F2F()F*F.F(F,F(F/F(F(*0F.F(F:F(I\"SGF&F(F%F()F*F'F(F,F(F/F(F(*,)F=F.F( F2F(F;F(F,F(F/F(F(*,I#F1GF&F(F9F(F;F(F,F(F/F(!\"\"*0F.F(FBF(F:F(F=F(F; F(F,F(F/F(F(*,FBF(F@F(F;F(F,F(F/F(FC*,F9F(F@F(F%F(F,F(F/F(F(*,F:F(F2F( F>F(F,F(F0F(F(*,F:F(F%F(F)F(F,F(F0F(F(*,F:F(F%F(F>F(F,F(F/F(F(**F:F(F) F(F,F(F/F(F(*,F=F(F2F(F>F(F,F(F0F(FC*,F=F(F%F(F)F(F,F(F0F(FC*,F=F(F%F( F>F(F,F(F/F(FC**F=F(F)F(F,F(F/F(FC*.F9F(F=F(F%F(F*F(F,F(F0F(F(*,F9F(F= F(F*F(F,F(F/F(F(*.F:F(F@F(F%F(F*F(F,F(F0F(FC*,F:F(F@F(F*F(F,F(F/F(FC*. F.F(F:F(F2F(F>F(F-F(F0F(FC*.F.F(F:F(F%F(F)F(F-F(F0F(FC*0F.F(F9F(F=F(F% F(F*F(F-F(F0F(FC**F9F(F;F(F,F(F0F(F(*.F.F(F:F(F=F(F;F(F,F(F0F(FC**F@F( F;F(F,F(F0F(F(**F9F(F%F(F;F(F-F(FC**F9F(F;F(F-F(F0F(FC*,F:F(F=F(F%F(F; F(F-F(F(*,F:F(F=F(F;F(F-F(F0F(F(*(F9F(F%F(F;F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ",P*.\"\"#\"\"\"I\"RG6\"F%)I\"wGF'F$F%)I\"xGF'F$F%)I\"yGF' F$F%)I\"zGF'F$F%F%*.F$F%I\"SGF'F%F)F%)F+\"\"$F%F,F%F.F%F%*.F$F%I#F1GF' F%F&F%F*F%F,F%F.F%!\"\"*.F$F%F5F%F1F%F*F%F,F%F.F%F%*.F$F%F&F%)F1F$F%F) F%F,F%F.F%F%**F(F%F2F%F,F%F/F%F%**F)F%)F+\"\"%F%F,F%F/F%F%**F)F%F2F%F, F%F.F%F%*(FF%F-F%F0F%F1**F:F%F*F%F-F%F0F%F1* *F:F%F>F%F-F%F/F%F1*.F)F%F=F%F'F%F>F%F-F%F0F%F%**F=F%F*F%F-F%F0F%F%**F =F%F>F%F-F%F/F%F%*,F9F%F=F%F+F%F-F%F0F%F1*,F:F%F@F%F+F%F-F%F0F%F%*.F,F %F:F%F'F%F>F%F.F%F0F%F%*,F)F%F:F%F*F%F.F%F0F%F%*.F)F%F9F%F=F%F+F%F.F%F 0F%F%*(F9F%F;F%F.F%F%**F:F%F=F%F;F%F.F%F1*&F9F%F;F%F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "*(),***I\"wG6\"\"\"\")I\"xGF'\"\"#F(I\"yGF'F(I\"zGF'F (F(*()F*\"\"$F(F,F(F-F(F(*()I\"SGF'F+F(F,F(F-F(F(*&F3F(F*F(!\"\"F+F(,, *(\"\"%F()F&F0F()F-F+F(F5**F8F(I#F1GF'F(F&F(F:F(F(*$)F&F+F(F(*(F+F(F&F (F-F(F5*$F:F(F(F(,6**F>F(F)F()F,F+F(F:F(F5**FF(F?F(F(*.\"#CF (I#F1GF&F()F-F1F(FF(F?F(F6*.\"\")F(FEF()F-F'F(FBF(F>F(F?F(F6*.\"# ?F(F3F(FAF(F$F(F>F(F?F(F(*.FHF(F3F(FFF(F0F(F>F(F?F(F(*.F9F()FEF'F(F-F( FF(F?F(F(*,F=F(FNF(FBF(F>F(F?F(F(*0\"#;F(FEF(F3F(FIF(F$F(F>F(F?F( F6*,FHF()F4F=F(FFF(F>F(F?F(F(*.FQF(F4F(FAF(F0F(F>F()F.F=F(F(*.FHF(F4F( FFF(FF(FUF(F(*.FKF(F4F(FFF(F0F(F>F(F?F(F(*.F9F(F4F(FIF(FF(F? F(F(*.F=F(FNF(F3F(F$F(F>F(F?F(F(*0FHF(FEF(F4F(FIF(F0F(F>F(FUF(F6*0F9F( FEF(F4F(F-F(F0F(F>F(F?F(F6*.F=F(FEF(F4F(FF(F?F(F6*.FHF()F4F1F(FFF (F%F(F>F(FUF(F(*.F9F(FhnF(FIF(F%F(F>F(F?F(F(*.\"#OF(F4F(FAF(F0F(F)F(FU F(F6*.FQF(F4F(FFF(FF()F.F1F(F6*,\"#AF(FFF (FF(FUF(F(*,\"#5F(FIF(FBF(F>F(FUF(F(*,F'F(F-F(FBF(F>F(F?F(F(*.F=F (FEF(FhnF(F%F(F>F(F?F(F6*0FDF(FEF(F4F(FIF(F0F(F)F(FUF(F(*.F=F(FEF(FIF( FF(F^oF(F(*.F'F(FEF(F-F(FBF(F>F(F^oF(F(*.\"#=F(FEF(F-F(FF(FU F(F6*,\"\"'F(FEF(FBF(F>F(FUF(F6*,F'F(FEF(FF(F?F(F(*.FDF(FhnF(FFF( F%F(F)F(FUF(F6*.F1F(F3F(FFF(F$F(F>F(F^oF(F6*,F3F(FIF(F0F(F>F(F^oF(F6*. \"#9F(F3F(FIF(F$F(F>F(FUF(F(*,F3F(F-F(F$F(F>F(F?F(F(**F3F(F0F(F>F(F?F( F(*,FQF(FAF(FF(F^oF(F(*.F[pF(FEF(F3F(F$F(F>F(FUF(F6*.FHF(FEF(FIF(FF(F^oF(F6*(FSF(F>F(F?F(F(*.FQF(F3F(FFF(F$F(F)F(F^oF(F6*.FDF(F3F(FIF (F$F(F)F(FUF(F6*.F1F(F4F(FFF(F0F(F>F()F.F'F(F6*,F4F(FIF(FF(FcqF(F 6*.F'F(F4F(F-F(FF(F^oF(F6*.F;F(F4F(F-F(F0F(F>F(FUF(F(*,F1F(F4F(F< F(F>F(FUF(F(*,F'F(F4F(F0F(F>F(F?F(F6*,FQF(FAF(FF(FcqF(F6*.F'F(FhnF(F-F(F%F(F>F(F^oF(F6*,F1F(FhnF( F%F(F>F(FUF(F(*.FDF(F3F(FFF(F$F(F*F(F^oF(F(*.F[pF(F4F(FFF(F0F(F)F(FcqF (F(*.F'F(F4F(FIF(FF(FcqF(F6*,F'F(F-F(FBF(F>F(FcqF(F6*,F[ pF(F-F(FF(F^oF(F(**F'F(FBF(F>F(F^oF(F(**F'F(FF(FUF(F6*0F'F(F EF(F4F(F-F(F0F(F)F(FcqF(F6*.F[pF(FEF(F4F(F0F(F)F(F^oF(F(*.F1F(FhnF(FIF (F%F(F)F(FcqF(F(*,F1F(FhnF(F%F(F)F(FUF(F6*.F'F(F3F(F-F(F$F(F>F(FcqF(F6 *,F'F(F3F(F$F(F>F(F^oF(F(*.FHF(F4F(FFF(F0F(F*F(FcqF(F(*.F9F(F4F(FIF(F0 F(F*F(F^oF(F(*,F1F(FFF(F " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 222 63 "Check that eq2,eq3,eq4 and eq5 are valid for some series y=Y(z)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "serR;\n" }{MPLTEXT 1 0 1 "#" }{MPLTEXT 1 0 54 "subfirst:=A->collect(subs(R=z*RR,S=z*SS,A ),z,factor);\n" }{MPLTEXT 1 0 54 "serRn:=n->sum(coeff(serR,z,i)*z^i,i= 0..n)+z^(n+1)*RR;\n" }{MPLTEXT 1 0 57 "serF1n:=n->sum(coeff(serF1,z,i) *z^i,i=0..n)+z^(n+1)*FF1;\n" }{MPLTEXT 1 0 95 "subfirstn:=(n,A)->colle ct(subs(R=serRn(n),S=subs(w=0,RR=SS,serRn(n)),F1=serF1n(n),A),z,factor );" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%serRG6\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"nG6\"F%6$I)operatorGF%I&arrowGF%F %,&-I$sumG6$%*protectedGI(_syslibGF%6$*&-I&coeffGF-6%I%serRGF%I\"zGF%I \"iGF%\"\"\")F5F6F7/F6;\"\"!F$F7*&)F5,&F$F7F7F7F7I#RRGF%F7F7F%F%F%" }} {PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"nG6\"F%6$I)operatorGF%I&arrowGF%F %,&-I$sumG6$%*protectedGI(_syslibGF%6$*&-I&coeffGF-6%I&serF1GF%I\"zGF% I\"iGF%\"\"\")F5F6F7/F6;\"\"!F$F7*&)F5,&F$F7F7F7F7I$FF1GF%F7F7F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$I\"nG6\"I\"AGF%F%6$I)operatorGF%I& arrowGF%F%-I(collectG6$%*protectedGI(_syslibGF%6%-I%subsGF-6&/I\"RGF%- I&serRnGF%6#F$/I\"SGF%-F16%/I\"wGF%\"\"!/I#RRGF%I#SSGF%F5/I#F1GF%-I'se rF1nGF%F7F&I\"zGF%I'factorGF,F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 91 "We first che ck that there is indeed a solution y=Y(z) for eq2 (there seems to be 2 of them)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "collect(eq2,z, factor);\n" }{MPLTEXT 1 0 50 "collect(subfirstn(2,eq2),z,factor): seri es(%,z,3);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 63 "collect(subs(w=z*WW ,subfirstn(2,eq2)),z,factor): series(%,z,4);" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 69 "collect(subs(w=y*z+z^2*WW,subfirstn(3,eq2)),z,factor) : series(%,z,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*()I\"yG6\"\"\"#\" \"\",4**F'F(I\"RGF&F()I\"wGF&F'F()I\"xGF&F'F(!\"\"**F'F(I\"SGF&F(F-F() F/\"\"$F(F0**F'F(I#F1GF&F(F+F(F.F(F(**F'F(F6F(F2F(F.F(F0**F'F(F+F()F2F 'F(F-F(F0*&F-F(F3F(F0*$)F/\"\"%F(F0**F'F(F+F(F2F(F/F(F0*&F9F(F/F(F(F() I\"zGF&F'F(F0**F/F(F%F(,8*(F,F(F.F(F%F(F(*(F-F(F3F(F%F(F(*,F'F(F+F(F2F (F-F(F%F(F(*(F9F(F-F(F%F(F0*(F'F(F,F(F.F(F0*(F'F(F-F(F3F(F0**F=F(F+F(F 2F(F-F(F0**F'F(F+F(F/F(F%F(F(**F'F(F2F(F/F(F%F(F0*(F'F(F+F(F/F(F0*&F2F (F/F(F(F(FAF(F(*(F-F(F.F(,(*(F'F(F+F(F%F(F(*&F2F(F%F(F0*&F'F(F+F(F0F(F 0" }}{PARA 11 "" 1 "" {XPPMATH 20 "++I\"zG6\",$**I%serRGF$\"\"\"I\"wGF $F()I\"xGF$\"\"#F(,&I\"yGF$F(F,!\"\"F(F/\"\"!*(F+F(F.F(,0*()F)F,F(F*F( F.F(F(*(F)F()F+\"\"$F(F.F(F(*()F'F,F(F)F(F.F(F(*(F,F(F4F(F*F(F/*(F,F(F )F(F6F(F/*(\"\"%F(F9F(F)F(F/*&F'F(F+F(F/F(F(*()F.F,F(,&*(F,F(F'F(F)F(F (F+F(F(,(*&F)F(F*F(F(*$F6F(F(*$F9F(F(F(F,-I\"OG%*protectedG6#F(F7" }} {PARA 11 "" 1 "" {XPPMATH 20 "++I\"zG6\",$*(I%serRGF$\"\"\")I\"xGF$\" \"#F(,(*&I#WWGF$F(I\"yGF$F(F(*&F+F(F.F(!\"\"F/F(F(F1F(*(F*F(F/F(,.*(F. F()F*\"\"$F(F/F(F(*(F.F()F'F+F(F/F(F(*(F+F(F.F(F5F(F1*&F5F(F/F(F(*(\" \"%F(F.F(F8F(F1*&F8F(F/F(F(F(F+*(F/F(F.F(,,F4F(**F+F(F'F(F5F(F/F(F(F9F 1*(F+F()F'F6F(F/F(F(F:F(F(F6-I\"OG%*protectedG6#F(F<" }}{PARA 11 "" 1 "" {XPPMATH 20 "+-I\"zG6\",$**I%serRGF$\"\"\")I\"xGF$\"\"#F(I\"yGF$F(, &F,F(F(!\"\"F(F.F(,$*&F*F(,.*&)F*\"\"$F()F,F4F(F.*&)F'F+F(F5F(F.*&F3F( )F,F+F(F(**I#WWGF$F(F'F(F*F(F,F(F(*(F4F(F7F(F9F(F(**F+F(F;F(F'F(F*F(F. F(F.F+*&F,F(,2*(F;F()F*\"\"%F(F,F(F(**F+F(F'F(F3F(F9F(F(F2F(**F;F(F7F( F*F(F,F(F(*(F+F(F;F(FAF(F.*(F+F()F'F4F(F9F(F(F8F.**FBF(F;F(F7F(F*F(F.F (F4*&F9F(,,**F+F(F;F(F'F(F3F(F(**F+F(F;F(F3F(F,F(F(**F+F(F'F(F)F(F9F(F (*(F+F(F;F(FGF(F(*(F4F(F;F(F3F(F.F(FB-I\"OG%*protectedG6#F(\"\"&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 56 "So there is a unique solution of the form W(z)=yz+O(z^2) " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "collect(subs(w=y*z+z^3* WW,subfirstn(4,eq2)),z,factor): series(%,z,6);" }{MPLTEXT 1 0 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 "+/I\"zG6\",$**I%serRGF$\"\"\")I\"xGF$\" \"#F(I\"yGF$F(,&F,F(F(!\"\"F(F.F(*(F*F()F,F+F(,**&)F*\"\"$F(F,F(F(*&)F 'F+F(F,F(F(*$F3F(F.*&F4F(F6F(F.F(F+,.**F+F(F'F(F3F()F,F4F(F(*&F3F()F, \"\"%F(F(*(F+F()F'F4F(F;F(F(*&F3F(F;F(F.**I#WWGF$F(F'F(F)F(F,F(F.**F+F (FCF(F'F(F)F(F(F4*(F*F(F,F(,,*(FCF(F3F(F,F(F(**F+F(F'F(F*F(F;F(F(*(FCF (F6F(F,F(F(*(F+F(FCF(F3F(F.*(F>F(FCF(F6F(F.F(F>*(F0F(FCF(,**(F+F(F'F(F 3F(F(*(F+F(F3F(F,F(F(*&F+F(F@F(F(*&F4F(F3F(F.F(\"\"&-I\"OG%*protectedG 6#F(\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 49 "So this solution starts as W(z)=yz+y^3xz ^3+O(z^4)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 38 "This series automatically cancels eq3." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "collect(eq3,z,factor);\n" }{MPLTEXT 1 0 51 "collect(subfirstn(2,eq3),z,factor): series(%,z,3); \n" }{MPLTEXT 1 0 69 "collect(subs(w=y*z+z^2*WW,subfirstn(3,eq3)),z,fa ctor): series(%,z,6);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*()I\"yG6\"\" \"#\"\"\",4*(\"\"$F()I\"wGF&F'F()I\"xGF&\"\"%F(F(*(F0F(F-F()F/\"\"&F(F (*$)F/\"\"'F(F(**F'F()I\"RGF&F'F(F-F()F/F'F(F(**F'F(F9F(I\"SGF&F()F/F+ F(F(**F'F()F " 0 "" {MPLTEXT 1 0 23 "c ollect(eq4,z,factor);\n" }{MPLTEXT 1 0 51 "collect(subfirstn(2,eq3),z, factor): series(%,z,3);\n" }{MPLTEXT 1 0 69 "collect(subs(w=y*z+z^3*WW ,subfirstn(4,eq3)),z,factor): series(%,z,7);" }{MPLTEXT 1 0 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 ",***)I\"yG6\"\"\"$\"\"\",**&)I\"wGF&\"\" #F()I\"xGF&F-F(!\"\"*&I#F1GF&F(F.F(F(*&)I\"SGF&F-F(F,F(F0*&F5F(F/F(F0F (,(*&F,F(F.F(F(*$)F/F'F(F(*$F4F(F(F()I\"zGF&F'F(F(**)F%F-F(F/F(,<**F5F (F+F(F.F(F%F(F(**F5F(F,F(F:F(F%F(F(*()F5F'F(F,F(F%F(F(**F'F(F5F(F+F(F. F(F0**F-F(F5F(F,F(F:F(F0*(F,F(F:F(F%F(F(*&)F/\"\"%F(F%F(F(*(F2F(F5F(F. F(F(*(F'F(FDF(F,F(F0*(F4F(F/F(F%F(F(*&F,F(F:F(F0*$FIF(F0*(F-F(F4F(F/F( F0F()F=F-F(F0**F.F(F%F(,2*(F+F(F.F(F%F(F(FGF(**F-F(F4F(F,F(F%F(F(F*F0F NF0*(F'F(F4F(F,F(F0*(F5F(F/F(F%F(F(F6F0F(F=F(F(**F,F(F:F(,&F%F(F(F0F(F 5F(F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "++I\"zG6\",$*&)I%serRGF$\"\"#\" \"\")I\"xGF$F)F*!\"\"\"\"!,$*,F)F*F(F*F,F*I\"yGF$F*,(*(F)F*I\"wGF$F*F+ F*F**$)F,\"\"$F*F**$F'F*F*F*F*F*,$*()F1F)F*,(*&F4F*F+F*F*F5F*F8F*F*,(* (F7F*F4F*F+F*F*F5F*F8F*F*F-F)-I\"OG%*protectedG6#F*F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "+3I\"zG6\",$*&)I%serRGF$\"\"#\"\"\")I\"xGF$F)F*!\"\" \"\"!,$*,F)F*F(F*F,F*I\"yGF$F*,&*$)F,\"\"$F*F**$F'F*F*F*F*F*,$*&)F1F)F *,**$)F,\"\"'F*F**(F)F*F'F*F4F*F**$)F(\"\"%F*F**(FAF*F(F*F4F*F-F*F-F), $**FAF*)F1F5F*F2F*F+F*F-F5*(F4F*F1F*,&*(F5F*F,F*FEF*F-*(FAF*I#WWGF$F*F (F*F*F*FA*&F+F*,,**FAF*FJF*F4F*F9F*F-**FAF*FJF*F'F*F9F*F-*(I#RRGF$F*F( F*F1F*F**(I#SSGF$F*F(F*F1F*F-*(F)F*FPF*F(F*F-F*\"\"&,$*(F,F*F1F*,2**F= F*FJF*F4F*F9F*F**(FPF*F4F*F1F*F**(FRF*F4F*F1F*F-*(FPF*F'F*F1F*F**(F)F* FPF*F4F*F-*(FRF*F'F*F1F*F-*(FAF*FPF*F'F*F-*(F)F*FRF*F'F*F-F*F-F=-I\"OG %*protectedG6#F*\"\"(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 49 "Next we check we took the correct pieces of res1:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "res1; \n" }{MPLTEXT 1 0 1 "F" }{MPLTEXT 1 0 16 "alse:=op(2,res1)" }{MPLTEXT 1 0 1 ";" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 16 " collect(subs(w=y" }{MPLTEXT 1 0 9 "*z+z^2*WW" }{MPLTEXT 1 0 47 ",subfi rstn(3,False)),z,factor): series(%,z,4);\n" }{MPLTEXT 1 0 1 "\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "*(),***I\"wG6\"\"\"\")I\"xGF'\"\"#F(I\"y GF'F(I\"zGF'F(F(*()F*\"\"$F(F,F(F-F(F(*()I\"SGF'F+F(F,F(F-F(F(*&F3F(F* F(!\"\"F+F(,,*(\"\"%F()F&F0F()F-F+F(F5**F8F(I#F1GF'F(F&F(F:F(F(*$)F&F+ F(F(*(F+F(F&F(F-F(F5*$F:F(F(F(,6**F>F(F)F()F,F+F(F:F(F5**F " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT 217 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT 217 76 "Next we eliminate w (standing fo r a series in eq4 and eq5) from eq4 and eq5 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "res3:=factor(resultant(eq4,eq5,w))/(2*y^5*x^10*z^ 10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "**,B*,\"\"%\"\"\"I#F1G6\"F&)I\"x GF(F%F&)I\"yGF(F%F&)I\"zGF(F%F&F&*()I\"SGF(F%F&F+F&F-F&F&*,F%F&F1F&)F* \"\"$F&F+F&F-F&!\"\"*,\"\"#F&)F1F4F&F*F&F+F&)F.F4F&F&**F%F&F)F&F+F&F9F &F5*,F%F&F8F&F*F&)F,F4F&F9F&F5**)F1F7F&)F*F7F&F+F&)F.F7F&F&**F%F&F)F&F F&F?F&FF&F?F&)F,F7F&F@F&F&*,F7F&F 1F&F3F&FF&F?F&FF&F*F&FEF&F.F&F5*(F)F&F,F&F.F&F&*,F7F&F>F&F*F&F,F&F.F& F&*(F1F&F?F&F,F&F&*&F1F&F?F&F5F7F&),D*.F%F&F'F&F>F&F*F&FF&F*F&FF&F*F&FEF&F@F&F&*,F%F&F1F&F?F&FF&)F*FCF&FF&F)F&FF&FOF&FF&F)F&FF&FOF&F,F&F@F&F5*,F7F&F1 F&FgpF&FEF&F.F&F&*,F7F&F1F&FOF&FEF&F@F&F&*,FfpF&F0F&F?F&F,F&F@F&F5*,F7 F&F8F&F3F&FEF&F.F&F&**F%F&F8F&F3F&F.F&F&*(F>F&F)F&F,F&F5F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 82 "Since our final equation res3 factorize we need to find what i s the correct factor" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "nop s(res3);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 185 "r1 := -x^5*y^3*z^3+F 1*x^3*y^3*z^3-S^2*x^2*y^3*z^3+S*x^3*y^3*z^2-S*x^2*y^3*z^3+S^3*y^3*z^2- x^3*y^3*z^2-S^3*y^2*z^2-x^4*y^2*z+x^3*y^2*z^2-2*S^2*x*y^2*z+x^4*y*z+2* S^2*x*y*z+S*x^2*y-S*x^2;\n" }{MPLTEXT 1 0 224 "r2 := 4*F1*x^4*y^4*z^4+ S^4*y^4*z^4-4*S*x^3*y^4*z^4+2*S^3*x*y^4*z^3-4*x^4*y^4*z^3-4*S^3*x*y^3* z^3+S^2*x^2*y^4*z^2+4*x^4*y^3*z^3-6*S^2*x^2*y^3*z^2+6*S^2*x^2*y^2*z^2- 2*S*x^3*y^3*z+6*S*x^3*y^2*z-4*S*x^3*y*z+x^4*y^2-2*x^4*y+x^4;\n" } {MPLTEXT 1 0 376 "r3 := -4*x^9*y^3*z^4+4*F1*x^7*y^3*z^4-12*S^2*x^6*y^3 *z^4+4*F1*S^2*x^4*y^3*z^4-12*S^4*x^3*y^3*z^4-4*S^6*y^3*z^4+12*S*x^7*y^ 2*z^3-x^8*y^3*z^2-2*x^7*y^3*z^3-x^6*y^3*z^4-4*F1*S*x^5*y^2*z^3+24*S^3* x^4*y^2*z^3-2*S^2*x^5*y^3*z^2-2*S^2*x^4*y^3*z^3+12*S^5*x*y^2*z^3-S^4*x ^2*y^3*z^2-12*S^2*x^5*y*z^2+2*S*x^6*y^2*z+2*S*x^5*y^2*z^2-12*S^4*x^2*y *z^2+2*S^3*x^3*y^2*z+4*S^3*x^3*z-S^2*x^4*y;\n" }{MPLTEXT 1 0 236 "r4 : = 4*F1*S^2*x*y^3*z^3+4*F1*S*x^2*y^3*z^2-2*S^3*y^3*z^3-x^3*y^3*z^3-8*F1 *S*x^2*y^2*z^2+F1*x^3*y^3*z-5*S^2*x*y^3*z^2-4*F1*x^3*y^2*z+6*S^2*x*y^2 *z^2-4*S*x^2*y^3*z+4*F1*x^3*y*z+10*S*x^2*y^2*z-x^3*y^3-6*S*x^2*y*z+4*x ^3*y^2-5*x^3*y+2*x^3;\n" }{MPLTEXT 1 0 30 "factor(res3-r1^2*r2*r3*r4^2 );\n" }{MPLTEXT 1 0 26 "subser(r1):series(%,z,6);\n" }{MPLTEXT 1 0 26 "subser(r2):series(%,z,6);\n" }{MPLTEXT 1 0 26 "subser(r3):series(%,z, 6);\n" }{MPLTEXT 1 0 25 "subser(r4):series(%,z,6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",@*()I\"xG6\"\"\" &\"\"\")I\"yGF&\"\"$F()I\"zGF&F+F(!\"\"**I#F1GF&F()F%F+F(F)F(F,F(F(**) I\"SGF&\"\"#F()F%F5F(F)F(F,F(F.**F4F(F1F(F)F()F-F5F(F(**F4F(F6F(F)F(F, F(F.*()F4F+F(F)F(F8F(F(*(F1F(F)F(F8F(F.*(F;F()F*F5F(F8F(F.*()F%\"\"%F( F>F(F-F(F.*(F1F(F>F(F8F(F(*,F5F(F3F(F%F(F>F(F-F(F.*(F@F(F*F(F-F(F(*,F5 F(F3F(F%F(F*F(F-F(F(*(F4F(F6F(F*F(F(*&F4F(F6F(F." }}{PARA 11 "" 1 "" {XPPMATH 20 ",B*,\"\"%\"\"\"I#F1G6\"F%)I\"xGF'F$F%)I\"yGF'F$F%)I\"zGF' F$F%F%*()I\"SGF'F$F%F*F%F,F%F%*,F$F%F0F%)F)\"\"$F%F*F%F,F%!\"\"*,\"\"# F%)F0F3F%F)F%F*F%)F-F3F%F%**F$F%F(F%F*F%F8F%F4*,F$F%F7F%F)F%)F+F3F%F8F %F4**)F0F6F%)F)F6F%F*F%)F-F6F%F%**F$F%F(F%F;F%F8F%F%*,\"\"'F%F=F%F>F%F ;F%F?F%F4*,FBF%F=F%F>F%)F+F6F%F?F%F%*,F6F%F0F%F2F%F;F%F-F%F4*,FBF%F0F% F2F%FDF%F-F%F%*,F$F%F0F%F2F%F+F%F-F%F4*&F(F%FDF%F%*(F6F%F(F%F+F%F4*$F( F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",P**\"\"%\"\"\")I\"xG6\"\"\"*F%) I\"yGF(\"\"$F%)I\"z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}}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "+%I\"zG6\"-I\"OG%*protectedG6# \"\"\"\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "+/I\"zG6\"*&)I\"xGF$\"\"% \"\"\"),&I\"yGF$F)F)!\"\"\"\"#F)\"\"!,$*,F.F))F'\"\"&F))F,F.F)F+F),&F, F)F.F-F)F-F.,$**F.F)F&F))F,F(F)F+F)F-\"\"$,$*()F'\"\"'F)F8F),&*$F4F)F) F.F-F)F-F(,$**F.F)F2F))F,F3F),(*&F.F)F4F)F)*&F9F)F,F)F-F.F)F)F-F3-I\"O G%*protectedG6#F)F=" }}{PARA 11 "" 1 "" {XPPMATH 20 "+'I\"zG6\",$*()I \"xGF$\"\"'\"\"\")I\"yGF$\"\"$F*),&F,F*F*!\"\"\"\"#F*F0\"\"%-I\"OG%*pr otectedG6#F*F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "+/I\"zG6\",$*()I\"xGF$ \"\"$\"\"\",&I\"yGF$F*\"\"#!\"\"F*),&F,F*F*F.F-F*F.\"\"!,$*,F-F*)F(\" \"%F*)F,F-F*F0F*,&*&F-F*F,F*F*F)F.F*F.F-,$*,F-F*F'F*F,F*F0F*,(*&F-F*)F ,F)F*F**&F)F*F6F*F.F-F*F*F.F),$*()F(\"\"&F*)F,FBF*,&*&F5F*F,F*F*FBF.F* F.F5,$**F-F*F4F*F=F*,,*&\"\"'F*)F,F5F*F**&\"#5F*F=F*F.F>F*F8F.F5F*F*F. FB-I\"OG%*protectedG6#F*FJ" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 217 41 "So r1 is the true equation satisfied by S" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "r1;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",@*()I\"xG6\"\"\"&\"\"\")I\"yGF&\"\"$F()I\"zGF&F+F(!\"\"* *I#F1GF&F()F%F+F(F)F(F,F(F(**)I\"SGF&\"\"#F()F%F5F(F)F(F,F(F.**F4F(F1F (F)F()F-F5F(F(**F4F(F6F(F)F(F,F(F.*()F4F+F(F)F(F8F(F(*(F1F(F)F(F8F(F.* (F;F()F*F5F(F8F(F.*()F%\"\"%F(F>F(F-F(F.*(F1F(F>F(F8F(F(*,F5F(F3F(F%F( F>F(F-F(F.*(F@F(F*F(F-F(F(*,F5F(F3F(F%F(F*F(F-F(F(*(F4F(F6F(F*F(F(*&F4 F(F6F(F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "eqS:=factor(res ultant(r1,eqF1,F1))/z^4;" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ",bam**\"#k\"\"\")I\"xG6\"\"#:F%)I\"yGF(\"\"*F%)I\"zGF(\"# 5F%F%*,\"$#>F%)I\"SGF(\"\"#F%)F'\"#7F%F*F%F-F%F%*,F1F%)F3\"\"%F%)F'F,F %F*F%F-F%F%*,F1F%F3F%)F'\"#8F%F*F%)F.F,F%!\"\"*,F1F%F3F%F5F%F*F%F-F%F% 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}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "#just checking\n" }{MPLTEXT 1 0 35 "factor(subs(y=1,S=F,eqS))/x^6/ z^4;\n" }{MPLTEXT 1 0 9 "%+eqFalg;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",d p*(\"#k\"\"\")I\"xG6\"\"\"*F%)I\"zGF(\"\"'F%F%**\"$#>F%)I\"FGF(\"\"#F% )F'F,F%F*F%F%**F.F%)F0\"\"%F%)F'\"\"$F%F*F%F%**F.F%F0F%)F'\"\"(F%)F+\" \"&F%!\"\"**F.F%F0F%F2F%F*F%F%*(F$F%)F0F,F%F*F%F%**\"$%QF%)F0F7F%)F'F5 F%F;F%F=**FBF%FCF%F6F%F*F%F%*(\"#'*F%F9F%F;F%F=**F.F%)F0F " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "algeqtoseries(eqS,z,S,5,true);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 49 "solve((y-1)*_Z^2+(-x*y^3+2*x*y^2)*_Z-x^2*y^4,_Z);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 50 "factor(subs(S=x^2*y*z+x*y^2*z^2+z^3*SS,eq S))/z^5:\n" }{MPLTEXT 1 0 29 "algeqtoseries(%,z,SS,8,true);" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "7#++I\"zG6\"*&)I\"xG6\"\" \"#\"\"\"I\"yG6\"\"\"\"\"\"\"-I'RootOfG6$%*protectedGI(_syslibG6\"6#,( *&,&I\"yG6\"\"\"\"\"\"\"!\"\"\"\"\")I#_ZG6$%*protectedGI(_syslibG6\"\" \"#\"\"\"\"\"\"*&,&*&I\"xG6\"\"\"\")I\"yG6\"\"\"$\"\"\"!\"\"*(\"\"#\" \"\"I\"xG6\"\"\"\")I\"yG6\"\"\"#\"\"\"\"\"\"\"\"\"I#_ZG6$%*protectedGI (_syslibG6\"\"\"\"\"\"\"*&)I\"xG6\"\"\"#\"\"\")I\"yG6\"\"\"%\"\"\"!\" \"\"\"#*&)I\"xG6\"\"\"$\"\"\")I\"yG6\"\"\"$\"\"\"\"\"$-I\"OG%*protecte dG6#\"\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$*&I\"xG6\"\"\"\")I \"yGF%\"\"#F&,$*(F$F&F'F&,&F(F&F&!\"\"F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "7#+5I\"zG6\"*&)I\"xG6\"\"\"$\"\"\")I\"yG6\"\"\"$\"\"\"\" \"!,$*(\"\"$\"\"\")I\"xG6\"\"\"#\"\"\")I\"yG6\"\"\"%\"\"\"\"\"\"\"\"\" *(,**(\"\"#\"\"\")I\"xG6\"\"\"$\"\"\")I\"yG6\"\"\"#\"\"\"\"\"\"*&\"\"# \"\"\")I\"yG6\"\"\"#\"\"\"\"\"\"I\"yG6\"\"\"\"\"\"\"\"\"\"\"\"\"I\"xG6 \"\"\"\")I\"yG6\"\"\"$\"\"\"\"\"#,$*(\"#5\"\"\")I\"xG6\"\"\"$\"\"\")I \"yG6\"\"\"'\"\"\"\"\"\"\"\"$*(,**(\"\"&\"\"\")I\"xG6\"\"\"$\"\"\")I\" yG6\"\"\"#\"\"\"\"\"\"*&\"#:\"\"\")I\"yG6\"\"\"#\"\"\"\"\"\"*&\"\"&\" \"\"I\"yG6\"\"\"\"\"\"\"\"\"%\"\"\"\"\"\")I\"xG6\"\"\"#\"\"\")I\"yG6\" \"\"&\"\"\"\"\"%*(,0*(\"#N\"\"\")I\"xG6\"\"\"$\"\"\")I\"yG6\"\"\"&\"\" \"\"\"\"*&\"\"(\"\"\")I\"yG6\"\"\"&\"\"\"\"\"\"*&\"\"'\"\"\")I\"yG6\" \"\"%\"\"\"\"\"\"*&\"\"(\"\"\")I\"yG6\"\"\"$\"\"\"\"\"\"*&\"\"%\"\"\") I\"yG6\"\"\"#\"\"\"\"\"\"*&\"\"%\"\"\"I\"yG6\"\"\"\"\"\"\"\"\"%\"\"\" \"\"\"I\"xG6\"\"\"\")I\"yG6\"\"\"$\"\"\"\"\"&*(,**(\"#9\"\"\")I\"xG6\" \"\"$\"\"\")I\"yG6\"\"\"#\"\"\"\"\"\"*&\"#%)\"\"\")I\"yG6\"\"\"#\"\"\" \"\"\"*&\"#@\"\"\"I\"yG6\"\"\"\"\"\"\"\"#:\"\"\"\"\"\")I\"xG6\"\"\"$\" \"\")I\"yG6\"\"\"(\"\"\"\"\"',$**\"\"#\"\"\",0*(\"#j\"\"\")I\"xG6\"\" \"$\"\"\")I\"yG6\"\"\"&\"\"\"\"\"\"*&\"#U\"\"\")I\"yG6\"\"\"&\"\"\"\" \"\"*&\"#G\"\"\")I\"yG6\"\"\"%\"\"\"\"\"\"*&\"#G\"\"\")I\"yG6\"\"\"$\" \"\"\"\"\"*&\"#7\"\"\")I\"yG6\"\"\"#\"\"\"\"\"\"*&\"#5\"\"\"I\"yG6\"\" \"\"\"\"\"\"\")\"\"\"\"\"\")I\"xG6\"\"\"#\"\"\")I\"yG6\"\"\"&\"\"\"\" \"\"\"\"(-I\"OG%*protectedG6#\"\"\"\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$,$*(I\"xG6\"\"\"\")I\"yGF&\"\"#F',&F)F 'F'!\"\"F,F,*&F%F'F(F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "# eqSS:=factor(subs(x=x/(y*z),factor(subs(S=x^2*z*y+SS,eqS))))*y^6*z^7;" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "#" }{MPLTEXT 1 0 18 "z0:=(432)^ (-1/6);\n" }{MPLTEXT 1 0 1 "#" }{MPLTEXT 1 0 37 "eqxHH:=factor(subs(SS =HH,z=z0,eqS));\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "### factor(subs(z=t^(1/3),su bs(x=x*z,subs(S" }{MPLTEXT 1 0 16 "=St,eqS/z^7))));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 10 "W e denote " }{TEXT 217 78 "G(x,y)=S(x,y,z0). In the paper, this series \+ is denoted S(x,y) (with a bold S)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 221 68 "Derivation \+ of the algebraic equation for the series G(x,y)=S(x,y,z0)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 61 "We now set z =z0 (radius of convergence of S). EqS factorizes." }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 4 "z0:=" }{MPLTEXT 1 0 13 "(432)^(-1/6);" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 33 "eqx:=factor(subs(S=G,z=z0,eqS)); \n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 312 "eqx1 := -2*y^3*x^5*sqrt(3) *2^(1/3)-2*G^2*sqrt(3)*2^(1/3)*x^2*y^3-12*2^(2/3)*x^4*y^2*sqrt(3)-24*x *G^2*sqrt(3)*2^(2/3)*y^2-2*y^3*x^2*G*sqrt(3)*2^(1/3)+12*x^4*sqrt(3)*2^ (2/3)*y+24*x*G^2*sqrt(3)*2^(2/3)*y+12*y^3*x^3*G-3*y^3*x^3*sqrt(3)+12*G ^3*y^3-6*x^3*y^3-12*G^3*y^2+72*x^2*G*2^(1/3)*y+12*x^3*y^2-72*x^2*G*2^( 1/3);\n" }{MPLTEXT 1 0 315 "eqx2 := -8*y^3*x^5*sqrt(3)*2^(1/3)-8*G^2*s qrt(3)*2^(1/3)*x^2*y^3-48*2^(2/3)*x^4*y^2*sqrt(3)-96*x*G^2*sqrt(3)*2^( 2/3)*y^2-8*y^3*x^2*G*sqrt(3)*2^(1/3)+48*x^4*sqrt(3)*2^(2/3)*y+96*x*G^2 *sqrt(3)*2^(2/3)*y+48*y^3*x^3*G+15*y^3*x^3*sqrt(3)+48*G^3*y^3-24*x^3*y ^3-48*G^3*y^2+288*x^2*G*2^(1/3)*y+48*x^3*y^2-288*x^2*G*2^(1/3);" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 7 "factor(" }{MPLTEXT 1 0 4 "eqx+" } {MPLTEXT 1 0 25 "1/279936*sqrt(3)*2^(1/3)*" }{MPLTEXT 1 0 14 "eqx1^2*e qx2);\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*&#\"\"\"\"$K%F%)F&#\"\"&\"\"'F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*,#\"\"\"\"'O*z#F%)\"\"##F%\"\"$F%)F*#F%F(F%,@*,\"\")F% )I\"yG6\"F*F%)I\"xGF2\"\"&F%F'F%F+F%!\"\"*.F/F%)I\"GGF2F(F%F+F%F'F%)F4 F(F%F0F%F6*,\"#[F%)F(#F(F*F%)F4\"\"%F%)F1F(F%F+F%F6*.\"#'*F%F4F%F8F%F+ F%F=F%FAF%F6*.F/F%F0F%F:F%F9F%F+F%F'F%F6*,FF%F+F%F3F%F/F%F8F%F'F%F%**F7F% F&F%)F+F)F%F4F%F%**F)F%F/F%FCF%F&F%F1*(F7F%)F4F)F%F&F%F%*(\"\"'F%FCF%F &F%F1*(F7F%FFF%F " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 34 "We determine which factor is true." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "eqx2/4-eqx1;" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 42 "algeqtoseries(subs(y=1,eqx1),x,G,3,true);\n" } {MPLTEXT 1 0 10 "evalf(%);\n" }{MPLTEXT 1 0 42 "algeqtoseries(subs(y=1 ,eqx2),x,G,3,true);\n" }{MPLTEXT 1 0 9 "evalf(%);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$**#\"#F\"\"%\"\"\")\"\"$#F'\"\"#F')I \"xG6\"F)F')I\"yGF.F)F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$++I\"xG6 \"!\"\"\"\"!,$**#\"\"\"\"\"%F+,&\"\"#F+*$)\"\"$#F+F.F+F+F+)F.#F.F1F+F0 F+F+F+,$**#F1\"\")F+F-F+)F.#F+F1F+,&F/F+F.F&F+F+F.-I\"OG%*protectedG6# F+F1++F$,&*&#F1F,F+F3F+F&*(F2F+F3F+F0F+F+F+,$*&F7F+F9F+F+F.#F+F8F1F\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$++ I\"xG6\"$!\"\"\"\"!F($!+(\\)eM6!#5\"\"\"$\"+pRB[KF+\"\"#-I\"OG%*protec tedG6#F,\"\"$++F$$\"+C\"=H'G!\"*F,$!+dRB[KF+F/$!++DJ?L!#6F4F0\"\"%" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 71 "eqx1 is the correct equation since eqx2 leads to negativ e coefficients." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "collect( eqx1,G,factor);\n" }{MPLTEXT 1 0 1 "#" }{MPLTEXT 1 0 9 "latex(%);" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 27 "#collect(eqx2/4,G,factor);\n" } {MPLTEXT 1 0 1 "#" }{MPLTEXT 1 0 9 "latex(%);" }{MPLTEXT 1 0 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 ",***\"#7\"\"\")I\"yG6\"\"\"#F%,&F'F%F%! \"\"F%)I\"GGF(\"\"$F%F%*0F)F%)F)#F%F.F%)F.#F%F)F%,(*&F&F%I\"xGF(F%F%*( F$F%F0F%F'F%F%*&F$F%F0F%F+F%F6F%F'F%)F-F)F%F+**F)F%,**()F'F.F%F0F%F2F% F%*(\"\"'F%F6F%F=F%F+*(\"#OF%F0F%F'F%F+*&FAF%F0F%F%F%)F6F)F%F-F%F+*.F3 F%F0F%F2F%,.**F)F%F&F%)F)#F)F.F%F2F%F%**\"\"%F%F'F%FGF%F2F%F+*(F.F%FGF %F&F%F%*(FJF%FCF%F&F%F%**\"#CF%F0F%F6F%F'F%F%*(FNF%F0F%F6F%F+F%F'F%)F6 F.F%F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 268 "eqG:=12*y^2*(y-1 )*G^3-2*2^(1/3)*sqrt(3)*(x*y^2+12*2^(1/3)*y-12*2^(1/3))*x*y*G^2-(2*(y^ 3*2^(1/3)*sqrt(3)-6*x*y^3-36*2^(1/3)*y+36*2^(1/3)))*x^2*G-(1/2)*2^(1/3 )*sqrt(3)*(2*sqrt(3)*2^(2/3)*y^2-4*sqrt(3)*2^(2/3)*y+3*2^(2/3)*y^2+4*x ^2*y^2+24*2^(1/3)*x*y-24*2^(1/3)*x)*y*x^3;" }{MPLTEXT 1 0 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 ",***\"#7\"\"\")I\"yG6\"\"\"#F%,&F'F%F%! \"\"F%)I\"GGF(\"\"$F%F%*0F)F%)F)#F%F.F%)F.#F%F)F%,(*&F&F%I\"xGF(F%F%*( F$F%F0F%F'F%F%*&F$F%F0F%F+F%F6F%F'F%)F-F)F%F+**F)F%,**()F'F.F%F0F%F2F% F%*(\"\"'F%F6F%F=F%F+*(\"#OF%F0F%F'F%F+*&FAF%F0F%F%F%)F6F)F%F-F%F+*.F3 F%F0F%F2F%,.**F)F%F&F%)F)#F)F.F%F2F%F%**\"\"%F%F'F%FGF%F2F%F+*(F.F%FGF %F&F%F%*(FJF%FCF%F&F%F%**\"#CF%F0F%F6F%F'F%F%*(FNF%F0F%F6F%F+F%F'F%)F6 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 0 268 "eqG:=12*y^2*(y-1)*G^3-2*2^(1/3)*sqrt(3)*(x*y^2+12*2^ (1/3)*y-12*2^(1/3))*x*y*G^2-(2*(y^3*2^(1/3)*sqrt(3)-6*x*y^3-36*2^(1/3) *y+36*2^(1/3)))*x^2*G-(1/2)*2^(1/3)*sqrt(3)*(2*sqrt(3)*2^(2/3)*y^2-4*s qrt(3)*2^(2/3)*y+3*2^(2/3)*y^2+4*x^2*y^2+24*2^(1/3)*x*y-24*2^(1/3)*x)* y*x^3;" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 105 "-------------------------- ---------------------------------------------------------------------- --------\n" }{TEXT 217 103 "From here we analyze the behaviour of [x^n ]G=[x^n]G(x,y,z0) using \"analytic combinatrorics\" techniques." }} {PARA 0 "" 0 "" {TEXT 217 104 "--------------------------------------- -----------------------------------------------------------------" }} {PARA 0 "" 0 "" {TEXT 217 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 221 14 "Asymptotic of " }{TEXT 221 22 "[x^n]G=[x^n]G(x,y,z0) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 145 "We first want to determine the radius o f convergence x0(y) of H(x,y). It is a root of the discriminant \"disc \" of eqG, hence a root of Dx=Dx1*Dx2." }{TEXT 217 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 92 "We now start the singularity analysis. F irst we determine the radius of convergence of G(x)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "disc:=factor(resultant(eqG,diff(eqG,G),G))/(-432*2^(1/3)*(y-1)*x ^6*y^8);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 16 "Dx1:=op(1,disc);" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 22 "Dx2:=op(1,op(2,disc));" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 12 "Dx:=Dx1*Dx2;" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,<**\"\"#\"\"\")I\"yG6\"\"\"$F&)F%# F%F*F&)F*#F&F%F&F&*,\"#7F&F-F&)F%#F&F*F&I\"xGF)F&F'F&F&**F%F&)F(F%F&F+ F&F-F&!\"\"*(\"\"*F&F'F&F+F&F6*,F0F&F-F&F1F&F3F&F5F&F6**\"#=F&F1F&F3F& F'F&F6*(\"\"%F&)F3F%F&F'F&F6**F;F&F(F&F+F&F-F&F6**F0F&F1F&F3F&F5F&F&*( F;F&F+F&F-F&F&*(\"#FF&F+F&F(F&F&**\"\"'F&F1F&F3F&F(F&F&*&F;F&F+F&F6F&) ,,*(F%F&F3F&F(F&F&**F%F&F-F&F1F&F(F&F&*(F%F&F1F&F-F&F6*(\"\"&F&F1F&F(F &F&*&FEF&F1F&F6F*F&" }}{PARA 11 "" 1 "" {XPPMATH 20 ",<**\"\"#\"\"\")I \"yG6\"\"\"$F%)F$#F$F)F%)F)#F%F$F%F%*,\"#7F%F,F%)F$#F%F)F%I\"xGF(F%F&F %F%**F$F%)F'F$F%F*F%F,F%!\"\"*(\"\"*F%F&F%F*F%F5*,F/F%F,F%F0F%F2F%F4F% F5**\"#=F%F0F%F2F%F&F%F5*(\"\"%F%)F2F$F%F&F%F5**F:F%F'F%F*F%F,F%F5**F/ F%F0F%F2F%F4F%F%*(F:F%F*F%F,F%F%*(\"#FF%F*F%F'F%F%**\"\"'F%F0F%F2F%F'F %F%*&F:F%F*F%F5" }}{PARA 11 "" 1 "" {XPPMATH 20 ",,*(\"\"#\"\"\"I\"xG6 \"F%I\"yGF'F%F%**F$F%)\"\"$#F%F$F%)F$#F%F+F%F(F%F%*(F$F%F-F%F*F%!\"\"* (\"\"&F%F-F%F(F%F%*&\"\"'F%F-F%F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&, <**\"\"#\"\"\")I\"yG6\"\"\"$F&)F%#F%F*F&)F*#F&F%F&F&*,\"#7F&F-F&)F%#F& F*F&I\"xGF)F&F'F&F&**F%F&)F(F%F&F+F&F-F&!\"\"*(\"\"*F&F'F&F+F&F6*,F0F& F-F&F1F&F3F&F5F&F6**\"#=F&F1F&F3F&F'F&F6*(\"\"%F&)F3F%F&F'F&F6**F;F&F( F&F+F&F-F&F6**F0F&F1F&F3F&F5F&F&*(F;F&F+F&F-F&F&*(\"#FF&F+F&F(F&F&**\" \"'F&F1F&F3F&F(F&F&*&F;F&F+F&F6F&,,*(F%F&F3F&F(F&F&**F%F&F-F&F1F&F(F&F &*(F%F&F1F&F-F&F6*(\"\"&F&F1F&F(F&F&*&FEF&F1F&F6F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 222 44 "non-necessary: slightly simplified equations" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "factor(subs(\{x=z0*x,G=z0^3*x*G\},36*eqx1/x^3)):\n" }{MPLTEXT 1 0 29 "eqx1bis:=collect(%,G,factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**()I\"yG6\"\"\"#\"\"\",&F%F(F(!\"\"F()I\"GGF&\"\"$F(F(* (,(*(\"#CF()F-#F(F'F(F%F(F(*&F1F(F2F(F**&I\"xGF&F(F$F(F(F(F%F()F,F'F(F ***\"#7F(F2F(,**$)F%F-F(F**(F9F(F2F(F%F(F(*&F9F(F2F(F**&F6F(F " 0 "" {MPLTEXT 1 0 50 "discbis:=factor(subs(x=z0*x,Dx)/(-(2/9)*sqrt(3))); " }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,<**\"\"*\"\"\")\"\"$#F&\"\"#F&I\" xG6\"F&)I\"yGF,F(F&F&*&)F+F*F&F-F&F&**\"\"'F&F'F&F+F&)F.F*F&!\"\"*(F2F &F'F&F-F&F4*(\"#=F&F+F&F-F&F4**F(F&F+F&F.F&F'F&F4*(F2F&F3F&F'F&F&*(F7F &F+F&F3F&F&*&\"#FF&F-F&F&*(\"#aF&F'F&F.F&F&*&F>F&F'F&F4*&\"#\")F&F.F&F 4F>F&F&,,*(\"\"&F&F'F&F.F&F&*&F+F&F.F&F&*&F2F&F'F&F4*&F2F&F.F&F&F2F4F& " }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 93 "Just \+ a guess for the critical value of y: a root common to Dx1 and Dx2: (th is gives y_critic=" }{TEXT 217 28 "2/11-2/11*3^(1/2)=0.77599..." } {TEXT 217 43 " and below we prove this is indeed correct)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "factor(resultant(Dx1,Dx2,x));\n" } {MPLTEXT 1 0 13 "\{solve(%,y)\};" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 10 "evalf(%);\n" }{MPLTEXT 1 0 9 "ycritic:=" }{MPLTEXT 1 0 21 "12/11-( 2/11)*sqrt(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*,#\"\"%\"%J8\"\"\") \"\"##F)\"\"$F',&*&\"#IF')F+#F'F)F'F'\"#PF'F')I\"yG6\"F)F'),(*&\"#6F'F 3F'F'\"#7!\"\"*&F)F'F/F'F'F+F'F:" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$\" \"!,&#\"#7\"#6\"\"\"*&#\"\"#F'F()\"\"$#F(F+F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$$\"\"!F$$\"+Bw!*fx!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&#\"#7\"#6\"\"\"*&#\"\"#F%F&)\"\"$#F&F)F&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 63 "Cand idates for the radius of convergence of G (the roots of Dx)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "sigma:=solve(Dx2,x);\n" } {MPLTEXT 1 0 52 "#simplify(%-2^(-2/3)*((2*sqrt(3)+6)/y-2*sqrt(3)-5));" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$**#\"\"\"\"\"#F%)F&#F%\"\"$F%,**(F& F%I\"yG6\"F%)F)F$F%F%*&F&F%F.F%!\"\"*&\"\"&F%F,F%F%\"\"'F0F%F,F0F0" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "solDx1:=\{solve(Dx1,x)\};\n " }{MPLTEXT 1 0 30 "tau1:=mafactor(op(1,solDx1));\n" }{MPLTEXT 1 0 29 "tau2:=mafactor(op(2,solDx1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$,$*( #\"\"\"\"\"%F&,.**\"\"'F&)I\"yG6\"\"\"#F&)F.#F&\"\"$F&)F1#F&F.F&!\"\"* *F*F&F2F&F/F&F,F&F&*(\"\"*F&F/F&F+F&F&*(F*F&F/F&F,F&F4*$),4*(\"$`\"F&) F,F'F&)F.#F.F1F&F&*(\"$C$F&)F,F1F&F?F&F4**\"$+\"F&F>F&F?F&F2F&F4*(\"$) >F&F?F&F+F&F&**\"$s\"F&FCF&F?F&F2F&F&**\"$3\"F&F+F&F?F&F2F&F4**\"#OF&F ,F&F?F&F2F&F&*(FMF&F?F&F,F&F4*&F7F&F?F&F&F3F&F&*&F1F&F/F&F4F&F+F4F4,$* (F%F&,.F)F&F5F4F6F4F8F&F9F&FPF&F&F+F4F&" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*(#\"\"\"\"\"%F%,.**\"\"'F%)I\"yG6\"\"\"#F%)F-#F%\"\"$F%)F0#F%F- F%!\"\"**F)F%F1F%F.F%F+F%F%*(\"\"*F%F.F%F*F%F%*(F)F%F.F%F+F%F3*$),$*,# F%\"%(>#F%)F-#F-F0F%,&*&\"$+\"F%F1F%F%\"$`\"F3F%,&F+F%F%F3F%),(*&\"#8F %F+F%F%F0F%*&F&F%F1F%F%F0F%F3F2F%F%*&F0F%F.F%F3F%F*F3F3" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*(#\"\"\"\"\"%F%,.**\"\"'F%)I\"yG6\"\"\"#F%)F-#F% \"\"$F%)F0#F%F-F%F%**F)F%F1F%F.F%F+F%!\"\"*(\"\"*F%F.F%F*F%F4*(F)F%F.F %F+F%F%*$),$*,#F%\"%(>#F%)F-#F-F0F%,&*&\"$+\"F%F1F%F%\"$`\"F4F%,&F+F%F %F4F%),(*&\"#8F%F+F%F%F0F%*&F&F%F1F%F%F0F%F4F2F%F%*&F0F%F.F%F%F%F*F4F% " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 75 "tau1 and tau 2 do not collid e inside (0,1). They collide at 0 and 1 though." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "factor(discrim(Dx1,x));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 22 "\{solve(%,y)\};evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*.#\"\"%\"%(>#\"\"\")\"\"##F)\"\"$F',&*&\"$+\"F')F+#F'F )F'F'\"$`\"!\"\"F')I\"yG6\"F)F',&F4F'F'F2F'),(*&\"#8F'F4F'F'F+F'*&F%F' F/F'F'F+F'F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%\"\"!\"\"\",&#\"\"$\"# 8!\"\"*&#\"\"%F(F$)F'#F$\"\"#F$F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%$ !+5%zqj(!#5$\"\"!F'$\"\"\"F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 222 39 " Plots of sigma, tau_1, tau_2 (not used)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plot(si gma,y=0..1);\n" }{MPLTEXT 1 0 19 "plot(tau1,y=0..1);\n" }{MPLTEXT 1 0 18 "plot(tau2,y=0..1);" }{MPLTEXT 1 0 34 "plot(\{tau2,sigma\},y=0..1,x =0..10);" }{MPLTEXT 1 0 1 "\n" }}{PARA 13 "" 1 "" {TEXT 223 0 "" } {GLPLOT2D 400.0 400.0 400.0 {PLOTDATA 2 "6*-%'CURVESG6$7iy7$$\"/DJq:+V ;!#<$\"1bkc1w>GO!#67$$\".D19.gG$!#;$\"2LR)y+A$Q\"=!#77$$\"/v$4r/!H\\!# <$\"2kq&>KP/47!#77$$\"-D\"G1?d'!#:$\"1FO**y\\\\m!*!#77$$\"1*\\i:&y+:#) !#>$\"1nb@t&H@D(!#77$$\".v=U4!e)*!#;$\"0>Igj_D/'!#67$$\"0v=#*4,,:\"!#< $\"14#p%4\"p&y^!#77$$\"-Dc7S98!#9$\"2DZ[XY\"eIX!#87$$\"2(\\7G89qy9!#>$ \"1oX0u5fES!#77$$\"2)*\\7.d,Ik\"!#>$\"1V%f;w)RBO!#77$$\"2)\\PMF $\"2#oqUUT^$H$!#87$$\".vV)=gr>!#:$\"1bn1$H5'=I!#77$$\"0D19/-f8#!#<$\"0 .3OM**fy#!#67$$\"/vV)>--I#!#;$\"1kiyH&=me#!#77$$\"0voaN-XY#!#<$\"2$oSZ C=#QT#!#87$$\",D^-)GE!#8$\"2l*eK2ZiiA!#87$$\"2&\\7`pE5$z#!#>$\"1/GJ'[: #H@!#77$$\"2&*\\il#GSdH!#>$\"2P%*y?^H1,#!#87$$\"2&\\Pf$)Hq@J!#>$\"2N' \\(>FEX!>!#87$$\"2&**\\iSJ+'G$!#>$\"22Q\")eNL!4=!#87$$\"2&\\il(H..X$!# >$\"2GGDK+NEs\"!#87$$\"2&*\\(oaMg9O!#>$\"2P>li/\"4W;!#87$$\"2'\\(=$\"2<)fg)*pPs:!#87$$\"-voP?VR!#9$\"2u.&e@\"Rm]\"!#87$$\"0D\"yDR]2 T!#<$\"2(odsu-;Y9!#87$$\"/D\"G3/=F%!#;$\"2XnbokL.R\"!#87$$\"0vV)RU5OW! 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The radius of convergence is sigma" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "yspe:=1;\n" }{MPLTEXT 1 0 45 "subspe:=A->factor(simplify(subs(y=yspe,A))):\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "eqGs pe:=subspe(eqG);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*,#\"\"\"\"\"#F% )\"\"$F$F%)F&#F%F(F%,.*,\"\"%F%I\"GG6\"F%F'F%)F&#F&F(F%I\"xGF/F%!\"\"* *F&F%F'F%F0F%F2F%F3*(F(F%F0F%F2F%F%*&F-F%)F2F(F%F%*&F-F%)F.F&F%F%*&F-F %F.F%F%F%)F2F&F%F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "#" } {MPLTEXT 1 0 35 "discy1:=factor(discrim(eqGspe,G));\n" }{MPLTEXT 1 0 8 "discy1:=" }{MPLTEXT 1 0 13 "subspe(disc);" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 19 "\{solve(discy1,x)\};\n" }{MPLTEXT 1 0 9 "evalf(%);" } }{PARA 11 "" 1 "" {XPPMATH 20 ",$*(\"\"%\"\"\")I\"xG6\"\"\"#F%),&*&F)F %F'F%!\"\"*$)F)#F%\"\"$F%F%F1F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$ \"\"!,$*&#\"\"\"\"\"#F')F(#F'\"\"$F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$$\"\"!F$$\"+]_g*H'!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 41 "So the singularity is 2^(-2 /3)=0.62996..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "\{solve(d iscy1,x)\};\n" }{MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$\"\"!,$*&#\"\"\"\"\"#F')F(#F'\"\"$F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$$\"\"!F$$\"+]_g*H'!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 63 "Now it only \+ remains to see which of sigma or tau_i is equal to " }{TEXT 217 8 "2^( -2/3)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "subspe(Dx1);\n" } {MPLTEXT 1 0 21 "solve(%,x);evalf(%);\n" }{MPLTEXT 1 0 13 "subspe(Dx2) ;\n" }{MPLTEXT 1 0 20 "solve(%,x);evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*&\"\"%\"\"\")I\"xG6\"\"\"#F%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"!F#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"\"!F$F#" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"\"#\"\"\"I\"xG6\"F%F%*$)F$#F%\" \"$F%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*&#\"\"\"\"\"#F%)F&#F%\" \"$F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+]_g*H'!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 39 "So sigma(1)=2^\{-2/3\} is the singularit y" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "sigmaspe:=2^(-2/3);\n" }{MPLTEXT 1 0 10 "evalf (%);\n" }{MPLTEXT 1 0 33 "factor(subs(x=sigmaspe,eqGspe));\n" } {MPLTEXT 1 0 30 "Gsigmaspe:=op(1,\{solve(%,G)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*&#\"\"\"\"\"#F%)F&#F%\"\"$F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+]_g*H'!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*(#\"#= \"%J8\"\"\",&*&\"#IF')\"\"$#F'\"\"#F'F'\"#P!\"\"F',0*&\"\")F')I\"GG6\" F,F'F'*(\"#;F')F5F.F'F+F'F0*&\"\"%F'F9F'F0*(F;F'F5F'F+F'F'*&F*F'F5F'F' *&F;F'F+F'F0F'F'F'F0" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**&#\"\"\"\"\"' F%),(\"#WF%*&\"#@F%)\"\"$#F%\"\"#F%F%*&F-F%),&\"$R$F%*&\"$#>F%F,F%F%F. F%F%#F%F-F%!\"\"*(F&F%,&*&#F%\"#=F%F,F%F7#F%\"\"*F7F%F'F7F%*&#F/F-F%F, F%F%F$F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "numer(factor(su bs(x=sigmaspe-xx,G=Gsigmaspe-GG,eqGspe)));op(4,%);\n" }{MPLTEXT 1 0 21 "puiseux(%,xx=0,GG,3);" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$**)\"\"##\"\"\"\"\"$F')F(#F'F%F',**,F%F'F)F')F%#F%F(F'I #GGG6\"F'I#xxGF0F'!\"\"*(F(F'F$F')F1F%F'F'*&F%F')F1F(F'F2*&F%F')F/F%F' F'F'),&*$F$F'F'*&F%F'F1F'F2F%F'F2" }}{PARA 11 "" 1 "" {XPPMATH 20 ",** ,\"\"#\"\"\")\"\"$#F%F$F%)F$#F$F'F%I#GGG6\"F%I#xxGF,F%!\"\"*(F'F%)F$#F %F'F%)F-F$F%F%*&F$F%)F-F'F%F.*&F$F%)F+F$F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "<#,&**#\"\"\"\"\"#\"\"\")\"\"$#\"\"\"\"\"#\"\"\")\"\"##\" \"#\"\"$\"\"\"I#xxG6\"\"\"\"\"\"\"*$)I#xxG6\"#\"\"$\"\"#\"\"\"\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT 222 14 "Case y_critic=" }{XPPEDIT 2 0 "Typesetting:-mrow(Types etting:-mfrac(Typesetting:-mn(\"12\", mathvariant = \"normal\"), Types etting:-mn(\"11\", mathvariant = \"normal\"), linethickness = \"1\", d enomalign = \"center\", numalign = \"center\", bevelled = \"false\"), \+ Typesetting:-mo(\"−\", mathvariant = \"normal\", fence = \"false \", separator = \"false\", stretchy = \"false\", symmetric = \"false\" , largeop = \"false\", movablelimits = \"false\", accent = \"false\", \+ lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mrow (Typesetting:-mfrac(Typesetting:-mn(\"2\", mathvariant = \"normal\"), \+ Typesetting:-mn(\"11\", mathvariant = \"normal\"), linethickness = \"1 \", denomalign = \"center\", numalign = \"center\", bevelled = \"false \"), Typesetting:-mo(\"⁢\", mathvariant = \"normal\", f ence = \"false\", separator = \"false\", stretchy = \"false\", symmetr ic = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mr ow(Typesetting:-msqrt(Typesetting:-mn(\"3\", mathvariant = \"normal\") )), Typesetting:-mi(\"\", italic = \"true\", mathvariant = \"italic\") ), Typesetting:-mi(\"\", italic = \"true\", mathvariant = \"italic\")) ;" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6&-I&mfracGF$ 6(-I#mnGF$6$Q#12F'/%,mathvariantGQ'normalF'-F/6$Q#11F'F2/%.linethickne ssGQ\"1F'/%+denomalignGQ'centerF'/%)numalignGF=/%)bevelledGQ&falseF'-I #moGF$6-Q(−F'F2/%&fenceGFB/%*separatorGFB/%)stretchyGFB/%*symmet ricGFB/%(largeopGFB/%.movablelimitsGFB/%'accentGFB/%'lspaceGQ,0.222222 2emF'/%'rspaceGFW-F#6&-F,6(-F/6$Q\"2F'F2F5F8F;F>F@-FD6-Q1&InvisibleTim es;F'F2FGFIFKFMFOFQFS/FVQ&0.0emF'/FYF_o-F#6#-I&msqrtGF$6#-F/6$Q\"3F'F2 -I#miGF$6%Q!F'/%'italicGQ%trueF'/F3Q'italicF'Fio" }{TEXT 222 1 "." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "yspe:=12/11-(2/11)*sqrt(3); \n" }{MPLTEXT 1 0 45 "subspe:=A->factor(simplify(subs(y=yspe,A))):\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&#\"#7\"#6\"\"\"*&#\"\"#F%F&)\"\"$#F& F)F&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "eqGspe:=subspe (eqG);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*(#\"#s\"%J8\"\"\",&*&\"#I F')\"\"$#F'\"\"#F'F'\"#P!\"\"F',:**F.F')F.#F.F,F')I\"xG6\"\"\"%F'F+F'F 0*,F8F'F6F')I\"GGF7F.F'F+F'F3F'F0*&F3F'F5F'F0*(F.F')F.#F'F,F')F6\"\"&F 'F'**F.F'F3F'F6F'F:F'F0**F.F'F>F')F6F.F'F:F'F'*,F8F'FDF'F;F'F+F'F>F'F' **F8F'F;F')F6F,F'F+F'F0**\"#:F'FDF'F;F'F>F'F'*(F.F'FGF'F+F'F0*&F.F')F; F,F'F'*$FGF'F'F'F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "#disc y1:=factor(discrim(eqGspe,G));\n" }{MPLTEXT 1 0 22 "discy1:=subspe(dis c);\n" }{MPLTEXT 1 0 1 "[" }{MPLTEXT 1 0 18 "solve(discy1,x)];\n" } {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$**#\"$K% \"(h:x\"\"\"\",&*&\"%?AF')\"\"$#F'\"\"#F'F'\"%pS!\"\"F',(*(\"\"%F'F+F' )F.#F'F,F'F'*&\"#>F'F4F'F0*&\"\")F'I\"xG6\"F'F0F'),(*&F+F'F4F'F'*&F.F' F4F'F'*&F.F'F:F'F0F3F'F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "7',&*(#\"\" \"\"\"#F&)\"\"$F%F&)F'#F&F)F&F&*&#\"#>\"\")F&F*F&!\"\",&F$F&*$F*F&F&F1 F1F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "7'$!+e))=,>!\"*$\"+'oW5N#F%F&F&F &" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 29 "So the singularity is 2.35..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "[solve(discy1,x)];\n" }{MPLTEXT 1 0 9 "evalf( %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "7\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "subspe(Dx1);\n" }{MPLTEXT 1 0 21 "solve(%,x);evalf(%);\n" }{MPLTEXT 1 0 13 "subspe(Dx2 );\n" }{MPLTEXT 1 0 20 "solve(%,x);evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$**#\"\"'\"%J8\"\"\",&*&\"#PF')\"\"$#F'\"\"#F'F'\"#!*!\" \"F',(*&)F.#F'F,F'F+F'F'*&F.F'F3F'F'*&F.F'I\"xG6\"F'F0F',(*(\"\"%F'F3F 'F+F'F'*&\"#>F'F3F'F0*&\"\")F'F7F'F0F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&*(#\"\"\"\"\"#F&)F'#F&\"\"$F&)F*F%F&F&*$F(F&F&,&F$F&*&#\"#>\" \")F&F(F&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+'oW5N#!\"*$!+e)) =,>F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*(#\"\"#\"#6\"\"\",&\"\"'!\" \"*$)\"\"$#F'F%F'F'F',(*&)F%#F'F-F'F,F'F'*&F%F'F1F'F'*&F%F'I\"xG6\"F'F *F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*(#\"\"\"\"\"#F%)F&#F%\"\"$F% )F)F$F%F%*$F'F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+'oW5N#!\"*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 217 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "sigmasp e:=(1/2)*2^(1/3)*sqrt(3)+2^(1/3);\n" }{MPLTEXT 1 0 33 "factor(subs(x=s igmaspe,eqGspe));\n" }{MPLTEXT 1 0 1 "[" }{MPLTEXT 1 0 12 "solve(%,G)] ;" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 19 "Gsigmaspe:=op(3,%);" }} {PARA 11 "" 1 "" {XPPMATH 20 ",&*(#\"\"\"\"\"#F%)\"\"$F$F%)F&#F%F(F%F% *$F)F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*(#\"#=\"%J8\"\"\",&*&\"#I F')\"\"$#F'\"\"#F'F'\"#P!\"\"F'),(*&F.F'I\"GG6\"F'F'F,F0*&F.F'F+F'F0F, F'F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "7%,&#\"\"$\"\"#\"\"\"*$)F%#F'F&F 'F'F#F#" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&#\"\"$\"\"#\"\"\"*$)F$#F&F% F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "numer(factor(subs(x =sigmaspe-xx,G=Gsigmaspe-GG,eqGspe)));op(2,%);\n" }{MPLTEXT 1 0 22 "pu iseux(%,xx=0,GG,1);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*(\"#O\"\"\", >**\"\"'F%)\"\"##F*\"\"$F%)F,#F%F*F%)I#xxG6\"\"\"%F%!\"\"*,F2F%F)F%F-F %F0F%)I#GGGF1F*F%F3*(\"#=F%F)F%F/F%F3*(F2F%)F*#F%F,F%)F0\"\"&F%F%**F2F %F)F%F0F%F5F%F%**F2F%F:F%)F0F*F%F5F%F3*,\"\")F%F:F%F@F%F-F%F6F%F3**FBF %F-F%F6F%)F0F,F%F%**F(F%F:F%F@F%F6F%F%**\"#:F%F:F%F-F%F@F%F3*(\"#CF%F- F%FDF%F%*&F2F%)F6F,F%F%*(\"#EF%F:F%F@F%F3*&\"#aF%FDF%F%F%,&*&\"#IF%F-F %F%\"#PF3F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",>**\"\"'\"\"\")\"\"##F '\"\"$F%)F)#F%F'F%)I#xxG6\"\"\"%F%!\"\"*,F/F%F&F%F*F%F-F%)I#GGGF.F'F%F 0*(\"#=F%F&F%F,F%F0*(F/F%)F'#F%F)F%)F-\"\"&F%F%**F/F%F&F%F-F%F2F%F%**F /F%F7F%)F-F'F%F2F%F0*,\"\")F%F7F%F=F%F*F%F3F%F0**F?F%F*F%F3F%)F-F)F%F% **F$F%F7F%F=F%F3F%F%**\"#:F%F7F%F*F%F=F%F0*(\"#CF%F*F%FAF%F%*&F/F%)F3F )F%F%*(\"#EF%F7F%F=F%F0*&\"#aF%FAF%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "<#*&)*&I#xxG6\"\"\"\",&*(#\"#:\"\"%\"\"\")\"\"$#\"\"\"\"\"#\"\"\")\" \"##\"\"\"\"\"$\"\"\"\"\"\"*&#\"#8\"\"#\"\"\")\"\"##\"\"\"\"\"$\"\"\" \"\"\"!\"\"#\"\"#\"\"$\"\"\",&*(#\"#:\"\"%\"\"\")\"\"$#\"\"\"\"\"#\"\" \")\"\"##\"\"\"\"\"$\"\"\"\"\"\"*&#\"#8\"\"#\"\"\")\"\"##\"\"\"\"\"$\" \"\"\"\"\"\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 222 54 "Singularity analysis (-> asymptotic coeffs of islands)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 217 122 "We first study the singularity a t sigma. At that point, G has value Gsigma (since two branches have th e same value Gsigma)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "fac tor(subs(x=sigma,eqG));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 14 "[solve (%,G)];\n" }{MPLTEXT 1 0 1 "G" }{MPLTEXT 1 0 16 "sigma:=op(1,%);\n" } {MPLTEXT 1 0 13 "#subs(y=1,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$**# \"\"\"\"\"%F%,2*(\"#7F%I\"GG6\"F%)I\"yGF+\"\"#F%F%*(\"#ZF%)\"\"$#F%F.F %F,F%F%*(F)F%F*F%F-F%!\"\"*(\"#'*F%F1F%F-F%F5*&\"#sF%F,F%F%*&\"#[F%F1F %F%*&\"$W\"F%F-F%F5F9F%F%),0*(F.F%F*F%F,F%F%*(\"\"(F%F1F%F,F%F5*(\"#=F %F1F%F-F%F%*&\"#6F%F,F%F5*&F)F%F1F%F5*&\"#CF%F-F%F%F)F5F.F%)F-F2F5F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "7%,$*(#\"\"\"\"\"#F&,.*(\"\"(F&)\"\"$F %F&)I\"yG6\"F'F&F&*(\"#=F&F+F&F.F&!\"\"*&\"#6F&F-F&F&*&\"#7F&F+F&F&*& \"#CF&F.F&F2F6F&F&F-F2F&F#,$**#F&F6F&,.*(\"#ZF&F+F&F-F&F&*(\"#'*F&F+F& F.F&F2*&\"#sF&F-F&F&*&\"#[F&F+F&F&*&\"$W\"F&F.F&F2FBF&F&,&F.F&F&F2F2F. F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*(#\"\"\"\"\"#F%,.*(\"\"(F%)\" \"$F$F%)I\"yG6\"F&F%F%*(\"#=F%F*F%F-F%!\"\"*&\"#6F%F,F%F%*&\"#7F%F*F%F %*&\"#CF%F-F%F1F5F%F%F,F1F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "numer(factor(subs(x=sigma-xx,G=Gsigma-GG,eqG))):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 37 "eqGxsigma:=collect(%,\{xx,GG\},factor);" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 22 "puiseux(%,xx=0,GG,2);\n" } {MPLTEXT 1 0 1 "#" }{MPLTEXT 1 0 35 "singGsigma:=map(mafactor,op(1,%)) ;\n" }{MPLTEXT 1 0 1 "#" }{MPLTEXT 1 0 23 "map(mafactor,op(2,%%));" } {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ",0**\"#C\"\"\")I\"yG 6\"\"\"%F%,&F'F%F%!\"\"F%)I#GGGF(\"\"$F%F+*&,(*,F)F%)F.#F%\"\"#F%)F4#F %F.F%)I#xxGF(F4F%)F'\"\"&F%F+*.#F)\"#8F%)F4#F4F.F%,&*&F:F%F2F%F%\"\"'F %F%)F'F.F%,,*(\"#EF%F2F%F'F%F%*&F=F%)F'F4F%F%*&F$F%F2F%F+*&\"#yF%F'F%F +\"#gF%F%F8F%F+*,#F4\"%t:F%,&*&\"#*)F%F2F%F%\"$Q\"F%F%,(*&F=F%F'F%F%\" #=F+*&F4F%F2F%F%F%FHF%),(*&\"#6F%F'F%F%\"#7F+FWF%F4F%F%F%)F-F4F%F%*&,( *(F$F%)F8F.F%F9F%F%*.#Ffn\"#BF%F5F%,&FAF%FenF%F%FHF%,0*(F=F%F2F%FHF%F% *&\"#YF%FCF%F%*(\"#AF%F2F%F'F%F%*&\"#$*F%FHF%F+*&\"#IF%F2F%F+*&FioF%F' F%F+\"#mF%F%F7F%F%*2#FB\"%*G$F%F>F%,&*&\"$G\"F%F2F%F%\"$H#F%F%,,*(\"#; F%F2F%F'F%F%*&F^oF%FHF%F%*&\"#?F%F2F%F+*&F[pF%F'F%F+\"#[F%F%FTF%FYF%F' F%F8F%F%F%F-F%F%*,F)F%F2F%F5F%)F8F:F%F9F%F%*.#F4F=F%F>F%F@F%FCF%,,*(\" #MF%F2F%F'F%F%*&\"#lF%FHF%F%FIF+*&\"$]\"F%F'F%F+FLF%F%)F8F)F%F%*.#F4\" &hB\"F%,&*&\"$J$F%F2F%F%\"$S&F%F%,,*(\"$!GF%F2F%F'F%F%*&\"$j#F%FHF%F%* &\"$/&F%F2F%F+*&\"$C*F%F'F%F+\"%K5F%F%,(*&\"#ZF%F'F%F+FjpF%*&F)F%F2F%F %F%FHF%F[oF%F+*.#F.\"%xoF%F5F%,&*&\"$0'F%F2F%F%\"%Q5F%F%FTF%)FcpF4F%F7 F%F%" }}{PARA 206 "" 1 "" {TEXT 224 0 "" }{HYPERLNK 224 "Error, (in al gcurves/g_evala) unable to execute add" 4 "http://www.maplesoft.com/su pport/help/errors/view.aspx?path=Error,%20(in%20algcurves%2Fg_evala)%2 0unable%20to%20execute%20add" "" }{TEXT 224 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 95 "Annoy ing: Maple does not want to do puiseux. I don't know why. We will reso rt to Newton polygon" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "lde gree(coeff(eqGxsigma,xx,0),GG);\n" }{MPLTEXT 1 0 34 "coeff(coeff(eqGxs igma,xx,0),GG,2);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 19 "evalf(solve( %,y));\n" }{MPLTEXT 1 0 34 "ldegree(coeff(eqGxsigma,xx,1),GG);" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 34 "ldegree(coe ff(eqGxsigma,xx,2),GG);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 35 "coeff( coeff(eqGxsigma,xx,2),GG,0);\n" }{MPLTEXT 1 0 18 "evalf(solve(%,y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*,#\"\"#\"%t:\"\"\",&*&\"#*)F')\"\"$#F'F%F'F'\"$Q\"F'F',(*&\"#8F'I \"yG6\"F'F'\"#=!\"\"*&F%F'F+F'F'F')F2F%F'),(*&\"#6F'F2F'F'\"#7F5F6F'F% F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'$\"+Ig9=6!\"*$\"\"!F'F&$\"+Bw! *fx!#5F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*,#\"\"$\"%xo\" \"\")\"\"##F'F%F',&*&\"$0'F')F%#F'F)F'F'\"%Q5F'F',(*&\"#8F'I\"yG6\"F'F '\"#=!\"\"*&F)F'F.F'F'F'),,*(\"#;F'F.F'F4F'F'*&\"#BF')F4F)F'F'*&\"#?F' F.F'F7*&\"#mF'F4F'F7\"#[F'F)F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'$ \"+Ig9=6!\"*$\"+;m\"p;\"F%$\"+BpVx\\!#5F&F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "collect(subs(GG=a*xx,eqGxsigma),xx,factor);\n" } {MPLTEXT 1 0 15 "coeff(%,xx,2);\n" }{MPLTEXT 1 0 14 "\{solve(%,a)\};\n " }{MPLTEXT 1 0 12 "c1:=op(1,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**, \"\"%\"\"\")\"\"$#F%\"\"#F%)F)#F%F'F%)I#xxG6\"\"\"&F%)I\"yGF.F/F%F%*.F )F%F&F%F*F%,0*,F)F%F&F%)F)#F)F'F%I\"aGF.F%)F1F)F%F%**\"#5F%F&F%F*F%F8F %F%*(F)F%)F7F)F%F8F%!\"\"**F:F%F&F%F*F%F1F%F=*(\"#DF%F*F%F8F%F%*(\"#UF %F*F%F1F%F=*&\"#7F%F*F%F%F%)F1F'F%)F-F$F%F%**F)F%,L*,F:F%F&F%F5F%FF%F&F%F5F%F8F%F%*(FapF%F5F%F8F%F%**FBF%F&F%F5F%F1F%F=*(\"#AF%F8F% F7F%F%**F$F%F&F%F7F%F1F%F%*(FDF%F5F%F1F%F=*(FNF%F5F%F&F%F%*(FNF%F7F%F1 F%F=*&FDF%F5F%F=F)F%)F-F)F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$**#\" \"\"\"%YJF%,&*&\"#*)F%)\"\"$#F%\"\"#F%F%\"$Q\"F%F%,(*&\"#8F%I\"yG6\"F% F%\"#=!\"\"*&F-F%F*F%F%F%),4**\"#>F%F*F%)F-#F-F+F%)F2F-F%F%*(F4F%F;F%F =F%F%**\"#UF%F*F%F;F%F2F%F5*(\"#AF%F=F%I\"aGF3F%F%**\"\"%F%F*F%FCF%F2F %F%*(\"#7F%F;F%F2F%F5*(\"#CF%F;F%F*F%F%*(FIF%FCF%F2F%F5*&FGF%F;F%F5F-F %F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "<#,$*,#\"\"\"\"\"#\"\"\")\"\"##\" \"#\"\"$\"\"\",.*(\"#>\"\"\")\"\"$#\"\"\"\"\"#\"\"\")I\"yG6\"\"\"#\"\" \"\"\"\"*(\"#U\"\"\")\"\"$#\"\"\"\"\"#\"\"\"I\"yG6\"\"\"\"!\"\"*&\"#= \"\"\")I\"yG6\"\"\"#\"\"\"\"\"\"*&\"#C\"\"\")\"\"$#\"\"\"\"\"#\"\"\"\" \"\"*&\"#7\"\"\"I\"yG6\"\"\"\"!\"\"\"#7!\"\"\"\"\"I\"yG6\"!\"\",(*&\"# 6\"\"\"I\"yG6\"\"\"\"\"\"\"\"#7!\"\"*&\"\"#\"\"\")\"\"$#\"\"\"\"\"#\" \"\"\"\"\"!\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*,#\"\"\"\"\"# F%)F&#F&\"\"$F%,.*(\"#>F%)F)F$F%)I\"yG6\"F&F%F%*(\"#UF%F-F%F/F%!\"\"*& \"#=F%F.F%F%*&\"#CF%F-F%F%*&\"#7F%F/F%F3F9F3F%F/F3,(*&\"#6F%F/F%F%F9F3 *&F&F%F-F%F%F3F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "subs(GG =c1*xx+KK," }{MPLTEXT 1 0 12 "eqGxsigma):\n" }{MPLTEXT 1 0 24 "numer(f actor(%))/2/y^2:\n" }{MPLTEXT 1 0 27 "collect(%,\{xx,KK\},factor);\n" }{MPLTEXT 1 0 22 "puiseux(%,xx=0,KK,0):\n" }{MPLTEXT 1 0 17 "map(mafac tor,%);\n" }{MPLTEXT 1 0 20 "singGsigma:=op(2,%);" }{MPLTEXT 1 0 1 "\n " }}{PARA 11 "" 1 "" {XPPMATH 20 ",.*,\"#7\"\"\",&I\"yG6\"F%F%!\"\"F%) F'\"\"#F%),(*&\"#6F%F'F%F%F$F)*&F+F%)\"\"$#F%F+F%F%F2F%)I#KKGF(F2F%F)* &,(*.F+F%F1F%)F+#F%F2F%)F'F2F%F,F%)I#xxGF(F+F%F)*0#F+\"#fF%)F+#F+F2F%, &*&\"#HF%F1F%F%\"#CF%F%,0*&\"$O#F%F;F%F%*&\"$\"oF%F*F%F)*(\"#yF%F1F%F* F%F%*&\"$a'F%F'F%F%*(\"$c\"F%F1F%F'F%F)\"$;#F)*&\"#%)F%F1F%F%F%F'F%)F- F+F%F=F%F%**#F%\"%t:F%,&*&\"#*)F%F1F%F%\"$Q\"F%F%,(*&\"#8F%F'F%F%\"#=F )F0F%F%)F-\"\"&F%F%F%)F5F+F%F%*&,&*0FFF%,&*$F1F%F%F\\oF%F%,(*&F2F%F'F% F%FboF%F2F)F%F&F%F*F%FUF%)F=F2F%F%*4#\"\"'\"$V\"F%F9F%,&*&\"$=#F%F1F%F %\"$T%F%F%,,*(\"#EF%F1F%F'F%F%*&\"#LF%F*F%F%*&\"#KF%F1F%F)*&\"#')F%F'F %F)\"#sF%F%FgnF%F-F%F&F%F'F%F#F%,&\"%O\")F%*&\"%vVF%F1F%F%F%)FgnF2F%F;F%FeoF%F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$,$*.#\"\"\"\"&w)=F&,&*&\"#*)F&)\"\"$#F&\"\"#F&F& \"$Q\"F&F&,(*&\"#8F&I\"yG6\"F&F&\"#=!\"\"*&F.F&F+F&F&F&),(*&\"#6F&F3F& F&\"#7F6F7F&F.F&,&F3F&F&F6F6)F3F.F6F&,$*0#F&\"0h\"HSy)zS(F&,&\")dF#*HF &*&\"(O!yrF&F+F&F&F&)F0\"\"%F&)F3\"\"'F&)F9\"#5F6)\"'^5;F-F&)*,,&\"%j6 F6*&\"%'3$F&F+F&F&F&I#xxGF4F&)F9\"\"&F&)F0F.F6)F3F,F6#F,F.F&F&" }} {PARA 11 "" 1 "" {XPPMATH 20 ",$*0#\"\"\"\"0h\"HSy)zS(F%,&\")dF#*HF%*& \"(O!yrF%)\"\"$#F%\"\"#F%F%F%),(*&\"#8F%I\"yG6\"F%F%\"#=!\"\"*&F.F%F+F %F%\"\"%F%)F3\"\"'F%),(*&\"#6F%F3F%F%\"#7F6F7F%\"#5F6)\"'^5;F-F%)*,,& \"%j6F6*&\"%'3$F%F+F%F%F%I#xxGF4F%)F<\"\"&F%)F0F.F6)F3F,F6#F,F.F%F%" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 145 "The singular behaviour of G at x=sigma is given by \"s ingGsigma\". Indeed he other value is not going to 0 (it is the branch not equal to Gsigma).\n" }{TEXT 217 79 "It is only real for y>y_criti c. In that range it gives coefficients in n^(-5/2)" }}{PARA 0 "" 0 "" {TEXT 217 133 "Thus the singularity sigma is impossible for ytau1(y) for y>ycritic and si ngularity is decreasing)." }}{PARA 0 "" 0 "" {TEXT 217 47 "It remains \+ to understand singularity at x=tau1" }}{PARA 0 "" 0 "" {TEXT 217 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "#f actor(subs(x=tau1,eqG));\n" }{MPLTEXT 1 0 1 "#" }{MPLTEXT 1 0 14 "[sol ve(%,G)]:\n" }{MPLTEXT 1 0 1 "#" }{MPLTEXT 1 0 15 "Gtau:=op(1,%):\n" } {MPLTEXT 1 0 13 "#subs(y=1,%);" }{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 217 51 "We now want to understand t he situation at x=tau1.\n" }{TEXT 217 175 "Unfortunately maple does no t handle well the substitution x=tau1 and the preceding treatment beco mes impossible. Instead we will proceed by elimation of tau1 using res ultants." }}{PARA 0 "" 0 "" {TEXT 217 1 "\n" }{TEXT 217 192 "We denote Gtau=G(x=tau1), and we set x=tau1-xx, and G=Gtau-GG in eqG. We obtain an equation eqGGxx for GG, xx, We then determine the possible singula rity behaviors using Newton polygon method." }}{PARA 0 "" 0 "" {TEXT 217 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "-factor(resultan t(eqG,Dx1,x))/576;\n" }{MPLTEXT 1 0 27 "subs(G=Gtau,op(1,op(2,%)));" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 8 "eqGtau:=" }{MPLTEXT 1 0 23 "colle ct(%,Gtau,factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "**,D*(\"#;\"\"\")I \"GG6\"\"\"#F&)I\"yGF)\"\"%F&F&**\"$/#F&F(F&)\"\"$#F&F*F&F+F&F&**\"$!= F&F(F&F0F&)F,F1F&!\"\"*(\"$)GF&F(F&F+F&F6*(\"#[F&F0F&F+F&F&**\"#7F&F(F &F0F&)F,F*F&F&*(F8F&F(F&F5F&F&*(\"$#>F&F5F&F0F&F6*&\"\"*F&F+F&F6**\"#O F&F(F&F0F&F,F&F6*(\"#sF&F0F&F=F&F6*&\"$1$F&F5F&F&*(\"$k)F&F0F&F,F&F&*& \"$N\"F&F=F&F&*&\"$['F&F0F&F6*&\"%G " 0 "" {MPLTEXT 1 0 57 "eqGG:=factor(resultant(subs(G=Gtau-GG,eqG),eqGtau,Gta u)):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 " eqGGxx:=resultant(subs(x=x-xx,eqGG),Dx1,x):\n" }{MPLTEXT 1 0 34 "eqGGx x:=collect(%,\{GG,xx\},factor):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 97 "We nww apply Newton's method: we first investigate which are the non-zer o coefficients of eqGGxx." }{TEXT 217 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "ldegree(coeff(eqGGxx,xx,0),GG);\n" }{MPLTEXT 1 0 32 "coeff(coeff(eqGGxx,xx,0),GG,4);\n" }{MPLTEXT 1 0 14 "\{fsolve(%,y)\}; " }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 32 "ldegree(coeff(eqGGxx,xx,1),GG );\n" }{MPLTEXT 1 0 32 "ldegree(coeff(eqGGxx,xx,2),GG);\n" }{MPLTEXT 1 0 32 "coeff(coeff(eqGGxx,xx,2),GG,0);\n" }{MPLTEXT 1 0 14 "\{fsolve( %,y)\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*4#\"-sAMoqe\"1>zB>'>!zD\"\"\",&*&\",!e2c.WF')\"\"$#F' \"\"#F'F'\",J&4>Fw!\"\"F',,*&\"#BF')I\"yG6\"F.F'F'*(\"\"'F'F+F'F5F'F'* &\"#FF'F5F'F'\"#=F0*&\"#UF'F+F'F'F',8*(\"&aR&F'F+F')F5\"\"%F'F'*&\"'` \"o#F')F5\"\"&F'F'*(\"'Em5F')F5F,F'F+F'F0*&\"'*Q0&F'FAF'F0*(\"'S2DF'F+ F'F4F'F'*&\"'ss@F'FIF'F'*(\"&)GOF'F+F'F5F'F0*&\"'9-QF'F4F'F'*&\"'k&Q\" F'F+F'F0*&\"&KT)F'F5F'F0\"'i]BF0F'),(*&\"#6F'F5F'F'\"#7F0*&F.F'F+F'F'F .F'),(*&\"#8F'F5F'F'F,F'*&FBF'F+F'F'FBF'),,*(\"#EF'F+F'F5F'F'*&FfnF'F4 F'F'*&\"\")F'F+F'F'*&\"#NF'F5F'F'\"#:F'FBF'),&F5F'F'F0F8F')F5\"#XF'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "<)$!+RzI&*o!\"*$!+4%zqj(!#5$!+o^[/QF( $\"\"!F,$\"+Aw!*fxF($\"+\\XiB$*F($\"\"\"F," }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*6#\",G4=hW#\"3Fw_!*)fi%\\5\"\"\")\"\"##F'\"\"$ F',&*&\"-ec=bV:F')F+#F'F)F'F'\"-HN-^tE!\"\"F',8*(\"&aR&F'F/F')I\"yG6\" \"\"%F'F'*&\"'`\"o#F')F7\"\"&F'F'*(\"'Em5F')F7F+F'F/F'F2*&\"'*Q0&F'F6F 'F2*(\"'S2DF'F/F')F7F)F'F'*&\"'ss@F'F@F'F'*(\"&)GOF'F/F'F7F'F2*&\"'9-Q F'FEF'F'*&\"'k&Q\"F'F/F'F2*&\"&KT)F'F7F'F2\"'i]BF2F'),(*&\"#6F'F7F'F' \"#7F2*&F)F'F/F'F'F+F'),(*&\"#8F'F7F'F'F+F'*&F9F'F/F'F'F+F'),,*&\"#BF' FEF'F'*(\"\"'F'F/F'F7F'F'*&\"#FF'F7F'F'\"#=F2*&\"#UF'F/F'F'F+F'),,*(\" #EF'F/F'F7F'F'*&FTF'FEF'F'*&\"\")F'F/F'F'*&\"#NF'F7F'F'\"#:F'F9F'),&F7 F'F'F2F=F')F7F`oF'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "<)$!+RzI&*o!\"*$ !+4%zqj(!#5$!+o^[/QF($\"\"!F,$\"+Aw!*fxF($\"+\\XiB$*F($\"\"\"F," }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 126 "Great: the Newton polygon method works uniformly in y on the interval (0,y_criti c) and shows a singularity of square-root type" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 105 "--------- ---------------------------------------------------------------------- -------------------------\n" }{TEXT 217 4 "Next" }{TEXT 217 168 " we a nalyze the asymptotic weight of oceans, that is,the other piece thatt \+ appears when cutting the percolated triangulation into two pieces (int erface at the origin). " }}{PARA 0 "" 0 "" {TEXT 217 171 "Let F=U(x,y, z0) = g.f. of tringulations with boundary (not necessirily simple) cou nted according to the number of external edges= \\sum x^\{length\}y^\{ external edges]z0^edges" }}{PARA 0 "" 0 "" {TEXT 217 104 "------------ ---------------------------------------------------------------------- ----------------------" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 217 46 "We denote U(x,y,z) the generating function of " }{TEXT 217 146 " tringulations with boundary (not necessirily simple) counte d according to the number of external edges= \\sum x^\{length\}y^\{ext ernal edges]z^edges." }}{PARA 0 "" 0 "" {TEXT 217 22 "We also denote F (x,y)=" }{TEXT 217 70 "U(x,y,z0). This series is denoted U(x,y) in the paper (with a bold U)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 221 53 "Algebraic equation for U(x,y ,z) and F(x,y)=U(x,y,z0) " }{TEXT 221 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 217 58 "We start with the functional equa tion eqUfun for U, where " }{TEXT 217 18 "U1=[x^1]U(x,y,z)=y" }{TEXT 217 14 "[x^1]U(x,1,z)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 " eqUfun:=1+y*x^2*z*U^2+z*(U-1-x*U1)/(x*y)+(y-1)*2*z*(y*x^2*z*U)*(U-1-y* x^2*z*U^2)/(x*y)+(y^2-1)*z*(x^4*y^2*z^2*U^3)/(x*y)+(y-1)*z*(x^2*y*z*U) /(x*y)-U;\n" }{MPLTEXT 1 0 25 "eqUfun:=numer(factor(%));" }}{PARA 11 " " 1 "" {XPPMATH 20 ",0\"\"\"F#**I\"yG6\"F#)I\"xGF&\"\"#F#I\"zGF&F#)I\" UGF&F)F#F#**F*F#,(*&I#U1GF&F#F(F#!\"\"F,F#F#F1F#F(F1F%F1F#*.F)F#,&F%F# F#F1F#)F*F)F#F(F#F,F#,(**F+F#F'F#F%F#F*F#F1F,F#F#F1F#F#*,,&*$)F%F)F#F# F#F1F#)F*\"\"$F#)F(F " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 13 "We denote TT=" }{TEXT 217 17 "U(x,1,z) and TT1=" }{TEXT 217 179 "U(x,1,z ). By specialization of eqUfun we get a functional equation for TT. We then proceed to solve it by the quadratic method. This gives us the a lgebraic equation eqTT1 for TT1 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "eqTT:=subs(U=TT,U1=TT1,y=1,eqUfun);\n" }{MPLTEXT 1 0 15 "disc rim(%,TT);\n" }{MPLTEXT 1 0 14 "discrim(%,x);\n" }{MPLTEXT 1 0 17 "eqT T1:=%/256/z^7;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",.*()I#TTG6\"\"\"#\"\" \")I\"xGF&\"\"$F(I\"zGF&F(F(*(I$TT1GF&F(F*F(F,F(!\"\"*&F%F(F*F(F/*&F%F (F,F(F(F*F(F,F/" }}{PARA 11 "" 1 "" {XPPMATH 20 ",.**\"\"%\"\"\"I$TT1G 6\"F%)I\"xGF'F$F%)I\"zGF'\"\"#F%F%*(F$F%F(F%F+F%!\"\"*(F$F%)F)\"\"$F%F *F%F%*$)F)F,F%F%*(F,F%F)F%F+F%F.*$F*F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*(\"$c#\"\"\",0*(\"#kF%)I$TT1G6\"\"\"$F%)I\"zGF+\"\"&F%F%*(\"#'* F%)F*\"\"#F%)F.\"\"%F%!\"\"*&\"#FF%F-F%F6*(\"#IF%F*F%)F.F,F%F%*&F2F%F. F%F%*$)F.F3F%F%F*F6F%)F.\"\"(F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",0* (\"#k\"\"\")I$TT1G6\"\"\"$F%)I\"zGF(\"\"&F%F%*(\"#'*F%)F'\"\"#F%)F+\" \"%F%!\"\"*&\"#FF%F*F%F3*(\"#IF%F'F%)F+F)F%F%*&F/F%F+F%F%*$)F+F0F%F%F' F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "algeqtoseries(eqTT1,z ,TT1,8,true);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "7#++ I\"zG6\"\"\"\"\"\"#\"\"%\"\"&\"#K\"\")-I\"OG%*protectedG6#\"\"\"\"#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 26 "We eliminate U1=yTT1 from " } {TEXT 217 60 " eqUfun using eqTT1. This gives an algebraic equation fo r U." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "subs(U1=y*TT1,eqUfu n);\n" }{MPLTEXT 1 0 40 "eqU:=factor(resultant(%,eqTT1,TT1))/z^2;" }} {PARA 11 "" 1 "" {XPPMATH 20 ",<**)I\"UG6\"\"\"$\"\"\")I\"xGF&\"\"%F() I\"yGF&F+F()I\"zGF&F'F(F(*,\"\"#F(F$F(F)F()F-F'F(F.F(!\"\"**F$F(F)F()F -F1F(F.F(F(**)F%F1F()F*F'F(F5F(F/F(F(*,F1F(F7F()F*F1F(F5F()F/F1F(F(*,F 1F(F7F(F:F(F-F(F;F(F3**F%F(F:F(F5F(F;F(F3**F%F(F:F(F-F(F;F(F(**I$TT1GF &F(F*F(F-F(F/F(F3*(F%F(F*F(F-F(F3*&F%F(F/F(F(*&F*F(F-F(F(F/F3" }} {PARA 11 "" 1 "" {XPPMATH 20 ",bel*,\"#k\"\"\")I\"UG6\"\"\"*F%)I\"xGF( \"#7F%)I\"yGF(F,F%)I\"zGF(F,F%!\"\"*,\"$%QF%F&F%F*F%)F.\"#6F%F/F%F%*, \"$g*F%F&F%F*F%)F.\"#5F%F/F%F1*,\"%!G\"F%F&F%F*F%)F.F)F%F/F%F%*,F7F%F& F%F*F%)F.\"\")F%F/F%F1*,F3F%F&F%F*F%)F.\"\"(F%F/F%F%*,F$F%F&F%F*F%)F. \"\"'F%F/F%F1*,\"$#>F%)F'F?F%)F+F5F%F8F%)F0F9F%F1*,F3F%FHF%)F+F9F%F8F% )F0F5F%F1*,\"$o(F%FHF%FIF%FF%FHF%FLF%FF%FJF%F1*,\"%SQF%FHF%FLF%F>F%FM F%F1*,F7F%FSF%FLF%FF%FMF%F%*,FGF%FHF%FIF%FDF%FJF%F1*,FQF%FHF%FLF%FD F%FMF%F1*,FQF%FSF%FLF%FAF%FMF%F1*,F3F%FHF%FLF%)F.\"\"&F%FMF%F%*,F7F%FS F%FLF%FDF%FMF%F%*,FGF%FSF%)F+F)F%FF%)F0F?F %F1*,FGF%FSF%FLF%FjnF%FMF%F1*,\"%O:F%FSF%F^oF%F>F%F_oF%F1*,F7F%FSF%)F+ F?F%F>F%FJF%F1*,\"#'*F%)F'FEF%F^oF%FF%F_oF%F%*,F7F%FioF%FfoF%F>F%FJF%F%*,FGF%FSF%FLF%FDF%FaoF%F1*,\"%sI F%FSF%F^oF%FDF%F_oF%F1*,\"%gdF%FSF%FfoF%FDF%FJF%F1*,\"%GF%FJF%F1*,F7F %FSF%F^oF%FjnF%F_oF%F%*,FWF%FSF%FfoF%FjnF%FJF%F%**FioF%F^oF%FF%F]qF %F%*,\"$![F%FioF%F^oF%FjnF%F_oF%F1*,FWF%FioF%FfoF%FjnF%FJF%F1*,FUF%Fip F%FfoF%FDF%FJF%F1*,FEF%FioF%F^oF%FAF%F]qF%F1*,F3F%FioF%FfoF%FAF%)F0FBF %F%*,F7F%FioF%FfoF%FbqF%FJF%F%*,FOF%FioF%)F+FBF%FAF%FaoF%F%*,FOF%FipF% FfoF%FjnF%FJF%F%*,\"#gF%FioF%F^oF%FDF%F]qF%F1*,FUF%FioF%FfoF%FDF%F[rF% F1*,F\\pF%FioF%F^rF%FDF%FaoF%F1*,F;F%FioF%)F+FEF%FDF%F_oF%F1*,FGF%FipF %FfoF%FAF%F[rF%F1*,FGF%FipF%FfoF%FbqF%FJF%F1*,FOF%FipF%F^rF%FAF%FaoF%F 1**FioF%F^oF%FjnF%F]qF%F1*,FOF%FioF%FfoF%FjnF%F[rF%F%*,\"%3YF%FioF%F^r 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%FitF%F`vF%F\\tF%FaoF%F%*,FGF%F]xF%FguF%FbqF%F[rF%F%*,FGF%F]xF%F`vF%Fb qF%FaoF%F%*,FEF%FcsF%FerF%FbqF%FdwF%F%*,FduF%FcsF%FguF%FbqF%F`tF%F1*,F 3F%FcsF%F`vF%FbqF%FbtF%F1*,FcqF%FitF%FerF%FjnF%FdwF%F%*,FEF%FitF%FguF% FjnF%F`tF%F%*,FUF%FitF%F`vF%FbwF%FaoF%F1*,FhoF%F]xF%FguF%F\\tF%F[rF%F1 *,F3F%F]xF%F`vF%F\\tF%FaoF%F1*,FEF%FcsF%FguF%F\\tF%F`tF%F1*,FOF%FcsF%F `vF%F\\tF%FbtF%F%*,FOF%FcsF%)F+F]tF%F\\tF%F]qF%F%*,F]tF%FitF%FerF%FbqF %FdwF%F1*,FduF%FitF%FguF%FbqF%F`tF%F%*,F_sF%FitF%F`vF%FbqF%FbtF%F%**F] xF%FguF%FjnF%F`tF%F1*,FGF%F]xF%F`vF%FbwF%FaoF%F%*,F7F%FcsF%FcyF%FbwF%F ]qF%F1*,F3F%FcsF%)F+F_tF%FbwF%F[rF%F1*,FEF%FitF%FguF%F\\tF%F`tF%F%*,FU F%FitF%F`vF%F\\tF%FbtF%F1*,FdoF%FitF%FcyF%F\\tF%F]qF%F1*,\"#GF%F]xF%Fg uF%FbqF%F`tF%F1*,F_wF%F]xF%F`vF%FbqF%FbtF%F1*,F3F%FcsF%F[zF%F.F%F[rF%F %*,FQF%FitF%FcyF%FbwF%F]qF%F%*,F7F%FitF%F[zF%FbwF%F[rF%F%**F]xF%FguF%F \\tF%F`tF%F1*,\"$W%F%F]xF%F`vF%F\\tF%FbtF%F%*,F7F%F]xF%FcyF%F\\tF%F]qF %F%*,FfvF%F'F%F`vF%FbqF%FbtF%F%*,F_tF%FitF%FguF%FbqF%F0F%F%*,FcqF%FitF %F`vF%FbqF%F_xF%F%*,F7F%FitF%F[zF%F.F%F[rF%F1*,FUF%F]xF%FcyF%FbwF%F]qF %F1*,FOF%F]xF%F[zF%FbwF%F[rF%F1*,FfvF%F'F%F`vF%F\\tF%FbtF%F1*,FGF%F'F% FcyF%F\\tF%F]qF%F1*,FEF%FitF%F`vF%F\\tF%F_xF%F1*,FarF%FitF%FcyF%F\\tF% FdwF%F%**F]xF%FguF%FbqF%F0F%F1*,FcqF%F]xF%F`vF%FbqF%F_xF%F1*,FOF%F]xF% F[zF%F.F%F[rF%F%*,FGF%F'F%FcyF%FbwF%F]qF%F%*,FGF%F'F%F[zF%FbwF%F[rF%F% **\"#FF%FcyF%F\\tF%F]qF%F%*,FcqF%FitF%FcyF%FbwF%FdwF%F%*,FGF%FitF%F[zF %FbwF%F`tF%F1*,FEF%F]xF%F`vF%F\\tF%F_xF%F%*,FbvF%F]xF%FcyF%F\\tF%FdwF% F1**F'F%F`vF%FbqF%F_xF%F%*,FGF%F'F%F[zF%F.F%F[rF%F1*,FGF%FitF%F+F%F.F% FbtF%F%*,FEF%F]xF%FcyF%FbwF%FdwF%F1*,F3F%F]xF%F[zF%FbwF%F`tF%F%**F'F%F `vF%F\\tF%F_xF%F1*,F`zF%F'F%FcyF%F\\tF%FdwF%F%*(F$F%FitF%F]qF%F1*,FfqF %F]xF%F+F%F.F%FbtF%F1*,F_tF%F'F%FcyF%FbwF%FdwF%F%*,FiwF%F'F%F[zF%FbwF% F`tF%F1*(FcyF%F\\tF%FdwF%F%*(F]xF%FcyF%F\\tF%F1*(FGF%F]xF%F]qF%F%*,F3F %F'F%F+F%F.F%FbtF%F%**FfvF%F[zF%FbwF%F`tF%F%*,F_tF%F]xF%F[zF%FbwF%F0F% F%*(F'F%FcyF%F\\tF%F%*(FGF%F'F%F]qF%F1**FhoF%F+F%F.F%FbtF%F1**F]xF%F+F %F.F%F_xF%F1*,F]tF%F'F%F[zF%FbwF%F0F%F1*&F$F%F]qF%F%*,F_tF%F'F%F+F%F.F %F_xF%F%*(F[zF%FbwF%F0F%F%*(F+F%F.F%F_xF%F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 217 56 "We now specialize to z=z0 to get \+ an equation for F(x,y)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 " -factor(subs(z=z0,U=F,eqU))*11664;\n" }{MPLTEXT 1 0 14 "eq1:=op(1,%); \n" }{MPLTEXT 1 0 20 "eq2:=op(1,op(2,%%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,<**\"\"%\"\"\")I\"FG6\"\"\"$F&)I\"xGF)F%F&)I\"yGF)F%F& F&**\"\")F&F'F&F+F&)F.F*F&!\"\"**F%F&F'F&F+F&)F.\"\"#F&F&*.F0F&)F(F5F& )F,F5F&F4F&)F*#F&F5F&)F5#F5F*F&F&*,\"#CF&F7F&)F5#F&F*F&)F,F*F&F4F&F&*. F0F&F7F&F9F&F;F&F8F&F.F&F2*.F%F&F(F&F8F&F4F&F9F&F;F&F2*.F%F&F(F&F;F&F9 F&F8F&F.F&F&*,\"#[F&F(F&F9F&F,F&F.F&F2**F>F&F9F&F,F&F.F&F&*(F>F&F(F&F? F&F&*(\"#XF&F,F&F.F&F2*&F>F&F?F&F2F&),<*(F'F&F+F&F-F&F&**F5F&F'F&F+F&F 1F&F2*(F'F&F+F&F4F&F&*.F5F&F7F&F8F&F4F&F9F&F;F&F&*,\"\"'F&F7F&F?F&FAF& F4F&F&*.F5F&F7F&F9F&F;F&F8F&F.F&F2*,F(F&F8F&F4F&F9F&F;F&F2*,F(F&F;F&F9 F&F8F&F.F&F&*,\"#7F&F(F&F9F&F,F&F.F&F2**FSF&F9F&F,F&F.F&F&*(FSF&F(F&F? F&F&*(\"\"*F&F,F&F.F&F&*&FSF&F?F&F2F5F&" }}{PARA 11 "" 1 "" {XPPMATH 20 ",<**\"\"%\"\"\")I\"FG6\"\"\"$F%)I\"xGF(F$F%)I\"yGF(F$F%F%**\"\")F% F&F%F*F%)F-F)F%!\"\"**F$F%F&F%F*F%)F-\"\"#F%F%*.F/F%)F'F4F%)F+F4F%F3F% )F)#F%F4F%)F4#F4F)F%F%*,\"#CF%F6F%)F4#F%F)F%)F+F)F%F3F%F%*.F/F%F6F%F8F %F:F%F7F%F-F%F1*.F$F%F'F%F7F%F3F%F8F%F:F%F1*.F$F%F'F%F:F%F8F%F7F%F-F%F %*,\"#[F%F'F%F8F%F+F%F-F%F1**F=F%F8F%F+F%F-F%F%*(F=F%F'F%F>F%F%*(\"#XF %F+F%F-F%F1*&F=F%F>F%F1" }}{PARA 11 "" 1 "" {XPPMATH 20 ",<*()I\"FG6\" \"\"$\"\"\")I\"xGF&\"\"%F()I\"yGF&F+F(F(**\"\"#F(F$F(F)F()F-F'F(!\"\"* (F$F(F)F()F-F/F(F(*.F/F()F%F/F()F*F/F(F3F()F'#F(F/F()F/#F/F'F(F(*,\"\" 'F(F5F()F/#F(F'F()F*F'F(F3F(F(*.F/F(F5F(F7F(F9F(F6F(F-F(F1*,F%F(F6F(F3 F(F7F(F9F(F1*,F%F(F9F(F7F(F6F(F-F(F(*,\"#7F(F%F(F7F(F*F(F-F(F1**F " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 82 "The equation factorizes so we need to determine which fa ctor is an equation for F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "algeqtoseries(eq1,x,F,7,true);\n" }{MPLTEXT 1 0 9 "evalf(%);" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 30 "algeqtoseries(eq2,x,F,7,true);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7#+3I\"xG6\"\"\"\"\"\"!,$**#\"\"\"\"#;\"\"\"I\"yG6\"\"\" \",&*&\"\")\"\"\")\"\"$#\"\"\"\"\"#\"\"\"\"\"\"\"#:\"\"\"\"\"\")\"\"## \"\"#\"\"$\"\"\"\"\"\"\"\"\",$*,#\"\"\"\"#C\"\"\")\"\"##\"\"\"\"\"$\" \"\")\"\"$#\"\"\"\"\"#\"\"\"I\"yG6\"\"\"\",(*(\"#C\"\"\"I\"yG6\"\"\"\" )\"\"$#\"\"\"\"\"#\"\"\"\"\"\"*&\"#T\"\"\"I\"yG6\"\"\"\"\"\"\"\"\"%\" \"\"\"\"\"\"\"\"\"\"#,$*(#\"\"\"\"#;\"\"\")I\"yG6\"\"\"#\"\"\",**(\"# \")\"\"\"I\"yG6\"\"\"\")\"\"$#\"\"\"\"\"#\"\"\"\"\"\"*&\"#:\"\"\")\"\" $#\"\"\"\"\"#\"\"\"\"\"\"*&\"$S\"\"\"\"I\"yG6\"\"\"\"\"\"\"\"#C\"\"\" \"\"\"\"\"\"\"\"$,$**#\"\"\"\"$%Q\"\"\")I\"yG6\"\"\"#\"\"\",,*(\"%nB\" \"\")I\"yG6\"\"\"#\"\"\")\"\"$#\"\"\"\"\"#\"\"\"\"\"\"*(\"%&=\"\"\"\"I \"yG6\"\"\"\")\"\"$#\"\"\"\"\"#\"\"\"\"\"\"*&\"%'4%\"\"\")I\"yG6\"\"\" #\"\"\"\"\"\"*&\"%K?\"\"\"I\"yG6\"\"\"\"\"\"\"\"#k\"\"\"\"\"\")\"\"## \"\"#\"\"$\"\"\"\"\"\"\"\"%,$**#\"\"\"\"$%Q\"\"\")I\"yG6\"\"\"$\"\"\") \"\"##\"\"\"\"\"$\"\"\",.*&\"%Uj\"\"\")I\"yG6\"\"\"#\"\"\"\"\"\"*&\"&: 3\"\"\"\"I\"yG6\"\"\"\"\"\"\"*(\"%kO\"\"\")I\"yG6\"\"\"#\"\"\")\"\"$# \"\"\"\"\"#\"\"\"\"\"\"\"$+'\"\"\"*(\"%Si\"\"\"I\"yG6\"\"\"\")\"\"$#\" \"\"\"\"#\"\"\"\"\"\"*&\"$?$\"\"\")\"\"$#\"\"\"\"\"#\"\"\"\"\"\"\"\"\" \"\"\"\"\"&,$*(#\"\"\"\"%/B\"\"\")I\"yG6\"\"\"$\"\"\",0*(\"&+%G\"\"\") I\"yG6\"\"\"$\"\"\")\"\"$#\"\"\"\"\"#\"\"\"\"\"\"*(\"'_G;\"\"\")I\"yG6 \"\"\"#\"\"\")\"\"$#\"\"\"\"\"#\"\"\"!\"\"*&\"&.#\\\"\"\")I\"yG6\"\"\" $\"\"\"\"\"\"*(\"&S1#\"\"\"I\"yG6\"\"\"\")\"\"$#\"\"\"\"\"#\"\"\"!\"\" *&\"']@G\"\"\")I\"yG6\"\"\"#\"\"\"!\"\"*&\"&Xg$\"\"\"I\"yG6\"\"\"\"!\" \"*&\"$?$\"\"\")\"\"$#\"\"\"\"\"#\"\"\"!\"\"\"\"\"!\"\"\"\"'-I\"OG%*pr otectedG6#\"\"\"\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "7#+3I\"xG6\"$\" \"\"\"\"!\"\"!,$*&$\"+B\"=H'G!\"*\"\"\"I\"yG6\"\"\"\"\"\"\"\"\"\",$*($ \"+spp#4*!#6\"\"\"I\"yG6\"\"\"\",&*&$\"+R>#pD)!\")\"\"\"I\"yG6\"\"\"\" \"\"\"$\"\"%\"\"!\"\"\"\"\"\"\"\"\"\"\"#,$*($\"++++]i!#6\"\"\")I\"yG6 \"\"\"#\"\"\",&*&$\"+a6'H!G!\"(\"\"\"I\"yG6\"\"\"\"\"\"\"$\"+7i2)*\\! \")\"\"\"\"\"\"\"\"\"\"\"$,$*($\"+2p&Q8%!#7\"\"\")I\"yG6\"\"\"#\"\"\", (*&$\"+jUw&>)!\"'\"\"\")I\"yG6\"\"\"#\"\"\"\"\"\"*&$\"+2-[%3%!\"'\"\" \"I\"yG6\"\"\"\"\"\"\"$\"#k\"\"!\"\"\"\"\"\"\"\"\"\"\"%,$*($\"+,W/\"G$ !#7\"\"\")I\"yG6\"\"\"$\"\"\",(*&$\"+;M#)o7!\"&\"\"\")I\"yG6\"\"\"#\" \"\"\"\"\"*&$\"+/(*Hi@!\"&\"\"\"I\"yG6\"\"\"\"\"\"\"$\"+fiDa6!\"'\"\" \"\"\"\"\"\"\"\"\"&,$*($\"+yxFSV!#8\"\"\")I\"yG6\"\"\"$\"\"\",**&$\"+& HC$R)*!\"&\"\"\")I\"yG6\"\"\"$\"\"\"\"\"\"*&$\"+#Qz@k&!\"%\"\"\")I\"yG 6\"\"\"#\"\"\"!\"\"*&$\"+oGXzr!\"&\"\"\"I\"yG6\"\"\"\"!\"\"$\"+'eiDa&! \"(!\"\"\"\"\"!\"\"\"\"'-I\"OG%*protectedG6#\"\"\"\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "7#+3I\"xG6\"\"\"\"\"\"!,$**#\"\"\"\"\"%\"\"\"I\"yG6 \"\"\"\",&\"\"$!\"\"*&\"\"#\"\"\")\"\"$#\"\"\"\"\"#\"\"\"\"\"\"\"\"\") \"\"##\"\"#\"\"$\"\"\"\"\"\"\"\"\",$*,#\"\"\"\"\"'\"\"\")\"\"##\"\"\" \"\"$\"\"\")\"\"$#\"\"\"\"\"#\"\"\"I\"yG6\"\"\"\",(*(\"\"'\"\"\"I\"yG6 \"\"\"\")\"\"$#\"\"\"\"\"#\"\"\"\"\"\"*&\"#5\"\"\"I\"yG6\"\"\"\"!\"\" \"\"\"\"\"\"\"\"\"\"\"\"\"\"#,$*(#\"\"\"\"\"%\"\"\")I\"yG6\"\"\"#\"\" \",**(\"#F\"\"\"I\"yG6\"\"\"\")\"\"$#\"\"\"\"\"#\"\"\"\"\"\"*&\"\"$\" \"\")\"\"$#\"\"\"\"\"#\"\"\"!\"\"*&\"#Y\"\"\"I\"yG6\"\"\"\"!\"\"\"\"' \"\"\"\"\"\"\"\"\"\"\"$,$**#\"\"\"\"#C\"\"\")I\"yG6\"\"\"#\"\"\",,*(\" $L$\"\"\")I\"yG6\"\"\"#\"\"\")\"\"$#\"\"\"\"\"#\"\"\"\"\"\"*(\"#p\"\" \"I\"yG6\"\"\"\")\"\"$#\"\"\"\"\"#\"\"\"!\"\"*&\"$!e\"\"\")I\"yG6\"\" \"#\"\"\"!\"\"*&\"$;\"\"\"\"I\"yG6\"\"\"\"\"\"\"\"\"%!\"\"\"\"\")\"\"# #\"\"#\"\"$\"\"\"!\"\"\"\"%,$**#\"\"\"\"#C\"\"\")I\"yG6\"\"\"$\"\"\") \"\"##\"\"\"\"\"$\"\"\",.*(\"%W9\"\"\")I\"yG6\"\"\"#\"\"\")\"\"$#\"\" \"\"\"#\"\"\"\"\"\"*(\"$?%\"\"\"I\"yG6\"\"\"\")\"\"$#\"\"\"\"\"#\"\"\" !\"\"*&\"%'\\#\"\"\")I\"yG6\"\"\"#\"\"\"!\"\"*&\"#?\"\"\")\"\"$#\"\"\" \"\"#\"\"\"\"\"\"*&\"$N(\"\"\"I\"yG6\"\"\"\"\"\"\"\"#I!\"\"\"\"\"\"\" \"\"\"&,$*(#\"\"\"\"$W\"\"\"\")I\"yG6\"\"\"$\"\"\",0*(\"&o%Q\"\"\")I\" yG6\"\"\"$\"\"\")\"\"$#\"\"\"\"\"#\"\"\"\"\"\"*(\"&c[\"\"\"\")I\"yG6\" \"\"#\"\"\")\"\"$#\"\"\"\"\"#\"\"\"!\"\"*&\"&\"om\"\"\")I\"yG6\"\"\"$ \"\"\"!\"\"*(\"%S6\"\"\"I\"yG6\"\"\"\")\"\"$#\"\"\"\"\"#\"\"\"\"\"\"*& \"&]c#\"\"\")I\"yG6\"\"\"#\"\"\"\"\"\"*&\"%D?\"\"\"I\"yG6\"\"\"\"!\"\" *&\"#?\"\"\")\"\"$#\"\"\"\"\"#\"\"\"!\"\"\"\"\"!\"\"\"\"'-I\"OG%*prote ctedG6#\"\"\"\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "7#+3I\"xG6\"$\"\" \"\"\"!\"\"!,$*&$\"+%[)yT=!#5\"\"\"I\"yG6\"\"\"\"\"\"\"\"\"\",$*($\"+* yyqj$!#5\"\"\"I\"yG6\"\"\"\",&*&$\")&[I#R!\")\"\"\"I\"yG6\"\"\"\"\"\" \"$\"\"\"\"\"!\"\"\"\"\"\"\"\"\"\"\"#,$*($\"+++++D!#5\"\"\")I\"yG6\"\" \"#\"\"\",&*&$\")#=Pl(!\")\"\"\"I\"yG6\"\"\"\"\"\"\"$\"*wv%Q!)!\"*\"\" \"\"\"\"\"\"\"\"\"$,$*($\"+^5<9m!#6\"\"\")I\"yG6\"\"\"#\"\"\",(*&$\")4 3FK!\"(\"\"\")I\"yG6\"\"\"#\"\"\"!\"\"*&$\")e]6N!\"(\"\"\"I\"yG6\"\"\" \"!\"\"$\"\"%\"\"!!\"\"\"\"\"!\"\"\"\"%,$*($\"+U5n\\_!#6\"\"\")I\"yG6 \"\"\"$\"\"\",(*&$\"(n83&!\"'\"\"\")I\"yG6\"\"\"#\"\"\"\"\"\"*&$\")1mQ v!\"(\"\"\"I\"yG6\"\"\"\"\"\"\"$\"*;;5k%!\")\"\"\"\"\"\"\"\"\"\"\"&,$* ($\"+WWWWp!#7\"\"\")I\"yG6\"\"\"$\"\"\",**&$\"(_pC&!\"&\"\"\")I\"yG6\" \"\"$\"\"\"!\"\"*&$\"(!oM\")!\"&\"\"\")I\"yG6\"\"\"#\"\"\"!\"\"*&$\")z ?Y]!\"'\"\"\"I\"yG6\"\"\"\"!\"\"$\"+;;5kM!\")!\"\"\"\"\"!\"\"\"\"'-I\" OG%*protectedG6#\"\"\"\"\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 89 "Some coeffs of gf(eq1) ar e negative so eq2 is the correct equation. We call it eqF below " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 233 "eqF:=F^3*x^4*y^4-2*F^3*x^4* y^3+F^3*x^4*y^2+2*F^2*x^2*y^2*sqrt(3)*2^(2/3)+6*F^2*2^(1/3)*x^3*y^2-2* F^2*sqrt(3)*2^(2/3)*x^2*y-F*x^2*y^2*sqrt(3)*2^(2/3)+F*2^(2/3)*sqrt(3)* x^2*y-12*F*sqrt(3)*x*y+6*sqrt(3)*x*y+6*2^(1/3)*F+9*x*y-6*2^(1/3);\n" } }{PARA 11 "" 1 "" {XPPMATH 20 ",<*()I\"FG6\"\"\"$\"\"\")I\"xGF&\"\"%F( )I\"yGF&F+F(F(**\"\"#F(F$F(F)F()F-F'F(!\"\"*(F$F(F)F()F-F/F(F(*.F/F()F %F/F()F*F/F(F3F()F'#F(F/F()F/#F/F'F(F(*,\"\"'F(F5F()F/#F(F'F()F*F'F(F3 F(F(*.F/F(F5F(F7F(F9F(F6F(F-F(F1*,F%F(F6F(F3F(F7F(F9F(F1*,F%F(F9F(F7F( F6F(F-F(F(*,\"#7F(F%F(F7F(F*F(F-F(F1**F " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4072 "eqU := -64*U^9*x^12*y^12*z ^12+384*U^9*x^12*y^11*z^12-960*U^9*x^12*y^10*z^12+1280*U^9*x^12*y^9*z^ 12-960*U^9*x^12*y^8*z^12+384*U^9*x^12*y^7*z^12-64*U^9*x^12*y^6*z^12-19 2*U^8*x^11*y^10*z^10-384*U^8*x^10*y^10*z^11+768*U^8*x^11*y^9*z^10+1920 *U^8*x^10*y^9*z^11+192*U^7*x^10*y^10*z^11-1152*U^8*x^11*y^8*z^10-3840* U^8*x^10*y^8*z^11-960*U^7*x^10*y^9*z^11+768*U^8*x^11*y^7*z^10+3840*U^8 *x^10*y^7*z^11+1920*U^7*x^10*y^8*z^11-192*U^8*x^11*y^6*z^10-1920*U^8*x ^10*y^6*z^11-1920*U^7*x^10*y^7*z^11+384*U^8*x^10*y^5*z^11+960*U^7*x^10 *y^6*z^11+192*U^7*x^9*y^9*z^9-192*U^7*x^10*y^8*z^8-192*U^7*x^10*y^5*z^ 11-1536*U^7*x^9*y^8*z^9-960*U^7*x^8*y^8*z^10-96*U^6*x^9*y^9*z^9+384*U^ 7*x^10*y^7*z^8+3456*U^7*x^9*y^7*z^9+3840*U^7*x^8*y^7*z^10+768*U^6*x^9* y^8*z^9+960*U^6*x^8*y^8*z^10-192*U^7*x^10*y^6*z^8-3072*U^7*x^9*y^6*z^9 -5760*U^7*x^8*y^6*z^10-1728*U^6*x^9*y^7*z^9-3840*U^6*x^8*y^7*z^10-192* U^5*x^8*y^8*z^10+960*U^7*x^9*y^5*z^9+3840*U^7*x^8*y^5*z^10-U^6*x^9*y^9 *z^6+1536*U^6*x^9*y^6*z^9+5760*U^6*x^8*y^6*z^10+768*U^5*x^8*y^7*z^10-9 60*U^7*x^8*y^4*z^10+4*U^6*x^9*y^8*z^6-480*U^6*x^9*y^5*z^9-3840*U^6*x^8 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F%FbtF%F1*,F_tF%F'F%FcyF%FbwF%FdwF%F%*,FiwF%F'F%F[zF%FbwF%F`tF%F1*(Fcy F%F\\tF%FdwF%F%*(F]xF%FcyF%F\\tF%F1*(FGF%F]xF%F]qF%F%*,F3F%F'F%F+F%F.F %FbtF%F%**FfvF%F[zF%FbwF%F`tF%F%*,F_tF%F]xF%F[zF%FbwF%F0F%F%*(F'F%FcyF %F\\tF%F%*(FGF%F'F%F]qF%F1**FhoF%F+F%F.F%FbtF%F1**F]xF%F+F%F.F%F_xF%F1 *,F]tF%F'F%F[zF%FbwF%F0F%F1*&F$F%F]qF%F%*,F_tF%F'F%F+F%F.F%F_xF%F%*(F[ zF%FbwF%F0F%F%*(F+F%F.F%F_xF%F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 232 "eqF:=F^3*x^4*y^4-2*F^3*x^4*y^3+F^3*x^4*y^2+2*F^2*x^2*y^2*sqrt (3)*2^(2/3)+6*F^2*2^(1/3)*x^3*y^2-2*F^2*sqrt(3)*2^(2/3)*x^2*y-F*x^2*y^ 2*sqrt(3)*2^(2/3)+F*2^(2/3)*sqrt(3)*x^2*y-12*F*sqrt(3)*x*y+6*sqrt(3)*x *y+6*2^(1/3)*F+9*x*y-6*2^(1/3);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ",<*()I\"FG6\"\"\"$\"\"\")I\"xGF&\"\"%F()I\"yGF&F+F(F(** \"\"#F(F$F(F)F()F-F'F(!\"\"*(F$F(F)F()F-F/F(F(*.F/F()F%F/F()F*F/F(F3F( )F'#F(F/F()F/#F/F'F(F(*,\"\"'F(F5F()F/#F(F'F()F*F'F(F3F(F(*.F/F(F5F(F7 F(F9F(F6F(F-F(F1*,F%F(F6F(F3F(F7F(F9F(F1*,F%F(F9F(F7F(F6F(F-F(F(*,\"#7 F(F%F(F7F(F*F(F-F(F1**F " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 221 69 "Assymptotic of [x^n]F=[x^n]F(x,y,z0) (-> asymptotic coef fs of oceans)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 89 "We now need to do the singularity analysis to get the asymptot ic of the coefficients of F" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "DiscF:=factor(result ant(eqF,diff(eqF,F),F))/(-3*(y-1)^2*y^6*x^9/50531*(1701+956*sqrt(3))); \n" }{MPLTEXT 1 0 32 "DF1:=collect(op(1,%),x,factor);\n" }{MPLTEXT 1 0 11 "#latex(%);\n" }{MPLTEXT 1 0 39 "DF2:=collect(op(1,op(2,%%)),x,fa ctor);\n" }{MPLTEXT 1 0 11 "#latex(%);\n" }{MPLTEXT 1 0 12 "DF:=DF1*DF 2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,<**\"\")\"\"\"I\"yG6\"F&)\"\"## F*\"\"$F&)F,#F&F*F&F&*(\"#OF&F)F&F'F&F&*(F%F&F)F&F-F&!\"\"*&F0F&F)F&F2 *,\"#IF&F-F&)F*#F&F,F&I\"xGF(F&)F'F*F&F2**\"#sF&F6F&F8F&F9F&F&*,\"#[F& F-F&F6F&F8F&F'F&F&**\"#gF&F6F&F8F&F'F&F2*(\"#BF&)F8F*F&)F'F,F&F&**\"\" 'F&F-F&FBF&F9F&F2*(\"#tF&FBF&F9F&F2**F=F&F-F&FBF&F'F&F&*(\"#KF&FBF&F'F &F&F&),,*(\"#5F&F6F&F-F&F&*(F*F&F-F&F8F&F&*(\"#8F&F8F&F'F&F2*&\"#7F&F6 F&F2*&\"\"&F&F8F&F2F,F&" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*(I\"yG6\" \"\"\",,*(\"\"'F&F$F&)\"\"$#F&\"\"#F&F&*&\"#BF&)F$F-F&!\"\"*&\"#[F&F*F &F1*&\"#tF&F$F&F&\"#KF1F&)I\"xGF%F-F&F1*.F-F&)F-#F&F+F&,&*&\"\"&F&F*F& F&\"#7F1F&F$F&,&*&F-F&F*F&F&*&F+F&F$F&F&F&F8F&F1**\"\"%F&)F-#F-F+F&,&F AF&\"\"*F&F&,&F$F&F&F1F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*&,(*&\" \"#\"\"\")\"\"$#F'F&F'F'*&\"#8F'I\"yG6\"F'!\"\"\"\"&F/F'I\"xGF.F'F'*( \"#5F')F&#F'F)F'F(F'F'*&\"#7F'F4F'F/" }}{PARA 11 "" 1 "" {XPPMATH 20 " *&,(*(I\"yG6\"\"\"\",,*(\"\"'F'F%F')\"\"$#F'\"\"#F'F'*&\"#BF')F%F.F'! \"\"*&\"#[F'F+F'F2*&\"#tF'F%F'F'\"#KF2F')I\"xGF&F.F'F2*.F.F')F.#F'F,F' ,&*&\"\"&F'F+F'F'\"#7F2F'F%F',&*&F.F'F+F'F'*&F,F'F%F'F'F'F9F'F2**\"\"% F')F.#F.F,F',&FBF'\"\"*F'F',&F%F'F'F2F'F'F',(*&,(FBF'*&\"#8F'F%F'F2F?F 2F'F9F'F'*(\"#5F'F;F'F+F'F'*&F@F'F;F'F2F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "solve(resultant(DF1,DF2,x),y);evalf(%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%,&#\"\"\"\"#6!\"\"*&#\"\"#F&F%)\"\"$#F%F*F%F %F#F#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$\"+yB4SA!#5F#F#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 121 "As we can see these equations (for the \+ singularity of F) are the same as the equation for the growth rate of \+ the weights." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "Dr1;\n" } {MPLTEXT 1 0 4 "Dr2;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*(I\"pG6\"\"\" \",,*(\"\"'F&)\"\"$#F&\"\"#F&F$F&F&*&\"#BF&)F$F-F&!\"\"*&\"#[F&F*F&F1* &\"#tF&F$F&F&\"#KF1F&)I\"xGF%F-F&F1*.F-F&)F-#F&F+F&,&*&\"\"&F&F*F&F&\" #7F1F&F$F&,&*&F+F&F$F&F&*&F-F&F*F&F&F&F8F&F1**\"\"%F&)F-#F-F+F&,&FBF& \"\"*F&F&,&F$F&F&F1F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*&,(*&\"\"# \"\"\")\"\"$#F'F&F'!\"\"*&\"#8F'I\"pG6\"F'F'\"\"&F'F'I\"xGF/F'F'*(\"#5 F')F&#F'F)F'F(F'F+*&\"#7F'F4F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "subs(y=p,DF1)-Dr1;\n" }{MPLTEXT 1 0 26 "factor(subs(y=p,DF2)+ Dr2);" }}{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 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "sF:=solve(DF2,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 ", $**\"\"#\"\"\",&*&\"\"&F%)\"\"$#F%F$F%F%\"\"'!\"\"F%)F$#F%F*F%,(*&F$F% F)F%F%*&\"#8F%I\"yG6\"F%F-F(F-F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 221 41 "Case y=1. T he radius of convergence is sF" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "yspe:=1;\n" }{MPLTEXT 1 0 45 "subspe:=A->factor(simplify(subs(y=y spe,A))):\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "eqFspe:=subspe(eqF);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*(#\"\"$\"\"#\"\"\")F&#F'F%F',.*(\"\"%F')I\"FG6\"F&F') I\"xGF/F%F'F'*,F,F'F.F')F%#F'F&F')F&#F&F%F'F1F'!\"\"**F&F'F3F'F5F'F1F' F'*(F%F'F5F'F1F'F'*&F,F'F.F'F'F,F7F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "#discy1:=factor(discrim(eqGspe,G));\n" }{MPLTEXT 1 0 23 "discyF:=subspe(DiscF);\n" }{MPLTEXT 1 0 19 "\{solve(discyF,x)\};\n " }{MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*,\"#O \"\"\",&\"$6#!\"\"*&\"$y\"F%)\"\"$#F%\"\"#F%F%F%,(*&)F.#F%F,F%F+F%F%*& F,F%F1F%F%*&\"\"'F%I\"xG6\"F%F%F%F6F%),(F0F%*$F1F%F(*&F.F%F6F%F(F,F%F% " }}{PARA 11 "" 1 "" {XPPMATH 20 "<%\"\"!,&*(#\"\"\"\"\"'F')\"\"##F'\" \"$F')F,#F'F*F'!\"\"*&F.F'F)F'F/,&*(F.F'F)F'F-F'F'F0F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%$!+RSoO**!#5$\"\"!F'$\"+56j6YF%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "subspe(DF1);solve(%,x);evalf(%);\n" } {MPLTEXT 1 0 32 "subspe(DF2);solve(%,x);evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "*(,&*&\"\"(\"\"\")\"\"$#F&\"\"#F&F&F(!\"\"F&I\"xG6\"F&,(* &)F*#F&F(F&F'F&F&*&F(F&F0F&F&*&\"\"'F&F,F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"!,&*(#\"\"\"\"\"'F')\"\"##F'\"\"$F')F,#F'F*F'!\"\"* &F.F'F)F'F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"\"!F$$!+RSoO**!#5" } }{PARA 11 "" 1 "" {XPPMATH 20 ",$*&,&*$)\"\"$#\"\"\"\"\"#F)F)\"\"*!\" \"F),(*&)F*#F)F'F)F&F)F)*$F/F)F,*&F*F)I\"xG6\"F)F,F)F," }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*(#\"\"\"\"\"#F%)F&#F%\"\"$F%)F)F$F%F%*&F$F%F'F%! \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+56j6Y!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "subspe(sF); evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*(#\"\"\"\"\"#F%)F&#F%\"\"$F%)F)F$F%F%*&F$F%F'F%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+56j6Y!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "sFspe:=(1/2)*sqrt(3)*2^(1/3)-(1/2)*2^(1/3);evalf( %);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*(#\"\"\"\"\"#F%)F&#F%\"\"$F%)F )F$F%F%*&F$F%F'F%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+56j6Y!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 41 "So the singularity is 2^(-2/3) =0.46.. =sF" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 47 "The singularity analysis below is non-necessary" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "sF;\n" }{MPLTEXT 1 0 23 "simplify(subs(y=1,%));\n" } {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$**\"\"#\" \"\",&*&\"\"&F%)\"\"$#F%F$F%F%\"\"'!\"\"F%)F$#F%F*F%,(*&F$F%F)F%F%*&\" #8F%I\"yG6\"F%F-F(F-F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*(,&*&\"\" &\"\"\")\"\"$#F'\"\"#F'F'\"\"'!\"\"F')F+#F'F)F',&*$F(F'F'\"\"*F-F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+96j6Y!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "factor(subs(x=sFspe,eqFspe));\n" }{MPLTEXT 1 0 27 "FsFspe:=op(1,\{solve(%,F)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$* *#\"\"$\"\"%\"\"\")\"\"##F'F%F',&*&F%F')F%#F'F)F'F'\"\"&!\"\"F'),(*&F) F'I\"FG6\"F'F'*$F-F'F0F'F0F)F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*& #\"\"\"\"\"#F%)\"\"$F$F%F%F$F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "numer(factor(subs(x=sFspe-xx,F=FsFspe-FF,eqFspe)))/(3*2^(1/3)) ;\n" }{MPLTEXT 1 0 22 "puiseux(%,xx=0,FF,2);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ",B*&\"\"%\"\"\")I#xxG6\"\"\"$F%!\"\"*,\"\"#F%)F)#F%F,F%I# FFGF(F%)F,#F,F)F%F'F%F%*,\"\"'F%)F/F,F%F0F%F-F%F'F%F%*,F3F%F4F%)F,#F%F )F%F-F%)F'F,F%F%*(F)F%F6F%F8F%F%*(F$F%F/F%F&F%F%*(F$F%F4F%F&F%F**(F,F% F-F%F&F%F**(F3F%F4F%F-F%F%*&\"#5F%F4F%F***F)F%F6F%F-F%F8F%F%**F$F%F-F% F/F%F&F%F%**F3F%F/F%F0F%F'F%F***\"#7F%F/F%F6F%F8F%F***FDF%F4F%F0F%F'F% F***F3F%F4F%F6F%F8F%F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "<#,&*&,&\"&o_( \"\"\"*&\"&cM%\"\"\")\"\"$#\"\"\"\"\"#\"\"\"\"\"\"\"\"\")*&I#xxG6\"\" \"\",&\"$%>\"\"\"*&\"$7\"\"\"\")\"\"$#\"\"\"\"\"#\"\"\"\"\"\"!\"\"#\" \"$\"\"#\"\"\"\"\"\"*(,&*&\"$F'\"\"\")\"\"##\"\"#\"\"$\"\"\"\"\"\"*(\" $i$\"\"\")\"\"##\"\"#\"\"$\"\"\")\"\"$#\"\"\"\"\"#\"\"\"\"\"\"\"\"\"I# xxG6\"\"\"\",&\"$%>\"\"\"*&\"$7\"\"\"\")\"\"$#\"\"\"\"\"#\"\"\"\"\"\"! \"\"\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 217 122 "We first study the singularity a t sigma. At that point, G has value Gsigma (since two branches have th e same value Gsigma)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "fac tor(subs(x=sF,eqF));\n" }{MPLTEXT 1 0 14 "[solve(%,F)];\n" }{MPLTEXT 1 0 13 "Fs:=op(1,%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*.\"\"#\"\" \")F$#F%\"\"$F%,&\"%pD!\"\"*&\"%![\"F%)F(#F%F$F%F%F%,2*&I\"FG6\"F%)I\" yGF3F$F%F%*(\"\"(F%F4F%F.F%F%*(F$F%F2F%F5F%F+*&F5F%F.F%F+*&\"#6F%F4F%F %F2F%*&\"\"%F%F5F%F+F%F+F%),**(\"#7F%F2F%F5F%F%*(\"\"&F%F5F%F.F%F+*$F. F%F+*&\"\"'F%F5F%F+F$F%),(*&F$F%F.F%F%*&\"#8F%F5F%F+FCF+F=F+F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "7%,$*(#\"\"\"\"#7F&,(*(\"\"&F&I\"yG6\"F& )\"\"$#F&\"\"#F&F&*$F-F&F&*&\"\"'F&F+F&F&F&F+!\"\"F&F#,$*&,,*(\"\"(F&) F+F0F&F-F&F&*&F+F&F-F&F4*&\"#6F&F:F&F&*&\"\"%F&F+F&F4F&F4F&,(*$F:F&F&* &F0F&F+F&F4F&F&F4F4" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*(#\"\"\"\"#7F% ,(*(\"\"&F%I\"yG6\"F%)\"\"$#F%\"\"#F%F%*$F,F%F%*&\"\"'F%F*F%F%F%F*!\" \"F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "numer(factor(subs(x =sF-xx,F=Fs-FF,eqF))):\n" }{MPLTEXT 1 0 34 "eqFFs:=collect(%,\{xx,FF\} ,factor);\n" }{MPLTEXT 1 0 22 "puiseux(%,xx=0,FF,2);\n" }{MPLTEXT 1 0 36 "#singGsigma:=map(mafactor,op(1,%));\n" }{MPLTEXT 1 0 24 "#map(mafa ctor,op(2,%%));" }}{PARA 11 "" 1 "" {XPPMATH 20 ",.*&,,*,\"$w&\"\"\"), &I\"yG6\"F'F'!\"\"\"\"#F')F*\"\"$F'),(*&F-F')F/#F'F-F'F'*&\"#8F'F*F'F, \"\"&F,\"\"%F')I#xxGF+F8F'F,*0\"%3YF')F-#F'F/F',&*&F7F'F3F'F'\"\"'F,F' F(F')F1F/F'F.F')F:F/F'F,*0\"&s9%F')F-#F-F/F',&\"#PF,*&\"#?F'F3F'F'F'F( F')F1F-F'F.F')F:F-F'F'*.\"'#f5\"F',&\"$A&F,*&\"$0$F'F3F'F'F'F1F'F(F'F. F'F:F'F,*,\"'))e;F'F=F',&\"%pDF,*&\"%![\"F'F3F'F'F'F(F'F.F'F'F')I#FFGF +F/F'F'*&,,*.#\"$W\"F6F',&F@F'FAF'F')F*F-F'F(F')F1F7F'F9F'F,*.\"%cMF'F =F',(*$F\\oF'F'F*F,F'F'F'F\\oF'F0F'FCF'F,*0#\"%_6\"#BF'FFF',&\"#FF,*& \"#HF'F3F'F'F',,*(F7F'F*F'F3F'F'*&\"#YF'F\\oF'F,*&F8F'F3F'F'*&\"#6F'F* F'F'\"#GF,F'F\\oF'FBF'FMF'F'*.#\"&[w#F6F',&\"#))F,*&\"#VF'F3F'F'F',,*( \"\"(F'F*F'F3F'F'*&\"#EF'F\\oF'F,F2F'*&F-F'F*F'F,\"\")F,F'FLF'F\\oF'F: F'F,*.#\"&CQ\"\"$@\"F'F=F',&\"%EJF,*&\"%x=F'F3F'F'F'F1F'),(F_pF,F'F,F2 F'F-F'F\\oF'F,F')FenF-F'F'*&,**.#\"#O\"$p\"F',&FJF'FIF'F'F*F'F(F')F1FA F'F9F'F,*0#\"$)GF6F'F=F'F[oF',&FaoF'F'F'F'F*F'F]oF'FCF'F,*0#Fer\"$\\$F 'FFF',&\"#>F'F2F'F',,*(\"#eF'F*F'F3F'F'*&FirF'F\\oF'F,*&\"#9F'F3F'F'*& \"$-#F'F*F'F,\"$L\"F,F'F*F'F0F'FMF'F'*0#\"%/BF`pF',&*&F[qF'F3F'F'\"#CF ,F',&F^qF'F'F'F'FiqF'F*F'FBF'F:F'F'F'FenF'F'*,#F'\"%(>#F',&FQF'FRF'F'F (F')F1F[qF'F9F'F,*.#FjsF`rF'F=F'FarF',(FaoF'F*F'F'F'F'FbrF'FCF'F,*.#\" #[F6F'FFF'F[oF')F[tF-F'F]oF'FMF'F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 95 "Annoying: Ma ple does not want to do puiseux. I don't know why. We will resort to N ewton polygon" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "ldegree(co eff(eqFFs,xx,0),FF);\n" }{MPLTEXT 1 0 31 "coeff(coeff(eqFFs,xx,0),FF,2 );\n" }{MPLTEXT 1 0 19 "evalf(solve(%,y));\n" }{MPLTEXT 1 0 31 "ldegre e(coeff(eqFFs,xx,1),FF);\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 31 "lde gree(coeff(eqFFs,xx,2),FF);\n" }{MPLTEXT 1 0 31 "coeff(coeff(eqFFs,xx, 2),FF,0);\n" }{MPLTEXT 1 0 18 "evalf(solve(%,y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*.#\"&CQ\"\"$@\" \"\"\")\"\"##F'\"\"$F',&\"%EJ!\"\"*&\"%x=F')F+#F'F)F'F'F',(*&F)F'F1F'F '*&\"#8F'I\"yG6\"F'F.\"\"&F.F'),(*&\"#6F'F7F'F.F'F.F4F'F)F')F7F)F'F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6'$!+'Hg9=\"!#5$\"\"!F'F&$\"+yB4SAF%F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*,#\"#[\"#8\"\"\")\"\"##F )\"\"$F',&*&\"\"&F')F+#F'F)F'F'\"\"'F'F'),&*&F)F'I\"yG6\"F'F'F'F'F)F') ,(*&F)F'F/F'F'*&F&F'F5F'!\"\"F.F;F.F'F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)$!+++++]!#5F#$!+'Hg9=\"F%F&F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "collect(subs(FF=a*xx,eqFFs),xx,factor);\n" }{MPLTEXT 1 0 15 "coeff(%,xx,2);\n" }{MPLTEXT 1 0 14 "\{solve(%,a)\};\n" } {MPLTEXT 1 0 12 "c1:=op(1,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",.*.\"$ w&\"\"\"),&I\"yG6\"F%F%!\"\"\"\"#F%)F(\"\"$F%),(*&F+F%)F-#F%F+F%F%*&\" #8F%F(F%F*\"\"&F*\"\"%F%)I\"aGF)F-F%)I#xxGF)\"\"(F%F**4\"#sF%)F+#F%F-F %,&*&F5F%F1F%F%\"\"'F*F%,.**\"#?F%)F(F+F%F1F%)F+#F+F-F%F%**F6F%F(F%F1F %FGF%F%*(\"#PF%FGF%FFF%F%*(\"#5F%F(F%FGF%F%*$FGF%F%*(\"#kF%F8F%F(F%F%F %)F8F+F%FFF%F&F%)F/F-F%)F:FBF%F**2\"#=F%FGF%,&FKF**&FEF%F1F%F%F%,T**\" %_NF%FGF%F8F%)F(F5F%F%**\"%#f#F%FGF%F8F%)F(F6F%F***\"%)o#F%FGF%F8F%F,F %F%**\"$k)F%FGF%F8F%FFF%F%*,\"%?>F%FGF%F8F%FenF%F1F%F%*,\"%O:F%F1F%FGF %F8F%FhnF%F**,F`oF%F1F%FGF%F8F%F,F%F%*,\"$%QF%F1F%FGF%F8F%FFF%F%**\"#' *F%F(F%FGF%F8F%F%**\"%![\"F%F1F%F>F%)F(FBF%F%*(\"%=RF%F>F%FenF%F**(\"$ ^$F%F>F%FhnF%F%**\"%kAF%F1F%F>F%FenF%F**(\"%pDF%F>F%FhoF%F%*(\"$'zF%F> F%F,F%F%*(\"$$=F%F>F%FFF%F%*(\"%/BF%FQF%FhnF%F%*(\"%3YF%FQF%F,F%F***\" $/\"F%F1F%F>F%FFF%F%**\"\")F%F(F%F1F%F>F%F%*(FUF%F>F%F(F%F%*(FfpF%FFF% FQF%F%**\"$3#F%F1F%F>F%FhnF%F%**\"$k%F%F,F%F1F%F>F%F%*$F>F%F%F%F8F%F(F %)F/F+F%)F:F5F%F%**,&\"$A&F**&\"$0$F%F1F%F%F%,`pF%F%*,\"'_^9F%F1F%FGF% FQF%FhnF%F%*,\"&gX$F%F1F%FGF%FQF%F,F%F%*,\"'_F?F%F1F%F>F%F8F%FhnF%F%*, F_rF%F1F%F>F%F8F%F,F%F%*,FfpF%F1F%F>F%F8F%FFF%F%*,\"'SiUF%F1F%F>F%F8F% )F(F;F%F%*,\"'[/?F%F1F%F>F%F8F%FhoF%F%*,\"'+3YF%F1F%F>F%F8F%FenF%F%** \"'s)R(F%F>F%F8F%FfrF%F%*(\"&;@#F%F1F%FhoF%F**(\"&3V$F%F1F%FenF%F%*(\" &Sb\"F%F1F%FhnF%F%**\"'w\\tF%FGF%FQF%FhoF%F%**\"'3)>\"F%FGF%FQF%FenF%F ***\"'g8NF%F>F%F8F%FhoF%F%*(\"'Sh5F%F1F%)F(F\\qF%F%*(\"'#f5\"F%F,F%F7F %F%*,\"'3ySF%F1F%FGF%FQF%FhoF%F%*,\"&c)*)F%F1F%FGF%FQF%FenF%F**(F]tF%F 7F%FenF%F%*(\"'%=@#F%F7F%FhnF%F**(\"#7F%F1F%F(F%F%*(\"%#*GF%F,F%F1F%F% **\"'#R$GF%FGF%FQF%FhnF%F%**\"&_v)F%FGF%FQF%F,F%F%**\"'3Q!)F%F>F%F8F%F enF%F%**\"%;#*F%FGF%FQF%FFF%F%**\"'?rNF%F>F%F8F%FhnF%F%**\"&CU'F%F>F%F 8F%F,F%F%**\"%gdF%F>F%F8F%FFF%F%*(\"$w#F%F1F%FFF%F%*&\"'`Q=F%F[tF%F%** \"$)GF%F>F%F8F%F(F%F%*(\"'_q8F%F1F%FfrF%F**&\"#GF%F(F%F%*&\"&!QfF%FenF %F%*&\"&7$QF%FhoF%F**&\"'ctBF%FfrF%F**&\"%?]F%F,F%F%*&\"&!*o#F%FhnF%F% *&\"$'\\F%FFF%F%F%F/F%)F:F6F%F**,\"#CF%F>F%,&*&FgoF%F1F%F%F`pF*F%,`pF% F%*,\"&o.\"F%F1F%FGF%FQF%FhnF%F%*,\"%cMF%F1F%FGF%FQF%F,F%F%*,\"&S)pF%F 1F%F>F%F8F%FhnF%F%*,\"&%e5F%F1F%F>F%F8F%F,F%F%*,\"$['F%F1F%F>F%F8F%FFF %F%*,\"''4*RF%F1F%F>F%F8F%FfrF%F%*,\"'7RIF%F1F%F>F%F8F%FhoF%F%*,\"'[7@ F%F1F%F>F%F8F%FenF%F%**\"'#H#pF%F>F%F8F%FfrF%F%*(\"'oN?F%F1F%FhoF%F%*( \"'345F%F1F%FenF%F%*(\"&!)f#F%F1F%FhnF%F%**\"';'R#F%FGF%FQF%FhoF%F%** \"&G0$F%FGF%FQF%FenF%F***\"'#**H&F%F>F%F8F%FhoF%F%FisF%*(\"%7pF%F,F%F7 F%F%*,\"'%yM\"F%F1F%FGF%FQF%FhoF%F%*,\"&#>CF%F1F%FGF%FQF%FenF%F**(FbyF %F7F%FenF%F%*(\"&CQ\"F%F7F%FhnF%F*FetF%*(\"%#z$F%F,F%F1F%F%**\"&[w#F%F GF%FQF%FhnF%F%**\"&W4\"F%FGF%FQF%F,F%F%**\"'S8PF%F>F%F8F%FenF%F%**\"%_ 6F%FGF%FQF%FFF%F%**\"';c7F%F>F%F8F%FhnF%F%**\"&+2#F%F>F%F8F%F,F%F%**\" %+=F%F>F%F8F%FFF%F%*(\"$7$F%F1F%FFF%F%FiuF%**\"#%)F%F>F%F8F%F(F%F%*(\" 'o8=F%F1F%FfrF%F%*&\"#JF%F(F%F%*&\"'&)\\XF%FhnF%F%*&\"$'eF%FFF%F%F%)F:F- F%F%*.#F_w\"$@\"F%F>F%,&\"%EJF**&\"%x=F%F1F%F%F%F/F%),8**\"$3\"F%F1F%F GF%F,F%F%**\"#')F%FFF%F1F%FGF%F%*(\"$m\"F%FGF%F,F%F%**FEF%F(F%F1F%FGF% F%*(\"$6\"F%FGF%FFF%F%*(F+F%FGF%F1F%F%*(FftF%F(F%FGF%F%**\"#[F%F1F%F8F %F(F%F%*(\"$k#F%F8F%FFF%F*FNF**(F_wF%F8F%F(F%F*F+F%)F:F+F%F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*,#\"#C\"$@\"\"\"\")\"\"##F'\"\"$F',&\"%EJ! \"\"*&\"%x=F')F+#F'F)F'F'F',(*&F)F'F1F'F'*&\"#8F'I\"yG6\"F'F.\"\"&F.F' ),8**\"$3\"F'F1F')F)#F)F+F')F7F+F'F'**\"#')F')F7F)F'F1F'F>F'F'*(\"$m\" F'F>F'F@F'F'**\"#?F'F7F'F1F'F>F'F'*(\"$6\"F'F>F'FCF'F'*(F)F'F>F'F1F'F' *(\"#7F'F7F'F>F'F'**\"#[F'F1F'I\"aGF8F'F7F'F'*(\"$k#F'FOF'FCF'F.*$F>F' F.*(F%F'FOF'F7F'F.F)F'F." }}{PARA 11 "" 1 "" {XPPMATH 20 "<#,$*,#\"\" \"\"#C\"\"\")\"\"##\"\"#\"\"$\"\"\",2*(\"$3\"\"\"\")I\"yG6\"\"\"$\"\" \")\"\"$#\"\"\"\"\"#\"\"\"\"\"\"*(\"#')\"\"\")\"\"$#\"\"\"\"\"#\"\"\") I\"yG6\"\"\"#\"\"\"\"\"\"*&\"$m\"\"\"\")I\"yG6\"\"\"$\"\"\"\"\"\"*(\"# ?\"\"\")\"\"$#\"\"\"\"\"#\"\"\"I\"yG6\"\"\"\"\"\"\"*&\"$6\"\"\"\")I\"y G6\"\"\"#\"\"\"\"\"\"*&\"\"#\"\"\")\"\"$#\"\"\"\"\"#\"\"\"\"\"\"*&\"#7 \"\"\"I\"yG6\"\"\"\"\"\"\"\"\"\"!\"\"\"\"\"I\"yG6\"!\"\",(*&\"#6\"\"\" I\"yG6\"\"\"\"!\"\"\"\"\"!\"\"*&\"\"#\"\"\")\"\"$#\"\"\"\"\"#\"\"\"\" \"\"!\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*,#\"\"\"\"#CF%)\"\" ##F(\"\"$F%,2*(\"$3\"F%)I\"yG6\"F*F%)F*#F%F(F%F%*(\"#')F%F1F%)F/F(F%F% *&\"$m\"F%F.F%F%*(\"#?F%F1F%F/F%F%*&\"$6\"F%F5F%F%*&F(F%F1F%F%*&\"#7F% F/F%F%F%!\"\"F%F/F?,(*&\"#6F%F/F%F?F%F?F " 0 "" {MPLTEXT 1 0 25 "subs(FF=c1*xx+KK,eqFFs):\n" }{MPLTEXT 1 0 44 " numer(factor(%))/(1588410*sqrt(3)+2750039):\n" }{MPLTEXT 1 0 27 "colle ct(%,\{xx,KK\},factor);\n" }{MPLTEXT 1 0 22 "puiseux(%,xx=0,KK,0):\n" }{MPLTEXT 1 0 27 "map(mafactor,simplify(%));\n" }{MPLTEXT 1 0 17 "sing Fs:=op(2,%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ",2*&,,*0#\"%cM\"%J8\" \"\",&\"(R+v#!\"\"*&\"(5%)e\"F))\"\"$#F)\"\"#F)F)F)),&I\"yG6\"F)F)F,F2 F)),(*&\"#6F)F5F)F,F)F,*&F2F)F/F)F)F0F))F5F0F)),(F;F)*&\"#8F)F5F)F,\" \"&F,\"\"%F))I#xxGF6FBF)F,*2#\"&[w#F(F))F2#F)F0F),&*&\")b1GBF)F/F)F)\" )%QE.%F,F)F3F)F7F))F>F0F)Fx8\"F)F/F)F)\"*Vg0(>F,F)F3F))F>F2F)F7F)F\"zm\"F)F/F)F)F)F>F)F3F)F7F)F\"\"'F))FDFAF)F,*2#\"$k)\"&\"F,*&\"#9F)F/F)F)F)FapF)F3F )FbpF))F>FAF)FCF)F,*2#\"&O2#\"%xWF)FHF),&*&\"(Q!yAF)F/F)F)\"(:%[RF,F), 0*&\"#PF)FF,\"F)F/F)F)\",C'e4aF)FbpF))F8FAF)F)F))FdoF2F)F)*&,,*4#\"#OF:F )FHF),&*&\"#aF)F/F)F)\"#$)F,F)F8F)F5F)F3F))F_pF2F))F>FarF))FDFdpF)F,*6 #\"$;#\"$`#F)FSF),&*&\"%X8F)F/F)F)\"%YCF,F)F8F),(*&F^uF)F5F)F,\"\"*F,F ;F)F)F_pF)F5F)F3F))F>\"\"(F)FepF)F,*0#\"#s\"&L?#F),&*&\"&Im)F)F/F)F)\" '>O9F,F),4*(\"%#\\&F)FF,F)F8F)F5F)FdqF)FOF)F)*2#\"%%=&\" %FerF))FDF`wF)F)*2#F)FgvF)FHF),&*&\"#fF)F/F)F)\"#CF, F),(\"#&*F,*&\"#IF)F/F)F)*&FgvF)F5F)F,F)F3F)FavF))F>F^wF)FcvF)F,*0#F^w \"$n\"F)FSF),&*&\"$7\"F)F/F)F)\"#(*F)F),4*(\"%;8F)Fe 7F),&*&\"&`y#F)F/F)F)\"&%=cF,F),8*(\"&#Q**F)F/F)F^xF)F)*&Fi^lF))F5FAF) F,*(\"'wr5F)FF)F^xF)F,*(\"&))\\&F)F/F)FbpF)F)*&\"'y6 AF)F\"F)FbpF)F,*&\"%#=\"F)F/F)F)*&\" &Bl#F)F5F)F,\"%\\G: /\"QG0>mU7F&,&\"4L$Q,lF\"*=jGF&*&\"4/Y\\\\=,jIl\"F&F+F&F&F&)F0\"\")F&I #xxGF5F&)F9F7F6,&\"+MNVs?F6*&\"+f^Y'>\"F&F+F&F&F&)\"'`J[F-F&)**FJF&FHF &FIF&FFF6F-F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*2#\"\"\"\"@_x7o>G: /\"QG0>mU7F%,&\"4L$Q,lF\"*=jGF%*&\"4/Y\\\\=,jIl\"F%)\"\"$#F%\"\"#F%F%F %),(*&F.F%F+F%F%*&\"#8F%I\"yG6\"F%!\"\"\"\"&F6\"\")F%I#xxGF5F%),(*&\"# 6F%F4F%F6F%F6F1F%F7F6,&\"+MNVs?F6*&\"+f^Y'>\"F%F+F%F%F%)\"'`J[F-F%)**F >F%F9F%F:F%F/F6F-F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "so lve((-2072433534+1196465159*sqrt(3))*(-1+2*sqrt(3)-11*y)^5>0,y);" }} {PARA 11 "" 1 "" {XPPMATH 20 "-I*RealRangeG6$%*protectedGI(_syslibG6\" 6$-I%OpenGF$6#,&#\"\"\"\"#6!\"\"*&#\"\"#F/F.)\"\"$#F.F3F.F.I)infinityG F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 99 "The singular behaviour of \+ F at x=sF is given by \"singFs\". Indeed he other value is not going t o 0.\n" }{TEXT 217 81 "It is only real for y>1-y_critic. In that range it gives coefficients in n^(-5/2)" }}{PARA 0 "" 0 "" {TEXT 217 133 "T hus the singularity sF is impossible for y>1-y_critic. Hence the singu larity for y in (1-y_critic,1] is one of the roots tF of DF1. " }} {PARA 0 "" 0 "" {TEXT 217 172 "The fact that the singularity at y=1 is sF ensures that sigma is the radius of G for all y=[1-y_critic,1] (be cause sF(y)>tF(y) for y>ycritic and singularity is decreasing)." }} {PARA 0 "" 0 "" {TEXT 217 46 "It remains to understand singularity at \+ x=tF." }}}{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 217 188 "We denote Ftau=F(x=tF) , and we set x=tF-xx, and F=Ftau-FF in eqF. We obtain an equation eqFF xx for FF, xx, We then determine the possible singularity behaviors us ing Newton polygon method." }}{PARA 0 "" 0 "" {TEXT 217 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "factor(resultant(eqF,DF1,x)) /(36*2^(2/3)*(744*sqrt(3)+1393));\n" }{MPLTEXT 1 0 28 "subs(F=Ftau,op( 1,op(2,%)));\n" }{MPLTEXT 1 0 31 "eqFtau:=collect(%,Ftau,factor);" }} {PARA 11 "" 1 "" {XPPMATH 20 "*(,<*&)I\"FG6\"\"\"#\"\"\")I\"yGF'F(F)F) **\"#FF)F&F))\"\"$#F)F(F)F*F)F)*(F(F)F%F)F+F)!\"\"*(\"#WF)F&F)F*F)F2** \"#@F)F&F)F.F)F+F)F2*(F/F)F.F)F*F)F)*$F%F)F)*(\"#MF)F&F)F+F)F)*&F(F)F* F)F2*(\"\"*F)F.F)F+F)F2*&\"\")F)F&F)F2*&\"\"%F)F+F)F)\"#;F)F)),D*(FBF) F%F))F+F/F)F)**\"#7F)F&F)F.F)FFF)F)*(\"#[F)F%F)F*F)F2*(\"#kF)F&F)FFF)F 2**\"#CF)F&F)F.F)F*F)F2*(\"\"'F)FFF)F.F)F)*(FJF)F%F)F+F)F)*(\"$#>F)F&F )F*F)F)*&\"#>F)FFF)F)*(\"#OF)F.F)F*F)F2*&FBF)F%F)F2*(FSF)F&F)F+F)F2*& \"#dF)F*F)F2*(\"#%)F)F.F)F+F)F)*&FLF)F&F)F)*&FJF)F+F)F)FLF2F(F))F+FAF) " }}{PARA 11 "" 1 "" {XPPMATH 20 ",D*(\"#;\"\"\")I%FtauG6\"\"\"#F%)I\" yGF(\"\"$F%F%**\"#7F%F'F%)F,#F%F)F%F*F%F%*(\"#[F%F&F%)F+F)F%!\"\"*(\"# kF%F'F%F*F%F4**\"#CF%F'F%F/F%F3F%F4*(\"\"'F%F*F%F/F%F%*(F2F%F&F%F+F%F% *(\"$#>F%F'F%F3F%F%*&\"#>F%F*F%F%*(\"#OF%F/F%F3F%F4*&F$F%F&F%F4*(F=F%F 'F%F+F%F4*&\"#dF%F3F%F4*(\"#%)F%F/F%F+F%F%*&F6F%F'F%F%*&F2F%F+F%F%F6F4 " }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*(\"#;\"\"\"),&I\"yG6\"F%F%!\"\"\" \"$F%)I%FtauGF)\"\"#F%F%**#\"\"%\"$H#F%,&F$F**&F+F%)F+#F%F.F%F%F%,0*( \"#[F%F5F%)F(F.F%F%*&F2F%)F(F+F%F**(\"$W\"F%F5F%F(F%F**&\"$9(F%F:F%F%* &F9F%F5F%F%*&\"$o(F%F(F%F*\"$c#F%F%F-F%F***#F%\"$`#F%,&\"#>F%*&\"\"'F% F5F%F%F%,,*(FKF%F5F%F(F%F%*&\"#BF%F:F%F*FAF**&\"#tF%F(F%F%\"#KF*F%,(*& \"#6F%F(F%F*F$F**&\"#7F%F5F%F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "eqFF:=factor(resultant(subs(F=Ftau-FF,eqF),eqFtau,Fta u)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "eqFFxx:=resultant(s ubs(x=x-xx,eqFF),DF1,x):\n" }{MPLTEXT 1 0 34 "eqFfxx:=collect(%,\{FF,x x\},factor):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 97 "We now apply Newton's method: we first i nvestigate which are the non-zero coefficients of eqFFxx." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "ldegree(coeff(eqFFxx,xx,0),FF);\n" }{MPLTEXT 1 0 40 "factor(coeff(coeff(eqFFxx,xx,0),FF,4));\n" }{MPLTEXT 1 0 21 "\{solve(%)\}:evalf(%);\n" }{MPLTEXT 1 0 32 "ldegree(coeff(eqF Fxx,xx,1),FF);\n" }{MPLTEXT 1 0 40 "factor(coeff(coeff(eqFFxx,xx,1),FF ,2));\n" }{MPLTEXT 1 0 21 "\{solve(%)\}:evalf(%);\n" }{MPLTEXT 1 0 32 "ldegree(coeff(eqFFxx,xx,2),FF);\n" }{MPLTEXT 1 0 40 "factor(coeff(coe ff(eqFFxx,xx,2),FF,0));\n" }{MPLTEXT 1 0 14 "\{fsolve(%,y)\};" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 ", $*6#\".WXoOT<\"\"-Bi!H5x#\"\"\")\"\"##F'\"\"$F',&*&\"./G#R*3n\"F')F+#F 'F)F'F'\".xq;OU*G!\"\"F',,*(\"\"'F'F/F'I\"yG6\"F'F'*&\"#BF')F6F)F'F2*& \"#[F'F/F'F2*&\"#tF'F6F'F'\"#KF2F',,*(\"&!*=\"F'F/F'F6F'F'*&\"&$=!)F'F :F'F2*&\"%[kF'F/F'F2*&\"&Da*F'F6F'F'\"&3-\"F2F'),(*&\"#6F'F6F'F2F'F2*& F)F'F/F'F'F)F'),(*&\"#8F'F6F'F2\"#;F'*&\"\"%F'F/F'F'FUF'),**(F)F'F/F'F 6F'F'*$F:F'F2*&F5F'F6F'F'F)F2FUF')F6\"#5F'),&F6F'F'F2\"#9F'F2" }} {PARA 11 "" 1 "" {XPPMATH 20 "<,$\"\"!F$$\"*SoE;#!\"*$\"+U'Gq;#!#5$\"+ yB4SAF*$\"\"\"F$$\"+QeAI7F'$\"+TzqjwG\"=F'F.*&$ \"+#o'H68F'F.^#F.F.!\"\",&F6F.F8F." }}{PARA 11 "" 1 "" {XPPMATH 20 "\" \"#" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*6#\"-ob3nn9\"0(R*QTaS9)\"\"\", &*&\"1K)*)3ej#*R&F')\"\"$#F'\"\"#F'F'\"1$\\B.H*z^$*!\"\"F',,*(\"&!*=\" F'F+F'I\"yG6\"F'F'*&\"&$=!)F')F4F.F'F0*&\"%[kF'F+F'F0*&\"&Da*F'F4F'F' \"&3-\"F0F',,*(\"\"'F'F+F'F4F'F'*&\"#BF'F8F'F0*&\"#[F'F+F'F0*&\"#tF'F4 F'F'\"#KF0F',@*(\"&?c\"F'F+F')F4F@F'F'*&\"%yeF')F4\"\"(F'F'*(\"'Mr7F'F +F')F4\"\"&F'F0*&\"&e7\"F'FKF'F0*(\"'k*>%F'F+F')F4\"\"%F'F'*&\"'9Z6F'F RF'F0*(\"'BVtF')F4F,F'F+F'F0*&\"'a#p&F'FXF'F'*(\"'scqF'F+F'F8F'F'*&\"( #zL6F'FhnF'F0*(\"'GwG\"=F'F.*&$\"+#o'H68F'F.^#F.F.!\"\",&F9F.F;F.,&]3FF041E227BE93B BF.*&]3FED9E8B3F015F81F.F>F.F?,&FBF.FCF.,&]3FF865219700D48FF.*&]3FE3B9 7CA09D0BE5F.F>F.F?,&FGF.FHF." }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" } }{PARA 11 "" 1 "" {XPPMATH 20 ",$*6#\"+;wkdI\"-\"))*orWB\"\"\")\"\"##F )\"\"$F',&*&\",5'\\A#>\"F')F+#F'F)F'F'\",\"H\\,l?!\"\"F',,*(\"&!*=\"F' F/F'I\"yG6\"F'F'*&\"&$=!)F')F6F)F'F2*&\"%[kF'F/F'F2*&\"&Da*F'F6F'F'\"& 3-\"F2F'),(*&\"#8F'F6F'F2\"#;F'*&\"\"%F'F/F'F'F+F'),(*&\"#6F'F6F'F2F'F 2*&F)F'F/F'F'F+F'),**(F)F'F/F'F6F'F'*$F:F'F2*&\"\"'F'F6F'F'F)F2FFF'),, *(FQF'F/F'F6F'F'*&\"#BF'F:F'F2*&\"#[F'F/F'F2*&\"#tF'F6F'F'\"#KF2\"\"&F '),&F6F'F'F2\"\"(F')F6FJF'F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "<*$\"\"! F$$\"+'RoE;#!#5$\"+T'Gq;#F'$\"+yB4SAF'$\"\"\"F$$\"+QeAI7!\"*$\"+Tzqj " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 217 124 "So Newtons method tells us the s ingularity is in X^\{1/2\} unless y=yy1 or yy2. It remains to settle t he cases y=yy1 or y=yy2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "[solve(2*sqrt(3)*y-y^2+6*y-2)];evalf(%);yy1:=op(1,%%);\n" }{MPLTEXT 1 0 87 "[solve(11890*sqrt(3)*y-80183*y^2-6448*sqrt(3)+95425*y-10208,y) ];evalf(%);yy2:=op(1,%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$,(*$)\"\" $#\"\"\"\"\"#F(F(F&F(*$),&\"#5F(*&\"\"'F(F%F(F(F'F(!\"\",(F$F(F&F(F*F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "7$$\"*SoE;#!\"*$\"+wZ$yC*F%" }} {PARA 11 "" 1 "" {XPPMATH 20 ",(*$)\"\"$#\"\"\"\"\"#F'F'F%F'*$),&\"#5F '*&\"\"'F'F$F'F'F&F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$,(*&#\"%X f\"&$=!)\"\"\")\"\"$#F(\"\"#F(F(#\"&Da*\"'m.;F(*&#\"\"*F/F(),&\")\\ZBx F(*&\"(WI[#F(F)F(F(F+F(!\"\",(F$F(F-F(F0F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$$\"+U'Gq;#!#5$\"+QeAI7!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*&#\"%Xf\"&$=!)\"\"\")\"\"$#F'\"\"#F'F'#\"&Da*\"'m.;F'* &#\"\"*F.F'),&\")\\ZBxF'*&\"(WI[#F'F(F'F'F*F'!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "eqFFxxy1:=collect(subs(y=yy1,eqFFxx),\{FF,x x\},factor):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "ldegree(coeff(eqFFxxy1,xx,0),FF);\n " }{MPLTEXT 1 0 34 "coeff(coeff(eqFFxxy1,xx,0),FF,8):\n" }{MPLTEXT 1 0 15 "evalf[100](%);\n" }{MPLTEXT 1 0 34 "ldegree(coeff(eqFFxxy1,xx,1) ,FF);\n" }{MPLTEXT 1 0 34 "coeff(coeff(eqFFxxy1,xx,1),FF,6):\n" } {MPLTEXT 1 0 15 "evalf[100](%);\n" }{MPLTEXT 1 0 34 "ldegree(coeff(eqF Fxxy1,xx,2),FF);\n" }{MPLTEXT 1 0 34 "coeff(coeff(eqFFxxy1,xx,2),FF,4) :\n" }{MPLTEXT 1 0 15 "evalf[100](%);\n" }{MPLTEXT 1 0 34 "ldegree(coe ff(eqFFxxy1,xx,3),FF);\n" }{MPLTEXT 1 0 34 "coeff(coeff(eqFFxxy1,xx,3) ,FF,2):\n" }{MPLTEXT 1 0 15 "evalf[100](%);\n" }{MPLTEXT 1 0 34 "ldegr ee(coeff(eqFFxxy1,xx,4),FF);\n" }{MPLTEXT 1 0 34 "coeff(coeff(eqFFxxy1 ,xx,4),FF,%):\n" }{MPLTEXT 1 0 15 "evalf[100](%);\n" }{MPLTEXT 1 0 1 " \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"Pf@pyb2pKw()z%>lbr2R!#N" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "$!N#\\F\"RVg\" )z.^i%=%zy0GH^8#*H'!#M" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "$\"NVj(G,aJH0 " 0 "" {MPLTEXT 1 0 53 "eqFFxxy2:=collect(subs(y=yy2 ,eqFFxx),\{FF,xx\},factor):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "eqFFxxy2fac:=factor( eqFFxxy2):" }}{PARA 207 "" 1 "" {TEXT 225 0 "" }{HYPERLNK 225 "Warning , computation interrupted" 4 "http://www.maplesoft.com/support/help/e rrors/view.aspx?path=Warning,%20%20computation%20interrupted" "" } {TEXT 225 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "ldegree(coeff(eqFFxxy2,xx,0),FF);\n" }{MPLTEXT 1 0 34 "coeff(coeff(eqFFxxy2,xx,0),FF,%):\n" }{MPLTEXT 1 0 15 "evalf[100]( %);\n" }{MPLTEXT 1 0 34 "ldegree(coeff(eqFFxxy2,xx,1),FF);\n" } {MPLTEXT 1 0 34 "coeff(coeff(eqFFxxy2,xx,1),FF,%):\n" }{MPLTEXT 1 0 15 "evalf[100](%);\n" }{MPLTEXT 1 0 34 "ldegree(coeff(eqFFxxy2,xx,2),F F);\n" }{MPLTEXT 1 0 34 "coeff(coeff(eqFFxxy2,xx,2),FF,%):\n" } {MPLTEXT 1 0 15 "evalf[100](%);\n" }{MPLTEXT 1 0 34 "ldegree(coeff(eqF Fxxy2,xx,3),FF);\n" }{MPLTEXT 1 0 34 "coeff(coeff(eqFFxxy2,xx,3),FF,%) :\n" }{MPLTEXT 1 0 15 "evalf[100](%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"bpa9y$)>#yhnzU@ao70()3&e .3Y@\"4O)ohrvl.\"Hn6KA<:s#yp%!#')" }}{PARA 11 "" 1 "" {XPPMATH 20 "\" \"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"^pC&*3%zi7P6')z\\w%))Gq^MUV_&p ?x5YDYeR:-tc#=\\8'4C9!#\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "$!]pQuZmt(QC;3L_`Zm#fK]$ek!*4S'3WeRFhEGM 2]lrQlG5!#!)" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "$!\\pYz)R*R7uh5*p#[qX0V&4mT<$)\\\"f%e3p))Gc(H+x?w1FN`(! #y" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 179 "Newton leaves 2 possibilities FF=X^1/2 or F=X, bu t this second possibility would have a negative coefficient. Which is \+ not possible (F=c/(tf-xx) is not possible with c negative).\n" }{TEXT 217 39 "So it is in X^(1/2) also for y1 and y2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 105 "------------------------------------------------------- -------------------------------------------------\n" }{TEXT 217 84 "Ne xt we study the derivative K=diff(S,z)|z=z0 -> average number of edges in midlands" }{TEXT 217 1 "\n" }{TEXT 217 72 "and the derivative L=di ff(U,z)|z=z0 -> average number of edges in oceans" }}{PARA 0 "" 0 "" {TEXT 217 104 "------------------------------------------------------- -------------------------------------------------" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 221 31 "an alysis of K(x,y)=S_z(x,y,z0) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 222 91 "Some side computations f or y=1 (about the expectation and variance of number of reef edges)" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "eqG;\n" }{MPLTEXT 1 0 21 "fa ctor(subs(y=1,%));\n" }{MPLTEXT 1 0 77 "eqG1:=-4*G*sqrt(3)*2^(2/3)*x-2 *sqrt(3)*2^(2/3)*x+3*2^(2/3)*x+4*x^3+4*G^2+4*G;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",***\"#7\"\"\")I\"yG6\"\"\"#F%,&F'F%F%!\"\"F%)I\"GGF(\"\" $F%F%*0F)F%)F.#F%F)F%)F)#F%F.F%,(*&I\"xGF(F%F&F%F%*(F$F%F2F%F'F%F%*&F$ F%F2F%F+F%F6F%F'F%)F-F)F%F+**F)F%,**()F'F.F%F0F%F2F%F%*(\"\"'F%F6F%F=F %F+*(\"#OF%F2F%F'F%F+*&FAF%F2F%F%F%)F6F)F%F-F%F+*.F1F%F0F%F2F%,.**F)F% F&F%F0F%)F)#F)F.F%F%**\"\"%F%F'F%F0F%FGF%F+*(F.F%FGF%F&F%F%*(FJF%FCF%F &F%F%**\"#CF%F2F%F6F%F'F%F%*(FNF%F2F%F6F%F+F%F'F%)F6F.F%F+" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*,#\"\"\"\"\"#F%)\"\"$F$F%)F&#F%F(F%,.*,\"\"%F %I\"GG6\"F%F'F%)F&#F&F(F%I\"xGF/F%!\"\"**F&F%F'F%F0F%F2F%F3*(F(F%F0F%F 2F%F%*&F-F%)F2F(F%F%*&F-F%)F.F&F%F%*&F-F%F.F%F%F%)F2F&F%F3" }}{PARA 11 "" 1 "" {XPPMATH 20 ",.*,\"\"%\"\"\"I\"GG6\"F%)\"\"$#F%\"\"#F%)F+#F+F )F%I\"xGF'F%!\"\"**F+F%F(F%F,F%F.F%F/*(F)F%F,F%F.F%F%*&F$F%)F.F)F%F%*& F$F%)F&F+F%F%*&F$F%F&F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "eqGdiffy:=collect(subs(G(y)=G,subs(diff(G(y),y)=Gy,diff(subs(G=G(y ),eqG),y))),G,factor);\n" }{MPLTEXT 1 0 21 "factor(subs(y=1,%));\n" } {MPLTEXT 1 0 29 "factor(resultant(%,eqG1,G));\n" }{MPLTEXT 1 0 191 "eq Gy1:=8*Gy*2^(2/3)*sqrt(3)*x^3+18*x^3*2^(2/3)+12*2^(1/3)*x^4+8*x^2*Gy^2 -8*Gy*2^(2/3)*sqrt(3)+12*Gy*sqrt(3)*2^(1/3)*x-24*Gy*2^(1/3)*x+18*sqrt( 3)*2^(1/3)*x-36*x^2*sqrt(3)-21*2^(1/3)*x+90*x^2;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ",***\"#7\"\"\"I\"yG6\"F%,&*&\"\"$F%F&F%F%\"\"#!\"\"F%)I \"GGF'F*F%F%*&,,*(\"#OF%)F&F*F%I#GyGF'F%F%*(F2F%)F&F+F%F4F%F,*,\"\"'F% )I\"xGF'F+F%F6F%)F*#F%F+F%)F+#F%F*F%F,*,\"#[F%F:F%F&F%F;F%)F+#F+F*F%F, **\"#CF%F;F%FAF%F:F%F%F%)F.F+F%F%*.F+F%F;F%F=F%,.*,F*F%F;F%FAF%F9F%F6F %F,**F+F%F4F%F:F%F3F%F%**FDF%F=F%F4F%F6F%F%**FDF%F=F%F4F%F&F%F,*(F*F%F :F%F6F%F%*(F$F%F:F%F;F%F,F%F:F%F.F%F,*&,6*,F8F%)F:F*F%F6F%F=F%F;F%F%*, F+F%F4F%F;F%F=F%F3F%F%*,FDF%F;F%FAF%F9F%F&F%F%**F$F%F4F%F:F%F3F%F,**F$ F%F;F%FAF%F9F%F,**\"\"*F%F;F%F:F%F6F%F%**\"#sF%F=F%F4F%F&F%F,*(\"#=F%F :F%F6F%F%*(FYF%F4F%F=F%F%*(FDF%F:F%F&F%F,F%F9F%F," }}{PARA 11 "" 1 "" {XPPMATH 20 ",<*&\"#7\"\"\")I\"GG6\"\"\"$F%F%*,\"#CF%)F'\"\"#F%I\"xGF( F%)F-#F-F)F%)F)#F%F-F%!\"\"*,\"\"'F%F,F%)F.F-F%)F-#F%F)F%F1F%F3*(\"#OF %F'F%)F.F)F%F%*.\"\"%F%F'F%I#GyGF(F%F1F%F7F%F6F%F3**\"#sF%F6F%F'F%F7F% F%*,F5F%F'F%F7F%F1F%F6F%F3**F$F%F/F%)F.F=F%F1F%F3**F5F%F7F%)F.\"\"&F%F 1F%F3*,F-F%F>F%F1F%F7F%F6F%F3*(F$F%F>F%F;F%F%*(\"\"*F%F;F%F1F%F3*&F5F% F;F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*,\"#[\"\"\")\"\"##F'\"\"$F% ,8*,\"\")F%F&F%I#GyG6\"F%)F)#F%F'F%)I\"xGF.F)F%F%*(\"#=F%F1F%F&F%F%*( \"#7F%)F'#F%F)F%)F2\"\"%F%F%*(F,F%)F2F'F%)F-F'F%F%**F,F%F-F%F/F%F&F%! \"\"*,F6F%F-F%F/F%F7F%F2F%F%**\"#CF%F-F%F7F%F2F%F?**F4F%F/F%F7F%F2F%F% *(\"#OF%F " 0 "" {MPLTEXT 1 0 42 "factor(resultant(eqG,eqGdiffy,G))/(-432);\n" } {MPLTEXT 1 0 9 "eq2Gdiffy" }{MPLTEXT 1 0 10 ":=op(2,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "*,)\"\"##\"\"\"\"\"$F&,\\w**\"$k)F&)F$#F$F'F&)I#GyG 6\"F'F&)I\"yGF/\"\"'F&!\"\"**\"$;#F&F+F&F-F&)F1\"\"%F&F3**\"%#f#F&F.F& )F'#F&F$F&)I\"xGF/F$F&F&**\"$3\"F&F-F&F+F&)F1\"\"*F&F3**F5F&F-F&F+F&)F 1\"\")F&F&**F5F&F-F&F+F&)F1\"\"(F&F&**\"$c(F&F-F&F+F&)F1\"\"&F&F&**\"% g@F&F.F&F " 0 "" {MPLTEXT 1 0 103 "eqGdiffyy:=collect(subs(Gy(y)=Gy,s ubs(diff(Gy(y),y)=Gyy,diff(subs(Gy=Gy(y),eq2Gdiffy),y))),Gy,factor);\n " }{MPLTEXT 1 0 25 "factor(subs(y=1,%))/x^2;\n" }{MPLTEXT 1 0 30 "fact or(resultant(%,eqGy1,Gy));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 477 "eq Gyy1:=-96*Gyy*2^(1/3)*x^4+336*sqrt(3)*x^5-528*Gyy*2^(2/3)*x^3+192*Gyy* 2^(2/3)+48*Gyy*sqrt(3)*2^(1/3)*x^4+32*Gyy*sqrt(3)*2^(2/3)*x^6-552*sqrt (3)*2^(1/3)*x+1440*Gyy*x^2-32*Gyy*sqrt(3)*x^5-832*Gyy*sqrt(3)*x^2-816* Gyy*2^(1/3)*x-3276*sqrt(3)*2^(2/3)*x^3+2256*x^2*sqrt(3)+1488*x^4*sqrt( 3)*2^(1/3)+8*Gyy^2*2^(1/3)*x^4+232*Gyy*sqrt(3)*2^(2/3)*x^3+288*Gyy*sqr t(3)*2^(1/3)*x-16*Gyy^2*x^5+4632*x^3*2^(2/3)-96*2^(1/3)*x^7-276*2^(2/3 )*x^6-3684*2^(1/3)*x^4+324*2^(1/3)*x-1584*x^5-5229*x^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**,\"#7\"\"\",H**\"#=F%)\"\"$#F%\"\"#F%)F,#F,F*F%) I\"yG6\"\"\"%F%F%*,\"$3\"F%)F,#F%F*F%F)F%I\"xGF1F%F/F%F%**\"#IF%F)F%F- F%)F0F*F%!\"\"*(\"#\")F%F-F%F/F%F;*,\"$!=F%F)F%F5F%F7F%F:F%F;**\"$i\"F %F5F%F7F%F/F%F;*(\"#OF%)F7F,F%F/F%F;**\"$9\"F%)F0F,F%F)F%F-F%F;*(\"#jF %F-F%F:F%F%*,\"#sF%F)F%F5F%F7F%FGF%F%**\"$A#F%F5F%F7F%F:F%F%*(\"#GF%FD F%F:F%F%**\"$)>F%F0F%F)F%F-F%F%*(\"$*=F%F-F%FGF%F%**F9F%F5F%F7F%FGF%F; *(FKF%F-F%F)F%F;*(\"$V#F%F0F%F-F%F;**F9F%F5F%F7F%F0F%F;*&FKF%F-F%F%F%, &F0F%F%F;F%F:F%)I#GyGF1F*F%F%**F2F%,`q*,FAF%I$GyyGF1F%F)F%F-F%F:F%F%*. 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" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*&\"\")\"\"\"),&*&\"\"#F%I\"xG6\"F %!\"\"*$)F)#F%\"\"$F%F%F0F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "<#,$*&# \"\"\"\"\"#\"\"\")\"\"##\"\"\"\"\"$\"\"\"\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "<\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$**\"\")\"\"\")\"\" ##F'\"\"$F%,&*&F'F%I\"xG6\"F%!\"\"*$)F'#F%F)F%F%F%),**()F)#F%F'F%F0F%F ,F%F.*(F'F%F&F%F5F%F%*&\"\"'F%)F,F'F%F%*(F9F%F0F%F,F%F%F'F%F%" }} {PARA 11 "" 1 "" {XPPMATH 20 "<%,$*&#\"\"\"\"\"#F&)F'#F&\"\"$F&F&,(*(# F&\"#7F&)F*F%F&F(F&F&F$!\"\"*&,$*&F-F&^#F&F&F&F&),&*&\"#RF&)F'#F'F*F&F 0*(\"#gF&F9F&F/F&F&F%F&F0,(F,F&F$F0*&F3F&F5F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<#$\"+]_g*H'!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$**\"\" )\"\"\")\"\"##F'\"\"$F%),6**F$F%)I\"xG6\"F)F%F&F%)F)#F%F'F%F%**\"\"%F% )F.F3F%F0F%)F'#F%F)F%F%*(\"#XF%F-F%F&F%!\"\"*(\"#OF%F5F%F4F%F9*&F;F%)F .\"\"&F%F9**\"#sF%F0F%F5F%F.F%F%*(\"$O\"F%)F.F'F%F0F%F9*&\"#[F%F&F%F%* (\"$c\"F%F5F%F.F%F9*&\"$G#F%FCF%F%F'F%,&*&F'F%F.F%F9*$F5F%F%F%F%" }} {PARA 11 "" 1 "" {XPPMATH 20 "<$$\"+]_g*H'!#5$\"+R(*HavF%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 61 "Great we see that the only possible dominant singularity for " }{TEXT 217 35 "Gyy is 2^\{-2/3\}. The same as for G." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "puiseux(eqG1,x=(1/2)*2^(1/3),G,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "<#,**&#\"\"\"\"\"#\"\"\")\"\"$#\"\"\" \"\"#\"\"\"\"\"\"#\"\"\"\"\"#!\"\"**#\"\"\"\"\"#\"\"\",&I\"xG6\"\"\"\" *&#\"\"\"\"\"#\"\"\")\"\"##\"\"\"\"\"$\"\"\"!\"\"\"\"\")\"\"##\"\"#\" \"$\"\"\")\"\"$#\"\"\"\"\"#\"\"\"\"\"\"*$),&I\"xG6\"!\"\"*&#\"\"\"\"\" #\"\"\")\"\"##\"\"\"\"\"$\"\"\"\"\"\"#\"\"$\"\"#\"\"\"\"\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 217 49 "Singularity in X^\{-3/2\} hence c oeffs in n^\{-5/2\}." }{TEXT 217 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "collect(eqGyy1,Gyy,factor);\n" }{MPLTEXT 1 0 36 "fact or(subs(Gyy=1/GG,GG^2*eqGyy1));\n" }{MPLTEXT 1 0 32 "puiseux(%,x=(1/2) *2^(1/3),GG,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(**\"\")\"\"\")I\"xG 6\"\"\"%F%,&*&\"\"#F%F'F%!\"\"*$)F,#F%\"\"$F%F%F%)I$GyyGF(F,F%F%*.F)F% )F,#F,F1F%)F1#F%F,F%,2**F,F%)F'F1F%F5F%F7F%F%*(F1F%F;F%F5F%F-*&F)F%)F' \"\"&F%F-*(F$F%F5F%F7F%F%**\"#=F%F7F%F/F%F'F%F-*(\"#CF%)F'F,F%F7F%F%*( \"#OF%F/F%F'F%F%*&\"#KF%FEF%F-F%F*F%F3F%F%**#F1F,F%F/F%F'F%,:**\"$7\"F %F5F%F&F%F7F%F%*(\"$G&F%F5F%F&F%F-*(\"$%=F%F/F%F>F%F-*&\"#kF%)F'\"\"'F %F-**\"$_(F%F7F%F5F%F'F%F%**\"%%=#F%F7F%F/F%FEF%F-*(\"$#**F%F;F%F7F%F% *(\"%VF%F5F%)F1\"\"(F%F7*(\"%otF%F5F%)F 1\"\"%F%F7*(FHF%F-F%FMF%F7*(\"$['F%F5F%F1F%F%*(\"$%QF%F-F%F'F%F%*(\"%K ;F%F-F%F1F%F7**\"$_&F%F5F%F'F%F0F%F7**\"%k#*F%F5F%F'F%F6F%F%**\"%%e\"F %F5F%F:F%F " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 66 "The expectation E[#reef-edge^2] is quadratic in n=bound ary length." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "eqS:=c ollect(eqS,S,factor);\n" }{MPLTEXT 1 0 88 "eqSdiff:=collect(subs(S(z)= S,subs(diff(S(z),z)=DS,diff(subs(S=S(z),eqS),z))),S,factor);\n" }} {PARA 11 "" 1 "" {XPPMATH 20 ",6*,\"#k\"\"\")I\"yG6\"\"\"'F%)I\"zGF(\" \"(F%),&F'F%F%!\"\"\"\"$F%)I\"SGF(\"\"*F%F/*0\"$#>F%I\"xGF(F%)F'\"\"&F %)F+F)F%)F.\"\"#F%,(*(F6F%)F'F;F%)F+F;F%F%*&F;F%F'F%F%F;F/F%)F2\"\")F% F%*0F5F%)F6F;F%)F'\"\"%F%)F+F8F%F.F%,4*(FDF%FEF%)F+FFF%F%*(F6F%FEF%F?F %F%*&FEF%)F+F0F%F/**F0F%F6F%)F'F0F%F?F%F%*&FOF%FMF%F%**FFF%F6F%F>F%F?F %F/*&F8F%F>F%F%*&\"#5F%F'F%F/F8F%F%)F2F,F%F/*,)F6F0F%FOF%FJF%,L**F$F%F WF%F&F%F9F%F%**\"$w&F%FDF%F&F%FJF%F%**\"$%QF%F6F%F&F%FGF%F/**FgnF%FDF% F7F%FJF%F/**FgnF%F6F%F7F%FGF%F%**F5F%FDF%FEF%FJF%F/*(\"#'*F%F&F%FMF%F% **\"$g*F%F6F%F7F%F?F%F%*(\"%_6F%F7F%FMF%F/**F^oF%F6F%FEF%F?F%F/*(\"%;? 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"<&I#DSG6\"I\"xGF$I\"yGF$I\"zGF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "seq (nops(op(i," }{MPLTEXT 1 0 7 "eqalgDS" }{MPLTEXT 1 0 8 ")),i=1.." } {MPLTEXT 1 0 13 "nops(eqalgDS)" }{MPLTEXT 1 0 2 ");" }}{PARA 11 "" 1 " " {XPPMATH 20 "6*\"\"\"\"\"#F$F$F$F$F$\"%e\\" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "op(1,eqalgDS)*op(2,eqalgDS)*op(3,eqalgDS)*op(4,eq algDS)*op(5,eqalgDS)*op(6,eqalgDS)*op(7,eqalgDS);\n" }{MPLTEXT 1 0 20 "eqalgDS2:=eqalgDS/%:" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*0\"'W@E\"\" \")I\"xG6\"\"#sF%)I\"yGF(\"$9\"F%)I\"zGF(\"#lF%),&*&\"$K%F%)F.\"\"'F%F %F%!\"\"\"\"*F%),&F+F%F%F6\"#@F%),<**\"\")F%)F'\"\"$F%)F+F@F%)F.\"\"%F %F%**\"#6F%)F'\"\"#F%FAF%)F.FGF%F%**\"#7F%F'F%FAF%)F.F@F%F%*&FAF%FBF%F %**FJF%FFF%)F+FGF%FHF%F6**FJF%F'F%FNF%FKF%F6*(F@F%F'F%FAF%F%*(F@F%FAF% F.F%F6*(\"\"(F%F'F%FNF%F6*(\"#8F%FNF%F.F%F%*(FCF%F'F%F+F%F%*(\"#=F%F+F %F.F%F6*&F>F%F.F%F%FGF%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 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\"\"\"\"\"**\"()o1N\"\"\")I\"yG6\"\"\"#\"\"\")\"\"$#\"\"\"\"\"#\"\"\") \"\"##\"\"#\"\"$\"\"\"\"\"\"*(\"(Sa-$\"\"\")\"\"##\"\"#\"\"$\"\"\")I\" yG6\"\"\"$\"\"\"!\"\"**\"(O&pz\"\"\"I\"yG6\"\"\"\")\"\"$#\"\"\"\"\"#\" \"\")\"\"##\"\"#\"\"$\"\"\"!\"\"*(\"(!3of\"\"\")\"\"##\"\"#\"\"$\"\"\" )I\"yG6\"\"\"#\"\"\"\"\"\"*(\"('p$*[\"\"\")\"\"##\"\"#\"\"$\"\"\")\"\" $#\"\"\"\"\"#\"\"\"\"\"\"*(\"(gTs(\"\"\"I\"yG6\"\"\"\")\"\"##\"\"#\"\" $\"\"\"!\"\"*&\"(wp)=\"\"\")\"\"##\"\"#\"\"$\"\"\"\"\"\"\"\"\"I#_ZG6$% *protectedGI(_syslibG6\"\"\"\"\"\"\"*(\"%O^\"\"\")\"\"$#\"\"\"\"\"#\" \"\")I\"yG6\"\"\"'\"\"\"\"\"\"*(\"&[+*\"\"\")\"\"$#\"\"\"\"\"#\"\"\")I \"yG6\"\"\"&\"\"\"!\"\"*&\"%>r\"\"\")I\"yG6\"\"\"'\"\"\"\"\"\"*(\"&[;) \"\"\")\"\"$#\"\"\"\"\"#\"\"\")I\"yG6\"\"\"%\"\"\"\"\"\"*&\"'()38\"\" \")I\"yG6\"\"\"&\"\"\"!\"\"*(\"(Kg:\"\"\"\")I\"yG6\"\"\"$\"\"\")\"\"$# \"\"\"\"\"#\"\"\"\"\"\"*&\"&Gb(\"\"\")I\"yG6\"\"\"%\"\"\"\"\"\"*(\"(%) zD\"\"\"\")\"\"$#\"\"\"\"\"#\"\"\")I\"yG6\"\"\"#\"\"\"!\"\"*&\"(!3[(* 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\"##\"\"\"\"\"$\"\"\")I\"yG6\"\"\"%\"\"\"!\"\"**\"&3a\"\"\"\")\"\"$#\" \"\"\"\"#\"\"\")\"\"##\"\"\"\"\"$\"\"\")I\"yG6\"\"\"#\"\"\"!\"\"*(\"%_ u\"\"\")\"\"##\"\"\"\"\"$\"\"\")I\"yG6\"\"\"$\"\"\"!\"\"**\"%/&*\"\"\" I\"yG6\"\"\"\")\"\"$#\"\"\"\"\"#\"\"\")\"\"##\"\"\"\"\"$\"\"\"\"\"\"*( \"&?^\"\"\"\")\"\"##\"\"\"\"\"$\"\"\")I\"yG6\"\"\"#\"\"\"\"\"\"*(\"%7p \"\"\")\"\"##\"\"\"\"\"$\"\"\"I\"yG6\"\"\"\"!\"\"\"\"\")I#_ZG6$%*prote ctedGI(_syslibG6\"\"\"#\"\"\"\"\"\"*&,>**\"$[\"\"\"\")\"\"$#\"\"\"\"\" #\"\"\")\"\"##\"\"#\"\"$\"\"\")I\"yG6\"\"\"'\"\"\"\"\"\"**\"$+*\"\"\") \"\"$#\"\"\"\"\"#\"\"\")\"\"##\"\"#\"\"$\"\"\")I\"yG6\"\"\"&\"\"\"\"\" \"*(\"$i\"\"\"\")\"\"##\"\"#\"\"$\"\"\")I\"yG6\"\"\"'\"\"\"!\"\"**\"%K A\"\"\")\"\"$#\"\"\"\"\"#\"\"\")\"\"##\"\"#\"\"$\"\"\")I\"yG6\"\"\"%\" \"\"!\"\"*(\"%q9\"\"\")\"\"##\"\"#\"\"$\"\"\")I\"yG6\"\"\"&\"\"\"!\"\" **\"%;c\"\"\")\"\"$#\"\"\"\"\"#\"\"\")\"\"##\"\"#\"\"$\"\"\")I\"yG6\" \"\"$\"\"\"\"\"\"*(\"&!G<\"\"\")\"\"##\"\"#\"\"$\"\"\")I\"yG6\"\"\"%\" \"\"\"\"\"**\"&[K\"\"\"\")I\"yG6\"\"\"#\"\"\")\"\"$#\"\"\"\"\"#\"\"\") \"\"##\"\"#\"\"$\"\"\"\"\"\"*(\"&%y<\"\"\")\"\"##\"\"#\"\"$\"\"\")I\"y G6\"\"\"$\"\"\"!\"\"**\"&Ci%\"\"\"I\"yG6\"\"\"\")\"\"$#\"\"\"\"\"#\"\" \")\"\"##\"\"#\"\"$\"\"\"!\"\"*(\"&cB#\"\"\")\"\"##\"\"#\"\"$\"\"\")I \"yG6\"\"\"#\"\"\"!\"\"*(\"&7&G\"\"\")\"\"##\"\"#\"\"$\"\"\")\"\"$#\" \"\"\"\"#\"\"\"\"\"\"*(\"&g`%\"\"\"I\"yG6\"\"\"\")\"\"##\"\"#\"\"$\"\" \"\"\"\"*&\"&O2#\"\"\")\"\"##\"\"#\"\"$\"\"\"!\"\"\"\"\"I#_ZG6$%*prote ctedGI(_syslibG6\"\"\"\"\"\"\"*(\"$k&\"\"\")\"\"$#\"\"\"\"\"#\"\"\")I \"yG6\"\"\"'\"\"\"!\"\"*(\"$7$\"\"\")\"\"$#\"\"\"\"\"#\"\"\")I\"yG6\" \"\"&\"\"\"!\"\"*&\"$\")*\"\"\")I\"yG6\"\"\"'\"\"\"\"\"\"*(\"%oA\"\"\" )\"\"$#\"\"\"\"\"#\"\"\")I\"yG6\"\"\"%\"\"\"\"\"\"*&\"%(G\"\"\"\")I\"y G6\"\"\"&\"\"\"\"\"\"*(\"&+i\"\"\"\")I\"yG6\"\"\"$\"\"\")\"\"$#\"\"\" \"\"#\"\"\"!\"\"*&\"%GE\"\"\")I\"yG6\"\"\"%\"\"\"!\"\"*(\"&CQ\"\"\"\") \"\"$#\"\"\"\"\"#\"\"\")I\"yG6\"\"\"#\"\"\"\"\"\"*&\"&_(e\"\"\")I\"yG6 \"\"\"$\"\"\"\"\"\"*&\"&Cq&\"\"\")I\"yG6\"\"\"#\"\"\"!\"\"\"\"\"-I\"OG %*protectedG6#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "factor(subs(y=1,eq1));" }{MPLTEXT 1 0 28 "algeqtoseries(%,x,K,1,tr ue);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 22 "factor(subs(y=1,eq2));" } {MPLTEXT 1 0 28 "algeqtoseries(%,x,K,1,true);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*(\"#k\"\"\",.*(\"\"*F%)I\"xG6\"\"\"$F%)\"\"##F.F,F%!\" \"**\"#7F%I\"KGF+F%F-F%F*F%F%**\"\"'F%F3F%)F.#F%F,F%)F*F.F%F0*(F.F%)F3 F.F%F6F%F%*&F:F%F*F%F0*&F(F%F8F%F%F%)F*\"\"%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "7#+'I\"xG6\"-I'RootOfG6$%*protectedGI(_syslibG6\"6#,(*&\" \"*\"\"\")\"\"##\"\"#\"\"$\"\"\"\"\"\"*(\"#C\"\"\")\"\"##\"\"\"\"\"$\" \"\"I#_ZG6$%*protectedGI(_syslibG6\"\"\"\"\"\"\"*&\"\"%\"\"\")I#_ZG6$% *protectedGI(_syslibG6\"\"\"#\"\"\"\"\"\"\"\"\"-I\"OG%*protectedG6#\" \"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*(\"\"#\"\"\",.*(\"#=F%)I \"xG6\"\"\"$F%)F$#F$F,F%!\"\"**\"\"'F%I\"KGF+F%F-F%F*F%F%**\"#7F%F2F%) F$#F%F,F%)F*F$F%F/*&)F2F$F%F5F%F%*(F$F%F9F%F*F%F/*&\"#OF%F7F%F/F%)F*\" \"%F%F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "7#+'I\"xG6\"-I'RootOfG6$%*pro tectedGI(_syslibG6\"6#,(*$)I#_ZG6$%*protectedGI(_syslibG6\"\"\"#\"\"\" \"\"\"*(\"\"'\"\"\")\"\"##\"\"\"\"\"$\"\"\"I#_ZG6$%*protectedGI(_sysli bG6\"\"\"\"\"\"\"*&\"#=\"\"\")\"\"##\"\"#\"\"$\"\"\"!\"\"\"\"\"-I\"OG% *protectedG6#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "solve(9*2^(2/3)+24*2^(1/3)*_Z+4*_Z^2);\n" }{MPLTEXT 1 0 10 "evalf(%); \n" }{MPLTEXT 1 0 40 "solve(_Z^2+6*2^(1/3)*_Z-18*2^(2/3),_Z);\n" } {MPLTEXT 1 0 9 "evalf(%);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&*&\"\"$\"\"\")\"\"##F&F%F&!\"\"*(#F%F(F&F'F&)F%#F&F(F &F&,&F$F*F+F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$!*SAR1&!\"*$!+gS8`qF %" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&*&\"\"$\"\"\")\"\"##F&F%F&!\"\" *(F%F&F'F&)F%#F&F(F&F&,&F$F*F+F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$ \"+p'ypw#!\"*$!+(\\]E.\"!\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 30 " So eq2 is the correct equation" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2249 "eqK:=(1/2*(2*x *y+2*y*sqrt(3)*2^(1/3)-2*sqrt(3)*2^(1/3)+5*2^(1/3)*y-6*2^(1/3)))*(2*sq rt(3)*2^(2/3)*y^3+12*sqrt(3)*2^(1/3)*x*y^3-2*y^2*sqrt(3)*2^(2/3)-12*sq rt(3)*2^(1/3)*x*y^2-9*2^(2/3)*y^3-18*2^(1/3)*x*y^3-4*x^2*y^3-18*y*sqrt (3)*2^(2/3)+12*2^(1/3)*x*y^2+18*sqrt(3)*2^(2/3)+27*y*2^(2/3)+6*2^(1/3) *x*y-18*2^(2/3))*(y-1)*y^2*K^3+(1/2*(2*x*y+2*y*sqrt(3)*2^(1/3)-2*sqrt( 3)*2^(1/3)+5*2^(1/3)*y-6*2^(1/3)))*(-x*y^2+12*2^(1/3)*y-12*2^(1/3))*(2 *sqrt(3)*2^(2/3)*y^3+12*sqrt(3)*2^(1/3)*x*y^3-2*y^2*sqrt(3)*2^(2/3)-12 *sqrt(3)*2^(1/3)*x*y^2-9*2^(2/3)*y^3-18*2^(1/3)*x*y^3-4*x^2*y^3-18*y*s qrt(3)*2^(2/3)+12*2^(1/3)*x*y^2+18*sqrt(3)*2^(2/3)+27*y*2^(2/3)+6*2^(1 /3)*x*y-18*2^(2/3))*x*y*K^2+6*2^(1/3)*(-8*sqrt(3)*2^(1/3)*x^3*y^6-303* 2^(1/3)*y^5+540*2^(1/3)*y^4-1080*y*sqrt(3)*2^(1/3)-4*sqrt(3)*x*y^7+232 *sqrt(3)*x*y^5+672*sqrt(3)*x*y^3-120*y^3*sqrt(3)*2^(1/3)-2*sqrt(3)*2^( 1/3)*y^7-26*sqrt(3)*2^(1/3)*y^5-756*sqrt(3)*x*y^4+792*sqrt(3)*2^(1/3)* y^2-8*sqrt(3)*2^(1/3)*y^6-144*sqrt(3)*x*y+4*sqrt(3)*2^(2/3)*x^2*y^7-18 *sqrt(3)*2^(2/3)*x^2*y^6+98*sqrt(3)*2^(2/3)*x^2*y^5+8*sqrt(3)*2^(1/3)* x^3*y^5-132*sqrt(3)*2^(2/3)*x^2*y^4+48*sqrt(3)*2^(2/3)*x^2*y^3+2^(1/3) *y^7+14*2^(1/3)*y^6-18*2^(1/3)*y^3-450*2^(1/3)*y^2+12*sqrt(3)*2^(1/3)* y^4+10*x*y^7-9*2^(2/3)*x^2*y^7+27*2^(2/3)*x^2*y^6+28*2^(1/3)*x^3*y^6-2 22*2^(2/3)*x^2*y^5-54*2^(1/3)*x^3*y^5+348*2^(2/3)*x^2*y^4+24*2^(1/3)*x ^3*y^4-108*2^(2/3)*x^2*y^3-36*2^(2/3)*x^2*y^2+4*x^4*y^6+420*x*y^4+360* x*y^3+216*2^(1/3)*y+53*x*y^6-339*x*y^5-936*x*y^2+432*x*y+432*sqrt(3)*2 ^(1/3))*x^2*K+(9/2)*2^(2/3)*(-576*sqrt(3)*x-16*sqrt(3)*2^(1/3)*x^3*y^4 +16*sqrt(3)*2^(1/3)*x^3*y^3-240*sqrt(3)*2^(2/3)*x^2*y^2+96*sqrt(3)*2^( 2/3)*x^2*y-6*2^(1/3)*y^5+25*2^(1/3)*y^4-12*sqrt(3)*x*y^5+456*sqrt(3)*x *y^3+44*y^3*sqrt(3)*2^(1/3)-4*sqrt(3)*2^(1/3)*y^5-56*sqrt(3)*x*y^4-24* sqrt(3)*2^(1/3)*y^2+4*sqrt(3)*2^(1/3)*y^6+1440*sqrt(3)*x*y+16*sqrt(3)* 2^(2/3)*x^2*y^5-40*sqrt(3)*2^(2/3)*x^2*y^4+168*sqrt(3)*2^(2/3)*x^2*y^3 -7*2^(1/3)*y^6-588*2^(1/3)*y^3+1152*2^(1/3)*y^2-20*sqrt(3)*2^(1/3)*y^4 -1248*sqrt(3)*x*y^2-36*2^(2/3)*x^2*y^5+60*2^(2/3)*x^2*y^4+56*2^(1/3)*x ^3*y^4-360*2^(2/3)*x^2*y^3-96*2^(1/3)*x^3*y^3+624*2^(2/3)*x^2*y^2+48*2 ^(1/3)*x^3*y^2-288*2^(2/3)*x^2*y+8*x^4*y^4+220*x*y^4-768*x*y^3-576*2^( 1/3)*y-4*sqrt(3)*x*y^6+8*x*y^6+36*x*y^5+792*x*y^2-288*x*y)*y*x^3;" }} {PARA 11 "" 1 "" {XPPMATH 20 ",**.#\"\"\"\"\"#F%,,*(F&F%I\"xG6\"F%I\"y GF*F%F%**F&F%F+F%)\"\"$F$F%)F&#F%F.F%F%*(F&F%F-F%F/F%!\"\"*(\"\"&F%F/F %F+F%F%*&\"\"'F%F/F%F2F%,<**F&F%F-F%)F&#F&F.F%)F+F.F%F%*,\"#7F%F-F%F/F %F)F%F;F%F%**F&F%)F+F&F%F-F%F9F%F2*,F=F%F-F%F/F%F)F%F?F%F2*(\"\"*F%F9F %F;F%F2**\"#=F%F/F%F)F%F;F%F2*(\"\"%F%)F)F&F%F;F%F2**FDF%F+F%F-F%F9F%F 2**F=F%F/F%F)F%F?F%F%*(FDF%F9F%F-F%F%*(\"#FF%F+F%F9F%F%**F6F%F/F%F)F%F +F%F%*&FDF%F9F%F2F%,&F+F%F%F2F%F?F%)I\"KGF*F.F%F%*0F$F%F'F%,(*&F)F%F?F %F2*(F=F%F/F%F+F%F%*&F=F%F/F%F2F%F7F%F)F%F+F%)FQF&F%F%*,F6F%F/F%,dp**F 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Moreover, whe n x=rho, two branches disapear at infinity. So the non-infinite branch is non-singular. So H_rho=infty." }{TEXT 217 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 222 15 "Case y_critic= " } {XPPEDIT 2 0 "Typesetting:-mrow(Typesetting:-mfrac(Typesetting:-mn(\"1 2\", mathvariant = \"normal\"), Typesetting:-mn(\"11\", mathvariant = \+ \"normal\"), linethickness = \"1\", denomalign = \"center\", numalign \+ = \"center\", bevelled = \"false\"), 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:-mrow(Typesetting:-mfrac(Typesetting:- mn(\"2\", mathvariant = \"normal\"), Typesetting:-mn(\"11\", mathvaria nt = \"normal\"), linethickness = \"1\", denomalign = \"center\", numa lign = \"center\", bevelled = \"false\"), Typesetting:-mo(\"&Invisible Times;\", mathvariant = \"normal\", fence = \"false\", separator = \"f alse\", stretchy = \"false\", symmetric = \"false\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mrow(Typesetting:-msqrt(Typesetting :-mn(\"3\", mathvariant = \"normal\"))), Typesetting:-mi(\"\", italic \+ = \"true\", mathvariant = \"italic\")), Typesetting:-mi(\"\", italic = \"true\", mathvariant = \"italic\"));" "-I%mrowG6#/I+modulenameG6\"I, TypesettingGI(_syslibGF'6&-I&mfracGF$6(-I#mnGF$6$Q#12F'/%,mathvariantG Q'normalF'-F/6$Q#11F'F2/%.linethicknessGQ\"1F'/%+denomalignGQ'centerF' /%)numalignGF=/%)bevelledGQ&falseF'-I#moGF$6-Q(−F'F2/%&fenceGFB/ %*separatorGFB/%)stretchyGFB/%*symmetricGFB/%(largeopGFB/%.movablelimi tsGFB/%'accentGFB/%'lspaceGQ,0.2222222emF'/%'rspaceGFW-F#6&-F,6(-F/6$Q \"2F'F2F5F8F;F>F@-FD6-Q1⁢F'F2FGFIFKFMFOFQFS/FVQ&0.0emF' /FYF_o-F#6#-I&msqrtGF$6#-F/6$Q\"3F'F2-I#miGF$6%Q!F'/%'italicGQ%trueF'/ F3Q'italicF'Fio" }{TEXT 222 1 "." }}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 0 6 "yspe:=" }{XPPEDIT 19 1 "Typesetting:-mrow(Typesetting: -mfrac(Typesetting:-mn(\"12\", mathvariant = \"normal\"), Typesetting: -mn(\"11\", mathvariant = \"normal\"), linethickness = \"1\", denomali gn = \"center\", numalign = \"center\", bevelled = \"false\"), Typeset ting:-mo(\"−\", mathvariant = \"normal\", fence = \"false\", sep arator = \"false\", stretchy = \"false\", symmetric = \"false\", large op = \"false\", movablelimits = \"false\", accent = \"false\", lspace \+ = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mfrac(Types etting:-mn(\"2\", mathvariant = \"normal\"), Typesetting:-mn(\"11\", m athvariant = \"normal\"), linethickness = \"1\", denomalign = \"center \", numalign = \"center\", bevelled = \"false\"), Typesetting:-mo(\"&I nvisibleTimes;\", mathvariant = \"normal\", fence = \"false\", separat or = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \" 0.0em\", rspace = \"0.0em\"), Typesetting:-msqrt(Typesetting:-mn(\"3\" , mathvariant = \"normal\")), Typesetting:-mo(\";\", mathvariant = \"n ormal\", fence = \"false\", separator = \"true\", stretchy = \"false\" , symmetric = \"false\", largeop = \"false\", movablelimits = \"false \", accent = \"false\", lspace = \"0.0em\", rspace = \"0.2777778em\"), Typesetting:-mspace(height = \"0.0ex\", width = \"0.0em\", depth = \" 0.0ex\", linebreak = \"newline\"), Typesetting:-mo(\" \", mathvariant \+ = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"f alse\", symmetric = \"false\", largeop = \"false\", movablelimits = \" false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), \+ Typesetting:-mi(\"evalf\", italic = \"true\", mathvariant = \"italic\" ), Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mi(\"%\", itali c = \"true\", mathvariant = \"italic\")), mathvariant = \"normal\"), T ypesetting:-mo(\";\", mathvariant = \"normal\", fence = \"false\", sep arator = \"true\", stretchy = \"false\", symmetric = \"false\", largeo p = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.2777778em\"));" "-I%mrowG6#/I+modulenameG6\"I ,TypesettingGI(_syslibGF'6--I&mfracGF$6(-I#mnGF$6$Q#12F'/%,mathvariant GQ'normalF'-F/6$Q#11F'F2/%.linethicknessGQ\"1F'/%+denomalignGQ'centerF '/%)numalignGF=/%)bevelledGQ&falseF'-I#moGF$6-Q(−F'F2/%&fenceGFB /%*separatorGFB/%)stretchyGFB/%*symmetricGFB/%(largeopGFB/%.movablelim itsGFB/%'accentGFB/%'lspaceGQ,0.2222222emF'/%'rspaceGFW-F,6(-F/6$Q\"2F 'F2F5F8F;F>F@-FD6-Q1⁢F'F2FGFIFKFMFOFQFS/FVQ&0.0emF'/FYF ]o-I&msqrtGF$6#-F/6$Q\"3F'F2-FD6-Q\";F'F2FG/FJQ%trueF'FKFMFOFQFSF\\o/F YQ,0.2777778emF'-I'mspaceGF$6&/%'heightGQ&0.0exF'/%&widthGF]o/%&depthG Fap/%*linebreakGQ(newlineF'-FD6-Q\"~F'F2FGFIFKFMFOFQFSF\\oF^o-I#miGF$6 %Q&evalfF'/%'italicGFio/F3Q'italicF'-I(mfencedGF$6$-F#6#-F]q6%Q\"%F'F` qFbqF2Feo" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 45 "subspe:=A->factor(si mplify(subs(y=yspe,A))):\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&#\"#7\"# 6\"\"\"*&#\"\"#F%F&)\"\"$#F&F)F&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+Bw!*fx!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "eqKspe:=s ubspe(eqK);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*(#\"$K%\")rr[>\"\"\",& \"&*QF'FEF 'FfnF'F5F'F-F'F**,FTF'FEF'F?F'F5F'F-F'F'*,\"$C$F'FBF'F?F'F-F'F;F'F'*,F VF'F4F'F-F'F;F'F:F'F'*&\"%S9F'FfnF'F**&\"%)o&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 217 31 "We now do singularity analysis:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "rhospe:=simplify(subspe(s igma));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*(#\"\"\"\"\"#F%)\"\"$F$F %)F&#F%F(F%F%*$F)F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "fa ctor(puiseux(eqKspe,x=rhospe,K,0));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "<#,$*(\"\"#\"\"\"),$**#\"\"$\"\"#\"\"\")\"\"##\"\"#\"\"$\"\"\",&\"#X \"\"\"*&\"#E\"\"\")\"\"$#\"\"\"\"\"#\"\"\"\"\"\"\"\"\",(*&)\"\"$#\"\" \"\"\"#\"\"\")\"\"##\"\"\"\"\"$\"\"\"\"\"\"*&\"\"#\"\"\")\"\"##\"\"\" \"\"$\"\"\"\"\"\"*&\"\"#\"\"\"I\"xG6\"\"\"\"!\"\"\"\"\"!\"\"#\"\"\"\" \"$\"\"\",(*&)\"\"$#\"\"\"\"\"#\"\"\")\"\"##\"\"\"\"\"$\"\"\"\"\"\"*& \"\"#\"\"\")\"\"##\"\"\"\"\"$\"\"\"\"\"\"*&\"\"#\"\"\"I\"xG6\"\"\"\"! \"\"!\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 217 198 "The singularity is in X^\{-2/3\} . This type of singularity indicates coefficients in n^(-1/3) =n^(4/3) *n^(-5/3). So the average number of edges in islands with reef edges w eighted y=y_critic is n^4/3." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 50 "We examine the behavior for y>y_critic (rho=sig ma)" }{TEXT 217 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "sigma ;\n" }{MPLTEXT 1 0 25 "puiseux(eqK,x=sigma,K,0);" }{MPLTEXT 1 0 0 "" } }{PARA 11 "" 1 "" {XPPMATH 20 ",$**#\"\"\"\"\"#F%)F&#F%\"\"$F%,**(F&F% )F)F$F%I\"yG6\"F%F%*&F&F%F,F%!\"\"*&\"\"&F%F-F%F%\"\"'F0F%F-F0F0" }} {PARA 11 "" 1 "" {XPPMATH 20 "<$*&,6**\"#_\"\"\")\"\"$#F'\"\"#F')F+#F+ F)F')I\"yG6\"\"\"%F'!\"\"**\"%K;F'F(F'F,F')F/F)F'F'*(\"$p\"F'F,F'F.F'F '**\"%kWF')F/F+F'F(F'F,F'F2*(\"%3MF'F,F'F5F'F'**\"%?VF'F/F'F(F'F,F'F'* (\"&G;\"F'F,F'F:F'F2*(\"%S9F'F,F'F(F'F2*(\"&'47F'F/F'F,F'F'*&\"%KSF'F, F'F2F',0*(\"$w\"F'F5F'F(F'F'*&\"$%[F'F.F'F'*(\"$o$F'F(F'F:F'F2*&\"%S:F 'F5F'F2*(\"$#>F'F(F'F/F'F'*&\"%!o\"F'F:F'F'*&\"$C'F'F/F'F2F2**)*(,&I\" xGF0F'**F*F')F+#F'F)F',**(F+F'F(F'F/F'F'*&F+F'F(F'F2*&\"\"&F'F/F'F'\" \"'F2F'F/F2F'F',.**\"%e9F'F:F'F(F'F,F'F2**\"%[LF'F/F'F(F'F,F'F'*(\"%TA F'F,F'F:F'F2*(\"%W>F'F,F'F(F'F2*(\"%g[F'F/F'F,F'F'*&\"%#f#F'F,F'F2F2,( *(\"\")F'F(F'F/F'F'*&\"#WF'F:F'F'*&\"#[F'F/F'F2F'F*F'F^oF'F[pF2FYF2" } }}{EXCHG {PARA 0 "" 0 "" {TEXT 217 124 "The singularity is in X^(-1/2) , so the coeffs are in n^(-1/2) =n^2*n^(-5/2). So the average number o f edges in islands with" }{TEXT 217 11 " reef edges" }{TEXT 217 28 " \+ weighted y>y_critic is n^2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 217 157 "Remains to analyze the situation for y " 0 "" {MPLTEXT 1 0 50 "eqKxx:=factor(resultant(subs(x=x-xx,eqK),Dx1,x)):\n" }{MPLTEXT 1 0 17 "degree(eqKxx,K);\n" }{MPLTEXT 1 0 31 "eqKKxx:=collec t(subs(K=1/KK,KK^" }{MPLTEXT 1 0 15 "degree(eqKxx,K)" }{MPLTEXT 1 0 19 "*eqKxx),KK,factor):" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "puisK:=puiseux(eqKKxx,xx=0,K K,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$*&)*&I#xxG6\"\"\"\"-I'RootOfG 6$%*protectedGI(_syslibGF'6#,2*&,P*&\"*^j+M&F()I\"yGF'\"\"$F(F(*&\"(V# G>v#F(F5F(F(*&\")gOKVF()F5\"\"*F(F(*&\"*3#3 )=\"F()F5\"\")F(F(*&\")N*>`\"F()F5\"\"'F(F(*&\"(hF/(F()F5\"#5F(F(*&\"* ?Xyf&F()F6#F(FGF(F?*(\"*eiDh\"F(F4F(FfnF(F(*(\"*+dP!oF(FfnF(FFF(F(*(\" *O%R\"y%F(FfnF(F5F(F(*(\"(+bT$F(FfnF(FXF(F(*(\")uL6EF(FfnF(FLF(F(*(\") GG]uF(FfnF(FPF(F(*(\"*EO&e>F(FfnF(FBF(F(*(\")s\"39%F(FfnF(FTF(F?*(\"*a o&y?F(FfnF(F=F(F?*(\"*wrg5)F(FfnF()F5\"\"%F(F?\"+O:`L;F?*&\"+\\Hg68F(F \\pF(F?F()I#_ZGF+FGF(F(*&,N**\"(C8z\"F(FfnF()FG#F(F6F(F9F(F(**\"(_8P'F (FfnF(FgpF(FXF(F?**\"(ki+*F(FfnF(FgpF(FLF(F?**\"(CAb(F(FgpF(FfnF(FPF(F (**\"(G8p(F(FfnF(FgpF(FBF(F(**\")Wj6EF(FfnF(FgpF(FTF(F(**\"(O#HBF(FfnF (FgpF(F=F(F(**\")Cu7;F(FfnF(FgpF(F\\pF(F?**\")!yY`$F(F4F(FfnF(FgpF(F?* *\")CJ'4#F(F5F(FfnF(FgpF(F(*(\"'EvzF(FgpF(FFF(F(*(\")K3&z#F(FgpF(F\\pF (F?*(\")m`FhF(FgpF(F4F(F?**\"'S#3%F(FfnF(FgpF(FFF(F(*(\")/nGOF(FgpF(F5 F(F(*(\"(c=.%F(FgpF(F=F(F(*(\"(u!\\7F(FgpF(F9F(F?*(\"(mNS$F(FgpF(FXF(F (*(\")q#R7$F(FgpF(FLF(F?*(\")A`g=F(FgpF(FPF(F(*(\")%pYL\"F(FgpF(FBF(F( *(\")uGCXF(FgpF(FTF(F(F(FbpF(F(**\"'ODF(FgpF(FFF(F? *(\"'+ICF(FgpF(F5F(F(F(FbpF(F(*(\"&[X$F(FesF(FPF(F(**\"&G4\"F(FfnF(Fes F(FPF(F(**\"&C]\"F(FfnF(FesF(FBF(F(*(\"%oAF(FesF(FTF(F?*(\"&W4\"F(FesF (F=F(F?*(\"&+3\"F(FesF(F\\pF(F(**\"%3YF(FfnF(FesF(F\\pF(F(**\"&!GNF(Ff nF(FesF(FTF(F?**\"%onF(FfnF(FesF(F=F(F(*(\"&Ot#F(FesF(FBF(F?" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 103 "The singularity is in X^(1/2) unless the coeff is 0. As we see below the coeff is not 0 for y " 0 "" {MPLTEXT 1 0 18 "subs(_Z=0,puisK);\n" }{MPLTEXT 1 0 165 "map(s implify,\{solve(-172536*sqrt(3)*2^(2/3)*y^9-3360*sqrt(3)*2^(2/3)*y^10+ 290920*sqrt(3)*2^(2/3)*y^11+504324*2^(2/3)*y^11-6564*2^(2/3)*y^10-2986 56*2^(2/3)*y^9,y)\});\n" }{MPLTEXT 1 0 10 "evalf(%);\n" }{MPLTEXT 1 0 9 "ycritic;\n" }{MPLTEXT 1 0 10 "evalf(%);\n" }{MPLTEXT 1 0 251 "map(s implify,\{solve(34548*2^(2/3)*y^8+10928*sqrt(3)*2^(2/3)*y^8+15024*sqrt (3)*2^(2/3)*y^7-2268*2^(2/3)*y^6-10944*2^(2/3)*y^5+10800*2^(2/3)*y^4+4 608*sqrt(3)*2^(2/3)*y^4-35280*sqrt(3)*2^(2/3)*y^6+6768*sqrt(3)*2^(2/3) *y^5-27336*2^(2/3)*y^7,y)\});evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$*&)*&I#xxG6\"\"\"\"-I'RootOfG6$%*protectedGI(_syslibGF'6#,.**\"'OD F(F;*(\"'c')HF(F6F(F8F(F;F;F4F(F)F(-F*6#,6*(\"&[X$F(F6F()F9\"\")F(F( **\"&G4\"F(F2F(F6F(FOF(F(**\"&C]\"F(F2F(F6F()F9\"\"(F(F(*(\"%oAF(F6F() F9\"\"'F(F;*(\"&W4\"F(F6F()F9\"\"&F(F;*(\"&+3\"F(F6F()F9\"\"%F(F(**\"% 3YF(F2F(F6F(F[oF(F(**\"&!GNF(F2F(F6F(FYF(F;**\"%onF(F2F(F6F(FgnF(F(*( \"&Ot#F(F6F(FUF(F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%\"\"!,$*(\"\"'\" \"\",&*&\"%.%*F')\"\"$#F'\"\"#F'F'\"&7j\"F'F',&*&\"&IF(F'F+F'F'\"'\"3E \"F'!\"\"F',$*(F,F',&*&\"&E&=F'F+F'F'\"&x?$F'F'F0F4F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%$!+3%zqj(!#5$\"\"!F'$\"+Aw!*fxF%" }}{PARA 11 "" 1 " " {XPPMATH 20 ",&#\"#7\"#6\"\"\"*&#\"\"#F%F&)\"\"$#F&F)F&!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "$\"+Bw!*fx!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%\"\"!,&#\"\"$\"#8!\"\"*&#\"\"%F'\"\"\")F&#F,\"\"#F,F(,& #\"#7\"#6F,*&#F/F3F,F-F,F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%$!+5%zqj (!#5$\"\"!F'$\"+Bw!*fxF%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 188 "So KK is in X^(1/2) and K is in X^\{-1/2\}. So its \+ coeffs are in n^\{-1/2\} =n* n^\{-3/2\}. So the average number of edg es in islands with reef edges weighted by y " 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 217 83 "The algebr aic equation for K (from which the asymptotic of coeffecient is obtain ed)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2249 "eqK:=(1/2*(2*x*y+2 *y*sqrt(3)*2^(1/3)-2*sqrt(3)*2^(1/3)+5*2^(1/3)*y-6*2^(1/3)))*(2*sqrt(3 )*2^(2/3)*y^3+12*sqrt(3)*2^(1/3)*x*y^3-2*y^2*sqrt(3)*2^(2/3)-12*sqrt(3 )*2^(1/3)*x*y^2-9*2^(2/3)*y^3-18*2^(1/3)*x*y^3-4*x^2*y^3-18*y*sqrt(3)* 2^(2/3)+12*2^(1/3)*x*y^2+18*sqrt(3)*2^(2/3)+27*y*2^(2/3)+6*2^(1/3)*x*y -18*2^(2/3))*(y-1)*y^2*K^3+(1/2*(2*x*y+2*y*sqrt(3)*2^(1/3)-2*sqrt(3)*2 ^(1/3)+5*2^(1/3)*y-6*2^(1/3)))*(-x*y^2+12*2^(1/3)*y-12*2^(1/3))*(2*sqr t(3)*2^(2/3)*y^3+12*sqrt(3)*2^(1/3)*x*y^3-2*y^2*sqrt(3)*2^(2/3)-12*sqr t(3)*2^(1/3)*x*y^2-9*2^(2/3)*y^3-18*2^(1/3)*x*y^3-4*x^2*y^3-18*y*sqrt( 3)*2^(2/3)+12*2^(1/3)*x*y^2+18*sqrt(3)*2^(2/3)+27*y*2^(2/3)+6*2^(1/3)* x*y-18*2^(2/3))*x*y*K^2+6*2^(1/3)*(-8*sqrt(3)*2^(1/3)*x^3*y^6-303*2^(1 /3)*y^5+540*2^(1/3)*y^4-1080*y*sqrt(3)*2^(1/3)-4*sqrt(3)*x*y^7+232*sqr t(3)*x*y^5+672*sqrt(3)*x*y^3-120*y^3*sqrt(3)*2^(1/3)-2*sqrt(3)*2^(1/3) *y^7-26*sqrt(3)*2^(1/3)*y^5-756*sqrt(3)*x*y^4+792*sqrt(3)*2^(1/3)*y^2- 8*sqrt(3)*2^(1/3)*y^6-144*sqrt(3)*x*y+4*sqrt(3)*2^(2/3)*x^2*y^7-18*sqr t(3)*2^(2/3)*x^2*y^6+98*sqrt(3)*2^(2/3)*x^2*y^5+8*sqrt(3)*2^(1/3)*x^3* y^5-132*sqrt(3)*2^(2/3)*x^2*y^4+48*sqrt(3)*2^(2/3)*x^2*y^3+2^(1/3)*y^7 +14*2^(1/3)*y^6-18*2^(1/3)*y^3-450*2^(1/3)*y^2+12*sqrt(3)*2^(1/3)*y^4+ 10*x*y^7-9*2^(2/3)*x^2*y^7+27*2^(2/3)*x^2*y^6+28*2^(1/3)*x^3*y^6-222*2 ^(2/3)*x^2*y^5-54*2^(1/3)*x^3*y^5+348*2^(2/3)*x^2*y^4+24*2^(1/3)*x^3*y ^4-108*2^(2/3)*x^2*y^3-36*2^(2/3)*x^2*y^2+4*x^4*y^6+420*x*y^4+360*x*y^ 3+216*2^(1/3)*y+53*x*y^6-339*x*y^5-936*x*y^2+432*x*y+432*sqrt(3)*2^(1/ 3))*x^2*K+(9/2)*2^(2/3)*(-576*sqrt(3)*x-16*sqrt(3)*2^(1/3)*x^3*y^4+16* sqrt(3)*2^(1/3)*x^3*y^3-240*sqrt(3)*2^(2/3)*x^2*y^2+96*sqrt(3)*2^(2/3) *x^2*y-6*2^(1/3)*y^5+25*2^(1/3)*y^4-12*sqrt(3)*x*y^5+456*sqrt(3)*x*y^3 +44*y^3*sqrt(3)*2^(1/3)-4*sqrt(3)*2^(1/3)*y^5-56*sqrt(3)*x*y^4-24*sqrt (3)*2^(1/3)*y^2+4*sqrt(3)*2^(1/3)*y^6+1440*sqrt(3)*x*y+16*sqrt(3)*2^(2 /3)*x^2*y^5-40*sqrt(3)*2^(2/3)*x^2*y^4+168*sqrt(3)*2^(2/3)*x^2*y^3-7*2 ^(1/3)*y^6-588*2^(1/3)*y^3+1152*2^(1/3)*y^2-20*sqrt(3)*2^(1/3)*y^4-124 8*sqrt(3)*x*y^2-36*2^(2/3)*x^2*y^5+60*2^(2/3)*x^2*y^4+56*2^(1/3)*x^3*y ^4-360*2^(2/3)*x^2*y^3-96*2^(1/3)*x^3*y^3+624*2^(2/3)*x^2*y^2+48*2^(1/ 3)*x^3*y^2-288*2^(2/3)*x^2*y+8*x^4*y^4+220*x*y^4-768*x*y^3-576*2^(1/3) *y-4*sqrt(3)*x*y^6+8*x*y^6+36*x*y^5+792*x*y^2-288*x*y)*y*x^3;" } {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**.#\"\"\"\"\"#F%,, *(F&F%I\"xG6\"F%I\"yGF*F%F%**F&F%)\"\"$F$F%)F&#F%F.F%F+F%F%*(F&F%F/F%F -F%!\"\"*(\"\"&F%F/F%F+F%F%*&\"\"'F%F/F%F2F%,<**F&F%)F+F.F%)F&#F&F.F%F -F%F%*,\"#7F%F-F%F/F%F)F%F9F%F%**F&F%)F+F&F%F:F%F-F%F2*(\"\"*F%F9F%F:F %F2*,F=F%F-F%F/F%F)F%F?F%F2**\"#=F%F/F%F)F%F9F%F2*(\"\"%F%)F)F&F%F9F%F 2**FDF%F+F%F:F%F-F%F2**F=F%F/F%F)F%F?F%F%*(FDF%F:F%F-F%F%*(\"#FF%F:F%F +F%F%**F6F%F/F%F)F%F+F%F%*&FDF%F:F%F2F%,&F+F%F%F2F%F?F%)I\"KGF*F.F%F%* 0F$F%F'F%,(*&F?F%F)F%F2*(F=F%F/F%F+F%F%*&F=F%F/F%F2F%F7F%F)F%F+F%)FQF& F%F%*,F6F%F/F%,dp*(\"#`F%F)F%)F+F6F%F%*(\"#5F%F)F%)F+\"\"(F%F%*(FFF%)F )FFF%FfnF%F%*,\"\")F%F-F%F/F%)F)F.F%)F+F4F%F%*,\"$K\"F%F-F%F:F%FGF%)F+ FFF%F2*,\"#[F%F-F%F:F%FGF%F9F%F%*,F^oF%F-F%F/F%F_oF%FfnF%F2*,FFF%F-F%F :F%FGF%FinF%F%*,FDF%F-F%F:F%FGF%FfnF%F2*,\"#)*F%F-F%F:F%FGF%F`oF%F%*( \"$O*F%F?F%F)F%F2*&F/F%FinF%F%*(\"#9F%F/F%FfnF%F%**\"#EF%F-F%F/F%F`oF% F2**F=F%F-F%F/F%FcoF%F%**FAF%F:F%FGF%FinF%F2**\"#aF%F/F%F_oF%F`oF%F2** \"$[$F%F:F%FGF%FcoF%F%**\"#GF%F/F%F_oF%FfnF%F%**\"#CF%F/F%F_oF%FcoF%F% **\"$A#F%F:F%FGF%F`oF%F2**\"$3\"F%F:F%FGF%F9F%F2**\"#OF%F:F%FGF%F?F%F2 **\"$?\"F%F9F%F/F%F-F%F2**FLF%F:F%FGF%FfnF%F%**\"$c(F%F-F%F)F%FcoF%F2* *\"$K#F%F-F%F)F%F`oF%F%**\"$s'F%F-F%F)F%F9F%F%*(\"$;#F%F/F%F+F%F%**FFF %F-F%F)F%FinF%F2*(\"$K%F%F/F%F-F%F%*(F_rF%F)F%F+F%F%**\"$W\"F%F-F%F)F% F+F%F2*(FDF%F/F%F9F%F2*(\"$]%F%F/F%F?F%F2**F&F%F/F%F-F%FinF%F2**F^oF%F /F%F-F%FfnF%F2*(\"$?%F%F)F%FcoF%F%*(\"$g$F%F)F%F9F%F%**\"$#zF%F?F%F/F% F-F%F%**\"%!3\"F%F-F%F/F%F+F%F2*(\"$.$F%F/F%F`oF%F2*(\"$S&F%F/F%FcoF%F %*(\"$R$F%F)F%F`oF%F2F%FGF%FQF%F%*,#FAF&F%F:F%,\\p*(F^oF%F)F%FfnF%F%*, \"#SF%F-F%F:F%FGF%FcoF%F2*,\"$o\"F%F-F%F:F%FGF%F9F%F%*,\"#;F%F-F%F/F%F _oF%FcoF%F2*,F_tF%F-F%F/F%F_oF%F9F%F%*,\"$S#F%F-F%F:F%FGF%F?F%F2*,\"#' *F%F-F%F:F%FGF%F+F%F%*,F_tF%F-F%F:F%FGF%F`oF%F%**FFF%F-F%F)F%FfnF%F2*( F]sF%F?F%F)F%F%*(F^oF%F\\oF%FcoF%F%*(FjnF%F/F%FfnF%F2**FFF%F-F%F/F%F`o F%F2**\"#?F%F-F%F/F%FcoF%F2**\"$)GF%F:F%FGF%F+F%F2**FeoF%F/F%F_oF%F?F% F%**\"#gF%F:F%FGF%FcoF%F%**\"#cF%F/F%F_oF%FcoF%F%**FaqF%F:F%FGF%F`oF%F 2**F[sF%F:F%FGF%F9F%F2**\"$C'F%F:F%FGF%F?F%F%**FdtF%F/F%F_oF%F9F%F2** \"#WF%F9F%F/F%F-F%F%**FcuF%F-F%F)F%FcoF%F2**F=F%F-F%F)F%F`oF%F2**\"$c% F%F-F%F)F%F9F%F%*(\"$w&F%F/F%F+F%F2**\"%[7F%F-F%F)F%F?F%F2*(F^uF%F)F%F +F%F2**\"%S9F%F-F%F)F%F+F%F%*(\"$)eF%F/F%F9F%F2*(\"%_6F%F/F%F?F%F%*(F` vF%F-F%F)F%F2**FFF%F/F%F-F%FfnF%F%*(\"$?#F%F)F%FcoF%F%*(\"$o(F%F)F%F9F %F2**F[qF%F?F%F/F%F-F%F2*(F6F%F/F%F`oF%F2*(\"#DF%F/F%FcoF%F%*(FaqF%F)F %F`oF%F%F%F+F%F_oF%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "B3 :=coeff(eqK,K,3)/(y^2*(y-1));\n" }{MPLTEXT 1 0 20 "B1:=coeff(eqK,K,1); \n" }{MPLTEXT 1 0 19 "B0:=coeff(eqK,K,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*(#\"\"\"\"\"#F%,<**F&F%)I\"yG6\"\"\"$F%)F&#F&F,F%)F,F$ F%F%*,\"#7F%F/F%)F&#F%F,F%I\"xGF+F%F)F%F%**F&F%)F*F&F%F-F%F/F%!\"\"*( \"\"*F%F)F%F-F%F7*,F1F%F/F%F2F%F4F%F6F%F7**\"#=F%F2F%F4F%F)F%F7*(\"\"% F%)F4F&F%F)F%F7**F " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 221 30 "analysis of L(x,y)=U_z(x,y,z0)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "e qU:=collect(eqU,U,factor);\n" }{MPLTEXT 1 0 88 "eqUdiff:=collect(subs( U(z)=U,subs(diff(U(z),z)=DU,diff(subs(U=U(z),eqU),z))),U,factor);\n" } }{PARA 11 "" 1 "" {XPPMATH 20 ",6*.\"#k\"\"\")I\"xG6\"\"#7F%)I\"yGF(\" \"'F%)I\"zGF(F)F%),&F+F%F%!\"\"F,F%)I\"UGF(\"\"*F%F1*0\"$#>F%)F'\"#5F% )F+\"\"&F%)F.F8F%)F0\"\"%F%,(*&F'F%F+F%F%*(\"\"#F%F+F%F.F%F%*&FAF%F.F% F1F%)F3\"\")F%F1*0F6F%)F'FDF%)F+F=F%)F.FDF%)F0FAF%,8*()F'FAF%FGF%)F.\" \"$F%F%**FNF%FLF%)F+FNF%FMF%F1**FNF%FLF%)F+FAF%FMF%F%*(FLF%F+F%FMF%F1* (F'F%FPF%F.F%F%*&FLF%FRF%F1**F,F%F'F%FRF%F.F%F1*(F:F%FRF%)F.FAF%F1**F: F%F'F%F+F%F.F%F%*(F8F%F+F%FXF%F%*&F:F%FXF%F1F%)F3\"\"(F%F%*,)F'F,F%FPF %)F.F,F%,V**\"#'*F%)F'FNF%F*F%FMF%F%**\"$o(F%F^oF%F9F%FMF%F1**\"$g*F%F LF%F9F%)F.F=F%F1**\"%GF%F'F%F+F%FXF%F%*(FgoF%F+F%FMF%F%*&F]qF%FMF%F1F%)F3F ,F%F1*.FAF%)F'F=F%FRF%FcoF%,\\o**F]oF%FiqF%F*F%FjnF%F%**FcpF%FiqF%F9F% FjnF%F1**\"$w&F%FiqF%FGF%FjnF%F%**FcpF%FiqF%FPF%FjnF%F1**F]oF%FiqF%F9F %FMF%F%**F]oF%FiqF%FRF%FjnF%F%**FcpF%F^oF%F9F%FcoF%F%**\"$)GF%FiqF%FGF %FMF%F1**FeoF%F^oF%FGF%FcoF%F1**FboF%FLF%FGF%)F.F:F%F1**F6F%FiqF%FPF%F MF%F%**\"%/BF%F^oF%FPF%FcoF%F%**\"%!)GF%FLF%FPF%FgrF%F%*&FiqF%F9F%F%** FAF%F^oF%F9F%F.F%F%**FboF%F^oF%FRF%FcoF%F1**F\\sF%FLF%FRF%FgrF%F1*(FAF %FiqF%FGF%F1**F,F%F^oF%FGF%F.F%F1**F]oF%FLF%FGF%FXF%F%**FboF%FLF%F+F%F grF%F%*&FiqF%FPF%F%**\"#!*F%F^oF%FPF%F.F%F1**F^rF%FLF%FPF%FXF%F1**F^rF %F'F%FPF%FMF%F1**FAF%F^oF%FRF%F.F%F1**F^rF%FLF%FRF%FXF%F%**FjoF%F'F%FR F%FMF%F%*(F_pF%FRF%FcoF%F%**FboF%F'F%F+F%FMF%F1*(FboF%F+F%FcoF%F1*&F_p F%FcoF%F%F%)F3F:F%F1*,FLF%F+F%FXF%,`o**F6F%)F'F:F%F*F%FjnF%F%**F`oF%Fe tF%F9F%FjnF%F1**F`oF%FiqF%F9F%)F.FgnF%F1**FboF%FetF%FGF%FjnF%F%**FjrF% FiqF%FGF%FhtF%F%**FAF%FetF%F*F%FMF%F%**FcpF%FetF%FPF%FjnF%F1**FjrF%Fiq F%FPF%FhtF%F1**F,F%FetF%F9F%FMF%F1**F6F%FiqF%F9F%FcoF%F%**F`oF%FiqF%FR F%FhtF%F%**FgsF%FetF%FGF%FMF%F1**FipF%FiqF%FGF%FcoF%F1**FeoF%F^oF%FGF% FgrF%F1**FAF%FetF%FPF%FMF%F1**FipF%FiqF%FPF%FcoF%F%**FaqF%F^oF%FPF%Fgr F%F%**FdqF%FLF%FPF%FjnF%F%**FAF%FiqF%F9F%F.F%F%**F\\sF%F^oF%FRF%FgrF%F 1**FgoF%FLF%FRF%FjnF%F1*&FetF%FGF%F1**FDF%FiqF%FGF%F.F%F1**F,F%F^oF%FG 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"7#+'I\"xG6\",$*&$\"+p'ypw#!\"*\"\"\"I\"yG6\"\"\"\"\"\"\"\"\"\"-I\" OG%*protectedG6#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 217 30 "So eq2 is the correct equation" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "#collect(eq2,L,factor);\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1925 "eqL:=(10*sqrt(3)*2^(1/3)+2*x*sqrt(3)-13*x*y-12*2^(1/3)-5*x)*(8*y *sqrt(3)*2^(2/3)+36*y*2^(2/3)-8*2^(2/3)*sqrt(3)-36*2^(2/3)-30*sqrt(3)* 2^(1/3)*x*y^2+72*2^(1/3)*x*y^2+48*sqrt(3)*2^(1/3)*x*y-60*2^(1/3)*x*y+2 3*x^2*y^3-73*x^2*y^2-6*sqrt(3)*x^2*y^2+32*x^2*y+48*sqrt(3)*x^2*y)*y^2* (y-1)^2*x^4*L^3-4*sqrt(3)*(y*sqrt(3)*2^(1/3)-sqrt(3)*2^(1/3)+6*x*y)*(1 0*sqrt(3)*2^(1/3)+2*x*sqrt(3)-13*x*y-12*2^(1/3)-5*x)*(8*y*sqrt(3)*2^(2 /3)+36*y*2^(2/3)-8*2^(2/3)*sqrt(3)-36*2^(2/3)-30*sqrt(3)*2^(1/3)*x*y^2 +72*2^(1/3)*x*y^2+48*sqrt(3)*2^(1/3)*x*y-60*2^(1/3)*x*y+23*x^2*y^3-73* x^2*y^2-6*sqrt(3)*x^2*y^2+32*x^2*y+48*sqrt(3)*x^2*y)*y*x^2*L^2-(24/11* (74*sqrt(3)-81))*(-89*x^5*y^5+219*x^5*y^4-143*x^5*y^3+936*sqrt(3)*2^(1 /3)*x*y^2-912*sqrt(3)*2^(1/3)*x*y+346*x^3*y^4*2^(2/3)*sqrt(3)+126*2^(2 /3)*x^3*y^2*sqrt(3)-1200*2^(2/3)*x^3*y^3*sqrt(3)-218*2^(1/3)*x^4*y^5*s qrt(3)+858*2^(1/3)*x^4*y^4*sqrt(3)-1302*2^(1/3)*x^4*y^3*sqrt(3)-202*2^ (1/3)*x^4*y^2*sqrt(3)+4*x^5*y^2*sqrt(3)+108*x^5*y^4*sqrt(3)+24*2^(2/3) *sqrt(3)+11*x^5*y^6-132*x^5*y^3*sqrt(3)+672*sqrt(3)*x^2*y^3+20*x^5*y^5 *sqrt(3)+8*2^(2/3)*sqrt(3)*x^3*y-384*2^(1/3)*x^4*y^5+1254*2^(1/3)*x^4* y^4-1212*2^(1/3)*x^4*y^3-336*sqrt(3)*x^2*y^2-24*y*sqrt(3)*2^(2/3)-90*2 ^(1/3)*x^4*y^2+2*x^5*y^2-192*sqrt(3)*x^2*y-24*sqrt(3)*2^(1/3)*x-600*2^ (2/3)*x^3*y^3+1812*2^(2/3)*x^3*y^2+48*2^(2/3)*x^3*y-144*y*2^(2/3)+4584 *x^2*y^2-12*2^(1/3)*x-828*x^3*y^4*2^(2/3)+204*2^(1/3)*x*y^2-4020*x^2*y ^3+300*x^2*y-192*2^(1/3)*x*y+144*2^(2/3))*L-(144/37*(132*sqrt(3)+203)) *(3816-66*2^(1/3)*x^2+2232*y*sqrt(3)+318*2^(2/3)*x-4800*sqrt(3)*x^2*y^ 2*2^(1/3)+8958*x*y^2*sqrt(3)*2^(2/3)-2556*sqrt(3)*x^2*2^(1/3)*y-8772*x *y*sqrt(3)*2^(2/3)+620*sqrt(3)*x^2*y^3*2^(1/3)-2232*sqrt(3)-111*x^3*y^ 4-516*x^3*y^3*sqrt(3)-186*2^(2/3)*sqrt(3)*x-48*x^3*y*sqrt(3)+40*sqrt(3 )*x^2*2^(1/3)+2940*x^3*sqrt(3)*y^2-5517*y^2*x^3+1251*x^3*y^3+57*x^3*y- 1134*x^2*y^3*2^(1/3)+8142*x^2*y^2*2^(1/3)-15702*x*y^2*2^(2/3)+4506*x^2 *y*2^(1/3)+15384*x*2^(2/3)*y-3816*y)*x*y;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**.,,*(\"#5\"\"\")\"\"$#F'\"\"#F')F+#F'F)F'F'*(F+F'I\"xG 6\"F'F(F'F'*(\"#8F'F/F'I\"yGF0F'!\"\"*&\"#7F'F,F'F4*&\"\"&F'F/F'F4F',< **\"\")F'F3F'F(F')F+#F+F)F'F'*(\"#OF'F3F'F#F'FgoF')F3 FYF'F'*(\"$V\"F'FgoF'FNF'F4*(F_oF'FgoF')F3FRF'F'*,\"$Y$F')F/F)F'F[pF'F F'F(F'FMF' F3F'F4**\"$+'F'F " 0 "" {MPLTEXT 1 0 28 "Dlx:=factor(subs(y=1-y,Dx));" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,@**\"\"#\"\"\")\"\"$#F&F%F&)F%#F%F(F&)I\"yG6\"F(F&F&*, \"#7F&F'F&)F%#F&F(F&I\"xGF.F&F,F&F&**\"\"%F&)F-F%F&F'F&F*F&!\"\"*(\"\" *F&F*F&F,F&F7*,\"#CF&F'F&F1F&F3F&F6F&F7**\"#=F&F1F&F3F&F,F&F7*(F5F&)F3 F%F&F,F&F7**\"#;F&F-F&F'F&F*F&F7*(\"#FF&F*F&F6F&F&*,F0F&F'F&F1F&F3F&F- F&F&**\"#UF&F1F&F3F&F6F&F&*(F0F&F?F&F6F&F&**F;F&F1F&F3F&F-F&F7*(F0F&F? F&F-F&F7*&F5F&F?F&F&F&,,**F%F&F-F&F'F&F1F&F&*(\"\"&F&F1F&F-F&F&*(F%F&F 3F&F-F&F&*$F1F&F&*&F%F&F3F&F7F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "DF;\n" }{MPLTEXT 1 0 7 "factor(" }{MPLTEXT 1 0 29 "coeff(eqL,L, 3)/DF);latex(%);\n" }{MPLTEXT 1 0 7 "factor(" }{MPLTEXT 1 0 19 "coeff( eqL,L,2)/DF);" }{MPLTEXT 1 0 9 "latex(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,(*(I\"yG6\"\"\"\",,*(\"\"'F')\"\"$#F'\"\"#F'F%F'F'*&\" #BF')F%F.F'!\"\"*&\"#[F'F+F'F2*&\"#tF'F%F'F'\"#KF2F')I\"xGF&F.F'F2*.F. F')F.#F'F,F',&*&\"\"&F'F+F'F'\"#7F2F'F%F',&*&F.F'F+F'F'*&F,F'F%F'F'F'F 9F'F2**\"\"%F')F.#F.F,F',&FBF'\"\"*F'F',&F%F'F'F2F'F'F',(*&,(FBF'*&\"# 8F'F%F'F2F?F2F'F9F'F'*(\"#5F'F+F'F;F'F'*&F@F'F;F'F2F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "*()I\"yG6\"\"\"#\"\"\"),&F$F'F'!\"\"F&F')I\"xGF%\"\"% F'" }}{PARA 6 "" 1 "" {TEXT 226 38 "\{y\}^\{2\} \\left( y-1 \\right) ^ \{2\}\{x\}^\{4\}" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*,\"\"%\"\"\")\"\" $#F%\"\"#F%,(*(I\"yG6\"F%F&F%)F)#F%F'F%F%*&F&F%F.F%!\"\"*(\"\"'F%I\"xG F-F%F,F%F%F%F,F%)F4F)F%F1" }}{PARA 6 "" 1 "" {TEXT 226 70 "-4\\,\\sqrt \{3\} \\left( y\\sqrt \{3\}\\sqrt [3]\{2\}-\\sqrt \{3\}\\sqrt [3]\{2 \}+6\\,x" }{TEXT 226 1 "\n" }{TEXT 226 18 "y \\right) y\{x\}^\{2\}" }} }{EXCHG {PARA 0 "" 0 "" {TEXT 217 191 "We know that rho is the same as for H, that is, a sol of DiscF (hence DF). Moreover, when x=rho, two \+ branches disapear at infinity. So the non-infinite branch is non-singu lar. So H_rho=infty." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 222 8 "Case y= " }{XPPEDIT 2 0 "Typesetting:-mrow(Typesetting:- mo(\"&uminus0;\", mathvariant = \"normal\", fence = \"false\", separat or = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \" 0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mfrac(Typesetti ng:-mrow(Typesetting:-mn(\"1\", mathvariant = \"normal\")), Typesettin g:-mrow(Typesetting:-mn(\"11\", mathvariant = \"normal\")), linethickn ess = \"1\", denomalign = \"center\", numalign = \"center\", bevelled \+ = \"false\"), 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:-mfrac(Typesetting:-mn(\"2\", mathvariant = \"normal\"), Typesetti ng:-mn(\"11\", mathvariant = \"normal\"), linethickness = \"1\", denom align = \"center\", numalign = \"center\", bevelled = \"false\"), Type setting:-mo(\"⁢\", mathvariant = \"normal\", fence = \" false\", separator = \"false\", stretchy = \"false\", symmetric = \"fa lse\", largeop = \"false\", movablelimits = \"false\", accent = \"fals e\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-msqrt(Types etting:-mn(\"3\", mathvariant = \"normal\")));" "-I%mrowG6#/I+modulena meG6\"I,TypesettingGI(_syslibGF'6(-I#moGF$6-Q*&uminus0;F'/%,mathvarian tGQ'normalF'/%&fenceGQ&falseF'/%*separatorGF4/%)stretchyGF4/%*symmetri cGF4/%(largeopGF4/%.movablelimitsGF4/%'accentGF4/%'lspaceGQ,0.2222222e mF'/%'rspaceGFC-I&mfracGF$6(-F#6#-I#mnGF$6$Q\"1F'F/-F#6#-FL6$Q#11F'F// %.linethicknessGFN/%+denomalignGQ'centerF'/%)numalignGFX/%)bevelledGF4 -F,6-Q\"+F'F/F2F5F7F9F;F=F?FAFD-FG6(-FL6$Q\"2F'F/FQFTFVFYFen-F,6-Q1&In visibleTimes;F'F/F2F5F7F9F;F=F?/FBQ&0.0emF'/FEFco-I&msqrtGF$6#-FL6$Q\" 3F'F/" }{TEXT 222 1 "." }}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 6 "y spe:=" }{XPPEDIT 19 1 "Typesetting:-mrow(Typesetting:-mi(\"\", italic \+ = \"true\", mathvariant = \"italic\"), Typesetting:-mi(\"\", italic = \+ \"true\", mathvariant = \"italic\"), Typesetting:-mi(\"\", italic = \" true\", mathvariant = \"italic\"));" "-I%mrowG6#/I+modulenameG6\"I,Typ esettingGI(_syslibGF'6%-I#miGF$6%Q!F'/%'italicGQ%trueF'/%,mathvariantG Q'italicF'F+F+" }{MPLTEXT 1 0 22 "-1/11+(2/11)*sqrt(3);\n" }{MPLTEXT 1 0 45 "subspe:=A->factor(simplify(subs(y=yspe,A))):\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&#\"\"\"\"#6!\"\"*&#\"\"#F%F$)\"\"$#F$F)F$F$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "eqLspe:=factor(subspe(eqL))/ (-144/67094329753)/(2708360*sqrt(3)-9065829);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",fo*,\"']d5\"\"\")I\"LG6\"\"\"#F%)F)#F)\"\"$F%)F,#F%F)F%) I\"xGF(\"\"%F%F%*,\"&)R?F%F&F%)F)#F%F,F%F-F%)F0\"\"&F%F%*,\"'#f?$F%F'F %F4F%F-F%F/F%F%*(\"%7vF%F&F%)F0\"\"'F%F%*(\"%+%*F%)F'F,F%F/F%F%*(\"';a QF%F4F%)F0F,F%F%*(\"([YQ'F%F'F%)F0F)F%F%*,\"%+sF%F@F%F*F%F-F%F6F%F%*( \")G(35\"F%F*F%FFF%F%*,\"(oxb%F%F'F%F-F%F4F%F0F%F%*(\"')oB$F%F'F%F*F%F %*,\"%5'F%F@F%F-F%F/ F%FU**\"'!)*3#F%F&F%F*F%F/F%FU**\"&[H%F%F'F%F4F%F/F%F%*(\"'/4]F%F/F%F- F%FU*(\"&;z\"F%F'F%F6F%FU*(\"$8$F%F@F%)F0\"\"(F%FU**\"())3'RF%F'F%F4F% F0F%FU**\"(#>)4\"F%F'F%F*F%F-F%FU*&\"(Cqi'F%F0F%FU*&\"(on!=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 217 31 "We now do singularity analysis:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "rhospe:=simplify(subspe(s F));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*(\"\"$\"\"\")F$#F%\"\"#F%)F (#F%F$F%F%*&\"\"%F%F)F%!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "factor(puiseux(eqLspe,x=rhospe,L,0));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "<#,$*&),$**\"#7\"\"\")\"\"##\"\"#\"\"$\"\"\",&\"#:!\"\"*& \"#9\"\"\")\"\"$#\"\"\"\"\"#\"\"\"\"\"\"\"\"\",(*(\"\"$\"\"\")\"\"$#\" \"\"\"\"#\"\"\")\"\"##\"\"\"\"\"$\"\"\"\"\"\"*&\"\"%\"\"\")\"\"##\"\" \"\"\"$\"\"\"!\"\"I\"xG6\"!\"\"\"\"\"!\"\"#\"\"\"\"\"$\"\"\",(*(\"\"$ \"\"\")\"\"$#\"\"\"\"\"#\"\"\")\"\"##\"\"\"\"\"$\"\"\"\"\"\"*&\"\"%\" \"\")\"\"##\"\"\"\"\"$\"\"\"!\"\"I\"xG6\"!\"\"!\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 198 "The singularity is in X^\{-2/3\}. This type of singularity in dicates coefficients in n^(-1/3) =n^(4/3)*n^(-5/3). So the average num ber of edges in islands with reef edges weighted y=y_critic is n^4/3." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 50 "We exami ne the behavior for y>y_critic (rho=sigma)" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 4 "sF;\n" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 23 "puiseux(eqL,x=sF,L,0));" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$**\"\"#\" \"\")F$#F%\"\"$F%,&*&\"\"&F%)F(#F%F$F%F%\"\"'!\"\"F%,(*&F$F%F,F%F%*&\" #8F%I\"yG6\"F%F/F+F/F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$,$*,\"\"' \"\"\"I\"yG6\"F&)\"\"##F*\"\"$F&,2*(\"$r#F&)F'F,F&)F,#F&F*F&F&*(\"$n\" F&F1F&)F'F*F&!\"\"*&\"$T&F&F0F&F&*(\"#ZF&F1F&F'F&F6*&\"$`\"F&F5F&F6*& \"#LF&F1F&F6*&\"$$=F&F'F&F&\"#xF&F&,4*(\"#WF&F0F&F1F&F&*&\"$@\"F&)F'\" \"%F&F6*(\"#%)F&F1F&F5F&F6*&\"$?#F&F0F&F&*(\"#OF&F1F&F'F&F&*&\"#!*F&F5 F&F6*&FHF&F1F&F&*&FHF&F'F&F&\"#8F6F6F6,$*2F,F&)F*#F&F,F&F1F&),&F%F&*$F 1F&F6F2F&)*(,,*(\"#5F&F1F&FVF&F&*(F*F&I\"xGF(F&F1F&F&*(FSF&F[oF&F'F&F6 *&\"#7F&FVF&F6*&\"\"&F&F[oF&F6F&,(*&\"#6F&F'F&F6F&F6*&F*F&F1F&F&F&,(Fd oF&*&FSF&F'F&F6F`oF6F6F2F&FeoF&FgnF6FaoF6F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 170 "The singularity is in X^(-1/2), so the coeffs are in n ^(-1/2) =n^2*n^(-5/2). So the average number of edges in ocean with o uter edges weighted y>-1/11+2sqr(3)/11 is n^2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 85 "Remains to analyze the situati on for y<1-ycritic (rho=tF). We know that Lrho=infty. ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "eqLxx:=factor(resultant(subs(x=x-xx ,eqL),DF1,x)):\n" }{MPLTEXT 1 0 17 "degree(eqLxx,L);\n" }{MPLTEXT 1 0 65 "eqLLxx:=collect(subs(L=1/LL,LL^degree(eqLxx,L)*eqLxx),LL,factor):" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "puisL:=puiseux(eqLLxx,xx=0,LL,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$*&)*&I#xxG6\"\"\"\"-I'RootOfG6$%*protectedGI(_syslibGF '6#,F*&,<\"')[T#F(*&\"'.IvF()I\"yGF'\"\"'F(F(*&\"([+.$F()F6\"\"&F(!\" \"*&\"(idG'F()F6\"\"%F(F(*&\"(o\"[\")F()F6\"\"$F(F<*&\"(zq`&F()F6\"\"# F(F(*&\"'+#f#F(F6F(F(*&\"'cI8F()FD#F(FHF(F(*(\"'g7jF(FMF(F:F(F(*(\"(SJ !GF(FMF(F?F(F<*(\"()QyQF(FCF(FMF(F(*(\"(W]I#F(FMF(FGF(F<*(\"'gVBF(FMF( F6F(F*(\"'+_?F()FH#F(FDF()F6\"\"(F(F(*(\"'+AxF(Fi nF(F5F(F<*(\"(3SB\"F(FinF(F:F(F(*(\"(%y;>F(FinF(F?F(F<*(\"(/O7#F(FinF( FCF(F(*(\"'3UwF(FinF(FGF(F<*(\"'GPVF(FinF(F6F(F(**\"&ma'F(FMF(FinF(F[o F(F(**\"'1.=F(FMF(FinF(F5F(F<**\"'cT6F(FMF(FinF(F:F(F(**\"';^KF(FMF(Fi nF(F?F(F<**\"'otdF(FCF(FMF(FinF(F(**\"'KC;F(FMF(FinF(FGF(F<**\"'S-@F(F 6F(FMF(FinF(F(F(FZF(F(*(\"&k&RF()FH#FHFDF()F6\"\")F(F(**\"&+9#F(FMF(Fi pF(F[qF(F(**\"'g2:F(FMF(FipF(F[oF(F<*(\"'WIzF(FipF(F5F(F(*(\"(+[B\"F(F ipF(F:F(F<*(\"(+46\"F(FipF(F?F(F(*(\"'W/:F(FipF(FGF(F(*(\"'C+dF(FipF(F CF(F<*(\"&%y9F(F6F(FipF(F<**\"'S'H$F(FMF(FipF(FCF(F<**\"'SSjF(FMF(FipF (F?F(F(**\"'W0RF(FGF(F(*&\")%Q [W\"F(F6F(F<*&\"'3m>F(FMF(F(\"'W,RF(F(FYF(F(*&,>*(\"'s\"4)F(FinF(F5F(F <*(\"(STx\"F(FinF(F:F(F(*(\"(crY\"F(FinF(F?F(F<*(\"'o]uF(FinF(FCF(F<*( \"(3c4#F(FinF(FGF(F(*(\"'C!H(F(FinF(F6F(F<**\"'KVDF(FMF(FinF(F5F(F(** \"'K`9F(FMF(FinF(F:F(F(**\"'3x$)F(FMF(FinF(F?F(F<**\"'?;9F(FCF(FMF(Fin F(F<**\"(;/:\"F(FMF(FinF(FGF(F(**\"'o*R%F(F6F(FMF(FinF(F<*&\"&_\"\\F(F inF(F(*(\"&7D$F(FMF(FinF(F(F(FZF(F(*(\"&,Q\"F(FipF(F5F(F(*(\"&/q(F(Fip F(F:F(F<*(\"'E'p\"F(FipF(F?F(F(*(\"&8o*F(FipF(FGF(F(*(\"'CB=F(FipF(FCF (F<*(\"&WH#F(F6F(FipF(F<**\"'wi5F(FMF(FipF(FCF(F<**\"&'z%*F(FMF(FipF(F ?F(F(**\"%3hF(FMF(FipF(F5F(F(**\"&_'RF(FMF(FipF(F:F(F<*&\"%K?F(FipF(F( *(\"%C5F(FipF(FMF(F(**\"&cs&F(FGF(FMF(FipF(F(**\"&cK\"F(F6F(FMF(FipF(F <" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 103 "The singularity is in X^(1/2) unless the coeff is \+ 0. As we see below the coeff is not 0 for y " 0 "" {MPLTEXT 1 0 18 "subs(_Z=0,puisL);\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 386 "map(simplify,\{solve(39564*2^(2/3)*y^8+21400 *sqrt(3)*2^(2/3)*y^8-150760*sqrt(3)*2^(2/3)*y^7+793044*2^(2/3)*y^6-123 4800*2^(2/3)*y^5+1110900*2^(2/3)*y^4+150444*2^(2/3)*y^2-570024*2^(2/3) *y^3-14784*y*2^(2/3)-329640*sqrt(3)*2^(2/3)*y^3+634040*sqrt(3)*2^(2/3) *y^4+441744*sqrt(3)*2^(2/3)*y^6-696320*sqrt(3)*2^(2/3)*y^5-274344*2^(2 /3)*y^7+88480*sqrt(3)*y^2*2^(2/3)-8944*sqrt(3)*y*2^(2/3),y)\});\n" } {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$*&)*&I#xx G6\"\"\"\"-I'RootOfG6$%*protectedGI(_syslibGF'6#,B*(\"&k&RF()\"\"##F3 \"\"$F()I\"yGF'\"\")F(F(**\"&+9#F()F5#F(F3F(F2F(F6F(F(**\"'g2:F(F;F(F2 F()F7\"\"(F(!\"\"*(\"'WIzF(F2F()F7\"\"'F(F(*(\"(+[B\"F(F2F()F7\"\"&F(F A*(\"(+46\"F(F2F()F7\"\"%F(F(*(\"'W/:F(F2F()F7F3F(F(*(\"'C+dF(F2F()F7F 5F(FA*(\"&%y9F(F7F(F2F(FA**\"'S'H$F(F;F(F2F(FSF(FA**\"'SSjF(F;F(F2F(FL F(F(**\"'W*( \"&,Q\"F(F2F(FDF(F(*(\"&/q(F(F2F(FHF(FA*(\"'E'p\"F(F2F(FLF(F(*(\"&8o*F (F2F(FPF(F(*(\"'CB=F(F2F(FSF(FA*(\"&WH#F(F7F(F2F(FA**\"'wi5F(F;F(F2F(F SF(FA**\"&'z%*F(F;F(F2F(FLF(F(**\"%3hF(F;F(F2F(FDF(F(**\"&_'RF(F;F(F2F (FHF(FA*&\"%K?F(F2F(F(*(\"%C5F(F2F(F;F(F(**\"&cs&F(FPF(F;F(F2F(F(**\"& cK\"F(F7F(F;F(F2F(FA" }}{PARA 11 "" 1 "" {XPPMATH 20 "<&\"\"!\"\"\",&# F$\"#6!\"\"*&#\"\"#F'F$)\"\"$#F$F+F$F$,&#\"#;\"#8F$*&#\"\"%F2F$F,F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<&$\"\"!F$$\"+yB4SA!#5$\"\"\"F$$\"+Tz qj " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 217 174 "So LL is in X^(1/2) and L is in X^\{-1/2\}. So its coeffs are in n^\{-1/2\} =n* n^\{-3/2\}. So the average number of edges in ocea ns with outer edges weighted y<1-ycritic is in n." }}}{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 217 83 "The algebraic equation for L (from which the asym ptotic of coeffecient is obtained)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1925 "eqL:=(10*sqrt(3)*2^(1/3)+2*x*sqrt(3)-13*x*y-12*2^(1 /3)-5*x)*(8*y*sqrt(3)*2^(2/3)+36*y*2^(2/3)-8*2^(2/3)*sqrt(3)-36*2^(2/3 )-30*sqrt(3)*2^(1/3)*x*y^2+72*2^(1/3)*x*y^2+48*sqrt(3)*2^(1/3)*x*y-60* 2^(1/3)*x*y+23*x^2*y^3-73*x^2*y^2-6*sqrt(3)*x^2*y^2+32*x^2*y+48*sqrt(3 )*x^2*y)*y^2*(y-1)^2*x^4*L^3-4*sqrt(3)*(y*sqrt(3)*2^(1/3)-sqrt(3)*2^(1 /3)+6*x*y)*(10*sqrt(3)*2^(1/3)+2*x*sqrt(3)-13*x*y-12*2^(1/3)-5*x)*(8*y *sqrt(3)*2^(2/3)+36*y*2^(2/3)-8*2^(2/3)*sqrt(3)-36*2^(2/3)-30*sqrt(3)* 2^(1/3)*x*y^2+72*2^(1/3)*x*y^2+48*sqrt(3)*2^(1/3)*x*y-60*2^(1/3)*x*y+2 3*x^2*y^3-73*x^2*y^2-6*sqrt(3)*x^2*y^2+32*x^2*y+48*sqrt(3)*x^2*y)*y*x^ 2*L^2-(24/11*(74*sqrt(3)-81))*(-89*x^5*y^5+219*x^5*y^4-143*x^5*y^3+936 *sqrt(3)*2^(1/3)*x*y^2-912*sqrt(3)*2^(1/3)*x*y+346*x^3*y^4*2^(2/3)*sqr t(3)+126*2^(2/3)*x^3*y^2*sqrt(3)-1200*2^(2/3)*x^3*y^3*sqrt(3)-218*2^(1 /3)*x^4*y^5*sqrt(3)+858*2^(1/3)*x^4*y^4*sqrt(3)-1302*2^(1/3)*x^4*y^3*s qrt(3)-202*2^(1/3)*x^4*y^2*sqrt(3)+4*x^5*y^2*sqrt(3)+108*x^5*y^4*sqrt( 3)+24*2^(2/3)*sqrt(3)+11*x^5*y^6-132*x^5*y^3*sqrt(3)+672*sqrt(3)*x^2*y ^3+20*x^5*y^5*sqrt(3)+8*2^(2/3)*sqrt(3)*x^3*y-384*2^(1/3)*x^4*y^5+1254 *2^(1/3)*x^4*y^4-1212*2^(1/3)*x^4*y^3-336*sqrt(3)*x^2*y^2-24*y*sqrt(3) *2^(2/3)-90*2^(1/3)*x^4*y^2+2*x^5*y^2-192*sqrt(3)*x^2*y-24*sqrt(3)*2^( 1/3)*x-600*2^(2/3)*x^3*y^3+1812*2^(2/3)*x^3*y^2+48*2^(2/3)*x^3*y-144*y *2^(2/3)+4584*x^2*y^2-12*2^(1/3)*x-828*x^3*y^4*2^(2/3)+204*2^(1/3)*x*y ^2-4020*x^2*y^3+300*x^2*y-192*2^(1/3)*x*y+144*2^(2/3))*L-(144/37*(132* sqrt(3)+203))*(3816-66*2^(1/3)*x^2+2232*y*sqrt(3)+318*2^(2/3)*x-4800*s qrt(3)*x^2*y^2*2^(1/3)+8958*x*y^2*sqrt(3)*2^(2/3)-2556*sqrt(3)*x^2*2^( 1/3)*y-8772*x*y*sqrt(3)*2^(2/3)+620*sqrt(3)*x^2*y^3*2^(1/3)-2232*sqrt( 3)-111*x^3*y^4-516*x^3*y^3*sqrt(3)-186*2^(2/3)*sqrt(3)*x-48*x^3*y*sqrt (3)+40*sqrt(3)*x^2*2^(1/3)+2940*x^3*sqrt(3)*y^2-5517*y^2*x^3+1251*x^3* y^3+57*x^3*y-1134*x^2*y^3*2^(1/3)+8142*x^2*y^2*2^(1/3)-15702*x*y^2*2^( 2/3)+4506*x^2*y*2^(1/3)+15384*x*2^(2/3)*y-3816*y)*x*y;" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**.,,*(\"#5\"\"\")\"\"##F'\"\"$F ')F+#F'F)F'F'*(F)F'F,F'I\"xG6\"F'F'*(\"#8F'F/F'I\"yGF0F'!\"\"*&\"#7F'F (F'F4*&\"\"&F'F/F'F4F',<**\"\")F'F3F')F)#F)F+F'F,F'F'*(\"#OF'F#F'FipF'FgoF'F'*(F)F'FipF'F DF'F'*(F^oF'FF'F(F'F/F'F3F'F4**F^oF'F/ F'F(F'F,F'F4**F^oF'F3F'F " 0 "" {MPLTEXT 1 0 12 "indets(eqL);" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%I\"LG6\"I\"xGF$I\"yGF$" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 38 "b3:=coeff(eqL,L,3)/(x^4*y^2*(y-1)^2);\n" }{MPLTEXT 1 0 82 "b2:=coeff(eqL,L,2)/(y*x^2)/(-4*sqrt(3)*(y*sqrt(3)*2^(1/3)-sqrt (3)*2^(1/3)+6*x*y));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 20 "b1:=coeff (eqL,L,1);\n" }{MPLTEXT 1 0 19 "b0:=coeff(eqL,L,0);" }{MPLTEXT 1 0 0 " " }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,,*(\"#5\"\"\")\"\"##F&\"\"$F&)F*# F&F(F&F&*(F(F&F+F&I\"xG6\"F&F&*(\"#8F&F.F&I\"yGF/F&!\"\"*&\"#7F&F'F&F3 *&\"\"&F&F.F&F3F&,<**\"\")F&F2F&)F(#F(F*F&F+F&F&*(\"#OF&F;F&F2F&F&*(F: F&F;F&F+F&F3*&F>F&F;F&F3*,\"#IF&F+F&F'F&F.F&)F2F(F&F3**\"#sF&F'F&F.F&F CF&F&*,\"#[F&F+F&F'F&F.F&F2F&F&**\"#gF&F'F&F.F&F2F&F3*(\"#BF&)F.F(F&)F 2F*F&F&**\"\"'F&F+F&FLF&FCF&F3*(\"#tF&FLF&FCF&F3**FGF&F+F&FLF&F2F&F&*( \"#KF&FLF&F2F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,,*(\"#5\"\"\") \"\"##F&\"\"$F&)F*#F&F(F&F&*(F(F&F+F&I\"xG6\"F&F&*(\"#8F&F.F&I\"yGF/F& !\"\"*&\"#7F&F'F&F3*&\"\"&F&F.F&F3F&,<**\"\")F&F2F&)F(#F(F*F&F+F&F&*( \"#OF&F;F&F2F&F&*(F:F&F;F&F+F&F3*&F>F&F;F&F3*,\"#IF&F+F&F'F&F.F&)F2F(F &F3**\"#sF&F'F&F.F&FCF&F&*,\"#[F&F+F&F'F&F.F&F2F&F&**\"#gF&F'F&F.F&F2F &F3*(\"#BF&)F.F(F&)F2F*F&F&**\"\"'F&F+F&FLF&FCF&F3*(\"#tF&FLF&FCF&F3** FGF&F+F&FLF&F2F&F&*(\"#KF&FLF&F2F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*(#\"#C\"#6\"\"\",&*&\"#uF')\"\"$#F'\"\"#F'F'\"#\")!\"\"F',^p**\"% a7F')F.#F'F,F')I\"xG6\"\"\"%F')I\"yGF8F9F'F'**\"$%QF'F4F'F6F')F;\"\"&F 'F0**\"%77F'F4F'F6F')F;F,F'F0**\"#!*F'F4F'F6F')F;F.F'F0**\"$+'F')F.#F. F,F')F7F,F'FBF'F0**\"%7=F'FHF'FJF'FEF'F'**\"#[F'FHF'FJF'F;F'F'**\"$G)F 'FJF'F:F'FHF'F0**\"$K\"F')F7F?F'FBF'F+F'F0**F9F'FSF'FEF'F+F'F'**\"$3\" F'FSF'F:F'F+F'F'**\"#?F'FSF'F>F'F+F'F'**\"$s'F'F+F')F7F.F'FBF'F'*,\"$7 *F'F+F'F4F'F7F'F;F'F0*(\"$V\"F'FSF'FBF'F0*(F&F'FSF')F;\"\"'F'F'*(\"#*) F'FSF'F>F'F0*(\"$>#F'FSF'F:F'F'*(F.F'FSF'FEF'F'*(F%F'FHF'F+F'F'*&\"$W \"F'FHF'F'**\"$#>F'F4F'F7F'F;F'F0**F%F'F7F'F4F'F+F'F0**F%F'F;F'FHF'F+F 'F0*,\"$e)F'F4F'F6F'F:F'F+F'F'*,\"%-8F'F4F'F6F'FBF'F+F'F0*,\"$-#F'F4F' F6F'FEF'F+F'F0*,\"\")F'FHF'F+F'FJF'F;F'F'*,\"$Y$F'FJF'F:F'FHF'F+F'F'*, \"$E\"F'FHF'FJF'FEF'F+F'F'*,\"%+7F'FHF'FJF'FBF'F+F'F0*,\"$=#F'F4F'F6F' F>F'F+F'F0*(\"%%e%F'FenF'FEF'F'**\"$/#F'F4F'F7F'FEF'F'**\"$O$F'F+F'Fen F'FEF'F0**FfoF'F+F'FenF'F;F'F0*(\"#7F'F4F'F7F'F0*(\"%?SF'FenF'FBF'F0*( FdoF'FHF'F;F'F0*,\"$O*F'F+F'F4F'F7F'FEF'F'*(\"$+$F'FenF'F;F'F'F'F0" }} {PARA 11 "" 1 "" {XPPMATH 20 ",$*,#\"$W\"\"#P\"\"\",&*&\"$K\"F')\"\"$# F'\"\"#F'F'\"$.#F'F',T\"%;QF'**\"%M6F')I\"xG6\"F.F')I\"yGF6F,F')F.#F'F ,F'!\"\"**\"%U\")F'F4F')F8F.F'F9F'F'**\"&-d\"F'F5F'F>F')F.#F.F,F'F;** \"%1XF'F4F'F8F'F9F'F'**\"&%Q:F'F5F'FAF'F8F'F'**\"#[F')F5F,F'F8F'F+F'F; **\"#SF'F+F'F4F'F9F'F'**\"%SHF'FIF'F+F'F>F'F'*(\"#mF'F9F'F4F'F;*(\"#dF 'FIF'F8F'F'*(\"$6\"F'FIF')F8\"\"%F'F;*(\"%^7F'FIF'F7F'F'*(\"%F'F;**\"$;&F'F+F'FIF'F7F'F;*(\"$=$F'FAF'F5F'F'*,\"%+[F'F+F'F4F'F>F'F9 F'F;*,\"%e*)F'F5F'F>F'F+F'FAF'F'*,\"%cDF'F+F'F4F'F9F'F8F'F;*,\"%s()F'F 5F'F8F'F+F'FAF'F;*,\"$?'F'F+F'F4F'F7F'F9F'F'**\"$'=F'F+F'FAF'F5F'F;*( \"%KAF'F8F'F+F'F'*&F1F'F8F'F;*&FeoF'F+F'F;F'F5F'F8F'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 "" {MPLTEXT 1 0 0 "" }}}{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 }