{VERSION 6 0 "SUN SPARC SOLARIS" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 " " 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Error" 7 8 1 {CSTYLE "" -1 -1 "" 0 1 255 0 255 1 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 31 "Solving equations by res ultant." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f:=x^2+y^2+z^2-3 :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "g:=x*y+y*z+z*x-3:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "h:=x*y*z+x+y+z-4:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "What are the common solutions (x,y ,z) =(alpha,beta,gamma) of the equations:" }}{PARA 0 "" 0 "" {TEXT -1 3 "f=0" }}{PARA 0 "" 0 "" {TEXT -1 3 "g=0" }}{PARA 0 "" 0 "" {TEXT -1 4 "h=0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 104 "Well, for f and g to have a common root x=alpha, we need that the following vanishes for y=beta,z=gamma:" }}{PARA 0 "" 0 "" {TEXT -1 21 "Rfg=resultant(f,g,x)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 97 "Furthermore g and h must have a common root x=alph a, and h and f must have a common root x=alpha." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "So y=beta,z=gamma must sa tisfy the following equations:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 16 "resultant(f,g,x)" }}{PARA 0 "" 0 "" {TEXT -1 16 "resultant(g,h,x)" }}{PARA 0 "" 0 "" {TEXT -1 17 "resultan t(h,f,x)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "Note that these are only necessary conditions on y and z, they're not sufficient conditions. Namely:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 149 "f,g,h have common root x=something im plies that f,g have common root, g,h have common root and h,f have com mon root. The converse is not true, see:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "f:=x*(x-1) g:=(x-2)*(x-1) and h:= x*(x-2)." }}{PARA 0 "" 0 "" {TEXT -1 136 "Now f,g have a common root \+ x=1, g,h have a common root x=2, and h,f have a common root x=0, but n evertheless f,g,h have no common root." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 99 "Anyway, we obtain the following neces sary (but maybe not sufficient) conditions on y=beta, z=gamma." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Rfg:=resultant(f,g,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$RfgG,4*(\"\"$\"\"\")%\"yG\"\"#F()% \"zGF+F(F(*(\"#7F(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(F0*&F'F(F,F(F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Rgh:=resultant(g,h,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$RghG,0*&\"\"%\"\"\"%\"yGF(!\"\"*&F'F(%\"zGF( F**$)F)\"\"#F(F(*(F'F(F)F(F,F(F(*$)F,F/F(F(*&F.F(F2F(F*\"\"$F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Rhf:=resultant(h,f,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$RhfG,8\"#8\"\"\"*&\"\")F'%\"yGF'!\" \"*&F)F'%\"zGF'F+*&\"\"#F')F*F/F'F'*(\"\"%F'F*F'F-F'F+*&F/F')F-F/F'F'* &)F*F2F'F4F'F'*(F/F')F*\"\"$F'F-F'F'*&)F-F2F'F0F'F'*(F/F')F-F9F'F*F'F' *(F9F'F0F'F4F'F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "Now from thes e equations we compute the following necessary conditions on z=gamma. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Rfg_gh:=resultant(Rfg,R gh,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'Rfg_ghG*&),&%\"zG\"\"\"F) !\"\"\"\"%F),<*$)F(\"#7F)F)*&F+F))F(\"#6F)F)*&\"#5F))F(F4F)F)*&F+F))F( \"\"*F)F)*&\"\"(F))F(\"\")F)F)*&\"#cF))F(F:F)F**&\"#?F))F(\"\"'F)F**& \"$_\"F))F(\"\"&F)F**&\"$:%F))F(F+F)F)*&\"$)eF))F(\"\"$F)F**&\"%%>\"F) )F(\"\"#F)F)*&\"%g7F)F(F)F*\"$T%F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Rgh_hf:=resultant(Rgh,Rhf,y);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'Rgh_hfG*&),&%\"zG\"\"\"F)!\"\"\"\"%F),<*$)F(\"#7F)F) *&F+F))F(\"#6F)F**&\"#9F))F(\"#5F)F**&F/F))F(\"\"*F)F)*&\"$B#F))F(\"\" )F)F)*&F=F))F(\"\"(F)F**&\"$W'F))F(\"\"'F)F**&\"$w(F))F(\"\"&F)F**&\"% nFF))F(F+F)F)*&\"%C:F))F(\"\"$F)F**&\"$M$F))F(\"\"#F)F**&\"$_#F)F(F)F) \"#\\F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Rhf_fg:=result ant(Rhf,Rfg,y);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'Rhf_fgG,P*$)%\"z G\"#C\"\"\"F**&\"#;F*)F(\"#AF*!\"\"*&\"$i\"F*)F(\"#?F*F**&F,F*)F(\"#>F *F**&\"%O6F*)F(\"#=F*F/*&\"$O$F*)F(\"#F*)F(\"#5F*F**&\"'[c@F*)F(\"\"*F*F /*&\"'.z9F*)F(\"\")F*F**&\"'Ci6F*)F(\"\"(F*F/*&\"&%)>(F*)F(\"\"'F*F**& \"&wH'F*)F(\"\"&F*F**&\"&eD)F*)F(\"\"%F*F/*&\"&#zzF*)F(\"\"$F*F/*&\"'? R;F*)F(\"\"#F*F**&\"&/:*F*F(F*F/\"&*o " 0 "" {MPLTEXT 1 0 20 "gcd(Rfg_gh, Rgh_hf);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&),&%\"zG\"\"\"F'!\"\"\"\"%F',4*$)F&\"\")F'F'*&F)F')F&\"\"(F'F'*&F )F')F&\"\"'F'F'*&\"#?F')F&\"\"&F'F(*&\"#EF')F&F)F'F(*&\"#GF')F&\"\"$F' F'*&\"$+\"F')F&\"\"#F'F'*&\"$S\"F'F&F'F(\"#\\F'F'" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 14 "gcd(%,Rhf_fg);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&),&%\"zG\"\"\"F'!\"\"\"\"%F',4*$)F&\"\")F'F'*&F)F')F&\"\"(F'F' *&F)F')F&\"\"'F'F'*&\"#?F')F&\"\"&F'F(*&\"#EF')F&F)F'F(*&\"#GF')F&\"\" $F'F'*&\"$+\"F')F&\"\"#F'F'*&\"$S\"F'F&F'F(\"#\\F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&),&%\"zG\"\"\"F'!\"\"\"\")F'),(*$)F&\"\"#F'F'*&\"\"%F'F&F'F'\" \"(F'F.F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "v:=\{solve(%) \};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG<%\"\"\",&\"\"#!\"\"*&\" \"$#F&F(^#F&F&F&,&F(F)*&^#F)F&F+F,F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "So for any solution (alpha,beta,gamma) of \{f,g,h\} we have tha t gamma must be:" }}{PARA 0 "" 0 "" {TEXT -1 19 "1, -2 +/- sqrt(-3)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 160 "If we \+ select one of these values, say: -2+sqrt(-3), how can we check if ther e exists a solution (alpha, beta, -2+sqrt(-3)), and if it exists, how \+ do we find it?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "gamma:=-2 +sqrt(-3);" }}{PARA 8 "" 1 "" {TEXT -1 58 "Error, attempting to assign to `gamma` which is protected\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "Can't use the name gamma because Maple uses that for something els e, so we'll call it Gamma." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Gamma:=-2+sqrt(-3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&GammaG,& \"\"#!\"\"*&\"\"$#\"\"\"F&^#F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "subs(z=Gamma,\{Rfg,Rgh,Rhf\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<%,4*(\"\"$\"\"\")%\"yG\"\"#F'),&F*!\"\"*&F&#F'F*^#F'F' F'F*F'F'*(\"#7F'F)F'F,F'F-\"\"*F'*$)F)\"\"%F'F'*(F*F')F)F&F'F,F'F'*(F* F')F,F&F'F)F'F'*$)F,F6F'F'*&F&F'F(F'F-*&F&F'F+F'F-,0*&F6F'F)F'F-\"#6F' *&^#!\"%F'F&F/F'*$F(F'F'*(F6F'F)F'F,F'F'*$F+F'F'*&F(F'F+F'F-,8\"#HF'*& \"\")F'F)F'F-*&^#!\")F'F&F/F'*&F*F'F(F'F'*(F6F'F)F'F,F'F-*&F*F'F+F'F'* &F5F'F+F'F'*(F*F'F8F'F,F'F'*&F " 0 "" {MPLTEXT 1 0 15 "gcd(%[1],%[2]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,,*(%\"yG\"\"\"\"\"$#F&\"\"#^#F&F&F&*& ^#!\"\"F&F'F(F&F)F-F%F&*$)F%F)F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "gcd(%,%%[3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*( %\"yG\"\"\"\"\"$#F&\"\"#^#F&F&F&*&^#!\"\"F&F'F(F&F)F-F%F&*$)F%F)F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#*&,&%\"yG\"\"\"F&!\"\"F&,(F%F&*&\"\"$#F&\"\"#^#F &F&F&F,F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "solve(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"\",&\"\"#!\"\"*&^#F&F#\"\"$#F#F%F# " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "So, when z=-2+sqrt(-3), then \+ we must have either y=1 or y=-2-sqrt(-3). Lets just take one of them: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "beta:=-2-sqrt(-3);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG,&\"\"#!\"\"*&^#F'\"\"\"\"\"$# F*F&F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "subs(z=Gamma,y=be ta,\{f,g,h\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<%,(*(%\"xG\"\"\",& \"\"#!\"\"*&^#F*F'\"\"$#F'F)F'F',&F)F**&F-F.^#F'F'F'F'F'F&F'\"\")F*,** $)F&F)F'F'*$)F(F)F'F'*$)F/F)F'F'F-F*,**&F&F'F(F'F'*&F(F'F/F'F'*&F/F'F& F'F'F-F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "gcd(%[1],%[2]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"xG\"\"\"F%!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "gcd(%,%%[3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"xG\"\"\"F%!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "If z=-2+sqrt(-3) and y=-2-sqrt(-3) then f,g,h indeed have a co mmon solution x=1. So we find a solution:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "(1, -2-sqrt(-3), -2+sqrt(3))." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 118 "Of cours e, we had different choices for z and for y. If we try all possibiliti es, we will find all solutions of f,g,h." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "solve( \{f,g,h\} );" }}{PARA 12 "" 1 "" {XPPMATH 20 " 6&<%/%\"zG\"\"\"/%\"yGF&/%\"xGF&<%F$/F*,&\"\"%!\"\"-%'RootOfG6#,(*$)%# _ZG\"\"#F&F&*&F.F&F6F&F&\"\"(F&F//F(F0<%F,F'/F%F0<%F " 0 "" {MPLTEXT 1 0 19 "_EnvExplicit:=true;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-_EnvExpl icitG%%trueG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "and then solve wi ll try to use radical (square roots and such) instead of RootOf's when ever it can." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "solve( \{f, g,h\} );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6)<%/%\"zG\"\"\"/%\"yGF&/%\" xGF&<%/F*,&\"\"#!\"\"*&^#F/F&\"\"$#F&F.F&/F(,&F.F/*&F2F3^#F&F&F&F$<%/F *F5/F(F-F$<%F,/F%F5F'<%F9/F%F-F'<%F:FF)" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 181 "Here Maple's solve command solved 3 equations in \+ 3 unknowns. Following what we did above, we could have found all these solutions ourselves, using only the following Maple commands:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 " (i) r esultant, gcd, factor, subs," }}{PARA 0 "" 0 "" {TEXT -1 163 " (ii) and applying Maple's solve command only to univariate polynomials\n \+ (see the other worksheet on what it means to solve a univari ate polynomial)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 288 "Also, if we are OK with representing algebraic numbers i n RootOf notation instead of radicals (square roots and such), then we wouldn't even have needed part (ii) because then if f is some irreduc ible polynomial in Q[x], then we can simply represent a solution of f \+ by typing RootOf(f,x)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Now lets try a harder example" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 19 "f:=x^2+y^2+x*z^2-3:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "g:=y^2+z^2+y*x^2-3:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "h:=z^2+x^2+z*y^2-1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "R1:=resultant(f,g,x); R2:=resultant(g,h,x); R3:=resul tant(h,f,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#R1G,@*&)%\"yG\"\"$ \"\"\")%\"zG\"\"%F*F**&)F,\"\"'F*F(F*F**(F)F*F(F*F+F*!\"\"*&F0F*)F,\" \"#F*F2*&\"#=F*F(F*F2\"\"*F**&\"#7F*F'F*F**(F5F*)F(F5F*F4F*F**(F0F*F(F *F4F*F**&F)F*F%#R2G*$),.*$)%\" yG\"\"#\"\"\"!\"\"*$)%\"zGF+F,F-\"\"$F,*&F*F,F/F,F,F*F-*&)F*F1F,F0F,F, F+F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#R3G,8*$)%\"zG\"\"'\"\"\"F** &\"\"%F*)F(\"\"#F*F**(F.F*)%\"yGF.F*)F(\"\"$F*F**(F.F*F0F*F-F*!\"\"F,F **(F,F*F(F*F0F*F**&F,F*F0F*F5*&F0F*)F(\"\"&F*F**&)F1F,F*F-F*F**(F.F*F< F*F(F*F5*$F " 0 "" {MPLTEXT 1 0 71 "T1:=res ultant(R1,R2,y): T2:=resultant(R2,R3,y): T3:=resultant(R3,R1,y):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "factor(gcd( gcd(T1,T2) ,T3)) ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*&,&%\"zG\"\"\"F&F&F&,L*$)F%\"#?F &F&*&\"\"$F&)F%\"#>F&!\"\"*$)F%\"#=F&F/*&\"#6F&)F%\"# " 0 "" {MPLTEXT 1 0 42 "r1, r2:=RootOf(op(1,%)) , RootOf(op(2,%));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>6$%#r1G%#r2G6$!\"\"-%'RootOfG6#,L*$)%#_ZG\"#?\"\"\"F1* &\"\"$F1)F/\"#>F1F(*$)F/\"#=F1F(*&\"#6F1)F/\"#F1F(*&\"$v\"F1)F/\"\"%F1F1*&\"#!*F1)F/F3F1F(*&\"#WF1)F/FRF1F1* &F:F1F/F1F(F1F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "subs(z=r 1,[R1,R2,R3]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "gcd(gcd(% [1],%[2]),%[3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"yG\"\"\"F%!\" \"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Since the above polynomial \+ has degree 1 in y, it is easy to solve." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "subs(z=r2,[R1,R2,R3]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "gcd(gcd(%[1],%[2]),%[3]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 280 "I used a colon instead of a semi-colon because this is a very long expression, too long to print (because this RootOf appears \+ in there many times, and each time it occupies a lot of screen space). Lets replace that RootOf by a single letter alpha before printing it \+ to the screen:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "subs(r2=a lpha,%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,L#\"4J!oV8w='='p\"1:(=qj1 Q`#!\"\"%\"yG\"\"\"*&#\"6*QsuV*4L?)=7\"0&RO#\\u/\\\"F)*$)%&alphaG\"\"' F)F)F)*&#\"49\\EP9Eir=#F&F)*$)F0\"#>F)F)F)*&#\"6'Q&Qg'H&>yQO\"F&F)*$)F 0\"#5F)F)F'*&#\"6xC:rwY&Qy5UF&F)*$)F0\"#6F)F)F'*&#\"6KyCchw=`nB\"F&F)* $)F0\"#9F)F)F'*&#\"4PB2=lAu@z$F&F)*$)F0\"#:F)F)F'*&#\"6c%Hv()3PHmXKF&F )*$)F0\"#7F)F)F)*&#\"6U\"oRrg(epN9\"F&F)*$)F0\"#8F)F)F)*&#\"64')pp))pt @]^#F&F)*$)F0\"\"&F)F)F)*&#\"6mKL79]'z%*G?F&F)*$)F0\"\"$F)F)F)*&#\"7J \")R*HHUuW*z;F&F)*$)F0\"\"(F)F)F'*&#\"6k;f$=EMNF)f\"F&F)*$)F0\"\")F)F) F'*&#\"6OBZ@C>y^