{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple P lot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "h1:=(r*u2^2+2*R-u1^2)/(2*(1-r));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#h1G*&,(*&%\"rG\"\"\")%#u2G\"\"#F)F )*&F,F)%\"RGF)F)*$)%#u1GF,F)!\"\"F),&F,F)*&F,F)F(F)F2F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Q1:=u1*h1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#Q1G*(%#u1G\"\"\",(*&%\"rGF')%#u2G\"\"#F'F'*&F-F'%\"R GF'F'*$)F&F-F'!\"\"F',&F-F'*&F-F'F*F'F2F2" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 14 "Q2:=u2*(1-h1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#Q2G*&%#u2G\"\"\",&F'F'*&,(*&%\"rGF')F&\"\"#F'F'*&F.F'%\"RGF'F'*$)%#u 1GF.F'!\"\"F',&F.F'*&F.F'F,F'F4F4F4F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "S:=u1^2*h1-r*u2^2*h1+1/2*r*u2^2+1/2*(1-r)*h1^2;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG,**(%#u1G\"\"#,(*&%\"rG\"\"\")%# u2GF(F,F,*&F(F,%\"RGF,F,*$)F'F(F,!\"\"F,,&F(F,*&F(F,F+F,F3F3F,**F+F,F. F(F)F,F4F3F3*(F(F3F+F,F.F(F,**F(F3,&F,F,F+F3F,F)F(F4!\"#F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "Letting U and V be the critical values o f u1^2 and u2^2 respectively then the criticality parabola is given by " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "U:=U0-alpha*s*(r*s-2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"UG,&%#U0G\"\"\"*(%&alphaGF'%\"sGF' ,&*&%\"rGF'F*F'F'\"\"#!\"\"F'F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "V:=V0-alpha*s*(2*r+s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" VG,&%#V0G\"\"\"*(%&alphaGF'%\"sGF',&*&\"\"#F'%\"rGF'F'F*F'F'!\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "where U0 and V0 are the coordinate s of the vertex of the parabola and" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "alpha:=r*(1-r)/(3*(1+r^2)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %&alphaG,$**\"\"$!\"\"%\"rG\"\"\",&F*F*F)F(F*,&F*F**$)F)\"\"#F*F*!\"#F *" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "We now calculate the vertex. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "The criticality condition is " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "eqn1:=0=(3^(1/2)*U0-3^(1/2)*r*V 0)^2+(6*(1-r)-8*R)*U0+(8*r*R-2*r*(1-r))*V0+4*(R^2-(1-r)*R);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/\"\"!,,*$),&*&\"\"$#\"\"\"\"\"#%#U0 GF.F.*(F,F-%\"rGF.%#V0GF.!\"\"F/F.F.*&,(\"\"'F.*&F7F.F2F.F4*&\"\")F.% \"RGF.F4F.F0F.F.*&,&*(F:F.F2F.F;F.F.*(F/F.F2F.,&F.F.F2F4F.F4F.F3F.F.*& \"\"%F.)F;F/F.F.*(FBF.F@F.F;F.F4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "The axis of symmetry is given by" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "eqn2:=3*(1+r^2)*(U0-r*V0)-4*R*(1+r^2)+(3+r^2)*(1-r)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/,(*(\"\"$\"\"\",&F)F)*$)%\"rG\"\"#F )F)F),&%#U0GF)*&F-F)%#V0GF)!\"\"F)F)*(\"\"%F)%\"RGF)F*F)F3*&,&F(F)F+F) F),&F)F)F-F3F)F)\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Now solv e eqn1 and eqn2 for U0 and V0" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "so lve(\{eqn1,eqn2\},\{U0,V0\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<$/%# V0G,$*,\"#7!\"\",B*&\"\"&\"\"\")%\"rG\"\"'F-F-*&\"#5F-)F/F,F-F)*(F(F-% \"RGF-F3F-F-*&\"#BF-)F/\"\"%F-F-*(F9F-)F5\"\"#F-F8F-F-*(F(F-F5F-F8F-F) *&\"#OF-)F/\"\"$F-F)*(\"#CF-F5F-F@F-F-*(\"\")F-F;F-)F/F " 0 "" {MPLTEXT 1 0 148 "U0:=factor(-1/12*(-3+4*R-4*r*R-r^2+4*R^2 +8*R*r^2+8*R^2*r^2-8*R*r^3+4*R^2*r^4+4*R*r^4-4*R*r^5-4*r^3+3*r^4-2*r^5 +r^6+6*r)/(-1-2*r^2-r^4+r+2*r^3+r^5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#U0G,$*,\"#7!\"\",.*$)%\"rG\"\"$\"\"\"F.*(\"\"#F.%\"RGF.)F,F0F .F(*$F2F.F(F,F(*&F0F.F1F.F(F.F.F.,.F*F.*(F0F.F1F.F2F.F(F3F(*&F-F.F,F.F .*&F0F.F1F.F(F-F(F.,&F,F.F.F(F(,&F.F.F3F.!\"#F(" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 159 "V0:=factor(-1/12*(5*r^6+12*R*r^5-10*r^5+23*r^ 4+4*R^2*r^4-12*R*r^4+24*R*r^3-36*r^3-24*R*r^2+27*r^2+8*R^2*r^2+12*r*R- 18*r-12*R+9+4*R^2)/(1+r^2)/r/(r^3-r^2+r-1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#V0G,$*.\"#7!\"\",.*&\"\"&\"\"\")%\"rG\"\"$F,F,*(\"\" #F,%\"RGF,)F.F1F,F,*&F+F,F3F,F(*&F/F,F.F,F,F/F(*&F1F,F2F,F,F,,.*$F-F,F ,*(F1F,F2F,F3F,F,*$F3F,F(*&F/F,F.F,F,F/F(*&F1F,F2F,F,F,,&F,F,F:F,!\"#F .F(,&F.F,F,F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Hence" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "U;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#,&*,\"#7!\"\",.*$)%\"rG\"\"$\"\"\"F,*(\"\"#F,%\"RGF,)F*F.F,F&*$F0F,F &F*F&*&F.F,F/F,F&F,F,F,,.F(F,*(F.F,F/F,F0F,F&F1F&*&F+F,F*F,F,*&F.F,F/F ,F&F+F&F,,&F*F,F,F&F&,&F,F,F1F,!\"#F&*.F+F&F*F,,&F,F,F*F&F,F8F9%\"sGF, ,&*&F*F,F " 0 "" {MPLTEXT 1 0 2 "V; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*.\"#7!\"\",.*&\"\"&\"\"\")%\"rG \"\"$F*F**(\"\"#F*%\"RGF*)F,F/F*F**&F)F*F1F*F&*&F-F*F,F*F*F-F&*&F/F*F0 F*F*F*,.*$F+F*F**(F/F*F0F*F1F*F**$F1F*F&*&F-F*F,F*F*F-F&*&F/F*F0F*F*F* ,&F*F*F8F*!\"#F,F&,&F,F*F*F&F&F&*.F-F&F,F*,&F*F*F,F&F*F;F<%\"sGF*,&*&F /F*F,F*F*F@F*F*F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "Feasible reg ion" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Th e lower boundary is" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "u2squared_mi n:=u1^2/r-2*R/r;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%.u2squared_minG, &*&%#u1G\"\"#%\"rG!\"\"\"\"\"*(F(F+%\"RGF+F)F*F*" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 21 "The upper boundary is" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "u2squared_max:=u1^2/r+2/r*(1-r-R);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%.u2squared_maxG,&*&%#u1G\"\"#%\"rG!\"\"\"\"\"*(F(F+ F)F*,(F+F+F)F*%\"RGF*F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Now \+ assign values for r and R: r = 0.1, R = -0.1 (corresponding to case 1) " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "r:=0.1;R:=-0.1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG$\"\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG$!\"\"F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "u2s quared_min;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&$\"\"\"F&F&)%#u1G\" \"#F&F&$\"+++++?!\"*F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "u 2squared_max;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&$\"\"\"F&F&)%#u1G \"\"#F&F&$\"#?\"\"!F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "wi th(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoo rds has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "plot1:=plot([t,subs(u1^2=t,u2squared_min),t=0..0.1],colour=black):" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "plot2:=plot([t,subs(u1^2=t ,u2squared_max),t=0..0.1],colour=black):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Hence the feasible region is" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "display([plot1,plot2],labels=[u1squared,u2squared]); 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