{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "2D Output" -1 20 "Times" 1 12 0 0 255 1 2 2 2 2 2 2 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 2 0 2 0 2 2 0 1 }{PSTYLE "Map le 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 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "Maple Plot" -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 2 0 2 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Lecture #1 - Math 541" }} {PARA 0 "" 0 "" {TEXT -1 67 "Genetic Control by Repression - Analysis \+ of Characteristic Equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Equilibrium Analysis" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 38 "f := x -> a1/(1+k*x^n) - (b1*b2/a2)*x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(,&*&%#a 1G\"\"\",&F/F/*&%\"kGF/)9$%\"nGF/F/!\"\"F/*&*(%#b1GF/%#b2GF/F4F/F/%#a2 GF6F6F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "a1:=2;a2:=1; b1:=1;b2:=1;n:=4;k:=0.1;R:=2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a1 G\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a2G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b1G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #b2G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG$\"\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"\"F%,&F%F%*&$F%!\" \"F%)%\"xG\"\"%F%F%F)\"\"#F+F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "x2e := fsolve(f(x)=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% $x2eG$\"+0>(3U\"!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "x1e := (b2/a2)*x2e;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$x1eG$\"+0>(3U\" !\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "f1 := x -> a1/(1+k* x^n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1GR6#%\"xG6\"6$%)operator G%&arrowGF(*&%#a1G\"\"\",&F.F.*&%\"kGF.)9$%\"nGF.F.!\"\"F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "plot(\{f1(x),(b1*b2*x/a2)\}, x=0..2);" }}{PARA 13 "" 1 "" {GLPLOT2D 191 191 191 {PLOTDATA 2 "6&-%'C URVESG6$7S7$$\"\"!F)F(7$$\"39LLLL3VfV!#>F+7$$\"3'pmm;H[D:)F-F/7$$\"3LL LLe0$=C\"!#=F27$$\"3ILLL3RBr;F4F67$$\"3Ymm;zjf)4#F4F97$$\"3=LL$e4;[\\# F4F<7$$\"3p****\\i'y]!HF4F?7$$\"3,LL$ezs$HLF4FB7$$\"3_****\\7iI_PF4FE7 $$\"3#pmmm@Xt=%F4FH7$$\"3QLLL3y_qXF4FK7$$\"3i******\\1!>+&F4FN7$$\"3() ******\\Z/NaF4FQ7$$\"3'*******\\$fC&eF4FT7$$\"3ELL$ez6:B'F4FW7$$\"3Smm m;=C#o'F4FZ7$$\"3-mmmm#pS1(F4Fgn7$$\"3]****\\i`A3vF4Fjn7$$\"3slmmm(y8! zF4F]o7$$\"3V++]i.tK$)F4F`o7$$\"39++](3zMu)F4Fco7$$\"3#pmm;H_?<*F4Ffo7 $$\"3emm;zihl&*F4Fio7$$\"39LLL3#G,***F4F\\p7$$\"3FapF_t7$$\"3/++v.Uac>FapFbt7$$\"\"#F)Fet-%'COLOURG6&%$R GBG$\"#5!\"\"F(F(-F$6$7S7$F(Fet7$F+$\"3]\"zZwF*****>Fap7$F/$\"3uJ35l6* ***>Fap7$F2$\"3KJ@-PC&***>Fap7$F6$\"3;6bk#*R%)**>Fap7$F9$\"3s[+5_@h**> Fap7$F<$\"3')Rry4bA**>Fap7$F?$\"3%RVLn^w&)*>Fap7$FB$\"3AR4)RfXv*>Fap7$ FE$\"3O\">'*y-Vg*>Fap7$FH$\"3%[um07qQ*>Fap7$FK$\"3sE\"y$3.J\"*>Fap7$FN $\"3Ifa4e)ev)>Fap7$FQ$\"3[0&))e2*p#)>Fap7$FT$\"3C(z<.14o(>Fap7$FW$\"3+ ea**>**Gq>Fap7$FZ$\"3qS]'R(G!4'>Fap7$Fgn$\"3ElRFhwS^>Fap7$Fjn$\"3mw^xl $)RQ>Fap7$F]o$\"3ehiq4*p\\#>Fap7$F`o$\"3%p:'\\`@,3>Fap7$Fco$\"3c`V;toc *)=Fap7$Ffo$\"3Ic]H[*4y'=Fap7$Fio$\"33'*ocEx[X=Fap7$F\\p$\"3\")3gZmP$) ==Fap7$F_p$\"3c6fWpUG)y\"Fap7$Fcp$\"3%zdrXj?$f)o\"Fap7$F\\q$\"3g:p)yn)=\\;Fap7$F_q$\"3$G&)*HhLL4;F ap7$Fbq$\"3%GKg6OeGc\"Fap7$Feq$\"3$=H:(e^K>:Fap7$Fhq$\"3o'y=!*=a7Z\"Fa p7$F[r$\"3!)H;#eno)o!z5Fap7$Fcs$\"36XypglfI5Fap7$F fs$\"3#p^VGP;%p)*F47$Fis$\"3;T\"\\Ac)>\"Q*F47$F\\t$\"3O3Gt'y^n&*)F47$F _t$\"3r-Vm3Y,=&)F47$Fbt$\"3a@/_[kC7\")F47$Fet$\"3i\"p2Bp2Bp(F4-Fht6&Fj tF(F[uF(-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;F(Fet%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "df1 := diff(f1(x),x);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$df1G,$*&*$)%\"xG\"\"$\"\"\"F+*$),& F+F+*&$F+!\"\"F+)F)\"\"%F+F+\"\"#F+F1$!\")F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "df1b := subs(x=x2e,df1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%df1bG$!+\">c#e6!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "Next we would like to fin d some roots to the characteristic equation. We let" }}{PARA 0 "" 0 " " {TEXT -1 19 "lambda = mu + I*vu." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "ce := (mu+I*nu+b1)*(mu+I*nu +b2)-a2*df1b*exp(-mu*R)*(cos(nu*R)-I*sin(nu*R));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ceG,&*$),(%#muG\"\"\"*&^#F*F*%#nuGF*F*F*F*\"\"#F*F** ($\"+\">c#e6!\"*F*-%$expG6#,$F)!\"#F*,&-%$cosG6#,$F-F.F**&^#!\"\"F*-%$ sinGF;F*F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "rce := co eff(ce,I,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$rceG,&*$),&%#muG\" \"\"F*F*\"\"#F*F**($\"+\">c#e6!\"*F*-%$expG6#,$F)!\"#F*-%$cosG6#,$%#nu GF+F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "ice := coeff(ce, I,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$iceG,&*&,&%#muG\"\"\"F)F)F )%#nuGF)\"\"#*($\"+\">c#e6!\"*F)-%$expG6#,$F(!\"#F)-%$sinG6#,$F*F+F)! \"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "fsolve(\{rce=0,ice= 0\},\{mu,nu\},\{mu=-1..0,nu=0..3\});" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#<$/%#nuG$\"+'ykN!)*!#5/%#muG$!+*y.B%>F(" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 50 "fsolve(\{rce=0,ice=0\},\{mu,nu\},\{mu=-10..0,nu=1.. 6\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%#muG$!+!pNtb&!#5/%#nuG$\" +;O/bR!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "fsolve(\{rce= 0,ice=0\},\{mu,nu\},\{mu=-10..0,nu=4..8\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%#muG$!+_ " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "15 1 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }