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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "%" }}{PARA 0 "" 0 "" {TEXT 
-1 33 "% Tese de Doutorado - COPPE/UFRJ." }}{PARA 0 "" 0 "" {TEXT -1 
1 "%" }}{PARA 0 "" 0 "" {TEXT -1 38 "% Rio de Janeiro, 16 de julho de \+
2003." }}{PARA 0 "" 0 "" {TEXT -1 1 "%" }}{PARA 0 "" 0 "" {TEXT -1 34 
"% Autor: Jose Paulo V. S. da Cunha" }}{PARA 0 "" 0 "" {TEXT -1 1 "%" 
}}{PARA 0 "" 0 "" {TEXT -1 69 "% Quadro comparativo de matrizes de gan
ho de alta frequencia (K=KpSp)" }}{PARA 0 "" 0 "" {TEXT -1 32 "% em re
lacao a quatro condicoes:" }}{PARA 0 "" 0 "" {TEXT -1 1 "%" }}{PARA 0 
"" 0 "" {TEXT -1 10 "% 1)  K>0;" }}{PARA 0 "" 0 "" {TEXT -1 20 "% 2)  \+
-K eh Hurwitz;" }}{PARA 0 "" 0 "" {TEXT -1 50 "% 3) Menores de K posit
ivos (Delta1>0 e Delta2>0)." }}{PARA 0 "" 0 "" {TEXT -1 101 "% 4) -K s
atisfaz as condicoes para modo deslizante com lei decontrole pela func
ao sinal estabelecidas" }}{PARA 0 "" 0 "" {TEXT -1 54 "%     num Teore
ma de Hsu, Kaszkurewicz e Bhaya (2000)." }}{PARA 0 "" 0 "" {TEXT -1 1 
"%" }}{PARA 0 "" 0 "" {TEXT -1 18 "% Primeira matriz:" }}{PARA 0 "" 0 
"" {TEXT -1 1 "%" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Kp1:=matrix(2,2
,[1,2*alpha,-2,alpha]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Kp1G-%'m
atrixG6#7$7$\"\"\",$%&alphaG\"\"#7$!\"#F," }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 1 "%" }}{PARA 0 "" 0 "" {TEXT -1 49 "% Usando-se a seguinte m
atriz de pre-compensacao:" }}{PARA 0 "" 0 "" {TEXT -1 1 "%" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 28 "Sp1:=matrix(2,2,[1,-2,2,1]);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%$Sp1G-%'matrixG6#7$7$\"\"\"!\"#7$\"\"#F*" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "%" }}{PARA 0 "" 0 "" {TEXT -1 72 "%
 tem-se a seguinte matriz de ganho de alta frequencia que eh simetrica
:" }}{PARA 0 "" 0 "" {TEXT -1 1 "%" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
20 "K1:=evalm(Kp1&*Sp1);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#K1G-%'matrix
G6#7$7$,&\"\"\"F+%&alphaG\"\"%,&!\"#F+F,\"\"#7$F.,&F-F+F,F+" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "%" }}{PARA 0 "" 0 "" {TEXT -1 31 "%
 Cujos menores principais sao:" }}{PARA 0 "" 0 "" {TEXT -1 1 "%" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "Delta11:=K1[1,1];Delta12:=linalg[de
t](evalm(K1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Delta11G,&\"\"\"F
&%&alphaG\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Delta12G,$%&alpha
G\"#D" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "%" }}{PARA 0 "" 0 "" 
{TEXT -1 76 "% Uma vez que K1 eh simetrica, as quatro condicoes tem os
 mesmos requisitos." }}{PARA 0 "" 0 "" {TEXT -1 1 "%" }}{PARA 0 "" 0 "
" {TEXT -1 84 "% Ambos os menores principais sao positivos para alpha>
0, conforme o grafico abaixo:" }}{PARA 0 "" 0 "" {TEXT -1 1 "%" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "plot([Delta11,Delta12],alpha=-0.5..
0.5);" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 13 "" 1 
"" {INLPLOT "6&-%'CURVESG6$7S7$$!1+++++++]!#;$!\"\"\"\"!7$$!1LLLe%G?y%
F*$!1LLLLQ6G\"*F*7$$!1mmT&esBf%F*$!1mmmT.\\p$)F*7$$!1LL$3s%3zVF*$!1LLL
$))Qj^(F*7$$!1LL$e/$QkTF*$!1KLL$=Kvl'F*7$$!1nmT5=q]RF*$!1ommTs!G!eF*7$
$!1LL3_>f_PF*$!1MLL3yO5]F*7$$!1++vo1YZNF*$!1+++vE%)*=%F*7$$!1LL3-OJNLF
*$!1MLL3WDTLF*7$$!1++v$*o%Q7$F*$!1,++vvQ&\\#F*7$$!1nmm\"RFj!HF*$!1mmmm
&4`i\"F*7$$!1LL$e4OZr#F*$!1KLLLQW*e)!#<7$$!1+++v'\\!*\\#F*$\"1I#******
*H,Q!#>7$$!1+++DwZ#G#F*$\"1(*******\\*3q)F_o7$$!1+++D.xt?F*$\"1++++(=
\\q\"F*7$$!1LL3-TC%)=F*$\"1nmm\"fBIY#F*7$$!1nmm\"4z)e;F*$\"1LLLLO[kLF*
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$!1nmm;1J\\5F*$\"1JLLLvv-eF*7$$!1)***\\(=[jL)F_o$\"1,++D2YlmF*7$$!1***
*\\iXg#G'F_o$\"1+++v\"ep[(F*7$$!1lmmT&Q(RTF_o$\"1MLL$e/TM)F*7$$!1nm;/'
=><#F_o$\"1LLLeDBJ\"*F*7$$!1EMLLe*e$\\Feo$\"1mmm;kD!)**F*7$$\"1em;zRQb
@F_o$\"1mm;f`@'3\"!#:7$$\"1&***\\(=>Y2%F_o$\"1++]nZ)H;\"Fgs7$$\"1hmm\"
zXu9'F_o$\"1mmmJy*eC\"Fgs7$$\"1'******\\y))G)F_o$\"1+++S^bJ8Fgs7$$\"1*
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$\"1++D\"oK0e%F*$\"1++]sI@KGFgs7$$\"1++v=5s#y%F*$\"1++]2%)38HFgs7$$\"1
+++++++]F*$\"\"$F--%'COLOURG6&%$RGBG$\"#5F,F-F--F$6$7S7$F($!1++++++]7!
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gs7$FR$!1L$3_+%GQ$)Fgs7$FW$!1,]PMsh4yFgs7$Ffn$!1nm;z%=eE(Fgs7$F[o$!1LL
eR-%oy'Fgs7$Fao$!1++](=CwC'Fgs7$Fgo$!1++]iS>1dFgs7$F\\p$!1++]7eU%=&Fgs
7$Fap$!1M$3_D51r%Fgs7$Ffp$!1nm;Hx>ZTFgs7$F[q$!1omm;M\"*pOFgs7$F`q$!1+]
(oH=Z6$Fgs7$Feq$!1ommTlFBEFgs7$Fjq$!1**\\(o/(3%3#Fgs7$F_r$!1+]iS6lq:Fg
s7$Fdr$!1mmTNY$\\.\"Fgs7$Fir$!1omT5lzHaF*7$F^s$!1cLLeR(RB\"F_o7$Fcs$\"
1Ym\"z%*f%)Q&F*7$Fis$\"1**\\(oza'=5Fgs7$F^t$\"1lm\"zWho`\"Fgs7$Fct$\"1
*****\\i>A2#Fgs7$Fht$\"1(**\\i:jff#Fgs7$F]u$\"1)*\\7`>r-JFgs7$Fbu$\"1)
**\\P4q`m$Fgs7$Fgu$\"1KLLeM%4<%Fgs7$F\\v$\"1****\\P4v5ZFgs7$Fav$\"1m;z
Wn*)*>&Fgs7$Ffv$\"1****\\7BmMdFgs7$F[w$\"1J$ek.NyB'Fgs7$F`w$\"1**\\i:b
zjnFgs7$Few$\"1LL$3-=!ysFgs7$Fjw$\"1**\\7G8O;yFgs7$F_x$\"1omm;*\\[L)Fg
s7$Fdx$\"1lmT&Qz]'))Fgs7$Fix$\"1M$ekG=4R*Fgs7$F^y$\"1*****\\i4T()*Fgs7
$Fcy$\"1L$3F9!zU5Fg[l7$Fhy$\"1nmmT>K#4\"Fg[l7$F]z$\"1+DJqJ8X6Fg[l7$Fbz
$\"1+voa-o&>\"Fg[l7$Fgz$\"1++++++]7Fg[l-F\\[l6&F^[lF-F_[lF--%+AXESLABE
LSG6$Q&alpha6\"%!G-%%VIEWG6$;$!\"&F,$\"\"&F,%(DEFAULTG" 2 324 324 324 
2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 10030 10061 10056 
10074 0 0 0 20530 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "%" }}{PARA 0 "" 0 "" {TEXT -1 94 "%
 To develop a more intereting  plant HFG matrix for this example, lets
 consider the following" }}{PARA 0 "" 0 "" {TEXT -1 19 "% generic matr
ices:" }}{PARA 0 "" 0 "" {TEXT -1 1 "%" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 60 "Kp:=matrix(2,2,[k1,k2,k3,k4]);Sp:=matrix(2,2,[s1,s2,s3,s4]);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#KpG-%'matrixG6#7$7$%#k1G%#k2G7$%#
k3G%#k4G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#SpG-%'matrixG6#7$7$%#s1
G%#s2G7$%#s3G%#s4G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "%" }}{PARA 
0 "" 0 "" {TEXT -1 12 "% which give" }}{PARA 0 "" 0 "" {TEXT -1 1 "%" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "K:=evalm(Kp&*Sp);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"KG-%'matrixG6#7$7$,&*&%#k1G\"\"\"%#s1GF-F-*&%#
k2GF-%#s3GF-F-,&*&F,\"\"\"%#s2GF-F-*&F0F4%#s4GF-F-7$,&*&%#k3GF-F.F4F-*
&%#k4GF-F1F4F-,&*&F;F4F5F4F-*&F=F4F7F4F-" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 1 "%" }}{PARA 0 "" 0 "" {TEXT -1 100 "% which can be used to \+
conclude that symmetry is achieved if (k1*s2+k2*s4=k3*s1+k4*s3) is sat
isfied." }}{PARA 0 "" 0 "" {TEXT -1 1 "%" }}{PARA 0 "" 0 "" {TEXT -1 
1 "%" }}{PARA 0 "" 0 "" {TEXT -1 68 "% Keeping this in mind, the folow
ing plant HFG matrix was developed:" }}{PARA 0 "" 0 "" {TEXT -1 1 "%" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Kp2:=matrix(2,2,[1,2*alpha^2,-2,a
lpha]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Kp2G-%'matrixG6#7$7$\"\"
\",$*$)%&alphaG\"\"#\"\"\"F/7$!\"#F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 1 "%" }}{PARA 0 "" 0 "" {TEXT -1 13 "% which gives" }}{PARA 0 "" 0 
"" {TEXT -1 1 "%" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "K2:=evalm(Kp2&*
Sp);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#K2G-%'matrixG6#7$7$,&%#s1G
\"\"\"*&)%&alphaG\"\"#\"\"\"%#s3GF,F0,&%#s2GF,*&F.F1%#s4GF,F07$,&F+!\"
#*&F/F,F2F1F,,&F4F9*&F/F1F6F1F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 
"%" }}{PARA 0 "" 0 "" {TEXT -1 93 "% thus there does not exists any fi
xed nonsingular Sp such that this matrix remains symmetric" }}{PARA 0 
"" 0 "" {TEXT -1 21 "% for all Real alpha." }}{PARA 0 "" 0 "" {TEXT 
-1 1 "%" }}{PARA 0 "" 0 "" {TEXT -1 51 "% Quatro condicoes sao avaliad
as para a matriz Kp2:" }}{PARA 0 "" 0 "" {TEXT -1 1 "%" }}{PARA 0 "" 
0 "" {TEXT -1 11 "% 1) Kp2>0:" }}{PARA 0 "" 0 "" {TEXT -1 1 "%" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "tr2:=linalg[trace](Kp2);d2:=simplif
y(linalg[det](Kp2)-(((Kp2[1,2]-Kp2[2,1])^2)/4));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$tr2G,&\"\"\"F&%&alphaGF&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#d2G,*%&alphaG\"\"\"*$)F&\"\"#\"\"\"F**$)F&\"\"%F+!\"
\"F/F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "% que sao positivos par
a 0.5248885987<alpha<1.490216120, como se observa no grafico abaixo." 
}}{PARA 0 "" 0 "" {TEXT -1 1 "%" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "
plot([tr2,d2],alpha=-2..2);fsolve(d2,alpha);" }}{PARA 13 "" 1 "" 
{INLPLOT "6&-%'CURVESG6$7S7$$!\"#\"\"!$!\"\"F*7$$!1LLL$Q6G\">!#:$!1LLL
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$3yO5]\"F0$!1MLL3yO5]F37$$!1++]nU)*=9F0$!1+++vE%)*=%F37$$!1LL$3WDTL\"F
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$$!1ommm9'=(eF3$\"1KLLL&Q\"GTF37$$!1,++v#\\N)\\F3$\"1*****\\s]k,&F37$$
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$$\"1mm\"H3XL7\"F0$\"1c-(eG^Z0\"F07$Ffw$\"1EuQoBtP5F07$$\"1mm\"zM]v?\"
F0$\"1kEEBTBw**F37$F[x$\"1(HRo[6ZK*F37$F`x$\"1RL6FkIwsF37$Fex$\"1WAVe.
#[%RF37$Fjx$!1d=kk:(H\"zF`o7$F_y$!1)\\j#QS3!e'F37$Fdy$!174Zj9Q8:F07$$
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F*-%+AXESLABELSG6$Q&alpha6\"%!G-%%VIEWG6$;F(Fhz%(DEFAULTG" 2 317 317 
317 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 10030 10061 10056 
10074 0 0 0 20530 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 0 8333 0 0 0 0 0 0 }
}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+()f))[_!#5$\"+?h@!\\\"!\"*" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "%" }}{PARA 0 "" 0 "" {TEXT -1 21 "%
 2) -Kp2 eh Hurwitz:" }}{PARA 0 "" 0 "" {TEXT -1 1 "%" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 30 "tr2;Delta22:=linalg[det](Kp2);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#,&\"\"\"F$%&alphaGF$" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%(Delta22G,&%&alphaG\"\"\"*$)F&\"\"#\"\"\"\"\"%" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 1 "%" }}{PARA 0 "" 0 "" {TEXT -1 99 "% A part
ir dos graficos abaixo conclui-se que -1<alpha<-0.25 ou alpha>0 para q
ue -Kp2 seja Hurwitz." }}{PARA 0 "" 0 "" {TEXT -1 1 "%" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 58 "plot([tr2,Delta22],alpha=-1.2..0.6);fsolve(Del
ta22,alpha);" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6$7S7$$!1++++++
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{XPPMATH 20 "6$$!+++++D!#5\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
33 "%\n% 3) Kp2 tem menores positivos:" }}{PARA 0 "" 0 "" {TEXT -1 1 "
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{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Delta21G\"\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,&%&alphaG\"\"\"*$)F$\"\"#\"\"\"\"\"%" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 103 "%\n% A partir dos graficos abaixo conclui-se q
ue alpha<-0.25 ou alpha>0 para que ambos os menores de Kp2" }}{PARA 0 
"" 0 "" {TEXT -1 18 "% sejam positivos." }}{PARA 0 "" 0 "" {TEXT -1 1 
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{XPPMATH 20 "6$$!+++++D!#5\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 
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" {MPLTEXT 1 0 60 "HKB2:=Kp2[1,1]/abs(Kp2[1,2])+Kp2[2,2]/abs(Kp2[2,1])
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"" {TEXT -1 1 "%" }}{PARA 0 "" 0 "" {TEXT -1 108 "% A partir dos grafi
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%!G-%%VIEWG6$;$\"\"#Fa^l$\"#DFa^l%(DEFAULTG" 2 370 370 370 2 0 1 0 2 
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"" {XPPMATH 20 "6#$!\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$!+++
++D!#5\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "%" }}{PARA 0 "" 0 "
" {TEXT -1 33 "% Outra matriz HFG eh a seguinte:" }}{PARA 0 "" 0 "" 
{TEXT -1 1 "%" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Kp3:=matrix(2,2,[a
lpha,-2,2*alpha^2,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Kp3G-%'ma
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0 "" 0 "" {TEXT -1 1 "%" }}{PARA 0 "" 0 "" {TEXT -1 13 "% which gives
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25 "K3_geral:=evalm(Kp3&*Sp);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)K3
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "%" }}{PARA 0 "" 0 "" {TEXT -1 93 "%
 thus there does not exists any fixed nonsingular Sp such that this ma
trix remains symmetric" }}{PARA 0 "" 0 "" {TEXT -1 21 "% for all Real \+
alpha." }}{PARA 0 "" 0 "" {TEXT -1 1 "%" }}{PARA 0 "" 0 "" {TEXT -1 
51 "% Quatro condicoes sao avaliadas para a matriz Kp3:" }}{PARA 0 "" 
0 "" {TEXT -1 1 "%" }}{PARA 0 "" 0 "" {TEXT -1 11 "% 1) Kp3>0:" }}
{PARA 0 "" 0 "" {TEXT -1 1 "%" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "tr
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91 "% que sao positivos para 0.5248885987<alpha<1.490216120, como se o
bserva no grafico abaixo." }}{PARA 0 "" 0 "" {TEXT -1 1 "%" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 44 "plot([tr3,d3],alpha=-2..2);fsolve(d3,alpha)
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(DEFAULTG" 2 317 317 317 2 0 1 0 2 9 0 4 2 1.000000 45.000000 
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1 0 0 0 0 0 0 0 0 0 0 0 }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+()f))[_!
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0 "" 0 "" {TEXT -1 21 "% 2) -Kp3 eh Hurwitz:" }}{PARA 0 "" 0 "" {TEXT 
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{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Delta32G,&%&alphaG\"\"\"*$)F&\"\"#
\"\"\"\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "%" }}{PARA 0 "" 0 "
" {TEXT -1 99 "% A partir dos graficos abaixo conclui-se que -1<alpha<
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0 0 0 0 }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$!+++++D!#5\"\"!" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 33 "%\n% 3) Kp3 tem menores positivos:" }}
{PARA 0 "" 0 "" {TEXT -1 1 "%" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "De
lta31:=Kp3[1,1];Delta32;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Delta31
G%&alphaG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%&alphaG\"\"\"*$)F$\"\"
#\"\"\"\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "%\n% A partir dos
 graficos abaixo conclui-se que alpha>0 para que ambos os menores de K
p3" }}{PARA 0 "" 0 "" {TEXT -1 18 "% sejam positivos." }}{PARA 0 "" 0 
"" {TEXT -1 1 "%" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "plot([Delta31,D
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{INLPLOT "6&-%'CURVESG6$7S7$$!1+++++++g!#;F(7$$!1LLLe%G?y&F*F,7$$!1mmT
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L3_>f_ZF*F;7$$!1++vo1YZXF*F>7$$!1LL3-OJNVF*FA7$$!1++v$*o%Q7%F*FD7$$!1m
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`pFhp7$$\"1=****\\i_QQ!#=F[q7$$\"1%***\\7y%3T#F`pF_q7$$\"1$****\\P![hY
F`pFbq7$$\"1HLLLQx$o'F`pFeq7$$\"1+++]P+V))F`pFhq7$$\"1mm\"zpe*z5F*F[r7
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s7$$\"1LLe9tOcFF*Fcs7$$\"1+++]Qk\\HF*Ffs7$$\"1LL$3dg6<$F*Fis7$$\"1nmmm
xGpLF*F\\t7$$\"1++D\"oK0e$F*F_t7$$\"1++v=5s#y$F*Fbt7$$\"1+++++++SF*Fet
-%'COLOURG6&%$RGBG$\"#5!\"\"\"\"!F^u-F$6$7S7$F($\"1+++++++%)F*7$F,$\"1
!Gd)yFr!f(F*7$F/$\"1)GJ#o)zu\"pF*7$F2$\"1:xs_it%>'F*7$F5$\"1vH!=&y&R]&
F*7$F8$\"1x%[ebxI&[F*7$F;$\"1u&>J:gAG%F*7$F>$\"1y7yW()HCPF*7$FA$\"1t,G
4Sm#=$F*7$FD$\"15m[(Q)fyEF*7$FG$\"1auo%3Iu>#F*7$FJ$\"1(\\T)3'p\\!=F*7$
FM$\"1UqCw(*G)R\"F*7$FP$\"1c-X=hQF5F*7$FS$\"1>CS$z_X0(F`p7$FV$\"1UmxS^
,LWF`p7$FY$\"1m>uz6w*o\"F`p7$Ffn$!1\"G@Si9C;$F]q7$Fin$!1V<T$zLGG#F`p7$
F\\o$!16eL_,T%p$F`p7$F_o$!1@1P$f\"[()[F`p7$Fbo$!1o,&*3hnxcF`p7$Feo$!1;
*)f+.XUhF`p7$Fho$!1A$3R]%pXiF`p7$F[p$!1cfRRVx4gF`p7$F^p$!1O2E+g4$Q&F`p
7$Fbp$!1vUd,`(4_%F`p7$Fep$!1gu?^s')eKF`p7$Fhp$!16&*[GS+%f\"F`p7$F[q$\"
1sw3&Qju*QF]q7$F_q$\"1N9]*HNLk#F`p7$Fbq$\"1'RgkMc1`&F`p7$Feq$\"1!QR&Rr
oq%)F`p7$Fhq$\"1c+HO_4(>\"F*7$F[r$\"16!*f8I[Y:F*7$F^r$\"1-\"eE]*\\j>F*
7$Far$\"1i*3KI/$*Q#F*7$Fdr$\"1'3!*p9N!pGF*7$Fgr$\"1]ug0CHsLF*7$Fjr$\"1
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\"1F(\\QR='3()F*7$Fbt$\"1<s-TBJ1&*F*7$Fet$\"1++++++S5!#:-Fht6&FjtF^uF[
uF^u-%+AXESLABELSG6$Q&alpha6\"%!G-%%VIEWG6$;$!\"'F]u$\"\"%F]u%(DEFAULT
G" 2 324 324 324 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 10030 
10061 10056 10074 0 0 0 20530 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 4 3333 
0 0 0 0 0 0 }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$!+++++D!#5\"\"!" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "%" }}{PARA 0 "" 0 "" {TEXT -1 52 "%
 4) -Kp3 obedece as condicoes para modo deslizante:" }}{PARA 0 "" 0 "
" {TEXT -1 1 "%" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "HKB3:=Kp3[1,1]/a
bs(Kp3[1,2])+Kp3[2,2]/abs(Kp3[2,1]);Delta32;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%HKB3G,&*&\"\"\"F'*$)-%$absG6#%&alphaG\"\"#F'!\"\"#\"
\"\"\"\"#F-F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%&alphaG\"\"\"*$)F$
\"\"#\"\"\"\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "%" }}{PARA 0 "
" 0 "" {TEXT -1 108 "% A partir dos graficos abaixo conclui-se que -1<
alpha<-0.25 ou alpha>0 para que -Kp3 satisfaca as condicoes" }}{PARA 
0 "" 0 "" {TEXT -1 23 "% para modo deslizante." }}{PARA 0 "" 0 "" 
{TEXT -1 1 "%" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "plot([HKB3,Delta3
2],alpha=-2..-0.2);plot([HKB3,Delta32],alpha=0.2..2.5);fsolve(HKB3,alp
ha);fsolve(Delta32,alpha);" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6
$7Y7$$!\"#\"\"!$!1++++++]()!#;7$$!1++]A^wg>!#:$!1BPxa,I.&)F-7$$!1+]Plq
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jv$\"1/(>Ov\"oV5F17$F_w$\"1$)y%GYA5!*)F-7$Fdw$\"1X\"o\"=fT5vF-7$Fiw$\"
19D3w&y?<'F-7$F^x$\"1'=LplCn*\\F-7$Fcx$\"1!f>(o.15RF-7$Fhx$\"1?FLWdaZH
F-7$F]y$\"1*HGU5%>k@F-7$Fby$\"1gjWBfY&Q\"F-7$F\\z$\"1Q6uzJ:nzFfs7$Ffz$
\"1)4J(Qdf5GFfs7$F`[l$!1@#=ZQV:/\"Ffs7$Fe\\l$!1)*************RFfs-Fj\\
l6&F\\]lF*F]]lF*-%+AXESLABELSG6$Q&alpha6\"%!G-%%VIEWG6$;F($F)F_]l%(DEF
AULTG" 2 370 370 370 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 
10030 10061 10056 10074 0 0 0 20530 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 0 
-24864 0 0 0 0 0 0 }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6$7gn7$$\"
1+++++++?!#;$\"1++++++g7!#97$$\"1n\"H#=omi?F*$\"1Eck)[8b=\"F-7$$\"1M$e
kjL`7#F*$\"1fRV#\\Xv6\"F-7$$\"1+voa/+)=#F*$\"1U$fM-g`0\"F-7$$\"1nm\"HF
n1D#F*$\"1Q*[(*3@K)**!#:7$$\"1+]P44+wBF*$\"1S:tg#*fv*)FB7$$\"1ML$eaM8]
#F*$\"1;@=oC`;\")FB7$$\"1</wsmQ5EF*$\"1#>/*4bAouFB7$$\"1+vo*zQ%>FF*$\"
1wGrc&zp*oFB7$$\"1%e9m#4\\GGF*$\"1')[y)eU6R'FB7$$\"1n;a`IaPHF*$\"1x?fT
W=TfFB7$$\"1+D\"y4CG=$F*$\"1Tipcaz%4&FB7$$\"1LL3U^5GMF*$\"1f5vhY.EWFB7
$$\"1LLLo?,vOF*$\"1;V/Du)e)QFB7$$\"1LLe%**=>#RF*$\"1chpq]wYMFB7$$\"1n;
/OeQ8WF*$\"1U\\i78n(y#FB7$$\"1L$3-^Q!p[F*$\"1-\"*>Yf[_BFB7$$\"1+](=YS3
M&F*$\"1)*[80k\"*>?FB7$$\"1M$3_ry(GeF*$\"14**\\_R7j<FB7$$\"1+]PW@::jF*
$\"1P-pWG[p:FB7$$\"1nm;**pW:oF*$\"1%f*=]7><9FB7$$\"1KLezp5csF*$\"1@b0K
LX78FB7$$\"1++]Zd=_xF*$\"1)G8H51'>7FB7$$\"1++]i9I]#)F*$\"1W&=_I!3Z6FB7
$$\"1****\\_#G.t)F*$\"1%e+N*R_#4\"FB7$$\"1L$3_cQi;*F*$\"1?3oL)3M0\"FB7
$$\"1nm;*3yXo*F*$\"1zbR&zGt,\"FB7$$\"1nmmlzO75FB$\"1^<**3uTS**F*7$$\"1
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REe6FB$\"1!Ge@N\"G=&*F*7$$\"1+D1&4+b?\"FB$\"1&41R%46o%*F*7$$\"1n;a8gya
7FB$\"1=jM(pl&\\%*F*7$$\"1nT5se/+8FB$\"1$fW*e/ge%*F*7$$\"1L$eRuk)[8FB$
\"1u'\\,$zU#\\*F*7$$\"1L3_JQd*R\"FB$\"1(eOtOV/b*F*7$$\"1+DJTirV9FB$\"1
&z)[M!\\uh*F*7$$\"1L$3KD\"R\"\\\"FB$\"1f*e`;2\\q*F*7$$\"1++]0UkS:FB$\"
1^'4-hR(4)*F*7$$\"1+]P5'G))e\"FB$\"1hMMz-$[#**F*7$$\"1+vo*\\\\aj\"FB$
\"1[YTZ>m/5FB7$$\"1+]i[S@(o\"FB$\"1!z]e`\\#>5FB7$$\"1mm;)zEPt\"FB$\"11
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FB7$$\"1+v=U_5p?FB$\"1_!>;SU8:\"FB7$$\"1LLL#>1o6#FB$\"1eb(zg))*p6FB7$$
\"1L$eMI(el@FB$\"1T\"o`s3%*=\"FB7$$\"1nTN#[kR@#FB$\"1O<%H%*))*37FB7$$
\"1++]&3=%eAFB$\"1'RcacRsA\"FB7$$\"1n;HJpO4BFB$\"1GP(zBO%[7FB7$$\"1LLL
'=O\\N#FB$\"1SWaRwin7FB7$$\"1+vo;D_.CFB$\"1:$3@c7$)G\"FB7$$\"1+DJMe-]C
FB$\"1\\C0'y4$38FB7$$\"1+++++++DFB$\"1++++++I8FB-%'COLOURG6&%$RGBG$\"#
5!\"\"\"\"!Fb^l-F$6$7S7$F($\"1+++++++OF*7$FI$\"1M]!*\\V+/]F*7$Fgn$\"1F
g5In?*Q'F*7$Fao$\"14Y:)3n)G\")F*7$F[p$\"1v7YV)\\u+\"FB7$F`p$\"1&4:^cd/
A\"FB7$Fep$\"10ld\"H0_V\"FB7$Fjp$\"1Zbv>r1v;FB7$F_q$\"1ia&RKk=%>FB7$Fd
q$\"1m;qy5wEAFB7$Fiq$\"1kY&=Td&RDFB7$F^r$\"1^Y#zBa;$GFB7$Fcr$\"1&o*HHR
2zJFB7$Fhr$\"1'=N^6Hxa$FB7$F]s$\"1()f5\"3y<#RFB7$Fbs$\"1NL*Q.@uF%FB7$F
gs$\"1]L_>**4?ZFB7$F\\t$\"1F*)HdN#>6&FB7$Fat$\"1oZc(y9re&FB7$Fft$\"1N9
b_P:DgFB7$F[u$\"1WpW*)ecClFB7$F`u$\"1&GX=,A%=qFB7$Feu$\"1<3msxt_vFB7$F
ju$\"1*\\//&H_g!)FB7$F_v$\"1`M$R84mi)FB7$Fdv$\"1dC.r9![B*FB7$Fiv$\"1m/
.$e#)4y*FB7$F^w$\"1i$zLFQ)Q5F-7$Fcw$\"1!HExC)\\.6F-7$Fhw$\"1hfH-Mjo6F-
7$F]x$\"1()*zj(HUL7F-7$Fbx$\"1DN@.zR28F-7$Fgx$\"1j\"*3Chpv8F-7$F\\y$\"
1Ws!=cH0X\"F-7$Fay$\"17+,zP/?:F-7$Ffy$\"18\"HM(\\*yf\"F-7$F[z$\"1zLK^]
\"Hn\"F-7$F`z$\"1+*Hy@lJv\"F-7$Fez$\"1MTLK`VL=F-7$Fjz$\"1ny^D\"*Q>>F-7
$F_[l$\"1JOf,!GS+#F-7$Fd[l$\"12e!z2mC4#F-7$Fi[l$\"1=d))R>0#=#F-7$F^\\l
$\"1%R:\"3F-mAF-7$Fc\\l$\"1ilYUp?kBF-7$Fh\\l$\"12UJ&R$y`CF-7$F]]l$\"1v
#epW?6b#F-7$Fb]l$\"1:2&*=K0YEF-7$Fg]l$\"1++++++]FF--F\\^l6&F^^lFb^lF_^
lFb^l-%+AXESLABELSG6$Q&alpha6\"%!G-%%VIEWG6$;$\"\"#Fa^l$\"#DFa^l%(DEFA
ULTG" 2 370 370 370 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 
10030 10061 10056 10074 0 0 0 20530 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 
-16401 -10204 0 0 0 0 0 0 }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!\"\"\"\"
!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$!+++++D!#5\"\"!" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 1 "%" }}{PARA 0 "" 0 "" {TEXT -1 33 "% Usando
-se Sp3 abaixo, obtem-se:" }}{PARA 0 "" 0 "" {TEXT -1 1 "%" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 48 "Sp3:=matrix(2,2,[0,1,-1,0]);K3:=evalm(Kp3&*
Sp3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Sp3G-%'matrixG6#7$7$\"\"!
\"\"\"7$!\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#K3G-%'matrixG6#7
$7$\"\"#%&alphaG7$!\"\",$*$)F+F*\"\"\"F*" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 1 "%" }}{PARA 0 "" 0 "" {TEXT -1 50 "% Quatro condicoes sao a
valiadas para a matriz K3:" }}{PARA 0 "" 0 "" {TEXT -1 1 "%" }}{PARA 
0 "" 0 "" {TEXT -1 10 "% 1) K3>0:" }}{PARA 0 "" 0 "" {TEXT -1 1 "%" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "tr33:=linalg[trace](K3);d33:=simpli
fy(linalg[det](K3)-(((K3[1,2]-K3[2,1])^2)/4));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%tr33G,&\"\"#\"\"\"*$)%&alphaGF&\"\"\"F&" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%$d33G,(%&alphaG#\"\"\"\"\"#*$)F&F)\"\"\"#\"#:
\"\"%#!\"\"F/F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "% que sao posi
tivos para alpha<-0.3333333333 ou alpha>0.2, como se observa no grafic
o abaixo." }}{PARA 0 "" 0 "" {TEXT -1 1 "%" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 66 "plot([tr33,d33],alpha=-2..2);fsolve(tr33,alpha);fsolv
e(d33,alpha);" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6$7S7$$!\"#\"
\"!$\"#5F*7$$!1LLL$Q6G\">!#:$\"1g\">kx%p<$*F07$$!1nm;M!\\p$=F0$\"1*p^#
3Nw[()F07$$!1LLL))Qj^<F0$\"10b^dDWO\")F07$$!1LLL=Kvl;F0$\"19dxovY\\vF0
7$$!1nm;C2G!e\"F0$\"1\\aMMVd%*pF07$$!1LL$3yO5]\"F0$\"1$**G[$GA1lF07$$!
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Q&\\7F0$\"1)G*)H@%pA^F07$$!1nmmc4`i6F0$\"1JmT]k&Hq%F07$$!1LLLQW*e3\"F0
$\"1JlSiMLeVF07$$!1,+++()>'***!#;$\"1Mw**o(z%)*RF07$$!1++++0\"*H\"*Fbo
$\"10-w9`5nOF07$$!1++++83&H)Fbo$\"1%>saZnhP$F07$$!1LLL3k(p`(Fbo$\"1laf
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lF*F][lF*-%+AXESLABELSG6$Q&alpha6\"%!G-%%VIEWG6$;F(Fgz%(DEFAULTG" 2 
317 317 317 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 10030 
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0 0 0 0 }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$!+LLLLL!#5$\"+++++?F%" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "%" }}{PARA 0 "" 0 "" {TEXT -1 20 "%
 2) -K3 eh Hurwitz:" }}{PARA 0 "" 0 "" {TEXT -1 1 "%" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 31 "tr33;Delta323:=linalg[det](K3);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#,&\"\"#\"\"\"*$)%&alphaGF$\"\"\"F$" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%)Delta323G,&%&alphaG\"\"\"*$)F&\"\"#\"\"\"\"\"%
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "%" }}{PARA 0 "" 0 "" {TEXT -1 
96 "% A partir dos graficos abaixo conclui-se que alpha<-0.25 ou alpha
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "plot([tr33,Delta323],alpha=-1.2..0.
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" {XPPMATH 20 "6$$!+++++D!#5\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
32 "%\n% 3) K3 tem menores positivos:" }}{PARA 0 "" 0 "" {TEXT -1 1 "%
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{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)Delta313G\"\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,&%&alphaG\"\"\"*$)F$\"\"#\"\"\"\"\"%" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 103 "%\n% A partir dos graficos abaixo conclui-se q
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"" 0 "" {TEXT -1 18 "% sejam positivos." }}{PARA 0 "" 0 "" {TEXT -1 1 
"%" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "plot([Delta313,Delta323],alph
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$;$!\"'F`u$\"\"%F`u%(DEFAULTG" 2 324 324 324 2 0 1 0 2 9 0 4 2 
1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20530 0 
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{XPPMATH 20 "6$$!+++++D!#5\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 
"%" }}{PARA 0 "" 0 "" {TEXT -1 51 "% 4) -K3 obedece as condicoes para \+
modo deslizante:" }}{PARA 0 "" 0 "" {TEXT -1 1 "%" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 58 "HKB33:=K3[1,1]/abs(K3[1,2])+K3[2,2]/abs(K3[2,1]);De
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-1 1 "%" }}{PARA 0 "" 0 "" {TEXT -1 104 "% A partir dos graficos abaix
o conclui-se que alpha<-0.25 ou alpha>0 para que -K3 satisfaca as cond
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{PARA 0 "" 0 "" {TEXT -1 1 "%" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "p
lot([HKB33,Delta323],alpha=-2..-0.2);plot([HKB33,Delta323],alpha=0.2..
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{XPPMATH 20 "6$$!+++++D!#5\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 
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{TEXT -1 1 "%" }}{PARA 0 "" 0 "" {TEXT -1 62 "% Quatro condicoes sao a
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{TEXT -1 1 "%" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "tr4:=linalg[trace]
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0 "" 0 "" {TEXT -1 51 "% Quatro condicoes sao avaliadas para a matriz \+
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