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{SECT 0 {EXCHG {PARA 4 "" 0 "" {TEXT 270 14 "Ejercicios de " }}{PARA 
4 "" 0 "" {TEXT 271 22 "(8) Estabilidad lineal" }}{PARA 4 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 269 50 "Ejercicio 08-27  (del EX
AMEN ORDINARIO DE 19ENE12)" }}{PARA 4 "" 0 "" {TEXT -1 55 "\nDe entre \+
los m\351todos impl\355citos de 2 etapas de tablero" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 260 18 "       0   |      " }
{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT 262 9 "        0" }}{PARA 
4 "" 0 "" {TEXT 261 18 "       1  |       " }{XPPEDIT 18 0 "alpha;" "6
#%&alphaG" }{TEXT 263 9 "        0" }}{PARA 4 "" 0 "" {TEXT -1 14 "   \+
  ---------" }{TEXT 256 24 "------------------------" }}{PARA 4 "" 0 "
" {TEXT 257 17 "          |      " }{XPPEDIT 18 0 "1/2;" "6#*&\"\"\"F$
\"\"#!\"\"" }{TEXT 258 10 "          " }{XPPEDIT 18 0 "1/2;" "6#*&\"\"
\"F$\"\"#!\"\"" }{TEXT 259 1 " " }}{PARA 4 "" 0 "" {TEXT -1 40 "se bus
caran los que sean de mayor orden." }}{PARA 4 "" 0 "" {TEXT -1 1 " " }
}{PARA 4 "" 0 "" {TEXT -1 40 "(hasta aqu\355 es Ejercicio del cap\355t
ulo 7)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 73 
"Calc\372lese la funci\363n de estabilidad de la familia general de lo
s m\351todos " }}{PARA 4 "" 0 "" {TEXT -1 21 "del tablero y, para  " }
{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 55 " = 2 ,  dib\372jese \+
el dominio de estabilidad y disc\372tase " }}{PARA 4 "" 0 "" {TEXT -1 
28 "la A-estabilidad  del m\351todo" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 
"interface(labeling=false):" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 
37 "# desarrollo de la verdadera solucion" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 14 "f0:=f(x,y(x)):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 44 "f1:=subs(diff(y(x),x)=f(x,y(x)),diff(f0,x)):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "f2:=subs(diff(y(x),x)=f(x,y(
x)),diff(f1,x)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "f3:=sub
s(diff(y(x),x)=f(x,y(x)),diff(f2,x)):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 43 "f0:=subs(\{y(x)=y0,x=x0\},f0):f0:=expand(f0):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "f1:=subs(\{y(x)=y0,x=x0\},f1
):f1:=expand(f1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "f2:=su
bs(\{y(x)=y0,x=x0\},f2):f2:=expand(f2):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 43 "f3:=subs(\{y(x)=y0,x=x0\},f3):f3:=expand(f3):" }}}
{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 23 "# desarrollo del m\351todo" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "c[1]:=0:c[2]:=1:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "a[1,1]:=alpha:a[1,2]:=0:a[2,
1]:=alpha:a[2,2]:=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "b[1
]:=1/2:b[2]:=1/2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "k10:=u
napply(f(x0+c[1]*h,y0+a[1,1]*h*kp10(h)+a[1,2]*h*kp20(h)),h):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "k20:=unapply(f(x0+c[2]*h,y0+
a[2,1]*h*kp10(h)+a[2,2]*h*kp20(h)),h):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "k100:=k10(0):k100:=expand(k100):" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 32 "k200:=k20(0):k200:=expand(k200):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Fi0:=expand(b[1]*k100+b[2]*k200):" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "k11:=D(k10):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "k21:=D(k20):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 66 "k110:=subs(\{kp10(0)=k100,kp20(0)=k200\},k11(0
)):k110:=expand(k110):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "k
210:=subs(\{kp10(0)=k100,kp20(0)=k200\},k21(0)):k210:=expand(k210):" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Fi1:=expand(b[1]*k110+b[2]
*k210):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "k12:=D(k11):" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "k22:=D(k21):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "k120:=subs(\{D(kp10)(0)=k110,D(kp20
)(0)=k210,kp10(0)=k100,kp20(0)=k200\},k12(0)):k120:=expand(k120):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "k220:=subs(\{D(kp10)(0)=k110
,D(kp20)(0)=k210,kp10(0)=k100,kp20(0)=k200\},k22(0)):k220:=expand(k220
):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Fi2:=expand(b[1]*k120
+b[2]*k220):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "k13:=D(k12)
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "k23:=D(k22):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "k130:=subs(\{(D@@2)(kp10)(0
)=k120,(D@@2)(kp20)(0)=k220,D(kp10)(0)=k110,D(kp20)(0)=k210,kp10(0)=k1
00,kp20(0)=k200\},k13(0)):k130:=expand(k130):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 140 "k230:=subs(\{(D@@2)(kp10)(0)=k120,(D@@2)(kp20)(
0)=k220,D(kp10)(0)=k110,D(kp20)(0)=k210,kp10(0)=k100,kp20(0)=k200\},k2
3(0)):k230:=expand(k230):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
33 "Fi3:=expand(b[1]*k130+b[2]*k230):" }}}{EXCHG {PARA 0 "" 0 "" 
{MPLTEXT 0 21 24 "# consistencia y orden 1" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 47 "cond1a:=coeff(Fi0,f(x0,y0))=coeff(f0,f(x0,y0));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond1aG/\"\"\"F&" }}}{EXCHG {PARA 
0 "" 0 "" {MPLTEXT 0 21 29 "# condiciones para el orden 2" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "cond2a:=coeff(Fi1,D[1](f)(x0,y0))=c
oeff((1/2)*f1,D[1](f)(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'c
ond2aG/#\"\"\"\"\"#F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "co
nd2b:=coeff(coeff(Fi1,D[2](f)(x0,y0)),f(x0,y0))=coeff(coeff((1/2)*f1,D
[2](f)(x0,y0)),f(x0,y0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond2b
G/%&alphaG#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 14 "or
den 2 para  " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{MPLTEXT 0 21 29 "
 = 1/2 (o sea, LOBATTO IIIb) " }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 
21 29 "# condiciones para el orden 3" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 92 "cond3a:=coeff(Fi2,D[1,1](f)(x0,y0))=coeff((1/3)*f2,D[
1,1](f)(x0,y0));subs(alpha=1/2,cond3a);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%'cond3aG/#\"\"\"\"\"##F'\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#/#\"\"\"\"\"##F%\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
124 "cond3b:=coeff(coeff(Fi2,D[1,2](f)(x0,y0)),f(x0,y0))=coeff(coeff((
1/3)*f2,D[1,2](f)(x0,y0)),f(x0,y0));subs(alpha=1/2,cond3b);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%'cond3bG/%&alphaG#\"\"#\"\"$" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/#\"\"\"\"\"##F&\"\"$" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 128 "cond3c:=coeff(coeff(Fi2,D[2,2](f)(x0,y0)),f(x0,y0
)^2)=coeff(coeff((1/3)*f2,D[2,2](f)(x0,y0)),f(x0,y0)^2);subs(alpha=1/2
,cond3c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3cG/*$)%&alphaG\"
\"#\"\"\"#F*\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/#\"\"\"\"\"%#F%
\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "cond3d:=coeff(coe
ff(Fi2,D[2](f)(x0,y0)),D[1](f)(x0,y0))=coeff(coeff((1/3)*f2,D[2](f)(x0
,y0)),D[1](f)(x0,y0));subs(alpha=1/2,cond3d);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'cond3dG/\"\"!#\"\"\"\"\"$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/\"\"!#\"\"\"\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 124 "cond3e:=coeff(coeff(Fi2,D[2](f)(x0,y0)^2),f(x0,y0))=
coeff(coeff((1/3)*f2,D[2](f)(x0,y0)^2),f(x0,y0));subs(alpha=1/2,cond3e
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cond3eG/,$*&\"\"#\"\"\")%&alp
haGF(F)F)#F)\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/#\"\"\"\"\"##F%
\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 36 "no hay orden 3 para
 ningun valor de " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{MPLTEXT 0 
21 2 "  " }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 24 "# funcion de est
abilidad" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f:=(x,y)->lambd
a*y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"yG6\"6$%)ope
ratorG%&arrowGF)*&%'lambdaG\"\"\"9%F/F)F)F)" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 64 "ka1:=unapply(f(x0+c[1]*h,y0+a[1,1]*h*kp1(h)+a[1,2]
*h*kp2(h)),h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ka1Gf*6#%\"hG6\"6
$%)operatorG%&arrowGF(*&%'lambdaG\"\"\",&%#y0GF.*(%&alphaGF.9$F.-%$kp1
G6#F3F.F.F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "ka2:=u
napply(f(x0+c[2]*h,y0+a[2,1]*h*kp1(h)+a[2,2]*h*kp2(h)),h);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%$ka2Gf*6#%\"hG6\"6$%)operatorG%&arrowGF(*&
%'lambdaG\"\"\",&%#y0GF.*(%&alphaGF.9$F.-%$kp1G6#F3F.F.F.F(F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "test:=solve(\{k1=subs(\{kp1(
h)=k1,kp2(h)=k2\},ka1(h)),k2=subs(\{kp1(h)=k1,kp2(h)=k2\},ka2(h))\},\{
k1,k2\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%testG<$/%#k1G,$*(%'lam
bdaG\"\"\"%#y0GF+,&F+!\"\"*(F*F+%&alphaGF+%\"hGF+F+F.F./%#k2GF(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "k1:=subs(test,k1);k2:=subs(t
est,k2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k1G,$*(%'lambdaG\"\"\"%
#y0GF(,&F(!\"\"*(F'F(%&alphaGF(%\"hGF(F(F+F+" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#k2G,$*(%'lambdaG\"\"\"%#y0GF(,&F(!\"\"*(F'F(%&alphaG
F(%\"hGF(F(F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "y1:=expa
nd(y0+h*(b[1]*k1+b[2]*k2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G,
&%#y0G\"\"\"**%\"hGF'%'lambdaGF'F&F',&F'!\"\"*(F*F'%&alphaGF'F)F'F'F,F
," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "y1:=collect(subs(lambd
a=z/h,y1),y0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G*&,&\"\"\"F'*&
%\"zGF',&F'!\"\"*&F)F'%&alphaGF'F'F+F+F'%#y0GF'" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 41 "r:=unapply(collect(simplify(y1/y0),z),z);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rGf*6#%\"zG6\"6$%)operatorG%&arrow
GF(*&,&\"\"\"!\"\"*&,&%&alphaGF.F.F/F.9$F.F.F.,&F.F/*&F3F.F2F.F.F/F(F(
F(" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 22 "# A-estabilidad para  \+
" }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{MPLTEXT 0 21 10 " = 2      " 
}}}{EXCHG {PARA 4 "" 0 "" {TEXT 264 29 "En general, como hemos visto \+
" }}{PARA 4 "" 0 "" {TEXT 266 4 "    " }{XPPEDIT 18 0 "r(z) = (1+(1-al
pha)*z)/(1-alpha*z);" "6#/-%\"rG6#%\"zG*&,&\"\"\"F**&,&F*F*%&alphaG!\"
\"F*F'F*F*F*,&F*F**&F-F*F'F*F.F." }{TEXT 265 3 "   " }}{PARA 4 "" 0 "
" {TEXT 268 9 "y, para  " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT 
267 6 " = 2 ," }}{PARA 4 "" 0 "" {TEXT 274 5 "     " }{XPPEDIT 18 0 "r
(z) = (1-z)/(1-2*z);" "6#/-%\"rG6#%\"zG*&,&\"\"\"F*F'!\"\"F*,&F*F**&\"
\"#F*F'F*F+F+" }{TEXT 273 3 "   " }{TEXT -1 0 "" }}{PARA 4 "" 0 "" 
{TEXT 272 39 " Pondremos que  z=x+i y .              " }}{PARA 4 "" 0 
"" {TEXT -1 4 "Como" }}{PARA 4 "" 0 "" {TEXT -1 29 "     1 -  z  = 1 -
 x  -  i  y" }}{PARA 4 "" 0 "" {TEXT -1 1 "y" }}{PARA 4 "" 0 "" {TEXT 
-1 39 "     1  - 2 z  = 1  - 2  x - 2  i  y , " }}{PARA 4 "" 0 "" 
{TEXT -1 8 "resulta " }}{PARA 4 "" 0 "" {TEXT -1 7 "     | " }
{XPPEDIT 18 0 "1-z;" "6#,&\"\"\"F$%\"zG!\"\"" }{TEXT -1 27 " | ^2 = (1
 - x )^2 +  y^2  " }}{PARA 4 "" 0 "" {TEXT -1 2 "y " }}{PARA 4 "" 0 "
" {TEXT -1 50 "     | 1  - 2 z | ^2 = (1  - 2  x )^2 +  4  y^2   " }}
{PARA 4 "" 0 "" {TEXT -1 46 "por lo que la curva  | r(z) |^2 = 1 se es
cribe" }}{PARA 4 "" 0 "" {TEXT -1 56 "     (1 - x )^2 +  y^2   =   (1 \+
 - 2  x )^2 +  4  y^2   " }}{PARA 4 "" 0 "" {TEXT -1 5 "o sea" }}
{PARA 4 "" 0 "" {TEXT -1 32 "    3 x^2 - 2 x + 3 y^2 =  0    " }}
{PARA 4 "" 0 "" {TEXT -1 12 "     x^2 -  " }{XPPEDIT 18 0 "2/3;" "6#*&
\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 14 " x + y^2 = 0  " }}{PARA 4 "" 0 "
" {TEXT -1 11 "La curva es" }}{PARA 4 "" 0 "" {TEXT -1 12 "      ( x -
 " }{XPPEDIT 18 0 "1/3;" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT -1 13 " )^2 +
 y^2 = " }{XPPEDIT 18 0 "1/9;" "6#*&\"\"\"F$\"\"*!\"\"" }{TEXT -1 3 " \+
  " }}{PARA 4 "" 0 "" {TEXT -1 37 "o sea, una circunferencia de centro
  " }{XPPEDIT 18 0 "1/3;" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT -1 5 " = ( \+
" }{XPPEDIT 18 0 "1/3;" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT -1 18 " , 0 ) \+
  y radio  " }{XPPEDIT 18 0 "1/3;" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT -1 
22 " que pasa por  (0,0)  " }}{PARA 4 "" 0 "" {TEXT -1 41 "Como  z=-1=
(-1,0)  verifica  | r(z) | =| " }{XPPEDIT 18 0 "2/3;" "6#*&\"\"#\"\"\"
\"\"$!\"\"" }{TEXT -1 46 " | < 1  resulta que el dominio de estabilida
d " }}{PARA 4 "" 0 "" {TEXT -1 91 "es el exterior de la circunferencia
 y que el dominio de estabilida contiene (sobradamente) " }}{PARA 4 "
" 0 "" {TEXT -1 3 "al " }{TEXT 275 70 "semiplano de parte real negativ
a; por lo tanto, el m\351todo es A-estable" }{TEXT -1 0 "" }}{PARA 0 "
" 0 "" {GLPLOT2D 378 378 378 {PLOTDATA 2 "6+-%'POINTSG6$7$$!\"&!\"\"$!
#6F)-%'SYMBOLG6#%&POINTG-F$6$7$F'$\"#6F)F,-F$6$7$$\"\"$\"\"!F3F,-F$6$7
$F8F*F,-%)POLYGONSG6$7U7$$\"\"#F:$F:F:7$$\"+,Z6#*>!\"*$\"+OBL`7!#57$$
\"+hJeo>FI$\"+t))*o[#FL7$$\"+'[w(H>FI$\"+FbC\"o$FL7$$\"+!o1j(=FI$\"+Tn
`<[FL7$$\"+%*p,4=FI$\"+CD&y(eFL7$$\"+F'o*G<FI$\"+g5ZXoFL7$$\"+!*RUP;FI
$\"+HC80xFL7$$\"+&zEe`\"FI$\"+a#zKW)FL7$$\"+#HzdU\"FI$\"+B0F[!*FL7$$\"
+%*p,48FI$\"+l^c5&*FL7$$\"+98Q(=\"FI$\"+3D(G#)*FL7$$\"+>0zi5FI$\"+%Gn-
)**FL7$$\"+.[4s$*FLF\\q7$$\"+ao=E\")FL$\"+2D(G#)*FL7$$\"+e+$)4pFL$\"+j
^c5&*FL7$$\"+(32Au&FL$\"+E0F[!*FL7$$\"+a?tTYFL$\"+d#zKW)FL7$$\"+2,wDOF
L$\"+JC80xFL7$$\"+<PJ5FFL$\"+]5ZXoFL7$$\"+]+$)4>FL$\"+9D&y(eFL7$$\"+&>
LpB\"FL$\"+Kn`<[FL7$$\"*Q^B-(FL$\"+>bC\"o$FL7$$\"*(QoTJFL$\"+l))*o[#FL
7$$\")')H&)yFL$\"+IBL`7FL7$FE$!+:w1-T!#>7$$\")()H&)yFL$!+QBL`7FL7$$\"*
*QoTJFL$!+t))*o[#FL7$$\"*T^B-(FL$!+FbC\"o$FL7$$\"+*>LpB\"FL$!+Sn`<[FL7
$$\"+a+$)4>FL$!+?D&y(eFL7$$\"+BPJ5FFL$!+c5ZXoFL7$$\"+8,wDOFL$!+PC80xFL
7$$\"+h?tTYFL$!+i#zKW)FL7$$\"+%42Au&FL$!+H0F[!*FL7$$\"+m+$)4pFL$!+m^c5
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