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{SECT 0 {EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 15 "limitfiveq3.mws" }}}
{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{MPLTEXT 0 21 143 "The special case v(theta)=1 ,  diff(Q[infinify](the
ta),theta)=0 for consistency and diff(v(theta),theta) = 0 diff(Q[infin
ity](theta),theta$2)=0" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 64 "No
tation diff(v(thtea),theta$n)=v[n] diff(Q(thtea),theta$n)=Q[n]" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 67 
"The following substitutions for P(t,theta) and Q(t,theta) are made:" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 435 "P(t,theta) = P[infinity]
(theta)-v(theta)*ln(t)+1/4*exp(P[infinity](theta))^2*diff(Q[infinity](
theta),`$`(theta,2))^2*t^2*ln(t)^2+V[1](theta)*t^2+(exp(2*P[infinity](
theta))*psi[Q](theta)*diff(Q[infinity](theta),`$`(theta,2))-1/4*diff(Q
[infinity](theta),`$`(theta,2))^2-1/4*diff(v(theta),`$`(theta,2)))*t^2
*ln(t),Q(t,theta) = Q[infinity](theta)+psi[Q](theta)*t^(2*v(theta))+1/
2*diff(Q[infinity](theta),`$`(theta,2))*t^(2*v(theta))*ln(t);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"PG6$%\"tG%&thetaG,,-&F%6#%)infini
tyG6#F(\"\"\"*&-%\"vGF.F/-%#lnG6#F'F/!\"\"*&#F/\"\"%F/**)-%$expG6#F*\"
\"#F/)-%%diffG6$-&%\"QGF,F.-%\"$G6$F(F?F?F/)F'F?F/)F3F?F/F/F/*&-&%\"VG
6#F/F.F/FJF/F/*(,(*(-F=6#,$*&F?F/F*F/F/F/-&%$psiG6#FFF.F/FAF/F/*&#F/F9
F/*$F@F/F/F6*&#F/F9F/-FB6$F1FGF/F6F/FJF/F3F/F//-FFF&,(FDF/*&FXF/)F',$*
&F?F/F1F/F/F/F/*&#F/F?F/*(FAF/FaoF/F3F/F/F/" }}}{EXCHG {PARA 0 "" 0 "
" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "resta
rt:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "grtw();" }}{PARA 6 "
" 1 "" {TEXT -1 25 "Scalar invariant library." }}{PARA 6 "" 1 "" 
{TEXT -1 28 "Last modified 25 March 1997." }}{PARA 6 "" 1 "" {TEXT -1 
25 "`Differential Invariants`" }}{PARA 6 "" 1 "" {TEXT -1 29 "`Last mo
dified Jan. 20, 1995`" }}{PARA 6 "" 1 "" {TEXT -1 41 "`Basis/tetrad re
lated object definitions`" }}{PARA 6 "" 1 "" {TEXT -1 31 "`Last modifi
ed 23 January 2001`" }}{PARA 6 "" 1 "" {TEXT -1 25 "`Last built 27 May
, 1999`" }}{PARA 6 "" 1 "" {TEXT -1 25 "`Last built 27 May, 1999`" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%=GRTensorII~Version~1.79~(R4)G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%06~February~2001G" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#%ZDeveloped~by~Peter~Musgrave,~Denis~Pollney~and~Kayl
l~LakeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%DCopyright~1994-2001~by~th
e~authors.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%inLatest~version~avail
able~from:~http://grtensor.phy.queensu.ca/G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%4c:/Grtii(6)/MetricsG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "qload(gowdy);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2D
efault~spacetimeG%&gowdyG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%9For~the
~gowdy~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,CoordinatesG" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"xG6#%#upG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/)%#x~G%\"aG-%'vectorG6#7&%\"tG%&thetaG%#x1G%#x2G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Line~elementG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*$)%$~dsG\"\"#\"\"\",,**-%$expG6#,$*&#F(F'F(-%&gammaG6
$%\"tG%&thetaGF(!\"\"F(F4#F6F'%#~dGF()F4%#2~GF(F6**F+F(F4F7F8F()F5F:F(
F(**F4F(-F,6#-%\"PGF3F(F8F()%#x1GF:F(F(*2F'F(F4F(F>F(-%\"QGF3F(F8F()FC
%\"~GF(%#d~GF()%#x2GFHF(F(*(,&*(F4F(F>F()FEF'F(F(*&F4F(F>F6F(F(F8F()FK
F:F(F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%,ConstraintsG7&/-%%diffG6$
-%&gammaG6$%\"tG%&thetaGF-,**(F-\"\"\")-%$expG6#-%\"PGF,\"\"#F1)-F(6$-
%\"QGF,F-F8F1!\"\"*&F-F1)-F(6$F6F-F8F1F>*(F-F1F2F1)-F(6$F<F.F8F1F>*&F-
F1)-F(6$F6F.F8F1F>/-F(6$F*F.,&**F8F1F-F1FIF1FAF1F>*,F8F1F-F1F2F1F:F1FE
F1F>/-F(6$F6-%\"$G6$F-F8,(-F(6$F6-FU6$F.F8F1*&FAF1F-F>F>*&-F46#,$*&F8F
1F6F1F1F1,&*$F9F1F1*$FDF1F>F1F1/-F(6$F<FT,*-F(6$F<FZF1*&F:F1F-F>F>*(F8
F1FAF1F:F1F>*(F8F1FIF1FEF1F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 15 "grcalc(WeylSq);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G
$\"#%*!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "gralter(_,13,
6,7);" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component simplification of a GR
TensorII object:" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" 
{TEXT -1 64 "Applying routine `Apply constraints repeatedly` to object
 WeylSq" }}{PARA 6 "" 1 "" {TEXT -1 40 "Applying routine expand to obj
ect WeylSq" }}{PARA 6 "" 1 "" {TEXT -1 40 "Applying routine factor to \+
object WeylSq" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"#Z!\"
$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 453 "grmap(_,subs,P(t,thet
a) = P[infinity](theta)-v(theta)*ln(t)+1/4*exp(P[infinity](theta))^2*d
iff(Q[infinity](theta),`$`(theta,2))^2*t^2*ln(t)^2+V[1](theta)*t^2+(ex
p(2*P[infinity](theta))*psi[Q](theta)*diff(Q[infinity](theta),`$`(thet
a,2))-1/4*diff(Q[infinity](theta),`$`(theta,2))^2-1/4*diff(v(theta),`$
`(theta,2)))*t^2*ln(t),Q(t,theta) = Q[infinity](theta)+psi[Q](theta)*t
^(2*v(theta))+1/2*diff(Q[infinity](theta),`$`(theta,2))*t^(2*v(theta))
*ln(t),`x`);" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying routine subs to \+
WeylSq" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {MPLTEXT 0 21 57 "Apply consistency relation but keep all ot
her derivatives" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "grmap(_,
subs,diff(Q[infinity](theta),theta$4)=Q[4],`x`);" }}{PARA 6 "" 1 "" 
{TEXT -1 31 "Applying routine subs to WeylSq" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 53 "grmap(_,subs,diff(Q[infinity](theta),theta$3)=0,
`x`);" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying routine subs to WeylSq
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "grmap(_,subs,diff(Q[inf
inity](theta),theta$2)=0,`x`);" }{MPLTEXT 0 21 0 "" }}{PARA 6 "" 1 "" 
{TEXT -1 31 "Applying routine subs to WeylSq" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 51 "grmap(_,subs,diff(Q[infinity](theta),theta)=0,`x
`);" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying routine subs to WeylSq" }
}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 46 "Make sure you keep all deri
vatives of v(theta)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "grma
p(_,subs,diff(v(theta),theta$4)=v[4],`x`);" }}{PARA 6 "" 1 "" {TEXT 
-1 31 "Applying routine subs to WeylSq" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 46 "grmap(_,subs,diff(v(theta),theta$3)=v[3],`x`);" }}
{PARA 6 "" 1 "" {TEXT -1 31 "Applying routine subs to WeylSq" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "grmap(_,subs,diff(v(theta),t
heta$2)=v[2],`x`);" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying routine su
bs to WeylSq" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "grmap(_,sub
s,diff(v(theta),theta$1)=0,`x`);" }}{PARA 6 "" 1 "" {TEXT -1 31 "Apply
ing routine subs to WeylSq" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
29 "grmap(_,subs,v(theta)=1,`x`);" }}{PARA 6 "" 1 "" {TEXT -1 31 "Appl
ying routine subs to WeylSq" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
15 "gralter(_,6,7);" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component simplifi
cation of a GRTensorII object:" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}
{PARA 6 "" 1 "" {TEXT -1 40 "Applying routine expand to object WeylSq
" }}{PARA 6 "" 1 "" {TEXT -1 40 "Applying routine factor to object Wey
lSq" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"$\\#!\"$" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "core:=simplify(limit(factor(
grcomponent(WeylSq,[])/(t*log(t)^2*exp(gamma(t,theta)))),t=0));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%coreG,$*&\"\"$\"\"\")&%\"vG6#\"\"#F
-F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "factor(subs(v[2]=
diff(v(theta),theta$2),Q[3]=diff(Q[infinity](theta),theta$3),core)*t*l
og(t)^2*exp(gamma(t,theta)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*,
\"\"$\"\"\")-%%diffG6$-%\"vG6#%&thetaG-%\"$G6$F.\"\"#F2F&%\"tGF&)-%#ln
G6#F3F2F&-%$expG6#-%&gammaG6$F3F.F&F&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "latex(%);" }}{PARA 6 "" 1 "" {TEXT -1 65 "3\\, \\left(
 \{\\frac \{d^\{2\}\}\{d\{\\theta\}^\{2\}\}\}v \\left( \\theta \\right
) " }}{PARA 6 "" 1 "" {TEXT -1 67 " \\right) ^\{2\}t \\left( \\ln  \\l
eft( t \\right)  \\right) ^\{2\}\{e^\{\\gamma" }}{PARA 6 "" 1 "" 
{TEXT -1 27 " \\left( t,\\theta \\right) \}\}" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 20 "kernelopts(cputime);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"%h=!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 9 }{VIEWOPTS 1 1 0 1 1 
1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }