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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "Coordinate calculation of \+
the Ricci and Weyl tensors from a basis" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "grtw();" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#%=GRTensorII~Version~1.79~(R6)G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%02~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#%4e:/Grtii(6)/MetricsG" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 18 "We load the bases." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "
qload(mix1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Default~spacetimeG%
%mix1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%8For~the~mix1~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&%&ThetaG%$PhiG%$PsiG%\"TG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%4Basis~inner~productG" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#-%$etaG6$%$bupGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%$etaG%$(a
)G\"\"\")%!G%$(b)GF(-%'matrixG6#7&7&F(\"\"!F1F17&F1F(F1F17&F1F1F(F17&F
1F1F1!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%=Basis~(covariant~compo
nents)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#w1G6#%#dnG" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/&%'omega1G6#%\"aG-%'vectorG6#7&,$*&*&-%$expG6#-
F'6#%\"TG\"\"\"-%%sqrtG6#,&F5F5*$)%$PsiG\"\"#F5!\"\"F5F5*$-F76#,&F5F5*
$)%&ThetaGF=F5F>F5F>F>*(F/F5F<F5-F76#FBF5\"\"!FI" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%#w2G6#%#dnG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%'om
ega2G6#%\"aG-%'vectorG6#7&,$*&*&-%$expG6#-%\"bG6#%\"TG\"\"\"%$PsiGF6F6
*$-%%sqrtG6#,&F6F6*$)%&ThetaG\"\"#F6!\"\"F6FAFA,$*(F/F6-F:6#F<F6-F:6#,
&F6F6*$)F7F@F6FAF6FA\"\"!FK" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#w3G6
#%#dnG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%'omega3G6#%\"aG-%'vectorG
6#7&\"\"!*&%&ThetaG\"\"\"-%$expG6#-%\"cG6#%\"TGF/*&F0F/*$-%%sqrtG6#,&F
/F/*$)%$PsiG\"\"#F/!\"\"F/FAF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#w
4G6#%#dnG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%'omega4G6#%\"aG-%'vect
orG6#7&\"\"!F,F,-%$expG6#,(-F'6#%\"TG\"\"\"-%\"bGF2F4-%\"cGF2F4" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%coMixmaster~metric~(e.g.~MTW~Box~30.1
,~Theta~=~cos(theta),~Psi~=~sin(psi))G" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 50 "From the bases we calculate the metric components." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "grcalc(g(dn,dn,pdn),g(up,up),detg);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"#]!\"$" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "gralter(_,6,7);" }}{PARA 6 "" 1 "" 
{TEXT -1 48 "Component simplification of a GRTensorII object:" }}
{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 46 "Applying \+
routine expand to object g(dn,dn,pdn)" }}{PARA 6 "" 1 "" {TEXT -1 42 "
Applying routine expand to object g(up,up)" }}{PARA 6 "" 1 "" {TEXT 
-1 38 "Applying routine expand to object detg" }}{PARA 6 "" 1 "" 
{TEXT -1 46 "Applying routine factor to object g(dn,dn,pdn)" }}{PARA 
6 "" 1 "" {TEXT -1 42 "Applying routine factor to object g(up,up)" }}
{PARA 6 "" 1 "" {TEXT -1 38 "Applying routine factor to object detg" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"$S\"!\"$" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 101 "Though not needed for the following calc
ulations, it is sometimes convenient to see the line element." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "grcalc(ds);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%*CPU~Time~G$\"#5!\"$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "grdisplay(_);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%8Fo
r~the~mix1~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Line~eleme
ntG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/*$)%$~dsG\"\"#\"\"\",.*&*(,(*$
)-%$expG6#-%\"aG6#%\"TGF'F(!\"\"*&F.F()%$PsiGF'F(F(*&)-F06#-%\"bGF4F'F
(F8F(F6F(%#~dGF()%&ThetaG%#2~GF(F(,&F6F(*$)FBF'F(F(F6F(*.F'F(,&*(F.F(-
%%sqrtG6#,&F(F(*$F8F(F6F(F9F(F6*(F;F(F9F(FJF(F(F(F@F()FB%\"~GF(%#d~GF(
)%$PhiGFQF(F(*(,0F7F(*(F.F(F8F(FFF(F6*$F;F(F(F:F6*&F;F(FFF(F6*(F;F(FFF
(F8F(F(*&FFF()-F06#-%\"cGF4F'F(F(F(F@F()FTFCF(F(*&*0F'F(FBF(FfnF(F@F(F
SF(FRF()F9FQF(F(*$-FK6#FMF(F6F(*&*(FfnF(F@F()F9FCF(F(,&F6F(FNF(F6F6*()
-F06#,(F2F(F>F(FinF(F'F(F@F()F5FCF(F6" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 84 "We now proceed to calculate the coordinate components of \+
the Ricci and Weyl tensors." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "grca
lc(Chr(dn,dn,up));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"
$T\"!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "gralter(_,6,7);
" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component simplification of a GRTenso
rII object:" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT 
-1 47 "Applying routine expand to object Chr(dn,dn,up)" }}{PARA 6 "" 
1 "" {TEXT -1 47 "Applying routine factor to object Chr(dn,dn,up)" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"$!>!\"$" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "grcalcalter(R(dn,dn),6);" }}{PARA 
6 "" 1 "" {TEXT -1 50 "Simplification will be applied during calculati
on." }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 42 "Ap
plying routine expand to object R(dn,dn)" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/%*CPU~Time~G$\"#S!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 15 "gralter(_,6,7);" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component simpli
fication of a GRTensorII object:" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}
{PARA 6 "" 1 "" {TEXT -1 42 "Applying routine expand to object R(dn,dn
)" }}{PARA 6 "" 1 "" {TEXT -1 42 "Applying routine factor to object R(
dn,dn)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"$5\"!\"$" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "grmap(_,autoAlias,`x`);" }}
{PARA 6 "" 1 "" {TEXT -1 38 "Applying routine autoAlias to R(dn,dn)" }
}{PARA 7 "" 1 "" {TEXT -1 51 "Warning, alias or macro b[T] defined in \+
terms of b\n" }}{PARA 7 "" 1 "" {TEXT -1 51 "Warning, alias or macro a
[T] defined in terms of a\n" }}{PARA 7 "" 1 "" {TEXT -1 51 "Warning, a
lias or macro c[T] defined in terms of c\n" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 13 "grdisplay(_);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%8
For~the~mix1~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0Covarian
t~RicciG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"RG6$%#dnGF&" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#/&%#R~G6#*&%&ThetaG\"\"\"%&ThetaGF),$*&,<*&)
-%$expG6#%\"aG\"\"#F))-F16#%\"cG\"\"%F)F)*()-F16#%\"bGF4F))%$PsiGF4F)F
5F)F)*&F/F))F<F9F)F)*(F/F)FBF)F?F)!\"\"**F4F)F;F)F?F)&F>6$%\"TGFHF)FD*
(F/F)F?F)F5F)FD*()F0F9F)F;F)F?F)F)*(F4F)F/F)&F3FGF)FD*&)F0\"\"'F)F?F)F
)*&)F<FPF)F?F)FD**F4F)F/F)F;F))F6F4F)FD*$FOF)FD**F4F)F/F)F?F)FMF)F)F)*
,F;F)FTF),&%&ThetaGF)F)FDF),&FYF)F)F)F)F/F)FD#F)F4" }}{PARA 12 "" 1 "
" {XPPMATH 20 "6#/&%#R~G6#*&%&ThetaG\"\"\"%$PhiGF),$*&**%$PsiGF)),&F.F
)F)!\"\"\"\"#F)),&F.F)F)F)F2F),2*&)-%$expG6#%\"aG\"\"%F))-F96#%\"bGF2F
)F)*&F=F))-F96#%\"cGF<F)F)*(F2F))F8F2F)&F;6$%\"TGFJF)F)*$)F8\"\"'F)F)*
&FGF)FBF)F1*$)F>FMF)F1*&FGF))F>F<F)F1*(F2F)F=F)&F@FIF)F1F)F)**FGF)F=F)
),$*&F0F)F4F)F1#\"\"$F2F))FCF2F)F1#F1F2" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#/&%#R~G6#*&%$PhiG\"\"\"%$PhiGF),$*&,fn*()-%$expG6#%\"bG\"\"#F))%
&ThetaGF4F))-F16#%\"cG\"\"%F)!\"\"**F4F)F/F)F5F)&F36$%\"TGF@F)F)*()F0F
;F)F5F))F8F4F)F)**F4F))-F16#%\"aGF4F)F/F)FCF)F<*(FEF)FBF))%$PsiGF4F)F)
*(F4F)F/F)F>F)F<**F5F)F7F)FJF)FEF)F<**F5F)F7F)FJF)F/F)F)**F5F)F/F))FFF
;F)FJF)F)**F5F)FBF)FEF)FJF)F<*&FPF)F/F)F)*(F5F)FPF)FCF)F)*(FPF)F/F)FJF
)F<*&)FF\"\"'F)FJF)F<*&)F0FWF)FJF)F)*(FEF)FJF)F7F)F)*(F/F)FJF)F7F)F<*$
FYF)F<*&F/F)F7F)F)*(F5F)F/F)FPF)F<*(FVF)F5F)FJF)F)*(FYF)F5F)FJF)F<*&FY
F)F5F)F)*&F5F))F8FWF)F<**F4F)F/F)FJF)F>F)F)**F4F)FEF)FJF)&FHF?F)F<**F4
F)F5F)FCF)&F:F?F)F<*,F4F)FEF)FJF)F5F)F`oF)F)*,F4F)F/F)F5F)FJF)F>F)F<F)
*(F/F)FCF)FEF)F<#F<F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#R~G6#*&%$
PhiG\"\"\"%$PsiGF),$*&**%&ThetaGF),&%$PsiGF)F)!\"\"F),&F0F)F)F)F),,*$)
-%$expG6#%\"bG\"\"%F)F)*$)-F76#%\"cGF:F)F1*$)-F76#%\"aGF:F)F)*&\"\"#F)
&F?6$%\"TGFIF)F1*(FFF))FBFFF))F6FFF)F1F)F)*(FLF)),$*&F/F)F2F)F1#\"\"$F
FF)FKF)F1#F)FF" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#R~G6#*&%$PsiG\"
\"\"%$PsiGF),$*&,,*$)-%$expG6#%\"bG\"\"%F)F)*$)-F16#%\"cGF4F)!\"\"*$)-
F16#%\"aGF4F)F)*&\"\"#F)&F96$%\"TGFDF)F:*(FAF))F=FAF))F0FAF)F:F)**FFF)
FGF),&%$PsiGF)F)F:F),&FJF)F)F)F)F:#F)FA" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/&%#R~G6#*&%\"TG\"\"\"%\"TGF),.&%\"aG6$%\"TGF/!\"\"&%\"bGF.F0&%
\"cGF.F0*(\"\"#F)&F-6#F/F)&F2F8F)F)*(F6F)&F4F8F)F7F)F)*(F6F)F;F)F9F)F)
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "grcalcalter(Ricciscalar
,6);" }}{PARA 6 "" 1 "" {TEXT -1 50 "Simplification will be applied du
ring calculation." }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" 
{TEXT -1 46 "Applying routine expand to object tRicciscalar" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"\"!F&" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 15 "gralter(_,6,7);" }}{PARA 6 "" 1 "" {TEXT -1 
48 "Component simplification of a GRTensorII object:" }}{PARA 6 "" 1 "
" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 45 "Applying routine expand
 to object Ricciscalar" }}{PARA 6 "" 1 "" {TEXT -1 45 "Applying routin
e factor to object Ricciscalar" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*C
PU~Time~G$\"#I!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "grcal
calter(C(dn,dn,dn,dn),6);" }}{PARA 6 "" 1 "" {TEXT -1 50 "Simplificati
on will be applied during calculation." }}{PARA 6 "" 1 "" {TEXT -1 0 "
" }}{PARA 6 "" 1 "" {TEXT -1 48 "Applying routine expand to object R(d
n,dn,dn,dn)" }}{PARA 6 "" 1 "" {TEXT -1 46 "Applying routine expand to
 object tRicciscalar" }}{PARA 6 "" 1 "" {TEXT -1 48 "Applying routine \+
expand to object C(dn,dn,dn,dn)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*
CPU~Time~G$\"$?#!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "gra
lter(_,6,7);" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component simplification \+
of a GRTensorII object:" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 
1 "" {TEXT -1 48 "Applying routine expand to object C(dn,dn,dn,dn)" }}
{PARA 6 "" 1 "" {TEXT -1 48 "Applying routine factor to object C(dn,dn
,dn,dn)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"%#p\"!\"$" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "grmap(_,autoAlias,`x`);" 
}}{PARA 6 "" 1 "" {TEXT -1 44 "Applying routine autoAlias to C(dn,dn,d
n,dn)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "grdisplay(_);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%8For~the~mix1~spacetime:G" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%/Covariant~WeylG" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#/&%#C~G6#**%&ThetaG\"\"\"%$PhiGF)%&ThetaGF)%$PhiGF),$*&
,dq**)-%$expG6#%\"aG\"\"#F)&F56#%\"TGF))-F36#%\"bGF6F)&F=F8F)!\"%*0\"
\"%F)F:F))%$PsiGF6F)F>F))%&ThetaGF6F))-F36#%\"cGF6F)&FIF8F)!\"\"*0FAF)
F1F)FBF)F7F)FDF)FFF)FJF)F)*&)F2\"\"'F)F:F)FK*()F2FAF)F:F)FFF)FK*(F6F)F
QF))F;FAF)F)*(F1F)FSF)FFF)FK**F6F)F1F))FGFAF)F:F)F)*&F1F))F;FOF)FK**F1
F)F:F)FDF)&F=6$F9F9F)FK**F1F)FDF)FFF)&FIFenF)F)*,F6F)F1F)F:F)FDF)FgnF)
F)**F:F)FDF)F1F)&F5FenF)FK**FDF)FFF)F1F)FjnF)F)*,F6F)FDF)FZF)F1F)FFF)F
K*(F:F)F1F)FjnF)F)*(F1F)F:F)FZF)F)**F6F)F1F)F:F)FgnF)FK*,FDF)FFF)F:F)F
BF)FZF)F)*,FDF)FFF)F1F)FBF)FjnF)FK*.F6F)FDF)FZF)F1F)FFF)FBF)F)*.F6F)FB
F)FDF)FjnF)FFF)F:F)FK*,F1F)FBF)FDF)FFF)FgnF)FK*,F:F)FBF)FDF)FFF)FgnF)F
)*,\"\"$F)F1F)FSF)FDF)FFF)F)*,FgoF)F1F)F:F)FDF)FVF)FK*,F6F)FQF)FBF)FDF
)FVF)FK**FNF)FBF)FDF)FFF)F)**FXF)FBF)FDF)FFF)FK*,F6F)FSF)FBF)FDF)FVF)F
)**F6F)FQF)FDF)FVF)F)*(FNF)F:F)FDF)F)*(FNF)FDF)FFF)FK**F6F)FQF)FSF)FDF
)FK*(FXF)F1F)FDF)F)*.FgoF)FQF)FBF)FDF)FFF)F:F)F)*.FgoF)F1F)FSF)FBF)FDF
)FFF)FK*(F1F)FDF))FGFOF)FK**F1F)FBF)FDF)FepF)F)**F:F)FBF)FDF)FepF)FK*.
FAF)F1F)F7F)F:F)FDF)F>F)F)*.FAF)F1F)F7F)FDF)FFF)FJF)FK*.F6F)FJF)F>F)F:
F)F1F)FDF)FK*.F6F)FJF)F7F)F:F)F1F)FDF)FK*,F6F)FJF)F>F)F:F)F1F)F)*,F6F)
FJF)F7F)F:F)F1F)F)*0F6F)F7F)F>F)FBF)FDF)FFF)F:F)F)*0F6F)FJF)F7F)FBF)FD
F)FFF)F:F)F)*0F6F)FJF)F>F)FBF)FDF)F1F)FFF)FK*.F6F)FJF)F>F)FDF)F1F)FFF)
F)*.F6F)F7F)F>F)FDF)F1F)FFF)F)*0F6F)F7F)F>F)FDF)F1F)FFF)FBF)FKF)*,F:F)
F1F),&FEF)F)FKF),&FEF)F)F)F)FFF)FK#F)FO" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#/&%#C~G6#**%&ThetaG\"\"\"%$PhiGF)%&ThetaGF)%$PsiGF),$*&**,&%$Psi
GF)F)!\"\"F),&F1F)F)F)F),\\o*()-%$expG6#%\"aG\"\"#F))-F86#%\"bGF;F))-F
86#%\"cGF;F)F2**\"\"$F)F6F))F=\"\"%F))F1F;F)F2*&)F7FGF)F<F)F2*(F;F)F6F
)FFF)F)**FEF)FJF)F<F)FHF)F)*(F;F)FJF)F@F)F)*&)F7\"\"'F)FHF)F)*&)F=FPF)
FHF)F2**F;F)FJF)FHF)F@F)F2**F;F)FFF)FHF)F@F)F)*&F6F))FAFGF)F2*(F6F)FHF
)FVF)F)*(F<F)FHF)FVF)F2*$FOF)F2**F;F)F6F)&F:6#%\"TGF)&F?FfnF)F)**FGF)F
6F)FenF)&FCFfnF)F2*,F;F)F6F)FHF)FenF)FhnF)F2*,FGF)F6F)FHF)FenF)FjnF)F)
*,F;F)F<F)FHF)FhnF)FenF)F)*,FGF)F<F)FHF)FhnF)FjnF)F2*(F6F)FHF)&F:6$Fgn
FgnF)F2*(F<F)FHF)&F?FaoF)F)*(F6F)FHF)&FCFaoF)F2*(F<F)FHF)FeoF)F)**F;F)
F6F)FHF)FcoF)F)**F;F)F<F)FHF)F`oF)F2*&F6F)F`oF)F)*(F;F)F6F)FcoF)F2*&F6
F)FeoF)F)**F;F)F6F)FjnF)FhnF)F)*,F;F)F6F)FHF)FjnF)FhnF)F2*,F;F)F<F)FHF
)FjnF)FenF)F)F)%&ThetaGF)F)*,F<F)F6F),&F_pF)F)F)F),&F_pF)F)F2F)),$*&F0
F)F3F)F2#FEF;F)F2#F2FP" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%#C~G6#**%
&ThetaG\"\"\"%$PhiGF)%&ThetaGF)%\"TGF),$*&*,%$PsiGF)%&ThetaGF),&F0F)F)
!\"\"F),&F0F)F)F)F),2*&)-%$expG6#%\"bG\"\"#F)&F;6#%\"TGF)!\"$*(\"\"$F)
)-F96#%\"aGF<F)&FFF>F)F)*&)-F96#%\"cGF<F)FGF)F3*&F=F)FIF)F)*(F<F)&FLF>
F)FCF)F3*&F=F)FCF)F3*&F7F)FGF)F)*(F<F)FOF)F7F)F)F)F)*(-%%sqrtG6#,$*&F2
F)F4F)F3F),&F1F)F)F3F),&F1F)F)F)F)F3#F)F<" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#/&%#C~G6#**%&ThetaG\"\"\"%$PhiGF)%$PhiGF)%$PsiGF),$*&*(
,J*&)-%$expG6#%\"aG\"\"#F))-F46#%\"bG\"\"%F)!\"$*&F2F))-F46#%\"cGF<F)F
)*$)F9\"\"'F)!\"\"*(F7F))F3F<F))F@F7F)FF*&F2F)&F66$%\"TGFMF)FF*&)F9F7F
)&F;FLF)F)*(F7F)F2F)FPF)F)*&FOF)F?F)FF*&F2F)&FBFLF)FF*$)F3FEF)F)*&FOF)
FTF)F)*(F7F)F8F)FIF)F)**F7F)F2F)&F66#FMF)&F;FenF)FF**F7F)FOF)FfnF)FZF)
F)**F<F)FOF)FfnF)&FBFenF)FF*(\"\"$F)FHF)FOF)F)*(F7F)FOF)FKF)FF**F7F)FO
F)FinF)FZF)F)**F7F)F2F)FinF)FfnF)FF**F<F)F2F)FZF)FinF)F)F)%&ThetaGF)%$
PsiGF)F)*&FOF)F2F)FF#FFFE" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%#C~G6#
**%&ThetaG\"\"\"%$PhiGF)%$PhiGF)%\"TGF),$*&%&ThetaGF),B*()-%$expG6#%\"
bG\"\"#F))%$PsiGF7F)&F66#%\"TGF)!\"$**\"\"$F))-F46#%\"aGF7F)F8F)&FCF;F
)F)*(F8F))-F46#%\"cGF7F)FDF)!\"\"*(F8F)F:F)FFF)F)**F7F)F8F)&FIF;F)F@F)
FJ*(F8F)F:F)F@F)FJ*(FDF)F2F)F8F)F)**F7F)F8F)FMF)F2F)F)*(F?F)F2F)F:F)F)
*(F7F)F2F)FDF)FJ*&FMF)F2F)FJ*&FMF)F@F)F)*(F?F)FFF)FMF)FJ*(F7F)FFF)FDF)
F)*&F:F)FFF)F)*&F:F)F@F)FJF)#FJF7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/
&%#C~G6#**%&ThetaG\"\"\"%$PhiGF)%$PsiGF)%\"TGF),$*&,0*&)-%$expG6#%\"bG
\"\"#F)&%\"aG6#%\"TGF)!\"\"*&&F5F9F))-F36#F8F6F)F;*&F>F)F7F)F)*&F1F)F=
F)F)*&)-F36#%\"cGF6F)F7F)F)*(F6F)FDF)&FGF9F)F;*&F=F)FDF)F)F)*$-%%sqrtG
6#,$*&,&%$PsiGF)F)F;F),&FRF)F)F)F)F;F)F;#F;F6" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#/&%#C~G6#**%&ThetaG\"\"\"%$PsiGF)%&ThetaGF)%$PsiGF),$*&
,\\o*()-%$expG6#%\"aG\"\"#F))-F36#%\"bGF6F))-F36#%\"cGF6F)!\"\"**\"\"$
F)F1F))F8\"\"%F))%$PsiGF6F)F?*&)F2FCF)F7F)F?*(F6F)F1F)FBF)F)**FAF)FGF)
F7F)FDF)F)*(F6F)FGF)F;F)F)*&)F2\"\"'F)FDF)F)*&)F8FMF)FDF)F?**F6F)FGF)F
DF)F;F)F?**F6F)FBF)FDF)F;F)F)*&F1F))F<FCF)F?*(F1F)FDF)FSF)F)*(F7F)FDF)
FSF)F?*$FLF)F?**F6F)F1F)&F56#%\"TGF)&F:FYF)F)**FCF)F1F)FXF)&F>FYF)F?*,
F6F)F1F)FDF)FXF)FenF)F?*,FCF)F1F)FDF)FXF)FgnF)F)*,F6F)F7F)FDF)FenF)FXF
)F)*,FCF)F7F)FDF)FenF)FgnF)F?*(F1F)FDF)&F56$FZFZF)F?*(F7F)FDF)&F:F^oF)
F)*(F1F)FDF)&F>F^oF)F?*(F7F)FDF)FboF)F)**F6F)F1F)FDF)F`oF)F)**F6F)F7F)
FDF)F]oF)F?*&F1F)F]oF)F)*(F6F)F1F)F`oF)F?*&F1F)FboF)F)**F6F)F1F)FgnF)F
enF)F)*,F6F)F1F)FDF)FgnF)FenF)F?*,F6F)F7F)FDF)FgnF)FXF)F)F)*.F7F)F1F),
&%&ThetaGF)F)F?F),&F^pF)F)F)F),&FEF)F)F?F),&FEF)F)F)F)F?#F?FM" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#C~G6#**%&ThetaG\"\"\"%$PsiGF)%&The
taGF)%\"TGF),$*&*&%$PsiGF),2*&)-%$expG6#%\"bG\"\"#F)&F76#%\"TGF)!\"$*(
\"\"$F))-F56#%\"aGF8F)&FBF:F)F)*&)-F56#%\"cGF8F)FCF)!\"\"*&F9F)FEF)F)*
(F8F)&FHF:F)F?F)FI*&F9F)F?F)FI*&F3F)FCF)F)*(F8F)FLF)F3F)F)F)F)*&,&%&Th
etaGF)F)FIF),&FRF)F)F)F)FI#FIF8" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%
#C~G6#**%&ThetaG\"\"\"%$PsiGF)%$PhiGF)%$PsiGF),$*&**,&%$PsiGF)F)!\"\"F
),&F1F)F)F)F),J*&)-%$expG6#%\"aG\"\"#F))-F86#%\"bG\"\"%F)!\"$*&F6F))-F
86#%\"cGF@F)F)*$)F=\"\"'F)F2*(F;F))F7F@F))FDF;F)F2*&F6F)&F:6$%\"TGFPF)
F2*&)F=F;F)&F?FOF)F)*(F;F)F6F)FSF)F)*&FRF)FCF)F2*&F6F)&FFFOF)F2*$)F7FI
F)F)*&FRF)FWF)F)*(F;F)F<F)FLF)F)**F;F)F6F)&F:6#FPF)&F?FhnF)F2**F;F)FRF
)FinF)FgnF)F)**F@F)FRF)FinF)&FFFhnF)F2*(\"\"$F)FKF)FRF)F)*(F;F)FRF)FNF
)F2**F;F)FRF)F\\oF)FgnF)F)**F;F)F6F)F\\oF)FinF)F2**F@F)F6F)FgnF)F\\oF)
F)F)F1F)F)*(FRF)),$*&F0F)F3F)F2#F^oF;F)F6F)F2#F)FI" }}{PARA 12 "" 1 "
" {XPPMATH 20 "6#/&%#C~G6#**%&ThetaG\"\"\"%$PsiGF)%$PhiGF)%\"TGF),$*&,
@*()%$PsiG\"\"#F)&%\"bG6#%\"TGF))-%$expG6#%\"cGF3F)F)**F3F)F1F)&F<F6F)
)-F:6#F5F3F)F)*&)-F:6#%\"aGF3F)&FFF6F)!\"\"*(FGF)F?F)F1F)F)**\"\"$F)F?
F)F1F)F4F)FH*&F>F)F?F)FH**F3F)F1F)F>F)FCF)FH*(F3F)F?F)F4F)F)*(F1F)F4F)
FCF)FH**FKF)FCF)F1F)FGF)F)*&F>F)FCF)F)*&F8F)FGF)F)*&F?F)FGF)FH*(F1F)F8
F)FGF)FH*&F8F)F>F)FHF)*$-%%sqrtG6#,$*&,&F2F)F)FHF),&F2F)F)F)F)FHF)FH#F
HF3" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%#C~G6#**%&ThetaG\"\"\"%\"TGF
)%&ThetaGF)%\"TGF),$*&,hn*()-%$expG6#%\"aG\"\"#F))-F36#%\"bGF6F))-F36#
%\"cGF6F)!\"#*&)F2\"\"%F)F7F)F)*&F1F))F8FBF)F)*&FAF)F;F)F)*(F6F))F2\"
\"'F))%$PsiGF6F)F)*(F6F))F8FHF)FIF)!\"\"*(FAF)FIF)F;F)FM*(FDF)FIF)F;F)
F)*&F1F))F<FBF)F)*(F1F)FIF)FQF)FM*(F7F)FIF)FQF)F)*&F6F)FGF)FM**F6F)F1F
)&F56#%\"TGF)&F:FWF)FM**F6F)F1F)FVF)&F>FWF)FM*,F6F)F1F)FIF)FVF)FYF)F)*
,F6F)F1F)FIF)FVF)FenF)F)*,F6F)F7F)FIF)FYF)FVF)FM*,F6F)F7F)FIF)FYF)FenF
)FM**F6F)F1F)FIF)&F56$FXFXF)FM**F6F)F7F)FIF)&F:F\\oF)F)*(F1F)FIF)&F>F
\\oF)F)*(F7F)FIF)F`oF)FM*(F1F)FIF)F^oF)F)*(F7F)FIF)F[oF)FM*(F6F)F1F)F[
oF)F)*&F1F)F^oF)FM*&F1F)F`oF)FM**FBF)F1F)FenF)FYF)F)*,FBF)F1F)FIF)FenF
)FYF)FM*,FBF)F7F)FIF)FenF)FVF)F)F)*&,&%&ThetaGF)F)FMF),&F\\pF)F)F)F)FM
#F)FH" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%#C~G6#**%&ThetaG\"\"\"%\"T
GF)%$PhiGF)%$PsiGF),$*&,@*&)-%$expG6#%\"cG\"\"#F)&F56#%\"TGF)F)**\"\"$
F))-F36#%\"aGF6F))%$PsiGF6F)&F?F8F)F)**F;F))-F36#%\"bGF6F)F@F)&FGF8F)!
\"\"*&F7F)FDF)FI*&FHF)F1F)FI*&FDF)FHF)F)*(F@F)FHF)F1F)F)*&FHF)F<F)F)*(
F@F)F1F)FBF)FI*(F6F)F<F)FBF)FI*&F7F)F<F)F)**F6F)F@F)F7F)FDF)F)**F6F)F@
F)F7F)F<F)FI*(F@F)FHF)F<F)FI*(FBF)FDF)F@F)F)F)*$-%%sqrtG6#,$*&,&FAF)F)
FIF),&FAF)F)F)F)FIF)FI#FIF6" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%#C~G
6#**%&ThetaG\"\"\"%\"TGF)%$PhiGF)%\"TGF),$*&**%$PsiGF)),&F0F)F)!\"\"\"
\"#F)),&F0F)F)F)F4F),F*$)-%$expG6#%\"aG\"\"'F)F4*&)F:\"\"%F))-F;6#%\"c
GF4F)F3*&)F:F4F))FCFAF)F3**FAF)FGF)&FE6#%\"TGF)&%\"bGFKF)F3*&)-F;6#FNF
AF)FBF)F)**F4F)FGF)&F=FKF)FJF)F)*&FGF)&FE6$FLFLF)F)**FAF))FQF4F)FJF)FT
F)F)*&FGF)&FNFWF)F)**F4F)FYF)FMF)FJF)F3**F4F)FYF)FMF)FTF)F3*&F4F))FQF>
F)F3*&FYF)FVF)F3*&FYF)&F=FWF)F3*(F4F)FYF)FenF)F)*(F4F)FGF)F\\oF)F3*&FY
F)FHF)F)**F4F)FGF)FTF)FMF)F)F)F)*$),$*&F2F)F6F)F3#\"\"$F4F)F3#F3F>" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%#C~G6#**%$PhiG\"\"\"%$PsiGF)%$PhiGF
)%$PsiGF),$*&*(,&%&ThetaGF)F)!\"\"F),&F1F)F)F)F),\\o*()-%$expG6#%\"aG
\"\"#F))-F86#%\"bGF;F))-F86#%\"cGF;F)F)**\"\"$F)F6F))F=\"\"%F))%$PsiGF
;F)F2*(F;F))F7FGF)F<F)F2*&F6F)FFF)F)**FEF)FKF)F<F)FHF)F)*(F;F)FFF)F@F)
F2*&)F7\"\"'F)FHF)F)*&)F=FQF)FHF)F2**F;F)FKF)FHF)F@F)F2**F;F)FFF)FHF)F
@F)F)**F;F)F<F)&F?6#%\"TGF)&F:FXF)F2**FGF)F<F)FWF)&FCFXF)F)*(F6F)FHF))
FAFGF)F)*(F<F)FHF)FhnF)F2*$FSF)F)*,F;F)F6F)FHF)FZF)FWF)F2*,FGF)F6F)FHF
)FZF)FfnF)F)*,F;F)F<F)FHF)FWF)FZF)F)*,FGF)F<F)FHF)FWF)FfnF)F2*&F<F)Fhn
F)F)*(F6F)FHF)&F:6$FYFYF)F2*(F<F)FHF)&F?FboF)F)*(F6F)FHF)&FCFboF)F2*(F
<F)FHF)FfoF)F)**F;F)F6F)FHF)FdoF)F)**F;F)F<F)FHF)FaoF)F2*&F<F)FdoF)F2*
&F<F)FfoF)F2*(F;F)F<F)FaoF)F)*,F;F)F6F)FHF)FfnF)FWF)F2*,F;F)F<F)FHF)Ff
nF)FZF)F)**F;F)F<F)FfnF)FZF)F2F)F)**F6F)F<F),&FIF)F)F2F),&FIF)F)F)F)F2
#F)FQ" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#C~G6#**%$PhiG\"\"\"%$PsiG
F)%$PhiGF)%\"TGF),$**%$PsiGF),&%&ThetaGF)F)!\"\"F),&F1F)F)F)F),2*&)-%$
expG6#%\"bG\"\"#F)&F:6#%\"TGF)!\"$*(\"\"$F))-F86#%\"aGF;F)&FEF=F)F)*&)
-F86#%\"cGF;F)FFF)F2*&F<F)FHF)F)*(F;F)&FKF=F)FBF)F2*&F<F)FBF)F2*&F6F)F
FF)F)*(F;F)FNF)F6F)F)F)#F)F;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%#C~
G6#**%$PhiG\"\"\"%\"TGF)%$PhiGF)%\"TGF),`s*()%&ThetaG\"\"#F))-%$expG6#
%\"cG\"\"%F))-F46#%\"aGF1F)#!\"\"\"\"'*&#F)F>F)*()-F46#%\"bGF1F)F/F)&F
;6$%\"TGFHF)F)F=**#F)\"\"$F)F8F)FBF))F3F1F)F)*&#F)FKF)*,&FE6#FHF)F8F))
%$PsiGF1F)F/F)&F;FQF)F)F=*,#F)F>F)F/F)F2F)FRF)F8F)F)*&#F)F>F)**F/F)FLF
))FCF7F)FRF)F)F=*&#F)F>F)**F/F)F2F)FRF)FBF)F)F=*,FVF)F/F)FLF))F9F7F)FR
F)F)*&#F)F>F)*&FinF)FBF)F)F=*&#F)F>F)*&F8F)FZF)F)F=**FVF)F/F)FZF)F8F)F
)*&#F)F>F)*(F/F)FinF)FLF)F)F=*&#F)F>F)*&FZF)FLF)F)F=*(FJF))F9F>F)FRF)F
)*&#F)FKF)*&)FCF>F)FRF)F)F=*&#F)F>F)*(FinF)FRF)FLF)F)F=**FVF)FZF)FRF)F
LF)F)**FJF)FBF)FPF)FTF)F)**FJF)FBF)FPF)&F6FQF)F)*&#F)F>F)*(F8F)FRF)F2F
)F)F=**FVF)FBF)FRF)F2F)F)*&FJF)F\\pF)F)*,FJF)F8F)FRF)FTF)FPF)F)*,FJF)F
8F)FRF)FTF)FcpF)F)*&#F)FKF)**FBF)FRF)FPF)FTF)F)F=*&#F)FKF)**FBF)FRF)FP
F)FcpF)F)F=*.FJF)FTF)FBF)F/F)FRF)FPF)F)*&#F)FKF)**FTF)FBF)F/F)FPF)F)F=
*,FJF)FTF)F/F)FLF)FcpF)F)*,FJF)FPF)F/F)FLF)FcpF)F)*&#F)F>F)*&FBF)F2F)F
)F=*&#F)FKF)*(F8F)FRF)FFF)F)F=**FJF)FBF)FRF)&FEFGF)F)**FVF)F8F)FRF)&F6
FGF)F)*&#F)F>F)*(FBF)FRF)F`rF)F)F=**FVF)F8F)FRF)F^rF)F)*&#F)F>F)*(FBF)
FRF)FFF)F)F=*&#F)FKF)*&FBF)F^rF)F)F=*(FVF)FBF)F`rF)F)*(FVF)FBF)FFF)F)*
&#F)FKF)**FBF)F/F)FPF)FcpF)F)F=*&#F)FKF)*,F8F)FRF)F/F)FTF)FcpF)F)F=*.F
JF)FBF)F/F)FRF)FPF)FcpF)F)**FVF)F/F)FBF)FinF)F)*,FVF)FBF)F/F)FRF)F`rF)
F)*&#F)F>F)**F8F)FRF)F/F)F^rF)F)F=*,FVF)FBF)F/F)FRF)FFF)F)**FJF)FBF)F/
F)F^rF)F)*&#F)FKF)*(F/F)FLF)F`rF)F)F=*&#F)F>F)*(FBF)F/F)F`rF)F)F=**FVF
)F/F)FLF)FFF)F)**FVF)F/F)FLF)F^rF)F)*,FJF)F8F)FRF)F/F)FFF)F)*&#F)FKF)*
*FBF)F/F)FRF)F^rF)F)F=*&#F)F>F)**F8F)FRF)F/F)F`rF)F)F=*&#F)FKF)*(FhoF)
F/F)FRF)F)F=**FJF)F\\pF)F/F)FRF)F)*&#F)FKF)*&F\\pF)F/F)F)F=*(FJF)F/F))
F3F>F)F)*&#F1FKF)**F8F)FRF)FcpF)FPF)F)F=*,#F1FKF)FBF)FRF)FcpF)FTF)F)*,
FguF)FBF)F/F)FcpF)FTF)F)*&#F1FKF)**F/F)FLF)FTF)FPF)F)F=*&#F1FKF)*(FBF)
FcpF)FTF)F)F=*.FguF)F8F)FRF)F/F)FcpF)FPF)F)*&#F1FKF)*,FBF)F/F)FRF)FcpF
)FTF)F)F=" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%#C~G6#**%$PhiG\"\"\"%
\"TGF)%$PsiGF)%\"TGF),$*&*,)-%$expG6#%\"cG\"\"#F),&%$PsiGF)F)!\"\"F),&
F7F)F)F)F),:*$)-F26#%\"bG\"\"%F)F)*&)F=F5F)F0F)F)*&)-F26#%\"aGF5F)F0F)
F)*&F5F))F1F@F)F8&F?6$%\"TGFLF8*$)FEF@F)F)*(F@F)&FG6#FLF)&F?FQF)F)*&F5
F)&F4FKF)F)&FGFKF8*(F5F)&F4FQF)FRF)F8*(F5F)FWF)FPF)F8*(F5F)FDF)FBF)F8F
)%&ThetaGF)F)*$),$*&F6F)F9F)F8#\"\"$F5F)F8#F)\"\"'" }}{PARA 12 "" 1 "
" {XPPMATH 20 "6#/&%#C~G6#**%$PsiG\"\"\"%\"TGF)%$PsiGF)%\"TGF),$*&*&)-
%$expG6#%\"cG\"\"#F),:*$)-F26#%\"bG\"\"%F)F)*&)F9F5F)F0F)F)*&)-F26#%\"
aGF5F)F0F)F)*&F5F))F1F<F)!\"\"&F;6$%\"TGFIFF*$)FAF<F)F)*(F<F)&FC6#FIF)
&F;FNF)F)*&F5F)&F4FHF)F)&FCFHFF*(F5F)&F4FNF)FOF)FF*(F5F)FTF)FMF)FF*(F5
F)F@F)F>F)FFF)F)*&,&%$PsiGF)F)FFF),&FYF)F)F)F)FF#F)\"\"'" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "18 20" 0 }{VIEWOPTS 1 
1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }