Topics in numerical relativity : the periodic standing-wave approximation, the stability of constraints in free evolution, and the spin of dynamical black holes

(1)

2)9:; !!8<= *+> 1 7?@! &%5AB5 C ! DEA-FGH7"52 !!<5+.

*+> 1I5AB.8%J+K$?+ML*+G

NPORQ!SUTVSCWFX

Y$ZR[\]^`_Bab\dc

egfh1i!jlkgTVi!monqp1msrPmsmutQ&fPkPvowFkxORQ:yzQ!{op1Tsj|QtQ&fRk}S

w~vojkVOPQQ!qj|QQvow

vo#kvojvowRhFOPTumVvqSvRORX

imsTuw~voj|fRTViegf1SUkgTskxp1kQ=vowFNQOf1vomVvqoX

h1iSiPQ&f1i i!msTsw~vojfPTVi

o

(2)

yv!WRQjlkQ&f

(3)

NROPTVSMkxORQ!SUTVSM#vfP Qj|fPSCfRpPtQjlTs#im!j|QmVi!kTFTskxX#kxORQi!kkgQt1kokvSUkxpCRX1Tf1SUkgQTfSkVOPQ!voj X5vowqjli FTskikgTVvf/pCSUTf1

fRpPtQjlTs#i!mFPTVSU j QkgTikgTVvfRNRORQdvqimvow1kxORQ=rPQmsPokxORQSUkxpCPX8vowMqjli FTskikgTVvf1i!m-RXdfPi!tTs#S`Tf2#iSUQ!S OPQj|Q

SlXdttQkj|XJj|Q!Rp1#kgTVvf8vojC1Qj kgp1j~W1i!kgTVvf8kVOPQ!voj|X$i!j|Q7f1vokdCvqSSUTWRmsQrFf1i!msmuXJSUQQtSk}vWRQG#vtTfP kvw}jp1TskgTVvfR

i!k7muQ!iSUkzw}voj=kVOPQi!j OPQkxXdCimP1j v!WRmsQtEvow7kVOPQTfPSl1TsjUi!mzif1 #vqi!msQ!SU#Q&fR#QvowWRTfPij|X2WPmsiO1vomsQSlXFSUkgQt+S

NROPTVSzkVOPQ!SUTVSz1j|Q!SUQ&fPkSkxORj QQGQ1TVSvqPQ!SzkVO1i!kFQ!i!O WPQ!i!jMSvtQ:j QmVi!kgTVvf1S ORTu(k}v5kVOPTVSSUkvoj X

OPi!1kgQjUSif11j Q!SUQ&fRk1j|Q RTVvpCSUmXPpWRmsTVS ORQ! (vojTfB&vomsmsi&WPvojUi!kgTVvf TskVO yzTuOPij hPj Ts Q$if1

vokVOPQjUSvfkVOPQGSv&i!msmsQ! "!# %$'&(*),+-# %+/.10324),56

h8790-iP1j v:TutikgTVvfJw~vojFWRTf1i!j X5Tf1Sl1Tsjlim;NROPQi!P1j v:

TutikgTVvf(TVSzkv)WPi!mVifR#Qvp1kdvoTf1/jUiPTVikgTVvf 8TskxO(TfP#vtTfP$jUiPTVikgTVvfFSUki#WPTumsTTf1 kxORQ:vojWRTskPif1t+i/dTf1

kVOPQ1j v!WPmuQtSUkikgTVvfPij|XTf@i5j vok}i!kTf1w}jli!tQ< O1i1kQjlS7JifP=Ji!P1muXkVOPQtQkxOPvq+k}v$kVOPQ1j v!WRmsQt vow

#v~voj WRTskgTfP/O1i!jlqQ!S#vpP1msQ!k}v if1vfPmuTfPQ!i!jSU&i!mVi!jrPQmV8Tf8kxORj QQPTutQ&f1SUTVvfPS

OPi!1kgQjUS?>A@ifPCB#vfP Qj|fJkVOPQ`SUk}i#WRTsmsTskxX vowq#vf1SUkgjUi!TfPkdrPQmV1S1TfJ&vfRQ&fRkTsvf1i!m!fFpPtQjlTu&i!mj|QmVi!kTRTskxX

SUTut2p1mVikgTVvfPS OPi!1kgQj'> i!mVSv1j|Q RTVvpCSUmX/PpWRmsTsS OPQ! (vojqTf#vomsmVi#WPvojUi!kTVvf TskVOkxORQ imukQO$fFpPtQjlT

#imoj|QmVi!kTRTskxX$dj vpPPimsvf1 TskVO TsO1i!QmD8vomVSUkPif1ECi 8jQ&fR#QGFGTsCPQjl-1j|Q!SUQ&fPkSzi:tQkVO1vqw~vojTuttQ

PTVi!kQmuX=#vojlj|Q#kgTf1 RTsvomVikgTVvfPSvowq#vf1SUkgjUi!TfPkSMi!w}kgQjRkVOPQX:O1i RQ`ij TVSUQ&fH O1i1kQjlS8@ifPCB71j Q!SUQ&fRktQkVO1vq1S

kvJI!CitLKGi (i!X$&vfPSUkjliTfPk FTVvomVi!kTVvfPSRXqf1itTs#i!msmuXTfk(vJSl1Q#TsrP&vfRkQ:k}S OPi!1kgQjM@ 1j Q FTVvpCSUmuX

PpWRmsTVS ORQ! (voj)TfJ&vomsmsi&WPvojUi!kgTVvf 8TskxOJkVOPQ imukQO5fRpPtQjlTs#imj|QmVikgTFTskxX5qjlvpP$if1NECi jQ&fR#QOFGTV1PQjU

1j|Q!SUQ&fPkSi5rPjUSUk;~voj 1Qj7msTfRQ!ijmuX2PQ!dQ&fRQjUi!kQ)SlXdttQkgjlTu:ORXd1Qj~W1vomuTs=j|Q1j|Q!SUQ&fPkikgTVvfvowL1TfPSUkQTf'SGQ!{opCi6

kgTVvf1SGTf qQ&fRQjUi!msTQ!(O1i!jtvfPTs)-ipCqQ< j|Q1j Q!SUQ&fRk}i!kTsvfTVS1j|Q!SUQ&fPkgQ!kVO1i!kSUki&WRTsmuTQ!SGi!msm&vfPSUkjliTfRk}Sl

TfR#mupCPTfPBkVO1vqSUQkVO1i!kiP1Q!ij OPQ&f kxORQ2SlXSUkgQt TVSjlTukkgQ&fDTfDrPjlSUkP}vojlPQj=w~voj t

O1i1kQjQB(1j|Q!SUQ&fPkS

i qQ&fPQjUi!msTi!kTVvfvowCkxORQFGTV1PQjP7oORQQmR}NQp'-vomVSSqX8Q Pvomp1kTVvf+SlXSUkQtSkxOPikP1j v RTVPQ!Stp1O8}Tt1j v RQ!

SUki&WRTsmsTukgXqNRORTVSzTVSzTf FQ!SUkTs-i!kgQ! TukxOfFpPtQj Ts&i!mdSUTut2p1mVikgTVvfPS`vowPi:SUTfPdmuQWRmVi!TOPvomsQ:SlCi QkTtQ<

ndTf1i!msmuX COPi!1kgQj:J1j|Q!SUQ&fPkS (voj+Tf 1jlvqqj|Q!SS=kv2Tut21msQtQ&fRk7#vqPQ$kv#imsp1mVi!kQ/kVOPQSl1Tf vowMWRmVi!T

O1vomuQ!STffRpPtQjlTs#imoSUTutp1mVi!kgTVvf1SqNPORTVSj|Q!{qp1Tuj Q!Szi Qmsm~PQrFfPQ!qQ&fPQjlimsT%oi!kTsvfvowFkVOPQG#vfP Q1kvow'I!j voki6

kgTVvf(qQ&fPQjlikvojUSKGvf8kvCvomVvqqTs#imdk(v~SlRORQjQ!SkVO1i!kti!X5fPvok-O1i FQGifFX$kgjp1QUFGTsmsmsTfP FQ#kvojUSe-1jQ!SUQ&fRki

(4)

NROPTVSkVOPQ!SUTVS (vp1mV8f1vokOPi FQ`WPQQ&fCvqSSUTWRmsQ 8TskxOPvp1kPkxORQOPQmuvowPi=msij qQGfFpPtWRQjzvowR1Q!v1msQif1et Tf

i:SUQ&fPkgTutQ&fRk}i!mqt+vqvqRoSv5e t dvoTfPJkv kgj XJkvfPi!tQ:iStifFXiSzeR&ifH

ndTsjUSUkifPBw~voj|Qt+vqSUkxPekVO1if82t2X@i FTVSvojlS E1QQ E1TfP-WPmsvt1tXqp1Ts1QPtQ&fPkvojg1Q!PTskvojx1i FQjUSi!jX

if1i Rvo&i!kgQ O1v)O1iSzSUQQ&ftQGkxORjlvpCO8tvqSUkRvow-tX (vojJv RQjMkxORQGmViSUkr FQGXQ!i!jUSlif1Pp1kdpP TskVO

t+voj|Q1i6dTfRQ!SSkxOPif2eCS O1vp1ms(O1i RQ:Q:1Q kgQ!OPTt kvqyzTsO1i!jl(hPjlTs Qot2X8kQ!i!OPQjxow}jTsQ&f1R&vomsmsi&WPvojUi!kvoj

if1TV1vomu OPvrPjUSUkq1jlv FQ!$kv)tQkVO1i!kdRORXSUTs#SzikdTskSPWRQ!SUk-S O1vp1mVt+i/dQpCSmVipCOifPJFGTu8NRO1vojfPQkVOPQ

Sl1Tsj TskgpCi!mFmuQ!iPQjvow1kxORQNoi1Tsj7qjlvpP+ifPvowCi5SUTVfPTsrP#ifRkMwjUi!#kgTVvfvowCkxORQjQmsikgTFTskgX$&vt2tpRfRTskgX Fi)t+if

w}j vt O1vt e vp1mV2O1i RQ=msQ!i!j|fPQ!tp1Otvoj QqTuwvfRmuXe 2O1ikxORQQ&fPQj oX8kvNdQQpPLMNRO1if8-Si!mVSv

dv)kv p1jlk p1kmsQjCif1NFGQ&fCE1TWdWRj QOPkxw}vojMSUQjRTf15vfJtX #ifP1TsCi!X5if1PQw}Q&fPSUQ`#vttTskgkQ!Slif1i!mVSv

v

ffCvXP67ORTsjlmuQXND8it1kvfRif185vfFfPi)j TVSU&vomumuw}vojMOPQm 8TskxOiRtTfPTVSUkgjUi!kTRQGt+i!kgkQjlS

ezi!mVSv8kxOPif kVOPQ/j Q!SUkMvowt2X#vomsmVi#W1vojlikvojUSldW1vokVOBi!k kiO+ifPBi!k

i!mskgQOet mup1qXkv$WRQ/i#WPmuQ/kv

SiX@kxOPikkVOPQj|Qi!j Q/k}vqvtifFX vow`kVOPQtkvmsTVSUkxPWFp1kkVOPQj|Qi!j Q$iw}Q kVO1i!kzetpCSUkVfk`v FQjlmVvqv?e`kVO1if8

Di!jJ7ORQQmRif1 D8i!jUi!mVhPwQTswwQjw~voj7w}j|Q!{qp1Q&fRkC1TsSU pCSSUTVvfPSifP+i FTs#Q-if1w}vojWRj Tf1qTfP 8TskxO+kxORQt

kv i!mskQO kxORQrFfPQ!SUk7&vqPQkVO1i!kO1iS)Q FQjWRQQ&f jlTukkgQ&f w}vojGfFpPtQj Ts&i!mj|QmVi!kTRTskxXe`i!mVSvkVO1if8 OPj TVS

1QQkmuQdw~voj7TVfPTskgTf18tX(TfRkQj|Q!SUk1TftikVOPQtikgTs#imjQmsikgTFTskgX -if11Q&f1j vtmsQX dw~voj7kgQ!iOPTfPtQ)tvqSUk

vow $O1i!keLofPv i#WPvp1kM#vtPp1kgQjUSif1fRpPtQjlTs#SeQ!Sl1Q TVimumuX (ifRkMkv8kxOPif ( ij TVi jUif1vojxFw}voj7ORQj

w}j TsQ&f11S ORTuif1OPQjPTfPw}Q kgTVvpCSCCiSSUTVvf/w~voj1SU TsQ&fR#Q&w~vojPw}j|Q!{op1Q&fPkgmuXj|QtTf1PTf1tQ7kxOPikeq&if/1vROFXSUTs&Sl

kVO1i!kFeRS OPvp1mV1vRORXSUTs#SlifPkVO1i!kFe (ifRkFkv CvROFXFSUTu&SoeRQfRQQ!1Q!1msQ&fRkgX$vowRj|QtTfPPTf1

e1kxOPif $tXwQmsmVv j QmsikgTFTskgXSUkgpCPQ&fRk}Sw}vojkxORQTsjw}jlTuQ&f11S OPTuifP2#vtjliPQjXFjlSCvXmuQ

ORpP

v

OPQ&fFdn-ifPq-FGQP1Qm E1Tu-ECv RQmVi! QoDif1PQmuoDi!kgkxORQ (Slqh1ifR-7qi Pv &7pPXdp Cij kTsp1mVi!jlmuX! )Q!vowgw}j|QXif1

e}mXFi TskVO $O1vtedvokk}v(k}i!mR@imVvokvowRORXSUTs#Sif1 i!m(iXS=TfRkQ&fPPQ! k}v2&vomsmsi&WPvojUi!kgQvf SvtQkVOPTfP?

DiXWPQmVikgQjeRi!mVSv)kVO1if8/kxORQGvojldifPTQjUSifP8ikgkQ&fPPQQ!SvowFkVOPQGNi!1TsjMqjUi8SUkgpCPQ&fRkmupRfRORQ!Slw~vojSUTukkgTf1

SvCikgTsQ&fRkmuX ORQ&f(eFkgjlTsQ!k}v i!jlop1QkVO1i!kFkTVSUkvojzkVOPQ!voj XJTVSzj|QmsQ PifPkdw}vojziSUkgj vRORXSUTs#S

e}wde-kxORTf8 kvqv=O1i!jlJi&WPvp1kdTskxe-#if8SUkij kdk}v FTsQ >kxORQ1jlTfRkTf1 vowkVOPTsSkxORQ!SUTVSziSif$Tut2CvojlkifRkdtvtQ&fRk

TfkxORQSUk}voj|X$vow-tX/msTswQSve (ifPkdkvkxOPif /i:wQ >1Q!v1msQGvfiG1QjlSvf1i!mmsQ FQmqiS Qmsm*ondTsjlSUkg!t2X/tvokxORQj

if1 t2XWRj vokxORQjgPkVOPQ t+vqSUkTut2CvojlkifRk1Q!v1msQ/Tf tXmsTsw}QJif1BkVOPQ)WRQ!SUk1Q!v1msQ/e RQ/Q RQjUofPv fRCif1

OPijmsQ!SGifPV7oijliOFw}vojt+i/dTf1(kxORQt Sv$O1iPPX/ei!mVSvkxOPif (t2XQ:kQ&fPPQ!@w}i!tTsmuX FCi!jlkgTsp1mVij muXtX

(5)

Q!Rp1&i!kgTVvfH

eqkVO1if8tX5vomV$w}j TsQ&f1 voQ 7qvqSi!w~vojCimumkVOPQ`kTtQ Q6RQ`Sl1Q&fPkOPifPqTf15vp1kx1voTf1:fPvokxORTf1oif1i!mVSv

Di!jq8Tsq |vqSUQRORDi!jlTsQ fRkxOPvfRX qif1 i#WRjlTsQmsmuQw}vojkgj Q!i!kgTf15tQGmsT-Q:itQtWRQjvowRkxORQw~itTsmuX

eqit qjUi!kQwgp1mw~vojPkVOPQzOPQmu$vowdi7fRpPt8WRQj1vowqw~ifPkiSUkgTsif1 TfPSl1TsjlTfPkQ!i!OPQjUSl&iwQ Hvow O1vt #vtQkv

tTf12TttQ!PTVi!kgQmuX iPQDvfPj voQ Tut Fp1jUS OfPQjx OPij msTsQ p1T 1Tsmsm-7p1kVOPQj mVif1R TRdQ 7oimVit+vfFFi!j|Qm

Fp1O1ij

ekVO1if8 CSi6hPQ-iO7qi!mVvfPQ&f ifP kVOPQOE

hFOPTumO1i!jtvfPTs!PQ!Sl1TskgQ7kVOPQzw}i kkxOPikkxORQX RQkxp1j|fRQ!

imskgQO

TfRk}vkVOPQ5SUQ#vf1WRQ!SUk1kVOPTfP(i#WPvp1k1msTFTf18TfkxORTVSkv fHei!mVSv$kVO1if87okgQ FQ)yzQTsORif1vokVOPQj1QjUSvfeFQ

fPQ FQjGtQkxw}vojG1jlv FTf1kv2tQkxOPikkVOPQj|Qi!jQJSUkgTsmsmTfPkgQjQ!SUkgTf1kVOPTf1dSw~vojpCS)kv1viS)mVvfP+iS Q6j|Q

msQ RQjMQ&f1vpCO(kv rFfP(kVOPQt

NROPQ(vojkVO1i!keMOPi FQPp1kkv (i!jl+kVOPTVSkVOPQ!SUTVSl1if1+Tf w}i kMimsmRvowkxORQ voj(eQ RQj`Q:1Q kMkv(CvFTVS

PQ!PTs&i!kgQ!kv kxORQtQt+voj|X8vowRtX$w~i!kxORQj 9@Q:msTFQ!(kvqqQkxORQjzw~vojzi#WPvp1kRi)PQ#iPQif1(iOPimswdeP#if'kFSi!X

kVO1i!kMkgTutQiSGdvqvqRRe (vfkSiX2Tuk (iSWPiRWFp1kCpRfRmsTRdQ/ifFX2vokxORQj`1QjlTsvq vowCtX(msTswQFeHof1v kVO1i!kMTsk

(6)

!"# %$

&"(')%*"+,

-&"('./# -$

021 *3 0

6 NROPQG1QjlTsvq1TuSUkifP1Tf1 (i FQi!P1j v:TutikgTVvfw~vojMfPvfPmsTfRQ!ijSU&i!mVi!jrPQmV1S QQQ Q

6 NROPQ:SUki&WRTsmsTukgX$vowR#vf1SUkgjUi!TfPkrPQmV1STf(w}j|QQQ Pvomp1kTVvf Q Q QQQ Q @

6 1kTt+i!mq&vfPSUkjliTfRkF1jv54sQ#kgTVvfw}vojCOFX-1Qj~WPvomsTsQ Rvomup1kgTVvf2SlXSUkQtS QQQ Q

6 fPQ qQ&fPQjlimsT%Q!JOPij|t+vfPTuQ Rvomup1kTsvf2SlXFSUkgQt Q Q QQQ Q

6 vf1SUkgjUi!TfRkP1i!t21Tf15Tf(kVOPQ FU7oNQ Rvomup1kTsvf2SlXFSUkgQt+S Q QQQ Q ,>

6 NROPQ:Sl1Tf2vowFPXqfPi!tTs&i!mWRmVi!TO1vomuQ!S Q QQQ Q Q QQQ Q A@

67%3(98":"; 0<

=

.7:>89 ?@ A#9A@%(B%C9$?D988"+3-% E3F$?9G$?/HI:"+,9BJ@ !39K9#9L !?B

9# = 0

egfRkjvqRp1#kgTVvf QQQ QQQ Q QQQ Q Q QQQ Q d

hPQjlTsvq1TuSvomup1kgTVvf1SlqSUk}if1PTfP (i FQ!Slif1tvq1Qmo1j v!WRmsQt+S Q QQQ Q B

pPtQjlTs#imqTt1msQtQ&fPkikgTVvf2if18j Q!Slp1muk}S QQQ Q Q QQQ Q

> vfR#mupCSUTsvf1S QQQ QQQ Q QQQ Q Q QQQ Q ><>

@ qfPv msQ!1qQtQ&fRk}S QQ QQQ Q QQQ Q Q QQQ Q >@

67%3(98":"; MN

O .7:P89 " D@ "(B C/$?Q8"8"+(-, !3F,(" ,9K9#9"R9J@TS,@98UK(3T@%9#TS

9AV :W 98,+93 E :? MX

(7)

6 Ci-dj vpRf1QQ QQQ Q QQQ Q Q QQQ Q >Y

6 vfPmuTfPQ!i!jt+vqPQmo1j v!WRmsQt Q QQQ Q Q QQQ Q @

6 p1kmuTfPQ=ifPSlpPtt+i!j X Q QQQ Q Q QQQ Q @d

1i!1kgQ!8&vqvoj PTf1i!kgQ!S Q QQQ Q QQQ Q Q QQQ Q @<

)Q&fRQjUi!mdi1i1kQ!#vqvojlPTfPikgQ!S Q QQQ Q Q QQQ Q @<

Sl1Q#TurPiCi1kgQ!&vqvoj PTf1i!kQSlXSUkgQt oN Q Q QQQ Q @6>

yzQ!{op1TsjQtQ&fPkS7w}voji1i!1kgQ!&vqvoj PTf1i!kgQ!S QQ Q Q QQQ Q @q

71Q kjlimtQkVO1vq1S TskVO2i1i1kQ!#vqvojlPTfPikgQ!S QQQ Q Q QQQ Q @Y

> DvqPQmVSif1tQkVO1vq1S Q QQQ Q QQQ Q Q QQQ Q B6@

@ pPtQjlTs#imqj|Q!Slp1mskS QQQ QQQ Q QQQ Q Q QQQ Q !

B vfR#mupCSUTsvf1S QQQ QQQ Q QQQ Q Q QQQ Q 9Y

qfPv msQ!1qQtQ&fRk}S QQ QQQ Q QQQ Q Q QQQ Q

Y P1Q&f1PTR: voQwrP TsQ&fRk}Szw~voji1i1kQ!#vqvoj1Tf1i!kQ!S Q Q QQQ Q

P1Q&f1PTR: NPORQ:SUk}if11i!jl8Sl1Q kjlimtQkVO1vqw~vojkVOPQG1PTutQ&f1SUTVvfPimdmsTfRQ!ijSU&i!mVi!jrPQmV Y

P1Q&f1PTR: Qk}i!TsmVSvowRkVOPQQTVqQ&fPSl1Q#kgjUi!mtQkVO1vq Q Q Q QQQ Q

67%3(98":";

M U8"#% E9?3 "8+ 3C'G(A:";89G3# /$?3*"# (C ;? ! <

>- egfRkjvqRp1#kgTVvf QQQ QQQ Q QQQ Q Q QQQ Q o

>- 1kgTut+i!md&vfPSUkjliTfRk1jv54sQ kTVvf Q QQQ Q Q QQQ Q

>- 7o#imsijMrPQmV1STf( p1jRQ!SlCi!#QkgTutQ Q QQQ Q Q QQQ Q 6

>- DvqPTsrPQ!SU&i!mVi!j (i FQSlXFSUkgQt Q QQQ Q Q QQQ Q 6

>- vf1SUkgjUi!TfRkF1j Q!SUQjRTf1:WPvpRf11ij|X/&vfP1TukTVvfPS Q Q QQQ Q <B

>- 1kTt+i!mq&vfPSUkjliTfRkF1jv54sQ#kgTVvf Q QQQ Q Q QQQ Q ,Y

>-> pPtQjlTs#imqj|Q!Slp1mskS QQQ QQQ Q QQQ Q Q QQQ Q

>->- NQ!SUkgTf1)WPvpRf11ij|X/&vfP1TukTVvfPS Q QQQ Q Q QQQ Q ,>

>->- NQ!SUkgTf1/&vfPSUkjliTfPk1j v54sQ kTsvf Q QQQ Q Q QQQ Q

>->- 1kTtTTf1/&vfPSUkjliTfPk1j v54sQ kTsvf QQQ Q Q QQQ Q &

>-@ TVSUpCSSUTVvf QQQ QQQ Q QQQ Q Q QQQ Q &q

>-B qfPv msQ!1qQtQ&fRk}S QQ QQQ Q QQQ Q Q QQQ Q &Y

67%3(98":"; 0

=

RB3T@999+:9G E3 >/$?3*"# (C ;? 0

O,O

@ egfRkjvqRp1#kgTVvf QQQ QQQ Q QQQ Q Q QQQ Q ,<

@ )Q&fRQjUi!msTQ!$O1i!j t+vfRTsQ Rvomup1kgTVvf2SlXSUkQt QQQ Q Q QQQ Q ,6>

@

(8)

@ vf1SUkgjUi!TfRkP1i!t21Tf1 Q Q QQQ Q Q QQQ Q ,Y

@ 8Q rPjlSUkP}vojlPQjqQ&fRQjUi!msTQ!JO1i!j t+vfRTsQ Rvomup1kgTVvf(SlXSUkQt Q Q QQQ Q ,

@ > CvpRf11i!jXJ&vfP1TukTVvfPS Q QQQ Q QQQ Q Q QQQ Q T>

@ >- ndTsjUSUk;~voj PQj#vf1SUkgjUi!TfPkQ Pvomup1kgTVvf2SlXSUkgQt Q Q Q QQQ Q T>

@ >- vf1SUkgjUi!TfRkPg1j Q!SUQjRTfP:W1vpRfP1ij|X$#vf1PTskgTVvf1S Q Q QQQ Q T>@

@ >- hFORXSUTs#imWPvpRfPCi!jXJ#vf1PTskgTVvf1S Q QQQ Q Q QQQ Q T>B

@ >-> 9@QmsmRgCvqSUQ!RfRQ!SS QQQ Q QQQ Q Q QQQ Q T>d

@ @ pPtQjlTs#imqj|Q!Slp1mskS QQQ QQQ Q QQQ Q Q QQQ Q T>Y

@ B qfPv msQ!1qQtQ&fRk}S QQ QQQ Q QQQ Q Q QQQ Q ,@d

67%3(98":";

0

=

N 3 "?@ !8% % :L. /$L3 *3C ;L#9 E 0

B egfRkjvqRp1#kgTVvf QQQ QQQ Q QQQ Q Q QQQ Q ,@<@

B e}mumupCSUkjlikgTVvf2vowPi:SUTut21msQGt+vqPQmdSlXSUkQt Q QQQ Q Q QQQ Q ,@Y

B NROPQGt+vqPTsrPQ!FU7oN(Q Pvomp1kTVvf2SlXSUkgQt Q QQQ Q Q QQQ Q AB

B> D$Xd1QjWPvomsTu#TskxX8vowkxORQ&vfPSUkjliTfRkRQ Rvomup1kTsvf QQQ Q Q QQQ Q AB/>

B@ 7oki&WRTsmuTskgX$vowR#vf1SUkgjUi!TfPkrPQmV1SpRf1PQjwjQQGQ Rvomup1kgTVvf Q Q QQQ Q AB6@

B@ NojUif1S FQjUSUQ FQ k}vojM&vfPSUkjliTfRktvq1Q!S QQQ Q Q QQQ Q ABo

B@ ECvf1qTskxpC1Tf1i!m-#vf1SUkgjUi!TfRkt+vqPQ!S QQQ Q Q QQQ Q AB,Y

BB OPvqvqSUTf1JCi!jUitQkgQjUS Q QQQ Q QQQ Q Q QQQ Q AB

B pPtQjlTs#imqkgQ!SUkS QQQ QQQ Q QQQ Q Q QQQ Q !

BY TVSUpCSSUTVvf QQQ QQQ Q QQQ Q Q QQQ Q !@

B qfPv msQ!1otQ&fRk}S QQ QQQ Q QQQ Q Q QQQ Q !/B

67%3(98":"; 0,<,<

< .7:W8 (' L; E K93#:3# 0 X

< egfRkjvqRp1#kgTVvf QQQ QQQ Q QQQ Q Q QQQ Q 5Y

< P1j v:TutikgQUFGTsmsmsTf1 FQ k}vojlSQ Q QQQ Q Q QQQ Q 5Yq

< voj|t+i!msTikgTVvf QQQ QQQ Q QQQ Q Q QQQ Q 5Y6

<> OPvqvqSUTf1$kVOPQ:iP1jlv:Tut+i!kgQGFGTsmsmsTf1 FQ k}vojlS QQQ Q Q QQQ Q 5Y6@

<@ pPtQjlTs#imqj|Q!Slp1mskS QQQ QQQ Q QQQ Q Q QQQ Q 5Y<B

<@ )Q&fRQjUi!msTQ!(SlRORQjlTs#i!mO1i!j tvfPTs#SV QQQ Q Q QQQ Q 5Yo

<@ )Q&fRQjUi!msTQ!j vokikgTVvf(qQ&fPQjlikvojUS QQQ Q Q QQQ Q 5Yo

<B 9 vojJTf81jvqqj Q!SS QQQ QQQ Q QQQ Q Q QQQ Q 5Yo

(9)

6 vf1SUkgjUi!TfRkf1voj t

||

c

||

:=

R

δ

ij

C

i

C

jd

3

x

1/2

w}vojGkxORQ$pRf1SUki#WPmuQ$SU#imsij (i FQSlXSUkQt TskVO

γ

=

−

1

if1$1j v54sQ kTsvf1Szi!kQ FQj X5kTutQpRfPTskSoNPORQ FQjlkgTs&i!moSU#i!msTf15TVSmuTfPQ!i!jgkvQtROPi6 SUTQJkVOPQ/jUi1TV@&vfPSUkjliTfRkqjlv kVOH)NROPQ!SUQJj Q!Slp1muk}S&itQ wjlvt iSUTt1msQJrFfPTskgQ~PTswgw}Qj|Q&fP Q

fRpPtQjlTs#imF&vqPQPSv8kVOPQXij|Q SUQ#vf18~voj PQj`#vf FQjlqQ&fRkTf@kVOPQ/qjlTsBSlCi!#Tf1 O1i1kQjO>

1j Q!SUQ&fRk}SMPQk}i!TsmsQ!/fFpPtQj Ts&i!mokgQ!SUk}SM&i!jljTsQ!vp1kWFX/t2X &v}i!p1kVO1vojlSl!pCSUTf1 iw~i!jPt+voj|QGi!#p'

jUi!kQCSUQpC1vqSl1Q kjlim&vqPQ< QQQ Q QQQ Q Q QQQ Q

NROPQ)h87<9;Svomp1kTVvfTVS`tQ!ifRkPk}v5WRQ)ifiP1jlv:Tut+i!kTsvfkv$kVOPQ)ROFXSUTs&i!mRSlCi! QkTutQ)vfRmuX(Tf

i:muTutTskgQ!j Q!qTVvf QQQ QQQ Q QQQ Q Q QQQ Q B

NROPQ I!msTsoORkP X-msTf1PQj6K=SUQCi!jUi!kTf1 kxORQ)QmsmsT1kTsifPORXd1Qj~W1vomuTsGj Q!qTVvfPS`vow1kVOPQ1jlv!WRmsQt Tf8

kQjlSUQ#kSkVOPQ5mVi!jlqQ5SlRORQjlTs#i!m1Slp1jlw~i! Q5vf ORTsOfRpPtQj Ts#imdWPvpRfPCi!j X(#vf1PTskgTVvf1S:i!j QTt

CvqSUQ!H QQQ QQQ QQQ Q QQQ Q Q QQQ Q q

NROPQ

Ψ

rPQmV@w~vojk(vjlvoki!kTfPCvoTfPkSvp1j| Q!STf@kxORQ Q!{opCikvojlTVi!mR1mVifPQ<NROPQ rPQmV1SS O1v f

ij|QGf1vfPmuTfPQ!i!jSvomp1kTVvfPSGvow'M{dS ->og @goifP g TskVO

a

Ω = 0

.

3

if1

λ

=

−

1

1n-voj #msij TskgX #kxORQ

ϕ

}i FQjUiqQzTVS1j Qtv RQ!/ikQ!iO jliPTupCSh1i!jlkS

iif1

WPvowqkVOPQ7r1qp1j|Q7SO1v :!j|Q

Sl1Q#kgTFQmuX kVOPQzf1vfPmuTfPQ!i!jCvp1k}dvoTfP5ifPJTfP-voTf15Svomup1kgTVvf1Soh1i!jlk

&oTsSCkVOPQSUk}if1PTfP-(i RQ

Svomup1kgTVvfR7if1 Ci!jlk

1TVS5kxORQvp1k}dvoTfP Svomup1kgTVvf Q:kgjUi! kQ! wjlvt Tsk 2NRORQ RQj kTs#i!mzSU&i!msQ

qTRQ!S rPQmV0SUkgj|Q&f1qkxO

ij~WRTskjlij|XBpRfRTskS:if1 kxORQ(OPvojlTvfPkim7#vqvojlPTfPikgQ!SJi!j|Q(#v}j vokikgTf1

i!jlkgQ!SUTVif&vqvoj PTf1i!kgQ!STf8pRfPTuk}S7vow

a

kVOPQ:PTVSUkifR#QvowPi:Svp1j|#QGwjlvt kVOPQj vok}i!kgTVvfi/:TVS >o > ':kgjUi!#kgQ!0vp1k-voTf1 f1vfPmuTfPQ!i!j (i FQ!S PSkgj|p1Qvp1k}dvoTfPDf1vfRmsTfRQ!ij (i FQ!Sn-voj

λ

=

0

,

−

2

,

−

10

Ψ

0

= 0

.

15

a

Ω = 0

.

3

TukxOi

180

×

20

×

32

qjlTVH1NRORQrPQmV

Ψ

TVSS OPv f@iSi

wpRfR#kgTVvf$vowdjMi!mVvfP5ijUiPTVim!msTfPQkVOPj vpCO$kVOPQSvp1j Q`CvoTfPkx!T;Q<!i!mVvf1kVOPQ

θ

=

π/

2

ϕ

= 0

msTfRQ< vfPkgTfRpCvpCS/ p1j RQ!S5S OPv #vtPp1ki!kTVvfPimzj|Q!Slp1mskSw~voj5vp1kdvoTf1 (i RQ!STVSU j|QkQ

CvoTfPkSl7w~voj5kVOPQ8f1vfRmsTfPQ!i!j5t+vqPQmVSl`SO1v kVOPQ2i!P1j v:TutikgQ(vp1k}dvoTfP (i RQ!S Q:kgjUi! kQ!

(10)

@ NROPQSitQtvq1QmsS7iSTfn-Ts?# >#WFp1k-Tf8kVOPQGj|Q!qTVvfvowkxORQSvp1j Q!S STfndTV #->=&vfRkTfRp'

vpCS p1jRQ!SMS O1v kVOPQ#vtPp1ki!kTVvfPSvowkVOPQkgjp1Qf1vfPmuTfPQ!i!jvp1kdvoTf1 (i RQ!Slif1PTVSU j|QkQ

CvoTfPkSS O1v kVOPQvp1kdvoTf1 (i RQiP1jlv:Tut+i!kgTVvfQ:kjli kgQ!w}jvt kVOPQfPvfPmsTfRQ!ijSUk}if1PTfP

(i RQSvomup1kgTVvf QQQ QQQ Q QQQ Q Q QQQ Q >

N (v5SUQkSvowR&vgt+v FTf1 i!jlkgQ!SUTVif&vqvoj PTf1i!kgQ!S QQQ Q Q QQQ Q @<

)Q!vtQkj TszWPiSUTVSzw~vojkVOPQN i1i!1kgQ!&vqvojPTf1i!kgQ!S Q Q QQQ Q @6>

1i!1kgQ!&vqvoj PTf1i!kgQ!STfkVOPQ

e

x,

e

y

1mVifPQif18kVOPj|QQ~PTutQ&fPSUTVvf1i!mq#vqvojlPTfPikgQSlp1jlw}i Q!S @<@ > 1jlj voj=Tf kVOPQ#vtPp1kgQ!Dvp1k}dvoTfP msTfRQ!ij)Svomup1kTsvf0iS5i2wgpRfP kTsvf vowkVOPQmVvo#ikgTVvf vowkVOPQ

vp1kQj`WPvpRf11i!j X5yzQ!Slp1muk}Si!j|Q$SOPv f@WPvokVO w}voj:SUkjliTVORkw}voj(i!jl@nd)Tf i1i!1kgQ!@&vqvoj

PTf1i!kQ!SifPw~voj7kxORQQTVqQ&f1Sl1Q kjlim-tQkVO1vqR-Q:1mVi!TfRQ!TfkVOPQ5kgQ:kx TskVO@vfRmuXt+vf1vCvomuQ

ifP{qpCiPj|pPCvomuQ:kgQj tS -Q1k Q Q QQQ Q Q QQQ Q @

@ NROPQQTVqQ&f FQ#kvojUSzw}vojzi

16

×

32

qjlTV&vt2Cij|Q!8k}v5kVOPQ:&voj j Q!SlCvfPPTf1 &vfRkTfRpPpPt QTVqQ&f8 wpRfR#kgTVvf1SlkxORQSlROPQj Ts&i!mCOPij|t+vfRTs&S8NROPQ#vfPkgTfRpCvpCS5p1jFQ!SSO1v kVOPQSlROPQjlTu&i!mMO1i!j

t+vfRTs&S kVOPQ:1ikiCvoTfPkS`i!j QkxORQ:#vtCvfRQ&fPkS`vowFkxORQ:QTsdQ&fRQ kvojUS QQQ Q B

B NROPQ

`

Ri!mup1Q!S=vowzkVOPQJPTVSU j QkgQ ifPop1mVijUECi1mVi!#TVif@vf i

16

×

32

qj TVB#vtCi!jQ! TukxOBkVOPQ TfPkgQ!qQj

`

Pi!mup1Q!SvowRkVOPQ:&vfRkTfRpPpPt ifPop1mVijLEMi1mVi!#TsifqNROPQQTVqQ&fRQ k}vojlS`vowFkxORQ=PTVSU j|QkQ ifPop1mVijECi!1msi TVifJij|Q7fPvokFPQ!qQ&fPQjlikgQ!Sv5iG#mpCSUkQjMvowRSUQ FQjUi!m

`

Pi!mup1Q!Svow-kxORQGQTVqQ&f1Sl1Q kjlimtQkVO1vq&voj j Q!SlCvfPCSzkv i=SUTfPqmsQ

`

Ri!mup1QvowRkVOPQ&vfRkTfRpPpPt 1j v!WPmuQtV QQQ Q B6

NROPQ

χ

PQ1Q&f1PQ&fP QvowCkVOPQQTVqQ&fPSl1Q#kgjUi!mtvq1Q)&voQwgrP#TuQ&fPkSNRORQ SvomsTVp1jFQ5S O1v (SkVOPQ &voQwgrP TsQ&fPk:vowkVOPQtvq1Q

Y20

kxOPik:TVSSlXdttQkgjlTui#W1vp1k:kVOPQ

e

Z

i/:TsS)kVOPQ@1iS ORQ!b p1jRQ S O1v (S=kxORQj Q!i!mPCijkzvow

Y22

/exf+WPvokVO #iSUQ!S=kVOPQJ1msvok`S OPv (S)kVOPQ#voQwgrP#TsQ&fRk}S=PTRTs1Q!WFX

kxORQGt+vf1vCvomuQ=#voQwrP TsQ&fRk QQQ Q QQQ Q Q QQQ Q B<B

Y vtCi!jlTsSvf0vowQ:i kGif1QTsdQ&fPSl1Q#kgjUi!mmsTfRQ!ij/vp1k-voTf1 Svomup1kgTVvf1SNRORQSvomsTV p1jRQ

S O1v (SkVOPQ Q:oi!#kMSvomup1kgTVvfR1Tf@kxORQ (i FQ j Q!qTVvfFR#vtPp1kgQ! wjlvtif+TfPrFfRTskQJSUQjlTuQ!SGNROPQ

vokxORQjRp1jFQ!SCS OPv HkxORQ`j|Q!Slp1mskovow&vt2Pp1k}i!kTsvf TskVOJiqjlTV TskVO

nχ

×

nΘ

×

nΦ

12001

×

16

×

32

ifP

χmin/a

= 0

.

2

χmax/a

= 75

CyzQ!Slp1mskSGi!j Q=S O1v f TskVO

`max

= 3

t+vf1vCvomuQ ifP

{opCiPjpPCvomsQ/tvqPQ!S dQ1kzifP

`max

= 5

t+vfPvCvomsQC{opCiPjpPCvomsQCifP+ORQ:oiPQ&iCvomsQ! NROPQ)j|Q!Slp1mskSi!j Q=S O1v f+ik

Θ = 0

iSi5wgpRfP kTVvfvow

r

dkVOPQ5PTVSUkifR#Q=w}j vt kVOPQ=#Q&fRkQj`vow1kxORQ

&vfRr1op1jUi!kTVvfJ QQQ QQQ Q QQQ Q Q QQQ Q

8Q!i!j}Svp1j QrPQmV1Szw~vojCf1vfPmuTfPQ!i!jt+vqPQmVS QQQ Q Q QQQ Q A>

n-vojMvp1kdvoTf1f1vfRmsTfRQ!ij (i RQ!SlkVOPQSUQ&f1SUTskgTFTskgX/vow-kVOPQjliPTVi!kTsvf$kvPQkiTsmsSvow-Svp1j Q`t2p1mR

(11)

vtCi!jlTsSvfBvowi$&vt2Pp1kQ!Bvp1k}dvoTfP2f1vfRmsTfPQ!i!jSvomup1kgTVvf #vfPkgTfRpCvpCS: p1j RQ!x1ifPBif

i!P1j v:Tt+i!kTVvf k}v+kVOPQ(vp1k-voTf1 Svomup1kgTVvf Q:kjli kQ! wjlvt kxORQ2SUkif1PTf1 (i RQ(Svomup1kgTVvf

1i!k}i kgXd1Q5CvoTfRk}S =yzQ!Slp1msk}S:i!jQ S OPv fBw~vojiJkxXd1Ts#imFfPvfPmsTfRQ!ijSU#i!mVij`t+vqPQmu TukxO@Ci/

jUitQkgQjUS

Ω = 0

.

3

λ

=

−

25

Ψ

0

= 0

.

15

χmin/a

= 0

.

05

χmax/a

= 200

qw}voji5qjlTs TskVO

nχ

×

nΘ

×

nΦ

12001

×

16

×

32

$yzQ!Slp1mskS5ij|QS OPv f i!mVvf1kVOPQ

e

Z

i/:TVS$NROPQSvomup1kgTVvf TVS1mVvokgkQ! iSGi/wgpRfP kTVvf@vow

e

Z

FkxORQ PTVSUk}ifP Q5w}j vtkVOPQ/voj TVqTf i!mVvf1kxORQ/i6:TVSkxORjlvpCO@kVOPQ Svp1j Q<NROPQQ:kgjUi! kQ!JCvoTfPkSTfkxORQ (i RQGvfPQi!j Qi:j|Q!Slp1mskvowFkgj Q!i!kgTf1kVOPQ (i FQ!SziSmsTf8

Q!ijoNPORQ:Slt+i!msmR~PTVSUkifP Q1msvokPS O1v (S7kxORQ`WPmuQ&f1PTf1/j Q!qTVvf2if1kVOPQTfFfRQjzjQ!qTVvf(Tf OPTuO

kxORQ:SUk}if1PTfP (i FQSvomup1kTsvf(TVSpCSUQ!2iSzifi!P1j v:TutikgTVvfw~vojkVOPQ:vp1k-voTf1$Svomp1kTVvf Q

& vtCi!jlTsSvfJvowokxORQif1i!muX-kgTs (i FQ!SCif15kVOPQ7&vt2Pp1kQ! (i RQ!SPw}voj

a

Ω = 0

.

5

χmin

= 0

.

05

a

ifP

χmax

= 70

a

if1

a2

= 0

i!k

χmin

= 0

NRORQ:tvf1vCvomsQ=TVS`SlpWRkgjUi!#kgQ!Tf2kxORQ=r1op1j|Q)vf kxORQjlTVOPkk}v i!msmVv w}vojzi:#vtCi!jlTsSvf(vowR#vtPp1kgQ!2if1if1i!muX-kTs (i RQit1msTukgpCPQ!S- Q Yo

, fifPop1mVij qj TV TukxO

nΘ

= 5

ifP

nΦ

= 6

=jlTVDCvoTfPkSl`CvoTfPkSik OPTsO i Rimup1Q w~voj

Ψ

TVS#vtPp1kgQ!RRi!j Q=&vffPQ kQ!WFX2SvomsTV+msTfPQ!S7NRORQ CiS OPQ!2msTfPQ!SGQ:kQ&fPkxORQ5qjlTV+kv

IROPifRk}vtKCvoTfPkSfRQQ!PQ!@w}vojkVOPQ5#vtPp1kikgTVvfGn-vojkVOPQ nd)ITt1msQtQ&fPkikgTVvfBvowkVOPQ

ECi!1mVi! TVifBi!kzCvoTfPk

A

kVOPQ Rimp1Qvow

Ψ

i!kCvoTfPk

A

0

TVS:fRQQ!1Q!H PX@kVOPQ8SlXdttQkj|X vow`kVOPQ RORXSUTs#imd1j v!WPmuQtqkVOPTVS Pi!mup1Q:&ifWPQ=j Q1msi Q!$WFX8kVOPQ Pi!mup1Q)ikRCvoTfRk

B

$OPTsO &7vfkVOPQ qjlTVHL7oTutTsmVi!jlmuX8kVOPQRi!mup1Qi!k

B

0

OPQ&f2fPQQ!PQ!Rq&if8WPQ)j Q1mVi! Q!WFXkxOPik1i!kPCvoTfPk

A

-kVOPQ Pi!mup1Q:i!k

C

0

&ifJWRQ=j|Q1mVi!#Q!/WFXkVO1i!kPi!k

C

okVOPQ Pi!mup1Q=ik

D

0

#if$WPQj|Q1mVi!#Q!JWFXkVO1i!kPik

D

ifPSv w}vojlkVOH QQQ QQQ Q QQQ Q Q QQQ Q q

>- vf1SUkgjUi!TfRk RTVvomsikgTVvf1S`w~vojQ Pvomp1kTVvfPS TskVO

γ1

=

γ2

= 0

qw}j|QQATf1=W1vpRfP1ij|X8&vfPPTskTsvf1Sl ifP+fPv(#vf1SUkgjUi!TfRk1jlv54sQ kgTVvf1S=hPmVvokgkgQ! i!j Q5jliPTVi!mPj Q!Svomp1kTVvfPS

Nr

= 21

31

. . .

61

Pi!msm

p1jRQ!SmuTsQ:vfk}v2vowRQ!i!OvokVOPQj Q QQQ Q Q QQQ Q A@

>- vf FQj dQ&fR#QG1mVvokPw}vojzkxORQ=Q Rvomup1kgTVvf(1j Q!SUQ&fRkQ!Tf(ndTV >-<-hPmsvokkgQ!i!jQ:PTswgw}Qj|Q&fP Q!Sw}j vt

kxORQ:Svomup1kTsvf TskVOjUiPTVimj Q!Svomup1kgTVvf

Nr

= 81

Q Q QQQ QQQ A@ >-

vf1SUkgjUi!TfRk RTsvomVikgTVvfPSMw~vojFQ Pvomup1kgTVvfPS TskVO

γ1

=

γ2

= 0

#&vfPSUkjliTfPk;1j|Q!SUQjFTf1`WPvpRf11ij|X &vfPPTskTsvf1Sldif1f1v &vfPSUkjliTfPkR1j v54sQ#kgTVvf17qvomsTV( p1j RQ!Si!j QfPvojt+i!msTQ!JWRX$kxORQ:{qpCifPkgTskxX

||∇

u

(

t

)

||

ORTsmsQ kVOPQ 1iS ORQ!+ p1j RQ!SGi!j Q:f1voj timuTQ!(WFX

||∇

u

(0)

||

`Q#i!X2vowkVOPQ)fPvoj timR TikgTVvf2w~i!#kvoj

||∇

u

(

t

)

||

jUi!kxORQjkxOPifqjlv 8kxOvow1kxORQ)#vf1SUkgjUi!TfPkS&ipCSUQ!SkxORQ5qjlv kVO2TfkxORQ OPTVORQ!SUkPjQ!Svomup1kgTVvf2SvomsTs2p1jFQ!Sl $OPTsO8O1i RQ#vf1SUkifRkRj vpRf18~vowgw3msQ RQm&vfPSUkjliTfPkFTVvomVi/

(12)

>-> vf FQj dQ&fR#Q$vow`Q Rvomup1kgTVvf1S S OPv fDTf ndTV >--JhPmsvokkgQ!Dij|QPTswgw}Qj|Q&fP Q!SGw}j vt kVOPQ8Q Rv

mup1kgTVvf TskVO

N

= 81

OPTsO TVSOPQ&fR#Qw}vojlkVO@kxORQ j QwQj Q&fR#Q5Svomp1kTVvf

uR

7qvomuTV p1jFQ!Sij|Q f1voj|t+i!msTQ!WFX

||

u

(

t

)

||

OPTumsQkVOPQCiS OPQ!/ p1j RQ!SCi!j Qf1voj|t+i!msTQ!WFX

||

u

(0)

||

Q#iX vowqkxORQ f1voj|t+i!msTikgTVvfw~i!#kvoj

||

u

(

t

)

||

#i!pCSUQ!SkVOPQdj v kVO+TfkVOPQ)OPTsoORQ!SUkPj Q!Svomp1kTVvf+SvomsTV+p1jFQ!Sl w~voj OPTsO

||

δu

(

t

)

||

TVSz#vf1SUkifRkPi!kjvpRf11vowgwmsQ FQm; Q Q QQQ QQQ >-@ vf1SUkgjUi!TfRk RTVvomsikgTVvf1Sw}vojPQ Pvomp1kTVvfPS TskVO

γ2

=

−

1

/M

&#vf1SUkgjUi!TfRkd1j|Q!SUQjFTfPWPvpRf11i!j X &vfPPTskTsvf1Sl!if1TskVO1vp1k&vfPSUkjliTfRk1jv54sQ kTVvfNPORQzTf1SUQkoS O1v (SCPTswwQjQ&fR#Q!S

||

δu

(

t

)

||

/

||

u

(

t

)

||

w}j vt kxORQ)j|Qw}Qj|Q&fP QSvomup1kgTVvf@vowPndTV >->1RNROPQ: p1j RQ!S7msQ RQm-vowgw1ikFmVikgQ)kgTutQ!SWRQ&ipCSUQW1vokVO

fRpPtQjUi!k}vojifP2PQ&f1vtTf1i!kvojzqjlv Q:CvfPQ&fPkgTVi!msmuX$ik-kVOPQ:Si!tQjUi!kQ!S QQQ Q

>-B vf1SUkgjUi!TfRk FTVvomVi!kgTVvf1S8w~voj Q Rvomup1kgTVvf1S TskVO

γ1

=

γ2

= 0

w}j|QQATfPWPvpRf11i!j XD#vf1PTR kTsvf1SlifP iSUTf1qmsQ#vf1SUkgjUi!TfRk1j v54sQ kTsvf i!k

t

= 20

M

TukxO

Λ = 2

/M

h1voTfPkS5SO1v

||

C

(

t

)

||

/

||∇

u

(

t

)

||

iwkQjCQ!iO8kgTutQ=SUkgQLoNPORQGTf1SUQk1mVvok}SzkVOPQ:SitQ:1ikivf2i:msTfPQ!i!jSU#imsQ<C & >- vf FQj dQ&fR#Q)vowMQ Rvomup1kgTVvf1SGSOPv f+Tf+ndTV />1BMhPmVvokgkQ!+i!jQ=PTswgw}Qj|Q&fP Q!S7wjlvt kVOPQQ Rvomup'

kTsvf TskVO

Nr

= 81

QQQ Q Q Q Q QQQ QQQ &d

>-Y yiPTVi!mz1jlvorPmuQ!S$vow

h

ψ

i10

ifP

h

CiC

i

i00

w}voj5kxORQQ Rvomup1kTsvfbvowndTV 8>-B NPORQSvomsTs0msTfPQ!S j Q1j|Q!SUQ&fPkkgTutQ!S

t/M

= 20

, . . . ,

25

NROPQ(1iSORQ!DmuTfPQ(j Q1j|Q!SUQ&fPkSkVOPQ(SUk}i!kgQ 4pCSUkWRQw~voj|Q kxORQ$#vf1SUkgjUi!TfPk1j v54sQ kTsvf i!k

t/M

= 20

5NRORQi!jlj v (S:Tf1PTs#ikgQ$kVOPQJmVvo&i!kgTVvf vowkVOPQ f1vf8 Slt+vqvokVOFfRQ!SS7Tf

ψ

QQQ QQQ Q QQQ Q Q QQQ Q &

>- Tswgw}Qj|Q&fP Q!S7WPQkQQ&fBQ Rvomup1kTsvf1S TukxO kTtQSUkgQ

∆

t

ifPBkVOPQJjQwQj Q&fR#Q/Svomup1kgTVvf

uR

vow

ndTV >->qCikCr8:Q!@Q Rvomup1kTsvf kTutQ!S

t0

G Pvomup1kgTVvfPSpCSUQ

γ1

=

γ2

= 0

w}j|QQATfP WPvpRf11ij|X &vfPPTskTsvf1Slqif18#vf1SUkgjUi!TfPk-1jlv54uQ#kgTVvf TskVO

Λ = 2

/M

i!w}kgQjQ!i!OkTtQ:SUkQP

∆

T

= ∆

t

6 >- vf1SUkgjUi!TfRk FTVvomVi!kTsvf1S

||

C

(

t

)

||

/

||∇

u

(

t

)

||

w}voj=Q Rvomup1kgTVvf1STskVO

γ1

= 0

ifP

γ2

=

−

1

/M

&vfPSUkjliTfPko1j|Q!SUQjFTf1GWPvpRfPCi!j X&vfP1TukTVvfPSlifPJ#vf1SUkgjUi!TfPkq1j v54sQ#kgTVvf TskVO

Λ =

√

2

/M

Q RQj|X

∆

T

= 2

M

exf1SUQkFS O1v (SkxORQ:Si!tQ=1i!k}i TukxO(rFfRQjkTtQj Q!Svomp1kTVvf QQQ &< >- Tswgw}Qj|Q&fP Q!S:wjlvt kxORQj|Qw}Qj|Q&fP QSvomup1kgTVvf

uR

vow`ndTV >->qzw}voj=kVOPQ8Q Rvomup1kgTVvf1S S O1v $fDTf

ndTV ,>- QQ QQQ QQQ Q QQQ Q Q QQQ Q &6>

>-& Tswgw}Qj|Q&fP Q!S

||

δu

(

t

)

||

/

||

u

(

t

)

||

wjlvt kVOPQJjQwQj Q&fR#Q Svomp1kTVvf

uR

vowndTV >->i!j Q 1mVvokgkQ! w}voj PTswwQj Q&fRkO1voTu#Q!S7vow

Λ

Pvomp1kTVvfPS TukxO

γ1

= 0

if1

γ2

=

−

1

/M

#vf1SUkgjUi!TfRkPg1j Q!SUQjRTfP WPvpRf11ij|X/&vfP1TukTVvfPSlo#vf1SUkgjUi!TfPk1jlv54sQ kgTVvfQ FQj X

∆

T

= 2

M

QQQ QQQ &<@

>-, Pvomup1kgTVvfPS 8TskxO

γ1

= 0

if1

γ2

=

−

1

/M

d#vf1SUkgjUi!TfPkC1j Q!SUQjRTf1 WPvpRf11i!j X(&vfPPTskTsvf1S ifP &vfPSUkjliTfPk1jlv54sQ kgTVvf Q FQj X

∆

T

Tswgw}Qj|Q&fP Q!S=w}jvtkxORQ8j QwQj Q&fR#Q$Svomup1kTsvf

uR

vow ndTV ,>->q-ik

t0

= 100

M

w~voj1TuwwQj Q&fRk-O1voTu#Q!Svow

∆

T

if1

Λ

QQQ Q QQQ <B

>-T> 7qvomsTVDp1jFQ msQwki/:TVS`S O1v (S/kxORQ(jlikgTVv@vowkgTutQ2Sl1Q&fPk`Tf QmsmsT1kTs$SvomFQ!S/kv kgTutQ2Sl1Q&fRk

TfBkxORQ OFXd1QjWPvomsTs5Q Rvomup1kgTVvf #vqPQ<5iS ORQ!Bp1jFQ

jlTVOPki/:TsSS O1v (S=kxORQ$jUi!kgTVv2vowkgTutQ

(13)

@ Pvomup1kgTVvf vow&vfPSUkjliTfRk FTVvomVi!kTsvf1S5w~vojQ7oO(i!jSUORTsmV TfPTukTVi!m1iki E1Qw}k7r1qp1j|QSO1v S

Q Pvomp1kTVvfPSMpCSUTfP Pi!jlTsvpCS Rimup1Q!SCvowdkxORQ`#vf1SUkgjUi!TfRk1it1TfP:Ci!jUitQkgQjUS

γ0

if1

γ2

pCSUTfP fRpPtQjlTs#imzj|Q!Svomup1kgTVvf

{

Nr, Lmax}

=

{

13

,

7

}

yzTVOPkr1op1j|Q(SO1v (S/kxORQ(msvf1BkgTutQ!SU&i!msQ Q Pvomp1kTVvf2vowkxORQ:Si!tQ=1i!k}iGw}vojkxORj QQPTswgw}Qj|Q&fPkfFpPtQj Ts&i!mqj|Q!Svomup1kTsvf1S QQQ Q T>Y

@ Pvomup1kgTVvf/vow&vfPSUkjliTfRk FTVvomVi!kTVvfPSCw}voj?FGQj jFTfRTskTsim1iki TskVOJSl1Tf5Ci!jUitQkgQj

~a

= (0

.

1

,

0

.

2

,

0

.

3)

w~vojSUQ RQjlimfFpPtQj Ts&i!mqj|Q!Svomup1kTsvf1S Q QQQ Q Q QQQ Q T>

@ Pvomup1kgTVvfvow87oO (i!jSUOPTsmVTfRTskTsim-1iki:1Qj kgp1j~WRQ!$WFX8i=qjUi RTski!kTVvfPim (i FQPp1msSUQ TskVO

i!t21msTskxpCPQ

10

−

3

E1Qw}kr1qp1j|Q:PQ1Ts#kS&vfPSUkjliTfRkFTVvomVi!kTVvfPSi!k Rij TVvpCSfRpPtQj Ts#imdj|Q!Svomup' kTsvf1SlifPkVOPQj TVOPkFr1op1j|QS O1v (S

Ψ

4

i FQjUidQ!8v FQjkVOPQ:vp1kgQjPWPvpRf11i!j X$vowFkVOPQ#vtPp'

k}i!kgTVvf1i!m-CvtiTfi!kPiSUTf1qmsQ:fFpPtQjTs&i!m-j Q!Svomup1kgTVvf87qvomuTVp1jFQ!SpCSUQ:w}j|QQATf1=W1vpRfP1ij|X

&vfPPTskTsvf1Sif11iS OPQ!p1jFQ!S7pCSUQ=&vfPSUkjliTfPk;1j|Q!SUQjFTf1if1(RORXSUTs#imWPvpRf11ij|X8#vf8

PTskTsvf1S QQ QQQ QQQ Q QQQ Q Q QQQ Q ,@

B voj tvowGkVOPQ2Qjljvoj

||

δu

||

/

||

u

||

7j|QmVi!kTFQk}v kxORQj|Qw}Qj|Q&fP QSvomup1kgTVvfRvf0ir8:Q! 1vt+i!Tf Q:kgQ&f1PTfPJw}j vt tTfPTut2pPt #vqvojlPTfPikgQ:jUi1TpCS

1

.

9

M

kv ti6:TutpPt

41

.

9

M

NROPQ=CvtiTf TVSWRjlvdQ&fTfPkv8QTVOPkMSOPQmsmsSQ!iO+vowMkxORTsqfRQ!SS

5

M

if1jUiPTVi!mFj|Q!Svomup1kTsvf

Nr

-OPvqSUQ&f vf w~vp1j1TuwwQj Q&fRkCjpRf1SiS

Nr

= 8

,

11

,

14

,

17

zNRORQ/#vf1SUkgjUi!TfRkz1i!t21Tf18kgQj t+S1j|Q!SUQ&fPkgQ!+Tf imumqvowRkVOPQ!SUQ:SUTutp1msikgTVvf1SO1i FQ

γ5

= 0

.

6

VQ Q Q QQQ QQQ ! B vf1SUkgjUi!TfRk-f1voj|t

||C||

/

||

∂u

||

w~vojkxORQ=SitQGjpRf1Sz1mVvokgkQ!8Tf(ndTV AB QQQ QQQ !A B 1jlj voj1fPvoj t

||

δu

||

/

||

u

||

w~vojMj|pRfPS7Tf8kxORQ:#vf1SUkgjUi!TfRkP}Cit1Q!SlXSUkgQt TskVO2vp1kQjRWPvpRf11ij|X ik

r

=

>d6

M

,Bq6

M

KYo<

M

!o<

M

NROPQmVvf1 kQj|t qjlv kVOJvowokVOPQQjlj voj-fPvoj t vo p1jlS Q:CvfRQ&fPkgTVimumuX2vf@i5kTtQ!SU#imsQ1j vCvojlkgTVvfPimRkvkVOPQ S{opCij|Q5vowCkxORQ5&vqvoj PTf1i!kgQ)CvqSUTskgTVvf vow

kxORQ:vp1kQjRW1vpRfP1ij|X/ QQ QQQ Q QQQ Q Q QQQ Q !A>

B> vf1SUkgjUi!TfRkdfPvoj t

||C||

/

||

∂u

||

vowkVOPQSitQjpRf1SiSkVO1vqSUQGTfn-Ts?,B-NROPQG#vf1SUkgjUi!TfRk}Szqj v iS

t

1/2

pRfPkgTsm1Q RQ&fRkgpCi!msmuXPjlTFQ&f Q:CvfRQ&fPkgTVimumuXWFXkVOPQJv FQjUi!msm1mVvqSS)vowi!#p1jUi!XPQtvf8

SUkjlikgQ!8TfndTV AB QQQ QQQ Q QQQ Q Q QQQ Q !A>

B@

vf1SUkgjUi!TfRkFf1voj t

||C||

/

||

∂u

||

vow1iGWRmVi!T O1vomsQ)SUTutp1msikgTVvf TskVO

Rmax

= 41

.

9

Nr

= 17

Tf

Q!iOvowFQTVOPkRSlpWP1vt+i!TfPSgifPk(v PTswgw}Qj|Q&fPkWPvpRf11ij|X$#vf1PTskgTVvfPS QQQ Q !@

< Tswgw}Qj|Q&fP QWRQkQQ&f kxORQ&vt2Pp1kQ! QTsdQ&fPi!mup1Qw}voj)if

n

×

n

qjlTs ifP i

2

n

×

2

n

dj TV w~vojkVOPQ 1j v!WPmuQt vowzqQ&fPQjlimsT%Q!VI

l

= 1

00

SlROPQj Ts#imRO1i!jtvfPTs#S:5i!ki/CvoTfRk}S:ij|Q i!k

n

=

5

,

10

,

15

,

20

,

25

&NROPQ:SU#imuTf15TVSzmVvqdi!jlTskVORtTsGvfJWPvokVO(i6:Q!SlifP8kxORQ=iPCi!j Q&fRk

(14)

<

L2

fPvoj t

||

δy

||2

:=

H

(

y2n

−

yn

)

2

dA

1/2

dvowCkxORQPTswgw}Qj|Q&fP Q`WPQkQQ&f2kVOPQ)#vtPp1kgQ!2O1i!j t+vfRTs

y

vf@if

n

×

n

qjlTV+if1@i

2

n

×

2

n

qjlTs@w}voj`kVOPQ1j v!WRmsQt vowMqQ&fPQjlimsT%Q!I

l

= 1

K SlROPQj Ts&i!mOPij|t+vfPTu&S:iki CvoTfPkS=i!jQ

n

= 5

,

10

,

15

,

20

,

25

zNPORQ/SU&i!msTfP(TVSGmVvqdi!jlTskVORtTs vf(WPvokVO+i/:Q!S vfRQj qQ&fP Q5i!P1Q!i!jUS7k}vWPQSUmsTVOPkgmuXSUmVv QjkVO1if

1

/n

2

kVOPTVSti!XJWRQ5Rp1Q k}v5TfRkQj|CvomVi!kTVvf8Qjlj vojUSlvoj1QjO1iCSQjljvojUSw}j vt kVOPQ#vqvojlPTfPikgQSUTfPqp1msij TskTuQ!S7i!kFkVOPQGCvomsQ!S/5Y

< Tswgw}Qj|Q&fP QWRQk8QQ&fkxORQ@&vt2Pp1kQ!,QTVqQ&f Rimup1Q

λ

w}voj8if

n

×

n

qjlTV,ifPi

2

n

×

2

n

qjlTV w~voj=kxORQ81jlv!WRmsQtEvowqQ&fPQjlimsT%Q! j vok}i!kgTVvf qQ&fPQjlikvojUS2NROPQSU#i!msTf1@TVSmVvqdi!jlTskVORtTs

vfDWPvokVOi/:Q!SlifP kVOPQQj jlvoj #vf FQjlqQ!S$iS

1

/n

2

NROPQ(1mVvok=TfP mupCPQ!S Pi!mup1Q!S/w~voj

n

=

15

,

20

,

25

,

30

,

35

QQQ QQQ Q QQQ Q Q QQQ Q 5Y <>

L2

f1voj tvowzkVOPQ$1TuwwQjQ&fR#Q:WRQkQQ&f kxORQJ#vtPp1kgQ! qQ&fPQjlikgTfP2wpRfR#kgTVvf

z

w~vojqQ&fRQjUi!mR TQ!(j vok}i!kgTVvfdQ&fRQjUi!k}vojlS`vfif

n

×

n

qjlTV2ifPi

2

n

×

2

n

dj TV2w}voj

n

= 15

,

20

,

25

,

30

,

35

vf FQj dQ&fR#Q7idiTfJiP1Q!i!jUSPk}v`WRQSUmuTVOPkgmuX/S O1i!msmVv Qj1kVO1if

1

/n

2

NROPQ 8TsmVPmuX Ri!j X-Tf1=j Q! p1jRQ`j QmsikgQ!SMkviGSvomup1kgTVvfw}voj $OPTsO8kVOPQQTsdQ&fPi!mup1Q`TVSzSlt+i!msm

i!PCi!j|Q&fPkgmuX#vf FQjlqTfPk}v

i!P1j v:Tt+i!kQmuX

λ

∼

.

095

x-ifPkxORQj Qi!P1Q!i!jUSkv WRQ5if+TVSSlp1QTukxOkxORQ)fPQ!i!jPQ!qQ&fPQjliX vowkVOPTVSQTVqQ&f Rimp1Q 8TskxO kVOPQ

λ

= 0

QTVqQ&f Rimp1Q/#vojlj|Q!SlCvf1PTfP2kv

z

= const

GNPORQ Pi!jlTR ikgTVvfPS)vowkxORQj|Q!Bp1jFQ1kxORQj Qw}voj QRti!XWPQJRp1Q$kvTsmsmC#vf1PTskgTVvfPTfP@vow7kVOPQJQTVqQ&f FQ#kvoj

(15)

NROPQ5j|Q!Pp1 kgTVvfw}i kvoj`w}vojvp1k-voTf1 (i FQ!SGRp1QkvkxORQ)f1vfPmuTfPQ!i!jlTskxX`nvojimumR&iSUQ!Sl

Ψ

0

=

0

.

15

)NROPQ$SUQ&vfP &vompPt2f j QwQjUSGk}v C{? )NROPTVS:Q!{qpCi!kgTVvf TsS)SvomFQ! w~voj

Rlin

CSUkgTR t+i!kQ pCSUQ!SkxORTVS Ri!mup1Q$vow

Rlin

Tf

exp (

−

√

−

λRlin

)

MSUkgTutikgQ/TVS=kVOPQ/jQ!Rp1 kTsvf w~i!#kvoj w~vpRfP2w}j vt i:fRpPtQjlTs#i!mSvomp1kTVvf+vow8C{? 6 NROPQ:mViSUkP&vomupPt(f+qTFQ!SkxORQ)j|Q!Slp1msk}S7w}j vt

8Q kvf8yi!ROPSvf #vtPp1ki!kTVvf 8TskxO

a

Ω = 0

.

3

TukxO kVOPQJvp1kQj7WPvpRfPCi!j Xik

r

= 30

a

ifP TskVO2i

r, θ, ϕ

qj TV2vow

120

×

20

×

32

QQQ Q Q QQQ Q >-

vf FQj dQ&fR#QvowrFfPTskgQPTswgw}Qj|Q&fP Q#vtPp1ki!kTVvfPS vfPmsTfPQ!i!j1vp1k}dvoTfPSvomup1kgTVvf1SCij|Qz#vt

Pp1kQ! TukxO(r FQ:1TuwwQj Q&fRkqj TV(j|Q!Svomup1kgTVvf1Szw}voj

λ

=

−

10

Ψ

0

= 0

.

15

a

Ω = 0

.

3

if1vp1kgQj WPvpRf11ij|X i!k

r

= 30

a

f

L

2

fPvoj t TVS#vtPp1kgQ!$w~voj1kVOPQ1TuwwQj Q&fR#QCWRQkQQ&fJkxORQGSvomup1kgTVvf w~vojdj TV

k

if1(qjlTV

k

+ 1

oNRORTVSzTVSzj QCvoj kQ!8iSzkxORQ I1jlj voj6Kw~vojdj TV

k

QQ QQQ >d

vf FQj dQ&fR#Q w}vojjlvoki!kTfP2msTfPQ!i!jt+vqPQmVS msmPtvq1QmsSGOPi FQ

a

Ω = 0

.

3

λ

= 0

1if1+pCSUQ vp1k}dvoTfPWPvpRf11i!jXB&vfPPTskTsvf1S$i!k

χmax

= 50

a

+NRORQ(#vtPp1kgQ! t+vf1vCvomuQ2kvBSvp1j| Q SUkj|Q&f1qkVOTfP1Q:

γQeff

/Q

TVSpRfPTskxXTfkxORQQ:oi!#kSvomup1kgTVvfHqNPORQQI!k(v5jQ!qTVvfKG#vtPp1ki!kTVvf j Qki!Tf1Szi!msmt2p1mskTCvomsQ!S7w}voj

χ <

3

a

QQQ Q QQQ QQQ QQQ vf FQj dQ&fR#Q8w}voj)j vok}i!kgTf1f1vfRmsTfPQ!i!j)tvq1QmsS mumt+vqPQmVS)O1i RQ

a

Ω = 0

.

3

λ

=

−

25

Ψ

0

= 0

.

15

if1JimumpCSUQvp1k}dvoTfPW1vpRfP1ij|X5&vfPPTskTsvf1Si!k

χmax

= 50

a

!NPORQUIkv=j|Q!dTsvfHK tQkVO1vqj Qki!Tf1Szi!msmotp1mukTuCvomuQ!Sw}voj

χ <

3

a

QQ Q QQQ QQQ QQQ A NROPQ jUiPTVikgTVvfj Q!Rp1 kTVvf+w~i!#kvojRp1Q5kv(kVOPQf1vfRmsTfRQ!ijTskgXn-vojGi!msmP#iSUQ!Sl

Ψ

0

= 0

.

15

FifP

a

Ω = 0

.

3

1NRORQSUQ#vf1#vomupPt2fj|Qw}QjUSkvCC{ @ NROPQ=kxORTsj &vompPt2fdTRQ!S7kxORQ)j|Q!Pp1 kTsvf8w~i! k}vojlSC1j|Q!SUQ&fPkgQ!$Tf$h1i!1Qj1e #NRORQmViSUk-#vomupPt(fqTFQ!SkxORQj|Q!Slp1msk}SvowkVOPQQTsdQ&fPSl1Q#kgjUi!m

&vt2Pp1k}i!kgTVvf 8TskxO

χmin

= 0

.

3

a

TukxO vp1k}dvoTfP(W1vpRfP1ij|X+&vfPPTskTsvf1Si!k

χmax

= 50

a

`max

= 3

if1i=qjTV 8TskxO

nχ

= 8001

nΘ

= 16

(16)

> NROPQj|Q!Rp1#kgTVvf8w}i kvojUSw}voj1fPvfPmsTfRQ!ijvp1k}dvoTfP (i FQ!S kVOPQPQ jQ!iSUQ`Tfit1msTskxpCPQRp1Qk}v

f1vfRmsTfPQ!i!j5Qwgw}Q k}S n-voj

a

Ω = 0

.

3

t+vqPQmVSlkVOPQw~i!#kvojUS i!j Q8#vtCi!j|Q!Dw}vojkVOPQ(PTsj Q kgmuX &vt2Pp1kQ! vp1k}dvoTfP Svomp1kTVvfPS8ifP w~voj/vp1kdvoTf1DSvomup1kgTVvf1SQ:kjli kgQ!w}j vtIf1vfRmsTfPQ!i!j

SUk}if1PTfP (i RQ!SJSvomup1kTsvf1S NROPQ

λ

= 0

j|Q!Slp1mskSJTfPPTs&i!kgQmsTfRQ!ij5tvq1QmsS$Tf OPTsO kxORQ j Q!Rp1 kTsvf w~i!#kvojTVSpRfPTskxX(WFX+PQrFfPTukTVvf5NRORQ Rimp1Q866B/>$w~vpRfPBw}vojGkxORQJQ:kjli kQ! Sv

mup1kgTVvf@qTFQ!SGifTfP1Tu&i!kTsvf@vowkVOPQ)fFpPtQjTs&i!mRi! p1jliXvowkVOPQ)Q:kgjUi!#kgTVvf1j vo Q!Pp1j|Q< msm

j Q!Slp1muk}S Qj|Q &vt2Pp1kQ! TskVO

`max

= 3

χmin

= 0

.

2

a

RifP

χmax

= 50

a

vfBiJdjTV 8TskxO

nχ

×

nΘ

×

nΦ

= 8001

×

16

×

32

NRORQJj|Q!Rp1#kgTVvf@w}i k}voj (iS#vtPp1kgQ!WFXki/-Tf12kVOPQ jUi!kTsv$vowCkxORQ5{opCiPjpPCvomsQ)#vtCvfRQ&fPkS mVSvJmsTVSUkgQ!+ij|Q=kVOPQ)Q!SUkTutikgQ! Rimp1Q!Svow

Rlin

dkVOPQ PTVSUk}ifP Qw}j vt kVOPQ=Svp1j Q!SCWRQXFvfP $OPTsO2kxORQGfPvfPmsTfPQ!i!jzQwwQ#kSzij|QSlpPP1j|Q!SSUQ!Pif1Q!SS

kTt+i!kQ!S:vowkxORQ$j|Q!Pp1 kgTVvf w~i!#kvojUS`WPiSUQ!BvfBkxORQ$Q!SUkgTut+i!kQ!S:vow

Rlin

7QQ C{ -@ n-voj &vfPSUTVSUkgQ&fPX TskVONi&WRmsQU :kVOPQ:Svp1j|#QSUkgj Q&fPqkxOOPiSCWRQQ&fki6dQ&fk}v:WRQ)<>Y QQ Q 9Y

@ NROPQQjlj vojUS 1TuwwQj Q&fR#Q!Swjlvt ifPimuX-kgTsSvomp1kTVvfP1vowRkxORQ:tvf1vCvomsQ R{opCiPj|pPCvomsQ)i!P1jv:T

t+i!kTsvfw}vojPTswwQjQ&fRk Rimp1Q!S`vowPkVOPQ)Svp1j Q)Sl1QQ!

a

Ω

FNROPQ)v RQjlimsmdQj jlvojTsS`kVOPQPTswgw}Qj|Q&fP Q vowFkVOPQ#vtPp1kikgTVvf1i!mdif1ifPimXkgTsSvomup1kTsvf2ik

χmax

= 70

a

!kxORQ i FQGQjlj vojMTsSkVOPQ:PTswgw}Qj Q&fP QvowRkVOPQ&vt2Pp1k}i!kgTVvf1i!mdifPifPimuX-kgTs`1Q!i/-}kv1Q!i/ i!t21msTskxpCPQ!S QQQ Q Y,Y

B NROPQ)Qwgw}Q kRvfkVOPQ i FQit1msTukgpCPQvow1kVOPQ=&vfP1TukTVvfPSGvf

a2

ifP

da2/dχ

i!k

χmin

= 0

.

3

a

NROPQ5j vomsQ vowkVOPQ

a2

kgQjt #ifWRQ)pRf1PQjUSUkvqvq iSGif+Qwgw}Q kTRQ)S OPTswkvowkVOPQ5 Q&fPkgQjvowSU&i!mVi!j

(17)

e}k3S7if(Q: TskTf1$kgTutQ:kv)WRQ=i=j|QmVikgTFTVSUk)fP Q}Q!SvokgQjlTs#vfP Q1kS`Slp1OiSCWRmVi!T O1vomsQ!S`ifP2qjUi FTsk}i!kgTVvf1i!m

msQ&fPSUQ!S7i!j|Q`f1v wgp1msmuX/pRf1PQjUSUkvqvq(kv5Q:TVSUk-TfkVOPQQIjQ!i!m;K (voj mVHqNRORQ!vojQkgTs#imifPQ:1QjlTutQ&fPkim&vqSltvomR

vqoX+O1i RQQ&fRkQj|Q! kVOPQ$tiTf1SUkgj Q!itV! )jli FTskikgTVvf1i!mR(i FQiSUkgjlvfPvtX TVS5i#W1vp1kkvw~vomsmsv f1 tvqSUk

Tut2CvojlkifRkzw~vojGkVOPTVS:kxORQ!SUTVSlCkxORQ .-+ )1Svomup1kTsvf vow 1TfPSUkQTf'S)rPQmsBQ!{opCikgTVvfPS=rFf1i!msmuX+SUQQt+S=kvWRQJi!'

1j vqiOPTfPJvp1jdjliSlL

n-vojt+ifRXX-Q!i!jUSlQ:oi!#kPSvomp1kTVvfPSvow81Tf1SUkgQTf'SQ!{opCi!kTVvfPS Qj|Q:vfPmuX(i RiTsmsi&WRmsQTf2#iSUQ!S7vowPQ:kgj QtQ

SlXdttQkj|X/voji!mVqQ WRjUi!TsSl1Q#TsimsTukgX " 3i!jlTVvpCSM1Qj kgp1j~WPikgTFQiP1jlvqi!OPQ!S Qj QPQ RQmVv1Q!R1j v FTVPTf1

if Q&f1voj tvpCS$it+vpRfRkGvowGTf1SUTsoORkGTfPkvBkVOPQ8f1i!kxp1j Q(vowqjUi RTuk}i!kTsvf1i!mPXqfPi!tTs&SlWFp1kkVOPQ(kjp1muX &S(; !+/. 0

"#PXqfPi!tTs&SO1i FQ2j|Qt+i!TfPQ! w}ijwjlvtEj Q!i!O pRfPkgTsm RQj|X j|Q#Q&fRkmuX@NRORQSUkxpRfFfRTf1 #vt1muQ:TskxX if1

f1vfRmsTfRQ!ij TskgX vowzkVOPQJrPQmVBQ!{opCi!kTsvf1S:Pp1kS=kVOPQ$RXdfPi!tTs#S)vowSUkgjlvfPqjUi RTski!kTVvfPim1rPQmV1S=vf i/1Q!1Q!SUki!m

kVO1i!k#if8vfPmuXWRQj|Q!iOPQ!kvq1i!X TukxOkVOPQORQmu(vowF#i!j Qwgp1m#vtPp1kgQjSUTut2p1mVikgTVvfPSqNROPTsSTsSkxORQ:j|Q!qTutQvow

+ %$))A(3 %5< (?

NROPQrPQms(vowdfRpPtQjlTs#imjQmVi!kgTRTukgXOPiS1j vqqj Q!SSUQ!Tf8Slt+i!msmqSUkgQCS SvtQkTutQ!S7w}voj(i!jPSvtQkgTutQ!S

WPi (i!jl v FQji RQj|XmVvfP1QjlTVvqBvowzkTutQ<5ekMO1iS&vtQ$kv2j|Q!{qp1Tuj Q/iSUTut2p1mskifRQ!vpCS=#vtt+if1 vow

SUkikgQ~vowP}kVOPQ~i!jlkM&vt2Pp1k}i!kgTVvf1i!muPfRpPtQjlTs#imCifP if1i!muX-kTs#i!m1tQkxOPvq1SlSvtQkTtQ!S)kv2kVOPQ/pRfPw~vojkgpRfPikgQ

Q: mupCSUTVvf0vowGRORXSUTs#im`Tf1SUTsoORk NRORTVSdij difRkgpCif Qwgw~vojlkxzO1v Q FQjxTVS/rFf1i!msmuXBWRQ!qTfFfRTf1 kv CiXDvowgw

TskVO(tp1mskgTu1msQtQkVO1vq1Sf1v i Ri!TsmVi#WPmuQGw~vojO1if1PmsTf1JkVOPQ=i!j OPQkxXdCim1j v!WRmsQt vowfFpPtQjlTu&i!mdj QmVi!kgTRTukgX

WRTf1i!jXWRmVi!TO1vomuQ!S "BqA!oT>q,@6B!/,Y!q& 3

NROPTVSkVOPQ!SUTVSTVSQ RTVPQ&fR#Q/vowzkVOPTVSjQ Rvomup1kgTVvfBvowzkVOPQ rPQmVRPTf kVO1i!kkVOPQ5ROPTsmsvqSvRORX PjlTFTfP2kVOPQ (voj

#ifWRQPTFTVPQ!SvtQ $O1i!k- msQ!ifPmuXTfPkvkxORj QQRO1iSUQ!SWRQw~voj|Q

OPi!1kgQjUSP:ifPCg!Rp1jlTf1

O1i1kgQjUS >A@

if1 BxifP(i!w}kgQj

(18)

D*D! .2 1 +.2235467"8 F)9: !! AF5$!+G M

.

OPi!1kgQjUS$ifP $j|QmVi!kQkvkv(i51jvqdjli!tSl1Q!ijOPQ!iPQ!Tf@j|Q Q&fPkPX-Q!ijlSWRX(yzTsO1i!j @hPjTs Q %o VF#i!msmsQ!

kVOPQ4< %! # "$L&(*),+-# %+/.10324),56 h879b-i!P1j v:Tt+i!kTVvfNPORQt+vqSUkFi!t8WPTukTVvpCSvowRkVOPTsSz1j v54sQ kSzvojlTVqTfPimddvqi!mVS

SUkgQttQ! w}j vt $O1i!kTf j|QkgjlvqSl1Q kmsvqvS1msTRdQ7if5Q:kgj|QtQ1Q!SSUTutTVSlt j Q!di!jlPTfPkxORQfPQ!i!jkQj|tH1j vqSl1Q#kS

w~voj1kVOPQ#vf FQ&fPkgTVvfPimqiP1jvqiO/k}vfRpPtQj Ts#imj QmVi!kgTRTukgXTfRkQ!qjUi!kgTf1Q Rvomup1kgTVvf8Q!{opCi!kTVvfPSTfkTutQ< )fPQ

t+i4VvojzPTswgrPp1mskgX TskVO(kVO1i!kRiP1jlvqi!OJO1iSzk}v 1vTskVO(&vt21QkTfPJkgTutQ!SU#i!msQ!S

kkxORQ=S OPvojlkRQ&fPRkVOPQj|QTVS

kVOPQ7jlTfPdTf1)kgTutQ!SU#i!msQ7vowqkxORQ`TfPPTRTsPpCi!mO1vomsQ!SlvowdvojlPQj

GM/c

3

kkxORQ`mVvfP)Q&f1R&kxORQjQTVS1kxORQ`kgTutQ!SU&i!msQ

vowzkVOPQ/Q!SSUQ&fPkgTVim1RXqf1itTu&S7kVOPQ/TfPSl1TsjUi!m;=NROPTVSTVS:SUQ RQjlimPkgTutQ!S=qj|Q!ikgQjkVO1if Q RQ&f kVOPQJvojWRTski!mR1QjlTVvq

p

r

3

/GM

w~voji7WRTfPij|XSUQCi!jUi!kTsvf8vowRjliPTupCS

r

dewRif8Q Rvomup1kgTVvf#vq1QGTsSkvWRQ&vtQfRpPtQj Ts#imsmX/pRfPSUk}i/ WRmsQoTuk Tsmsm1vJSvJvfkVOPQ=jlTfPqTf1$kgTutQ!SU&i!msQow}voj OPTuOkxORQfFpPtQj Ts&i!mdkTtQSUkgQ2TVSi1i!1kgQ!Rj|Q&f1PQjlTf1

kVOPQtp1O(mVvf1qQj }kgQj t ROFXFSUTu&i!m-PXqfPi!tTs&S7vp1kPvowPj|Q!iOH RQ&fTuwPkxORQ:#vqPQ=j|Qt+i!TfPS7SUki#WPmuQ ifP8fPv (i/

1i!XSl!t+vqSUkRvowRkxORQt 1vqgkxORQGtTVSltikgOvowRkxORQQ Pvomup1kgTVvf(kgTutQ=SUkgQw}jvt kxORQGt+voj|Q:TfPkgQj Q!SUkgTf1 TfPSl1TsjUi!m

kgTutQ!SU#imuQ1j Q!SUQ&fRk}Sif8TfRQwrPQ&fR XJvowRkVOPQ:SUkifPCi!j 8i!P1j vqiOvowFTfRkQ!qjUi!kgTf1/kxORQGQ Pvomup1kgTVvf(Q!{opCi!kTVvfPS7Tf

kgTutQ<

NROPQ1QjTVvqPTs)SUkif1PTf1 (i FQ iP1jlv:Tut+i!kTsvf+TVf1voj|Q!SkVOPQ/S O1voj kQjkgTutQ!SU&i!msQ<`egfw~i!#kCTskCTVf1voj|Q!S4)

kgTutQ!SU#imuQ!SoNRORQvoj WRTsk-TVSiP1jlv:Tut+i!kQ!$iS1WRQTfP Q:i kgmuX51QjlTVvqPTsvojTf8w~i! kgQ:i kgmuX&S()(P %!+)ATfiG#v

j vok}i!kTf1w}jUitQvowq#vqvoj 1Tf1i!kQ!S )fR#QkVOPTVSCif1Si!k;TVSPTutCvqSUQ!RkVOPQw~vp1j ~PTutQ&f1SUTVvfPimQ Rvomup1kgTVvf 1j v!WPmuQt

SUTut21msTsrPQ!Skv5i`WPvpRfPCi!j X1Pi!mup1Q`1j v!WRmsQt vf8kVOPQGkVOPj|QQ~PTutQ&fPSUTVvf1i!mtifRTsw~voms8vow FGTsmumsTf1/vojWRTskS )fP Q

Slp1OBiJSUkikgTVvfPij|X+Svomup1kgTVvfBTVSGw~vpRfPPRkxORQJSUmsv =CSUQp1mVi!jTfPSl1TsjUi!mP#if2WRQ TfRw}Qjlj|Q!@iPTVi#WPikgTs#imumuX NROPTVS

i1Tsi&WPi!kTsiSSlpPt1kgTVvf2WRj|Q!i6-S1v fFRvowM&vp1jlSUQ $OPQ&f+kVOPQOPvomsQ!SrFf1i!msmuX(1mupRf1qQ5kv (i!jl+vfRQ ifPvokxORQjg

WFp1kkVOPQj|QGTVSiSUTVfPTsrP#ifPkF1QjTVvqvowFkgTutQ t2p1mskgTu1msQvojWRTskS %6! dTf ORTsOkxORQTf1Sl1Tsjlimj QtiTfPSzSUmVv

WFp1kFkVOPQQ Rvomup1kTsvf(TVSzk}vqv)f1vfRmsTfRQ!ijzw}vojiGCvqSUkP8Q kvfPTVif(Q:Cif1SUTVvf

e}kf1v iP1Q!i!jUSPkxOPikqkVOPTsSmVi!kQ7Tf1Sl1TujUi!mSUkiqQ`&if)WRQ7O1if1PmsQ! RQj|X QmumWFX)tvq1QjfJQ Pvomup1kgTVvf$&vqPQ!Sl

Sv kVOPQ2h879 i!P1j v:TutikgTVvfTVS5f1v mVvfPqQj fRQ#Q!SSi!j X iSJi+-vsWRQkQQ&fwjlvtKQ!i!jlmuXBCvqSUkP8Q kvfPTVif

#imsp1mVi!kTsvf1S=kvmVi!kgQ/fRpPtQjlTs#i!mC1mupRfPqQ8SUTutp1msikgTVvf1SCDv Q RQjxCTsk&vp1ms &vfR#QTPi#WRmuX@SUkgTsmsmPWRQ/pCSUQwp1m;

NROPQ$WPiSUTs$TfPQwgrP#TuQ&fPX CvqSUQ! WFX kxORQ(tTVSlt+i!kO vowGkTtQ!SU#imsQ!S j|Qt+i!Tf1S5Q FQ&f Tf t+vqPQj|f fFpPtQjlTu&i!m

#vq1Q!S yzQt+v FTf1+Tsk Tsmumj|Q!{qp1Tuj Q/kxORQTut21msQtQ&fRk}i!kTsvfDvow7SUk}i#WRmsQ$Tt1msTs TskkgTutQ}SUkQP1TfP+ims-vojTskVORt+Sl

i(SlpW1SUkifPkgTVimQww}vojlkw~voj)Slp1OD#vt1muTs&i!kgQ! &vqPQ!S2e}k7TVSCvqSSUTWRmsQkVO1i!kvfR#Q8kVOPQ8t+i!OPTfRQj X TVSrFf1i!msmuX

! "$#&%!'()**,+.-/0$!$12314$*5678%14 9:<;= "-(>9!= "5(9.!=7!(!14(>:?.! ()+* 1

7* &!12=2%!@A$7:?ABC%14 $D** &? &7 E.!:?$+* (F +!GH!/, "1'9'IJ@KL7M!:?$* 0(=5$14NA ":F%A'(F,*A@

9O*P#@7$AP +P/"!0!(!$1,.!Q!$6GH!/RA &!/1S-!$!&M6TU=71S!75:?$.! (!5 5!/V*!Q-!W +6!12:F "

/0-!/0H$!(O*:F%! * &= +C2C:F:F*14V6@2N7@/0H +0R'$12*5/C "14.:F!9/U:F%$:?$'(=IX&L7M"Y'A@2-!

;N! "14PN.Z*!;R[/C Z! E+.-14*1Y*N.!H:F-!2=:F%1A@P +! &1\13](,7 V* "14M

^$_ !/W/*U:? "143%14+* 1Y:? ":?SB$*!\!%12AU9!\+7$1414(OOH`-!!A +*:?A +0a "-'/0 "!/b*;=$5@c "b

-* &!M&Td9!Z!/14( +0.!3%!$12 0(7!(!/b.;=5%%14 #@:?* &RY*\:?.! (U9W;RN &!3*12!+7 "14:?H7!(/0b

;=5E -7 &87 , &!U;N*8 "-7/C "/=;=5W]6*MeU!V.W(! &!$ &!VAA-'E.!V/012567* &!b.;=5Ef-#=

!]67g7 [!+*$1.!O127R +H2!!/0R +\9!(/?$!$14/MNh014 :i*Z &!RAd! "%!O' F+7$1W*O1'N +!$!/CN +3*R 1'97

(19)

TfB1mVi!#Q/k}vO1if1PmsQ$kVOPQJmVi!kQJTf1Sl1Tsjlim TskVO h87<9 tQkxOPvq1SlMkVOPQTuj=QwgrP#TuQ&fPXt+iX2WRQ$#vt1QkgTskgTRQ TskVO

Tut21msTs TskRQ Rvomup1kTsvfH

f1vokVOPQjGCvqSSUTWRmsQJCi!Xvowgwzw~vojh879 &i!msp1mVi!kTVvfPS=TVS=kxOPikzkVOPQX+#vp1mV 1j v RTs1Q/TVPQ!i!mCTfPTskgTVi!m1iki$w}voj

#vf FQ&fPkgTVvf1i!moQ Rvomup1kTsvf1S-1TfPSUkQTfSkVOPQ!voj|XJ1QrFfRQ!SCf1vokvfPmX$kVOPQQ Rvomup1kgTVvf(vowqQ!vtQkgj X WFp1k-i!mVSv $S!+10

&(P ) "+ ( )(P %!+ &kVO1i!kdt2pCSUkoWRQGSi!kgTVSUrPQ!kxORjlvpCOPvp1kFQ!i!O$SlCi!kTVi!moSUmuTs#Q<NROPQGTfPTskgTVi!mq1iki!w}j vt ORTsOif

Q Rvomup1kTsvf+WRQ!qTfPSlRtpCSUkzSi!kgTVSUwXkxORQ/#vf1SUkgjUi!TfRkzQ!{qpCi!kgTVvf1Sw~vojkVOPQJSUTut2p1mVikgTVvfBkv$WRQ5RORXSUTs#imumuXtQ!if

TfPdwgp1m;Rvoj if1#vomsmVi#WPvojUi!k}vojlS

SUQQQ<? %6@ `O1i FQSUQkkVOPQ!SUQ(#vf1SUkgjUi!TfPkGQ!{opCi!kTVvfPS$iSJif QmsmsT1kTs

SlXSUkQt2PitQ&fPi&WRmsQkvSUk}if11i!jl2fFpPtQj Ts&i!mRkgQOfPTV{op1Q!S OPv Q FQjgFkxORQjQ5i!j|Q5SlpWRkmsQkgTsQ!SGj|QmVikgQ!kv$O1v

kVOPQX@S OPvp1mV+WRQ$SvomRQ! w~voj46)A(3 "$ ),GSlCikgTVi!m1&vfRr1qp1jlikgTVvfPSvowRORXSUTs#imCTfRkQj|Q!SUk =NRORQ$Q!{opCikgTVvfPS=Tf

RvomRQ8wj QQmX@Sl1Q#Tur1i&WRmsQ1i!k}iCjQmVi!kgTf1kv+i$WPi!T-qj vpRf1 #vfPw~voj|t+i!mzqQ!vtQkgj XBvf kVOPQ8TfPTskgTVi!mzSlCikgTVi!m

SUmsTs Q<:NROPQ!SUQ wj QQ 1i!k}i/&if2WRQ OPvqSUQ&fBik TsmsmuCif1@kVOPQ QmsmsTu1kgTsQ!{opCi!kTsvf1S Tsmsm11jlv FTVPQ i$Svomp1kTVvfBw}voj

ifRX j|Q!iSvfPi&WRmsQ!11i!k}i OPvqvqSUTf1(kxORQ$S!$ (Mwj QQ51i!k}iw}voji/qTRQ&fRORXSUTs#im1SUTukgpCi!kTsvf@TVS`f1vokMQ!iSlX7exf

Ci!jlkgTs p1msijx1kxORQJf1i!TFQ ifP SUkifPCi!j CO1voTu#QMi!1i!k7#vfPw}vojtimdQ!vtQkj|X#POPiSGWRQQ&f S O1v $f kv8WRQ$Tf

#vtCi!kTWRmsQ TskVOBjlvoki!kTVvfPim < vojGj|Q#kgTsmsTfPQ!i!j ">d Mt+vokgTVvf vowzQ RQ&fBSUTf1qmsQ5WRmVi!OPvomsQ!S 71TfFfRTf1voj

WPvqvqSUkQ! #vfPr1op1jUi!kgTVvf1SkVO1i!k&vtQ8w}j vt 1i!k7&vfRw~voj timqQ!vtQkgj XBij|Q5fPQ Q!SSij TsmuX i!##vtCifPTsQ!@WFX

iSlpWPSUkifRkTsimit+vpRfRkGvow I 4upRf8 jUiPTVikgTVvfFK(pRfFRORXSUTs#im7qjUi FTsk}i!kgTVvf1i!m (i RQ!S)1j|Q!SUQ&fPkvfPmuX k}v imumVv

kVOPQSlCi! QkTutQ=kvWRQ:TfPTukTVi!msmuX#vfPw}voj timsmX 1ik1NRORTVS4upRf 8jUiPTVikgTVvf2{op1TsdmuX8jUiPTVi!kQ!STf2kxORQ)Q Rvomup1kgTVvf

if18&vfRk}itTf1i!kQ!SzkVOPQ:SUTutp1msikgTVvf(kVO1i!k (iSTfPkgQ&f1PQ!H

vojlj|Q#kgTfP kVOPTsSz1j v!WRmsQt vow,4upRf8JjUi1TsikgTVvf 8Tsmsm

j|Q!{qp1Tuj QfPQ O1voTs Q!S:vowz&vfRw~voj timC1i!k}idWRQkgkQjj|Q1j|Q!SUQ&fPkgTf1kxORQ kTV1i!mCSUkjp1#kxp1j|QJvowzkVOPQ)WRTf1i!jXWRmVi!T

O1vomuQ!S'ECv RQmsi Q "d# O1iSWRQ!opRfiSlXSUkQtikgTsSUkxpCPX8vow1tvqPTsrPQ!TfRTskTsim-&vfRw~voj|t+i!mFqQ!vtQkgjX dif1

O1iSw~vpRfP1jlvtTVSUTf1SUTVfPS ECQkgkgTf18kVOPQ5#vfPw~voj|t+i!mPqQ!vtQkgjX2SUTut21muX$WPQ i SlpP1Qj|CvqSUTskTsvf vowk(vTf

PTFTVRpCimWPvqvqSUkQ!(WRmVi!TJOPvomsQ qQ!vtQkgjlTsQ!SlqORQ=O1iSSlpWPSUk}ifPkgTVi!msmuX(j|Q!Pp1 Q!2kVOPQ 4upRf8jUiPTVi!kTVvfTf8WRTf1i!jX

Q Rvomup1kTsvf1SldCij kTu p1mVi!jlmX8TskS7OPTVO }w}j|Q!{op1Q&fPX8#vtCvfPQ&fRk}SyzQ!Slp1mskSw}j vt h879K&i!msp1mVi!kTVvfPSl OPTuO+imR

j|Q!iRX(j Q1j|Q!SUQ&fPkMkxORQJQ:i kMSlCi QkTtQJdQ!vtQkj|X+iCij kwjlvt kVOPQJSUmVv SUQ p1msijTf1Sl1Tsjlimu1#vp1mV jQ!Rp1 Q

kVOPTVS@4upRf $jUiPTVi!kTVvftp1Owp1j kxORQjgTswdfPQQ!PQ!H

OPi!1kgQjUS/ifP=51j Q!SUQ&fRkPkxORQj Q!Slp1muk}SvowPtvqSUkvow1tX (voj TskVOkVOPQ)1Qj TVvqPTs:SUk}if1PTfP (i RQ=&vom

mVi#WPvojUi!kTVvf CvokxOj|QmVikgQ:k}vJik}vX8tvq1QmdpCSUQ!kv$mViX1v fTVPQ!iS`ifP(t+i!OPTfRQj XJWRQw~voj|Q)i!kgk}i!TdTfP$kVOPQ

wp1mumQ!{opCikgTVvfPSzvowqQ&fRQjUi!mj|QmVikgTFTskxX/NROPTsSk}vX tvq1QmTVSvowWRTf1i!jXCvoTfPk-OPij qQ!SM#vpP1msQ!k}v)i`f1vfPmuTfPQ!i!j

SU#imVi!jrPQms(vf Tf8-v (SSdT-SlCi QkgTutQ

ψ

+

λF

(

ψ

) = Source

,

6

ORQjQ

F

TVS=SvtQ f1vfPmuTfPQ!i!j=wgpRfP kTsvf vowkVOPQ$SU&i!mVi!jGrPQmVR

λ

TsS)iw}j|QQmuXSl1Q#Tur1i&WRmsQJ&vfPSUk}ifPkxMifPBkVOPQ Svp1j|#QTVS5k(v #v~voj~WRTskTf1 CvoTfPkO1i!jlqQ!Slj Q1j|Q!SUQ&fPkgQ!@WFX PQmski2wgpRfP kTVvfPS 8TskxOif1op1mVi!j FQmVvo TskgX

Ω

(20)

#vf1PTskgTVvf(kxOPikkxORQrPQmVS O1vp1mV(jlvokikgQGjlTsdTs1mX TukxOkVOPQ:Svp1j|#Q

∂tψ

=

−

Ω

∂φψ.

6o

/SUTfP$kxORTVSQ!{opCi!kTVvfFdkxORQ)kgTutQ1Qj TRikgTFQ!S`iP1Q!ij Tf1/Tf

6!Ri!j|Q=j|Q1mVi!#Q! TskVO+if1op1mVi!jPQjlTRi!kTFQ!S

NROPTVS`msQ!i FQ!Si)Pp1j|QmuXSlCi!kgTVimPTswgw}Qj|Q&fPkgTVimdQ!{opCi!kTVvfFWRp1k1vfRQ5vowMi p1j TVvpCSSvojlk1TukPTVS7f1vok1Pp1j|QmuX8QmsmsT1kTs

vojGORXd1Qj WPvomsTsA$egf ifBTfRkQj TVvoj)j|Q!qTVvfR ORQj QJkxORQ FQ#kvoj=kVO1i!kqQ&fPQjlikgQ!S=kVOPQ8SlXdt2tQkgj X

∂t

+ Ω

∂φ

MTVS kgTutQmsT-QkVOPQ5j|Q!Pp1 Q!Q!{opCikgTVvfTVSGQmsmsTu1kgTsA p1k}SUTs1Q vowMkxORTVSGj Q!qTVvfkxORQ SlXdttQkgj X FQ#kvoj`TsS:SlCi! QmsTRdQ

if1 kVOPQj|Q!Rp1#Q!@Q!{opCikgTVvf TVS:OFX-1Qj~WPvomsTsAJNPORTVS=tTR:Q! O1i!jUi! kQjGj|Q1j Q!SUQ&fRk}SkVOPQrPjlSUkti4svoj=#vt1msT

#ikgTVvf vowzkxORQJtQkxOPvqPMSUTfR#Q$imum 8QmsmR3ofPv fBfFpPtQjTs&i!m1tQkVO1vq1S=w}vojGQwrP TsQ&fPkgmuXSvomRTf1+PTswgw}Qj|Q&fPkgTVim

Q!{opCikgTVvfPSiSSlpPtQ=dmsv!W1i!mqORXd1Qj WPvomsTs TskxXvojzQmsmsT1kTs TskxX/Fn-vojkxORTVS`j|Q!iSvfR 8Q)iP1jlvqi!OPQ!8kxORTVSORQmsTs#imumuX1

j|Q!Pp1 Q! (i RQ5Q!{opCi!kTsvf RQj|X2#ip1kTVvpCSUmX#-WFXPTsj Q k1rFfPTskgQ~PTswgw}Qj|Q&fP Q5PTVSU j QkgTikgTVvfe jlvokgQt+vqSUkvow

kVOPTVSfRpPtQjlTs#i!mq&vqPQ<

eqpCSUQ!SUkifPCi!j $SUQ#vf18~voj PQjCrFfRTskgQ~PTswgw}Qj|Q&fP Q`SUkgQ&fP TsmVSMk}v)j|Q!Rp1#Q`kxORQ`Ci!jlkgTVi!mPTswgw}Qj|Q&fPkgTVi!mQ!{qpCi!kgTVvf

kv iSlXSUkgQt vow#vpP1msQ! f1vfPmuTfPQ!i!ji!mVqQ WRjUi!Ts Q!{qpCi!kgTVvf1SNROPQ8rPQmV (iS5PQrFfPQ!DvfDi8tQ!S O vowqjlTV

CvoTfRk}Slif1kVOPQ)1TuwwQj Q&fRkTVi!mqQ!{opCikgTVvf (iSi!P1j v:Tt+i!kQ!(ikPQ!i!Oqj TV2CvoTfRk1pCSUTf1PTswwQj Q&fR#Q!Svow1kVOPQ

rPQmV Ri!mup1Q!S=i!kkVOPQ$dTRQ&f qjlTV@CvoTfPk7if1 Tsk}SGfPQTVOdW1vojlS ewzkxORQj QJij|Q

n

qjlTs CvoTfRk}SlMkxORTVS=j|Q!Slp1mskS:Tf i SlXSUkQtIvow

n

&vpP1muQ!i!mVqQ WRjUi!Ts Q!{qpCi!kgTVvf1S w~voj=kxORQ

n

pRf qfPv f Rimup1Q!S5vowkxORQ(rPQmsLNROPQ!SUQ8#vpP1msQ! Q!{opCikgTVvfPS Qj Q)kxORQ&f@SvomFQ!(WRX(i 8Q 8k}vf }yiRO1Svf SUOPQtQFTf OPTuOkxORQ5Q!{qpCi!kgTVvf1SGi!j Q=TskgQjUi!kTFQmuX

msTfRQ!ijTQ!+ifP SvomRQ!Rj Q!Slp1mskgTfPTf Slp1# Q!SSUTFQmuXWPQkgkgQjiP1jlv:Tut+i!kgTVvf1Skv8kxORQ5kjp1Q/Svomup1kTsvfHNROPQ!SUQ

msTfRQ!ijTQ!SlXSUkQtS Qj|QOPifP1muQ!ik1Q!i!OSUkgQpCSUTfP8SUk}if11i!jl E h F x PmuTWRjUi!jlTsQ!S7w~vojkxORQ5PTsj|Q#k

fRpPtQjlTs#i!mMSvomp1kTVvf vowmuTfPQ!i!jSlXSUkQtS CQ&fCj vtmuQXiSSUTVSUkgQ! qj Q!i!kgmuX(TfBkxORQ/TfPkgQjlw~i! Tf12vowtX&vqPQ

TskVO(kVOPQUE h F j vp1kgTfPQ!S

S QmsmRiS 8jlTskgTf1kVOPQ=&vqPQ)1QrFfRTf1kVOPQ)PTVSU#j|QkgTQ!Q!{opCikgTVvfPSGifP2kVOPQ 8Q 8k}vf }yiRO1SvfTskQjli6

kgTVvfRdeMimsSv jvokQ=&vqPQ=k}v/&i!msp1mVikgQ)kVOPQ I!Q:i kK:Svomup1kgTVvf@vow1kVOPQmuTfPQ!i!j

λ

= 0

R1j v!WRmsQt w}j vt i5SUQjlTsQ!S Q:Cif1SUTsvf@Tf RvomFTf12SlROPQj Ts&i!mOPij|t+vfPTu&S:ifPCQ!SSUQmPwgpRfP kTsvf1SGEMi!kgQjQmsQtQ&fPkS:vowMkxORQ OPi!1kgQjgTf

RvomRTf1$kVOPQ:1jlvo Q!Rp1j QGw}vojzQ:kgjUi!#kgTf1Jif=I!vp1k-voTf1 K)Svomup1kgTVvfwjlvt i ISUkifPPTf1 i FQ<K:Svomup1kgTVvfF-ij|Q

t+vqSUkgmuX2Rp1Q)kv$kVOPQ)Qwgw~voj k}S`vowPtX#v~ip1kxOPvojUSMNROPQ=kQ:k1vowMO1i1kQj iS 8jlTskgkQ&f8WFX8yzTsO1i!jl2hPjlTs Q)w}voj

PpWRmsTs#ikgTVvf(iS xvow$OPTsOe(iSzmsTVSUkgQ!2iSzi&v}i!p1kVO1vojxoif18TVSzj|Q1jlTfRkQ! FQjWPi!kTut

OPi!1kgQj:j|QrFfPQ!SkVOPQtQkxOPvq1S7vow-O1i1kQj1)kv)1j v RTVPQGSUTsofRTsrP#ifRkmuXJTut21jlv FQ!JfRpPtQjlTs#imj Q!Slp1muk}S

yi!kxORQjkxOPif@SvomRTf1(vp1jQ!{opCikgTVvf1STf SlRORQjlTs#i!mR&vqvoj PTf1i!kgQ!SlQ(vojdQ! Tf@if i1i!1kgQ!#vqvojlPTfPikgQ

SlXSUkQt kVO1i!kqTVSCt2p1O5WRQkkgQj1Slp1TskgQ!$kv=kVOPQSUkgjp1#kxp1jQ`vow-izWRTf1i!jXOPij qQ7#vfPr1op1jlikgTVvfHoNROPTVSMi1i1kQ! #v

voj 1Tf1i!kQGSlXSUkgQt 1j v RTVPQ!8ip1k}vtikgTs7jQrFfRQtQ&fRkFvowkxORQfRpPtQj Ts#imqqjTV$fPQ!i!jkVOPQGk(v Svp1j QO1i!jlqQ!Sl

ORTsmsQ$j|Q!Rp1#Tf12kvSUkif11ij SlRORQjlTs#i!mC#vqvoj 1Tf1i!kQ!Sw~i!ji i!X fPw~vojkgpRfPikgQmuX Q$w}i Q! SlpWPkgmsQJ&vt=

1msTs#i!kTVvfPS 8TskxO2kVOPQ=i!P1muTs&i!kgTVvf(vowRtX$rFfPTskgQ~PTswgw}Qj|Q&fP QG#vq1QTf(kVOPTsSiCi1kgQ!#vqvoj 1Tf1i!kQ:SlXSUkQt2qif1

(21)

iCi1kgQ! &vqvoj PTf1i!kQ!S 9@Q(#i!msmsQ! kVOPTVS)fPQ iP1j vqiORMkxORQ I!QTVqQ&fPSl1Q#kgjUi!mMtQkxOPvqLK e}k`TVS5i-Tf1 vow

#vt1j vtTVSUQ WRQkQQ&f iSl1Q kjlimCQ:Cif1SUTVvf Tf vp1j:Sl1Q TVi!msmuX1~i1i!1kgQ!Bif1op1mVi!j&vqvoj PTf1i!kgQ!SgCif1 i

rFfPTukQ}1TuwwQj Q&fR#Q tQkVO1vqHNRORQ8Sl1Q#kgjUi!mPQ#vtCvqSUTskgTVvf iS=tiPQJfPvok`TfPkv&vfRQ&fRkTVvfPimSlROPQjlTu&i!m

O1i!jtvfPTs#SloWRp1k1jlikVOPQj`vfPkvkxORQQTVqQ&f-WPiSUTVSGvowCkxORQ5PTVSU#j|QkgToi!kgTVvf+vowCkVOPQ if1op1mVi!j7msi!1mVi! TVifvf@vp1j7#v

voj 1Tf1i!kQ8qj TVHNROPTVS5Sl1Q kgjUi!mj|Q1j Q!SUQ&fRk}i!kTsvf iSkVOPQ&f kjpRfP#ikgQ! kvrPmskQj)vp1k7ROFXSUTs&i!msmuX Tsj j QmuQ PifPk

S O1voj kP}PTVSUk}ifP Q w}Q!i!kgp1j|Q!S ECvqvqSUQmuXSl1Q!i6dTfPqFkxORTVStQ!if1SGkVOPQ QTVqQ&f1Sl1Q kjlimF&vqPQ TVSi/rFfRTskgQ~PTswgw}Qj|Q&fP Q

#vq1Q8TskxO+i!p1kvt+i!kgTs:t2p1mskgTuCvomsQ rPmskgQjlTfP?NPORTVSi!p1kvt+i!kgTsrPmskQjTf1p1j Q!kVOPQ)fFpPtQjTs&i!mRTfPSUk}i#WRTsmsTskgTsQ!S

1j|Q!SUQ&fPk-Tf8tXJt+voj|QG&vfRQ&fRkTsvf1i!m-rFfRTskgQ~PTswgw}Qj|Q&fP Q#vq1Q<

NROPQ=Tut1msQtQ&fPkikgTVvfifP2pCSUQvowCkVOPQ)QTsdQ&fPSl1Q#kgjUi!m-&vqPQ iS`imutvqSUk1Q&fPkgTsj QmX8&i!jlj TsQ!(vp1kFWFX1Q&f8

4VitTf Cj vtmuQX if1 yzTsO1i!j DhPjTs Q<2NPORQt+vqSUk`TutCvoj k}ifPk`QmsQtQ&fRk}S vowO1i1kgQj 2w~voj=kxORTVS5kxORQ!SUTVS ij|Q

kVOPQj|Qw~voj|QGkxOPvqSUQ=kVO1i!kFj|QmVi!kQGkv5&vt2Cij TVSvf TskVOtX$rFfPTskgQ~PTswgw}Qj|Q&fP Q#vq1QCi!jlkgTs p1msij muX$kVOPQ#vfPrPj|t+i/

kgTVvf2vowR QjlkiTf8j Q!Slp1mskSw}jvt O1i1kQjCTf8Noi#WPmuQ >ePi!mVSv5&i!jljTsQ!vp1kRSvtQ=if1i!muX-kTs#i!m (voj/jQmuQ PifPk

w~voj`OPi!1kgQj PSlp1O iS TfCijlimsmuQmTskVO vokVOPQjUSlRw~voj jvqSSS}OPQ-Tf1d#imsp1mVi!kTf18kxORQ &voQwgrP#TuQ&fPkS`fPQ

Q!SSi!j X kv Q:1jQ!SS5kVOPQ(ORQmsTs#imumuX1}j|Q!Pp1 Q! (i FQ(Q!{qpCi!kgTVvf0Tf0vp1j/i1i!1kgQ! #vqvoj 1Tf1i!kQ!S NROPQkgQ:kvow

O1i1kQjU (iS jTskgkQ&f+WFX@yzTsOPij hPj Ts#QMif1BPpWRmsTsS OPQ! iS x xMvow OPTsO e (iS)i#v~ip1kVO1vojJe}kTVS

j|Q1jlTfRkQ! FQjWPikgTut

iSO1i1kQj-7oTfR#Q$kxORQ$PpWRmsTs#ikgTVvfDvow`kxORQ (voj@1j|Q!SUQ&fPkgQ! Tf OPi!1kgQjUS=2if1 kVOPQ/OPQmsTs#i!msmuX1}j|Q!Rp1#Q! SU#i!mVij

(i RQ1j v!WRmsQt O1iSWPQQ&f Tf FQ!SUkgTVdikgQ! wp1jlkVOPQjWFX vokVOPQjlS RvqS ORTV1i2Qk`im* "> &vt2Cij|Q! vp1j)rFfRTskgQ

PTswgw}Qj|Q&fP Q$if1 QTVqQ&fPSl1Q#kgjUi!m#vqPQ!S TskVODvokVOPQjUS5j|Q#Q&fRkmX 1Q FQmVv1Q! i!k Tsm(ip'dQQSlXSUkgQt+i!kTu&i!msmuX

TfRQ!SUkgTVdikgTfP if Q:CifPPQ! mViSSJvowf1vfRmsTfRQ!ij TskTuQ!SlzWRQkgkQjj|Q1j|Q!SUQ&fPkgTf1 kxORQfPvfPmsTfRQ!ij kgQj tSJkvWRQ

Q:1Q kgQ!@Tf qQ&fRQjUi!mPjQmVi!kgTRTukgX ECip if1@hPjlTs Q o PQ RQmVv1Q! i/t2p1mskgTV1v�

Topics in numerical relativity : the periodic standing-wave approximation, the stability of constraints in free evolution, and the spin of dynamical black holes

NPORQ!SUTVSCWFX

 NROPQ

0

0

B NROPQ

χmin/a

0

χmin/a

0

0

0

0

Nr, Lmax}

0

0

= const

0

0

0

γQeff

0

0

) = Source

NROPQ