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Qtv;2M M/ +`#QM +QMi`B#miBQMbc i?2 `2KBMBM; Kbb 7`+iBQM- b /2`Bp2/ BM i?2 T`2b2Mi@/v bQH` T?QiQbT?2`2- Bb ;Bp2M #v i?2 ?v/`Q;2M UX = 0.7381V M/ ?2HBmK UY = 0.2485V +QMi2MibX PM i?2 Qi?2` ?M/- "B; "M; Mm+H2QbvMi?2bBb T`2/B+ib i?2 2`Hv +QbKB+ THbK iQ #2 2bb2MiBHHv 7`22 Q7 K2iHb- r?B+? ?2M+2 Kmbi ?p2 #22M bvMi?2bBx2/ Hi2` b i?2 }`bi TQTmHiBQM Q7 bi`b 2p2MimHHv TT2`2/X h?2`27Q`2-i?2 i2``2bi`BH M/ bQH` #mM/M+2 Q7 +?2KB+H 2H2K2Mib HBF2 +`#QM- MBi`Q;2M-Qtv;2M Q` bBHB+QM Ĝ r?B+? `2 7mM/K2MiH 7Q` HB72 QM 1`i? M/ T`2/QKBMi2 BM i?2 iKQbT?2`2 M/ +`mbi Q7 Qm` THM2i Ĝ +M #2 2tTHBM2/ QMHv #v BMpQFBM; b2[m2M+2 Q7 bi2HH` ;2M2`iBQMb r?B+? BM+`2bBM;Hv TQHHmi2/ i?2 BMi2`bi2HH` K2/BmK 7`QK r?B+?- #2ir22M 4 M/ 5 :v` ;Q- i?2 amM M/ 1`i? Q`B;BMi2/X
ai2HH` BMi2`BQ`b `2T`2b2Mi i?2 #2bi TH+2 BM i?2 lMBp2`b2 r?2`2 iQ }M/ i?2 `B;?i T?vbB+H +QM/BiBQMb 7Q` 7mbBQM `2+iBQMb iQ Q++m`c BM T`BM+BTH2- #v K2Mb Q7 i?Bb FBM/ Q7 K2+?MBbK UBX2X ?v/`QbiiB+ Q` 2tTHQbBp2 i?2`KQMm+H2` #m`MBM; rBi?BM bi`bV-QM2 +M 2tTHBM i?2 Q`B;BM Q7 HH i?2 +?2KB+H 2H2K2Mib 7`QK +`#QM iQ B`QMX h?2 HB;?i2` +?2KB+H 2H2K2Mib- BM T`iB+mH` ?v/`Q;2M M/ ?2HBmK- r2`2 HKQbi 2MiB`2Hv bvMi?2bBx2/ /m`BM; i?2 T`BKQ`/BH Mm+H2QbvMi?2bBb M/ Hi2` /2bi`Qv2/ BMbB/2 bi`b Ĝ bi2HH` ;2M2`iBQM #v bi2HH` ;2M2`iBQM Ĝ 7Q` i?2 T`Q/m+iBQM Q7 i?2 ?2pB2` +?2KB+H 2H2K2MibX *?2KB+H 2H2K2Mib rBi? iQKB+ r2B;?i A > 56 +MMQi #2 bvMi?2bBx2/ pB i?2`KQMm+H2` #m`MBM;- bBM+2 i?Bb T`Q+2bb Bb MQi +QMp2MB2Mi 7Q` i?2B` T`Q/m+iBQM BM i2`Kb Q7 2M2`;v #HM+2X AM T`iB+mH`- 7Q` A > 56- i?2 #BM/BM; 2M2`;v T2` Mm+H2QM HKQbi +QMiBMmQmbHv /BKBMBb?2b M/ i?2 Mm+H2` `2+iBQMb rQmH/ `2[mB`2 H`;2 2M2`;v +QMbmKTiBQM i i?2 2tT2Mb2 Q7 i?2 2MpB`QMK2Mi iQ Q++m`X h?2`27Q`2-M Qi?2` T?vbB+H T`Q+2bb Kmbi #2 BMpQF2/ 7Q` i?2 T`Q/m+iBQM Q7 i?2 ?2pv 2H2K2Mib r?B+? HB2 BM i?2 T2`BQ/B+ i#H2 #2vQM/ B`QMX
h?2 KBM T?vbB+H K2+?MBbK 7Q` i?2 Mm+H2QbvMi?2bBb Q7 i?2 +?2KB+H 2H2K2Mib ?2pB2` i?M B`QM Bb ;Bp2M #v i?2 bQ@+HH2/ dzM2mi`QM@+Tim`2 T`Q+2bbǴ BMbB/2 bi`b-MK2Hv #v b2`B2b Q7 M2mi`QM@+Tim`2b QMiQ M BM+`2bBM;Hv ?2pB2` iQKB+ Mm+H2mb-7QHHQr2/ #v β− /2+vb Q7 i?2 +Tim`2/ M2mi`QMbX A7 i?2 M2mi`QM@+Tim`2b T`Q+22/
rBi? M ?B;?2` `i2 rBi? `2bT2+i iQ i?2 β− /2+vb U+QM/BiBQM Q7 ?B;? M2mi`QM
~mt2bV- i?2M i?2 `2+iBQMb ;Bp2 `Bb2 iQ M `@T`Q+2bb 2H2K2Mi- r?2`2 i?2 H2ii2` dz`Ǵ biM/b 7Q` dz`TB/Ƕc Qi?2`rBb2- i?2 `2+iBQMb ;Bp2 `Bb2 iQ M b@T`Q+2bb 2H2K2Mi- r?2`2 dzbǴ Q#pBQmbHv biM/b 7Q` dzbHQrǴX 1+? M2mi`QM@+Tim`2 2H2K2Mi +M ?p2 #Qi? M b@ M/ `@T`Q+2bb +?MM2H Q7 Mm+H2QbvMi?2bBbc 2m`QTBmK Bb T`2/QKBMMiHv T`Q/m+2/ b M `@T`Q+2bb 2H2K2Mi- r?BH2 #`BmK Bb KQbiHv M b@T`Q+2bb 2H2K2MiX
+QKTmi2/ Qp2` /Bb+`2i2 i?`22@/BK2MbBQMH ;`B/ Q7 /Bz2`2Mi K2iHHB+BiB2b- 2z2+iBp2 i2KT2`im`2b- M/ bm`7+2 ;`pBiB2bX L2p2`i?2H2bb- QM2 +M 2bBHv Tbb 7`QK i?2 (m, g, T2z) bT+2 Q7 i?2 bi2HH` bT2+i` iQ i?2 KQ`2 i?2Q`2iB+H (m, Z, τ) bT+2 #v
mbBM; bi2HH` 2pQHmiBQM`v i`+Fb Q` BbQ+?`QM2bX AM+B/2MiHHv- 7m`i?2` T`K2i2` r?B+? Bb mbmHHv iF2M BMiQ ++QmMi iQ +QKTmi2 2p2M KQ`2 T`2+Bb2 bi2HH` bT2+i` Bb ;Bp2M #v i?2 [α/62] #mM/M+2 `iBQ BM i?2 bi`X
"v bi`iBM; 7`QK i?2 /Bbi`B#miBQM Q7 i?2 7mM/K2MiH bi2HH` T`K2i2`b BM ;HtB2b- r2 +M BMi`Q/m+2 i?`22 7mM/K2MiH BM;`2/B2Mib 7Q` ;Htv 7Q`KiBQM M/ 2pQHmiBQM KQ/2Hb- b 7QHHQrbX
Ç h?2 /Bbi`B#miBQM Q7 i?2 bi2HH` ;2b `2~2+ib i?2 ;Htv bi` 7Q`KiBQM ?BbiQ`vX Ç h?2 /Bbi`B#miBQM Q7 i?2 bi2HH` Kbb2b Bb /2i2`KBM2/ #v i?2 bQ@+HH2/ BMBiBH
Kbb 7mM+iBQMX
Ç h?2 /Bbi`B#miBQM Q7 i?2 bi2HH` K2iHHB+BiB2b Bb /B`2+i +QMb2[m2M+2 Q7 i?2
;Htv +?2KB+H 2pQHmiBQMX
h?2 i?`22 7Q`2K2MiBQM2/ BM;`2/B2Mib /Q MQi HB2 i i?2 bK2 H2p2H BM i?2 KQ/2Hbc BM T`iB+mH`- i?2 +?2KB+H 2pQHmiBQM +M #2 +QKTmi2/ QMHv B7 r2 FMQr i?2 ;Htv bi` 7Q`KiBQM ?BbiQ`v M/ BMBiBH Kbb 7mM+iBQMX AM 7+i- iQ +QKTmi2 i?2 ;Htv +?2KB+H 2pQHmiBQM- r2 ?p2 iQ `2+QMbi`m+i i?2 #B`i? iBK2 M/ i?2 MmK#2` Q7 HH i?2 bi`b r?B+? `2 /vBM; i 2+? iBK2 Q7 i?2 ;Htv 2pQHmiBQM, i?2 iBK2 Q7 7Q`KiBQM Q7 i?2 /vBM; bi`b +QK2b 7`QK i?2 Tbi ;Htv bi` 7Q`KiBQM ?BbiQ`v- r?BH2 i?2B` MmK#2` +QK2b 7`QK i?2 bbmK2/ BMBiBH Kbb 7mM+iBQMX h?Bb Bb /m2 iQ i?2 7+i i?i Ĝ b i?2 bi`b /B2 Ĝ i?2v 2tT2H K2iHb BMiQ i?2 ;Htv BMi2`bi2HH` K2/BmK M/ ?2M+2 +`m+BHHv +QMi`B#mi2 iQ i?2 ;Htv +?2KB+H 2pQHmiBQMX *?2KB+H 2pQHmiBQM KQ/2Hb 2Mi2` BM i?2 #`Q/ `2 Q7 bim/B2b- r?B+? Bb +HH2/ BM i?2 HBi2`im`2 b ;Htv bi`Q@
`+?2QHQ;vc `Qm;?Hv bT2FBM;- r2 bi`m;;H2 iQ `2+QMbi`m+i i?2 Tbi 2pQHmiBQM`v
?BbiQ`v Q7 ;HtB2b- bi`iBM; 7`QK i?2B` Q#b2`p2/ T?vbB+H M/ +?2KB+H T`QT2`iB2b i i?2 T`2b2Mi iBK2X
q2 `2K`F QM i?2 7+i i?i MmK2`B+H +Q/2b Q7 +?2KB+H 2pQHmiBQM +M T`QpB/2 Ĝ BM T`BM+BTH2 Ĝ +QKTH2i2 T?vbB+H /2b+`BTiBQM Q7 i?2 ;Htv 2pQHmiBQMc #v QMHv bi`iBM; 7`QK i?2 bbmKTiBQM Q7 ;Htv ;b Kbb bb2K#Hv ?BbiQ`v- i?2v `2 +T#H2 Q7 T`2/B+iBM; i?2 `mM Q7 i?2 ;Htv bi2HH` M/ ;b Kbb2b rBi? iBK2- b r2HH b i?2 ;Htv bi` 7Q`KiBQM ?BbiQ`v- ?2M+2 MQi QMHv i?2 2pQHmiBQM Q7 i?2 ;Htv +?2KB+H #mM/M+2b rBi? iBK2X
h?2 bi` 7Q`KiBQM ?BbiQ`v Bb `2;mHi2/ #v HH i?2 T?vbB+H T`Q+2bb2b r?B+? Hi2` i?2 ;b +QMi2Mi rBi?BM ;HtB2b- M/ +?2KB+H 2pQHmiBQM Bib2H7 Kmbi #2 BM+Hm/2/ BM i?2b2 T`Q+2bb2bX b i?2 bi` 7Q`KiBQM T`Q+22/b- i?2 ;b Bb +QMiBMmQmbHv HQ+F2/ mT BMiQ bi`bX h?2M- b i?2 bi`b /B2- T`i Q7 i?2 dzHQ+F2/Ǵ ;b Bb `2@BMD2+i2/ BMiQ i?2 ;Htv BMi2`bi2HH` K2/BmKX 6`QK i?2 2tT2HH2/ ;b #v /vBM; bi`b- 7m`i?2` bi`b BM i?2 7mim`2 rBHH 7Q`K- +mbBM; +QMiBMmQmb `2+v+HBM; Q7 i?2 ;Htv BMi2`bi2HH` K2/BmKX am+? `2+v+HBM; ?b M BM~m2M+2 QM i?2 ;Htv bi` 7Q`KiBQM ?BbiQ`v-r?B+? Kmbi #2 iF2M BMiQ ++QmMiX AM bmKK`v- i?2 +?2KB+H 2pQHmiBQM Bb bi`B+iHv +QmTH2/ rBi? i?2 ;Htv bi` 7Q`KiBQM ?BbiQ`v, QM2 +MMQi 7mHHv mM/2`biM/ i?2 7Q`K2` rBi?Qmi T`2+Bb2Hv FMQrBM; i?2 Hii2`- M/ pB+2p2`bX
RXkXj aBKTH2 M/ +QKTQbBi2 bi2HH` TQTmHiBQMb
G2i mb bbmK2 i?i ;Htv G 2tT2`B2M+2/ bi` 7Q`KiBQM +iBpBiv Qp2` }MBi2 T2`BQ/ Q7 iBK2 ∆ta6 = [t1, t2] BM i?2 TbiX i 2p2`v b?Q`i iBK2 BMi2`pH ∆taaS '
∆ta6 /m`BM; i?2 bi` 7Q`KiBQM +iBpBiv- bBKTH2 bi2HH` TQTmHiBQM Bb bbmK2/ iQ Q`B;BMi2- MK2Hv TQTmHiBQM Q7 bi`b r?B+? r2`2 #Q`M /m`BM; #m`bi Q7 bi` 7Q`KiBQM UMK2Hv b `2bTQMb2 iQ .B`+ /2Hi 7mM+iBQM BM i?2 bi` 7Q`KiBQM +iBpBivV- HH +?`+i2`Bx2/ #v i?2 bK2 ;2 M/ K2iHHB+BivX h?2 #2bi 2tKTH2b Q7 bBKTH2 bi2HH` TQTmHiBQMb BM i?2 lMBp2`b2 `2 i?2 QH/2bi ;HQ#mH` +Hmbi2`b BM i?2 :Htv ?HQ- r?B+? ?Qbi bi`b rBi? M2`Hv +QMbiMi ;2 M/ K2iHHB+Biv- HBF2Hv Q`B;BMiBM; 7`QK bBM;H2 #m`biv 2TBbQ/2 Q7 bi` 7Q`KiBQMX 6`QK i?2Q`2iB+H TQBMi Q7 pB2r- bBKTH2 bi2HH` TQTmHiBQMb `2 `2T`2b2Mi2/ #v i?2 bi2HH` BbQ+?`QM2b BM i?2 +QHQ`@K;MBim/2 /B;`KX
q2 +M /2}M2 ;Htv b HBM2` +QK#BMiBQM Q7 bBKTH2 bi2HH` TQTmHiBQMb rBi? +QMiBMmQmbHv BM+`2bBM; ;2b M/ /Bz2`2Mi K2iHHB+BiB2bX 6Q` i?Bb `2bQM- ;Ht@ B2b `2 /2}M2/ b +QKTQbBi2 bi2HH` TQTmHiBQMbX "v p`vBM; i?2 r2B;?i Q7 Mv ;Bp2M bBKTH2 bi2HH` TQTmHiBQM BM i?2 HBM2` +QK#BMiBQM- i?2 ;Htv +M #2 bB;MB}+MiHv /Bz2`2Mi- i Mv iBK2 Q7 Bib 2pQHmiBQMX am+? r2B;?ib- waaS- BM i?2 HBM2` +QK#BM@
iBQM /2}M2 BM mMB[m2 rv #Qi? i?2 ;Htv bi` 7Q`KiBQM ?BbiQ`v Ĝ bBM+2 r2 +M `2+QMbi`m+i ?Qr KMv bi`b 7Q`K b 7mM+iBQM Q7 iBK2 UM/ ?2M+2 i?2 T`2b2Mi@/v /Bbi`B#miBQM Q7 i?2 bi2HH` ;2b- B7 r2 FMQr i?2 bi2HH` HB72iBK2bV Ĝ M/ i?2 K2iH@ HB+Biv 2pQHmiBQM Q7 i?2 ;Htv BMi2`bi2HH` K2/BmK Ĝ r?B+? Bb `2T`2b2Mi2/ #v i?2 K2iHHB+Biv Q7 i?2 bi`b i i?2B` #B`i?X
AM bmKK`v- 7`QK i?2Q`2iB+H TQBMi Q7 pB2r- Mv ;Bp2M ;Htv T?vbB+H T`QT@ 2`iv- ˆOG- +M #2 2tT`2bb2/ BM i?2 7QHHQrBM; rv, ˆ OG(t) = % ti<t
waaS(ti) ˆOaaS&t − ti, Z(ti)', URX9V
r?2`2 i?2 bmK Bb K/2 Qp2` HH i?2 iBK2b tiQ7 bi` 7Q`KiBQM- r?B+? `2 T`BQ` iQ i?2
+m``2Mi QM2c waaS(ti)`2T`2b2Mib i?2 r2B;?i bbQ+Bi2/ iQ i?2 bi2HH` TQTmHiBQM #Q`M
i tic }MHHv- ˆOaaS(τ, Z) Bb i?2 ;Bp2M T?vbB+H T`QT2`iv- b T`2/B+i2/ 7Q` bBKTH2
bi2HH` TQTmHiBQM rBi? ;2 τ M/ K2iHHB+Biv Z = Z(τ)X q2 `2KBM/ i?2 `2/2` i?i 2[miBQM URX9V /2b+`B#2b mM`2bQHp2/ ;HtB2b- r?2`2 QMHv i?2 +mKmHiBp2 2z2+i Q7 i?2 bi`b HBp2 i i?2 T`2b2Mi iBK2 +M #2 bim/B2/X
AM 2[miBQM URX9V- i?2 Kbb Q7 i?2 bi`b Ur?B+? Bb i?2 i?B`/ 7mM/K2MiH bi2HH` T`K2i2` #2vQM/ i?2 bi2HH` ;2 M/ K2iHHB+BivV Bb ?B//2M BM i?2 +QKTmiiBQM Q7 i?2 [mMiBiv ˆOaaS(τ, Z)X o2`v ;2M2`HHv- ˆOaaS +M #2 +QKTmi2/ b 7QHHQrb,
ˆ
OaaS(τ, Z) =% mk
f⋆(mk, τ, Z) ˆO⋆(mk, τ, Z), URX8V
r?2`2 i?2 [mMiBiB2b f⋆(mk, τ, Z) T?vbB+HHv `2T`2b2Mib i?2 iQiH MmK#2` Q7 bi`b
BM i?2 bBKTH2 bi2HH` TQTmHiBQM rBi? Kbb mk- ;2 τ M/ K2iHHB+Biv Z- rBi?
ˆ
O⋆(mk, τ, Z) #2BM; i?2B` +Q``2bTQM/BM; T`2/B+i2/ T?vbB+H T`QT2`iv mM/2` 2tKB@
MiBQM- b ;Bp2M Ĝ 7Q` 2tKTH2 Ĝ #v bi2HH` BbQ+?`QM2bX
RXkX "bB+ BM;`2/B2Mib
`2T`2b2Mib i?2 +QMi`B#miBQM Q7 HH i?2 bi`b rBi? Kbb m- K2iHHB+Biv Z M/ ;2 τ iQ i?2 [mMiBiv ˆOaaS(τ, Z)Ĝ Bb bbmK2/ iQ /2T2M/ QMHv mTQM i?2 Kbb- m- Q7 i?2 bi`X
h?Bb Bb p2`v bi`QM; bbmKTiBQMX AM T`+iB+2- Bi K2Mb i?i i?2 Kbb /Bbi`B#miBQM 7mM+iBQM Q7 i?2 bi`b Q7 bBKTH2 bi2HH` TQTmHiBQM Bb mMBp2`bH 7mM+iBQM M/ ?2M+2 Bi /Q2b MQi /2T2M/ mTQM i?2 TH+2 Q` iBK2 Q7 #B`i? Q7 i?2 bi`b Q7 i?2 bBKTH2 bi2HH` TQTmHiBQM X
"v BMb2`iBM; 2[miBQM URX8V BMiQ 2[miBQM URX9V M/ #v bbmKBM; i?i f⋆ =
f⋆(m)- r2 +M 2p2MimHHv }M/ i?2 7QHHQrBM; 2[miBQM 7Q` i?2 ;2M2`B+ ;Htv T?vbB+H
T`QT2`iv- ˆOG, ˆ OG(t) = % ti<t % mk waaS(ti) f⋆(mk) ˆO⋆ & mk, t − ti, Z(ti)'. URXeV
"v HQQFBM; i 2[miBQM URXeV- i?2 [mMiBiv f⋆(mk)T?vbB+HHv `2T`2b2Mib i?2 7`+iBQM
Q7 waaS(ti)+QMi`B#mi2/ #v i?2 bi`b rBi? Kbb mkX
G2i mb bbmK2 i?i ˆOG BM 2[miBQM URXeV Bb i?2 iQiH ;Htv bi2HH` Kbb- M⋆X
>BbiQ`B+HHv- i?Bb T`iB+mH` +b2 Bb mb2/ iQ /2}M2 i?2 MQ`KHBxiBQM Q7 i?2 [mMiBiv f⋆ M/ i?2 T?vbB+H /BK2MbBQMb Q7 waaSX AM T`iB+mH`- #v ++Q`/BM;Hv bbmKBM;
ˆ O⋆(mk, τ, Z) = mk- r2 ?p2 i?2 7QHHQrBM; 2[miBQM, M⋆(t) = ( % ti<t waaS(ti) ) × ( % mk f⋆(mk) mk ) . URXdV
AM Q`/2` iQ ?p2 T?vbB+HHv +QMbBbi2Mi `2bmHib r2 ?p2 iQ bbmK2 r?i 7QHHQrbX Ç waaS(ti) T?vbB+HHv `2T`2b2Mib i?2 iQiH Kbb Q7 bi`b r?B+? r2`2 #Q`M i ti
-MK2Hv i?2 iQiH Kbb Q7 i?2 bBKTH2 bi2HH` TQTmHiBQM rBi? ;2 τ = t − ti
M/ K2iHHB+Biv Z(ti)X h?2`27Q`2- Bi ?b i?2 /BK2MbBQMb Q7 Kbb U?2M+2 i?2
mMBib Q7 J!VX
Ç h?2 [mMiBiv f⋆(mk) Bb /B`2+iHv T`QTQ`iBQMH iQ i?2 MmK#2` Q7 bi`b r?B+?
r2`2 #Q`M rBi? Kbb mk BM bBKTH2 bi2HH` TQTmHiBQMX Ai Bb MQ`KHBx2/ bm+?
i?i *mkmkf⋆(mk) = 1X
b r2 rBHH b22 BM i?2 7QHHQrBM; /Bb+mbbBQM- i?2 [mMiBiB2b waaS M/ f⋆ `2 /B`2+iHv
`2Hi2/ iQ i?2 bi` 7Q`KiBQM `i2 M/ BMBiBH Kbb 7mM+iBQM- `2bT2+iBp2Hvc i?2B` T`Q/m+i BM 2[miBQM URXeV Bb HbQ bi`B+iHv `2Hi2/ iQ i?2 bQ@+HH2/ dzbi2HH` #B`i?`i2 7mM+iBQMǴ- r?B+? r2 rBHH BMi`Q/m+2 BM r?i 7QHHQrbX
7m`i?2` Ĝ p2`v bm#iH2 Ĝ bbmKTiBQM Bb K/2- r?B+? Bb MQi `2+Q;MBx2/ BM i?2 HBi2`im`2, f⋆ b?QmH/ /2T2M/ Ĝ BM T`BM+BTH2 Ĝ HbQ QM i?2 Q#b2`p#H2 ˆO⋆- #mi r2 Hrvb M2;H2+i i?Bb
2z2+iX 6Q` 2tKTH2- i?Bb K2Mb i?i r2 bm#iHv bbmK2 i?i i?2 bi`b BM ;Bp2M Kbb `M;2 m1≤ m ≤ m2 +QMi`B#mi2 iQ Mv T?vbB+H [mMiBiv Q7 i?2B` bBKTH2 bi2HH` TQTmHiBQM rBi?
Hrvb i?2 bK2 KQmMi UBi +QmH/ #2 i?2 "@#M/ HmKBMQbBiv- i?2 A@#M/ QM2- Q` i?2 iQiH bi2HH` Kbb Bib2H7VX h?Bb Bb MQi Q#pBQmbX
h?2 bi2HH` #B`i?`i2 7mM+iBQM
h?2 MmK#2` Q7 bi`b r?B+? r2`2 #Q`M BM i?2 iBK2 BMi2`pH t, t + dt M/ rBi? Kbb BM i?2 `M;2 m, m + dm `2T`2b2Mib- #v /2}MBiBQM- i?2 bQ@+HH2/ bi2HH` #B`i?`i2
7mM+iBQM- B(t, m)X h?Bb [mMiBiv Bb bbmK2/ iQ #2 i?2 T`Q/m+i Q7 i?2 BMBiBH Kbb
7mM+iBQM- φ(m)- rBi? i?2 ;Htv bi` 7Q`KiBQM `i2- ψ(t)- ++Q`/BM; iQ i?2 7QHHQrBM; 7Q`KmH,
B(t, m) = ψ(t) φ(m) dt dm. URX3V h?2`2 Bb MQ T?vbB+H `2bQM 7Q` bm+? 7+iQ`BxiBQM BM i?2 iBK2 M/ Kbb /2T2M@ /2M+2 Q7 i?2 bi2HH` #B`i?`i2 7+iBQM- 2t+2Ti Qm` B;MQ`M+2 #Qmi i?2 bi` 7Q`KiBQM T`Q+2bb BM ;HtB2bX h?2 T?vbB+H /2}MBiBQMb Q7 bi` 7Q`KiBQM `i2 M/ BMBiBH Kbb 7mM+iBQM `2 i?2 7QHHQrBM; QM2bX
Ai Bb /2}M2/ b i?2 KQmMi Q7 Kbb rBi?BM i?2 ;Htv BMi2`bi2HH` K2/BmK r?B+? Bb +QMp2`i2/ BMiQ bi`b T2` mMBi iBK2X Ai Bb mbm@ HHv 2tT`2bb2/ BM i2`Kb Q7 J!v`−1X ai` 7Q`KiBQM `i2 Bb Q#b2`p2/ iQ +Q`@
`2Hi2 BM ;HtB2b rBi? KMv 7mM/K2MiH T?vbB+H [mMiBiB2b- bm+? b i?2 iQiH ;Htv bi2HH` Kbb M/ i?2 p2`;2 K2iHHB+Biv Q7 i?2 ;Htv BMi2`bi2H@ H` K2/BmKX L2p2`i?2H2bb- i?2 #2bi 2tKTH2 Q7 +Q``2HiBQM Bb i?2 bQ@+HH2/
a+?KB/i@E2MMB+mii Hr Ua+?KB/i-RN8NcE2MMB+mii-RNN3V #2ir22M i?2 /2MbBiv Q7 bi` 7Q`KiBQM `i2 M/ i?2 ;Htv ;b Kbb /2MbBivX >BbiQ`B+HHv- #b2/ QM i?2 bm;;2biBQMb Q7 i?2 2`Hv rQ`F #v a+?KB/i URN8NV- ;Htv 7Q`KiBQM M/ 2pQHmiBQM KQ/2Hb ?p2 Hrvb bbmK2/ i?2 7QHHQrBM; 2KTB`B+H Hr 7Q` KQ/@ 2HBM; i?2 bi` 7Q`KiBQM T`Q+2bb, ψ(r, t) ∝ ν ρ;(r, t)k- r?2`2 ρ;`2T`2b2Mib i?2
pQHmK2 UQ` bm`7+2V ;b Kbb /2MbBiv Q7 i?2 ;Htv BMi2`bi2HH` K2/BmK- i?2 BM/2t k HB2b BM i?2 BMi2`pH [1, 2]- M/ ν `2T`2b2Mib i?2 bQ@+HH2/ bi` 7Q`KiBQM
2{+B2M+v- r?Qb2 /BK2MbBQMb `2 i?2 BMp2`b2 Q7 iBK2X b
7Q`2K2MiBQM2/-i?Bb Hr rb Q`B;BMHHv T`QTQb2/ #va+?KB/i URN8NV iQ }i Q#b2`piBQMH /iX Gi2`- E2MMB+mii URNN3V /2`Bp2/ k = 1.4 ± 0.15 7Q` bi`@7Q`KBM; bTB`H M/ bi`#m`bi ;HtB2b BM i?2 GQ+H lMBp2`b2X Pi?2` T`K2i`BxiBQMb Q7 i?2 bi` 7Q`KiBQM Hr ?p2 #22M 2tTHQ`2/ BM i?2 HBi2`im`2c i?2v KF2 mb2 Ĝ 7Q` 2t@ KTH2 Ĝ Q7 i?2 /vMKB+H iBK2b+H2 Q` i?2 M;mH` `QiiBQM bT22/ U"QBbbB2` -kyRjVX
Ai `2T`2b2Mib i?2 Kbb bT2+i`mK rBi? r?B+? bi`b /Bbi`B#mi2 BM Kbb i i?2B` #B`i?X h?2 BMBiBH Kbb 7mM+iBQM Bb mbmHHv MQ`KH@ Bx2/ BM Kbb bm+? i?i+100J!
0.1J! dm m φ(m)X JQbi Q7 i?2 BMBiBH Kbb 7mM+iBQMb
r?B+? ?p2 #22M T`QTQb2/ BM i?2 HBi2`im`2 iQ }i aQH` M2B;?#Q`?QQ/ /i `2 /2}M2/ b bBM;H2@ Q` KmHiB@bHQT2 TQr2` Hr Q7 i?2 7Q`K, φ(m) ∝ m−(1+x)X
h?2 KQbi 7KQmb BMBiBH Kbb 7mM+iBQM Bb i?2 QM2 #v aHT2i2` URN88V- r?B+? bbmK2b x = 1.35X h?2 bbmKTiBQM Q7 /Bz2`2Mi BMBiBH Kbb 7mM+iBQMb +M bi`QM;Hv z2+i i?2 +?2KB+H 2pQHmiBQM Q7 ;HtB2b- bBM+2 i?Bb [mMiBiv /2i2`@ KBM2b i?2 `2HiBp2 MmK#2`b Q7 bi`b r?B+? HB2 BM /Bz2`2Mi Kbb `M;2b M/ ?2M+2 i?2 KQmMi Q7 i?2 `2biBimiBQM Q7 +?2KB+H 2H2K2Mib #v /vBM; bi`bX h?2 BMBiBH Kbb 7mM+iBQM Bb Q#b2`p2/ iQ #2 bi`QM;Hv #Bb2/ iQr`/b HQr@Kbb bi`b- bBM+2 i?2v ?p2 i?2 H`;2bi HB72iBK2b M/ ?2M+2 i?2 H`;2bi T`Q##BHBiv iQ #2 Q#b2`p2/ i i?2 T`2b2Mi iBK2X
RXkX "bB+ BM;`2/B2Mib 6mM/K2MiH ;Htv BMi2;`i2/ T`QT2`iB2b
"v bi`iBM; 7`QK i?2 ;Htv bi` 7Q`KiBQM ?BbiQ`v M/ bbmKBM; T`iB+mH` BMB@ iBH Kbb 7mM+iBQM- QM2 +M +QKTmi2 i?2 i2KTQ`H 2pQHmiBQM Q7 KMv 7mM/K2MiH ;Htv T`QT2`iB2bX AM r?i 7QHHQrb- bQK2 2tKTH2b `2 ;Bp2M iQ BHHmbi`i2 i?BbX
h?2 iQiH ;Htv bi2HH` KbbX "v bbmKBM; i?i mKt(t′, t)`2T`2b2Mib i?2 Kbb
Q7 bi` ?pBM; HB72iBK2 τ = |t − t′| UHH i?2 bi`b #Q`M i t′- ?pBM; Kbb H`;2`
i?M mKt- `2 H`2/v /2/ i tV- i?2 `mM Q7 i?2 iQiH ;Htv bi2HH` Kbb rBi?
iBK2 +M #2 +QKTmi2/ #v K2Mb Q7 i?2 7QHHQrBM; 2[miBQM, M⋆(t) = t , 0 dt′ψ(t′) mKt(t′ ,t) , 0.1J! dm φ(m) m, URXNV
r?2`2 i?2 BMi2;`H Qp2` Kbb `2T`2b2Mib i?2 Kbb 7`+iBQM Q7 i?2 bi`b r?B+? 7Q`K2/ i i?2 iBK2 t′ M/ +M #2 biBHH Q#b2`p2/ b HBp2 i i?2 iBK2 t- MK2Hv HH i?2 bi`b
rBi? Kbb m ≤ mKt(t′, t)- #Q`M i i?2 iBK2 t′- +M #2 biBHH Q#b2`p2/ b HBp2 i i?2
iBK2 tX h?2 7mM+iBQMH #2?pBQ` Q7 mKt Bb bBKTHv ;Bp2M #v i?2 BMp2`b2 7mM+iBQM Q7
i?2 bi2HH` HB72iBK2bX
h?2 ;Htv BMi2;`i2/ bT2+i`mKX aBKBH`Hv- i?2 BMi2;`i2/ ;Htv bT2+i`mK i
i?2 iBK2 t +M #2 +QKTmi2/ b 7QHHQrbX Fλ(t) = t , 0 dt′ ψ(t′ ) mKt(t′ ,t) , 0.1J! dm φ(m) fλ & m, t′ , Z(t′ )', URXRyV r?2`2 fλ &
m, t′, Z(t′)' Bb i?2 bT2+i`mK Q7 bi` rBi? Kbb m- ;2 |t−t′|- M/ BMBiBH
K2iHHB+Biv Z(t′)X AM 2[miBQM URXRyV- i?2 BMi2;`i2/ ;Htv bT2+i`mK Bb +QKTmi2/
#v bmKKBM; mT QMHv i?2 +QMi`B#miBQM Q7 i?2 bi`b r?B+? +M #2 biBHH Q#b2`p2/ b HBp2 i i?2 iBK2 tX
h?2 `i2 Q7 /2i? Q7 bi`bX 7mM/K2MiH [mMiBiv 7Q` +?2KB+H 2pQHmiBQM KQ/2Hb
Bb i?2 `i2 Q7 /2i? Q7 i?2 bi`b b 7mM+iBQM Q7 i?2 ;Htv HB72iBK2X h?Bb [mMiBiv +M #2 +QKTmi2/ B7 r2 FMQr i?2 #2?pBQ` Q7 i?2 ;Htv bi` 7Q`KiBQM `i2 b 7mM+iBQM Q7 iBK2- i?2 BMBiBH Kbb 7mM+iBQM M/ i?2 bi2HH` HB72iBK2bX AM T`iB+mH`-i?2 `i2 Q7 /2i? Q7 T`iB+mH`-i?2 bi`b ?pBM; Kbb BM T`iB+mH`-i?2 `M;2 m1 ≤ m ≤ m2- i i?2 iBK2
t Q7 i?2 ;Htv 2pQHmiBQM- Bb ;Bp2M #v i?2 7QHHQrBM; 7Q`KmH, R(t) =
m2
,
m1
dm φ(m) ψ(t − τm), URXRRV
r?2`2 Ĝ rBi? i?2 Tm`TQb2 Q7 bBKTHB7vBM; i?2 7Q`KmH Ĝ r2 ?p2 bbmK2/ i?i i?2 ;Htv ?b mM/2`;QM2 +QMiBMmQmb bi` 7Q`KiBQM +iBpBiv Qp2` Bib 2MiB`2 HB72iBK2
M/ i?i i?2 iBK2 t Bb bm+? i?i mKt(0, t) < m1X q2 `2KBM/ i?2 `2/2` i?i i?2
[mMiBiv τm BM 2[miBQM URXRRV `2T`2b2Mib i?2 HB72iBK2 Q7 bi` rBi? Kbb mX
h?2 T?vbB+H K2MBM; Q7 2[miBQM URXRRV Bb i?i 2+? bBM;H2 iBK2 t′ < t Q7 i?2
;Htv 2pQHmiBQM Bb mMBpQ+HHv bbQ+Bi2/ iQ T`iB+mH` bi`- r?B+? rb #Q`M i i?2 iBK2 t′ M/ /B2b i i?2 iBK2 t- ?pBM; Kbb m bm+? i?i Bib +Q``2bTQM/BM;
HB72iBK2 Bb τm = |t − t′|X ai`b rBi? p2`v bKHH bi2HH` Kbb2b `2 bbQ+Bi2/ iQ
i?2 2`HB2bi bi;2b Q7 i?2 ;Htv 2pQHmiBQM- r?BH2 KbbBp2 bi`b +M QMHv +QK2 7`QK 2TQ+?b r?B+? `2 p2`v +HQb2 iQ i?2 T`2b2Mi iBK2X
h?2 KQMQiQMB+ U+QMiBMmQmbHv /2+`2bBM;V `2HiBQM #2ir22M i?2 bi2HH` HB72iBK2b M/ bi2HH` Kbb HHQrb mb iQ `2r`Bi2 i?2 BMi2;`H Qp2` Kbb BM 2[miBQM URXRRV b M BMi2;`H Qp2` HH i?2 T`2pBQmb iBK2b Q7 bi` 7Q`KiBQMc BM 7+i- b 7Q`2K2MiBQM2/-2+? bi` /vBM; i i?2 iBK2 t Q`B;BMi2/ i ;Bp2M T`2pBQmb iBK2 t′ < t- r?B+?
Bb mMBpQ+HHv /2i2`KBM2/ #v i?2 BMBiBH bi2HH` KbbX AM T`iB+mH`- r2 +M `2r`Bi2 2[miBQM URXRRV b 7QHHQrbX R(t) = τ2 , τ1 dτ- -dm(τ ) dτ -- φ(m) ψ(t − τ ), URXRkV r?2`2 i?2 [mMiBiv m(τ) `2T`2b2Mib i?2 BMp2`b2 bi2HH` HB72iBK2b- MK2Hv i?2 Kbb Q7 bi` rBi? HB72iBK2 τX lbmHHv- +?2KB+H 2pQHmiBQM KQ/2Hb +QKTmi2 i?2 [mMiBiv R(t)#v mbBM; 2[miBQM URXRkVX
h?2 bi2HH` `2biBimiBQM `i2 Q7 +?2KB+H 2H2K2MibX G2i mb bbmK2 i?i pX(m, Z)
`2T`2b2Mib i?2 bi2HH` Mm+H2QbvMi?2iB+ vB2H/b Q7 i?2 +?2KB+H 2H2K2Mi X- MK2Hv i?2 Kbb BM i?2 7Q`K Q7 i?2 +?2KB+H 2H2K2Mi X r?B+? Bb 7`2b?Hv bvMi?2bBx2/ #v bi` rBi? Kbb m M/ K2iHHB+Biv Z- M/ Hi2` 2D2+i2/ BMiQ i?2 ;Htv BMi2`bi2HH` K2/BmK i i?2 bi2HH` /2i? Ub22 HbQ a2+iBQMRXkX9VX h?2 `i2 Q7 `2biBimiBQM Q7 i?2 M2rHv T`Q/m+2/ [mMiBiv Q7 X- #v HH i?2 bi`b rBi? Kbb BM i?2 `M;2 m1≤ m ≤
m2- i i?2 iBK2 t- +M #2 +QKTmi2/ #v K2Mb Q7 i?2 7QHHQrBM; 7Q`KmH,
RX(t) = m2
,
m1
dm φ(m) ψ&t − τm'pX&m, Z(t − τm)'. URXRjV
AM 2[miBQM URXRjV- i?2 BMBiBH Kbb 7mM+iBQM THvb 7mM/K2MiH `QH2- iQ;2i?2` rBi? i?2 bi2HH` HB72iBK2bX AM T`iB+mH`- i?2 BMBiBH Kbb 7mM+iBQM ;Bp2b /Bz2`2Mi r2B;?ib iQ bi`b UM/ ?2M+2 bi2HH` vB2H/bV #2HQM;BM; iQ /Bz2`2Mi Kbb `M;2b- r?2`2b i?2 bi2HH` HB72iBK2b `2;mHi2 i?2 /2Hv iBK2 rBi? r?B+? bi`b 2M`B+? i?2 BMi2`bi2HH` K2/BmK i i?2B` /2i?X GQr@Kbb bi`b ?p2 i?2 ?B;?2bi r2B;?ib BM i?2 BMBiBH Kbb 7mM+iBQM #mi HbQ i?2 H`;2bi ivTB+H HB72iBK2bc ?2M+2 HQr@Kbb bi`b `2 2tT2+i2/ iQ ?2pBHv TQHHmi2 i?2 BMi2`bi2HH` K2/BmK rBi? i?2B` Mm+H2QbvMi?2iB+ T`Q/m+ib QM HQM; ivTB+H iBK2@b+H2bX PM i?2 Qi?2` ?M/- KbbBp2 bi`b 2M`B+? i?2 BMi2`bi2HH` K2/BmK QM p2`v b?Q`i ivTB+H iBK2 b+H2b 7`QK i?2 bi` 7Q`KiBQM 2p2MiX
AM +?2KB+H 2pQHmiBQM KQ/2Hb- Bi Bb +mbiQK`v iQ bbmK2,
RXkX "bB+ BM;`2/B2Mib
i i?2 iBK2 t Udz.Ǵ biM/b 7Q` dz/2/ǴV Q`- 2[mBpH2MiHv- i?2 KtBKmK Kbb Q7 i?2 bi`b HBp2 i i?2 iBK2 tX h?Bb [mMiBiv +QMiBMmQmbHv /2+`2b2b b 7mM+iBQM Q7 i?2 ;Htv HB72iBK2X
Ç m2 = 100J!- r?B+? Bb i?2 bbmK2/ KtBKmK bi2HH` Kbb i?i +M #2
7Q`K2/ BM i?2 bi` 7Q`KiBQM T`Q+2bbX
"v +QKT`BM; 2[miBQM URXRjV rBi? 2[miBQM URXRyV- HvBM; i i?2 #bBb Q7 +?2KB@ +H M/ bT2+i`Q@T?QiQK2i`B+ ;Htv 2pQHmiBQM`v +Q/2b- `2bT2+iBp2Hv- r2 +M TT`2@ +Bi2 i?i i?2 TQTmHiBQM Q7 bi`b +QMi`B#miBM; iQ i?2 ;Htv +?2KB+H 2M`B+?K2Mi `2 +QKTH2K2Mi`v rBi? `2bT2+i iQ i?2 bi`b +QMi`B#miBM; iQ i?2 ;Htv BMi2;`i2/ HB;?iX AM T`iB+mH`- +?2KB+H 2pQHmiBQM Bb /2i2`KBM2/ #v HH i?2 bi`b /vBM; i i?2 iBK2 t- r?BH2 i?2 BMi2;`i2/ ;Htv HB;?i Bb ;Bp2M #v HH i?2 bi`b biBHH HBp2 i tX
RXkX9 h?2 Mm+H2QbvMi?2iB+ bi2HH` vB2H/b
b H`2/v MiB+BTi2/ BM i?2 /Bb+mbbBQM #Qp2- i?2 KQmMi Q7 Kbb Q7 i?2 M2rHv 7Q`K2/ +?2KB+H 2H2K2Mi X- b bvMi?2bBx2/ #v bi` /m`BM; Bib p`BQmb 2pQHmiBQM`v T?b2b- Bb +HH2/ bi2HH` vB2H/ Q7 i?2 +?2KB+H 2H2K2Mi XX h?2 bbmK2/ bi2HH` vB2H/ /i#b2 Bb T`Q##Hv i?2 KQbi BKTQ`iMi [mMiBiv 7Q` i?2 /2p2HQTK2Mi Q7 M ++m`i2 ;Htv +?2KB+H 2pQHmiBQM KQ/2HX
ai2HH` vB2H/b `2 T`QpB/2/ #v bi2HH` 2pQHmiBQM`v KQ/2Hbc KMv ;`QmTb BM i?2 bi`QMQKB+H +QKKmMBiv +QKT2i2 BM i?2 /2p2HQTK2Mi Q7 i?2b2 KQ/2HbX h?2 +QK@ TH2tBiv M/ ++m`+v Q7 bi2HH` KQ/2Hb ?p2 `2+?2/ MQr/vb p2`v ?B;? H2p2Hb- bBM+2 i?2v `2 +T#H2 Q7 iFBM; BMiQ ++QmMi KQbi Q7 i?2 `2H2pMi T?vbB+H T`Q+2bb2b BM~m@ 2M+BM; i?2 T`Q/m+iBQM M/ /BzmbBQM Q7 i?2 +?2KB+H 2H2K2Mib rBi?BM bi`b- iQ;2i?2` rBi? p2`v ++m`i2 i`2iK2Mi Q7 i?2 i?2`KQ/vMKB+b Q7 i?2 bi2HH` BMi2`BQ`bX 6Q` 2tKTH2- i?2 KQbi /pM+2/ bi2HH` KQ/2Hb iF2 BMiQ ++QmMi i?2 +QK#BM2/ 2z2+i Q7 bi2HH` `QiiBQM- +QMp2+iBQM M/ Kbb HQbbX JQbi Q7 i?2 bi2HH` KQ/2Hb `2 #H2 iQ +QKTmi2 i?2 2tTHQbBp2 Mm+H2QbvMi?2bBb Q7 i?2 +?2KB+H 2H2K2Mib- #v bbmKBM; T`iB+mH` amT2`MQp 2tTHQbBQM K2+?MBbK M/ #v BM+Hm/BM; +QKT`2?2MbBp2 M2i@ rQ`F Q7 Mm+H2` `2+iBQMb r?B+? `2 bQHp2/ iQ;2i?2` rBi? i?2 /Bz2`2MiBH 2[miBQMb 7Q` i?2 bi2HH` 2pQHmiBQMX
Hi?Qm;? HQi Q7 T`Q;`2bb ?b #22M K/2 BM i?2 /2p2HQTK2Mi Q7 i?2 bi2HH` KQ/2Hb BM i?2 Hbi /2+/2b- i?2`2 Bb biBHH MQi +QMb2Mbmb BM i?2 HBi2`im`2 #Qmi i?2 T`2+Bb2 pHm2b Q7 i?2 Mm+H2QbvMi?2iB+ bi2HH` vB2H/bc i?2 ivTB+H /Bz2`2M+2 BM i?2 }MH +?2KB+H #mM/M+2b- b BMi`Q/m+2/ #v i?2 bbmKTiBQMb Q7 /Bz2`2Mi bi2HH` vB2H/ +QKTBHiBQMb- Bb Q7 i?2 Q`/2` Q7 ∼ 0.2 /2tX Hi?Qm;? i?2 mM+2`iBMiv Bb biBHH MQi M2;HB;B#H2 7Q` KQbi Q7 i?2 +?2KB+H 2H2K2Mib- i?2 KDQ`Biv Q7 i?2 rQ`Fb BM i?2 HBi2`im`2 ;`22 #Qmi i?2 T`Q/m+iBQM K2+?MBbK M/ bBi2 Q7 i?2 p`BQmb +?2KB+H 2H2K2Mib rBi?BM bi`bX L2p2`i?2H2bb- +QM+2`MBM; i?Bb TQBMi- i?2`2 `2 biBHH 72r UBK@ TQ`iMiV QT2M T`Q#H2Kb- bm+? b i?2 T`BK`v MBi`Q;2M T`Q/m+iBQM BM KbbBp2 bi`b-i p2`v HQr K2iHHB+Bivc i?2 1m Mm+H2QbvMi?2bBb U`2+2MiHv- M Hi2`Mbi`b-iBp2 b+2M`BQ rBi? `2bT2+i iQ i?2 +HbbB+H `@T`Q+2bb BM KbbBp2 bi`b ?b #22M T`QTQb2/- MK2Hv i?2 1m Mm+H2QbvMi?2bBb BM M2mi`QM bi` K2`;2`bVc i?2 +?2KB+H 2H2K2Mib T`Q/m+2/ #v i?2 }`bi SQTmHiBQM AAA Q7 bi`bX AM bmKK`v- i?2`2 `2 +?2KB+H 2H2K2Mib UHBF2 Qtv;2MV 7Q` r?B+? bi2HH` KQ/2Hb +QMp2`;2 iQr`/b M ;`22K2Mi 7Q` i?2B` bi2HH`
vB2H/bc 7Q` Qi?2` +?2KB+H 2H2K2Mib UHBF2 iBiMBmK- M/ bQK2 Qi?2` α@2H2K2MibV i?2 bBimiBQM Bb biBHH +`BiB+H M/ i?2B` bi2HH` vB2H/b `2 MQi `2HB#H2X
*?2KB+H 2H2K2Mib +M #2 T`Q/m+2/ #v bi`b 2Bi?2` /m`BM; i?2 ?v/`QbiiB+ i?2`@ KQMm+H2` #m`MBM;b Ui?Bb Bb pHB/ #Qi? 7Q` HQr@ M/ BMi2`K2/Bi2@Kbb bi`b M/ KbbBp2 bi`bV- Q` i?`Qm;? i?2 }MH 2tTHQbBp2 i?2`KQMm+H2` #m`MBM;b- Q++m``BM; b KbbBp2 bi`b 2p2MimHHv 2tTHQ/2 b +Q`2@+QHHTb2 amT2`MQp2X Hi?Qm;? i?2 `2H2b2 Q7 i?2 +?2KB+H 2H2K2Mib BMiQ i?2 ;Htv BMi2`bi2HH` K2/BmK KB;?i #2 +QMiBMmQmb b i?2 bi` 2pQHp2b iQr`/b i?2 Hi2bi 2pQHmiBQM`v T?b2b M/ bi2HH` rBM/b /2@ p2HQT- +?2KB+H 2pQHmiBQM KQ/2Hb bbmK2 i?i i?2 +?2KB+H 2M`B+?K2Mi Q++m`b i i?2 dzbi2HH` /2i?Ǵ- MK2Hv i i?2 2M/ Q7 i?2 bi2HH` HB72iBK2- BM +mKmHiBp2 rvX b 7Q`2K2MiBQM2/- i?2 bi2HH` vB2H/b `2 T`QpB/2/ BM i?2 HBi2`im`2 b 7mM+iBQMb Q7 i?2 bi2HH` Kbb M/ K2iHHB+BivX h?2`2 Bb biBHH bQK2 +QM7mbBQM BM i?2 HBi2`im`2 #Qmi i?2 2t+i /2}MBiBQM Q7 dzbi2HH` vB2H/Ǵc BM T`iB+mH` Ĝ Hi?Qm;? i?2 Q`B;BMH rQ`Fb /2}M2 i?2 bi2HH` vB2H/b #v BM+Hm/BM; QMHv i?2 7`2b? +?2KB+H 2H2K2Mib b M2rHv 7Q`K2/ #v i?2 bi` Ĝ KMv rQ`Fb /2}M2 Bi b i?2 iQiH KQmMi Q7 X r?B+? Bb 2tT2HH2/ #v i?2 bi` i Bib /2i?X h?2 irQ [mMiBiB2b `2 +QM+2TimHHv /Bz2`2Mic BM i?Bb h?2bBb- r2 rBHH +QM7Q`K iQ i?2 Q`B;BMH /2}MBiBQMX
h?2 7QHHQrBM; 2[miBQM 2tTHBMb ?Qr iQ `2+Qp2` i?2 bi2HH` vB2H/ Q7 i?2 +?2KB+H 2H2K2Mi X 7`QK i?2 iQiH KQmMi Q7 X r?B+? Bb 2tT2HH2/ #v i?2 bi`,
pX(m, Z) = M2D,X(m, Z) − ZX&m − M`2K(m, Z)', URXR9V
r?2`2 pX `2T`2b2Mib i?2 bi2HH` vB2H/ Q7 Xc i?2 [mMiBiv M2D,X Bb i?2 iQiH KQmMi
Q7 X 2D2+i2/ #v i?2 bi` i Bib /2i?c ZX Bb i?2 #mM/M+2 #v Kbb Q7 X BM i?2
Q`B;BMH ;b KBtim`2 7`QK r?B+? i?2 bi` Q`B;BMi2/c }MHHv- M`2K `2T`2b2Mib i?2
bi2HH` `2KMMi- b H27i #v i?2 bi` i Bib /2i?X h?2 b2+QM/ //2M/ BM i?2 `B;?i@ ?M/ bB/2 Q7 2[miBQM URXR9V `2T`2b2Mib i?2 Q`B;BMH [mMiBiv Q7 X r?B+? Bb 2D2+i2/ rBi?Qmi #2BM; Mm+H2`Hv T`Q+2bb2/ #v i?2 bi`X
AM +?2KB+H 2pQHmiBQM KQ/2Hb- bi`b `2 #`Q/Hv bm#@/BpB/2/ BMiQ irQ KBM +Hbb2b- ++Q`/BM; iQ i?2B` BMBiBH Kbb M/ ?2M+2 i?2B` }MH 7i2, HQr@ M/ BMi2`K2/Bi2@Kbb bi`b U0.8 ! m ! 8 J!V- r?B+? 2M/ i?2B` HBp2b b r?Bi2 /r`7bc
KbbBp2 bi`b Um > 8 J!V- 2tTHQ/BM; b +Q`2@+QHHTb2 amT2`MQp2 M/ H2pBM;
M2mi`QM bi` Q` #H+F ?QH2 b `2KMMiX q2 `2KBM/ i?2 `2/2` i?i bi`b rBi? m ! 0.8J! ?p2 HB72iBK2b 2[mH Q` H`;2` i?M i?2 >m##H2 iBK2 M/ ?2M+2 i?2v
THv K`;BMH `QH2 BM +?2KB+H 2pQHmiBQM KQ/2Hb- bBM+2 i?2v /Q MQi +QMi`B#mi2 iQ i?2 ;H+iB+ +?2KB+H 2M`B+?K2MiX
AM r?i 7QHHQrb- r2 b+?2KiB+HHv bmKK`Bx2 i?2 +QMi`B#miBQM Q7 i?2 irQ +Hbb2b iQ i?2 +?2KB+H 2M`B+?K2Mi Q7 ;HtB2bX
GQr@ M/ BMi2`K2/Bi2@Kbb bi`b
h?2 bi`b rBi? 0.8 ! m ! 8 J! `2 i?2 KQbi BKTQ`iMi T`Q/m+2`b Q7 4>2- 12 *-13*- 14L-17P M/ b@T`Q+2bb 2H2K2Mib HBF2 "- G- a` M/ w` BM i?2 lMBp2`b2X GQr@
M/ BMi2`K2/Bi2@Kbb bi`b THv HbQ 7mM/K2MiH `QH2 BM i?2 /mbi T`Q/m+iBQM-7`QK i?2 bvKTiQiB+ ;BMi #`M+? iQ i?2 THM2i`v M2#mH T?b2X h?2B` iKQbT?2`2 bi`ib #2BM; p2`v /Bzmb2 /m`BM; i?2 bvKTiQiB+ ;BMi #`M+? T?b2- HQM; r?B+?
Q`B;BMHHv T`2b2Mi BM i?2 bi` i Bib #B`i?c BM 7+i- i?2 * M/ P Mm+H2B Ĝ 7i2` #2BM; T`Q/m+2/ #v i?2 bi` i?`Qm;? i?2 i`BTH2@α T`Q+2bb BM i?2 BMM2` >2@#m`MBM; xQM2 Ĝ KB;`i2 iQ i?2 Qmi2` >@#m`MBM; b?2HH r?2`2 i?2v `2 mb2/ iQ bvMi?2bBx2 dzT`BK`vǴ LX L2p2`i?2H2bb-14L ?b KQbiHv b2+QM/`v Q`B;BM BM HQr@ M/ BMi2`K2/Bi2@Kbb
bi`b- bBM+2 BM i?2 KDQ`Biv Q7 i?2 +b2b i?2 *LP +v+H2 BMpQHp2b * M/ P Mm+H2B r?B+? r2`2 H`2/v T`2b2Mi BM i?2 bi`b i Bib #B`i?X
AM bmKK`v- i?2 bi2HH` vB2H/b Q7 i?2 b2+QM/`v 2H2K2Mib b+H2 rBi? i?2 BMBiBH K2iHHB+Biv Q7 i?2 bi`- bBM+2 i?2B` Mm+H2QbvMi?2bBb +`m+BHHv /2T2M/b mTQM i?2 BMBiBH KQmMi Q7 K2iHb BM i?2 bi` Ui?2v +i b b22/b 7Q` i?2 T`Q/m+iBQM Q7 i?2 b2+QM/`v 2H2K2MibV- r?BH2 i?2 bi2HH` vB2H/b Q7 i?2 T`BK`v 2H2K2Mib /Q MQi +Q``2Hi2 rBi? K2iHHB+Biv- bBM+2 i?2 K2iHb r?B+? KB;?i #2 M2+2bb`v 7Q` i?2B` T`Q/m+iBQM `2 bvMi?2bBx2/ #v i?2 bi` Bib2H7X
"2vQM/ MBi`Q;2M- i?2 Qi?2` p2`v BKTQ`iMi +?2KB+H 2H2K2Mi r?B+? Bb bvM@ i?2bBx2/ #v HQr@ M/ BMi2`K2/Bi2@Kbb bi`b Bb +`#QMX AM T`iB+mH`- i?2 KQbi BKTQ`iMi +`#QM T`Q/m+2`b `2 i?2 bi`b rBi? Kbb BM i?2 `M;2 1 ! m ! 3 J!
-r?B+? Ĝ BM+B/2MiHHv Ĝ HbQ bB;MB}+MiHv +QMi`B#mi2 iQ i?2 BMBiBH Kbb 7mM+iBQMX ai`b rBi?BM i?i Kbb `M;2 `2 HbQ `2bTQMbB#H2 7Q` i?2 T`Q/m+iBQM Q7 " M/ Qi?2` ?2pv b@T`Q+2bb 2H2K2Mib- HBF2 a` Q` GX
AM bmKK`v,
Ç GQr@Kbb bi`b rBi? Kbb BM i?2 `M;2 0.08 ! m ! 0.5 J! QMHv B;MBi2 > BM
i?2B` +Q`2X h?2v /B2 b >2@r?Bi2 /r`7b M/ i?2B` ?2HBmK +Q`2 Bb bmTTQ`i2/ ;BMbi 7m`i?2` +QHHTb2 #v i?2 2H2+i`QM@/2;2M2`+v T`2bbm`2X h?2B` +Q`2 #2@ +QK2b 2H2+i`QM@/2;2M2`i2 #27Q`2 `2+?BM; i?2 M2+2bb`v i2KT2`im`2b 7Q` i?2 >2 #m`MBM;X
Ç GQr@Kbb bi`b rBi? Kbb BM i?2 `M;2 0.5 ! m ! 2.25 J! 2M/ i?2B` HBp2b b
*P r?Bi2 /r`7bX "27Q`2 H2pBM; i?2 `2/ ;BMi #`M+?- i?2v `2 +?`+i2`@ Bx2/ #v ?B;?Hv 2H2+i`QM@/2;2M2`i2 ?2HBmK +Q`2X i i?2 iBT Q7 i?2 `2/ ;BMi #`M+?- MK2Hv i i?2 iBK2 Q7 i?2 >2 B;MBiBQM- i?2 ?2HBmK +Q`2 ?b ;`QrM mT iQ pHm2 r?B+? Bb T`2/B+i2/ iQ #2 Hrvb `QmM/ ∼ 0.48@0.50 J!X h?2M- b
}`bi i`BTH2@α `2+iBQMb bi`i BM i?2 +Q`2- i?2v +mb2 bm//2M BM+`2b2 BM i?2 +2Mi`H i2KT2`im`2 Q7 i?2 bi`- r?B+? Bb MQi ++QKTMB2/ #v M 2tTMbBQM Q7 i?2 +Q`2- bBM+2 Bi Bb biBHH ?B;?Hv 2H2+i`QM@/2;2M2`i2X h?2`27Q`2- bi2HH` KQ/@ 2Hb T`2/B+i bQ`i Q7 TQbBiBp2 722/#+F #2ir22M i?2 +2Mi`H i2KT2`im`2 M/ i?2 i`BTH2@α `2+iBQM `i2c BM T`iB+mH`- i?2v T`2/B+i dzi?2`KH `mMrvǴ iQ;2i?2` rBi? p2`v H`;2 `2H2b2 Q7 Mm+H2` 2M2`;v Qp2` p2`v b?Q`i KQmMi Q7 iBK2 U72r b2+QM/bVX _Qm;?Hv bT2FBM;- bTBF2 Bb T`2/B+i2/ iQ Q++m` BM i?2 HB;?i +m`p2 Q7 i?2b2 bi`b Ui?i Bb i?2 bQ@+HH2/ ?2HBmK ~b?V- b i?2v bi`i #m`MBM; ?2HBmK BM i?2B` +Q`2 M/ ?2M+2 bi`i KQpBM; iQr`/b i?2 #Hm2 ?Q`BxQMiH #`M+?X
Ç AMi2`K2/Bi2@Kbb bi`b rBi? Kbb BM i?2 `M;2 2.25 ! m ! 8 J! 2M/ i?2B`
HBp2b b *P r?Bi2 /r`7bX h?2v `2 T`2/B+i2/ iQ H2p2 i?2 `2/ ;BMi #`M+? rBi? MQM@/2;2M2`i2 ?2HBmK +Q`2- rBi?Qmi Mv ?2HBmK ~b?X
RXkX "bB+ BM;`2/B2Mib 20 30 40 50 60 Progenitor mass [M!] 0 5 10 15 20 Ej e c te d m a s s [M ! ]
Yields: Romano et al. (2010) Z = 0 .0 Z = 1.0 × 10−5 Z = 1.0 × 10−3 Z = 4.0 × 10−3 Z = 2.0 × 10−2
6B;m`2 RXe, AM i?Bb };m`2- r2 `2TQ`i i?2 2D2+i2/ Kbb Q7 Qtv;2M b 7mM+iBQM Q7 i?2 T`Q;2MBiQ`
Kbb- 7Q` /Bz2`2Mi BMBiBH bi2HH` K2iHHB+BiB2bX h?Bb b2i Q7 bi2HH` vB2H/b Bb 7`QK _QKMQ 2i HX
UkyRyVX h?2 bQHB/ HBM2 BM #H+F +Q``2bTQM/b iQ i?2 bi2HH` vB2H/b i Z = 1.0×10−10c i?2 /b?2/
HBM2 BM `2/ iQ Z = 1.0 × 10−5c i?2 #H+F /Qii2/ HBM2 iQ Z = 1.0 × 10−3c i?2 /b?2/@/Qii2/ HBM2
BM #Hm2 iQ Z = 3 × 10−3- M/ i?2 ;`22M b[m`2b iQ i?2 bi2HH` vB2H/b i Z = 2.0 × 10−2X
:`Q2M2r2;2M URNNdV T`QpB/2/ /2iBH2/ +QKTBHiBQM Q7 bi2HH` vB2H/b 7Q` bi`b rBi? Kbb BM i?2 `M;2 0.8 ≤ m ≤ 8 J! M/ K2iHHB+BiB2b Z = 0.001-
0.004-0.008- 0.02- M/ 0.04X KQ`2 `2+2Mi +QKTBHiBQM Q7 bi2HH` vB2H/b 7Q` HQr@ M/ BMi2`K2/Bi2@Kbb bi`b Bb i?2 QM2 #vE`Fb UkyRyV- r?B+? Bb bbmK2/ #v HKQbi HH i?2 +m``2Mi +?2KB+H 2pQHmiBQM KQ/2Hb r?B+? `2 +T#H2 Q7 `2T`Q/m+BM; i?2 JBHFv qv +?2KB+H #mM/M+2 Tii2`Mb U_QKMQ 2i HX- kyRyVX 6BMHHv- `2+2Mi bi2HH` vB2H/ +QKTBHiBQM Bb i?2 QM2 T`QpB/2/ #v i?2 SQHQ o2Mim`Ƕb ;`QmT i i?2 bi`QMQKB+H P#b2`piQ`v Q7 _QK2 Ub22- 7Q` 2tKTH2- o2Mim` 2i HX kyRjV- r?B+? ?b #22M bbmK2/ #voBM+2MxQ 2i HXUkyRe#V iQ `2T`Q/m+2 i?2 L M/ P #mM/M+2b b Q#b2`p2/ BM bKTH2 Q7 aHQM .B;BiH aFv am`p2v ;HtB2bX
AM 6B;X RX8- r2 b?Qr ?Qr i?2 bi2HH` vB2H/b Q7 o2Mim` 2i HX UkyRjV 7Q` *-P- M/ T`BK`v M/ b2+QM/`v L p`v b 7mM+iBQMb Q7 i?2 BMBiBH bi2HH` Kbb- 7Q` /Bz2`2Mi K2iHHB+BiB2bX h?2 bi2HH` vB2H/ Q7 ;Bp2M +?2KB+H 2H2K2Mi X Bb /2}M2/ b i?2 2D2+i2/ KQmMi Q7 Kbb Q7 i?2 M2rHv 7Q`K2/ XX "v HQQFBM; i i?2 };m`2- i?2 bi2HH` vB2H/b Q7 b2+QM/`v L BM+`2b2 rBi? K2iHHB+Biv- #v K2Mb Q7 i?2 +QMbmKTiBQM Q7 i?2 * M/ P Mm+H2B Q`B;BMHHv T`2b2Mi BM i?2 bi`X AM 7+i- /m`BM; i?2 *LP +v+H2- i?2 ;HQ#H #mM/M+2 Q7 i?2 *LP Mm+H2B `2KBMb +QMbiMi- r?BH2 i?2 `2HiBp2 #mM/M+2b Q7 2+? *LP 2H2K2Mi +M bB;MB}+MiHv p`vX "v HQQFBM; i 6B;X RX8 -i?2 bi2HH` T`Q/m+2`b Q7 T`BK`v L ?p2 Kbb2b BM -i?2 `M;2 #2ir22M ∼ 3 J! M/
6J!X 6m`i?2`KQ`2- i?2 T`Q/m+iBQM Q7 T`BK`v L /Q2b MQi b?Qr Mv +Q``2HiBQM
rBi? i?2 +QMbmKTiBQM Q7 i?2 Q`B;BMH * M/ P BM i?2 bi`- b 2tT2+i2/X
15 20 25 30 35 40 Progenitor mass [M!] 0 2 4 6 8 10 Ej e c te d m a s s [M ! ]
Yields: Nomoto et al. (2013) Z = 0 .0 Z = 1.0 × 10−3 Z = 4.0 × 10−3 Z = 8.0 × 10−3 Z = 2.0 × 10−2 Z = 5.0 × 10−2
6B;m`2 RXd, AM i?Bb };m`2- r2 b?Qr ?Qr i?2 2D2+i2/ Kbb Q7 Qtv;2M p`v b 7mM+iBQM Q7
i?2 T`Q;2MBiQ` Kbb- 7Q` /Bz2`2Mi BMBiBH bi2HH` K2iHHB+BiB2b- r?2M bbmKBM; i?2LQKQiQ 2i HX
UkyRjV bi2HH` vB2H/bX h?2 bQHB/ HBM2 BM #H+F +Q``2bTQM/b iQ i?2 bi2HH` vB2H/b i Z = 0.0c i?2 /Qii2/ HBM2 BM #H+F iQ Z = 1.0 × 10−3c i?2 /b?2/@/Qii2/ HBM2 BM #Hm2 iQ Z = 4.0 × 10−3c i?2
bQHB/ HBM2 BM K;2Mi iQ Z = 8.0 × 10−3c i?2 bQHB/ HBM2 BM ;`22M iQ Z = 2.0 × 10−2- M/ i?2
bQHB/ HBM2 BM +vM iQ i?2 bi2HH` vB2H/b i Z = 5.0 × 10−2X
JbbBp2 bi`b
JbbBp2 bi`b `2 /2}M2/ b ?pBM; Kbb m " 8 J!- Hi?Qm;? i?2 HBKBiBM; Kbb Bb
biBHH ?B;?Hv mM+2`iBMX AM bmKK`v,
Ç ai`b rBi? Kbb 8 ! m ! 10 J! ;Bp2 `Bb2 iQ i?2 bQ@+HH2/ dzbmT2` bvKTiQiB+
;BMi #`M+? bi`bǴ M/ /2p2HQT M 2H2+i`QM@/2;2M2`i2 P@L2@J; +Q`2c b i?2 +Q`2 `2+?2b +`BiB+H /2MbBiv Q7 i?2 Q`/2` Q7 ∼ 4 × 109; +K−3- +Q``2@
bTQM/BM; iQ Kbb `QmM/ ∼ 1.38 J!- 2@+Tim`2b QMiQ i?2 J; Mm+H2B +mb2
i?2 /2;2M2`i2 T`2bbm`2 iQ /2+`2b2 M/ i?2 +Q`2 iQ +QHHTb2c i?2M- i?2 P 2tTHQbBp2Hv B;MBi2b- ;BpBM; 2p2MimHHv `Bb2 iQ M 2@+Tim`2 amT2`MQpX
Ç ai`b rBi? Kbb BM i?2 `M;2 10 J! ! m < Mq_(Z) ;Bp2 `Bb2 iQ hvT2 AA
amT2`MQp2 M/ `2 i?2 KQbi BKTQ`iMi K2iH@T`Q/m+2`b BM i?2 lMBp2`b2-/2bTBi2 i?2 7+i i?i i?2B` +QMi`B#miBQM BM i?2 BMBiBH Kbb 7mM+iBQM Bb Km+? KBMQ` rBi? `2bT2+i iQ HQr@Kbb bi`bX Mq_ Bb i?2 KBMBKmK Kbb 7Q` i?2
7Q`KiBQM Q7 qQH7@_v2i bi`- r?B+? Bb ?B;?Hv mM+2`iBM bBM+2 Bi /2T2M/b QM i?2 T`2b+`BTiBQMb #Qmi i?2 bi2HH` Kbb HQbb- r?B+? BM im`M /2T2M/ #Qi? QM i?2 BMBiBH bi2HH` Kbb M/ QM i?2 K2iHHB+Biv Ui?2 ?B;?2` i?2 K2iHHB+Biv-i?2 HQr2` im`Mb Qmi iQ #2 Mq_VX JbbBp2 bi`b- #27Q`2 2tTHQ/BM; b hvT2 AA
RXkX "bB+ BM;`2/B2Mib
7`QK i?2 bm``QmM/BM; aB #m`MBM; b?2HH- i?2 2H2+i`QM@/2;2M2`+v T`2bbm`2 +M@ MQi T`2p2Mi MvKQ`2 i?2 +Q`2 7`QK i?2 ;`pBiiBQMH +QHHTb2X h?2 B`QM +Q`2 Ĝ b Bi +QHHTb2b M/ `2+?2b Hrvb ?B;?2` /2MbBiB2b Ĝ Tbb2b i?`Qm;? i?2 bi;2b Q7 dzT?QiQ@/BbBMi2;`iBQMǴ Q7 i?2 B`QM Mm+H2B M/ α@T`iB+H2b- M/ }MHHv dzM2m@ i`QMBxiBQMǴX i i?2 2M/- i?2b2 bi`b +M H2p2 2Bi?2` M2mi`QM bi` Q` Ĝ B7 i?2 /2;2M2`+v T`2bbm`2 Q7 i?2 M2mi`QMb +MMQi bmbiBM i?2 +Q`2 Ui?Bb Q++m`b 7Q` +Q`2 Kbb2b " 3 J!V Ĝ #H+F ?QH2 b `2KMMibX .m`BM; i?2 pBQH2Mi
+Q`2@+QHHTb2- amT2`MQp 2tTHQbBQM Q7 hvT2 AA iF2b TH+2X h?2v +M 2M`B+? i?2 BMi2`bi2HH` K2/BmK 2Bi?2` i?`Qm;? bi2HH` rBM/b- b i?2 bi` 2pQHp2b BM Bib dzMQ`KHǴ 2pQHmiBQM`v bi;2b- Q` i i?2 p2`v }MH 2tTHQbBp2 bi;2b- b i?2 #m`MBM; b?2HHb `2 T`iHv #HQrM rv #v i?2 b?Q+Frp2b Q`B;BMiBM; 7`QK i?2 bQ@+HH2/ dz+Q`2 #QmM+2Ǵ- r?B+? iF2b TH+2 r?2M i?2 +QHHTbBM; +Q`2 `2+?2b bQ 2ti`2K2Hv ?B;? /2MbBiB2b i?i i?2 bi`QM; Mm+H2` 7Q`+2b bm//2MHv ?Hi i?2 +QHHTb2c i?2 Kbb 2tTmHbBQM Bb Hi2` `2BM7Q`+2/ #v ?B;?@2M2`;v M2mi`BMQbX h?2 HBKBiBM; BMBiBH Kbb Q7 i?2 bi` 7Q` i?2 7Q`KiBQM Q7 #H+F ?QH2 Bb biBHH ?B;?Hv mM+2`iBMc Bi Bb i?Qm;?i iQ #2 Q7 i?2 Q`/2` Q7 ∼ 40 J!X h?2 T`2b2M+2 Q7
#H+F ?QH2 b `2KMMi KB;?i +mb2 i?2 7HH#+F Q7 bm#biMiBH T`i Q7 i?2 2tT2HH2/ Ki2`BH- ?2M+2 bi`QM;Hv `2/m+BM; i?2 bi2HH` vB2H/bX *QM+2`MBM; i?2 Mm+H2QbvMi?2bBb Q7 i?2 +?2KB+H 2H2K2Mib- 16P Bb KQbiHv T`Q/m+2/ /m`BM; i?2
?v/`QbiiB+ >2 #m`MBM;c 23L /m`BM; i?2 ?v/`QbiiB+ *- L2 M/ > #m`MBM;bc 24J; M/27H /m`BM; i?2 ?v/`QbiiB+ * M/ L2 #m`MBM;bc 28aB M/40* +M
#2 T`Q/m+2/ #Qi? /m`BM; i?2 ?v/`QbiiB+ M/ /m`BM; i?2 2tTHQbBp2 P #m`M@ BM;c }MHHv- +?2KB+H 2H2K2Mib HBF2 48hB- 52*`- 55JM- M/ 5662 `2 KBMHv
bvMi?2bBx2/ /m`BM; i?2 2tTHQbBp2 aB #m`MBM;X 7`+iBQM Q7 i?2 M2mi`QMb T`Q@ /m+2/ #v i?2 B`QM T?QiQ@/BbBMi2;`iBQM +QmH/ #2 bm++2bbBp2Hv BMpQHp2/ BM i?2 T`Q/m+iBQM Q7 `@T`Q+2bb 2H2K2Mib HBF2 1mX
Ç ai`b rBi? Kbb BM i?2 `M;2 Mq_(Z) ! m ≤ 100 J! mM/2`;Q p2`v BM@
i2Mb2 bi2HH` rBM/b r?B+? `2KQp2 i?2B` 2ti2`MH >@`B+? 2Mp2HQT2bX h?2v #2@ +QK2 qQH7@_v2i bi`b M/ `2 2tT2+i2/ iQ /B2 #v 2p2MimHHv 2tTHQ/BM; b hvT2 A#f+ amT2`MQp2- r?B+? `2 +m``2MiHv i?Qm;?i iQ #2 /B`2+iHv HBMF2/ iQ HQM; :KK _v "m`bib- #2+mb2 Q7 i?2 Q#b2`p2/ bBKBH`Biv Q7 i?2B` bT2+i`X h?2b2 bi`b +QMi`B#mi2 iQ i?2 ;Htv +?2KB+H 2M`B+?K2Mi KBMHv rBi? 4 >2-12*-13*-22L2- 14L- M/18P- +QM+2`MBM; i?2 ?v/`QbiiB+ +?2KB+H 2H2K2Mibc
i?2 ?2pB2` +?2KB+H 2H2K2Mib `2 bvMi?2bBx2/ /m`BM; i?2 }MH amT2`MQp 2tTHQbBQMX
h?2 KQbi rB/2Hv mb2/ +QKTBHiBQM Q7 bi2HH` vB2H/b 7Q` KbbBp2 bi`b Bb i?2 QM2 #v qQQbH2v q2p2`URNN8VX `2+2Mi +QKTBHiBQM Q7 bi2HH` vB2H/b 7Q` KbbBp2 bi`b Bb i?2 QM2 bbmK2/ #v_QKMQ 2i HXUkyRyV- r?2`2 i?2 >2- *- L M/ P bi2HH` vB2H/b `2 iF2M 7`QK i?2 `2bmHib Q7 i?2 :2M2p bi2HH` KQ/2Hb Q7 KbbBp2 bi`b- iFBM; BMiQ ++QmMi i?2 +QK#BM2/ 2z2+i Q7 Kbb HQbb M/ `QiiBQM- #mi QMHv bbmKBM; ?v/`QbiiB+ #m`MBM; UJ2vM2i J2/2`- kyykc >B`b+?B 2i HX- kyy8c >B`b+?B- kyydc 1Fbi`ƺK 2i HX-kyy3Vc 7Q` ?2pB2` 2H2K2Mib-_QKMQ 2i HXUkyRyV bbmK2 i?2 bi2HH` vB2H/b Q7 KbbBp2 bi`b #vEQ#vb?B 2i HXUkyyeV- r?B+? /Q BM+Hm/2 biM/`/ Kbb HQbb M/ i?2 2tTHQbBp2 Mm+H2QbvMi?2bBb #mi MQi i?2 2z2+i Q7 `QiiBQMX
h?2 Kbb HQbb /`Bp2M #v `QiiBQM im`Mb Qmi iQ #2 T`iB+mH`Hv BKTQ`iMi i
HKQbi bQH` K2iHHB+Biv M/ #Qp2 BM /2T`2bbBM; i?2 Qtv;2M bi2HH` vB2H/b Q7 i?2 KQbi KbbBp2 bi`b UM " 30@40 M!VX AM 7+i- Kbb HQbb BM+`2b2b rBi? bi2HH` K2iHHB+Biv
M/ bi`b Q7 ?B;? K2iH +QMi2Mi HQQb2 >- >2- #mi HbQ *- i?`Qm;? `/BiBp2Hv HBM2 /`Bp2M rBM/bX h?2`27Q`2- i?2 * T`Q/m+iBQM Bb BM+`2b2/ #v Kbb HQbb r?2`2b i?2 Qtv;2M T`Q/m+iBQM Bb /2+`2b2/- bBM+2 T`i Q7 * r?B+? rQmH/ ?p2 #22M i`Mb7Q`K2/ BMiQ P- Bb HQbi 7`QK i?2 bi` UJ2/2`- RNNkVX 6BMHHv- i?2 2z2+i Q7 `QiiBQM Bb iQ T`Q/m+2 KBtBM; M/ 2M?M+2b Kbb HQbb- rBi? i?2 2{+B2M+v Q7 i?2 KBtBM; T`Q+2bb #2BM; H`;2` i HQr2` K2iHHB+BiB2b Ub22 HbQ *?BTTBMB 2i HX kyy3- M/ `272`2M+2b i?2`2BMVX
M Qi?2` +QKTBHiBQM Q7 bi2HH` vB2H/b Q7 KbbBp2 bi`b Bb i?2 QM2 #v LQKQiQ 2i HX UkyRjV- r?B+? Bb `2pB2r- +QK#BMBM; i?2 `2bmHib Q7 i?2 rQ`Fb #v LQKQiQ 2i HX kyyec EQ#vb?B 2i HX kyye- kyRRX 6BMHHv- M Qi?2` ;`QmT rQ`FBM; QM bi2HH` 2pQHmiBQM`v KQ/2Hb Q7 KbbBp2 bi`b Bb i?2 QM2 Q7 J`+Q *?B2{ M/ SQHQ GBKQM;B i i?2 bi`QMQKB+H P#b2`piQ`v Q7 _QK2- r?Q BM+Hm/2 BM i?2B` KQ/2Hb i?2 +QK@ #BM2/ 2z2+i Q7 Kbb M/ `QiiBQM M/ iF2 BMiQ ++QmMi HbQ i?2 }MH 2tTHQbBp2 Mm+H2QbvMi?2bBb U*?B2{ GBKQM;B-kyRjVX
AM 6B;m`2b RXe M/ RXd- r2 b?Qr ?Qr i?2 Qtv;2M bi2HH` vB2H/b Q7 _QKMQ 2i HX UkyRyV M/ LQKQiQ 2i HX UkyRjV- `2bT2+iBp2Hv- p`v b 7mM+iBQMb Q7 i?2 BMBiBH bi2HH` Kbb M/ 7Q` /Bz2`2Mi K2iHHB+BiB2bX h?2 :2M2p bi2HH` vB2H/b `2 pBH#H2 QMHv mT 60 M! M/ r2 i?2`27Q`2 bbmK2 BM Qm` biM/`/ +b2 i?i i?2 vB2H/b 7`QK
60 iQ 100 M! `2 +QMbiMiX PM i?2 Qi?2` ?M/- i?2 bi2HH` vB2H/b Q7 KbbBp2 bi`b
Q7EQ#vb?B 2i HXUkyye- BM+Hm/2/ BM _QKMQ 2i HX kyRy- 7Q` i?2 2H2K2Mib ?2pB2` i?M Qtv;2MV M/ LQKQiQ 2i HX UkyRjV `2 pBH#H2 QMHv mT iQ 40 M! M/ i?mb
r2 F22T i?2K +QMbiMi 7Q` bi`b rBi? H`;2` BMBiBH KbbX q2 `2K`F QM i?2 7+i i?i p2`v KbbBp2 bi`b `2 2tT2+i2/ iQ H2p2 #H+F ?QH2 b `2KMMic i?2`27Q`2- bB;MB}+i?2`27Q`2-Mi 7`i?2`27Q`2-+iBQM Q7 i?2 bi2HHi?2`27Q`2-` Mm+H2Qbvi?2iB+ T`Q/m+ib BM i?2 2D2+ii?2`27Q`2- Q7 p2`v KbbBp2 bi`b Kv 2p2MimHHv 7HH #+F QMiQ i?2 #H+F ?QH2X h?Bb T`Q+2bb KB;?i +mb2 `2/m+iBQM Q7 i?2 bi2HH` vB2H/b Q7 p2`v KbbBp2 bi`bX
RXkX8 hvT2 A amT2`MQp2
hvT2 A amT2`MQp2 THv 7mM/K2MiH `QH2 #Qi? BM i?2 TQHHmiBQM M/ BM i?2 ?2iBM; Q7 i?2 BMi2`bi2HH` K2/BmK Q7 ;HtB2bX 1+? hvT2 A amT2`MQp `2H2b2b M KQmMi Q7 2M2`;v r?B+? Bb ∼ 10512`;c KQ`2Qp2`- hvT2 A amT2`MQp2 `2 `2bTQMbB#H2 7Q`
i?2 Mm+H2QbvMi?2bBb Q7 KQbi Q7 i?2 B`QM M/ B`QM@T2F 2H2K2Mib BM ;HtB2b- iQ;2i?2` rBi? bB;MB}+Mi KQmMi Q7 bBHB+QM- +H+BmK M/ Qi?2` HB;?i 2H2K2MibX h?2`27Q`2-i?2 2z2+i Q7 hvT2 A amT2`MQp2 Kmbi #2 iF2M BMiQ ++QmMi #v h?2`27Q`2-i?2 ;Htv 2pQHmiBQM KQ/2Hb r?B+? BK i #2BM; ++m`i2 M/ +QKTH2i2X
hvT2 A amT2`MQp2 `2 Q#b2`p2/ iQ 2tTHQ/2 BM HH FBM/ Q7 ;HtB2b- 7`QK i?2 2`Hv iQ i?2 Hi2 KQ`T?QHQ;B+H ivT2b- i p`BM+2 rBi? hvT2 AA amT2`MQp2 r?B+? `2 T`272`2MiBHHv Q#b2`p2/ iQ 2tTHQ/2 BM i?2 Hi2@ivT2 bi`@7Q`KBM; QM2bX 6m`i?2`@ KQ`2- b hvT2 A amT2`MQp2 2tTHQ/2- i?2v /Q MQi H2p2 Mv `2KMMi M/ HKQbi HH i?2 Ki2`BH Bb 2tTHQbBp2Hv bvMi?2bBx2/ iQ 2p2MimHHv T`Q/m+2 H`;2 KQmMib Q7 B`QMc BM hvT2 AA amT2`MQp2- i?2 KDQ`Biv Q7 i?2 B`QM `2KBMb dzHQ+F2/ mTǴ BM i?2 `2KMMiX h?i 2tTHBMb r?v hvT2 AA amT2`MQp2 /Q MQi +QMi`B#mi2 iQ i?2 B`QM +?2KB+H 2M`B+?K2Mi b hvT2 A amT2`MQp2X
RXkX "bB+ BM;`2/B2Mib
hvT2 A amT2`MQp2 ?p2 #22M 2ti2MbBp2Hv mb2/ b biM/`/ +M/H2b BM i?2 Tbi #2+mb2 i?2v /BbTHv M2`Hv mMBp2`bH HB;?i +m`p2 T`Q}H2- r?B+? Bb TQr2`2/ #v i?2 /2+v Q7 HKQbi ∼ 0.6 J! Q7 56LB BMiQ 56*Q UBi iF2b M2`Hv 9 /vbV- r?B+?
2p2MimHHv /2+vb BMiQ 5662 7i2` `Qm;?Hv 114 /vbX 6`QK M Q#b2`piBQMH TQBMi
Q7 pB2r- i?2 bT2+i` Q7 hvT2 A amT2`MQp2 /Q MQi b?Qr Mv 2pB/2M+2 Q7 > Q` >2 HBM2b M/ Ĝ i i?2 T2F Q7 i?2 HB;?i +m`p2 Ĝ i?2v +H2`Hv /BbTHv bBHB+QM M/ bmHT?m` HBM2bc Hi2` QM- bi`QM; B`QM HBM2b TT2`X
hrQ Hi2`MiBp2 KQ/2Hb ?p2 #22M T`QTQb2/ BM i?2 HBi2`im`2 iQ 2tTHBM i?2 Q`B;BM Q7 hvT2 A amT2`MQp2X "Qi? b+2M`BQb BMpQHp2 M U2H2+i`QM@/2;2M2`i2V *P r?Bi2 /r`7- #2HQM;BM; iQ #BM`v bvbi2K M/ 2t+22/BM; Ĝ i +2`iBM TQBMi Q7 Bib 2pQHmiBQM Ĝ i?2 *?M/`b2F?` Kbb HBKBic i?Bb ?TT2Mb 2Bi?2` #2+mb2 Q7 i?2 Kbb i`Mb72` 7`QK KBM b2[m2M+2 Q` `2/@;BMi +QKTMBQM Ui?Bb Bb i?2 bQ@ +HH2/ bBM;H2@/2;2M2`i2 b+2M`BQV- Q` #2+mb2 Q7 i?2 +QH2b+2M+2 rBi? M Qi?2` *P r?Bi2 /r`7- r?B+? HQbi M;mH` KQK2MimK pB ;`pBiiBQMH rp2 2KBbbBQM M/ 2p2MimHHv +QHHB/2/ rBi? i?2 +QKTMBQM- mM/2`;QBM; i?2 bK2 T`Q+2bb U/Qm#H2@
/2;2M2`i2 b+2M`BQVX AM #Qi? +b2b- /2~;`iBQM `i?2` i?M /2iQMiBQM Bb M22/2/
iQ B;MBi2 +`#QM M/ T`QT;i2 i?2 #m`MBM; 7`QMic BM 7+i- /2~;`iBQM Bb i?2 QMHv K2+?MBbK rBi? r?B+? i?2`2 Bb bB;MB}+Mi Mm+H2QbvMi?2bBb Q7 +?2KB+H 2H2K2Mib HB;?i2` i?M i?2 B`QM@T2F QM2b- BM ;`22K2Mi rBi? r?i Bb Q#b2`p2/ BM i?2 bT2+i` Q7 HH hvT2 A amT2`MQp2X
AM bmKK`v- i?2 K2+?MBbK 7Q` i?2 2tTHQbBQM Q7 hvT2 A amT2`MQp2 BMpQHp2b HQr@ M/ BMi2`K2/Bi2@Kbb bi`b- r?B+? Kmbi ?p2 Kbb2b BM i?2 `M;2 0.8 ! m ! 8J!- BM Q`/2` iQ ?p2 *P r?Bi2 /r`7 BM >m##H2 iBK2X h?2`27Q`2- hvT2 A
amT2`MQp2 +M 2tTHQ/2 Qp2` H`;2 `M;2 Q7 ivTB+H iBK2@b+H2b 7`QK i?2 bi` 7Q`KiBQM 2p2Mi- 7`QK KBMBKmK iBK2 /2Hv r?B+? Bb M2`Hv ∼ 30 Jv` Ui?Bb Bb HKQbi i?2 HB72iBK2 Q7 M 8 J! bi`V iQ >m##H2 iBK2X AM i?2 bBM;H2@/2;2M2`i2
b+2M`BQ- r2 ?p2 irQ HQr@ M/ BMi2`K2/Bi2@Kbb bi`b, i?2 KQbi KbbBp2 QM2 Bb i?2 bQ@+HH2/ T`BK`v bi`- r?B+? #2+QK2b M 2H2+i`QM@/2;2M2`i2 *P r?Bi2 /r`7 M/ ++`2i2b Ki2`BH 7`QK i?2 +QKTMBQMc i?2 H2bb KbbBp2 QM2 Bb i?2 bQ@+HH2/
b2+QM/`v bi`- r?B+? 2pQHp2b KQ`2 bHQrHv M/ 722/b i?2 UT`BK`vV *P r?Bi2 /r`7
rBi? Bib bi2HH` rBM/X h?2 KBM T`K2i2` r?B+? `2;mHi2b i?2 dz+HQ+FǴ 7Q` i?2 2tTHQbBQM Bb ;Bp2M #v i?2 Kbb Q7 i?2 b2+QM/`v bi`- bBM+2 i?2 Kbb T`BK`BHv /2i2`KBM2b i?2 HB72iBK2 Q7 i?2 bi`X
h?2 hvT2 A amT2`MQp `i2 BM ;HtB2b +M #2 i?2Q`2iB+HHv `2+Qp2`2/ B7 r2 FMQr i?2 ;Htv bi` 7Q`KiBQM ?BbiQ`v M/ i?2 /Bbi`B#miBQM 7mM+iBQM Q7 i?2 /2Hv iBK2b rBi? r?B+? hvT2 A amT2`MQp2 2tTHQ/2 bBM+2 i?2 7Q`KiBQM Q7 i?2 Q`B;BMH #B@ M`v bvbi2KX JMv .2Hv hBK2 .Bbi`B#miBQMb ?p2 #22M T`QTQb2/ BM i?2 HBi2`im`2 iQ `2T`Q/m+2 i?2 hvT2 A amT2`MQp `i2 BM ;HtB2bX
"v /2}MBiBQM- i?2 .2Hv hBK2 .Bbi`B#miBQM Q7 hvT2 A amT2`MQp2- .h.A(τ
i?2 ;Htv bi` 7Q`KiBQM `i2 rBi? bmBi#H2 .2Hv hBK2 .Bbi`B#miBQM- b 7QHHQrbX RA(t) = CA min(t,τ2) , τ1 dτ.h.A(τ ) ψ(t − τ), URXR8V
r?2`2 τ1 M/ τ2 `2 bmBi#H2 pHm2b /2T2M/BM; QM i?2 /QTi2/ b+2M`BQ 7Q` i?2
.h.X h?2 MQ`KHBxiBQM +QMbiMi CA Bb `2Hi2/ iQ i?2 7`+iBQM Q7 bi`b BM i?2
#BM`v bvbi2Kb ;BpBM; `Bb2 iQ hvT2 A amT2`MQp2 M/ Bi +M p`v 7`QK ;Htv iQ ;Htv M/ Bb mbmHHv /2i2`KBM2/ #v }iiBM; i?2 Q#b2`p2/ T`2b2Mi@/v hvT2 A amT2`MQp `i2 BM ;HtB2b- r?2M Q#b2`p2/X
AM oBM+2MxQ- Jii2m++B aTBiQMB UkyReV- r2 ?p2 T`QTQb2/ M Hi2`MiBp2 K2i?Q/ iQ +QKTmi2 i?2 +QMbiMi CA BM i?2 KQ/2Hb- T`QpB/BM; M 2t+2HH2Mi ;`22@
K2Mi rBi? i?2 Q#b2`p2/ +?2KB+H #mM/M+2 Tii2`Mb Q7 i?2 JBHFv qvX AM T`iB+@ mH`- Q#b2`piBQMb bm;;2bi M BMi2;`i2/ MmK#2` Q7 hvT2 A amT2`MQp2 BM ;HtB2b-r?B+? Bb ∼ 1 amT2`MQp T2` M⋆ = 103J! Q7 bi2HH` Kbb 7Q`K2/- rBi? b+ii2`
Q7 #Qmi 2@10 `QmM/ i?Bb pHm2 U"2HH 2i HX- kyyjc JQx 2i HX- kyR9VX h?2`27Q`2-i?2 MQ`KHBxiBQM +QMbiMib Q7 h?2`27Q`2-i?2 .2Hv hBK2 .Bbi`B#miBQMb +M #2 +?Qb2M bQ b iQ 7mH}HH i?Bb +`Bi2`BmKc BM T`iB+mH`- i?2 +QMbiMi CA BM 2[miBQM URXR8V +M #2
+QKTmi2/ #v `2[mB`BM; i?2 7QHHQrBM; +QMbi`BMi, +tG 0 dt ′R A(t′) +tG 0 dt ′ψ(t′) = 2amT2`MQp2 103J ! , URXReV
r?2`2 i?2 MmK2`iQ` `2T`2b2Mib i?2 BMi2;`i2/ MmK#2` Q7 hvT2 A amT2`MQp2 M/ i?2 /2MQKBMiQ` i?2 iQiH ;Htv bi2HH` Kbb 7Q`K2/X
AM r?i 7QHHQrb- r2 #`B2~v bmKK`Bx2 i?2 KBM rvb r?B+? ?p2 #22M T`QTQb2/ BM i?2 HBi2`im`2 7Q` KQ/2HBM; i?2 hvT2 A amT2`MQp `i2 BM ;HtB2bX
.Qm#H2 /2;2M2`i2 b+2M`BQ ++Q`/BM; iQ:`2;;BQUkyy8V- i?2 /Qm#H2 /2;2M2`i2 b+2M`BQ Ĝ BM r?B+? hvT2 A amT2`MQp2 Q`B;BMi2 7`QK #BM`v bvbi2Kb Q7 r?Bi2 /r`7b HQbBM; M;mH` KQK2MimK pB ;`pBiiBQMH rp2 2KBbbBQM Ĝ +M #2 KQ/2H2/ #v bbmKBM; .h.A(τ ) ∝ 1/τ Ub22 HbQ hQiMB 2i HX kyy3V- r?B+? T`QpB/2b HKQbi
i?2 bK2 }MH `2bmHib 7Q` i?2 hvT2 A amT2`MQp `i2 BM ;HtB2b b i?2 .2Hv hBK2 .Bbi`B#miBQM T`QTQb2/ #va+?ƺM`B+? "BMM2v UkyyNVX
"BKQ/H .2Hv hBK2 .Bbi`B#miBQM M Qi?2` rB/2Hv mb2/ .2Hv hBK2 .Bbi`B#miBQM
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