HAL Id: inria-00072003
https://hal.inria.fr/inria-00072003
Submitted on 23 May 2006
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents
A Statistical Model of Skewed Associativity
Pierre Michaud
To cite this version:
Pierre Michaud. A Statistical Model of Skewed Associativity. [Research Report] RR-4582, INRIA.
2002. �inria-00072003�
ISRN INRIA/RR--4582--FR+ENG
a p p o r t
d e r e c h e r c h e
THÈME 1
A Statistical Model of Skewed Associativity
Pierre Michaud
N° 4582
Octobre 2002
+,.-0/1/2-43,.57698;:=<
> ?A@CBDEGFIHJ1KLD1MONQPRD STKLUQKVSW@CBD1K
XZYW[]\VD SG^%_TX`
HaM]bAb*[cYdSaeQDYWD1fg?hD1YWfg?ADji;kl0mOnco4FqpGf SL[OrAYLDo]scsco&FqoOmbAMOtOD1K
uRvw1xCy7zh{OxO| > ?A}~KRbAM]b;DCYbhYLD2KdD1i0SWKMKdSWM]SL}KVSW}~f1M]TB[QeQD1T[]KLD SL McKLKL[Qf }M7SL}~}SVUc}f(SW}BfCMcfg?h}iAt
MOiAeKLOD1D2e McKLKL[Qf }M7SW}}SVUO }SW?MOiDCBbh?AMcKd}K[OiKLOD1D2eMOKWKd[Qf }M7SW}}SVUODKd?h[7#SW?AM7SKdDCSd
McKLKL[Qf }M7SW}}SVU}K%ih[]SDCfC}D1i0S%?hD1iRSW?hD&[cYL}~iht]KLD SKL}1D&}~K f ~[cKLDSL[SL?hDfCMcfg?hD&Kd}~CDcD&YLDCDCY%SL[
SW?h}K9MOKSL?hD&Q*$¡ 7¢L£]0¤O¥ ¦C§ QbAYL[crhD1B¨$DaKd?h[7SW?AM7S9}f(SL}~B f1MOfg?h}~ihtG}~K=iA[]S9MbAYWMcf(SL}fCMOQKd[cNQSW}[ci
SW[SW?hDNhiA}SG[cYL}~iht]KdDCSabhYW[Orh~DCB#DC}SL?hD1Y1MOSW?h[ONhtc?}~f SL}~B©f1MOfg?h}~ihtD1BNhM7SWD1KaNh9McKLKL[Qf }M7SL}~}SVU
[cY%[OYW0}~ihtO KLD SgKBNAfg?ªM]YWtODCYSW?AM]i«SW?hD¬}~f SL}~BrANQ®;D1YR}SWKLDCVjpi«SW?hD¬[OSL?hD1Y?AM]i*e«D¬KL?h[7
SW?AM7So7¯ M°U¬KLODC%D1eMOKWKL[fC}~M]SL}~}SVUD1BNhM7SWD1K&Nh~McKLKL[Qf }M7SW}}SVU¨[OY%[OYW0}~ihtO KLD SgKNAb±SW[?AM] SL?hD
f1MOfg?hDaKL}~CDO0M]iAeSW?AM7S²]¯ M°UKLOD1D2e McKLKL[Qf }M7SW}}SVU}~KZMOB[0KVSZD2³cNA}7M]~DCi0SZSW[&Nh~;McKLKL[Qf }M7SL}~}SVUc]}¯Dc
KLOD1D2eMOKWKd[Qf }M7SW}}SVU}KaMbhYgMOf SL}fCM]KL[O~NQSL}~[OiSL[SW?hDNhih}Sa%[OYW0}~ihtO KLD S%bhYL[crh~DCB¨
´µ0¶¸·W¹º y7»wO| bhYW[Qf D1KWKL[OY2f1MOfg?hDcfC[OiQ¼A}f(SgKCZKLOD1D2eMOKWKd[Qf }M7SW}}SVUO$KLD SL McKLKL[Qf }M7SW}}SVUO0}f(SW}B
f1MOfg?h}~iht*KdSWM]SL}KVSW}~f1M]B[QeQDC
w | ^D SLSLDGJ SLN*eQDbAYLJ2KdD1icSWDTNhiRB[QeQ@1D&KdSWM7SW}~KdSL}³0NhDehDTSWYL[c}~KBJ SL?A[ehD1K%eQDYWJ1KL[O~NQSW}[cieQD1K
fC[OiQ¼*}SgKehM]iAKaD2K BJ1B[O}~YLD2KfCMOfg?AD&ehD1KbhYW[Qf D2KLKLDCNhYgK9MOKWKL[fC}~M]SL}~}SWJGb*M]YDCi*KdD1Brh~D1K1QDf1MOfg?hDeQD
}f(SW}BD1K19D SMOKWKL[fC}~M]SL}~}SWJrh}~MO}~KLJCDcT[ONAKB[ci0SLYW[OiAK³cNADRMcKLKL[Qf }M7SL}~}SWJbAM]YDCiAKLDCBrh~D1KiD2KVS
b*MOKNhihDBJ SL?A[ehDDCf1MOfCD~[OYgKL³0NhDDih[cBrhYWDe[crQ\VD SgK! KVSW[QfgODCY&ehMOiAKGDfCMcfg?hDD2KVSbhYW[Qfg?hDeQD
M¨SgM]}~~DReQN f1MOfg?hDc4^DCMihJ1fCD1KWKd}SLD1YWMO}SMOKWKd[QfC}~M]SL}~0}SLJfC[OBbh~@ SWDO9³0Nh}¡D2KVSeQ}fC}~D" BD SLSLYWDRD1i
[D1NhYLDZD1ibhYgM7SW}~³0NhDc#T[ONAKB[Oi0SLYW[OiAK³0NhD¡~D fCMOfg?AD¡eQD%0}f(SW}BD2Kb;DCYWBD S9eQDKL}~BNh~DCYMOKWKL[fC}~M]SL}~}SWJ
fC[OBbh~@ SWDb;[ONhY Nhiih[OBrhYWDGe$[Orh\VD1f(SgK%r*D2M]NAfC[ONhbbhN*KtcYWMOiAe³0NhDGDf1MOfg?hDGeQD}f(SW}BD&NA} B%CBDO
^D1b*D1iAehMOicS2 f DCSdSLDBJ SW?h[QeQDihDYLJ2Kd[cNQSbAMcKGDbhYW[Orh~@CBDO'&(McKLKL[Qf }M7SW}}SLJrA}~MO}~KLJCDcDCi4YWDC7MOiAfg?hDO
YWJ1KL[ONhS¡bhYgM7SL}³0NhDCBD1icS%DGbhYW[Orh~@CBD'=D1~DTD2KVS bhYW[fg?ADTeQDGMOKWKd[QfC}~M]SL}~0}SLJf [OBbh~@ SWDM°cD1faKLDCNh~DCBD1icS
SWYL[c}~K r*M]iAf1KC
)!º x2w · {+* w| bAYL[Qf D2KLKLDCNAY1fCMcfg?hDOhfC[OiQ¼A}SWK1AMcKLKL[Qf }M7SL}~}SWJOB[QeQ@C~DKVSgM7SW}~KdSL}³0NhD
= ¡
^%Mcfg?h}~iht¬}KM4rAMOKL}~fSLD2fg?hih}³cNAD[OTf [OBbhNhSLDCYM]Ygfg?h}SLD2f(SLNAYLDcN*KdD2e[OYBM]iUeQ}®¸DCYWDCi0SbhNhYWb*[0KdD2KC
D1ihDCYgM]~~UKLb*D2M]}iAtAcMfCMOfg?AD}~KMeQD1eh}~f1M7SLD2eBDCB[OYWU?h}fg?SLYW}~D1K9SW[?h[c~eM&%[OYW}ihtO KLD S9[OeAM7SWM
}~i0SL[M~}B}SLD2eKdbAMcf Dc¡> ?hDehM]SWMBM°URr;DMOf SLNAMO bhYW[OtOYgM]B eAM7SWMAhbhYW[OtcYWMOB }iAKdSLYWNAf SL}~[OiAK1AMOeAeQYLD2KLK
SWYWMOiAKL~M]SL}~[OiAK11rAYWMOiAfg?bhYWD1eQ}f(SW}[ciAK12BD1B[cYLUeQD1b*D1iAeQD1iAf UbhYWD1eh}~f SL}~[OiAK1°[OY?AM7SWDCOD1Y1=^[OYWYLD2Kdb;[OiAeQ
}~ihtcUc$SW?hDf1MOfg?hDBM°Ur*DRM4ehM7SgM¬fCMcfg?hDO=M]i}~iAKdSLYWNAf(SW}[cifCMOfg?ADO=M¬> &GM%>GD SWfO~¬>[¬r*D
tcDCihD1YWMO.%D&}~$KLb*D2M]}~i SL?h}KaKdSLNAeQUR[]9M%[OYW}iAt]KdDCS[]a C§ .¦(
`[cBD SL}~BD1KMfCMOfg?ADG}KaiA[]STM]rADSW[?h[OeSL?hD?h[O~D[]ZM%[OYW0}~ihtO KLD S2QrhNQST[cihU MbAM]YLSa[]=}S1
!iSL?A}~KaKdSLN*eQUOh[crQ\VD1f SWK%SW?AM7SSW?hDfCMcfg?hDGf1M]ihiA[]S?h[Oe¨M]YWDYWD D1YLYWD1eSL[McK#"¦g¦(0¤ C§ .¦($!VeQD1MO~UO
MRfCMOfg?ADBNAKdSB}~ih}B}~CDSL?hDi0NABr;DCYT[O9B}KWKd}~iht[crQ\VD1f SWK1A?h}~DbhYW[70}eQ}~ihtMOKaMOKdSM]i¬McfCfCD1KWKTMOK
b;[cKWKL}rh~DSL[[crQ\VD1f SWK~U}iht¨}~iSL?hDf1MOfg?hDc&%a[7%DCcDCYGSL?hD2KdDSV%[¨YLD2³0Nh}YWDCBD1icSgKM]YWDM]i0SL}~ih[OB}f]R_
McKVS¡McfCfCD1KWK=SL}~BDO0?h}~fg?}~Bbh}~D1K¡?AM]YgeQ M]YWD%KL}~BbA}f }SVUO0}~KZtODCiADCYgM]~U[crQSWMO}iAD1eM7S9SW?hDTfC[cKdSZ[] Kd[cBD
iNhBr*D1Y[OB}KWKd}~iht4[crQ\VD1f SWK1('}KWKd}~iht¬[crQ\VD1f SWKBM°Ur;DeQNhD¨KL}Bbh~USL[¬SW?hDR%[OYW}iAt]KdDCSr*D1}iht
M]YWtOD1Y&SL?AMOiSL?ADRfCMcfg?hDO¬> ?hD1U±BM°UM]Kd[r;DehNhDSL[4SL?hDR?AMOKL?h}~ihtNhiAf SL}~[OiN*KdD2eSL[4McfCf D2KLKSL?hD
f1MOfg?hDc}i¨?h}fg?fCMOKLDSL?hD1UM]YWDfCMO~D1e)g 7*%C B}~KWKLD1K1
`D1OD1YWMOKL[O~NQSL}~[OiAKT?AM°ODr;DCD1i¬bhYW[Ob;[cKLD1e SW[YLD1B[7ODf [ciQ¼A}f(SGB}KWKdD2KC`QD SdMOKWKL[fC}~M]SL}~}SVU}~KTSL?hD
B[0KVS=}~ehDC~UNAKdD2eKL[O~NQSW}[ci ! SZf [ciAKL}~KdSWK[];Kdbh~}SdSL}~ihtGSL?AD f1MOfg?hD}i0SW[GKLDCcDCYgM]0rAM]ih¸7D2MOfg?r*M]ihbhYW[]
}eQ}~ihtMb;[cKWKd}~rh~DG~[QfCM]SL}~[Oi[cY MOi[crQ\VD1f S MOiAer*D1}ihtMOf1f D2KLKLD1eR}SW?RSW?hDKLMOBDG?AMcKd?h}~ihtNhiAf(SW}[ci
DKL?h[7}~i±SW?h}KKVSWNAeQU4SL?AM]S1$?A}~DD ®¸D1f SL}~ODM7SKL[O~}ihtSL?ADf [Oih¼A}~f SB}KWKbhYW[Orh~DCB ?ADCiSL?hD
%[OYW}iAt]KdDCSa}~KTBNAfg?4KLBM]~D1YaSW?AM]iSW?hDf1MOfg?hDc;KLD SL McKLKL[Qf }M7SW}}SVUR}KTih[OScDCYWURD fC}D1icSG?hDCi4SL?hD
%[OYW}iAt]KdDCS%KL}1DG}Kf ~[cKLDSL[SL?hDf1MOfg?hDKL}1DO=±D&YWD D1Y SL[SL?h}KMOK%SW?hDQA¡ 7¢L£]0¤O¥ ¦ §CbhYW[Orh~DCB¨
+}~f SL}~BfCMcfg?h}~iht-,m/.$?AMOKar*D1DCi¨bhYW[Ob;[cKLD1e}iE00OsMOKMKL[O~NQSW}[ciRSW[SL?hDfC[OiQ¼*}~f SaB}KWKbhYW[Orh~DCB
}~ieQ}~YLD2f(SBM]bhb;D1efCMcfg?hD1K11! Sf [ciAKd}KdSWKG[]MOeheh}ihtM¨KLBMO=NA~U0 McKLKL[Qf }M7SL}~ODrhNQ®¸DCYSL[¨?h[Oe±SL?hD
2}~f SL}~BK3[]f [Oih¼A}~f SWK}~iSL?hDRBMO}if1MOfg?hDc±DRKL?h[7 }iSL?A}~KKVSWNAeQU¬SW?AM7S}f(SW}BfCMOfg?A}iht¨D1BNQ
M7SWD1KNh~$MOKWKL[fC}~M]SL}~}SVU[cY%%[OYW}iAt]KdDCSWKBNAfg? ~MOYLtcDCYSL?AMOiRSW?hD&}~f SL}~B rhNQ®¸DCY}SWKLDCVAfC[Oi54AYWB}iAt
bAYLD10}~[ON*Kd~UbhNhrA}Kd?AD1e¨D PQb;DCYW}BD1icSgM][crAKdD1YL7M]SL}~[OiAK16%a[7%DCcDCY20}f(SW}BfCMcfg?h}iAt}KaiA[]STMbhYgMOf SL}fCM]
KL[O~NQSW}[ciRSW[SW?hDNhih}Sa%[OYW0}~ihtO KLD S bhYW[Orh~DCB¨
`cDC%D1eMOKWKL[fC}~M]SL}~}SVU¬ MOK}i0SLYW[QeQNAfCD1e}~iE070O²¨MOKM]i}~BbAYL[7cDCBDCi0S[7OD1YKLD SdMOKWKL[fC}~M]SL}~}SVUc
> ?AD r*MOKL}~f}~ehD1Mf [ciAKd}KdSWK[]G}iAehD PQ}iAt±SW?hDrAMOihQK}SL?ªeQ}®¸DCYWDCi0S?AMOKL?h}~iht4NAiAf(SW}[ciAKC`cDC%D1e
McKLKL[Qf }M7SW}}SVU MOKZ}~ih}SL}M]~UbhYW[Ob;[cKLD1e[cY¡}~iAKdSLYWNAf(SW}[ciAKMOiAeehM]SWMf1MOfg?hD2K8,~E1s/. 9! S MOK¡~M]SLDCY KL?h[7i
SW?AM7S¨KdcDC%D1eMOKWKL[fC}~M]SL}~}SVUc%NAih}~ODKdDCSdMOKWKd[QfC}~M]SL}~0}SVUO }KRYgM7SW?hDCY}i*KdD1iAKd}SL}~OD4SL[ KdbAM]SL}M]G[Qf1M]~}SVU
DC®;D2f(SgK:,o. ¸MOiAe}KTDCf }~DCi0ST[cYTBM]iU SVUb*D2Ka[]Z[crQ\VD1f SWK;,EcE<. ='[cYLD1[7ODCY2AKLDCcDCYgM] }~BbhD1BDCi0SWMOrhD
YWDCbA~Mcf DCBD1icS b;[O~}~fC}D2KD1YLDGbhYW[Ob;[cKLD1eQiAD1M]YW~UMOK tO[[Qe MOK &$H?>@,~EOE<.
> ?h}KKdSLNAeQU ?h}tc?h~}tc?cSgKSW?hD MOf SSW?AM7SSW?hD¨D f }~DCiAfCU[]&KdcDC%D1eMOKWKd[QfC}~M]SL}~0}SVU4}K}iA?hDCYWDCi0SL~U
KdSWM]SL}KVSW}~f1M] M]iAe eQ[D1Kih[]SYLD1U[Oi M]iUfg?*M]YgMOf(SWDCYW}~KdSL}fCK[]SL?AD MObhbh~}~f1M7SL}~[Oi*KC±±D¨Kd?A[7 SW?AM7So°
M°UKdcDC%D1eMOKWKL[fC}~M]SL}~}SVUDCBNh~M]SLD2KNA McKLKL[Qf }M7SW}}SVU±[cY[cYL}~iht]KdDCSWKNhb SW[?AM]TSW?hDfCMOfg?AD
KL}~CDO*M]iAeSL?*M7S²]¯ M°URKLODC%D1eQ McKLKL[Qf }M7SL}~}SVU}KaM]~B[cKdSaD1³0Nh}~7M]~DCi0SSL[NAMOKWKd[Qf }M7SW}}SVUO¡> ?*M7Sa}KC
KLOD1D2eMOKWKd[Qf }M7SW}}SVU}KaMbhYgMOf SL}fCM]KL[O~NQSL}~[OiSL[SW?hDNhih}Sa%[OYW0}~ihtO KLD S%bhYL[crh~DCB¨
`D2f(SW}[ci oeQD1KWf YW}~r*D2K[ONhYM]bhbhYW[cMcfg?4SW[¨SL?hDbhYW[Orh~DCB M]iAebhYWDC~}~B}~iAMOYLU4McKLKLNhBbQSW}[ciAK&[OYSL?hD
B[QeQD1.`D2f(SL}~[Oi*KR²h l MOiAemeQD2KLfCYL}~r;D¨SL?hD±B[QeQD1a[cYRKdDCSdMOKWKd[QfC}~M]SL}~0}SVUO%}~f SL}~B f1MOfg?h}~ihtM]i*e
KLOD1D2eMOKWKd[Qf }M7SW}}SVURYWD1KLb*D2f(SW}cDC~UO¸}SW?M[Qf NAK&[ci¬SL?ADNhiA}S%[OYW0}~ihtO KLD STbAYL[crhD1B¨`D1f SL}~[Oim
bAYL[crhD1B¨ ?h}fg?}KKLb*D2f }4*fSL[¬KLOD1D2e McKLKL[Qf }M7SW}}SVUOM]iAe%DKVSWNAeQU¬SW?hYWDCDeQ}®;D1YLD1icSbA~Mcf DCBD1icS
MOtc[OYW}SW?hBKC
= ' = ) $#
_TSW?h[ONAtO?4B[OSL}~°M]SLD2erU¬MB}~fCYL[0M]Ygfg?h}SLD1f SLNhYWD³0NhD2KVSW}[ci ;SL?h}K&KVSWNAeQUBM]cD1Kih[McKLKLNhBbQSL}~[Oi*KT[ci
KLb;D1f }4*fbhYW[Ob;DCYLSL}~D1K[]SL?AD¨MObhbh~}~f1M7SW}[ciAK}~OD Kdb*M7SL}M]MOiAeSLD1Bb*[cYWMO%~[QfCMO}SVU(9ihD1}SW?hDCY[Oi SL?hD
SVUb;D[]a[crQ\VD1f SWK1¨±D MOeQ[cbQSM¬KdSWM]SL}f]SL}~BDC~D1KWKb;[O}~icS[]a}D1 tc}cDCiM¨%[OYW0}~ihtO KLD SKL}1DOD
fC[OiAKL}eQDCYM]~¡SW?hDb;[cKWKL}rh~DfC[Oi54*tONhYgM7SW}[ciAK[cYcDCUQKRf [ciAKVSW}SWNQSL}~iht±SL?hD %[OYW0}~ihtO KLD SMcKD2³cN*M]~U
~}~ODC~UOM]iAe±DeQDCSLD1YLB}~ihDM]iM]bhbhYW[°PQ}BM]SL}~[Oi4[O¡SW?hDM°OD1YWMOtODiNhBr*D1Y[]B}KLKL}~iht¨[OrQ\VD2f(SWK1±D
ehDI4AihDGSW?hD1 °§C¢ 1¤c§:"¦g¦(0¤=g¢7 ¯ 7z McKZSW?hDM°OD1YWMOtODaiNhBr*D1Y%[OB}KLKL}iAt[crQ\VD1f SWK eQ}~0}eQD2e
rUSW?hD%[OYW0}~ihtO KLD S Kd}~CD! "#$#&%(')
DMOKWKLNhBD&SL?hDf1MOfg?hD}KaKLbh}ST}~i0SL[+*jrAMOihQKC*jr;DC}~ihtSW?hD °¦g¦C ].2-O§ ¤O¢L§g§ -,¡MOfg?¨rAMOih
. / 0%#&#*1'=}~KGMOf1f D2KLKLD1eSL?AYL[cNhtO?¬MR?AMOKL?h}~ihtNhiAf SL}~[OiAK324Z?A}~fg?¬BMObODCUQKT[Oi0SL[Rf1MOfg?hD[QfCM]SL}~[OiAK
[ciSL?hDrAM]iA;9! 5*768%OcSL?hDGfCMcfg?hDa}K¡KWM]}eSL[r;D 7¢L§C.¥" 99h§7:,MOfg?[Orh\VD1f(S%}iSL?hD[cYL}~iht]KdDCS
?*MOK;* b*[0KLKL}~rhD&~[QfCM7SW}[ciAK1Q[OihD&[Oi D1Mcfg?r*M]ih¸
> ?hDbhYW[Orh~DCB [] KdSLNAeQU}~ihtSW?hDT[QfCfCNhbAMOiAf U[] Mf1MOfg?hDBM°Ur*D0}~DC%D1eMcKZMrAMOKd¯}~iQ NhYLi*K<MO°M
r*M]~~Kd¯}~iQ rh}~iAKbhYL[crh~DCB¨_ rAM]~%}~KMOKWKd[QfC}~M]SLD1eSW[±M±Kd}~ihtO~DR[crQ\VD1f S}~iSL?ADR%[OYW0}~ihtO KLD S2rhNhSMOi
[crQ\VD1f STBM°URr;DMcKLKL[Qf }M7SLD2eSL[RKLDCcDCYgM] rAMOK1a_ai¬NhYWitOD1ihDCYgM]~~URYWDCbhYWD1KLDCi0SWKMRfCMOfg?AD[Qf1M7SL}~[Oi$A[OY
M¨KLD S[O%~[QfCM7SW}[ciAK1M]i*e±?*MOK&YW[0[cB [OYMOiNhih~}B}SLD1eiNhBr*D1Y[] rAMOKCDbhNhS}i0SL[D2MOfg?NhYWi
SW?hDrAMOK f [OYWYWD1KLb*[ciAeQ}~ihtSW[SL?hD[crQ\VD1f SWK%SL?*M7S { º
#*.»
r;DGbA~Mcf D1e}~iSW?hDfCMcfg?hDG[QfCM]SL}~[Oi=[cYaKdDCS [O
~[QfCM]SL}~[OiAK>=MOKWKd[QfC}~M]SLD1eSW[SL?hDNhYWi =±DTMcKLKLNhBDTMO*b*[0KLKL}rADTfC[Oi54*tONhYgM7SW}[ciAK=[]rAMOKZ}~i0SL[NhYWiAKM]YWD
D2³0Nh}bAYL[crAM]rh~DO> ?hDBMObhbh}~iht[]%MfCMcfg?hDbhYW[OrAD1B [cicSW[KLNAfg?¬rAMOKd¯}~iQ NhYLi*KB[ehDC=YLD2³0Nh}YWD1KaSL?hD
[c~[7}~ihtMcKLKLNhBbQSW}[ciAK
EOM]~ b;[cKWKd}~rh~DG%[OYW}iAt]KdDCSWK[]?OD1UQK }i= "@#&#ACB/%D'$MOYLD&D1³0Nh}~bhYW[OrAMOrh~D
oQ%[OYD1ODCYWUfCMcfg?hD~[QfCM]SL}~[OiFEGH "#$#IJBK%('¯¡SL?hD¬KL}1D[OTSW?hD¨bhYLD1}BMOtOD2MLON
4HP
ERQ6TSVUXW
2 4 P
UYQZ6[E]\D1³0NAM]K1A_^DI
²h1A_^DIa`J?
> ?hD 4AYWKdSTMOKWKLNhBbQSL}~[Oi }~KTbAM]YLSa[]=[ONAYaM]YWtONhBDCi0S2AMOK %DM]}~B SW[Kd?A[7SL?*M7SaSW?hDiAM7SWNhYWD[]=SL?hD
DCfCMOfCU[]9KdcDC%D1eMOKWKL[fC}~M]SL}~}SVU}~KKdSWM]SL}KVSW}~f1M] MOiAeeh[0D2K ih[]Sa~}~DG}~iM]bhbA}fCM]SL}~[OiAK fg?AMOYWMcf(SLD1YL}KdSL}fCK1
> ?AD¬KdD2f [Oi*e MOiAe SL?h}~YWe«McKLKLNhBbQSL}~[Oi*KC%}.DO~%SL?AM]SRM]~abhYWDC}~BM]tOD2K?AM°OD SL?AD4KLMOBD¨KL}~CD¬M]i*e SW?AM7S
Ab^cI }~K BNAfg? tOYWD1M7SWDCYSL?AMOid?9Q}~K ?hD1YLDG[cYSW?hDD1³0Nh}~bhYW[OrAMOrh}~}SVU[]=KdDCSWK []OD1UKSW[SWYWMOiAKL~M]SLDG}i0SL[
D2³0Nh}bAYL[crAM]rh}~~}SVU[]$r*M]~~Kd¯}~iQ NhYWiAK¡fC[Oi54AtcNhYgM7SL}~[Oi*KC_aKd[}Bbh~}~fC}S?hD1YLD}~K¡SL?ADGMcKLKLNhBbQSL}~[OiSL?AM]S-24
}K%MtO[[Qe?AMcKd?A}ihtNhiAf(SW}[ci Q}¯DcQM]iRNhirh}~McKdD2e[OiADO=> ?hDTSW?h}~YWeMcKLKLNhBbQSW}[ci}~KtOD1ihDCYgM]~~USWYLNhDT[OY
B[0KVSabhYgMOf SL}fCMO$fCMcfg?hD&bhYL[crh~DCBKC*MOK SW?hDOD1UKLbAMOfCD}~Ka[]SLD1i¬KdD1OD1YWMO[OYgeQDCYgK []=BM]tOiA}SWNAeQDM]YWtOD1Y
SW?AM]iSL?AD[cYL}~iht]KdDCS1
egfMhcikj_lnm5oqpco>rnstiZl
Cvuwyxnxz{|k}~
ln
s C
ln(stigrj3lnti
C(
ligm o
Ccy
ig
z
igl
Ct iZmlngi
>
i:hcigjmto>i
! " #%$
!iRSL?h}K KdD2f(SW}[ci D}i0SLYW[QeQNAfCDTSW?hDT[cYLBNhMSL?*M7S }~¸r*DGNAKLD1eRMcKZSW?hDGrAMOKL}~K[]KdNhr*KdD2³cNADCi0S KdSWM]SL}KV
SW}~f1M]B[QeQDCK1
DMOKWKdNhBDI eQ}KVSW}iAtONh}Kd?*M]rh~DNAYLiAKGM]i*eM? eQ}KdSL}~ihtONh}KL?AM]rh~DrAM]~~K1_ rAM]~=}~KGBM]bhb;D1e±[Oi0SW[
M±YgM]iAeQ[cB NhYWi «±DMOKWKLNhBDD2MOfg? NhYLi fCMOi ?h[c~e M]i Nhih~}~B}SLD2e i0NABr;DCY[]GrAM]~KC > ?hDCYWDM]YWD
I'&±eQ}KdSL}~iAf(SfC[Oi54*tONhYgM7SW}[ciAKCDRMcKLKLNhBDM]~qI(&f [ci54AtONAYWM]SL}~[OiAK&M]YWDD1³0Nh}~bhYW[OrAMOrh~Dd'¬M7PQ%DC~
%[cSWCBM]ihi KVSgM7SW}~KdSL}fCK ,² .(9D&KLD1MOYWfg?SL?hDM°OD1YWMOtODiNhBr*D1Y)%*a[O$NhYWiAK ?h[OeQ}~ihtD PhMOf SL~U,+rAM]~KC
}SL?-+. "vSL?hDf1MOKLD/+6 "T}~K=KL[OBD SW}BD1KYWD DCYWYWD1e&SW[}~iSW?hD~}SLDCYgM7SWNhYWDMOKSW?hD#7¦W¦( ; Cy9G]
9¸¢L §" ,² . DKL?h[7}~i_abhb;DCi*eQ}P_ SL?*M7S1°[cY Ia`G%M]iAev?d` %c0)%*ZM]bAbhYL[°PQ}~BM7SLD2K MaX=[c}~KWKd[ci
eh}~KdSLYW}rANQSL}~[Oi
)%*1 I P &
2 Q *
+354 L76
8
VEy
'¬[OYWDC[7OD1Y1=[cY+I M]YWtOD DCih[cNhtO?$ZSL?AD¬eQ}~KdSLYW}~rhNQSL}~[Oi []SL?hD¨iNhBr*D1Y[]&NhYLi*K?A[OeQ}iAt+4rAM]~K}~K
fC[OiAfCDCi0SLYgM7SWD1eM]YW[ONhiAeRSL?ADBD2M]i hSL?AM]Sa}~K1QB[cKdSTf [OiG4AtONhYgM7SW}[ciAK M]YWDf ~[cKLDTSW[SL?hDM°cDCYgM]tODGf [ci54At]
NAYWM]SL}~[Oi
9 :;</=#>?@A>$B C= /
DA[OYWBNhMOK eQD1YL}~OD2eRMOiAM]~U0SL}fCM]~~U}iSL?h}KaKdSLNAehURM]YWDB[0KVSWURMcKdUBbQSW[]SL}f&M]bhbAYL[°PQ}~BM]SL}~[OiAK1$D}~
N*KdD '[ci0SLD4^%M]YW[KL}BNhM7SL}~[Oi*KSL[7M]~}~eAM7SLDDCPQb*D1YL}~BDCi0SWMO~USL?AD1KLD¬M]bhbhYW[°PQ}BM]SL}~[OiAK1¡}¯DcKL?h[7
SW?AM7S2ZNhiAehDCYSL?hD¨MOKWKLNhBbQSL}~[OiAK[]SW?hD B[QeQD1.M]iAe[OYYLD2M]~}~KdSL}fRf1MOfg?hD Kd}~CD2KC=SW?hDR[cYLBNhMOKM]YWD
YWD1McKd[ciAM]rAUTMcfCfCNhYWM]SLDc !iM]~O[cNhY'[cicSWD%^%M]YW~[aKd}~BNhM7SW}[ciAK11%D¡eQYgM°F%y"FEZYgM]i*eQ[OB f [OiG4AtONhYgM7SW}[ciAK1
MOiAe f [OBbhNhSLDM°OD1YWMOtODa7MONhD2K YW[OB SW?hD1KLDf [OiG4AtONhYgM7SW}[ciAK1
G
%5h$$% &('g*)
ª}SL?KdDCSdMOKWKd[Qf }M7SW}}SVUO0M]~hSL?ADTrAMOihQK¡M]YWDa}~iAeQDCPD2e}SL?SW?hDKWM]BDa?AMcKd?A}ihtNhiAf(SW}[ci %aD1iAf DTSL?hD
KLD S=[];b*[0KLKL}~rhD%~[f1M7SW}[ciAK[OY9M]i[OrQ\VD2f(S<¦C§C ?~ 7¯ 7A¦(][OY9KL?h[OYLS>$}KfC[OBbh~D SWDC~UeQD SWDCYWB}ihD2erU
}SWK%~[QfCM7SW}[ciR[ciMbAM]YLSL}f NhM]Y%rAMOih¸hKLM°UrAMOihEO & DCS<I r;DSW?hDGSL[OSWM]¸iNhBr*D1Y%[OfCMcfg?hDG[QfCM]SL}~[OiAK
MOiAe?4SW?hDiNhBr;DCY[O[Orh\VD1f(SgK}~iRSW?hD[cYL}~iht]KLD S1=> ?hDiNhBr*D1Y[]eh}~KdSL}~iAf(SGKdDCSWK [][Qf1M7SL}~[Oi*K }~K
I-H6[I^t*$!iSL?ADYLD1BMO}iA}iht[O$SL?A}~KaKdSLN*eQUOh%DeQDI4*ihD&SL?hD%[OYW}iAt]KdDCSd¯SL[]fCMcfg?hD KL}1DYWM]SL}~[
I 6 ?
I
DfCMOiBMObRSW?hDKdDCSdMOKWKd[QfC}~M]SL}~0}SVUbhYW[Orh~DCB [ci0SL[MrAM]~~Kd }iQ NhYWiAK%bhYW[Orh~DCB McK[O~[7aK19> ?hDCYWD&}~K
[cihDGrAM]~¸b*D1Y%[crQ\VD1f S MOiAe[cihDGNhYLib*D1YKdDCS%[O$[Qf1M7SL}~[Oi*KCZ`[D?AM°OD?±eQ}KdSL}~ihtONh}KL?AM]rh~DTr*M]~~K M]i*e
I-H¡eh}~KdSL}~ihtONA}~KL?AM]rADNhYWiAK1DYWD1f1M]~SL?AM]SZD1MOfg?NhYWi?AMOK=YW[[OBj[cY¡M]iNhih~}~B}SLD2eiNhBr;DCYZ[O;r*M]~~K>(
DA[OYNhYLi*Kaf [cicSgM]}~ih}~ihtJ+rAM]~Ka}SL?'+LK[**%DfC[ONhi0S+bB=*B}KLKL}iAt[OrQ\VD2f(SgKC8>Kd}~ihtJDA[OYWBNhM¨EO
YWDCbA~Mcf }~ihtI }SL?MIMH2Q%D[OrhSWM]}~iSL?hDiNhBr*D1Y[]=Nhih[QfCfCNhbh}~D1e fCMOfg?ADG~[QfCM]SL}~[OiAK
)º x µ zQy *º º » µ *
I E &o l Gn E E &o l Gn E
sAE2s shs]l o sAscscm]² sAscsOshE sAscsOsOs sAscsOsOs shs]lcnOl shsOs s shsOsOs0o shsOsOscs shsOscsOs
sAoOs shs0AE0 sAs0o]s sAscshE1n sAscsOsOs sAscsOsOs shs0O² shscoQE0 shsOscoOs shsOsOscs shsOscsOs
sA²cs sh~E1²c²0 sAsOlc²O² sAscs E sAscsOs]l sAscsOsOs sh~E1² E shs]lOl 0 shsOs 0 shsOsOs0m shsOscsOs
sAl0s sh~E7lAE sAsE1² sAsAE1nO² sAscscoQE sAscsOsOs sh~E]mOn shs ]o shshE0 shsOscocm shsOscshE
sAmOs shoQEcE sAE2sco sAsc²cm 0 sAscs sAscsOscm shoQE1²AE sh~E1sO² shsO² shsOsOnOl shsOscsOn
sA s sho7l sAE2²]lcn sAs0m 00 sAsAE1n sAscsO²hE sho7lcncs sh~E1² m shs E°m shsco]sc² shsOsc²0
sA]s sho 0 sAE 0O² sAscnOn sAsc² 7n sAsAEOE1s sho]nOscn sh~E7sc² shs0Os0m shsO²0 shshE°o]n
sAncs sh²hE2scm sAoOsO²co sAE°oQE2o sAs l0m sAs0o]nhE sh²hEOE sho]s]l0² sh~E2o]²0o shs n shsO²cs0
sA0cs sh²O²700 sAoO² sAE°m n sAs70Onco sAs0m sh²]lcs sho]² 0 sh~E2m]n sh~E1sOs shs sOs
Ecscs sh² 7² sAo 0O² sAE0O²On sAE2² E sAs70 l sh² 0 sho7s sh~E0cm]l sh~E1²0 shs070co
EcE2s sh²00o sA²cshE sAoO²Os0 sAE 0cm sAE1l0o sh²0O²0m sh²Osco]l sho]²cocm sh~E1nhE°m sh~EClm
EcoOs shlAE7s sA²c²co]s sAo nOs sAoco]²Os sAE0O²cm shlAE sh²O²O²cs sho 0c² shoOo7l sh~E00m]²
Ec²cs shlc²70 sA² E sA²cs]lcs sAo m]n sAo]lc² shlOlcsOl sh² oco sh²OscmhE sho Oo sho7lmQE
Ecl0s shl ECl sA²cn0O² sA²c²On sA²cs 7² sAo0hE2m shl E0 sh²0Oscs sh²O²00m sh²OsOnc² sho 00oOm
EcmOs shlcnAE sAl*E2m sA²E1² sA²Ol m sA²c² s shlcncohE shlAE ² sh² ]oco sh²]lOo sh²O² l
>M]rADE+Z`D SL McKLKL[Qf }M7SW}}SVU":d
P
* I Q[OrQSgM]}~ihD1e}SL? '[Oi0SWD&^%M]YW~[KL}BNh~M]SL}~[OiAK¡[Oi¨o7lcsfCMOfg?AD
~[QfCM]SL}~[OiAKb~D SbAM]YLS>%M]i*e [crQSWMO}ihD2eR}SL? DA[OYWBNhMOKoM]iAe¨²dYW}~tO?0SbAM]YLS>
LON
*
I H PI *Q
*
+3 4 L
P* B+Q
T[]SWDSW?AM7S&SW?h}~KM]bhbhYW[°PQ}BM]SL}~[Oi¬}K&BD1M]iA}ihtONh9[OiAU[cYIMHv` %O¸SW?AM7S}K15* I±DehDI4AihD
SW?hD 7§C¢ C¤0§Q; Cy9;§=g¢ C¯ 7 z McK%SL?hDiNhBr*D1Ya[]=Nhih[Qf1f Nhbh}~D1e¨f1MOfg?hD&~[f1M7SW}[ciAKeQ}~0}eQD2e
rUSW?hD&SL[OSWMOi0NABr;DCYa[OfCMcfg?hD&[Qf1M7SL}~[Oi*KI!Dh[OYaMKLD SL McKLKL[Qf }M7SL}~ODGfCMcfg?hDOSW?hD ¬}~K
O
P
* I Q!1
4
L
LON
* PI *Q
*
+3
P
%1B
+
* Q .o
> ?ADb¬f1M]i¨r*D[crQSWMO}iAD1eRYL[cB SL?hD O
P
* I Q-6K%1B
%1B O
P
* I Q
I ²
>M]rAD2KEKL?h[7aK SW?hDR4[crQSWMO}iAD1e¨}SL?'DA[OYWBNhMOKToM]iAe²[cYT7M]YW}~[ONAKT°MONAD1Ka[O
I
MOiAed* ! S
MO~KL[KL?h[7aK=SL?AD3d[OrhSWM]}~ihD2e}SW?&'[Oi0SLD&^%M]YW[KL}BNhM7SL}~[Oi*K9[OiRo]lcsfCMcfg?hD~[f1M7SW}[ciAKC ! SfCMOi
r;D[Or*KdD1YLcD1eSW?AM7SDA[OYWBNhMobAYL[7}eQD1K MYLD2MOKL[OiAMOrh~UMOfCfCNhYgM7SLD&M]bAbhYL[°PQ}~BM7SL}~[Oi$
DfCMOieQ}~KdSL}~ihtcNh}~KL?¨SL?AYLD1D}i0SLD1YLD2KVSW}iAtYLD1tO}~[OiAK%[cY I 7M]~NhD2K I KQ I %OAMOiAe I f ~[cKLD&SL[
%cDA[OY I KQ*SL?hD]d±[O¡MReQ}~YLD2f(SL¯BM]bAb*D2efCMOfg?AD}~KGf ~[cKLDSW[SL?*M7S&[]¡MNh~~UcMOKWKL[fC}~M]SL}~OD[cihDO
?A}~fg?¨fCMOir;D[OrQSgM]}~ihD1eYW[OB DA[OYWBNhM²
P
I Qq6
" [OY I %
%1B
N
[OY I K[%
!i¨[OSL?hD1Y[cYWehK10SL?hDr*D1ihDI4hSa[O}~iAfCYLD2MOKL}ihtSL?ADMOKWKd[Qf }M7SW}}SVUReQ}~B}ih}KL?hD1Ka³0Nh}fgUMOK %DG}~iAfCYLD2MOKLD
SW?hD%[OYW0}~ihtO KLD S Kd}~CD&r;DCUO[ciAe I 68%ODh[cY I %cSL?hD ¬fCMOir;DM]bAbhYL[°PQ}~BM7SLD2eRMcK
d
P
*I Q
PI
*Q
P
* %yQ 3
l
}¯DO~OSW?hDRr;DC?*M°OD1KZ}~ODTMb*[cUih[cB}M]*}i I []$ehDCtOYWDCD1*$!ibAM]YLSL}f NhM]Y2
P
% I Q=?AMOKM#4AYgKVS
ehDCYW}7M7SW}cDT?h}fg?}K%b*[0Kd}SL}~OD&M7S I 6C"A9> ?h}~K%%D1MO0iAD1KWKZ[]eQ}YWD1f Sd BM]bhb;D1efCMOfg?AD1KZ}K%D1¸ih[7i
SW?h[ONAtO? ih[]Sa}~i0SLNh}SL}~ODchM]iAe }Kf ~[cKLDC~UYLD1~M]SLD2eSW[SW?hDKL[]fCM]~~D1erA}YLSL?AeAM°UbAM]YgMOeh[°P¸
! i3
t
mi
KF#"b<C rj
$%$
$ o>
lno>r
$
io>m
<C m
o>ml
lnm;o'&syr
lntrni
>)(+*y,(+"y
o C
.-0/
~ i ~
lnrnr ml1&stro i2-43
~
ogj
o>mm
glo lpVl
j3ro i
$<C6587
9;:
iZo>t
$+<
l1&<o
l
<C_
ig
&1im
i
i
$~
l
&
%$
i: o>
iZr
o
l
<C m 7
0 0.1 0.2 0.3 0.4
1 2 4 8 16 32 64
D}~tONhYWDE+Z`DCSdMOKWKd[Qf }M7SW}}SVU Zd4MOKMNhiAf SL}~[Oi¨[] * [OY I 68%+Dh[cYLBNhMOKaoM]iAe ²
µ x ¹º y h· w µ x y º v *µ !itOD1ihDCYgM]¯SV[O¯ M°U4KLD SdMOKWKL[fC}~M]SL}~}SVU±}~KKdNhfC}D1i0S[OY
B[0KVSbhYW[Orh~DCBK&}~iO[O~}iAt M %[OYW0}~ihtO KLD SGBNAfg?KLBMO~DCYGSW?AM]iSL?ADfCMOfg?ADO1%a[7%DCOD1Y1YWD1KLNhSWK&[ci
>M]rADEKd?h[7 SW?AM7S%?hDCiSW?hDT%[OYW}ihtO KLD SZ}KZ[OSL?hDGKLMOBDa[OYgeQDCY¡[]BM]tOiA}SWNAeQDMcK9SW?hDf1MOfg?hDaKL}1DO
P I Q¡}KaKL}tcih}4*fCMOicS29> ?hD&[cYWKdSafCMOKLDG}~K I 6 %c
D}~tONhYWDE KL?h[7aK SW?hDf NhYWOD%[]ASL?hD1dMOKMaNhi*f(SL}~[Oi[] * [cY I 6 %c7[OrQSgM]}~ihD1e}SW?-DA[OYWBNhMOK
oMOiAe¨²h¡> ?hD ¬eQD2f YWD1McKdD2KKd~[7UMcK1*j}~iAf YWD1McKdD2K_.MOK~[OiAtMcK1I ` * q,ZcDCi}SL?d*76A
SW?hD }KRKdSL}~GKd}~tOiA}4*f1M]i0S1 M]YW[ONAiAeªmR `QD SdMOKWKL[fC}~M]SL}~}SVU}Kih[OSM]iªDCf }~DCi0SRKd[cNhSL}~[Oi [OY
YWDCB[7}~iht±fC[OiQ¼A}f(SgK?hDCi SL?hD¨%[OYW}ihtO KLD SKd}~CD }~KfC[0KdDSL[±SL?AD¨f1MOfg?hD¨KL}1DOD¨YLDCDCYSL[SL?A}~K
bAYL[crhD1B MOK%SW?hDQA¡ 7¢L£]0¤O¥ ¦C§ bAYL[crhD1B¨
zQx x µ º » µ
*T»
º µ w º
x
µ
*z *
.µ w
z *
{czQx
.º { zhy°zA{]x
µ y w1x
{Ow
`[cBDDCB
bA}YW}~f1M];ih[7D2eQtODM]r;[ONQS fCMOfg?AD1KZ}K%ih[]S%D PQbhM]}~ihD1eRrUSW?hDGB[QeQD1.DA[OY%D PhM]Bbh~DOcSL?hDG%DC~ ih[7i
YWNh~Da[]¸SW?0NABr 0?h}fg?KVSgM7SLD2K9SW?AM7SMo° %M°UKLD SL McKLKL[Qf }M7SW}cD%}~iAKdSLYWNAf(SW}[ci7eAM7SWMf1MOfg?hD}KZD2³0Nh}7MOD1icS
SW[MeQ}YWD1f Sd BM]bhb;D1ef1MOfg?hDTSV}~fCDTM]YWtOD1Y1c}~Kih[]Sf [ci54AYWBD1erUSL?hDGB[QeQDC¯9> ?h}~K%KL?h[7aKZSW?AM7S KdN*fg?
2YWNhDI3 eQD1b*D1iAehK[Oi«M]bAbh}fCM]SL}~[Oi fg?AM]YgMOf SLDCYW}KVSW}~f1KbAYL[crAM]rh~UKLbAM7SW}~MO[QfCMO}SVU}~i SW?h}~Kf1MOKLDy M]i*e
KL?h[cNh~eih[OSar*D&SgM]OD1i McK MbhYL[cb*D1YdSVU[O=KLD SL McKLKL[Qf }M7SW}}SVU}~i¨tOD1ihDCYgM]¯
Z
_!}~f SL}~B rhNh®;D1Y}KTMKLBM]~ Nh~~UcMOKWKL[fC}~M]SL}~ODf1MOfg?hDc*M]~[OiAtcKL}~eQD&SW?hDBMO}i¬f1MOfg?hDcA?h}fg?¬bhNhYWb;[cKLD
}KRSL[?h[c~eªSL?hD±[OrQ\VD2f(SWKRSL?*M7S MOYLD 2}f(SL}~B;3¬[]f [ciQ¼A}f(SgK}~iSL?AD4BM]}~i fCMcfg?hDO +}~f SL}~B fCMcfg?h}iAt
MOKTbhYW[Ob;[cKLD1e¨rU a[cYLBM]ic[ONAbhbh} ,m .=McKMOi4M]SLD1YLiAM]SL}~ODSW[ KLD SL McKLKL[Qf }M7SW}}SVUO8! S&}~Kb;[cKWKd}~rh~DSL[
tcDCihD1YWMO}~CD}~f SL}~B f1MOfg?h}~iht SL[¬M¨KLD SdMOKWKL[fC}~M]SL}~ODBM]}~if1MOfg?hDc&%a[7%DCcDCY20}f(SW}BfCMcfg?h}iAt BMOOD1K
KLDCi*KdD[Oih~U[OYM4~[7 MOKWKd[Qf }M7SW}}SVU4BM]}~i f1MOfg?hD M]iAe M¬}f(SL}~B rhNh®;D1YBNAfg? KLBM]~D1YSW?AM]i SL?hD
BMO}i¨fCMcfg?hDO
X9M]YgM]BDCSLDCY
I
KVSW}~ eQD1ih[]SWD1K¡SL?hDGYWM]SL}~[[O SL?hD&%[OYW}ihtO KLD S¡KL}~CD_?¬[7OD1YZSL?ADBM]}~if1MOfg?hDGKd}~CDbI
D}~icSWYL[QeQN*f DGSL?hDYgM7SW}[[OSW?hD}~f SL}~B rhNQ®¸DCYaKL}~CD I[7ODCY%SW?hDBM]}~i¨f1MOfg?hDc
]6 I
I
'¬}~KWKd}~iht[Orh\VD1f(SgKM]YWDGSL?hD[crQ\VD1f SWK%SL?*M7SaihD1}SW?hDCYSW?hDBM]}~ifCMOfg?ADGiA[OY SL?AD0}f(SW}B rhNQ®¸DCYTf1M]i ?h[Oe
> ?ADGfC[OYWYLD2Kdb;[OiAeh}ihtM°cDCYgM]tcDB}KWKd}~ihtYWMcf(SL}~[OiR}~¸r*D&eQDCiA[]SLD2e Z> ?ADih[OSWM]SL}~[Oi
P
* I Q
YWD D1YWKSW[SL?hD YgMOf(SW}[ci [O[crQ\VD1f SWKSL?*M7SSL?ADBMO}i«fCMcfg?hD¨}~KiA[]SM]rh~D¨SL[?h[c~e ! S}K[OrhSWM]}~ihD2e
}SL? Dh[cYLBNh~McK%oM]iAe²h9_TKWKLNhB}ihtB[cKdS%rAMOKd¯}~iQ NhYLi*K¡f [ci54AtcNhYWM]SL}~[OiAKM]YWDfC[0KdDTSL[SL?hD&M°OD1YWMOtOD
fC[Oi54*tONhYgM7SW}[ci 0SL?hD fCMOir;DM]bAbhYL[°PQ}~BM7SLD2eRMcK
P* I Q d
P"
P
* I QZB
I Q .m
> ?AD}i0SWDCYWD1KdSL}~ihtbAM]YgM]BDCSLDCY&[cY}f(SL}~B f1MOfg?h}~iht}KSW?hD7M]~NhD I P
Qr*D1[7 ?h}fg?SL?hD }~K
iNh~¯Z> ?h}~Kf1M]i r*DehD SLD1YLB}~ihD1e¨McK%SL?hDKL[O~NQSL}~[Oi []SL?ADGD2³0NAM7SW}[ci
d
P
*I Q:B
I
6["
DA[OYaMeQ}~YLD2f(SL¯BM]bAb*D2ef1MOfg?hDcQD&tODCSaMKd}~BbhDMObhbhYW[°P}~BM7SW}[ciRNAKL}iAt DA[OYWBNhMl}SW? *[68%
I P Q
DA[OY}~iAKdSWMOiAf Dc}SL?=6 "@#"@%ODtcD S I "#$% A !i[]SW?hDCY[cYWehK1$[ciSL?h}KDCPhM]Bbh~DO0}f(SW}B
f1MOfg?h}~iht¬D1BNhM7SWD1KNhMcKLKL[Qf }M7SL}~}SVU±[OY[cYL}~iht]KLD SWKNhbSL[F% 4SL}~BD2KSW?hDR}f(SW}B rhNh®;D1YKL}1DO
%T[7D1OD1Y1c[OY I K I QSW?hD }iAfCYLD2MOKLD1K%OD1YLUMOKdS }SL? I
+}~f SL}~BfCMcfg?h}~iht&}~K9cDCYWUD fC}D1icS9[cYZYWDCB[7}iht 2rh}YLSL?*ehM°UbAMOYWMceQ[°P 3¡B}KWKdD2K=DCiAfC[ONhi0SLD1YLD2e}SL?
eh}YWD1f Sd BMObhbh}~ihtA$%T[7D1ODCY200}f(SW}BfCMcfg?h}iAt}K ih[]SaMbhYgMOf SL}fCM]KL[O~NQSL}~[OiSL[SW?hD&Nhih}Sa[cYL}~iht]KdDCS
bAYL[crhD1B¨MD}tcNhYWDoKL?h[7aKSL?hD+dMOK&MNhiAf SL}~[Oi[]¬[OY I 6 %MOiAe¬[cYMeh}YWD1f Sd BMObhb;D1e
o7¯ M°UM]i*elO¯ M°UKLD SdMOKWKL[fC}~M]SL}~ODGBM]}~i¨f1MOfg?hDc ! SaYWD1³0Nh}~YLD2KMl] %M°UKLD SL McKLKL[Qf }M7SL}~ODGBM]}~ifCMOfg?AD
MOiAeM40}f(SW}B rANQ®;D1Y¨E]m¨SL?hD BM]}~i fCMcfg?hDKL}~CD[cYSW?hDd SW[4r;D1f [cBD}~iAKL}tcih}4;fCM]i0S2`N*fg?
fC[Oi54*tONhYgM7SW}[ciR}K ih[[cihtOD1Y}iSL?hDKLbh}~YL}Sa[]}~f SL}~BfCMcfg?h}~ihtA
+A$% &('g;)
`QODC%D1eQ McKLKL[Qf }M7SL}~}SVU MOK}i0SWYL[QeQNAfCD1erU_aiAeQYWJ`D1CihD2f-,~E1sQ9o%EOE<. MOKMOi}BbhYW[7OD1BDCi0S[7OD1Y
KLD SL McKLKL[Qf }M7SW}}SVUO_ * %M°U KdcDC%D1eMOKWKd[QfC}~M]SL}~ODf1MOfg?hD±f [Oi*Kd}KVSgK[]* SgM]rh~D1K[] I^t* D1icSWYL}~D1K
D2MOfg? _TK&}~iSL?hDf1MOKLD[O%KLD SL McKLKL[Qf }M7SL}~}SVUc M]i[OrQ\VD2f(S?AMcK* b;[cKWKL}rh~DfCMOfg?AD~[QfCM]SL}~[OiAK1[OihD[ci
D2MOfg?rAM]iA;&%T[7D1OD1Y1NhiA}~ODKLD SL McKLKL[Qf }M7SL}~}SVUcD1Mcfg?±rAMOih . 0%#&#*1'¡}~iM¨KdcDC%D1eMOKWKd[QfC}~M]SL}~OD
f1MOfg?hD}KZ}~iAeQDCPQD1e}SL?RM&?AMcKd?A}ihtNhiAf(SW}[ci]2
4
eQ}®¸DCYWDCi0S¡YW[OB!SL?AM]S[]¸SW?hDa[]SW?hDCYrAM]iAK1=_ªBM7\V[OY
fC[OiAKLD1³0NhD1iAf D[OSL?h}KKdcDC}~iht[]SW?hD}~iAeQD PQ}~iht¬NAiAf(SW}[ciAK}KSW?AM7SSL?hDtO~[OrAMO¡fCMcfg?hD[QfCfCNhbAMOiAf U
ehDCb;DCiAehK[OiR?h}~fg?R[]SL?AD3* b;[cKWKL}rh~DT~[QfCM]SL}~[OiAK¡}~Kfg?h[cKLDCi[cY%MOi[OrQ\VD2f(S2DYWD D1YZSW[SL?A}~K%MOK¡SL?hD
0 0.1 0.2 0.3 0.4
0.001 0.01 0.1 1
w=1 w=2 w=4
D}tcNhYLDo ^NhYLcD1K%[]SL?AD McKMNhiAf(SW}[ci¨[] [OY I 68%
# " #%$ #%/= < # $ < !## @ C=<# #%
> ?ADbh~Mcf D1BD1i0SZbhYL[crh~DCB! MOKZ}YWYLD1D17M]i0S=[OY¡KLD SL McKLKL[Qf }M7SL}~}SVUc Dh[OYKdcDC%D1eMOKWKd[QfC}~M]SL}~ODf1MOfg?hD2KC
}S&}~K&M f YWNAf }M]bhYW[Orh~DCB¨ !i±SL?h}K&KVSWNAeQUO$D}~Zf [ciAKd}eQD1YTSW?hYLD1DbA~Mcf DCBD1icSGBD SL?A[eAK G[OYgeQDCYWD1eQ
r*M]ihQK=bhMOfCDCBDCi0S< c³0NAMOKL}¯[cbQSL}~BM]bhMOf D1BDCi0S3 (cM]iAe}SLDCYgM7SW}cDYgM]i*eQ[OBbA~Mcf DCBD1icS
0> ?AD4AYgKVS¡bA~Mcf DCBD1icS%M]~tO[OYW}SL?hB bhYL[7}eQD1K¡M]iD PhMOBbADa[O[OihDC¯bAMcKLKZBDCSL?h[Qe9> ?hDGKdD2f [ciAe
bA~Mcf DCBD1icSaMOtc[OYW}SW?hB¨0[ciRSW?hD&[]SW?hDCY?AMOiAeQbhYW[7}~ehD1K MbAYWMcf(SL}fCMO¸[7%DCY r;[ONhiAeR[OY SW?hD d[]M
KLOD1D2eMOKWKd[Qf }M7SW}cDf1MOfg?hDc > ?ADSL?h}~Yge¨bh~Mcf D1BD1i0SM]~tO[cYL}SL?hB#f [cYLYWD1KLb*[ciAehK%SW[SL?hDbhYWMcf(SW}~f1M]$f1MOKLD
,o. AM]i*e fC[OiOD1YLtcD1KZSL[7 M]YgeRM]i¨[cbQSL}~BM]bhMOfCDCBDCi0S1
9 !## C= =F B 00A A! #%$
DfCMOiBM]bSL?hDKLODC%D1eQ McKLKL[Qf }M7SL}~}SVUGbhYW[Orh~DCBj[ci0SL[MGrAMOKd¯}~iQ NhYLi*K$bhYW[OrAD1BMOK[O~[7aK1> ?ADCYWD
}K[OihDaNAYLib;DCY fCMcfg?hDa~[QfCM7SW}[ciMD2MOfg?NhYWi?AMcKZYW[0[cB![cYMOiNhih~}B}SLD2eiNhBr;DCY[] rAMOK>(9> ?hDKLD S
[]¸M]~QNhYWiAK=}~K=b*M]YLSL}SL}~[OihD2e}~icSW[b*«eh}~K\V[O}~i0S9KLNhrAKLD SgK=[] I+^*«NhYLi*KC7D2MOfg?KdNhr*KdDCS9fC[OYWYLD2Kdb;[Oi*eQ}iAtaSL[
MeQ}®¸DCYWDCi0SarAM]iA;_j[cYL}~iht]KLD S [] ?±[crQ\VD1f SWK }KaMOKWKd[QfC}~M]SLD1eRSL[+* rAMOKd¯}~iQ NhYLi*Kf [OiG4AtONhYgM7SW}[ciAK1
[cihDf [OiG4AtONhYgM7SW}[ci¬b;DCYrAMOih¸ ,¡MOfg?[Orh\VD1f(S&}K&YLD1bhYLD2KdD1i0SLD1e¬r0U* rAMOK1}.DO~ [OiADrAMO9b;DCY&rAMOih
fC[Oi54*tONhYgM7SW}[ci TDMOYLDKVSWNAeQU}~ihtSW?hDtc[crAM]=f [ci54AtcNhYWM]SL}~[OiYWD1KLNhSL}~ihtYW[OB#SL?ADfC[OBrA}iAM]SL}~[Oi¬[O
SW?hD1KLDb*jb*D1Yd rAMOihf [ci54AtONAYWM]SL}~[OiAK1> ?AD&SL[]SgM]iNhBr*D1Y[]=tO~[OrAMO¸fC[Oi54*tONhYgM7SW}[ciAK%}~K
I
*"!
&
# :
iOtro>gi&1i
C(
$c
rni&=lnmylnm
lC @$ &
iOstmsto>r
$
igtro>gi&1i
C(
$c
rni& lC
o>
igm
7 : iOstmso>r
$
igtro>gi&1i
CD
$c
rni+&1mOyi o>rnm
~ l 3
i o>
o m
&1i
cy
ig
mo
$
iZo>ggigmmi &
%$
i $ igVsti
CD rj
o
C_>
i $m 7 i
C<C i:o;&
<Ct
mikpDi
$
o>r
cV
ig
m &stm Z
i<ikpVln
ig
$ &
i<o>
i o
c(
Ct<C
3
%$
o>grni
$
igtro>gi&1i
CD
rnlnkj
rnlnhci%$
A&q $ lnim
lnti
C(
lj
i
cy
ig
$ii $i C
gi
i rnio>m
$ i(sti
CD rj 7 l
3
s $ m o ln l
CD O>
pylni
~Z
o>rnr
cV
ig
m5o
$
i('ig(so>r) o C
i $ iZlnm
Cy $ igtro>gi&1i
C(
(stigm
l
<C7
)º x µ zQy *º º » µ *
I &o G² l &o G² l
sh~E1s shsOsOs shsOsOscs shsOscsOs shsOscs0 sAscsOsOs sAscsOsOs
sho]s shsOscm shsOsOsAE shsOscsOs shsOs s sAscsOshE sAscsOsOs
sh²Os shshE 0 shsOsOs70 shsOscsOs shshE sAscshE1s sAscsOsOs
shlcs shsO²]l shsOs]lo shsOscsco shsO²0m]n sAscs]l sAscsOsO²
shm]s shscm]ncn shshE2o0 shsOsAE shs shE sAsAE1²cm sAscsco]s
sh s shsOn shsco]n70 shsOs shsOn70co sAs0o 0On sAscsOnco
sh7s sh~E2o]s0m shscmOo shscohECl sh~E2ohE sAs0m]²0 sAs0oOo
shnOs sh~E2mOmOs shsOn]lo shs]l E sh~E2m m sAscncm]² sAsOl]m
sh0Os sh~E0Os sh~E2o]s0m shsOnAE1s sh~E00o]² sAE°oOo]s sAscnco]²
EOsOs shoOo7l sh~E s sh~E2oco 0 shoOoOnO² sAE oQE sAE°o7l0o
EO~E1s sho ²cs sho]scocm sh~E n sho ²On sAoOsO² sAE7sOs
EOo]s sho 0 sho7lOl*E shoQE°m]n sho 0cnO² sAo]l0mQE sAohE n
EO²Os sh²O²Os70 sho]n]l sho E sh²O²AECl sAoOncm7l sAo o7l
EOlcs sh² ohE sh²co]²0o sh²Os0mOo sh² ²Os sA²0o]²On sA²cscm
EOm]s sh²0coco sh²cm 0Ol sh²]lm sh²00o]n sA²0m 00 sA²Ol s
>M]rADao =`OD1D2eMOKWKd[Qf }M7SW}}SVU [crQSWMO}iAD1e}SW?&'[Oi0SLDT^%MOYL~[Kd}~BNhM7SW}[ciAK=[Oio7l0s&fCMOfg?AD
~[QfCM]SL}~[OiAKb~D SbAM]YLS>%M]i*e [crQSWMO}iAD1e}SW? DA[OYWBNhM YW}tc?cSbAMOYdSc
DMOKWKdNABDM]~=SL?AD1KLDtc[crAM]9fC[Oi54AtcNhYgM7SL}~[Oi*KMOYLDD1³0Nh}~bhYW[OrAMOrhDc:!iMOeheh}SW}[ci4SW[SW?hDf [ciAeQ}SL}~[OiAK
~}KVSWD1e}~i`D2f(SW}[cio [cYD2MOfg??AMcKd?h}~ihtTNhiAf(SW}[ci24;KdD1bAM]YgM7SWDC~UO2SL?h}KMOKWKdNhBbQSW}[ci[]hD2³0Nh}bAYL[crAM]rh}~~}SVU
YWD1³0Nh}~YWD1KSL?AM]S[OYD1ODCYWU+* Nhbh~D[]9fCMcfg?hD~[QfCM]SL}~[OiAK
PE N
(#$#&# E Q(
4 N 2 LON 4HP
E 4Q! 6 A
P
I^t*Q
` ?
}SL?CA#SW?hD¬Kd}~CD[OTSW?hD¬OD1U KLbAMOfCDO Dh[cYKd?h[cYdS2%D¨}~TKWM°USL?*M7SSL?AD?AMOKL?h}~ihtNhiAf(SW}[ciAK]24
MOYLD 2[OYLSL?A[OtO[ciAM]3( pYLSL?h[ctO[Oi*M]~}SVU}KaB[cYLDYWD1KdSLYW}~f SL}~OD&SW?AM]iSW?hD* §C¢(¥7£ ]¦9A§C¢g¦g ]bAYL[cb*D1YdSVU
McKehDI4AihD2e}~i ,o. iAMOBDC~UO9SW?AM7S
P2 LON
4 PE 4QM2 LON
PE
QQ+6
2
[cY .6 %a[7%DCcDCY2
[cYdSW?h[Otc[OiAMO}SVU }Bbh~}D2KG}i0SLD1Yd rAMOiheQ}~KLb;DCYgKd}~[Oi ª}SW?4SW?h}~K&McKLKLNhBbQSL}~[Oi±[]%[OYLSL?h[ctO[Oi*M]~}SVUchSL?AD*
b;DCYL rAM]ihr*M]~~Kd¯}~iQ NhYWiAKf [ci54AtONAYWM]SL}~[OiAK?AM°OD&}~iAeQD1b*D1iAeQDCi0STKdSWM]SL}KVSW}~f1KC
# # % @ " #%$ #%/=! #"%$'&
> ?AD&p %X MOtc[OYW}SW?hB }K%M[OiAD bAMOKWK¡MOtc[OYW}SW?hB¨DGMcKLKLNhBDSW?AM7S%SL?hDGrAMOihQKMOYLDaiNhBr*D1YLD2eYW[OB
%%SW[*0MOiAeSL?hDT[OrQ\VD2f(SWK9MOYLD iNhBr*D1YLD2eYW[OB % SL[?9=> ?hDp %XM]~tO[OYW}SL?hBjSWYL}~D1K9SL[bhMOfCD%[crQ\VD1f SWK
( 6 % #$#&#?±KLD1³0NhD1icSW}~MO~UO9! SL?hD~[QfCM]SL}~[Oi[OY[Orh\VD1f(S
(
[Oi rAMOihM%&}~KaDCBbQSVUOQ%D&bhMOf D[crQ\VD1f S
(
[ci
r*M]ih!%c DCKdD%D~[[O4M]SGr*M]ih h;! SL?hD~[QfCM7SW}[ci4[cY&[OrQ\VD2f(S ( [ci4rAMOih }K&DCBbQSVUO$DbhMOf D
