Classical Mechanics Problems - UNLV Physics

The problems and answers can be posted on the web in pdf format. The problems have been suggested mainly by Goldstein problems, but have all been writ...

5 downloads 875 Views 236KB Size
     !  "$#&%('*),+.-/+10&230546 798:6(; '*<=;>"$0@?.#:4 ACBD0FE:A GH0&IKJL0&M('*<=NPO05
W W

W W W

W

W

W W W

7XNY4A ?30ZE:A #F[\4C#RB ^7 ](_.`*'*;C8('aE1[Lbc;05`*dfeg? ]h_.`*'*;C8(0@)i jZkRk(l +1#RE1](#:46 l

 

  !"$#%'&)(+*$,-.0/132546879;:<>=?<+@BA"C+D-EFG@H@$I?JK@BCL:M"=QC+E;N[fa@^DU=Y^@BU.S%<+N+E-:.M^h"CO@$F"UEV?<g-F"A"NoXG=0TBEa=UUF_EEMPfCO:.NON+EM1F|c;V E$Z}Y^:T$EM N+X_=NN+X%E>:S"E0=<JK@BCah"CO@$F"UEV?Y^@BUS"<+NOE:M‚25†COS‚ˆS%:.NO:@$MG9-gGh}ZGu$‡%g_h"C+@BF"UEV‰r$Z^4=0T$E=QNjVc?C+ES_=DNO:.@BM =M%S…N+X"ENOX"E<+@BA"C+D-EN+EHN+<=M"S @$NOX"ECPCOEJKECOEM"DE<J5@$UU@f‹NOX"E'D-@$MHN+EMHNO<ZwŒi$E-M"ECl=QUD:Nl=QN+:@$M :<^A"<+AG=QUU.cHg E$Z iGZgHw;C[s[|Ž$Ž3JK@BCwLC[J5I$EM]gHh}Z"|ŽBŽ"Z wN;N+X"EEM"S8@QJN+X"EFG@H@BI{=COE'N[f@ =h%hGEM%S":DE<Z^\X%E^zGC+D-@T$EC=i$EP=QM"S NOX"EL@BC+S"ECO:M"i@J N+X%E>h%C+@$F%U.E-V <F|c?DXG=h"NOEC+<;:<M"@$NcBENzGMG=QUZ}L7XHcH<+:D<˜LE-hG=CON+V E-M|Nk@Jo—LEf™6 EH:D@P\EDXJK@BC:N+<fX%@'X"E-U.h_ES š_:.iBXHN[s[N+E
NhE:A 0FE:A ; ›

rqˆUEV?EMHNl=QC+c 7CO@$F"UEV?:M"E-V{=NO:D< ¤  ):.iB:S¢a@HS"c ˆdHAG=NO:.@BM"<@J6 @$NO:@$M ¥t¦ <+D-:.UU=NO:@$M"< § \X"EP4U=ˆdHAG=QN+:@$M ¨HX"EEN :



ut6 A%U.NO:h"UEs 4X%@$:DE>7 C+@$F%U.E-V~w;M"<+fECL\=QF"UE<

0@d 054C0FE(<=05; w;C[JKIBEM]gGY Zor‡ § Ž"gH !"0/'KQ !"* *B(L&! 0 ƒ ƒL2—;Ef a@$COIoj]w;D=S"E-V :DP7C+E
::

8(#:?UA=- l `*0ZB0ZE:A #:4 6P7!4SNR_.`*0FBD;#RE()PAC8(0#F[\4C#RE1[1'*#REUNY4&B ]h`*#ZA '*NhE ›

J ]h` A '*?.`*0=eB› 8(NY'*<@0D7 4CNR_/`*0ZB; Ž$Ž%rLdHVA"UNaŽBŽ"r0Ž$Ž?r'rr;E==M%S N+:V?E$g$:[Z E$Zg

 

   

}2   

    !#" 9%$ Ž 

=QC+E^
 

   

Ž$Ž%rLdHVA"UNaŽBŽ$†BŽ$Žu ¤ †1V @HS"EC=NOE>NOX":M"IH:M"iGj}M"@$M s œ}=i$C=M"iB:=MG=QF"UEJK@$CODE< G® $°³ 2KY^@H†0s+u"r9 †"Z\X"EPœ}=i$C=M"iBE^EdHAG=NO:.@BM"<D=QM{F_E^fC+:N+NOEM8:M NOX"E^JK@BC+V

 

   

& )' (* 3( * 5$ 4 . & " (,-/+ . 021 ( -.

j

=B9 M%ET$EC0Z F_9 fX"EM N+X"E-C+E=C+E;JKCO:DN+:@$M_=U%JK@$CODE<Z D09 fX%EM œ]=Qi$C=M"iB:`=QM{2ƒ<+:M"DE NOX"Ec8D0=M]ž NF_E'S"E-C+:T$ESJKCO@$V =i$E-M"ECl=QU:.–-ES8h_@$N+E-M|NO:=U9@BC>=COEPM"@BNLf=MHN+ES:.MN+X"Eœ]=Qi$Cl=QM"i$:=QMJK@BCL<+@BV E COE=
7698;:<

=4 .

;> 8;8?6A@

Ž$Ž%rLdHVA"UNaŽBŽHŽ$Ž?rLE=
 

   

r

u

4XG=h%NZrŒˆUEV EMHN=C+c 7 C+@$F%U.E-V <=M"S N+X%E3œ}=iBCl=M%i$:=M n"@BC+VA"U=NO:@M GZw;M{:V?hG@BC+Nl=QMHNJK@$CODEPNOXG=ND=QM FGE^:M"DUA"S"E-S{:M8=œ]=iBCl=QM"i$:=M?TH:`== =B9 NOX"E^JK@BC+DE^@QJWMG=QN+A"COE$Z F_9 NOX"EPM"@BC+V =UHJK@BC+DEBZ D09 NOX"EPIH:M"ENO:.D)JKC+:DNO:@$M?JK@$CODE$Z S_9 NOX"E'œp@$C+E-M|NO–;JK@$CODE$Z E09 NOX"EPhG=Q<+
.](`*` >E(; I0@4

 8 ?¬ :  6A@  8;@  ¬ :x:.<0j

7!4SNR_.`*0FBD;

Ž$Ž%rLdBJKA%U.U]ŽBŽ"r0Ž$Žr ¤ ŽE=
 

   

­

$ _& 2& " L9 

fX"E-C+E ­ :<V @BV EMHNOA"V8g0@JD@$A%C+<+EBgG=M%S :<IH:.M%EN+:DLE-M"EC+iBc|Z˜;@'M"@BNL=NOX":M"IH:M"iGj}V{=Qi$M":N+A%S"E@J46”TBEDNO@$C G® $°³ 2KY^@H†0s+u$‡"Z uB9 u"ZœpEN  F_E>NOX"E^DEMHN+E-C@JWV =
 

   

    

¨HX%@f‹NOXG=N

. . 1

8 @

    $!  # "  

1

u

r

JKCO@$Vb=M =C+F%:.NOCl=QC+c@$CO:i$:M]Z ED=QUU

 %$ .

   . "  .



fX"E-C+E   Z'& \X%EC+E^=C+EN[f@PNOC+:DI|<=0T$=:U=F%U.E@BM"ELD=QMJK@$COE<+E-E>A%<+:M"iGjp@$M"E;:.NOX":M"IH:M"iGj}—L:EU<+E-MŸœ}=iBCl=M%i$ELE-d|A_=N+:@BM G® $°³ 2KY^@H†0s+†$Ž"Z § 9 †"Z\X"EPœ}=i$C=M"iBE^EdHAG=NO:.@BM f:N+X iBEM"EC=U:–ES JK@BC+DE^EHh"U:D:N+Uc
 

   

& ' (  (  5$ 4 . & " (,-+ . 0 1 ( -/.

:<EdHA":T$=UEMHNN+@NOX"E^—L:EU<+EM œ]=iBCl=QM"i$E;EdHAG=NO:.@BM

7 CO@T$E^N+X%:.<0Z w;U.
( + u (  $ 4 .  (,-+ . 1 ( -/. ( * + u ( * $ (,-/+ . 1 ( -/.

Ž

=T$=QU.:S =QU.NOEC+M_=N+:TBELJ5@$C+V JK@$CN+X"Eœ]=iBCl=QM"i$E;EdHAG=NO:.@BM f:.NOX NOX"E'œ]=Qi$C=M"iB:`=QM(

4X_=h"N0ZarŒˆUEV E-M|N=COc 7C+@BF"UEV <=M"S NOX"EPœ]=iBCl=QM"i$:=M n"@BC+VA"U=QN+:@$M

†

Ž$Ž%rLdBJKA%U.U]ŽBŽ © $Ž Žr ¤ ŽE=
 

   

* $  1 

*

*

¢A"NL=1M"@$MHs[<+N=M"S_=C+S8œ}=i$C=M"iB:=M  D0=M FGE^DCOE=NOES8FHc =S%S":M"i'œ}=i$C=M"iBEaE-d|A_=N+:@BM"<Z}\X"EM%@$MHs[<+N=M"SG=QC+S1JK@$COV <V :iBX|N}FGEV?@$COEaD@BMHT$EM":EMHNNO@PA"
- 5& & "

*

* $ * & & 2 -"  " 9 

fX"E-C+E N+X"E A"M"
JKA%M"DN+:@BM{:M =œ]=iBCl=QM"i$:=M =QM"S8=UU_c$@$A f=MHN:.V?V ES%:`=QN+EUcP:<N+X"E1EdHAG=NO:.@BM"<@JV @BN+:@$M(

2& & "

Ž$Ž%rLdBJKA%U.U]Ž%ruBŽ$Žr ¤ ŽE=h%U`=QM"ENo<=M"<=:CoCOE<+:<+N=M"D-E$Z \X"E1E<+D=QhGETBEU@HD:N[c :E-<+D=QhGET$EU@HD:N[cJKCO@$VbNOX"E'ˆ=QC+N+X?JKCO@$VbNOX":<JK@$COVA"U= 

 

   

Ž$Ž%rLdBJKA%U.U]Ž%r†BŽ$ŽPu ¤ ŽV @HS"EC=NOE>NOX":M"IH:M"iGj}C+@HDI$ENU=QA"M"DX G® $°³ 2KY^@H†0s+†"r$Zr0†$9 ¥ Z¨HX%@f™N+XG=QNN+X"E^EdHAG=QN+:@$M?@JV @$NO:@$M1JK@$C=U=A"M"DX":M"i1C+@HDI$EN;
 

   

 & & "

$ 1  1   & &  " 

fX"E-C+E:.<;N+X"EEHXG=QA"<+N^<+h_EES…@QJN+X"EE ED-N+:M"i?JKA"EU[Z^4@$M"
 "2 ¥ Ž
)$ 

& & " $ 1



Ž$Ž%rLdBJKA%U.U]Ž%r ¥ $Ž ŽPu ¤ ŽV @HS"EC=NOE>NOX":M"IH:M"iGj %…EF_ECJK@$CODE G® $°³ 2KY^@H†0s+†$u"Zr ¥ 9iBEM"EC=U:–ES hG@BN+EMHNO:`=QU[gHT$EU@HD:N[cBsS"E-hGEM%S"EMHNLh_@$NOEMHN+:=U § Zœp@$M"i_goU@$M"i8=iB@ @$Mt=?D@$MHNO:.M%EMHN^J=C0goJ[=CP=0fR=0c %…EFGE-CPh"C+@BhG@B<+ES„:Mq
 

   

 $ 1  1 ' + 1 !  0#" "

fX"E-C+E :
4 . $ 1 (( /- $ .  & & " ' (,( /- $ + . 0 

;EC+E' :<N+X"EPiBEM"EC=U:–ES?JK@BC+DE'=QM"S " :<N+X"E^iBEM"EC=U:–ES8D@H@BC+S":MG=QN+E$Z &  eS"@BM]ž NaIHM"@f™@QJ =QM|c?M":‘JKN[cf=0c@J]z_M"S":M"iN+X":<h_@$N+E-M|NO:=U[Z]ªBA"<+N;<+N=C+Nf:NOX{JK@$( C $ :.<TBEC+c?<+:V h%U.EBZ

8;@ 



4XG=h%NZrŒˆUEV EMHN=C+c 7 C+@$F%U.E-V <=M"S N+X%E3œ}=iBCl=M%i$:=M n"@BC+VA"U=NO:@M

Ž$Ž%rLdBJKA%U.U]Ž%r § $Ž Žr ¤ ŽE=9 '˜;@ cB@$A M%EES8NO@D@$M":M8=M%<+fEC+:M"iNOX":
 

   

 !





Ž$Ž%rLdBJKA%U.U]ŽBu$ŽBŽ$ŽPu ¤ ŽV @HS"EC=NOE>NOX":M"IH:M"iGj}D@$V?h"UEœ]=iBCl=QM"i$:=M G® $°³ 2KY^@H†0s+†$†"Z uBŽ$9fE:C+S?JK@BC+DE ‡"Z¨"=0ccB@$A XG=0TBE>NOX"E^Cl=QN+X"EC;D@$V?h"UE rs[S":V EM"
 

   

* $ fX"E-C+E

r ru



3

 +     +  } 2  9 1

2 9



:.<<+@BV E;hG@B<+:N+:T$E;S"EzGM%:.NOE^D@$M"E-|h%U.:D:NEdHAG=QN+:@$M?@JV @$NO:@$MJK@BCN+X":
A%<+NN+X%EP@$COS":MG=COc 4=C+NOE<+:=QM



+



+

Ž$Ž%rLdBJKA%U.U]ŽBu"r0Ž$ŽPu ¤ ŽV @HS"EC=NOE>NOX":M"IH:M"iGj}C+@BNl=NO:@$MG=QUGwN[f@H@|S%@$:S]Z G® $°³ 2KY^@H†0s+†$†"Z u%r9 rŽ"Z a@BAXG=0T$EL=>JKC+:DNO:.@BM"UE<+<<+A%C[J=QDE^f:.NOX =^N+:MHcX"@$UEf:N+X =;J5C+:DN+:@BM"UE<+<ES"iBE$Z\X"ECOE3=QC+E;N[fa@Ph_@$:MHNV =<+
X QZ ¦ M%EPV{=Q<+<:<S"@$M%EPN+X"E1X"@$UEPNOX"EP@$NOX"EC;:.<;<+U:S":M"i@$M N+X"E N=F"UE$ Z %™XG=QNP=C+ENOX"EN[f@{iBEM"EC=U:–ES…D@H@BC+S":MG=QN+E<0! Z %…C+:N+ES"@fM N+X"E œ}=iBCl=M%i$:=Mq=QM"SzGM"S N+X"E œ}=i$C=M"iBE EdHAG=QN+:@$M%<Z ES%A"DE NOX"Eh"C+@BF"UEV”NO@8@$M"EuM%Sq@BC+S"EC1S" : ECOEMHN+:=UkEdHAG=NO:.@BM‚=QM"S @$F"N=:MqNOX"EzGCO<+NP:MHN+EiBCl=QU@J NOX"EPEdHAG=QN+:@$M]Z

 

    



Ž$Ž%rLdBJKA%U.U]ŽBu"rBrŽPu ¤ ŽV @HS"EC=NOE>NOX":M"IH:M"iGj}wN[fa@H@HSHs[U:.IBEV{=QDlX%:.M%E G® $°³ 2KY^@H†0s+†$†"Z u%r9S"@BA"F"UEL:M"D-U.:M"E^h"CO@$F"UEV g$wN[f@H@HS"@$:S V =DX":M"E r$r$ZwyNl=QA"NU:i$XHN]=QM"SN+X"ECO@$h_ELhG=Q<+:M"D-U.:M"EBZ n%@$COVA"U=N+EkN+X"E^œ]=iBCl=QM"i$:=M1JK@BCN+X"E;<+cH<+NOEV~=QM"S<[email protected]N+X"E^œ]=Qi$Cl=QM"i$EE-d|A_=N+:@BM"<JK@BCN+X"E^=DD-EUECl=QN+:@$M"<@J]F_@$NOX V =<+
 

   









8(#:?UA=- j #RBD'*` A N1E ; 7!4C'aEh<='*?.`*0D#\Eh)PJ NR4S0HNhE AC8(03#Z[\4C#RE1[\'*#\E(NR4&B ](`*#FA '*NhE ›

J ]h` A '*?.`*0=eB› 8(NY'*<@0D7 4CNR_/`*0ZB; Ž$ŽBuPdHVA"UNaŽBŽ"r0Ž$Ž?rL ¤ E==QV :UN+@$M}ž <h"CO:M"D:h"UE G® $°³ 2KY^@H†0s+†|9 r$Z^=V :UN+@BM]ž <h"C+:M"D:h"UE^:
 

   

Ž$ŽBuPdHVA"UNaŽBŽ$uBŽ$Ž?r ¤ †1E=
 

   

. $ 3( -/+ *.  * $ ur   +  1 2  9  

Y^:TBEM NOX"E'œ]=Qi$C=M"iB:`=QM fXG=QN:< =B9  F_9  D09  S_9 2 E09 2

ž <D@$M "A i|=QN+E^V @BV EMHN+A%V  Z

 Z +  +Z 1 ((

((, + 9-9-ZZ 1 



Ž$ŽBuPdHVA"UNaŽBŽ$†BŽ$Ž?r'rPu1E=
 

    . . -/+ . & & " -/. -.

-/.

.](`*` e >E(; I^054H7!4SNR_.`*0FBD; Ž$ŽBuPdBJKA%U.U]ŽBŽ$uBŽ$ŽPu ¤ ŽV @HS"EC=NOE>NOX":M"IH:M"iGj}D@$M A"i|=QN+E;C+@$N=NO:.@BM V @$V?EMHN+A"V ¤

4XG=h%NZu“>=V?:U.NO@$M]ž <7CO:.M%D:h"UE3=QM"S86 @$COE^@$M N+X"EPœ}=i$C=M"iB:=M{n%@$COVA"U=NO:.@M ¥

 

   

G® $°³ 2KY^@H†0s ¥ †"Z uB9 r$Z¨HX%@f N+XG=QNLN+X%E'D@BM A"iH=NOEPV @BV EMHN+A%VŒJK@$CL= C+Eh%C+E
$   

1

  !    $





*

fX"E-C+E :<]N+X%E@$COS":MG=COc>V?EDXG=M%:.D0=U|=QM"i$A%U`=QCoV @BV EMHN+A%V8Z]2—;@$NOE :<}A"<+AG=QUU.c;A"<+E-SPJK@BCW=QM"i$A"U=QCoV @$V?EMHN+A"V8g F"A%NX"EC+E1faE^M"EE-S JK@$CLœ}=i$C=M"iB:=M]Z 9^e5JN+X"EPh_@$NOEMHN+:=Uo:<NOX"EPœo@$COEMHN+–^J5@$C+D-E>h_@$NOEMHN+:=U[g

*

$ $ - 2 1


$    -

 

¬ ¯

 9



  -  ¬ 



fX"E-C+E  :.<NOX"E^DXG=COi$EP@BM{h_=C+NO:DUEOZ ¨ :.M%DE8:
 $ 



fX"E-C+E  :< = COEU=N+:TBE{D-@|@BC+S":MG=QN+E8NOXG=NfE8M"E-T$EC?M"EES™NO@…V EMHNO:.@BM¡=Qi|=:M„:.MyNOX":<h"C+@BF"UEVqZy—;@$N+EN+XG=QN  <EoEDN+:TBEUc U:I$E<+X%:vJ5N+:M"iPNOX"EL@BC+:i$:M@J]NOX"E! D@H@$COS":MG=NOE<@$CU:I$EV?@TH:.M%i>=UU_N+X"E" D@H@BC+S":MG=QN+E< T$=QC+cH:M"i ?:   :MJ5@$C+V =NO:.@BM]Z Ž$ŽBuPdBJKA%U.U]ŽBŽ$†BŽ$Žr ¤ ŽE=
 

   

Ž$ŽBuPdBJKA%U.U]ŽBŽHŽ$ŽPu ¤ ŽV @HS"EC=NOE>NOX":M"IH:M"iGj}i$E@HS"E-<+:DE-|NOC+EVA%V h_=N+X%@BC+:i$:MG9-ZW¨ X"@f‹NOX":<Z &  ˜;@:NN+X"E^E=Q<+c f=0cHZ

 

   

8;@ 

Ž$ŽBuPdBJKA%U.U]ŽBŽ ¤ $Ž ŽPu ¤ ŽV @HS"EC=NOE>NOX":M"IH:M"iGj}V :M":V:v–-:.M%i3=QDN+:@BM{X_=C+S f=0c G® $°³ 2KY^@H†0s ¥ GZ ¤ 9 GZwŒhG=QC+N+:DUE<+A"F EDNPNO@{h_@$NOEMHN+:=U 2  9   2fX"ECOEb:<^= D-@$M"h%C+@TH:S"ES8FHc N+X"E EM%Sth_@$:MHN+<Z?e[V h_@$@BN+X"E-CLD@BM"<+NOCl=:MHNZ %…EBž COE^S"@$:M"iN+X":<h"CO@$F"UEV“N+X"EPX_=C+S f=0c|gB@JD@$A%C+<+EBZ

 

    " " $

$

$ " " 1

$

$

Ž$ŽBuPdBJKA%U.U]Ž%r$r0Ž$ŽPu ¤ ŽV @HS"EC=NOE>NOX":M"IH:M"iGj}D@$UU:<+:@$M =QM"S8œ]=Qi$Cl=QM"i$:=QM G® $°³ 2KY^@H†0s ¥$¤ ZrBr9 ¤ Z4@BU.U:<+:@BM"<=C+E;ET$E-M|NO<:MfX":DlX?<+NOC+@$M%i':MHN+EC=DNO:@$MJ5@$C+D-E=QU.U}N+X"E JK@BC+DE-<g_EHDEh"N)JK@$CNOX"EPD@BUU.:NOX"ED@$UU:<+:@$M]Ze[M h"C+:M"D-:.h%U.E1<+@$V?E^@JaNOX"E'D-@$UU:.
 

   

*

*

*

*

4XG=Qh"NZu“>=QV :UN+@BM]ž < 7 C+:M"D:h"UEP=M"S86 @BC+E^@$M NOX"EPœ]=iBCl=QM"i$:=M n"@BC+VA"U=QN+:@$M §

D=QM"M"@BNaCOE=S":UcF_E>:M"D-U.A%S"ES :M?N+X"E^œ]=iBCl=QM"i$:=M}ZWœpENaNOX"E^i$EM%ECl=QU.:–E-S{D-@$UU:.:<

- . ,4 .

& ' ( * (* 5$ 4 .  & " ( -/+ . 0 1 ( /- . ¨ X"@f‹NOXG=NNOX"E^DXG=M"iBE^:.M NOX"E^D@$M A"i|=QN+E^V @BV EMHN+A%V . :MŸ=1D@BU.U:<+:@BM ETBEMHN:.<  . $   .  fX"E-C+E   .   :<N+X"E^D@BUU.:D-@$UU:.COE<+A%U.N;:<E"=DNOU.c NOC+A"EP:‘J -. :.<;DcHDU:DLJ5@$C * =M"S :
JKA%M"DN+:@BMHsU:I$E^=M"S?@$NOX"ECJ5@$C+D-E
Ž$ŽBuPdBJKA%U.U]Ž%r†BŽ$ŽPu ¤ ŽV @HS"EC=NOE>NOX":M"IH:M"iGj}D=Nš_c|:M"i@  X"@H@$h G® $°³ 2KY^@H†0s ¥$¥ Zr0†$94=QN+A"UUA"< ¥ Z4=QN+A"UUA"<N+X"ED0=N:V?E=
 

   

8;@ 

)$

Ž$ŽBuPdBJKA%U.U]Ž%r ¥ $Ž ŽPu ¤ ŽV @HS"EC=NOE>NOX":M"IH:M"iGj}SG=V?hGE-S XG=QC+V @BM":D@$E-d|A_=N+:@BM{@QJWV?@$N+:@BM]g:S"EMHN+:‘JKcNOX"EPNOX"E^EdHAG=NO:@$M @JV @$NO:@$M]Z F_9{˜;@HE<NOX"EP<+cHNOCl=QM"<[JK@BC+V =N+:@BMG9

 

   

* $

'

-+ 1

- $ 

-0 

 

J5@$CN+X%E3œ}=iBCl=M%i$:=M]Zn}:M"S NOX"EPEdHAG=QN+:@$M?@JV @BN+:@$M:M{NOEC+V?
 

   

$



 - -+

$



- 

4XG=h%NZu“>=V?:U.NO@$M]ž <7CO:.M%D:h"UE3=QM"S86 @$COE^@$M N+X"EPœ}=i$C=M"iB:=M{n%@$COVA"U=NO:.@M ©

NOX":M"IH:M"iGj}F"U@HDlI =QM"S faE-S"i$E G® $°³ 2KY^@H†0s ¥$§ Z uBŽ$9S"@BA"F"UELD-@$MHN+C=:MHNh"C+@BF"UEV ‡"Zw F"U@HDI @QJ V =fES"iBEP:<@$M =X"@$CO:.–-@$MHNl=QUoh"U=M"E^N+X_=N:.=QM"S V{=IBELNOX"EPfES"i$E^=D-DEUECl=QN+:@$M @JNOX"E'F%U.@HDI S"@fM N+X"E fES"iBE^:.# < <+: M :M N+X%EPU:.V?:NWNOXG= N    Ž"Z D09{n}:M"S N+X%E>V?@$
 

   

8(#:?UA=- O8(0 › 0ZE:A 4S#:` ( NR4S<=0H7!4CNY_.`*0FB ›

J ]h` A '*?.`*0=eB› 8(NY'*<@0D7 4CNR_/`*0ZB; Ž$ŽB†PdHVA"UNaŽBŽ"r0Ž$Ž?rL u1E=
 

    







Ž$ŽB†PdHVA"UNaŽBŽ$uBŽ$Ž?rL8r;E=
 

    $  + + $ + $ + $ $   



Ž$ŽB†PdHVA"UNaŽBŽ © Ž$Ž?rL ¤ E=
 

   

.](`*` e >E(; I^054H7!4SNR_.`*0FBD; Ž$ŽB†PdBJKA%U.U]ŽBŽ"r0Ž$Žr ¤ ŽE=
 

    

‡

 $ 1   "   

rŽ

4XG=Qh"NZ† \X"E'4EMHN+C=Un"@BC+DEP7CO@$F"UEV JK@BCqh_=C+NO:DUE ?fX"ECOE :<…=M :MHN+E-i$EC0Z \X%E¡S%Cl=ixJK@$CODEyS"Eh_EM"S"EM%DE™@$M " iBEM"EC=U:–=NO:.@BM]gHF"A"N:N:
JK@BC@$A"CD=Q<+EP


$ 1 

$ 1

r u u

 r



   



   







V{=0c™F_E¡= A"M":V h_@$CONl=QM|N

fX"E-C+EPNOX"E^N[fa@<+:S"ENOX":M"IH:M"iGj}<[email protected]:M"iP£^Eh%U.E-Cž
 

   

"

 $ r #_ 2+r %1 D@$ < 9 $ r   2Or % D@$l<9

fX"E-C+E  :<h_EC+:X"EU:@$M?S":<+N=M"DEBZ \X%E'V?E=M8=M%@$V{=QUc 2NOX"E'V?@$£^Eh%U.E-CLF_@$A"M"S @BC+F":Nh_EC+:@HS]j

"

 $ u   ™$ u   # 

\X%E>E-DDEMHN+CO:D =QM"@$V =Uc 2NOX"EPf:US{=QM"S DCl=Q–c{=QM"@$V =UUcP:
"

$

#_2Or

1 %D@B< 9 

:<gH@JD@BA"C+
"

"

 $ r #_ 2+r %1 D@$ < 9 $ r   2Or % D@$l<9

fX"E-C+E  :<h_EC+:X"EU:@$M?S":<+N=M"DEBZ\X"E M"EES?M"@$NFGE;N+X"@BA"i$XHN@QJW=<hG@B<+:N+:@BM:.M=M%i$UEL=N=UU[ZoL@fETBECg NO@qr
" $ 1 "
e[N:<S"ECO:T$ES FHc E$Z i_Z.gHY^@_rŽ$Žr0Ž$u"Z \X%E'X":
"

"





"



4XG=h%NZ†Œ\X"E4EMHN+Cl=QUWn%@$CODEP7C+@BF"UEV

rBr

e[M…=S%S":N+:@$M8NO@ N+X"E6 ES":ET$=QUa=M%@$V{=QU:.E-<g"ež TBEPS"EzGM%ESt= JK@$A"CON+X…=M"@BV{=QU.cHgHEdHAG=QM|NP=QM"@$V =Uc82NOX"EV @B<+N EdHAG=QF"UE =QM"@$V =Uc"9Z;\X":<^:.<^NOX"E =M"iBA"U=C;hG@B<+:N+:@$M V E0=<+A%C+ESJKC+@BV N+X"EE-V h"N[c?JK@HDA"<^@QJRNOX"EEUU:.h%N+:D=U}@$COF":N f:N+XNOX"EL–-EC+@=NN+X"Eh_ECO:.X%EU:.@BMq2ƒN+X"E;C+E=QUGh_EC+:X"E-U.:@BMG9Z\X"EE-d|A_=MHNR=QM"@$V =Uc^:<Nl=QI$EMA"h?:M =^U=N+E-Ch"C+@BF"UEV :M NOX":<DXG=h"NOECL@QJa4687Z =B9{n}:M"S N+X%E>COEU=NO:.@BM F_EN[fEEM D@$< =QM"S D-@$< ¡=QM"S?zGM"S :N+COEU=NO:.@BM :.M „Cl=M%i$E  Ž +Z E09{¨ X"@ftNOXG=N  =M%S ;XG=0TBEL=L@BM"Es[N+@Qs@BM"EC+EU=QN+:@$M:M'NOX"E  C=M"iBE  Ž  Z}¨HX"@$A%U.SN+X%EC+E;FGEL=QM|c=VF":i$A%:.N[c :M h"CO:.M%D:h"UE>:M?S"EN+E-C+V :M":M"i=A"M":dHA"E 1:.M NOX"EPCl=QM"i$E  Ž _JKC+@BV  :.M NOX"E^Cl=M%i$E  Ž   ˜;@HE
"



"

"











" "



e[M NOEC+V?<@J  £^Eh%U.E-Cž


"

"



$ 1  " 

$%<+:M]2  " 





"

"



9

"$



¨ X"@f NOXG=N  :.<=UfR=0cH<^i$COE=NOECPNOXG=M…–E-C+@GgE-|D-Eh"N=N  €Ž =M"S  2ƒ:Z E$Zgoh_ECO:.X%EU:.@BMt=M"S…=h%X"EU:@$MG9-Z &   EšGE-DNL@BM{NOX"E^@$CO:.iB:MG=U_£^Eh"UECž <;EdHAG=NO:.@BM]Z iH9{¨ X" @f‹NOXG=N  VA%<+N:.M?i$EM%ECl=QU]F_EPN+COE=NOESq=E-h"UECž
8;@ 

8;@ 

"

"

"

Ž$ŽB†PdBJKA%U.U]Ž%rŽBŽ$ŽPu ¤ ŽV @HS"EC=NOE>NOX":M"IH:M"iGj}D@$V?EN[s[h"U=M%ENaD-@$UU:.E-M"@$A"iBXqE-M"EC+iBcqNO@ ENOX"EPh"U=QM"EN[s[D@BV END@BUU.:
#_g      g    go=M% S # Z\C+c NO@zGM"S = COE==QM"S E ED-N+:@$(M  F_9{n}:M"SyN+X%E8<+h_ED:=U:–ESyEHh"COE<+
 

   

$

$

Ž$ŽB†PdBJKA%U.U]Ž%r$r0Ž$ŽPu ¤ ŽV @HS"EC=NOE>NOX":M"IH:M"iGj}D@$UU=h%<+:M"iP£^Eh"UEC;@$COF":N G® $°³ 2KY^@H†0sru © Zr$r9

 

   

ru

4XG=Qh"NZ† \X"E'4EMHN+C=Un"@BC+DEP7CO@$F"UEV

"

GZ¨"=0ccB@$A?XG=S8=1FG@BA"M"S £^Eh"UEC@$COF":Nl=QUo<+cH<+NOEVqZe[M =M :M"
"

"

"

"   $    u  fX"E-C+E  :<N+X"E;h"C+Es " @BC+F":N=U"h_EC+:@HS @QJNOX"ELh%U`=QM"EN0Z &  8;@  ¢EJK@$COEJKA%<+<+:M"i=C+@BA"M"Sf:N+XEdHAG=QN+:@$M"


Ž$ŽB†PdBJKA%U.U]ŽBŽ"rBr$rLu ¤ ŽV @HS"EC=NOE>NOX":M"IH:M"iGjYP=A%<+<žHU=0f‹=QM"S :M"M"ECLˆ=QC+NOX ¤ Z a@BAHs+=UUGCOEV E-VFGECYP=QA"<+<0žHU`=0fxJK@$CNOX"E^i$Cl=0TH:Nl=QN+:@$M_=U%zGEUS…2:[Z E$ZgBJK@$CODE^hGEC;A"M":NV{=
&

 & ¬ $ 1      

fX"E-C+E :–-EC+@?@JWh_@$NOEMHN+:=U=N " ‹Ž%Z D09 %…CO:N+E…S"@fM N+X"E„–EC+@Qs =QM"i$A%U`=QC[s[V @BV EMHN+A%V œ]=iBCl=QM"i$:=MŒ2:[Z E$ZgPNOX"E™œ]=Qi$Cl=QM"i$:=QM™JK@BC‚=xh"A"C+E-U.c C=S":=QU V?@$NO:.@BMG9)JK@$C= hG=QC+NO:.D-U.E @QJLV{=E-d|A_=N+:@BM{@QJWV?@$N+:@BM@JWNOX"E^hG=CON+:DUE  E09 % XG=N;:.<;N+X"Ei$E-M"ECl=QU<+@BUA"N+:@$M@JaNOX"E'E-d|A_=N+:@BM8@JV @BN+:@$M tY^:T$E=M8EHh"COE<+
a@BAS"@$M]ž NM"EES NO@'
8

8;@  $ 



8 ;

$

8?6A@  





Ž$ŽB†PdBJKA%U.U]Ž%r ¥ $Ž ŽPu ¤ ŽV @HS"EC=NOE>NOX":M"IH:M"iGj}EdHAG=MHNh"CO@$F"UEV G® $°³ 2KY^@H†0sru$‡%Zr ¥ 9 ¥ Z\X"E1DC+:V E^@QJR7N+@BUEVc82D:COD=8r0Ž$Ž or § ŽPw^˜>9)fR=Q=QNaNOX"E^C+E=QUo@$C+F%:.NOE-d|A_=MHN =Q< =ŸS"EJKED-N@J>X%:.<1V @HS"EU
 

   

4XG=h%NZ†Œ\X"E4EMHN+Cl=QUWn%@$CODEP7C+@BF"UEV

r0†

f@$COI8N+@8E-U.:V?:.M_=N+E1N+X"EEdHAG=QMHNZ?wLM"S…@QJRD@BA"C+
@$A"C;V :<+NO@ S"E-V @$M%<+Nl=QN+EN+XG=QN^N+X"E=M"iBA"U=C;V @$NO:@$M @J = h%U`=QM"ENC+EU=QN+:T$ENO@8N+X"E?EV h%N[c JK@HDA"<:Mt=M…EUU:h"N+:D=QU@$COF":NP:<':M"S"E-EStA"M%:vJ5@$C+VŠN+@ z_C+<+N@$COS"EC:M <+V{=QUU 0Z œpENž <D=UU"NOX"ELEV?h"N[c^JK@HDA%=QM"i$A%U`=QCD@H@$C+S%:.M_=N+E;V E=Q<+A"COESJKCO@$VŒN+X"E;EdHAG=MHN0g$NOX"ELEdHAG=QMHN =QM"@$V =Uc "2 9-Z e[M =MHctD=Q<+E$g;N+X%E‚¨HN=NOEq˜;EhG=CON+V?EMHN f:UUS":V?E=M…=M%@$V{=QUc  Z \X"E-M z_M"S N+X"EPCOEU=NO:.@BM @QJ E-d|A_=MHNL=M"@BV{=QU.c1N+@N+COA"EP=M"@BV{=QU.UcHZ

8;@ 

"

"

Ž$ŽB†PdBJKA%U.U]ŽBu"r0Ž$Žr ¤ ŽN+@BA"i$X N+X%:.M%I|:M"i_j]h%C+EDE-<+<+:M"i?£^Eh"UECL@BC+F":N G® $°³ 2KY^@H†0sr†$Ž%Z u"r9 § Z¨"=0ccB@$A8=COE^i$:T$EM NOX"E^hG@BN+EMHN+:=QU    2" 9

"

 

   

%$ 1

"

"

 

=B9{¨ X"@fbN+X_=NN+X%E{F_@$A%M"St@BC+F":Nl=QU<[email protected]%N+:@$M JK@BC " 2C=S":A"<1JKCO@$V‰NOX"E?JK@$CODE DEMHN+EC9:< =M„EUU:.h%<+EJ5A"M"DNO:.@BM @QJPh"<+EA%S"@™=M"iBU.E hG=QCl=QV ENOEC  Z \X"E  hG=QCl=QV ENOEC :<+M}ž NCOE=UUc¡=QM‹=M%i$UE @$C{= N+:V EBgF"A"N?:< <+@BV E J5A"M"DNO:.@BMq@QJRE-:.NOX"EC0Z ˆHh"C+E-<+<EUU.:h"NOX":<^:.<^NOX"Eh"C+E-DE<+h%U`=QM"EN0Z S_9{¨ X"@f‹NOXG=NE-|h%C+E
8 @ 

8 @  $ "

+

$

1 " 1 

    $ 1 u      

E09 \X%EhGECO:X"EU:@$M…@J>6ŸEC+DA%C+ct=S%T$=M"DE-6ŸEC+DA%C+což <'V?@$N+:@BM =QFG@TBEPfXG=QNLD=QM8FGE=D-D@$A"MHNOES8J5@$CLFHc h"U=M%ENl=QC+c hGE-C+N+A%C+FG=QN+:@$M"<0ZPn}:.M%S8N+X"ECl=QN+:@@J  %2 #H9NOXG=N;i$:T$E-< NOX":E-|NOCl=h_EC+:X"E-U.:@BM{=S%T$=M"DE^f=
8?69@ )





Ž$ŽB†PdBJKA%U.U]ŽBu$†BŽ$Žr ¤ ŽE=NOX"ELˆ=QC+N+X}ž <V =<+<2:[Z E$Zg  9}A"<+:M"i^@BM"Uc NOX"E[email protected]@f:M"iS_=Nl=%j]ˆ=QC+N+X1<+:S"ECOE=UHc$E0=CW† ¥$¤ Z u ¤$¥ † ¥$¤B¤$¥ S_=0c|<0g0V E=QM>¨HA"M s ˆ=QC+NOX'S":
 

   





#

8(#:?UA=- ' [1'*)N()U6H'aE(0FBD#FA '*<=; ›



J ]h` A '*?.`*0=eB› 8(NY'*<@0D7 4CNR_/`*0ZB; .](`*` e >E(; I^054H7!4SNR_.`*0FBD;

r

8(#:?UA=- ' [1'*)N()U6 ›





](#FA '*NhEh; NRd J N:A '*NhE

J ]h` A '*?.`*0=eB› 8(NY'*<@0D7 4CNR_/`*0ZB; .](`*` e >E(; I^054H7!4SNR_.`*0FBD;



8(#:?UA=-  ; <='*`*`*#FA ' NhEh; ›

J ]h` A '*?.`*0=eB› 8(NY'*<@0D7 4CNR_/`*0ZB; .](`*` e >E(; I^054H7!4SNR_.`*0FBD;



8(#:?UA=- O8(0 › `*# ; ; '*<=#:`XJ 05NYd Qh?05<='*# ` 0@`*#FA ' %(' A 6 ›



J ]h` A '*?.`*0=eB› 8(NY'*<@0D7 4CNR_/`*0ZB; .](`*` >E(; I0@4

7!4SNR_.`*0FBD;



8(#:?UA=- O8(0 #RBD'*` A N1E('*#RE ( NR4&B ](`*#FA '*NhE ›

J ]h` A '*?.`*0=eB› 8(NY'*<@0D7 4CNR_/`*0ZB; Ž$Ž © dHVA"UNaŽBŽ"r0Ž$Ž?rL ¤ E==QV :UN+@$M%:`=QMPJ5@$C+VA%U`=QN+:@$M r$ZœoEN0ž <}h"U=0c Hl* B(  Wn%@$C %rŽ$Ž%gNOX"ER=QM"<+fEC}:.<0jowLMP=QU.NOEC+M_=N+:TBEJK@$COVA"U=NO:.@BMa@QJG=S"T$=QM"DESPD-U`=Q<+V?EV @BC+coj_>=QV :UN+@$M%:`=QMPJ5@$C+VA%U`= u"Z\X"E^^=V :UN+@BM":=MD=M F_EPD@BM"<+NOC+A"DNOESJKC+@BVbN+X"EPœ}=i$C=M"iB:=M A%<+:M"iN+X"E^J5@$C+VA%U`=%j =B 9    Z F_ 9    Z D0 9 

Z S_ 9  r Z   2 9 Z E0 9 

$ + -+ 1 * $ -+ 1 * $ (3* (9- + $ * $ * & - # " & "

Ž$Ž © dHVA"UNaŽBŽ$†BŽ$Ž?rL †1E==V?:U.NO@$M":=QM'COA"UE †"Ze5J2+r91S"Ez_M":M"iqCOEU=NO:.@BM"N+X%E{iBEM"EC=U:–ES„D@H@$C+S%:.M_=N+E-<S"@qM%@$NS"Eh_EM"SyEHh"U:D:N+Ucq@BMtNO:.V?E =QM"S‹25u$91N+X"E h_@$NOEMHN+:=Uo:<@$COS":MG=COc82:[Z E$ZgHM"@$NTBEU@HD:N[c$s[S"Eh_EM"S%EMHNl9g%N+X"E-M8N+X"E1>=V?:U.NO@$M":=0M:<i$:T$EM?FHcj =B 9   Z F_ 9   Z D0 9    Z S_ 9  PZ E0 9  PZ

$ 1 $ $

$ 1 $

Ž$Ž © dHVA"UNaŽBŽHŽ$Žu r ¤ V @HS"EC=NOE>V?EV @BC+coj_>=QV :UN+@$M%:`=QM'D-@$M"
( $  ("

NOX"EM =B9 M%@$N+X%:.M%iPJKA%C+N+X%ECLD=QM FGEP:MHJ5EC+COES8=F_@$A"NNOX"E^<+cH<+NOEVqZ F_9{=QUUo@$N+X%ECL==QUi$EF"C="Z D09 

Ž =M%S8N+X"E1>=V?:U.NO@$M":=QM  ¬   •=D@BM"<+N=MHNL@QJaN+X%EPV @BN+:@$M]gHF%A"NL:N M%EDE=QV :UN+@$M%:`=QM  Œ=D@BM"<+N=MHN@JN+X"E^V @BN+:@$M}gF"A"N:N  8M"EDE-<+=QV :UN+@$M%:`=QM  q=D@BM"<+N=MHN@JN+X"E^V?@$N+:@BM]gF"A"N:N  M"EDE-<+
& & " $ & & " $ & & " $

?8 69@ 8  >  28?6A@ r©

:<

28?6A@

4X_=h"N0Z © \X"E^>=QV :UN+@$M%:`=QM n%@$COVA"U=NO:.@BM

.](`*` >E(; I0@4

r0‡

7!4SNR_.`*0FBD;

Ž$Ž © dBJKA%U.U]ŽBŽ"r0Ž$ŽPu ¤ ŽV @HS"EC=NOE>NOX":M"IH:M"iGj}i$EM%ECl=QUW=D-N+:@$M?:.MHNOEi$C=M"S G® $°³ 2KY^@H†0s+† ¥ u%Zr9@BCV{=0cHFGE OV @HS":‘zGES œ}=iBCl=M%i$:=M r$Z\X"E =DNO:@$M…:MHN+EiBCl=M%St@JLNOX"E V?@HS":‘zGES >=V?:U.NO@$M":=QM8:<+M]ž NP:M…i$E-M"ECl=QUL=8œ]=Qi$C=M"iB:`=QM™2   2 9 9-g$F"A%N M"@BMHs[<+Nl=QM"SG=QC+S8œ]=Qi$Cl=QM"i$:=QMq2 

 

   

* $ 1

& -  " & " ; $   - +  1  2 -    " 9  & 2 &- "   " 9 

* $ 

91M"@$C=

1

fX"E-C+ELˆ:M"f@$A%U.S COED@$iBM":–EPNOXG=NNOEC+V Z\X"EPiBEM"EC=U =QDN+:@$M :MHN+EiBCl=QM"S :<N+X%E'œ]=Qi$Cl=QM"i$:=QM82<+N=M"S_=C+S @BCM"@$MHs[<+N=M"S_=C+S_9fX"EM"ETBE& C  :<M"@BNL=^JKA%M"DN+:@BM{@QJWNOX"E   Z =B9{¨ X"@ftNOXG=NN+X%ERœ}=i$C=M"iBEaE-d|A_=N+:@BM"<}JK@$UU@f8J5C+@$V NOX"Ei$EM"E-Cl=Uo=D-N+:@$MP:MHN+E-i$Cl=QM"S:.M1N+X"ED=Q<+EfX"E-C+E  ‹Ž%Z e[M NOX":<D=E-d|A_=N+:@BM"<JK@BCL=iBEM"EC=UW=QDN+:@BM8:M|NOEi$C=M"S :M N+X"E D0=<+E1fX"EC+ E    Z D09{¨ X"@f™N+X_=NN+X%E>h_=C+NOiBEM"EC=U =QDN+:@BM{:MHN+E-i$Cl=QM"S fX%EC+ E JK@$Ch_=C+NP25=$9=QM" S 5 J $ @  C G h  = O C P N  2 G F 9 Z 

5$

*



$ 1 -

,$ -

5$

$

"

Ž$Ž © dBJKA%U.U]ŽBŽ$uBŽ$Žr ¤ ŽE==QV :UvNO@$M":=M G® $°³ 2KY^@H†0s+† ¥ u%Z u$9 u"Z )ED=UU"NOX"ELM%@$MHs[<+N=M"SG=QC+S œ}=i$C=M"iB:=M 2VcP@fM:S":@$<+cHM"D-Cl=NO:DMG=V?E9}:.
 

    * $ 1 - #"

*  $ * & & 2 -" #" 9 

f X"E-C+E'2 9k:=V?:U.NO@$M":=QM   D@$COC+E=QV :UN+@$M}ž
*

*

8?6A@  

Ž$Ž © dBJKA%U.U]ŽBŽ$†BŽ$ŽPu ¤ ŽV @HS"EC=NOE>NOX":M"IH:M"iGjYP=V?:U.NO@M":=M G® $°³ 2KY^@H†0s+† ¥ u%Z †$9 †"Z\X"E Y1=V :UN+@BM":=M 2  93:.<1U.:IBE'NOX"E>=QV :UN+@$M%:`=QM]g"E-|D-Eh"NN+X"E?N+@$N=UNO:V EPS"E-C+:T$=NO:.TBE¨HX%@f X"@f N+@?D@$MHN+COA"DN^N+X%E YP=QV :UN+@BM":=0MJKC+@BV N+X"Eœ]=Qi$Cl=QM"i$:=QM{=M%SqS"E-C+:T$E'Y1=V :UN+@BM]ž < EdHAG=QN+:@$M%<j   =M"S  

 

    - +  + #" -

-

- $ 9((  + 

8?6A@ )

$ 1 (( - +  

Y1=V :UN+@BM fR=Q=QV :UN+@$M}ž <=V?:U.NO@$M <+A%h"h"C+E-<+<+E-S Q= U.V?@$
Ž$Ž © dBJKA%U.U]ŽBŽHŽ$Žr ¤ ŽE=E-:.iBEMHT$=UA"E<  E:iBEMHT$EDNO@$CO<  =M%St=M :.MHTBEC+
 

   











 g]
NOXG=N^N+X%EE:i$EMHT$=UA"E-<

u$Ž

4XG=Qh"NZ © \X"EP>=QV :UN+@BM":=M'n%@$COVA"U=N+:@BM

Ž$Ž © dBJKA%U.U]Ž%ruBŽ$ŽPu ¤ ŽV @HS"EC=NOE>NOX":M"IH:M"iGj}>=QV :UN+@$M%:`=0Mu0s[F_@|S%c DEMHNOCl=UpJK@$CODE G® $°³ 2KY^@H†0s+† ¥ †%Zru$9 ¤ ZœpENž NOX"E'u sF_@HS"cDEMHN+C=U}JK@BC+DEPh%C+@$F%U.E-V =QF{:M":NO:.@1A"MHN+:U]fE^XG=0T$E^C+E-S"A"DES :NN+@dHAG=QS"Cl=QN+A"COE<'2ƒ:Z E$ZgHNO@ :MHN+EiBCl=QN+:@$M"<9Z =B! 9 @$A8=COEPi$:T$E-M{N[f@hG=QC+N+:DUE<^=DNO:M"i@$M"UcA"M%S"ECP=DEMHN+C=U}JK@BC+DEPNOXG=NE0=DX8EHECON+<^@$M N+X%EP@$NOX"ECZ %…CO:.NOE S%@fM{NOX"EPœ]=Qi$Cl=QM"i$:=QMJK@BCaNOX"EP<+cH46?s[C+E-U`=QN+:T$E;D@H@$COS":MG=NOE=QV :UN+@$M%:`=QM^:.MN+E-C+V @QJN+X"E>=QV :UN+@$M%:`=QM =COEP<+Eh_=Cl=QN+EUc8D-@$M"@QJN+X"EV @BN+:@$M]Z \X%EM8=C+iBA"E^N+XG=QNaNOX"E^N[fa@hG=QC+N+<D0=M FGE^NOC+E=QN+ESq=Q=QV :UN+@BM":=M h%C+@$F%U.E-V~:M NOEC+V?<@JhG@BU`=QCaD-@|@BC+S":MG=QN+EPD0=M"@BM":D=UoT$=QC+:=F"UE<0Z -9{¨ X"@f N+XG=QNy@$M"E D=M%@$M":D=QU{=QV :UN+@BM":=M]g=M%S~NOX|A%<¡:NO< D-@$C+COE<+h_@$M%S":M"i^D@$M A"i|=QN+EV @$V?EMHN+A"Vy:<]D-@$M"@BF"Nl=Q:M"ES]Z

 

   

Ž$Ž © dBJKA%U.U]Ž%rHŽ$ŽPu ¤ ŽV @HS"EC=NOE>NOX":M"IH:M"iGj}>=QV :UN+@$M%:`=0MD@BM"<+NOC+A"DNO:@$Mq†QC+S C+A"UE G® $°³ 2KY^@H†0s+† ¥ _Zr|9}D@BMHN+C+A%DN+:@$MŸJKC+@BVbN+X"EP†QC+S E=
 

    

* $ # +   % +  ! + +    + +   + 1

 

  

fX"E-C+E #_g % g ! g ]g g_=M"S  =COE>D-@$M"=V?:U.NO@$M":=0M]Z_e[<;N+X"E^>=QV :UN+@BM":=MN+X"E^N+@BNl=QUoEM"ECOi$c  D09 % XG=NdHAG=QM|NO:N+:E<=C+E^@BF|TH:@$A%<D@$M"V?@$NO:.@B(M  S_9 %…CO:N+E^@$A"N^=V?:.UNO@$M]ž <EdHAG=NO:@$M"<)JK@$CN+X"E1>=V?:U.NO@$M":=QM]ZowLCOEPN+X%EC+E'=QMHc @$NOX"EC;:.S%EMHN+:‘z=F%U.E;D@$M%<+Nl=QMHN+<^@J NOX"E V?@$NO:.@B(M  %…C+:N+E@BA"NPN+X%E<+@$UA"NO:@$M"<^@JLNOX"EEdHAG=QN+:@$M"<0! Z @$A…S"@BM]ž N>X_=0T$EN+@ F_E NO@|@ EHh"U:D:NP:‘JN+XG=QN D-C+E=QN+E
4X_=h"N0Z © \X"E^>=QV :UN+@$M%:`=QM n%@$COVA"U=NO:.@BM

u%r

Ž$Ž © dBJKA%U.U]Ž%r ¤ $Ž ŽP† ¤ ŽXG=COS N+X":M"IH:M"iGj}V E-<+<+c?>=QV :UN+@$M%:`=0M G® $°³ 2KY^@H†0s+† ¥ _Zr ¤ 9 § ZY^:TBEM NOX"E'œ]=Qi$C=M"iB:`=QM

 

   

* $ - +   #  - +  % -    - +  - +  1 ! -  

zGM%S{NOX"EP^=V :UN+@BM":=0M =QM"S N+X"E^hG=QC+NO:.D-A"U=C>=QV :UN+@BM]ž NOX":M"IH:M"iGj}S":<+iBA":<+ES8¨H ¦ G® $°³ 2KY^@H†0s+† ¥ _Zr ¥ 9 © Z a@BA{=QC+EPiB:.TBEM{=^=V?:.UNO@$M":=0M@J@$M"E^S"E-i$C+E-EP@J}JKC+EE-S"@$V j

 

   

   $ u 1 % -      % u -     2  %    9   u-  -

-+

B= 9{n}:M"S N+X%E>D-@$C+COE<+h_@$M%S":M"i{œ}=i$C=M"iB:=M:M N+ECOV <@J =QM"S Z F_9{n}:M"S8=M?EdHA":T$=UEMHNœ]=iBCl=QM"i$:=M  NOXG=N:<EHh"U:D:N+UcN+:V E s:M"S"E-hGEM%S"EMHNZ D09 % XG=N :=QV :UN+@BM":=M   :M N+ECOV < @J :N+< h"C+@BhGE-CtD=QM"@$M":D=QU'T$=QC+:=F%U.E-< D-@$C+COE<+h_@$M%S":M"i NO@   % XG=N=<=V{=QN+N+E-C@JD=NOEi$@BC+c{=QC+E   =QM"S  f:N+X COE<+h_EDN^NO@  =M"S ež V A"<+NU@H@$IH:M"i^JK@BCL=<+EMHNOEM"DE=
*

*

-

  =QM"S

-

&  8;@

- 

8(#:?UA=- ^› #RE(N1E('*<=#:` O4S#RE(; d NY4&B#@A '*NhEh; ›

J ]h` A '*?.`*0=eB› 8(NY'*<@0D7 4CNR_/`*0ZB; Ž$ŽB‡PdHVA"UNaŽBŽ"r0Ž$Ž?rL8r;E=
  -+ =B9 F_9 D09 S_9 E09 %

%

%

%

%

+ 1  $   4  1

 & 2 4 & "   # " 9 

XG=NEdHAG=QN+:@$M?VA"<+N=1D=M"@BM":D=U}N+C=M"< JK@$COV{=NO:@$M=QV :UN+@BM]ž =QV :UN+@BM]ž


Ž$ŽB‡PdHVA"UNaŽBŽ$uBŽ$Ž?r'rPu1E=
.](`*` e >E(; I^054H7!4SNR_.`*0FBD; Ž$ŽB‡PdBJKA%U.U]ŽBŽ"r0Ž$ŽPu ¤ ŽV @HS"EC=NOE>NOX":M"IH:M"iGj}D@$V?h"UED=M"@BM":D=UoNOCl=QM"<Z G® $°³ 2KY^@H†0s[|u"rBZr9 r$Z\X"E^@BM"Es[S"Ei$COEEs[@Js5JKCOEES"@BV D@$V?h"UEN+C=M"< JK@$COV{=NO:@$M

 

   

4$ -   

5$ 4 

:NOX":M"IH:M"iGj}D=M%@$M":D=QU.:N[cT$ECO:‘zGES G® $°³ 2KY^@H†0s[|u$u%Z |9 u"ZY^:TBEM NOX"EPN+C=M"< JK@$COV{=QN+:@$M

 

   

45$

U.M

'


 - 0 

TBEC+:‘JKcS":COEDN+Uc N+X_=N:Na:
 $ -

D@BN





4X_=h"N0Z‡b4=M"@BM":D=Uo\C=M"< JK@$COV{=NO:@$M"<

uB†

Ž$ŽB‡PdBJKA%U.U]Ž%rŽBŽ$ŽPu ¤ ŽV @HS"EC=NOE>NOX":M"IH:M"iGj}D=M%@$M":D=QUoN+Cl=QM"<[J5@$C+V =NO:.@BM G® $°³ 2KY^@H†0s[|u$†%ZrŽ$9}D@BM"<+NOC+A"D-N+:M"i{=iBEM"EC=NO:.M%iJKA"M"DNO:@$M †"Z a@BA{=QC+EPiB:.TBEM NOX"EPNOCl=M%<[JK@BC+V{=QN+:@$M

 

   

5$ -  

45$ -  

B= 9{n}:M"S N+X%E>:MHT$E-C+<+E1N+Cl=QM"<[J5@$C+V =NO:.@BM]Z a@BA h_=C+NO:=UoS"ECO:T$=N+:TBE<@JN+X"EP@BUS T$=CO:=F"UE< =QM"S  f:NOX{COE<+h_EDN^NO@N+X"E^M"Ef @$M%E< Œ=M%S Z D09{n}:M"SŠN+X"E D@$M"S%:.NO:@$M”A%M"S"EC fX":DX N+X%E M%Ef T$=QC+:=F%U.E-<™=Q<>= <+:V h"UE)JKA"M"DNO:@$M8@QJk= V :HN+A"COEL@QJa@$US8=M"SM"Ef T$=CO:=F"UE
4

-

8;@ 

Ž$ŽB‡PdBJKA%U.U]ŽBu$uBŽ$ŽPu ¤ ŽV @HS"EC=NOE>NOX":M"IH:M"iGj}D=M%@$M":D=QUoN+Cl=QM"<[J5@$C+V =NO:.@BM G® $°³ 2KY^@H†0s[|u ¤ Z u$u$9u sS"E-i$C+E-EP@JJKCOEES"@BV~DcHDU:D^D=M%@$M":D=QUoN+Cl=QM"<[J5@$C+V =NO:.@BM GZ4@BM"<+:S"ECNOX"EP^=V :UN+@BM":=M      2  9  u fX"E-C+EPNOX"EPS"cHMG=QV :D=U%d|A_=MHN+:N+:E
 

   

 $ '  1-   0

 - -  

4  $ -  

4  $ -  - 

}n :M"S8=U=V?:U.NO@$M":=QMt:M‹NOEC+V?< @J'NOX"E…M"Ef•D=QM"@$M":D=QU^T$=C+:=QF"UE<Z“¢aE…D=QC+EJ5A"UUPN+X"E-C+Et:<8=M E==V?:U.NO@$M":=QM DcHDU:D :M?FG@BN+X =QM"S Ze5JWcB@$A D=QM]ž NzGM"S :NgHc$@BA"C=M"=QV :UN+@BM":=M @JLNOX"E hG=QC+N 2D9=M%<+fECgk<+@$UTBEJK@$CPNOX"E  ž <'=QM"S   ž
74 

4 - -

-





4

8(#:?UA=- l k #RBD'*` A NhE1e +1# <=NY_.'O8h0@NY46P#RE() C< A '*NhE\e >E1[1`*0!#:4S'*# _/`*0@; ›

J ]h` A '*?.`*0=eB› 8(NY'*<@0D7 4CNR_/`*0ZB; Ž"r0ŽPdHVA"UNaŽBŽ"r0Ž$Ž?r'rr;E==QV :UN+@BMHsªH=D@BF":"EdHAG=QN+:@$M r$Z\X"E^=V?:U.NO@$MHs[ª|=QD@$F%:"EdHAG=NO:@$M :<j =B9 ^=V?:.UNO@$M]ž <^h"C+:M"D-:.h%U.E?JKA%M"DN+:@BMtfX":DXy:.<N+X%E8i$EM%ECl=QN+:M"iJKA"M"DNO:@$My@JP=8D=QM"@$M%:.D0=UN+C=M"< JK@$COV{=QN+:@$M…NO@ D0=M"@BM":D=UoT$=QC+:=F"UE<N+X_=NL=COE'=UU Z ¬ F_9 ^=V?:.UNO@$M]ž <h"C+:M"D-:.h%U.EBZ D09 ^=V?:.UNO@$M]ž <^h"C+:M"D-:.h%U.E?JKA%M"DN+:@BMtfX":DXy:.<N+X%E8i$EM%ECl=QN+:M"iJKA"M"DNO:@$My@JP=8D=QM"@$M%:.D0=UN+C=M"< JK@$COV{=QN+:@$M…NO@ D0=M"@BM":D=UoT$=QC+:=F"UE<N+X_=NL=COE'=UU Z ¬ S_9 NOX"EPNO:.V?Es[S"Eh_EM"S"EMHNdHAG=QM|NOA"VbV?EDXG=M":D=QUGf=0T$E;JKA%M"DN+:@BM]Z E09 ^=V?:.UNO@$M]ž <^h"C+:M"D-:.h%U.E?JKA%M"DN+:@BMtfX":DXy:.<N+X%E8i$EM%ECl=QN+:M"iJKA"M"DNO:@$My@JP=8D=QM"@$M%:.D0=UN+C=M"< JK@$COV{=QN+:@$M…NO@ D0=M"@BM":D=UoT$=QC+:=F"UE<N+X_=NL=COE'=UU ¬  ¬ ¬ Z

> 698 @ 8;@

8?6A@ >,6A8 @ 8 @

> 8?6A8 > : 6  8 @

Ž"r0ŽPdHVA"UNaŽBŽ$uBŽ$Ž?rL †1E==QV :UN+@$M}ž <DXG=QCl=D-N+ECO:.@BMN+:V EBg^=V?:.UNO@$M]ž <]h%C+:M"D:h"UEkJKA"M"DNO:@$MD=QMFGEfCO:.NON+EM fX"E-C+E =B9 F_9 D09 S_9 E09

    

2 -   " 9%$ 2 -  9 1  ")

:<>=QV :UN+@$M}ž <h"CO:M"D:h"UEP=M"S  :.<NOX"E^zGM%E>=QV :UN+@$M}ž <DXG=QCl=D-N+ECO:.=QV :UN+@$M%:`=QM'E-d|A_=U<Z :<>=QV :UN+@$M}ž <h"CO:M"D:h"UE;JKA"M"DNO:@$M{=QM"S  N+X"E^D@BM"<+N=MHNLNOX"E^>=QV :UN+@$M%:`=QM'E-d|A_=U<Z :<>=QV :UN+@$M}ž <A"M%h"C+:M"D:h"UES?JKA%M"DN+:@BM{=M%S  NOX"EPD@BM"<+N=MHNN+X%EP>=QV :UN+@$M%:`=0MS"@HE<;M"@$NEdHAG=QU[Z

Ž"r0ŽPdHVA"UNaŽBŽ$†BŽ$Ž?rL u1E=
7 -/.  .

-/.

 -  (     ' -.  (  . $ -/. 0%"



. :MHN+@



fX"E-C+E^N+X"E^h"CO:V E<COEJKECNO@ =QUUGD=QM"@$M%:.D0=UohG=Q:.CO=QV :UN+@BM":=M EdHAG=QU<g"=M%S{NOX"EP^=V :UN+@BM'DXG=QCl=QDN+ECO:<+N+:D^J5A"M"DNO:.@BM :.<h%C+E
-.

 2 -  9 $   2 - # 9   . 2 - .  W9 

NOX"EM8D@H@BC+S":MG=QN+E :<j =B9 :M"
4XG=h%NZrŽ >=V?:U.NO@$MHs[ª|=QD@$F%:"\X"E@BC+c{=QM"S wLDNO:@$MHs[wLM"iBUE>«=C+:=QF"UE<



Ž"r0ŽPdHVA"UNaŽBŽHŽ$Ž?rL ¤ E=
$

.](`*` e >E(; I^054H7!4SNR_.`*0FBD;

8(#:?UA=- lYl › `*#:;; '*<=#:`›8(#:NR; ›

J ]h` A '*?.`*0=eB› 8(NY'*<@0D7 4CNR_/`*0ZB; .](`*` >E(; I0@4

7!4SNR_.`*0FBD;



8(#:?UA=- l:j › #RE(NhE('*<=#:` 7X054AC](4S_.#FA '*NhE O8(05NR4 6 ›

J ]h` A '*?.`*0=eB› 8(NY'*<@0D7 4CNR_/`*0ZB; .](`*` e >E(; I^054H7!4SNR_.`*0FBD;



8(#:?UA=- l 3 #Z[\4C#RE1[\'*#\E #RE() #RBD'*` A N1E('*#RE ( NR4&B ](`*#FA '*NhE(;d NR4 › NhE:A 'aE1](Nh](;>Q16(; A 0FBD;^#RE()  '*0@`*).; ›

J ]h` A '*?.`*0=eB› 8(NY'*<@0D7 4CNR_/`*0ZB; .](`*` e >E(; I^054H7!4SNR_.`*0FBD;





?.?0FE().' M l ^› `*#:; ; '*<=# ` J 05
8 3 

0® \X":<'EdHAG=QN+:@$MyNOXG=M…T$=QC+:=F%U.E-<g=M"S…
…® 6 ­®

 3 9 o­% ®"³³-± G¯³ (   (    ( $ (   r (   r (  $  

(  ( (  " ( " " ( "
(  ( (  " ( " " ( " 0 " <+:M ( ( 0 "


$    u  
_¯W®"² _®  ®   ; "¯° : ® o± > ± o±    r   D-c|D-U.:D  r  $    . $   $ ! r  8=QMHN+:DcHDU:D   .   Ž  @$NOX"EC+f:
"¯G± "¯³)" ¯° :  3

" ¯_®7(  ' % 0 ± _ ¯)^®$ ³± _ ¯ ³ * $  1 * $  1 $ * * $ *  & 2 & - " .  " 9 *  $ *     2 -.  -+ .  " 9 D-@$<





& ' (3* (3* & " ,( - + . 0 1 ( - . $‹Ž

& ' (3* (3* & " ,( - + . 0 1 ( - . $ 4 . u$‡



& ' ( ( & " ,( - + .0 1 ( - . $ 4 .

†$Ž

w;h"hGE-M"S":qrŒ4U=<+
 ¥

3 $²Q® ³ "¯°  ®H¯ 0± &[³ 

]2  9 $ 1   2 9 $ - 2  

 $ 1    

 $ 1   $         1 " > ®H¯ 

&  $²® )

 5®

$  1   

u

r





 

 #   $



u

9



   

r



    

U:M"E=QCJK@BC+DE ¬

 9

œo@BC+EMHN+–;JK@BC+DE  L=0cHUE:i$X]ž <S":
 $ 1    "    

 & ¬ $ 1      $ 1  "

"



§

$

%$ ur    $ $ - 2 1

2 9







:MHT$E-C+<+E s


$           $    

 $      $  $         & $ r $  + " &" u    "   " +  $ } 2 " 9  "    $ } 2 " 9 1 1 " +"   ur  "  +   $  ur  " +   ur     $  " $ 1

$ %

u

  ! 

r

r

 "



$ 1 2 1u

 $ r #_ 2+r %1 D@B < 9 $ r   2+r " D@B< 9

$

D-@$<  r -D @$<

1

1

Nl=QM



$ 1

u





1





r

"     "   

$

u #

   $ r  u     $ r 1   # $   #    $ " $ 1 % <+:M]2 k 9 ' $ r   N=M ' u 0 r  u 0 1

1  9 $  2+r   9 $ 1 u #    $ "   $‹u    # " $#_2+r 1 " D@B< 9  "

#_2Or

D-@$<

r9

i$C=0T|:N=N+:@BM



w;h"hGE-M"S":qrŒ4U=<+
&…±   _¯W± "¯  3   0± "¯  $   -+ 1 * &

©

$ 1 (( * $ * B2 -  " 9 2* :‘J N+X%EM   $ *  $  =M"S  $   -+ 1 +

†%r

- + $ ( (   ( (  " $ 1 (( * "  $ . 2 -  " 9 -/+ . 2*  $ .  2 -  " 9 -/+ . - +   *  $   2 -  " 9 - +   *  $ .  2 -  " 9 -/+ . * $ *  $ . $  2 -  " 9 -/+ . - +  1 * 

 $ 

* 2 -  - +  " 9%$ *  2 -  " 9 '  2 - #" 9  ur '  @2 -  " 9 '  N+X"E-M ­ $5@ '  =M"S  $ ur 2 ­  1  9/@  2 ­ 1 9 1 * $2 - #" 9 &  $ (  $ (3*  $    - + …2 -   #" 9  &  2 -    " 9 & "  1 &" (" 1 (" &" "   > "¯%G¯±[² & @

"¯³ 3  ± G ¯³ :‘J

‡



   - +  1 …2 -   #" 9 3$  4 +  1

&

2

=$ 

4  "9 

$  

4  $ - 

4   #" 9  &&  " $   ( (  "  ( 5$   2 -  4  " 9   $ 1 (( 4   $ (  ( (    5$   2 -    " 9 1 4    4 $ ( $ ((      $ 1 (( 4   5$   2   4 #" 9  -    $-  1 (    5$   2    #" 9  -    1 4    $-  1 ( (    4  $ ((       $   2 -  " 9    2 - # " 9 4 . $  . 2 -  " 9  . $ (( - .     ( ( -/ . ­ $    $ 2 ­  "9   1 &…±   _¯  G² ± @  ®

  ' -  ( (  - #"  ( (  " ‹ $ Ž $  2 -  #" 9 0  2-  "9   -+ 1 …

r0Ž

$  4+ 1

…2

  $ 

†$u

w;h"hGE-M"S":qrŒ4U=<+
$ (( - $  2 -   " 9 $  2 -  9 1   $  4 +  $ (( $ Žr

$ 





(

- $ -2  ( $ ( "   $ " 4 $   $ "

  -   (      ' /- .  (  . $ - . 0#"

(



"9



$

 2

 "9

 ' -  ((- 0 $  ( $  "  $ "

(







$

 2 -  9%$   2 -   9   . 2 -/.  9  ' - .  ( -/ . . 0 $ . 

"



?.?0FE().' M j J ](` A '*?/`*0&e$›8(NY'a<=0D7 4CNR_/`*0ZB > E(; I^054 O# _/`*0@;

8 3 

0® n%@$CLNOX"@$=QM"<+fEC^h"U=DE-<>@BM‚=QMq=M%<+fEC^Nl=F%U.E=M"S @T$ECOU=0c|<;:NL@$M<+N+A%S"EMHN'=M%<+fEC^Nl=F%U.E-< ¦ M" =M%S{dHA":DIHUc:.S%EMHN+:‘zGE  $” = Br Z u%Z %† Z _ Z ¤ Z

F

D

S

  {  Š ] 

E

=

F

D

S

¦

¦

¦

¦

¦

¥ Z ¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

§ Z ¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

© Z "‡ Z rŽ%Z ¦

¦

¦

¦

¦

E

†$†

†

w;h"hGE-M"S":{u€6 A"UN+:h"UEs+4X"@B:.D-E>7CO@$F"UEV€wLM"  $” = $r Z u"Z "† Z G Z ¤ Z ¥ Z § Z © Z ‡"Z r0Ž"Z

F

D

S

E

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

=

¦

K 

  {  Š ]  rBr$Z r0u"Z r0†"Z r-GZ r¤ Z r¥ Z r§ Z r© Z r0‡"Z uBŽ"Z

F

D

S

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

¦

E