Post on 07-Jul-2016
description
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5;)%0,6>+9,)6P6(/=>) /" ⊆ ) /25;=1) :9) ;329/@>) =5) PO"9 ) <54A1<9/) 949<9;29438) B9) ?38</PO!9 ) =5) 6"! ∈ ) *1=1) /<) ;32/2) =5) PO!9 ) 949<9;254) =/89) =389:05;B9) 949<9;25451/! ∈ ) 081;) /041=/A1/) 96) *=9:2) 949<9;2) :9) </1) ;5<9C29) C1) ,6'&,3%') 451) !) 081;) 96PO"9 ):9);5<9C29)1</G1;9/)081;)9)/)<54A1<11)"6*28/G9<) /29;A1/) /:508/) B1:21;=A191) =/89) :9) ?/=9) F;289) :1<U3454) 9) O?5;=A19E
/041=/A19P)C1);32/A1/) PEO!9 )=/89)89089@1;2>)X/43/89/)?5;=A191))9))F;)05;=254)!6
M^
(9)/42?94E)9:29)9X1B9;2)=>95∈ P PEO #" × )1/8) 6PO #!9 ∈
YfQ#_D*c+#6)_94/A1/) B1/G3;/4>)δ) ) F;28J3)<54A1<9)/) 89089@1;2>) 3) ?5;=A19) 09=/89)3);5<1<)?5;=A19),8%3*,-.)OB9?1;12>)09)/P6)(/=>) ^PO $/9 = ):05;9<)=>)?5;=A1/)9)9:29)3)?5;=A19)->3)*'3*.)09)/6
Y)?5;=A19) H/9 →I )=3;:1B98/2>)=/)3)894/A19)F;289)/)C1)H)/89)3)1;X98:>) E!−9=/89)9:29)3)894/A19)F;289) POH9 )C1)/)B9?1;12>)081;
6EE ! 9$!9!$ ∈⇔∈ −
(/=>) <54A1<9/) #) 9:29) 3) 0/829) /) 451) PPEOOPO /9#/9 ⊆ ) /25;=1) <54A1<9/KPOIN #!9! ∈ ) :9) ;5<9C29) ,6'&,3%'5 +%-,0+>-.) 081;) 9) /)<54A1<11)#) C1) :9) ;329/@>
P6O! #9 −
YU:98X><)=>)894/A1/) !−9 );5)9:29E)F;)G9;98/4E)3)/041=/A196) !−9 9:29)3)/041=/A19;5</1)B/=>):9)X981?1=>)1<041=/A1/
6C1POEPO M!M!MM!! $$!!$!9$!9 ≠⇒≠==
[5;=A1/) O/041=/A1/P) 9) 09;285) =/89) !−9 ) 9:29) 232) 3) ?5;=A19) :9) ;5<9C29) 923-1,%I,23,7>-.) :/5) ,3=%-1,%6) Y) ?5;=A19) O/041=/A19P) =/89) 9:29) F;) /=94/C1) 21<0) 1;]9=A19) C1901<38?1:<) :9) ;5<9C29)I,=%-1,%) O:/5) /041=/A19) U1]9=21X>P6)(/=>) H/9 →I ) 9:29) 3
U1]9=A19)/25;=1)C1) /H9 →− I! )9:29)B9)/:9<9;9/)3)U1]9=A196[19) /=5<) 9) 3) /041=/A19) /)<54A1<11)/) F;)<54A1<9/)H) C1)&) 3) /041=/A19) /) 451)H) F;
<54A1<9/)A6) &32><) =5) 9& ) ) /041=/A1/) =/89) ?/=9) :>J1) =389:05;B>) 381=>851) 949<9;2E/! ∈ )949<9;254) PPOO !9& )B1;)A6)(/=>)9)C1)&):5;2)U1]9=A11E)/25;=1) 9& ) )9:29)3)U1]9=A19
6PO !!! −−− = &99& ))
(/=>)9)9:29)3)/041=/A19)/25;=1) !−99 ) )9:29)3)/041=/A19)C1)/;5<9)/041=/A1/)1B9;J
21=>)/)<54A1<11)H)F;)9/)F;:>C1E)F;)21<0)=9) 99 )!− )9:29)3)894/A19)F;)/6)(/=>)F;:>)9)9:29
U15;1X3=>) /25;=1) !−99 ) ) =Z2) C1) 99 )!− ) :5;2) /041=/A11) 1B9;21=9E) 081</) /) <54A1<11PEO/9 )/)B35/)/)<54A1<11)/6*<1;21<)/1=1)C1)58<>23/8949)08308192>A1)G9;98/49)/49)?5;=A11438I
!°6) PROPO #9"9#" ⊂⇒⊂M°6) RPO ∅≠⇔∅≠ "9"
L°6) PROPOPO #9"9#"9 ¿¿ =g°6) PROPOPO #9"9#"9 ¡¡ ="°6) PROPO !! #9"9#" −− ⊂⇒⊂W°6) PROPOPO !!! #9"9#"9 −−− = ¿¿\°6) PROPOPO !!! #9"9#"9 −−− = ¡¡a°6) P6OPO !! "9"9 −− = !!
M!
YfQ#_D*c+#6)79;285)/041=/A11) :9)</1)52141@9/@>)C1)/429)B9;5<181) C1) ;32/A11E) F;/?/8/) =9438) B/29) /1=16) (9) 9H9<045E) /041=/A1149) <54A1<11)N*) /) ;5<989438) ;/258/49F;28J3)<54A1<9)H) :9);5<9:=) D,+2+,) =5) X/4381) F;5 H6) Q9) ?343:9C29) ;32/A1/) PO3!3→:/5E)</1)?89=X9;2E) *N∈33! KN )O:/5)F;=>E)</1):1<045E) KP6N 3!
!6g6"?/(90--":.2%-9.0--<"'4,3-(-)&
[19) R→/9 I )3)/041=/A19)/)5;91)<54A1<1)/)F;)<54A1<9/)R)/);5<989438)89/49C1)?19) 6/" ⊂
Q9):05;9)=>)/041=/A1/) 9)9:29)6,3>+'*.) :/5)6.+&,3,*.5 ,39%+,>+)09)<54A1<9/)"B/=>)<54A1<9/)X/4381438) R⊂PO"9 )9:29)3)<54A1<9)<1;38/2>6)%/8G1;9/)1;?9813/8>
PO1;? "9 ) /) <54A1<11) PO"9 ) :9) ;5<9C29) 6'+&,3%'5 ,39%+,>'+.5 /) /041=/A1915 95 09
<54A1<9/5")C1):9);329/@>) P6O1;? !9"! ∈
)*C/B/8
P6O1;?PO1;? "9!9"!
=∈
&5<>854) PO1;? !96"! ∈
= )9:29)=/8/=2981@/2)B9)58<>23/8949)B35>)08308192>A1I
!°6) PEO!96 < )09;285)381=9) R"! ∈M°6)B/=>) Eα<6 )9H1:2>) "! ∈ )/:2?94)=/) 6PO α<!9Q9):05;9)=>)/041=/A1/)9)9:29)6'=>+'*.):/5)6.+&,3,*.5)20%+,>+)09)<54A1<9/)"
B/=>) 1</G1;9/) PO"9 ) 9:29) 3)<54A1<9)</]38/2>6)%/8G1;9/) :509813/8>) PO:50 "9 ) /<54A1<11) PO"9 ):9);5<9C29)6'+&,3%'5)20%+,>'+.5/)/041=/A19159509)<54A1<9/5")C1):9
;329/@>) P6O:50 !9"! ∈
*C/B/8P6O:50PO:50 "9!9
"!=
∈
&5<>854) PO:50 !9@"! ∈
= )9:29)=/8/=2981@/2)B9)58<>23/8949)B35>)08308192>A1I
!°6) EPO @!9 £ )09;285)381=9) R"! ∈M°6)B/=>) E@<α )9H1:2>) "! ∈ )/:2?94)=/) P6O!9<α(/=>)/041=/A1/)9)9:29)<1;38/2>)O</]38/2>P)09)B3<9;154):>5)B9)B9?1;1A19)/E):9
:05;9E)</1):1<045E)=>)9)9:29)<1;38/2>)O</]38/2>PE)?>8>)/)</1):09=1?1=/)<54A1<9/)09=/89) /89) /=9/:2>) 08308192/29E) 1/8) ;5<>854) PO1;? !9
/! ∈) PPO:50O !9
/! ∈) :9) ;5<9C29)6'+F
&,3%'5,39%+,>'+.)O6'+&,3%'5)20%+,>'+.P)/)?5;=A191596Q9):05;9)=>)/041=/A1/)9)9:29)6.+&,3,*.50%562(1,6%'5"E)B/=>)9:29)C1)<1;38/2>)C1
</]38/2>)09)"E)/B1=>)B/=>)<54A1<9/)X/4381438) PO"9 )9:29)<>8G1;12>6*89)43=)1;9G/412/29/
P6O:50PO1;? !9!9"!"! ∈∈
<
MM
.949)B35>)</8G1;1):5;2)9G/49)B/=>)C1);5</1)B/=>)/041=/A1/)9)9:29)=3;:2/;2>)09)"6(/=>) /041=/A1/) 9) 9:29)<>8G1;12>) 09)/) :9) :05;9E) </1) :1<045E) =>) 9) 9:29)6.+F
&,3,*.E)?>8>)/42>):09=1?1=/A196YU:98X><)=>)B/=>)/)9:29)3)<54A1<9)B9);5<989)89/49)C1)B/=>) 9)9:29)<1;38/2>E
/25;=1)G8/?1=54)451)9):9)/?4>)F;)F;289G1<9):125/2)B9/:508/)B890291) 66$ = )(/=>)/041=/A1/9)9:29)</]38/2>E)G8/?1=54)451)9)9:29):125/2)F;)F;289G1<9):5U)B89/02/) 6@$ = )(/=>)9)9:29<>8G1;12>)G8/?1=54):>5):9)/?4>)F;)F;289G1<9)=5081;:)F;289)B8902949) 6$ = )C1) 6@$ =
7_Y7Y,+c+#6)"0(,-'1,'5 R→/9 I 5%)*%56.+&,3,*.G58'-.5D,5326',58'-.5%!,)*.235326.+5 ^>J 5')*9%(5K3-L*5 J!9 <PO 50%3*+25>+,-%5 6/! ∈
?%6>3)*+'1,%6) *X9<E) F;28J/B9X>8E) KEIPONPO /!!9/9 ∈= ) 1/8) <54A1<9/) PO/99:29) <>8G1;12>) B/=>) C1) ;5</1) B/=>) 9H1:2>) 5;) ;5<>8) ^>J ) /:2?94) =/) 09;285) 381=9
PEO/9$ ∈ ) :>) /1U>) 43=) 1;9G/412/29/) 6J$ < ) (/8) PO!9$ = ) C1) B9=1E) 09;285) 381=9E/! ∈ )/X9<) EPO J!9 < )=99/)=9)289U51/)B9<3;:28/26(1;) /=9/:2>) 08308192/29) 89@542>) =>) /041=/A1/) 9) 9:29)<>8G1;12>) B/=>) C1) ;5</1
B/=>)<3B5454):>5) 9 )9:29)3)/041=/A19)<>8G1;12>6#h#%7$#6)!6)[19) 6=3:PO !!9 = )*=9/:2>)?5;=A19)9:29)B9?1;12>)09)R)C1)9:29)<>8J
G1;12>)O09)RPE)B93/89=9) 6!=3: <! )*X9<
!=3:1;? −=∈
!! R
6!=3::50 =∈
!! R
M6) [19) EdgPO M!!9 −= ) B9?1;12>) 09)
−=
L
MEL
M" ) C1) <>8G1;12>) 09) /=9:2
1;298X/46)%/8G1;149)451)9):5;2
6Mdg:50
^dg1;?
M
M
=−=
=−=
∈
∈
!@
!6
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Q9)X981?1=>)1<9B1/2)=>)B/=>) R→/9 I )C1) R→/& I ):5;2)/041=/A11)<>8G1;129E/25;=1)/041=/A1149) &99&9 i1POE R∈αα+ ):5;2)B9)/:9<9;9/)<>8G1;1296
*041=/A1/)9)9:29)<1;38/2>)B/=>)C1);5</1)B/=>) 9− )9:29)</]38/2>6)*X9<
PP6OO:50PO1;? !9!9/!/!
−−=∈∈
*041=/A1/)9)9:29)</]38/2>)B/=>)C1);5</1)B/=>) 9− )9:29)<1;38/2>)C1)/X9<
PP6OO1;?PO:50 !9!9/!/!
−−=∈∈
(/=>) 9) 9:29) 3) /041=/A19) /) 5;91) <54A1<1)/) F;) R ) 9H1:2>) OF;) PR ) 949<9;2949
PO1;? !9/! ∈
)C1) PO:50 !9/! ∈
)B9?1;129)4/)?94)=/)F;)=/@54)/041=/A11438)=5)X/4381)F;)R)O5;B9);5
9H1:2>)F;232B9/5;/)/=9:29)949<9;29P6
ML
#49<9;254) R∈6 ):9);5<9C29)6'+&,3%'5,39%+,>'+.)/)/041=/A191)9)09)<54A1<9//)OC1):9);329/@>) PPO1;? !96
/! ∈= )B/=>) PO!96 £ )09;285)381=9) /! ∈ )C1)B/=>)09;285
381=9) 6>α )9H1:2>) /! ∈ )/:2?94)F;=Z2) 6PO α<!96 <
V;) <3B) /;/43G) 949<9;254) R∈@ ) :9) ;5<9C29) 6'+&,3%'5 )20%+,>'+.) /
/041=/A191) 9) 09)<54A1<9/)/) OC1) :9) ;329/@>) PPO:50 !9@/! ∈
= ) B/=>) @!9 £PO ) 09;285
381=9) /! ∈ )C1)B/=>)09;285)381=9) E@<α )9H1:2>) /! ∈ )/:2?94)F;=Z2) 6PO @!9 <<α*041=/A1/) 9) :9) ;5<9C29) 6.+&,3,*.5 ,39%+,>+) O89:09=21X) :5098138P) B/=>
∞−>∈
PO1;? !9/!
)O89:09=21X) P6PO:50 ∞<∈
!9/!
*041=/A1149)<>8G1;129)1;?98138)C1):5098138):9);5<9:=)6.+&,3,*%6)(/=>
∞−=∈
PO1;? !9/!
)C1) ∞=∈
PO:50 !9/!
/041=/A1/)9):9);5<9C29)3%6.+&,3,*.6YfQ#_D*c+#6)(/=>)/041=/A1/)9);5)/89)X/4381)89/49)C1)1/)X/4381)F;28J5;):0/A15)=5M³3 )B1<9;:15;1E);5)</1)/5):9;:);3A15;149)B9)/041=/A19)<1;38/2>):/5)</]38/2>)C1
;1=1) ;3A15;149) B9) </8G1;9) 1;?9813/8>) :/5) :509813/8>) OB93/89=9) F;28J5;) /:9<9;9/:0/A15);5):9)B9?1;9C29)3)894/A19)B9)38B1;9P6
!6"6"@2.0-/")+2+%+3-9
[19)/)3)<54A1<9);9X1B>)C1)f)3)<54A1<9)B9)0>8A1)/49)451)/E)=/89)/89)58<>23/J8949)08308192>A1I
!°6)B/=>) :,," ∈KN )9:29)3)?/<1419)3/89=/89) O;9X1B>P)B9)<54A1<1)/0/8A1;Z;B) 451
fE)/25;=1) R":,
,"∈
∈ f
M°6)B/=>) 3""" EEE M! ! )/0/8A1;)451)fE)/25;=1)#3
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!=∈ f;
L°6) f∈/ )C1) 6f∈∅(/=>):9)B>)<54A1<9/ f)=5)08308192>A149)B9)</1):5:):9):05;9)=>)09)<54A1<9/)/):J/
B/2)3)*>0>(>&,%)τ)O:/5):J/)B9?1;12)3))*+2-*2+.5*>0>(>&,-.P6).5)/429)=5X1;29E)3)230343G19/:508/)<54A1<11);9X1B9)/)9:29E)081;)B9?1;1A19E)3)?/<1419)f)=5)08308192>A149)!°)j)L°6
%54A1<9/)/)09;285)=/89):J/)B/2)3)230343G19)τ)O09;285)=/89):J/)B9?1;12)3):285=J258>)230343G1=>P):9);5<9C29))0'1,25*>0>(>&,-6)#49<9;2949)5;51):0/A15)230343G1=):9;5<9:=)023-*%6)%54A1<149)/0/8A1;Z;B)451)f):9);5<9:=)62(1,6,58%)-C,)%6)/)9:29)3<54A1<9)B9:=b1:>)C1)B9)/:9<9;9/)<54A1<9/)X1B>)9:29)3)<54A1<9)B9:=b1:>6
V;28J3)<54A1<9) :9) 032) 1;283B5=9) B1X98:9) 230343G116) .9/)</1) :1<04>) 9:29) 23J0343G1/)?38</2>)B1;)<54A1<9/)/)F;:>C1)C1)B1;)<54A1<9/)X1B>6
k;) /42) 9H9<045) B9) 230343G19) F4) =3;:212519) ?/<141/) 2525838) 0>8A1438) 451) /6*=9/:2>)230343G19):9);5<9C29)230343G19)8,)-+%*.6
Mg
(#[+&+c+#6)M5 0'+*%5 #5 '5 232,5 )0'1,25 *>0>(>&,-5 )%5 326%D*%5 62(1,6%5 K3-C,).G8'-.562(1,6%'5 #" != 5%)*%5>562(1,6%58%)-C,).6
V;28J5;):0/A15)230343G1=)h)<54A1<149)F;=b1:9)/5)58<>23/8949)08308192>A1I!°6)B/=>) :,,# ∈KN )9:29)3)?/<1419)3/89=/89)O;9X1B>P)B9)<54A1<1)F;=b1:9E)/25;=1
#:,,#
∈)9:29)3)<54A1<9)F;=b1:>R
M°6) B/=>) 3### EEE M! ! ) :5;2) <54A1<1) F;=b1:9E) /25;=1) "3
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!=) 9:29) 3) <54A1<9
F;=b1:>RL°6)<54A1<149 ∅ )C1)/):5;2)<54A1<1)F;=b1:96(9<3;:28><)08308192>A149)!°)j)L°)/49)<54A1<1438)F;=b1:96!°6) [19) ,# ) 3) <54A1<9) F;=b1:>6) *25;=1E) F;) U/@/) B9?1;1A191) B/29) </1) :5:E
<54A1<9/) ,, #" != )9:29)3)<54A1<9)B9:=b1:>)C1)B9)/:9<9;9/) ":,
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(1;)9G/412/29/
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(1;) =949) B9) </1) :5:) 89@542>) =>) B/=>) F;28J3) <54A1<9) /) :J/5) B9?1;12) S:5UJ
<54A1<149) F;=b1:9T) /25;=1) /) 9:29) 5;) :0/A15) 230343G1=6) V;28J/B9X>8E) 3) <54A1<9
/" ⊂ )X/)?1E)081;)B9?1;1A19E)3)<54A1<9)B9:=b1:>)B/=>) "! )9:29)F;=b1:>6
V;) 49G>258>) =5) 08308192/29/) L°) /) <54A1<1438) F;=b1:9) ?/=9<) 3U:98X/A1/) =>:0/A154)230343G1=)/)C1)<54A1<9/)∅ ):5;2E)F;)/=94/C1)21<0E)<54A1<1)F;=b1:9)C1)<54A1<1B9:=b1:96) V;28J5;) :0/A15) F;@9:28/2) =5) 230343G1/) B1:=892>E) 381=9) <54A1<9) 9:29) F;
/=94/C1)21<0)C1)B9:=b1:>)C1)F;=b1:>6
A&9-(4).)&."/(/-"2/(9)6)[19) ^! )5;)05;=2)3/89=/89)/4):0/A15451)230343G1=)/6
781;) 8%9,3,1,%E) >+,-%5 62(1,6%) E/B ⊂ ) 0%3*+25 -'+%5 %!,)*.5 >5 62(1,6%5 8%)-C,).5 !B
')*9%(5-.5 E!^ BB! ⊂∈ 5%)*%57%-,3.*'*%5'5023-*2(2,5 6^!
V;)0/821=54/8E)381=9) 1;298X/4)B9:=b1:) PEO I' )=/89)=3;A1;9)05;=254)!)9:29)3)X9J
=1;>2/29)/)451)!6
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J;#A#7?* ,D(%5 =∞→
''
6
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#E* I&L&L&L&&6 7'
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--
'6 '' ⋅−++−+−= "
J'"#-&(#+K*94*#7#&B*8B*;%7)7%(4*<):&*>):A#54:&#(4,+,U,a,*9B*<4*#7#&4*8B*;%7)(*#75':%8
'6'
-
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/&0+!+ (+1*)2&10+3&4+ "R + 5&+ #+ (+ 6*472&0+306&4&'8+%0+!+ 7*+9$)(:&+ :0$)0+ ;<$*+ 6*472&0:0$)8+306&4&'8+%0+!=-
><'0+ 7*4(<7*'+ 78+ 9$)($:0$+ 6*472&0&+ #+ + ?4+ %*47'*)+ =@@@; ". "$$$$ != + 3&4+!0<'0+ =;$# + ;<$*+ ==-@@@; ". "$$$# ! + A$78+ &4'0:%:0'81+ $:B*104'*)+ $+ 7$+ 907'(:+ ?4<%$2&*)+907'(:&$)+!@+$'*47&+<%*401+78+#+0<'0+(+#%"&'()*+),-.*/)*0,+(,1(-.*0)&23+(,-.-A0($:070+9$:&$C&)$+907'(:&$)8+$+0<'0+07D&9$)04'8+7*+"+9$:&$C&)0+:0$)0+ "$$$ @@@ ". !
;7$:0+ :0%:0E&4'8+ 7((:3(4$'0)0+ )*&+ $=@+ 6*472&$+ R→!# F + <0+ 1$&+ 4*105'0+ #%"&'()+),-.*/)*"*0,+(,1(-)*+),-)-
GHI>JKLMN>-+O*47'0)0+3&4+<%$2&*)+907'(:&$)+ "R + <0+4('0$E8+$30<0(:&@+%04':*<&1%)&6&7$:0$+<7:&0:&&@+%:&4+ =@; 4$ + ;?4+ )(7+30+ ==-@; ". $$$ = + P4+7(4<07&428@+9$)($:0$6*472&0&+ RR →⊂ "F !# +?4+%*47'*)+ !4$ ∈=@; +<0+4('0$E8+ =-@; 4$#
"#$%&'*( -42&'4/5-+ /&0+ R→!# F + 5&+ @!$ ∈ + 7*+ =@@@@; ". "$$$$ != + *430%:0<*%*401+7((:3(4$'0)0+ "$$$ @@@ ," ! +6&Q$'0-+R*)2&10$+ S@@FT ... !$$$! ∈∈= R30%&430@+?4+B040:$)@+30+9$)(:&)0+ "$$$ @@@ ," ! +$)0<0-+O04':*+9$)(:&)0+ "$$$ @@@ ," !
6&Q$'0@+ 6*472&$+ ;30+ (+ <&4B*:8+ 9$:&$C&)8@ =.$ + =@@@;@F ".... "$$$#$!# !→→R<0+ 4*105'0+ #%"&'()* 5,+'(,-.* $+ 6*472&0&+ #-+ O04':*+ 6&07$:0+ <&<'01+ 30+ 4*10:0
"$$$ @@@ ," ! + %04':*+ 7$:0+1*)2&10$+ .! + 4*+ 0<'0+ 9&38+ <0+ 306&405'0+ 7U'0+ (+ 6*472&0%$:2&$)8+ -.# +P4+1(3+$<01848'(:+<0+306&40<7+6*472&&)0+%$:2&$)0+ -@@@ ," "### !
!,
>V>ROWX-+ /&0+ ""Y=@; 4$4$# −−= + 306&4&'8+ %0+ 70:7*)+ 7*+ 704':*)+ ?4
(:&B&40$+6+$+<&<'01*)*&+30+7((:3(4$'0+7$:'0E&040+:07'$4B*)$:0+;?4+%)$4=+5&+:$E$+,@$3&78+%0
S-YF=@T; "" £4$4$! +=
O04':*+ @Z=4 + (C2&401+ 6*472&$+ %$:2&$)8+ ". Y=; $$# −= + 306&4&'8+ %0
7(1%$7'*)+ [-,@,\−
O04':*+"
.=4 + (C2&401+ 6*472&$+ %$:2&$)8+ ". !
,]=; $$# −= +306&4&'8+ %0
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"
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624+'%#/(+#$%&'*'+ =-;@F "!!# RR ⊂→
A>/N_NMN>-+R*)2&10$+%*47'0)(:+3&4+<%$2&*)+ @.+"R +30+6(:1$+ ==;@@@; . $#$$ "!
*430+ !$$$$ " ∈= =@@@; ". ! +<0+4*105'0+7+,#(&%-+6*472&0&+;:0$)0+30+"+9$:&$C&)0=+#-L5$3$:@+1*)2&10$
=S;@F=@T; $#8!$8$9 =∈=
:0%:0E&4'8+B:$6&7*)+6*472&0&+ -F RR →⊂ "!# +>7*$2&$+ =;$#8 = <0+4*105'0+)&%,'(,7+,#(&%-%(+6*472&0&+#-
A0+ 0Q01%)*@+ B:$6&7*)+ *40&+ 6*472&&+ #+ 30+ 3(*8+ 9$:&$C&)0+ $@+ 4+ 306&4&'8+ %0+ (
1*)2&10+ "R⊂! +;3&4+%)$4=+0<'0+(+1*)2&10+3&4+ ,R +5&+$4*10
=S-@;@=@;F=@@T; 4$#8!4$84$ =∈
`:$6&7*)+*40&+6*472&&+:0$)0+30+3(*8+9$:&$C&)0+:0$)0+0<'0+(+:%5+,#,'.+?4+<%$2&*)
7*+':0&+3&104<&*4&+ -,R +>7*$2&$+ =@; 4$#8 = +0<'0+07*$2&$+B:$6&7*)*&-
"#$%&'*( 3*%,12'4/5( )*( 342'47'/5( 3*%,12'4/5-+ /&0+ 1*)2&10$+ "! R∈ + 5&6*472&&)0+:0$)0+ ;### @@@ ". ! +306&4&'0+%0+!-+/&0@+30+$<01040$@+ =;@=@;=@; ". $#$#$# ;!
9$)(:&)0+6*472&&)(:+ =@@"@.; ;(#( != +?4+%*47'*)+ -!$ ∈ +I0+%($'0+7(4<&30:$+78+<&<'0a
1*)+ 70)(:+;+ 4*10:0+ =;@@=;=@; ". $#$#$# ;! + 7(4<'&'*&0+ 7((:3(4$'0)0+ *4*&+ %*47'
3&4+ <%$2&*)+ -;R + A$78+ 6&078:*&+ %*47'+ !$ ∈ + 6$701+ <8a&+ 7(:0<%*438+ %*47'*)
@==;@=@;=@;; ".;
; $#$#$# R∈! + (C2&401+ (+ 6*472&0+ #+ 306&4&'8+ %0+ !+ 7*+ 9$)(:&+ ?4
Rm + %0+ 7$:0+ (+ 4('81+ ;!# R→F + <$*+ =-@@@; ". ;#### != + GC<0:981+ 78+ $'U'
$:B*104'*)+ $+ 7U'+ 5&+ 9$)(:&)0+ ==;@@=;;=; . $#$#$# ;!= + $)0+ 6*472&0&+ +#+ <*4'+ 907'(:&
!!
;:0<%07'&9+3&4+ "R +5&+ =-;R +/*472&$+306&4&'8+$&7&+<0+<%*40+78+0<'0+(+ +#%"&'()*0)&23<
+(,-.* /)* 0,+(,1(-.* 0)&23+(,-.-+ /*472&&)0+ :0$)0 ;### @@@ ". ! + <*4'+ 7(1%(404'0)06*472&0&+907'(:&$)0+#-
A0($:070+ =@@@@; ". "$$$$ != + 9$)(:&)0+ =;$# + $)0+ 6*472&0&+ #+ <0+ <7:&*
=@@@; ". "$$$# ! + 5&+ 307&+ 1$&+ %*'01+ <%*40+ 78+ #+ 0<'0+ (+ #%"&'()* 0)&23+(,-.* /)* "0,+(,1(-)*+),-)-
!"#$%&''( )*( +*,)&''( $#%-#( .#( /%$'%0'-1( /#)23$'%-14( 5*,)&'#( )36"*714( 5*,)&'#',/#$71
/&0+<%$2&*)+907'(:&$)+ "R +5&+!+(+<*C1*)2&10+$+)*&+ -"R +I8+7(4<&30:81+6*472&&)0+#@+7+306&4&'0+%0+!+7*+9$)(:&+:0$)0-
O:&4+ /)#("('()=+ <*1$+ 7# + + $+ 6*472&&)(:+ #+ 5&+ 7+ 0<'0+ 6*472&$+ :0$)8+ 306&4&'8+ %01*)2&10$+!@+%:&4+0B$)&'$'0$
-@=;=;==;; !$$7$#$7# ∈+=+
O:(3*<*)+α+#+ $)+ 6*472&0&+ #+ 7*+ 4*18:*)+ :0$)+ α+ 0<'0+ 6*472&$+ :0$)8+ 306&4&'8+ %01*)2&10$+!+%:&4+0B$)&'$'0$
=-;==;; $#$# α=αR*)2&10$+ '*'*:(:+ 6*472&&)(:+ 306&4&'0+ %0+ !+ 7*+ 9$)(:&+ ?4+ R + 0<'0+ *4+ :5,'(%
0)&23+(,-+?4+:$%(:'+7*+(%0:$2&&)0+ 7# + +5&+ @#α 700$+70+<0+90:&6&78+&103&$'-
"#$%&'4( %1.-#85-+ /&0+ 1*)2&1&)0+ RR ⊂⊂ >! " @ + 5&+ 6*472&&)0+ @F >!# →-F R→>7 +O:&4+7(4<&30:$:0$+6*472&0&+#+*4*&+0)0104'+($:07$:0+ !$ ∈ +?&+7(:0<%*430
0)0104'*)+ -=; >$# ∈ + L%(&+ 0)0104'*)*&+ >$# ∈=; + ?&+ 7(:0<%*430+ %:&4+ 6*472&$+ 70)0104'*)+ -==;; R∈$#7 +L<'60)@+7*+$b*'(:*)+7(:0<%(430420)(:+ ==;;=; $#7$#$ →→<0+ <'$C&)05'0+ (+ 7(:0<%(430428+ x g f x→ ( ( )) + 3&:07'+ 30+ )$+!+ )$+ R @+ 4*1&'8+ #%"&'()&3;5%:.+$+)*&+7+7*+#@+4('$'8+ -#7 " +L5$3$:@
==;;==;; $#7$#7 ="
%04':*+(:&70+ -=@@@; ". !$$$$ " ∈= !
GHI>JKLMN>?* G%0:$2&$+ 30+ 7(1%*40:0+ $+ 3(*8+ 6*472&&+ :0$)0+ 30+ 9$:&$C&)8907'(:&$)8+ ;30+1$&+1*)'0+9$:&$C&)0+ :0$)0=+4*+ $:0+ <04<-+ P4':a$3098:@+ $1C0)0+ 6*472&&)*U43+9$)(:&+?4+1*)2&1&+3&4+R@+*4$+3&4':0+0)0+':0C*&0+<8+6&0+306&4&'8+%0+(+1*)2&103&4+R@+5&+307&+$70$<'$+0<'0+(+6*472&0+:0$)8+30+(+<&4B*:8+9$:&$C&)8+:0$)8-
"#$%&'4( '$3*285-+ /&0+ RR ⊂→⊂ @!# "F + (+ $%)&7$2&0+ C&b07'&98-+ I0+ %($'0
307&+<'$C&)&+(+7(:0<%(430428+ =@@; . "$$#8 !→ +7*+ @=@@@; ". @8$$$# " ∈=! +30+ )$
0)0104'0)0+ 1*)2&1&&+ @R⊂@ + )$+ 0)0104'0)0+ 1*)2&1&&+ -"! R⊂ + L70$<'8+ 7(:0<a%(430428+306&405'0+#%"&'(,*("0)+:.+$+)*&+#@+4('$'8+ -.−# +L5$3$:@+6*472&$+ !@# →− F.
$:0+?4+%*47'*)+ @8 ∈ +9$)($:0$+ -=@@@;=; ".. !$$$8# " ∈=− !
K$)(:&)0+ =;. 8# − + <*4'+09&304'@+907'(:&@+307&+ .−# + 0<'0+(+ 6*472&0+907'(:&$)8+30
9$:&$C&)8+:0$)8+ =-; R⊂@ +/*472&$+&490:<8+ .−# +0<'0+'('+(+C&b072&0+5&+ -=; .. ## =−−
!]
G%0:$2&$+30+7(1%*40:0+$+6*472&&)(:+#+5&+ .−# +$:0+<04<-+I0+$:$'8+7*+*5*:&428+78$*+)(7+0B$)&'82&)0F
@=@@@;==;;==;; "... 8$$$#8##8## " === −− !"
%04':*+(:&70+ c@8 ∈=@@@@;=;==@@@;;=@@@=;; ".
.".
.".
.""" $$$8#$$$##$$$## !!!" === −−−
%04':*+(:&70+ -=@@@; ". !$$$ " ∈!
,-"-+9'.',*:(;1$,'$#',4,*
9'.',*-+/&0+1*)2&10$+ @"! R⊂ + Z$ +*4+%*47'+30+$7*1*)$:0+%04':*+1*)2&10$!+5&+6*472&$+:0$)8+ -F R→!#
A>/N_NMN>-+A%;.+%-* +),-+α+):2)* -(;(2,* #%"&'()(+ #+ B"* 5%"&2%-+ Z$ +/,&.* 5)"2+%3+(&)*0)&(".2,2)+C+,*-%(+α+)$(:2.*3*0)&(".2,2)+D+,*-%(+ Z$ +,:2#)-*B"&E2*3+(&,+)*,+*#(
@!D$ ¡∈ + @Z$$ ≠ +:.*,0);+ -=; C$# ∈ +I7:&01=-;)&1
Z
$#$$→
=α
F+3538('((-)+*:18'($:0+7(4<'&'*&0+306&4&2&&+07D&9$)04'0+$)0+)&1&'0&-OJGOGdNMNL+ .-+ α+ 0<'0+ )&1&'$+ 6*472&0&+ #+ ?4+ %*47'*)+ @Z$ + 3$78+ 5&+ 4*1$&+ 3$78@
%04':*+(:&70+5&:+ *N∈ST "$ +30+%*47'0+3&4+!+7(490:B04'+78':0+ @Z$ +$901α=
∞→=;)&1 "
"$# -
OJGOGdNMNL+"-+α+0<'0+)&1&'$+6*472&0&+#++?4+%*47'*)+3&<'$42$'+ @Z$ +3$78+5&+4*1$&3$78@+%04':*+(:&70+4*18:+ @Z>ε +0Q&<'8+*4+4*18:+ @Z=; >εδ +$<'60)+?47U'@+(:&7$:0+$:
6&+ Z$$ ≠ +3&4+!+7*+ =;Z εδ− <$$ +<8+$901+ -=; ε<α−$#A01(4<':$2&$+ $70<'(:+5+3538('((+ <0+ 6$70+ 7$+ 5&+ ?4+ 7$E*)+ 6*472&&)(:+ :0$)0+ 30+ (
<&4B*:8+9$:&$C&)8+:0$)8-GHI>JKLMNN-+.°-+A$78+ %*401+ ?4+ 09&30428+ 7((:3(4$'0)0+ %*47'0)(:+ 3&4
@"R + 6(:1*)8:&)0+ 30+ 1$&+ <*<+ <0+ ':$4<7:&*+ 7(:0<%*4E8'(:-+ L<'60)@+ 3$78!$$$$ " ∈= =@@@; ". ! + 5&+ =@@@; ZZ
"Z.Z "$$$$ != + $'*47&+ 7(43&2&$+ Z$$→ + 0<'0
07D&9$)04'8+ 7*+ 7(43&2&&)0+ -@@@ ZZ""
Z.. "" $$$$$$ →→→ ! + A0+ $700$@+ ?4+ )(7+ 30
α=→
=;)&1Z
$#$$
+<7:&01
-=@@@;)&1 ".
Z
Z""
Z..
α=
→
→
→"
"$"$
$$
$$$$$ !
!
P4+ %$:'&7*)$:@+ 3$78+ 6*472&$+ #+ 0<'0+ 306&4&'8+ %0+ 1*)2&10$+ @"R⊂! + 4('81+ 7*=@; 4$ + *4+ %*47'+ ($:07$:0+ 3&4+ !+ 5&+ 7*+ =@; ZZ 4$ + *4+ %*47'+ 30+ $7*1*)$:0+ %04':*
1*)2&10$+!-+I7:&01-=@;)&1
Z
Z
α=→→
4$#44$$
!e
I0+ <%*40+ 78+ $70$<'$+ 0<'0+ )&1&'$+ 6*472&0&+ #+ 7U43+ $+ 5&+ 4+ '&43+ &430%04304';3$:+ <&1*)'$4=+ 78':0+ Z$ + 5&@+ :0<%07'&9+ -Z4 + P4+ $70<'+ 7$E@+ 5+3538('(,+ "+ <0+ ':$4<7:&0
$<'60)F+α+0<'0+)&1&'$+6*472&0&+#+?4+%*47'*)+ =@; ZZ 4$ +3$78+%04':*+(:&70+4*18:+ @Z>ε0Q&<'8+*4+4*18:+ @Z=; >εδ +$<'60)+?47U'+(:&7$:0+$:+6&+ =@;=@; ZZ 4$4$ ≠ +3&4+ "R⊂! +7*
=;@=; ZZ εδ<−εδ<− 44$$ +<8+$901+ -=@; ε<α−4$#
"°-+ P4+70)0+30+1$&+<*<+$1+%:0<*%*<+78+ )&1&'$+α+ $+ 6*472&0&+ RR →⊂ "!# F0<'0+ 6&4&'8-+A$78+ ∞+=α + <$*+ ∞− + $'*47&+ %*'01+ )*$+ 7$+ 907&48'82&+ $)0+ )*&+ ∞+ + 5&@
:0<%07'&9@+ @∞− +&4'0:9$)0)0+ =@; ∞+ε +5&+ =@@; ε∞− +7*+ε+$:C&':$:+4*+40$%8:$'+%(E&'&9-+P4$70<'+ 7$E+ 7(43&2&$+ 7$+ =;$# + <8+ $%$:2&48+ 907&48'82&&+ =@; ∞+ε + 0<'0+ 07D&9$)04'8+ 7*
@=; ε>$# +&$:+7(43&2&$+7$+ =;$# +<8+$%$:2&48+907&48'82&&++ =@; ε∞− +0<'0+07D&9$)04'8+7*-=; ε<$# +F+3538('(,+"+<0+6(:1*)0$E8+$<'60)F
/*472&$+ RR →⊂ "!# F +$:0+)&1&'$+ ∞+ + ?4+%*47'*)+ @Z$ +3$78+5&+4*1$&+3$78
%04':*+(:&70+ Z>ε +7(:0<%*430+ @Z=; >εδ +$<'60)+?47U'+(:&7$:0+$:+6&+ @@Z !$$$ ∈≠ +7*
=;Z εδ<− $$ +<8+$901+ -=; ε>$#
/*472&$+ RR →⊂ "!# F + $:0+ )&1&'$+ ∞− + ?4+ %*47'*)+ @Z$ + 3$78+ %04':*+ (:&70
Z>ε + 7(:0<%*430+ @Z=; >εδ + $<'60)+ ?47U'+ (:&7$:0+ $:+ 6&+ @@Z !$$$ ∈≠ + 7*
=;Z εδ<− $$ +<8+$901+ -=; ε<$#,°-+f($'0+%:(%:&0'82&)0+ )&1&'0)(:+30+ 6*472&&+ :0$)0+30+(+ <&4B*:8+9$:&$C&)8+ :0$)8
<0+%8<':0$E8+ 5&+ %04':*+ 6*472&&)0+ :0$)0+30+1$&+1*)'0+9$:&$C&)0+ :0$)0-+A0+ $<01040$@%:(%:&0'82&)0+)&1&'0)(:+30+6*472&&+:0$)0@+7$:0+4*+&1%)&78+:0)$2&$+30+(:3&40+5&+%:(3*<*)<0+%8<':0$E8+5&+%04':*+6*472&&)0+907'(:&$)0-
L5$3$:@+ 3$78+ #+ 0<'0+ (+ #%"&'()* 0)&23+(,-.+ 306&4&'8+ %0+1*)2&10$+ @"! R⊂ + 7*
9$)(:&+ ?4+ @;R + &$:+ Z$ + 0<'0+ *4+ %*47'+ 30+ $7*1*)$:0+ %04':*+ 1*)2&10$+ !@+ )&1&'$
6*472&0&*#++?4+%*47'*)* Z$ +<0+306&405'0+$<'60)F
K07'(:*)+#∈ ;R +0<'0+-(;(2,*#%"&'()(*0)&23+(,-)*#++?4+%*47'*)+ @Z$ +3$78+%04':*
(:&70+907&48'$'0+C+$+ )*&+#+ ;?4+ =;R +0Q&<'8+(+907&48'$'0+D+$+ )*&+ Z$ + ;?4+ ="R +$<'60)
?47U'+(:&7$:0+$:+6&+ @!D$ ¡∈ +7*+ @Z$$ ≠ +<8+$901+ -=; C$# ∈F+35+38('(,+"+3&4+7$E*)+6*472&&)(:+:0$)0+309&40+$7*1F/*472&$+ ;"!# RR →⊂F + $:0+ )&1&'$+ #+ ?4+ %*47'*)+ Z$ + 3$78+ 5&+ 4*1$&+ 3$78
%04':*+(:&70+ Z>ε +0Q&<'8+*4+4*18:+ @Z=; >εδ +$<'60)+?47U'+(:&7$:0+$:+6&+ Z$$ ≠ +3&4+!@
&40B$)&'$'0$+ =;Z εδ<− $$ +&1%)&78+ -=; ε<− #$#
G,&.*)$(:2.*-(;(2,*%")(*#%"&'((*0)&23+(,-)*B"2+<%"*5%"&2=*),*):2)*%"(&.?K07'(:*)+ #∈ ;R + 0<'0+ )&1&'$+ 6*472&0&+ #+ ?4+ %*47'*)+ Z$ + 3$78+ 5&+ 4*1$&+ 3$78
-Z=;)&1Z
=−→
#$#$$
GC<0:981+78+%04':*+6*472&&)0+907'(:&$)0+<0+%8<':0$E8+306&4&2&&)0+3$'0+?4+7$E*)6*472&&)(:+:0$)0@+7*+7(43&2&$+7$+1(3*)*)+<8+6&0+?4)(7*&'+7*+4(:1$-
!g
F+3538('(,+ *:18'($:0+ :03*70+ <'*3&*)+ )&1&'0)(:+ 6*472&0&+ 907'(:&$)0+ )$+ 70)+ $))&1&'0)(:+7(1%(404'0)(:+<$)0+:0$)0-
OJGOGdNMN>-+H()* #%"&'(,+ ;"!# RR →⊂F + 5&+ ;### @@@ ". ! + &3;53")"2)-)* )(
+),-)@+&%+ -@@"@.@F "(!# "( !=→⊂ RR +H()@+/)*,:);)"),@+ -=@@@; ".
;; R∈= #!###
C)&23+%-+#+):2)*-(;(2,*#%"&'()(+#+B"*5%"&2%-* Z$ +/,&.*I(*"%;,(*/,&.
-@@"@.@=;)&1Z
;($# (($$
!# ==→
>V>ROW>-+.-+/&0+ "=@;@=@; R∈+= 4$4$4$# +5&+ -=@; "R∈1,
=-@;=;)&1=@;)&1 1,#1,4$4$#14,$
14,$
=+=+=→→
→→
"-+/&0+ "=@;@=@; R∈= 4$4$4$# +5&+ -=@; "R∈1,
=-@;)&1=@;)&1 1,#,1$44$#14,$
14,$
===→→
→→
,-+/&0+ Z@=@;@=@; " >∈= $4$$4$# 4 R +5&+ -Z@=@; " >∈ ,1, R
=-@;)&1=@;)&1 1,#,$4$# 14
14,$
14,$
===→→
→→
;1$,'$#',4,*4( +#$%&''/12( 2*4/*( )*( 342'47'/5( 3*%,12'4/5-+ /&0+ 6*472&$RR →⊂ "!# F +5&+*4+%*47'+ -Z !$ ∈
A>/N_NMN>-+H%"&'(,*#*):2)*&3"2("%.*B"*5%"&2%-+ Z$ +/,&.*5)"2+%*3+(&)*0)&(".2,2)
C+,*-%(+ =; Z$# +)$(:2.*3*0)&(".2,2)*D*,*-%(* Z$ +,:2#)-*B"&E2@+3+(&,+)*,+*#(+ !D$ ¡∈:.*,0);+ -=; C$# ∈
R$&+1042&(481+$)'0+3(*8+/)#("('((*$)0+7(4'&4*&'82&&@+07D&9$)04'0+7*+$70$<'$-OJGOGdNMNL+ .-+ /*472&$+ #+ 0<'0+ &3"2("%.* B"* 5%"&2%-* @Z$ + 3$78+ 5&+ 4*1$&+ 3$78@
%04':*+(:&70++5&:+ @@ST !$$ """ ∈∈ *N7(490:B04'+78':0+ @Z$ +$901
=-;=;)&1 Z$#$# ""
=∞→
OJGOGdNMNL+ "-+ /*472&$+ #+ 0<'0+ &3"2("%.* B"* 5%"&2%-+ @Z$ + 3$78+ 5&+ 4*1$&+ 3$78@%04':*+ (:&70+ 4*18:+ @Z>ε + 0Q&<'8+ *4+ 4*18:+ @Z=; >εδ + $<'60)+ ?47U'@+ (:&7$:0+ $:+ 6&
=;@ Z εδ<−∈ $$!$ +&1%)&78+ -=;=; Z ε<− $#$#
A$78+ Z$ + 0<'0+ %*47'+ 30+ $7*1*)$:0+ %04':*+1*)2&10$+"! R⊂ + <0+ %($'0+ 3$+ (
306&4&2&0+$+7(4'&4*&'82&&+7*+$b*'(:*)+)&1&'0&-A>/N_NMN>-+ /*472&$+ #+ 0<'0+ &3"2("%.* ?4+ %*47'*)+ Z$ + 3$78+ 5&+ 4*1$&+ 3$78
=@;=;)&1 ZZ
$#$#$$
=→
+$3&78+3$78+5&+4*1$&+3$78F
-Z=;=;)&1 ZZ
=−→
$#$#$$
!^
A$78+ 6*472&$+ #+ 0<'0+ 7(4'&4*8+ ?4+ 6&07$:0+ %*47'+ 3&4+ !@+ <0+ <%*40+ 78+ #+ 0<'07(4'&4*8+%0+;%-'(;),*!-
GHI>JKLMNN-+.-+ A$78+ %*401+ ?4+ 09&30428+ 7((:3(4$'0)0+ ZZ"
Z. @@@ "$$$ ! + $)0
%*47'*)*&+ @Z !$ ∈ + $'*47&+ 6*472&$+ #* 0<'0+ 7(4'&4*8+ ?4+ !$$$$ " ∈= =@@@; ZZ"
Z.Z !
3$78+ 5&+ 4*1$&+ 3$78@+ %04':*+ (:&70+ 4*18:+ :0$)+ @Z>ε + 0Q&<'8+ *4+ 4*18:Z=; >εδ + $<'60)+ ?47U'+ (:&7$:0+ $:+ 6&+ %*47'*)+ !$$$$ " ∈= =@@; ". ! + 7*
@Z$$ ≠ + &40B$)&'82&)0+ =;@=@;=@; ZZ""
Z.. εδ<−εδ<−εδ<− "" $$$$$$ ! + &1%)&78
-=@@@;=@@@; ZZ"
Z.". ε<− "" $$$#$$$# !!
>V>ROWX-+/*472&$+#+3$'8+%:&4
""
"" =$:77(<;=@;
4$
4$4$#
++=
306&4&'8+%0+1*)2&10$=SZ@ZT;hS.F=@T; """ £4$J4$! +∈=
0<'0+7(4'&4*8+%0+!-"-+A06&4&2&$+ 7(4'&4*&'82&&+ 6*472&&)(:+ :0$)0+ 30+ 9$:&$C&)8+ 907'(:&$)8+ ;30+ "
9$:&$C&)0+:0$)0=+<0+0Q'&430+5&+%04':*+6*472&&)0+907'(:&$)0-/&0+ 6*472&$+ ;"!# RR →⊂F + 5&+ *4+ %*47'+ -Z !$ ∈ + A06&4&2&$+ 3$'8+ %:&4
%:(%(E&2&$+"+%04':*+6*472&$+ RR →⊂ "!# F +<0+6(:1*)0$E8+$<'60)F
/*472&$+907'(:&$)8+ ;"!# RR →⊂F +0<'0+7(4'&4*8+?4+%*47'*)+ @Z !$ ∈ +3$78+5&4*1$&+ 3$78@+ %04':*+ (:&70+ 4*18:+ @Z>ε + 0Q&<'8+ *4+ 4*18:+ Z=; >εδ + $<'60)+ ?47U'@
(:&7$:0+$:+6&+ =@;@ Z εδ<−∈ $$!$ +&1%)&78+ -=;=; Z ε<− $#$#
<21-2'*,5&'(4/*(+#$%&''/12(2*4/*(%1$,'$#*-+O:(%:&0'82&)0+6*472&&)(:+:0$)0+30+(<&4B*:8+ 9$:&$C&)8+ :0$)8+ :81U4+ 9$)$C&)0+ 5&+ %04':*+ 6*472&&)0+ :0$)0+ 30+ 1$&+ 1*)'09$:&$C&)0-+/&0+6*472&$+ RR →⊂ "!# F +5&+ -Z !$ ∈
.-+A$78+6*472&$+#+0<'0+7(4'&4*8+?4+%*47'*)+ !$ ∈Z +;<$*+%0+1*)2&10$+!=+$'*47&# +0<'0+7(4'&4*8+?4+%*47'*)+ Z$ +;:0<%07'&9+%0+1*)2&10$+!=-
"-+A$78+6*472&$+#+0<'0+7(4'&4*8+?4+ Z$ +5&+ @Z=; Z ≠$# +0Q&<'8+(+907&48'$'0+D+$+)*&
Z$ +%0+7$:0+6*472&$+#+0<'0+3&60:&'8+30+E0:(-,-+A$78+6*472&$+#+0<'0+7(4'&4*8+?4+%*47'*)+ @Z$ +0Q&<'8+(+907&48'$'0+D+$+)*&+ Z$
%0+7$:0+6*472&$+#+0<'0+18:B&4&'8-!-+A$78+ RR →⊂ "!# F + 5&+ RR →⊂ "!7 F + <*4'+ 6*472&&+ 7(4'&4*0+ ?4
%*47'*)+ Z$ +;<$*+%0+1*)2&10$+!=+&$:+α+0<'0+*4+4*18:+:0$)+$'*47&+6*472&&)0+ 7# + +5&#α +<*4'+7(4'&4*0+?4+%*47'*)+ Z$ +;:0<%07'&9+%0+1*)2&10$+!=-]-+/&0+1*)2&1&)0+ "! R⊂ +5&+ -R⊂> +A$78+6*472&$+ >!# →F +0<'0+7(4'&4*8+?4
%*47'*)+ @Z !$ ∈ + &$:+ 6*472&$+ R→>7 F + 0<'0+ 7(4'&4*8+ ?4+ %*47'*)+ + @=; ZZ >$74 ∈=$'*47&+ 6*472&$+ 7(1%*<8+ R→!#7 F" + 0<'0+ 7(4'&4*8+ ?4+ %*47'*)+ -Z !$ ∈ + A$78+ #0<'0+ 7(4'&4*8+%0+1*)2&10$+!+ 5&+7+ 0<'0+ 7(4'&4*8+%0+1*)2&10$+>+ $'*47&+ #7 " + 0<'07(4'&4*8+%0+1*)2&10$+!-
!Y
,-,-+=*2'34,*(-42&'4/*
I8+7(4<&30:81@+%04':*+<&1%)&6&7$:0$+0Q%*40:&&@+6*472&$+ -F " RR →⊂!# +/&0=@; ZZ 4$ +*4+%*47'+&4'0:&(:+$)+1*)2&1&&+ "R⊂! +5&+ =@; 4$ +*4+%*47'+($:07$:0+3&4+!-A>/N_NMN>-+H%"&'(,* #* ,+)* B"* 5%"&2%-* =@; ZZ 4$ +/)+(0,2.* 5,+'(,-.* B"* +,53+2* &%
0,+(,1(-,*$=*/,&.*-(;(2,
Z
ZZZ =@;=@;)&1
Z $$
4$#4$#$$ −
−→
)$(:2.* I(* ):2)* #("(2.-+ K(;(2,* B":.I(* :)* "%;)I2)* /)+(0,2,* 5,+'(,-.* B"* +,53+2* &%0,+(,1(-,* $* ,* #%"&'()(* #* B"* 5%"&2%-+ =@; ZZ 4$ + I(* :)* "32),8.+ =@; ZZ 4$#$′ + :,%
$
4$#
∂∂ =@; ZZ +:,%*B"&.+ =-@; ZZ 4$#G$ +L5$3$:@
-=@;=@;
)&1=@;Z
ZZZZZ
Z $$
4$#4$#4$#
$$$ −
−=′→
P4+1(3+$<01848'(:@+ <%*401+78+ #%"&'(,* #* ,+)* B"*5%"&2%-+ =@@; ZZ 4$ +/)+(0,2.5,+'(,-.*B"*+,53+2*&%*0,+(,1(-.*4@+/,&.*-(;(2,
Z
ZZZ =@;=@;)&1
Z 44
4$#4$#44 −
−→
)$(:2.* I(* ):2)* #("(2.-+ K(;(2,* B":.I(* :)* "%;)I2)* /)+(0,2,* 5,+'(,-.* B"* +,53+2* &%0,+(,1(-,* 4* ,* #%"&'()(* #* B"* 5%"&2%-+ =@; ZZ 4$ + I(* :)* "32),8.+ =@; ZZ 4$#4′ + :,%
4
4$#
∂∂ =@; ZZ *:,%*B"&.+ =-@; ZZ 4$#G4 +L5$3$:@
-=@;=@;
)&1=@;Z
ZZZZZ
Z 44
4$#4$#4$#
444 −
−=′→
A>/N_NMN>-+ /*472&$+ #+ $:0+ /)+(0,2.* 5,+'(,-.+ ?4+ :$%(:'+ 7*+ $+ ;:0<%07'&9+ 4=+ 5)* 3:%1;%-'(;)* @!L ⊂ + 3$78+ #+ $:0+ 30:&9$'8+ %$:2&$)8+ ?4+ :$%(:'+ 7*+ $+ ;:0<%07'&9+ 4=+ ?46&07$:0+ %*47'+ -=@; L4$ ∈ + A$78+ 6*472&$+ #+ $:0+ 30:&9$'8+ %$:2&$)8+ ?4+ :$%(:'+ 7*+ $
;:0<%07'&9+ 7*+4=+ %0+1*)2&10$+L+ $'*47&+ 6*472&$+ $# ′ ;:0<%07'&9+ =4# ′ + 306&4&'8+ %0+L+ <04*105'0+/)+(0,2,*5,+'(,-.*,*#%"&'()(*#*B"*+,53+2*&%*$*;:0<%07'&9+B"*+,53+2+7*+4=+5&+<0
4('0$E8+ $# ′ +<$*+ $#
∂∂+<$*+?478+ #G$ +
+:0<%07'&9 4# ′ ++<$*+ 4#
∂∂+ -?478<$*
#G4
O:$7'&7@+ 30:&9$'$+ %$:2&$)8+ $# ′ + <0+ 7$)7*)0$E8+ 7(4<&30:U43+ ?4+ =@; 4$# + %0+ 47(4<'$4'+5&+30:&9U43+%0+ #+ +7$+(+6*472&0+30+(+<&4B*:8+9$:&$C&)8@+$-+A0:&9$'$+%$:2&$)8
4# ′ +<0+7$)7*)0$E8+7(4<&30:U43+?4+ =@; 4$# +%0+$+7(4<'$4'+5&+30:&9U43+%0+#+7$+(+6*472&030+(+<&4B*:8+9$:&$C&)8@+4-
>V>ROW>-+.-++/&0+6*472&&)0+#=*7*306&4&'0+%0+ "R +%:&4F
-=@;@=@; $44$74$4$# =+=
]Z
L901-=@;@=@;@.=@;@.=@; $4$744$74$#4$# 4$4$ =′=′=′=′
"-+/&0+6*472&$+#+306&4&'8+%0+ "R +%:&4
-$:7'B=;=@; ""
4
$4$4$# +=
L901
44
$$
4$
44$
4
$$4$#$ +=
+++=′ $:7'B"=;$:7'B"=@;
""""
-$:7'B"=;$:7'B"=@;""
"" $4
$4
4$
$4$
4
$44$#4 −=
++−=′
,-+/&0+6*472&$+#+306&4&'8+%0+<01&%)$4*)+30<7D&<+ Z>$ +%:&4
-=@; ", 4$4$4$# −=L901
."",=@; −−=′ 4$ 4$4$4$#
-)4"=@; , $$4$4$# 44 −=′
53$6*-%(-*'(8%9$%,9#("#,2$*(+*,)&''-#($#%-#(.#(.3*1(/%$'%0'-#($#%-#4/&0+6*472&$+ RR →⊂ "F !# +5&+ =@; ZZ 4$ +*4+%*47'+ &4'0:&(:+1*)2&1&&+!-+I8+$)0B01
*4+ %*47'+ $:C&':$:+ =@; 4$ + 3&4':a(+ 907&48'$'0+ C+ $+ )*&+ =@; ZZ 4$ + 5&@+ (3$'8+ $)0<@+ <8
1042&401+6&Q+%*47'*)+ =@; 4$ -+I8+%:0<*%*401+78+0Q&<'8+30:&9$'0)0+%$:2&$)0+ $# ′ +5&+ 4# ′
%0+907&48'$'0$+C+$+)*&+ =@; ZZ 4$ +5&+<8+7(4<&30:81+0B$)&'$'0$
=@;=@;=@;=@;=@;=@; ZZZZZZ 4$#4$#4$#4$#4$#4$# −+−=− ;.=
/&0+ C4% ∈=@; + 5&+ C0$ ∈=@; Z + 3(*8+ %*47'0+ $:C&':$:0+ 3&4+ 907&48'$'0$+ C+ $
%*47'*)*&+ =-@; ZZ 4$ +I8+4('81
=-@;=;=c@;=; Z 0$#04%#% =ψ=ϕ ;"=
A0($:070+30:&9$'0)0+%$:2&$)0+ $# ′ +5&+ 4# ′ +0Q&<'8+%0+907&48'$'0$+C@+6*472&&)0+ϕ+5&+ψ+<*4'30:&9$C&)0+5&+$901
=-@;=;=c@;=; Z 0$#04%#% 4$ ′=ψ′′=ϕ′ ;,=
L%)&7U43+ '0(:01$+7:05'0:&)(:+ 6&4&'0+ 6*472&&)(:+ϕ+ 5&+ψ@+ 303*701+78+ 0Q&<'8+ *4%*47'+ =@; Z $$∈ξ +5&+*4+%*47'+ =@; Z 44∈η +$<'60)+?47U'
==;;=;=;=c=;;=;=; ZZZZ 4444$$$$ 4$ −ηψ′=ψ−ψ−ξϕ′=ϕ−ϕ
$3&78@+*'&)&EU43+;"=+5&+;,=@
==;@;=@;=@; ZZ $$4#4$#4$# $ −ξ′=−5&
=-=;@;=@;=@; ZZZZZ 44$#4$#4$# 4 −η′=−
].
>B$)&'$'0$+;.=+309&40
=-=;@;==;@;=@;=@; ZZZZZ 44$#$$4#4$#4$# 4$ −η′+−ξ′=−
L70$<'8+ :0)$2&0+ <0+ 4*105'0+ #3+;%-,* -%(* K,7+,"7)* 5)"2+%* #%"&'((-)* +),-)* /)/3%.*0,+(,1(-)*+),-)-
L<$3$:@+ /,&.* /)+(0,2)-)* 5,+'(,-)+ $# ′ + I(+ 4# ′ + ,-)* #%"&'()(+ RR →⊂ "F !#
)$(:2.*5)*3* 0)&(".2,2)*C* ,* 5%"&2%-%(+ =@; ZZ 4$ + ("2)+(3+* -%(+!@+,2%"&(* 5)"2+%* 3+(&)
5%"&2* C4$ ∈=@; +)$(:2.*%"*"%;.++ξ+&%5+(":* B"2+)+ Z$ +I(*$* I(*%"*"%;.+*η* &%5+(":B"2+)* Z4 *I(*4*,:2#)-*B"&E2**,+)*-3&*#3+;%-,*-%(*K,7+,"7)?
X'&)&EU43+6(:1*)$+)*&+W$B:$4B0+301(4<':81f>GJ>RL-+G,&.*#%"&'(,*#**,+)*/)+(0,2)*5,+'(,-)*;.+7("(2)*B"2+<3*0)&(".2,2)*C
,*-%(* =@@; ZZ 4$ *("2)+(3+*-%(*!@*,2%"&(*#*):2)*&3"2("%.*B"*5%"&2%-* =-@; ZZ 4$
GMN6AOPJLQRM?+ /*472&&)0+ $# ′ + 5&+ 4# ′ + 6&&43+ 18:B&4&'0+ %0+ C@+ 0Q&<'8+ *4+ 4*18:@Z>K +$<'60)+?47U'
@=@;@=@; K4$#K4$# 4$ ££ ′′
%04':*+(:&70+ -=@; C4$ ∈A&4+6(:1*)$+)*&+W$B:$4B0+:0E*)'8
-=;
=@;=;=@;=@;=@;
ZZ
ZZZZ
44$$K
44$#$$4#4$#4$# 4$
−+−
−⋅η′+−⋅ξ′−
£
££
A$:
Z==;)&1 ZZ
Z
Z
=−+−→→
44$$44$$
5&+307&=@;=@;)&1 ZZ
Z
Z
4$#4$#44$$
=→→
$3&78+6*472&$+#+0<'0+7(4'&4*8+?4+%*47'*)+ =-@; ZZ 4$
=*2'34,*/*(-42&'4/*(4/*(#$*'( +#$%&''( 2*4/*()*(!( 342'47'/*( 2*4/*-+ /&0+ $7*16*472&$+ =,;F > RR "!# " →⊂ +5&+ =@@@; ZZ
"Z.Z "$$$$ != +*4+%*47'+&4'0:&(:+1*)2&1&&+!-
A>/N_NMN>-+ H%"&'(,* #* ,+)* /)+(0,2.* 5,+'(,-.* B"* 5%"&2%-* Z$ * B"* +,53+2* &%
0,+(,1(-,* S$ */,&.
Z
ZZ"
Z.
ZZ.
Z.
Z"
Z. =@@@;=@@@@@@@;
)&1SS
""SSS
$$ $$
$$$#$$$$$$#
S −−+−
→
!!!
)$(:2.* I(* ):2)* #("(2.-+ K(;(2,* B":.I(* :)* "%;)I2)* /)+(0,2,* 5,+'(,-.* ,* #%"&'()(* #* B"
+,53+2* &%* S$ * I(* :)* "32),8.* =@@@; ZZ"
Z. "$ $$$#
S!′ * :,%*
S
"
$
$$$#
∂∂ =@@@; ZZ
"Z. !
* :,%* B"&.
=-@@; ZZ. "$ $$#G
S!
]"
/*472&$+ RR →⊂ "!# F +$:0+"+30:&9$'0+%$:2&$)0+?4+%*47'*)+ -=@@@; ZZ"
Z. !$$$ " ∈!
>V>ROW>-+.-+A$78+ #+ 0<'0+(+ 6*472&0+ :0$)8+30+ ':0&+ 9$:&$C&)0+ :0$)0+ 306&4&'8+ %0+ (1*)2&10+ ,R⊂! + 5&+ 3$78+ ;,@* 1@* &T+ 0<'0+ *4+ %*47'+ &4'0:&(:+ 1*)2&1&&+ !@+ 70)0+ ':0&30:&9$'0+$)0+)*&+#+<*4'F
,$
&1,#&1$#&1,#G
$
&1,#&1,#
,$$$ −
−==∂
∂=′→
=@@;=@@;)&1=@@;
=@@;=@@;
14
&1,#&4,#&1,#G
4
&1,#&1,#
1444 −
−==∂
∂=′→
=@@;=@@;)&1=@@;
=@@;=@@;
&8
&1,#81,#&1,#G
8
&1,#&1,#
&888 −
−==∂
∂=′→
=@@;=@@;)&1=@@;
=@@;=@@;
%04':*+(:&70+ -=@@; !84$ ∈"-+A0:&9$'0)0+%$:2&$)0+$)0+6*472&0&+ RR →,F# +306&4&'8+30
"""=@@; 84$84$# ++=<*4'
"""""""""cc
84$
8#
84$
4#
84$
$# 84$
++=′
++=′
++=′
7*+ =-Z@Z@Z;=@@; ≠84$A0($:070+ 30:&9$'0)0+ %$:2&$)0+ $)0+ 6*472&0&+ =";F >RR "!# " →⊂ + <*4'+ 7D&$:
30:&9$'0)0+ 6*472&&)(:+%$:2&$)0+$)0+ )*&+ #+ ;7$:0+ <*4'+ 6*472&&+ :0$)0+30+(+ <&4B*:8+9$:&$C&)8:0$)8=@+:0E*)'8+78+(%0:$2&&)0+$)B0C:&70+0607'*$'0+$<*%:$+6*472&&)(:+:0$)0+30+"+9$:&$C&)0:0$)0+7$:0+$*+30:&9$'0+%$:2&$)0+7(43*7+'('+)$+6*472&&+7$:0+$*+30:&9$'0+%$:2&$)0-+J0B*)&)030+30:&9$:0+$+<*10&@+3&60:0420&@+%:(3*<*)*&@+7U'*)*&+6*472&&)(:+30+"+9$:&$C&)0+:0$)0+7$:0$*+30:&9$'0+%$:2&$)0+<*4'+$70)0$5&+7$+5&+%04':*+6*472&&)0+:0$)0+30+(+9$:&$C&)8+:0$)8-
f($'0+%:(%:&0'82&)0+:0)$'&90+)$+30:&9$'0)0+%$:2&$)0+$)0+6*472&&)(:+:0$)0+30+3(*89$:&$C&)0+:0$)0+:81U4+$3098:$'0+5&+%04':*+30:&9$'0)0+%$:2&$)0+$)0+6*472&&)(:+:0$)0+301$&+1*)'0+9$:&$C&)0-
=*2'34,*/*( -42&'4/*( 4/*( #$*'( +#$%&''( 3*%,12'4/*-+ /&0+ ;"!# RR →⊂F + (
6*472&0+ 907'(:&$)8+ $)0+ 78:0&+ 7(1%(404'0+ :0$)0+ ;### @@@ ". ! + $*+ 30:&9$'0
%$:2&$)0+ ?4+ :$%(:'+ 7*+ ==@@@;;@@@ ".". !$$$$$$$ "" ∈= !! + ?4':a*4+ %*47'
@@@; Z"
Z.Z !$$$ = + -N4'=@ Z !$" ∈!
H%"&'(,* #* ,+)* /)+(0,2.* 5,+'(,-.* B"* +,53+2* &%* S$ * B"* 5%"&2%-* Z$ * /,&.* 23,2)
&3;53")"2)-)*+),-)* ;### @@@ ". ! *,%*/)+(0,2.*5,+'(,-.*B"*+,53+2*&%* S$ *B"*5%"&2%-
Z$ ?*G)+(0,2,*5,+'(,-.*S$
$#
∂∂ =; Z * ):2)* 0)&23+%-*/("* ;R * &,+)*,+)*/+)52* &3;53")"2)
/)+(0,2)-)*5,+'(,-)*B"*+,53+2*&%* S$ *,-)*#%"&'((-3+* -@@@ ". ;### ! +A07&
-=;
@@=;
@=;=; ZZ"Z.Z ;
S
;
SSS $
$#
$
$#
$
$#
$
$#R∈
∂
∂∂
∂∂
∂=∂
∂!
],
f($'0+%:(%:&0'82&)0+ :0)$'&90+ )$+ 30:&9$'0)0+ %$:2&$)0+ $)0+ 6*472&&)(:+ :0$)0+ 30+1$&1*)'0+ 9$:&$C&)0+ :81U4+ $3098:$'0+ 5&+ %04':*+ 30:&9$'0)0+ %$:2&$)0+ $)0+ 6*472&&)(:907'(:&$)0-
>V>ROWX-+/*472&$+ ,,F RR →# +306&4&'8+%:&4
++++++=
.@.
@.
=;"""""" 4$
8
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4
84
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48
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4
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++++−
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48
84
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8
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=*2'34,*(-42&'4/*()*(12)'$(8#-*2'12-+I8+7(4<&30:81+6*472&$+ RR →⊂ "F !#5&+ <8+ %:0<*%*401+ 78+ 1*)2&10$+ !+ 0<'0+ 30<7D&<8-+ P4+ $70$<'8+ &%('0E8+ 30:&9$'0)0%$:2&$)0+ $# ′ +5&+ 4# ′ +0Q&<'8+?4+6&07$:0+%*47'+ -=@; !4$ ∈ +#*1+30:&9$'0)0+%$:2&$)0+ $# ′ +5&
4# ′ +<*4'+306&4&'0+%0+ @"R⊂! +?4<0$148+78+0)0+<*4'+6*472&&+30+3(*8+9$:&$C&)0+5&+307&
<0+%($'0+%*40+%:(C)01$+)$(:2)"')(*/)+(0,2)-3+*5,+'(,-)*,-)*#%"&'((-3++ $# ′ +5&+ 4# ′ -
A$78+30:&9$'0)0+%$:2&$)0+$)0+6*472&&)(:+ $# ′ +5&+ 4# ′ +0Q&<'8@+0)0+<0+4*10<7+/)+(0,2)5,+'(,-)*/)*3+/("%-*/3(*$)0+6*472&0&+#*5&+<0+4('0$E8
c=;"
"
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∂∂
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∂∂=′′
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∂∂
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∂∂∂=′′
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4
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## 444
/*472&&)0+ 4$# ′′ +5&+ 4$# ′′ +<0+4*10<7+30:&9$'0+%$:2&$)0+;($2)+30+(:3&4*)+3(&-
GHI>JKLMN>-+ A0:&9$'0)0+ %$:2&$)0+ $# ′ + 5&+ 4# ′ + <0+ 4*10<7+ /)+(0,2)* 5,+'(,-)* /)3+/("%-*B"2E(+$)0+6*472&0&+#-
G+6*472&0+30+3(*8+9$:&$C&)0+%($'0+$90$+3(*8+30:&9$'0+%$:2&$)0+30+(:3&4*)+?4'U&5&+ %$':*+ 30:&9$'0+ %$:2&$)0+ 30+ (:3&4*)+ 3(&-+ G+ 6*472&0+ 30+ ':0&+ 9$:&$C&)0+ $@+4@+8+ %($'0$90$+4(*8+30:&9$'0+%$:2&$)0+30+(:3&4*)+3(&@+4('$'0
-@@@@@@@@ """ 848$8844$4$84$$ ######### ′′′′′′′′′′′′′′′′′′
]!
P4+B040:$)@+(+6*472&0+:0$)8+30+"+9$:&$C&)0+:0$)0+ RR →⊂ "!# F +%($'0+$90$+"30:&9$'0+ %$:2&$)0+ 30+ (:3&4*)+ ?4'U&+ 5&+ ""+ 30:&9$'0+ %$:2&$)0+ 30+ (:3&4*)+ 3(&
=-@@"@.@; "SU#SU$$
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>V>ROWX-+/&0+6*472&$+ RR →⊂ "F !# +3$'8+%:&4
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!
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4$
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J01$:781+78+ -$44$ ## ′′=′′I0+ 306&40<7+ ?4+1(3+ $<01848'(:+ 30:&9$'0)0+ %$:2&$)0+ 30+ (:3&4*)+ ':0&@+ 7$+ 6&&43
30:&9$'0)0+ %$:2&$)0+ $)0+ 30:&9$'0)(:+ %$:2&$)0+ 30+ (:3&4*)+ 3(&-+ W$+ 60)+ <0+ 306&40<730:&9$'0)0+%$:2&$)0+30+*4+(:3&4+S+($:07$:0+$)0+*40&+6*472&&+:0$)0+30+"+9$:&$C&)0+:0$)0-
A$78+ RR →⊂ "!# F + 5&+ !$$$ " N4'=@@; . ∈= ! + 30:&9$'$+ %$:2&$)8+ 30+ (:3&4*)+ S+ $
6*472&0&+#@+?4+7$:0+<0+30:&90$E8+%$:2&$)+30+ .α +(:&+?4+:$%(:'+7*+ @.$ +30++ "α +?4+:$%(:'
7*+ "$ +5-$-1-3-@+30+ "α +?4+:$%(:'+7*+ =; ". S$ "" =α++α+α ! +<0+4('0$E8
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# ααα !+<$*+ -
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∂!
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L901c"c", ","" 4$$#$44$# 4$ −=′−=′
c"c!,c!,c"e ""
"""" $#$4$#$4$#4$4#
4$44$$−=′′−=′′−=′′−=′′
c!c!ec!ece "", $#4$#4$#4# 4$$4$4$$ −=′′′−=′′′−=′′′=′′′
-Zc!c!c!e ,"" =′′′−=′′′−=′′′−=′′′ 4$44$4$4 #$#$#4$#
J01$:781+ 78+ $44$ ## ′′=′′ + 5&+ -@ """" $44$44$$4$4$4$###### ′′′=′′′=′′′′′′=′′′=′′′/ + #*+ $)'0
7*9&4'0@+?4+30:&9$'0)0+%$:2&$)0+1&Q'0+30+(:3&4*)+3(&+<$*+':0&+$)0+6*472&0&+3$'0@+4*+$:0&1%(:'$428+ (:3&40$+ ?4+ 7$:0+ <0+ 30:&90$E8+ ?4+ :$%(:'+ 7*+ 9$:&$C&)0)0+ $+ 5&+ 4@+ 7&+ 4*1$&"%;.+%-*/)*3+(+30+7$:0+<0+30:&90$E8+?4+:$%(:'+7*+6&07$:0+9$:&$C&)8+?4+%$:'0-
]]
P4+B040:$)@+30:&9$'0)0+%$:2&$)0+1&Q'0+ 4$# ′′ +5&+ $4# ′′ +4*+<*4'+0B$)0-+A0+$<01040$@?4+B040:$)@+30:&9$'0)0+%$:2&$)0+1&Q'0+30+(:3&4*)+':0&+ "" @@ $4$4$4$ ### ′′′′′′′′′ +4*+<*4'+0B$)0+5&
4&7&+$44$44$
### "" @@ ′′′′′′′′′ +4*+<*4'+0B$)0-
X:18'($:0$+'0(:018++%0+7$:0+4*+(+301(4<':81+<'$C&)05'0+7(43&2&&+<*6&7&04'0%04':*+0B$)&'$'0$+30:&9$'0)(:+%$:2&$)0+1&Q'0-
>*12*.4(/#'(?%@A42B-+G,&.*#%"&'(,+ RR →⊂ "F !# +,+)*/)+(0,2)*5,+'(,-)
;($2)*/)*3+/("%-*/3(* 4$# ′′ *I(+ $4# ′′ +B"2+<3*0)&(".2,2)*C*,*%"%(*5%"&2* !4$ N4'=@; ZZ ∈ *I(
/,&.* 4$# ′′ *I(* $4# ′′ *:%"2*&3"2("%)*B"*5%"&2%-+ =@; ZZ 4$ *,2%"&(
=-@;=@; ZZZZ 4$#4$# $44$ ′′=′′
GHI>JKLMNN-+ .°-+ f0(:01$+ )*&+ I7Di$:E+ :81U40+ $3098:$'8+ 5&+ %04':*+ 6*472&&)0:0$)0+<$*+907'(:&$)0+30+':0&+<$*+1$&+1*)'0+9$:&$C&)0+:0$)0-
"°-+ G+ '0(:018+ $<01848'($:0$+ 0<'0+ 9$)$C&)8+ 5&+ %04':*+ 30:&9$'0)0+ %$:2&$)0+ 30(:3&4+<*%0:&(:-+A0+0Q01%)*@+3$78+30:&9$'0)0+%$:2&$)0+1&Q'0+$)0+*40&+6*472&&+:0$)0+303(*8+9$:&$C&)0@+?4+7$:0+9$:&$C&)0)0+?4+:$%(:'+7*+7$:0+<0+30:&90$E8+&4'0:9&4+30+$70)$5&4*18:+ 30+ (:&@+ 0Q&<'8+ %0+ (+ 907&48'$'0+ $+ %*47'*)*&+ =@; ZZ 4$ + 5&+ <*4'+ 7(4'&4*0+ ?4
%*47'*)+ =@@; ZZ 4$ + $'*47&+ $70<'0+ 30:&9$'0+ %$:2&$)0+ <*4'+ 0B$)0+ ?4+ =-@; ZZ 4$ + P4
%$:'&7*)$:@+ 3$78+ $4$4$## ′′′′′′ @" + 5&+ "$4
# ′′′ + 0Q&<'8+ %0+ (+ 907&48'$'0+ $+ %*47'*)*&+ =@; ZZ 4$ + 5&
<*4'+7(4'&4*0+?4+%*47'*)+ =@@; ZZ 4$ +$'*47&
=-@;=@;=@; ZZZZZZ "" 4$#4$#4$#$4$4$4$′′′=′′′=′′′
,-!-(='+*2*$&'47'/',4,*4(#$*'(+#$%&''()*(!(342'47'/*(2*4/*
A>/N_NMN>-+/&0+6*472&$ @F " RR →⊂!# +5&+ =@; ZZ 4$ +*4+%*47'+&4'0:&(:+1*)2&1&&
!-+/*472&$+#+0<'0+/(#)+)"'(,1(-.*B"*5%"&2%-+ =@; ZZ 4$ +3$78+0Q&<'8+3(*8+4*10:0+:0$)0+α5&+β+5&+(+6*472&0+ @F " RR →⊂ϕ ! +7(4'&4*8+5&+4*)8+?4+%*47'*)+ =@@; ZZ 4$ +$3&78
@Z=@;=@;)&1 ZZ
Z
Z
=ϕ=ϕ→→
4$4$44$$
$<'60)+?47U'@+%04':*+(:&70+%*47'+ @=@; !4$ ∈ +<8+0Q&<'0+0B$)&'$'0$
-=;=;=@;=;=;=@;=@; "Z
"ZZZZZ 44$$4$44$$4$#4$# −+−ϕ+−β+−α=−
A$78+ 1*)2&10$+ !+ 0<'0+ 30<7D&<8+ 5&+ 3$78+ 6*472&$+ #+ 0<'0+ 3&60:042&$C&)8?4+ 6&07$:0+ %*47'+ $)+ 1*)2&1&&+ !@+ <%*401+ 78+ #%"&'(,* #* ):2)* /(#)+)"'(,1(-.* 5);%-'(;),*!-
!"
#$%&'()*+,-.+/+01'$1)*0'+$'++)23,)%3&1
45656576576557657766 89
899999 !!""!"!"!"!"## −+−=−==
:()10$1*&;)%3&1<+$7)$=1.+&1&$1)-'+%)01'$)1,)>$/+%+&)>+/$'$%<+1?+.+&1&$1)/(%0<+$+)$>$2+%$
45765656576576 9999 #!"!!""!"$!"$ ϕ+−β+−α=−@2+>$%&
49576>.+,
9
9
=→→
!"!!""
A$31'$0$
7565675656 89
899
89
899 !!""!!!!"""" −+−−−+−− <<
1().30)+%$=1.+&;<+.$)) 4B576>
7B576>
99 <<!"
!!
!"
"" −−
!"#$%&%%'#()(*+,('-(#.,/'$%0(,(#&%+1%2%.+.(+'0/#)&%(%'!!#CD#DEFGFH)B4)%&'() $*+',-&) $) ./0.)#-$.1.+,-&2-3() 4+) 5*+'0*3) 6!" F%&576 99 ∈
&0*+'-).&)&1.)#.1-7&0.)5&1,-&3.)4+)5*+'0*3) 576 99 !" )I+) 7576 99 α=′ !"$" ) 4576 99 β=′ !"$!89&3-0&0.&) 51-+) '&1.) /.) #.$-+.:0.) #-$.1.+,-&2-3-0&0.&) $*+',-.-) $) /.) /'1-.
( ) 457655676556765767 99999999 #!"!!!"$""!"$!"$!"$ !" ϕ+−′+−′=−%.;<+/01&,-.4)A10;)J%)$=1.+&1&$1
89
899999 56565765656576576 !!""!"!!""!"$!"$ −+−ϕ+−β+−α=−
.(;,)) 79!! = )3?<+%$,
457656576576 999999 ""!"""!"$!"$ −ϕ+−α=−
A$)1+0+7)-'+%)J,-;'<+'$1)0() 799 ≠− "" )'$K(.&;
4576576576
9
99
9
999
""
""!"
""
!"$!"$
−−
ϕ+α=−−
L'$0M%>).1).+,+&;)-$%&'() 79"" → )>$>(0$,
576576576
.+, 999
999
9
!"$""
!"$!"$"
""′=α=
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→
>$31'$0$
49576576.+,
576.+,576.+,
999
9
99
9
99
9
99
=ϕ=ϕ=
=−−
ϕ=−−
ϕ
→
→→
!"!"
""
""!"
""
""!"
""
""""
H,)1';&1&)>$0+)0;)/(%0<+1)$)1'$)J%)-(%0&(.) 576 99 !" )>$'+21&;)-1'<+1.;)J%)'1-3'&
0()")I+) 4576 99 α=′ !"$"
!N
O%) ,3>) 1%1.3=7) >10;) J%) $=1.+&1&$1) >$) >$/+%+<+$) 1) >+/$'$%<+1?+.+&;<++) .(;,79"" = )3?<+%$,
457656576576 999999 !!!"!!!"$!"$ −ϕ+−β=−
O,-;'<+%>)1+0+)-'+%) 99 ≠− !! )I+)&'$0M%>).1).+,+&;7)>$>(0$,
576576576
.+, 999
999
9
!"$!!
!"$!"$!
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−−
→7
0$$1)0$)1'1&;)0;)/(%0<+1)$)1'$)J%)-(%0&(.) 576 99 !" )>$'+21&;)-1'<+1.;)J%)'1-3'&)0()!)I+
4576 99 β=′ !"$!=1<5<>-,-&)$*&$)>$,3%*&'1&;4
:DPQ@:FPGR4)%&'()$*+',-&)$)./0.)#-$.1.+,-&2-3()5.);*3,-;.&)67)&0*+'-).&)&1.#.1-7&0.)5&1,-&3.) "$ ′ )I+) !$ ′ )-$)64
#'3-3K+<+1) '$0+-'30;) %() $*&$) 1>$2;'1&;4) @S+*&;) /(%0<++) 01'$) 1() >$'+21&$-1'<+1.$7) >1') 01'$)%() *(%&)>+/$'$%<+1?+.$4)H0$1*&1) J%*$1,%;) 0;) -$%&'() /(%0<++.$) >$>3(;)*1(),1+),(.&$)21'+1?+.$7)>+/$'$%<+1?+.+&1&$1)I+)>$'+21?+.+&1&$1)-1'<+1.;)%()*(%&%3<+(%+)$0T+21.$%&$4
#CD#DEFGFH)84)%&'()$*+',-&)$)./0.)#-$.1.+,-&2-3()4+)5*+'0*3) 576 99 !" )&0*+'-).&./0.)'<+0-+*()4+)&'./0)5*+'04
%.;<+/01&,-.4)U+$)$=1.+&1&$1
#!"!!""!"$!"$ 5765656576576 9999 ϕ+−β+−α=−
-'+%) 01'$) 1,) >$/+%+&) >+/$'$%<+1?+.+&1&$1) /(%0<+$+) $) J%) -(%0&(.) 4F%&576 99 6!" ∈A$31'$0$)J%)-(%0&(.) 5776 99 !" )&3<+)&$',$%++)>+%)-1'&$1)>'$1-&;)1)10$*&$+)$=1.+&;<+)1(.+,+&1)K$'37)'$K(.&;)0;
[ ] 9576576.+, 99
9
9
=−→→
!"$!"$!!""
*1(576576.+, 99
9
9
!"$!"$!!""
=→→
7
0$$1)0$)1'1&;)0;)/(%0<+1)$)$*&$)03%&+%(;)J%)-(%0&(.) 5476 99 !"
:DPQ@:FPGR4)%&'()$*+',-&)$)./0.)#-$.1.+,-&2-3()5.);*3,-;.&)6)&0*+'-).&)./0.'<+0-+*()5.)64
C$0+-'301) %() $*&$) 1>$2;'1&;4) D) /(%0<+$) 03%&+%(;) %() $*&$) J%) ,3>) %$0$*1'>+/$'$%<+1?+.;4
DVQ@CWHGFF4)B°4)U+$) /(%0<+1) 5X6Y >RR +6$ + →⊂ ) I+) 57776 998
9B9 +"""" !=
(%)-(%0&)+%&$'+3')1.),(.<+,++)64
!Z
U(%0<+1)$)$*&$)>+/$'$%<+1?+.;)J%)-(%0&(.) 5776 99B9 +""" != )>10;)$S+*&;)%(,$'$.$
'$1.$) 57787B6 +-- !=α ) I+) 3) /(%0<+$)ϕ) >$/+%+&;) -$)67) 03%&+%(;) I+) %(.;) J%) -(%0&(.577776 99
89B +""" ! )1>+0;
795777657776.+, 998
9B8B
9
988
9BB
=ϕ=ϕ
→
→→ ++
""
""""
""""""
++
!!
!
1*&/$.)J%0M&)-$%&'()3'+0$)-(%0&) 757776 8B 6"""" + ∈= ! )*;)12$,)$=1.+&1&$1
#""""""""$"""$ +
+
----++ 57776565777657776
B8B
9998
9B8B ∑
=ϕ+−α=− !!!
(%>$
456B
89∑=
−=+
--- ""#
8°4)U+$) /(%0<+1)2$0&3'+1.;) ;6$ RR →⊂ 8Y ) I+) /+$) ;$$ 77B ! ) 03,-3%$%&$.$*1.$) '$1.$4) Q-(%$,) 0;) /(%0<+1) 2$0&3'+1.;) $) $*&$) >+/$'$%<+1?+.;) J%) -(%0&(.
7F%&576 99 6!" ∈ )>10;)&31&$)03,-3%$%&$.$) ;$$$ 777 8B ! )*(%&)>+/$'$%<+1?+.$)J%)10$*&
-(%0&4)HI1>1'7) /(%0<+1)2$0&3'+1.;) $)$*&$)>+/$'$%<+1?+.;) J%)-(%0&(.) 576 99 !" )>10;)1(.30)$=1.+&;<+.$
;-#!"!!""!"$!"$ ----- 7787B75765656576576 9999 !=ϕ+−β+−α=− 7
(%>$) 7576 6!" ∈ )+1') 57776577776 8B8B ;; βββ=βααα=α !! )*(%&)>3+)2$0&3'+)>+%;R ) I+) ) 57776 8B ;ϕϕϕ=ϕ ! ) $*&$) 3) /(%0<+$) 2$0&3'+1.;) >$/+%+&;) -$),(.<+,$1)6) 0(
21.3'+) J%) 7;R ) 03%&+%(;) J%) 576 99 !" ) I+) %(.;) J%) 5476 99 !" ) 6@2+>$%&7) 03,-3%$%&$.$
'$1.$) ;ϕϕϕ 777 8B ! )1.$)/(%0<+$+)ϕ)*(%&)03%&+%($)I+)%(.$)J%)-(%0&(.) 576 99 !" 54A+%)03%>+<++.$)%$0$*1'$)-$%&'()>+/$'$%<+1?+.+&1&$1)(%$+)/(%0<++)'$1.$7)/3',(.1&$
J%)-'3-3K+<+1)B7)'$K(.&;
)!
!"$
"
!"$ --
-- ∂
∂=β
∂∂
=α576
7576 9999 7
1>+0;
7576576
77576
5776 999999BB "
!"$
"
!"$
"
!"$ ;; ∂
∂=
∂∂
∂∂=αα=α !!
4576576
77576
5776 999999BB !
!"$
!
!"$
!
!"$ ;; ∂
∂=
∂
∂∂
∂=ββ=β !!
:3%*+>$'1<++)1*$,;%;&31'$)*$)/10)I+)1*(-'1)/(%0<++.3')2$0&3'+1.$)>$)+)21'+1?+.$'$1.$)0()21.3'+)J%) 4;R
!"
#$!$!"#$%&%'(#)*)!+'%#!$+',(##!-%!!!.)&#)/#*%!&%)*%
%&' ()*+,-./&0' 12*(3,4' 56 7 RR →⊂!" ' 8.' (4/.' )' 8/.+282*.0' -,1./.*9
3,4:,;&'<*'82*(=2;' >5? @@ #$ ',*=./,)/'02;3,0,,'!$'AB,+=&'-.(,'.C4;,=4=.4
5>5?>>?5?>>?5?>5?>5? @@@@@@@@ %#$###$"$$#$"#$"#$" #$ ϕ+−′+−′=−2*-.
$>?>?5@>5?>5?;,0 7@
7@@@
@
@
##$$%#$#$
##$$
−+−==ϕ=ϕ→→
D,1./.*34' >5?>5? @@ #$"#$" − ' +.' *20.E=.' &'()*('(+, "-.&/0(0, ", &1'(23-.4
56*1+'(, &'()*('071'' @$$ − ' E,' @## − ' +7(, 8+'0+907(71', $' E,' #$' D.)4/.(.' 12*(3,4
5@→ϕ ' (F*-' @$$ → ' E,' 5@##→ ' (/.E=./.4' >5?>5? @@ #$"#$" − ' 8)4=.' 1,' 48/)B,9
04=&'(2'"-.&/0+,70.0+'6' >>?5?>>?5? @@@@@@ ###$"$$#$" #$ −′+−′ '-.1,*,=&'8.*=/2')/,(.
$>5? 7R∈#$ 'G(.4+=&' 12*(3,.' ;,*,4/&' +.' *20.E=.' %0"('(./0+7+, "-.&/0(0, "' <*' 82*(=2;
>5? @@ #$ 'E,'+.'*)=.4H&' >$5?- @@ #$" 'GE4-4/
>$>?5?>>?5?>5I5?- @@@@@@@@ ###$"$$#$"#$#$" #$ −′+−′=D.)4/.(.
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F2+%'242()6'/3)/6+9278()9+%92)*-34://.2)&/3),2,G'-.)&'269)5-39)&2'/H%G/.2)62,-.:/,2%)"()'27-.98)48)$/)*-34:/%)&)2592)&2'/H%G/.8)62)")$/()<3)6.-5() ?@?@ $#$& = )$/
?>@?@ = $#$& = ) I+3*+',) 92+'2,2/) .-/) J+..2() 2D/598) -3) 6-349) ?((@ = $$∈ξ ) &%48
$$ <= )@5%-) ?((@ =$$∈ξ )&%48) $$ >= ?)%59*2.)<34K9) >=?@ =ξ′& )F%'
α−−−=
=α−−−+−
−−
−′′′−+′′−−′′−+′−′=′
−+
−+−
B?B@
B?B@?@B
L
?@?@A
?@
?@?@A
?@?@
?AB@
?@
?@AL?@
?@AB
?@AB
?@?@?@
%!!
%!!
!!
'$%'#!
'$
'$%'#!
'$'#
!
'$
'#'$
'#'$
'#'$
'#'#'&
!
!
$/
>=?@?@A?@
?@ B?B@ =αξ−−ξξ−=ξ′ −+ %!!
$%#!
$&
F2)%/4/)+G:/32,
>?@A?@ ?B@
B
ξξ−=α ++−
!%!
#%!
$
J27-.98
>?@A
?@?@?@ ?B@
B= ξξ−−= +
+−!
%!%
! #%!
$$$$(
M-K3&) (B+= !% )+G:/32,
>?@?AB@
?@?@ ?B@
B= ξ
+−= +
+!
!
! #!
$$$(
C-G)%42%598)*+',8() !( )52)3-,2$92))*+',-.-,/.01&)1!&*>
F2+%'242) ξ )2592)4-6'/35)<39'2) =$ )$/)$()2D/598)-3)3-,8') B= <θ< )%59*2.)<34K9
?>@ == $$$ −θ+=ξ ) F%48) 3+98,) (=$$2 −= ) %9-34/) ?(B=@= <θ<θ+=ξ 2$ ) /%'*+',-.%).-/)0%1.+')&2)+'&/3-.)!()4+'256-3789+%'2)*-34:/2/)'2%.2)&2)+)H%'/%G/.8)#()<36-349-.) (=$ )52)54'/2
?(@?AB@
?@A
?@AB
?@?@ =?B@
B
=?@
=== 2$#!
2$#
!
2$#
2$#2$# !
!!
!θ+
++++′+=+ +
+!
-3&2)'259-.) !( )2592)54'/5)<3)*+',%).-/)M%E'%3E2>
!N
O3)6%'9/4-.%'()&%48) "∈= )$/)&%48).-8,) (== =$ )*+',-.%).-/)0%1.+')4-)'259-..-/)M%E'%3E2)&2H/32
?B=@?(@?AB@
?=@A
?=@AB
?=@?@ ?B@B
?@ <θ<θ+
+++′+= ++
$#!
$#
!
$#
$#$# !
!!
!
!
C-G)%42%598)*+',8()*+',-.%).-/)0%1.+')@<3)6-349-.)=()4-)'259-.).-/)M%E'%3E2?)52),%/3-,2$92)#3)4,-1.-,/.516.01,)/!>
!"#$%#%#&'()*%#%+) ?>(@ 7$(! )P/2)&/3)3+-) RR →⊂ L; "# )$/)6-349-.)/392Q
'/+') >?(@ == "7$ ∈ ) C8) 6'25-6-32,) 48) *-34:/%) #) *+'*. 8/#*)*!9/1:/-;. 8*) n +1 ) 3)/<!')=3. >*6/!;'1'*) ?) 1. @,!6',-,/) ?(@ == 7$ ) A/. +;. -,;4) ?(@ BB 7$ ) 1):/')1). <!. >*6/=!;'1'*1)?>
C8) 4+35/&2'8,) *-34://.2) R→RB(=S;$ ) $/) (RB(=S; R→7 ) &2*/3/92) 6239'-+'/42) RB(=S∈' )6'/3)'2.%://.2
'$$$'$ ?@?@ =B= −+=
'777'7 ?@?@ =B= −+=
$/)*-34:/%)4+,6-58)&)&2*/3/98)6239'-) RB(=S∈' )6'/3)2E%./9%92%
?>?@(?@@??@?(@@?@ =B==B= '777'$$$#'7'$#'& −+−+==
TG52'H8,)48)*-34:/%)4+,6-58)&)2592)&/*2'23:/%G/.8)&2) B+! )+'/)62)S=()BR)$/
>B((L(B??(@?(@@?@&?@&?@& +=
∂∂+
∂∂= !%'7'$#'7
7'$
$'&
%
% !
F%') '77'7'$$'$ &?@?@&(&?@?@& =B=B −=−= )$/)&24/
>&??@?(@@?@?@?@& =B=B%
%
% ''7'$#777
$$$
'&
−
∂∂+−
∂∂=
#$%&%'
??(@?(@@?@?@&?@&
=B=B '7'$#777
$$$'
'&%
%
%
−
∂∂+−
∂∂=
/%'
>B((B?((@?@?@?=@&
===B=B +=
−
∂∂+−
∂∂=
∂!%7$#77
7$$
$'
&%
%
%
!
U2) &2) %.98) 6%'92() &2+%'242) *-34:/%) &) 2592) &2'/H%G/.8) &2) B+! ) +'/) 62) S=() BR(6-92,)54'/2)*+',-.%).-/)V%4)M%-'/3)&2)+'&/3-.)!()4+'256-3789+%'2)*-34:/2/)&)@&2)+H%'/%G/.8?>)#H2,
?>B=@?(@?AB@
?=@A
?=@AL
?=@AB
?=@?@ ?B@B
?@L
<θ<θ+
+++′′+′+= ++
'&!
'&
!
'&
'&'
&'& !!
!!
!
!W
TG52'H8,) 48) ?(@??=@?(=@@?=@ == 7$#7$#& == ) /%') == ??B@?(B@@?B@ 7$#&
?>(@ BB 7$#=F/3)*+',-.%).-/)V%4)M%-'/3)'27-.98
?>@?AB@
B?=@
AB
?=@ALB
?=@ABB
?=@?B@ ?B@?@ θ+
+++′′+′+= +!! &!
&!
&&&& !
F24/
??>@?(@@?@?@?AB@
B
?(@?@?@A
B
?(@?@?@ABB
?(@?B@?(@
B
=B=B
===B=B
===B=B==BB
θθ
−
∂∂+−
∂∂
++
+
−
∂∂+−
∂∂+
+
−
∂∂+−
∂∂+==
+
7$#777
$$$!
7$#777
$$$!
7$#777
$$$
7$#&7$#
!
!
!
!
F%48)3+98,) ?@?(@ θ=ηθ=ξ 7$ )$/):/32,)52%,%)48)6-349-.) ?(@ BB 7$ )%)*+59)%.25
%'G/9'%') <3) H24/389%92%) ?. %) 6-349-.-/. ?(@ == 7$ ) '27-.98) 48) 6239'-) +'/42) 6-349
?7$ ∈?(@ ) 2D/598) -3) 6-349) ?(@ ηξ ) @62) 52E,239-.) 42) -32$92) ?(@ == 7$ ) 4-) ??(@ 7$
%59*2.)<34K9
?>(@?@?@?AB@
B
?(@?@?@A
B
?(@?@?@ABB
?(@?(@
B
==
====
======
ηξ
−∂∂+−
∂∂
++
+
−
∂∂+−
∂∂+
+
−
∂∂+−
∂∂+=
+
#777
$$$!
7$#777
$$$!
7$#777
$$$
7$#7$#
!
!
!
!
F2)%/4/)&2&-42,
?(@?@?@?AB@
B?(@
B
== ηξ
−
∂∂+−
∂∂
+=
+
#777
$$$!
7$(!
!
$/>=?(@./,
=
=
=→→
7$(!77$$
#,)&2,+359'%9) &24/) 48()816;. #,!69/1) RR →⊂ L; "# ) *+'*. 8/#*)*!9/1:/-;8*. B+! )3)/.<!')=3.>*6/!;'1'*.?.1.@,!6',-,/. "7$ ∈?(@ == )1',!6/B.@*!'),.3)/6*.@,!6'
?7$ ∈?(@ )*$/+';.,!.@,!6'. ?∈ηξ ?(@ )@@*.+*&4*!',-.61)*.,!*A'*.@,!6'*-*. ?(@ == 7$A/) ??(@ 7$ )1+'#*-.<!6C'
?>(@?@?@?AB@
B?(@
B
== ηξ
−
∂∂+−
∂∂
+=
+
#777
$$$!
7$(!
!
!"
#$%&'()*++,-.,-!"#$% &'(#)*"% RR →⊂ /0 +& -",-%.-,*/"0-%1",)*"2-%.-%3,.*4
('2%(%#3(0*('-%5(0,43%/-#*($0"0-%6%"%1'(#0'2'*% 1+23415 "" +78 ∈ -,-90'2%&3,:'2-*%2'*
;"723,%.-%3,.*('2%(<%#3,-91'(=$03",-%&'(#)*-*%&%%5(%1'(#0'2% 4115 "" 78 -9-%13"0-%1'(-9'>%&3,:"
4154156
.415 78.78(
78? (( ϕ=
'(.-% &'(#)*"- RR →⊂ϕ /0 + - -90-% #3(0*('$% @*% ('2$% 5(% 1'(#0'2% 415 "" 78 <% *",
,4545415 /"
/" 778878. −+−=
72-89:3;<=>9:1-."#$%&'(#)*"- RR →⊂ /0 +& -",-%.-,*/"0-%1",)*"2-%.-%3,.*('2
.3*% #3(0*('-% 5(0,43% /-#*($0"0-% 6% "% 1'(#0'2'*% 4115 "" 78 - "0'(#*% 1-(0,'% 3,*#-678 ∈415 -",-%23#%&3,:'2"
?,454@5415/.
45415
4455415
/45415
/.
45415
45415
415415
/"
/"
/"/
""/
""""
//
"/""
/
"""
"""
""
778878
777
78&7788
78
78&88
8
78&
777
78&88
8
78&78&78&
−+−ϕ+
+
−
∂∂+−−
∂∂∂+−
∂∂+
+−∂
∂+−∂
∂+=
/,-AB:C=>9->=;-D9E>B:-F93G-9;<;-8H23:=-I=2<J;;>H-:H9>H-FH-FB=G-K9:;9L;>H-:H9>H5M;- N2-89:3;<=>9:-8H23:=- I=2<J;;>H- :H9>H-FH-B-K9:;9L;>G- :H9>G4- OH-8B93H- O<:;H- N2-CBF9OHCG2G3B:-M;-8H23:=-I=2<J;;>H-:H9>H-O9=-KH<3B:;9>H-FH- 4P5 ≥(( -K9:;9L;>H-:H9>H,-QH9OHCH2H9-3HB:HCH>H-8:HRH2393H-C9;-O=O-:GCS2-9FHKG:93H-M;-8H23:=-I=2<J;;>H-FH-C9;C=>3H-K9:;9L;>H,
P,!,!"#$%&'&(&!)*+,-..(/%!%&0(&!1&! 4/5 >(( !20%.03.(&!%&0(&
A;H- RR →⊂ /0 +& -B-I=2<J;H-:H9>G-FH-FB=G-K9:;9L;>H-:H9>H-81%71-FHI;2;3G-8H
C=>J;CH9- /R⊂+ -M;- 415 "" 78 -=2-8=2<3-9>-C=>J;C;;-+,
Q&A+T+*+&,- A'(#0'2- +78 ∈415 "" - -90-% 1'(#0% .-% :"8*:% 23#"2% 5,-2"0*/4% "2&'(#)*-*% &<% ."#$% -8*90$% 3% /-#*($0"0-% 6% "% "#-90'*% 1'(#0<% "90&-2% 5(#B0% 1-(0,'% 3,*#-
4,154151415 "" 78&78&+678 <¡∈ -A'(#0'2% +78 ∈415 "" - -90-% 1'(#0% .-% :*(*:23#"2% C,-2"0*/D% "2% &'(#)*-*% &<% ."#$% -8*90$% 3% /-#*($0"0-% 6% "% 9"<% "90&-2% 5(#B0<% 1-(0,'3,*#-% 4,154151415 "" 78&78&+678 >¡∈
U=2<3H>H-FH-C9V;C->B<9>-M;-FH-C;2;C->B<9>-OH-2=CHO<-1'(#0-%.-%-80,-:%23#"2C,-2"0*/D-9>H-I=2<J;H;-&,
!.
Q9<G- 415 "" 78 - HO3H- =2- 8=2<3- FH- C9V;C- >B<9>- 5C;2;C- >B<9>4- 9>- I=2<J;H;- &1
93=2<;- 415 "" 78& - OH-2=CHM3H-:"8*:%23#"2- 5:*(*:%23#"24%9>- I=2<J;H;,-W9V;CH>H-M;C;2;CH>H->B<9>H-5:H>93;KH4-OH-2=CHO<--80,-:-%23#"2-%5,-2"0*/-4-9>H-I=2<J;H;-&,
A;H- 415 "" 78 -=2-8=2<3-FH-HV3:HC-5>B<9>4-9>-I=2<J;H;-&1-;23H:;B:-C=>J;C;;-+,-%G
8:HO=8=2HC-<G-I=2<J;9-&-9:H-FH:;K93H-89:J;9>H- 8& ′ -M;- 7& ′ -N2-8=2<3=>- 4,15 "" 78 -)3=2<;1
I=2<J;9-89:J;9>G- 10"
RR →⊂ϕ 7+ -FHI;2;3G-8H-C=>J;CH9
X41510Y ""+7888+7 ∈∈= R
HO3H- FH:;K9L;>G- N2- 8=2<3=>- ,"8 - )KHC- 41545 "78&8 =ϕ - M;- 4,1545 """ 78&8 8′=ϕ′QHB9:H<H- "8 -HO3H-=2-8=2<3-;23H:;B:-C=>J;C;;-+-M;-=2-8=2<3-FH-HV3:HC-9>-I=2<J;H;-ϕ1N2-L9R9- 3HB:HCH;- >=;-AH:C931-FH:;K939- ϕ′ -9- I=2<J;H;-ϕ- OH-92=>H9RG- N2-8=2<3=>- 1"8
9F;<G- ,"45 " =ϕ′ 8 -'HR=>3G- ,"415 "" =′ 78&8ZB2O;FH:S2F-98B;-I=2<J;9-89:J;9>G- 10
"RR →⊂ψ 8+ -FHI;2;3G-8H-C=>J;CH9
X41510Y ""+7877+8 ∈∈= R
OH-9:93G-<G- ,"415 "" =′ 78&7)M9F9:1-9:H->B<-1,313=*)*"0!"#$% &'(#)*"% &% ",-% .-,*/"0-% 1",)*"2-% 5(0,4'(% 1'(#0% .-% -80,-:- 4115 "" 78
*(0-,*3,%:'2)*:**%+<%.-,*/"0-2-%1",)*"2-% 8& ′ -M;- 7& ′ -9-%"('2-"=$%5(%"#-90%1'(#0<-9F;<G
,"4151"415 """" =′=′ 78&78& 78
#$%&'()*++,-.,-U:B8BR;J;9-FHCB2O3:93G-9;<;-HO3H-9FHKG:93G-M;-F9<G- 415 "" 78 -2=HO3H-8=2<3- ;23H:;B:-C=>J;C;;-+1-F9:-+-<B2J;2H-FB=G-OH[CH23H1-=2=>-89:9>H>-<=-9V9#81- ;9:- <H>G>9>3- <=- 9V9- #71- <9:H- OH- ;23H:OH<3H9RG- N2- 415 "" 78 - 9O3IH>- <9- 8=2<3=>
415 "" 78 -OG-2=-OH-9I>H->9-2;<;-=29-F;2-HV3:HC;3GJ;>H-<H>B:-FB=G-OH[CH23H,
/,- A;H- 415 "" 78 - =2- 8=2<3- ;23H:;B:-C=>J;C;;-+,- Q9<G- 415 "" 78 - HO3H- =2- 8=2<3FH- HV3:HC- 9>- I=2<J;H;- &- M;- F9<G- &- HO3H- F;IH:H2J;9L;>G- N2- 9<HO3- 8=2<31- 93=2<;
1"415F "" =78& -<HH9-<H-:HR=>3G-;CHF;93-F;2-H[9>;393H9
778&878&78& 78 F415F415415F """""" ′+′==2FH
,"4151"415 """" =′=′ 78&78& 78
P,-E(%1'(#0% 415 "" 78 - *(0-,*3,%:'2)*:**%+<% 1-(0,'% #",-% 1"415F "" =78& - O9=1
<HH9-<H- HO3H- 9<H>9M;- >=<:=1- 8H23:=- <9:H- "4151"415 """" =′=′ 78&78& 78 - 9-%(':-@0-
1'(#0% 90")*3(",% "2% &'(#)*-*% &F- )M9F9:1- F9<G- 415 "" 78 - HO3H- =2- 8=2<3- O39J;B29:- 9>
I=2<J;H;- &1- I=2<J;9- &- HO3H-F;IH:H2J;9L;>G- N2- 415 "" 78 - M;- 9:H- FH:;K93H- 89:J;9>H- 2=>H- N29<HO3-8=2<3,
!/
Q;2-1,313=*)*"-FHCB2O3:93G-C9;-O=O-:HR=>3G-<G-3,*#-%1'(#0%.-%-80,-:%523#"24.*(%*(0-,*3,'2%:'2)*:**%+<%5(%#",-%&'(#)*"%&%-90-%.*&-,-()*">*2$<%-90-%1'(#0%90")*3(","2% &'(#)*-*% &F% G8*90$% 5(9$% 1'(#0-% 90")*3(",-% "2-% &'(#)*-*% &% #",-% ('% 9'(0% 1'(#0-% .--80,-:-5>B<9>41-9M9-<=C-:HR=>3G-M;-F;2-=:CG3B9:H>H-HVHC8>H,
&\&WU]&,-.°,-A;H-I=2<J;9-&-FHI;2;3G-8H- /R -8:;2-H[9>;393H9
//415 7878& −=
M;-8=2<3=>- ,4"1"5 /R∈ -Z9><=>GC- 78 && ′′ 1 -M;- df .
1"F^F/F/F1/1/ =−=−=′=′ &7788&7&8& 78 --N2-5"1-"4
M;,"4"1"51"4"1"5 =′=′ 78 &&
A=2<J;9- &- 9:H-FH:;K93H-89:J;9>H- <B23;2=H- N2- 5"1- "41- FH<;- HO3H- F;IH:H2J;9L;>G- N29<HO3-8=2<3,-)M9F9:-8=2<3=>-5"1-"4-HO3H-8=2<3-O39J;B29:-9>-I=2<J;H;,-%G-<H:<H3GC-F9<G9<HO39-HO3H-8=2<3-FH-HV3:HC,
UH23:=-B:;<H-8=2<3- 4"158 -9KHC- 4,"1"5"4"15 / &88& >>= -UH23:=-B:;<H-8=2<3
411"5 7 -9KHC- 4,"1"5"41"5 / &77& <<−= -QH<;- 4"1"5415 &78& − -2=-8GO3:H9RG-9<H_
>9M;- OHC2- 8H23:=- B:;<H- 1415 /R∈78 - 9F;<G- I=2<J;9- &- 2=- 9:H- N2- 8=2<3=>- 5"1- "4- 2;<;C;2;C-5>B<9>41-2;<;-C9V;C-5>B<9>4,
/°,-A;H-I=2<J;9-&--FHI;2;3G-8H- /R -8:;2
,/415 // 87878& +−=
)KHC- 778878&7&8& 78 F/F4//5415F1/1// −+=−=′+=′ - M;- "F =& - 8H23:=
,"1. =−= 78 -U=2<3=>-5`.1-"4-2=-HO3H-8=2<3-FH-HV3:HC1-FHB9:H<H
//// 4.5./4"1.5415 78878&78& −+=++−=−− -M;- "4.5 // >−+ 78
F9<G- "4.51. // <−+>+ 7878 -F9<G- ,. 78 <+a,-Q;2-BLOH:K9J;;>H-/-M;-P-:HR=>3G-<G1-F9<G-+-HO3H-B-C=>J;CH-FHO<b;OG-M;-F9<G
I=2<J;9-&-HO3H-F;IH:H2J;9L;>G-8H-C=>J;CH9-+1-3B93H-OB>=J;;>H-O;O3HC=>=;
"415"415 =′=′ 78&78& 78
O=23- O39J;B29:H,- U=2<3H>H- FH- HV3:HC- >B<9>- 5I;;2F- 8=2<3H- O39J;B29:H4- OH- 9I>G- 8:;23:HOB>=J;;>H-9<HO3=;-O;O3HC1-F9:-2=-03"0--OB>=J;;>H-O;O3HC=>=;-FH-C9;-O=O-O=23-8=2<3H-FHHV3:HC,
UH23:=-9-;FH23;I;<9-8:;23:H-8=2<3H>H-O39J;B29:H-'(-2--8=2<3H-FH-HV3:HC1-3:HL=;HOG-=3;>;RGC-FH:;K93H>H-89:J;9>H-FH-B:F;2=>-FB;-9>H-I=2<J;H;-&,
A;H-FH<;- 415 "" 78 -=2-1'(#0%90")*3(",-9>-I=2<J;H;- ,0 / RR →⊂+& -%G-8:HO=_8=2HC-<G-I=2<J;9-&%",-%.-,*/"0-%1",)*"2-%.-%3,.*('2%.3*%#3(0*('--N23:_B-KH<;2G393H-69-8=2<3=>=;- 4,15 "" 78
!P
Z=C-- 415 "" 78 - -HO3H--=2--8=2<3- -O39J;B29:1- - ,"4151"415 """" =′=′ 78&78& 78
QHB9:H<H-I=2<J;9-&-9:H-FH:;K93H-89:J;9>H-FH-B:F;2=>-FB;-<B23;2=H-8H-KH<;2G393H9-6-9>=;- 4115 "" 78 - 8=3HC- O<:;H- IB:C=>9- >=;- D9E>B:- FH- B:F;2=>- FB;1- <B:HO8=2RG3B9:H
I=2<J;H;-&1-N2-8=2<3=>- 4,15 "" 78 -UH23:=-B:;<H- 678 ∈415 -9KHC
415415/
.?44515
4454515/44515@/
.
4451544515415415
//"""/
""""/
"""/
""""""""
78.787778&
778878&8878&
7778&8878&78&78&
7
878
78
ϕ+−′′+
+−−′′+−′′+
+−′+−′+=
=2FH- "415415>;C ""
"
"
=ϕ=ϕ→→
78787788
-M;- ,4545415 /"
/" 778878. −+−=
%G- 2B3GC- 4,1541154115 """""" // 78N&>78&" 7788′′=′′=′′= - AB:C=>9- >=;
D9E>B:-OH-C9;-O<:;H
415415/
.
?454455/45@/
.415415
/
/"""
/"""
78.78
77#7788>88"78&78&
ϕ+
+−+−−+−=−
FHB9:H<H,"415^"415 """" =′=′ 78&78& 78
UH23:=- 441554115415 "" 6787878 ∈≠ - 9KHC- "415 ≠78. - FH<;- 8=3HC- 2B39
,415
4151415
415 ""
78.
7778/
78.
8878'
−=
−= -QH-9;<;1-8H23:=-B:;<H-8=2<3- 4115415 "" 7878 ≠
BLJ;2HC- ;FH23;393H9- 1.415415 // ≡+ 78/78' - <9:H- 9:93G- <G- B:;<H- 8BR;J;H- 9:- B<=89
8=2<3=>- 415 78 -N2-KH<;2G393H9-6-9-8=2<3=>=;- 415 "" 78 -8=2<3=>- 415 /' -FHO<:;H-=2-<H:<<=-<H23:=>-N2-B:;[;2H-FH-:9RG-.,-)M9F9:
4,15?415415415415/415@/
.
415415
///
""
78.7878#/78/78>'78"'
78&78&
ϕ+++=
=−
%HC2=>- F;IH:H2JH;- 415415 "" 78&78& − - HO3H- 9<H>9M;- <=- OHC2=>- HV8:HO;H;- F;289:923HRH-F;2-CHCL:=>-F:H83-9>-9<HO3H;-H[9>;3GJ;,
%G- 8:HO=8=2HC- <G- ,"/>− >"# - 72- 9<HO3- <9R- 3:;2BC=>- #>" +α+α // - 9:H
:GFG<;2;- <BC8>HVH1- 9F;<G- 2=- OH- 92=>H9RG- 8H23:=- 2;<;- B- K9>B9:H- 5:H9>G4- 9- >=;- α,QHB9:H<H- .415415 // =+ 78/78' -:HR=>3G-<G-M;
415415415/415 // 78#/78/78>'78"' ++
!a
2=-OH-92=>H9RG-N2-2;<;-=2-8=2<3- 441541155 78/78' -M;-9:H-9<H>9M;-OHC2-<=-",-W9;
C=>31--F;2-<B2F;J;9-<G- 1"415>;C
"
"
=ϕ→→
787788
-FHF=<HC-<G-HV;O3G-=2-F;O<-<;:<=>9:-6-<=
<H23:=>-N2- 415 "" 78 -M;-:9R9-O=I;<;H23-FH-C;<G1-9O3IH>-N2<S31-8H23:=-B:;<H- 1415 678 ∈
4,15415415/415415 // 78#/78/78>'78"'78 ++<ϕ
QH<;-N2-KH<;2G393H9-6-HV8:HO;9
415415415415/415 // 7878#/78/78>'78"' ϕ+++
9:H-9<H>9M;-OHC2-<=- 4115415415/415 // 78#/78/78>'78"' ++ -9F;<G-9:H-9<H>9M;-OHC2<=-",
)M9F9:1-N2-;8B3HR9- "/>− >"# -5O9=- 4"4?15@415415 /"""""" // >′′−′′′′ 78&78&78& 7878
F9<G- ">" -93=2<;- "415415 "" >− 78&78& -M;-FH<;- 415 "" 78 -HO3H-=2-8=2<3-FH-C;2;C
>B<9>^-F9<G- 1"<" -93=2<;- "415415 "" <− 78&78& -M;-FH<;- 415 "" 78 -HO3H-=2-8=2<3-FHC9V;C->B<9>,
%G-<B2O;FH:GC-9<=C-<9R=>-<S2F- 1"/<− >"# -9F;<G
,"4?15@415415 /"""""" // <′′−′′′′ 78&78&78& 7878
D:;2BC=>- #>" +α+α // - 9:H- :GFG<;2;- :H9>H- M;- F;O3;2<3H1- FH<;- 9:H- =2- 92=C;3OHC2-8H23:=-K9>B:;-9>H->=;-α-<=8:;2OH-N23:H-:GFG<;2;-M;-OHC2-<B23:9:-8H23:=-K9>B:;9>H->=;-α%-N2-9I9:9-:GFG<;2;>B:,-'HR=>3G-<G
415415
415/
415
415 /
/
78/#78/
78'>
78/
78'"
++
;9- 93S3- K9>B:;- O3:;<3- 2H[93;KH- <S3- M;- K9>B:;- O3:;<3- 8BR;3;KH,- U:;2- =:C9:H1- N2- B:;<HKH<;2G393H-9-8=2<3=>=;- 415 "" 78 -F;IH:H2J9- 415415 "" 78&78& − - ;9-K9>B:;-2H[93;KH-M;K9>B:;-8BR;3;KH1-FH<;- 415 "" 78 -2=-HO3H-=2-8=2<3-FH-HV3:HC,
)C-FHCB2O3:93-FH<;- 0-3,-:"- N2-L9R9-<G:H;9- :H<=2B9M3HC-8=2<3H-FH- HV3:HC5>B<9>4-8:;23:H-8=2<3H>H-O39J;B29:H,
D&#'&Wc,-!"#$% 415 "" 78 --90-%'(%1'(#0%90")*3(",%"2%&'(#)*-*%&%@*%."#$%&'(#)*"%&",-%.-,*/"0-%1",)*"2-%.-%3,.*('2%.3*%#3(0*('-%5(0,43%/-#*($0"0-%"%1'(#0'2'*- 415 "" 78"0'(#*0
),-!"#$- 4151"4?15@415415 ""/
"""""" // 7878&78&78& 7878>′′−′′′′ - -90-% '(% 1'(#0
.-%-80,-:%23#"2%"2%&'(#)*-*%&%@*%"(':-H
415 "" 78 --90-%'(%1'(#0%.-%:*(*:%23#"21-F9<G- ^"415 ""/ >′′ 78&8
415 "" 78 --90-%'(%1'(#0%.-%:"8*:%23#"21-F9<G- ,"415 ""/ <′′ 78&8
!d
$,-!"#$- 4151"4?15@415415 ""/
"""""" // 7878&78&78& 7878 <′′−′′′′ -('%-90-%'(%1'(#0
.-%-80,-:%"2%&'(#)*-*%&F&\&WU]e,-A;H
,4151P415 /PP R∈++= 78877878&
)I>GC-8=2<3H>H-O39J;B29:H,-UH23:=-9<H9O39-:HRB>KGC-O;O3HC=>
=+=′=+=′
,"PP415
"PP415/
/
8778&
7878&
7
8
#LJ;2HC- OB>=J;;>H- 5"1- "4- M;- 5`.1- `.4,- Z9><=>GC- 9<=C- FH:;K93H>H- 89:J;9>H- FHB:F;2=>-FB;
,f415^P415^f415 // 778&78&878& 7788 =′′=′′=′′
QH-9;<;"4"1"5^P4"1"5^"4"1"5 // ==′′=′′
7788 &&&
M;,f4.1.5^P4.1.5^f4.1.5 // −=−−′′=−−′′−=−−′′
7788 &&&
)KHC1"g4?"1"5@4"1"54"1"5 /
// <−=′′−′′′′ 7878 &&&
FH<;-5"1-"4-2=-HO3H-8=2<3-FH-HV3:HC,-)8B;
------------------------ 1"/!gPf4?.1.5@4.1.54.1.5 /// >=−=−−′′−−−′′−−′′ 7878 &&&
FH<;- 5`.1- `.4- HO3H- 8=2<3- FH- HV3:HC,- QHB9:H<H- 1"f4.1.5/ >−=−−′′8& - :HR=>3G- <G
5`.1-`.4-HO3H-8=2<3-FH-C9V;C,
#$%&'()*+&,-Q9<G- "4?15@415415 /"""""" // =′′−′′′′ 78&78&78& 7878 - 2=- 8=3HC- 9I;:_
C9-<G-8=2<3=>- 415 "" 78 -HO3H-8=2<3-FH-HV3:HC-9>-I=2<J;H;-&,-72-=2H>H-<9R=:;- 415 "" 78HO3H-8=2<3-FH-HV3:HC1-N2-9>3H-<9R=:;-2=-HO3H,-72-9<H9O3G-O;3=9J;H-HO3H-;2F;<93-9-OH-I9<H=2-O3=F;=-F;:H<3-9>-OHC2=>=;-F;IH:H2JH;- 4,15415 "" 78&78& −
&\&WU]e,-A;H-I=2<J;9-&-FHI;2;3G-N2-B:;<H-8=2<3-- /415 R∈78 -8:;2-:H>9J;9
,415 a/ 7878& +=)KHC
Pa415^/415 778&878& 78 =′=′
M;- "1" =′=′ 78 && - N2- 8=2<3=>- 5"1- "4,- QH<;- 5"1- "4- HO3H- 8=2<3- O39J;B29:- 9>- I=2<J;H;
<B2O;FH:93H,-)8B;1./415^"4151/415 /
// 778&78&78& 7788 =′′=′′=′′
1"4?"1"5@4"1"54"1"5 /// =′′−′′′′ 7878 &&&
FH<;-2=-8=3HC-9I;:C9-2;C;<-FHO8:H-8=2<3=>-5"1-"4,
!f
%G-BLOH:KGC-N2OG-<G- "4"1"5 =& -M;- 1"415 >78& -8H23:=- 41"1"5415 ≠78 -9F;<G"4"1"5415 >− &78& -M;-FH<;-5"1-"4-HO3H-=2-8=2<3-FH-C;2;C-9>-I=2<J;H;,
"#$%&'&! (/,0(&! 4&+$%*! )*+,-..(&! %&0(&! 1&- 4P5 >(( - 20%.03.(&! %&0(&,- A;H
RR →⊂ (+& 0 -B-I=2<J;H-:H9>G-FH- 4P5 >(( -K9:;9L;>H-:H9>H-FHI;2;3G-8H- (+ R⊂ -M;
41115 ""/
"." (8888 != -=2-8=2<3-9>-C=>J;C;;-+,
Q&A+T+*+&,-A'(#0'2% +8888 ( ∈= 41115 ""/
"." ! - -90-% '(% 1'(#0% .-% :"8*:% 23#"2
C,-91-#0*/%.-%:*(*:%23#"2D%"2%&'(#)*-*%&<%."#$%-8*90$%3%/-#*($0"0-%6%"%"#-90'*%1'(#0<"90&-2%5(#B0<%1-(0,'%3,*#-% +68888 ( ¡∈= 41115 /. ! -9$%"/-:
4111541115 ""/
"./. (( 888&888& !! <
,-91-#0*/4,111541115 ""
/"./. (( 888&888& !! >
Q9<G- 41115 ""/
"." (8888 != - HO3H- =2- 8=2<3- FH-C9V;C- >B<9>- 5:HO8H<3;K-C;2;C
>B<9>4- 9>- I=2<J;H;1- 93=2<;- 41115 ""/
". (888& ! - OH- 2=CHM3H- :"8*:% 23#"2- 5:HO8H<3;K
:*(*:% 23#"24- 9>- I=2<J;H;- &,- Q9<G- 41115 ""/
"." (8888 != - HO3H- =2- 8=2<3- O39J;B29:
F;2- ;23H:;B:=>- C=>J;C;;- +1- 93=2<;- 1"4111545 ""/
"." =′=′ (88 888&8&
**! ,11/1. (* !=
A'(#0'2- +8 ∈" - -90-% '(% 1'(#0% 90")*3(",% "2% &'(#)*-*- &1- ."#$% &'(#)*"% &% -90-
.*&-,-()*">*2$%5(- "8 -@*%."#$- ,"45F " =8&
U=2<3H>H- O39J;B29:H- 9>H- I=2<J;H;- 10 RR →⊂ (+& - =2FH- +- HO3H- B- C=>J;CHFHO<b;OG1-OH-FH3H:C;2G-:HRB>KS2F-O;O3HC=>-FH-H<=9J;;
=′
=′
=′
,"41115
,
"41115
"41115
/.
/.
/.
/
.
(8
(8
(8
888&
888&
888&
(!
!!!!!!!!!
!
!
%H- FHCB2O3:H9RG1- N2- 9<H>9M;- CBF- <9- 8H23:=- I=2<J;;>H- FH- FB=G- K9:;9L;>H1=:CG3B9:H9- 0-3,-:$- <9:H- 2H- 8H:C;3H- OG- ;FH23;I;<GC- 8=2<3H- FH- HV3:HC- 8:;23:H8=2<3H>H-O39J;B29:H-9>H-I=2<J;H;-FH-C9;-C=>3H-K9:;9L;>H,
D&#'&Wc,- A;H- 41115 ""/
"." (8888 != - =2- 8=2<3- O39J;B29:- 9>- I=2<J;H;- ⊂+& 0
,RR →⊂ ( -%G-8:HO=8=2HC-<G-I=2<J;9-&-9:H-FH:;K93H-FH-B:F;2=>-FB;-<B23;2=H-N23:_BKH<;2G393H-B9:H<9:H-9-8=2<3=>=;- ,41115 ""
/"." +8888 ( ∈= !
),-Q9<G- ∑=
′′(
I*I*(88 ''888&
I*.1
""/
". 141115 ! -=2FH- 1
"
.
88' ***
−= 145 /"
.
/*
(
** 88. ∑
=−= 2
=-OH-92=>H9RG1-93=2<;- "8 --HO3H-=2-8=2<3-FH-HV3:HC-9>-I=2<J;H;-&-M;-92=CH0
41115 ""/
"." (8888 != -HO3H-=2-8=2<3-FH-C;2;C->B<9>-F9<G- ^"45
.1"∑
=>′′
(
I*I*88 ''8&
I*
!!
41115 ""/
"." (8888 != -HO3H-=2-8=2<3-FH-C9V;C->B<9>-F9<G- ,"45
.1"∑
=<′′
(
I*I*88 ''8&
I*
$,- Q9<G- ∑=
′′(
I*I*(8 ''888&
I8*.1
""/
". 41115
1! - ;9- N2- B:;<H- KH<;2G393H- 9- 8=2<3=>=;
41115 ""/
"." (8888 != -93S3-K9>B:;-8BR;3;KH-<S3-M;-K9>B:;-2H[93;KH1-8=2<3=>- "8 -2=-HO3H
8=2<3-FH-HV3:HC,
P,h,!5&$/10!,&(/%!'0.!'.,.!46$%0$&
%G- 8:HO=8=2HC- <G- O=23- <=2BO<=3H- K9>B:;>H- I=2<J;H;- 458&7 = - N2- 92=C;3H
8=2<3H- 9>H- FBCH2;=>=;- FH- FHI;2;J;H1- 9F;<G- O=23- <=2BO<=3H- 8H:H<b;>H- FH- K9>B:;
111/1.4154115141154115 //.. (*8&7787878 **(( !! == -IG:G-9-I;-<=2BO<=3G-I=2<J;9
&,- e:CG:;C- OG- 98:BV;CGC- 458& - 59F;<G- OG- HV8:;CGC- >H[G3=:9- F;23:H- K9:;9L;>H>H
8-M;-74-8:;23:_B-I=2<J;H-8B>;2BC;9>G- 458A: -FH-[:9F=>- ,(: <
A;H- ,11/1.145 (*78A **: !== -&:B:;>H-<9:H-OH-I9<-F9<G-OH-N2>B<=;HO<-K9>B:;>H
45 *8& -8:;2- 45 *: 8A -O=23- ,11/1.41545 (*8A8&77- *:**** !=−=−=Q9<G-OH-;9-F:H83-#,*0-,*'%.-%:*(*:*=",-%"%-,3,**%.-%"1,38*:",--O=C9-3=3=:B:
9L93H:;>B:- ∑=
(
*** --
.
11 - BLOH:KGC- <G- 9<H9O39- <B2F=<H- >9- B- 9>H[H:H- [:HM;3G- 9- I=2<J;H;
1:A -FHB9:H<H- (--- 111 /. ! -8B3-9KH9-OHC2H-F;IH:;3H-M;-FH<;-O=C9->B:-9>[HL:;<G-2=
HV8:;CG- <B:H<3- H:B9:H9- 3B39>G,- W9;- ;2F;<93- 9:- I;- OG- OH- <B2O;FH:H- ,.
∑=
(
**- - 72OG1
FHB9:H<H- I=2<J;9- CBF=>- 2=- HO3H- FH:;K9L;>G- N2- B:;[;2H- OH- FH3H:C;2G- :*(*:':'29':-*%1$0,"0-23,%-,3,**
∑ ∑= =
−=(
*
(
***:* 8&8A-
. .
// ,4?545@
QH-9;<;1-:-03."%i#-23,%:"*%:*#*%1$0,"0-j,A;H- 145 ."
::: 8"8""8A +++= ! -=2FH-<BHI;<;H2J;;- :""" 111 ." ! -OH-FH3H:C;2G
F;2-<B2F;J;9-<9-∑=
(
**-
.
/ -OG-I;H-C;2;CG,
%H-<B2O;FH:G-I=2<J;9
,?545@41115.
/."." ∑
=+++−=
(
*
:*:**: 8"8""8&"""J !!
!h
ZB2F;J;;>H-FH-C;2;C-N2-:98B:3-<=-K9:;9L;>H>H- :""" 111 ." ! -O=230
1"11"1"."
=∂∂=
∂∂=
∂∂
:"
J
"
J
"
J!
O9=
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.."
.."
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=−+++
=−+++
=−+++
∑
∑
∑
=
=
=
*:*:*
(
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(
**
*:*:*
(
*
8&8"8""8
8&8"8""8
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!
"""
!
!
)<HO3-O;O3HC-FH-H<=9J;;-OH-O<:;H-O=L-IB:C9-(3,:"2$-5k9=OO4
=+++
=+++
=+++
∑ ∑ ∑ ∑
∑ ∑ ∑ ∑
∑ ∑ ∑
= = = =
+
= = = =
+
= = =
(
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8&88"8"8"
8&88"8"8"
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45
45
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!
""""
!
!
Q9<G- K9>B:;>H- (888 111 /. ! - O=23- F;O3;2<3H- 93=2<;- O;O3HC=>- FH- H<=9J;;- 9:HOB>=J;H-=2;<G1-FH<;-I=2<J;9- :A -HO3H-=2;<-FH3H:C;293G,
#$%&'()*+&,-72->B<=>-I=2<J;H;-8B>;2BC;9>H- :A -OH-8B93H->=9-B-I=2<J;H-K-B9:H<9:H,WH3BF9- FH- 98:BV;C9:H- 9- =2H;- I=2<J;;- 2H<=2BO<=3H- &- 8:;23:_B- I=2<J;H- K- OH
2=CHM3H-"I'90",-1-K-I;;2F-&'(#)*"%.-%"I'90",-,-UH23:=-9-9>H[H-<S3-C9;-<B:H<3-I=2<J;9FH-9l=O39:H-HO3H-2H<HO9:-OG-OH-:H8:HR;23H-N2-8>92-8=2<3H>H- (*78 ** 11/1.4115 != -M;OG-OH-3:9OHRH-8:;23:H-M;-8:;2-=2H>H-F;2-9<HO3H-8=2<3H-B-<=:LG-9O3IH>-N2<S3-H:B:;>H-8BR;_3;KH- M;- 2H[93;KH- OG- OH- <BC8H2OHRH,-U:B<HFH=>- 9<HO391- 2=C;3-"I'90",-% K,"&*#$- 8H:_C;3H-9I>9:H9-0*1'2'*-FH-<=:LG-;2F;<93-8H23:=-9l=O39:H,-72-O<B8=>-"I'90$,**%"("2*0*#--9F93H>B:-HO3H-2H<HO9:-OG-OH-9>H9[G-C9;-C=>3H-<=:LH-M;-98B;-F;23:H-9<HO3H9-OG-OH-:HJ;2G18H23:=-=3;>;R9:H9-CH3BFH;-<H>B:-C9;-C;<;-8G3:93H-5O9=-9-9>3H;-CH3BFH-FH-98:BV;C9:H9-I=2<J;H;-&%41-9<HH9-<G:H;9-N;-<B:HO8=2FH-O=C9-8G3:93H>B:-9L93H:;>B:-C;2;CG,
)U]+Z)*+&,- 'HR=>393=>- =2=;- HV8H:;CH23- HO3H- N2O<:;O- N23:_B- 39LH>G- FH- K9>B:;,11/1.4115 (*78 ** != - %G- OH- HV8:;CH- >H[G3=:9- F;23:H- K9:;9L;>H>H- 8- M;- 7- 8:;23:_B
I=2<J;H- FH- [:9F=>- N23S;- FH- IB:C9- 1>"87 += - K9>B:;>H- "- M;- >- I;;2F- FH3H:C;293H
!g
9O3IH>--N2<S3--9L93H:H9-I9JG--FH-;2IB:C9J;9--I=:2;R93G-FH-CGO=:G3B:;1-9L93H:H-F93G-FH
145 /
.
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∑=
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*** >"87>"J
.
/ ,45415
ZB2F;J;;>H-2H<HO9:H-FH-C;2;C-F9=
1"415
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∑ ∑= =
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= /;&/ DE)&6DE %"%4 $$
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$$
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V°,/<&1/ '4(546$/Q/ 54>*)'./ 0./3$0./ *N∈$$" 9: /-;1-+)$+ &'(+,-+ ")$05''+ 02$1'$)-C
)$'"2(.+02$3-(4-$1+/-+)$+'$1-(37*+02./701/e7C/Dd/061(-+2+")$05'-+"9+71)$0'
∫ ∫=∞→
D
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7*108&$/)&6&'./"/42'4/34/$246414$/0(1'&1*./%4/e7C/Dd,/]4>*)'./0./ *N∈$"$ C /;&/"/2*1'
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$)4/7*108&&)(5/ *N∈$"$ C C/$2'74)/0$/F1'5I*1/%*10'/ dCeB D7% ∈ /2./$@46C/34/4a46%)*C
,BDE B =%>$ /#*6/;&5*)/34/1*6454/ D9E: B%>$ /42'4/0(1@45J41'C/&$5/;&5*)/ 9: $> ′ /42'4/*1&I7(56/ 0(1@45J41'/ 0.'54/ "/ E$@46/ DC$$ "> =′ / 54>*)'./ 0./ ;&5*)/ 9: $> / 0(1@45J4/ *1&7(560.'54/(/7*108&4/345&@$S&)./>/;&/ ,"> =′ /H;$3$5C/7*108&$/>/42'4/(/%5&6&'&@./$/)*&/",
!QV
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∫∫ =−→−==D
7$$
D
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A>CAC>AC> Q! -9#9#9# " ,;+-+3+(>5,$+,"-
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S
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-7&-)4-'&$'&,81*$0-+&724'-)*/$<#-#$(4C
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/SR>0S>
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593;-54387)B<4:-L1-)9)('3*#'(")$')B-31</)25-2;17)B<46284/3-NIK) 5-0) 5-) 3/) ?4;B/3-) ?-) I234) C909;-L1-) 2'3&%("() 9) 23/:417) ;/:7) ?-) I234
B<96-3417)?-)02)201543-62=)5931<2)/3-4)12O-)C4<-L14@,-./017) 57) >3191?-2/32) -O4;17) /3) -%52+ "0+ %&'(")$$0%(+ ,$#"#-$"('=) 52<-
('&('1$#2.+',%(2*0+,$#"#-$"(+&'+-"('+70+5*&%(2.+*#+&"(2'#'(+&'#2(*+"+&*2'"+!'#',$-$"4'+ %+ "#*3$2.+ 5*3.+4'+!"#$+ &'+ -"('+ %+ "('+ *#+ "02+ &"(2'#'(@) (-0):24) C<-56-31=) /32;-:-3-2)59;1)-;1-)2;4:4021)5/)?9IP3?2)2C-<-317)B02;2:-31/0/4)C4323542<=)?-L4)Q)>3<-204121-) Q) 4%!:#4") -;1-) 3/:24) *#"+ 4$#2('+ -%3&%#'#2'0') 59;1/0/4) /3-4) 9B-<2844C4323542<-@
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!!"
#$%&'()*)++!"
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01+ %234*%*536+ 71+ %$2'3532*)+ !/+ 8&4%*53+ 83+ (+ 4*61+ 83+ 9$5&+ "-+ %3+ 7$23+ (%)$43$:1+ %$2'3532*)*&+!,+ %35'2*+ (+ %32&($81+ 83+ '&6%+ #;+ <5+ 7(58&=&&+ %234'$9&)&'3.+ >$?&53)3+%32&($83&+83+'&6%+#+%$2'3532*)+!/+%2&63@'3+4*6$+?&5$)1+"$A"-;+#B+7$23;+<5+6(85$'*2$);+4$'&4?$73+7(58&=&$+"+A"-;+#B+C+"-.
DEFGHGIGE.+%&'()*+$ ;J +++ →× RRR, $+$(-./*$0+12+./$ 3'$ ++ ×∈ RRB;A - #"34'3+,A"-;+#B$ 4/$ '&5/6#/$ 7289'7-$ (2./4:&';-#2+./$ :1+4-.**$ 4&5/*$ "-$ :/$ 7&.+#+$ #<7+(-=
$B+,$+75*#/$7/.*0+#/$:+.)*+1/$7/$2.7*'&1$3'#9*$3'$.+:2.#$(&$"-$6*$#<$*+.
-B;A
;-B;A -
-
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∂>∂
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#",
"
#",K
A,$/4#/$4#.*(#$(./4(-#2+./$3'$.+:2.#$(&$>*/(+./$7*'#./$+.?&5/'#/1/$"-$6*$#BK9B+ .-B;-AB-;A - == #,",
DEFGHGIGE.+%&'()*+$ ;J +++ →× RRR" $+$ (-./*$ 0+12+./$ 3'$ ++ ×∈ RRB;A - #"34'3+ "$A"-;+ #B;+ 4/$ '&5/6#/$ 4&5-$ A0+12+./B$ >*'+1-$ 4+&$ 4&5-$ A0+12+./B$ ./0/'*#-:+.#/'/.&1&*$!/<$(+./$+$:1+4+#$4&5+$"-$:/'#.&$:/.*2+7+$7/$#*5:$#<$7+(-=
/.+"$+./$7/.*0+#/$:+.)*+1/$7/$2.7*'&1$3'#9*$(2'#*'&/$3'$.+:2.#$(&$"-$6*$#<$*+.
-B;A
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-
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,.+ .-B;-A;B-;A -- == #""""L5+9$:$+$734'(2+8(*1+83?&5&=&&+/?+1*#+#/+
--- B;AB;A "#""#", −= A/B
7/>*'/6#/$7289'7+$(2./4:&';-#2+./$:1+4-.**$4&5/*$"-$:/$:/.*2+7+$7/$#*5:$#@
!M-
DEFGHGIGE.+,+(-$ #$N$/$ +'$ 6*$ "-$A$/--$ &'*#-)*$ 52'/#+./$ A*.6.B;+ 7289'7+(2./4:&';-#2+./$4/$'&5/6#/$:.2(/'#.+03+5('3$:1+7*+:K+ B./;/--A,: =
DEFGHGIGE.+,+(-$#+N+/++'$6*+0-+N+/+&@5@<$7289'7+$(2./4:&';-#2+./$4/$'&5/6#/
7289'7-$&'*#+.-$+'&+1-.+03+5('3$:1+7*+*K+ ./--
B/;/A:
,* ==
!"#"$%&'($"()*(+,$&&)-&'(+"+"+A"-;+#B01+%234*%*536+71+43+%)$43$:1+4*6$+"-+%3+8*2$'$+#;+83O35&58+<5+?&5$)+"+A"-;+#B.
F&3+ B+C+-K+ 8$71+ %)$4$635'*)+ 4*63&+ "-+ 43+ ?$73+ %3+ %32&($8$+ 83+ '&6%+ #+P+B;+ $'*57&O$)($23$+ ?&5$)1+ 34'3+"+A"-;+ #+P+BB.+Q$2&$=&$+ 4*63&+ ?&5$)3+"+ <5+ &5'32O$)*)+ 83+ '&6%+ 83)*5R&63+B+34'3
B;;AB;A -- #""B#"" −+&$2
.-B;AB;AB;AB;A
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-³#">
#
#""
B
#""B#""B
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∂=−+→
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B;A8
B;A8-
- #">#
#"" = A,B
$+7123&+4()*=&3+%$2'&7*)$21;+<5+7(58&=&&)3+ ;-B;-A;B-;A -- == #"""" +34'3
∫ αα+=#
">"#""-
--- .8B;AB;A A!B
L5+%$2'&7*)$2;+8$71+"-+N+/+*.6.+@&+#+N+/+$5;
∫ αα+=+=+=/
-
.8B;/A//B/;/A/B/;/A >*,"
DEFGHGIGE.+C2'4#+'#+$&+N+/+P+*$4/$'&5/6#/$>+(#2.$7/$>.&(#*>*(+./$+'&+1-D$&T+/+4/'&5/6#/$>+(#2.$7/$+(#&+1*;+./$+'&+1-.
UV0EWQXIGE.+ER$)&'$'3$+ A/B+ $2$'1+ 71+ "+A"-;+#B+ @&+ ,+A"-;+#B+ 8&?321+ %2&5'2Y(
7(54'$5'1.+ D37&;+ <5+ &%('3:$+ ;8
B;A8
8
B;A8.;7' --
- #
#",
#
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4()*=&$+37*$=&3&+8&?3235=&$)3
;-B;AB;;A8
B;A8--
- >= #">#">#
#",AMB
7$23+4$'&4?$73+7(58&=&&)3+,+A"-;+-B+N+,+A-;+#B+N+-;+$8&71
∫ αα=#
">#",-
-- .8B;AB;A AZB
!M/
L5+%$2'&7*)$2;
∫ αα==/
-
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,-.,.+#$%&'()1345-02
DEFGHGIGE.+!1+4+./+$4&5/*$"-+4/$/>/(#&/+;-$3'$./?*5$7/$7289'7-$4*5:1-$7+(-:/$#2+#-$:/.*2+7+$7/$:1+4+./$#<$4&5+$*'*)*+1-$"-+'&$4/$527*>*(-.
[2&5+*26$23;+<5+23R&6+83+8(9S581+4&6%)1
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E7*$=&&)3+8&?3235=&$)3+A,B+@&+AMB+43+472&*
B.AB;AB;A
--- #?"7#
#"7,
7#
#"7" ==
L5+7(58&=&&)3+&5&=&$)3+8$'3+23:*)'1
.BAB;A-
-- αα= ∫ 7?"#",#
A]B
.BA/B;A-
--
αα+= ∫
#
7?"#"" A^B
L5+%$2'&7*)$2;+8$71+?A#B+N+*+(9=&536
.B;A -- *#"#", = A"B
B./AB;A -- *#"#"" += A/-B
%2.5&1/1/+ A"B+ 6*+ A/-B+ 4&'#$ &#*1*;+#/$ 3'$ :.+(#*(-$ :/'#.&$ 7/#/.5*'+./+$ 0+12.**>*'+1/$3'$:1+4+5/'#/1/$3'$./?*5$7/$7289'7-$4*5:1-.
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.,$%/+")/#&+&0(#")1')),2"$(3&/'&)-&'('4&($"/.+D$71+ 4*6$+ "-+ 34'3+ %)$4$'1+ %35'2*+ %32&($8$+ 83+ '&6%+ #+ A/E:.*5+#$ 3'$ +'*B
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;/--
B;A ---
:#"*#"#", == A"′B
!M,
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./--
/B/AB;A ---
+=+= :#
"*#"#"" A/-′B
,.+F&3+5+*5+5*612+83+%12=&+3R$)3+$)3+$5*)*&+$.<.J+8$71+5+N+/;+$'*57&+/+$5+N+/$5K+ 8$71+5+N+,;+ $'*57&+ /+ $5+N+,+ 43634'23K+ 8$71+5+N+ M;+ $'*57&+ /+ $5+N+M+ '2&634'23K8$71+5+N+/,;+$'*57&+/+$5+N+/,+)*5&+A7*+7(5O35=&$+71+?&37$23+)*51+$23+7S'3+!-+83+:&)3BK8$71+5+N+!\-;+$'*57&+/+$5+N+!\-+:&)3+A7(5O35=&(5$)B.+F&3+#5+N+5#;+8*2$'$+(%32$=&*5&&
A%)$4$635'*)*&B+<5+5+?2$7=&*5&+83+$5K+837&+ .5
## 5=
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;/--
B;A --- 5
#:"
5
#*"#", 55 == A//B
./--
//B;A ---
+=
+=
5
#:"
5
#*"#"" 55 A/,B
!.+D$71+4*6$+"-+43+%)$43$:1+%3+8*2$'$+#+<5+23R&6+83+8(9S581+4&6%)1+@&+8$71
∑=
θ='
FF#
/
+ $.<.+ %3+ ?&37$23+ %32&($81+ Fθ + %)$4$23$+ 43+ ?$73+ 7*+ %2(735'*)+ $5*$)
;/-- FF *: = +$'*57&J
∑ ∑ ∑= = =
θ=θ=θ='
F
'
F
'
FFFFFF :
"*"",#",
/ / /
---- ;
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./--/
//B;A/
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--
θ+=
θ+= ∑∑
==
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FFF
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$B$"A/--+---+*.6.K+,;Z+$5&B+8$71+:+N+M-K
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7B$:;+8$71+"-+N+/--+---+*.6.;+#+N+M+$5&$"A"-;+#B+N+,\-+---+*.6."21&)*/@$$B+F()(4&58+A/,B+$O36
*.6.---,--/,/--
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-- .. +=+ F!%′E
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++++++,-$M0)/>4$-,.)&'$-&,<('&)+&,4$<'>,5,$(,$'&>,+E/$+$%*$7,),&>*.+),$('$$ A]494 =.5.$$$$$ L'A>AP4#9N#!9#T_(8L'A>A]49NN]";]"PF8L'A>AN49T"9"!_#"+8 !#9 === 555
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++++++,-$$+8$ A'A>;P9TPT]94!8#P49:844: ;P
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,#*)(2#>/#*/%70*+#*,-.)/*,%(<*"#*,-($/<+,-.)/-VWJ0XYZ420-+/&9+5.A+>;>*7;?+7C+67(9'*)+;-+;6';+>;+$7;;$I&+9$'*=C+7*+>(@M9>$
5';(=;'&7<+ ;-+ '=;@*&;+ 6C+ =;%=;E&9';+ >(@M9>$+ %;+ %;=&($>$+ >;+ '&?%+ %M9C+ )$+ 67$>;9:C7(=;6%*9EC'($=;+7$%&'$)*)*&+67(9'$'<+;F$)*$'C+7*+*9+$9*?&'+%=(7;9'A-
/$7C+67=&;?+5.A+6*@+H(=?$
-% ;;; += 5,A
%*9;?+D9+;F&>;9:C+H$%'*)+7C+,-.)/(4*#,/#*#A%4*-(*+.36)+%*%9#$#)/0*8%4.$''*%-/(%4#;%*%*-%"'/%4(4('*+',".)'3'4*4%*,-%+#)&0*;<+ *"#*+($%/%* ∗θ−θ=/ +>'*-(*"$.-#)/(4*+#,-.)/+<+N+.KK=-+[=(7;9'*)+>;+67(9'+<+%($';+H&+;G$)+6$*+9*+%=(7;9'*)+>;+;?&6&*9;+$%()&:;&+"+N.KK'-
!"#
$%&'()*+%,*%)*,(+ -'+%&'.&/01(,(2+%)+ 345+ 62+')027(2+ !"#$%& '()*#$(+8+9/1%2+ ,*(/2:)*(,(+%,/2+,;/&<102,:=+6%&'()*+/,>1&',*+62+')027(2+!"#$%&"#,-'"*(+8
?@ABCBDB@8+.#$!%($%(&/&!-&$0,-1%-&2#34$25&($0(+5&0$*%('5&2-&!"#$%+6,)+'(%5($0(+5&2-&!"#$%8
E/1'+ ,',*&F12+ %)+ G&HI'G,+ 610;*=+ 71+ G&HI'G,+ %&0;)6=+ G2&62H10+ !"#$%0+!*,6+0+71+!"#$%0+&"#,60!8
44848+!"#$%&'()*+,
?@ABCBDB@8+7("5&2#34$2(&(8-'-$%5&"(6*%(+0+0*&("%0(++3!"#$%(%5+9(&-!%-&"(+"0:+(%5&"(&2#34$25&!*,6+5J+(%0$"*-'()*0$-(&-!%-&K2-&!"#$%&!*,6+0L8
A12+ 9"J+!+ M+ N,*&,/2,+ 6%&'()*)1+ 610;*)8+ ?,%=+ ;/&%2'()*+ ,'),*+ G2+ 6%&'(+ 26(2;+O+PQQ/+ 71+ G,%=+ &;2/,>1)'2,+ 62+ G26.=7&,/=+ %)+ ,%2*,71+ ;/&%2'(+ ,'),*+ G2+ 6%&'(+ ;2G)/,(,+%J+,()'%1J+;&(/1N1(+G2.1'1>121+G2+0,1+6)6J
++++++++
++++++++++++++++
+
?,%=+&;2/,>1)
%J+1,/+ <
$
<
% θ= ∑=P
+71+-
JPQQ << /; = +,N20
U1%1+$
<
% θ= ∑=P
?1'+ 3P5+ /2:)*
?,%=+>1'20+62,0,+%
E/1'+)/0,/2+ 9
62+2<;/10=+-'+,'1J+6
9"@&!+O
++ JPQQ
%;9%/9 ((
⋅=⋅⋅ ++++++++++G,%=+%+62+2<;/10=+-'+,'1R 3!5
++ JPQQ ,
%;9
,
%/9 ,
(,
( ⋅⋅=⋅⋅ +G,%=+%+62+2<;/10=+-'+./,%>1)'1
+++++++++++++++++++++++++++++++++++++++++2F,*2+3,5+G2+,'+3%,58 3S5
'2,+')+62+2.2%()2,:=+%)+,%2*,71+;/&%2'(+,'),*+G2+6%&'(+;2+G)/,(,
'+;2/1&,G,+G2+(10;+ <θ +62+&;2/2,:=+%)+;/&%2'()*+,'),*+G2+6%&'(
<<
$
<
(<<
$
<(!" ;
9/99 θ=θ= ∑∑
== PPJ PQQ
8 3T5
< +26(2+-=6'*,(%&>$&($*?
(=+ J"( 999 −= + 6,)J+ -'+ %,:)*+ 6%&'()*)1+ 610;*)J+ 8J !"( 999 −=
=+ /%99 (!" =J +3;/10,+.&/0)*=+3!55J+&H>1'20
8P /%
99( +
= 3"5
!"J +%,*%)*,(+-'+.)'%>12+G2+N,*&,/2,+.1',*=+,+&;2/,>1)'11+9J+G,%=+%
2+&H>1'2+-'*&%)1'G+9(+G,(=+G2+3"5+-'+3!58+V2:)*(=
8PQQPJ ;%
9;%
/%
9/%9 !" +
=+
= 3W5
!"X
!"9 J + %,*%)*,(+;&(/1N1(+ .&/0)*21+ 3W5+ 62+')027(2+!"#$%& !*,6+0&'()*#$(+8+ Y*+ N&0
'&(,+ 853J'!"9 + Y'+ %&'62%1'>=J+ 8#',0+(& 3"5+ 2-8*$-1%-& "(6*%(+0+& !"#$%(%& 9(& 6-$%'0& !"#$%
!*,6+0&'()*#$(+8Y'+;,/(1%)*,/J+G,%=+9+O+P+)808+71+%+O+P+,'+.&/0)*,+3W5+G2N1'2
8P
53J /
/9 '
!" +=
?@ABCBDB@8+.#$!%($%(
/
/0
+=P
3W′5
!-&$0,-1%-&!"#$%&0$*%('+6,)+0$*%(%-&2-&!"#$%89H62/N=0+%=+ 6%&'()*+)'1(,/+26(2+ 2F,*+%)+N,*&,/2,+,%(),*=+ ,+G&HI':11+ ,'),*2
)'1(,/2+/Z?@ABCBDB@8+ .#$!%($%(+ 0/A −=+= − P5P3 P + !-& $0,-1%-& 8("%#'& 2-& ("%0(+*B('-
!(0&8("%#'&2-&!"#$%8Y'+;/,%(1%=+ P[P −+ /% +71+3W5+62+/2G)%2+*,
JJ 9/%9 !" = 3#5
%+.11'G+-=6'*,(%&>$&($*8+?,%=+ J,
%% ,= +.&*&61'G+3S5+&H>1'20
,
9;%9 ,
!" PQQJ = 8 3#′5
$%&'()*+ 610;*)+ %,*%)*,(+ ;&(/1N1(+ 3#5+ 6,)+ 3#′5+ 3G2%1+ -'+ 1;&(2:,+ P[P −+ /% 5+ 62
')027(2+!"#$%&!*,6+0&"#,-'"*(+8+Y*+'&(=0+ 853J"!"9
?1'+%&'G1>1,+ 53J"!"( 999 −= +71+3#5+/2:)*(=
8QP5JP3 >−−= /%/%99( 3X5
C"-(!%5&8#',0+5&2-8*$-1%-&"(6*%(+0+&!"#$%(%&9(J+ >$&"(B0+&!"#$%0+0*&"#,-'"*(+8?2&,/2%2+/%&\+P+G2G)%20+%=+6%&'()*+610;*)+%&02/%1,*+')+62+,;*1%=+;2'(/)+;2/1&,G20,/1+G2+(10;8
!"#$%&'()*%%+,(%-%#&+ 53J'!"9 +.%+ 53
J"!"9
P5+ J53J53J
"!"
'!" 99 < + %22,+ %2+ ,/,(=+ %=+ 53
J"!"9 + %&'N1'2+ H='%11+ %,/2+ %)0;=/=+ G&%)]
02'(2*2+G2+;*,(=J+;2+%I'G+ 53J'!"9 +%&'N1'2+G2>1'=(&/)*)1+G&%)02'(2*&/+3;&*1>2158
45+?1'+3!5+71+ 3S5+ /2:)*(=+%=+ 53J'!"9 + /2;/2:1'(=+G&HI'G,+610;*=+%&/26;)':=(&,/2
N,*&/11+6%&'(,(2+9(+;2+G)/,(,+G2+(10;+%+G2+*,+6%&'(,/2+71+;I'=+*,+6%,G2'>,+;&*1>21J+%,*]%)*,(=+%)+;/&%2'()*+,'),*+G2+6%&'(+;+O+PQQ/8+?1'+3#5+71+3#′5+/2:)*(=+%= 53
J"!"9 +/2;/2:1'(=
G&HI'G,+ 610;*=+ %&/26;)':=(&,/2+ N,*&/11+ .1',*2+ 3%,;1(,*)*)1+ '&01',*5+ 9J+ (&(+ ;2;2/1&,G,+G2+(10;+%+71+%)+,%2*,71+;/&%2'(+,'),*+G2+6%&'(+;8
!WQ
!5+?1'+3W5+71+3#5+&H>1'20
JP
53J53
J /%
99
"!"'
!" += 3PQ5
G2+)'G2
J53J
53J
53J /%999 '
!"'!"
"!" ⋅=− 3PQ′5
%22,+ %2+ ,/,(=+ %=+ G,%=+ 6%&'()*+ 610;*)+ /,>1&',*+ 6],/+ ;*,6,+ ;2+ G)/,(,+ %+ %)+ G&HI'G=610;*=+ 71+ %)+ ;/&%2'()*+ ,'),*+ ;+ O+ PQQ/J+ ,()'%1+ G&HI'G,+ %&/26;)':=(&,/2+ ;*,6,]02'()*)1+26(2+2F,*=+%)+G1.2/2'>,+G1'(/2+6%&'()*+610;*)+%&02/%1,*+71+%2*+/,>1&',*8
S5+?1'+3W5J+(/2%I'G+*,+*101(=+;2'(/)+ ∞→% +&H>1'20
8*10 53J 99 '!"
%=
∞→3PP5
E/1'+ )/0,/2J+ G,%=+ ;2/1&,G,+ G2+ (10;+ %+ ;I'=+ *,+ 6%,G2'>=+ 26(2+ .&,/(2+ 0,/2J
99 '!" →53J +71+G2%1+ Q→(9 +3N2:1+71+3"558
/'0)*%%+1#2('+93+ ∗9 +.%+9(A12+9+%,;1(,*)*+'&01',*+%,/2+62+&H>1'2+;/1'+;*,6,/2,+6)021+9Q+;2+G)/,(,+θ+%)
;/&%2'()*+6+O+PQQ*J+-'+/2F10+G2+G&HI'G=+610;*=8+U)+*&%+2F,*1(=>1*2Z
JQ5JP35JP3 QQ θ<θ<θ+=θ+= ∗∗∗ *99*99
G1'+%,/2+/2:)*(=
8P
5P3∗
∗
θ+θ+=
*
*99
Y'*&%)1'G+,%2,6(=+2<;/2612+;2'(/)+9+-'+3"5+71+/26;2%(1N+3X5+&H>1'20
J5P53P3
5P3
/%*
*99( +θ+
θ+=∗
∗3P45
6-$%'0&!"#$%0+&!*,6+0&'()*#$(++71
JP
5P53P3∗
∗
θ+−θ+=
*
/%*99( 3P!5
6-$%'0&!"#$%0+&!*,6+0&"#,-'"*(+8
9^$@V_UDB@8+ ?1'+ 3P45+ G2G)%20+ J∗< 99( + G,%=+ 8P ∗θ+
>*
*/ + Y'+ %,:+ %&'(/,/
8∗99( ³ + U',*&FJ+ G1'+ 3P!5+ /2:)*(=+ ∗< 99( + G,%=+θ+
>*
*/P
+ 71+ ∗99( ³ + -'+ %,:
%&'(/,/8+Y'+;/,%(1%=+26(2+;&61H1*=+&/1%,/2+G1'(/2+,%26(2+61(),>118@`@aEbc8+ A12+ θQ+ 0&02'()*+ -'+ %,/2+ ,+ .&6(+ %)0;=/,(=+ &+ ;&*1>=+ *,+ ;/2>)*
8J)80P4QQQQ =9 + 6%,G2'(=+PQ+ *)'1+0,1+ (I/:1)+ %)+ ;/&%2'()*+ ,'),*+ G2+ PQd8+E/26)]
!WP
;)'I'G+%=+;&626&/)*+;&*1>21+62+;/2:1'(=+*,+6%&'(,/2+%)+!+*)'1+-',1'(2+G2+6%,G2'>=J+6=62+G2(2/01'2Z
,5+N,*&,/2,+'&01',*=+,+;&*1>21+*,+6%,G2'>=+395RH5+N,*&,/2,+.1',*=+*,+6%&'(,/2+3%)/6)*+;&*1>215+3 ∗9 5R%5+ N,*&,/2,+ 6%&'(,(=+ ,+ ;&*1>21+ 39(5J+ ,;*1%I'G+ .12+ 6%&'()*+ 610;*)+ /,>1&',*J+ .12
6%&'()*+ 610;*)+ %&02/%1,*+ %)+ )')*+ G1'+ ;/&%2'(2*2+ ,'),*2+ G2+ 6%&'(+ ;P+ O+ #dR+ ;4+ OO+PQdR+;!+O+P4d8
9#+0)*-?+,5+ 8)80P!QQQP4
PQPJQPP4QQQRPQ =
⋅+=
+= 9
,
%*99 ,
H5+ 8)80P4WQQP4
WPJQPP4QQQ5RP3Q =
⋅+=θ+= ∗∗∗ 9*99
%5+E2'(/)+6%&'(+610;*)+/,>1&',*+62+,;*1%=+3"5R+/2:)*(=
===
=d8P4;2'(/))808!TXJP4"4P
dRPQ;2'(/))808X4"JP4"#4
dR#;2'(/))808QX#JP4WST
;
;
;
9(
E2'(/)+6%&'(+610;*)+%&02/%1,*+ 5RP3 /%99( −= +/2:)*(=
===
=d8P4;2'(/))808P4"PQ
dRPQ;2'(/))808P4"WT
dR#;2'(/))808P4WSQ
;
;
;
9(
9H62/N=0+%=+;2'(/)+ JdJ# (99; <= ∗ +%22,+%2+-'62,0'=+%=+G2>1'=(&/)*+;&*1>21;/1027(2J+ ;/1'+ NI':,/2,+ ,%26(21,J+ &+ 6)0=+ 0,1+ 0,/2+ G2%I(+ N,*&,/2,+ 21+ *,+ G,(,6%&'(=/11J+.,;(+*$(2,*!*3*+8
448!8+!"#$%&"#)*,'
?@ABCBDB@8+7("5&2#34$2(&(8-'-$%5&"(6*%(+0+0*&("%0(++3!"#$%(%5+9(&-!%-&"(+"0:+(%5&"(&2#34$25&"#,60!5J+(%0$"*-'()*0$-(&-!%-&K2-&!"#$%&"#,60!L8
C&(=0+6%&'()*+%&0;)6+%)+ 8J ""9
?2&,/2%2+-'+/2F10+G2+G&HI'G=+%&0;)6=
%( /99 5P3 += 3PS5
6,)
J5P3 %( /99 −+= 3PT5
1,/+ J(" 999 −= +-'+H,:,+G2.1'1>121+G2+0,1+6)6+/2:)*(=
e85P3PfeP5Pf3J%%
("" /9/99 −+−=−+= 3P"5
!"#
$%&'(!()*+),(&%(-%.(*/*,()012.-%+(34(%4.(5.( ,6
"
#
"
! θ= ∑=
(.%2(34(1)2.7%8%(8)(+.-1
"θ (*)(71)2)%9'(&/(127&)4+/:(%4/%:(8)(*&74+( ,,6,6;; #"$% "" == (%+/4&.
<=6>66=6> 6
66,
θ+−=
−θ+= −
==∏∏ ""
#
"""
#
"&'' $($(( >6"=
?4( .17+)9%( &'( 87@A48%( /4.+%2'( %4/%:'( 8)( *&74+( $( )*+)( -.&'( :/'-$!$ ! +≈+ 6=6> (5.(B72-/:):)(>6C=(8)D.4
=>,, 6)*'&'' (
$!
($!$!(( =
+== < >6E=
FG%:.+'H.:)( >6C=( 5.( >6"=( 8)B.4)*&( *'+#!,-. '+/0,*. )&12+#&-,( 1)( &%2)( 3:( 47+'-
<=>,)''( (FG%:.+'H.:)(>6E=(8)B.4)*&(*'+#!,-.'+/0,*.'+/3)'2&-,(47+%+( <=>,
'''(
!"#$%&&'()*+"'(&(,&' ∗( '()'-.)/&%&&#"'$0#&-1+&&'2-.)*3#3&'-.4032I.)J(;( K( D%:7%2)%( .4.H.%:'( >8)( &/-1'2%2)=( %( 17:.H).,( &%2)( )*+)( *&%8)4+'( :%
-7-)4+/:(θL0(M(6;;2( K(127&)4+/:(8)()D%:/%2)(%(17:.H).L
∗θ ( K(-7-)4+/:(34(&%2)(*)(*&74+)%9'(17:.H%,(&/(&748.H.%( <θ<θ∗
N/(:7&()G%:.+'H.:)
,=6>,=6> ;;∗θ∗θ +=+= 2((2(( >6O=
8)(/48)
<=6>=6> !2(2(( +=+= ∗θ−θ∗ ∗ >#;=
P.4A48( *)%-%( 8)( >6Q=( 2)9/:+'( &',( 4#. '+#52122-3. &0-2'6)22. *'+#!,-,2. '+/0,*,4#!)3.'&02!&-,-.*'+#!&!.(&..72.',)*,-.0+-2132. ∗( &,.-+'.)3-&122-3J
,6
6!
& $
2((
++= ∗ >#6=
03#!),.*'+#!,-.'+/0,*.)&12+#&-(5.
,6
=6>
$!
2((
!
& ++= ∗ >##=
03#!),.*'+#!,-.'+/0,*.'+/3)'2&-<$.4(>#6=(2)9/:+'(&'( ∗((& £ (8%&'( ,2$ ³ .%2(8.4(>##=(8)8/&)-(&'( ∗((& £ (8%&'
<6=6>
!
2$
! −+³
$FIRSRPRF<( 8+#*!&#!&(!
2$
! 6=6>;
−+= . ',. 0)+0)23!&!3&. =>=> ;; $($(& ∗= ( *3
#,/37!3.0)&9.&-.0)+'3#!,-,2.53.*'+#!<
FTFUVWX<(Y(17:.H'(%2)(D%:7%2)%( 8)( )-.*./4)( </<-6;Z; =( ( 5.( )*+)( *&%8)4+'1)*+)(Q(%4.(&/(127&)4+/:(%4/%:(0(M(E[<($%&'(*&74+%2)%(17:.H).(*)(B%&)(&/(127&)4+/:8)(6;[(*'(*)(%B:),(%+A+(34(&%9/:(*&74+/:/.(*.-1:/(&A+(5.(34(&):(%:(*&74+/:/.(&7-1/*,&)(*/-'(12.-)5+)(17*)*72/:().J(%=(:%(*&%8)4H'L(@=(&/(87.(%4.(34%.4+)(8)(*&%8)4H'L(&=&/(7(\/-'+%+)(8)(%4(34%.4+)(8)(*&%8)4H'<
(+-,123<(%=( </<-6ZCO!,#E;,E=>66; QZ =+=(@=(!(M(#(%4.L($(M(;,6,
=
=<1)4+2/=6Q><&B>/<-<#;C,6#6Z!
=L1)4+2/=O>&B<>/<-<Z6Z,6Z!OO
=L*%/1)4+2/=6Q>=,C><&B></<-Z,6##ZZ
=>,
=>,
=>,
=>,
)''
'*'
'''
)*'
&
(
(
((
(
&=(#
6=! (%4.L($(M(;,6,
=
=
<1)4+2//<-<,Z#Z,6Z;;E
L1)4+2//<-<,C6C,6!OQE
L1)4+2/<,/<-C,6!OO!
=>,
=>,
=>,
=>,
)''
'*'
'''
)*'
&
(
(
((
(
Y]^F_`NPRF<( N1:.&%2)%( *&74+/:/.( *.-1:/( *%/( %( *&74+/:/.( &7-1/*( 8)1.48),2)B)2.48/a4)(:%(+.-1,(87%2(8)(:/4G.-)%(1)2.7%8).( ∗θ−θ=! (8)(:%(*&74+%2)(:%(*&%8)4H'5.( 4/( 8)1.48)( 8)( )0+2)-.+'H.:)( .4+)2D%:/:/.( b<,c θθ∗ ( U%.( -/:+,( 8)7%2)&)( &%1.+%:/:*&74+%+((&(*)(&%:&/:)%9'(34(B/4&H.)(8)(&%1.+%:/:(47-.4%:(((B7:7*.48(/4%(8.4(B72-/:):)J
7@*)2D'-(&',(8%&'(!
&))%( &)( )01:.&'( 12*&/2+<
FTFUVWX<(Y( 887@A48%(%B)2)4+'(&6;[,(8/1'(Q(%4.(8)%4/-.+)( 2)8/&)2.( 312)9.4+'(1)4+2/(%&e
!"!
=&(
,1)4+2/,\=^>6
L1)4+2/=,6>
L*%/1)4+2/,6
=>,
+a
=>,
=>,
=>,
)''
'*'
'''
)*'
(
($!(
(($!
(
+
−+
(d(6(%4,(%+/4&.
,=6>=6>=6> 6 !$($!($!( −− +<+<−
)B)2.4H%( @'4&.:72( 1)4+2/( *&74+/:( *.-1:/( &7-)2&.%:,( 1)( +)2-)4
%+72.)( </<-6;C; =( ( +2)@/.)( 2%-@/2*%+'( .4+)G2%:( 3-12)/4'( &/%:&/:%+',( 34( 2)G.-(8)(87@A48'(&7-1/*',( &/(127&)4+/:(%4/%:(8)(:%(8%+%(348%+72'2..<(?4(&74+2%&+/:(8)(3-12/-/+(*/4+(12)D'9/+)(5.4( &%9( 8)( 2%-@/2*%2)( 34%.4+)( 8)( *&%8)4H'<( $%&'( 8)@.+72/:( *).+%2)%(8%+72.).(8/1'(Z,Q(%4.(&%2)(%2(+2)@/.(*'(B.)(127&)4+/:(%4/%:
!"Z
8)( ,6;;,=>, $%( ''' = (1)4+2/(&%(8)@.+72/:(*'(1:'+)%*&',(B%H'(8)(D%:7%2)%(8%+72.).(:%(8%+%
*&74+'2..,(7(*/-'J(%=(-%.(-%2)L(@=()G%:'L(&=(-%.(-.&'<(+-,123:( `%:7%2)%( 8%+72.).( :%( *&%8)4H'( >!(M(Q(%4.=( )*+)( ,=6>; !2(( += ( %8.&'
((M(6C6;Q6;( /<-<( `%:7%2)%( 8%+72.).( &/(#
6( %4.( 34%.4+)( 8)( *&%8)4H'( )*+)( >*7:/H.%
2%H.74%:'=( </<-6Q!"!;Q=∗( ($)7%2)&)( ∗= ((& (1)4+2/(12%G/:(127&)4+/:/.(8)(*&74+
,6=6>
; !
2$
! −+= (%8.&'($;(M(;,;O"C6"C(2)9/:+'J
%=( 8%&'( ,;$$ < ( %+/4&.( ∗> ((& ( 5.,( 34( &74*)&.4H',( 8)@.+72/:( 2%-@/2*)%9'( 7*/-'(-%.(-%2)(8)&A+(D%:7%2)%(8%+72.).(:%(8%+%(*&74+'2..(%&)*+).%L
@=(8%&'( ,;$$ = (%+/4&.( L∗= ((&&=(8%&'( ,;$$ > (%+/4&.( <∗< ((&Y]^F_`NPRF<( $%&'( *)( &%:&/:)%9'( *&74+/:( &7-)2&.%:( *.-1:/( &/( 127&)4+/:
%(M(6;;$,(.4)G%:.+'H.:)(.-1:.&%+)(34(%871+%2)%(8)&.9.).(12.D.48(-'2.-)%(127&)4+/:/.%4/%:(8)(*&74+(2'-A4(4)-78.B.&%+),(8%2(12%G/:(8)(*&74+( ;$ (*)(-78.B.&'<(?4+2a%8)D'2,
=6>=6>=6> $!2($!(( !& −+=−= ∗ ( 5.(8.4( &748.H.%( ∗((& £ ( 2)9/:+'(
!
!
2!
2$
=6>
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./
"
8
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!!∑
== A"C
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