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L
∫
6
]^_h[hagZq_gh
I NU =
+ LNU ⋅ H − LNU
⋅
Ω
U
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Hg^hihegy_lm]heijbgyluc\>@^h π ?keblhqdZgZ[ex^_gbygZoh^blky\gmljbh[t_fZ
hdjm`_ggh]h ih\_joghklvx 6 lh Ω = π ^ey j_]meyjghc qZklb ih\_joghklb Ω = π <
^\mf_jguo aZ^ZqZo lhqd_ gZ[ex^_gby khhl\_lkl\m_l iehkdbc m]he Λ gZ dhglmj_ b lh]^Z
Ω = Λ >eylhq_dgZj_]meyjghcqZklbdhglmjZ Λ = π GZ ih\_joghklb b^_Zevgh]h ijh\h^gbdZ jbk fZ]gblgh_ ihe_ b iehlghklv lhdZ
F\yaZgu]jZgbqgufmkeh\b_f
ρ = - ρ QL × +
L
L
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LZdbfh[jZahfbabihemqZ_fbgl_]jZevgh_mjZ\g_gb_hlghkbl_evghiehlghklb
∫ Q × - ρ × U ⋅ I NU G6 = - ρ L
L
6
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ijZ\e_gbyfk_^bgbqgufb\_dlhjZfb b ! ehdZevghckbkl_fudhhj^bgZl
- = - τ τ ϕ ⋅ + - ϕ τ ϕ ⋅ ! AZl_f^hfgh`ZykdZeyjghihhq_j_^bgZ_^bgbqgu_\_dlhju L b ! L \lhqd_gZ[ex^_gby
L ihemqbfbgl_]jZevgh_mjZ\g_gb_^eydhfihg_gliehlghklblhdZ
∫ > -
τ L
⋅ QL × × U + - ϕ L ⋅ QL × ! × U @ ⋅ I NU G6 = -Lτ 6
τ ϕ ϕ ∫ > - !L ⋅ QL × × U + - !L ⋅ QL × ! × U@ ⋅ I NU G6 = -L
6
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\u]ey^bl[he__]jhfha^dhq_f^eyfZ]gblgh]hihey>@
(!L = −
 - L ⋅ U ⋅ U  L
 − LNU
 N - L × U × U  L
 −  +
 − L −  ⋅ H G6 
∫
Ωξ F 6 
U
U
NU 
 NU 
N U
<uibr_f]jZgbqgh_mkeh\b_^eywe_dljbq_kdh]hihey\lhqd_gZ[ex^_gby L ρ = L GLY- ρ QL ⋅ (
L
L
& Bgl_]jZevgh_ mjZ\g_gb_ hlghkbl_evgh iehlghklb lhdZihemqZ_fh_ba b nhjfm
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e__h]jZgbqbfkyjZkkfhlj_gb_f[he__ijhklh]hbgl_]jZevgh]hmjZ\g_gbyihemq_ggh]hba
jZkkfhlj_gbyfZ]gblgh]hihey
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IjbZdkbZevghckbff_ljbbehdZevgmxdhhj^bgZlm τ \u[_j_f\^hevh[jZamxs_cnb
]mju \jZs_gby f_jb^bZgZ k _^bgbqguf \_dlhjhf ZehdZevgZydhhj^bgZlZ ϕ kh\iZ^_lk
m]eh\hcdhhj^bgZlhcpbebg^jbq_kdhckbkl_fubf_xs_c\wlhclhqd_gZijZ\e_gb__^bgbqgh
]h \_dlhjZ ! Dhfihg_glu ih\_joghklghc iehlghklb lhdZ ]Zjfhgbq_kdb aZ\bkyl hl m]eZ ϕ JZkdeZ^u\Zy\_dlhjiehlghklblhdZihhjlh]hgZevgufgZijZ\e_gbyf b ! ihemqZ_f
- τ ϕ = - τ τ ⋅ FRV Pϕ ⋅ + - ϕ τ ⋅ VLQ Pϕ ⋅ ! ]^_ P ±ZabfmlZevgh_qbkeh - τ b - ϕ −nmgdpbblhevdhdhhj^bgZlu τ J_amevlZl\_dlhjguohi_jZpbc\pbebg^jbq_kdhckbkl_f_k_^bgbqgufb\_dlhjZfbba
ihdZaZg\Ijbeh`_gbbBgl_]jbjh\Zgb_ihiehsZ^bkwe_f_glhf G6 = 5 ⋅ Gτ ⋅ Gϕ ijb\h
^bl d bgl_]jbjh\Zgbx \^hev h[jZamxs_c / ih iZjZf_ljbq_kdhc dhhj^bgZl_ τ bkdhfuo
nmgdpbc - τ b - ϕ by^jZ * µη y\eyxsbokynmgdpbyfb τ b τ L Bgl_]jbjh\Zgb_ihm]em
ϕ \dexq_gh\\uqbke_gb_dhfihg_gly^jZ
∫ > -
τ τ ⋅ * ττ τ τ L + - ϕ ⋅ * τϕ τ τ L @Gτ = - τ τ L /
τ ϕτ ϕ ϕϕ τ ∫ > - τ ⋅ * τ τ L + - ⋅ * τ τ L @Gτ = - τ L /
<_jogbcbg^_dky^jZgZijbf_j * τϕ τ τ L ihdZau\Z_l ijh_dpbx\_ebqbgufZ]gbl
gh]hihey\lhqd_kdhhj^bgZlhc τ L kha^Z\Z_fh]h_^bgbqghciehlghklvx ϕ cdhfihg_glulh
dZkdhhj^bgZlhc τ π
* ττ = 5 ∫ I NU ⋅ − 5L FRV γ = + ' FRV ϕ ⋅ FRV Pϕ ⋅ Gϕ
π
* τϕ = − 5 ∫ I NU ⋅ VLQ Pϕ ⋅ VLQ ϕ ⋅ Gϕ
π
* ϕτ = 5 ∫ I NU ⋅ 5L FRV γ =L FRV γ 5 − ' FRV γ 5L ⋅ VLQ Pϕ ⋅ VLQ ϕ ⋅ Gϕ
π
* ϕϕ = 5 ∫ I NU ⋅ 5 FRV γ = L − 'L FRV ϕ ⋅ FRV Pϕ ⋅ Gϕ
A^_kv I NU babh[hagZq_gh
U = 5 + 5L − 55L FRV ϕ + = L − = ' = 5 FRV γ = + = L − = FRV γ 5
'L = 5L FRV γ
L =
− = L − = FRV γ
L 5
Dhhj^bgZlu lhqdb bgl_]jbjh\Zgby 5 = 5 τ = = = τ b gZijZ\eyxsb_ dhkbgmku we_f_glZ
ih\_joghklbdhkyf FRV γ 5 FRV γ = y\eyxlkynmgdpbyfb τ \_ebqbgukbg^_dkhf L nmgdpbb
dhhj^bgZlulhqdbgZ[ex^_gby τ L jbkM^\h_gb_\hagbdZ_l\k\yabkl_fqlhkmq_lhfq_l
gh]hbebg_q_lgh]hoZjZdl_jZih^ugl_]jZevghcnmgdpbbijh\h^bfbgl_]jbjh\Zgb_g_ihihe
ghfmm]emŒZ\ij_^_eZohl ^h π = L 5L γ =L γ 5L =
L
5
Jbk <aZbfgh_ jZkiheh`_gb_ gZ dhglmj_ / lhqdb
gZ[ex^_gby L k dhhj^bgZlZfb = L 5L b lhqdb bgl_]
/
Uτ ϕ Q
= 5 γ = γ 5 jbjh\Zgbykdhhj^bgZlZfb = 5 τ γ γ= 5
Kbff_ljbqgu_lbiudhe_[Zgbc
>ey^bihevguob\ukrbolbih\dhe_[Zgbcdh]^Z P ≠ \j_ahgZlhjZoijhba\hevghc
nhjfukpbebg^jbq_kdhckbff_ljb_cih\_^_gb_ihe_cg_\hafh`ghdeZkkbnbpbjh\Zlvih ( beb + -lbiZf dhe_[Zgbc dZd wlh bf__l f_klh ijb jZkkfhlj_gbb j_ahgZlhjh\ ki_pbZevghc
nhjfupbebg^jZbebkn_juGh^eykbff_ljbqguolbih\ P = lZdZydeZkkbnbdZpbybf__l
f_klhKemqZcdh]^Zkms_kl\m_llhevdhf_jb^bZgguclhd - τ khhl\_lkl\m_l ( lbiZfdhe_
[Zgbcdh]^Zkms_kl\m_llhevdhZabfmlZevguclhd - ϕ khhl\_lkl\m_l + lbiZf
Bgl_]jZevgh_mjZ\g_gb_bf__lke_^mxsbc\b^
∫ - τ ⋅ * τ τ Gτ = - τ L
L
/
]^_^ey ( lbih\h[hagZqbf - = τ * = * τ ^ey + lbih\ - = ϕ * = * ϕ Y^jZ * τ b * ϕ ihemqbfba * ττ b * ϕϕ ijb P = π
* τ = 5 ∫ I NU ⋅ − 5L FRV γ = − ' FRV ϕ Gϕ π
* ϕ = 5 ∫ I NU ⋅ 5 FRV γ =L − 'L FRV ϕ Gϕ >\mf_jgZyaZ^ZqZ\^_dZjlh\hckbkl_f_dhhj^bgZlJ_]meyjguc\hegh\h^
AZ^ZqZ h djblbq_kdbo lbiZo dhe_[Zgbc \ j_]meyjghf \hegh\h^_ jZkiZ^Z_lky gZ aZ^Zqb
^ey ( b + lbih\
: >ey ( lbih\ kms_kl\m_l lhevdh dhfihg_glZ iehlghklb lhdZ - = τ gZijZ\e_ggZy
\^hev\hegh\h^Zbkhhl\_lkl\_gghdhfihg_gluihe_c ( = + ; + < H[hagZqbfq_j_a τ [ \ iZjZf_ljbq_kdmxdhhj^bgZlmlhqdbih\_joghklbBgl_]jZev
gh_ij_^klZ\e_gb_^eywe_dljbq_kdh]hihey ( = aZibku\Z_lkyq_j_anmgdpbx=jbgZ
L
* NU = − + NU y\eyxsmxkynmg^Zf_glZevgufj_r_gb_ff_jgh]hmjZ\g_gby=_evf]hevpZ>@
∆* + N * = δ U Ijh^hevgZy dhfihg_glZ we_dljbq_kdh]h ihey \ujZ`Z_lky bgl_]jZehf \^hev h[jZamxs_c / \hegh\h^Zbbf__l\b^
πωµ ( = ρ L = L
Ω
/
∫ - τ ⋅ * NU Gτ π
mqblu\Z_ldhgnb]mjZpbxih\_joghklb\lhqd_gZ[ex^_gbydZdb\\ujZ
Ω
`_gbb JZ^bmk\_dlhj U τ τ L = !L τ L − !τ gZijZ\e_ghlbklhqgbdZiheydlhqd_gZ[ex
]^_fgh`bl_ev
^_gbyNmgdpby=Zgd_ey
+ NU = - + L<NU \ujZ`Z_lkyq_j_anmgdpbb;_kk_eybhibku\Z_l\hegmjZkoh^ysmxkyhlbklhqgbdZ
DZdb\ur_nhjfmebjm_f\gmlj_ggxxaZ^Zqmwe_dljh^bgZfbdbhlghkbl_evghfZ]gbl
gh]hiheydhlhjh_gZoh^bfbamjZ\g_gbyFZdk\_eeZ
+=−
URW ( Lωµ
=jZgbqgh_mkeh\b_ Q L × + L = - L ^eyfZ]gblgh]hiheygZb^_Zevghijh\h^ys_cih\_jo
ghklb\lhqd_gZ[ex^_gbykdhhj^bgZlhc τ L ijb\h^bldbgl_]jZevghfmmjZ\g_gbxhlgh
kbl_evghih\_joghklghciehlghklblhdZ
∫ - τ ⋅ * τ τ Gτ = - τ L
/
y^jhdhlhjh]hbf__l\b^
* τ τ L =
FRV γ
L ;
b FRV γ
hkyf ; b < L <
L

LπN \ − \ τ [ − [τ 
+ NU FRV γ ; L L
− FRV γ <L L
Ω
U τ τ L U τ τ L 

y\eyxlky gZijZ\eyxsbfb dhkbgmkZfb we_f_glZ ih\_joghklb \ lhqd_ τ L d
U τ τ L = [L − [τ + \L − \ τ Nmgdpby=Zgd_ey
+ NU = - NU + L< NU \ujZ`_ggZyq_j_anmgdpbb;_kk_eyihy\ey_lkyihke_^bnn_j_gpbZevguohi_jZpbc
;>eydjblbq_kdbo + lbih\\\hegh\h^_kms_kl\m_llhevdhihi_j_qgZydhk_\hceb
gbb\hegh\h^Zdhfihg_glZiehlghklblhdZ -τ τ gZijZ\e_ggZy\^heviZjZf_ljbq_kdhcdhhj
^bgZlu τ bdhfihg_gluihe_c + = ( ; (< Bgl_]jZevgh_\ujZ`_gb_^ey\uqbke_gbyfZ]
gblgh]hiheybf__l\b^
+ = !L = ∫ - τ ⋅ * τ τ L Gτ /
]^_
* τ τ L =
L πN 
\ − \ τ [ − [τ 
− FRV γ τ L
+ FRV γ ; τ L
 Ω
U τ τ L U τ τ L 

ke_^m_ljZaebqZlvk
Ij_h[jZah\Zgb_bgl_]jZevgh]hmjZ\g_gbydkbkl_f_ebg_cguomjZ\g_gbc
<f_lh^_dheehdZpbc>@ bgl_]jZevgh_mjZ\g_gb_ \uihegy_lkylhqgh\^bkdj_l
ghf 1 qbke_lhq_dih\_joghklb\lhqdZodheehdZpbbF_`^mlhqdZfb^eybkdhfhcnmgdpbb
- ij_^iheZ]Z_lkydZdZyeb[hZiijhdkbfZpbyWlhiha\hey_lij_h[jZah\Zlvbgl_]jZebamjZ\
g_gbydkmff_
∫ - τ * τ τ
1
,
/
Gτ = ∑ - M $LM M =
bbgl_]jZevgh_mjZ\g_gb_dkbkl_f_ 1 ebg_cguomjZ\g_gbc
 $ ττ  τϕ $
$ ϕτ   - τ   - τ 

=
 $ ϕϕ  - ϕ  - ϕ 
A^_kv $ −ih^fZljbpujZaf_jhf 1 × 1 Z - −ih^\_dlhju^ebghc 1 >eyiehkdhcaZ^Zqbb^eykbff_ljbqguolbih\dhe_[ZgbcZdkbZevghkbff_ljbqghcaZ
^ZqbqbkehmjZ\g_gbcjZ\ghqbkemlhq_ddheehdZpbb 1 $ ⋅ - = - ]^_ - ijbgbfZ_lagZq_gb_ - τ beb - ϕ GZb[he__ ijhklhc \b^ dhwnnbpb_glu $LM fZljbpu bf_xl ijb ZiijhdkbfZpbb - eb
g_cghcnmgdpb_c\ij_^_eZo M ]hwe_f_glZih\_joghklb
ghc\ij_^_eZo
GZijbf_jijb
ihklhyg
ihemqbfwe_f_glih^fZljbpu $ $LM = ∫ * ττ L Gτ /M
]^_ * ττ L ih^klZ\ey_fbabebba
Klhqdbaj_gbyf_lh^Zdhg_qguowe_f_glh\dmkhqghihklhyggZybebg_cgZyZiijhdkb
fZpbywd\b\Ze_glgZij_^klZ\e_gbxbkdhfhcnmgdpbbjy^hfijyfhm]hevguobeblj_m]hevguo
bgl_jiheypbhgguonmgdpbc
GZ 6 fbgl_j\Ze_^ebghc K6 bagZq_gbyfb - 6 b - 6 + gZdhgpZobgl_j\ZeZkdhhj^b
gZlZfb τ 6 b τ 6 + nmgdpbyij_^klZ\e_gZ\ujZ`_gb_f
- 6 τ =
beb\fZljbqghf\b^_
]^_ -
6 τ 6 + − τ
τ −τ 6
-6 +
- 6 + K6
K6
- τ = &τ ⋅ - 6 ±\_dlhjklhe[_p 1 agZq_gbcnmgdpbb\maeZok_ldb
& &
&τ =  − &

 −
&  
_klv e_glhqgZy fZljbpZ bgl_jiheypbhgguo nmgdpbc jZaf_jhf 1 × 1 + >ey dmkhqgh
ihklhygghcZiijhdkbfZpbb &τ \ujh`^Z_lky\_^bgbqgmx^bZ]hgZevgmxfZljbpmjZaf_jhf
1 -RQJ b $GDPV >@ ijh^_fhgkljbjh\Zeb ijbf_g_gb_ dmkhqghihklhygghc ZiijhdkbfZ
pbb^eyf_jghcaZ^ZqbjZkq_lZj_ahgZlhjh\f_lh^hfbgl_]jZevguomjZ\g_gbcLhqghklvjZk
q_lh\j_ahgZgkghcqZklhlukhklZ\eyeZJZg__ZgZeh]bqgZylhqghklvkbkihevah\Zgb
_fdmkhqghihklhygghcZiijhdkbfZpbbihemq_gZZ\lhjhf^ZgghcjZ[hluijbjZkq_l_hk_kbf
f_ljbqguo j_ahgZlhjh\ M\_ebq_gb_ jZaf_jh\ k_ldb g_ ijb\h^beh d m\_ebq_gbx lhqghklb
>eymemqr_gbylhqghklbklZehg_h[oh^bfufjZkkfhlj_lvZiijhdkbfZpbx[he__\ukhdh]hih
jy^dZ
L_ogbdZkieZcgZiijhdkbfZpbb^eyj_r_gbybgl_]jZevgh]hmjZ\g_gby
<i_j\u_l_ogbdmdm[bq_kdh]hkieZcgZ^eyj_r_gbydjZ_\hcaZ^ZqbmjZ\g_gbyEZieZkZ
f_lh^hfbgl_]jZevguomjZ\g_gbcjZajZ[hlZeB\Zgh\>@LhqghklvjZkq_lZihl_gpbZeZ\l_klh
\uoaZ^ZqZoij_\urZeZgZihjy^hdlhqghklvihemq_ggmxbkihevah\Zgb_fiZjZ[hebq_kdhcZi
ijhdkbfZpbb b gZ q_luj_ ihjy^dZ ij_\urZeZ lhqghklv dmkhqghihklhygghc ZiijhdkbfZpbb
ijbl_o`_jZaf_jZojZkq_lghck_ldb<ukhdZylhqghklvj_amevlZlh\kbkihevah\Zgb_fkieZcgZ
ih\b^bfhfmk\yaZgZkl_fqlhdm[bq_kdbckieZcgh[eZ^Z_lk\hckl\hffbgbfZevghcdjb\bagu
kj_^b^jm]bobgl_jihebjmxsbonmgdpbcckl_i_gb>@
KieZcg fh`gh jZkkfZljb\Zlv dZd h^ghf_jguc dhg_qguc we_f_gl k [hevrbf qbkehf
\gmlj_ggbomaeh\ 1 ?]hm^h[ghaZibkZlvq_j_a[_ajZaf_jgu_iZjZf_ljujbk
τ −τ6
K6
τ −τ6
=
K6
Z6 = −
Z6 Z6 = Z6 > Z6 − @
Z6 = Z6 > Z6 − @
τ 6 ±dhhj^bgZlZmaeZ K6 ±^ebgZbgl_j\ZeZ
τ
K6 − τ6 Z6 6
Z6 K6 6 +
Jbk JZkiheh`_gb_ maeh\ gZ ]jZgbqghc ebgbb
h[eZklb b h[hagZq_gby dhhj^bgZl ^ey hij_^_e_
gby kieZcgZ Z6 ± ehdZevgZy dhhj^bgZlZ baf_
gyxsZykyhl ^h τ 6 + AgZq_gb_nmgdpbbgZ 6 fbgl_j\Ze_jZ\gh
- 6 τ = - 6 Z6 τ + - 6 +Z6 τ + 0 6 Z6 τ + 0 6 +Z6 τ ]^_ dhwnnbpb_glu 0 6 \ujZ`Zxlky q_j_a agZq_gby nmgdpbb - 6 \ maeZo < fZljbqghf \b^_
dhwnnbpb_glujZ\gu
0 = + −'- A^_kv + b ' −e_glhqgu_fZljbpujZaf_jhf 1 × 1 - _klv\_dlhjklhe[_pbj_amevlZlu
mfgh`_gbygZg_]hlZd`_y\eyxlkyklhe[pZfb
−

 
 µ
λ

 
 µ
λ
+=
 


µ 1 −
λ1 − 


−


]^_ λ6 =
K
µ 6 = − λ6 bh[hagZq_gh α 6 = 6 − α6 + K6
 
 α + 
'=
 



− α
α + α + α − α
α + α α 1 − + IjbjZ\ghf_jghck_ld_we_f_glufZljbpijhklu_
− α 1 −




 

α 1 − + α 1 − 




 − 

  
 

 ' =  
+ =  










− 

Mkeh\by gZ ]jZgbpZo kieZcgZ ij_^hklZ\eyxl k\h[h^m \u[hjZ < ^Zgguo \ujZ`_gbyo
dhwnnbpb_glu klhe[pZ '- '- = b '- 1 = qlh khhl\_lkl\m_l jZ\_gkl\m gmex \lhjhc
ijhba\h^ghcnmgdpbbgZdhgpZokieZcgZ
KieZcgaZibr_f\fZljbqghcnhjf_
- τ = : τ + :
τ 5 - = &τ - A^_kv - τ −\_dlhjklhe[_pZiijhdkbfbjm_fuonmgdpbcjZaf_jhf 1 − (
)
 Z τ Z τ 
 Z τ Z τ 




: τ =   : τ =   

Z1− τ Z1 − 
Z1 − τ Z1 − 


_klv fZljbpu iZjZf_ljh\ jZaf_jhf 1 × 1 − 5 = + − ' − ijhba\_^_gb_ fZljbp ba
FZljbpZ &τ jZaf_jhf 1 × 1 − \hlebqb_hliheghklvxaZiheg_gZ
Ijbih^klZgh\d_\bgl_]jZeijboh^bfdkmff_bgl_]jZeh\ih 1 − bgl_j\Z
eZf /6 1 −  1

 ∑ &VM τ - L *6 τ τ L Gτ τ
τ
τ
τ
*
G
=
∑
L
∫/
∫


V = / 6  M =

I_j_klZ\ey_ff_klZfbkmffbjh\Zgb_ih V b M ∫
- τ * τ τ L Gτ =
/
 1 −


 τ
τ
τ
τ
&
*
G
∑
M ∑ ∫ VM
6
L


M =
 6 = /6

1
IhemqZ_fdhwnnbpb_glu $LM kbkl_fuebg_cguomjZ\g_gbc
$LM =
1 −
∑ ∫&
6 = / 6
VM
τ *6 τ τ L Gτ AZibr_fdhwnnbpb_gluy\ghq_j_awe_f_glufZljbp & b 5 1 −
(
)
$LM µν = , LMµν ε M1 + , LMµν ε M1 + ∑ 5 MV ,LVµν + 5 M V +, LVµν 6 =
A^_kv 1 b 1 −ghf_jZgZqZevghcbdhg_qghclhq_dkieZcgZ>eyh^bghqgh]hkieZcgZ
1 = 1 = 1 Bg^_dku µ ν ijh[_]ZxlagZq_gby ϕ b τ kbf\he ε jZ\_g
 L ≠ M
ε LM =   L = M
<\oh^bl\uqbke_gb_bgl_]jZeh\dhlhju_fh`ghaZibkZlvh[sbf\ujZ`_gb_f
, NLV =
τ 6 +
∫ω
N V
τ *6 τ τ L Gτ τ6
Bg^_dk N ijbgbfZ_l agZq_gby hl ^h b bg^_dkbjm_l nmgdpbb ω
µη N τ \ \ujZ`_gbyo bnmgdpbb * = * µν \u[bjZ_f
>ey g_kbff_ljbqguo lbih\ dhe_[Zgbc bg^_dkbjm_f , = ,
ba\ujZ`_gbc>eykbff_ljbqguolbih\ih^klZ\ey_fnmgdpbb * ba\ujZ`_gbcbeb
^eyiehkdhcaZ^Zqbba\ujZ`_gbcbebBg^_dk V mdZau\Z_lgZijbgZ^e_`ghklv
dhhj^bgZlbgl_j\Zem τ 6 τ 6 + LZdbf h[jZahf dhwnnbpb_glu $ kbkl_fu ebg_cguo mjZ\g_gbc \dexqZxl \uqbke_
gb_lbih\bgl_]jZeh\^eyg_kbff_ljbqguofh^dhe_[Zgbcbeblbih\^eykbff_ljbq
guofh^biehkdhcaZ^Zqb
IjhklZy ]_hf_ljby gZijbf_j kn_jZ fh`_l [ulv ij_^klZ\e_gZ h^gbf dhg_qguf
kieZcgwe_f_glhfKeh`gZy]_hf_ljbyhibku\Z_lkyg_kdhevdbfbkieZcgZfbijbgZ^e_`Zsbfb
_kl_kl\_ggufmqZkldZf]_hf_ljbb −^m]ZfijyfufAgZq_gbykieZcgh\\kf_`guolhqdZo]jZ
gbpu krb\Zxlky l_ klZ\blky mkeh\b_ g_ij_ju\ghklb iehlghklb lhdZ - =jZgbqgu_ lhqdb
we_f_glh\ ijbgZ^e_`Zl kf_`guf kieZcgZf Wlh khhl\_lkl\m_l lhfm qlh fZljbpZ 5 ba
P khklhysZybaih^fZljbp 5 jZaf_jhf 1 P × 1 P i_j_dju\Z_lkyk\hbfbm]eh\ufbdhwn
nbpb_glZfb
>jm]hc ijb_f jZkkfhlj_gguc B\Zgh\uf>@ khklhbl\lhf qlhh^gZblZ`_kf_`gZy
lhqdZgZijbf_j M ijbgZ^e_`ZsZy\lhjhfmkieZcgmbg^_dkbjm_lkydZd M + Mkeh\b_krb
\Zgby - M = - M + hagZqZ_l qlh \ khhl\_lkl\mxs_c L hc kljhd_ fZljbpu $ [m^_l $LM = $L M + = − Z\k_hklZevgu_we_f_glujZ\gugmexIhjy^hdkbkl_fum\_ebqb\Z_lkygZqbkeh
kf_`guolhq_dFZljbpZ 5 khklhblbag_i_j_k_dZxsbokyih^fZljbp 5
P <uqbke_gb_bgl_]jZeh\
Ijb \uqbke_gbb bgl_]jZeh\ ih mqZkldZf L b L + jbk ijbe_`Zsbf lhqd_ gZ
[ex^_gby V = L dh]^Z dhhj^bgZlu ϕ → b τ → τ L g_h[oh^bfh mqblu\Zlv qlh ih^ugl_
]jZevgu_nmgdpbbbf_xlbgl_]jbjm_fmxhkh[_gghklvIjb ϕ → banmgdpbcdhg_qgu
fb hklZxlky * ττ b * ϕϕ Ijb τ → τ L hkh[_gghklv ijbkmlkl\m_l \ bgl_]jZeZo ,LL b , L L − QZklbbgl_]jZeh\kfgh`bl_e_f τ − τ L hkh[_gghklvxg_h[eZ^Zxl
* τ τ L Jbk Ih^ugl_]jZevgZy nmgdpby k hkh[_g
ghklvxijb V = L b__ZkbfilhlbdZ
FRV γ =L
OQτ − τ L 5L
τ 6 − , LL =
τ L +
∫Z
τ τ L * τ τ L Gτ =
τ L +
τL
, L L −
τL
τ L +
∫ * τ τ Gτ − ∫
L
τL
τL
τL
V =L
τ6 τ 6 + τ −τL
* τ τ L Gτ
KL
τL
τ −τ
= ∫ Z τ τ L * τ τ L Gτ = ∫ * τ τ L Gτ − ∫ L
* τ τ L Gτ
KL −
τ L −
τ L −
τ L −
τ
H[hagZqbfbg^_dkhf
bgl_]jZeukhkh[_gghklvx
, ττ = ∫ * ττ τ τ L Gτ
/
,
ϕϕ = ∫ * ϕϕ τ τ L Gτ
/
KlZg^Zjlgucijb_f\uqbke_gbybgl_]jZeh\khkh[_gghklvxaZdexqZ_lky\bkihevah\Zgbb
Zkbfilhlbdb ih^ugl_]jZevghc nmgdpbb GZ i_j\hf wlZi_ \uql_fb^h[Z\bfnmgdpbb=jbgZ
Φ b Φ dhevpZkhklZpbhgZjguflhdhf
, =
∫ (*τ τ − Φ
L
/
H^ghkeZ]Z_fh_ Φ
)
τ τ L + Φ τ τ L Gτ \ujZabfdZdZkbfilhlbdmih^ugl_]jZevghcnmgdpbb
Φ τ τ L = * τ τ L ijb P = ϕ → τ → τ L <lhjmxqZklv Φ ijhbgl_]jbjm_fihm]em ϕ b\ujZabfq_j_aihegu_weebilbq_kdb_bgl_
]jZeui_j\h]h (η b\lhjh]h .η jh^Zbkihevamyba\_klgh_\ujZ`_gb_^eynmgdpbb=jbgZ
dhevpZklhdhfIhemqbfihe_agu_nhjfmeu^ey\klj_qZxsbokybgl_]jZeh\
π


O + 55L
 − .η +
(η  O


FRV ϕ
∫ U Gϕ = 55 O + 55
L
L
π
Gϕ
∫U
=
O O + 55L
(η ]^_ η =
O
U = 5 + 5L − 55, FRV ϕ + = − = L jbk
O + 55,
<lhjhc wlZi Hkh[_gghklv \ Φ k\yaZgZ k weebilbq_kdbf bgl_]jZehf .η b ghkbl
eh]Zjbnfbq_kdbc oZjZdl_j <uql_f qbke_gguc bgl_]jZe hl Zkbfilhlbq_kdh]h ij_^_eZ
Φ τ τ L τ →τ L = −
OLP τ →τ L Φ = −
FRV γ =L OQO 5L
b
^h[Z\bf
ZgZeblbq_kdbc
ij_^_e
FRV γ =L  KL

OQ −  5L


Ko_fZ\uqbke_gbybgl_]jZeh\khkh[_gghklvxke_^mxsZy
(
)
, = ∫ * τ τ L − * τ τ L P = ϕ → τ →τ L + Φ τ − Φ τ → τ L Gτ + OLP τ →τ L Φ /
<uibr_f\ujZ`_gby^eybgl_]jZeh\k\u^_e_gghchkh[_gghklvx
, ττ =
 55L FRV γ = − ' O + 55L 5L

 ττ '

(η + .η  *
τ
τ

+
L
∫
O
5L
π /
O + 55L 



FRV γ =L FRV γ = L  KL

OQ O  Gτ −
+
 OQ −  5L
5L  

, ϕϕ =
 5 FRV γ =L − 'L O + 55L 5L
'

 ϕϕ 

(η + L .η  τ
τ
*
+
L
∫

π /
5L
O
O + 55L 



FRV γ =L FRV γ =L  KL

+
OQ O  Gτ −
 OQ −  5L
5L  

]^_h[hagZq_gh
ττ *
π
= 5 ∫ I NU (− 5 L FRV γ
=
− ' FRV ϕ )Gϕ
*
ϕϕ π
(
= 5 ∫ I NU 5 FRV γ
L =
)
− ' L FRV ϕ Gϕ
 + LNU − LNU

FRV Pϕ −
H
Ω  U
ΩU U
+ 5 − 5 L FRV γ 5
I NU = I NU −
' = 5 FRV γ
=
' L = 5 L FRV γ
L =
=
− 5 − 5 L FRV γ

 
L 5
IZjZf_lj O → τ − τ L ihdZaZggZjbkHklZevgu_h[hagZq_gbyl_`_qlhb\b
γ
]
=
γ5
5
5L τ
= − =L O
/
JbkH[hagZq_gbyijb\uqbke_gbbbgl_]jZeh\khkh[_gghklvx
>ey kbff_ljbqguo ( b + lbih\ dhe_[Zgbc P = bgl_]jZeu k hkh[_gghklvx \u
qbkeyxlky ih nhjfmeZf b khhl\_lkl\_ggh Ih^ugl_]jZevgu_ \ujZ`_gby b iehkdhcaZ^Zqbg_h[eZ^Zxlhkh[_gghklvx
Ijb\uqbke_gbbbgl_]jZeh\\oh^ysbo\kmffmbkihevamxlkyd\Z^jZlmjgu_
nhjfmeu=ZmkkZ>@H[hagZqbfih^ugl_]jZevgu_nmgdpbbq_j_a I τ b I τ ϕ >eyhl
^_evgh]h V ]hbgl_j\ZeZ^ebghc O6 = τ 6 + − τ 6 ihemqZ_f
τ 6 +
∫
τ6
π
I τ G τ ∫ I τ ϕ G ϕ =
O6 π
Q


\
I
τ
∑
 L L ∑ \ M I τ L ϕ M  L = 
M =

Q
O
π
[ M + τ L = 6 [L + Z \L [L ± lZ[ebqgu_ agZq_gby \_kh\ b Z[kpbkk d\Z^jZ
lmjguo nhjfme ihjy^hd dhlhjuo Q b Q \u[bjZHlky bkoh^y ba g_h[oh^bfhc lhqghklb \u
]^_ ϕ M =
qbke_gbc
Ihbkdj_ahgZgkguoqZklhl
&bkl_fZh^ghjh^guoebg_cguomjZ\g_gbc
($N − , )- = ddhlhjhcijb\h^blkyjZkkfZljb\Z_fZykbkl_fZbf__lj_r_gb_dh]^Z
GHW ($N − , ) = ]^_ , ±_^bgbqgZy^bZ]hgZevgZyfZljbpZIhke_hij_^_e_gbydhjg_c N = N nmgdpbb
I N = GHW ($N − , ) b khhl\_lkl\_ggh \uqbke_gby fZljbpu $ N ba fh`ghgZclb\_dlhjj_r_gby - >ey
wlh]hh^bgbawe_f_glh\ - kqblZ_fba\_klgufbj_rZ_fmf_gvr_ggmxgZ_^bgbpmg_h^ghjh^
gmxkbkl_fmhlghkbl_evghhklZevguowe_f_glh\ - >jm]hcih^oh^hkgh\ZggZZgZeba_kh[kl\_gguoagZq_gbc λ fZljbpuMjZ\g_gb_hlgh
kbl_evgh λ aZibku\Z_fke_^mxsbfh[jZahf
($N − , )- = λ N - Ihgylgh qlh ijb N = N fbgbfZevgh_ kh[kl\_ggh_agZq_gb_ λPLQ N fZljbpu kh]eZkgh
^he`ghh[jZsZlvky\gmevZkh[kl\_gguc\_dlhj - [m^_lj_r_gb_fkbkl_fu:e]hjblfhij_
^_e_gby λPLQ hkgh\ZggZh[jZlguobl_jZpbyo>@
$ N − , - Q + = - Q Ijb[hevrhfqbke_bl_jZpbc Q hlghr_gb_^ebg\_dlhjh\ ^_ebfwlhhlghr_gb_ke_^mxsbfh[jZahf
Q b -
Q +
koh^blkyd λPLQ Hij_
- L Q ∑
Q +
L = - L
A^_kv 1 ±jZaf_jghklv\_dlhjZ - >Zggh_hij_^_e_gb_g_kljh]hjZ\gh λPLQ ghhghihe_agh
Q
N =
λPLQ
1
l_f qlh kh^_j`bl agZd b jZ\gh gmex\lhqd_j_ahgZgkZ<u[hjgZqZevgh]hagZq_gby\_dlhjZ
- agZq_gbyg_bf__lb\[ebab N = N ]^_ λPLQ N → bl_jZpbbkoh^ylky[ukljh
Ihke_\uqbke_gbydhwnnbpb_glh\fZljbpu $N − , hgZijb\h^blkyf_lh^hfbkdex
q_gby=ZmkkZd\_jog_clj_m]hevghcnhjf_>@Ihke_q_]hj_r_gb_kbkl_fubbl_jZpbb
aZgbfZxlf_gvr__fZrbggh_\j_fyq_f\j_fy\uqbke_gbykZfhcfZljbpu
LZdbfh[jZahfijb N = N ^he`gu\uihegylkyke_^mxsb_mkeh\by
Z GHW $ N − , = \ 5H λPLQ N = k ,P λPLQ N = Qbke_ggu_bkke_^h\ZgbyihdZaZebke_^mxsb_hkh[_gghklbnmgdpbc
Fh^mev^_l_jfbgZglZ\[ebabj_ahgZgkZklZgh\blkyfZeufghg_kljh]hgme_\ufWlhmdZ
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i_j_oh^_q_j_a N = N i_j\Zyijhba\h^gZyf_gy_lagZdjbkWlhaZljm^gy_lihbkdfb
gbfmfZlZdhc9h[jZaghcnmgdpbb
< kemqZ_ dh]^Z j_ahgZgku [ebadb d \ujh`^_gbx ih\_^_gb_ ^_l_jfbgZglZ klZgh\blky
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j_ahgZgkZijbwlhfiZ^Z_l
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bf_xlky jZaju\u Z \[ebab j_ahgZgkZ wlZ nmgdpby [ebadZ d ebg_cghc Dhjgb nmgdpbb
,P λPLQ N gZ^_`gh \uqbkeyxlky \ l_klh\uo aZ^ZqZo k \ukhdhc lhqghklvx jZ\gu
ZgZeblbq_kdbfj_ahgZgkgufagZq_gbyfbbkihevamxlky^eyhij_^_e_gbyj_ahgZgkZ
,PλPLQ
(
+ GHW$N,
úñ$
Jbk>\Z[ebadhjZkiheh`_gguoj_ahgZgkZPbebg^jbq_kdbcj_ahgZlhj5 f+ P
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JZkkfhljbfh^ghf_jgucj_ahgZlhjh[jZah\Zgguc^\mfyiZjZee_evgufbijh\h^ysbfb
iehkdhklyfbIh\_joghklguclhdiehlghklvx - b - gZijZ\e_g\^hevhkb ] jbk
- -
+ + + + =
<
G
;
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mjZ\g_gbc
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; + +; = −
+ ; + + ; = -
Dhfihg_glufZ]gblguoihe_cgZiehkdhklyobba\_klgu^eyh^ghf_jghciehkdhcaZ^Zqb
+ ; = +; = - HLNG = − - HLNG
+ ; =
+ ;
I_j_g_kyijZ\u_qZklb\e_\hi_j_c^_fdh^ghjh^ghckbkl_f_mjZ\g_gbc
HLNG
HLNG -
= beb\fZljbqghf\b^_ % ⋅ - = -
Wlhckbkl_f_khhl\_lkl\m_loZjZdl_jbklbq_kdh_mjZ\g_gb_
− λ HLNG
= H LNG − λ
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GHW % = − HL NG
λ = ± HL NG ]jZnbdbdhlhjuoihdZaZgugZjbkJ_ahgZgkgu_qZklhlukhhl\_lkl\mxldhjgyfnmgdpbc
,P λ N 5H λ N GHW %N l_
N j_a G = Qπ ÿ&) $)$
% ⋅ - Q + = - Q j_amevlZldhlhjh]h^Z_lke_^mxs__
λPLQ

 -Q
- Q  λ
= OLP Q → ∞  Q + + Q +  = 
-   λ
 -

π
π
π
≤ NG ≤
≤ NG ≤
$EVGHW
,Pλ1 , ,Pλ2
5Hλ1 5Hλ2
<b^ghqlhlhqdbj_ahgZgkZm^h[ghhij_^_eylvihdhjgyffgbfhcqZklbkh[kl\_gguoagZ
q_gbckbkl_fu.
λ1
λ2
λ1
NGπ)
λ2
NGπ)
NGπ)
Jbk>_l_jfbgZglbkh[kl\_ggu_agZq_gbyh^ghf_jgh]hj_ahgZlhjZ@bjghc
ebgb_c\u^_e_gj_amevlZlbl_jZpbhggh]hijhp_kkZ\uqbke_gby λ PLQ <uqbke_gb_ihe_cbiZjZf_ljh\j_ahgZlhjh\
J_amevlZlhf jZkq_lZ fh^u y\eyxlky __ qZklhlZ b dhfihg_glu iehlghklb lhdZ \ maeZo
k_ldb Ihey b iZjZf_lju j_ahgZlhjZ jZkkqblu\Zxlky hl^_evghc ijhp_^mjhc <_ebqbgu k\y
aZggu_ k ih\_joghklvx \uqbkeyxlky \_kvfZ [ukljh Wlh gZijy`_gghklb fZ]gblguo b we_d
ljbq_kdbo ihe_c gZ ih\_joghklb ihl_jb \ kl_gdZo >ey jZkq_lZ gZdhie_gghc wg_j]bb
µ
: = ∫ + G 9 bkihevam_lkyihe_agZynhjfmeZij_h[jZah\Zgbyh[t_fgh]hbgl_]jZeZ\bgl_
]jZeihih\_joghklb>@
µ ∫ + G9 =
µ + − ξ ( U ⋅ Q G 6 ∫
A^_kv ghjfZev Q we_f_glZ ih\_joghklb gZijZ\e_gZ gZjm`m JZ^bmk\_dlhj U gZijZ\e_g ba
ijhba\hevghclhqdbgZijbf_jbagZqZeZdhhj^bgZl<f_jghcaZ^Zq_bgl_]jbjh\Zgb_\_^_l
kyihdhhj^bgZl_ τ <_ebqbgZ U ⋅ Q = 5 FRV γ = − = FRV γ 5 y\ey_lkynmgdpb_c τ >eyjZkq_lZihe_c\lhqdZo\gmljbj_ahgZlhjZg_h[oh^bfhbgl_]jbjh\Zgb_ih\k_cih
\_joghklbihwlhfmihkljh_gb_dZjlbguihe_c\h[t_f_aZgbfZ_lagZqbl_evgh_\j_fy
;ukljuf ihjlj_lhf fh^u fh`_l kem`blv dZjlbgZ jZkij_^_e_gby ih\_joghklghc iehl
ghklblhdZGZjbkihdZaZgihjlj_lfh^upbebg^jbq_kdh]hj_ahgZlhjZ<\_jomihkljh_gu
ebgbbjZ\gh]hmjh\gy - = FRQVW gZiehkdhklbdhhj^bgZlghckbkl_fu ϕ − τ ]^_ ϕ baf_gy_l
kyhl ^h π b τ hl ^h / / −^ebgZh[jZamxs_cEbgbbmjh\gyihdZau\ZxlgZijZ\e_
gb_ fZ]gblgh]h ihey gZ ih\_joghklb Z ]jZ^b_gl mdZau\Z_l gZijZ\e_gb_ \_dlhjZ iehlghklb
lhdZ
<uibr_f \ujZ`_gby^ey\uqbke_gbydhfihg_glihe_c ihemqZ_fuobabgl_]jZevguo
nhjfmeb
:dkbZevgZy kbff_ljby Ijhba\hevgu_ lbiu dhe_[Zgbc FZ]gblgh_ ihe_ \ lhqd_ k
dhhj^bgZlZfb 5L = L ihemqZ_f kmffbjh\Zgb_f dhfihg_gl kha^Z\Z_fuo ϕ b τ khklZ\eyx
sbfbih\_joghklghciehlghklblhdZy\eyxsbfbkynmgdpbyfbdhhj^bgZl 5 τ = τ + 5 = + 5ϕ + + 5τ + ϕ = + ϕϕ + + ϕτ + = = + =ϕ + + =τ Dhfihg_glufZ]gblgh]hiheyihemqZ_fu_babf_xl\b^
π
+ 5ϕ = ∫ - ϕ τ 5 = L − = G τ ∫ I NU FRV Pϕ FRV ϕ G ϕ /
π
+ 5τ = −∫ - τ τ 5' G τ ∫ I NU VLQ Pϕ VLQ ϕ G ϕ /
π
+ ϕϕ = ∫ - ϕ τ 5 = L − = G τ ∫ I NU VLQ Pϕ VLQ ϕ G ϕ /
+
τ ϕ
= ∫ -
τ π
τ 5 G τ ∫ 5L FRV γ = − ' FRV ϕ I NU FRV Pϕ G ϕ /
π
+ =ϕ = ∫ - ϕ τ 5 G τ ∫ 5 − 5L FRV ϕ I NU FRV Pϕ G ϕ /
+
τ =
= 5L ∫ -
τ π
τ 5 FRV γ 5 G τ ∫ I NU VLQ Pϕ VLQ ϕ G ϕ /
A^_kv I NU jZ\gh j_Zevghc qZklb dhfie_dkghc nmgdpbb ijb agZq_gbb Ω = π ' b U h[hagZq_gu\Bg^_dkhf L h[hagZqbfdhhj^bgZlulhqdb\dhlhjhcjZkkfZljb\Z_lkyihe_
<ujZ`_gb_ ^ey \uqbke_gby we_dljbq_kdh]h ihey \u]ey^bl keh`g__ <_dlhj we_d
ljbq_kdh]hihey ( jZaeZ]Z_lkyihhjlh]hgZevgufgZijZ\e_gbyf U b θ kn_jbq_kdhckbkl_fu
dhhj^bgZlDhfihg_glujZaeh`_gbyjZkkfZljb\ZxlkydZdkmffZihe_ckha^Z\Z_fuo ϕ b τ dhfihg_glZfbih\_joghklghciehlghklblhdZBlZdh_jZaeh`_gb_g_h[oh^bfhijh^_eZlv^ey
dZ`^hcbadhfihg_glihey\pbebg^jbq_kdhckbkl_f_dhhj^bgZl
(5 = (5ϕU + (5ϕθ + (5τU + (5τθ (ϕ = (ϕϕU + (ϕϕθ + (ϕτU + (ϕτθ (= = (=ϕU + (=ϕθ + (=τU + (=τθ LZdbfh[jZahfihemqZ_fbgl_]jZeh\^ey\uqbke_gbywe_dljbq_kdh]hihey
(
ϕU 5
= − 5L ∫ /
ϕ π
5 G τ ∫ I NU 5L − 5 FRV ϕ VLQ Pϕ VLQ ϕ G ϕ
π
(ϕϕU = 5L ∫ - ϕ 5 G τ ∫ I NU FRV Pϕ VLQ ϕ G ϕ
/
π
( =ϕU = − 5L ∫ - ϕ 5'= G τ ∫ I NU VLQ Pϕ VLQ ϕ G ϕ
/
π
( 5τU = ∫ - τ 5 G τ ∫ '= FRV γ = − 5 − 5L FRV ϕ FRV γ 5 5L − 5 FRV ϕ I NU FRV Pϕ G ϕ
/
π
(ϕτU = ∫ - τ 5 G τ ∫ '= FRV γ = − 5 − 5L FRV ϕ FRV γ 5 I NU VLQ Pϕ VLQ ϕ G ϕ
/
π
( =τU = ∫ - τ 5'= G τ ∫ '= FRV γ = − 5 − 5L FRV ϕ FRV γ 5 I NU FRV Pϕ G ϕ
/
π
( 5ϕθ = ∫ - ϕ 5 G τ ∫ ' = + 5 5 − 5L FRV ϕ I NU VLQ P ϕ VLQ ϕ G ϕ
/
π
(ϕϕθ = ∫ - ϕ 5 G τ ∫ 5L − 5 FRV ϕ 5 − 5L FRV ϕ − ' = FRV ϕ I NU FRV P ϕ G ϕ /
(
ϕθ =
= − 5L ∫ -
ϕ π
5' = G τ ∫ I NU VLQ P ϕ VLQ ϕ G ϕ
/
π
( 5τθ = ∫ - τ 5 G τ ∫ '= 5L FRV γ = − ' FRV ϕ − 55L FRV γ 5 VLQ ϕ I NU FRV Pϕ G ϕ
/
π
(ϕτθ = ∫ - τ 5 G τ ∫ 5L FRV γ 5 5L − 5 FRV ϕ + '= ' I NU VLQ Pϕ VLQ ϕ G ϕ
/
(
τθ =
= −∫ -
τ /
H[hagZq_gh
π
5 G τ ∫ 5L FRV γ = 5L − 5 FRV ϕ + ' 5 − 5L FRV ϕ I NU FRV Pϕ G ϕ
'= τ = = L − = τ 'τ = 5 FRV γ = + = L − = FRV γ 5
I NU =


FRV NU + VLQ NU 

πξ FU  NU

I NU =
 
N

−  FRV NU + VLQ NU 

 πξ FU   N U
NU


U ϕ τ = 5 + 5L − 55L FRV ϕ + = L − = Dhhj^bgZlulhqdbih\_joghklb = b 5 bgZdehgu FRV γ = FRV γ 5 we_f_glZih\_jogh
klbdhkyfy\eyxlkynmgdpbyfbdhhj^bgZlu τ >eyhibkZgbynmgdpbb - τ aZ^ZgghckieZcghfbkihevam_lkydm[bq_kdh_ij_^klZ\e_
gb_>@ ihemqZ_fh_ba dh]^ZagZq_gby - L \maeZok_ldbm`_ba\_klgubdhwnnbpb_glu
0 L \uqbke_gukh]eZkghGZdZ`^hfih^ugl_j\Ze_ >τ 6 τ 6 + @ - τ = - 6 + E6 τ − τ 6 + F6 τ − τ 6 + G 6 τ − τ 6 ]^_
E6 =
- 6 + − - 6
− K6 0 6 + + 0 6 K6
F6 = 0 6 0 6 + − 0 6
K6
G6 =
:dkbZevgZykbff_ljby ( lbidhe_[Zgbc P = Kms_kl\m_llhevdh - τ −dhfih
g_glZih\_joghklghciehlghklblhdZ + ϕ −dhfihg_glZfZ]gblgh]hiheyb (5 = (5τU + (5τθ b
( = = ( =τU + ( =τθ −dhfihg_gluwe_dljbq_kdh]hihey
+ϕ = +
τ ϕ
= ∫ /
τ π
τ 5 G τ ∫ 5L FRV γ = − ' FRV ϕ I NU G ϕ Khhl\_lkl\mxsb_ \ujZ`_gby ^ey \uqbke_gby dhfihg_gl we_dljbq_kdh]h ihey \u[bjZ_f ba
\ur_ijb\_^_gguoh[sbo\ujZ`_gbcijbagZq_gbb P = :dkbZevgZykbff_ljby + lbiudhe_[Zgbc P = Kms_kl\m_llhevdh - ϕ −dhf
ihg_glZ ih\_joghklgh]h lhdZ + 5 = + 5ϕ b + = = + =ϕ − dhfihg_glu fZ]gblgh]h ihey b
(ϕ = (ϕϕU + (ϕϕθ −dhfihg_glZwe_dljbq_kdh]hihey<ujZ`_gby^eydhfihg_glihemqZ_fba
h[sbo\ujZ`_gbcijb P = >_dZjlh\Zy^\mf_jgZykbkl_fZdhhj^bgZl ( lbiuKms_kl\m_lijh^hevgZydhfih
g_glZwe_dljbq_kdh]hihey ( = \uqbkey_fZydZdj_ZevgZyqZklvbihi_j_qgh_fZ]gblgh_
ihe_kdhfihg_glZfb + ; b + < \uqbkey_fufbba
( = [L \ L =
ωµ - τ ⋅ - NU G τ
∫/
+ ; [L \ L =
\ − \ τ N
G τ - τ ⋅ < NU L
∫
/
U
+ < [L \ L = −
[ − [τ N
Gτ
- τ ⋅ < NU L
∫
/
U
]^_ U = U [L \ L τ = [L − [ τ + \L − \ τ >_dZjlh\Zykbkl_fZdhhj^bgZl + lbiuKms_kl\mxldhfihg_gluihe_c + = ( ; (< <uqbke_gb_fZ]gblgh]hiheyijhba\h^blkyihnhjfmeZfNhjfmeu^eyjZkq_lZ
we_dljbq_kdh]hiheyihemqZ_fbamjZ\g_gbyFZdk\_eeZ
= URW +
(
Lωξ Ijb\uqbke_gbbihe_ci_j_oh^bfhldhfie_dkguo\_ebqbgd^_ckl\bl_evguf
( ; [L \L =
=
- τ ⋅ * ; [L \L τ Gτ
∫/
=
(< [L \L = − ∫ - τ ⋅ *< [L \L τ Gτ
/
Nmgdpbb=jbgZ^eywe_dljbq_kdh]hiheyjZ\gu

∂ U
∂U 
∂U
∂ U 
∂U 
 FRV γ ;
<′NU FRV
FRV
* ; =  FRV γ ;
<
NU
N
+
−
−
γ
γ
\
\
∂\L 
∂\L
∂[L ∂\L 
∂[L 
∂\L


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∂ U 
∂U 
∂U
∂U 
<′NU  FRV γ ;
− FRV γ \
− FRV γ \
*< =  FRV γ ;
< NU + N
 ∂[L ∂\ L
∂[L 
∂\L
∂[L 
∂[L 

]^_ U = U [L \L τ dZdb\ur_ = = µ Ijhba\h^gZynmgdpbb;_kk_eyjZ\gZ
ε
<′NU = < NU −
5
Ebgbbwe_dljbq_kdh]hihey\
iehkdhklb 5 − = j_ahgZlhjZ
EbgbbfZ]gblgh]hiheygZih\_joghklb
ϕ
Ebgbbmjh\g_c
π
-
< NU NU
- = FRQVW
τ
Lhj_p
Pbebg^j
Lhj_p
/
Dhfihg_gluiehlghklblhdZ
-
= ϕ - τ τ
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úñ$
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F=p
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1 = × /RJ)))
6XSHU/DQV
1 = × 6XSHU/DQV
850(/7
1 = × 13 = /RJ)))
0$;:(// 1 = 0$;:(// 1 = )úñ$
)0ñ$
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1 = × 6XSHU/DQV
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L
L
L = 7; L + 7< M + 7= N I
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l_f_dhhj^bgZl
; = FRV γ 5 ⋅ FRV φ
7< = FRV γ 5 ⋅ VLQ φ 7= = FRV γ =
5; = 5L − 5 ⋅ FRV ϕ
5< = − 5 ⋅ VLQ ϕ
5= = = L − =
Φ ; = − VLQ ϕ
Φ < = FRV ϕ I
Φ = = 7;
L = FRV γ 5
L 7< L = 7=
L = FRV γ =
L L
Φ; = Φ<
L = Φ=
L =
1;
L = − FRV γ =
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L = FRV γ 5
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FRV γ 5 τ FRV γ = τ y\eyxlkynmgdpbyfbehdZevghcdhhj^bgZlu τ Zdhhj^bgZlu khhl\_l
FRV γ 5 L = FRV γ 5 τ L 5L = 5 τ L = L = = τ L kl\mxsb_
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