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JZa^_e L_ieh\hc j_`bf a^Zgby \k_]h qZkh\

Hkgh\u l_hjbb l_iehh[f_gZ

L_hjby l_iehh[f_gZ wlh gZmdZ h ijhp_kkZo i_j_ghkZ l_iehlu Jy^ \Z`guo ijhp_kkh\ ijh_dlbjh\Zgby b kljhbl_evkl\Z a^Zgbc b khhjm`_gbc j_rZxlky gZ hkgh\_ l_hjbb l_iehh[f_gZ AgZgb_ aZdhgh\ l_iehh[f_gZ iha\hey_l bg`_g_jm kljhbl_ex m\yaZlv lhesbgm b fZl_jbZe h]jZ`^Zxs_c dhgkljmdpbb HD k hlhibl_evgufb ijb[hjZfb HI jZajZ[hlZlv gh\u_ kljhbl_evgu_ fZl_jbZeu b dhgkljmdpbb [he__ wdhghfbqgu_ b kihkh[gu_ gZ^_`gh aZsbsZlv q_eh\_dZ hl oheh^Z `Zju b ^jm]b_ \hijhku dhlhju_ \hagbdZxl \ ijhp_kk_ jZa\blby kljhbl_evghc l_ogbdb

<b^u i_j_^Zqb l_iehlu

L_iehh[f_g ij_^klZ\ey_l keh`guc ijhp_kk dhlhjuc fh`gh jZkqe_gblv gZ jy^ ijhkluo ijhp_kkh\ JZaebqZxl we_f_glZjguo ijbgpbibZevgh hlebqguo h^bg hl ^jm]h]h ijhp_kkh\ l_iehijh\h^ghklv dhg\_dpby b l_ieh\h_ baemq_gb_

Ijhp_kk l_iehijh\h^ghklb ijhbkoh^bl ijb g_ihkj_^kl\_gghf khijbdhkgh\_gbbfhe_dme Zlhfh\ b k\h[h^guo we_dljhgh\ khijh\h`^Zxsbfky h[f_ghf wg_j]bb b bo l_ieh\h]h ^\b`_gby l_iehijh\h^ghklv \ `b^dbo b hkh[_ggh \ ]Zahh[jZaguo kj_^Zo g_agZqbl_evgZ

L\_j^u_ l_eZ h[eZ^Zxl jZaebqghc l_iehijh\h^ghklvx L_eZ k fZehc l_iehijh\h^ghklvx gZau\Zxl l_iehbaheypbhggufb

Ijhp_kk dhg\_dpbb ijhbkoh^bl ebrv \ `b^dhklyo b ]ZaZo b ij_^klZ\ey_l kh[hc i_j_ghk l_iehlu \ j_amevlZl_ i_j_f_s_gby b i_j_f_rb\Zgby qZklbp `b^dhklb beb ]ZaZ Dhg\_dpby \k_]^Z khijh\h`^Z_lky l_iehijh\h^ghklvx

?keb i_j_f_s_gb_ qZklbp `b^dhklb beb ]ZaZ h[mkeh\e_gh jZaghklvx bo iehlghkl_c !

lh lZdh_ i_j_f_s_gb_ gZau\Zxl _kl_kl\_gghc dhg\_dpb_c Ijb _kl_kl\_gghc dhg\_dpbb gZ]j_lu_ h[t_fu l_iehghkbl_ey ih^gbfZxlky hoeZ^b\rb_ky himkdZxlky GZijbf_j HI kbkl_fu p_gljZevgh]h hlhie_gby khijbdZkZxlky k \ha^mohf i_j_^Z_lky hl ijb[hjZ \ ^jm]b_ qZklb ihf_s_gby

?keb `b^dhklv beb ]Za @ b = i_j_f_sZ_lky k ihfhsvx gZkhkZ lh lZdh_ i_j_f_s_gb_ gZau\Zxl \ugm`^_gghc dhg\_dpb_c L_iehh[f_g \ wlhf kemqZ_ ijhbkoh^bl agZqbl_evgh bgl_gkb\g__ q_f ijb _kl_kl\_gghc dhg\_dpbb

Ijhp_kk l_ieh\h]h baemq_gby khklhbl \ i_j_ghk_ l_iehlu hl h^gh]h l_eZ d ^jm]hfm we_dljhfZ]gblgufb \hegZfb \hagbdZxsbfb \ j_amevlZl_ keh`guo fhe_dmeyjguo b Zlhfguo \hafms_gbc EmqbklZy wg_j]by \hagbdZ_l \ l_eZo aZ kq_l ^jm]bo \b^h\ wg_j]bb ]eZ\guf h[jZahf l_ieh\hc We_dljhfZ]gblgu_ \hegu jZkijhkljZgyxlky hl ih\_joghklb l_eZ \h \k_ klhjhgu <klj_qZy gZ k\h_f imlb ^jm]b_ l_eZ

emqbklZy wg_j]by fh`_l bfb qZklbqgh ih]ehsZlvky ij_\jZsZykv kgh\Z \ l_iehlmm\_ebqb\ZyWl_eZ

L_iehijh\h^ghklv

AZdhg Nmjv_ ] y\eyxlky hkgh\guf aZdhghf l_iehijh\h^ghklb mklZgZ\eb\Zxsbf ijyfmx ijhihjpbhgZevgmx aZ\bkbfhklv f_`^m ih\_joghklghc iehlghklvx l_ieh\h]h ihlhdZ b l_fi_jZlmjguf ]jZ^b_glhf JUDG

T = −λ JUDG∂ W

]^_ dhwnnbpb_gl l_iehijh\h^ghklb <l fžK

AgZd fbgmk mdZau\Z_l gZ lh qlh \_dlhj l_ieh\h]h ihlhdZ gZijZ\e_g \ klhjhgm ijhlb\hiheh`gmx JUDG GWIjhba\h^gZy l_fi_jZlmju i_ji_g^bdmeyjgZy d bahl_jfbq_kdhc ih\_joghklb gZau\Z_lky

JUDGGW =

∂ W

]jZ^ f

 

 

∂ Q

Dhwnnbpb_gl l_iehijh\h^ghklb jZ\_g m^_evghfm l_ieh\hfm ihlhdm T ijb JUDG l_fi_jZlmju W \ ž& gZ _^bgbpm ^ebgu beb lhesbgu

Dhwnnbpb_gl l_iehijh\h^ghklb y\ey_lky \Z`ghc l_iehnbabq_kdhc oZjZdl_jbklbdhc \_s_kl\Z q_f [hevr_ dhwnnbpb_gl l_iehijh\h^ghklb l_f [hevr_c l_iehijh\h^ghklvx h[eZ^Z_l fZl_jbZe \_s_kl\h Dhwnnbpb_gl aZ\bkbl hl ijbjh^u \_s_kl\Z _]h kljmdlmju \eZ`ghklb gZebqby ijbf_k_c l_fi_jZlmjuWb ^jm]bo nZdlhjh\ < ijZdlbq_kdbo jZkq_lZo dhwnnbpb_gl kljhbl_evguo fZl_jbZeh\ g_h[oh^bfh ijbgbfZlv ih KGBI,, ´Kljhbl_evgZy l_iehl_ogbdZ´

JZkkfhljbf h^ghkehcgmx jbk iehkdmx kl_gdm lhesbghc / dhwnnbpb_gl l_iehijh\h^ghklb dhlhjhc ihklhyg_gFRQVWb jZ\_g

Jbk Lbohfbjh\ klj jbk Jbk Lbohfbjh\ klj jbk

JZkij_^_e_gb_ W \ h^ghkehcghc iehkdhc JZkij_^_e_gb_ W \ fgh]hkehcghc iehkdhc

kl_gd_ kl_gd_

L_fi_jZlmjZ W gZ ]jZgbpZo kl_gdb W b W ijbq_f W W L_iehlZ jZkijhkljZgy_lky lhevdh \^hev hkb;Ijb wlbo mkeh\byo l_fi_jZlmjgh_ ihe_ [m^_l h^ghf_jguf b bahl_jfbq_kdb_ ih\_joghklb ih\_joghklb h^bgZdh\uo l_fi_jZlmj [m^ml iehkdbfb b iZjZee_evgufb ih\_joghklyf kl_gdb

Ijhbgl_]jbjh\Z\ mjZ\g_gb_ Nmjv_ [_a \u\h^Z hdhgqZl_evgh ihemqbf

T =

λ

W − W <l f

δ

 

 

Ba mjZ\g_gby \b^gh qlh ih\_joghklgZy iehlghklv l_ieh\h]h ihlhdZ aZ\bkbl hl

l_fi_jZlmjgh]h i_j_iZ^Z ∆ W = W − W

l _ T =

λ

∆ W beb

δ

 

 

∆ W

 

 

∆ W

 

 

T =

 

 

=

 

δ

 

 

 

 

λ

5

 

 

 

 

 

 

 

 

 

 

 

Hlghr_gb_ / 5 gZau\Z_lky l_jfbq_kdbf khhlghr_gb_f kl_gdb AgZy ih\_joghklgmx iehlghklv l_ieh\h]h ihlhdZTfh`gh hij_^_eblv h[s__ dhebq_kl\h l_iehlu i_j_^Z\Z_fh_ aZ qZk q_j_a ih\_joghklv kl_gdb)ih nhjfme_

]^_ Gkj±kj_^gbc ^bZf_lj pbebg^jZ

4 = T )

Lh _klv h[s__ dhebq_kl\h l_iehlu i_j_^Z\Z_fh_ q_j_a h^ghkehcgmx iehkdmx kl_gdm ijhihjpbhgZevgh iehlghklb l_ieh\h]h ihlhdZ gZ f Tb iehsZ^b ih\_joghklb kl_gdb)

JZkkfhljbf l_iehijh\h^ghklv iehkdhc fgh]hkehcghc kl_gdb khklhys_c ba Q fZl_jbZevguo keh_\ iehlgh ijbe_]Zxsbo h^bg d ^jm]hfm jbk

DZ`^uc kehc bf__l aZ^Zggmx lhesbgm / b dhwnnbpb_gl l_iehijh\h^ghklb

Fgh]hkehcgufb y\eyxlky kl_gu b i_j_djulby djmighiZg_evguo b dbjibqguo a^Zgbc Ijb klZpbhgZjghf l_ieh\hf j_`bf_ l_ieh\u_ ihlhdb ijhoh^ysb_ q_j_a dZ`^uc ba keh_\ kl_gdb h^bgZdh\u

Ijhkmffbjh\Z\ l_ieh\u_ ihlhdb q_j_a dZ`^uc kehc fh`gh aZibkZlv [_a \u\h^Z

T

=

 

 

W − W

 

 

=

∆ W

 

δ

 

 

δ

 

 

δ

 

 

 

 

 

+

 

+

 

5

 

 

 

λ

 

λ

 

λ

 

 

 

 

 

 

 

 

 

 

 

]^_ ¨W±l_fi_jZlmjguc i_j_oh^ ¨W W W

5 5 5 5 « 5Q ±h[s__ l_jfbq_kdh_ khijhlb\e_gb_ fgh]hkehcghc kl_gdb jZ\gh_ kmff_ l_jfbq_kdbo khijhlb\e_gbc hl^_evguo keh_\ KmffZjguc l_ieh\hc ihlhd 4 <l q_j_a fgh]hkehcgmx iehkdmx kl_gdm hij_^_ey_lky ih nhjfme_

4 = W − WQ+ )

Q 5

L=

>ey jZkq_lZ l_ieh\h]h ihlhdZ q_j_a fgh]hkehcgmx beb h^ghkehcgmx pbebg^jbq_kdmx kl_gdb _keb lhesbgZ __ ih kjZ\g_gbx k ^bZf_ljhf G g_agZqbl_evgZ fh`gh ihevah\Zlvky nhjfmehc ^ey iehkdhc kl_gdb

Ijb wlhf iehsZ^v ) ih^kqblu\Z_lky )kj = π Gkj O

f_`^m gZjm`ghc b \gmlj_gg_c ih\_joghklyfb _]h kl_gdb

Dhg\_dlb\guc l_iehh[f_g

L_ieh\hc ihlhd 4 <l i_j_^Z\Z_fuc ijb dhg\_dlb\ghf l_iehh[f_g_ hij_^_ey_lky ih nhjfme_ GvxlhgZ

4 = α d ) W` − Wkl

]^_ .±dhwnnbpb_gl l_iehhl^Zqb

W`±l_fi_jZlmjZ l_iehghkbl_eyž&Wkl ±l_fi_jZlmjZ ih\_joghklb kl_gdbž&)±ih\_joghklv khijbdhkgh\_gby l_iehghkbl_ey kh kl_gdhc f

Ijbgy\ ) f ihemqbf ih\_joghklgmx iehlghklv l_ieh\h]h ihlhdZ T <l f

T = α W` − Wkl

Ba mjZ\g_gby fh`gh hij_^_eblv jZaghklv W` Wkl b l_fi_jZlmjm Wkl

=^_ .±l_jfbq_kdh_ khijhlb\e_gb_ kl_gdb ijb l_iehhl^Zq_ khijbdhkgh\_gb_f Nhjfmehc fh`gh ihevah\Zlvky dZd ijb l_iehhl^Zq_ hl `b^dhklb beb ]ZaZ @ beb = d kl_gd_ lZd b ijb l_iehhl^Zq_ hl kl_gdb d @ beb = `b^dhklb beb ]Zam

Dhwnnbpb_gl l_iehhl^Zqb . <l f ž&ij_^klZ\ey_l kh[hc dhebq_kl\h l_iehlu ijhoh^ys_c \ _^bgbpm \j_f_gb hl `b^dhklb ]ZaZ d kl_gd_ beb gZh[hjhl q_j_a f ih\_joghklb ) ijb jZaghklb W `b^dhklb ]ZaZ b kl_gdbž& ¨W` b kl ž&

Dhwnnbpb_gl l_iehhl^Zqb . \ hlebqbb hl ±hq_gv keh`gZy \_ebqbgZ dhlhjZy mqblu\Z_l ke_^mxsb_ nZdlhju h[kem`b\Zxsb_ ijhl_dZgb_ dhg\_dlb\gh]h l_iehh[f_gZ oZjZdl_j ^\b`_gby `b^dhklb beb ]ZaZ eZfbgZjgh_ beb lmj[me_glgh_ b ijbjh^Z _]h \hagbdgh\_gby kdhjhklv ^\b`_gby `b^dhklb beb ]ZaZ±9 nbabq_kdb_ iZjZf_lju @ beb =l_iehijh\h^ghklv \yadhklv iehlghklv ! l_iehijh\h^ghklv &S dhwnnbpb_gl h[t_fgh]h jZkrbj_gby W` b W] b Wkl nhjfm N b ebg_cgu_ jZaf_ju hfu\Z_fuo @ b = ih\_joghkl_c O O W «

LZdbf h[jZahf l h

α = I 9 µ ρ & S β W` Wkl N O O O

>ey hij_^_e_gby \_ebqbgu . ^ey jZaebqguo kemqZ_\ dhg\_dlb\gh]h l_iehh[f_gZ ij_^eZ]Zxlky wfibjbq_kdb_ nhjfmeu bf_xsb_ h^gZdh h]jZgbq_ggmx h[eZklv ijbf_g_gby AgZqbl_evgh emqrbc j_amevlZl ^Z_l hij_^_e_gb_ \_ebqbgu . k bkihevah\Zgb_f djbl_jb_\ ih^h[by±[_ajZaf_jguo khhlghr_gbc iZjZf_ljh\ oZjZdl_jbamxsbo nbabq_kdbc ijhp_kk

Ijb\_^_f g_dhlhju_ djbl_jbb ih^h[by ^ey hij_^_e_gby . b bo djZldmx oZjZdl_jbklbdm

1X =

 

α

O

djbl_jbc Gmkk_evlZ

 

 

 

λ

 

 

 

 

 

 

 

 

 

3U =

 

µ & S J

djbl_jbc IjZg^ley

 

 

 

 

 

 

 

 

 

 

λ

 

 

 

 

5H =

9 O ρ

 

djbl_jbc J_cghev^kZ

 

 

 

µ

 

 

 

 

 

 

 

 

 

*U =

 

J O

 

β

W djbl_jbc =jZk]hnZ

 

 

ν

 

 

 

 

 

 

 

 

 

IZjZf_lju \oh^ysb_ \ wlb \ujZ`_gby±kfhljb \ gZqZe_ dhgki_dlZ e_dpbb

DZdb_ `_ y\e_gby ijhp_kku oZjZdl_jbamxl wlb djbl_jbb"

Djbl_jbc Gmkk_evlZ 1X ±beb djbl_jbc l_iehhl^Zqb oZjZdl_jbam_l bgl_gkb\ghklv l_iehhl^Zqb gZ ]jZgbp_ `b^dhklv ]Za±l\_j^h_ l_eh b \k_]^Z y\ey_lky bkdhfhc \_ebqbghc

Djbl_jbc J_cghev^kZ 5H ±ij_^klZ\ey_l kh[hc hlghr_gb_ kbe bg_jpbb d kbeZf \gmlj_gg_]h lj_gby b oZjZdl_jbam_l ]b^jh^bgZfbq_kdbc j_`bf ^\b`_gby Ijb 5H

±eZfbgZjgh_ ijb 5H! ±lmj[me_glgh_ l_q_gb_

Djbl_jbc IjZg^ley 3U ±oZjZdl_jbam_l nbabq_kdb_ k\hckl\Z @ b = b kihkh[ghklv jZkijhkljZg_gby l_iehlu \ @ b =

Djbl_jbc =jZk]hnZ *U ±oZjZdl_jbam_l ih^t_fgu_ kbeu \hagbdZxsb_ \ @ beb = \ke_^kl\b_ jZaghklb iehlghklb bo qZklbp b \uau\Zxsbo lZd gZau\Z_fmx l g k\h[h^gmx dhg\_dpbx

< h[s_f kemqZ_ djbl_jbZevgZy aZ\bkbfhklv dhg\_dlb\gh]h l_iehh[f_gZ bf__l \b^

1X = I 5H*U 3U

>ey \gmlj_ggbo ih\_joghkl_c h]jZ`^Zxsbo dhgkljmdpbc HD hlZieb\Z_fh]h a^Zgby djbl_jbZevgh_ mjZ\g_gb_ bf__l \b^

1X = *U 3U

>ey hij_^_e_gby dhwnnbpb_glZ l_iehhl^Zqb ih mjZ\g_gbx ih^kqblu\Zxlky qbkeh\u_ agZq_gby djbl_jb_\ \oh^ysbo \ ijZ\mx qZklv jZ\_gkl\Z hij_^_ey_l djbl_jbc 1X b ba g_]h±dhwnnbpb_gl l_iehhl^Zqb α

L_iehh[f_g baemq_gb_f

L_iehh[f_g baemq_gb_f ijhbkoh^bl f_`^m l_eZfb gZoh^ysbfbky gZ [hevrbo beb [ebadbo jZkklhygbyo ^jm] hl ^jm]Z Ijbf_j baemq_gb_ KhegpZ gZ A_fex < l_ogbd_ l_iehh[f_g baemq_gb_f bf__l f_klh \ dhleZo kbkl_fZo hlhie_gby KH a^Zgbc \ kmrbevguo Z]j_]ZlZo b l i Ijb \ukhdbo l_fi_jZlmjZo l_iehh[f_g baemq_gb_f ij_h[eZ^Z_l gZ^ hklZevgufb \b^Zfb l_iehh[f_gZ b bf__l \ k\yab k wlbf \Z`gh_ agZq_gb_ Emq_bkimkdZgb_ l_ieZ \ ijhkljZgkl\_ fh`_l [ulv f [ jZ\ghf_jguf beb gZijZ\e_gguf EmqbklZy wg_j]by bkimkdZ_lky gZ dZdh_ eb[h l_ieh \ aZ\bkbfhklb hl _]h nbabq_kdbo k\hckl\ nhjfu b khklhygby ih\_joghklb ijb wlhf qZklbqgh ih]ehsZ_lky wlbf l_iehf b i_j_oh^bl \ l_ieh\mx beb ^jm]b_ nhjfu wg_j]bb Z hklZevgZy qZklv hljZ`Z_lky b qZklbqgh ijhoh^bl q_j_a g_]h jbk

Jbk Ko_fZ jZkij_^_e_gby emqbklhc wg_j]bb iZ^Zxs_c gZ l_eh

H[hagZqbf q_j_a 4 ±h[s__ dhebq_kl\h emqbklhc wg_j]bb iZ^Zxs_c gZ l_eh Z q_j_a 4$ 45 b 4'±khhl\_lkl\_ggh dhebq_kl\h emqbklhc wg_j]bb ih]ehs_gghc 4$ hljZ`_gghc45 b ijhr_^r_c q_j_a g_]h 4' L _

4 = 4$ + 45 + 4'

JZa^_eb\ h[_ qZklb jZ\_gkl\Z gZ 4 ihemqbf

4 = 4$ = 45 = 4' beb = $ + 5 + ' 4 4 4 4

]^_ :±ih]ehsZl_evgZy kihkh[ghklv l_eZ 5±hljZ`Zl_evgZy '±ijhimkdgZy

?keb : l _ 5 ' ±lh l_eh iheghklvx ih]ehsZ_l \k_ iZ^Zxsb_ gZ g_]h l_ieh\u_ emqb b gZau\Zxlky Z[khexlgh q_jgufb Ijbf_ju g_nlygZy kZ`Z :

\h^Z b e_^ : ±

?keb 5 ±lh l_eh gZau\Z_lky a_jdZevguf _keb hljZ`_gb_ ijZ\bevgh_ beb Z[khexlgh [_euf _keb hljZ`_gb_ jZkk_yggh_±^bnnmagh_ Ijbf_ju ahehlh f_^v±5 ±± hljZ`_gb_ a_jdZevgh_ [_eZy rlmdZlmjdZ±5±hljZ`_gb_ ^bnnmagh_

?keb' ±lh l_eh gZau\Z_lky Z[khexlgh ijhgbpZ_fuf ijhajZqguf beb ^bZl_jfbqguf Ijbf_ju \ha^mo±ijZdlbq_kdb ijhajZqgZy kj_^Z hdhggh_ kl_deh± ijhimkdZ_l k\_lh\u_ emqb b ihqlb g_ijhajZqgh ^ey mevljZnbhe_lh\uo b ^ebggh\hegh\uo \heg bgnjZdjZkgh]h baemq_gby

Hkgh\gu_ aZdhgu l_ieh\h]h baemq_gbyAZdhg Kl_nZgZ ;hevpfZgZ±mklZgZ\eb\Z_l k\yav baemqZl_evghc kihkh[ghklb Z[khexlgh q_jgh]h l_eZ k l_fi_jZlmjhc 7 .

( = & 7 .

]^_ K ±dhwnnbpb_gl baemq_gby Z[khexlgh q_jgh]h l_eZAZdhg EZf[_jlZ±fZdkbfZevgh_PD[baemq_gb_ _^bgbpu ih\_joghklb ihbkoh^bl i_ji_g^bdmeyjgh d g_c

4ϕ = 4 FRVϕ

Wlh \ujZ`_gb_ hij_^_ey_l aZ\bkbfhklv baemqZ_fhc l_ieh\hc wg_j]bb hl __ gZijZ\e_gby

AZdhg Dbjo]hnZ hij_^_ey_l qlh hlghr_gb_ baemqZl_evghc kihkh[ghklb l_eZ (d _]h ih]ehsZl_evghc kihkh[ghklb : ^ey \k_o l_e h^bgZdh\h b jZ\gh baemqZl_evghc kihkh[ghklb Z[khexlgh q_jgh]h l_eZ ? ijb lhc `_ l_fi_jZlmj_ W 7 b aZ\bkbl lhevdh hl

W 7 l _ ? : = ? = I 7

Dhjhldh jZkkfhljbf kemqZc l_iehh[f_gZ baemq_gb_f f_`^m ^\mfy kl_gdZfb bf_xsbfb [hevrmx ih\_joghklv b jZkiheh`_ggufb iZjZee_evgh gZ g_[hevrhf jZkklhygbb jbk

Jbk Ko_fZ l_ieh\uo ihlhdh\ f_`^m ^\mfy iZjZee_evgufb kl_gdZfb ijb l_iehh[f_g_ baemq_gb_f

 

= &

 

 

7

 

 

7

 

 

 

4

 

 

 

 

 

 

 

)

 

 

 

ij

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

]^_ &ij ijb\_^_gguc dhwnnbpb_gl baemq_gby <lf °K

Kij

=

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

+

 

+

 

 

 

 

 

 

 

 

 

 

 

 

 

K

K

 

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