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L 7Zdbfh[jZ ahfjZkkfZljb\Z_f\heguk\hegh\ufqbkehf N gZijZ\e_ggu_hlbklhqgbdZiheyIZjZf_lj Ω y\ey_lkyl_e_kgufm]ehfih^dhlhjuf\b^_gbkke_^m_fuch[t_fbalhqdbgZ[ex^_gby>@ Hg^hihegy_lm]heijbgyluc\>@^h π ?keblhqdZgZ[ex^_gbygZoh^blky\gmljbh[t_fZ hdjm`_ggh]h ih\_joghklvx 6 lh Ω = π êy 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 GhjfZev d ih\_joghklb Q L gZijZ\e_gZ gZjm`m ijh\h^gbdZ < m]eh\hc lhqd_ ghjfZev fh`_l ijbgZê_`Zlvex[hcbakf_`guoih\_joghkl_c V = ; ρL Ω ρ U - Q + < Jbk=jZgbqgh_mkeh\b_ Jbk Ijbf_j dh]^Z lhqdZ gZ[ex^_gby L gZoh ^blky\h\gmlj_gg_fm]emih\_joghklb 6 lhdZ LZdbfh[jZahfbabihemqZ_fbgl_]jZevgh_mjZ\g_gb_hlghkbl_evghiehlghklb ∫ Q × - ρ × U ⋅ I NU G6 = - ρ L L 6 <_dlhj iehlghklb lhdZ gZ ih\_joghklb ijh\h^gbdZ jZaeh`bf ih hjlh]hgZevguf gZ 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_êydhfihg_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 <gmlj_ggxxaZ^Zqmwe_dljh^bgZfbdbfh`ghknhjfmebjh\Zlvhlghkbl_evgheb[hfZ] gblgh]h ihey eb[h we_dljbq_kdh]h ihey Bgl_]jZevgh_ \ujZ`_gb_ êy we_dljbq_kdh]h ihey \u]ey^bl[he__]jhfha^dhq_fêyfZ]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_êywe_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 ebjm_lkyZgZeh]bqghbgl_]jZevghfmmjZ\g_gbx k[he__keh`gufbij_h[jZah\Zgbyfb>Z e__h]jZgbqbfkyjZkkfhlj_gb_f[he__ijhklh]hbgl_]jZevgh]hmjZ\g_gbyihemq_ggh]hba jZkkfhlj_gbyfZ]gblgh]hihey :dkbZevghkbff_ljbqgZyaZ^ZqZ 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 JZkdeZû\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_jZihûgl_]jZevghcnmgdpbbijh\h^bfbgl_]jbjh\Zgb_g_ihihe ghfmm]emZ\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_juGhêykbff_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 / ]^_êy ( lbih\h[hagZqbf - = τ * = * τ êy + 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 êy ( 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_êywe_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 êyfZ]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_êy\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_`^mlhqdZfbêybkdhfhcnmgdpbb - ij_îheZ]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^\_dlhjuêbghc 1 >eyiehkdhcaZ^Zqbbêykbff_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_êbghc 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 pbbêyf_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_ogbdZkieZcgZiijhdkbfZpbbêyj_r_gbybgl_]jZevgh]hmjZ\g_gby <i_j\u_l_ogbdmdm[bq_kdh]hkieZcgZêyj_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 ±êbgZbgl_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 êy 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 êyiehkdhcaZ^Zqbba\ujZ`_gbcbebBg^_dk V mdZau\Z_lgZijbgZê_`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\êyg_kbff_ljbqguofh^dhe_[Zgbcbeblbih\êykbff_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_kdhevdbfbkieZcgZfbijbgZê_`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\ ijbgZê_`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 ijbgZê_`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 ihûgl_ ]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 Ihûgl_]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 ihûgl_]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 \ujZabfdZdZkbfilhlbdmihûgl_]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_êynmgdpbb=jbgZ dhevpZklhdhfIhemqbfihe_agu_nhjfmeuêy\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`_gbyêybgl_]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 Ihûgl_]jZevgu_ \ujZ`_gby b iehkdhcaZ^Zqbg_h[eZ^Zxlhkh[_gghklvx Ijb\uqbke_gbbbgl_]jZeh\\oh^ysbo\kmffmbkihevamxlkyd\Z^jZlmjgu_ nhjfmeu=ZmkkZ>@H[hagZqbfihûgl_]jZevgu_nmgdpbbq_j_a I τ b I τ ϕ >eyhl ^_evgh]h V ]hbgl_j\ZeZêbghc 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]hcihôh^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_êbg\_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 au\Z_lgZlhqlhfgbfu_b^_ckl\bl_evgu_qZklb_]hklZgh\ylkygme_\ufbg_ijbh^ghfb lhf`_agZq_gbb N Fh^mev^_l_jfbgZglZ\^Zebhlj_ahgZgkZ\_^_lk_[yfhghlhgghGh\[ebabj_ahgZgkZijb 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 [ebadbfd8h[jZaghfmkhkeZ[h\ujZ`_ggufbfbgbfmfZfbjbkLhqghklvhij_^_e_gby j_ahgZgkZijbwlhfiZ^Z_l Ohjhrbfbk\hckl\Zfbh[eZ^Z_lih\_^_gb_fgbfhcqZklb ,P λPLQ N F_`^mj_ahgZgkZfb 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_gbyfbbkihevamxlkyêyhij_^_e_gbyj_ahgZgkZ ,PλPLQ ( + GHW$N, úñ$ Jbk>\Z[ebadhjZkiheh`_gguoj_ahgZgkZPbebg^jbq_kdbcj_ahgZlhj5 f+ P :gZeblbq_kdb_qZklhlu )( = F=p ) G = F=p Qbke_ggucjZkq_l )( = F=p )G = F=p :gZeblbq_kdbcl_klêybeexkljZpbbf_lh^ZihbkdZj_ahgZgkguoqZklhl JZkkfhljbfh^ghf_jgucj_ahgZlhjh[jZah\Zgguc^\mfyiZjZee_evgufbijh\h^ysbfb iehkdhklyfbIh\_joghklguclhdiehlghklvx - b - gZijZ\e_g\^hevhkb ] jbk - - + + + + = < G ; JbkH^ghf_jgucj_ahgZlhjJZkkfZljb\Zxlkylhevdhiehkdb_\hegu Bgl_]jZevgZy nhjfmeZ êy fZ]gblgh]h ihey ijb\h^bl d kbkl_f_ ^\mo ebg_cguo mjZ\g_gbc + - ; + +; = − + ; + + ; = - Dhfihg_glufZ]gblguoihe_cgZiehkdhklyobba\_klguêyh^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 Ihemqbf^_l_jfbgZglbkh[kl\_ggh_agZq_gb_ 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 fhû 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 fhû fh`_l kem`blv dZjlbgZ jZkij_^_e_gby ih\_joghklghc iehl ghklblhdZGZjbkihdZaZgihjlj_lfhûpbebg^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 / / −êbgZh[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`_gbyêy\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_ êy \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^_eZlvêy 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\êy\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`^hfihûgl_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 êy \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`_gbyêydhfihg_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^blkyihnhjfmeZfNhjfmeuêyjZkq_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=jbgZêywe_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   ∂ U ∂ 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 Jbk KjZ\g_gb_ jZkkqblZgguo dZjlbg ihe_c \ h[t_f_ j_ahgZlhjZ b gZ ih \_joghklbLbidhe_[Zgbc ( \pbebg^jbq_kdhfj_ahgZlhj_DZjlbgZgZih \_joghklbkljhblky[ukljhbfh`_lkem`blvihjlj_lhffhû Ijh]jZffZ0$;:(//bl_klh\u_jZkq_lu GZ hkgh\_ baeh`_ggh]h f_lh^Z jZajZ[hlZgZ ijh]jZffZ k ]jZnbq_kdbf bgl_jn_ckhf jZ[hlZxsZyih^06'26beb:,1'2:6Ij_^_evgucjZaf_jfZljbpu × khhl\_lkl \m_lk_ld_kmaeZfbêyfh^kZabfmlZevgufb\ZjbZpbyfbbmaeZfbêykbff_ljbqguo fh^ Ijh]jZffZ kdZgbjm_l k rZ]hf ih qZklhl_ \ aZ^Zgghf ^bZiZahg_ qZklhl <j_fy jZkq_lZ fZljbpumdZaZggh]hij_^_evgh]hjZaf_jZbj_r_gb_kbFl_fumjZ\g_gbc\h^ghclhqd_qZklhlu khklZ\ey_lhlh^ghc^hg_kdhevdbofbgmlêy3HQWLXPF=pbaZ\bkblhllhqghklbbgl_]jb jh\Zgby hij_^_ey_fhc\u[hjhfihjy^dZd\Z^jZlmjguonhjfme=ZmkkZ >eyhij_^_e_gby j_ahgZgkghc fhû g_h[oh^bfh ijhba\_klb jZkq_l \−lhqdZoqZklhluKmf_gvr_gb_fjZa f_jghklb k_ldb \j_fy jZkq_lZ khdjZsZ_lky d\Z^jZlbqgh >ey ]_hf_ljbb ijhklhc nhjfu pb ebg^jZ kn_ju ^hklZlhqgh k_ldb k − maeZfb b \j_fy jZkq_lZ \ h^ghc lhqd_qZklhlukh klZ\ey_l g_kdhevdh k_dmg^ K_ldZ k maeZfb gZ ih\_joghklb khhl\_lkl\m_l ijbf_jgh ijh kljZgkl\_gghc k_ld_ k lukyqZfb maeh\ êy ijh]jZff 850(/7 683(5),6+ beb 6XSHU/DQV>@ L_klh\u_ jZkq_lu ijh\h^bebkv êy kn_jbq_kdh]h b pbebg^jbq_kdh]h j_ahgZlhjh\ Z lZd`_ êy pbebg^jbq_kdh]h b ijyfhm]hevgh]h \hegh\h^h\ Ih\_^_gb_ ^_l_jfbgZglZ b kh[kl \_ggh]hagZq_gbyfZljbpu\^bZiZahg_qZklhlho\Zlu\Zxs_f^bihevguofh^pbebg^jbq_ kdh]hj_ahgZlhjZihdZaZghgZjbk ,PλPLQ GHW$N, 5HλPLQ úñ$ JbkJ_amevlZljZkq_lZpbebg^jbq_kdh]hj_ahgZlhjZ 5 = P + = P >bihevgu_ fhû P = J_ahgZgkgu_qZklhlukhhl\_lkl\mxldhjgyfnmgdpbb ,P λPLQ H[eZklvk[ebadh jZkiheh`_ggufbfh^Zfb\jZchg_F=pihdZaZgZ[he__ih^jh[ghgZjbk >eybeexkljZpbblhqghklb\lZ[ebp_ijb\_^_guj_amevlZlujZkq_lh\j_ahgZgkguoqZk lhl^bihevguofh^kn_jbq_kdh]hj_ahgZlhjZjZ^bmkhf kfkcfhûihxbkjZ\g_ gb_bokZgZeblbq_kdbfbagZq_gbyfbK_ldZ 1 = Hrb[dZg_ij_\urZ_l − Ghf_jfhû :gZeblbq_kdh_ agZq_gb_ qZklhlu F=p JZkq_lF=p Ijh\h^behkvkjZ\g_gb_lhqghklbjZkq_lh\j_ahgZgkguoqZklhl^hfh^ba\_klgu fbijh]jZffZfb850(/76XSHU/DQVêyg_kbff_ljbqguofh^b6XSU)LVKêykbff_ljbqguo fh^jbkIjbmdZaZgguogZjbkmgd_jZaf_jZok_lhd\uqbkebl_evgu_j_kmjkub\j_fyjZk q_lh\jZagufbijh]jZffZfb[ueb[ebadbLhqghklvjZkq_lh\j_ahgZgkguoqZklhlpbebg^jb q_kdh]hj_ahgZlhjZkhklZ\beZ − kn_jbq_kdh]h− − Lhqghklvg_iZ^ZeZkm\_ebq_gb_f qZklhlubghf_jZfhûijbjZaf_jZok_ldb 1 = − maeh\ 6XSHU)LVK 1 = × /RJ))) 6XSHU/DQV 1 = × 6XSHU/DQV 850(/7 1 = × 13 = /RJ))) 0$;:(// 1 = 0$;:(// 1 = )úñ$ )0ñ$ Kbff_ljbqgu_fhû P = G_kbff_ljbqgu_fhû P = Pbebg^jbq_kdbcj_ahgZlhj 5 = f / = f 6XSHU )LVK 1 /RJ)) ) 6XSHU/DQV 1 = × 6XSHU/DQV 1 = × 0$;:(// 1 = 13 = = × 850(/7 /RJ) )) 0$;:(// 1 = )0ñ$ )úñ$ Kbff_ljbqgu_fhû P = G_kbff_ljbqgu_fhû P = Kn_jbq_kdbcj_ahgZlhj 5 = kf JbkLhqghklvijh]jZff850(/76XSHU)LVK6XSHU/DQVb0$;:(//DZ`^ZylhqdZkhhl\_l kl\m_lj_ahgZgkghcqZklhl_ ) :gZeblbq_kdh_agZq_gb_qZklhlu ) IhdZaZgujZaf_juk_lhd Mf_gvr_gb_lhqghklbjZkq_lh\j_ahgZgkghcqZklhlukmf_gvr_gb_fjZaf_jZk_ldbih dZaZghgZjbkG_dhlhju_\ukrb_lbiudhe_[ZgbcklZgh\ylkyg_jZaebqbfuWlhijhbkoh ^blmlZdbolbih\êydhlhjuoqbkeZlhq_dkieZcgZg_^hklZlhqghêyZiijhdkbfZpbbj_adh baf_gyxsbokyihe_c\^hevdZdh]heb[hmqZkldZih\_joghklb\^ZgghfkemqZ_\^hevlhjpZbeb h[_qZcdbpbebg^jZ>eygbarbolbih\dhe_[ZgbclhqghklvhklZ_lky\ukhdhc^Z`_ijbjZaf_ jZok_ldb± <ujh`^_ggu_fhûij_^klZ\eyxljZkq_lgu_ljm^ghklbêy\kydh]hf_lh^Z\uqbke_gbc Ki_pbnbdZijh]jZffu0$;:(//aZdexqZ_lky\kdZgbjh\ZgbbihqZklhl_\aZ^Zgghf^bZiZ ahg_?kebrZ]kdZgbjh\Zgby[hevr_jZkklhygbyf_`^mqZklhlZfb[ebadbofh^lh_klvhiZk ghklv ijhkdhqblv fh^m MdZaZgb_f gZ gZebqb_ [ebadbo fh^ kem`bl ih\_^_gb_ fh^mey ^_l_j fbgZglZfZljbpu\aZ\bkbfhklbhlqZklhluNmgdpbyi_j_klZ_l[ulv9h[jZaghc\[ebabj_ ahgZgkZbklZgh\blky[ebadhcd8h[jZaghc <ur_ gZjbk ihdZaZgijbf_j^\mo[ebadbofh^\pbebg^jbq_kdhfj_ahgZlhj_5 f/ f<lZdbof_klZog_h[oh^bfhkdZgbjh\Zlvk[he__f_edbfrZ]hfihqZklhl_ 1 /RJ)) ) /RJ)) ) 1 )úñ$ )úñ$ JbkIZ^_gb_lhqghklb0$;:(//ijbmf_gvr_gbbjZaf_jZk_ldbG_dRlhju_\ukrb_lbiudhe_ [ZgbcklZgh\ylkyg_jZaebqbfuhj^bgZlZjZ\gZPbebg^jbq_kdbcj_ahgZlhj 5 = f / = f G_kbff_ljbqgu_lbiu P = Lhqdbkhhl\_lkl\mxlj_ahgZgkgufqZklhlZf ) AZdexq_gb_ F_lh^ bgl_]jZevguo mjZ\g_gbc êy ^\mf_jguo aZ^Zq ^Ze \hafh`ghklv bkihevah\Zlv ZiijhdkbfZpbxj_r_gbyh^ghf_jgufkieZcghfWlhgZjy^mkZddmjZlgufbgl_]jbjh\Zgb_fb f_lh^hfjZkq_lZj_ahgZgkghcqZklhluiha\hebehihemqblv\ukhdmxlhqghklvj_r_gby\l_k lh\uoaZ^ZqZoAZiZkihlhqghklb^Z_lgZ^_`ghklvêyZgZebaZj_ahgZgkh\h[eZkl_cbf_xsbo keh`gmx]_hf_ljbx G_ij_^klZ\ey_l[hevrhcljm^ghklbgZjy^mkwe_dljbq_kdbfblhdZfb\\_klb\jZkkfhl j_gb_fZ]gblgu_lhdbbkhhl\_lkl\_ggh]jZgbqgh_mkeh\b_êywe_dljbq_kdh]hiheyb^Ze__ gZwlhchkgh\_\\_klb\aZ^Zqmebg_cgu_^bwe_dljbdbbn_jjhfZ]g_lbdb Wlhl ihôh^ ]h^blky lZd`_ êy j_r_gby aZ^Zqb baemq_gby êy q_]h g_h[oh^bfh ^h[Z \blvj_abklb\gu_]jZgbqgu_mkeh\byb]jZgbqgu_mkeh\byêybaemqZl_eyAZ^Zqm`_baemq_ gbyhldjuluoZgl_ggbgl_]jZevguff_lh^hflhevdhbfh`ghj_rZlv AZfZgqb\uf y\ey_lky ijbf_gblv f_lh^ bgl_]jZevguo mjZ\g_gbc êy f_jguo aZ^Zq we_dljh^bgZfbdbbkihevamxêyZiijhdkbfZpbbj_r_gby^\mf_jguckieZcgbeb^jm]mx^\m f_jgmxZiijhdkbfZpbx\ukhdh]hihjy^dZ Ijbeh`_gb_ AZibr_f\ujZ`_gbyêy_^bgbqguo\_dlhjh\\^_dZjlh\hckbkl_f_dhhj^bgZl = 7; L + 7< M + 7= N U = 5; L + 5< M + 5= N ! = Φ ; L + Φ< M + Φ = N L L L L = 7; L + 7< M + 7= N I !L = Φ L; L + Φ<L M + Φ L= N QL = 1 ;L L + 1<L M + 1 =L N <ujZabfdhfihg_glu_^bgbqguo\_dlhjh\q_j_abodhhj^bgZlu\pbebg^jbq_kdhckbk 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 = 1< L = 1= L = FRV γ 5 L L GZijZ\eyxsb_ dhkbgmku we_f_glZ ih\_joghklb d hkyf 5 b = h[hagZq_gu FRV γ 5 FRV γ = FRV γ 5L FRV γ 5L jbkBg^_dk L khhl\_lkl\m_llhqd_gZ[ex^_gbykdhhj^bgZlhc τ L Dhhj^bgZlu khhl\_lkl\mxsb_ lhqd_ bklhqgbdZ 5 = 5 τ = = = τ b dhkbgmku m]eh\ 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_ lhqd_ gZ[ex^_gby FRV γ 5 L = FRV γ = τ L _klv nmgdpbb τ L M]he ϕ hlkqblu\Z_lky hl m]eh\hc dhhj^bgZlu lhqdb gZ[ex^_gbyêydhlhjhcijbgylh ϕ L = Kibkhdebl_jZlmju >@ &RPSXWHU &RGHV IRU 3DUWLFOH $FFHOHUDWRU 'HVLJQDQG$QDO\VLV$&RPSHQGLXP/$85/RV$OD PRV$FFHOHUDWRU&RGH*URXS/RV$ODPRV6HFRQG(GLWLRQ0D\ >@ . +ROEDFK 5) +ROVLQJHU 683(5),6+ ± $ &RPSXWHU 3URJUDP IRU (YDOXDWLRQ RI 5) &DYLWLHV ZLWK &\OLQGULFDO6\PPHWU\3DUWLFOH$FFHOHUDWRUV9ROSS >@ %0)RPHOHWDO$QHZFRGHIRUHYDOXDWLRQRIWKHHOHFWURPDJQHWLFILHOGVDQGUHVRQDQFHIUHTXHQFLHVRID[L V\PPHWULF5)FDYLWLHV3DUWLFOH$FFHOHUDWYS >@ :>=jb]hjv_\b^jWe_dljhggZyl_ogbdZK_jWe_dljhgbdZK<Q\uik >@ :=>Zcdh\kdbcb^jIj_ijbglBN<WIjhl\bgh >@ 0 GH -RQJ ) $GDPV&DYLW\ 5) 0RGH $QDOLV\V8VLQJD%RXQGDU\,QWHJUDO0HWKRG3URFRIWKH 3DUWLFOH$FFHOHUDWRU&RQI3$&9ROSS >@ <YB\Zgh\FFDZjebg_j<?L_jy_\<IYdh\e_\Ijbf_g_gb_f_lh^Z]jZgbqguobgl_]jZev guomjZ\g_gbcêyjZkq_lZ\ukhdhqZklhlguoj_ahgZlhjh\@mjgZe\uqbkebl_evghcfZl_fZlbdbb fZl_fZlbq_kdhcnbabdb<uiFk >@ <YB\Zgh\<IBevbgJ_r_gb_kf_rZgguodjZ_\uoaZ^ZqêymjZ\g_gbyEZieZkZf_lh^hfbgl_ ]jZevguo mjZ\g_gbc Lbih\u_ ijh]jZffu j_r_gby aZ^Zq fZl_fZlbq_kdhc nbabdb Kb[bjkdh_ hl^_ e_gb_:GKKKJGh\hkb[bjkd >@ :Ih^`h?Fbee_j<dg<uqbkebl_evgu_f_lhû\we_dljh^bgZfbd_FFbjk >@ <<GbdhevkdbcWe_dljh^bgZfbdZbjZkijhkljZg_gb_jZ^bh\hegFGZmdZ >@ :N<_jeZgv<KKbabdh\Bl_]jZevgu_mjZ\g_gbyDb_\GZmdh\Z>mfdZ >@ :>=jb]hjv_\<;Ygd_\bqJ_ahgZlhjubj_ahgZlhjgu_aZf_êyxsb_kbkl_fuK<QFJZ^bhb k\yav >@ <K<eZ^bfbjh\MjZ\g_gbyfZl_fZlbq_kdhcnbabdbFGZmdZ >@ E>EZg^Zm?FEbnrbpWe_dljh^bgZfbdZkiehrguokj_^FGZmdZk >@ >`NhjkZclFFZevdhevfDFhme_jFZrbggu_f_lhûfZl_fZlbq_kdbo\uqbke_gbcFFbj >@ Mbedbgkhg JZcgr KijZ\hqgbd Ze]hjblfh\ gZ yaud_ :E=HE Ebg_cgZy Ze]_[jZ F FZrbgh kljh_gb_ >@ F:[jZfh\bpBKlb]ZgKijZ\hqgbdihki_pbZevgufnmgdpbyfFGZmdZk JmdhibkvihklmibeZbxey]h^Z <?L_jy_\ JZa\blb_f_lh^Zbgl_]jZevguomjZ\g_gbcêyjZkq_lZK<Qj_ahgZlhjh\ Hjb]bgZefZd_lih^]hlh\e_gkihfhsvxkbkl_fu:25' J_^ZdlhjG<?`_eZL_ogbq_kdbcj_^ZdlhjG<Hjeh\Z BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB IhîbkZghdi_qZlbNhjfZloHnk_lgZyi_qZlv I_qeMqbaêLbjZÀZdZaBg^_dk EJ BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB =GPJNBgklblmlnbabdb\ukhdbowg_j]bc Ijhl\bghFhkdh\kdhch[e Bg^_dk IJ?IJBGLBN<W

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