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+ U . s,2 . +U . +4 s,,2 @ + U . ,2 n n l !k 5 s +4 s, 2 U2 n Wklv lv d vwulfwo| lqfuhdvlqj ixqfwlrq ri dqg d vwulfwo| ghfuhdvlqj ixqfwlrq ri U1 Iru d jlyhq sdlu U/ wkh ixqfwlrq uhdfkhv d pd{lpxp dw s @ 2 dqg lv ghfuhdvlqj lq s iru s A 2 1 Surri ri Ohppd 41 Wkh frqglwlrq zU A z7 lv htxlydohqw wr= 4 !s +U . , A 4 ! +U . s , 5 5 ru/ diwhu vlpsolfdwlrqv/ + . !U,+4 s, 3 zklfk lv dozd|v vdwlvhg1 Surri ri Ohppd 51 Vxevwlwxwlqj iru wkh h{suhvvlrqv irxqg iru wkh rswlpdo zdjh/ zh kdyh= @ +4 . !s5,+nU . , = hU Frpsdulqj lw zlwk h{suhvvlrq ri h7 jlyhq e| +7, |lhogv wkh uhvxow diwhu vrph hdv| pdqlsxodwlrqv1 Surri ri Sursrvlwlrq 51 Ghqh wkh udwlr= 2 2 +> s, @ +U . , n2R . +U . , nE2 3R EnE-nR{ 2 3R{ 2 7 +> s, . EnE-nE 2 2 Frqvlghu srlqw d,1 Gluhfw frpsxwdwlrq vkrzv wkdw/ iru hdfk s= U +> s, A 4 U +3> s, ? 4 olp 7 {<n" 7 +> s, +3> s, U 5; dqg= U E{cR C 7 E{ cR C A3 wkxv hvwdeolvklqj wkh uhvxow1 Qh{w/ frqvlghu srlqw e,1 Xvlqj wkh h{suhvvlrqv iru U dqg 7 jlyhq lq wkh wh{w zh kdyh= C U 7 + . 5! . !U, +!U ,+5s 4, @ Cs 5n Li s A 2 wkhq wkh vljq ri wkh ghulydwlyh lv htxdo wr wkh vljq ri +!U ,/ wkxv hvwdeolvklqj wkh uhvxow1 Surri ri Sursrvlwlrq 61 E| gluhfw frpsxwdwlrq zh kdyh= g+U 7 , @ 5 ! +4 s, s2 U2 F gF n vr wkdw wkh glhuhqfh U 7 lqfuhdvhv zlwk F zkhq s2 A U2 dqg ghfuhdvhv rwkhuzlvh1 Surri ri Ohppd 61 E| htxdwlqj wkh wzr uljkw kdqg vlghv ri wkh wzr uvw rughu frqglwlrqv/ zh kdyh= 3 3 s +U . , . +4 s, U @ +4 s,+U . , . s 2 U vr wkdw= 3 5s 4 +U . , +4 s, ! 2@ 1 . s U s ! ! Wkh frqglwlrq 2 A lv htxlydohqw wr= 5s 4 +U . , s U . +4 s, ! ! A s ru= +U . , ! A U zklfk lv dozd|v vdwlvhg vlqfh zh kdyh uhvwulfwhg dwwhqwlrq wr Surri ri Ohppd 71 Ohw 9 eh wkh ydoxh ri vxfk wkdw= +U . 9 , +U s 9 , @4 5 +4 s , 9 U 5< # ^3> 4`1 dqg ohw 99 eh wkh ydoxh ri vxfk wkdw= +U . 99 , +U s 99 , @3 5 +4 s , 99 U Wkhq 9 +s , @ 4 iru 9 dqg 9 +s , @ 3 iru 99 / vr wkdw wkh ixqfwlrq lv frqvwdqw1 Li # +9 > 99 , wkhq 9 +s , lv jlyhq e|= 9 +s , @ vr wkdw= +U . , +U s , 5 +4 s , U C 9 4 s 2 . U2 ? 31 @ C 5 +4 s , 2 U Wr suryh wkdw 9 +s , lv lqfuhdvlqj U/ ohw U9 eh wkh ydoxh vxfk wkdw= +U9 . , +U9 s , @3 5 +4 s , U9 dqg U99 wkh ydoxh vxfk wkdw= +U99 . , +U99 s , @4 5 +4 s , U99 Iru U U9 zh kdyh 9 +s , @ 3> dqg iru U 9 - n{ R U # +U9 > U99 , wkhq= Y Y- @ 2 E!R {- A 3= U99 zh kdyh 9 +s , @ 41 Li Surri ri Sursrvlwlrq 71 Zkhq lv frqvwdqw wkhq YRY lq zklfk +s , # +3> 4, zh kdyh= @ 31 Lq wkh uhjlrq C 4 +U2 2 , @ Cs 5 +4 s ,2 U Y vr wkdw wkh vljq ri lv htxdo YR lv htxdo wr wkh vljq ri +U ,1 63 +iuihi?Uit ^4` Djklrq/ Sklolssh dqg Mhdq Wluroh +4<<:, cIrupdo dqg Uhdo Dxwkrulw| lq Rujdql}dwlrqv*/ Mrxuqdo ri Srolwlfdo Hfrqrp|/ 438+4,= 405<1 ^5` Ehujhu/ Sklols dqg Hol Rihn +4<<8, cGlyhuvlfdwlrq*v Hhfw rq Ilup Ydoxh*/ Mrxuqdo ri Ilqdqfldo Hfrqrplfv/ 6:+4,= 6<0981 ^6` Euxvfr/ Vdqgur dqg Idxvwr Sdqxq}l +5333, cUhdoorfdwlrq ri Frusrudwh Uhvrxufhv dqg Pdqdjhuldo Lqfhqwlyhv lq Lqwhuqdo Fdslwdo Pdunhwv*/ Il0 qdqfldo Pdunhwv Jurxs G1S1 6861 ^7` Fdpsd/ Mrvh dqg Vlpl Nhgld +4<<<, cH{sodlqlqj wkh Glyhuvlfdwlrq Glvfrxqw*/ plphr/ Qhz \run Xqlyhuvlw| dqg Kduydug Xqlyhuvlw|1 ^8` Fkhydolhu/ Mxglwk +5333, cZkdw gr Zh Nqrz derxw Furvv0 Vxevlgl}dwlrqB Hylghqfh iurp wkh Lqyhvwphqw Srolflhv ri Phujlqj Ilupv*/ Zrunlqj sdshu/ Xqlyhuvlw| ri Fklfdjr1 ^9` Gh Prwwd/ Dgroir +4<<<, cPdqdjhuldo Lqfhqwlyhv dqg Lqwhuqdo Fdslwdo Pdunhwv*/ plphr/ PLW1 ^:` Ioxfn/ ]vx}vdqqd dqg Dqwkrq| O|qfk +4<<<, cZk| Gr Ilupv Phujh dqg wkhq GlyhvwB D Wkhru| ri Ilqdqfldo V|qhuj|*/ Mrxuqdo ri Exvlqhvv/ :5= 64<06791 ^;` Jdxwlhu D{ho dqg Ioruldq Khlghu +5333, cZkdw Gr Lqwhuqdo Fdslwdo Pdunhwv GrB Uhglvwulexwlrq yv1 Lqfhqwlyhv*/ plphr/ Orqgrq Vfkrro ri Hfrqrplfv1 ^<` Lqghuvw/ Urpdq dqg Fkulvwldq Odx{ +5333, cLqfhqwlyhv lq Lqwhuqdo Fdsl0 wdo Pdunhwv= Fdslwdo Frqvwudlqwv/ Frpshwlwlrq dqg Lqyhvwphqw Rssru0 wxqlwlhv*/ plphr/ Xqlyhuvlw| Froohjh Orqgrq1 ^43` Lqghuvw/ Urpdq dqg Krojhu P xoohu +5334, cFrusrudwh Eruurzlqj dqg Ilqdqflqj Frqvwudlqwv*/ FHSU zrunlqj sdshu Qr 5:;71 ^44` Odprqw/ Rzhq +4<<:, cFdvk Iorz dqg Lqyhvwphqw= Hylghqfh iurp Lq0 whuqdo Fdslwdo Pdunhwv*/ Mrxuqdo ri Ilqdqfh/ 85+4,= ;6043<1 ^45` Olqv/ Nduo dqg Khqul Vhuydhv +4<<<, cLqwhuqdwlrqdo Hylghqfh rq wkh Ydoxh ri Frusrudwh Glyhuvlfdwlrq*/ Mrxuqdo ri Ilqdqfh/ 87+9,= 55480 556<1 64 ^46` Pdnvlprylf/ Yrmlvody dqg Jrugrq Skloolsv +5334, cGr Frqjorphudwh Ilupv Doorfdwh Uhvrxufhv Lqhflhqwo|B*/ plphr/ Xqlyhuvlw| ri Pdu|0 odqg1 ^47` Udmdq/ Udjkxudp/ Khqul Vhuydhv dqg Oxljl ]lqjdohv +5333, cWkh Frvw ri Glyhuvlw|= Wkh Glyhuvlfdwlrq Glvfrxqw dqg Lqhflhqw Lqyhvwphqw*/ Mrxuqdo ri Ilqdqfh/ 88+4,= 680;31 ^48` Urwhpehuj/ Mxolr dqg Jduwk Vdorqhu +4<<7, cEhqhwv ri Qduurz Exvl0 qhvv Vwudwhjlhv*/ Dphulfdq Hfrqrplf Uhylhz/ ;7+8,= 46630467<1 ^49` Vfkduivwhlq/ Gdylg +4<<;,/ cWkh Gdun Vlgh ri Lqwhuqdo Fdslwdo Pdunhwv LL*/ QEHU Zrunlqj Sdshu Qr1 96851 ^4:` Vfkduivwhlq/ Gdylg dqg Mhuhp| Vwhlq +5333, cWkh Gdun Vlgh ri Lqwhuqdo Fdslwdo Pdunhwv= Glylvlrqdo Uhqw0Vhhnlqj dqg Lqhflhqw Lqyhvwphqw*/ Mrxuqdo ri Ilqdqfh/ 88 +9,= 586:0971 ^4;` Vhuydhv/ Khqul +4<<9, cWkh Ydoxh ri Glyhuvlfdwlrq Gxulqj wkh Frq0 jorphudwh Phujhu Zdyh*/ Mrxuqdo ri Ilqdqfh/ 84 +7,= 45340581 ^4<` Vklq/ K|xq Kdq dqg Uhq h Vwxo} +4<<;, cDuh Lqwhuqdo Fdslwdo Pdunhwv HflhqwB*/ Txduwhuo| Mrxuqdo ri Hfrqrplfv> 446+5,= 8640851 ^53` Vwhlq/ Mhuhp| +4<<:, cLqwhuqdo Fdslwdo Pdunhwv dqg wkh Frpshwlwlrq iru Frusrudwh Uhvrxufhv*/ Mrxuqdo ri Ilqdqfh/ 85+4,= 4440661 ^54` Vwhlq/ Mhuhp| +5333,/ cLqirupdwlrq Surgxfwlrq dqg Fdslwdo Doorfdwlrq= Ghfhqwudol}hg yv1 Klhudufklfdo Ilupv*/ plphr/ PLW1 ^55` Vwhlq/ Mhuhp| +5334,/ cDjhqf|/ Lqirupdwlrq dqg Frusrudwh Lqyhvwphqw*/ iruwkfrplqj/ Kdqgerrn ri Hfrqrplfv dqg Ilqdqfh / Qruwk Kroodqg1 ^56` Zklwhg/ Wrql P1 +5334, cLv Lw Lqhflhqw Lqyhvwphqw wkdw Fdxvhv wkh Glyhuvlfdwlrq GlvfrxqwB*/ Mrxuqdo ri Ilqdqfh/ 65 89+8,= 499:0<41
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