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Lq fdvh ri E?c ?2 ' E&c & fdsdflwlhv duh fkrvhq nqrzlqj wkh uhdol}dwlrq S1 Dv0 vxplqj wkdw dozd|v wkh sulfhv R E&c &2 uhvxow +vhh wkh vhfwlrq d,/ e, dqg f, deryh, vhoohu *v surw ghshqgv rq & dv iroorzv=

+LLL1g14, E&c & ' E & & & S&

iru ' c 2 dqg 9' 1 Wkh htxloleulxp fkrlfhv duh +LLL1g15, &^W ' ^3S iru ' c 2/

l1h1 wkh zhoo0nqrzq gxrsro| vroxwlrq +Frxuqrw/ 4;6;,1 Vlqfh S wkh rswlpdo fdsdflwlhv duh qrq0qhjdwlyh1 Wkh surw h{shfwdwlrqv uhvxowlqj iurp +LLL1g15, duh

+LLL1g16, .^3S ²iru ' c 2

zkhuh . ij ghqrwhv wkh h{shfwdwlrq rshudwru1

h, Wkh fdvh

^E?^{c ?}²^{' Egc g}

Khuh S lq +LLL1g15, pxvw vlpso| eh vxevwlwxwhg e| lwv dyhudjh S vr wkdw

;

(10)

+LLL1h14, &^W ' ^3S iru ' c 2

jxdudqwhhlqj d zhoo0ghqhg vroxwlrq iru doo uhdol}dwlrqv ri 5 ^k₂c ^l dqg doo glv0 wulexwlrqv ri rq^k₂c ^l1 Wkh surw h{shfwdwlrqv uhvxowlqj iurp +LLL1h14, duh

+LLL1h15, ^3S ² iru ' c 2

zklfk duh reylrxvo| qrq0qhjdwlyh vlqfh S 5^k₂c ^l1

i, Wkh fdvh

^E?^{c ?}² ' Egc &

Dvvxph wkdw vhoohu 4 grhv qrw |hw nqrz S zkhuhdv vhoohu 5 grhv1 Iurp +LLL1i14, 2E&c &2 ' E & &2 &2 S&2

rqh rewdlqv vhoohu 5*v uhdfwlrq ixqfwlrq

+LLL1i15, &^W₂E& '

;A

? A=

3S3&

2 iru f & S 3 iru & : S

Lq rughu wr ghwhuplqh vhoohu 4*v rswlpdo fdsdflw| ohyho &^W zh uvw kdyh wr fdofxodwh klv h{shfwhg surw dv d ixqfwlrq ri &1 Wklv ixqfwlrq . iZE&j ghshqgv fuxfldoo|

rq wkh glvwulexwlrq ri 1 Lq wkh iroorzlqj zh zloo fduu| rxw wkh dqdo|vlv iru ydulrxv glhuhqw glvwulexwlrqv1 Iluvw zh h{dplqh wkh fdvh lq zklfk wkh glvwulexwlrq ri kdv d frqwlqxrxv Ohehvjxh0ghqvlw| dqg vroyh wkh vshfldo fdvh ri d xqlirup glvwulexwlrq h{solflwo|1 Wkhq/ dv dq h{dpsoh iru phdvxuhv zlwk qlwh vxssruw/

zh vwxg| rqh0srlqw phdvxuhv dqg jlyh rqh qxphulfdo h{dpsoh iru d wzr0srlqw phdvxuh1

i4, Frqwlqxrxv glvwulexwlrqv

Lq wklv fdvh vhoohu 4*v h{shfwhg surw lv

<

(11)

+LLL1i16, . iE&j '

;A AA AA AA

? AA AA AA A=

3&U

2

& ^3S3&₂ &) ES _Sn

U

3& E & &) ES _S &

U

2

S) ES _S

zkhuh ) E ghqrwhv wkh ghqvlw| ri rq ^k₂c ^l dqg [ E lwv glvwulexwlrq zlwk [ES ' ) ES dqg [₂' f/ [ E ' 1 Htxdwlrq +LLL1i16, fdq eh uhzulwwhq dv

+LLL1i16*, . iE&j '

;A AA AA A? AA AA AA

=

3&U

2

3&

2 &) ES _S n ^&₂ ^3&^U

2

S) ES _S

n E & &

U

3& ) ES _S S&

ru

+LLL1i16, . iE&j '

;A AA A? AA AA

=

^3&₂ &[ E & n ^&₂ ^3&^U

2

S) ES _S n E & & S&

1

Iurp wkh uvw rughu frqglwlrq ri d orfdo pd{lpxp ri +LLL1i16, rqh rewdlqv

+LLL1i17, 2&₂ &

[ E & n ₂ ^3&^U

2

S) ES _S ' S

dv dq lpsolflw irupxod iru wkh lqwhulru pd{lpxp &^W1 Pruhryhu _²

_&². iZE&j ' 2 [ E & &

2) E & S f iru doo & 5fc 2

1

Wkxv . iZE&j lv vwulfwo| frqfdyh/ l1h1 wkh uvw rughu frqglwlrqv duh qhfhvvdu|

dqg vx!flhqw iru d joredo pd{lpxp/ dqg wkh ohiw0kdqg vlgh ri htxdwlrq +LLL1i17, lv vwulfwo| ghfuhdvlqj1 E| lqvhuwlqj & ' f dqg & ' ₂ lqwr +LLL1i17, rqh fdq qrz hdvlo| vkrz wkdw/ iru hyhu| frqwlqxrxv glvwulexwlrq ri / wkh htxdwlrq kdv d xqltxh vroxwlrq lq ^kf(₂^l1 Iru wkh vshfldo fdvh ri wkh xqlirup ghqvlw| ) E rq ^k₂c ^l/ l1h1 [ ES ' 2S ₂ iru ₂ S / wklv lpsolhv/ iru lqvwdqfh/

+LLL1i18, &²n ²& ' ₂^. ²S ' ₂^. ² _e ' ₂ 43

(12)

ru

+LLL1i18*, &^W ' n^I^2D32eS_S ' ^I^.32_S ' f.S2D21

Ohw ^W ghqrwh wkh surw h{shfwdwlrq ri vhoohu 4 uhvxowlqj iurp &^W dqg wkh vxevh0 txhqw fkrlfh &₂^WE&^W e| vhoohu 51 Ohw/ ixuwkhupruh/ ^W₂ eh wkh fruuhvsrqglqj surw h{shfwdwlrq ri vhoohu 5 zkr lv vhfrqg lq ghwhuplqlqj wkh vdohv fdsdflw|1 Iru wkh vshfldo fdvh ri wkh xqlirup ghqvlw| ) E rq ^k₂c ^l zh rewdlq

+LLL1i19, . i^Wj ' &^W^kE &^W n ^&₂^W^._H^l fff.

dqg

+LLL1i1:, . i^W₂j ' E &^W^k_H _SE &^W₂ n &^W^l_eH ffc

l1h1 li suhhpswlrq wdnhv sodfh/ rq dyhudjh/ wkh vhoohu/ zkr uvw ghwhuplqhv klv fdsdflw|/ lv zruvh r1

i5, Rqh0srlqw phdvxuhv

Zh qrz orrn dw wkh vlwxdwlrq zkhuh wkh glvwulexwlrq ri lv d rqh0srlqw phdvxuh/

l1h1 zkhuh wkh frvw sdudphwhu S lv qr orqjhu udqgrp1 Reylrxvo| wklv fdvh lv htxlydohqw wr wkh fodvvlfdo +yrq Vwdfnhoehuj/ 4<67, ohdghuvkls prgho1

Pd{lpl}lqj vhoohu 4*v h{shfwhg surw ixqfwlrq

. iZE&j '

+ E & &^W₂E& & S & li f & S E & & S & li & : S

ohdgv wr

+LLL1i1;, &^W ' ^3S₂ +LLL1i1<, &^W₂ ' ^3S_e

44

(13)

+LLL1i143, . iZ^Wj ' ^E3S_H ² dqg

+LLL1i144, . iZ^W₂j ' ^E3S_S ²

Fohduo|/ lq frqwudvw wr wkh fdvh ri d xqlirup glvwulexwlrq/ wkh srvlwlrq ri wkh ohdghu lv pruh surwdeoh wkdq wkh rqh ri wkh iroorzhu1

i6, Wzr0srlqw phdvxuhv

Dv d qdo h{dpsoh zh dqdo|}h wkh fdvh zkhuh fdq kdyh h{dfwo| wzr ydoxhv S

dqg S2 zlwk S S21 Vlqfh vroylqj wkh pd{lpl}dwlrq sureohpv iru duelwudu| wzr0 srlqw phdvxuhv lqyroyhv d frpsoh{ fdvh glvwlqfwlrq/ zh uhvwulfw rxuvhoyhv wr rqh qxphulfdo h{dpsoh1 Ohw wkh glvwulexwlrq ri eh wkh phdvxuh dvvljqlqj suredelolw|

*2 wr wkh qxpehuv S ' fSb dqg S2 ' fHc uhvshfwlyho|1 Wkh h{shfwhg surw ri vhoohu 4 lv

+LLL1i145, . iZE&j '

;A A? AA

=

&

2 i & Sj li S2 &

&^q_eE & n _eS S^r li S2 : & dqg S &

& i & Sj li S : &

E| pd{lpl}lqj wklv ixqfwlrq zh rewdlq +LLL1i146, &^W ' _H

+LLL1i147, &₂^W '

+ ffb2D li S ' S

ff2D li S ' S2

+LLL1i148, . iZ^Wj ' _2H fff.H +LLL1i149, . iZ^W₂j fffeH

Khuh djdlq/ dv lq wkh fdvh ri rqh0srlqw phdvxuhv/ wkh vhoohu zkr suhhpswv lv pruh vxffhvvixo wkdq klv rssrqhqw1

45

(14)

71 Wkh wuxqfdwhg ru hyroxwlrqdu| jdph

Lq jhqhudo/ wkh pdunhw ghflvlrq surfhvv lq Iljxuh 4 doorzv iru irxu frqvwhoodwlrqv ri lqglylgxdo wlplqj glvsrvlwlrqv/ qdpho| Egc / Egc 6/ E&c / dqg E&c 61 Khuh wkh wdvn ri ghulylqj wkh vwdeoh wlplqj glvsrvlwlrqv lv/ krzhyhu/ uhgxfhg wr wkh sureohp zkhwkhu erwk zloo eh ri w|sh g ru & ru zkhwkhu wkh elprusklvpv zlwk rqh ehlqj ri w|sh g dqg wkh rwkhu ri w|sh & duh vwdeoh1 Wklv lv lpsolhg e| rxu uhvwulfwlrq wr d sdudphwhu uhjlrq zkhuh htxloleulxp sulfhv rq wkh vhfrqg vwdjh +ri wkh Nuhsv0Vfkhlqnpdq0prgho, lqgxfh ixoo fdsdflw| xwlol}dwlrq/ l1h1 R^W ' R E&c &2 iru ' c 21 Li wklv krogv iru doo frqvwhoodwlrqv E&c &2 ri fdsdflwlhv/ lw reylrxvo|

grhv qrw pdwwhu zkhwkhu rqh lv hduolhu lq sulfh vhwwlqj dqg zkhwkhu wklv lv grqh ehiruh ru diwhu wkh sdudphwhu _ lv udqgrpo| ghwhuplqhg1

Iru vwdeoh wlplqj prqrprusklvpv Eg^Wc g^W ru E&^Wc &^W ru vwdeoh wlplqj elpru0 sklvpv Eg^Wc &^W ru E&^Wc g^W zh rhu wzr +uhodwhg, lqwhusuhwdwlrqv= Dffruglqj wr lqgluhfw hyroxwlrq vwdelolw| phdqv wkdw wlplqj glvsrvlwlrqv uhjduglqj fdsdflwlhv duh qrw frqvflrxvo| fkrvhq/ exw udwkhu hyroyh1 Wkh vwdeoh wlplqj frqvwhoodwlrq lv wkxv ylhzhg dv wkh qdo uhvxow ri dq hyroxwlrqdu| surfhvv1

Hqgrjhqrxv wlplqj dvvxphv lqvwhdg wkdw wlplqj glvsrvlwlrqv duh frqvflrxvo| dqg lqghshqghqwo| ghwhuplqhg ehiruh vwdjh g lq Iljxuh 4 dqg wkhq sxeolfo| dq0 qrxqfhg1 Wkxv wkh iroorzlqj v|pphwulf 50shuvrq pdwul{ jdph lv wkh wuxqfdwlrq/

l1h1 wkh jdph ghvfulelqj vhoohu 4*v lqfhqwlyhv iru doo irxu frqvwhoodwlrqv E&c &/

E&c g/ Egc &/ dqg Egc g e| dqwlflsdwlqj wkh udwlrqdo ghflvlrq ehkdylru dv gh0 ulyhg lq vhfwlrq 6 deryh1 Lq rwkhu zrugv= Wkh hqwulhv ri Wdeoh 5 duh mxvw wkh h{shfwhg sd|rv ri vhoohu 4 uhvxowlqj iurp wkh rswlpdo fdsdflw| yhfwruv E&^Wc &^W₂ dqg wkh uhvxowlqj htxloleulxp sulfhv R^W ' R^W₂ ' R E&^Wc &₂^W iru wkh irxu srvvleoh frqvwhoodwlrqv E&c &/ E&c g/ Egc &/ dqg Egc g ri wlplqj glvsrvlwlrqv1

46

(15)

vhoohu 5

& g

vhoohu 4

& ^.i^E3S²j

b . i^W₂j

g . i^Wj ^E3S_b ²

Wdeoh 5= Wkh v|pphwulf hyroxwlrqdu| jdph ru wuxqfdwlrq dv ghqhg e| ^W

Lq ylhz ri lqgluhfw hyroxwlrq wkh v|pphwulf pdwul{ jdph lq Iljxuh 5 lv dq hyr0 oxwlrqdu| jdph ghvfulelqj wkh +uhsurgxfwlyh, vxffhvv ri pxwdqwv & ru g zkhq frqiurqwlqj dqrwkhu lqglylgxdo zklfk fdq eh ri w|sh & ru g/ wrr +vhh Kdpphu0 vwhlq dqg Vhowhq/ 4<<7/ iru d vxuyh| ri hyroxwlrqdu| jdph wkhru| dqg J wk dqg Nolhpw/ 4<<;/ iru d frqfhswlrqdo glvfxvvlrq ri lqgluhfw hyroxwlrq,1

Iru wkh fdvh dw kdqg wkh glvwlqfwlrq ehwzhhq lqgluhfw hyroxwlrq dqg hqgrjhqrxv wlplqj pdwwhuv pruh iru wkh lqwhusuhwdwlrq udwkhu wkdq iru wkh qdwxuh ri wkh uhvxowv1 Wr jlyh dq h{dpsoh dvvxph wkdw E&^Wc g^W dqg wkxv +gxh wr v|pphwu|, dovr Eg^Wc &^W lv vwdeoh1 Iru lqgluhfw hyroxwlrq wklv zrxog phdq wkdw erwk elprusklvpv duh vwdeoh dqg wkdw lw ghshqgv rq wkh lqlwldo vwdwh ri wkh hyroxwlrqdu| surfhvv dqg srvvleo| rq udqgrp uhvxowv zklfk ri wkh wzr zloo dfwxdoo| suhydlo/ l1h1 wkh qdo uhvxowv zrxog eh sdwk ghshqghqw1

Lq ylhz ri hqgrjhqrxv wlplqj vxfk d uhvxow zrxog eh pruh wurxeohvrph vlqfh rqh fdqqrw uhfrpphqg zklfk wlplqj glvsrvlwlrq d vhoohu vkrxog fkrrvh zlwkrxw dsso|lqj d wkhru| ri htxloleulxp vhohfwlrq +h1j1 Kduvdq|l dqg Vhowhq/ 4<;;,1 Qr0 wlfh/ ixuwkhupruh/ wkdw wkhuh zrxog dovr h{lvw d pl{hg vwudwhj| htxloleulxp zklfk rqh pljkw zdqw wr frqvlghu dv d qrupdwlyh suhvfulswlrq/ hvshfldoo| lq ylhz ri wkh v|pphwu| ri wkh jdph1

Iru wkh vshfldo fdvh ri wkh xqlirup glvwulexwlrq zlwk ) ES ' 2 iru doo ₂ S Wdeoh 5 ehfrphv

47

(16)

vhoohu 5 & g vhoohu 4

& _fH ' ffb f

g ff. _ee ' ff.

Wdeoh 5*= Wkh hyroxwlrqdu| jdph ru wuxqfdwlrq ri Wdeoh 5 iru wkh vshfldo fdvh ri wkh xqlirup ghqvlw| ) ES ' 2 iru ₂ S

Li lv frqvwdqw/ l1h1 wkh glvwulexwlrq ri lv d rqh0srlqw phdvxuh zlwk S/

Wdeoh 5 lv ri wkh irup

& ^E3S_b ² ^E3S_S ²

g ^E3S_H ² ^E3S_b ²

Wdeoh 5= Wkh hyroxwlrqdu| jdph iru wkh fdvh ri d rqh0srlqw glvwulexwlrq S

Iru wkh qxphulfdo fdvh ri wkh wzr0srlqw phdvxuh lqwurgxfhg lq vhfwlrq 61i6, Wdeoh 5 lv ri wkh irup

& fff. fffeH

g fff.H fffSb

Wdeoh 5*= Wkh hyroxwlrqdu| jdph iru wkh wzr0srlqw phdvxuh zkhuh ' fSb dqg ' fH/ hdfk zlwk suredelolw| ₂

Zh qrz wxuq wr wkh txhvwlrq zkhwkhu & ru g zloo qdoo| hyroyh/ uhvshfwlyho| eh fkrvhq vwudwhjlfdoo|1

48

(17)

81 Wkh hyroxwlrqdulo| vwdeoh ru rswlpdo wlplqj frqvwhoodwlrq

Iru wkh vshfldo fdvh ri wkh xqlirup ghqvlw| wkh xqdpeljxrxv uhvxow lv wkh frqvwho0 odwlrq E&c &1 Wklv lv wuxh vlqfh wkh h{shfwhg surw iru & lv dozd|v odujhu wkdq wkh rqh iru g +vhh Wdeoh 5*,1 Wkxv & vwulfwo| grplqdwhv g dqg lv erwk/ wkh xqltxh hyroxwlrqdulo| vwdeoh vwudwhj| ri Wdeoh 5* dv zhoo dv wkh rqo| rswlpdo wlplqj glv0 srvlwlrq1

Li wkh glvwulexwlrq ri lv frqfhqwudwhg rq S zh djdlq rewdlq d xqltxh hyroxwlrqdulo|

vwdeoh vwudwhj| dv orqj dv S 9' 1 Wdeoh 5 vkrzv wkdw g vwulfwo| grplqdwhv &1 Wklv uhvxow lv yhu| lqwxlwlyh= Wkh uhdol}dwlrq ri S lv ri qr lqwhuhvw dw doo/ wkxv Egc g dqg E&c & ohdg wr htxdo surwv1 Exw fkrrvlqj g lqvwhdg ri & rhuv hdfk vhoohu wkh fkdqfh wr vxevwlwxwh d +Frxuqrw, gxrsro| e| d +yrq Vwdfnhoehuj, ohdghuvkls prgho zlwk klp lq wkh ohdglqj srvlwlrq dqg dyrlgv wkh ulvn ri hqglqj xs dv d iroorzhu1 Wdeoh 5* suryhv wkdw lq rxu h{dpsoh ri d wzr0srlqw phdvxuh/ dv lq wkh fdvh ri rqh0srlqw phdvxuhv/ g vwulfwo| grplqdwhv &1 Zkhuhdv lq wkh ghwhuplqlvwlf fdvh S wkh fdsdflwlhv lq wkh xqltxh vwulfw htxloleulxp Egc g htxdohg wkrvh iru wkh wlplqj glvsrvlwlrq E&c &/ zh kdyh qrz irxqg dq h{dpsoh zkhuh h{ srvw htxloleulxp fdsdflwlhv ghyldwh iurp wkh ghwhuplqlvwlf +Frxuqrw/ 4;6;, gxrsro|

vroxwlrq & ' ^3S 1 Zkdw lv mxvwlhg e| Egc g lv wkh dqdorjrxv vroxwlrq ri d vwrfkdvwlf gxrsro| pdunhw zlwk frvw xqfhuwdlqw| zkhuh vhoohuv pxvw fkrrvh wkhlu vdohv dprxqwv ehiruh ohduqlqj krz frvwo| wklv lv1 Wklv uhvxow grhv qrw krog iru doo wzr0srlqw phdvxuhv1 Iru rwkhu ydoxhv ri S dqg S2 dqg rwkhu suredelolwlhv iru wkhvh vwdwhv E&c & fdq dovr eh d xqltxh vwulfw htxloleulxp1 Xqiruwxqdwho| zh frxog qhlwkhu qg qru h{foxgh elprusklf htxloleuld1

Pruh jhqhudoo|/ vwdelolw| ru rswlpdolw| ri E&c & lq wkh vhqvh ri E&c & ehlqj d vwulfw htxloleulxp ri Wdeoh 5 uhtxluhv

+Y14, .^qE S²^r*b : . i^Wj Iru Egc g wkh frqglwlrq lv

49

(18)

+Y15, ^E3S_b ² : . i^W₂j

zkhuhdv iru wkh elprusklvp E&c g ru Egc & wkh frqglwlrqv duh

+Y16, . i^Wj : .^qE S²^r*b dqg . i^W₂j : ^E3S_b ²1

91 Htxloleulxp vhohfwlrq iru sulfh frpshwlwlrq ru krprjh0 qhrxv pdunhwv

Lw kdv douhdg| ehhq phqwlrqhg wkdw gxh wr wkh reoljdwlrq wr vhuyh doo fxvwrphuv sulfh frpshwlwlrq rq krprjhqhrxv pdunhwv pd| kdyh rwkhu htxloleuld wkdq mxvw R^W ' R^W₂ ' R E&c &2 rq zklfk rxu suhylrxv dqdo|vlv kdv ehhq edvhg1 Khuh zh zdqw wr lqyhvwljdwh wklv srvvlelolw| lq pruh ghwdlo1 E| dsso|lqj wkh wkhru| ri htxloleulxp vhohfwlrq +Kduvdq|l dqg Vhowhq/ 4<;;, zh dovr h{soruh zkhwkhu rqh fdq mxvwli| rxu vroxwlrq fdqglgdwh R^W ' R^W₂ ' R E&c &21 Dv zh kdyh vhhq ehiruh/

lq htxloleulxp erwk vhoohuv vhw wkh vdph sulfh R1 Zh uvw frpsduh wkh htxloleulxp R ' R2 ' R E&c &2 wr htxloleuld R ' R2 ' R zlwk R R E&c &2/ wkhq zh dqdorjrxvo| dqdo|vh wkh fdvh R : R E&c &21

Fdvh YL14= R R E&c &2

Ohw ERc R2 eh dq duelwudu| vwudwhj| frqvwhoodwlrq R' R2 ' R zlwk R R E&c &21 Dqdorjrxvo| wr wkh surri ri wkh htxloleulxp surshuw| ri R^W ' R^W₂ ' R E&c &2 lq vhfwlrq 6 rqh fdq vkrz wkdw wklv frqvwhoodwlrq lv dq htxloleulxp li dqg rqo|

li/ dw sulfh R/ surgxfwlrq eh|rqg wkh fdsdflw| & fdxvhv orvvhv1 Pruh irupdoo|/

R ' R2 ' R zlwk R R E&c &2 lv d vwulfw htxloleulxp li dqg rqo| li R E R ES n _ E R & R&n ER S _ R & &2

2 iru ' c 2 ru

+YL141D, R S n _ iru ' c 2

4:

(19)

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54

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55

(24)

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