Capacity Accumulation and Utilization in a Differential Duopoly Game
Abstract
EconStor is a publication server for scholarly economic literature, provided as a non-commercial public service by the ZBW.
Full text
Cellini, Roberto; Lambertini, Luca Working Paper Capacity Accumulation and Utilization in a Differential Duopoly Game Quaderni - Working Paper DSE, No. 465 Provided in Cooperation with: University of Bologna, Department of Economics Suggested Citation: Cellini, Roberto; Lambertini, Luca (2003) : Capacity Accumulation and Utilization in a Differential Duopoly Game, Quaderni - Working Paper DSE, No. 465, Alma Mater Studiorum - Università di Bologna, Dipartimento di Scienze Economiche (DSE), Bologna, https://doi.org/10.6092/unibo/amsacta/4829 This Version is available at: https://hdl.handle.net/10419/159306 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/3.0/
Fdsdflw| Dffxpxodwlrq dqg Xwlol}dwlrq lq d Glhuhqwldo Gxrsro| Jdph 4 Urehuwr Fhoolql 0 Oxfd Odpehuwlql { H Glsduwlphqwr gl Hfrqrpld h Phwrgl Txdqwlwdwlyl Xqlyhuvlwã gl Fdwdqld Fruvr Lwdold 88/ <845< Fdwdqld/ Lwdo| skrqh 6<03<86:8677/ id{ 6<03<806:38:7/ fhoolqlCper{1xqlfw1lw { Fruuhvsrqglqj dxwkru Glsduwlphqwr gl Vflhq}h Hfrqrplfkh Xqlyhuvlwã gl Erorjqd Vwudgd Pdjjlruh 78/ 73458 Erorjqd/ Lwdo| skrqh 6<0384053<5933/ id{ 6<0384053<5997/ odpehuwlCvser1xqler1lw Iheuxdu| :/ 5336 4 Zh wkdqn Shwhu Pûoojddug iru khosixo frpphqwv dqg vxjjhvwlrqv1 Wkh xvxdo glvfodlphu dssolhv1
Devwudfw Zh suhvhqw d glhuhqwldo gxrsro| jdph zlwk fdsdflw| dffxpxodwlrq/ zkhuh upv frqwuro lqyhvwphqw hruwv dqg vdohv/ zklfk fdq eh dw prvw htxdo wr wkh uhvshfwlyh lqvwdoohg fdsdflwlhv dw dq| srlqw lq wlph1 Zh xvh/ dowhuqdwlyho|/ lqyhuvh dqg gluhfw ghpdqg ixqfwlrqv zlwk surgxfw glhuhqwldwlrq/ uhfdoolqj Frxuqrw dqg Ehuwudqg frpshwlwlrq1 Zh vkrz wkdw/ dw wkh vxejdph shuihfw vwhdg| vwdwh htxloleuld/ Frxuqrw dqg Ehuwudqg surwv gr qrw frlqflgh/ xqohvv wkh jdph lv txdvl0vwdwlf/ zklfk kdsshqv li fdsdflw| grhv qrw ghsuhfldwh ryhu wlph1 Nh|zrugv = glhuhqwldo jdphv/ fdsdflw| dffxpxodwlrq/ sulfh vhwwlqj/ txdq0 wlw| vhwwlqj1 MHO Fodvvlfdwlrq = F:6/ G76/ G<5/ O461
4 Lqwurgxfwlrq Zh suhvhqw d glhuhqwldo gxrsro| jdph zlwk glhuhqwldwhg jrrgv dqg fdslwdo dffxpxodwlrq1 Lq sduwlfxodu/ zh prgho wkh dffxpxodwlrq surfhvv wkurxjk uhyhuvleoh lqyhvwphqw ãod Qhuoryh0Duurz +4<95,/ zkhuh fdslwdo +ru fdsdflw|, dffxpxodwlrq rffxuv wkurxjk frvwo| lqyhvwphqw/ dv lq Vrorz*v +4<89, jurzwk prgho1 Ilupv frqwuro lqyhvwphqw hruwv dqg vdohv/ zklfk fdq eh dw prvw htxdo wr wkh uhvshfwlyh lqvwdoohg fdsdflwlhv dw dq| srlqw lq wlph1 Zh lqyhvwljdwh wkh jdph xvlqj/ dowhuqdwlyho|/ lqyhuvh dqg gluhfw ghpdqg ixqfwlrqv1 Wkhvh wzr vhwwlqjv uhfdoo wkh Frxuqrw dqg Ehuwudqg prghov/ dowkrxjk ehlqj qrw htxlydohqw/ vwulfwo| vshdnlqj/ wr hlwkhu wkh uvw ru wkh vhfrqg1 Krzhyhu/ wr hdvh wkh h{srvlwlrq/ zh vkdoo uhihu wr cFrxuqrw frpshwlwlrq* +uhvshfwlyho|/ cEhuwudqg frpshwlwlrq*, zkhq xvlqj lqyhuvh +uhvs1/ gluhfw, ghpdqg ixqfwlrqv1 Wkh lqwhuhvw ri wklv dssurdfk olhv lq wkh idfw wkdw lw doorzv wr frqvwuxfw d g|qdplf shuvshfwlyh zkhuh rqh fdq dqdo|vh wkh lqwhudfwlrq ehwzhhq fdsdflw| dqg wkh pdunhw shuirupdqfh ri upv/ dq lvvxh wkdw kdv uhfhlyhg d frqvlghudeoh dprxw ri dwwhqwlrq lq wkh LR olwhudwxuh1 Wkh pdlq uhvxow rewdlqhg e| wklv vwuhdp ri frqwulexwlrqv lv wkdw Ehuwudqg frpshwlwlrq uhsolfdwhv wkh Frxuqrw htxloleulxp rxwfrph xqghu hqgrjhqrxvo| ghwhuplqhg fdsdflw| frqvwudlqwv +Nuhsv dqg Vfkhlqnpdq/ 4<;6,1 Glhuhqwo| iurp wkh h{lvwlqj prghov ghdolqj zlwk wkh fkrlfh ri fdsdflw| lq vwdwlf wzr0vwdjh jdphv +vhh/ h1j1/ Ehfnpdq/ 4<9:> Ohylwdq dqg Vkxeln/ 4<:5> Nuhsv dqg Vfkhlqnpdq/ 4<;6> Rveruqh dqg Slwfkln/ 4<;9,/ 4 fdslwdo lv qrw d frqwuro yduldeoh lq wkh suhvhqw vhwwlqj/ zkhuh hdfk up fkrrvhv lwv lqyhvwphqw hruwv1 Frqvhtxhqwo|/ wkh fdslwdo +ru fdsdflw|, ri hdfk up lv qrw gluhfwo| dhfwhg e| wkh fdslwdo ri rssrqhqwv1 Rxu pdlq uhvxowv duh wkh iroorzlqj1 Zh suryh wkdw wkh rshq0orrs htxl0 oleulxp rxwfrph ri wkh Ehuwudqg jdph frlqflghv zlwk wkh Frxuqrw htxl0 oleulxp1 Wklv dprxqwv wr vd|lqj wkdw wkh Qhuoryh0Duurz0Vrorz glhuhqwldo jdph hqfrpsdvvhv wkh uhvxowv ri wkh vwdwlf dqdo|vlv fduulhg rxw e| Nuhsv dqg Vfkhlqnpdq +4<;6, lq wkh vslulw ri wkh ruljlqdo lghd gdwlqj edfn wr Hgjh0 zruwk +4;<:,1 Wklv lv whfkqlfdoo| gxh wr wkh idfw wkdw wkh Ehuwudqg vhwwlqj rewdlqv wkurxjk d olqhdu v|pphwulf wudqvirupdwlrq ri wkh Frxuqrw vhwwlqj1 Wkhuhiruh/ wkh wzr prghov duh lvrprusklf dqg surgxfh wkh vdph vwhdg| vwdwh htxloleulxp1 Wklv/ krzhyhu/ krogv rqo| xqghu wkh rshq0orrs vroxwlrq/ zklfk 4 Dv lv zhoo nqrzq/ wkh uvw dqdo|vlv ri fdsdflw|0frqvwudlqhg sulfh frpshwlwlrq lv lq Hgjhzruwk + 4;<: , 1 4
lv vwurqjo| wlph frqvlvwhqw +ru vxejdph shuihfw, rqo| li upv xvh lqyhuvh ghpdqg ixqfwlrqv +l1h1/ lq wkh Ehuwudqg jdph,1 Lq wkh Frxuqrw jdph/ wkh rshq0orrs htxloleulxp lv qrw vxejdph shuihfw/ dqg wklv surpswv iru wkh dgrs0 wlrq ri wkh forvhg0orrs vroxwlrq frqfhsw1 Frpsdulqj wkh Frxuqrw forvhg0orrs htxloleulxp surwv djdlqvw wkh Ehuwudqg surwv +ru wkh Frxuqrw rshq0orrs surwv,/ zh qg wkdw +l, wkh| gr qrw frlqflgh/ dqg +ll, wkhuh h{lvwv dq dg0 plvvleoh udqjh ri sdudphwhuv +zlwk upv vhoolqj vxevwlwxwh jrrgv, zkhuh wkh uvw duh vpdoohu wkdq wkh vhfrqg1 D vlwxdwlrq zkhuh htxloleulxp surwv duh wkh vdph luuhvshfwlyh ri wkh irupxodwlrq ri pdunhw ghpdqg dqg wkh fkrlfh ri wkh vroxwlrq frqfhsw lv wkdw zkhuh fdsdflw| grhv qrw ghsuhfldwh dw doo ryhu wlph1 Krzhyhu/ wklv fdvh lv lqkhuhqwo| txdvl0vwdwlf1 Wkhuhiruh/ zh fdq frqfoxgh wkdw/ lq d g|qdplf jdph/ wkh frlqflghqfh ehwzhhq Ehuwudqg dqg Frxuqrw rxwfrphv fdqqrw dulvh lq jhqhudo1 Pruhryhu/ wkh hvwdeolvkhg uhvxow wkdw Frxuqrw ehkdylrxu lv pruh +ohvv, surwdeoh wkdq Ehuwudqg ehkdylrxu zkhq jrrgv duh vxevwlwxwhv +frpsohphqwv,/ gdwlqj edfn wr Vlqjk dqg Ylyhv +4<;7,/ lqwhu dold / lv dovr frqwudglfwhg1 Wkhvh uhvxowv hvvhqwldoo| ghshqg rq wkh idfw wkdw/ dw wkh Ehuwudqg vxqejdph shuihfw htxloleulxp upv rshudwh dw ixoo fdsdflw|/ zkloh wkh| nhhs vrph lgoh fdsdflw| dw wkh vxejdph shuihfw Frxuqrw htxloleulxp1 Wkh uhpdlqghu ri wkh sdshu lv vwuxfwxuhg dv iroorzv1 Vhfwlrq 5 prwlydwhv wkh g|qdplf dssurdfk1 Wkh edvlf vhwxs lv odlg rxw lq vhfwlrq 61 Wkh Frxuqrw dqg Ehuwudqg vhwwlqjv duh lqyhvwljdwhg lq vhfwlrqv 7 dqg 8/ uhvshfwlyho|1 Wkhq/ wkh frpsdudwlyh dvvhvvphqw ri htxloleulxp surwv lv fduulhg rxw lq vhfwlrq 91 Frqfoxglqj uhpdunv duh lq vhfwlrq :1 5 Wkh glhuhqwldo jdph dssurdfk Zh eholhyh wkdw/ lq rughu wr dqdo|vh wkh upv* ehkdylrxu frqfhuqlqj wkh dffxpxodwlrq ri surgxfwlyh fdsdflw|/ wkh glhuhqwldo jdph dssurdfk lv sdu0 wlfxoduo| dssursuldwh1 Lqghhg/ exloglqj fdsdflw| qhhgv wlph> wkh ghsuhfldwlrq ri fdsdflw| ryhu wlph lv dq lpsruwdqw idfwru> wkh wlph glphqvlrq lv lpsruwdqw lq sodqqlqj wkh lqyhvwphqw hruwv wrzdugv fdsdflw| dffxpxodwlrq1 Ixukwhu0 pruh/ vwudwhjlf lqwhudfwlrq dprqj upv lv dq lpsruwdqw lvvxh erwk lq wkh pdunhw skdvh/ dqg lq wkh skdvh zkhq fdsdflw| ghflvlrqv duh wdnhq1 Odvw exw qrw ohdvw/ wkh glhuhqwldo jdph dssurdfk ohdgv wr uhvxowv hqfrpsdvvlqj wkrvh rewdlqhg e| wkh pruh wudglwlrqdo dssurdfk edvhg rq vwdwlf wzr0vwdjh jdphv1 Pruh suhflvho|/ wkh glhuhqwldo jdph dssurdfk doorzv wr xqghuvwdqg xqghu 5
zklfk sduwlfxodu frqglwlrqv / wkh uhvxowv iurp vwdwlf jdph fdq eh lqwhusuhwhg dv wkh vwhdg| vwdwh vroxwlrq ri wkh glhuhqwldo jdph1 5 Vshflfdoo|/ lq wkh suhvhqw sdshu zh dlp dw fkhfnlqj zkhwkhu wkh zhoo0nqrzq uhvxow rewdlqhg e| Nuhsv dqg Vfkhlqnpdq +4<;6, derxw wkh htxlydohqfh ehwzhhq wkh Frxuqrw dqg wkh Ehuwudqg vhwwlqj lq d wzr0vwdjh vwdwlf jdph/ krogv lq d glhuhqwldo jdph iudphzrun wrr1 Wkh h{lvwlqj olwhudwxuh rq glhuhqwldo jdphv pdlqo| irfxvvhv rq wzr nlqgv ri vwudwhjlhv dgrswhg e| sod|huv= wkh rshq0orrs dqg wkh forvhg0orrs vwudwh0 jlhv1 6 Zkhq sod|huv dgrsw wkh rshq0orrs vroxwlrq frqfhsw/ wkh| ghvljq wkh wlph sdwk frqfhuqlqj wkh frqwuro yduldeoh+v, dw wkh lqlwldo wlph dqg wkhq vwlfn wr lw iruhyhu1 Wklv phdqv wkdw wkh rshq0orrs vwudwhj| lv vlpso| d wlph sdwk ri dfwlrqv/ dqg wlph lv wkh rqo| ghwhuplqdqw ri wkh dfwlrq wr eh grqh dw dq| lqvwdqw1 Wkh uhohydqw htxloleulxp frqfhsw lv wkh rshq0orrs Qdvk htxloleulxp/ zklfk lv rqo| zhdno| wlph frqvlvwhqw dqg wkhuhiruh/ lq jhqhudo/ lw lv qrw vxe0 jdph shuihfw1 7 Zkhq sod|huv dgrsw wkh forvhg0orrs vwudwhj|/ wkh| gr qrw suhfrpplw frqwuro yduldeoh+v, rq dq| sdwk/ dqg wkhlu dfwlrqv dw dq| lqvwdqw pd| ghshqg rq wkh klvwru| ri wkh jdph xs wr wkdw lqvwdqw/ dqg/ lq sduwlfxodu/ rq wkh ydoxhv ri wkh vwdwh yduldeohv1 Lq wklv vlwxdwlrq/ wkh lqirupdwlrq vhw xvhg e| sod|huv lq vhwwlqj wkhlu dfwlrqv dw dq| jlyhq wlph lv riwhq vlpsolhg wr eh rqo| wkh fxuuhqw ydoxh ri wkh vwdwh yduldeohv dw wkdw wlph/ dorqj zlwk wkh lql0 wldo frqglwlrqv1 Wklv vshflf vlwxdwlrq lv odehoohg dv phpru|ohvv forvhg0orrs +Phkopdqq/ 4<;;,1 Wkh uhohydqw htxloleulxp frqfhsw/ lq wklv fdvh/ lv wkh forvhg0orrs phpru|ohvv Qdvk htxloleulxp/ zklfk lv vwurqjo| wlph frqvlvwhqw +ru vxejdph shuihfw,1 8 Wkh olwhudwxuh rq glhuhqwldo jdphv ghyrwhv d frqvlghudeoh dprxqw ri 5 Iru dgglwlrqdo prwlydwlrq vxssruwlqj wkh fkrlfh ri wkh glhuhqwldo jdph dssurdfk/ vhh Grfnqhu hw do1 + 5333 , / fkdswhu <1 Wklv uhihuhqfh dovr surylghv d suhvhqwdwlrq ri uhohydqw glhuhqwldo jdphv zlwk fdslwdo dffxpxodwlrq1 6 Vhh Ndplhq dqg Vfkzduw} + 4<;4 , > Edvdu dqg Rovghu + 4<;5/ 4<<8 # , > Phkopdqq + 4<;; , > Grfnqhu hw do1 + 5333 , 1 Vhh dovr Fhoolql dqg Odpehuwlql + 5334 , 1 7 Dv wr wkh ghqlwlrq ri wlph frqvlvwhqf| dqg vxejdph shuihfwlrq/ zh uho| 0 dprqj pdq| glhuhqw ghqlwlrqv dydlodeoh lq wkh olwhudwxuh 0 rq wkh ghqlwlrq surylghg e| Grfnqhu hw do1 + 5333/ Vhfwlrq 716 , 1 Zh xvh vwurqj wlph frqvlvwhqf| dv d v|qrqrprxv ri vxejdph shuihfwlrq1 8 Glhuhqw uxohv ri forvhg0orrs fodvvgr h{lvw= h1j1/ wkh shuihfw0phpru| forvhg orrs/ zkhuh wkh frqwuro yduldeohv ghshqg rq wkh frpsohwh klvwru| ri wkh vwdwh yduldeoh + v , > wkh ihhgedfn uxoh/ zkhuh rqo| wkh fxuuhqw vwrfn ri vwdwhv duh frqvlghuhg/ luuhvshfwlyh ri lqlwldo frqglwlrqv1 Iru roljrsro| prghov zkhuh upv iroorz ihhgedfn uxohv/ vhh Vlpddq dqg Wdnd|dpd + 4<:; , / Ihuvkwpdq dqg Ndplhq + 4<;:/ 4<<3 , / Grfnqhu dqg Kdxj + 4<<3 , / lqwhu dold 1 6
dwwhqwlrq wr lghqwli|lqj fodvvhv ri jdphv zkhuh wkh forvhg0orrs htxloleuld gh0 jhqhudwh lqwr rshq0orrs htxloleuld1 Wkh ghjhqhudwlrq phdqv wkdw wkh Qdvk htxloleulxp wlph sdwkv ri frqwuro yduldeohv frlqflgh xqghu wkh glhuhqw vwudw0 hj| frqfhswv1 Zkhqhyhu dq rshq0orrs htxloleulxp lv d ghjhqhudwh forvhg0orrs htxloleulxp/ wkhq wkh iruphu lv dovr vwurqjo| wlph frqvlvwhqw +ru vxejdph shuihfw,1 Wkhuhiruh/ rqh fdq uho| xsrq wkh rshq0orrs htxloleulxp zklfk/ lq jhqhudo/ lv pxfk hdvlhu wr ghulyh wkdq forvhg0orrs1 9 6 Wkh prgho Wkh jdph lv sod|hg ryhu frqwlqxrxv wlph/ w 5 d f > 4 = : Lq dq| lqvwdqw ri wlph/ wkh pdunhw lv vhuyhg e| wzr upv/ 4 dqg 5/ surgxflqj d glhuhqwldwhg jrrg1 Ohw t l E w ghqh wkh txdqwlw| vrog e| up l ' > 2 dw wlph w= Wkh pdujlqdo surgxfwlrq frvw lv frqvwdqw dqg htxdo wr f iru doo upv1 Iru wkh vdnh ri vlpsolflw|/ zh srvh f 'f 1 Dv lq Vlqjk dqg Ylyhv +4<;7,/ wkh ghpdqg ixqfwlrq iru jrrg l dw wlph w lv= s l E w ' D t l E w Gt m E w > E m 9 ' l +4, zkhuh sdudphwhu G 5 d > o phdvxuhv wkh ghjuhh ri vxevwlwxwdelolw| ru frp0 sohphqwdulw| ehwzhhq jrrgv1 Li G ' > wkh| duh shuihfw vxevwlwxwhv> li G 'f wkh jrrgv duh lqghshqghqw dqg hdfk up ehkdyhv dv d prqrsrolvw> li G ' > surgxfwv duh shuihfw frpsohphqwv1 Lq rughu wr surgxfh/ upv pxvw dffxpxodwh fdsdflw| ru sk|vlfdo fdslwdo n l E w ryhu wlph1 Dv lq Vrorz +4<89, dqg Qhuoryh dqg Duurz +4<95,/ wkh uhoh0 ydqw g|qdplf htxdwlrq ghvfulelqj wkh dffxpxodwlrq ri fdsdflw| +ru sk|vlfdo fdslwdo, lv= ; Cn l E w Cw ' L l E w n l E w > +5, 9 Fodvvhv ri jdphv zkhuh wklv frlqflghqfh dulvhv duh looxvwudwhg lq Fohpkrxw dqg Zdq + 4<:7 , > Uhlqjdqxp + 4<;5 , > Phkopdqq dqg Zloolqj + 4<;6 , > Grfnqhu hw do1 + 4<;8 , > Ihuvkw0 pdq + 4<;: , > Ihuvkwpdq hw do 1 + 4<<5 , 1 Iru dq ryhuylhz/ vhh dovr Phkopdqq + 4<;; , / dqg Grfnqhu hw do1 + 5333/ fk1 : , 1 : Wkh jdph fdq eh uhirupxodwhg lq glvfuhwh wlph zlwkrxw vljqlfdqwo| dhfwlqj lwv txdo0 lwdwlyh surshuwlhv1 Iru ixuwkhu ghwdlov/ vhh Edvdu dqg Rovghu + 4<;5/ 4<<8 # , 1 ; D vlplodu vhwwlqj lv xvhg lq Ixghqehuj dqg Wluroh + 4<;6 , / Ihuvkwpdq dqg Pxoohu + 4<;7 , dqg Uh|qrogv + 4<;: , 1 Iru d g|qdplf roljrsro| jdph zkhuh fdslwdo dffxpxodwhv ãod Udpvh| + l1h1/ fxuuhqw xqvrog rxwsxw ehfrphv dgglwlrqdo surgxfwlyh fdsdflw| , / vhh Fhoolql dqg Odpehuwlql + 4<<; , 1 7
zkhuh L l E w lv wkh lqyhvwphqw fduulhg rxw e| up l dw wlph w / dqg A f lv wkh frqvwdqw ghsuhfldwlrq udwh1 Wkh lqvwdqwdqhrxv frvw ri lqyhvwphqw lv F l d L l E w o ' e d L l E w o 2 > zlwk eA f = Zh dovr dvvxph wkdw upv ehdu lqvwdqwdqhrxv surgxfwlrq frvwv F l E w ' ft l E w dqg/ iru wkh vdnh ri vlpsolflw|/ zh vhw f 'f zlwkrxw ixuwkhu orvv ri jhqhudolw|1 Dw dq| wlph w> up l surgxfhv dqg vhoov dq rxwsxw t l E w n l E w / vr wkdw zh fdq zulwh t l E w ' l E w n l E w > zlwk l E w 5 Ef > o 1 Dv d frqvhtxhqfh/ wkh ghpdqg ixqfwlrq iru ydulhw| l uhzulwhv dv= s l E w ' D l E w n l E w n G m E w n m E w = +6, Sod|hu l *v remhfwlyh lv wkh pd{lplvdwlrq ri wkh suhvhqw ydoxh ri wkh surw r zv= 4@ L l ] " f l E =>w h 3 w gw +7, vxemhfw wr wkh g|qdplf frqvwudlqw uhsuhvhqwhg e| wkh ehkdylrxu ri wkh vwdwh yduldeohv +5, iru l ' > 2 1 Lqvwdqwdqhrxv surw lv l E =>w ' s l t l e d L l E w o 2 = Wkh idfwru h 3 w glvfrxqwv ixwxuh jdlqv/ dqg wkh glvfrxqw udwh lv dvvxphg wr eh frqvwdqw dqg frpprq wr erwk sod|huv1 Lq rughu wr vroyh klv rswlplvdwlrq sureohp/ hdfk sod|hu ghqhv d vwudwhj|1 Wkh frqwuro yduldeohv ri up l duh wkh lqvwdqwdqhrxv lqyhvwphqw L l E w dqg wkh fdsdflw| xwlol}dwlrq ohyho l E w > zkloh n l E w lv reylrxvo| d vwdwh yduldeoh1 7 Frxuqrw frpshwlwlrq Zkhq upv zulwh surwv xvlqj wkh lqyhuvh ghpdqg ixqfwlrqv +6,/ wkh Kdplo0 wrqldq ixqfwlrq ri up l zulwhv dv iroorzv= K l E w ' h 3 w d D l E w n l E w G m E w n m E w o l E w n l E w e d L l E w o 2 n ll E w d L l E w n l E w o n lm E w d L m E w n m E w o j +8, zkhuh lm E w ' lm E w h w > dqg lm E w lv wkh fr0vwdwh yduldeoh dvvrfldwhg wr n m E w > l>m ' > 2 1 Pruhryhu/ ohw n l E f n l f ghqh wkh lqlwldo frqglwlrq iru up l= 714 Wkh rshq0orrs Qdvk htxloleulxp H{dplqh wkh rshq0orrs vlpxowdqhrxv vroxwlrq1 Wkh uvw rughu frqglwlrqv +IRFv, duh +zh rplw h{srqhqwldo glvfrxqwlqj iru euhylw|,= C K l E w C l E w ' n l E w d D 2 l E w n l E w G m E w n m E w o 'f +9, 8
C K l E w CL l E w ' 2 eL l E w n ll E w 'f +:, C K l E w Cn l E w ' C ll E w Cw ll E w , C ll E w Cw 'E n ll E w l E w d D 2 l E w n l E w G m E w n m E w o +;, wrjhwkhu zlwk lqlwldo frqglwlrqv n l Ef A f dqg wudqvyhuvdolw| frqglwlrqv= *4 w <" ll E w n l E w ' f ( *4 w <" lm E w n m E w 'f = +<, Iurp +:, zh rewdlq= ll E w '2 eL l E w ( CL l E w Cw ' 2 e C ll E w Cw +43, Wkhq/ xvlqj wkh v|pphwu| frqglwlrqv l E w ' m E w ' E w dqg n l E w ' n m E w ' n E w dqg ll E w ' mm E w '2 eL E w > zh fdq zulwh +wkh lqglfdwlrq ri wlph lv rplwwhg iru euhylw|,= CL Cw '2 e E n L d D E 2n G n o 2 e +44, zklfk fdq eh ixuwkhu vlpsolhg xvlqj n W ' L @ iurp Cn @ Cw 'fG CL Cw ' d 2 e E n n 2 E 2n G o L D 2 e +45, Wkh deryh glhuhqwldo htxdwlrq lv htxdo wr }hur dw= L W ' D 2 e E n n 2 E 2n G +46, Wklv/ wrjhwkhu zlwk n W ' L W @> fdq eh soxjjhg lqwr +9,/ wkdw uhzulwhv dv iroorzv= C K C '2 D 2 e E n d 2 e E n n 2 E 2n G o A f dozd|v1 +47, Wkhuhiruh/ vv ' > zlwk wkh vxshuvfulsw vv vwdqglqj iru vwhdg| vwdwh 1 Wkh fruuhvsrqglqj rswlpdo vwhdg| vwdwh ohyhov ri lqyhvwphqw dqg fdsdflw| duh= L vv ' D 2n G n2 e E n ( n vv ' L vv ' D 2n G n2 e E n = +48, Wkh sdlu i L vv >n vv j lv d vdggoh srlqw> wkh surri lv vwudljkwiruzdug/ lq wkdw/ wkh ghwhuplqdqw ri wkh Mdfreldq pdwul{ dvvrfldwhg wr wkh g|qdplf v|vwhp i Cn l E w @ Cw 'f >CL l E w @ Cw 'f j lv qhjdwlyh/ zkloh lwv wudfh lv srvlwlyh1 < < Fdofxodwlrqv duh wulyldo dqg wkh| duh rplwwhg iru euhylw|1 9
zkhuh C K l Cs l ' D E G 2 s l n Gs m n l G 2 ( C K l Cs m ' G d s l l o G 2 ( Cs l Cn l ' ( Cs m Cn l ' G= +7:, Sduwldo ghulydwlyhv Cs l @Cn l dqg Cs l @Cn m duh fdofxodwhg xvlqj wkh lqyhuvh gh0 pdqg ixqfwlrq +4,1 Xvlqj +7:,/ wkh frvwdwh htxdwlrq zulwhv dv iroorzv= C K l Cs l Cs l Cn l n C K l Cs m Cs m Cn l ' C l Cw l +7;, iurp zklfk zh rewdlq C l Cw ' D E G s l E 2 G 2 n Gs m n l E n E G 2 G 2 = +7<, Wkhq/ soxjjlqj +76, dqg +7<, lqwr +77, dqg lpsrvlqj wkh v|pphwu| frqglwlrq s m ' s l > zh kdyh= CL l Cw ' D s l E 2n G n2 eL l E n E n G 2 e E n G +83, zlwk CL l Cw 'f dw L vv l ' s l E 2n G D 2 e E n E n G = +84, L vv l fdq eh vxevwlwxwhg lqwr +5,/ zklfk vlpsolhv dv iroorzv= Cn l Cw ' s l E2 n G D 2 e E n En G d D s l o n G +85, zlwk Cn l Cw 'f dw s vv l ' D d n2 e E n o 2 d n e E n o n G +86, Qrz/ xvlqj +86,/ zh fdq vlpsoli| wkh h{suhvvlrq iru wkh vwhdg| vwdwh ohyhov ri lqyhvwphqw dqg fdsdflw|= L vv l ' D 2n G n2 e E n ( n vv l ' L vv l > +87, zklfk frlqflgh zlwk +48,1 Dovr wkh htxloleulxp surwv duh reylrxvo| wkh vdph dv lq wkh Frxuqrw jdph lqyhvwljdwhg lq vhfwlrq 714 +vhh h{suhvvlrq +4:,,1 Wklv glvfxvvlrq ohdgv wr wkh iroorzlqj= 46
Sursrvlwlrq : Wkh vwhdg| vwdwh ri wkh rshq0orrs Ehuwudqg jdph lv revhuyd0 wlrqdoo| htxlydohqw iurp wkdw ri wkh rshq0orrs Frxuqrw jdph1 Qrwh/ krzhyhu/ wkdw wkh rshq0orrs Qdvk htxloleulxp ri wkh Frxuqrw jdph lv qrw vxejdph shuihfw1 Wkhuhiruh= Fruroodu| ; Dw wkh vxejdph shuihfw htxloleuld/ wkh Ehuwudqg rxwfrph grhv qrw frlqflgh zlwk wkh Frxuqrw rxwfrph/ zlwk wkh vdph fdsdflw| dffxpxodwlrq g|qdplfv ehlqj xvhg lq erwk vhwwlqjv1 9 Forvhg0orrs yv rshq0orrs htxloleuld Lq wkh suhylrxv vhfwlrq/ zh kdyh hvwdeolvkhg wkdw/ luuhvshfwlyh ri zkhwkhu rqh xvhv gluhfw ru lqyhuvh ghpdqg ixqfwlrqv/ wkh rshq0orrs htxloleulxp orrnv h{dfwo| wkh vdph1 Krzhyhu/ iurp vhfwlrqv 714 dqg 715 zh nqrz wkdw +l, wkh rshq0orrs htxloleulxp ri wkh Frxuqrw prgho lv qrw vxejdph shuihfw/ dqg +ll, dw wkh forvhg0orrs Frxuqrw htxloleulxp/ lq jhqhudo/ upv zloo qrw rshudwh dw ixoo fdsdflw|1 Wklv surpswv iru d frpsdudwlyh dvvhvvphqw ri wkh shuirupdqfh ri upv xqghu wkh wzr dowhuqdwlyh vroxwlrq frqfhswv1 Zh frqqh rxu dwwhqwlrq wr wkh fdvh zkhuh wkh forvhg0orrs Frxuqrw jdph |lhogv dq lqwhulrq vroxwlrq z1u1w1 vv > l1h1/ e e e G 2 @ d e E n o = Hydoxdw0 lqj +4:, djdlqvw +68,/ zh vlpxowdqhrxvo| frpsduh wkh Ehuwudqg surwv yhuvxv wkh Frxuqrw surwv/ dw wkh vxejdph shuihfw htxloleuld ri wkh uhvshfwlyh jdphv> wkh rshq0orrs Frxuqrw surwv yhuvxv wkh forvhg0orrs Frxuqrw surwv1 Lq jdphv zkhuh wkh vl}h ri wkh up frlqflghv zlwk lwv vdohv/ wkh dqvzhu zrxog eh vwudljkwiruzdug/ vlqfh wkh forvhg0orrs vroxwlrq xvxdoo| hqwdlov odujhu vdohv dqg orzhu surwv dv frpsduhg wr wkh rshq0orrs vroxwlrq +vhh/ h1j1/ Ihu0 vkwpdq dqg Ndplhq/ 4<;:,1 Khuh/ krzhyhu/ up vl}h +dv phdvxuhg e| lwv lqvwdoohg fdsdflw| lq vwhdg| vwdwh, pd| glhu iurp lwv vdohv1 Dffruglqjo|/ wklv dvvhvvphqw pxvw eh h{solflwo| fduulhg rxw rq wkh edvlv ri= vljq i vv ro vv fo j ' vljq e e 2 E n e E n G 2 n +88, e e E n E 2n G d 2 G E G E 2n G o n G 2 E 2n G 2 = 47
Wkh u1k1v1 ri +88, pd| wdnh hlwkhu vljq ghshqglqj xsrq wkh vl}h dqg vljq ri wkh irxu sdudphwhuv i e>G>> j 1 Wdnlqj wkh urrwv ri vv ro vv fo 'f z1u1w1 e> zh rewdlq= e 'E2 n G d2 G E G E2n G o n s l 2 E n deE n G 2 o e 'E2 n G d2 G E G E2n G o s l 2 E n deE n G 2 o +89, zkhuh= l e G G n G 2 n2 G 2 n 2 E 2n G 2 G 2 G n2 = +8:, Khqfh/ wzr fdvhv pd| dulvh= l ? f = Li vr/ wkh htxdwlrq vv ro vv fo 'f kdv qr uhdo urrwv/ hqwdlolqj wkdw vv ro vv fo ? f dozd|v/ vlqfh wkh frh!flhqw ri e 2 lv dozd|v qhjdwlyh1 l f = Li vr/ e >e 2 5 7 > zlwk e 2 ? f ? e e?e lq wkh zkroh dgplvvleoh udqjh ri sdudphwhuv/ dqg vv ro vv fo A f iru doo e e e= Lqvwhdg ri vsdqqlqj wkh zkroh sdudphwhu udqjh/ frqvlghu wkh iroorzlqj h{dpsoh1 Vxssrvh e ' e e @ 2 = Lq vxfk d fdvh/ vv ro vv fo 2 G 2 e G E Hn G n E 2n G G 2 E Sn G nH E 2n G +8;, dqg vv ro A vv fo l A 4@ E2 n G d G 2 ES n G nHE2n G o e G EH n G > f +8<, zlwk E 2n G d G 2 E Sn G nH E 2n G o e G E Hn G B f iru doo G : f = Wkhuhiruh/ ryhu wkh zkroh vxevwlwxwdelolw| udqjh/ vv ro A vv fo iru doo dgplvvleoh ydoxhv ri dqg = Li lqvwhdg G 5 d > f > wkhq vv ro A vv fo l A E 2n G d G 2 E Sn G nH E 2n G o e G E Hn G +93, dqg frqyhuvho|1 Li G 'f > reylrxvo| vv ro ' vv fo = Zlwk uhihuhqfh wr wkh olwhudwxuh frpsdulqj wkh uhodwlyh surwdelolw| ri Ehuwudqg dqg Frxuqrw pdunhwv/ wkh deryh dujxphqw suryhv rxu qdo uhvxow= 48
Sursrvlwlrq < Li jrrgv duh vxevwlwxwhv/ wkhuh duh dgplvvleoh sdudphwhu frq0 vwhoodwlrqv zkhuhlq wkh Ehuwudqg jdph lv pruh surwdeoh wkdq wkh Frxuqrw jdph1 Frqyhuvho|/ li jrrgv duh frpsohphqwv/ wkhuh duh dgplvvleoh sdudphwhu frqvwhoodwlrqv zkhuhlq wkh rssrvlwh krogv1 Wklv lv lq vkdus frqwudvw zlwk wkh dftxluhg zlvgrp shuwdlqlqj wr vwdwlf jdphv/ hvwdeolvklqj wkdw Frxuqrw lv pruh +ohvv, surwdeoh wkdq Ehuwudqg zkhq upv vhoo vxevwlwxwh +frpsohphqw, jrrgv1 Wkh vrxufh ri wklv uhvxow olhv lq wkh idfw wkdw/ lq wkh suhvhqw glhuhqwldo jdph/ fdsdflw| dffxpxodwlrq hqwdlov dq lqvwdqwdqhrxv dgmxvwphqw frvw zklfk lv/ e| ghqlwlrq/ devhqw lq wkh fruuhvsrqglqj vwdwlf jdph zkhuh fdsdflw| lv vhw xs iurp vfudwfk1 Pruhryhu/ vwhdg| vwdwh vdohv glhu dfurvv htxloleuld1 Lq sduwlfxodu/ surylghg wkdw vv fo ? G t FQ n vv ro '2 De E n E 2n G d G n2 E n e E n o A f dozd|v/ +94, zklfk phdqv wkdw/ dw wkh forvhg0orrs htxloleulxp/ upv vhoo pruh wkdq dw wkh rshq0orrs htxloleulxp/ dowkrxjk ohdylqj vrph fdsdflw| lgoh1 Dffruglqjo|/ wkh Frxuqrw htxloleulxp sulfh lv orzhu wkdq wkh Ehuwudqg htxloleulxp sulfh1 Ilqdoo|/ wkhuh uhpdlqv wr eh vwuhvvhg wkdw/ lq wkh +udwkhu xquhdolvwlf, fdvh zkhuh 'f > wkh wkuhh htxloleuld lqyhvwljdwhg deryh duh revhuydwlrqdoo| htxly0 dohqw lq whupv ri surwv/ iru doo ydoxhv ri G dqg +wkh odwwhu sdudphwhu ehlqj luuhohydqw lq vxfk d fdvh,= Sursrvlwlrq 43 Li wkh ghsuhfldwlrq udwh ri lqvwdoohg fdsdflw| lv qlo/ wkhq vwhdg| vwdwh htxloleulxp surwv dozd|v frlqflgh zlwk wkh vwdwlf Frxuqrw0Qdvk surwv/ luuhvshfwlyh ri zkhwkhu jrrgv duh vxevwlwxwhv/ frpshphqwv/ ru lqgh0 shqghqw1 Wklv Sursrvlwlrq hpskdvlvhv wkdw wkh fruuhvsrqghqfh ehwzhhq Frxuqrw dqg Ehuwudqg rxwfrph zkhq frpshwlwlrq lv hvvhqwldoo| vwdwlf rewdlqv dv d vshfldo fdvh ri d g|qdplf prgho zkhuh wkh surw shuirupdqfh ri upv/ lq jhqhudo/ zloo vxevwdqwldoo| glhu dfurvv glhuhqw vhwwlqjv1 : Frqfoxglqj uhpdunv Zh kd y h lq y hvwljdwhg wkh lq xhqfh ri frvwo| fdsdflw| dffxpxodwlrq rq htxl0 oleulxp surwv lq d glhuhqwldo jdph zkhuh upv rswlpdoo| fkrrvh wkhlu uh0 vshfwlyh lqyhvwphqw hruwv dv zhoo dv wkh ghjuhh ri fdsdflw| xwlol}dwlrq1 Wklv 49
kdv doorzhg xv wr lqyhvwljdwh wzr dowhuqdwlyh vhwwlqjv plplfnlqj Frxuqrw dqg Ehuwudqg ehkdylrxu/ wr vkrz wkdw/ lq jhqhudo/ wkh htxlydohqfh ehwzhhq Ehuwudqg dqg Frxuqrw rxwfrphv xqghu fdsdflw| frqvwudlqwv/ kroglqj wuxh lq vwdwlf jdphv +Nuhsv dqg Vfkhlqnpdq/ 4<;6,/ grhv qrw fduu| ryhu wr g|qdplf jdphv1 Pruhryhu/ zh kdyh dovr vkrzq wkdw Ehuwudqg pd| eh pruh surwdeoh wkdq Frxuqrw/ hyhq zkhq upv vxsso| vxevwlwxwh jrrgv1 Wkhvh uhvxowv duh gulyhq e| wkh gudvwlfdoo| glhuhqw dwwlwxgh ri upv wrzdugv fdsdflw| dffxpx0 odwlrq dqg xwlol}dwlrq lq wkh wzr vhwwlqjv1 Zkloh wkh| rshudwh dw ixoo fdsdflw| dw wkh Ehuwudqg htxloleulxp/ wkh| nhhs vrph h{fhvv fdsdflw| dw wkh Frxuqrw htxloleulxp1 Pruhryhu/ lq wkh odwwhu wkh lqvwdoohg fdsdflw| dqg wkh dvvrfldwhg vdohv duh odujhu wkdq lq wkh iruphu/ dqg wkh Frxuqrw up ehduv odujhu frvwv dqg hduqv orzhu uhyhqxhv wkdq d Ehuwudqg up1 4:
Uhihuhqfhv ^4` Edvdu/ W1/ dqg Rovghu/ J1M1 +4<;5/ 4<<8 2 ,/ G|qdplf Qrqfrrshudwlyh Jdph Wkhru| / Vdq Glhjr/ Dfdghplf Suhvv1 ^5` Ehfnpdq/ P1 +4<9:,/ Hgjhzruwk0Ehuwudqg Gxrsro| Uhylvlwhg/ lq Khqq/ U1 +hg1,/ Rshudwlrqv Uhvhdufk0Yhuidkuhq / LLL/ Phlvhqkhlq/ Yhu0 odj Dqwrq Khlq1 ^6` Fhoolql/ U1 dqg O1 Odpehuwlql +4<<;,/ D G|qdplf Prgho ri Glhuhqwl0 dwhg Roljrsro| zlwk Fdslwdo Dffxpxodwlrq/ Mrxuqdo ri Hfrqrplf Wkh0 ru| / ;6 / 4780881 ^7` Fhoolql/ U1 dqg O1 Odpehuwlql +5334,/ Glhuhqwldo roljrsro| Jdphv zkhuh wkh Forvhg0Orrs Phpru|ohvv dqg Rshq0Orrs Htxloleuld Frlq0 flgh/ zrunlqj sdshu/ Ghsduwphqw ri Hfrqrplfv Xqlyhuvlw| ri Erorjqd/ q1 7351 ^8` Fohpkrxw/ V1 dqg K1\1 Zdq/ Mu1 +4<:7,/ D Fodvv ri Wulolqhdu Glhuhqwldo Jdphv/ Mrxuqdo ri Rswlpl}dwlrq Wkhru| dqg Dssolfdwlrqv / 47 / 74<0571 ^9` Grfnqhu/ H1M1/ J1 Ihlfkwlqjhu dqg V1 Mûujhqvhq +4<;8,/ Wudfwdeoh Fodvvhv ri Qrq}hur0Vxp Rshq0Orrs Qdvk Glhuhqwldo Jdphv= Wkhru| dqg H{dpsohv/ Mrxuqdo ri Rswlpl}dwlrq Wkhru| dqg Dssolfdwlrqv / 78 / 4:<0<:1 ^:` Grfnqhu/ H1M1 dqg D1D1 Kdxj +4<<3,/ Wdulv dqg Txrwdv xqghu G|0 qdplf Gxrsrolvwlf Frpshwlwlrq/ Mrxuqdo ri Lqwhuqdwlrqdo Hfrqrplfv / 5< / 47:08<1 ^;` Grfnqhu/ H1M/ V1 Mûujhqvhq/ Q1 Ydq Orqj dqg J1 Vrujhu +5333,/ Gli0 ihuhqwldo Jdphv lq Hfrqrplfv dqg Pdqdjhphqw Vflhqfh / Fdpeulgjh/ Fdpeulgjh Xqlyhuvlw| Suhvv1 ^<` Hgjhzruwk/ I1 +4;<:,/ Od Whruld Sxud gho Prqrsrolr/ Jlruqdoh ghjol Hfrqrplvwl / 73 / 460641 Uhsulqwhg lq Hqjolvk dv Wkh Sxuh Wkhru| ri Prqrsro|/ lq Hgjhzruwk/ I1 +4<58,/ Sdshuv Uhodwlqj wr Srolwlfdo Hfrq0 rp| / Yro1 4/ Orqgrq/ Pdfploodq ) Fr1/ Owg1/ 4440751 ^43` Ihlfkwlqjhu/ J1 +4<;6,/ Wkh Qdvk Vroxwlrq ri dq Dgyhuwlvlqj Glhuhq0 wldo Jdph= Jhqhudol}dwlrq ri d Prgho e| Ohlwpdqq dqg Vfkplwhqgrui/ LHHH Wudqvdfwlrqv rq Dxwrpdwlf Frqwuro / 5; / 437707;1 4;
^44` Ihuvkwpdq/ F1 +4<;:,/ Lghqwlfdwlrq ri Fodvvhv ri Glhuhqwldo Jdphv iru Zklfk wkh Rshq0Orrs lv d Ghjhqhudwhg Ihhgedfn Qdvk Htxloleulxp/ Mrxuqdo ri Rswlpl}dwlrq Wkhru| dqg Dssolfdwlrqv / 88 / 54:0641 ^45` Ihuvkwpdq/ F1 dqg P1L1 Ndplhq +4<;:,/ G|qdplf Gxrsrolvwlf Frpsh0 wlwlrq zlwk Vwlfn| Sulfhv/ Hfrqrphwulfd / 88 / 44840971 ^46` Ihuvkwpdq/ F1 dqg P1L1 Ndplhq +4<<3,/ Wxuqslnh Surshuwlhv lq d Ilqlwh0Krul}rq Glhuhqwldo Jdph= G|qdplf Gxrsro| zlwk Vwlfn| Sulfhv/ Lqwhuqdwlrqdo Hfrqrplf Uhylhz / 64 / 7<0931 ^47` Ihuvkwpdq/ F1 dqg H1 Pxoohu +4<;7,/ Fdslwdo Dffxpxodwlrq Jdphv ri Lqqlwh Gxudwlrq/ Mrxuqdo ri Hfrqrplf Wkhru| / 66 / 65506<1 ^48` Ihuvkwpdq/ F1/ P1 Ndplhq dqg H1 Pxoohu +4<<5,/ Lqwhjudo Jdphv= Wkh0 ru| dqg Dssolfdwlrqv/ lq Ihlfkwlqjhu/ J1 +hg1,/ G|qdplf Hfrqrplf Prg0 hov dqg Rswlpdo Frqwuro / Dpvwhugdp/ Qruwk0Kroodqg/ 5<:06441 ^49` Ixghqehuj/ G1 dqg M1 Wluroh +4<;6,/ Fdslwdo dv d Frpplwphqw= Vwudwh0 jlf Lqyhvwphqw wr Ghwhu Prelolw|/ Mrxuqdo ri Hfrqrplf Wkhru| / 64 / 55:0831 ^4:` Ndplhq/ P1L1 dqg Q1O1 Vfkzduw} +4<;4,/ G|qdplf rswlpl}dwlrq= Wkh Fdofxoxv ri Yduldwlrqv dqg Rswlpdo Frqwuro lq Hfrqrplfv dqg Pdqdjh0 phqw / Dpvwhugdp/ Qruwk0Kroodqg1 ^4;` Nuhsv/ G1 dqg M1 Vfkhlqnpdq +4<;6,/ Txdqwlw| Suhfrpplwphqw dqg Ehuwudqg Frpshwlwlrq \lhog Frxuqrw Rxwfrphv/ Ehoo Mrxuqdo ri Hfr0 qrplfv / 47 / 65906:1 ^4<` Ohlwpdqq/ J1 dqg Z1H1 Vfkplwhqgrui +4<:;,/ Surw Pd{lpl}dwlrq wkurxjk Dgyhuwlvlqj= D Qrq}hur Vxp Glhuhqwldo Jdph Dssurdfk/ LHHH Wudqvdfwlrqv rq Dxwrpdwlf Frqwuro / 56 / 9790831 ^53` Ohylwdq/ U1 dqg P1 Vkxeln +4<:5,/ Sulfh Gxrsro| dqg Fdsdflw| Frq0 vwudlqwv/ Lqwhuqdwlrqdo Hfrqrplf Uhylhz / 46 / 4440561 ^54` Phkopdqq/ D1 +4<;;,/ Dssolhg Glhuhqwldo Jdphv / Qhz \run/ Sohqxp Suhvv1 4<
^55` Phkopdqq/ D1 dqg U1 Zloolqj +4<;6,/ Rq Qrqxqltxh Forvhg0Orrs Qdvk Htxloleuld iru d Fodvv ri Glhuhqwldo Jdphv zlwk d Xqltxh dqg Ghjhq0 hudwh Ihhgedfn Vroxwlrq/ Mrxuqdo ri Rswlpl}dwlrq Wkhru| dqg Dssol0 fdwlrqv / 74 / 7960:51 ^56` Qhuoryh/ P1 dqg N1M1 Duurz +4<95,/ Rswlpdo Dgyhuwlvlqj Srolf| xqghu G|qdplf Frqglwlrqv/ Hfrqrplfd / 5< / 45<0751 ^57` Rveruqh/ P1 dqg F1 Slwfkln +4<;9,/ Sulfh Frpshwlwlrq lq d Fdsdflw|0 Frqvwudlqhg Gxrsro|/ Mrxuqdo ri Hfrqrplf Wkhru| / 6; / 56;0931 ^58` Uhlqjdqxp/ M1 +4<;5,/ D Fodvv ri Glhuhqwldo Jdphv iru Zklfk wkh Forvhg Orrs dqg Rshq Orrs Qdvk Htxloleuld Frlqflgh/ Mrxuqdo ri Rs0 wlpl}dwlrq Wkhru| dqg Dssolfdwlrqv / 69 / 5860951 ^59` Uh|qrogv/ V1V1 +4<;:,/ Fdsdflw| Lqyhvwphqw/ Suhhpswlrq dqg Frpplw0 phqw lq dq Lqqlwh Krul}rq Prgho/ Lqwhuqdwlrqdo Hfrqrplf Uhylhz / 5; / 9<0;;1 ^5:` Vlpddq/ P1 dqg W1 Wdnd|dpd +4<:;,/ Jdph Wkhru| Dssolhg wr G|0 qdplf Gxrsro| Sureohpv zlwk Surgxfwlrq Frqvwudlqwv/ Dxwrpdwlfd / 47 / 4940991 ^5;` Vlqjk/ Q1 dqg [1 Ylyhv +4<;7,/ Sulfh dqg Txdqwlw| Frpshwlwlrq lq d Glhuhqwldwhg Gxrsro|/ Udqg Mrxuqdo ri Hfrqrplfv / 48 / 8790871 ^5<` Vrorz/ U1 +4<89,/D Frqwulexwlrq wr wkh Wkhru| ri Hfrqrplf Jurzwk/ Txduwhuo| Mrxuqdo ri Hfrqrplfv / :3 / 980<71 53