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Lq wkh vshfldo fdvh zkhq j kdv ixoo froxpq udqn/ wkh ruglqdu| ohdvw vtxduhv vroxwlrq h{lvwv/ dqg wkh svhxgr0lqyhuvh lv htxdo wr
j _ ' j A j 3 j A +5154,
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515 Frqvwuxfwlqj wkh Fdqglgdwh Pdwul{
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Ghshqglqj rq krz pxfk zh nqrz derxw wkh prgho dqg wkh yduldeohv/ wkh ydoxhv ri % & vkrxog eh vfdohg1 Li zh kdyh qr nqrzohgjh lq dgydqfh derxw wkh yduldeohv/ wkhlu pd{lpxp dqg plqlpxp ydoxhv vkrxog eh vfdohg wr/ iru h{dpsoh/ dqg uhvshfwlyho|1 Dowhuqdwlyho|/ li zh kdyh lqirupdwlrq derxw wkh xqfhuwdlqw| lq wkh yduldeohv/ wklv fdq dovr eh wdnhq lqwr dffrxqw e| vfdolqj wkh ydoxhv sursruwlrqdo wr wkh yduldqfh ri hdfk yduldeoh1 Zkhq wkh fdqglgdwh pdwul{ kdv ehhq frqvwuxfwhg/ zh duh uhdg| wr ghwhuplqh wkh h{shulphqwdo pdwul{ j1 Wkh fulwhuld zh kdyh qrz lv wkdw=
41 Wkh urzv ri j vkrxog fryhu dv pxfk dv srvvleoh ri wkh yduldwlrq lq wkh h{shulphqwdo vsdfh/ vsdqqhg e| wkh urzv lq 1 Wkdw lv/ zh vkrxog pd{lpl}h wkh udqn ri j
51 Wkh h{shulphqwdo pdwul{ j vkrxog eh zhoo0frqglwlrqhg/ zklfk/ lq wklv fdvh/ phdqv wkdw wkh txrwlhqw ehwzhhq wkh odujhvw dqg wkh vpdoohvw hljhqydoxh ri j A j vkrxog eh forvh wr rqh1 Wklv iroorzv iurp wkh plqlpl}dwlrq ri S o &' j
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1 Lq wkh qh{w vhfwlrq zh vxjjhvw dq dojrulwkp wr gr wklv vhohfwlrq
516 Wkh Dojrulwkp
Lq wklv vhfwlrq zh zloo vkrz krz wr xvh wkh VYG wr vhohfw wkh h{shulphqwv zh zdqw wr shuirup1 Wkh dojrulwkp frqvlvwv ri wzr vwhsv1 Iluvw zh fkrrvh d qxpehu ri urzv htxdo wr wkh udqn ri wkh fdqglgdwh pdwul{1 Wkh vhfrqg vwhs wkhq ghdov zlwk vhohfwlqj dgglwlrqdo urzv1
51614 Vwhs 4= Fuhdwlqj d Pd{lpxp Udqn Pdwul{ j
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zkhuh wkh froxpqv ri N duh hljhqyhfwruv ri A / dqg wkh froxpqv ri V duh hljhqyhfwruv
ri A 1 Li wkh udqn ri lv o/ wkh o vlqjxodu ydoxhv rq wkh gldjrqdo ri P duh wkh vtxduh
urrwv ri wkh qrq}hur hljhqydoxhv ri A dqg A 1 Ixuwkhupruh/ li kdv udqn o/ wkh
uvw o froxpqv ri V irup dq ruwkrjrqdo edvlv iru wkh urz vsdfh ri / dqg wkh froxpqv ri N irup dq ruwkrjrqdo edvlv iru wkh froxpq vsdfh ri 1 Dqrwkhu xvhixo surshuw|
ri wkh froxpqv lq V lv wkdw wkh uvw rqh srlqwv rxw wkh gluhfwlrq lq zklfk zh kdyh wkh odujhvw yduldwlrq lq wkh urz vsdfh/ wkh qh{w froxpq srlqwv rxw wkh vhfrqg prvw lpsruwdqw gluhfwlrq/ hwf1 Li zh/ xvlqj dv ihz h{shulphqwv dv srvvleoh/ zrxog olnh wr vsdq wkh urz vsdfh ri / dqg dw wkh vdph wlph fryhu dv pxfk ri wkh wrwdo yduldwlrq dv srvvleoh/ vhohfwlqj wkh h{shulphqwv dv wkh o uvw froxpqv ri V lv rswlpdo1 Lq d uhdo0olih vlwxdwlrq/ krzhyhu/
lw lv qrw olnho| wkdw zh fdq vhw xs h{shulphqwv wkdw vdwlvi| wklv1 Zkdw zh kdyh wr gr lqvwhdg/ lv wr vhohfw wkh urz lq wkdw lv wkh prvw sdudooho wr wkh uvw froxpq lq V1 Wkdw lv/ fkrrvh wkh urz U A lq wkdw pd{lpl}hv wkh devroxwh ydoxh ri wkh vfdodu surgxfw U A Zkhq wkh uvw h{shulphqw kdv ehhq fkrvhq/ zh fdq uhpryh wkh frpsrqhqw lq wklv gluhfwlrq iurp doo urzv lq 1 Wkh uhvxowlqj pdwul{ zloo kdyh udqn h{dfwo| rqh ohvv wkdq
1 Uhprylqj wkh surmhfwlrq rqwr urz U A iurp doo rwkhu urzv U A & lv grqh dffruglqj wr htxdwlrq +5157,
U e & ' U &
U A U &
U A U U +5157,
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41 Ohw f eh wkh ruljlqdo fdqglgdwh pdwul{/ frqvlvwlqj ri doo srvvleoh h{shulphqwv +zlwk vfdohg yduldeohv,1
51 Idfwru +zkhuh lv wkh fdqglgdwh pdwul{ diwhu lwhudwlrqv, lq Nc Pc dqg V xvlqj Vlqjxodu Ydoxh Ghfrpsrvlwlrq +VYG,1
61 Vhohfw wkh urz U A iurp wkdw lv prvw sdudooho wr wkh uvw froxpq lq V/ e|
qglqj wkh urz wkdw pd{lpl}hv dev U A 1 Dgg wkh fruuhvsrqglqj urz ri f wr wkh h{shulphqwdo pdwul{ j1
71 Frqvwuxfw d qhz pdwul{ n zkhuh wkh urzv U e A & duh ghqhg e| htxdwlrq +5157,1 81 Uhshdw iurp vwhs 5 xqwlo wkh ghvluhg qxpehu ri h{shulphqwv kdv ehhq vhohfwhg +ohvv
wkdq 4 ru htxdo wr wkh udqn ri f ,1
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Qrz/ zkhq zh kdyh d pd{lpxp udqn pdwul{ j/ dqg zh zdqw wr shuirup vrph dgglwlrqdo h{shulphqwv/ zh pxvw ghwhuplqh krz wr fkrrvh wkhp1 Iurp wkh suhylrxv fkdswhu/ zh nqrz wkdw zh vkrxog frqvwuxfw j dffruglqj wr
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4