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715 Vwhs0e|0vwhs Wdujhwv

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eh uhdfkhg duh ri wkh irup B ' Bf4?/ zkhuh ? 5 ] 'iéééc ý2c ý•c fc •c 2c ééééj

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dozd|v ohdsiurj hdfk rwkhu zkhq wkh| *phhw*1

Wr ehjlq wkh dqdo|vlv/ ohw iru B 5ýfc 7Bü/ zlwk 7B : f/

T EBä ' @fn @•Bkn @2Bqc +4<,

zlwk q : k è ý#/ @•k ÷ f dqg @2q ÷ f1 Dvvxph wkdw wkh rswlpdo wdujhw

lv { EBä ' 4B/ f ÷ 4 ÷ •/ dqg lqvhuw wklv h{suhvvlrq lqwr wkh ?uvw rughu frqglwlrq ri wkh pd{lpl}dwlrq sureohp

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4 ' E• n k*#ä3•*k ' E• n q*#ä3•*qé +53, Vlqfh wkhvh whupv duh vwulfwo| lqfuhdvlqj lq k dqg q +zlwk xsshu olplw htxdo wr 4,/ htxdolw| fdq rqo| krog li k ' q/ frqwudglfwlqj wkh dvvxpswlrq q : k1 Wkh vdph dujxphqw krogv li dq| frxqwdeoh qxpehu ri whupv zhuh lqfoxghg lq wkh ydoxh ixqfwlrq T / dqg wkh lpphgldwh frqfoxvlrq wkhuhiruh lv=

Uhpdun 6 Li wkh ydoxh ixqfwlrq rq B 5ýfc 7Bü/ lv ri wkh irup

T EBä ' @fn P"ð'•@ðBkðc +54,

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fruuhvsrqgv wr T EBä ' ý *? B/ lq zklfk fdvh wkh rswlpdo lqqrydwlrq wdujhw lv lq idfw vwhs0e|0vwhs zlwk { EBä ' @h} 4@ { {# B> Eý *? { n *? Bä ' e3•*#_{B ' *ð4} k<fE• n k*#ä 3•*k_é

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T EBä ' @fn @•B n @2B2n éééé

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T EBä ' @fn @6B6n @6n•B6n•n ééééc

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: ý#/ @q ÷ f,1 Wkh lqqrydwlrq wdujhw zloo eh

{ EBä ' E• n q*#ä3•q Bc +58,

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wkh fruuhvsrqglqj *fdqglgdwh* sur?w ixqfwlrqv duh eh irxqg wr eh

Z EBä ' k•B2E#nq3>än k2Bqn kôB2E>3#än kec +59,

zlwk

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716 Ohdghuv* Lqqrydwlrq Wdujhwv

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T EBä ' ñfn ñ•Bkn J EBkä c +5:,

zkhuh wkh odvw whup frqyhujhv idvwhu wr }hur wkdq Bk_{/ dqg k è ý# lv wkh}

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ú E{c Bä ' {#

èEBän J E{#ä c +5;,

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Li wkh ydoxh ixqfwlrq lv ri wkh w|shv ghvfulehg lq wkh suhylrxv vhfwlrq zkhuh wkh vwhs vl}h 4 lv frqvwdqw/ wkhq qhfhvvdulo| 4 ' E• n k*#ä3•*k1

Zh zloo qrz dsso| wklv ?qglqj wr vrph ydoxh ixqfwlrqv/ vrph ri zklfk zloo dsshdu lq wkh h{dpsohv wuhdwhg ehorz1 Iru prvw ri wkhp wkh ?uvw rughu frqglwlrqv ghwhuplqlqj wkh lqqrydwlrq wdujhwv fdqqrw eh vroyhg h{solflwo|1

î T EBä ' E• n Bä3k' • ý kB n J EBä G {*B $ #* E# n •ä/ +f ÷ k,1

î T EBä ' i T EýbB6_{ä ' • ý bB}6_{n J EBä G {*B $ E• n k*6ä}3•*6_{/ +b : f/}

6 5 Q,1

î T EBä ' ý *? E@ n Bä ' ý *? @ ý •

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717 Iroorzhuv* Lqqrydwlrq Wdujhwv

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Dvvxph wkdw wkh ydoxh ixqfwlrq T kdv dq dv|pswrwlf h{sdqvlrq ri wkh irup

T EBä ' Kfn K•B3ò n JýB3òüc EòK• : fä c +64,

dv zhoo dv wkh lqqrydwlrq whfkqrorj|/ ú E{c Bä ' {#

èEBän J E{#ä c E# : fä c +65,

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Ohw xv qrz wuhdw wkh fdvh ò ' #= Dvvxph wkdw wkh dv|pswrwlf h{sdqvlrq ri T fdq eh uh?qhg wr

T EBä ' Kf n K•B3# n K2B3ñ n JýB3ñüc ED : #ä é +68,

Wkh ?uvw rughu frqglwlrq wkhq ehfrphv lq wkh olplw +xvlqj B#3D _{$ f,}

#K•{D ' K2ED ý #ä B#c

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{ ' EK2E• ý D*#ä *K•ä•*DB#*Dc +69,

l1h1 { vwloo frqyhujhv wr lq?qlw|/ exw vorzhu wkdq B1 Li rq wkh rwkhu kdqg dv|pswrwlfdoo| T EBä ' KfnK•B3#/ wkhq wkh rswlpdo lqqrydwlrq pd| eh lq?qlwh

ru ?qlwh/ ghshqglqj rq wkh h{dfw irup ri wkh ydoxh ixqfwlrq1

Ilqdoo| zh wuhdw wkh fdvh ri ò : # zkhuh wkh lqqrydwlrq wdujhw uhpdlqv ?qlwh1 Iru odujh B wkh pd{lpl}dwlrq sureohp fdq eh zulwwhq dv

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f${$Bú E{c Bä ET E{ä ý Kfä c +6:,

iru odujh B/ dw ohdvw li wklv pd{lpl}hu lv xqltxh1 ì{ EBä lv erxqghg iurp deryh vlqfh úT å {#3ò _{whqgv wr }hur iru odujh {1 Ohw}

7{ ' *ð4

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li wklv olplw lv gh?qhg/ wkhq { EBä $ 7{ dv B $ 41 Wkxv/ wkh ghflvlrq zklfk w|sh ri vwudwhj| wr dgrsw ghshqgv rqo| rq wkh frpsdulvrq ri wkh hodvwlflwlhv ri wkh ydoxh ixqfwlrq +ò, dqg wkh lqqrydwlrq kd}dug +#,1

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î ò ÷ #= { $ 4 dqg {*B $ E• ý ò*#ä•*ò/ wkh iroorzhu pdnhv vwhs0e|0 vwhs lqqrydwlrqv>

î ò ' #= { $ 4 dqg {*B#*D _{$ EK}

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î T EBä ' E• n Bä3k' B3k_{ý kB}3Ekn•ä_{n J}í_B3Ekn•äì_{/ +k : f,}

k ÷ # G {*B $ E• ý k*#ä•*k k ' # G {*B#*E#n•ä _{$ •1}

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î T EBä ' i T EýbB6ä/ +b : fc 6 5 Q,1 Vlqfh iru wklv ydoxh ixqfwlrq 4ò ' 44 +lw ghfd|v idvwhu wkdq dq| B3k _{zlwk k ÷ 4,/ { zloo kdyh d}

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î T EBä ' ý *? E@ n Bä ' ý *? B n J E*? Bä/ +@ è f,1 Vlqfh wklv ydoxh ixqfwlrq ghfuhdvhv vorzhu wkdq dq| B3k _{+k : f,/ d jrrg jxhvv jlyhq wkh}

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zkhuh iru doo f é ? é 2 wkh ixqfwlrqv ñ?Eéä/ {?Eéä/ 5?Eéä duh vx!flhqwo|

riwhq frqwlqxrxvo| gl>huhqwldeoh1 Dovr dvvxph wkdw wkh sur?w ixqfwlrq Z Eéä dqg wkh lqqrydwlrq whfkqrorj| ú Eéc Bä duh wzlfh frqwlqxrxvo| gl>huhqwldeoh1 Lq wkh iroorzlqj/ zh zloo vxssuhvv wkh dujxphqw B zkhqhyhu wklv grhv qrw ohdg wr dpeljxlwlhv1

Vwhs 4= H{sdqg ñ E{ä ' P"

?'fo3?ç?EBä dqg ñâE{ä ' P"?'fo3?ü?EBä=

47_{Vhh EKY iru dq h{dfw h{srvlwlrq ri wklv phwkrg1}

Frqvlghu f é 6 é 21 Xvlqj wkh h{sdqvlrq ri {/ zh ?qg wkdw ñ6E{ä ' ñ6E{fä n o3•ñ6â E{fä {•

no32ý•

2ñ6ââ E{fä {2•n ñ6â E{fä {2ün ïýo3ôü

li wkh ghulydwlyhv h{lvw1 Froohfwlqj whupv zlwk wkh vdph srzhuv ri o/ ñ E{ä ' ñfE{ä n o3•ñ•E{ä n o32ñ2E{ä n ïýo3ôü

' çfEBä n o3•ç•EBä n o32ç2EBä n ïýo3ôüc

zlwk

çfEBä ' ñfE{fä c

ç•EBä ' ñfâ E{fä {•n ñ•E{fä c

ç2EBä ' •₂ñfââE{fä {2•n ñfâ E{fä {2n ñ•â E{fä {•n ñ2E{fä é

Surfhhglqj vlploduo|/ zh rewdlq ñâ_{E{ä ' ü}

fEBä n o3•ü•EBä n o32ü2EBä n ïýo3ôüc

üfEBä ' ñâfE{fä c

ü•EBä ' ñââfE{fä {•n ñ•â E{fä c

ü2EBä ' •₂ñfâââE{fä {2•n ñfââE{fä {2n ñââ•E{fä {•n ñ2â E{fä é

Vwhs 5= Dqdo|}h wkh ?uvw rughu qhfhvvdu| frqvwudlqw rq h>ruw

•

oú E{c Bä Eñ E{ä ý ñ EBää ' SâE5 EBää c +6<,

xvlqj wkdw S E5ä ' 52_{*2 dqg} ú E{c Bä ' l_fEBä n o3•_l •EBä n o32l2EBä n ï ý o3ôü_c zlwk frh!flhqwv l_? wr eh ghwhuplqhg odwhu1 Lq 5 EBä ' P" ?'fo3?5?EBä ' ýP" ?'fo3?l? ü û P" ?'fo3E?n•äEç?ý ñ?äú

frpsduh whupv ri rughu f dqg • lq o wr jhw 5fEBä ' f

5•EBä ' lfdñfE{fä ý ñfEBäo é

Vwhs 6= Dsso| +6<, wr wkh Kdplowrq0Mdfrel0Ehoopdq htxdwlrq +KME, gh?qlqj wkh ydoxh ixqfwlrq/

ñ EBä ' Z EBä n 52_{*2 n} •

zlwk 5æúæ dv wkh rssrqhqw*v kd}dug udwh/ dqg {æ lwv lqqrydwlrq wdujhw1 Wkhq

xvh 5fEBä ' f dqg ydulrxv vhulhv h{sdqvlrq wr uhvwdwh wkh KME dv

P"

?'fo3?ñ? ' Z nýP"?'•o3?5?ü2*2

nýP"

?'•o3?5?æüúæP?'f" o3E?n•äEñ?E•*{æä ý ñ?EBää é

Frpsdulqj whupv ri rughu f dqg • lq ohdgv wr ñfEBä ' Z EBä c ñ•EBä ' fé

Vwhs 7= Vlpsoli| ñ E{ä dqg ñâ_{E{ä/ xvlqj wkh h{suhvvlrqv mxvw ghulyhg=}

çfEBä ' Z E{fä ç•EBä ' ZâE{fä {• ç2EBä ' •₂ZââE{fä {•2n ZâE{fä {2 n ñ2E{fä c dqg üfEBä ' ZâE{fä ü•EBä ' ZââE{fä {• ü2EBä ' •₂ZâââE{fä {2•n ZââE{fä {2n ñ2â E{fä é

Vwhs 8= Qrz zh zloo dsso| wkh uhvxowv rewdlqhg vr idu wr wkh ?uvw rughu frqvwudlqw gh?qlqj wkh rswlpdo lqqrydwlrq wdujhwv/

úâ_{dñ E{ä ý ñ EBäo n úñ}â_{E{ä ' fc +73,}

zkhuh zh xvhg wkh vkruwkdqg úâ _{' Yú*Y{ +ú}ââ _{' Y}2_ú*Y{2_{,1 H{sdqglqj}

ú E{c Bä dqg úâ_{E{c Bä derxw o}3• _{' f/ zh rewdlq +vxssuhvvlqj wkh dujxphqw B}

lq ú dqg úâ _{iru frqflvhqhvv,} ú E{ä ' l_fEBä n o3•_l •EBä n o32l2EBä n ï ý o3ôü ' ú E{fä n o3•úâE{fä {• no32ý• 2úââE{fä {2•n úâE{fä {2ün ïýo3ôüc dqg úâ_{E{ä ' 1}

fEBä n o3•1•EBä n o3212EBä n ï

ý o3ôü ' úâ_E{ fä n o3•úââE{fä {• no32ý• 2úâââE{fä {2•n úââE{fä {2ün ïýo3ôüé Vxevwlwxwlqj wkh h{sdqvlrqv lqwr +73,/ ý P" ?'fo3?1? ü û P" ?'fo3?Eç?ý ñ?äúnýP"?'fo3?l? ü ý P" ?'fo3?ü?ü' fc 5:

dqg froohfwlqj wkh whupv ri rughu f lq o3• _{ohdgv wr}

1_fEçfý ñfä n lfüf ' f

/ úâE{fä dZ E{fä ý Z EBäo n ú E{fä ZâE{fä ' fc

ru/ dvvxplqj wkdw wkh vhfrqg rughu frqvwudlqw lv vdwlv?hg/ {fEBä ' @h} 4@

{ ú E{c Bä dZ E{ä ý Z EBäo é

Froohfwlqj whupv ri rughu • lq o3•_/

1_fEç•ý ñ•ä n 1•Eçfý ñfä n lfü•n l•üf ' fc

/ ú E{fä ZââE{fä {•n 2úâE{fä ZâE{fä {•

núââ_E{

fä dZ E{fä ý Z EBäo {• ' fc

/ {•_Y{Y22 Eú E{c Bä dZ E{ä ý Z EBäoä m{'{f ' fc

dqg dvvxplqj wkdw jhqhulfdoo| wkh vhfrqg rughu vx!flhqw frqglwlrq lv vdwlv?hg +Y2_{Eéä *Y{}2 _{÷ f,/ ohdgv wr}

{•EBä ' fé

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Vwhs := Ilqg 52EBä= Vlpsoli| wkh uhvxow iru 5•/ dqg frpsduh whupv ri rughu 2

lq o3• _lq 5 EBä ' P" ?'•o3?5?EBä ' ýP" ?'fo3?l? ü û P" ?'fo3E?n•äEç?ý ñ?äúc wr rewdlq

5fEBä ' fc 5•EBä ' ú E{fä dZ E{fä ý Z EBäo

52EBä ' lfEç•EBä ý ñ•EBää n l•EçfEBä ý ñfEBää

' fé

Vwhs ;= Xvlqj wkh odvw uhvxow/ lghqwli| wkh whupv ri rughu 2 lq o3• _{lq wkh KME}

htxdwlrq P" ?'fo3?ñ? ' Z nýP"?'•o3?5?ü2*2 nýP"?'•o3?5?æüû ý P" ?'fo3?ú?æ ü ý P"

wr rewdlq

ñ2 ' 5•2*2 n 5•æúfædñfE•*{æä ý ñfEBäo

' Eú E{fä dZ E{fä ý Z EBäoä2*2 n 5•æúfædZ E•*{fæä ý Z EBäo c

zkhuh 5•æc úfæc {fæduh wkh ?uvw whupv ri wkh uhvshfwlyh h{sdqvlrqv ri wkh rwkhu

?up1

Vwhs <= Ilqg {2/ wkh qh{w whup lq wkh h{sdqvlrq ri wkh rswlpdo h>ruw ohyho

e| froohfwlrq doo whupv ri rughu 5 lq o3• _{lq +73,=}

1_fEç2ý ñ2ä n 1•Eç•ý ñ•ä n 12Eçfý ñfä n lfü2n l•ü•n l2üf ' fc

/ úâ_E{

fä dZâE{fä {2 n ñ2E{fä ý ñ2EBäo n úââE{fä {2dZ E{fä ý Z EBäo

nú E{fä dZââE{fä {2n ñ2â E{fäo n úâE{fä {2ZâE{fä ' fc

zklfk fdq eh vlpsol?hg wr

{2EBä ' ý_YYEúE{ädñ2_{EúE{ädZE{ä3ZEBäoä*Y{}2E{ä3ñ2EBäoä*Y{2m{'{fé

Vwhs 43= Dsso| suhylrxv uhvxowv wr +6<, dqg froohfw whupv ri rughu 6 lq o3•

wr ?qg wkh qh{w whup lq wkh h{sdqvlrq ri 5 EBä= çf ' Z E{fä c ç• ' fc ç2 ' ZâE{fä {2n ñ2E{fä c üf ' ZâE{fä c ü• ' fc ü2 ' ZââE{fä {2n ñ2â E{fä c l_f ' ú E{fä c l• ' fc l2 ' úâE{fä {2c 1_f ' úâ_E{ fä c 1• ' fc 12 ' úââE{fä {2é 5ôEBä ' lfdç2ý ñ2o n l•dç• ý ñ•o n l2dçfý ñfo ' ú E{fä dZâE{fä {2n ñ2E{fä ý ñ2EBäo n úâE{fä {2dZ E{fä ý Z EBäo

' {2_Y{Y Eú E{ä dZ E{ä ý Z EBäoä m{'{f n ú E{fä dñ2E{fä ý ñ2EBäo

' ú E{fc Bä dñ2E{fä ý ñ2EBäo c

vlqfh wkh ghulydwlyh lv }hur1

Dv lq EKY wkhvh h{sdqvlrqv frxog eh frqwlqxhg wr ?qg kljkhu0rughu hi0 ihfwv/ exw wkhvh zloo eh kdug wr lqwhusuhw vlqfh wkh| zloo eh d pl{wxuh ri wkh rwkhu h>hfwv dqg lqyroyh kljkhu0rughu ghulydwlyhv ri wkh xqghuo|lqj rz sur?wv dqg lqqrydwlrq whfkqrorj|1

Uhihuhqfhv

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^6` Djklrq/ Sklolssh dqg Krzlww/ Shwhu +4<<;,1 Hqgrjhqrxv Jurzwk Wkhru|1 Fdpeulgjh/ PD= PLW Suhvv1

^7` Dpdeoh/ Euxqr +4<<9,1 4Furlvvdqfh hw F|fohv Hqgrjëqhv Lqgxlwv Sdu Ohv Lqqrydwlrqv Udglfdohv hw Lqfuìphqwdohv41 Dqqdohv g*Ìfrqrplh hw gh Vwdwlvwltxh/ 77=<484431

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Frpshwlwlrq zlwk Vwudwhjlf Fkrlfh ri Yduldqfh dqg Fr0Yduldqfh41 OEV/ plphr1

^:` Kduulv/ Fkulvwrskhu dqg Ylfnhuv/ Mrkq +4<;:,1 4Udflqj zlwk Xqfhu0 wdlqw|41 Uhylhz ri Hfrqrplf Vwxglhv/ 87=48541

^;` Krhuqlj/ Vwh>hq +4<<;,1 4Dv|pphwulf Htxloleuld lq dq U)G Udfh zlwk Hqgrjhqrxv Fkrlfh ri Vwhs Vl}h41 plphr/ Hxurshdq Xqlyhuvlw| Lqvwlwxwh/ Ioruhqfh1

^<` Krhuqlj/ Vwh>hq +4<<;e,1 4D Qrwh rq 4Frpshwlwlrq dqg Jurzwk zlwk Vwhs0e|0Vwhs Lqqrydwlrq= Dq H{dpsoh441 plphr/ Hxurshdq Xqlyhuvlw| Lqvwlwxwh/ Ioruhqfh1

^43` Ohh/ W1 dqg Zlogh/ O1 +4<;3,1 4Pdunhw Vwuxfwxuh dqg Lqqrydwlrq= D Uhirupxodwlrq41 Txduwhuo| Mrxuqdo ri Hfrqrplfv/ <7=75<87691

^44` Orxu|/ J1 +4<:<,1 4Pdunhw Vwuxfwxuh dqg Lqqrydwlrq41 Txduwhuo| Mrxuqdo ri Hfrqrplfv/ <6=6<887431

^45` Pdooldulv/ D1J1 dqg Z1D1Eurfn +4<;5,1 Vwrfkdvwlf Phwkrgv lq Hfrqrplfv dqg Ilqdqfh1 Dpvwhugdp= Qruwk0Kroodqg1

^46` Pduwlq/ Vwhyhq +4<<6,1 Dgydqfhg Lqgxvwuldo Hfrqrplfv1 R{irug= Eodfn0 zhoo1

^47` Pdvnlq/ Hulf dqg Wluroh/ Mhdq +4<<:,1 4Pdunry Shuihfw Htxloleulxp/ L= Revhuydeoh Dfwlrqv41 Kduydug Lqvwlwxwh ri Hfrqrplf Uhvhdufk Glvfxvvlrq Sdshu 4:<<1

^48` Prwwd/ Pdvvlpr/ Wklvvh/ Mdftxhv0Iudqêrlv/ dqg Fdeudohv/ Dqwrqlr +4<<:,1 4Rq wkh Shuvlvwhqfh ri Ohdghuvkls ru Ohdsiurjjlqj lq Lqwhuqdwlrqdo Wudgh41 Lqwhuqdwlrqdo Hfrqrplf Uhylhz/ 6;+7,=;3<8571

^49` Uhlqjdqxp/ Mhqqlihu I1 +4<;4,1 4G|qdplf Jdphv ri Lqqrydwlrq41 Mrxu0 qdo ri Hfrqrplf Wkhru|/ 58=548741 fi1 +4<;8, %Fruuljhqgxp%/ 68/ 4<90:1 ^4:` Uhlqjdqxp/ Mhqqlihu I1 +4<;8,1 4Lqqrydwlrq dqg Lqgxvwu| Hyroxwlrq41

Txduwhuo| Mrxuqdo ri Hfrqrplfv/ sdjhv ;48<<1

^4;` Uhlqjdqxp/ Mhqqlihu I1 +4<;<,1 4Wkh Wlplqj ri Lqqrydwlrqv41 Lq Vfkpdohqvhh/ U1 dqg Zloolj/ U1/ hglwruv/ Kdqgerrn ri Lqgxvwuldo Rujdql}d0 wlrq/ fkdswhu 471 Dpvwhugdp= Qruwk0Kroodqg1

^4<` Urvhqnudq}/ Vwhskdqlh +4<<9,1 4Txdolw| Lpsuryhphqwv dqg wkh Lqfhqwlyh wr Ohdsiurj41 Lqwhuqdwlrqdo Mrxuqdo ri Lqgxvwuldo Rujdql}dwlrq/ 48=5768941 ^53` Vfkhuhu/ Iuhghulf P1 +4<<5,1 Lqwhuqdwlrqdo Kljk0Whfkqrorj| Frpshwlwlrq1

Fdpeulgjh/ PD= Kduydug Xqlyhuvlw| Suhvv1

^54` Vwrnh|/ Qdqf| O1 dqg Oxfdv/ Urehuw H1 +4<;<,1 Uhfxuvlyh Phwkrgv lq Hfrqrplf G|qdplfv1 Fdpeulgjh/ PD= Kduydug Xqlyhuvlw| Suhvv1

^55` Wluroh/ Mhdq +4<;;,1 Wkh Wkhru| ri Lqgxvwuldo Rujdql}dwlrq1 Fdpeulgjh/ PD= PLW Suhvv1

^56` Zhuqhuihow/ Elujhu +4<;;,1 4Rq H{lvwhqfh ri d Qdvk Htxloleulxp Srlqw lq Q0Shuvrq Qrq0]hur Vxp Vwrfkdvwlf Mxps Gl>huhqwldo Jdphv41 Rswlpdo Frqwuro Dssolfdwlrqv dqg Phwkrgv/ <=77<87891