[David C Lay] Linear Algebra and Its Applications(BookZZ.org)

' 0 6 3 5 )

& % * 5 * 0 /

/LQHDU$OJHEUD DQG,WV$SSOLFDWLRQV 'DYLG& /D\ 8QLYHUVLW\RI0DU\ODQG³&ROOHJH3DUN

Addison-Wesley

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
,6%1 ,6%1

7RP\ZLIH /LOOLDQ DQGRXUFKLOGUHQ &KULVWLQD 'HERUDK DQG0HOLVVD ZKRVH VXSSRUW HQFRXUDJHPHQW DQGIDLWKIXO SUD\HUVPDGHWKLVERRNSRVVLEOH

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³$VVHWVIRU8QGHUJUDG XDWH0DWKHPDWLFV ZLWK'&DUOVRQ & 5 -RKQVRQ DQG$ ' 3RUWHU 3URIHVVRU/D\KDVUHFHLYHGIRXUXQLYHUVLW\DZDUGVIRUWHDFKLQJH[FHOOHQFH LQFOXG LQJ LQ WKHWLWOHRI'LVWLQJXLVKHG6FKRODU²7HDFKHURIWKH8QLYHUVLW\RI0DU\ODQG ,Q KHZDVJLYHQRQHRIWKH0DWKHPDWLFDO$VVRFLDWLRQRI$PHULFD·V$ZDUGVIRU 'LVWLQJXLVKHG&ROOHJHRU8QLYHUVLW\7HDFKLQJRI0DWKHPDWLFV +HKDVEHHQHOHFWHG E\WKHXQLYHUVLW\VWXGHQWVWRPHPEHUVKLSLQ$OSKD/DPEGD'HOWD1DWLRQDO6FKRODVWLF +RQRU6RFLHW\DQG*ROGHQ.H\1DWLRQDO+RQRU6RFLHW\ ,Q $XURUD8QLYHUVLW\ FRQIHUUHGRQKLPWKH2XWVWDQGLQJ$OXPQXVDZDUG /D\LVDPHPEHURIWKH$PHULFDQ 0DWKHPDWLFDO6RFLHW\ WKH&DQDGLDQ0DWKHPDWLFDO6RFLHW\ WKH,QWHUQDWLRQDO/LQHDU $OJHEUD6RFLHW\ WKH0DWKHPDWLFDO$VVRFLDWLRQRI$PHULFD 6LJPD;L DQGWKH6RFLHW\ IRU,QGXVWULDODQG$SSOLHG0DWKHPDWLFV 6LQFH KHKDVVHUYHGVHYHUDOWHUPVRQWKH QDWLRQDOERDUGRIWKH$VVRFLDWLRQRI&KULVWLDQVLQWKH0DWKHPDWLFDO6FLHQFHV

JW

&RQWHQWV 3UHIDFH L[ $ 1RWHWR6WXGHQWV

[Y

$IBQUFS /LQHDU(TXDWLRQVLQ/LQHDU$OJHEUD

*/530%6$503: &9".1-& /LQHDU0RGHOVLQ(FRQRPLFVDQG(QJLQHHULQJ 6\VWHPVRI/LQHDU(TXDWLRQV 5RZ5HGXFWLRQDQG(FKHORQ)RUPV 9HFWRU(TXDWLRQV 7KH0DWUL[(TXDWLRQ A[ D E 6ROXWLRQ6HWVRI/LQHDU6\VWHPV $SSOLFDWLRQVRI/LQHDU6\VWHPV /LQHDU,QGHSHQGHQFH ,QWURGXFWLRQWR/LQHDU7UDQVIRUPDWLRQV 7KH0DWUL[RID/LQHDU7UDQVIRUPDWLRQ /LQHDU0RGHOVLQ%XVLQHVV 6FLHQFH DQG(QJLQHHULQJ 6XSSOHPHQWDU\([HUFLVHV

$IBQUFS 0DWUL[$OJHEUD

*/530%6$503: &9".1-& &RPSXWHU0RGHOVLQ$LUFUDIW'HVLJQ 0DWUL[2SHUDWLRQV 7KH,QYHUVHRID0DWUL[ &KDUDFWHUL]DWLRQVRI,QYHUWLEOH0DWULFHV 3DUWLWLRQHG0DWULFHV 0DWUL[)DFWRUL]DWLRQV 7KH/HRQWLHI,QSXW²2XWSXW0RGHO $SSOLFDWLRQVWR&RPSXWHU*UDSKLFV 6XEVSDFHVRI Rn 'LPHQVLRQDQG5DQN 6XSSOHPHQWDU\([HUFLVHV

$IBQUFS 'HWHUPLQDQWV

*/530%6$503: &9".1-& 5DQGRP3DWKVDQG'LVWRUWLRQ ,QWURGXFWLRQWR'HWHUPLQDQWV 3URSHUWLHVRI'HWHUPLQDQWV

W

WJ

$POUFOUT

&UDPHU·V5XOH 9ROXPH DQG/LQHDU7UDQVIRUPDWLRQV 6XSSOHPHQWDU\([HUFLVHV

$IBQUFS 9HFWRU6SDFHV

*/530%6$503: &9".1-& 6SDFH)OLJKWDQG&RQWURO6\VWHPV 9HFWRU6SDFHVDQG6XEVSDFHV 1XOO6SDFHV &ROXPQ6SDFHV DQG/LQHDU7UDQVIRUPDWLRQV /LQHDUO\,QGHSHQGHQW6HWV %DVHV &RRUGLQDWH6\VWHPV 7KH'LPHQVLRQRID9HFWRU6SDFH 5DQN &KDQJHRI%DVLV $SSOLFDWLRQVWR'LIIHUHQFH(TXDWLRQV $SSOLFDWLRQVWR0DUNRY&KDLQV 6XSSOHPHQWDU\([HUFLVHV

$IBQUFS (LJHQYDOXHVDQG(LJHQYHFWRUV

*/530%6$503: &9".1-& '\QDPLFDO6\VWHPVDQG6SRWWHG2ZOV (LJHQYHFWRUVDQG(LJHQYDOXHV 7KH&KDUDFWHULVWLF(TXDWLRQ 'LDJRQDOL]DWLRQ (LJHQYHFWRUVDQG/LQHDU7UDQVIRUPDWLRQV &RPSOH[(LJHQYDOXHV 'LVFUHWH'\QDPLFDO6\VWHPV $SSOLFDWLRQVWR'LIIHUHQWLDO(TXDWLRQV ,WHUDWLYH(VWLPDWHVIRU(LJHQYDOXHV 6XSSOHPHQWDU\([HUFLVHV

$IBQUFS 2UWKRJRQDOLW\DQG/HDVW6TXDUHV

*/530%6$503: &9".1-& 7KH1RUWK$PHULFDQ'DWXP DQG*36 1DYLJDWLRQ ,QQHU3URGXFW /HQJWK DQG2UWKRJRQDOLW\ 2UWKRJRQDO6HWV 2UWKRJRQDO3URMHFWLRQV 7KH*UDP²6FKPLGW3URFHVV /HDVW6TXDUHV3UREOHPV $SSOLFDWLRQVWR/LQHDU0RGHOV ,QQHU3URGXFW6SDFHV $SSOLFDWLRQVRI,QQHU3URGXFW6SDFHV 6XSSOHPHQWDU\([HUFLVHV

$POUFOUT WJJ

$IBQUFS 6\PPHWULF0DWULFHVDQG4XDGUDWLF)RUPV */530%6$503: &9".1-& 0XOWLFKDQQHO,PDJH3URFHVVLQJ 'LDJRQDOL]DWLRQRI6\PPHWULF0DWULFHV 4XDGUDWLF)RUPV &RQVWUDLQHG2SWLPL]DWLRQ 7KH6LQJXODU9DOXH'HFRPSRVLWLRQ $SSOLFDWLRQVWR,PDJH3URFHVVLQJDQG6WDWLVWLFV 6XSSOHPHQWDU\([HUFLVHV

$IBQUFS 7KH*HRPHWU\RI9HFWRU6SDFHV */530%6$503: &9".1-& 7KH3ODWRQLF6ROLGV $IÀQH&RPELQDWLRQV $IÀQH,QGHSHQGHQFH &RQYH[&RPELQDWLRQV +\SHUSODQHV 3RO\WRSHV &XUYHVDQG6XUIDFHV

$IBQUFS 2SWLPL]DWLRQ2QOLQH */530%6$503: &9".1-& 7KH%HUOLQ$LUOLIW 0DWUL[*DPHV /LQHDU3URJUDPPLQJ³*HRPHWULF0HWKRG /LQHDU3URJUDPPLQJ³6LPSOH[0HWKRG 'XDOLW\

$IBQUFS )LQLWH6WDWH0DUNRY&KDLQV2QOLQH */530%6$503: &9".1-& *RRJOHDQG0DUNRY&KDLQV ,QWURGXFWLRQDQG([DPSOHV 7KH6WHDG\6WDWH9HFWRUDQG*RRJOH·V3DJH5DQN &RPPXQLFDWLRQ&ODVVHV &ODVVLÀFDWLRQRI6WDWHVDQG3HULRGLFLW\ 7KH)XQGDPHQWDO0DWUL[ 0DUNRY&KDLQVDQG%DVHEDOO6WDWLVWLFV

WJJJ

$POUFOUT

$SSHQGL[HV $ %

8QLTXHQHVVRIWKH5HGXFHG(FKHORQ)RUP &RPSOH[1XPEHUV $

*ORVVDU\ $ $QVZHUVWR2GG1XPEHUHG([HUFLVHV ,QGH[ , 3KRWR&UHGLWV 3

$

$

3UHIDFH 7KHUHVSRQVHRIVWXGHQWVDQGWHDFKHUVWRWKHÀ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

8)"5h4 /&8 */ 5)*4 &%*5*0/ 7KHPDLQJRDORIWKLVUHYLVLRQZDVWRXSGDWHWKHH[HUFLVHVDQGSURYLGHDGGLWLRQDOFRQ WHQW ERWKLQWKHERRNDQGRQOLQH 0RUHWKDQSHUFHQWRIWKHH[HUFLVHVDUHQHZRUXSGDWHG HVSHFLDOO\WKHFRPSXWD WLRQDOH[HUFLVHV 7KHH[HUFLVHVHWVUHPDLQRQHRIWKHPRVWLPSRUWDQWIHDWXUHVRIWKLV ERRN DQGWKHVHQHZH[HUFLVHVIROORZWKHVDPHKLJKVWDQGDUGRIWKHH[HUFLVHVHWVRI WKHSDVWWKUHHHGLWLRQV 7KH\DUHFUDIWHGLQDZD\WKDWUHWHOOVWKHVXEVWDQFHRIHDFK RIWKHVHFWLRQVWKH\IROORZ GHYHORSLQJWKHVWXGHQWV·FRQÀGHQFHZKLOHFKDOOHQJLQJ WKHPWRSUDFWLFHDQGJHQHUDOL]HWKHQHZLGHDVWKH\KDYHMXVWHQFRXQWHUHG 7ZHQW\À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ÀYH PLQXWHFODVVSHULRGV $ VHFRQGFRXUVHLQOLQHDUDOJHEUDDSSOLFDWLRQVW\SLFDOO\EHJLQVZLWKDVXEVWDQWLDO UHYLHZRINH\LGHDVIURPWKHÀUVWFRXUVH ,ISDUWRI&KDSWHULVLQWKHÀUVWFRXUVH WKHVHFRQGFRXUVHFRXOGLQFOXGHDEULHIUHYLHZRIVHFWLRQVWRDQGWKHQDIRFXVRQ WKHJHRPHWU\LQVHFWLRQVDQG 7KDWZRXOGOHDGQDWXUDOO\LQWRWKHRQOLQHFKDSWHUV DQG ZKLFKKDYHEHHQXVHGZLWK&KDSWHUDWDQXPEHURIVFKRROVIRUWKHSDVW ÀYH\HDUV 7KH 6WXG\*XLGH ZKLFKKDVDOZD\VEHHQDQLQWHJUDOSDUWRIWKHERRN KDVEHHQXS GDWHGWRFRYHUWKHQHZ&KDSWHU $VZLWKSDVWHGLWLRQV WKH 6WXG\*XLGH LQFRUSRUDWHV JY

Y

1SFGBDF

GHWDLOHGVROXWLRQVWRHYHU\WKLUGRGGQXPEHUHGH[HUFLVHDVZHOODVVROXWLRQVWRHYHU\ RGGQXPEHUHGZULWLQJH[HUFLVHIRUZKLFKWKHWH[WRQO\SURYLGHVDKLQW 7ZRQHZFKDSWHUVDUHQRZDYDLODEOHRQOLQH DQGFDQEHXVHGLQDVHFRQGFRXUVH &KDSWHU 2SWLPL]DWLRQ &KDSWHU )LQLWH6WDWH0DUNRY&KDLQV $QDFFHVVFRGHLVUHTXLUHGDQGLVDYDLODEOHWRTXDOLÀHGDGRSWHUV )RUPRUHLQIRUPD WLRQ YLVLW ZZZSHDUVRQKLJKHUHGFRPLUF RUFRQWDFW\RXU3HDUVRQUHSUHVHQWDWLYH 3RZHU3RLQW VOLGHVDUHQRZDYDLODEOHIRUWKHFRUHVHFWLRQVRIWKHWH[W DOVRLQ FOXGHGDUHÀJXUHVIURPWKHWH[W

%*45*/$5*7& '&"563&4 (DUO\,QWURGXFWLRQRI.H\&RQFHSWV 0DQ\IXQGDPHQWDOLGHDVRIOLQHDUDOJHEUDDUHLQWURGXFHGZLWKLQWKHÀUVWVHYHQOHFWXUHV LQWKHFRQFUHWHVHWWLQJRI Rn DQGWKHQJUDGXDOO\H[DPLQHGIURPGLIIHUHQWSRLQWVRIYLHZ /DWHUJHQHUDOL]DWLRQVRIWKHVHFRQFHSWVDSSHDUDVQDWXUDOH[WHQVLRQVRIIDPLOLDULGHDV YLVXDOL]HGWKURXJKWKHJHRPHWULFLQWXLWLRQGHYHORSHGLQ&KDSWHU $ PDMRUDFKLHYHPHQW RIWKLVWH[WLVWKDWWKHOHYHORIGLIÀFXOW\LVIDLUO\HYHQWKURXJKRXWWKHFRXUVH

$ 0RGHUQ9LHZRI0DWUL[0XOWLSOLFDWLRQ *RRGQRWDWLRQLVFUXFLDO DQGWKHWH[WUHÁHFWVWKHZD\VFLHQWLVWVDQGHQJLQHHUVDFWXDOO\ XVHOLQHDUDOJHEUDLQSUDFWLFH 7KHGHÀQLWLRQVDQGSURRIVIRFXVRQWKHFROXPQVRIDPD WUL[UDWKHUWKDQRQWKHPDWUL[HQWULHV $ FHQWUDOWKHPHLVWRYLHZDPDWUL[²YHFWRUSURGXFW A[ DVDOLQHDUFRPELQDWLRQRIWKHFROXPQVRI A 7KLVPRGHUQDSSURDFKVLPSOLÀHVPDQ\ DUJXPHQWV DQGLWWLHVYHFWRUVSDFHLGHDVLQWRWKHVWXG\RIOLQHDUV\VWHPV

/LQHDU7UDQVIRUPDWLRQV /LQHDUWUDQVIRUPDWLRQVIRUPD´WKUHDGµWKDWLVZRYHQLQWRWKHIDEULFRIWKHWH[W 7KHLU XVHHQKDQFHVWKHJHRPHWULFÁDYRURIWKHWH[W ,Q&KDSWHU IRULQVWDQFH OLQHDUWUDQV IRUPDWLRQVSURYLGHDG\QDPLFDQGJUDSKLFDOYLHZRIPDWUL[²YHFWRUPXOWLSOLFDWLRQ

(LJHQYDOXHVDQG'\QDPLFDO6\VWHPV (LJHQYDOXHVDSSHDUIDLUO\HDUO\LQWKHWH[W LQ&KDSWHUVDQG %HFDXVHWKLVPDWHULDO LVVSUHDGRYHUVHYHUDOZHHNV VWXGHQWVKDYHPRUHWLPHWKDQXVXDOWRDEVRUEDQGUHYLHZ WKHVHFULWLFDOFRQFHSWV (LJHQYDOXHVDUHPRWLYDWHGE\DQGDSSOLHGWRGLVFUHWHDQGFRQ WLQXRXVG\QDPLFDOV\VWHPV ZKLFKDSSHDULQ6HFWLRQV DQG DQGLQÀYH VHFWLRQVRI&KDSWHU 6RPHFRXUVHVUHDFK&KDSWHU DIWHUDERXWÀYHZHHNVE\FRYHULQJ 6HFWLRQV DQGLQVWHDGRI&KDSWHU 7KHVHWZRRSWLRQDOVHFWLRQVSUHVHQWDOOWKH YHFWRUVSDFHFRQFHSWVIURP&KDSWHU QHHGHGIRU&KDSWHU

2UWKRJRQDOLW\DQG/HDVW6TXDUHV3UREOHPV 7KHVHWRSLFVUHFHLYHDPRUHFRPSUHKHQVLYHWUHDWPHQWWKDQLVFRPPRQO\IRXQGLQEHJLQ QLQJWH[WV 7KH/LQHDU$OJHEUD&XUULFXOXP6WXG\*URXSKDVHPSKDVL]HGWKHQHHGIRU DVXEVWDQWLDOXQLWRQRUWKRJRQDOLW\DQGOHDVWVTXDUHVSUREOHPV EHFDXVHRUWKRJRQDOLW\ SOD\VVXFKDQLPSRUWDQWUROHLQFRPSXWHUFDOFXODWLRQVDQGQXPHULFDOOLQHDUDOJHEUDDQG EHFDXVHLQFRQVLVWHQWOLQHDUV\VWHPVDULVHVRRIWHQLQSUDFWLFDOZRUN

1SFGBDF

YJ

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

$ 6WURQJ*HRPHWULF(PSKDVLV (YHU\PDMRUFRQFHSWLQWKHFRXUVHLVJLYHQDJHRPHWULFLQWHUSUHWDWLRQ EHFDXVHPDQ\ VWXGHQWVOHDUQEHWWHUZKHQWKH\FDQYLVXDOL]HDQLGHD 7KHUHDUHVXEVWDQWLDOO\PRUH GUDZLQJVKHUHWKDQXVXDO DQGVRPHRIWKHÀJXUHVKDYHQHYHUEHIRUHDSSHDUHGLQDOLQHDU DOJHEUDWH[W

([DPSOHV 7KLVWH[WGHYRWHVDODUJHUSURSRUWLRQRILWVH[SRVLWRU\PDWHULDOWRH[DPSOHVWKDQGR PRVWOLQHDUDOJHEUDWH[WV 7KHUHDUHPRUHH[DPSOHVWKDQDQLQVWUXFWRUZRXOGRUGLQDULO\ SUHVHQWLQFODVV %XWEHFDXVHWKHH[DPSOHVDUHZULWWHQFDUHIXOO\ ZLWKORWVRIGHWDLO VWXGHQWVFDQUHDGWKHPRQWKHLURZQ

7KHRUHPVDQG3URRIV ,PSRUWDQWUHVXOWVDUHVWDWHGDVWKHRUHPV 2WKHUXVHIXOIDFWVDUHGLVSOD\HGLQWLQWHGER[HV IRUHDV\UHIHUHQFH 0RVWRIWKHWKHRUHPVKDYHIRUPDOSURRIV ZULWWHQZLWKWKHEHJLQQLQJ VWXGHQWLQPLQG ,QDIHZFDVHV WKHHVVHQWLDOFDOFXODWLRQVRIDSURRIDUHH[KLELWHGLQD FDUHIXOO\FKRVHQH[DPSOH 6RPHURXWLQHYHULÀFDWLRQVDUHVDYHGIRUH[HUFLVHV ZKHQWKH\ ZLOOEHQHÀWVWXGHQWV

3UDFWLFH3UREOHPV $ IHZFDUHIXOO\VHOHFWHG3UDFWLFH3UREOHPVDSSHDUMXVWEHIRUHHDFKH[HUFLVHVHW &RP SOHWHVROXWLRQVIROORZWKHH[HUFLVHVHW 7KHVHSUREOHPVHLWKHUIRFXVRQSRWHQWLDOWURXEOH VSRWVLQWKHH[HUFLVHVHWRUSURYLGHD´ZDUPXSµIRUWKHH[HUFLVHV DQGWKHVROXWLRQV RIWHQFRQWDLQKHOSIXOKLQWVRUZDUQLQJVDERXWWKHKRPHZRUN

([HUFLVHV 7KHDEXQGDQWVXSSO\RIH[HUFLVHVUDQJHVIURPURXWLQHFRPSXWDWLRQVWRFRQFHSWXDOTXHV WLRQVWKDWUHTXLUHPRUHWKRXJKW $ JRRGQXPEHURILQQRYDWLYHTXHVWLRQVSLQSRLQWFRQ FHSWXDOGLIÀFXOWLHVWKDW, KDYHIRXQGRQVWXGHQWSDSHUVRYHUWKH\HDUV (DFKH[HUFLVH VHWLVFDUHIXOO\DUUDQJHGLQWKHVDPHJHQHUDORUGHUDVWKHWH[W KRPHZRUNDVVLJQPHQWV DUHUHDGLO\DYDLODEOHZKHQRQO\SDUWRIDVHFWLRQLVGLVFXVVHG $ QRWDEOHIHDWXUHRIWKH H[HUFLVHVLVWKHLUQXPHULFDOVLPSOLFLW\ 3UREOHPV´XQIROGµTXLFNO\ VRVWXGHQWVVSHQG OLWWOHWLPHRQQXPHULFDOFDOFXODWLRQV 7KHH[HUFLVHVFRQFHQWUDWHRQWHDFKLQJXQGHUVWDQG LQJUDWKHUWKDQPHFKDQLFDOFDOFXODWLRQV 7KHH[HUFLVHVLQWKH )RXUWK(GLWLRQ PDLQWDLQ WKHLQWHJULW\RIWKHH[HUFLVHVIURPWKHWKLUGHGLWLRQ ZKLOHSURYLGLQJIUHVKSUREOHPVIRU VWXGHQWVDQGLQVWUXFWRUV ([HUFLVHVPDUNHGZLWKWKHV\PERO>0@ DUHGHVLJQHGWREHZRUNHGZLWKWKHDLGRID ´0DWUL[SURJUDPµDFRPSXWHUSURJUDP VXFKDV 0$7/$% 0DSOH70 0DWKHPDWLFD 0DWK&DG RU'HULYH70 RUDSURJUDPPDEOHFDOFXODWRUZLWKPDWUL[FDSDELOLWLHV VXFKDV WKRVHPDQXIDFWXUHGE\7H[DV,QVWUXPHQWV

YJJ 1SFGBDF

7UXH)DOVH4XHVWLRQV 7RHQFRXUDJHVWXGHQWVWRUHDGDOORIWKHWH[WDQGWRWKLQNFULWLFDOO\ , KDYHGHYHORSHG VLPSOHWUXHIDOVHTXHVWLRQVWKDWDSSHDULQVHFWLRQVRIWKHWH[W MXVWDIWHUWKHFRPSXWD WLRQDOSUREOHPV 7KH\FDQEHDQVZHUHGGLUHFWO\IURPWKHWH[W DQGWKH\SUHSDUHVWXGHQWV IRUWKHFRQFHSWXDOSUREOHPVWKDWIROORZ 6WXGHQWVDSSUHFLDWHWKHVH TXHVWLRQV³DIWHU WKH\JHWXVHGWRWKHLPSRUWDQFHRIUHDGLQJWKHWH[WFDUHIXOO\ %DVHGRQFODVVWHVWLQJ DQGGLVFXVVLRQVZLWKVWXGHQWV , GHFLGHGQRWWRSXWWKHDQVZHUVLQWKHWH[W 7KH 6WXG\ *XLGH WHOOVWKHVWXGHQWVZKHUHWRÀQGWKHDQVZHUVWRWKHRGGQXPEHUHGTXHVWLRQV $Q DGGLWLRQDOWUXHIDOVHTXHVWLRQVPRVWO\DWWKHHQGVRIFKDSWHUV WHVWXQGHUVWDQGLQJ RIWKHPDWHULDO 7KHWH[WGRHVSURYLGHVLPSOH7) DQVZHUVWRPRVWRIWKHVHTXHVWLRQV EXWLWRPLWVWKHMXVWLÀFDWLRQVIRUWKHDQVZHUVZKLFKXVXDOO\UHTXLUHVRPHWKRXJKW

:ULWLQJ([HUFLVHV $QDELOLW\WRZULWHFRKHUHQWPDWKHPDWLFDOVWDWHPHQWVLQ(QJOLVKLVHVVHQWLDOIRUDOOVWX GHQWVRIOLQHDUDOJHEUD QRWMXVWWKRVHZKRPD\JRWRJUDGXDWHVFKRROLQPDWKHPDWLFV 7KHWH[WLQFOXGHVPDQ\H[HUFLVHVIRUZKLFKDZULWWHQMXVWLÀFDWLRQLVSDUWRIWKHDQVZHU &RQFHSWXDOH[HUFLVHVWKDWUHTXLUHDVKRUWSURRIXVXDOO\FRQWDLQKLQWVWKDWKHOSDVWXGHQW JHWVWDUWHG )RUDOORGGQXPEHUHGZULWLQJH[HUFLVHV HLWKHUDVROXWLRQLVLQFOXGHGDW WKHEDFNRIWKHWH[WRUDKLQWLVSURYLGHGDQGWKHVROXWLRQLVJLYHQLQWKH 6WXG\*XLGH GHVFULEHGEHORZ

&RPSXWDWLRQDO7RSLFV 7KHWH[WVWUHVVHVWKHLPSDFWRIWKHFRPSXWHURQERWKWKHGHYHORSPHQWDQGSUDFWLFHRI OLQHDUDOJHEUDLQVFLHQFHDQGHQJLQHHULQJ )UHTXHQW1XPHULFDO1RWHVGUDZDWWHQWLRQ WRLVVXHVLQFRPSXWLQJDQGGLVWLQJXLVKEHWZHHQWKHRUHWLFDOFRQFHSWV VXFKDVPDWUL[ LQYHUVLRQ DQGFRPSXWHULPSOHPHQWDWLRQV VXFKDV/8 IDFWRUL]DWLRQV

8&# 4611035 7KLV:HEVLWHDW ZZZSHDUVRQKLJKHUHGFRPOD\ FRQWDLQVVXSSRUWPDWHULDOIRUWKHWH[W ERRN )RUVWXGHQWV WKH:HEVLWHFRQWDLQV UHYLHZVKHHWV DQG SUDFWLFHH[DPV ZLWK VROXWLRQV WKDWFRYHUWKHPDLQWRSLFVLQWKHWH[W 7KH\FRPHGLUHFWO\IURPFRXUVHV, KDYHWDXJKWLQSDVW\HDUV (DFKUHYLHZVKHHWLGHQWLÀHVNH\GHÀQLWLRQV WKHRUHPV DQG VNLOOVIURPDVSHFLÀHGSRUWLRQRIWKHWH[W

$SSOLFDWLRQVE\&KDSWHUV 7KH:HEVLWHDOVRFRQWDLQVVHYHQ&DVH6WXGLHV ZKLFKH[SDQGWRSLFVLQWURGXFHGDWWKH EHJLQQLQJRIHDFKFKDSWHU DGGLQJUHDOZRUOGGDWDDQGRSSRUWXQLWLHVIRUIXUWKHUH[SOR UDWLRQ ,QDGGLWLRQ PRUHWKDQ$SSOLFDWLRQ3URMHFWVHLWKHUH[WHQGWRSLFVLQWKHWH[WRU LQWURGXFHQHZDSSOLFDWLRQV VXFKDVFXELFVSOLQHV DLUOLQHÁLJKWURXWHV GRPLQDQFHPDWUL FHVLQVSRUWVFRPSHWLWLRQ DQGHUURUFRUUHFWLQJFRGHV 6RPHPDWKHPDWLFDODSSOLFDWLRQV DUHLQWHJUDWLRQWHFKQLTXHV SRO\QRPLDOURRWORFDWLRQ FRQLFVHFWLRQV TXDGULFVXUIDFHV DQGH[WUHPDIRUIXQFWLRQVRIWZRYDULDEOHV 1XPHULFDOOLQHDUDOJHEUDWRSLFV VXFKDV FRQGLWLRQQXPEHUV PDWUL[IDFWRUL]DWLRQV DQGWKH45 PHWKRGIRUÀQGLQJHLJHQYDOXHV DUHDOVRLQFOXGHG :RYHQLQWRHDFKGLVFXVVLRQDUHH[HUFLVHVWKDWPD\LQYROYHODUJHGDWD VHWVDQGWKXVUHTXLUHWHFKQRORJ\IRUWKHLUVROXWLRQ

*HWWLQJ6WDUWHGZLWK7HFKQRORJ\ ,I\RXUFRXUVHLQFOXGHVVRPHZRUNZLWK0$7/$%0DSOH 0DWKHPDWLFD RU7, FDO FXODWRUV \RXFDQUHDGRQHRIWKHSURMHFWVRQWKH:HEVLWHIRUDQLQWURGXFWLRQWRWKH

1SFGBDF

YJJJ

WHFKQRORJ\ ,QDGGLWLRQ WKH 6WXG\*XLGH SURYLGHVLQWURGXFWRU\PDWHULDOIRUÀUVWWLPH XVHUV

'DWD)LOHV +XQGUHGVRIÀOHVFRQWDLQGDWDIRUDERXWQXPHULFDOH[HUFLVHVLQWKHWH[W &DVH6WXG LHV DQG$SSOLFDWLRQ3URMHFWV 7KHGDWDDUHDYDLODEOHDW ZZZSHDUVRQKLJKHUHGFRPOD\ LQDYDULHW\RIIRUPDWV³IRU0$7/$%0DSOH 0DWKHPDWLFD DQGWKH7, JUDSKLFFDOFXODWRUV %\DOORZLQJVWXGHQWVWRDFFHVVPDWULFHVDQGYHFWRUVIRUDSDUWLFXODU SUREOHPZLWKRQO\DIHZNH\VWURNHV WKHGDWDÀOHVHOLPLQDWHGDWDHQWU\HUURUVDQGVDYH WLPHRQKRPHZRUN

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ÁRSV LWHUDWLYHPHWKRGV DQGWKH69'DQGD IHZSURMHFWVH[SORUHDSSOLFDWLRQVVXFKDV/DJUDQJHLQWHUSRODWLRQDQG0DUNRYFKDLQV

4611-&.&/54 6WXG\*XLGH $ SULQWHGYHUVLRQRIWKH 6WXG\*XLGH LVDYDLODEOHDWORZFRVW , ZURWHWKLV *XLGH WR EHDQLQWHJUDOSDUWRIWKHFRXUVH $QLFRQ SG LQWKHWH[WGLUHFWVVWXGHQWVWRVSHFLDO VXEVHFWLRQVRIWKH *XLGH WKDWVXJJHVWKRZWRPDVWHUNH\FRQFHSWVRIWKHFRXUVH 7KH *XLGH VXSSOLHVDGHWDLOHGVROXWLRQWRHYHU\WKLUGRGGQXPEHUHGH[HUFLVH ZKLFKDOORZV VWXGHQWVWRFKHFNWKHLUZRUN $ FRPSOHWHH[SODQDWLRQLVSURYLGHGZKHQHYHUDQRGG QXPEHUHGZULWLQJH[HUFLVHKDVRQO\D´+LQWµLQWKHDQVZHUV )UHTXHQW´:DUQLQJVµ LGHQWLI\FRPPRQHUURUVDQGVKRZKRZWRSUHYHQWWKHP 0$7/$% ER[HVLQWURGXFH FRPPDQGVDVWKH\DUHQHHGHG $SSHQGL[HVLQWKH 6WXG\*XLGH SURYLGHFRPSDUDEOH LQIRUPDWLRQDERXW0DSOH 0DWKHPDWLFD DQG7, JUDSKLQJFDOFXODWRUV,6%1

,QVWUXFWRU¶V(GLWLRQ )RUWKHFRQYHQLHQFHRILQVWUXFWRUV WKLVVSHFLDOHGLWLRQLQFOXGHVEULHIDQVZHUVWRDOO H[HUFLVHV $ 1RWHWRWKH,QVWUXFWRU DWWKHEHJLQQLQJRIWKHWH[WSURYLGHVDFRPPHQWDU\ RQWKHGHVLJQDQGRUJDQL]DWLRQRIWKHWH[W WRKHOSLQVWUXFWRUVSODQWKHLUFRXUVHV ,WDOVR GHVFULEHVRWKHUVXSSRUWDYDLODEOHIRULQVWUXFWRUV ,6%1

,QVWUXFWRU¶V7HFKQRORJ\0DQXDOV (DFKPDQXDOSURYLGHVGHWDLOHGJXLGDQFHIRULQWHJUDWLQJDVSHFLÀFVRIWZDUHSDFNDJHRU JUDSKLFFDOFXODWRUWKURXJKRXWWKHFRXUVH ZULWWHQE\IDFXOW\ZKRKDYHDOUHDG\XVHGWKH WHFKQRORJ\ZLWKWKLVWH[W 7KHIROORZLQJPDQXDOVDUHDYDLODEOHWRTXDOLÀHGLQVWUXFWRUV WKURXJKWKH3HDUVRQ,QVWUXFWRU5HVRXUFH&HQWHU ZZZSHDUVRQKLJKHUHGFRPLUF 0$7 /$% ,6%1 0DSOH,6%1 0DWKHPDWLFD,6%1 DQGWKH7,C,6%1

YJW

1SFGBDF

"$,/08-&%(.&/54 , DPLQGHHGJUDWHIXOWRPDQ\JURXSVRISHRSOHZKRKDYHKHOSHGPHRYHUWKH\HDUVZLWK YDULRXVDVSHFWVRIWKLVERRN , ZDQWWRWKDQN,VUDHO*RKEHUJDQG5REHUW(OOLVIRUPRUHWKDQÀIWHHQ\HDUVRI UHVHDUFKFROODERUDWLRQ ZKLFKJUHDWO\VKDSHGP\YLHZRIOLQHDUDOJHEUD $QG LWKDV EHHQDSULYLOHJHWREHDPHPEHURIWKH/LQHDU$OJHEUD&XUULFXOXP6WXG\*URXSDORQJ ZLWK'DYLG&DUOVRQ &KDUOHV-RKQVRQ DQG'XDQH3RUWHU 7KHLUFUHDWLYHLGHDVDERXW WHDFKLQJOLQHDUDOJHEUDKDYHLQÁXHQFHGWKLVWH[WLQVLJQLÀFDQWZD\V , VLQFHUHO\WKDQNWKHIROORZLQJUHYLHZHUVIRUWKHLUFDUHIXODQDO\VHVDQGFRQVWUXFWLYH VXJJHVWLRQV 5DIDO$EODPRZLF] 7HQQHVVHH7HFKQRORJLFDO8QLYHUVLW\ %ULDQ(%ODQN :DVKLQJWRQ8QLYHUVLW\LQ6W /RXLV 9DKLG'DEEDJKLDQ$EGRO\ 6LPRQ)UDVHU8QLYHUVLW\ -DPHV/+DUWPDQ 7KH&ROOHJHRI:RRVWHU 5LFKDUG3.XEHOND 6DQ-RVH6WDWH8QLYHUVLW\ 0DUWLQ1LNRORY 8QLYHUVLW\RI&RQQHFWLFXW ,O\D06SLWNRYVN\ &ROOHJHRI:LOOLDP 0DU\

-RKQ$ORQJL 1RUWKZHVWHUQ8QLYHUVLW\ 6WHYHQ%HOOHQRW )ORULGD6WDWH8QLYHUVLW\ +HUPDQ*ROOZLW]HU 'UH[HO8QLYHUVLW\ 'DYLG5.LQFDLG 7KH8QLYHUVLW\RI7H[DVDW$XVWLQ 'RXJODV%0HDGH 8QLYHUVLW\RI6RXWK&DUROLQD 7LP2OVRQ 8QLYHUVLW\RI)ORULGD $OEHUW/9LWWHU,,, 7XODQH8QLYHUVLW\

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³JLYLQJZLVHFRXQVHODQGHQFRXUDJHPHQW³*UHJ7RELQ SXEOLVKHU /DXULH 5RVDWRQH IRUPHUHGLWRU DQG:LOOLDP+RIIPDQ FXUUHQWHGLWRU 7KDQN\RXDOOVRPXFK 'DYLG&/D\

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ÀQGHLWKHUWKHGHVLUHGH[SODQDWLRQRUDWOHDVWD JRRGKLQW
1XPHULFDO1RWHV , KRSH\RXUHDGWKH1XPHULFDO1RWHVLQWKHWH[W HYHQLI\RXDUHQRWXVLQJDFRPSXWHURU JUDSKLFFDOFXODWRUZLWKWKHWH[W ,QUHDOOLIH PRVWDSSOLFDWLRQVRIOLQHDUDOJHEUDLQYROYH QXPHULFDOFRPSXWDWLRQVWKDWDUHVXEMHFWWRVRPHQXPHULFDOHUURU HYHQWKRXJKWKDWHUURU PD\EHH[WUHPHO\VPDOO 7KH1XPHULFDO1RWHVZLOOZDUQ\RXRISRWHQWLDOGLIÀFXOWLHVLQ XVLQJOLQHDUDOJHEUDODWHULQ\RXUFDUHHU DQGLI\RXVWXG\WKHQRWHVQRZ \RXDUHPRUH OLNHO\WRUHPHPEHUWKHPODWHU ,I\RXHQMR\UHDGLQJWKH1XPHULFDO1RWHV \RXPD\ZDQWWRWDNHDFRXUVHODWHULQ QXPHULFDOOLQHDUDOJHEUD %HFDXVHRIWKHKLJKGHPDQGIRULQFUHDVHGFRPSXWLQJSRZHU FRPSXWHUVFLHQWLVWVDQGPDWKHPDWLFLDQVZRUNLQQXPHULFDOOLQHDUDOJHEUDWRGHYHORS IDVWHUDQGPRUHUHOLDEOHDOJRULWKPVIRUFRPSXWDWLRQV DQGHOHFWULFDOHQJLQHHUVGHVLJQ IDVWHUDQGVPDOOHUFRPSXWHUVWRUXQWKHDOJRULWKPV 7KLVLVDQH[FLWLQJÀHOG DQG\RXU ÀUVWFRXUVHLQOLQHDUDOJHEUDZLOOKHOS\RXSUHSDUHIRULW

YW

YWJ

" /PUF UP 4UVEFOUT

6WXG\*XLGH 7R KHOS \RX VXFFHHG LQ WKLV FRXUVH , VXJJHVW WKDW \RX SXUFKDVH WKH 6WXG\ *XLGH ZZZP\SHDUVRQVWRUHFRP 1RWRQO\ZLOOLWKHOS\RXOHDUQOLQHDU DOJHEUD LWDOVRZLOOVKRZ\RXKRZWRVWXG\PDWKHPDWLFV $WVWUDWHJLFSRLQWVLQ\RXU WH[WERRN DQLFRQ SG ZLOOGLUHFW\RXWRVSHFLDOVXEVHFWLRQVLQWKH 6WXG\*XLGH HQWLWOHG ´0DVWHULQJ/LQHDU$OJHEUD&RQFHSWVµ 7KHUH\RXZLOOÀQGVXJJHVWLRQVIRUFRQVWUXFWLQJ HIIHFWLYHUHYLHZVKHHWVRINH\FRQFHSWV 7KHDFWRISUHSDULQJWKHVKHHWVLVRQHRI WKHVHFUHWVWRVXFFHVVLQWKHFRXUVH EHFDXVH\RXZLOOFRQVWUXFW OLQNVEHWZHHQLGHDV 7KHVHOLQNVDUHWKH´JOXHµ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³0$7/$% 0DSOH 0DWKHPDWLFD RU D 7, JUDSKLQJFDOFXODWRU³\RXFDQVDYHPDQ\KRXUVRIKRPHZRUNWLPH 7KH 6WXG\*XLGH LV \RXU ÓDE PDQXDOµ WKDW H[SODLQV KRZ WR XVH HDFK RI WKHVH PDWUL[ XWLOLWLHV ,W LQWURGXFHVQHZFRPPDQGVZKHQWKH\DUHQHHGHG

/LQHDU(TXDWLRQVLQ /LQHDU$OJHEUD

*/530%6$503: &9".1-&

/LQHDU0RGHOVLQ(FRQRPLFV DQG(QJLQHHULQJ ,WZDVODWHVXPPHULQ +DUYDUG3URIHVVRU:DVVLO\ /HRQWLHIZDVFDUHIXOO\IHHGLQJWKHODVWRIKLVSXQFKHG FDUGVLQWRWKHXQLYHUVLW\·V0DUN,, FRPSXWHU 7KHFDUGV FRQWDLQHGHFRQRPLFLQIRUPDWLRQDERXWWKH86HFRQRP\ DQGUHSUHVHQWHGDVXPPDU\RIPRUHWKDQSLHFHV RILQIRUPDWLRQSURGXFHGE\WKH86%XUHDXRI/DERU 6WDWLVWLFVDIWHUWZR\HDUVRI LQWHQVLYH ZRUN /HRQWLHIKDG GLYLGHGWKH86HFRQRP\LQWR´VHFWRUVµ VXFKDVWKH FRDOLQGXVWU\ WKHDXWRPRWLYHLQGXVWU\ FRPPXQLFDWLRQV DQGVRRQ )RUHDFKVHFWRU KHKDGZULWWHQDOLQHDUHTXDWLRQ WKDWGHVFULEHGKRZWKHVHFWRUGLVWULEXWHGLWVRXWSXWWR WKHRWKHUVHFWRUVRIWKHHFRQRP\ %HFDXVHWKH0DUN,, RQHRIWKHODUJHVWFRPSXWHUVRILWVGD\ FRXOGQRWKDQGOH WKHUHVXOWLQJV\VWHPRIHTXDWLRQVLQXQNQRZQV /HRQWLHIKDGGLVWLOOHGWKHSUREOHPLQWRDV\VWHPRI HTXDWLRQVLQXQNQRZQV 3URJUDPPLQJWKH0DUN,, FRPSXWHUIRU/HRQWLHI·V HTXDWLRQVKDGUHTXLUHGVHYHUDOPRQWKVRIHIIRUW DQGKH ZDVDQ[LRXVWRVHHKRZORQJWKHFRPSXWHUZRXOGWDNH WRVROYHWKHSUREOHP 7KH0DUN,, KXPPHGDQGEOLQNHG IRUKRXUVEHIRUHÀQDOO\SURGXFLQJDVROXWLRQ :HZLOO GLVFXVVWKHQDWXUHRIWKLVVROXWLRQLQ6HFWLRQVDQG /HRQWLHI ZKRZDVDZDUGHGWKH1REHO3UL]H LQ(FRQRPLF6FLHQFH RSHQHGWKHGRRUWRDQHZHUD LQPDWKHPDWLFDOPRGHOLQJLQHFRQRPLFV +LVHIIRUWV

DW+DUYDUGLQPDUNHGRQHRIWKHÀUVWVLJQLÀFDQW XVHVRIFRPSXWHUVWRDQDO\]HZKDWZDVWKHQDODUJH VFDOHPDWKHPDWLFDOPRGHO 6LQFHWKDWWLPH UHVHDUFKHUV LQPDQ\RWKHUÀ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ÀFDQGEXVLQHVVÀHOGVWKDQDQ\ RWKHUXQGHUJUDGXDWHPDWKHPDWLFVVXEMHFW 7KHPDWHULDOLQ WKLVWH[WSURYLGHVWKHIRXQGDWLRQIRUIXUWKHUZRUNLQPDQ\ LQWHUHVWLQJDUHDV +HUHDUHDIHZSRVVLELOLWLHV RWKHUVZLOO EHGHVFULEHGODWHU !

2LOH[SORUDWLRQ :KHQDVKLSVHDUFKHVIRURIIVKRUH RLOGHSRVLWV LWVFRPSXWHUVVROYHWKRXVDQGVRI VHSDUDWHV\VWHPVRIOLQHDUHTXDWLRQV HYHU\GD\ 7KH

$)"15&3

/LQHDU(TXDWLRQVLQ/LQHDU$OJHEUD

VHLVPLFGDWDIRUWKHHTXDWLRQVDUHREWDLQHGIURP XQGHUZDWHUVKRFNZDYHVFUHDWHGE\H[SORVLRQV IURPDLUJXQV 7KHZDYHVERXQFHRIIVXEVXUIDFH URFNVDQGDUHPHDVXUHGE\JHRSKRQHVDWWDFKHGWR PLOHORQJFDEOHVEHKLQGWKHVKLS !

/LQHDUSURJUDPPLQJ 0DQ\LPSRUWDQWPDQDJHPHQW GHFLVLRQVWRGD\DUHPDGHRQWKHEDVLVRIOLQHDU SURJUDPPLQJ PRGHOVWKDWXWLOL]H KXQGUHGV RI YDULDEOHV 7KH DLUOLQH LQGXVWU\ IRU LQVWDQFH

HPSOR\VOLQHDUSURJUDPVWKDWVFKHGXOHÁLJKWFUHZV PRQLWRUWKHORFDWLRQVRIDLUFUDIW RUSODQWKHYDULHG VFKHGXOHVRIVXSSRUWVHUYLFHVVXFKDVPDLQWHQDQFH DQGWHUPLQDORSHUDWLRQV !

(OHFWULFDOQHWZRUNV (QJLQHHUVXVHVLPXODWLRQ VRIWZDUHWRGHVLJQHOHFWULFDOFLUFXLWVDQGPLFURFKLSV LQYROYLQJPLOOLRQVRIWUDQVLVWRUV 6XFKVRIWZDUH UHOLHVRQOLQHDUDOJHEUDWHFKQLTXHVDQGV\VWHPVRI OLQHDUHTXDWLRQV WEB

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

4:45&.4 0' -*/&"3 &26"5*0/4 $ OLQHDUHTXDWLRQ LQWKHYDULDEOHV x1 ; : : : ; xn LVDQHTXDWLRQWKDWFDQEHZULWWHQLQWKH IRUP a1 x1 C a2 x2 C ! ! ! C an xn D b

ZKHUH b DQGWKH FRHIÀFLHQWV a1 ; : : : ; an DUHUHDORUFRPSOH[QXPEHUV XVXDOO\NQRZQ LQDGYDQFH 7KHVXEVFULSW n PD\EHDQ\SRVLWLYHLQWHJHU ,QWH[WERRNH[DPSOHVDQG H[HUFLVHV n LVQRUPDOO\EHWZHHQDQG ,QUHDOOLIHSUREOHPV n PLJKWEHRU RUHYHQODUJHU 7KHHTXDWLRQV !p " 4x1 " 5x2 C 2 D x1 DQG x2 D 2 6 " x1 C x3 DUHERWKOLQHDUEHFDXVHWKH\FDQEHUHDUUDQJHGDOJHEUDLFDOO\DVLQHTXDWLRQ p 3x1 " 5x2 D "2 DQG 2x1 C x2 " x3 D 2 6 7KHHTXDWLRQV

4x1 " 5x2 D x1 x2

DQG

p x2 D 2 x1 " 6

p DUHQRWOLQHDUEHFDXVHRIWKHSUHVHQFHRI x1 x2 LQWKHÀUVWHTXDWLRQDQG x1 LQWKHVHFRQG $ V\VWHPRIOLQHDUHTXDWLRQV RUD OLQHDUV\VWHP LVDFROOHFWLRQRIRQHRUPRUH OLQHDUHTXDWLRQVLQYROYLQJWKHVDPHYDULDEOHV³VD\ x1 ; : : : ; xn $QH[DPSOHLV 2x1 " x2 C 1:5x3 D 8 " 4x3 D "7 x1

6\VWHPVRI/LQHDU(TXDWLRQV

$ VROXWLRQ RIWKHV\VWHPLVDOLVW .s1 ; s2 ; : : : ; sn / RIQXPEHUVWKDWPDNHVHDFKHTXDWLRQD WUXHVWDWHPHQWZKHQWKHYDOXHV s1 ; : : : ; sn DUHVXEVWLWXWHGIRU x1 ; : : : ; xn UHVSHFWLYHO\ )RU LQVWDQFH .5; 6:5; 3/ LVDVROXWLRQRIV\VWHP EHFDXVH ZKHQWKHVHYDOXHVDUHVXEVWLWXWHG LQ IRU x1 ; x2 ; x3 UHVSHFWLYHO\ WKHHTXDWLRQVVLPSOLI\WR 8 D 8 DQG "7 D "7 7KHVHWRIDOOSRVVLEOHVROXWLRQVLVFDOOHGWKH VROXWLRQVHW RIWKHOLQHDUV\VWHP 7ZR OLQHDUV\VWHPVDUHFDOOHG HTXLYDOHQW LIWKH\KDYHWKHVDPHVROXWLRQVHW 7KDWLV HDFK VROXWLRQRIWKHÀUVWV\VWHPLVDVROXWLRQRIWKHVHFRQGV\VWHP DQGHDFKVROXWLRQRIWKH VHFRQGV\VWHPLVDVROXWLRQRIWKHÀUVW )LQGLQJWKHVROXWLRQVHWRIDV\VWHPRIWZROLQHDUHTXDWLRQVLQWZRYDULDEOHVLVHDV\ EHFDXVHLWDPRXQWVWRÀQGLQJWKHLQWHUVHFWLRQRIWZROLQHV $ W\SLFDOSUREOHPLV

x1 " 2x2 D "1 "x1 C 3x2 D 3 7KHJUDSKVRIWKHVHHTXDWLRQVDUHOLQHV ZKLFKZHGHQRWHE\ `1 DQG `2 $ SDLURIQXPEHUV .x1 ; x2 / VDWLVÀHV ERWK HTXDWLRQVLQWKHV\VWHPLIDQGRQO\LIWKHSRLQW .x1 ; x2 / OLHVRQERWK `1 DQG `2 ,QWKHV\VWHPDERYH WKHVROXWLRQLVWKHVLQJOHSRLQW .3; 2/ DV\RXFDQHDVLO\ YHULI\ 6HH)LJ x2 2

l2

3

x1

l1

'*(63& ([DFWO\RQHVROXWLRQ

2IFRXUVH WZROLQHVQHHGQRWLQWHUVHFWLQDVLQJOHSRLQW³WKH\FRXOGEHSDUDOOHO RU WKH\FRXOGFRLQFLGHDQGKHQFH´LQWHUVHFWµDWHYHU\SRLQWRQWKHOLQH )LJXUHVKRZVWKH JUDSKVWKDWFRUUHVSRQGWRWKHIROORZLQJV\VWHPV D

x1 " 2x2 D "1 "x1 C 2x2 D 3

E

x1 " 2x2 D "1 "x1 C 2x2 D 1

x2

x2

2

l2

2

3 l1

x1

3

x1

l1 (a)

(b)

'*(63& D 1RVROXWLRQ E ,QÀQLWHO\PDQ\VROXWLRQV

)LJXUHVDQGLOOXVWUDWHWKHIROORZLQJJHQHUDOIDFWDERXWOLQHDUV\VWHPV WREH YHULÀHGLQ6HFWLRQ

$)"15&3


$ V\VWHPRIOLQHDUHTXDWLRQVKDV QRVROXWLRQ RU H[DFWO\RQHVROXWLRQ RU LQÀQLWHO\PDQ\VROXWLRQV $ V\VWHPRIOLQHDUHTXDWLRQVLVVDLGWREH FRQVLVWHQW LILWKDVHLWKHURQHVROXWLRQRU LQÀQLWHO\PDQ\VROXWLRQV DV\VWHPLV LQFRQVLVWHQW LILWKDVQRVROXWLRQ

0DWUL[1RWDWLRQ 7KHHVVHQWLDOLQIRUPDWLRQRIDOLQHDUV\VWHPFDQEHUHFRUGHGFRPSDFWO\LQDUHFWDQJXODU DUUD\FDOOHGD PDWUL[ *LYHQWKHV\VWHP

x1 " 2x2 C x3 D 2x2 " 8x3 D

0 8

"4x1 C 5x2 C 9x3 D "9 ZLWKWKHFRHIÀFLHQWVRIHDFKYDULDEOHDOLJQHGLQFROXPQV WKHPDWUL[ 2 3 1 "2 1 4 0 2 "8 5 "4 5 9

LVFDOOHGWKH FRHIÀFLHQWPDWUL[ RU PDWUL[RIFRHIÀFLHQWV RIWKHV\VWHP DQG 2 3 1 "2 1 0 4 0 2 "8 8 5 "4 5 9 "9

LVFDOOHGWKH DXJPHQWHGPDWUL[ RIWKHV\VWHP 7KHVHFRQGURZKHUHFRQWDLQVD]HUREH FDXVHWKHVHFRQGHTXDWLRQFRXOGEHZULWWHQDV 0 ! x1 C 2x2 " 8x3 D 8 $QDXJPHQWHG PDWUL[RIDV\VWHPFRQVLVWVRIWKHFRHIÀFLHQWPDWUL[ZLWKDQDGGHGFROXPQFRQWDLQLQJ WKHFRQVWDQWVIURPWKHULJKWVLGHVRIWKHHTXDWLRQV 7KH VL]H RIDPDWUL[WHOOVKRZPDQ\URZVDQGFROXPQVLWKDV 7KHDXJPHQWHGPDWUL[ DERYHKDVURZVDQGFROXPQVDQGLVFDOOHGD 3 # 4 UHDGÉ\µ PDWUL[ ,I m DQG n DUHSRVLWLYHLQWHJHUV DQ m ! n PDWUL[ LVDUHFWDQJXODUDUUD\RIQXPEHUVZLWK m URZVDQG n FROXPQV 7KHQXPEHURIURZVDOZD\VFRPHVÀUVW 0DWUL[QRWDWLRQZLOO VLPSOLI\WKHFDOFXODWLRQVLQWKHH[DPSOHVWKDWIROORZ

6ROYLQJD/LQHDU6\VWHP 7KLVVHFWLRQDQGWKHQH[WGHVFULEHDQDOJRULWKP RUDV\VWHPDWLFSURFHGXUH IRUVROYLQJ OLQHDUV\VWHPV 7KHEDVLFVWUDWHJ\LV WRUHSODFHRQHV\VWHPZLWKDQHTXLYDOHQWV\VWHP LH RQHZLWKWKHVDPHVROXWLRQVHW WKDWLVHDVLHUWRVROYH 5RXJKO\VSHDNLQJ XVHWKH x1 WHUPLQWKHÀUVWHTXDWLRQRIDV\VWHPWRHOLPLQDWH WKH x1 WHUPVLQWKHRWKHUHTXDWLRQV 7KHQXVHWKH x2 WHUPLQWKHVHFRQGHTXDWLRQWR HOLPLQDWHWKH x2 WHUPVLQWKHRWKHUHTXDWLRQV DQGVRRQ XQWLO\RXÀQDOO\REWDLQDYHU\ VLPSOHHTXLYDOHQWV\VWHPRIHTXDWLRQV 7KUHHEDVLFRSHUDWLRQVDUHXVHGWRVLPSOLI\DOLQHDUV\VWHP 5HSODFHRQHHTXDWLRQ E\WKHVXPRILWVHOIDQGDPXOWLSOHRIDQRWKHUHTXDWLRQ LQWHUFKDQJHWZRHTXDWLRQV DQG PXOWLSO\DOOWKHWHUPVLQDQHTXDWLRQE\DQRQ]HURFRQVWDQW $IWHUWKHÀUVWH[DPSOH \RX ZLOOVHHZK\WKHVHWKUHHRSHUDWLRQVGRQRWFKDQJHWKHVROXWLRQVHWRIWKHV\VWHP


&9".1-& 6ROYHV\VWHP 40-65*0/ 7KHHOLPLQDWLRQSURFHGXUHLVVKRZQKHUHZLWKDQGZLWKRXWPDWUL[QRWDWLRQ DQGWKHUHVXOWVDUHSODFHGVLGHE\VLGHIRUFRPSDULVRQ x1 " 2x2 C x3 D 0 2x2 " 8x3 D 8 "4x1 C 5x2 C 9x3 D "9

2

3 1 "2 1 0 4 0 2 "8 8 5 "4 5 9 "9

.HHS x1 LQWKHÀUVWHTXDWLRQDQGHOLPLQDWHLWIURPWKHRWKHUHTXDWLRQV 7RGRVR DGG WLPHVHTXDWLRQWRHTXDWLRQ $IWHUVRPHSUDFWLFH WKLVW\SHRIFDOFXODWLRQLVXVXDOO\ SHUIRUPHGPHQWDOO\

4 ! ŒHTXDWLRQ!W C ŒHTXDWLRQ!W

ŒQHZHTXDWLRQ!W

4x1 " 8x2 C 4x3 D 0 "4x1 C 5x2 C 9x3 D "9 " 3x2 C 13x3 D "9

7KHUHVXOWRIWKLVFDOFXODWLRQLVZULWWHQLQSODFHRIWKHRULJLQDOWKLUGHTXDWLRQ

x1 " 2x2 C x3 D 0 2x2 " 8x3 D 8 " 3x2 C 13x3 D "9

2

3 1 "2 1 0 40 2 "8 8 5 0 "3 13 "9

1RZ PXOWLSO\HTXDWLRQE\ 1=2 LQRUGHUWRREWDLQDVWKHFRHIÀFLHQWIRU x2 7KLV FDOFXODWLRQZLOOVLPSOLI\WKHDULWKPHWLFLQWKHQH[WVWHS

x1 " 2x2 C x3 D 0 x2 " 4x3 D 4 " 3x2 C 13x3 D "9

2

3 1 "2 1 0 40 1 "4 4 5 0 "3 13 "9

8VHWKH x2 LQHTXDWLRQWRHOLPLQDWHWKH "3x2 LQHTXDWLRQ 7KH´PHQWDOµFRPSXWDWLRQ LV 3 ! ŒHTXDWLRQ!W 3x2 " 12x3 D 12 C ŒHTXDWLRQ!W "3x2 C 13x3 D "9

x3 D

>QHZHTXDWLRQ!W

3

7KHQHZV\VWHPKDVD WULDQJXODU IRUPñ

x1 " 2x2 C x3 D 0 x2 " 4x3 D 4 x3 D 3

2

1 "2 1 40 1 "4 0 0 1

3 0 45 3

(YHQWXDOO\ \RXZDQWWRHOLPLQDWHWKH "2x2 WHUPIURPHTXDWLRQ EXWLWLVPRUHHIÀFLHQW WRXVHWKH x3 LQHTXDWLRQÀUVW WRHOLPLQDWHWKH "4x3 DQG Cx3 WHUPVLQHTXDWLRQVDQG 7KHWZR´PHQWDOµFDOFXODWLRQVDUH

4 ! ŒHT !W C ŒHT !W

ŒQHZHT !W

4x3 D 12 x2 " 4x3 D 4

x2

D 16

"1 ! ŒHT !W C ŒHT !W

ŒQHZHT !W

" x3 D "3

x1 " 2x2 C x3 D

x1 " 2x2

ñ 7KHLQWXLWLYHWHUP WULDQJXODU ZLOOEHUHSODFHGE\DSUHFLVHWHUPLQWKHQH[WVHFWLRQ

0

D "3

$)"15&3


,WLVFRQYHQLHQWWRFRPELQHWKHUHVXOWVRIWKHVHWZRRSHUDWLRQV

x1 " 2x2 x2

D "3 D 16 x3 D 3

2

1 "2 40 1 0 0

0 0 1

3 "3 16 5 3

1RZ KDYLQJFOHDQHGRXWWKHFROXPQDERYHWKH x3 LQHTXDWLRQ PRYHEDFNWRWKH x2 LQ HTXDWLRQDQGXVHLWWRHOLPLQDWHWKH "2x2 DERYHLW %HFDXVHRIWKHSUHYLRXVZRUNZLWK x3 WKHUHLVQRZQRDULWKPHWLFLQYROYLQJ x3 WHUPV $GGWLPHVHTXDWLRQWRHTXDWLRQ DQGREWDLQWKHV\VWHP 8 2 3 ˆ D 29 1 0 0 29
7KHZRUNLVHVVHQWLDOO\GRQH ,WVKRZVWKDWWKHRQO\VROXWLRQRIWKHRULJLQDOV\VWHPLV .29; 16; 3/ +RZHYHU VLQFHWKHUHDUHVRPDQ\FDOFXODWLRQVLQYROYHG LWLVDJRRGSUDFWLFH WRFKHFNWKHZRUN 7RYHULI\WKDW .29; 16; 3/ LV DVROXWLRQ VXEVWLWXWHWKHVHYDOXHVLQWR WKHOHIWVLGHRIWKHRULJLQDOV\VWHP DQGFRPSXWH (29, 16, 3)

(DFKRIWKHRULJLQDOHTXDWLRQV GHWHUPLQHVDSODQHLQWKUHH GLPHQVLRQDOVSDFH 7KHSRLQW .29; 16; 3/ OLHVLQDOOWKUHHSODQHV

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

5RZRSHUDWLRQVFDQEHDSSOLHGWRDQ\PDWUL[ QRWPHUHO\WRRQHWKDWDULVHVDVWKH DXJPHQWHGPDWUL[RIDOLQHDUV\VWHP 7ZRPDWULFHVDUHFDOOHG URZHTXLYDOHQW LIWKHUH LVDVHTXHQFHRIHOHPHQWDU\URZRSHUDWLRQVWKDWWUDQVIRUPVRQHPDWUL[LQWRWKHRWKHU ,WLVLPSRUWDQWWRQRWHWKDWURZRSHUDWLRQVDUH UHYHUVLEOH ,IWZRURZVDUHLQWHU FKDQJHG WKH\FDQEHUHWXUQHGWRWKHLURULJLQDOSRVLWLRQVE\DQRWKHULQWHUFKDQJH ,ID URZLVVFDOHGE\DQRQ]HURFRQVWDQW c WKHQPXOWLSO\LQJWKHQHZURZE\ 1=c SURGXFHV WKHRULJLQDOURZ )LQDOO\ FRQVLGHUDUHSODFHPHQWRSHUDWLRQLQYROYLQJWZRURZV³VD\ URZVDQG³DQGVXSSRVHWKDW c WLPHVURZLVDGGHGWRURZWRSURGXFHDQHZURZ 7RÚHYHUVHµWKLVRSHUDWLRQ DGG "c WLPHVURZWRQHZ URZDQGREWDLQWKHRULJLQDO URZ 6HH([HUFLVHV²DWWKHHQGRIWKLVVHFWLRQ ò $ FRPPRQSDUDSKUDVHRIURZUHSODFHPHQWLV´$GGWRRQHURZDPXOWLSOHRIDQRWKHUURZµ


$WWKHPRPHQW ZHDUHLQWHUHVWHGLQURZRSHUDWLRQVRQWKHDXJPHQWHGPDWUL[RID V\VWHPRIOLQHDUHTXDWLRQV 6XSSRVHDV\VWHPLVFKDQJHGWRDQHZRQHYLDURZRSHUD WLRQV %\FRQVLGHULQJHDFKW\SHRIURZRSHUDWLRQ \RXFDQVHHWKDWDQ\VROXWLRQRIWKH RULJLQDOV\VWHPUHPDLQVDVROXWLRQRIWKHQHZV\VWHP &RQYHUVHO\ VLQFHWKHRULJLQDO V\VWHPFDQEHSURGXFHGYLDURZRSHUDWLRQVRQWKHQHZV\VWHP HDFKVROXWLRQRIWKHQHZ V\VWHPLVDOVRDVROXWLRQRIWKHRULJLQDOV\VWHP 7KLVGLVFXVVLRQMXVWLÀHVWKHIROORZLQJ VWDWHPHQW ,IWKHDXJPHQWHGPDWULFHVRIWZROLQHDUV\VWHPVDUHURZHTXLYDOHQW WKHQWKHWZR V\VWHPVKDYHWKHVDPHVROXWLRQVHW 7KRXJK([DPSOHLVOHQJWK\ \RXZLOOÀQGWKDWDIWHUVRPHSUDFWLFH WKHFDOFXODWLRQV JRTXLFNO\ 5RZRSHUDWLRQVLQWKHWH[WDQGH[HUFLVHVZLOOXVXDOO\EHH[WUHPHO\HDV\WR SHUIRUP DOORZLQJ\RXWRIRFXVRQWKHXQGHUO\LQJFRQFHSWV 6WLOO \RXPXVWOHDUQWR SHUIRUPURZRSHUDWLRQVDFFXUDWHO\EHFDXVHWKH\ZLOOEHXVHGWKURXJKRXWWKHWH[W 7KHUHVWRIWKLVVHFWLRQVKRZVKRZWRXVHURZRSHUDWLRQVWRGHWHUPLQHWKHVL]HRID VROXWLRQVHW ZLWKRXWFRPSOHWHO\VROYLQJWKHOLQHDUV\VWHP

([LVWHQFHDQG8QLTXHQHVV4XHVWLRQV 6HFWLRQ ZLOOVKRZZK\DVROXWLRQVHWIRUDOLQHDUV\VWHPFRQWDLQVHLWKHUQRVROXWLRQV RQHVROXWLRQ RULQÀQLWHO\PDQ\VROXWLRQV $QVZHUVWRWKHIROORZLQJWZRTXHVWLRQVZLOO GHWHUPLQHWKHQDWXUHRIWKHVROXWLRQVHWIRUDOLQHDUV\VWHP 7RGHWHUPLQHZKLFKSRVVLELOLW\LVWUXHIRUDSDUWLFXODUV\VWHP ZHDVNWZRTXHVWLRQV 580 '6/%".&/5"- 26&45*0/4 "#065 " -*/&"3 4:45&. ,VWKHV\VWHPFRQVLVWHQW WKDWLV GRHVDWOHDVWRQHVROXWLRQ H[LVW" ,IDVROXWLRQH[LVWV LVLWWKH RQO\ RQH WKDWLV LVWKHVROXWLRQ XQLTXH" 7KHVHWZRTXHVWLRQVZLOODSSHDUWKURXJKRXWWKHWH[W LQPDQ\GLIIHUHQWJXLVHV 7KLV VHFWLRQDQGWKHQH[WZLOOVKRZKRZWRDQVZHUWKHVHTXHVWLRQVYLDURZRSHUDWLRQVRQWKH DXJPHQWHGPDWUL[

&9".1-& 'HWHUPLQHLIWKHIROORZLQJV\VWHPLVFRQVLVWHQW x1 " 2x2 C x3 D 0 2x2 " 8x3 D 8 "4x1 C 5x2 C 9x3 D "9 40-65*0/ 7KLVLVWKHV\VWHPIURP([DPSOH 6XSSRVHWKDWZHKDYHSHUIRUPHGWKH URZRSHUDWLRQVQHFHVVDU\WRREWDLQWKHWULDQJXODUIRUP 2 3 x1 " 2x2 C x3 D 0 1 "2 1 0 40 1 "4 4 5 x2 " 4x3 D 4 0 0 1 3 x D3 3

$WWKLVSRLQW ZHNQRZ x3 :HUHZHWRVXEVWLWXWHWKHYDOXHRI x3 LQWRHTXDWLRQ ZH FRXOGFRPSXWH x2 DQGKHQFHFRXOGGHWHUPLQH x1 IURPHTXDWLRQ 6RDVROXWLRQH[LVWV WKHV\VWHPLVFRQVLVWHQW ,QIDFW x2 LVXQLTXHO\GHWHUPLQHGE\HTXDWLRQVLQFH x3 KDV

$)"15&3


RQO\RQHSRVVLEOHYDOXH DQG x1 LVWKHUHIRUHXQLTXHO\GHWHUPLQHGE\HTXDWLRQ 6RWKH VROXWLRQLVXQLTXH

&9".1-& 'HWHUPLQHLIWKHIROORZLQJV\VWHPLVFRQVLVWHQW x2 " 4x3 D 8

2x1 " 3x2 C 2x3 D 1

5x1 " 8x2 C 7x3 D 1 40-65*0/ 7KHDXJPHQWHGPDWUL[LV 2

0 1 "4 4 2 "3 2 5 "8 7

3 8 15 1

7RREWDLQDQ x1 LQWKHÀUVWHTXDWLRQ LQWHUFKDQJHURZVDQG 2

2 "3 2 40 1 "4 5 "8 7

3 1 85 1

7RHOLPLQDWHWKH 5x1 WHUPLQWKHWKLUGHTXDWLRQ DGG "5=2 WLPHVURZWRURZ 2

3 2 "3 2 1 40 1 "4 8 5 0 "1=2 2 "3=2

1H[W XVHWKH x2 WHUPLQWKHVHFRQGHTXDWLRQWRHOLPLQDWHWKH ".1=2/x2 WHUPIURPWKH WKLUGHTXDWLRQ $GG 1=2 WLPHVURZWRURZ 2

2 "3 2 40 1 "4 0 0 0

3 1 8 5 5=2

7KHDXJPHQWHGPDWUL[LVQRZLQWULDQJXODUIRUP 7RLQWHUSUHWLWFRUUHFWO\ JREDFNWR HTXDWLRQQRWDWLRQ

2x1 " 3x2 C 2x3 D 1 x2 " 4x3 D 8 0 D 5=2 7KLVV\VWHPLVLQFRQVLVWHQW EHFDXVHWKHUHLVQRSRLQWWKDWOLHV LQDOOWKUHHSODQHV

7KHHTXDWLRQ 0 D 5=2 LVDVKRUWIRUPRI 0x1 C 0x2 C 0x3 D 5=2 7KLVV\VWHPLQWULDQ JXODUIRUPREYLRXVO\KDVDEXLOWLQFRQWUDGLFWLRQ 7KHUHDUHQRYDOXHVRI x1 ; x2 ; x3 WKDW VDWLVI\ EHFDXVHWKHHTXDWLRQ 0 D 5=2 LVQHYHUWUXH 6LQFH DQG KDYHWKHVDPH VROXWLRQVHW WKHRULJLQDOV\VWHPLVLQFRQVLVWHQWLH KDVQRVROXWLRQ 3D\FORVHDWWHQWLRQWRWKHDXJPHQWHGPDWUL[LQ ,WVODVWURZLVW\SLFDORIDQ LQFRQVLVWHQWV\VWHPLQWULDQJXODUIRUP


/6.&3*$"- /05& ,QUHDOZRUOGSUREOHPV V\VWHPVRIOLQHDUHTXDWLRQVDUHVROYHGE\DFRPSXWHU )RU DVTXDUHFRHIÀFLHQWPDWUL[ FRPSXWHUSURJUDPVQHDUO\DOZD\VXVHWKHHOLPLQDWLRQ DOJRULWKPJLYHQKHUHDQGLQ6HFWLRQ PRGLÀHGVOLJKWO\IRULPSURYHGDFFXUDF\ 7KHYDVWPDMRULW\RIOLQHDUDOJHEUDSUREOHPVLQEXVLQHVVDQGLQGXVWU\DUH VROYHGZLWKSURJUDPVWKDWXVH ÁRDWLQJSRLQWDULWKPHWLF 1XPEHUVDUHUHSUHVHQWHG DVGHFLPDOV ˙:d1 ! ! ! dp # 10r ZKHUH r LVDQLQWHJHUDQGWKHQXPEHU p RIGLJLWVWR WKHULJKWRIWKHGHFLPDOSRLQWLVXVXDOO\EHWZHHQDQG $ULWKPHWLFZLWKVXFK QXPEHUVW\SLFDOO\LVLQH[DFW EHFDXVHWKHUHVXOWPXVWEHURXQGHGRUWUXQFDWHG WR WKHQXPEHURIGLJLWVVWRUHG ´5RXQGRIIHUURUµLVDOVRLQWURGXFHGZKHQDQXPEHU VXFKDV 1=3 LVHQWHUHGLQWRWKHFRPSXWHU VLQFHLWVGHFLPDOUHSUHVHQWDWLRQPXVWEH DSSUR[LPDWHGE\DÀQLWHQXPEHURIGLJLWV )RUWXQDWHO\ LQDFFXUDFLHVLQÁRDWLQJ SRLQWDULWKPHWLFVHOGRPFDXVHSUREOHPV 7KHQXPHULFDOQRWHVLQWKLVERRNZLOO RFFDVLRQDOO\ZDUQRILVVXHVWKDW\RXPD\QHHGWRFRQVLGHUODWHULQ\RXUFDUHHU

13"$5*$& 130#-&.4 7KURXJKRXWWKHWH[W SUDFWLFHSUREOHPVVKRXOGEHDWWHPSWHGEHIRUHZRUNLQJWKHH[HU FLVHV 6ROXWLRQVDSSHDUDIWHUHDFKH[HUFLVHVHW 6WDWHLQZRUGVWKHQH[WHOHPHQWDU\URZRSHUDWLRQWKDWVKRXOGEHSHUIRUPHGRQWKH V\VWHPLQRUGHUWRVROYHLW >0RUHWKDQRQHDQVZHULVSRVVLEOHLQD @ D x1 C 4x2 " 2x3 x2 " 7x3 5x3 x3

C 8x4 C 2x4 " x4 C 3x4

D 12 D "4 D 7 D "5

E x1 " 3x2 C 5x3 " 2x4 D

0

x2 C 8x3

D "4

2x3

D

3

x4 D

1

7KHDXJPHQWHGPDWUL[RIDOLQHDUV\VWHPKDVEHHQWUDQVIRUPHGE\URZRSHUDWLRQV LQWRWKHIRUPEHORZ 'HWHUPLQHLIWKHV\VWHPLVFRQVLVWHQW 2 3 1 5 2 "6 40 4 "7 2 5 0 0 5 0 ,V .3; 4; "2/ DVROXWLRQRIWKHIROORZLQJV\VWHP"

5x1 " x2 C 2x3 D 7 "2x1 C 6x2 C 9x3 D 0 "7x1 C 5x2 " 3x3 D "7 )RUZKDWYDOXHVRI h DQG k LVWKHIROORZLQJV\VWHPFRQVLVWHQW"

2x1 " x2 D h "6x1 C 3x2 D k

$)"15&3


&9&3$*4&4 6ROYHHDFKV\VWHPLQ([HUFLVHV²E\XVLQJHOHPHQWDU\URZ RSHUDWLRQVRQWKHHTXDWLRQVRURQWKHDXJPHQWHGPDWUL[ )ROORZ WKHV\VWHPDWLFHOLPLQDWLRQSURFHGXUHGHVFULEHGLQWKLVVHFWLRQ

x1 C 5x2 D

7

"2x1 " 7x2 D "5

3x1 C 6x2 D "3

5x1 C 7x2 D 10

)LQGWKHSRLQW .x1 ; x2 / WKDWOLHVRQWKHOLQH x1 C 2x2 D 4 DQG RQWKHOLQH x1 " x2 D 1 6HHWKHÀJXUH

2

3 1 0 0

x2 C 5x3 D "4

6ROYHWKHV\VWHPVLQ([HUFLVHV²

x1 C 4x2 C 3x3 D "2

2x1 C 7x2 C x3 D "2

x2

x1 – x2 = 1

3 0 "2 "7 0 3 67 7 1 0 25 0 1 "2

1 60 6 4 0 0

x1 " 5x2 C 4x3 D "3

2x1 " 7x2 C 3x3 D "2

"2x1 C x2 C 7x3 D "1 x1

x1

x1 + 2x2 = 4

2x1

,Q([HUFLVHV² WKHDXJPHQWHGPDWUL[RIDOLQHDUV\VWHPKDV EHHQUHGXFHGE\URZRSHUDWLRQVWRWKHIRUPVKRZQ ,QHDFKFDVH FRQWLQXHWKHDSSURSULDWHURZRSHUDWLRQVDQGGHVFULEHWKHVROXWLRQ VHWRIWKHRULJLQDOV\VWHP 2 3 1 7 3 "4 60 1 "1 37 7 6 40 0 0 15 0 0 1 "2 2 3 1 "5 4 0 0 60 1 0 1 07 7 6 40 0 3 0 05 0 0 0 2 0 2 3 1 "1 0 0 "5 60 1 "2 0 "7 7 7 6 40 0 1 "3 25 0 0 0 1 4

8 7

x2 C 5x3 D "2

)LQGWKHSRLQWRILQWHUVHFWLRQRIWKHOLQHV x1 C 2x2 D "13 DQG 3x1 " 2x2 D 1 &RQVLGHUHDFKPDWUL[LQ([HUFLVHVDQGDVWKHDXJPHQWHGPDWUL[ RIDOLQHDUV\VWHP 6WDWHLQZRUGVWKHQH[WWZRHOHPHQWDU\URZ RSHUDWLRQVWKDWVKRXOGEHSHUIRUPHGLQWKHSURFHVVRIVROYLQJWKH V\VWHP 2 3 1 "4 "3 0 7 60 1 4 0 67 7 6 40 0 1 0 25 0 0 0 1 "5 2 3 1 "6 4 0 "1 60 2 "7 0 47 7 6 40 0 1 2 "3 5 0 0 4 1 2

" 3x3 D

2x1 C 2x2 C 9x3 D

" 6x3 D "8

x2 C 2x3 D

3

3x1 C 6x2 " 2x3 D "4 'HWHUPLQHLIWKHV\VWHPVLQ([HUFLVHVDQGDUHFRQVLVWHQW 'RQRWFRPSOHWHO\VROYHWKHV\VWHPV

x1 " 6x2

D5

x2 " 4x3 C x4 D 0

"x1 C 6x2 C x3 C 5x4 D 3 "x2 C 5x3 C 4x4 D 0

2x1 3x2 C 3x3

" 4x4 D "10 D

0

"3x1 C 2x2 C 3x3 C x4 D

5

x3 C 4x4 D "1

'R WKH WKUHH OLQHV 2x1 C 3x2 D "1 6x1 C 5x2 D 0 DQG 2x1 " 5x2 D 7 KDYHDFRPPRQSRLQWRILQWHUVHFWLRQ" ([ SODLQ 'RWKHWKUHHSODQHV 2x1 C 4x2 C 4x3 D 4 x2 " 2x3 D "2 DQG 2x1 C 3x2 D 0 KDYHDWOHDVWRQHFRPPRQSRLQWRILQWHU VHFWLRQ" ([SODLQ ,Q([HUFLVHV² GHWHUPLQHWKHYDOXHV RI h VXFKWKDWWKH PDWUL[LVWKHDXJPHQWHGPDWUL[RIDFRQVLVWHQWOLQHDUV\VWHP # $ # $ 1 h 4 1 h "5 3 6 8 2 "8 6 # $ # $ 1 4 "2 "4 12 h 3 h "6 2 "6 "3 ,Q ([HUFLVHV DQG NH\ VWDWHPHQWV IURP WKLV VHFWLRQ DUH HLWKHUTXRWHGGLUHFWO\ UHVWDWHGVOLJKWO\EXWVWLOOWUXH RUDOWHUHG LQVRPHZD\WKDWPDNHVWKHPIDOVHLQVRPHFDVHV 0DUNHDFK VWDWHPHQW7UXHRU)DOVH DQG MXVWLI\ \RXUDQVZHU ,IWUXH JLYHWKH

DSSUR[LPDWHORFDWLRQZKHUHDVLPLODUVWDWHPHQWDSSHDUV RUUHIHU WRDGHÀQLWLRQRUWKHRUHP ,IIDOVH JLYHWKHORFDWLRQRIDVWDWHPHQW WKDWKDVEHHQTXRWHGRUXVHGLQFRUUHFWO\ RUFLWHDQH[DPSOHWKDW VKRZVWKHVWDWHPHQWLVQRWWUXHLQDOOFDVHV 6LPLODUWUXHIDOVH TXHVWLRQVZLOODSSHDULQPDQ\VHFWLRQVRIWKHWH[W D (YHU\HOHPHQWDU\URZRSHUDWLRQLVUHYHUVLEOH E $ 5 # 6 PDWUL[KDVVL[URZV

F 7KHVROXWLRQVHWRIDOLQHDUV\VWHPLQYROYLQJYDULDEOHV x1 ; : : : ; xn LVDOLVWRIQXPEHUV .s1 ; : : : ; sn / WKDWPDNHVHDFK HTXDWLRQLQWKHV\VWHPDWUXHVWDWHPHQWZKHQWKHYDOXHV s1 ; : : : ; sn DUHVXEVWLWXWHGIRU x1 ; : : : ; xn UHVSHFWLYHO\ G 7ZRIXQGDPHQWDOTXHVWLRQVDERXWDOLQHDUV\VWHPLQYROYH H[LVWHQFHDQGXQLTXHQHVV D 7ZRPDWULFHVDUHURZHTXLYDOHQWLIWKH\KDYHWKHVDPH QXPEHURIURZV E (OHPHQWDU\URZRSHUDWLRQVRQDQDXJPHQWHGPDWUL[QHYHU FKDQJHWKHVROXWLRQVHWRIWKHDVVRFLDWHGOLQHDUV\VWHP F 7ZRHTXLYDOHQWOLQHDUV\VWHPVFDQKDYHGLIIHUHQWVROXWLRQ VHWV G $ FRQVLVWHQWV\VWHPRIOLQHDUHTXDWLRQVKDVRQHRUPRUH VROXWLRQV )LQG DQ HTXDWLRQ LQYROYLQJ g h DQG k WKDW PDNHV WKLVDXJPHQWHGPDWUL[FRUUHVSRQGWRDFRQVLVWHQWV\VWHP 2 3 1 "4 7 g 4 0 3 "5 h5 "2 5 "9 k

$QLPSRUWDQWFRQFHUQLQWKHVWXG\RIKHDWWUDQVIHULVWRGHWHUPLQH WKHVWHDG\VWDWHWHPSHUDWXUHGLVWULEXWLRQRIDWKLQSODWHZKHQWKH WHPSHUDWXUHDURXQGWKHERXQGDU\LVNQRZQ $VVXPHWKHSODWH VKRZQLQWKHÀJXUHUHSUHVHQWVDFURVVVHFWLRQRIDPHWDOEHDP ZLWKQHJOLJLEOHKHDWÁRZLQWKHGLUHFWLRQSHUSHQGLFXODUWRWKH SODWH /HW T1 ; : : : ; T4 GHQRWHWKHWHPSHUDWXUHVDWWKHIRXULQWHULRU QRGHVRIWKHPHVKLQWKHÀJXUH 7KHWHPSHUDWXUHDWDQRGHLV DSSUR[LPDWHO\HTXDOWRWKHDYHUDJHRIWKHIRXUQHDUHVWQRGHV³WR WKHOHIW DERYH WRWKHULJKW DQGEHORZ )RULQVWDQFH

T1 D .10 C 20 C T2 C T4 /=4;

10° 10°

2x1 C 4x2 D f

cx1 C dx2 D g

ax1 C bx2 D f

,Q([HUFLVHV² ÀQGWKHHOHPHQWDU\URZRSHUDWLRQWKDWWUDQV IRUPVWKHÀUVWPDWUL[LQWRWKHVHFRQG DQGWKHQÀQGWKHUHYHUVH URZRSHUDWLRQWKDWWUDQVIRUPVWKHVHFRQGPDWUL[LQWRWKHÀUVW 2 3 2 3 0 "2 5 3 "1 6 3 "5 5 4 1 3 "5 5 4 1 3 "1 6 0 "2 5 2 3 2 3 1 3 "4 1 3 "4 6 5 4 0 "2 65 4 0 "2 0 "5 10 0 1 "2 2 3 2 3 1 "2 1 0 1 "2 1 0 5 "2 8 5 4 0 5 "2 85 4 0 4 "1 3 "6 0 7 "1 "6 2 3 2 3 1 2 "5 0 1 2 "5 0 1 "3 "2 5 4 0 1 "3 "2 5 4 0 0 4 "12 7 0 0 0 15

6XSSRVHWKHV\VWHPEHORZLVFRQVLVWHQWIRUDOOSRVVLEOHYDOXHV RI f DQG g :KDWFDQ\RXVD\DERXWWKHFRHIÀFLHQWV c DQG d " -XVWLI\\RXUDQVZHU

6XSSRVH a b c DQG d DUHFRQVWDQWVVXFKWKDW a LVQRW]HUR DQGWKHV\VWHPEHORZLVFRQVLVWHQWIRUDOOSRVVLEOHYDOXHVRI f DQG g :KDWFDQ\RXVD\DERXWWKHQXPEHUV a b c DQG d " -XVWLI\\RXUDQVZHU


4T1 " T2 " T4 D 30

RU

20°

20°

1

2

4

3

30°

30°

40° 40°

:ULWHDV\VWHPRIIRXUHTXDWLRQVZKRVHVROXWLRQJLYHVHVWL PDWHVIRUWKHWHPSHUDWXUHV T1 ; : : : ; T4 6ROYHWKHV\VWHPRIHTXDWLRQVIURP([HUFLVH >+LQW 7R VSHHGXSWKHFDOFXODWLRQV LQWHUFKDQJHURZVDQGEHIRUH VWDUWLQJÚHSODFHµRSHUDWLRQV@

cx1 C dx2 D g

&RQVWUXFWWKUHHGLIIHUHQWDXJPHQWHGPDWULFHVIRUOLQHDUV\V WHPVZKRVHVROXWLRQVHWLV x1 D 3 x2 D "2 x3 D "1

3 6HH)UDQN0:KLWH +HDWDQG0DVV7UDQVIHU 5HDGLQJ 0$ $GGLVRQ:HVOH\3XEOLVKLQJ SS ²

40-65*0/4 50 13"$5*$& 130#-&.4 D )RU´KDQGFRPSXWDWLRQµ WKHEHVWFKRLFHLVWRLQWHUFKDQJHHTXDWLRQVDQG $QRWKHUSRVVLELOLW\LVWRPXOWLSO\HTXDWLRQE\ 1=5 2U UHSODFHHTXDWLRQE\ LWVVXPZLWK "1=5 WLPHVURZ ,QDQ\FDVH GRQRWXVHWKH x2 LQHTXDWLRQ WR HOLPLQDWHWKH 4x2 LQHTXDWLRQ :DLWXQWLODWULDQJXODUIRUPKDVEHHQUHDFKHGDQG WKH x3 WHUPVDQG x4 WHUPVKDYHEHHQHOLPLQDWHGIURPWKHÀUVWWZRHTXDWLRQV

$)"15&3


E 7KHV\VWHPLVLQWULDQJXODUIRUP )XUWKHUVLPSOLÀFDWLRQEHJLQVZLWKWKH x4 LQWKH IRXUWKHTXDWLRQ 8VHWKH x4 WRHOLPLQDWHDOO x4 WHUPVDERYHLW 7KHDSSURSULDWH VWHSQRZLVWRDGGWLPHVHTXDWLRQWRHTXDWLRQ $IWHUWKDW PRYHWRHTXDWLRQ PXOWLSO\LWE\ 1=2 DQGWKHQXVHWKHHTXDWLRQWRHOLPLQDWHWKH x3 WHUPVDERYH LW 7KHV\VWHPFRUUHVSRQGLQJWRWKHDXJPHQWHGPDWUL[LV

x1 C 5x2 C 2x3 D "6 4x2 " 7x3 D 2 5x3 D 0

(3, 4, –2)

6LQFH .3; 4; "2/ VDWLVÀHVWKHÀUVW WZRHTXDWLRQV LWLVRQWKHOLQHRI WKHLQWHUVHFWLRQRIWKHÀUVWWZR SODQHV 6LQFH .3; 4; "2/ GRHVQRW VDWLVI\DOOWKUHHHTXDWLRQV LWGRHV QRWOLHRQDOOWKUHHSODQHV

7KHWKLUGHTXDWLRQPDNHV x3 D 0 ZKLFKLVFHUWDLQO\DQDOORZDEOHYDOXHIRU x3 $IWHU HOLPLQDWLQJWKH x3 WHUPVLQHTXDWLRQVDQG \RXFRXOGJRRQWRVROYHIRUXQLTXH YDOXHVIRU x2 DQG x1 +HQFHDVROXWLRQH[LVWV DQGLWLVXQLTXH &RQWUDVWWKLVVLWXDWLRQ ZLWKWKDWLQ([DPSOH ,WLVHDV\WRFKHFNLIDVSHFLÀFOLVWRIQXPEHUVLVDVROXWLRQ 6HW x1 D 3 x2 D 4 DQG x3 D "2 DQGÀQGWKDW

5.3/ " .4/ C 2."2/ D 15 " 4 " 4 D 7 "2.3/ C 6.4/ C 9."2/ D "6 C 24 " 18 D 0 "7.3/ C 5.4/ " 3."2/ D "21 C 20 C 6 D 5

$OWKRXJKWKHÀUVWWZRHTXDWLRQVDUHVDWLVÀHG WKHWKLUGLVQRW VR .3; 4; "2/ LVQRWD VROXWLRQRIWKHV\VWHP 1RWLFHWKHXVHRISDUHQWKHVHVZKHQPDNLQJWKHVXEVWLWXWLRQV 7KH\DUHVWURQJO\UHFRPPHQGHGDVDJXDUGDJDLQVWDULWKPHWLFHUURUV :KHQWKHVHFRQGHTXDWLRQLVUHSODFHGE\LWVVXPZLWKWLPHVWKHÀUVWHTXDWLRQ WKH V\VWHPEHFRPHV

2x1 " x2 D h 0 D k C 3h

,I k C 3h LVQRQ]HUR WKHV\VWHPKDVQRVROXWLRQ 7KHV\VWHPLVFRQVLVWHQWIRUDQ\ YDOXHVRI h DQG k WKDWPDNH k C 3h D 0

308 3&%6$5*0/ "/% &$)&-0/ '03.4 7KLVVHFWLRQUHÀQHVWKHPHWKRGRI6HFWLRQ LQWRDURZUHGXFWLRQDOJRULWKPWKDWZLOO HQDEOHXVWRDQDO\]HDQ\V\VWHPRIOLQHDUHTXDWLRQVñ %\XVLQJRQO\WKHÀUVWSDUWRI WKHDOJRULWKP ZHZLOOEHDEOHWRDQVZHUWKHIXQGDPHQWDOH[LVWHQFHDQGXQLTXHQHVV TXHVWLRQVSRVHGLQ6HFWLRQ 7KHDOJRULWKPDSSOLHVWRDQ\PDWUL[ ZKHWKHURUQRWWKHPDWUL[LVYLHZHGDVDQ DXJPHQWHGPDWUL[IRUDOLQHDUV\VWHP 6RWKHÀUVWSDUWRIWKLVVHFWLRQFRQFHUQVDQ DUELWUDU\UHFWDQJXODUPDWUL[DQGEHJLQVE\LQWURGXFLQJWZRLPSRUWDQWFODVVHVRIPDWULFHV WKDWLQFOXGHWKH´WULDQJXODUµPDWULFHVRI6HFWLRQ ,QWKHGHÀQLWLRQVWKDWIROORZ D QRQ]HUR URZRUFROXPQLQDPDWUL[PHDQVDURZRUFROXPQWKDWFRQWDLQVDWOHDVWRQH QRQ]HURHQWU\ D OHDGLQJHQWU\ RIDURZUHIHUVWRWKHOHIWPRVWQRQ]HURHQWU\LQDQRQ]HUR URZ ñ 7KHDOJRULWKPKHUHLVDYDULDQWRIZKDWLVFRPPRQO\FDOOHG *DXVVLDQHOLPLQDWLRQ $ VLPLODUHOLPLQDWLRQ PHWKRGIRUOLQHDUV\VWHPVZDVXVHGE\&KLQHVHPDWKHPDWLFLDQVLQDERXW %& 7KHSURFHVVZDVXQNQRZQ LQ:HVWHUQFXOWXUHXQWLOWKHQLQHWHHQWKFHQWXU\ ZKHQDIDPRXV*HUPDQPDWKHPDWLFLDQ &DUO)ULHGULFK*DXVV GLVFRYHUHGLW $ *HUPDQHQJLQHHU :LOKHOP-RUGDQ SRSXODUL]HGWKHDOJRULWKPLQDQWH[WRQJHRGHV\

%&'*/*5*0/

5RZ5HGXFWLRQDQG(FKHORQ)RUPV

$ UHFWDQJXODUPDWUL[LVLQ HFKHORQIRUP RU URZHFKHORQIRUP LILWKDVWKH IROORZLQJWKUHHSURSHUWLHV $OOQRQ]HURURZVDUHDERYHDQ\URZVRIDOO]HURV (DFKOHDGLQJHQWU\RIDURZLVLQDFROXPQWRWKHULJKWRIWKHOHDGLQJHQWU\RI WKHURZDERYHLW $OOHQWULHVLQDFROXPQEHORZDOHDGLQJHQWU\DUH]HURV ,IDPDWUL[LQHFKHORQIRUPVDWLVÀHVWKHIROORZLQJDGGLWLRQDOFRQGLWLRQV WKHQLWLV LQ UHGXFHGHFKHORQIRUP RU UHGXFHGURZHFKHORQIRUP 7KHOHDGLQJHQWU\LQHDFKQRQ]HURURZLV (DFKOHDGLQJLVWKHRQO\QRQ]HURHQWU\LQLWVFROXPQ $Q HFKHORQPDWUL[ UHVSHFWLYHO\ UHGXFHGHFKHORQPDWUL[ LVRQHWKDWLVLQHFKHORQ IRUPUHVSHFWLYHO\ UHGXFHGHFKHORQIRUP 3URSHUW\VD\VWKDWWKHOHDGLQJHQWULHVIRUP DQ HFKHORQ ´VWHSOLNHµ SDWWHUQWKDWPRYHVGRZQDQGWRWKHULJKWWKURXJKWKHPDWUL[ 3URSHUW\LVDVLPSOHFRQVHTXHQFHRISURSHUW\ EXWZHLQFOXGHLWIRUHPSKDVLV 7KH´WULDQJXODUµPDWULFHVRI6HFWLRQ VXFKDV 2 3 2 3 2 "3 2 1 1 0 0 29 40 1 "4 8 5 DQG 4 0 1 0 16 5 0 0 0 5=2 0 0 1 3 DUHLQHFKHORQIRUP ,QIDFW WKHVHFRQGPDWUL[LVLQUHGXFHGHFKHORQIRUP +HUHDUH DGGLWLRQDOH[DPSOHV

&9".1-& 7KHIROORZLQJPDWULFHVDUHLQHFKHORQIRUP 7KHOHDGLQJHQWULHV PD\KDYHDQ\QRQ]HURYDOXH WKHVWDUUHGHQWULHV$ PD\KDYHDQ\YDOXHLQFOXGLQJ]HUR 2 3 2 3 0 $ $ $ $ $ $ $ $ $ $ $ 60 $ $ $ $ $ $7 0 0 6 7 60 $ $7 6 7; 6 0 $ $ $ $ $7 0 0 0 6 7 40 5 0 0 0 40 $ $ $ $5 0 0 0 0 0 0 0 0 $ 0 0 0 0 0 0 0 0

7KHIROORZLQJPDWULFHVDUHLQUHGXFHGHFKHORQIRUPEHFDXVHWKHOHDGLQJHQWULHVDUH·V DQGWKHUHDUH·VEHORZ DQGDERYH HDFKOHDGLQJ 2 3 2 3 0 1 $ 0 0 0 $ $ 0 $ 1 0 $ $ 60 1 0 0 $ $ 0 $7 0 0 6 7 60 1 $ $7 6 7; 6 0 1 0 $ $ 0 $7 0 0 0 6 7 40 5 0 0 0 40 $ 0 $5 0 0 0 0 1 $ 0 0 0 0 1 $ 0 0 0 0 0 0 0 0

$Q\QRQ]HURPDWUL[PD\EH URZUHGXFHG WKDWLV WUDQVIRUPHGE\HOHPHQWDU\URZ RSHUDWLRQV LQWRPRUHWKDQRQHPDWUL[LQHFKHORQIRUP XVLQJGLIIHUHQWVHTXHQFHVRIURZ RSHUDWLRQV +RZHYHU WKHUHGXFHGHFKHORQIRUPRQHREWDLQVIURPDPDWUL[LVXQLTXH 7KHIROORZLQJWKHRUHPLVSURYHGLQ$SSHQGL[$ DWWKHHQGRIWKHWH[W

5)&03&.

6OJRVFOFTT PG UIF 3FEVDFE &DIFMPO 'PSN (DFKPDWUL[LVURZHTXLYDOHQWWRRQHDQGRQO\RQHUHGXFHGHFKHORQPDWUL[

$)"15&3


,IDPDWUL[ A LVURZHTXLYDOHQWWRDQHFKHORQPDWUL[ U ZHFDOO U DQHFKHORQIRUP RUURZHFKHORQIRUP RI A LI U LVLQUHGXFHGHFKHORQIRUP ZHFDOO U WKHUHGXFHG HFKHORQIRUPRI A >0RVWPDWUL[SURJUDPVDQGFDOFXODWRUVZLWKPDWUL[FDSDELOLWLHV XVHWKHDEEUHYLDWLRQ55() IRUUHGXFHGURZ HFKHORQIRUP 6RPHXVH5() IRUURZ HFKHORQIRUP@

3LYRW3RVLWLRQV :KHQURZRSHUDWLRQVRQDPDWUL[SURGXFHDQHFKHORQIRUP IXUWKHUURZRSHUDWLRQVWR REWDLQWKHUHGXFHGHFKHORQIRUPGRQRWFKDQJHWKHSRVLWLRQVRIWKHOHDGLQJHQWULHV 6LQFH WKHUHGXFHGHFKHORQIRUPLVXQLTXH WKHOHDGLQJHQWULHVDUHDOZD\VLQWKHVDPHSRVLWLRQV LQDQ\HFKHORQIRUPREWDLQHGIURPDJLYHQPDWUL[ 7KHVHOHDGLQJHQWULHVFRUUHVSRQGWR OHDGLQJ·VLQWKHUHGXFHGHFKHORQIRUP

%&'*/*5*0/

$ SLYRWSRVLWLRQ LQDPDWUL[ A LVDORFDWLRQLQ A WKDWFRUUHVSRQGVWRDOHDGLQJ LQWKHUHGXFHGHFKHORQIRUPRI A $ SLYRWFROXPQ LVDFROXPQRI A WKDWFRQWDLQV DSLYRWSRVLWLRQ ,Q([DPSOH WKHVTXDUHV LGHQWLI\WKHSLYRWSRVLWLRQV 0DQ\IXQGDPHQWDO FRQFHSWVLQWKHÀUVWIRXUFKDSWHUVZLOOEHFRQQHFWHGLQRQHZD\RUDQRWKHUZLWKSLYRW SRVLWLRQVLQDPDWUL[

&9".1-& 5RZUHGXFHWKHPDWUL[ A EHORZWRHFKHORQIRUP DQGORFDWHWKHSLYRW FROXPQVRI A

2

3 0 "3 "6 4 9 6 "1 "2 "1 3 17 7 AD6 4 "2 "3 0 3 "1 5 1 4 5 "9 "7

40-65*0/ 8VHWKHVDPHEDVLFVWUDWHJ\DVLQ6HFWLRQ 7KHWRSRIWKHOHIWPRVW QRQ]HURFROXPQLVWKHÀUVWSLYRWSRVLWLRQ $ QRQ]HURHQWU\ RU SLYRW PXVWEHSODFHG LQWKLVSRVLWLRQ $ JRRGFKRLFHLVWRLQWHUFKDQJHURZVDQGEHFDXVHWKHPHQWDO FRPSXWDWLRQVLQWKHQH[WVWHSZLOOQRWLQYROYHIUDFWLRQV 2

3LYRW

3 1 ! 4 5 "9 "7 6 "1 "2 "1 3 17 6 7 4 "2 "3 0 3 "1 5 0 "3 "6 4 9 " 3LYRWFROXPQ

&UHDWH]HURVEHORZWKHSLYRW E\DGGLQJPXOWLSOHVRIWKHÀUVWURZWRWKHURZVEHORZ DQGREWDLQPDWUL[ EHORZ 7KHSLYRWSRVLWLRQLQWKHVHFRQGURZPXVWEHDVIDUOHIW DVSRVVLEOH³QDPHO\ LQWKHVHFRQGFROXPQ &KRRVHWKHLQWKLVSRVLWLRQDVWKHQH[W SLYRW 3LYRW 2 3 1 4 5 "9 "7 60 2 ! 4 "6 "6 7 6 7 40 5 10 "15 "15 5 4 9 0 "3 "6 " 1H[WSLYRWFROXPQ


$GG "5=2 WLPHVURZWRURZ DQGDGG 3=2 WLPHVURZWRURZ 2 3 1 4 5 "9 "7 60 2 4 "6 "6 7 6 7 40 0 0 0 05 0 0 0 "5 0

7KHPDWUL[LQ LVGLIIHUHQWIURPDQ\HQFRXQWHUHGLQ6HFWLRQ 7KHUHLVQRZD\WR FUHDWHDOHDGLQJHQWU\LQFROXPQ :HFDQ·WXVHURZRUEHFDXVHGRLQJVRZRXOG GHVWUR\WKHHFKHORQDUUDQJHPHQWRIWKHOHDGLQJHQWULHVDOUHDG\SURGXFHG +RZHYHU LI ZHLQWHUFKDQJHURZVDQG ZHFDQSURGXFHDOHDGLQJHQWU\LQFROXPQ 2

1 60 6 40 0

"

4 2 0 0 "

3LYRW 3 5 "9 "7 4 "6 "6 7 7 0 "5 ! 0 5 0 0 0

2 6

0 *HQHUDOIRUP 6 4

$

0 0

0 0

$ $ 0 0

" 3LYRWFROXPQV

$ $ 0

3 $ $7 7 $5 0

7KHPDWUL[LVLQHFKHORQIRUPDQGWKXVUHYHDOVWKDWFROXPQV DQGRI A DUHSLYRW FROXPQV 3LYRWSRVLWLRQV 2 3 0 !"3 "6 4 9 6 "1 "2 !"1 3 17 7 AD6 4 "2 "3 0 3 !"1 5 1 4 5 "9 "7 "

"

"

3LYRWFROXPQV

$ SLYRW DVLOOXVWUDWHGLQ([DPSOH LVDQRQ]HURQXPEHULQDSLYRWSRVLWLRQWKDWLV XVHGDVQHHGHGWRFUHDWH]HURVYLDURZRSHUDWLRQV 7KHSLYRWVLQ([DPSOHZHUH DQG "5 1RWLFHWKDWWKHVHQXPEHUVDUHQRWWKHVDPHDVWKHDFWXDOHOHPHQWVRI A LQWKH KLJKOLJKWHGSLYRWSRVLWLRQVVKRZQLQ :LWK([DPSOHDVDJXLGH ZHDUHUHDG\WRGHVFULEHDQHIÀFLHQWSURFHGXUHIRU WUDQVIRUPLQJDPDWUL[LQWRDQHFKHORQRUUHGXFHGHFKHORQPDWUL[ &DUHIXOVWXG\DQG PDVWHU\RIWKLVSURFHGXUHQRZZLOOSD\ULFKGLYLGHQGVODWHULQWKHFRXUVH

7KH5RZ5HGXFWLRQ$OJRULWKP 7KHDOJRULWKPWKDWIROORZVFRQVLVWVRIIRXUVWHSV DQGLWSURGXFHVDPDWUL[LQHFKHORQ IRUP $ ÀIWKVWHSSURGXFHVDPDWUL[LQUHGXFHGHFKHORQIRUP :HLOOXVWUDWHWKHDOJRULWKP E\DQH[DPSOH

&9".1-& $SSO\HOHPHQWDU\URZRSHUDWLRQVWRWUDQVIRUPWKHIROORZLQJPDWUL[ ÀUVWLQWRHFKHORQIRUPDQGWKHQLQWRUHGXFHGHFKHORQIRUP 2 3 0 3 "6 6 4 "5 4 3 "7 8 "5 8 95 3 "9 12 "9 6 15

40-65*0/

45&1 %HJLQZLWKWKHOHIWPRVWQRQ]HURFROXPQ 7KLVLVDSLYRWFROXPQ 7KHSLYRW SRVLWLRQLVDWWKHWRS

$)"15&3


2

0 3 "6 6 4 3 "7 8 "5 3 "9 12 "9 " 3LYRWFROXPQ

4 8 6

3 "5 95 15

45&1 6HOHFWDQRQ]HURHQWU\LQWKHSLYRWFROXPQDVDSLYRW ,IQHFHVVDU\ LQWHUFKDQJH URZVWRPRYHWKLVHQWU\LQWRWKHSLYRWSRVLWLRQ ,QWHUFKDQJHURZVDQG :HFRXOGKDYHLQWHUFKDQJHGURZVDQGLQVWHDG 2

3 !"9 4 3 "7 0 3

3LYRW

12 "9 8 "5 "6 6

6 8 4

3 15 95 "5

45&1 8VHURZUHSODFHPHQWRSHUDWLRQVWRFUHDWH]HURVLQDOOSRVLWLRQVEHORZWKHSLYRW $VDSUHOLPLQDU\VWHS ZHFRXOGGLYLGHWKHWRSURZE\WKHSLYRW %XWZLWKWZR·VLQ FROXPQ LWLVMXVWDVHDV\WRDGG "1 WLPHVURZWRURZ 2

3 !"9 40 2 0 3

3LYRW

12 "9 "4 4 "6 6

6 2 4

3 15 "6 5 "5

45&1 &RYHURULJQRUH WKHURZFRQWDLQLQJWKHSLYRWSRVLWLRQDQGFRYHUDOOURZV LIDQ\ DERYHLW $SSO\VWHSV²WRWKHVXEPDWUL[WKDWUHPDLQV 5HSHDWWKHSURFHVVXQWLO WKHUHDUHQRPRUHQRQ]HURURZVWRPRGLI\ :LWKURZFRYHUHG VWHSVKRZVWKDWFROXPQLVWKHQH[WSLYRWFROXPQ IRUVWHS VHOHFWDVDSLYRWWKH´WRSµHQWU\LQWKDWFROXPQ 2

3LYRW

3 3 "9 12 "9 6 15 40 2 ! "4 4 2 "6 5 0 3 "6 6 4 "5 " 1HZSLYRWFROXPQ

)RUVWHS ZHFRXOGLQVHUWDQRSWLRQDOVWHSRIGLYLGLQJWKH´WRSµURZRIWKHVXEPDWUL[E\ WKHSLYRW ,QVWHDG ZHDGG "3=2 WLPHVWKH´WRSµURZWRWKHURZEHORZ 7KLVSURGXFHV 2

3 "9 40 2 0 0

12 "9 "4 4 0 0

3 6 15 2 "6 5 1 4


:KHQZHFRYHUWKHURZFRQWDLQLQJWKHVHFRQGSLYRWSRVLWLRQIRUVWHS ZHDUHOHIWZLWK DQHZVXEPDWUL[KDYLQJRQO\RQHURZ 2

3 "9 40 2 0 0

12 "9 "4 4 0 0

3 6 15 2 "6 5 1! 4

3LYRW

6WHSV²UHTXLUHQRZRUNIRUWKLVVXEPDWUL[ DQGZHKDYHUHDFKHGDQHFKHORQIRUPRI WKHIXOOPDWUL[ ,IZHZDQWWKHUHGXFHGHFKHORQIRUP ZHSHUIRUPRQHPRUHVWHS 45&1 %HJLQQLQJZLWKWKHULJKWPRVWSLYRWDQGZRUNLQJXSZDUGDQGWRWKHOHIW FUHDWH ]HURVDERYHHDFKSLYRW ,IDSLYRWLVQRW PDNHLWE\DVFDOLQJRSHUDWLRQ

7KHULJKWPRVWSLYRWLVLQURZ &UHDWH]HURVDERYHLW DGGLQJVXLWDEOHPXOWLSOHVRIURZ WRURZVDQG 2 3 ! 5RZ C ."6/ ! URZ 3 "9 12 "9 0 "9 ! 5RZ C ."2/ ! URZ 40 5 2 "4 4 0 "14 0 0 0 0 1 4 7KHQH[WSLYRWLVLQURZ 6FDOHWKLVURZ GLYLGLQJE\WKHSLYRW 2

3 "9 12 "9 40 1 "2 2 0 0 0 0

3 0 "9 0 "7 5 1 4

! 5RZVFDOHGE\

1 2

&UHDWHD]HURLQFROXPQE\DGGLQJWLPHVURZWRURZ 2

3 40 0

0 "6 1 "2 0 0

9 2 0

3 0 "72 0 "7 5 1 4

! 5RZ C .9/ ! URZ

)LQDOO\ VFDOHURZ GLYLGLQJE\WKHSLYRW 2

1 40 0

0 "2 1 "2 0 0

3 2 0

3 0 "24 0 "7 5 1 4

! 5RZVFDOHGE\

1 3

7KLVLVWKHUHGXFHGHFKHORQIRUPRIWKHRULJLQDOPDWUL[ 7KHFRPELQDWLRQRIVWHSV²LVFDOOHGWKH IRUZDUGSKDVH RIWKHURZUHGXFWLRQ DOJRULWKP 6WHS ZKLFKSURGXFHVWKHXQLTXHUHGXFHGHFKHORQIRUP LVFDOOHGWKH EDFNZDUGSKDVH

/6.&3*$"- /05& ,QVWHSDERYH DFRPSXWHUSURJUDPXVXDOO\VHOHFWVDVDSLYRWWKHHQWU\LQD FROXPQKDYLQJWKHODUJHVWDEVROXWHYDOXH 7KLVVWUDWHJ\ FDOOHG SDUWLDOSLYRWLQJ LVXVHGEHFDXVHLWUHGXFHVURXQGRIIHUURUVLQWKHFDOFXODWLRQV

$)"15&3


6ROXWLRQVRI/LQHDU6\VWHPV 7KHURZUHGXFWLRQDOJRULWKPOHDGVGLUHFWO\WRDQH[SOLFLWGHVFULSWLRQRIWKHVROXWLRQVHW RIDOLQHDUV\VWHPZKHQWKHDOJRULWKPLVDSSOLHGWRWKHDXJPHQWHGPDWUL[RIWKHV\VWHP 6XSSRVH IRUH[DPSOH WKDWWKHDXJPHQWHGPDWUL[RIDOLQHDUV\VWHPKDVEHHQ FKDQJHGLQWRWKHHTXLYDOHQW UHGXFHG HFKHORQIRUP 2 3 1 0 "5 1 40 1 1 45 0 0 0 0 7KHUHDUHWKUHHYDULDEOHVEHFDXVHWKHDXJPHQWHGPDWUL[KDVIRXUFROXPQV 7KH DVVRFLDWHGV\VWHPRIHTXDWLRQVLV

x1

" 5x3 D 1 x2 C x3 D 4 0 D0

7KHYDULDEOHV x1 DQG x2 FRUUHVSRQGLQJWRSLYRWFROXPQVLQWKHPDWUL[DUHFDOOHG EDVLF YDULDEOHVò 7KHRWKHUYDULDEOH x3 LVFDOOHGD IUHHYDULDEOH :KHQHYHUDV\VWHP LV FRQVLVWHQW DV LQ WKH VROXWLRQ VHWFDQEH GHVFULEHG H[SOLFLWO\E\VROYLQJWKH UHGXFHG V\VWHPRIHTXDWLRQVIRUWKHEDVLFYDULDEOHVLQWHUPVRI WKHIUHHYDULDEOHV 7KLVRSHUDWLRQLVSRVVLEOHEHFDXVHWKHUHGXFHGHFKHORQIRUPSODFHV HDFKEDVLFYDULDEOHLQRQHDQGRQO\RQHHTXDWLRQ ,Q VROYHWKHÀUVWHTXDWLRQIRU x1 DQGWKHVHFRQGIRU x2 ,JQRUHWKHWKLUGHTXDWLRQ LWRIIHUVQRUHVWULFWLRQRQWKHYDULDEOHV 8 ˆ
&9".1-& )LQGWKHJHQHUDOVROXWLRQRIWKHOLQHDUV\VWHPZKRVHDXJPHQWHGPD WUL[KDVEHHQUHGXFHGWR

2

1 40 0

6 0 0

3 2 "5 "2 "4 2 "8 "1 3 5 0 0 1 7

40-65*0/ 7KHPDWUL[LVLQHFKHORQIRUP EXWZHZDQWWKHUHGXFHGHFKHORQIRUP EHIRUHVROYLQJIRUWKHEDVLFYDULDEOHV 7KHURZUHGXFWLRQLVFRPSOHWHGQH[W 7KHV\PERO % EHIRUHDPDWUL[LQGLFDWHVWKDWWKHPDWUL[LVURZHTXLYDOHQWWRWKHSUHFHGLQJPDWUL[ 2 3 2 3 1 6 2 "5 "2 "4 1 6 2 "5 0 10 40 0 2 "8 "1 3 5 % 4 0 0 2 "8 0 10 5 0 0 0 0 1 7 0 0 0 0 1 7 2 3 2 3 1 6 2 "5 0 10 1 6 0 3 0 0 %4 0 0 1 "4 0 5 5 % 4 0 0 1 "4 0 5 5 0 0 0 0 1 7 0 0 0 0 1 7 ò 6RPHWH[WVXVHWKHWHUP OHDGLQJYDULDEOHV EHFDXVHWKH\FRUUHVSRQGWRWKHFROXPQVFRQWDLQLQJOHDGLQJ HQWULHV


7KHUHDUHÀYHYDULDEOHVEHFDXVHWKHDXJPHQWHGPDWUL[KDVVL[FROXPQV 7KHDVVRFLDWHG V\VWHPQRZLV x1 C 6x2 C 3x4 D0 D5 x3 " 4x4 x5 D 7 7KHSLYRWFROXPQVRIWKHPDWUL[DUH DQG VRWKHEDVLFYDULDEOHVDUH x1 x3 DQG x5 7KHUHPDLQLQJYDULDEOHV x2 DQG x4 PXVWEHIUHH 6ROYHIRUWKHEDVLFYDULDEOHVWR REWDLQWKHJHQHUDOVROXWLRQ 8 ˆ x1 D "6x2 " 3x4 ˆ ˆ ˆ ˆ ˆ
3DUDPHWULF'HVFULSWLRQVRI6ROXWLRQ6HWV 7KHGHVFULSWLRQVLQ DQG DUH SDUDPHWULFGHVFULSWLRQV RIVROXWLRQVHWVLQZKLFK WKHIUHHYDULDEOHVDFWDVSDUDPHWHUV 6ROYLQJDV\VWHP DPRXQWVWRÀQGLQJDSDUDPHWULF GHVFULSWLRQRIWKHVROXWLRQVHWRUGHWHUPLQLQJWKDWWKHVROXWLRQVHWLVHPSW\ :KHQHYHUDV\VWHPLVFRQVLVWHQWDQGKDVIUHHYDULDEOHV WKHVROXWLRQVHWKDVPDQ\ SDUDPHWULFGHVFULSWLRQV )RULQVWDQFH LQV\VWHP ZHPD\DGGWLPHVHTXDWLRQWR HTXDWLRQDQGREWDLQWKHHTXLYDOHQWV\VWHP

x1 C 5x2 D 21 x2 C x3 D 4 :HFRXOGWUHDW x2 DVDSDUDPHWHUDQGVROYHIRU x1 DQG x3 LQWHUPVRI x2 DQGZHZRXOG KDYHDQDFFXUDWHGHVFULSWLRQRIWKHVROXWLRQVHW +RZHYHU WREHFRQVLVWHQW ZHPDNHWKH DUELWUDU\ FRQYHQWLRQRIDOZD\VXVLQJWKHIUHHYDULDEOHVDVWKHSDUDPHWHUVIRUGHVFULELQJ DVROXWLRQVHW 7KHDQVZHUVHFWLRQDWWKHHQGRIWKHWH[WDOVRUHÁHFWVWKLVFRQYHQWLRQ :KHQHYHUDV\VWHPLVLQFRQVLVWHQW WKHVROXWLRQVHWLVHPSW\ HYHQZKHQWKHV\VWHP KDVIUHHYDULDEOHV ,QWKLVFDVH WKHVROXWLRQVHWKDV QR SDUDPHWULFUHSUHVHQWDWLRQ

%DFN6XEVWLWXWLRQ &RQVLGHUWKHIROORZLQJV\VWHP ZKRVHDXJPHQWHGPDWUL[LVLQHFKHORQIRUPEXWLV QRW LQUHGXFHGHFKHORQIRUP

x1 " 7x2 C 2x3 " 5x4 C 8x5 D 10 x2 " 3x3 C 3x4 C x5 D "5 x4 " x5 D 4 $ FRPSXWHUSURJUDPZRXOGVROYHWKLVV\VWHPE\EDFNVXEVWLWXWLRQ UDWKHUWKDQE\FRP SXWLQJWKHUHGXFHGHFKHORQIRUP 7KDWLV WKHSURJUDPZRXOGVROYHHTXDWLRQIRU x4 LQ WHUPVRI x5 DQGVXEVWLWXWHWKHH[SUHVVLRQIRU x4 LQWRHTXDWLRQ VROYHHTXDWLRQIRU x2 DQGWKHQVXEVWLWXWHWKHH[SUHVVLRQVIRU x2 DQG x4 LQWRHTXDWLRQDQGVROYHIRU x1 2XUPDWUL[IRUPDWIRUWKHEDFNZDUGSKDVHRIURZUHGXFWLRQ ZKLFKSURGXFHVWKHUH GXFHGHFKHORQIRUP KDVWKHVDPHQXPEHURIDULWKPHWLFRSHUDWLRQVDVEDFNVXEVWLWXWLRQ %XWWKHGLVFLSOLQHRIWKHPDWUL[IRUPDWVXEVWDQWLDOO\UHGXFHVWKHOLNHOLKRRGRIHUURUV

$)"15&3


GXULQJKDQGFRPSXWDWLRQV 7KHEHVWVWUDWHJ\LVWRXVHRQO\WKH UHGXFHG HFKHORQIRUP WRVROYHDV\VWHP 7KH 6WXG\*XLGH WKDWDFFRPSDQLHVWKLVWH[WRIIHUVVHYHUDOKHOSIXO VXJJHVWLRQVIRUSHUIRUPLQJURZRSHUDWLRQVDFFXUDWHO\DQGUDSLGO\

/6.&3*$"- /05& ,Q JHQHUDO WKH IRUZDUG SKDVH RI URZ UHGXFWLRQ WDNHV PXFK ORQJHU WKDQ WKH EDFNZDUGSKDVH $QDOJRULWKPIRUVROYLQJDV\VWHPLVXVXDOO\PHDVXUHGLQÁRSV RUÁRDWLQJSRLQWRSHUDWLRQV $ ÁRS LVRQHDULWKPHWLFRSHUDWLRQC; "; $; = RQWZRUHDOÁRDWLQJSRLQWQXPEHUV )RUDQ n # .n C 1/ PDWUL[ WKHUHGXFWLRQ WRHFKHORQIRUPFDQWDNH 2n3 =3 C n2 =2 " 7n=6 ÁRSVZKLFKLVDSSUR[LPDWHO\ 2n3 =3 ÁRSVZKHQ n LVPRGHUDWHO\ODUJH³VD\ n & 30/ ,QFRQWUDVW IXUWKHU UHGXFWLRQWRUHGXFHGHFKHORQIRUPQHHGVDWPRVW n2 ÁRSV

([LVWHQFHDQG8QLTXHQHVV4XHVWLRQV $OWKRXJKDQRQUHGXFHGHFKHORQIRUPLVDSRRUWRROIRUVROYLQJDV\VWHP WKLVIRUPLV MXVWWKHULJKWGHYLFHIRUDQVZHULQJWZRIXQGDPHQWDOTXHVWLRQVSRVHGLQ6HFWLRQ

&9".1-& 'HWHUPLQHWKHH[LVWHQFHDQGXQLTXHQHVVRIWKHVROXWLRQVWRWKHV\VWHP 3x2 " 6x3 C 6x4 C 4x5 D "5 3x1 " 7x2 C 8x3 " 5x4 C 8x5 D 9 3x1 " 9x2 C 12x3 " 9x4 C 6x5 D 15 40-65*0/ 7KHDXJPHQWHGPDWUL[RIWKLVV\VWHPZDVURZUHGXFHGLQ([DPSOHWR 2 3 3 "9 12 "9 6 15 40 2 "4 4 2 "6 5 0 0 0 0 1 4

7KHEDVLFYDULDEOHVDUH x1 x2 DQG x5 WKHIUHHYDULDEOHVDUH x3 DQG x4 7KHUHLVQR HTXDWLRQVXFKDV 0 D 1 WKDWZRXOGLQGLFDWHDQLQFRQVLVWHQWV\VWHP VRZHFRXOGXVH EDFNVXEVWLWXWLRQWRÀQGDVROXWLRQ %XWWKH H[LVWHQFH RIDVROXWLRQLVDOUHDG\FOHDU LQ $OVR WKHVROXWLRQLV QRWXQLTXH EHFDXVHWKHUHDUHIUHHYDULDEOHV (DFKGLIIHUHQW FKRLFHRI x3 DQG x4 GHWHUPLQHVDGLIIHUHQWVROXWLRQ 7KXVWKHV\VWHPKDVLQÀQLWHO\PDQ\ VROXWLRQV :KHQDV\VWHPLVLQHFKHORQIRUPDQGFRQWDLQVQRHTXDWLRQRIWKHIRUP 0 D b ZLWK b QRQ]HUR HYHU\QRQ]HURHTXDWLRQFRQWDLQVDEDVLFYDULDEOHZLWKDQRQ]HURFRHIÀFLHQW (LWKHUWKHEDVLFYDULDEOHVDUHFRPSOHWHO\GHWHUPLQHGZLWKQRIUHHYDULDEOHV RUDWOHDVW RQHRIWKHEDVLFYDULDEOHVPD\EHH[SUHVVHGLQWHUPVRIRQHRUPRUHIUHHYDULDEOHV ,Q WKHIRUPHUFDVH WKHUHLVDXQLTXHVROXWLRQ LQWKHODWWHUFDVH WKHUHDUHLQÀQLWHO\PDQ\ VROXWLRQVRQHIRUHDFKFKRLFHRIYDOXHVIRUWKHIUHHYDULDEOHV 7KHVHUHPDUNVMXVWLI\WKHIROORZLQJWKHRUHP ó 7UDGLWLRQDOO\ D ÁRS ZDVRQO\DPXOWLSOLFDWLRQRUGLYLVLRQ EHFDXVHDGGLWLRQDQGVXEWUDFWLRQWRRNPXFKOHVV WLPHDQGFRXOGEHLJQRUHG 7KHGHÀQLWLRQRI ÁRS JLYHQKHUHLVSUHIHUUHGQRZ DVDUHVXOWRIDGYDQFHVLQ FRPSXWHUDUFKLWHFWXUH 6HH*ROXEDQG9DQ/RDQ 0DWUL[&RPSXWDWLRQV QGHG %DOWLPRUH 7KH-RKQV +RSNLQV3UHVV SS ²

5)&03&.


&YJTUFODF BOE 6OJRVFOFTT 5IFPSFN $ OLQHDUV\VWHPLVFRQVLVWHQWLIDQGRQO\LIWKHULJKWPRVWFROXPQRIWKHDXJPHQWHG PDWUL[LV QRW DSLYRWFROXPQ³WKDWLV LIDQGRQO\LIDQHFKHORQIRUPRIWKH DXJPHQWHGPDWUL[KDV QR URZRIWKHIRUP

Œ0 !!! 0 b !

ZLWK b QRQ]HUR

,IDOLQHDUV\VWHPLVFRQVLVWHQW WKHQWKHVROXWLRQVHWFRQWDLQVHLWKHUL DXQLTXH VROXWLRQ ZKHQWKHUHDUHQRIUHHYDULDEOHV RULL LQÀQLWHO\PDQ\VROXWLRQV ZKHQ WKHUHLVDWOHDVWRQHIUHHYDULDEOH 7KHIROORZLQJSURFHGXUHRXWOLQHVKRZWRÀ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

13"$5*$& 130#-&.4 )LQGWKHJHQHUDOVROXWLRQRIWKHOLQHDUV\VWHPZKRVHDXJPHQWHGPDWUL[LV # $ 1 "3 "5 0 0 1 1 3 )LQGWKHJHQHUDOVROXWLRQRIWKHV\VWHP

x1 " 2x2 " x3 C 3x4 D 0 "2x1 C 4x2 C 5x3 " 5x4 D 3 3x1 " 6x2 " 6x3 C 8x4 D 2

&9&3$*4&4 ,Q([HUFLVHVDQG GHWHUPLQHZKLFKPDWULFHVDUHLQUHGXFHG HFKHORQIRUPDQGZKLFKRWKHUVDUHRQO\LQHFKHORQIRUP 2

1 D 4 0 0

0 1 0

0 0 1

3

0 05 1

2

1 E 4 0 0

0 1 0

1 1 0

3

0 05 1

2

1 60 F 6 40 0

0 1 0 0

0 1 0 0

3 0 07 7 05 1

2

1 60 G 6 40 0

1 2 0 0

0 0 0 0

1 2 3 0

3 1 27 7 35 4

$)"15&3 2

1 D 4 0 0 2 0 61 6 F 4 0 0 2 0 60 6 G 4 0 0


0 1 0

1 1 0

0 2 0 0

0 0 1 0

1 0 0 0

1 1 0 0

3 1 15 0 3 0 07 7 05 1

1 1 0 0

2

1 E 4 0 0

0 2 0

0 0 1

3 0 05 1

3 1 17 7 15 0

5RZUHGXFHWKHPDWULFHVLQ([HUFLVHVDQGWRUHGXFHGHFKHORQ IRUP &LUFOHWKHSLYRWSRVLWLRQVLQWKHÀQDOPDWUL[DQGLQWKH RULJLQDOPDWUL[ DQGOLVWWKHSLYRWFROXPQV 2 3 2 3 1 2 4 8 1 2 4 5 4 6 85 4 5 45 4 2 4 2 3 6 9 12 4 5 4 2 'HVFULEH WKH SRVVLEOH HFKHORQ IRUPV RI D QRQ]HUR 2 # 2 PDWUL[ 8VHWKHV\PEROV $ DQG DVLQWKHÀUVWSDUWRI ([DPSOH

5HSHDW([HUFLVHIRUDQRQ]HUR 3 # 2 PDWUL[ )LQGWKHJHQHUDOVROXWLRQVRIWKHV\VWHPVZKRVHDXJPHQWHGPD WULFHVDUHJLYHQLQ([HUFLVHV² # $ # $ 1 3 4 7 1 "3 0 "5 3 9 7 6 "3 7 0 9 # $ # $ 0 1 "2 3 1 "2 "1 4 1 "3 4 "6 "2 4 "5 6 2 3 2 3 1 0 "9 0 4 3 "2 4 0 60 1 3 0 "1 7 7 05 4 9 "6 12 6 40 0 0 1 "7 5 6 "4 8 0 0 0 0 0 1 2 3 1 "3 0 "1 0 "2 60 1 0 0 "4 17 7 6 40 0 0 1 9 45 0 0 0 0 0 0 2 3 1 0 "5 0 "8 3 60 1 4 "1 0 67 7 6 40 0 0 0 1 05 0 0 0 0 0 0

([HUFLVHVDQGXVHWKHQRWDWLRQRI([DPSOHIRUPDWULFHV LQHFKHORQIRUP 6XSSRVHHDFKPDWUL[UHSUHVHQWVWKHDXJPHQWHG PDWUL[IRUDV\VWHPRIOLQHDUHTXDWLRQV ,QHDFKFDVH GHWHUPLQHLI WKHV\VWHPLVFRQVLVWHQW ,IWKHV\VWHPLVFRQVLVWHQW GHWHUPLQHLI WKHVROXWLRQLVXQLTXH 2 3 $ $ $ $ $5 D 4 0 0 0 0 0 2 3 0 $ $ $ 0 $ $5 E 4 0 0 0 0 0

2

D 4 0 0 2

E 4 0 0

$ 0 $ 0 0

3 $ $5

$

0

$ $

3 $ $5 $

,Q([HUFLVHVDQG GHWHUPLQHWKHYDOXHV RI h VXFKWKDWWKH PDWUL[LVWKHDXJPHQWHGPDWUL[RIDFRQVLVWHQWOLQHDUV\VWHP # $ # $ 1 "1 4 1 "3 1 "2 3 h h 6 "2 ,Q([HUFLVHVDQG FKRRVH h DQG k VXFKWKDWWKHV\VWHPKDVD QRVROXWLRQ E DXQLTXHVROXWLRQ DQGF PDQ\VROXWLRQV *LYH VHSDUDWHDQVZHUVIRUHDFKSDUW

x1 C hx2 D 2

4x1 C 8x2 D k

x1 " 3x2 D 1

2x1 C hx2 D k

,Q([HUFLVHVDQG PDUNHDFKVWDWHPHQW7UXHRU)DOVH -XVWLI\ HDFKDQVZHU D ,QVRPHFDVHV DPDWUL[PD\EHURZUHGXFHGWRPRUH WKDQRQHPDWUL[LQUHGXFHGHFKHORQIRUP XVLQJGLIIHUHQW VHTXHQFHVRIURZRSHUDWLRQV E 7KHURZUHGXFWLRQDOJRULWKPDSSOLHVRQO\WRDXJPHQWHG PDWULFHVIRUDOLQHDUV\VWHP F $ EDVLF YDULDEOH LQ D OLQHDU V\VWHP LV D YDULDEOH WKDW FRUUHVSRQGVWRDSLYRWFROXPQLQWKHFRHIÀFLHQWPDWUL[ G )LQGLQJDSDUDPHWULFGHVFULSWLRQRIWKHVROXWLRQVHWRID OLQHDUV\VWHPLVWKHVDPHDV VROYLQJ WKHV\VWHP H ,IRQHURZLQDQHFKHORQIRUPRIDQDXJPHQWHGPDWUL[ LV Œ 0 0 0 5 0 ! WKHQWKHDVVRFLDWHGOLQHDUV\VWHPLV LQFRQVLVWHQW D 7KHUHGXFHGHFKHORQIRUPRIDPDWUL[LVXQLTXH E ,IHYHU\FROXPQRIDQDXJPHQWHGPDWUL[FRQWDLQVDSLYRW WKHQWKHFRUUHVSRQGLQJV\VWHPLVFRQVLVWHQW F 7KHSLYRWSRVLWLRQVLQDPDWUL[GHSHQGRQZKHWKHUURZ LQWHUFKDQJHVDUHXVHGLQWKHURZUHGXFWLRQSURFHVV G $ JHQHUDOVROXWLRQRIDV\VWHPLVDQH[SOLFLWGHVFULSWLRQ RIDOOVROXWLRQVRIWKHV\VWHP H :KHQHYHUDV\VWHPKDVIUHHYDULDEOHV WKHVROXWLRQVHW FRQWDLQVPDQ\VROXWLRQV 6XSSRVHWKHFRHIÀFLHQWPDWUL[RIDOLQHDUV\VWHPRIIRXU HTXDWLRQVLQIRXUYDULDEOHVKDVDSLYRWLQHDFKFROXPQ ([ SODLQZK\WKHV\VWHPKDVDXQLTXHVROXWLRQ 6XSSRVHDV\VWHPRIOLQHDUHTXDWLRQVKDVD 3 # 5 DXJPHQWHG PDWUL[ZKRVHÀIWKFROXPQLVQRWDSLYRWFROXPQ ,VWKH V\VWHPFRQVLVWHQW" :K\RUZK\QRW " 4 7UXHIDOVHTXHVWLRQVRIWKLVW\SHZLOODSSHDULQPDQ\VHFWLRQV 0HWKRGV IRUMXVWLI\LQJ\RXUDQVZHUVZHUHGHVFULEHGEHIRUH([HUFLVHVDQGLQ 6HFWLRQ

6XSSRVHWKHFRHIÀFLHQWPDWUL[RIDV\VWHPRIOLQHDUHTXDWLRQV KDVDSLYRWSRVLWLRQLQHYHU\URZ ([SODLQZK\WKHV\VWHPLV FRQVLVWHQW 6XSSRVHD 3 # 5 FRHIÀFLHQW PDWUL[IRUDV\VWHPKDVWKUHH SLYRWFROXPQV ,VWKHV\VWHPFRQVLVWHQW" :K\RUZK\QRW" 5HVWDWHWKHODVWVHQWHQFHLQ7KHRUHPXVLQJWKHFRQFHSWRI SLYRWFROXPQV ´,IDOLQHDUV\VWHPLVFRQVLVWHQW WKHQWKH µ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ÀFV\VWHPRIWKUHHHTXDWLRQVLQWZRXQNQRZQV 6XSSRVHDQ n # .n C 1/ PDWUL[LVURZUHGXFHGWRUHGXFHG HFKHORQ IRUP $SSUR[LPDWHO\ ZKDW IUDFWLRQ RI WKH WRWDO QXPEHURIRSHUDWLRQVÁRSV LVLQYROYHGLQWKHEDFNZDUG SKDVHRIWKHUHGXFWLRQZKHQ n D 20" ZKHQ n D 200" 6XSSRVHH[SHULPHQWDOGDWDDUHUHSUHVHQWHGE\DVHWRISRLQWVLQWKH SODQH $Q LQWHUSRODWLQJSRO\QRPLDO IRUWKHGDWDLVDSRO\QRPLDO ZKRVH JUDSK SDVVHV WKURXJK HYHU\ SRLQW ,Q VFLHQWLÀF ZRUN


VXFKDSRO\QRPLDOFDQEHXVHG IRUH[DPSOH WRHVWLPDWHYDOXHV EHWZHHQWKHNQRZQGDWDSRLQWV $QRWKHUXVHLVWRFUHDWHFXUYHVIRU JUDSKLFDOLPDJHVRQDFRPSXWHUVFUHHQ 2QHPHWKRGIRUÀQGLQJDQ LQWHUSRODWLQJSRO\QRPLDOLVWRVROYHDV\VWHPRIOLQHDUHTXDWLRQV WEB

)LQGWKHLQWHUSRODWLQJSRO\QRPLDO p.t/ D a0 C a1 t C a2 t 2 IRUWKHGDWD .1; 6/ .2; 15/ .3; 28/ 7KDWLV ÀQG a0 a1 DQG a2 VXFKWKDW

a0 C a1 .1/ C a2 .1/2 D 6 a0 C a1 .2/ C a2 .2/2 D 15 a0 C a1 .3/ C a2 .3/2 D 28 >0@ ,QDZLQGWXQQHOH[SHULPHQW WKHIRUFHRQDSURMHFWLOH GXHWRDLUUHVLVWDQFHZDVPHDVXUHGDWGLIIHUHQWYHORFLWLHV 9HORFLW\IWVHF )RUFHOE )LQGDQLQWHUSRODWLQJSRO\QRPLDOIRUWKHVHGDWDDQGHVWLPDWH WKHIRUFHRQWKHSURMHFWLOHZKHQWKHSURMHFWLOHLVWUDYHOLQJ DWIWVHF 8VH p.t/ D a0 C a1 t C a2 t 2 C a3 t 3 C a4 t 4 C a5 t 5 :KDWKDSSHQVLI\RXWU\WRXVHDSRO\QRPLDORIGHJUHH OHVVWKDQ" 7U\DFXELFSRO\QRPLDO IRULQVWDQFH

5 ([HUFLVHVPDUNHGZLWKWKHV\PERO>0@ DUHGHVLJQHGWREHZRUNHGZLWK WKHDLGRID´0DWUL[SURJUDPµDFRPSXWHUSURJUDP VXFKDV 0$7/$% 0DSOH70 0DWKHPDWLFD 0DWK&DG RU'HULYH70 RUD SURJUDPPDEOHFDOFXODWRUZLWKPDWUL[FDSDELOLWLHV VXFKDVWKRVH PDQXIDFWXUHGE\7H[DV,QVWUXPHQWVRU+HZOHWW3DFNDUG

40-65*0/4 50 13"$5*$& 130#-&.4 7KHUHGXFHGHFKHORQIRUPRIWKHDXJPHQWHGPDWUL[DQGWKHFRUUHVSRQGLQJV\VWHP DUH # $ x1 " 2x3 D 9 1 0 "2 9 DQG 0 1 1 3 x2 C x3 D 3

7KHJHQHUDOVROXWLRQRIWKH V\VWHPRIHTXDWLRQVLVWKHOLQHRI LQWHUVHFWLRQRIWKHWZRSODQHV

7KHEDVLFYDULDEOHVDUH x1 DQG x2 DQGWKHJHQHUDOVROXWLRQLV 8 ˆ
1RWH ,WLVHVVHQWLDOWKDWWKHJHQHUDOVROXWLRQGHVFULEHHDFKYDULDEOH ZLWKDQ\SDUDP HWHUVFOHDUO\LGHQWLÀHG 7KHIROORZLQJVWDWHPHQWGRHV QRW GHVFULEHWKHVROXWLRQ 8 ˆ

$)"15&3


5RZUHGXFHWKHV\VWHP·VDXJPHQWHGPDWUL[ 2 3 2 1 "2 "1 3 0 1 "2 "1 3 4 "2 4 5 "5 35 % 40 0 3 1 3 "6 "6 8 2 0 0 "3 "1 2 1 "2 "1 3 % 40 0 3 1 0 0 0 0

3 0 35 2 3 0 35 5

7KLVHFKHORQPDWUL[VKRZVWKDWWKHV\VWHPLV LQFRQVLVWHQW EHFDXVHLWVULJKWPRVW FROXPQLVDSLYRWFROXPQ WKHWKLUGURZFRUUHVSRQGVWRWKHHTXDWLRQ 7KHUH LVQRQHHGWRSHUIRUPDQ\PRUHURZRSHUDWLRQV 1RWHWKDWWKHSUHVHQFHRIWKHIUHH YDULDEOHVLQWKLVSUREOHPLVLUUHOHYDQWEHFDXVHWKHV\VWHPLVLQFRQVLVWHQW

7&$503 &26"5*0/4 ,PSRUWDQWSURSHUWLHVRIOLQHDUV\VWHPVFDQEHGHVFULEHGZLWKWKHFRQFHSWDQGQRWDWLRQ RIYHFWRUV 7KLVVHFWLRQFRQQHFWVHTXDWLRQVLQYROYLQJYHFWRUVWRRUGLQDU\V\VWHPVRI HTXDWLRQV 7KHWHUP YHFWRU DSSHDUVLQDYDULHW\RIPDWKHPDWLFDODQGSK\VLFDOFRQWH[WV ZKLFKZHZLOOGLVFXVVLQ&KDSWHU ´9HFWRU6SDFHVµ 8QWLOWKHQ YHFWRU ZLOOPHDQDQ RUGHUHGOLVWRIQXPEHUV 7KLVVLPSOHLGHDHQDEOHVXVWRJHWWRLQWHUHVWLQJDQGLPSRUWDQW DSSOLFDWLRQVDVTXLFNO\DVSRVVLEOH

9HFWRUVLQ R2 $ PDWUL[ZLWKRQO\RQHFROXPQLVFDOOHGD FROXPQYHFWRU RUVLPSO\D YHFWRU ([DPSOHV RIYHFWRUVZLWKWZRHQWULHVDUH # $ # $ # $ 3 :2 w1 XD ; YD ; ZD w2 "1 :3 ZKHUH w1 DQG w2 DUHDQ\UHDOQXPEHUV 7KHVHWRIDOOYHFWRUVZLWKWZRHQWULHVLVGHQRWHG E\ R2 UHDGÚWZRµ 7KH R VWDQGVIRUWKHUHDOQXPEHUVWKDWDSSHDUDVHQWULHVLQWKH YHFWRUV DQGWKHH[SRQHQWLQGLFDWHVWKDWHDFKYHFWRUFRQWDLQVWZRHQWULHVñ 2 7ZRYHFWRUVLQ # $ # $R DUH HTXDO LIDQGRQO\LIWKHLUFRUUHVSRQGLQJHQWULHVDUHHTXDO 4 7 7KXV DQG DUH QRW HTXDO EHFDXVHYHFWRUVLQ R2 DUH RUGHUHGSDLUV RIUHDO 7 4 QXPEHUV *LYHQWZRYHFWRUV X DQG Y LQ R2 WKHLU VXP LVWKHYHFWRU X C Y REWDLQHGE\DGGLQJ FRUUHVSRQGLQJHQWULHVRI X DQG Y )RUH[DPSOH # $ # $ # $ # $ 1 2 1C2 3 C D D "2 5 "2 C 5 3 *LYHQDYHFWRU X DQGDUHDOQXPEHU c WKH VFDODUPXOWLSOH RI X E\ c LVWKHYHFWRU c X REWDLQHGE\PXOWLSO\LQJHDFKHQWU\LQ X E\ c )RULQVWDQFH # $ # $ # $ 3 3 15 LI X D DQG c D 5; WKHQ c X D 5 D "1 "1 "5 ñ 0RVWRIWKHWH[WFRQFHUQVYHFWRUVDQGPDWULFHVWKDWKDYHRQO\UHDOHQWULHV +RZHYHU DOOGHÀQLWLRQVDQG WKHRUHPVLQ&KDSWHUV ² DQGLQPRVWRIWKHUHVWRIWKHWH[W UHPDLQYDOLGLIWKHHQWULHVDUHFRPSOH[ QXPEHUV &RPSOH[YHFWRUVDQGPDWULFHVDULVHQDWXUDOO\ IRUH[DPSOH LQHOHFWULFDOHQJLQHHULQJDQGSK\VLFV

9HFWRU(TXDWLRQV

7KHQXPEHU c LQ c X LVFDOOHGD VFDODU LWLVZULWWHQLQOLJKWIDFHW\SHWRGLVWLQJXLVKLWIURP WKHEROGIDFHYHFWRU X 7KHRSHUDWLRQVRIVFDODUPXOWLSOLFDWLRQDQGYHFWRUDGGLWLRQFDQEHFRPELQHG DVLQ WKHIROORZLQJH[DPSOH

&9".1-& *LYHQ X D

#

40-65*0/ 4X D

$ # $ 1 2 DQG Y D ÀQG 4X ."3/Y DQG 4X C ."3/Y "2 "5 #

$ 4 ; "8

DQG

4X C ."3/Y D

#

."3/Y D

#

"6 15

$

$ # $ # $ 4 "6 "2 C D "8 15 7

6RPHWLPHV WKLVWH[WPD\ZULWHDFROXPQ # IRUFRQYHQLHQFHDQGDOVRWRVDYHVSDFH $ 3 YHFWRUVXFKDV LQWKHIRUP .3; "1/ ,QWKLVFDVH WKHSDUHQWKHVHVDQGWKHFRPPD "1 % & GLVWLQJXLVKWKHYHFWRU .3; "1/ IURPWKH 1 # 2 URZPDWUL[ 3 "1 ZULWWHQZLWKEUDFNHWV DQGQRFRPPD 7KXV # $ % & 3 ¤ 3 "1 "1 EHFDXVHWKHPDWULFHVKDYHGLIIHUHQWVKDSHV HYHQWKRXJKWKH\KDYHWKHVDPHHQWULHV

*HRPHWULF'HVFULSWLRQVRI R2 &RQVLGHUDUHFWDQJXODUFRRUGLQDWHV\VWHPLQWKHSODQH %HFDXVHHDFKSRLQWLQWKHSODQH LVGHWHUPLQHGE\DQRUGHUHGSDLURIQXPEHUV ZHFDQLGHQWLI\DJHRPHWULFSRLQW .a; b/ # $ a ZLWKWKHFROXPQYHFWRU 6RZHPD\UHJDUG R2 DVWKHVHWRIDOOSRLQWVLQWKHSODQH b 6HH)LJ x2

x2

(2, 2)

(2, 2)

x1

x1 (– 2, –1)

(3, –1)

'*(63& 9HFWRUVDVSRLQWV

(– 2, –1)

(3, –1)

'*(63& 9HFWRUVZLWKDUURZV

#

$ 3 7KHJHRPHWULFYLVXDOL]DWLRQRIDYHFWRUVXFKDV LVRIWHQDLGHGE\LQFOXGLQJ "1 DQDUURZGLUHFWHGOLQHVHJPHQW IURPWKHRULJLQ .0; 0/ WRWKHSRLQW .3; "1/ DVLQ)LJ ,QWKLVFDVH WKHLQGLYLGXDOSRLQWVDORQJWKHDUURZLWVHOIKDYHQRVSHFLDOVLJQLÀFDQFHò 7KHVXPRIWZRYHFWRUVKDVDXVHIXOJHRPHWULFUHSUHVHQWDWLRQ 7KHIROORZLQJUXOH FDQEHYHULÀHGE\DQDO\WLFJHRPHWU\ ò ,QSK\VLFV DUURZVFDQUHSUHVHQWIRUFHVDQGXVXDOO\DUHIUHHWRPRYHDERXWLQVSDFH 7KLVLQWHUSUHWDWLRQRI YHFWRUVZLOOEHGLVFXVVHGLQ6HFWLRQ

$)"15&3


1BSBMMFMPHSBN 3VMF GPS "EEJUJPO ,I X DQG Y LQ R2 DUHUHSUHVHQWHGDVSRLQWVLQWKHSODQH WKHQ X C Y FRUUHVSRQGVWR WKHIRXUWKYHUWH[RIWKHSDUDOOHORJUDPZKRVHRWKHUYHUWLFHVDUH X DQG Y 6HH )LJ x2 u+v u v x1

0 '*(63& 7KHSDUDOOHORJUDPUXOH

&9".1-& 7KHYHFWRUV X D LQ)LJ

# $ # $ # $ 2 "6 "4 Y D DQG X C Y D DUHGLVSOD\HG 2 1 3 x2

u+v

3 u

v –6

2

x1

'*(63&

7KHQH[WH[DPSOHLOOXVWUDWHVWKHIDFWWKDWWKHVHWRIDOOVFDODUPXOWLSOHVRIRQHÀ[HG QRQ]HURYHFWRULVDOLQHWKURXJKWKHRULJLQ .0; 0/ $ 3 'LVSOD\WKHYHFWRUV X 2X DQG " 23 X RQDJUDSK "1 # $ # $ 6 "2 40-65*0/ 6HH)LJ ZKHUH X 2X D DQG " 23 X D DUHGLVSOD\HG 7KH "2 2=3 DUURZIRU 2X LVWZLFHDVORQJDVWKHDUURZIRU X DQGWKHDUURZVSRLQWLQWKHVDPH GLUHFWLRQ 7KHDUURZIRU " 23 X LVWZRWKLUGVWKHOHQJWKRIWKHDUURZIRU X DQGWKHDUURZV SRLQWLQRSSRVLWHGLUHFWLRQV ,QJHQHUDO WKHOHQJWKRIWKHDUURZIRU c X LV jcj WLPHVWKH

&9".1-& /HW X D

#

x2

x2 – –2 u 3

0u

x1

x1 u

u 2u

Typical multiples of u '*(63&

The set of all multiples of u

9HFWRU(TXDWLRQV

OHQJWKRIWKHDUURZIRU X >5HFDOOWKDWWKHOHQJWKRIWKHOLQHVHJPHQWIURP .0; 0/ WR .a; b/ p LV a2 C b 2 :HVKDOOGLVFXVVWKLVIXUWKHULQ&KDSWHU @

x3

9HFWRUVLQ R3

2a

a

x2

x1 '*(63&

9HFWRUVLQ R3 DUH 3 # 1 FROXPQPDWULFHVZLWKWKUHHHQWULHV 7KH\DUHUHSUHVHQWHGJHR PHWULFDOO\E\SRLQWVLQDWKUHHGLPHQVLRQDOFRRUGLQDWHVSDFH 2 ZLWKDUURZVIURPWKHRUL 3 2 JLQVRPHWLPHVLQFOXGHGIRUYLVXDOFODULW\ 7KHYHFWRUV D D 4 3 5 DQG 2D DUHGLVSOD\HG 4 LQ)LJ

9HFWRUVLQ Rn

6FDODUPXOWLSOHV

,I n LVDSRVLWLYHLQWHJHU Rn UHDGÚQµ GHQRWHVWKHFROOHFWLRQRIDOOOLVWVRU RUGHUHG nWXSOHV RI n UHDOQXPEHUV XVXDOO\ZULWWHQDV n # 1 FROXPQPDWULFHV VXFKDV 2 3 u1 6 u2 7 6 7 XD6 : 7 4 :: 5 un

7KHYHFWRUZKRVHHQWULHVDUHDOO]HURLVFDOOHGWKH ]HURYHFWRU DQGLVGHQRWHGE\ 7KHQXPEHURIHQWULHVLQ ZLOOEHFOHDUIURPWKHFRQWH[W (TXDOLW\RIYHFWRUVLQ Rn DQGWKHRSHUDWLRQVRIVFDODUPXOWLSOLFDWLRQDQGYHFWRU DGGLWLRQLQ Rn DUHGHÀQHGHQWU\E\HQWU\MXVWDVLQ R2 7KHVHRSHUDWLRQVRQYHFWRUV KDYHWKHIROORZLQJSURSHUWLHV ZKLFKFDQEHYHULÀHGGLUHFWO\IURPWKHFRUUHVSRQGLQJ SURSHUWLHVIRUUHDOQXPEHUV 6HH3UDFWLFH3UREOHPDQG([HUFLVHVDQGDWWKHHQG RIWKLVVHFWLRQ "MHFCSBJD 1SPQFSUJFT PG Rn )RUDOO X; Y; Z LQ Rn DQGDOOVFDODUV c DQG d L LL LLL LY

x2 v

XCYDYCX .X C Y/ C Z D X C .Y C Z/ XCDCXDX X C ."X/ D "X C X D ZKHUH "X GHQRWHV ."1/X

Y YL YLL YLLL

c.X C Y/ D c X C c Y .c C d /X D c X C d X c.d X/ D .cd /.X/ 1X D X

x1 u –v u–v '*(63&

9HFWRUVXEWUDFWLRQ

)RUVLPSOLFLW\RIQRWDWLRQ DYHFWRUVXFKDV X C ."1/Y LVRIWHQZULWWHQDV X " Y )LJXUH VKRZV X " Y DVWKHVXPRI X DQG "Y

/LQHDU&RPELQDWLRQV *LYHQYHFWRUV Y1 ; Y2 ; : : : ; Yp LQ Rn DQGJLYHQVFDODUV c1 ; c2 ; : : : ; cp WKHYHFWRU \ GHÀQHG E\ \ D c1 Y1 C ! ! ! C cp Yp LVFDOOHGD OLQHDUFRPELQDWLRQ RI Y1 ; : : : ; Yp ZLWK ZHLJKWV c1 ; : : : ; cp 3URSHUW\LL DERYHSHUPLWVXVWRRPLWSDUHQWKHVHVZKHQIRUPLQJVXFKDOLQHDUFRPELQDWLRQ 7KH

$)"15&3


ZHLJKWVLQDOLQHDUFRPELQDWLRQFDQEHDQ\UHDOQXPEHUV LQFOXGLQJ]HUR )RUH[DPSOH VRPHOLQHDUFRPELQDWLRQVRIYHFWRUV Y1 DQG Y2 DUH p 3 Y1 C Y2 ; 12 Y1 .D 12 Y1 C 0Y2 /; DQG .D 0Y1 C 0Y2 /

&9".1-& )LJXUHLGHQWLÀHVVHOHFWHGOLQHDUFRPELQDWLRQVRI Y1 D

#

"1 1

$

DQG # $ 2 Y2 D 1RWHWKDWVHWVRISDUDOOHOJULGOLQHVDUHGUDZQWKURXJKLQWHJHUPXOWLSOHVRI 1 Y1 DQG Y2 (VWLPDWHWKHOLQHDUFRPELQDWLRQVRI Y1 DQG Y2 WKDWJHQHUDWHWKHYHFWRUV X DQG Z –3 v1 + v2 2

3v1 w

3v2 2v2

2v1

u

v1 v1 – v2 – v2

v2 0

– 2v1 + v2

– v1

– 2v2

– 2v1

'*(63& /LQHDUFRPELQDWLRQVRI Y1 DQG Y2

40-65*0/ 7KHSDUDOOHORJUDPUXOHVKRZVWKDW X LVWKHVXPRI 3Y1 DQG "2Y2 WKDWLV X D 3Y1 " 2Y2

3v1 w

2v1 v1 0 –v2

7KLVH[SUHVVLRQIRU X FDQEHLQWHUSUHWHGDVLQVWUXFWLRQVIRUWUDYHOLQJIURPWKHRULJLQ WR X DORQJWZRVWUDLJKWSDWKV )LUVW WUDYHOXQLWVLQWKH Y1 GLUHFWLRQWR 3Y1 DQGWKHQ WUDYHO "XQLWVLQWKH Y2 GLUHFWLRQSDUDOOHOWRWKHOLQHWKURXJK Y2 DQG 1H[W DOWKRXJK WKHYHFWRU Z LVQRWRQDJULGOLQH Z DSSHDUVWREHDERXWKDOIZD\EHWZHHQWZRSDLUVRI JULGOLQHV DWWKHYHUWH[RIDSDUDOOHORJUDPGHWHUPLQHGE\ .5=2/Y1 DQG ."1=2/Y2 6HH )LJ 7KXVDUHDVRQDEOHHVWLPDWHIRU Z LV Z D 52 Y1 " 12 Y2

'*(63&

7KHQH[WH[DPSOHFRQQHFWVDSUREOHPDERXWOLQHDUFRPELQDWLRQVWRWKHIXQGDPHQWDO H[LVWHQFHTXHVWLRQVWXGLHGLQ6HFWLRQVDQG 2 3 2 3 2 3 1 2 7 &9".1-& /HW D1 D 4 "2 5 D2 D 4 5 5 DQG E D 4 4 5 'HWHUPLQHZKHWKHU "5 6 "3 E FDQEHJHQHUDWHGRUZULWWHQ DVDOLQHDUFRPELQDWLRQRI D1 DQG D2 7KDWLV GHWHUPLQH ZKHWKHUZHLJKWV x1 DQG x2 H[LVWVXFKWKDW

x1 D1 C x2 D2 D E

,IYHFWRUHTXDWLRQ KDVDVROXWLRQ ÀQGLW

40-65*0/ 8VHWKHGHÀQLWLRQVRIVFDODUPXOWLSOLFDWLRQDQGYHFWRUDGGLWLRQWRUHZULWH WKHYHFWRUHTXDWLRQ 2 3 2 3 2 3 1 2 7 x1 4 "2 5 C x2 4 5 5 D 4 4 5 "5 6 "3 "

D1

"

D2

"

E

ZKLFKLVWKHVDPHDV

9HFWRU(TXDWLRQV

2

3 2 3 2 3 x1 2x2 7 4 "2x1 5 C 4 5x2 5 D 4 4 5 "5x1 6x2 "3 2

DQG

3 2 3 x1 C 2x2 7 4 "2x1 C 5x2 5 D 4 4 5 "5x1 C 6x2 "3

x1 C 2x2 D 7 "2x1 C 5x2 D 4 "5x1 C 6x2 D "3

7KHYHFWRUVRQWKHOHIWDQGULJKWVLGHVRI DUHHTXDOLIDQGRQO\LIWKHLUFRUUHVSRQGLQJ HQWULHVDUHERWKHTXDO 7KDWLV x1 DQG x2 PDNHWKHYHFWRUHTXDWLRQ WUXHLIDQGRQO\ LI x1 DQG x2 VDWLVI\WKHV\VWHP

7RVROYHWKLVV\VWHP URZUHGXFHWKHDXJPHQWHGPDWUL[RIWKHV\VWHPDVIROORZVó 2 3 2 3 2 3 2 3 1 2 7 1 2 7 1 2 7 1 0 3 4 "2 5 4 5 % 4 0 9 18 5 % 4 0 1 2 5 % 4 0 1 2 5 "5 6 "3 0 16 32 0 16 32 0 0 0

7KHVROXWLRQRI LV x1 D 3 DQG x2 D 2 +HQFH E LVDOLQHDUFRPELQDWLRQRI D1 DQG D2 ZLWKZHLJKWV x1 D 3 DQG x2 D 2 7KDWLV 2 3 2 3 2 3 1 2 7 34 "2 5 C 24 5 5 D 4 4 5 "5 6 "3

2EVHUYHLQ([DPSOHWKDWWKHRULJLQDOYHFWRUV D1 D2 DQG E DUHWKHFROXPQVRIWKH DXJPHQWHGPDWUL[WKDWZHURZUHGXFHG 2 3 1 2 7 4 "2 5 45 "5 6 "3 "

" "

D1

D2

E

)RUEUHYLW\ ZULWHWKLVPDWUL[LQDZD\WKDWLGHQWLÀHVLWVFROXPQV³QDPHO\

Œ D1 D2 E !

,WLVFOHDUKRZWRZULWHWKLVDXJPHQWHGPDWUL[LPPHGLDWHO\IURPYHFWRUHTXDWLRQ ZLWKRXWJRLQJWKURXJKWKHLQWHUPHGLDWHVWHSVRI([DPSOH 7DNHWKHYHFWRUVLQWKH RUGHULQZKLFKWKH\DSSHDULQ DQGSXWWKHPLQWRWKHFROXPQVRIDPDWUL[DVLQ 7KHGLVFXVVLRQDERYHLVHDVLO\PRGLÀHGWRHVWDEOLVKWKHIROORZLQJIXQGDPHQWDOIDFW $ YHFWRUHTXDWLRQ

x1 D1 C x2 D2 C ! ! ! C xn Dn D E KDVWKHVDPHVROXWLRQVHWDVWKHOLQHDUV\VWHPZKRVHDXJPHQWHGPDWUL[LV % & D1 D2 !!! Dn E

,QSDUWLFXODU E FDQEHJHQHUDWHGE\DOLQHDUFRPELQDWLRQRI D1 ; : : : ; Dn LIDQGRQO\ LIWKHUHH[LVWVDVROXWLRQWRWKHOLQHDUV\VWHPFRUUHVSRQGLQJWRWKHPDWUL[ ó 7KHV\PERO " EHWZHHQPDWULFHVGHQRWHVURZHTXLYDOHQFH6HFWLRQ

$)"15&3


2QHRIWKHNH\LGHDVLQOLQHDUDOJHEUDLVWRVWXG\WKHVHWRIDOOYHFWRUVWKDWFDQEH JHQHUDWHGRUZULWWHQDVDOLQHDUFRPELQDWLRQRIDÀ[HGVHW fY1 ; : : : ; Yp g RIYHFWRUV

%&'*/*5*0/

,I Y1 ; : : : ; Yp DUHLQ Rn WKHQWKHVHWRIDOOOLQHDUFRPELQDWLRQVRI Y1 ; : : : ; Yp LVGHQRWHGE\ 6SDQ fY1 ; : : : ; Yp g DQGLVFDOOHGWKH VXEVHWRI Rn VSDQQHG RU JHQHUDWHG E\ Y1 ; : : : ; Yp 7KDWLV 6SDQ fY1 ; : : : ; Yp g LVWKHFROOHFWLRQRIDOO YHFWRUVWKDWFDQEHZULWWHQLQWKHIRUP

c1 Y1 C c2 Y2 C ! ! ! C cp Yp ZLWK c1 ; : : : ; cp VFDODUV

$VNLQJZKHWKHUDYHFWRU E LVLQ 6SDQ fY1 ; : : : ; Yp g DPRXQWVWRDVNLQJZKHWKHUWKH YHFWRUHTXDWLRQ x1 Y1 C x2 Y2 C ! ! ! C xp Yp D E KDVDVROXWLRQ RU HTXLYDOHQWO\ DVNLQJZKHWKHUWKHOLQHDUV\VWHPZLWKDXJPHQWHGPDWUL[ Œ Y1 ! ! ! Yp E ! KDVDVROXWLRQ 1RWH WKDW 6SDQ fY1 ; : : : ; Yp g FRQWDLQV HYHU\ VFDODU PXOWLSOH RI Y1 IRU H[DP SOH VLQFH c Y1 D c Y1 C 0Y2 C ! ! ! C 0Yp ,QSDUWLFXODU WKH]HURYHFWRUPXVWEHLQ 6SDQ fY1 ; : : : ; Yp g

$ *HRPHWULF'HVFULSWLRQRI6SDQ fYg DQG6SDQ fX Yg

/HW Y EHDQRQ]HURYHFWRULQ R3 7KHQ 6SDQ fYg LVWKHVHWRIDOOVFDODUPXOWLSOHVRI Y ZKLFKLVWKHVHWRISRLQWVRQWKHOLQHLQ R3 WKURXJK Y DQG 6HH)LJ ,I X DQG Y DUHQRQ]HURYHFWRUVLQ R3 ZLWK Y QRWDPXOWLSOHRI X WKHQ 6SDQ fX; Yg LV WKHSODQHLQ R3 WKDWFRQWDLQV X Y DQG ,QSDUWLFXODU 6SDQ fX; Yg FRQWDLQVWKHOLQHLQ R3 WKURXJK X DQG DQGWKHOLQHWKURXJK Y DQG 6HH)LJ x3

x3

Span{v} 5u

v

3u u

x2 x1 '*(63& 6SDQ fYg DVDOLQH

WKURXJKWKHRULJLQ

2

x1

v

2v

3v

x2

'*(63& 6SDQ fX; Yg DVD SODQHWKURXJKWKHRULJLQ

3 2 3 2 3 1 5 "3 &9".1-& /HW D1 D 4 "2 5 D2 D 4 "13 5 DQG E D 4 8 5 3 "3 1 3 6SDQ fD1 ; D2 g LVDSODQHWKURXJKWKHRULJLQLQ R ,V E LQWKDWSODQH"

7KHQ

9HFWRU(TXDWLRQV

40-65*0/ 'RHVWKHHTXDWLRQ x1 D1 C x2 D2 D E KDYHDVROXWLRQ" 7RDQVZHUWKLV URZ UHGXFHWKHDXJPHQWHGPDWUL[ Œ D1 D2 E ! 2 3 2 3 2 3 1 5 "3 1 5 "3 1 5 "3 4 "2 "13 8 5 % 4 0 "3 2 5 % 4 0 "3 2 5 3 "3 1 0 "18 10 0 0 "2

7KHWKLUGHTXDWLRQLV 0 D "2 ZKLFKVKRZVWKDWWKHV\VWHPKDVQRVROXWLRQ 7KHYHFWRU HTXDWLRQ x1 D1 C x2 D2 D E KDVQRVROXWLRQ DQGVR E LV QRW LQ 6SDQ fD1 ; D2 g

/LQHDU&RPELQDWLRQVLQ$SSOLFDWLRQV 7KHÀQDOH[DPSOHVKRZVKRZVFDODUPXOWLSOHVDQGOLQHDUFRPELQDWLRQVFDQDULVHZKHQ DTXDQWLW\VXFKDV´FRVWµLVEURNHQGRZQLQWRVHYHUDOFDWHJRULHV 7KHEDVLFSULQFLSOHIRU WKHH[DPSOHFRQFHUQVWKHFRVWRISURGXFLQJVHYHUDOXQLWVRIDQLWHPZKHQWKHFRVWSHU XQLWLVNQRZQ ' ( ' ( ' ( QXPEHU FRVW WRWDO ! D RIXQLWV SHUXQLW FRVW

&9".1-& $ FRPSDQ\PDQXIDFWXUHVWZRSURGXFWV )RUZRUWKRISURGXFW %WKHFRPSDQ\VSHQGVRQPDWHULDOV RQODERU DQGRQRYHUKHDG )RU ZRUWKRISURGXFW&WKHFRPSDQ\VSHQGVRQPDWHULDOV RQODERU DQG RQRYHUKHDG /HW 2 3 2 3 :45 :40 E D 4 :25 5 DQG F D 4 :30 5 :15 :15

7KHQ E DQG F UHSUHVHQWWKH´FRVWVSHUGROODURILQFRPHµIRUWKHWZRSURGXFWV

D :KDWHFRQRPLFLQWHUSUHWDWLRQFDQEHJLYHQWRWKHYHFWRU 100E" E 6XSSRVHWKHFRPSDQ\ZLVKHVWRPDQXIDFWXUH x1 GROODUVZRUWKRISURGXFW% DQG x2 GROODUVZRUWKRISURGXFW&*LYHDYHFWRUWKDWGHVFULEHVWKHYDULRXVFRVWVWKH FRPSDQ\ZLOOKDYHIRUPDWHULDOV ODERU DQGRYHUKHDG

40-65*0/ D &RPSXWH

2

3 2 3 :45 45 100E D 1004 :25 5 D 4 25 5 :15 15

7KH YHFWRU 100E OLVWV WKH YDULRXV FRVWV IRU SURGXFLQJ ZRUWK RI SURGXFW %³QDPHO\ IRUPDWHULDOV IRUODERU DQGIRURYHUKHDG E 7KHFRVWVRIPDQXIDFWXULQJ x1 GROODUVZRUWKRI% DUHJLYHQE\WKHYHFWRU x1 E DQG WKHFRVWVRIPDQXIDFWXULQJ x2 GROODUVZRUWKRI& DUHJLYHQE\ x2 F +HQFHWKHWRWDO FRVWVIRUERWKSURGXFWVDUHJLYHQE\WKHYHFWRU x1 E C x2 F

13"$5*$& 130#-&.4 3URYHWKDW X C Y D Y C X IRUDQ\ X DQG Y LQ Rn )RUZKDWYDOXHV RI h ZLOO \ EHLQ 6SDQfY1 ; Y2 ; Y3 g LI 2 3 2 3 2 3 1 5 "3 Y1 D 4 "1 5; Y2 D 4 "4 5; Y3 D 4 1 5; "2 "7 0

DQG

2

3 "4 \ D 4 35 h

$)"15&3


&9&3$*4&4 ,Q([HUFLVHVDQG FRPSXWH X C Y DQG X " 2Y # $ # $ # $ # $ "1 "3 3 2 X D YD X D YD 2 "1 2 "1 ,Q([HUFLVHVDQG GLVSOD\WKHIROORZLQJYHFWRUVXVLQJDUURZV RQDQ [\JUDSK X Y "Y "2Y X C Y X " Y DQG X " 2Y 1RWLFH WKDW X " Y LVWKHYHUWH[RIDSDUDOOHORJUDPZKRVHRWKHUYHUWLFHVDUH X DQG "Y X DQG Y DVLQ([HUFLVH

X DQG Y DVLQ([HUFLVH

,Q([HUFLVHVDQG ZULWHDV\VWHPRIHTXDWLRQVWKDWLVHTXLYDOHQW WRWKHJLYHQYHFWRUHTXDWLRQ 2 3 2 3 2 3 3 5 2 x1 4 "2 5 C x2 4 0 5 D 4 "3 5 8 "9 8 x1

#

$ # $ # $ # $ 3 7 "2 0 C x2 C x3 D "2 3 1 0

8VHWKHDFFRPSDQ\LQJÀJXUHWRZULWHHDFKYHFWRUOLVWHGLQ([HU FLVHVDQGDVDOLQHDUFRPELQDWLRQRI X DQG Y ,VHYHU\YHFWRU LQ R2 DOLQHDUFRPELQDWLRQRI X DQG Y"

d

b

c

u

2v

v

a

0

–v – 2v

w y

–u

x

z

9HFWRUV D E F DQG G 9HFWRUV Z [ \ DQG ] ,Q([HUFLVHVDQG ZULWHDYHFWRUHTXDWLRQWKDWLVHTXLYDOHQWWR WKHJLYHQV\VWHPRIHTXDWLRQV

x2 C 5x3 D 0

4x1 C 6x2 " x3 D 0

"x1 C 3x2 " 8x3 D 0

3x1 " 2x2 C 4x3 D 3

"2x1 " 7x2 C 5x3 D 1 5x1 C 4x2 " 3x3 D 2

,Q([HUFLVHVDQG GHWHUPLQHLI E LVDOLQHDUFRPELQDWLRQRI D1 D2 DQG D3 2 3 2 3 2 3 2 3 1 0 5 2 D1 D 4 "2 5 D2 D 4 1 5 D3 D 4 "6 5 E D 4 "1 5 0 2 8 6 2 3 2 3 2 3 2 3 1 "2 "6 11 D1 D 4 0 5 D2 D 4 3 5 D3 D 4 7 5 E D 4 "5 5 1 "2 5 9

,Q([HUFLVHVDQG GHWHUPLQHLI E LVDOLQHDUFRPELQDWLRQRI WKHYHFWRUVIRUPHGIURPWKHFROXPQVRIWKHPDWUL[ A 2 3 2 3 1 "4 2 3 3 5 5 E D 4 "7 5 A D 4 0 "2 8 "4 "3 2 3 2 3 1 0 5 2 1 "6 5 E D 4 "1 5 A D 4 "2 0 2 8 6 2 3 2 3 2 3 1 "5 3 /HW D1 D 4 3 5 D2 D 4 "8 5 DQG E D 4 "5 5 )RUZKDW "1 2 h YDOXHV RI h LV E LQWKHSODQHVSDQQHGE\ D1 DQG D2 " 2 3 2 3 2 3 1 "2 h /HW Y1 D 4 0 5 Y2 D 4 1 5 DQG \ D 4 "3 5 )RUZKDW 7 "2 "5 YDOXHV RI h LV \ LQWKHSODQHJHQHUDWHGE\ Y1 DQG Y2 " ,Q([HUFLVHVDQG OLVWÀYHYHFWRUVLQ 6SDQ fY1 ; Y2 g )RUHDFK YHFWRU VKRZWKHZHLJKWVRQ Y1 DQG Y2 XVHGWRJHQHUDWHWKHYHFWRU DQGOLVWWKHWKUHHHQWULHVRIWKHYHFWRU 'RQRWPDNHDVNHWFK 2 3 2 3 3 "4 Y1 D 4 1 5 Y2 D 4 0 5 2 1 2 3 2 3 1 "2 Y1 D 4 1 5 Y2 D 4 3 5 "2 0 *LYHDJHRPHWULFGHVFULSWLRQRI 2 3 2 3 6SDQ fY1 ; Y2 g IRUWKHYHFWRUV 8 12 Y1 D 4 2 5 DQG Y2 D 4 3 5 "6 "9

*LYHDJHRPHWULFGHVFULSWLRQRI 6SDQ fY1 ; Y2 g IRUWKHYHFWRUV LQ([HUFLVH # $ # $ # $ 2 2 h DQG Y D 6KRZ WKDW LV LQ /HW X D "1 1 k 6SDQ fX; Yg IRUDOO h DQG k &RQVWUXFWD 3 # 3 PDWUL[ A ZLWKQRQ]HURHQWULHV DQGDYHFWRU E LQ R3 VXFKWKDW E LV QRW LQWKHVHWVSDQQHGE\WKHFROXPQV RI A ,Q([HUFLVHVDQG PDUNHDFKVWDWHPHQW7UXHRU)DOVH -XVWLI\ HDFKDQVZHU # $ "4 LV Œ "4 3 ! D $QRWKHUQRWDWLRQIRUWKHYHFWRU 3 # $ "2 DQG E 7KH SRLQWV LQ WKH SODQH FRUUHVSRQGLQJ WR 5 # $ "5 OLHRQDOLQHWKURXJKWKHRULJLQ 2 F $QH[DPSOHRIDOLQHDUFRPELQDWLRQRIYHFWRUV Y1 DQG Y2 LVWKHYHFWRU 12 Y1

9HFWRU(TXDWLRQV

G 7KHVROXWLRQVHWRIWKHOLQHDUV\VWHPZKRVHDXJPHQWHG PDWUL[LV Œ D1 D2 D3 E ! LVWKHVDPHDVWKHVROXWLRQ VHWRIWKHHTXDWLRQ x1 D1 C x2 D2 C x3 D3 D E

HDFKWRQRI% EXUQHG WKHSODQWSURGXFHVPLOOLRQ%WX JRIVXOIXUGLR[LGH DQG JRISDUWLFXODWHPDWWHU D +RZPXFKKHDWGRHVWKHVWHDPSODQWSURGXFHZKHQLW EXUQV x1 WRQVRI$ DQG x2 WRQVRI%"

D :KHQ X DQG Y DUHQRQ]HURYHFWRUV 6SDQ fX; Yg FRQWDLQV RQO\WKHOLQHWKURXJK X DQGWKHRULJLQ DQGWKHOLQHWKURXJK Y DQGWKHRULJLQ

E 6XSSRVHWKHRXWSXWRIWKHVWHDPSODQWLVGHVFULEHGE\ DYHFWRUWKDWOLVWVWKHDPRXQWVRIKHDW VXOIXUGLR[LGH DQGSDUWLFXODWHPDWWHU ([SUHVVWKLVRXWSXWDVDOLQHDU FRPELQDWLRQRIWZRYHFWRUV DVVXPLQJWKDWWKHSODQWEXUQV x1 WRQVRI$ DQG x2 WRQVRI%

H 7KHVHW 6SDQ fX; Yg LVDOZD\VYLVXDOL]HGDVDSODQHWKURXJK WKHRULJLQ

E $Q\OLVWRIÀYHUHDOQXPEHUVLVDYHFWRULQ R5 F $VNLQJ ZKHWKHU WKH OLQHDU V\VWHP FRUUHVSRQGLQJ WR DQDXJPHQWHGPDWUL[ Œ D1 D2 D3 E ! KDVDVROXWLRQ DPRXQWVWRDVNLQJZKHWKHU E LVLQ 6SDQ fD1 ; D2 ; D3 g G 7KHYHFWRU Y UHVXOWVZKHQDYHFWRU X " Y LVDGGHGWRWKH YHFWRU Y

H 7KH ZHLJKWV c1 ; : : : ; cp LQ D OLQHDU FRPELQDWLRQ c1 Y1 C ! ! ! C cp Yp FDQQRWDOOEH]HUR 2 3 2 3 1 0 "4 4 3 "2 5 DQG E D 4 1 5 'HQRWHWKH /HW A D 4 0 "2 6 3 "4 FROXPQVRI A E\ D1 D2 D3 DQGOHW W D 6SDQ fD1 ; D2 ; D3 g D ,V E LQ fD1 ; D2 ; D3 g" +RZPDQ\YHFWRUVDUHLQ fD1 ; D2 ; D3 g"

F >0@ 2YHUDFHUWDLQWLPHSHULRG WKHVWHDPSODQWSURGXFHG PLOOLRQ%WXRIKHDW JRIVXOIXUGLR[LGH DQG JRISDUWLFXODWHPDWWHU 'HWHUPLQHKRZPDQ\WRQV RIHDFKW\SHRIFRDOWKHVWHDPSODQWPXVWKDYHEXUQHG ,QFOXGHDYHFWRUHTXDWLRQDVSDUWRI\RXUVROXWLRQ /HW Y1 ; : : : ; Yk EH SRLQWV LQ R3 DQG VXSSRVH WKDW IRU j D 1; : : : ; k DQREMHFWZLWKPDVV mj LVORFDWHGDWSRLQW Yj 3K\VLFLVWVFDOOVXFKREMHFWV SRLQWPDVVHV 7KHWRWDOPDVVRI WKHV\VWHPRISRLQWPDVVHVLV

m D m1 C ! ! ! C mk 7KH FHQWHURIJUDYLW\ RU FHQWHURIPDVV RIWKHV\VWHPLV

E ,V E LQ W " +RZPDQ\YHFWRUVDUHLQ W "

F 6KRZWKDW D1 LVLQ W >+LQW 5RZRSHUDWLRQVDUHXQQHF HVVDU\@ 2 3 2 3 2 0 6 10 8 5 5 OHW E D 4 3 5 DQGOHW W EH /HW A D 4 "1 1 "2 1 7 WKHVHWRIDOOOLQHDUFRPELQDWLRQVRIWKHFROXPQVRI A D ,V E LQ W "

YD

&RPSXWHWKHFHQWHURIJUDYLW\RIWKHV\VWHPFRQVLVWLQJRIWKH IROORZLQJSRLQWPDVVHVVHHWKHÀJXUH

3RLQW

E 6KRZWKDWWKHVHFRQGFROXPQRI A LVLQ W

Y1 Y2 Y3 Y4

$ PLQLQJFRPSDQ\KDVWZRPLQHV 2QHGD\·VRSHUDWLRQ DW PLQH SURGXFHV RUH WKDW FRQWDLQV PHWULF WRQV RI FRSSHUDQGNLORJUDPVRIVLOYHU ZKLOHRQHGD\·VRSHUDWLRQ DW PLQH SURGXFHV RUH WKDW FRQWDLQV PHWULF # WRQV $ RI 30 DQG FRSSHUDQGNLORJUDPVRIVLOYHU /HW Y1 D 600 # $ 40 7KHQ Y1 DQG Y2 UHSUHVHQWWKH´RXWSXWSHUGD\µ Y2 D 380 RIPLQHDQGPLQH UHVSHFWLYHO\ D :KDWSK\VLFDOLQWHUSUHWDWLRQFDQEHJLYHQWRWKHYHFWRU 5Y1 " E 6XSSRVHWKHFRPSDQ\RSHUDWHVPLQHIRU x1 GD\VDQG PLQHIRU x2 GD\V :ULWHDYHFWRUHTXDWLRQZKRVH VROXWLRQ JLYHV WKH QXPEHU RI GD\V HDFK PLQH VKRXOG RSHUDWHLQRUGHUWRSURGXFHWRQVRIFRSSHUDQG NLORJUDPVRIVLOYHU 'RQRWVROYHWKHHTXDWLRQ

1 Œm1 Y1 C ! ! ! C mk Yk ! m

0DVV

D .2; "2; 4/ D ."4; 2; 3/ D .4; 0; "2/ D .1; "6; 0/

J J J J

x3 v2

v1

0 x1 v3

v4

x2

F >0@ 6ROYHWKHHTXDWLRQLQE $ VWHDPSODQWEXUQVWZRW\SHVRIFRDO DQWKUDFLWH$ DQG ELWXPLQRXV% )RUHDFKWRQRI$ EXUQHG WKHSODQWSURGXFHV PLOOLRQ%WXRIKHDW JUDPV J RIVXOIXUGLR[LGH DQG JRISDUWLFXODWHPDWWHUVROLGSDUWLFOHSROOXWDQWV )RU

/HW Y EH WKH FHQWHU RI PDVV RI D V\VWHP RI SRLQW PDVVHV ORFDWHG DW Y1 ; : : : ; Yk DV LQ ([HUFLVH ,V Y LQ 6SDQ fY1 ; : : : ; Yk g" ([SODLQ


$)"15&3

$ WKLQWULDQJXODUSODWHRIXQLIRUPGHQVLW\DQGWKLFNQHVVKDV YHUWLFHVDW Y1 D .0; 1/ Y2 D .8; 1/ DQG Y3 D .2; 4/ DVLQWKH ÀJXUHEHORZ DQGWKHPDVVRIWKHSODWHLVJ x2 4

VROXWLRQ" ,VWKHVROXWLRQXQLTXH" 8VHWKHÀJXUHWRH[SODLQ \RXUDQVZHUV x2

v3

v3 Metal Plate

v1 b

v1

v2 8

x1

D )LQGWKH .x; y/FRRUGLQDWHVRIWKHFHQWHURIPDVVRIWKH SODWH 7KLVÉDODQFHSRLQWµRIWKHSODWHFRLQFLGHVZLWK WKHFHQWHURIPDVVRIDV\VWHPFRQVLVWLQJRIWKUHHJUDP SRLQWPDVVHVORFDWHGDWWKHYHUWLFHVRIWKHSODWH E 'HWHUPLQHKRZWRGLVWULEXWHDQDGGLWLRQDOPDVVRIJ DWWKHWKUHHYHUWLFHVRIWKHSODWHWRPRYHWKHEDODQFH SRLQWRIWKHSODWHWR .2; 2/ >+LQW /HW w1 w2 DQG w3 GHQRWHWKHPDVVHVDGGHGDWWKHWKUHHYHUWLFHV VRWKDW w1 C w2 C w3 D 6@

&RQVLGHUWKHYHFWRUV Y1 Y2 Y3 DQG E LQ R2 VKRZQLQWKH ÀJXUH 'RHVWKHHTXDWLRQ x1 Y1 C x2 Y2 C x3 Y3 D E KDYHD

0

x1 v2

8VH WKH YHFWRUV X D .u1 ; : : : ; un / Y D .v1 ; : : : ; vn / DQG Z D .w1 ; : : : ; wn / WRYHULI\WKHIROORZLQJDOJHEUDLFSURSHU WLHVRI Rn D .X C Y/ C Z D X C .Y C Z/ E c.X C Y/ D c X C c Y IRUHDFKVFDODU c

8VHWKHYHFWRU X D .u1 ; : : : ; un / WRYHULI\WKHIROORZLQJDOJH EUDLFSURSHUWLHVRI Rn D X C ."X/ D ."X/ C X D E c.d X/ D .cd /X IRUDOOVFDODUV c DQG d

40-65*0/4 50 13"$5*$& 130#-&.4 7DNHDUELWUDU\YHFWRUV X D .u1 ; : : : ; un / DQG Y D .v1 ; : : : ; vn / LQ Rn DQGFRPSXWH X C Y D .u1 C v1 ; : : : ; un C vn / D .v1 C u1 ; : : : ; vn C un / DYCX

h= 9 v1 Span {v1, v2, v3}

v2 h= 5 v 3 h= 1

2

3 "4 7KHSRLQWV 4 3 5 OLHRQDOLQH h WKDWLQWHUVHFWVWKHSODQHZKHQ h D 5

'HÀQLWLRQRIYHFWRUDGGLWLRQ

&RPPXWDWLYLW\RIDGGLWLRQLQ R 'HÀQLWLRQRIYHFWRUDGGLWLRQ

7KHYHFWRU \ EHORQJVWR 6SDQ fY1 ; Y2 ; Y3 g LIDQGRQO\LIWKHUHH[LVWVFDODUV x1 ; x2 ; x3 VXFKWKDW 2 3 2 3 2 3 2 3 1 5 "3 "4 x1 4 "1 5 C x2 4 "4 5 C x3 4 1 5 D 4 3 5 "2 "7 0 h 7KLVYHFWRUHTXDWLRQLVHTXLYDOHQWWRDV\VWHPRIWKUHHOLQHDUHTXDWLRQVLQWKUHH XQNQRZQV ,I\RXURZUHGXFHWKHDXJPHQWHGPDWUL[IRUWKLVV\VWHP \RXÀQGWKDW 2 3 2 3 2 3 1 5 "3 "4 1 5 "3 "4 1 5 "3 "4 4 "1 "4 1 3 5 % 4 0 1 "2 "1 5 % 4 0 1 "2 "1 5 "2 "7 0 h 0 3 "6 h " 8 0 0 0 h"5

7KHV\VWHPLVFRQVLVWHQWLIDQGRQO\LIWKHUHLVQRSLYRWLQWKHIRXUWKFROXPQ 7KDW LV h " 5 PXVWEH 6R \ LVLQ 6SDQ fY1 ; Y2 ; Y3 g LIDQGRQO\LI h D 5

5HPHPEHU 7KHSUHVHQFHRIDIUHHYDULDEOHLQDV\VWHPGRHVQRWJXDUDQWHHWKDWWKH V\VWHPLVFRQVLVWHQW

5)& ."53*9 &26"5*0/ "Y C $ IXQGDPHQWDOLGHDLQOLQHDUDOJHEUDLVWRYLHZDOLQHDUFRPELQDWLRQRIYHFWRUVDVWKH SURGXFWRIDPDWUL[DQGDYHFWRU 7KHIROORZLQJGHÀQLWLRQSHUPLWVXVWRUHSKUDVHVRPH RIWKHFRQFHSWVRI6HFWLRQ LQQHZZD\V

%&'*/*5*0/

7KH0DWUL[(TXDWLRQ A[ D E

,I A LVDQ m # n PDWUL[ ZLWKFROXPQV D1 ; : : : ; Dn DQGLI [ LVLQ Rn WKHQWKH SURGXFWRI A DQG[ GHQRWHGE\ A[ LV WKHOLQHDUFRPELQDWLRQRIWKHFROXPQV RI A XVLQJWKHFRUUHVSRQGLQJHQWULHVLQ[DVZHLJKWV WKDWLV 2 3 x1 % &6 : 7 A[ D D1 D2 ! ! ! Dn 4 :: 5 D x1 D1 C x2 D2 C ! ! ! C xn Dn xn 1RWHWKDW A[ LVGHÀQHGRQO\LIWKHQXPEHURIFROXPQVRI A HTXDOVWKHQXPEHURIHQWULHV LQ [

&9".1-&

2 3 $ 4 # $ # $ # $ 1 2 "1 4 5 1 2 "1 D 3 D4 C3 C7 0 "5 3 0 "5 3 7 # $ # $ # $ # $ 4 6 "7 3 D C C D 0 "15 21 6 2 3 2 3 2 3 2 3 2 3 2 3 2 "3 # $ 2 "3 8 "21 "13 4 D 44 8 5 C 74 0 5 D 4 32 5 C 4 0 5 D 4 32 5 E 4 8 0 5 7 "5 2 "5 2 "20 14 "6 #

&9".1-& )RU Y1 ; Y2 ; Y3 LQ Rm ZULWHWKHOLQHDUFRPELQDWLRQ 3Y1 " 5Y2 C 7Y3 DV DPDWUL[WLPHVDYHFWRU

40-65*0/ 3ODFH Y1 ; Y2 ; Y3 LQWRWKHFROXPQVRIDPDWUL[ A DQGSODFHWKHZHLJKWV "5 DQGLQWRDYHFWRU [ 7KDWLV 3 2 3 % & 3Y1 " 5Y2 C 7Y3 D Y1 Y2 Y3 4 "5 5 D A[ 7

6HFWLRQ VKRZHGKRZWRZULWHDV\VWHPRIOLQHDUHTXDWLRQVDVDYHFWRUHTXDWLRQ LQYROYLQJDOLQHDUFRPELQDWLRQRIYHFWRUV )RUH[DPSOH WKHV\VWHP

x1 C 2x2 " x3 D 4 "5x2 C 3x3 D 1

LVHTXLYDOHQWWR

x1

# $ # $ # $ # $ 1 2 "1 4 C x2 C x3 D 0 "5 3 1

$VLQ([DPSOH WKHOLQHDUFRPELQDWLRQRQWKHOHIWVLGHLVDPDWUL[WLPHVDYHFWRU VR WKDW EHFRPHV 2 3 # $ x1 # $ 1 2 "1 4 5 4 x2 D 0 "5 3 1 x3 (TXDWLRQ KDVWKHIRUP A[ D E 6XFKDQHTXDWLRQLVFDOOHGD PDWUL[HTXDWLRQ WRGLVWLQJXLVKLWIURPDYHFWRUHTXDWLRQVXFKDVLVVKRZQLQ 1RWLFHKRZWKHPDWUL[LQ LVMXVWWKHPDWUL[RIFRHIÀFLHQWVRIWKHV\VWHP 6LPLODUFDOFXODWLRQVVKRZWKDWDQ\V\VWHPRIOLQHDUHTXDWLRQV RUDQ\YHFWRUHTXDWLRQ VXFKDV FDQEHZULWWHQDVDQHTXLYDOHQWPDWUL[HTXDWLRQLQWKHIRUP A[ D E 7KLV VLPSOHREVHUYDWLRQZLOOEHXVHGUHSHDWHGO\WKURXJKRXWWKHWH[W +HUHLVWKHIRUPDOUHVXOW

$)"15&3


5)&03&.

,I A LVDQ m # n PDWUL[ ZLWKFROXPQV D1 ; : : : ; Dn DQGLI E LVLQ Rm WKHPDWUL[ HTXDWLRQ A[ D E KDVWKHVDPHVROXWLRQVHWDVWKHYHFWRUHTXDWLRQ

x1 D1 C x2 D2 C ! ! ! C xn Dn D E

ZKLFK LQWXUQ KDVWKHVDPHVROXWLRQVHWDVWKHV\VWHPRIOLQHDUHTXDWLRQVZKRVH DXJPHQWHGPDWUL[LV % & D1 D2 ! ! ! Dn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³E\URZUHGXFLQJWKHDXJPHQWHGPDWUL[ 2WKHUPHWKRGVRIVROXWLRQZLOO EHGLVFXVVHGODWHU

([LVWHQFHRI6ROXWLRQV 7KHGHÀQLWLRQRI A[ OHDGVGLUHFWO\WRWKHIROORZLQJXVHIXOIDFW 7KHHTXDWLRQ A[ D E KDVDVROXWLRQLIDQGRQO\LI E LVDOLQHDUFRPELQDWLRQRIWKH FROXPQVRI A

6HFWLRQ FRQVLGHUHGWKHH[LVWHQFHTXHVWLRQ ´,V E LQ 6SDQ fD1 ; : : : ; Dn g"µ (TXLY DOHQWO\ ´,V A[ D E FRQVLVWHQW"µ $ KDUGHUH[LVWHQFHSUREOHPLVWRGHWHUPLQHZKHWKHU WKHHTXDWLRQ A[ D E LVFRQVLVWHQW IRUDOO SRVVLEOH E 2

3 2 3 1 3 4 b1 &9".1-& /HW A D 4 "4 2 "6 5 DQG E D 4 b2 5 ,VWKHHTXDWLRQ A[ D E b3 "3 "2 "7 FRQVLVWHQWIRUDOOSRVVLEOH b1 ; b2 ; b3 "

40-65*0/ 5RZUHGXFHWKHDXJPHQWHGPDWUL[IRU A[ D E 2 3 2 3 1 3 4 b1 1 3 4 b1 4 "4 2 "6 b2 5 % 4 0 14 10 b2 C 4b1 5 "3 "2 "7 b3 0 7 5 b3 C 3b1 2 3 1 3 4 b1 5 b2 C 4b1 % 4 0 14 10 0 0 0 b3 C 3b1 " 12 .b2 C 4b1 /

7KH WKLUG HQWU\ LQ FROXPQ HTXDOV b1 " 12 b2 C b3 7KH HTXDWLRQ A[ D E LV QRW FRQVLVWHQWIRUHYHU\ E EHFDXVHVRPHFKRLFHVRI E FDQPDNH b1 " 12 b2 C b3 QRQ]HUR

x3


7KHUHGXFHGPDWUL[LQ([DPSOHSURYLGHVDGHVFULSWLRQRIDOO E IRUZKLFKWKH HTXDWLRQ A[ D E LV FRQVLVWHQW 7KHHQWULHVLQ E PXVWVDWLVI\

b1 " 12 b2 C b3 D 0 , a 3} {a , a 2 Span 1 x1

0

x2

'*(63&

7KHFROXPQVRI A D Œ D1 D2 D3 ! VSDQDSODQH WKURXJK

5)&03&.

7KLVLVWKHHTXDWLRQRIDSODQHWKURXJKWKHRULJLQLQ R3 7KHSODQHLVWKHVHWRIDOOOLQHDU FRPELQDWLRQVRIWKHWKUHHFROXPQVRI A 6HH)LJ 7KHHTXDWLRQ A[ D E LQ([DPSOHIDLOVWREHFRQVLVWHQWIRUDOO E EHFDXVHWKH HFKHORQIRUPRI A KDVDURZRI]HURV ,I A KDGDSLYRWLQDOOWKUHHURZV ZHZRXOG QRWFDUHDERXWWKHFDOFXODWLRQVLQWKHDXJPHQWHGFROXPQEHFDXVHLQWKLVFDVHDQHFKHORQ IRUPRIWKHDXJPHQWHGPDWUL[FRXOGQRWKDYHDURZVXFKDV Œ 0 0 0 1 ! ,QWKHQH[WWKHRUHP WKHVHQWHQFH´7KHFROXPQVRI A VSDQ Rm µPHDQVWKDW HYHU\ E LQ m R LVDOLQHDUFRPELQDWLRQRIWKHFROXPQVRI A ,QJHQHUDO DVHWRIYHFWRUV fY1 ; : : : ; Yp g LQ Rm VSDQV RU JHQHUDWHV Rm LIHYHU\YHFWRULQ Rm LVDOLQHDUFRPELQDWLRQRI Y1 ; : : : ; Yp ³WKDWLV LI 6SDQ fY1 ; : : : ; Yp g D Rm

/HW A EHDQ m # n PDWUL[ 7KHQWKHIROORZLQJVWDWHPHQWVDUHORJLFDOO\HTXLYDOHQW 7KDWLV IRUDSDUWLFXODU A HLWKHUWKH\DUHDOOWUXHVWDWHPHQWVRUWKH\DUHDOOIDOVH D E F G

)RUHDFK E LQ Rm WKHHTXDWLRQ A[ D E KDVDVROXWLRQ (DFK E LQ Rm LVDOLQHDUFRPELQDWLRQRIWKHFROXPQVRI A 7KHFROXPQVRI A VSDQ Rm A KDVDSLYRWSRVLWLRQLQHYHU\URZ

7KHRUHPLVRQHRIWKHPRVWXVHIXOWKHRUHPVLQWKLVFKDSWHU 6WDWHPHQWVD E DQGF DUHHTXLYDOHQWEHFDXVHRIWKHGHÀQLWLRQRI A[ DQGZKDWLWPHDQVIRUD VHWRIYHFWRUVWRVSDQ Rm 7KHGLVFXVVLRQDIWHU([DPSOHVXJJHVWVZK\D DQGG DUHHTXLYDOHQW DSURRILVJLYHQDWWKHHQGRIWKHVHFWLRQ 7KHH[HUFLVHVZLOOSURYLGH H[DPSOHVRIKRZ7KHRUHPLVXVHG :DUQLQJ 7KHRUHPLVDERXWD FRHIÀFLHQWPDWUL[ QRWDQDXJPHQWHGPDWUL[ ,IDQ DXJPHQWHGPDWUL[ Œ A E ! KDVDSLYRWSRVLWLRQLQHYHU\URZ WKHQWKHHTXDWLRQ A[ D E PD\RUPD\QRWEHFRQVLVWHQW

&RPSXWDWLRQRI A[ 7KHFDOFXODWLRQVLQ([DPSOHZHUHEDVHGRQWKHGHÀQLWLRQRIWKHSURGXFWRIDPDWUL[ A DQGDYHFWRU [ 7KHIROORZLQJVLPSOHH[DPSOHZLOOOHDGWRDPRUHHIÀFLHQWPHWKRGIRU FDOFXODWLQJWKHHQWULHVLQ A[ ZKHQZRUNLQJSUREOHPVE\KDQG 2

3 2 3 2 3 4 x1 &9".1-& &RPSXWH A[ ZKHUH A D 4 "1 5 "3 5 DQG [ D 4 x2 5 x3 6 "2 8

$)"15&3


40-65*0/ )URPWKHGHÀQLWLRQ 2 32 3 2 3 2 3 2 3 2 3 4 x1 2 3 4 4 "1 5 "3 54 x2 5 D x1 4 "1 5 C x2 4 5 5 C x3 4 "3 5 x3 6 "2 8 6 "2 8 2 3 2 3 2 3 2x1 3x2 4x3 D 4 "x1 5 C 4 5x2 5 C 4 "3x3 5 6x1 "2x2 8x3 2 3 2x1 C 3x2 C 4x3 D 4 "x1 C 5x2 " 3x3 5 6x1 " 2x2 C 8x3

7KHÀUVWHQWU\LQWKHSURGXFW A[ LVDVXPRISURGXFWVVRPHWLPHVFDOOHGD GRWSURGXFW XVLQJWKHÀUVWURZRI A DQGWKHHQWULHVLQ [ 7KDWLV 2 32 3 2 3 2 3 4 x1 2x1 C 3x2 C 4x3 4 54 x2 5 D 4 5 x3

7KLVPDWUL[VKRZVKRZWRFRPSXWHWKHÀUVWHQWU\LQ A[ GLUHFWO\ ZLWKRXWZULWLQJGRZQ DOOWKHFDOFXODWLRQVVKRZQLQ 6LPLODUO\ WKHVHFRQGHQWU\LQ A[ FDQEHFDOFXODWHGDW RQFHE\PXOWLSO\LQJWKHHQWULHVLQWKHVHFRQGURZRI A E\WKHFRUUHVSRQGLQJHQWULHVLQ [ DQGWKHQVXPPLQJWKHUHVXOWLQJSURGXFWV 2 32 3 2 3 x1 4 "1 5 "3 54 x2 5 D 4 "x1 C 5x2 " 3x3 5 x3 /LNHZLVH WKHWKLUGHQWU\LQ A[ FDQEHFDOFXODWHGIURPWKHWKLUGURZRI A DQGWKHHQWULHV LQ [ 3PX7FDUPS 3VMF GPS $PNQVUJOH "Y ,IWKHSURGXFW A[ LVGHÀQHG WKHQWKH i WKHQWU\LQ A[ LVWKHVXPRIWKHSURGXFWVRI FRUUHVSRQGLQJHQWULHVIURPURZ i RI A DQGIURPWKHYHFWRU [

&9".1-&

2 3 $ 4 # $ # $ 1 2 "1 4 5 1 ! 4 C 2 ! 3 C ."1/ ! 7 3 D 3 D D 0 "5 3 0 ! 4 C ."5/ ! 3 C 3 ! 7 6 7 2 3 2 3 2 3 2 "3 # $ 2 ! 4 C ."3/ ! 7 "13 4 8 ! 4 C 0 ! 7 5 D 4 32 5 D4 E 4 8 0 5 7 "5 2 ."5/ ! 4 C 2 ! 7 "6 2 32 3 2 3 2 3 1 0 0 r 1!r C0!sC0!t r F 4 0 1 0 54 s 5 D 4 0 ! r C 1 ! s C 0 ! t 5 D 4 s 5 0 0 1 t 0!r C0!sC1!t t #

%\GHÀQLWLRQ WKHPDWUL[LQ([DPSOHF ZLWK·VRQWKHGLDJRQDODQG·VHOVHZKHUH LVFDOOHGDQ LGHQWLW\PDWUL[ DQGLVGHQRWHGE\ I 7KHFDOFXODWLRQLQSDUWF VKRZVWKDW I [ D [ IRUHYHU\ [ LQ R3 7KHUHLVDQDQDORJRXV n # n LGHQWLW\PDWUL[ VRPHWLPHV ZULWWHQDV In $VLQSDUWF In [ D [ IRUHYHU\ [ LQ Rn


3URSHUWLHVRIWKH0DWUL[±9HFWRU3URGXFW A[ 7KHIDFWVLQWKHQH[WWKHRUHPDUHLPSRUWDQWDQGZLOOEHXVHGWKURXJKRXWWKHWH[W 7KH SURRIUHOLHVRQWKHGHÀQLWLRQRI A[ DQGWKHDOJHEUDLFSURSHUWLHVRI Rn

5)&03&.

,I A LVDQ m # n PDWUL[ X DQG Y DUHYHFWRUVLQ Rn DQG c LVDVFDODU WKHQ D A.X C Y/ D AX C AY E A.c X/ D c.AX/

1300' )RUVLPSOLFLW\ WDNH n D 3 A D Œ D1 D2 D3 ! DQG X Y LQ R3 7KHSURRIRI WKHJHQHUDOFDVHLVVLPLODU )RU i D 1; 2; 3 OHW ui DQG vi EHWKH i WKHQWULHVLQ X DQG Y UHVSHFWLYHO\ 7RSURYHVWDWHPHQWD FRPSXWH A.X C Y/ DVDOLQHDUFRPELQDWLRQRIWKH FROXPQVRI A XVLQJWKHHQWULHVLQ X C Y DVZHLJKWV 2 3 u1 C v1 A.X C Y/ D Œ D1 D2 D3 !4 u2 C v2 5 u3 C v3 # # # D .u1 C v1 /D1 C .u2 C v2 /D2 C .u3 C v3 /D3 " " "

(QWULHVLQ X C Y &ROXPQVRI A

D .u1 D1 C u2 D2 C u3 D3 / C .v1 D1 C v2 D2 C v3 D3 / D AX C AY

7RSURYHVWDWHPHQWE FRPSXWH A.c X/ DVDOLQHDUFRPELQDWLRQRIWKHFROXPQVRI A XVLQJWKHHQWULHVLQ c X DVZHLJKWV 2 3 cu1 A.c X/ D Œ D1 D2 D3 !4 cu2 5 D .cu1 /D1 C .cu2 /D2 C .cu3 /D3 cu3

D c.u1 D1 / C c.u2 D2 / C c.u3 D3 / D c.u1 D1 C u2 D2 C u3 D3 / D c.AX/

/6.&3*$"- /05& 7RRSWLPL]HDFRPSXWHUDOJRULWKPWRFRPSXWH A[ WKHVHTXHQFHRIFDOFXODWLRQV VKRXOGLQYROYHGDWDVWRUHGLQFRQWLJXRXVPHPRU\ORFDWLRQV 7KHPRVWZLGHO\ XVHGSURIHVVLRQDODOJRULWKPVIRUPDWUL[FRPSXWDWLRQVDUHZULWWHQLQ)RUWUDQ D ODQJXDJHWKDWVWRUHVDPDWUL[DVDVHWRIFROXPQV 6XFKDOJRULWKPVFRPSXWH A[ DVDOLQHDUFRPELQDWLRQRIWKHFROXPQVRI A ,QFRQWUDVW LIDSURJUDPLVZULWWHQLQ WKHSRSXODUODQJXDJH&ZKLFKVWRUHVPDWULFHVE\URZV A[ VKRXOGEHFRPSXWHG YLDWKHDOWHUQDWLYHUXOHWKDWXVHVWKHURZVRI A

1300' 0' 5)&03&. $VZDVSRLQWHGRXWDIWHU7KHRUHP VWDWHPHQWVD E DQG F DUHORJLFDOO\HTXLYDOHQW 6R LWVXIÀFHVWRVKRZIRUDQDUELWUDU\PDWUL[ A WKDWD DQGG DUHHLWKHUERWKWUXHRUERWKIDOVH 7KDWZLOOWLHDOOIRXUVWDWHPHQWVWRJHWKHU /HW U EHDQHFKHORQIRUPRI A *LYHQ E LQ Rm ZHFDQURZUHGXFHWKHDXJPHQWHG PDWUL[ Œ A E ! WRDQDXJPHQWHGPDWUL[ Œ U G ! IRUVRPH G LQ Rm ŒA E! % !!! % ŒU

G!

$)"15&3


,IVWDWHPHQWG LVWUXH WKHQHDFKURZRI U FRQWDLQVDSLYRWSRVLWLRQDQGWKHUHFDQEH QRSLYRWLQWKHDXJPHQWHGFROXPQ 6R A[ D E KDVDVROXWLRQIRUDQ\ E DQGD LVWUXH ,IG LVIDOVH WKHODVWURZRI U LVDOO]HURV /HW G EHDQ\YHFWRUZLWKDLQLWVODVWHQWU\ 7KHQ Œ U G ! UHSUHVHQWVDQ LQFRQVLVWHQW V\VWHP 6LQFHURZRSHUDWLRQVDUHUHYHUVLEOH Œ U G ! FDQEHWUDQVIRUPHGLQWRWKHIRUP Œ A E ! 7KHQHZV\VWHP A[ D E LVDOVR LQFRQVLVWHQW DQGD LVIDOVH

13"$5*$& 130#-&.4

3 2 3 3 1 5 "2 0 "7 6 "2 7 7 4 9 5 ,WFDQEHVKRZQ 9 "5 5 S D 6 /HW A D 4 "3 1 4 0 5 DQG E D 4 "8 "1 7 0 "4 WKDW S LVDVROXWLRQRI A[ D E 8VHWKLVIDFWWRH[KLELW E DVDVSHFLÀFOLQHDU FRPELQDWLRQRIWKHFROXPQVRI A # $ # $ # $ 2 5 4 "3 XD DQG Y D 9HULI\7KHRUHPD LQWKLVFDVH /HW A D 3 1 "1 5 E\FRPSXWLQJ A.X C Y/ DQG AX C AY 2

3

2

&9&3$*4&4 &RPSXWHWKHSURGXFWVLQ([HUFLVHV²XVLQJD WKHGHÀQLWLRQ DV LQ([DPSOH DQGE WKHURZ²YHFWRUUXOHIRUFRPSXWLQJ A[ ,ID SURGXFWLVXQGHÀQHG H[SODLQZK\ 2 32 3 2 3 $ "4 2 3 1 # 5 6 54 "2 5 4 3 5 4 1 "1 0 1 7 1 2 3 2 3 $ # $ 1 1 2 # "2 1 3 "4 425 15 4 "3 3 3 2 1 1 6 1 ,Q([HUFLVHV² XVHWKHGHÀQLWLRQRI A[ WRZULWHWKHPDWUL[ HTXDWLRQDVDYHFWRUHTXDWLRQ RUYLFHYHUVD 2 3 2 # $ # $ 7 1 2 "3 1 6 6 "1 7 D "4 "2 "3 1 "1 4 1 5 1 "1 2 3 2 3 2 "3 # "21 $ 6 3 6 17 27 7 "3 6 7 6 4 8 "5 5 5 D 4 "49 5 "2 1 11 2 3 2 3 2 3 2 3 4 "5 7 6 6 "1 7 6 37 6 "8 7 6 "8 7 7 6 7 6 7 6 7 x1 6 4 7 5 C x2 4 "5 5 C x3 4 0 5 D 4 0 5 "4 1 2 "7 # $ # $ # $ # $ # $ 2 "1 "4 0 5 ´1 C ´2 C ´3 C ´4 D "4 5 3 2 12

,Q([HUFLVHVDQG ZULWHWKHV\VWHPÀUVWDVDYHFWRUHTXDWLRQ DQGWKHQDVDPDWUL[HTXDWLRQ

5x1 C x2 " 3x3 D 8

2x2 C 4x3 D 0

4x1 " x2 D 8

5x1 C 3x2 D 2 3x1 " x2 D 1

*LYHQ A DQG E LQ([HUFLVHVDQG ZULWHWKHDXJPHQWHGPDWUL[ IRU WKH OLQHDU V\VWHP WKDW FRUUHVSRQGV WR WKH PDWUL[ HTXDWLRQ A[ D E 7KHQVROYHWKHV\VWHPDQGZULWHWKHVROXWLRQDVDYHFWRU 2 3 2 3 1 3 "4 "2 5 2 5 E D 4 4 5 A D 4 1 "3 "7 6 12 2 3 2 3 1 2 "1 1 2 5 E D 4 2 5 A D 4 "3 "4 5 2 3 "3 2 3 2 3 0 3 "5 6 5 ,V X LQWKHSODQHLQ /HW X D 4 4 5 DQG A D 4 "2 4 1 1 R3 VSDQQHGE\WKHFROXPQVRI A" 6HHWKHÀJXUH :K\RU ZK\QRW" u?

u?

Plane spanned by the columns of A

Where is u?

2

3 2 3 4 2 5 "1 1 "1 5 ,V X LQWKHVXEVHW /HW X D 4 "1 5 DQG A D 4 0 4 1 2 0 RI R3 VSDQQHGE\WKHFROXPQVRI A" :K\RUZK\QRW"

$ # $ 3 "1 b1 DQG E D 6KRZWKDWWKHHTXDWLRQ b2 "9 3 A[ D E GRHV QRW KDYH D VROXWLRQ IRU DOO SRVVLEOH E DQG GHVFULEH WKH VHW RI DOO E IRU ZKLFK A[ D E GRHV KDYH D VROXWLRQ

/HW A D

#

5HSHDWWKHUHTXHVWVIURP([HUFLVHZLWK 2 3 2 3 1 "2 "1 b1 2 0 5; DQG E D 4 b2 5: A D 4 "2 4 "1 3 b3

D (YHU\PDWUL[HTXDWLRQ A[ D E FRUUHVSRQGVWRDYHFWRU HTXDWLRQZLWKWKHVDPHVROXWLRQVHW E ,IWKHHTXDWLRQ A[ D E LVFRQVLVWHQW WKHQ E LVLQWKHVHW VSDQQHGE\WKHFROXPQVRI A F $Q\OLQHDUFRPELQDWLRQRIYHFWRUVFDQDOZD\VEHZULWWHQ LQWKHIRUP A[ IRUDVXLWDEOHPDWUL[ A DQGYHFWRU [ G ,IWKHFRHIÀFLHQWPDWUL[ A KDVDSLYRWSRVLWLRQLQHYHU\ URZ WKHQWKHHTXDWLRQ A[ D E LVLQFRQVLVWHQW

([HUFLVHV²UHIHUWRWKHPDWULFHV A DQG B EHORZ 0DNH DSSURSULDWHFDOFXODWLRQVWKDWMXVWLI\\RXUDQVZHUVDQGPHQWLRQDQ DSSURSULDWHWKHRUHP 2 3 2 3 1 3 0 3 1 4 1 2 6 "1 "1 "1 6 17 1 3 "4 7 7 B D 60 7 AD6 4 0 "4 5 4 2 "8 0 2 6 75 2 0 3 "1 2 9 5 "7 +RZPDQ\URZVRI A FRQWDLQDSLYRWSRVLWLRQ" 'RHVWKH HTXDWLRQ A[ D E KDYHDVROXWLRQIRUHDFK E LQ R4 "

&DQHYHU\YHFWRULQ R4 EHZULWWHQDVDOLQHDUFRPELQDWLRQRI WKHFROXPQVRIWKHPDWUL[ B DERYH" 'RWKHFROXPQVRI B VSDQ R3 " &DQHDFKYHFWRULQ R4 EHZULWWHQDVDOLQHDUFRPELQDWLRQRI WKHFROXPQVRIWKHPDWUL[ A DERYH" 'RWKHFROXPQVRI A VSDQ R4 " 'RWKHFROXPQVRI B VSDQ R4 " 'RHVWKHHTXDWLRQ B [ D \ KDYHDVROXWLRQIRUHDFK \ LQ R4 " 2 3 2 3 2 3 1 0 1 6 07 6 "1 7 6 07 7 6 7 6 7 /HW Y1 D 6 'RHV 4 "1 5 Y2 D 4 0 5 Y3 D 4 0 5 0 1 "1

fY1 ; Y2 ; Y3 g VSDQ R4 " :K\ RUZK\QRW" 2 3 2 3 2 3 0 0 4 /HW Y1 D 4 0 5 Y2 D 4 "3 5 Y3 D 4 "2 5 "3 9 "6


'RHV

fY1 ; Y2 ; Y3 g VSDQ R3 " :K\ RUZK\QRW"

,Q([HUFLVHVDQG PDUNHDFKVWDWHPHQW7UXHRU)DOVH -XVWLI\ HDFKDQVZHU D 7KHHTXDWLRQ A[ D E LVUHIHUUHGWRDVD YHFWRUHTXDWLRQ

E $ YHFWRU E LVDOLQHDUFRPELQDWLRQRIWKHFROXPQVRID PDWUL[ A LIDQGRQO\LIWKHHTXDWLRQ A[ D E KDVDWOHDVW RQHVROXWLRQ F 7KHHTXDWLRQ A[ D E LVFRQVLVWHQWLIWKHDXJPHQWHGPD WUL[ Œ A E ! KDVDSLYRWSRVLWLRQLQHYHU\URZ G 7KHÀUVWHQWU\LQWKHSURGXFW A[ LVDVXPRISURGXFWV H ,IWKHFROXPQVRIDQ m # n PDWUL[ A VSDQ Rm WKHQWKH HTXDWLRQ A[ D E LVFRQVLVWHQWIRUHDFK E LQ Rm

I ,I A LVDQ m # n PDWUL[DQGLIWKHHTXDWLRQ A[ D E LV LQFRQVLVWHQWIRUVRPH E LQ Rm WKHQ A FDQQRWKDYHDSLYRW SRVLWLRQLQHYHU\URZ

H 7KHVROXWLRQVHWRIDOLQHDUV\VWHPZKRVHDXJPHQWHG PDWUL[LV Œ D1 D2 D3 E ! LVWKHVDPHDVWKHVROXWLRQVHW RI A[ D E LI A D Œ D1 D2 D3 !

I ,I A LVDQ m # n PDWUL[ZKRVHFROXPQVGRQRWVSDQ Rm WKHQWKHHTXDWLRQ A[ D E LVFRQVLVWHQWIRUHYHU\ E LQ Rm 2 32 3 2 3 4 "3 1 "3 "7 5 54 "1 5 D 4 "3 5 8VH WKLV 1RWH WKDW 4 5 "2 "6 2 "3 2 10 IDFWDQGQRURZRSHUDWLRQV WRÀQGVFDODUV c1 c2 c3 VXFK 2 3 2 3 2 3 2 3 "7 4 "3 1 WKDW 4 "3 5 D c1 4 5 5 C c2 4 "2 5 C c3 4 5 5 10 "6 2 "3 2 3 2 3 2 3 7 3 5 /HW X D 4 2 5 Y D 4 1 5 DQG Z D 4 1 5 ,W FDQ EH 5 3 1 VKRZQWKDW 2X " 3Y " Z D 8VHWKLVIDFWDQGQRURZ RSHUDWLRQV WR ÀQG x1 DQG x2 WKDW VDWLVI\ WKH HTXDWLRQ 2 3 2 3 7 3 # $ 5 x 42 1 5 1 D 4 1 5 x2 5 3 1

5HZULWHWKHQXPHULFDO PDWUL[HTXDWLRQEHORZLQV\PEROLF IRUPDVDYHFWRUHTXDWLRQ XVLQJV\PEROV Y1 ; Y2 ; : : : IRUWKH YHFWRUVDQG c1 ; c2 ; : : : IRUVFDODUV 'HÀQHZKDWHDFKV\PERO UHSUHVHQWV XVLQJWKHGDWDJLYHQLQWKHPDWUL[HTXDWLRQ 3 2 "3 # $6 1 7 # $ 7 "3 5 "4 9 7 6 11 6 27 D 7 5 8 1 "2 "4 6 "11 4 "1 5 2

/HW T1 T2 T3 DQG Y UHSUHVHQWYHFWRUVLQ R5 DQGOHW x1 x2 DQG x3 GHQRWHVFDODUV :ULWHWKHIROORZLQJYHFWRUHTXDWLRQDV DPDWUL[HTXDWLRQ ,GHQWLI\DQ\V\PEROV\RXFKRRVHWRXVH

x1 T1 C x2 T2 C x3 T3 D Y

&RQVWUXFW D 3 # 3 PDWUL[ QRW LQ HFKHORQ IRUP ZKRVH FROXPQVVSDQ R3 6KRZWKDWWKHPDWUL[\RXFRQVWUXFWKDV WKHGHVLUHGSURSHUW\ &RQVWUXFW D 3 # 3 PDWUL[ QRW LQ HFKHORQ IRUP ZKRVH FROXPQVGR QRW VSDQ R3 6KRZWKDWWKHPDWUL[\RXFRQVWUXFW KDVWKHGHVLUHGSURSHUW\ /HW A EHD 3 # 2 PDWUL[ ([SODLQZK\WKHHTXDWLRQ A[ D E FDQQRWEHFRQVLVWHQWIRUDOO E LQ R3 *HQHUDOL]H\RXUDU JXPHQWWRWKHFDVHRIDQDUELWUDU\ A ZLWKPRUHURZVWKDQ FROXPQV

$)"15&3

/LQHDU(TXDWLRQVLQ/LQHDU$OJHEUD 2

7 6 "5 6 4 6 "7 2 10 6 "8 6 4 "7 3

&RXOGDVHWRIWKUHHYHFWRUVLQ R4 VSDQDOORI R4 " ([SODLQ :KDWDERXW n YHFWRUVLQ Rm ZKHQ n LVOHVVWKDQ m" 6XSSRVH A LVD 4 # 3 PDWUL[DQG E LVDYHFWRULQ R4 ZLWK WKHSURSHUW\WKDW A[ D E KDVDXQLTXHVROXWLRQ :KDWFDQ \RXVD\DERXWWKHUHGXFHGHFKHORQIRUPRI A" -XVWLI\\RXU DQVZHU /HW A EHD 3 # 4 PDWUL[ OHW Y1 DQG Y2 EHYHFWRUVLQ R3 DQG OHW Z D Y1 C Y2 6XSSRVH Y1 D AX1 DQG Y2 D AX2 IRUVRPH YHFWRUV X1 DQG X2 LQ R4 :KDWIDFWDOORZV\RXWRFRQFOXGH WKDWWKHV\VWHP A[ D Z LVFRQVLVWHQW" 1RWH X1 DQG X2 GHQRWHYHFWRUV QRWVFDODUHQWULHVLQYHFWRUV

2

5 6 "7 6 4 11 "3

/HW A EHD 5 # 3 PDWUL[ OHW \ EHDYHFWRULQ R3 DQGOHW ] EH DYHFWRULQ R5 6XSSRVH A\ D ] :KDWIDFWDOORZV\RXWR FRQFOXGHWKDWWKHV\VWHP A[ D 5] LVFRQVLVWHQW"

"7 4 11 "1

3 8 "9 7 7 75 15

1 4 "6 "10 "5 "1 10 12

11 "6 "7 "3 "4 6 5 6 "9 4 "7 2

2

4 "5 6 3 "7 6 4 5 "6 9 1 3 6 "3 7 7 "8 5 12

"1 "4 "1 10

3 8 27 7 45 7

3 12 "9 7 7 "3 5 7

>0@ )LQGDFROXPQRIWKHPDWUL[LQ([HUFLVHWKDWFDQEH GHOHWHGDQG\HWKDYHWKHUHPDLQLQJPDWUL[FROXPQVVWLOOVSDQ R4

6XSSRVH A LVD 4 # 4 PDWUL[DQG E LVDYHFWRULQ R ZLWKWKH SURSHUW\WKDW A[ D E KDVDXQLTXHVROXWLRQ ([SODLQZK\WKH FROXPQVRI A PXVWVSDQ R4 4

>0@ )LQGDFROXPQRIWKHPDWUL[LQ([HUFLVHWKDWFDQEH GHOHWHGDQG\HWKDYHWKHUHPDLQLQJPDWUL[FROXPQVVWLOOVSDQ R4 &DQ\RXGHOHWHPRUHWKDQRQHFROXPQ"

>0@ ,Q([HUFLVHV² GHWHUPLQHLIWKHFROXPQVRIWKHPDWUL[ VSDQ R4 SG

2 "5 "3 4 10 "2 9 2

.BTUFSJOH -JOFBS "MHFCSB $PODFQUT 4QBO m

WEB

40-65*0/4 50 13"$5*$& 130#-&.4 7KHPDWUL[HTXDWLRQ 3 2 3 2 3 3 1 5 "2 0 6 "7 7 "2 7 4 4 "3 1 9 "5 56 95 4 05 D 4 "8 "1 7 0 "4 2

LVHTXLYDOHQWWRWKHYHFWRUHTXDWLRQ 2

3 2 3 2 3 2 3 2 3 1 5 "2 0 "7 34 "3 5 " 24 1 5 C 04 9 5 " 44 "5 5 D 4 9 5 4 "8 "1 7 0

ZKLFKH[SUHVVHV E DVDOLQHDUFRPELQDWLRQRIWKHFROXPQVRI A # $ # $ # $ 4 "3 1 XCYD C D "1 5 4 # $# $ # $ # $ 2 5 1 2 C 20 22 A.X C Y/ D D D 3 1 4 3C4 7 # $# $ # $# $ 2 5 4 2 5 "3 AX C AY D C 3 1 "1 3 1 5 # $ # $ # $ 3 19 22 D C D 11 "4 7

6ROXWLRQ6HWVRI/LQHDU6\VWHPV

40-65*0/ 4&54 0' -*/&"3 4:45&.4 6ROXWLRQVHWVRIOLQHDUV\VWHPVDUHLPSRUWDQWREMHFWVRIVWXG\LQOLQHDUDOJHEUD 7KH\ ZLOODSSHDUODWHULQVHYHUDOGLIIHUHQWFRQWH[WV 7KLVVHFWLRQXVHVYHFWRUQRWDWLRQWRJLYH H[SOLFLWDQGJHRPHWULFGHVFULSWLRQVRIVXFKVROXWLRQVHWV

+RPRJHQHRXV/LQHDU6\VWHPV $ V\VWHPRIOLQHDUHTXDWLRQVLVVDLGWREH KRPRJHQHRXV LILWFDQEHZULWWHQLQWKH IRUP A[ D ZKHUH A LVDQ m # n PDWUL[DQG LVWKH]HURYHFWRULQ Rm 6XFKD V\VWHP A[ D DOZD\V KDVDWOHDVWRQHVROXWLRQ QDPHO\ [ D WKH]HURYHFWRULQ Rn / 7KLV]HURVROXWLRQLVXVXDOO\FDOOHGWKH WULYLDOVROXWLRQ )RUDJLYHQHTXDWLRQ A[ D ; WKHLPSRUWDQWTXHVWLRQLVZKHWKHUWKHUHH[LVWVD QRQWULYLDOVROXWLRQ WKDWLV DQRQ]HUR YHFWRU [ WKDWVDWLVÀHV A[ D : 7KH([LVWHQFHDQG8QLTXHQHVV7KHRUHPLQ6HFWLRQ 7KHRUHP OHDGVLPPHGLDWHO\WRWKHIROORZLQJIDFW 7KHKRPRJHQHRXVHTXDWLRQ A[ D KDVDQRQWULYLDOVROXWLRQLIDQGRQO\LIWKH HTXDWLRQKDVDWOHDVWRQHIUHHYDULDEOH

&9".1-& 'HWHUPLQHLIWKHIROORZLQJKRPRJHQHRXVV\VWHPKDVDQRQWULYLDO VROXWLRQ 7KHQGHVFULEHWKHVROXWLRQVHW

3x1 C 5x2 " 4x3 D 0 "3x1 " 2x2 C 4x3 D 0 6x1 C x2 " 8x3 D 0 40-65*0/ /HW A EHWKHPDWUL[RIFRHIÀFLHQWVRIWKHV\VWHPDQGURZUHGXFHWKH DXJPHQWHGPDWUL[ Œ A ! WRHFKHORQIRUP 2 3 2 3 2 3 3 5 "4 0 3 5 "4 0 3 5 "4 0 4 "3 "2 4 05 % 40 3 0 05 % 40 3 0 05 6 1 "8 0 0 "9 0 0 0 0 0 0

n{v

Spa

6LQFH x3 LVDIUHHYDULDEOH A[ D KDVQRQWULYLDOVROXWLRQVRQHIRUHDFKFKRLFHRI x3 7RGHVFULEHWKHVROXWLRQVHW FRQWLQXHWKHURZUHGXFWLRQRI Œ A ! WR UHGXFHG HFKHORQ IRUP 2 3 x1 " 43 x3 D 0 1 0 " 43 0 40 1 0 05 D0 x2 0 0 0 0 0 D0

}

x3

v 0

x1

'*(63&

x2

6ROYHIRUWKHEDVLFYDULDEOHV x1 DQG x2 DQGREWDLQ x1 D 43 x3 x2 D 0 ZLWK x3 IUHH $VD YHFWRU WKHJHQHUDOVROXWLRQRI A[ D KDVWKHIRUP 2 3 24 3 243 243 x1 x 3 3 3 3 6 7 6 7 6 7 6 7 [ D 4 x2 5 D 4 0 5 D x3 4 0 5 D x3 Y; ZKHUH Y D 4 0 5

x3

x3

1

1

+HUH x3 LVIDFWRUHGRXWRIWKHH[SUHVVLRQIRUWKHJHQHUDOVROXWLRQYHFWRU 7KLVVKRZVWKDW HYHU\VROXWLRQRI A[ D LQWKLVFDVHLVDVFDODUPXOWLSOHRI Y 7KHWULYLDOVROXWLRQLV REWDLQHGE\FKRRVLQJ x3 D 0: *HRPHWULFDOO\ WKHVROXWLRQVHWLVDOLQHWKURXJK LQ R3 6HH)LJ

$)"15&3


1RWLFHWKDWDQRQWULYLDOVROXWLRQ [ FDQKDYHVRPH]HURHQWULHVVRORQJDVQRWDOORI LWVHQWULHVDUH]HUR

&9".1-& $ VLQJOHOLQHDUHTXDWLRQFDQEHWUHDWHGDVDYHU\VLPSOHV\VWHPRI HTXDWLRQV 'HVFULEHDOOVROXWLRQVRIWKHKRPRJHQHRXV´V\VWHPµ

10x1 " 3x2 " 2x3 D 0

x1 x3

v

X x2

'*(63&

40-65*0/ 7KHUHLVQRQHHGIRUPDWUL[QRWDWLRQ 6ROYHIRUWKHEDVLFYDULDEOH x1 LQ WHUPVRIWKHIUHHYDULDEOHV 7KHJHQHUDOVROXWLRQLV x1 D :3x2 C :2x3 ZLWK x2 DQG x3 IUHH $VDYHFWRU WKHJHQHUDOVROXWLRQLV 2 3 2 3 2 3 2 3 x1 :3x2 C :2x3 :3x2 :2x3 5 D 4 x2 5 C 4 0 5 x2 [ D 4 x2 5 D 4 x3 x3 0 x3 2 3 2 3 :3 :2 D x2 4 1 5 C x3 4 0 5 ZLWK x2 x3 IUHH 0 1 "

u

"

Y

7KLVFDOFXODWLRQVKRZVWKDWHYHU\VROXWLRQRI LVDOLQHDUFRPELQDWLRQRIWKHYHFWRUV X DQG Y VKRZQLQ 7KDWLV WKHVROXWLRQVHWLV 6SDQ fX; Yg 6LQFHQHLWKHU X QRU Y LVD VFDODUPXOWLSOHRIWKHRWKHU WKHVROXWLRQVHWLVDSODQHWKURXJKWKHRULJLQ 6HH)LJ ([DPSOHV DQG DORQJ ZLWK WKH H[HUFLVHV LOOXVWUDWH WKH IDFW WKDW WKH VROX WLRQVHWRIDKRPRJHQHRXVHTXDWLRQ A[ D FDQDOZD\VEHH[SUHVVHGH[SOLFLWO\DV 6SDQ fY1 ; : : : ; Yp g IRUVXLWDEOHYHFWRUV Y1 ; : : : ; Yp ,IWKHRQO\VROXWLRQLVWKH]HURYHFWRU WKHQWKHVROXWLRQVHWLV 6SDQ fg ,IWKHHTXDWLRQ A[ D KDVRQO\RQHIUHHYDULDEOH WKHVROXWLRQVHWLVDOLQHWKURXJKWKHRULJLQ DVLQ)LJ $ SODQHWKURXJKWKHRULJLQ DVLQ)LJ SURYLGHVDJRRGPHQWDOLPDJHIRUWKHVROXWLRQVHWRI A[ D ZKHQWKHUH DUHWZRRUPRUHIUHHYDULDEOHV 1RWH KRZHYHU WKDWDVLPLODUÀJXUHFDQEHXVHGWR YLVXDOL]H 6SDQ fX; Yg HYHQZKHQ X DQG Y GRQRWDULVHDVVROXWLRQVRI A[ D : 6HH)LJ LQ6HFWLRQ

3DUDPHWULF9HFWRU)RUP 7KHRULJLQDOHTXDWLRQ IRUWKHSODQHLQ([DPSOHLVDQ LPSOLFLW GHVFULSWLRQRIWKH SODQH 6ROYLQJWKLVHTXDWLRQDPRXQWVWRÀQGLQJDQ H[SOLFLW GHVFULSWLRQRIWKHSODQHDV WKHVHWVSDQQHGE\ X DQG Y (TXDWLRQ LVFDOOHGD SDUDPHWULFYHFWRUHTXDWLRQ RIWKH SODQH 6RPHWLPHVVXFKDQHTXDWLRQLVZULWWHQDV [ D sX C t Y

.s; t LQ R/

WRHPSKDVL]HWKDWWKHSDUDPHWHUVYDU\RYHUDOOUHDOQXPEHUV ,Q([DPSOH WKHHTXDWLRQ [ D x3 Y ZLWK x3 IUHH RU [ D t Y ZLWK t LQ R LVDSDUDPHWULFYHFWRUHTXDWLRQRIDOLQH :KHQHYHUDVROXWLRQVHWLVGHVFULEHGH[SOLFLWO\ZLWKYHFWRUVDVLQ([DPSOHVDQG ZH VD\WKDWWKHVROXWLRQLVLQ SDUDPHWULFYHFWRUIRUP

6ROXWLRQVRI1RQKRPRJHQHRXV6\VWHPV :KHQDQRQKRPRJHQHRXVOLQHDUV\VWHPKDVPDQ\VROXWLRQV WKHJHQHUDOVROXWLRQFDQEH ZULWWHQLQSDUDPHWULFYHFWRUIRUPDVRQHYHFWRUSOXVDQDUELWUDU\OLQHDUFRPELQDWLRQRI YHFWRUVWKDWVDWLVI\WKHFRUUHVSRQGLQJKRPRJHQHRXVV\VWHP


&9".1-& 'HVFULEHDOOVROXWLRQVRI A[ D E ZKHUH 2

3 2 3 3 5 "4 7 A D 4 "3 "2 4 5 DQG E D 4 "1 5 6 1 "8 "4 40-65*0/ +HUH A LVWKHPDWUL[RIFRHIÀFLHQWVIURP([DPSOH 5RZRSHUDWLRQVRQ Œ A E ! SURGXFH 2 3 2 3 3 5 "4 7 1 0 " 43 "1 x1 " 43 x3 D "1 4 "3 "2 4 "1 5 % 4 0 1 0 2 5; D 2 x2 6 1 "8 "4 0 0 0 0 0 D 0

7KXV x1 D "1 C 43 x3 x2 D 2 DQG x3 LVIUHH $VDYHFWRU WKHJHQHUDOVROXWLRQRI A[ D E KDVWKHIRUP 2 3 2 3 2 3 24 3 2 3 243 x1 "1 C 43 x3 "1 x "1 3 3 3 6 7 6 7 6 7 6 7 6 7 6 7 [ D 4 x2 5 D 4 2 5 D 4 2 5 C 4 0 5 D 4 2 5 C x3 4 0 5 x3 x3 x3 1 0 0 "

S

7KHHTXDWLRQ [ D S C x3 Y RU ZULWLQJ t DVDJHQHUDOSDUDPHWHU v+p p v

[ D S C tY

t LQ R

"

Y

GHVFULEHVWKHVROXWLRQVHWRI A[ D E LQSDUDPHWULFYHFWRUIRUP 5HFDOOIURP([DPSOH WKDWWKHVROXWLRQVHWRI A[ D KDVWKHSDUDPHWULFYHFWRUHTXDWLRQ [ D tY

t LQ R

>ZLWKWKHVDPH Y WKDWDSSHDUVLQ @ 7KXVWKHVROXWLRQVRI A[ D E DUHREWDLQHGE\ DGGLQJWKHYHFWRU S WRWKHVROXWLRQVRI A[ D 7KHYHFWRU S LWVHOILVMXVWRQHSDUWLFXODU $GGLQJ S WR Y WUDQVODWHV Y WR Y C S VROXWLRQRI A[ D E >FRUUHVSRQGLQJWR t D 0 LQ @ '*(63&

L+p

L

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x = b p + tv

'*(63&

Ax = 0

p

7UDQVODWHGOLQH

v

tv

'*(63& 3DUDOOHOVROXWLRQVHWVRI A[ D E DQG

A[ D

7KHUHODWLRQEHWZHHQWKHVROXWLRQVHWVRI A[ D E DQG A[ D VKRZQLQ)LJ JHQHUDOL]HVWRDQ\ FRQVLVWHQW HTXDWLRQ A[ D E DOWKRXJKWKHVROXWLRQVHWZLOOEHODUJHU WKDQDOLQHZKHQWKHUHDUHVHYHUDOIUHHYDULDEOHV 7KHIROORZLQJWKHRUHPJLYHVWKHSUHFLVH VWDWHPHQW 6HH([HUFLVHIRUDSURRI

$)"15&3


5)&03&.

6XSSRVHWKHHTXDWLRQ A[ D E LVFRQVLVWHQWIRUVRPHJLYHQ E DQGOHW S EHD VROXWLRQ 7KHQWKHVROXWLRQVHWRI A[ D E LVWKHVHWRIDOOYHFWRUVRIWKHIRUP Z D S C Yh ZKHUH Yh LVDQ\VROXWLRQRIWKHKRPRJHQHRXVHTXDWLRQ A[ D 7KHRUHPVD\VWKDWLI A[ D E KDVDVROXWLRQ WKHQWKHVROXWLRQVHWLVREWDLQHGE\ WUDQVODWLQJWKHVROXWLRQVHWRI A[ D XVLQJDQ\SDUWLFXODUVROXWLRQ S RI A[ D E IRU WKHWUDQVODWLRQ )LJXUHLOOXVWUDWHVWKHFDVHLQZKLFKWKHUHDUHWZRIUHHYDULDEOHV (YHQ ZKHQ n > 3 RXUPHQWDOLPDJHRIWKHVROXWLRQVHWRIDFRQVLVWHQWV\VWHP A[ D E ZLWK E ¤ LVHLWKHUDVLQJOHQRQ]HURSRLQWRUDOLQHRUSODQHQRWSDVVLQJWKURXJKWKHRULJLQ

Ax = b Ax = 0

p

'*(63& 3DUDOOHOVROXWLRQVHWVRI

A[ D E DQG A[ D

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

13"$5*$& 130#-&.4 (DFKRIWKHIROORZLQJHTXDWLRQVGHWHUPLQHVDSODQHLQ R3 'RWKHWZRSODQHV LQWHUVHFW" ,IVR GHVFULEHWKHLULQWHUVHFWLRQ

x1 C 4x2 " 5x3 D 0 2x1 " x2 C 8x3 D 9 :ULWHWKHJHQHUDOVROXWLRQRI 10x1 " 3x2 " 2x3 D 7 LQSDUDPHWULFYHFWRUIRUP DQG UHODWHWKHVROXWLRQVHWWRWKHRQHIRXQGLQ([DPSOH


&9&3$*4&4 ,Q([HUFLVHV² GHWHUPLQHLIWKHV\VWHPKDVDQRQWULYLDOVROXWLRQ 7U\WRXVHDVIHZURZRSHUDWLRQVDVSRVVLEOH

2x1 " 5x2 C 8x3 D 0

"2x1 " 7x2 C x3 D 0

x1 " 2x2 C 3x3 D 0

"2x1 " 3x2 " 4x3 D 0

4x1 C 2x2 C 7x3 D 0

2x1 " 4x2 C 9x3 D 0

"3x1 C 4x2 " 8x3 D 0

"2x1 C 5x2 C 4x3 D 0

5x1 " 3x2 C 2x3 D 0

"3x1 " 4x2 C 2x3 D 0

,Q([HUFLVHVDQG IROORZWKHPHWKRGRI([DPSOHVDQG WRZULWHWKHVROXWLRQVHWRIWKHJLYHQKRPRJHQHRXVV\VWHPLQ SDUDPHWULFYHFWRUIRUP

2x1 C 2x2 C 4x3 D 0

"4x1 " 4x2 " 8x3 D 0

x1 C 2x2 " 3x3 D 0

2x1 C x2 " 3x3 D 0

" 3x2 " 3x3 D 0

"1x1 C x2

D0

,Q([HUFLVHV² GHVFULEHDOOVROXWLRQVRI A[ D LQSDUDPHWULF YHFWRUIRUP ZKHUH A LVURZHTXLYDOHQWWRWKHJLYHQPDWUL[ # $ # $ 1 3 "3 7 1 "3 "8 5 0 1 "4 5 0 1 2 "4 # $ # $ 3 "6 6 "1 "4 0 "4 "2 4 "2 2 "8 0 8 2 3 1 "4 "2 0 3 "5 60 0 1 0 0 "1 7 7 6 40 0 0 0 1 "4 5 0 0 0 0 0 0 2 3 1 "2 3 "6 5 0 60 0 0 1 4 "6 7 7 6 40 0 0 0 0 15 0 0 0 0 0 0

6XSSRVHWKHVROXWLRQVHWRIDFHUWDLQV\VWHPRIOLQHDUHTXD WLRQVFDQEHGHVFULEHGDV x1 D 5 C 4x3 x2 D "2 " 7x3 ZLWK x3 IUHH 8VHYHFWRUVWRGHVFULEHWKLVVHWDVDOLQHLQ R3

6XSSRVH WKH VROXWLRQ VHW RI D FHUWDLQ V\VWHP RI OLQHDU HTXDWLRQV FDQ EH GHVFULEHG DV x1 D 5x4 x2 D 3 " 2x4 x3 D 2 C 5x4 ZLWK x4 IUHH 8VHYHFWRUVWRGHVFULEHWKLVVHW DVDÓLQHµLQ R4 'HVFULEH DQG FRPSDUH WKH VROXWLRQ VHWV RI x1 C 5x2 " 3x3 D 0 DQG x1 C 5x2 " 3x3 D "2

'HVFULEH DQG FRPSDUH WKH VROXWLRQ VHWV RI x1 " 2x2 C 3x3 D 0 DQG x1 " 2x2 C 3x3 D 4

)ROORZWKHPHWKRGRI([DPSOHWRGHVFULEHWKHVROXWLRQVRI WKHIROORZLQJV\VWHPLQSDUDPHWULFYHFWRUIRUP $OVR JLYH DJHRPHWULFGHVFULSWLRQRIWKHVROXWLRQVHWDQGFRPSDUHLWWR WKDWLQ([HUFLVH

$VLQ([HUFLVH GHVFULEHWKHVROXWLRQVRIWKHIROORZLQJ V\VWHPLQSDUDPHWULFYHFWRUIRUP DQGSURYLGHDJHRPHWULF FRPSDULVRQZLWKWKHVROXWLRQVHWLQ([HUFLVH

x1 C 2x2 " 3x3 D

5

2x1 C x2 " 3x3 D 13

"x1 C x2

D "8

,Q([HUFLVHVDQG ÀQGWKHSDUDPHWULFHTXDWLRQRIWKHOLQH WKURXJK D SDUDOOHOWR E # $ # $ # $ # $ "2 "5 3 "7 D D ED D D ED 0 3 "2 6 ,Q([HUFLVHVDQG ÀQGDSDUDPHWULFHTXDWLRQRIWKHOLQH M WKURXJK S DQG T >+LQW M LVSDUDOOHOWRWKHYHFWRU T " S 6HHWKH ÀJXUHEHORZ@ # $ # $ # $ # $ 3 4 "3 0 S D TD S D TD "3 1 2 "3 x2 q–p M

–p q

x1

0

p

,Q([HUFLVHVDQG PDUNHDFKVWDWHPHQW7UXHRU)DOVH -XVWLI\ HDFKDQVZHU D $ KRPRJHQHRXVHTXDWLRQLVDOZD\VFRQVLVWHQW E 7KHHTXDWLRQ A[ D JLYHVDQH[SOLFLWGHVFULSWLRQRILWV VROXWLRQVHW F 7KHKRPRJHQHRXVHTXDWLRQ A[ D KDVWKHWULYLDOVR OXWLRQLIDQGRQO\LIWKHHTXDWLRQKDVDWOHDVWRQHIUHH YDULDEOH G 7KHHTXDWLRQ [ D S C t Y GHVFULEHVDOLQHWKURXJK Y SDU DOOHOWR S H 7KHVROXWLRQVHWRI A[ D E LVWKHVHWRIDOOYHFWRUVRI WKHIRUP Z D S C Yh ZKHUH Yh LVDQ\VROXWLRQRIWKH HTXDWLRQ A[ D D $ KRPRJHQHRXVV\VWHPRIHTXDWLRQVFDQEHLQFRQVLVWHQW E ,I [ LVDQRQWULYLDOVROXWLRQRI A[ D WKHQHYHU\HQWU\LQ [ LVQRQ]HUR

2x1 C 2x2 C 4x3 D

8

F 7KHHIIHFWRIDGGLQJ S WRDYHFWRULVWRPRYHWKHYHFWRULQ DGLUHFWLRQSDUDOOHOWR S

"4x1 " 4x2 " 8x3 D "16 " 3x2 " 3x3 D

12

G 7KHHTXDWLRQ A[ D E LVKRPRJHQHRXVLIWKH]HURYHFWRU LVDVROXWLRQ

$)"15&3


H ,I A[ D E LVFRQVLVWHQW WKHQWKHVROXWLRQVHWRI A[ D E LVREWDLQHGE\WUDQVODWLQJWKHVROXWLRQVHWRI A[ D

3URYH7KHRUHP D 6XSSRVH S LVDVROXWLRQRI A[ D E VRWKDW AS D E /HW Yh EHDQ\VROXWLRQRIWKHKRPRJHQHRXVHTXDWLRQ A[ D DQGOHW Z D S C Yh 6KRZWKDW Z LVDVROXWLRQRI A[ D E E /HW Z EHDQ\VROXWLRQRI A[ D E DQGGHÀQH Yh D Z " S 6KRZWKDW Yh LVDVROXWLRQRI A[ D 7KLVVKRZVWKDW HYHU\VROXWLRQRI A[ D E KDVWKHIRUP Z D S C Yh ZLWK S DSDUWLFXODUVROXWLRQRI A[ D E DQG Yh DVROXWLRQRI A[ D

6XSSRVH A LVWKH 3 # 3 ]HUR PDWUL[ZLWKDOO]HURHQWULHV 'HVFULEHWKHVROXWLRQVHWRIWKHHTXDWLRQ A[ D

6XSSRVH A[ D E KDVDVROXWLRQ ([SODLQZK\WKHVROXWLRQLV XQLTXHSUHFLVHO\ZKHQ A[ D KDVRQO\WKHWULYLDOVROXWLRQ ,Q([HUFLVHV² D GRHVWKHHTXDWLRQ A[ D KDYHDQRQWULY LDOVROXWLRQDQGE GRHVWKHHTXDWLRQ A[ D E KDYHDWOHDVWRQH VROXWLRQIRUHYHU\SRVVLEOH E" A LVD 3 # 3 PDWUL[ZLWKWKUHHSLYRWSRVLWLRQV

A LVD 4 # 4 PDWUL[ZLWKWKUHHSLYRWSRVLWLRQV A LVD 2 # 5 PDWUL[ZLWKWZRSLYRWSRVLWLRQV A LVD 3 # 2 PDWUL[ZLWKWZRSLYRWSRVLWLRQV

,I E ¤ FDQWKHVROXWLRQVHWRI A[ D E EHDSODQHWKURXJK WKHRULJLQ" ([SODLQ 2 3 1 &RQVWUXFWD 3 # 3 QRQ]HURPDWUL[ A VXFKWKDWWKHYHFWRU 4 1 5 1 LVDVROXWLRQRI A[ D

&RQVWUXFW 2 3 D 3 # 3 QRQ]HUR PDWUL[ A VXFK WKDW WKH YHFWRU 2 4 "1 5 LVDVROXWLRQRI A[ D 1 2 3 "1 "3 *LYHQ A D 4 7 21 5 ÀQG RQH QRQWULYLDO VROXWLRQ RI "2 "6 A[ D E\LQVSHFWLRQ >+LQW 7KLQNRIWKHHTXDWLRQ A[ D ZULWWHQDVDYHFWRUHTXDWLRQ@ 2 3 3 "2 4 5 ÀQG RQH QRQWULYLDO VROXWLRQ RI *LYHQ A D 4 "6 12 "8 A[ D E\LQVSHFWLRQ

&RQVWUXFWD 2 # 2 PDWUL[ A VXFKWKDWWKHVROXWLRQVHWRIWKH HTXDWLRQ A[ D LVWKHOLQHLQ R2 WKURXJK .4; 1/ DQGWKH RULJLQ 7KHQ ÀQGDYHFWRU E LQ R2 VXFKWKDWWKHVROXWLRQ VHWRI A[ D E LV QRW DOLQHLQ R2 SDUDOOHOWRWKHVROXWLRQVHW RI A[ D :K\GRHVWKLV QRW FRQWUDGLFW7KHRUHP"

/HW A EHDQ m # n PDWUL[DQGOHW Z EHDYHFWRULQ Rn WKDW VDWLVÀHVWKHHTXDWLRQ A[ D 6KRZWKDWIRUDQ\VFDODU c WKHYHFWRU c Z DOVRVDWLVÀHV A[ D >7KDWLV VKRZWKDW A.c Z/ D @

/HW A EHDQ m # n PDWUL[ DQGOHW Y DQG Z EHYHFWRUVLQ Rn ZLWKWKHSURSHUW\WKDW AY D DQG AZ D ([SODLQ ZK\ A.Y C Z/ PXVWEHWKH]HURYHFWRU 7KHQH[SODLQZK\ A.c Y C d Z/ D IRUHDFKSDLURIVFDODUV c DQG d 6XSSRVH A LVD 3 # 3 PDWUL[DQG E LVDYHFWRULQ R3 VXFKWKDW WKHHTXDWLRQ A[ D E GRHV QRW KDYHDVROXWLRQ 'RHVWKHUH H[LVWDYHFWRU \ LQ R3 VXFKWKDWWKHHTXDWLRQ A[ D \ KDVD XQLTXHVROXWLRQ" 'LVFXVV

40-65*0/4 50 13"$5*$& 130#-&.4 5RZUHGXFHWKHDXJPHQWHGPDWUL[ #

1 4 "5 2 "1 8

0 9

$

%

#

1 4 0 "9

x1

"5 18

0 9

$

C 3x3 D 4 x2 " 2x3 D "1

%

#

1 0

0 3 4 1 "2 "1

$

7KXV x1 D 4 " 3x3 ; x2 D "1 C 2x3 ZLWK x3 IUHH 7KHJHQHUDOVROXWLRQLQSDUDPHW ULFYHFWRUIRUPLV 2 3 2 3 2 3 2 3 x1 4 " 3x3 4 "3 4 x2 5 D 4 "1 C 2x3 5 D 4 "1 5 C x3 4 2 5 x3 x3 0 1 "

S

"

Y

7KHLQWHUVHFWLRQRIWKHWZRSODQHVLVWKHOLQHWKURXJK S LQWKHGLUHFWLRQRI Y

$SSOLFDWLRQVRI/LQHDU6\VWHPV

% & % & 7KHDXJPHQWHGPDWUL[ 10 "3 "2 7 LVURZHTXLYDOHQWWR 1 ":3 ":2 :7 DQGWKHJHQHUDOVROXWLRQLV x1 D :7 C :3x2 C :2x3 ZLWK x2 DQG x3 IUHH 7KDWLV 2

3 2 3 2 3 2 3 2 3 x1 :7 C :3x2 C :2x3 :7 :3 :2 5 D 4 0 5 C x2 4 1 5 C x3 4 0 5 x2 [ D 4 x2 5 D 4 x3 x3 0 0 1

D

S

C

x2 X C

x3 Y

7KHVROXWLRQVHWRIWKHQRQKRPRJHQHRXVHTXDWLRQ A[ D E LVWKHWUDQVODWHGSODQH S C 6SDQ fX; Yg ZKLFKSDVVHVWKURXJK S DQGLVSDUDOOHOWRWKHVROXWLRQVHWRIWKH KRPRJHQHRXVHTXDWLRQLQ([DPSOH

"11-*$"5*0/4 0' -*/&"3 4:45&.4
$ +RPRJHQHRXV6\VWHPLQ(FRQRPLFV WEB

7KHV\VWHPRIHTXDWLRQVLQYDULDEOHV PHQWLRQHGLQWKLVFKDSWHU·VLQWURGXFWLRQ LVQRZNQRZQDVD/HRQWLHI´LQSXW²RXWSXWµRU´SURGXFWLRQµ PRGHOñ 6HFWLRQ ZLOO H[DPLQHWKLVPRGHOLQPRUHGHWDLO ZKHQPRUHWKHRU\DQGEHWWHUQRWDWLRQDUHDYDLODEOH )RUQRZ ZHORRNDWDVLPSOHU´H[FKDQJHPRGHOµ DOVRGXHWR/HRQWLHI 6XSSRVHDQDWLRQ·VHFRQRP\LVGLYLGHGLQWRPDQ\VHFWRUV VXFKDVYDULRXVPDQX IDFWXULQJ FRPPXQLFDWLRQ HQWHUWDLQPHQW DQGVHUYLFHLQGXVWULHV 6XSSRVHWKDWIRUHDFK VHFWRUZHNQRZLWVWRWDORXWSXWIRURQH\HDUDQGZHNQRZH[DFWO\KRZWKLVRXWSXWLV GLYLGHGRU´H[FKDQJHGµDPRQJWKHRWKHUVHFWRUVRIWKHHFRQRP\ /HWWKHWRWDOGROODU YDOXHRIDVHFWRU·VRXWSXWEHFDOOHGWKH SULFH RIWKDWRXWSXW /HRQWLHISURYHGWKH IROORZLQJUHVXOW 7KHUHH[LVW HTXLOLEULXPSULFHV WKDWFDQEHDVVLJQHGWRWKHWRWDORXWSXWVRIWKH YDULRXVVHFWRUVLQVXFKDZD\WKDWWKHLQFRPHRIHDFKVHFWRUH[DFWO\EDODQFHVLWV H[SHQVHV 7KHIROORZLQJH[DPSOHVKRZVKRZWRÀQGWKHHTXLOLEULXPSULFHV

&9".1-& 6XSSRVHDQHFRQRP\FRQVLVWVRIWKH&RDO (OHFWULFSRZHU DQG6WHHO VHFWRUV DQGWKHRXWSXWRIHDFKVHFWRULVGLVWULEXWHGDPRQJWKHYDULRXVVHFWRUVDVVKRZQ LQ7DEOH RQSDJH ZKHUHWKHHQWULHVLQDFROXPQUHSUHVHQWWKHIUDFWLRQDOSDUWVRID VHFWRU·VWRWDORXWSXW 7KHVHFRQGFROXPQRI7DEOH IRULQVWDQFH VD\VWKDWWKHWRWDORXWSXWRIWKH(OHFWULF VHFWRULVGLYLGHGDVIROORZV WR&RDO WR6WHHO DQGWKHUHPDLQLQJWR (OHFWULF (OHFWULFWUHDWVWKLVDVDQH[SHQVHLWLQFXUVLQRUGHUWRRSHUDWHLWVEXVLQHVV 6LQFHDOORXWSXWPXVWEHWDNHQLQWRDFFRXQW WKHGHFLPDOIUDFWLRQVLQHDFKFROXPQPXVW VXPWR ñ 6HH:DVVLO\:/HRQWLHI ´,QSXW²2XWSXW(FRQRPLFVµ 6FLHQWLÀF$PHULFDQ 2FWREHU SS ²

$)"15&3

/LQHDU(TXDWLRQVLQ/LQHDU$OJHEUD .1

'HQRWHWKHSULFHVLH GROODUYDOXHV RIWKHWRWDODQQXDORXWSXWVRIWKH&RDO (OHFWULF DQG6WHHOVHFWRUVE\ p& p( DQG p6 UHVSHFWLYHO\ ,ISRVVLEOH ÀQGHTXLOLEULXP SULFHVWKDWPDNHHDFKVHFWRU·VLQFRPHPDWFKLWVH[SHQGLWXUHV

Electric

.4 Coal .6

.2

5"#-&

" 4JNQMF &DPOPNZ

'LVWULEXWLRQRI2XWSXWIURP

.5

Steel .6

&RDO

(OHFWULF

6WHHO

3XUFKDVHGE\

&RDO

(OHFWULF

6WHHO

.4

.2

40-65*0/ $ VHFWRUORRNVGRZQDFROXPQWRVHHZKHUHLWVRXWSXWJRHV DQGLWORRNV DFURVVDURZWRVHHZKDWLWQHHGVDVLQSXWV )RULQVWDQFH WKHÀUVWURZRI7DEOH VD\VWKDW&RDOUHFHLYHVDQGSD\VIRU RIWKH(OHFWULFRXWSXWDQGRIWKH6WHHO RXWSXW 6LQFHWKHUHVSHFWLYHYDOXHVRIWKHWRWDORXWSXWVDUH p( DQG p6 &RDOPXVWVSHQG :4p( GROODUVIRULWVVKDUHRI(OHFWULF·VRXWSXWDQG :6p6 IRULWVVKDUHRI6WHHO·VRXWSXW 7KXV&RDO·VWRWDOH[SHQVHVDUH :4p( C :6p6 7RPDNH&RDO·VLQFRPH p& HTXDOWRLWV H[SHQVHV ZHZDQW p& D :4p( C :6p6 7KHVHFRQGURZRIWKHH[FKDQJHWDEOHVKRZVWKDWWKH(OHFWULFVHFWRUVSHQGV :6p& IRUFRDO :1p( IRUHOHFWULFLW\ DQG :2p6 IRUVWHHO +HQFHWKHLQFRPHH[SHQVHUHTXLUHPHQW IRU(OHFWULFLV p( D :6p& C :1p( C :2p6 )LQDOO\ WKHWKLUGURZRIWKHH[FKDQJHWDEOHOHDGVWRWKHÀQDOUHTXLUHPHQW

p6 D :4p& C :5p( C :2p6

7RVROYHWKHV\VWHPRIHTXDWLRQV DQG PRYHDOOWKHXQNQRZQVWRWKHOHIW VLGHVRIWKHHTXDWLRQVDQGFRPELQHOLNHWHUPV >)RULQVWDQFH RQWKHOHIWVLGHRI ZULWH p( " :1p( DV :9p( @

p& " :4p( " :6p6 D 0 ":6p& C :9p( " :2p6 D 0 ":4p& " :5p( C :8p6 D 0 5RZUHGXFWLRQLVQH[W 2 1 ":4 ":6 4 ":6 :9 ":2 ":4 ":5 :8

)RUVLPSOLFLW\KHUH GHFLPDOVDUHURXQGHGWRWZRSODFHV 3 2 3 2 3 0 1 ":4 ":6 0 1 ":4 ":6 0 0 5 % 4 0 :66 ":56 0 5 % 4 0 :66 ":56 0 5 0 0 ":66 :56 0 0 0 0 0 2 3 2 3 1 ":4 ":6 0 1 0 ":94 0 % 4 0 1 ":85 0 5 % 4 0 1 ":85 0 5 0 0 0 0 0 0 0 0


7KHJHQHUDOVROXWLRQLV p& D :94p6 p( D :85p6 DQG p6 LVIUHH 7KHHTXLOLEULXPSULFH YHFWRUIRUWKHHFRQRP\KDVWKHIRUP 2 3 2 3 2 3 p& :94p6 :94 S D 4 p( 5 D 4 :85p6 5 D p6 4 :85 5 p6 p6 1

$Q\QRQQHJDWLYH FKRLFHIRU p6 UHVXOWVLQDFKRLFHRIHTXLOLEULXPSULFHV )RULQVWDQFH LIZHWDNH p6 WREHRUPLOOLRQ WKHQ p& D 94 DQG p( D 85 7KHLQFRPHVDQG H[SHQGLWXUHVRIHDFKVHFWRUZLOOEHHTXDOLIWKHRXWSXWRI&RDOLVSULFHGDWPLOOLRQ WKDWRI(OHFWULFDWPLOOLRQ DQGWKDWRI6WHHODWPLOOLRQ

%DODQFLQJ&KHPLFDO(TXDWLRQV &KHPLFDO HTXDWLRQV GHVFULEH WKH TXDQWLWLHV RI VXEVWDQFHV FRQVXPHG DQG SURGXFHG E\FKHPLFDOUHDFWLRQV )RULQVWDQFH ZKHQSURSDQHJDVEXUQV WKHSURSDQH&3 +8 FRPELQHVZLWKR[\JHQ22 WRIRUPFDUERQGLR[LGH&22 DQGZDWHU+2 2 DFFRUGLQJ WRDQHTXDWLRQRIWKHIRUP

.x1 /&3 +8 C .x2 /22 ! .x3 /&22 C .x4 /+2 2

7RÉDODQFHµWKLVHTXDWLRQ DFKHPLVWPXVWÀQGZKROHQXPEHUV x1 ; : : : ; x4 VXFKWKDWWKH WRWDOQXPEHUVRIFDUERQ& K\GURJHQ+ DQGR[\JHQ2 DWRPVRQWKHOHIWPDWFKWKH FRUUHVSRQGLQJQXPEHUVRIDWRPVRQWKHULJKWEHFDXVHDWRPVDUHQHLWKHUGHVWUR\HGQRU FUHDWHGLQWKHUHDFWLRQ $ V\VWHPDWLFPHWKRGIRUEDODQFLQJFKHPLFDOHTXDWLRQVLVWRVHWXSDYHFWRUHTXDWLRQ WKDWGHVFULEHVWKHQXPEHUVRIDWRPVRIHDFKW\SHSUHVHQWLQDUHDFWLRQ 6LQFHHTXDWLRQ LQYROYHVWKUHHW\SHVRIDWRPVFDUERQ K\GURJHQ DQGR[\JHQ FRQVWUXFWDYHFWRULQ R3 IRUHDFKUHDFWDQWDQGSURGXFWLQ WKDWOLVWVWKHQXPEHUVRI´DWRPVSHUPROHFXOHµ DVIROORZV 2 3 2 3 2 3 2 3 3 0 1 0 ! &DUERQ &3 +8W 4 8 5; 22W 4 0 5; &22W 4 0 5; +2 2W 4 2 5 ! +\GURJHQ 0 2 2 1 ! 2[\JHQ 7REDODQFHHTXDWLRQ WKHFRHIÀFLHQWV x1 ; : : : ; x4 PXVWVDWLVI\ 2 3 2 3 2 3 2 3 3 0 1 0 x1 4 8 5 C x2 4 0 5 D x3 4 0 5 C x4 4 2 5 0 2 2 1

7RVROYH PRYHDOOWKHWHUPVWRWKHOHIWFKDQJLQJWKHVLJQVLQWKHWKLUGDQGIRXUWK YHFWRUV 2 3 2 3 2 3 2 3 2 3 3 0 "1 0 0 x1 4 8 5 C x2 4 0 5 C x3 4 0 5 C x4 4 "2 5 D 4 0 5 0 2 "2 "1 0 5RZUHGXFWLRQRIWKHDXJPHQWHGPDWUL[IRUWKLVHTXDWLRQOHDGVWRWKHJHQHUDOVROXWLRQ

x1 D 14 x4 ; x2 D 54 x4 ; x3 D 34 x4 ; ZLWK x4 IUHH

6LQFHWKHFRHIÀFLHQWVLQDFKHPLFDOHTXDWLRQPXVWEHLQWHJHUV WDNH x4 D 4 LQZKLFK FDVH x1 D 1 x2 D 5 DQG x3 D 3 7KHEDODQFHGHTXDWLRQLV &3 +8 C 522 ! 3&22 C 4+2 2

7KHHTXDWLRQZRXOGDOVREHEDODQFHGLI IRUH[DPSOH HDFKFRHIÀFLHQWZHUHGRXEOHG )RU PRVWSXUSRVHV KRZHYHU FKHPLVWVSUHIHUWRXVHDEDODQFHGHTXDWLRQZKRVHFRHIÀFLHQWV DUHWKHVPDOOHVWSRVVLEOHZKROHQXPEHUV

$)"15&3


1HWZRUN)ORZ WEB

x1 30 x2 '*(63&

$ MXQFWLRQ RUQRGH

6\VWHPVRIOLQHDUHTXDWLRQVDULVHQDWXUDOO\ZKHQVFLHQWLVWV HQJLQHHUV RUHFRQRPLVWV VWXG\WKHÁRZRIVRPHTXDQWLW\WKURXJKDQHWZRUN )RULQVWDQFH XUEDQSODQQHUVDQG WUDIÀFHQJLQHHUVPRQLWRUWKHSDWWHUQRIWUDIÀFÁRZLQDJULGRIFLW\VWUHHWV (OHFWULFDO HQJLQHHUVFDOFXODWHFXUUHQWÁRZWKURXJKHOHFWULFDOFLUFXLWV $QGHFRQRPLVWVDQDO\]H WKHGLVWULEXWLRQRISURGXFWVIURPPDQXIDFWXUHUVWRFRQVXPHUVWKURXJKDQHWZRUNRI ZKROHVDOHUV DQG UHWDLOHUV )RU PDQ\ QHWZRUNV WKH V\VWHPV RI HTXDWLRQV LQYROYH KXQGUHGVRUHYHQWKRXVDQGVRIYDULDEOHVDQGHTXDWLRQV $ QHWZRUN FRQVLVWVRIDVHWRISRLQWVFDOOHG MXQFWLRQV RU QRGHV ZLWKOLQHVRUDUFV FDOOHG EUDQFKHV FRQQHFWLQJVRPHRUDOORIWKHMXQFWLRQV 7KHGLUHFWLRQRIÁRZLQHDFK EUDQFKLVLQGLFDWHG DQGWKHÁRZDPRXQWRUUDWH LVHLWKHUVKRZQRULVGHQRWHGE\D YDULDEOH 7KHEDVLFDVVXPSWLRQRIQHWZRUNÁRZLVWKDWWKHWRWDOÁRZLQWRWKHQHWZRUNHTXDOV WKHWRWDOÁRZRXWRIWKHQHWZRUNDQGWKDWWKHWRWDOÁRZLQWRDMXQFWLRQHTXDOVWKHWRWDO ÁRZRXWRIWKHMXQFWLRQ )RUH[DPSOH )LJ VKRZVXQLWVÁRZLQJLQWRDMXQFWLRQ WKURXJKRQHEUDQFK ZLWK x1 DQG x2 GHQRWLQJWKHÁRZVRXWRIWKHMXQFWLRQWKURXJKRWKHU EUDQFKHV 6LQFHWKHÁRZLV´FRQVHUYHGµDWHDFKMXQFWLRQ ZHPXVWKDYH x1 C x2 D 30 ,QDVLPLODUIDVKLRQ WKHÁRZDWHDFKMXQFWLRQLVGHVFULEHGE\DOLQHDUHTXDWLRQ 7KH SUREOHPRIQHWZRUNDQDO\VLVLVWRGHWHUPLQHWKHÁRZLQHDFKEUDQFKZKHQSDUWLDO LQIRUPDWLRQVXFKDVWKHÁRZLQWRDQGRXWRIWKHQHWZRUN LVNQRZQ

&9".1-& 7KHQHWZRUNLQ)LJ VKRZVWKHWUDIÀFÁRZLQYHKLFOHVSHUKRXU RYHUVHYHUDORQHZD\VWUHHWVLQGRZQWRZQ%DOWLPRUHGXULQJDW\SLFDOHDUO\DIWHUQRRQ 'HWHUPLQHWKHJHQHUDOÁRZSDWWHUQIRUWKHQHWZRUN 100

x3 Calvert St.

South St. N

300

Lombard St. B

C

x2 300

400

x4 x5

Pratt St. A

D

600

x1 500

'*(63& %DOWLPRUHVWUHHWV

40-65*0/ :ULWHHTXDWLRQVWKDWGHVFULEHWKHÁRZ DQGWKHQÀQGWKHJHQHUDOVROXWLRQ RIWKHV\VWHP /DEHOWKHVWUHHWLQWHUVHFWLRQVMXQFWLRQV DQGWKHXQNQRZQÁRZVLQWKH EUDQFKHV DVVKRZQLQ)LJ $WHDFKLQWHUVHFWLRQ VHWWKHÁRZLQHTXDOWRWKHÁRZRXW ,QWHUVHFWLRQ $ % & '

)ORZLQ

)ORZRXW

300 C 500 D x1 C x2

x2 C x4 D 300 C x3

100 C 400 D x4 C x5 x1 C x5 D 600


$OVR WKHWRWDOÁRZLQWRWKHQHWZRUN .500 C 300 C 100 C 400/ HTXDOVWKHWRWDOÁRZ RXWRIWKHQHWZRUN .300 C x3 C 600/ ZKLFKVLPSOLÀHVWR x3 D 400 &RPELQHWKLV HTXDWLRQZLWKDUHDUUDQJHPHQWRIWKHÀUVWIRXUHTXDWLRQVWRREWDLQWKHIROORZLQJV\VWHP RIHTXDWLRQV x1 C x2 D 800 D 300 x2 " x3 C x4 x4 C x5 D 500 C x5 D 600 x1 D 400 x3 5RZUHGXFWLRQRIWKHDVVRFLDWHGDXJPHQWHGPDWUL[OHDGVWR

x1

C x5 D 600 " x5 D 200 x2 D 400 x3 x4 C x5 D 500

7KHJHQHUDOÁRZSDWWHUQIRUWKHQHWZRUNLVGHVFULEHGE\ 8 ˆ x1 D 600 " x5 ˆ ˆ ˆ ˆ ˆ
$ QHJDWLYHÁRZLQDQHWZRUNEUDQFKFRUUHVSRQGVWRÁRZLQWKHGLUHFWLRQRSSRVLWH WRWKDWVKRZQRQWKHPRGHO 6LQFHWKHVWUHHWVLQWKLVSUREOHPDUHRQHZD\ QRQHRIWKH YDULDEOHVKHUHFDQEHQHJDWLYH 7KLVIDFWOHDGVWRFHUWDLQOLPLWDWLRQVRQWKHSRVVLEOH YDOXHVRIWKHYDULDEOHV )RULQVWDQFH x5 ' 500 EHFDXVH x4 FDQQRWEHQHJDWLYH 2WKHU FRQVWUDLQWVRQWKHYDULDEOHVDUHFRQVLGHUHGLQ3UDFWLFH3UREOHP

13"$5*$& 130#-&.4 6XSSRVHDQHFRQRP\KDVWKUHHVHFWRUV $JULFXOWXUH 0LQLQJ DQG0DQXIDFWXULQJ $JULFXOWXUHVHOOVRILWVRXWSXWWR0LQLQJDQGWR0DQXIDFWXULQJ DQGUHWDLQV WKHUHVW 0LQLQJVHOOVRILWVRXWSXWWR$JULFXOWXUHDQGWR0DQXIDFWXULQJ DQGUHWDLQVWKHUHVW 0DQXIDFWXULQJVHOOVRILWVRXWSXWWR$JULFXOWXUHDQGWR 0LQLQJ DQGUHWDLQVWKHUHVW 'HWHUPLQHWKHH[FKDQJHWDEOHIRUWKLVHFRQRP\ ZKHUH WKHFROXPQVGHVFULEHKRZWKHRXWSXWRIHDFKVHFWRULVH[FKDQJHGDPRQJWKHWKUHH VHFWRUV &RQVLGHUWKHQHWZRUNÁRZVWXGLHGLQ([DPSOH 'HWHUPLQHWKHSRVVLEOHUDQJHRI YDOXHVRI x1 DQG x2 >+LQW 7KHH[DPSOHVKRZHGWKDW x5 ' 500 :KDWGRHVWKLV LPSO\DERXW x1 DQG x2 " $OVR XVHWKHIDFWWKDW x5 & 0@

$)"15&3


&9&3$*4&4 6XSSRVHDQHFRQRP\KDVRQO\WZRVHFWRUV *RRGVDQG6HU YLFHV (DFK\HDU *RRGVVHOOVRILWVRXWSXWWR6HUYLFHV DQGNHHSVWKHUHVW ZKLOH6HUYLFHVVHOOVRILWVRXWSXWWR *RRGVDQGUHWDLQVWKHUHVW )LQGHTXLOLEULXPSULFHVIRUWKH DQQXDORXWSXWVRIWKH*RRGVDQG6HUYLFHVVHFWRUVWKDWPDNH HDFKVHFWRU·VLQFRPHPDWFKLWVH[SHQGLWXUHV Goods

Services .8

.2

.3 .7

)LQGDQRWKHUVHWRIHTXLOLEULXPSULFHVIRUWKHHFRQRP\LQ ([DPSOH 6XSSRVH WKH VDPH HFRQRP\ XVHG -DSDQHVH \HQLQVWHDGRIGROODUVWRPHDVXUHWKHYDOXHVRIWKHYDULRXV VHFWRUV·RXWSXWV :RXOGWKLVFKDQJHWKHSUREOHPLQDQ\ZD\" 'LVFXVV &RQVLGHUDQHFRQRP\ZLWKWKUHHVHFWRUV )XHOVDQG3RZHU 0DQXIDFWXULQJ DQG6HUYLFHV )XHOVDQG3RZHUVHOOV RILWVRXWSXWWR0DQXIDFWXULQJ WR6HUYLFHV DQGUHWDLQV WKHUHVW 0DQXIDFWXULQJVHOOVRILWVRXWSXWWR)XHOVDQG 3RZHU WR6HUYLFHV DQGUHWDLQVWKHUHVW 6HUYLFHVVHOOV WR)XHOVDQG3RZHU WR0DQXIDFWXULQJ DQGUHWDLQV WKHUHVW D &RQVWUXFWWKHH[FKDQJHWDEOHIRUWKLVHFRQRP\ E 'HYHORSDV\VWHPRIHTXDWLRQVWKDWOHDGVWRSULFHVDW ZKLFKHDFKVHFWRU·VLQFRPHPDWFKHVLWVH[SHQVHV 7KHQ ZULWHWKHDXJPHQWHGPDWUL[WKDWFDQEHURZUHGXFHGWR ÀQGWKHVHSULFHV F >0@ )LQGDVHWRIHTXLOLEULXPSULFHVZKHQWKHSULFHIRU WKH6HUYLFHVRXWSXWLVXQLWV 6XSSRVH DQ HFRQRP\ KDV IRXU VHFWRUV 0LQLQJ /XPEHU (QHUJ\ DQG7UDQVSRUWDWLRQ 0LQLQJVHOOVRILWVRXWSXW WR/XPEHU WR(QHUJ\ DQGUHWDLQVWKHUHVW /XPEHU VHOOVRILWVRXWSXWWR0LQLQJ WR(QHUJ\ WR 7UDQVSRUWDWLRQ DQGUHWDLQVWKHUHVW (QHUJ\VHOOVRILWV RXWSXWWR0LQLQJ WR/XPEHU WR7UDQVSRUWDWLRQ DQGUHWDLQVWKHUHVW 7UDQVSRUWDWLRQVHOOVRILWVRXWSXWWR 0LQLQJ WR/XPEHU WR(QHUJ\ DQGUHWDLQVWKHUHVW D &RQVWUXFWWKHH[FKDQJHWDEOHIRUWKLVHFRQRP\ E >0@ )LQGDVHWRIHTXLOLEULXPSULFHVIRUWKHHFRQRP\ $QHFRQRP\KDVIRXUVHFWRUV $JULFXOWXUH 0DQXIDFWXULQJ 6HUYLFHV DQG7UDQVSRUWDWLRQ $JULFXOWXUHVHOOVRILWV RXWSXWWR0DQXIDFWXULQJ WR6HUYLFHV WR7UDQV SRUWDWLRQ DQGUHWDLQVWKHUHVW 0DQXIDFWXULQJVHOOVRILWV RXWSXWWR$JULFXOWXUH WR6HUYLFHV WR7UDQVSRUWD WLRQ DQGUHWDLQVWKHUHVW 6HUYLFHVVHOOVRILWVRXWSXWWR $JULFXOWXUH WR0DQXIDFWXULQJ WR7UDQVSRUWDWLRQ

DQGUHWDLQVWKHUHVW 7UDQVSRUWDWLRQVHOOVRILWVRXWSXW WR$JULFXOWXUH WR0DQXIDFWXULQJ WR6HUYLFHV DQG UHWDLQVWKHUHVW D &RQVWUXFWWKHH[FKDQJHWDEOHIRUWKLVHFRQRP\ E >0@ )LQGDVHWRIHTXLOLEULXPSULFHVIRUWKHHFRQRP\LI WKHYDOXHRI7UDQVSRUWDWLRQLVSHUXQLW F 7KH6HUYLFHVVHFWRUODXQFKHVDVXFFHVVIXO´HDWIDUPIUHVKµ FDPSDLJQ DQGLQFUHDVHVLWVVKDUHRIWKHRXWSXWIURPWKH $JULFXOWXUDOVHFWRUWR ZKHUHDVWKHVKDUHRI$JUL FXOWXUDOSURGXFWLRQJRLQJWR0DQXIDFWXULQJIDOOVWR &RQVWUXFWWKHH[FKDQJHWDEOHIRUWKLVQHZHFRQRP\ G >0@ )LQGDVHWRIHTXLOLEULXPSULFHVIRUWKLVQHZHFRQRP\ LIWKHYDOXHRI7UDQVSRUWDWLRQ LVVWLOO SHUXQLW :KDWHIIHFWKDVWKH´HDWIDUPIUHVKµFDPSDLJQKDGRQWKH HTXLOLEULXPSULFHVIRUWKHVHFWRUVLQWKLVHFRQRP\" %DODQFHWKHFKHPLFDOHTXDWLRQVLQ([HUFLVHV²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ÀGHUHDFWVYLROHQWO\ZLWKZDWHUWRIRUPERULFDFLG DQGK\GURJHQVXOÀGHJDVWKHVPHOORIURWWHQHJJV 7KH XQEDODQFHGHTXDWLRQLV %2 63 C +2 2 ! +3 %23 C +2 6 >0@ ,ISRVVLEOH XVHH[DFWDULWKPHWLFRUDUDWLRQDOIRUPDWIRU FDOFXODWLRQVLQEDODQFLQJWKHIROORZLQJFKHPLFDOUHDFWLRQ 3E16 C &U0Q2 28 ! 3E3 24 C &U2 23 C 0Q22 C 12 >0@ 7KHFKHPLFDOUHDFWLRQEHORZFDQEHXVHGLQVRPHLQ GXVWULDOSURFHVVHV VXFKDVWKHSURGXFWLRQRIDUVHQH$V+3 8VHH[DFWDULWKPHWLFRUDUDWLRQDOIRUPDWIRUFDOFXODWLRQVWR EDODQFHWKLVHTXDWLRQ 0Q6 C $V2 &U10 235 C +2 624 ! +0Q24 C $V+3 C &U63 212 C +2 2

/LQHDU,QGHSHQGHQFH

)LQGWKHJHQHUDOÁRZSDWWHUQRIWKHQHWZRUNVKRZQLQWKH ÀJXUH $VVXPLQJWKDWWKHÁRZVDUHDOOQRQQHJDWLYH ZKDWLV WKHVPDOOHVWSRVVLEOHYDOXHIRU x4 " B

x2 A

x1

VKRZQLQWKHÀJXUH )ORZUDWHVDUHLQFDUVPLQXWH E 'HVFULEHWKHJHQHUDOWUDIÀFSDWWHUQZKHQWKHURDGZKRVH ÁRZLV x5 LVFORVHG F :KHQ x5 D 0 ZKDWLVWKHPLQLPXPYDOXHRI x4 "

100 x3

x4

90

D )LQGWKHJHQHUDOÁRZSDWWHUQRIWKHQHWZRUNVKRZQLQWKH ÀJXUH E $VVXPLQJWKDWWKHÁRZPXVWEHLQWKHGLUHFWLRQVLQGL FDWHG ÀQGWKHPLQLPXPÁRZVLQWKHEUDQFKHVGHQRWHG E\ x2 x3 x4 DQG x5

80 60

A

40

x2

x1 E

x3

B

x5

C x6

x4

D

D

100 x2

C

x3

90

,QWHUVHFWLRQVLQ(QJODQGDUHRIWHQFRQVWUXFWHGDVRQHZD\ ÚRXQGDERXWVµ VXFKDVWKHRQHVKRZQLQWKHÀJXUH $VVXPH WKDWWUDIÀFPXVWWUDYHOLQWKHGLUHFWLRQVVKRZQ )LQGWKH JHQHUDO VROXWLRQ RI WKH QHWZRUN ÁRZ )LQG WKH VPDOOHVW SRVVLEOHYDOXHIRU x6

100 60 90

x1

A

B

x6

80

x2

F x5

20

B

x4

x5

80

30

x1

A

80

C

40

C

x4 E

D

70 100

x3

80 90

D )LQGWKHJHQHUDOWUDIÀFSDWWHUQRIWKHIUHHZD\QHWZRUN

40-65*0/4 50 13"$5*$& 130#-&.4 :ULWHWKHSHUFHQWDJHVDVGHFLPDOV 6LQFHDOORXWSXWPXVWEHWDNHQLQWRDFFRXQW HDFK FROXPQPXVWVXPWR 7KLVIDFWKHOSVWRÀOOLQDQ\PLVVLQJHQWULHV 'LVWULEXWLRQRI2XWSXWIURP $JULFXOWXUH

0LQLQJ

0DQXIDFWXULQJ

3XUFKDVHGE\

$JULFXOWXUH

0LQLQJ

0DQXIDFWXULQJ

6LQFH x5 ' 500 WKH HTXDWLRQV ' DQG $ IRU x1 DQG x2 LPSO\ WKDW x1 & 100 DQG x2 ' 700 7KHIDFWWKDW x5 & 0 LPSOLHVWKDW x1 ' 600 DQG x2 & 200 6R 100 ' x1 ' 600 DQG 200 ' x2 ' 700

-*/&"3 */%&1&/%&/$& 7KHKRPRJHQHRXVHTXDWLRQVLQ6HFWLRQ FDQEHVWXGLHGIURPDGLIIHUHQWSHUVSHFWLYH E\ZULWLQJWKHPDVYHFWRUHTXDWLRQV ,QWKLVZD\ WKHIRFXVVKLIWVIURPWKHXQNQRZQ VROXWLRQVRI A[ D WRWKHYHFWRUVWKDWDSSHDULQWKHYHFWRUHTXDWLRQV

$)"15&3


)RULQVWDQFH FRQVLGHUWKHHTXDWLRQ 2 3 2 3 2 3 2 3 1 4 2 0 x1 4 2 5 C x2 4 5 5 C x3 4 1 5 D 4 0 5 3 6 0 0

7KLV HTXDWLRQ KDV D WULYLDO VROXWLRQ RI FRXUVH ZKHUH x1 D x2 D x3 D 0 6HFWLRQ WKHPDLQLVVXHLVZKHWKHUWKHWULYLDOVROXWLRQLVWKH RQO\RQH

%&'*/*5*0/

$V LQ

$QLQGH[HGVHWRIYHFWRUV fY1 ; : : : ; Yp g LQ Rn LVVDLGWREH OLQHDUO\LQGHSHQGHQW LIWKHYHFWRUHTXDWLRQ x1 Y1 C x2 Y2 C ! ! ! C xp Yp D

KDVRQO\WKHWULYLDOVROXWLRQ 7KHVHW fY1 ; : : : ; Yp g LVVDLGWREH OLQHDUO\GHSHQGHQW LIWKHUHH[LVWZHLJKWV c1 ; : : : ; cp QRWDOO]HUR VXFKWKDW

c1 Y1 C c2 Y2 C ! ! ! C cp Yp D

(TXDWLRQ LVFDOOHGD OLQHDUGHSHQGHQFHUHODWLRQ DPRQJ Y1 ; : : : ; Yp ZKHQWKH ZHLJKWVDUHQRWDOO]HUR $QLQGH[HGVHWLVOLQHDUO\GHSHQGHQWLIDQGRQO\LILWLVQRW OLQHDUO\LQGHSHQGHQW )RUEUHYLW\ ZHPD\VD\WKDW Y1 ; : : : ; Yp DUHOLQHDUO\GHSHQGHQW ZKHQ ZH PHDQ WKDW fY1 ; : : : ; Yp g LV D OLQHDUO\ GHSHQGHQW VHW :H XVH DQDORJRXV WHUPLQRORJ\IRUOLQHDUO\LQGHSHQGHQWVHWV 2 3 2 3 2 3 1 4 2 &9".1-& /HW Y1 D 4 2 5 Y2 D 4 5 5 DQG Y3 D 4 1 5 3 6 0 D 'HWHUPLQHLIWKHVHW fY1 ; Y2 ; Y3 g LVOLQHDUO\LQGHSHQGHQW E ,ISRVVLEOH ÀQGDOLQHDUGHSHQGHQFHUHODWLRQDPRQJ Y1 Y2 DQG Y3

40-65*0/ D :HPXVWGHWHUPLQHLIWKHUHLVDQRQWULYLDOVROXWLRQRIHTXDWLRQ DERYH 5RZRSHU DWLRQVRQWKHDVVRFLDWHGDXJPHQWHGPDWUL[VKRZWKDW 2 3 2 3 1 4 2 0 1 4 2 0 42 5 1 0 5 % 4 0 "3 "3 0 5 3 6 0 0 0 0 0 0

&OHDUO\ x1 DQG x2 DUHEDVLFYDULDEOHV DQG x3 LVIUHH (DFKQRQ]HURYDOXHRI x3 GHWHUPLQHVDQRQWULYLDOVROXWLRQRI +HQFH Y1 ; Y2 ; Y3 DUHOLQHDUO\GHSHQGHQWDQG QRWOLQHDUO\LQGHSHQGHQW E 7RÀQGDOLQHDUGHSHQGHQFHUHODWLRQDPRQJ Y1 Y2 DQG Y3 FRPSOHWHO\URZUHGXFH WKHDXJPHQWHGPDWUL[DQGZULWHWKHQHZV\VWHP 2 3 x1 " 2x3 D 0 1 0 "2 0 40 1 1 05 x2 C x3 D 0 0 0 0 0 0D0

7KXV x1 D 2x3 x2 D "x3 DQG x3 LVIUHH &KRRVHDQ\QRQ]HURYDOXHIRU x3 ³VD\ x3 D 5 7KHQ x1 D 10 DQG x2 D "5 6XEVWLWXWHWKHVHYDOXHVLQWRHTXDWLRQ DQG REWDLQ 10Y1 " 5Y2 C 5Y3 D

7KLVLVRQHRXWRILQÀQLWHO\PDQ\ SRVVLEOHOLQHDUGHSHQGHQFHUHODWLRQVDPRQJ Y1 Y2 DQG Y3

/LQHDU,QGHSHQGHQFH

/LQHDU,QGHSHQGHQFHRI0DWUL[&ROXPQV 6XSSRVHWKDWZHEHJLQZLWKDPDWUL[ A D Œ D1 ! ! ! Dn ! LQVWHDGRIDVHWRIYHFWRUV 7KH PDWUL[HTXDWLRQ A[ D FDQEHZULWWHQDV

x1 D1 C x2 D2 C ! ! ! C xn Dn D (DFKOLQHDUGHSHQGHQFHUHODWLRQDPRQJWKHFROXPQVRI A FRUUHVSRQGVWRDQRQWULYLDO VROXWLRQRI A[ D 7KXVZHKDYHWKHIROORZLQJLPSRUWDQWIDFW 7KHFROXPQVRIDPDWUL[ A DUHOLQHDUO\LQGHSHQGHQWLIDQGRQO\LIWKHHTXDWLRQ A[ D KDV RQO\ WKHWULYLDOVROXWLRQ 2

0 &9".1-& 'HWHUPLQH LI WKH FROXPQV RI WKH PDWUL[ A D 4 1 5 OLQHDUO\LQGHSHQGHQW

3 1 4 2 "1 5 DUH 8 0

40-65*0/ 7RVWXG\ A[ D URZUHGXFHWKHDXJPHQWHGPDWUL[ 2 3 2 3 2 0 1 4 0 1 2 "1 0 1 2 "1 41 2 "1 0 5 % 4 0 1 4 0 5 % 4 0 1 4 5 8 0 0 0 "2 5 0 0 0 13

3 0 05 0

$WWKLVSRLQW LWLVFOHDUWKDWWKHUHDUHWKUHHEDVLFYDULDEOHVDQGQRIUHHYDULDEOHV 6R WKHHTXDWLRQ A[ D KDVRQO\WKHWULYLDOVROXWLRQ DQGWKHFROXPQVRI A DUHOLQHDUO\ LQGHSHQGHQW

6HWVRI2QHRU7ZR9HFWRUV $ VHWFRQWDLQLQJRQO\RQHYHFWRU³VD\ Y³LVOLQHDUO\LQGHSHQGHQWLIDQGRQO\LI Y LV QRWWKH]HURYHFWRU 7KLVLVEHFDXVHWKHYHFWRUHTXDWLRQ x1 Y D KDVRQO\WKHWULYLDO VROXWLRQZKHQ Y ¤ 7KH]HURYHFWRULVOLQHDUO\GHSHQGHQWEHFDXVH x1 D KDVPDQ\ QRQWULYLDOVROXWLRQV 7KHQH[WH[DPSOHZLOOH[SODLQWKHQDWXUHRIDOLQHDUO\GHSHQGHQWVHWRIWZRYHFWRUV

&9".1-& 'HWHUPLQHLIWKHIROORZLQJVHWVRIYHFWRUVDUHOLQHDUO\LQGHSHQGHQW # $ # $ 3 6 D Y1 D Y2 D 1 2

# $ # $ 3 6 E Y1 D Y2 D 2 2

40-65*0/ D 1RWLFHWKDW Y2 LVDPXOWLSOHRI Y1 QDPHO\ Y2 D 2Y1 +HQFH "2Y1 C Y2 D ZKLFK VKRZVWKDW fY1 ; Y2 g LVOLQHDUO\GHSHQGHQW E 7KHYHFWRUV Y1 DQG Y2 DUHFHUWDLQO\ QRW PXOWLSOHVRIRQHDQRWKHU &RXOGWKH\EH OLQHDUO\GHSHQGHQW" 6XSSRVH c DQG d VDWLVI\

c Y1 C d Y2 D ,I c ¤ 0 WKHQZHFDQVROYHIRU Y1 LQWHUPVRI Y2 QDPHO\ Y1 D ."d=c/Y2 7KLV UHVXOWLVLPSRVVLEOHEHFDXVH Y1 LV QRW DPXOWLSOHRI Y2 6R c PXVWEH]HUR 6LPLODUO\ d PXVWDOVREH]HUR 7KXV fY1 ; Y2 g LVDOLQHDUO\LQGHSHQGHQWVHW

$)"15&3


7KHDUJXPHQWVLQ([DPSOHVKRZWKDW\RXFDQDOZD\VGHFLGH E\LQVSHFWLRQ ZKHQD VHWRIWZRYHFWRUVLVOLQHDUO\GHSHQGHQW 5RZRSHUDWLRQVDUHXQQHFHVVDU\ 6LPSO\FKHFN ZKHWKHUDWOHDVWRQHRIWKHYHFWRUVLVDVFDODUWLPHVWKHRWKHU 7KHWHVWDSSOLHVRQO\WR VHWVRI WZR YHFWRUV

x2 (6, 2) (3, 1) x1

$ VHWRIWZRYHFWRUV fY1 ; Y2 g LVOLQHDUO\GHSHQGHQWLIDWOHDVWRQHRIWKHYHFWRUVLV DPXOWLSOHRIWKHRWKHU 7KHVHWLVOLQHDUO\LQGHSHQGHQWLIDQGRQO\LIQHLWKHURIWKH YHFWRUVLVDPXOWLSOHRIWKHRWKHU

Linearly dependent x2 (3, 2)

(6, 2)

,QJHRPHWULFWHUPV WZRYHFWRUVDUHOLQHDUO\GHSHQGHQWLIDQGRQO\LIWKH\OLHRQWKH VDPHOLQHWKURXJKWKHRULJLQ )LJXUHVKRZVWKHYHFWRUVIURP([DPSOH x1 Linearly independent '*(63&

6HWVRI7ZRRU0RUH9HFWRUV 7KHSURRIRIWKHQH[WWKHRUHPLVVLPLODUWRWKHVROXWLRQRI([DPSOH 'HWDLOVDUHJLYHQ DWWKHHQGRIWKLVVHFWLRQ

5)&03&.

$IBSBDUFSJ[BUJPO PG -JOFBSMZ %FQFOEFOU 4FUT $QLQGH[HGVHW S D fY1 ; : : : ; Yp g RIWZRRUPRUHYHFWRUVLVOLQHDUO\GHSHQGHQWLI DQGRQO\LIDWOHDVWRQHRIWKHYHFWRUVLQ S LVDOLQHDUFRPELQDWLRQRIWKHRWKHUV ,Q IDFW LI S LVOLQHDUO\GHSHQGHQWDQG Y1 ¤ WKHQVRPH Yj ZLWK j > 1 LVDOLQHDU FRPELQDWLRQRIWKHSUHFHGLQJYHFWRUV Y1 ; : : : ; Yj #1 :DUQLQJ 7KHRUHPGRHV QRW VD\WKDW HYHU\ YHFWRULQDOLQHDUO\GHSHQGHQWVHWLVD OLQHDUFRPELQDWLRQRIWKHSUHFHGLQJYHFWRUV $ YHFWRULQDOLQHDUO\GHSHQGHQWVHWPD\ IDLOWREHDOLQHDUFRPELQDWLRQRIWKHRWKHUYHFWRUV 6HH3UDFWLFH3UREOHP 2 3 2 3 3 1 &9".1-& /HW X D 4 1 5 DQG Y D 4 6 5 'HVFULEHWKHVHWVSDQQHGE\ X DQG Y 0 0 DQGH[SODLQZK\DYHFWRU Z LVLQ 6SDQ fX; Yg LIDQGRQO\LI fX; Y; Zg LVOLQHDUO\GHSHQGHQW

40-65*0/ 7KHYHFWRUV X DQG Y DUHOLQHDUO\LQGHSHQGHQWEHFDXVHQHLWKHUYHFWRULVD PXOWLSOHRIWKHRWKHU DQGVRWKH\VSDQDSODQHLQ R3 6HH6HFWLRQ ,QIDFW 6SDQ fX; Yg LVWKH x1 x2 SODQHZLWK x3 D 0/ ,I Z LVDOLQHDUFRPELQDWLRQRI X DQG Y WKHQ fX; Y; Zg LVOLQHDUO\GHSHQGHQW E\7KHRUHP &RQYHUVHO\ VXSSRVHWKDW fX; Y; Zg LVOLQHDUO\GHSHQGHQW %\7KHRUHP VRPHYHFWRULQ fX; Y; Zg LVDOLQHDUFRPELQDWLRQ RIWKHSUHFHGLQJYHFWRUVVLQFH X ¤ / 7KDWYHFWRUPXVWEH Z VLQFH Y LVQRWDPXOWLSOH RI X 6R Z LVLQ 6SDQ fX; Yg 6HH)LJ x3

x1

u

x3

v

x2

w

Linearly dependent, w in Span{u, v} '*(63& /LQHDUGHSHQGHQFHLQ R3

x1

u

w

v

Linearly independent, w not in Span{u, v}

x2

/LQHDU,QGHSHQGHQFH

([DPSOHJHQHUDOL]HVWRDQ\VHW fX; Y; Zg LQ R3 ZLWK X DQG Y OLQHDUO\LQGHSHQGHQW 7KHVHW fX; Y; Zg ZLOOEHOLQHDUO\GHSHQGHQWLIDQGRQO\LI Z LVLQWKHSODQHVSDQQHGE\ X DQG Y 7KHQH[WWZRWKHRUHPVGHVFULEHVSHFLDOFDVHVLQZKLFKWKHOLQHDUGHSHQGHQFHRID VHWLVDXWRPDWLF 0RUHRYHU 7KHRUHPZLOOEHDNH\UHVXOWIRUZRUNLQODWHUFKDSWHUV

5)&03&. p

*

n *

*

* * *

* * *

* * *

* * *

'*(63&

,I p > n WKHFROXPQVDUHOLQHDUO\ GHSHQGHQW x2

,IDVHWFRQWDLQVPRUHYHFWRUVWKDQWKHUHDUHHQWULHVLQHDFKYHFWRU WKHQWKHVHW LVOLQHDUO\GHSHQGHQW 7KDWLV DQ\VHW fY1 ; : : : ; Yp g LQ Rn LVOLQHDUO\GHSHQGHQWLI p > n

1300' /HW A D Œ Y1 ! ! ! Yp ! 7KHQ A LV n # p DQGWKHHTXDWLRQ A[ D FRUUH VSRQGVWRDV\VWHPRI n HTXDWLRQVLQ p XQNQRZQV ,I p > n WKHUHDUHPRUHYDULDEOHV WKDQHTXDWLRQV VRWKHUHPXVWEHDIUHHYDULDEOH +HQFH A[ D KDVDQRQWULYLDOVROXWLRQ DQGWKHFROXPQVRI A DUHOLQHDUO\GHSHQGHQW 6HH)LJ IRUDPDWUL[YHUVLRQRIWKLV WKHRUHP :DUQLQJ 7KHRUHPVD\VQRWKLQJDERXWWKHFDVHLQZKLFKWKHQXPEHURIYHFWRUVLQ WKHVHWGRHV QRW H[FHHGWKHQXPEHURIHQWULHVLQHDFKYHFWRU

(–2, 2) (2, 1) x1 (4, –1)

'*(63&

$ OLQHDUO\GHSHQGHQWVHWLQ R2

5)&03&.

# $ # $ # $ 2 4 "2 &9".1-& 7KHYHFWRUV DUHOLQHDUO\GHSHQGHQWE\7KHRUHP 1 "1 2 EHFDXVHWKHUHDUHWKUHHYHFWRUVLQWKHVHWDQGWKHUHDUHRQO\WZRHQWULHVLQHDFKYHFWRU 1RWLFH KRZHYHU WKDWQRQHRIWKHYHFWRUVLVDPXOWLSOHRIRQHRIWKHRWKHUYHFWRUV 6HH )LJ

,IDVHW S D fY1 ; : : : ; Yp g LQ Rn FRQWDLQVWKH]HURYHFWRU WKHQWKHVHWLVOLQHDUO\ GHSHQGHQW

1300' %\UHQXPEHULQJWKHYHFWRUV ZHPD\VXSSRVH Y1 D 7KHQWKHHTXDWLRQ 1Y1 C 0Y2 C ! ! ! C 0Yp D VKRZVWKDW S LVOLQHDUO\GHSHQGHQW

&9".1-& 'HWHUPLQHE\LQVSHFWLRQLIWKHJLYHQVHWLVOLQHDUO\GHSHQGHQW 2 3 2 3 2 3 2 3 1 2 3 4 D 4 7 5 4 0 5 4 1 5 4 1 5 6 9 5 8

40-65*0/

2 3 2 3 2 3 2 0 1 E 4 3 5 4 0 5 4 1 5 5 0 8

2

3 2 3 "2 3 6 4 7 6 "6 7 7 6 7 F 6 4 6 5 4 "9 5 10 15

D 7KHVHWFRQWDLQVIRXUYHFWRUV HDFKRIZKLFKKDVRQO\WKUHHHQWULHV 6RWKHVHWLV OLQHDUO\GHSHQGHQWE\7KHRUHP E 7KHRUHPGRHVQRWDSSO\KHUHEHFDXVHWKHQXPEHURIYHFWRUVGRHVQRWH[FHHGWKH QXPEHURIHQWULHVLQHDFKYHFWRU 6LQFHWKH]HURYHFWRULVLQWKHVHW WKHVHWLVOLQHDUO\ GHSHQGHQWE\7KHRUHP F &RPSDUHWKHFRUUHVSRQGLQJHQWULHVRIWKHWZRYHFWRUV 7KHVHFRQGYHFWRUVHHPVWR EH "3=2 WLPHVWKHÀUVWYHFWRU 7KLVUHODWLRQKROGVIRUWKHÀUVWWKUHHSDLUVRIHQWULHV EXWIDLOVIRUWKHIRXUWKSDLU 7KXVQHLWKHURIWKHYHFWRUVLVDPXOWLSOHRIWKHRWKHU DQG KHQFHWKH\DUHOLQHDUO\LQGHSHQGHQW

$)"15&3

SG


.BTUFSJOH -JOFBS *OEFQFOEFODF m

,Q JHQHUDO \RX VKRXOG UHDG D VHFWLRQ WKRURXJKO\ VHYHUDO WLPHV WR DEVRUE DQ LPSRUWDQWFRQFHSWVXFKDVOLQHDULQGHSHQGHQFH 7KHQRWHVLQWKH 6WXG\*XLGH IRU WKLVVHFWLRQZLOOKHOS\RXOHDUQWRIRUPPHQWDOLPDJHVRINH\LGHDVLQOLQHDUDOJHEUD )RULQVWDQFH WKHIROORZLQJSURRILVZRUWKUHDGLQJFDUHIXOO\EHFDXVHLWVKRZVKRZWKH GHÀQLWLRQRIOLQHDULQGHSHQGHQFHFDQEH XVHG

1300' 0' 5)&03&. $IBSBDUFSJ[BUJPO PG -JOFBSMZ %FQFOEFOU 4FUT

,IVRPH Yj LQ S HTXDOVDOLQHDUFRPELQDWLRQRIWKHRWKHUYHFWRUV WKHQ Yj FDQEH VXEWUDFWHGIURPERWKVLGHVRIWKHHTXDWLRQ SURGXFLQJDOLQHDUGHSHQGHQFHUHODWLRQ ZLWKDQRQ]HURZHLJKW ."1/ RQ Yj >)RULQVWDQFH LI Y1 D c2 Y2 C c3 Y3 WKHQ D ."1/Y1 C c2 Y2 C c3 Y3 C 0Y4 C ! ! ! C 0Yp @ 7KXV S LVOLQHDUO\GHSHQGHQW &RQYHUVHO\ VXSSRVH S LVOLQHDUO\GHSHQGHQW ,I Y1 LV]HUR WKHQLWLVDWULYLDO OLQHDUFRPELQDWLRQRIWKHRWKHUYHFWRUVLQ S 2WKHUZLVH Y1 ¤ DQGWKHUHH[LVWZHLJKWV c1 ; : : : ; cp QRWDOO]HUR VXFKWKDW c1 Y1 C c2 Y2 C ! ! ! C cp Yp D

/HW j EHWKHODUJHVWVXEVFULSWIRUZKLFK cj ¤ 0 ,I j D 1 WKHQ c1 Y1 D ZKLFKLV LPSRVVLEOHEHFDXVH Y1 ¤ 6R j > 1 DQG

c1 Y1 C ! ! ! C cj Yj C 0Yj C1 C ! ! ! C 0Yp D cj Yj D "c1 Y1 " ! ! ! " cj #1 Yj #1 ) * ) * cj #1 c1 Yj D " Y1 C ! ! ! C " Yj #1 cj cj

13"$5*$& 130#-&.4 2 3 2 3 2 3 2 3 3 "6 0 3 /HW X D 4 2 5 Y D 4 1 5 Z D 4 "5 5 DQG ] D 4 7 5 "4 7 2 "5

$UHWKHVHWV fX; Yg; fX; Zg; fX; ]g; fY; Zg; fY; ]g DQG fZ; ]g HDFKOLQHDUO\LQGHSHQ GHQW" :K\RUZK\QRW" 'RHVWKHDQVZHUWR3UREOHPLPSO\WKDW fX; Y; Z; ]g LVOLQHDUO\LQGHSHQGHQW" 7RGHWHUPLQHLI fX; Y; Z; ]g LVOLQHDUO\GHSHQGHQW LVLWZLVHWRFKHFNLI VD\ Z LVD OLQHDUFRPELQDWLRQRI X Y DQG ]" ,V fX; Y; Z; ]g OLQHDUO\GHSHQGHQW"

&9&3$*4&4 ,Q([HUFLVHV² GHWHUPLQHLIWKHYHFWRUVDUHOLQHDUO\LQGHSHQ GHQW -XVWLI\HDFKDQVZHU 2 3 2 3 2 3 2 3 2 3 2 3 5 7 9 0 0 "1 4 2 5 4 0 5 4 3 5 4 0 5 4 2 5 4 4 5 0 "6 "8 3 "8 1

#

$ # $ 2 "4 "3 6

#

$ # $ "1 "3 3 "9

,Q([HUFLVHV² GHWHUPLQHLIWKHFROXPQVRIWKHPDWUL[IRUPD OLQHDUO\LQGHSHQGHQWVHW -XVWLI\HDFKDQVZHU

2

3 0 "3 9 6 2 1 "7 7 7 6 4 "1 4 "5 5 1 "4 "2 2

1 4 "3 5 4 "2 "7 "4 "5 7

2

3 0 15 5

3 "4 "3 0 6 0 "1 57 7 6 4 1 1 "5 5 2 1 "10 2

1 "2 3 4 "6 4 "2 0 1 "1

3 2 25 3

,Q ([HUFLVHV DQG D IRU ZKDW YDOXHV RI h LV Y3 LQ 6SDQ fY1 ; Y2 g DQGE IRUZKDWYDOXHVRI h LV fY1 ; Y2 ; Y3 g OLQHDUO\ GHSHQGHQW" -XVWLI\HDFKDQVZHU

2

3 2 3 2 3 1 "3 5 Y1 D 4 "3 5 Y2 D 4 9 5 Y3 D 4 "7 5 2 "6 h 2 3 2 3 2 3 1 "3 2 Y1 D 4 "3 5 Y2 D 4 9 5 Y3 D 4 "5 5 "5 15 h

,Q([HUFLVHV² ÀQGWKHYDOXHV RI h IRUZKLFKWKHYHFWRUV DUHOLQHDUO\ GHSHQGHQW -XVWLI\HDFKDQVZHU 2 3 2 3 2 3 2 3 2 3 2 3 2 4 "2 3 "6 9 4 "2 5 4 "6 5 4 2 5 4 "6 5 4 4 5 4 h 5 4 7 h 1 "3 3 2 3 2 3 2 3 2 3 2 3 2 3 1 "2 3 1 "3 2 4 5 5 4 "9 5 4 h 5 4 "2 5 4 7 5 4 1 5 "3 6 "9 "4 6 h

A LVD 4 # 2 PDWUL[ A D ŒD1 D1

/LQHDU,QGHSHQGHQFH

D2 ! DQG D2 LVQRWDPXOWLSOHRI

A LVD 4 # 3 PDWUL[ A D ŒD1 D2 D3 ! VXFKWKDW fD1 ; D2 g LV OLQHDUO\LQGHSHQGHQWDQG D3 LVQRWLQ 6SDQ fD1 ; D2 g

+RZPDQ\SLYRWFROXPQVPXVWD 6 # 4 PDWUL[KDYHLILWV FROXPQVDUHOLQHDUO\LQGHSHQGHQW" :K\" +RZPDQ\SLYRWFROXPQVPXVWD 4 # 6 PDWUL[KDYHLILWV FROXPQVVSDQ R4 " :K\" &RQVWUXFW 3 # 2 PDWULFHV A DQG B VXFKWKDW A[ D KDVD QRQWULYLDOVROXWLRQ EXW B [ D KDVRQO\WKHWULYLDOVROXWLRQ D )LOOLQWKHEODQNLQWKHIROORZLQJVWDWHPHQW ´,I A LV DQ m # n PDWUL[ WKHQ WKH FROXPQV RI A DUH OLQHDUO\ SLYRWFROXPQVµ LQGHSHQGHQWLIDQGRQO\LI A KDV E ([SODLQZK\WKHVWDWHPHQWLQD LVWUXH

'HWHUPLQHE\LQVSHFWLRQZKHWKHUWKHYHFWRUVLQ([HUFLVHV² DUHOLQHDUO\ LQGHSHQGHQW -XVWLI\HDFKDQVZHU 2 3 2 3 # $ # $ # $ # $ 2 "3 5 2 1 "1 4 "4 5 4 6 5 1 8 3 7 8 "12 2 3 2 3 2 3 # $ # $ # $ # $ 5 0 "7 3 "1 3 7 4 "3 5 4 0 5 4 2 5 4 5 5 1 "1 0 4 2 3 2 3 2 3 2 3 2 3 "8 2 1 "2 0 4 12 5 4 "3 5 4 4 5 4 5 5 4 0 5 "4 "1 "7 3 0

([HUFLVHVDQGVKRXOGEHVROYHG ZLWKRXWSHUIRUPLQJURZ RSHUDWLRQV >+LQW :ULWH A[ D DVDYHFWRUHTXDWLRQ@ 2 3 2 3 5 6 "5 1 "4 7 7 *LYHQ A D 6 4 "3 "1 "4 5 REVHUYHWKDWWKHWKLUGFROXPQ 1 0 1 LVWKHVXPRIWKHÀUVWWZRFROXPQV )LQGDQRQWULYLDOVROXWLRQ RI A[ D 2 3 4 3 "5 4 5 REVHUYHWKDWWKHÀUVWFROXPQ *LYHQ A D 4 "2 "2 "2 "3 7 PLQXVWKUHHWLPHVWKHVHFRQGFROXPQHTXDOVWKHWKLUGFROXPQ )LQGDQRQWULYLDOVROXWLRQRI A[ D

D 7KHFROXPQVRIDPDWUL[ A DUHOLQHDUO\LQGHSHQGHQWLIWKH HTXDWLRQ A[ D KDVWKHWULYLDOVROXWLRQ

(DFKVWDWHPHQWLQ([HUFLVHV²LVHLWKHUWUXHLQDOOFDVHV RUIDOVHIRUDWOHDVWRQHH[DPSOH ,IIDOVH FRQVWUXFWDVSHFLÀF H[DPSOHWRVKRZWKDWWKHVWDWHPHQWLVQRWDOZD\VWUXH 6XFK DQH[DPSOHLVFDOOHGD FRXQWHUH[DPSOH WRWKHVWDWHPHQW ,ID VWDWHPHQW LV WUXH JLYH D MXVWLÀFDWLRQ 2QH VSHFLÀF H[DPSOH FDQQRWH[SODLQZK\DVWDWHPHQWLVDOZD\VWUXH
,Q([HUFLVHVDQG PDUNHDFKVWDWHPHQW7UXHRU)DOVH -XVWLI\ HDFKDQVZHURQWKHEDVLVRIDFDUHIXOUHDGLQJRIWKHWH[W

E ,I S LVDOLQHDUO\GHSHQGHQWVHW WKHQHDFKYHFWRULVDOLQHDU FRPELQDWLRQRIWKHRWKHUYHFWRUVLQ S

F 7KHFROXPQVRIDQ\ 4 # 5 PDWUL[DUHOLQHDUO\GHSHQGHQW

G ,I [ DQG \ DUHOLQHDUO\LQGHSHQGHQW DQGLI f[; \; ]g LV OLQHDUO\GHSHQGHQW WKHQ ] LVLQ 6SDQ f[; \g

,I Y1 ; : : : ; Y4 DUHLQ R4 DQG Y3 D 2Y1 C Y2 WKHQ fY1 ; Y2 ; Y3 ; Y4 g LVOLQHDUO\GHSHQGHQW

D ,I X DQG Y DUH OLQHDUO\ LQGHSHQGHQW DQG LI Z LV LQ 6SDQ fX; Yg WKHQ fX; Y; Zg LVOLQHDUO\GHSHQGHQW

,I Y1 DQG Y2 DUHLQ R4 DQG Y2 LVQRWDVFDODUPXOWLSOHRI Y1 WKHQ fY1 ; Y2 g LVOLQHDUO\LQGHSHQGHQW

E ,IWKUHHYHFWRUVLQ R3 OLHLQWKHVDPHSODQHLQ R3 WKHQ WKH\DUHOLQHDUO\GHSHQGHQW

,I Y1 ; : : : ; Y5 DUHLQ R5 DQG Y3 D WKHQ fY1 ; Y2 ; Y3 ; Y4 ; Y5 g LV OLQHDUO\GHSHQGHQW

F ,IDVHWFRQWDLQVIHZHUYHFWRUVWKDQWKHUHDUHHQWULHVLQWKH YHFWRUV WKHQWKHVHWLVOLQHDUO\LQGHSHQGHQW

,I Y1 Y2 Y3 DUHLQ R3 DQG Y3 LV QRW DOLQHDUFRPELQDWLRQRI Y1 ; Y2 WKHQ fY1 ; Y2 ; Y3 g LVOLQHDUO\LQGHSHQGHQW

G ,IDVHWLQ Rn LVOLQHDUO\GHSHQGHQW WKHQWKHVHWFRQWDLQV PRUHWKDQ n YHFWRUV ,Q([HUFLVHV² GHVFULEHWKHSRVVLEOHHFKHORQIRUPVRIWKH PDWUL[ 8VHWKHQRWDWLRQRI([DPSOHLQ6HFWLRQ A LVD 2 # 2 PDWUL[ZLWKOLQHDUO\GHSHQGHQWFROXPQV A LVD 3 # 3 PDWUL[ZLWKOLQHDUO\LQGHSHQGHQWFROXPQV

,I Y1 ; : : : ; Y4 DUHLQ R4 DQG fY1 ; Y2 ; Y3 g LVOLQHDUO\GHSHQGHQW WKHQ fY1 ; Y2 ; Y3 ; Y4 g LVDOVROLQHDUO\GHSHQGHQW ,I fY1 ; : : : ; Y4 g LVDOLQHDUO\LQGHSHQGHQWVHWRIYHFWRUVLQ R4 WKHQ fY1 ; Y2 ; Y3 g LVDOVROLQHDUO\LQGHSHQGHQW >+LQW 7KLQN DERXW x1 Y1 C x2 Y2 C x3 Y3 C 0 ! Y4 D @

6XSSRVH A LVDQ m # n PDWUL[ZLWKWKHSURSHUW\WKDWIRUDOO E LQ Rm WKHHTXDWLRQ A[ D E KDVDWPRVWRQHVROXWLRQ 8VHWKH

$)"15&3

/LQHDU(TXDWLRQVLQ/LQHDU$OJHEUD 2

3 12 10 "6 8 4 "14 6 "7 "6 4 "5 "7 97 6 7 7 9 9 "9 9 9 "18 A D 6 6 7 4 "4 "3 "1 0 "8 15 8 7 "5 6 1 "11

GHÀQLWLRQRIOLQHDULQGHSHQGHQFHWRH[SODLQZK\WKHFROXPQV RI A PXVWEHOLQHDUO\LQGHSHQGHQW 6XSSRVHDQ m # n PDWUL[ A KDV n SLYRWFROXPQV ([SODLQ ZK\IRUHDFK E LQ Rm WKHHTXDWLRQ A[ D E KDVDWPRVWRQH VROXWLRQ >+LQW ([SODLQZK\ A[ D E FDQQRWKDYHLQÀQLWHO\ PDQ\VROXWLRQV@

>0@ :LWK A DQG B DVLQ([HUFLVH VHOHFWDFROXPQ Y RI A WKDWZDVQRWXVHGLQWKHFRQVWUXFWLRQRI B DQGGHWHUPLQHLI Y LVLQWKHVHWVSDQQHGE\WKHFROXPQVRI B 'HVFULEH\RXU FDOFXODWLRQV

>0@ ,Q([HUFLVHVDQG XVHDVPDQ\FROXPQVRI A DVSRVVLEOH WRFRQVWUXFWDPDWUL[ B ZLWKWKHSURSHUW\WKDWWKHHTXDWLRQ B [ D KDVRQO\WKHWULYLDOVROXWLRQ 6ROYH B [ D WRYHULI\\RXUZRUN 2 3 3 "4 10 7 "4 6 "5 "3 "7 "11 15 7 7 A D 6 4 4 3 5 2 15 8 "7 23 4 15 x3

40-65*0/4 50 13"$5*$& 130#-&.4 Span{u, v, z}

w

x1

>0@ 5HSHDW([HUFLVHZLWKWKHPDWULFHV A DQG B IURP ([HUFLVH 7KHQJLYHDQH[SODQDWLRQIRUZKDW\RXGLVFRYHU DVVXPLQJWKDW B ZDVFRQVWUXFWHGDVVSHFLÀHG

x2

*/530%6$5*0/ 50 -*/&"3 53"/4'03."5*0/4 7KHGLIIHUHQFHEHWZHHQDPDWUL[HTXDWLRQ A[ D E DQGWKHDVVRFLDWHGYHFWRUHTXDWLRQ x1 D1 C ! ! ! C xn Dn D E LVPHUHO\DPDWWHURIQRWDWLRQ +RZHYHU DPDWUL[HTXDWLRQ A[ D E FDQDULVHLQOLQHDUDOJHEUDDQGLQDSSOLFDWLRQVVXFKDVFRPSXWHUJUDSKLFVDQG VLJQDOSURFHVVLQJ LQDZD\WKDWLVQRWGLUHFWO\FRQQHFWHGZLWKOLQHDUFRPELQDWLRQVRI YHFWRUV 7KLVKDSSHQVZKHQZHWKLQNRIWKHPDWUL[ A DVDQREMHFWWKDW´DFWVµRQDYHFWRU [ E\PXOWLSOLFDWLRQWRSURGXFHDQHZYHFWRUFDOOHG A[ )RULQVWDQFH WKHHTXDWLRQV 3 2 3 2 1 # $ 1 # $ # $ # $ 7 7 4 "3 1 3 6 4 "3 1 3 6 617 D 5 6 47 D 0 DQG 2 0 5 1 415 8 2 0 5 1 4 "1 5 0 1 3 "

A

"

[

"

E

"

A

"

X

"

VD\WKDWPXOWLSOLFDWLRQE\ A WUDQVIRUPV [ LQWR E DQGWUDQVIRUPV X LQWRWKH]HURYHFWRU 6HH)LJ

,QWURGXFWLRQWR/LQHDU7UDQVIRUPDWLRQV

multiplication by A

x

multiplication

0

by A

u !

b

0

4

!2

'*(63& 7UDQVIRUPLQJYHFWRUVYLDPDWUL[

PXOWLSOLFDWLRQ

)URPWKLVQHZSRLQWRIYLHZ VROYLQJWKHHTXDWLRQ A[ D E DPRXQWVWRÀQGLQJ DOOYHFWRUV [ LQ R4 WKDWDUHWUDQVIRUPHGLQWRWKHYHFWRU E LQ R2 XQGHUWKH´DFWLRQµRI PXOWLSOLFDWLRQE\ A 7KHFRUUHVSRQGHQFHIURP [ WR A[ LVD IXQFWLRQ IURPRQHVHWRIYHFWRUVWRDQRWKHU 7KLVFRQFHSWJHQHUDOL]HVWKHFRPPRQQRWLRQRIDIXQFWLRQDVDUXOHWKDWWUDQVIRUPVRQH UHDOQXPEHULQWRDQRWKHU $ WUDQVIRUPDWLRQ RU IXQFWLRQ RU PDSSLQJ T IURP Rn WR Rm LVDUXOHWKDWDVVLJQV WRHDFKYHFWRU [ LQ Rn DYHFWRU T .[/ LQ Rm 7KHVHW Rn LVFDOOHGWKH GRPDLQ RI T DQG Rm LVFDOOHGWKH FRGRPDLQ RI T 7KHQRWDWLRQ T W Rn ! Rm LQGLFDWHVWKDWWKHGRPDLQ RI T LV Rn DQGWKHFRGRPDLQLV Rm )RU [ LQ Rn WKHYHFWRU T .[/ LQ Rm LVFDOOHGWKH LPDJH RI [ XQGHUWKHDFWLRQRI T 7KHVHWRIDOOLPDJHV T .[/ LVFDOOHGWKH UDQJH RI T 6HH)LJ T

T(x)

x

Ra

ng

!

n

!m Domain

e

Codomain

'*(63& 'RPDLQ FRGRPDLQ DQGUDQJH

RI T W Rn ! Rm

7KHQHZ WHUPLQRORJ\LQWKLV VHFWLRQLVLPSRUWDQWEHFDXVHDG\QDPLFYLHZRI PDWUL[²YHFWRUPXOWLSOLFDWLRQLVWKHNH\WRXQGHUVWDQGLQJVHYHUDOLGHDVLQOLQHDUDOJHEUD DQGWREXLOGLQJPDWKHPDWLFDOPRGHOVRISK\VLFDOV\VWHPVWKDWHYROYHRYHUWLPH 6XFK G\QDPLFDOV\VWHPV ZLOOEHGLVFXVVHGLQ6HFWLRQV DQGDQGWKURXJKRXW &KDSWHU

0DWUL[7UDQVIRUPDWLRQV 7KHUHVWRIWKLVVHFWLRQIRFXVHVRQPDSSLQJVDVVRFLDWHGZLWKPDWUL[PXOWLSOLFDWLRQ )RU HDFK [ LQ Rn T .[/ LVFRPSXWHGDV A[ ZKHUH A LVDQ m # n PDWUL[ )RUVLPSOLFLW\ ZH VRPHWLPHVGHQRWHVXFKD PDWUL[WUDQVIRUPDWLRQ E\ [ 7! A[ 2EVHUYHWKDWWKHGRPDLQ RI T LV Rn ZKHQ A KDV n FROXPQVDQGWKHFRGRPDLQRI T LV Rm ZKHQHDFKFROXPQRI A KDV m HQWULHV 7KHUDQJHRI T LVWKHVHWRIDOOOLQHDUFRPELQDWLRQVRIWKHFROXPQVRI A EHFDXVHHDFKLPDJH T .[/ LVRIWKHIRUP A[

$)"15&3


2

3 2 3 2 3 # $ 1 "3 3 3 2 &9".1-& /HW A D 4 3 5 5 X D E D 4 2 5 F D 4 2 5 DQG "1 "1 7 "5 5 GHÀQHDWUDQVIRUPDWLRQ T W R2 ! R3 E\ T .[/ D A[ VR WKDW 2 3 2 3 1 "3 # $ x1 " 3x2 x T .[/ D A[ D 4 3 5 5 1 D 4 3x1 C 5x2 5 x2 "x1 C 7x2 "1 7

x2

u = –12   

x1

T

D )LQG T .X/ WKHLPDJHRI X XQGHUWKHWUDQVIRUPDWLRQ T E )LQGDQ [ LQ R2 ZKRVHLPDJHXQGHU T LV E F ,VWKHUHPRUHWKDQRQH [ ZKRVHLPDJHXQGHU T LV E" G 'HWHUPLQHLI F LVLQWKHUDQJHRIWKHWUDQVIRUPDWLRQ T

40-65*0/

x3

D &RPSXWH x2

x1

 5 T(u) =  1  – 9 

2

3 2 3 $ 1 "3 # 5 2 T . X / D AX D 4 3 5 5 D 4 15 "1 "1 7 "9

E 6ROYH T .[/ D E IRU [ 7KDWLV VROYH A[ D E RU 2 3 2 3 1 "3 # $ 3 x 4 3 55 1 D 4 25 x2 "1 7 "5

8VLQJWKHPHWKRGGLVFXVVHGLQ6HFWLRQ URZUHGXFHWKHDXJPHQWHGPDWUL[ 2 3 2 3 2 3 2 3 1 "3 3 1 "3 3 1 "3 3 1 0 1:5 4 3 5 2 5 % 4 0 14 "7 5 % 4 0 1 ":5 5 % 4 0 1 ":5 5 "1 7 "5 0 4 "2 0 0 0 0 0 0 # $ 1:5 7KHLPDJHRIWKLV [ XQGHU T LVWKH +HQFH x1 D 1:5 x2 D ":5 DQG [ D ":5 JLYHQYHFWRU E F $Q\ [ ZKRVHLPDJHXQGHU T LV E PXVWVDWLVI\HTXDWLRQ )URP LWLVFOHDUWKDW HTXDWLRQ KDVDXQLTXHVROXWLRQ 6RWKHUHLVH[DFWO\RQH [ ZKRVHLPDJHLV E G 7KHYHFWRU F LVLQWKHUDQJHRI T LI F LVWKHLPDJHRIVRPH [ LQ R2 WKDWLV LI F D T .[/ IRUVRPH [ 7KLVLVMXVWDQRWKHUZD\RIDVNLQJLIWKHV\VWHP A[ D F LVFRQVLVWHQW 7R ÀQGWKHDQVZHU URZUHGXFHWKHDXJPHQWHGPDWUL[ 2 3 2 3 2 3 2 3 1 "3 3 1 "3 3 1 "3 3 1 "3 3 4 3 5 2 5 % 4 0 14 "7 5 % 4 0 1 2 5 % 4 0 1 25 "1 7 5 0 4 8 0 14 "7 0 0 "35 7KHWKLUGHTXDWLRQ 0 D "35 VKRZVWKDWWKHV\VWHPLVLQFRQVLVWHQW 6R F LV QRW LQ WKHUDQJHRI T

7KHTXHVWLRQLQ([DPSOHF LVD XQLTXHQHVV SUREOHPIRUDV\VWHPRIOLQHDU HTXDWLRQV WUDQVODWHGKHUHLQWRWKHODQJXDJHRIPDWUL[WUDQVIRUPDWLRQV ,V E WKHLPDJH RID XQLTXH [ LQ Rn " 6LPLODUO\ ([DPSOHG LVDQ H[LVWHQFH SUREOHP 'RHVWKHUH H[LVW DQ [ ZKRVHLPDJHLV F" 7KHQH[WWZRPDWUL[WUDQVIRUPDWLRQVFDQEHYLHZHGJHRPHWULFDOO\ 7KH\UHLQIRUFH WKHG\QDPLFYLHZRIDPDWUL[DVVRPHWKLQJWKDWWUDQVIRUPVYHFWRUVLQWRRWKHUYHFWRUV 6HFWLRQ FRQWDLQVRWKHULQWHUHVWLQJH[DPSOHVFRQQHFWHGZLWKFRPSXWHUJUDSKLFV

2

x3

0 x2 x1

3 1 0 0 &9".1-& ,I A D 4 0 1 0 5 0 0 0 SRLQWVLQ R3 RQWRWKH x1 x2 SODQHEHFDXVH 2 3 2 x1 1 0 4 x2 5 7! 4 0 1 x3 0 0

6HH)LJ

'*(63&

$ SURMHFWLRQWUDQVIRUPDWLRQ


WKHQ WKH WUDQVIRUPDWLRQ [ 7! A[ SURMHFWV 32 3 2 3 0 x1 x1 0 54 x2 5 D 4 x2 5 x3 0 0

#

$ 1 3 &9".1-& /HW A D 7KHWUDQVIRUPDWLRQ T W R2 ! R2 GHÀ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À[HG 6KHDUWUDQVIRUPDWLRQVDSSHDULQSK\VLFV JHRORJ\ DQGFU\VWDOORJUDSK\

sheep x2

x2 T 2

2

2 sheared sheep

x1

2

8

x1

'*(63& $ VKHDUWUDQVIRUPDWLRQ

/LQHDU7UDQVIRUPDWLRQV 7KHRUHPLQ6HFWLRQ VKRZVWKDWLI A LV m # n WKHQWKHWUDQVIRUPDWLRQ [ 7! A[ KDV WKHSURSHUWLHV A.X C Y/ D AX C AY DQG A.c X/ D cAX

IRUDOO X; Y LQ Rn DQGDOOVFDODUV c 7KHVHSURSHUWLHV ZULWWHQLQIXQFWLRQQRWDWLRQ LGHQWLI\ WKHPRVWLPSRUWDQWFODVVRIWUDQVIRUPDWLRQVLQOLQHDUDOJHEUD

%&'*/*5*0/

$ WUDQVIRUPDWLRQRUPDSSLQJ T LV OLQHDU LI L T .X C Y/ D T .X/ C T .Y/ IRUDOO X; Y LQWKHGRPDLQRI T LL T .c X/ D cT .X/ IRUDOOVFDODUV c DQGDOO X LQWKHGRPDLQRI T (YHU\PDWUL[WUDQVIRUPDWLRQLVDOLQHDUWUDQVIRUPDWLRQ ,PSRUWDQWH[DPSOHVRI OLQHDUWUDQVIRUPDWLRQVWKDWDUHQRWPDWUL[WUDQVIRUPDWLRQVZLOOEHGLVFXVVHGLQ&KDSWHUV DQG

$)"15&3


/LQHDUWUDQVIRUPDWLRQV SUHVHUYHWKHRSHUDWLRQVRIYHFWRUDGGLWLRQDQGVFDODUPXO WLSOLFDWLRQ 3URSHUW\L VD\VWKDWWKHUHVXOW T .X C Y/ RIÀUVWDGGLQJ X DQG Y LQ Rn DQG WKHQDSSO\LQJ T LVWKHVDPHDVÀUVWDSSO\LQJ T WR X DQGWR Y DQGWKHQDGGLQJ T .X/ DQG T .Y/ LQ Rm 7KHVHWZRSURSHUWLHVOHDGHDVLO\WRWKHIROORZLQJXVHIXOIDFWV ,I T LVDOLQHDUWUDQVIRUPDWLRQ WKHQ

T ./ D

T .c X C d Y/ D cT .X/ C d T .Y/

DQG IRUDOOYHFWRUV X Y LQWKHGRPDLQRI T DQGDOOVFDODUV c; d 3URSHUW\ IROORZVIURPFRQGLWLRQLL LQWKHGHÀQLWLRQ EHFDXVH T ./ D T .0X/ D 0T .X/ D 3URSHUW\ UHTXLUHVERWKL DQGLL

T .c X C d Y/ D T .c X/ C T .d Y/ D cT .X/ C d T .Y/

2EVHUYHWKDW LIDWUDQVIRUPDWLRQVDWLVÀHV IRUDOO X Y DQG c; d LWPXVWEHOLQHDU 6HW c D d D 1 IRUSUHVHUYDWLRQRIDGGLWLRQ DQGVHW d D 0 IRUSUHVHUYDWLRQRIVFDODU PXOWLSOLFDWLRQ 5HSHDWHGDSSOLFDWLRQRI SURGXFHVDXVHIXOJHQHUDOL]DWLRQ

T .c1 Y1 C ! ! ! C cp Yp / D c1 T .Y1 / C ! ! ! C cp T .Yp /

,QHQJLQHHULQJDQGSK\VLFV LVUHIHUUHGWRDVD VXSHUSRVLWLRQSULQFLSOH 7KLQN RI Y1 ; : : : ; Yp DVVLJQDOVWKDWJRLQWRDV\VWHPDQG T .Y1 /; : : : ; T .Yp / DVWKHUHVSRQVHVRI WKDWV\VWHPWRWKHVLJQDOV 7KHV\VWHPVDWLVÀHVWKHVXSHUSRVLWLRQSULQFLSOHLIZKHQHYHU DQLQSXWLVH[SUHVVHGDVDOLQHDUFRPELQDWLRQRIVXFKVLJQDOV WKHV\VWHP·VUHVSRQVHLV WKHVDPH OLQHDUFRPELQDWLRQRIWKHUHVSRQVHVWRWKHLQGLYLGXDOVLJQDOV :HZLOOUHWXUQ WRWKLVLGHDLQ&KDSWHU

&9".1-& *LYHQDVFDODU r GHÀQH T W R2 ! R2 E\ T .[/ D r [ T LVFDOOHGD

FRQWUDFWLRQ ZKHQ 0 ' r ' 1 DQGD GLODWLRQ ZKHQ r > 1 /HW r D 3 DQGVKRZWKDW T LVDOLQHDUWUDQVIRUPDWLRQ

40-65*0/ /HW X Y EHLQ R2 DQGOHW c; d EHVFDODUV 7KHQ T .c X C d Y/ D 3.c X C d Y/

D 3c X C 3d Y D c.3X/ C d.3Y/

)

'HÀQLWLRQRI T 9HFWRUDULWKPHWLF

D cT .X/ C d T .Y/

7KXV T LVDOLQHDUWUDQVIRUPDWLRQEHFDXVHLWVDWLVÀHV 6HH)LJ x2

T

T(u)

x2

u x1

'*(63& $ GLODWLRQWUDQVIRUPDWLRQ

x1


&9".1-& 'HÀQHDOLQHDUWUDQVIRUPDWLRQ T W R2 ! R2 E\

$# $ # $ 0 "1 x1 "x2 D x1 1 0 x2 # $ # $ # $ 4 2 6 )LQGWKHLPDJHVXQGHU T RI X D YD DQG X C Y D 1 3 4

T .[/ D

40-65*0/

T .X / D

#

0 "1 1 0

#

$# $ # $ # $# $ # $ 4 "1 0 "1 2 "3 D ; T .Y/ D D ; 1 4 1 0 3 2 # $# $ # $ 0 "1 6 "4 T .X C Y / D D 1 0 4 6

1RWHWKDW T .X C Y/ LVREYLRXVO\HTXDOWR T .X/ C T .Y/ ,WDSSHDUVIURP)LJ WKDW T URWDWHV X Y DQG X C Y FRXQWHUFORFNZLVHDERXWWKHRULJLQWKURXJK 90ı ,QIDFW T WUDQVIRUPVWKHHQWLUHSDUDOOHORJUDPGHWHUPLQHGE\ X DQG Y LQWRWKHRQHGHWHUPLQHGE\ T .X/ DQG T .Y/ 6HH([HUFLVH x2

T(u + v)

T

T(u)

u+v

v T(v) u

x1

'*(63& $ URWDWLRQWUDQVIRUPDWLRQ

7KHÀQDOH[DPSOHLVQRWJHRPHWULFDO LQVWHDG LWVKRZVKRZDOLQHDUPDSSLQJFDQ WUDQVIRUPRQHW\SHRIGDWDLQWRDQRWKHU

&9".1-& $ FRPSDQ\PDQXIDFWXUHVWZRSURGXFWV % DQG&8VLQJGDWDIURP ([DPSOHLQ6HFWLRQ ZHFRQVWUXFWD´XQLWFRVWµPDWUL[ U D Œ E F ! ZKRVH FROXPQVGHVFULEHWKH´FRVWVSHUGROODURIRXWSXWµIRUWKHSURGXFWV 2

3URGXFW % &

:45 U D 4 :25 :15

3 :40 0DWHULDOV :35 5 /DERU :15 2YHUKHDG

/HW [ D .x1 ; x2 / EHD´SURGXFWLRQµYHFWRU FRUUHVSRQGLQJWR x1 GROODUVRISURGXFW% DQG x2 GROODUVRISURGXFW&DQGGHÀQH T W R2 ! R3 E\ 2 3 2 3 2 3 :45 :40 7RWDOFRVWRIPDWHULDOV 5 T .[/ D U [ D x1 4 :25 5 C x2 4 :30 5 D 4 7RWDOFRVWRIODERU 7RWDOFRVWRIRYHUKHDG :15 :15

7KHPDSSLQJ T WUDQVIRUPVDOLVWRISURGXFWLRQTXDQWLWLHVPHDVXUHGLQGROODUV LQWRD OLVWRIWRWDOFRVWV 7KHOLQHDULW\RIWKLVPDSSLQJLVUHÁHFWHGLQWZRZD\V ,ISURGXFWLRQLVLQFUHDVHGE\DIDFWRURI VD\ IURP [ WR 4[ WKHQWKHFRVWVZLOO LQFUHDVHE\WKHVDPHIDFWRU IURP T .[/ WR 4T .[/

$)"15&3


,I [ DQG \ DUHSURGXFWLRQYHFWRUV WKHQWKHWRWDOFRVWYHFWRUDVVRFLDWHGZLWKWKH FRPELQHG SURGXFWLRQ [ C \ LV SUHFLVHO\ WKH VXP RI WKH FRVW YHFWRUV T .[/ DQG T .\/

13"$5*$& 130#-&.4 6XSSRVH T W R5 ! R2 DQG T .[/ D A[ IRUVRPHPDWUL[ A DQGIRUHDFK [ LQ R5 +RZ PDQ\URZVDQGFROXPQVGRHV A KDYH" # $ 1 0 /HW A D *LYHDJHRPHWULFGHVFULSWLRQRIWKHWUDQVIRUPDWLRQ [ 7! A[ 0 "1 7KHOLQHVHJPHQWIURP WRDYHFWRU X LVWKHVHWRISRLQWVRIWKHIRUP t X ZKHUH 0 ' t ' 1 6KRZWKDWDOLQHDUWUDQVIRUPDWLRQ T PDSVWKLVVHJPHQWLQWRWKHVHJPHQW EHWZHHQ DQG T .X/

&9&3$*4&4 #

$ 0 /HW A D DQGGHÀQH T W R2 ! R2 E\ T .[/ D A[ 2 # $ # $ 1 a DQG Y D )LQGWKHLPDJHVXQGHU T RI X D "3 b 21 3 2 3 2 3 0 0 3 a 3 1 5 X D 4 6 5 DQG Y D 4 b 5 0 /HW A D 4 0 3 1 "9 c 0 0 3 3 3 'HÀQH T W R ! R E\ T .[/ D A[ )LQG T .X/ DQG T .Y/

2 0

,Q([HUFLVHV² ZLWK T GHÀQHGE\ T .[/ D A[ ÀQGDYHFWRU [ ZKRVHLPDJHXQGHU T LV E DQGGHWHUPLQHZKHWKHU [ LVXQLTXH 2 2 3 3 1 0 "3 "2 1 6 5 E D 4 3 5 A D 4 "3 2 "2 "1 "1 2 3 2 3 1 "2 3 "6 1 "3 5 E D 4 "4 5 A D 4 0 2 "5 6 "5 # $ # $ 1 "5 "7 "2 A D ED "3 7 5 "2 2 3 2 3 1 "3 2 1 6 3 "8 7 6 67 8 7 E D 6 7 A D 6 40 4 35 1 25 1 0 8 10

/HW A EHD 6 # 5 PDWUL[ :KDWPXVW a DQG b EHLQRUGHUWR GHÀQH T W Ra ! Rb E\ T .[/ D A["

+RZPDQ\URZVDQGFROXPQVPXVWDPDWUL[ A KDYHLQRUGHU WRGHÀQHDPDSSLQJIURP R5 LQWR R7 E\WKHUXOH T .[/ D A[" )RU([HUFLVHVDQG ÀQGDOO [ LQ R4 WKDWDUHPDSSHGLQWRWKH ]HURYHFWRUE\WKHWUDQVIRUPDWLRQ [ 7! A[ IRUWKHJLYHQPDWUL[ A 2 3 1 "3 5 "5 1 "3 55 A D 4 0 2 "4 4 "4

2

3 61 A D 6 40 1

2

2 0 1 4

10 2 2 10

3 "6 "4 7 7 35 8

3 "1 /HW E D 4 1 5 DQGOHW A EHWKHPDWUL[LQ([HUFLVH ,V E 0 LQWKHUDQJHRIWKHOLQHDUWUDQVIRUPDWLRQ [ 7! A[" :K\RU ZK\QRW" 2 3 "1 6 37 7 /HW E D 6 4 "1 5 DQGOHW A EHWKHPDWUL[LQ([HUFLVH ,V 4 E LQWKHUDQJHRIWKHOLQHDUWUDQVIRUPDWLRQ [ 7! A[" :K\RU ZK\QRW"

,Q([HUFLVHV² XVHDUHFWDQJXODUFRRUGLQDWHV\VWHPWRSORW # $ # $ 5 "2 YD DQGWKHLULPDJHVXQGHUWKHJLYHQWUDQVIRU XD 2 4 PDWLRQ T 0DNHDVHSDUDWHDQGUHDVRQDEO\ODUJHVNHWFKIRUHDFK H[HUFLVH 'HVFULEHJHRPHWULFDOO\ZKDW T GRHVWRHDFKYHFWRU [ LQ R2 # $# $ "1 0 x1 T .[/ D x2 0 "1 # $# $ 2 0 x1 T .[/ D 0 2 x2 # $# $ 0 1 x1 T .[/ D x2 1 0 # $# $ 0 0 x1 T .[/ D 0 2 x2 /HW T W R2 ! R2 EHDOLQHDUWUDQVIRUPDWLRQWKDWPDSV X D # $ # $ # $ # $ 3 4 3 "1 LQWR DQGPDSV Y D LQWR 8VHWKHIDFW 4 1 3 3 WKDW T LVOLQHDUWRÀQGWKHLPDJHVXQGHU T RI 2X 3Y DQG 2X C 3Y

7KHÀJXUHVKRZVYHFWRUV X Y DQG Z DORQJZLWKWKHLPDJHV T .X/ DQG T .Y/ XQGHUWKHDFWLRQRIDOLQHDUWUDQVIRUPDWLRQ T W R2 ! R2 &RS\WKLVÀJXUHFDUHIXOO\ DQGGUDZWKHLPDJH T .Z/ DVDFFXUDWHO\DVSRVVLEOH >+LQW )LUVW ZULWH Z DVD OLQHDUFRPELQDWLRQRI X DQG Y@ x2

x2

w

T(v)

u

x1

v

x1 T(u)

# $ # $ # $ # $ 1 0 2 "1 H2 D \1 D DQG \2 D DQG 0 1 5 6 OHW T W R2 ! R2 EHDOLQHDUWUDQVIRUPDWLRQWKDWPDSV # $ #H1 LQWR $ 5 x1 \1 DQGPDSV H2 LQWR \2 )LQGWKHLPDJHVRI DQG x2 "3 # $ # $ # $ x1 "3 7 Y1 D DQG Y2 D DQG OHW /HW [ D x2 5 "2 T W R2 ! R2 EHDOLQHDUWUDQVIRUPDWLRQWKDWPDSV [ LQWR x1 Y1 C x2 Y2 )LQGDPDWUL[ A VXFKWKDW T .[/ LV A[ IRUHDFK [

/HW H1 D

,Q([HUFLVHVDQG PDUNHDFKVWDWHPHQW7UXHRU)DOVH -XVWLI\ HDFKDQVZHU D $ OLQHDUWUDQVIRUPDWLRQLVDVSHFLDOW\SHRIIXQFWLRQ E ,I A LVD 3 # 5 PDWUL[DQG T LVDWUDQVIRUPDWLRQGHÀQHG E\ T .[/ D A[ WKHQWKHGRPDLQRI T LV R3

F ,I A LVDQ m # n PDWUL[ WKHQWKHUDQJHRIWKHWUDQVIRUPD WLRQ [ 7! A[ LV Rm G (YHU\OLQHDUWUDQVIRUPDWLRQLVDPDWUL[WUDQVIRUPDWLRQ H $ WUDQVIRUPDWLRQ T LVOLQHDULIDQGRQO\LI

T .c1 Y1 C c2 Y2 / D c1 T .Y1 / C c2 T .Y2 /

IRUDOO Y1 DQG Y2 LQWKHGRPDLQRI T DQGIRUDOOVFDODUV c1 DQG c2 D 7KHUDQJHRIWKHWUDQVIRUPDWLRQ [ 7! A[ LVWKHVHWRIDOO OLQHDUFRPELQDWLRQVRIWKHFROXPQVRI A E (YHU\PDWUL[WUDQVIRUPDWLRQLVDOLQHDUWUDQVIRUPDWLRQ F ,I T W R ! R LVDOLQHDUWUDQVIRUPDWLRQDQGLI F LVLQ Rm WKHQDXQLTXHQHVVTXHVWLRQLV´,V F LQWKHUDQJHRI T "µ n

m

G $ OLQHDUWUDQVIRUPDWLRQSUHVHUYHVWKHRSHUDWLRQVRIYHFWRU DGGLWLRQDQGVFDODUPXOWLSOLFDWLRQ H $ OLQHDUWUDQVIRUPDWLRQ T W Rn ! Rm DOZD\VPDSVWKH RULJLQRI Rn WRWKHRULJLQRI Rm : 'HÀQH f W R ! R E\ f .x/ D mx C b D 6KRZWKDW f LVDOLQHDUWUDQVIRUPDWLRQZKHQ b D 0

E )LQGDSURSHUW\RIDOLQHDUWUDQVIRUPDWLRQWKDWLVYLRODWHG ZKHQ b ¤ 0 F :K\LV f FDOOHGDOLQHDUIXQFWLRQ"


$Q DIÀQHWUDQVIRUPDWLRQ T W Rn ! Rm KDVWKHIRUP T .[/ D A[ C E ZLWK A DQ m # n PDWUL[ DQG E LQ Rm 6KRZ WKDW T LV QRW DOLQHDUWUDQVIRUPDWLRQZKHQ E ¤ $IÀQH WUDQVIRUPDWLRQVDUHLPSRUWDQWLQFRPSXWHUJUDSKLFV *LYHQ Y ¤ DQG S LQ Rn WKHOLQHWKURXJK S LQWKHGLUHFWLRQRI Y KDVWKHSDUDPHWULFHTXDWLRQ [ D S C t Y 6KRZWKDWDOLQHDU WUDQVIRUPDWLRQ T W Rn ! Rn PDSVWKLVOLQHRQWRDQRWKHUOLQH RURQWRDVLQJOHSRLQWD GHJHQHUDWHOLQH D 6KRZWKDWWKHOLQHWKURXJKYHFWRUV S DQG T LQ Rn PD\EH ZULWWHQLQWKHSDUDPHWULFIRUP [ D .1 " t/S C t T 5HIHU WRWKHÀJXUHZLWK([HUFLVHVDQGLQ6HFWLRQ E 7KHOLQHVHJPHQWIURP S WR T LVWKHVHWRISRLQWVRIWKH IRUP .1 " t/S C t T IRU 0 ' t ' 1 DVVKRZQLQWKHÀJXUH EHORZ 6KRZWKDWDOLQHDUWUDQVIRUPDWLRQ T PDSVWKLV OLQHVHJPHQWRQWRDOLQHVHJPHQWRURQWRDVLQJOHSRLQW (t = 0) p (1 – t)p + tq

T(q) x

T(x) T(p)

(t = 1) q

/HW X DQG Y EHOLQHDUO\LQGHSHQGHQWYHFWRUVLQ R3 DQGOHW P EHWKHSODQHWKURXJK X Y DQG 7KHSDUDPHWULFHTXDWLRQ RI P LV [ D s X C t Y ZLWK s; t LQ R 6KRZWKDWDOLQHDU WUDQVIRUPDWLRQ T W R3 ! R3 PDSV P RQWRDSODQHWKURXJK RURQWRDOLQHWKURXJK RURQWRMXVWWKHRULJLQLQ R3 :KDW PXVWEHWUXHDERXW T .X/ DQG T .Y/ LQRUGHUIRUWKHLPDJHRI WKHSODQH P WREHDSODQH" /HW X DQG Y EHYHFWRUVLQ Rn ,WFDQEHVKRZQWKDWWKHVHW P RI DOOSRLQWVLQWKHSDUDOOHORJUDPGHWHUPLQHGE\ X DQG Y KDVWKH IRUP aX C b Y IRU 0 ' a ' 1 0 ' b ' 1 /HW T W Rn ! Rm EHDOLQHDUWUDQVIRUPDWLRQ ([SODLQZK\WKHLPDJHRIDSRLQW LQ P XQGHUWKHWUDQVIRUPDWLRQ T OLHVLQWKHSDUDOOHORJUDP GHWHUPLQHGE\ T .X/ DQG T .Y/ /HW T W R2 ! R2 EHWKHOLQHDUWUDQVIRUPDWLRQWKDWUHÁHFWV HDFKSRLQWWKURXJKWKH x2 D[LV 0DNHWZRVNHWFKHVVLPLODU WR )LJ WKDW LOOXVWUDWH SURSHUWLHV L DQG LL RI D OLQHDU WUDQVIRUPDWLRQ 6XSSRVHYHFWRUV Y1 ; : : : ; Yp VSDQ Rn DQGOHW T W Rn ! Rn EH D OLQHDU WUDQVIRUPDWLRQ 6XSSRVH T .Yi / D IRU i D 1; : : : ; p 6KRZWKDW T LVWKH]HURWUDQVIRUPDWLRQ 7KDWLV VKRZWKDWLI [ LVDQ\YHFWRULQ Rn WKHQ T .[/ D

/HW T W Rn ! Rm EH D OLQHDU WUDQVIRUPDWLRQ DQG OHW fY1 ; Y2 ; Y3 g EHDOLQHDUO\GHSHQGHQWVHWLQ Rn ([SODLQZK\ WKHVHW fT .Y1 /; T .Y2 /; T .Y3 /g LVOLQHDUO\GHSHQGHQW ,Q([HUFLVHV² FROXPQYHFWRUVDUHZULWWHQDVURZV VXFKDV [ D .x1 ; x2 / DQG T .[/ LVZULWWHQDV T .x1 ; x2 / 6KRZ WKDW WKH WUDQVIRUPDWLRQ T GHÀQHG E\ T .x1 ; x2 / D .x1 " 2jx2 j; x1 " 4x2 / LVQRWOLQHDU 6KRZ WKDW WKH WUDQVIRUPDWLRQ T GHÀQHG E\ T .x1 ; x2 / D .x1 " 2x2 ; x1 " 3; 2x1 " 5x2 / LVQRWOLQHDU

$)"15&3


/HW T W R3 ! R3 EHWKHWUDQVIRUPDWLRQWKDWUHÁHFWVHDFK YHFWRU [ D .x1 ; x2 ; x3 / WKURXJK WKH SODQH x3 D 0 RQWR T .[/ D .x1 ; x2 ; "x3 / 6KRZWKDW T LVDOLQHDUWUDQVIRUPD WLRQ >6HH([DPSOHIRULGHDV@ /HW T W R3 ! R3 EHWKHWUDQVIRUPDWLRQWKDWSURMHFWVHDFK YHFWRU [ D .x1 ; x2 ; x3 / RQWRWKHSODQH x2 D 0 VR T .[/ D .x1 ; 0; x3 / 6KRZWKDW T LVDOLQHDUWUDQVIRUPDWLRQ /HW T W Rn ! Rm EHDOLQHDUWUDQVIRUPDWLRQ 6XSSRVH fX; Yg LVDOLQHDUO\LQGHSHQGHQWVHW EXW fT .X/; T .Y/g LVDOLQHDUO\ GHSHQGHQWVHW 6KRZWKDW T .[/ D KDVDQRQWULYLDOVROXWLRQ >+LQW 8VH WKH IDFW WKDW c1 T .X/ C c2 T .Y/ D IRU VRPH ZHLJKWV c1 DQG c2 QRWERWK]HUR@ >0@ ,Q([HUFLVHVDQG WKHJLYHQPDWUL[GHWHUPLQHVDOLQHDU WUDQVIRUPDWLRQ T )LQGDOO [ VXFKWKDW T .[/ D

2

3 2 3 3 5 "5 3 4 "7 0 6 5 "8 7 0 07 7 47 7 7 6 4 6 "8 4 1 35 6 45 3 "6 "4 9 "7 "2 0 2 3 8 6 77 7 >0@ /HW E D 6 4 5 5 DQGOHW A EHWKHPDWUL[LQ([HUFLVH "3 ,V E LQWKHUDQJHRIWKHWUDQVIRUPDWLRQ [ 7! A[" ,IVR ÀQG DQ [ ZKRVHLPDJHXQGHUWKHWUDQVIRUPDWLRQLV E 2 3 4 6 "4 7 7 >0@ /HW E D 6 4 "4 5 DQGOHW A EHWKHPDWUL[LQ([HUFLVH "7 ,V E LQWKHUDQJHRIWKHWUDQVIRUPDWLRQ [ 7! A[" ,IVR ÀQG DQ [ ZKRVHLPDJHXQGHUWKHWUDQVIRUPDWLRQLV E

2 6 "7 6 4 "3 "9

SG

.BTUFSJOH -JOFBS 5SBOTGPSNBUJPOT m

40-65*0/4 50 13"$5*$& 130#-&.4 x2

A PXVWKDYHÀYHFROXPQVIRU A[ WREHGHÀQHG A PXVWKDYHWZRURZVIRUWKH FRGRPDLQRI T WREH R2

Au x

v

x1 Av u

Ax

3ORWVRPHUDQGRPSRLQWVYHFWRUV RQJUDSKSDSHUWRVHHZKDWKDSSHQV $ SRLQWVXFK DV .4; 1/ PDSVLQWR .4; "1/ 7KHWUDQVIRUPDWLRQ [ 7! A[ UHÁHFWVSRLQWVWKURXJKWKH x D[LVRU x1 D[LV /HW [ D t X IRUVRPH t VXFKWKDW 0 ' t ' 1 6LQFH T LVOLQHDU T .t X/ D t T .X/ ZKLFK LVDSRLQWRQWKHOLQHVHJPHQWEHWZHHQ DQG T .X/

7KHWUDQVIRUPDWLRQ [ 7! A[

5)& ."53*9 0' " -*/&"3 53"/4'03."5*0/ :KHQHYHUDOLQHDUWUDQVIRUPDWLRQ T DULVHVJHRPHWULFDOO\RULVGHVFULEHGLQZRUGV ZH XVXDOO\ZDQWDÍRUPXODµIRU T .[/ 7KHGLVFXVVLRQWKDWIROORZVVKRZVWKDWHYHU\OLQHDU WUDQVIRUPDWLRQIURP Rn WR Rm LVDFWXDOO\DPDWUL[WUDQVIRUPDWLRQ [ 7! A[ DQGWKDW LPSRUWDQWSURSHUWLHVRI T DUHLQWLPDWHO\UHODWHGWRIDPLOLDUSURSHUWLHVRI A 7KHNH\WR ÀQGLQJ A LVWRREVHUYHWKDW T LVFRPSOHWHO\GHWHUPLQHGE\ZKDWLWGRHVWRWKHFROXPQV RIWKH n # n LGHQWLW\PDWUL[ In #

x2 e 2 = 01 

1 e 1 = 0 

x1

$ # $ # $ 1 0 1 0 &9".1-& 7KHFROXPQVRI I2 D DUH H1 D DQG H2 D 6XS 0 1 0 1 SRVH T LVDOLQHDUWUDQVIRUPDWLRQIURP R2 LQWR R3 VXFKWKDW 2 3 2 3 5 "3 T .H1 / D 4 "7 5 DQG T .H2 / D 4 8 5 2 0

:LWKQRDGGLWLRQDOLQIRUPDWLRQ ÀQGDIRUPXODIRUWKHLPDJHRIDQDUELWUDU\ [ LQ R2

7KH0DWUL[RID/LQHDU7UDQVIRUPDWLRQ

40-65*0/ :ULWH [D

#

x1 x2

$

# $ # $ 1 0 D x1 C x2 D x1 H1 C x2 H2 0 1

6LQFH T LVD OLQHDU WUDQVIRUPDWLRQ

T .[/ D x1 T .H1 / C x2 T .H2 / 2 3 2 3 2 3 5 "3 5x1 " 3x2 D x1 4 "7 5 C x2 4 8 5 D 4 "7x1 C 8x2 5 2x1 C 0 2 0

7KHVWHSIURPHTXDWLRQ WRHTXDWLRQ H[SODLQVZK\NQRZOHGJHRI T .H1 / DQG T .H2 / LVVXIÀFLHQWWRGHWHUPLQH T .[/ IRUDQ\ [ 0RUHRYHU VLQFH H[SUHVVHV T .[/ DV DOLQHDUFRPELQDWLRQRIYHFWRUV ZHFDQSXWWKHVHYHFWRUVLQWRWKHFROXPQVRIDPDWUL[ A DQGZULWH DV # $ % & x1 T .[/ D T .H1 / T . H2 / D A[ x2

5)&03&.

/HW T W Rn ! Rm EHDOLQHDUWUDQVIRUPDWLRQ 7KHQWKHUHH[LVWVDXQLTXHPDWUL[ A VXFKWKDW T .[/ D A[ IRUDOO [ LQ Rn

,QIDFW A LVWKH m # n PDWUL[ZKRVH j WKFROXPQLVWKHYHFWRU T .Hj / ZKHUH Hj LV WKH j WKFROXPQRIWKHLGHQWLW\PDWUL[LQ Rn % & T .H n / A D T .H1 / ! ! !

1300' :ULWH [ D In [ D Œ H1 ! ! ! Hn ![ D x1 H1 C ! ! ! C xn Hn DQGXVHWKHOLQHDULW\ RI T WRFRPSXWH T .[/ D T .x1 H1 C ! ! ! C xn Hn / D x1 T .H1 / C ! ! ! C xn T .Hn / 2 3 x1 % &6 : 7 T .Hn / 4 :: 5 D A[ D T .H1 / ! ! ! xn

7KHXQLTXHQHVVRI A LVWUHDWHGLQ([HUFLVH 7KHPDWUL[ A LQ LVFDOOHGWKH VWDQGDUGPDWUL[IRUWKHOLQHDUWUDQVIRUPD WLRQ T :HNQRZQRZWKDWHYHU\OLQHDUWUDQVIRUPDWLRQIURP Rn WR Rm FDQEHYLHZHGDV DPDWUL[WUDQVIRUPDWLRQ DQGYLFHYHUVD 7KHWHUP OLQHDUWUDQVIRUPDWLRQ IRFXVHVRQD SURSHUW\RIDPDSSLQJ ZKLOH PDWUL[WUDQVIRUPDWLRQ GHVFULEHVKRZVXFKDPDSSLQJLV LPSOHPHQWHG DV([DPSOHV DQGLOOXVWUDWH

&9".1-& )LQGWKHVWDQGDUGPDWUL[ A IRUWKHGLODWLRQWUDQVIRUPDWLRQ T .[/ D 3[ IRU [ LQ R2

$)"15&3


40-65*0/ :ULWH # $ 3 T .H1 / D 3H1 D DQG 0

# $ 0 T .H2 / D 3H2 D 3

# # # $ 3 0 AD 0 3

&9".1-& /HW T W R2 ! R2 EHWKHWUDQVIRUPDWLRQWKDWURWDWHVHDFKSRLQWLQ

R2 DERXWWKHRULJLQWKURXJKDQDQJOH ' ZLWKFRXQWHUFORFNZLVHURWDWLRQIRUDSRVLWLYH DQJOH :HFRXOGVKRZJHRPHWULFDOO\WKDWVXFKDWUDQVIRUPDWLRQLVOLQHDU 6HH)LJ LQ 6HFWLRQ )LQGWKHVWDQGDUGPDWUL[ A RIWKLVWUDQVIRUPDWLRQ # $ # $ # $ # $ 1 FRV ' 0 " VLQ ' 40-65*0/ URWDWHVLQWR DQG URWDWHVLQWR 6HH)LJ VLQ ' FRV ' 0 1 %\7KHRUHP # $ FRV ' " VLQ ' AD VLQ ' FRV ' ([DPSOHLQ6HFWLRQ LVDVSHFLDOFDVHRIWKLVWUDQVIRUPDWLRQ ZLWK ' D "=2

(– sin ϕ, cos ϕ)

x2

(0, 1)

ϕ ϕ

(cos ϕ, sin ϕ) (1, 0)

x1

'*(63& $ URWDWLRQWUDQVIRUPDWLRQ

*HRPHWULF/LQHDU7UDQVIRUPDWLRQVRI R2

x2

0 1

1 0 '*(63&

7KHXQLWVTXDUH

x1

([DPSOHVDQGLOOXVWUDWHOLQHDUWUDQVIRUPDWLRQVWKDWDUHGHVFULEHGJHRPHWULFDOO\ 7DEOHV ² LOOXVWUDWH RWKHU FRPPRQ JHRPHWULF OLQHDU WUDQVIRUPDWLRQV RI WKH SODQH %HFDXVHWKHWUDQVIRUPDWLRQVDUHOLQHDU WKH\DUHGHWHUPLQHGFRPSOHWHO\E\ZKDWWKH\ GRWRWKHFROXPQVRI I2 ,QVWHDGRIVKRZLQJRQO\WKHLPDJHVRI H1 DQG H2 WKHWDEOHV VKRZZKDWDWUDQVIRUPDWLRQGRHVWRWKHXQLWVTXDUH)LJ 2WKHU WUDQVIRUPDWLRQV FDQ EH FRQVWUXFWHG IURP WKRVH OLVWHG LQ 7DEOHV ² E\ DSSO\LQJRQHWUDQVIRUPDWLRQDIWHUDQRWKHU )RULQVWDQFH DKRUL]RQWDOVKHDUFRXOGEH IROORZHGE\DUHÁHFWLRQLQWKH x2 D[LV 6HFWLRQ ZLOOVKRZWKDWVXFKD FRPSRVLWLRQ RIOLQHDUWUDQVIRUPDWLRQVLVOLQHDU $OVR VHH([HUFLVH

([LVWHQFHDQG8QLTXHQHVV4XHVWLRQV 7KHFRQFHSWRIDOLQHDUWUDQVIRUPDWLRQSURYLGHVDQHZZD\WRXQGHUVWDQGWKHH[LVWHQFH DQGXQLTXHQHVVTXHVWLRQVDVNHGHDUOLHU 7KHWZRGHÀQLWLRQVIROORZLQJ7DEOHV²JLYH WKHDSSURSULDWHWHUPLQRORJ\IRUWUDQVIRUPDWLRQV

5"#-&


3FáFDUJPOT

7UDQVIRUPDWLRQ

,PDJHRIWKH8QLW6TXDUH

5HÁHFWLRQWKURXJK WKH x1 D[LV

x2

1 0

6WDQGDUG0DWUL[ # $ 1 0 0 "1

x1

 0 –1

5HÁHFWLRQWKURXJK WKH x2 D[LV

#

x2

0 1

0 1

$

x1

–1  0

5HÁHFWLRQWKURXJK WKHOLQH x2 D x1

"1 0

#

x2 x 2 = x1

0 1

1 0

–1  0

1 0

$

x1

x2

5HÁHFWLRQWKURXJK WKHOLQH x2 D "x1

0 1

#

0 "1 "1 0

$

#

"1 0 0 "1

$

x1 x2 = – x1

 0 –1

5HÁHFWLRQWKURXJK WKHRULJLQ

x2

–1  0

x1

 0 –1

$)"15&3


5"#-&

$POUSBDUJPOT BOE &YQBOTJPOT

7UDQVIRUPDWLRQ +RUL]RQWDO FRQWUDFWLRQ DQGH[SDQVLRQ

,PDJHRIWKH8QLW6TXDUH x2

x2

0 1

0 1 x1

k 0

x1

k 0

0
9HUWLFDO FRQWUDFWLRQ DQGH[SDQVLRQ

6WDQGDUG0DWUL[ # $ k 0 0 1

k>1

x2

#

x2

0 k

1 0

0 k

$

0 k x1

1 0 0
5"#-&

x1

1 0 k>1

4IFBST

7UDQVIRUPDWLRQ


+RUL]RQWDOVKHDU

x2

6WDQGDUG0DWUL[ # $ 1 k 0 1

x2 k 1

k 1

1 0

k

x1

k<0

9HUWLFDOVKHDU

x1

1 0

k k>0

x2

#

x2

0 1

0 1

1 k

k

x1

x1

1 k

k k<0

k>0

1 k

0 1

$

5"#-&


1SPKFDUJPOT

7UDQVIRUPDWLRQ


6WDQGDUG0DWUL[ # $ 1 0 0 0

x2

3URMHFWLRQRQWR WKH x1 D[LV

0 0

3URMHFWLRQRQWR WKH x2 D[LV

x1

1 0

#

x2

0 0

0 1

$

0 1 x1

0 0

%&'*/*5*0/

$ PDSSLQJ T W Rn ! Rm LVVDLGWREH RQWR Rm LIHDFK E LQ Rm LVWKHLPDJHRI DWOHDVWRQH [ LQ Rn (TXLYDOHQWO\ T LVRQWR Rm ZKHQWKHUDQJHRI T LVDOORIWKHFRGRPDLQ Rm 7KDWLV T PDSV Rn RQWR Rm LI IRUHDFK E LQWKHFRGRPDLQ Rm WKHUHH[LVWVDWOHDVWRQHVROXWLRQ RI T .[/ D E ´'RHV T PDS Rn RQWR Rm "µ LVDQH[LVWHQFHTXHVWLRQ 7KHPDSSLQJ T LV QRW RQWRZKHQWKHUHLVVRPH E LQ Rm IRUZKLFKWKHHTXDWLRQ T .[/ D E KDVQRVROXWLRQ 6HH)LJ

in

ma

Do

T

in

ma

Do

R

an

T

Ra

ng

e

ge

!

!m

n

!

T is not onto !m

!m

n

T is onto !m m

'*(63& ,VWKHUDQJHRI T DOORI R "

%&'*/*5*0/

$ PDSSLQJ T W Rn ! Rm LVVDLGWREH RQHWRRQH LIHDFK E LQ Rm LVWKHLPDJH RI DWPRVWRQH [ LQ Rn

$)"15&3


(TXLYDOHQWO\ T LVRQHWRRQHLI IRUHDFK E LQ Rm WKHHTXDWLRQ T .[/ D E KDVHLWKHU DXQLTXHVROXWLRQRUQRQHDWDOO ´,V T RQHWRRQH"µ LVDXQLTXHQHVVTXHVWLRQ 7KH PDSSLQJ T LV QRW RQHWRRQHZKHQVRPH E LQ Rm LVWKHLPDJHRIPRUHWKDQRQHYHFWRU LQ Rn ,IWKHUHLVQRVXFK E WKHQ T LVRQHWRRQH 6HH)LJ in ma

Do

Ra

ng

T

0

!

n

Ra

ng

T

in ma

e

Do

0

0 !m

!

e 0

n

!m

T is not one-to-one

T is one-to-one

'*(63& ,VHYHU\ E WKHLPDJHRIDWPRVWRQHYHFWRU"

SG

.BTUFSJOH &YJTUFODF BOE 6OJRVFOFTT m

7KH SURMHFWLRQWUDQVIRUPDWLRQVVKRZQLQ7DEOHDUH QRW RQHWRRQHDQGGR QRW PDS R2 RQWR R2 7KHWUDQVIRUPDWLRQVLQ7DEOHV DQGDUHRQHWRRQH DQG GRPDS R2 RQWR R2 2WKHUSRVVLELOLWLHVDUHVKRZQLQWKHWZRH[DPSOHVEHORZ ([DPSOHDQGWKHWKHRUHPVWKDWIROORZVKRZKRZWKHIXQFWLRQSURSHUWLHVRIEHLQJ RQHWRRQHDQGPDSSLQJRQWRDUHUHODWHGWRLPSRUWDQWFRQFHSWVVWXGLHGHDUOLHULQWKLV FKDSWHU

&9".1-& /HW T EHWKHOLQHDUWUDQVIRUPDWLRQZKRVHVWDQGDUGPDWUL[LV 2

1 "4 8 A D 4 0 2 "1 0 0 0

3 1 35 5

'RHV T PDS R4 RQWR R3 " ,V T DRQHWRRQHPDSSLQJ"

40-65*0/ 6LQFH A KDSSHQVWREHLQHFKHORQIRUP ZHFDQVHHDWRQFHWKDW A KDVD SLYRWSRVLWLRQLQHDFKURZ %\7KHRUHPLQ6HFWLRQ IRUHDFK E LQ R3 WKHHTXDWLRQ A[ D E LVFRQVLVWHQW ,QRWKHUZRUGV WKHOLQHDUWUDQVIRUPDWLRQ T PDSV R4 LWVGRPDLQ RQWR R3 +RZHYHU VLQFHWKHHTXDWLRQ A[ D E KDVDIUHHYDULDEOHEHFDXVHWKHUHDUH IRXUYDULDEOHVDQGRQO\WKUHHEDVLFYDULDEOHV HDFK E LVWKHLPDJHRIPRUHWKDQRQH [ 7KDWLV T LV QRW RQHWRRQH

5)&03&.

/HW T W Rn ! Rm EHDOLQHDUWUDQVIRUPDWLRQ 7KHQ T LVRQHWRRQHLIDQGRQO\LI WKHHTXDWLRQ T .[/ D KDVRQO\WKHWULYLDOVROXWLRQ

1300' 6LQFH T LVOLQHDU T ./ D ,I T LVRQHWRRQH WKHQWKHHTXDWLRQ T .[/ D KDVDWPRVWRQHVROXWLRQDQGKHQFHRQO\WKHWULYLDOVROXWLRQ ,I T LVQRWRQHWRRQH WKHQ WKHUHLVD E WKDWLVWKHLPDJHRIDWOHDVWWZRGLIIHUHQWYHFWRUVLQ Rn ³VD\ X DQG Y 7KDW LV T .X/ D E DQG T .Y/ D E %XWWKHQ VLQFH T LVOLQHDU T . X " Y / D T .X / " T .Y / D E " E D 7KHYHFWRU X " Y LVQRW]HUR VLQFH X ¤ Y +HQFHWKHHTXDWLRQ T .[/ D KDVPRUHWKDQ RQHVROXWLRQ 6R HLWKHUWKHWZRFRQGLWLRQVLQWKHWKHRUHPDUHERWKWUXHRUWKH\DUHERWK IDOVH

5)&03&.


/HW T W Rn ! Rm EHDOLQHDUWUDQVIRUPDWLRQDQGOHW A EHWKHVWDQGDUGPDWUL[IRU T 7KHQ D T PDSV Rn RQWR Rm LIDQGRQO\LIWKHFROXPQVRI A VSDQ Rm E T LVRQHWRRQHLIDQGRQO\LIWKHFROXPQVRI A DUHOLQHDUO\LQGHSHQGHQW

1300' D %\7KHRUHPLQ6HFWLRQ WKHFROXPQVRI A VSDQ Rm LIDQGRQO\LIIRUHDFK E LQ Rm WKHHTXDWLRQ A[ D E LVFRQVLVWHQW³LQRWKHUZRUGV LIDQGRQO\LIIRUHYHU\ E WKHHTXDWLRQ T .[/ D E KDVDWOHDVWRQHVROXWLRQ 7KLVLVWUXHLIDQGRQO\LI T PDSV Rn RQWR Rm E 7KHHTXDWLRQV T .[/ D DQG A[ D DUHWKHVDPHH[FHSWIRUQRWDWLRQ 6R E\ 7KHRUHP T LVRQHWRRQHLIDQGRQO\LI A[ D KDVRQO\WKHWULYLDOVROXWLRQ 7KLVKDSSHQVLIDQGRQO\LIWKHFROXPQVRI A DUHOLQHDUO\LQGHSHQGHQW DVZDVDOUHDG\ QRWHGLQWKHER[HGVWDWHPHQW LQ6HFWLRQ 6WDWHPHQWD LQ7KHRUHPLVHTXLYDOHQWWRWKHVWDWHPHQW´T PDSV Rn RQWR Rm LIDQGRQO\LIHYHU\YHFWRULQ Rm LVDOLQHDUFRPELQDWLRQRIWKHFROXPQVRI Aµ 6HH 7KHRUHPLQ6HFWLRQ ,QWKHQH[WH[DPSOHDQGLQVRPHH[HUFLVHVWKDWIROORZ FROXPQYHFWRUVDUHZULWWHQLQ URZV VXFKDV [ D .x1 ; x2 / DQG T .[/ LVZULWWHQDV T .x1 ; x2 / LQVWHDGRIWKHPRUHIRUPDO T ..x1 ; x2 //

x2 e2 e1

x1

T T

x3

&9".1-& /HW T .x1 ; x2 / D .3x1 C x2 5x1 C 7x2 x1 C 3x2 / 6KRZWKDW T LVD RQHWRRQHOLQHDUWUDQVIRUPDWLRQ 'RHV T PDS R2 RQWR R3 "

40-65*0/ :KHQ [ DQG T .[/ DUHZULWWHQDVFROXPQYHFWRUV \RXFDQGHWHUPLQHWKH VWDQGDUGPDWUL[RI T E\LQVSHFWLRQ YLVXDOL]LQJWKHURZ²YHFWRUFRPSXWDWLRQRIHDFK HQWU\LQ A[ 2

3 2 3x1 C x2 " T .[/ D 4 5x1 C 7x2 5 D 4 " x1 C 3x2 "

a2 a1 Span{a1, a 2}

x1

7KHWUDQVIRUPDWLRQ T LVQRW RQWR R3

A

3 2 " # $ 3 x "5 1 D 45 x2 " 1

3 1 # $ x 75 1 x2 3

6R T LVLQGHHGDOLQHDUWUDQVIRUPDWLRQ ZLWKLWVVWDQGDUGPDWUL[ A VKRZQLQ 7KH FROXPQVRI A DUHOLQHDUO\LQGHSHQGHQWEHFDXVHWKH\DUHQRWPXOWLSOHV %\7KHRUHP E T LVRQHWRRQH 7RGHFLGHLI T LVRQWR R3 H[DPLQHWKHVSDQRIWKHFROXPQVRI A 6LQFH A LV 3 # 2 WKHFROXPQVRI A VSDQ R3 LIDQGRQO\LI A KDVSLYRWSRVLWLRQV E\7KHRUHP 7KLVLVLPSRVVLEOH VLQFH A KDVRQO\FROXPQV 6RWKHFROXPQVRI A GR QRWVSDQ R3 DQGWKHDVVRFLDWHGOLQHDUWUDQVIRUPDWLRQLVQRWRQWR R3

13"$5*$& 130#-&. /HW T W R2 ! R2 EHWKHWUDQVIRUPDWLRQWKDWÀUVWSHUIRUPVDKRUL]RQWDOVKHDUWKDWPDSV H2 LQWR H2 " :5H1 EXWOHDYHV H1 XQFKDQJHG DQGWKHQUHÁHFWVWKHUHVXOWWKURXJKWKH x2 D[LV $VVXPLQJWKDW T LVOLQHDU ÀQGLWVVWDQGDUGPDWUL[ >+LQW 'HWHUPLQHWKHÀQDO ORFDWLRQRIWKHLPDJHVRI H1 DQG H2 @

$)"15&3


&9&3$*4&4 ,Q([HUFLVHV² DVVXPHWKDW T LVDOLQHDUWUDQVIRUPDWLRQ )LQG WKHVWDQGDUGPDWUL[RI T

x2 a1

T W R2 ! R4 T .H1 / D .3; 1; 3; 1/ DQG T .H2 / D ."5; 2; 0; 0/ ZKHUH H1 D .1; 0/ DQG H2 D .0; 1/

T W R3 ! R2 T .H1 / D .1; 4/ T .H2 / D ."2; 9/ DQG T .H3 / D .3; "8/ ZKHUH H1 H2 DQG H3 DUHWKHFROXPQVRI WKH 3 # 3 LGHQWLW\PDWUL[ T W R2 ! R2 LVDYHUWLFDOVKHDUWUDQVIRUPDWLRQWKDWPDSV H1 LQWR H1 " 3H2 EXWOHDYHV H2 XQFKDQJHG

T W R2 ! R2 LVDKRUL]RQWDOVKHDUWUDQVIRUPDWLRQWKDWOHDYHV H1 XQFKDQJHGDQGPDSV H2 LQWR H2 C 2H1 T W R2 ! R2 URWDWHVSRLQWVDERXWWKHRULJLQ WKURXJK "=2 UDGLDQVFRXQWHUFORFNZLVH

T W R2 ! R2 URWDWHV SRLQWV DERXW WKH RULJLQ WKURXJK "3"=2 UDGLDQVFORFNZLVH

T W R2 ! R2 ÀUVW URWDWHV SRLQWV WKURXJK "3"=4 UDGLDQV FORFNZLVH DQGWKHQUHÁHFWVSRLQWVWKURXJKWKHKRUL]RQWDO p p x1 D[LV >+LQW T .H1 / D ."1= 2; 1= 2/@ T W R2 ! R2 ÀUVWSHUIRUPVDKRUL]RQWDOVKHDUWKDWWUDQV IRUPV H2 LQWR H2 C 2H1 OHDYLQJ H1 XQFKDQJHG DQGWKHQUH ÁHFWVSRLQWVWKURXJKWKHOLQH x2 D "x1 T W R ! R ÀUVWUHÁHFWVSRLQWVWKURXJKWKHKRUL]RQWDO x1 D[LVDQGWKHQURWDWHVSRLQWV ""=2 UDGLDQV 2

2

T W R2 ! R2 ÀUVWUHÁHFWVSRLQWVWKURXJKWKHKRUL]RQWDO x1 D[LVDQGWKHQUHÁHFWVSRLQWVWKURXJKWKHOLQH x2 D x1

$ OLQHDU WUDQVIRUPDWLRQ T W R2 ! R2 ÀUVW UHÁHFWV SRLQWV WKURXJKWKH x1 D[LVDQGWKHQUHÁHFWVSRLQWVWKURXJKWKH x2 D[LV 6KRZWKDW T FDQDOVREHGHVFULEHGDVDOLQHDUWUDQVIRU PDWLRQWKDWURWDWHVSRLQWVDERXWWKHRULJLQ :KDWLVWKHDQJOH RIWKDWURWDWLRQ"

6KRZ WKDW WKH WUDQVIRUPDWLRQ LQ ([HUFLVH LV PHUHO\ D URWDWLRQDERXWWKHRULJLQ :KDWLVWKHDQJOHRIWKHURWDWLRQ" /HW T W R2 ! R2 EHWKHOLQHDUWUDQVIRUPDWLRQVXFKWKDW T .H1 / DQG T .H2 / DUHWKHYHFWRUVVKRZQLQWKHÀJXUH 8VLQJWKH ÀJXUH VNHWFKWKHYHFWRU T .2; 1/ x2

T(e1)

a2

x1

,Q([HUFLVHVDQG ÀOOLQWKHPLVVLQJHQWULHVRIWKHPDWUL[ DVVXPLQJWKDWWKHHTXDWLRQKROGVIRUDOOYDOXHVRIWKHYDULDEOHV 2 32 3 2 3 ‹ ‹ ‹ x1 2x1 " 4x2 ‹ ‹ 54 x2 5 D 4 x1 " x3 5 4 ‹ ‹ ‹ ‹ x3 "x2 C 3x3 2 3 2 3 ‹ ‹ # $ 3x1 " 2x2 x 1 ‹5 4 ‹ D 4 x1 C 4x2 5 x2 ‹ ‹ x2 ,Q([HUFLVHV² VKRZWKDW T LVDOLQHDUWUDQVIRUPDWLRQE\ ÀQGLQJDPDWUL[WKDWLPSOHPHQWVWKHPDSSLQJ 1RWHWKDW x1 ; x2 ; : : : DUHQRWYHFWRUVEXWDUHHQWULHVLQYHFWRUV T .x1 ; x2 ; x3 ; x4 / D .x1 C 2x2 ; 0; 2x2 C x4 ; x2 " x4 / T .x1 ; x2 / D .x1 C 4x2 ; 0; x1 " 3x2 ; x1 / T .x1 ; x2 ; x3 / D .x1 " 5x2 C 4x3 ; x2 " 6x3 / T .x1 ; x2 ; x3 ; x4 / D 3x1 C 4x3 " 2x4 1RWLFH T W R4 ! R/ /HW T W R2 ! R2 EH D OLQHDU WUDQVIRUPDWLRQ VXFK WKDW T .x1 ; x2 / D .x1 C x2 ; 4x1 C 5x2 / )LQG [ VXFKWKDW T .[/ D .3; 8/ /HW T W R2 ! R3 EH D OLQHDU WUDQVIRUPDWLRQ ZLWK T .x1 ; x2 / D .2x1 " x2 ; "3x1 C x2 ; 2x1 " 3x2 / )LQG [ VXFK WKDW T .[/ D .0; "1; "4/ ,Q([HUFLVHVDQG PDUNHDFKVWDWHPHQW7UXHRU)DOVH -XVWLI\ HDFKDQVZHU D $ OLQHDUWUDQVIRUPDWLRQ T W Rn ! Rm LVFRPSOHWHO\GH WHUPLQHGE\LWVHIIHFWRQWKHFROXPQVRIWKH n # n LGHQWLW\ PDWUL[ E ,I T W R2 ! R2 URWDWHVYHFWRUVDERXWWKHRULJLQWKURXJK DQDQJOH ' WKHQ T LVDOLQHDUWUDQVIRUPDWLRQ

T(e 2 ) x1

/HW T W R2 ! R2 EHDOLQHDUWUDQVIRUPDWLRQZLWKVWDQGDUG PDWUL[ A D Œ D1 D2 ! ZKHUH D1 DQG D2 DUHVKRZQLQWKH ÀJXUHDWWKHWRSRIFROXPQ 8VLQJWKHÀJXUH GUDZWKH # $ 1 XQGHUWKHWUDQVIRUPDWLRQ T LPDJHRI "2

F :KHQWZROLQHDUWUDQVIRUPDWLRQVDUHSHUIRUPHGRQHDIWHU DQRWKHU WKHFRPELQHGHIIHFWPD\QRWDOZD\VEHDOLQHDU WUDQVIRUPDWLRQ G $ PDSSLQJ T W Rn ! Rm LVRQWR Rm LIHYHU\YHFWRU [ LQ Rn PDSVRQWRVRPHYHFWRULQ Rm H ,I A LVD 3 # 2 PDWUL[ WKHQWKHWUDQVIRUPDWLRQ [ 7! A[ FDQQRWEHRQHWRRQH D ,I A LVD 4 # 3 PDWUL[ WKHQWKHWUDQVIRUPDWLRQ [ 7! A[ PDSV R3 RQWR R4

E (YHU\OLQHDUWUDQVIRUPDWLRQIURP Rn WR Rm LVDPDWUL[ WUDQVIRUPDWLRQ F 7KHFROXPQVRIWKHVWDQGDUGPDWUL[IRUDOLQHDUWUDQVIRU PDWLRQIURP Rn WR Rm DUHWKHLPDJHVRIWKHFROXPQVRI WKH n # n LGHQWLW\PDWUL[XQGHU T

G $ PDSSLQJ T W Rn ! Rm LVRQHWRRQHLIHDFKYHFWRULQ Rn PDSVRQWRDXQLTXHYHFWRULQ Rm

H 7KHVWDQGDUGPDWUL[RIDKRUL]RQWDOVKHDUWUDQVIRUPDWLRQ # $ a 0 ZKHUH a DQG d IURP R2 WR R2 KDVWKHIRUP 0 d DUH ˙1 ,Q([HUFLVHV² GHWHUPLQHLIWKHVSHFLÀHGOLQHDUWUDQVIRUPD WLRQLVD RQHWRRQHDQGE RQWR -XVWLI\HDFKDQVZHU 7KHWUDQVIRUPDWLRQLQ([HUFLVH

9HULI\WKHXQLTXHQHVVRI A LQ7KHRUHP /HW T W Rn ! Rm EHDOLQHDUWUDQVIRUPDWLRQVXFKWKDW T .[/ D B [ IRUVRPH m # n PDWUL[ B 6KRZWKDWLI A LVWKHVWDQGDUGPDWUL[IRU T WKHQ A D B >+LQW 6KRZWKDW A DQG B KDYHWKHVDPH FROXPQV@ /HW S W Rp ! Rn DQG T W Rn ! Rm EHOLQHDUWUDQVIRUPD WLRQV 6KRZWKDWWKHPDSSLQJ [ 7! T .S.[// LVDOLQHDUWUDQV IRUPDWLRQIURP Rp WR Rm >+LQW &RPSXWH T .S.c X C d Y// IRU X; Y LQ Rp DQGVFDODUV c DQG d -XVWLI\HDFKVWHSRI WKHFRPSXWDWLRQ DQGH[SODLQZK\WKLVFRPSXWDWLRQJLYHVWKH GHVLUHGFRQFOXVLRQ@ ,IDOLQHDUWUDQVIRUPDWLRQ T W Rn ! Rm PDSV Rn RQWR Rm FDQ\RXJLYHDUHODWLRQEHWZHHQ m DQG n" ,I T LVRQHWRRQH ZKDWFDQ\RXVD\DERXW m DQG n" :K\LVWKHTXHVWLRQ´,VWKHOLQHDUWUDQVIRUPDWLRQ T RQWR"µ DQH[LVWHQFHTXHVWLRQ"

7KHWUDQVIRUPDWLRQLQ([HUFLVH 7KHWUDQVIRUPDWLRQLQ([HUFLVH 7KHWUDQVIRUPDWLRQLQ([HUFLVH ,Q([HUFLVHVDQG GHVFULEHWKHSRVVLEOHHFKHORQIRUPVRIWKH VWDQGDUGPDWUL[IRUDOLQHDUWUDQVIRUPDWLRQ T 8VHWKHQRWDWLRQRI ([DPSOHLQ6HFWLRQ T W R3 ! R4 LVRQHWRRQH


T W R4 ! R3 LVRQWR

/HW T W Rn ! Rm EH D OLQHDU WUDQVIRUPDWLRQ ZLWK A LWV VWDQGDUGPDWUL[ &RPSOHWHWKHIROORZLQJVWDWHPHQWWRPDNH SLYRW LWWUXH ´T LVRQHWRRQHLIDQGRQO\LI A KDV FROXPQVµ ([SODLQZK\WKHVWDWHPHQWLVWUXH >+LQW /RRN LQWKHH[HUFLVHVIRU6HFWLRQ @ /HW T W Rn ! Rm EH D OLQHDU WUDQVIRUPDWLRQ ZLWK A LWV VWDQGDUGPDWUL[ &RPSOHWHWKHIROORZLQJVWDWHPHQWWRPDNH LWWUXH ´T PDSV Rn RQWR Rm LIDQGRQO\LI A KDV SLYRWFROXPQVµ )LQGVRPHWKHRUHPVWKDWH[SODLQZK\WKH VWDWHPHQWLVWUXH

>0@ ,Q([HUFLVHV² OHW T EHWKHOLQHDUWUDQVIRUPDWLRQZKRVH VWDQGDUGPDWUL[LVJLYHQ ,Q([HUFLVHVDQG GHFLGHLI T LV DRQHWRRQHPDSSLQJ ,Q([HUFLVHVDQG GHFLGHLI T PDSV R5 RQWR R5 -XVWLI\\RXUDQVZHUV 2 3 2 3 "5 6 "5 "6 7 5 9 "9 6 8 6 5 3 "3 87 6 4 "4 7 7 7 6 6 4 2 4 4 9 5 "12 5 8 0 75 "3 2 7 "12 "6 "6 6 5 2 3 4 "7 3 7 5 6 6 "8 7 5 12 "8 6 7 6 6 "7 10 "8 "9 14 7 7 4 3 "5 4 2 "6 5 "5 6 "6 "7 3 2 3 9 43 5 6 "1 6 14 15 "7 "5 47 6 7 6 6 "8 "6 12 "5 "9 7 7 4 "5 "6 "4 9 85 13 14 15 3 11

40-65*0/ 50 13"$5*$& 130#-&. WEB

)ROORZZKDWKDSSHQVWR H1 DQG H2 6HH)LJ )LUVW H1 LVXQDIIHFWHGE\WKHVKHDUDQG WKHQLVUHÁHFWHGLQWR "H1 6R T .H1 / D "H1 6HFRQG H2 JRHVWR H2 " :5H1 E\WKHVKHDU x2

x2

x2

–.5  1

0 1

1 0

x1

Shear transformation

.5 1

1 0

x1

–1  0

Reflection through the x2-axis

'*(63& 7KHFRPSRVLWLRQRIWZRWUDQVIRUPDWLRQV

x1

$)"15&3


WUDQVIRUPDWLRQ 6LQFHUHÁHFWLRQWKURXJKWKH x2 D[LVFKDQJHV H1 LQWR "H1 DQGOHDYHV H2 XQFKDQJHG WKHYHFWRU H2 " :5H1 JRHVWR H2 C :5H1 6R T .H2 / D H2 C :5H1 7KXVWKH VWDQGDUGPDWUL[RI T LV # $ % & % & "1 :5 T .H1 / T .H2 / D "H1 H2 C :5H1 D 0 1

-*/&"3 .0%&-4 */ #64*/&44 4$*&/$& "/% &/(*/&&3*/( 7KHPDWKHPDWLFDOPRGHOVLQWKLVVHFWLRQDUHDOO OLQHDU WKDWLV HDFKGHVFULEHVDSUREOHP E\PHDQVRIDOLQHDUHTXDWLRQ XVXDOO\LQYHFWRURUPDWUL[IRUP 7KHÀUVWPRGHOFRQFHUQV QXWULWLRQEXWDFWXDOO\LVUHSUHVHQWDWLYHRIDJHQHUDOWHFKQLTXHLQOLQHDUSURJUDPPLQJ SUREOHPV 7KHVHFRQGPRGHOFRPHVIURPHOHFWULFDOHQJLQHHULQJ 7KHWKLUGPRGHO LQWURGXFHVWKHFRQFHSWRID OLQHDUGLIIHUHQFHHTXDWLRQ DSRZHUIXOPDWKHPDWLFDOWRROIRU VWXG\LQJG\QDPLFSURFHVVHVLQDZLGHYDULHW\RIÀHOGVVXFKDVHQJLQHHULQJ HFRORJ\ HFRQRPLFV WHOHFRPPXQLFDWLRQV DQGWKHPDQDJHPHQWVFLHQFHV /LQHDUPRGHOVDUH LPSRUWDQWEHFDXVHQDWXUDOSKHQRPHQDDUHRIWHQOLQHDURUQHDUO\OLQHDUZKHQWKHYDULDEOHV LQYROYHGDUHKHOGZLWKLQUHDVRQDEOHERXQGV $OVR OLQHDUPRGHOVDUHPRUHHDVLO\DGDSWHG IRUFRPSXWHUFDOFXODWLRQWKDQDUHFRPSOH[QRQOLQHDUPRGHOV $V\RXUHDGDERXWHDFKPRGHO SD\DWWHQWLRQWRKRZLWVOLQHDULW\UHÁHFWVVRPH SURSHUW\RIWKHV\VWHPEHLQJPRGHOHG

&RQVWUXFWLQJD1XWULWLRXV:HLJKW/RVV'LHW WEB

7KHIRUPXODIRUWKH&DPEULGJH'LHW DSRSXODUGLHWLQWKHV ZDVEDVHGRQ\HDUV RI UHVHDUFK $ WHDP RI VFLHQWLVWV KHDGHG E\ 'U $ODQ + +RZDUG GHYHORSHG WKLV GLHWDW&DPEULGJH8QLYHUVLW\DIWHUPRUHWKDQHLJKW\HDUVRIFOLQLFDOZRUNZLWKREHVH SDWLHQWVñ 7KHYHU\ORZFDORULHSRZGHUHGIRUPXODGLHWFRPELQHVDSUHFLVHEDODQFH RIFDUERK\GUDWH KLJKTXDOLW\SURWHLQ DQGIDW WRJHWKHUZLWKYLWDPLQV PLQHUDOV WUDFH HOHPHQWV DQGHOHFWURO\WHV 0LOOLRQVRISHUVRQVKDYHXVHGWKHGLHWWRDFKLHYHUDSLGDQG VXEVWDQWLDOZHLJKWORVV 7RDFKLHYHWKHGHVLUHGDPRXQWVDQGSURSRUWLRQVRIQXWULHQWV 'U +RZDUGKDGWR LQFRUSRUDWHDODUJHYDULHW\RIIRRGVWXIIVLQWKHGLHW (DFKIRRGVWXIIVXSSOLHGVHYHUDORI WKHUHTXLUHGLQJUHGLHQWV EXWQRWLQWKHFRUUHFWSURSRUWLRQV )RULQVWDQFH QRQIDWPLON ZDVDPDMRUVRXUFHRISURWHLQEXWFRQWDLQHGWRRPXFKFDOFLXP 6RVR\ÁRXUZDVXVHGIRU SDUWRIWKHSURWHLQEHFDXVHVR\ÁRXUFRQWDLQVOLWWOHFDOFLXP +RZHYHU VR\ÁRXUFRQWDLQV SURSRUWLRQDOO\WRRPXFKIDW VRZKH\ZDVDGGHGVLQFHLWVXSSOLHVOHVVIDWLQUHODWLRQWR FDOFLXP 8QIRUWXQDWHO\ ZKH\FRQWDLQVWRRPXFKFDUERK\GUDWH: : : : 7KHIROORZLQJH[DPSOHLOOXVWUDWHVWKHSUREOHPRQDVPDOOVFDOH /LVWHGLQ7DEOH DUHWKUHHRIWKHLQJUHGLHQWVLQWKHGLHW WRJHWKHUZLWKWKHDPRXQWVRIFHUWDLQQXWULHQWV VXSSOLHGE\JUDPVJ RIHDFKLQJUHGLHQWò

&9".1-& ,ISRVVLEOH ÀQGVRPHFRPELQDWLRQRIQRQIDWPLON VR\ÁRXU DQGZKH\ WRSURYLGHWKHH[DFWDPRXQWVRISURWHLQ FDUERK\GUDWH DQGIDWVXSSOLHGE\WKHGLHWLQ RQHGD\7DEOH ñ 7KHÀUVWDQQRXQFHPHQWRIWKLVUDSLGZHLJKWORVVUHJLPHQZDVJLYHQLQWKH ,QWHUQDWLRQDO-RXUQDORI2EHVLW\ ² ò ,QJUHGLHQWVLQWKHGLHWDVRI QXWULHQWGDWDIRULQJUHGLHQWVDGDSWHGIURP86'$ $JULFXOWXUDO +DQGERRNV1RDQG

/LQHDU0RGHOVLQ%XVLQHVV 6FLHQFH DQG(QJLQHHULQJ

5"#-& $PRXQWVJ 6XSSOLHGSHU JRI,QJUHGLHQW 1RQIDWPLON

6R\ÁRXU

:KH\

$PRXQWVJ 6XSSOLHGE\ &DPEULGJH'LHWLQ2QH'D\

3URWHLQ

&DUERK\GUDWH

1XWULHQW

)DW

40-65*0/ /HW x1 x2 DQG x3 UHVSHFWLYHO\ GHQRWHWKHQXPEHURIXQLWVJ RI WKHVHIRRGVWXIIV 2QHDSSURDFKWRWKHSUREOHPLVWRGHULYHHTXDWLRQVIRUHDFKQXWULHQW VHSDUDWHO\ )RULQVWDQFH WKHSURGXFW ' ( ' ( x1 XQLWVRI SURWHLQSHUXQLW ! QRQIDWPLON RIQRQIDWPLON JLYHVWKHDPRXQWRISURWHLQVXSSOLHGE\ x1 XQLWVRIQRQIDWPLON 7RWKLVDPRXQW ZH ZRXOGWKHQDGGVLPLODUSURGXFWVIRUVR\ÁRXUDQGZKH\DQGVHWWKHUHVXOWLQJVXPHTXDO WRWKHDPRXQWRISURWHLQZHQHHG $QDORJRXVFDOFXODWLRQVZRXOGKDYHWREHPDGHIRU HDFKQXWULHQW $ PRUHHIÀFLHQWPHWKRG DQGRQHWKDWLVFRQFHSWXDOO\VLPSOHU LVWRFRQVLGHUD ´QXWULHQWYHFWRUµIRUHDFKIRRGVWXIIDQGEXLOGMXVWRQHYHFWRUHTXDWLRQ 7KHDPRXQWRI QXWULHQWVVXSSOLHGE\ x1 XQLWVRIQRQIDWPLONLVWKHVFDODUPXOWLSOH '

6FDODU ( ' 9HFWRU ( x1 XQLWVRI QXWULHQWVSHUXQLW ! D x1 D1 QRQIDWPLON RIQRQIDWPLON

ZKHUH D1 LVWKHÀUVWFROXPQLQ7DEOH /HW D2 DQG D3 EHWKHFRUUHVSRQGLQJYHFWRUV IRUVR\ÁRXUDQGZKH\ UHVSHFWLYHO\ DQGOHW E EHWKHYHFWRUWKDWOLVWVWKHWRWDOQXWULHQWV UHTXLUHGWKHODVWFROXPQRIWKHWDEOH 7KHQ x2 D2 DQG x3 D3 JLYHWKHQXWULHQWVVXSSOLHG E\ x2 XQLWVRIVR\ÁRXUDQG x3 XQLWVRIZKH\ UHVSHFWLYHO\ 6RWKHUHOHYDQWHTXDWLRQLV

x1 D1 C x2 D2 C x3 D3 D E

5RZUHGXFWLRQRIWKHDXJPHQWHGPDWUL[IRUWKHFRUUHVSRQGLQJV\VWHPRIHTXDWLRQV VKRZVWKDW 2 3 2 3 36 51 13 33 1 0 0 :277 4 52 34 74 45 5 % ! ! ! % 4 0 1 0 :392 5 0 7 1 3 0 0 1 :233

7RWKUHHVLJQLÀFDQWGLJLWV WKHGLHWUHTXLUHVXQLWVRIQRQIDWPLON XQLWVRI VR\ÁRXU DQGXQLWVRIZKH\LQRUGHUWRSURYLGHWKHGHVLUHGDPRXQWVRISURWHLQ FDUERK\GUDWH DQGIDW

,WLVLPSRUWDQWWKDWWKHYDOXHVRI x1 x2 DQG x3 IRXQGDERYHDUHQRQQHJDWLYH 7KLVLV QHFHVVDU\IRUWKHVROXWLRQWREHSK\VLFDOO\IHDVLEOH +RZFRXOG\RXXVH ":233 XQLWVRI ZKH\ IRULQVWDQFH" :LWKDODUJHQXPEHURIQXWULHQWUHTXLUHPHQWV LWPD\EHQHFHVVDU\ WRXVHDODUJHUQXPEHURIIRRGVWXIIVLQRUGHUWRSURGXFHDV\VWHPRIHTXDWLRQVZLWKD ´QRQQHJDWLYHµVROXWLRQ 7KXVPDQ\ PDQ\GLIIHUHQWFRPELQDWLRQVRIIRRGVWXIIVPD\ QHHGWREHH[DPLQHGLQRUGHUWRÀQGDV\VWHPRIHTXDWLRQVZLWKVXFKDVROXWLRQ ,Q IDFW WKHPDQXIDFWXUHURIWKH&DPEULGJH'LHWZDVDEOHWRVXSSO\QXWULHQWVLQSUHFLVH DPRXQWVXVLQJRQO\LQJUHGLHQWV 7KHGLHWFRQVWUXFWLRQSUREOHPOHDGVWRWKH OLQHDU HTXDWLRQ EHFDXVHWKHDPRXQW RIQXWULHQWVVXSSOLHGE\HDFKIRRGVWXIIFDQEHZULWWHQDVDVFDODUPXOWLSOHRIDYHFWRU DV LQ 7KDWLV WKHQXWULHQWVVXSSOLHGE\DIRRGVWXIIDUH SURSRUWLRQDO WRWKHDPRXQWRI

$)"15&3


WKHIRRGVWXIIDGGHGWRWKHGLHWPL[WXUH $OVR HDFKQXWULHQWLQWKHPL[WXUHLVWKH VXP RI WKHDPRXQWVIURPWKHYDULRXVIRRGVWXIIV 3UREOHPVRIIRUPXODWLQJVSHFLDOL]HGGLHWVIRUKXPDQVDQGOLYHVWRFNRFFXUIUH TXHQWO\ 8VXDOO\WKH\DUHWUHDWHGE\OLQHDUSURJUDPPLQJWHFKQLTXHV 2XUPHWKRGRI FRQVWUXFWLQJYHFWRUHTXDWLRQVRIWHQVLPSOLÀHVWKHWDVNRIIRUPXODWLQJVXFKSUREOHPV

/LQHDU(TXDWLRQVDQG(OHFWULFDO1HWZRUNV WEB

&XUUHQWÁRZLQDVLPSOHHOHFWULFDOQHWZRUNFDQEHGHVFULEHGE\DV\VWHPRIOLQHDU HTXDWLRQV $ YROWDJHVRXUFHVXFKDVDEDWWHU\IRUFHVDFXUUHQWRIHOHFWURQVWRÁRZ WKURXJKWKHQHWZRUN :KHQWKHFXUUHQWSDVVHVWKURXJKDUHVLVWRUVXFKDVDOLJKWEXOERU PRWRU VRPHRIWKHYROWDJHLV´XVHGXSµ E\2KP·VODZ WKLVÝROWDJHGURSµDFURVVD UHVLVWRULVJLYHQE\ V D RI ZKHUHWKHYROWDJH V LVPHDVXUHGLQ YROWV WKHUHVLVWDQFH R LQ RKPV GHQRWHGE\ # DQG WKHFXUUHQWÁRZ I LQ DPSHUHV DPSV IRUVKRUW 7KHQHWZRUNLQ)LJ FRQWDLQVWKUHHFORVHGORRSV 7KHFXUUHQWVÁRZLQJLQORRSV DQGDUHGHQRWHGE\ I1 ; I2 DQG I3 UHVSHFWLYHO\ 7KHGHVLJQDWHGGLUHFWLRQVRIVXFK ORRSFXUUHQWV DUHDUELWUDU\ ,IDFXUUHQWWXUQVRXWWREHQHJDWLYH WKHQWKHDFWXDOGLUHFWLRQ RIFXUUHQWÁRZLVRSSRVLWHWRWKDWFKRVHQLQWKHÀJXUH ,IWKHFXUUHQWGLUHFWLRQVKRZQLV DZD\IURPWKHSRVLWLYHORQJHU VLGHRIDEDWWHU\ DURXQGWRWKHQHJDWLYHVKRUWHU VLGH WKHYROWDJHLVSRVLWLYH RWKHUZLVH WKHYROWDJHLVQHJDWLYH &XUUHQWÁRZLQDORRSLVJRYHUQHGE\WKHIROORZLQJUXOH ,*3$))0''h4 70-5"(& -"8 7KHDOJHEUDLFVXPRIWKH RI YROWDJHGURSVLQRQHGLUHFWLRQDURXQGDORRSHTXDOV WKHDOJHEUDLFVXPRIWKHYROWDJHVRXUFHVLQWKHVDPHGLUHFWLRQDURXQGWKHORRS

&9".1-& 'HWHUPLQHWKHORRSFXUUHQWVLQWKHQHWZRUNLQ)LJ 30 volts 4Ω A

4Ω

I1

B

3Ω 1Ω

C

1Ω

5 volts 1Ω

1Ω

I2

I3

D 1Ω

40-65*0/ )RUORRS WKHFXUUHQW I1 ÁRZVWKURXJKWKUHHUHVLVWRUV DQGWKHVXPRIWKH RI YROWDJHGURSVLV 4I1 C 4I1 C 3I1 D .4 C 4 C 3/I1 D 11I1 &XUUHQWIURPORRSDOVRÁRZVLQSDUWRIORRS WKURXJKWKHVKRUW EUDQFK EHWZHHQ A DQG B 7KHDVVRFLDWHG RI GURSWKHUHLV 3I2 YROWV +RZHYHU WKHFXUUHQWGLUHFWLRQIRU WKHEUDQFK $% LQORRSLVRSSRVLWHWRWKDWFKRVHQIRUWKHÁRZLQORRS VRWKHDOJHEUDLF VXPRIDOO RI GURSVIRUORRSLV 11I1 " 3I2 6LQFHWKHYROWDJHLQORRSLV C30 YROWV .LUFKKRII·VYROWDJHODZLPSOLHVWKDW

11I1 " 3I2 D 30 20 volts '*(63&

7KHHTXDWLRQIRUORRSLV

"3I1 C 6I2 " I3 D 5

7KHWHUP "3I1 FRPHVIURPWKHÁRZRIWKHORRSFXUUHQWWKURXJKWKHEUDQFK $% ZLWK DQHJDWLYHYROWDJHGURSEHFDXVHWKHFXUUHQWÁRZWKHUHLVRSSRVLWHWRWKHÁRZLQORRS 7KHWHUP 6I2 LVWKHVXPRIDOOUHVLVWDQFHVLQORRS PXOWLSOLHGE\WKHORRSFXUUHQW 7KH WHUP "I3 D "1 ! I3 FRPHVIURPWKHORRSFXUUHQWÁRZLQJWKURXJKWKHRKPUHVLVWRU LQEUDQFK &' LQWKHGLUHFWLRQRSSRVLWHWRWKHÁRZLQORRS 7KHORRSHTXDWLRQLV

"I2 C 3I3 D "25


1RWHWKDWWKHYROWEDWWHU\LQEUDQFK &' LVFRXQWHGDVSDUWRIERWKORRSDQGORRS EXWLWLV "5 YROWVIRUORRSEHFDXVHRIWKHGLUHFWLRQFKRVHQIRUWKHFXUUHQWLQORRS 7KHYROWEDWWHU\LVQHJDWLYHIRUWKHVDPHUHDVRQ 7KHORRSFXUUHQWVDUHIRXQGE\VROYLQJWKHV\VWHP

11I1 " 3I2 D 30 5 "3I1 C 6I2 " I3 D " I2 C 3I3 D "25

5RZRSHUDWLRQVRQWKHDXJPHQWHGPDWUL[OHDGWRWKHVROXWLRQ I1 D DPSV I2 D DPS DQG I3 D "8 DPSV 7KHQHJDWLYHYDOXHRI I3 LQGLFDWHVWKDWWKHDFWXDOFXUUHQW LQORRSÁRZVLQWKHGLUHFWLRQRSSRVLWHWRWKDWVKRZQLQ)LJ ,WLVLQVWUXFWLYHWRORRNDWV\VWHP DVDYHFWRUHTXDWLRQ 2 3 2 3 2 3 2 3 11 "3 0 30 I1 4 "3 5 C I2 4 6 5 C I3 4 "1 5 D 4 5 5 0 "1 3 "25 "

U1

"

U2

"

U3

"

Y

7KHÀUVWHQWU\RIHDFKYHFWRUFRQFHUQVWKHÀUVWORRS DQGVLPLODUO\IRUWKHVHFRQGDQG WKLUGHQWULHV 7KHÀUVWUHVLVWRUYHFWRU U1 OLVWVWKHUHVLVWDQFHLQWKHYDULRXVORRSVWKURXJK ZKLFKFXUUHQW I1 ÁRZV $ UHVLVWDQFHLVZULWWHQQHJDWLYHO\ZKHQ I1 ÁRZVDJDLQVWWKH ÁRZGLUHFWLRQLQDQRWKHUORRS ([DPLQH)LJ DQGVHHKRZWRFRPSXWHWKHHQWULHVLQ U1 WKHQGRWKHVDPHIRU U2 DQG U3 7KHPDWUL[IRUPRIHTXDWLRQ 2 3 I1 RL D Y; ZKHUH R D Œ U1 U2 U3 ! DQG L D 4 I2 5 I3

SURYLGHVDPDWUL[YHUVLRQRI2KP·VODZ ,IDOOORRSFXUUHQWVDUHFKRVHQLQWKHVDPH GLUHFWLRQVD\ FRXQWHUFORFNZLVH WKHQDOOHQWULHVRIIWKHPDLQGLDJRQDORI R ZLOOEH QHJDWLYH 7KHPDWUL[HTXDWLRQ RL D Y PDNHVWKHOLQHDULW\RIWKLVPRGHOHDV\WRVHHDWDJODQFH )RULQVWDQFH LIWKHYROWDJHYHFWRULVGRXEOHG WKHQWKHFXUUHQWYHFWRUPXVWGRXEOH $OVR D VXSHUSRVLWLRQSULQFLSOH KROGV 7KDWLV WKHVROXWLRQRIHTXDWLRQ LVWKHVXPRIWKH VROXWLRQVRIWKHHTXDWLRQV 2 3 2 3 2 3 30 0 0 RL D 4 0 5; RL D 4 5 5; DQG RL D 4 0 5 0 0 "25

(DFKHTXDWLRQKHUHFRUUHVSRQGVWRWKHFLUFXLWZLWKRQO\RQHYROWDJHVRXUFHWKHRWKHU VRXUFHVEHLQJUHSODFHGE\ZLUHVWKDWFORVHHDFKORRS 7KHPRGHOIRUFXUUHQWÁRZLV OLQHDU SUHFLVHO\EHFDXVH2KP·VODZDQG.LUFKKRII·VODZDUHOLQHDU 7KHYROWDJHGURS DFURVVDUHVLVWRULV SURSRUWLRQDO WRWKHFXUUHQWÁRZLQJWKURXJKLW2KP DQGWKH VXP RI WKHYROWDJHGURSVLQDORRSHTXDOVWKHVXPRIWKHYROWDJHVRXUFHVLQWKHORRS.LUFKKRII /RRSFXUUHQWVLQDQHWZRUNFDQEHXVHGWRGHWHUPLQHWKHFXUUHQWLQDQ\EUDQFKRI WKHQHWZRUN ,IRQO\RQHORRSFXUUHQWSDVVHVWKURXJKDEUDQFK VXFKDVIURP B WR D LQ)LJ WKHEUDQFKFXUUHQWHTXDOVWKHORRSFXUUHQW ,IPRUHWKDQRQHORRSFXUUHQW SDVVHVWKURXJKDEUDQFK VXFKDVIURP A WR B WKHEUDQFKFXUUHQWLVWKHDOJHEUDLFVXP RIWKHORRSFXUUHQWVLQWKHEUDQFK.LUFKKRII·VFXUUHQWODZ )RULQVWDQFH WKHFXUUHQWLQ EUDQFK $% LV I1 " I2 D 3 " 1 D DPSV LQWKHGLUHFWLRQRI I1 7KHFXUUHQWLQEUDQFK &' LV I2 " I3 D DPSV

$)"15&3


'LIIHUHQFH(TXDWLRQV ,QPDQ\ÀHOGVVXFKDVHFRORJ\ HFRQRPLFV DQGHQJLQHHULQJ DQHHGDULVHVWRPRGHO PDWKHPDWLFDOO\DG\QDPLFV\VWHPWKDWFKDQJHVRYHUWLPH 6HYHUDOIHDWXUHVRIWKHV\VWHP DUHHDFKPHDVXUHGDWGLVFUHWHWLPHLQWHUYDOV SURGXFLQJDVHTXHQFHRIYHFWRUV [0 [1 [2 ; : : : : 7KHHQWULHVLQ [k SURYLGHLQIRUPDWLRQDERXWWKH VWDWH RIWKHV\VWHPDWWKHWLPH RIWKH k WKPHDVXUHPHQW ,IWKHUHLVDPDWUL[ A VXFKWKDW [1 D A[0 [2 D A[1 DQG LQJHQHUDO [k C1 D A[k

IRU k D 0; 1; 2; : : :

WKHQ LVFDOOHGD OLQHDUGLIIHUHQFHHTXDWLRQ RU UHFXUUHQFHUHODWLRQ *LYHQVXFK DQHTXDWLRQ RQHFDQFRPSXWH [1 [2 DQGVRRQ SURYLGHG [0 LVNQRZQ 6HFWLRQV DQG DQGVHYHUDOVHFWLRQVLQ&KDSWHU ZLOOGHYHORSIRUPXODVIRU [k DQGGHVFULEH ZKDWFDQKDSSHQWR [k DV k LQFUHDVHVLQGHÀQLWHO\ 7KHGLVFXVVLRQEHORZLOOXVWUDWHVKRZ DGLIIHUHQFHHTXDWLRQPLJKWDULVH $ VXEMHFWRILQWHUHVWWRGHPRJUDSKHUVLVWKHPRYHPHQWRISRSXODWLRQVRUJURXSVRI SHRSOHIURPRQHUHJLRQWRDQRWKHU 7KHVLPSOHPRGHOKHUHFRQVLGHUVWKHFKDQJHVLQWKH SRSXODWLRQRIDFHUWDLQFLW\DQGLWVVXUURXQGLQJVXEXUEVRYHUDSHULRGRI\HDUV )L[DQLQLWLDO\HDU³VD\ ³DQGGHQRWHWKHSRSXODWLRQVRIWKHFLW\DQGVXEXUEV WKDW\HDUE\ r0 DQG s0 UHVSHFWLYHO\ /HW [0 EHWKHSRSXODWLRQYHFWRU # $ r &LW\SRSXODWLRQ [0 D 0 s0 6XEXUEDQSRSXODWLRQ )RUDQGVXEVHTXHQW\HDUV GHQRWHWKHSRSXODWLRQVRIWKHFLW\DQGVXEXUEVE\WKH YHFWRUV # $ # $ # $ r r r [1 D 1 ; [2 D 2 ; [3 D 3 ; : : : s1 s2 s3 2XUJRDOLVWRGHVFULEHPDWKHPDWLFDOO\KRZWKHVHYHFWRUVPLJKWEHUHODWHG 6XSSRVHGHPRJUDSKLFVWXGLHVVKRZWKDWHDFK\HDUDERXWRIWKHFLW\·VSRSXODWLRQ PRYHVWRWKHVXEXUEVDQGUHPDLQVLQWKHFLW\ ZKLOHRIWKHVXEXUEDQSRSXODWLRQ PRYHVWRWKHFLW\DQGUHPDLQVLQWKHVXEXUEV 6HH)LJ City

Suburbs .05

.95

.97 .03

'*(63& $QQXDOSHUFHQWDJHPLJUDWLRQEHWZHHQFLW\DQGVXEXUEV

$IWHU\HDU WKHRULJLQDO r0 SHUVRQVLQWKHFLW\DUHQRZGLVWULEXWHGEHWZHHQFLW\DQG VXEXUEVDV # $ # $ :95r0 :95 5HPDLQLQFLW\ D r0 :05r0 0RYHWRVXEXUEV :05 7KH s0 SHUVRQVLQWKHVXEXUEVLQDUHGLVWULEXWHG\HDUODWHUDV # $ :03 0RYHWRFLW\ s0 5HPDLQLQVXEXUEV :97


7KHYHFWRUVLQ DQG DFFRXQWIRUDOORIWKHSRSXODWLRQLQó 7KXV # $ # $ # $ # $# $ r1 :95 :03 :95 :03 r0 D r0 C s0 D s1 :05 :97 :05 :97 s0 7KDWLV [1 D M [0

ZKHUH M LVWKH PLJUDWLRQPDWUL[ GHWHUPLQHGE\WKHIROORZLQJWDEOH )URP &LW\ 6XEXUEV

7R

:95 :05

&LW\ 6XEXUEV

#

:03 :97

$

(TXDWLRQ GHVFULEHVKRZWKHSRSXODWLRQFKDQJHVIURPWR ,IWKHPLJUDWLRQ SHUFHQWDJHVUHPDLQFRQVWDQW WKHQWKHFKDQJHIURPWRLVJLYHQE\ [2 D M [1 DQGVLPLODUO\IRUWRDQGVXEVHTXHQW\HDUV ,QJHQHUDO [k C1 D M [k

IRU k D 0; 1; 2; : : :

7KHVHTXHQFHRIYHFWRUV f[0 ; [1 ; [2 ; : : :g GHVFULEHVWKHSRSXODWLRQRIWKHFLW\VXEXUEDQ UHJLRQRYHUDSHULRGRI\HDUV

&9".1-& &RPSXWHWKHSRSXODWLRQRIWKHUHJLRQMXVWGHVFULEHGIRUWKH\HDUV DQG JLYHQWKDWWKHSRSXODWLRQLQZDVLQWKHFLW\DQG LQWKHVXEXUEV # $ 600;000 40-65*0/ 7KHLQLWLDOSRSXODWLRQLQLV [0 D )RU 400;000 # $# $ # $ :95 :03 600;000 582;000 [1 D D :05 :97 400;000 418;000 )RU [2 D M [1 D

#

:95 :05

:03 :97

$#

582;000 418;000

$

D

#

565;440 434;560

$

7KHPRGHOIRUSRSXODWLRQPRYHPHQWLQ LV OLQHDU EHFDXVHWKHFRUUHVSRQGHQFH [k 7! [k C1 LVDOLQHDUWUDQVIRUPDWLRQ 7KHOLQHDULW\GHSHQGVRQWZRIDFWV WKHQXPEHU RISHRSOHZKRFKRVHWRPRYHIURPRQHDUHDWRDQRWKHULV SURSRUWLRQDO WRWKHQXPEHURI SHRSOHLQWKDWDUHD DVVKRZQLQ DQG DQGWKHFXPXODWLYHHIIHFWRIWKHVHFKRLFHV LVIRXQGE\ DGGLQJ WKHPRYHPHQWRISHRSOHIURPWKHGLIIHUHQWDUHDV

13"$5*$& 130#-&. )LQGDPDWUL[ A DQGYHFWRUV [ DQG E VXFKWKDWWKHSUREOHPLQ([DPSOHDPRXQWVWR VROYLQJWKHHTXDWLRQ A[ D E ó )RUVLPSOLFLW\ ZHLJQRUHRWKHULQÁXHQFHVRQWKHSRSXODWLRQVXFKDVELUWKV GHDWKV DQGPLJUDWLRQLQWRDQG RXWRIWKHFLW\VXEXUEDQUHJLRQ

$)"15&3


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ÀEHU DQG JRIIDW 2QHVHUYLQJRI&ULVSL[ VXSSOLHVFDORULHV JRISURWHLQ JRIÀEHU DQG JRI IDW D 6HWXSDPDWUL[ B DQGDYHFWRU X VXFKWKDW B X JLYHVWKH DPRXQWVRIFDORULHV SURWHLQ ÀEHU DQGIDWFRQWDLQHGLQ DPL[WXUHRIWKUHHVHUYLQJVRI6KUHGGHG:KHDWDQGWZR VHUYLQJVRI&ULVSL[

FODVVLFDO 0DF DQG &KHHVH WR $QQLH·V :KROH :KHDW 6KHOOVDQG:KLWH&KHGGDU :KDWSURSRUWLRQVRIVHUYLQJV RIHDFKIRRGVKRXOGVKHXVHWRPHHWWKHVDPHJRDOVDVLQ SDUWD " 7KH&DPEULGJH'LHWVXSSOLHV JRIFDOFLXPSHUGD\ LQ DGGLWLRQWRWKHQXWULHQWVOLVWHGLQWKH7DEOHIRU([DPSOH 7KHDPRXQWVRIFDOFLXPSHUXQLW J VXSSOLHGE\WKH WKUHHLQJUHGLHQWVLQWKH&DPEULGJH'LHWDUHDVIROORZV J IURPQRQIDWPLON JIURPVR\ÁRXU DQG JIURPZKH\ $QRWKHULQJUHGLHQWLQWKHGLHWPL[WXUHLVLVRODWHGVR\SURWHLQ ZKLFKSURYLGHVWKHIROORZLQJQXWULHQWVLQHDFKXQLW JRI SURWHLQ JRIFDUERK\GUDWH JRIIDW DQG JRIFDOFLXP D 6HWXSDPDWUL[HTXDWLRQZKRVHVROXWLRQGHWHUPLQHVWKH DPRXQWVRIQRQIDWPLON VR\ÁRXU ZKH\ DQGLVRODWHG VR\SURWHLQQHFHVVDU\WRVXSSO\WKHSUHFLVHDPRXQWVRI SURWHLQ FDUERK\GUDWH IDW DQGFDOFLXPLQWKH&DPEULGJH 'LHW 6WDWHZKDWWKHYDULDEOHVLQWKHHTXDWLRQUHSUHVHQW E >0@ 6ROYHWKHHTXDWLRQLQD DQGGLVFXVV\RXUDQVZHU ,Q([HUFLVHV² ZULWHDPDWUL[HTXDWLRQWKDWGHWHUPLQHVWKHORRS FXUUHQWV >0@ ,I0$7/$% RUDQRWKHUPDWUL[SURJUDPLVDYDLODEOH VROYHWKHV\VWHPIRUWKHORRSFXUUHQWV 20 V 1Ω

3Ω

D >0@ ,IVKHZDQWVWROLPLWKHUOXQFKWRFDORULHVEXW JHW JRISURWHLQDQG JRIÀEHU ZKDWSURSRUWLRQVRI VHUYLQJVRI0DFDQG&KHHVH EURFFROL DQGFKLFNHQVKRXOG VKHXVH" E >0@ 6KHIRXQGWKDWWKHUHZDVWRRPXFKEURFFROLLQWKH SURSRUWLRQVIURPSDUWD VRVKHGHFLGHGWRVZLWFKIURP

I2

4Ω

1Ω

20 V

I2

40 V

I3

10 V

I4

3Ω

4Ω

4Ω

7Ω 30 V

I2 2Ω

2Ω 3Ω

4Ω

I1

1Ω 2Ω

40 V

1Ω

4Ω 4Ω

20 V I4

2Ω 1Ω

2Ω

I3

10 V

4Ω

I1

1Ω

2Ω 1Ω

3Ω 30 V

30 V

10 V

$IWHUWDNLQJDQXWULWLRQFODVV DELJ$QQLH·V 0DFDQG&KHHVH IDQ GHFLGHVWR LPSURYHWKHOHYHOVRISURWHLQDQG ÀEHULQ KHUIDYRULWHOXQFKE\DGGLQJEURFFROLDQGFDQQHGFKLFNHQ 7KHQXWULWLRQDOLQIRUPDWLRQIRUWKHIRRGVUHIHUUHGWRLQWKLV H[HUFLVHDUHJLYHQLQWKHWDEOHEHORZ 1XWULWLRQ,QIRUPDWLRQSHU6HUYLQJ 1XWULHQW 0DFDQG&KHHVH %URFFROL &KLFNHQ 6KHOOV &DORULHV 3URWHLQJ )LEHUJ

I1

5Ω

E >0@ 6XSSRVHWKDW\RXZDQWDFHUHDOZLWKPRUHÀEHUWKDQ &ULVSL[EXWIHZHUFDORULHVWKDQ6KUHGGHG:KHDW ,VLW SRVVLEOHIRUDPL[WXUHRIWKHWZRFHUHDOVWRVXSSO\ FDORULHV JRISURWHLQ JRIÀEHU DQG JRI IDW" ,IVR ZKDWLVWKHPL[WXUH"

1Ω

I4

10 V

5Ω 6Ω

I3 20 V

3Ω


50 V

40 V

1Ω I1

3Ω

I4 4Ω

1Ω

2Ω 3Ω

I5 3Ω

1Ω

1Ω

3Ω

I2

I3

2Ω

2Ω

30 V

>0@ %XGJHW 5HQW$ &DULQ:LFKLWD .DQVDVKDVDÁHHWRI DERXWFDUV DWWKUHHORFDWLRQV $ FDUUHQWHGDWRQHORFDWLRQ PD\EHUHWXUQHGWRDQ\RIWKHWKUHHORFDWLRQV 7KHYDULRXV IUDFWLRQVRIFDUVUHWXUQHGWRWKHWKUHHORFDWLRQVDUHVKRZQLQ WKHPDWUL[EHORZ 6XSSRVHWKDWRQ0RQGD\WKHUHDUHFDUV DWWKHDLUSRUWRUUHQWHGIURPWKHUH FDUVDWWKHHDVWVLGH RIÀFH DQGFDUVDWWKHZHVWVLGHRIÀFH :KDWZLOOEHWKH DSSUR[LPDWHGLVWULEXWLRQRIFDUVRQ:HGQHVGD\" &DUV5HQWHG)URP $LUSRUW (DVW :HVW 2 3 :97 :05 :10 4:00 :90 :055 :03 :05 :85

5HWXUQHG7R $LUSRUW (DVW :HVW

20 V

,QDFHUWDLQUHJLRQ DERXWRIDFLW\·VSRSXODWLRQPRYHV WRWKHVXUURXQGLQJVXEXUEVHDFK\HDU DQGDERXWRIWKH VXEXUEDQSRSXODWLRQPRYHVLQWRWKHFLW\ ,Q WKHUHZHUH UHVLGHQWVLQWKHFLW\DQGLQWKHVXEXUEV 6HWXSDGLIIHUHQFHHTXDWLRQWKDWGHVFULEHVWKLVVLWXDWLRQ ZKHUH [0 LVWKHLQLWLDOSRSXODWLRQLQ 7KHQHVWLPDWH WKH SRSXODWLRQV LQ WKH FLW\ DQG LQ WKH VXEXUEV WZR \HDUV ODWHU LQ ,JQRUHRWKHUIDFWRUVWKDWPLJKWLQÁXHQFHWKH SRSXODWLRQVL]HV ,QDFHUWDLQUHJLRQ DERXWRIDFLW\·VSRSXODWLRQPRYHV WRWKHVXUURXQGLQJVXEXUEVHDFK\HDU DQGDERXWRIWKH VXEXUEDQSRSXODWLRQPRYHVLQWRWKHFLW\ ,Q WKHUHZHUH UHVLGHQWVLQWKHFLW\DQGLQWKHVXEXUEV 6HWXSDGLIIHUHQFHHTXDWLRQWKDWGHVFULEHVWKLVVLWXDWLRQ ZKHUH [0 LVWKHLQLWLDOSRSXODWLRQLQ 7KHQHVWLPDWHWKH SRSXODWLRQVLQWKHFLW\DQGLQWKHVXEXUEVWZR\HDUVODWHU LQ ,Q WKHSRSXODWLRQRI&DOLIRUQLDZDV DQG WKHSRSXODWLRQOLYLQJLQWKH8QLWHG6WDWHVEXW RXWVLGH &DOL IRUQLDZDV 'XULQJWKH\HDU LWLVHVWLPDWHGWKDW SHUVRQVPRYHGIURP&DOLIRUQLDWRHOVHZKHUHLQWKH 8QLWHG6WDWHV ZKLOHSHUVRQVPRYHGWR&DOLIRUQLD IURPHOVHZKHUHLQWKH8QLWHG6WDWHV D 6HWXSWKHPLJUDWLRQPDWUL[IRUWKLVVLWXDWLRQ XVLQJÀYH GHFLPDOSODFHVIRUWKHPLJUDWLRQUDWHVLQWRDQGRXWRI &DOLIRUQLD /HW\RXUZRUNVKRZKRZ\RXSURGXFHGWKH PLJUDWLRQPDWUL[ E >0@ &RPSXWHWKHSURMHFWHGSRSXODWLRQVLQWKH\HDU IRU&DOLIRUQLDDQGHOVHZKHUHLQWKH8QLWHG6WDWHV DVVXP LQJWKDWWKHPLJUDWLRQUDWHVGLGQRWFKDQJHGXULQJWKH \HDUSHULRG 7KHVHFDOFXODWLRQVGRQRWWDNHLQWRDFFRXQW ELUWKV GHDWKV RUWKHVXEVWDQWLDOPLJUDWLRQRISHUVRQVLQWR &DOLIRUQLDDQGHOVHZKHUHLQWKH8QLWHG6WDWHVIURPRWKHU FRXQWULHV 4 0LJUDWLRQGDWDVXSSOLHGE\WKH'HPRJUDSKLF5HVHDUFK8QLWRIWKH

&DOLIRUQLD6WDWH'HSDUWPHQWRI)LQDQFH

>0@ /HW M DQG [0 EHDVLQ([DPSOH D &RPSXWHWKHSRSXODWLRQYHFWRUV [k IRU k D 1; : : : ; 20 'LVFXVVZKDW\RXÀQG E 5HSHDWSDUWD ZLWKDQLQLWLDOSRSXODWLRQRILQ WKHFLW\DQGLQWKHVXEXUEV :KDWGR\RXÀQG" >0@ 6WXG\KRZFKDQJHVLQERXQGDU\WHPSHUDWXUHVRQDVWHHO SODWHDIIHFWWKHWHPSHUDWXUHVDWLQWHULRUSRLQWVRQWKHSODWH D %HJLQE\HVWLPDWLQJWKHWHPSHUDWXUHV T1 T2 T3 T4 DW HDFKRIWKHVHWVRIIRXUSRLQWVRQWKHVWHHOSODWHVKRZQLQ WKHÀJXUH ,QHDFKFDVH WKHYDOXHRI Tk LVDSSUR[LPDWHG E\WKHDYHUDJHRIWKHWHPSHUDWXUHVDWWKHIRXUFORVHVW SRLQWV 6HH([HUFLVHVDQGLQ6HFWLRQ ZKHUH WKHYDOXHVLQGHJUHHV WXUQRXWWREH .20; 27:5; 30; 22:5/ +RZLVWKLVOLVWRIYDOXHVUHODWHGWR\RXUUHVXOWVIRUWKH SRLQWVLQVHWD DQGVHWE " E :LWKRXW PDNLQJ DQ\ FRPSXWDWLRQV JXHVV WKH LQWHULRU WHPSHUDWXUHVLQD ZKHQWKHERXQGDU\WHPSHUDWXUHVDUH DOOPXOWLSOHGE\ &KHFN\RXUJXHVV F )LQDOO\ PDNHDJHQHUDOFRQMHFWXUHDERXWWKHFRUUHVSRQ GHQFHIURPWKHOLVWRIHLJKWERXQGDU\WHPSHUDWXUHVWRWKH OLVWRIIRXULQWHULRUWHPSHUDWXUHV

0º 0º

Plate A

Plate B

20º

20º

0º

1

2

4

3

20º

20º

(a)

0º

10º

0º

10º

0º

1

2

4

3

10º

10º

(b)

40º 40º

$)"15&3


40-65*0/ 50 13"$5*$& 130#-&. 2

36 A D 4 52 0

3 13 74 5; 1:1

51 34 7

2

3 x1 [ D 4 x2 5; x3

2

3 33 E D 4 45 5 3

$)"15&3 4611-&.&/5"3: &9&3$*4&4 0DUNHDFKVWDWHPHQW7UXHRU)DOVH -XVWLI\HDFKDQVZHU ,I WUXH FLWHDSSURSULDWHIDFWVRUWKHRUHPV ,IIDOVH H[SODLQZK\ RUJLYHDFRXQWHUH[DPSOHWKDWVKRZVZK\WKHVWDWHPHQWLVQRW WUXHLQHYHU\FDVH D (YHU\PDWUL[LVURZHTXLYDOHQWWRDXQLTXHPDWUL[LQ HFKHORQIRUP E $Q\V\VWHPRI n OLQHDUHTXDWLRQVLQ n YDULDEOHVKDVDW PRVW n VROXWLRQV F

,IDV\VWHPRIOLQHDUHTXDWLRQVKDVWZRGLIIHUHQWVROX WLRQV LWPXVWKDYHLQÀQLWHO\PDQ\VROXWLRQV

G ,IDV\VWHPRIOLQHDUHTXDWLRQVKDVQRIUHHYDULDEOHV WKHQ LWKDVDXQLTXHVROXWLRQ H

I

,I DQ DXJPHQWHG PDWUL[ Œ A E ! LV WUDQVIRUPHG LQWR Œ C G ! E\HOHPHQWDU\URZRSHUDWLRQV WKHQWKHHTXD WLRQV A[ D E DQG C [ D G KDYHH[DFWO\WKHVDPHVROX WLRQVHWV ,IDV\VWHP A[ D E KDVPRUHWKDQRQHVROXWLRQ WKHQVR GRHVWKHV\VWHP A[ D

J ,I A LVDQ m # n PDWUL[DQGWKHHTXDWLRQ A[ D E LV FRQVLVWHQWIRUVRPH E WKHQWKHFROXPQVRI A VSDQ Rm

K ,IDQDXJPHQWHGPDWUL[ Œ A E ! FDQEHWUDQVIRUPHGE\ HOHPHQWDU\URZRSHUDWLRQVLQWRUHGXFHGHFKHORQIRUP WKHQWKHHTXDWLRQ A[ D E LVFRQVLVWHQW

R ,I A LVDQ m # n PDWUL[ LIWKHHTXDWLRQ A[ D E KDVDW OHDVWWZRGLIIHUHQWVROXWLRQV DQGLIWKHHTXDWLRQ A[ D F LVFRQVLVWHQW WKHQWKHHTXDWLRQ A[ D F KDVPDQ\VROX WLRQV S ,I A DQG B DUHURZHTXLYDOHQW m # n PDWULFHVDQGLIWKH FROXPQVRI A VSDQ Rm WKHQVRGRWKHFROXPQVRI B T ,IQRQHRIWKHYHFWRUVLQWKHVHW S D fY1 ; Y2 ; Y3 g LQ R3 LV DPXOWLSOHRIRQHRIWKHRWKHUYHFWRUV WKHQ S LVOLQHDUO\ LQGHSHQGHQW U

,I fX; Y; Zg LVOLQHDUO\LQGHSHQGHQW WKHQ X Y DQG Z DUH QRWLQ R2

V

,QVRPHFDVHV LWLVSRVVLEOHIRUIRXUYHFWRUVWRVSDQ R5

W

,I X DQG Y DUHLQ Rm WKHQ "X LVLQ 6SDQfX; Yg

X ,I X Y DQG Z DUHQRQ]HURYHFWRUVLQ R2 WKHQ Z LVDOLQHDU FRPELQDWLRQRI X DQG Y Y

,I Z LVDOLQHDUFRPELQDWLRQRI X DQG Y LQ Rn WKHQ X LVD OLQHDUFRPELQDWLRQRI Y DQG Z

Z 6XSSRVHWKDW Y1 Y2 DQG Y3 DUHLQ R5 Y2 LVQRWDPXOWLSOH RI Y1 DQG Y3 LVQRWDOLQHDUFRPELQDWLRQRI Y1 DQG Y2 7KHQ fY1 ; Y2 ; Y3 g LVOLQHDUO\LQGHSHQGHQW [ $ OLQHDUWUDQVIRUPDWLRQLVDIXQFWLRQ

L

,IPDWULFHV A DQG B DUHURZHTXLYDOHQW WKH\KDYHWKH VDPHUHGXFHGHFKHORQIRUP

\

,I A LVD 6 # 5 PDWUL[ WKHOLQHDUWUDQVIRUPDWLRQ [ 7! A[ FDQQRWPDS R5 RQWR R6

M

7KHHTXDWLRQ A[ D KDVWKHWULYLDOVROXWLRQLIDQGRQO\ LIWKHUHDUHQRIUHHYDULDEOHV

]

,I A LVDQ m # n PDWUL[ZLWK m SLYRWFROXPQV WKHQWKH OLQHDUWUDQVIRUPDWLRQ [ 7! A[ LVDRQHWRRQHPDSSLQJ

N ,I A LVDQ m # n PDWUL[DQGWKHHTXDWLRQ A[ D E LVFRQ VLVWHQWIRUHYHU\ E LQ Rm WKHQ A KDV m SLYRWFROXPQV O

,IDQ m # n PDWUL[ A KDVDSLYRWSRVLWLRQLQHYHU\URZ WKHQWKHHTXDWLRQ A[ D E KDVDXQLTXHVROXWLRQIRUHDFK E LQ Rm

P ,IDQ n # n PDWUL[ A KDV n SLYRWSRVLWLRQV WKHQWKH UHGXFHGHFKHORQIRUPRI A LVWKH n # n LGHQWLW\PDWUL[

Q ,I 3 # 3 PDWULFHV A DQG B HDFKKDYHWKUHHSLYRWSRVL WLRQV WKHQ A FDQEHWUDQVIRUPHGLQWR B E\HOHPHQWDU\ URZRSHUDWLRQV

/HW a DQG b UHSUHVHQWUHDOQXPEHUV 'HVFULEHWKHSRVVLEOH VROXWLRQVHWVRIWKHOLQHDU HTXDWLRQ ax D b >+LQW 7KH QXPEHURIVROXWLRQVGHSHQGVXSRQ a DQG b @ 7KHVROXWLRQV .x; y; ´/ RIDVLQJOHOLQHDUHTXDWLRQ

ax C by C c´ D d

IRUPDSODQHLQ R3 ZKHQ a b DQG c DUHQRWDOO]HUR &RQVWUXFW VHWVRIWKUHHOLQHDUHTXDWLRQVZKRVHJUDSKVD LQWHUVHFWLQ DVLQJOHOLQH E LQWHUVHFWLQDVLQJOHSRLQW DQGF KDYHQR

&KDSWHU6XSSOHPHQWDU\([HUFLVHV

SRLQWVLQFRPPRQ 7\SLFDOJUDSKVDUHLOOXVWUDWHGLQWKHÀJXUH

F 'HÀQHDQDSSURSULDWHOLQHDUWUDQVIRUPDWLRQ T XVLQJWKH PDWUL[LQE DQGUHVWDWHWKHSUREOHPLQWHUPVRI T 'HVFULEHWKHSRVVLEOHHFKHORQIRUPVRIWKHPDWUL[ A 8VHWKH QRWDWLRQRI([DPSOHLQ6HFWLRQ D A LVD 2 # 3 PDWUL[ZKRVHFROXPQVVSDQ R2

Three planes intersecting in a line (a)

Three planes intersecting in a point (b)

E A LVD 3 # 3 PDWUL[ZKRVHFROXPQVVSDQ R3 # $ 5 :ULWH WKH YHFWRU DV WKH VXP RI WZR YHFWRUV 6 RQH RQ WKH OLQH f.x; y/ W y D 2xg DQG RQH RQ WKH OLQH f.x; y/ W y D x=2g

/HW D1 ; D2 DQG E EHWKHYHFWRUVLQ R2 VKRZQLQWKHÀJXUH DQG OHW A D ŒD1 D2 ! 'RHVWKHHTXDWLRQ A[ D E KDYHDVROXWLRQ" ,IVR LVWKHVROXWLRQXQLTXH" ([SODLQ x2 b a1 Three planes with no intersection (c)

Three planes with no intersection (c')

a2

6XSSRVHWKHFRHIÀFLHQWPDWUL[RIDOLQHDUV\VWHPRIWKUHH HTXDWLRQV LQ WKUHH YDULDEOHV KDV D SLYRW SRVLWLRQ LQ HDFK FROXPQ ([SODLQZK\WKHV\VWHPKDVDXQLTXHVROXWLRQ 'HWHUPLQH h DQG k VXFKWKDWWKHVROXWLRQVHWRIWKHV\VWHP L LVHPSW\ LL FRQWDLQVDXQLTXHVROXWLRQ DQGLLL FRQWDLQV LQÀQLWHO\PDQ\VROXWLRQV D

x1 C 3x2 D k

4x1 C hx2 D 8

E "2x1 C hx2 D

x1

1

6x1 C kx2 D "2

&RQVLGHUWKHSUREOHPRIGHWHUPLQLQJZKHWKHUWKHIROORZLQJ V\VWHPRIHTXDWLRQVLVFRQVLVWHQW

4x1 " 2x2 C 7x3 D "5 8x1 " 3x2 C 10x3 D "3

D 'HÀQHDSSURSULDWHYHFWRUV DQGUHVWDWHWKHSUREOHPLQ WHUPVRIOLQHDUFRPELQDWLRQV 7KHQVROYHWKDWSUREOHP E 'HÀQHDQDSSURSULDWHPDWUL[ DQGUHVWDWHWKHSUREOHP XVLQJWKHSKUDVH´FROXPQVRI Aµ F 'HÀQHDQDSSURSULDWHOLQHDUWUDQVIRUPDWLRQ T XVLQJWKH PDWUL[LQE DQGUHVWDWHWKHSUREOHPLQWHUPVRI T &RQVLGHUWKHSUREOHPRIGHWHUPLQLQJZKHWKHUWKHIROORZLQJ V\VWHPRIHTXDWLRQVLVFRQVLVWHQWIRUDOO b1 b2 b3

2x1 " 4x2 " 2x3 D b1

"5x1 C x2 C x3 D b2 7x1 " 5x2 " 3x3 D b3

D 'HÀQHDSSURSULDWHYHFWRUV DQGUHVWDWHWKHSUREOHPLQ WHUPVRI 6SDQ fY1 ; Y2 ; Y3 g 7KHQVROYHWKDWSUREOHP

E 'HÀQHDQDSSURSULDWHPDWUL[ DQGUHVWDWHWKHSUREOHP XVLQJWKHSKUDVH´FROXPQVRI Aµ

&RQVWUXFWD 2 # 3 PDWUL[ A QRWLQHFKHORQIRUP VXFKWKDW WKHVROXWLRQRI A[ D LVDOLQHLQ R3

&RQVWUXFWD 2 # 3 PDWUL[ A QRWLQHFKHORQIRUP VXFKWKDW WKHVROXWLRQRI A[ D LVDSODQHLQ R3

:ULWHWKH UHGXFHG HFKHORQIRUPRID 3 # 3 PDWUL[ A VXFK WKDW 2 WKH3 ÀUVW2 WZR 3 FROXPQV RI A DUH SLYRW FROXPQV DQG 3 0 A4 "2 5 D 4 0 5 1 0 '# $ # $( 1 a ; LV 'HWHUPLQHWKHYDOXHV RI a VXFKWKDW a aC2 OLQHDUO\LQGHSHQGHQW

,QD DQGE VXSSRVHWKHYHFWRUVDUHOLQHDUO\LQGHSHQGHQW :KDWFDQ\RXVD\DERXWWKHQXPEHUV a; : : : ; f " -XVWLI\\RXU DQVZHUV >+LQW 8VHDWKHRUHPIRUE @ 2 3 2 3 2 3 2 3 2 3 2 3 a b d a b d 617 6c7 6 e 7 6 7 6 7 6 4 5 4 5 4 5 0 c e E 4 5 4 5 4 7 D 0 1 f 5 0 0 f 0 0 1

8VH7KHRUHPLQ6HFWLRQ WRH[SODLQZK\WKHFROXPQVRI WKHPDWUL[ A DUHOLQHDUO\LQGHSHQGHQW 2 3 1 0 0 0 62 5 0 07 7 AD6 43 6 8 05 4 7 9 10 ([SODLQ ZK\ D VHW fY1 ; Y2 ; Y3 ; Y4 g LQ R5 PXVW EH OLQHDUO\ LQGHSHQGHQWZKHQ fY1 ; Y2 ; Y3 g LVOLQHDUO\LQGHSHQGHQWDQG Y4 LV QRW LQ 6SDQ fY1 ; Y2 ; Y3 g

6XSSRVH fY1 ; Y2 g LVDOLQHDUO\LQGHSHQGHQWVHWLQ Rn 6KRZ WKDW fY1 ; Y1 C Y2 g LVDOVROLQHDUO\LQGHSHQGHQW

$)"15&3


6XSSRVH Y1 ; Y2 ; Y3 DUHGLVWLQFWSRLQWVRQRQHOLQHLQ R3 7KH OLQHQHHGQRWSDVVWKURXJKWKHRULJLQ 6KRZWKDW fY1 ; Y2 ; Y3 g LVOLQHDUO\GHSHQGHQW /HW T W Rn ! Rm EHDOLQHDUWUDQVIRUPDWLRQ DQGVXSSRVH T .X/ D Y 6KRZWKDW T ."X/ D "Y

/HW T W R ! R EH WKH OLQHDU WUDQVIRUPDWLRQ WKDW UH ÁHFWV HDFK YHFWRU WKURXJK WKH SODQH x2 D 0 7KDW LV T .x1 ; x2 ; x3 / D .x1 ; "x2 ; x3 / )LQGWKHVWDQGDUGPDWUL[RI T 3

3

/HW A EHD 3 # 3 PDWUL[ZLWKWKHSURSHUW\WKDWWKHOLQHDU WUDQVIRUPDWLRQ [ 7! A[ PDSV R3 RQWR R3 ([SODLQZK\WKH WUDQVIRUPDWLRQPXVWEHRQHWRRQH $ *LYHQVURWDWLRQ LVDOLQHDUWUDQVIRUPDWLRQIURP Rn WR Rn XVHGLQFRPSXWHUSURJUDPVWRFUHDWHD]HURHQWU\LQDYHFWRU XVXDOO\DFROXPQRIDPDWUL[ 7KHVWDQGDUGPDWUL[RID *LYHQVURWDWLRQLQ R2 KDVWKHIRUP # $ a "b ; a2 C b 2 D 1 b a # $ # $ 4 5 )LQG a DQG b VXFKWKDW LVURWDWHGLQWR 3 0 x2 (4, 3)

(5, 0)

$ *LYHQVURWDWLRQLQ R2 WEB

x1

7KHIROORZLQJHTXDWLRQGHVFULEHVD*LYHQVURWDWLRQLQ R3 )LQG a DQG b 2

a 40 b

32 3 2 p 3 0 "b 2 2 5 1 0 54 3 5 D 4 3 5 ; 0 a 4 0

a2 C b 2 D 1

$ ODUJH DSDUWPHQW EXLOGLQJ LV WR EH EXLOW XVLQJ PRGXODU FRQVWUXFWLRQ WHFKQLTXHV 7KH DUUDQJHPHQW RI DSDUWPHQWV RQDQ\SDUWLFXODUÁRRULVWREHFKRVHQIURPRQHRIWKUHH EDVLFÁRRUSODQV 3ODQ$ KDVDSDUWPHQWVRQRQHÁRRU LQFOXGLQJWKUHHEHGURRPXQLWV WZREHGURRPXQLWV DQG RQHEHGURRPXQLWV (DFKÁRRURISODQ% LQFOXGHVWKUHH EHGURRPXQLWV WZREHGURRPXQLWV DQGRQHEHGURRP XQLWV (DFKÁRRURISODQ& LQFOXGHVWKUHHEHGURRPXQLWV WZREHGURRPXQLWV DQGRQHEHGURRPXQLWV 6XSSRVHWKH EXLOGLQJFRQWDLQVDWRWDORI x1 ÁRRUVRISODQ$ x2 ÁRRUVRI SODQ%DQG x3 ÁRRUVRISODQ& 2 3 3 D :KDWLQWHUSUHWDWLRQFDQEHJLYHQWRWKHYHFWRU x1 4 7 5" 8 E :ULWHDIRUPDOOLQHDUFRPELQDWLRQ RIYHFWRUVWKDWH[ SUHVVHV WKH WRWDO QXPEHUV RI WKUHH WZR DQG RQH EHGURRPDSDUWPHQWVFRQWDLQHGLQWKHEXLOGLQJ F >0@ ,VLWSRVVLEOHWRGHVLJQWKHEXLOGLQJZLWKH[DFWO\ WKUHHEHGURRPXQLWV WZREHGURRPXQLWV DQGRQH EHGURRPXQLWV" ,IVR LVWKHUHPRUHWKDQRQHZD\WRGR LW" ([SODLQ\RXUDQVZHU

0DWUL[$OJHEUD

*/530%6$503: &9".1-&

&RPSXWHU0RGHOVLQ$LUFUDIW'HVLJQ 7RGHVLJQWKHQH[WJHQHUDWLRQRIFRPPHUFLDODQGPLOLWDU\ DLUFUDIW HQJLQHHUVDW%RHLQJ·V3KDQWRP:RUNVXVH' PRGHOLQJDQGFRPSXWDWLRQDOÁXLGG\QDPLFV&)' 7KH\ VWXG\WKHDLUÁRZDURXQGDYLUWXDODLUSODQHWRDQVZHU LPSRUWDQWGHVLJQTXHVWLRQVEHIRUHSK\VLFDOPRGHOVDUH FUHDWHG 7KLVKDVGUDVWLFDOO\UHGXFHGGHVLJQF\FOHWLPHV DQGFRVW³DQGOLQHDUDOJHEUDSOD\VDFUXFLDOUROHLQWKH SURFHVV

RULJLQDOZLUHIUDPHPRGHO %R[HVLQWKLVJULGOLHHLWKHU FRPSOHWHO\LQVLGHRUFRPSOHWHO\RXWVLGHWKHSODQH RUWKH\ LQWHUVHFWWKHVXUIDFHRIWKHSODQH 7KHFRPSXWHUVHOHFWV WKHER[HVWKDWLQWHUVHFWWKHVXUIDFHDQGVXEGLYLGHVWKHP UHWDLQLQJRQO\WKHVPDOOHUER[HVWKDWVWLOOLQWHUVHFWWKH VXUIDFH 7KHVXEGLYLGLQJSURFHVVLVUHSHDWHGXQWLOWKHJULG LVH[WUHPHO\ÀQH $ W\SLFDOJULGFDQLQFOXGHRYHU ER[HV

7KHYLUWXDODLUSODQHEHJLQVDVDPDWKHPDWLFDO´ZLUH IUDPHµPRGHOWKDWH[LVWVRQO\LQFRPSXWHUPHPRU\DQG RQJUDSKLFVGLVSOD\WHUPLQDOV $ PRGHORID%RHLQJ LVVKRZQ 7KLVPDWKHPDWLFDOPRGHORUJDQL]HVDQG LQÁXHQFHVHDFKVWHSRIWKHGHVLJQDQGPDQXIDFWXUHRIWKH DLUSODQH³ERWKWKHH[WHULRUDQGLQWHULRU 7KH&)' DQDO\VLV FRQFHUQVWKHH[WHULRUVXUIDFH

7KHSURFHVVIRUÀQGLQJWKHDLUÁRZDURXQGWKHSODQH LQYROYHVUHSHDWHGO\VROYLQJDV\VWHPRIOLQHDUHTXDWLRQV A[ D E WKDWPD\LQYROYHXSWRPLOOLRQHTXDWLRQVDQG YDULDEOHV 7KHYHFWRU E FKDQJHVHDFKWLPH EDVHGRQGDWD IURPWKHJULGDQGVROXWLRQVRISUHYLRXVHTXDWLRQV 8VLQJ WKHIDVWHVWFRPSXWHUVDYDLODEOHFRPPHUFLDOO\ D3KDQWRP :RUNVWHDPFDQVSHQGIURPDIHZKRXUVWRVHYHUDOGD\V VHWWLQJXSDQGVROYLQJDVLQJOHDLUÁRZSUREOHP $IWHUWKH WHDPDQDO\]HVWKHVROXWLRQ WKH\PD\PDNHVPDOOFKDQJHV WRWKHDLUSODQHVXUIDFHDQGEHJLQWKHZKROHSURFHVVDJDLQ 7KRXVDQGVRI&)' UXQVPD\EHUHTXLUHG

$OWKRXJKWKHÀQLVKHGVNLQRIDSODQHPD\VHHP VPRRWK WKHJHRPHWU\RIWKHVXUIDFHLVFRPSOLFDWHG ,Q DGGLWLRQWRZLQJVDQGDIXVHODJH DQDLUFUDIWKDVQDFHOOHV VWDELOL]HUV VODWV ÁDSV DQGDLOHURQV 7KHZD\DLUÁRZV DURXQGWKHVHVWUXFWXUHVGHWHUPLQHVKRZWKHSODQHPRYHV WKURXJKWKHVN\ (TXDWLRQVWKDWGHVFULEHWKHDLUÁRZDUH FRPSOLFDWHG DQGWKH\PXVWDFFRXQWIRUHQJLQHLQWDNH HQJLQHH[KDXVW DQGWKHZDNHVOHIWE\WKHZLQJVRIWKH SODQH 7RVWXG\WKHDLUÁRZ HQJLQHHUVQHHGDKLJKO\UHÀQHG GHVFULSWLRQRIWKHSODQH·VVXUIDFH $ FRPSXWHUFUHDWHVDPRGHORIWKHVXUIDFHE\ÀUVW VXSHULPSRVLQJDWKUHHGLPHQVLRQDOJULGRIÉR[HVµRQWKH

7KLVFKDSWHUSUHVHQWVWZRLPSRUWDQWFRQFHSWVWKDW DVVLVWLQWKHVROXWLRQRIVXFKPDVVLYHV\VWHPVRIHTXDWLRQV !

3DUWLWLRQHG PDWULFHV $ W\SLFDO &)' V\VWHP RIHTXDWLRQVKDVD´VSDUVHµFRHIÀFLHQWPDWUL[ ZLWKPRVWO\]HURHQWULHV *URXSLQJWKHYDULDEOHV FRUUHFWO\OHDGVWRDSDUWLWLRQHGPDWUL[ZLWKPDQ\ ]HUREORFNV 6HFWLRQLQWURGXFHVVXFKPDWULFHV DQGGHVFULEHVVRPHRIWKHLUDSSOLFDWLRQV

$)"15&3 !

0DWUL[$OJHEUD

0DWUL[IDFWRUL]DWLRQV (YHQZKHQZULWWHQZLWK SDUWLWLRQHGPDWULFHV WKHV\VWHPRIHTXDWLRQVLV FRPSOLFDWHG 7RIXUWKHUVLPSOLI\WKHFRPSXWDWLRQV WKH&)' VRIWZDUHDW%RHLQJXVHVZKDWLVFDOOHG DQ/8 IDFWRUL]DWLRQRIWKHFRHIÀFLHQWPDWUL[ 6HFWLRQGLVFXVVHV/8 DQGRWKHUXVHIXOPDWUL[ IDFWRUL]DWLRQV )XUWKHUGHWDLOVDERXWIDFWRUL]DWLRQV DSSHDUDWVHYHUDOSRLQWVODWHULQWKHWH[W

7RDQDO\]HDVROXWLRQRIDQDLUÁRZV\VWHP HQJLQHHUV ZDQWWRYLVXDOL]HWKHDLUÁRZRYHUWKHVXUIDFHRIWKHSODQH 7KH\XVHFRPSXWHUJUDSKLFV DQGOLQHDUDOJHEUDSURYLGHV WKHHQJLQHIRUWKHJUDSKLFV 7KHZLUHIUDPHPRGHORIWKH SODQH·VVXUIDFHLVVWRUHGDVGDWDLQPDQ\PDWULFHV 2QFHWKH LPDJHKDVEHHQUHQGHUHGRQDFRPSXWHUVFUHHQ HQJLQHHUV FDQFKDQJHLWVVFDOH ]RRPLQRURXWRIVPDOOUHJLRQV DQG URWDWHWKHLPDJHWRVHHSDUWVWKDWPD\EHKLGGHQIURPYLHZ (DFKRIWKHVHRSHUDWLRQVLVDFFRPSOLVKHGE\DSSURSULDWH

0RGHUQ&)' KDVUHYROXWLRQL]HGZLQJGHVLJQ 7KH%RHLQJ %OHQGHG:LQJ%RG\LVLQGHVLJQIRUWKH\HDURUVRRQHU

PDWUL[PXOWLSOLFDWLRQV 6HFWLRQH[SODLQVWKHEDVLF LGHDV WEB

2XUDELOLW\WRDQDO\]HDQGVROYHHTXDWLRQVZLOOEHJUHDWO\HQKDQFHGZKHQZHFDQSHUIRUP DOJHEUDLFRSHUDWLRQVZLWKPDWULFHV )XUWKHUPRUH WKHGHÀQLWLRQVDQGWKHRUHPVLQWKLV FKDSWHUSURYLGHVRPHEDVLFWRROVIRUKDQGOLQJWKHPDQ\DSSOLFDWLRQVRIOLQHDUDOJHEUD WKDWLQYROYHWZRRUPRUHPDWULFHV )RUVTXDUHPDWULFHV WKH,QYHUWLEOH0DWUL[7KHRUHP LQ6HFWLRQ WLHVWRJHWKHUPRVWRIWKHFRQFHSWVWUHDWHGHDUOLHULQWKHWH[W 6HFWLRQV DQGH[DPLQHSDUWLWLRQHGPDWULFHVDQGPDWUL[IDFWRUL]DWLRQV ZKLFKDSSHDULQPRVW PRGHUQXVHVRIOLQHDUDOJHEUD 6HFWLRQV DQGGHVFULEHWZRLQWHUHVWLQJDSSOLFDWLRQV RIPDWUL[DOJHEUD WRHFRQRPLFVDQGWRFRPSXWHUJUDSKLFV

."53*9 01&3"5*0/4 ,I A LVDQ m ! n PDWUL[³WKDWLV DPDWUL[ZLWK m URZVDQG n FROXPQV³WKHQWKHVFDODU HQWU\LQWKH i WKURZDQG j WKFROXPQRI A LVGHQRWHGE\ aij DQGLVFDOOHGWKH .i; j /HQWU\ RI A 6HH)LJ )RULQVWDQFH WKH .3; 2/HQWU\LVWKHQXPEHU a32 LQWKHWKLUGURZ VHFRQG FROXPQ (DFKFROXPQRI A LVDOLVWRI m UHDOQXPEHUV ZKLFKLGHQWLÀHVDYHFWRULQ Rm 2IWHQ WKHVHFROXPQVDUHGHQRWHGE\ D1 ; : : : ; Dn DQGWKHPDWUL[ A LVZULWWHQDV ! " A D D1 D 2 " " " D n

2EVHUYHWKDWWKHQXPEHU aij LVWKH i WKHQWU\IURPWKHWRS RIWKH j WKFROXPQYHFWRU Dj 7KH GLDJRQDOHQWULHV LQDQ m ! n PDWUL[ A D Œ aij ! DUH a11 ; a22 ; a33 ; : : : ; DQGWKH\ IRUPWKH PDLQGLDJRQDO RI A $ GLDJRQDOPDWUL[ LVDVTXDUH n ! n PDWUL[ZKRVH QRQGLDJRQDOHQWULHVDUH]HUR $QH[DPSOHLVWKH n ! n LGHQWLW\PDWUL[ In $Q m ! n PDWUL[ZKRVHHQWULHVDUHDOO]HURLVD ]HURPDWUL[ DQGLVZULWWHQDV 7KHVL]HRID]HUR PDWUL[LVXVXDOO\FOHDUIURPWKHFRQWH[W

Row i

a11

Column j a1 j

a1n

a i1

ai j

a in

am1

am j

am n

a1

aj

an

0DWUL[2SHUDWLRQV

= A

'*(63& 0DWUL[QRWDWLRQ

6XPVDQG6FDODU0XOWLSOHV 7KHDULWKPHWLFIRUYHFWRUVGHVFULEHGHDUOLHUKDVDQDWXUDOH[WHQVLRQWRPDWULFHV :H VD\WKDWWZRPDWULFHVDUH HTXDO LIWKH\KDYHWKHVDPHVL]HLH WKHVDPHQXPEHURI URZVDQGWKHVDPHQXPEHURIFROXPQV DQGLIWKHLUFRUUHVSRQGLQJFROXPQVDUHHTXDO ZKLFKDPRXQWVWRVD\LQJWKDWWKHLUFRUUHVSRQGLQJHQWULHVDUHHTXDO ,I A DQG B DUH m ! n PDWULFHV WKHQWKH VXP A C B LVWKH m ! n PDWUL[ZKRVHFROXPQVDUHWKHVXPV RIWKHFRUUHVSRQGLQJFROXPQVLQ A DQG B 6LQFHYHFWRUDGGLWLRQRIWKHFROXPQVLVGRQH HQWU\ZLVH HDFKHQWU\LQ A C B LVWKHVXPRIWKHFRUUHVSRQGLQJHQWULHVLQ A DQG B 7KH VXP A C B LVGHÀQHGRQO\ZKHQ A DQG B DUHWKHVDPHVL]H

&9".1-& /HW AD 7KHQ

#

4 #1

0 3

$ 5 ; 2

BD

ACB D

#

#

1 3

1 5

$ 1 ; 7

5 2

1 8

6 9

C D

$

#

2 #3 0 1

$

EXW A C C LVQRWGHÀQHGEHFDXVH A DQG C KDYHGLIIHUHQWVL]HV ,I r LVDVFDODUDQG A LVDPDWUL[ WKHQWKH VFDODUPXOWLSOH rA LVWKHPDWUL[ZKRVH FROXPQVDUH r WLPHVWKHFRUUHVSRQGLQJFROXPQVLQ A $VZLWKYHFWRUV #A VWDQGVIRU .#1/A DQG A # B LVWKHVDPHDV A C .#1/B

&9".1-& ,I A DQG B DUHWKHPDWULFHVLQ([DPSOH WKHQ # 1 2B D 2 3 # 4 A # 2B D #1

1 5 0 3

1 7

$

$

D

#

2 6

# 5 2 # 2 6

2 10

2 14

2 10

2 14

$

$

D

#

2 #2 3 #7 #7 #12

$

,WZDVXQQHFHVVDU\LQ([DPSOH WRFRPSXWH A # 2B DV A C .#1/2B EHFDXVHWKH XVXDOUXOHVRIDOJHEUDDSSO\WRVXPVDQGVFDODUPXOWLSOHVRIPDWULFHV DVWKHIROORZLQJ WKHRUHPVKRZV

5)&03&.

/HW A; B DQG C EHPDWULFHVRIWKHVDPHVL]H DQGOHW r DQG s EHVFDODUV D A C B D B C A E .A C B/ C C D A C .B C C / F A C 0 D A

G r.A C B/ D rA C rB H .r C s/A D rA C sA I r.sA/ D .rs/A

$)"15&3

0DWUL[$OJHEUD

(DFKHTXDOLW\LQ7KHRUHP LVYHULÀHGE\VKRZLQJWKDWWKHPDWUL[RQWKHOHIWVLGHKDV WKHVDPHVL]HDVWKHPDWUL[RQWKHULJKWDQGWKDWFRUUHVSRQGLQJFROXPQVDUHHTXDO 6L]H LVQRSUREOHPEHFDXVH A B DQG C DUHHTXDOLQVL]H 7KHHTXDOLW\RIFROXPQVIROORZV LPPHGLDWHO\IURPDQDORJRXVSURSHUWLHVRIYHFWRUV )RULQVWDQFH LIWKH j WK FROXPQVRI A B DQG C DUH Dj Ej DQG Fj UHVSHFWLYHO\ WKHQWKH j WK FROXPQVRI .A C B/ C C DQG A C .B C C / DUH

.Dj C Ej / C Fj

DQG

Dj C .Ej C Fj /

UHVSHFWLYHO\ 6LQFHWKHVHWZRYHFWRUVXPVDUHHTXDOIRUHDFK j SURSHUW\E LVYHULÀHG %HFDXVHRIWKHDVVRFLDWLYHSURSHUW\RIDGGLWLRQ ZHFDQVLPSO\ZULWH A C B C C IRUWKHVXP ZKLFKFDQEHFRPSXWHGHLWKHUDV .A C B/ C C RUDV A C .B C C / 7KH VDPHDSSOLHVWRVXPVRIIRXURUPRUHPDWULFHV

0DWUL[0XOWLSOLFDWLRQ :KHQDPDWUL[ B PXOWLSOLHVDYHFWRU [ LWWUDQVIRUPV [ LQWRWKHYHFWRU B [ ,IWKLVYHFWRU LVWKHQPXOWLSOLHGLQWXUQE\DPDWUL[ A WKHUHVXOWLQJYHFWRULV A.B [/ 6HH)LJ Multiplication

Multiplication

by B

by A

x

A(Bx)

Bx

'*(63& 0XOWLSOLFDWLRQE\ B DQGWKHQ A

7KXV A.B [/ LVSURGXFHGIURP [ E\D FRPSRVLWLRQ RIPDSSLQJV³WKHOLQHDUWUDQVIRU PDWLRQVVWXGLHGLQ6HFWLRQ 2XUJRDOLVWRUHSUHVHQWWKLVFRPSRVLWHPDSSLQJDV PXOWLSOLFDWLRQE\DVLQJOHPDWUL[ GHQRWHGE\ $% VRWKDW

A.B [/ D .AB/[

6HH)LJ Multiplication

Multiplication

by B

by A

x

Bx

A(Bx)

Multiplication by AB

'*(63& 0XOWLSOLFDWLRQE\ $%

,I A LV m ! n B LV n ! p DQG [ LVLQ Rp GHQRWHWKHFROXPQVRI B E\ E1 ; : : : ; Ep DQGWKHHQWULHVLQ [ E\ x1 ; : : : ; xp 7KHQ

B [ D x1 E1 C " " " C xp Ep %\WKHOLQHDULW\RIPXOWLSOLFDWLRQE\ A

A.B [/ D A.x1 E1 / C " " " C A.xp Ep / D x1 AE1 C " " " C xp AEp

0DWUL[2SHUDWLRQV

7KHYHFWRU A.B [/ LVDOLQHDUFRPELQDWLRQRIWKHYHFWRUV AE1 ; : : : ; AEp XVLQJWKHHQWULHV LQ [ DVZHLJKWV ,QPDWUL[QRWDWLRQ WKLVOLQHDUFRPELQDWLRQLVZULWWHQDV

A.B [/ D Œ AE1 AE2 " " " AEp ![ 7KXVPXOWLSOLFDWLRQE\ Œ AE1 AE2 " " " AEp ! WUDQVIRUPV [ LQWR A.B [/ :HKDYH IRXQGWKHPDWUL[ZHVRXJKW

%&'*/*5*0/

,I A LVDQ m ! n PDWUL[ DQGLI B LVDQ n ! p PDWUL[ZLWKFROXPQV E1 ; : : : ; Ep WKHQWKHSURGXFW AB LVWKH m ! p PDWUL[ZKRVHFROXPQVDUH AE1 ; : : : ; AEp 7KDW LV ! " ! " AB D A E1 E2 " " " Ep D AE1 AE2 " " " AEp 7KLVGHÀQLWLRQPDNHVHTXDWLRQ WUXHIRUDOO [ LQ Rp (TXDWLRQ SURYHVWKDWWKH FRPSRVLWHPDSSLQJLQ)LJ LVDOLQHDUWUDQVIRUPDWLRQDQGWKDWLWVVWDQGDUGPDWUL[LV AB 0XOWLSOLFDWLRQRIPDWULFHVFRUUHVSRQGVWRFRPSRVLWLRQRIOLQHDUWUDQVIRUPDWLRQV

&9".1-& &RPSXWH AB ZKHUH A D

#

$ # 2 3 4 3 DQG B D 1 #5 1 #2

40-65*0/ :ULWH B D Œ E1 E2 E3 ! DQGFRPSXWH #

2 3 AE1 D 1 #5 # $ 11 D #1

$# $ 4 ; 1

#

2 3 AE 2 D 1 #5 # $ 0 D 13

7KHQ

AB D AŒ E1 E2 E3 ! D

#

!

$#

$ 3 ; #2

11 #1

! ! $ 0 21 13 #9 "

"

AE 1

AE 2

AE 3

"

#

2 3 AE 3 D 1 #5 # $ 21 D #9

$ 6 3 $# $ 6 3

1RWLFHWKDWVLQFHWKHÀUVWFROXPQRI AB LV AE1 ; WKLVFROXPQLVDOLQHDUFRPELQDWLRQ RIWKHFROXPQVRI A XVLQJWKHHQWULHVLQ E1 DVZHLJKWV $ VLPLODUVWDWHPHQWLVWUXHIRU HDFKFROXPQRI AB: (DFKFROXPQRI AB LVDOLQHDUFRPELQDWLRQRIWKHFROXPQVRI A XVLQJZHLJKWV IURPWKHFRUUHVSRQGLQJFROXPQRI B

2EYLRXVO\ WKHQXPEHURIFROXPQVRI A PXVWPDWFKWKHQXPEHURIURZVLQ B LQ RUGHUIRUDOLQHDUFRPELQDWLRQVXFKDV AE1 WREHGHÀQHG $OVR WKHGHÀQLWLRQRI AB VKRZVWKDW AB KDVWKHVDPHQXPEHURIURZVDV$ DQGWKHVDPHQXPEHURIFROXPQV DV%

&9".1-& ,I A LVD 3 ! 5 PDWUL[DQG B LVD 5 ! 2 PDWUL[ ZKDWDUHWKHVL]HVRI AB DQG BA LIWKH\DUHGHÀQHG"

$)"15&3

0DWUL[$OJHEUD

40-65*0/ 6LQFH A KDVFROXPQVDQG B KDVURZV WKHSURGXFW AB LVGHÀQHGDQG LVD 3 ! 2 PDWUL[ A B 3 AB 32 2 3 $ $ $ $ $ $ $ $ $ 7 4 $ $ $ $ $ 56 6$ $7 D 4$ $5 6 $ $ $ $ $ 6$ $7 $ $ 7 4$ $5 $ $ 2

3!5

" "0DWFK

5!2

3!2

" "

6L]HRI AB

7KHSURGXFW BA LV QRW GHÀQHGEHFDXVHWKHFROXPQVRI B GRQRWPDWFKWKHURZV RI A 7KHGHÀQLWLRQRI AB LVLPSRUWDQWIRUWKHRUHWLFDOZRUNDQGDSSOLFDWLRQV EXWWKH IROORZLQJUXOHSURYLGHVDPRUHHIÀFLHQWPHWKRGIRUFDOFXODWLQJWKHLQGLYLGXDOHQWULHVLQ AB ZKHQZRUNLQJVPDOOSUREOHPVE\KDQG

308$0-6./ 36-& '03 $0.165*/( "# ,IWKHSURGXFW AB LVGHÀQHG WKHQWKHHQWU\LQURZ i DQGFROXPQ j RI AB LVWKH VXPRIWKHSURGXFWVRIFRUUHVSRQGLQJHQWULHVIURPURZ i RI A DQGFROXPQ j RI B ,I .AB/ij GHQRWHVWKH .i; j /HQWU\LQ AB DQGLI A LVDQ m ! n PDWUL[ WKHQ

.AB/ij D ai1 b1j C ai2 b2j C " " " C ai n bnj

7RYHULI\WKLVUXOH OHW B D Œ E1 " " " Ep ! &ROXPQ j RI AB LV AEj DQGZHFDQ FRPSXWH AEj E\WKHURZ²YHFWRUUXOHIRUFRPSXWLQJ A[ IURP6HFWLRQ 7KH i WK HQWU\ LQ AEj LVWKHVXPRIWKHSURGXFWVRIFRUUHVSRQGLQJHQWULHVIURPURZ i RI A DQGWKH YHFWRU Ej ZKLFKLVSUHFLVHO\WKHFRPSXWDWLRQGHVFULEHGLQWKHUXOHIRUFRPSXWLQJWKH .i; j /HQWU\RI AB

&9".1-& 8VHWKHURZ²FROXPQUXOHWRFRPSXWHWZRRIWKHHQWULHVLQ AB IRUWKH PDWULFHVLQ([DPSOH $QLQVSHFWLRQRIWKHQXPEHUVLQYROYHGZLOOPDNHLWFOHDUKRZ WKHWZRPHWKRGVIRUFDOFXODWLQJ AB SURGXFHWKHVDPHPDWUL[

40-65*0/ 7RÀQGWKHHQWU\LQURZ DQGFROXPQ RI AB FRQVLGHUURZ RI A DQG FROXPQ RI B 0XOWLSO\FRUUHVSRQGLQJHQWULHVDQGDGGWKHUHVXOWV DVVKRZQEHORZ $# # # 2 3 4 3 AB D 1 #5 1 #2

! $ # 6 ! D 3 !

! !

2.6/ C 3.3/ !

$

D

#

! !

! !

21 !

$

)RUWKHHQWU\LQURZ DQGFROXPQ RI AB XVHURZRI A DQGFROXPQ RI B #

2 3 # 1 #5

$#

! $ # 4 3 6 ! D 1 #2 3 !

! 1.3/ C #5.#2/

21 !

$

D

#

! !

! 13

21 !

$

0DWUL[2SHUDWLRQV

&9".1-& )LQGWKHHQWULHVLQWKHVHFRQGURZRI AB ZKHUH 2

3 2 #5 0 6 #1 3 #4 7 7 AD6 4 6 #8 #7 5; #3 0 9

2

3 4 #6 B D 47 15 3 2

40-65*0/ %\WKHURZ²FROXPQUXOH WKHHQWULHVRIWKHVHFRQGURZRI AB FRPHIURP URZ RI A DQGWKHFROXPQVRI B 2 3 ! ! 3 2 #5 0 2 4 #6 7 #6 #1 3 #4 74 6 15 4 6 #8 #7 5 7 3 2 #3 0 9 2 3 2 3 ! ! ! ! 6 # 4 C 21 # 12 6 6C3#87 17 7D6 5 7 D6 4 5 4! ! ! !5 ! ! ! !

1RWLFHWKDWVLQFH([DPSOH UHTXHVWHGRQO\WKHVHFRQGURZRI AB ZHFRXOGKDYH ZULWWHQMXVWWKHVHFRQGURZRI A WRWKHOHIWRI B DQGFRPSXWHG 3 2 ! " 4 #6 ! " #1 3 #4 4 7 1 5 D 5 1 3 2

7KLVREVHUYDWLRQDERXWURZVRI AB LVWUXHLQJHQHUDODQGIROORZVIURPWKHURZ²FROXPQ UXOH /HWURZi .A/ GHQRWHWKH i WKURZRIDPDWUL[ A 7KHQ URZi .AB/ D URZi .A/ " B

3URSHUWLHVRI0DWUL[0XOWLSOLFDWLRQ 7KHIROORZLQJWKHRUHPOLVWVWKHVWDQGDUGSURSHUWLHVRIPDWUL[PXOWLSOLFDWLRQ 5HFDOOWKDW Im UHSUHVHQWVWKH m ! m LGHQWLW\PDWUL[DQG Im [ D [ IRUDOO [ LQ Rm

5)&03&.

/HW A EHDQ m ! n PDWUL[ DQGOHW B DQG C KDYHVL]HVIRUZKLFKWKHLQGLFDWHG VXPVDQGSURGXFWVDUHGHÀQHG

A.BC / D .AB/C A.B C C / D AB C AC .B C C /A D BA C CA r.AB/ D .rA/B D A.rB/ IRUDQ\VFDODU r H Im A D A D AIn

D E F G

DVVRFLDWLYHODZRIPXOWLSOLFDWLRQ OHIWGLVWULEXWLYHODZ ULJKWGLVWULEXWLYHODZ

LGHQWLW\IRUPDWUL[PXOWLSOLFDWLRQ

1300' 3URSHUWLHVE ²H DUHFRQVLGHUHGLQWKHH[HUFLVHV 3URSHUW\ D IROORZVIURP WKHIDFWWKDWPDWUL[PXOWLSOLFDWLRQFRUUHVSRQGVWRFRPSRVLWLRQRIOLQHDUWUDQVIRUPDWLRQV ZKLFKDUHIXQFWLRQV DQGLWLVNQRZQRUHDV\WRFKHFN WKDWWKHFRPSRVLWLRQRIIXQF WLRQVLVDVVRFLDWLYH +HUHLVDQRWKHUSURRIRID WKDWUHVWVRQWKH´FROXPQGHÀQLWLRQµRI WKHSURGXFWRIWZRPDWULFHV /HW C D Œ F1 " " " Fp !

$)"15&3

0DWUL[$OJHEUD

%\WKHGHÀQLWLRQRIPDWUL[PXOWLSOLFDWLRQ

BC D Œ B F1 " " " B Fp !

A.BC / D Œ A.B F1 / " " " A.B Fp / ! 5HFDOOIURPHTXDWLRQ WKDWWKHGHÀQLWLRQRI AB PDNHV A.B [/ D .AB/[ IRUDOO [ VR

A.BC / D Œ .AB/F1 " " " .AB/Fp ! D .AB/C 7KHDVVRFLDWLYHDQGGLVWULEXWLYHODZVLQ7KHRUHPV DQGVD\HVVHQWLDOO\WKDWSDLUV RISDUHQWKHVHVLQPDWUL[H[SUHVVLRQVFDQEHLQVHUWHGDQGGHOHWHGLQWKHVDPHZD\DVLQ WKHDOJHEUDRIUHDOQXPEHUV ,QSDUWLFXODU ZHFDQZULWH ABC IRUWKHSURGXFW ZKLFK FDQEHFRPSXWHGHLWKHUDV A.BC / RUDV .AB/C ñ 6LPLODUO\ DSURGXFW ABCD RIIRXU PDWULFHVFDQEHFRPSXWHGDV A.BCD/ RU .ABC /D RU A.BC /D DQGVRRQ ,WGRHVQRW PDWWHUKRZZHJURXSWKHPDWULFHVZKHQFRPSXWLQJWKHSURGXFW VRORQJDVWKHOHIWWR ULJKWRUGHURIWKHPDWULFHVLVSUHVHUYHG 7KHOHIWWRULJKWRUGHULQSURGXFWVLVFULWLFDOEHFDXVH AB DQG BA DUHXVXDOO\QRW WKHVDPH 7KLVLVQRWVXUSULVLQJ EHFDXVHWKHFROXPQVRI AB DUHOLQHDUFRPELQDWLRQV RIWKHFROXPQVRI A ZKHUHDVWKHFROXPQVRI BA DUHFRQVWUXFWHGIURPWKHFROXPQVRI B 7KHSRVLWLRQRIWKHIDFWRUVLQWKHSURGXFW AB LVHPSKDVL]HGE\VD\LQJWKDW A LV ULJKWPXOWLSOLHG E\ B RUWKDW B LV OHIWPXOWLSOLHG E\ A ,I AB D BA ZHVD\WKDW A DQG B FRPPXWH ZLWKRQHDQRWKHU #

$ # 5 1 2 &9".1-& /HW A D DQG B D 3 #2 4 QRWFRPPXWH 7KDWLV YHULI\WKDW AB ¤ BA

40-65*0/ AB D BA D

# #

$ 14 3 #2 #6 $# $ # $ 0 5 1 10 2 D 3 3 #2 29 #2

5 1 3 #2 2 4

$#

2 4

0 3

$

$ 0 6KRZWKDWWKHVHPDWULFHVGR 3

D

#

([DPSOHLOOXVWUDWHVWKHÀUVWRIWKHIROORZLQJOLVWRILPSRUWDQWGLIIHUHQFHVEHWZHHQ PDWUL[DOJHEUDDQGWKHRUGLQDU\DOJHEUDRIUHDOQXPEHUV 6HH([HUFLVHV²IRUH[DP SOHVRIWKHVHVLWXDWLRQV :$51,1*6 ,QJHQHUDO AB ¤ BA 7KH FDQFHOODWLRQ ODZV GR QRW KROG IRU PDWUL[ PXOWLSOLFDWLRQ 7KDW LV LI AB D AC WKHQLWLV QRW WUXHLQJHQHUDOWKDW B D C 6HH([HUFLVH ,IDSURGXFW AB LVWKH]HURPDWUL[ \RX FDQQRW FRQFOXGHLQJHQHUDOWKDWHLWKHU A D 0 RU B D 0 6HH([HUFLVH

3RZHUVRID0DWUL[ WEB

,I A LVDQ n ! n PDWUL[DQGLI k LVDSRVLWLYHLQWHJHU WKHQ Ak GHQRWHVWKHSURGXFWRI k ñ :KHQ B LVVTXDUHDQG C KDVIHZHUFROXPQVWKDQ A KDVURZV LWLVPRUHHIÀFLHQWWRFRPSXWH A.BC / WKDQ .AB/C

0DWUL[2SHUDWLRQV

FRSLHVRI A

Ak D „ A ƒ‚ """A … k

,I A LVQRQ]HURDQGLI [ LVLQ Rn ; WKHQ Ak [ LVWKHUHVXOWRIOHIWPXOWLSO\LQJ [ E\ A UHSHDWHGO\ k WLPHV ,I k D 0; WKHQ A0 [ VKRXOGEH [ LWVHOI 7KXV A0 LVLQWHUSUHWHGDVWKH LGHQWLW\PDWUL[ 0DWUL[SRZHUVDUHXVHIXOLQERWKWKHRU\DQGDSSOLFDWLRQV6HFWLRQV DQGODWHULQWKHWH[W

7KH7UDQVSRVHRID0DWUL[ *LYHQDQ m ! n PDWUL[ A WKH WUDQVSRVH RI A LVWKH n ! m PDWUL[ GHQRWHGE\ AT ZKRVHFROXPQVDUHIRUPHGIURPWKHFRUUHVSRQGLQJURZVRI A

&9".1-& /HW AD

#

2

3 #5 2 B D 4 1 #3 5; 0 4

$

a c

b ; d

7KHQ

AT D

5)&03&.

#

a b

$

c ; d

BT D

#

#5 1 2 #3

C D $

0 ; 4

#

1 #3

CT

1 1 5 #2

1 7

$

2

3 1 #3 61 57 7 D6 4 1 #2 5 1 7

/HW A DQG B GHQRWHPDWULFHVZKRVHVL]HVDUHDSSURSULDWHIRUWKHIROORZLQJVXPV DQGSURGXFWV D E F G

.AT /T D A .A C B/T D AT C B T )RUDQ\VFDODU r .rA/T D rAT .AB/T D B TAT

3URRIVRID ²F DUHVWUDLJKWIRUZDUGDQGDUHRPLWWHG )RUG VHH([HUFLVH 8VXDOO\ .AB/T LVQRWHTXDOWR ATB T HYHQZKHQ A DQG B KDYHVL]HVVXFKWKDWWKH SURGXFW ATB T LVGHÀQHG 7KHJHQHUDOL]DWLRQRI7KHRUHPG WRSURGXFWVRIPRUHWKDQWZRIDFWRUVFDQEH VWDWHGLQZRUGVDVIROORZV 7KHWUDQVSRVHRIDSURGXFWRIPDWULFHVHTXDOVWKHSURGXFWRIWKHLUWUDQVSRVHVLQ WKH UHYHUVH RUGHU

7KHH[HUFLVHVFRQWDLQQXPHULFDOH[DPSOHVWKDWLOOXVWUDWHSURSHUWLHVRIWUDQVSRVHV

$)"15&3

0DWUL[$OJHEUD

/6.&3*$"- /05&4 7KHIDVWHVWZD\WRREWDLQ AB RQDFRPSXWHUGHSHQGVRQWKHZD\LQZKLFK WKHFRPSXWHUVWRUHVPDWULFHVLQLWVPHPRU\ 7KHVWDQGDUGKLJKSHUIRUPDQFH DOJRULWKPV VXFKDVLQ/$3$&.FDOFXODWH AB E\FROXPQV DVLQRXUGHÀQLWLRQ RIWKHSURGXFW $ YHUVLRQRI/$3$&. ZULWWHQLQ&FDOFXODWHV AB E\URZV 7KHGHÀQLWLRQRI AB OHQGVLWVHOIZHOOWRSDUDOOHOSURFHVVLQJRQDFRPSXWHU 7KHFROXPQVRI B DUHDVVLJQHGLQGLYLGXDOO\RULQJURXSVWRGLIIHUHQWSURFHV VRUV ZKLFKLQGHSHQGHQWO\DQGKHQFHVLPXOWDQHRXVO\FRPSXWHWKHFRUUHVSRQG LQJFROXPQVRI AB

13"$5*$& 130#-&.4 6LQFHYHFWRUVLQ Rn PD\EHUHJDUGHGDV n ! 1 PDWULFHV WKHSURSHUWLHVRIWUDQVSRVHV LQ7KHRUHP DSSO\WRYHFWRUV WRR /HW # $ # $ 1 #3 5 AD DQG [ D #2 4 3 &RPSXWH .A[/T [TAT [[T DQG [T[ ,V AT[T GHÀQHG" /HW A EHD 4 ! 4 PDWUL[DQGOHW [ EHDYHFWRULQ R4 :KDWLVWKHIDVWHVWZD\WR FRPSXWH A2 [" &RXQWWKHPXOWLSOLFDWLRQV

&9&3$*4&4 ,Q([HUFLVHV DQG FRPSXWHHDFKPDWUL[VXPRUSURGXFWLILWLV GHÀQHG ,IDQH[SUHVVLRQLVXQGHÀQHG H[SODLQZK\ /HW # $ # $ 2 0 #1 7 #5 1 AD ; BD ; 4 #5 2 1 #4 #3 # $ # $ # $ 1 2 3 5 #5 C D ; DD ; ED #2 1 #1 4 3 #2A B # 2A AC CD A C 3B 2C # 3E DB EC ,QWKHUHVWRIWKLVH[HUFLVHVHWDQGLQWKRVHWRIROORZ DVVXPHWKDW HDFKPDWUL[H[SUHVVLRQLVGHÀQHG 7KDWLV WKHVL]HVRIWKHPDWULFHV DQGYHFWRUV LQYROYHG´PDWFKµDSSURSULDWHO\ # $ 2 #5 &RPSXWH 3I2 # A DQG .3I2 /A /HW A D 3 #2 &RPSXWH A # 5I3 DQG .5I3 /A ZKHUH 2 3 5 #1 3 3 #6 5: A D 4 #4 #3 1 2

,Q([HUFLVHVDQG FRPSXWHWKHSURGXFW AB LQWZRZD\V D E\ WKHGHÀQLWLRQ ZKHUH AE1 DQG AE2 DUHFRPSXWHGVHSDUDWHO\ DQG E E\WKHURZ²FROXPQUXOHIRUFRPSXWLQJ AB 2 3 # $ #1 3 4 #2 4 5; B D A D 4 2 #2 3 5 #3

2

3 # $ 4 #3 1 4 5 5; B D A D 4 #3 3 #2 0 1

,IDPDWUL[ A LV 5 ! 3 DQGWKHSURGXFW AB LV 5 ! 7 ZKDWLV WKHVL]HRI B " +RZPDQ\URZVGRHV B KDYHLI BC LVD 5 ! 4 PDWUL[" # $ # $ 2 3 1 9 DQG B D :KDWYDOXHV /HW A D #1 1 #3 k RI k LIDQ\ ZLOOPDNH AB D BA" # $ # $ 3 #6 #1 1 BD DQG C D /HW A D #1 2 3 4 # $ #3 #5 9HULI\WKDW AB D AC DQG\HW B ¤ C 2 1 2 3 2 3 1 2 3 5 0 0 4 5 5 DQG D D 4 0 3 0 5 &RP /HW A D 4 2 3 5 6 0 0 2 SXWH AD DQG DA: ([SODLQKRZWKHFROXPQVRUURZVRI A FKDQJHZKHQ A LVPXOWLSOLHGE\ D RQWKHULJKWRURQWKHOHIW )LQGD 3 ! 3 PDWUL[ B; QRWWKHLGHQWLW\PDWUL[RUWKH]HUR PDWUL[ VXFKWKDW AB D BA # $ 3 #6 /HW A D &RQVWUXFWD 2 ! 2 PDWUL[ B VXFKWKDW #2 4 AB LVWKH]HURPDWUL[ 8VHWZRGLIIHUHQWQRQ]HURFROXPQV IRU B:

/HW U1 ; : : : ; Up EH YHFWRUV LQ Rn DQG OHW Q EH DQ m ! n PDWUL[ :ULWHWKHPDWUL[ Œ QU1 " " " QUp ! DVD SURGXFW RI WZRPDWULFHVQHLWKHURIZKLFKLVDQLGHQWLW\PDWUL[ /HW U EHWKH 3 ! 2 FRVWPDWUL[GHVFULEHGLQ([DPSOHLQ 6HFWLRQ 7KHÀUVWFROXPQRI U OLVWVWKHFRVWVSHUGROODURI RXWSXWIRUPDQXIDFWXULQJSURGXFW B; DQGWKHVHFRQGFROXPQ OLVWVWKHFRVWVSHUGROODURIRXWSXWIRUSURGXFW C: 7KHFRVWV DUHFDWHJRUL]HGDVPDWHULDOV ODERU DQGRYHUKHDG /HW T1 EHDYHFWRULQ R2 WKDWOLVWVWKHRXWSXWPHDVXUHGLQGROODUV RISURGXFWV% DQG& PDQXIDFWXUHGGXULQJWKHÀUVWTXDUWHU RIWKH\HDU DQGOHW T2 ; T3 ; DQG T4 EHWKHDQDORJRXVYHFWRUV WKDWOLVWWKHDPRXQWVRISURGXFWV% DQG& PDQXIDFWXUHGLQ WKHVHFRQG WKLUG DQGIRXUWKTXDUWHUV UHVSHFWLYHO\ *LYHDQ HFRQRPLFGHVFULSWLRQRIWKHGDWDLQWKHPDWUL[ UQ; ZKHUH Q D Œ T1 T2 T3 T4 !: ([HUFLVHVDQGFRQFHUQDUELWUDU\PDWULFHV A B DQG C IRU ZKLFKWKHLQGLFDWHGVXPVDQGSURGXFWVDUHGHÀQHG 0DUNHDFK VWDWHPHQW7UXHRU)DOVH -XVWLI\HDFKDQVZHU D ,I A DQG B DUH 2 ! 2 PDWULFHVZLWKFROXPQV D1 D2 DQG E1 E2 UHVSHFWLYHO\ WKHQ AB D Œ D1 E1 D2 E2 !

E (DFK FROXPQ RI AB LV D OLQHDU FRPELQDWLRQ RI WKH FROXPQVRI B XVLQJZHLJKWVIURPWKHFRUUHVSRQGLQJFRO XPQRI A F AB C AC D A.B C C /

G AT C B T D .A C B/T

H 7KHWUDQVSRVHRIDSURGXFWRIPDWULFHVHTXDOVWKHSURGXFW RIWKHLUWUDQVSRVHVLQWKHVDPHRUGHU D 7KHÀUVWURZRI AB LVWKHÀUVWURZRI A PXOWLSOLHGRQWKH ULJKWE\ B E ,I A DQG B DUH 3 ! 3 PDWULFHVDQG B D Œ E1 WKHQ AB D Œ AE1 C AE2 C AE3 !

E2

E 3 !

F ,I A LVDQ n ! n PDWUL[ WKHQ .A2 /T D .AT /2 G .ABC /T D C T AT B T

H 7KHWUDQVSRVHRIDVXPRIPDWULFHVHTXDOVWKHVXPRIWKHLU WUDQVSRVHV # $ # $ 1 #3 #3 #11 DQG AB D GHWHUPLQHWKH ,I A D #3 5 1 17 ÀUVWDQGVHFRQG FROXPQVRI B

6XSSRVHWKHWKLUGFROXPQRI B LVDOO]HURV :KDWFDQEHVDLG DERXWWKHWKLUGFROXPQRI AB " 6XSSRVHWKHWKLUGFROXPQRI B LVWKHVXPRIWKHÀUVWWZR FROXPQV :KDWFDQEHVDLGDERXWWKHWKLUGFROXPQRI AB " :K\" 6XSSRVHWKHÀUVWWZRFROXPQV E1 DQG E2 RI B DUHHTXDO :KDWFDQEHVDLGDERXWWKHFROXPQVRI AB " :K\" 6XSSRVHWKHODVWFROXPQRI AB LVHQWLUHO\]HURVEXW B LWVHOI KDVQRFROXPQRI]HURV :KDWFDQEHVDLGDERXWWKHFROXPQV RI A"

0DWUL[2SHUDWLRQV

6KRZWKDWLIWKHFROXPQVRI B DUHOLQHDUO\GHSHQGHQW WKHQ VRDUHWKHFROXPQVRI AB 6XSSRVH CA D In WKH n ! n LGHQWLW\PDWUL[ 6KRZWKDWWKH HTXDWLRQ A[ D KDVRQO\WKHWULYLDOVROXWLRQ ([SODLQZK\ A FDQQRWKDYHPRUHFROXPQVWKDQURZV 6XSSRVH A LVD 3 ! n PDWUL[ZKRVHFROXPQVVSDQ R3 : ([SODLQ KRZWRFRQVWUXFWDQ n ! 3 PDWUL[ D VXFKWKDW AD D I3 :

6XSSRVH A LVDQ m ! n PDWUL[DQGWKHUHH[LVW n ! m PDWULFHV C DQG D VXFK WKDW CA D In DQG AD D Im : 3URYH WKDW m D n DQG C D D: >+LQW 7KLQNDERXWWKHSURGXFW CAD @

6XSSRVH AD D Im WKH m ! m LGHQWLW\PDWUL[ 6KRZWKDW IRUDQ\ E LQ Rm WKHHTXDWLRQ A[ D E KDVDVROXWLRQ >+LQW 7KLQNDERXWWKHHTXDWLRQ AD E D E:@([SODLQZK\ A FDQQRW KDYHPRUHURZVWKDQFROXPQV

,Q([HUFLVHVDQG YLHZYHFWRUVLQ Rn DV n ! 1 PDWULFHV )RU X DQG Y LQ Rn WKHPDWUL[SURGXFW XT Y LVD 1 ! 1 PDWUL[ FDOOHGWKH VFDODUSURGXFW RU LQQHUSURGXFW RI X DQG Y ,WLVXVXDOO\ZULWWHQ DVDVLQJOHUHDOQXPEHUZLWKRXWEUDFNHWV 7KHPDWUL[SURGXFW XYT LVDQ n ! n PDWUL[ FDOOHGWKH RXWHUSURGXFW RI X DQG Y 7KH SURGXFWV XT Y DQG XYT ZLOODSSHDUODWHULQWKHWH[W 2 3 2 3 #3 a /HW X D 4 2 5 DQG Y D 4 b 5 &RPSXWH XT Y YT X XYT #5 c DQG YXT ,I X DQG Y DUHLQ Rn KRZDUH XT Y DQG YT X UHODWHG" +RZDUH XYT DQG YXT UHODWHG" 3URYH7KHRUHPE DQGF 8VHWKHURZ²FROXPQUXOH 7KH .i; j /HQWU\LQ A.B C C / FDQEHZULWWHQDV

ai1 .b1j C c1j / C " " " C ai n .bnj C cnj / RU n X

kD1

ai k .bkj C ckj /

3URYH7KHRUHPG >+LQW 7KH .i; j /HQWU\LQ .rA/B LV .rai1 /b1j C " " " C .rai n /bnj @ 6KRZWKDW Im A D A ZKHUH A LVDQ m ! n PDWUL[ $VVXPH Im [ D [ IRUDOO [ LQ Rm 6KRZWKDW AIn D A ZKHQ A LVDQ m ! n PDWUL[ >+LQW 8VH WKHFROXPQ GHÀQLWLRQRI AIn @ 3URYH7KHRUHPG >+LQW &RQVLGHUWKH j WK URZRI .AB/T @ *LYHDIRUPXODIRU .AB [/T ZKHUH [ LVDYHFWRUDQG A DQG B DUHPDWULFHVRIDSSURSULDWHVL]HV >0@ 5HDGWKHGRFXPHQWDWLRQIRU\RXUPDWUL[SURJUDP DQG ZULWHWKHFRPPDQGVWKDWZLOOSURGXFHWKHIROORZLQJPDWULFHV ZLWKRXWNH\LQJLQHDFKHQWU\RIWKHPDWUL[ D $ 4 ! 5 PDWUL[RI]HURV E $ 5 ! 3 PDWUL[RIRQHV

F 7KH 5 ! 5 LGHQWLW\PDWUL[

G $ 4 ! 4 GLDJRQDOPDWUL[ ZLWKGLDJRQDOHQWULHV

$)"15&3

0DWUL[$OJHEUD

$ XVHIXOZD\WRWHVWQHZLGHDVLQPDWUL[DOJHEUD RUWRPDNH FRQMHFWXUHV LV WR PDNH FDOFXODWLRQV ZLWK PDWULFHV VHOHFWHG DW UDQGRP &KHFNLQJDSURSHUW\IRUDIHZPDWULFHVGRHVQRWSURYH WKDWWKHSURSHUW\KROGVLQJHQHUDO EXWLWPDNHVWKHSURSHUW\PRUH EHOLHYDEOH $OVR LIWKHSURSHUW\LVDFWXDOO\IDOVH PDNLQJDIHZ FDOFXODWLRQVPD\KHOSWRGLVFRYHUWKLV >0@ :ULWHWKHFRPPDQGV WKDWZLOOFUHDWHD 5 ! 6 PDWUL[ ZLWKUDQGRPHQWULHV ,QZKDWUDQJHRIQXPEHUVGRWKHHQWULHV OLH" 7HOOKRZWRFUHDWHD 4 ! 4 PDWUL[ZLWKUDQGRPLQWHJHU HQWULHVEHWZHHQ #9 DQG >+LQW ,I x LVDUDQGRPQXPEHU VXFKWKDW 0 < x < 1 WKHQ #9:5 < 19.x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

ZD\IRUWKUHHSDLUVRIUDQGRP 4 ! 4 PDWULFHV 5HSRUW\RXU FRQFOXVLRQV >0@ 8VHDWOHDVWWKUHHSDLUVRIUDQGRP 4 ! 4 PDWULFHV A DQG B WR WHVW WKH HTXDOLWLHV .A C B/T D AT C B T DQG .AB/T D B TAT DVZHOODV .AB/T D ATB T 6HH([HUFLVH 5HSRUW\RXUFRQFOXVLRQV >1RWH 0RVWPDWUL[SURJUDPV XVH A0 IRU AT @ >0@ /HW 2 0 60 6 S D6 60 40 0

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

3 0 07 7 07 7 15 0

&RPSXWH S k IRU k D 2; : : : ; 6

>0@ 'HVFULEHLQZRUGVZKDWKDSSHQVZKHQ A5 A10 A20 DQG A30 DUHFRPSXWHGIRU 2 3 1=4 1=2 1=4 1=3 1=6 5 A D 4 1=2 1=4 1=6 7=12

40-65*0/4 50 13"$5*$& 130#-&.4 # $# $ # $ ! " 1 #3 5 #4 A[ D D 6R .A[/T D #4 2 $OVR #2 4 3 2 $ # ! " 1 #2 ! " T T D #4 2 : [A D 5 3 #3 4

7KHTXDQWLWLHV .A[/T DQG [TAT DUHHTXDO E\7KHRUHP G 1H[W # $ # $ " 5 ! 25 15 T 5 3 D [[ D 3 15 9 # $ ! " 5 [T [ D 5 3 D Œ 25 C 9 ! D 34 3

$ 1 ! 1 PDWUL[VXFKDV [T[ LVXVXDOO\ZULWWHQZLWKRXWWKHEUDFNHWV )LQDOO\ AT[T LV QRWGHÀQHG EHFDXVH [T GRHVQRWKDYHWZRURZVWRPDWFKWKHWZRFROXPQVRI AT 7KHIDVWHVWZD\WRFRPSXWH A2 [ LVWRFRPSXWH A.A[/ 7KHSURGXFW A[ UHTXLUHV PXOWLSOLFDWLRQV IRUHDFKHQWU\ DQG A.A[/ UHTXLUHVPRUH ,QFRQWUDVW WKH SURGXFW A2 UHTXLUHVPXOWLSOLFDWLRQV IRUHDFKRIWKHHQWULHVLQ A2 $IWHUWKDW A2 [ WDNHVPRUHPXOWLSOLFDWLRQV IRUDWRWDORI

5)& */7&34& 0' " ."53*9 0DWUL[DOJHEUDSURYLGHVWRROVIRUPDQLSXODWLQJPDWUL[HTXDWLRQVDQGFUHDWLQJYDULRXV XVHIXOIRUPXODVLQZD\VVLPLODUWRGRLQJRUGLQDU\DOJHEUDZLWKUHDOQXPEHUV 7KLV VHFWLRQLQYHVWLJDWHVWKHPDWUL[DQDORJXHRIWKHUHFLSURFDO RUPXOWLSOLFDWLYHLQYHUVH RI DQRQ]HURQXPEHU

7KH,QYHUVHRID0DWUL[

5HFDOOWKDWWKHPXOWLSOLFDWLYHLQYHUVHRIDQXPEHUVXFKDVLVRU 5"1 7KLV LQYHUVHVDWLVÀHVWKHHTXDWLRQV

5"1 " 5 D 1

DQG

5 " 5"1 D 1

7KHPDWUL[JHQHUDOL]DWLRQUHTXLUHV ERWK HTXDWLRQVDQGDYRLGVWKHVODQWHGOLQHQRWDWLRQ IRUGLYLVLRQ EHFDXVHPDWUL[PXOWLSOLFDWLRQLVQRWFRPPXWDWLYH )XUWKHUPRUH DIXOO JHQHUDOL]DWLRQLVSRVVLEOHRQO\LIWKHPDWULFHVLQYROYHGDUHVTXDUHñ $Q n ! n PDWUL[ A LVVDLGWREH LQYHUWLEOH LIWKHUHLVDQ n ! n PDWUL[ C VXFKWKDW

CA D I

DQG

AC D I

ZKHUH I D In ; WKH n ! n LGHQWLW\PDWUL[ ,QWKLVFDVH C LVDQ LQYHUVH RI A ,QIDFW C LVXQLTXHO\GHWHUPLQHGE\ A EHFDXVHLI B ZHUHDQRWKHULQYHUVHRI A WKHQ B D BI D B.AC / D .BA/C D IC D C: 7KLVXQLTXHLQYHUVHLVGHQRWHGE\ A"1 VRWKDW

A"1 A D I

DQG

AA"1 D I

$ PDWUL[WKDWLV QRW LQYHUWLEOHLVVRPHWLPHVFDOOHGD VLQJXODUPDWUL[ DQGDQLQYHUWLEOH PDWUL[LVFDOOHGD QRQVLQJXODUPDWUL[ #

$ # $ 2 5 #7 #5 &9".1-& ,I A D DQG C D WKHQ #3 #7 3 2 # $# $ # $ 2 5 #7 #5 1 0 AC D D DQG #3 #7 3 2 0 1 # $# $ # $ #7 #5 2 5 1 0 CA D D 3 2 #3 #7 0 1

7KXV C D A"1 +HUHLVDVLPSOHIRUPXODIRUWKHLQYHUVHRID 2 ! 2 PDWUL[ DORQJZLWKDWHVWWRWHOO LIWKHLQYHUVHH[LVWV

5)&03&.

/HW A D

#

a c

$ b ,I ad # bc ¤ 0 WKHQ A LVLQYHUWLEOHDQG d # $ 1 d #b "1 A D a ad # bc #c

,I ad # bc D 0 WKHQ A LVQRWLQYHUWLEOH

7KHVLPSOHSURRIRI7KHRUHP LVRXWOLQHGLQ([HUFLVHV DQG 7KHTXDQWLW\ ad # bc LVFDOOHGWKH GHWHUPLQDQW RI A DQGZHZULWH GHW A D ad # bc 7KHRUHPVD\VWKDWD 2 ! 2 PDWUL[ A LVLQYHUWLEOHLIDQGRQO\LI GHW A ¤ 0 ñ 2QHFRXOGVD\WKDWDQ m # n PDWUL[ A LVLQYHUWLEOHLIWKHUHH[LVW n # m PDWULFHV C DQG D VXFKWKDW CA D In DQG AD D Im : +RZHYHU WKHVHHTXDWLRQVLPSO\WKDW A LVVTXDUHDQG C D D: 7KXV A LVLQYHUWLEOH DVGHÀQHGDERYH 6HH([HUFLVHV²LQ6HFWLRQ

$)"15&3

0DWUL[$OJHEUD

&9".1-& )LQGWKHLQYHUVHRI A D

#

3 5

$ 4 6

40-65*0/ 6LQFH GHW A D 3.6/ # 4.5/ D #2 ¤ 0 A LVLQYHUWLEOH DQG # $ # $ # $ 1 6 #4 6=.#2/ #4=.#2/ #3 2 A"1 D D D #5=.#2/ 3=.#2/ 5=2 #3=2 #2 #5 3 ,QYHUWLEOHPDWULFHVDUHLQGLVSHQVDEOHLQOLQHDUDOJHEUD³PDLQO\IRUDOJHEUDLFFDOFX ODWLRQVDQGIRUPXODGHULYDWLRQV DVLQWKHQH[WWKHRUHP 7KHUHDUHDOVRRFFDVLRQVZKHQ DQLQYHUVHPDWUL[SURYLGHVLQVLJKWLQWRDPDWKHPDWLFDOPRGHORIDUHDOOLIHVLWXDWLRQ DV LQ([DPSOH EHORZ

5)&03&.

,I A LVDQLQYHUWLEOH n ! n PDWUL[ WKHQIRUHDFK E LQ Rn WKHHTXDWLRQ A[ D E KDV WKHXQLTXHVROXWLRQ [ D A"1 E

1300' 7DNHDQ\ E LQ Rn $ VROXWLRQH[LVWVEHFDXVHLI A"1 E LVVXEVWLWXWHGIRU [ WKHQ A[ D A.A"1 E/ D .AA"1 /E D I E D E 6R A"1 E LVDVROXWLRQ 7RSURYHWKDWWKH VROXWLRQLVXQLTXH VKRZWKDWLI X LVDQ\VROXWLRQ WKHQ X; LQIDFW PXVWEH A"1 E ,QGHHG LI AX D E ZHFDQPXOWLSO\ERWKVLGHVE\ A"1 DQGREWDLQ A"1 AX D A"1 E;

I X D A"1 E;

X D A"1 E

DQG

&9".1-& $ KRUL]RQWDOHODVWLFEHDPLVVXSSRUWHGDWHDFKHQGDQGLVVXEMHFWHGWR IRUFHVDWSRLQWV DVVKRZQLQ)LJ /HW I LQ R3 OLVWWKHIRUFHVDWWKHVHSRLQWV DQG OHW \ LQ R3 OLVWWKHDPRXQWVRIGHÁHFWLRQWKDWLV PRYHPHQW RIWKHEHDPDWWKHWKUHH SRLQWV 8VLQJ+RRNH·VODZIURPSK\VLFV LWFDQEHVKRZQWKDW \ D DI ZKHUH D LVD ÁH[LELOLW\PDWUL[ ,WVLQYHUVHLVFDOOHGWKH VWLIIQHVVPDWUL[ 'HVFULEHWKH SK\VLFDOVLJQLÀFDQFHRIWKHFROXPQVRI D DQG D "1 #1   y1 

f1

#2   y2 

#3

f2

  y3 

f3

'*(63& 'HÁHFWLRQRIDQHODVWLFEHDP

40-65*0/ :ULWH I3 D Œ H1 H2 H3 ! DQGREVHUYHWKDW

D D DI3 D Œ D H1 D H2 D H3 !

,QWHUSUHWWKHYHFWRU H1 D .1; 0; 0/ DVDXQLWIRUFHDSSOLHGGRZQZDUGDWSRLQWRQWKH EHDPZLWK]HURIRUFHDWWKHRWKHUWZRSRLQWV 7KHQ D H1 ; WKHÀUVWFROXPQRI D; OLVWV WKHEHDPGHÁHFWLRQVGXHWRDXQLWIRUFHDWSRLQW 6LPLODUGHVFULSWLRQVDSSO\WRWKH VHFRQGDQGWKLUGFROXPQVRI D: 7RVWXG\WKHVWLIIQHVVPDWUL[ D "1 REVHUYHWKDWWKHHTXDWLRQ I D D "1 \ FRPSXWHVD IRUFHYHFWRU I ZKHQDGHÁHFWLRQYHFWRU \ LVJLYHQ :ULWH

D "1 D D "1 I3 D Œ D "1 H1 D "1 H2 D "1 H3 !

1RZLQWHUSUHW H1 DVDGHÁHFWLRQYHFWRU 7KHQ D "1 H1 OLVWVWKHIRUFHVWKDWFUHDWHWKH GHÁHFWLRQ 7KDWLV WKHÀUVWFROXPQRI D "1 OLVWVWKHIRUFHVWKDWPXVWEHDSSOLHGDWWKH

7KH,QYHUVHRID0DWUL[

WKUHHSRLQWVWRSURGXFHDXQLWGHÁHFWLRQDWSRLQWDQG]HURGHÁHFWLRQVDWWKHRWKHUSRLQWV 6LPLODUO\ FROXPQVDQGRI D "1 OLVWWKHIRUFHVUHTXLUHGWRSURGXFHXQLWGHÁHFWLRQVDW SRLQWVDQG UHVSHFWLYHO\ ,QHDFKFROXPQ RQHRUWZRRIWKHIRUFHVPXVWEHQHJDWLYH SRLQWXSZDUG WRSURGXFHDXQLWGHÁHFWLRQDWWKHGHVLUHGSRLQWDQG]HURGHÁHFWLRQVDW WKHRWKHUWZRSRLQWV ,IWKHÁH[LELOLW\LVPHDVXUHG IRUH[DPSOH LQLQFKHVRIGHÁHFWLRQ SHUSRXQGRIORDG WKHQWKHVWLIIQHVVPDWUL[HQWULHVDUHJLYHQLQSRXQGVRIORDGSHULQFK RIGHÁHFWLRQ 7KHIRUPXODLQ7KHRUHP LVVHOGRPXVHGWRVROYHDQHTXDWLRQ A[ D E QXPHULFDOO\ EHFDXVHURZUHGXFWLRQRI Œ A E ! LVQHDUO\DOZD\VIDVWHU 5RZUHGXFWLRQLVXVXDOO\ PRUHDFFXUDWH WRR ZKHQFRPSXWDWLRQVLQYROYHURXQGLQJRIIQXPEHUV 2QHSRVVLEOH H[FHSWLRQLVWKH 2 ! 2 FDVH ,QWKLVFDVH PHQWDOFRPSXWDWLRQVWRVROYH A[ D E DUH VRPHWLPHVHDVLHUXVLQJWKHIRUPXODIRU A"1 DVLQWKHQH[WH[DPSOH

&9".1-& 8VHWKHLQYHUVHRIWKHPDWUL[ A LQ([DPSOH WRVROYHWKHV\VWHP 3x1 C 4x2 D 3 5x1 C 6x2 D 7

40-65*0/ 7KLVV\VWHPLVHTXLYDOHQWWR A[ D E VR # $# $ # $ #3 2 3 5 "1 [DA ED D 5=2 #3=2 7 #3 7KHQH[WWKHRUHPSURYLGHVWKUHHXVHIXOIDFWVDERXWLQYHUWLEOHPDWULFHV

5)&03&.

D ,I A LVDQLQYHUWLEOHPDWUL[ WKHQ A"1 LVLQYHUWLEOHDQG

.A"1 /"1 D A

E ,I A DQG B DUH n ! n LQYHUWLEOHPDWULFHV WKHQVRLV AB DQGWKHLQYHUVHRI AB LVWKHSURGXFWRIWKHLQYHUVHVRI A DQG B LQWKHUHYHUVHRUGHU 7KDWLV

.AB/"1 D B "1 A"1

F ,I A LVDQLQYHUWLEOHPDWUL[ WKHQVRLV AT DQGWKHLQYHUVHRI AT LVWKHWUDQVSRVH RI A"1 7KDWLV .AT /"1 D .A"1 /T

1300' 7RYHULI\VWDWHPHQWD ÀQGDPDWUL[ C VXFKWKDW A"1 C D I

DQG

CA"1 D I

,QIDFW WKHVHHTXDWLRQVDUHVDWLVÀHGZLWK A LQSODFHRI C +HQFH A"1 LVLQYHUWLEOH DQG A LVLWVLQYHUVH 1H[W WRSURYHVWDWHPHQWE FRPSXWH

.AB/.B "1 A"1 / D A.BB "1 /A"1 D AIA"1 D AA"1 D I

$ VLPLODUFDOFXODWLRQVKRZVWKDW .B "1 A"1 /.AB/ D I )RUVWDWHPHQWF XVH7KHRUHP G UHDGIURPULJKWWROHIW .A"1 /T AT D .AA"1 /T D I T D I 6LPLODUO\ AT .A"1 /T D I T D I +HQFH AT LVLQYHUWLEOH DQGLWVLQYHUVHLV .A"1 /T 7KHIROORZLQJJHQHUDOL]DWLRQRI7KHRUHPE LVQHHGHGODWHU 7KHSURGXFWRI n ! n LQYHUWLEOHPDWULFHVLVLQYHUWLEOH DQGWKHLQYHUVHLVWKH SURGXFWRIWKHLULQYHUVHVLQWKHUHYHUVHRUGHU

$)"15&3

0DWUL[$OJHEUD

7KHUHLVDQLPSRUWDQWFRQQHFWLRQEHWZHHQLQYHUWLEOHPDWULFHVDQGURZRSHUDWLRQV WKDWOHDGVWRDPHWKRGIRUFRPSXWLQJLQYHUVHV $VZHVKDOOVHH DQLQYHUWLEOHPDWUL[ A LVURZHTXLYDOHQWWRDQLGHQWLW\PDWUL[ DQGZHFDQÀQG A"1 E\ ZDWFKLQJWKHURZ UHGXFWLRQRI A WR I

(OHPHQWDU\0DWULFHV $Q HOHPHQWDU\PDWUL[ LVRQHWKDWLVREWDLQHGE\SHUIRUPLQJDVLQJOHHOHPHQWDU\URZ RSHUDWLRQRQDQLGHQWLW\PDWUL[ 7KHQH[WH[DPSOHLOOXVWUDWHVWKHWKUHHNLQGVRIHOHPHQ WDU\PDWULFHV

&9".1-& /HW 2

1 E1 D 4 0 #4

0 1 0

3 0 0 5; 1

2

0 1 E2 D 4 1 0 0 0 2 a b e A D 4d g h

3 0 0 5; 1 3 c f 5 i

2

1 E3 D 4 0 0

0 1 0

3 0 0 5; 5

&RPSXWH E1 A E2 A DQG E3 A DQGGHVFULEHKRZWKHVHSURGXFWVFDQEHREWDLQHGE\ HOHPHQWDU\URZRSHUDWLRQVRQ A

40-65*0/ 9HULI\WKDW 2 a E1 A D 4 d g # 4a

b e h # 4b

2

3 c f 5; i # 4c

a E3 A D 4 d 5g

b e 5h

2

d E2 A D 4 a g 3 c f 5: 5i

e b h

3 f c 5; i

$GGLWLRQRI #4 WLPHVURZ RI A WRURZ SURGXFHV E1 A 7KLVLVDURZUHSODFHPHQW RSHUDWLRQ $QLQWHUFKDQJHRIURZV DQGRI A SURGXFHV E2 A DQGPXOWLSOLFDWLRQRI URZRI A E\SURGXFHV E3 A /HIWPXOWLSOLFDWLRQWKDWLV PXOWLSOLFDWLRQRQWKHOHIW E\ E1 LQ([DPSOH KDVWKH VDPHHIIHFWRQDQ\ 3 ! n PDWUL[ ,WDGGV #4 WLPHVURZ WRURZ ,QSDUWLFXODU VLQFH E1 " I D E1 ZHVHHWKDW E1 LWVHOI LVSURGXFHGE\WKLVVDPHURZRSHUDWLRQRQWKHLGHQWLW\ 7KXV([DPSOH LOOXVWUDWHVWKHIROORZLQJJHQHUDOIDFWDERXWHOHPHQWDU\PDWULFHV 6HH ([HUFLVHV DQG ,IDQHOHPHQWDU\URZRSHUDWLRQLVSHUIRUPHGRQDQ m ! n PDWUL[ A WKHUHVXOWLQJ PDWUL[FDQEHZULWWHQDV EA ZKHUHWKH m ! m PDWUL[ E LVFUHDWHGE\SHUIRUPLQJ WKHVDPHURZRSHUDWLRQRQ Im

6LQFHURZRSHUDWLRQVDUHUHYHUVLEOH DVVKRZQLQ6HFWLRQ HOHPHQWDU\PDWULFHV DUHLQYHUWLEOH IRULI E LVSURGXFHGE\DURZRSHUDWLRQRQ I WKHQWKHUHLVDQRWKHUURZ RSHUDWLRQRIWKHVDPHW\SHWKDWFKDQJHV E EDFNLQWR I +HQFHWKHUHLVDQHOHPHQWDU\ PDWUL[ F VXFKWKDW FE D I 6LQFH E DQG F FRUUHVSRQGWRUHYHUVHRSHUDWLRQV EF D I WRR

7KH,QYHUVHRID0DWUL[

(DFKHOHPHQWDU\PDWUL[ E LVLQYHUWLEOH 7KHLQYHUVHRI E LVWKHHOHPHQWDU\PDWUL[ RIWKHVDPHW\SHWKDWWUDQVIRUPV E EDFNLQWR I 2

1 &9".1-& )LQGWKHLQYHUVHRI E1 D 4 0 #4

40-65*0/ 7RWUDQVIRUP E1 LQWR I DGG C4 PDWUL[WKDWGRHVWKLVLV 2 1 E1"1 D 4 0 C4

0 1 0

3 0 0 5 1

WLPHVURZ WRURZ 7KHHOHPHQWDU\ 3 0 0 1 05 0 1

7KHIROORZLQJWKHRUHPSURYLGHVWKHEHVWZD\WRÝLVXDOL]HµDQLQYHUWLEOHPDWUL[ DQGWKHWKHRUHPOHDGVLPPHGLDWHO\WRDPHWKRGIRUÀQGLQJWKHLQYHUVHRIDPDWUL[

5)&03&.

$Q n ! n PDWUL[ A LVLQYHUWLEOHLIDQGRQO\LI A LVURZHTXLYDOHQWWR In DQGLQ WKLVFDVH DQ\VHTXHQFHRIHOHPHQWDU\URZRSHUDWLRQVWKDWUHGXFHV A WR In DOVR WUDQVIRUPV In LQWR A"1

1300' 6XSSRVHWKDW A LVLQYHUWLEOH 7KHQ VLQFHWKHHTXDWLRQ A[ D E KDVDVROXWLRQ IRUHDFK E 7KHRUHP A KDVDSLYRWSRVLWLRQLQHYHU\URZ7KHRUHPLQ6HFWLRQ %HFDXVH A LVVTXDUH WKH n SLYRWSRVLWLRQVPXVWEHRQWKHGLDJRQDO ZKLFKLPSOLHVWKDW WKHUHGXFHGHFKHORQIRUPRI A LV In 7KDWLV A % In 1RZVXSSRVH FRQYHUVHO\ WKDW A % In 7KHQ VLQFHHDFKVWHSRIWKHURZUHGXFWLRQ RI A FRUUHVSRQGVWROHIWPXOWLSOLFDWLRQE\DQHOHPHQWDU\PDWUL[ WKHUHH[LVWHOHPHQWDU\ PDWULFHV E1 ; : : : ; Ep VXFKWKDW A % E1 A % E2 .E1 A/ % " " " % Ep .Ep"1 " " " E1 A/ D In 7KDWLV

Ep " " " E1 A D In

6LQFHWKHSURGXFW Ep " " " E1 RILQYHUWLEOHPDWULFHVLVLQYHUWLEOH OHDGVWR

.Ep " " " E1 /"1 .Ep " " " E1 /A D .Ep " " " E1 /"1 In A D .Ep " " " E1 /"1

7KXV A LVLQYHUWLEOH DVLWLVWKHLQYHUVHRIDQLQYHUWLEOHPDWUL[7KHRUHP $OVR "1

A"1 D Œ .Ep " " " E1 /"1 !

D Ep " " " E1

7KHQ A"1 D Ep " " " E1 " In ZKLFKVD\VWKDW A"1 UHVXOWVIURPDSSO\LQJ E1 ; : : : ; Ep VXF FHVVLYHO\WR In 7KLVLVWKHVDPHVHTXHQFHLQ WKDWUHGXFHG A WR In

$Q$OJRULWKPIRU)LQGLQJ $± ,IZHSODFH A DQG I VLGHE\VLGHWRIRUPDQDXJPHQWHGPDWUL[ Œ A I ! WKHQURZ RSHUDWLRQVRQWKLVPDWUL[SURGXFHLGHQWLFDORSHUDWLRQVRQ A DQGRQ I %\7KHRUHP HLWKHUWKHUHDUHURZRSHUDWLRQVWKDWWUDQVIRUP A WR In DQG In WR A"1 RUHOVH A LVQRW LQYHUWLEOH

$)"15&3

0DWUL[$OJHEUD

"-(03*5). '03 '*/%*/( "m 5RZUHGXFHWKHDXJPHQWHGPDWUL[ Œ A I ! ,I A LVURZHTXLYDOHQWWR I WKHQ Œ A I ! LVURZHTXLYDOHQWWR Œ I A"1 ! 2WKHUZLVH A GRHVQRWKDYHDQLQYHUVH 2

3 2 3 5 LILWH[LVWV 8

0 1 &9".1-& )LQGWKHLQYHUVHRIWKHPDWUL[ A D 4 1 0 4 #3 40-65*0/

2

0 1 2 1 0 ŒA I ! D 41 0 3 0 1 4 #3 8 0 0 2 1 0 3 0 1 % 40 1 2 1 0 0 #3 #4 0 #4 2 1 0 3 0 1 1 0 % 40 1 2 0 0 1 3=2 #2 2 1 0 0 #9=2 7 4 % 4 0 1 0 #2 0 0 1 3=2 #2

3 2 0 1 0 05 % 40 1 1 4 #3 3 2 0 1 0 05 % 40 1 1 0 0 3 0 0 5 1=2 3 #3=2 #1 5 1=2

3 2 8

0 1 0

1 0 0

3 2 2

0 1 1 0 3 #4

3 0 05 1 3 0 05 1

7KHRUHPVKRZV VLQFH A % I WKDW A LVLQYHUWLEOH DQG 2 3 #9=2 7 #3=2 4 #1 5 A"1 D 4 #2 3=2 #2 1=2

,WLVDJRRGLGHDWRFKHFNWKHÀQDODQVZHU 2 32 3 2 0 1 2 #9=2 7 #3=2 1 4 #1 5 D 4 0 AA"1 D 4 1 0 3 5 4 #2 4 #3 8 3=2 #2 1=2 0 ,WLVQRWQHFHVVDU\WRFKHFNWKDW A"1 A D I VLQFH A LVLQYHUWLEOH

0 1 0

3 0 05 1

$QRWKHU9LHZRI0DWUL[,QYHUVLRQ 'HQRWHWKHFROXPQVRI In E\ H1 ; : : : ; Hn 7KHQURZUHGXFWLRQRI Œ A I ! WR Œ I FDQEHYLHZHGDVWKHVLPXOWDQHRXVVROXWLRQRIWKH n V\VWHPV

A[ D H1 ;

A[ D H 2 ;

:::;

A[ D Hn

A"1 !

ZKHUHWKH´DXJPHQWHGFROXPQVµRIWKHVHV\VWHPVKDYHDOOEHHQSODFHGQH[WWR A WRIRUP Œ A H1 H2 " " " Hn ! D Œ A I ! 7KHHTXDWLRQ AA"1 D I DQGWKHGHÀQLWLRQRIPDWUL[ PXOWLSOLFDWLRQVKRZWKDWWKHFROXPQVRI A"1 DUHSUHFLVHO\WKHVROXWLRQVRIWKHV\VWHPV LQ 7KLVREVHUYDWLRQLVXVHIXOEHFDXVHVRPHDSSOLHGSUREOHPVPD\UHTXLUHÀQGLQJ RQO\RQHRUWZRFROXPQVRI A"1 ,QWKLVFDVH RQO\WKHFRUUHVSRQGLQJV\VWHPVLQ QHHGEHVROYHG

7KH,QYHUVHRID0DWUL[

/6.&3*$"- /05& ,QSUDFWLFDOZRUN A"1 LVVHOGRPFRPSXWHG XQOHVVWKHHQWULHVRI A"1 DUHQHHGHG &RPSXWLQJERWK A"1 DQG A"1 E WDNHVDERXWWKUHHWLPHVDVPDQ\DULWKPHWLF RSHUDWLRQVDVVROYLQJ A[ D E E\URZUHGXFWLRQ DQGURZUHGXFWLRQPD\EHPRUH DFFXUDWH

WEB

13"$5*$& 130#-&.4 8VHGHWHUPLQDQWVWRGHWHUPLQHZKLFKRIWKHIROORZLQJPDWULFHVDUHLQYHUWLEOH # $ # $ # $ 3 #9 4 #9 6 #9 D E F 2 6 0 5 #4 6 2 3 1 #2 #1 6 5 LILWH[LVWV )LQGWKHLQYHUVHRIWKHPDWUL[ A D 4 #1 5 5 #4 5

&9&3$*4&4 )LQGWKHLQYHUVHVRIWKHPDWULFHVLQ([HUFLVHV² # $ # $ 8 6 3 2 5 4 8 5

#

7 3 #6 #3

$

#

2 #4 4 #6

$

8VHWKHLQYHUVHIRXQGLQ([HUFLVHWRVROYHWKHV\VWHP

8x1 C 6x2 D

2

5x1 C 4x2 D #1

7x1 C 3x2 D #9

#

D )LQG A!1 ; DQGXVHLWWRVROYHWKHIRXUHTXDWLRQV

A[ D E 1 ;

E ,I A DQG B DUH n ! n DQGLQYHUWLEOH WKHQ A!1 B !1 LVWKH LQYHUVHRI AB # $ a b DQG ab # cd ¤ 0 WKHQ A LVLQYHUWLEOH F ,I A D c d G ,I A LV DQ LQYHUWLEOH n ! n PDWUL[ WKHQ WKH HTXDWLRQ A[ D E LVFRQVLVWHQWIRU HDFK E LQ Rn

A[ D E 2 ;

D ,I A LVLQYHUWLEOH WKHQHOHPHQWDU\URZRSHUDWLRQVWKDW UHGXFH A WRWKHLGHQWLW\ In DOVRUHGXFH A!1 WR In E ,I A LVLQYHUWLEOH WKHQWKHLQYHUVHRI A!1 LV A LWVHOI

4

$ # $ # $ # $ 1 2 #1 1 2 E1 D E2 D E3 D 5 12 3 #5 6 # $ 3 DQG E4 D 5

/HW A D

D ,Q RUGHU IRU D PDWUL[ B WR EH WKH LQYHUVH RI A WKH HTXDWLRQV AB D I DQG BA D I PXVWERWKEHWUXH

H (DFKHOHPHQWDU\PDWUL[LVLQYHUWLEOH

8VHWKHLQYHUVHIRXQGLQ([HUFLVHWRVROYHWKHV\VWHP

#6x1 # 3x2 D

,Q([HUFLVHVDQG PDUNHDFKVWDWHPHQW7UXHRU)DOVH -XVWLI\ HDFKDQVZHU

A[ D E 3 ;

A[ D E 4

E 7KH IRXU HTXDWLRQV LQ SDUW D FDQ EH VROYHG E\ WKH VDPH VHW RI URZ RSHUDWLRQV VLQFH WKH FRHIÀFLHQW PD WUL[ LV WKH VDPH LQ HDFK FDVH 6ROYH WKH IRXU HTXD WLRQVLQSDUW D E\URZUHGXFLQJWKHDXJPHQWHGPDWUL[ Œ A E1 E2 E3 E4 ! 6XSSRVH P LVLQYHUWLEOHDQG A D PBP !1 6ROYHIRU B LQ WHUPVRI A

F $ SURGXFWRILQYHUWLEOH n ! n PDWULFHVLVLQYHUWLEOH DQG WKHLQYHUVHRIWKHSURGXFWLVWKHSURGXFWRIWKHLULQYHUVHV LQWKHVDPHRUGHU G ,I A LVDQ n ! n PDWUL[DQG A[ D Hj LVFRQVLVWHQWIRU HYHU\ j 2 f1; 2; : : : ; ng; WKHQ A LV LQYHUWLEOH 1RWH H1 ; : : : ; Hn UHSUHVHQWWKHFROXPQVRIWKHLGHQWLW\PDWUL[ H ,I A FDQEHURZUHGXFHGWRWKHLGHQWLW\PDWUL[ WKHQ A PXVWEHLQYHUWLEOH /HW A EHDQLQYHUWLEOH n ! n PDWUL[ DQGOHW B EHDQ n ! p PDWUL[ 6KRZWKDWWKHHTXDWLRQ AX D B KDVDXQLTXHVROX WLRQ A!1 B: 8VHPDWUL[DOJHEUDWRVKRZWKDWLI A LVLQYHUWLEOHDQG D VDWLVÀHV AD D I WKHQ D D A!1 :

6XSSRVH AB D AC ZKHUH B DQG C DUH n ! p PDWULFHVDQG A LVLQYHUWLEOH 6KRZWKDW B D C ,VWKLVWUXH LQJHQHUDO ZKHQ A LVQRWLQYHUWLEOH"

$)"15&3

0DWUL[$OJHEUD

6XSSRVH .B # C /D D 0 ZKHUH B DQG C DUH m ! n PDWULFHV DQG D LVLQYHUWLEOH 6KRZWKDW B D C

/HW A EHDQLQYHUWLEOH n ! n PDWUL[ DQGOHW B EHDQ n ! p PDWUL[ ([SODLQZK\ A!1 B FDQEHFRPSXWHGE\URZUHGXF WLRQ ,I Œ A

B ! % """ % Œ I

X ! WKHQ X D A!1 B

,I A LVODUJHUWKDQ 2 ! 2; WKHQURZUHGXFWLRQRI ŒA PXFKIDVWHUWKDQFRPSXWLQJERWK A!1 DQG A!1 B:

B! LV

6XSSRVH A DQG B DUH n ! n PDWULFHV B LVLQYHUWLEOH DQG AB LVLQYHUWLEOH 6KRZWKDW A LVLQYHUWLEOH >+LQW /HW C D AB DQGVROYHWKLVHTXDWLRQIRU A@ 6XSSRVH A B DQG C DUHLQYHUWLEOH n ! n PDWULFHV 6KRZ WKDW ABC LVDOVRLQYHUWLEOHE\SURGXFLQJDPDWUL[ D VXFK WKDW .ABC /D D I DQG D.ABC / D I

6ROYHWKHHTXDWLRQ AB D BC IRU A DVVXPLQJWKDW A B DQG C DUHVTXDUHDQG B LVLQYHUWLEOH

,I A; B; DQG C DUH n ! n LQYHUWLEOHPDWULFHV GRHVWKHHTXD WLRQ C !1 .A C X /B !1 D In KDYHDVROXWLRQ X " ,IVR ÀQG LW 6XSSRVH A B DQG X DUH n ! n PDWULFHVZLWK A X DQG A # AX LQYHUWLEOH DQGVXSSRVH

.A # AX/!1 D X !1 B

D ([SODLQZK\ B LVLQYHUWLEOH E 6ROYHHTXDWLRQ IRU X ,IDPDWUL[QHHGVWREHLQYHUWHG H[SODLQZK\WKDWPDWUL[LVLQYHUWLEOH ([SODLQZK\WKHFROXPQVRIDQ n ! n PDWUL[ A DUHOLQHDUO\ LQGHSHQGHQWZKHQ A LVLQYHUWLEOH ([SODLQZK\WKHFROXPQVRIDQ n ! n PDWUL[ A VSDQ Rn ZKHQ A LVLQYHUWLEOH >+LQW 5HYLHZ7KHRUHP LQ6HFWLRQ @ 6XSSRVH A LV n ! n DQGWKHHTXDWLRQ A[ D KDVRQO\WKH WULYLDOVROXWLRQ ([SODLQZK\ A KDV n SLYRWFROXPQVDQG A LV URZHTXLYDOHQWWR In %\7KHRUHP WKLVVKRZVWKDW A PXVW EHLQYHUWLEOH 7KLVH[HUFLVHDQG([HUFLVH ZLOOEHFLWHGLQ 6HFWLRQ 6XSSRVH A LV n ! n DQGWKHHTXDWLRQ A[ D E KDVDVROXWLRQ IRUHDFK E LQ Rn ([SODLQZK\ A PXVWEHLQYHUWLEOH >+LQW ,V A URZHTXLYDOHQWWR In ‹! # $ a b ([HUFLVHVDQGSURYH7KHRUHPIRU A D c d 6KRZWKDWLI ad # bc D 0 WKHQWKHHTXDWLRQ A[ D KDV PRUHWKDQRQHVROXWLRQ :K\GRHVWKLVLPSO\WKDW A LVQRW LQYHUWLEOH" >+LQW )LUVW FRQVLGHU a D b D 0:# 7KHQ $ LI a DQG #b b DUHQRWERWK]HUR FRQVLGHUWKHYHFWRU [ D @ a

6KRZWKDWLI ad # bc ¤ 0 WKHIRUPXODIRU A!1 ZRUNV

([HUFLVHVDQGSURYHVSHFLDOFDVHVRIWKHIDFWVDERXWHOHPHQ WDU\PDWULFHVVWDWHGLQWKHER[IROORZLQJ([DPSOH +HUH A LVD 3 ! 3 PDWUL[DQG I D I3 : $ JHQHUDOSURRIZRXOGUHTXLUHVOLJKWO\ PRUHQRWDWLRQ

/HW A EHD 3 ! 3 PDWUL[ D 8VH HTXDWLRQ IURP 6HFWLRQ WR VKRZ WKDW URZi .A/ D URZi .I / " A IRU i D 1; 2; 3 E 6KRZWKDWLIURZVDQGRI A DUHLQWHUFKDQJHG WKHQWKH UHVXOWPD\EHZULWWHQDV EA ZKHUH E LVDQHOHPHQWDU\ PDWUL[IRUPHGE\LQWHUFKDQJLQJURZVDQGRI I:

F 6KRZWKDWLIURZRI A LVPXOWLSOLHGE\ WKHQWKHUHVXOW PD\EHZULWWHQDV EA; ZKHUH E LVIRUPHGE\PXOWLSO\LQJ URZRI I E\ 6XSSRVHURZRI A LVUHSODFHGE\ URZ2 .A/ # 3 " URZ1 .A/ 6KRZWKDWWKHUHVXOWLV EA ZKHUH E LVIRUPHGIURP I E\ UHSODFLQJ URZ2 .I / E\ URZ2 .I / # 3 " URZ1 .A/ )LQGWKHLQYHUVHVRIWKHPDWULFHVLQ([HUFLVHV² LIWKH\H[LVW 8VHWKHDOJRULWKPLQWURGXFHGLQWKLVVHFWLRQ # $ # $ 1 #3 3 6 4 #9 4 7 2

3 1 0 #2 1 45 4 #3 2 #3 4

2

3 1 2 #1 35 4 #4 #7 #2 #6 4

8VHWKHDOJRULWKPIURPWKLVVHFWLRQWRÀQGWKHLQYHUVHVRI 2 3 2 3 1 0 0 0 1 0 0 61 1 0 07 7: 41 1 0 5 DQG 6 41 1 1 05 1 1 1 1 1 1 1

/HW A EHWKHFRUUHVSRQGLQJ n ! n PDWUL[ DQGOHW B EHLWV LQYHUVH *XHVVWKHIRUPRI B; DQGWKHQVKRZWKDW AB D I

5HSHDWWKHVWUDWHJ\RI([HUFLVHWRJXHVVWKHLQYHUVH B RI 2 3 1 0 0 """ 0 62 2 0 07 6 7 3 3 3 0 7: AD6 6 : 7 : : :: :: 5 4 :: n n n """ n

6KRZWKDW AB D I 2 3 #1 #7 #3 15 6 5 )LQGWKHWKLUGFROXPQRI A!1 /HW A D 4 2 1 3 2 ZLWKRXWFRPSXWLQJ WKHRWKHUFROXPQV 2

#25 >0@ /HW A D 4 536 154

#9 185 52

3 #27 537 5 )LQGWKHVHFRQGDQG 143

WKLUGFROXPQVRI A!1 ZLWKRXWFRPSXWLQJ WKHÀUVWFROXPQ 2

1 /HW A D 4 1 1

3 2 3 5 &RQVWUXFWD 2 ! 3 PDWUL[ C E\WULDODQG 5

HUURU XVLQJRQO\ # DQGDVHQWULHV VXFK WKDW CA D I2 &RPSXWH AC DQGQRWHWKDW AC ¤ I3 :

/HW A D

#

1 #1 1 0 1 #1

$ 0 &RQVWUXFW D 4 ! 2 PDWUL[ 1

D XVLQJRQO\DQGDVHQWULHV VXFK WKDW AD D I2 : ,VLW SRVVLEOHWKDW CA D I4 IRUVRPH 4 ! 2 PDWUL[ C " :K\RU ZK\QRW" >0@ /HW 2 :011 D D 4 :003 :001

:003 :009 :003

3 :001 :003 5 :011

EHDÁH[LELOLW\PDWUL[ ZLWKÁH[LELOLW\PHDVXUHGLQLQFKHVSHU SRXQG 6XSSRVHWKDWIRUFHVRI DQG OEDUHDSSOLHGDW SRLQWV DQG UHVSHFWLYHO\ LQ)LJ RI([DPSOH )LQG WKHFRUUHVSRQGLQJGHÁHFWLRQV

>0@ &RPSXWHWKHVWLIIQHVVPDWUL[ D !1 IRU D LQ([HUFLVH /LVWWKHIRUFHVQHHGHGWRSURGXFHDGHÁHFWLRQRILQDW SRLQW ZLWK]HURGHÁHFWLRQVDWWKHRWKHUSRLQWV

&KDUDFWHUL]DWLRQVRI,QYHUWLEOH0DWULFHV

>0@ /HW 2 :0130 6 :0050 6 DD4 :0020 :0010

:0050 :0100 :0040 :0020

:0020 :0040 :0100 :0050

3 :0010 :0020 7 7 :0050 5 :0130

EHDÁH[LELOLW\PDWUL[IRUDQHODVWLFEHDPVXFKDVWKHRQHLQ ([DPSOH ZLWKIRXUSRLQWVDWZKLFKIRUFHLVDSSOLHG 8QLWV DUHFHQWLPHWHUVSHUQHZWRQRIIRUFH 0HDVXUHPHQWVDWWKH IRXUSRLQWVVKRZGHÁHFWLRQVRI DQG FP 'HWHUPLQHWKHIRUFHVDWWKHIRXUSRLQWV

>0@ :LWK D DVLQ([HUFLVH GHWHUPLQHWKHIRUFHVWKDW SURGXFHDGHÁHFWLRQRI FPDWWKHVHFRQGSRLQWRQWKH EHDP ZLWK]HURGHÁHFWLRQVDWWKHRWKHUWKUHHSRLQWV +RZLV WKHDQVZHUUHODWHGWRWKHHQWULHVLQ D !1 " >+LQW )LUVWDQVZHU WKHTXHVWLRQZKHQWKHGHÁHFWLRQLV FPDWWKHVHFRQGSRLQW@

40-65*0/4 50 13"$5*$& 130#-&.4 # $ 3 #9 D GHW D 3 " 6 # .#9/ " 2 D 18 C 18 D 36 7KHGHWHUPLQDQWLVQRQ]HUR VR 2 6 WKHPDWUL[LVLQYHUWLEOH # $ 4 #9 E GHW D 4 " 5 # .#9/ " 0 D 20 ¤ 0 7KHPDWUL[LVLQYHUWLEOH 0 5 # $ 6 #9 F GHW D 6 " 6 # .#9/.#4/ D 36 # 36 D 0 7KHPDWUL[LVQRWLQYHUWLEOH #4 6 2 3 1 #2 #1 1 0 0 6 0 1 05 Œ A I ! % 4 #1 5 5 #4 5 0 0 1 2 3 1 #2 #1 1 0 0 % 40 3 5 1 1 05 0 6 10 #5 0 1 2 3 1 #2 #1 1 0 0 % 40 3 5 1 1 05 0 0 0 #7 #2 1 6R Œ A I ! LVURZHTXLYDOHQWWRDPDWUL[RIWKHIRUP Œ B D ! ZKHUH B LVVTXDUH DQGKDVDURZRI]HURV )XUWKHUURZRSHUDWLRQVZLOOQRWWUDQVIRUP B LQWR I VRZH VWRS A GRHVQRWKDYHDQLQYHUVH

$)"3"$5&3*;"5*0/4 0' */7&35*#-& ."53*$&4 7KLVVHFWLRQSURYLGHVDUHYLHZRIPRVWRIWKHFRQFHSWVLQWURGXFHGLQ&KDSWHU LQ UHODWLRQWRV\VWHPVRI n OLQHDUHTXDWLRQVLQ n XQNQRZQVDQGWR VTXDUH PDWULFHV 7KH PDLQUHVXOWLV7KHRUHP

$)"15&3

0DWUL[$OJHEUD

5)&03&.

5IF *OWFSUJCMF .BUSJY 5IFPSFN /HW A EHDVTXDUH n ! n PDWUL[ 7KHQWKHIROORZLQJVWDWHPHQWVDUHHTXLYDOHQW 7KDWLV IRUDJLYHQ A WKHVWDWHPHQWVDUHHLWKHUDOOWUXHRUDOOIDOVH D E F G H I J K L M N O

(a) ( j)

( b) (c)

(d)

A LVDQLQYHUWLEOHPDWUL[ A LVURZHTXLYDOHQWWRWKH n ! n LGHQWLW\PDWUL[ A KDV n SLYRWSRVLWLRQV 7KHHTXDWLRQ A[ D KDVRQO\WKHWULYLDOVROXWLRQ 7KHFROXPQVRI A IRUPDOLQHDUO\LQGHSHQGHQWVHW 7KHOLQHDUWUDQVIRUPDWLRQ [ 7! A[ LVRQHWRRQH 7KHHTXDWLRQ A[ D E KDVDWOHDVWRQHVROXWLRQIRUHDFK E LQ Rn 7KHFROXPQVRI A VSDQ Rn 7KHOLQHDUWUDQVIRUPDWLRQ [ 7! A[ PDSV Rn RQWR Rn 7KHUHLVDQ n ! n PDWUL[ C VXFKWKDW CA D I 7KHUHLVDQ n ! n PDWUL[ D VXFKWKDW AD D I AT LVDQLQYHUWLEOHPDWUL[

)LUVW ZHQHHGVRPHQRWDWLRQ ,IWKHWUXWKRIVWDWHPHQWD DOZD\VLPSOLHVWKDW VWDWHPHQWM LVWUXH ZHVD\WKDWD LPSOLHV M DQGZULWH D ) M 7KHSURRIZLOO HVWDEOLVKWKH´FLUFOHµRILPSOLFDWLRQVVKRZQLQ)LJ ,IDQ\RQHRIWKHVHÀYHVWDWHPHQWV LVWUXH WKHQVRDUHWKHRWKHUV )LQDOO\ WKHSURRIZLOOOLQNWKHUHPDLQLQJVWDWHPHQWVRI WKHWKHRUHPWRWKHVWDWHPHQWVLQWKLVFLUFOH

'*(63&

(k) (g)

(a) (g)

(h)

(i)

(d)

(e)

(f )

(a)

(l)

1300' ,IVWDWHPHQWD LVWUXH WKHQ A"1 ZRUNVIRU C LQM VR D ) M 1H[W M ) G E\([HUFLVH LQ6HFWLRQ 7XUQEDFNDQGUHDGWKHH[HUFLVH $OVR G ) F E\ ([HUFLVHLQ6HFWLRQ ,I A LVVTXDUHDQGKDV n SLYRWSRVLWLRQV WKHQWKHSLYRWV PXVWOLHRQWKHPDLQGLDJRQDO LQZKLFKFDVHWKHUHGXFHGHFKHORQIRUPRI A LV In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´7KHHTXDWLRQ A[ D E KDVD XQLTXH VROXWLRQIRUHDFK E LQ Rn µ 7KLVVWDWHPHQW FHUWDLQO\LPSOLHVE DQGKHQFHLPSOLHVWKDW A LVLQYHUWLEOH 7KHQH[WIDFWIROORZVIURP7KHRUHP DQG([HUFLVH LQ6HFWLRQ /HW A DQG B EHVTXDUHPDWULFHV ,I AB D I WKHQ A DQG B DUHERWKLQYHUWLEOH ZLWK B D A"1 DQG A D B "1


7KH,QYHUWLEOH0DWUL[7KHRUHPGLYLGHVWKHVHWRIDOO n ! n PDWULFHVLQWRWZRGLVMRLQW FODVVHV WKHLQYHUWLEOHQRQVLQJXODU PDWULFHV DQGWKHQRQLQYHUWLEOHVLQJXODU PDWULFHV (DFKVWDWHPHQWLQWKHWKHRUHPGHVFULEHVDSURSHUW\RIHYHU\ n ! n LQYHUWLEOHPDWUL[ 7KH QHJDWLRQ RIDVWDWHPHQWLQWKHWKHRUHPGHVFULEHVDSURSHUW\RIHYHU\ n ! n VLQJXODU PDWUL[ )RULQVWDQFH DQ n ! n VLQJXODUPDWUL[LV QRW URZHTXLYDOHQWWR In GRHV QRW KDYH n SLYRWSRVLWLRQV DQGKDVOLQHDUO\ GHSHQGHQW FROXPQV 1HJDWLRQVRIRWKHUVWDWHPHQWV DUHFRQVLGHUHGLQWKHH[HUFLVHV

&9".1-& 8VHWKH,QYHUWLEOH0DWUL[7KHRUHPWRGHFLGHLI A LVLQYHUWLEOH 2

40-65*0/

2

3 1 0 #2 1 #2 5 AD4 3 #5 #1 9

3 2 1 0 #2 1 1 45 % 40 A % 40 0 #1 #1 0

3 0 #2 1 45 0 3

6R A KDVWKUHHSLYRWSRVLWLRQVDQGKHQFHLVLQYHUWLEOH E\WKH,QYHUWLEOH0DWUL[7KHRUHP VWDWHPHQWF SG

&YQBOEFE 5BCMF GPS UIF *.5 m

7KH SRZHURIWKH,QYHUWLEOH0DWUL[7KHRUHPOLHVLQWKHFRQQHFWLRQVLWSURYLGHV DPRQJVRPDQ\LPSRUWDQWFRQFHSWV VXFKDVOLQHDULQGHSHQGHQFHRIFROXPQVRIDPDWUL[ A DQGWKHH[LVWHQFHRIVROXWLRQVWRHTXDWLRQVRIWKHIRUP A[ D E ,WVKRXOGEHHPSKD VL]HG KRZHYHU WKDWWKH,QYHUWLEOH0DWUL[7KHRUHP DSSOLHVRQO\WRVTXDUHPDWULFHV )RU H[DPSOH LIWKHFROXPQVRID 4 ! 3 PDWUL[DUHOLQHDUO\LQGHSHQGHQW ZHFDQQRWXVHWKH ,QYHUWLEOH0DWUL[7KHRUHPWRFRQFOXGHDQ\WKLQJDERXWWKHH[LVWHQFHRUQRQH[LVWHQFHRI VROXWLRQVWRHTXDWLRQVRIWKHIRUP A[ D E

,QYHUWLEOH/LQHDU7UDQVIRUPDWLRQV 5HFDOOIURP6HFWLRQWKDWPDWUL[PXOWLSOLFDWLRQFRUUHVSRQGVWRFRPSRVLWLRQRIOLQHDU WUDQVIRUPDWLRQV :KHQDPDWUL[ A LVLQYHUWLEOH WKHHTXDWLRQ A"1 A[ D [ FDQEHYLHZHG DVDVWDWHPHQWDERXWOLQHDUWUDQVIRUPDWLRQV 6HH)LJ Multiplication by A x

Ax Multiplication by A–1

'*(63& A!1 WUDQVIRUPV A[ EDFNWR [

$ OLQHDUWUDQVIRUPDWLRQ T W Rn ! Rn LVVDLGWREH LQYHUWLEOH LIWKHUHH[LVWVDIXQF WLRQ S W Rn ! Rn VXFKWKDW

S.T .[// D [ T .S.[// D [

IRUDOO [ LQ Rn IRUDOO [ LQ Rn

7KHQH[WWKHRUHPVKRZVWKDWLIVXFKDQ S H[LVWV LWLVXQLTXHDQGPXVWEHDOLQHDU WUDQVIRUPDWLRQ :HFDOO S WKH LQYHUVH RI T DQGZULWHLWDV T "1

$)"15&3

0DWUL[$OJHEUD

5)&03&.

/HW T W Rn ! Rn EHDOLQHDUWUDQVIRUPDWLRQDQGOHW A EHWKHVWDQGDUGPDWUL[IRU T 7KHQ T LVLQYHUWLEOHLIDQGRQO\LI A LVDQLQYHUWLEOHPDWUL[ ,QWKDWFDVH WKH OLQHDUWUDQVIRUPDWLRQ S JLYHQE\ S.[/ D A"1 [ LVWKHXQLTXHIXQFWLRQVDWLVI\LQJ HTXDWLRQV .1/ DQG .2/

1300' 6XSSRVHWKDW T LVLQYHUWLEOH 7KHQ VKRZVWKDW T LVRQWR Rn IRULI E LVLQ Rn DQG [ D S.E/ WKHQ T .[/ D T .S.E// D E VRHDFK E LVLQWKHUDQJHRI T 7KXV A LV LQYHUWLEOH E\WKH,QYHUWLEOH0DWUL[7KHRUHP VWDWHPHQWL &RQYHUVHO\ VXSSRVHWKDW A LVLQYHUWLEOH DQGOHW S.[/ D A"1 [ 7KHQ S LVDOLQHDU WUDQVIRUPDWLRQ DQG S REYLRXVO\VDWLVÀHV DQG )RULQVWDQFH S.T .[// D S.A[/ D A"1 .A[/ D [ 7KXV T LVLQYHUWLEOH 7KHSURRIWKDW S LVXQLTXHLVRXWOLQHGLQ([HUFLVH

&9".1-& :KDWFDQ\RXVD\DERXWDRQHWRRQHOLQHDUWUDQVIRUPDWLRQ T IURP Rn LQWR Rn " 40-65*0/ 7KHFROXPQVRIWKHVWDQGDUGPDWUL[ A RI T DUHOLQHDUO\LQGHSHQGHQWE\ 7KHRUHPLQ6HFWLRQ 6R A LVLQYHUWLEOH E\WKH,QYHUWLEOH0DWUL[7KHRUHP DQG T PDSV Rn RQWR Rn $OVR T LVLQYHUWLEOH E\7KHRUHP

/6.&3*$"- /05&4

WEB

,QSUDFWLFDOZRUN \RXPLJKWRFFDVLRQDOO\HQFRXQWHUD´QHDUO\VLQJXODUµRU LOO FRQGLWLRQHG PDWUL[³DQLQYHUWLEOHPDWUL[WKDWFDQEHFRPHVLQJXODULIVRPHRI LWVHQWULHVDUHFKDQJHGHYHUVRVOLJKWO\ ,QWKLVFDVH URZUHGXFWLRQPD\SURGXFH IHZHUWKDQ n SLYRWSRVLWLRQV DVDUHVXOWRIURXQGRIIHUURU $OVR URXQGRIIHUURU FDQVRPHWLPHVPDNHDVLQJXODUPDWUL[DSSHDUWREHLQYHUWLEOH 6RPH PDWUL[ SURJUDPV ZLOO FRPSXWH D FRQGLWLRQ QXPEHU IRU D VTXDUH PDWUL[ 7KHODUJHUWKHFRQGLWLRQQXPEHU WKHFORVHUWKHPDWUL[LVWREHLQJVLQJXODU 7KHFRQGLWLRQQXPEHURIWKHLGHQWLW\PDWUL[LV $ VLQJXODUPDWUL[KDVDQ LQÀQLWHFRQGLWLRQQXPEHU ,QH[WUHPHFDVHV DPDWUL[SURJUDPPD\QRWEHDEOH WRGLVWLQJXLVKEHWZHHQDVLQJXODUPDWUL[DQGDQLOOFRQGLWLRQHGPDWUL[ ([HUFLVHV²VKRZWKDWPDWUL[FRPSXWDWLRQVFDQSURGXFHVXEVWDQWLDOHUURU ZKHQDFRQGLWLRQQXPEHULVODUJH

13"$5*$& 130#-&.4 2 3 2 3 4 3 4 5 LVLQYHUWLEOH 'HWHUPLQHLI A D 4 2 2 3 4 6XSSRVHWKDWIRUDFHUWDLQ n ! n PDWUL[ A VWDWHPHQW J RIWKH,QYHUWLEOH0DWUL[ 7KHRUHPLV QRW WUXH :KDWFDQ\RXVD\DERXWHTXDWLRQVRIWKHIRUP A[ D E" 6XSSRVHWKDW A DQG B DUH n ! n PDWULFHVDQGWKHHTXDWLRQ AB [ D KDVDQRQWULYLDO VROXWLRQ :KDWFDQ\RXVD\DERXWWKHPDWUL[ AB "


&9&3$*4&4 8QOHVV RWKHUZLVH VSHFLÀHG DVVXPH WKDW DOO PDWULFHV LQ WKHVH H[HUFLVHVDUH n ! n 'HWHUPLQHZKLFKRIWKHPDWULFHVLQ([HU FLVHV ²DUHLQYHUWLEOH 8VHDVIHZFDOFXODWLRQVDVSRVVLEOH -XVWLI\\RXUDQVZHUV # $ # $ 5 7 #4 2 #3 #6 6 #3 2 3 2 3 3 0 0 #5 1 4 05 0 05 4 #3 #4 4 0 8 5 #3 1 4 9 2 3 2 3 3 0 #3 1 #3 #6 0 45 4 35 4 2 4 0 #4 0 7 #3 6 0 2 3 2 3 #1 #3 0 1 3 4 7 4 6 3 60 5 8 #3 7 1 4 67 7 7 6 6 4 #2 #6 40 3 25 0 2 85 0 #1 2 1 0 0 0 1 2 3 4 0 #3 #7 6 #6 9 9 97 6 7 >0@ 4 7 #5 10 19 5 #1 2 4 #1 2 3 5 3 1 7 9 66 4 2 8 #8 7 6 7 7 5 3 10 97 >0@ 6 6 7 49 6 4 #9 #5 5 8 5 2 11 4

,Q([HUFLVHVDQG WKHPDWULFHVDUHDOO n ! n (DFKSDUW RIWKHH[HUFLVHVLVDQ LPSOLFDWLRQ RIWKHIRUP´,I h VWDWHPHQW i WKHQ h VWDWHPHQW iµ 0DUNDQLPSOLFDWLRQDV7UXHLIWKHWUXWKRI h VWDWHPHQW i DOZD\V IROORZVZKHQHYHU h VWDWHPHQW i KDSSHQV WREHWUXH $QLPSOLFDWLRQLV)DOVHLIWKHUHLVDQLQVWDQFHLQZKLFK h VWDWHPHQW i LVIDOVHEXW h VWDWHPHQW i LVWUXH -XVWLI\HDFK DQVZHU D ,IWKHHTXDWLRQ A[ D KDVRQO\WKHWULYLDOVROXWLRQ WKHQ A LVURZHTXLYDOHQWWRWKH n ! n LGHQWLW\PDWUL[

E ,IWKHFROXPQVRI A VSDQ Rn WKHQWKHFROXPQVDUHOLQ HDUO\LQGHSHQGHQW F ,I A LVDQ n ! n PDWUL[ WKHQWKHHTXDWLRQ A[ D E KDVDW OHDVWRQHVROXWLRQIRUHDFK E LQ Rn

G ,IWKHHTXDWLRQ A[ D KDVDQRQWULYLDOVROXWLRQ WKHQ A KDVIHZHUWKDQ n SLYRWSRVLWLRQV H ,I AT LVQRWLQYHUWLEOH WKHQ A LVQRWLQYHUWLEOH D ,IWKHUHLVDQ n ! n PDWUL[ D VXFKWKDW AD D I WKHQ DA D I

E ,IWKHOLQHDUWUDQVIRUPDWLRQ [ 7! A[ PDSV Rn LQWR Rn WKHQWKHURZUHGXFHGHFKHORQIRUPRI A LV I

F ,IWKHFROXPQVRI A DUHOLQHDUO\LQGHSHQGHQW WKHQWKH FROXPQVRI A VSDQ Rn

G ,IWKHHTXDWLRQ A[ D E KDVDWOHDVWRQHVROXWLRQIRUHDFK E LQ Rn WKHQWKHWUDQVIRUPDWLRQ [ 7! A[ LVQRWRQHWRRQH

H ,IWKHUHLVD E LQ Rn VXFKWKDWWKHHTXDWLRQ A[ D E LV FRQVLVWHQW WKHQWKHVROXWLRQLVXQLTXH

$Q m ! n XSSHUWULDQJXODUPDWUL[ LVRQHZKRVHHQWULHV EHORZ WKHPDLQGLDJRQDODUH·VDVLQ([HUFLVH :KHQ LVDVTXDUHXSSHUWULDQJXODUPDWUL[LQYHUWLEOH" -XVWLI\\RXU DQVZHU $Q m ! n ORZHUWULDQJXODUPDWUL[ LVRQHZKRVHHQWULHV DERYH WKHPDLQGLDJRQDODUH·VDVLQ([HUFLVH :KHQ LVDVTXDUHORZHUWULDQJXODUPDWUL[LQYHUWLEOH" -XVWLI\\RXU DQVZHU ,VLWSRVVLEOHIRUD 4 ! 4 PDWUL[WREHLQYHUWLEOHZKHQLWV FROXPQVGRQRWVSDQ R4 " :K\RUZK\QRW" ,IDQ n ! n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n FDQWKHFROXPQVRIWKH n ! n PDWUL[ C VSDQ Rn " :K\RUZK\QRW"

,I n ! n PDWULFHV E DQG F KDYHWKHSURSHUW\WKDW EF D I; WKHQ E DQG F FRPPXWH ([SODLQZK\ $VVXPHWKDW F LVDQ n ! n PDWUL[ ,IWKHHTXDWLRQ F [ D \ LVLQFRQVLVWHQWIRUVRPH \ LQ Rn ; ZKDWFDQ\RXVD\DERXWWKH HTXDWLRQ F [ D ‹ :K\" ,IDQ n ! n PDWUL[ G FDQQRWEHURZUHGXFHGWR In ; ZKDWFDQ \RXVD\DERXWWKHFROXPQVRI G " :K\" 9HULI\WKHER[HGVWDWHPHQWSUHFHGLQJ([DPSOH ([SODLQ ZK\ WKH FROXPQV RI A2 VSDQ Rn ZKHQHYHU WKH FROXPQVRIDQ n ! n PDWUL[ A DUHOLQHDUO\LQGHSHQGHQW

/HW A DQG B EH n ! n PDWULFHV 6KRZWKDWLI AB LVLQYHUWLEOH VRLV A +LQW 7KHUHLVDPDWUL[ W VXFKWKDW ABW D I: :K\"@ /HW A DQG B EH n ! n PDWULFHV 6KRZWKDWLI AB LVLQYHUWLEOH VRLV B

,I A LVDQ n ! n PDWUL[DQGWKHWUDQVIRUPDWLRQ [ 7! A[ LV RQHWRRQH ZKDWHOVHFDQ\RXVD\DERXWWKLVWUDQVIRUPDWLRQ" -XVWLI\\RXUDQVZHU

$)"15&3

0DWUL[$OJHEUD

,I A LV DQ n ! n PDWUL[ DQG WKH HTXDWLRQ A[ D E KDV PRUHWKDQRQHVROXWLRQIRUVRPH E; WKHQWKHWUDQVIRUPDWLRQ [ 7! A[ LVQRWRQHWRRQH :KDWHOVHFDQ\RXVD\DERXWWKLV WUDQVIRUPDWLRQ" -XVWLI\\RXUDQVZHU 6XSSRVH A LVDQ n ! n PDWUL[ZLWKWKHSURSHUW\WKDWWKH HTXDWLRQ A[ D E KDVDWOHDVWRQHVROXWLRQIRUHDFK E LQ Rn :LWKRXWXVLQJ7KHRUHPV RU H[SODLQZK\HDFKHTXDWLRQ A[ D E KDVLQIDFWH[DFWO\RQHVROXWLRQ 6XSSRVH A LVDQ n ! n PDWUL[ZLWKWKHSURSHUW\WKDWWKHHTXD WLRQ A[ D KDVRQO\WKHWULYLDOVROXWLRQ :LWKRXWXVLQJWKH ,QYHUWLEOH0DWUL[7KHRUHP H[SODLQGLUHFWO\ZK\WKHHTXDWLRQ A[ D E PXVWKDYHDVROXWLRQIRUHDFK E LQ Rn ,Q([HUFLVHVDQG T LVDOLQHDUWUDQVIRUPDWLRQIURP R2 LQWR R2 6KRZWKDW T LVLQYHUWLEOHDQGÀQGDIRUPXODIRU T !1 T .x1 ; x2 / D .#5x1 C 9x2 ; 4x1 # 7x2 / T .x1 ; x2 / D .2x1 # 8x2 ; #2x1 C 7x2 / /HW T W Rn ! Rn EHDQLQYHUWLEOHOLQHDUWUDQVIRUPDWLRQ ([ SODLQZK\ T LVERWKRQHWRRQHDQGRQWR Rn : 8VHHTXDWLRQV DQG 7KHQJLYHDVHFRQGH[SODQDWLRQXVLQJRQHRU PRUHWKHRUHPV 6XSSRVHDOLQHDUWUDQVIRUPDWLRQ T W Rn ! Rn KDVWKHSURS HUW\WKDW T .X/ D T .Y/ IRUVRPHSDLURIGLVWLQFWYHFWRUV X DQG Y LQ Rn : &DQ T PDS Rn RQWR Rn " :K\RUZK\QRW" 6XSSRVH T DQG U DUHOLQHDUWUDQVIRUPDWLRQVIURP Rn WR Rn VXFKWKDW T .U.[// D [ IRUDOO [ LQ Rn ,VLWWUXHWKDW U.T .[// D [ IRUDOO [ LQ Rn " :K\RUZK\QRW" /HW T W Rn ! Rn EH DQ LQYHUWLEOH OLQHDU WUDQVIRUPDWLRQ DQGOHW S DQG U EHIXQFWLRQVIURP Rn LQWR Rn VXFKWKDW S.T .[// D [ DQG U.T .[// D [ IRUDOO [ LQ Rn 6KRZWKDW U.Y/ D S.Y/ IRUDOO Y LQ Rn 7KLVZLOOVKRZWKDW T KDVD XQLTXHLQYHUVH DVDVVHUWHGLQ7KHRUHP >+LQW *LYHQDQ\ Y LQ Rn ZHFDQZULWH Y D T .[/ IRUVRPH [ :K\" &RPSXWH S.Y/ DQG U.Y/@ /HW T EHDOLQHDUWUDQVIRUPDWLRQWKDWPDSV Rn RQWR Rn 6KRZ WKDW T !1 H[LVWVDQGPDSV Rn RQWR Rn : ,V T !1 DOVRRQHWR RQH" 6XSSRVH T DQG S VDWLVI\WKHLQYHUWLELOLW\HTXDWLRQV DQG ZKHUH T LVDOLQHDUWUDQVIRUPDWLRQ 6KRZGLUHFWO\WKDW S LVDOLQHDUWUDQVIRUPDWLRQ >+LQW *LYHQ X; Y LQ Rn OHW [ D S.X/ \ D S.Y/ 7KHQ T .[/ D X T .\/ D Y :K\" $S SO\ S WRERWKVLGHVRIWKHHTXDWLRQ T .[/ C T .\/ D T .[ C \/ $OVR FRQVLGHU T .c [/ D cT .[/@

>0@ 6XSSRVHDQH[SHULPHQWOHDGVWRWKHIROORZLQJV\VWHPRI HTXDWLRQV

4:5x1 C 3:1x2 D 19:249 1:6x1 C 1:1x2 D 6:843

D 6ROYHV\VWHP DQGWKHQVROYHV\VWHP EHORZ LQ ZKLFKWKHGDWDRQWKHULJKWKDYHEHHQURXQGHGWRWZR GHFLPDOSODFHV ,QHDFKFDVH ÀQGWKH H[DFW VROXWLRQ

4:5x1 C 3:1x2 D 19:25 1:6x1 C 1:1x2 D 6:84

E 7KHHQWULHVLQV\VWHP GLIIHUIURPWKRVHLQV\VWHP E\OHVVWKDQ )LQGWKHSHUFHQWDJHHUURUZKHQXVLQJ WKHVROXWLRQRI DVDQDSSUR[LPDWLRQIRUWKHVROXWLRQRI F 8VHDPDWUL[SURJUDPWRSURGXFHWKHFRQGLWLRQQXPEHURI WKHFRHIÀFLHQWPDWUL[LQ ([HUFLVHV²VKRZKRZWRXVHWKHFRQGLWLRQQXPEHURID PDWUL[ A WR HVWLPDWH WKH DFFXUDF\ RI D FRPSXWHG VROXWLRQ RI A[ D E ,IWKHHQWULHVRI A DQG E DUHDFFXUDWHWRDERXW r VLJQLÀFDQW GLJLWVDQGLIWKHFRQGLWLRQQXPEHURI A LVDSSUR[LPDWHO\ 10 k ZLWK k DSRVLWLYHLQWHJHU WKHQWKHFRPSXWHGVROXWLRQRI A[ D E VKRXOG XVXDOO\EHDFFXUDWHWRDWOHDVW r # k VLJQLÀFDQWGLJLWV >0@ /HW A EHWKHPDWUL[LQ([HUFLVH )LQGWKHFRQGLWLRQ QXPEHURI A &RQVWUXFWDUDQGRPYHFWRU [ LQ R4 DQGFRPSXWH E D A[ 7KHQXVHDPDWUL[SURJUDPWRFRPSXWHWKHVROXWLRQ [1 RI A[ D E 7RKRZPDQ\GLJLWVGR [ DQG [1 DJUHH" )LQG RXWWKHQXPEHURIGLJLWVWKHPDWUL[SURJUDPVWRUHVDFFXUDWHO\ DQGUHSRUWKRZPDQ\GLJLWVRIDFFXUDF\DUHORVWZKHQ [1 LV XVHGLQSODFHRIWKHH[DFWVROXWLRQ [ >0@ 5HSHDW([HUFLVHIRUWKHPDWUL[LQ([HUFLVH >0@ 6ROYHDQHTXDWLRQ A[ D E IRUDVXLWDEOH E WRÀQGWKHODVW FROXPQRIWKHLQYHUVHRIWKH ÀIWKRUGHU+LOEHUWPDWUL[ 2 3 1 1=2 1=3 1=4 1=5 6 1=2 1=3 1=4 1=5 1=6 7 6 7 1=3 1=4 1=5 1=6 1=7 7 AD6 6 7 4 1=4 1=5 1=6 1=7 1=8 5 1=5 1=6 1=7 1=8 1=9 +RZPDQ\GLJLWVLQHDFKHQWU\RI [ GR\RXH[SHFWWREH FRUUHFW" ([SODLQ >1RWH 7KHH[DFWVROXWLRQLV #12600 #88200 @

>0@ 6RPHPDWUL[SURJUDPV VXFKDV0$7/$%KDYHDFRP PDQGWRFUHDWH+LOEHUWPDWULFHVRIYDULRXVVL]HV ,ISRVVLEOH XVHDQLQYHUVHFRPPDQGWRFRPSXWHWKHLQYHUVHRIDWZHOIWK RUGHURUODUJHU+LOEHUWPDWUL[ A &RPSXWH AA!1 5HSRUW ZKDW\RXÀQG SG

.BTUFSJOH 3FWJFXJOH BOE 3FáFDUJOH m

3DUWLWLRQHG0DWULFHV

40-65*0/4 50 13"$5*$& 130#-&.4 7KHFROXPQVRI A DUHREYLRXVO\OLQHDUO\GHSHQGHQWEHFDXVHFROXPQVDQGDUH PXOWLSOHVRIFROXPQ +HQFH A FDQQRWEHLQYHUWLEOH E\WKH,QYHUWLEOH0DWUL[ 7KHRUHP ,IVWDWHPHQWJ LV QRW WUXH WKHQWKHHTXDWLRQ A[ D E LVLQFRQVLVWHQWIRUDWOHDVWRQH E LQ Rn $SSO\WKH,QYHUWLEOH0DWUL[7KHRUHPWRWKHPDWUL[ AB LQSODFHRI A 7KHQVWDWHPHQW G EHFRPHV AB [ D KDVRQO\WKHWULYLDOVROXWLRQ 7KLVLVQRWWUXH 6R AB LVQRW LQYHUWLEOH

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

&9".1-& 7KHPDWUL[ 2

3 A D 4 #5 #8

0 2 #6

#1 4 3

5 0 1

9 #3 7

3 #2 15 #4

FDQDOVREHZULWWHQDVWKH 2 ! 3 SDUWLWLRQHG RU EORFN PDWUL[ # $ A11 A12 A13 AD A21 A22 A23 ZKRVHHQWULHVDUHWKH EORFNV RU VXEPDWULFHV # $ # $ 3 0 #1 5 9 A11 D ; A12 D ; #5 2 4 0 #3 ! " ! " A21 D #8 #6 3 ; A22 D 1 7 ;

A13 D

#

#2 1

$

! " A23 D #4

&9".1-& :KHQDPDWUL[ A DSSHDUVLQDPDWKHPDWLFDOPRGHORIDSK\VLFDO V\VWHPVXFKDVDQHOHFWULFDOQHWZRUN DWUDQVSRUWDWLRQV\VWHP RUDODUJHFRUSRUDWLRQ LWPD\EHQDWXUDOWRUHJDUG A DVDSDUWLWLRQHGPDWUL[ )RULQVWDQFH LIDPLFURFRPSXWHU FLUFXLWERDUGFRQVLVWVPDLQO\RIWKUHH9/6, YHU\ODUJHVFDOHLQWHJUDWHG PLFURFKLSV WKHQWKHPDWUL[IRUWKHFLUFXLWERDUGPLJKWKDYHWKHJHQHUDOIRUP 2 3 A11 A12 A13 6 7 A D 4 A21 A22 A23 5 A31 A32 A33 7KHVXEPDWULFHVRQWKH´GLDJRQDOµRI A³QDPHO\ A11 A22 DQG A33 ³FRQFHUQWKHWKUHH 9/6, FKLSV ZKLOHWKHRWKHUVXEPDWULFHVGHSHQGRQWKHLQWHUFRQQHFWLRQVDPRQJWKRVH PLFURFKLSV

$)"15&3

0DWUL[$OJHEUD

$GGLWLRQDQG6FDODU0XOWLSOLFDWLRQ ,IPDWULFHV A DQG B DUHWKHVDPHVL]HDQGDUHSDUWLWLRQHGLQH[DFWO\WKHVDPHZD\ WKHQLWLVQDWXUDOWRPDNHWKHVDPHSDUWLWLRQRIWKHRUGLQDU\PDWUL[VXP A C B ,QWKLV FDVH HDFKEORFNRI A C B LVWKHPDWUL[ VXPRIWKHFRUUHVSRQGLQJEORFNVRI A DQG B 0XOWLSOLFDWLRQRIDSDUWLWLRQHGPDWUL[E\DVFDODULVDOVRFRPSXWHGEORFNE\EORFN

0XOWLSOLFDWLRQRI3DUWLWLRQHG0DWULFHV 3DUWLWLRQHGPDWULFHVFDQEHPXOWLSOLHGE\WKHXVXDOURZ²FROXPQUXOHDVLIWKHEORFN HQWULHVZHUHVFDODUV SURYLGHGWKDWIRUDSURGXFW AB WKHFROXPQSDUWLWLRQRI A PDWFKHV WKHURZSDUWLWLRQRI B

&9".1-& /HW 2

2 #3 1 4 A D 1 5 #2 0 #4 #2

3

# $ 0 #4 3 #1 5 D A11 A12 ; A21 A22 7 #1

2

6 6 #2 6 #3 BD6 6 4 #1 5

3 4 17 # $ 7 7 7 D B1 7 B2 35

2

7KHFROXPQVRI A DUHSDUWLWLRQHGLQWRDVHWRIFROXPQVDQGWKHQDVHWRI FROXPQV 7KHURZVRI B DUHSDUWLWLRQHGLQWKHVDPHZD\³LQWRDVHWRIURZVDQG WKHQDVHWRIURZV :HVD\WKDWWKHSDUWLWLRQVRI A DQG B DUH FRQIRUPDEOH IRU EORFN PXOWLSOLFDWLRQ ,WFDQEHVKRZQWKDWWKHRUGLQDU\SURGXFW AB FDQEHZULWWHQDV 2 3 # $# $ # $ #5 4 A11 A12 B1 A11 B1 C A12 B2 AB D D D 4 #6 2 5 A21 A22 B2 A21 B1 C A22 B2 2 1

,WLVLPSRUWDQWIRUHDFKVPDOOHUSURGXFWLQWKHH[SUHVVLRQIRU AB WREHZULWWHQZLWK WKHVXEPDWUL[IURP A RQWKHOHIW VLQFHPDWUL[PXOWLSOLFDWLRQLVQRWFRPPXWDWLYH )RU LQVWDQFH 3 2 # $ # $ 6 4 2 #3 1 4 15 12 #2 1 5 D A11 B1 D 1 5 #2 2 #5 #3 7 # $# $ # $ 0 #4 #1 3 #20 #8 A12 B2 D D 3 #1 5 2 #8 7 +HQFHWKHWRSEORFNLQ AB LV

A11 B1 C A12 B2 D

#

15 12 2 #5

$

#

#20 #8 C #8 7

$

D

#

#5 #6

4 2

$

7KHURZ²FROXPQUXOHIRUPXOWLSOLFDWLRQRIEORFNPDWULFHVSURYLGHVWKHPRVWJHQHUDO ZD\WRUHJDUGWKHSURGXFWRIWZRPDWULFHV (DFKRIWKHIROORZLQJYLHZVRIDSURGXFW KDVDOUHDG\EHHQGHVFULEHGXVLQJVLPSOHSDUWLWLRQVRIPDWULFHV WKHGHÀQLWLRQRI A[ XVLQJWKHFROXPQVRI A WKHFROXPQGHÀQLWLRQRI AB WKHURZ²FROXPQUXOHIRU FRPSXWLQJ AB DQG WKHURZVRI AB DVSURGXFWVRIWKHURZVRI A DQGWKHPDWUL[ B $ ÀIWKYLHZRI AB DJDLQXVLQJSDUWLWLRQV IROORZVLQ7KHRUHP EHORZ 7KHFDOFXODWLRQVLQWKHQH[WH[DPSOHSUHSDUHWKHZD\IRU7KHRUHP +HUH FROk .A/ LVWKH k WKFROXPQRI A DQG URZk .B/ LVWKH k WKURZRI B

3DUWLWLRQHG0DWULFHV

&9".1-& /HW A D

#

2 $ a 2 DQG B D 4 c 5 e

#3 1 1 #4

3 b d 5 9HULI\WKDW f

AB D FRO1 .A/ URZ1 .B/ C FRO2 .A/ URZ2 .B/ C FRO3 .A/ URZ3 .B/

40-65*0/ (DFKWHUPDERYHLVDQ RXWHUSURGXFW 6HH([HUFLVHV DQGLQ6HF WLRQ %\WKHURZ²FROXPQUXOHIRUFRPSXWLQJDPDWUL[SURGXFW # $ # $ " #3 ! #3a #3b a b D FRO1 .A/ URZ1 .B/ D 1 a b # $ # $ " 1 ! c d c d D FRO2 .A/ URZ2 .B/ D #4 #4c #4d # $ # $ " 2 ! 2e 2f e f D FRO3 .A/ URZ3 .B/ D 5 5e 5f 7KXV

3 X k D1

FROk .A/ URZk .B/ D

#

#3a C c C 2e a # 4c C 5e

#3b C d C 2f b # 4d C 5f

$

7KLVPDWUL[LVREYLRXVO\ AB 1RWLFHWKDWWKH .1; 1/HQWU\LQ AB LVWKHVXPRIWKH .1; 1/ HQWULHVLQWKHWKUHHRXWHUSURGXFWV WKH .1; 2/HQWU\LQ AB LVWKHVXPRIWKH .1; 2/HQWULHV LQWKHWKUHHRXWHUSURGXFWV DQGVRRQ

5)&03&.

$PMVNO3PX &YQBOTJPO PG "# ,I A LV m ! n DQG B LV n ! p WKHQ

2

3 URZ1 .B/ 6 URZ2 .B/ 7 6 7 AB D Œ FRO1 .A/ FRO2 .A/ " " " FROn .A/ ! 6 7 :: 4 5 :

URZn .B/

D FRO1 .A/ URZ1 .B/ C " " " C FROn .A/ URZn .B/

1300' )RUHDFKURZLQGH[ i DQGFROXPQLQGH[ j WKH .i; j /HQWU\LQ FROk .A/ URZk .B/ LVWKHSURGXFWRI aik IURP FROk .A/ DQG bkj IURP URZk .B/ +HQFHWKH .i; j /HQWU\LQWKH VXPVKRZQLQHTXDWLRQ LV ai1 b1j .k D 1/

C

ai2 b2j .k D 2/

C """ C

ai n bnj .k D n/

7KLVVXPLVDOVRWKH .i; j /HQWU\LQ AB E\WKHURZ²FROXPQUXOH

,QYHUVHVRI3DUWLWLRQHG0DWULFHV 7KHQH[WH[DPSOHLOOXVWUDWHVFDOFXODWLRQVLQYROYLQJLQYHUVHVDQGSDUWLWLRQHGPDWULFHV

&9".1-& $ PDWUL[RIWKHIRUP

#

A11 A12 AD 0 A22

$

LVVDLGWREH EORFNXSSHUWULDQJXODU $VVXPHWKDW A11 LV p ! p A22 LV q ! q DQG A LV LQYHUWLEOH )LQGDIRUPXODIRU A"1

$)"15&3

0DWUL[$OJHEUD

40-65*0/ 'HQRWH A"1 E\ B DQGSDUWLWLRQ B VRWKDW # $# $ # $ A11 A12 B11 B12 Ip 0 D 0 A22 B21 B22 0 Iq

7KLVPDWUL[HTXDWLRQSURYLGHVIRXUHTXDWLRQVWKDWZLOOOHDGWRWKHXQNQRZQEORFNV B11 ; : : : ; B22 &RPSXWHWKHSURGXFWRQWKHOHIWVLGHRIHTXDWLRQ DQGHTXDWHHDFKHQWU\ ZLWKWKHFRUUHVSRQGLQJEORFNLQWKHLGHQWLW\PDWUL[RQWKHULJKW 7KDWLV VHW

A11 B11 C A12 B21 A11 B12 C A12 B22 A22 B21 A22 B22

D Ip D0 D0 D Iq

%\LWVHOI HTXDWLRQ GRHVQRWVKRZWKDW A22 LVLQYHUWLEOH +RZHYHU VLQFH A22 LV VTXDUH WKH,QYHUWLEOH0DWUL[7KHRUHPDQG WRJHWKHUVKRZWKDW A22 LVLQYHUWLEOHDQG "1 "1 B22 D A22 1H[W OHIWPXOWLSO\ERWKVLGHVRI E\ A22 DQGREWDLQ "1 B21 D A22 0D0

VRWKDW VLPSOLÀHVWR

A11 B11 C 0 D Ip

"1 6LQFH A11 LVVTXDUH WKLVVKRZVWKDW A11 LVLQYHUWLEOHDQG B11 D A11 )LQDOO\ XVHWKHVH UHVXOWVZLWK WRÀQGWKDW

7KXV

"1 A11 B12 D #A12 B22 D #A12 A22 "1

A

D

"

A11 A12 0 A22

#"1

D

"

DQG

"1 "1 B12 D #A11 A12 A22

"1 1 "1 A11 #A" 11 A12 A22 "1 0 A22

#

$ EORFNGLDJRQDOPDWUL[ LVDSDUWLWLRQHGPDWUL[ZLWK]HUREORFNVRIIWKHPDLQ GLDJRQDORIEORFNV 6XFKDPDWUL[LVLQYHUWLEOHLIDQGRQO\LIHDFKEORFNRQWKHGLDJRQDO LVLQYHUWLEOH 6HH([HUFLVHV DQG

/6.&3*$"- /05&4 :KHQ PDWULFHV DUH WRR ODUJH WR ÀW LQ D FRPSXWHU·V KLJKVSHHG PHPRU\ SDUWLWLRQLQJSHUPLWVWKHFRPSXWHUWRZRUNZLWKRQO\WZRRUWKUHHVXEPDWULFHV DWDWLPH )RULQVWDQFH RQHOLQHDUSURJUDPPLQJUHVHDUFKWHDPVLPSOLÀHG DSUREOHPE\SDUWLWLRQLQJWKHPDWUL[LQWRURZVDQGFROXPQV 7KH SUREOHP·VVROXWLRQWRRNDERXWPLQXWHVRQD&UD\VXSHUFRPSXWHU 6RPHKLJKVSHHGFRPSXWHUV SDUWLFXODUO\WKRVHZLWKYHFWRUSLSHOLQHDUFKLWHF WXUH SHUIRUPPDWUL[FDOFXODWLRQVPRUHHIÀFLHQWO\ZKHQWKHDOJRULWKPVXVH SDUWLWLRQHGPDWULFHV 3URIHVVLRQDOVRIWZDUHIRUKLJKSHUIRUPDQFHQXPHULFDOOLQHDUDOJHEUD VXFKDV /$3$&.PDNHVLQWHQVLYHXVHRISDUWLWLRQHGPDWUL[FDOFXODWLRQV

ñ 7KHVROXWLRQWLPHGRHVQ·WVRXQGWRRLPSUHVVLYHXQWLO\RXOHDUQWKDWHDFKRIWKHEORFNFROXPQVFRQWDLQHG DERXWLQGLYLGXDOFROXPQV 7KHRULJLQDOSUREOHPKDGHTXDWLRQVDQGRYHUYDULDEOHV 1HDUO\PLOOLRQRIWKHPRUHWKDQELOOLRQHQWULHVLQWKHPDWUL[ZHUHQRQ]HUR 6HH5REHUW(%L[E\HW DO ´9HU\/DUJH6FDOH/LQHDU3URJUDPPLQJ $ &DVH6WXG\LQ&RPELQLQJ,QWHULRU3RLQWDQG6LPSOH[ 0HWKRGVµ 2SHUDWLRQV5HVHDUFK QR ² ò 7KHLPSRUWDQFHRIEORFNPDWUL[DOJRULWKPVIRUFRPSXWHUFDOFXODWLRQVLVGHVFULEHGLQ 0DWUL[&RPSXWDWLRQV UGHG E\*HQH+*ROXEDQG&KDUOHV)YDQ/RDQ%DOWLPRUH -RKQV+RSNLQV8QLYHUVLW\3UHVV

3DUWLWLRQHG0DWULFHV

7KHH[HUFLVHVWKDWIROORZJLYHSUDFWLFHZLWKPDWUL[DOJHEUDDQGLOOXVWUDWHW\SLFDO FDOFXODWLRQVIRXQGLQDSSOLFDWLRQV

13"$5*$& 130#-&.4 # $ I 0 6KRZWKDW LVLQYHUWLEOHDQGÀQGLWVLQYHUVH A I ! " &RPSXWH X TX ZKHUH X LVSDUWLWLRQHGDV X1 X2

&9&3$*4&4 ,Q([HUFLVHV² DVVXPHWKDWWKHPDWULFHVDUHSDUWLWLRQHGFRQ IRUPDEO\IRUEORFNPXOWLSOLFDWLRQ &RPSXWHWKHSURGXFWVVKRZQ LQ([HUFLVHV ² # $# $ # $# $ I 0 A B E 0 P Q E I C D 0 F R S # $# $ # $# $ 0 I A B I 0 W X I 0 C D #E I Y Z ,Q([HUFLVHV² À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

I 4 A21 A31

32 0 0 B11 I 0 54 B21 0 I B31

3 2 B12 C11 B22 5 D 4 0 B32 0

7KHLQYHUVHRI 2

I 0 4A I B D

3 0 05 I

LV

)LQG P Q DQG R

2

I 4P Q

0 I R

3 0 0 5: I

3 C12 C22 5 C32

,Q([HUFLVHVDQG PDUNHDFKVWDWHPHQW7UXHRU)DOVH -XVWLI\ HDFKDQVZHU D ,I A D Œ A1 A2 ! DQG B D Œ B1 B2 ! ZLWK A1 DQG A2 WKH VDPH VL]HV DV B1 DQG B2 UHVSHFWLYHO\ WKHQ A C B D Œ A1 C B1 A2 C B2 ! # $ # $ A11 A12 B1 DQG B D WKHQWKHSDUWLWLRQV E ,I A D A21 A22 B2 RI A DQG B DUHFRQIRUPDEOHIRUEORFN PXOWLSOLFDWLRQ # $ A1 DQG D ,I A1 A2 B1 DQG B2 DUH n ! n PDWULFHV A D A2 B D Œ B1 B2 ! WKHQWKHSURGXFW BA LVGHÀ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k JLYLQJLQIRUPDWLRQDWGLIIHUHQWWLPHVDERXWKRZ WKHSUREH·VSRVLWLRQFRPSDUHVZLWKLWVSODQQHGWUDMHFWRU\ /HW Xk EHWKHPDWUL[ Œ[1 " " " [k ! 7KHPDWUL[ Gk D Xk XkT LV FRPSXWHGDVWKHUDGDUGDWDDUHDQDO\]HG :KHQ [kC1 DUULYHV DQHZ GkC1 PXVWEHFRPSXWHG 6LQFHWKHGDWDYHFWRUVDUULYH DWKLJKVSHHG WKHFRPSXWDWLRQDOEXUGHQFRXOGEHVHYHUH %XW SDUWLWLRQHG PDWUL[ PXOWLSOLFDWLRQ KHOSV WUHPHQGRXVO\ &RPSXWHWKHFROXPQ²URZH[SDQVLRQVRI Gk DQG GkC1 DQG GHVFULEHZKDWPXVWEHFRPSXWHGLQRUGHUWR XSGDWH Gk WR IRUP GkC1

$)"15&3

0DWUL[$OJHEUD

$VVXPH A # sIn LVLQYHUWLEOHDQGYLHZ DVDV\VWHPRIWZR PDWUL[HTXDWLRQV 6ROYHWKHWRSHTXDWLRQIRU [ DQGVXEVWLWXWH LQWRWKHERWWRPHTXDWLRQ 7KHUHVXOWLVDQHTXDWLRQRIWKH IRUP W .s/X D \ ZKHUH W .s/ LVDPDWUL[WKDWGHSHQGVRQ s W .s/ LVFDOOHGWKH WUDQVIHUIXQFWLRQ RIWKHV\VWHPEHFDXVH LWWUDQVIRUPVWKHLQSXW X LQWRWKHRXWSXW \ )LQG W .s/ DQG GHVFULEHKRZLWLVUHODWHGWRWKHSDUWLWLRQHG V\VWHPPDWUL[ RQ WKHOHIWVLGHRI 6HH([HUFLVH

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m ! n GDWDPDWUL[VXFKWKDW X T X LVLQYHUWLEOH DQGOHW M D Im # X.X T X/!1 X T $GGDFROXPQ [0 WRWKH GDWDDQGIRUP ! " [0 W D X

&RPSXWH W T W 7KH .1; 1/HQWU\LV X T X 6KRZWKDWWKH 6FKXUFRPSOHPHQW([HUFLVH RI X T X FDQEHZULWWHQ LQ WKH IRUP [T0 M [0 ,W FDQ EH VKRZQ WKDW WKH TXDQWLW\ .[T0 M [0 /!1 LVWKH .2; 2/HQWU\LQ .W T W /!1 7KLVHQWU\ KDV D XVHIXO VWDWLVWLFDO LQWHUSUHWDWLRQ XQGHU DSSURSULDWH K\SRWKHVHV

,QWKHVWXG\RIHQJLQHHULQJFRQWURORISK\VLFDOV\VWHPV DVWDQGDUG VHWRIGLIIHUHQWLDOHTXDWLRQVLVWUDQVIRUPHGE\/DSODFHWUDQVIRUPV LQWRWKHIROORZLQJV\VWHPRIOLQHDUHTXDWLRQV # $# $ # $ A # sIn B [ D \ C Im X

ZKHUH A LV n ! n B LV n ! m C LV m ! n DQG s LVDYDULDEOH 7KHYHFWRU X LQ Rm LVWKH´LQSXWµWRWKHV\VWHP \ LQ Rm LVWKH ´RXWSXWµ DQG [ LQ Rn LVWKH´VWDWHµYHFWRU $FWXDOO\ WKHYHFWRUV [ X DQG \ DUHIXQFWLRQVRI s EXWWKLVGRHVQRWDIIHFWWKHDOJHEUDLF FDOFXODWLRQVLQ([HUFLVHV DQG

6XSSRVHWKHWUDQVIHUIXQFWLRQ W .s/ LQ([HUFLVHLVLQYHUW LEOHIRUVRPH s ,WFDQEHVKRZQWKDWWKHLQYHUVHWUDQVIHU IXQFWLRQ W .s/!1 ZKLFKWUDQVIRUPVRXWSXWVLQWRLQSXWV LVWKH 6FKXUFRPSOHPHQWRI A # BC # sIn IRUWKHPDWUL[EHORZ )LQGWKLV6FKXUFRPSOHPHQW 6HH([HUFLVH # $ A # BC # sIn B #C Im D 9HULI\WKDW A2 D I ZKHQ A D

#

$ 1 0 2 #1

E 8VHSDUWLWLRQHGPDWULFHVWRVKRZWKDW M 2 D I ZKHQ 2 3 1 0 0 0 6 2 #1 0 07 7 M D6 41 0 #1 05 0 1 #2 1

*HQHUDOL]HWKHLGHDRI([HUFLVHE\FRQVWUXFWLQJD 6!6 2 3 A 0 0 PDWUL[ M D 4 0 B 0 5 VXFKWKDW M 2 D I: 0DNH C D C 0 D QRQ]HUR 2 ! 2 PDWUL[ 6KRZWKDW\RXUFRQVWUXFWLRQZRUNV 8VHSDUWLWLRQHGPDWULFHVWRSURYHE\LQGXFWLRQWKDWWKHSURG XFWRIWZRORZHUWULDQJXODUPDWULFHVLVDOVRORZHUWULDQJXODU >+LQW $ .k C 1/ ! .k C 1/ PDWUL[ A1 FDQEHZULWWHQLQWKH IRUPEHORZ ZKHUH a LVDVFDODU Y LVLQ Rk DQG A LVD k ! k ORZHUWULDQJXODUPDWUL[ 6HHWKH 6WXG\*XLGH IRUKHOSZLWK LQGXFWLRQ@ # $ a T 5IF 1SJODJQMF PG A1 D SG *OEVDUJPO m Y A 8VH SDUWLWLRQHG PDWULFHV WR SURYH E\ LQGXFWLRQ WKDW IRU n D 2; 3; : : : ; WKH n ! n PDWUL[ A VKRZQEHORZLVLQYHUWLEOH DQG B LVLWVLQYHUVH 2 3 1 0 0 """ 0 61 1 0 07 6 7 6 1 1 1 0 7 AD6 : 7 : :: 4 :: 5 1 1 1 """ 1 2 3 1 0 0 """ 0 6 #1 1 0 07 6 7 6 0 #1 1 07 BD6 7 6 : 7 :: :: 4 :: 5 : : 0 ::: #1 1

)RU WKH LQGXFWLRQ VWHS DVVXPH A DQG B DUH .k C 1/ ! .k C 1/ PDWULFHV DQGSDUWLWLRQ A DQG B LQDIRUP VLPLODUWRWKDWGLVSOD\HGLQ([HUFLVH :LWKRXWXVLQJURZUHGXFWLRQ ÀQGWKHLQYHUVHRI 2 3 1 2 0 0 0 63 5 0 0 07 6 7 0 0 2 0 07 AD6 6 7 40 0 0 7 85 0 0 0 5 6

>0@ )RUEORFNRSHUDWLRQV LWPD\EHQHFHVVDU\WRDFFHVVRU HQWHUVXEPDWULFHVRIDODUJHPDWUL[ 'HVFULEHWKHIXQFWLRQV RUFRPPDQGVRIDPDWUL[SURJUDPWKDWDFFRPSOLVKWKHIRO ORZLQJWDVNV 6XSSRVH A LVD 20 ! 30 PDWUL[ D 'LVSOD\ WKH VXEPDWUL[ RI A IURP URZV WR DQG FROXPQVWR E ,QVHUWD 5 ! 10 PDWUL[ B LQWR A EHJLQQLQJDWURZDQG FROXPQ

0DWUL[)DFWRUL]DWLRQV

$ A 0 0 AT >1RWH ,WPD\QRWEHQHFHVVDU\WRVSHFLI\WKH]HUREORFNV LQ C @

F &UHDWH D 50 ! 50 PDWUL[RIWKHIRUP C D

#

>0@ 6XSSRVHPHPRU\RUVL]HUHVWULFWLRQVSUHYHQWDPDWUL[ SURJUDPIURPZRUNLQJZLWKPDWULFHVKDYLQJPRUHWKDQ URZVDQGFROXPQV DQGVXSSRVHVRPHSURMHFWLQYROYHV 50 ! 50 PDWULFHV A DQG B 'HVFULEHWKHFRPPDQGVRURS HUDWLRQVRIWKHPDWUL[SURJUDPWKDWDFFRPSOLVKWKHIROORZLQJ WDVNV D &RPSXWH A C B E &RPSXWH AB

F 6ROYH A[ D E IRUVRPHYHFWRU E LQ R50 DVVXPLQJWKDW A FDQ EH SDUWLWLRQHG LQWR D 2 ! 2 EORFN PDWUL[ ŒAij ! ZLWK A11 DQLQYHUWLEOH 20 ! 20 PDWUL[ A22 DQLQYHUWLEOH 30 ! 30 PDWUL[ DQG A12 D]HURPDWUL[ >+LQW 'HVFULEH DSSURSULDWHVPDOOHUV\VWHPVWRVROYH ZLWKRXWXVLQJDQ\ PDWUL[LQYHUVHV@

40-65*0/4 50 13"$5*$& 130#-&.4 # $ # $ I 0 W X ,I LVLQYHUWLEOH LWVLQYHUVHKDVWKHIRUP 9HULI\WKDW A I Y Z # $# $ # $ I 0 W X W X D A I Y Z AW C Y AX C Z

6R W X Y Z PXVWVDWLVI\ W D I X D 0 AW C Y D 0 DQG AX C Z D I ,W IROORZVWKDW Y D #A DQG Z D I +HQFH # $# $ # $ I 0 I 0 I 0 D A I #A I 0 I

7KHSURGXFWLQWKHUHYHUVHRUGHULVDOVRWKHLGHQWLW\ VRWKHEORFNPDWUL[LVLQYHUW # $ I 0 LEOH DQGLWVLQYHUVHLV
."53*9 '"$503*;"5*0/4 $ IDFWRUL]DWLRQ RIDPDWUL[ A LVDQHTXDWLRQWKDWH[SUHVVHV A DVDSURGXFWRIWZRRUPRUH PDWULFHV :KHUHDVPDWUL[PXOWLSOLFDWLRQLQYROYHVD V\QWKHVLV RIGDWDFRPELQLQJWKH HIIHFWVRIWZRRUPRUHOLQHDUWUDQVIRUPDWLRQVLQWRDVLQJOHPDWUL[ PDWUL[IDFWRUL]DWLRQ LVDQ DQDO\VLV RIGDWD ,QWKHODQJXDJHRIFRPSXWHUVFLHQFH WKHH[SUHVVLRQRI A DVD SURGXFWDPRXQWVWRD SUHSURFHVVLQJ RIWKHGDWDLQ A RUJDQL]LQJWKDWGDWDLQWRWZRRU PRUHSDUWVZKRVHVWUXFWXUHVDUHPRUHXVHIXOLQVRPHZD\ SHUKDSVPRUHDFFHVVLEOHIRU FRPSXWDWLRQ

$)"15&3

0DWUL[$OJHEUD

0DWUL[IDFWRUL]DWLRQVDQG ODWHU IDFWRUL]DWLRQVRIOLQHDUWUDQVIRUPDWLRQVZLOODSSHDU DWDQXPEHURINH\SRLQWVWKURXJKRXWWKHWH[W 7KLVVHFWLRQIRFXVHVRQDIDFWRUL]DWLRQ WKDWOLHVDWWKHKHDUWRIVHYHUDOLPSRUWDQWFRPSXWHUSURJUDPVZLGHO\XVHGLQDSSOLFD WLRQV VXFKDVWKHDLUÁRZSUREOHPGHVFULEHGLQWKHFKDSWHULQWURGXFWLRQ 6HYHUDORWKHU IDFWRUL]DWLRQV WREHVWXGLHGODWHU DUHLQWURGXFHGLQWKHH[HUFLVHV

7KH/8 )DFWRUL]DWLRQ 7KH/8 IDFWRUL]DWLRQ GHVFULEHGEHORZ LVPRWLYDWHGE\WKHIDLUO\FRPPRQLQGXVWULDO DQGEXVLQHVVSUREOHPRIVROYLQJDVHTXHQFHRIHTXDWLRQV DOOZLWKWKHVDPHFRHIÀFLHQW PDWUL[ A[ D E1 ; A[ D E2 ; : : : ; A[ D Ep 6HH([HUFLVH IRUH[DPSOH $OVRVHH6HFWLRQ ZKHUHWKHLQYHUVHSRZHUPHWKRG LVXVHGWRHVWLPDWHHLJHQYDOXHVRIDPDWUL[E\VROYLQJHTXDWLRQVOLNHWKRVHLQVHTXHQFH RQHDWDWLPH :KHQ A LVLQYHUWLEOH RQHFRXOGFRPSXWH A"1 DQGWKHQFRPSXWH A"1 E1 A"1 E2 DQGVRRQ +RZHYHU LWLVPRUHHIÀFLHQWWRVROYHWKHÀUVWHTXDWLRQLQVHTXHQFH E\ URZUHGXFWLRQDQGREWDLQDQ/8 IDFWRUL]DWLRQRI A DWWKHVDPHWLPH 7KHUHDIWHU WKH UHPDLQLQJHTXDWLRQVLQVHTXHQFH DUHVROYHGZLWKWKH/8 IDFWRUL]DWLRQ $WÀUVW DVVXPHWKDW A LVDQ m ! n PDWUL[WKDWFDQEHURZUHGXFHGWRHFKHORQ IRUP ZLWKRXWURZLQWHUFKDQJHV /DWHU ZHZLOOWUHDWWKHJHQHUDOFDVH 7KHQ A FDQ EHZULWWHQLQWKHIRUP A D LU ZKHUH L LVDQ m ! m ORZHUWULDQJXODUPDWUL[ZLWK·V RQWKHGLDJRQDODQG U LVDQ m ! n HFKHORQIRUPRI A )RULQVWDQFH VHH)LJ 6XFK DIDFWRUL]DWLRQLVFDOOHGDQ /8 IDFWRUL]DWLRQ RI A 7KHPDWUL[ L LVLQYHUWLEOHDQGLV FDOOHGD XQLW ORZHUWULDQJXODUPDWUL[ 1 * A= * *

0 1 * *

0 0 1 *

*

0 0 0 0 0 1 0

0 0

L

* * 0 0

* * 0

* * * 0

U

'*(63& $Q/8 IDFWRUL]DWLRQ

%HIRUHVWXG\LQJKRZWRFRQVWUXFW L DQG U ZHVKRXOGORRNDWZK\WKH\DUHVR XVHIXO :KHQ A D LU WKHHTXDWLRQ A[ D E FDQEHZULWWHQDV L.U [/ D E :ULWLQJ \ IRU U [ ZHFDQÀQG [ E\VROYLQJWKH SDLU RIHTXDWLRQV

L\ D E U[ D \

)LUVWVROYH L\ D E IRU \; DQGWKHQVROYH U [ D \ IRU [ 6HH)LJ (DFKHTXDWLRQLV HDV\WRVROYHEHFDXVH L DQG U DUHWULDQJXODU

&9".1-& ,WFDQEHYHULÀHGWKDW 2

3 2 3 #7 #2 2 1 0 6 #3 7 6 #1 5 1 0 1 7D6 AD6 4 6 #4 0 #5 5 4 2 #5 #9 5 #5 12 #3 8

0 0 1 3

32 3 0 3 #7 #2 2 6 07 27 76 0 #2 #1 7 D LU 4 5 0 0 0 #1 1 5 1 0 0 0 #1


Multiplication by A x

b

Multiplication by U

y

Multiplication by L

'*(63& )DFWRUL]DWLRQRIWKHPDSSLQJ [ 7! A[

2

3 #9 6 57 7 8VHWKLV/8 IDFWRUL]DWLRQRI A WRVROYH A[ D E ZKHUH E D 6 4 7 5 11

40-65*0/ 7KHVROXWLRQRI L\ D E QHHGVRQO\PXOWLSOLFDWLRQVDQGDGGLWLRQV EH FDXVHWKHDULWKPHWLFWDNHVSODFHRQO\LQFROXPQ 7KH]HURVEHORZHDFKSLYRWLQ L DUH FUHDWHGDXWRPDWLFDOO\E\WKHFKRLFHRIURZRSHUDWLRQV 2 3 2 3 1 0 0 0 #9 1 0 0 0 #9 6 #1 6 ! " ! " 1 0 0 57 1 0 0 #4 7 7 % 60 7D I L E D6 \ 4 2 #5 5 4 5 1 0 7 0 0 1 0 5 #3 8 3 1 11 0 0 0 1 1

7KHQ IRU U [ D \ WKHÉDFNZDUGµSKDVHRIURZUHGXFWLRQUHTXLUHV GLYLVLRQV PXO WLSOLFDWLRQV DQG DGGLWLRQV )RULQVWDQFH FUHDWLQJWKH]HURVLQFROXPQ RI Œ U \ ! UHTXLUHVGLYLVLRQLQURZ DQGPXOWLSOLFDWLRQ²DGGLWLRQSDLUVWRDGGPXOWLSOHVRIURZ WRWKHURZVDERYH 2 3 2 3 2 3 3 #7 #2 2 #9 1 0 0 0 3 3 6 0 #2 #1 6 6 7 ! " 2 #4 7 1 0 0 47 7 % 60 7; [ D 6 4 7 U \ D6 40 4 #6 5 0 #1 1 5 5 4 0 0 1 0 #6 5 0 0 0 #1 1 0 0 0 1 #1 #1

7RÀQG [ UHTXLUHVDULWKPHWLFRSHUDWLRQV RU´ÁRSVµÁRDWLQJSRLQWRSHUDWLRQV H[FOXGLQJWKHFRVWRIÀQGLQJ L DQG U ,QFRQWUDVW URZUHGXFWLRQRI Œ A E ! WR Œ I [ ! WDNHV RSHUDWLRQV

7KHFRPSXWDWLRQDOHIÀFLHQF\RIWKH/8 IDFWRUL]DWLRQGHSHQGVRQNQRZLQJ L DQG U 7KHQH[WDOJRULWKPVKRZVWKDWWKHURZUHGXFWLRQRI A WRDQHFKHORQIRUP U DPRXQWVWR DQ/8 IDFWRUL]DWLRQEHFDXVHLWSURGXFHV L ZLWKHVVHQWLDOO\QRH[WUDZRUN $IWHUWKHÀUVW URZUHGXFWLRQ L DQG U DUHDYDLODEOHIRUVROYLQJDGGLWLRQDOHTXDWLRQVZKRVHFRHIÀFLHQW PDWUL[LV A

$Q/8 )DFWRUL]DWLRQ$OJRULWKP 6XSSRVH A FDQEHUHGXFHGWRDQHFKHORQIRUP U XVLQJRQO\URZUHSODFHPHQWVWKDWDGGD PXOWLSOHRIRQHURZWRDQRWKHUURZ EHORZLW ,QWKLVFDVH WKHUHH[LVWXQLWORZHUWULDQJXODU HOHPHQWDU\PDWULFHV E1 ; : : : ; Ep VXFKWKDW

Ep " " " E1 A D U 7KHQ ZKHUH

A D .Ep " " " E1 /"1 U D LU L D .Ep " " " E1 /"1

$)"15&3

0DWUL[$OJHEUD

,WFDQEHVKRZQWKDWSURGXFWVDQGLQYHUVHVRIXQLWORZHUWULDQJXODUPDWULFHVDUHDOVRXQLW ORZHUWULDQJXODU )RULQVWDQFH VHH([HUFLVH 7KXV L LVXQLWORZHUWULDQJXODU 1RWHWKDWWKHURZRSHUDWLRQVLQHTXDWLRQ ZKLFKUHGXFH A WR U DOVRUHGXFH WKH L LQHTXDWLRQ WR I EHFDXVH Ep " " " E1 L D .Ep " " " E1 /.Ep " " " E1 /"1 D I 7KLV REVHUYDWLRQLVWKHNH\WR FRQVWUXFWLQJ L "-(03*5). '03 "/ -6 '"$503*;"5*0/ 5HGXFH A WRDQHFKHORQIRUP U E\DVHTXHQFHRIURZUHSODFHPHQWRSHUDWLRQV LISRVVLEOH 3ODFHHQWULHVLQ L VXFKWKDWWKH VDPHVHTXHQFHRIURZRSHUDWLRQV UHGXFHV L WR I

6WHSLVQRWDOZD\VSRVVLEOH EXWZKHQLWLV WKHDUJXPHQWDERYHVKRZVWKDWDQ/8 IDFWRUL]DWLRQH[LVWV ([DPSOHZLOOVKRZKRZWRLPSOHPHQWVWHS %\FRQVWUXFWLRQ L ZLOOVDWLVI\ .Ep " " " E1 /L D I XVLQJWKHVDPH E1 ; : : : ; Ep DVLQHTXDWLRQ 7KXV L ZLOOEHLQYHUWLEOH E\WKH,QYHUWLEOH 0DWUL[7KHRUHP ZLWK .Ep " " " E1 / D L"1 )URP L"1 A D U DQG A D LU 6R VWHS ZLOOSURGXFHDQDFFHSWDEOH L

&9".1-& )LQGDQ/8 IDFWRUL]DWLRQRI 2

3 2 4 #1 5 #2 6 #4 #5 3 #8 1 7 7 AD6 4 2 #5 #4 1 85 #6 0 7 #3 1

40-65*0/ 6LQFH A KDVIRXUURZV L VKRXOGEH 4 ! 4 7KHÀUVWFROXPQRI L LVWKHÀUVW FROXPQRI A GLYLGHGE\WKHWRSSLYRWHQWU\ 2 3 1 0 0 0 6 #2 1 0 07 7 LD6 4 1 1 05 #3 1

&RPSDUHWKHÀUVWFROXPQVRI A DQG L 7KHURZRSHUDWLRQVWKDWFUHDWH]HURVLQWKH ÀUVWFROXPQRI A ZLOODOVRFUHDWH]HURVLQWKHÀUVWFROXPQRI L 7RPDNHWKLVVDPH FRUUHVSRQGHQFHRIURZRSHUDWLRQVRQ A KROGIRUWKHUHVWRI L ZDWFKDURZUHGXFWLRQRI A WRDQHFKHORQIRUP U 7KDWLV KLJKOLJKWWKHHQWULHV LQHDFKPDWUL[WKDWDUHXVHGWR GHWHUPLQHWKHVHTXHQFHRIURZRSHUDWLRQVWKDWWUDQVIRUP A LQWR U >6HHWKHKLJKOLJKWHG HQWULHVLQHTXDWLRQ @ 2 3 2 3 2 4 #1 5 #2 2 4 #1 5 #2 6 #4 #5 6 3 1 2 #3 7 3 #8 1 7 7 % 60 7 D A1 AD6 4 2 #5 #4 1 8 5 4 0 #9 #3 #4 10 5 0 12 4 12 #5 #6 0 7 #3 1 2 3 2 3 2 4 #1 5 #2 2 4 #1 5 #2 60 6 3 1 2 #3 7 3 1 2 #3 7 7 % 60 7DU % A2 D 6 40 2 15 40 0 0 2 15 0 0 5 0 0 0 4 7 0 0 0 0


7KHKLJKOLJKWHGHQWULHVDERYHGHWHUPLQHWKHURZUHGXFWLRQRI A WR U $WHDFKSLYRW FROXPQ GLYLGHWKHKLJKOLJKWHGHQWULHVE\WKHSLYRWDQGSODFHWKHUHVXOWLQWR L 2 3 2 2 3 6 #4 7 3 # $ 6 7 4 2 54 #9 5 2 ! " 5 4 12 #6 &2 2 #

1 6 #2 6 4 1 #3

&3 #

1 #3 4

&2 &5 # #3

1 2

7 7 5 1

DQG

2

1 0 6 #2 1 LD6 4 1 #3 #3 4

0 0 1 2

$QHDV\FDOFXODWLRQYHULÀHVWKDWWKLV L DQG U VDWLVI\ LU D A

SG

3 0 07 7 05 1

,QSUDFWLFDOZRUN URZLQWHUFKDQJHVDUHQHDUO\DOZD\VQHHGHG EHFDXVHSDUWLDOSLY RWLQJLVXVHGIRUKLJKDFFXUDF\ 5HFDOOWKDWWKLVSURFHGXUHVHOHFWV DPRQJWKHSRVVLEOH FKRLFHVIRUDSLYRW DQHQWU\LQWKHFROXPQKDYLQJWKHODUJHVWDEVROXWHYDOXH 7RKDQGOH URZLQWHUFKDQJHV WKH/8 IDFWRUL]DWLRQDERYHFDQEHPRGLÀHGHDVLO\WRSURGXFHDQ L WKDW LV SHUPXWHGORZHUWULDQJXODU LQWKHVHQVHWKDWDUHDUUDQJHPHQWFDOOHGDSHUPXWDWLRQ RIWKHURZVRI L FDQPDNH L .XQLW/ ORZHUWULDQJXODU 7KHUHVXOWLQJ SHUPXWHG/8 IDFWRUL]DWLRQ VROYHV A[ D E LQWKHVDPHZD\DVEHIRUH H[FHSWWKDWWKHUHGXFWLRQRI Œ L E ! WR Œ I \ ! IROORZVWKHRUGHURIWKHSLYRWVLQ L IURPOHIWWRULJKW VWDUWLQJZLWK WKHSLYRWLQWKHÀUVWFROXPQ $ UHIHUHQFHWRDQ´/8 IDFWRUL]DWLRQµXVXDOO\LQFOXGHVWKH SRVVLELOLW\WKDW L PLJKWEHSHUPXWHGORZHUWULDQJXODU )RU GHWDLOV VHHWKH 6WXG\*XLGH

1FSNVUFE -6 'BDUPSJ[BUJPOT m

/6.&3*$"- /05&4 7KHIROORZLQJRSHUDWLRQFRXQWVDSSO\WRDQ n ! n GHQVHPDWUL[ A ZLWKPRVW HQWULHVQRQ]HUR IRU n PRGHUDWHO\ODUJH VD\ n ' 30 &RPSXWLQJDQ/8 IDFWRUL]DWLRQRI A WDNHVDERXW 2n3 =3 ÁRSVDERXWWKHVDPH DVURZUHGXFLQJ Œ A E !/ ZKHUHDVÀQGLQJ A"1 UHTXLUHVDERXW 2n3 ÁRSV

WEB

6ROYLQJ L\ D E DQG U [ D \ UHTXLUHVDERXW 2n2 ÁRSV EHFDXVHDQ\ n ! n WULDQJXODUV\VWHPFDQEHVROYHGLQDERXW n2 ÁRSV 0XOWLSOLFDWLRQRI E E\ A"1 DOVRUHTXLUHVDERXW 2n2 ÁRSV EXWWKHUHVXOWPD\ QRWEHDVDFFXUDWHDVWKDWREWDLQHGIURP L DQG U EHFDXVHRIURXQGRIIHUURU ZKHQFRPSXWLQJERWK A"1 DQG A"1 E/ ,I A LVVSDUVHZLWKPRVWO\]HURHQWULHV WKHQ L DQG U PD\EHVSDUVH WRR ZKHUHDV A"1 LVOLNHO\WREHGHQVH ,QWKLVFDVH DVROXWLRQRI A[ D E ZLWKDQ /8 IDFWRUL]DWLRQLV PXFK IDVWHUWKDQXVLQJ A"1 6HH([HUFLVH

$ 0DWUL[)DFWRUL]DWLRQLQ(OHFWULFDO(QJLQHHULQJ 0DWUL[IDFWRUL]DWLRQLVLQWLPDWHO\UHODWHGWRWKHSUREOHPRIFRQVWUXFWLQJDQHOHFWULFDO QHWZRUNZLWKVSHFLÀHGSURSHUWLHV 7KHIROORZLQJGLVFXVVLRQJLYHVMXVWDJOLPSVHRIWKH FRQQHFWLRQEHWZHHQIDFWRUL]DWLRQDQGFLUFXLWGHVLJQ ñ 6HH6HFWLRQLQ $SSOLHG/LQHDU$OJHEUD UGHG E\%HQ1REOHDQG-DPHV:'DQLHO(QJOHZRRG&OLIIV 1-3UHQWLFH+DOO 5HFDOOWKDWIRURXUSXUSRVHV D ÁRS LV C " # RU $

$)"15&3

0DWUL[$OJHEUD

6XSSRVHWKHER[LQ)LJ UHSUHVHQWVVRPHVRUWRIHOHFWULFFLUFXLW ZLWKDQLQSXW # $ v1 DQGRXWSXW 5HFRUGWKHLQSXWYROWDJHDQGFXUUHQWE\ ZLWKYROWDJH v LQYROWVDQG i1 # $ v FXUUHQW i LQDPSV DQGUHFRUGWKHRXWSXWYROWDJHDQGFXUUHQWE\ 2 )UHTXHQWO\ WKH i2 # $ # $ v v2 WUDQVIRUPDWLRQ 1 7! LVOLQHDU 7KDWLV WKHUHLVDPDWUL[ A FDOOHGWKH WUDQVIHU i1 i2 PDWUL[ VXFKWKDW # $ # $ v2 v DA 1 i2 i1 i1 input terminals

i2 electric circuit

v1

v2

output terminals

'*(63& $ FLUFXLWZLWKLQSXWDQGRXWSXW

WHUPLQDOV

)LJXUHVKRZVD ODGGHUQHWZRUN ZKHUHWZRFLUFXLWVWKHUHFRXOGEHPRUH DUH FRQQHFWHGLQVHULHV VRWKDWWKHRXWSXWRIRQHFLUFXLWEHFRPHVWKHLQSXWRIWKHQH[WFLUFXLW 7KHOHIWFLUFXLWLQ)LJ LVFDOOHGD VHULHVFLUFXLW ZLWKUHVLVWDQFH R1 LQRKPV i1 v1

i2 R1

i2 v2

A series circuit

i3 R2

v3

A shunt circuit

'*(63& $ ODGGHUQHWZRUN

7KHULJKWFLUFXLWLQ)LJLVD VKXQWFLUFXLW ZLWKUHVLVWDQFH R2 8VLQJ2KP·VODZDQG .LUFKKRII·VODZV RQHFDQVKRZWKDWWKHWUDQVIHUPDWULFHVRIWKHVHULHVDQGVKXQWFLUFXLWV UHVSHFWLYHO\ DUH # $ # $ 1 #R1 1 0 DQG 0 1 1 #1=R2 7UDQVIHUPDWUL[ RIVHULHVFLUFXLW

7UDQVIHUPDWUL[ RIVKXQWFLUFXLW

&9".1-& D &RPSXWHWKHWUDQVIHUPDWUL[RIWKHODGGHUQHWZRUNLQ)LJ # $ 1 #8 E 'HVLJQDODGGHUQHWZRUNZKRVHWUDQVIHUPDWUL[LV #:5 5 40-65*0/ D /HW A1 DQG A2 EHWKHWUDQVIHUPDWULFHVRIWKHVHULHVDQGVKXQWFLUFXLWV UHVSHFWLYHO\ 7KHQDQLQSXWYHFWRU [ LVWUDQVIRUPHGÀUVWLQWR A1 [ DQGWKHQLQWR A2 .A1 [/ 7KHVHULHV FRQQHFWLRQRIWKHFLUFXLWVFRUUHVSRQGVWRFRPSRVLWLRQRIOLQHDUWUDQVIRUPDWLRQV DQG WKHWUDQVIHUPDWUL[RIWKHODGGHUQHWZRUNLVQRWHWKHRUGHU # $# $ # $ 1 0 1 #R1 1 #R1 A2 A1 D D 1 0 1 #1=R2 1 C R1 =R2 #1=R2


$ 1 #8 LQWRWKHSURGXFWRIWUDQVIHUPDWULFHV DVLQHTXD #:5 5 WLRQ ORRN IRU R1 DQG R2 LQ)LJ WRVDWLVI\ # $ # $ 1 #R1 1 #8 D #1=R2 1 C R1 =R2 #:5 5

E 7RIDFWRUWKHPDWUL[

#

)URPWKH .1; 2/HQWULHV R1 D 8 RKPV DQGIURPWKH .2; 1/HQWULHV 1=R2 D :5 RKP DQG R2 D 1=:5 D 2 RKPV :LWKWKHVHYDOXHV WKHQHWZRUNLQ)LJ KDVWKHGHVLUHG WUDQVIHUPDWUL[

$ QHWZRUNWUDQVIHUPDWUL[VXPPDUL]HVWKHLQSXW²RXWSXWEHKDYLRUWKHGHVLJQVSHF LÀFDWLRQV RIWKHQHWZRUNZLWKRXWUHIHUHQFHWRWKHLQWHULRUFLUFXLWV 7RSK\VLFDOO\EXLOG DQHWZRUNZLWKVSHFLÀHGSURSHUWLHV DQHQJLQHHUÀUVWGHWHUPLQHVLIVXFKDQHWZRUNFDQ EHFRQVWUXFWHGRU UHDOL]HG 7KHQWKHHQJLQHHUWULHVWRIDFWRUWKHWUDQVIHUPDWUL[LQWR PDWULFHVFRUUHVSRQGLQJWRVPDOOHUFLUFXLWVWKDWSHUKDSVDUHDOUHDG\PDQXIDFWXUHGDQG UHDG\IRUDVVHPEO\ ,QWKHFRPPRQFDVHRIDOWHUQDWLQJFXUUHQW WKHHQWULHVLQWKHWUDQVIHU PDWUL[DUHXVXDOO\UDWLRQDOFRPSOH[YDOXHGIXQFWLRQV 6HH([HUFLVHV DQGLQ 6HFWLRQ DQG([DPSOH LQ6HFWLRQ $ VWDQGDUGSUREOHPLVWRÀQGD PLQLPDO UHDOL]DWLRQ WKDWXVHVWKHVPDOOHVWQXPEHURIHOHFWULFDOFRPSRQHQWV

13"$5*$& 130#-&.

2

2 6 6 6 )LQGDQ/8 IDFWRUL]DWLRQRI A D 6 6 2 4 4 #6

#4 #9 #7 #2 3

3 #2 3 #5 8 7 7 #3 9 7 7 >1RWH ,WZLOOWXUQRXWWKDW A #2 #1 5 3 4

KDVRQO\WKUHHSLYRWFROXPQV VRWKHPHWKRGRI([DPSOH ZLOOSURGXFHRQO\WKHÀUVW WKUHHFROXPQVRI L 7KHUHPDLQLQJWZRFROXPQVRI L FRPHIURP I5 @

&9&3$*4&4 ,Q([HUFLVHV² VROYHWKHHTXDWLRQ A[ D E E\XVLQJWKH/8 IDFWRUL]DWLRQJLYHQIRU A ,Q([HUFLVHV DQG DOVRVROYH A[ D E E\RUGLQDU\URZUHGXFWLRQ 2 3 2 3 3 #7 #2 #7 5 1 5; E D 4 5 5 A D 4 #3 6 #4 0 2 2 32 3 1 0 0 3 #7 #2 1 0 54 0 #2 #1 5 A D 4 #1 2 #5 1 0 0 #1 2 3 2 3 2 #6 4 2 8 0 5; E D 4 #4 5 A D 4 #4 0 #4 6 6 2 32 3 1 0 0 2 #6 4 1 0 54 0 #4 85 A D 4 #2 0 1 1 0 0 #2 2 3 2 3 2 #4 2 6 5 2 5; E D 4 0 5 A D 4 #4 6 #9 1 6

2

1 0 1 A D 4 #2 3 #1 2

1 #1 A D 4 1 #3 3 7 2 1 0 1 A D 41 3 #5 2

32 0 2 #4 0 54 0 #3 1 0 0

3 2 65 1

3 2 3 2 0 1 5; E D 4 #5 5 5 7 32 3 0 1 #1 2 0 54 0 #2 #1 5 1 0 0 #6

3 2 3 1 #2 #2 #3 1 6 3 #9 7 667 0 #9 7; E D 6 7 A D 6 4 #1 405 2 4 75 #3 #6 26 2 3 2 32 3 1 0 0 0 1 #2 #2 #3 6 3 6 1 0 07 6 07 76 0 #3 7 AD6 4 #1 5 4 0 1 0 0 0 2 45 #3 4 #2 1 0 0 0 1

$)"15&3

0DWUL[$OJHEUD

2

3 2 3 1 3 2 0 1 6 #2 #3 #4 6 7 12 7 7; E D 6 #2 7 A D 6 4 3 4 #1 5 0 4 #36 5 #5 #3 #8 49 2 2 32 1 0 0 0 1 3 2 6 #2 76 0 1 0 0 3 0 76 AD6 4 3 #3 1 0 54 0 0 #2 #5 4 #1 1 0 0 0

3 0 12 7 7 05 1

)LQGDQ/8 IDFWRUL]DWLRQRIWKHPDWULFHVLQ([HUFLVHV ²ZLWK L XQLWORZHUWULDQJXODU 1RWHWKDW0$7/$% ZLOOXVXDOO\SURGXFH DSHUPXWHG/8 IDFWRUL]DWLRQEHFDXVHLWXVHVSDUWLDOSLYRWLQJIRU QXPHULFDODFFXUDF\ # $ # $ 2 5 6 4 #3 #4 12 5 2 3 2 3 3 1 2 #5 0 4 0 #4 5 2 #5 5 4 #9 4 10 9 9 14 10 10 16 2 3 2 3 3 7 2 2 3 2 45 95 4 6 19 4 4 13 #3 #2 3 #6 5 4 2 3 2 3 1 3 #5 #3 1 3 1 5 6 #1 #5 6 5 8 47 20 6 31 7 7 7 6 6 4 4 4 #2 #1 #1 #4 5 2 #5 #7 5 #2 #4 7 5 #1 7 1 7 2 3 2 #3 4 2 3 6 #4 7 2 0 5 2 8 #7 6 7 6 4 5 #6 3 #13 #3 6 6 #5 14 7 7 4 #6 4 9 16 17 9 #12 5 8 #6 19 :KHQ A LV LQYHUWLEOH 0$7/$% ÀQGV A!1 E\ IDFWRULQJ A D LU ZKHUH L PD\EHSHUPXWHGORZHUWULDQJXODU LQ YHUWLQJ L DQG U DQGWKHQFRPSXWLQJ U !1 L!1 8VHWKLV PHWKRGWRFRPSXWHWKHLQYHUVHRI A LQ([HUFLVH $SSO\ WKHDOJRULWKPLQ6HFWLRQ WR L DQGWR U )LQG A!1 DVLQ([HUFLVH XVLQJ A IURP([HUFLVH /HW A EHDORZHUWULDQJXODU n ! n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²SURYLGHDJOLPSVHRIVRPHZLGHO\XVHGPDWUL[ IDFWRUL]DWLRQV VRPHRIZKLFKDUHGLVFXVVHGODWHULQWKHWH[W

5HGXFHG LU )DFWRUL]DWLRQ :LWK A DVLQWKH3UDFWLFH3URE OHP ÀQGD 5 ! 3 PDWUL[ B DQGD 3 ! 4 PDWUL[ C VXFKWKDW A D BC *HQHUDOL]HWKLVLGHDWRWKHFDVHZKHUH A LV m ! n A D LU DQG U KDVRQO\WKUHHQRQ]HURURZV 5DQN)DFWRUL]DWLRQ 6XSSRVHDQ m ! n PDWUL[ A DGPLWVD IDFWRUL]DWLRQ A D CD ZKHUH C LV m ! 4 DQG D LV 4 ! n D 6KRZWKDW A LVWKHVXPRIIRXURXWHUSURGXFWV 6HH 6HFWLRQ E /HW m D 400 DQG n D 100 ([SODLQZK\DFRPSXWHUSUR JUDPPHUPLJKWSUHIHUWRVWRUHWKHGDWDIURP A LQWKHIRUP RIWZRPDWULFHV C DQG D QR )DFWRUL]DWLRQ 6XSSRVH A D QR ZKHUH Q DQG R DUH n ! n R LVLQYHUWLEOHDQGXSSHUWULDQJXODU DQG Q KDVWKH SURSHUW\WKDW QT Q D I 6KRZWKDWIRUHDFK E LQ Rn WKH HTXDWLRQ A[ D E KDVDXQLTXHVROXWLRQ :KDWFRPSXWDWLRQV ZLWK Q DQG R ZLOOSURGXFHWKHVROXWLRQ" WEB

6LQJXODU 9DOXH 'HFRPSRVLWLRQ 6XSSRVH A D UDV T ZKHUH U DQG V DUH n ! n PDWULFHVZLWKWKHSURSHUW\WKDW U T U D I DQG V T V D I DQGZKHUH D LVDGLDJRQDOPDWUL[ ZLWKSRVLWLYHQXPEHUV "1 ; : : : ; "n RQWKHGLDJRQDO 6KRZWKDW A LVLQYHUWLEOH DQGÀQGDIRUPXODIRU A!1 6SHFWUDO)DFWRUL]DWLRQ 6XSSRVHD 3 ! 3 PDWUL[ A DGPLWV DIDFWRUL]DWLRQDV A D PDP !1 ZKHUH P LVVRPHLQYHUWLEOH 3 ! 3 PDWUL[DQG D LVWKHGLDJRQDOPDWUL[ 2 3 2 0 0 3 05 D D 40 0 0 1

6KRZWKDWWKLVIDFWRUL]DWLRQLVXVHIXOZKHQFRPSXWLQJKLJK SRZHUVRI A )LQGIDLUO\VLPSOHIRUPXODVIRU A2 A3 DQG Ak k DSRVLWLYHLQWHJHU XVLQJ P DQGWKHHQWULHVLQ D

'HVLJQWZRGLIIHUHQWODGGHUQHWZRUNVWKDWHDFKRXWSXW YROWV DQG DPSVZKHQWKHLQSXWLV YROWVDQG DPSV 6KRZWKDWLIWKUHHVKXQWFLUFXLWVZLWKUHVLVWDQFHV R1 R2 R3 DUHFRQQHFWHGLQVHULHV WKHUHVXOWLQJQHWZRUNKDVWKHVDPH WUDQVIHUPDWUL[DVDVLQJOHVKXQWFLUFXLW )LQGDIRUPXODIRU WKHUHVLVWDQFHLQWKDWFLUFXLW D &RPSXWHWKHWUDQVIHUPDWUL[RIWKHQHWZRUNLQWKHÀJXUH EHORZ i1 v1

i2 R1

#

i2 v2

i3 R2

i3 v3

i4 R3

v4

$ 3 #12 E /HW A D 'HVLJQ D ODGGHU QHWZRUN #1=3 5=3 ZKRVHWUDQVIHUPDWUL[LV A E\ÀQGLQJDVXLWDEOHPDWUL[ IDFWRUL]DWLRQRI A

)LQG D GLIIHUHQW IDFWRUL]DWLRQ RI WKH WUDQVIHU PDWUL[ A LQ ([HUFLVH DQGWKHUHE\GHVLJQDGLIIHUHQWODGGHUQHWZRUN ZKRVHWUDQVIHUPDWUL[LV A >0@ &RQVLGHUWKHKHDWSODWHLQWKHIROORZLQJÀJXUHUHIHUWR ([HUFLVHLQ6HFWLRQ

5° 5°

0°

0°

0°

0°

1

3

5

7

2

4

6

8

10°

10°

10°

10°


E 8VHWKH/8 IDFWRUL]DWLRQWRVROYH A[ D E

F 2EWDLQ A!1 DQGQRWHWKDW A!1 LVDGHQVHPDWUL[ZLWKQR EDQGVWUXFWXUH :KHQ A LVODUJH L DQG U FDQEHVWRUHG LQPXFKOHVVVSDFHWKDQ A!1 7KLVIDFWLVDQRWKHUUHDVRQ IRUSUHIHUULQJWKH/8 IDFWRUL]DWLRQRI A WR A!1 LWVHOI

>0@ 7KHEDQGPDWUL[ A VKRZQEHORZFDQEHXVHGWRHVWLPDWH WKHXQVWHDG\FRQGXFWLRQRIKHDWLQDURGZKHQWKHWHPSHUD WXUHVDWSRLQWV p1 ; : : : ; p4 RQWKHURGFKDQJHZLWKWLPH

20° 20°

∆x

∆x p1

7KHVROXWLRQWRWKHVWHDG\VWDWHKHDWÁRZSUREOHPIRUWKLV SODWHLVDSSUR[LPDWHGE\WKHVROXWLRQWRWKHHTXDWLRQ A[ D E ZKHUH E D .5; 15; 0; 10; 0; 10; 20; 30/ DQG 2 3 4 #1 #1 6 #1 7 4 0 #1 6 7 6 #1 7 0 4 #1 #1 6 7 6 7 #1 #1 4 0 #1 7 AD6 6 7 #1 0 4 #1 #1 6 7 6 7 #1 #1 4 0 #1 6 7 4 #1 0 4 #1 5 #1 #1 4 WEB

7KHPLVVLQJHQWULHVLQ A DUH]HURV 7KHQRQ]HURHQWULHVRI A OLHZLWKLQDEDQGDORQJWKHPDLQGLDJRQDO 6XFK EDQG PDWULFHV RFFXU LQ D YDULHW\ RI DSSOLFDWLRQV DQG RIWHQ DUH H[WUHPHO\ODUJHZLWKWKRXVDQGVRIURZVDQGFROXPQVEXW UHODWLYHO\QDUURZEDQGV D 8VHWKHPHWKRGLQ([DPSOHWRFRQVWUXFWDQ/8 IDFWRU L]DWLRQRI A DQGQRWHWKDWERWKIDFWRUVDUHEDQGPDWULFHV ZLWKWZRQRQ]HURGLDJRQDOVEHORZRUDERYHWKHPDLQ GLDJRQDO &RPSXWH LU # A WRFKHFN\RXUZRUN

∆x p2

∆x p3

∆x p4

7KHFRQVWDQW C LQWKHPDWUL[GHSHQGVRQWKHSK\VLFDOQDWXUH RIWKHURG WKHGLVWDQFH #x EHWZHHQWKHSRLQWVRQWKHURG DQGWKHOHQJWKRIWLPH #t EHWZHHQVXFFHVVLYHWHPSHUDWXUH PHDVXUHPHQWV 6XSSRVHWKDWIRU k D 0; 1; 2; : : : ; DYHFWRU Wk LQ R4 OLVWVWKHWHPSHUDWXUHVDWWLPH k#t ,IWKHWZRHQGVRIWKH URGDUHPDLQWDLQHGDW 0ı WKHQWKHWHPSHUDWXUHYHFWRUVVDWLVI\ WKHHTXDWLRQ AWkC1 D Wk .k D 0; 1; : : :/ ZKHUH 2 3 .1 C 2C / #C 6 #C 7 .1 C 2C / #C 7 AD6 4 #C .1 C 2C / #C 5 #C .1 C 2C / D )LQGWKH/8 IDFWRUL]DWLRQRI A ZKHQ C D 1 $ PDWUL[ VXFKDV A ZLWKWKUHHQRQ]HURGLDJRQDOVLVFDOOHGD WULGL DJRQDO PDWUL[ 7KH L DQG U IDFWRUV DUH ELGLDJRQDO PDWULFHV

E 6XSSRVH C D 1 DQG W0 D .10; 15; 15; 10/T 8VHWKH/8 IDFWRUL]DWLRQRI A WRÀQGWKHWHPSHUDWXUHGLVWULEXWLRQV W1 W2 W3 DQG W4 2 6HH%LVZD1'DWWD 1XPHULFDO/LQHDU$OJHEUDDQG$SSOLFDWLRQV 3DFLÀF *URYH &$%URRNV&ROH SS ²

40-65*0/ 50 13"$5*$& 130#-&. 2

3 2 3 2 #4 #2 3 2 #4 #2 3 6 6 #9 #5 60 87 3 1 #1 7 6 7 6 7 6 7 6 A D 6 2 #7 #3 9 7 % 6 0 #3 #1 6 7 7 4 4 #2 #2 #1 5 40 6 2 #7 5 0 #9 #3 13 #6 3 3 4 2 3 2 3 2 #4 #2 3 2 #4 #2 3 60 60 3 1 #1 7 3 1 #1 7 6 7 6 7 7 % 60 5 0 0 0 0 0 57 %6 6 7 6 7DU 40 40 0 0 #5 5 0 0 05 0 0 0 10 0 0 0 0

'LYLGHWKHHQWULHVLQHDFKKLJKOLJKWHGFROXPQE\WKHSLYRWDWWKHWRS 7KHUHVXOWLQJ FROXPQVIRUPWKHÀUVWWKUHHFROXPQVLQWKHORZHUKDOIRI L 7KLVVXIÀFHVWRPDNHURZ UHGXFWLRQRI L WR I FRUUHVSRQGWRUHGXFWLRQRI A WR U 8VHWKHODVWWZRFROXPQVRI I5

$)"15&3

0DWUL[$OJHEUD

WRPDNH L XQLWORZHUWULDQJXODU 2

3 2 2 3 6 67 3 2 6 7 3 6 2 76 #3 7 5 6 76 7 4 4 54 6 54 #5 5 10 #9 #6 &2 2 #

1 6 3 6 6 1 6 4 2 #3

&3 #

&5 #

1 #1 2 #3

1 #1 2

3

7 7 """ 7 7 5

2

1 0 0 6 3 1 0 6 1 #1 1 LD6 6 4 2 2 #1 #3 #3 2

0 0 0 1 0

3 0 07 7 07 7 05 1

5)& -&0/5*&' */165 065165 .0%&WEB

/LQHDUDOJHEUDSOD\HGDQHVVHQWLDOUROHLQWKH1REHOSUL]H²ZLQQLQJZRUNRI:DVVLO\ /HRQWLHI DVPHQWLRQHGDWWKHEHJLQQLQJRI&KDSWHU 7KHHFRQRPLFPRGHOGHVFULEHG LQWKLVVHFWLRQLVWKHEDVLVIRUPRUHHODERUDWHPRGHOVXVHGLQPDQ\SDUWVRIWKHZRUOG 6XSSRVHDQDWLRQ·VHFRQRP\LVGLYLGHGLQWR n VHFWRUVWKDWSURGXFHJRRGVRUVHUYLFHV DQGOHW [ EHD SURGXFWLRQYHFWRU LQ Rn WKDWOLVWVWKHRXWSXWRIHDFKVHFWRUIRURQH \HDU $OVR VXSSRVHDQRWKHUSDUWRIWKHHFRQRP\FDOOHGWKH RSHQVHFWRU GRHVQRW SURGXFHJRRGVRUVHUYLFHVEXWRQO\FRQVXPHVWKHP DQGOHW G EHD ÀQDOGHPDQGYHFWRU RU ELOORIÀQDOGHPDQGV WKDWOLVWVWKHYDOXHVRIWKHJRRGVDQGVHUYLFHVGHPDQGHG IURPWKHYDULRXVVHFWRUVE\WKHQRQSURGXFWLYHSDUWRIWKHHFRQRP\ 7KHYHFWRU G FDQ UHSUHVHQWFRQVXPHUGHPDQG JRYHUQPHQWFRQVXPSWLRQ VXUSOXVSURGXFWLRQ H[SRUWV RU RWKHUH[WHUQDOGHPDQGV $VWKHYDULRXVVHFWRUVSURGXFHJRRGVWRPHHWFRQVXPHUGHPDQG WKHSURGXFHUV WKHPVHOYHVFUHDWHDGGLWLRQDO LQWHUPHGLDWHGHPDQG IRUJRRGVWKH\QHHGDVLQSXWVIRU WKHLURZQSURGXFWLRQ 7KHLQWHUUHODWLRQVEHWZHHQWKHVHFWRUVDUHYHU\FRPSOH[ DQGWKH FRQQHFWLRQEHWZHHQWKHÀQDOGHPDQGDQGWKHSURGXFWLRQLVXQFOHDU /HRQWLHIDVNHGLI WKHUHLVDSURGXFWLRQOHYHO [ VXFKWKDWWKHDPRXQWVSURGXFHGRU´VXSSOLHGµ ZLOOH[DFWO\ EDODQFHWKHWRWDOGHPDQGIRUWKDWSURGXFWLRQ VRWKDW 8 9 8 9 & < ÀQDO = < DPRXQW = % LQWHUPHGLDWH SURGXFHG D C GHPDQG GHPDQG : ; : ; [ G

7KHEDVLFDVVXPSWLRQRI/HRQWLHI·VLQSXW²RXWSXWPRGHOLVWKDWIRUHDFKVHFWRU WKHUHLV D XQLWFRQVXPSWLRQYHFWRU LQ Rn WKDWOLVWVWKHLQSXWVQHHGHG SHUXQLWRIRXWSXW RIWKH VHFWRU $OOLQSXWDQGRXWSXWXQLWVDUHPHDVXUHGLQPLOOLRQVRIGROODUV UDWKHUWKDQLQ TXDQWLWLHVVXFKDVWRQVRUEXVKHOV 3ULFHVRIJRRGVDQGVHUYLFHVDUHKHOGFRQVWDQW $VDVLPSOHH[DPSOH VXSSRVHWKHHFRQRP\FRQVLVWVRIWKUHHVHFWRUV³PDQXIDF WXULQJ DJULFXOWXUH DQGVHUYLFHV³ZLWKXQLWFRQVXPSWLRQYHFWRUV F1 F2 DQG F3 DVVKRZQ LQWKHWDEOHWKDWIROORZV

7KH/HRQWLHI,QSXW 2XWSXW0RGHO

,QSXWV&RQVXPHGSHU8QLWRI2XWSXW 3XUFKDVHGIURP

0DQXIDFWXULQJ

$JULFXOWXUH

6HUYLFHV

0DQXIDFWXULQJ

$JULFXOWXUH

6HUYLFHV

"

"

"

F1

F2

F3

&9".1-& :KDWDPRXQWVZLOOEHFRQVXPHGE\WKHPDQXIDFWXULQJVHFWRULILW GHFLGHVWRSURGXFHXQLWV"

40-65*0/ &RPSXWH

2

3 2 3 :50 50 100F1 D 1004 :20 5 D 4 20 5 :10 10

7RSURGXFHXQLWV PDQXIDFWXULQJZLOORUGHULH ´GHPDQGµ DQGFRQVXPHXQLWV IURPRWKHUSDUWVRIWKHPDQXIDFWXULQJVHFWRU XQLWVIURPDJULFXOWXUH DQGXQLWV IURPVHUYLFHV ,IPDQXIDFWXULQJGHFLGHVWRSURGXFH x1 XQLWVRIRXWSXW WKHQ x1 F1 UHSUHVHQWVWKH LQ WHUPHGLDWHGHPDQGV RIPDQXIDFWXULQJ EHFDXVHWKHDPRXQWVLQ x1 F1 ZLOOEHFRQVXPHGLQ WKHSURFHVVRIFUHDWLQJWKH x1 XQLWVRIRXWSXW /LNHZLVH LI x2 DQG x3 GHQRWHWKHSODQQHG RXWSXWVRIWKHDJULFXOWXUHDQGVHUYLFHVVHFWRUV x2 F2 DQG x3 F3 OLVWWKHLUFRUUHVSRQGLQJ LQWHUPHGLDWHGHPDQGV 7KHWRWDOLQWHUPHGLDWHGHPDQGIURPDOOWKUHHVHFWRUVLVJLYHQE\

fLQWHUPHGLDWHGHPDQGg D x1 F1 C x2 F2 C x3 F3 D C[

ZKHUH C LVWKH FRQVXPSWLRQPDWUL[ Œ F1 F2 F3 ! QDPHO\ 2 3 :50 :40 :20 C D 4 :20 :30 :10 5 :10 :10 :30

(TXDWLRQV DQG \LHOG/HRQWLHI·VPRGHO

5)& -&0/5*&' */165065165 .0%&- 03 130%6$5*0/ &26"5*0/ [ D C[ C G $PRXQW SURGXFHG

,QWHUPHGLDWH GHPDQG

)LQDO GHPDQG

(TXDWLRQ PD\DOVREHZULWWHQDV I [ # C [ D G RU

.I # C /[ D G

&9".1-& &RQVLGHUWKHHFRQRP\ZKRVHFRQVXPSWLRQPDWUL[LVJLYHQE\ 6XSSRVHWKHÀQDOGHPDQGLV XQLWVIRUPDQXIDFWXULQJ XQLWVIRUDJULFXOWXUH DQG XQLWVIRUVHUYLFHV )LQGWKHSURGXFWLRQOHYHO [ WKDWZLOOVDWLVI\WKLVGHPDQG

$)"15&3

0DWUL[$OJHEUD

40-65*0/ 7KHFRHIÀFLHQWPDWUL[LQ LV 2 3 2 1 0 0 :5 :4 1 0 5 # 4 :2 :3 I #C D 40 0 0 1 :1 :1

3 2 3 :2 :5 #:4 #:2 :1 5 D 4 #:2 :7 #:1 5 :3 #:1 #:1 :7

7RVROYH URZUHGXFHWKHDXJPHQWHGPDWUL[ 2 3 2 3 2 :5 #:4 #:2 50 5 #4 #2 500 1 4 #:2 :7 #:1 30 5 % 4 #2 7 #1 300 5 % " " " % 4 0 #:1 #:1 :7 20 #1 #1 7 200 0

0 1 0

0 0 1

3 226 119 5 78

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·VZRUWKRILQSXWVWR SURGXFHRQHXQLWRIRXWSXW

5)&03&.

/HW C EHWKHFRQVXPSWLRQPDWUL[IRUDQHFRQRP\ DQGOHW G EHWKHÀQDOGHPDQG ,I C DQG G KDYHQRQQHJDWLYHHQWULHVDQGLIHDFKFROXPQVXPRI C LVOHVVWKDQ WKHQ .I # C /"1 H[LVWVDQGWKHSURGXFWLRQYHFWRU [ D .I # C /"1 G

KDVQRQQHJDWLYHHQWULHVDQGLVWKHXQLTXHVROXWLRQRI [ D C[ C G 7KHIROORZLQJGLVFXVVLRQZLOOVXJJHVWZK\WKHWKHRUHPLVWUXHDQGZLOOOHDGWRD QHZZD\WRFRPSXWH .I # C /"1

$ )RUPXODIRU , ±& ± ,PDJLQHWKDWWKHGHPDQGUHSUHVHQWHGE\ G LVSUHVHQWHGWRWKHYDULRXVLQGXVWULHVDWWKH EHJLQQLQJRIWKH\HDU DQGWKHLQGXVWULHVUHVSRQGE\VHWWLQJWKHLUSURGXFWLRQOHYHOVDW [ D G ZKLFKZLOOH[DFWO\PHHWWKHÀQDOGHPDQG $VWKHLQGXVWULHVSUHSDUHWRSURGXFH G WKH\VHQGRXWRUGHUVIRUWKHLUUDZPDWHULDOVDQGRWKHULQSXWV 7KLVFUHDWHVDQLQWHUPHGLDWH GHPDQGRI C G IRULQSXWV 7RPHHWWKHDGGLWLRQDOGHPDQGRI C G WKHLQGXVWULHVZLOOQHHGDVDGGLWLRQDOLQSXWV WKHDPRXQWVLQ C.C G/ D C 2 G 2IFRXUVH WKLVFUHDWHVDVHFRQGURXQGRILQWHUPHGLDWH GHPDQG DQGZKHQWKHLQGXVWULHVGHFLGHWRSURGXFHHYHQPRUHWRPHHWWKLVQHZGHPDQG WKH\FUHDWHDWKLUGURXQGRIGHPDQG QDPHO\ C.C 2 G/ D C 3 G $QGVRLWJRHV 7KHRUHWLFDOO\ WKLVSURFHVVFRXOGFRQWLQXHLQGHÀQLWHO\ DOWKRXJKLQUHDOOLIHLWZRXOG QRWWDNHSODFHLQVXFKDULJLGVHTXHQFHRIHYHQWV :HFDQGLDJUDPWKLVK\SRWKHWLFDO VLWXDWLRQDVIROORZV


'HPDQG7KDW 0XVW%H0HW

,QSXWV1HHGHGWR 0HHW7KLV'HPDQG

G

CG

VWURXQG

CG

QGURXQG

2

C G

C.C G/ D C 2 G

UGURXQG

C 3G :: :

)LQDOGHPDQG

,QWHUPHGLDWHGHPDQG

C.C 2 G/ D C 3 G C.C 3 G/ D C 4 G :: :

7KHSURGXFWLRQOHYHO [ WKDWZLOOPHHWDOORIWKLVGHPDQGLV [ D G C C G C C 2G C C 3G C " " "

D .I C C C C 2 C C 3 C " " " /G

7RPDNHVHQVHRIHTXDWLRQ FRQVLGHUWKHIROORZLQJDOJHEUDLFLGHQWLW\

.I # C /.I C C C C 2 C " " " C C m / D I # C mC1

,WFDQEHVKRZQWKDWLIWKHFROXPQVXPVLQ C DUHDOOVWULFWO\OHVVWKDQ WKHQ I # C LVLQ YHUWLEOH C m DSSURDFKHVWKH]HURPDWUL[DV m JHWVDUELWUDULO\ODUJH DQG I # C mC1 ! I 7KLVIDFWLVDQDORJRXVWRWKHIDFWWKDWLIDSRVLWLYHQXPEHU t LVOHVVWKDQ WKHQ t m ! 0 DV m LQFUHDVHV 8VLQJHTXDWLRQ ZULWH

.I # C /"1 ( I C C C C 2 C C 3 C " " " C C m ZKHQWKHFROXPQVXPVRI C DUHOHVVWKDQ

7KHDSSUR[LPDWLRQLQ PHDQVWKDWWKHULJKWVLGHFDQEHPDGHDVFORVHWR .I # C /"1 DVGHVLUHGE\WDNLQJ m VXIÀFLHQWO\ODUJH ,QDFWXDOLQSXW²RXWSXWPRGHOV SRZHUVRIWKHFRQVXPSWLRQPDWUL[DSSURDFKWKH]HUR PDWUL[UDWKHUTXLFNO\ 6R UHDOO\SURYLGHVDSUDFWLFDOZD\WRFRPSXWH .I # C /"1 /LNHZLVH IRUDQ\ G WKHYHFWRUV C m G DSSURDFKWKH]HURYHFWRUTXLFNO\ DQG LVD SUDFWLFDOZD\WRVROYH .I # C /[ D G ,IWKHHQWULHVLQ C DQG G DUHQRQQHJDWLYH WKHQ VKRZVWKDWWKHHQWULHVLQ [ DUHQRQQHJDWLYH WRR

7KH(FRQRPLF,PSRUWDQFHRI(QWULHVLQ , ±& ± 7KHHQWULHVLQ .I # C /"1 DUHVLJQLÀFDQWEHFDXVHWKH\FDQEHXVHGWRSUHGLFWKRZWKH SURGXFWLRQ [ ZLOOKDYHWRFKDQJHZKHQWKHÀQDOGHPDQG G FKDQJHV ,QIDFW WKHHQWULHV LQFROXPQ j RI .I # C /"1 DUHWKH LQFUHDVHG DPRXQWVWKHYDULRXVVHFWRUVZLOOKDYHWR SURGXFHLQRUGHUWRVDWLVI\ DQLQFUHDVHRI XQLW LQWKHÀQDOGHPDQGIRURXWSXWIURP VHFWRU j 6HH([HUFLVH

/6.&3*$"- /05& ,QDQ\DSSOLHGSUREOHPQRWMXVWLQHFRQRPLFV DQHTXDWLRQ A[ D E FDQDOZD\V EHZULWWHQDV .I # C /[ D E ZLWK C D I # A ,IWKHV\VWHPLVODUJHDQG VSDUVH ZLWKPRVWO\]HURHQWULHV LWFDQKDSSHQWKDWWKHFROXPQVXPVRIWKHDEVROXWH YDOXHVLQ C DUHOHVVWKDQ ,QWKLVFDVH C m ! 0 ,I C m DSSURDFKHV]HUR TXLFNO\HQRXJK DQG ZLOOSURYLGHSUDFWLFDOIRUPXODVIRUVROYLQJ A[ D E DQGÀQGLQJ A"1

$)"15&3

0DWUL[$OJHEUD

13"$5*$& 130#-&. 6XSSRVHDQHFRQRP\KDVWZRVHFWRUV JRRGVDQGVHUYLFHV 2QHXQLWRIRXWSXWIURP JRRGVUHTXLUHVLQSXWVRI XQLWIURPJRRGVDQG XQLWIURPVHUYLFHV 2QHXQLWRI RXWSXWIURPVHUYLFHVUHTXLUHVLQSXWVRI XQLWIURPJRRGVDQG XQLWIURPVHUYLFHV 7KHUHLVDÀQDOGHPDQGRIXQLWVRIJRRGVDQGXQLWVRIVHUYLFHV 6HWXSWKH/HRQWLHI LQSXW²RXWSXWPRGHOIRUWKLVVLWXDWLRQ

&9&3$*4&4 Agriculture

Manufacturing

Services

Open Sector

([HUFLVHV ² UHIHU WR DQ HFRQRP\ WKDW LV GLYLGHG LQWR WKUHH VHFWRUV³PDQXIDFWXULQJ DJULFXOWXUH DQGVHUYLFHV )RUHDFKXQLW RIRXWSXW PDQXIDFWXULQJUHTXLUHV XQLWIURPRWKHUFRPSDQLHV LQWKDWVHFWRU XQLWIURPDJULFXOWXUH DQG XQLWIURPVHUYLFHV )RUHDFKXQLWRIRXWSXW DJULFXOWXUHXVHV XQLWRILWVRZQRXWSXW XQLWIURPPDQXIDFWXULQJ DQG XQLWIURPVHUYLFHV )RUHDFK XQLWRIRXWSXW WKHVHUYLFHVVHFWRUFRQVXPHV XQLWIURPVHUYLFHV XQLWIURPPDQXIDFWXULQJ EXWQRDJULFXOWXUDOSURGXFWV

'HWHUPLQHWKHSURGXFWLRQOHYHOVQHHGHGWRVDWLVI\DÀQDOGH PDQGRIXQLWVIRUPDQXIDFWXULQJ XQLWVIRUDJULFXOWXUH DQGXQLWVIRUVHUYLFHV

&RQVWUXFWWKHFRQVXPSWLRQPDWUL[IRUWKLVHFRQRP\ DQGGH WHUPLQHZKDWLQWHUPHGLDWHGHPDQGVDUHFUHDWHGLIDJULFXOWXUH SODQVWRSURGXFH XQLWV

8VH DQ LQYHUVH PDWUL[ WR GHWHUPLQH WKH SURGXFWLRQ OHYHO QHFHVVDU\WRVDWLVI\WKHÀQDOGHPDQG # $ # $ :2 :5 16 5HSHDW([HUFLVHZLWK C D DQG G D :6 :1 12

'HWHUPLQHWKHSURGXFWLRQOHYHOVQHHGHGWRVDWLVI\DÀQDO GHPDQGRIXQLWVIRUDJULFXOWXUH ZLWKQRÀQDOGHPDQGIRU WKHRWKHUVHFWRUV 'RQRWFRPSXWHDQLQYHUVHPDWUL[ 'HWHUPLQHWKHSURGXFWLRQOHYHOVQHHGHGWRVDWLVI\DÀQDO GHPDQGRIXQLWVIRUPDQXIDFWXULQJ ZLWKQRÀQDOGHPDQG IRUWKHRWKHUVHFWRUV 'RQRWFRPSXWHDQLQYHUVHPDWUL[

&RQVLGHUWKHSURGXFWLRQPRGHO [ D C [ C G IRUDQHFRQRP\ ZLWKWZRVHFWRUV ZKHUH # $ # $ :0 :5 50 C D ; GD :6 :2 30

/HW C DQG G EHDVLQ([HUFLVH D 'HWHUPLQHWKHSURGXFWLRQOHYHOQHFHVVDU\WRVDWLVI\DÀQDO GHPDQGIRUXQLWRIRXWSXWIURPVHFWRU E 8VHDQLQYHUVHPDWUL[WRGHWHUPLQHWKHSURGXFWLRQOHYHO # $ 51 QHFHVVDU\WRVDWLVI\DÀQDOGHPDQGRI 30

$ # $ # $ 51 50 1 D C WRH[SODLQKRZ 30 30 0 DQGZK\WKHDQVZHUVWRSDUWVD DQGE DQGWR([HUFLVH DUHUHODWHG

F 8VHWKHIDFWWKDW

#

/HW C EHDQ n ! n FRQVXPSWLRQPDWUL[ZKRVHFROXPQVXPV DUHOHVVWKDQ /HW [ EHWKHSURGXFWLRQYHFWRUWKDWVDWLVÀHV DÀQDOGHPDQG G DQGOHW #[ EHDSURGXFWLRQYHFWRUWKDW VDWLVÀHVDGLIIHUHQWÀQDOGHPDQG #G D 6KRZWKDWLIWKHÀQDOGHPDQGFKDQJHVIURP G WR G C #G WKHQWKHQHZSURGXFWLRQOHYHOPXVWEH [ C #[ 7KXV #[ JLYHVWKHDPRXQWVE\ZKLFKSURGXFWLRQPXVW FKDQJH LQ RUGHUWRDFFRPPRGDWHWKH FKDQJH #G LQGHPDQG E /HW #G EHWKHYHFWRULQ Rn ZLWKDVWKHÀUVWHQWU\DQG 0·VHOVHZKHUH ([SODLQZK\WKHFRUUHVSRQGLQJSURGXFWLRQ #[ LVWKHÀUVWFROXPQRI .I # C /!1 7KLVVKRZVWKDWWKH ÀUVWFROXPQRI .I # C /!1 JLYHVWKHDPRXQWVWKHYDULRXV VHFWRUVPXVWSURGXFHWRVDWLVI\DQLQFUHDVHRIXQLWLQWKH ÀQDOGHPDQGIRURXWSXWIURPVHFWRU 6ROYHWKH/HRQWLHISURGXFWLRQHTXDWLRQIRUDQHFRQRP\ZLWK WKUHHVHFWRUV JLYHQWKDW 2 3 2 3 :2 :2 :0 40 C D 4 :3 :1 :3 5 DQG G D 4 60 5 :1 :0 :2 80

7KHFRQVXPSWLRQPDWUL[ C IRUWKH86HFRQRP\LQ KDVWKHSURSHUW\WKDW HYHU\HQWU\ LQWKHPDWUL[ .I # C /!1 LV QRQ]HURDQGSRVLWLYH :KDWGRHVWKDWVD\DERXWWKHHIIHFW RIUDLVLQJWKHGHPDQGIRUWKHRXWSXWRIMXVWRQHVHFWRURIWKH HFRQRP\" 7KH/HRQWLHISURGXFWLRQHTXDWLRQ [ D C [ C G LVXVXDOO\ DFFRPSDQLHGE\DGXDO SULFHHTXDWLRQ T

SDC SCY

ZKHUH S LVD SULFHYHFWRU ZKRVHHQWULHVOLVWWKHSULFHSHUXQLW IRUHDFKVHFWRU·VRXWSXW DQG Y LVD YDOXHDGGHGYHFWRU ZKRVH HQWULHVOLVWWKHYDOXHDGGHGSHUXQLWRIRXWSXW 9DOXHDGGHG LQFOXGHVZDJHV SURÀW GHSUHFLDWLRQ HWF $QLPSRUWDQWIDFW LQHFRQRPLFVLVWKDWWKHJURVVGRPHVWLFSURGXFW*'3 FDQ EHH[SUHVVHGLQWZRZD\V

f JURVVGRPHVWLFSURGXFW g D ST G D YT [

9HULI\WKHVHFRQGHTXDOLW\ >+LQW &RPSXWH ST [ LQWZR ZD\V@ 1

:DVVLO\:/HRQWLHI ´7KH:RUOG(FRQRP\RIWKH

/HW C EH D FRQVXPSWLRQ PDWUL[ VXFK WKDW C m ! 0 DV m ! 1 DQG IRU m D 1; 2; : : : ; OHW Dm D I C C C " " " C C m )LQGDGLIIHUHQFHHTXDWLRQWKDWUHODWHV Dm DQG DmC1 DQGWKHUHE\REWDLQDQLWHUDWLYHSURFHGXUHIRUFRPSXWLQJIRU PXOD IRU .I # C /!1 >0@ 7KH FRQVXPSWLRQ PDWUL[ C EHORZ LV EDVHG RQ LQSXW²RXWSXWGDWDIRUWKH86HFRQRP\LQ ZLWKGDWD IRUVHFWRUVJURXSHGLQWRODUJHUVHFWRUV QRQPHWDO KRXVHKROGDQGSHUVRQDOSURGXFWV ÀQDOPHWDOSURGXFWV VXFK DV PRWRU YHKLFOHV EDVLF PHWDO SURGXFWV DQG PLQLQJ EDVLF QRQPHWDO SURGXFWV DQG DJULFXOWXUH HQHUJ\ VHUYLFHV DQG HQWHUWDLQPHQWDQGPLVFHOODQHRXV SURGXFWV )LQGWKHSURGXFWLRQOHYHOVQHHGHGWRVDWLVI\WKH ÀQDOGHPDQG G 8QLWVDUHLQPLOOLRQVRIGROODUV 2 3 :1588 :0064 :0025 :0304 :0014 :0083 :1594 6 :0057 :2645 :0436 :0099 :0083 :0201 :3413 7 6 7 6 :0264 :1506 :3557 :0139 :0142 :0070 :0236 7 6 7 6 :3299 :0565 :0495 :3636 :0204 :0483 :0649 7; 6 7 6 :0089 :0081 :0333 :0295 :3412 :0237 :0020 7 6 7 4 :1190 :0901 :0996 :1260 :1722 :2368 :3369 5 :0063 :0126 :0196 :0098 :0064 :0132 :0012 2

3 74;000 6 56;000 7 6 7 6 10;500 7 6 7 7 GD6 6 25;000 7 6 17;500 7 6 7 4 196;000 5 5;000

>0@ 7KH GHPDQG YHFWRU LQ ([HUFLVH LV UHDVRQDEOH IRU GDWD EXW/HRQWLHI·VGLVFXVVLRQRIWKHHFRQRP\LQWKH UHIHUHQFHFLWHGWKHUHXVHGDGHPDQGYHFWRUFORVHUWR GDWD G D .99640; 75548; 14444; 33501; 23527; 263985; 6526/

)LQGWKHSURGXFWLRQOHYHOVQHHGHGWRVDWLVI\WKLVGHPDQG >0@ 8VHHTXDWLRQ WRVROYHWKHSUREOHPLQ([HUFLVH 6HW [.0/ D G; DQGIRU k D 1; 2; : : : FRPSXWH [.k/ D G C C [.k!1/ +RZPDQ\VWHSVDUHQHHGHGWRREWDLQWKHDQVZHULQ([HU FLVH WRIRXUVLJQLÀFDQWÀJXUHV"

2 :DVVLO\:/HRQWLHI ´7KH6WUXFWXUHRIWKH86(FRQRP\µ 6FLHQWLÀF $PHULFDQ $SULO SS²

$)"15&3

0DWUL[$OJHEUD

40-65*0/ 50 13"$5*$& 130#-&. 7KHIROORZLQJGDWDDUHJLYHQ ,QSXWV1HHGHGSHU8QLWRI2XWSXW 3XUFKDVHGIURP

*RRGV

6HUYLFHV

([WHUQDO'HPDQG

*RRGV

6HUYLFHV

7KH/HRQWLHILQSXW²RXWSXWPRGHOLV [ D C [ C G ZKHUH # $ # $ :2 :4 20 C D ; GD :5 :3 30

"11-*$"5*0/4 50 $0.165&3 (3"1)*$4 &RPSXWHUJUDSKLFVDUHLPDJHVGLVSOD\HGRUDQLPDWHGRQDFRPSXWHUVFUHHQ $SSOLFD WLRQVRIFRPSXWHUJUDSKLFVDUHZLGHVSUHDGDQGJURZLQJUDSLGO\ )RULQVWDQFH FRPSXWHU DLGHGGHVLJQ&$' LVDQLQWHJUDOSDUWRIPDQ\HQJLQHHULQJSURFHVVHV VXFKDVWKH DLUFUDIWGHVLJQSURFHVVGHVFULEHGLQWKHFKDSWHULQWURGXFWLRQ 7KHHQWHUWDLQPHQWLQGXVWU\ KDVPDGHWKHPRVWVSHFWDFXODUXVHRIFRPSXWHUJUDSKLFV³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ÀOOLQFORVHGUHJLRQVERXQGHGE\WKHOLQHVDQGFXUYHV 2IWHQ FXUYHGOLQHV DUHDSSUR[LPDWHGE\VKRUWVWUDLJKWOLQHVHJPHQWV DQGDÀJXUHLVGHÀQHGPDWKHPDWLFDOO\ E\DOLVWRISRLQWV $PRQJWKHVLPSOHVW' JUDSKLFVV\PEROVDUHOHWWHUVXVHGIRUODEHOVRQWKHVFUHHQ 6RPHOHWWHUVDUHVWRUHGDVZLUHIUDPHREMHFWV RWKHUVWKDWKDYHFXUYHGSRUWLRQVDUHVWRUHG ZLWKDGGLWLRQDOPDWKHPDWLFDOIRUPXODVIRUWKHFXUYHV

&9".1-& 7KHFDSLWDOOHWWHU1 LQ)LJ LVGHWHUPLQHGE\HLJKWSRLQWV RU YHUWLFHV 6 5

8

3

x FRRUGLQDWH y FRRUGLQDWH

7

1 2 '*(63&

5HJXODU N

7KHFRRUGLQDWHVRIWKHSRLQWVFDQEHVWRUHGLQDGDWDPDWUL[ D

4

#

0 0

:5 0

:5 6:42

6 0

9HUWH[

6 8

5:5 8

5:5 1:58

$

0 8

DD

,QDGGLWLRQWR D LWLVQHFHVVDU\WRVSHFLI\ZKLFKYHUWLFHVDUHFRQQHFWHGE\OLQHV EXWZH RPLWWKLVGHWDLO 7KHPDLQUHDVRQJUDSKLFDOREMHFWVDUHGHVFULEHGE\FROOHFWLRQVRIVWUDLJKWOLQHVHJ PHQWVLVWKDWWKHVWDQGDUGWUDQVIRUPDWLRQVLQFRPSXWHUJUDSKLFVPDSOLQHVHJPHQWVRQWR RWKHUOLQHVHJPHQWV )RULQVWDQFH VHH([HUFLVH LQ6HFWLRQ 2QFHWKHYHUWLFHV

$SSOLFDWLRQVWR&RPSXWHU*UDSKLFV

WKDWGHVFULEHDQREMHFWKDYHEHHQWUDQVIRUPHG WKHLULPDJHVFDQEHFRQQHFWHGZLWKWKH DSSURSULDWHVWUDLJKWOLQHVWRSURGXFHWKHFRPSOHWHLPDJHRIWKHRULJLQDOREMHFW #

$ 1 :25 &9".1-& *LYHQ A D GHVFULEHWKHHIIHFWRIWKHVKHDUWUDQVIRUPD 0 1 WLRQ [ 7! A[ RQWKHOHWWHU1 LQ([DPSOH

40-65*0/ %\GHÀQLWLRQRIPDWUL[PXOWLSOLFDWLRQ WKHFROXPQVRIWKHSURGXFW AD FRQWDLQWKHLPDJHVRIWKHYHUWLFHVRIWKHOHWWHU1

8

6 5

# 0 AD D 0

3 7 1 2

:5 0

2:105 6:420

6 0

8 8

7:5 8

5:895 1:580

$

2 8

7KHWUDQVIRUPHGYHUWLFHVDUHSORWWHGLQ)LJ DORQJZLWKFRQQHFWLQJOLQHVHJPHQWVWKDW FRUUHVSRQGWRWKRVHLQWKHRULJLQDOÀJXUH

4

7KHLWDOLF N LQ)LJ ORRNVDELWWRRZLGH 7RFRPSHQVDWH VKULQNWKHZLGWKE\D VFDOHWUDQVIRUPDWLRQWKDWDIIHFWVWKH x FRRUGLQDWHVRIWKHSRLQWV

'*(63&

6ODQWHG N

&9".1-& &RPSXWHWKHPDWUL[RIWKHWUDQVIRUPDWLRQWKDWSHUIRUPVDVKHDUWUDQV IRUPDWLRQ DVLQ([DPSOH DQGWKHQVFDOHVDOO x FRRUGLQDWHVE\DIDFWRURI

40-65*0/ 7KHPDWUL[WKDWPXOWLSOLHVWKH x FRRUGLQDWHRIDSRLQWE\LV # $ :75 0 SD 0 1 '*(63&

&RPSRVLWHWUDQVIRUPDWLRQRI N

6RWKHPDWUL[RIWKHFRPSRVLWHWUDQVIRUPDWLRQLV # $# :75 0 1 SA D 0 1 0 # $ :75 :1875 D 0 1

:25 1

$

7KHUHVXOWRIWKLVFRPSRVLWHWUDQVIRUPDWLRQLVVKRZQLQ)LJ 7KHPDWKHPDWLFVRIFRPSXWHUJUDSKLFVLVLQWLPDWHO\FRQQHFWHGZLWKPDWUL[PXOWL SOLFDWLRQ 8QIRUWXQDWHO\ WUDQVODWLQJDQREMHFWRQDVFUHHQGRHVQRWFRUUHVSRQGGLUHFWO\ WRPDWUL[PXOWLSOLFDWLRQEHFDXVHWUDQVODWLRQLVQRWDOLQHDUWUDQVIRUPDWLRQ 7KHVWDQGDUG ZD\WRDYRLGWKLVGLIÀFXOW\LVWRLQWURGXFHZKDWDUHFDOOHG KRPRJHQHRXVFRRUGLQDWHV

+RPRJHQHRXV&RRUGLQDWHV x2 4 2

–4

–2

7UDQVODWLRQE\

# $ 4 3

2

4

x1

(DFKSRLQW .x; y/ LQ R2 FDQEHLGHQWLÀHGZLWKWKHSRLQW .x; y; 1/ RQWKHSODQHLQ R3 WKDWOLHVRQHXQLWDERYHWKH xy SODQH :HVD\WKDW .x; y/ KDV KRPRJHQHRXVFRRUGLQDWHV .x; y; 1/ )RULQVWDQFH WKHSRLQW .0; 0/ KDVKRPRJHQHRXVFRRUGLQDWHV .0; 0; 1/ +RPR JHQHRXVFRRUGLQDWHVIRUSRLQWVDUHQRWDGGHGRUPXOWLSOLHGE\VFDODUV EXWWKH\FDQEH WUDQVIRUPHGYLDPXOWLSOLFDWLRQE\ 3 ! 3 PDWULFHV

&9".1-& $ WUDQVODWLRQRIWKHIRUP .x; y/ 7! .x C h; y C k/ LVZULWWHQLQKR PRJHQHRXVFRRUGLQDWHVDV .x; y; 1/ 7! .x C h; y C k; 1/ 7KLVWUDQVIRUPDWLRQFDQEH FRPSXWHGYLDPDWUL[PXOWLSOLFDWLRQ 2 32 3 2 3 1 0 h x xCh 40 1 k 54 y 5 D 4 y C k 5 0 0 1 1 1

$)"15&3

0DWUL[$OJHEUD

&9".1-& $Q\OLQHDUWUDQVIRUPDWLRQRQ R2 LVUHSUHVHQWHGZLWKUHVSHFWWRKRPR # $ JHQHRXVFRRUGLQDWHVE\DSDUWLWLRQHGPDWUL[RIWKHIRUP PDWUL[ 7\SLFDOH[DPSOHVDUH 2 3 FRV ' # VLQ ' 0 4 VLQ ' FRV ' 05; 0 0 1 &RXQWHUFORFNZLVH URWDWLRQDERXWWKH RULJLQ DQJOH '

2

0 41 0

1 0 0

3 0 0 5; 1

5HÁHFWLRQ WKURXJK y D x

A 0

0 ZKHUH A LVD 2 ! 2 1

2

s 40 0

0 t 0

3 0 05 1

6FDOH x E\ s DQG y E\ t

&RPSRVLWH7UDQVIRUPDWLRQV 7KHPRYHPHQWRIDÀJXUHRQDFRPSXWHUVFUHHQRIWHQUHTXLUHVWZRRUPRUHEDVLFWUDQV IRUPDWLRQV 7KHFRPSRVLWLRQRIVXFKWUDQVIRUPDWLRQVFRUUHVSRQGVWRPDWUL[PXOWLSOLFD WLRQZKHQKRPRJHQHRXVFRRUGLQDWHVDUHXVHG

&9".1-& )LQGWKH 3 ! 3 PDWUL[WKDWFRUUHVSRQGVWRWKHFRPSRVLWHWUDQVIRUPD

WLRQRIDVFDOLQJE\ DURWDWLRQRI 90ı DERXWWKHRULJLQ DQGÀQDOO\DWUDQVODWLRQWKDW DGGV .#:5; 2/ WRHDFKSRLQWRIDÀJXUH Original Figure

After Scaling

After Rotating

After Translating

40-65*0/ ,I ' D $=2 WKHQ VLQ ' D 1 DQG FRV ' D 0 )URP([DPSOHVDQG ZH KDYH 2 3 2 32 3 x :3 0 0 x 4 y 5 ##6FDOH ####! 4 0 :3 0 54 y 5 1 0 0 1 1 2 32 32 3 0 #1 0 :3 0 0 x 5RWDWH ######! 4 1 0 0 54 0 :3 0 54 y 5 0 0 1 0 0 1 1 2 32 32 32 3 1 0 #:5 0 #1 0 :3 0 0 x 7UDQVODWH 0 0 54 0 :3 0 54 y 5 ######! 4 0 1 2 54 1 0 0 1 0 0 1 0 0 1 1 7KHPDWUL[IRUWKHFRPSRVLWHWUDQVIRUPDWLRQLV 2 32 32 3 1 0 #:5 0 #1 0 :3 0 0 40 1 2 54 1 0 0 54 0 :3 0 5 0 0 1 0 0 1 0 0 1 2 32 0 #1 #:5 :3 4 5 4 1 0 2 0 D 0 0 1 0

0 :3 0

3 2 3 0 0 #:3 #:5 0 5 D 4 :3 0 2 5 1 0 0 1

' &RPSXWHU*UDSKLFV 6RPHRIWKHQHZHVWDQGPRVWH[FLWLQJZRUNLQFRPSXWHUJUDSKLFVLVFRQQHFWHGZLWK PROHFXODUPRGHOLQJ :LWK' WKUHHGLPHQVLRQDO JUDSKLFV DELRORJLVWFDQH[DPLQHD VLPXODWHGSURWHLQPROHFXOHDQGVHDUFKIRUDFWLYHVLWHVWKDWPLJKWDFFHSWDGUXJPROHFXOH 7KHELRORJLVWFDQURWDWHDQGWUDQVODWHDQH[SHULPHQWDOGUXJDQGDWWHPSWWRDWWDFKLWWRWKH SURWHLQ 7KLVDELOLW\WR YLVXDOL]H SRWHQWLDOFKHPLFDOUHDFWLRQVLVYLWDOWRPRGHUQGUXJDQG FDQFHUUHVHDUFK ,QIDFW DGYDQFHVLQGUXJGHVLJQGHSHQGWRVRPHH[WHQWXSRQSURJUHVV


LQWKHDELOLW\RIFRPSXWHUJUDSKLFVWRFRQVWUXFWUHDOLVWLFVLPXODWLRQVRIPROHFXOHVDQG WKHLULQWHUDFWLRQVñ &XUUHQWUHVHDUFKLQPROHFXODUPRGHOLQJLVIRFXVHGRQ YLUWXDOUHDOLW\ DQHQYLURQ PHQWLQZKLFKDUHVHDUFKHUFDQVHHDQG IHHO WKHGUXJPROHFXOHVOLGHLQWRWKHSURWHLQ ,Q )LJ VXFKWDFWLOHIHHGEDFNLVSURYLGHGE\DIRUFHGLVSOD\LQJUHPRWHPDQLSXODWRU

'*(63& 0ROHFXODUPRGHOLQJLQYLUWXDOUHDOLW\

&RPSXWHU6FLHQFH'HSDUWPHQW 8QLYHUVLW\RI 1RUWK&DUROLQDDW&KDSHO+LOO 3KRWRE\%R 6WUDLQ

$QRWKHUGHVLJQIRUYLUWXDOUHDOLW\LQYROYHVDKHOPHWDQGJORYHWKDWGHWHFWKHDG KDQG DQG ÀQJHUPRYHPHQWV 7KHKHOPHWFRQWDLQVWZRWLQ\FRPSXWHUVFUHHQV RQHIRUHDFKH\H 0DNLQJWKLVYLUWXDOHQYLURQPHQWPRUHUHDOLVWLFLVDFKDOOHQJHWRHQJLQHHUV VFLHQWLVWV DQGPDWKHPDWLFLDQV 7KHPDWKHPDWLFVZHH[DPLQHKHUHEDUHO\RSHQVWKHGRRUWRWKLV LQWHUHVWLQJÀHOGRIUHVHDUFK

+RPRJHQHRXV ' &RRUGLQDWHV %\DQDORJ\ZLWKWKH' FDVH ZHVD\WKDW .x; y; ´; 1/ DUHKRPRJHQHRXVFRRUGLQDWHVIRU WKHSRLQW .x; y; ´/ LQ R3 ,QJHQHUDO .X; Y; Z; H / DUH KRPRJHQHRXVFRRUGLQDWHV IRU .x; y; ´/ LI H ¤ 0 DQG Y X Z y D ; DQG ´ D xD ; H H H (DFKQRQ]HURVFDODUPXOWLSOHRI .x; y; ´; 1/ JLYHVDVHWRIKRPRJHQHRXVFRRUGLQDWHVIRU .x; y; ´/ )RULQVWDQFH ERWK .10; #6; 14; 2/ DQG .#15; 9; #21; #3/ DUHKRPRJHQHRXV FRRUGLQDWHVIRU .5; #3; 7/ 7KHQH[WH[DPSOHLOOXVWUDWHVWKHWUDQVIRUPDWLRQVXVHGLQPROHFXODUPRGHOLQJWR PRYHDGUXJLQWRDSURWHLQPROHFXOH

&9".1-& *LYH 4 ! 4 PDWULFHVIRUWKHIROORZLQJWUDQVIRUPDWLRQV

D 5RWDWLRQDERXWWKH y D[LVWKURXJKDQDQJOHRI 30ı %\FRQYHQWLRQ DSRVLWLYHDQJOH LVWKHFRXQWHUFORFNZLVHGLUHFWLRQZKHQORRNLQJWRZDUGWKHRULJLQIURPWKHSRVLWLYH KDOIRIWKHD[LVRIURWDWLRQ³LQWKLVFDVH WKH y D[LV ñ 5REHUW3RRO ´&RPSXWLQJLQ6FLHQFHµ 6FLHQFH $SULO S

$)"15&3

0DWUL[$OJHEUD

E 7UDQVODWLRQE\WKHYHFWRU S D .#6; 4; 5/

40-65*0/ z e3

e1 x

'*(63&

e2 y

D )LUVW FRQVWUXFWWKH 3 ! 3 PDWUL[IRUWKHURWDWLRQ 7KHYHFWRU p H1 URWDWHVGRZQWRZDUG WKHQHJDWLYH ´D[LV VWRSSLQJDW .FRV 30ı ; 0; # VLQ 30ı / D . 3=2; 0; #:5/ 7KHYHFWRU H2 RQWKH y D[LVGRHVQRWPRYH EXW H3 RQWKH ´D[LVURWDWHVGRZQWRZDUGWKHSRVLWLYH p x D[LV VWRSSLQJDW .VLQ 30ı ; 0; FRV 30ı / D .:5; 0; 3=2/ 6HH)LJ )URP6HFWLRQ WKHVWDQGDUGPDWUL[IRUWKLVURWDWLRQLV 2p 3 3=2 0 :5 4 0 5 1 p0 3=2 #:5 0 6RWKHURWDWLRQPDWUL[IRUKRPRJHQHRXVFRRUGLQDWHVLV 2p 3=2 0 :5 6 0 1 p0 AD6 4 #:5 3=2 0 0 0 0

3 0 07 7 05 1

E :HZDQW .x; y; ´; 1/ WRPDSWR .x # 6; y C 4; ´ C 5; 1/ 7KHPDWUL[WKDWGRHVWKLVLV 2 3 1 0 0 #6 60 1 0 47 6 7 40 0 1 55 0 0 0 1

3HUVSHFWLYH3URMHFWLRQV $ WKUHHGLPHQVLRQDOREMHFWLVUHSUHVHQWHGRQWKHWZRGLPHQVLRQDOFRPSXWHUVFUHHQE\ SURMHFWLQJWKHREMHFWRQWRD YLHZLQJSODQH :HLJQRUHRWKHULPSRUWDQWVWHSV VXFKDV VHOHFWLQJWKHSRUWLRQRIWKHYLHZLQJSODQHWRGLVSOD\RQWKHVFUHHQ )RUVLPSOLFLW\ OHW WKH xy SODQHUHSUHVHQWWKHFRPSXWHUVFUHHQ DQGLPDJLQHWKDWWKHH\HRIDYLHZHULV DORQJWKHSRVLWLYH ´D[LV DWDSRLQW .0; 0; d / $ SHUVSHFWLYHSURMHFWLRQ PDSVHDFKSRLQW .x; y; ´/ RQWRDQLPDJHSRLQW .x % ; y % ; 0/ VRWKDWWKHWZRSRLQWVDQGWKHH\HSRVLWLRQ FDOOHGWKH FHQWHURISURMHFWLRQ DUHRQDOLQH 6HH)LJ D y (x*, y*, 0) x* x

0 (x, y, z)

0 z

x

d–z (0, 0, d ) z (a)

(b) "

'*(63& 3HUVSHFWLYHSURMHFWLRQRI .x; y; ´/ RQWR .x ; y " ; 0/


7KHWULDQJOHLQWKH x´SODQHLQ)LJ D LVUHGUDZQLQSDUWE VKRZLQJWKHOHQJWKV RIOLQHVHJPHQWV 6LPLODUWULDQJOHVVKRZWKDW

x x% D d d #´

x dx D d #´ 1 # ´=d

DQG

x% D

y% D

y 1 # ´=d

6LPLODUO\

8VLQJKRPRJHQHRXVFRRUGLQDWHV ZHFDQUHSUHVHQWWKHSHUVSHFWLYHSURMHFWLRQE\DPD ' ( y x ; ; 0; 1 6FDOLQJWKHVH WUL[ VD\ P :HZDQW .x; y; ´; 1/ WRPDSLQWR 1 # ´=d 1 # ´=d FRRUGLQDWHVE\ 1 # ´=d ZHFDQDOVRXVH .x; y; 0; 1 # ´=d / DVKRPRJHQHRXVFRRUGLQDWHV IRUWKHLPDJH 1RZLWLVHDV\WRGLVSOD\ P ,QIDFW 2 3 2 1 x 6y 7 60 7 6 P6 4 ´5 D 40 1 0

0 0 1 0 0 0 0 #1=d

32 3 2 0 x x 6y 7 6 07 y 76 7 D 6 054 ´5 4 0 1 1 1 # ´=d

3 7 7 5

&9".1-& /HW S EHWKHER[ZLWKYHUWLFHV .3; 1; 5/ .5; 1; 5/ .5; 0; 5/ .3; 0; 5/ .3; 1; 4/ .5; 1; 4/ .5; 0; 4/ DQG .3; 0; 4/ )LQGWKHLPDJHRI S XQGHUWKHSHUVSHFWLYHSUR MHFWLRQZLWKFHQWHURISURMHFWLRQDW .0; 0; 10/ 40-65*0/ /HW P EHWKHSURMHFWLRQPDWUL[ DQGOHW D EHWKHGDWDPDWUL[IRU S XVLQJ KRPRJHQHRXVFRRUGLQDWHV 7KHGDWDPDWUL[IRUWKHLPDJHRI S LV

2

1 60 PD D 6 40 0 2 3 6 1 D6 4 0 :5

0 0 1 0 0 0 0 #1=10

5 1 0 :5

5 0 0 :5

3 2

3 0 0 :5

0 3 61 07 7 6 05 45 1 1 3 1 0 :6

5 1 0 :6

9HUWH[

5 1 5 1

5 0 5 1

3 0 5 1

5 1 4 1

5 0 4 1

5 0 0 :6

3 3 07 7 05 :6

3 1 4 1

3 3 07 7 45 1

7RREWDLQ R3 FRRUGLQDWHV XVHHTXDWLRQ EHIRUH([DPSOH DQGGLYLGHWKHWRSWKUHH HQWULHVLQHDFKFROXPQE\WKHFRUUHVSRQGLQJHQWU\LQWKHIRXUWKURZ 2 6 42 0

S XQGHUWKHSHUVSHFWLYH WUDQVIRUPDWLRQ

WEB

9HUWH[

10 2 0

10 0 0

6 0 0

5 1:7 0

8:3 1:7 0

8:3 0 0

3 5 05 0

7KLVWH[W·VZHEVLWHKDVVRPHLQWHUHVWLQJDSSOLFDWLRQVRIFRPSXWHUJUDSKLFV LQFOXG LQJDIXUWKHUGLVFXVVLRQRISHUVSHFWLYHSURMHFWLRQV 2QHRIWKHFRPSXWHUSURMHFWVRQWKH ZHEVLWHLQYROYHVVLPSOHDQLPDWLRQ

$)"15&3

0DWUL[$OJHEUD

/6.&3*$"- /05& &RQWLQXRXVPRYHPHQWRIJUDSKLFDO' REMHFWVUHTXLUHVLQWHQVLYHFRPSXWDWLRQ ZLWK 4 ! 4 PDWULFHV SDUWLFXODUO\ ZKHQ WKH VXUIDFHV DUH UHQGHUHG WR DSSHDU UHDOLVWLF ZLWKWH[WXUHDQGDSSURSULDWHOLJKWLQJ +LJKHQGFRPSXWHUJUDSKLFV ERDUGV KDYH 4 ! 4 PDWUL[RSHUDWLRQVDQGJUDSKLFVDOJRULWKPVHPEHGGHGLQWKHLU PLFURFKLSVDQGFLUFXLWU\ 6XFKERDUGVFDQSHUIRUPWKHELOOLRQVRIPDWUL[PXOWLSOL FDWLRQVSHUVHFRQGQHHGHGIRUUHDOLVWLFFRORUDQLPDWLRQLQ' JDPLQJSURJUDPV

)XUWKHU5HDGLQJ -DPHV')ROH\ $QGULHVYDQ'DP 6WHYHQ.)HLQHU DQG-RKQ)+XJKHV &RPSXWHU *UDSKLFV 3ULQFLSOHVDQG3UDFWLFH UGHG %RVWRQ 0$$GGLVRQ:HVOH\ &KDSWHUVDQG

13"$5*$& 130#-&. 5RWDWLRQRIDÀJXUHDERXWDSRLQW S LQ R2 LVDFFRPSOLVKHGE\ÀUVWWUDQVODWLQJWKHÀJXUH E\ #S URWDWLQJDERXWWKHRULJLQ DQGWKHQWUDQVODWLQJEDFNE\ S 6HH)LJ &RQVWUXFW WKH 3 ! 3 PDWUL[WKDWURWDWHVSRLQWV #30ı DERXWWKHSRLQW .#2; 6/ XVLQJKRPRJHQHRXV FRRUGLQDWHV x2

x2

p

x2

p x1

(a) Original figure.

x2

p x1

(b) Translated to origin by – p.

p x1

(c) Rotated about the origin.

x1 (d) Translated back by p.

'*(63& 5RWDWLRQRIÀJXUHDERXWSRLQW S

&9&3$*4&4 :KDW 3 ! 3 PDWUL[ZLOOKDYHWKHVDPHHIIHFWRQKRPRJHQHRXV FRRUGLQDWHVIRU R2 WKDWWKHVKHDUPDWUL[ A KDVLQ([DPSOH"

5HÁHFWSRLQWVWKURXJKWKH x D[LV DQGWKHQURWDWH 45ı DERXW WKHRULJLQ

8VHPDWUL[PXOWLSOLFDWLRQWRÀQGWKHLPDJHRIWKHWULDQJOH # $ 4 2 5 ZLWKGDWDPDWUL[ D D XQGHUWKHWUDQVIRUPD 0 2 3 WLRQWKDWUHÁHFWVSRLQWVWKURXJKWKH y D[LV 6NHWFKERWKWKH RULJLQDOWULDQJOHDQGLWVLPDJH

5RWDWHSRLQWV 45ı DERXWWKHRULJLQ WKHQUHÁHFWWKURXJKWKH x D[LV

,Q([HUFLVHV² ÀQGWKH 3 ! 3 PDWULFHVWKDWSURGXFHWKHGH VFULEHGFRPSRVLWH' WUDQVIRUPDWLRQV XVLQJKRPRJHQHRXVFRRU GLQDWHV 7UDQVODWHE\ .2; 1/ DQGWKHQURWDWH 90ı DERXWWKHRULJLQ 7UDQVODWHE\ .#1; 4/ DQGWKHQVFDOHWKH x FRRUGLQDWHE\ DQGWKH y FRRUGLQDWHE\

5RWDWHSRLQWVWKURXJK 60ı DERXWWKHSRLQW .6; 8/ 5RWDWHSRLQWVWKURXJK 45ı DERXWWKHSRLQW .3; 7/ $ 2 ! 100 GDWDPDWUL[ D FRQWDLQVWKHFRRUGLQDWHVRI SRLQWV &RPSXWHWKHQXPEHURIPXOWLSOLFDWLRQVUHTXLUHGWR WUDQVIRUPWKHVHSRLQWVXVLQJWZRDUELWUDU\ 2 ! 2 PDWULFHV A DQG B &RQVLGHUWKHWZRSRVVLELOLWLHV A.BD/ DQG .AB/D 'LVFXVVWKHLPSOLFDWLRQVRI\RXUUHVXOWVIRUFRPSXWHUJUDSK LFVFDOFXODWLRQV

ò 6HH-DQ2]HU ´+LJK3HUIRUPDQFH*UDSKLFV%RDUGVµ 3& 0DJD]LQH 6HSWHPEHU SS ² $OVR ´7KH8OWLPDWH8SJUDGH*XLGH 0RYLQJ2Q8Sµ 3& 0DJD]LQH -DQXDU\ SS ²


&RQVLGHUWKHIROORZLQJJHRPHWULF' WUDQVIRUPDWLRQV D D GLODWLRQLQZKLFK x FRRUGLQDWHVDQG y FRRUGLQDWHVDUHVFDOHG E\WKHVDPHIDFWRU R DURWDWLRQ DQG T DWUDQVODWLRQ 'RHV D FRPPXWH ZLWK R" 7KDWLV LV D.R.[// D R.D.[// IRUDOO [ LQ R2 " 'RHV D FRPPXWHZLWK T " 'RHV R FRPPXWHZLWK T" $ URWDWLRQRQDFRPSXWHUVFUHHQLVVRPHWLPHVLPSOHPHQWHG DVWKHSURGXFWRIWZRVKHDUDQGVFDOHWUDQVIRUPDWLRQV ZKLFK FDQVSHHGXSFDOFXODWLRQVWKDWGHWHUPLQHKRZDJUDSKLFLPDJH DFWXDOO\DSSHDUVLQWHUPVRIVFUHHQSL[HOV 7KHVFUHHQFRQ VLVWVRIURZVDQGFROXPQVRIVPDOOGRWV FDOOHG SL[HOV 7KH ÀUVWWUDQVIRUPDWLRQ A1 VKHDUVYHUWLFDOO\DQGWKHQFRPSUHVVHV HDFKFROXPQRISL[HOV WKHVHFRQGWUDQVIRUPDWLRQ A2 VKHDUV KRUL]RQWDOO\DQGWKHQVWUHWFKHVHDFKURZRISL[HOV /HW 2 3 1 0 0 0 5 A1 D 4 VLQ ' FRV ' 0 0 1 2 3 VHF ' # WDQ ' 0 1 0 5 A2 D 4 0 0 0 1 DQGVKRZWKDWWKHFRPSRVLWLRQRIWKHWZRWUDQVIRUPDWLRQVLV DURWDWLRQLQ R2

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

6KRZWKDWWKHWUDQVIRUPDWLRQLQ([HUFLVH LVHTXLYDOHQWWR DURWDWLRQ DERXWWKHRULJLQIROORZHGE\DWUDQVODWLRQE\ S )LQG S :KDW YHFWRU LQ . 12 ; # 14 ; # 18 ; 241 /"

R3

KDV KRPRJHQHRXV FRRUGLQDWHV

$UH .1; #2; #3; 4/ DQG .10; #20; #30; 40/ KRPRJHQHRXVFR RUGLQDWHVIRUWKHVDPHSRLQWLQ R3 " :K\RUZK\QRW"

*LYHWKH 4 ! 4 PDWUL[WKDWURWDWHVSRLQWVLQ R3 DERXWWKH x D[LVWKURXJKDQDQJOHRI 60ı 6HHWKHÀJXUH z e3

e1

e2

x

y

*LYHWKH 4 ! 4 PDWUL[WKDWURWDWHVSRLQWVLQ R3 DERXWWKH ´D[LV WKURXJK DQ DQJOH RI #30ı DQG WKHQ WUDQVODWHV E\ S D .5; #2; 1/ /HW S EHWKHWULDQJOHZLWKYHUWLFHV .4:2; 1:2; 4/ .6; 4; 2/ DQG .2; 2; 6/ )LQGWKHLPDJHRI S XQGHUWKHSHUVSHFWLYHSURMHFWLRQ ZLWKFHQWHURISURMHFWLRQDW .0; 0; 10/ /HW S EHWKHWULDQJOHZLWKYHUWLFHV .7; 3; #5/ .12; 8; 2/ DQG .1; 2; 1/ )LQGWKHLPDJHRI S XQGHUWKHSHUVSHFWLYHSURMHFWLRQ ZLWKFHQWHURISURMHFWLRQDW .0; 0; 10/ ([HUFLVHVDQGFRQFHUQWKHZD\LQZKLFKFRORULVVSHFLÀHG IRUGLVSOD\LQFRPSXWHUJUDSKLFV $ FRORURQDFRPSXWHUVFUHHQ LVHQFRGHGE\WKUHHQXPEHUV .R; G; B/ WKDWOLVWWKHDPRXQWVRI HQHUJ\DQHOHFWURQJXQPXVWWUDQVPLWWRUHG JUHHQ DQGEOXH SKRVSKRUGRWVRQWKHFRPSXWHUVFUHHQ $ IRXUWKQXPEHUVSHFLÀHV WKHOXPLQDQFHRULQWHQVLW\RIWKHFRORU >0@ 7KHDFWXDOFRORUDYLHZHUVHHVRQDVFUHHQLVLQÁXHQFHG E\WKHVSHFLÀFW\SHDQGDPRXQWRISKRVSKRUVRQWKHVFUHHQ 6RHDFKFRPSXWHUVFUHHQPDQXIDFWXUHUPXVWFRQYHUWEHWZHHQ WKH .R; G; B/ GDWD DQG DQ LQWHUQDWLRQDO &,( VWDQGDUG IRU FRORU ZKLFKXVHVWKUHHSULPDU\FRORUV FDOOHG X Y DQG Z $ W\SLFDOFRQYHUVLRQIRUVKRUWSHUVLVWHQFHSKRVSKRUVLV 2 32 3 2 3 :61 :29 :150 R X 4 :35 :59 :063 54 G 5 D 4 Y 5 :04 :12 :787 B Z $ FRPSXWHUSURJUDPZLOOVHQGDVWUHDPRIFRORULQIRUPDWLRQ WRWKHVFUHHQ XVLQJVWDQGDUG&,( GDWD .X; Y; Z/ )LQGWKH HTXDWLRQWKDWFRQYHUWVWKHVHGDWDWRWKH .R; G; B/ GDWDQHHGHG IRUWKHVFUHHQ·VHOHFWURQJXQ

>0@ 7KHVLJQDOEURDGFDVWE\FRPPHUFLDOWHOHYLVLRQGHVFULEHV HDFKFRORUE\DYHFWRU .Y; I; Q/ ,IWKHVFUHHQLVEODFNDQG ZKLWH RQO\WKH Y FRRUGLQDWHLVXVHG 7KLVJLYHVDEHWWHU PRQRFKURPHSLFWXUHWKDQXVLQJ&,( GDWDIRUFRORUV 7KH FRUUHVSRQGHQFHEHWZHHQ <,4 DQGD´VWDQGDUGµ 5*% FRORULV JLYHQE\ 2

3 2 32 3 Y :299 :587 :114 R 4 I 5 D 4 :596 #:275 #:321 54 G 5 Q :212 #:528 :311 B

$)"15&3

0DWUL[$OJHEUD

$ VFUHHQPDQXIDFWXUHUZRXOGFKDQJHWKHPDWUL[HQWULHVWR ZRUNIRULWV 5*% VFUHHQV )LQGWKHHTXDWLRQWKDWFRQYHUWV

WKH <,4 GDWDWUDQVPLWWHGE\WKHWHOHYLVLRQVWDWLRQWRWKH 5*% GDWDQHHGHGIRUWKHWHOHYLVLRQVFUHHQ

40-65*0/ 50 13"$5*$& 130#-&. $VVHPEOH WKHp PDWULFHV ULJKWWROHIW IRU WKH WKUHH RSHUDWLRQV FRV.#30ı / D 3=2 DQG VLQ.#30ı / D #:5 ZHKDYH 5RWDWHDURXQG 7UDQVODWH WKHRULJLQ p EDFNE\ 2 32 p

1 40 0

8VLQJ S D .#2; 6/

7UDQVODWH 32 E\ #p 3

0 #2 3=2 0 1 0 2 p1=2 1 6 54 #1=2 3=2 0 54 0 1 #6 5 0 1 0 0 1 0 0 1 p 2p 3 3=2 p3 # 5 p1=2 D 4 #1=2 3=2 #3 3 C 5 5 0 0 1

46#41"$&4 0' Rn 7KLVVHFWLRQIRFXVHVRQLPSRUWDQWVHWVRIYHFWRUVLQ Rn FDOOHG VXEVSDFHV 2IWHQVXE VSDFHVDULVHLQFRQQHFWLRQZLWKVRPHPDWUL[ A; DQGWKH\SURYLGHXVHIXOLQIRUPDWLRQ DERXWWKHHTXDWLRQ A[ D E 7KHFRQFHSWVDQGWHUPLQRORJ\LQWKLVVHFWLRQZLOOEHXVHG UHSHDWHGO\WKURXJKRXWWKHUHVWRIWKHERRNñ

%&'*/*5*0/

D 7KH]HURYHFWRULVLQ H E )RUHDFK X DQG Y LQ H WKHVXP X C Y LVLQ H F )RUHDFK X LQ H DQGHDFKVFDODU c WKHYHFWRU c X LVLQ H

x3

,QZRUGV DVXEVSDFHLV FORVHG XQGHUDGGLWLRQDQGVFDODUPXOWLSOLFDWLRQ $V\RXZLOO VHHLQWKHQH[WIHZH[DPSOHV PRVWVHWVRIYHFWRUVGLVFXVVHGLQ&KDSWHUDUHVXEVSDFHV )RULQVWDQFH DSODQHWKURXJKWKHRULJLQLVWKHVWDQGDUGZD\WRYLVXDOL]HWKHVXEVSDFHLQ ([DPSOH 6HH)LJ

v1 v2

x1

$ VXEVSDFH RI Rn LVDQ\VHW H LQ Rn WKDWKDVWKUHHSURSHUWLHV

0

x2

'*(63&

&9".1-& ,I Y1 DQG Y2 DUHLQ Rn DQG H D 6SDQ fY1 ; Y2 g WKHQ H LVDVXEVSDFH

RI Rn 7RYHULI\WKLVVWDWHPHQW QRWHWKDWWKH]HURYHFWRULVLQ H EHFDXVH 0Y1 C 0Y2 LV DOLQHDUFRPELQDWLRQRI Y1 DQG Y2 / 1RZWDNHWZRDUELWUDU\YHFWRUVLQ H VD\

6SDQ fY1 ; Y2 g DVDSODQHWKURXJK WKHRULJLQ

7KHQ

X D s1 Y1 C s2 Y2

DQG

Y D t1 Y1 C t2 Y2

X C Y D .s1 C t1 /Y1 C .s2 C t2 /Y2

ZKLFKVKRZVWKDW X C Y LVDOLQHDUFRPELQDWLRQRI Y1 DQG Y2 DQGKHQFHLVLQ H $OVR IRU DQ\VFDODU c WKHYHFWRU c X LVLQ H EHFDXVH c X D c.s1 Y1 C s2 Y2 / D .cs1 /Y1 C .cs2 /Y2 ñ 6HFWLRQVDQGDUHLQFOXGHGKHUHWRSHUPLWUHDGHUVWRSRVWSRQHWKHVWXG\RIPRVWRUDOORIWKHQH[WWZR FKDSWHUVDQGWRVNLSGLUHFWO\WR&KDSWHU LIVRGHVLUHG 2PLW WKHVHWZRVHFWLRQVLI\RXSODQWRZRUNWKURXJK &KDSWHU EHIRUHEHJLQQLQJ&KDSWHU

6XEVSDFHVRI Rn

,I Y1 LVQRW]HURDQGLI Y2 LVDPXOWLSOHRI Y1 WKHQ Y1 DQG Y2 VLPSO\VSDQD OLQH WKURXJKWKHRULJLQ 6RDOLQHWKURXJKWKHRULJLQLVDQRWKHUH[DPSOHRIDVXEVSDFH

&9".1-& $ OLQH L QRW WKURXJKWKHRULJLQLV QRW DVXEVSDFH EHFDXVHLWGRHVQRW

x2

FRQWDLQWKHRULJLQ DVUHTXLUHG $OVR )LJ VKRZVWKDW L LVQRWFORVHGXQGHUDGGLWLRQ RUVFDODUPXOWLSOLFDWLRQ v1

v2 x1

} , v2

n{v 1

Spa

u!v

u

2w w

v L u ! v is not on L

Y1 ¤ Y2 D k Y1

L 2w is not on L

'*(63&

&9".1-& )RU Y1 ; : : : ; Yp LQ Rn WKHVHWRIDOOOLQHDUFRPELQDWLRQVRI Y1 ; : : : ; Yp LVDVXEVSDFHRI Rn 7KHYHULÀFDWLRQRIWKLVVWDWHPHQWLVVLPLODUWRWKHDUJXPHQWJLYHQ LQ([DPSOH :HVKDOOQRZUHIHUWR 6SDQ fY1 ; : : : ; Yp g DV WKHVXEVSDFHVSDQQHG RU JHQHUDWHG E\ Y1 ; : : : ; Yp 1RWHWKDW Rn LVDVXEVSDFHRILWVHOIEHFDXVHLWKDVWKHWKUHHSURSHUWLHVUHTXLUHGIRUD VXEVSDFH $QRWKHUVSHFLDOVXEVSDFHLVWKHVHWFRQVLVWLQJRIRQO\WKH]HURYHFWRULQ Rn 7KLVVHW FDOOHGWKH ]HURVXEVSDFH DOVRVDWLVÀHVWKHFRQGLWLRQVIRUDVXEVSDFH

&ROXPQ6SDFHDQG1XOO6SDFHRID0DWUL[ 6XEVSDFHVRI Rn XVXDOO\RFFXULQDSSOLFDWLRQVDQGWKHRU\LQRQHRIWZRZD\V ,QERWK FDVHV WKHVXEVSDFHFDQEHUHODWHGWRDPDWUL[

%&'*/*5*0/

x3

x2

Col A

0 b

x1

7KH FROXPQVSDFH RIDPDWUL[ A LVWKHVHW &RO A RIDOOOLQHDUFRPELQDWLRQVRIWKH FROXPQVRI A ,I A D Œ D1 " " " Dn ! ZLWK WKH FROXPQV LQ Rm WKHQ &RO A LV WKH VDPH DV 6SDQ fD1 ; : : : ; Dn g ([DPSOH VKRZVWKDWWKH FROXPQVSDFHRIDQ m ! n PDWUL[LVD VXEVSDFHRI Rm 1RWHWKDW &RO A HTXDOV Rm RQO\ZKHQWKHFROXPQVRI A VSDQ Rm 2WKHUZLVH &RO A LVRQO\SDUWRI Rm 2 3 2 3 1 #3 #4 3 &9".1-& /HW A D 4 #4 6 #2 5 DQG E D 4 3 5 'HWHUPLQHZKHWKHU E LV #3 7 6 #4 LQWKHFROXPQVSDFHRI A 40-65*0/ 7KHYHFWRU E LVDOLQHDUFRPELQDWLRQRIWKHFROXPQVRI A LIDQGRQO\LI E FDQEHZULWWHQDV A[ IRUVRPH [ WKDWLV LIDQGRQO\LIWKHHTXDWLRQ A[ D E KDVD VROXWLRQ 5RZUHGXFLQJWKHDXJPHQWHGPDWUL[ Œ A E ! 2 3 2 3 2 3 1 #3 #4 3 1 #3 #4 3 1 #3 #4 3 4 #4 6 #2 3 5 % 4 0 #6 #18 15 5 % 4 0 #6 #18 15 5 #3 7 6 #4 0 #2 #6 5 0 0 0 0 ZHFRQFOXGHWKDW A[ D E LVFRQVLVWHQWDQG E LVLQ &RO A

$)"15&3

0DWUL[$OJHEUD

7KHVROXWLRQRI([DPSOH VKRZVWKDWZKHQDV\VWHPRIOLQHDUHTXDWLRQVLVZULWWHQ LQWKHIRUP A[ D E WKHFROXPQVSDFHRI A LVWKHVHWRIDOO E IRUZKLFKWKHV\VWHPKDV DVROXWLRQ

%&'*/*5*0/

7KH QXOOVSDFH RIDPDWUL[ A LVWKHVHW 1XO A RIDOOVROXWLRQVRIWKHKRPRJHQHRXV HTXDWLRQ A[ D :KHQ A KDV n FROXPQV WKHVROXWLRQVRI A[ D EHORQJWR Rn DQGWKHQXOOVSDFH RI A LVDVXEVHWRI Rn ,QIDFW 1XO A KDVWKHSURSHUWLHVRID VXEVSDFH RI Rn

5)&03&.

7KHQXOOVSDFHRIDQ m ! n PDWUL[ A LVDVXEVSDFHRI Rn (TXLYDOHQWO\ WKH VHWRIDOOVROXWLRQVRIDV\VWHP A[ D RI m KRPRJHQHRXVOLQHDUHTXDWLRQVLQ n XQNQRZQVLVDVXEVSDFHRI Rn

1300' 7KH]HURYHFWRULVLQ 1XO A EHFDXVH A D 7RVKRZWKDW 1XO A VDWLVÀHVWKH RWKHUWZRSURSHUWLHVUHTXLUHGIRUDVXEVSDFH WDNHDQ\ X DQG Y LQ 1XO A 7KDWLV VXSSRVH AX D DQG AY D 7KHQ E\DSURSHUW\RIPDWUL[PXOWLSOLFDWLRQ A.X C Y/ D AX C AY D C D

7KXV X C Y VDWLVÀHV A[ D DQGVR X C Y LVLQ 1XO A $OVR IRUDQ\VFDODU c; A.c X/ D c.AX/ D c./ D ZKLFKVKRZVWKDW c X LVLQ 1XO A 7RWHVWZKHWKHUDJLYHQYHFWRU Y LVLQ 1XO A MXVWFRPSXWH AY WRVHHZKHWKHU AY LV WKH]HURYHFWRU %HFDXVH 1XO A LVGHVFULEHGE\DFRQGLWLRQWKDWPXVWEHFKHFNHGIRUHDFK YHFWRU ZHVD\WKDWWKHQXOOVSDFHLVGHÀQHG LPSOLFLWO\ ,QFRQWUDVW WKHFROXPQVSDFHLV GHÀQHG H[SOLFLWO\ EHFDXVHYHFWRUVLQ &RO A FDQEHFRQVWUXFWHGE\OLQHDUFRPELQDWLRQV IURPWKHFROXPQVRI A 7RFUHDWHDQH[SOLFLWGHVFULSWLRQRI 1XO A VROYHWKHHTXDWLRQ A[ D DQGZULWHWKHVROXWLRQLQSDUDPHWULFYHFWRUIRUP 6HH([DPSOH EHORZ ò

%DVLVIRUD6XEVSDFH %HFDXVHDVXEVSDFHW\SLFDOO\FRQWDLQVDQLQÀQLWHQXPEHURIYHFWRUV VRPHSUREOHPV LQYROYLQJDVXEVSDFHDUHKDQGOHGEHVWE\ZRUNLQJZLWKDVPDOOÀQLWHVHWRIYHFWRUVWKDW VSDQWKHVXEVSDFH 7KHVPDOOHUWKHVHW WKHEHWWHU ,WFDQEHVKRZQWKDWWKHVPDOOHVW SRVVLEOHVSDQQLQJVHWPXVWEHOLQHDUO\LQGHSHQGHQW x3

%&'*/*5*0/

$ EDVLV IRUDVXEVSDFH H RI Rn LVDOLQHDUO\LQGHSHQGHQWVHWLQ H WKDWVSDQV H

&9".1-& 7KHFROXPQVRIDQLQYHUWLEOH n ! n PDWUL[IRUPDEDVLVIRUDOORI Rn

e3

e2 e1 x1 '*(63&

7KHVWDQGDUGEDVLVIRU R3

x2

EHFDXVHWKH\DUHOLQHDUO\LQGHSHQGHQWDQGVSDQ Rn E\WKH,QYHUWLEOH0DWUL[7KHRUHP 2QHVXFKPDWUL[LVWKH n ! n LGHQWLW\PDWUL[ ,WVFROXPQVDUHGHQRWHGE\ H1 ; : : : ; Hn 2 3 2 3 2 3 1 0 0 607 617 6 ::: 7 7 6 7 6 7 H1 D 6 4 ::: 5 ; H2 D 4 ::: 5 ; : : : ; Hn D 4 0 5 0 0 1 7KHVHW fH1 ; : : : ; Hn g LVFDOOHGWKH VWDQGDUGEDVLV IRU Rn 6HH)LJ ò 7KHFRQWUDVWEHWZHHQ 1XO A DQG &RO A LVGLVFXVVHGIXUWKHULQ6HFWLRQ

6XEVSDFHVRI Rn

7KHQH[WH[DPSOHVKRZVWKDWWKHVWDQGDUGSURFHGXUHIRUZULWLQJWKHVROXWLRQVHWRI A[ D LQSDUDPHWULFYHFWRUIRUPDFWXDOO\LGHQWLÀHVDEDVLVIRU 1XO A 7KLVIDFWZLOOEH XVHGWKURXJKRXW&KDSWHU

&9".1-& )LQGDEDVLVIRUWKHQXOOVSDFHRIWKHPDWUL[ 2

3 1 #7 3 #1 5 8 #4

#3 6 #1 A D 4 1 #2 2 2 #4 5

40-65*0/ )LUVW ZULWHWKHVROXWLRQRI A[ D LQSDUDPHWULFYHFWRUIRUP 2 3 x1 # 2x2 # x4 C 3x5 D 0 1 #2 0 #1 3 0 ! " 4 5 A % 0 0 1 2 #2 0 ; x3 C 2x4 # 2x5 D 0 0 0 0 0 0 0 0D0

7KHJHQHUDOVROXWLRQLV x1 D 2x2 C x4 # 3x5 x3 D #2x4 C 2x5 ZLWK x2 x4 DQG x5 IUHH 2 3 2 3 2 3 2 3 2 3 x1 2x2 C x4 # 3x5 2 1 #3 6 x2 7 6 7 617 6 07 6 07 x2 6 7 6 7 6 7 6 7 6 7 6 x3 7 D 6 #2x4 C 2x5 7 D x2 6 0 7 C x4 6 #2 7 C x5 6 2 7 6 7 6 7 6 7 6 7 6 7 4 x4 5 4 5 405 4 15 4 05 x4 x5 x5 0 0 1 "

"

Y

X

"

Z

D x2 X C x4 Y C x5 Z

(TXDWLRQ VKRZVWKDW 1XO A FRLQFLGHVZLWKWKHVHWRIDOOOLQHDUFRPELQDWLRQVRI X Y; DQG Z: 7KDWLV fX; Y; Zg JHQHUDWHV 1XO A ,QIDFW WKLVFRQVWUXFWLRQRI X Y DQG Z DXWRPDWLFDOO\PDNHVWKHPOLQHDUO\LQGHSHQGHQW EHFDXVHHTXDWLRQ VKRZVWKDW D x2 X C x4 Y C x5 Z RQO\LIWKHZHLJKWV x2 x4 DQG x5 DUHDOO]HUR ([DPLQHHQWULHV DQGLQWKHYHFWRU x2 X C x4 Y C x5 Z 6R fX; Y; Zg LVD EDVLV IRU 1XO A )LQGLQJDEDVLVIRUWKHFROXPQVSDFHRIDPDWUL[LVDFWXDOO\OHVVZRUNWKDQÀQGLQJ DEDVLVIRUWKHQXOOVSDFH +RZHYHU WKHPHWKRGUHTXLUHVVRPHH[SODQDWLRQ /HW·VEHJLQ ZLWKDVLPSOHFDVH

&9".1-& )LQGDEDVLVIRUWKHFROXPQVSDFHRIWKHPDWUL[ 2

1 60 BD6 40 0

0 #3 5 1 2 #1 0 0 0 0 0 0

3 0 07 7 15 0

40-65*0/ 'HQRWHWKHFROXPQVRI B E\ E1 ; : : : ; E5 DQGQRWHWKDW E3 D #3E1 C 2E2 DQG E4 D 5E1 # E2 7KHIDFWWKDW E3 DQG E4 DUHFRPELQDWLRQVRIWKHSLYRWFROXPQVPHDQVWKDW DQ\FRPELQDWLRQRI E1 ; : : : ; E5 LVDFWXDOO\MXVWDFRPELQDWLRQRI E1 E2 DQG E5 ,QGHHG LI Y LVDQ\YHFWRULQ &RO B VD\ Y D c1 E1 C c2 E2 C c3 E3 C c4 E4 C c5 E5

WKHQ VXEVWLWXWLQJIRU E3 DQG E4 ZHFDQZULWH Y LQWKHIRUP

Y D c1 E1 C c2 E2 C c3 .#3E1 C 2E2 / C c4 .5E1 # E2 / C c5 E5

ZKLFKLVDOLQHDUFRPELQDWLRQRI E1 E2 DQG E5 6R fE1 ; E2 ; E5 g VSDQV &RO B $OVR E1 E2 DQG E5 DUHOLQHDUO\LQGHSHQGHQW EHFDXVHWKH\DUHFROXPQVIURPDQLGHQWLW\PDWUL[ 6RWKHSLYRWFROXPQVRI B IRUPDEDVLVIRU &RO B

$)"15&3

0DWUL[$OJHEUD

7KHPDWUL[ B LQ([DPSOH LVLQUHGXFHGHFKHORQIRUP 7RKDQGOHDJHQHUDOPDWUL[ A UHFDOOWKDWOLQHDUGHSHQGHQFHUHODWLRQVDPRQJWKHFROXPQVRI A FDQEHH[SUHVVHG LQWKHIRUP A[ D IRUVRPH [ ,IVRPHFROXPQVDUHQRWLQYROYHGLQDSDUWLFXODU GHSHQGHQFHUHODWLRQ WKHQWKHFRUUHVSRQGLQJHQWULHVLQ [ DUH]HUR :KHQ A LVURZ UHGXFHGWRHFKHORQIRUP B WKHFROXPQVDUHGUDVWLFDOO\FKDQJHG EXWWKHHTXDWLRQV A[ D DQG B [ D KDYHWKHVDPHVHWRIVROXWLRQV 7KDWLV WKHFROXPQVRI A KDYH H[DFWO\WKHVDPHOLQHDUGHSHQGHQFHUHODWLRQVKLSV DVWKHFROXPQVRI B

&9".1-& ,WFDQEHYHULÀHGWKDWWKHPDWUL[ A D Œ D1 D2

2

1 3 3 6 #2 #2 2 " " " D5 ! D 6 4 2 3 0 3 4 #1

3 2 #9 #8 27 7 7 15 11 #8

LVURZHTXLYDOHQWWRWKHPDWUL[ B LQ([DPSOH )LQGDEDVLVIRU &RO A

40-65*0/ )URP ([DPSOH WKH SLYRW FROXPQV RI A DUH FROXPQV DQG $OVR E3 D #3E1 C 2E2 DQG E4 D 5E1 # E2 6LQFHURZRSHUDWLRQVGRQRWDIIHFWOLQHDU GHSHQGHQFHUHODWLRQVDPRQJWKHFROXPQVRIWKHPDWUL[ ZHVKRXOGKDYH D3 D #3D1 C 2D2

DQG

D4 D 5D1 # D2

&KHFNWKDWWKLVLVWUXH %\WKHDUJXPHQWLQ([DPSOH D3 DQG D4 DUHQRWQHHGHGWR JHQHUDWHWKHFROXPQVSDFHRI A $OVR fD1 ; D2 ; D5 g PXVWEHOLQHDUO\LQGHSHQGHQW EHFDXVH DQ\GHSHQGHQFHUHODWLRQDPRQJ D1 D2 DQG D5 ZRXOGLPSO\WKHVDPHGHSHQGHQFHUHODWLRQ DPRQJ E1 E2 DQG E5 6LQFH fE1 ; E2 ; E5 g LVOLQHDUO\LQGHSHQGHQW fD1 ; D2 ; D5 g LVDOVR OLQHDUO\LQGHSHQGHQWDQGKHQFHLVDEDVLVIRU &RO A 7KHDUJXPHQWLQ([DPSOH FDQEHDGDSWHGWRSURYHWKHIROORZLQJWKHRUHP

5)&03&.

7KHSLYRWFROXPQVRIDPDWUL[ A IRUPDEDVLVIRUWKHFROXPQVSDFHRI A

:DUQLQJ %HFDUHIXOWRXVH SLYRWFROXPQVRI A LWVHOI IRUWKHEDVLVRI &RO A 7KH FROXPQVRIDQHFKHORQIRUP B DUHRIWHQQRWLQWKHFROXPQVSDFHRI A )RULQVWDQFH LQ([DPSOHV DQG WKHFROXPQVRI B DOOKDYH]HURVLQWKHLUODVWHQWULHVDQGFDQQRW JHQHUDWHWKHFROXPQVRI A

SG

.BTUFSJOH 4VCTQBDF $PM " /VM " #BTJT m

13"$5*$& 130#-&.4 2 3 2 3 1 #1 5 #7 /HW A D 4 2 0 7 5 DQG X D 4 3 5 ,V X LQ 1XO A" ,V X LQ &RO A" -XVWLI\ #3 #5 #3 2 HDFKDQVZHU 2 3 0 1 0 1 5 ÀQGDYHFWRULQ 1XO A DQGDYHFWRULQ &RO A *LYHQ A D 4 0 0 0 0 0

6XSSRVH DQ n ! n PDWUL[ A LVLQYHUWLEOH :KDWFDQ\RXVD\DERXW &RO A" $ERXW 1XO A"

6XEVSDFHVRI Rn

&9&3$*4&4 ([HUFLVHV²GLVSOD\VHWVLQ R2 $VVXPHWKHVHWVLQFOXGHWKH ERXQGLQJOLQHV ,QHDFKFDVH JLYHDVSHFLÀFUHDVRQZK\WKHVHW H LV QRW DVXEVSDFHRI R2 )RULQVWDQFH ÀQGWZRYHFWRUVLQ H ZKRVHVXPLV QRW LQ H RUÀQGDYHFWRULQ H ZLWKDVFDODUPXOWLSOH WKDWLVQRWLQ H 'UDZDSLFWXUH

/HW

2

3 2 3 2 3 2 #3 #4 Y1 D 4 #8 5 Y2 D 4 8 5 Y3 D 4 6 5 6 #7 #7 2 3 6 ! " S D 4 #10 5; DQG A D Y1 Y2 Y3 : 11

D +RZPDQ\YHFWRUVDUHLQ fY1 ; Y2 ; Y3 g"

E +RZPDQ\YHFWRUVDUHLQ &RO A" F ,V S LQ &RO A" :K\RUZK\QRW" /HW

3 2 3 2 3 #2 #2 0 Y1 D 4 0 5; Y2 D 4 3 5; Y3 D 4 #5 5; 6 3 5 2 3 #6 DQG S D 4 1 5 'HWHUPLQHLI S LVLQ &RO A ZKHUH A D 17 Œ Y 1 Y 2 Y 3 !

2

:LWK A DQG S DVLQ([HUFLVH GHWHUPLQHLI S LVLQ 1XO A 2 3 #5 :LWK X D 4 5 5 DQG A DVLQ([HUFLVH GHWHUPLQHLI X LV 3 LQ 1XO A

,Q([HUFLVHVDQG JLYHLQWHJHUV p DQG q VXFKWKDW 1XO A LVD VXEVSDFHRI Rp DQG &RO A LVDVXEVSDFHRI Rq 2 3 3 2 1 #5 1 75 A D 4 #9 #4 9 2 #5 1 2 3 1 2 3 6 4 5 77 6 7 7 #5 #1 0 A D 6 6 7 4 2 7 11 5 3 3 4

2

3 2 3 2 3 1 #2 #3 /HW Y1 D 4 3 5 Y2 D 4 #3 5 DQG Z D 4 #3 5 'HWHU #4 7 10 PLQHLI Z LVLQWKHVXEVSDFHRI R3 JHQHUDWHGE\ Y1 DQG Y2 2

3 2 3 2 3 1 4 5 6 #3 7 6 #4 7 6 #3 7 7 6 7 6 7 /HW Y1 D 6 4 2 5 Y2 D 4 5 5 Y3 D 4 6 5 DQG X D 3 7 5 2 3 #1 6 #7 7 4 6 7 4 #1 5 'HWHUPLQHLI X LVLQWKHVXEVSDFHRI R JHQHUDWHG 2 E\ fY1 ; Y2 ; Y3 g

)RU A DVLQ([HUFLVH ÀQGDQRQ]HURYHFWRULQ 1XO A DQGD QRQ]HURYHFWRULQ &RO A )RU A DVLQ([HUFLVH ÀQGDQRQ]HURYHFWRULQ 1XO A DQG DQRQ]HURYHFWRULQ &RO A 'HWHUPLQHZKLFKVHWVLQ([HUFLVHV²DUHEDVHVIRU R2 RU R3 -XVWLI\HDFKDQVZHU # $ # $ # $ # $ 4 16 #2 4 #2 #3 5 #10 2 3 2 3 2 3 2 3 2 3 2 3 0 5 6 1 3 5 4 0 5 4 0 5 4 3 5 4 1 5 4 #1 5 4 1 5 #2 4 2 #3 2 #4 2 3 2 3 2 3 2 3 2 3 2 3 3 6 1 3 #3 0 4 #8 5 4 2 5 4 #6 5 4 #6 5 4 7 5 4 7 5 1 #5 #7 7 5 9

0DWUL[$OJHEUD

$)"15&3

,Q([HUFLVHVDQG PDUNHDFKVWDWHPHQW7UXHRU)DOVH -XVWLI\ HDFKDQVZHU D $ VXEVSDFHRI Rn LVDQ\VHW H VXFKWKDWL WKH]HURYHFWRU LVLQ H; LL X; Y; DQG X C Y DUHLQ H DQGLLL c LVDVFDODU DQG c X LVLQ H E ,I Y1 ; : : : ; Yp DUHLQ R ; WKHQ6SDQ fY! 1 ; : : : ; Yp g LVWKHVDPH " """ Yp DVWKHFROXPQVSDFHRIWKHPDWUL[ Y1 n

F 7KHVHWRIDOOVROXWLRQVRIDV\VWHPRI m KRPRJHQHRXV HTXDWLRQVLQ n XQNQRZQVLVDVXEVSDFHRI Rm :

G 7KHFROXPQVRIDQLQYHUWLEOH n ! n PDWUL[IRUPDEDVLV IRU Rn H 5RZRSHUDWLRQVGRQRWDIIHFWOLQHDUGHSHQGHQFHUHODWLRQV DPRQJWKHFROXPQVRIDPDWUL[ D $ VXEVHW H RI Rn LVDVXEVSDFHLIWKH]HURYHFWRULVLQ H E ,I B LVDQHFKHORQIRUPRIDPDWUL[ A WKHQWKHSLYRW FROXPQVRI B IRUPDEDVLVIRU &RO A: F *LYHQYHFWRUV Y1 ; : : : ; Yp LQ Rn ; WKHVHWRIDOOOLQHDUFRP ELQDWLRQVRIWKHVHYHFWRUVLVDVXEVSDFHRI Rn : G /HW H EHDVXEVSDFHRI Rn : ,I [ LVLQ H DQG \ LVLQ Rn ; WKHQ [ C \ LVLQ H:

2

3 #1 63 1 6 A D 4 0 3 6 3 2 3 #1 60 2 %6 40 0 0 0

#3 3 9 9

#1 0 #1 #2

#3 6 0 0

0 0 #1 0

3 8 27 7 #4 5 6 3 6 #4 7 7 25 0

&RQVWUXFWD 3 ! 3 PDWUL[ A DQGDQRQ]HURYHFWRU E VXFKWKDW E LVLQ&RO A EXW E LVQRWWKHVDPHDVDQ\RQHRIWKHFROXPQV RI A &RQVWUXFWD 3 ! 3 PDWUL[ A DQGDYHFWRU E VXFKWKDW E LV QRW LQ&RO A &RQVWUXFWDQRQ]HUR 3 ! 3 PDWUL[ A DQGDQRQ]HURYHFWRU E VXFKWKDW E LVLQ 1XO A 6XSSRVHWKHFROXPQVRIDPDWUL[ A D ŒD1 " " " Dp ! DUHOLQHDUO\ LQGHSHQGHQW ([SODLQZK\ fD1 ; : : : ; Dp g LVDEDVLVIRU&RO A ,Q([HUFLVHV² UHVSRQGDVFRPSUHKHQVLYHO\DVSRVVLEOH DQG MXVWLI\\RXUDQVZHU 6XSSRVH F LVD 5 ! 5 PDWUL[ZKRVHFROXPQVSDFHLVQRWHTXDO WR R5 : :KDWFDQEHVDLGDERXW 1XO F "

H 7KHFROXPQVSDFHRIDPDWUL[ A LVWKHVHWRIVROXWLRQVRI A[ D E

,I B LVD 7 ! 7 PDWUL[DQG &RO B D R7 ZKDWFDQEHVDLGDERXW VROXWLRQVRIHTXDWLRQVRIWKHIRUP B [ D E IRU E LQ R7 "

([HUFLVHV²GLVSOD\DPDWUL[ A DQGDQHFKHORQIRUPRI A )LQGDEDVLVIRU &RO A DQGDEDVLVIRU 1XO A 2 3 2 3 4 5 9 #2 1 2 6 #5 5 1 12 5 % 4 0 1 5 #6 5 A D 4 6 3 4 8 #3 0 0 0 0

:KDWFDQEHVDLGDERXWWKHVKDSHRIDQ m ! n PDWUL[ A ZKHQ WKHFROXPQVRI A IRUPDEDVLVIRU Rm "

2

3 2 9 0 1 #2 7 25 % 40 0 6 #6 0 0

3 #6 A D 4 2 #4 3 #6 2

1 6 #1 A D 6 4 #2 3 2 1 60 %6 40 0

4 2 2 6 4 2 0 0

3 8 #3 #7 7 3 47 7 9 5 55 9 #5 #2 3 8 0 5 5 0 #1 7 7 0 1 45 0 0 0

5 3 0

3 4 65 0

,I C LVD 6 ! 6 PDWUL[DQG 1XO C LVWKH]HURVXEVSDFH ZKDW FDQEHVDLGDERXWVROXWLRQVRIHTXDWLRQVRIWKHIRUP C [ D E IRU E LQ R6 "

,I B LVD 5 ! 5 PDWUL[DQG 1XO B LV QRW WKH]HURVXEVSDFH ZKDWFDQEHVDLGDERXW &RO B " :KDWFDQEHVDLGDERXW 1XO C ZKHQ C LVD 6 ! 4 PDWUL[ZLWK OLQHDUO\LQGHSHQGHQWFROXPQV" >0@ ,Q([HUFLVHVDQG FRQVWUXFWEDVHVIRUWKHFROXPQVSDFH DQGWKHQXOOVSDFHRIWKHJLYHQPDWUL[ A -XVWLI\\RXUZRUN 2 3 3 #5 0 #1 3 6 #7 9 #4 9 #11 7 7 A D 6 4 #5 7 #2 5 #7 5 3 #7 #3 4 0 2 3 5 3 2 #6 #8 6 4 1 3 #8 #7 7 7 A D 6 4 5 1 4 5 19 5 #7 #5 #2 8 5 WEB

$PMVNO 4QBDF BOE /VMM 4QBDF

WEB

" #BTJT GPS $PM "

'LPHQVLRQDQG5DQN

40-65*0/4 50 13"$5*$& 130#-&.4 7RGHWHUPLQHZKHWKHU X LVLQ 1XO A VLPSO\FRPSXWH 2 32 3 2 3 1 #1 5 #7 0 AX D 4 2 0 7 54 3 5 D 4 0 5 #3 #5 #3 2 0

7KHUHVXOWVKRZVWKDW X LVLQ 1XO A: 'HFLGLQJZKHWKHU X LVLQ &RO A UHTXLUHVPRUH ZRUN 5HGXFHWKHDXJPHQWHGPDWUL[ ŒA X! WRHFKHORQIRUPWRGHWHUPLQHZKHWKHU WKHHTXDWLRQ A[ D X LVFRQVLVWHQW 2 3 2 3 2 3 1 #1 5 #7 1 #1 5 #7 1 #1 5 #7 4 2 0 7 3 5 % 4 0 2 #3 17 5 % 4 0 2 #3 17 5 #3 #5 #3 2 0 #8 12 #19 0 0 0 49 7KHHTXDWLRQ A[ D X KDVQRVROXWLRQ VR X LVQRWLQ &RO A

,QFRQWUDVWWR3UDFWLFH3UREOHP ÀQGLQJDYHFWRULQ 1XO A UHTXLUHVPRUHZRUN WKDQWHVWLQJZKHWKHUDVSHFLÀHGYHFWRULVLQ 1XO A +RZHYHU VLQFH A LVDOUHDG\ LQUHGXFHGHFKHORQIRUP WKHHTXDWLRQ A[ D VKRZVWKDWLI [ D .x1 ; x2 ; x3 /; WKHQ x2 D 0; x3 D 0; DQG x1 LVDIUHHYDULDEOH 7KXV DEDVLVIRU 1XO A LV Y D .1; 0; 0/: )LQGLQJMXVWRQHYHFWRULQ &RO A LVWULYLDO VLQFHHDFKFROXPQRI A LVLQ &RO A ,Q WKLVSDUWLFXODUFDVH WKHVDPHYHFWRU Y LVLQERWK 1XO A DQG &RO A )RUPRVW n ! n PDWULFHV WKH]HURYHFWRURI Rn LVWKHRQO\YHFWRULQERWK 1XO A DQG &RO A ,I A LVLQYHUWLEOH WKHQWKHFROXPQVRI A VSDQ Rn ; E\WKH,QYHUWLEOH0DWUL[7KHRUHP %\GHÀQLWLRQ WKHFROXPQVRIDQ\PDWUL[DOZD\VVSDQWKHFROXPQVSDFH VRLQWKLV FDVH &RO A LVDOORI Rn : ,QV\PEROV &RO A D Rn : $OVR VLQFH A LVLQYHUWLEOH WKH HTXDWLRQ A[ D KDVRQO\WKHWULYLDOVROXWLRQ 7KLVPHDQVWKDW 1XO A LVWKH]HUR VXEVSDFH ,QV\PEROV 1XO A D fg:

%*.&/4*0/ "/% 3"/, 7KLVVHFWLRQFRQWLQXHVWKHGLVFXVVLRQRIVXEVSDFHVDQGEDVHVIRUVXEVSDFHV EHJLQQLQJ ZLWKWKHFRQFHSWRIDFRRUGLQDWHV\VWHP 7KHGHÀQLWLRQDQGH[DPSOHEHORZVKRXOGPDNH DXVHIXOQHZWHUP GLPHQVLRQ VHHPTXLWHQDWXUDO DWOHDVWIRUVXEVSDFHVRI R3

&RRUGLQDWH6\VWHPV 7KHPDLQUHDVRQIRUVHOHFWLQJDEDVLVIRUDVXEVSDFH H; LQVWHDGRIPHUHO\DVSDQQLQJ VHW LVWKDWHDFKYHFWRULQ H FDQEHZULWWHQLQRQO\RQHZD\DVDOLQHDUFRPELQDWLRQRI WKHEDVLVYHFWRUV 7RVHHZK\ VXSSRVH B D fE1 ; : : : ; Ep g LVDEDVLVIRU H DQGVXSSRVH DYHFWRU [ LQ H FDQEHJHQHUDWHGLQWZRZD\V VD\ [ D c1 E1 C " " " C cp Ep

DQG

[ D d1 E1 C " " " C dp Ep

7KHQ VXEWUDFWLQJJLYHV D [ # [ D .c1 # d1 /E1 C " " " C .cp # dp /Ep

6LQFH B LVOLQHDUO\LQGHSHQGHQW WKHZHLJKWVLQ PXVWDOOEH]HUR 7KDWLV cj D dj IRU 1 ) j ) p; ZKLFKVKRZVWKDWWKHWZRUHSUHVHQWDWLRQVLQ DUHDFWXDOO\WKHVDPH

$)"15&3

0DWUL[$OJHEUD

%&'*/*5*0/

6XSSRVHWKHVHW B D fE1 ; : : : ; Ep g LVDEDVLVIRUDVXEVSDFH H )RUHDFK [ LQ H WKH FRRUGLQDWHVRI[UHODWLYHWRWKHEDVLV B DUHWKHZHLJKWV c1 ; : : : ; cp VXFKWKDW [ D c1 E1 C " " " C cp Ep ; DQGWKHYHFWRULQ Rp 2 3 c1 6 7 Œ[!B D 4 ::: 5

cp

LVFDOOHGWKH FRRUGLQDWHYHFWRURI[UHODWLYHWR B RUWKH B FRRUGLQDWHYHFWRU RI[

2 3 2 3 2 3 3 #1 3 &9".1-& /HW Y1 D 4 6 5 Y2 D 4 0 5 [ D 4 12 5 DQG B D fY1 ; Y2 g 7KHQ 2 1 7 B LVDEDVLVIRU H D 6SDQ fY1 ; Y2 g EHFDXVH Y1 DQG Y2 DUHOLQHDUO\LQGHSHQGHQW 'HWHU PLQHLI [ LVLQ H DQGLILWLV ÀQGWKHFRRUGLQDWHYHFWRURI [ UHODWLYHWR B

40-65*0/ ,I [ LVLQ H WKHQWKHIROORZLQJYHFWRUHTXDWLRQLVFRQVLVWHQW 2 3 2 3 2 3 3 #1 3 4 5 4 5 4 5 6 0 12 c1 C c2 D 2 1 7

7KHVFDODUV c1 DQG c2 LIWKH\H[LVW DUHWKH B FRRUGLQDWHVRI [ 5RZRSHUDWLRQVVKRZ WKDW 2 3 2 3 3 #1 3 1 0 2 46 0 12 5 % 4 0 1 3 5 2 1 7 0 0 0 # $ 2 7KXV c1 D 2 c2 D 3 DQG Œ [ !B D 7KHEDVLV B GHWHUPLQHVD´FRRUGLQDWHV\VWHPµ 3 RQ H ZKLFKFDQEHYLVXDOL]HGE\WKHJULGVKRZQLQ)LJ

3v2 x = 2v1 + 3v2

2v2 v2 0

v1

2v1

'*(63& $ FRRUGLQDWHV\VWHPRQDSODQH H LQ

R3 :

ñ ,WLVLPSRUWDQWWKDWWKHHOHPHQWVRI B DUHQXPEHUHGEHFDXVHWKHHQWULHVLQ Œ[!B GHSHQGRQWKHRUGHURIWKH YHFWRUVLQ B

'LPHQVLRQDQG5DQN

1RWLFHWKDWDOWKRXJKSRLQWVLQ H DUHDOVRLQ R3 WKH\DUHFRPSOHWHO\GHWHUPLQHGE\ WKHLUFRRUGLQDWHYHFWRUV ZKLFKEHORQJWR R2 7KHJULGRQWKHSODQHLQ)LJ PDNHV H ÓRRNµOLNH R2 7KHFRUUHVSRQGHQFH [ 7! Œ [ !B LVDRQHWRRQHFRUUHVSRQGHQFH EHWZHHQ H DQG R2 WKDWSUHVHUYHVOLQHDUFRPELQDWLRQV :HFDOOVXFKDFRUUHVSRQGHQFH DQ LVRPRUSKLVP DQGZHVD\WKDW H LV LVRPRUSKLF WR R2 ,QJHQHUDO LI B D fE1 ; : : : ; Ep g LVDEDVLVIRU H WKHQWKHPDSSLQJ [ 7! Œ[!B LVD RQHWRRQHFRUUHVSRQGHQFHWKDWPDNHV H ORRNDQGDFWWKHVDPHDV Rp HYHQWKRXJKWKH YHFWRUVLQ H WKHPVHOYHVPD\KDYHPRUHWKDQ p HQWULHV 6HFWLRQ KDVPRUHGHWDLOV

7KH'LPHQVLRQRID6XEVSDFH ,WFDQEHVKRZQWKDWLIDVXEVSDFH H KDVDEDVLVRI p YHFWRUV WKHQHYHU\EDVLVRI H PXVW FRQVLVWRIH[DFWO\ p YHFWRUV 6HH([HUFLVHV DQG 7KXVWKHIROORZLQJGHÀQLWLRQ PDNHVVHQVH

%&'*/*5*0/

7KH GLPHQVLRQ RIDQRQ]HURVXEVSDFH H GHQRWHGE\ GLP H LVWKHQXPEHURI YHFWRUVLQDQ\EDVLVIRU H 7KHGLPHQVLRQRIWKH]HURVXEVSDFH fg LVGHÀQHGWR EH]HUR 7KHVSDFH Rn KDVGLPHQVLRQ n (YHU\EDVLVIRU Rn FRQVLVWVRI n YHFWRUV $ SODQH WKURXJK LQ R3 LVWZRGLPHQVLRQDO DQGDOLQHWKURXJK LVRQHGLPHQVLRQDO

&9".1-& 5HFDOOWKDWWKHQXOOVSDFHRIWKHPDWUL[ A LQ([DPSOH LQ6HFWLRQ KDGDEDVLVRIYHFWRUV 6RWKHGLPHQVLRQRI 1XO A LQWKLVFDVHLV 2EVHUYHKRZHDFK EDVLVYHFWRUFRUUHVSRQGVWRDIUHHYDULDEOHLQWKHHTXDWLRQ A[ D 2XUFRQVWUXFWLRQ DOZD\VSURGXFHVDEDVLVLQWKLVZD\ 6R WRÀQGWKHGLPHQVLRQRI 1XO A VLPSO\LGHQWLI\ DQGFRXQWWKHQXPEHURIIUHHYDULDEOHVLQ A[ D 7KH UDQN RIDPDWUL[ A GHQRWHGE\ UDQN A LVWKHGLPHQVLRQRIWKHFROXPQVSDFH RI A 6LQFHWKHSLYRWFROXPQVRI A IRUPDEDVLVIRU &RO A WKHUDQNRI A LVMXVWWKHQXPEHU RISLYRWFROXPQVLQ A

&9".1-& 'HWHUPLQHWKHUDQNRIWKHPDWUL[ 2

3 2 5 #3 #4 8 64 7 #4 #3 9 7 7 AD6 46 9 #5 2 4 5 0 #9 6 5 #6

#

3LYRWFROXPQV

#

40-65*0/ 5HGXFH A WRHFKHORQIRUP 2 3 2 3 2 5 #3 #4 8 2 5 #3 #4 8 6 0 #3 6 2 5 #7 7 2 5 #7 7 7 % " " " % 6 0 #3 7 A%6 4 0 #6 40 4 14 #20 5 0 0 4 #6 5 0 #9 6 5 #6 0 0 0 0 0 #

%&'*/*5*0/

7KHPDWUL[ A KDV SLYRWFROXPQV VR UDQN A D 3 ò 7KH]HURVXEVSDFHKDV QR EDVLVEHFDXVHWKH]HURYHFWRUE\LWVHOIIRUPVDOLQHDUO\GHSHQGHQWVHW

$)"15&3

0DWUL[$OJHEUD

7KHURZUHGXFWLRQLQ([DPSOH UHYHDOVWKDWWKHUHDUHWZRIUHHYDULDEOHVLQ A[ D EHFDXVHWZRRIWKHÀYHFROXPQVRI A DUH QRW SLYRWFROXPQV 7KHQRQSLYRWFROXPQV FRUUHVSRQGWRWKHIUHHYDULDEOHVLQ A[ D 6LQFHWKHQXPEHURISLYRWFROXPQVSOXVWKH QXPEHURIQRQSLYRWFROXPQVLVH[DFWO\WKHQXPEHURIFROXPQV WKHGLPHQVLRQVRI &RO A DQG 1XO A KDYHWKHIROORZLQJXVHIXOFRQQHFWLRQ 6HHWKH5DQN7KHRUHPLQ6HFWLRQ IRUDGGLWLRQDOGHWDLOV

5)&03&.

5IF 3BOL 5IFPSFN ,IDPDWUL[ A KDV n FROXPQV WKHQ UDQN A C GLP 1XO A D n 7KHIROORZLQJWKHRUHPLVLPSRUWDQWIRUDSSOLFDWLRQVDQGZLOOEHQHHGHGLQ&KDS WHUV DQG 7KHWKHRUHPSURYHGLQ6HFWLRQ LVFHUWDLQO\SODXVLEOH LI\RXWKLQNRI D p GLPHQVLRQDOVXEVSDFHDVLVRPRUSKLFWR Rp 7KH,QYHUWLEOH0DWUL[7KHRUHPVKRZV WKDW p YHFWRUVLQ Rp DUHOLQHDUO\LQGHSHQGHQWLIDQGRQO\LIWKH\DOVRVSDQ Rp

5)&03&.

5IF #BTJT 5IFPSFN /HW H EHD p GLPHQVLRQDOVXEVSDFHRI Rn $Q\OLQHDUO\LQGHSHQGHQWVHWRIH[DFWO\ p HOHPHQWVLQ H LVDXWRPDWLFDOO\DEDVLVIRU H $OVR DQ\VHWRI p HOHPHQWVRI H WKDWVSDQV H LVDXWRPDWLFDOO\DEDVLVIRU H

5DQNDQGWKH,QYHUWLEOH0DWUL[7KHRUHP 7KH YDULRXV YHFWRU VSDFH FRQFHSWV DVVRFLDWHG ZLWK D PDWUL[ SURYLGH VHYHUDO PRUH VWDWHPHQWVIRUWKH,QYHUWLEOH0DWUL[7KHRUHP 7KH\DUHSUHVHQWHGEHORZWRIROORZWKH VWDWHPHQWVLQWKHRULJLQDOWKHRUHPLQ6HFWLRQ

5)&03&.

5IF *OWFSUJCMF .BUSJY 5IFPSFN DPOUJOVFE

/HW A EHDQ n ! n PDWUL[ 7KHQWKHIROORZLQJVWDWHPHQWVDUHHDFKHTXLYDOHQWWR WKHVWDWHPHQWWKDW A LVDQLQYHUWLEOHPDWUL[ P Q R S T U

7KHFROXPQVRI A IRUPDEDVLVRI Rn : &RO A D Rn GLP &RO A D n UDQN A D n 1XO A D fg GLP 1XO A D 0

1300' 6WDWHPHQWP LVORJLFDOO\HTXLYDOHQWWRVWDWHPHQWVH DQGK UHJDUGLQJOLQHDU LQGHSHQGHQFHDQGVSDQQLQJ 7KHRWKHUÀYHVWDWHPHQWVDUHOLQNHGWRWKHHDUOLHURQHVRI WKHWKHRUHPE\WKHIROORZLQJFKDLQRIDOPRVWWULYLDOLPSOLFDWLRQV J ) Q ) R ) S ) U ) T ) G

6WDWHPHQWJ ZKLFKVD\VWKDWWKHHTXDWLRQ A[ D E KDVDWOHDVWRQHVROXWLRQIRUHDFK E LQ Rn LPSOLHVVWDWHPHQWQ EHFDXVH &RO A LVSUHFLVHO\WKHVHWRIDOO E VXFKWKDW WKHHTXDWLRQ A[ D E LVFRQVLVWHQW 7KHLPSOLFDWLRQV Q ) R ) S IROORZIURPWKH GHÀQLWLRQVRI GLPHQVLRQ DQG UDQN ,IWKHUDQNRI A LV n WKHQXPEHURIFROXPQVRI A WKHQ GLP 1XO A D 0 E\WKH5DQN7KHRUHP DQGVR 1XO A D fg 7KXV S ) U ) T

SG

'LPHQVLRQDQG5DQN

$OVR VWDWHPHQWT LPSOLHVWKDWWKHHTXDWLRQ A[ D KDVRQO\WKHWULYLDOVROXWLRQ ZKLFK LVVWDWHPHQWG 6LQFHVWDWHPHQWVG DQGJ DUHDOUHDG\NQRZQWREHHTXLYDOHQWWRWKH VWDWHPHQWWKDW A LVLQYHUWLEOH WKHSURRI LVFRPSOHWH


/6.&3*$"- /05&4 0DQ\DOJRULWKPVGLVFXVVHGLQWKLVWH[WDUHXVHIXOIRUXQGHUVWDQGLQJFRQFHSWV DQGPDNLQJVLPSOHFRPSXWDWLRQVE\KDQG +RZHYHU WKHDOJRULWKPVDUHRIWHQ XQVXLWDEOHIRUODUJHVFDOHSUREOHPVLQUHDOOLIH 5DQNGHWHUPLQDWLRQLVDJRRGH[DPSOH ,WZRXOGVHHPHDV\WRUHGXFHDPDWUL[ WRHFKHORQIRUPDQGFRXQWWKHSLYRWV %XWXQOHVVH[DFWDULWKPHWLFLVSHUIRUPHG RQDPDWUL[ZKRVHHQWULHVDUHVSHFLÀHGH[DFWO\ URZRSHUDWLRQVFDQFKDQJHWKH # $ 5 7 DSSDUHQWUDQNRIDPDWUL[ )RULQVWDQFH LIWKHYDOXHRI x LQWKHPDWUL[ 5 x LVQRWVWRUHGH[DFWO\DVLQDFRPSXWHU WKHQWKHUDQNPD\EHRU GHSHQGLQJ RQZKHWKHUWKHFRPSXWHUWUHDWV x # 7 DV]HUR ,QSUDFWLFDODSSOLFDWLRQV WKHHIIHFWLYHUDQNRIDPDWUL[ A LVRIWHQGHWHUPLQHG IURPWKHVLQJXODUYDOXHGHFRPSRVLWLRQRI A WREHGLVFXVVHGLQ6HFWLRQ

WEB

13"$5*$& 130#-&.4 'HWHUPLQHWKHGLPHQVLRQRIWKHVXEVSDFH H RI R3 VSDQQHGE\WKHYHFWRUV Y1 ; Y2 ; DQG Y3 : )LUVW ÀQGDEDVLVIRU H 2 3 2 3 2 3 2 3 #1 Y1 D 4 #8 5; Y2 D 4 #7 5; Y3 D 4 6 5 6 #1 #7 &RQVLGHUWKHEDVLV

BD

# $ 3 IRU R ,I Œ [ !B D ZKDWLV [" 2

%#

$ # $& 1 :2 ; :2 1

2

&RXOG R3 SRVVLEO\FRQWDLQDIRXUGLPHQVLRQDOVXEVSDFH" ([SODLQ

&9&3$*4&4 ,Q([HUFLVHVDQG ÀQGWKHYHFWRU [ GHWHUPLQHGE\WKHJLYHQ FRRUGLQDWH YHFWRU Œ[!B DQG WKH JLYHQ EDVLV B: ,OOXVWUDWH \RXU DQVZHUZLWKDÀJXUH DVLQWKHVROXWLRQRI3UDFWLFH3UREOHP %# $ # $& # $ 1 2 3 B D ; ; Œ [ !B D 1 #1 2 B D

%#

$ # $& # $ #3 3 #1 ; ; Œ [ !B D 1 2 2

,Q([HUFLVHV² WKHYHFWRU [ LVLQDVXEVSDFH H ZLWKDEDVLV B D fE1 ; E2 g: )LQGWKH BFRRUGLQDWHYHFWRURI [

$ # $ # $ 2 #1 0 ; E2 D ;[ D #3 5 7 # $ # $ # $ 1 #2 1 E1 D ; E2 D ;[ D #5 3 9 2 3 2 3 2 3 1 #2 2 E1 D 4 4 5; E2 D 4 #7 5; [ D 4 9 5 #3 5 #7 2 3 2 3 2 3 #3 7 5 E1 D 4 2 5; E2 D 4 #3 5; [ D 4 0 5 #4 5 #2

E1 D

#

$)"15&3

0DWUL[$OJHEUD

# $ # $ # $ # $ 3 #1 7 4 ; E2 D ZD ; [D ; DQG 0 2 #2 1 B D fE1 ; E2 g: 8VH WKH ÀJXUH WR HVWLPDWH ŒZ!B DQG Œ[!B &RQÀUP\RXUHVWLPDWHRI Œ[!B E\XVLQJLWDQG fE1 ; E2 g WR FRPSXWH [

/HW E1 D

b2

x b1

0

w

# $ # $ # $ # $ 0 2 #2 2 /HW E1 D E2 D [D \D 2 1 3 4 # $ #1 DQG B D fE1 ; E2 g 8VHWKHÀJXUHWRHVWLPDWH ]D #2:5 Œ[!B Œ\!B DQG Œ]!B &RQÀUP\RXUHVWLPDWHVRI Œ\!B DQG Œ]!B E\XVLQJWKHPDQG fE1 ; E2 g WRFRPSXWH \ DQG ]

y x b1 b2 0 z

([HUFLVHV²GLVSOD\DPDWUL[ A DQGDQHFKHORQIRUPRI A )LQG EDVHVIRU&RO A DQG 1XO A DQGWKHQVWDWHWKHGLPHQVLRQVRIWKHVH VXEVSDFHV 2 3 2 3 1 3 2 #6 1 3 3 2 63 6 9 1 57 0 5 #7 7 7 % 60 7 A D 6 42 5 4 6 #1 9 0 0 0 55 5 15 0 14 0 0 0 0 2

1 #2 #1 6 2 #1 1 A D 6 4 #2 0 #2 3 1 4 2 1 #2 #1 60 1 1 %6 40 0 0 0 0 0

3 5 4 5 67 7 1 #6 5 1 5 3 2 0 0 37 7 1 05 0 1

2

2 6 3 6 A D 4 0 #3 2 1 60 %6 40 0 2 1 65 6 A D 4 4 3 2 1 60 %6 40 0

3 4 #5 2 #3 6 #8 3 #5 7 7 0 9 0 95 #6 #7 #3 #10 3 2 #5 1 #4 0 5 0 57 7 0 0 0 05 0 0 0 0 3 2 #4 4 6 1 #9 2 10 7 7 6 #9 12 15 5 4 #5 8 9 3 2 8 4 #6 2 3 4 #1 7 7 0 5 0 #5 5 0 0 0 0

,Q([HUFLVHVDQG ÀQGDEDVLVIRUWKHVXEVSDFHVSDQQHGE\ WKHJLYHQYHFWRUV :KDWLVWKHGLPHQVLRQRIWKHVXEVSDFH" 2 3 2 3 2 3 2 3 1 #3 2 #4 6 #3 7 6 9 7 6 #1 7 6 5 7 7 6 7 6 7 6 7 6 4 2 5; 4 #6 5; 4 4 5; 4 #3 5 #4 12 2 7 2 3 2 3 2 3 2 3 2 3 1 2 0 #1 3 6 #1 7 6 #3 7 6 #1 7 6 4 7 6 #7 7 7 6 7 6 7 6 7 6 7 6 4 #2 5; 4 #1 5; 4 3 5; 4 #7 5; 4 6 5 3 4 #2 7 #9 6XSSRVH D 4 ! 6 PDWUL[ A KDV IRXU SLYRW FROXPQV &RO A D R4 ‹ ,V 1XO A D R2 ‹ ([SODLQ\RXUDQVZHUV

,V

6XSSRVH D 4 ! 7 PDWUL[ A KDV WKUHH SLYRW FROXPQV ,V &RO A D R3 ‹ :KDWLVWKHGLPHQVLRQRI 1XO A" ([SODLQ\RXU DQVZHUV ,Q([HUFLVHVDQG PDUNHDFKVWDWHPHQW7UXHRU)DOVH -XVWLI\ HDFKDQVZHU +HUH A LVDQ m ! n PDWUL[ D ,I B D fY1 ; : : : ; Yp g LVDEDVLVIRUDVXEVSDFH H DQGLI [ D c1 Y1 C " " " C cp Yp ; WKHQ c1 ; : : : ; cp DUH WKH FRRUGL QDWHVRI [ UHODWLYHWRWKHEDVLV B: E (DFKOLQHLQ Rn LVDRQHGLPHQVLRQDOVXEVSDFHRI Rn

F 7KHGLPHQVLRQRI &RO A LVWKHQXPEHURISLYRWFROXPQV LQ A G 7KHGLPHQVLRQVRI &RO A DQG 1XO A DGGXSWRWKHQXPEHU RIFROXPQVLQ A H ,IDVHWRI p YHFWRUVVSDQVD p GLPHQVLRQDOVXEVSDFH H RI Rn ; WKHQWKHVHYHFWRUVIRUPDEDVLVIRU H D ,I B LVDEDVLVIRUDVXEVSDFH H; WKHQHDFKYHFWRULQ H FDQEHZULWWHQLQRQO\RQHZD\DVDOLQHDUFRPELQDWLRQRI WKHYHFWRUVLQ B: E 7KHGLPHQVLRQRI 1XO A LVWKHQXPEHURIYDULDEOHVLQWKH HTXDWLRQ A[ D : F 7KHGLPHQVLRQRIWKHFROXPQVSDFHRI A LV UDQN A:

G ,I B D fY1 ; : : : ; Yp g LVDEDVLVIRUDVXEVSDFH H RI Rn ; WKHQWKHFRUUHVSRQGHQFH [ 7! Œ[!B PDNHV H ORRNDQGDFW WKHVDPHDV Rp : H ,I H LVD p GLPHQVLRQDOVXEVSDFHRI Rn ; WKHQDOLQHDUO\ LQGHSHQGHQWVHWRI p YHFWRUVLQ H LVDEDVLVIRU H ,Q([HUFLVHV² MXVWLI\HDFKDQVZHURUFRQVWUXFWLRQ ,IWKHVXEVSDFHRIDOOVROXWLRQVRI A[ D KDVDEDVLVFRQ VLVWLQJRIWKUHHYHFWRUVDQGLI A LVD 5 ! 7 PDWUL[ ZKDWLVWKH UDQNRI A" :KDWLVWKHUDQNRID 6 ! 8 PDWUL[ZKRVHQXOOVSDFHLVWKUHH GLPHQVLRQDO" ,IWKHUDQNRID 9 ! 8 PDWUL[ A LV ZKDWLVWKHGLPHQVLRQRI WKHVROXWLRQVSDFHRI A[ D ‹

6KRZWKDWDVHW fY1 ; : : : ; Y5 g LQ Rn LVOLQHDUO\GHSHQGHQWLI GLP 6SDQ fY1 ; : : : ; Y5 g D 4

,ISRVVLEOH FRQVWUXFWD 3 ! 5 PDWUL[ A VXFKWKDW GLP 1XO A D 3 DQGGLP&RO A D 2 &RQVWUXFWD 3 ! 4 PDWUL[ZLWKUDQN

/HW A EH DQ n ! p PDWUL[ ZKRVH FROXPQ VSDFH LV p GLPHQVLRQDO ([SODLQZK\WKHFROXPQVRI A PXVWEHOLQHDUO\ LQGHSHQGHQW 6XSSRVHFROXPQV DQGRIDPDWUL[ A DUHOLQHDUO\ LQGHSHQGHQWEXWDUHQRWQHFHVVDULO\SLYRWFROXPQV DQGWKH UDQNRI A LV ([SODLQZK\WKHÀYHFROXPQVPHQWLRQHGPXVW EHDEDVLVIRUWKHFROXPQVSDFHRI A

E /HW C D Œ F1 " " " Fq ! ([SODLQZK\WKHUHLVDQRQ]HUR YHFWRU X VXFKWKDW C X D

F 8VH B DQG C WRVKRZWKDW AX D : 7KLVVKRZVWKDWWKH FROXPQVRI A DUHOLQHDUO\GHSHQGHQW

8VH([HUFLVHWRVKRZWKDWLI A DQG B DUHEDVHVIRUD VXEVSDFH W RI Rn WKHQ A FDQQRWFRQWDLQPRUHYHFWRUVWKDQ B; DQG FRQYHUVHO\ B FDQQRWFRQWDLQPRUHYHFWRUVWKDQ A: >0@ /HW H D 6SDQ fY1 ; Y2 g DQG B D fY1 ; Y2 g: 6KRZWKDW [ LV LQ H DQGÀQGWKH BFRRUGLQDWHYHFWRURI [ ZKHQ 2 3 2 3 2 3 15 14 16 6 #5 7 6 #10 7 6 07 7 6 7 6 7 Y1 D 6 4 12 5; Y2 D 4 13 5; [ D 4 11 5 7 17 #3

>0@ /HW H D 6SDQ fY1 ; Y2 ; Y3 g DQG B D fY1 ; Y2 ; Y3 g: 6KRZ WKDW B LVDEDVLVIRU H DQG [ LVLQ H DQGÀQGWKH BFRRUGLQDWH YHFWRURI [; ZKHQ 2 3 2 3 2 3 2 3 #6 8 #9 11 6 37 6 07 6 47 6 #2 7 7 6 7 6 7 6 7 Y1 D 6 4 #9 5; Y2 D 4 7 5; Y3 D 4 #8 5; [ D 4 17 5 4 #3 3 #8

&RQVWUXFW A D ŒY1 Y2 Y3 ! VRWKDWWKHVXEVSDFHVSDQQHGE\ Y1 ; Y2 ; Y3 LVWKHFROXPQ VSDFHRI A $ EDVLVIRUWKLVVSDFHLVSURYLGHGE\WKHSLYRWFROXPQVRI A 2 3 2 3 2 3 2 3 #1 2 3 #1 2 3 #1 A D 4 #8 #7 6 5 % 4 0 5 2 5 % 4 0 5 2 5 6 #1 #7 0 #10 #4 0 0 0

0 v2

v3

x

1

b2 b1 1

.BTUFSJOH %JNFOTJPO BOE 3BOL m

40-65*0/4 50 13"$5*$& 130#-&.4

Co v1

6XSSRVH YHFWRUV E1 ; : : : ; Ep VSDQ D VXEVSDFH W; DQG OHW fD1 ; : : : ; Dq g EHDQ\VHWLQ W FRQWDLQLQJPRUHWKDQ p YHF WRUV )LOOLQWKHGHWDLOVRIWKHIROORZLQJDUJXPHQWWRVKRZ WKDW fD1 ; : : : ; Dq g PXVW EH OLQHDUO\ GHSHQGHQW )LUVW OHW B D Œ E1 " " " Ep ! DQG A D Œ D1 " " " Dq !: D ([SODLQZK\IRUHDFKYHFWRU Dj WKHUHH[LVWVDYHFWRU Fj LQ Rp VXFKWKDW Dj D B Fj :

SG

lA

'LPHQVLRQDQG5DQN

7KHÀUVWWZRFROXPQVRI A DUHSLYRWFROXPQVDQGIRUPDEDVLVIRU H 7KXV GLP H D #2: $ 3 ,I Œ[!B D WKHQ [ LVIRUPHGIURPDOLQHDUFRPELQDWLRQRIWKHEDVLVYHFWRUVXVLQJ 2 ZHLJKWVDQG # $ # $ # $ 1 :2 3:4 [ D 3E1 C 2E2 D 3 C2 D :2 1 2:6 7KHEDVLV fE1 ; E2 g GHWHUPLQHVD FRRUGLQDWHV\VWHP IRU R2 LOOXVWUDWHGE\WKHJULGLQ WKHÀJXUH 1RWHKRZ [ LVXQLWVLQWKH E1 GLUHFWLRQDQGXQLWVLQWKH E2 GLUHFWLRQ

$)"15&3

0DWUL[$OJHEUD

$ IRXUGLPHQVLRQDOVXEVSDFHZRXOGFRQWDLQDEDVLVRIIRXUOLQHDUO\LQGHSHQGHQW YHFWRUV 7KLVLVLPSRVVLEOHLQVLGH R3 : 6LQFHDQ\OLQHDUO\LQGHSHQGHQWVHWLQ R3 KDV QRPRUHWKDQWKUHHYHFWRUV DQ\VXEVSDFHRI R3 KDVGLPHQVLRQQRPRUHWKDQ 7KH VSDFH R3 LWVHOILVWKHRQO\WKUHHGLPHQVLRQDOVXEVSDFHRI R3 2WKHUVXEVSDFHVRI R3 KDYHGLPHQVLRQ RU

$)"15&3 4611-&.&/5"3: &9&3$*4&4 $VVXPHWKDWWKHPDWULFHVPHQWLRQHGLQWKHVWDWHPHQWVEHORZ KDYHDSSURSULDWHVL]HV 0DUNHDFKVWDWHPHQW7UXHRU)DOVH -XVWLI\HDFKDQVZHU D ,I A DQG B DUH m ! n WKHQERWK AB T DQG ATB DUH GHÀQHG E ,I AB D C DQG C KDV FROXPQV WKHQ A KDV FROXPQV F

/HIWPXOWLSO\LQJDPDWUL[ B E\DGLDJRQDOPDWUL[ A ZLWK QRQ]HURHQWULHVRQWKHGLDJRQDO VFDOHVWKHURZVRI B

G ,I BC D BD WKHQ C D D

H I

,I AC D 0 WKHQHLWKHU A D 0 RU C D 0

,I A DQG B DUH n ! n WKHQ .A C B/.A # B/ D A2 # B 2

J $Q HOHPHQWDU\ n ! n PDWUL[ KDV HLWKHU n RU n C 1 QRQ]HURHQWULHV K 7KHWUDQVSRVHRIDQHOHPHQWDU\PDWUL[LVDQHOHPHQWDU\ PDWUL[ L

$QHOHPHQWDU\PDWUL[PXVWEHVTXDUH

M

(YHU\VTXDUHPDWUL[LVDSURGXFWRIHOHPHQWDU\PDWULFHV

N ,I A LV D 3 ! 3 PDWUL[ ZLWK WKUHH SLYRW SRVLWLRQV WKHUH H[LVW HOHPHQWDU\ PDWULFHV E1 ; : : : ; Ep VXFK WKDW Ep " " " E 1 A D I O

,I AB D I WKHQ A LVLQYHUWLEOH

P ,I A DQG B DUHVTXDUHDQGLQYHUWLEOH WKHQ AB LVLQYHUW LEOH DQG .AB/!1 D A!1 B !1

Q ,I AB D BA DQGLI A LVLQYHUWLEOH WKHQ A!1 B D BA!1

R ,I A LVLQYHUWLEOHDQGLI r ¤ 0 WKHQ .rA/!1 D rA!1 2 3 1 S ,I A LVD 3 ! 3 PDWUL[DQGWKHHTXDWLRQ A[ D 4 0 5 KDV 0 DXQLTXHVROXWLRQ WKHQ A LVLQYHUWLEOH # $ 4 5 )LQGWKHPDWUL[ C ZKRVHLQYHUVHLV C !1 D 6 7 2 3 0 0 0 0 0 5 6KRZWKDW A3 D 0 8VHPDWUL[ /HW A D 4 1 0 1 0 DOJHEUDWRFRPSXWHWKHSURGXFW .I # A/.I C A C A2 /

6XSSRVH An D 0 IRUVRPH n > 1 )LQGDQLQYHUVHIRU I # A 6XSSRVH DQ n ! n PDWUL[ A VDWLVÀHV WKH HTXDWLRQ A2 # 2A C I D 0 6KRZWKDW A3 D 3A # 2I DQG A4 D 4A # 3I # $ # $ 1 0 0 1 /HW A D BD 7KHVHDUH 3DXOLVSLQ 0 #1 1 0

PDWULFHV XVHG LQ WKH VWXG\ RI HOHFWURQ VSLQ LQ TXDQWXP PHFKDQLFV 6KRZWKDW A2 D I B 2 D I DQG AB D #BA 0DWULFHVVXFKWKDW AB D #BA DUHVDLGWR DQWLFRPPXWH 2 3 2 3 1 3 8 #3 5 4 11 5 DQG B D 4 1 5 5 &RPSXWH /HW A D 4 2 1 2 5 3 4 A!1 B ZLWKRXWFRPSXWLQJ A!1 >+LQW A!1 B LVWKHVROXWLRQ RIWKHHTXDWLRQ AX D B @ [ 7! A[ PDSV # )LQGDPDWUL[ $ # $A VXFKWKDWWKHWUDQVIRUPDWLRQ # $ # $ 1 2 1 3 DQG LQWR DQG UHVSHFWLYHO\ >+LQW 3 1 1 7 :ULWHDPDWUL[HTXDWLRQLQYROYLQJ A DQGVROYHIRU A@ # $ # $ 5 4 7 3 DQG B D )LQG A 6XSSRVH AB D #2 3 2 1 6XSSRVH A LVLQYHUWLEOH ([SODLQZK\ ATA LVDOVRLQYHUWLEOH 7KHQVKRZWKDW A!1 D .ATA/!1 AT /HW x1 ; : : : ; xn EHÀ[HGQXPEHUV 7KHPDWUL[EHORZ FDOOHG D 9DQGHUPRQGHPDWUL[ RFFXUVLQDSSOLFDWLRQVVXFKDVVLJQDO SURFHVVLQJ HUURUFRUUHFWLQJFRGHV DQGSRO\QRPLDOLQWHUSR ODWLRQ 2 3 1 x1 x12 """ x1n!1 61 x2 x22 """ x2n!1 7 6 7 V D 6 :: :: :: :: 7 4: : : : 5

1

xn

xn2

"""

xnn!1

*LYHQ \ D .y1 ; : : : ; yn / LQ Rn VXSSRVH F D .c0 ; : : : ; cn!1 / LQ Rn VDWLVÀHV V F D \ DQGGHÀQHWKHSRO\QRPLDO

p.t/ D c0 C c1 t C c2 t 2 C " " " C cn!1 t n!1

D 6KRZ WKDW p.x1 / D y1 ; : : : ; p.xn / D yn :H FDOO p.t/ DQ LQWHUSRODWLQJ SRO\QRPLDO IRU WKH SRLQWV .x1 ; y1 /; : : : ; .xn ; yn / EHFDXVHWKHJUDSKRI p.t/ SDVVHV WKURXJKWKHSRLQWV E 6XSSRVH x1 ; : : : ; xn DUHGLVWLQFWQXPEHUV 6KRZWKDWWKH FROXPQVRI V DUHOLQHDUO\LQGHSHQGHQW >+LQW +RZPDQ\ ]HURVFDQDSRO\QRPLDORIGHJUHH n # 1 KDYH"@

F 3URYH ´,I x1 ; : : : ; xn DUHGLVWLQFWQXPEHUV DQG y1 ; : : : ; yn DUHDUELWUDU\QXPEHUV WKHQWKHUHLVDQLQWHUSRODWLQJSRO\ QRPLDORIGHJUHH ) n # 1 IRU .x1 ; y1 /; : : : ; .xn ; yn /µ

/HW A D LU ZKHUH L LVDQLQYHUWLEOHORZHUWULDQJXODUPD WUL[DQG U LVXSSHUWULDQJXODU ([SODLQZK\WKHÀUVWFROXPQ RI A LVDPXOWLSOHRIWKHÀUVWFROXPQRI L +RZLVWKHVHFRQG FROXPQRI A UHODWHGWRWKHFROXPQVRI L"


*LYHQ X LQ Rn ZLWK XTX D 1 OHW P D XXT DQRXWHUSURGXFW DQG Q D I # 2P -XVWLI\VWDWHPHQWVD E DQGF E P T D P F Q2 D I D P 2 D P 7KH WUDQVIRUPDWLRQ [ 7! P [ LV FDOOHG D SURMHFWLRQ DQG [ 7! Q[ LVFDOOHGD +RXVHKROGHUUHÁHFWLRQ 6XFKUHÁHFWLRQV DUHXVHGLQFRPSXWHUSURJUDPVWRFUHDWHPXOWLSOH]HURVLQD YHFWRUXVXDOO\DFROXPQRIDPDWUL[ 2 3 2 3 0 1 /HW X D 4 0 5 DQG [ D 4 5 5 'HWHUPLQH P DQG Q DVLQ 1 3 ([HUFLVH DQGFRPSXWH P [ DQG Q[ 7KHÀJXUHVKRZVWKDW Q[ LVWKHUHÁHFWLRQRI [ WKURXJKWKH x1 x2 SODQH x3 Px

x u

x2 x ! Px

x1

Qx

$ +RXVHKROGHUUHÁHFWLRQWKURXJKWKHSODQH x3 D 0

6XSSRVH C D E3 E2 E1 B ZKHUH E1 E2 DQG E3 DUHHOHPHQ WDU\PDWULFHV ([SODLQZK\ C LVURZHTXLYDOHQWWR B /HW A EHDQ n ! n VLQJXODUPDWUL[ 'HVFULEHKRZWRFRQVWUXFW DQ n ! n QRQ]HURPDWUL[ B VXFKWKDW AB D 0: /HW A EHD 6 ! 4 PDWUL[DQG B D 4 ! 6 PDWUL[ 6KRZWKDWWKH 6 ! 6 PDWUL[ AB FDQQRWEHLQYHUWLEOH 6XSSRVH A LVD 5 ! 3 PDWUL[DQGWKHUHH[LVWVD 3 ! 5 PDWUL[ C VXFKWKDW CA D I3 6XSSRVHIXUWKHUWKDWIRUVRPHJLYHQ E LQ R5 WKHHTXDWLRQ A[ D E KDVDWOHDVWRQHVROXWLRQ 6KRZ WKDWWKLVVROXWLRQLVXQLTXH

>0@ &HUWDLQG\QDPLFDOV\VWHPVFDQEHVWXGLHGE\H[DPLQLQJ SRZHUVRIDPDWUL[ VXFKDVWKRVHEHORZ 'HWHUPLQHZKDW KDSSHQV WR Ak DQG B k DV k LQFUHDVHV IRU H[DPSOH WU\ k D 2; : : : ; 16/ 7U\WRLGHQWLI\ZKDWLVVSHFLDODERXW A DQG B ,QYHVWLJDWHODUJHSRZHUVRIRWKHUPDWULFHVRIWKLVW\SH DQGPDNHDFRQMHFWXUHDERXWVXFKPDWULFHV 2 3 2 3 :4 :2 :3 0 :2 :3 A D 4 :3 :6 :3 5; B D 4 :1 :6 :3 5 :3 :2 :4 :9 :2 :4 >0@ /HW An EHWKH n ! n PDWUL[ZLWK·VRQWKHPDLQGLDJRQDO IRU n D 4 DQG DQG DQG·VHOVHZKHUH &RPSXWH A!1 n PDNHDFRQMHFWXUHDERXWWKHJHQHUDOIRUPRI A!1 n IRUODUJHU YDOXHVRI n

This page intentionally left blank

'HWHUPLQDQWV

WEB

*/530%6$503: &9".1-&

5DQGRP3DWKVDQG'LVWRUWLRQ ,Q KLV DXWRELRJUDSKLFDO ERRN ´6XUHO\
,QRUGHUWRUHFRUGWKHDFWXDOURXWHVRIWKHDQWVDQGWR FRPPXQLFDWHWKHUHVXOWVWRRWKHUV LWLVFRQYHQLHQWWRXVH DUHFWDQJXODUPDSRIWKHJOREH 7KHUHDUHPDQ\ZD\VWR FUHDWHVXFKPDSV 2QHVLPSOHZD\LVWRXVHWKHORQJLWXGH DQGODWLWXGHRQWKHJOREHDV x DQG y FRRUGLQDWHVRQWKHPDS $VLVWKHFDVHZLWKDOOPDSV WKHUHVXOWLVQRWDIDLWKIXO UHSUHVHQWDWLRQRIWKHJOREH )HDWXUHVQHDUWKH´HTXDWRUµ ORRNPXFKWKHVDPHRQWKHJOREHDQGWKHPDS EXWUHJLRQV QHDUWKH´SROHVµRIWKHJOREHDUHGLVWRUWHG ,PDJHVRISRODU UHJLRQVDUHPXFKODUJHUWKDQWKHLPDJHVRIVLPLODUVL]HG UHJLRQVQHDUWKHHTXDWRU 7RÀWLQZLWKLWVVXUURXQGLQJVRQ WKHPDS WKHLPDJHRIDQDQWQHDURQHRIWKHSROHVVKRXOG EHODUJHUWKDQRQHQHDUWKHHTXDWRU +RZPXFKODUJHU" 6XUSULVLQJO\ ERWKWKHDQWSDWKDQGWKHDUHDGLVWRUWLRQ SUREOHPVDUHEHVWDQVZHUHGWKURXJKWKHXVHRIWKHGHWHUPL QDQW WKHVXEMHFWRIWKLVFKDSWHU ,QGHHG WKHGHWHUPLQDQW KDVVRPDQ\XVHVWKDWDVXPPDU\RIWKHDSSOLFDWLRQVNQRZQ LQWKHHDUO\·VÀOOHGDIRXUYROXPHWUHDWLVHE\7KRPDV 0XLU :LWKFKDQJHVLQHPSKDVLVDQGWKHJUHDWO\LQFUHDVHG VL]HVRIWKHPDWULFHVXVHGLQPRGHUQDSSOLFDWLRQV PDQ\ XVHVWKDWZHUHLPSRUWDQWWKHQDUHQRORQJHUFULWLFDOWRGD\ 1HYHUWKHOHVV WKHGHWHUPLQDQWVWLOOSOD\VDQLPSRUWDQWUROH

1 7KHVROXWLRQWRWKHDQWSDWKSUREOHPDQGWZRRWKHUDSSOLFDWLRQV FDQ EHIRXQGLQD-XQH 0DWKHPDWLFDO0RQWKO\ DUWLFOHE\$UWKXU %HQMDPLQDQG1DRPL&DPHURQ

$)"15&3

'HWHUPLQDQWV

%H\RQGLQWURGXFLQJWKHGHWHUPLQDQWLQ6HFWLRQ WKLVFKDSWHUSUHVHQWVWZRLPSRUWDQW LGHDV 6HFWLRQ GHULYHVDQLQYHUWLELOLW\FULWHULRQIRUDVTXDUHPDWUL[WKDWSOD\VDSLYRWDO UROHLQ&KDSWHU 6HFWLRQ VKRZVKRZWKHGHWHUPLQDQWPHDVXUHVWKHDPRXQWE\ZKLFK DOLQHDUWUDQVIRUPDWLRQFKDQJHVWKHDUHDRIDÀJXUH :KHQDSSOLHGORFDOO\ WKLVWHFKQLTXH DQVZHUVWKHTXHVWLRQRIDPDS·VH[SDQVLRQUDWHQHDUWKHSROHV 7KLVLGHDSOD\VDFULWLFDO UROHLQPXOWLYDULDEOHFDOFXOXVLQWKHIRUPRIWKH-DFRELDQ

*/530%6$5*0/ 50 %&5&3.*/"/54 5HFDOOIURP6HFWLRQ WKDWD 2 ! 2 PDWUL[LVLQYHUWLEOHLIDQGRQO\LILWVGHWHUPLQDQW LVQRQ]HUR 7RH[WHQGWKLVXVHIXOIDFWWRODUJHUPDWULFHV ZHQHHGDGHÀQLWLRQIRUWKH GHWHUPLQDQWRIDQ n ! n PDWUL[ :HFDQGLVFRYHUWKHGHÀQLWLRQIRUWKH 3 ! 3 FDVHE\ ZDWFKLQJZKDWKDSSHQVZKHQDQLQYHUWLEOH 3 ! 3 PDWUL[ A LVURZUHGXFHG &RQVLGHU A D Œaij ! ZLWK a11 ¤ 0 ,IZHPXOWLSO\WKHVHFRQGDQGWKLUGURZVRI A E\ a11 DQGWKHQVXEWUDFWDSSURSULDWHPXOWLSOHVRIWKHÀUVWURZIURPWKHRWKHUWZRURZV ZH ÀQGWKDW A LVURZHTXLYDOHQWWRWKHIROORZLQJWZRPDWULFHV 2 3 2 3 a11 a12 a13 a11 a12 a13 4 a11 a21 a11 a22 a11 a23 5 " 4 0 a11 a22 # a12 a21 a11 a23 # a13 a21 5 a11 a31 a11 a32 a11 a33 0 a11 a32 # a12 a31 a11 a33 # a13 a31

6LQFH A LVLQYHUWLEOH HLWKHUWKH .2; 2/HQWU\RUWKH .3; 2/HQWU\RQWKHULJKWLQ LV QRQ]HUR /HWXVVXSSRVHWKDWWKH .2; 2/HQWU\LVQRQ]HUR 2WKHUZLVH ZHFDQPDNHD URZLQWHUFKDQJHEHIRUHSURFHHGLQJ 0XOWLSO\URZE\ a11 a22 # a12 a21 DQGWKHQWRWKH QHZURZDGG #.a11 a32 # a12 a31 / WLPHVURZ 7KLVZLOOVKRZWKDW 2 3 a11 a12 a13 a11 a22 # a12 a21 a11 a23 # a13 a21 5 A"4 0 0 0 a11 " ZKHUH

" D a11 a22 a33 C a12 a23 a31 C a13 a21 a32 # a11 a23 a32 # a12 a21 a33 # a13 a22 a31

6LQFH A LVLQYHUWLEOH " PXVWEHQRQ]HUR 7KHFRQYHUVHLVWUXH WRR DVZHZLOOVHHLQ 6HFWLRQ :HFDOO " LQ WKH GHWHUPLQDQW RIWKH 3 ! 3 PDWUL[ A 5HFDOOWKDWWKHGHWHUPLQDQWRID 2 ! 2 PDWUL[ A D Œaij ! LVWKHQXPEHU GHW A D a11 a22 # a12 a21 )RUD 1 ! 1 PDWUL[³VD\ A D Œa11 !³ZHGHÀQH GHW A D a11 7RJHQHUDOL]HWKHGHÀQL WLRQRIWKHGHWHUPLQDQWWRODUJHUPDWULFHV ZH·OOXVH 2 ! 2 GHWHUPLQDQWVWRUHZULWHWKH 3 ! 3 GHWHUPLQDQW " GHVFULEHG DERYH 6LQFH WKH WHUPV LQ " FDQ EH JURXSHG DV .a11 a22 a33 # a11 a23 a32 / # .a12 a21 a33 # a12 a23 a31 / C .a13 a21 a32 # a13 a22 a31 / ! " ! " ! " a22 a23 a21 a23 a21 a22 " D a11 $ GHW # a12 $ GHW C a13 $ GHW a32 a33 a31 a33 a31 a32 )RUEUHYLW\ ZULWH

" D a11 $ GHW A11 # a12 $ GHW A12 C a13 $ GHW A13

ZKHUH A11 A12 DQG A13 DUHREWDLQHGIURP A E\GHOHWLQJWKHÀUVWURZDQGRQHRIWKH WKUHHFROXPQV )RUDQ\VTXDUHPDWUL[ A OHW Aij GHQRWHWKHVXEPDWUL[IRUPHGE\GHOHWLQJ

,QWURGXFWLRQWR'HWHUPLQDQWV

WKH i WKURZDQG j WKFROXPQRI A )RULQVWDQFH LI 2 3 1 #2 5 0 62 0 4 #1 7 7 AD6 43 1 0 75 0 4 #2 0

WKHQ A32 LVREWDLQHGE\FURVVLQJRXWURZ DQGFROXPQ 2 3 1 #2 5 0 62 0 4 #1 7 6 7 43 1 0 75 0 4 #2 0 VRWKDW

A32

2

3 1 5 0 D 4 2 4 #1 5 0 #2 0

:HFDQQRZJLYHD UHFXUVLYH GHÀQLWLRQRIDGHWHUPLQDQW :KHQ n D 3 GHW A LVGHÀQHG XVLQJGHWHUPLQDQWVRIWKH 2 ! 2 VXEPDWULFHV A1j DVLQ DERYH :KHQ n D 4 GHW A XVHVGHWHUPLQDQWVRIWKH 3 ! 3 VXEPDWULFHV A1j ,QJHQHUDO DQ n ! n GHWHUPLQDQWLV GHÀQHGE\GHWHUPLQDQWVRI .n # 1/ ! .n # 1/ VXEPDWULFHV

%&'*/*5*0/

)RU n % 2 WKH GHWHUPLQDQW RIDQ n ! n PDWUL[ A D Œaij ! LVWKHVXPRI n WHUPV RIWKHIRUP ˙a1j GHW A1j ZLWKSOXVDQGPLQXVVLJQVDOWHUQDWLQJ ZKHUHWKHHQWULHV a11 ; a12 ; : : : ; a1n DUHIURPWKHÀUVWURZRI A ,QV\PEROV GHW A D a11 GHW A11 # a12 GHW A12 C $ $ $ C .#1/1Cn a1n GHW A1n n X D .#1/1Cj a1j GHW A1j j D1

&9".1-& &RPSXWHWKHGHWHUPLQDQWRI 2

3 1 5 0 A D 4 2 4 #1 5 0 #2 0

40-65*0/ &RPSXWH GHW A D a11 GHW A11 # a12 GHW A12 C a13 GHW A13 ! " ! " ! " 4 #1 2 #1 2 4 GHW A D 1 $ GHW # 5 $ GHW C 0 $ GHW #2 0 0 0 0 #2 D 1.0 # 2/ # 5.0 # 0/ C 0.#4 # 0/ D #2

$QRWKHUFRPPRQQRWDWLRQIRUWKHGHWHUPLQDQWRIDPDWUL[XVHVDSDLURIYHUWLFDO OLQHVLQSODFHRIEUDFNHWV 7KXVWKHFDOFXODWLRQLQ([DPSOHFDQEHZULWWHQDV ˇ ˇ ˇ ˇ ˇ ˇ ˇ 4 #1 ˇ ˇ 2 #1 ˇ ˇ2 4 ˇˇ ˇ ˇ ˇ ˇ ˇ GHW A D 1ˇ # 5ˇ C 0ˇ D $ $ $ D #2 #2 0 ˇ 0 0ˇ 0 #2 ˇ 7RVWDWHWKHQH[WWKHRUHP LWLVFRQYHQLHQWWRZULWHWKHGHÀQLWLRQRI GHW A LQDVOLJKWO\ GLIIHUHQWIRUP *LYHQ A D Œaij ! WKH .i; j /FRIDFWRU RI A LVWKHQXPEHU Cij JLYHQE\

Cij D .#1/i Cj GHW Aij

7KHQ GHW A D a11 C11 C a12 C12 C $ $ $ C a1n C1n

$)"15&3

'HWHUPLQDQWV

7KLVIRUPXODLVFDOOHGD FRIDFWRUH[SDQVLRQDFURVVWKHÀUVWURZ RI A :HRPLWWKH SURRIRIWKHIROORZLQJIXQGDPHQWDOWKHRUHPWRDYRLGDOHQJWK\GLJUHVVLRQ

5)&03&.

7KHGHWHUPLQDQWRIDQ n ! n PDWUL[ A FDQEHFRPSXWHGE\DFRIDFWRUH[SDQVLRQ DFURVVDQ\URZRUGRZQDQ\FROXPQ 7KHH[SDQVLRQDFURVVWKH i WKURZXVLQJWKH FRIDFWRUVLQ LV GHW A D ai1 Ci1 C ai2 Ci2 C $ $ $ C ai n Ci n 7KHFRIDFWRUH[SDQVLRQGRZQWKH j WKFROXPQLV GHW A D a1j C1j C a2j C2j C $ $ $ C anj Cnj 7KHSOXVRUPLQXVVLJQLQWKH .i; j /FRIDFWRUGHSHQGVRQWKHSRVLWLRQRI aij LQWKH PDWUL[ UHJDUGOHVVRIWKHVLJQRI aij LWVHOI 7KHIDFWRU .#1/i Cj GHWHUPLQHVWKHIROORZLQJ FKHFNHUERDUGSDWWHUQRIVLJQV 2 3 C # C $$$ 6# 7 C # 6 7 6C 7 # C 4 5 :: :: : :

&9".1-& 8VHDFRIDFWRUH[SDQVLRQDFURVVWKHWKLUGURZWRFRPSXWH GHW A ZKHUH 2

40-65*0/ &RPSXWH

3 1 5 0 A D 4 2 4 #1 5 0 #2 0

GHW A D a31 C31 C a32 C32 C a33 C33

D .#1/3C1 a31 GHW A31 C .#1/3C2 a32 GHW A32 C .#1/3C3 a33 GHW A33 ˇ ˇ ˇ ˇ ˇ ˇ ˇ5 ˇ1 ˇ1 0 ˇˇ 0 ˇˇ 5 ˇˇ ˇ ˇ ˇ D 0ˇ # .#2/ˇ C 0ˇ 4 #1 ˇ 2 #1 ˇ 2 4ˇ D 0 C 2.#1/ C 0 D #2

7KHRUHPLVKHOSIXOIRUFRPSXWLQJWKHGHWHUPLQDQWRIDPDWUL[WKDWFRQWDLQVPDQ\ ]HURV )RUH[DPSOH LIDURZLVPRVWO\]HURV WKHQWKHFRIDFWRUH[SDQVLRQDFURVVWKDWURZ KDVPDQ\WHUPVWKDWDUH]HUR DQGWKHFRIDFWRUVLQWKRVHWHUPVQHHGQRWEHFDOFXODWHG 7KHVDPHDSSURDFKZRUNVZLWKDFROXPQWKDWFRQWDLQVPDQ\]HURV

&9".1-& &RPSXWH GHW A ZKHUH 2

3 3 #7 8 9 #6 60 2 #5 7 37 6 7 6 A D 60 0 1 5 07 7 40 0 2 4 #1 5 0 0 0 #2 0 40-65*0/ 7KHFRIDFWRUH[SDQVLRQGRZQWKHÀUVWFROXPQRI A KDVDOOWHUPVHTXDOWR ]HURH[FHSWWKHÀUVW 7KXV ˇ ˇ ˇ 2 #5 7 3 ˇˇ ˇ ˇ0 1 5 0 ˇˇ GHW A D 3 $ ˇˇ # 0 $ C21 C 0 $ C31 # 0 $ C41 C 0 $ C51 2 4 #1 ˇˇ ˇ0 ˇ0 0 #2 0 ˇ

,QWURGXFWLRQWR'HWHUPLQDQWV

+HQFHIRUWKZHZLOORPLWWKH]HURWHUPVLQWKHFRIDFWRUH[SDQVLRQ 1H[W H[SDQGWKLV 4 ! 4 GHWHUPLQDQWGRZQWKHÀUVWFROXPQ LQRUGHUWRWDNHDGYDQWDJHRIWKH]HURVWKHUH :HKDYH ˇ ˇ ˇ1 5 0 ˇˇ ˇ GHW A D 3 $ 2 $ ˇˇ 2 4 #1 ˇˇ ˇ 0 #2 0ˇ

7KLV 3 ! 3 GHWHUPLQDQWZDVFRPSXWHGLQ([DPSOHDQGIRXQGWRHTXDO #2 +HQFH GHW A D 3 $ 2 $ .#2/ D #12

7KHPDWUL[LQ([DPSOHZDVQHDUO\WULDQJXODU 7KHPHWKRGLQWKDWH[DPSOHLV HDVLO\DGDSWHGWRSURYHWKHIROORZLQJWKHRUHP

5)&03&.

,I A LVDWULDQJXODUPDWUL[ WKHQ GHW A LVWKHSURGXFWRIWKHHQWULHVRQWKHPDLQ GLDJRQDORI A 7KHVWUDWHJ\LQ([DPSOHRIORRNLQJIRU]HURVZRUNVH[WUHPHO\ZHOOZKHQDQHQWLUH URZRUFROXPQFRQVLVWVRI]HURV ,QVXFKDFDVH WKHFRIDFWRUH[SDQVLRQDORQJVXFKDURZ RUFROXPQLVDVXPRI]HURV 6RWKHGHWHUPLQDQWLV]HUR 8QIRUWXQDWHO\ PRVWFRIDFWRU H[SDQVLRQVDUHQRWVRTXLFNO\HYDOXDWHG

/6.&3*$"- /05& %\WRGD\·VVWDQGDUGV D 25 ! 25 PDWUL[LVVPDOO
([HUFLVHV²H[SORUHLPSRUWDQWSURSHUWLHVRIGHWHUPLQDQWV PRVWO\IRUWKH 2 ! 2 FDVH 7KHUHVXOWVIURP([HUFLVHV²ZLOOEHXVHGLQWKHQH[WVHFWLRQWRGHULYHWKH DQDORJRXVSURSHUWLHVIRU n ! n PDWULFHV

&9&3$*4&4

13"$5*$& 130#-&. ˇ ˇ ˇ 5 #7 2 2 ˇˇ ˇ ˇ 0 3 0 #4 ˇˇ &RPSXWH ˇˇ #5 #8 0 3 ˇˇ ˇ ˇ 0 5 0 #6 ˇ

&RPSXWH WKH GHWHUPLQDQWV LQ ([HUFLVHV ² XVLQJ D FRIDFWRU H[SDQVLRQDFURVVWKHÀUVWURZ ,Q([HUFLVHV² DOVRFRPSXWHWKH GHWHUPLQDQWE\DFRIDFWRUH[SDQVLRQGRZQWKHVHFRQGFROXPQ ˇ ˇ ˇ ˇ ˇ3 ˇ0 0 4 ˇˇ 5 1 ˇˇ ˇ ˇ 3 2 ˇˇ 0 ˇˇ ˇˇ 4 #3 ˇˇ 2 ˇ0 ˇ2 5 #1 ˇ 4 1ˇ

ˇ ˇ ˇ 2 #4 3 ˇˇ ˇ 1 2 ˇˇ ˇˇ 3 ˇ1 4 #1 ˇ ˇ ˇ2 ˇ ˇˇ 4 ˇ5

ˇ 3 #4 ˇˇ 0 5 ˇˇ 1 6ˇ

ˇ ˇ1 ˇ ˇˇ 2 ˇ3

3 1 4

ˇ 5 ˇˇ 1 ˇˇ 2ˇ

ˇ ˇ ˇ 5 #2 4 ˇˇ ˇ 3 #5 ˇˇ ˇˇ 0 ˇ 2 #4 7ˇ

'HWHUPLQDQWV

$)"15&3

ˇ ˇ4 ˇ ˇˇ 6 ˇ9

3 5 7


ˇ ˇ8 1 ˇ 0 ˇˇ 4 ˇ 3 #2


&RPSXWHWKHGHWHUPLQDQWVLQ([HUFLVHV²E\FRIDFWRUH[SDQ VLRQV $WHDFKVWHS FKRRVHDURZRUFROXPQWKDWLQYROYHVWKHOHDVW DPRXQWRIFRPSXWDWLRQ ˇ ˇ ˇ ˇ ˇ6 ˇ 1 #2 0 0 5 ˇˇ 5 2 ˇˇ ˇ ˇ ˇ1 ˇ0 7 2 #5 ˇˇ 0 3 0 ˇˇ ˇˇ ˇˇ ˇ 2 0 0 0 2 #6 #7 5 ˇˇ ˇ ˇ ˇ ˇ8 ˇ5 3 1 8ˇ 0 4 4ˇ ˇ ˇ ˇ ˇ ˇ3 ˇ4 5 #8 4 ˇˇ 0 0 0 ˇˇ ˇ ˇ ˇ 0 #2 ˇ 7 #1 3 #7 ˇˇ 0 0 ˇˇ ˇˇ ˇˇ ˇ 0 1 5ˇ 6 3 0 ˇˇ ˇ0 ˇ2 ˇ0 ˇ 5 #8 0 0 2ˇ 4 #3 ˇ ˇ ˇ ˇ4 0 #7 3 #5 ˇˇ ˇ ˇ0 0 2 0 0 ˇˇ ˇ 3 #6 4 #8 ˇˇ ˇˇ 7 ˇ5 0 5 2 #3 ˇˇ ˇ ˇ0 0 9 #1 2ˇ ˇ ˇ ˇ6 3 2 4 0 ˇˇ ˇ ˇ9 0 #4 1 0 ˇˇ ˇ 6 7 1 ˇˇ ˇˇ 8 #5 ˇ3 0 0 0 0 ˇˇ ˇ ˇ4 2 3 2 0ˇ

7KHH[SDQVLRQRID 3 ! 3 GHWHUPLQDQWFDQEHUHPHPEHUHGE\WKH IROORZLQJGHYLFH :ULWHDVHFRQGFRS\RIWKHÀUVWWZRFROXPQVWR WKHULJKWRIWKHPDWUL[ DQGFRPSXWHWKHGHWHUPLQDQWE\PXOWLSO\ LQJHQWULHVRQVL[GLDJRQDOV –

–

–

a11 a12 a13 a11 a12 a21 a22 a23 a21 a22 a 31 a 32 a 33 a 31 a 32 +

+

+

$GGWKHGRZQZDUGGLDJRQDOSURGXFWVDQGVXEWUDFWWKHXSZDUG SURGXFWV 8VHWKLVPHWKRGWRFRPSXWHWKHGHWHUPLQDQWVLQ([ HUFLVHV² :DUQLQJ 7KLVWULFNGRHVQRWJHQHUDOL]HLQDQ\ UHDVRQDEOHZD\WR 4 ! 4 RUODUJHUPDWULFHV ˇ ˇ ˇ ˇ ˇ3 ˇ0 0 4 ˇˇ 5 1 ˇˇ ˇ ˇ 3 2 ˇˇ 0 ˇˇ ˇˇ 2 ˇˇ 4 #3 ˇ0 ˇ ˇ 5 #1 2 4 1ˇ ˇ ˇ ˇ ˇ ˇ 2 #4 ˇ1 3 ˇˇ 3 5 ˇˇ ˇ ˇ 1 2 ˇˇ 1 1 ˇˇ ˇˇ 3 ˇˇ 2 ˇ1 ˇ3 4 #1 ˇ 4 2ˇ

,Q ([HUFLVHV ² H[SORUH WKH HIIHFW RI DQ HOHPHQWDU\ URZ RSHUDWLRQRQWKHGHWHUPLQDQWRIDPDWUL[ ,QHDFKFDVH VWDWHWKH URZRSHUDWLRQDQGGHVFULEHKRZLWDIIHFWVWKHGHWHUPLQDQW ! " ! " ! " ! " a b c d a b a b c d a b c d kc kd

" ! " 4 3 4 6 5 C 3k 6 C 4k ! " ! " a b a C kc b C kd c d c d 2 3 2 3 1 1 1 k k k 8 #4 5 4 #3 8 #4 5 4 #3 2 #3 2 2 #3 2 2 3 2 3 a b c 3 2 2 2 2 5 4 a b c5 4 3 6 5 6 6 5 6

!

3 5

&RPSXWHWKHGHWHUPLQDQWVRIWKHHOHPHQWDU\PDWULFHVJLYHQLQ ([HUFLVHV² 6HH6HFWLRQ 2 3 2 3 1 0 0 1 0 0 1 05 1 05 4 0 4 0 0 k 1 k 0 1 2 3 2 3 k 0 0 1 0 0 1 05 k 05 4 0 4 0 0 0 1 0 0 1 2 3 2 3 0 1 0 0 0 1 0 05 1 05 4 1 4 0 0 0 1 1 0 0 8VH([HUFLVHV²WRDQVZHUWKHTXHVWLRQVLQ([HUFLVHVDQG *LYHUHDVRQVIRU\RXUDQVZHUV

:KDWLVWKHGHWHUPLQDQWRIDQHOHPHQWDU\URZUHSODFHPHQW PDWUL[" :KDWLVWKHGHWHUPLQDQWRIDQHOHPHQWDU\VFDOLQJPDWUL[ZLWK k RQWKHGLDJRQDO" ,Q([HUFLVHV² YHULI\WKDW GHW EA D .GHW ! E/.GHW"A/ ZKHUH a b E LVWKHHOHPHQWDU\PDWUL[VKRZQDQG A D c d ! " ! " 0 1 1 0 1 0 0 k ! " ! " 1 k 1 0 0 1 k 1 ! " 3 1 /HW A D :ULWH 5A ,V GHW 5A D 5 GHW A" 4 2 ! " a b /HW A D DQGOHW k EHDVFDODU )LQGDIRUPXODWKDW c d UHODWHV GHW kA WR k DQG GHW A ,Q([HUFLVHVDQG A LVDQ n ! n PDWUL[ 0DUNHDFKVWDWHPHQW 7UXHRU)DOVH -XVWLI\HDFKDQVZHU D $Q n ! n GHWHUPLQDQW LV GHÀQHG E\ GHWHUPLQDQWV RI .n # 1/ ! .n # 1/ VXEPDWULFHV E 7KH .i; j /FRIDFWRURIDPDWUL[ A LVWKHPDWUL[ Aij RE WDLQHGE\GHOHWLQJIURP A LWV i WKURZDQG j WKFROXPQ

D 7KHFRIDFWRUH[SDQVLRQRI GHW A GRZQDFROXPQLVWKH QHJDWLYHRIWKHFRIDFWRUH[SDQVLRQDORQJDURZ

E 7KHGHWHUPLQDQWRIDWULDQJXODUPDWUL[LVWKHVXPRIWKH HQWULHVRQWKHPDLQGLDJRQDO ! " ! " 3 1 DQG Y D &RPSXWHWKHDUHDRIWKHSDU /HW X D 0 2 DOOHORJUDPGHWHUPLQHGE\ X Y X C Y DQG DQGFRPSXWH WKHGHWHUPLQDQWRI Œ X Y ! +RZGRWKH\FRPSDUH" 5HSODFH WKHÀUVWHQWU\RI Y E\DQDUELWUDU\QXPEHU x DQGUHSHDWWKH SUREOHP 'UDZDSLFWXUHDQGH[SODLQZKDW\RXÀQG ! " ! " a c DQG Y D ZKHUH a b c DUHSRVLWLYHIRU /HW u D b 0 VLPSOLFLW\ &RPSXWHWKHDUHDRIWKHSDUDOOHORJUDPGHWHU PLQHGE\ X Y X C Y DQG DQGFRPSXWHWKHGHWHUPLQDQWVRI WKHPDWULFHV Œ X Y ! DQG Œ Y X ! 'UDZDSLFWXUHDQGH[SODLQ ZKDW\RXÀQG >0@ ,VLWWUXHWKDW GHW.A C B/ D GHW A C GHW B " 7RÀQG RXW JHQHUDWHUDQGRP 5 ! 5 PDWULFHV A DQG B DQGFRPSXWH GHW.A C B/ # GHW A # GHW B 5HIHUWR([HUFLVHLQ6HF

3URSHUWLHVRI'HWHUPLQDQWV

WLRQ 5HSHDWWKHFDOFXODWLRQVIRUWKUHHRWKHUSDLUVRI n ! n PDWULFHV IRUYDULRXVYDOXHVRI n 5HSRUW\RXUUHVXOWV

>0@ ,V LW WUXH WKDW GHW AB D .GHW A/.GHW B/" ([SHULPHQW ZLWKIRXUSDLUVRIUDQGRPPDWULFHVDVLQ([HUFLVH DQG PDNHDFRQMHFWXUH

>0@ &RQVWUXFWDUDQGRP 4 ! 4 PDWUL[ A ZLWKLQWHJHUHQWULHV EHWZHHQ #9 DQG DQGFRPSDUH GHW A ZLWK GHW AT GHW.#A/ GHW.2A/ DQG GHW.10A/ 5HSHDWZLWKWZRRWKHUUDQGRP 4 ! 4 LQWHJHUPDWULFHV DQGPDNHFRQMHFWXUHVDERXWKRZWKHVHGH WHUPLQDQWVDUHUHODWHG 5HIHUWR([HUFLVH LQ6HFWLRQ 7KHQFKHFN\RXUFRQMHFWXUHVZLWKVHYHUDOUDQGRP 5 ! 5 DQG 6 ! 6 LQWHJHUPDWULFHV 0RGLI\\RXUFRQMHFWXUHV LIQHFHV VDU\ DQGUHSRUW\RXUUHVXOWV >0@ +RZ LV GHW A!1 UHODWHG WR GHW A" ([SHULPHQW ZLWK UDQGRP n ! n LQWHJHUPDWULFHVIRU n D 4 DQG DQGPDNH DFRQMHFWXUH 1RWH ,QWKHXQOLNHO\HYHQWWKDW\RXHQFRXQWHU DPDWUL[ZLWKD]HURGHWHUPLQDQW UHGXFHLWWRHFKHORQIRUP DQGGLVFXVVZKDW\RXÀQG

40-65*0/ 50 13"$5*$& 130#-&. 7DNHDGYDQWDJHRIWKH]HURV %HJLQZLWKDFRIDFWRUH[SDQVLRQGRZQWKHWKLUGFROXPQWR REWDLQD 3 ! 3 PDWUL[ ZKLFKPD\EHHYDOXDWHGE\DQH[SDQVLRQGRZQLWVÀUVWFROXPQ ˇ ˇ ˇ 5 #7 ˇ ˇ 2 2 ˇˇ ˇ ˇ 0 3 #4 ˇˇ ˇ 0 ˇ ˇ 3 0 #4 ˇ ˇ D .#1/1C3 2ˇ #5 #8 3 ˇˇ ˇ #5 #8 ˇ 0 3 ˇˇ ˇ ˇ 0 5 #6 ˇ ˇ 0 5 0 #6 ˇ ˇ ˇ ˇ 3 #4 ˇ 2C1 ˇ ˇ D 20 D 2 $ .#1/ .#5/ˇ 5 #6 ˇ 7KH .#1/2C1 LQWKHQH[WWRODVWFDOFXODWLRQFDPHIURPWKH .2; 1/SRVLWLRQRIWKH #5 LQ WKH 3 ! 3 GHWHUPLQDQW

1301&35*&4 0' %&5&3.*/"/54 7KHVHFUHWRIGHWHUPLQDQWVOLHVLQKRZWKH\FKDQJHZKHQURZRSHUDWLRQVDUHSHUIRUPHG 7KHIROORZLQJWKHRUHPJHQHUDOL]HVWKHUHVXOWVRI([HUFLVHV²LQ6HFWLRQ 7KH SURRILVDWWKHHQGRIWKLVVHFWLRQ

5)&03&.

3PX 0QFSBUJPOT /HW A EHDVTXDUHPDWUL[ D ,IDPXOWLSOHRIRQHURZRI A LVDGGHGWRDQRWKHUURZWRSURGXFHDPDWUL[ B WKHQ GHW B D GHW A E ,IWZRURZVRI A DUHLQWHUFKDQJHGWRSURGXFH B WKHQ GHW B D # GHW A F ,IRQHURZRI A LVPXOWLSOLHGE\ k WRSURGXFH B WKHQ GHW B D k $ GHW A 7KH IROORZLQJ H[DPSOHV VKRZ KRZ WR XVH 7KHRUHP WR ÀQG GHWHUPLQDQWV HIÀFLHQWO\

$)"15&3

'HWHUPLQDQWV

2

3 1 #4 2 &9".1-& &RPSXWH GHW A ZKHUH A D 4 #2 8 #9 5 #1 7 0

40-65*0/ 7KHVWUDWHJ\LVWRUHGXFH A WRHFKHORQIRUPDQGWKHQWRXVHWKHIDFWWKDW WKHGHWHUPLQDQWRIDWULDQJXODUPDWUL[LVWKHSURGXFWRIWKHGLDJRQDOHQWULHV 7KHÀUVW WZRURZUHSODFHPHQWVLQFROXPQGRQRWFKDQJHWKHGHWHUPLQDQW ˇ ˇ ˇ ˇ ˇ ˇ ˇ 1 #4 2 ˇˇ ˇˇ 1 #4 2 ˇˇ ˇˇ 1 #4 2 ˇˇ ˇ GHW A D ˇˇ #2 8 #9 ˇˇ D ˇˇ 0 0 #5 ˇˇ D ˇˇ 0 0 #5 ˇˇ ˇ #1 7 0 ˇ ˇ #1 7 0 ˇ ˇ 0 3 2 ˇ $QLQWHUFKDQJHRIURZVDQGUHYHUVHVWKHVLJQRIWKHGHWHUPLQDQW VR ˇ ˇ ˇ 1 #4 2 ˇˇ ˇ 2 ˇˇ D #.1/.3/.#5/ D 15 GHW A D #ˇˇ 0 3 ˇ0 0 #5 ˇ

$ FRPPRQXVHRI7KHRUHP F LQKDQGFDOFXODWLRQVLVWR IDFWRURXWDFRPPRQ PXOWLSOHRIRQHURZ RIDPDWUL[ )RULQVWDQFH ˇ ˇ ˇ ˇ ˇ & ˇ& & & ˇˇ & & ˇˇ ˇ ˇ ˇ 5k #2k 3k ˇˇ D k ˇˇ 5 #2 3 ˇˇ ˇ ˇ & ˇ ˇ & & & & &ˇ ZKHUHWKHVWDUUHGHQWULHVDUHXQFKDQJHG :HXVHWKLVVWHSLQWKHQH[WH[DPSOH 2

2 #8 6 3 #9 &9".1-& &RPSXWH GHW A ZKHUH A D 6 4 #3 0 1 #4

6 5 1 0

3 8 10 7 7 #2 5 6

40-65*0/ 7RVLPSOLI\WKHDULWKPHWLF ZHZDQWDLQWKHXSSHUOHIWFRUQHU :HFRXOG LQWHUFKDQJHURZVDQG ,QVWHDG ZHIDFWRURXWIURPWKHWRSURZ DQGWKHQSURFHHG ZLWKURZUHSODFHPHQWVLQWKHÀUVWFROXPQ ˇ ˇ ˇ ˇ ˇ 1 #4 ˇ 1 #4 3 4 ˇˇ 3 4 ˇˇ ˇ ˇ ˇ 3 #9 ˇ0 5 10 ˇˇ 3 #4 #2 ˇˇ GHW A D 2ˇˇ D 2ˇˇ ˇ 0 1 #2 ˇ 10 10 ˇˇ ˇ #3 ˇ 0 #12 ˇ 1 #4 ˇ ˇ 0 6 0 0 #3 2ˇ 1H[W ZHFRXOGIDFWRURXWDQRWKHUIURPURZ RUXVHWKHLQWKHVHFRQGFROXPQDVD SLYRW :HFKRRVHWKHODWWHURSHUDWLRQ DGGLQJWLPHVURZ WRURZ ˇ ˇ ˇ 1 #4 3 4 ˇˇ ˇ ˇ0 3 #4 #2 ˇˇ GHW A D 2ˇˇ 0 #6 2 ˇˇ ˇ0 ˇ0 0 #3 2 ˇ

)LQDOO\ DGGLQJ #1=2 WLPHVURZ WRURZ DQGFRPSXWLQJWKH´WULDQJXODUµGHWHUPLQDQW ZHÀQGWKDW ˇ ˇ ˇ 1 #4 3 4 ˇˇ ˇ ˇ0 3 #4 #2 ˇˇ GHW A D 2ˇˇ D 2 $ .1/.3/.#6/.1/ D #36 0 #6 2 ˇˇ ˇ0 ˇ0 0 0 1ˇ

U=

0 0 0

* 0 0

* * 0

U=

* 0 0

* * 0 0

* * *

6XSSRVHDVTXDUHPDWUL[ A KDVEHHQUHGXFHGWRDQHFKHORQIRUP U E\URZUHSODFH PHQWVDQGURZLQWHUFKDQJHV 7KLVLVDOZD\VSRVVLEOH 6HHWKHURZUHGXFWLRQDOJRULWKP LQ6HFWLRQ ,IWKHUHDUH r LQWHUFKDQJHV WKHQ7KHRUHPVKRZVWKDW

* *

6LQFH U LVLQHFKHORQIRUP LWLVWULDQJXODU DQGVR GHW U LVWKHSURGXFWRIWKH GLDJRQDOHQWULHV u11 ; : : : ; unn ,I A LVLQYHUWLEOH WKHHQWULHV ui i DUHDOOSLYRWVEHFDXVH A " In DQGWKH ui i KDYHQRWEHHQVFDOHGWR·V 2WKHUZLVH DWOHDVW unn LV]HUR DQGWKH SURGXFW u11 $ $ $ unn LV]HUR 6HH)LJ 7KXV

det U ≠ 0

0 0 0


0

det U = 0 '*(63&

7\SLFDOHFKHORQIRUPVRIVTXDUH PDWULFHV

GHW A D .#1/r GHW U

! 8 ˆ <.#1/r $ SURGXFWRI GHW A D SLYRWVLQ U ˆ : 0

ZKHQ A LVLQYHUWLEOH

ZKHQ A LVQRWLQYHUWLEOH

,WLVLQWHUHVWLQJWRQRWHWKDWDOWKRXJKWKHHFKHORQIRUP U GHVFULEHGDERYHLVQRWXQLTXH EHFDXVHLWLVQRWFRPSOHWHO\URZUHGXFHG DQGWKHSLYRWVDUHQRWXQLTXH WKH SURGXFW RIWKHSLYRWV LV XQLTXH H[FHSWIRUDSRVVLEOHPLQXVVLJQ )RUPXOD QRWRQO\JLYHVDFRQFUHWHLQWHUSUHWDWLRQRI GHW A EXWDOVRSURYHVWKH PDLQWKHRUHPRIWKLVVHFWLRQ

5)&03&.

$ VTXDUHPDWUL[ A LVLQYHUWLEOHLIDQGRQO\LI GHW A ¤ 0 7KHRUHPDGGVWKHVWDWHPHQW´GHW A ¤ 0µWRWKH,QYHUWLEOH0DWUL[7KHRUHP $ XVHIXOFRUROODU\LVWKDW GHW A D 0 ZKHQWKHFROXPQVRI A DUHOLQHDUO\GHSHQGHQW $OVR GHW A D 0 ZKHQWKH URZV RI A DUHOLQHDUO\GHSHQGHQW 5RZVRI A DUHFROXPQVRI AT DQGOLQHDUO\GHSHQGHQWFROXPQVRI AT PDNH AT VLQJXODU :KHQ AT LVVLQJXODU VRLV A E\WKH,QYHUWLEOH0DWUL[7KHRUHP ,QSUDFWLFH OLQHDUGHSHQGHQFHLVREYLRXVZKHQ WZRFROXPQVRUWZRURZVDUHWKHVDPHRUDFROXPQRUDURZLV]HUR 2 3 3 #1 2 #5 6 0 5 #3 #6 7 7 &9".1-& &RPSXWH GHW A ZKHUH A D 6 4 #6 7 #7 4 5 #5 #8 0 9

40-65*0/ $GGWLPHVURZWRURZWRREWDLQ 2 3 3 #1 2 #5 6 0 5 #3 #6 7 7D0 GHW A D GHW 6 4 0 5 #3 #6 5 #5 #8 0 9

EHFDXVHWKHVHFRQGDQGWKLUGURZVRIWKHVHFRQGPDWUL[DUHHTXDO

/6.&3*$"- /05&4

WEB

0RVWFRPSXWHUSURJUDPVWKDWFRPSXWH GHW A IRUDJHQHUDOPDWUL[ A XVHWKH PHWKRGRIIRUPXOD DERYH ,WFDQEHVKRZQWKDWHYDOXDWLRQRIDQ n ! n GHWHUPLQDQWXVLQJURZRSHUDWLRQV UHTXLUHVDERXW 2n3 =3 DULWKPHWLFRSHUDWLRQV $Q\PRGHUQPLFURFRPSXWHUFDQ FDOFXODWHD 25 ! 25 GHWHUPLQDQWLQDIUDFWLRQRIDVHFRQG VLQFHRQO\DERXW RSHUDWLRQVDUHUHTXLUHG

$)"15&3

'HWHUPLQDQWV

&RPSXWHUVFDQDOVRKDQGOHODUJH´VSDUVHµPDWULFHV ZLWKVSHFLDOURXWLQHVWKDWWDNH DGYDQWDJHRIWKHSUHVHQFHRIPDQ\]HURV 2IFRXUVH ]HURHQWULHVFDQVSHHGKDQGFRPSX WDWLRQV WRR 7KHFDOFXODWLRQVLQWKHQH[WH[DPSOHFRPELQHWKHSRZHURIURZRSHUDWLRQV ZLWKWKHVWUDWHJ\IURP6HFWLRQ RIXVLQJ]HURHQWULHVLQFRIDFWRUH[SDQVLRQV 2

3 0 1 2 #1 6 2 5 #7 3 7 7 &9".1-& &RPSXWH GHW A ZKHUH A D 6 4 0 3 6 25 #2 #5 4 #2

40-65*0/ $ JRRGZD\WREHJLQLVWRXVHWKHLQFROXPQ DVDSLYRW HOLPLQDWLQJ WKH #2 EHORZLW 7KHQXVHDFRIDFWRUH[SDQVLRQWRUHGXFHWKHVL]HRIWKHGHWHUPLQDQW IROORZHGE\DQRWKHUURZUHSODFHPHQWRSHUDWLRQ 7KXV ˇ ˇ ˇ0 ˇ ˇ ˇ ˇ 1 2 #1 ˇˇ ˇ ˇ1 ˇ1 2 #1 ˇˇ 2 #1 ˇˇ ˇ2 ˇ ˇ ˇ 5 #7 3 ˇ GHW A D ˇˇ D #2ˇˇ 3 6 2 ˇˇ D #2ˇˇ 0 0 5 ˇˇ 0 3 6 2 ˇˇ ˇ ˇ 0 #3 ˇ 0 #3 1ˇ 1ˇ ˇ0 0 #3 1 ˇ $QLQWHUFKDQJHRIURZVDQGZRXOGSURGXFHD´WULDQJXODUGHWHUPLQDQWµ $QRWKHU DSSURDFKLVWRPDNHDFRIDFWRUH[SDQVLRQGRZQWKHÀUVWFROXPQ ˇ ˇ ˇ 0 5 ˇˇ ˇ GHW A D .#2/.1/ˇ D #2 $ .15/ D #30 #3 1 ˇ

&ROXPQ2SHUDWLRQV

:HFDQSHUIRUPRSHUDWLRQVRQWKHFROXPQVRIDPDWUL[LQDZD\WKDWLVDQDORJRXVWRWKH URZRSHUDWLRQVZHKDYHFRQVLGHUHG 7KHQH[WWKHRUHPVKRZVWKDWFROXPQRSHUDWLRQV KDYHWKHVDPHHIIHFWVRQGHWHUPLQDQWVDVURZRSHUDWLRQV

5)&03&.

,I A LVDQ n ! n PDWUL[ WKHQ GHW AT D GHW A

1300' 7KHWKHRUHPLVREYLRXVIRU n D 1 6XSSRVHWKHWKHRUHPLVWUXHIRU k ! k GHWHUPLQDQWVDQGOHW n D k C 1 7KHQWKHFRIDFWRURI a1j LQ A HTXDOVWKHFRIDFWRU RI aj1 LQ AT EHFDXVHWKHFRIDFWRUVLQYROYH k ! k GHWHUPLQDQWV +HQFHWKHFRIDFWRU H[SDQVLRQRI GHW A DORQJWKHÀUVW URZ HTXDOVWKHFRIDFWRUH[SDQVLRQRI GHW AT GRZQWKH ÀUVW FROXPQ 7KDWLV A DQG AT KDYHHTXDOGHWHUPLQDQWV 7KXVWKHWKHRUHPLVWUXHIRU n D 1 DQGWKHWUXWKRIWKHWKHRUHPIRURQHYDOXHRI n LPSOLHVLWVWUXWKIRUWKHQH[WYDOXH RI n %\WKHSULQFLSOHRILQGXFWLRQ WKHWKHRUHPLVWUXHIRUDOO n % 1 %HFDXVHRI7KHRUHP HDFKVWDWHPHQWLQ7KHRUHPLVWUXHZKHQWKHZRUG URZ LV UHSODFHGHYHU\ZKHUHE\ FROXPQ 7RYHULI\WKLVSURSHUW\ RQHPHUHO\DSSOLHVWKHRULJLQDO 7KHRUHPWR AT $ URZRSHUDWLRQRQ AT DPRXQWVWRDFROXPQRSHUDWLRQRQ A &ROXPQRSHUDWLRQVDUHXVHIXOIRUERWKWKHRUHWLFDOSXUSRVHVDQGKDQGFRPSXWDWLRQV +RZHYHU IRUVLPSOLFLW\ZH·OOSHUIRUPRQO\URZRSHUDWLRQVLQQXPHULFDOFDOFXODWLRQV

'HWHUPLQDQWVDQG0DWUL[3URGXFWV 7KHSURRIRIWKHIROORZLQJXVHIXOWKHRUHPLVDWWKHHQGRIWKHVHFWLRQ $SSOLFDWLRQVDUH LQWKHH[HUFLVHV

5)&03&.


.VMUJQMJDBUJWF 1SPQFSUZ ,I A DQG B DUH n ! n PDWULFHV WKHQ GHW AB D .GHW A/.GHW B/

&9".1-& 9HULI\7KHRUHPIRU A D 40-65*0/ AB D

!

6 3

1 2

"!

4 1

!

6 3

" ! 1 4 DQG B D 2 1

3 2

"

!

D

25 14

20 13

" 3 2

"

DQG GHW AB D 25 $ 13 # 20 $ 14 D 325 # 280 D 45 6LQFH GHW A D 9 DQG GHW B D 5

.GHW A/.GHW B/ D 9 $ 5 D 45 D GHW AB :DUQLQJ $ FRPPRQPLVFRQFHSWLRQLVWKDW7KHRUHPKDVDQDQDORJXHIRU VXPV RI PDWULFHV +RZHYHU GHW.A C B/ LV QRW HTXDOWR GHW A C GHW B LQJHQHUDO

$ /LQHDULW\3URSHUW\RIWKH'HWHUPLQDQW)XQFWLRQ )RUDQ n ! n PDWUL[ A ZHFDQFRQVLGHU GHW A DVDIXQFWLRQRIWKH n FROXPQYHFWRUVLQ A :HZLOOVKRZWKDWLIDOOFROXPQVH[FHSWRQHDUHKHOGÀ[HG WKHQ GHW A LVD OLQHDU IXQFWLRQ RIWKDWRQHYHFWRU YDULDEOH 6XSSRVHWKDWWKH j WKFROXPQRI A LVDOORZHGWRYDU\ DQGZULWH # $ $$$ Dj !1 [ Dj C1 $$$ Dn A D D1 'HÀQHDWUDQVIRUPDWLRQ T IURP Rn WR R E\ # $$$ Dj !1 T .[/ D GHW D1

[

Dj C1

$$$

7KHQ

T .c [/ D cT .[/ IRUDOOVFDODUV c DQGDOO [ LQ Rn T .X C Y/ D T .X/ C T .Y/ IRUDOO X Y LQ Rn

Dn

$

3URSHUW\ LV7KHRUHPF DSSOLHGWRWKHFROXPQVRI A $ SURRIRISURSHUW\ IROORZVIURPDFRIDFWRUH[SDQVLRQRI GHW A GRZQWKH j WKFROXPQ 6HH([HUFLVH 7KLV PXOWL OLQHDULW\SURSHUW\RIWKHGHWHUPLQDQWWXUQVRXWWRKDYHPDQ\XVHIXOFRQVH TXHQFHVWKDWDUHVWXGLHGLQPRUHDGYDQFHGFRXUVHV

3URRIVRI7KHRUHPV DQG ,WLVFRQYHQLHQWWRSURYH7KHRUHPZKHQLWLVVWDWHGLQWHUPVRIWKHHOHPHQWDU\PDWULFHV GLVFXVVHGLQ6HFWLRQ :HFDOODQHOHPHQWDU\PDWUL[ E D URZUHSODFHPHQW PDWUL[ LI E LVREWDLQHGIURPWKHLGHQWLW\ I E\DGGLQJDPXOWLSOHRIRQHURZWRDQRWKHUURZ E LV DQ LQWHUFKDQJH LI E LVREWDLQHGE\LQWHUFKDQJLQJWZRURZVRI I DQG E LV DVFDOHE\ r LI E LVREWDLQHGE\PXOWLSO\LQJDURZRI I E\DQRQ]HURVFDODU r :LWKWKLVWHUPLQRORJ\ 7KHRUHPFDQEHUHIRUPXODWHGDVIROORZV

$)"15&3

'HWHUPLQDQWV

,I A LVDQ n ! n PDWUL[DQG E LVDQ n ! n HOHPHQWDU\PDWUL[ WKHQ GHW ($ D .GHW E/.GHW A/ ZKHUH

8 ˆ < 1 GHW E D #1 ˆ : r

LI E LVDURZUHSODFHPHQW LI E LVDQLQWHUFKDQJH LI E LVDVFDOHE\ r

1300' 0' 5)&03&. 7KHSURRILVE\LQGXFWLRQRQWKHVL]HRI A 7KHFDVHRID 2 ! 2 PDWUL[ZDVYHULÀHGLQ([HUFLVHV²RI6HFWLRQ 6XSSRVHWKHWKHRUHPKDV EHHQYHULÀHGIRUGHWHUPLQDQWVRI k ! k PDWULFHVZLWK k % 2 OHW n D k C 1 DQGOHW A EH n ! n 7KHDFWLRQRI E RQ A LQYROYHVHLWKHUWZRURZVRURQO\RQHURZ 6RZH FDQH[SDQG GHW EA DFURVVDURZWKDWLVXQFKDQJHGE\WKHDFWLRQRI E VD\ URZ i /HW Aij UHVSHFWLYHO\ Bij / EHWKHPDWUL[REWDLQHGE\GHOHWLQJURZ i DQGFROXPQ j IURP A UHVSHFWLYHO\ EA 7KHQWKHURZVRI Bij DUHREWDLQHGIURPWKHURZVRI Aij E\WKHVDPH W\SHRIHOHPHQWDU\URZRSHUDWLRQWKDW E SHUIRUPVRQ A 6LQFHWKHVHVXEPDWULFHVDUH RQO\ k ! k WKHLQGXFWLRQDVVXPSWLRQLPSOLHVWKDW GHW Bij D ˛ $ GHW Aij ZKHUH ˛ D 1 #1 RU r GHSHQGLQJRQWKHQDWXUHRI E 7KHFRIDFWRUH[SDQVLRQDFURVV URZ i LV GHW ($ D ai1 .#1/i C1 GHW Bi1 C $ $ $ C ai n .#1/i Cn GHW Bi n

D ˛ai1 .#1/i C1 GHW Ai1 C $ $ $ C ˛ai n .#1/i Cn GHW Ai n D ˛ $ GHW A

,QSDUWLFXODU WDNLQJ A D In ZHVHHWKDW GHW E D 1 #1 RU r GHSHQGLQJRQWKHQDWXUH RI E 7KXVWKHWKHRUHPLVWUXHIRU n D 2 DQGWKHWUXWKRIWKHWKHRUHPIRURQHYDOXHRI n LPSOLHVLWVWUXWKIRUWKHQH[WYDOXHRI n %\WKHSULQFLSOHRILQGXFWLRQ WKHWKHRUHP PXVWEHWUXHIRU n % 2 7KHWKHRUHPLVWULYLDOO\WUXHIRU n D 1

1300' 0' 5)&03&. ,I A LVQRWLQYHUWLEOH WKHQQHLWKHULV $% E\([HUFLVH LQ6HFWLRQ ,QWKLVFDVH GHW $% D .GHW A/.GHW B/ EHFDXVHERWKVLGHVDUH]HUR E\ 7KHRUHP ,I A LVLQYHUWLEOH WKHQ A DQGWKHLGHQWLW\PDWUL[ In DUHURZHTXLYDOHQWE\ WKH,QYHUWLEOH0DWUL[7KHRUHP 6RWKHUHH[LVWHOHPHQWDU\PDWULFHV E1 ; : : : ; Ep VXFKWKDW A D Ep Ep!1 $ $ $ E1 $ In D Ep Ep!1 $ $ $ E1 )RUEUHYLW\ ZULWH jAj IRU GHW A 7KHQUHSHDWHGDSSOLFDWLRQRI7KHRUHP DVUHSKUDVHG DERYH VKRZVWKDW

j$%j D jEp $ $ $ E1 Bj D jEp jjEp!1 $ $ $ E1 Bj D $ $ $ D jEp j $ $ $ jE1 jjBj D $ $ $ D jEp $ $ $ E1 jjBj D jAjjBj 13"$5*$& 130#-&.4 ˇ ˇ ˇ 1 #3 1 #2 ˇˇ ˇ ˇ 2 #5 #1 #2 ˇ ˇ LQDVIHZVWHSVDVSRVVLEOH &RPSXWH ˇˇ 5 1 ˇˇ ˇ 0 #4 ˇ #3 10 #6 8 ˇ


8VHDGHWHUPLQDQWWRGHFLGHLI Y1 Y2 Y3 DUHOLQHDUO\LQGHSHQGHQW ZKHQ 2 3 2 3 2 3 5 #3 2 Y1 D 4 #7 5; Y2 D 4 3 5; Y3 D 4 #7 5 9 #5 5

&9&3$*4&4 (DFKHTXDWLRQLQ([HUFLVHV²LOOXVWUDWHVDSURSHUW\RIGHWHUPL QDQWV 6WDWHWKHSURSHUW\ ˇ ˇ ˇ ˇ ˇ0 ˇ 1 #3 5 #2 ˇˇ 6 ˇˇ ˇ ˇ 6 ˇˇ D #ˇˇ 0 5 #2 ˇˇ ˇˇ 1 #3 ˇ 4 #1 ˇ 4 #1 8ˇ 8ˇ ˇ ˇ ˇ ˇ ˇ 2 #6 ˇ 1 #3 4 ˇˇ 2 ˇˇ ˇ ˇ 5 #2 ˇˇ D 2ˇˇ 3 5 #2 ˇˇ ˇˇ 3 ˇ1 ˇ ˇ 6 3 1 6 3ˇ ˇ ˇ ˇ ˇ ˇ1 3 #4 ˇˇ ˇˇ 1 3 #4 ˇˇ ˇ 0 #3 ˇˇ D ˇˇ 0 #6 5 ˇˇ ˇˇ 2 ˇ 5 #4 7 ˇ ˇ 5 #4 7ˇ ˇ ˇ ˇ ˇ ˇ1 2 3 ˇˇ ˇˇ 1 2 3 ˇˇ ˇ 5 #4 ˇˇ D ˇˇ 0 5 #4 ˇˇ ˇˇ 0 ˇ3 7 4ˇ ˇ0 1 #5 ˇ )LQG WKH GHWHUPLQDQWV LQ ([HUFLVHV ² E\ URZ UHGXFWLRQ WR HFKHORQIRUP ˇ ˇ ˇ ˇ ˇ 1 ˇ1 5 #6 ˇˇ 5 #3 ˇˇ ˇ ˇ 4 ˇˇ 3 ˇˇ ˇˇ 3 #3 ˇˇ #1 #4 ˇ #2 #7 ˇ 2 13 #7 ˇ 9ˇ ˇ ˇ ˇ ˇ ˇ 1 ˇ 1 3 0 2 ˇˇ 3 3 #4 ˇˇ ˇ ˇ ˇ #2 #5 ˇ 0 7 4 ˇˇ 1 2 #5 ˇˇ ˇˇ ˇˇ ˇ 3 5 2 1 2 5 4 #3 ˇˇ ˇ ˇ ˇ ˇ 1 #1 ˇ ˇ 2 #3 #3 #7 #5 2ˇ ˇ ˇ ˇ 1 #1 #3 0 ˇˇ ˇ ˇ 0 1 5 4 ˇˇ ˇˇ #1 2 8 5 ˇˇ ˇ ˇ 3 #1 #2 3ˇ ˇ ˇ ˇ 1 3 #1 0 #2 ˇˇ ˇ ˇ 0 2 #4 #1 #6 ˇˇ ˇ ˇ 2 3 9 ˇˇ ˇ #2 #6 ˇ 3 ˇ 7 #3 8 #7 ˇ ˇ ˇ 3 5 5 2 7ˇ

&RPELQHWKHPHWKRGVRIURZUHGXFWLRQDQGFRIDFWRUH[SDQVLRQWR FRPSXWHWKHGHWHUPLQDQWVLQ([HUFLVHV² ˇ ˇ ˇ ˇ ˇ 2 ˇ #1 5 #3 #1 ˇˇ 2 3 0 ˇˇ ˇ ˇ ˇ 3 ˇ 3 0 1 #3 ˇˇ 4 3 0 ˇˇ ˇˇ ˇˇ ˇ 0 #4 9ˇ 4 6 6 ˇˇ ˇ #6 ˇ 5 ˇ 4 10 #4 #1 ˇ ˇ 4 2 4 3ˇ ˇ ˇ ˇ ˇ ˇ 2 ˇ #3 #2 5 4 1 ˇˇ 1 #4 ˇˇ ˇ ˇ ˇ 4 ˇ 1 7 6 2 ˇˇ 3 0 #3 ˇˇ ˇˇ ˇˇ ˇ 6 #2 #4 0 #3 4 #2 8 ˇˇ ˇ ˇ ˇ ˇ #6 ˇ ˇ 7 7 0 3 #4 0 4ˇ

)LQGWKHGHWHUPLQDQWVLQ([HUFLVHV² ZKHUH ˇ ˇ ˇa b c ˇˇ ˇ ˇd e f ˇˇ D 7: ˇ ˇg h iˇ ˇ ˇ ˇ ˇ a ˇ a b c ˇˇ b c ˇ ˇ e f ˇˇ 3e 3f ˇˇ d ˇˇ 3d ˇ 5g ˇ g 5h 5i ˇ h i ˇ ˇ ˇ ˇ ˇa ˇ ˇ ˇ b cˇ h iˇ ˇ ˇg h i ˇˇ b c ˇˇ ˇˇ g ˇˇ a ˇd ˇd e f ˇ e f ˇ ˇ ˇ ˇ a ˇ b c ˇ ˇ 2e C b 2f C c ˇˇ ˇˇ 2d C a ˇ g ˇ h i ˇ ˇ ˇaCd bCe c C f ˇˇ ˇ e f ˇˇ ˇˇ d ˇ g h i ˇ

ˇ ˇ ˇ ˇ ˇ ˇ

,Q([HUFLVHV² XVHGHWHUPLQDQWVWRÀQGRXWLIWKHPDWUL[LV LQYHUWLEOH 2 3 2 3 2 3 0 5 0 #1 3 45 4 1 #3 #2 5 4 1 1 2 1 0 5 3 2 3 2 0 0 8 6 1 #7 #5 07 6 7 4 3 8 6 05 0 7 5 4 ,Q([HUFLVHV² XVHGHWHUPLQDQWVWRGHFLGHLIWKHVHWRIYHFWRUV LVOLQHDUO\LQGHSHQGHQW 2 3 2 3 2 3 2 3 2 3 2 3 4 #7 #3 7 #8 7 4 6 5 4 0 5 4 #5 5 4 #4 5 4 5 5 4 0 5 #7 2 6 #6 7 #5 2 3 2 3 2 3 2 3 3 2 #2 0 6 5 7 6 #6 7 6 #1 7 6 0 7 6 7 6 7 6 7 6 7 4 #6 5 4 0 5 4 3 5 4 0 5 4 7 0 #3

,Q([HUFLVHVDQG A DQG B DUH n ! n PDWULFHV 0DUNHDFK VWDWHPHQW7UXHRU)DOVH -XVWLI\HDFKDQVZHU

D $ URZUHSODFHPHQWRSHUDWLRQGRHVQRWDIIHFWWKHGHWHUPL QDQWRIDPDWUL[ E 7KHGHWHUPLQDQWRI A LVWKHSURGXFWRIWKHSLYRWVLQDQ\ HFKHORQIRUP U RI A PXOWLSOLHGE\ .#1/r ZKHUH r LVWKH QXPEHURIURZLQWHUFKDQJHVPDGHGXULQJURZUHGXFWLRQ IURP A WR U

$)"15&3

'HWHUPLQDQWV

F ,I WKH FROXPQV RI A DUH OLQHDUO\ GHSHQGHQW WKHQ GHW A D 0

G GHW.A C B/ D GHW A C GHW B

G GHW A

D ,IWZRURZLQWHUFKDQJHVDUHPDGHLQVXFFHVVLRQ WKHQWKH QHZGHWHUPLQDQWHTXDOVWKHROGGHWHUPLQDQW E 7KHGHWHUPLQDQWRI A LVWKHSURGXFWRIWKHGLDJRQDOHQWULHV LQ A F ,I GHW A LV]HUR WKHQWZRURZVRUWZRFROXPQVDUHWKH VDPH RUDURZRUDFROXPQLV]HUR G GHW A D .#1/ GHW A T

2

1 &RPSXWH GHW B 5 ZKHUH B D 4 1 1

LQWKHH[HUFLVHVDERYH WRFRPSXWH D GHW $% E GHW 5A

0 1 2

3 1 2 5 1

8VH7KHRUHPEXWQRW7KHRUHP WRVKRZWKDWLIWZRURZV RIDVTXDUHPDWUL[ A DUHHTXDO WKHQ GHW A D 0 7KHVDPHLV WUXHIRUWZRFROXPQV :K\"

,Q ([HUFLVHV ² PHQWLRQ DQ DSSURSULDWH WKHRUHP LQ \RXU H[SODQDWLRQ

1 GHW A )LQGDIRUPXODIRU GHW.rA/ ZKHQ A LVDQ n ! n PDWUL[

6KRZWKDWLI A LVLQYHUWLEOH WKHQ GHW A!1 D

/HW A DQG B EHVTXDUHPDWULFHV 6KRZWKDWHYHQWKRXJK $% DQG %$ PD\ QRW EH HTXDO LW LV DOZD\V WUXH WKDW GHW $% D GHW %$ /HW A DQG P EHVTXDUHPDWULFHV ZLWK P LQYHUWLEOH 6KRZ WKDW GHW.PAP !1 / D GHW A

/HW U EHDVTXDUHPDWUL[VXFKWKDW U TU D I 6KRZWKDW GHW U D ˙1

6XSSRVH WKDW A LV D VTXDUH PDWUL[ VXFK WKDW GHW A4 D 0 ([SODLQZK\ A FDQQRWEHLQYHUWLEOH

9HULI\WKDW GHW AB D .GHW A/.GHW B/ IRUWKHPDWULFHVLQ([HUFLVHV DQG 'RQRWXVH7KHRUHP ! " ! " 3 0 2 0 A D B D 6 1 5 4 ! " ! " 3 6 4 2 A D B D #1 #2 #1 #1

/HW A DQG B EH 3 ! 3 PDWULFHV ZLWK GHW A D 4 DQG GHW B D #3 8VHSURSHUWLHVRIGHWHUPLQDQWVLQWKHWH[WDQG

!1

H GHW A

F GHW B T

3

/HW A DQG B EH 4 ! 4 PDWULFHV ZLWK GHW A D #1 DQG GHW B D 2 &RPSXWH F GHW 2A D GHW $% E GHW B 5 G GHW ATA

H GHW B !1

%$ 9HULI\WKDW GHW A D GHW B C GHW C ZKHUH ! " ! " ! " aCe bCf a b e f AD ; BD ; C D c d c d c d ! " ! " 1 0 a b /HW A D DQG B D 6KRZWKDW 0 1 c d GHW.A C B/ D GHW A C GHW B LIDQGRQO\LI a C d D 0 9HULI\WKDW GHW A D GHW B C GHW C ZKHUH 2 3 a11 a12 u1 C v1 a22 u2 C v2 5; A D 4 a21 a31 a32 u3 C v3 2 3 2 a11 a12 u1 a11 a12 a22 u2 5; C D 4 a21 a22 B D 4 a21 a31 a32 u3 a31 a32

1RWH KRZHYHU WKDW A LV QRW WKHVDPHDV B C C

3 v1 v2 5 v3

5LJKWPXOWLSOLFDWLRQE\DQHOHPHQWDU\PDWUL[ E DIIHFWVWKH FROXPQV RI A LQWKHVDPHZD\WKDWOHIWPXOWLSOLFDWLRQDIIHFWV WKH URZV 8VH7KHRUHPVDQGDQGWKHREYLRXVIDFWWKDW E T LVDQRWKHUHOHPHQWDU\PDWUL[WRVKRZWKDW GHW $( D .GHW E/.GHW A/ 'RQRWXVH7KHRUHP

>0@ &RPSXWH GHW ATA DQG GHW AAT IRUVHYHUDOUDQGRP 4 ! 5 PDWULFHVDQGVHYHUDOUDQGRP 5 ! 6 PDWULFHV :KDWFDQ\RX VD\DERXW ATA DQG AAT ZKHQ A KDVPRUHFROXPQVWKDQ URZV" >0@ ,I GHW A LVFORVHWR]HUR LVWKHPDWUL[ A QHDUO\VLQJX ODU" ([SHULPHQWZLWKWKHQHDUO\VLQJXODU 4 ! 4 PDWUL[ A LQ ([HUFLVH RI6HFWLRQ &RPSXWHWKHGHWHUPLQDQWVRI A 10A DQG 0:1A ,QFRQWUDVW FRPSXWHWKHFRQGLWLRQQXPEHUV RIWKHVHPDWULFHV 5HSHDWWKHVHFDOFXODWLRQVZKHQ A LVWKH 4 ! 4 LGHQWLW\PDWUL[ 'LVFXVV\RXUUHVXOWV

40-65*0/4 50 13"$5*$& 130#-&.4 3HUIRUPURZUHSODFHPHQWVWRFUHDWH]HURVLQWKHÀUVWFROXPQDQGWKHQFUHDWHDURZ RI]HURV ˇ ˇ ˇ ˇ ˇ ˇ ˇ 1 #3 1 #2 ˇˇ ˇˇ 1 #3 1 #2 ˇˇ ˇˇ 1 #3 1 #2 ˇˇ ˇ ˇ 2 #5 #1 #2 ˇ ˇ 0 1 #3 2 ˇˇ ˇˇ 0 1 #3 2 ˇˇ ˇ ˇDˇ Dˇ D0 ˇ 0 #4 ˇ ˇ ˇ 5 1 ˇ ˇ 0 #4 5 1 ˇ ˇ 0 #4 5 1 ˇˇ ˇ ˇ #3 10 #6 8 ˇ ˇ 0 1 #3 2ˇ ˇ0 0 0 0ˇ

GHW Œ Y1 Y2

&UDPHU V5XOH 9ROXPH DQG/LQHDU7UDQVIRUPDWLRQV

ˇ ˇ ˇ ˇ ˇ 5 #3 2 ˇˇ ˇˇ 5 #3 2 ˇˇ ˇ Y3 ! D ˇˇ #7 3 #7 ˇˇ D ˇˇ #2 0 #5 ˇˇ ˇ 9 #5 5 ˇ ˇ 9 #5 5 ˇ ˇ ˇ ˇ ˇ ˇ #2 #5 ˇ ˇ 5 2 ˇˇ ˇ ˇ ˇ D #.#3/ˇ # .#5/ˇ 9 5ˇ #2 #5 ˇ

5RZDGGHG WRURZ &RIDFWRUVRI FROXPQ

D 3 $ .35/ C 5 $ .#21/ D 0 %\7KHRUHP WKHPDWUL[ Œ Y1 Y2 Y3 ! LVQRWLQYHUWLEOH 7KHFROXPQVDUHOLQHDUO\ GHSHQGHQW E\WKH,QYHUWLEOH0DWUL[7KHRUHP

$3".&3h4 36-& 70-6.& "/% -*/&"3 53"/4'03."5*0/4 7KLVVHFWLRQDSSOLHVWKHWKHRU\RIWKHSUHFHGLQJVHFWLRQVWRREWDLQLPSRUWDQWWKHRUHWLFDO IRUPXODVDQGDJHRPHWULFLQWHUSUHWDWLRQRIWKHGHWHUPLQDQW

&UDPHU¶V5XOH &UDPHU·VUXOHLVQHHGHGLQDYDULHW\RIWKHRUHWLFDOFDOFXODWLRQV )RULQVWDQFH LWFDQEH XVHGWRVWXG\KRZWKHVROXWLRQRI A[ D E LVDIIHFWHGE\FKDQJHVLQWKHHQWULHVRI E +RZHYHU WKHIRUPXODLVLQHIÀFLHQWIRUKDQGFDOFXODWLRQV H[FHSWIRU 2 ! 2 RUSHUKDSV 3 ! 3 PDWULFHV )RUDQ\ n ! n PDWUL[ A DQGDQ\ E LQ Rn OHW Ai .E/ EHWKHPDWUL[REWDLQHGIURP A E\UHSODFLQJFROXPQ i E\WKHYHFWRU E

$$$

E !

Ai .E/ D ŒD1

$$$

Dn !

FRO i

5)&03&.

$SBNFShT 3VMF /HW A EHDQLQYHUWLEOH n ! n PDWUL[ )RUDQ\ E LQ Rn WKHXQLTXHVROXWLRQ [ RI A[ D E KDVHQWULHVJLYHQE\

xi D

GHW Ai .E/ ; GHW A

i D 1; 2; : : : ; n

1300' 'HQRWHWKHFROXPQVRI A E\ D1 ; : : : ; Dn DQGWKHFROXPQVRIWKH n ! n LGHQWLW\ PDWUL[ I E\ H1 ; : : : ; Hn ,I A[ D E WKHGHÀQLWLRQRIPDWUL[PXOWLSOLFDWLRQVKRZVWKDW # $ # $ $$$ [ $ $ $ Hn D AH1 $$$ A[ $$$ AH n A $ Ii .[/ D A H1 # $ $$$ E $ $ $ Dn D Ai .E/ D D1 %\WKHPXOWLSOLFDWLYHSURSHUW\RIGHWHUPLQDQWV

.GHW A/.GHW Ii .[// D GHW Ai .E/

7KHVHFRQGGHWHUPLQDQWRQWKHOHIWLVVLPSO\ xi 0DNHDFRIDFWRUH[SDQVLRQDORQJWKH i WKURZ +HQFH .GHW A/ $ xi D GHW Ai .E/ 7KLVSURYHV EHFDXVH A LVLQYHUWLEOHDQG GHW A ¤ 0

&9".1-& 8VH&UDPHU·VUXOHWRVROYHWKHV\VWHP 3x1 # 2x2 D 6 #5x1 C 4x2 D 8

$)"15&3

'HWHUPLQDQWV

40-65*0/ 9LHZWKHV\VWHPDV A[ D E 8VLQJWKHQRWDWLRQLQWURGXFHGDERYH ! " ! " ! " 3 #2 6 #2 3 6 AD ; A1 .E/ D ; A 2 .E / D #5 4 8 4 #5 8 6LQFH GHW A D 2 WKHV\VWHPKDVDXQLTXHVROXWLRQ %\&UDPHU·VUXOH

24 C 16 GHW A1 .E/ D D 20 GHW A 2 24 C 30 GHW A2 .E/ x2 D D D 27 GHW A 2

x1 D

$SSOLFDWLRQWR(QJLQHHULQJ $ QXPEHURILPSRUWDQWHQJLQHHULQJSUREOHPV SDUWLFXODUO\LQHOHFWULFDOHQJLQHHULQJDQG FRQWUROWKHRU\ FDQEHDQDO\]HGE\ /DSODFHWUDQVIRUPV 7KLVDSSURDFKFRQYHUWVDQ DSSURSULDWHV\VWHPRIOLQHDUGLIIHUHQWLDOHTXDWLRQVLQWRDV\VWHPRIOLQHDUDOJHEUDLF HTXDWLRQVZKRVHFRHIÀFLHQWVLQYROYHDSDUDPHWHU 7KHQH[WH[DPSOHLOOXVWUDWHVWKHW\SH RIDOJHEUDLFV\VWHPWKDWPD\DULVH

&9".1-& &RQVLGHUWKHIROORZLQJV\VWHPLQZKLFK s LVDQXQVSHFLÀHGSDUDPHWHU 'HWHUPLQHWKHYDOXHVRI s IRUZKLFKWKHV\VWHPKDVDXQLTXHVROXWLRQ DQGXVH&UDPHU·V UXOHWRGHVFULEHWKHVROXWLRQ 3sx1 # 2x2 D 4 #6x1 C sx2 D 1

40-65*0/ 9LHZWKHV\VWHPDV A[ D E 7KHQ ! " ! " 3s #2 4 #2 AD ; A1 .E/ D ; #6 s 1 s

A2 .E/ D

!

3s #6

4 1

"

6LQFH GHW A D 3s 2 # 12 D 3.s C 2/.s # 2/ WKHV\VWHPKDVDXQLTXHVROXWLRQSUHFLVHO\ZKHQ s ¤ ˙2 )RUVXFKDQ s WKHVROXWLRQLV .x1 ; x2 / ZKHUH

x1 D x2 D

4s C 2 GHW A1 .E/ D GHW A 3.s C 2/.s # 2/

3s C 24 sC8 GHW A2 .E/ D D GHW A 3.s C 2/.s # 2/ .s C 2/.s # 2/

$ )RUPXODIRU $± &UDPHU·VUXOHOHDGVHDVLO\WRDJHQHUDOIRUPXODIRUWKHLQYHUVHRIDQ n ! n PDWUL[ A 7KH j WKFROXPQRI A!1 LVDYHFWRU [ WKDWVDWLVÀHV

A[ D Hj ZKHUH Hj LVWKH j WKFROXPQRIWKHLGHQWLW\PDWUL[ DQGWKH i WKHQWU\RI [ LVWKH .i; j /HQWU\ RI A!1 %\&UDPHU·VUXOH ˚ % GHW Ai .Hj / .i; j /HQWU\RI A!1 D xi D GHW A


5HFDOOWKDW Aj i GHQRWHVWKHVXEPDWUL[RI A IRUPHGE\GHOHWLQJURZ j DQGFROXPQ i $ FRIDFWRUH[SDQVLRQGRZQFROXPQ i RI Ai .Hj / VKRZVWKDW GHW Ai .Hj / D .#1/i Cj GHW Aj i D Cj i

ZKHUH Cj i LVDFRIDFWRURI A %\ WKH .i; j /HQWU\RI A!1 LVWKHFRIDFWRU Cj i GLYLGHG E\ GHW A >1RWHWKDWWKHVXEVFULSWVRQ Cj i DUHWKHUHYHUVHRI .i; j /@ 7KXV 2 3 C11 C21 $$$ Cn1 C22 $$$ Cn2 7 1 6 6 C12 7 A!1 D 6 :: :: :: 7 GHW A 4 : : :5

C1n

C2n

$$$

Cnn

7KHPDWUL[RIFRIDFWRUVRQWKHULJKWVLGHRI LVFDOOHGWKH DGMXJDWH RU FODVVL FDODGMRLQW RI A GHQRWHGE\ DGM A 7KHWHUP DGMRLQW DOVRKDVDQRWKHUPHDQLQJLQ DGYDQFHGWH[WVRQOLQHDUWUDQVIRUPDWLRQV 7KHQH[WWKHRUHPVLPSO\UHVWDWHV

5)&03&.

"O *OWFSTF 'PSNVMB /HW A EHDQLQYHUWLEOH n ! n PDWUL[ 7KHQ 1 A!1 D DGM A GHW A 2

3 2 1 3 &9".1-& )LQGWKHLQYHUVHRIWKHPDWUL[ A D 4 1 #1 1 5 1 4 #2

40-65*0/ 7KHQLQHFRIDFWRUVDUH ˇ ˇ ˇ ˇ ˇ #1 ˇ1 1 ˇˇ 1 ˇˇ ˇ C11 D Cˇˇ D #2; C D # 12 ˇ 1 #2 ˇ D 3; 4 #2 ˇ ˇ ˇ ˇ ˇ ˇ1 ˇ2 3 ˇˇ 3 ˇˇ ˇ C21 D #ˇˇ D 14; C D C 22 ˇ 1 #2 ˇ D #7; 4 #2 ˇ ˇ ˇ ˇ ˇ ˇ 1 ˇ2 3 ˇˇ 3 ˇˇ ˇ C31 D Cˇˇ D 4; C D # D 1; 32 ˇ1 #1 1 ˇ 1ˇ

ˇ ˇ ˇ 1 #1 ˇ ˇD5 C13 D Cˇˇ 1 4ˇ ˇ ˇ ˇ2 1 ˇˇ C23 D #ˇˇ D #7 1 4ˇ ˇ ˇ ˇ2 1 ˇˇ C33 D Cˇˇ D #3 1 #1 ˇ

7KHDGMXJDWHPDWUL[LVWKH WUDQVSRVH RIWKHPDWUL[RIFRIDFWRUV >)RULQVWDQFH C12 JRHV LQWKH .2; 1/ SRVLWLRQ@ 7KXV 2 3 #2 14 4 DGM A D 4 3 #7 1 5 5 #7 #3

:HFRXOGFRPSXWH GHW A GLUHFWO\ EXWWKHIROORZLQJFRPSXWDWLRQSURYLGHVDFKHFNRQ WKHFDOFXODWLRQVDERYH DQG SURGXFHV GHW A 2 32 3 2 3 #2 14 4 2 1 3 14 0 0 1 54 1 #1 1 5 D 4 0 14 0 5 D 14I .DGM A/ $ A D 4 3 #7 5 #7 #3 1 4 #2 0 0 14 6LQFH .DGM A/A D 14I 7KHRUHPVKRZVWKDW GHW A D 14 DQG 2 3 2 3 #2 14 4 #1=7 1 2=7 1 4 3 #7 1 5 D 4 3=14 #1=2 1=14 5 A!1 D 14 5 #7 #3 5=14 #1=2 #3=14

$)"15&3

'HWHUPLQDQWV

/6.&3*$"- /05&4 7KHRUHPLVXVHIXOPDLQO\IRUWKHRUHWLFDOFDOFXODWLRQV 7KHIRUPXODIRU A!1 SHUPLWVRQHWRGHGXFHSURSHUWLHVRIWKHLQYHUVHZLWKRXWDFWXDOO\FDOFXODWLQJLW ([FHSWIRUVSHFLDOFDVHV WKHDOJRULWKPLQ6HFWLRQ JLYHVDPXFKEHWWHUZD\WR FRPSXWH A!1 LIWKHLQYHUVHLVUHDOO\QHHGHG &UDPHU·VUXOHLVDOVRDWKHRUHWLFDOWRRO ,WFDQEHXVHGWRVWXG\KRZVHQVLWLYH WKHVROXWLRQRI A[ D E LVWRFKDQJHVLQDQHQWU\LQ E RULQ A SHUKDSVGXH WRH[SHULPHQWDOHUURUZKHQDFTXLULQJWKHHQWULHVIRU E RU A :KHQ A LVD 3 ! 3 PDWUL[ZLWK FRPSOH[ HQWULHV &UDPHU·VUXOHLVVRPHWLPHVVHOHFWHGIRUKDQG FRPSXWDWLRQEHFDXVHURZUHGXFWLRQRI Œ A E ! ZLWKFRPSOH[DULWKPHWLFFDQEH PHVV\ DQGWKHGHWHUPLQDQWVDUHIDLUO\HDV\WRFRPSXWH )RUDODUJHU n ! n PDWUL[ UHDORUFRPSOH[ &UDPHU·VUXOHLVKRSHOHVVO\LQHIÀFLHQW &RPSXWLQJMXVW RQH GHWHUPLQDQWWDNHVDERXWDVPXFKZRUNDVVROYLQJ A[ D E E\URZUHGXFWLRQ

'HWHUPLQDQWVDV$UHDRU9ROXPH ,QWKHQH[WDSSOLFDWLRQ ZHYHULI\WKHJHRPHWULFLQWHUSUHWDWLRQRIGHWHUPLQDQWVGHVFULEHG LQWKHFKDSWHULQWURGXFWLRQ $OWKRXJKDJHQHUDOGLVFXVVLRQRIOHQJWKDQGGLVWDQFHLQ Rn ZLOOQRWEHJLYHQXQWLO&KDSWHU ZHDVVXPHKHUHWKDWWKHXVXDO(XFOLGHDQFRQFHSWVRI OHQJWK DUHD DQGYROXPHDUHDOUHDG\XQGHUVWRRGIRU R2 DQG R3

5)&03&.

SG

1300' 7KH WKHRUHPLVREYLRXVO\WUXHIRUDQ\ 2 ! 2 GLDJRQDOPDWUL[ ˇ ! "ˇ & ' ˇ 0 ˇˇ DUHDRI ˇ GHW a D jad j D ˇ UHFWDQJOH 0 d ˇ

" (FPNFUSJD 1SPPG m

0  d   

y

 a    0

,I A LVD 2 ! 2 PDWUL[ WKHDUHDRIWKHSDUDOOHORJUDPGHWHUPLQHGE\WKHFROXPQVRI A LV jGHW Aj ,I A LVD 3 ! 3 PDWUL[ WKHYROXPHRIWKHSDUDOOHOHSLSHGGHWHUPLQHG E\WKHFROXPQVRI A LV jGHW Aj

x

6HH)LJ ,WZLOOVXIÀFHWRVKRZWKDWDQ\ 2 ! 2 PDWUL[ A D Œ D1 D2 ! FDQEHWUDQV IRUPHGLQWRDGLDJRQDOPDWUL[LQDZD\WKDWFKDQJHVQHLWKHUWKHDUHDRIWKHDVVRFLDWHG SDUDOOHORJUDPQRU jGHW Aj )URP6HFWLRQ ZHNQRZWKDWWKHDEVROXWHYDOXHRIWKH GHWHUPLQDQWLVXQFKDQJHGZKHQWZRFROXPQVDUHLQWHUFKDQJHGRUDPXOWLSOHRIRQH FROXPQLVDGGHGWRDQRWKHU $QGLWLVHDV\WRVHHWKDWVXFKRSHUDWLRQVVXIÀFHWRWUDQVIRUP A LQWRDGLDJRQDOPDWUL[ &ROXPQLQWHUFKDQJHVGRQRWFKDQJHWKHSDUDOOHORJUDPDWDOO 6RLWVXIÀFHVWRSURYHWKHIROORZLQJVLPSOHJHRPHWULFREVHUYDWLRQWKDWDSSOLHVWRYHFWRUV LQ R2 RU R3

'*(63&

$UHD D jad j

/HW D1 DQG D2 EH QRQ]HUR YHFWRUV 7KHQ IRU DQ\ VFDODU c WKH DUHD RI WKH SDUDOOHORJUDP GHWHUPLQHG E\ D1 DQG D2 HTXDOV WKH DUHD RI WKH SDUDOOHORJUDP GHWHUPLQHGE\ D1 DQG D2 C c D1 7RSURYHWKLVVWDWHPHQW ZHPD\DVVXPHWKDW D2 LVQRWDPXOWLSOHRI D1 IRURWKHU ZLVH WKHWZRSDUDOOHORJUDPVZRXOGEHGHJHQHUDWHDQGKDYH]HURDUHD ,I L LVWKHOLQH WKURXJK DQG D1 WKHQ D2 C L LVWKHOLQHWKURXJK D2 SDUDOOHOWR L DQG D2 C c D1 LVRQ WKLVOLQH 6HH)LJ 7KHSRLQWV D2 DQG D2 C c D1 KDYHWKHVDPHSHUSHQGLFXODUGLVWDQFH WR L +HQFHWKHWZRSDUDOOHORJUDPVLQ)LJ KDYHWKHVDPHDUHD VLQFHWKH\VKDUHWKH EDVHIURP WR D1 7KLVFRPSOHWHVWKHSURRIIRU R2

&UDPHU V5XOH 9ROXPH DQG/LQHDU7UDQVIRUPDWLRQV a2

a 2 + c a1

a2 + L

L a1

0

c a1

'*(63& 7ZRSDUDOOHORJUDPVRIHTXDODUHD z 0  0  c   

x

a 0 0  

 0  b    0

y

7KHSURRIIRU R3 LVVLPLODU 7KHWKHRUHPLVREYLRXVO\WUXHIRUD 3 ! 3 GLDJRQDO PDWUL[ 6HH)LJ $QGDQ\ 3 ! 3 PDWUL[ A FDQEHWUDQVIRUPHGLQWRDGLDJRQDOPDWUL[ XVLQJFROXPQRSHUDWLRQVWKDWGRQRWFKDQJH jGHW Aj 7KLQNDERXWGRLQJURZRSHUDWLRQV RQ AT 6RLWVXIÀFHVWRVKRZWKDWWKHVHRSHUDWLRQVGRQRWDIIHFWWKHYROXPHRIWKH SDUDOOHOHSLSHGGHWHUPLQHGE\WKHFROXPQVRI A $ SDUDOOHOHSLSHGLVVKRZQLQ)LJ DVDVKDGHGER[ZLWKWZRVORSLQJVLGHV ,WV YROXPHLVWKHDUHDRIWKHEDVHLQWKHSODQH 6SDQ fD1 ; D3 g WLPHVWKHDOWLWXGHRI D2 DERYH 6SDQ fD1 ; D3 g $Q\YHFWRU D2 C c D1 KDVWKHVDPHDOWLWXGHEHFDXVH D2 C c D1 OLHVLQWKH SODQH D2 C 6SDQ fD1 ; D3 g ZKLFKLVSDUDOOHOWR 6SDQ fD1 ; D3 g +HQFHWKHYROXPHRIWKH SDUDOOHOHSLSHGLVXQFKDQJHGZKHQ Œ D1 D2 D3 ! LVFKDQJHGWR Œ D1 D2 C c D1 D3 ! 7KXVDFROXPQUHSODFHPHQWRSHUDWLRQGRHVQRWDIIHFWWKHYROXPHRIWKHSDUDOOHOHSLSHG 6LQFHFROXPQLQWHUFKDQJHVKDYHQRHIIHFWRQWKHYROXPH WKHSURRILVFRPSOHWH

'*(63&

9ROXPH D jabcj a2

+

} ,a3 a1 { n

a

Sp

a2 0

} ,a3 a1 { n

a3

n{

a

Sp

} ,a3

a1

a2

+

a3

a Sp a 2 + ca 1 a 2

a1

} ,a3 a1 { n

a

Sp

a1

0

'*(63& 7ZRSDUDOOHOHSLSHGVRIHTXDOYROXPH

&9".1-& &DOFXODWHWKHDUHDRIWKHSDUDOOHORJUDPGHWHUPLQHGE\WKHSRLQWV .#2; #2/ .0; 3/ .4; #1/ DQG .6; 4/ 6HH)LJ D 40-65*0/ )LUVWWUDQVODWHWKHSDUDOOHORJUDPWRRQHKDYLQJWKHRULJLQDVDYHUWH[ )RU H[DPSOH VXEWUDFWWKHYHUWH[ .#2; #2/ IURPHDFKRIWKHIRXUYHUWLFHV 7KHQHZSDUDO OHORJUDPKDVWKHVDPHDUHD DQGLWVYHUWLFHVDUH .0; 0/ .2; 5/ .6; 1/ DQG .8; 6/ 6HH x2

x2

x1

(a)

x1

(b)

'*(63& 7UDQVODWLQJDSDUDOOHORJUDPGRHVQRWFKDQJHLWV

DUHD

$)"15&3

'HWHUPLQDQWV

)LJ E 7KLVSDUDOOHORJUDPLVGHWHUPLQHGE\WKHFROXPQVRI ! " 2 6 AD 5 1 6LQFH jGHW Aj D j#28j WKHDUHDRIWKHSDUDOOHORJUDPLV

/LQHDU7UDQVIRUPDWLRQV 'HWHUPLQDQWVFDQEHXVHGWRGHVFULEHDQLPSRUWDQWJHRPHWULFSURSHUW\RIOLQHDUWUDQV IRUPDWLRQVLQWKHSODQHDQGLQ R3 ,I T LVDOLQHDUWUDQVIRUPDWLRQDQG S LVDVHWLQWKH GRPDLQRI T OHW T .S / GHQRWHWKHVHWRILPDJHVRISRLQWVLQ S :HDUHLQWHUHVWHGLQKRZ WKHDUHDRUYROXPH RI T .S / FRPSDUHVZLWKWKHDUHDRUYROXPH RIWKHRULJLQDOVHW S )RUFRQYHQLHQFH ZKHQ S LVDUHJLRQERXQGHGE\DSDUDOOHORJUDP ZHDOVRUHIHUWR S DV DSDUDOOHORJUDP

5)&03&.

/HW T W R2 ! R2 EHWKHOLQHDUWUDQVIRUPDWLRQGHWHUPLQHGE\D 2 ! 2 PDWUL[ A ,I S LVDSDUDOOHORJUDPLQ R2 WKHQ

fDUHDRI T .S /g D jGHW Aj $ fDUHDRI Sg

fYROXPHRI T .S /g D jGHW Aj $ fYROXPHRI Sg

,I T LVGHWHUPLQHGE\D 3 ! 3 PDWUL[ A DQGLI S LVDSDUDOOHOHSLSHGLQ R3 WKHQ

1300' &RQVLGHUWKH 2 ! 2 FDVH ZLWK A D Œ D1 D2 ! $ SDUDOOHORJUDPDWWKHRULJLQLQ R2 GHWHUPLQHGE\YHFWRUV E1 DQG E2 KDVWKHIRUP S D fs1 E1 C s2 E2 W 0 ' s1 ' 1; 0 ' s2 ' 1g 7KHLPDJHRI S XQGHU T FRQVLVWVRISRLQWVRIWKHIRUP

T .s1 E1 C s2 E2 / D s1 T .E1 / C s2 T .E2 / D s1 AE1 C s2 AE2 ZKHUH 0 ' s1 ' 1 0 ' s2 ' 1 ,WIROORZVWKDW T .S / LVWKHSDUDOOHORJUDPGHWHUPLQHG E\WKHFROXPQVRIWKHPDWUL[ Œ AE1 AE2 ! 7KLVPDWUL[FDQEHZULWWHQDV AB ZKHUH B D Œ E1 E2 ! %\7KHRUHPDQGWKHSURGXFWWKHRUHPIRUGHWHUPLQDQWV

fDUHDRI T .S /g D jGHW ABj D jGHW Aj $ jGHW Bj D jGHW Aj $ fDUHDRI Sg

$QDUELWUDU\SDUDOOHORJUDPKDVWKHIRUP S C S ZKHUH S LVDYHFWRUDQG S LVDSDUDOOHOR JUDPDWWKHRULJLQ DVDERYH ,WLVHDV\WRVHHWKDW T WUDQVIRUPV S C S LQWR T .S/ C T .S/ 6HH([HUFLVH 6LQFHWUDQVODWLRQGRHVQRWDIIHFWWKHDUHDRIDVHW

fDUHDRI T .S C S/g D fDUHDRI T .S/ C T .S /g D fDUHDRI T .S /g D j GHW Aj $ fDUHDRI Sg D j GHW Aj $ fDUHDRI S C S g

7UDQVODWLRQ %\HTXDWLRQ 7UDQVODWLRQ

7KLVVKRZVWKDW KROGVIRUDOOSDUDOOHORJUDPVLQ R2 7KHSURRIRI IRUWKH 3 ! 3 FDVHLVDQDORJRXV


:KHQZHDWWHPSWWRJHQHUDOL]H7KHRUHPWRDUHJLRQLQ R2 RU R3 WKDWLVQRW ERXQGHGE\VWUDLJKWOLQHVRUSODQHV ZHPXVWIDFHWKHSUREOHPRIKRZWRGHÀQHDQG FRPSXWHLWVDUHDRUYROXPH 7KLVLVDTXHVWLRQVWXGLHGLQFDOFXOXV DQGZHVKDOORQO\ RXWOLQHWKHEDVLFLGHDIRU R2 ,I R LVDSODQDUUHJLRQWKDWKDVDÀQLWHDUHD WKHQ R FDQ EHDSSUR[LPDWHGE\DJULGRIVPDOOVTXDUHVWKDWOLHLQVLGH R %\PDNLQJWKHVTXDUHV VXIÀFLHQWO\VPDOO WKHDUHDRI R PD\EHDSSUR[LPDWHGDVFORVHO\DVGHVLUHGE\WKHVXP RIWKHDUHDVRIWKHVPDOOVTXDUHV 6HH)LJ

0

0

'*(63& $SSUR[LPDWLQJDSODQDUUHJLRQE\DXQLRQRIVTXDUHV

7KHDSSUR[LPDWLRQLPSURYHVDVWKHJULGEHFRPHVÀQHU

,I T LVDOLQHDUWUDQVIRUPDWLRQDVVRFLDWHGZLWKD 2 ! 2 PDWUL[ A WKHQWKHLPDJHRI DSODQDUUHJLRQ R XQGHU T LVDSSUR[LPDWHGE\WKHLPDJHVRIWKHVPDOOVTXDUHVLQVLGH R 7KHSURRIRI7KHRUHP VKRZVWKDWHDFKVXFKLPDJHLVDSDUDOOHORJUDPZKRVHDUHDLV jGHW Aj WLPHVWKHDUHDRIWKHVTXDUH ,I R0 LVWKHXQLRQRIWKHVTXDUHVLQVLGH R WKHQWKH DUHDRI T .R0 / LV jGHW Aj WLPHVWKHDUHDRI R0 6HH)LJ $OVR WKHDUHDRI T .R0 / LVFORVH WRWKHDUHDRI T .R/ $QDUJXPHQWLQYROYLQJDOLPLWLQJSURFHVVPD\EHJLYHQWRMXVWLI\ WKHIROORZLQJJHQHUDOL]DWLRQRI7KHRUHP

T

0

R'

0

T(R' )

'*(63& $SSUR[LPDWLQJ T .R/ E\DXQLRQRISDUDOOHORJUDPV

7KHFRQFOXVLRQVRI7KHRUHPKROGZKHQHYHU S LVDUHJLRQLQ R2 ZLWKÀQLWHDUHD RUDUHJLRQLQ R3 ZLWKÀQLWHYROXPH

&9".1-& /HW a DQG b EHSRVLWLYHQXPEHUV )LQGWKHDUHDRIWKHUHJLRQ E ERXQGHGE\WKHHOOLSVHZKRVHHTXDWLRQLV

x22 x12 C D1 a2 b2

'HWHUPLQDQWV

$)"15&3

u2

D

u1

1

x2

40-65*0/ :HFODLPWKDW E LVWKHLPDJHRIWKHXQLWGLVN D XQGHUWKHOLQHDUWUDQVIRU ! " ! " ! " a 0 u1 x1 PDWLRQ T GHWHUPLQHGE\WKHPDWUL[ A D EHFDXVHLI X D [D u2 x2 0 b DQG [ D AX WKHQ x1 x2 u1 D DQG u2 D a b ,WIROORZVWKDW X LVLQWKHXQLWGLVN ZLWK u21 C u22 ' 1 LIDQGRQO\LI [ LVLQ E ZLWK .x1 =a/2 C .x2 =b/2 ' 1 %\WKHJHQHUDOL]DWLRQRI7KHRUHP

T

b E

fDUHDRIHOOLSVHg D fDUHDRI T .D/g D jGHW Aj $ fDUHDRI Dg

x1

a

D ab $ $.1/2 D $ab

13"$5*$& 130#-&.

! " ! " 1 5 /HW S EHWKHSDUDOOHORJUDPGHWHUPLQHGE\WKHYHFWRUV E1 D DQG E2 D DQG 3 1 ! " 1 #:1 OHW A D &RPSXWHWKHDUHDRIWKHLPDJHRI S XQGHUWKHPDSSLQJ [ 7! A[ 0 2

&9&3$*4&4 8VH&UDPHU·VUXOHWRFRPSXWHWKHVROXWLRQVRIWKHV\VWHPVLQ ([HUFLVHV² 5x1 C 7x2 D 3

4x1 C x2 D 6

2x1 C 4x2 D 1

3x1 # 2x2 D

#5x1 C 3x2 D

#5x1 C 6x2 D #5

2x1 C x2

#3x1

7

C x3 D #8

x2 C 2x3 D #3

2x1 C x2 C x3 D

#x1 C

2x3 D

4 2

3x1 C x2 C 3x3 D #2

,Q([HUFLVHV² GHWHUPLQHWKHYDOXHVRIWKHSDUDPHWHU s IRU ZKLFKWKHV\VWHPKDVDXQLTXHVROXWLRQ DQGGHVFULEHWKHVROXWLRQ 6sx1 C 4x2 D

5

9x1 C 2sx2 D #2

sx1 # 2sx2 D #1

3x1 C 6sx2 D

4

3sx1 # 5x2 D 3

9x1 C 5sx2 D 2

2sx1 C

0 1 3

3 0 05 2

2

1 2 4 0 #3 0 0

3 4 15 3

6XSSRVHWKDWDOOWKHHQWULHVLQ A DUHLQWHJHUVDQG GHW A D 1 ([SODLQZK\DOOWKHHQWULHVLQ A!1 DUHLQWHJHUV

9

3x1 # x2 D #5

D

3 4 #1 #2

6KRZWKDWLI A LV 2 ! 2 WKHQ7KHRUHP JLYHVWKHVDPH IRUPXODIRU A!1 DVWKDWJLYHQE\7KHRUHP LQ6HFWLRQ

5x1 C 2x2 D 7

7

2

x2 D 1

3sx1 C 6sx2 D 2

,Q([HUFLVHV² FRPSXWHWKHDGMXJDWHRIWKHJLYHQPDWUL[ DQG WKHQXVH7KHRUHPWRJLYHWKHLQYHUVHRIWKHPDWUL[ 2 3 2 3 0 #2 #1 1 1 3 0 05 15 4 3 4 2 #2 #1 1 1 0 1 0 2 3 2 3 3 5 4 3 6 7 0 15 2 15 4 1 4 0 2 1 1 2 3 4

,Q([HUFLVHV² ÀQGWKHDUHDRIWKHSDUDOOHORJUDPZKRVH YHUWLFHVDUHOLVWHG .0; 0/ .5; 2/ .6; 4/ .11; 6/ .0; 0/ .#1; 3/ .4; #5/ .3; #2/ .#1; 0/ .0; 5/ .1; #4/ .2; 1/

.0; #2/ .6; #1/ .#3; 1/ .3; 2/ )LQGWKHYROXPHRIWKHSDUDOOHOHSLSHGZLWKRQHYHUWH[DW WKHRULJLQDQGDGMDFHQWYHUWLFHVDW .1; 0; #2/ .1; 2; 4/ DQG .7; 1; 0/ )LQGWKHYROXPHRIWKHSDUDOOHOHSLSHGZLWKRQHYHUWH[DW WKHRULJLQDQGDGMDFHQWYHUWLFHVDW .1; 4; 0/ .#2; #5; 2/ DQG .#1; 2; #1/ 8VHWKHFRQFHSWRIYROXPHWRH[SODLQZK\WKHGHWHUPLQDQWRI D 3 ! 3 PDWUL[ A LV]HURLIDQGRQO\LI A LVQRWLQYHUWLEOH 'R QRWDSSHDOWR7KHRUHP LQ6HFWLRQ >+LQW 7KLQNDERXW WKHFROXPQVRI A@ /HW T W Rm ! Rn EHDOLQHDUWUDQVIRUPDWLRQ DQGOHW S EHD YHFWRUDQG S DVHWLQ Rm 6KRZWKDWWKHLPDJHRI S C S XQGHU T LVWKHWUDQVODWHGVHW T .S/ C T .S/ LQ Rn


/HW S EH WKH SDUDOOHORJUDP GHWHUPLQHG E\ WKH YHFWRUV ! " ! " ! " #2 #2 6 #2 E1 D DQG E2 D DQGOHW A D 3 5 #3 2 &RPSXWH WKH DUHD RI WKH LPDJH RI S XQGHU WKH PDSSLQJ [ 7! A[ ! " ! " 4 0 5HSHDW ([HUFLVH ZLWK E1 D E2 D DQG #7 1 ! " 7 2 AD 1 1 )LQGDIRUPXODIRUWKHDUHDRIWKHWULDQJOHZKRVHYHUWLFHVDUH Y1 DQG Y2 LQ R2 /HW R EHWKHWULDQJOHZLWKYHUWLFHVDW .x1 ; y1 / .x2 ; y2 / DQG .x3 ; y3 / 6KRZWKDW 2 3 x1 y1 1 1 y2 15 fDUHDRIWULDQJOHg D GHW 4 x2 2 y3 1 x3

>+LQW 7UDQVODWH R WRWKHRULJLQE\VXEWUDFWLQJRQHRIWKH YHUWLFHV DQGXVH([HUFLVH@

/HW T W R3 ! R3 EHWKHOLQHDUWUDQVIRUPDWLRQGHWHUPLQHG 2 3 a 0 0 b 0 5 ZKHUH a b DQG c DUH E\WKHPDWUL[ A D 4 0 0 0 c SRVLWLYHQXPEHUV /HW S EHWKHXQLWEDOO ZKRVHERXQGLQJ VXUIDFHKDVWKHHTXDWLRQ x12 C x22 C x32 D 1 D 6KRZWKDW T .S / LVERXQGHGE\WKHHOOLSVRLGZLWKWKH x2 x2 x2 HTXDWLRQ 12 C 22 C 32 D 1 a b c E 8VHWKHIDFWWKDWWKHYROXPHRIWKHXQLWEDOOLV 4$=3 WRGHWHUPLQHWKHYROXPHRIWKHUHJLRQERXQGHGE\WKH HOOLSVRLGLQSDUWD

/HW S EHWKHWHWUDKHGURQLQ R3 ZLWKYHUWLFHVDWWKHYHFWRUV H1 H2 DQG H3 DQGOHW S 0 EHWKHWHWUDKHGURQZLWKYHUWLFHVDW YHFWRUV Y1 Y2 DQG Y3 6HHWKHÀJXUH x3 e3

x3 S

v3

x2

S'

v2

x2

e2 0

0 e1

x1

v1 x1

D 'HVFULEHDOLQHDUWUDQVIRUPDWLRQWKDWPDSV S RQWR S 0 E )LQGDIRUPXODIRUWKHYROXPHRIWKHWHWUDKHGURQ S 0 XVLQJ WKHIDFWWKDW

fYROXPHRI S g D .1=3/fDUHDRIEDVHg $ fKHLJKWg

>0@ 7HVWWKHLQYHUVHIRUPXODRI7KHRUHP IRUDUDQGRP 4 ! 4 PDWUL[ A 8VH\RXUPDWUL[SURJUDPWRFRPSXWHWKH FRIDFWRUVRIWKH 3 ! 3 VXEPDWULFHV FRQVWUXFWWKHDGMXJDWH DQGVHW B D .DGM A/=.GHW A/ 7KHQFRPSXWH B # LQY.A/ ZKHUH LQY.A/ LVWKHLQYHUVHRI A DVFRPSXWHGE\WKHPDWUL[ SURJUDP 8VHÁRDWLQJSRLQWDULWKPHWLFZLWKWKHPD[LPXP SRVVLEOHQXPEHURIGHFLPDOSODFHV 5HSRUW\RXUUHVXOWV >0@ 7HVW&UDPHU·VUXOHIRUDUDQGRP 4 ! 4 PDWUL[ A DQGD UDQGRP 4 ! 1 YHFWRU E &RPSXWHHDFKHQWU\LQWKHVROXWLRQRI A[ D E DQGFRPSDUHWKHVHHQWULHVZLWKWKHHQWULHVLQ A!1 E :ULWHWKHFRPPDQGRUNH\VWURNHV IRU\RXUPDWUL[SURJUDP WKDWXVHV&UDPHU·VUXOHWRSURGXFHWKHVHFRQGHQWU\RI [ >0@ ,I\RXUYHUVLRQRI0$7/$% KDVWKH IORSV FRPPDQG XVHLWWRFRXQWWKHQXPEHURIÁRDWLQJSRLQWRSHUDWLRQVWRFRP SXWH A!1 IRUDUDQGRP 30 ! 30 PDWUL[ &RPSDUHWKLVQXPEHU ZLWKWKHQXPEHURIÁRSVQHHGHGWRIRUP .DGM A/=.GHW A/

40-65*0/ 50 13"$5*$& 130#-&. ˇ ! "ˇ ˇ 1 5 ˇˇ 7KHDUHDRI S LV ˇˇ GHW D 14; DQG GHW A D 2 %\7KHRUHP WKHDUHDRIWKH 3 1 ˇ LPDJHRI S XQGHUWKHPDSSLQJ [ 7! A[ LV jGHW Aj $ fDUHDRI Sg D 2 $ 14 D 28

$)"15&3 4611-&.&/5"3: &9&3$*4&4 0DUNHDFKVWDWHPHQW7UXHRU)DOVH -XVWLI\HDFKDQVZHU $VVXPHWKDWDOOPDWULFHVKHUHDUHVTXDUH D ,I A LVD 2 ! 2 PDWUL[ZLWKD]HURGHWHUPLQDQW WKHQRQH FROXPQRI A LVDPXOWLSOHRIWKHRWKHU E ,I WZR URZV RI D 3 ! 3 PDWUL[ A DUH WKH VDPH WKHQ GHW A D 0

F ,I A LVD 3 ! 3 PDWUL[ WKHQ GHW 5A D 5 GHW A

G ,I A DQG B DUH n ! n PDWULFHV ZLWK GHW A D 2 DQG GHW B D 3 WKHQ GHW.A C B/ D 5 H ,I A LV n ! n DQG GHW A D 2 WKHQ GHW A3 D 6

I ,I B LVSURGXFHGE\LQWHUFKDQJLQJWZRURZVRI A WKHQ GHW B D GHW A

J ,I B LVSURGXFHGE\PXOWLSO\LQJURZRI A E\ WKHQ GHW B D 5 $ GHW A

$)"15&3

'HWHUPLQDQWV

K ,I B LV IRUPHG E\ DGGLQJ WR RQH URZ RI A D OLQHDU FRPELQDWLRQRIWKHRWKHUURZV WKHQ GHW B D GHW A L GHW AT D # GHW A

N GHW ATA % 0

O $Q\V\VWHPRI n OLQHDUHTXDWLRQVLQ n YDULDEOHVFDQEH VROYHGE\&UDPHU·VUXOH P ,I X DQG Y DUHLQ R2 DQG GHW Œ X Y ! D 10 WKHQWKHDUHD RIWKHWULDQJOHLQWKHSODQHZLWKYHUWLFHVDW X DQG Y LV R ,I A LVLQYHUWLEOH WKHQ GHW A

!1

D GHW A

S ,I A LVLQYHUWLEOH WKHQ .GHW A/.GHW A!1 / D 1 8VHURZRSHUDWLRQVWRVKRZWKDWWKHGHWHUPLQDQWVLQ([HUFLVHV² DUHDOO]HUR ˇ ˇ ˇ ˇ ˇ 12 ˇ1 13 14 ˇˇ a b C c ˇˇ ˇ ˇ 16 17 ˇˇ b a C c ˇˇ ˇˇ 1 ˇˇ 15 ˇ 18 ˇ1 19 20 ˇ c aCbˇ ˇ ˇ a ˇ ˇˇ a C x ˇaCy

b bCx bCy

ˇ c ˇˇ c C x ˇˇ cCyˇ

&RPSXWHWKHGHWHUPLQDQWVLQ([HUFLVHVDQG ˇ ˇ ˇ9 1 9 9 9 ˇˇ ˇ ˇ9 0 9 9 2 ˇˇ ˇ 0 0 5 0 ˇˇ ˇˇ 4 ˇ9 0 3 9 0 ˇˇ ˇ ˇ6 0 0 7 0ˇ ˇ ˇ4 ˇ ˇ0 ˇ ˇˇ 6 ˇ0 ˇ ˇ0

8 1 8 8 8

8 0 8 8 2

8 0 8 3 0

GHW T D .b # a/.c # a/.c # b/

/HW f .t/ D GHW V ZLWK x1 x2 x3 DOOGLVWLQFW ([SODLQZK\ f .t/ LVDFXELFSRO\QRPLDO VKRZWKDWWKHFRHIÀFLHQWRI t 3 LV QRQ]HUR DQGÀQGWKUHHSRLQWVRQWKHJUDSKRI f

M GHW.#A/ D # GHW A

Q ,I A3 D 0 WKHQ GHW A D 0

8VHURZRSHUDWLRQVWRVKRZWKDW

ˇ 5 ˇˇ 0 ˇˇ 7 ˇˇ 0 ˇˇ 0ˇ

6KRZWKDWWKHHTXDWLRQRIWKHOLQHLQ R2 WKURXJKGLVWLQFW SRLQWV .x1 ; y1 / DQG .x2 ; y2 / FDQEHZULWWHQDV 2 3 1 x y y1 5 D 0 x1 GHW 4 1 1 x2 y2

)LQGD 3 ! 3 GHWHUPLQDQWHTXDWLRQVLPLODUWRWKDWLQ([HUFLVH WKDWGHVFULEHVWKHHTXDWLRQRIWKHOLQHWKURXJK .x1 ; y1 / ZLWK VORSH m ([HUFLVHVDQGFRQFHUQGHWHUPLQDQWVRIWKHIROORZLQJ 9DQGHU PRQGHPDWULFHV 2 3 2 3 1 t t2 t3 2 1 a a 6 7 61 x1 x12 x13 7 6 7 7 T D 41 b b 2 5; V .t / D 6 61 2 37 x x x 2 4 5 2 2 1 c c2 1 x3 x32 x33

'HWHUPLQHWKHDUHDRIWKHSDUDOOHORJUDPGHWHUPLQHGE\WKH SRLQWV .1; 4/ .#1; 5/ .3; 9/ DQG .5; 8/ +RZFDQ\RXWHOO WKDWWKHTXDGULODWHUDOGHWHUPLQHGE\WKHSRLQWVLVDFWXDOO\D SDUDOOHORJUDP" 8VHWKHFRQFHSWRIDUHDRIDSDUDOOHORJUDPWRZULWHDVWDWH PHQWDERXWD 2 ! 2 PDWUL[ A WKDWLVWUXHLIDQGRQO\LI A LV LQYHUWLEOH 6KRZWKDWLI A LVLQYHUWLEOH WKHQ DGM A LVLQYHUWLEOH DQG 1 A .DGM A/!1 D GHW A >+LQW *LYHQPDWULFHV B DQG C ZKDWFDOFXODWLRQV ZRXOG VKRZWKDW C LVWKHLQYHUVHRI B‹! /HW A B C D DQG I EH n ! n PDWULFHV 8VHWKHGHÀQL WLRQRUSURSHUWLHVRIDGHWHUPLQDQWWRMXVWLI\WKHIROORZLQJ IRUPXODV 3DUW F LVXVHIXOLQDSSOLFDWLRQVRIHLJHQYDOXHV &KDSWHU ! " ! " A 0 I 0 D GHW A E GHW D GHW D D GHW 0 I C D ! " ! " A 0 A B F GHW D .GHW A/.GHW D/ D GHW C D 0 D /HW A B C DQG D EH n ! n PDWULFHVZLWK A LQYHUWLEOH D )LQGPDWULFHV X DQG Y WRSURGXFHWKHEORFN/8 IDFWRU L]DWLRQ ! " ! "! " A B I 0 A B D C D X I 0 Y DQGWKHQVKRZWKDW ! " A B D .GHW A/ $ GHW.D # &$!1 B/ GHW C D

E 6KRZWKDWLI AC D &$ WKHQ ! " A B D GHW.AD # CB/ GHW C D

/HW J EH WKH n ! n PDWUL[ RI DOO ·V DQG FRQVLGHU A D .a # b/I C bJ WKDWLV 2 3 a b b $$$ b 6b a b $$$ b7 6 7 6b b a $ $ $ b7 AD6: 7 : : : :: 6: :: :: :: 7 : 4: 5

b

b

b

$$$

a

&RQÀUPWKDW GHW A D .a # b/ Œa C .n # 1/b! DVIROORZV D 6XEWUDFWURZIURPURZ URZIURPURZ DQGVRRQ DQGH[SODLQZK\WKLVGRHVQRWFKDQJHWKHGHWHUPLQDQWRI WKHPDWUL[ n!1


E :LWKWKHUHVXOWLQJPDWUL[IURPSDUWD DGGFROXPQWR FROXPQ WKHQDGGWKLVQHZFROXPQWRFROXPQ DQGVR RQ DQGH[SODLQZK\WKLVGRHVQRWFKDQJHWKHGHWHUPLQDQW F )LQGWKHGHWHUPLQDQWRIWKHUHVXOWLQJPDWUL[IURPE /HW A EHWKHRULJLQDOPDWUL[JLYHQLQ([HUFLVH DQGOHW 2 3 a#b b b $$$ b 6 0 a b $$$ b7 6 7 6 0 b a $ $ $ b7 BD6 : :: :: :: 7 :: 6 : 7 : 4 : : : :5

0

2

b 6b 6 6 C D 6 b: 6: 4: b

b

b

b a b :: :

b b a :: :

$$$ $$$ $$$ :: :

b

b

$$$

$$$

3

a

b b7 7 b7 :: 7 7 :5

a

1RWLFHWKDW A B DQG C DUHQHDUO\WKHVDPHH[FHSWWKDWWKH ÀUVWFROXPQRI A HTXDOVWKHVXPRIWKHÀUVWFROXPQVRI B DQG C $ OLQHDULW\SURSHUW\ RIWKHGHWHUPLQDQWIXQFWLRQ GLVFXVVHGLQ6HFWLRQ VD\VWKDW GHW A D GHW B C GHW C 8VHWKLVIDFWWRSURYHWKHIRUPXODLQ([HUFLVHE\LQGXFWLRQ RQWKHVL]HRIPDWUL[ A >0@ $SSO\WKHUHVXOWRI([HUFLVHWRÀQGWKHGHWHUPLQDQWV RIWKHIROORZLQJPDWULFHV DQGFRQÀUP\RXUDQVZHUVXVLQJD PDWUL[SURJUDP 3 2 2 3 8 3 3 3 3 3 8 8 8 63 8 3 3 37 7 68 6 3 8 87 6 7 63 3 8 3 37 7 6 48 5 8 3 8 43 3 3 8 35 8 8 8 3 3 3 3 3 8

>0@ 8VHDPDWUL[SURJUDPWRFRPSXWHWKHGHWHUPLQDQWVRI WKHIROORZLQJPDWULFHV 3 2 2 3 1 1 1 1 1 1 1 61 2 2 27 7 6 41 2 25 41 2 3 35 1 2 3 1 2 3 4 2 3 1 1 1 1 1 61 2 2 2 27 6 7 61 2 3 3 37 6 7 41 2 3 4 45 1 2 3 4 5 8VHWKHUHVXOWVWRJXHVVWKHGHWHUPLQDQWRIWKHPDWUL[EHORZ DQGFRQÀUP\RXUJXHVVE\XVLQJURZRSHUDWLRQVWRHYDOXDWH WKDWGHWHUPLQDQW 2 3 1 1 1 $$$ 1 61 2 2 $$$ 27 6 7 61 2 3 $ $ $ 37 6: 7 : : : :: 6: :: :: :: 7 : 4: 5

1

2

3

$$$

n

>0@ 8VHWKHPHWKRGRI([HUFLVHWRJXHVVWKHGHWHUPLQDQW RI 2 3 1 1 1 $$$ 1 61 7 3 3 $$$ 3 6 7 61 7 3 6 $ $ $ 6 6: 7 :: :: :: :: 6: 7 : 4: 5 : : : 1 3 6 $$$ 3.n # 1/ -XVWLI\\RXUFRQMHFWXUH >+LQW 8VH([HUFLVHF DQGWKH UHVXOWRI([HUFLVH@


9HFWRU6SDFHV

*/530%6$503: &9".1-&

6SDFH)OLJKWDQG&RQWURO6\VWHPV 7ZHOYHVWRULHVKLJKDQGZHLJKLQJWRQV &ROXPELD URVH PDMHVWLFDOO\RIIWKHODXQFKLQJSDGRQDFRRO3DOP6XQGD\ PRUQLQJLQ$SULO $ SURGXFWRIWHQ\HDUV·LQWHQVLYH UHVHDUFKDQGGHYHORSPHQW WKHÀUVW86VSDFHVKXWWOHZDVD WULXPSKRIFRQWUROV\VWHPVHQJLQHHULQJGHVLJQ LQYROYLQJ PDQ\EUDQFKHVRIHQJLQHHULQJ³DHURQDXWLFDO FKHPLFDO HOHFWULFDO K\GUDXOLF DQGPHFKDQLFDO 7KHVSDFHVKXWWOH·VFRQWUROV\VWHPVDUHDEVROXWHO\ FULWLFDOIRUÁLJKW %HFDXVHWKHVKXWWOHLVDQXQVWDEOH DLUIUDPH LWUHTXLUHVFRQVWDQWFRPSXWHUPRQLWRULQJGXULQJ DWPRVSKHULFÁLJKW 7KHÁLJKWFRQWUROV\VWHPVHQGVD VWUHDPRIFRPPDQGVWRDHURG\QDPLFFRQWUROVXUIDFHVDQG VPDOOWKUXVWHUMHWV )LJXUH VKRZVDW\SLFDOFORVHG ORRSIHHGEDFNV\VWHPWKDWFRQWUROVWKHSLWFKRIWKHVKXWWOH

Commanded pitch rate Commanded pitch

+ –

K1 Pitch rate

+

+ –

GXULQJÁLJKW 7KHSLWFKLVWKHHOHYDWLRQ DQJOH RIWKHQRVH FRQH 7KHMXQFWLRQV\PEROV˝ VKRZZKHUHVLJQDOV IURPYDULRXVVHQVRUVDUHDGGHGWRWKHFRPSXWHUVLJQDOV ÁRZLQJDORQJWKHWRSRIWKHÀJXUH 0DWKHPDWLFDOO\ WKHLQSXWDQGRXWSXWVLJQDOVWRDQ HQJLQHHULQJV\VWHPDUHIXQFWLRQV ,WLVLPSRUWDQWLQ DSSOLFDWLRQVWKDWWKHVHIXQFWLRQVFDQEHDGGHG DVLQ )LJ DQGPXOWLSOLHGE\VFDODUV 7KHVHWZRRSHUDWLRQV RQIXQFWLRQVKDYHDOJHEUDLFSURSHUWLHVWKDWDUHFRPSOHWHO\ DQDORJRXVWRWKHRSHUDWLRQVRIDGGLQJYHFWRUVLQ Rn DQGPXOWLSO\LQJDYHFWRUE\DVFDODU DVZHVKDOOVHH LQ6HFWLRQV DQG )RUWKLVUHDVRQ WKHVHWRIDOO SRVVLEOHLQSXWVIXQFWLRQV LVFDOOHGD YHFWRUVSDFH 7KH PDWKHPDWLFDOIRXQGDWLRQIRUV\VWHPVHQJLQHHULQJUHVWV

Commanded pitch acceleration K2

Pitch rate error

+

+

–

Controller

Shuttle dynamics

G1(s)

G2(s)

Pitch acceleration error

Pitch

Accelerometer s2

Rate gyro s

Inertial measuring unit 1 '*(63& 3LWFKFRQWUROV\VWHPIRUWKHVSDFHVKXWWOH 6RXUFH $GDSWHGIURP 6SDFH6KXWWOH*1 & 2SHUDWLRQV 0DQXDO 5RFNZHOO,QWHUQDWLRQDO

$)"15&3

9HFWRU6SDFHV

RQYHFWRUVSDFHVRIIXQFWLRQV DQG&KDSWHUH[WHQGVWKH WKHRU\RIYHFWRUVLQ Rn WRLQFOXGHVXFKIXQFWLRQV /DWHURQ

\RXZLOOVHHKRZRWKHUYHFWRUVSDFHVDULVHLQHQJLQHHULQJ SK\VLFV DQGVWDWLVWLFV WEB

7KHPDWKHPDWLFDOVHHGVSODQWHGLQ&KDSWHUV DQGJHUPLQDWHDQGEHJLQWREORVVRP LQWKLVFKDSWHU 7KHEHDXW\DQGSRZHURIOLQHDUDOJHEUDZLOOEHVHHQPRUHFOHDUO\ZKHQ \RXYLHZ Rn DVRQO\RQHRIDYDULHW\RIYHFWRUVSDFHVWKDWDULVHQDWXUDOO\LQDSSOLHG SUREOHPV $FWXDOO\ DVWXG\RIYHFWRUVSDFHVLVQRWPXFKGLIIHUHQWIURPDVWXG\RI Rn LWVHOI EHFDXVH\RXFDQXVH\RXUJHRPHWULFH[SHULHQFHZLWK R2 DQG R3 WRYLVXDOL]HPDQ\ JHQHUDOFRQFHSWV %HJLQQLQJZLWKEDVLFGHÀQLWLRQVLQ6HFWLRQ WKHJHQHUDOYHFWRUVSDFHIUDPHZRUN GHYHORSVJUDGXDOO\WKURXJKRXWWKHFKDSWHU $ JRDORI6HFWLRQV ²LVWRGHPRQVWUDWH KRZFORVHO\RWKHUYHFWRUVSDFHVUHVHPEOH Rn 6HFWLRQRQUDQNLVRQHRIWKHKLJK SRLQWVRIWKHFKDSWHU XVLQJYHFWRUVSDFHWHUPLQRORJ\WRWLHWRJHWKHULPSRUWDQWIDFWVDERXW UHFWDQJXODUPDWULFHV 6HFWLRQ DSSOLHVWKHWKHRU\RIWKHFKDSWHUWRGLVFUHWHVLJQDOVDQG GLIIHUHQFHHTXDWLRQVXVHGLQGLJLWDOFRQWUROV\VWHPVVXFKDVLQWKHVSDFHVKXWWOH 0DUNRY FKDLQV LQ6HFWLRQ SURYLGHDFKDQJHRISDFHIURPWKHPRUHWKHRUHWLFDOVHFWLRQVRI WKHFKDSWHUDQGPDNHJRRGH[DPSOHVIRUFRQFHSWVWREHLQWURGXFHGLQ&KDSWHU

7&$503 41"$&4 "/% 46#41"$&4 0XFKRIWKHWKHRU\LQ&KDSWHUV DQGUHVWHGRQFHUWDLQVLPSOHDQGREYLRXVDOJH EUDLFSURSHUWLHVRI Rn OLVWHGLQ6HFWLRQ ,QIDFW PDQ\RWKHUPDWKHPDWLFDOV\VWHPV KDYHWKHVDPHSURSHUWLHV 7KHVSHFLÀFSURSHUWLHVRILQWHUHVWDUHOLVWHGLQWKHIROORZLQJ GHÀQLWLRQ

%&'*/*5*0/

$ YHFWRUVSDFH LVDQRQHPSW\VHW V RIREMHFWV FDOOHG YHFWRUV RQZKLFKDUHGH ÀQHGWZRRSHUDWLRQV FDOOHG DGGLWLRQ DQG PXOWLSOLFDWLRQE\VFDODUV UHDOQXPEHUV VXEMHFWWRWKHWHQD[LRPVRUUXOHV OLVWHGEHORZ 7KHD[LRPVPXVWKROGIRUDOO YHFWRUV X Y DQG Z LQ V DQGIRUDOOVFDODUV c DQG d

7KHVXPRI X DQG Y GHQRWHGE\ X C Y LVLQ V X C Y D Y C X .X C Y/ C Z D X C .Y C Z/ 7KHUHLVD ]HUR YHFWRU LQ V VXFKWKDW X C D X )RUHDFK X LQ V WKHUHLVDYHFWRU !X LQ V VXFKWKDW X C .!X/ D 7KHVFDODUPXOWLSOHRI X E\ c GHQRWHGE\ c X LVLQ V c.X C Y/ D c X C c Y .c C d /X D c X C d X c.d X/ D .cd /X 1X D X

ñ 7HFKQLFDOO\ V LVD UHDOYHFWRUVSDFH $OORIWKHWKHRU\LQWKLVFKDSWHUDOVRKROGVIRUD FRPSOH[YHFWRUVSDFH LQZKLFKWKHVFDODUVDUHFRPSOH[QXPEHUV :HZLOOORRNDWWKLVEULHÁ\LQ&KDSWHU 8QWLOWKHQ DOOVFDODUV DUHDVVXPHGWREHUHDO

9HFWRU6SDFHVDQG6XEVSDFHV

8VLQJRQO\WKHVHD[LRPV RQHFDQVKRZWKDWWKH]HURYHFWRULQ$[LRP LVXQLTXH DQGWKHYHFWRU !X FDOOHGWKH QHJDWLYH RI X LQ$[LRPLVXQLTXHIRUHDFK X LQ V 6HH([HUFLVHV DQG 3URRIVRIWKHIROORZLQJVLPSOHIDFWVDUHDOVRRXWOLQHGLQWKH H[HUFLVHV )RUHDFK X LQ V DQGVFDODU c

0X D c D !X D .!1/X

&9".1-& 7KHVSDFHV Rn ZKHUH n " 1 DUHWKHSUHPLHUH[DPSOHVRIYHFWRU VSDFHV 7KHJHRPHWULFLQWXLWLRQGHYHORSHGIRU R3 ZLOOKHOS\RXXQGHUVWDQGDQGYLVXDOL]H PDQ\FRQFHSWVWKURXJKRXWWKHFKDSWHU

v

'*(63&

3v

–v

&9".1-& /HW V EHWKHVHWRIDOODUURZVGLUHFWHGOLQHVHJPHQWV LQWKUHHGLPHQ VLRQDOVSDFH ZLWKWZRDUURZVUHJDUGHGDVHTXDOLIWKH\KDYHWKHVDPHOHQJWKDQGSRLQW LQWKHVDPHGLUHFWLRQ 'HÀQHDGGLWLRQE\WKHSDUDOOHORJUDPUXOHIURP6HFWLRQ DQGIRUHDFK Y LQ V GHÀQH c Y WREHWKHDUURZZKRVHOHQJWKLV jcj WLPHVWKHOHQJWKRI Y SRLQWLQJLQWKHVDPHGLUHFWLRQDV Y LI c " 0 DQGRWKHUZLVHSRLQWLQJLQWKHRSSRVLWH GLUHFWLRQ 6HH)LJ 6KRZWKDW V LVDYHFWRUVSDFH 7KLVVSDFHLVDFRPPRQPRGHO LQSK\VLFDOSUREOHPVIRUYDULRXVIRUFHV 40-65*0/ 7KHGHÀQLWLRQRI V LVJHRPHWULF XVLQJFRQFHSWVRIOHQJWKDQGGLUHFWLRQ 1R xy´FRRUGLQDWHV\VWHPLVLQYROYHG $QDUURZRI]HUROHQJWKLVDVLQJOHSRLQWDQG UHSUHVHQWVWKH]HURYHFWRU 7KHQHJDWLYHRI Y LV .!1/Y 6R$[LRPV DQGDUH HYLGHQW 7KHUHVWDUHYHULÀHGE\JHRPHWU\ )RULQVWDQFH VHH)LJV DQG v+u

v w

u

v

u

v+w

u u+v

u+v

u+v+w

'*(63& X C Y D Y C X

'*(63& .X C Y/ C Z D X C .Y C Z/

&9".1-& /HW S EHWKHVSDFHRIDOOGRXEO\LQÀQLWHVHTXHQFHVRIQXPEHUVXVXDOO\ ZULWWHQLQDURZUDWKHUWKDQDFROXPQ

f yk g D .: : : ; y!2 ; y!1 ; y0 ; y1 ; y2 ; : : :/ ,I f´k g LVDQRWKHUHOHPHQWRI S WKHQWKHVXP f yk g C f´k g LVWKHVHTXHQFH f yk C ´k g IRUPHGE\DGGLQJFRUUHVSRQGLQJWHUPVRI f yk g DQG f´k g 7KHVFDODUPXOWLSOH c f yk g LV WKHVHTXHQFH fcyk g 7KHYHFWRUVSDFHD[LRPVDUHYHULÀHGLQWKHVDPHZD\DVIRU Rn (OHPHQWVRI S DULVHLQHQJLQHHULQJ IRUH[DPSOH ZKHQHYHUDVLJQDOLVPHDVXUHGRU VDPSOHG DWGLVFUHWHWLPHV $ VLJQDOPLJKWEHHOHFWULFDO PHFKDQLFDO RSWLFDO DQGVRRQ 7KHPDMRUFRQWUROV\VWHPVIRUWKHVSDFHVKXWWOH PHQWLRQHGLQWKHFKDSWHULQWURGXFWLRQ XVHGLVFUHWHRUGLJLWDO VLJQDOV )RUFRQYHQLHQFH ZHZLOOFDOO S WKHVSDFHRIGLVFUHWH WLPH VLJQDOV $ VLJQDOPD\EHYLVXDOL]HGE\DJUDSKDVLQ)LJ

$)"15&3

9HFWRU6SDFHV

–5

0

5

10

'*(63& $ GLVFUHWHWLPHVLJQDO

&9".1-& )RU n " 0 WKHVHW Pn RISRO\QRPLDOVRIGHJUHHDWPRVW n FRQVLVWVRI DOOSRO\QRPLDOVRIWKHIRUP

S.t / D a0 C a1 t C a2 t 2 C # # # C an t n

ZKHUHWKHFRHIÀFLHQWV a0 ; : : : ; an DQGWKHYDULDEOH t DUHUHDOQXPEHUV 7KH GHJUHH RI S LVWKHKLJKHVWSRZHURI t LQ ZKRVHFRHIÀFLHQWLVQRW]HUR ,I S.t / D a0 ¤ 0 WKH GHJUHHRI S LV]HUR ,IDOOWKHFRHIÀFLHQWVDUH]HUR S LVFDOOHGWKH ]HURSRO\QRPLDO 7KH ]HURSRO\QRPLDOLVLQFOXGHGLQ Pn HYHQWKRXJKLWVGHJUHH IRUWHFKQLFDOUHDVRQV LVQRW GHÀQHG ,I S LVJLYHQE\ DQGLI T.t/ D b0 C b1 t C # # # C bn t n WKHQWKHVXP S C T LV GHÀQHGE\

.S C T/.t/ D S.t / C T.t / D .a0 C b0 / C .a1 C b1 /t C # # # C .an C bn /t n 7KHVFDODUPXOWLSOH c S LVWKHSRO\QRPLDOGHÀQHGE\

.c S/.t / D c S.t / D ca0 C .ca1 /t C # # # C .can /t n

7KHVHGHÀQLWLRQVVDWLVI\$[LRPV DQGEHFDXVH S C T DQG c S DUHSRO\QRPLDOV RIGHJUHHOHVVWKDQRUHTXDOWR n $[LRPV DQG²IROORZIURPSURSHUWLHVRIWKH UHDOQXPEHUV &OHDUO\ WKH]HURSRO\QRPLDODFWVDVWKH]HURYHFWRULQ$[LRP )LQDOO\ .!1/S DFWVDVWKHQHJDWLYHRI S VR$[LRPLVVDWLVÀHG 7KXV Pn LVDYHFWRUVSDFH 7KHYHFWRUVSDFHV Pn IRUYDULRXV n DUHXVHG IRULQVWDQFH LQVWDWLVWLFDOWUHQGDQDO\VLV RIGDWD GLVFXVVHGLQ6HFWLRQ

&9".1-& /HW V EHWKHVHWRIDOOUHDOYDOXHGIXQFWLRQVGHÀQHGRQDVHW D 7\SL FDOO\ D LVWKHVHWRIUHDOQXPEHUVRUVRPHLQWHUYDORQWKHUHDOOLQH )XQFWLRQVDUHDGGHG LQWKHXVXDOZD\ I C J LVWKHIXQFWLRQZKRVHYDOXHDW t LQWKHGRPDLQ D LV I.t/ C J.t/ /LNHZLVH IRUDVFDODU c DQGDQ I LQ V WKHVFDODUPXOWLSOH c I LVWKHIXQFWLRQZKRVHYDOXH DW t LV c I.t / )RULQVWDQFH LI D D R I.t / D 1 C VLQ 2t DQG J.t / D 2 C :5t WKHQ f+g f

.I C J/.t / D 3 C VLQ 2t C :5t

DQG

.2J/.t/ D 4 C t

7ZRIXQFWLRQVLQ V DUHHTXDOLIDQGRQO\LIWKHLUYDOXHVDUHHTXDOIRUHYHU\ t LQ D +HQFHWKH]HURYHFWRULQ V LVWKHIXQFWLRQWKDWLVLGHQWLFDOO\]HUR I.t / D 0 IRUDOO t DQG WKHQHJDWLYHRI I LV .!1/I $[LRPV DQGDUHREYLRXVO\WUXH DQGWKHRWKHUD[LRPV IROORZIURPSURSHUWLHVRIWKHUHDOQXPEHUV VR V LVDYHFWRUVSDFH

g

0 '*(63&

7KHVXPRIWZRYHFWRUV IXQFWLRQV

,WLVLPSRUWDQWWRWKLQNRIHDFKIXQFWLRQLQWKHYHFWRUVSDFH V RI([DPSOH DVD VLQJOHREMHFW DVMXVWRQH´SRLQWµRUYHFWRULQWKHYHFWRUVSDFH 7KHVXPRIWZRYHFWRUV I DQG J IXQFWLRQVLQ V RUHOHPHQWVRI DQ\ YHFWRUVSDFH FDQEHYLVXDOL]HGDVLQ)LJ EHFDXVHWKLVFDQKHOS\RXFDUU\RYHUWRDJHQHUDOYHFWRUVSDFHWKHJHRPHWULFLQWXLWLRQ \RXKDYHGHYHORSHGZKLOHZRUNLQJZLWKWKHYHFWRUVSDFH Rn 6HHWKH 6WXG\*XLGH IRU KHOSDV\RXOHDUQWRDGRSWWKLVPRUHJHQHUDOSRLQWRIYLHZ


6XEVSDFHV ,QPDQ\SUREOHPV DYHFWRUVSDFHFRQVLVWVRIDQDSSURSULDWHVXEVHWRIYHFWRUVIURPVRPH ODUJHUYHFWRUVSDFH ,QWKLVFDVH RQO\WKUHHRIWKHWHQYHFWRUVSDFHD[LRPVQHHGWREH FKHFNHG WKHUHVWDUHDXWRPDWLFDOO\VDWLVÀHG

%&'*/*5*0/

$ VXEVSDFH RIDYHFWRUVSDFH V LVDVXEVHW H RI V WKDWKDVWKUHHSURSHUWLHV D 7KH]HURYHFWRURI V LVLQ H E H LVFORVHGXQGHUYHFWRUDGGLWLRQ 7KDWLV IRUHDFK X DQG Y LQ H WKHVXP X C Y LVLQ H F H LVFORVHGXQGHUPXOWLSOLFDWLRQE\VFDODUV 7KDWLV IRUHDFK X LQ H DQGHDFK VFDODU c WKHYHFWRU c X LVLQ H 3URSHUWLHVD E DQGF JXDUDQWHHWKDWDVXEVSDFH H RI V LVLWVHOID YHFWRU VSDFH XQGHUWKHYHFWRUVSDFHRSHUDWLRQVDOUHDG\GHÀQHGLQ V 7RYHULI\WKLV QRWH WKDWSURSHUWLHVD E DQGF DUH$[LRPV DQG $[LRPV DQG²DUH DXWRPDWLFDOO\WUXHLQ H EHFDXVHWKH\DSSO\WRDOOHOHPHQWVRI V LQFOXGLQJWKRVHLQ H $[LRPLVDOVRWUXHLQ H EHFDXVHLI X LVLQ H WKHQ .!1/X LVLQ H E\SURSHUW\F DQG ZHNQRZIURPHTXDWLRQ RQSDJH WKDW .!1/X LVWKHYHFWRU !X LQ$[LRP 6RHYHU\VXEVSDFHLVDYHFWRUVSDFH &RQYHUVHO\ HYHU\YHFWRUVSDFHLVDVXEVSDFH RILWVHOIDQGSRVVLEO\RIRWKHUODUJHUVSDFHV 7KHWHUP VXEVSDFH LVXVHGZKHQDWOHDVW WZRYHFWRUVSDFHVDUHLQPLQG ZLWKRQHLQVLGHWKHRWKHU DQGWKHSKUDVH VXEVSDFHRI V LGHQWLÀHV V DVWKHODUJHUVSDFH 6HH)LJ

H 0 V '*(63&

$ VXEVSDFHRI V

&9".1-& 7KHVHWFRQVLVWLQJRIRQO\WKH]HURYHFWRULQDYHFWRUVSDFH V LVD VXEVSDFHRI V FDOOHGWKH ]HURVXEVSDFH DQGZULWWHQDV fg &9".1-& /HW P EHWKHVHWRIDOOSRO\QRPLDOVZLWKUHDOFRHIÀFLHQWV ZLWKRSHUD WLRQVLQ P GHÀQHGDVIRUIXQFWLRQV 7KHQ P LVDVXEVSDFHRIWKHVSDFHRIDOOUHDOYDOXHG IXQFWLRQVGHÀQHGRQ R $OVR IRUHDFK n " 0 Pn LVDVXEVSDFHRI P EHFDXVH Pn LVD VXEVHWRI P WKDWFRQWDLQVWKH]HURSRO\QRPLDO WKHVXPRIWZRSRO\QRPLDOVLQ Pn LVDOVR LQ Pn DQGDVFDODUPXOWLSOHRIDSRO\QRPLDOLQ Pn LVDOVRLQ Pn

x3

&9".1-& 7KHYHFWRUVSDFH R2 LV QRW DVXEVSDFHRI R3 EHFDXVH R2 LVQRWHYHQD

H x2 x1 '*(63&

7KH x1 x2 SODQHDVDVXEVSDFHRI R3

VXEVHWRI R3 7KHYHFWRUVLQ R3 DOOKDYHWKUHHHQWULHV ZKHUHDVWKHYHFWRUVLQ R2 KDYH RQO\WZR 7KHVHW 82 3 9 < s = H D 4 t 5 W s DQG t DUHUHDO : ; 0

LVDVXEVHWRI R3 WKDWÓRRNVµDQG´DFWVµOLNH R2 DOWKRXJKLWLVORJLFDOO\GLVWLQFWIURP R2 6HH)LJ 6KRZWKDW H LVDVXEVSDFHRI R3

40-65*0/ 7KH]HURYHFWRULVLQ H DQG H LVFORVHGXQGHUYHFWRUDGGLWLRQDQGVFDODU PXOWLSOLFDWLRQEHFDXVHWKHVHRSHUDWLRQVRQYHFWRUVLQ H DOZD\VSURGXFHYHFWRUVZKRVH WKLUGHQWULHVDUH]HURDQGVREHORQJWR H / 7KXV H LVDVXEVSDFHRI R3 ò 6RPHWH[WVUHSODFHSURSHUW\D LQWKLVGHÀQLWLRQE\WKHDVVXPSWLRQWKDW H LVQRQHPSW\ 7KHQD FRXOGEH GHGXFHGIURPF DQGWKHIDFWWKDW 0X D %XWWKHEHVWZD\WRWHVWIRUDVXEVSDFHLVWRORRNÀUVWIRUWKH]HUR YHFWRU ,I LVLQ H WKHQSURSHUWLHVE DQGF PXVWEHFKHFNHG ,I LV QRW LQ H WKHQ H FDQQRWEHD VXEVSDFHDQGWKHRWKHUSURSHUWLHVQHHGQRWEHFKHFNHG

$)"15&3

9HFWRU6SDFHV

&9".1-& $ SODQHLQ R3 QRW WKURXJKWKHRULJLQLVQRWDVXEVSDFHRI R3 EHFDXVH WKHSODQHGRHVQRWFRQWDLQWKH]HURYHFWRURI R3 6LPLODUO\ DOLQHLQ R2 QRW WKURXJKWKH RULJLQ VXFKDVLQ)LJ LV QRW DVXEVSDFHRI R2

x2

$ 6XEVSDFH6SDQQHGE\D6HW H x1 '*(63&

7KHQH[WH[DPSOHLOOXVWUDWHVRQHRIWKHPRVWFRPPRQZD\VRIGHVFULELQJDVXEVSDFH $VLQ&KDSWHU WKHWHUP OLQHDUFRPELQDWLRQ UHIHUVWRDQ\VXPRIVFDODUPXOWLSOHVRI YHFWRUV DQG 6SDQ fY1 ; : : : ; Yp g GHQRWHVWKHVHWRIDOOYHFWRUVWKDWFDQEHZULWWHQDVOLQHDU FRPELQDWLRQVRI Y1 ; : : : ; Yp

$ OLQHWKDWLVQRWDYHFWRUVSDFH

&9".1-& *LYHQ Y1 DQG Y2 LQDYHFWRUVSDFH V OHW H D 6SDQ fY1 ; Y2 g 6KRZ WKDW H LVDVXEVSDFHRI V

40-65*0/ 7KH]HURYHFWRULVLQ H VLQFH D 0Y1 C 0Y2 7RVKRZWKDW H LVFORVHG XQGHUYHFWRUDGGLWLRQ WDNHWZRDUELWUDU\YHFWRUVLQ H VD\ X D s1 Y1 C s2 Y2

DQG

Z D t1 Y1 C t2 Y2

%\$[LRPV DQGIRUWKHYHFWRUVSDFH V X C Z D .s1 Y1 C s2 Y2 / C .t1 Y1 C t2 Y2 / D .s1 C t1 /Y1 C .s2 C t2 /Y2

x3

6R X C Z LVLQ H )XUWKHUPRUH LI c LVDQ\VFDODU WKHQE\$[LRPV DQG

c X D c.s1 Y1 C s2 Y2 / D .cs1 /Y1 C .cs2 /Y2 ZKLFKVKRZVWKDW c X LVLQ H DQG H LVFORVHGXQGHUVFDODUPXOWLSOLFDWLRQ 7KXV H LVD VXEVSDFHRI V

v1 v2

x1

0

x2

'*(63&

$QH[DPSOHRIDVXEVSDFH

5)&03&.

,Q6HFWLRQ \RXZLOOVHHWKDWHYHU\QRQ]HURVXEVSDFHRI R3 RWKHUWKDQ R3 LWVHOI LVHLWKHU 6SDQ fY1 Y2 g IRUVRPHOLQHDUO\LQGHSHQGHQW Y1 DQG Y2 RU 6SDQ fYg IRU Y ¤ ,Q WKHÀUVWFDVH WKHVXEVSDFHLVDSODQHWKURXJKWKHRULJLQ LQWKHVHFRQGFDVH LWLVDOLQH WKURXJKWKHRULJLQ 6HH)LJ ,WLVKHOSIXOWRNHHSWKHVHJHRPHWULFSLFWXUHVLQPLQG HYHQIRUDQDEVWUDFWYHFWRUVSDFH 7KHDUJXPHQWLQ([DPSOHFDQHDVLO\EHJHQHUDOL]HGWRSURYHWKHIROORZLQJ WKHRUHP ,I Y1 ; : : : ; Yp DUHLQDYHFWRUVSDFH V WKHQ 6SDQ fY1 ; : : : ; Yp g LVDVXEVSDFHRI V :HFDOO 6SDQ fY1 ; : : : ; Yp g WKHVXEVSDFHVSDQQHG RU JHQHUDWHG E\ fY1 ; : : : ; Yp g *LYHQDQ\VXEVSDFH H RI V D VSDQQLQJ RU JHQHUDWLQJ VHW IRU H LVDVHW fY1 ; : : : ; Yp g LQ H VXFKWKDW H D 6SDQ fY1 ; : : : ; Yp g 7KHQH[WH[DPSOHVKRZVKRZWRXVH7KHRUHP

&9".1-& /HW H EHWKHVHWRIDOOYHFWRUVRIWKHIRUP .a ! 3b; b ! a; a; b/ ZKHUH a DQG b DUHDUELWUDU\VFDODUV 7KDWLV OHW H D f.a ! 3b; b ! a; a; b/ W a DQG b LQ Rg 6KRZWKDW H LVDVXEVSDFHRI R4


40-65*0/ :ULWHWKHYHFWRUVLQ H DVFROXPQYHFWRUV 7KHQDQDUELWUDU\YHFWRULQ H KDVWKHIRUP 2 3 2 3 2 3 a ! 3b 1 !3 6 b!a 7 6 7 6 7 6 7 D a6 !1 7 C b 6 1 7 4 a 5 4 15 4 05 b 0 1 !

!

Y1

Y2

7KLVFDOFXODWLRQVKRZVWKDW H D 6SDQ fY1 ; Y2 g ZKHUH Y1 DQG Y2 DUHWKHYHFWRUVLQGLFDWHG DERYH 7KXV H LVDVXEVSDFHRI R4 E\7KHRUHP ([DPSOHLOOXVWUDWHVDXVHIXOWHFKQLTXHRIH[SUHVVLQJDVXEVSDFH H DVWKHVHW RIOLQHDUFRPELQDWLRQVRIVRPHVPDOOFROOHFWLRQRIYHFWRUV ,I H D 6SDQ fY1 ; : : : ; Yp g ZHFDQWKLQNRIWKHYHFWRUV Y1 ; : : : ; Yp LQWKHVSDQQLQJVHWDV´KDQGOHVµWKDWDOORZXVWR KROGRQWRWKHVXEVSDFH H &DOFXODWLRQVZLWKWKHLQÀQLWHO\PDQ\YHFWRUVLQ H DUHRIWHQ UHGXFHGWRRSHUDWLRQVZLWKWKHÀQLWHQXPEHURIYHFWRUVLQWKHVSDQQLQJVHW

&9".1-& )RUZKDWYDOXHV RI h ZLOO \ EHLQWKHVXEVSDFHRI R3 VSDQQHGE\ Y1 ; Y2 ; Y3 LI

2

3 1 Y1 D 4 !1 5; !2

2

3 5 Y2 D 4 !4 5; !7

2

3 !3 Y3 D 4 1 5; 0

DQG

2

3 !4 \ D 4 35 h

40-65*0/ 7KLVTXHVWLRQLV3UDFWLFH3UREOHP LQ6HFWLRQ ZULWWHQKHUHZLWKWKH WHUP VXEVSDFH UDWKHUWKDQ 6SDQ fY1 ; Y2 ; Y3 g 7KHVROXWLRQWKHUHVKRZVWKDW \ LVLQ 6SDQ fY1 ; Y2 ; Y3 g LIDQGRQO\LI h D 5 7KDWVROXWLRQLVZRUWKUHYLHZLQJQRZ DORQJ ZLWK([HUFLVHV ²DQG²LQ6HFWLRQ $OWKRXJKPDQ\YHFWRUVSDFHVLQWKLVFKDSWHUZLOOEHVXEVSDFHVRI Rn LWLVLPSRUWDQW WRNHHSLQPLQGWKDWWKHDEVWUDFWWKHRU\DSSOLHVWRRWKHUYHFWRUVSDFHVDVZHOO 9HFWRU VSDFHVRIIXQFWLRQVDULVHLQPDQ\DSSOLFDWLRQV DQGWKH\ZLOOUHFHLYHPRUHDWWHQWLRQODWHU

13"$5*$& 130#-&.4

WEB

6KRZWKDWWKHVHW H RIDOOSRLQWVLQ R2 RIWKHIRUP .3s; 2 C 5s/ LVQRWDYHFWRUVSDFH E\VKRZLQJWKDWLWLVQRWFORVHGXQGHUVFDODUPXOWLSOLFDWLRQ )LQGDVSHFLÀFYHFWRU X LQ H DQGDVFDODU c VXFKWKDW c X LVQRWLQ H /HW W D 6SDQ fY1 ; : : : ; Yp g ZKHUH Y1 ; : : : ; Yp DUHLQDYHFWRUVSDFH V 6KRZWKDW Yk LVLQ W IRU 1 $ k $ p >+LQW )LUVWZULWHDQHTXDWLRQWKDWVKRZVWKDW Y1 LVLQ W 7KHQDGMXVW\RXUQRWDWLRQIRUWKHJHQHUDOFDVH@

&9&3$*4&4 /HW V EHWKHÀUVWTXDGUDQWLQWKH xy SODQH WKDWLV OHW !" # $ x V D W x " 0; y " 0 y D ,I X DQG Y DUHLQ V LV X C Y LQ V " :K\"

E )LQGDVSHFLÀFYHFWRU X LQ V DQGDVSHFLÀFVFDODU c VXFK

WKDW c X LV QRW LQ V 7KLVLVHQRXJKWRVKRZWKDW V LV QRW DYHFWRUVSDFH /HW W EHWKHXQLRQRIWKHÀUVWDQGWKLUGTXDGUDQWVLQWKH xy !" # $ x W xy " 0 SODQH 7KDWLV OHW W D y D ,I X LVLQ W DQG c LVDQ\VFDODU LV c X LQ W " :K\"

$)"15&3 E

9HFWRU6SDFHV

)LQGVSHFLÀFYHFWRUV X DQG Y LQ W VXFKWKDW X C Y LV QRWLQ W 7KLVLVHQRXJKWRVKRZWKDW W LV QRW DYHFWRU VSDFH

/HW H EHWKHVHWRISRLQWVLQVLGHDQGRQWKHXQLWFLUFOHLQ !" # $ x W x 2 C y 2 $ 1 )LQG WKH [\SODQH 7KDWLV OHW H D y DVSHFLÀFH[DPSOH³WZRYHFWRUVRUDYHFWRUDQGDVFDODU³WR VKRZWKDW H LVQRWDVXEVSDFHRI R2 &RQVWUXFWDJHRPHWULFÀJXUHWKDWLOOXVWUDWHVZK\DOLQHLQ R2 QRW WKURXJKWKHRULJLQLVQRWFORVHGXQGHUYHFWRUDGGLWLRQ ,Q([HUFLVHV² GHWHUPLQHLIWKHJLYHQVHWLVDVXEVSDFHRI Pn IRU DQDSSURSULDWHYDOXHRI n -XVWLI\\RXUDQVZHUV $OOSRO\QRPLDOVRIWKHIRUP S.t/ D at ZKHUH a LVLQ R 2

$OOSRO\QRPLDOVRIWKHIRUP S.t / D a C t 2 ZKHUH a LVLQ R $OOSRO\QRPLDOVRIGHJUHHDWPRVW ZLWKLQWHJHUVDVFRHIÀ FLHQWV $OOSRO\QRPLDOVLQ Pn VXFKWKDW S.0/ D 0

2

3 !2t /HW H EHWKHVHWRIDOOYHFWRUVRIWKHIRUP 4 5t 5 )LQGD 3t YHFWRU Y LQ R3 VXFKWKDW H D 6SDQ fYg :K\GRHVWKLVVKRZ WKDW H LVDVXEVSDFHRI R3 " 2 3 3t /HW H EHWKHVHWRIDOOYHFWRUVRIWKHIRUP 4 0 5 ZKHUH t !7t LVDQ\UHDOQXPEHU 6KRZWKDW H LVDVXEVSDFHRI R3 8VH WKHPHWKRGRI([HUFLVH 2 3 2b C 3c /HW W EHWKHVHWRIDOOYHFWRUVRIWKHIRUP 4 !b 5 2c ZKHUH b DQG c DUHDUELWUDU\ )LQGYHFWRUV X DQG Y VXFKWKDW W D 6SDQ fX; Yg :K\GRHVWKLVVKRZWKDW W LVDVXEVSDFH RI R3 " 2 3 2s C 4t 6 2s 7 7 /HW W EHWKHVHWRIDOOYHFWRUVRIWKHIRUP 6 4 2s ! 3t 5 5t 6KRZ WKDW W LV D VXEVSDFH RI R4 8VH WKH PHWKRGRI ([HUFLVH 2 3 2 3 2 3 2 3 1 2 4 3 /HW Y1 D 4 0 5 Y2 D 4 1 5 Y3 D 4 2 5 DQG Z D 4 1 5 !1 3 6 2 D ,V Z LQ fY1 ; Y2 ; Y3 g" +RZPDQ\YHFWRUVDUHLQ fY1 ; Y2 ; Y3 g"

E +RZPDQ\YHFWRUVDUHLQ 6SDQ fY1 ; Y2 ; Y3 g"

F ,V Z LQWKHVXEVSDFHVSDQQHGE\ fY1 ; Y2 ; Y3 g" :K\" 2 3 1 /HW Y1 ; Y2 ; Y3 EHDVLQ([HUFLVH DQGOHW Z D 4 3 5 ,V Z 14 LQWKHVXEVSDFHVSDQQHGE\ fY1 ; Y2 ; Y3 g" :K\"

,Q([HUFLVHV² OHW W EHWKHVHWRIDOOYHFWRUVRIWKHIRUP VKRZQ ZKHUH a b DQG c UHSUHVHQWDUELWUDU\UHDOQXPEHUV ,Q HDFKFDVH HLWKHUÀQGDVHW S RIYHFWRUVWKDWVSDQV W RUJLYHDQ H[DPSOHWRVKRZWKDW W LV QRW DYHFWRUVSDFH 2 3 2 3 2a C 3b 1 4 !1 5 4 3a ! 5b 5 2a ! 5b 3b C 2a 2 3 2 3 2a ! b 4a C 3b 6 3b ! c 7 6 7 0 7 7 6 6 4 3c ! a 5 4 a C 3b C c 5 3b 3b ! 2c

,IDPDVV m LVSODFHGDWWKHHQGRIDVSULQJ DQGLIWKHPDVVLV SXOOHGGRZQZDUGDQGUHOHDVHG WKHPDVV²VSULQJV\VWHPZLOO EHJLQWRRVFLOODWH 7KHGLVSODFHPHQW y RIWKHPDVVIURPLWV UHVWLQJSRVLWLRQLVJLYHQE\DIXQFWLRQRIWKHIRUP

y.t/ D c1 FRV !t C c2 VLQ !t

ZKHUH ! LVDFRQVWDQWWKDWGHSHQGVRQWKHVSULQJDQGWKHPDVV 6HHWKHÀJXUHEHORZ 6KRZWKDWWKHVHWRIDOOIXQFWLRQV GHVFULEHGLQ ZLWK ! À[HGDQG c1 c2 DUELWUDU\ LVDYHFWRU VSDFH

y

7KHVHWRIDOOFRQWLQXRXVUHDOYDOXHGIXQFWLRQVGHÀQHGRQD FORVHGLQWHUYDO Œa; b! LQ R LVGHQRWHGE\ C Œa; b! 7KLVVHWLV DVXEVSDFHRIWKHYHFWRUVSDFHRIDOOUHDOYDOXHGIXQFWLRQV GHÀQHGRQ Œa; b! D :KDWIDFWVDERXWFRQWLQXRXVIXQFWLRQVVKRXOGEHSURYHG LQRUGHUWRGHPRQVWUDWHWKDW C Œa; b! LVLQGHHGDVXEVSDFH DV FODLPHG" 7KHVH IDFWV DUH XVXDOO\ GLVFXVVHG LQ D FDOFXOXVFODVV E 6KRZWKDW fI LQ C Œa; b! W I .a/ D I .b/g LVDVXEVSDFHRI C Œa; b! )RUÀ[HGSRVLWLYHLQWHJHUV m DQG n WKHVHW Mm!n RIDOO m % n PDWULFHVLVDYHFWRUVSDFH XQGHUWKHXVXDORSHUDWLRQVRIDGGLWLRQ RIPDWULFHVDQGPXOWLSOLFDWLRQE\UHDOVFDODUV " # a b 'HWHUPLQHLIWKHVHW H RIDOOPDWULFHVRIWKHIRUP 0 d LVDVXEVSDFHRI M2!2 /HW F EHDÀ[HG 3 % 2 PDWUL[ DQGOHW H EHWKHVHWRIDOO PDWULFHV A LQ M2!4 ZLWKWKHSURSHUW\WKDW FA D 0 WKH]HUR PDWUL[LQ M3!4 / 'HWHUPLQHLI H LVDVXEVSDFHRI M2!4


,Q([HUFLVHVDQG PDUNHDFKVWDWHPHQW7UXHRU)DOVH -XVWLI\ HDFKDQVZHU D ,I I LVDIXQFWLRQLQWKHYHFWRUVSDFH V RIDOOUHDOYDOXHG IXQFWLRQVRQ R DQGLI I .t / D 0 IRUVRPH t WKHQ I LVWKH ]HURYHFWRULQ V E $ YHFWRULVDQDUURZLQWKUHHGLPHQVLRQDOVSDFH F $ VXEVHW H RIDYHFWRUVSDFH V LVDVXEVSDFHRI V LIWKH ]HURYHFWRULVLQ H

c D IRUHYHU\VFDODU c c D c. C / D c C c

H $QDORJVLJQDOVDUHXVHGLQWKHPDMRUFRQWUROV\VWHPVIRU WKHVSDFHVKXWWOH PHQWLRQHGLQWKHLQWURGXFWLRQWRWKH FKDSWHU D $ YHFWRULVDQ\HOHPHQWRIDYHFWRUVSDFH E ,I X LVDYHFWRULQDYHFWRUVSDFH V WKHQ .!1/X LVWKHVDPH DVWKHQHJDWLYHRI X F $ YHFWRUVSDFHLVDOVRDVXEVSDFH G R2 LVDVXEVSDFHRI R3 H $ VXEVHW H RIDYHFWRUVSDFH V LVDVXEVSDFHRI V LIWKH IROORZLQJFRQGLWLRQVDUHVDWLVÀHG L WKH]HURYHFWRURI V LVLQ H LL X Y DQG X C Y DUHLQ H DQGLLL c LVD VFDODUDQG c X LVLQ H ([HUFLVHV²VKRZKRZWKHD[LRPVIRUDYHFWRUVSDFH V FDQ EHXVHGWRSURYHWKHHOHPHQWDU\SURSHUWLHVGHVFULEHGDIWHUWKH GHÀQLWLRQRIDYHFWRUVSDFH )LOOLQWKHEODQNVZLWKWKHDSSURSULDWH D[LRPQXPEHUV %HFDXVHRI$[LRP $[LRPVDQGLPSO\ UHVSHFWLYHO\ WKDW C X D X DQG !X C X D IRUDOO X

c C .!c /DŒc C c ! C .!c /

c C .!c /Dc C Œc C .!c /! Dc C

C Z D .!X/ C Z D !X

E\$[LRP

D

E\$[LRP

E

E\$[LRP

F

)LOOLQWKHPLVVLQJD[LRPQXPEHUVLQWKHIROORZLQJSURRIWKDW 0X D IRUHYHU\ X LQ V

0X D .0 C 0/X D 0X C 0X

E\$[LRP

D

0X C .!0X/ D 0X C Œ0X C .!0X/! D 0X C D 0X

E\$[LRP

F

E\$[LRP

G

E\$[LRP

H

/HW X DQG Y EHYHFWRUVLQDYHFWRUVSDFH V DQGOHW H EHDQ\ VXEVSDFHRI V WKDWFRQWDLQVERWK X DQG Y ([SODLQZK\ H DOVRFRQWDLQV 6SDQ fX; Yg 7KLVVKRZVWKDW 6SDQ fX; Yg LVWKH VPDOOHVWVXEVSDFHRI V WKDWFRQWDLQVERWK X DQG Y /HW H DQG K EH VXEVSDFHV RI D YHFWRU VSDFH V 7KH LQWHUVHFWLRQ RI H DQG K ZULWWHQDV H \ K LVWKHVHWRI Y LQ V WKDWEHORQJWRERWK H DQG K 6KRZWKDW H \ K LV DVXEVSDFHRI V 6HHWKHÀJXUH *LYHDQH[DPSOHLQ R2 WRVKRZWKDWWKHXQLRQRIWZRVXEVSDFHVLVQRW LQJHQHUDO D VXEVSDFH

H!

K

0 H V

K

*LYHQVXEVSDFHV H DQG K RIDYHFWRUVSDFH V WKH VXP RI H DQG K ZULWWHQDV H C K LVWKHVHWRIDOOYHFWRUVLQ V WKDWFDQEHZULWWHQDVWKHVXPRIWZRYHFWRUV RQHLQ H DQG WKHRWKHULQ K WKDWLV

H C K D fZ Z

X Y IRUVRPH X LQ H

DQGVRPH Y LQ Kg

D 6KRZWKDW H C K LVDVXEVSDFHRI V

E 6KRZWKDW H LVDVXEVSDFHRI H C K DQG K LVDVXEVSDFH RI H C K

$GGWKHQHJDWLYHRI 0X WRERWKVLGHV

0X C .!0X/ D Œ0X C 0X! C .!0X/

E

6XSSRVH c X D IRUVRPHQRQ]HURVFDODU c 6KRZWKDW X D 0HQWLRQWKHD[LRPVRUSURSHUWLHV\RXXVH

&RPSOHWHWKHIROORZLQJSURRIWKDW !X LVWKH XQLTXHYHFWRU LQ V VXFK WKDW X C .!X/ D 6XSSRVH WKDW Z VDWLVÀHV X C Z D $GGLQJ !X WRERWKVLGHV ZHKDYH

Œ.!X/ C X! C Z D .!X/ C

D

3URYHWKDW .!1/X D !X >+LQW 6KRZWKDW X C .!1/X D 8VHVRPHD[LRPVDQGWKHUHVXOWVRI([HUFLVHV DQG@

&RPSOHWH WKH IROORZLQJ SURRI WKDW WKH ]HUR YHFWRU LV XQLTXH 6XSSRVH WKDW Z LQ V KDV WKH SURSHUW\ WKDW X C Z D Z C X D X IRUDOO X LQ V ,QSDUWLFXODU C Z D +HQFH Z D C Z D %XW C Z D Z E\$[LRP

.!X/ C ŒX C Z! D .!X/ C

E\$[LRP E\$[LRP

$GGWKHQHJDWLYHRI c WRERWKVLGHV

Dc

G $ VXEVSDFHLVDOVRDYHFWRUVSDFH

E\$[LRP

E

E\$[LRP

F

E\$[LRP

G

)LOOLQWKHPLVVLQJD[LRPQXPEHUVLQWKHIROORZLQJSURRIWKDW

6XSSRVH X1 ; : : : ; Xp DQG Y1 ; : : : ; Yq DUHYHFWRUVLQDYHFWRU VSDFH V DQGOHW

H D 6SDQ fX1 ; : : : ; Xp g DQG K D 6SDQ fY1 ; : : : ; Yq g 6KRZWKDW H C K D 6SDQ fX1 ; : : : ; Xp ; Y1 ; : : : ; Yq g

$)"15&3

9HFWRU6SDFHV

>0@ 6KRZ WKDW Z LV LQ WKH VXEVSDFH RI R4 VSDQQHG E\ Y1 ; Y2 ; Y3 ZKHUH 2 3 2 3 2 3 2 3 9 8 !4 !7 6 !4 7 6 !4 7 6 37 6 67 7 6 7 6 7 6 7 ZD6 4 !4 5; Y1 D 4 !3 5; Y2 D 4 !2 5; Y3 D 4 !5 5 7 9 !8 !18 >0@ 'HWHUPLQHLI \ LVLQWKHVXEVSDFHRI R4 VSDQQHGE\WKH FROXPQVRI A ZKHUH 2 3 2 3 !4 3 !5 !9 6 !8 7 6 8 7 !6 7 7 6 7 \D6 4 6 5; A D 4 !5 !8 35 !5 2 !2 !9

>0@ 7KH YHFWRU VSDFH H D 6SDQ f1; FRV2 t; FRV4 t; FRV6 t g FRQWDLQVDWOHDVWWZRLQWHUHVWLQJIXQFWLRQVWKDWZLOOEHXVHG

LQDODWHUH[HUFLVH I.t/ D 1 ! 8 FRV2 t C 8 FRV4 t

J.t/ D !1 C 18 FRV2 t ! 48 FRV4 t C 32 FRV6 t 6WXG\WKHJUDSKRI I IRU 0 $ t $ 2" DQGJXHVVDVLPSOHIRU PXODIRU I.t/ 9HULI\\RXUFRQMHFWXUHE\JUDSKLQJWKHGLIIHU HQFHEHWZHHQ 1 C I.t/ DQG\RXUIRUPXODIRU I.t/ +RSHIXOO\ \RXZLOOVHHWKHFRQVWDQWIXQFWLRQ 5HSHDWIRU J >0@ 5HSHDW([HUFLVHIRUWKHIXQFWLRQV I.t/ D 3 VLQ t ! 4 VLQ3 t

J.t/ D 1 ! 8 VLQ2 t C 8 VLQ4 t

K.t/ D 5 VLQ t ! 20 VLQ3 t C 16 VLQ5 t LQWKHYHFWRUVSDFH 6SDQ f1; VLQ t; VLQ2 t; : : : ; VLQ5 t g

40-65*0/4 50 13"$5*$& 130#-&.4 " # 3 7DNH DQ\ X LQ H ³VD\ X D ³DQG WDNH DQ\ c ¤ 1³VD\ c D 2 7 " # 6 cX D ,IWKLVLVLQ H WKHQWKHUHLVVRPH s VXFKWKDW 14 "

3s 2 C 5s

#

D

"

6 14

7KHQ

#

7KDWLV s D 2 DQG s D 12=5 ZKLFKLVLPSRVVLEOH 6R 2X LVQRWLQ H DQG H LVQRWD YHFWRUVSDFH Y1 D 1Y1 C 0Y2 C # # # C 0Yp 7KLV H[SUHVVHV Y1 DV D OLQHDU FRPELQDWLRQ RI Y1 ; : : : ; Yp VR Y1 LVLQ W ,QJHQHUDO Yk LVLQ W EHFDXVH Yk D 0Y1 C # # # C 0Yk !1 C 1Yk C 0Yk C1 C # # # C 0Yp

/6-- 41"$&4 $0-6./ 41"$&4 "/% -*/&"3 53"/4'03."5*0/4 ,QDSSOLFDWLRQVRIOLQHDUDOJHEUD VXEVSDFHVRI Rn XVXDOO\DULVHLQRQHRIWZRZD\V DV WKHVHWRIDOOVROXWLRQVWRDV\VWHPRIKRPRJHQHRXVOLQHDUHTXDWLRQVRU DVWKHVHW RIDOOOLQHDUFRPELQDWLRQVRIFHUWDLQVSHFLÀHGYHFWRUV ,QWKLVVHFWLRQ ZHFRPSDUHDQG FRQWUDVWWKHVHWZRGHVFULSWLRQVRIVXEVSDFHV DOORZLQJXVWRSUDFWLFHXVLQJWKHFRQFHSWRI DVXEVSDFH $FWXDOO\ DV\RXZLOOVRRQGLVFRYHU ZHKDYHEHHQZRUNLQJZLWKVXEVSDFHV HYHUVLQFH6HFWLRQ 7KHPDLQQHZIHDWXUHKHUHLVWKHWHUPLQRORJ\ 7KHVHFWLRQ FRQFOXGHVZLWKDGLVFXVVLRQRIWKHNHUQHODQGUDQJHRIDOLQHDUWUDQVIRUPDWLRQ

7KH1XOO6SDFHRID0DWUL[ &RQVLGHUWKHIROORZLQJV\VWHPRIKRPRJHQHRXVHTXDWLRQV

x1 ! 3x2 ! 2x3 D 0 !5x1 C 9x2 C x3 D 0

1XOO6SDFHV &ROXPQ6SDFHV DQG/LQHDU7UDQVIRUPDWLRQV

,QPDWUL[IRUP WKLVV\VWHPLVZULWWHQDV A[ D ZKHUH

AD

"

1 !3 !2 !5 9 1

#

5HFDOOWKDWWKHVHWRIDOO [ WKDWVDWLVI\ LVFDOOHGWKH VROXWLRQVHW RIWKHV\VWHP 2IWHQLWLVFRQYHQLHQWWRUHODWHWKLVVHWGLUHFWO\WRWKHPDWUL[ A DQGWKHHTXDWLRQ A[ D :HFDOOWKHVHWRI [ WKDWVDWLVI\ A[ D WKH QXOOVSDFH RIWKHPDWUL[ A

%&'*/*5*0/

7KH QXOOVSDFH RIDQ m % n PDWUL[ A ZULWWHQDV 1XO A LVWKHVHWRIDOOVROXWLRQV RIWKHKRPRJHQHRXVHTXDWLRQ A[ D ,QVHWQRWDWLRQ 1XO A D f[ W [ LVLQ Rn DQG A[ D g

$ PRUHG\QDPLFGHVFULSWLRQRI 1XO A LVWKHVHWRIDOO [ LQ Rn WKDWDUHPDSSHGLQWR WKH]HURYHFWRURI Rm YLDWKHOLQHDUWUDQVIRUPDWLRQ [ 7! A[ 6HH)LJ

lA 0

Nu

0

n

!m

!

'*(63&

2

3 5 &9".1-& /HW A EHWKHPDWUL[LQ DERYH DQGOHW X D 4 3 5 'HWHUPLQHLI !2 X EHORQJVWRWKHQXOOVSDFHRI A

40-65*0/ 7RWHVWLI X VDWLVÀHV AX D VLPSO\FRPSXWH AX D

"

3 " # " # 5 1 !3 !2 4 5! 9C4 0 35 D D !5 9 1 !25 C 27 ! 2 0 !2 2

#

7KXV X LVLQ 1XO A 7KHWHUP VSDFH LQ QXOOVSDFH LVDSSURSULDWHEHFDXVHWKHQXOOVSDFHRIDPDWUL[LVD YHFWRUVSDFH DVVKRZQLQWKHQH[WWKHRUHP

5)&03&.

7KHQXOOVSDFHRIDQ m % n PDWUL[ A LVDVXEVSDFHRI Rn (TXLYDOHQWO\ WKH VHWRIDOOVROXWLRQVWRDV\VWHP A[ D RI m KRPRJHQHRXVOLQHDUHTXDWLRQVLQ n XQNQRZQVLVDVXEVSDFHRI Rn

$)"15&3

9HFWRU6SDFHV

1300' &HUWDLQO\ 1XO A LVDVXEVHWRI Rn EHFDXVH A KDV n FROXPQV :HPXVWVKRZ WKDW 1XO A VDWLVÀHVWKHWKUHHSURSHUWLHVRIDVXEVSDFH 2IFRXUVH LVLQ 1XO A 1H[W OHW X DQG Y UHSUHVHQWDQ\WZRYHFWRUVLQ 1XO A 7KHQ AX D

DQG

AY D

7RVKRZWKDW X C Y LVLQ 1XO A ZHPXVWVKRZWKDW A.X C Y/ D 8VLQJDSURSHUW\RI PDWUL[PXOWLSOLFDWLRQ FRPSXWH

A.X C Y/ D AX C AY D C D 7KXV X C Y LVLQ 1XO A DQG 1XO A LVFORVHGXQGHUYHFWRUDGGLWLRQ )LQDOO\ LI c LVDQ\ VFDODU WKHQ A.c X/ D c.AX/ D c./ D ZKLFKVKRZVWKDW c X LVLQ 1XO A 7KXV 1XO A LVDVXEVSDFHRI Rn

&9".1-& /HW H EHWKHVHWRIDOOYHFWRUVLQ R4 ZKRVHFRRUGLQDWHV a b c d VDWLVI\WKHHTXDWLRQV a ! 2b C 5c D d DQG c ! a D b 6KRZWKDW H LVDVXEVSDFHRI R4

40-65*0/ 5HDUUDQJHWKHHTXDWLRQVWKDWGHVFULEHWKHHOHPHQWVRI H DQGQRWHWKDW H LVWKHVHWRIDOOVROXWLRQVRIWKHIROORZLQJV\VWHPRIKRPRJHQHRXVOLQHDUHTXDWLRQV a ! 2b C 5c ! d D 0 !a ! b C c D0 %\7KHRUHP H LVDVXEVSDFHRI R4 ,WLVLPSRUWDQWWKDWWKHOLQHDUHTXDWLRQVGHÀQLQJWKHVHW H DUHKRPRJHQHRXV 2WKHUZLVH WKHVHWRIVROXWLRQVZLOOGHÀQLWHO\ QRW EHDVXEVSDFHEHFDXVHWKH]HURYHFWRU LVQRWDVROXWLRQRIDQRQKRPRJHQHRXVV\VWHP $OVR LQVRPHFDVHV WKHVHWRIVROXWLRQV FRXOGEHHPSW\

$Q([SOLFLW'HVFULSWLRQRI 1XO A 7KHUHLVQRREYLRXVUHODWLRQEHWZHHQYHFWRUVLQ 1XO A DQGWKHHQWULHVLQ A :HVD\WKDW 1XO A LVGHÀQHG LPSOLFLWO\ EHFDXVHLWLVGHÀQHGE\DFRQGLWLRQWKDWPXVWEHFKHFNHG 1RH[SOLFLWOLVWRUGHVFULSWLRQRIWKHHOHPHQWVLQ 1XO A LVJLYHQ +RZHYHU VROYLQJ WKHHTXDWLRQ A[ D DPRXQWVWRSURGXFLQJDQ H[SOLFLW GHVFULSWLRQRI 1XO A 7KHQH[W H[DPSOHUHYLHZVWKHSURFHGXUHIURP6HFWLRQ

&9".1-& )LQGDVSDQQLQJVHWIRUWKHQXOOVSDFHRIWKHPDWUL[ 2

!3 6 !1 2 A D 4 1 !2 2 !4 5

3 1 !7 3 !1 5 8 !4

40-65*0/ 7KHÀUVWVWHSLVWRÀQGWKHJHQHUDOVROXWLRQRI A[ D LQWHUPVRIIUHH YDULDEOHV 5RZUHGXFHWKHDXJPHQWHGPDWUL[ Œ A ! WR UHGXFHG HFKHORQIRUPLQRUGHU WRZULWHWKHEDVLFYDULDEOHVLQWHUPVRIWKHIUHHYDULDEOHV 2 3 x1 ! 2x2 ! x4 C 3x5 D 0 1 !2 0 !1 3 0 40 0 1 2 !2 0 5; x3 C 2x4 ! 2x5 D 0 0 0 0 0 0 0 0D0


7KHJHQHUDOVROXWLRQLV x1 D 2x2 C x4 ! 3x5 x3 D !2x4 C 2x5 ZLWK x2 x4 DQG x5 IUHH 1H[W GHFRPSRVHWKHYHFWRUJLYLQJWKHJHQHUDOVROXWLRQLQWRDOLQHDUFRPELQDWLRQ RIYHFWRUVZKHUH WKHZHLJKWVDUHWKHIUHHYDULDEOHV 7KDWLV 2 3 2 3 2 3 2 3 2 3 x1 2x2 C x4 ! 3x5 2 1 !3 6 x2 7 6 7 617 6 07 6 07 x2 6 7 6 7 6 7 6 7 6 7 6 x3 7 D 6 !2x4 C 2x5 7 D x2 6 0 7 C x4 6 !2 7 C x5 6 2 7 6 7 6 7 6 7 6 7 6 7 4 x4 5 4 5 405 4 15 4 05 x4 x5 x5 0 0 1 " X

" Y

" Z

D x2 X C x4 Y C x5 Z

(YHU\OLQHDUFRPELQDWLRQRI X Y DQG Z LVDQHOHPHQWRI 1XO A 7KXV fX; Y; Zg LVD VSDQQLQJVHWIRU 1XO A 7ZRSRLQWVVKRXOGEHPDGHDERXWWKHVROXWLRQRI([DPSOH WKDWDSSO\WRDOO SUREOHPVRIWKLVW\SHZKHUH 1XO A FRQWDLQVQRQ]HURYHFWRUV :HZLOOXVHWKHVHIDFWV ODWHU 7KHVSDQQLQJVHWSURGXFHGE\WKHPHWKRGLQ([DPSOH LVDXWRPDWLFDOO\OLQHDUO\ LQGHSHQGHQWEHFDXVHWKHIUHHYDULDEOHVDUHWKHZHLJKWVRQWKHVSDQQLQJYHFWRUV )RU LQVWDQFH ORRNDWWKHQG WK DQGWKHQWULHVLQWKHVROXWLRQYHFWRULQ DQGQRWH WKDW x2 X C x4 Y C x5 Z FDQEH RQO\LIWKHZHLJKWV x2 ; x4 DQG x5 DUHDOO]HUR :KHQ 1XO A FRQWDLQVQRQ]HURYHFWRUV WKHQXPEHURIYHFWRUVLQWKHVSDQQLQJVHWIRU 1XO A HTXDOVWKHQXPEHURIIUHHYDULDEOHVLQWKHHTXDWLRQ A[ D

7KH&ROXPQ6SDFHRID0DWUL[ $QRWKHULPSRUWDQWVXEVSDFHDVVRFLDWHGZLWKDPDWUL[LVLWVFROXPQVSDFH 8QOLNHWKH QXOOVSDFH WKHFROXPQVSDFHLVGHÀQHGH[SOLFLWO\YLDOLQHDUFRPELQDWLRQV

%&'*/*5*0/

7KH FROXPQVSDFH RIDQ m % n PDWUL[ A ZULWWHQDV &RO A LVWKHVHWRIDOOOLQHDU FRPELQDWLRQVRIWKHFROXPQVRI A ,I A D Œ D1 # # # Dn ! WKHQ &RO A D 6SDQ fD1 ; : : : ; Dn g

6LQFH 6SDQ fD1 ; : : : ; Dn g LVDVXEVSDFH E\7KHRUHP WKHQH[WWKHRUHPIROORZVIURP WKHGHÀQLWLRQRI &RO A DQGWKHIDFWWKDWWKHFROXPQVRI A DUHLQ Rm

5)&03&.

7KHFROXPQVSDFHRIDQ m % n PDWUL[ A LVDVXEVSDFHRI Rm 1RWHWKDWDW\SLFDOYHFWRULQ &RO A FDQEHZULWWHQDV A[ IRUVRPH [ EHFDXVHWKH QRWDWLRQ A[ VWDQGVIRUDOLQHDUFRPELQDWLRQRIWKHFROXPQVRI A 7KDWLV &RO A D fE W E D A[ IRUVRPH [ LQ Rn g 7KHQRWDWLRQ A[ IRUYHFWRUVLQ &RO A DOVRVKRZVWKDW &RO A LVWKH UDQJH RIWKHOLQHDU WUDQVIRUPDWLRQ [ 7! A[ :HZLOOUHWXUQWRWKLVSRLQWRIYLHZDWWKHHQGRIWKHVHFWLRQ

$)"15&3

9HFWRU6SDFHV

&9".1-& )LQGDPDWUL[ A VXFKWKDW W D &RO A

x2

82 9 3 < 6a ! b = W D 4 a C b 5 W a b LQ R : ; !7a

x3 0

W

x1

40-65*0/ )LUVW ZULWH W DVDVHWRIOLQHDUFRPELQDWLRQV 8 2 9 82 3 2 3 3 2 39 6 !1 6 !1 = < = < W D a4 1 5 C b 4 1 5 W a b LQ R D 6SDQ 4 1 5; 4 1 5 : ; : ; !7 0 !7 0 2

3 6 !1 6HFRQG XVHWKHYHFWRUVLQWKHVSDQQLQJVHWDVWKHFROXPQVRI A /HW A D 4 1 1 5 !7 0 7KHQ W D &RO A DVGHVLUHG

5HFDOOIURP7KHRUHP LQ6HFWLRQWKDWWKHFROXPQVRI A VSDQ Rm LIDQGRQO\LI WKHHTXDWLRQ A[ D E KDVDVROXWLRQIRUHDFK E :HFDQUHVWDWHWKLVIDFWDVIROORZV 7KHFROXPQVSDFHRIDQ m % n PDWUL[ A LVDOORI Rm LIDQGRQO\LIWKHHTXDWLRQ A[ D E KDVDVROXWLRQIRUHDFK E LQ Rm

7KH&RQWUDVW%HWZHHQ 1XO A DQG &RO A ,WLVQDWXUDOWRZRQGHUKRZWKHQXOOVSDFHDQGFROXPQVSDFHRIDPDWUL[DUHUHODWHG ,Q IDFW WKHWZRVSDFHVDUHTXLWHGLVVLPLODU DV([DPSOHV ²ZLOOVKRZ 1HYHUWKHOHVV D VXUSULVLQJ FRQQHFWLRQ EHWZHHQ WKH QXOO VSDFH DQG FROXPQ VSDFH ZLOO HPHUJH LQ 6HFWLRQ DIWHUPRUHWKHRU\LVDYDLODEOH

&9".1-& /HW

2

2 4 !2 A D 4 !2 !5 7 3 7 !8

3 1 35 6

D ,IWKHFROXPQVSDFHRI A LVDVXEVSDFHRI Rk ZKDWLV k " E ,IWKHQXOOVSDFHRI A LVDVXEVSDFHRI Rk ZKDWLV k "

40-65*0/ D 7KHFROXPQVRI A HDFKKDYHWKUHHHQWULHV VR &RO A LVDVXEVSDFHRI Rk ZKHUH k D 3 E $ YHFWRU [ VXFKWKDW A[ LVGHÀQHGPXVWKDYHIRXUHQWULHV VR 1XO A LVDVXEVSDFHRI Rk ZKHUH k D 4 :KHQDPDWUL[LVQRWVTXDUH DVLQ([DPSOH WKHYHFWRUVLQ 1XO A DQG &RO A OLYH LQHQWLUHO\GLIIHUHQW´XQLYHUVHVµ )RUH[DPSOH QROLQHDUFRPELQDWLRQRIYHFWRUVLQ R3 FDQSURGXFHDYHFWRULQ R4 :KHQ A LVVTXDUH 1XO A DQG &RO A GRKDYHWKH]HURYHFWRU LQFRPPRQ DQGLQVSHFLDOFDVHVLWLVSRVVLEOHWKDWVRPHQRQ]HURYHFWRUVEHORQJWRERWK 1XO A DQG &RO A


&9".1-& :LWK A DVLQ([DPSOH ÀQGDQRQ]HURYHFWRULQ &RO A DQGDQRQ]HUR YHFWRULQ 1XO A

2

3 2 40-65*0/ ,WLVHDV\WRÀQGDYHFWRULQ &RO A $Q\FROXPQRI A ZLOOGR VD\ 4 !2 5 3 7RÀQGDQRQ]HURYHFWRULQ 1XO A URZUHGXFHWKHDXJPHQWHGPDWUL[ Œ A ! DQGREWDLQ 2

1 ŒA ! & 40 0

0 9 1 !5 0 0

0 0 1

3 0 05 0

7KXV LI [ VDWLVÀHV A[ D WKHQ x1 D !9x3 x2 D 5x3 x4 D 0 DQG x3 LVIUHH $V VLJQLQJDQRQ]HURYDOXHWR x3 ³VD\ x3 D 1³ZHREWDLQDYHFWRULQ 1XO A QDPHO\ [ D .!9; 5; 1; 0/ 2

3 2 3 3 3 6 !2 7 7 4 5 &9".1-& :LWK A DVLQ([DPSOH OHW X D 6 4 !1 5 DQG Y D !1 3 0

D 'HWHUPLQHLI X LVLQ 1XO A &RXOG X EHLQ &RO A" E 'HWHUPLQHLI Y LVLQ &RO A &RXOG Y EHLQ 1XO A"

40-65*0/ D $QH[SOLFLWGHVFULSWLRQRI 1XO A LVQRWQHHGHGKHUH 6LPSO\FRPSXWHWKHSURGXFW AX 3 2 2 3 2 3 2 3 3 2 4 !2 1 6 0 0 7 !2 7 4 5 ¤ 405 !3 AX D 4 !2 !5 7 3 56 D 4 !1 5 3 7 !8 6 3 0 0 2EYLRXVO\ X LV QRW DVROXWLRQRI A[ D VR X LVQRWLQ 1XO A $OVR ZLWKIRXUHQWULHV X FRXOGQRWSRVVLEO\EHLQ &RO A VLQFH &RO A LVDVXEVSDFHRI R3 E 5HGXFH Œ A Y ! WRDQHFKHORQIRUP 2 3 2 3 2 4 !2 1 3 2 4 !2 1 3 Œ A Y ! D 4 !2 !5 7 3 !1 5 & 4 0 1 !5 !4 !2 5 3 7 !8 6 3 0 0 0 17 1 $WWKLVSRLQW LWLVFOHDUWKDWWKHHTXDWLRQ A[ D Y LVFRQVLVWHQW VR Y LVLQ &RO A :LWK RQO\WKUHHHQWULHV Y FRXOGQRWSRVVLEO\EHLQ 1XO A VLQFH 1XO A LVDVXEVSDFHRI R4

7KHWDEOHRQSDJHVXPPDUL]HVZKDWZHKDYHOHDUQHGDERXW 1XO A DQG &RO A ,WHP LVDUHVWDWHPHQWRI7KHRUHPV DQGD LQ6HFWLRQ

.HUQHODQG5DQJHRID/LQHDU7UDQVIRUPDWLRQ 6XEVSDFHVRIYHFWRUVSDFHVRWKHUWKDQ Rn DUHRIWHQGHVFULEHGLQWHUPVRIDOLQHDU WUDQVIRUPDWLRQLQVWHDGRIDPDWUL[ 7RPDNHWKLVSUHFLVH ZHJHQHUDOL]HWKHGHÀQLWLRQ JLYHQLQ6HFWLRQ

$)"15&3

9HFWRU6SDFHV

$POUSBTU #FUXFFO /VM " BOE $PM " GPS BO N Y O .BUSJY " 1XO A

&RO A n

1XO A LVDVXEVSDFHRI R

&RO A LVDVXEVSDFHRI Rm

1XO A LVLPSOLFLWO\GHÀQHG WKDWLV \RXDUH JLYHQRQO\DFRQGLWLRQ .A[ D / WKDWYHF WRUVLQ 1XO A PXVWVDWLVI\

&RO A LVH[SOLFLWO\GHÀQHG WKDWLV \RXDUH WROGKRZWREXLOGYHFWRUVLQ &RO A

,WWDNHVWLPHWRÀQGYHFWRUVLQ 1XO A 5RZ RSHUDWLRQVRQ Œ A ! DUHUHTXLUHG

,WLVHDV\WRÀQGYHFWRUVLQ &RO A 7KH FROXPQV RI A DUH GLVSOD\HG RWKHUV DUH IRUPHGIURPWKHP

7KHUHLVQRREYLRXVUHODWLRQEHWZHHQ 1XO A DQGWKHHQWULHVLQ A

7KHUHLVDQREYLRXVUHODWLRQEHWZHHQ &RO A DQGWKHHQWULHVLQ A VLQFHHDFKFROXPQRI A LVLQ &RO A

$ W\SLFDOYHFWRU Y LQ 1XO A KDVWKHSURSHUW\ WKDW AY D

$ W\SLFDOYHFWRU Y LQ &RO A KDVWKHSURSHUW\ WKDWWKHHTXDWLRQ A[ D Y LVFRQVLVWHQW

1XO A D fg LIDQGRQO\LIWKHHTXDWLRQ A[ D KDVRQO\WKHWULYLDOVROXWLRQ

&RO A D Rm LIDQGRQO\LIWKHHTXDWLRQ A[ D E KDVDVROXWLRQIRUHYHU\ E LQ Rm

*LYHQDVSHFLÀFYHFWRU Y LWLVHDV\WRWHOOLI Y LVLQ 1XO A -XVWFRPSXWH AY

1XO A D fg LIDQGRQO\LIWKHOLQHDUWUDQV IRUPDWLRQ [ 7! A[ LVRQHWRRQH

%&'*/*5*0/

*LYHQDVSHFLÀFYHFWRU Y LWPD\WDNHWLPH WRWHOOLI Y LVLQ &RO A 5RZRSHUDWLRQVRQ Œ A Y ! DUHUHTXLUHG

&RO A D Rm LIDQGRQO\LIWKHOLQHDUWUDQV IRUPDWLRQ [ 7! A[ PDSV Rn RQWR Rm

$ OLQHDUWUDQVIRUPDWLRQ T IURPDYHFWRUVSDFH V LQWRDYHFWRUVSDFH W LVDUXOH WKDWDVVLJQVWRHDFKYHFWRU [ LQ V DXQLTXHYHFWRU T .[/ LQ W VXFKWKDW L T .X C Y/ D T .X/ C T .Y/ LL T .c X/ D cT .X/

IRUDOO X Y LQ V DQG IRUDOO X LQ V DQGDOOVFDODUV c

7KH NHUQHO RU QXOOVSDFH RIVXFKD T LVWKHVHWRIDOO X LQ V VXFKWKDW T .X/ D WKH]HURYHFWRULQ W / 7KH UDQJH RI T LVWKHVHWRIDOOYHFWRUVLQ W RIWKHIRUP T .[/ IRUVRPH [ LQ V ,I T KDSSHQVWRDULVHDVDPDWUL[WUDQVIRUPDWLRQ³VD\ T .[/ D A[ IRUVRPHPDWUL[ A³WKHQWKHNHUQHODQGWKHUDQJHRI T DUHMXVWWKHQXOOVSDFHDQGWKH FROXPQVSDFHRI A DVGHÀQHGHDUOLHU ,WLVQRWGLIÀFXOWWRVKRZWKDWWKHNHUQHORI T LVDVXEVSDFHRI V 7KHSURRILV HVVHQWLDOO\WKHVDPHDVWKHRQHIRU7KHRUHP $OVR WKHUDQJHRI T LVDVXEVSDFHRI W 6HH)LJ DQG([HUFLVH in

a om

D

T

l ne

r Ke 0

Ra

ng

e

0

W V

Kernel is a subspace of V

Range is a subspace of W

'*(63& 6XEVSDFHVDVVRFLDWHGZLWK

DOLQHDUWUDQVIRUPDWLRQ

,QDSSOLFDWLRQV DVXEVSDFHXVXDOO\DULVHVDVHLWKHUWKHNHUQHORUWKHUDQJHRIDQ DSSURSULDWHOLQHDUWUDQVIRUPDWLRQ )RULQVWDQFH WKHVHWRIDOOVROXWLRQVRIDKRPRJH QHRXVOLQHDUGLIIHUHQWLDOHTXDWLRQWXUQVRXWWREHWKHNHUQHORIDOLQHDUWUDQVIRUPDWLRQ


7\SLFDOO\ VXFKDOLQHDUWUDQVIRUPDWLRQLVGHVFULEHGLQWHUPVRIRQHRUPRUHGHULYDWLYHV RIDIXQFWLRQ 7RH[SODLQWKLVLQDQ\GHWDLOZRXOGWDNHXVWRRIDUDÀHOGDWWKLVSRLQW 6R ZHFRQVLGHURQO\WZRH[DPSOHV 7KHÀUVWH[SODLQVZK\WKHRSHUDWLRQRIGLIIHUHQWLDWLRQ LVDOLQHDUWUDQVIRUPDWLRQ

&9".1-& $BMDVMVT SFRVJSFE /HW V EHWKHYHFWRUVSDFHRIDOOUHDOYDOXHGIXQF WLRQV f GHÀQHGRQDQLQWHUYDO Œa; b! ZLWKWKHSURSHUW\WKDWWKH\DUHGLIIHUHQWLDEOHDQG WKHLUGHULYDWLYHVDUHFRQWLQXRXVIXQFWLRQVRQ Œa; b! /HW W EHWKHYHFWRUVSDFH C Œa; b! RIDOOFRQWLQXRXVIXQFWLRQVRQ Œa; b! DQGOHW D W V ! W EHWKHWUDQVIRUPDWLRQWKDW FKDQJHV f LQ V LQWRLWVGHULYDWLYH f 0 ,QFDOFXOXV WZRVLPSOHGLIIHUHQWLDWLRQUXOHVDUH

D.f C g/ D D.f / C D.g/

DQG

D.cf / D cD.f /

7KDWLV D LVDOLQHDUWUDQVIRUPDWLRQ ,WFDQEHVKRZQWKDWWKHNHUQHORI D LVWKHVHWRI FRQVWDQWIXQFWLRQVRQ Œa; b! DQGWKHUDQJHRI D LVWKHVHW W RIDOOFRQWLQXRXVIXQFWLRQV RQ Œa; b!

&9".1-& $BMDVMVT SFRVJSFE 7KHGLIIHUHQWLDOHTXDWLRQ y 00 C ! 2 y D 0

ZKHUH ! LVDFRQVWDQW LVXVHGWRGHVFULEHDYDULHW\RISK\VLFDOV\VWHPV VXFKDVWKH YLEUDWLRQRIDZHLJKWHGVSULQJ WKHPRYHPHQWRIDSHQGXOXP DQGWKHYROWDJHLQDQ LQGXFWDQFHFDSDFLWDQFHHOHFWULFDOFLUFXLW 7KHVHWRIVROXWLRQVRI LVSUHFLVHO\WKH NHUQHORIWKHOLQHDUWUDQVIRUPDWLRQWKDWPDSVDIXQFWLRQ y D f .t / LQWRWKHIXQFWLRQ f 00 .t / C ! 2 f .t / )LQGLQJDQH[SOLFLWGHVFULSWLRQRIWKLVYHFWRUVSDFHLVDSUREOHPLQ GLIIHUHQWLDOHTXDWLRQV 7KHVROXWLRQVHWWXUQVRXWWREHWKHVSDFHGHVFULEHGLQ([HUFLVH LQ6HFWLRQ

13"$5*$& 130#-&.4 82 3 9 < a = /HW W D 4 b 5 W a ! 3b ! c D 0 6KRZLQWZRGLIIHUHQWZD\VWKDW W LVD : ; c VXEVSDFHRI R3 8VHWZRWKHRUHPV 2 3 2 3 2 3 7 !3 5 2 7 /HW A D 4 !4 1 !5 5 Y D 4 1 5 DQG Z D 4 6 5 6XSSRVH\RXNQRZWKDW !5 2 !4 !1 !3 WKHHTXDWLRQV A[ D Y DQG A[ D Z DUHERWKFRQVLVWHQW :KDWFDQ\RXVD\DERXWWKH HTXDWLRQ A[ D Y C Z"

&9&3$*4&4 2

3 1 'HWHUPLQHLI Z D 4 3 5 LVLQ 1XO A ZKHUH !4 2 3 3 !5 !3 0 5: A D 4 6 !2 !8 4 1

2

3 1 'HWHUPLQHLI Z D 4 !1 5 LVLQ 1XO A ZKHUH 1 2 3 2 6 4 2 5 5: A D 4 !3 !5 !4 1

$)"15&3

9HFWRU6SDFHV

,Q([HUFLVHV² ÀQGDQH[SOLFLWGHVFULSWLRQRI 1XO A E\OLVWLQJ YHFWRUVWKDWVSDQWKHQXOOVSDFH " # 1 2 4 0 A D 0 1 3 !2 " # 1 !3 2 0 A D 0 0 3 0 2 3 1 !4 0 2 0 0 1 !5 05 A D 4 0 0 0 0 0 2 2 3 1 3 !4 !3 1 1 !3 1 05 A D 4 0 0 0 0 0 0

,Q([HUFLVHV² HLWKHUXVHDQDSSURSULDWHWKHRUHPWRVKRZWKDW WKHJLYHQVHW W LVDYHFWRUVSDFH RUÀQGDVSHFLÀFH[DPSOHWR WKHFRQWUDU\ 82 3 9 82 3 9 < a = < r = 4 b 5 W a C b C c D 2 4 s 5 W 3r ! 2 D 3s C t : ; : ; c t 82 3 9 82 3 9 p a ˆ > ˆ > ˆ > ˆ > <6 7 = <6 7 = q 7 p ! 3q D 4s b 7 3a C b D c 6 6 4 5 W 4 5 W r c 2p D s C 5r > a C b C 2c D 2d > ˆ ˆ ˆ > ˆ > : ; : ; s d 82 9 82 9 3 3 s ! 2t 3p ! 5q ˆ > ˆ > ˆ > ˆ > <6 = < = 6 4q 7 3 C 3s 7 6 7 W s t UHDO 7 W p q UHDO 6 4 4 5 5 3s C t p ˆ > ˆ > ˆ > ˆ > : ; : ; 2s qC1 82 9 82 9 3 3 < c ! 6d = < !s C 3t = 4 d 5 W c d UHDO 4 s ! 2t 5 W s t UHDO : ; : ; c 5s ! t ,Q([HUFLVHVDQG ÀQG A VXFKWKDWWKHJLYHQVHWLV &RO A 82 9 3 2s C t ˆ > ˆ > <6 = r ! s C 2t 7 6 7 4 W r s t UHDO 3r C s 5 ˆ > ˆ > : ; 2r ! s ! t 82 9 3 b!c ˆ > ˆ > <6 = 7 2b C 3d 7 W b c d UHDO 6 4 5 b C 3c ! 3d ˆ > ˆ > : ; cCd

)RUWKHPDWULFHVLQ([HUFLVHV² D ÀQG k VXFKWKDW 1XO A LV DVXEVSDFHRI Rk DQGE ÀQG k VXFKWKDW &RO A LVDVXEVSDFHRI Rk 2 3 2 3 6 !4 5 !2 3 6 !3 6 !1 27 0 !1 7 7 7 A D 6 A D 6 4 !9 4 0 !2 !2 5 65 9 !6 !5 7 2 " # 4 5 !2 6 0 A D 1 1 0 1 0 % & 2 0 !5 A D 1 !3

:LWK A DVLQ([HUFLVH ÀQGDQRQ]HURYHFWRULQ 1XO A DQG DQRQ]HURYHFWRULQ &RO A :LWK A DVLQ([HUFLVH ÀQGDQRQ]HURYHFWRULQ 1XO A DQG DQRQ]HURYHFWRULQ &RO A " # " # !2 4 2 /HW A D DQG Z D 'HWHUPLQHLI Z LVLQ !1 2 1 &RO A ,V Z LQ 1XO A" 2 3 2 3 10 !8 !2 !2 2 6 0 7 627 2 2 !2 7 DQG Z D 6 7 'HWHUPLQH /HW A D 6 4 1 !1 405 6 05 1 1 0 !2 2 LI Z LVLQ &RO A ,V Z LQ 1XO A" ,Q([HUFLVHVDQG A GHQRWHVDQ m % n PDWUL[ 0DUNHDFK VWDWHPHQW7UXHRU)DOVH -XVWLI\HDFKDQVZHU D 7KHQXOOVSDFHRI A LVWKHVROXWLRQVHWRIWKHHTXDWLRQ A[ D E 7KHQXOOVSDFHRIDQ m % n PDWUL[LVLQ Rm

F 7KH FROXPQ VSDFH RI A LV WKH UDQJH RI WKH PDSSLQJ [ 7! A[

G ,IWKHHTXDWLRQ A[ D E LVFRQVLVWHQW WKHQ &RO A LV Rm H 7KHNHUQHORIDOLQHDUWUDQVIRUPDWLRQLVDYHFWRUVSDFH

I &RO A LVWKHVHWRIDOOYHFWRUVWKDWFDQEHZULWWHQDV A[ IRU VRPH [ D $ QXOOVSDFHLVDYHFWRUVSDFH E 7KHFROXPQVSDFHRIDQ m % n PDWUL[LVLQ Rm F &RO A LVWKHVHWRIDOOVROXWLRQVRI A[ D E

G 1XO A LVWKHNHUQHORIWKHPDSSLQJ [ 7! A[

H 7KHUDQJHRIDOLQHDUWUDQVIRUPDWLRQLVDYHFWRUVSDFH I 7KHVHWRIDOOVROXWLRQVRIDKRPRJHQHRXVOLQHDUGLIIHUHQ WLDOHTXDWLRQLVWKHNHUQHORIDOLQHDUWUDQVIRUPDWLRQ ,WFDQEHVKRZQWKDWDVROXWLRQRIWKHV\VWHPEHORZLV x1 D 3 x2 D 2 DQG x3 D !1 8VHWKLVIDFWDQGWKHWKHRU\IURPWKLV VHFWLRQWRH[SODLQZK\DQRWKHUVROXWLRQLV x1 D 30 x2 D 20 DQG x3 D !10 2EVHUYHKRZWKHVROXWLRQVDUHUHODWHG EXW PDNHQRRWKHUFDOFXODWLRQV

x1 ! 3x2 ! 3x3 D 0

!2x1 C 4x2 C 2x3 D 0 !x1 C 5x2 C 7x3 D 0

&RQVLGHUWKHIROORZLQJWZRV\VWHPVRIHTXDWLRQV

5x1 C x2 ! 3x3 D 0

!9x1 C 2x2 C 5x3 D 1 4x1 C x2 ! 6x3 D 9

5x1 C x2 ! 3x3 D 0

!9x1 C 2x2 C 5x3 D 5 4x1 C x2 ! 6x3 D 45

,WFDQEHVKRZQWKDWWKHÀUVWV\VWHPKDVDVROXWLRQ 8VH WKLVIDFWDQGWKHWKHRU\IURPWKLVVHFWLRQWRH[SODLQZK\WKH VHFRQGV\VWHPPXVWDOVRKDYHDVROXWLRQ 0DNHQRURZ RSHUDWLRQV


3URYH7KHRUHPDVIROORZV *LYHQDQ m % n PDWUL[ A DQ HOHPHQWLQ &RO A KDVWKHIRUP A[ IRUVRPH [ LQ Rn /HW A[ DQG AZ UHSUHVHQWDQ\WZRYHFWRUVLQ &RO A D ([SODLQZK\WKH]HURYHFWRULVLQ &RO A E 6KRZWKDWWKHYHFWRU A[ C AZ LVLQ &RO A

F *LYHQDVFDODU c VKRZWKDW c.A[/ LVLQ &RO A

/HW T W V ! W EHDOLQHDUWUDQVIRUPDWLRQIURPDYHFWRU VSDFH V LQWRDYHFWRUVSDFH W 3URYHWKDWWKHUDQJHRI T LVDVXEVSDFHRI W >+LQW 7\SLFDOHOHPHQWVRIWKHUDQJH KDYHWKHIRUP T .[/ DQG T .Z/ IRUVRPH [ Z LQ V @ " # S.0/ )RULQVWDQFH LI 'HÀQH T W P2 ! R2 E\ T .S/ D S.1/ " # 3 S.t/ D 3 C 5t C 7t 2 WKHQ T .S/ D 15 D 6KRZ WKDW T LV D OLQHDU WUDQVIRUPDWLRQ >+LQW )RU DUELWUDU\SRO\QRPLDOV S T LQ P2 FRPSXWH T .S C T/ DQG T .c S/@ E )LQGDSRO\QRPLDO S LQ P2 WKDWVSDQVWKHNHUQHORI T DQG GHVFULEHWKHUDQJHRI T 'HÀQH "D OLQHDU WUDQVIRUPDWLRQ T W P2 ! R2 E\ # S.0/ T .S / D )LQGSRO\QRPLDOV S1 DQG S2 LQ P2 WKDW S.0/ VSDQWKHNHUQHORI T DQGGHVFULEHWKH UDQJHRI T

/HW M2!2 EH WKH YHFWRU VSDFH RI DOO 2 % 2 PDWULFHV T W#M2!2 ! M2!2 E\ T .A/ D A C AT ZKHUH DQGGHÀQH " a b AD c d D 6KRZWKDW T LVDOLQHDUWUDQVIRUPDWLRQ E /HW B EHDQ\HOHPHQWRI M2!2 VXFKWKDW B T D B )LQG DQ A LQ M2!2 VXFKWKDW T .A/ D B

F 6KRZWKDWWKHUDQJHRI T LVWKHVHWRI B LQ M2!2 ZLWKWKH SURSHUW\WKDW B T D B G 'HVFULEHWKHNHUQHORI T

&DOFXOXVUHTXLUHG 'HÀQH T W C Œ0; 1! ! C Œ0; 1! DVIROORZV )RU I LQ C Œ0; 1! OHW T .I/ EHWKHDQWLGHULYDWLYH ) RI I VXFK WKDW ).0/ D 0 6KRZWKDW T LVDOLQHDUWUDQVIRUPDWLRQ DQG GHVFULEHWKHNHUQHORI T 6HHWKHQRWDWLRQLQ([HUFLVHRI 6HFWLRQ

/HW V DQG W EHYHFWRUVSDFHV DQGOHW T W V ! W EHDOLQHDU WUDQVIRUPDWLRQ *LYHQDVXEVSDFH U RI V OHW T .U / GHQRWH WKHVHWRIDOOLPDJHVRIWKHIRUP T .[/ ZKHUH [ LVLQ U 6KRZ WKDW T .U / LVDVXEVSDFHRI W *LYHQ T W V ! W DVLQ([HUFLVH DQGJLYHQDVXEVSDFH Z RI W OHW U EHWKHVHWRIDOO [ LQ V VXFKWKDW T .[/ LVLQ Z 6KRZWKDW U LVDVXEVSDFHRI V >0@ 'HWHUPLQHZKHWKHU Z LVLQWKHFROXPQVSDFHRI A WKH QXOOVSDFHRI A RUERWK ZKHUH 2 3 2 3 1 7 6 !4 1 6 17 6 !5 !1 0 !2 7 7 6 7 ZD6 4 !1 5; A D 4 9 !11 7 !3 5 !3 19 !9 7 1 >0@ 'HWHUPLQHZKHWKHU Z LVLQWKHFROXPQVSDFHRI A WKH QXOOVSDFHRI A RUERWK ZKHUH 2 3 2 3 1 !8 5 !2 0 627 6 !5 2 1 !2 7 7 6 7 ZD6 4 1 5; A D 4 10 !8 6 !3 5 0 3 !2 1 0

>0@ /HW D1 ; : : : ; D5 GHQRWH WKH FROXPQV RI WKH PDWUL[ A ZKHUH 2 3 5 1 2 2 0 63 3 2 !1 !12 7 7; B D Œ D1 D2 D4 ! AD6 48 4 4 !5 12 5 2 1 1 0 !2 D ([SODLQZK\ D3 DQG D5 DUHLQWKHFROXPQVSDFHRI B

E )LQGDVHWRIYHFWRUVWKDWVSDQV 1XO A F /HW T W R5 ! R4 EHGHÀQHGE\ T .[/ D A[ ([SODLQZK\ T LVQHLWKHURQHWRRQHQRURQWR >0@ /HW H D 6SDQ fY1 ; Y2 g DQG K D 6SDQ fY3 ; Y4 g ZKHUH 2 3 2 3 2 3 2 3 5 1 2 0 Y1 D 4 3 5; Y2 D 4 3 5; Y3 D 4 !1 5; Y4 D 4 !12 5: 8 4 5 !28

7KHQ H DQG K DUH VXEVSDFHV RI R3 ,Q IDFW H DQG K DUHSODQHVLQ R3 WKURXJKWKHRULJLQ DQGWKH\LQWHUVHFW LQ D OLQH WKURXJK )LQG D QRQ]HUR YHFWRU Z WKDW JHQ HUDWHVWKDWOLQH >+LQW Z FDQEHZULWWHQDV c1 Y1 C c2 Y2 DQGDOVRDV c3 Y3 C c4 Y4 7REXLOG Z VROYHWKHHTXDWLRQ c1 Y1 C c2 Y2 D c3 Y3 C c4 Y4 IRUWKHXQNQRZQ cj ·V@

SG

.BTUFSJOH 7FDUPS 4QBDF 4VCTQBDF $PM " BOE /VM " m

40-65*0/4 50 13"$5*$& 130#-&.4 )LUVWPHWKRG W LVDVXEVSDFHRI R3 E\7KHRUHP EHFDXVH W LVWKHVHWRIDOOVROX WLRQVWRDV\VWHPRIKRPRJHQHRXVOLQHDUHTXDWLRQVZKHUHWKHV\VWHPKDVRQO\RQH HTXDWLRQ (TXLYDOHQWO\ W LVWKHQXOOVSDFHRIWKH 1 % 3 PDWUL[ A D Œ 1 !3 !1 !

$)"15&3

9HFWRU6SDFHV

6HFRQGPHWKRG 6ROYHWKHHTXDWLRQ a ! 3b ! c D 0 IRUWKHOHDGLQJYDULDEOH a LQ 2 3 3b C c WHUPVRIWKHIUHHYDULDEOHV b DQG c $Q\VROXWLRQKDVWKHIRUP 4 b 5 ZKHUH b c DQG c DUHDUELWUDU\ DQG 2 3 2 3 2 3 3b C c 3 1 4 b 5 D b4 1 5 C c4 0 5 c 0 1 " Y1

" Y2

7KLVFDOFXODWLRQVKRZVWKDW W D 6SDQ fY1 ; Y2 g 7KXV W LVDVXEVSDFHRI R3 E\ 7KHRUHP :HFRXOGDOVRVROYHWKHHTXDWLRQ a ! 3b ! c D 0 IRU b RU c DQGJHW DOWHUQDWLYHGHVFULSWLRQVRI W DVDVHWRIOLQHDUFRPELQDWLRQVRIWZRYHFWRUV %RWK Y DQG Z DUHLQ &RO A 6LQFH &RO A LVDYHFWRUVSDFH Y C Z PXVWEHLQ &RO A 7KDWLV WKHHTXDWLRQ A[ D Y C Z LVFRQVLVWHQW

-*/&"3-: */%&1&/%&/5 4&54 #"4&4 ,QWKLVVHFWLRQZHLGHQWLI\DQGVWXG\WKHVXEVHWVWKDWVSDQDYHFWRUVSDFH V RUDVXEVSDFH H DV´HIÀFLHQWO\µDVSRVVLEOH 7KHNH\LGHDLVWKDWRIOLQHDULQGHSHQGHQFH GHÀQHGDV LQ Rn $QLQGH[HGVHWRIYHFWRUV fY1 ; : : : ; Yp g LQ V LVVDLGWREH OLQHDUO\LQGHSHQGHQW LI WKHYHFWRUHTXDWLRQ c1 Y1 C c2 Y2 C # # # C cp Yp D

KDV RQO\ WKHWULYLDOVROXWLRQ c1 D 0; : : : ; cp D 0ñ 7KHVHW fY1 ; : : : ; Yp g LVVDLGWREH OLQHDUO\GHSHQGHQW LI KDVDQRQWULYLDOVROXWLRQ WKDWLV LIWKHUHDUHVRPHZHLJKWV c1 ; : : : ; cp QRWDOO]HUR VXFKWKDW KROGV ,QVXFKD FDVH LVFDOOHGD OLQHDUGHSHQGHQFHUHODWLRQ DPRQJ Y1 ; : : : ; Yp -XVWDVLQ Rn DVHWFRQWDLQLQJDVLQJOHYHFWRU Y LVOLQHDUO\LQGHSHQGHQWLIDQGRQO\LI Y ¤ $OVR DVHWRIWZRYHFWRUVLVOLQHDUO\GHSHQGHQWLIDQGRQO\LIRQHRIWKHYHFWRUV LVDPXOWLSOHRIWKHRWKHU $QGDQ\VHWFRQWDLQLQJWKH]HURYHFWRULVOLQHDUO\GHSHQGHQW 7KHIROORZLQJWKHRUHPKDVWKHVDPHSURRIDV7KHRUHP LQ6HFWLRQ

5)&03&.

$QLQGH[HGVHW fY1 ; : : : ; Yp g RIWZRRUPRUHYHFWRUV ZLWK Y1 ¤ LVOLQHDUO\ GHSHQGHQWLIDQGRQO\LIVRPH Yj ZLWK j > 1/ LVDOLQHDUFRPELQDWLRQRIWKH SUHFHGLQJYHFWRUV Y1 ; : : : ; Yj !1 7KHPDLQGLIIHUHQFHEHWZHHQOLQHDUGHSHQGHQFHLQ Rn DQGLQDJHQHUDOYHFWRUVSDFH LVWKDWZKHQWKHYHFWRUVDUHQRW nWXSOHV WKHKRPRJHQHRXVHTXDWLRQ XVXDOO\FDQQRW EHZULWWHQDVDV\VWHPRI n OLQHDUHTXDWLRQV 7KDWLV WKHYHFWRUVFDQQRWEHPDGHLQWR WKHFROXPQVRIDPDWUL[ A LQRUGHUWRVWXG\WKHHTXDWLRQ A[ D :HPXVWUHO\LQVWHDG RQWKHGHÀQLWLRQRIOLQHDUGHSHQGHQFHDQGRQ7KHRUHP

&9".1-& /HW S1 .t / D 1 S2 .t / D t DQG S3 .t/ D 4 ! t 7KHQ fS1 ; S2 ; S3 g LV OLQHDUO\GHSHQGHQWLQ P EHFDXVH S3 D 4S1 ! S2 ñ ,WLVFRQYHQLHQWWRXVH c1 ; : : : ; cp LQ IRUWKHVFDODUVLQVWHDGRI x1 ; : : : ; xp DVZHGLGLQ&KDSWHU

/LQHDUO\,QGHSHQGHQW6HWV %DVHV

&9".1-& 7KHVHW fVLQ t; FRV t g LVOLQHDUO\LQGHSHQGHQWLQ C Œ0; 1! WKHVSDFHRI

DOOFRQWLQXRXVIXQFWLRQVRQ 0 $ t $ 1 EHFDXVH VLQ t DQG FRV t DUHQRWPXOWLSOHVRIRQH DQRWKHU DVYHFWRUVLQ C Œ0; 1! 7KDWLV WKHUHLVQRVFDODU c VXFKWKDW FRV t D c # VLQ t IRU DOO t LQ Œ0; 1! /RRNDWWKHJUDSKVRI VLQ t DQG FRV t +RZHYHU fVLQ t FRV t; VLQ 2t g LV OLQHDUO\GHSHQGHQWEHFDXVHRIWKHLGHQWLW\ VLQ 2t D 2 VLQ t FRV t IRUDOO t

%&'*/*5*0/

/HW H EH D VXEVSDFH RI D YHFWRU VSDFH V B D fE1 ; : : : ; Ep g LQ V LVD EDVLV IRU H LI

$Q LQGH[HG VHW RI YHFWRUV

L B LVDOLQHDUO\LQGHSHQGHQWVHW DQG LL WKHVXEVSDFHVSDQQHGE\ B FRLQFLGHVZLWK H WKDWLV

H D 6SDQ fE1 ; : : : ; Ep g

7KHGHÀQLWLRQRIDEDVLVDSSOLHVWRWKHFDVHZKHQ H D V EHFDXVHDQ\YHFWRUVSDFH LVDVXEVSDFHRILWVHOI 7KXVDEDVLVRI V LVDOLQHDUO\LQGHSHQGHQWVHWWKDWVSDQV V 2EVHUYHWKDWZKHQ H ¤ V FRQGLWLRQLL LQFOXGHVWKHUHTXLUHPHQWWKDWHDFKRIWKH YHFWRUV E1 ; : : : ; Ep PXVWEHORQJWR H EHFDXVH 6SDQ fE1 ; : : : ; Ep g FRQWDLQV E1 ; : : : ; Ep DVVKRZQLQ6HFWLRQ

&9".1-& /HW A EHDQLQYHUWLEOH n % n PDWUL[³VD\ A D Œ D1 # # # Dn ! 7KHQ WKHFROXPQVRI A IRUPDEDVLVIRU Rn EHFDXVHWKH\DUHOLQHDUO\LQGHSHQGHQWDQGWKH\ VSDQ Rn E\WKH,QYHUWLEOH0DWUL[7KHRUHP &9".1-& /HW H1 ; : : : ; Hn EHWKHFROXPQVRIWKH n % n LGHQWLW\PDWUL[ In 7KDW

x3

LV

e3

2 3 1 607 6 7 H1 D 6 : 7; 4 :: 5

0

e2 e1 x1 '*(63&

7KHVWDQGDUGEDVLVIRU R3

x2

2 3 0 617 6 7 H 2 D 6 : 7; 4 :: 5

0

:::;

2 3 0 6 :: 7 6 7 Hn D 6 : 7 405 1

7KHVHW fH1 ; : : : ; Hn g LVFDOOHGWKH VWDQGDUGEDVLV IRU Rn )LJ 2

3 2 3 2 3 3 !4 !2 &9".1-& /HW Y1 D 4 0 5 Y2 D 4 1 5 DQG Y3 D 4 1 5 'HWHUPLQHLI !6 7 5 fY1 ; Y2 ; Y3 g LVDEDVLVIRU R3

40-65*0/ 6LQFHWKHUHDUHH[DFWO\WKUHHYHFWRUVKHUHLQ R3 ZHFDQXVHDQ\RIVHYHUDO PHWKRGVWRGHWHUPLQHLIWKHPDWUL[ A D Œ Y1 Y2 Y3 ! LVLQYHUWLEOH )RULQVWDQFH WZR URZUHSODFHPHQWVUHYHDOWKDW A KDVWKUHHSLYRWSRVLWLRQV 7KXV A LVLQYHUWLEOH $VLQ ([DPSOH WKHFROXPQVRI A IRUPDEDVLVIRU R3

&9".1-& /HW S D f1; t; t 2 ; : : : ; t n g 9HULI\WKDW S LVDEDVLVIRU Pn 7KLVEDVLV LVFDOOHGWKH VWDQGDUGEDVLV IRU Pn

40-65*0/ &HUWDLQO\ S VSDQV Pn 7RVKRZWKDW S LVOLQHDUO\LQGHSHQGHQW VXSSRVHWKDW c0 ; : : : ; cn VDWLVI\ c0 # 1 C c1 t C c2 t 2 C # # # C cn t n D .t / 7KLVHTXDOLW\PHDQVWKDWWKHSRO\QRPLDORQWKHOHIWKDVWKHVDPHYDOXHVDVWKH]HURSRO\ QRPLDORQWKHULJKW $ IXQGDPHQWDOWKHRUHPLQDOJHEUDVD\VWKDWWKHRQO\SRO\QRPLDO

$)"15&3

y

9HFWRU6SDFHV

LQ Pn ZLWKPRUHWKDQ n ]HURVLVWKH]HURSRO\QRPLDO 7KDWLV HTXDWLRQ KROGVIRUDOO t RQO\LI c0 D # # # D cn D 0 7KLVSURYHVWKDW S LVOLQHDUO\LQGHSHQGHQWDQGKHQFHLVD EDVLVIRU Pn 6HH)LJ

y = t2 y=t

3UREOHPVLQYROYLQJOLQHDULQGHSHQGHQFHDQGVSDQQLQJLQ Pn DUHKDQGOHGEHVWE\D WHFKQLTXHWREHGLVFXVVHGLQ6HFWLRQ

7KH6SDQQLQJ6HW7KHRUHP

y=1 t

'*(63&

$VZHZLOOVHH DEDVLVLVDQ´HIÀFLHQWµVSDQQLQJVHWWKDWFRQWDLQVQRXQQHFHVVDU\ YHFWRUV ,QIDFW DEDVLVFDQEHFRQVWUXFWHGIURPDVSDQQLQJVHWE\GLVFDUGLQJXQQHHGHG YHFWRUV

&9".1-& /HW 2

3 0 Y1 D 4 2 5; !1

7KHVWDQGDUGEDVLVIRU P2

2 3 2 Y2 D 4 2 5; 0

2

3 6 Y3 D 4 16 5; !5

DQG

H D 6SDQ fY1 ; Y2 ; Y3 g:

1RWHWKDW Y3 D 5Y1 C 3Y2 DQGVKRZWKDW 6SDQ fY1 ; Y2 ; Y3 g D 6SDQ fY1 ; Y2 g 7KHQÀQGD EDVLVIRUWKHVXEVSDFH H

40-65*0/ (YHU\YHFWRULQ 6SDQ fY1 ; Y2 g EHORQJVWR H EHFDXVH

x2

c1 Y1 C c2 Y2 D c1 Y1 C c2 Y2 C 0Y3

H

v1

1RZOHW [ EHDQ\YHFWRULQ H ³VD\ [ D c1 Y1 C c2 Y2 C c3 Y3 6LQFH Y3 D 5Y1 C 3Y2 ZHPD\VXEVWLWXWH [ D c1 Y1 C c2 Y2 C c3 .5Y1 C 3Y2 / D .c1 C 5c3 /Y1 C .c2 C 3c3 /Y2

v3

v2

x3 x1

7KXV [ LVLQ 6SDQ fY1 ; Y2 g VRHYHU\YHFWRULQ H DOUHDG\EHORQJVWR 6SDQ fY1 ; Y2 g :H FRQFOXGHWKDW H DQG 6SDQ fY1 ; Y2 g DUHDFWXDOO\WKHVDPHVHWRIYHFWRUV ,WIROORZVWKDW fY1 ; Y2 g LVDEDVLVRI H VLQFH fY1 ; Y2 g LVREYLRXVO\OLQHDUO\LQGHSHQGHQW 7KHQH[WWKHRUHPJHQHUDOL]HV([DPSOH

5)&03&.

5IF 4QBOOJOH 4FU 5IFPSFN /HW S D fY1 ; : : : ; Yp g EHDVHWLQ V DQGOHW H D 6SDQ fY1 ; : : : ; Yp g D ,IRQHRIWKHYHFWRUVLQ S ³VD\ Yk ³LVDOLQHDUFRPELQDWLRQRIWKHUHPDLQLQJ YHFWRUVLQ S WKHQWKHVHWIRUPHGIURP S E\UHPRYLQJ Yk VWLOOVSDQV H E ,I H ¤ fg VRPHVXEVHWRI S LVDEDVLVIRU H

1300' D %\UHDUUDQJLQJWKHOLVWRIYHFWRUVLQ S LIQHFHVVDU\ ZHPD\VXSSRVHWKDW Yp LVD OLQHDUFRPELQDWLRQRI Y1 ; : : : ; Yp!1 ³VD\ Yp D a1 Y1 C # # # C ap!1 Yp!1

[ D c1 Y1 C # # # C cp!1 Yp!1 C cp Yp

*LYHQDQ\ [ LQ H ZHPD\ZULWH

IRUVXLWDEOHVFDODUV c1 ; : : : ; cp 6XEVWLWXWLQJWKHH[SUHVVLRQIRU Yp IURP LQWR LWLVHDV\WRVHHWKDW [ LVDOLQHDUFRPELQDWLRQRI Y1 ; : : : ; Yp!1 7KXV fY1 ; : : : ; Yp!1 g VSDQV H EHFDXVH [ ZDVDQDUELWUDU\HOHPHQWRI H


E ,IWKHRULJLQDOVSDQQLQJVHW S LVOLQHDUO\LQGHSHQGHQW WKHQLWLVDOUHDG\DEDVLVIRU H 2WKHUZLVH RQHRIWKHYHFWRUVLQ S GHSHQGVRQWKHRWKHUVDQGFDQEHGHOHWHG E\SDUW D 6RORQJDVWKHUHDUHWZRRUPRUHYHFWRUVLQWKHVSDQQLQJVHW ZHFDQUHSHDWWKLV SURFHVVXQWLOWKHVSDQQLQJVHWLVOLQHDUO\LQGHSHQGHQWDQGKHQFHLVDEDVLVIRU H ,I WKHVSDQQLQJVHWLVHYHQWXDOO\UHGXFHGWRRQHYHFWRU WKDWYHFWRUZLOOEHQRQ]HURDQG KHQFHOLQHDUO\LQGHSHQGHQW EHFDXVH H ¤ fg

%DVHVIRU 1XO A DQG &RO A :HDOUHDG\NQRZKRZWRÀQGYHFWRUVWKDWVSDQWKHQXOOVSDFHRIDPDWUL[ A 7KH GLVFXVVLRQ LQ 6HFWLRQ SRLQWHG RXW WKDW RXU PHWKRG DOZD\V SURGXFHV D OLQHDUO\ LQGHSHQGHQWVHWZKHQ 1XO A FRQWDLQVQRQ]HURYHFWRUV 6R LQWKLVFDVH WKDWPHWKRG SURGXFHVD EDVLV IRU 1XO A 7KHQH[WWZRH[DPSOHVGHVFULEHDVLPSOHDOJRULWKPIRUÀQGLQJDEDVLVIRUWKH FROXPQVSDFH

&9".1-& )LQGDEDVLVIRU &RO B ZKHUH % B D E1

E2

###

2

1 & 60 E5 D 6 40 0

4 0 0 0

0 2 1 !1 0 0 0 0

3 0 07 7 15 0

40-65*0/ (DFKQRQSLYRWFROXPQRI B LVDOLQHDUFRPELQDWLRQRIWKHSLYRWFROXPQV ,QIDFW E2 D 4E1 DQG E4 D 2E1 ! E3 %\WKH6SDQQLQJ6HW7KHRUHP ZHPD\GLVFDUG E2 DQG E4 DQG fE1 ; E3 ; E5 g ZLOOVWLOOVSDQ &RO B /HW 82 3 2 3 2 39 1 0 0 > ˆ ˆ = <6 7 6 7 6 7> 0 1 7; 6 7; 6 0 7 S D fE 1 ; E 3 ; E 5 g D 6 4 5 4 5 4 5 0 0 1 > ˆ > ˆ ; : 0 0 0

6LQFH E1 ¤ 0 DQGQRYHFWRULQ S LVDOLQHDUFRPELQDWLRQRIWKHYHFWRUVWKDWSUHFHGHLW S LVOLQHDUO\LQGHSHQGHQW7KHRUHP 7KXV S LVDEDVLVIRU &RO B :KDWDERXWDPDWUL[ A WKDWLV QRW LQUHGXFHGHFKHORQIRUP" 5HFDOOWKDWDQ\ OLQHDUGHSHQGHQFHUHODWLRQVKLSDPRQJWKHFROXPQVRI A FDQEHH[SUHVVHGLQWKHIRUP A[ D ZKHUH [ LVDFROXPQRIZHLJKWV ,IVRPHFROXPQVDUHQRWLQYROYHGLQD SDUWLFXODUGHSHQGHQFHUHODWLRQ WKHQWKHLUZHLJKWVDUH]HUR :KHQ A LVURZUHGXFHG WRDPDWUL[ B WKHFROXPQVRI B DUHRIWHQWRWDOO\GLIIHUHQWIURPWKHFROXPQVRI A +RZHYHU WKHHTXDWLRQV A[ D DQG B [ D KDYHH[DFWO\WKHVDPHVHWRIVROXWLRQV ,I A D Œ D1 # # # Dn ! DQG B D Œ E1 # # # En ! WKHQWKHYHFWRUHTXDWLRQV

x1 D1 C # # # C xn Dn D

DQG

x1 E1 C # # # C xn En D

DOVRKDYHWKHVDPHVHWRIVROXWLRQV 7KDWLV WKHFROXPQVRI A KDYH H[DFWO\WKHVDPH OLQHDUGHSHQGHQFHUHODWLRQVKLSV DVWKHFROXPQVRI B

&9".1-& ,WFDQEHVKRZQWKDWWKHPDWUL[ % A D D1

D2

###

2

1 4 & 6 3 12 D5 D 6 42 8 5 20

0 1 1 2

3 2 !1 5 57 7 3 25 8 8

LVURZHTXLYDOHQWWRWKHPDWUL[ B LQ([DPSOH )LQGDEDVLVIRU &RO A

$)"15&3

9HFWRU6SDFHV

40-65*0/ ,Q([DPSOHZHVDZWKDW E2 D 4E1

DQG

E4 D 2E1 ! E3

D 2 D 4D 1

DQG

D4 D 2D1 ! D3

VRZHFDQH[SHFWWKDW

&KHFNWKDWWKLVLVLQGHHGWKHFDVH 7KXVZHPD\GLVFDUG D2 DQG D4 ZKHQVHOHFWLQJ DPLQLPDOVSDQQLQJVHWIRU &RO A ,QIDFW fD1 ; D3 ; D5 g PXVWEHOLQHDUO\LQGHSHQGHQW EHFDXVHDQ\OLQHDUGHSHQGHQFHUHODWLRQVKLSDPRQJ D1 D3 D5 ZRXOGLPSO\DOLQHDU GHSHQGHQFHUHODWLRQVKLSDPRQJ E1 E3 E5 %XWZHNQRZWKDW fE1 ; E3 ; E5 g LVDOLQHDUO\ LQGHSHQGHQWVHW 7KXV fD1 ; D3 ; D5 g LVDEDVLVIRU &RO A 7KHFROXPQVZHKDYHXVHGIRU WKLVEDVLVDUHWKHSLYRWFROXPQVRI A ([DPSOHVDQGLOOXVWUDWHWKHIROORZLQJXVHIXOIDFW

5)&03&.

7KHSLYRWFROXPQVRIDPDWUL[ A IRUPDEDVLVIRU &RO A

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

7ZR9LHZVRID%DVLV :KHQWKH6SDQQLQJ6HW7KHRUHPLVXVHG WKHGHOHWLRQRIYHFWRUVIURPDVSDQQLQJVHW PXVWVWRSZKHQWKHVHWEHFRPHVOLQHDUO\LQGHSHQGHQW ,IDQDGGLWLRQDOYHFWRULVGHOHWHG LWZLOOQRWEHDOLQHDUFRPELQDWLRQRIWKHUHPDLQLQJYHFWRUV DQGKHQFHWKHVPDOOHUVHW ZLOOQRORQJHUVSDQ V 7KXVDEDVLVLVDVSDQQLQJVHWWKDWLVDVVPDOODVSRVVLEOH $ EDVLVLVDOVRDOLQHDUO\LQGHSHQGHQWVHWWKDWLVDVODUJHDVSRVVLEOH ,I S LVDEDVLV IRU V DQGLI S LVHQODUJHGE\RQHYHFWRU³VD\ Z³IURP V WKHQWKHQHZVHWFDQQRWEH OLQHDUO\LQGHSHQGHQW EHFDXVH S VSDQV V DQG Z LVWKHUHIRUHDOLQHDUFRPELQDWLRQRIWKH HOHPHQWVLQ S

&9".1-& 7KHIROORZLQJWKUHHVHWVLQ R3 VKRZKRZDOLQHDUO\LQGHSHQGHQWVHW FDQEHHQODUJHGWRDEDVLVDQGKRZIXUWKHUHQODUJHPHQWGHVWUR\VWKHOLQHDULQGHSHQGHQFH RIWKHVHW $OVR DVSDQQLQJVHWFDQEHVKUXQNWRDEDVLV EXWIXUWKHUVKULQNLQJGHVWUR\V

WKHVSDQQLQJSURSHUW\ 82 3 2 39 2 = < 1 4 0 5; 4 3 5 : ; 0 0

/LQHDUO\LQGHSHQGHQW EXWGRHVQRWVSDQ R3

SG

.BTUFSJOH #BTJT m


82 3 2 3 2 39 2 4 = < 1 4 0 5; 4 3 5; 4 5 5 : ; 0 0 6 $ EDVLV IRU R3

82 3 2 3 2 3 2 39 2 4 7 = < 1 4 0 5; 4 3 5; 4 5 5; 4 8 5 : ; 0 0 6 9 6SDQV R3 EXWLV OLQHDUO\GHSHQGHQW

13"$5*$& 130#-&.4 2 3 2 3 1 !2 /HW Y1 D 4 !2 5 DQG Y2 D 4 7 5 'HWHUPLQHLI fY1 ; Y2 g LVDEDVLVIRU R3 ,V 3 !9 2 fY1 ; Y2 g DEDVLVIRU R " 2 3 2 3 2 3 2 3 1 6 2 !4 /HW Y1 D 4 !3 5 Y2 D 4 2 5 Y3 D 4 !2 5 DQG Y4 D 4 !8 5 )LQGDEDVLVIRU 4 !1 3 9 WKHVXEVSDFH W VSDQQHGE\ fY1 ; Y2 ; Y3 ; Y4 g 82 3 9 2 3 2 3 1 0 < s = /HW Y1 D 4 0 5 Y2 D 4 1 5 DQG H D 4 s 5 W s LQ R 7KHQHYHU\YHFWRULQ H : ; 0 0 0 LVDOLQHDUFRPELQDWLRQRI Y1 DQG Y2 EHFDXVH 2 3 2 3 2 3 s 1 0 4 s 5 D s4 0 5 C s4 1 5 0 0 0 ,V fY1 ; Y2 g DEDVLVIRU H "

&9&3$*4&4 'HWHUPLQHZKHWKHUWKHVHWVLQ([HUFLVHV²DUHEDVHVIRU R3 2IWKHVHWVWKDWDUH QRW EDVHV GHWHUPLQHZKLFKRQHVDUHOLQHDUO\ LQGHSHQGHQWDQGZKLFKRQHVVSDQ R3 -XVWLI\\RXUDQVZHUV 2 3 2 3 2 3 2 3 2 3 2 3 1 1 1 1 0 0 4 0 5 4 1 5 4 1 5 4 1 5 4 0 5 4 1 5 0 0 1 0 0 1 2

3 2 3 2 3 1 3 !2 4 0 5 4 1 5 4 !1 5 !3 !4 1 2

2

3 2 3 2 3 2 2 !8 4 !1 5 4 !3 5 4 5 5 1 2 4

3 2 3 2 3 2 3 2 3 2 3 3 !3 0 0 1 !4 4 !3 5 4 7 5 4 0 5 4 !3 5 4 2 5 4 3 5 0 0 0 5 !4 6 2 3 2 3 2 3 2 3 2 3 2 3 !2 6 1 0 2 0 4 3 5 4 !1 5 4 !2 5 4 3 5 4 !1 5 4 0 5 0 5 3 !1 5 !1

)LQGEDVHVIRUWKHQXOOVSDFHVRIWKHPDWULFHVJLYHQLQ([HUFLVHV DQG 5HIHUWRWKHUHPDUNVWKDWIROORZ([DPSOH LQ6HFWLRQ 2 3 2 3 1 0 !2 !2 1 1 !2 1 5 1 1 45 1 0 !1 !2 5 4 0 4 0 3 !1 !7 3 0 0 !8 0 16 )LQG D EDVLV IRU WKH VHW RI YHFWRUV LQ R3 LQ WKH SODQH x ! 3y C 2´ D 0 >+LQW 7KLQNRIWKHHTXDWLRQDVD´V\V WHPµRIKRPRJHQHRXVHTXDWLRQV@ )LQGDEDVLVIRUWKHVHWRIYHFWRUVLQ R2 RQWKHOLQH y D !3x ,Q([HUFLVHVDQG DVVXPHWKDW A LVURZHTXLYDOHQWWR B )LQGEDVHVIRU 1XO A DQG &RO A 2 3 2 3 !2 4 !2 !4 1 0 6 5 1 5 B D 4 0 2 5 35 A D 4 2 !6 !3 !3 8 2 !3 0 0 0 0

$)"15&3 2

1 61 6 A D 4 2 3 2 1 60 6 BD4 0 0

9HFWRU6SDFHV

3 2 3 !4 8 2 0 2 87 7 4 !3 10 95 6 0 6 9 3 2 0 2 5 0 3 !6 37 7 0 0 0 !7 5 0 0 0 0

,Q([HUFLVHV² ÀQGDEDVLVIRUWKHVSDFHVSDQQHGE\WKHJLYHQ YHFWRUV Y1 ; : : : ; Y5 2 3 2 3 2 3 2 3 2 3 1 0 2 2 3 6 0 7 6 1 7 6 !2 7 6 !1 7 6 !1 7 7 6 7 6 7 6 7 6 7 6 4 !2 5 4 2 5 4 !8 5 4 10 5 4 !6 5 3 3 0 3 9 2 3 2 3 2 3 2 3 2 3 1 !2 3 5 2 6 0 7 6 0 7 6 !1 7 6 !3 7 6 !1 7 7 6 7 6 7 6 7 6 7 6 4 0 5 4 0 5 4 1 5 4 3 5 4 1 5 1 2 !1 !4 0 2 3 2 3 2 3 2 3 2 3 2 4 !2 8 !8 6 0 7 6 0 7 6 !4 7 6 4 7 6 4 7 6 7 6 7 6 7 6 7 6 7 7 6 7 6 7 6 7 6 7 >0@ 6 6 !4 7 6 2 7 6 0 7 6 8 7 6 0 7 4 !6 5 4 !4 5 4 1 5 4 !3 5 4 0 5 0 4 !7 15 1 2 3 2 3 2 3 2 3 2 3 !3 3 0 6 !6 6 2 7 6 0 7 6 2 7 6 !2 7 6 3 7 6 7 6 7 6 7 6 7 6 7 7 6 7 6 7 6 7 6 7 >0@ 6 6 6 7 6 !9 7 6 !4 7 6 !14 7 6 0 7 4 0 5 4 0 5 4 0 5 4 0 5 4 !1 5 !7 6 !1 13 0 2 3 2 3 2 3 4 1 7 /HW Y1 D 4 !3 5 Y2 D 4 9 5 Y3 D 4 11 5 DQGDOVROHW 7 !2 6 H D 6SDQ fY1 ; Y2 ; Y3 g ,WFDQEHYHULÀHGWKDW 4Y1 C 5Y2 ! 3Y3 D 8VHWKLVLQIRUPDWLRQWRÀQGDEDVLVIRU H 7KHUHLV PRUHWKDQRQHDQVZHU 2 3 2 3 2 3 3 4 2 6 47 637 6 57 7 6 7 6 7 /HW Y1 D 6 4 !2 5 Y2 D 4 2 5 DQG Y3 D 4 !6 5 ,WFDQEH !5 4 !14 YHULÀHGWKDW 2Y1 ! Y2 ! Y3 D 8VHWKLVLQIRUPDWLRQWRÀQG DEDVLVIRU H D 6SDQ fY1 ; Y2 ; Y3 g ,Q([HUFLVHVDQG PDUNHDFKVWDWHPHQW7UXHRU)DOVH -XVWLI\ HDFKDQVZHU D $ VLQJOHYHFWRUE\LWVHOILVOLQHDUO\GHSHQGHQW

D $ OLQHDUO\LQGHSHQGHQWVHWLQDVXEVSDFH H LVDEDVLVIRU H E ,IDÀQLWHVHW S RIQRQ]HURYHFWRUVVSDQVDYHFWRUVSDFH V WKHQVRPHVXEVHWRI S LVDEDVLVIRU V F $ EDVLVLVDOLQHDUO\LQGHSHQGHQWVHWWKDWLVDVODUJHDV SRVVLEOH G 7KHVWDQGDUGPHWKRGIRUSURGXFLQJDVSDQQLQJVHWIRU 1XO A GHVFULEHGLQ6HFWLRQ VRPHWLPHVIDLOVWRSUR GXFHDEDVLVIRU 1XO A H ,I B LVDQHFKHORQIRUPRIDPDWUL[ A WKHQWKHSLYRW FROXPQVRI B IRUPDEDVLVIRU &RO A 6XSSRVH R4 D 6SDQ fY1 ; : : : ; Y4 g ([SODLQZK\ fY1 ; : : : ; Y4 g LVDEDVLVIRU R4 /HW B D fY1 ; : : : ; Yn g EHDOLQHDUO\LQGHSHQGHQWVHWLQ Rn ([SODLQZK\ B PXVWEHDEDVLVIRU Rn 2 3 2 3 2 3 1 0 0 /HW Y1 D 4 0 5 Y2 D 4 1 5 Y3 D 4 1 5 DQGOHW H EHWKH 1 1 0 VHWRIYHFWRUVLQ R3 ZKRVHVHFRQGDQGWKLUGHQWULHVDUHHTXDO 7KHQHYHU\YHFWRULQ H KDVDXQLTXHH[SDQVLRQDVDOLQHDU FRPELQDWLRQRI Y1 ; Y2 ; Y3 EHFDXVH 2 3 2 3 2 3 2 3 s 1 0 0 4 t 5 D s 4 0 5 C .t ! s/4 1 5 C s 4 1 5 t 1 1 0 IRUDQ\ s DQG t ,V fY1 ; Y2 ; Y3 g DEDVLVIRU H " :K\RUZK\ QRW"

,QWKHYHFWRUVSDFHRIDOOUHDOYDOXHGIXQFWLRQV ÀQGDEDVLV IRUWKHVXEVSDFHVSDQQHGE\ fVLQ t; VLQ 2t; VLQ t FRV t g /HW V EH WKH YHFWRUVSDFH RIIXQFWLRQV WKDWGHVFULEH WKH YLEUDWLRQRIDPDVV²VSULQJV\VWHP 5HIHUWR([HUFLVH LQ 6HFWLRQ )LQGDEDVLVIRU V 5/& FLUFXLW 7KHFLUFXLWLQWKHÀJXUHFRQVLVWVRIDUHVLVWRU R RKPV DQLQGXFWRUL KHQU\V DFDSDFLWRUC IDUDGV DQG DQLQLWLDOYROWDJHVRXUFH /HW b D R=.2L/ DQGVXSSRVH p R L DQG C KDYHEHHQVHOHFWHGVRWKDW b DOVRHTXDOV 1= LC 7KLVLVGRQH IRULQVWDQFH ZKHQWKHFLUFXLWLVXVHGLQD YROWPHWHU /HW v.t/ EH WKH YROWDJH LQ YROWV DW WLPH t PHDVXUHGDFURVVWKHFDSDFLWRU ,WFDQEHVKRZQWKDW v LV LQWKHQXOOVSDFH H RIWKHOLQHDUWUDQVIRUPDWLRQWKDWPDSV v.t/ LQWR Lv 00 .t/ C Rv 0 .t/ C .1=C /v.t/ DQG H FRQVLVWVRI DOOIXQFWLRQVRIWKHIRUP v.t/ D e "bt .c1 C c2 t/ )LQGDEDVLV IRU H

E ,I H D 6SDQ fE1 ; : : : ; Ep g WKHQ fE1 ; : : : ; Ep g LVDEDVLVIRU H F 7KHFROXPQVRIDQLQYHUWLEOH n % n PDWUL[IRUPDEDVLV IRU Rn G $ EDVLVLVDVSDQQLQJVHWWKDWLVDVODUJHDVSRVVLEOH H ,QVRPHFDVHV WKHOLQHDUGHSHQGHQFHUHODWLRQVDPRQJWKH FROXPQVRIDPDWUL[FDQEHDIIHFWHGE\FHUWDLQHOHPHQWDU\ URZRSHUDWLRQVRQWKHPDWUL[

R Voltage source

C L

([HUFLVHVDQGVKRZWKDWHYHU\EDVLVIRU Rn PXVWFRQWDLQ H[DFWO\ n YHFWRUV

/HW S D fY1 ; : : : ; Yk g EHDVHWRI k YHFWRUVLQ Rn ZLWK k < n 8VHDWKHRUHPIURP6HFWLRQWRH[SODLQZK\ S FDQQRWEH DEDVLVIRU Rn /HW S D fY1 ; : : : ; Yk g EHDVHWRI k YHFWRUVLQ Rn ZLWK k > n 8VHDWKHRUHPIURP&KDSWHU WRH[SODLQZK\ S FDQQRWEHD EDVLVIRU Rn ([HUFLVHVDQGUHYHDODQLPSRUWDQWFRQQHFWLRQEHWZHHQOLQ HDULQGHSHQGHQFHDQGOLQHDUWUDQVIRUPDWLRQVDQGSURYLGHSUDFWLFH XVLQJ WKH GHÀQLWLRQ RI OLQHDU GHSHQGHQFH /HW V DQG W EH YHFWRUVSDFHV OHW T W V ! W EHDOLQHDUWUDQVIRUPDWLRQ DQGOHW fY1 ; : : : ; Yp g EHDVXEVHWRI V 6KRZWKDWLI fY1 ; : : : ; Yp g LVOLQHDUO\GHSHQGHQWLQ V WKHQ WKH VHW RI LPDJHV fT .Y1 /; : : : ; T .Yp /g LV OLQHDUO\ GHSHQ GHQW LQ W 7KLV IDFW VKRZV WKDW LI D OLQHDU WUDQVIRUPD WLRQPDSVDVHW fY1 ; : : : ; Yp g RQWRDOLQHDUO\ LQGHSHQGHQW VHW fT .Y1 /; : : : ; T .Yp /g WKHQWKHRULJLQDOVHWLVOLQHDUO\LQGHSHQ GHQW WRREHFDXVHLWFDQQRWEHOLQHDUO\GHSHQGHQW 6XSSRVHWKDW T LVDRQHWRRQHWUDQVIRUPDWLRQ VRWKDWDQ HTXDWLRQ T .X/ D T .Y/ DOZD\VLPSOLHV X D Y 6KRZWKDWLI WKHVHWRILPDJHV fT .Y1 /; : : : ; T .Yp /g LVOLQHDUO\GHSHQGHQW WKHQ fY1 ; : : : ; Yp g LVOLQHDUO\GHSHQGHQW 7KLVIDFWVKRZVWKDW DRQHWRRQHOLQHDUWUDQVIRUPDWLRQPDSVDOLQHDUO\LQGHSHQ GHQWVHWRQWRDOLQHDUO\LQGHSHQGHQWVHW EHFDXVHLQWKLVFDVH WKHVHWRILPDJHVFDQQRWEHOLQHDUO\GHSHQGHQW &RQVLGHUWKHSRO\QRPLDOV S1 .t / D 1 C t 2 DQG S2 .t/ D 1 ! t 2 ,V fS1 ; S2 g DOLQHDUO\LQGHSHQGHQWVHWLQ P3 " :K\RUZK\ QRW" &RQVLGHUWKHSRO\QRPLDOV S1 .t/ D 1 C t S2 .t / D 1 ! t DQG S3 .t/ D 2 IRUDOO t %\LQVSHFWLRQ ZULWHDOLQHDUGHSHQ


GHQFHUHODWLRQDPRQJ S1 S2 DQG S3 7KHQÀQGDEDVLVIRU 6SDQ fS1 ; S2 ; S3 g /HW V EHDYHFWRUVSDFHWKDWFRQWDLQVDOLQHDUO\LQGHSHQ GHQWVHW fX1 ; X2 ; X3 ; X4 g 'HVFULEHKRZWRFRQVWUXFWDVHWRI YHFWRUV fY1 ; Y2 ; Y3 ; Y4 g LQ V VXFKWKDW fY1 ; Y3 g LVDEDVLVIRU 6SDQ fY1 ; Y2 ; Y3 ; Y4 g

>0@ /HW H D 6SDQ fX1 ; X2 ; X3 g DQG K D 6SDQ fY1 ; Y2 ; Y3 g ZKHUH 2 3 2 3 2 3 1 0 3 6 27 6 27 6 47 7 6 7 6 7 X1 D 6 4 0 5; X2 D 4 !1 5; X3 D 4 1 5; !1 1 !4 2 3 2 3 2 3 !2 2 !1 6 !2 7 6 37 6 47 7 6 7 6 7 Y1 D 6 4 !1 5; Y2 D 4 2 5; Y3 D 4 6 5 !6 !2 3

)LQGEDVHVIRU H K DQG H C K 6HH([HUFLVHVDQG LQ6HFWLRQ

>0@ 6KRZWKDW ft; VLQ t; FRV 2t; VLQ t FRV t g LVDOLQHDUO\LQGH SHQGHQWVHWRIIXQFWLRQVGHÀQHGRQ R 6WDUWE\DVVXPLQJWKDW

c1 # t C c2 # VLQ t C c3 # FRV 2t C c4 # VLQ t FRV t D 0

(TXDWLRQ PXVW KROG IRU DOO UHDO t VR FKRRVH VHYHUDO VSHFLÀFYDOXHVRI t VD\ t D 0; :1; :2/ XQWLO\RXJHWDV\VWHP RIHQRXJKHTXDWLRQVWRGHWHUPLQHWKDWDOOWKH cj PXVWEH]HUR >0@ 6KRZWKDW f1; FRV t; FRV2 t; : : : ; FRV6 t g LVDOLQHDUO\LQGH SHQGHQWVHWRIIXQFWLRQVGHÀQHGRQ R 8VHWKHPHWKRGRI ([HUFLVH 7KLVUHVXOWZLOOEHQHHGHGLQ([HUFLVH LQ 6HFWLRQ WEB

40-65*0/4 50 13"$5*$& 130#-&.4 /HW A D Œ Y1 Y2 ! 5RZRSHUDWLRQVVKRZWKDW 2

1 A D 4 !2 3

3 2 !2 1 75 & 40 !9 0

3 !2 35 0

1RWHYHU\URZRI A FRQWDLQVDSLYRWSRVLWLRQ 6RWKHFROXPQVRI A GRQRWVSDQ R3 E\7KHRUHP LQ6HFWLRQ +HQFH fY1 ; Y2 g LVQRWDEDVLVIRU R3 6LQFH Y1 DQG Y2 DUHQRWLQ R2 WKH\FDQQRWSRVVLEO\EHDEDVLVIRU R2 +RZHYHU VLQFH Y1 DQG Y2 DUHREYLRXVO\OLQHDUO\LQGHSHQGHQW WKH\DUHDEDVLVIRUDVXEVSDFHRI R3 QDPHO\ 6SDQ fY1 ; Y2 g

6HWXSDPDWUL[ A ZKRVHFROXPQVSDFHLVWKHVSDFHVSDQQHGE\ fY1 ; Y2 ; Y3 ; Y4 g DQG WKHQURZUHGXFH A WRÀQGLWVSLYRWFROXPQV 2 3 2 3 2 3 1 6 2 !4 1 6 2 !4 1 6 2 !4 A D 4 !3 2 !2 !8 5 & 4 0 20 4 !20 5 & 4 0 5 1 !5 5 4 !1 3 9 0 !25 !5 25 0 0 0 0

$)"15&3

9HFWRU6SDFHV

7KH ÀUVW WZR FROXPQV RI A DUH WKH SLYRW FROXPQV DQG KHQFH IRUP D EDVLV RI &RO A D W +HQFH fY1 ; Y2 g LVDEDVLVIRU W 1RWHWKDWWKHUHGXFHGHFKHORQIRUPRI A LVQRWQHHGHGLQRUGHUWRORFDWHWKHSLYRWFROXPQV 1HLWKHU Y1 QRU Y2 LVLQ H VR fY1 ; Y2 g FDQQRWEHDEDVLVIRU H ,QIDFW fY1 ; Y2 g LVD EDVLVIRUWKH SODQH RIDOOYHFWRUVRIWKHIRUP .c1 ; c2 ; 0/ EXW H LVRQO\D OLQH

$003%*/"5& 4:45&.4 $QLPSRUWDQWUHDVRQIRUVSHFLI\LQJDEDVLV B IRUDYHFWRUVSDFH V LVWRLPSRVHD ´FRRUGLQDWHV\VWHPµRQ V 7KLVVHFWLRQZLOOVKRZWKDWLI B FRQWDLQV n YHFWRUV WKHQ WKHFRRUGLQDWHV\VWHPZLOOPDNH V DFWOLNH Rn ,I V LVDOUHDG\ Rn LWVHOI WKHQ B ZLOO GHWHUPLQHDFRRUGLQDWHV\VWHPWKDWJLYHVDQHZÝLHZµRI V 7KHH[LVWHQFHRIFRRUGLQDWHV\VWHPVUHVWVRQWKHIROORZLQJIXQGDPHQWDOUHVXOW

5)&03&.

5IF 6OJRVF 3FQSFTFOUBUJPO 5IFPSFN /HW B D fE1 ; : : : ; En g EHDEDVLVIRUDYHFWRUVSDFH V 7KHQIRUHDFK [ LQ V WKHUH H[LVWVDXQLTXHVHWRIVFDODUV c1 ; : : : ; cn VXFKWKDW [ D c1 E1 C # # # C cn En

1300' 6LQFH B VSDQV V WKHUHH[LVWVFDODUVVXFKWKDW KROGV 6XSSRVH [ DOVRKDV WKHUHSUHVHQWDWLRQ [ D d1 E1 C # # # C dn En IRUVFDODUV d1 ; : : : ; dn 7KHQ VXEWUDFWLQJ ZHKDYH D [ ! [ D .c1 ! d1 /E1 C # # # C .cn ! dn /En

6LQFH B LVOLQHDUO\LQGHSHQGHQW WKHZHLJKWVLQ PXVWDOOEH]HUR 7KDWLV cj D dj IRU 1 $ j $ n

%&'*/*5*0/

6XSSRVH B D fE1 ; : : : ; En g LVDEDVLVIRU V DQG [ LVLQ V 7KH FRRUGLQDWHVRI [ UHODWLYHWRWKHEDVLV B RUWKH BFRRUGLQDWHVRI [ DUHWKHZHLJKWV c1 ; : : : ; cn VXFKWKDW [ D c1 E1 C # # # C cn En ,I c1 ; : : : ; cn DUHWKH B FRRUGLQDWHVRI [ WKHQWKHYHFWRULQ Rn 2 3 c1 % & 6 :: 7 [ B D 4 :5 cn

LVWKH FRRUGLQDWHYHFWRURI [ UHODWLYHWR B/ RUWKH BFRRUGLQDWHYHFWRURI [ 7KH % & PDSSLQJ [ 7! [ B LVWKH FRRUGLQDWHPDSSLQJGHWHUPLQHGE\ B/ñ ñ 7KHFRQFHSWRIDFRRUGLQDWHPDSSLQJDVVXPHVWKDWWKHEDVLV B LVDQLQGH[HGVHWZKRVHYHFWRUVDUHOLVWHGLQ VRPHÀ[HGSUHDVVLJQHGRUGHU 7KLVSURSHUW\PDNHVWKHGHÀQLWLRQRI Œ [ !B XQDPELJXRXV

&RRUGLQDWH6\VWHPV

&9".1-& &RQVLGHU D EDVLV B D fE1 ; E2 g IRU R2 ZKHUH E1 D

" # 1 DQG 0

" # " # 1 !2 2 E2 D 6XSSRVHDQ [ LQ R KDVWKHFRRUGLQDWHYHFWRU Œ [ !B D )LQG [ 2 3

40-65*0/ 7KH B FRRUGLQDWHVRI [ WHOOKRZWREXLOG [ IURPWKHYHFWRUVLQ B 7KDWLV " # " # " # 1 1 1 C3 D [ D .!2/E1 C 3E2 D .!2/ 0 2 6

&9".1-& 7KHHQWULHVLQWKHYHFWRU [ D

" # 1 DUHWKHFRRUGLQDWHVRI [ UHODWLYHWR 6

WKH VWDQGDUGEDVLV E D fH1 ; H2 g VLQFH " # " # " # 1 1 0 D1# C6# D 1 # H1 C 6 # H2 6 0 1

,I E D fH1 ; H2 g WKHQ Œ [ !E D [

$ *UDSKLFDO,QWHUSUHWDWLRQRI&RRUGLQDWHV $ FRRUGLQDWHV\VWHPRQDVHWFRQVLVWVRIDRQHWRRQHPDSSLQJRIWKHSRLQWVLQWKHVHW LQWR Rn )RUH[DPSOH RUGLQDU\JUDSKSDSHUSURYLGHVDFRRUGLQDWHV\VWHPIRUWKHSODQH ZKHQRQHVHOHFWVSHUSHQGLFXODUD[HVDQGDXQLWRIPHDVXUHPHQWRQHDFKD[LV )LJXUH VKRZVWKHVWDQGDUGEDVLV fH1 ; H2 g WKHYHFWRUV E1 .D H1 / DQG E2 IURP([DPSOH DQGWKH " # 1 7KHFRRUGLQDWHVDQGJLYHWKHORFDWLRQRI [ UHODWLYHWRWKHVWDQGDUG YHFWRU [ D 6 EDVLV XQLWLQWKH H1 GLUHFWLRQDQG XQLWVLQWKH H2 GLUHFWLRQ )LJXUH VKRZVWKHYHFWRUV E1 E2 DQG [ IURP)LJ *HRPHWULFDOO\ WKHWKUHH YHFWRUVOLHRQDYHUWLFDOOLQHLQERWKÀJXUHV +RZHYHU WKHVWDQGDUGFRRUGLQDWHJULG B LQ([DPSOH 7KH ZDVHUDVHGDQGUHSODFHGE\DJULGHVSHFLDOO\DGDSWHGWRWKHEDVLV " # !2 FRRUGLQDWHYHFWRU Œ [ !B D JLYHVWKHORFDWLRQRI [ RQWKLVQHZFRRUGLQDWHV\VWHP 3 !2 XQLWV LQWKH E1 GLUHFWLRQDQG XQLWVLQWKH E2 GLUHFWLRQ x

x

e2 0

b2 b1 = e1

'*(63& 6WDQGDUGJUDSK

SDSHU

b2 0

b1

'*(63& B JUDSKSDSHU

&9".1-& ,QFU\VWDOORJUDSK\ WKHGHVFULSWLRQRIDFU\VWDOODWWLFHLVDLGHGE\ FKRRVLQJDEDVLV fX; Y; Zg IRU R3 WKDWFRUUHVSRQGVWRWKUHHDGMDFHQWHGJHVRIRQH´XQLW FHOOµRIWKHFU\VWDO $QHQWLUHODWWLFHLVFRQVWUXFWHGE\VWDFNLQJWRJHWKHUPDQ\FRSLHVRI RQHFHOO 7KHUHDUHIRXUWHHQEDVLFW\SHVRIXQLWFHOOV WKUHHDUHGLVSOD\HGLQ)LJ ò ò $GDSWHGIURP 7KH6FLHQFHDQG(QJLQHHULQJRI0DWHULDOV WK(G E\'RQDOG5$VNHODQG%RVWRQ 3ULQGOH :HEHU 6FKPLGW S

$)"15&3

9HFWRU6SDFHV w w

w

0

0 v

u

0 v

u

u

(b) Body-centered cubic

(a) Simple monoclinic

v (c) Face-centered orthorhombic

'*(63& ([DPSOHVRIXQLWFHOOV

7KHFRRUGLQDWHVRIDWRPVZLWKLQWKHFU\VWDODUHJLYHQUHODWLYHWRWKHEDVLVIRUWKH ODWWLFH )RULQVWDQFH 2 3 1=2 4 1=2 5 1 LGHQWLÀHVWKHWRSIDFHFHQWHUHGDWRPLQWKHFHOOLQ)LJ F

&RRUGLQDWHVLQ Rn :KHQDEDVLV B IRU Rn LVÀ[HG WKH B FRRUGLQDWHYHFWRURIDVSHFLÀHG [ LVHDVLO\IRXQG DVLQWKHQH[WH[DPSOH " # " # " # 2 !1 4 &9".1-& /HW E1 D E2 D [D DQG B D fE1 ; E2 g )LQGWKH 1 1 5 FRRUGLQDWHYHFWRU Œ [ !B RI [ UHODWLYHWR B

40-65*0/ 7KH B FRRUGLQDWHV c1 c2 RI [ VDWLVI\ " # " # " # 2 !1 4 c1 C c2 D 1 1 5 E1

x

b2

b1

'*(63&

7KH BFRRUGLQDWHYHFWRURI [ LV .3; 2/

RU

"

E2

2 !1 1 1 E1

E2

#"

c1 c2

#

[

" # 4 D 5

[

7KLVHTXDWLRQFDQEHVROYHGE\URZRSHUDWLRQVRQDQDXJPHQWHGPDWUL[RUE\XVLQJ WKHLQYHUVHRIWKHPDWUL[RQWKHOHIW ,QDQ\FDVH WKHVROXWLRQLV c1 D 3 c2 D 2 7KXV [ D 3E1 C 2E2 DQG " # " # c 3 Œ [ !B D 1 D c2 2 6HH)LJ 7KH PDWUL[ LQ FKDQJHV WKH B FRRUGLQDWHV RI D YHFWRU [ LQWR WKH VWDQGDUG FRRUGLQDWHVIRU [ $QDQDORJRXVFKDQJHRIFRRUGLQDWHVFDQEHFDUULHGRXWLQ Rn IRU DEDVLV B D fE1 ; : : : ; En g /HW

PB D Œ E1 E2 # # # En !

&RRUGLQDWH6\VWHPV

7KHQWKHYHFWRUHTXDWLRQ [ D c1 E1 C c2 E2 C # # # C cn En LVHTXLYDOHQWWR [ D PB Œ [ !B

:HFDOO PB WKH FKDQJHRIFRRUGLQDWHVPDWUL[ IURP B WRWKHVWDQGDUGEDVLVLQ Rn /HIWPXOWLSOLFDWLRQE\ PB WUDQVIRUPVWKHFRRUGLQDWHYHFWRU Œ [ !B LQWR [ 7KHFKDQJHRI FRRUGLQDWHVHTXDWLRQ LVLPSRUWDQWDQGZLOOEHQHHGHGDWVHYHUDOSRLQWVLQ&KDSWHUV DQG 6LQFHWKHFROXPQVRI PB IRUPDEDVLVIRU Rn PB LVLQYHUWLEOHE\WKH,QYHUWLEOH 0DWUL[7KHRUHP /HIWPXOWLSOLFDWLRQE\ PB!1 FRQYHUWV [ LQWRLWV B FRRUGLQDWHYHFWRU

PB!1 [ D Œ [ !B

7KHFRUUHVSRQGHQFH [ 7! Œ [ !B SURGXFHGKHUHE\ PB!1 LVWKHFRRUGLQDWHPDSSLQJ PHQWLRQHGHDUOLHU 6LQFH PB!1 LVDQLQYHUWLEOHPDWUL[ WKHFRRUGLQDWHPDSSLQJLVDRQH WRRQHOLQHDUWUDQVIRUPDWLRQIURP Rn RQWR Rn E\WKH,QYHUWLEOH0DWUL[7KHRUHP 6HH DOVR7KHRUHP LQ6HFWLRQ 7KLVSURSHUW\RIWKHFRRUGLQDWHPDSSLQJLVDOVRWUXH LQDJHQHUDOYHFWRUVSDFHWKDWKDVDEDVLV DVZHVKDOOVHH

7KH&RRUGLQDWH0DSSLQJ &KRRVLQJDEDVLV B D fE1 ; : : : ; En g IRUDYHFWRUVSDFH V LQWURGXFHVDFRRUGLQDWHV\VWHP LQ V 7KHFRRUGLQDWHPDSSLQJ [ 7! Œ [ !B FRQQHFWVWKHSRVVLEO\XQIDPLOLDUVSDFH V WR WKHIDPLOLDUVSDFH Rn 6HH)LJ 3RLQWVLQ V FDQQRZEHLGHQWLÀHGE\WKHLUQHZ ´QDPHVµ [ ]B [x]B

x

!n

V

'*(63& 7KHFRRUGLQDWHPDSSLQJIURP V RQWR Rn

5)&03&.

/HW B D fE1 ; : : : ; En g EHDEDVLVIRUDYHFWRUVSDFH V 7KHQWKHFRRUGLQDWH PDSSLQJ [ 7! Œ [ !B LVDRQHWRRQHOLQHDUWUDQVIRUPDWLRQIURP V RQWR Rn

1300' 7DNHWZRW\SLFDOYHFWRUVLQ V VD\ X D c1 E1 C # # # C cn En Z D d1 E1 C # # # C dn En 7KHQ XVLQJYHFWRURSHUDWLRQV X C Z D .c1 C d1 /E1 C # # # C .cn C dn /En

$)"15&3

9HFWRU6SDFHV

,WIROORZVWKDW

2

3 2 3 2 3 c1 C d1 c1 d1 6 7 6 7 6 : : : 7 :: Œ X C Z !B D 4 5 D 4 :: 5 C 4 :: 5 D Œ X !B C Œ Z !B

cn C dn

cn

dn

6RWKHFRRUGLQDWHPDSSLQJSUHVHUYHVDGGLWLRQ ,I r LVDQ\VFDODU WKHQ

r X D r.c1 E1 C # # # C cn En / D .rc1 /E1 C # # # C .rcn /En 6R

2

3 2 3 rc1 c1 6 :: 7 6 :: 7 Œ r X !B D 4 : 5 D r 4 : 5 D rŒ X !B

rcn

cn

7KXVWKHFRRUGLQDWHPDSSLQJDOVRSUHVHUYHVVFDODUPXOWLSOLFDWLRQDQGKHQFHLVDOLQHDU WUDQVIRUPDWLRQ 6HH([HUFLVHV DQGIRUYHULÀFDWLRQWKDWWKHFRRUGLQDWHPDSSLQJLV RQHWRRQHDQGPDSV V RQWR Rn 7KHOLQHDULW\RIWKHFRRUGLQDWHPDSSLQJH[WHQGVWROLQHDUFRPELQDWLRQV MXVWDVLQ 6HFWLRQ ,I X1 ; : : : ; Xp DUHLQ V DQGLI c1 ; : : : ; cp DUHVFDODUV WKHQ

Œ c1 X1 C # # # C cp Xp !B D c1 Œ X1 !B C # # # C cp Œ Xp !B

SG

*TPNPSQIJD 7FDUPS 4QBDFT m

,QZRUGV VD\VWKDWWKH B FRRUGLQDWHYHFWRURIDOLQHDUFRPELQDWLRQRI X1 ; : : : ; Xp LV WKH VDPH OLQHDUFRPELQDWLRQRIWKHLUFRRUGLQDWHYHFWRUV 7KHFRRUGLQDWHPDSSLQJLQ7KHRUHP LVDQLPSRUWDQWH[DPSOHRIDQ LVRPRUSKLVP IURP V RQWR Rn ,QJHQHUDO DRQHWRRQHOLQHDUWUDQVIRUPDWLRQIURPDYHFWRUVSDFH V RQWRDYHFWRUVSDFH W LVFDOOHGDQ LVRPRUSKLVP IURP V RQWR W LVR IURPWKH*UHHN IRU´WKHVDPHµ DQG PRUSK IURPWKH*UHHNIRUÍRUPµRU´VWUXFWXUHµ 7KHQRWDWLRQDQG WHUPLQRORJ\IRU V DQG W PD\GLIIHU EXWWKHWZRVSDFHVDUHLQGLVWLQJXLVKDEOHDVYHFWRU VSDFHV (YHU\YHFWRUVSDFHFDOFXODWLRQLQ V LVDFFXUDWHO\UHSURGXFHGLQ W DQGYLFH YHUVD ,QSDUWLFXODU DQ\UHDOYHFWRUVSDFHZLWKDEDVLVRI n YHFWRUVLVLQGLVWLQJXLVKDEOH IURP Rn 6HH ([HUFLVHV DQG

&9".1-& /HW B EHWKHVWDQGDUGEDVLVRIWKHVSDFH P3 RISRO\QRPLDOV WKDWLV OHW B D f1; t; t 2 ; t 3 g $ W\SLFDOHOHPHQW S RI P3 KDVWKHIRUP

S.t/ D a0 C a1 t C a2 t 2 C a3 t 3 6LQFH S LVDOUHDG\GLVSOD\HGDVDOLQHDUFRPELQDWLRQRIWKHVWDQGDUGEDVLVYHFWRUV ZH FRQFOXGHWKDW 2 3 a0 6 a1 7 7 Œ S !B D 6 4 a2 5 a3 7KXVWKHFRRUGLQDWHPDSSLQJ S 7! Œ S !B LVDQLVRPRUSKLVPIURP P3 RQWR R4 $OOYHFWRU VSDFHRSHUDWLRQVLQ P3 FRUUHVSRQGWRRSHUDWLRQVLQ R4

,IZHWKLQNRI P3 DQG R4 DVGLVSOD\VRQWZRFRPSXWHUVFUHHQVWKDWDUHFRQQHFWHG YLDWKHFRRUGLQDWHPDSSLQJ WKHQHYHU\YHFWRUVSDFHRSHUDWLRQLQ P3 RQRQHVFUHHQLV H[DFWO\GXSOLFDWHGE\DFRUUHVSRQGLQJYHFWRURSHUDWLRQLQ R4 RQWKHRWKHUVFUHHQ 7KH YHFWRUVRQWKH P3 VFUHHQORRNGLIIHUHQWIURPWKRVHRQWKH R4 VFUHHQ EXWWKH\´DFWµDV YHFWRUVLQH[DFWO\WKHVDPHZD\ 6HH)LJ

&RRUGLQDWH6\VWHPV

'*(63& 7KHVSDFH P3 LVLVRPRUSKLFWR R4

&9".1-& 8VH FRRUGLQDWH YHFWRUV WR YHULI\ WKDW WKH SRO\QRPLDOV 1 C 2t 2 4 C t C 5t 2 DQG 3 C 2t DUHOLQHDUO\GHSHQGHQWLQ P2

40-65*0/ 7KHFRRUGLQDWHPDSSLQJIURP([DPSOH SURGXFHVWKHFRRUGLQDWHYHFWRUV .1; 0; 2/ .4; 1; 5/ DQG .3; 2; 0/ UHVSHFWLYHO\ :ULWLQJWKHVHYHFWRUVDVWKH FROXPQV RID PDWUL[ A ZHFDQGHWHUPLQHWKHLULQGHSHQGHQFHE\URZUHGXFLQJWKHDXJPHQWHGPDWUL[ IRU A[ D 2 3 2 3 1 4 3 0 1 4 3 0 40 1 2 05 & 40 1 2 05 2 5 0 0 0 0 0 0

7KHFROXPQVRI A DUHOLQHDUO\GHSHQGHQW VRWKHFRUUHVSRQGLQJSRO\QRPLDOVDUHOLQHDUO\ GHSHQGHQW ,QIDFW LWLVHDV\WRFKHFNWKDWFROXPQ RI A LVWLPHVFROXPQ PLQXV WLPHVFROXPQ 7KHFRUUHVSRQGLQJUHODWLRQIRUWKHSRO\QRPLDOVLV

3 C 2t D 2.4 C t C 5t 2 / ! 5.1 C 2t 2 / 7KHÀQDOH[DPSOHFRQFHUQVDSODQHLQ R3 WKDWLVLVRPRUSKLFWR R2

&9".1-& /HW

2 3 3 Y 1 D 4 6 5; 2

2

3 !1 Y 2 D 4 0 5; 1

2

3 3 [ D 4 12 5; 7

DQG B D fY1 ; Y2 g 7KHQ B LVDEDVLVIRU H D 6SDQ fY1 ; Y2 g 'HWHUPLQHLI [ LVLQ H DQG LILWLV ÀQGWKHFRRUGLQDWHYHFWRURI [ UHODWLYHWR B

40-65*0/ ,I [ LVLQ H WKHQWKHIROORZLQJYHFWRUHTXDWLRQLVFRQVLVWHQW 2 3 2 3 2 3 3 !1 3 c1 4 6 5 C c2 4 0 5 D 4 12 5 2 1 7

7KHVFDODUV c1 DQG c2 LIWKH\H[LVW DUHWKH B FRRUGLQDWHVRI [ 8VLQJURZRSHUDWLRQV ZHREWDLQ 2 3 2 3 3 !1 3 1 0 2 46 0 12 5 & 4 0 1 3 5 2 1 7 0 0 0

$)"15&3

9HFWRU6SDFHV

7KXV c1 D 2 c2 D 3 DQG Œ [ !B D LVVKRZQLQ)LJ

" # 2 7KHFRRUGLQDWHV\VWHPRQ H GHWHUPLQHGE\ B 3

3v2 x = 2v1 + 3v2

2v2 v2 0

v1

2v1

'*(63& $ FRRUGLQDWHV\VWHPRQDSODQH H LQ

R3

,IDGLIIHUHQWEDVLVIRU H ZHUHFKRVHQ ZRXOGWKHDVVRFLDWHGFRRUGLQDWHV\VWHPDOVR PDNH H LVRPRUSKLFWR R2 " 6XUHO\ WKLVPXVWEHWUXH :HVKDOOSURYHLWLQWKHQH[W VHFWLRQ

13"$5*$& 130#-&.4 2 3 2 3 2 3 2 3 1 !3 3 !8 /HW E1 D 4 0 5 E2 D 4 4 5 E3 D 4 !6 5 DQG [ D 4 2 5 0 0 3 3 3 D 6KRZWKDWWKHVHW B D fE1 ; E2 ; E3 g LVDEDVLVRI R E )LQGWKHFKDQJHRIFRRUGLQDWHVPDWUL[IURP B WRWKHVWDQGDUGEDVLV F :ULWHWKHHTXDWLRQWKDWUHODWHV [ LQ R3 WR Œ [ !B G )LQG Œ [ !B IRUWKH [ JLYHQDERYH

7KHVHW B D f1 C t; 1 C t 2 ; t C t 2 g LVDEDVLVIRU P2 )LQGWKHFRRUGLQDWHYHFWRURI S.t/ D 6 C 3t ! t 2 UHODWLYHWR B

&9&3$*4&4 ,Q ([HUFLVHV ² ÀQG WKH YHFWRU [ GHWHUPLQHG E\ WKH JLYHQ FRRUGLQDWHYHFWRU Œ [ !B DQGWKHJLYHQEDVLV B !" # " #$ " # 3 !4 5 B D ; Œ [ !B D !5 6 3 !" # " #$ " # 3 !4 !2 B D ; Œ [ !B D 2 1 5 82 3 2 3 2 39 2 3 1 5 4 = 1 < B D 4 !2 5; 4 0 5; 4 !3 5 Œ [ !B D 4 0 5 ; : 3 !2 0 !2 82 3 2 3 2 39 2 3 3 4 = !3 < !2 B D 4 2 5; 4 0 5; 4 !1 5 Œ [ !B D 4 2 5 ; : 0 2 3 !1

,Q([HUFLVHV² ÀQGWKHFRRUGLQDWHYHFWRU Œ [ !B RI [ UHODWLYHWR WKHJLYHQEDVLV B D fE1 ; : : : ; En g " # " # " # 1 3 !1 E1 D E2 D [D !2 !5 1 " # " # " # 1 2 !1 E1 D E2 D [D !4 !3 !6 2 3 2 3 2 3 2 3 1 !3 2 8 E1 D 4 !1 5 E2 D 4 4 5 E3 D 4 !2 5 [ D 4 !9 5 !3 9 4 6 2 3 2 3 2 3 2 3 1 2 1 0 E1 D 4 1 5 E2 D 4 0 5 E3 D 4 !1 5 [ D 4 0 5 3 8 3 !2

,Q([HUFLVHVDQG ÀQGWKHFKDQJHRIFRRUGLQDWHVPDWUL[IURP B WRWKHVWDQGDUGEDVLVLQ Rn !" # " #$ 1 2 B D !3 !5 82 3 2 3 2 39 2 1 = < 3 B D 4 0 5 4 2 5 4 !2 5 ; : 6 !4 3

,Q([HUFLVHVDQG XVHDQLQYHUVHPDWUL[WRÀQG Œ [ !B IRUWKH JLYHQ [ DQG B !" # " #$ " # 1 !3 2 B D ; ;[ D !2 5 !5 !" # " #$ " # 1 2 2 B D ; ;[ D !1 !1 3

7KHVHW B D f1 C t 2 ; t C t 2 ; 1 C 2t C t 2 g LVDEDVLVIRU P2 )LQGWKHFRRUGLQDWHYHFWRURI S.t / D 1 C 4t C 7t 2 UHODWLYH WR B 7KH VHW B D f1 ! t 2 ; t ! t 2 ; 2 ! t C t 2 g LV D EDVLV IRU P2 )LQGWKHFRRUGLQDWHYHFWRURI S.t/ D 1 C 3t ! 6t 2 UHODWLYH WR B ,Q([HUFLVHVDQG PDUNHDFKVWDWHPHQW7UXHRU)DOVH -XVWLI\ HDFKDQVZHU 8QOHVVVWDWHGRWKHUZLVH B LVDEDVLVIRUDYHFWRU VSDFH V D ,I [ LVLQ V DQGLI B FRQWDLQV n YHFWRUV WKHQWKH B FRRUGLQDWHYHFWRURI [ LVLQ Rn E ,I PB LVWKHFKDQJHRIFRRUGLQDWHVPDWUL[ WKHQ Œ[!B D PB [ IRU [ LQ V F 7KHYHFWRUVSDFHV P3 DQG R3 DUHLVRPRUSKLF

D ,I B LVWKHVWDQGDUGEDVLVIRU Rn WKHQWKH BFRRUGLQDWH YHFWRURIDQ [ LQ Rn LV [ LWVHOI E 7KHFRUUHVSRQGHQFH Œ [ !B 7! [ LVFDOOHGWKHFRRUGLQDWH PDSSLQJ F ,QVRPHFDVHV DSODQHLQ R3 FDQEHLVRPRUSKLFWR R2 " # " # " # 1 2 !3 Y2 D Y3 D VSDQ R2 7KHYHFWRUV Y1 D !3 !8 7 EXWGRQRW IRUPDEDVLV )LQGWZRGLIIHUHQWZD\VWRH[SUHVV " # 1 DVDOLQHDUFRPELQDWLRQRI Y1 Y2 Y3 1

/HW B D fE1 ; : : : ; En g EHDEDVLVIRUDYHFWRUVSDFH V ([SODLQ ZK\WKH BFRRUGLQDWHYHFWRUVRI E1 ; : : : ; En DUHWKHFROXPQV H1 ; : : : ; Hn RIWKH n % n LGHQWLW\PDWUL[

/HW S EHDÀQLWHVHWLQDYHFWRUVSDFH V ZLWKWKHSURSHUW\ WKDWHYHU\ [ LQ V KDVDXQLTXHUHSUHVHQWDWLRQDVDOLQHDU FRPELQDWLRQRIHOHPHQWVRI S 6KRZWKDW S LVDEDVLVRI V 6XSSRVH fY1 ; : : : ; Y4 g LVDOLQHDUO\GHSHQGHQWVSDQQLQJVHW IRU D YHFWRU VSDFH V 6KRZ WKDW HDFK Z LQ V FDQ EH H[SUHVVHGLQPRUHWKDQRQHZD\DVDOLQHDUFRPELQDWLRQRI Y1 ; : : : ; Y4 >+LQW /HW Z D k1 Y1 C # # # C k4 Y4 EHDQDUELWUDU\ YHFWRULQ V 8VHWKHOLQHDUGHSHQGHQFHRI fY1 ; : : : ; Y4 g WR

&RRUGLQDWH6\VWHPV

SURGXFHDQRWKHUUHSUHVHQWDWLRQRI Z DVDOLQHDUFRPELQDWLRQ RI Y1 ; : : : ; Y4 @ !" # " #$ 1 !2 /HW B D ; 6LQFHWKHFRRUGLQDWHPDSSLQJ !4 9

GHWHUPLQHGE\ B LVDOLQHDUWUDQVIRUPDWLRQIURP R2 LQWR R2 WKLVPDSSLQJPXVWEHLPSOHPHQWHGE\VRPH 2 % 2 PDWUL[ A )LQGLW >+LQW 0XOWLSOLFDWLRQE\ A VKRXOGWUDQVIRUPDYHFWRU [ LQWRLWVFRRUGLQDWHYHFWRU Œ [ !B @

/HW B D fE1 ; : : : ; En g EHDEDVLVIRU Rn 3URGXFHDGHVFULS WLRQRIDQ n % n PDWUL[ A WKDWLPSOHPHQWVWKHFRRUGLQDWH PDSSLQJ [ 7! Œ [ !B 6HH([HUFLVH ([HUFLVHV ² FRQFHUQ D YHFWRU VSDFH V D EDVLV B D fE1 ; : : : ; En g DQGWKHFRRUGLQDWHPDSSLQJ [ 7! Œ [ !B 6KRZ WKDW WKH FRRUGLQDWH PDSSLQJ LV RQHWRRQH +LQW 6XSSRVH Œ X !B D Œ Z !B IRUVRPH X DQG Z LQ V DQGVKRZ WKDW X D Z 6KRZWKDWWKHFRRUGLQDWHPDSSLQJLV RQWR Rn 7KDWLV JLYHQ DQ\ \ LQ Rn ZLWKHQWULHV y1 ; : : : ; yn SURGXFH X LQ V VXFKWKDW Œ X !B D \ 6KRZ WKDW D VXEVHW fX1 ; : : : ; Xp g LQ V LV OLQHDUO\ LQ GHSHQGHQW LI DQG RQO\ LI WKH VHW RI FRRUGLQDWH YHFWRUV fŒ X1 !B ; : : : ; Œ Xp !B g LVOLQHDUO\LQGHSHQGHQWLQ Rn +LQW 6LQFHWKHFRRUGLQDWHPDSSLQJLVRQHWRRQH WKHIROORZLQJ HTXDWLRQVKDYHWKHVDPHVROXWLRQV c1 ; : : : ; cp

c1 X1 C # # # C cp Xp D Œ c1 X1 C # # # C cp Xp !B D Œ !B

7KH]HURYHFWRULQ V 7KH]HURYHFWRULQ Rn

*LYHQYHFWRUV X1 ; : : : ; Xp DQG Z LQ V VKRZWKDW Z LVDOLQHDU FRPELQDWLRQRI X1 ; : : : ; Xp LIDQGRQO\LI Œ Z !B LVDOLQHDU FRPELQDWLRQRIWKHFRRUGLQDWHYHFWRUV Œ X1 !B ; : : : ; Œ Xp !B ,Q ([HUFLVHV ² XVH FRRUGLQDWH YHFWRUV WR WHVW WKH OLQHDU LQGHSHQGHQFHRIWKHVHWVRISRO\QRPLDOV ([SODLQ\RXUZRUN 1 C 2t 3 2 C t ! 3t 2 !t C 2t 2 ! t 3 1 ! 2t 2 ! t 3 t C 2t 3 1 C t ! 2t 2 .1 ! t/2 t ! 2t 2 C t 3 .1 ! t/3 .2 ! t/3 .3 ! t /2 1 C 6t ! 5t 2 C t 3 8VHFRRUGLQDWHYHFWRUVWRWHVWZKHWKHUWKHIROORZLQJVHWVRI SRO\QRPLDOVVSDQ P2 -XVWLI\\RXUFRQFOXVLRQV D 1 ! 3t C 5t 2 !3 C 5t ! 7t 2 !4 C 5t ! 6t 2 1 ! t 2 E 5t C t 2 1 ! 8t ! 2t 2 !3 C 4t C 2t 2 2 ! 3t

/HW S1 .t/ D 1 C t 2 S2 .t/ D t ! 3t 2 S3 .t/ D 1 C t ! 3t 2 D 8VHFRRUGLQDWHYHFWRUVWRVKRZWKDWWKHVHSRO\QRPLDOV IRUPDEDVLVIRU P2 E &RQVLGHUWKHEDVLV2B D f3S1 ; S2 ; S3 g IRU P2 )LQG T LQ P2 !1 JLYHQWKDW ŒT!B D 4 1 5 2

$)"15&3

9HFWRU6SDFHV

,Q([HUFLVHVDQG GHWHUPLQHZKHWKHUWKHVHWVRISRO\QRPLDOV IRUPDEDVLVIRU P3 -XVWLI\\RXUFRQFOXVLRQV >0@ 3 C 7t; 5 C t ! 2t 3 ; t ! 2t 2 ; 1 C 16t ! 6t 2 C 2t 3

VRPHDGGLWLRQDODWRPVPD\EHLQWKHXQLWFHOODWWKH RFWDKHGUDO DQG WHWUDKHGUDO VLWHVVRQDPHGEHFDXVHRIWKHJHRPHWULFREMHFWV IRUPHGE\DWRPVDWWKHVHORFDWLRQV

>0@ 5 ! 3t C 4t 2 C 2t 3 ; 9 C t C 8t 2 ! 6t 3 ; 6 ! 2t C 5t 2 ; t 3

w

>0@ /HW H D 6SDQ fY1 ; Y2 g DQG B D fY1 ; Y2 g 6KRZWKDW [ LV LQ H DQGÀQGWKH BFRRUGLQDWHYHFWRURI [ IRU 2 3 2 3 2 3 11 14 19 6 !5 7 6 !8 7 6 !13 7 7 6 7 6 7 Y1 D 6 4 10 5; Y2 D 4 13 5; [ D 4 18 5 7 10 15

>0@ /HW H D 6SDQ fY1 ; Y2 ; Y3 g DQG B D fY1 ; Y2 ; Y3 g 6KRZ WKDW B LVDEDVLVIRU H DQG [ LVLQ H DQGÀQGWKH BFRRUGLQDWH YHFWRURI [ IRU 2 3 2 3 2 3 2 3 !6 8 !9 4 6 47 6 !3 7 6 57 6 77 7 6 7 6 7 6 7 Y1 D 6 4 !9 5; Y2 D 4 7 5; Y3 D 4 !8 5; [ D 4 !8 5 4 !3 3 3

>0@ ([HUFLVHVDQGFRQFHUQWKHFU\VWDOODWWLFHIRUWLWDQLXP ZKLFKKDVWKHKH[DJRQDOVWUXFWXUHVKRZQRQWKHOHIWLQWKHDF 2 3 2 3 2 3 2:6 0 0 FRPSDQ\LQJÀJXUH 7KHYHFWRUV 4 !1:5 5 4 3 5 4 0 5 LQ R3 0 0 4:8 IRUPDEDVLVIRUWKHXQLWFHOOVKRZQRQWKHULJKW 7KHQXPEHUV KHUHDUH cQJVWURP XQLWV c D 10"8 FP ,QDOOR\VRIWLWDQLXP

0

v

u

7KHKH[DJRQDOFORVHSDFNHGODWWLFHDQGLWVXQLWFHOO 2 3 1=2 2QHRIWKHRFWDKHGUDOVLWHVLV 4 1=4 5 UHODWLYHWRWKHODWWLFH 1=6 EDVLV 'HWHUPLQHWKHFRRUGLQDWHVRIWKLVVLWHUHODWLYHWRWKH VWDQGDUGEDVLVRI R3 2 3 1=2 2QHRIWKHWHWUDKHGUDOVLWHVLV 4 1=2 5 'HWHUPLQHWKHFRRU 1=3 GLQDWHVRIWKLVVLWHUHODWLYHWRWKHVWDQGDUGEDVLVRI R3

40-65*0/4 50 13"$5*$& 130#-&.4 D ,WLVHYLGHQWWKDWWKHPDWUL[ PB D Œ E1 E2 E3 ! LVURZHTXLYDOHQWWRWKHLGHQWLW\ PDWUL[ %\WKH,QYHUWLEOH0DWUL[7KHRUHP PB LVLQYHUWLEOHDQGLWVFROXPQVIRUP DEDVLVIRU R3 2 3 1 !3 3 E )URPSDUWD WKHFKDQJHRIFRRUGLQDWHVPDWUL[LV PB D 4 0 4 !6 5 0 0 3

F [ D PB Œ [ !B G 7RVROYHWKHHTXDWLRQLQF LWLVSUREDEO\HDVLHUWRURZUHGXFHDQDXJPHQWHG PDWUL[WKDQWRFRPSXWH PB!1 2 3 2 3 1 !3 3 !8 1 0 0 !5 40 4 !6 2 5 & 4 0 1 0 2 5 0 0 3 3 0 0 1 1 PB

+HQFH

[

I

Œ [ !B

2

3 !5 Œ [B ! D 4 2 5 1

7KHFRRUGLQDWHVRI S.t/ D 6 C 3t ! t 2 ZLWKUHVSHFWWR B VDWLVI\

c1 .1 C t / C c2 .1 C t 2 / C c3 .t C t 2 / D 6 C 3t ! t 2

7KH'LPHQVLRQRID9HFWRU6SDFH

(TXDWLQJFRHIÀFLHQWVRIOLNHSRZHUVRI t ZHKDYH

c1 C c2 D 6 C c3 D 3 c1 c2 C c3 D !1

2

3 5 6ROYLQJ ZHÀQGWKDW c1 D 5 c2 D 1 c3 D !2 DQG Œ S !B D 4 1 5 !2

5)& %*.&/4*0/ 0' " 7&$503 41"$& 7KHRUHPLQ6HFWLRQLPSOLHVWKDWDYHFWRUVSDFH V ZLWKDEDVLV B FRQWDLQLQJ n YHFWRUVLVLVRPRUSKLFWR Rn 7KLVVHFWLRQVKRZVWKDWWKLVQXPEHU n LVDQLQWULQVLF SURSHUW\FDOOHGWKHGLPHQVLRQ RIWKHVSDFH V WKDWGRHVQRWGHSHQGRQWKHSDUWLFXODU FKRLFHRIEDVLV 7KHGLVFXVVLRQRIGLPHQVLRQZLOOJLYHDGGLWLRQDOLQVLJKWLQWRSURSHUWLHV RIEDVHV 7KHÀUVWWKHRUHPJHQHUDOL]HVDZHOONQRZQUHVXOWDERXWWKHYHFWRUVSDFH Rn

5)&03&.

,IDYHFWRUVSDFH V KDVDEDVLV B D fE1 ; : : : ; En g WKHQDQ\VHWLQ V FRQWDLQLQJ PRUHWKDQ n YHFWRUVPXVWEHOLQHDUO\GHSHQGHQW

1300' /HW fX1 ; : : : ; Xp g EHDVHWLQ V ZLWKPRUHWKDQ n YHFWRUV 7KHFRRUGLQDWHYHFWRUV Œ X1 !B ; : : : ; Œ Xp !B IRUPDOLQHDUO\GHSHQGHQWVHWLQ Rn EHFDXVHWKHUHDUHPRUHYHFWRUV p WKDQHQWULHVn LQHDFKYHFWRU 6RWKHUHH[LVWVFDODUV c1 ; : : : ; cp QRWDOO]HUR VXFK WKDW 2 3 0 6 :: 7 c1 Œ X1 !B C # # # C cp Œ Xp !B D 4 : 5 7KH]HURYHFWRULQ Rn 0

6LQFHWKHFRRUGLQDWHPDSSLQJLVDOLQHDUWUDQVIRUPDWLRQ 2 3 0 % & 6 :: 7 c1 X1 C # # # C cp Xp B D 4 : 5 0

7KH ]HUR YHFWRU RQ WKH ULJKW GLVSOD\V WKH n ZHLJKWV QHHGHG WR EXLOG WKH YHFWRU c1 X1 C # # # C cp Xp IURP WKH EDVLV YHFWRUV LQ B 7KDW LV c1 X1 C # # # C cp Xp D 0 # E1 C # # # C 0 # En D 6LQFH WKH ci DUH QRW DOO ]HUR fX1 ; : : : ; Xp g LV OLQHDUO\ GHSHQGHQWñ 7KHRUHPLPSOLHVWKDWLIDYHFWRUVSDFH V KDVDEDVLV B D fE1 ; : : : ; En g WKHQHDFK OLQHDUO\LQGHSHQGHQWVHWLQ V KDVQRPRUHWKDQ n YHFWRUV ñ 7KHRUHPDOVRDSSOLHVWRLQÀQLWHVHWVLQ V $QLQÀQLWHVHWLVVDLGWREHOLQHDUO\GHSHQGHQWLIVRPHÀQLWH VXEVHWLVOLQHDUO\GHSHQGHQW RWKHUZLVH WKHVHWLVOLQHDUO\LQGHSHQGHQW ,I S LVDQLQÀQLWHVHWLQ V WDNHDQ\ VXEVHW fX1 ; : : : ; Xp g RI S ZLWK p > n 7KHSURRIDERYHVKRZVWKDWWKLVVXEVHWLVOLQHDUO\GHSHQGHQW DQG KHQFHVRLV S

$)"15&3

9HFWRU6SDFHV

5)&03&.

,IDYHFWRUVSDFH V KDVDEDVLVRI n YHFWRUV WKHQHYHU\EDVLVRI V PXVWFRQVLVWRI H[DFWO\ n YHFWRUV

1300' /HW B1 EHDEDVLVRI n YHFWRUVDQG B2 EHDQ\RWKHUEDVLVRI V 6LQFH B1 LV DEDVLVDQG B2 LVOLQHDUO\LQGHSHQGHQW B2 KDVQRPRUHWKDQ n YHFWRUV E\7KHRUHP $OVR VLQFH B2 LVDEDVLVDQG B1 LVOLQHDUO\LQGHSHQGHQW B2 KDVDWOHDVW n YHFWRUV 7KXV B2 FRQVLVWVRIH[DFWO\ n YHFWRUV ,IDQRQ]HURYHFWRUVSDFH V LVVSDQQHGE\DÀQLWHVHW S WKHQDVXEVHWRI S LVD EDVLVIRU V E\WKH6SDQQLQJ6HW7KHRUHP ,QWKLVFDVH 7KHRUHP HQVXUHVWKDWWKH IROORZLQJGHÀQLWLRQPDNHVVHQVH

%&'*/*5*0/

,I V LVVSDQQHGE\DÀQLWHVHW WKHQ V LVVDLGWREH ÀQLWHGLPHQVLRQDO DQGWKH GLPHQVLRQ RI V ZULWWHQDV GLP V LVWKHQXPEHURIYHFWRUVLQDEDVLVIRU V 7KH GLPHQVLRQRIWKH]HURYHFWRUVSDFH fg LVGHÀQHGWREH]HUR ,I V LVQRWVSDQQHG E\DÀQLWHVHW WKHQ V LVVDLGWREH LQÀQLWHGLPHQVLRQDO

&9".1-& 7KHVWDQGDUGEDVLVIRU Rn FRQWDLQV n YHFWRUV VR GLP Rn D n 7KH

VWDQGDUGSRO\QRPLDOEDVLV f1; t; t 2 g VKRZVWKDW GLP P2 D 3 ,QJHQHUDO GLP Pn D n C 1 7KHVSDFH P RIDOOSRO\QRPLDOVLVLQÀQLWHGLPHQVLRQDO([HUFLVH 2 3 2 3 3 !1 &9".1-& /HW H D 6SDQ fY1 ; Y2 g ZKHUH Y1 D 4 6 5 DQG Y2 D 4 0 5 7KHQ 2 1 H LVWKHSODQHVWXGLHGLQ([DPSOH LQ6HFWLRQ $ EDVLVIRU H LV fY1 ; Y2 g VLQFH Y1 DQG Y2 DUHQRWPXOWLSOHVDQGKHQFHDUHOLQHDUO\LQGHSHQGHQW 7KXV GLP H D 2

3v2 2v2 v2 0

v1

2v1

&9".1-& )LQGWKHGLPHQVLRQRIWKHVXEVSDFH

82 9 3 a ! 3b C 6c ˆ > ˆ > <6 = 7 5a C 4d 6 7 H D 4 W a b c d LQ R b ! 2c ! d 5 ˆ > ˆ > : ; 5d

40-65*0/ ,WLVHDV\WRVHHWKDW H LVWKHVHWRIDOOOLQHDUFRPELQDWLRQVRIWKHYHFWRUV 2 3 2 3 2 3 2 3 1 !3 6 0 657 6 07 6 07 6 47 7 6 7 6 7 6 7 Y1 D 6 4 0 5; Y2 D 4 1 5; Y3 D 4 !2 5; Y4 D 4 !1 5 0 0 0 5

&OHDUO\ Y1 ¤ Y2 LVQRWDPXOWLSOHRI Y1 EXW Y3 LVDPXOWLSOHRI Y2 %\WKH6SDQQLQJ 6HW7KHRUHP ZHPD\GLVFDUG Y3 DQGVWLOOKDYHDVHWWKDWVSDQV H )LQDOO\ Y4 LVQRWD OLQHDUFRPELQDWLRQRI Y1 DQG Y2 6R fY1 ; Y2 ; Y4 g LVOLQHDUO\LQGHSHQGHQWE\7KHRUHP LQ6HFWLRQ DQGKHQFHLVDEDVLVIRU H 7KXV GLP H D 3

&9".1-& 7KHVXEVSDFHVRI R3 FDQEHFODVVLÀHGE\GLPHQVLRQ 6HH)LJ GLPHQVLRQDOVXEVSDFHV 2QO\WKH]HURVXEVSDFH GLPHQVLRQDOVXEVSDFHV $Q\VXEVSDFHVSDQQHGE\DVLQJOHQRQ]HURYHFWRU 6XFK VXEVSDFHVDUHOLQHVWKURXJKWKHRULJLQ


GLPHQVLRQDO VXEVSDFHV $Q\ VXEVSDFH VSDQQHG E\ WZR OLQHDUO\ LQGHSHQGHQW YHFWRUV 6XFKVXEVSDFHVDUHSODQHVWKURXJKWKHRULJLQ GLPHQVLRQDOVXEVSDFHV 2QO\ R3 LWVHOI $Q\WKUHHOLQHDUO\LQGHSHQGHQWYHFWRUV LQ R3 VSDQDOORI R3 E\WKH,QYHUWLEOH0DWUL[7KHRUHP x3

x3 3-dim 2-dim

0-dim 1-dim x2

x1 (a)

x2

x1 (b)

3

'*(63& 6DPSOHVXEVSDFHVRI R

6XEVSDFHVRID)LQLWH'LPHQVLRQDO6SDFH 7KHQH[WWKHRUHPLVDQDWXUDOFRXQWHUSDUWWRWKH6SDQQLQJ6HW7KHRUHP

5)&03&.

/HW H EH D VXEVSDFH RI D ÀQLWHGLPHQVLRQDO YHFWRU VSDFH V $Q\ OLQHDUO\ LQGHSHQGHQWVHWLQ H FDQEHH[SDQGHG LIQHFHVVDU\ WRDEDVLVIRU H $OVR H LV ÀQLWHGLPHQVLRQDODQG GLP H $ GLP V

1300' ,I H D fg WKHQFHUWDLQO\ GLP H D 0 $ GLP V 2WKHUZLVH OHW S D fX1 ; : : : ; Xk g EHDQ\OLQHDUO\LQGHSHQGHQWVHWLQ H ,I S VSDQV H WKHQ S LVDEDVLVIRU H 2WKHUZLVH WKHUHLVVRPH Xk C1 LQ H WKDWLVQRWLQ 6SDQ S %XWWKHQ fX1 ; : : : ; Xk ; Xk C1 g ZLOOEHOLQHDUO\LQGHSHQGHQW EHFDXVHQRYHFWRULQWKHVHWFDQEHDOLQHDUFRPELQDWLRQRI YHFWRUVWKDWSUHFHGHLWE\7KHRUHP 6RORQJDVWKHQHZVHWGRHVQRWVSDQ H ZHFDQFRQWLQXHWKLVSURFHVVRIH[SDQGLQJ S WRDODUJHUOLQHDUO\LQGHSHQGHQWVHWLQ H %XWWKHQXPEHURIYHFWRUVLQDOLQHDUO\ LQGHSHQGHQWH[SDQVLRQRI S FDQQHYHUH[FHHGWKHGLPHQVLRQRI V E\7KHRUHP 6RHYHQWXDOO\WKHH[SDQVLRQRI S ZLOOVSDQ H DQGKHQFHZLOOEHDEDVLVIRU H DQG GLP H $ GLP V :KHQWKHGLPHQVLRQRIDYHFWRUVSDFHRUVXEVSDFHLVNQRZQ WKHVHDUFKIRUDEDVLV LVVLPSOLÀHGE\WKHQH[WWKHRUHP ,WVD\VWKDWLIDVHWKDVWKHULJKWQXPEHURIHOHPHQWV WKHQRQHKDVRQO\WRVKRZHLWKHUWKDWWKHVHWLVOLQHDUO\LQGHSHQGHQWRUWKDWLWVSDQVWKH VSDFH 7KHWKHRUHPLVRIFULWLFDOLPSRUWDQFHLQQXPHURXVDSSOLHGSUREOHPVLQYROYLQJ GLIIHUHQWLDOHTXDWLRQVRUGLIIHUHQFHHTXDWLRQV IRUH[DPSOH ZKHUHOLQHDULQGHSHQGHQFH LVPXFKHDVLHUWRYHULI\WKDQVSDQQLQJ

5)&03&.

5IF #BTJT 5IFPSFN /HW V EHD p GLPHQVLRQDOYHFWRUVSDFH p " 1 $Q\OLQHDUO\LQGHSHQGHQWVHWRI H[DFWO\ p HOHPHQWVLQ V LVDXWRPDWLFDOO\DEDVLVIRU V $Q\VHWRIH[DFWO\ p HOHPHQWVWKDWVSDQV V LVDXWRPDWLFDOO\DEDVLVIRU V

$)"15&3

9HFWRU6SDFHV

1300' %\7KHRUHP DOLQHDUO\LQGHSHQGHQWVHW S RI p HOHPHQWVFDQEHH[WHQGHGWR DEDVLVIRU V %XWWKDWEDVLVPXVWFRQWDLQH[DFWO\ p HOHPHQWV VLQFH GLP V D p 6R S PXVWDOUHDG\EHDEDVLVIRU V 1RZVXSSRVHWKDW S KDV p HOHPHQWVDQGVSDQV V 6LQFH V LVQRQ]HUR WKH6SDQQLQJ6HW7KHRUHPLPSOLHVWKDWDVXEVHW S 0 RI S LVDEDVLVRI V 6LQFH GLP V D p S 0 PXVWFRQWDLQ p YHFWRUV +HQFH S D S 0

7KH'LPHQVLRQVRI 1XO A DQG &RO A 6LQFHWKHSLYRWFROXPQVRIDPDWUL[ A IRUPDEDVLVIRU &RO A ZHNQRZWKHGLPHQVLRQ RI &RO A DVVRRQDVZHNQRZWKHSLYRWFROXPQV 7KHGLPHQVLRQRI 1XO A PLJKWVHHPWR UHTXLUHPRUHZRUN VLQFHÀQGLQJDEDVLVIRU 1XO A XVXDOO\WDNHVPRUHWLPHWKDQDEDVLV IRU &RO A %XWWKHUHLVDVKRUWFXW /HW A EHDQ m % n PDWUL[ DQGVXSSRVHWKHHTXDWLRQ A[ D KDV k IUHHYDULDEOHV )URP6HFWLRQ ZHNQRZWKDWWKHVWDQGDUGPHWKRGRIÀQGLQJDVSDQQLQJVHWIRU 1XO A ZLOOSURGXFHH[DFWO\ k OLQHDUO\LQGHSHQGHQWYHFWRUV³VD\ X1 ; : : : ; Xk ³RQHIRUHDFK IUHHYDULDEOH 6R fX1 ; : : : ; Xk g LVDEDVLVIRU 1XO A DQGWKHQXPEHURIIUHHYDULDEOHV GHWHUPLQHVWKHVL]HRIWKHEDVLV /HWXVVXPPDUL]HWKHVHIDFWVIRUIXWXUHUHIHUHQFH 7KHGLPHQVLRQRI 1XO A LVWKHQXPEHURIIUHHYDULDEOHVLQWKHHTXDWLRQ A[ D DQGWKHGLPHQVLRQRI &RO A LVWKHQXPEHURISLYRWFROXPQVLQ A

&9".1-& )LQGWKHGLPHQVLRQVRIWKHQXOOVSDFHDQGWKHFROXPQVSDFHRI 2

!3 6 !1 2 A D 4 1 !2 2 !4 5

3 1 !7 3 !1 5 8 !4

40-65*0/ 5RZUHGXFHWKHDXJPHQWHGPDWUL[ Œ A ! WRHFKHORQIRUP 2

1 !2 40 0 0 0

2 1 0

3 !1 2 !2 0 0

3 0 05 0

7KHUHDUHWKUHHIUHHYDULDEOHV³x2 x4 DQG x5 +HQFHWKHGLPHQVLRQRI 1XO A LV $OVR GLP &RO A D 2 EHFDXVH A KDVWZRSLYRWFROXPQV

13"$5*$& 130#-&.4 'HFLGHZKHWKHUHDFKVWDWHPHQWLV7UXHRU)DOVH DQGJLYHDUHDVRQIRUHDFKDQVZHU +HUH V LVDQRQ]HURÀQLWHGLPHQVLRQDOYHFWRUVSDFH ,I GLP V D p DQGLI S LVDOLQHDUO\GHSHQGHQWVXEVHWRI V WKHQ S FRQWDLQVPRUH WKDQ p YHFWRUV ,I S VSDQV V DQGLI T LVDVXEVHWRI V WKDWFRQWDLQVPRUHYHFWRUVWKDQ S WKHQ T LV OLQHDUO\GHSHQGHQW


&9&3$*4&4 )RU HDFK VXEVSDFH LQ ([HUFLVHV ² D ÀQG D EDVLV IRU WKH VXEVSDFH DQGE VWDWHWKHGLPHQVLRQ 82 9 82 9 3 3 < s ! 2t = < 2a = 4 s C t 5 W s t LQ R 4 !4b 5 W a b LQ R : ; : ; 3t !2a 82 9 82 9 3 3 2c p C 2q ˆ > ˆ > ˆ > ˆ > <6 = <6 = 7 7 a ! b !p 6 7 W a b c LQ R 7 W p q LQ R 6 4 4 5 5 b ! 3c 3p ! q ˆ > ˆ > ˆ > ˆ > : ; : ; a C 2b pCq 82 9 3 p ! 2q ˆ > ˆ > <6 = 7 2p C 5r 7 W p q r LQ R 6 4 !2q C 2r 5 ˆ > ˆ > : ; !3p C 6r 82 9 3 3a ! c ˆ > ˆ > <6 = 7 !b ! 3c 6 7 4 W a b c LQ R !7a C 6b C 5c 5 ˆ > ˆ > : ; !3a C c f.a; b; c/ W a ! 3b C c D 0; b ! 2c D 0; 2b ! c D 0g f.a; b; c; d / W a ! 3b C c D 0g )LQGWKHGLPHQVLRQRIWKHVXEVSDFHRIDOOYHFWRUVLQ R3 ZKRVH ÀUVWDQGWKLUGHQWULHVDUHHTXDO

H RI R2 VSDQQHGE\ )LQGWKHGLPHQVLRQRIWKHVXEVSDFH " # " # " # 1 !2 !3 !5 10 15 ,Q ([HUFLVHV DQG ÀQG WKH GLPHQVLRQ RI WKH VXEVSDFH VSDQQHGE\WKHJLYHQYHFWRUV 2 3 2 3 2 3 2 3 1 3 !2 5 4 0 5 4 1 5 4 !1 5 4 2 5 2 1 1 2 2

3 2

3 2

3 2

3

1 !3 !2 !3 4 !2 5 4 !6 5 4 3 5 4 5 5 0 0 5 5

'HWHUPLQHWKHGLPHQVLRQVRI 1XO A DQG &RO A IRUWKHPDWULFHV VKRZQLQ([HUFLVHV² 2 3 1 !6 9 0 !2 60 1 2 !4 57 7 A D 6 40 0 0 5 15 0 0 0 0 0 2 3 1 2 !4 3 !2 6 0 60 0 0 1 0 !3 77 7 A D 6 40 0 0 0 1 4 !2 5 0 0 0 0 0 0 1 2 3 " # 1 2 3 0 0 3 2 0 1 0 1 5 A D A D 4 0 !6 5 0 0 0 1 0

2

1 !1 1 A D 4 0 0 0

3 0 35 1

2

1 A D 4 0 0

3 1 !1 2 05 0 0

,Q([HUFLVHVDQG V LVDYHFWRUVSDFH 0DUNHDFKVWDWHPHQW 7UXHRU)DOVH -XVWLI\HDFKDQVZHU D 7KH QXPEHU RI SLYRW FROXPQV RI D PDWUL[ HTXDOV WKH GLPHQVLRQRILWVFROXPQVSDFH E $ SODQHLQ R3 LVDWZRGLPHQVLRQDOVXEVSDFHRI R3 F 7KHGLPHQVLRQRIWKHYHFWRUVSDFH P4 LV G ,I GLP V D n DQG S LVDOLQHDUO\LQGHSHQGHQWVHWLQ V WKHQ S LVDEDVLVIRU V H ,I D VHW fY1 ; : : : ; Yp g VSDQV D ÀQLWHGLPHQVLRQDO YHFWRU VSDFH V DQGLI T LVDVHWRIPRUHWKDQ p YHFWRUVLQ V WKHQ T LVOLQHDUO\GHSHQGHQW D R2 LVDWZRGLPHQVLRQDOVXEVSDFHRI R3 E 7KHQXPEHURIYDULDEOHVLQWKHHTXDWLRQ A[ D HTXDOV WKHGLPHQVLRQRI 1XO A F $ YHFWRUVSDFHLVLQÀQLWHGLPHQVLRQDOLILWLVVSDQQHGE\ DQLQÀQLWHVHW G ,I GLP V D n DQGLI S VSDQV V WKHQ S LVDEDVLVRI V

H 7KHRQO\WKUHHGLPHQVLRQDOVXEVSDFHRI R3 LV R3 LWVHOI 7KHÀUVWIRXU+HUPLWHSRO\QRPLDOVDUH 2t !2 C 4t 2 DQG !12t C 8t 3 7KHVHSRO\QRPLDOVDULVHQDWXUDOO\LQWKHVWXG\ RIFHUWDLQLPSRUWDQWGLIIHUHQWLDOHTXDWLRQVLQPDWKHPDWLFDO SK\VLFV 6KRZWKDWWKHÀUVWIRXU+HUPLWHSRO\QRPLDOVIRUP DEDVLVRI P3 7KHÀUVWIRXU/DJXHUUHSRO\QRPLDOVDUH 1 ! t 2 ! 4t C t 2 DQG 6 ! 18t C 9t 2 ! t 3 6KRZWKDWWKHVHSRO\QRPLDOVIRUPD EDVLVRI P3 /HW B EHWKHEDVLVRI P3 FRQVLVWLQJRIWKH+HUPLWHSRO\QRPL DOVLQ([HUFLVH DQGOHW S.t/ D !1 C 8t 2 C 8t 3 )LQGWKH FRRUGLQDWHYHFWRURI S UHODWLYHWR B /HW B EH WKH EDVLV RI P2 FRQVLVWLQJ RI WKH ÀUVW WKUHH /DJXHUUH SRO\QRPLDOV OLVWHG LQ ([HUFLVH DQG OHW S.t/ D 5 C 5t ! 2t 2 )LQGWKHFRRUGLQDWHYHFWRURI S UHODWLYH WR B /HW S EHDVXEVHWRIDQ nGLPHQVLRQDOYHFWRUVSDFH V DQG VXSSRVH S FRQWDLQVIHZHUWKDQ n YHFWRUV ([SODLQZK\ S FDQQRWVSDQ V /HW H EHDQ nGLPHQVLRQDOVXEVSDFHRIDQ nGLPHQVLRQDO YHFWRUVSDFH V 6KRZWKDW H D V ([SODLQZK\WKHVSDFH P RIDOOSRO\QRPLDOVLVDQLQÀQLWH GLPHQVLRQDOVSDFH 2 6HH ,QWURGXFWLRQWR)XQFWLRQDO$QDO\VLV GHG E\$(7D\ORUDQG 'DYLG&/D\1HZ

$)"15&3

9HFWRU6SDFHV

6KRZWKDWWKHVSDFH C.R/ RIDOOFRQWLQXRXVIXQFWLRQVGHÀQHG RQWKHUHDOOLQHLVDQLQÀQLWHGLPHQVLRQDOVSDFH ,Q([HUFLVHVDQG V LVDQRQ]HURÀQLWHGLPHQVLRQDOYHFWRU VSDFH DQGWKHYHFWRUVOLVWHGEHORQJWR V 0DUNHDFKVWDWHPHQW 7UXHRU)DOVH -XVWLI\HDFKDQVZHU 7KHVHTXHVWLRQVDUHPRUH GLIÀFXOWWKDQWKRVHLQ([HUFLVHV DQG D ,I WKHUH H[LVWV D VHW fY1 ; : : : ; Yp g WKDW VSDQV V WKHQ GLP V $ p E ,IWKHUHH[LVWVDOLQHDUO\LQGHSHQGHQWVHW fY1 ; : : : ; Yp g LQ V WKHQ GLP V " p

F ,I GLP V D p WKHQWKHUHH[LVWVDVSDQQLQJVHWRI p C 1 YHFWRUVLQ V D ,IWKHUHH[LVWVDOLQHDUO\GHSHQGHQWVHW fY1 ; : : : ; Yp g LQ V WKHQ GLP V $ p E ,IHYHU\VHWRI p HOHPHQWVLQ V IDLOVWRVSDQ V WKHQ GLP V > p F ,I p " 2 DQG GLP V D p WKHQHYHU\VHWRI p ! 1 QRQ]HUR YHFWRUVLVOLQHDUO\LQGHSHQGHQW

>0@ $FFRUGLQJWR7KHRUHP DOLQHDUO\LQGHSHQGHQWVHW fY1 ; : : : ; Yk g LQ Rn FDQEHH[SDQGHGWRDEDVLVIRU Rn 2QH ZD\WRGRWKLVLVWRFUHDWH A D Œ Y1 # # # Yk H1 # # # Hn ! ZLWK H1 ; : : : ; Hn WKHFROXPQVRIWKHLGHQWLW\PDWUL[ WKHSLYRW FROXPQVRI A IRUPDEDVLVIRU Rn D 8VHWKHPHWKRGGHVFULEHGWRH[WHQGWKHIROORZLQJYHFWRUV WRDEDVLVIRU R5 2 3 2 3 2 3 !9 9 6 6 !7 7 6 47 6 77 6 7 6 7 6 7 7; Y2 D 6 1 7; Y3 D 6 !8 7 8 Y1 D 6 6 7 6 7 6 7 4 !5 5 4 65 4 55 7 !7 !7 E ([SODLQZK\WKHPHWKRGZRUNVLQJHQHUDO :K\DUHWKH RULJLQDOYHFWRUV Y1 ; : : : ; Yk LQFOXGHGLQWKHEDVLVIRXQGIRU &RO A" :K\LV &RO A D Rn "

>0@ /HW B D f1; FRV t; FRV2 t; : : : ; FRV6 t g DQG C D f1; FRV t; FRV 2t; : : : ; FRV 6t g $VVXPH WKH IROORZLQJ WULJRQRPHWULF LGHQWLWLHVVHH([HUFLVH LQ6HFWLRQ

([HUFLVHVDQGFRQFHUQÀQLWHGLPHQVLRQDOYHFWRUVSDFHV V DQG W DQGDOLQHDUWUDQVIRUPDWLRQ T W V ! W /HW H EHDQRQ]HURVXEVSDFHRI V DQGOHW T .H / EHWKHVHWRI LPDJHVRIYHFWRUVLQ H 7KHQ T .H / LVDVXEVSDFHRI W E\ ([HUFLVH LQ6HFWLRQ 3URYHWKDW GLP T .H / $ GLP H /HW H EH D QRQ]HUR VXEVSDFH RI V DQG VXSSRVH T LV D RQHWRRQH OLQHDU PDSSLQJ RI V LQWR W 3URYH WKDW GLP T .H / D GLP H ,I T KDSSHQVWREHDRQHWRRQHPDS SLQJRI V RQWR W WKHQ GLP V D GLP W ,VRPRUSKLFÀQLWH GLPHQVLRQDOYHFWRUVSDFHVKDYHWKHVDPHGLPHQVLRQ

FRV 2t FRV 3t FRV 4t FRV 5t FRV 6t

D !1 C 2 FRV2 t D !3 FRV t C 4 FRV3 t D 1 ! 8 FRV2 t C 8 FRV4 t D 5 FRV t ! 20 FRV3 t C 16 FRV5 t D !1 C 18 FRV2 t ! 48 FRV4 t C 32 FRV6 t

/HW H EHWKHVXEVSDFHRIIXQFWLRQVVSDQQHGE\WKHIXQFWLRQV LQ B 7KHQ B LVDEDVLVIRU H E\([HUFLVH LQ6HFWLRQ D :ULWHWKH BFRRUGLQDWHYHFWRUVRIWKHYHFWRUVLQ C DQG XVHWKHPWRVKRZWKDW C LVDOLQHDUO\LQGHSHQGHQWVHWLQ H E ([SODLQZK\ C LVDEDVLVIRU H

40-65*0/4 50 13"$5*$& 130#-&.4 )DOVH &RQVLGHUWKHVHW fg

7UXH %\WKH6SDQQLQJ6HW7KHRUHP S FRQWDLQVDEDVLVIRU V FDOOWKDWEDVLV S 0 7KHQ T ZLOOFRQWDLQPRUHYHFWRUVWKDQ S 0 %\7KHRUHP T LVOLQHDUO\GHSHQGHQW

3"/, :LWKWKHDLGRIYHFWRUVSDFHFRQFHSWV WKLVVHFWLRQWDNHVDORRN LQVLGH DPDWUL[DQG UHYHDOVVHYHUDOLQWHUHVWLQJDQGXVHIXOUHODWLRQVKLSVKLGGHQLQLWVURZVDQGFROXPQV )RULQVWDQFH LPDJLQHSODFLQJUDQGRPQXPEHUVLQWRD 40 % 50 PDWUL[ A DQG WKHQGHWHUPLQLQJERWKWKHPD[LPXPQXPEHURIOLQHDUO\LQGHSHQGHQWFROXPQVLQ A DQG WKHPD[LPXPQXPEHURIOLQHDUO\LQGHSHQGHQWFROXPQVLQ AT URZVLQ A 5HPDUNDEO\ WKHWZRQXPEHUVDUHWKHVDPH $VZH·OOVRRQVHH WKHLUFRPPRQYDOXHLVWKH UDQN RIWKH PDWUL[ 7RH[SODLQZK\ ZHQHHGWRH[DPLQHWKHVXEVSDFHVSDQQHGE\WKHURZVRI A

5DQN

7KH5RZ6SDFH ,I A LVDQ m % n PDWUL[ HDFKURZRI A KDV n HQWULHVDQGWKXVFDQEHLGHQWLÀHGZLWKD YHFWRULQ Rn 7KHVHWRIDOOOLQHDUFRPELQDWLRQVRIWKHURZYHFWRUVLVFDOOHGWKH URZ VSDFH RI A DQGLVGHQRWHGE\ 5RZ A (DFKURZKDV n HQWULHV VR 5RZ A LVDVXEVSDFH RI Rn 6LQFHWKHURZVRI A DUHLGHQWLÀHGZLWKWKHFROXPQVRI AT ZHFRXOGDOVRZULWH &RO AT LQSODFHRI 5RZ A

&9".1-& /HW 2

!2 6 1 AD6 4 3 1

!5 8 3 !5 11 !19 7 !13

3 0 !17 1 57 7 7 15 5 !3

DQG

U1 U2 U3 U4

D .!2; !5; 8; 0; !17/ D .1; 3; !5; 1; 5/ D .3; 11; !19; 7; 1/ D .1; 7; !13; 5; !3/

7KHURZVSDFHRI A LVWKHVXEVSDFHRI R5 VSDQQHGE\ fU1 ; U2 ; U3 ; U4 g 7KDWLV 5RZ A D 6SDQ fU1 ; U2 ; U3 ; U4 g ,WLVQDWXUDOWRZULWHURZYHFWRUVKRUL]RQWDOO\ KRZHYHU WKH\PD\ DOVREHZULWWHQDVFROXPQYHFWRUVLIWKDWLVPRUHFRQYHQLHQW ,IZHNQHZVRPHOLQHDUGHSHQGHQFHUHODWLRQVDPRQJWKHURZVRIPDWUL[ A LQ ([DPSOH ZHFRXOGXVHWKH6SDQQLQJ6HW7KHRUHPWRVKULQNWKHVSDQQLQJVHWWRD EDVLV 8QIRUWXQDWHO\ URZRSHUDWLRQVRQ A ZLOOQRWJLYHXVWKDWLQIRUPDWLRQ EHFDXVH URZRSHUDWLRQVFKDQJHWKHURZGHSHQGHQFHUHODWLRQV %XWURZUHGXFLQJ A LVFHUWDLQO\ ZRUWKZKLOH DVWKHQH[WWKHRUHPVKRZV

5)&03&.

,IWZRPDWULFHV A DQG B DUHURZHTXLYDOHQW WKHQWKHLUURZVSDFHVDUHWKHVDPH ,I B LVLQHFKHORQIRUP WKHQRQ]HURURZVRI B IRUPDEDVLVIRUWKHURZVSDFHRI A DVZHOODVIRUWKDWRI B

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ÀUVWURZODVW 7KXVWKHQRQ]HUR URZVRI B IRUPDEDVLVRIWKHFRPPRQ URZVSDFHRI B DQG A 7KHPDLQUHVXOWRIWKLVVHFWLRQLQYROYHVWKHWKUHHVSDFHV 5RZ A &RO A DQG 1XO A 7KHIROORZLQJH[DPSOHSUHSDUHVWKHZD\IRUWKLVUHVXOWDQGVKRZVKRZ RQH VHTXHQFHRI URZRSHUDWLRQVRQ A OHDGVWREDVHVIRUDOOWKUHHVSDFHV

&9".1-& )LQGEDVHVIRUWKHURZVSDFH WKHFROXPQVSDFH DQGWKHQXOOVSDFHRI WKHPDWUL[

2

!2 6 1 AD6 4 3 1

!5 8 3 !5 11 !19 7 !13

3 0 !17 1 57 7 7 15 5 !3

$)"15&3

9HFWRU6SDFHV

40-65*0/ 7RÀQGEDVHVIRUWKHURZVSDFHDQGWKHFROXPQVSDFH URZUHGXFH A WRDQ HFKHORQIRUP 2 3 1 3 !5 1 5 60 1 !2 2 !7 7 7 A&BD6 40 0 0 !4 20 5 0 0 0 0 0

%\7KHRUHP WKHÀUVWWKUHHURZVRI B IRUPDEDVLVIRUWKHURZVSDFHRI A DVZHOO DVIRUWKHURZVSDFHRI B 7KXV %DVLVIRU 5RZ A f.1; 3; !5; 1; 5/; .0; 1; !2; 2; !7/; .0; 0; 0; !4; 20/g )RUWKHFROXPQVSDFH REVHUYHIURP B WKDWWKHSLYRWVDUHLQFROXPQV DQG +HQFH FROXPQV DQGRI A QRW B IRUPDEDVLVIRU &RO A 82 3 2 3 2 39 !2 !5 0 > ˆ > ˆ <6 6 3 7 6 1 7= 17 6 7 6 7 6 7 %DVLVIRU &RO A 4 ; ; 3 5 4 11 5 4 7 5> ˆ > ˆ ; : 1 7 5

1RWLFHWKDWDQ\HFKHORQIRUPRI A SURYLGHVLQLWVQRQ]HURURZV DEDVLVIRU 5RZ A DQGDOVRLGHQWLÀHVWKHSLYRWFROXPQVRI A IRU &RO A +RZHYHU IRU 1XO A ZHQHHGWKH UHGXFHGHFKHORQIRUP )XUWKHUURZRSHUDWLRQVRQ B \LHOG 2

1 60 A&B&C D6 40 0

0 1 1 !2 0 0 0 0

3 0 1 0 37 7 1 !5 5 0 0

7KHHTXDWLRQ A[ D LVHTXLYDOHQWWR C [ D WKDWLV

x1 C

x3 x2 ! 2x3

C x5 D 0 C 3x5 D 0 x4 ! 5x5 D 0

6R x1 D !x3 ! x5 ; x2 D 2x3 ! 3x5 ; x4 D 5x5 ZLWK x3 DQG x5 IUHHYDULDEOHV 7KHXVXDO FDOFXODWLRQVGLVFXVVHGLQ6HFWLRQ VKRZWKDW 82 3 2 39 !1 !1 > ˆ > ˆ ˆ > ˆ 7 6 7> <6 6 2 7 6 !3 7= 6 7 7 6 %DVLVIRU 1XO A 6 1 7; 6 0 7 ˆ > ˆ 4 0 5 4 5 5> > ˆ > ˆ ; : 0 1

2EVHUYHWKDW XQOLNHWKHEDVLVIRU &RO A WKHEDVHVIRU 5RZ A DQG 1XO A KDYHQRVLPSOH FRQQHFWLRQZLWKWKHHQWULHVLQ A LWVHOIñ ñ ,WLVSRVVLEOHWRÀQGDEDVLVIRUWKHURZVSDFH 5RZ A WKDWXVHVURZVRI A )LUVWIRUP AT DQGWKHQURZ UHGXFHXQWLOWKHSLYRWFROXPQVRI AT DUHIRXQG 7KHVHSLYRWFROXPQVRI AT DUHURZVRI A DQGWKH\IRUP DEDVLVIRUWKHURZVSDFHRI A

5DQN

:DUQLQJ $OWKRXJKWKHÀUVWWKUHHURZVRI B LQ([DPSOH DUHOLQHDUO\LQGHSHQGHQW LWLVZURQJWRFRQFOXGHWKDWWKHÀUVWWKUHHURZVRI A DUHOLQHDUO\LQGHSHQGHQW ,QIDFW WKHWKLUGURZRI A LVWLPHVWKHÀUVWURZSOXVWLPHVWKHVHFRQGURZ 5RZRSHUDWLRQV PD\FKDQJHWKHOLQHDUGHSHQGHQFHUHODWLRQVDPRQJWKH URZV RIDPDWUL[

7KH5DQN7KHRUHP WEB

%&'*/*5*0/

7KHQH[WWKHRUHPGHVFULEHVIXQGDPHQWDOUHODWLRQVDPRQJWKHGLPHQVLRQVRI &RO A 5RZ A DQG 1XO A

7KH UDQN RI A LVWKHGLPHQVLRQRIWKHFROXPQVSDFHRI A

6LQFH 5RZ A LVWKHVDPHDV &RO AT WKHGLPHQVLRQRIWKHURZVSDFHRI A LVWKHUDQN RI A 7KHGLPHQVLRQRIWKHQXOOVSDFHLVVRPHWLPHVFDOOHGWKH QXOOLW\ RI A WKRXJKZH ZLOOQRWXVHWKLVWHUP $QDOHUWUHDGHUPD\KDYHDOUHDG\GLVFRYHUHGSDUWRUDOORIWKHQH[WWKHRUHPZKLOH ZRUNLQJWKHH[HUFLVHVLQ6HFWLRQ RUUHDGLQJ([DPSOH DERYH T

5)&03&.

5IF 3BOL 5IFPSFN 7KHGLPHQVLRQVRIWKHFROXPQVSDFHDQGWKHURZVSDFHRIDQ m % n PDWUL[ A DUH HTXDO 7KLVFRPPRQGLPHQVLRQ WKHUDQNRI A DOVRHTXDOVWKHQXPEHURISLYRW SRVLWLRQVLQ A DQGVDWLVÀHVWKHHTXDWLRQ UDQN A C GLP 1XO A D n

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

&9".1-& D ,I A LVD 7 % 9 PDWUL[ZLWKDWZRGLPHQVLRQDOQXOOVSDFH ZKDWLVWKHUDQNRI A" E &RXOGD 6 % 9 PDWUL[KDYHDWZRGLPHQVLRQDOQXOOVSDFH"

$)"15&3

9HFWRU6SDFHV

40-65*0/ D 6LQFH A KDVFROXPQV .UDQN A/ C 2 D 9 DQGKHQFH UDQN A D 7 E 1R ,ID 6 % 9 PDWUL[ FDOOLW B KDGDWZRGLPHQVLRQDOQXOOVSDFH LWZRXOGKDYHWR KDYHUDQN E\WKH5DQN7KHRUHP %XWWKHFROXPQVRI B DUHYHFWRUVLQ R6 DQGVR WKHGLPHQVLRQRI &RO B FDQQRWH[FHHG WKDWLV UDQN B FDQQRWH[FHHG 7KHQH[WH[DPSOHSURYLGHVDQLFHZD\WRYLVXDOL]HWKHVXEVSDFHVZHKDYHEHHQ VWXG\LQJ ,Q&KDSWHU ZHZLOOOHDUQWKDW 5RZ A DQG 1XO A KDYHRQO\WKH]HURYHFWRU LQFRPPRQDQGDUHDFWXDOO\´SHUSHQGLFXODUµWRHDFKRWKHU 7KHVDPHIDFWZLOODSSO\ WR 5RZ AT .D &RO A/ DQG 1XO AT 6R)LJ ZKLFKDFFRPSDQLHV([DPSOH FUHDWHV DJRRGPHQWDOLPDJHIRUWKHJHQHUDOFDVH 7KHYDOXHRIVWXG\LQJ AT DORQJZLWK A LV GHPRQVWUDWHGLQ([HUFLVH 2 3 3 0 !1 &9".1-& /HW A D 4 3 0 !1 5 ,WLVUHDGLO\FKHFNHGWKDW 1XO A LVWKH x2 4 0 5 D[LV 5RZ A LVWKH x1 x3 SODQH &RO A LVWKHSODQHZKRVHHTXDWLRQLV x1 ! x2 D 0 DQG 1XO AT LVWKHVHWRIDOOPXOWLSOHVRI .1; !1; 0/ )LJXUH VKRZV 1XO A DQG 5RZ A LQ WKHGRPDLQRIWKHOLQHDUWUDQVIRUPDWLRQ [ 7! A[ WKHUDQJHRIWKLVPDSSLQJ &RO A LV VKRZQLQDVHSDUDWHFRS\RI R3 DORQJZLWK 1XO AT x3

x3 A

0

Nu

T

lA

lA

x1

x2

A Row

0

x2

Nu

Co

lA

!3

x1 !3

'*(63& 6XEVSDFHVGHWHUPLQHGE\DPDWUL[ A

$SSOLFDWLRQVWR6\VWHPVRI(TXDWLRQV 7KH5DQN7KHRUHPLVDSRZHUIXOWRROIRUSURFHVVLQJLQIRUPDWLRQDERXWV\VWHPVRI OLQHDUHTXDWLRQV 7KHQH[WH[DPSOHVLPXODWHVWKHZD\DUHDOOLIHSUREOHPXVLQJOLQHDU HTXDWLRQVPLJKWEHVWDWHG ZLWKRXWH[SOLFLWPHQWLRQRIOLQHDUDOJHEUDWHUPVVXFKDV PDWUL[ VXEVSDFH DQGGLPHQVLRQ

&9".1-& $ VFLHQWLVWKDVIRXQGWZRVROXWLRQVWRDKRPRJHQHRXVV\VWHPRI HTXDWLRQVLQ YDULDEOHV 7KHWZRVROXWLRQVDUHQRWPXOWLSOHV DQGDOORWKHUVROXWLRQV FDQEHFRQVWUXFWHGE\DGGLQJWRJHWKHUDSSURSULDWHPXOWLSOHVRIWKHVHWZRVROXWLRQV &DQWKHVFLHQWLVWEH FHUWDLQ WKDWDQDVVRFLDWHGQRQKRPRJHQHRXVV\VWHPZLWKWKHVDPH FRHIÀFLHQWV KDVDVROXWLRQ"

40-65*0/

5DQN

5DQNDQGWKH,QYHUWLEOH0DWUL[7KHRUHP 7KH YDULRXV YHFWRU VSDFH FRQFHSWV DVVRFLDWHG ZLWK D PDWUL[ SURYLGH VHYHUDO PRUH VWDWHPHQWVIRUWKH,QYHUWLEOH0DWUL[7KHRUHP 7KHQHZVWDWHPHQWVOLVWHGKHUHIROORZ WKRVHLQWKHRULJLQDO,QYHUWLEOH0DWUL[7KHRUHPLQ6HFWLRQ

5)&03&.


/HW A EHDQ n % n PDWUL[ 7KHQWKHIROORZLQJVWDWHPHQWVDUHHDFKHTXLYDOHQWWR WKHVWDWHPHQWWKDW A LVDQLQYHUWLEOHPDWUL[ P Q R S T U

7KHFROXPQVRI A IRUPDEDVLVRI Rn &RO A D Rn GLP &RO A D n UDQN A D n 1XO A D fg GLP 1XO A D 0

1300' 6WDWHPHQWP LVORJLFDOO\HTXLYDOHQWWRVWDWHPHQWVH DQGK UHJDUGLQJOLQHDU LQGHSHQGHQFHDQGVSDQQLQJ 7KHRWKHUÀYHVWDWHPHQWVDUHOLQNHGWRWKHHDUOLHURQHVRI WKHWKHRUHPE\WKHIROORZLQJFKDLQRIDOPRVWWULYLDOLPSOLFDWLRQV .J/ ) .Q/ ) .R/ ) .S/ ) .U/ ) .T/ ) .G/

SG


6WDWHPHQWJ ZKLFKVD\VWKDWWKHHTXDWLRQ A[ D E KDVDWOHDVWRQHVROXWLRQIRUHDFK E LQ Rn LPSOLHVQ EHFDXVH &RO A LVSUHFLVHO\WKHVHWRIDOO E VXFKWKDWWKHHTXDWLRQ A[ D E LVFRQVLVWHQW 7KHLPSOLFDWLRQVQ ) R ) S IROORZIURPWKHGHÀQLWLRQVRIGLPHQVLRQ DQGUDQN ,IWKHUDQNRI A LV n WKHQXPEHURIFROXPQVRI A WKHQ GLP 1XO A D 0 E\WKH 5DQN7KHRUHP DQGVR 1XO A D fg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

$)"15&3

9HFWRU6SDFHV

/6.&3*$"- /05& 0DQ\DOJRULWKPVGLVFXVVHGLQWKLVWH[WDUHXVHIXOIRUXQGHUVWDQGLQJFRQFHSWV DQGPDNLQJVLPSOHFRPSXWDWLRQVE\KDQG +RZHYHU WKHDOJRULWKPVDUHRIWHQ XQVXLWDEOHIRUODUJHVFDOHSUREOHPVLQUHDOOLIH 5DQNGHWHUPLQDWLRQLVDJRRGH[DPSOH ,WZRXOGVHHPHDV\WRUHGXFHDPDWUL[ WRHFKHORQIRUPDQGFRXQWWKHSLYRWV %XWXQOHVVH[DFWDULWKPHWLFLVSHUIRUPHG RQDPDWUL[ZKRVHHQWULHVDUHVSHFLÀHGH[DFWO\ URZRSHUDWLRQVFDQFKDQJHWKH " # 5 7 DSSDUHQWUDQNRIDPDWUL[ )RULQVWDQFH LIWKHYDOXHRI x LQWKHPDWUL[ 5 x LVQRWVWRUHGH[DFWO\DVLQDFRPSXWHU WKHQWKHUDQNPD\EHRU GHSHQGLQJ RQZKHWKHUWKHFRPSXWHUWUHDWV x ! 7 DV]HUR ,QSUDFWLFDODSSOLFDWLRQV WKHHIIHFWLYHUDQNRIDPDWUL[ A LVRIWHQGHWHUPLQHG IURPWKHVLQJXODUYDOXHGHFRPSRVLWLRQRI A WREHGLVFXVVHGLQ6HFWLRQ 7KLV GHFRPSRVLWLRQLVDOVRDUHOLDEOHVRXUFHRIEDVHVIRU &RO A 5RZ A 1XO A DQG 1XO AT

WEB

13"$5*$& 130#-&.4 7KHPDWULFHVEHORZDUHURZHTXLYDOHQW 2 3 2 !1 1 !6 8 6 1 !2 !4 3 !2 7 7; AD6 4 !7 8 10 3 !10 5 4 !5 !7 0 4

,Q([HUFLVHV² DVVXPHWKDWWKHPDWUL[ A LVURZHTXLYDOHQWWR B :LWKRXWFDOFXODWLRQV OLVW UDQN A DQG GLP 1XO A 7KHQÀQGEDVHV IRU &RO A 5RZ A DQG 1XO A 2 3 1 !4 9 !7 2 !4 1 5 A D 4 !1 5 !6 10 7 2 3 1 0 !1 5 5 !6 5 B D 4 0 !2 0 0 0 0

1 62 A D 6 43 3 2 1 60 BD6 40 0

3 6 9 9 3 0 0 0

3 4 !1 2 6 0 !3 7 7 3 6 !3 5 0 9 0 3 4 !1 2 1 !1 17 7 0 0 !5 5 0 0 0

1 !2 !4 3 60 3 9 !12 BD6 40 0 0 0 0 0 0 0

)LQG UDQN A DQG GLP 1XO A )LQGEDVHVIRU &RO A DQG 5RZ A :KDWLVWKHQH[WVWHSWRSHUIRUPWRÀQGDEDVLVIRU 1XO A" +RZPDQ\SLYRWFROXPQVDUHLQDURZHFKHORQIRUPRI AT "

&9&3$*4&4

2

2

2

2 6 !6 6 !2 !3 6 6 A D 4 4 9 !12 !2 3 6 2 2 6 !6 60 3 0 BD6 40 0 0 0 0 0 2 1 1 !2 61 2 !3 6 0 A D 6 6 1 !1 4 1 !2 2 1 !2 1 2 1 1 !2 60 1 !1 6 0 0 1 BD6 6 40 0 0 0 0 0

6 !3 9 3 6 3 0 0

3 0 3 3 3 3 3 0

0 1 0 !2 0 1 1 !3 0 2 0 1 0 !3 1 !13 0 1 0 0

3 6 !6 7 7 12 5 !6 3

6 07 7 05 0 3 !2 !3 7 7 67 7 05 !1 3 !2 !1 7 7 !1 7 7 !1 5 1

3 !2 12 7 7 05 0

5DQN

,ID 4 % 7 PDWUL[ A KDVUDQN ÀQG GLP 1XO A GLP 5RZ A DQG UDQN AT

H ,I A DQG B DUHURZHTXLYDOHQW WKHQWKHLUURZVSDFHVDUH WKHVDPH

,ID 7 % 5 PDWUL[ A KDVUDQN ÀQG GLP 1XO A GLP 5RZ A DQG UDQN AT

6XSSRVHWKHVROXWLRQVRIDKRPRJHQHRXVV\VWHPRIÀYHOLQHDU HTXDWLRQVLQVL[XQNQRZQVDUHDOOPXOWLSOHVRIRQHQRQ]HUR VROXWLRQ :LOOWKHV\VWHPQHFHVVDULO\KDYHDVROXWLRQIRU HYHU\SRVVLEOHFKRLFHRIFRQVWDQWVRQWKHULJKWVLGHVRIWKH HTXDWLRQV" ([SODLQ

6XSSRVH D 4 % 7 PDWUL[ A KDV IRXU SLYRW FROXPQV &RO A D R4 " ,V 1XO A D R3 " ([SODLQ\RXUDQVZHUV

,V

6XSSRVHD 6 % 8 PDWUL[ A KDVIRXUSLYRWFROXPQV :KDWLV GLP 1XO A" ,V &RO A D R4 " :K\RUZK\QRW"

,IWKHQXOOVSDFHRID 4 % 6 PDWUL[ A LVGLPHQVLRQDO ZKDW LVWKHGLPHQVLRQRIWKHFROXPQVSDFHRI A" ,V &RO A D R3 " :K\RUZK\QRW"

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m % n PDWUL[ VWDWHPHQW7UXHRU)DOVH -XVWLI\HDFKDQVZHU

0DUN HDFK

D 7KHURZVSDFHRI A LVWKHVDPHDVWKHFROXPQVSDFHRI AT E ,I B LVDQ\HFKHORQIRUPRI A DQGLI B KDVWKUHHQRQ]HUR URZV WKHQWKHÀUVWWKUHHURZVRI A IRUP D EDVLVIRU 5RZ A F 7KHGLPHQVLRQVRIWKHURZVSDFHDQGWKHFROXPQVSDFH RI A DUHWKHVDPH HYHQLI A LVQRWVTXDUH G 7KHVXPRIWKHGLPHQVLRQVRIWKHURZVSDFHDQGWKHQXOO VSDFHRI A HTXDOVWKHQXPEHURIURZVLQ A H 2QDFRPSXWHU URZRSHUDWLRQVFDQFKDQJHWKHDSSDUHQW UDQNRIDPDWUL[ D ,I B LVDQ\HFKHORQIRUPRI A WKHQWKHSLYRWFROXPQVRI B IRUPDEDVLVIRUWKHFROXPQVSDFHRI A E 5RZRSHUDWLRQVSUHVHUYHWKHOLQHDUGHSHQGHQFHUHODWLRQV DPRQJWKHURZVRI A F 7KHGLPHQVLRQRIWKHQXOOVSDFHRI A LVWKHQXPEHURI FROXPQVRI A WKDWDUH QRW SLYRWFROXPQV G 7KHURZVSDFHRI AT LVWKHVDPHDVWKHFROXPQVSDFHRI A

6XSSRVHDQRQKRPRJHQHRXVV\VWHPRIVL[OLQHDUHTXDWLRQV LQHLJKWXQNQRZQVKDVDVROXWLRQ ZLWKWZRIUHHYDULDEOHV ,V LWSRVVLEOHWRFKDQJHVRPHFRQVWDQWVRQWKHHTXDWLRQV·ULJKW VLGHVWRPDNHWKHQHZV\VWHPLQFRQVLVWHQW" ([SODLQ 6XSSRVHDQRQKRPRJHQHRXVV\VWHPRIQLQHOLQHDUHTXDWLRQV LQWHQXQNQRZQVKDVDVROXWLRQIRUDOOSRVVLEOHFRQVWDQWVRQ WKHULJKWVLGHVRIWKHHTXDWLRQV ,VLWSRVVLEOHWRÀQGWZR QRQ]HURVROXWLRQVRIWKHDVVRFLDWHGKRPRJHQHRXVV\VWHPWKDW DUH QRW PXOWLSOHVRIHDFKRWKHU" 'LVFXVV ,VLVSRVVLEOHWKDWDOOVROXWLRQVRIDKRPRJHQHRXVV\VWHPRI WHQOLQHDUHTXDWLRQVLQWZHOYHYDULDEOHVDUHPXOWLSOHVRIRQH À[HGQRQ]HURVROXWLRQ" 'LVFXVV $ KRPRJHQHRXVV\VWHPRIWZHOYHOLQHDUHTXDWLRQVLQHLJKW XQNQRZQVKDVWZRÀ[HGVROXWLRQVWKDWDUHQRWPXOWLSOHVRI HDFKRWKHU DQGDOORWKHUVROXWLRQVDUHOLQHDUFRPELQDWLRQVRI WKHVHWZRVROXWLRQV &DQWKHVHWRIDOOVROXWLRQVEHGHVFULEHG ZLWKIHZHUWKDQWZHOYHKRPRJHQHRXVOLQHDUHTXDWLRQV" ,IVR KRZPDQ\" 'LVFXVV ,VLWSRVVLEOHIRUDQRQKRPRJHQHRXVV\VWHPRIVHYHQHTXD WLRQVLQVL[XQNQRZQVWRKDYHDXQLTXHVROXWLRQIRUVRPH ULJKWKDQGVLGHRIFRQVWDQWV" ,VLWSRVVLEOHIRUVXFKDV\VWHP WRKDYHDXQLTXHVROXWLRQIRUHYHU\ULJKWKDQGVLGH" ([SODLQ $ VFLHQWLVWVROYHVDQRQKRPRJHQHRXVV\VWHPRIWHQOLQHDU HTXDWLRQVLQWZHOYHXQNQRZQVDQGÀQGVWKDWWKUHHRIWKH XQNQRZQVDUHIUHHYDULDEOHV &DQWKHVFLHQWLVWEHFHUWDLQ WKDW LIWKHULJKWVLGHVRIWKHHTXDWLRQVDUHFKDQJHG WKHQHZ QRQKRPRJHQHRXVV\VWHPZLOOKDYHDVROXWLRQ" 'LVFXVV ,QVWDWLVWLFDOWKHRU\ DFRPPRQUHTXLUHPHQWLVWKDWDPDWUL[ EH RI IXOO UDQN 7KDW LV WKH UDQN VKRXOG EH DV ODUJH DV SRVVLEOH ([SODLQZK\DQ m % n PDWUL[ZLWKPRUHURZVWKDQ FROXPQVKDVIXOOUDQNLIDQGRQO\LILWVFROXPQVDUHOLQHDUO\ LQGHSHQGHQW ([HUFLVHV²FRQFHUQDQ m % n PDWUL[ A DQGZKDWDUHRIWHQ FDOOHGWKH IXQGDPHQWDOVXEVSDFHV GHWHUPLQHGE\ A :KLFK RI WKH VXEVSDFHV 5RZ A &RO A 1XO A 5RZ AT &RO AT DQG 1XO AT DUHLQ Rm DQGZKLFKDUHLQ Rn " +RZ PDQ\GLVWLQFWVXEVSDFHVDUHLQWKLVOLVW" -XVWLI\WKHIROORZLQJHTXDOLWLHV D GLP 5RZ A C GLP 1XO A D n 1XPEHURIFROXPQVRI A E GLP &RO A C GLP 1XO AT D m 1XPEHURIURZVRI A

8VH([HUFLVHWRH[SODLQZK\WKHHTXDWLRQ A[ D E KDVD VROXWLRQIRUDOO E LQ Rm LIDQGRQO\LIWKHHTXDWLRQ AT [ D KDVRQO\WKHWULYLDOVROXWLRQ

$)"15&3

9HFWRU6SDFHV

6XSSRVH A LV m % n DQG E LVLQ Rm :KDWKDVWREHWUXH DERXWWKHWZRQXPEHUV UDQN Œ A E ! DQG UDQN A LQRUGHUIRU WKHHTXDWLRQ A[ D E WREHFRQVLVWHQW" 5DQNPDWULFHVDUHLPSRUWDQWLQVRPHFRPSXWHUDOJRULWKPVDQG VHYHUDOWKHRUHWLFDOFRQWH[WV LQFOXGLQJWKHVLQJXODUYDOXHGHFRP SRVLWLRQLQ&KDSWHU ,WFDQEHVKRZQWKDWDQ m % n PDWUL[ A KDVUDQN LIDQGRQO\LILWLVDQRXWHUSURGXFW WKDWLV A D XYT IRUVRPH X LQ Rm DQG Y LQ Rn ([HUFLVHV ²VXJJHVWZK\WKLV SURSHUW\LVWUXH 2 3 2 3 2 a 9HULI\WKDW UDQN XYT $ 1 LI X D 4 !3 5 DQG Y D 4 b 5 5 c " # " 1 1 !3 )LQG Y LQ R3 VXFKWKDW /HW X D 2 2 !6

4 8

#

D XYT

/HW A EHDQ\ 2 % 3 PDWUL[VXFKWKDW UDQN A D 1 OHW X EHWKH ÀUVWFROXPQRI A DQGVXSSRVH X ¤ ([SODLQZK\WKHUH LVDYHFWRU Y LQ R3 VXFKWKDW A D XYT +RZFRXOGWKLV FRQVWUXFWLRQEHPRGLÀHGLIWKHÀUVWFROXPQRI A ZHUH]HUR" /HW A EHDQ m % n PDWUL[RI UDQN r > 0 DQGOHW U EHDQHFKH ORQIRUPRI A ([SODLQZK\WKHUHH[LVWVDQLQYHUWLEOHPDWUL[ E VXFKWKDW A D EU DQGXVHWKLVIDFWRUL]DWLRQWRZULWH A DVWKHVXPRI r UDQN 1 PDWULFHV >+LQW 6HH7KHRUHP LQ 6HFWLRQ @

2

3 7 !9 !4 5 3 !3 !7 6 !4 6 7 !2 !6 !5 57 6 7 7 5 !7 !6 5 !6 2 8 >0@ /HW A D 6 6 7 4 !3 5 8 !1 !7 !4 85 6 !8 !5 4 4 9 3 D &RQVWUXFWPDWULFHV C DQG N ZKRVHFROXPQVDUHEDVHVIRU &RO A DQG 1XO A UHVSHFWLYHO\ DQGFRQVWUXFWDPDWUL[ R ZKRVH URZV IRUPDEDVLVIRU 5RZ A

E &RQVWUXFW D PDWUL[ M ZKRVH FROXPQV IRUP D ED VLVIRU 1XO AT IRUPWKHPDWULFHV S D Œ RT N ! DQG T D Œ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

40-65*0/4 50 13"$5*$& 130#-&.4 A KDV WZR SLYRW FROXPQV VR UDQN A D 2 6LQFH A KDV FROXPQV DOWRJHWKHU GLP 1XO A D 5 ! 2 D 3 7KHSLYRWFROXPQVRI A DUHWKHÀUVWWZRFROXPQV 6RDEDVLVIRU &RO A LV 82 3 2 39 2 !1 > ˆ > ˆ <6 6 7= 17 7; 6 !2 7 fD1 ; D2 g D 6 4 !7 5 4 8 5> ˆ > ˆ ; : 4 !5

7KH QRQ]HUR URZV RI B IRUP D EDVLV IRU 5RZ A QDPHO\ f.1; !2; !4; 3; !2/ .0; 3; 9; !12; 12/g ,QWKLVSDUWLFXODUH[DPSOH LWKDSSHQVWKDWDQ\WZRURZVRI A IRUPDEDVLVIRUWKHURZVSDFH EHFDXVHWKHURZVSDFHLVWZRGLPHQVLRQDODQGQRQH RIWKHURZVRI A LVDPXOWLSOHRIDQRWKHUURZ ,QJHQHUDO WKHQRQ]HURURZVRIDQ HFKHORQIRUPRI A VKRXOGEHXVHGDVDEDVLVIRU 5RZ A QRWWKHURZVRI A LWVHOI )RU 1XO A WKHQH[WVWHSLVWRSHUIRUPURZRSHUDWLRQVRQ B WRREWDLQWKHUHGXFHG HFKHORQIRUPRI A SG

.BKPS 3FWJFX PG ,FZ $PODFQUT m

5DQN AT D UDQN A E\WKH5DQN7KHRUHP EHFDXVH &RO AT D 5RZ A 6R AT KDV WZRSLYRWSRVLWLRQV

&KDQJHRI%DVLV

$)"/(& 0' #"4*4 :KHQDEDVLV B LVFKRVHQIRUDQ nGLPHQVLRQDOYHFWRUVSDFH V WKHDVVRFLDWHGFRRUGLQDWH PDSSLQJRQWR Rn SURYLGHVDFRRUGLQDWHV\VWHPIRU V (DFK [ LQ V LVLGHQWLÀHGXQLTXHO\ E\LWV B FRRUGLQDWHYHFWRU Œ [ !B ñ ,QVRPHDSSOLFDWLRQV DSUREOHPLVGHVFULEHGLQLWLDOO\XVLQJDEDVLV B EXWWKH SUREOHP·VVROXWLRQLVDLGHGE\FKDQJLQJ B WRDQHZEDVLV C ([DPSOHVZLOOEHJLYHQLQ &KDSWHUV DQG (DFKYHFWRULVDVVLJQHGDQHZ C FRRUGLQDWHYHFWRU ,QWKLVVHFWLRQ ZHVWXG\KRZ Œ [ !C DQG Œ [ !B DUHUHODWHGIRUHDFK [ LQ V 7RYLVXDOL]HWKHSUREOHP FRQVLGHUWKHWZRFRRUGLQDWHV\VWHPVLQ)LJ ,Q)LJ D [ D 3E1 C E2 ZKLOHLQ)LJ E WKHVDPH [ LVVKRZQDV [ D 6F1 C 4F2 7KDWLV " # " # 3 6 Œ [ !B D DQG Œ [ !C D 1 4 2XUSUREOHPLVWRÀQGWKHFRQQHFWLRQEHWZHHQWKHWZRFRRUGLQDWHYHFWRUV ([DPSOH VKRZVKRZWRGRWKLV SURYLGHGZHNQRZKRZ E1 DQG E2 DUHIRUPHGIURP F1 DQG F2 b2

4 c2 c2 0

x

0 c 1

x b1

6c 1

3b1 (a)

(b)

'*(63& 7ZRFRRUGLQDWHV\VWHPVIRUWKHVDPHYHFWRUVSDFH

&9".1-& &RQVLGHUWZREDVHV B D fE1 ; E2 g DQG C D fF1 ; F2 g IRUDYHFWRUVSDFH V VXFKWKDW

E1 D 4F1 C F2

DQG

E2 D !6F1 C F2

6XSSRVH [ D 3E1 C E2 " # 3 7KDWLV VXSSRVH Œ [ !B D )LQG Œ [ !C 1

40-65*0/ $SSO\WKHFRRUGLQDWHPDSSLQJGHWHUPLQHGE\ C WR [ LQ 6LQFHWKH FRRUGLQDWHPDSSLQJLVDOLQHDUWUDQVIRUPDWLRQ Œ [ !C D Œ 3E1 C E2 !C

D 3Œ E1 !C C Œ E2 !C

:HFDQZULWHWKLVYHFWRUHTXDWLRQDVDPDWUL[HTXDWLRQ XVLQJWKHYHFWRUVLQWKHOLQHDU FRPELQDWLRQDVWKHFROXPQVRIDPDWUL[ " # % & 3 Œ E2 !C Œ [ !C D Œ E1 !C 1 ñ 7KLQNRI Œ [ !B DVD´QDPHµIRU [ WKDWOLVWVWKHZHLJKWVXVHGWREXLOG [ DVDOLQHDUFRPELQDWLRQRIWKHEDVLV YHFWRUVLQ B

$)"15&3

9HFWRU6SDFHV

7KLVIRUPXODJLYHV Œ [ !C RQFHZHNQRZWKHFROXPQVRIWKHPDWUL[ )URP " # " # 4 !6 Œ E1 !C D DQG Œ E2 !C D 1 1 7KXV SURYLGHVWKHVROXWLRQ

Œ [ !C D

"

4 !6 1 1

#" # " # 3 6 D 1 4

7KH C FRRUGLQDWHVRI [ PDWFKWKRVHRIWKH [ LQ)LJ 7KHDUJXPHQWXVHGWRGHULYHIRUPXOD FDQEHJHQHUDOL]HGWR\LHOGWKHIROORZLQJ UHVXOW 6HH([HUFLVHV DQG

5)&03&.

/HW B D fE1 ; : : : ; En g DQG C D fF1 ; : : : ; Fn g EHEDVHVRIDYHFWRUVSDFH V 7KHQ WKHUHLVDXQLTXH n % n PDWUL[ C PB VXFKWKDW

Œ [ !C D C PB Œ [ !B

7KHFROXPQVRI C PB DUHWKH C FRRUGLQDWHYHFWRUVRIWKHYHFWRUVLQWKHEDVLV B 7KDWLV % & P ŒE2 !C # # # ŒEn !C C B D ŒE1 !C 7KHPDWUL[ C PB LQ7KHRUHPLVFDOOHGWKH FKDQJHRIFRRUGLQDWHVPDWUL[IURP B WR C 0XOWLSOLFDWLRQE\ C PB FRQYHUWV B FRRUGLQDWHVLQWR C FRRUGLQDWHVò )LJXUH LOOXVWUDWHVWKHFKDQJHRIFRRUGLQDWHVHTXDWLRQ V x [ ]B

[ ]C [x]C

multiplication by

!n

[x]B

P

C←B

!n

'*(63& 7ZRFRRUGLQDWHV\VWHPVIRU V

7KHFROXPQVRI C PB DUHOLQHDUO\LQGHSHQGHQWEHFDXVHWKH\DUHWKHFRRUGLQDWH YHFWRUVRIWKHOLQHDUO\LQGHSHQGHQWVHW B 6HH([HUFLVH LQ6HFWLRQ 6LQFH C PB LVVTXDUH LWPXVWEHLQYHUWLEOH E\WKH,QYHUWLEOH0DWUL[7KHRUHP /HIWPXOWLSO\LQJERWK VLGHVRIHTXDWLRQ E\ . C PB /!1 \LHOGV

. C PB /!1 Œ [ !C D Œ [ !B ò 7RUHPHPEHUKRZWRFRQVWUXFWWKHPDWUL[ WKLQNRI

C

P Œ [ ! DVDOLQHDUFRPELQDWLRQRIWKHFROXPQVRI B B

P C B 7KHPDWUL[YHFWRUSURGXFWLVD C FRRUGLQDWHYHFWRU VRWKHFROXPQVRI YHFWRUV WRR

C

P

B

VKRXOGEH C FRRUGLQDWH

&KDQJHRI%DVLV

7KXV . C PB /!1 LVWKHPDWUL[WKDWFRQYHUWV C FRRUGLQDWHVLQWR B FRRUGLQDWHV 7KDWLV

. C PB /!1 D B P C

&KDQJHRI%DVLVLQ Rn ,I B D fE1 ; : : : ; En g DQG E LVWKH VWDQGDUGEDVLV fH1 ; : : : ; Hn g LQ Rn WKHQ ŒE1 !E D E1 DQGOLNHZLVHIRUWKHRWKHUYHFWRUVLQ B ,QWKLVFDVH E PB LVWKHVDPHDVWKHFKDQJHRI FRRUGLQDWHVPDWUL[ PB LQWURGXFHGLQ6HFWLRQ QDPHO\

PB D Œ E1 E2 # # # En ! 7RFKDQJHFRRUGLQDWHVEHWZHHQWZRQRQVWDQGDUGEDVHVLQ Rn ZHQHHG7KHRUHP 7KHWKHRUHPVKRZVWKDWWRVROYHWKHFKDQJHRIEDVLVSUREOHP ZHQHHGWKHFRRUGLQDWH YHFWRUVRIWKHROGEDVLVUHODWLYHWRWKHQHZEDVLV # " # " # " # !9 !5 1 3 E2 D F1 D F2 D DQGFRQ 1 !1 !4 !5 VLGHUWKHEDVHVIRU R2 JLYHQE\ B D fE1 ; E2 g DQG C D fF1 ; F2 g )LQGWKHFKDQJHRI FRRUGLQDWHVPDWUL[IURP B WR C

&9".1-& /HW E1 D

"

40-65*0/ 7KHPDWUL[ C PB LQYROYHVWKH C FRRUGLQDWHYHFWRUVRI E1 DQG E2 /HW " # " # x1 y1 Œ E 1 !C D DQG Œ E2 !C D 7KHQ E\GHÀQLWLRQ x2 y2 " # " # % & x1 % & y1 F1 F2 F F D E1 DQG D E2 1 2 x2 y2 7RVROYHERWKV\VWHPVVLPXOWDQHRXVO\ DXJPHQWWKHFRHIÀFLHQWPDWUL[ZLWK E1 DQG E2 DQGURZUHGXFH " # " # % & 1 3 !9 !5 1 0 6 4 F1 F2 E1 E2 D & 1 !1 !5 !3 !4 !5 0 1

7KXV

Œ E1 !C D

"

6 !5

#

DQG

Œ E2 !C D

"

4 !3

#

7KHGHVLUHGFKDQJHRIFRRUGLQDWHVPDWUL[LVWKHUHIRUH " # % & 6 4 P D Œ E1 ! Œ E2 !C D C B C !5 !3

2EVHUYHWKDWWKHPDWUL[ C PB LQ([DPSOH DOUHDG\DSSHDUHGLQ 7KLVLVQRW VXUSULVLQJEHFDXVHWKHÀUVWFROXPQRI C PB UHVXOWVIURPURZUHGXFLQJ Œ F1 F2 E1 ! WR Œ I Œ E1 !C ! DQGVLPLODUO\IRUWKHVHFRQGFROXPQRI C PB 7KXV

Œ F1 F2 E1 E2 ! & Œ I

C

P ! B

$QDQDORJRXVSURFHGXUHZRUNVIRUÀQGLQJWKHFKDQJHRIFRRUGLQDWHVPDWUL[EHWZHHQ DQ\WZREDVHVLQ Rn

$)"15&3

9HFWRU6SDFHV

# " # " # " # 1 !2 !7 !5 E2 D F1 D F2 D DQGFRQ !3 4 9 7 VLGHUWKHEDVHVIRU R2 JLYHQE\ B D fE1 ; E2 g DQG C D fF1 ; F2 g

&9".1-& /HW E1 D

"

D )LQGWKHFKDQJHRIFRRUGLQDWHVPDWUL[IURP C WR B E )LQGWKHFKDQJHRIFRRUGLQDWHVPDWUL[IURP B WR C

40-65*0/ D 1RWLFHWKDW B P C LVQHHGHGUDWKHUWKDQ C PB DQGFRPSXWH " # " % & 1 !2 !7 !5 1 E1 E2 F1 F2 D & 9 7 !3 4 0 6R

P B C D

"

5 6

3 4

0 1

#

5 6

3 4

#

E %\SDUWD DQGSURSHUW\ DERYHZLWK B DQG C LQWHUFKDQJHG " # " # 4 !3 2 !3=2 P D . P /!1 D 1 D C B B C !3 5=2 2 !6 5 $QRWKHUGHVFULSWLRQRIWKHFKDQJHRIFRRUGLQDWHVPDWUL[ C PB XVHVWKHFKDQJHRI FRRUGLQDWHPDWULFHV PB DQG PC WKDWFRQYHUW B FRRUGLQDWHVDQG C FRRUGLQDWHV UHVSHF WLYHO\ LQWRVWDQGDUGFRRUGLQDWHV 5HFDOOWKDWIRUHDFK [ LQ Rn

PB Œ[!B D [; 7KXV

PC Œ[!C D [;

DQG

Œ[!C D PC!1 [

Œ[!C D PC!1 [ D PC!1 PB Œ[!B

,Q Rn WKHFKDQJHRIFRRUGLQDWHVPDWUL[ C PB PD\EHFRPSXWHGDV PC!1 PB $FWXDOO\ IRUPDWULFHVODUJHUWKDQ 2 % 2 DQDOJRULWKPDQDORJRXVWRWKHRQHLQ([DPSOHLVIDVWHU WKDQFRPSXWLQJ PC!1 DQGWKHQ PC!1 PB 6HH([HUFLVHLQ6HFWLRQ

13"$5*$& 130#-&.4 /HW F D fI1 ; I2 g DQG G D fJ1 ; J2 g EHEDVHVIRUDYHFWRUVSDFH V DQGOHW P EHD PDWUL[ZKRVHFROXPQVDUH Œ I1 !G DQG Œ I2 !G :KLFKRIWKHIROORZLQJHTXDWLRQVLV VDWLVÀHGE\ P IRUDOO Y LQ V " L Œ Y !F D P Œ Y !G LL Œ Y !G D PŒ Y !F /HW B DQG C EHDVLQ([DPSOH 8VHWKHUHVXOWVRIWKDWH[DPSOHWRÀQGWKHFKDQJH RIFRRUGLQDWHVPDWUL[IURP C WR B

&9&3$*4&4 /HW B D fE1 ; E2 g DQG C D fF1 ; F2 g EHEDVHVIRUDYHFWRUVSDFH V DQGVXSSRVH E1 D 6F1 ! 2F2 DQG E2 D 9F1 ! 4F2 D )LQGWKHFKDQJHRIFRRUGLQDWHVPDWUL[IURP B WR C E )LQG Œ [ !C IRU [ D !3E1 C 2E2 8VHSDUWD

/HW B D fE1 ; E2 g DQG C D fF1 ; F2 g EHEDVHVIRUDYHFWRUVSDFH V DQGVXSSRVH E1 D !2F1 C 4F2 DQG E2 D 3F1 ! 6F2 D )LQGWKHFKDQJHRIFRRUGLQDWHVPDWUL[IURP B WR C E )LQG Œ [ !C IRU [ D 2E1 C 3E2

/HW U D fX1 ; X2 g DQG W D fZ1 ; Z2 g EHEDVHVIRU V DQGOHW P EHDPDWUL[ZKRVHFROXPQVDUH Œ X1 !W DQG ŒX2 !W :KLFK RIWKHIROORZLQJHTXDWLRQVLVVDWLVÀHGE\ P IRUDOO [ LQ V " L Œ [ !U D P Œ [ !W LL Œ [ !W D P Œ [ !U

/HW A D fD1 ; D2 ; D3 g DQG D D fG1 ; G2 ; G3 g EHEDVHVIRU V DQGOHW P D Œ ŒG1 !A ŒG2 !A ŒG3 !A ! :KLFKRIWKHIROORZ LQJHTXDWLRQVLVVDWLVÀHGE\ P IRUDOO [ LQ V " L Œ [ !A D P Œ [ !D LL Œ [ !D D P Œ [ !A /HW A D fD1 ; D2 ; D3 g DQG B D fE1 ; E2 ; E3 g EH EDVHV IRU D YHFWRU VSDFH V DQG VXSSRVH D1 D 4E1 ! E2 D2 D !E1 C E2 C E3 DQG D3 D E2 ! 2E3 D )LQGWKHFKDQJHRIFRRUGLQDWHVPDWUL[IURP A WR B E )LQG Œ [ !B IRU [ D 3D1 C 4D2 C D3

/HW D D fG1 ; G2 ; G3 g DQG F D fI1 ; I2 ; I3 g EH EDVHV IRU D YHFWRU VSDFH V DQG VXSSRVH I1 D 2G1 ! G2 C G3 I2 D 3G2 C G3 DQG I3 D !3G1 C 2G3 D )LQGWKHFKDQJHRIFRRUGLQDWHVPDWUL[IURP F WR D E )LQG Œ [ !D IRU [ D I1 ! 2I2 C 2I3

,Q([HUFLVHV² OHW B D fE1 ; E2 g DQG C D fF1 ; F2 g EHEDVHVIRU R2 ,QHDFKH[HUFLVH ÀQGWKHFKDQJHRIFRRUGLQDWHVPDWUL[IURP B WR C DQGWKHFKDQJHRIFRRUGLQDWHVPDWUL[IURP C WR B " # " # " # " # 7 !3 1 !2 E1 D E2 D F1 D F2 D 5 !1 !5 2 " # " # " # " # !1 1 1 1 E1 D E2 D F1 D F2 D 8 !7 2 1 " # " # " # " # 4 8 2 !2 E1 D E2 D F1 D F2 D 4 4 2 2 " # " # " # " # 6 4 4 3 E1 D E2 D F1 D F2 D !12 2 2 9 ,Q([HUFLVHVDQG B DQG C DUHEDVHVIRUDYHFWRUVSDFH V 0DUNHDFKVWDWHPHQW7UXHRU)DOVH -XVWLI\HDFKDQVZHU D 7KHFROXPQVRIWKHFKDQJHRIFRRUGLQDWHVPDWUL[ DUH BFRRUGLQDWHYHFWRUVRIWKHYHFWRUVLQ C

C

P

B

E ,I V D Rn DQG C LVWKH VWDQGDUG EDVLVIRU V WKHQ C P B LVWKHVDPHDVWKHFKDQJHRIFRRUGLQDWHVPDWUL[ PB LQWUR GXFHGLQ6HFWLRQ D 7KHFROXPQVRI C P B DUHOLQHDUO\LQGHSHQGHQW E ,I V D R2 B D fE1 ; E2 g DQG C D fF1 ; F2 g WKHQ URZ UHGXFWLRQRI Œ F1 F2 E1 E2 ! WR Œ I P ! SURGXFHVD PDWUL[ P WKDWVDWLVÀHV Œ [ !B D P Œ [ !C IRUDOO [ LQ V

,Q P2 ÀQGWKHFKDQJHRIFRRUGLQDWHVPDWUL[IURPWKHEDVLV B D f1 ! 2t C t 2 ; 3 ! 5t C 4t 2 ; 2t C 3t 2 g WR WKH VWDQGDUG EDVLV C D f1; t; t 2 g 7KHQÀQGWKH BFRRUGLQDWHYHFWRUIRU !1 C 2t

,Q P2 ÀQGWKHFKDQJHRIFRRUGLQDWHVPDWUL[IURPWKHED VLV B D f1 ! 3t 2 ; 2 C t ! 5t 2 ; 1 C 2t g WRWKHVWDQGDUGEDVLV 7KHQZULWH t 2 DVDOLQHDUFRPELQDWLRQRIWKHSRO\QRPLDOVLQ B

&KDQJHRI%DVLV

([HUFLVHVDQGSURYLGHDSURRIRI7KHRUHP )LOOLQD MXVWLÀFDWLRQIRUHDFKVWHS *LYHQ Y LQ V WKHUHH[LVWVFDODUV x1 ; : : : ; xn VXFKWKDW Y D x1 E1 C x2 E2 C # # # C xn En EHFDXVHD $SSO\WKHFRRUGLQDWHPDSSLQJGHWHU PLQHGE\WKHEDVLV C DQGREWDLQ

ŒY !C D x 1 ŒE 1 !C C x2 ŒE 2 !C C # # # C x n ŒE n !C EHFDXVHE % Œ Y !C D Œ E1 !C

7KLVHTXDWLRQPD\EHZULWWHQLQWKHIRUP 2 3 x1 &6 : 7 Œ E2 !C # # # Œ En !C 4 :: 5 .8/ xn

E\WKHGHÀQLWLRQRIF 7KLVVKRZVWKDWWKHPDWUL[ P VKRZQLQ VDWLVÀHV ŒY!C D P ŒY!B IRUHDFK Y LQ V C B C B EHFDXVHWKHYHFWRURQWKHULJKWVLGHRI LVG 6XSSRVH Q LVDQ\PDWUL[VXFKWKDW

ŒY!C D QŒY!B

IRUHDFK Y LQ V

.9/

6HW Y D E1 LQ 7KHQ VKRZVWKDW ŒE1 !C LVWKHÀUVW 6LPLODUO\ IRU k D 2; : : : ; n FROXPQRI Q EHFDXVHD EHFDXVHF 7KLV WKH k WKFROXPQRI Q LVE P VKRZVWKDWWKHPDWUL[ C B GHÀQHGE\ LQ7KHRUHPLV WKHRQO\PDWUL[WKDWVDWLVÀHVFRQGLWLRQ >0@ /HW B D f[0 ; : : : ; [6 g DQG C D f\0 ; : : : ; \6 g ZKHUH [k LV WKHIXQFWLRQ FRVk t DQG \k LVWKHIXQFWLRQ FRV kt ([HUFLVH LQ6HFWLRQ VKRZHGWKDWERWK B DQG C DUHEDVHVIRUWKH YHFWRUVSDFH H D 6SDQ f[0 ; : : : ; [6 g % & ### Œ \6 !B DQGFDOFXODWH P "1 D 6HW P D Œ \0 !B

E ([SODLQZK\WKHFROXPQVRI P "1 DUHWKH C FRRUGLQDWH YHFWRUVRI [0 ; : : : ; [6 7KHQXVHWKHVHFRRUGLQDWHYHFWRUV WRZULWHWULJRQRPHWULFLGHQWLWLHVWKDWH[SUHVVSRZHUVRI FRV t LQWHUPVRIWKHIXQFWLRQVLQ C 6HHWKH 6WXG\*XLGH

>0@ &DOFXOXVUHTXLUHG 5HFDOOIURPFDOFXOXVWKDWLQWHJUDOV VXFKDV Z .5 FRV3 t ! 6 FRV4 t C 5 FRV5 t ! 12 FRV6 t/ dt .10/ DUHWHGLRXVWRFRPSXWH 7KHXVXDOPHWKRGLVWRDSSO\LQWH JUDWLRQE\SDUWVUHSHDWHGO\DQGXVHWKHKDOIDQJOHIRUPXOD 8VHWKHPDWUL[ P RU P "1 IURP([HUFLVH WRWUDQVIRUP WKHQFRPSXWHWKHLQWHJUDO 3 7KHLGHDIRU([HUFLVHV DQGDQGÀYHUHODWHGH[HUFLVHVLQHDUOLHU VHFWLRQVFDPHIURPDSDSHUE\-DFN:5RJHUV -U RI$XEXUQ8QLYHUVLW\ SUHVHQWHGDWDPHHWLQJRIWKH,QWHUQDWLRQDO/LQHDU$OJHEUD6RFLHW\ $XJXVW 6HH´$SSOLFDWLRQVRI/LQHDU$OJHEUDLQ&DOFXOXVµ $PHULFDQ0DWKHPDWLFDO0RQWKO\

$)"15&3

>0@ /HW 2

9HFWRU6SDFHV

E )LQGDEDVLV fZ1 ; Z2 ; Z3 g IRU R3 VXFKWKDW P LVWKHFKDQJH RIFRRUGLQDWHVPDWUL[IURP fY1 ; Y2 ; Y3 g WR fZ1 ; Z2 ; Z3 g

3

1 2 !1 !5 0 5 P D 4 !3 4 6 1 2 3 2 3 2 3 !2 !8 !7 Y1 D 4 2 5 Y2 D 4 5 5 Y3 D 4 2 5 3 2 6

/HW B D fE1 ; E2 g C D fF1 ; F2 g DQG D D fG1 ; G2 g EHEDVHV IRUDWZRGLPHQVLRQDOYHFWRUVSDFH D :ULWHDQHTXDWLRQWKDWUHODWHVWKHPDWULFHV C P B D P C DQG P -XVWLI\\RXUUHVXOW D

D )LQG D EDVLV fX1 ; X2 ; X3 g IRU R3 VXFK WKDW P LV WKH FKDQJHRIFRRUGLQDWHV PDWUL[ IURP fX1 ; X2 ; X3 g WR WKH EDVLV fY1 ; Y2 ; Y3 g >+LQW :KDWGRWKHFROXPQVRI C P B UHSUHVHQW"@

B

E >0@ 8VHDPDWUL[SURJUDPHLWKHUWRKHOS\RXÀQGWKH HTXDWLRQRUWRFKHFNWKHHTXDWLRQ\RXZULWH :RUNZLWK WKUHHEDVHVIRU R2 6HH([HUFLVHV²

40-65*0/4 50 13"$5*$& 130#-&.4 6LQFHWKHFROXPQVRI P DUH G FRRUGLQDWHYHFWRUV DYHFWRURIWKHIRUP P [ PXVWEH D G FRRUGLQDWHYHFWRU 7KXV P VDWLVÀHVHTXDWLRQLL 7KHFRRUGLQDWHYHFWRUVIRXQGLQ([DPSOH VKRZWKDW " # % & 4 !6 P D Œ E1 ! Œ E2 !C D C B C 1 1 " 1 P D . P /!1 D 1 B C C B 10 !1

+HQFH

6 4

#

D

"

:1 !:1

:6 :4

#

"11-*$"5*0/4 50 %*''&3&/$& &26"5*0/4 1RZ WKDW SRZHUIXO FRPSXWHUV DUH ZLGHO\ DYDLODEOH PRUH DQG PRUH VFLHQWLÀF DQG HQJLQHHULQJSUREOHPVDUHEHLQJWUHDWHGLQDZD\WKDWXVHVGLVFUHWH RUGLJLWDO GDWDUDWKHU WKDQFRQWLQXRXVGDWD 'LIIHUHQFHHTXDWLRQVDUHRIWHQWKHDSSURSULDWHWRROWRDQDO\]H VXFKGDWD (YHQZKHQDGLIIHUHQWLDOHTXDWLRQLVXVHGWRPRGHODFRQWLQXRXVSURFHVV D QXPHULFDOVROXWLRQLVRIWHQSURGXFHGIURPDUHODWHGGLIIHUHQFHHTXDWLRQ 7KLVVHFWLRQKLJKOLJKWVVRPHIXQGDPHQWDOSURSHUWLHVRIOLQHDUGLIIHUHQFHHTXDWLRQV WKDWDUHEHVWH[SODLQHGXVLQJOLQHDUDOJHEUD

'LVFUHWH7LPH6LJQDOV 7KHYHFWRUVSDFH S RIGLVFUHWHWLPHVLJQDOVZDVLQWURGXFHGLQ6HFWLRQ $ VLJQDO LQ S LVDIXQFWLRQGHÀQHGRQO\RQWKHLQWHJHUVDQGLVYLVXDOL]HGDVDVHTXHQFHRIQXPEHUV VD\ fyk g )LJXUH VKRZVWKUHHW\SLFDOVLJQDOVZKRVHJHQHUDOWHUPVDUH .:7/k 1k DQG .!1/k UHVSHFWLYHO\ yk = .7k

– 2 –1

0

yk = 1k

1

2

'*(63& 7KUHHVLJQDOVLQ S

–2 – 1

0

yk = (–1)k

1

2

–2

0

2

$SSOLFDWLRQVWR'LIIHUHQFH(TXDWLRQV

'LJLWDOVLJQDOVREYLRXVO\DULVHLQHOHFWULFDODQGFRQWUROV\VWHPVHQJLQHHULQJ EXW GLVFUHWHGDWDVHTXHQFHVDUHDOVRJHQHUDWHGLQELRORJ\ SK\VLFV HFRQRPLFV GHPRJUDSK\ DQGPDQ\RWKHUDUHDV ZKHUHYHUDSURFHVVLVPHDVXUHG RU VDPSOHG DWGLVFUHWHWLPH LQWHUYDOV :KHQDSURFHVVEHJLQVDWDVSHFLÀFWLPH LWLVVRPHWLPHVFRQYHQLHQWWRZULWH DVLJQDODVDVHTXHQFHRIWKHIRUP .y0 ; y1 ; y2 ; : : :/ 7KHWHUPV yk IRU k < 0 HLWKHUDUH DVVXPHGWREH]HURRUDUHVLPSO\RPLWWHG

&9".1-& 7KHFU\VWDOFOHDUVRXQGVIURPDFRPSDFWGLVFSOD\HUDUHSURGXFHG IURPPXVLFWKDWKDVEHHQVDPSOHGDWWKHUDWHRIWLPHVSHUVHFRQG 6HH)LJ $W HDFKPHDVXUHPHQW WKHDPSOLWXGHRIWKHPXVLFVLJQDOLVUHFRUGHGDVDQXPEHU VD\ yk 7KHRULJLQDOPXVLFLVFRPSRVHGRIPDQ\GLIIHUHQWVRXQGVRIYDU\LQJIUHTXHQFLHV \HW WKHVHTXHQFH fyk g FRQWDLQVHQRXJKLQIRUPDWLRQWRUHSURGXFHDOOWKHIUHTXHQFLHVLQWKH VRXQGXSWRDERXWF\FOHVSHUVHFRQG KLJKHUWKDQWKHKXPDQHDUFDQVHQVH

y

t '*(63& 6DPSOHGGDWDIURPDPXVLFVLJQDO

/LQHDU,QGHSHQGHQFHLQWKH6SDFH S RI6LJQDOV 7RVLPSOLI\QRWDWLRQ ZHFRQVLGHUDVHWRIRQO\WKUHHVLJQDOVLQ S VD\ fuk g fvk g DQG fwk g 7KH\DUHOLQHDUO\LQGHSHQGHQWSUHFLVHO\ZKHQWKHHTXDWLRQ

c1 uk C c2 vk C c3 wk D 0

IRUDOO k

LPSOLHVWKDW c1 D c2 D c3 D 0 7KHSKUDVHÍRUDOO k µPHDQVIRUDOOLQWHJHUV³SRVLWLYH QHJDWLYH DQG]HUR 2QHFRXOGDOVRFRQVLGHUVLJQDOVWKDWVWDUWZLWK k D 0 IRUH[DPSOH LQZKLFKFDVH ÍRUDOO k µZRXOGPHDQIRUDOOLQWHJHUV k " 0 6XSSRVH c1 c2 c3 VDWLVI\ 7KHQHTXDWLRQ KROGVIRUDQ\WKUHHFRQVHFXWLYH YDOXHVRI k VD\ k k C 1 DQG k C 2 7KXV LPSOLHVWKDW

c1 uk C1 C c2 vk C1 C c3 wk C1 D 0

IRUDOO k

c1 uk C2 C c2 vk C2 C c3 wk C2 D 0

IRUDOO k

DQG

SG

5IF $BTPSBUJ 5FTU m

+HQFH c1 c2 c3 VDWLVI\ 2 uk 4 uk C1 uk C2

vk vk C1 vk C2

32 3 2 3 wk c1 0 wk C1 54 c2 5 D 4 0 5 wk C2 c3 0

IRUDOO k

7KH FRHIÀFLHQWPDWUL[LQWKLVV\VWHPLVFDOOHGWKH &DVRUDWLPDWUL[ RIWKHVLJQDOV DQG WKHGHWHUPLQDQWRIWKHPDWUL[LVFDOOHGWKH &DVRUDWLDQ RI fuk g fvk g DQG fwk g ,I WKH&DVRUDWLPDWUL[LVLQYHUWLEOHIRUDWOHDVWRQHYDOXHRI k WKHQ ZLOOLPSO\WKDW c1 D c2 D c3 D 0 ZKLFKZLOOSURYHWKDWWKHWKUHHVLJQDOVDUHOLQHDUO\LQGHSHQGHQW

$)"15&3

9HFWRU6SDFHV

&9".1-& 9HULI\WKDW 1k .!2/k DQG 3k DUHOLQHDUO\LQGHSHQGHQWVLJQDOV

–4

k

–2 –2

7KHVLJQDOV 1k .!2/k DQG 3k

.!2/k .!2/k C1 .!2/k C2

3 3k 3k C1 5 3k C2

5RZRSHUDWLRQVFDQVKRZIDLUO\HDVLO\WKDWWKLVPDWUL[LVDOZD\VLQYHUWLEOH +RZHYHU LW LVIDVWHUWRVXEVWLWXWHDYDOXHIRU k ³VD\ k D 0³DQGURZUHGXFHWKHQXPHULFDOPDWUL[ 2 3 2 3 2 3 1 1 1 1 1 1 1 1 1 4 1 !2 3 5 & 4 0 !3 2 5 & 4 0 !3 25 1 4 9 0 3 8 0 0 10 7KH &DVRUDWL PDWUL[ LV LQYHUWLEOH IRU k D 0 LQGHSHQGHQW

6R 1k .!2/k DQG 3k DUH OLQHDUO\

,ID&DVRUDWLPDWUL[LVQRWLQYHUWLEOH WKHDVVRFLDWHGVLJQDOVEHLQJWHVWHGPD\RU PD\QRWEHOLQHDUO\GHSHQGHQW 6HH([HUFLVH +RZHYHU LWFDQEHVKRZQWKDWLI WKHVLJQDOVDUHDOOVROXWLRQVRIWKH VDPH KRPRJHQHRXVGLIIHUHQFHHTXDWLRQGHVFULEHG EHORZ WKHQHLWKHUWKH&DVRUDWLPDWUL[LVLQYHUWLEOHIRUDOO k DQGWKHVLJQDOVDUHOLQHDUO\ LQGHSHQGHQW RUHOVHWKH&DVRUDWLPDWUL[LVQRWLQYHUWLEOHIRUDOO k DQGWKHVLJQDOVDUH OLQHDUO\GHSHQGHQW $ QLFHSURRIXVLQJOLQHDUWUDQVIRUPDWLRQVLVLQWKH 6WXG\*XLGH

/LQHDU'LIIHUHQFH(TXDWLRQV *LYHQVFDODUV a0 ; : : : ; an ZLWK a0 DQG an QRQ]HUR DQGJLYHQDVLJQDO f´k g WKHHTXDWLRQ

a0 yk Cn C a1 yk Cn!1 C # # # C an!1 yk C1 C an yk D ´k

IRUDOO k

LVFDOOHGD OLQHDUGLIIHUHQFHHTXDWLRQ RU OLQHDUUHFXUUHQFHUHODWLRQ RIRUGHU n )RU VLPSOLFLW\ a0 LVRIWHQWDNHQHTXDOWR ,I f´k g LVWKH]HURVHTXHQFH WKHHTXDWLRQLV KRPRJHQHRXV RWKHUZLVH WKHHTXDWLRQLV QRQKRPRJHQHRXV

&9".1-& ,QGLJLWDOVLJQDOSURFHVVLQJ DGLIIHUHQFHHTXDWLRQVXFKDV GH VFULEHVD OLQHDUÀOWHU DQG a0 ; : : : ; an DUHFDOOHGWKH ÀOWHUFRHIÀFLHQWV ,I fyk g LVWUHDWHG DVWKHLQSXWDQG f´k g DVWKHRXWSXW WKHQWKHVROXWLRQVRIWKHDVVRFLDWHGKRPRJHQHRXV HTXDWLRQDUHWKHVLJQDOVWKDWDUHÀOWHUHG RXW DQGWUDQVIRUPHGLQWRWKH]HURVLJQDO /HWXV IHHGWZRGLIIHUHQWVLJQDOVLQWRWKHÀOWHU

:35yk C2 C :5yk C1 C :35yk D ´k p +HUHLVDQDEEUHYLDWLRQIRU 2=4 7KHÀUVWVLJQDOLVFUHDWHGE\VDPSOLQJWKH FRQWLQXRXVVLJQDO y D FRV." t =4/ DWLQWHJHUYDOXHVRI t DVLQ)LJ D 7KHGLVFUHWH VLJQDOLV fyk g D f: : : ; FRV.0/; FRV."=4/; FRV.2"=4/; FRV.3"=4/; : : :g p )RUVLPSOLFLW\ ZULWH ˙:7 LQSODFHRI ˙ 2=2 VRWKDW fyk g D f : : : ; 1; :7; 0; !:7; !1; !:7; 0; :7; 1; :7; 0; : : :g "

2

40-65*0/ 7KH&DVRUDWLPDWUL[LV 2 k 1 4 1k C1 1k C2

kD0

7DEOHVKRZVDFDOFXODWLRQRIWKHRXWSXWVHTXHQFH f´k g ZKHUH :35.:7/ LVDQDEEUHYLDWLRQ p p IRU . 2=4/. 2=2/ D :25 7KHRXWSXWLV fyk g VKLIWHGE\RQHWHUP

y

y

πt  y = cos –– 4 

1

$SSOLFDWLRQVWR'LIIHUHQFH(TXDWLRQV 3πt  y = cos –––  4 

1

2

1

t

1 –1

t

2 –1

(a)

(b)

'*(63& 'LVFUHWHVLJQDOVZLWKGLIIHUHQWIUHTXHQFLHV

5"#-& k

$PNQVUJOH UIF 0VUQVU PG B 'JMUFS

yk ykC1 ykC2

:35yk

C :5ykC1 C :35ykC2 D ´k

:35.!:7/

:5.!:7/ :35.!1/

!:7

!:7

!:7

!1

!:7

:35.!:7/ :5.!1/ :35.!:7/

!:7

:35.!1/ :5.!:7/

:35.!:7/

:: :

!1

!:7

!:7 :: :

!1

!1

!:7 :: :

$ GLIIHUHQW LQSXW VLJQDO LV SURGXFHG IURP WKH KLJKHU IUHTXHQF\ VLJQDO y D FRV.3" t =4/ VKRZQLQ)LJ E 6DPSOLQJDWWKHVDPHUDWHDVEHIRUHSURGXFHVDQHZ LQSXWVHTXHQFH "

fwk g D f: : : ; 1; !:7; 0; :7; !1; :7; 0; !:7; 1; !:7; 0; : : :g kD0

:KHQ fwk g LVIHGLQWRWKHÀOWHU WKHRXWSXWLVWKH]HURVHTXHQFH 7KHÀOWHU FDOOHGD ORZSDVVÀOWHU OHWV fyk g SDVVWKURXJK EXWVWRSVWKHKLJKHUIUHTXHQF\ fwk g ,QPDQ\DSSOLFDWLRQV DVHTXHQFH f´k g LVVSHFLÀHGIRUWKHULJKWVLGHRIDGLIIHUHQFH HTXDWLRQ DQGD fyk g WKDWVDWLVÀHV LVFDOOHGD VROXWLRQ RIWKHHTXDWLRQ 7KHQH[W H[DPSOHVKRZVKRZWRÀQGVROXWLRQVIRUDKRPRJHQHRXVHTXDWLRQ

&9".1-& 6ROXWLRQVRIDKRPRJHQHRXVGLIIHUHQFHHTXDWLRQRIWHQKDYHWKHIRUP yk D r k IRUVRPH r )LQGVRPHVROXWLRQVRIWKHHTXDWLRQ yk C3 ! 2yk C2 ! 5yk C1 C 6yk D 0

IRUDOO k

40-65*0/ 6XEVWLWXWH r IRU yk LQWKHHTXDWLRQDQGIDFWRUWKHOHIWVLGH k

r k C3 ! 2r k C2 ! 5r k C1 C 6r k D 0 k

3

2

r .r ! 2r ! 5r C 6/ D 0

k

r .r ! 1/.r C 2/.r ! 3/ D 0

6LQFH LVHTXLYDOHQWWR r k VDWLVÀHVWKHGLIIHUHQFHHTXDWLRQ LIDQGRQO\LI r VDWLVÀHV 7KXV 1k .!2/k DQG 3k DUHDOOVROXWLRQVRI )RULQVWDQFH WRYHULI\WKDW 3k LVDVROXWLRQRI FRPSXWH

3k C3 ! 2 # 3k C2 ! 5 # 3k C1 C 6 # 3k

D 3k .27 ! 18 ! 15 C 6/ D 0

IRUDOO k

$)"15&3

9HFWRU6SDFHV

,QJHQHUDO DQRQ]HURVLJQDO r k VDWLVÀHVWKHKRPRJHQHRXVGLIIHUHQFHHTXDWLRQ

yk Cn C a1 yk Cn!1 C # # # C an!1 yk C1 C an yk D 0

IRUDOO k

LIDQGRQO\LI r LVDURRWRIWKH DX[LOLDU\HTXDWLRQ

r n C a1 r n!1 C # # # C an!1 r C an D 0 :HZLOOQRWFRQVLGHUWKHFDVHLQZKLFK r LVDUHSHDWHGURRWRIWKHDX[LOLDU\HTXDWLRQ :KHQWKHDX[LOLDU\HTXDWLRQKDVD FRPSOH[URRW WKHGLIIHUHQFHHTXDWLRQKDVVROXWLRQV RIWKHIRUP s k FRV k! DQG s k VLQ k! IRUFRQVWDQWV s DQG ! 7KLVKDSSHQHGLQ([DPSOH

6ROXWLRQ6HWVRI/LQHDU'LIIHUHQFH(TXDWLRQV *LYHQ a1 ; : : : ; an FRQVLGHUWKHPDSSLQJ T W S ! S WKDWWUDQVIRUPVDVLJQDO fyk g LQWRD VLJQDO fwk g JLYHQE\

wk D yk Cn C a1 yk Cn!1 C # # # C an!1 yk C1 C an yk ,WLVUHDGLO\FKHFNHGWKDW T LVD OLQHDU WUDQVIRUPDWLRQ 7KLVLPSOLHVWKDWWKHVROXWLRQVHW RIWKHKRPRJHQHRXVHTXDWLRQ

yk Cn C a1 yk Cn!1 C # # # C an!1 yk C1 C an yk D 0

IRUDOO k

LVWKHNHUQHORI T WKHVHWRIVLJQDOVWKDW T PDSVLQWRWKH]HURVLJQDO DQGKHQFHWKH VROXWLRQVHWLVD VXEVSDFH RI S $Q\OLQHDUFRPELQDWLRQRIVROXWLRQVLVDJDLQDVROXWLRQ 7KHQH[WWKHRUHP DVLPSOHEXWEDVLFUHVXOW ZLOOOHDGWRPRUHLQIRUPDWLRQDERXWWKH VROXWLRQVHWVRIGLIIHUHQFHHTXDWLRQV

5)&03&.

,I an ¤ 0 DQGLI f´k g LVJLYHQ WKHHTXDWLRQ

yk Cn C a1 yk Cn!1 C # # # C an!1 yk C1 C an yk D ´k

IRUDOO k

KDVDXQLTXHVROXWLRQZKHQHYHU y0 ; : : : ; yn!1 DUHVSHFLÀHG

1300' ,I y0 ; : : : ; yn!1 DUHVSHFLÀHG XVH WR GHÀQH yn D ´0 ! Œ a1 yn!1 C # # # C an!1 y1 C an y0 ! $QGQRZWKDW y1 ; : : : ; yn DUHVSHFLÀHG XVH WRGHÀQH ynC1 ,QJHQHUDO XVHWKH UHFXUUHQFHUHODWLRQ ynCk D ´k ! Œ a1 yk Cn!1 C # # # C an yk ! WRGHÀQH ynCk IRU k " 0 7RGHÀQH yk IRU k < 0 XVHWKHUHFXUUHQFHUHODWLRQ

yk D

1 1 ´k ! Œ yk Cn C a1 yk Cn!1 C # # # C an!1 yk C1 ! an an

7KLVSURGXFHVDVLJQDOWKDWVDWLVÀHV &RQYHUVHO\ DQ\VLJQDOWKDWVDWLVÀHV IRUDOO k FHUWDLQO\VDWLVÀHV DQG VRWKHVROXWLRQRI LVXQLTXH

5)&03&.

7KHVHW H RIDOOVROXWLRQVRIWKH nWKRUGHUKRPRJHQHRXVOLQHDUGLIIHUHQFHHTXDWLRQ

yk Cn C a1 yk Cn!1 C # # # C an!1 yk C1 C an yk D 0 LVDQ nGLPHQVLRQDOYHFWRUVSDFH

IRUDOO k


1300' $VZDVSRLQWHGRXWHDUOLHU H LVDVXEVSDFHRI S EHFDXVH H LVWKHNHUQHO RIDOLQHDUWUDQVIRUPDWLRQ )RU fyk g LQ H OHW F fyk g EHWKHYHFWRULQ Rn JLYHQE\ .y0 ; y1 ; : : : ; yn!1 / ,WLVUHDGLO\YHULÀHGWKDW F W H ! Rn LVDOLQHDUWUDQVIRUPDWLRQ *LYHQDQ\YHFWRU .y0 ; y1 ; : : : ; yn!1 / LQ Rn 7KHRUHPVD\VWKDWWKHUHLVDXQLTXH VLJQDO fyk g LQ H VXFKWKDW F fyk g D .y0 ; y1 ; : : : ; yn!1 / 7KLVPHDQVWKDW F LVDRQH WRRQH OLQHDU WUDQVIRUPDWLRQ RI H RQWR Rn WKDW LV F LV DQ LVRPRUSKLVP 7KXV GLP H D GLP Rn D n 6HH([HUFLVH LQ6HFWLRQ

&9".1-& )LQGDEDVLVIRUWKHVHWRIDOOVROXWLRQVWRWKHGLIIHUHQFHHTXDWLRQ yk C3 ! 2yk C2 ! 5yk C1 C 6yk D 0

IRUDOO k

40-65*0/ 2XUZRUNLQOLQHDUDOJHEUDUHDOO\SD\VRIIQRZ :HNQRZIURP([DPSOHV DQGWKDW 1k .!2/k DQG 3k DUHOLQHDUO\LQGHSHQGHQWVROXWLRQV ,QJHQHUDO LWFDQEH GLIÀFXOWWRYHULI\GLUHFWO\WKDWDVHWRIVLJQDOV VSDQV WKHVROXWLRQVSDFH %XWWKDWLVQR SUREOHPKHUHEHFDXVHRIWZRNH\WKHRUHPV³7KHRUHP ZKLFKVKRZVWKDWWKHVROXWLRQ VSDFHLVH[DFWO\WKUHHGLPHQVLRQDO DQGWKH%DVLV7KHRUHPLQ6HFWLRQ ZKLFKVD\V WKDWDOLQHDUO\LQGHSHQGHQWVHWRI n YHFWRUVLQDQ nGLPHQVLRQDOVSDFHLVDXWRPDWLFDOO\ DEDVLV 6R 1k .!2/k DQG 3k IRUPDEDVLVIRUWKHVROXWLRQVSDFH 7KHVWDQGDUGZD\WRGHVFULEHWKH´JHQHUDOVROXWLRQµRIWKHGLIIHUHQFHHTXDWLRQ LVWRH[KLELWDEDVLVIRUWKHVXEVSDFHRIDOOVROXWLRQV 6XFKDEDVLVLVXVXDOO\FDOOHGD IXQGDPHQWDOVHWRIVROXWLRQV RI ,QSUDFWLFH LI\RXFDQÀQG n OLQHDUO\LQGHSHQGHQW VLJQDOVWKDWVDWLVI\ WKH\ZLOODXWRPDWLFDOO\VSDQWKH nGLPHQVLRQDOVROXWLRQVSDFH DVH[SODLQHGLQ([DPSOH

1RQKRPRJHQHRXV(TXDWLRQV 7KHJHQHUDOVROXWLRQRIWKHQRQKRPRJHQHRXVGLIIHUHQFHHTXDWLRQ

yk Cn C a1 yk Cn!1 C # # # C an!1 yk C1 C an yk D ´k

IRUDOO k

FDQEHZULWWHQDVRQHSDUWLFXODUVROXWLRQRI SOXVDQDUELWUDU\OLQHDUFRPELQDWLRQRI DIXQGDPHQWDOVHWRIVROXWLRQVRIWKHFRUUHVSRQGLQJKRPRJHQHRXVHTXDWLRQ 7KLV IDFWLVDQDORJRXVWRWKHUHVXOWLQ6HFWLRQ VKRZLQJWKDWWKHVROXWLRQVHWVRI A[ D E DQG A[ D DUHSDUDOOHO %RWKUHVXOWVKDYHWKHVDPHH[SODQDWLRQ 7KHPDSSLQJ [ 7! A[ LVOLQHDU DQGWKHPDSSLQJWKDWWUDQVIRUPVWKHVLJQDO fyk g LQWRWKHVLJQDO f´k g LQ LV OLQHDU 6HH([HUFLVH

&9".1-& 9HULI\WKDWWKHVLJQDO yk D k 2 VDWLVÀHVWKHGLIIHUHQFHHTXDWLRQ yk C2 ! 4yk C1 C 3yk D !4k

IRUDOO k

7KHQÀQGDGHVFULSWLRQRIDOOVROXWLRQVRIWKLVHTXDWLRQ

40-65*0/ 6XEVWLWXWH k 2 IRU yk RQWKHOHIWVLGHRI .k C 2/2 !4.k C 1/2 C 3k 2

D .k 2 C 4k C 4/ ! 4.k 2 C 2k C 1/ C 3k 2 D !4k

6R k 2 LVLQGHHGDVROXWLRQRI 7KHQH[WVWHSLVWRVROYHWKHKRPRJHQHRXVHTXDWLRQ

yk C2 ! 4yk C1 C 3yk D 0 7KHDX[LOLDU\HTXDWLRQLV

r 2 ! 4r C 3 D .r ! 1/.r ! 3/ D 0

k2

$)"15&3

an k + Sp 2

3k

9HFWRU6SDFHV

k k} {1 , 3

k , 3k }

Span{1

1k

'*(63&

6ROXWLRQVHWVRIGLIIHUHQFH HTXDWLRQV DQG

7KHURRWVDUH r D 1; 3 6RWZRVROXWLRQVRIWKHKRPRJHQHRXVGLIIHUHQFHHTXDWLRQDUH 1k DQG 3k 7KH\DUHREYLRXVO\QRWPXOWLSOHVRIHDFKRWKHU VRWKH\DUHOLQHDUO\LQGHSHQGHQW VLJQDOV %\7KHRUHP WKHVROXWLRQVSDFHLVWZRGLPHQVLRQDO VR 3k DQG 1k IRUPDEDVLV IRUWKHVHWRIVROXWLRQVRIHTXDWLRQ 7UDQVODWLQJWKDWVHWE\DSDUWLFXODUVROXWLRQRI WKHQRQKRPRJHQHRXVHTXDWLRQ ZHREWDLQWKHJHQHUDOVROXWLRQRI

k 2 C c1 1k C c2 3k ;

k 2 C c1 C c2 3k

RU

)LJXUH JLYHVDJHRPHWULFYLVXDOL]DWLRQRIWKHWZRVROXWLRQVHWV (DFKSRLQWLQWKHÀJXUH FRUUHVSRQGVWRRQHVLJQDOLQ S

5HGXFWLRQWR6\VWHPVRI)LUVW2UGHU(TXDWLRQV $ PRGHUQZD\WRVWXG\DKRPRJHQHRXV nWKRUGHUOLQHDUGLIIHUHQFHHTXDWLRQLVWRUHSODFH LWE\DQHTXLYDOHQWV\VWHPRIÀUVWRUGHUGLIIHUHQFHHTXDWLRQV ZULWWHQLQWKHIRUP [k C1 D A[k

IRUDOO k

ZKHUHWKHYHFWRUV [k DUHLQ Rn DQG A LVDQ n % n PDWUL[ $ VLPSOHH[DPSOHRIVXFKDYHFWRUYDOXHG GLIIHUHQFHHTXDWLRQZDVDOUHDG\VWXGLHG LQ6HFWLRQ )XUWKHUH[DPSOHVZLOOEHFRYHUHGLQ6HFWLRQV DQG

&9".1-& :ULWHWKHIROORZLQJGLIIHUHQFHHTXDWLRQDVDÀUVWRUGHUV\VWHP yk C3 ! 2yk C2 ! 5yk C1 C 6yk D 0 40-65*0/ )RUHDFK k VHW

IRUDOO k

2

3 yk [k D 4 yk C1 5 yk C2

7KHGLIIHUHQFHHTXDWLRQVD\VWKDW yk C3 D !6yk C 5yk C1 C 2yk C2 VR 3 2 2 3 2 32 3 0 C yk C1 C 0 yk C1 0 1 0 yk 6 7 [k C1 D 4 yk C2 5 D 4 0 C 0 C yk C2 5 D 4 0 0 1 54 yk C1 5 yk C3 yk C2 !6 5 2 !6yk C 5yk C1 C 2yk C2 7KDWLV

[k C1 D A[k ,QJHQHUDO WKHHTXDWLRQ

IRUDOO k;

ZKHUH

2

0 AD4 0 !6

yk Cn C a1 yk Cn!1 C # # # C an!1 yk C1 C an yk D 0 FDQEHUHZULWWHQDV [k C1 D A[k IRUDOO k ZKHUH 2 2 3 0 1 0 yk 6 0 0 1 6 : 6 yk C1 7 6 : 6 7 [k D 6 7; A D 6 : :: 6 4 5 : 4 0 0 0 yk Cn!1 !an !an!1 !an!2

3 0 15 2

1 0 5

IRUDOO k

:::

0 0 :: :

3

7 7 7 : 7 7 1 5 # # # !a1

::


)XUWKHU5HDGLQJ +DPPLQJ 5: 'LJLWDO)LOWHUV UGHG(QJOHZRRG&OLIIV 1-3UHQWLFH+DOO SS ² .HOO\ :* DQG$&3HWHUVRQ 'LIIHUHQFH(TXDWLRQV QGHG6DQ'LHJR +DUFRXUW $FDGHPLF3UHVV 0LFNHQV 5( 'LIIHUHQFH(TXDWLRQV QGHG1HZ
13"$5*$& 130#-&. ,WFDQEHVKRZQWKDWWKHVLJQDOV 2k 3k VLQ k" DQG 3k FRV k" DUHVROXWLRQVRI 2 2

yk C3 ! 2yk C2 C 9yk C1 ! 18yk D 0 6KRZWKDWWKHVHVLJQDOVIRUPDEDVLVIRUWKHVHWRIDOOVROXWLRQVRIWKHGLIIHUHQFHHTXDWLRQ

&9&3$*4&4 9HULI\WKDWWKHVLJQDOVLQ([HUFLVHVDQGDUHVROXWLRQVRIWKH DFFRPSDQ\LQJGLIIHUHQFHHTXDWLRQ 2k ; .!4/k I ykC2 C 2ykC1 ! 8yk D 0 5k ; .!5/k I ykC2 ! 25yk D 0

6KRZWKDWWKHVLJQDOVLQ([HUFLVHV²IRUPDEDVLVIRUWKHVROXWLRQ VHWRIWKHDFFRPSDQ\LQJGLIIHUHQFHHTXDWLRQ 7KHVLJQDOVDQGHTXDWLRQLQ([HUFLVH 7KHVLJQDOVDQGHTXDWLRQLQ([HUFLVH

,Q([HUFLVHV² ÀQGDEDVLVIRUWKHVROXWLRQVSDFHRIWKH GLIIHUHQFHHTXDWLRQ 3URYHWKDWWKHVROXWLRQV\RXÀQGVSDQWKH VROXWLRQVHW ykC2 ! ykC1 C 29 yk D 0

ykC2 ! 5ykC1 C 6yk D 0

6ykC2 C ykC1 ! 2yk D 0 ykC2 ! 25yk D 0 ([HUFLVHV DQG FRQFHUQ D VLPSOH PRGHO RI WKH QDWLRQDO HFRQRP\GHVFULEHGE\WKHGLIIHUHQFHHTXDWLRQ

YkC2 ! a.1 C b/YkC1 C abYk D 1

.!2/k k.!2/k ykC2 C 4ykC1 C 4yk D 0 ' ( k ' ( 4k FRV k" 4 VLQ k" ykC2 C 16yk D 0 2 2

+HUH Yk LVWKHWRWDOQDWLRQDOLQFRPHGXULQJ\HDU k a LVDFRQVWDQW OHVVWKDQ FDOOHGWKH PDUJLQDOSURSHQVLW\WRFRQVXPH DQG b LV DSRVLWLYH FRQVWDQWRIDGMXVWPHQW WKDWGHVFULEHVKRZFKDQJHVLQ FRQVXPHUVSHQGLQJDIIHFWWKHDQQXDOUDWHRISULYDWHLQYHVWPHQW

1k 2k .!2/k ykC3 ! ykC2 ! 4ykC1 C 4yk D 0

)LQGWKHJHQHUDOVROXWLRQRIHTXDWLRQ ZKHQ a D :9 DQG b D 49 :KDWKDSSHQVWR Yk DV k LQFUHDVHV" >+LQW )LUVWÀQGD SDUWLFXODUVROXWLRQRIWKHIRUP Yk D T ZKHUH T LVDFRQVWDQW FDOOHGWKHHTXLOLEULXPOHYHORIQDWLRQDOLQFRPH@

,Q ([HUFLVHV ² DVVXPH WKH VLJQDOV OLVWHG DUH VROXWLRQV RI WKHJLYHQGLIIHUHQFHHTXDWLRQ 'RWKHVLJQDOVIRUPDEDVLVIRU WKHVROXWLRQVSDFHRIWKHHTXDWLRQ" -XVWLI\\RXUDQVZHUVXVLQJ DSSURSULDWHWKHRUHPV

.!1/k 2k 3k ykC3 ! 4ykC2 C 1ykC1 C 6yk D 0 ' ( k ' ( 2k 5k FRV k" 5 VLQ k" 2 2 ykC3 ! 2ykC2 C 25ykC1 ! 50yk D 0

.!2/k k.!2/k 3k ykC3 C ykC2 ! 8ykC1 ! 12yk D 0 .!1/ 2 ykC3 ! 3ykC2 C 4yk D 0 k

k

3k .!2/k ykC4 ! 13ykC2 C 36yk D 0

)LQGWKHJHQHUDOVROXWLRQRIHTXDWLRQ ZKHQ a D :9 DQG b D :5 1 )RUH[DPSOH

VHH 'LVFUHWH'\QDPLFDO6\VWHPV E\-DPHV76DQGHIXU 2[IRUG &ODUHQGRQ3UHVV SS ² 7KHRULJLQDO DFFHOHUDWRUPXOWLSOLHUPRGHO LVDWWULEXWHGWRWKHHFRQRPLVW3$ 6DPXHOVRQ

$)"15&3

9HFWRU6SDFHV

$ OLJKWZHLJKWFDQWLOHYHUHGEHDPLVVXSSRUWHGDW N SRLQWVVSDFHG IWDSDUW DQGDZHLJKWRIOELVSODFHGDWWKHHQGRIWKH EHDP IWIURPWKHÀUVWVXSSRUW DVLQWKHÀJXUH /HW yk EH WKHEHQGLQJPRPHQWDWWKH k WKVXSSRUW 7KHQ y1 D 5000 IWOE 6XSSRVHWKHEHDPLVULJLGO\DWWDFKHGDWWKH N WKVXSSRUWDQGWKH EHQGLQJPRPHQWWKHUHLV]HUR ,QEHWZHHQ WKHPRPHQWVVDWLVI\ WKH WKUHHPRPHQWHTXDWLRQ

ykC2 C 4ykC1 C yk D 0 10'

500 lb

IRU k D 1; 2; : : : ; N ! 2

10'

1 –1

1

2

3

N

y1

y2

y3

yN

)LQGWKHJHQHUDOVROXWLRQRIGLIIHUHQFHHTXDWLRQ -XVWLI\ \RXUDQVZHU )LQGWKHSDUWLFXODUVROXWLRQRI WKDWVDWLVÀHVWKH ERXQGDU\ FRQGLWLRQV y1 D 5000 DQG yN D 0 7KHDQVZHULQYROYHV N :KHQDVLJQDOLVSURGXFHGIURPDVHTXHQFHRIPHDVXUHPHQWV PDGH RQ D SURFHVV D FKHPLFDO UHDFWLRQ D ÁRZ RI KHDW WKURXJKDWXEH DPRYLQJURERWDUP HWF WKHVLJQDOXVXDOO\ FRQWDLQVUDQGRP QRLVH SURGXFHGE\PHDVXUHPHQWHUURUV $ VWDQGDUGPHWKRGRISUHSURFHVVLQJWKHGDWDWRUHGXFHWKHQRLVH LVWRVPRRWKRUÀOWHUWKHGDWD 2QHVLPSOHÀOWHULVD PRYLQJ DYHUDJH WKDWUHSODFHVHDFK yk E\LWVDYHUDJHZLWKWKHWZR DGMDFHQWYDOXHV IRU k D 1; 2; : : :

6XSSRVHDVLJQDO yk IRU k D 0; : : : ; 14 LV

9; 5; 7; 3; 2; 4; 6; 5; 7; 6; 8; 10; 9; 5; 7 8VHWKHÀOWHUWRFRPSXWH ´1 ; : : : ; ´13 0DNHDEURNHQOLQH JUDSKWKDWVXSHULPSRVHVWKHRULJLQDOVLJQDODQGWKHVPRRWKHG VLJQDO /HW fyk g EHWKHVHTXHQFHSURGXFHGE\VDPSOLQJWKHFRQWLQX RXVVLJQDO 2 FRV "4t C FRV 3"4 t DW t D 0; 1; 2; : : : ; DVVKRZQLQ WKHÀJXUH 7KHYDOXHVRI yk EHJLQQLQJZLWK k D 0 DUH

3; :7; 0; !:7; !3; !:7; 0; :7; 3; :7; 0; : : : p ZKHUHLVDQDEEUHYLDWLRQIRU 2=2 D &RPSXWHWKHRXWSXWVLJQDO f´k g ZKHQ fyk g LVIHGLQWRWKH ÀOWHULQ([DPSOH E ([SODLQKRZDQGZK\WKHRXWSXWLQSDUWD LVUHODWHGWR WKHFDOFXODWLRQVLQ([DPSOH

t

1 2

6DPSOHGGDWDIURP 2 FRV

10'

C 13 yk C 13 yk"1 D ´k

πt ––– 3πt  y = 2 cos –– 4  + cos  4 

.15/

%HQGLQJPRPHQWVRQDFDQWLOHYHUHGEHDP

1 y 3 kC1

y

"t 4

C FRV

3" t 4

([HUFLVHVDQGUHIHUWRDGLIIHUHQFHHTXDWLRQRIWKHIRUP ykC1 ! ayk D b IRUVXLWDEOHFRQVWDQWV a DQG b $ ORDQRIKDVDQLQWHUHVWUDWHRISHUPRQWKDQGD PRQWKO\SD\PHQWRI 7KHORDQLVPDGHDWPRQWK k D 0 DQGWKHÀUVWSD\PHQWLVPDGHRQHPRQWKODWHU DW k D 1 )RU k D 0; 1; 2; : : : ; OHW yk EHWKHXQSDLGEDODQFHRIWKHORDQMXVW DIWHUWKH k WKPRQWKO\SD\PHQW 7KXV

y1 D 10;000 C .:01/10;000 ! 450 1HZ %DODQFH ,QWHUHVW 3D\PHQW EDODQFH GXH DGGHG D :ULWHDGLIIHUHQFHHTXDWLRQVDWLVÀHGE\ fyk g

E >0@ &UHDWHDWDEOHVKRZLQJ k DQGWKHEDODQFH yk DWPRQWK k /LVWWKHSURJUDPRUWKHNH\VWURNHV\RXXVHGWRFUHDWH WKHWDEOH F >0@ :KDWZLOO k EHZKHQWKHODVWSD\PHQWLVPDGH" +RZ PXFKZLOOWKHODVWSD\PHQWEH" +RZPXFKPRQH\GLGWKH ERUURZHUSD\LQWRWDO" $WWLPH k D 0 DQLQLWLDOLQYHVWPHQWRILVPDGHLQWRD VDYLQJVDFFRXQWWKDWSD\VLQWHUHVWSHU\HDUFRPSRXQGHG PRQWKO\ 7KHLQWHUHVWUDWHSHUPRQWKLV (DFKPRQWK DIWHUWKHLQLWLDOLQYHVWPHQW DQDGGLWLRQDOLVDGGHGWR WKHDFFRXQW )RU k D 0; 1; 2; : : : ; OHW yk EHWKHDPRXQWLQWKH DFFRXQWDWWLPH k MXVWDIWHUDGHSRVLWKDVEHHQPDGH D :ULWHDGLIIHUHQFHHTXDWLRQVDWLVÀHGE\ fyk g E >0@ &UHDWHDWDEOHVKRZLQJ k DQGWKHWRWDODPRXQWLQWKH VDYLQJVDFFRXQWDWPRQWK k IRU k D 0 WKURXJK /LVW \RXUSURJUDPRUWKHNH\VWURNHV\RXXVHGWRFUHDWHWKH WDEOH

F >0@ +RZPXFKZLOOEHLQWKHDFFRXQWDIWHUWZR\HDUVWKDW LV PRQWKV IRXU\HDUV DQGÀYH\HDUV" +RZPXFKRI WKHÀYH\HDUWRWDOLVLQWHUHVW" ,Q([HUFLVHV² VKRZWKDWWKHJLYHQVLJQDOLVDVROXWLRQRI WKHGLIIHUHQFHHTXDWLRQ 7KHQÀQGWKHJHQHUDOVROXWLRQRIWKDW GLIIHUHQFHHTXDWLRQ yk D k 2 I ykC2 C 3ykC1 ! 4yk D 7 C 10k yk D 1 C kI ykC2 ! 6ykC1 C 5yk D !4 yk D k ! 2I ykC2 ! 4yk D 8 ! 3k yk D 1 C 2kI ykC2 ! 25yk D !48k ! 20

:ULWHWKHGLIIHUHQFHHTXDWLRQVLQ([HUFLVHV DQGDVÀUVWRUGHU V\VWHPV [kC1 D A[k IRUDOO k ykC4 C 3ykC3 ! 8ykC2 C 6ykC1 ! 2yk D 0 ykC3 ! 5ykC2 C 8yk D 0

:KDWLVWKHRUGHURIWKHIROORZLQJGLIIHUHQFHHTXDWLRQ" ([ SODLQ\RXUDQVZHU

ykC3 C a1 ykC2 C a2 ykC1 C a3 yk D 0

/HW yk D k 2 DQG ´k D 2kjkj $UH WKH VLJQDOV fyk g DQG f´k g OLQHDUO\LQGHSHQGHQW" (YDOXDWHWKHDVVRFLDWHG&DVRUDWL PDWUL[ C.k/ IRU k D 0 k D !1 DQG k D !2 DQGGLVFXVV \RXUUHVXOWV

/HW a DQG b EHQRQ]HURQXPEHUV 6KRZWKDWWKHPDSSLQJ T GHÀQHGE\ T fyk g D fwk g ZKHUH

wk D ykC2 C aykC1 C byk

LVDOLQHDUWUDQVIRUPDWLRQIURP S LQWR S /HW V EHDYHFWRUVSDFH DQGOHW T W V ! V EHDOLQHDU WUDQVIRUPDWLRQ *LYHQ ] LQ V VXSSRVH [p LQ V VDWLVÀHV T .[p / D ] DQG OHW X EH DQ\ YHFWRU LQ WKH NHUQHO RI T 6KRZWKDW X C [p VDWLVÀHVWKHQRQKRPRJHQHRXVHTXDWLRQ T .[/ D ] /HW S0 EH WKH YHFWRU VSDFH RI DOO VHTXHQFHV RI WKH IRUP .y0 ; y1 ; y2 ; : : :/ DQGGHÀQHOLQHDUWUDQVIRUPDWLRQV T DQG D IURP S0 LQWR S0 E\

/HW f g DQG h EHOLQHDUO\LQGHSHQGHQWIXQFWLRQVGHÀQHGIRU DOOUHDOQXPEHUV DQGFRQVWUXFWWKUHHVLJQDOVE\VDPSOLQJWKH YDOXHVRIWKHIXQFWLRQVDWWKHLQWHJHUV

uk D f .k/;

vk D g.k/;

0XVWWKHVLJQDOVEHOLQHDUO\LQGHSHQGHQWLQ S" 'LVFXVV

,VWKHIROORZLQJGLIIHUHQFHHTXDWLRQRIRUGHU" ([SODLQ

ykC3 C 5ykC2 C 6ykC1 D 0

$SSOLFDWLRQVWR0DUNRY&KDLQV

wk D h.k/

T .y0 ; y1 ; y2 ; : : :/ D .y1 ; y2 ; y3 ; : : :/

D.y0 ; y1 ; y2 ; : : :/ D .0; y0 ; y1 ; y2 ; : : :/

6KRZWKDW TD D I WKHLGHQWLW\WUDQVIRUPDWLRQRQ S0 DQG \HW DT ¤ I

40-65*0/ 50 13"$5*$& 130#-&. ([DPLQHWKH&DVRUDWLPDWUL[ 2 2k 6 6 C.k/ D 6 2k C1 4 2k C2

3k VLQ k" 2 3k C1 VLQ .k C21/" 3k C2 VLQ .k C22/"

3k FRV k" 2

3

7 7 3k C1 FRV .k C21/" 7 5 3k C2 FRV .k C22/"

6HW k D 0 DQGURZUHGXFHWKHPDWUL[WRYHULI\WKDWLWKDVWKUHHSLYRWSRVLWLRQVDQGKHQFH LVLQYHUWLEOH 2 3 2 3 1 0 1 1 0 1 !2 5 C.0/ D 4 2 3 0 5 & 4 0 3 4 0 !9 0 0 !13

7KH&DVRUDWLPDWUL[LVLQYHUWLEOHDW k D 0 VRWKHVLJQDOVDUHOLQHDUO\LQGHSHQGHQW 6LQFHWKHUHDUHWKUHHVLJQDOV DQGWKHVROXWLRQVSDFH H RIWKHGLIIHUHQFHHTXDWLRQKDV GLPHQVLRQ 7KHRUHP WKHVLJQDOVIRUPDEDVLVIRU H E\WKH%DVLV7KHRUHP

"11-*$"5*0/4 50 ."3,07 $)"*/4 7KH0DUNRYFKDLQVGHVFULEHGLQWKLVVHFWLRQDUHXVHGDVPDWKHPDWLFDOPRGHOVRID ZLGHYDULHW\RIVLWXDWLRQVLQELRORJ\ EXVLQHVV FKHPLVWU\ HQJLQHHULQJ SK\VLFV DQG HOVHZKHUH ,QHDFKFDVH WKHPRGHOLVXVHGWRGHVFULEHDQH[SHULPHQWRUPHDVXUHPHQW WKDWLVSHUIRUPHGPDQ\WLPHVLQWKHVDPHZD\ ZKHUHWKHRXWFRPHRIHDFKWULDORIWKH H[SHULPHQWZLOOEHRQHRIVHYHUDOVSHFLÀHGSRVVLEOHRXWFRPHV DQGZKHUHWKHRXWFRPH RIRQHWULDOGHSHQGVRQO\RQWKHLPPHGLDWHO\SUHFHGLQJWULDO )RUH[DPSOH LIWKHSRSXODWLRQRIDFLW\DQGLWVVXEXUEVZHUHPHDVXUHGHDFK\HDU WKHQDYHFWRUVXFKDV " # :60 [0 D :40

$)"15&3

9HFWRU6SDFHV

FRXOGLQGLFDWHWKDWRIWKHSRSXODWLRQOLYHVLQWKHFLW\DQGLQWKHVXEXUEV 7KH GHFLPDOVLQ [0 DGGXSWREHFDXVHWKH\DFFRXQWIRUWKHHQWLUHSRSXODWLRQRIWKHUHJLRQ 3HUFHQWDJHVDUHPRUHFRQYHQLHQWIRURXUSXUSRVHVKHUHWKDQSRSXODWLRQWRWDOV $ YHFWRUZLWKQRQQHJDWLYHHQWULHVWKDWDGGXSWRLVFDOOHGD SUREDELOLW\YHFWRU $ VWRFKDVWLFPDWUL[ LVDVTXDUHPDWUL[ZKRVHFROXPQVDUHSUREDELOLW\YHFWRUV $ 0DUNRY FKDLQ LVDVHTXHQFHRISUREDELOLW\YHFWRUV [0 ; [1 ; [2 ; : : : WRJHWKHUZLWKDVWRFKDVWLF PDWUL[ P VXFKWKDW [1 D P [0 ;

[2 D P [1 ;

[3 D P [2 ;

:::

7KXVWKH0DUNRYFKDLQLVGHVFULEHGE\WKHÀUVWRUGHUGLIIHUHQFHHTXDWLRQ [k C1 D P [k

IRU k D 0; 1; 2; : : :

:KHQ D 0DUNRY FKDLQ RI YHFWRUV LQ Rn GHVFULEHV D V\VWHP RU D VHTXHQFH RI H[SHULPHQWV WKHHQWULHVLQ [k OLVW UHVSHFWLYHO\ WKHSUREDELOLWLHVWKDWWKHV\VWHPLVLQ HDFKRI n SRVVLEOHVWDWHV RUWKHSUREDELOLWLHVWKDWWKHRXWFRPHRIWKHH[SHULPHQWLVRQH RI n SRVVLEOHRXWFRPHV )RUWKLVUHDVRQ [k LVRIWHQFDOOHGD VWDWHYHFWRU

&9".1-& 6HFWLRQH[DPLQHGDPRGHOIRUSRSXODWLRQPRYHPHQWEHWZHHQD FLW\DQGLWVVXEXUEV 6HH)LJ 7KHDQQXDOPLJUDWLRQEHWZHHQWKHVHWZRSDUWVRIWKH PHWURSROLWDQUHJLRQZDVJRYHUQHGE\WKH PLJUDWLRQPDWUL[ M )URP &LW\ 6XEXUEV " #

:95 MD :05

:03 :97

7R &LW\ 6XEXUEV

7KDWLV HDFK\HDURIWKHFLW\SRSXODWLRQPRYHVWRWKHVXEXUEV DQGRIWKH VXEXUEDQSRSXODWLRQPRYHVWRWKHFLW\ 7KHFROXPQVRI M DUHSUREDELOLW\YHFWRUV VR M LVDVWRFKDVWLFPDWUL[ 6XSSRVHWKHSRSXODWLRQRIWKHUHJLRQLVLQ WKHFLW\DQGLQWKHVXEXUEV 7KHQWKHLQLWLDOGLVWULEXWLRQRIWKHSRSXODWLRQLQWKH UHJLRQLVJLYHQE\ [0 LQ DERYH :KDWLVWKHGLVWULEXWLRQRIWKHSRSXODWLRQLQ" ,Q" City

Suburbs .05

.95

.97 .03

'*(63& $QQXDOSHUFHQWDJHPLJUDWLRQEHWZHHQFLW\DQGVXEXUEV

40-65*0/ ,Q([DPSOHRI6HFWLRQ ZHVDZWKDWDIWHURQH\HDU WKHSRSXODWLRQ " # 600;000 FKDQJHGWR YHFWRU 400;000 " #" # " # :95 :03 600;000 582;000 D :05 :97 400;000 418;000


,IZHGLYLGHERWKVLGHVRIWKLVHTXDWLRQE\WKHWRWDOSRSXODWLRQRI PLOOLRQ DQGXVHWKH IDFWWKDW kM [ D M.k [/ ZHÀQGWKDW " #" # " # :95 :03 :600 :582 D :05 :97 :400 :418 "

# :582 7KHYHFWRU [1 D JLYHVWKHSRSXODWLRQGLVWULEXWLRQLQ 7KDWLV RI :418 WKHUHJLRQOLYHGLQWKHFLW\DQGOLYHGLQWKHVXEXUEV 6LPLODUO\ WKHSRSXODWLRQ GLVWULEXWLRQLQLVGHVFULEHGE\DYHFWRU [2 ZKHUH " #" # " # :95 :03 :582 :565 [ 2 D M [1 D D :05 :97 :418 :435

&9".1-& 6XSSRVHWKHYRWLQJUHVXOWVRIDFRQJUHVVLRQDOHOHFWLRQDWDFHUWDLQ YRWLQJSUHFLQFWDUHUHSUHVHQWHGE\DYHFWRU [ LQ R3 2 3 YRWLQJ'HPRFUDWLF' [ D 4 YRWLQJ5HSXEOLFDQ5 5 YRWLQJ/LEHUWDULDQ/

6XSSRVHZHUHFRUGWKHRXWFRPHRIWKHFRQJUHVVLRQDOHOHFWLRQHYHU\WZR\HDUVE\DYHFWRU RIWKLVW\SHDQGWKHRXWFRPHRIRQHHOHFWLRQGHSHQGVRQO\RQWKHUHVXOWVRIWKHSUHFHGLQJ HOHFWLRQ 7KHQWKHVHTXHQFHRIYHFWRUVWKDWGHVFULEHWKHYRWHVHYHU\WZR\HDUVPD\EH D0DUNRYFKDLQ $VDQH[DPSOHRIDVWRFKDVWLFPDWUL[ P IRUWKLVFKDLQ ZHWDNH 2 '

)URP 5

:70 P D 4 :20 :10

:10 :80 :10

/ 3 :30 :30 5 :40

7R ' 5 /

7KHHQWULHVLQWKHÀUVWFROXPQ ODEHOHG'GHVFULEHZKDWWKHSHUVRQVYRWLQJ'HPRFUDWLF LQRQHHOHFWLRQZLOOGRLQWKHQH[WHOHFWLRQ +HUHZHKDYHVXSSRVHGWKDWZLOOYRWH' DJDLQLQWKHQH[WHOHFWLRQ ZLOOYRWH5DQGZLOOYRWH/6LPLODULQWHUSUHWDWLRQV KROGIRUWKHRWKHUFROXPQVRI P $ GLDJUDPIRUWKLVPDWUL[LVVKRZQLQ)LJ .70

.80 Democratic vote

.20

Republican vote

.10

.30

.10 .10

.30

Libertarian vote .40 '*(63& 9RWLQJFKDQJHVIURPRQHHOHFWLRQWRWKH

QH[W

$)"15&3

9HFWRU6SDFHV

,IWKH´WUDQVLWLRQµSHUFHQWDJHVUHPDLQFRQVWDQWRYHUPDQ\\HDUVIURPRQHHOHFWLRQ WRWKHQH[W WKHQWKHVHTXHQFHRIYHFWRUVWKDWJLYHWKHYRWLQJRXWFRPHVIRUPVD0DUNRY FKDLQ 6XSSRVHWKHRXWFRPHRIRQHHOHFWLRQLVJLYHQE\ 2

3 :55 [0 D 4 :40 5 :05

'HWHUPLQHWKHOLNHO\RXWFRPHRIWKHQH[WHOHFWLRQDQGWKHOLNHO\RXWFRPHRIWKHHOHFWLRQ DIWHUWKDW

40-65*0/ 7KHRXWFRPHRIWKHQH[WHOHFWLRQLVGHVFULEHGE\WKHVWDWHYHFWRU [1 DQG WKDWRIWKHHOHFWLRQDIWHUWKDWE\ [2 ZKHUH 2

:70 [1 D P [0 D 4 :20 :10 2

:70 [2 D P [1 D 4 :20 :10

:10 :80 :10 :10 :80 :10

32 3 2 3 :30 :55 :440 :30 54 :40 5 D 4 :445 5 :40 :05 :115

32 3 2 3 :30 :440 :3870 :30 54 :445 5 D 4 :4785 5 :40 :115 :1345

ZLOOYRWH' ZLOOYRWH5 ZLOOYRWH/ ZLOOYRWH' ZLOOYRWH5 ZLOOYRWH/

7RXQGHUVWDQGZK\ [1 GRHVLQGHHGJLYHWKHRXWFRPHRIWKHQH[WHOHFWLRQ VXSSRVH SHUVRQVYRWHGLQWKH´ÀUVWµHOHFWLRQ ZLWKYRWLQJ'YRWLQJ5DQGYRWLQJ/ 6HHWKHSHUFHQWDJHVLQ [0 ,QWKHQH[WHOHFWLRQ RIWKHZLOOYRWH' DJDLQ 10 RIWKHZLOOVZLWFKIURP5 WR'DQGRIWKHZLOOVZLWFKIURP/ WR'7KXV WKHWRWDO' YRWHZLOOEH

:70.550/ C :10.400/ C :30.50/ D 385 C 40 C 15 D 440

7KXVRIWKHYRWHQH[WWLPHZLOOEHIRUWKH' FDQGLGDWH 7KHFDOFXODWLRQLQ LV HVVHQWLDOO\WKHVDPHDVWKDWXVHGWRFRPSXWHWKHÀUVWHQWU\LQ [1 $QDORJRXVFDOFXODWLRQV FRXOGEHPDGHIRUWKHRWKHUHQWULHVLQ [1 IRUWKHHQWULHVLQ [2 DQGVRRQ

3UHGLFWLQJWKH'LVWDQW)XWXUH 7KHPRVWLQWHUHVWLQJDVSHFWRI 0DUNRYFKDLQVLVWKHVWXG\RIDFKDLQ·VORQJWHUP EHKDYLRU )RULQVWDQFH ZKDWFDQEHVDLGLQ([DPSOH DERXWWKHYRWLQJDIWHUPDQ\ HOHFWLRQVKDYHSDVVHGDVVXPLQJWKDWWKHJLYHQVWRFKDVWLFPDWUL[FRQWLQXHVWRGHVFULEH WKHWUDQVLWLRQSHUFHQWDJHVIURPRQHHOHFWLRQWRWKHQH[W " 2U ZKDWKDSSHQVWRWKH SRSXODWLRQ GLVWULEXWLRQ LQ ([DPSOH ´LQ WKH ORQJ UXQµ" %HIRUH DQVZHULQJ WKHVH TXHVWLRQV ZHWXUQWRDQXPHULFDOH[DPSOH 2

3 2 3 :5 :2 :3 1 &9".1-& /HW P D 4 :3 :8 :3 5 DQG [0 D 4 0 5 &RQVLGHUDV\VWHPZKRVH :2 0 :4 0 VWDWHLVGHVFULEHGE\WKH0DUNRYFKDLQ [k C1 D P [k IRU k D 0; 1; : : : :KDWKDSSHQVWR WKHV\VWHPDVWLPHSDVVHV" &RPSXWHWKHVWDWHYHFWRUV [1 ; : : : ; [15 WRÀQGRXW

40-65*0/

2

:5 [1 D P [0 D 4 :3 :2 2 :5 [2 D P [1 D 4 :3 :2 2 :5 [3 D P [2 D 4 :3 :2

:2 :8 0 :2 :8 0 :2 :8 0


32 3 2 3 :3 1 :5 :3 54 0 5 D 4 :3 5 :4 0 :2 32 3 2 3 :3 :5 :37 :3 54 :3 5 D 4 :45 5 :4 :2 :18 32 3 2 3 :3 :37 :329 :3 54 :45 5 D 4 :525 5 :4 :18 :146

7KHUHVXOWVRIIXUWKHUFDOFXODWLRQVDUHVKRZQEHORZ ZLWKHQWULHVURXQGHGWRIRXURUÀYH VLJQLÀFDQWÀJXUHV 2 3 2 3 2 3 2 3 :3133 :3064 :3032 :3016 [4 D 4 :5625 5; [5 D 4 :5813 5; [6 D 4 :5906 5; [7 D 4 :5953 5 :1242 :1123 :1062 :1031 2 3 2 3 2 3 2 3 :3008 :3004 :3002 :3001 [8 D 4 :5977 5; [9 D 4 :5988 5; [10 D 4 :5994 5; [11 D 4 :5997 5 :1016 :1008 :1004 :1002 2 3 2 3 2 3 2 3 :30005 :30002 :30001 :30001 [12 D 4 :59985 5; [13 D 4 :59993 5; [14 D 4 :59996 5; [15 D 4 :59998 5 :10010 :10005 :10002 :10001 2 3 :3 7KHVH YHFWRUV VHHP WR EH DSSURDFKLQJ T D 4 :6 5 7KH SUREDELOLWLHV DUH KDUGO\ :1 FKDQJLQJIURPRQHYDOXHRI k WRWKHQH[W 2EVHUYHWKDWWKHIROORZLQJFDOFXODWLRQLV H[DFWZLWKQRURXQGLQJHUURU 3 2 2 32 3 2 3 :15 C :12 C :03 :5 :2 :3 :3 :30 6 7 P T D 4 :3 :8 :3 54 :6 5 D 4 :09 C :48 C :03 5 D 4 :60 5 D T :2 0 :4 :1 :10 :06 C 0 C :04

:KHQWKHV\VWHPLVLQVWDWH T WKHUHLVQRFKDQJHLQWKHV\VWHPIURPRQHPHDVXUHPHQW WRWKHQH[W

6WHDG\6WDWH9HFWRUV ,I P LVDVWRFKDVWLFPDWUL[ WKHQD VWHDG\VWDWHYHFWRU RU HTXLOLEULXPYHFWRU IRU P LV DSUREDELOLW\YHFWRU T VXFKWKDW PT D T ,WFDQEHVKRZQWKDWHYHU\VWRFKDVWLFPDWUL[KDVDVWHDG\VWDWHYHFWRU ,Q([DPSOH T LVDVWHDG\VWDWHYHFWRUIRU P # :375 LVDVWHDG\VWDWHYHFWRUIRUWKH :625 SRSXODWLRQPLJUDWLRQPDWUL[ M LQ([DPSOH EHFDXVH " #" # " # " # :95 :03 :375 :35625 C :01875 :375 MT D D D DT :05 :97 :625 :01875 C :60625 :625

&9".1-& 7KHSUREDELOLW\YHFWRU T D

"

$)"15&3

9HFWRU6SDFHV

,IWKHWRWDOSRSXODWLRQRIWKHPHWURSROLWDQUHJLRQLQ([DPSOHLVPLOOLRQ WKHQ T IURP([DPSOHZRXOGFRUUHVSRQGWRKDYLQJSHUVRQVLQWKHFLW\DQG LQ WKH VXEXUEV $W WKH HQG RI RQH \HDU WKH PLJUDWLRQ RXW RI WKH FLW\ ZRXOG EH .:05/.375;000/ D 18;750 SHUVRQV DQGWKHPLJUDWLRQ LQWR WKHFLW\IURPWKHVXEXUEV ZRXOGEH .:03/.625;000/ D 18;750 SHUVRQV $VDUHVXOW WKHSRSXODWLRQLQWKHFLW\ZRXOG UHPDLQWKHVDPH 6LPLODUO\ WKHVXEXUEDQSRSXODWLRQZRXOGEHVWDEOH 7KHQH[WH[DPSOHVKRZVKRZWR ÀQG DVWHDG\VWDWHYHFWRU

&9".1-& /HW P D

"

:6 :4

# :3 )LQGDVWHDG\VWDWHYHFWRUIRU P :7

40-65*0/ )LUVW VROYHWKHHTXDWLRQ P [ D [ P[ ! [ D P[ ! I[ D .P ! I /[ D )RU P DVDERYH

P !I D

"

:6 :4

5HFDOOIURP6HFWLRQWKDW I [ D [

# " :3 1 ! :7 0

0 1

#

D

"

!:4 :3 :4 !:3

#

7RÀQGDOOVROXWLRQVRI .P ! I /[ D URZUHGXFHWKHDXJPHQWHGPDWUL[ " # " # " # !:4 :3 0 !:4 :3 0 1 !3=4 0 & & :4 !:3 0 0 0 0 0 0 0 " # 3=4 7KHQ x1 D 34 x2 DQG x2 LVIUHH 7KHJHQHUDOVROXWLRQLV x2 1

"

3=4 1

#

1H[W FKRRVHDVLPSOHEDVLVIRUWKHVROXWLRQVSDFH 2QHREYLRXVFKRLFHLV " # 3 EXWDEHWWHUFKRLFHZLWKQRIUDFWLRQVLV Z D FRUUHVSRQGLQJWR x2 D 4/ 4 )LQDOO\ ÀQGDSUREDELOLW\YHFWRULQWKHVHWRIDOOVROXWLRQVRI P [ D [ 7KLVSURFHVV LVHDV\ VLQFHHYHU\VROXWLRQLVDPXOWLSOHRIWKHVROXWLRQ Z DERYH 'LYLGH Z E\WKHVXP RILWVHQWULHVDQGREWDLQ " # 3=7 TD 4=7 $VDFKHFN FRPSXWH " 6=10 PT D 4=10

3=10 7=10

#"

3=7 4=7

#

D

"

18=70 C 12=70 12=70 C 28=70

#

D

"

30=70 40=70

#

DT

7KHQH[WWKHRUHPVKRZVWKDWZKDWKDSSHQHGLQ([DPSOH LVW\SLFDORIPDQ\ VWRFKDVWLFPDWULFHV :HVD\WKDWDVWRFKDVWLFPDWUL[LV UHJXODU LIVRPHPDWUL[SRZHU P k FRQWDLQVRQO\VWULFWO\SRVLWLYHHQWULHV )RU P LQ([DPSOH 2 3 :37 :26 :33 P 2 D 4 :45 :70 :45 5 :18 :04 :22

6LQFHHYHU\HQWU\LQ P 2 LVVWULFWO\SRVLWLYH P LVDUHJXODUVWRFKDVWLFPDWUL[ $OVR ZHVD\WKDWDVHTXHQFHRIYHFWRUV f[k W k D 1; 2; : : :g FRQYHUJHV WRDYHFWRU T DV k ! 1 LIWKHHQWULHVLQ [k FDQEHPDGHDVFORVHDVGHVLUHGWRWKHFRUUHVSRQGLQJ HQWULHVLQ T E\WDNLQJ k VXIÀFLHQWO\ODUJH

5)&03&.


,I P LVDQ n % n UHJXODUVWRFKDVWLFPDWUL[ WKHQ P KDVDXQLTXHVWHDG\VWDWHYHFWRU T )XUWKHU LI [0 LVDQ\LQLWLDOVWDWHDQG [k C1 D P [k IRU k D 0; 1; 2; : : : ; WKHQWKH 0DUNRYFKDLQ f[k g FRQYHUJHVWR T DV k ! 1

7KLVWKHRUHPLVSURYHGLQVWDQGDUGWH[WVRQ0DUNRYFKDLQV 7KHDPD]LQJSDUWRI WKHWKHRUHPLVWKDWWKHLQLWLDOVWDWHKDVQRHIIHFWRQWKHORQJWHUPEHKDYLRURIWKH0DUNRY FKDLQ
&9".1-& ,Q([DPSOH ZKDWSHUFHQWDJHRIWKHYRWHUVDUHOLNHO\WRYRWHIRUWKH 5HSXEOLFDQFDQGLGDWHLQVRPHHOHFWLRQPDQ\\HDUVIURPQRZ DVVXPLQJWKDWWKHHOHFWLRQ RXWFRPHVIRUPD0DUNRYFKDLQ"

40-65*0/ )RUFRPSXWDWLRQVE\KDQG WKH ZURQJ DSSURDFKLVWRSLFNVRPHLQLWLDO YHFWRU [0 DQGFRPSXWH [1 ; : : : ; [k IRUVRPHODUJHYDOXHRI k
!:3 :1 :3 Œ .P ! I / ! D 4 :2 !:2 :3 :1 :1 !:6

3 0 05 0

5HFDOOIURPHDUOLHUZRUNZLWKGHFLPDOVWKDWWKHDULWKPHWLFLVVLPSOLÀHGE\PXOWLSO\LQJ HDFKURZE\ñ 2

!3 1 3 4 2 !2 3 1 1 !6

3 2 0 1 05 & 40 0 0

0 !9=4 1 !15=4 0 0

3 0 05 0

7KHJHQHUDOVROXWLRQRI .P ! I /[ D LV x1 D 94 x3 ; x2 D 15 x DQG x3 LVIUHH &KRRVLQJ 4 3 x3 D 4 ZHREWDLQDEDVLVIRUWKHVROXWLRQVSDFHZKRVHHQWULHVDUHLQWHJHUV DQGIURP WKLVZHHDVLO\ÀQGWKHVWHDG\VWDWHYHFWRUZKRVHHQWULHVVXPWR 2

3 9 Z D 4 15 5; 4

DQG

2

3 2 3 9=28 :32 T D 4 15=28 5 ' 4 :54 5 4=28 :14

7KHHQWULHVLQ T GHVFULEHWKHGLVWULEXWLRQRIYRWHVDWDQHOHFWLRQWREHKHOGPDQ\\HDUV IURPQRZDVVXPLQJWKHVWRFKDVWLFPDWUL[FRQWLQXHVWRGHVFULEHWKHFKDQJHVIURPRQH HOHFWLRQWRWKHQH[W 7KXV HYHQWXDOO\ DERXW 54 RIWKHYRWHZLOOEHIRUWKH5HSXEOLFDQ FDQGLGDWH

ñ :DUQLQJ 'RQ·WPXOWLSO\RQO\ P E\ ,QVWHDG PXOWLSO\WKHDXJPHQWHGPDWUL[IRUHTXDWLRQ .P ! I /[ D E\

$)"15&3

9HFWRU6SDFHV

/6.&3*$"- /05&
DQG LQJHQHUDO

[k D P k [ 0

IRU k D 0; 1; : : :

7RFRPSXWHDVSHFLÀFYHFWRUVXFKDV [3 IHZHUDULWKPHWLFRSHUDWLRQVDUHQHHGHG WRFRPSXWH [1 [2 DQG [3 UDWKHUWKDQ P 3 DQG P 3 [0 +RZHYHU LI P LVVPDOO³VD\ 30 % 30³WKHPDFKLQHFRPSXWDWLRQWLPHLVLQVLJQLÀFDQWIRUERWKPHWKRGV DQGD FRPPDQGWRFRPSXWH P 3 [0 PLJKWEHSUHIHUUHGEHFDXVHLWUHTXLUHVIHZHUKXPDQ NH\VWURNHV

13"$5*$& 130#-&.4 6XSSRVHWKHUHVLGHQWVRIDPHWURSROLWDQUHJLRQPRYHDFFRUGLQJWRWKHSUREDELOLWLHV LQWKHPLJUDWLRQPDWUL[ M LQ([DPSOHDQGDUHVLGHQWLVFKRVHQ´DWUDQGRPµ 7KHQ DVWDWHYHFWRUIRUDFHUWDLQ\HDUPD\EHLQWHUSUHWHGDVJLYLQJWKHSUREDELOLWLHVWKDWWKH SHUVRQLVDFLW\UHVLGHQWRUDVXEXUEDQUHVLGHQWDWWKDWWLPH " # 1 D 6XSSRVHWKHSHUVRQFKRVHQLVDFLW\UHVLGHQWQRZ VRWKDW [0 D :KDWLVWKH 0 OLNHOLKRRGWKDWWKHSHUVRQZLOOOLYHLQWKHVXEXUEVQH[W\HDU" E :KDWLVWKHOLNHOLKRRGWKDWWKHSHUVRQZLOOEHOLYLQJLQWKHVXEXUEVLQWZR\HDUV" " # " # :6 :2 :3 /HW P D DQG T D ,V T DVWHDG\VWDWHYHFWRUIRU P " :4 :8 :7 :KDWSHUFHQWDJHRIWKHSRSXODWLRQLQ([DPSOH ZLOOOLYHLQWKHVXEXUEVDIWHUPDQ\ \HDUV"

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

ZLWKDSUREDELOLW\RI DQGLWZLOOFKRRVHWKHRWKHUIRRGV RQWKHQH[WWULDOZLWKHTXDOSUREDELOLWLHVRI D :KDWLVWKHVWRFKDVWLFPDWUL[IRUWKLVVLWXDWLRQ" E ,IWKHDQLPDOFKRRVHVIRRGRQDQLQLWLDOWULDO ZKDWLV WKHSUREDELOLW\WKDWLWZLOOFKRRVHIRRGRQWKHVHFRQG WULDODIWHUWKHLQLWLDOWULDO"

E *LYHWKHLQLWLDOVWDWHYHFWRU F :KDWSHUFHQWDJHRIWKHOLVWHQHUVZLOOEHOLVWHQLQJWRWKH PXVLFVWDWLRQDW $0 DIWHUWKHVWDWLRQEUHDNVDW DQG $0 " $ ODERUDWRU\DQLPDOPD\HDWDQ\RQHRIWKUHHIRRGVHDFKGD\ /DERUDWRU\UHFRUGVVKRZWKDWLIWKHDQLPDOFKRRVHVRQHIRRG RQRQHWULDO LWZLOOFKRRVHWKHVDPHIRRGRQWKHQH[WWULDO

2QDQ\JLYHQGD\ DVWXGHQWLVHLWKHUKHDOWK\RULOO 2I WKHVWXGHQWVZKRDUHKHDOWK\WRGD\ ZLOOEHKHDOWK\

WRPRUURZ 2IWKHVWXGHQWVZKRDUHLOOWRGD\ ZLOOVWLOO EHLOOWRPRUURZ D :KDWLVWKHVWRFKDVWLFPDWUL[IRUWKLVVLWXDWLRQ" E 6XSSRVHRIWKHVWXGHQWVDUHLOORQ0RQGD\ :KDW IUDFWLRQRUSHUFHQWDJHRIWKHVWXGHQWVDUHOLNHO\WREHLOO RQ7XHVGD\" 2Q:HGQHVGD\" F ,IDVWXGHQWLVKHDOWK\WRGD\ ZKDWLVWKHSUREDELOLW\WKDW KHRUVKHZLOOEHKHDOWK\WZRGD\VIURPQRZ" 7KHZHDWKHULQ&ROXPEXVLVHLWKHUJRRG LQGLIIHUHQW RUEDG RQDQ\JLYHQGD\ ,IWKHZHDWKHULVJRRGWRGD\ WKHUHLVD FKDQFHLWZLOOEHJRRGWRPRUURZ DFKDQFHLWZLOO EHLQGLIIHUHQW DQGDFKDQFHLWZLOOEHEDG ,IWKHZHDWKHU LVLQGLIIHUHQWWRGD\ WKHUHLVDFKDQFHLWZLOOEHJRRG WRPRUURZ DQGDFKDQFHLWZLOOEHLQGLIIHUHQW )LQDOO\ LIWKHZHDWKHULVEDGWRGD\ WKHUHLVDFKDQFHLWZLOOEH JRRGWRPRUURZDQGDFKDQFHLWZLOOEHLQGLIIHUHQW D :KDWLVWKHVWRFKDVWLFPDWUL[IRUWKLVVLWXDWLRQ" E 6XSSRVHWKHUHLVDFKDQFHRIJRRGZHDWKHUWRGD\ DQGDFKDQFHRILQGLIIHUHQWZHDWKHU :KDWDUHWKH FKDQFHVRIEDGZHDWKHUWRPRUURZ" F 6XSSRVHWKHSUHGLFWHGZHDWKHUIRU0RQGD\LVLQ GLIIHUHQWZHDWKHUDQGEDGZHDWKHU :KDWDUHWKH FKDQFHVIRUJRRGZHDWKHURQ:HGQHVGD\" ,Q([HUFLVHV² À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ÀF VWXGHQWLVLOO" 'RHVLWPDWWHULIWKDWSHUVRQLVLOOWRGD\" 5HIHUWR([HUFLVH ,QWKHORQJUXQ KRZOLNHO\LVLWIRUWKH ZHDWKHULQ&ROXPEXVWREHJRRGRQDJLYHQGD\" >0@ 7KH'HPRJUDSKLF5HVHDUFK8QLWRIWKH&DOLIRUQLD6WDWH 'HSDUWPHQWRI)LQDQFHVXSSOLHGGDWDIRUWKHIROORZLQJPL JUDWLRQPDWUL[ ZKLFKGHVFULEHVWKHPRYHPHQWRIWKH8QLWHG 6WDWHVSRSXODWLRQGXULQJ ,Q DERXWRIWKH


WRWDOSRSXODWLRQOLYHGLQ&DOLIRUQLD :KDWSHUFHQWDJHRIWKH WRWDOSRSXODWLRQZRXOGHYHQWXDOO\OLYHLQ&DOLIRUQLDLIWKH OLVWHGPLJUDWLRQSUREDELOLWLHVZHUHWRUHPDLQFRQVWDQWRYHU PDQ\\HDUV" )URP " &$ 5HVWRI86 # :9821 :0029 :0179 :9971

7R &DOLIRUQLD 5HVWRI86

>0@ ,Q'HWURLW +HUW]5HQW$ &DUKDVDÁHHWRIDERXW FDUV 7KHSDWWHUQRIUHQWDODQGUHWXUQORFDWLRQVLVJLYHQE\ WKHIUDFWLRQVLQWKHWDEOHEHORZ 2QDW\SLFDOGD\ DERXWKRZ PDQ\FDUVZLOOEHUHQWHGRUUHDG\WRUHQWIURPWKHGRZQWRZQ ORFDWLRQ" &DUV5HQWHGIURP &LW\ 'RZQ 0HWUR $LUSRUW WRZQ $LUSRUW 2 3 :09 :01 :90 4:01 :015 :90 :90 :09 :09

5HWXUQHGWR &LW\$LUSRUW 'RZQWRZQ 0HWUR$LUSRUW

/HW P EHDQ n % n VWRFKDVWLFPDWUL[ 7KHIROORZLQJDUJXPHQW VKRZVWKDWWKHHTXDWLRQ P [ D [ KDVDQRQWULYLDOVROXWLRQ ,Q IDFW DVWHDG\VWDWHVROXWLRQH[LVWVZLWKQRQQHJDWLYHHQWULHV $ SURRI LV JLYHQ LQ VRPH DGYDQFHG WH[WV -XVWLI\ HDFK DVVHUWLRQEHORZ 0HQWLRQDWKHRUHPZKHQDSSURSULDWH D ,IDOOWKHRWKHUURZVRI P ! I DUHDGGHGWRWKHERWWRP URZ WKHUHVXOWLVDURZRI]HURV E 7KHURZVRI P ! I DUHOLQHDUO\GHSHQGHQW

F 7KHGLPHQVLRQRIWKHURZVSDFHRI P ! I LVOHVVWKDQ n G P ! I KDVDQRQWULYLDOQXOOVSDFH

6KRZ WKDW HYHU\ 2 % 2 VWRFKDVWLF PDWUL[ KDV DW OHDVW RQH VWHDG\VWDWHYHFWRU $Q\VXFKPDWUL[FDQEHZULWWHQLQWKH " # 1!˛ ˇ ZKHUH ˛ DQG ˇ DUHFRQVWDQWV IRUP P D ˛ 1!ˇ EHWZHHQDQG 7KHUHDUHWZROLQHDUO\LQGHSHQGHQWVWHDG\ VWDWHYHFWRUVLI ˛ D ˇ D 0 2WKHUZLVH WKHUHLVRQO\RQH /HW S EHWKH 1 % n URZPDWUL[ZLWKDLQHDFKFROXPQ

S DŒ1 1

### 1 !

D ([SODLQZK\DYHFWRU [ LQ Rn LVDSUREDELOLW\YHFWRULIDQG RQO\LILWVHQWULHVDUHQRQQHJDWLYHDQG S [ D 1 $ 1 % 1 PDWUL[VXFKDVWKHSURGXFW S [ LVXVXDOO\ZULWWHQZLWKRXW WKHPDWUL[EUDFNHWV\PEROV E /HW P EH DQ n % n VWRFKDVWLF PDWUL[ SP D S

([SODLQ ZK\

F /HW P EH DQ n % n VWRFKDVWLF PDWUL[ DQG OHW [ EH D SUREDELOLW\YHFWRU 6KRZWKDW P [ LVDOVRDSUREDELOLW\ YHFWRU

8VH([HUFLVHWRVKRZWKDWLI P LVDQ n % n VWRFKDVWLF PDWUL[ WKHQVRLV P 2

$)"15&3

9HFWRU6SDFHV

>0@ ([DPLQHSRZHUVRIDUHJXODUVWRFKDVWLFPDWUL[ D &RPSXWH P k IRU k D 2; 3; 4; 5 ZKHQ 2 3 :3355 :3682 :3067 :0389 6 :2663 :2723 :3277 :5451 7 7 P D6 4 :1935 :1502 :1589 :2395 5 :2047 :2093 :2067 :1765

'LVSOD\FDOFXODWLRQVWRIRXUGHFLPDOSODFHV :KDWKDS SHQVWRWKHFROXPQVRI P k DV k LQFUHDVHV" &RPSXWHWKH VWHDG\VWDWHYHFWRUIRU P

E &RPSXWH Q k 2 :97 QD4 0 :03

IRU k D 10; 20; : : : ; 80 ZKHQ 3 :05 :10 :90 :05 5 :05 :85

6WDELOLW\ IRU Qk WR IRXU GHFLPDO SODFHV PD\ UHTXLUH k D 116 RUPRUH &RPSXWHWKHVWHDG\VWDWHYHFWRUIRU

Q &RQMHFWXUHZKDWPLJKWEHWUXHIRUDQ\UHJXODUVWRFKDV WLFPDWUL[ F 8VH7KHRUHPWRH[SODLQZKDW\RXIRXQGLQSDUWVD DQGE >0@ &RPSDUHWZRPHWKRGVIRUÀQGLQJWKHVWHDG\VWDWHYHFWRU T RIDUHJXODUVWRFKDVWLFPDWUL[ P FRPSXWLQJ T DVLQ ([DPSOH RU FRPSXWLQJ P k IRUVRPHODUJHYDOXHRI k DQGXVLQJRQHRIWKHFROXPQVRI P k DVDQDSSUR[LPDWLRQIRU T >7KH 6WXG\*XLGH GHVFULEHVDSURJUDP QXOEDVLV WKDWDOPRVW DXWRPDWHVPHWKRG @ ([SHULPHQWZLWKWKHODUJHVWUDQGRPVWRFKDVWLFPDWULFHV \RXUPDWUL[SURJUDPZLOODOORZ DQGXVH k D 100 RUVRPH RWKHUODUJHYDOXH )RUHDFKPHWKRG GHVFULEHWKHWLPH \RX QHHGWRHQWHUWKHNH\VWURNHVDQGUXQ\RXUSURJUDP 6RPH YHUVLRQV RI 0$7/$% KDYH FRPPDQGV IORSV DQG WLF : : : WRF WKDWUHFRUGWKHQXPEHURIÁRDWLQJSRLQWRSHUDWLRQV DQGWKHWRWDOHODSVHGWLPH0$7/$% XVHV &RQWUDVWWKH DGYDQWDJHVRIHDFKPHWKRG DQGVWDWHZKLFK\RXSUHIHU

40-65*0/4 50 13"$5*$& 130#-&.4 D 6LQFHRIWKHFLW\UHVLGHQWVZLOOPRYHWRWKHVXEXUEVZLWKLQRQH\HDU WKHUHLV DFKDQFHRIFKRRVLQJVXFKDSHUVRQ :LWKRXWIXUWKHUNQRZOHGJHDERXWWKH SHUVRQ ZHVD\WKDWWKHUHLVDFKDQFHWKHSHUVRQZLOOPRYHWRWKHVXEXUEV 7KLVIDFWLVFRQWDLQHGLQWKHVHFRQGHQWU\RIWKHVWDWHYHFWRU [1 ZKHUH " #" # " # :95 :03 1 :95 [1 D M [0 D D :05 :97 0 :05 E 7KHOLNHOLKRRGWKDWWKHSHUVRQZLOOEHOLYLQJLQWKHVXEXUEVDIWHUWZR\HDUVLV EHFDXVH " #" # " # :95 :03 :95 :904 [2 D M [1 D D :05 :97 :05 :096

7KHVWHDG\VWDWHYHFWRUVDWLVÀHV P [ D [ 6LQFH " #" # " # :6 :2 :3 :32 PT D D ¤T :4 :8 :7 :68

WEB

ZHFRQFOXGHWKDW T LV QRW WKHVWHDG\VWDWHYHFWRUIRU P M LQ([DPSOHLVDUHJXODUVWRFKDVWLFPDWUL[EHFDXVHLWVHQWULHVDUHDOOVWULFWO\ SRVLWLYH 6RZHPD\XVH7KHRUHP :HDOUHDG\NQRZWKHVWHDG\VWDWHYHFWRU IURP([DPSOH 7KXVWKHSRSXODWLRQGLVWULEXWLRQYHFWRUV [k FRQYHUJHWR " # :375 TD :625 (YHQWXDOO\RIWKHSRSXODWLRQZLOOOLYHLQWKHVXEXUEV

$)"15&3 4611-&.&/5"3: &9&3$*4&4 0DUNHDFKVWDWHPHQW7UXHRU)DOVH -XVWLI\HDFKDQVZHU ,IWUXH FLWHDSSURSULDWHIDFWVRUWKHRUHPV ,IIDOVH H[SODLQ ZK\RUJLYHDFRXQWHUH[DPSOHWKDWVKRZVZK\WKHVWDWHPHQW LVQRWWUXHLQHYHU\FDVH ,QSDUWVD ²I Y1 ; : : : ; Yp DUH

YHFWRUVLQDQRQ]HURÀQLWHGLPHQVLRQDOYHFWRUVSDFH V DQG S D f Y 1 ; : : : ; Y p g D 7KHVHWRIDOOOLQHDUFRPELQDWLRQVRI Y1 ; : : : ; Yp LVDYHFWRU VSDFH


E ,I fY1 ; : : : ; Yp"1 g VSDQV V WKHQ S VSDQV V F

,I fY1 ; : : : ; Yp"1 g LVOLQHDUO\LQGHSHQGHQW WKHQVRLV S

G ,I S LVOLQHDUO\LQGHSHQGHQW WKHQ S LVDEDVLVIRU V H I

,I 6SDQ S D V WKHQVRPHVXEVHWRI S LVDEDVLVIRU V

,I GLP V D p DQG 6SDQ S D V WKHQ S FDQQRWEHOLQHDUO\ GHSHQGHQW

J $ SODQHLQ R3 LVDWZRGLPHQVLRQDOVXEVSDFH K 7KHQRQSLYRWFROXPQVRIDPDWUL[DUHDOZD\VOLQHDUO\ GHSHQGHQW L

5RZRSHUDWLRQVRQDPDWUL[ A FDQFKDQJHWKHOLQHDU GHSHQGHQFHUHODWLRQVDPRQJWKHURZVRI A

M

5RZRSHUDWLRQVRQDPDWUL[FDQFKDQJHWKHQXOOVSDFH

N 7KHUDQNRIDPDWUL[HTXDOVWKHQXPEHURIQRQ]HURURZV O

,IDQ m % n PDWUL[ A LVURZHTXLYDOHQWWRDQHFKHORQPD WUL[ U DQGLI U KDV k QRQ]HURURZV WKHQWKHGLPHQVLRQ RIWKHVROXWLRQVSDFHRI A[ D LV m ! k

P ,I B LVREWDLQHGIURPDPDWUL[ A E\VHYHUDOHOHPHQWDU\ URZRSHUDWLRQV WKHQ UDQN B D UDQN A

Q 7KHQRQ]HURURZVRIDPDWUL[ A IRUPDEDVLVIRU 5RZ A

R ,IPDWULFHV A DQG B KDYHWKHVDPHUHGXFHGHFKHORQIRUP WKHQ 5RZ A D 5RZ B

S ,I H LVDVXEVSDFHRI R3 WKHQWKHUHLVD 3 % 3 PDWUL[ A VXFKWKDW H D &RO A

SURRIRIWKH6SDQQLQJ6HW7KHRUHP6HFWLRQ WRSURGXFH DEDVLVIRU H ([SODLQKRZWRVHOHFWDSSURSULDWHPHPEHUV RI S 6XSSRVH S1 S2 S3 S4 DUHVSHFLÀFSRO\QRPLDOVWKDWVSDQD WZRGLPHQVLRQDOVXEVSDFH H RI P5 'HVFULEHKRZRQHFDQ ÀQGDEDVLVIRU H E\H[DPLQLQJWKHIRXUSRO\QRPLDOVDQG PDNLQJDOPRVWQRFRPSXWDWLRQV :KDWZRXOG\RXKDYHWRNQRZDERXWWKHVROXWLRQVHWRID KRPRJHQHRXVV\VWHPRIOLQHDUHTXDWLRQVLQYDULDEOHV LQ RUGHU WR NQRZ WKDW HYHU\ DVVRFLDWHG QRQKRPRJHQHRXV HTXDWLRQKDVDVROXWLRQ" 'LVFXVV /HW H EHDQ nGLPHQVLRQDOVXEVSDFHRIDQ nGLPHQVLRQDO YHFWRUVSDFH V ([SODLQZK\ H D V /HW T W Rn ! Rm EHDOLQHDUWUDQVIRUPDWLRQ D :KDWLVWKHGLPHQVLRQRIWKHUDQJHRI T LI T LVDRQHWR RQHPDSSLQJ" ([SODLQ E :KDWLVWKHGLPHQVLRQRIWKHNHUQHORI T VHH6HFWLRQ LI T PDSV Rn RQWR Rm " ([SODLQ /HW S EHDPD[LPDOOLQHDUO\LQGHSHQGHQWVXEVHWRIDYHFWRU VSDFH V 7KDWLV S KDVWKHSURSHUW\WKDWLIDYHFWRUQRWLQ S LVDGMRLQHGWR S WKHQWKHQHZVHWZLOOQRORQJHUEHOLQHDUO\ LQGHSHQGHQW 3URYHWKDW S PXVWEHDEDVLVIRU V >+LQW :KDW LI S ZHUHOLQHDUO\LQGHSHQGHQWEXWQRWDEDVLVRI V ‹!

T ,I A LV m % n DQG UDQN A D m WKHQWKHOLQHDUWUDQVIRU PDWLRQ [ 7! A[ LVRQHWRRQH

/HW S EHDÀQLWHPLQLPDOVSDQQLQJVHWRIDYHFWRUVSDFH V 7KDWLV S KDVWKHSURSHUW\WKDWLIDYHFWRULVUHPRYHGIURP S WKHQWKHQHZVHWZLOOQRORQJHUVSDQ V 3URYHWKDW S PXVW EHDEDVLVIRU V

V

([HUFLVHV²GHYHORSSURSHUWLHVRIUDQNWKDWDUHVRPHWLPHV QHHGHGLQDSSOLFDWLRQV $VVXPHWKHPDWUL[ A LV m % n

U

W

,I A LV m % n DQGWKHOLQHDUWUDQVIRUPDWLRQ [ 7! A[ LV RQWR WKHQ UDQN A D m $ FKDQJHRIFRRUGLQDWHVPDWUL[LVDOZD\VLQYHUWLEOH

,I B D fE1 ; : : : ; En g DQG C D fF1 ; : : : ; Fn g DUHEDVHVIRUD YHFWRUVSDFH V WKHQWKH j WKFROXPQRIWKHFKDQJHRI FRRUGLQDWHVPDWUL[ C P B LVWKHFRRUGLQDWHYHFWRU ŒFj !B

)LQGDEDVLVIRUWKHVHWRIDOOYHFWRUVRIWKHIRUP 2 3 a ! 2b C 5c 6 2a C 5b ! 8c 7 6 7 4 !a ! 4b C 7c 5 : %HFDUHIXO 3a C b C c 2 3 2 3 2 3 !2 1 b1 /HW X1 D 4 4 5 X2 D 4 2 5 E D 4 b2 5 DQG b3 !6 !5 W D 6SDQ fX1 ; X2 g )LQGDQ LPSOLFLW GHVFULSWLRQRI W WKDW LV ÀQGDVHWRIRQHRUPRUHKRPRJHQHRXVHTXDWLRQVWKDW FKDUDFWHUL]HWKHSRLQWVRI W >+LQW :KHQLV E LQ W ‹! ([SODLQZKDWLVZURQJZLWKWKHIROORZLQJGLVFXVVLRQ /HW I.t/ D 3 C t DQG J.t/ D 3t C t 2 DQGQRWHWKDW J.t/ D t I.t/ 7KHQ fI; Jg LVOLQHDUO\GHSHQGHQWEHFDXVH J LVDPXOWLSOHRI I &RQVLGHU WKH SRO\QRPLDOV S1 .t / D 1 C t S2 .t/ D 1 ! t S3 .t/ D 4 S4 .t/ D t C t 2 DQG S5 .t / D 1 C 2t C t 2 DQG OHW H EH WKH VXEVSDFH RI P5 VSDQQHG E\ WKH VHW S D fS1 ; S2 ; S3 ; S4 ; S5 g 8VHWKHPHWKRGGHVFULEHGLQWKH

6KRZIURPSDUWVD DQGE WKDW UDQN $% FDQQRWH[FHHGWKH UDQN RI A RU WKH UDQN RI B ,Q JHQHUDO WKH UDQN RI D SURGXFWRIPDWULFHVFDQQRWH[FHHGWKHUDQNRIDQ\IDFWRULQ WKHSURGXFW D 6KRZWKDWLI B LV n % p WKHQ UDQN $% $ UDQN A >+LQW ([SODLQZK\HYHU\YHFWRULQWKHFROXPQVSDFHRI $% LVLQ WKHFROXPQVSDFHRI A@ E 6KRZWKDWLI B LV n % p WKHQ UDQN $% $ UDQN B >+LQW 8VHSDUWD WRVWXG\ UDQN.$%/T @ 6KRZ WKDW LI P LV DQ LQYHUWLEOH m % m PDWUL[ WKHQ UDQN PA D UDQN A >+LQW $SSO\([HUFLVHWR PA DQG P "1 .PA/@ 6KRZWKDWLI Q LVLQYHUWLEOH WKHQ UDQN AQ D UDQN A >+LQW 8VH([HUFLVHWRVWXG\ UDQN.AQ/T @ /HW A EHDQ m % n PDWUL[ DQGOHW B EHDQ n % p PDWUL[ VXFKWKDW $% D 0 6KRZWKDW UDQN A C UDQN B $ n >+LQW 2QHRIWKHIRXUVXEVSDFHV 1XO A &RO A 1XO B DQG &RO B LV FRQWDLQHGLQRQHRIWKHRWKHUWKUHHVXEVSDFHV@ ,I A LVDQ m % n PDWUL[RI UDQN r WKHQD UDQNIDFWRUL]DWLRQ RI A LVDQHTXDWLRQRIWKHIRUP A D CR ZKHUH C LVDQ m % r PDWUL[RIUDQN r DQG R LVDQ r % n PDWUL[RIUDQN r

$)"15&3

9HFWRU6SDFHV

6XFKDIDFWRUL]DWLRQDOZD\VH[LVWV([HUFLVHLQ6HFWLRQ *LYHQDQ\WZR m % n PDWULFHV A DQG B XVHUDQN IDFWRUL]DWLRQVRI A DQG B WRSURYHWKDW UDQN.A C B/ $ UDQN A C UDQN B

>+LQW :ULWH A C B DVWKHSURGXFWRIWZRSDUWLWLRQHGPDWUL FHV@ $ VXEPDWUL[ RIDPDWUL[ A LVDQ\PDWUL[WKDWUHVXOWVIURP GHOHWLQJVRPHRUQR URZVDQGRUFROXPQVRI A ,WFDQEH VKRZQWKDW A KDV UDQN r LIDQGRQO\LI A FRQWDLQVDQLQYHUWLEOH r % r VXEPDWUL[DQGQRODUJHUVTXDUHVXEPDWUL[LVLQYHUWLEOH 'HPRQVWUDWHSDUWRIWKLVVWDWHPHQWE\H[SODLQLQJD ZK\ DQ m % n PDWUL[ A RI UDQN r KDVDQ m % r VXEPDWUL[ A1 RI UDQN r DQGE ZK\ A1 KDVDQLQYHUWLEOH r % r VXEPDWUL[ A2 7KHFRQFHSWRIUDQNSOD\VDQLPSRUWDQWUROHLQWKHGHVLJQRI HQJLQHHULQJFRQWUROV\VWHPV VXFKDVWKHVSDFHVKXWWOHV\VWHP PHQWLRQHGLQWKLVFKDSWHU·VLQWURGXFWRU\H[DPSOH $ VWDWHVSDFH PRGHO RIDFRQWUROV\VWHPLQFOXGHVDGLIIHUHQFHHTXDWLRQRIWKH IRUP [kC1 D A[k C B Xk

IRU k D 0; 1; : : :

###

ZKHUH A LV n % n B LV n % m f[k g LVDVHTXHQFHRI´VWDWHYHFWRUVµ LQ Rn WKDWGHVFULEHWKHVWDWHRIWKHV\VWHPDWGLVFUHWHWLPHV DQG fXk g LVD FRQWURO RU LQSXW VHTXHQFH 7KHSDLU .A; B/ LVVDLGWREH FRQWUROODEOH LI UDQN Œ B

%$A2 B

An"1 B ! D n

7KHPDWUL[WKDWDSSHDUVLQ LVFDOOHGWKH FRQWUROODELOLW\PDWUL[ IRUWKHV\VWHP ,I .A; B/ LVFRQWUROODEOH WKHQWKHV\VWHPFDQEH FRQWUROOHG RUGULYHQIURPWKHVWDWH WRDQ\VSHFLÀHGVWDWH Y LQ Rn LQDWPRVW n VWHSV VLPSO\E\FKRRVLQJDQDSSURSULDWHFRQWURO VHTXHQFHLQ Rm 7KLVIDFWLVLOOXVWUDWHGLQ([HUFLVHIRU n D 4

DQG m D 2 )RUDIXUWKHUGLVFXVVLRQRIFRQWUROODELOLW\ VHHWKLV WH[W·VZHEVLWH&DVH6WXG\IRU&KDSWHU WEB

6XSSRVH A LVD 4 % 4 PDWUL[DQG B LVD 4 % 2 PDWUL[ DQGOHW X0 ; : : : ; X3 UHSUHVHQWDVHTXHQFHRILQSXWYHFWRUVLQ R2 D 6HW [0 D FRPSXWH [1 ; : : : ; [4 IURPHTXDWLRQ DQG ZULWHDIRUPXODIRU [4 LQYROYLQJWKHFRQWUROODELOLW\PDWUL[ M DSSHDULQJLQHTXDWLRQ 1RWH 7KHPDWUL[ M LV FRQVWUXFWHGDVDSDUWLWLRQHGPDWUL[ ,WVRYHUDOOVL]HKHUH LV 4 % 8 E 6XSSRVH .A; B/ LVFRQWUROODEOHDQG Y LVDQ\YHFWRULQ R4 ([SODLQZK\WKHUHH[LVWVDFRQWUROVHTXHQFH X0 ; : : : ; X3 LQ R2 VXFKWKDW [4 D Y

'HWHUPLQHLIWKHPDWUL[SDLUVLQ([HUFLVHV²DUHFRQWUROODEOH 2 3 2 3 :9 1 0 0 0 5 B D 4 1 5 A D 4 0 !:9 0 0 :5 1 2 3 2 3 :8 !:3 0 1 1 5 B D 4 1 5 A D 4 :2 :5 0 0 !:5 0 2 3 2 3 0 1 0 0 1 6 0 6 7 0 1 0 7 7 B D 6 0 7 >0@ A D 6 4 0 5 4 0 0 1 05 !2 !4:2 !4:8 !3:6 !1 2 3 2 3 0 1 0 0 1 6 0 6 7 0 1 0 7 7 B D 6 0 7 >0@ A D 6 4 0 4 05 0 0 1 5 !1 !13 !12:2 !1:5 !1

(LJHQYDOXHVDQG (LJHQYHFWRUV

*/530%6$503: &9".1-&

'\QDPLFDO6\VWHPVDQG6SRWWHG2ZOV ,Q WKHQRUWKHUQVSRWWHGRZOEHFDPHWKHFHQWHURI DQDWLRQZLGHFRQWURYHUV\RYHUWKHXVHDQGPLVXVHRIWKH PDMHVWLFIRUHVWVLQWKH3DFLÀF1RUWKZHVW (QYLURQPHQ WDOLVWVFRQYLQFHGWKHIHGHUDOJRYHUQPHQWWKDWWKHRZOZDV WKUHDWHQHGZLWKH[WLQFWLRQLIORJJLQJFRQWLQXHGLQWKHROG JURZWKIRUHVWVZLWKWUHHVRYHU\HDUVROG ZKHUHWKH RZOVSUHIHUWROLYH 7KHWLPEHULQGXVWU\ DQWLFLSDWLQJ WKHORVVRIWRMREVDVDUHVXOWRIQHZ JRYHUQPHQWUHVWULFWLRQVRQORJJLQJ DUJXHGWKDWWKHRZO VKRXOGQRWEHFODVVLÀHGDVD´WKUHDWHQHGVSHFLHVµDQGFLWHG DQXPEHURISXEOLVKHGVFLHQWLÀFUHSRUWVWRVXSSRUWLWVFDVH &DXJKWLQWKHFURVVÀUHRIWKHWZROREE\LQJJURXSV PDWKHPDWLFDOHFRORJLVWVLQWHQVLÀHGWKHLUGULYHWRXQGHU VWDQGWKHSRSXODWLRQG\QDPLFVRIWKHVSRWWHGRZO 7KH OLIHF\FOHRIDVSRWWHGRZOGLYLGHVQDWXUDOO\LQWRWKUHH VWDJHV MXYHQLOHXSWR \HDUROG VXEDGXOWWR \HDUV DQGDGXOWRYHU \HDUV 7KHRZOVPDWHIRUOLIHGXULQJ WKHVXEDGXOWDQGDGXOWVWDJHV EHJLQWREUHHGDVDGXOWV DQGOLYHIRUXSWR \HDUV (DFKRZOSDLUUHTXLUHVDERXW KHFWDUHV VTXDUHPLOHV IRULWVRZQKRPHWHUULWRU\ $ FULWLFDOWLPHLQWKHOLIHF\FOHLVZKHQWKHMXYHQLOHVOHDYH WKHQHVW 7RVXUYLYHDQGEHFRPHDVXEDGXOW DMXYHQLOH PXVWVXFFHVVIXOO\ÀQGDQHZKRPHUDQJHDQGXVXDOO\D PDWH

1 ´7KH*UHDW6SRWWHG2ZO:DUµ

SS ²

5HDGHU·V'LJHVW 1RYHPEHU

$ ÀUVWVWHSLQVWXG\LQJWKHSRSXODWLRQG\QDPLFVLVWR PRGHOWKHSRSXODWLRQDW\HDUO\LQWHUYDOV DWWLPHVGHQRWHG E\ k D 0; 1; 2; : : : : 8VXDOO\ RQHDVVXPHVWKDWWKHUHLVD UDWLRRIPDOHVWRIHPDOHVLQHDFKOLIHVWDJHDQGFRXQWVRQO\ WKHIHPDOHV 7KHSRSXODWLRQDW\HDU k FDQEHGHVFULEHG E\DYHFWRU [k D .jk ; sk ; ak / ZKHUH jk sk DQG ak DUHWKH QXPEHUVRIIHPDOHVLQWKHMXYHQLOH VXEDGXOW DQGDGXOW VWDJHV UHVSHFWLYHO\ 8VLQJDFWXDOÀHOGGDWDIURPGHPRJUDSKLFVWXGLHV 5/DPEHUVRQDQGFRZRUNHUVFRQVLGHUHGWKHIROORZLQJ VWDJHPDWUL[PRGHO 2

3 2 jk C1 0 4 sk C1 5 D 4 :18 ak C1 0

0 0 :71

32 3 :33 jk 0 54 sk 5 ak :94

+HUHWKHQXPEHURIQHZMXYHQLOHIHPDOHVLQ\HDU k C 1 LV WLPHVWKHQXPEHURIDGXOWIHPDOHVLQ\HDU k EDVHG RQWKHDYHUDJHELUWKUDWHSHURZOSDLU $OVR RIWKH MXYHQLOHVVXUYLYHWREHFRPHVXEDGXOWV DQGRIWKH VXEDGXOWVDQGRIWKHDGXOWVVXUYLYHWREHFRXQWHGDV DGXOWV 7KHVWDJHPDWUL[PRGHOLVDGLIIHUHQFHHTXDWLRQRIWKH IRUP [k C1 D A[k 6XFKDQHTXDWLRQLVRIWHQFDOOHGD 2 5 + /DPEHUVRQ 5 0F.HOYH\ % 5 1RRQ DQG& 9RVV ´$ '\QDPLF $QDO\VLVRIWKH9LDELOLW\RIWKH1RUWKHUQ6SRWWHG2ZOLQD)UDJPHQWHG )RUHVW(QYLURQPHQWµ &RQVHUYDWLRQ%LRORJ\ ² $OVR D SULYDWHFRPPXQLFDWLRQIURP3URIHVVRU/DPEHUVRQ

$)"15&3

(LJHQYDOXHVDQG(LJHQYHFWRUV

G\QDPLFDOV\VWHP RUD GLVFUHWHOLQHDUG\QDPLFDO V\VWHP EHFDXVHLWGHVFULEHVWKHFKDQJHVLQDV\VWHPDV WLPHSDVVHV 7KHMXYHQLOHVXUYLYDOUDWHLQWKH/DPEHUVRQVWDJH PDWUL[LVWKHHQWU\DIIHFWHGPRVWE\WKHDPRXQWRIROG JURZWKIRUHVWDYDLODEOH $FWXDOO\ RIWKHMXYHQLOHV QRUPDOO\VXUYLYHWROHDYHWKHQHVW EXWLQWKH:LOORZ &UHHNUHJLRQRI&DOLIRUQLDVWXGLHGE\/DPEHUVRQDQGKLV FROOHDJXHV RQO\RIWKHMXYHQLOHVWKDWOHIWWKHQHVWZHUH DEOHWRÀQGQHZKRPHUDQJHV 7KHUHVWSHULVKHGGXULQJWKH VHDUFKSURFHVV

$ VLJQLÀFDQWUHDVRQIRUWKHIDLOXUHRIRZOVWRÀQGQHZ KRPHUDQJHVLVWKHLQFUHDVLQJIUDJPHQWDWLRQRIROGJURZWK WLPEHUVWDQGVGXHWRFOHDUFXWWLQJRIVFDWWHUHGDUHDVRQ WKHROGJURZWKODQG :KHQDQRZOOHDYHVWKHSURWHFWLYH FDQRS\RIWKHIRUHVWDQGFURVVHVDFOHDUFXWDUHD WKHULVNRI DWWDFNE\SUHGDWRUVLQFUHDVHVGUDPDWLFDOO\ 6HFWLRQ ZLOO VKRZWKDWWKHPRGHOGHVFULEHGDERYHSUHGLFWVWKHHYHQWXDO GHPLVHRIWKHVSRWWHGRZO EXWWKDWLIRIWKHMXYHQLOHV ZKRVXUYLYHWROHDYHWKHQHVWDOVRÀQGQHZKRPHUDQJHV WKHQWKHRZOSRSXODWLRQZLOOWKULYH WEB

7KHJRDORIWKLVFKDSWHULVWRGLVVHFWWKHDFWLRQRIDOLQHDUWUDQVIRUPDWLRQ [ 7! A[ LQWR HOHPHQWVWKDWDUHHDVLO\YLVXDOL]HG ([FHSWIRUDEULHIGLJUHVVLRQLQ6HFWLRQ DOO PDWULFHVLQWKHFKDSWHUDUHVTXDUH 7KHPDLQDSSOLFDWLRQVGHVFULEHGKHUHDUHWRGLVFUHWH G\QDPLFDOV\VWHPV LQFOXGLQJWKHVSRWWHGRZOVGLVFXVVHGDERYH +RZHYHU WKHEDVLF FRQFHSWV³HLJHQYHFWRUVDQGHLJHQYDOXHV³DUHXVHIXOWKURXJKRXWSXUHDQGDSSOLHGPDWK HPDWLFV DQGWKH\DSSHDULQVHWWLQJVIDUPRUHJHQHUDOWKDQZHFRQVLGHUKHUH (LJHQYDOXHV DUHDOVRXVHGWRVWXG\GLIIHUHQWLDOHTXDWLRQVDQG FRQWLQXRXV G\QDPLFDOV\VWHPV WKH\ SURYLGHFULWLFDOLQIRUPDWLRQLQHQJLQHHULQJGHVLJQ DQGWKH\DULVHQDWXUDOO\LQÀHOGVVXFK DVSK\VLFVDQGFKHPLVWU\

&*(&/7&$5034 "/% &*(&/7"-6&4 $OWKRXJKDWUDQVIRUPDWLRQ [ 7! A[ PD\PRYHYHFWRUVLQDYDULHW\RIGLUHFWLRQV LWRIWHQ KDSSHQVWKDWWKHUHDUHVSHFLDOYHFWRUVRQZKLFKWKHDFWLRQRI A LVTXLWHVLPSOH !

" ! " ! " 3 !2 !1 2 &9".1-& /HW A D XD DQG Y D 7KHLPDJHVRI X DQG 1 0 1 1 Y XQGHUPXOWLSOLFDWLRQE\ A DUHVKRZQLQ)LJ ,QIDFW AY LVMXVW 2Y 6R A RQO\ ´VWUHWFKHVµ RUGLODWHV Y x2 Av u Au

v

1 1

x1

'*(63& (IIHFWVRIPXOWLSOLFDWLRQE\ A

$VDQRWKHUH[DPSOH UHDGHUVRI6HFWLRQ ZLOOUHFDOOWKDWLI A LVDVWRFKDVWLFPDWUL[ WKHQWKHVWHDG\VWDWHYHFWRU T IRU A VDWLVÀHVWKHHTXDWLRQ A[ D [ 7KDWLV AT D 1 " T

(LJHQYHFWRUVDQG(LJHQYDOXHV

7KLVVHFWLRQVWXGLHVHTXDWLRQVVXFKDV

A[ D 2[

RU

A[ D !4[

ZKHUHVSHFLDOYHFWRUVDUHWUDQVIRUPHGE\ A LQWRVFDODUPXOWLSOHVRIWKHPVHOYHV

%&'*/*5*0/

$Q HLJHQYHFWRU RIDQ n # n PDWUL[ A LVDQRQ]HURYHFWRU [ VXFKWKDW A[ D ![ IRUVRPHVFDODU ! $ VFDODU ! LVFDOOHGDQ HLJHQYDOXH RI A LIWKHUHLVDQRQWULYLDO VROXWLRQ [ RI A[ D ![ VXFKDQ [ LVFDOOHGDQ HLJHQYHFWRUFRUUHVSRQGLQJWR ! ,WLVHDV\WRGHWHUPLQHLIDJLYHQYHFWRULVDQHLJHQYHFWRURIDPDWUL[ ,WLVDOVRHDV\ WRGHFLGHLIDVSHFLÀHGVFDODULVDQHLJHQYDOXH

&9".1-& /HW A D

x2 Au

YHFWRUVRI A"

20 Av

40-65*0/ v

–30 –10

u

– 20

AX D !4X EXW AY ¤ !Y

30

x1

AX D AY D

! !

1 5 1 5

!

" ! " ! " 6 6 3 XD DQG Y D $UH X DQG Y HLJHQ 2 !5 !2

1 5

" ! " !24 6 D !4 D !4X 20 !5 "! " ! " ! " 6 3 !9 3 D ¤! 2 !2 11 !2

6 2

"!

6 !5

"

D

!

7KXV X LVDQHLJHQYHFWRUFRUUHVSRQGLQJWRDQHLJHQYDOXH .!4/ EXW Y LVQRWDQHLJHQYHFWRU RI A EHFDXVH AY LVQRWDPXOWLSOHRI Y

&9".1-& 6KRZWKDWLVDQHLJHQYDOXHRIPDWUL[ A LQ([DPSOH DQGÀQGWKH FRUUHVSRQGLQJHLJHQYHFWRUV 40-65*0/ 7KHVFDODULVDQHLJHQYDOXHRI A LIDQGRQO\LIWKHHTXDWLRQ A[ D 7[

KDVDQRQWULYLDOVROXWLRQ %XW LVHTXLYDOHQWWR A[ ! 7[ D RU

.A ! 7I /[ D

7RVROYHWKLVKRPRJHQHRXVHTXDWLRQ IRUPWKHPDWUL[ ! " ! " ! " 1 6 7 0 !6 6 A ! 7I D ! D 5 2 0 7 5 !5 7KHFROXPQVRI A ! 7I DUHREYLRXVO\OLQHDUO\GHSHQGHQW VR KDVQRQWULYLDOVROX WLRQV 7KXV LV DQHLJHQYDOXHRI A 7RÀQGWKHFRUUHVSRQGLQJHLJHQYHFWRUV XVHURZ RSHUDWLRQV ! " ! " !6 6 0 1 !1 0 $ 5 !5 0 0 0 0 ! " 1 7KHJHQHUDOVROXWLRQKDVWKHIRUP x2 (DFKYHFWRURIWKLVIRUPZLWK x2 ¤ 0 LVDQ 1 HLJHQYHFWRUFRUUHVSRQGLQJWR ! D 7 ñ 1RWHWKDWDQHLJHQYHFWRUPXVWEH QRQ]HUR E\GHÀQLWLRQ EXWDQHLJHQYDOXHPD\EH]HUR 7KHFDVHLQZKLFK WKHQXPEHULVDQHLJHQYDOXHLVGLVFXVVHGDIWHU([DPSOH

$)"15&3


:DUQLQJ $OWKRXJKURZUHGXFWLRQZDVXVHGLQ([DPSOHWRÀQGHLJHQYHFWRUV LW FDQQRWEHXVHGWRÀQGHLJHQYDOXHV $QHFKHORQIRUPRIDPDWUL[ A XVXDOO\GRHV QRW GLVSOD\WKHHLJHQYDOXHVRI A 7KHHTXLYDOHQFHRIHTXDWLRQV DQG REYLRXVO\KROGVIRUDQ\ ! LQSODFHRI ! D 7 7KXV ! LVDQHLJHQYDOXHRIDQ n # n PDWUL[ A LIDQGRQO\LIWKHHTXDWLRQ

.A ! !I /[ D

KDVDQRQWULYLDOVROXWLRQ 7KHVHWRI DOO VROXWLRQVRI LVMXVWWKHQXOOVSDFHRIWKHPDWUL[ A ! !I 6RWKLVVHWLVD VXEVSDFH RI Rn DQGLVFDOOHGWKH HLJHQVSDFH RI A FRUUHVSRQGLQJ WR ! 7KHHLJHQVSDFHFRQVLVWVRIWKH]HURYHFWRUDQGDOOWKHHLJHQYHFWRUVFRUUHVSRQGLQJ WR ! ([DPSOHVKRZVWKDWIRUPDWUL[ A LQ([DPSOH WKHHLJHQVSDFHFRUUHVSRQGLQJWR ! D 7 FRQVLVWVRI DOO PXOWLSOHVRI .1; 1/ ZKLFKLVWKHOLQHWKURXJK .1; 1/ DQGWKHRULJLQ )URP([DPSOH \RXFDQFKHFNWKDWWKHHLJHQVSDFHFRUUHVSRQGLQJWR ! D !4 LVWKH OLQHWKURXJK .6; !5/ 7KHVHHLJHQVSDFHVDUHVKRZQLQ)LJ DORQJZLWKHLJHQYHFWRUV .1; 1/ DQG .3=2; !5=4/ DQGWKHJHRPHWULFDFWLRQRIWKHWUDQVIRUPDWLRQ [ 7! A[ RQHDFK HLJHQVSDFH x2

Multiplication by 7 Eigenspace for λ = 7

2

x1

2

Multiplication by – 4

Eigenspace for λ = – 4 (6, –5)

'*(63& (LJHQVSDFHVIRU ! D !4 DQG ! D 7

2

4 !1 &9".1-& /HW A D 4 2 1 2 !1 WKHFRUUHVSRQGLQJHLJHQVSDFH 40-65*0/ )RUP

2

4 !1 A ! 2I D 4 2 1 2 !1

3 6 6 5 $QHLJHQYDOXHRI A LV )LQGDEDVLVIRU 8

3 2 6 2 65!40 8 0

0 2 0

3 2 0 2 !1 0 5 D 4 2 !1 2 2 !1

DQGURZUHGXFHWKHDXJPHQWHGPDWUL[IRU .A ! 2I /[ D 2 3 2 2 !1 6 0 2 !1 6 4 2 !1 6 05 $ 40 0 0 2 !1 6 0 0 0 0

3 0 05 0

3 6 65 6


$WWKLVSRLQW LWLVFOHDUWKDWLVLQGHHGDQHLJHQYDOXHRI A EHFDXVHWKHHTXDWLRQ .A ! 2I /[ D KDVIUHHYDULDEOHV 7KHJHQHUDOVROXWLRQLV 2

3 2 3 2 3 x1 1=2 !3 4 x2 5 D x2 4 1 5 C x3 4 0 5; x3 0 1

x2 DQG x3 IUHH

7KHHLJHQVSDFH VKRZQLQ)LJ LVDWZRGLPHQVLRQDOVXEVSDFHRI R3 $ EDVLVLV 82 3 2 39 !3 = < 1 4 2 5; 4 0 5 : ; 0 1 x3

x3 Multiplication by A

Eigen

space

for l !

Eigen 2

space

for l !

2

'*(63& A DFWVDVDGLODWLRQRQWKHHLJHQVSDFH

/6.&3*$"- /05& ([DPSOHVKRZVDJRRGPHWKRGIRUPDQXDOFRPSXWDWLRQRIHLJHQYHFWRUVLQ VLPSOHFDVHVZKHQDQHLJHQYDOXHLVNQRZQ 8VLQJDPDWUL[SURJUDPDQGURZ UHGXFWLRQWRÀQGDQHLJHQVSDFHIRUDVSHFLÀHGHLJHQYDOXH XVXDOO\ZRUNV WRR EXWWKLVLVQRWHQWLUHO\UHOLDEOH 5RXQGRIIHUURUFDQOHDGRFFDVLRQDOO\WRDUHGXFHG HFKHORQIRUPZLWKWKHZURQJQXPEHURISLYRWV 7KHEHVWFRPSXWHUSURJUDPV FRPSXWHDSSUR[LPDWLRQVIRUHLJHQYDOXHVDQGHLJHQYHFWRUVVLPXOWDQHRXVO\ WR DQ\GHVLUHGGHJUHHRIDFFXUDF\ IRUPDWULFHVWKDWDUHQRWWRRODUJH 7KHVL]H RIPDWULFHVWKDWFDQEHDQDO\]HGLQFUHDVHVHDFK\HDUDVFRPSXWLQJSRZHUDQG VRIWZDUHLPSURYH

7KHIROORZLQJWKHRUHPGHVFULEHVRQHRIWKHIHZVSHFLDOFDVHVLQZKLFKHLJHQYDOXHV FDQEHIRXQGSUHFLVHO\ &DOFXODWLRQRIHLJHQYDOXHVZLOODOVREHGLVFXVVHGLQ6HFWLRQ

5)&03&.

7KHHLJHQYDOXHVRIDWULDQJXODUPDWUL[DUHWKHHQWULHVRQLWVPDLQGLDJRQDO

1300' )RUVLPSOLFLW\ FRQVLGHUWKH 3 # 3 FDVH ,I A LVXSSHUWULDQJXODU WKHQ A ! !I

$)"15&3


KDVWKHIRUP

2

3 2 3 a11 a12 a13 ! 0 0 a22 a23 5 ! 4 0 ! 05 A ! !I D 4 0 0 0 a33 0 0 ! 2 3 a12 a13 a11 ! ! a22 ! ! a23 5 D4 0 0 0 a33 ! !

7KHVFDODU ! LVDQHLJHQYDOXHRI A LIDQGRQO\LIWKHHTXDWLRQ .A ! !I /[ D KDVD QRQWULYLDOVROXWLRQ WKDWLV LIDQGRQO\LIWKHHTXDWLRQKDVDIUHHYDULDEOH %HFDXVHRIWKH ]HURHQWULHVLQ A ! !I LWLVHDV\WRVHHWKDW .A ! !I /[ D KDVDIUHHYDULDEOHLIDQG RQO\LIDWOHDVWRQHRIWKHHQWULHVRQWKHGLDJRQDORI A ! !I LV]HUR 7KLVKDSSHQVLIDQG RQO\LI ! HTXDOVRQHRIWKHHQWULHV a11 a22 a33 LQ A )RUWKHFDVHLQZKLFK A LVORZHU WULDQJXODU VHH([HUFLVH 2

3 2 3 6 !8 4 &9".1-& /HW A D 4 0 0 6 5 DQG B D 4 !2 0 0 2 5 XHVRI A DUH DQG 7KHHLJHQYDOXHVRI B DUHDQG

0 1 3

3 0 0 5 7KHHLJHQYDO 4

:KDWGRHVLWPHDQIRUDPDWUL[ A WRKDYHDQHLJHQYDOXHRI VXFKDVLQ([DPSOH " 7KLVKDSSHQVLIDQGRQO\LIWKHHTXDWLRQ

A[ D 0[

KDVDQRQWULYLDOVROXWLRQ %XW LVHTXLYDOHQWWR A[ D ZKLFKKDVDQRQWULYLDOVROXWLRQ LIDQGRQO\LI A LVQRWLQYHUWLEOH 7KXV LVDQHLJHQYDOXHRI A LIDQGRQO\LI A LVQRW LQYHUWLEOH 7KLVIDFWZLOOEHDGGHGWRWKH,QYHUWLEOH0DWUL[7KHRUHPLQ6HFWLRQ 7KHIROORZLQJLPSRUWDQWWKHRUHPZLOOEHQHHGHGODWHU ,WVSURRILOOXVWUDWHVDW\SLFDO FDOFXODWLRQZLWKHLJHQYHFWRUV

5)&03&.

,I Y1 ; : : : ; Yr DUHHLJHQYHFWRUVWKDWFRUUHVSRQGWRGLVWLQFWHLJHQYDOXHV !1 ; : : : ; !r RIDQ n # n PDWUL[ A WKHQWKHVHW fY1 ; : : : ; Yr g LVOLQHDUO\LQGHSHQGHQW

1300' 6XSSRVH fY1 ; : : : ; Yr g LVOLQHDUO\GHSHQGHQW 6LQFH Y1 LVQRQ]HUR 7KHRUHPLQ 6HFWLRQVD\VWKDWRQHRIWKHYHFWRUVLQWKHVHWLVDOLQHDUFRPELQDWLRQRIWKHSUHFHGLQJ YHFWRUV /HW p EHWKHOHDVWLQGH[VXFKWKDW YpC1 LVDOLQHDUFRPELQDWLRQRIWKHSUHFHGLQJ OLQHDUO\LQGHSHQGHQW YHFWRUV 7KHQWKHUHH[LVWVFDODUV c1 ; : : : ; cp VXFKWKDW c 1 Y 1 C " " " C cp Y p D Y p C 1

c1 AY1 C " " " C cp AYp D AYpC1 c1 !1 Y1 C " " " C cp !p Yp D !pC1 YpC1

c1 .!1 ! !pC1 /Y1 C " " " C cp .!p ! !pC1 /Yp D

0XOWLSO\LQJERWKVLGHVRI E\ A DQGXVLQJWKHIDFWWKDW AYk D !k Yk IRUHDFK k ZH REWDLQ

0XOWLSO\LQJERWKVLGHVRI E\ !pC1 DQGVXEWUDFWLQJWKHUHVXOWIURP ZHKDYH 6LQFH fY1 ; : : : ; Yp g LVOLQHDUO\LQGHSHQGHQW WKHZHLJKWVLQ DUHDOO]HUR %XWQRQHRI WKHIDFWRUV !i ! !pC1 DUH]HUR EHFDXVHWKHHLJHQYDOXHVDUHGLVWLQFW +HQFH ci D 0 IRU i D 1; : : : ; p %XWWKHQ VD\VWKDW YpC1 D ZKLFKLVLPSRVVLEOH +HQFH fY1 ; : : : ; Yr g FDQQRWEHOLQHDUO\GHSHQGHQWDQGWKHUHIRUHPXVWEHOLQHDUO\LQGHSHQGHQW


(LJHQYHFWRUVDQG'LIIHUHQFH(TXDWLRQV 7KLVVHFWLRQFRQFOXGHVE\VKRZLQJKRZWRFRQVWUXFWVROXWLRQVRIWKHÀUVWRUGHUGLIIHU HQFHHTXDWLRQGLVFXVVHGLQWKHFKDSWHULQWURGXFWRU\H[DPSOH [ k C1 D A[ k

.k D 0; 1; 2; : : :/

.k D 1; 2; : : :/

,I A LVDQ n # n PDWUL[ WKHQ LVD UHFXUVLYH GHVFULSWLRQRIDVHTXHQFH f[k g LQ Rn $ VROXWLRQ RI LVDQH[SOLFLWGHVFULSWLRQRI f[k g ZKRVHIRUPXODIRUHDFK [k GRHVQRW GHSHQGGLUHFWO\RQ A RURQWKHSUHFHGLQJWHUPVLQWKHVHTXHQFHRWKHUWKDQWKHLQLWLDO WHUP [0 7KHVLPSOHVWZD\WREXLOGDVROXWLRQRI LVWRWDNHDQHLJHQYHFWRU [0 DQGLWV FRUUHVSRQGLQJHLJHQYDOXH ! DQGOHW [k D !k [0 7KLVVHTXHQFHLVDVROXWLRQEHFDXVH

A[k D A.!k [0 / D !k .A[0 / D !k .![0 / D !k C1 [0 D [k C1 /LQHDUFRPELQDWLRQVRIVROXWLRQVLQWKHIRUPRIHTXDWLRQ DUHVROXWLRQV WRR 6HH ([HUFLVH

13"$5*$& 130#-&.4

2

6 !3 ,VDQHLJHQYDOXHRI A D 4 3 0 2 2

3 1 5 5" 6

,I [ LVDQHLJHQYHFWRURI A FRUUHVSRQGLQJWR ! ZKDWLV A3 [" 6XSSRVHWKDW E1 DQG E2 DUHHLJHQYHFWRUVFRUUHVSRQGLQJWRGLVWLQFWHLJHQYDOXHV !1 DQG !2 UHVSHFWLYHO\ DQGVXSSRVHWKDW E3 DQG E4 DUHOLQHDUO\LQGHSHQGHQWHLJHQYHFWRUV FRUUHVSRQGLQJWRDWKLUGGLVWLQFWHLJHQYDOXH !3 'RHVLWQHFHVVDULO\IROORZWKDW fE1 ; E2 ; E3 ; E4 g LVDOLQHDUO\LQGHSHQGHQWVHW" >+LQW &RQVLGHUWKHHTXDWLRQ c1 E1 C c2 E2 C .c3 E3 C c4 E4 / D @

&9&3$*4&4 ,V ! D 2 DQHLJHQYDOXHRI

!

,V ! D !3 DQHLJHQYDOXHRI

" 2 " :K\RUZK\QRW" 8

3 3 !

!1 6

" 4 " :K\RUZK\QRW" 9

! " ! " 1 1 !1 DQHLJHQYHFWRURI " ,IVR ÀQGWKHHLJHQ 3 6 !4 YDOXH ! " ! " !1 5 2 ,V DQ HLJHQYHFWRU RI " ,I VR ÀQG WKH 1 3 6 HLJHQYDOXH

,V

2

3 2 3 3 !4 3 3 ,V 4 !2 5 DQHLJHQYHFWRURI 4 2 !3 !2 5" ,IVR ÀQG 1 !1 0 !2 WKHHLJHQYDOXH

2

3 2 3 1 3 6 7 2 7 5" ,IVR ÀQGWKH ,V 4 !2 5 DQHLJHQYHFWRURI 4 3 2 5 6 4 HLJHQYDOXH 2 3 3 0 !1 3 1 5" ,IVR ÀQGRQH ,V ! D 4 DQHLJHQYDOXHRI 4 2 !3 4 5 FRUUHVSRQGLQJHLJHQYHFWRU 2 3 4 !2 3 3 5" ,IVR ÀQGRQH ,V ! D 1 DQHLJHQYDOXHRI 4 0 !1 !1 2 !2 FRUUHVSRQGLQJHLJHQYHFWRU

,Q([HUFLVHV² ÀQGDEDVLVIRUWKHHLJHQVSDFHFRUUHVSRQGLQJ WRHDFKOLVWHGHLJHQYDOXH ! " 3 0 A D ! D 1 2 1

$)"15&3

" 2 ! D !5 1 ! " 1 !3 AD ! D !1 !4 5 ! " 4 1 AD ! D 3 3 6 2 3 4 0 1 1 0 5 ! D 1 A D 4 !2 !2 0 1 2 3 4 0 !1 0 3 5 ! D 3 A D 43 2 !2 5 2 3 !4 1 1 2 5 ! D !5 A D 4 2 !3 3 3 !2 2 3 5 0 !1 0 61 3 0 07 7 ! D 4 AD6 4 2 !1 3 05 4 !2 !2 4

A D

!


!4 3

)LQGWKHHLJHQYDOXHVRIWKHPDWULFHVLQ([HUFLVHVDQG 2 3 2 3 0 0 0 5 0 0 3 45 0 05 4 0 4 0 0 0 !2 !1 0 3 2 3 1 2 3 2 3 5 ÀQGRQHHLJHQYDOXH ZLWKQRFDO )RU A D 4 1 1 2 3 FXODWLRQ -XVWLI\\RXUDQVZHU :LWKRXWFDOFXODWLRQ ÀQGRQHHLJHQYDOXHDQGWZROLQHDUO\ 2 3 2 2 2 2 2 5 -XVWLI\ LQGHSHQGHQWHLJHQYHFWRUVRI A D 4 2 2 2 2 \RXUDQVZHU ,Q([HUFLVHVDQG A LVDQ n # n PDWUL[ 0DUNHDFKVWDWHPHQW 7UXHRU)DOVH -XVWLI\HDFKDQVZHU D ,I A[ D ![ IRUVRPHYHFWRU [ WKHQ ! LVDQHLJHQYDOXHRI A E $ PDWUL[ A LVQRWLQYHUWLEOHLIDQGRQO\LILVDQHLJHQ YDOXHRI A F $ QXPEHU c LVDQHLJHQYDOXHRI A LIDQGRQO\LIWKH HTXDWLRQ .A ! cI /[ D KDVDQRQWULYLDOVROXWLRQ

G )LQGLQJDQHLJHQYHFWRURI A PD\EHGLIÀFXOW EXWFKHFN LQJZKHWKHUDJLYHQYHFWRULVLQIDFWDQHLJHQYHFWRULV HDV\ H 7RÀQGWKHHLJHQYDOXHVRI A UHGXFH A WRHFKHORQIRUP D ,I A[ D ![ IRUVRPHVFDODU ! WKHQ [ LVDQHLJHQYHFWRURI A E ,I Y1 DQG Y2 DUHOLQHDUO\LQGHSHQGHQWHLJHQYHFWRUV WKHQ WKH\FRUUHVSRQGWRGLVWLQFWHLJHQYDOXHV

F $ VWHDG\VWDWHYHFWRUIRUDVWRFKDVWLFPDWUL[LVDFWXDOO\DQ HLJHQYHFWRU G 7KHHLJHQYDOXHVRIDPDWUL[DUHRQLWVPDLQGLDJRQDO H $QHLJHQVSDFHRI A LVDQXOOVSDFHRIDFHUWDLQPDWUL[ ([SODLQZK\D 2 # 2 PDWUL[FDQKDYHDWPRVWWZRGLVWLQFW HLJHQYDOXHV ([SODLQZK\DQ n # n PDWUL[FDQKDYHDWPRVW n GLVWLQFWHLJHQYDOXHV &RQVWUXFWDQH[DPSOHRID 2 # 2 PDWUL[ZLWKRQO\RQHGLVWLQFW HLJHQYDOXH /HW ! EHDQHLJHQYDOXHRIDQLQYHUWLEOHPDWUL[ A 6KRZWKDW !!1 LVDQHLJHQYDOXHRI A!1 >+LQW 6XSSRVHDQRQ]HUR [ VDWLVÀHV A[ D ![@

6KRZWKDWLI A2 LVWKH]HURPDWUL[ WKHQWKHRQO\HLJHQYDOXH RI A LV 6KRZWKDW ! LVDQHLJHQYDOXHRI A LIDQGRQO\LI ! LVDQ HLJHQYDOXHRI AT >+LQW )LQGRXWKRZ A ! !I DQG AT ! !I DUHUHODWHG@ 8VH([HUFLVHWRFRPSOHWHWKHSURRIRI7KHRUHP IRUWKH FDVHLQZKLFK A LVORZHUWULDQJXODU &RQVLGHUDQ n # n PDWUL[ A ZLWKWKHSURSHUW\WKDWWKHURZ VXPV DOO HTXDO WKH VDPH QXPEHU s 6KRZ WKDW s LV DQ HLJHQYDOXHRI A >+LQW )LQGDQHLJHQYHFWRU@ &RQVLGHUDQ n # n PDWUL[ A ZLWKWKHSURSHUW\WKDWWKHFROXPQ VXPV DOO HTXDO WKH VDPH QXPEHU s 6KRZ WKDW s LV DQ HLJHQYDOXHRI A >+LQW 8VH([HUFLVHVDQG@ ,Q([HUFLVHVDQG OHW A EHWKHPDWUL[RIWKHOLQHDUWUDQV IRUPDWLRQ T :LWKRXWZULWLQJ A ÀQGDQHLJHQYDOXHRI A DQG GHVFULEHWKHHLJHQVSDFH T LVWKHWUDQVIRUPDWLRQRQ R2 WKDWUHÁHFWVSRLQWVDFURVVVRPH OLQHWKURXJKWKHRULJLQ T LVWKHWUDQVIRUPDWLRQRQ R3 WKDWURWDWHVSRLQWVDERXWVRPH OLQHWKURXJKWKHRULJLQ /HW X DQG Y EHHLJHQYHFWRUVRIDPDWUL[ A ZLWKFRUUHVSRQGLQJ HLJHQYDOXHV ! DQG " DQGOHW c1 DQG c2 EHVFDODUV 'HÀQH [k D c1 !k X C c2 "k Y

.k D 0; 1; 2; : : :/

D :KDWLV [kC1 E\GHÀQLWLRQ" E &RPSXWH A[k IURPWKHIRUPXODIRU [k DQGVKRZWKDW A[k D [kC1 7KLVFDOFXODWLRQZLOOSURYHWKDWWKHVH TXHQFH f[k g GHÀQHGDERYHVDWLVÀHVWKHGLIIHUHQFHHTXD WLRQ [kC1 D A[k .k D 0; 1; 2; : : :/

'HVFULEHKRZ\RXPLJKWWU\WREXLOGDVROXWLRQRIDGLIIHUHQFH HTXDWLRQ [kC1 D A[k .k D 0; 1; 2; : : :/ LI\RXZHUHJLYHQWKH LQLWLDO [0 DQGWKLVYHFWRUGLGQRWKDSSHQWREHDQHLJHQYHFWRU RI A >+LQW +RZPLJKW\RXUHODWH [0 WRHLJHQYHFWRUVRI A"@ /HW X DQG Y EHWKHYHFWRUVVKRZQLQWKHÀJXUH DQGVXSSRVH X DQG Y DUHHLJHQYHFWRUVRID 2 # 2 PDWUL[ A WKDWFRUUHVSRQG WRHLJHQYDOXHVDQG UHVSHFWLYHO\ /HW T W R2 ! R2 EH WKHOLQHDUWUDQVIRUPDWLRQJLYHQE\ T .[/ D A[ IRUHDFK [ LQ R2 DQGOHW Z D X C Y 0DNHDFRS\RIWKHÀJXUH DQGRQ

WKHVDPHFRRUGLQDWHV\VWHP FDUHIXOO\SORWWKHYHFWRUV T .X/ T .Y/ DQG T .Z/ x2

12 4 2 1

1 11 3

12 6 8 6 6 6 16 4 0 8 2 !23 6 !10 6 6 6 11 4 !27 !5

!90 !49 !52 !30 !41

2

v u

2

x1

5HSHDW([HUFLVH DVVXPLQJ X DQG Y DUHHLJHQYHFWRUVRI A WKDWFRUUHVSRQGWRHLJHQYDOXHV !1 DQG UHVSHFWLYHO\ >0@ ,Q([HUFLVHV² XVHDPDWUL[SURJUDPWRÀQGWKHHLJHQ YDOXHVRIWKHPDWUL[ 7KHQXVHWKHPHWKRGRI([DPSOHZLWKD URZUHGXFWLRQURXWLQHWRSURGXFHDEDVLVIRUHDFKHLJHQVSDFH

7KH&KDUDFWHULVWLF(TXDWLRQ 2

3 4 45 7

5 67 6 4 4 3

30 15 12 10 15

57 12 5 31 !15

30 15 0 22 15

!9 !10 !3 !27 !5

!2 !4 !4 !1 3

30 15 7 7 20 7 7 10 5 7

!15 2 !19 25 1

3 2 !4 2 !4 7 7 2 05 1 !3

3 !59 !22 7 7 !15 7 7 !37 5 31

40-65*0/4 50 13"$5*$& 130#-&.4 7KHQXPEHULVDQHLJHQYDOXHRI A LIDQGRQO\LIWKHHTXDWLRQ .A ! 5I /[ D KDVD QRQWULYLDOVROXWLRQ )RUP 2 3 2 3 2 3 6 !3 1 5 0 0 1 !3 1 55 A ! 5I D 4 3 0 5 5 ! 4 0 5 0 5 D 4 3 !5 2 2 6 0 0 5 2 2 1 DQGURZUHGXFHWKHDXJPHQWHGPDWUL[ 2 3 2 1 !3 1 0 1 !3 1 4 3 !5 5 05 $ 40 4 2 2 2 1 0 0 8 !1

3 2 0 1 !3 1 05 $ 40 4 2 0 0 0 !5

3 0 05 0

$WWKLVSRLQW LWLVFOHDUWKDWWKHKRPRJHQHRXVV\VWHPKDVQRIUHHYDULDEOHV 7KXV A ! 5I LVDQLQYHUWLEOHPDWUL[ ZKLFKPHDQVWKDWLV QRW DQHLJHQYDOXHRI A ,I [ LVDQHLJHQYHFWRURI A FRUUHVSRQGLQJWR ! WKHQ A[ D ![ DQGVR

A2 [ D A.![/ D !A[ D !2 [

$JDLQ A3 [ D A.A2 [/ D A.!2 [/ D !2 A[ D !3 [ 7KHJHQHUDOSDWWHUQ Ak [ D !k [ LVSURYHGE\LQGXFWLRQ
c1 E1 C c2 E2 C .c3 E3 C c4 E4 / D

LPSOLHV c1 D c2 D 0 %XWWKHQ c3 DQG c4 PXVWDOVREH]HURVLQFH E3 DQG E4 DUH OLQHDUO\LQGHSHQGHQW +HQFHDOOWKHFRHIÀFLHQWVLQWKHRULJLQDOHTXDWLRQPXVWEH ]HUR DQGWKHYHFWRUV E1 E2 E3 DQG E4 DUHOLQHDUO\LQGHSHQGHQW

5)& $)"3"$5&3*45*$ &26"5*0/ 8VHIXOLQIRUPDWLRQDERXWWKHHLJHQYDOXHVRIDVTXDUHPDWUL[ A LVHQFRGHGLQDVSHFLDO VFDODUHTXDWLRQFDOOHGWKHFKDUDFWHULVWLFHTXDWLRQRI A $ VLPSOHH[DPSOHZLOOOHDGWR WKHJHQHUDOFDVH

$)"15&3


&9".1-& )LQGWKHHLJHQYDOXHVRI A D

!

" 2 3 3 !6

40-65*0/ :HPXVWÀQGDOOVFDODUV ! VXFKWKDWWKHPDWUL[HTXDWLRQ .A ! !I /[ D KDVDQRQWULYLDOVROXWLRQ %\WKH,QYHUWLEOH0DWUL[7KHRUHPLQ6HFWLRQ WKLVSUREOHP LVHTXLYDOHQWWRÀQGLQJDOO ! VXFKWKDWWKHPDWUL[ A ! !I LV QRW LQYHUWLEOH ZKHUH ! " ! " ! " 2 3 ! 0 2!! 3 A ! !I D ! D 3 !6 0 ! 3 !6 ! ! %\7KHRUHPLQ6HFWLRQ WKLVPDWUL[IDLOVWREHLQYHUWLEOHSUHFLVHO\ZKHQLWV GHWHUPLQDQWLV]HUR 6RWKHHLJHQYDOXHVRI A DUHWKHVROXWLRQVRIWKHHTXDWLRQ ! " 2!! 3 GHW.A ! !I / D GHW D0 3 !6 ! ! 5HFDOOWKDW GHW

!

a c

b d

"

D ad ! bc

6R GHW.A ! !I / D .2 ! !/.!6 ! !/ ! .3/.3/

D !12 C 6! ! 2! C !2 ! 9

D !2 C 4! ! 21 D .! ! 3/.! C 7/

,I GHW.A ! !I / D 0 WKHQ ! D 3 RU ! D !7 6RWKHHLJHQYDOXHVRI A DUHDQG !7 7KHGHWHUPLQDQWLQ([DPSOH WUDQVIRUPHGWKHPDWUL[HTXDWLRQ .A ! !I /[ D ZKLFKLQYROYHV WZR XQNQRZQV .! DQG [/ LQWRWKHVFDODUHTXDWLRQ !2 C 4! ! 21 D 0 ZKLFKLQYROYHVRQO\RQHXQNQRZQ 7KHVDPHLGHDZRUNVIRU n # n PDWULFHV +RZHYHU EHIRUHWXUQLQJWRODUJHUPDWULFHV ZHVXPPDUL]HWKHSURSHUWLHVRIGHWHUPLQDQWVQHHGHG WRVWXG\HLJHQYDOXHV

'HWHUPLQDQWV /HW A EH DQ n # n PDWUL[ OHW U EH DQ\ HFKHORQ IRUP REWDLQHG IURP A E\ URZ UHSODFHPHQWVDQGURZLQWHUFKDQJHVZLWKRXWVFDOLQJ DQGOHW r EHWKHQXPEHURIVXFK URZLQWHUFKDQJHV 7KHQWKH GHWHUPLQDQW RI A ZULWWHQDV GHW A LV .!1/r WLPHVWKH SURGXFWRIWKHGLDJRQDOHQWULHV u11 ; : : : ; unn LQ U ,I A LVLQYHUWLEOH WKHQ u11 ; : : : ; unn DUHDOO SLYRWV EHFDXVH A $ In DQGWKH ui i KDYHQRWEHHQVFDOHGWR·V 2WKHUZLVH DW OHDVW unn LV]HUR DQGWKHSURGXFW u11 " " " unn LV]HUR 7KXVñ 8 ˆ <.!1/r " GHW A D ˆ : 0;

SURGXFWRI SLYRWVLQ U

!

;

ZKHQ A LVLQYHUWLEOH

ZKHQ A LVQRWLQYHUWLEOH

ñ )RUPXOD ZDVGHULYHGLQ6HFWLRQ 5HDGHUVZKRKDYHQRWVWXGLHG&KDSWHU PD\XVHWKLVIRUPXODDV WKHGHÀQLWLRQRI GHW A ,WLVDUHPDUNDEOHDQGQRQWULYLDOIDFWWKDWDQ\HFKHORQIRUP U REWDLQHGIURP A ZLWKRXWVFDOLQJJLYHVWKHVDPHYDOXHIRU GHW A

7KH&KDUDFWHULVWLF(TXDWLRQ

2

3 1 5 0 &9".1-& &RPSXWH GHW A IRU A D 4 2 4 !1 5 0 !2 0

40-65*0/ 7KHIROORZLQJURZUHGXFWLRQXVHVRQHURZLQWHUFKDQJH 2

3 2 3 2 3 1 5 0 1 5 0 1 5 0 0 5 $ 4 0 !2 0 5 D U1 A $ 4 0 !6 !1 5 $ 4 0 !2 0 !2 0 0 !6 !1 0 0 !1

6R GHW A HTXDOV .!1/1 .1/.!2/.!1/ D !2 7KHIROORZLQJDOWHUQDWLYHURZUHGXFWLRQ DYRLGVWKHURZLQWHUFKDQJHDQGSURGXFHVDGLIIHUHQWHFKHORQIRUP 7KHODVWVWHSDGGV !1=3 WLPHVURZ WRURZ 2

3 2 1 5 0 1 5 A $ 4 0 !6 !1 5 $ 4 0 !6 0 !2 0 0 0

3 0 !1 5 D U2 1=3

7KLVWLPH GHW A LV .!1/0 .1/.!6/.1=3/ D !2 WKHVDPHDVEHIRUH )RUPXOD IRUWKHGHWHUPLQDQWVKRZVWKDW A LVLQYHUWLEOHLIDQGRQO\LI GHW A LV QRQ]HUR 7KLVIDFW DQGWKHFKDUDFWHUL]DWLRQRILQYHUWLELOLW\IRXQGLQ6HFWLRQ FDQEH DGGHGWRWKH,QYHUWLEOH0DWUL[7KHRUHP

5)&03&.


/HW A EHDQ n # n PDWUL[ 7KHQ A LVLQYHUWLEOHLIDQGRQO\LI V 7KHQXPEHULV QRW DQHLJHQYDOXHRI A W 7KHGHWHUPLQDQWRI A LV QRW ]HUR

x3 a2

a3

x1

x2

a1

'*(63&

5)&03&.

:KHQ A LVD 3 # 3 PDWUL[ j GHW Aj WXUQVRXWWREHWKH YROXPH RIWKHSDUDOOHOHSLSHG GHWHUPLQHGE\WKHFROXPQV D1 D2 D3 RI A DVLQ)LJ 6HH6HFWLRQIRUGHWDLOV 7KLVYROXPHLV QRQ]HUR LIDQGRQO\LIWKHYHFWRUV D1 D2 D3 DUHOLQHDUO\LQGHSHQGHQW LQ ZKLFKFDVHWKHPDWUL[ A LVLQYHUWLEOH ,IWKHYHFWRUVDUHQRQ]HURDQGOLQHDUO\GHSHQGHQW WKH\OLHLQDSODQHRUDORQJDOLQH 7KHQH[WWKHRUHPOLVWVIDFWVQHHGHGIURP6HFWLRQVDQG 3DUWD LVLQFOXGHG KHUHIRUFRQYHQLHQWUHIHUHQFH 1SPQFSUJFT PG %FUFSNJOBOUT /HW A DQG B EH n # n PDWULFHV

A LVLQYHUWLEOHLIDQGRQO\LI GHW A ¤ 0 GHW AB D .GHW A/.GHW B/ GHW AT D GHW A ,I A LVWULDQJXODU WKHQ GHW A LVWKHSURGXFWRIWKHHQWULHVRQWKHPDLQGLDJRQDO RI A H $ URZUHSODFHPHQWRSHUDWLRQRQ A GRHVQRWFKDQJHWKHGHWHUPLQDQW $ URZ LQWHUFKDQJHFKDQJHVWKHVLJQRIWKHGHWHUPLQDQW $ URZVFDOLQJDOVRVFDOHVWKH GHWHUPLQDQWE\WKHVDPHVFDODUIDFWRU

D E F G

$)"15&3


7KH&KDUDFWHULVWLF(TXDWLRQ 7KHRUHP D VKRZV KRZ WRGHWHUPLQHZKHQDPDWUL[RIWKHIRUP A ! !I LV QRW LQYHUWLEOH 7KHVFDODUHTXDWLRQ GHW.A ! !I / D 0 LVFDOOHGWKH FKDUDFWHULVWLFHTXDWLRQ RI A DQGWKHDUJXPHQWLQ([DPSOHMXVWLÀHVWKHIROORZLQJIDFW $ VFDODU ! LVDQHLJHQYDOXHRIDQ n # n PDWUL[ A LIDQGRQO\LI ! VDWLVÀHVWKH FKDUDFWHULVWLFHTXDWLRQ GHW.A ! !I / D 0

&9".1-& )LQGWKHFKDUDFWHULVWLFHTXDWLRQRI 2

3 5 !2 6 !1 60 3 !8 0 7 7 AD6 40 0 5 45 0 0 0 1

40-65*0/ )RUP A ! !I DQGXVH7KHRUHP G 2 5!! !2 6 0 3 !! GHW.A ! !I / D GHW 6 4 0 0 0 0

6 !8 5!! 0

D .5 ! !/.3 ! !/.5 ! !/.1 ! !/

3 !1 0 7 7 4 5 1!!

7KHFKDUDFWHULVWLFHTXDWLRQLV

.5 ! !/2 .3 ! !/.1 ! !/ D 0 RU

.! ! 5/2 .! ! 3/.! ! 1/ D 0 ([SDQGLQJWKHSURGXFW ZHFDQDOVRZULWH

!4 ! 14!3 C 68!2 ! 130! C 75 D 0 ,Q([DPSOHVDQG GHW .A ! !I / LVDSRO\QRPLDOLQ ! ,WFDQEHVKRZQWKDWLI A LV DQ n # n PDWUL[ WKHQ GHW .A ! !I / LVDSRO\QRPLDORIGHJUHH n FDOOHGWKH FKDUDFWHULVWLF SRO\QRPLDO RI A 7KHHLJHQYDOXHLQ([DPSOH LVVDLGWRKDYH PXOWLSOLFLW\ EHFDXVH .! ! 5/ RFFXUV WZRWLPHVDVDIDFWRURIWKHFKDUDFWHULVWLFSRO\QRPLDO ,QJHQHUDO WKHDOJHEUDLF PXOWLSOLFLW\ RIDQHLJHQYDOXH ! LVLWVPXOWLSOLFLW\DVDURRWRIWKHFKDUDFWHULVWLFHTXDWLRQ

&9".1-& 7KHFKDUDFWHULVWLFSRO\QRPLDORID 6 # 6 PDWUL[LV !6 ! 4!5 ! 12!4 )LQGWKHHLJHQYDOXHVDQGWKHLUPXOWLSOLFLWLHV

40-65*0/ )DFWRUWKHSRO\QRPLDO !6 ! 4!5 ! 12!4 D !4 .!2 ! 4! ! 12/ D !4 .! ! 6/.! C 2/ 7KHHLJHQYDOXHVDUHPXOWLSOLFLW\ PXOWLSOLFLW\ DQG !2 PXOWLSOLFLW\


SG

'BDUPSJOH B 1PMZOPNJBM m

:HFRXOGDOVROLVWWKHHLJHQYDOXHVLQ([DPSOH DV 0; 0; 0; 0; 6 DQG !2 VRWKDWWKH HLJHQYDOXHVDUHUHSHDWHGDFFRUGLQJWRWKHLUPXOWLSOLFLWLHV %HFDXVHWKHFKDUDFWHULVWLFHTXDWLRQIRUDQ n # n PDWUL[LQYROYHVDQ nWKGHJUHH SRO\QRPLDO WKHHTXDWLRQKDVH[DFWO\ n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ÀFXOW XQOHVVWKHPDWUL[LVFDUHIXOO\FKRVHQ 6HHWKH1XPHULFDO1RWHVDWWKHHQGRIWKLV VHFWLRQ

6LPLODULW\ 7KHQH[WWKHRUHPLOOXVWUDWHVRQHXVHRIWKHFKDUDFWHULVWLFSRO\QRPLDO DQGLWSURYLGHV WKHIRXQGDWLRQIRUVHYHUDOLWHUDWLYHPHWKRGVWKDW DSSUR[LPDWH HLJHQYDOXHV ,I A DQG B DUH n # n PDWULFHV WKHQ A LV VLPLODU WR B LI WKHUH LV DQ LQYHUWLEOH PDWUL[ P VXFKWKDW P !1 $3 D B RU HTXLYDOHQWO\ A D PBP !1 :ULWLQJ Q IRU P !1 ZHKDYH Q!1 BQ D A 6R B LVDOVRVLPLODUWR A DQGZHVD\VLPSO\WKDW A DQG B DUHVLPLODU &KDQJLQJ A LQWR P !1 $3 LVFDOOHGD VLPLODULW\WUDQVIRUPDWLRQ

5)&03&.

,I n # n PDWULFHV A DQG B DUHVLPLODU WKHQWKH\KDYHWKHVDPHFKDUDFWHULVWLF SRO\QRPLDODQGKHQFHWKHVDPHHLJHQYDOXHVZLWKWKHVDPHPXOWLSOLFLWLHV

1300' ,I B D P !1 $3 WKHQ B ! !I D P !1 $3 ! !P !1 P D P !1 .$3 ! !P / D P !1 .A ! !I /P 8VLQJWKHPXOWLSOLFDWLYHSURSHUW\ E LQ7KHRUHP ZHFRPSXWH GHW.B ! !I / D GHWŒP !1 .A ! !I /P #

D GHW.P !1 / " GHW.A ! !I / " GHW.P /

6LQFH GHW.P !1 / " GHW.P / D GHW.P !1P / D GHW I D 1 ZH VHH IURP HTXDWLRQ WKDW GHW.B ! !I / D GHW.A ! !I / :$51,1*6 7KHPDWULFHV

!

2 0

1 2

"

DQG

!

1 0

0 2

"

DUHQRWVLPLODUHYHQWKRXJKWKH\KDYHWKHVDPHHLJHQYDOXHV 6LPLODULW\LVQRWWKHVDPHDVURZHTXLYDOHQFH ,I A LVURZHTXLYDOHQWWR B WKHQ B D ($ IRUVRPHLQYHUWLEOHPDWUL[ E 5RZRSHUDWLRQVRQDPDWUL[ XVXDOO\FKDQJHLWVHLJHQYDOXHV

$)"15&3


$SSOLFDWLRQWR'\QDPLFDO6\VWHPV (LJHQYDOXHVDQGHLJHQYHFWRUVKROGWKHNH\WRWKHGLVFUHWHHYROXWLRQRIDG\QDPLFDO V\VWHP DVPHQWLRQHGLQWKHFKDSWHULQWURGXFWLRQ

&9".1-& /HW A D

!

:95 :05

QDPLFDOV\VWHPGHÀQHGE\ [k C1

" :03 $QDO\]HWKHORQJWHUPEHKDYLRURIWKHG\ :97 ! " :6 D A[k .k D 0; 1; 2; : : :/ ZLWK [0 D :4

40-65*0/ 7KHÀUVWVWHSLVWRÀQGWKHHLJHQYDOXHVRI A DQGDEDVLVIRUHDFKHLJHQVSDFH 7KHFKDUDFWHULVWLFHTXDWLRQIRU A LV ! " :95 ! ! :03 0 D GHW D .:95 ! !/.:97 ! !/ ! .:03/.:05/ :05 :97 ! ! D !2 ! 1:92! C :92

%\WKHTXDGUDWLFIRUPXOD

p

p 1:92 ˙ :0064 .1:92/2 ! 4.:92/ D !D 2 2 1:92 ˙ :08 D 1 RU :92 D 2 ,WLVUHDGLO\FKHFNHGWKDWHLJHQYHFWRUVFRUUHVSRQGLQJWR ! D 1 DQG ! D :92 DUHPXOWLSOHV RI ! " ! " 3 1 Y1 D DQG Y2 D 5 !1 1:92 ˙

UHVSHFWLYHO\ 7KHQH[WVWHSLVWRZULWHWKHJLYHQ [0 LQWHUPVRI Y1 DQG Y2 7KLVFDQEHGRQHEHFDXVH fY1 ; Y2 g LVREYLRXVO\DEDVLVIRU R2 :K\" 6RWKHUHH[LVWZHLJKWV c1 DQG c2 VXFKWKDW ! " c [0 D c1 Y1 C c2 Y2 D Œ Y1 Y2 # 1 c2

,QIDFW

!

c1 c2

"

"!1 ! " 3 1 :60 D Œ Y1 Y2 # [0 D 5 !1 :40 ! "! " ! " 1 !1 !1 :60 :125 D D :225 !8 !5 3 :40 !1

!

%HFDXVH Y1 DQG Y2 LQ DUHHLJHQYHFWRUVRI A ZLWK AY1 D Y1 DQG AY2 D :92Y2 ZH HDVLO\FRPSXWHHDFK [k [1 D A[0 D c1 AY1 C c2 AY2 D c1 Y1 C c2 .:92/Y2 [2 D A[1 D c1 AY1 C c2 .:92/AY2

8VLQJOLQHDULW\RI [ 7! A[

Y1 DQG Y2 DUHHLJHQYHFWRUV

D c1 Y1 C c2 .:92/2 Y2

DQGVRRQ ,QJHQHUDO [k D c1 Y1 C c2 .:92/k Y2

.k D 0; 1; 2; : : :/

8VLQJ c1 DQG c2 IURP

! " ! " 3 1 [k D :125 C :225.:92/k 5 !1

.k D 0; 1; 2; : : :/


7KLVH[SOLFLWIRUPXODIRU [k JLYHVWKHVROXWLRQRIWKHGLIIHUHQFHHTXDWLRQ [k C1 D A[k ! " :375 $V k ! 1 .:92/k WHQGVWR]HURDQG [k WHQGVWR D :125Y1 :625 7KHFDOFXODWLRQVLQ([DPSOH KDYHDQLQWHUHVWLQJDSSOLFDWLRQWRD0DUNRYFKDLQ GLVFXVVHGLQ6HFWLRQ 7KRVHZKRUHDGWKDWVHFWLRQPD\UHFRJQL]HWKDWPDWUL[ A LQ([DPSOH DERYHLVWKHVDPHDVWKHPLJUDWLRQPDWUL[ M LQ6HFWLRQ [0 LVWKH LQLWLDOSRSXODWLRQGLVWULEXWLRQEHWZHHQFLW\DQGVXEXUEV DQG [k UHSUHVHQWVWKHSRSXODWLRQ GLVWULEXWLRQDIWHU k \HDUV 7KHRUHP LQ6HFWLRQ VWDWHGWKDWIRUDPDWUL[VXFKDV A WKHVHTXHQFH [k WHQGV WRDVWHDG\VWDWHYHFWRU 1RZZHNQRZ ZK\ WKH [k EHKDYHWKLVZD\ DWOHDVWIRUWKH PLJUDWLRQPDWUL[ 7KHVWHDG\VWDWHYHFWRULV :125Y1 DPXOWLSOHRIWKHHLJHQYHFWRU Y1 DQGIRUPXOD IRU [k VKRZVSUHFLVHO\ZK\ [k ! :125Y1

/6.&3*$"- /05&4 &RPSXWHUVRIWZDUHVXFKDV0DWKHPDWLFDDQG0DSOHFDQXVHV\PEROLFFDOFX ODWLRQVWRÀQGWKHFKDUDFWHULVWLFSRO\QRPLDORIDPRGHUDWHVL]HGPDWUL[ %XW WKHUHLVQRIRUPXODRUÀQLWHDOJRULWKPWRVROYHWKHFKDUDFWHULVWLFHTXDWLRQRID JHQHUDO n # n PDWUL[IRU n % 5 7KHEHVWQXPHULFDOPHWKRGVIRUÀQGLQJHLJHQYDOXHVDYRLGWKHFKDUDFWHULVWLF SRO\QRPLDOHQWLUHO\ ,QIDFW 0$7/$% ÀQGVWKHFKDUDFWHULVWLFSRO\QRPLDO RIDPDWUL[ A E\ÀUVWFRPSXWLQJWKHHLJHQYDOXHV !1 ; : : : ; !n RI A DQGWKHQ H[SDQGLQJWKHSURGXFW .! ! !1 /.! ! !2 / " " " .! ! !n / 6HYHUDOFRPPRQDOJRULWKPVIRUHVWLPDWLQJWKHHLJHQYDOXHVRIDPDWUL[ A DUHEDVHGRQ7KHRUHP 7KHSRZHUIXO 45 DOJRULWKP LVGLVFXVVHGLQWKH H[HUFLVHV $QRWKHUWHFKQLTXH FDOOHG -DFREL·VPHWKRG ZRUNVZKHQ A D AT DQGFRPSXWHVDVHTXHQFHRIPDWULFHVRIWKHIRUP

A1 D A

Ak C1 D Pk!1 Ak Pk

DQG

.k D 1; 2; : : :/

(DFKPDWUL[LQWKHVHTXHQFHLVVLPLODUWR A DQGVRKDVWKHVDPHHLJHQYDOXHV DV A 7KHQRQGLDJRQDOHQWULHVRI Ak C1 WHQGWR]HURDV k LQFUHDVHV DQGWKH GLDJRQDOHQWULHVWHQGWRDSSURDFKWKHHLJHQYDOXHVRI A 2WKHUPHWKRGVRIHVWLPDWLQJHLJHQYDOXHVDUHGLVFXVVHGLQ6HFWLRQ

13"$5*$& 130#-&. )LQGWKHFKDUDFWHULVWLFHTXDWLRQDQGHLJHQYDOXHVRI A D

&9&3$*4&4 )LQGWKHFKDUDFWHULVWLFSRO\QRPLDODQGWKHUHDOHLJHQYDOXHVRIWKH PDWULFHVLQ([HUFLVHV² ! " ! " 2 7 !4 !1 7 2 6 1

!

!4 6

2 7

"

!

8 3

2 3

"

! !

8 4 5 !4

4 8

"

3 4

!

" 1 !4 4 2

"

! !

9 !2 2 5

"

!4 2

3 1

"

([HUFLVHV²UHTXLUHWHFKQLTXHVIURP6HFWLRQ )LQGWKH FKDUDFWHULVWLFSRO\QRPLDORIHDFKPDWUL[ XVLQJHLWKHUDFRIDFWRU H[SDQVLRQRUWKHVSHFLDOIRUPXODIRU 3 # 3 GHWHUPLQDQWVGHVFULEHG

$)"15&3


SULRU WR ([HUFLVHV ² LQ 6HFWLRQ >1RWH )LQGLQJ WKH FKDUDFWHULVWLFSRO\QRPLDORID 3 # 3 PDWUL[LVQRWHDV\WRGRZLWK MXVWURZRSHUDWLRQV EHFDXVHWKHYDULDEOH ! LVLQYROYHG@ 2 3 2 3 4 0 !1 3 1 1 4 !1 5 5 05 4 0 4 0 1 0 2 !2 0 7 2 3 2 3 3 0 0 !1 0 2 1 45 1 05 4 2 4 3 1 0 4 0 1 2 2 3 2 3 6 !2 0 4 0 !1 9 05 0 45 4 !2 4 !1 5 8 3 0 2 3

)RUWKHPDWULFHVLQ([HUFLVHV² OLVWWKHUHDOHLJHQYDOXHV UHSHDWHGDFFRUGLQJWRWKHLUPXOWLSOLFLWLHV 2 3 2 3 5 5 0 2 3 0 0 0 60 66 2 !3 67 2 0 07 7 7 6 6 40 40 0 3 !2 5 3 6 05 0 0 0 5 2 3 3 !5 2 3 3 0 0 0 0 6 !5 1 0 0 07 6 7 6 8 0 0 07 6 3 7 4 0 !7 2 1 05 !4 1 9 !2 3

,WFDQEHVKRZQWKDWWKHDOJHEUDLFPXOWLSOLFLW\RIDQHLJHQ YDOXH ! LVDOZD\VJUHDWHUWKDQRUHTXDOWRWKHGLPHQVLRQRIWKH HLJHQVSDFHFRUUHVSRQGLQJWR ! )LQG h LQWKHPDWUL[ A EHORZ VXFKWKDWWKHHLJHQVSDFHIRU ! D 4 LVWZRGLPHQVLRQDO 2 3 4 2 3 3 60 2 h 37 7 AD6 40 0 4 14 5 0 0 0 2 /HW A EHDQ n # n PDWUL[ DQGVXSSRVH A KDV n UHDOHLJHQYDO XHV !1 ; : : : ; !n UHSHDWHGDFFRUGLQJWRPXOWLSOLFLWLHV VRWKDW GHW .A ! !I / D .!1 ! !/.!2 ! !/ " " " .!n ! !/ ([SODLQZK\ GHW A LVWKHSURGXFWRIWKH n HLJHQYDOXHVRI A 7KLVUHVXOWLVWUXHIRUDQ\VTXDUHPDWUL[ZKHQFRPSOH[ HLJHQYDOXHVDUHFRQVLGHUHG 8VHDSURSHUW\RIGHWHUPLQDQWVWRVKRZWKDW A DQG AT KDYH WKHVDPHFKDUDFWHULVWLFSRO\QRPLDO ,Q([HUFLVHVDQG A DQG B DUH n # n PDWULFHV 0DUNHDFK VWDWHPHQW7UXHRU)DOVH -XVWLI\HDFKDQVZHU D 7KHGHWHUPLQDQWRI A LVWKHSURGXFWRIWKHGLDJRQDOHQWULHV LQ A E $QHOHPHQWDU\URZRSHUDWLRQRQ A GRHVQRWFKDQJHWKH GHWHUPLQDQW F .GHW A/.GHW B/ D GHW

%$G ,I ! C 5 LVDIDFWRURIWKHFKDUDFWHULVWLFSRO\QRPLDORI A WKHQLVDQHLJHQYDOXHRI A

D ,I A LV 3 # 3 ZLWKFROXPQV D1 D2 D3 WKHQ GHW A HTXDOV WKHYROXPHRIWKHSDUDOOHOHSLSHGGHWHUPLQHGE\ D1 D2 D3 E GHW AT D .!1/ GHW A

F 7KHPXOWLSOLFLW\RIDURRW r RIWKHFKDUDFWHULVWLFHTXDWLRQ RI A LVFDOOHGWKHDOJHEUDLFPXOWLSOLFLW\RI r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c Y2

F )RU k D 1; 2; : : : ; GHÀQH [k D Ak [0 &RPSXWH [1 DQG [2 DQGZULWHDIRUPXODIRU [k 7KHQVKRZWKDW [k ! Y1 DV k LQFUHDVHV ! " a b 8VH IRUPXOD IRU D GHWHUPLQDQW /HW A D c d JLYHQEHIRUH([DPSOH WRVKRZWKDW GHW A D ad ! bc &RQVLGHUWZRFDVHV a ¤ 0 DQG a D 0 2 3 2 3 2 3 :5 :2 :3 :3 1 /HW A D 4 :3 :8 :3 5 Y1 D 4 :6 5 Y2 D 4 !3 5 :2 0 :4 :1 2 2 3 2 3 !1 1 Y3 D 4 0 5 DQG Z D 4 1 5 1 1 D 6KRZWKDW Y1 Y2 Y3 DUHHLJHQYHFWRUVRI A >1RWH A LVWKH VWRFKDVWLFPDWUL[VWXGLHGLQ([DPSOH RI6HFWLRQ @ E /HW [0 EHDQ\YHFWRULQ R3 ZLWKQRQQHJDWLYHHQWULHVZKRVH VXPLV ,Q6HFWLRQ [0 ZDVFDOOHGDSUREDELOLW\ YHFWRU ([SODLQZK\WKHUHDUHFRQVWDQWV c1 c2 c3 VXFK WKDW [0 D c1 Y1 C c2 Y2 C c3 Y3 &RPSXWH ZT [0 DQGGH GXFHWKDW c1 D 1 F )RU k D 1; 2; : : : ; GHÀQH [k D Ak [0 ZLWK [0 DVLQSDUW E 6KRZWKDW [k ! Y1 DV k LQFUHDVHV

>0@ &RQVWUXFWDUDQGRPLQWHJHUYDOXHG 4 # 4 PDWUL[ A DQG YHULI\WKDW A DQG AT KDYHWKHVDPHFKDUDFWHULVWLFSRO\QRPLDO WKHVDPHHLJHQYDOXHVZLWKWKHVDPHPXOWLSOLFLWLHV 'R A DQG AT KDYHWKHVDPHHLJHQYHFWRUV" 0DNHWKHVDPHDQDO\VLV RID 5 # 5 PDWUL[ 5HSRUWWKHPDWULFHVDQG\RXUFRQFOXVLRQV >0@ &RQVWUXFWDUDQGRPLQWHJHUYDOXHG 4 # 4 PDWUL[ A D 5HGXFH A WRHFKHORQIRUP U ZLWKQRURZVFDOLQJ DQGXVH U LQIRUPXOD EHIRUH([DPSOH WRFRPSXWH GHW A ,I A KDSSHQVWREHVLQJXODU VWDUWRYHUZLWKDQHZUDQGRP PDWUL[

'LDJRQDOL]DWLRQ

F /LVWWKHPDWUL[ A DQG WRIRXUGHFLPDOSODFHV OLVWWKH SLYRWVLQ U DQGWKHHLJHQYDOXHVRI A &RPSXWH GHW A ZLWK \RXUPDWUL[SURJUDP DQGFRPSDUHLWZLWKWKHSURGXFWV \RXIRXQGLQD DQGE 2 3 !6 28 21 >0@ /HW A D 4 4 !15 !12 5 )RUHDFKYDOXHRI a LQ !8 a 25

WKHVHW f32; 31:9; 31:8; 32:1; 32:2g FRPSXWHWKH FKDUDFWHULV WLFSRO\QRPLDORI A DQGWKHHLJHQYDOXHV ,QHDFKFDVH FUHDWH DJUDSKRIWKHFKDUDFWHULVWLFSRO\QRPLDO p.t/ D GHW .A ! tI / IRU 0 & t & 3 ,ISRVVLEOH FRQVWUXFWDOOJUDSKVRQRQHFRRU GLQDWHV\VWHP 'HVFULEHKRZWKHJUDSKVUHYHDOWKHFKDQJHV LQWKHHLJHQYDOXHVDV a FKDQJHV

E &RPSXWHWKHHLJHQYDOXHVRI A DQGWKHSURGXFWRIWKHVH HLJHQYDOXHVDVDFFXUDWHO\DVSRVVLEOH

40-65*0/ 50 13"$5*$& 130#-&. 7KHFKDUDFWHULVWLFHTXDWLRQLV

!

1!! 0 D GHW.A ! !I / D GHW 4

!4 2!!

"

D .1 ! !/.2 ! !/ ! .!4/.4/ D !2 ! 3! C 18

)URPWKHTXDGUDWLFIRUPXOD

p

p 3 ˙ !63 .!3/2 ! 4.18/ D !D 2 2 ,W LV FOHDU WKDW WKH FKDUDFWHULVWLF HTXDWLRQ KDV QR UHDO VROXWLRQV VR A KDV QR UHDO HLJHQYDOXHV 7KHPDWUL[ A LVDFWLQJRQWKHUHDOYHFWRUVSDFH R2 DQGWKHUHLVQRQRQ]HUR YHFWRU Y LQ R2 VXFKWKDW AY D !Y IRUVRPHVFDODU ! 3˙

%*"(0/"-*;"5*0/ ,QPDQ\FDVHV WKHHLJHQYDOXH²HLJHQYHFWRULQIRUPDWLRQFRQWDLQHGZLWKLQDPDWUL[ A FDQ EHGLVSOD\HGLQDXVHIXOIDFWRUL]DWLRQRIWKHIRUP A D 3'3!1 ZKHUH D LVDGLDJRQDO PDWUL[ ,QWKLVVHFWLRQ WKHIDFWRUL]DWLRQHQDEOHVXVWRFRPSXWH Ak TXLFNO\IRUODUJH YDOXHVRI k DIXQGDPHQWDOLGHDLQVHYHUDODSSOLFDWLRQVRIOLQHDUDOJHEUD /DWHU LQ 6HFWLRQV DQG WKHIDFWRUL]DWLRQZLOOEHXVHGWRDQDO\]HDQG GHFRXSOH G\QDPLFDO V\VWHPV 7KHIROORZLQJH[DPSOHLOOXVWUDWHVWKDWSRZHUVRIDGLDJRQDOPDWUL[DUHHDV\WR FRPSXWH ! " ! "! " ! 2 " 5 0 5 0 5 0 5 0 &9".1-& ,I D D WKHQ D 2 D D 0 3 0 3 0 3 0 32 DQG ! "! 2 " ! 3 " 5 0 5 0 5 0 D 3 D DD 2 D D 0 3 0 32 0 33 ,QJHQHUDO

Dk D

!

5k 0

0 3k

"

IRU k % 1

,I A D 3'3!1 IRUVRPHLQYHUWLEOH P DQGGLDJRQDO D WKHQ Ak LVDOVRHDV\WR FRPSXWH DVWKHQH[WH[DPSOHVKRZV

$)"15&3


" 7 2 )LQGDIRUPXODIRU Ak JLYHQWKDW A D 3'3!1 !4 1 ! " ! " 1 1 5 0 P D DQG D D !1 !2 0 3

&9".1-& /HW A D ZKHUH

!

40-65*0/ 7KHVWDQGDUGIRUPXODIRUWKHLQYHUVHRID 2 # 2 PDWUL[\LHOGV ! " 2 1 P !1 D !1 !1 7KHQ E\DVVRFLDWLYLW\RIPDWUL[PXOWLSOLFDWLRQ

A2 D .3'3!1 /.3'3!1 / D 3' .P !1 P / DP !1 D 3''3!1 „ ƒ‚ … 2

D 3' P

!1

D

$JDLQ

!

1 1 !1 !2

"!

I

52 0

0 32

"!

2 1 !1 !1

"

A3 D .3'3!1 /A2 D .3'P !1 /PD 2 P !1 D 3''2 P !1 D 3'3 P !1 „ ƒ‚ … I

,QJHQHUDO IRU k % 1

Ak D 3'k P !1 D D

!

!

1 1 !1 !2

2 " 5k ! 3k 2 " 3k ! 2 " 5k

"!

5k 0

5k ! 3k 2 " 3k ! 5k

"

0 3k

"!

2 1 !1 !1

"

$ VTXDUHPDWUL[ A LVVDLGWREH GLDJRQDOL]DEOH LI A LVVLPLODUWRDGLDJRQDOPDWUL[ WKDWLV LI A D 3'3!1 IRUVRPHLQYHUWLEOHPDWUL[ P DQGVRPHGLDJRQDOPDWUL[ D 7KHQH[WWKHRUHPJLYHVDFKDUDFWHUL]DWLRQRIGLDJRQDOL]DEOHPDWULFHVDQGWHOOVKRZWR FRQVWUXFWDVXLWDEOHIDFWRUL]DWLRQ

5)&03&.

5IF %JBHPOBMJ[BUJPO 5IFPSFN $Q n # n PDWUL[ A LVGLDJRQDOL]DEOHLIDQGRQO\LI A KDV n OLQHDUO\LQGHSHQGHQW HLJHQYHFWRUV ,QIDFW A D 3'3!1 ZLWK D DGLDJRQDOPDWUL[ LIDQGRQO\LIWKHFROXPQVRI P DUH n OLQHDUO\LQGHSHQGHQWHLJHQYHFWRUVRI A ,QWKLVFDVH WKHGLDJRQDOHQWULHV RI D DUHHLJHQYDOXHVRI A WKDWFRUUHVSRQG UHVSHFWLYHO\ WRWKHHLJHQYHFWRUVLQ P ,QRWKHUZRUGV A LVGLDJRQDOL]DEOHLIDQGRQO\LIWKHUHDUHHQRXJKHLJHQYHFWRUVWR IRUPDEDVLVRI Rn :HFDOOVXFKDEDVLVDQ HLJHQYHFWRUEDVLV RI Rn

1300' )LUVW REVHUYHWKDWLI P LVDQ\ n # n PDWUL[ZLWKFROXPQV Y1 ; : : : ; Yn DQGLI D LVDQ\GLDJRQDOPDWUL[ZLWKGLDJRQDOHQWULHV !1 ; : : : ; !n WKHQ $3 D AŒ Y1 Y2 " " " Yn # D Œ AY1 AY2 " " " AYn # ZKLOH

2

!1 6 0 6 3' D P 6 :: 4 : 0

0 !2 :: :

""" """

0 0 :: :

0

"""

!n

3

7 7 7 D Œ !1 Y1 !2 Y2 " " " !n Yn # 5

'LDJRQDOL]DWLRQ

1RZVXSSRVH A LVGLDJRQDOL]DEOHDQG A D 3'3!1 7KHQULJKWPXOWLSO\LQJWKLVUHODWLRQ E\ P ZHKDYH $3 D 3' ,QWKLVFDVH HTXDWLRQV DQG LPSO\WKDW

Œ AY1 AY2 " " " AYn # D Œ !1 Y1 !2 Y2 " " " !n Yn #

(TXDWLQJFROXPQV ZHÀQGWKDW

AY1 D !1 Y1 ;

AY2 D !2 Y2 ;

:::;

AYn D !n Yn

6LQFH P LVLQYHUWLEOH LWVFROXPQV Y1 ; : : : ; Yn PXVWEHOLQHDUO\LQGHSHQGHQW $OVR VLQFH WKHVHFROXPQVDUHQRQ]HUR WKHHTXDWLRQVLQ VKRZWKDW !1 ; : : : ; !n DUHHLJHQYDOXHVDQG Y1 ; : : : ; Yn DUHFRUUHVSRQGLQJHLJHQYHFWRUV 7KLVDUJXPHQWSURYHVWKH´RQO\LIµSDUWVRI WKHÀUVWDQGVHFRQGVWDWHPHQWV DORQJZLWKWKHWKLUGVWDWHPHQW RIWKHWKHRUHP )LQDOO\ JLYHQDQ\ n HLJHQYHFWRUV Y1 ; : : : ; Yn XVHWKHPWRFRQVWUXFWWKHFROXPQVRI P DQGXVHFRUUHVSRQGLQJHLJHQYDOXHV !1 ; : : : ; !n WRFRQVWUXFW D %\HTXDWLRQV ² $3 D 3' 7KLVLVWUXHZLWKRXWDQ\FRQGLWLRQRQWKHHLJHQYHFWRUV ,I LQIDFW WKH HLJHQYHFWRUVDUHOLQHDUO\LQGHSHQGHQW WKHQ P LVLQYHUWLEOHE\WKH,QYHUWLEOH0DWUL[ 7KHRUHP DQG $3 D 3' LPSOLHVWKDW A D 3'3!1

'LDJRQDOL]LQJ0DWULFHV &9".1-& 'LDJRQDOL]HWKHIROORZLQJPDWUL[ LISRVVLEOH 2

3 1 3 3 A D 4 !3 !5 !3 5 3 3 1

7KDWLV ÀQGDQLQYHUWLEOHPDWUL[ P DQGDGLDJRQDOPDWUL[ D VXFKWKDW A D 3'3!1

40-65*0/ 7KHUHDUHIRXUVWHSVWRLPSOHPHQWWKHGHVFULSWLRQLQ7KHRUHP 6WHS )LQGWKHHLJHQYDOXHVRI$ $VPHQWLRQHGLQ6HFWLRQ WKHPHFKDQLFVRIWKLV VWHSDUHDSSURSULDWHIRUDFRPSXWHUZKHQWKHPDWUL[LVODUJHUWKDQ 2 # 2 7RDYRLG XQQHFHVVDU\GLVWUDFWLRQV WKHWH[WZLOOXVXDOO\VXSSO\LQIRUPDWLRQQHHGHGIRUWKLVVWHS ,QWKHSUHVHQWFDVH WKHFKDUDFWHULVWLFHTXDWLRQWXUQVRXWWRLQYROYHDFXELFSRO\QRPLDO WKDWFDQEHIDFWRUHG

0 D GHW .A ! !I / D !!3 ! 3!2 C 4

D !.! ! 1/.! C 2/2

7KHHLJHQYDOXHVDUH ! D 1 DQG ! D !2

6WHS )LQGWKUHHOLQHDUO\LQGHSHQGHQWHLJHQYHFWRUVRI$ 7KUHH YHFWRUVDUHQHHGHG EHFDXVH A LVD 3 # 3 PDWUL[ 7KLVLVWKHFULWLFDOVWHS ,ILWIDLOV WKHQ7KHRUHPVD\V WKDW A FDQQRWEHGLDJRQDOL]HG 7KHPHWKRGLQ6HFWLRQ SURGXFHVDEDVLVIRUHDFK HLJHQVSDFH 2 3 1 %DVLVIRU ! D 1W Y1 D 4 !1 5 1 2 3 2 3 !1 !1 %DVLVIRU ! D !2W Y2 D 4 1 5 DQG Y3 D 4 0 5 0 1

$)"15&3


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

&9".1-& 'LDJRQDOL]HWKHIROORZLQJPDWUL[ LISRVVLEOH 2

3 2 4 3 A D 4 !4 !6 !3 5 3 3 1

40-65*0/ 7KHFKDUDFWHULVWLFHTXDWLRQRI A WXUQVRXWWREHH[DFWO\WKHVDPHDVWKDWLQ ([DPSOH 0 D GHW .A ! !I / D !!3 ! 3!2 C 4 D !.! ! 1/.! C 2/2 7KH HLJHQYDOXHV DUH ! D 1 DQG ! D !2 +RZHYHU LW LV HDV\ WR YHULI\ WKDW HDFK HLJHQVSDFHLVRQO\RQHGLPHQVLRQDO 2 3 1 %DVLVIRU ! D 1W Y1 D 4 !1 5 1 2 3 !1 %DVLVIRU ! D !2W Y2 D 4 1 5 0 7KHUHDUHQRRWKHUHLJHQYDOXHV DQGHYHU\HLJHQYHFWRURI A LVDPXOWLSOHRIHLWKHU Y1 RU Y2 +HQFHLWLVLPSRVVLEOHWRFRQVWUXFWDEDVLVRI R3 XVLQJHLJHQYHFWRUVRI A %\ 7KHRUHP A LV QRW GLDJRQDOL]DEOH

7KH IROORZLQJ WKHRUHP SURYLGHV D VXIÀFLHQW FRQGLWLRQ IRU D PDWUL[ WR EH GLDJRQDOL]DEOH

5)&03&.

$Q n # n PDWUL[ZLWK n GLVWLQFWHLJHQYDOXHVLVGLDJRQDOL]DEOH

'LDJRQDOL]DWLRQ

1300' /HW Y1 ; : : : ; Yn EHHLJHQYHFWRUVFRUUHVSRQGLQJWRWKH n GLVWLQFWHLJHQYDOXHVRI DPDWUL[ A 7KHQ fY1 ; : : : ; Yn g LVOLQHDUO\LQGHSHQGHQW E\7KHRUHP LQ6HFWLRQ +HQFH A LVGLDJRQDOL]DEOH E\7KHRUHP ,WLVQRW QHFHVVDU\ IRUDQ n # n PDWUL[WRKDYH n GLVWLQFWHLJHQYDOXHVLQRUGHUWR EHGLDJRQDOL]DEOH 7KH 3 # 3 PDWUL[LQ([DPSOH LVGLDJRQDOL]DEOHHYHQWKRXJKLWKDV RQO\WZRGLVWLQFWHLJHQYDOXHV

&9".1-& 'HWHUPLQHLIWKHIROORZLQJPDWUL[LVGLDJRQDOL]DEOH 2

3 5 !8 1 A D 40 0 75 0 0 !2

40-65*0/ 7KLVLVHDV\ 6LQFHWKHPDWUL[LVWULDQJXODU LWVHLJHQYDOXHVDUHREYLRXVO\ DQG !2 6LQFH A LVD 3 # 3 PDWUL[ZLWKWKUHHGLVWLQFWHLJHQYDOXHV A LVGLDJRQDOL]DEOH

0DWULFHV:KRVH(LJHQYDOXHV$UH1RW'LVWLQFW ,IDQ n # n PDWUL[ A KDV n GLVWLQFWHLJHQYDOXHV ZLWKFRUUHVSRQGLQJHLJHQYHFWRUV Y1 ; : : : ; Yn DQGLI P D Œ Y1 " " " Yn # WKHQ P LVDXWRPDWLFDOO\LQYHUWLEOHEHFDXVHLWVFROXPQV DUHOLQHDUO\LQGHSHQGHQW E\7KHRUHP :KHQ A LVGLDJRQDOL]DEOHEXWKDVIHZHUWKDQ n GLVWLQFWHLJHQYDOXHV LWLVVWLOOSRVVLEOHWREXLOG P LQDZD\WKDWPDNHV P DXWRPDWLFDOO\ LQYHUWLEOH DVWKHQH[WWKHRUHPVKRZVñ

5)&03&.

/HW A EHDQ n # n PDWUL[ZKRVHGLVWLQFWHLJHQYDOXHVDUH !1 ; : : : ; !p

D )RU 1 & k & p WKHGLPHQVLRQRIWKHHLJHQVSDFHIRU !k LVOHVVWKDQRUHTXDOWR WKHPXOWLSOLFLW\RIWKHHLJHQYDOXH !k E 7KHPDWUL[ A LVGLDJRQDOL]DEOHLIDQGRQO\LIWKHVXPRIWKHGLPHQVLRQVRI WKHHLJHQVSDFHVHTXDOV n DQGWKLVKDSSHQVLIDQGRQO\LIL WKHFKDUDFWHULVWLF SRO\QRPLDOIDFWRUVFRPSOHWHO\LQWROLQHDUIDFWRUVDQGLL WKHGLPHQVLRQRIWKH HLJHQVSDFHIRUHDFK !k HTXDOVWKHPXOWLSOLFLW\RI !k F ,I A LVGLDJRQDOL]DEOHDQG Bk LVDEDVLVIRUWKHHLJHQVSDFHFRUUHVSRQGLQJWR !k IRUHDFK k WKHQWKHWRWDOFROOHFWLRQRIYHFWRUVLQWKHVHWV B1 ; : : : ; Bp IRUPVDQ HLJHQYHFWRUEDVLVIRU Rn

&9".1-& 'LDJRQDOL]HWKHIROORZLQJPDWUL[ LISRVVLEOH 2

3 5 0 0 0 6 0 5 0 07 7 AD6 4 1 4 !3 0 5 !1 !2 0 !3 ñ 7KHSURRIRI7KHRUHPLVVRPHZKDWOHQJWK\EXWQRWGLIÀFXOW )RULQVWDQFH VHH6 )ULHGEHUJ $ ,QVHO DQG / 6SHQFH /LQHDU$OJHEUD WKHG(QJOHZRRG&OLIIV 1-3UHQWLFH+DOO 6HFWLRQ

$)"15&3


40-65*0/ 6LQFH A LVDWULDQJXODUPDWUL[ WKHHLJHQYDOXHVDUHDQG !3 HDFKZLWK PXOWLSOLFLW\ 8VLQJWKHPHWKRGLQ6HFWLRQ ZHÀQGDEDVLVIRUHDFKHLJHQVSDFH 2 3 2 3 !8 !16 6 47 6 47 7 6 7 %DVLVIRU ! D 5W Y1 D 6 4 1 5 DQG Y2 D 4 0 5 0 1 2 3 2 3 0 0 607 607 6 7 6 %DVLVIRU ! D !3W Y3 D 4 5 DQG Y4 D 4 7 1 05 0 1 7KH VHW fY1 ; : : : ; Y4 g LV OLQHDUO\ LQGHSHQGHQW E\ 7KHRUHP Œ Y1 " " " Y4 # LVLQYHUWLEOH DQG A D 3'3!1 ZKHUH 2 3 2 !8 !16 0 0 5 6 4 7 60 4 0 0 7 DQG D D 6 P D6 4 1 40 0 1 05 0 1 0 1 0

13"$5*$& 130#-&.4

WEB

&9&3$*4&4 ,Q([HUFLVHVDQG OHW A D PDP !1 DQGFRPSXWH A4 ! " ! " 5 7 2 0 P D DD 2 3 0 1 P D

" ! 2 1 DD 3 0

1 2

0 3

"

,Q([HUFLVHVDQG XVHWKHIDFWRUL]DWLRQ A D PDP !1 WRFRP SXWH Ak ZKHUH k UHSUHVHQWVDQDUELWUDU\SRVLWLYHLQWHJHU ! " ! "! "! " a 0 1 0 a 0 1 0 D 2.a ! b/ b 2 1 0 b !2 1

!

1 !6 2 !6

"

D

!

3 !2 2 !1

"!

3 0 0 0 5 0 07 7 0 !3 0 5 0 0 !3

" 4 !3 &RPSXWH A ZKHUH A D 2 !1 ! " ! " ! " !3 12 3 2 /HW A D Y1 D DQG Y2 D 6XSSRVH\RXDUHWROGWKDW Y1 DQG !2 7 1 1 Y2 DUHHLJHQYHFWRUVRI A 8VHWKLVLQIRUPDWLRQWRGLDJRQDOL]H A /HW A EHD 4 # 4 PDWUL[ZLWKHLJHQYDOXHV DQG !2 DQGVXSSRVH\RXNQRZWKDW WKHHLJHQVSDFHIRU ! D 3 LVWZRGLPHQVLRQDO 'R\RXKDYHHQRXJKLQIRUPDWLRQWR GHWHUPLQHLI A LVGLDJRQDOL]DEOH" 8

!

6R WKH PDWUL[ P D

!3 0 0 !2

"!

!1 !2

2 3

"

,Q([HUFLVHVDQG WKHPDWUL[ A LVIDFWRUHGLQWKHIRUP 3'3!1 8VHWKH'LDJRQDOL]DWLRQ7KHRUHPWRÀQGWKHHLJHQYDOXHVRI A DQG DEDVLVIRUHDFKHLJHQVSDFH

!

2

2 A D 4 1 !1 2 1 D 4 !1 0 2 3 A D 4 !3 0 2 3 D 40 1

3 !1 !1 4 15 !1 2 32 32 !1 0 3 0 0 0 1 !1 54 0 2 0 54 !1 !1 1 0 0 3 !1 3 0 0 4 95 0 3 32 32 0 !1 3 0 0 0 1 !3 54 0 4 0 54 !3 0 0 0 0 3 !1

3 !1 !1 !1 !1 5 !1 0

0 1 0

3 1 95 3

'LDJRQDOL]HWKHPDWULFHVLQ([HUFLVHV² LISRVVLEOH 7KHUHDO HLJHQYDOXHVIRU([HUFLVHV²DQGDUHLQFOXGHGEHORZWKH PDWUL[ ! " ! " 1 0 3 2 6 !1 0 3

!

" 2 !1 1 4 2 0 1 42 1 3 3 ! D !1; 2 2 2 4 1 3 !1 !2 ! D 1; 2 0 !1 4 1 2 !1 !1 ! D 0;

2

2 4 2 2 2

0 2 2

5 !3 60 3 6 4 0 0 0 0

3 1 25 2 5 3 !1 !1 5 2 5 3 !1 15 0 1 3 0 05 2

3 0 9 1 !2 7 7 2 05 0 2

!

1 4 2

3 41 1 2

3 2

"

1 3 1 ! D 2;

5

3 1 15 3

3 0 !2 3 25 0 3 ! D 2; 3 2 3 1 2 !3 42 5 !2 5 1 3 1 !D0 2 3 2 !2 !2 4 3 !3 !2 5 2 !2 !2 ! D !2; !1; 0 2 3 3 0 0 0 60 2 0 07 6 7 40 0 2 05 1 0 0 3

2 41 0

,Q([HUFLVHVDQG A B P DQG D DUH n # n PDWULFHV 0DUNHDFKVWDWHPHQW7UXHRU)DOVH -XVWLI\HDFKDQVZHU 6WXG\ 7KHRUHPVDQGDQGWKHH[DPSOHVLQWKLVVHFWLRQFDUHIXOO\EHIRUH \RXWU\WKHVHH[HUFLVHV D A LVGLDJRQDOL]DEOHLI A D 3'3!1 IRUVRPHPDWUL[ D DQG VRPHLQYHUWLEOHPDWUL[ P E ,I Rn KDVDEDVLVRIHLJHQYHFWRUVRI A WKHQ A LVGLDJR QDOL]DEOH F A LVGLDJRQDOL]DEOHLIDQGRQO\LI A KDV n HLJHQYDOXHV FRXQWLQJPXOWLSOLFLWLHV G ,I A LVGLDJRQDOL]DEOH WKHQ A LVLQYHUWLEOH D A LVGLDJRQDOL]DEOHLI A KDV n HLJHQYHFWRUV E ,I A LVGLDJRQDOL]DEOH WKHQ A KDV n GLVWLQFWHLJHQYDOXHV F ,I $3 D 3' ZLWK D GLDJRQDO WKHQWKHQRQ]HURFROXPQV RI P PXVWEHHLJHQYHFWRUVRI A G ,I A LVLQYHUWLEOH WKHQ A LVGLDJRQDOL]DEOH A LVD 5 # 5 PDWUL[ZLWKWZRHLJHQYDOXHV 2QHHLJHQVSDFH LV WKUHHGLPHQVLRQDO DQG WKH RWKHU HLJHQVSDFH LV WZR GLPHQVLRQDO ,V A GLDJRQDOL]DEOH" :K\" A LVD 3 # 3 PDWUL[ZLWKWZRHLJHQYDOXHV (DFKHLJHQVSDFH LVRQHGLPHQVLRQDO ,V A GLDJRQDOL]DEOH" :K\"

'LDJRQDOL]DWLRQ

A LVD 4 # 4 PDWUL[ZLWKWKUHHHLJHQYDOXHV 2QHHLJHQVSDFH LVRQHGLPHQVLRQDO DQGRQHRIWKHRWKHUHLJHQVSDFHVLVWZR GLPHQVLRQDO ,VLWSRVVLEOHWKDW A LV QRW GLDJRQDOL]DEOH" -XVWLI\\RXUDQVZHU A LVD 7 # 7 PDWUL[ZLWKWKUHHHLJHQYDOXHV 2QHHLJHQVSDFHLV WZRGLPHQVLRQDO DQGRQHRIWKHRWKHUHLJHQVSDFHVLVWKUHH GLPHQVLRQDO ,VLWSRVVLEOHWKDW A LV QRW GLDJRQDOL]DEOH" -XVWLI\\RXUDQVZHU 6KRZWKDWLI A LVERWKGLDJRQDOL]DEOHDQGLQYHUWLEOH WKHQVR LV A!1 6KRZWKDWLI A KDV n OLQHDUO\LQGHSHQGHQWHLJHQYHFWRUV WKHQ VRGRHV AT >+LQW 8VHWKH'LDJRQDOL]DWLRQ7KHRUHP@ $ IDFWRUL]DWLRQ A D 3'3!1 LVQRWXQLTXH 'HPRQVWUDWHWKLV ! " 3 0 XVH IRUWKHPDWUL[ A LQ([DPSOH :LWK D1 D 0 5 WKHLQIRUPDWLRQLQ ([DPSOH WRÀQGDPDWUL[ P1 VXFKWKDW A D P1 D1 P1!1

:LWK A DQG D DVLQ([DPSOH ÀQGDQLQYHUWLEOH P2 XQHTXDO WRWKH P LQ([DPSOH VXFKWKDW A D P2 '3!1 2

&RQVWUXFWDQRQ]HUR 2 # 2 PDWUL[WKDWLVLQYHUWLEOHEXWQRW GLDJRQDOL]DEOH &RQVWUXFWDQRQGLDJRQDO 2 # 2 PDWUL[WKDWLVGLDJRQDOL]DEOH EXWQRWLQYHUWLEOH >0@ 'LDJRQDOL]H WKH PDWULFHV LQ ([HUFLVHV ² 8VH \RXU PDWUL[SURJUDP·VHLJHQYDOXHFRPPDQGWRÀQGWKHHLJHQYDOXHV DQGWKHQFRPSXWHEDVHVIRUWKHHLJHQVSDFHVDVLQ6HFWLRQ 2 3 9 !4 !2 !4 6 !56 32 !28 44 7 7 6 4 !14 !14 6 !14 5 42 !33 21 !45 2 3 4 !9 !7 8 2 6 !7 !9 0 7 14 7 6 7 7 5 10 5 !5 !10 6 6 7 4 !2 3 7 0 45 !3 !13 !7 10 11 2 3 13 !12 9 !15 9 6 6 !5 9 !15 97 6 7 6 !12 !5 6 97 6 6 7 4 6 !12 9 !8 95 !6 12 12 !6 !2 2 3 24 !6 2 6 2 6 72 51 9 !99 97 6 7 7 0 !63 15 63 63 6 6 7 4 72 15 9 !63 95 0 63 21 !63 !27

$)"15&3


40-65*0/4 50 13"$5*$& 130#-&.4 GHW .A ! !I / D !2 ! 3! C 2 D .! ! 2/.!! ! "1/ 7KHHLJHQYDOXHVDUHDQG DQG ! " 3 1 WKHFRUUHVSRQGLQJHLJHQYHFWRUVDUH Y1 D DQG Y2 D 1H[W IRUP 2 1

P D

!

" 1 ; 1

3 2

DD

6LQFH A D 3'3!1

A8 D 3'8 P !1 D

&RPSXWH AY1 D

!

" 0 ; 1

2 0

3 2

1 1

3 2

1 1

"! "!

!3 12 !2 7

P !1 D

DQG

28 0

!

1 !1 !2 3

"

" 1 !1 !2 3 "! " 0 1 !1 1 !2 3

0 18

256 0 ! " 766 !765 D 510 !509 D

!

!

!

"!

"! " ! " 3 3 D D 1 " Y1 DQG 1 1

AY2 D

!

!3 12 !2 7

"! " ! " 2 6 D D 3 " Y2 1 3

6R Y1 DQG Y2 DUHHLJHQYHFWRUVIRUWKHHLJHQYDOXHVDQG UHVSHFWLYHO\ 7KXV !1

A D 3'3 ;

SG

.BTUFSJOH &JHFOWBMVF BOE &JHFOTQBDF m

ZKHUH

P D

!

3 1

2 1

"

DQG

DD

!

1 0

0 3

"

&*(&/7&$5034 "/% -*/&"3 53"/4'03."5*0/4 7KHJRDORIWKLVVHFWLRQLVWRXQGHUVWDQGWKHPDWUL[IDFWRUL]DWLRQ A D 3'3!1 DVD VWDWHPHQWDERXWOLQHDUWUDQVIRUPDWLRQV :HVKDOOVHHWKDWWKHWUDQVIRUPDWLRQ [ 7! A[ LVHVVHQWLDOO\WKHVDPHDVWKHYHU\VLPSOHPDSSLQJ X 7! D X ZKHQYLHZHGIURPWKH SURSHUSHUVSHFWLYH $ VLPLODULQWHUSUHWDWLRQZLOODSSO\WR A DQG D HYHQZKHQ D LVQRW DGLDJRQDOPDWUL[ 5HFDOOIURP6HFWLRQ WKDWDQ\OLQHDUWUDQVIRUPDWLRQ T IURP Rn WR Rm FDQEH LPSOHPHQWHGYLDOHIWPXOWLSOLFDWLRQE\DPDWUL[ A FDOOHGWKH VWDQGDUGPDWUL[ RI T 1RZZHQHHGWKHVDPHVRUWRIUHSUHVHQWDWLRQIRUDQ\OLQHDUWUDQVIRUPDWLRQEHWZHHQWZR ÀQLWHGLPHQVLRQDOYHFWRUVSDFHV

(LJHQYHFWRUVDQG/LQHDU7UDQVIRUPDWLRQV

7KH0DWUL[RID/LQHDU7UDQVIRUPDWLRQ /HW V EHDQ nGLPHQVLRQDOYHFWRUVSDFH OHW W EHDQ mGLPHQVLRQDOYHFWRUVSDFH DQG OHW T EHDQ\OLQHDUWUDQVIRUPDWLRQIURP V WR W 7RDVVRFLDWHDPDWUL[ZLWK T FKRRVH RUGHUHG EDVHV B DQG C IRU V DQG W UHVSHFWLYHO\ *LYHQDQ\ [ LQ V WKHFRRUGLQDWHYHFWRU Œ [ #B LVLQ Rn DQGWKHFRRUGLQDWHYHFWRURI LWVLPDJH Œ T .[/ #C LVLQ Rm DVVKRZQLQ)LJ V

W

T

T(x)

x

[x]B

[T(x)]C

!n

!m

'*(63& $ OLQHDUWUDQVIRUPDWLRQIURP V WR W

7KHFRQQHFWLRQEHWZHHQ Œ [ #B DQG Œ T .[/ #C LVHDV\WRÀQG /HW fE1 ; : : : ; En g EHWKH EDVLV B IRU V ,I [ D r1 E1 C " " " C rn En WKHQ 2 3 r1 6 :: 7 Œ[#B D 4 : 5

rn

DQG

T .[/ D T .r1 E1 C " " " C rn En / D r1 T .E1 / C " " " C rn T .En /

EHFDXVH T LVOLQHDU 1RZ VLQFHWKHFRRUGLQDWHPDSSLQJIURP W WR Rm LVOLQHDU 7KHRUHP LQ6HFWLRQ HTXDWLRQ OHDGVWR

Œ T .[/ #C D r1 Œ T .E1 / #C C " " " C rn Œ T .En / #C

6LQFH C FRRUGLQDWHYHFWRUVDUHLQ Rm WKHYHFWRUHTXDWLRQ FDQEHZULWWHQDVDPDWUL[ HTXDWLRQ QDPHO\ x

T

Œ T .[/ #C D M Œ [ #B

T(x)

ZKHUH

[x]B '*(63&

Multiplication by M

[T(x)]C

# M D Œ T .E1 / #C

Œ T .E2 / #C

"""

Œ T .En / #C

$

7KHPDWUL[ M LVDPDWUL[UHSUHVHQWDWLRQRI T FDOOHGWKH PDWUL[IRU 7 UHODWLYHWRWKH EDVHV B DQG C 6HH)LJ (TXDWLRQ VD\VWKDW VRIDUDVFRRUGLQDWHYHFWRUVDUHFRQFHUQHG WKHDFWLRQRI T RQ [ PD\EHYLHZHGDVOHIWPXOWLSOLFDWLRQE\ M

&9".1-& 6XSSRVH B D fE1 ; E2 g LVDEDVLVIRU V DQG C D fF1 ; F2 ; F3 g LVDEDVLV IRU W /HW T W V ! W EHDOLQHDUWUDQVIRUPDWLRQZLWKWKHSURSHUW\WKDW T .E1 / D 3F1 ! 2F2 C 5F3

DQG

)LQGWKHPDWUL[ M IRU T UHODWLYHWR B DQG C

T .E2 / D 4F1 C 7F2 ! F3

$)"15&3


40-65*0/ 7KH C FRRUGLQDWHYHFWRUVRIWKH LPDJHV RI E1 DQG E2 DUH 2 2 3 3 3 4 Œ T .E1 / #C D 4 !2 5 DQG Œ T .E2 / #C D 4 7 5 5 !1 ! ! 3 3 4 75 M D 4 !2 5 !1

+HQFH

2

,I B DQG C DUHEDVHVIRUWKHVDPHVSDFH V DQGLI T LVWKHLGHQWLW\WUDQVIRUPDWLRQ T .[/ D [ IRU [ LQ V WKHQPDWUL[ M LQ LVMXVWDFKDQJHRIFRRUGLQDWHVPDWUL[VHH 6HFWLRQ

/LQHDU7UDQVIRUPDWLRQVIURP V LQWR V x

[x]B '*(63&

T

Multiplication by [T]B

T(x)

,QWKHFRPPRQFDVHZKHUH W LVWKHVDPHDV V DQGWKHEDVLV C LVWKHVDPHDV B WKH PDWUL[ M LQ LVFDOOHGWKH PDWUL[IRU 7 UHODWLYHWR B RUVLPSO\WKH BPDWUL[IRU 7 DQGLVGHQRWHGE\ Œ T #B 6HH)LJ 7KH B PDWUL[IRU T W V ! V VDWLVÀHV

Œ T .[/ #B D Œ T #B Œ [ #B ;

[T(x)]B

IRUDOO [ LQ V

&9".1-& 7KHPDSSLQJ T W P2 ! P2 GHÀQHGE\ T .a0 C a1 t C a2 t 2 / D a1 C 2a2 t LVDOLQHDUWUDQVIRUPDWLRQ &DOFXOXVVWXGHQWVZLOOUHFRJQL]H T DVWKHGLIIHUHQWLDWLRQ RSHUDWRU D )LQGWKH B PDWUL[IRU T ZKHQ B LVWKHEDVLV f1; t; t 2 g E 9HULI\WKDW Œ T .S/ #B D Œ T #B Œ S #B IRUHDFK S LQ P2

40-65*0/ D &RPSXWHWKHLPDJHVRIWKHEDVLVYHFWRUV

T .1/ D 0 T .t / D 1

7KH]HURSRO\QRPLDO 7KHSRO\QRPLDOZKRVHYDOXHLVDOZD\V

2

T .t / D 2t 7KHQZULWHWKH B FRRUGLQDWHYHFWRUVRI T .1/ T .t / DQG T .t 2 / ZKLFKDUHIRXQGE\ LQVSHFWLRQLQWKLVH[DPSOH DQGSODFHWKHPWRJHWKHUDVWKH B PDWUL[IRU T 2 3 2 3 2 3 0 1 0 Œ T .1/ #B D 4 0 5; Œ T .t / #B D 4 0 5; Œ T .t 2 / #B D 4 2 5 0 0 0 ! 0 Œ T #B D 4 0 0 2

! 1 0 0

! 3 0 25 0


E )RUDJHQHUDO S.t/ D a0 C a1 t C a2 t 2

3 a1 Œ T .S/ #B D Œ a1 C 2a2 t #B D 4 2a2 5 0 2 32 3 0 1 0 a0 D 4 0 0 2 54 a1 5 D Œ T #B Œ S #B a2 0 0 0

6HH)LJ

2

T a0 + a1t + a 2t 2

!2

a0 a1 a2 "3

WEB

a1 + 2a2 t

!2

Multiplication by [T ]B

a1 2a2 0

"3

'*(63& 0DWUL[UHSUHVHQWDWLRQRIDOLQHDU

WUDQVIRUPDWLRQ

/LQHDU7UDQVIRUPDWLRQVRQ Rn ,QDQDSSOLHGSUREOHPLQYROYLQJ Rn DOLQHDUWUDQVIRUPDWLRQ T XVXDOO\DSSHDUVÀUVWDV DPDWUL[WUDQVIRUPDWLRQ [ 7! A[ ,I A LVGLDJRQDOL]DEOH WKHQWKHUHLVDEDVLV B IRU Rn FRQVLVWLQJRIHLJHQYHFWRUVRI A 7KHRUHP EHORZVKRZVWKDW LQWKLVFDVH WKH B PDWUL[ IRU T LVGLDJRQDO 'LDJRQDOL]LQJ A DPRXQWVWRÀQGLQJDGLDJRQDOPDWUL[UHSUHVHQWDWLRQ RI [ 7! A[

5)&03&.

%JBHPOBM .BUSJY 3FQSFTFOUBUJPO 6XSSRVH A D 3'3!1 ZKHUH D LVDGLDJRQDO n # n PDWUL[ ,I B LVWKHEDVLVIRU Rn IRUPHGIURPWKHFROXPQVRI P WKHQ D LVWKH B PDWUL[IRUWKHWUDQVIRUPDWLRQ [ 7! A[

1300' 'HQRWHWKHFROXPQVRI P E\ E1 ; : : : ; En VRWKDW B D fE1 ; : : : ; En g DQG P D Œ E1 " " " En # ,QWKLVFDVH P LVWKHFKDQJHRIFRRUGLQDWHVPDWUL[ PB GLVFXVVHGLQ 6HFWLRQ ZKHUH P Œ [ #B D [ DQG Œ [ #B D P !1 [ ,I T .[/ D A[ IRU [ LQ Rn WKHQ # $ " " " Œ T .En / #B Œ T #B D Œ T .E1 / #B # $ """ Œ AEn #B D Œ AE1 #B

D ŒP DP DP !1

!1

!1 !1

AE1 " " " P

!1

AEn #

AŒ E1 " " " En #

6LQFH T .[/ D A[

&KDQJHRIFRRUGLQDWHV 0DWUL[PXOWLSOLFDWLRQ

$3

6LQFH A D 3'3 ZHKDYH Œ T #B D P

'HÀQLWLRQRI Œ T #B

!1

$3 D D

$)"15&3


" 7 2 )LQGD !4 1 EDVLV B IRU R2 ZLWKWKHSURSHUW\WKDWWKH B PDWUL[IRU T LVDGLDJRQDOPDWUL[

&9".1-& 'HÀQH T W R2 ! R2 E\ T .[/ D A[ ZKHUH A D

!

40-65*0/ )URP([DPSOHLQ6HFWLRQ ZHNQRZWKDW A D 3'3!1 ZKHUH ! " ! " 1 1 5 0 P D DQG D D !1 !2 0 3

7KHFROXPQVRI P FDOOWKHP E1 DQG E2 DUHHLJHQYHFWRUVRI A %\7KHRUHP D LVWKH B PDWUL[IRU T ZKHQ B D fE1 ; E2 g 7KHPDSSLQJV [ 7! A[ DQG X 7! D X GHVFULEHWKH VDPHOLQHDUWUDQVIRUPDWLRQ UHODWLYHWRGLIIHUHQWEDVHV

6LPLODULW\RI0DWUL[5HSUHVHQWDWLRQV 7KHSURRIRI7KHRUHP GLGQRWXVHWKHLQIRUPDWLRQWKDW D ZDVGLDJRQDO +HQFH LI A LVVLPLODUWRDPDWUL[ C ZLWK A D P CP !1 WKHQ C LVWKH B PDWUL[IRUWKH WUDQVIRUPDWLRQ [ 7! A[ ZKHQWKHEDVLV B LVIRUPHGIURPWKHFROXPQVRI P 7KH IDFWRUL]DWLRQ A D P CP !1 LVVKRZQLQ)LJ x

Multiplication by A

Ax Multiplication by P

Multiplication by P –1 [x]B

Multiplication by C

[Ax]B

'*(63& 6LPLODULW\RIWZRPDWUL[UHSUHVHQWDWLRQV

A D 3&3!1

&RQYHUVHO\ LI T W Rn ! Rn LVGHÀQHGE\ T .[/ D A[ DQGLI B LVDQ\EDVLVIRU R WKHQWKH B PDWUL[IRU T LVVLPLODUWR A ,QIDFW WKHFDOFXODWLRQVLQWKHSURRIRI 7KHRUHP VKRZWKDWLI P LVWKHPDWUL[ZKRVHFROXPQVFRPHIURPWKHYHFWRUVLQ B WKHQ ŒT #B D P !1 $3 7KXV WKHVHWRIDOOPDWULFHVVLPLODUWRDPDWUL[ A FRLQFLGHVZLWK WKHVHWRIDOOPDWUL[UHSUHVHQWDWLRQVRIWKHWUDQVIRUPDWLRQ [ 7! A[ n

!

" ! " ! " 4 !9 3 2 &9".1-& /HW A D E1 D DQG E2 D 7KHFKDUDFWHULVWLF 4 !8 2 1 SRO\QRPLDORI A LV .! C 2/2 EXWWKHHLJHQVSDFHIRUWKHHLJHQYDOXH !2 LVRQO\RQH GLPHQVLRQDO VR A LVQRWGLDJRQDOL]DEOH +RZHYHU WKHEDVLV B D fE1 ; E2 g KDVWKH SURSHUW\WKDWWKH B PDWUL[IRUWKHWUDQVIRUPDWLRQ [ 7! A[ LVDWULDQJXODUPDWUL[FDOOHG WKH -RUGDQIRUP RI Añ )LQGWKLV B PDWUL[

E2 # WKHQWKH B PDWUL[LV P !1 $3 &RPSXWH ! "! " ! " 4 !9 3 2 !6 !1 $3 D D 4 !8 2 1 !4 0 ! "! " ! " !1 2 !6 !1 !2 1 !1 P $3 D D 2 !3 !4 0 0 !2

40-65*0/ ,I P D Œ E1

1RWLFHWKDWWKHHLJHQYDOXHRI A LVRQWKHGLDJRQDO ñ (YHU\VTXDUHPDWUL[ A LVVLPLODUWRDPDWUL[LQ-RUGDQIRUP 7KHEDVLVXVHGWRSURGXFHD-RUGDQIRUP FRQVLVWVRIHLJHQYHFWRUVDQGVRFDOOHG´JHQHUDOL]HGHLJHQYHFWRUVµRI A 6HH&KDSWHURI $SSOLHG/LQHDU $OJHEUD UGHG (QJOHZRRG&OLIIV 1-3UHQWLFH+DOO E\%1REOHDQG-:'DQLHO


/6.&3*$"- /05& $QHIÀFLHQWZD\WRFRPSXWHD B PDWUL[ P !1 $3 LVWRFRPSXWH $3 DQGWKHQWRURZ UHGXFHWKHDXJPHQWHGPDWUL[ Œ P $3 # WR Œ I P !1 $3 # $ VHSDUDWHFRPSXWDWLRQ RI P !1 LVXQQHFHVVDU\ 6HH([HUFLVH LQ6HFWLRQ

13"$5*$& 130#-&.4 )LQG T .a0 C a1 t C a2 t 2 / LI T LVWKHOLQHDUWUDQVIRUPDWLRQIURP P2 WR P2 ZKRVH PDWUL[UHODWLYHWR B D f1; t; t 2 g LV 2 3 3 4 0 Œ T #B D 4 0 5 !1 5 1 !2 7

/HW A B DQG C EH n # n PDWULFHV 7KHWH[WKDVVKRZQWKDWLI A LVVLPLODUWR B WKHQ B LVVLPLODUWR A 7KLVSURSHUW\ WRJHWKHUZLWKWKHVWDWHPHQWVEHORZ VKRZVWKDW ´VLPLODUWRµLVDQ HTXLYDOHQFHUHODWLRQ 5RZHTXLYDOHQFHLVDQRWKHUH[DPSOHRIDQ HTXLYDOHQFHUHODWLRQ 9HULI\SDUWVD DQGE D A LVVLPLODUWR A E ,I A LVVLPLODUWR B DQG B LVVLPLODUWR C WKHQ A LVVLPLODUWR C

&9&3$*4&4 /HW B D fE1 ; E2 ; E3 g DQG D D fG1 ; G2 g EHEDVHVIRUYHFWRU VSDFHV V DQG W UHVSHFWLYHO\ /HW T W V ! W EHDOLQHDU WUDQVIRUPDWLRQZLWKWKHSURSHUW\WKDW

T .E1 / D 3G1 ! 5G2 ; T .E2 / D !G1 C 6G2 ; T .E3 / D 4G2 )LQGWKHPDWUL[IRU T UHODWLYHWR B DQG D

/HW D D fG1 ; G2 g DQG B D fE1 ; E2 g EHEDVHVIRUYHFWRUVSDFHV V DQG W UHVSHFWLYHO\ /HW T W V ! W EHDOLQHDUWUDQVIRU PDWLRQZLWKWKHSURSHUW\WKDW

T .G1 / D 3E1 ! 3E2 ; T .G2 / D !2E1 C 5E2 )LQGWKHPDWUL[IRU T UHODWLYHWR D DQG B

/HW E D fH1 ; H2 ; H3 g EH WKH VWDQGDUG EDVLV IRU R3 OHW B D fE1 ; E2 ; E3 g EHDEDVLVIRUDYHFWRUVSDFH V DQGOHW T W R3 ! V EHDOLQHDUWUDQVIRUPDWLRQZLWKWKHSURSHUW\WKDW

T .x1 ; x2 ; x3 / D .2x3 ! x2 /E1 ! .2x2 /E2 C .x1 C 3x3 /E3 D &RPSXWH T .H1 / T .H2 / DQG T .H3 / E &RPSXWH ŒT .H1 /#B ŒT .H2 /#B DQG ŒT .H3 /#B

F )LQGWKHPDWUL[IRU T UHODWLYHWR E DQG B

/HW B D fE1 ; E2 ; E3 g EHDEDVLVIRUDYHFWRUVSDFH V DQGOHW T W V ! R2 EHDOLQHDUWUDQVIRUPDWLRQZLWKWKHSURSHUW\WKDW ! " 2x1 ! 3x2 C x3 T .x1 E1 C x2 E2 C x3 E3 / D !2x1 C 5x3 )LQGWKHPDWUL[IRU T UHODWLYHWR B DQGWKHVWDQGDUGEDVLVIRU R2

/HW T W P2 ! P3 EHWKHWUDQVIRUPDWLRQWKDWPDSVDSRO\QR PLDO S.t/ LQWRWKHSRO\QRPLDO .t C 3/S.t/ D )LQGWKHLPDJHRI S.t/ D 3 ! 2t C t 2 E 6KRZWKDW T LVDOLQHDUWUDQVIRUPDWLRQ

F )LQGWKHPDWUL[IRU T UHODWLYHWRWKHEDVHV f1; t; t 2 g DQG f1; t; t 2 ; t 3 g /HW T W P2 ! P4 EHWKHWUDQVIRUPDWLRQWKDWPDSVDSRO\QR PLDO S.t/ LQWRWKHSRO\QRPLDO S.t/ C 2t 2 S.t/ D )LQGWKHLPDJHRI S.t/ D 3 ! 2t C t 2 E 6KRZWKDW T LVDOLQHDUWUDQVIRUPDWLRQ

F )LQGWKHPDWUL[IRU T UHODWLYHWRWKHEDVHV f1; t; t 2 g DQG f1; t; t 2 ; t 3 ; t 4 g $VVXPHWKHPDSSLQJ T W P2 ! P2 GHÀQHGE\

T .a0 C a1 t C a2 t 2 / D 3a0 C .5a0 ! 2a1 /t C .4a1 C a2 /t 2 LVOLQHDU )LQGWKHPDWUL[UHSUHVHQWDWLRQRI T UHODWLYHWRWKH EDVLV B D f1; t; t 2 g /HW B D fE1 ; E2 ; E3 g EHDEDVLVIRUDYHFWRUVSDFH V )LQG T .4E1 ! 3E2 / ZKHQ T LVDOLQHDUWUDQVIRUPDWLRQIURP V WR V ZKRVHPDWUL[UHODWLYHWR B LV 2

0 Œ T #B D 4 2 1

3 0 1 1 !2 5 3 1

$)"15&3

(LJHQYDOXHVDQG(LJHQYHFWRUV 2

3 S.!1/ 'HÀQH T W P2 ! R3 E\ T .S/ D 4 S.0/ 5 S.1/ D )LQGWKHLPDJHXQGHU T RI S.t/ D 5 C 3t

E 6KRZWKDW T LVDOLQHDUWUDQVIRUPDWLRQ

F )LQGWKHPDWUL[IRU T UHODWLYHWRWKHEDVLV f1; t; t 2 g IRU P2 DQGWKHVWDQGDUGEDVLVIRU R3 2 3 S.!2/ 6 S.3/ 7 7 'HÀQH T W P3 ! R4 E\ T .S/ D 6 4 S.1/ 5 S.0/ D 6KRZWKDW T LVDOLQHDUWUDQVIRUPDWLRQ E )LQGWKHPDWUL[IRU T UHODWLYHWRWKHEDVLV f1; t; t 2 ; t 3 g IRU P3 DQGWKHVWDQGDUGEDVLVIRU R4 ,Q([HUFLVHVDQG ÀQGWKH BPDWUL[IRUWKHWUDQVIRUPDWLRQ [ 7! A[ ZKHUH B D fE1 ; E2 g ! " ! " ! " !4 !1 !1 !1 E1 D E2 D A D 6 1 2 1 ! " ! " ! " !6 !2 0 !1 A D E1 D E2 D 4 0 1 2 ,Q([HUFLVHV² GHÀQH T W R2 ! R2 E\ T .[/ D A[ )LQGD EDVLV B IRU R2 ZLWKWKHSURSHUW\WKDW ŒT #B LVGLDJRQDO ! " ! " 0 1 2 3 A D A D !3 4 3 2 ! " ! " 1 2 4 !2 A D A D 3 !4 !1 5 ! " ! " 4 1 1 /HW A D DQG B D fE1 ; E2 g IRU E1 D !1 2 !1 ! " !1 'HÀQH T W R2 ! R2 E\ T .[/ D A[ E2 D 2 D 9HULI\WKDW E1 LVDQHLJHQYHFWRURI A EXWWKDW A LVQRW GLDJRQDOL]DEOH E )LQGWKH BPDWUL[IRU T

'HÀQH T W R3 ! R3 E\ T .[/ D A[ ZKHUH A LV D 3 # 3 PDWUL[ZLWKHLJHQYDOXHV DQG !2 'RHVWKHUHH[LVWDEDVLV B IRU R3 VXFKWKDWWKH BPDWUL[IRU T LVDGLDJRQDOPDWUL[" 'LVFXVV 9HULI\WKHVWDWHPHQWVLQ([HUFLVHV² 7KHPDWULFHVDUHVTXDUH ,I A LV LQYHUWLEOH DQG VLPLODU WR B WKHQ B LV LQYHUWLEOH DQG A!1 LVVLPLODUWR B !1 >+LQW P !1 $3 D B IRUVRPH LQYHUWLEOH P ([SODLQZK\ B LVLQYHUWLEOH 7KHQÀQGDQ LQYHUWLEOH Q VXFKWKDW Q!1 A!1 Q D B !1 @

,I A LVVLPLODUWR B WKHQ A2 LVVLPLODUWR B 2

,I B LVVLPLODUWR A DQG C LVVLPLODUWR A WKHQ B LVVLPLODU WR C

,I A LVGLDJRQDOL]DEOHDQG B LVVLPLODUWR A WKHQ B LVDOVR GLDJRQDOL]DEOH ,I B D P !1 $3 DQG [ LVDQHLJHQYHFWRURI A FRUUHVSRQGLQJ WRDQHLJHQYDOXH ! WKHQ P !1 [ LVDQHLJHQYHFWRURI B FRUUH VSRQGLQJDOVRWR ! ,I A DQG B DUHVLPLODU WKHQWKH\KDYHWKHVDPHUDQN >+LQW 5HIHUWR6XSSOHPHQWDU\([HUFLVHV DQGLQ&KDSWHU @ 7KH WUDFH RIDVTXDUHPDWUL[ A LVWKHVXPRIWKHGLDJRQDO HQWULHVLQ A DQGLVGHQRWHGE\ WU A ,WFDQEHYHULÀHGWKDW WU .F G/ D WU .GF / IRU DQ\ WZR n # n PDWULFHV F DQG G 6KRZWKDWLI A DQG B DUHVLPLODU WKHQ WU A D WU B ,WFDQEHVKRZQWKDWWKHWUDFHRIDPDWUL[ A HTXDOVWKHVXPRI WKHHLJHQYDOXHVRI A 9HULI\WKLVVWDWHPHQWIRUWKHFDVHZKHQ A LVGLDJRQDOL]DEOH /HW V EH Rn ZLWKDEDVLV B D fE1 ; : : : ; En g OHW W EH Rn ZLWKWKHVWDQGDUGEDVLV GHQRWHGKHUHE\ E DQGFRQVLGHUWKH LGHQWLW\WUDQVIRUPDWLRQ I W Rn ! Rn ZKHUH I.[/ D [ )LQG WKHPDWUL[IRU I UHODWLYHWR B DQG E :KDWZDVWKLVPDWUL[ FDOOHGLQ6HFWLRQ " /HW V EHDYHFWRUVSDFHZLWKDEDVLV B D fE1 ; : : : ; En g OHW W EHWKHVDPHVSDFH V ZLWKDEDVLV C D fF1 ; : : : ; Fn g DQGOHW I EHWKHLGHQWLW\WUDQVIRUPDWLRQ I W V ! W )LQGWKHPDWUL[ IRU I UHODWLYHWR B DQG C :KDWZDVWKLVPDWUL[FDOOHGLQ 6HFWLRQ " /HW V EHDYHFWRUVSDFHZLWKDEDVLV B D fE1 ; : : : ; En g )LQG WKH BPDWUL[IRUWKHLGHQWLW\WUDQVIRUPDWLRQ I W V ! V >0@ ,Q([HUFLVHVDQG ÀQGWKH BPDWUL[IRUWKHWUDQVIRUPD WLRQ [ 7! A[ ZKHUH B D fE1 ; E2 ; E3 g 2 3 6 !2 !2 1 !2 5 A D 4 3 2 !2 2 2 3 2 3 2 3 1 2 !1 E1 D 4 1 5 E2 D 4 1 5 E3 D 4 !1 5 1 3 0 2 3 !7 !48 !16 14 6 5 A D 4 1 !3 !45 !19 2 3 2 3 2 3 !3 !2 3 E1 D 4 1 5 E2 D 4 1 5 E3 D 4 !1 5 !3 !3 0

>0@ /HW T EHWKHWUDQVIRUPDWLRQZKRVHVWDQGDUGPDWUL[LV JLYHQEHORZ )LQGDEDVLVIRU R4 ZLWKWKHSURSHUW\WKDW Œ T #B LVGLDJRQDO 2 3 !6 4 0 9 6 !3 0 1 67 7 AD6 4 !1 !2 1 05 !4 4 0 7

&RPSOH[(LJHQYDOXHV

40-65*0/4 50 13"$5*$& 130#-&.4 /HW S.t / D a0 C a1 t C a2 t 2 DQGFRPSXWH 2 32 3 2 3 3 4 0 a0 3a0 C 4a1 5 5a1 ! a2 Œ T .S/ #B D Œ T #B Œ S #B D 4 0 5 !1 54 a1 5 D 4 a2 a0 ! 2a1 C 7a2 1 !2 7 6R T .S/ D .3a0 C 4a1 / C .5a1 ! a2 /t C .a0 ! 2a1 C 7a2 /t 2

D A D .I /!1 AI VR A LVVLPLODUWR A E %\K\SRWKHVLV WKHUHH[LVWLQYHUWLEOHPDWULFHV P DQG Q ZLWKWKHSURSHUW\WKDW B D P !1 $3 DQG C D Q!1 BQ 6XEVWLWXWHWKHIRUPXODIRU B LQWRWKHIRUPXOD IRU C DQGXVHDIDFWDERXWWKHLQYHUVHRIDSURGXFW

C D Q!1 BQ D Q!1 .P !1 $3/Q D .34/!1 A.34/ 7KLVHTXDWLRQKDVWKHSURSHUIRUPWRVKRZWKDW A LVVLPLODUWR C

$0.1-&9 &*(&/7"-6&4 6LQFHWKHFKDUDFWHULVWLFHTXDWLRQRIDQ n # n PDWUL[LQYROYHVDSRO\QRPLDORIGHJUHH n WKHHTXDWLRQDOZD\VKDVH[DFWO\ n URRWV FRXQWLQJPXOWLSOLFLWLHV SURYLGHGWKDWSRVVLEO\ FRPSOH[URRWVDUHLQFOXGHG 7KLVVHFWLRQVKRZVWKDWLIWKHFKDUDFWHULVWLFHTXDWLRQRI DUHDOPDWUL[ A KDVVRPHFRPSOH[URRWV WKHQWKHVHURRWVSURYLGHFULWLFDOLQIRUPDWLRQ DERXW A 7KHNH\LVWROHW A DFWRQWKHVSDFH C n RI nWXSOHVRIFRPSOH[QXPEHUVñ 2XULQWHUHVWLQ C n GRHVQRWDULVHIURPDGHVLUHWR´JHQHUDOL]HµWKHUHVXOWVRIWKH HDUOLHUFKDSWHUV DOWKRXJKWKDWZRXOGLQIDFWRSHQXSVLJQLÀFDQWQHZDSSOLFDWLRQVRI OLQHDUDOJHEUDò 5DWKHU WKLVVWXG\RIFRPSOH[HLJHQYDOXHVLVHVVHQWLDOLQRUGHUWRXQFRYHU ´KLGGHQµLQIRUPDWLRQDERXWFHUWDLQPDWULFHVZLWKUHDOHQWULHVWKDWDULVHLQDYDULHW\RI UHDOOLIHSUREOHPV 6XFKSUREOHPVLQFOXGHPDQ\UHDOG\QDPLFDOV\VWHPVWKDWLQYROYH SHULRGLFPRWLRQ YLEUDWLRQ RUVRPHW\SHRIURWDWLRQLQVSDFH 7KH PDWUL[ HLJHQYDOXH²HLJHQYHFWRU WKHRU\ DOUHDG\ GHYHORSHG IRU Rn DSSOLHV HTXDOO\ZHOOWR C n 6RDFRPSOH[VFDODU ! VDWLVÀHV GHW.A ! !I / D 0 LIDQGRQO\LI WKHUHLVDQRQ]HURYHFWRU [ LQ C n VXFKWKDW A[ D ![ :HFDOO ! DFRPSOH[ HLJHQYDOXH DQG [ DFRPSOH[ HLJHQYHFWRU FRUUHVSRQGLQJWR ! " 0 !1 WKHQ WKH OLQHDU WUDQVIRUPDWLRQ [ 7! A[ RQ R2 1 0 URWDWHVWKHSODQHFRXQWHUFORFNZLVHWKURXJKDTXDUWHUWXUQ 7KHDFWLRQRI A LVSHULRGLF VLQFHDIWHUIRXUTXDUWHUWXUQV DYHFWRULVEDFNZKHUHLWVWDUWHG 2EYLRXVO\ QRQRQ]HUR YHFWRULVPDSSHGLQWRDPXOWLSOHRILWVHOI VR A KDVQRHLJHQYHFWRUVLQ R2 DQGKHQFHQR UHDOHLJHQYDOXHV ,QIDFW WKHFKDUDFWHULVWLFHTXDWLRQRI A LV

&9".1-& ,I A D

!

!2 C 1 D 0 ñ 5HIHUWR$SSHQGL[% IRUDEULHIGLVFXVVLRQRIFRPSOH[QXPEHUV 0DWUL[DOJHEUDDQGFRQFHSWVDERXW UHDOYHFWRUVSDFHVFDUU\RYHUWRWKHFDVHZLWKFRPSOH[HQWULHVDQGVFDODUV ,QSDUWLFXODU A.c [ C d \/ D cA[ C dA\ IRU A DQ m " n PDWUL[ZLWKFRPSOH[HQWULHV [ \ LQ C n DQG c d LQ C ò $ VHFRQGFRXUVHLQOLQHDUDOJHEUDRIWHQGLVFXVVHVVXFKWRSLFV 7KH\DUHRISDUWLFXODULPSRUWDQFHLQ HOHFWULFDOHQJLQHHULQJ

$)"15&3


7KHRQO\URRWVDUHFRPSOH[ ! D i DQG ! D !i +RZHYHU LIZHSHUPLW A WRDFWRQ C 2 WKHQ ! "! " ! " ! " 0 !1 1 i 1 D Di 1 0 !i 1 !i ! "! " ! " ! " 0 !1 1 !i 1 D D !i 1 0 i 1 i ! " ! " 1 1 7KXV i DQG !i DUHHLJHQYDOXHV ZLWK DQG DVFRUUHVSRQGLQJHLJHQYHFWRUV $ !i i PHWKRGIRU ÀQGLQJ FRPSOH[HLJHQYHFWRUVLVGLVFXVVHGLQ([DPSOH 7KHPDLQIRFXVRIWKLVVHFWLRQZLOOEHRQWKHPDWUL[LQWKHQH[WH[DPSOH

&9".1-& /HW A D IRUHDFKHLJHQVSDFH

!

:5 :75

" !:6 )LQGWKHHLJHQYDOXHVRI A DQGÀQGDEDVLV 1:1

40-65*0/ 7KHFKDUDFWHULVWLFHTXDWLRQRI A LV ! " :5 ! ! !:6 0 D GHW D .:5 ! !/.1:1 ! !/ ! .!:6/.:75/ :75 1:1 ! !

D !2 ! 1:6! C 1 p )URPWKHTXDGUDWLFIRUPXOD ! D 12 Œ1:6 ˙ .!1:6/2 ! 4# D :8 ˙ :6i )RUWKHHLJHQ YDOXH ! D :8 ! :6i FRQVWUXFW ! " ! " :5 !:6 :8 ! :6i 0 A ! .:8 ! :6i /I D ! :75 1:1 0 :8 ! :6i ! " !:3 C :6i !:6 D :75 :3 C :6i 5RZUHGXFWLRQRIWKHXVXDODXJPHQWHGPDWUL[LVTXLWHXQSOHDVDQWE\KDQGEHFDXVHRIWKH FRPSOH[DULWKPHWLF +RZHYHU KHUHLVDQLFHREVHUYDWLRQWKDWUHDOO\VLPSOLÀHVPDWWHUV 6LQFH :8 ! :6i LVDQHLJHQYDOXH WKHV\VWHP

.!:3 C :6i /x1 ! :6x2 D 0 :75x1 C .:3 C :6i /x2 D 0

KDVDQRQWULYLDOVROXWLRQZLWK x1 DQG x2 SRVVLEO\FRPSOH[QXPEHUV 7KHUHIRUH ERWK HTXDWLRQVLQ GHWHUPLQHWKHVDPHUHODWLRQVKLSEHWZHHQ x1 DQG x2 DQGHLWKHUHTXDWLRQ FDQEHXVHGWRH[SUHVVRQHYDULDEOHLQWHUPVRIWKHRWKHUó 7KHVHFRQGHTXDWLRQLQ OHDGVWR

:75x1 D .!:3 ! :6i /x2 x1 D .!:4 ! :8i /x2 &KRRVH x2 D 5 WRHOLPLQDWHWKHGHFLPDOV DQGREWDLQ x1 D !2 ! 4i $ EDVLVIRUWKH HLJHQVSDFHFRUUHVSRQGLQJWR ! D :8 ! :6i LV ! " !2 ! 4i Y1 D 5 ó $QRWKHUZD\WRVHHWKLVLVWRUHDOL]HWKDWWKHPDWUL[LQHTXDWLRQ LVQRWLQYHUWLEOH VRLWVURZVDUHOLQHDUO\ GHSHQGHQWDVYHFWRUVLQ C 2 / DQGKHQFHRQHURZLVDFRPSOH[ PXOWLSOHRIWKHRWKHU

&RPSOH[(LJHQYDOXHV

$QDORJRXVFDOFXODWLRQVIRU ! D :8 C :6i SURGXFHWKHHLJHQYHFWRU ! " !2 C 4i Y2 D 5 $VDFKHFNRQWKHZRUN FRPSXWH ! "! " ! " :5 !:6 !2 C 4i !4 C 2i AY2 D D D .:8 C :6i /Y2 :75 1:1 5 4 C 3i 6XUSULVLQJO\ WKHPDWUL[ A LQ([DPSOH GHWHUPLQHVDWUDQVIRUPDWLRQ [ 7! A[ WKDW LVHVVHQWLDOO\DURWDWLRQ 7KLVIDFWEHFRPHVHYLGHQWZKHQDSSURSULDWHSRLQWVDUHSORWWHG

&9".1-& 2QHZD\WRVHHKRZPXOWLSOLFDWLRQE\WKHPDWUL[ A LQ([DPSOH DIIHFWVSRLQWVLVWRSORWDQDUELWUDU\LQLWLDOSRLQW³VD\ [0 D .2; 0/³DQGWKHQWRSORW VXFFHVVLYHLPDJHVRIWKLVSRLQWXQGHUUHSHDWHGPXOWLSOLFDWLRQVE\ A 7KDWLV SORW ! "! " ! " :5 !:6 2 1:0 [ 1 D A[ 0 D D :75 1:1 0 1:5 ! "! " ! " :5 !:6 1:0 !:4 [ 2 D A[ 1 D D :75 1:1 1:5 2:4 [ 3 D A[ 2 ; : : :

)LJXUHVKRZV [0 ; : : : ; [8 DVODUJHUGRWV 7KHVPDOOHUGRWVDUHWKHORFDWLRQVRI [9 ; : : : ; [100 7KHVHTXHQFHOLHVDORQJDQHOOLSWLFDORUELW x2 x2

x3

x1

x4

x0 x5

x1

x6 x7

x8

'*(63& ,WHUDWHVRIDSRLQW [0

XQGHUWKHDFWLRQRIDPDWUL[ZLWKD FRPSOH[HLJHQYDOXH

2IFRXUVH )LJ GRHVQRWH[SODLQ ZK\ WKHURWDWLRQRFFXUV 7KHVHFUHWWRWKHURWDWLRQ LVKLGGHQLQWKHUHDODQGLPDJLQDU\SDUWVRIDFRPSOH[HLJHQYHFWRU

5HDODQG,PDJLQDU\3DUWVRI9HFWRUV 7KHFRPSOH[FRQMXJDWHRIDFRPSOH[YHFWRU [ LQ C n LVWKHYHFWRU [ LQ C n ZKRVHHQWULHV DUHWKHFRPSOH[FRQMXJDWHVRIWKHHQWULHVLQ [ 7KH UHDO DQG LPDJLQDU\SDUWV RID FRPSOH[YHFWRU [ DUHWKHYHFWRUV 5H [ DQG ,P [ LQ Rn IRUPHGIURPWKHUHDODQGLPDJLQDU\ SDUWVRIWKHHQWULHVRI [

$)"15&3


2

3 2 3 2 3 3!i 3 !1 &9".1-& ,I [ D 4 i 5 D 4 0 5 C i 4 1 5 WKHQ 2 C 5i 2 5 2 3 2 3 2 3 2 3 2 3 3 !1 3 !1 3Ci 5H [ D 4 0 5; ,P [ D 4 1 5; DQG [ D 4 0 5 ! i 4 1 5 D 4 !i 5 2 5 2 5 2 ! 5i

,I B LVDQ m # n PDWUL[ZLWKSRVVLEO\FRPSOH[HQWULHV WKHQ B GHQRWHVWKHPDWUL[ ZKRVHHQWULHVDUHWKHFRPSOH[FRQMXJDWHVRIWKHHQWULHVLQ B 3URSHUWLHVRIFRQMXJDWHV IRUFRPSOH[QXPEHUVFDUU\RYHUWRFRPSOH[PDWUL[DOJHEUD

r [ D r [;

B [ D B [;

BC D B C ;

DQG

rB D r B

(LJHQYDOXHVDQG(LJHQYHFWRUVRID5HDO0DWUL[ 7KDW$FWVRQ C n /HW A EHDQ n # n PDWUL[ZKRVHHQWULHVDUHUHDO 7KHQ A[ D A[ D A[ ,I ! LVDQ HLJHQYDOXHRI A DQG [ LVDFRUUHVSRQGLQJHLJHQYHFWRULQ C n WKHQ

A[ D A[ D ![ D ![ +HQFH ! LVDOVRDQHLJHQYDOXHRI A ZLWK [ DFRUUHVSRQGLQJHLJHQYHFWRU 7KLVVKRZVWKDW ZKHQ A LVUHDO LWVFRPSOH[HLJHQYDOXHVRFFXULQFRQMXJDWHSDLUV +HUHDQGHOVHZKHUH ZHXVHWKHWHUP FRPSOH[HLJHQYDOXH WRUHIHUWRDQHLJHQYDOXH ! D a C bi ZLWK b ¤ 0

&9".1-& 7KHHLJHQYDOXHVRIWKHUHDOPDWUL[LQ([DPSOH DUHFRPSOH[FRQ MXJDWHV QDPHO\ :8 ! :6i DQG :8 C :6i 7KH FRUUHVSRQGLQJ HLJHQYHFWRUV IRXQG LQ ([DPSOH DUHDOVRFRQMXJDWHV ! " ! " !2 ! 4i !2 C 4i Y1 D DQG Y2 D D Y1 5 5 7KHQH[WH[DPSOHSURYLGHVWKHEDVLFÉXLOGLQJEORFNµIRUDOOUHDO 2 # 2 PDWULFHV ZLWKFRPSOH[HLJHQYDOXHV " a !b ZKHUH a DQG b DUHUHDODQGQRWERWK]HUR WKHQWKH b a HLJHQYDOXHVRI C DUH p! D a ˙ bi 6HHWKH3UDFWLFH3UREOHPDWWKHHQGRIWKLVVHFWLRQ $OVR LI r D j!j D a2 C b 2 WKHQ ! " ! "! " a=r !b=r r 0 FRV ' ! VLQ ' C Dr D VLQ ' FRV ' b=r a=r 0 r

&9".1-& ,I C D

Im z (a, b) r

b ϕ a

'*(63&

Re z

!

ZKHUH ' LVWKHDQJOHEHWZHHQWKHSRVLWLYH x D[LVDQGWKHUD\IURP .0; 0/ WKURXJK .a; b/ 6HH)LJ DQG$SSHQGL[ %7KHDQJOH ' LVFDOOHGWKH DUJXPHQW RI ! D a C bi 7KXV WKHWUDQVIRUPDWLRQ [ 7! C [ PD\EHYLHZHGDVWKHFRPSRVLWLRQRIDURWDWLRQWKURXJKWKH DQJOH ' DQGDVFDOLQJE\ j!j VHH)LJ )LQDOO\ ZHDUHUHDG\WRXQFRYHUWKHURWDWLRQWKDWLVKLGGHQZLWKLQDUHDOPDWUL[ KDYLQJDFRPSOH[HLJHQYDOXH

&RPSOH[(LJHQYDOXHV

Scaling Ax

x2

x Rotation ϕ

x1 '*(63& $ URWDWLRQIROORZHGE\D

VFDOLQJ

" ! " :5 !:6 !2 ! 4i ! D :8 ! :6i DQG Y1 D DVLQ :75 1:1 5 ([DPSOH $OVR OHW P EHWKH 2 # 2 UHDOPDWUL[ ! " # $ !2 !4 ,P Y1 D P D 5H Y1 5 0 DQGOHW ! "! "! " ! " 1 0 4 :5 !:6 !2 !4 :8 !:6 !1 C D P $3 D D 5 0 :6 8 20 !5 !2 :75 1:1

&9".1-& /HW A D

!

%\ ([DPSOH C LV D SXUH URWDWLRQ EHFDXVH j!j2 D .:8/2 C .:6/2 D 1 )URP C D P !1 $3 ZHREWDLQ ! " :8 !:6 A D P CP !1 D P P !1 :6 8 +HUHLVWKHURWDWLRQ´LQVLGHµ A 7KHPDWUL[ P SURYLGHVDFKDQJHRIYDULDEOH VD\ [ D P X 7KHDFWLRQRI A DPRXQWVWRDFKDQJHRIYDULDEOHIURP [ WR X IROORZHGE\ DURWDWLRQ DQGWKHQDUHWXUQWRWKHRULJLQDOYDULDEOH 6HH)LJ 7KHURWDWLRQSURGXFHV DQHOOLSVH DVLQ)LJ LQVWHDGRIDFLUFOH EHFDXVHWKHFRRUGLQDWHV\VWHPGHWHUPLQHG E\WKHFROXPQVRI P LVQRWUHFWDQJXODUDQGGRHVQRWKDYHHTXDOXQLWOHQJWKVRQWKHWZR D[HV x

A

Change of P –1 variable u

Ax P

C Rotation

Change of variable

Cu

'*(63& 5RWDWLRQGXHWRDFRPSOH[HLJHQYDOXH

7KHQH[WWKHRUHPVKRZVWKDWWKHFDOFXODWLRQVLQ([DPSOH FDQEHFDUULHGRXWIRU DQ\ 2 # 2 UHDOPDWUL[ A KDYLQJDFRPSOH[HLJHQYDOXH ! 7KHSURRIXVHVWKHIDFWWKDW LIWKHHQWULHVLQ A DUHUHDO WKHQ A.5H [/ D 5H A[ DQG A.,P [/ D ,P A[ DQGLI [ LVDQ HLJHQYHFWRUIRUDFRPSOH[HLJHQYDOXH WKHQ 5H [ DQG ,P [ DUHOLQHDUO\LQGHSHQGHQWLQ R2 6HH([HUFLVHVDQG 7KHGHWDLOVDUHRPLWWHG

5)&03&.

/HW A EHDUHDO 2 # 2 PDWUL[ZLWKDFRPSOH[HLJHQYDOXH ! D a ! bi b ¤ 0 DQG DQDVVRFLDWHGHLJHQYHFWRU Y LQ C 2 7KHQ ! " a !b !1 A D 3&P ; ZKHUH P D Œ 5H Y ,P Y # DQG C D b a

$)"15&3


7KH SKHQRPHQRQ GLVSOD\HG LQ ([DPSOH SHUVLVWV LQ KLJKHU GLPHQVLRQV )RU LQVWDQFH LI A LVD 3 # 3 PDWUL[ZLWKDFRPSOH[HLJHQYDOXH WKHQWKHUHLVDSODQHLQ R3 RQZKLFK A DFWVDVDURWDWLRQSRVVLEO\FRPELQHGZLWKVFDOLQJ (YHU\YHFWRULQWKDW SODQHLVURWDWHGLQWRDQRWKHUSRLQWRQWKHVDPHSODQH :HVD\WKDWWKHSODQHLV LQYDULDQW XQGHU A

x3

x3 x2

x0 x1

w0 x1

w7

w1 w8

w2 x2

'*(63&

,WHUDWHVRIWZRSRLQWVXQGHUWKH DFWLRQRID 3 # 3 PDWUL[ZLWKD FRPSOH[HLJHQYDOXH

2

3 :8 !:6 0 0 5 KDV HLJHQYDOXHV :8 ˙ :6i DQG &9".1-& 7KH PDWUL[ A D 4 :6 :8 0 0 1:07 $Q\YHFWRU Z0 LQWKH x1 x2 SODQHZLWKWKLUGFRRUGLQDWH LVURWDWHGE\ A LQWR DQRWKHUSRLQWLQWKHSODQH $Q\YHFWRU [0 QRWLQWKHSODQHKDVLWV x3 FRRUGLQDWHPXOWLSOLHG E\ 7KHLWHUDWHVRIWKHSRLQWV Z0 D .2; 0; 0/ DQG [0 D .2; 0; 1/ XQGHUPXOWLSOLFDWLRQ E\ A DUHVKRZQLQ)LJ

13"$5*$& 130#-&. 6KRZWKDWLI a DQG b DUHUHDO WKHQWKHHLJHQYDOXHVRI A D ! " ! " 1 1 FRUUHVSRQGLQJHLJHQYHFWRUV DQG !i i

&9&3$*4&4 /HWHDFKPDWUL[LQ([HUFLVHV²DFWRQ C 2 )LQGWKHHLJHQYDOXHV DQGDEDVLVIRUHDFKHLJHQVSDFHLQ C 2 ! " ! " 1 !2 3 !3 1 3 3 3 ! " ! " 5 1 1 !2 !8 1 1 3 ! " ! " 3 1 7 !5 !2 5 1 3 ,Q([HUFLVHV² XVH([DPSOH WROLVWWKHHLJHQYDOXHVRI A ,QHDFKFDVH WKHWUDQVIRUPDWLRQ [ 7! A[ LVWKHFRPSRVLWLRQRID URWDWLRQDQGDVFDOLQJ *LYHWKHDQJOH ' RIWKHURWDWLRQ ZKHUH !$ < ' & $ DQGJLYHWKHVFDOHIDFWRU r p " !p " ! 3 !1 3 3 p p3 3 3 1 !3 3 ! " ! " 0 2 0 :5 !2 0 !:5 0 p " ! p " ! ! 3 3 ! 3 p1 p 3 3 !1 ! 3 ,Q([HUFLVHV² ÀQGDQLQYHUWLEOHPDWUL[ P DQGDPDWUL[ C ! " a !b RIWKHIRUP VXFKWKDWWKHJLYHQPDWUL[KDVWKHIRUP b a !1 A D P CP ! " ! " 1 !2 3 !3 1 3 1 1

! ! !

!

"

!11 !4 20 5

0 !2

5 2

1:52 !:7 :56 :4

"

"

a !b b a

! ! !

"

4 !2 1 6 3 !5 2 5

DUH a ˙ bi ZLWK

" "

!3 !8 4 5

"

,Q([DPSOH VROYHWKHÀUVWHTXDWLRQLQ IRU x!2 LQWHUPVRI " 2 x1 DQGIURPWKDWSURGXFHWKHHLJHQYHFWRU \ D !1 C 2i IRUWKHPDWUL[ A 6KRZWKDWWKLV \ LVDFRPSOH[ PXOWLSOHRI WKHYHFWRU Y1 XVHGLQ([DPSOH /HW A EHDFRPSOH[RUUHDO n # n PDWUL[ DQGOHW [ LQ C n EH DQHLJHQYHFWRUFRUUHVSRQGLQJWRDQHLJHQYDOXH ! LQ C 6KRZ WKDWIRUHDFKQRQ]HURFRPSOH[VFDODU " WKHYHFWRU "[ LVDQ HLJHQYHFWRURI A &KDSWHUZLOOIRFXVRQPDWULFHV A ZLWKWKHSURSHUW\WKDW AT D A ([HUFLVHV DQGVKRZWKDWHYHU\HLJHQYDOXHRIVXFKDPDWUL[ LVQHFHVVDULO\UHDO /HW A EHDQ n # n UHDOPDWUL[ZLWKWKHSURSHUW\WKDW AT D A OHW [ EHDQ\YHFWRULQ C n DQGOHW q D [TA[ 7KHHTXDOLWLHV EHORZVKRZWKDW q LVDUHDOQXPEHUE\YHULI\LQJWKDW q D q *LYHDUHDVRQIRUHDFKVWHS

q D [T A[ D [T A[ D [TA[ D .[T A[/T D [T AT [ D q D E F G H

/HW A EHDQ n # n UHDOPDWUL[ZLWKWKHSURSHUW\WKDW AT D A 6KRZWKDWLI A[ D ![ IRUVRPHQRQ]HURYHFWRU [ LQ C n WKHQ LQIDFW ! LVUHDODQGWKHUHDOSDUWRI [ LVDQHLJHQYHFWRURI A >+LQW &RPSXWH [TA[ DQGXVH([HUFLVH $OVR H[DPLQH WKHUHDODQGLPDJLQDU\SDUWVRI A[@ /HW A EHDUHDO n # n PDWUL[ DQGOHW [ EHDYHFWRULQ C n 6KRZWKDW 5H.A[/ D A.5H [/ DQG ,P.A[/ D A.,P [/

/HW A EHDUHDO 2 # 2 PDWUL[ZLWKDFRPSOH[HLJHQYDOXH ! D a ! bi b ¤ 0 DQGDQDVVRFLDWHGHLJHQYHFWRU Y LQ C 2 D 6KRZ WKDW A.5H Y/ D a 5H Y C b ,P Y DQG A.,P Y/ D !b 5H Y C a ,P Y >+LQW :ULWH Y D 5H Y C i ,P Y DQG FRPSXWH AY@ E 9HULI\WKDWLI P DQG C DUHJLYHQDVLQ7KHRUHP WKHQ $3 D P C

'LVFUHWH'\QDPLFDO6\VWHPV

>0@ ,Q([HUFLVHVDQG ÀQGDIDFWRUL]DWLRQRIWKHJLYHQ PDWUL[ A LQWKHIRUP A D P CP !1 ZKHUH C LVDEORFNGLDJRQDO PDWUL[ZLWK 2 # 2 EORFNVRIWKHIRUPVKRZQLQ([DPSOH )RU HDFKFRQMXJDWHSDLURIHLJHQYDOXHV XVHWKHUHDODQGLPDJLQDU\ SDUWVRIRQHHLJHQYHFWRULQ C 4 WRFUHDWHWZRFROXPQVRI P 2 3 26 33 23 20 6 !6 !8 !1 !13 7 7 A D 6 4 !14 !19 !16 35 !20 !20 !20 !14 2

7 6 !20 A D 6 4 0 10

11 !40 !5 28

20 !86 !10 60

3 17 !74 7 7 !10 5 53

40-65*0/ 50 13"$5*$& 130#-&. 5HPHPEHUWKDWLWLVHDV\WRWHVWZKHWKHUDYHFWRULVDQHLJHQYHFWRU 7KHUHLVQRQHHGWR H[DPLQHWKHFKDUDFWHULVWLFHTXDWLRQ &RPSXWH

A[ D !

!

a !b b a

"!

1 !i

"

D

!

a C bi b ! ai

"

!

1 D .a C bi / !i

"

" 1 7KXV LVDQHLJHQYHFWRUFRUUHVSRQGLQJWR ! D a C bi )URPWKHGLVFXVVLRQLQWKLV !i ! " 1 VHFWLRQ PXVWEHDQHLJHQYHFWRUFRUUHVSRQGLQJWR ! D a ! bi i

%*4$3&5& %:/".*$"- 4:45&.4 (LJHQYDOXHVDQGHLJHQYHFWRUVSURYLGHWKHNH\WRXQGHUVWDQGLQJWKHORQJWHUPEHKDYLRU RU HYROXWLRQ RIDG\QDPLFDOV\VWHPGHVFULEHGE\DGLIIHUHQFHHTXDWLRQ [k C1 D A[k 6XFKDQHTXDWLRQZDVXVHGWRPRGHOSRSXODWLRQPRYHPHQWLQ6HFWLRQ YDULRXV 0DUNRYFKDLQV LQ6HFWLRQ DQG WKH VSRWWHGRZOSRSXODWLRQLQWKH LQWURGXFWRU\ H[DPSOHIRUWKLVFKDSWHU 7KHYHFWRUV [k JLYHLQIRUPDWLRQDERXWWKHV\VWHPDVWLPH GHQRWHGE\ k SDVVHV ,QWKHVSRWWHGRZOH[DPSOH IRULQVWDQFH [k OLVWHGWKHQXPEHUV RIRZOVLQWKUHHDJHFODVVHVDWWLPH k 7KHDSSOLFDWLRQVLQWKLVVHFWLRQIRFXVRQHFRORJLFDOSUREOHPVEHFDXVHWKH\DUHHDVLHU WRVWDWHDQGH[SODLQWKDQ VD\ SUREOHPVLQSK\VLFVRUHQJLQHHULQJ +RZHYHU G\QDPLFDO V\VWHPVDULVHLQPDQ\VFLHQWLÀFÀHOGV )RULQVWDQFH VWDQGDUGXQGHUJUDGXDWHFRXUVHV LQFRQWUROV\VWHPVGLVFXVVVHYHUDODVSHFWVRIG\QDPLFDOV\VWHPV 7KHPRGHUQ VWDWH VSDFH GHVLJQPHWKRGLQVXFKFRXUVHVUHOLHVKHDYLO\RQPDWUL[DOJHEUDñ 7KH VWHDG\VWDWH UHVSRQVH RIDFRQWUROV\VWHPLVWKHHQJLQHHULQJHTXLYDOHQWRIZKDWZHFDOOKHUHWKH ÓRQJWHUPEHKDYLRUµRIWKHG\QDPLFDOV\VWHP [k C1 D A[k ñ 6HH* ) )UDQNOLQ - ' 3RZHOO DQG$ (PDPL1DHLPL )HHGEDFN&RQWURORI'\QDPLF6\VWHPV WKHG 8SSHU6DGGOH5LYHU 1-3UHQWLFH+DOO 7KLVXQGHUJUDGXDWHWH[WKDVDQLFHLQWURGXFWLRQWRG\QDPLF PRGHOV&KDSWHU 6WDWHVSDFHGHVLJQLVFRYHUHGLQ&KDSWHUV DQG

$)"15&3


8QWLO([DPSOH ZHDVVXPHWKDW A LVGLDJRQDOL]DEOH ZLWK n OLQHDUO\LQGHSHQ GHQWHLJHQYHFWRUV Y1 ; : : : ; Yn DQGFRUUHVSRQGLQJHLJHQYDOXHV !1 ; : : : ; !n )RUFRQYH QLHQFH DVVXPHWKHHLJHQYHFWRUVDUHDUUDQJHGVRWKDW j!1 j % j!2 j % " " " % j!n j 6LQFH fY1 ; : : : ; Yn g LVDEDVLVIRU Rn DQ\LQLWLDOYHFWRU [0 FDQEHZULWWHQXQLTXHO\DV [0 D c1 Y1 C " " " C cn Yn

7KLV HLJHQYHFWRUGHFRPSRVLWLRQ RI [0 GHWHUPLQHVZKDWKDSSHQVWRWKHVHTXHQFH f[k g 7KHQH[WFDOFXODWLRQJHQHUDOL]HVWKHVLPSOHFDVHH[DPLQHGLQ([DPSOH RI6HFWLRQ 6LQFHWKH Yi DUHHLJHQYHFWRUV [1 D A[0 D c1 AY1 C " " " C cn AYn D c1 !1 Y1 C " " " C cn !n Yn ,QJHQHUDO [k D c1 .!1 /k Y1 C " " " C cn .!n /k Yn

.k D 0; 1; 2; : : :/

7KHH[DPSOHVWKDWIROORZLOOXVWUDWHZKDWFDQKDSSHQLQ DV k ! 1

$ 3UHGDWRU±3UH\6\VWHP 'HHSLQWKHUHGZRRGIRUHVWVRI&DOLIRUQLD GXVN\IRRWHGZRRGUDWVSURYLGHXSWRRI WKHGLHWIRUWKHVSRWWHGRZO WKHPDLQSUHGDWRURIWKHZRRGUDW ([DPSOHXVHVDOLQHDU G\QDPLFDOV\VWHPWRPRGHOWKHSK\VLFDOV\VWHPRIWKHRZOVDQGWKHUDWV $GPLWWHGO\ WKHPRGHOLVXQUHDOLVWLFLQVHYHUDOUHVSHFWV EXWLWFDQSURYLGHDVWDUWLQJSRLQWIRUWKH VWXG\RIPRUHFRPSOLFDWHGQRQOLQHDUPRGHOVXVHGE\HQYLURQPHQWDOVFLHQWLVWV " Ok Rk ZKHUH k LVWKHWLPHLQPRQWKV Ok LVWKHQXPEHURIRZOVLQWKHUHJLRQVWXGLHG DQG Rk LVWKHQXPEHURIUDWVPHDVXUHGLQWKRXVDQGV 6XSSRVH

&9".1-& 'HQRWHWKHRZODQGZRRGUDWSRSXODWLRQVDWWLPH k E\ [k D

Ok C1 D .:5/Ok C .:4/Rk Rk C1 D !p " Ok C .1:1/Rk

!

ZKHUH p LVDSRVLWLYHSDUDPHWHUWREHVSHFLÀHG 7KH .:5/Ok LQWKHÀUVWHTXDWLRQVD\V WKDWZLWKQRZRRGUDWVIRUIRRG RQO\KDOIRIWKHRZOVZLOOVXUYLYHHDFKPRQWK ZKLOHWKH .1:1/Rk LQWKHVHFRQGHTXDWLRQVD\VWKDWZLWKQRRZOVDVSUHGDWRUV WKHUDWSRSXODWLRQ ZLOOJURZE\SHUPRQWK ,IUDWVDUHSOHQWLIXO WKH .:4/Rk ZLOOWHQGWRPDNHWKH RZOSRSXODWLRQULVH ZKLOHWKHQHJDWLYHWHUP !p " Ok PHDVXUHVWKHGHDWKVRIUDWVGXHWR SUHGDWLRQE\RZOV ,QIDFW 1000p LVWKHDYHUDJHQXPEHURIUDWVHDWHQE\RQHRZOLQ RQHPRQWK 'HWHUPLQHWKHHYROXWLRQRIWKLVV\VWHPZKHQWKHSUHGDWLRQSDUDPHWHU p LV

40-65*0/ :KHQ p D :104 WKH HLJHQYDOXHV RI WKH FRHIÀFLHQW PDWUL[ A IRU WKH HTXDWLRQVLQ WXUQRXWWREH !1 D 1:02 DQG !2 D :58 &RUUHVSRQGLQJHLJHQYHFWRUV DUH ! " ! " 10 5 Y1 D ; Y2 D 13 1 $QLQLWLDO [0 FDQEHZULWWHQDV [0 D c1 Y1 C c2 Y2 7KHQ IRU k % 0 [k D c1 .1:02/k Y1 C c2 .:58/k Y2 ! " ! " k 10 k 5 D c1 .1:02/ C c2 .:58/ 13 1


$V k ! 1 .:58/k UDSLGO\DSSURDFKHV]HUR $VVXPH c1 > 0 7KHQ IRUDOOVXIÀFLHQWO\ ODUJH k [k LVDSSUR[LPDWHO\WKHVDPHDV c1 .1:02/k Y1 DQGZHZULWH ! " 10 [k ' c1 .1:02/k 13 7KHDSSUR[LPDWLRQLQ LPSURYHVDV k LQFUHDVHV DQGVRIRUODUJH k ! " ! " 10 10 D .1:02/c1 .1:02/k ' 1:02[k [k C1 ' c1 .1:02/k C1 13 13

7KHDSSUR[LPDWLRQLQ VD\VWKDWHYHQWXDOO\ERWKHQWULHVRI [k WKHQXPEHUVRIRZOV DQGUDWV JURZE\DIDFWRURIDOPRVWHDFKPRQWK DPRQWKO\JURZWKUDWH %\ [k LVDSSUR[LPDWHO\DPXOWLSOHRI .10; 13/ VRWKHHQWULHVLQ [k DUHQHDUO\LQWKHVDPH UDWLRDVWR 7KDWLV IRUHYHU\RZOVWKHUHDUHDERXWWKRXVDQGUDWV ([DPSOHLOOXVWUDWHVWZRJHQHUDOIDFWVDERXWDG\QDPLFDOV\VWHP [k C1 D A[k LQ ZKLFK A LV n # n LWVHLJHQYDOXHVVDWLVI\ j!1 j % 1 DQG 1 > j!j j IRU j D 2; : : : ; n DQG Y1 LVDQHLJHQYHFWRUFRUUHVSRQGLQJWR !1 ,I [0 LVJLYHQE\HTXDWLRQ ZLWK c1 ¤ 0 WKHQ IRUDOOVXIÀFLHQWO\ODUJH k [k C1 ' !1 [k DQG [k ' c1 .!1 /k Y1

7KH DSSUR[LPDWLRQV LQ DQG FDQ EH PDGH DV FORVH DV GHVLUHG E\ WDNLQJ k VXIÀFLHQWO\ODUJH %\ WKH [k HYHQWXDOO\JURZDOPRVWE\DIDFWRURI !1 HDFKWLPH VR !1 GHWHUPLQHVWKHHYHQWXDOJURZWKUDWHRIWKHV\VWHP $OVR E\ WKHUDWLRRIDQ\WZR HQWULHVLQ [k IRUODUJH k LVQHDUO\WKHVDPHDVWKHUDWLRRIWKHFRUUHVSRQGLQJHQWULHVLQ Y1 7KHFDVHLQZKLFK !1 D 1 LVLOOXVWUDWHGLQ([DPSOH LQ6HFWLRQ

*UDSKLFDO'HVFULSWLRQRI6ROXWLRQV :KHQ A LV 2 # 2 DOJHEUDLFFDOFXODWLRQVFDQEHVXSSOHPHQWHGE\DJHRPHWULFGHVFULSWLRQ RIDV\VWHP·VHYROXWLRQ :HFDQYLHZWKHHTXDWLRQ [k C1 D A[k DVDGHVFULSWLRQRI ZKDWKDSSHQVWRDQLQLWLDOSRLQW [0 LQ R2 DVLWLVWUDQVIRUPHGUHSHDWHGO\E\WKHPDSSLQJ [ 7! A[ 7KHJUDSKRI [0 ; [1 ; : : : LVFDOOHGD WUDMHFWRU\ RIWKHG\QDPLFDOV\VWHP

&9".1-& 3ORWVHYHUDOWUDMHFWRULHVRIWKHG\QDPLFDOV\VWHP [k C1 D A[k ZKHQ AD

!

:80 0

0 :64

"

! " 1 40-65*0/ 7KHHLJHQYDOXHVRI A DUHDQG ZLWKHLJHQYHFWRUV Y1 D DQG 0 ! " 0 Y2 D ,I [0 D c1 Y1 C c2 Y2 WKHQ 1 ! " ! " k 1 k 0 [k D c1 .:8/ C c2 .:64/ 0 1

2IFRXUVH [k WHQGVWR EHFDXVH .:8/k DQG .:64/k ERWKDSSURDFKDV k ! 1 %XW WKH ZD\ [k JRHVWRZDUG LVLQWHUHVWLQJ )LJXUH RQSDJH VKRZVWKHÀUVWIHZWHUPV RIVHYHUDOWUDMHFWRULHVWKDWEHJLQDWSRLQWVRQWKHERXQGDU\RIWKHER[ZLWKFRUQHUVDW .˙3; ˙3/ 7KHSRLQWVRQHDFKWUDMHFWRU\DUHFRQQHFWHGE\DWKLQFXUYH WRPDNHWKH WUDMHFWRU\HDVLHUWRVHH

$)"15&3


x0

x2

x0 x1

x0

3

x1 x2

x1

x2

x2

3

x1

'*(63& 7KHRULJLQDVDQDWWUDFWRU

,Q([DPSOH WKHRULJLQLVFDOOHGDQ DWWUDFWRU RIWKHG\QDPLFDOV\VWHPEHFDXVH DOOWUDMHFWRULHVWHQGWRZDUG 7KLVRFFXUVZKHQHYHUERWKHLJHQYDOXHVDUHOHVVWKDQ LQPDJQLWXGH 7KHGLUHFWLRQRIJUHDWHVWDWWUDFWLRQLVDORQJWKHOLQHWKURXJK DQGWKH HLJHQYHFWRU Y2 IRUWKHHLJHQYDOXHRIVPDOOHUPDJQLWXGH ,QWKHQH[WH[DPSOH ERWKHLJHQYDOXHVRI A DUHODUJHUWKDQLQPDJQLWXGH DQG LVFDOOHGD UHSHOOHU RIWKHG\QDPLFDOV\VWHP $OOVROXWLRQVRI [k C1 D A[k H[FHSWWKH FRQVWDQW ]HURVROXWLRQDUHXQERXQGHGDQGWHQGDZD\IURPWKHRULJLQò

&9".1-& 3ORWVHYHUDOW\SLFDOVROXWLRQVRIWKHHTXDWLRQ [k C1 D A[k ZKHUH AD

!

1:44 0

0 1:2

"

40-65*0/ 7KHHLJHQYDOXHVRI A DUHDQG ,I [0 D

!

! " ! " 1 k 0 C c2 .1:2/ [k D c1 .1:44/ 0 1

" c1 WKHQ c2

k

%RWKWHUPVJURZLQVL]H EXWWKHÀ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ò 7KHRULJLQLVWKHRQO\SRVVLEOHDWWUDFWRURUUHSHOOHULQD OLQHDU G\QDPLFDOV\VWHP EXWWKHUHFDQEHPXOWLSOH DWWUDFWRUVDQGUHSHOOHUVLQDPRUHJHQHUDOG\QDPLFDOV\VWHPIRUZKLFKWKHPDSSLQJ [k 7! [kC1 LVQRWOLQHDU ,QVXFKDV\VWHP DWWUDFWRUVDQGUHSHOOHUVDUHGHÀQHGLQWHUPVRIWKHHLJHQYDOXHVRIDVSHFLDOPDWUL[ZLWK YDULDEOHHQWULHV FDOOHGWKH -DFRELDQPDWUL[ RIWKHV\VWHP


x2

x1

'*(63& 7KHRULJLQDVDUHSHOOHU

&9".1-& 3ORWVHYHUDOW\SLFDOVROXWLRQVRIWKHHTXDWLRQ \k C1 D D \k ZKHUH DD

!

2:0 0

0 0:5

"

:HZULWH D DQG \ KHUHLQVWHDGRI A DQG [ EHFDXVHWKLVH[DPSOHZLOOEHXVHGODWHU 6KRZWKDWDVROXWLRQ f\k g LVXQERXQGHGLILWVLQLWLDOSRLQWLVQRWRQWKH x2 D[LV ! " c 40-65*0/ 7KHHLJHQYDOXHVRI D DUHDQG ,I \0 D 1 WKHQ c2 ! " ! " 1 0 C c2 .:5/k \k D c1 2k 0 1 ,I \0 LVRQWKH x2 D[LV WKHQ c1 D 0 DQG \k ! DV k ! 1 %XWLI \0 LVQRWRQWKH x2 D[LV WKHQWKHÀUVWWHUPLQWKHVXPIRU \k EHFRPHVDUELWUDULO\ODUJH DQGVR f\k g LVXQERXQGHG )LJXUHVKRZVWHQWUDMHFWRULHVWKDWEHJLQQHDURURQWKH x2 D[LV x2 x0

x1 x2

x3 x1

x3

x2 x1 x0

'*(63& 7KHRULJLQDVDVDGGOHSRLQW

$)"15&3


&KDQJHRI9DULDEOH 7KHSUHFHGLQJWKUHHH[DPSOHVLQYROYHGGLDJRQDOPDWULFHV 7RKDQGOHWKHQRQGLDJRQDO FDVH ZHUHWXUQIRUDPRPHQWWRWKH n # n FDVHLQZKLFKHLJHQYHFWRUVRI A IRUPD EDVLV fY1 ; : : : ; Yn g IRU Rn /HW P D Œ Y1 " " " Yn # DQGOHW D EHWKHGLDJRQDOPDWUL[ ZLWKWKHFRUUHVSRQGLQJHLJHQYDOXHVRQWKHGLDJRQDO *LYHQDVHTXHQFH f[k g VDWLVI\LQJ [k C1 D A[k GHÀQHDQHZVHTXHQFH f\k g E\ \k D P !1 [k ;

[ k D P \k

RUHTXLYDOHQWO\;

6XEVWLWXWLQJWKHVHUHODWLRQVLQWRWKHHTXDWLRQ [k C1 D A[k DQGXVLQJWKHIDFWWKDW A D 3'3!1 ZHÀQGWKDW

P \k C1 D $3\k D .3'3!1 /P \k D 3'\k

/HIWPXOWLSO\LQJERWKVLGHVE\ P !1 ZHREWDLQ

\k C1 D D \k

,IZHZULWH \k DV \.k/ DQGGHQRWHWKHHQWULHVLQ \.k/ E\ y1 .k/; : : : ; yn .k/ WKHQ 2 3 2 32 3 y1 .k C 1/ !1 0 """ 0 y1 .k/ 6 7 6 7 :: 7 6 6 6 y2 .k C 1/ 7 6 0 7 : 7 !2 6 7D6 7 6 y2 .k/ 7 :: 6 7 6 :: 7 6 :: 7 :: 4 5 4 : : : 0 54 : 5 yn .k C 1/ yn .k/ 0 """ 0 !n

7KHFKDQJHRIYDULDEOHIURP [k WR \k KDV GHFRXSOHG WKHV\VWHPRIGLIIHUHQFHHTXDWLRQV 7KHHYROXWLRQRI y1 .k/ IRUH[DPSOH LVXQDIIHFWHGE\ZKDWKDSSHQVWR y2 .k/; : : : ; yn .k/ EHFDXVH y1 .k C 1/ D !1 " y1 .k/ IRUHDFK k 7KHHTXDWLRQ [k D P \k VD\VWKDW \k LVWKHFRRUGLQDWHYHFWRURI [k ZLWKUHVSHFWWR WKHHLJHQYHFWRUEDVLV fY1 ; : : : ; Yn g :HFDQGHFRXSOHWKHV\VWHP [k C1 D A[k E\PDNLQJ FDOFXODWLRQVLQWKHQHZHLJHQYHFWRUFRRUGLQDWHV\VWHP :KHQ n D 2 WKLVDPRXQWVWR XVLQJJUDSKSDSHUZLWKD[HVLQWKHGLUHFWLRQVRIWKHWZRHLJHQYHFWRUV

&9".1-& 6KRZWKDWWKHRULJLQLVDVDGGOHSRLQWIRUVROXWLRQVRI [k C1 D A[k ZKHUH

AD

!

1:25 !:75

!:75 1:25

"

)LQGWKHGLUHFWLRQVRIJUHDWHVWDWWUDFWLRQDQGJUHDWHVWUHSXOVLRQ

40-65*0/ 8VLQJVWDQGDUGWHFKQLTXHV ! "ZHÀQGWKDW!A "KDVHLJHQYDOXHVDQG ZLWK 1 1 FRUUHVSRQGLQJHLJHQYHFWRUV Y1 D DQG Y2 D UHVSHFWLYHO\ 6LQFH j2j > 1 !1 1 DQG j:5j < 1 WKHRULJLQLVDVDGGOHSRLQWRIWKHG\QDPLFDOV\VWHP ,I [0 D c1 Y1 C c2 Y2 WKHQ [k D c1 2k Y1 C c2 .:5/k Y2

7KLVHTXDWLRQORRNVMXVWOLNHHTXDWLRQ LQ([DPSOH ZLWK Y1 DQG Y2 LQSODFHRIWKH VWDQGDUGEDVLV 2QJUDSKSDSHU GUDZD[HVWKURXJK DQGWKHHLJHQYHFWRUV Y1 DQG Y2 6HH)LJ 0RYHPHQWDORQJWKHVHD[HVFRUUHVSRQGVWRPRYHPHQWDORQJWKHVWDQGDUGD[HVLQ)LJ ,Q)LJ WKHGLUHFWLRQRIJUHDWHVW UHSXOVLRQ LVWKHOLQHWKURXJK DQGWKHHLJHQYHFWRU Y1 ZKRVHHLJHQYDOXHLVJUHDWHUWKDQLQPDJQLWXGH ,I [0 LVRQWKLVOLQH WKH c2 LQ LV]HUR DQG [k PRYHVTXLFNO\DZD\IURP 7KHGLUHFWLRQRIJUHDWHVW DWWUDFWLRQ LVGHWHUPLQHG E\WKHHLJHQYHFWRU Y2 ZKRVHHLJHQYDOXHLVOHVVWKDQLQPDJQLWXGH $ QXPEHURIWUDMHFWRULHVDUHVKRZQLQ)LJ :KHQWKLVJUDSKLVYLHZHGLQWHUPVRI WKHHLJHQYHFWRUD[HV WKHSLFWXUHÓRRNVµHVVHQWLDOO\WKHVDPHDVWKHSLFWXUHLQ)LJ


y

x0

x3 x2

x1

v2 x0

x1 x x2 x3 v1


&RPSOH[(LJHQYDOXHV :KHQDUHDO 2 # 2 PDWUL[ A KDVFRPSOH[HLJHQYDOXHV A LVQRWGLDJRQDOL]DEOHZKHQ DFWLQJRQ R2 / EXWWKHG\QDPLFDOV\VWHP [k C1 D A[k LVHDV\WRGHVFULEH ([DPSOH RI6HFWLRQ LOOXVWUDWHGWKHFDVHLQZKLFKWKHHLJHQYDOXHVKDYHDEVROXWHYDOXH 7KH LWHUDWHVRIDSRLQW [0 VSLUDOHGDURXQGWKHRULJLQDORQJDQHOOLSWLFDOWUDMHFWRU\ ,I A KDVWZRFRPSOH[HLJHQYDOXHVZKRVHDEVROXWHYDOXHLVJUHDWHUWKDQ WKHQ LV DUHSHOOHUDQGLWHUDWHVRI [0 ZLOOVSLUDORXWZDUGDURXQGWKHRULJLQ ,IWKHDEVROXWHYDOXHV RIWKHFRPSOH[HLJHQYDOXHVDUHOHVVWKDQ WKHQWKHRULJLQLVDQDWWUDFWRUDQGWKHLWHUDWHV RI [0 VSLUDOLQZDUGWRZDUGWKHRULJLQ DVLQWKHIROORZLQJH[DPSOH

&9".1-& ,WFDQEHYHULÀHGWKDWWKHPDWUL[ !

" :8 :5 AD !:1 1:0 ! " 1 ( 2i KDVHLJHQYDOXHV :9 ˙ :2i ZLWKHLJHQYHFWRUV )LJXUH RQSDJH VKRZV 1 ! " ! " 0 3 WKUHHWUDMHFWRULHVRIWKHV\VWHP [k C1 D A[k ZLWKLQLWLDOYHFWRUV DQG 2:5 0 ! " 0 !2:5

6XUYLYDORIWKH6SRWWHG2ZOV 5HFDOOIURPWKLVFKDSWHU·VLQWURGXFWRU\H[DPSOHWKDWWKHVSRWWHGRZOSRSXODWLRQLQWKH :LOORZ&UHHNDUHDRI&DOLIRUQLDZDVPRGHOHGE\DG\QDPLFDOV\VWHP [k C1 D A[k LQ ZKLFKWKHHQWULHVLQ [k D .jk ; sk ; ak / OLVWHGWKHQXPEHUVRIIHPDOHVDWWLPH k LQWKH MXYHQLOH VXEDGXOW DQGDGXOWOLIHVWDJHV UHVSHFWLYHO\ DQG A LVWKHVWDJHPDWUL[ 2 3 0 0 :33 0 05 A D 4 :18 0 :71 :94

$)"15&3

(LJHQYDOXHVDQG(LJHQYHFWRUV x2

x0

x1

x3 x3

x2

x1

x2

x2 x1

x3

x0

x1

x0

'*(63& 5RWDWLRQDVVRFLDWHGZLWKFRPSOH[

HLJHQYDOXHV

0$7/$% VKRZV WKDW WKH HLJHQYDOXHV RI A DUH DSSUR[LPDWHO\ !1 D :98 !2 D !:02 C :21i DQG !3 D !:02 ! :21i 2EVHUYH WKDW DOO WKUHH HLJHQYDOXHV DUH OHVVWKDQLQPDJQLWXGH EHFDXVH j!2 j2 D j!3 j2 D .!:02/2 C .:21/2 D :0445 )RUWKHPRPHQW OHW A DFWRQWKHFRPSOH[YHFWRUVSDFH C 3 7KHQ EHFDXVH A KDV WKUHHGLVWLQFWHLJHQYDOXHV WKHWKUHHFRUUHVSRQGLQJHLJHQYHFWRUVDUHOLQHDUO\LQGHSHQGHQW DQGIRUPDEDVLVIRU C 3 'HQRWHWKHHLJHQYHFWRUVE\ Y1 Y2 DQG Y3 7KHQWKHJHQHUDO VROXWLRQRI [k C1 D A[k XVLQJYHFWRUVLQ C 3 KDVWKHIRUP [k D c1 .!1 /k Y1 C c2 .!2 /k Y2 C c3 .!3 /k Y3

,I [0 LVDUHDOLQLWLDOYHFWRU WKHQ [1 D A[0 LVUHDOEHFDXVH A LVUHDO 6LPLODUO\ WKH HTXDWLRQ [k C1 D A[k VKRZVWKDWHDFK [k RQWKHOHIWVLGHRI LVUHDO HYHQWKRXJK LWLVH[SUHVVHGDVDVXPRIFRPSOH[YHFWRUV +RZHYHU HDFKWHUPRQWKHULJKWVLGH RI LVDSSURDFKLQJWKH]HURYHFWRU EHFDXVHWKHHLJHQYDOXHVDUHDOOOHVVWKDQLQ PDJQLWXGH 7KHUHIRUHWKHUHDOVHTXHQFH [k DSSURDFKHVWKH]HURYHFWRU WRR 6DGO\ WKLV PRGHOSUHGLFWVWKDWWKHVSRWWHGRZOVZLOOHYHQWXDOO\DOOSHULVK ,VWKHUHKRSHIRUWKHVSRWWHGRZO" 5HFDOOIURPWKHLQWURGXFWRU\H[DPSOHWKDWWKH HQWU\LQWKHPDWUL[ A LQ FRPHVIURPWKHIDFWWKDWDOWKRXJKRIWKHMXYHQLOH RZOVOLYHORQJHQRXJKWROHDYHWKHQHVWDQGVHDUFKIRUQHZKRPHWHUULWRULHV RQO\ RIWKDWJURXSVXUYLYHWKHVHDUFKDQGÀQGQHZKRPHUDQJHV 6HDUFKVXUYLYDOLVVWURQJO\ LQÁXHQFHGE\WKHQXPEHURIFOHDUFXWDUHDVLQWKHIRUHVW ZKLFKPDNHWKHVHDUFKPRUH GLIÀFXOWDQGGDQJHURXV 6RPHRZOSRSXODWLRQVOLYHLQDUHDVZLWKIHZRUQRFOHDUFXWDUHDV ,WPD\EHWKDW DODUJHUSHUFHQWDJHRIWKHMXYHQLOHRZOVWKHUHVXUYLYHDQGÀQGQHZKRPHUDQJHV 2I FRXUVH WKHSUREOHPRIWKHVSRWWHGRZOLVPRUHFRPSOH[WKDQZHKDYHGHVFULEHG EXWWKH ÀQDOH[DPSOHSURYLGHVDKDSS\HQGLQJWRWKHVWRU\

&9".1-& 6XSSRVHWKHVHDUFKVXUYLYDOUDWHRIWKHMXYHQLOHRZOVLV VRWKH .2; 1/HQWU\LQWKHVWDJHPDWUL[ A LQ LVLQVWHDGRI :KDWGRHVWKHVWDJHPDWUL[ PRGHOSUHGLFWDERXWWKLVVSRWWHGRZOSRSXODWLRQ" 40-65*0/ 1RZWKHHLJHQYDOXHVRI A WXUQRXWWREHDSSUR[LPDWHO\ !1 D 1:01 !2 D !:03 C :26i DQG !3 D !:03 ! :26i $QHLJHQYHFWRUIRU !1 LVDSSUR[LPDWHO\ Y1 D .10; 3; 31/ /HW Y2 DQG Y3 EHFRPSOH[ HLJHQYHFWRUVIRU !2 DQG !3 ,QWKLVFDVH HTXDWLRQ


EHFRPHV [k D c1 .1:01/k Y1 C c2 .!:03 C :26i /k Y2 C c3 .!:03 ! :26i /k Y3

$V k ! 1 WKHVHFRQGWZRYHFWRUVWHQGWR]HUR 6R [k EHFRPHVPRUHDQGPRUHOLNH WKHUHDO YHFWRU c1 .1:01/k Y1 7KHDSSUR[LPDWLRQVLQHTXDWLRQV DQG IROORZLQJ ([DPSOH DSSO\ KHUH $OVR LW FDQ EH VKRZQ WKDW WKH FRQVWDQW c1 LQ WKH LQLWLDO GHFRPSRVLWLRQRI [0 LVSRVLWLYHZKHQWKHHQWULHVLQ [0 DUHQRQQHJDWLYH 7KXVWKHRZO SRSXODWLRQZLOOJURZVORZO\ ZLWKDORQJWHUPJURZWKUDWHRI 7KHHLJHQYHFWRU Y1 GHVFULEHVWKHHYHQWXDOGLVWULEXWLRQRIWKHRZOVE\OLIHVWDJHV IRUHYHU\DGXOWV WKHUH ZLOOEHDERXWMXYHQLOHVDQGVXEDGXOWV

)XUWKHU5HDGLQJ )UDQNOLQ *) -'3RZHOO DQG0/:RUNPDQ 'LJLWDO&RQWURORI'\QDPLF6\VWHPV UGHG 5HDGLQJ 0$$GGLVRQ:HVOH\ 6DQGHIXU -DPHV7 'LVFUHWH'\QDPLFDO6\VWHPV³7KHRU\DQG$SSOLFDWLRQV 2[IRUG 2[IRUG8QLYHUVLW\3UHVV 7XFKLQVN\ 3KLOLS 0DQDJHPHQWRID%XIIDOR+HUG 80$3 0RGXOH /H[LQJWRQ 0$&20$3

13"$5*$& 130#-&.4 7KHPDWUL[ A EHORZKDVHLJHQYDOXHV 23 DQG 13 ZLWKFRUUHVSRQGLQJHLJHQYHFWRUV Y1 Y2 DQG Y3 2 3 2 3 2 3 2 3 7 !2 0 !2 2 1 14 !2 6 2 5; Y1 D 4 2 5; Y2 D 4 1 5; Y3 D 4 2 5 AD 9 0 2 5 1 2 !2 2

3 1 )LQGWKHJHQHUDOVROXWLRQRIWKHHTXDWLRQ [k C1 D A[k LI [0 D 4 11 5 !2 :KDWKDSSHQVWRWKHVHTXHQFH f[k g LQ3UDFWLFH3UREOHPDV k ! 1"

&9&3$*4&4 /HW A EHD 2 # 2 PDWUL[ZLWKHLJHQYDOXHVDQG 1=3 DQG ! " ! " 1 !1 FRUUHVSRQGLQJHLJHQYHFWRUV Y1 D DQG Y2 D /HW 1 1 f[k g EHDVROXWLRQRIWKHGLIIHUHQFHHTXDWLRQ [kC1 D A[k ! " 9 [0 D 1 D &RPSXWH [1 D A[0 >+LQW
6XSSRVHWKHHLJHQYDOXHVRID 3 # 3 PDWUL[ A DUH 4=5 DQG 2 3 2 3 1 2 3=5 ZLWKFRUUHVSRQGLQJHLJHQYHFWRUV 4 0 5 4 1 5 DQG !3 !5 2 3 2 3 !3 !2 4 !3 5 /HW [0 D 4 !5 5 )LQGWKHVROXWLRQRIWKHHTXDWLRQ 7 3 [kC1 D A[k IRUWKHVSHFLÀHG [0 DQGGHVFULEHZKDWKDSSHQV DV k ! 1

$)"15&3


,Q([HUFLVHV² DVVXPHWKDWDQ\LQLWLDOYHFWRU [0 KDVDQHLJHQ YHFWRUGHFRPSRVLWLRQVXFKWKDWWKHFRHIÀFLHQW c1 LQHTXDWLRQ RIWKLVVHFWLRQLVSRVLWLYH 'HWHUPLQHWKHHYROXWLRQRIWKHG\QDPLFDOV\VWHPLQ([DP SOH ZKHQWKHSUHGDWLRQSDUDPHWHU p LVLQHTXDWLRQ *LYHDIRUPXODIRU [k :/ 'RHVWKHRZOSRSXODWLRQJURZRU GHFOLQH" :KDWDERXWWKHZRRGUDWSRSXODWLRQ" 'HWHUPLQHWKHHYROXWLRQRIWKHG\QDPLFDOV\VWHPLQ([DPSOH ZKHQWKHSUHGDWLRQSDUDPHWHU p LV :125 *LYHDIRUPXOD IRU [k $VWLPHSDVVHV ZKDWKDSSHQVWRWKHVL]HVRIWKHRZO DQGZRRGUDWSRSXODWLRQV" 7KHV\VWHPWHQGVWRZDUGZKDWLV VRPHWLPHVFDOOHGDQXQVWDEOHHTXLOLEULXP :KDWGR\RXWKLQN PLJKWKDSSHQWRWKHV\VWHPLIVRPHDVSHFWRIWKHPRGHOVXFK DVELUWKUDWHVRUWKHSUHGDWLRQUDWH ZHUHWRFKDQJHVOLJKWO\" ,QROGJURZWKIRUHVWVRI'RXJODVÀU WKHVSRWWHGRZOGLQHV PDLQO\RQÁ\LQJVTXLUUHOV 6XSSRVHWKHSUHGDWRU²SUH\PDWUL[ ! " :4 :3 6KRZWKDWLI IRUWKHVHWZRSRSXODWLRQVLV A D !p 1:2 WKHSUHGDWLRQSDUDPHWHU p LV ERWKSRSXODWLRQVJURZ (VWLPDWHWKHORQJWHUPJURZWKUDWHDQGWKHHYHQWXDOUDWLRRI RZOVWRÁ\LQJVTXLUUHOV 6KRZWKDWLIWKHSUHGDWLRQSDUDPHWHU p LQ([HUFLVH LV ERWKWKHRZOVDQGWKHVTXLUUHOVZLOOHYHQWXDOO\SHULVK )LQGD YDOXHRI p IRUZKLFKSRSXODWLRQVRIERWKRZOVDQGVTXLUUHOV WHQGWRZDUGFRQVWDQWOHYHOV :KDWDUHWKHUHODWLYHSRSXODWLRQ VL]HVLQWKLVFDVH" /HW A KDYHWKHSURSHUWLHVGHVFULEHGLQ([HUFLVH D ,VWKHRULJLQDQDWWUDFWRU DUHSHOOHU RUDVDGGOHSRLQWRI WKHG\QDPLFDOV\VWHP [kC1 D A[k "

E )LQGWKHGLUHFWLRQVRIJUHDWHVWDWWUDFWLRQDQGRUUHSXOVLRQ IRUWKLVG\QDPLFDOV\VWHP F 0DNHDJUDSKLFDOGHVFULSWLRQRIWKHV\VWHP VKRZLQJWKH GLUHFWLRQV RI JUHDWHVW DWWUDFWLRQ RU UHSXOVLRQ ,QFOXGH D URXJK VNHWFK RI VHYHUDO W\SLFDO WUDMHFWRULHV ZLWKRXW FRPSXWLQJVSHFLÀFSRLQWV

'HWHUPLQH WKH QDWXUH RI WKH RULJLQ DWWUDFWRU UHSHOOHU RU VDGGOHSRLQW IRUWKHG\QDPLFDOV\VWHP [kC1 D A[k LI A KDV WKHSURSHUWLHVGHVFULEHGLQ([HUFLVH )LQGWKHGLUHFWLRQVRI JUHDWHVWDWWUDFWLRQRUUHSXOVLRQ ,Q([HUFLVHV² FODVVLI\WKHRULJLQDVDQDWWUDFWRU UHSHOOHU RUVDGGOHSRLQWRIWKHG\QDPLFDOV\VWHP [kC1 D A[k )LQGWKH GLUHFWLRQVRIJUHDWHVWDWWUDFWLRQDQGRUUHSXOVLRQ ! " ! " 1:7 !:3 :3 :4 A D A D !1:2 :8 !:3 1:1 3

2QHRIWKHOLPLWDWLRQVRIWKHPRGHOLQ([DPSOHLVWKDWWKHUHDOZD\V H[LVWLQLWLDOSRSXODWLRQYHFWRUV [0 ZLWKSRVLWLYHHQWULHVVXFKWKDWWKH FRHIÀFLHQW c1 LVQHJDWLYH 7KHDSSUR[LPDWLRQ LVVWLOOYDOLG EXWWKH HQWULHVLQ [k HYHQWXDOO\EHFRPHQHJDWLYH

" :6 1:4 ! " ! " :8 :3 1:7 :6 A D A D !:4 1:5 !:4 :7 2 3 2 3 :4 0 :2 :1 /HW A D 4 :3 :8 :3 5 7KHYHFWRU Y1 D 4 :6 5 LVDQ :3 :2 :5 :3 HLJHQYHFWRUIRU A DQGWZRHLJHQYDOXHVDUHDQG &RQ VWUXFWWKHVROXWLRQRIWKHG\QDPLFDOV\VWHP [kC1 D A[k WKDW VDWLVÀHV [0 D .0; :3; :7/ :KDWKDSSHQVWR [k DV k ! 1"

A D

!

:4 !:4

:5 1:3

"

A D

!

:5 !:3

>0@ 3URGXFHWKHJHQHUDOVROXWLRQRIWKHG\QDPLFDOV\VWHP [kC1 D A[k ZKHQ A LVWKHVWRFKDVWLFPDWUL[IRUWKH+HUW] 5HQW$ &DUPRGHOLQ([HUFLVHRI6HFWLRQ &RQVWUXFWDVWDJHPDWUL[PRGHOIRUDQDQLPDOVSHFLHVWKDWKDV WZROLIHVWDJHV MXYHQLOHXSWR\HDUROG DQGDGXOW 6XSSRVH WKHIHPDOHDGXOWVJLYHELUWKHDFK\HDUWRDQDYHUDJHRI IHPDOHMXYHQLOHV (DFK\HDU RIWKHMXYHQLOHVVXUYLYH WREHFRPHDGXOWVDQGRIWKHDGXOWVVXUYLYH )RU k % 0 OHW [k D .jk ; ak / ZKHUHWKHHQWULHVLQ [k DUHWKHQXPEHUVRI IHPDOHMXYHQLOHVDQGIHPDOHDGXOWVLQ\HDU k D &RQVWUXFWWKHVWDJHPDWUL[ A VXFKWKDW [kC1 D A[k IRU k % 0

E 6KRZWKDWWKHSRSXODWLRQLVJURZLQJ FRPSXWHWKHHYHQ WXDOJURZWKUDWHRIWKHSRSXODWLRQ DQGJLYHWKHHYHQWXDO UDWLRRIMXYHQLOHVWRDGXOWV F >0@ 6XSSRVHWKDWLQLWLDOO\WKHUHDUHMXYHQLOHVDQG DGXOWVLQWKHSRSXODWLRQ 3URGXFHIRXUJUDSKVWKDWVKRZ KRZWKHSRSXODWLRQFKDQJHVRYHUHLJKW\HDUV D WKH QXPEHURIMXYHQLOHV E WKHQXPEHURIDGXOWV F WKH WRWDOSRSXODWLRQ DQGG WKHUDWLRRIMXYHQLOHVWRDGXOWV HDFK\HDU :KHQGRHVWKHUDWLRLQG VHHPWRVWDELOL]H" ,QFOXGHDOLVWLQJRIWKHSURJUDPRUNH\VWURNHVXVHGWR SURGXFHWKHJUDSKVIRUF DQGG

$ KHUGRI$PHULFDQEXIIDORELVRQ FDQEHPRGHOHGE\DVWDJH PDWUL[VLPLODUWRWKDWIRUWKHVSRWWHGRZOV 7KHIHPDOHVFDQEH GLYLGHGLQWRFDOYHVXSWR\HDUROG \HDUOLQJVWR\HDUV DQGDGXOWV 6XSSRVHDQDYHUDJHRIIHPDOHFDOYHVDUH ERUQHDFK\HDUSHUDGXOWIHPDOHV 2QO\DGXOWVSURGXFH RIIVSULQJ (DFK\HDU DERXWRIWKHFDOYHVVXUYLYH RIWKH\HDUOLQJVVXUYLYH DQGRIWKHDGXOWVVXUYLYH )RU k % 0 OHW [k D .ck ; yk ; ak / ZKHUHWKHHQWULHVLQ [k DUHWKH QXPEHUVRIIHPDOHVLQHDFKOLIHVWDJHDW\HDU k D &RQVWUXFWWKHVWDJHPDWUL[ A IRUWKHEXIIDORKHUG VXFK WKDW [kC1 D A[k IRU k % 0 E >0@ 6KRZWKDWWKHEXIIDORKHUGLVJURZLQJ GHWHUPLQH WKHH[SHFWHGJURZWKUDWHDIWHUPDQ\\HDUV DQGJLYHWKH H[SHFWHGQXPEHUVRIFDOYHVDQG\HDUOLQJVSUHVHQWSHU DGXOWV

$SSOLFDWLRQVWR'LIIHUHQWLDO(TXDWLRQV

40-65*0/4 50 13"$5*$& 130#-&.4 7KHÀUVWVWHSLVWRZULWH [0 DVDOLQHDUFRPELQDWLRQRI Y1 Y2 DQG Y3 5RZUHGXFWLRQ RI Œ Y1 Y2 Y3 [0 # SURGXFHVWKHZHLJKWV c1 D 2 c2 D 1 DQG c3 D 3 VRWKDW [0 D 2Y1 C 1Y2 C 3Y3 6LQFHWKHHLJHQYDOXHVDUH 23 DQG 13 WKHJHQHUDOVROXWLRQLV % &k % &k 2 1 k Y2 C 3 " Y3 [k D 2 " 1 Y1 C 1 " 3 3 2 3 2 3 2 3 % &k 2 % &k !2 1 2 1 415C3" 4 25 D 24 2 5 C 3 3 1 2 !2

$V k ! 1 WKHVHFRQGDQGWKLUGWHUPVLQ WHQGWRWKH]HURYHFWRU DQG 2 3 % &k % &k !4 2 1 [k D 2Y1 C Y2 C 3 Y3 ! 2Y1 D 4 4 5 3 3 2

"11-*$"5*0/4 50 %*''&3&/5*"- &26"5*0/4 7KLVVHFWLRQGHVFULEHVFRQWLQXRXVDQDORJXHVRIWKHGLIIHUHQFHHTXDWLRQVVWXGLHGLQ 6HFWLRQ ,QPDQ\DSSOLHGSUREOHPV VHYHUDOTXDQWLWLHVDUHYDU\LQJFRQWLQXRXVO\ LQWLPH DQGWKH\DUHUHODWHGE\DV\VWHPRIGLIIHUHQWLDOHTXDWLRQV

x10 D a11 x1 C " " " C a1n xn x20 D a21 x1 C " " " C a2n xn :: : 0 xn D an1 x1 C " " " C ann xn

+HUH x1 ; : : : ; xn DUHGLIIHUHQWLDEOHIXQFWLRQVRI t ZLWKGHULYDWLYHV x10 ; : : : ; xn0 DQGWKH aij DUHFRQVWDQWV 7KHFUXFLDOIHDWXUHRIWKLVV\VWHPLVWKDWLWLV OLQHDU 7RVHHWKLV ZULWHWKH V\VWHPDVDPDWUL[GLIIHUHQWLDOHTXDWLRQ [0 .t/ D A[.t/ ZKHUH

2

3 x1 .t / 6 : 7 [.t / D 4 :: 5 ;

xn .t /

2

3 x10 .t / 6 : 7 [0 .t / D 4 :: 5 ;

xn0 .t /

DQG

2

a11 6 : A D 4 :: an1

""" """

3 a1n :: 7 : 5

ann

$ VROXWLRQ RIHTXDWLRQ LVDYHFWRUYDOXHGIXQFWLRQWKDWVDWLVÀHV IRUDOO t LQVRPH LQWHUYDORIUHDOQXPEHUV VXFKDV t % 0 (TXDWLRQ LV OLQHDU EHFDXVHERWKGLIIHUHQWLDWLRQRIIXQFWLRQVDQGPXOWLSOLFDWLRQRI YHFWRUVE\DPDWUL[DUHOLQHDUWUDQVIRUPDWLRQV 7KXV LI X DQG Y DUHVROXWLRQVRI [0 D A[ WKHQ c X C d Y LVDOVRDVROXWLRQ EHFDXVH

.c X C d Y/0 D c X0 C d Y0 D cAX C dAY D A.c X C d Y/

$)"15&3


(QJLQHHUVFDOOWKLVSURSHUW\ VXSHUSRVLWLRQ RIVROXWLRQV $OVR WKHLGHQWLFDOO\]HUR IXQFWLRQLVDWULYLDO VROXWLRQRI ,QWKHWHUPLQRORJ\RI&KDSWHU WKHVHWRIDOO VROXWLRQVRI LVD VXEVSDFH RIWKHVHWRIDOOFRQWLQXRXVIXQFWLRQVZLWKYDOXHVLQ Rn 6WDQGDUG WH[WVRQGLIIHUHQWLDOHTXDWLRQVVKRZ WKDWWKHUHDOZD\VH[LVWVZKDWLV FDOOHGD IXQGDPHQWDOVHWRIVROXWLRQV WR ,I A LV n # n WKHQWKHUHDUH n OLQHDUO\ LQGHSHQGHQWIXQFWLRQVLQDIXQGDPHQWDOVHW DQGHDFKVROXWLRQRI LVDXQLTXHOLQHDU FRPELQDWLRQRIWKHVH n IXQFWLRQV 7KDWLV DIXQGDPHQWDOVHWRIVROXWLRQVLVD EDVLV IRU WKHVHWRIDOOVROXWLRQVRI DQGWKHVROXWLRQVHWLVDQ nGLPHQVLRQDOYHFWRUVSDFHRI IXQFWLRQV ,IDYHFWRU [0 LVVSHFLÀHG WKHQWKH LQLWLDOYDOXHSUREOHP LVWRFRQVWUXFWWKH XQLTXH IXQFWLRQ [ VXFKWKDW [0 D A[ DQG [.0/ D [0 :KHQ A LVDGLDJRQDOPDWUL[ WKHVROXWLRQVRI FDQEHSURGXFHGE\HOHPHQWDU\ FDOFXOXV )RULQVWDQFH FRQVLGHU " # " #" # x10 .t / 3 0 x1 .t / D x20 .t / 0 !5 x2 .t / WKDWLV

x10 .t / x20 .t /

D 3x1 .t/ D !5x2 .t/

7KHV\VWHP LVVDLGWREH GHFRXSOHG EHFDXVHHDFKGHULYDWLYHRIDIXQFWLRQGHSHQGV RQO\RQWKHIXQFWLRQLWVHOI QRWRQVRPHFRPELQDWLRQRU´FRXSOLQJµRIERWK x1 .t/ DQG x2 .t / )URPFDOFXOXV WKHVROXWLRQVRI DUH x1 .t / D c1 e 3t DQG x2 .t / D c2 e !5t IRUDQ\ FRQVWDQWV c1 DQG c2 (DFKVROXWLRQRIHTXDWLRQ FDQEHZULWWHQLQWKHIRUP ! " ! " ! " ! " x1 .t/ c e 3t 1 3t 0 !5t D 1 !5t D c1 e C c2 e c2 e x2 .t/ 0 1 7KLVH[DPSOHVXJJHVWVWKDWIRUWKHJHQHUDOHTXDWLRQ [0 D A[ DVROXWLRQPLJKWEHD OLQHDUFRPELQDWLRQRIIXQFWLRQVRIWKHIRUP [.t/ D Ye !t

IRUVRPHVFDODU ! DQGVRPHÀ[HGQRQ]HURYHFWRU Y >,I Y D WKHIXQFWLRQ [.t/ LV LGHQWLFDOO\]HURDQGKHQFHVDWLVÀHV [0 D A[@ 2EVHUYHWKDW [0 .t / D !Ye !t

A[.t / D AYe

!t

%\FDOFXOXV VLQFH Y LVDFRQVWDQWYHFWRU 0XOWLSO\LQJERWKVLGHVRI E\ A

6LQFH e !t LVQHYHU]HUR [0 .t/ ZLOOHTXDO A[.t / LIDQGRQO\LI !Y D AY WKDWLV LIDQG RQO\LI ! LVDQHLJHQYDOXHRI A DQG Y LVDFRUUHVSRQGLQJHLJHQYHFWRU 7KXVHDFK HLJHQYDOXH²HLJHQYHFWRUSDLUSURYLGHVDVROXWLRQ RI [0 D A[ 6XFKVROXWLRQVDUH VRPHWLPHVFDOOHG HLJHQIXQFWLRQV RIWKHGLIIHUHQWLDOHTXDWLRQ (LJHQIXQFWLRQVSURYLGH WKHNH\WRVROYLQJV\VWHPVRIGLIIHUHQWLDOHTXDWLRQV

R1

&9".1-& 7KHFLUFXLWLQ)LJ FDQEHGHVFULEHGE\WKHGLIIHUHQWLDOHTXDWLRQ +

C1 R2 +

C2 '*(63&

"

x10 .t / x20 .t /

#

D

"

!.1=R1 C 1=R2 /=C1 1=.R2 C2 /

1=.R2 C1 / !1=.R2 C2 /

#"

x1 .t/ x2 .t/

#

ZKHUH x1 .t / DQG x2 .t / DUHWKHYROWDJHVDFURVVWKHWZRFDSDFLWRUVDWWLPH t 6XSSRVH UHVLVWRU R1 LV RKP R2 LV RKPV FDSDFLWRU C1 LV IDUDG DQG C2 LV IDUDG DQG VXSSRVHWKHUHLVDQLQLWLDOFKDUJHRI YROWVRQFDSDFLWRU C1 DQG YROWVRQFDSDFLWRU C2 )LQGIRUPXODVIRU x1 .t / DQG x2 .t / WKDWGHVFULEHKRZWKHYROWDJHVFKDQJHRYHUWLPH


! " x1 .t/ 40-65*0/ /HW A GHQRWHWKHPDWUL[GLVSOD\HGDERYH DQGOHW [.t / D )RUWKH x2 .t/ ! " ! " !1:5 :5 5 GDWDJLYHQ A D DQG [.0/ D 7KHHLJHQYDOXHVRI A DUH !1 D !:5 1 !1 4 DQG !2 D !2 ZLWKFRUUHVSRQGLQJHLJHQYHFWRUV ! " ! " 1 !1 Y1 D DQG Y2 D 2 1

7KHHLJHQIXQFWLRQV [1 .t / D Y1 e !1 t DQG [2 .t/ D Y2 e !2 t ERWKVDWLVI\ [0 D A[ DQGVRGRHV DQ\OLQHDUFRPELQDWLRQRI [1 DQG [2 6HW ! " ! " 1 !:5t !1 !2t !1 t !2 t e C c2 e [.t / D c1 Y1 e C c2 Y2 e D c1 2 1

DQGQRWHWKDW [.0/ D c1 Y1 C c2 Y2 6LQFH Y1 DQG Y2 DUHREYLRXVO\OLQHDUO\LQGHSHQGHQW DQGKHQFHVSDQ R2 c1 DQG c2 FDQEHIRXQGWRPDNH [.0/ HTXDOWR [0 ,QIDFW WKHHTXDWLRQ ! " ! " ! " 1 !1 5 c1 C c2 D 2 1 4 "

"

Y1

"

Y2

[0

OHDGVHDVLO\WR c1 D 3 DQG c2 D !2 7KXVWKHGHVLUHGVROXWLRQRIWKHGLIIHUHQWLDOHTXDWLRQ [0 D A[ LV ! " ! " 1 !:5t !1 !2t [.t/ D 3 e !2 e 2 1 RU # ! " " !:5t x1 .t / 3e C 2e !2t D x2 .t / 6e !:5t ! 2e !2t )LJXUH VKRZVWKHJUDSK RU WUDMHFWRU\ RI [.t/ IRU t % 0 DORQJZLWKWUDMHFWRULHVIRU VRPHRWKHULQLWLDOSRLQWV 7KHWUDMHFWRULHVRIWKHWZRHLJHQIXQFWLRQV [1 DQG [2 OLHLQWKH HLJHQVSDFHVRI A 7KH IXQFWLRQV [1 DQG [2 ERWK GHFD\ WR ]HUR DV t ! 1 EXW WKH YDOXHV RI [2 GHFD\IDVWHUEHFDXVHLWVH[SRQHQWLVPRUHQHJDWLYH 7KHHQWULHVLQWKHFRUUHVSRQGLQJ HLJHQYHFWRU Y2 VKRZWKDWWKHYROWDJHVDFURVVWKHFDSDFLWRUVZLOOGHFD\WR]HURDVUDSLGO\ DVSRVVLEOHLIWKHLQLWLDOYROWDJHVDUHHTXDOLQPDJQLWXGHEXWRSSRVLWHLQVLJQ

4

x0

v1 v2 5

'*(63& 7KHRULJLQDVDQDWWUDFWRU

$)"15&3


,Q)LJ WKHRULJLQLVFDOOHGDQ DWWUDFWRU RU VLQN RIWKHG\QDPLFDOV\VWHPEHFDXVH DOOWUDMHFWRULHVDUHGUDZQLQWRWKHRULJLQ 7KHGLUHFWLRQRIJUHDWHVWDWWUDFWLRQLVDORQJWKH WUDMHFWRU\RIWKHHLJHQIXQFWLRQ [2 DORQJWKHOLQHWKURXJK DQG Y2 / FRUUHVSRQGLQJWR WKHPRUHQHJDWLYHHLJHQYDOXH ! D !2 7UDMHFWRULHVWKDWEHJLQDWSRLQWVQRWRQWKLVOLQH EHFRPHDV\PSWRWLFWRWKHOLQHWKURXJK DQG Y1 EHFDXVHWKHLUFRPSRQHQWVLQWKH Y2 GLUHFWLRQGHFD\VRUDSLGO\ ,IWKHHLJHQYDOXHVLQ([DPSOH ZHUHSRVLWLYHLQVWHDGRIQHJDWLYH WKHFRUUHVSRQGLQJ WUDMHFWRULHVZRXOGEHVLPLODULQVKDSH EXWWKHWUDMHFWRULHVZRXOGEHWUDYHUVHG DZD\ IURP WKHRULJLQ ,QVXFKDFDVH WKHRULJLQLVFDOOHGD UHSHOOHU RU VRXUFH RIWKHG\QDPLFDO V\VWHP DQGWKHGLUHFWLRQRIJUHDWHVWUHSXOVLRQLVWKHOLQHFRQWDLQLQJWKHWUDMHFWRU\RIWKH HLJHQIXQFWLRQFRUUHVSRQGLQJWRWKHPRUHSRVLWLYHHLJHQYDOXH

&9".1-& 6XSSRVHDSDUWLFOHLVPRYLQJLQDSODQDUIRUFHÀHOGDQGLWVSRVLWLRQ YHFWRU [ VDWLVÀHV [0 D A[ DQG [.0/ D [0 ZKHUH ! " ! " 4 !5 2:9 AD ; [0 D !2 1 2:6 6ROYHWKLVLQLWLDOYDOXHSUREOHPIRU t % 0 DQGVNHWFKWKHWUDMHFWRU\RIWKHSDUWLFOH 40-65*0/ 7KHHLJHQYDOXHVRI A WXUQRXWWREH !1 D 6 DQG !2 D !1 ZLWKFRUUHVSRQG LQJHLJHQYHFWRUV Y1 D .!5; 2/ DQG Y2 D .1; 1/ )RUDQ\FRQVWDQWV c1 DQG c2 WKHIXQFWLRQ ! " ! " !5 6t 1 !t !1 t !2 t [.t/ D c1 Y1 e C c2 Y2 e D c1 e C c2 e 2 1 LVDVROXWLRQRI [0 D A[ :HZDQW c1 DQG c2 WRVDWLVI\ [.0/ D [0 WKDWLV ! " ! " ! " ! "! " ! " !5 1 2:9 !5 1 c1 2:9 c1 C c2 D RU D 2 1 2:6 2 1 c2 2:6 &DOFXODWLRQVVKRZWKDW c1 D !3=70 DQG c2 D 188=70 DQGVRWKHGHVLUHGIXQFWLRQLV ! " ! " !3 !5 6t 188 1 !t [.t / D e C e 2 70 70 1 7UDMHFWRULHVRI [ DQGRWKHUVROXWLRQVDUHVKRZQLQ)LJ

,Q)LJ WKHRULJLQLVFDOOHGD VDGGOHSRLQW RIWKHG\QDPLFDOV\VWHPEHFDXVH VRPHWUDMHFWRULHVDSSURDFKWKHRULJLQDWÀUVWDQGWKHQFKDQJHGLUHFWLRQDQGPRYHDZD\ IURPWKHRULJLQ $ VDGGOHSRLQWDULVHVZKHQHYHUWKHPDWUL[ A KDVERWKSRVLWLYHDQG QHJDWLYHHLJHQYDOXHV 7KHGLUHFWLRQRIJUHDWHVWUHSXOVLRQLVWKHOLQHWKURXJK Y1 DQG FRUUHVSRQGLQJWRWKHSRVLWLYHHLJHQYDOXH 7KHGLUHFWLRQRIJUHDWHVWDWWUDFWLRQLVWKHOLQH WKURXJK Y2 DQG FRUUHVSRQGLQJWRWKHQHJDWLYHHLJHQYDOXH

v1

x0 v2



'HFRXSOLQJD'\QDPLFDO6\VWHP 7KHIROORZLQJGLVFXVVLRQVKRZVWKDWWKHPHWKRGRI([DPSOHVDQGSURGXFHVD IXQGDPHQWDOVHWRIVROXWLRQVIRUDQ\G\QDPLFDOV\VWHPGHVFULEHGE\ [0 D A[ ZKHQ A LV n # n DQGKDV n OLQHDUO\LQGHSHQGHQWHLJHQYHFWRUV WKDWLV ZKHQ A LVGLDJRQDOL]DEOH 6XSSRVHWKHHLJHQIXQFWLRQVIRU A DUH Y 1 e !1 t ;

: : : ; Y n e !n t

ZLWK Y1 ; : : : ; Yn OLQHDUO\LQGHSHQGHQWHLJHQYHFWRUV /HW P D Œ Y1 " " " Yn # DQGOHW D EHWKHGLDJRQDOPDWUL[ZLWKHQWULHV !1 ; : : : ; !n VRWKDW A D 3'3!1 1RZPDNHD FKDQJH RIYDULDEOH GHÀQLQJDQHZIXQFWLRQ \ E\ \.t / D P !1 [.t /

RU HTXLYDOHQWO\;

[.t / D P \.t/

7KHHTXDWLRQ [.t/ D P \.t/ VD\VWKDW \.t/ LVWKHFRRUGLQDWHYHFWRURI [.t/ UHODWLYHWRWKH HLJHQYHFWRUEDVLV 6XEVWLWXWLRQRI P \ IRU [ LQWKHHTXDWLRQ [0 D A[ JLYHV

d .P \/ D A.P \/ D .3'3!1 /P \ D 3'\ dt 6LQFH P LVDFRQVWDQWPDWUL[ WKHOHIWVLGHRI LV P \0 /HIWPXOWLSO\ERWKVLGHVRI E\ P !1 DQGREWDLQ \0 D D \ RU 2 0 3 2 32 3 y1 .t / !1 0 """ 0 y1 .t/ 6 0 7 6 7 :: 7 6 6 6 y .t / 7 6 0 7 : 7 !2 6 2 7D6 7 6 y2 .t/ 7 6 :: 7 6 :: 7 6 :: 7 :: 4 : 5 4 : : 0 54 : 5 yn0 .t / yn .t/ 0 """ 0 !n

7KHFKDQJHRIYDULDEOHIURP [ WR \ KDV GHFRXSOHG WKHV\VWHPRIGLIIHUHQWLDOHTXDWLRQV EHFDXVHWKHGHULYDWLYHRIHDFKVFDODUIXQFWLRQ yk GHSHQGVRQO\RQ yk 5HYLHZWKHDQDO RJRXVFKDQJHRIYDULDEOHVLQ6HFWLRQ 6LQFH y10 D !1 y1 ZHKDYH y1 .t/ D c1 e !1 t ZLWKVLPLODUIRUPXODVIRU y2 ; : : : ; yn 7KXV 2 3 2 3 c1 e !1 t c1 6 :: 7 6 :: 7 \.t/ D 4 : 5 ; ZKHUH 4 : 5 D \.0/ D P !1 [.0/ D P !1 [0

cn e !n t

cn

7RREWDLQWKHJHQHUDOVROXWLRQ [ RIWKHRULJLQDOV\VWHP FRPSXWH [.t / D P \.t / D Œ Y1 " " " Yn # \.t/

D c1 Y1 e !1 t C " " " C cn Yn e !n t

7KLVLVWKHHLJHQIXQFWLRQH[SDQVLRQFRQVWUXFWHGDVLQ([DPSOH

&RPSOH[(LJHQYDOXHV ,QWKHQH[WH[DPSOH DUHDOPDWUL[ A KDVDSDLURIFRPSOH[HLJHQYDOXHV ! DQG ! ZLWK DVVRFLDWHGFRPSOH[HLJHQYHFWRUV Y DQG Y 5HFDOOIURP6HFWLRQ WKDWIRUDUHDOPDWUL[ FRPSOH[HLJHQYDOXHVDQGDVVRFLDWHGHLJHQYHFWRUVFRPHLQFRQMXJDWHSDLUV 6RWZR VROXWLRQVRI [0 D A[ DUH [1 .t / D Ye !t

DQG

[2 .t / D Ye !t

,WFDQEHVKRZQWKDW [2 .t / D [1 .t / E\XVLQJDSRZHUVHULHVUHSUHVHQWDWLRQIRUWKH FRPSOH[H[SRQHQWLDOIXQFWLRQ $OWKRXJKWKHFRPSOH[HLJHQIXQFWLRQV [1 DQG [2 DUH FRQYHQLHQWIRUVRPHFDOFXODWLRQVSDUWLFXODUO\LQHOHFWULFDOHQJLQHHULQJ UHDOIXQFWLRQV

$)"15&3


DUHPRUHDSSURSULDWHIRUPDQ\SXUSRVHV )RUWXQDWHO\ WKHUHDODQGLPDJLQDU\SDUWVRI [1 DUHUHDO VROXWLRQVRI [0 D A[ EHFDXVHWKH\DUHOLQHDUFRPELQDWLRQVRIWKHVROXWLRQVLQ 1 1 5H.Ye !t / D Œ [1 .t/ C [1 .t / #; ,P.Ye !t / D Œ [1 .t / ! [1 .t / # 2 2i 7RXQGHUVWDQGWKHQDWXUHRI 5H.Ye !t / UHFDOOIURPFDOFXOXVWKDWIRUDQ\QXPEHU x WKHH[SRQHQWLDOIXQFWLRQ e x FDQEHFRPSXWHGIURPWKHSRZHUVHULHV

1 2 1 x C " " " C xn C " " " 2Š nŠ ZKHQ ! LVFRPSOH[

ex D 1 C x C 7KLVVHULHVFDQEHXVHGWRGHÀQH e !t

1 1 .!t /2 C " " " C .!t /n C " " " 2Š nŠ %\ZULWLQJ ! D a C bi ZLWK a DQG b UHDO DQGXVLQJVLPLODUSRZHUVHULHVIRUWKHFRVLQH DQGVLQHIXQFWLRQV RQHFDQVKRZWKDW e !t D 1 C .!t / C

e .aCbi/t D e at " e ibt D e at .FRV bt C i VLQ bt /

+HQFH Ye !t D .5H Y C i ,P Y/ " e at .FRV bt C i VLQ bt /

D Œ .5H Y/ FRV bt ! .,P Y/ VLQ bt #e at

C iŒ .5H Y/ VLQ bt C .,P Y/ FRV bt #e at

6RWZRUHDOVROXWLRQVRI [0 D A[ DUH

\1 .t/ D 5H [1 .t/ D Œ .5H Y/ FRV bt ! .,P Y/ VLQ bt # e at

\2 .t/ D ,P [1 .t/ D Œ .5H Y/ VLQ bt C .,P Y/ FRV bt # e at ,WFDQEHVKRZQWKDW \1 DQG \2 DUHOLQHDUO\LQGHSHQGHQWIXQFWLRQVZKHQ b ¤ 0 ñ

&9".1-& 7KHFLUFXLWLQ)LJ FDQEHGHVFULEHGE\WKHHTXDWLRQ "

R1

+

C R2 iL L '*(63&

iL0 vC0

#

D

"

!R2 =L !1=L 1=C !1=.R1 C /

#"

iL vC

#

ZKHUH iL LVWKHFXUUHQWSDVVLQJWKURXJKWKHLQGXFWRU L DQG vC LVWKHYROWDJHGURSDFURVV WKHFDSDFLWRU C 6XSSRVH R1 LV RKPV R2 LV RKP C LV IDUDG DQG L LV KHQU\ )LQGIRUPXODVIRU iL DQG vC LIWKHLQLWLDOFXUUHQWWKURXJKWKHLQGXFWRULV DPSHUHVDQG WKHLQLWLDOYROWDJHDFURVVWKHFDSDFLWRULV YROWV ! " ! " !2 !2:5 3 40-65*0/ )RU WKH GDWD JLYHQ A D DQG [0 D 7KH PHWKRG 10 !2 3 GLVFXVVHGLQ6HFWLRQSURGXFHVWKHHLJHQYDOXH ! D !2 C 5i DQGWKHFRUUHVSRQGLQJ ! " i HLJHQYHFWRU Y1 D 7KHFRPSOH[VROXWLRQVRI [0 D A[ DUHFRPSOH[OLQHDUFRPEL 2 QDWLRQVRI ! " ! " i .!2C5i/t !i .!2!5i/t [1 .t / D e DQG [2 .t / D e 2 2 ñ 6LQFH [2 .t/ LVWKHFRPSOH[FRQMXJDWHRI [1 .t/ WKHUHDODQGLPDJLQDU\SDUWVRI [2 .t/ DUH \1 .t / DQG !\2 .t/ UHVSHFWLYHO\ 7KXVRQHFDQXVHHLWKHU [1 .t / RU [2 .t/ EXWQRWERWK WRSURGXFHWZRUHDOOLQHDUO\LQGHSHQGHQW VROXWLRQVRI [0 D A[

1H[W XVHHTXDWLRQ WRZULWH [1 .t / D

x0


! " i !2t e .FRV 5t C i VLQ 5t / 2

7KHUHDODQGLPDJLQDU\SDUWVRI [1 SURYLGHUHDOVROXWLRQV ! " ! " ! VLQ 5t !2t FRV 5t !2t \1 .t/ D e ; \2 .t/ D e 2 FRV 5t 2 VLQ 5t 6LQFH \1 DQG \2 DUHOLQHDUO\LQGHSHQGHQWIXQFWLRQV WKH\IRUPDEDVLVIRUWKHWZR GLPHQVLRQDOUHDOYHFWRUVSDFHRIVROXWLRQVRI [0 D A[ 7KXVWKHJHQHUDOVROXWLRQLV ! " ! " ! VLQ 5t !2t FRV 5t !2t [.t/ D c1 e C c2 e 2 FRV 5t 2 VLQ 5t ! " ! " ! " ! " 3 0 1 3 7RVDWLVI\ [.0/ D ZHQHHG c1 C c2 D ZKLFKOHDGVWR c1 D 1:5 DQG 3 2 0 3 c2 D 3 7KXV ! " ! " ! VLQ 5t !2t FRV 5t !2t [.t/ D 1:5 e C3 e 2 FRV 5t 2 VLQ 5t RU

'*(63&

7KHRULJLQDVDVSLUDOSRLQW

!

iL .t/ vC .t /

"

D

!

" !1:5 VLQ 5t C 3 FRV 5t !2t e 3 FRV 5t C 6 VLQ 5t

6HH)LJ ,Q)LJ WKHRULJLQLVFDOOHGD VSLUDOSRLQW RIWKHG\QDPLFDOV\VWHP 7KHURWDWLRQ LVFDXVHGE\WKHVLQHDQGFRVLQHIXQFWLRQVWKDWDULVHIURPDFRPSOH[HLJHQYDOXH 7KH WUDMHFWRULHVVSLUDOLQZDUGEHFDXVHWKHIDFWRU e !2t WHQGVWR]HUR 5HFDOOWKDW !2 LVWKHUHDO SDUWRIWKHHLJHQYDOXHLQ([DPSOH :KHQ A KDVDFRPSOH[HLJHQYDOXHZLWKSRVLWLYH UHDOSDUW WKHWUDMHFWRULHVVSLUDORXWZDUG ,IWKHUHDOSDUWRIWKHHLJHQYDOXHLV]HUR WKH WUDMHFWRULHVIRUPHOOLSVHVDURXQGWKHRULJLQ

13"$5*$& 130#-&.4 $ UHDO 3 # 3 PDWUL[ A KDVHLJHQYDOXHV !:5 :2 C :3i DQG :2 ! :3i ZLWKFRUUHVSRQGLQJ HLJHQYHFWRUV 2 3 2 3 2 3 1 1 C 2i 1 ! 2i Y1 D 4 !2 5; Y2 D 4 4i 5 ; DQG Y3 D 4 !4i 5 1 2 2

,V A GLDJRQDOL]DEOHDV A D 3'3!1 XVLQJFRPSOH[PDWULFHV" :ULWHWKHJHQHUDOVROXWLRQRI [0 D A[ XVLQJFRPSOH[HLJHQIXQFWLRQV DQGWKHQÀQG WKHJHQHUDOUHDOVROXWLRQ 'HVFULEHWKHVKDSHVRIW\SLFDOWUDMHFWRULHV

&9&3$*4&4 $ SDUWLFOHPRYLQJLQDSODQDUIRUFHÀHOGKDVDSRVLWLRQYHFWRU [ WKDWVDWLVÀHV [0 D A[ 7KH 2 # 2 PDWUL[ A KDVHLJHQYDOXHV ! " !3 DQG ZLWKFRUUHVSRQGLQJHLJHQYHFWRUV Y1 D DQG 1

!

" !1 Y2 D )LQGWKHSRVLWLRQRIWKHSDUWLFOHDWWLPH t 1 ! " !6 DVVXPLQJWKDW [.0/ D 1

$)"15&3


/HW A EHD 2 # 2 PDWUL[ZLWKHLJHQYDOXHV !3 DQG! !1 ! " " DQG !1 1 FRUUHVSRQGLQJHLJHQYHFWRUV Y1 D DQG Y2 D /HW 1 1 [.t/ EHWKHSRVLWLRQRIDSDUWLFOHDWWLPH t 6ROYHWKHLQLWLDO ! " 2 0 YDOXHSUREOHP [ D A[ [.0/ D 3

,Q([HUFLVHV² VROYHWKHLQLWLDOYDOXHSUREOHP [0 .t/ D A[.t/ IRU t % 0 ZLWK [.0/ D .3; 2/ &ODVVLI\WKHQDWXUHRIWKHRULJLQ DVDQDWWUDFWRU UHSHOOHU RUVDGGOHSRLQWRIWKHG\QDPLFDOV\VWHP GHVFULEHGE\ [0 D A[ )LQGWKHGLUHFWLRQVRIJUHDWHVWDWWUDFWLRQ DQGRUUHSXOVLRQ :KHQWKHRULJLQLVDVDGGOHSRLQW VNHWFKW\SLFDO WUDMHFWRULHV ! " ! " 2 3 !2 !5 A D A D !1 !2 1 4 ! " ! " 7 !1 1 !2 A D A D 3 3 3 !4 ,Q([HUFLVHVDQG PDNHDFKDQJHRIYDULDEOHWKDWGHFRXSOHV WKH HTXDWLRQ [0 D A[ :ULWH WKH HTXDWLRQ [.t/ D P \.t/ DQG VKRZWKHFDOFXODWLRQWKDWOHDGVWRWKHXQFRXSOHGV\VWHP \0 D D \ VSHFLI\LQJ P DQG D A DVLQ([HUFLVH

A DVLQ([HUFLVH 0

,Q ([HUFLVHV ² FRQVWUXFW WKH JHQHUDO VROXWLRQ RI [ D A[ LQYROYLQJFRPSOH[HLJHQIXQFWLRQVDQGWKHQREWDLQWKHJHQHUDOUHDO VROXWLRQ 'HVFULEHWKHVKDSHVRIW\SLFDOWUDMHFWRULHV ! " ! " !3 2 3 1 A D A D !1 !1 !2 1 ! " ! " !3 !9 !7 10 A D A D 2 3 !4 5 ! " ! " 4 !3 !2 1 A D A D 6 !2 !8 2 2 3 !8 !12 !6 1 25 >0@ A D 4 2 7 12 5

2

3 !6 !11 16 5 !4 5 >0@ A D 4 2 !4 !5 10 2 3 30 64 23 >0@ A D 4 !11 !23 !9 5 6 15 4 2 3 53 !30 !2 >0@ A D 4 90 !52 !3 5 20 !10 2

>0@ )LQGIRUPXODVIRUWKHYROWDJHV v1 DQG v2 DVIXQFWLRQVRI WLPH t IRUWKHFLUFXLWLQ([DPSOH DVVXPLQJWKDW R1 D 1=5 RKP R2 D 1=3 RKP C1 D 4 IDUDGV C2 D 3 IDUDGV DQGWKH LQLWLDOFKDUJHRQHDFKFDSDFLWRULV YROWV >0@ )LQGIRUPXODVIRUWKHYROWDJHV v1 DQG v2 IRUWKHFLUFXLWLQ ([DPSOH DVVXPLQJWKDW R1 D 1=15 RKP R2 D 1=3 RKP C1 D 9 IDUDGV C2 D 2 IDUDGV DQGWKHLQLWLDOFKDUJHRQHDFK FDSDFLWRULV YROWV >0@ )LQGIRUPXODVIRUWKHFXUUHQW iL DQGWKHYROWDJH vC IRUWKHFLUFXLWLQ([DPSOH DVVXPLQJWKDW R1 D 1 RKP R2 D :125 RKP C D :2 IDUDG L D :125 KHQU\ WKHLQLWLDO FXUUHQWLV DPS DQGWKHLQLWLDOYROWDJHLV YROWV >0@ 7KHFLUFXLWLQWKHÀJXUHLVGHVFULEHGE\WKHHTXDWLRQ " # " #" # iL0 0 1=L iL D vC0 vC !1=C !1=.RC / ZKHUH iL LVWKHFXUUHQWWKURXJKWKHLQGXFWRU L DQG vC LVWKH YROWDJHGURSDFURVVWKHFDSDFLWRU C )LQGIRUPXODVIRU iL DQG vC ZKHQ R D :5 RKP C D 2:5 IDUDGV L D :5 KHQU\ WKHLQLWLDO FXUUHQWLV DPS DQGWKHLQLWLDOYROWDJHLV YROWV R +

C L

40-65*0/4 50 13"$5*$& 130#-&.4
3 2 3 2 3 1 1 C 2i 1 ! 2i [.t/ D c1 4 !2 5e !:5t C c2 4 4i 5 e .:2C:3i/t C c3 4 !4i 5 e .:2!:3i/t 1 2 2

7KHVFDODUV c1 c2 c3 KHUHFDQEHDQ\FRPSOH[QXPEHUV 7KHÀUVWWHUPLQ [.t / LVUHDO 7ZRPRUHUHDOVROXWLRQVFDQEHSURGXFHGXVLQJWKHUHDODQGLPDJLQDU\SDUWVRIWKH

VHFRQGWHUPLQ [.t/

,WHUDWLYH(VWLPDWHVIRU(LJHQYDOXHV

2

3 1 C 2i 4 4i 5 e :2t .FRV :3t C i VLQ :3t / 2

7KHJHQHUDOUHDOVROXWLRQKDVWKHIROORZLQJIRUP ZLWK UHDO VFDODUV c1 c2 c3 2 3 2 3 2 3 1 FRV :3t ! 2 VLQ :3t VLQ :3t C 2 FRV :3t 5 e :2t C c3 4 5 e :2t !4 VLQ :3t 4 FRV :3t c1 4 !2 5e !:5t C c2 4 2 FRV :3t 2 VLQ :3t 1

$Q\VROXWLRQZLWK c2 D c3 D 0 LVDWWUDFWHGWRWKHRULJLQEHFDXVHRIWKHQHJDWLYH H[SRQHQWLDOIDFWRU 2WKHUVROXWLRQVKDYHFRPSRQHQWVWKDWJURZZLWKRXWERXQG DQG WKHWUDMHFWRULHVVSLUDORXWZDUG %HFDUHIXOQRWWRPLVWDNHWKLVSUREOHPIRURQHLQ6HFWLRQ 7KHUHWKHFRQGLWLRQ IRUDWWUDFWLRQWRZDUG ZDVWKDWDQHLJHQYDOXHEHOHVVWKDQLQPDJQLWXGH WRPDNH j!jk ! 0 +HUHWKHUHDOSDUWRIWKHHLJHQYDOXHPXVWEHQHJDWLYH WRPDNH e !t ! 0

*5&3"5*7& &45*."5&4 '03 &*(&/7"-6&4 ,QVFLHQWLÀFDSSOLFDWLRQVRIOLQHDUDOJHEUD HLJHQYDOXHVDUHVHOGRPNQRZQSUHFLVHO\ )RUWXQDWHO\ DFORVHQXPHULFDODSSUR[LPDWLRQLVXVXDOO\TXLWHVDWLVIDFWRU\ ,QIDFW VRPH DSSOLFDWLRQVUHTXLUHRQO\DURXJKDSSUR[LPDWLRQWRWKHODUJHVWHLJHQYDOXH 7KHÀUVW DOJRULWKPGHVFULEHGEHORZFDQZRUNZHOOIRUWKLVFDVH $OVR LWSURYLGHVDIRXQGDWLRQ IRUDPRUHSRZHUIXOPHWKRGWKDWFDQJLYHIDVWHVWLPDWHVIRURWKHUHLJHQYDOXHVDVZHOO

7KH3RZHU0HWKRG 7KHSRZHUPHWKRGDSSOLHVWRDQ n # n PDWUL[ A ZLWKD VWULFWO\GRPLQDQWHLJHQYDOXH !1 ZKLFKPHDQVWKDW !1 PXVWEHODUJHULQDEVROXWHYDOXHWKDQDOOWKHRWKHUHLJHQYDOXHV ,QWKLVFDVH WKHSRZHUPHWKRGSURGXFHVDVFDODUVHTXHQFHWKDWDSSURDFKHV !1 DQGD YHFWRUVHTXHQFHWKDWDSSURDFKHVDFRUUHVSRQGLQJHLJHQYHFWRU 7KHEDFNJURXQGIRUWKH PHWKRGUHVWVRQWKHHLJHQYHFWRUGHFRPSRVLWLRQXVHGDWWKHEHJLQQLQJRI6HFWLRQ $VVXPHIRUVLPSOLFLW\WKDW A LVGLDJRQDOL]DEOHDQG Rn KDVDEDVLVRIHLJHQYHFWRUV Y1 ; : : : ; Yn DUUDQJHGVRWKHLUFRUUHVSRQGLQJHLJHQYDOXHV !1 ; : : : ; !n GHFUHDVHLQVL]H ZLWK WKHVWULFWO\GRPLQDQWHLJHQYDOXHÀUVW 7KDWLV #

j!1 j > j!2 j % j!3 j % " " " % j!n j

6WULFWO\ODUJHU

$VZHVDZLQHTXDWLRQ RI6HFWLRQ LI [ LQ Rn LVZULWWHQDV [ D c1 Y1 C " " " C cn Yn WKHQ Ak [ D c1 .!1 /k Y1 C c2 .!2 /k Y2 C " " " C cn .!n /k Yn .k D 1; 2; : : :/ $VVXPH c1 ¤ 0 7KHQ GLYLGLQJE\ .!1 /k % &k % &k 1 !2 !n k A [ D c1 Y1 C c2 Y 2 C " " " C cn Yn .!1 /k !1 !1

.k D 1; 2; : : :/

)URPLQHTXDOLW\ WKHIUDFWLRQV !2 =!1 ; : : : ; !n =!1 DUHDOOOHVVWKDQLQPDJQLWXGHDQG VRWKHLUSRZHUVJRWR]HUR +HQFH

.!1 /!k Ak [ ! c1 Y1

DV k ! 1

$)"15&3


7KXV IRUODUJH k DVFDODUPXOWLSOHRI Ak [ GHWHUPLQHVDOPRVWWKHVDPH GLUHFWLRQ DVWKH HLJHQYHFWRU c1 Y1 6LQFHSRVLWLYHVFDODUPXOWLSOHVGRQRWFKDQJHWKHGLUHFWLRQRIDYHFWRU Ak [ LWVHOISRLQWVDOPRVWLQWKHVDPHGLUHFWLRQDV Y1 RU !Y1 SURYLGHG c1 ¤ 0 " ! " ! " 1:8 :8 4 !:5 Y1 D DQG [ D 7KHQ A KDV :2 1:2 1 1 HLJHQYDOXHVDQG DQGWKHHLJHQVSDFHIRU !1 D 2 LVWKHOLQHWKURXJK DQG Y1 )RU k D 0; : : : ; 8 FRPSXWH Ak [ DQGFRQVWUXFWWKHOLQHWKURXJK DQG Ak [ :KDWKDSSHQVDV k LQFUHDVHV"

&9".1-& /HW A D

!

40-65*0/ 7KHÀUVWWKUHHFDOFXODWLRQVDUH ! "! " ! " 1:8 :8 !:5 !:1 A[ D D :2 1:2 1 1:1 ! "! " ! " 1:8 :8 !:1 :7 A2 [ D A.A[/ D D :2 1:2 1:1 1:3 ! "! " ! " 1:8 :8 :7 2:3 3 2 A [ D A.A [/ D D :2 1:2 1:3 1:7 $QDORJRXVFDOFXODWLRQVFRPSOHWH7DEOH 5"#-& k k

A [

!

!:5 1

*UFSBUFT PG B 7FDUPS " !

!:1 1:1

" !

:7 1:3

" !

2:3 1:7

" !

5:5 2:5

" !

11:9 4:1

" !

24:7 7:3

" !

50:3 13:7

" !

101:5 26:5

"

7KHYHFWRUV [ A[; : : : ; A4 [ DUHVKRZQLQ)LJ 7KHRWKHUYHFWRUVDUHJURZLQJ WRRORQJWRGLVSOD\ +RZHYHU OLQHVHJPHQWVDUHGUDZQVKRZLQJWKHGLUHFWLRQVRIWKRVH YHFWRUV ,QIDFW WKHGLUHFWLRQVRIWKHYHFWRUVDUHZKDWZHUHDOO\ZDQWWRVHH QRWWKHYHF WRUVWKHPVHOYHV 7KHOLQHVVHHPWREHDSSURDFKLQJWKHOLQHUHSUHVHQWLQJWKHHLJHQVSDFH VSDQQHGE\ Y1 0RUHSUHFLVHO\ WKHDQJOHEHWZHHQWKHOLQHVXEVSDFH GHWHUPLQHGE\ Ak [ DQGWKHOLQHHLJHQVSDFH GHWHUPLQHGE\ Y1 JRHVWR]HURDV k ! 1 x2

A4 x Ax x

A3x

A2 x 1

Eigenspace v1

1

4

10

x1

'*(63& 'LUHFWLRQVGHWHUPLQHGE\ [ A[ A2 [; : : : ; A7 [

7KHYHFWRUV .!1 /!k Ak [ LQ DUHVFDOHGWRPDNHWKHPFRQYHUJHWR c1 Y1 SURYLGHG c1 ¤ 0 :HFDQQRWVFDOH Ak [ LQWKLVZD\EHFDXVHZHGRQRWNQRZ !1 %XWZHFDQVFDOH HDFK Ak [ WRPDNHLWVODUJHVWHQWU\D ,WWXUQVRXWWKDWWKHUHVXOWLQJVHTXHQFH f[k g ZLOO FRQYHUJHWRDPXOWLSOHRI Y1 ZKRVHODUJHVWHQWU\LV )LJXUH VKRZVWKHVFDOHGVHTXHQFH


IRU([DPSOH 7KHHLJHQYDOXH !1 FDQEHHVWLPDWHGIURPWKHVHTXHQFH f[k g WRR :KHQ [k LVFORVHWRDQHLJHQYHFWRUIRU !1 WKHYHFWRU A[k LVFORVHWR !1 [k ZLWKHDFKHQWU\LQ A[k DSSUR[LPDWHO\ !1 WLPHVWKHFRUUHVSRQGLQJHQWU\LQ [k %HFDXVHWKHODUJHVWHQWU\LQ [k LV WKHODUJHVWHQWU\LQ A[k LVFORVHWR !1 &DUHIXOSURRIVRIWKHVHVWDWHPHQWVDUH RPLWWHG x2 2

Ax 1 x = x0 x1

A3x A2 x x2 x3

Eigenspace

x4 Multiple of v1 1

4

x1

'*(63& 6FDOHGPXOWLSOHVRI [ A[ A2 [; : : : ; A7 [

5)& 108&3 .&5)0% '03 &45*."5*/( " 453*$5-: %0.*/"/5 &*(&/7"-6& 6HOHFWDQLQLWLDOYHFWRU [0 ZKRVHODUJHVWHQWU\LV )RU k D 0; 1; : : : ; D &RPSXWH A[k E /HW "k EHDQHQWU\LQ A[k ZKRVHDEVROXWHYDOXHLVDVODUJHDVSRVVLEOH F &RPSXWH [k C1 D .1="k /A[k )RUDOPRVWDOOFKRLFHVRI [0 WKHVHTXHQFH f"k g DSSURDFKHVWKHGRPLQDQW HLJHQYDOXH DQGWKHVHTXHQFH f[k g DSSURDFKHVDFRUUHVSRQGLQJHLJHQYHFWRU !

" ! " 6 5 0 &9".1-& $SSO\WKHSRZHUPHWKRGWR A D ZLWK [0 D 6WRS 1 2 1 ZKHQ k D 5 DQGHVWLPDWHWKHGRPLQDQWHLJHQYDOXHDQGDFRUUHVSRQGLQJHLJHQYHFWRU RI A

40-65*0/ &DOFXODWLRQVLQWKLVH[DPSOHDQGWKHQH[WZHUHPDGHZLWK0$7/$% ZKLFKFRPSXWHVZLWKGLJLWDFFXUDF\ DOWKRXJKZHVKRZRQO\DIHZVLJQLÀFDQWÀJXUHV KHUH 7REHJLQ FRPSXWH A[0 DQGLGHQWLI\WKHODUJHVWHQWU\ "0 LQ A[0 ! "! " ! " 6 5 0 5 A[0 D D ; "0 D 5 1 2 1 2 6FDOH A[0 E\ 1="0 WRJHW [1 FRPSXWH A[1 DQGLGHQWLI\WKHODUJHVWHQWU\LQ A[1 ! " ! " 1 1 5 1 [1 D A[0 D D :4 "0 5 2 ! "! " ! " 6 5 1 8 A[1 D D ; "1 D 8 1 2 :4 1:8 6FDOH A[1 E\ 1="1 WRJHW [2 FRPSXWH A[2 DQGLGHQWLI\WKHODUJHVWHQWU\LQ A[2 ! " ! " 1 1 8 1 [2 D A[1 D D :225 "1 8 1:8 ! "! " ! " 6 5 1 7:125 A[2 D D ; "2 D 7:125 1 2 :225 1:450

$)"15&3


6FDOH A[2 E\ 1="2 WRJHW [3 DQGVRRQ 7KHUHVXOWVRI0$7/$% FDOFXODWLRQVIRUWKH ÀUVWÀYHLWHUDWLRQVDUHDUUDQJHGLQ7DEOH 5"#-& k [k

A[ k "k

5IF 1PXFS .FUIPE GPS &YBNQMF

! " 0 1 ! " 5 2

!

!

1 :4

"

8 1:8

"

!

!

1 :225

"

7:125 1:450

"

!

!

1 :2035

"

7:0175 1:4070

"

!

!

1 :2005

"

7:0025 1:4010

"

!

!

1 :20007

"

7:00036 1:40014

"

7KHHYLGHQFHIURP7DEOH VWURQJO\VXJJHVWVWKDW f[k g DSSURDFKHV .1; :2/ DQG f"k g DSSURDFKHV ,IVR WKHQ .1; :2/ LVDQHLJHQYHFWRUDQGLVWKHGRPLQDQWHLJHQYDOXH 7KLV LVHDVLO\YHULÀHGE\FRPSXWLQJ ! " ! 1 6 A D :2 1

5 2

"!

1 :2

"

D

!

7 1:4

"

! " 1 D7 :2

7KHVHTXHQFH f"k g LQ([DPSOH FRQYHUJHGTXLFNO\WR !1 D 7 EHFDXVHWKHVHFRQG HLJHQYDOXHRI A ZDVPXFKVPDOOHU ,QIDFW !2 D 1 ,QJHQHUDO WKHUDWHRIFRQYHUJHQFH GHSHQGVRQWKHUDWLR j!2 =!1 j EHFDXVHWKHYHFWRU c2 .!2 =!1 /k Y2 LQHTXDWLRQ LVWKHPDLQ VRXUFHRIHUURUZKHQXVLQJDVFDOHGYHUVLRQRI Ak [ DVDQHVWLPDWHRI c1 Y1 7KHRWKHU IUDFWLRQV !j =!1 DUHOLNHO\WREHVPDOOHU ,I j!2 =!1 j LVFORVHWR WKHQ f"k g DQG f[k g FDQFRQYHUJHYHU\VORZO\ DQGRWKHUDSSUR[LPDWLRQPHWKRGVPD\EHSUHIHUUHG :LWKWKHSRZHUPHWKRG WKHUHLVDVOLJKWFKDQFHWKDWWKHFKRVHQLQLWLDOYHFWRU [ ZLOOKDYHQRFRPSRQHQWLQWKH Y1 GLUHFWLRQZKHQ c1 D 0 %XWFRPSXWHUURXQGLQJ HUURUVGXULQJWKHFDOFXODWLRQVRIWKH [k DUHOLNHO\WRFUHDWHDYHFWRUZLWKDWOHDVWDVPDOO FRPSRQHQWLQWKHGLUHFWLRQRI Y1 ,IWKDWRFFXUV WKH [k ZLOOVWDUWWRFRQYHUJHWRDPXOWLSOH RI Y1

7KH,QYHUVH3RZHU0HWKRG 7KLVPHWKRGSURYLGHVDQDSSUR[LPDWLRQIRU DQ\ HLJHQYDOXH SURYLGHGDJRRGLQLWLDO HVWLPDWH ˛ RIWKHHLJHQYDOXH ! LVNQRZQ ,QWKLVFDVH ZHOHW B D .A ! ˛I /!1 DQGDSSO\ WKHSRZHUPHWKRGWR B ,WFDQEHVKRZQWKDWLIWKHHLJHQYDOXHVRI A DUH !1 ; : : : ; !n WKHQ WKHHLJHQYDOXHVRI B DUH

1 ; !1 ! ˛

1 ; !2 ! ˛

:::;

1 !n ! ˛

DQGWKHFRUUHVSRQGLQJHLJHQYHFWRUVDUHWKHVDPHDVWKRVHIRU A 6HH([HUFLVHV DQG 6XSSRVH IRUH[DPSOH WKDW ˛ LVFORVHUWR !2 WKDQWRWKHRWKHUHLJHQYDOXHVRI A 7KHQ 1=.!2 ! ˛/ ZLOOEHDVWULFWO\GRPLQDQWHLJHQYDOXHRI B ,I ˛ LVUHDOO\FORVHWR !2 WKHQ 1=.!2 ! ˛/ LV PXFK ODUJHUWKDQWKHRWKHUHLJHQYDOXHVRI B DQGWKHLQYHUVHSRZHU PHWKRGSURGXFHVDYHU\UDSLGDSSUR[LPDWLRQWR !2 IRUDOPRVWDOOFKRLFHVRI [0 7KH IROORZLQJDOJRULWKPJLYHVWKHGHWDLOV


5)& */7&34& 108&3 .&5)0% '03 &45*."5*/( "/ &*(&/7"-6& ! 0' " 6HOHFWDQLQLWLDOHVWLPDWH ˛ VXIÀFLHQWO\FORVHWR ! 6HOHFWDQLQLWLDOYHFWRU [0 ZKRVHODUJHVWHQWU\LV )RU k D 0; 1; : : : ; D 6ROYH .A ! ˛I /\k D [k IRU \k E /HW "k EHDQHQWU\LQ \k ZKRVHDEVROXWHYDOXHLVDVODUJHDVSRVVLEOH F &RPSXWH &k D ˛ C .1="k / G &RPSXWH [k C1 D .1="k /\k )RUDOPRVWDOOFKRLFHVRI [0 WKHVHTXHQFH f&k g DSSURDFKHVWKHHLJHQYDOXH ! RI A DQGWKHVHTXHQFH f[k g DSSURDFKHVDFRUUHVSRQGLQJHLJHQYHFWRU 1RWLFHWKDW B RUUDWKHU .A ! ˛I /!1 GRHVQRWDSSHDULQWKHDOJRULWKP ,QVWHDGRI FRPSXWLQJ .A ! ˛I /!1 [k WRJHWWKHQH[WYHFWRULQWKHVHTXHQFH LWLVEHWWHUWR VROYH WKHHTXDWLRQ .A ! ˛I /\k D [k IRU \k DQGWKHQVFDOH \k WRSURGXFH [k C1 / 6LQFHWKLV HTXDWLRQIRU \k PXVWEHVROYHGIRUHDFK k DQ/8 IDFWRUL]DWLRQRI A ! ˛I ZLOOVSHHGXS WKHSURFHVV

&9".1-& ,WLVQRWXQFRPPRQLQVRPHDSSOLFDWLRQVWRQHHGWRNQRZWKHVPDOOHVW HLJHQYDOXHRIDPDWUL[ A DQGWRKDYHDWKDQGURXJKHVWLPDWHVRIWKHHLJHQYDOXHV 6XSSRVH DQGDUHHVWLPDWHVIRUWKHHLJHQYDOXHVRIWKHPDWUL[ A EHORZ )LQG WKHVPDOOHVWHLJHQYDOXH DFFXUDWHWRVL[GHFLPDOSODFHV 2 3 10 !8 !4 A D 4 !8 13 4 5 !4 5 4

40-65*0/ 7KHWZRVPDOOHVWHLJHQYDOXHVVHHPFORVHWRJHWKHU VRZHXVHWKHLQYHUVH SRZHUPHWKRGIRU A ! 1:9I 5HVXOWVRID0$7/$% FDOFXODWLRQDUHVKRZQLQ7DEOH +HUH [0 ZDVFKRVHQDUELWUDULO\ \k D .A ! 1:9I /!1 [k "k LVWKHODUJHVWHQWU\LQ \k &k D 1:9 C 1="k DQG [k C1 D .1="k /\k $VLWWXUQVRXW WKHLQLWLDOHLJHQYDOXHHVWLPDWH ZDVIDLUO\JRRG DQGWKHLQYHUVHSRZHUVHTXHQFHFRQYHUJHGTXLFNO\ 7KHVPDOOHVW HLJHQYDOXHLVH[DFWO\ 5"#-& k [k

\k

"k &k

5IF *OWFSTF 1PXFS .FUIPE 2

3

2

3

2

3

2

2 3 1 415 1 2 3 4:45 4 :50 5 7:76

:5736 4 :0646 5 1 2 3 5:0131 4 :0442 5 9:9197

:5054 4 :0045 5 1 2 3 5:0012 4 :0031 5 9:9949

:5004 4 :0003 5 1 2 3 5:0001 4 :0002 5 9:9996

3 :50003 4 :00002 5 1 2 3 5:000006 4 :000015 5 9:999975

,IDQHVWLPDWHIRUWKHVPDOOHVWHLJHQYDOXHRIDPDWUL[LVQRWDYDLODEOH RQHFDQVLPSO\ WDNH ˛ D 0 LQWKHLQYHUVHSRZHUPHWKRG 7KLVFKRLFHRI ˛ ZRUNVUHDVRQDEO\ZHOOLIWKH VPDOOHVWHLJHQYDOXHLVPXFKFORVHUWR]HURWKDQWRWKHRWKHUHLJHQYDOXHV

$)"15&3


7KHWZRDOJRULWKPVSUHVHQWHGLQWKLVVHFWLRQDUHSUDFWLFDOWRROVIRUPDQ\VLPSOH VLWXDWLRQV DQGWKH\SURYLGHDQLQWURGXFWLRQWRWKHSUREOHPRIHLJHQYDOXHHVWLPDWLRQ $ PRUHUREXVWDQGZLGHO\XVHGLWHUDWLYHPHWKRGLVWKH45 DOJRULWKP )RULQVWDQFH LWLV WKHKHDUWRIWKH0$7/$% FRPPDQG HLJ$ ZKLFKUDSLGO\FRPSXWHVHLJHQYDOXHVDQG HLJHQYHFWRUVRI A $ EULHIGHVFULSWLRQRIWKH45 DOJRULWKPZDVJLYHQLQWKHH[HUFLVHV IRU6HFWLRQ )XUWKHUGHWDLOVDUHSUHVHQWHGLQPRVWPRGHUQQXPHULFDODQDO\VLVWH[WV

13"$5*$& 130#-&. +RZFDQ\RXWHOOLIDJLYHQYHFWRU [ LVDJRRGDSSUR[LPDWLRQWRDQHLJHQYHFWRURID PDWUL[ A" ,ILWLV KRZZRXOG\RXHVWLPDWHWKHFRUUHVSRQGLQJHLJHQYDOXH" ([SHULPHQW ZLWK 2 3 2 3 5 8 4 1:0 A D 4 8 3 !1 5 DQG [ D 4 !4:3 5 4 !1 2 8:1

&9&3$*4&4 ,Q([HUFLVHV² WKHPDWUL[ A LVIROORZHGE\DVHTXHQFH f[k g SURGXFHGE\WKHSRZHUPHWKRG 8VHWKHVHGDWDWRHVWLPDWHWKH ODUJHVWHLJHQYDOXHRI A DQGJLYHDFRUUHVSRQGLQJHLJHQYHFWRU ! " 4 3 A D 1 2 ! " ! " ! " ! " ! " 1 1 1 1 1 ; ; ; ; 0 :25 :3158 :3298 :3326 ! " 1:8 !:8 A D !3:2 4:2 ! " ! " ! " ! " ! " 1 !:5625 !:3021 !:2601 !:2520 ; ; ; ; 0 1 1 1 1 !

" :5 :2 A D :4 :7 ! " ! " ! " ! " ! " 1 1 :6875 :5577 :5188 ; ; ; ; 0 :8 1 1 1 ! " 4:1 !6 A D 3 !4:4 ! " ! " ! " ! " ! " 1 1 1 1 1 ; ; ; ; 1 :7368 :7541 :7490 :7502 ! " ! " 15 16 1 /HW A D 7KHYHFWRUV [; : : : ; A5 [ DUH !20 !21 1 ! " ! " ! " ! " ! " 31 !191 991 !4991 24991 ; ; ; ; : !41 241 !1241 6241 !31241 )LQGDYHFWRUZLWKDLQWKHVHFRQGHQWU\WKDWLVFORVHWR DQHLJHQYHFWRURI A 8VHIRXUGHFLPDOSODFHV &KHFN\RXU HVWLPDWH DQGJLYHDQHVWLPDWHIRUWKHGRPLQDQWHLJHQYDOXH RI A

" !2 !3 5HSHDW([HUFLVH XVLQJWKHIROORZ 6 7 LQJVHTXHQFH [ A[; : : : ; A5 [ ! " ! " ! " ! " ! " ! " 1 !5 !29 !125 !509 !2045 ; ; ; ; ; 1 13 61 253 1021 4093

/HW A D

!

>0@ ([HUFLVHV²UHTXLUH0$7/$% RURWKHUFRPSXWDWLRQDODLG ,Q([HUFLVHV DQG XVHWKHSRZHUPHWKRGZLWKWKH [0 JLYHQ /LVW f[k g DQG f"k g IRU k D 1; : : : ; 5 ,Q([HUFLVHV DQG OLVW "5 DQG "6 ! " ! " 6 7 1 [0 D A D 8 5 0 ! " ! " 2 1 1 A D [0 D 4 5 0 2 3 2 3 8 0 12 1 1 5 [0 D 4 0 5 A D 4 1 !2 0 3 0 0 2 3 2 3 1 2 !2 1 1 9 5 [0 D 4 0 5 A D 4 1 0 1 9 0

$QRWKHUHVWLPDWHFDQEHPDGHIRUDQHLJHQYDOXHZKHQDQDSSUR[ LPDWHHLJHQYHFWRULVDYDLODEOH 2EVHUYHWKDWLI A[ D ![ WKHQ [TA[ D [T .![/ D !.[T [/ DQGWKH 5D\OHLJKTXRWLHQW

R.[/ D

[TA[ [T [

HTXDOV ! ,I [ LVFORVHWRDQHLJHQYHFWRUIRU ! WKHQWKLVTXRWLHQW LVFORVHWR ! :KHQ A LVDV\PPHWULFPDWUL[ .AT D A/ WKH 5D\OHLJKTXRWLHQW R.[k / D .[Tk A[k /=.[Tk [k / ZLOOKDYHURXJKO\ WZLFHDVPDQ\GLJLWVRIDFFXUDF\DVWKHVFDOLQJIDFWRU "k LQWKH SRZHUPHWKRG 9HULI\WKLVLQFUHDVHGDFFXUDF\LQ([HUFLVHV DQG E\FRPSXWLQJ "k DQG R.[k / IRU k D 1; : : : ; 4

" ! " 2 1 [0 D 2 0 ! " ! " !3 2 1 A D [0 D 2 0 0

A D

!

5 2


>0@ /HW A EH DV LQ ([HUFLVH 8VH WKH LQYHUVH SRZHU PHWKRGZLWK [0 D .1; 0; 0/ WRHVWLPDWHWKHHLJHQYDOXHRI A QHDU ˛ D !1:4 ZLWKDQDFFXUDF\WRIRXUGHFLPDOSODFHV >0@ ,Q([HUFLVHVDQG ÀQGD WKHODUJHVWHLJHQYDOXHDQGE WKHHLJHQYDOXHFORVHVWWR]HUR ,QHDFKFDVH VHW [0 D .1; 0; 0; 0/ DQGFDUU\RXWDSSUR[LPDWLRQVXQWLOWKHDSSUR[LPDWLQJVHTXHQFH VHHPVDFFXUDWHWRIRXUGHFLPDOSODFHV ,QFOXGHWKHDSSUR[LPDWH HLJHQYHFWRU 2 3 10 7 8 7 6 7 5 6 57 7 A D 6 4 8 6 10 95 7 5 9 10 2 3 1 2 3 2 6 2 12 13 11 7 7 A D 6 4 !2 3 0 25 4 5 7 2

([HUFLVHVDQGDSSO\WRD 3 # 3 PDWUL[ A ZKRVHHLJHQYDOXHV DUHHVWLPDWHGWREH !4 DQG ,IWKHHLJHQYDOXHVFORVHWRDQG !4 DUHNQRZQ WR KDYH GLIIHUHQWDEVROXWHYDOXHV ZLOOWKHSRZHUPHWKRGZRUN" ,V LWOLNHO\WREHXVHIXO" 6XSSRVHWKHHLJHQYDOXHVFORVHWRDQG !4 DUHNQRZQWRKDYH H[DFWO\WKHVDPHDEVROXWHYDOXH 'HVFULEHKRZRQHPLJKW REWDLQDVHTXHQFHWKDWHVWLPDWHVWKHHLJHQYDOXHFORVHWR 6XSSRVH A[ D ![ ZLWK [ ¤ /HW ˛ EHDVFDODUGLIIHUHQW IURPWKHHLJHQYDOXHVRI A DQGOHW B D .A ! ˛I /!1 6XE WUDFW ˛ [ IURPERWKVLGHVRIWKHHTXDWLRQ A[ D ![ DQGXVH DOJHEUDWRVKRZWKDW 1=.! ! ˛/ LVDQHLJHQYDOXHRI B ZLWK [ DFRUUHVSRQGLQJHLJHQYHFWRU

$ FRPPRQPLVFRQFHSWLRQLVWKDWLI A KDVDVWULFWO\GRPLQDQW HLJHQYDOXH WKHQ IRUDQ\VXIÀFLHQWO\ODUJHYDOXHRI k WKH YHFWRU Ak [ LVDSSUR[LPDWHO\HTXDOWRDQHLJHQYHFWRURI A )RUWKHWKUHHPDWULFHVEHORZ VWXG\ZKDWKDSSHQVWR Ak [ ZKHQ [ D .:5; :5/ DQGWU\WRGUDZJHQHUDOFRQFOXVLRQVIRU D 2 # 2 PDWUL[ ! " ! " :8 0 1 0 D A D E A D 0 :2 0 :8 ! " 8 0 F A D 0 2

6XSSRVH " LVDQHLJHQYDOXHRIWKH B LQ([HUFLVH DQGWKDW [ LVDFRUUHVSRQGLQJHLJHQYHFWRU VRWKDW .A ! ˛I /!1 [ D "[ 8VHWKLVHTXDWLRQWRÀQGDQHLJHQYDOXHRI A LQWHUPVRI " DQG ˛ >1RWH " ¤ 0 EHFDXVH B LVLQYHUWLEOH@ >0@ 8VHWKHLQYHUVHSRZHUPHWKRGWRHVWLPDWHWKHPLGGOH HLJHQYDOXHRIWKH A LQ([DPSOH ZLWKDFFXUDF\WRIRXU GHFLPDOSODFHV 6HW [0 D .1; 0; 0/

40-65*0/ 50 13"$5*$& 130#-&. )RUWKHJLYHQ A DQG [

2

32 3 2 3 5 8 4 1:00 3:00 A[ D 4 8 3 !1 54 !4:30 5 D 4 !13:00 5 4 !1 2 8:10 24:50

,I A[ LVQHDUO\DPXOWLSOHRI [ WKHQWKHUDWLRVRIFRUUHVSRQGLQJHQWULHVLQWKHWZRYHFWRUV VKRXOGEHQHDUO\FRQVWDQW 6RFRPSXWH

f HQWU\LQ A[ g ) f HQWU\LQ [ g D f UDWLR g 3:00 1:00 3:000 !13:00 !4:30 3:023 24:50 8:10 3:025

WEB

(DFKHQWU\LQ A[ LVDERXWWLPHVWKHFRUUHVSRQGLQJHQWU\LQ [ VR [ LVFORVHWRDQ HLJHQYHFWRU $Q\RIWKHUDWLRVDERYHLVDQHVWLPDWHIRUWKHHLJHQYDOXH 7RÀYHGHFLPDO SODFHV WKHHLJHQYDOXHLV

$)"15&3


$)"15&3 4611-&.&/5"3: &9&3$*4&4 7KURXJKRXWWKHVHVXSSOHPHQWDU\H[HUFLVHV A DQG B UHSUHVHQW VTXDUHPDWULFHVRIDSSURSULDWHVL]HV 0DUNHDFKVWDWHPHQWDV7UXHRU)DOVH -XVWLI\HDFKDQVZHU D ,I A LVLQYHUWLEOHDQGLVDQHLJHQYDOXHIRU A WKHQLV DOVRDQHLJHQYDOXHRI A!1 E ,I A LVURZHTXLYDOHQWWRWKHLGHQWLW\PDWUL[ I WKHQ A LV GLDJRQDOL]DEOH F

,I A FRQWDLQVDURZRUFROXPQRI]HURV WKHQLVDQ HLJHQYDOXHRI A

G (DFKHLJHQYDOXHRI A LVDOVRDQHLJHQYDOXHRI A2 H

(DFKHLJHQYHFWRURI A LVDOVRDQHLJHQYHFWRURI A2

I

(DFKHLJHQYHFWRURIDQLQYHUWLEOHPDWUL[ A LVDOVRDQ HLJHQYHFWRURI A!1

J (LJHQYDOXHVPXVWEHQRQ]HURVFDODUV K (LJHQYHFWRUVPXVWEHQRQ]HURYHFWRUV L

7ZRHLJHQYHFWRUVFRUUHVSRQGLQJWRWKHVDPHHLJHQYDOXH DUHDOZD\VOLQHDUO\GHSHQGHQW

M

6LPLODUPDWULFHVDOZD\VKDYHH[DFWO\WKHVDPHHLJHQ YDOXHV

N 6LPLODUPDWULFHVDOZD\VKDYHH[DFWO\WKHVDPHHLJHQ YHFWRUV O

7KHVXPRIWZRHLJHQYHFWRUVRIDPDWUL[ A LVDOVRDQ HLJHQYHFWRURI A

P 7KH HLJHQYDOXHV RI DQ XSSHU WULDQJXODU PDWUL[ A DUH H[DFWO\WKHQRQ]HURHQWULHVRQWKHGLDJRQDORI A Q 7KH PDWULFHV A DQG AT KDYH WKH VDPH HLJHQYDOXHV FRXQWLQJPXOWLSOLFLWLHV R ,ID 5 # 5 PDWUL[ A KDVIHZHUWKDQGLVWLQFWHLJHQYDOXHV WKHQ A LVQRWGLDJRQDOL]DEOH S 7KHUHH[LVWVD 2 # 2 PDWUL[WKDWKDVQRHLJHQYHFWRUVLQ R2 T ,I A LVGLDJRQDOL]DEOH WKHQWKHFROXPQVRI A DUHOLQHDUO\ LQGHSHQGHQW U

$ QRQ]HUR YHFWRU FDQQRW FRUUHVSRQG WR WZR GLIIHUHQW HLJHQYDOXHVRI A

V

$ VTXDUH PDWUL[ A LVLQYHUWLEOHLIDQGRQO\LIWKHUHLVD FRRUGLQDWHV\VWHPLQZKLFKWKHWUDQVIRUPDWLRQ [ 7! A[ LVUHSUHVHQWHGE\DGLDJRQDOPDWUL[

W

,I HDFK YHFWRU Hj LQ WKH VWDQGDUG EDVLV IRU Rn LV DQ HLJHQYHFWRURI A WKHQ A LVDGLDJRQDOPDWUL[

X ,I A LVVLPLODUWRDGLDJRQDOL]DEOHPDWUL[ B WKHQ A LV DOVRGLDJRQDOL]DEOH Y

,I A DQG B DUHLQYHUWLEOH n # n PDWULFHV WKHQ AB LV VLPLODUWR BA

Z $Q n # n PDWUL[ZLWK n OLQHDUO\LQGHSHQGHQWHLJHQYHF WRUVLVLQYHUWLEOH

[ ,I A LVDQ n # n GLDJRQDOL]DEOHPDWUL[ WKHQHDFKYHFWRU LQ Rn FDQEHZULWWHQDVDOLQHDUFRPELQDWLRQRIHLJHQYHF WRUVRI A 6KRZWKDWLI [ LVDQHLJHQYHFWRURIWKHPDWUL[SURGXFW AB DQG B [ ¤ WKHQ B [ LVDQHLJHQYHFWRURI BA 6XSSRVH [ LVDQHLJHQYHFWRURI A FRUUHVSRQGLQJWRDQHLJHQ YDOXH ! D 6KRZWKDW [ LVDQHLJHQYHFWRURI 5I ! A :KDWLVWKH FRUUHVSRQGLQJHLJHQYDOXH" E 6KRZWKDW [ LVDQHLJHQYHFWRURI 5I ! 3A C A2 :KDWLV WKHFRUUHVSRQGLQJHLJHQYDOXH" 8VHPDWKHPDWLFDOLQGXFWLRQWRVKRZWKDWLI ! LVDQHLJHQYDOXH RIDQ n # n PDWUL[ A ZLWK [ DFRUUHVSRQGLQJHLJHQYHFWRU WKHQ IRUHDFKSRVLWLYHLQWHJHU m !m LVDQHLJHQYDOXHRI Am ZLWK [ DFRUUHVSRQGLQJHLJHQYHFWRU ,I p.t/ D c0 C c1 t C c2 t 2 C " " " C cn t n GHÀQH p.A/ WREH WKHPDWUL[IRUPHGE\UHSODFLQJHDFKSRZHURI t LQ p.t / E\ WKHFRUUHVSRQGLQJSRZHURI A ZLWK A0 D I 7KDWLV

p.A/ D c0 I C c1 A C c2 A2 C " " " C cn An 6KRZWKDWLI ! LVDQHLJHQYDOXHRI A WKHQRQHHLJHQYDOXHRI p.A/ LV p.!/ ! " 2 0 6XSSRVH A D 3'3!1 ZKHUH P LV 2 # 2 DQG D D 0 7 D /HW B D 5I ! 3A C A2 6KRZWKDW B LVGLDJRQDOL]DEOH E\ÀQGLQJDVXLWDEOHIDFWRUL]DWLRQRI B

E *LYHQ p.t/ DQG p.A/ DVLQ([HUFLVH VKRZWKDW p.A/ LV GLDJRQDOL]DEOH 6XSSRVH A LVGLDJRQDOL]DEOHDQG p.t/ LVWKHFKDUDFWHULVWLF SRO\QRPLDORI A 'HÀQH p.A/ DVLQ([HUFLVH DQGVKRZ WKDW p.A/ LVWKH]HURPDWUL[ 7KLVIDFW ZKLFKLVDOVRWUXHIRU DQ\ VTXDUHPDWUL[ LVFDOOHGWKH &D\OH\²+DPLOWRQWKHRUHP D /HW A EHDGLDJRQDOL]DEOH n # n PDWUL[ 6KRZWKDWLIWKH PXOWLSOLFLW\RIDQHLJHQYDOXH ! LV n WKHQ A D !I ! " 3 1 E 8VHSDUWD WRVKRZWKDWWKHPDWUL[ A D LVQRW 0 3 GLDJRQDOL]DEOH 6KRZWKDW I ! A LVLQYHUWLEOHZKHQDOOWKHHLJHQYDOXHVRI A DUHOHVVWKDQLQPDJQLWXGH >+LQW :KDWZRXOGEHWUXHLI I ! A ZHUHQRWLQYHUWLEOH"@ 6KRZWKDWLI A LVGLDJRQDOL]DEOH ZLWKDOOHLJHQYDOXHVOHVV WKDQLQPDJQLWXGH WKHQ Ak WHQGVWRWKH]HURPDWUL[DV k ! 1 >+LQW &RQVLGHU Ak [ ZKHUH [ UHSUHVHQWVDQ\RQH RIWKHFROXPQVRI I @ /HW X EHDQHLJHQYHFWRURI A FRUUHVSRQGLQJWRDQHLJHQYDOXH ! DQGOHW H EHWKHOLQHLQ Rn WKURXJK X DQGWKHRULJLQ D ([SODLQZK\ H LVLQYDULDQWXQGHU A LQWKHVHQVHWKDW A[ LVLQ H ZKHQHYHU [ LVLQ H


E /HW K EHDRQHGLPHQVLRQDOVXEVSDFHRI Rn WKDWLVLQYDUL DQWXQGHU A ([SODLQZK\ K FRQWDLQVDQHLJHQYHFWRURI A ! " A X 8VHIRUPXOD IRUWKHGHWHUPLQDQW /HW G D 0 B LQ6HFWLRQWRH[SODLQ ZK\ GHW G D .GHW A/.GHW B/ )URP WKLV GHGXFHWKDWWKHFKDUDFWHULVWLFSRO\QRPLDORI G LVWKH SURGXFWRIWKHFKDUDFWHULVWLFSRO\QRPLDOVRI A DQG B

8VH([HUFLVHWRÀQGWKHHLJHQYDOXHVRIWKHPDWULFHVLQ([HU FLVHV DQG 2 3 3 !2 8 5 !2 5 A D 4 0 0 !4 3 2 3 1 5 !6 !7 62 4 5 27 7 A D 6 40 0 !7 !4 5 0 0 3 1 /HW J EH WKH n # n PDWUL[ RI DOO ·V DQG FRQVLGHU A D .a ! b/I C bJ WKDWLV 2 3 a b b """ b 6b a b """ b7 6 7 6b b a """ b7 AD6 : :: :: :: 7 :: 6 : 7 : 4 : : : :5

b

b

b

"""

a

8VHWKHUHVXOWVRI([HUFLVHLQWKH6XSSOHPHQWDU\([HUFLVHV IRU&KDSWHUWRVKRZWKDWWKHHLJHQYDOXHVRI A DUH a ! b DQG a C .n ! 1/b :KDWDUHWKHPXOWLSOLFLWLHVRIWKHVHHLJHQYDO XHV"

$SSO\WKHUHVXOWRI([HUFLVHWRÀQGWKHHLJHQYDOXHVRIWKH 2 3 7 3 3 3 3 2 3 63 1 2 2 7 3 3 37 6 7 3 3 7 3 37 1 2 5 DQG 6 PDWULFHV 4 2 6 7 43 2 2 1 3 3 7 35 3 3 3 3 7 ! " a11 a12 5HFDOOIURP([HUFLVHLQ6HFWLRQ /HW A D a21 a22 WKDW WU A WKHWUDFHRI A LVWKHVXPRIWKHGLDJRQDOHQWULHV LQ A 6KRZWKDWWKHFKDUDFWHULVWLFSRO\QRPLDORI A LV

!2 ! .WU A/! C GHW A 7KHQVKRZWKDWWKHHLJHQYDOXHVRID 2 # 2 PDWUL[ A DUHERWK % & WU A 2 UHDOLIDQGRQO\LI GHW A & 2 ! " :4 !:3 ([SODLQ ZK\ Ak DSSURDFKHV /HW A D :4 1:2 ! " !:5 !:75 DV k ! 1 1:0 1:50 ([HUFLVHV²FRQFHUQWKHSRO\QRPLDO

p.t/ D a0 C a1 t C " " " C an!1 t n!1 C t n

DQGDQ n # n PDWUL[ Cp FDOOHGWKH FRPSDQLRQPDWUL[ RI p 2 3 0 1 0 """ 0 6 0 0 1 0 7 6 : :: 7 6 : 7 Cp D 6 : : 7 6 7 4 0 0 0 1 5 !a1 !a2 """ !an!1 !a0

:ULWHWKHFRPSDQLRQPDWUL[ Cp IRU p.t/ D 6 ! 5t C t 2 DQG WKHQÀQGWKHFKDUDFWHULVWLFSRO\QRPLDORI Cp /HW p.t/ D .t ! 2/.t ! 3/.t ! 4/ D !24 C 26t ! 9t 2 C t 3 :ULWHWKHFRPSDQLRQPDWUL[IRU p.t/ DQGXVHWHFKQLTXHV IURP&KDSWHUWRÀQGLWVFKDUDFWHULVWLFSRO\QRPLDO 8VHPDWKHPDWLFDOLQGXFWLRQWRSURYHWKDWIRU n % 2

GHW.Cp ! !I / D .!1/n .a0 C a1 ! C " " " C an!1 !n!1 C !n /

D .!1/n p.!/

>+LQW ([SDQGLQJE\FRIDFWRUVGRZQWKHÀUVWFROXPQ VKRZ WKDW GHW .Cp ! !I / KDVWKHIRUP .!!/B C .!1/n a0 ZKHUH B LVDFHUWDLQSRO\QRPLDOE\WKHLQGXFWLRQDVVXPSWLRQ @ /HW p.t/ D a0 C a1 t C a2 t 2 C t 3 DQGOHW ! EHD]HURRI p D :ULWHWKHFRPSDQLRQPDWUL[IRU p E ([SODLQ ZK\ !3 D !a0 ! a1 ! ! a2 !2 DQG VKRZ WKDW .1; !; !2 / LVDQHLJHQYHFWRURIWKHFRPSDQLRQPDWUL[IRU p /HW p EHWKHSRO\QRPLDOLQ([HUFLVH DQGVXSSRVHWKH HTXDWLRQ p.t/ D 0 KDVGLVWLQFWURRWV !1 !2 !3 /HW V EH WKH9DQGHUPRQGHPDWUL[ 2 3 1 1 1 6 !2 !3 7 V D 4 !1 5 !21 !22 !23

7KHWUDQVSRVHRI V ZDVFRQVLGHUHGLQ6XSSOHPHQWDU\([HU FLVH LQ&KDSWHU 8VH([HUFLVH DQGDWKHRUHPIURP WKLVFKDSWHUWRGHGXFHWKDW V LVLQYHUWLEOHEXWGRQRWFRPSXWH V !1 / 7KHQH[SODLQZK\ V !1 Cp V LVDGLDJRQDOPDWUL[

>0@ 7KH 0$7/$% FRPPDQG URRWVS FRPSXWHV WKH URRWVRIWKHSRO\QRPLDOHTXDWLRQ p.t/ D 0 5HDGD0$7/$% PDQXDO DQGWKHQGHVFULEHWKHEDVLFLGHDEHKLQGWKHDOJRULWKP IRUWKH URRWV FRPPDQG >0@ 8VHDPDWUL[SURJUDPWRGLDJRQDOL]H 2 3 !3 !2 0 7 !1 5 A D 4 14 !6 !3 1

LISRVVLEOH 8VHWKHHLJHQYDOXHFRPPDQGWRFUHDWHWKHGLDJ RQDOPDWUL[ D ,IWKHSURJUDPKDVDFRPPDQGWKDWSURGXFHV HLJHQYHFWRUV XVHLWWRFUHDWHDQLQYHUWLEOHPDWUL[ P 7KHQ FRPSXWH $3 ! 3' DQG 3'3!1 'LVFXVV\RXUUHVXOWV 2 3 !8 5 !2 0 6 !5 2 1 !2 7 7 >0@ 5HSHDW([HUFLVHIRU A D 6 4 10 !8 6 !3 5 3 !2 1 0


2UWKRJRQDOLW\DQG /HDVW6TXDUHV

*/530%6$503: &9".1-&

7KH1RUWK$PHULFDQ'DWXP DQG*36 1DYLJDWLRQ ,PDJLQHVWDUWLQJDPDVVLYHSURMHFWWKDW\RXHVWLPDWHZLOO WDNHWHQ\HDUVDQGUHTXLUHWKHHIIRUWVRIVFRUHVRISHRSOH WRFRQVWUXFWDQGVROYHDE\V\VWHP RIOLQHDUHTXDWLRQV 7KDWLVH[DFWO\ZKDWWKH1DWLRQDO *HRGHWLF6XUYH\GLGLQ ZKHQLWVHWRXWWRXSGDWH WKH1RUWK$PHULFDQ'DWXP1$' ³DQHWZRUNRI SUHFLVHO\ORFDWHGUHIHUHQFHSRLQWVWKDWVSDQWKHHQWLUH1RUWK $PHULFDQFRQWLQHQW WRJHWKHUZLWK*UHHQODQG +DZDLL WKH 9LUJLQ,VODQGV 3XHUWR5LFR DQGRWKHU&DULEEHDQLVODQGV 7KHUHFRUGHGODWLWXGHVDQGORQJLWXGHVLQWKH1$' PXVWEHGHWHUPLQHGWRZLWKLQDIHZFHQWLPHWHUVEHFDXVH WKH\IRUPWKHEDVLVIRUDOOVXUYH\V PDSV OHJDOSURSHUW\ ERXQGDULHV DQGOD\RXWVRIFLYLOHQJLQHHULQJSURMHFWV VXFKDVKLJKZD\VDQGSXEOLFXWLOLW\OLQHV +RZHYHU PRUHWKDQQHZSRLQWVKDGEHHQDGGHGWRWKH GDWXPVLQFHWKHODVWDGMXVWPHQWLQ DQGHUURUVKDG JUDGXDOO\DFFXPXODWHGRYHUWKH\HDUV GXHWRLPSUHFLVH PHDVXUHPHQWV DQG VKLIWV LQ WKH HDUWK·V FUXVW 'DWD JDWKHULQJIRUWKH1$' UHDGMXVWPHQWZDVFRPSOHWHGLQ 7KHV\VWHPRIHTXDWLRQVIRUWKH1$' KDGQRVROXWLRQ LQWKHRUGLQDU\VHQVH EXWUDWKHUKDGD OHDVWVTXDUHV VROXWLRQ ZKLFKDVVLJQHGODWLWXGHVDQGORQJLWXGHVWRWKH UHIHUHQFHSRLQWVLQDZD\WKDWFRUUHVSRQGHGEHVWWRWKH PLOOLRQREVHUYDWLRQV 7KHOHDVWVTXDUHVVROXWLRQZDVIRXQG LQE\VROYLQJDUHODWHGV\VWHPRIVRFDOOHG

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ÀQGLQJDXVHU·VSRVLWLRQRQWKHJURXQG UHODWLYHWRHVWDEOLVKHGPDSV :KHQDFDUGULYHURUD PRXQWDLQFOLPEHU WXUQVRQD*36 UHFHLYHU WKHUHFHLYHU PHDVXUHVWKHUHODWLYHDUULYDOWLPHVRIVLJQDOVIURPDW OHDVWWKUHHVDWHOOLWHV 7KLVLQIRUPDWLRQ WRJHWKHUZLWKWKH WUDQVPLWWHGGDWDDERXWWKHVDWHOOLWHV·ORFDWLRQVDQGPHVVDJH WLPHV LVXVHGWRDGMXVWWKH*36 UHFHLYHU·VWLPHDQGWR GHWHUPLQHLWVDSSUR[LPDWHORFDWLRQRQWKHHDUWK *LYHQ LQIRUPDWLRQIURPDIRXUWKVDWHOOLWH WKH*36 UHFHLYHUFDQ HYHQHVWDEOLVKLWVDSSUR[LPDWHDOWLWXGH 1 $ PDWKHPDWLFDOGLVFXVVLRQRIWKHVROXWLRQVWUDWHJ\DORQJZLWKGHWDLOV RIWKHHQWLUH1$' SURMHFW DSSHDUVLQ 1RUWK$PHULFDQ'DWXPRI &KDUOHV56FKZDU]HG 1DWLRQDO*HRGHWLF6XUYH\ 1DWLRQDO2FHDQLF DQG$WPRVSKHULF$GPLQLVWUDWLRQ12$$ 3URIHVVLRQDO3DSHU126

$)"15&3

2UWKRJRQDOLW\DQG/HDVW6TXDUHV

%RWKWKH1$' DQG*36 SUREOHPVDUHVROYHGE\ÀQGLQJ DYHFWRUWKDW´DSSUR[LPDWHO\VDWLVÀHVµDQLQFRQVLVWHQW

V\VWHPRIHTXDWLRQV $ FDUHIXOH[SODQDWLRQRIWKLVDSSDUHQW FRQWUDGLFWLRQZLOOUHTXLUHLGHDVGHYHORSHGLQWKHÀUVWÀYH VHFWLRQVRIWKLVFKDSWHU WEB

,QRUGHUWRÀQGDQDSSUR[LPDWHVROXWLRQWRDQLQFRQVLVWHQWV\VWHPRIHTXDWLRQVWKDWKDV QRDFWXDOVROXWLRQ DZHOOGHÀQHGQRWLRQRIQHDUQHVVLVQHHGHG 6HFWLRQ LQWURGXFHV WKHFRQFHSWVRIGLVWDQFHDQGRUWKRJRQDOLW\LQDYHFWRUVSDFH 6HFWLRQV DQGVKRZ KRZRUWKRJRQDOLW\FDQEHXVHGWRLGHQWLI\WKHSRLQWZLWKLQDVXEVSDFH W WKDWLVQHDUHVW WRDSRLQW \ O\LQJRXWVLGHRI W %\WDNLQJ W WREHWKHFROXPQVSDFHRIDPDWUL[ 6HFWLRQ GHYHORSVDPHWKRGIRUSURGXFLQJDSSUR[LPDWHÓHDVWVTXDUHVµ VROXWLRQV IRULQFRQVLVWHQWOLQHDUV\VWHPV VXFKDVWKHV\VWHPVROYHGIRUWKH1$' UHSRUW 6HFWLRQ SURYLGHVDQRWKHURSSRUWXQLW\WRVHHRUWKRJRQDOSURMHFWLRQVDWZRUN FUHDWLQJDPDWUL[IDFWRUL]DWLRQZLGHO\XVHGLQQXPHULFDOOLQHDUDOJHEUD 7KHUHPDLQLQJ VHFWLRQVH[DPLQHVRPHRIWKHPDQ\OHDVWVTXDUHVSUREOHPVWKDWDULVHLQDSSOLFDWLRQV LQFOXGLQJWKRVHLQYHFWRUVSDFHVPRUHJHQHUDOWKDQ Rn

*//&3 130%6$5 -&/(5) "/% 035)0(0/"-*5: *HRPHWULFFRQFHSWVRIOHQJWK GLVWDQFH DQGSHUSHQGLFXODULW\ ZKLFKDUHZHOONQRZQIRU R2 DQG R3 DUHGHÀQHGKHUHIRU Rn 7KHVHFRQFHSWVSURYLGHSRZHUIXOJHRPHWULFWRROV IRUVROYLQJPDQ\DSSOLHGSUREOHPV LQFOXGLQJWKHOHDVWVTXDUHVSUREOHPVPHQWLRQHG DERYH $OOWKUHHQRWLRQVDUHGHÀQHGLQWHUPVRIWKHLQQHUSURGXFWRIWZRYHFWRUV

7KH,QQHU3URGXFW ,I X DQG Y DUHYHFWRUVLQ Rn WKHQZHUHJDUG X DQG Y DV n ! 1 PDWULFHV 7KHWUDQVSRVH XT LVD 1 ! n PDWUL[ DQGWKHPDWUL[SURGXFW XT Y LVD 1 ! 1 PDWUL[ ZKLFKZHZULWHDV DVLQJOHUHDOQXPEHUDVFDODU ZLWKRXWEUDFNHWV 7KHQXPEHU XT Y LVFDOOHGWKH LQQHU SURGXFW RI X DQG Y DQGRIWHQLWLVZULWWHQDV X ! Y 7KLVLQQHUSURGXFW PHQWLRQHGLQWKH H[HUFLVHVIRU6HFWLRQ LVDOVRUHIHUUHGWRDVD GRWSURGXFW ,I 2

3 u1 6 u2 7 6 7 XD6 : 7 4 :: 5

un

DQG

2

3 v1 6 v2 7 6 7 YD6 : 7 4 :: 5

vn

WKHQWKHLQQHUSURGXFWRI X DQG Y LV

Œ u1 u2

2

3 v1 6 v2 7 6 7 " " " un ! 6 : 7 D u1 v1 C u2 v2 C " " " C un vn 4 :: 5

vn

,QQHU3URGXFW /HQJWK DQG2UWKRJRQDOLW\

2

3 2 3 2 3 &9".1-& &RPSXWH X ! Y DQG Y ! X IRU X D 4 #5 5 DQG Y D 4 2 5 #1 #3 40-65*0/ 2 3 3 X ! Y D XT Y D Œ 2 #5 #1 !4 2 5 D .2/.3/ C .#5/.2/ C .#1/.#3/ D #1 #3 2 3 2 Y ! X D YT X D Œ 3 2 #3 !4 #5 5 D .3/.2/ C .2/.#5/ C .#3/.#1/ D #1 #1

,WLVFOHDUIURPWKHFDOFXODWLRQVLQ([DPSOH ZK\ X ! Y D Y ! X 7KLVFRPPXWDWLYLW\ RIWKHLQQHUSURGXFWKROGVLQJHQHUDO 7KHIROORZLQJSURSHUWLHVRIWKHLQQHUSURGXFW DUHHDVLO\GHGXFHGIURPSURSHUWLHVRIWKHWUDQVSRVHRSHUDWLRQLQ6HFWLRQ 6HH ([HUFLVHV DQGDWWKHHQGRIWKLVVHFWLRQ

5)&03&.

/HW X Y DQG Z EHYHFWRUVLQ Rn DQGOHW c EHDVFDODU 7KHQ D E F G

X!Y D Y!X .X C Y/! Z D X ! Z C Y ! Z .c X/! Y D c.X ! Y/ D X !.c Y/ X ! X $ 0 DQG X ! X D 0 LIDQGRQO\LI X D

3URSHUWLHVE DQGF FDQEHFRPELQHGVHYHUDOWLPHVWRSURGXFHWKHIROORZLQJXVHIXO UXOH

.c1 X1 C " " " C cp Xp /! Z D c1 .X1 ! Z/ C " " " C cp .Xp ! Z/

7KH/HQJWKRID9HFWRU ,I Y LVLQ Rn ZLWKHQWULHV v1 ; : : : ; vn WKHQWKHVTXDUHURRWRI Y ! Y LVGHÀQHGEHFDXVH Y ! Y LVQRQQHJDWLYH

%&'*/*5*0/

x2

√  a2 + b2

|a|

0

! " a 6XSSRVH Y LVLQ R VD\ Y D ,IZHLGHQWLI\ Y ZLWKDJHRPHWULFSRLQWLQWKH b SODQH DVXVXDO WKHQ kYk FRLQFLGHVZLWKWKHVWDQGDUGQRWLRQRIWKHOHQJWKRIWKHOLQH VHJPHQWIURPWKHRULJLQWR Y 7KLVIROORZVIURPWKH3\WKDJRUHDQ7KHRUHPDSSOLHGWRD WULDQJOHVXFKDVWKHRQHLQ)LJ $ VLPLODUFDOFXODWLRQZLWKWKHGLDJRQDORIDUHFWDQJXODUER[VKRZVWKDWWKHGHÀQLWLRQ RIOHQJWKRIDYHFWRU Y LQ R3 FRLQFLGHVZLWKWKHXVXDOQRWLRQRIOHQJWK )RUDQ\VFDODU c WKHOHQJWKRI c Y LV jcj WLPHVWKHOHQJWKRI Y 7KDWLV 2

(a, b) |b|

7KH OHQJWK RU QRUP RI Y LVWKHQRQQHJDWLYHVFDODU kYk GHÀQHGE\ q p kYk D Y ! Y D v12 C v22 C " " " C vn2 ; DQG kYk2 D Y ! Y

x1

'*(63&

,QWHUSUHWDWLRQRI kYk DVOHQJWK

kc Yk D jcjkYk 7RVHHWKLV FRPSXWH kc Yk2 D .c Y/! .c Y/ D c 2 Y ! Y D c 2 kYk2 DQGWDNHVTXDUHURRWV

$)"15&3


$ YHFWRUZKRVHOHQJWKLVLVFDOOHGD XQLWYHFWRU ,IZH GLYLGH DQRQ]HURYHFWRU Y E\LWVOHQJWK³WKDWLV PXOWLSO\E\ 1=kYk³ZHREWDLQDXQLWYHFWRU X EHFDXVHWKHOHQJWK RI X LV .1=kYk/kYk 7KHSURFHVVRIFUHDWLQJ X IURP Y LVVRPHWLPHVFDOOHG QRUPDOL]LQJ Y DQGZHVD\WKDW X LV LQWKHVDPHGLUHFWLRQ DV Y 6HYHUDOH[DPSOHVWKDWIROORZXVHWKHVSDFHVDYLQJQRWDWLRQIRUFROXPQ YHFWRUV

&9".1-& /HW Y D .1; #2; 2; 0/ )LQGDXQLWYHFWRU X LQWKHVDPHGLUHFWLRQDV Y 40-65*0/ )LUVW FRPSXWHWKHOHQJWKRI Y kYk2 D Y ! Y D .1/2 C .#2/2 C .2/2 C .0/2 D 9 p kYk D 9 D 3

7KHQ PXOWLSO\ Y E\ 1=kYk WRREWDLQ

x2

1

3 2 3 1 1=3 6 7 1 1 1 6 #2 7 7 D 6 #2=3 7 XD YD YD 6 4 5 4 2 2=3 5 kYk 3 3 0 0

W

x

x1

1

y

x1

1

D

1 9

C

4 9

C

4 9

C0D1

YHFWRU ] WKDWLVDEDVLVIRU W

x2

z

7RFKHFNWKDW kXk D 1 LWVXIÀFHVWRVKRZWKDW kXk2 D 1 # $2 # $2 # $2 kXk2 D X ! X D 13 C # 23 C 23 C .0/2

&9".1-& /HW W EHWKHVXEVSDFHRI R2 VSDQQHGE\ [ D . 23 ; 1/ )LQGDXQLW

(a)

1

2

(b) '*(63&

1RUPDOL]LQJDYHFWRUWRSURGXFHD XQLWYHFWRU

40-65*0/ W FRQVLVWVRIDOOPXOWLSOHVRI [ DVLQ)LJ D $Q\QRQ]HURYHFWRULQ W LVDEDVLVIRU W 7RVLPSOLI\WKHFDOFXODWLRQ ´VFDOHµ [ WRHOLPLQDWHIUDFWLRQV 7KDWLV PXOWLSO\ [ E\WRJHW ! " 2 \D 3 p 1RZFRPSXWH k\k2 D 22 C 32 D 13 k\k D 13 DQGQRUPDOL]H \ WRJHW ! " ! p " 1 2 2=p13 ]D p D 3 3= 13 13 p p 6HH)LJ E $QRWKHUXQLWYHFWRULV .#2= 13; #3= 13/

'LVWDQFHLQ Rn :HDUHUHDG\QRZWRGHVFULEHKRZFORVHRQHYHFWRULVWRDQRWKHU 5HFDOOWKDWLI a DQG b DUHUHDOQXPEHUV WKHGLVWDQFHRQWKHQXPEHUOLQHEHWZHHQ a DQG b LVWKHQXPEHU ja # bj 7ZRH[DPSOHVDUHVKRZQLQ)LJ 7KLVGHÀQLWLRQRIGLVWDQFHLQ R KDVDGLUHFWDQDORJXH LQ Rn a 1

2

b 3

4 5 6 6 units apart

7

8

a 9

|2 – 8| = |– 6| = 6 or |8 – 2| = |6| = 6 '*(63& 'LVWDQFHVLQ R

–3 –2 –1

b 0 1 2 7 units apart

3

4

5

|(– 3) – 4| = |– 7| = 7 or |4 – (–3)| = |7| = 7

%&'*/*5*0/


)RU X DQG Y LQ Rn WKH GLVWDQFHEHWZHHQXDQGY ZULWWHQDV GLVW .X; Y/ LVWKH OHQJWKRIWKHYHFWRU X # Y 7KDWLV GLVW .X; Y/ D kX # Yk

,Q R2 DQG R3 WKLVGHÀQLWLRQRIGLVWDQFHFRLQFLGHVZLWKWKHXVXDOIRUPXODVIRUWKH (XFOLGHDQGLVWDQFHEHWZHHQWZRSRLQWV DVWKHQH[WWZRH[DPSOHVVKRZ

&9".1-& &RPSXWHWKHGLVWDQFHEHWZHHQWKHYHFWRUV X D .7; 1/ DQG Y D .3; 2/ 40-65*0/ &DOFXODWH

! " ! " ! " 7 3 4 # D 1 2 #1 p p kX # Yk D 42 C .#1/2 D 17

X#YD

7KHYHFWRUV X Y DQG X # Y DUHVKRZQLQ)LJ :KHQWKHYHFWRU X # Y LVDGGHG WR Y WKHUHVXOWLV X 1RWLFHWKDWWKHSDUDOOHORJUDPLQ)LJ VKRZVWKDWWKHGLVWDQFHIURP X WR Y LVWKHVDPHDVWKHGLVWDQFHIURP X # Y WR x2 v

||u – v|| u

1

x1

1

u–v

–v '*(63& 7KHGLVWDQFHEHWZHHQ X DQG Y LV

WKHOHQJWKRI X # Y

&9".1-& ,I X D .u1 ; u2 ; u3 / DQG Y D .v1 ; v2 ; v3 / WKHQ

p GLVW .X; Y/ D kX # Yk D .X # Y/!.X # Y/ p D .u1 # v1 /2 C .u2 # v2 /2 C .u3 # v3 /2

2UWKRJRQDO9HFWRUV ||u – v||

u

v ||u – (– v)|| 0 –v '*(63&

7KHUHVWRIWKLVFKDSWHUGHSHQGVRQWKHIDFWWKDWWKHFRQFHSWRISHUSHQGLFXODUOLQHVLQ RUGLQDU\(XFOLGHDQJHRPHWU\KDVDQDQDORJXHLQ Rn &RQVLGHU R2 RU R3 DQGWZROLQHVWKURXJKWKHRULJLQGHWHUPLQHGE\YHFWRUV X DQG Y 7KHWZROLQHVVKRZQLQ)LJ DUHJHRPHWULFDOO\SHUSHQGLFXODULIDQGRQO\LIWKHGLVWDQFH IURP X WR Y LVWKHVDPHDVWKHGLVWDQFHIURP X WR #Y 7KLVLVWKHVDPHDVUHTXLULQJWKH VTXDUHVRIWKHGLVWDQFHVWREHWKHVDPH 1RZ 2

Œ GLVW .X; #Y/ ! D kX # .#Y/k2 D kX C Yk2 D .X C Y/! .X C Y/ D X !.X C Y/ C Y ! .X C Y/ D X!X C X!Y C Y!X C Y!Y 2

2

D kXk C kYk C 2X ! Y

7KHRUHP E 7KHRUHP D E 7KHRUHP D

$)"15&3


7KHVDPHFDOFXODWLRQVZLWK Y DQG #Y LQWHUFKDQJHGVKRZWKDW

ŒGLVW .X; Y/!2 D kXk2 C k # Yk2 C 2X ! .#Y/ D kXk2 C kYk2 # 2X ! Y

7KHWZRVTXDUHGGLVWDQFHVDUHHTXDOLIDQGRQO\LI 2X ! Y D #2X ! Y ZKLFKKDSSHQVLIDQG RQO\LI X ! Y D 0 7KLVFDOFXODWLRQVKRZVWKDWZKHQYHFWRUV X DQG Y DUHLGHQWLÀHGZLWKJHRPHWULF SRLQWV WKHFRUUHVSRQGLQJOLQHVWKURXJKWKHSRLQWVDQGWKHRULJLQDUHSHUSHQGLFXODULI DQGRQO\LI X ! Y D 0 7KHIROORZLQJGHÀQLWLRQJHQHUDOL]HVWR Rn WKLVQRWLRQRISHUSHQ GLFXODULW\RU RUWKRJRQDOLW\ DVLWLVFRPPRQO\FDOOHGLQOLQHDUDOJHEUD

%&'*/*5*0/

7ZRYHFWRUV X DQG Y LQ Rn DUH RUWKRJRQDO WRHDFKRWKHU LI X ! Y D 0 2EVHUYHWKDWWKH]HURYHFWRULVRUWKRJRQDOWRHYHU\YHFWRULQ Rn EHFDXVH T Y D 0 IRUDOO Y 7KHQH[WWKHRUHPSURYLGHVDXVHIXOIDFWDERXWRUWKRJRQDOYHFWRUV 7KHSURRIIRO ORZVLPPHGLDWHO\IURPWKHFDOFXODWLRQLQ DERYHDQGWKHGHÀQLWLRQRIRUWKRJRQDOLW\ 7KHULJKWWULDQJOHVKRZQLQ)LJ SURYLGHVDYLVXDOL]DWLRQRIWKHOHQJWKVWKDWDSSHDULQ WKHWKHRUHP

5)&03&. u+v ||v|| u

||u + v||

5IF 1ZUIBHPSFBO 5IFPSFN 7ZRYHFWRUV X DQG Y DUHRUWKRJRQDOLIDQGRQO\LI kX C Yk2 D kXk2 C kYk2

2UWKRJRQDO&RPSOHPHQWV 7RSURYLGHSUDFWLFHXVLQJLQQHUSURGXFWV ZHLQWURGXFHDFRQFHSWKHUHWKDWZLOOEHRIXVH LQ6HFWLRQ DQGHOVHZKHUHLQWKHFKDSWHU ,IDYHFWRU ] LVRUWKRJRQDOWRHYHU\YHFWRU LQDVXEVSDFH W RI Rn WKHQ ] LVVDLGWREH RUWKRJRQDOWR W 7KHVHWRIDOOYHFWRUV ] WKDWDUHRUWKRJRQDOWR W LVFDOOHGWKH RUWKRJRQDOFRPSOHPHQW RI W DQGLVGHQRWHGE\ W ? DQGUHDGDV´W SHUSHQGLFXODUµRUVLPSO\´W SHUSµ

v ||u|| 0 '*(63&

&9".1-& /HW W EHDSODQHWKURXJKWKHRULJLQLQ R3 DQGOHW L EHWKHOLQH w 0

z

L

W '*(63&

$ SODQHDQGOLQHWKURXJK DV RUWKRJRQDOFRPSOHPHQWV

WKURXJKWKHRULJLQDQGSHUSHQGLFXODUWR W ,I ] DQG Z DUHQRQ]HUR ] LVRQ L DQG Z LVLQ W WKHQWKHOLQHVHJPHQWIURP WR ] LVSHUSHQGLFXODUWRWKHOLQHVHJPHQWIURP WR Z WKDWLV ] ! Z D 0 6HH)LJ 6RHDFKYHFWRURQ L LVRUWKRJRQDOWRHYHU\ Z LQ W ,QIDFW L FRQVLVWVRI DOO YHFWRUVWKDWDUHRUWKRJRQDOWRWKH Z·VLQ W DQG W FRQVLVWVRI DOOYHFWRUVRUWKRJRQDOWRWKH ]·VLQ L 7KDWLV

LDW?

DQG

W D L?

7KHIROORZLQJWZRIDFWVDERXW W ? ZLWK W DVXEVSDFHRI Rn DUHQHHGHGODWHU LQWKHFKDSWHU 3URRIVDUHVXJJHVWHGLQ([HUFLVHV DQG ([HUFLVHV ²SURYLGH H[FHOOHQWSUDFWLFHXVLQJSURSHUWLHVRIWKHLQQHUSURGXFW $ YHFWRU [ LVLQ W ? LIDQGRQO\LI [ LVRUWKRJRQDOWRHYHU\YHFWRULQDVHWWKDW VSDQV W W ? LVDVXEVSDFHRI Rn


7KHQH[WWKHRUHPDQG([HUFLVH YHULI\WKHFODLPVPDGHLQ6HFWLRQ FRQFHUQLQJ WKHVXEVSDFHVVKRZQLQ)LJ $OVRVHH([HUFLVH LQ6HFWLRQ A

0

0

T

Nu

lA Nu

lA

Co

wA Ro

lA

'*(63& 7KHIXQGDPHQWDOVXEVSDFHVGHWHUPLQHG E\DQ m ! n PDWUL[ A

5)&03&.

/HW A EHDQ m ! n PDWUL[ 7KHRUWKRJRQDOFRPSOHPHQWRIWKHURZVSDFHRI A LV WKHQXOOVSDFHRI A DQGWKHRUWKRJRQDOFRPSOHPHQWRIWKHFROXPQVSDFHRI A LV WKHQXOOVSDFHRI AT

.5RZ A/? D 1XO A

DQG

.&RO A/? D 1XO AT

1300' 7KHURZ²FROXPQUXOHIRUFRPSXWLQJ A[ VKRZVWKDWLI [ LVLQ 1XO A WKHQ [ LV RUWKRJRQDOWRHDFKURZRI A ZLWKWKHURZVWUHDWHGDVYHFWRUVLQ Rn / 6LQFHWKHURZV RI A VSDQWKHURZVSDFH [ LVRUWKRJRQDOWR 5RZ A &RQYHUVHO\ LI [ LVRUWKRJRQDOWR 5RZ A WKHQ [ LVFHUWDLQO\RUWKRJRQDOWRHDFKURZRI A DQGKHQFH A[ D 7KLVSURYHV WKHÀUVWVWDWHPHQWRIWKHWKHRUHP 6LQFHWKLVVWDWHPHQWLVWUXHIRUDQ\PDWUL[ LWLVWUXH IRU AT 7KDWLV WKHRUWKRJRQDOFRPSOHPHQWRIWKHURZVSDFHRI AT LVWKHQXOOVSDFHRI AT 7KLVSURYHVWKHVHFRQGVWDWHPHQW EHFDXVH 5RZ AT D &RO A

$QJOHVLQ R2 DQG R3 2SWLRQDO ,I X DQG Y DUHQRQ]HURYHFWRUVLQHLWKHU R2 RU R3 WKHQWKHUHLVDQLFHFRQQHFWLRQEHWZHHQ WKHLULQQHUSURGXFWDQGWKHDQJOH # EHWZHHQWKHWZROLQHVHJPHQWVIURPWKHRULJLQWRWKH SRLQWVLGHQWLÀHGZLWK X DQG Y 7KHIRUPXODLV X ! Y D kXk kYk FRV #

7RYHULI\WKLVIRUPXODIRUYHFWRUVLQ R2 FRQVLGHUWKHWULDQJOHVKRZQLQ)LJ ZLWKVLGHV RIOHQJWKV kXk kYk DQG kX # Yk %\WKHODZRIFRVLQHV

kX # Yk2 D kXk2 C kYk2 # 2kXk kYk FRV #

(u1, u2) ||u – v|| ||u||

(v1, v2)

! ||v||

'*(63& 7KHDQJOHEHWZHHQWZRYHFWRUV

$)"15&3


ZKLFKFDQEHUHDUUDQJHGWRSURGXFH & 1% kXk kYk FRV # D kXk2 C kYk2 # kX # Yk2 2 & 1% 2 D u1 C u22 C v12 C v22 # .u1 # v1 /2 # .u2 # v2 /2 2 D u1 v1 C u2 v2 D X!Y

7KHYHULÀFDWLRQIRU R3 LVVLPLODU :KHQ n > 3 IRUPXOD PD\EHXVHGWR GHÀQH WKH DQJOHEHWZHHQWZRYHFWRUVLQ Rn ,QVWDWLVWLFV IRULQVWDQFH WKHYDOXHRI FRV # GHÀQHG E\ IRUVXLWDEOHYHFWRUV X DQG Y LVZKDWVWDWLVWLFLDQVFDOOD FRUUHODWLRQFRHIÀFLHQW

13"$5*$& 130#-&.4

2 3 2 3 " ! " 4=3 5 #2 #3 /HW D D ED F D 4 #1 5 DQG G D 4 6 5 1 1 2=3 #1 ' ( D!E D!E &RPSXWH DQG D D!D D!D )LQGDXQLWYHFWRU X LQWKHGLUHFWLRQRI F 6KRZWKDW G LVRUWKRJRQDOWR F 8VHWKHUHVXOWVRI3UDFWLFH3UREOHPV DQGWRH[SODLQZK\ G PXVWEHRUWKRJRQDOWR WKHXQLWYHFWRU X !

&9&3$*4&4 &RPSXWHWKHTXDQWLWLHVLQ([HUFLVHV²XVLQJWKHYHFWRUV 2 3 2 3 ! " ! " 3 6 #1 4 XD YD Z D 4 #1 5 [ D 4 #2 5 2 6 #5 3 Y!X X ! X Y ! X DQG X!X

1 Z Z!Z ) X!Y * Y!Y

[!Z Z ! Z [ ! Z DQG Z!Z

Y

kZk

1 X X!X )["Z* ["[

[

k[k

,Q([HUFLVHV² ÀQGDXQLWYHFWRULQWKHGLUHFWLRQRIWKHJLYHQ YHFWRU 2 3 ! " #6 #30 4 4 5 40 #3 2

3 7=4 4 1=2 5 1

!

)LQGWKHGLVWDQFHEHWZHHQ [ D

8=3 2

!

10 #3

"

"

DQG \ D

!

" #1 #5

2

3 2 3 0 #4 )LQGWKHGLVWDQFHEHWZHHQ X D 4 #5 5 DQG ] D 4 #1 5 2 8

'HWHUPLQHZKLFKSDLUVRIYHFWRUVLQ([HUFLVHV²DUHRUWKRJ RQDO 2 3 2 3 ! " ! " 12 2 8 #2 ED X D 4 3 5 Y D 4 #3 5 D D #5 #3 #5 3 2 3 2 3 2 3 2 3 3 #4 #3 1 6 27 6 17 6 77 6 #8 7 7 6 7 6 7 6 7 X D 6 4 #5 5 Y D 4 #2 5 \ D 4 4 5 ] D 4 15 5 0 6 0 #7

,Q([HUFLVHVDQG DOOYHFWRUVDUHLQ Rn 0DUNHDFKVWDWHPHQW 7UXHRU)DOVH -XVWLI\HDFKDQVZHU D Y ! Y D kYk2

E )RUDQ\VFDODU c X ! .c Y/ D c.X ! Y/

F ,IWKHGLVWDQFHIURP X WR Y HTXDOVWKHGLVWDQFHIURP X WR #Y WKHQ X DQG Y DUHRUWKRJRQDO

G )RUDVTXDUHPDWUL[ A YHFWRUVLQ &RO A DUHRUWKRJRQDOWR YHFWRUVLQ 1XO A H ,I YHFWRUV Y1 ; : : : ; Yp VSDQ D VXEVSDFH W DQG LI [ LV RUWKRJRQDOWRHDFK Yj IRU j D 1; : : : ; p WKHQ [ LVLQ W ?

D X ! Y # Y ! X D 0

E )RUDQ\VFDODU c kc Yk D ckYk

F ,I [ LVRUWKRJRQDOWRHYHU\YHFWRULQDVXEVSDFH W WKHQ [ LVLQ W ? G ,I kXk2 C kYk2 D kX C Yk2 WKHQ X DQG Y DUHRUWKRJRQDO H )RUDQ m ! n PDWUL[ A YHFWRUVLQWKHQXOOVSDFHRI A DUH RUWKRJRQDOWRYHFWRUVLQWKHURZVSDFHRI A

8VHWKHWUDQVSRVHGHÀQLWLRQRIWKHLQQHUSURGXFWWRYHULI\ SDUWVE DQGF RI7KHRUHP 0HQWLRQWKHDSSURSULDWHIDFWV IURP&KDSWHU /HW X D .u1 ; u2 ; u3 / ([SODLQ ZK\ X ! X $ 0 :KHQ LV X ! X D 0" 2 3 2 3 2 #7 /HW X D 4 #5 5 DQG Y D 4 #4 5 &RPSXWHDQGFRPSDUH #1 6 X ! Y kXk2 kYk2 DQG kX C Yk2 'RQRWXVH WKH3\WKDJRUHDQ 7KHRUHP

9HULI\WKH SDUDOOHORJUDPODZ IRUYHFWRUV X DQG Y LQ Rn


/HW W EHDVXEVSDFHRI Rn DQGOHW W ? EHWKHVHWRIDOO YHFWRUVRUWKRJRQDOWR W 6KRZWKDW W ? LVDVXEVSDFHRI Rn XVLQJWKHIROORZLQJVWHSV D 7DNH ] LQ W ? DQGOHW X UHSUHVHQWDQ\HOHPHQWRI W 7KHQ ] ! X D 0 7DNHDQ\VFDODU c DQGVKRZWKDW c ] LV RUWKRJRQDOWR X 6LQFH X ZDVDQDUELWUDU\HOHPHQWRI W WKLVZLOOVKRZWKDW c ] LVLQ W ? E 7DNH ]1 DQG ]2 LQ W ? DQGOHW X EHDQ\HOHPHQWRI W 6KRZWKDW ]1 C ]2 LVRUWKRJRQDOWR X :KDWFDQ\RX FRQFOXGHDERXW ]1 C ]2 " :K\" F )LQLVKWKHSURRIWKDW W ? LVDVXEVSDFHRI Rn 6KRZWKDWLI [ LVLQERWK W DQG W ? WKHQ [ D >0@ &RQVWUXFWDSDLU X Y RIUDQGRPYHFWRUVLQ R4 DQGOHW 2 3 :5 :5 :5 :5 6 :5 :5 #:5 #:5 7 7 AD6 4 :5 #:5 :5 #:5 5 :5 #:5 #:5 :5

D 'HQRWH WKH FROXPQV RI A E\ D1 ; : : : ; D4 &RP SXWH WKH OHQJWK RI HDFK FROXPQ DQG FRPSXWH D1 ! D2 D1 ! D3 ; D1 ! D4 ; D2 ! D3 ; D2 ! D4 DQG D3 ! D4

kX C Yk2 C kX # Yk2 D 2kXk2 C 2kYk2 ! " ! " a x /HW Y D 'HVFULEHWKHVHW H RIYHFWRUV WKDWDUH b y RUWKRJRQDOWR Y >+LQW &RQVLGHU Y D DQG Y ¤ @ 2 3 5 /HW X D 4 #6 5 DQGOHW W EHWKHVHWRIDOO [ LQ R3 VXFKWKDW 7 X ! [ D 0 :KDWWKHRUHP LQ&KDSWHU FDQEHXVHGWRVKRZWKDW W LVDVXEVSDFHRI R3 " 'HVFULEH W LQJHRPHWULFODQJXDJH

G 5HSHDWSDUWVE DQGF IRUWZRRWKHUSDLUVRIUDQGRP YHFWRUV :KDWGR\RXFRQMHFWXUHDERXWWKHHIIHFWRI A RQ YHFWRUV"

6XSSRVHDYHFWRU \ LVRUWKRJRQDOWRYHFWRUV X DQG Y 6KRZ WKDW \ LVRUWKRJRQDOWRWKHYHFWRU X C Y

>0@ *HQHUDWHUDQGRPYHFWRUV [ \ DQG Y LQ R4 ZLWKLQWHJHU HQWULHVDQG Y ¤ DQGFRPSXWHWKHTXDQWLWLHV

6XSSRVH \ LVRUWKRJRQDOWR X DQG Y 6KRZWKDW \ LVRU WKRJRQDOWRHYHU\ Z LQ 6SDQ fX; Yg >+LQW $QDUELWUDU\ Z LQ 6SDQ fX; Yg KDVWKHIRUP Z D c1 X C c2 Y 6KRZWKDW \ LV RUWKRJRQDOWRVXFKDYHFWRU Z@ w u 0 y

{u

an

Sp

v } v ,

/HW W D 6SDQ fY1 ; : : : ; Yp g 6KRZWKDWLI [ LVRUWKRJRQDOWR HDFK Yj IRU 1 % j % p WKHQ [ LVRUWKRJRQDOWRHYHU\YHFWRU LQ W

E &RPSXWHDQGFRPSDUHWKHOHQJWKVRI X AX Y DQG AY

F 8VHHTXDWLRQ LQWKLVVHFWLRQWRFRPSXWHWKHFRVLQHRI WKHDQJOHEHWZHHQ X DQG Y &RPSDUHWKLVZLWKWKHFRVLQH RIWKHDQJOHEHWZHHQ AX DQG AY

) [!Y *

Y!Y

Y;

) \!Y *

Y!Y

Y;

.[ C \/! Y .10[/! Y Y; Y Y!Y Y!Y

5HSHDWWKHFRPSXWDWLRQVZLWKQHZUDQGRPYHFWRUV [ DQG \ :KDWGR\RXFRQMHFWXUHDERXWWKHPDSSLQJ [ 7! T .[/ D ) [!Y * Y IRU Y ¤ " 9HULI\\RXUFRQMHFWXUHDOJHEUDLFDOO\ Y!Y 2 3 #6 3 #27 #33 #13 6 6 #5 25 28 14 7 6 7 8 #6 34 38 18 7 >0@ /HW A D 6 6 7 &RQVWUXFW 4 12 #10 50 41 23 5 14 #21 49 29 33 D PDWUL[ N ZKRVH FROXPQV IRUP D EDVLV IRU 1XO A DQG FRQVWUXFWDPDWUL[ R ZKRVH URZV IRUPDEDVLVIRU 5RZ A VHH 6HFWLRQIRUGHWDLOV 3HUIRUPDPDWUL[FRPSXWDWLRQZLWK N DQG R WKDWLOOXVWUDWHVDIDFWIURP7KHRUHP

$)"15&3


40-65*0/4 50 13"$5*$& 130#-&.4 ' ( ! " 7 7 D!E D!E #14=5 D ! E D 7 D ! D D 5 +HQFH D DQG DD DD 7=5 D!D 5 D!D 5 2 3 4 p 6FDOH F PXOWLSO\LQJE\WRJHW \ D 4 #3 5 &RPSXWH k\k2 D 29 DQG k\k D 29 2 p 3 2 4=p29 1 7KHXQLWYHFWRULQWKHGLUHFWLRQRIERWK F DQG \ LV X D \ D 4 #3=p29 5 k\k 2= 29 G LVRUWKRJRQDOWR F EHFDXVH 2 3 2 3 5 4=3 2 20 #6# D0 G ! F D 4 6 5 ! 4 #1 5 D 3 3 #1 2=3 G LVRUWKRJRQDOWR X EHFDXVH X KDVWKHIRUP k F IRUVRPH k DQG G ! X D G ! .k F/ D k.G ! F/ D k.0/ D 0

035)0(0/"- 4&54 $ VHWRIYHFWRUV fX1 ; : : : ; Xp g LQ Rn LVVDLGWREHDQ RUWKRJRQDOVHW LIHDFKSDLURIGLVWLQFW YHFWRUVIURPWKHVHWLVRUWKRJRQDO WKDWLV LI Xi ! Xj D 0 ZKHQHYHU i ¤ j

u3

&9".1-& 6KRZWKDW fX1 ; X2 ; X3 g LVDQRUWKRJRQDOVHW ZKHUH 2 3 3 X1 D 4 1 5; 1

x3

2

3 #1 X 2 D 4 2 5; 1

2

3 #1=2 X3 D 4 #2 5 7=2

40-65*0/ &RQVLGHUWKHWKUHHSRVVLEOHSDLUVRIGLVWLQFWYHFWRUV QDPHO\ fX1 ; X2 g fX1 ; X3 g DQG fX2 ; X3 g u2 u1 x1

x2

'*(63&

5)&03&.

X1 ! X2 D 3.#1/ C 1.2/ C 1.1/ D 0 # $ # $ X1 ! X3 D 3 # 12 C 1.#2/ C 1 72 D 0 # $ # $ X2 ! X3 D #1 # 12 C 2.#2/ C 1 72 D 0

(DFKSDLURIGLVWLQFWYHFWRUVLVRUWKRJRQDO DQGVR fX1 ; X2 ; X3 g LVDQRUWKRJRQDOVHW 6HH )LJ WKHWKUHHOLQHVHJPHQWVWKHUHDUHPXWXDOO\SHUSHQGLFXODU ,I S D fX1 ; : : : ; Xp g LVDQRUWKRJRQDOVHWRIQRQ]HURYHFWRUVLQ Rn WKHQ S LV OLQHDUO\LQGHSHQGHQWDQGKHQFHLVDEDVLVIRUWKHVXEVSDFHVSDQQHGE\ S

1300' ,I D c1 X1 C " " " C cp Xp IRUVRPHVFDODUV c1 ; : : : ; cp WKHQ 0 D ! X1 D .c1 X1 C c2 X2 C " " " C cp Xp /! X1 D .c1 X1 /! X1 C .c2 X2 /! X1 C " " " C .cp Xp /! X1 D c1 .X1 ! X1 / C c2 .X2 ! X1 / C " " " C cp .Xp ! X1 / D c1 .X1 ! X1 /

EHFDXVH X1 LVRUWKRJRQDOWR X2 ; : : : ; Xp 6LQFH X1 LVQRQ]HUR X1 ! X1 LVQRW]HURDQGVR c1 D 0 6LPLODUO\ c2 ; : : : ; cp PXVWEH]HUR 7KXV S LVOLQHDUO\LQGHSHQGHQW

%&'*/*5*0/

2UWKRJRQDO6HWV

$Q RUWKRJRQDOEDVLV IRUDVXEVSDFH W RI Rn LVDEDVLVIRU W WKDWLVDOVRDQ RUWKRJRQDOVHW 7KHQH[WWKHRUHPVXJJHVWVZK\DQRUWKRJRQDOEDVLVLVPXFKQLFHUWKDQRWKHUEDVHV 7KHZHLJKWVLQDOLQHDUFRPELQDWLRQFDQEHFRPSXWHGHDVLO\

5)&03&.

/HW fX1 ; : : : ; Xp g EHDQRUWKRJRQDOEDVLVIRUDVXEVSDFH W RI Rn )RUHDFK \ LQ W WKHZHLJKWVLQWKHOLQHDUFRPELQDWLRQ \ D c1 X1 C " " " C cp Xp DUHJLYHQE\

cj D

\ ! Xj Xj ! Xj

.j D 1; : : : ; p/

1300' $VLQWKHSUHFHGLQJSURRI WKHRUWKRJRQDOLW\RI fX1 ; : : : ; Xp g VKRZVWKDW \ ! X1 D .c1 X1 C c2 X2 C " " " C cp Xp /! X1 D c1 .X1 ! X1 / 6LQFH X1 ! X1 LVQRW]HUR WKHHTXDWLRQDERYHFDQEHVROYHGIRU c1 7RÀQG cj IRU j D 2; : : : ; p FRPSXWH \ ! Xj DQGVROYHIRU cj 3 &9".1-& 7KHVHW 2 S D3fX1 ; X2 ; X3 g LQ([DPSOH LVDQRUWKRJRQDOEDVLVIRU R

6 ([SUHVVWKHYHFWRU \ D 4 1 5 DVDOLQHDUFRPELQDWLRQRIWKHYHFWRUVLQ S #8 40-65*0/ &RPSXWH

\ ! X1 D 11; X1 ! X1 D 11;

\ ! X2 D #12; X2 ! X2 D 6;

\ ! X3 D #33 X3 ! X3 D 33=2

%\7KHRUHP \D

\ ! X1 \ ! X2 \ ! X3 X1 C X2 C X3 X1 ! X1 X2 ! X 2 X3 ! X3

11 #12 #33 X1 C X2 C X3 11 6 33=2 D X1 # 2X2 # 2X3 D

1RWLFHKRZHDV\LWLVWRFRPSXWHWKHZHLJKWVQHHGHGWREXLOG \ IURPDQRUWKRJRQDO EDVLV ,IWKHEDVLVZHUHQRWRUWKRJRQDO LWZRXOGEHQHFHVVDU\WRVROYHDV\VWHPRIOLQHDU HTXDWLRQVLQRUGHUWRÀQGWKHZHLJKWV DVLQ&KDSWHU :HWXUQQH[WWRDFRQVWUXFWLRQWKDWZLOOEHFRPHDNH\VWHSLQPDQ\FDOFXODWLRQV LQYROYLQJRUWKRJRQDOLW\ DQGLWZLOOOHDGWRDJHRPHWULFLQWHUSUHWDWLRQRI7KHRUHP

$Q2UWKRJRQDO3URMHFWLRQ *LYHQDQRQ]HURYHFWRU X LQ Rn FRQVLGHUWKHSUREOHPRIGHFRPSRVLQJDYHFWRU \ LQ Rn LQWRWKHVXPRIWZRYHFWRUV RQHDPXOWLSOHRI X DQGWKHRWKHURUWKRJRQDOWR X :HZLVK WRZULWH \ D \O C ]

$)"15&3

z = y – yˆ

0


y

yˆ = !u

'*(63&

)LQGLQJ ˛ WRPDNH \ # \O RUWKRJRQDOWR X

u

ZKHUH \O D ˛ X IRUVRPHVFDODU ˛ DQG ] LVVRPHYHFWRURUWKRJRQDOWR X 6HH)LJ *LYHQ DQ\VFDODU ˛ OHW ] D \ # ˛ X VRWKDW LVVDWLVÀHG 7KHQ \ # \O LVRUWKRJRQDOWR X LIDQG RQO\LI 0 D .\ # ˛ X/! X D \ ! X # .˛ X/! X D \ ! X # ˛.X ! X/ \!X \!X 7KDWLV LVVDWLVÀHGZLWK ] RUWKRJRQDOWR X LIDQGRQO\LI ˛ D DQG \O D X X!X X!X 7KHYHFWRU \O LVFDOOHGWKH RUWKRJRQDOSURMHFWLRQRI\RQWRX DQGWKHYHFWRU ] LVFDOOHG WKH FRPSRQHQWRI\RUWKRJRQDOWRX ,I c LVDQ\QRQ]HURVFDODUDQGLI X LVUHSODFHGE\ c X LQWKHGHÀQLWLRQRI \O WKHQWKH RUWKRJRQDOSURMHFWLRQRI \ RQWR c X LVH[DFWO\WKHVDPHDVWKHRUWKRJRQDOSURMHFWLRQRI \ RQWR X ([HUFLVH +HQFHWKLVSURMHFWLRQLVGHWHUPLQHGE\WKH VXEVSDFH L VSDQQHG E\ X WKHOLQHWKURXJK X DQG 6RPHWLPHV \O LVGHQRWHGE\ SURML \ DQGLVFDOOHGWKH RUWKRJRQDOSURMHFWLRQRI\RQWR L 7KDWLV \O D SURML \ D

\!X X X!X

! " ! " 7 4 DQG X D )LQGWKHRUWKRJRQDOSURMHFWLRQRI \ 6 2 RQWR X 7KHQZULWH \ DVWKHVXPRIWZRRUWKRJRQDOYHFWRUV RQHLQ 6SDQ fXg DQGRQH RUWKRJRQDOWR X

&9".1-& /HW \ D

40-65*0/ &RPSXWH

! " ! " 7 4 ! D 40 6 2 ! " ! " 4 4 X!X D ! D 20 2 2

\!X D

7KHRUWKRJRQDOSURMHFWLRQRI \ RQWR X LV

! " ! " 40 \!X 4 8 \O D XD XD2 D 2 4 X!X 20

DQGWKHFRPSRQHQWRI \ RUWKRJRQDOWR X LV ! " ! " ! " 7 8 #1 \ # \O D # D 6 4 2 7KHVXPRIWKHVHWZRYHFWRUVLV \ 7KDWLV ! " ! " ! " 7 8 #1 D C 6 4 2 " \

" \O

" .\ # \O /

7KLVGHFRPSRVLWLRQRI \ LVLOOXVWUDWHGLQ)LJ 1RWH ,IWKHFDOFXODWLRQVDERYHDUH FRUUHFW WKHQ fO\; \ # \O g ZLOOEHDQRUWKRJRQDOVHW $VDFKHFN FRPSXWH ! " ! " 8 #1 \O !.\ # \O / D ! D #8 C 8 D 0 4 2 6LQFHWKHOLQHVHJPHQWLQ)LJ EHWZHHQ \ DQG \O LVSHUSHQGLFXODUWR L E\FRQVWUXF WLRQRI \O WKHSRLQWLGHQWLÀHGZLWK \O LVWKHFORVHVWSRLQWRI L WR \ 7KLVFDQEHSURYHG IURPJHRPHWU\ :HZLOODVVXPHWKLVIRU R2 QRZDQGSURYHLWIRU Rn LQ6HFWLRQ

x2

2UWKRJRQDO6HWV

y

6

L = Span{u} yˆ

3 y – yˆ

u

1

8

x1

'*(63& 7KHRUWKRJRQDOSURMHFWLRQRI \ RQWRD

OLQH L WKURXJKWKHRULJLQ

&9".1-& )LQGWKHGLVWDQFHLQ)LJ IURP \ WR L 40-65*0/ 7KHGLVWDQFHIURP \ WR L LVWKHOHQJWKRIWKHSHUSHQGLFXODUOLQHVHJPHQW IURP \ WRWKHRUWKRJRQDOSURMHFWLRQ \O 7KLVOHQJWKHTXDOVWKHOHQJWKRI \ # \O 7KXVWKH GLVWDQFHLV p p k\ # \O k D .#1/2 C 22 D 5

$ *HRPHWULF,QWHUSUHWDWLRQRI7KHRUHP

7KHIRUPXODIRUWKHRUWKRJRQDOSURMHFWLRQ \O LQ KDVWKHVDPHDSSHDUDQFHDVHDFKRIWKH WHUPVLQ7KHRUHP 7KXV7KHRUHP GHFRPSRVHVDYHFWRU \ LQWRDVXPRIRUWKRJRQDO SURMHFWLRQVRQWRRQHGLPHQVLRQDOVXEVSDFHV ,WLVHDV\WRYLVXDOL]HWKHFDVHLQZKLFK W D R2 D 6SDQ fX1 ; X2 g ZLWK X1 DQG X2 RUWKRJRQDO $Q\ \ LQ R2 FDQEHZULWWHQLQWKHIRUP \ ! X1 \ ! X2 \D X1 C X2 X 1 ! X1 X2 ! X2

7KHÀUVWWHUPLQ LVWKHSURMHFWLRQRI \ RQWRWKHVXEVSDFHVSDQQHGE\ X1 WKHOLQH WKURXJK X1 DQGWKHRULJLQ DQGWKHVHFRQGWHUPLVWKHSURMHFWLRQRI \ RQWRWKHVXEVSDFH VSDQQHGE\ X2 7KXV H[SUHVVHV \ DVWKHVXPRILWVSURMHFWLRQVRQWRWKHRUWKRJRQDO D[HVGHWHUPLQHGE\ X1 DQG X2 6HH)LJ u2 yˆ 2 = projection onto u2

y

0 yˆ 1 = projection onto u1 u1 '*(63& $ YHFWRUGHFRPSRVHGLQWR

WKHVXPRIWZRSURMHFWLRQV

7KHRUHP GHFRPSRVHVHDFK \ LQ 6SDQ fX1 ; : : : ; Xp g LQWRWKHVXPRI p SURMHFWLRQV RQWRRQHGLPHQVLRQDOVXEVSDFHVWKDWDUHPXWXDOO\RUWKRJRQDO

$)"15&3


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

y u

'*(63&

2UWKRQRUPDO6HWV $ VHW fX1 ; : : : ; Xp g LVDQ RUWKRQRUPDOVHW LILWLVDQRUWKRJRQDOVHWRIXQLWYHFWRUV ,I W LVWKHVXEVSDFHVSDQQHGE\VXFKDVHW WKHQ fX1 ; : : : ; Xp g LVDQ RUWKRQRUPDOEDVLV IRU W VLQFHWKHVHWLVDXWRPDWLFDOO\OLQHDUO\LQGHSHQGHQW E\7KHRUHP 7KHVLPSOHVWH[DPSOHRIDQRUWKRQRUPDOVHWLVWKHVWDQGDUGEDVLV fH1 ; : : : ; Hn g IRU Rn $Q\QRQHPSW\VXEVHWRI fH1 ; : : : ; Hn g LVRUWKRQRUPDO WRR +HUHLVDPRUHFRPSOL FDWHGH[DPSOH

&9".1-& 6KRZWKDW fY1 ; Y2 ; Y3 g LVDQRUWKRQRUPDOEDVLVRI R3 ZKHUH 2

p 3 3=p11 6 7 Y1 D 4 1= 11 5; p 1= 11

40-65*0/ &RPSXWH

'*(63&

2

p 3 #1=p66 6 7 Y3 D 4 #4= 66 5 p 7= 66

7KXV fY1 ; Y2 ; Y3 g LVDQRUWKRJRQDOVHW $OVR

v3

x1

p 3 #1=p6 6 7 Y2 D 4 2= 6 5; p 1= 6

p p p Y1 ! Y2 D #3= 66 C 2= 66 C 1= 66 D 0 p p p Y1 ! Y3 D #3= 726 # 4= 726 C 7= 726 D 0 p p p Y2 ! Y3 D 1= 396 # 8= 396 C 7= 396 D 0

x3

v1

2

v2 x2

Y1 ! Y1 D 9=11 C 1=11 C 1=11 D 1 Y2 ! Y2 D 1=6 C 4=6 C 1=6 D 1 Y3 ! Y3 D 1=66 C 16=66 C 49=66 D 1

ZKLFKVKRZVWKDW Y1 Y2 DQG Y3 DUHXQLWYHFWRUV 7KXV fY1 ; Y2 ; Y3 g LVDQRUWKRQRUPDOVHW 6LQFHWKHVHWLVOLQHDUO\LQGHSHQGHQW LWVWKUHHYHFWRUVIRUPDEDVLVIRU R3 6HH)LJ

2UWKRJRQDO6HWV

:KHQWKHYHFWRUVLQDQRUWKRJRQDOVHWRIQRQ]HURYHFWRUVDUH QRUPDOL]HG WRKDYH XQLWOHQJWK WKHQHZYHFWRUVZLOOVWLOOEHRUWKRJRQDO DQGKHQFHWKHQHZVHWZLOOEHDQRU WKRQRUPDOVHW 6HH([HUFLVH ,WLVHDV\WRFKHFNWKDWWKHYHFWRUVLQ)LJ ([DPSOH DUHVLPSO\WKHXQLWYHFWRUVLQWKHGLUHFWLRQVRIWKHYHFWRUVLQ)LJ ([DPSOH 0DWULFHVZKRVHFROXPQVIRUPDQRUWKRQRUPDOVHWDUHLPSRUWDQWLQDSSOLFDWLRQVDQG LQFRPSXWHUDOJRULWKPVIRUPDWUL[FRPSXWDWLRQV 7KHLUPDLQSURSHUWLHVDUHJLYHQLQ 7KHRUHPV DQG

5)&03&.

$Q m ! n PDWUL[ U KDVRUWKRQRUPDOFROXPQVLIDQGRQO\LI U T U D I

1300' 7RVLPSOLI\QRWDWLRQ ZHVXSSRVHWKDW U KDVRQO\WKUHHFROXPQV HDFKDYHFWRU LQ Rm 7KHSURRIRIWKHJHQHUDOFDVHLVHVVHQWLDOO\WKHVDPH /HW U D Œ X1 X2 X3 ! DQGFRPSXWH 2 T3 2 T 3 X1 X1 X1 XT1 X2 XT1 X3 & 6 6 7% 7 U TU D 4 XT2 5 X1 X2 X3 D 4 XT2 X1 XT2 X2 XT2 X3 5 XT3

XT3 X1 XT3 X2 XT3 X3

7KHHQWULHVLQWKHPDWUL[DWWKHULJKWDUHLQQHUSURGXFWV XVLQJWUDQVSRVHQRWDWLRQ 7KH FROXPQVRI U DUHRUWKRJRQDOLIDQGRQO\LI XT1 X2 D XT2 X1 D 0;

XT1 X3 D XT3 X1 D 0;

XT2 X3 D XT3 X2 D 0

7KHFROXPQVRI U DOOKDYHXQLWOHQJWKLIDQGRQO\LI XT1 X1 D 1;

XT2 X2 D 1;

XT3 X3 D 1

7KHWKHRUHPIROORZVLPPHGLDWHO\IURP ²

5)&03&.

/HW U EHDQ m ! n PDWUL[ZLWKRUWKRQRUPDOFROXPQV DQGOHW [ DQG \ EHLQ Rn 7KHQ D kU [k D k[k E .U [/!.U \/ D [ ! \ F .U [/! .U \/ D 0 LIDQGRQO\LI [ ! \ D 0 3URSHUWLHVD DQGF VD\WKDWWKHOLQHDUPDSSLQJ [ 7! U [ SUHVHUYHVOHQJWKVDQG RUWKRJRQDOLW\ 7KHVHSURSHUWLHVDUHFUXFLDOIRUPDQ\FRPSXWHUDOJRULWKPV 6HH([HU FLVH IRUWKHSURRIRI7KHRUHP 2

3 p !p " 1=p2 2=3 2 6 7 &9".1-& /HW U D 4 1= 2 #2=3 5 DQG [ D 1RWLFHWKDW U KDVRU 3 0 1=3 WKRQRUPDOFROXPQVDQG 3 2 p p ! p " 1= 2 ! " 2=3 p 1 0 1= 2 1= 2 0 T 5 4 U U D 1= 2 #2=3 D 0 1 2=3 #2=3 1=3 0 1=3

9HULI\WKDW kU [k D k[k

$)"15&3


40-65*0/

2

p 3 2 3 1=p2 2=3 ! p " 3 2 U [ D 4 1= 2 #2=3 5 D 4 #1 5 3 1 0 1=3 p p kU [k D 9 C 1 C 1 D 11 p p k[k D 2 C 9 D 11

7KHRUHPVDQGDUHSDUWLFXODUO\XVHIXOZKHQDSSOLHGWR VTXDUH PDWULFHV $Q RUWKRJRQDOPDWUL[ LVDVTXDUHLQYHUWLEOHPDWUL[ U VXFKWKDW U !1 D U T %\7KHRUHP VXFKDPDWUL[KDVRUWKRQRUPDOFROXPQVñ ,WLVHDV\WRVHHWKDWDQ\ VTXDUH PDWUL[ZLWK RUWKRQRUPDOFROXPQVLVDQRUWKRJRQDOPDWUL[ 6XUSULVLQJO\ VXFKDPDWUL[PXVWKDYH RUWKRQRUPDO URZV WRR 6HH([HUFLVHV DQG 2UWKRJRQDOPDWULFHVZLOODSSHDU IUHTXHQWO\LQ&KDSWHU

&9".1-& 7KHPDWUL[

2

p p p 3 3=p11 #1=p6 #1=p66 6 7 U D 4 1= 11 2= 6 #4= 66 5 p p p 1= 6 7= 66 1= 11

LVDQRUWKRJRQDOPDWUL[EHFDXVHLWLVVTXDUHDQGEHFDXVHLWVFROXPQVDUHRUWKRQRUPDO E\([DPSOH 9HULI\WKDWWKHURZVDUHRUWKRQRUPDO WRR

13"$5*$& 130#-&.4 p " ! ! p " #1=p5 2=p5 /HW X1 D DQG X2 D 6KRZWKDW fX1 ; X2 g LVDQRUWKRQRUPDO 2= 5 1= 5 EDVLVIRU R2 DQG)LJ &RPSXWHWKHRUWKRJRQDOSURMHFWLRQ \O RI /HW \ DQG L EHDVLQ([DPSOH ! " 2 \ RQWR L XVLQJ X D LQVWHDGRIWKH X LQ([DPSOH 1 ! p " #3 2 /HW U DQG [ EHDVLQ([DPSOH DQGOHW \ D 9HULI\WKDW U [ !U \ D [ ! \ 6

&9&3$*4&4 ,Q([HUFLVHV² GHWHUPLQHZKLFKVHWVRIYHFWRUVDUHRUWKRJRQDO 2 3 2 3 2 3 2 3 2 3 2 3 #1 5 3 1 0 #5 4 4 5 4 2 5 4 #4 5 4 #2 5 4 1 5 4 #2 5 #3 1 #7 1 2 1 2

3 2 3 2 3 2 #6 3 4 #7 5 4 #3 5 4 1 5 #1 9 #1

2

3 2 3 2 3 2 0 4 4 #5 5 4 0 5 4 #2 5 #3 0 6

2

3 2 3 2 3 3 #1 3 6 #2 7 6 3 7 6 8 7 6 7 6 7 6 7 4 1 5 4 #3 5 4 7 5 3 4 0

2

3 2 3 2 3 5 #4 3 6 #4 7 6 1 7 6 3 7 7 6 7 6 7 6 4 0 5 4 #3 5 4 5 5 3 8 #1

,Q([HUFLVHV² VKRZWKDW fX1 ; X2 g RU fX1 ; X2 ; X3 g LVDQRUWKRJ RQDOEDVLVIRU R2 RU R3 UHVSHFWLYHO\ 7KHQH[SUHVV [ DVDOLQHDU FRPELQDWLRQRIWKH X·V ! " ! " ! " 2 6 9 X2 D DQG [ D X1 D #3 4 #7

ñ $ EHWWHUQDPHPLJKWEH RUWKRQRUPDOPDWUL[ DQGWKLVWHUPLVIRXQGLQVRPHVWDWLVWLFVWH[WV +RZHYHU RUWKRJRQDOPDWUL[ LVWKHVWDQGDUGWHUPLQOLQHDUDOJHEUD

! " ! " ! " 3 #2 #6 X2 D DQG [ D 1 6 3 2 3 2 3 2 3 2 3 1 #1 2 8 X1 D 4 0 5 X2 D 4 4 5 X3 D 4 1 5 DQG [ D 4 #4 5 1 1 #2 #3 2 3 2 3 2 3 2 3 3 2 1 5 X1 D 4 #3 5 X2 D 4 2 5 X3 D 4 1 5 DQG [ D 4 #3 5 0 #1 4 1 ! " 1 &RPSXWHWKHRUWKRJRQDOSURMHFWLRQRI RQWRWKHOLQH 7 ! " #4 DQGWKHRULJLQ WKURXJK 2 ! " 1 &RPSXWHWKHRUWKRJRQDOSURMHFWLRQRI RQWRWKHOLQH #1 ! " #1 DQGWKHRULJLQ WKURXJK 3 ! " ! " 2 4 /HW \ D DQG X D :ULWH \ DVWKHVXPRIWZR 3 #7 RUWKRJRQDOYHFWRUV RQHLQ 6SDQ fXg DQGRQHRUWKRJRQDOWR X ! " ! " 2 7 DQG X D :ULWH \ DVWKHVXPRIDYHFWRU /HW \ D 6 1 LQ 6SDQ fXg DQGDYHFWRURUWKRJRQDOWR X ! " ! " 3 8 DQG X D &RPSXWHWKHGLVWDQFHIURP \ /HW \ D 1 6 WRWKHOLQHWKURXJK X DQGWKHRULJLQ ! " ! " #3 1 /HW \ D DQG X D &RPSXWHWKHGLVWDQFHIURP \ 9 2 WRWKHOLQHWKURXJK X DQGWKHRULJLQ

2UWKRJRQDO6HWV

X1 D

E ,I \ LVDOLQHDUFRPELQDWLRQRIQRQ]HURYHFWRUVIURPDQ RUWKRJRQDOVHW WKHQWKHZHLJKWVLQWKHOLQHDUFRPELQDWLRQ FDQEHFRPSXWHGZLWKRXWURZRSHUDWLRQVRQDPDWUL[

F ,IWKHYHFWRUVLQDQRUWKRJRQDOVHWRIQRQ]HURYHFWRUVDUH QRUPDOL]HG WKHQVRPHRIWKHQHZYHFWRUVPD\QRWEH RUWKRJRQDO

,Q([HUFLVHV² GHWHUPLQHZKLFKVHWVRIYHFWRUVDUHRUWKRQRU PDO ,IDVHWLVRQO\RUWKRJRQDO QRUPDOL]HWKHYHFWRUVWRSURGXFH DQRUWKRQRUPDOVHW 2 3 2 3 2 3 2 3 1=3 #1=2 0 0 4 1=3 5 4 0 5 4 1 5 4 #1 5 1=3 1=2 0 0 2 3 2 3 ! " ! " #2=3 1=3 #:6 :8 4 1=3 5 4 2=3 5 :8 :6 2=3 0 p 3 2 2 p 3 2 3 0p 1=p10 3=p10 4 3=p20 5 4 #1=p20 5 4 #1=p2 5 1= 2 3= 20 #1= 20 p p 2 3 2 3 2 3 1=p18 1= 2 #2=3 0p 5 4 1=3 5 4 4=p18 5 4 #2=3 #1= 2 1= 18

,Q([HUFLVHVDQG DOOYHFWRUVDUHLQ Rn 0DUNHDFKVWDWHPHQW 7UXHRU)DOVH -XVWLI\HDFKDQVZHU D 1RWHYHU\OLQHDUO\LQGHSHQGHQWVHWLQ Rn LVDQRUWKRJRQDO VHW

G $ PDWUL[ ZLWK RUWKRQRUPDO FROXPQV LV DQ RUWKRJRQDO PDWUL[ H ,I L LVDOLQHWKURXJK DQGLI \O LVWKHRUWKRJRQDOSURMHFWLRQ RI \ RQWR L WKHQ kO\k JLYHVWKHGLVWDQFHIURP \ WR L D 1RWHYHU\RUWKRJRQDOVHWLQ Rn LVOLQHDUO\LQGHSHQGHQW E ,IDVHW S D fX1 ; : : : ; Xp g KDVWKHSURSHUW\WKDW Xi ! Xj D 0 ZKHQHYHU i ¤ j WKHQ S LVDQRUWKRQRUPDOVHW

F ,IWKHFROXPQVRIDQ m ! n PDWUL[ A DUHRUWKRQRUPDO WKHQ WKHOLQHDUPDSSLQJ [ 7! A[ SUHVHUYHVOHQJWKV

G 7KHRUWKRJRQDOSURMHFWLRQRI \ RQWR Y LVWKHVDPHDVWKH RUWKRJRQDOSURMHFWLRQRI \ RQWR c Y ZKHQHYHU c ¤ 0 H $QRUWKRJRQDOPDWUL[LVLQYHUWLEOH

3URYH7KHRUHP >+LQW )RUD FRPSXWH kU [k2 RUSURYH E ÀUVW@ 6XSSRVH W LV D VXEVSDFH RI Rn VSDQQHG E\ n QRQ]HUR RUWKRJRQDOYHFWRUV ([SODLQZK\ W D Rn /HW U EHDVTXDUHPDWUL[ZLWKRUWKRQRUPDOFROXPQV ([SODLQ ZK\ U LVLQYHUWLEOH 0HQWLRQWKHWKHRUHPV\RXXVH /HW U EHDQ n ! n RUWKRJRQDOPDWUL[ 6KRZWKDWWKHURZVRI U IRUPDQRUWKRQRUPDOEDVLVRI Rn /HW U DQG V EH n ! n RUWKRJRQDOPDWULFHV ([SODLQZK\ UV LVDQRUWKRJRQDOPDWUL[ >7KDWLV H[SODLQZK\ UV LV LQYHUWLEOHDQGLWVLQYHUVHLV .UV /T @ /HW U EHDQRUWKRJRQDOPDWUL[ DQGFRQVWUXFW V E\LQWHU FKDQJLQJVRPHRIWKHFROXPQVRI U ([SODLQZK\ V LVDQ RUWKRJRQDOPDWUL[ 6KRZWKDWWKHRUWKRJRQDOSURMHFWLRQRIDYHFWRU \ RQWRDOLQH L WKURXJKWKHRULJLQLQ R2 GRHVQRWGHSHQGRQWKHFKRLFH RIWKHQRQ]HUR X LQ L XVHGLQWKHIRUPXODIRU \O 7RGR WKLV VXSSRVH \ DQG X DUHJLYHQDQG \O KDVEHHQFRPSXWHGE\ IRUPXOD LQWKLVVHFWLRQ 5HSODFH X LQWKDWIRUPXODE\ c X ZKHUH c LVDQXQVSHFLÀHGQRQ]HURVFDODU 6KRZWKDWWKHQHZ IRUPXODJLYHVWKHVDPH \O

/HW fY1 ; Y2 g EHDQRUWKRJRQDOVHWRIQRQ]HURYHFWRUV DQGOHW c1 c2 EHDQ\QRQ]HURVFDODUV 6KRZWKDW fc1 Y1 ; c2 Y2 g LVDOVR DQRUWKRJRQDOVHW 6LQFHRUWKRJRQDOLW\RIDVHWLVGHÀQHGLQ WHUPVRISDLUVRIYHFWRUV WKLVVKRZVWKDWLIWKHYHFWRUVLQ DQRUWKRJRQDOVHWDUHQRUPDOL]HG WKHQHZVHWZLOOVWLOOEH RUWKRJRQDO *LYHQ X ¤ LQ Rn OHW L D 6SDQ fXg 6KRZWKDWWKHPDS SLQJ [ 7! SURML [ LVDOLQHDUWUDQVIRUPDWLRQ

*LYHQ X ¤ LQ Rn OHW L D 6SDQ fXg )RU \ LQ Rn WKH UHÁHFWLRQRI\LQ L LVWKHSRLQW UHÁL \ GHÀQHGE\

$)"15&3


UHÁL \ D 2! SURML \ # \ 6HH WKH ÀJXUH ZKLFK VKRZV WKDW UHÁL \ LV WKH VXP RI \O D SURML \ DQG \O # \ 6KRZWKDWWKHPDSSLQJ \ 7! UHÁL \ LVDOLQHDUWUDQVIRUPDWLRQ x2

y L = Span{u} yˆ

y – yˆ

u

ref l L y x1

yˆ – y

7KHUHÁHFWLRQRI \ LQDOLQHWKURXJKWKHRULJLQ

2

#6 6 #1 6 6 3 6 6 6 AD6 6 2 6 6 #3 6 4 #2 1

#3 2 6 #3 #1 6 #1 2

6 1 3 6 2 3 2 1

3 1 #6 7 7 #2 7 7 #1 7 7 37 7 27 7 #3 5 6

>0@ ,QSDUWVD ²G OHW U EHWKHPDWUL[IRUPHGE\QRUPDO L]LQJHDFKFROXPQRIWKHPDWUL[ A LQ([HUFLVH D &RPSXWH U TU DQG U U T +RZGRWKH\GLIIHU" E *HQHUDWH D UDQGRP YHFWRU \ LQ R8 DQG FRPSXWH S D U U T\ DQG ] D \ # S ([SODLQZK\ S LVLQ &RO A 9HULI\WKDW ] LVRUWKRJRQDOWR S F 9HULI\WKDW ] LVRUWKRJRQDOWRHDFKFROXPQRI U

>0@ 6KRZWKDWWKHFROXPQVRIWKHPDWUL[ A DUHRUWKRJRQDO E\PDNLQJDQDSSURSULDWHPDWUL[FDOFXODWLRQ 6WDWHWKHFDO FXODWLRQ\RXXVH

G 1RWLFHWKDW \ D S C ] ZLWK S LQ &RO A ([SODLQZK\ ] LV LQ .&RO A/? 7KHVLJQLÀFDQFHRIWKLVGHFRPSRVLWLRQRI \ ZLOOEHH[SODLQHGLQWKHQH[WVHFWLRQ

40-65*0/4 50 13"$5*$& 130#-&.4 7KHYHFWRUVDUHRUWKRJRQDOEHFDXVH X1 ! X2 D #2=5 C 2=5 D 0 7KH\DUHXQLWYHFWRUVEHFDXVH

p p kX1 k2 D .#1= 5/2 C .2= 5/2 D 1=5 C 4=5 D 1 p p kX2 k2 D .2= 5/2 C .1= 5/2 D 4=5 C 1=5 D 1 ,QSDUWLFXODU WKHVHW fX1 ; X2 g LVOLQHDUO\LQGHSHQGHQW DQGKHQFHLVDEDVLVIRU R2 VLQFH WKHUHDUHWZRYHFWRUVLQWKHVHW ! " ! " 7 2 :KHQ \ D DQG X D 6 1 ! " ! " ! " 20 2 \!X 2 8 \O D XD D4 D 1 1 4 X!X 5

SG

.BTUFSJOH 0SUIPHPOBM #BTJT m

7KLVLVWKHVDPH \O IRXQGLQ([DPSOH 7KHRUWKRJRQDOSURMHFWLRQGRHVQRWVHHPWR GHSHQGRQWKH X FKRVHQRQWKHOLQH 6HH([HUFLVH 2 p 3 2 3 1=p2 2=3 ! p " 1 #3 2 U \ D 4 1= 2 #2=3 5 D 4 #7 5 6 2 0 1=3 2 3 !p " 3 2 $OVR IURP([DPSOH [ D DQG U [ D 4 #1 5 +HQFH 3 1

U [ !U \ D 3 C 7 C 2 D 12;

DQG

[ ! \ D #6 C 18 D 12

2UWKRJRQDO3URMHFWLRQV

035)0(0/"- 130+&$5*0/4 7KHRUWKRJRQDOSURMHFWLRQRIDSRLQWLQ R2 RQWRDOLQHWKURXJKWKHRULJLQKDVDQLPSRUWDQW DQDORJXHLQ Rn *LYHQDYHFWRU \ DQGDVXEVSDFH W LQ Rn WKHUHLVDYHFWRU \O LQ W VXFK WKDW \O LVWKHXQLTXHYHFWRULQ W IRUZKLFK \ # \O LVRUWKRJRQDOWR W DQG \O LV WKHXQLTXHYHFWRULQ W FORVHVWWR \ 6HH)LJ 7KHVHWZRSURSHUWLHVRI \O SURYLGHWKH NH\WRÀQGLQJOHDVWVTXDUHVVROXWLRQVRIOLQHDUV\VWHPV PHQWLRQHGLQWKHLQWURGXFWRU\ H[DPSOHIRUWKLVFKDSWHU 7KHIXOOVWRU\ZLOOEHWROGLQ6HFWLRQ 7RSUHSDUHIRUWKHÀUVWWKHRUHP REVHUYHWKDWZKHQHYHUDYHFWRU \ LVZULWWHQDVD OLQHDUFRPELQDWLRQRIYHFWRUV X1 ; : : : ; Xn LQ Rn WKHWHUPVLQWKHVXPIRU \ FDQEHJURXSHG LQWRWZRSDUWVVRWKDW \ FDQEHZULWWHQDV

y

\ D ]1 C ]2 W '*(63&

0

yˆ

ZKHUH ]1 LVDOLQHDUFRPELQDWLRQRIVRPHRIWKH Xi DQG ]2 LVDOLQHDUFRPELQDWLRQRI WKHUHVWRIWKH Xi 7KLVLGHDLVSDUWLFXODUO\XVHIXOZKHQ fX1 ; : : : ; Xn g LVDQRUWKRJRQDO EDVLV 5HFDOOIURP6HFWLRQ WKDW W ? GHQRWHVWKHVHWRIDOOYHFWRUVRUWKRJRQDOWRD VXEVSDFH W

&9".1-& /HW fX1 ; : : : ; X5 g EHDQRUWKRJRQDOEDVLVIRU R5 DQGOHW \ D c1 X1 C " " " C c5 X5 &RQVLGHUWKHVXEVSDFH W D 6SDQ fX1 ; X2 g DQGZULWH \ DVWKHVXPRIDYHFWRU ]1 LQ W DQGDYHFWRU ]2 LQ W ?

40-65*0/ :ULWH \ D c1 X1 C c2 X2 C c3 X3 C c4 X4 C c5 X5 „ ƒ‚ … „ ƒ‚ … ]1 ]2

ZKHUH

]1 D c1 X1 C c2 X2

LVLQ 6SDQ fX1 ; X2 g

DQG

]2 D c3 X3 C c4 X4 C c5 X5

LVLQ 6SDQ fX3 ; X4 ; X5 g:

7RVKRZWKDW ]2 LVLQ W ? LWVXIÀFHVWRVKRZWKDW ]2 LVRUWKRJRQDOWRWKHYHFWRUVLQWKH EDVLV fX1 ; X2 g IRU W 6HH6HFWLRQ 8VLQJSURSHUWLHVRIWKHLQQHUSURGXFW FRPSXWH ]2 ! X1 D .c3 X3 C c4 X4 C c5 X5 /! X1 D c3 X3 ! X1 C c4 X4 ! X1 C c5 X5 ! X1 D0 EHFDXVH X1 LVRUWKRJRQDOWR X3 X4 DQG X5 $ VLPLODUFDOFXODWLRQVKRZVWKDW ]2 ! X2 D 0 7KXV ]2 LVLQ W ? 7KHQH[WWKHRUHPVKRZVWKDWWKHGHFRPSRVLWLRQ \ D ]1 C ]2 LQ([DPSOH FDQEH FRPSXWHGZLWKRXWKDYLQJDQRUWKRJRQDOEDVLVIRU Rn ,WLVHQRXJKWRKDYHDQRUWKRJRQDO EDVLVRQO\IRU W

$)"15&3


5)&03&.

5IF 0SUIPHPOBM %FDPNQPTJUJPO 5IFPSFN /HW W EHDVXEVSDFHRI Rn 7KHQHDFK \ LQ Rn FDQEHZULWWHQXQLTXHO\LQWKH IRUP \ D \O C ] ZKHUH \O LVLQ W DQG ] LVLQ W ? ,QIDFW LI fX1 ; : : : ; Xp g LVDQ\RUWKRJRQDOEDVLV RI W WKHQ \ ! Xp \ ! X1 \O D X1 C " " " C Xp X 1 ! X1 Xp ! Xp

DQG ] D \ # \O

7KHYHFWRU \O LQ LVFDOOHGWKH RUWKRJRQDOSURMHFWLRQRI \ RQWR W DQGRIWHQLV ZULWWHQDV SURMW \ 6HH)LJ :KHQ W LVDRQHGLPHQVLRQDOVXEVSDFH WKHIRUPXODIRU \O PDWFKHVWKHIRUPXODJLYHQLQ6HFWLRQ y

z = y – yˆ

0

yˆ = projW y

W '*(63& 7KHRUWKRJRQDOSURMHFWLRQRI \

RQWR W

1300' /HW fX1 ; : : : ; Xp g EHDQ\RUWKRJRQDOEDVLVIRU W DQGGHÀQH \O E\ ñ 7KHQ \O LVLQ W EHFDXVH \O LVDOLQHDUFRPELQDWLRQRIWKHEDVLV X1 ; : : : ; Xp /HW ] D \ # \O 6LQFH X1 LVRUWKRJRQDOWR X2 ; : : : ; Xp LWIROORZVIURP WKDW '

( \ ! X1 ] ! X1 D .\ # \O /! X1 D \ ! X1 # X1 ! X 1 # 0 # " " " # 0 X1 ! X1 D \ ! X1 # \ ! X1 D 0

7KXV ] LVRUWKRJRQDOWR X1 6LPLODUO\ ] LVRUWKRJRQDOWRHDFK Xj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ñ :HPD\DVVXPHWKDW W LVQRWWKH]HURVXEVSDFH IRURWKHUZLVH W ? D Rn DQG LVVLPSO\ \ D C \ 7KHQH[WVHFWLRQZLOOVKRZWKDWDQ\QRQ]HURVXEVSDFHRI Rn KDVDQRUWKRJRQDOEDVLV


2

3 2 3 2 3 2 #2 1 &9".1-& /HW X1 D 4 5 5 X2 D 4 1 5 DQG \ D 4 2 5 2EVHUYHWKDW fX1 ; X2 g #1 1 3 LVDQRUWKRJRQDOEDVLVIRU W D 6SDQ fX1 ; X2 g :ULWH \ DVWKHVXPRIDYHFWRULQ W DQG DYHFWRURUWKRJRQDOWR W

40-65*0/ 7KHRUWKRJRQDOSURMHFWLRQRI \ RQWR W LV \ ! X1 \ ! X2 \O D X1 C X2 X 1 ! X1 X2 ! X2 2 3 2 3 2 3 2 3 2 3 2 #2 2 #2 #2=5 94 3 94 15 55C 4 15 D 55C 4 15 D 4 2 5 D 30 #1 6 30 30 1 #1 1 1=5 2 3 2 3 2 3 1 #2=5 7=5 \ # \O D 4 2 5 # 4 2 5 D 4 0 5 3 1=5 14=5

$OVR

7KHRUHP HQVXUHVWKDW \ # \O LVLQ W ? 7RFKHFNWKHFDOFXODWLRQV KRZHYHU LWLVDJRRG LGHDWRYHULI\WKDW \ # \O LVRUWKRJRQDOWRERWK X1 DQG X2 DQGKHQFHWRDOORI W 7KH GHVLUHGGHFRPSRVLWLRQRI \ LV 2 3 2 3 2 3 1 #2=5 7=5 \ D 425 D 4 2 5C4 0 5 3 1=5 14=5

$ *HRPHWULF,QWHUSUHWDWLRQRIWKH2UWKRJRQDO3URMHFWLRQ :KHQ W LVDRQHGLPHQVLRQDOVXEVSDFH WKHIRUPXOD IRU SURMW \ FRQWDLQVMXVWRQH WHUP 7KXV ZKHQ GLP W > 1 HDFKWHUPLQ LVLWVHOIDQRUWKRJRQDOSURMHFWLRQRI \ RQWRDRQHGLPHQVLRQDOVXEVSDFHVSDQQHGE\RQHRIWKH X·VLQWKHEDVLVIRU W )LJXUH LOOXVWUDWHVWKLVZKHQ W LVDVXEVSDFHRI R3 VSDQQHGE\ X1 DQG X2 +HUH \O 1 DQG \O 2 GHQRWH WKHSURMHFWLRQVRI \ RQWRWKHOLQHVVSDQQHGE\ X1 DQG X2 UHVSHFWLYHO\ 7KHRUWKRJRQDO SURMHFWLRQ \O RI \ RQWR W LVWKHVXPRIWKHSURMHFWLRQVRI \ RQWRRQHGLPHQVLRQDOVXE VSDFHVWKDWDUHRUWKRJRQDOWRHDFKRWKHU 7KHYHFWRU \O LQ)LJ FRUUHVSRQGVWRWKHYHFWRU \ LQ)LJ RI6HFWLRQ EHFDXVHQRZLWLV \O WKDWLVLQ W y u2 yˆ 2 y . u1 y . u2 ––––– ––––– u1 + u yˆ = u . . u = yˆ 1 + yˆ 2 u 1 1 2 u2 2

0 yˆ 1 u1

'*(63& 7KHRUWKRJRQDOSURMHFWLRQRI \ LVWKHVXPRI

LWVSURMHFWLRQVRQWRRQHGLPHQVLRQDOVXEVSDFHVWKDWDUH PXWXDOO\RUWKRJRQDO

$)"15&3


3URSHUWLHVRI2UWKRJRQDO3URMHFWLRQV ,I fX1 ; : : : ; Xp g LVDQRUWKRJRQDOEDVLVIRU W DQGLI \ KDSSHQVWREHLQ W WKHQWKH IRUPXODIRU SURMW \ LVH[DFWO\WKHVDPHDVWKHUHSUHVHQWDWLRQRI \ JLYHQLQ7KHRUHP LQ6HFWLRQ ,QWKLVFDVH SURMW \ D \ ,I \ LVLQ W D 6SDQ fX1 ; : : : ; Xp g WKHQ SURMW \ D \ 7KLVIDFWDOVRIROORZVIURPWKHQH[WWKHRUHP

5)&03&.

5IF #FTU "QQSPYJNBUJPO 5IFPSFN /HW W EHDVXEVSDFHRI Rn OHW \ EHDQ\YHFWRULQ Rn DQGOHW \O EHWKHRUWKRJRQDO SURMHFWLRQRI \ RQWR W 7KHQ \O LVWKHFORVHVWSRLQWLQ W WR \ LQWKHVHQVHWKDW IRUDOO Y LQ W GLVWLQFWIURP \O

k\ # \O k < k\ # Yk

7KHYHFWRU \O LQ7KHRUHP LVFDOOHG WKHEHVWDSSUR[LPDWLRQWR \ E\HOHPHQWVRI W /DWHUVHFWLRQVLQWKHWH[WZLOOH[DPLQHSUREOHPVZKHUHDJLYHQ \ PXVWEHUHSODFHG RU DSSUR[LPDWHG E\DYHFWRU Y LQVRPHÀ[HGVXEVSDFH W 7KHGLVWDQFHIURP \ WR Y JLYHQ E\ k\ # Yk FDQEHUHJDUGHGDVWKH´HUURUµRIXVLQJ Y LQSODFHRI \ 7KHRUHP VD\VWKDW WKLVHUURULVPLQLPL]HGZKHQ Y D \O ,QHTXDOLW\ OHDGVWRDQHZSURRIWKDW \O GRHVQRWGHSHQGRQWKHSDUWLFXODURUWKRJR QDOEDVLVXVHGWRFRPSXWHLW ,IDGLIIHUHQWRUWKRJRQDOEDVLVIRU W ZHUHXVHGWRFRQVWUXFW DQRUWKRJRQDOSURMHFWLRQRI \ WKHQWKLVSURMHFWLRQZRXOGDOVREHWKHFORVHVWSRLQWLQ W WR \ QDPHO\ \O

1300' 7DNH Y LQ W GLVWLQFWIURP \O 6HH)LJ 7KHQ \O # Y LVLQ W %\WKH2UWKRJRQDO 'HFRPSRVLWLRQ7KHRUHP \ # \O LVRUWKRJRQDOWR W ,QSDUWLFXODU \ # \O LVRUWKRJRQDO WR \O # Y ZKLFKLVLQ W 6LQFH \ # Y D .\ # \O / C .O\ # Y/ WKH3\WKDJRUHDQ7KHRUHPJLYHV

k\ # Yk2 D k\ # \O k2 C kO\ # Yk2 6HHWKHFRORUHGULJKWWULDQJOHLQ)LJ 7KHOHQJWKRIHDFKVLGHLVODEHOHG 1RZ kO\ # Yk2 > 0 EHFDXVH \O # Y ¤ DQGVRLQHTXDOLW\ IROORZVLPPHGLDWHO\ y ˆ ||y – y|| yˆ 0 W

||yˆ – v||

||y – v||

v

'*(63& 7KHRUWKRJRQDOSURMHFWLRQRI \

RQWR W LVWKHFORVHVWSRLQWLQ W WR \


2

3 2 3 2 3 2 #2 1 &9".1-& ,I X1 D 4 5 5 X2 D 4 1 5 \ D 4 2 5 DQG W D 6SDQ fX1 ; X2 g #1 1 3 DVLQ([DPSOH WKHQ WKHFORVHVWSRLQWLQ W WR \ LV 2 3 #2=5 \ ! X1 \ ! X2 \O D X1 C X2 D 4 2 5 X 1 ! X1 X2 ! X2 1=5

&9".1-& 7KHGLVWDQFHIURPDSRLQW \ LQ Rn WRDVXEVSDFH W LVGHÀQHGDVWKH

GLVWDQFHIURP \ WRWKHQHDUHVWSRLQWLQ W )LQGWKHGLVWDQFHIURP \ WR W D 6SDQ fX1 ; X2 g ZKHUH 2 3 2 3 2 3 #1 5 1 \ D 4 #5 5; X1 D 4 #2 5; X2 D 4 2 5 10 1 #1

40-65*0/ %\WKH%HVW$SSUR[LPDWLRQ7KHRUHP WKHGLVWDQFHIURP \ WR W LV k\ # \O k ZKHUH \O D SURMW \ 6LQFH fX1 ; X2 g LVDQRUWKRJRQDOEDVLVIRU W 2 3 2 3 2 3 5 1 #1 15 #21 14 7 #2 5 # 4 2 5 D 4 #8 5 \O D X1 C X2 D 30 6 2 2 #1 1 4 2 3 2 3 2 3 #1 #1 0 \ # \O D 4 #5 5 # 4 #8 5 D 4 3 5 10 4 6 k\ # \O k2 D 32 C 62 D 45 p p 7KHGLVWDQFHIURP \ WR W LV 45 D 3 5

7KHÀQDOWKHRUHPLQWKLVVHFWLRQVKRZVKRZIRUPXOD IRU SURMW \ LVVLPSOLÀHG ZKHQWKHEDVLVIRU W LVDQRUWKRQRUPDOVHW

5)&03&.

,I fX1 ; : : : ; Xp g LVDQRUWKRQRUPDOEDVLVIRUDVXEVSDFH W RI Rn WKHQ SURMW \ D .\ ! X1 /X1 C .\ ! X2 /X2 C " " " C .\ ! Xp /Xp

,I U D Œ X1 X2 " " " Xp ! WKHQ

SURMW \ D U U T\

IRUDOO \ LQ Rn

1300' )RUPXOD IROORZVLPPHGLDWHO\IURP LQ7KHRUHP $OVR VKRZV WKDW SURMW \ LVDOLQHDUFRPELQDWLRQRIWKHFROXPQVRI U XVLQJWKHZHLJKWV \ ! X1 \ ! X2 ; : : : ; \ ! Xp 7KHZHLJKWVFDQEHZULWWHQDV XT1 \; XT2 \; : : : ; XTp \ VKRZLQJWKDWWKH\ DUHWKHHQWULHVLQ U T\ DQGMXVWLI\LQJ WEB

6XSSRVH U LVDQ n ! p PDWUL[ZLWKRUWKRQRUPDOFROXPQV DQGOHW W EHWKHFROXPQ VSDFHRI U 7KHQ

U T U [ D Ip [ D [ T

U U \ D SURMW \

IRUDOO [ LQ Rp n

IRUDOO \ LQ R

7KHRUHP 7KHRUHP

,I U LVDQ n ! n VTXDUH PDWUL[ZLWKRUWKRQRUPDOFROXPQV WKHQ U LVDQ RUWKRJRQDO PDWUL[ WKHFROXPQVSDFH W LVDOORI Rn DQG U U T\ D I \ D \ IRUDOO \ LQ Rn $OWKRXJKIRUPXOD LVLPSRUWDQWIRUWKHRUHWLFDOSXUSRVHV LQSUDFWLFHLWXVXDOO\ LQYROYHVFDOFXODWLRQVZLWKVTXDUHURRWVRIQXPEHUVLQWKHHQWULHVRIWKH Xi / )RUPXOD LVUHFRPPHQGHGIRUKDQGFDOFXODWLRQV

$)"15&3


13"$5*$& 130#-&. 2 3 2 3 2 3 #7 #1 #9 /HW X1 D 4 1 5 X2 D 4 1 5 \ D 4 1 5 DQG W D 6SDQ fX1 ; X2 g 8VHWKHIDFW 4 #2 6 WKDW X1 DQG X2 DUHRUWKRJRQDOWRFRPSXWH SURMW \

&9&3$*4&4 ,Q([HUFLVHVDQG \RXPD\DVVXPHWKDW fX1 ; : : : ; X4 g LVDQ RUWKRJRQDOEDVLVIRU R4 2 3 2 3 2 3 2 3 0 3 1 5 6 17 657 6 07 6 #3 7 7 6 7 6 7 6 7 X1 D 6 4 #4 5 X2 D 4 1 5 X3 D 4 1 5 X4 D 4 #1 5 #1 1 #4 1 2 3 10 6 #8 7 7 [D6 4 2 5 :ULWH [ DVWKHVXPRIWZRYHFWRUV RQHLQ 0 6SDQ fX1 ; X2 ; X3 g DQGWKHRWKHU LQ 6SDQ fX4 g 2 3 2 3 2 3 2 3 1 #2 1 #1 627 6 17 6 17 6 17 7 6 7 6 7 6 7 X1 D 6 4 1 5 X2 D 4 #1 5 X3 D 4 #2 5 X4 D 4 1 5 1 1 #1 #2 2 3 4 6 57 7 YD6 4 #3 5 :ULWH Y DVWKHVXPRI WZRYHFWRUV RQHLQ 3 6SDQ fX1 g DQGWKHRWKHU LQ 6SDQ fX2 ; X3 ; X4 g ,Q([HUFLVHV² YHULI\WKDW fX1 ; X2 g LVDQRUWKRJRQDOVHW DQGWKHQ ÀQGWKHRUWKRJRQDOSURMHFWLRQRI \ RQWR 6SDQ fX1 ; X2 g 2 3 2 3 2 3 #1 1 #1 \ D 4 4 5 X1 D 4 1 5 X2 D 4 1 5 3 0 0 2 3 2 3 2 3 6 3 #4 \ D 4 3 5 X1 D 4 4 5 X2 D 4 3 5 #2 0 0 2 3 2 3 2 3 #1 3 1 \ D 4 2 5 X1 D 4 #1 5 X2 D 4 #1 5 6 2 #2 2 3 2 3 2 3 6 #4 0 \ D 4 4 5 X1 D 4 #1 5 X2 D 4 1 5 1 1 1

,Q([HUFLVHV² OHW W EHWKHVXEVSDFHVSDQQHGE\WKH X·V DQG ZULWH \ DVWKHVXPRIDYHFWRULQ W DQGDYHFWRURUWKRJRQDOWR W 2 3 2 3 2 3 1 1 5 \ D 4 3 5 X1 D 4 3 5 X2 D 4 1 5 #2 4 5

2

3 2 3 2 3 #1 1 #1 \ D 4 4 5 X1 D 4 1 5 X2 D 4 3 5 3 1 #2 2 3 2 3 2 3 2 3 4 1 #1 #1 6 37 617 6 37 6 07 7 6 7 6 7 6 7 \ D 6 4 3 5 X1 D 4 0 5 X2 D 4 1 5 X3 D 4 1 5 #1 1 #2 1 2 3 2 3 2 3 2 3 3 1 1 0 647 6 17 607 6 #1 7 6 7 6 7 6 7 6 7 \ D 4 5 X1 D 4 X D X D 5 05 2 415 3 4 15 6 #1 1 #1

,Q([HUFLVHVDQG ÀQGWKHFORVHVWSRLQWWR \ LQWKHVXEVSDFH W VSDQQHGE\ Y1 DQG Y2 2 3 2 3 2 3 3 3 1 617 6 17 6 #1 7 7 6 7 6 7 \ D 6 4 5 5 Y1 D 4 #1 5 Y2 D 4 1 5 1 1 #1 2 3 2 3 2 3 3 1 #4 6 #1 7 6 #2 7 6 17 7 6 7 6 7 \ D 6 4 1 5 Y1 D 4 #1 5 Y2 D 4 0 5 13 2 3

,Q([HUFLVHVDQG ÀQGWKHEHVWDSSUR[LPDWLRQWR ] E\YHFWRUV RIWKHIRUP c1 Y1 C c2 Y2 2 3 2 3 2 3 3 2 1 6 #7 7 6 #1 7 6 17 7 6 7 6 7 ] D 6 4 2 5 Y1 D 4 #3 5 Y2 D 4 0 5 3 1 #1 2 3 2 3 2 3 2 2 5 6 47 6 07 6 #2 7 7 6 7 6 7 ] D 6 4 0 5 Y1 D 4 #1 5 Y2 D 4 4 5 #1 2 #3 2 3 2 3 2 3 5 #3 #3 /HW \ D 4 #9 5 X1 D 4 #5 5 X2 D 4 2 5 )LQG WKH 5 1 1 GLVWDQFHIURP \ WRWKH SODQHLQ R3 VSDQQHGE\ X1 DQG X2

/HW \ Y1 DQG Y2 EHDVLQ([HUFLVH )LQGWKHGLVWDQFHIURP \ WRWKHVXEVSDFHRI R4 VSDQQHGE\ Y1 DQG Y2 2 3 2 3 2 3 4 2=3 #2=3 /HW \ D 4 8 5 X1 D 4 1=3 5 X2 D 4 2=3 5 DQG 1 2=3 1=3 W D 6SDQ fX1 ; X2 g

D /HW U D Œ X1

T

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n 0DUN HDFKVWDWHPHQW7UXHRU)DOVH -XVWLI\HDFKDQVZHU D ,I ] LV RUWKRJRQDO WR X1 DQG WR X2 DQG LI W D 6SDQ fX1 ; X2 g WKHQ ] PXVWEHLQ W ? E )RUHDFK \ DQGHDFKVXEVSDFH W WKHYHFWRU \ # SURMW \ LVRUWKRJRQDOWR W

F 7KHRUWKRJRQDOSURMHFWLRQ \O RI \ RQWRDVXEVSDFH W FDQ VRPHWLPHVGHSHQGRQWKHRUWKRJRQDOEDVLVIRU W XVHGWR FRPSXWH \O

G ,I \ LVLQDVXEVSDFH W WKHQWKHRUWKRJRQDOSURMHFWLRQRI \ RQWR W LV \ LWVHOI

H ,IWKHFROXPQVRIDQ n ! p PDWUL[ U DUHRUWKRQRUPDO WKHQ U U T\ LVWKHRUWKRJRQDOSURMHFWLRQRI \ RQWRWKHFROXPQ VSDFHRI U

X2 ! &RPSXWH U TU DQG U U T

E &RPSXWH SURMW \ DQG .U U /\ p " ! " ! 7 1=p10 X1 D DQG W D 6SDQ fX1 g /HW \ D 9 #3= 10 D /HW U EHWKH 2 ! 1 PDWUL[ZKRVHRQO\FROXPQLV X1 &RPSXWH U TU DQG U U T


D ,I W LVDVXEVSDFHRI Rn DQGLI Y LVLQERWK W DQG W ? WKHQ Y PXVWEHWKH]HURYHFWRU E ,QWKH2UWKRJRQDO'HFRPSRVLWLRQ7KHRUHP HDFKWHUPLQ IRUPXOD IRU \O LVLWVHOIDQRUWKRJRQDOSURMHFWLRQRI \ RQWRDVXEVSDFHRI W F ,I \ D ]1 C ]2 ZKHUH ]1 LVLQDVXEVSDFH W DQG ]2 LVLQ W ? WKHQ ]1 PXVWEHWKHRUWKRJRQDOSURMHFWLRQRI \ RQWR W G 7KHEHVWDSSUR[LPDWLRQWR \ E\HOHPHQWVRIDVXEVSDFH W LVJLYHQE\WKHYHFWRU \ # SURMW \

H ,I DQ n ! p PDWUL[ U KDV RUWKRQRUPDO FROXPQV WKHQ U U T[ D [ IRUDOO [ LQ Rn

/HW A EHDQ m ! n PDWUL[ 3URYHWKDWHYHU\YHFWRU [ LQ Rn FDQEHZULWWHQLQWKHIRUP [ D S C X ZKHUH S LVLQ 5RZ A DQG X LVLQ 1XO A $OVR VKRZWKDWLIWKHHTXDWLRQ A[ D E LVFRQVLVWHQW WKHQWKHUHLVDXQLTXH S LQ 5RZ A VXFKWKDW AS D E /HW W EH D VXEVSDFH RI Rn ZLWK DQ RUWKRJRQDO EDVLV fZ1 ; : : : ; Zp g DQGOHW fY1 ; : : : ; Yq g EHDQRUWKRJRQDOEDVLVIRU W ? D ([SODLQZK\ fZ1 ; : : : ; Zp ; Y1 ; : : : ; Yq g LVDQRUWKRJRQDO VHW E ([SODLQZK\WKHVHWLQSDUWD VSDQV Rn F 6KRZWKDW GLP W C GLP W ? D n

>0@ /HW U EHWKH 8 ! 4 PDWUL[LQ([HUFLVH LQ6HFWLRQ )LQGWKHFORVHVWSRLQWWR \ D .1; 1; 1; 1; 1; 1; 1; 1/ LQ &RO U :ULWH WKH NH\VWURNHV RU FRPPDQGV \RX XVH WR VROYH WKLV SUREOHP >0@ /HW U EHWKHPDWUL[LQ([HUFLVH )LQGWKHGLVWDQFH IURP E D .1; 1; 1; 1; #1; #1; #1; #1/ WR &RO U

40-65*0/ 50 13"$5*$& 130#-&. &RPSXWH SURMW \ D

88 #2 \ ! X1 \ ! X2 X1 C X2 D X1 C X2 X 1 ! X1 X2 ! X2 66 6

2 3 #7 44 15# D 3 4

2 3 2 3 #1 #9 14 15 D 4 15 D \ 3 #2 6

,QWKLVFDVH \ KDSSHQVWREHDOLQHDUFRPELQDWLRQRI X1 DQG X2 VR \ LVLQ W 7KHFORVHVW SRLQWLQ W WR \ LV \ LWVHOI


$)"15&3

5)& (3". 4$).*%5 130$&44 7KH *UDP²6FKPLGW SURFHVV LV D VLPSOH DOJRULWKP IRU SURGXFLQJ DQ RUWKRJRQDO RU RUWKRQRUPDOEDVLVIRUDQ\QRQ]HURVXEVSDFHRI Rn 7KHÀUVWWZRH[DPSOHVRIWKHSURFHVV DUHDLPHGDWKDQGFDOFXODWLRQ 2 3 2 3 3 1 &9".1-& /HW W D 6SDQ f[1 ; [2 g ZKHUH [1 D 4 6 5 DQG [2 D 4 2 5 &RQ 0 2 VWUXFWDQRUWKRJRQDOEDVLV fY1 ; Y2 g IRU W

x3

v2

W x2

0 p x1

v1 = x1 '*(63&

&RQVWUXFWLRQRIDQRUWKRJRQDO EDVLV fY1 ; Y2 g

x2

40-65*0/ 7KHVXEVSDFH W LVVKRZQLQ)LJ DORQJZLWK [1 [2 DQGWKHSURMHFWLRQ S RI [2 RQWR [1 7KHFRPSRQHQWRI [2 RUWKRJRQDOWR [1 LV [2 # S ZKLFKLVLQ W EHFDXVH LWLVIRUPHGIURP [2 DQGDPXOWLSOHRI [1 /HW Y1 D [1 DQG 2 3 2 3 2 3 1 3 0 15 [2 ! [1 [1 D 4 2 5 # 4 6 5 D 4 0 5 Y2 D [2 # S D [2 # [1 ! [1 45 0 2 2 7KHQ fY1 ; Y2 g LVDQRUWKRJRQDOVHWRIQRQ]HURYHFWRUVLQ W 6LQFH GLP W D 2 WKHVHW fY1 ; Y2 g LVDEDVLVIRU W 7KHQH[WH[DPSOHIXOO\LOOXVWUDWHVWKH*UDP²6FKPLGWSURFHVV 6WXG\LWFDUHIXOO\ 2 3 2 3 2 3 1 0 0 617 617 607 7 6 7 6 7 &9".1-& /HW [1 D 6 4 1 5 [2 D 4 1 5 DQG [3 D 4 1 5 7KHQ f[1 ; [2 ; [3 g LV 1 1 1 FOHDUO\OLQHDUO\LQGHSHQGHQWDQGWKXVLVDEDVLVIRUDVXEVSDFH W RI R4 &RQVWUXFWDQ RUWKRJRQDOEDVLVIRU W

40-65*0/ 6WHS /HW Y1 D [1 DQG W1 D 6SDQ f[1 g D 6SDQ fY1 g

6WHS /HW Y2 EHWKHYHFWRUSURGXFHGE\VXEWUDFWLQJIURP [2 LWVSURMHFWLRQRQWRWKH VXEVSDFH W1 7KDWLV OHW Y2 D [2 # SURMW1 [2 [2 ! Y1 D [2 # Y1 6LQFH Y1 D [1 Y !Y 2 3 1 12 3 2 3 0 1 #3=4 6 1 7 3 6 1 7 6 1=4 7 7 6 7 6 7 D6 4 1 5 # 4 4 1 5 D 4 1=4 5 1 1 1=4

$VLQ([DPSOH Y2 LVWKHFRPSRQHQWRI [2 RUWKRJRQDOWR [1 DQG fY1 ; Y2 g LVDQ RUWKRJRQDOEDVLVIRUWKHVXEVSDFH W2 VSDQQHGE\ [1 DQG [2 6WHS0 RSWLRQDO ,IDSSURSULDWH VFDOH Y2 WRVLPSOLI\ODWHUFRPSXWDWLRQV 6LQFH Y2 KDV IUDFWLRQDOHQWULHV LWLVFRQYHQLHQWWRVFDOHLWE\DIDFWRURIDQGUHSODFH fY1 ; Y2 g E\WKH RUWKRJRQDOEDVLV 2 3 2 3 1 #3 617 6 17 0 7 6 7 Y1 D 6 4 1 5; Y 2 D 4 1 5 1 1

7KH*UDP±6FKPLGW3URFHVV

6WHS /HW Y3 EHWKHYHFWRUSURGXFHGE\VXEWUDFWLQJIURP [3 LWVSURMHFWLRQRQWRWKH VXEVSDFH W2 8VHWKHRUWKRJRQDOEDVLV fY1 ; Y02 g WRFRPSXWHWKLVSURMHFWLRQRQWR W2 3URMHFWLRQRI [3 RQWR Y1

3URMHFWLRQRI [3 RQWR Y02

!

SURMW2 [3 D

!

[3 ! Y02 0 Y Y02 ! Y02 2

[3 ! Y1 Y1 C Y1 ! Y1

2 3 2 3 2 3 1 #3 0 6 7 6 7 26 1 7 7 C 2 6 1 7 D 6 2=3 7 D 6 4 5 4 5 4 1 2=3 5 4 1 12 1 1 2=3

7KHQ Y3 LVWKHFRPSRQHQWRI [3 RUWKRJRQDOWR W2 QDPHO\ 2 3 2 3 2 3 0 0 0 6 0 7 6 2=3 7 6 #2=3 7 7 6 7 6 7 Y3 D [3 # SURMW2 [3 D 6 4 1 5 # 4 2=3 5 D 4 1=3 5 1 2=3 1=3

6HH)LJ IRUDGLDJUDPRIWKLVFRQVWUXFWLRQ 2EVHUYHWKDW Y3 LVLQ W EHFDXVH [3 DQG SURMW2 [3 DUHERWKLQ W 7KXV fY1 ; Y02 ; Y3 g LVDQRUWKRJRQDOVHWRIQRQ]HURYHFWRUV DQGKHQFHDOLQHDUO\LQGHSHQGHQWVHWLQ W 1RWHWKDW W LVWKUHHGLPHQVLRQDOVLQFHLW ZDVGHÀQHGE\DEDVLVRIWKUHHYHFWRUV +HQFH E\WKH%DVLV7KHRUHPLQ6HFWLRQ fY1 ; Y02 ; Y3 g LVDQRUWKRJRQDOEDVLVIRU W v3

x3 v'2

0

v1 W2 = Span{v1, v'2}

projW x 3 2

'*(63& 7KHFRQVWUXFWLRQRI Y3 IURP [3

DQG W2

7KHSURRIRIWKHQH[WWKHRUHPVKRZVWKDWWKLVVWUDWHJ\UHDOO\ZRUNV 6FDOLQJRI YHFWRUVLVQRWPHQWLRQHGEHFDXVHWKDWLVXVHGRQO\WRVLPSOLI\KDQGFDOFXODWLRQV

5)&03&.

5IF (SBN4DINJEU 1SPDFTT *LYHQDEDVLV f[1 ; : : : ; [p g IRUDQRQ]HURVXEVSDFH W RI Rn GHÀQH Y1 D [1

[2 ! Y1 Y1 Y1 ! Y1 [3 ! Y1 [3 ! Y2 Y3 D [3 # Y1 # Y2 Y1 ! Y1 Y2 ! Y2 :: : [p ! Y1 [p ! Y 2 [ p ! Y p !1 Yp D [p # Y1 # Y2 # " " " # Yp!1 Y1 ! Y1 Y2 ! Y2 Yp!1 ! Yp!1 Y2 D [2 #

7KHQ fY1 ; : : : ; Yp g LVDQRUWKRJRQDOEDVLVIRU W ,QDGGLWLRQ 6SDQ fY1 ; : : : ; Yk g D 6SDQ f[1 ; : : : ; [k g

IRU 1 % k % p

$)"15&3


1300' )RU 1 % k % p OHW Wk D 6SDQ f[1 ; : : : ; [k g 6HW Y1 D [1 VRWKDW 6SDQ fY1 g D 6SDQ f[1 g 6XSSRVH IRU VRPH k < p ZH KDYH FRQVWUXFWHG Y1 ; : : : ; Yk VR WKDW fY1 ; : : : ; Yk g LVDQRUWKRJRQDOEDVLVIRU Wk 'HÀQH Yk C1 D [k C1 # SURMWk [k C1

%\WKH2UWKRJRQDO'HFRPSRVLWLRQ7KHRUHP Yk C1 LVRUWKRJRQDOWR Wk 1RWHWKDW SURMWk [k C1 LVLQ Wk DQGKHQFHDOVRLQ Wk C1 6LQFH [k C1 LVLQ Wk C1 VRLV Yk C1 EHFDXVH Wk C1 LVDVXEVSDFHDQGLVFORVHGXQGHUVXEWUDFWLRQ )XUWKHUPRUH Yk C1 ¤ EHFDXVH [k C1 LVQRWLQ Wk D 6SDQ f[1 ; : : : ; [k g +HQFH fY1 ; : : : ; Yk C1 g LVDQRUWKRJRQDOVHWRI QRQ]HURYHFWRUVLQWKH .k C 1/GLPHQVLRQDOVSDFH Wk C1 %\WKH%DVLV7KHRUHPLQ6HF WLRQ WKLVVHWLVDQRUWKRJRQDOEDVLVIRU Wk C1 +HQFH Wk C1 D 6SDQ fY1 ; : : : ; Yk C1 g :KHQ k C 1 D p WKHSURFHVVVWRSV 7KHRUHP VKRZVWKDWDQ\QRQ]HURVXEVSDFH W RI Rn KDVDQRUWKRJRQDOEDVLV EH FDXVHDQRUGLQDU\EDVLV f[1 ; : : : ; [p g LVDOZD\VDYDLODEOHE\7KHRUHP LQ6HFWLRQ DQGWKH*UDP²6FKPLGWSURFHVVGHSHQGVRQO\RQWKHH[LVWHQFHRIRUWKRJRQDOSURMHFWLRQV RQWRVXEVSDFHVRI W WKDWDOUHDG\KDYHRUWKRJRQDOEDVHV

2UWKRQRUPDO%DVHV $Q RUWKRQRUPDO EDVLV LV FRQVWUXFWHG HDVLO\ IURP DQ RUWKRJRQDO EDVLV fY1 ; : : : ; Yp g VLPSO\QRUPDOL]HLH ´VFDOHµ DOOWKH Yk :KHQZRUNLQJSUREOHPVE\KDQG WKLVLV HDVLHUWKDQQRUPDOL]LQJHDFK Yk DVVRRQDVLWLVIRXQGEHFDXVHLWDYRLGVXQQHFHVVDU\ ZULWLQJRIVTXDUHURRWV

&9".1-& ([DPSOH FRQVWUXFWHGWKHRUWKRJRQDOEDVLV 2 3 3 Y 1 D 4 6 5; 0

2 3 0 Y2 D 4 0 5 2

$QRUWKRQRUPDOEDVLVLV 2 3 2 p 3 3 1= 5 1 1 4 5 4 p 5 6 D 2= 5 X1 D Y1 D p kY1 k 45 0 0

2 3 0 1 X2 D Y2 D 4 0 5 kY2 k 1

45 )DFWRUL]DWLRQRI0DWULFHV WEB

,IDQ m ! n PDWUL[ A KDVOLQHDUO\LQGHSHQGHQWFROXPQV [1 ; : : : ; [n WKHQDSSO\LQJWKH *UDP²6FKPLGWSURFHVVZLWKQRUPDOL]DWLRQV WR [1 ; : : : ; [n DPRXQWVWR IDFWRULQJ A DV GHVFULEHGLQWKHQH[WWKHRUHP 7KLVIDFWRUL]DWLRQLVZLGHO\XVHGLQFRPSXWHUDOJRULWKPV IRUYDULRXVFRPSXWDWLRQV VXFKDVVROYLQJHTXDWLRQVGLVFXVVHGLQ6HFWLRQ DQG ÀQGLQJHLJHQYDOXHVPHQWLRQHGLQWKHH[HUFLVHVIRU6HFWLRQ

7KH*UDP±6FKPLGW3URFHVV

5)&03&.

5IF 23 'BDUPSJ[BUJPO ,I A LVDQ m ! n PDWUL[ZLWKOLQHDUO\LQGHSHQGHQWFROXPQV WKHQ A FDQEHIDFWRUHG DV A D 45 ZKHUH Q LVDQ m ! n PDWUL[ZKRVHFROXPQVIRUPDQRUWKRQRUPDO EDVLVIRU &RO A DQG R LVDQ n ! n XSSHUWULDQJXODULQYHUWLEOHPDWUL[ZLWKSRVLWLYH HQWULHVRQLWVGLDJRQDO

1300' 7KHFROXPQVRI A IRUPDEDVLV f[1 ; : : : ; [n g IRU &RO A &RQVWUXFWDQRUWKRQRU PDOEDVLV fX1 ; : : : ; Xn g IRU W D &RO A ZLWKSURSHUW\ LQ7KHRUHP 7KLVEDVLVPD\ EHFRQVWUXFWHGE\WKH*UDP²6FKPLGWSURFHVVRUVRPHRWKHUPHDQV /HW Q D Œ X1 X2 " " " X n !

)RU k D 1; : : : ; n; [k LVLQ 6SDQ f[1 ; : : : ; [k g D 6SDQ fX1 ; : : : ; Xk g 6RWKHUHDUHFRQ VWDQWV r1k ; : : : ; rkk VXFKWKDW [k D r1k X1 C " " " C rkk Xk C 0! Xk C1 C " " " C 0! Xn

:HPD\DVVXPHWKDW rkk $ 0 ,I rkk < 0 PXOWLSO\ERWK rkk DQG Xk E\ #1 7KLVVKRZV WKDW [k LVDOLQHDUFRPELQDWLRQRIWKHFROXPQVRI Q XVLQJDVZHLJKWVWKHHQWULHVLQWKH YHFWRU 2 3 r1k 6 :: 7 6 : 7 6 7 6 7 Uk D 6 rkk 7 6 0 7 6 : 7 4 :: 5 0 7KDWLV [k D QUk IRU k D 1; : : : ; n /HW R D Œ U1 " " " Un ! 7KHQ

A D Œ [1 " " " [n ! D Œ QU1 " " " QUn ! D 45

7KHIDFWWKDW R LVLQYHUWLEOHIROORZVHDVLO\IURPWKHIDFWWKDWWKHFROXPQVRI A DUHOLQHDUO\ LQGHSHQGHQW([HUFLVH 6LQFH R LVFOHDUO\XSSHUWULDQJXODU LWVQRQQHJDWLYHGLDJRQDO HQWULHVPXVWEHSRVLWLYH 2

1 61 &9".1-& )LQGD45 IDFWRUL]DWLRQRI A D 6 41 1

0 1 1 1

3 0 07 7 15 1

40-65*0/ 7KHFROXPQVRI A DUHWKHYHFWRUV [1 [2 DQG [3 LQ([DPSOH $Q RUWKRJRQDOEDVLVIRU &RO A D 6SDQ f[1 ; [2 ; [3 g ZDVIRXQGLQWKDWH[DPSOH 2 3 2 3 2 3 1 #3 0 617 6 17 6 #2=3 7 0 7 6 7 6 7 Y1 D 6 4 1 5; Y2 D 4 1 5; Y3 D 4 1=3 5 1 1 1=3

7RVLPSOLI\WKHDULWKPHWLFWKDWIROORZV VFDOH Y3 E\OHWWLQJ Y03 D 3Y3 7KHQQRUPDOL]H WKHWKUHHYHFWRUVWRREWDLQ X1 X2 DQG X3 DQGXVHWKHVHYHFWRUVDVWKHFROXPQVRI Q 2 3 p 1=2 #3=p12 0p 6 7 #2=p6 7 1=p12 6 1=2 QD6 7 1=p6 5 1=p12 4 1=2 1= 6 1=2 1= 12

$)"15&3


%\FRQVWUXFWLRQ WKHÀUVW k FROXPQVRI Q DUHDQRUWKRQRUPDOEDVLVRI 6SDQ f[1 ; : : : ; [k g )URPWKHSURRIRI7KHRUHP A D 45 IRUVRPH R 7RÀQG R REVHUYHWKDW QTQ D I EHFDXVHWKHFROXPQVRI Q DUHRUWKRQRUPDO +HQFH

QTA D QT .45/ D IR D R DQG 2

1=2 1=2 1=2 p p p 4 R D #3= 12 1= p12 1=p12 1= 6 0 #2= 6 2 3 2 3=2 1 p p D 4 0 3= 12 2=p12 5 0 0 2= 6

2 3 1 1=2 p 61 1=p12 56 41 1= 6 1

0 1 1 1

3 0 07 7 15 1

/6.&3*$"- /05&4 :KHQWKH*UDP²6FKPLGWSURFHVVLVUXQRQDFRPSXWHU URXQGRIIHUURUFDQ EXLOGXSDVWKHYHFWRUV Xk DUHFDOFXODWHG RQHE\RQH )RU j DQG k ODUJHEXW XQHTXDO WKHLQQHUSURGXFWV XTj Xk PD\QRWEHVXIÀFLHQWO\FORVHWR]HUR 7KLV ORVVRIRUWKRJRQDOLW\FDQEHUHGXFHGVXEVWDQWLDOO\E\UHDUUDQJLQJWKHRUGHU RIWKHFDOFXODWLRQV +RZHYHU DGLIIHUHQWFRPSXWHUEDVHG45 IDFWRUL]DWLRQLV XVXDOO\SUHIHUUHGWRWKLVPRGLÀHG*UDP²6FKPLGWPHWKRGEHFDXVHLW\LHOGVD PRUHDFFXUDWHRUWKRQRUPDOEDVLV HYHQWKRXJKWKHIDFWRUL]DWLRQUHTXLUHVDERXW WZLFHDVPXFKDULWKPHWLF 7RSURGXFHD45 IDFWRUL]DWLRQRIDPDWUL[ A DFRPSXWHUSURJUDPXVXDOO\ OHIWPXOWLSOLHV A E\DVHTXHQFHRIRUWKRJRQDOPDWULFHVXQWLO A LVWUDQVIRUPHG LQWRDQXSSHUWULDQJXODUPDWUL[ 7KLVFRQVWUXFWLRQLVDQDORJRXVWRWKHOHIW PXOWLSOLFDWLRQE\HOHPHQWDU\PDWULFHVWKDWSURGXFHVDQ/8 IDFWRUL]DWLRQRI A

13"$5*$& 130#-&.

2 3 2 3 1 1=3 /HW W D 6SDQ f[1 ; [2 g ZKHUH [1 D 4 1 5 DQG [2 D 4 1=3 5 &RQVWUXFWDQRUWKRQRU 1 #2=3 PDOEDVLVIRU W

&9&3$*4&4 ,Q([HUFLVHV² WKHJLYHQVHWLVDEDVLVIRUDVXEVSDFH W 8VH WKH*UDP²6FKPLGWSURFHVVWRSURGXFHDQRUWKRJRQDOEDVLVIRU W 2 3 2 3 2 3 2 3 3 8 0 5 4 0 5 4 5 5 4 4 5 4 6 5 #1 #6 2 #7

2

3 2 3 2 4 4 #5 5 4 #1 5 1 2 2 3 2 3 1 7 6 #4 7 6 #7 7 7 6 7 6 4 0 5 4 #4 5 1 1

2

3 2 3 3 #3 4 #4 5 4 14 5 5 #7 2 3 2 3 3 #5 6 #1 7 6 9 7 7 6 7 6 4 2 5 4 #9 5 #1 3

ñ 6HH )XQGDPHQWDOVRI0DWUL[&RPSXWDWLRQV E\'DYLG6 :DWNLQV1HZ
7KH*UDP±6FKPLGW3URFHVV )LQGDQRUWKRQRUPDOEDVLVRIWKHVXEVSDFHVSDQQHGE\WKH YHFWRUVLQ([HUFLVH )LQGDQRUWKRQRUPDOEDVLVRIWKHVXEVSDFHVSDQQHGE\WKH YHFWRUVLQ([HUFLVH )LQGDQRUWKRJRQDOEDVLVIRUWKHFROXPQVSDFHRIHDFKPDWUL[LQ ([HUFLVHV² 2 3 2 3 3 #5 1 #1 6 6 6 1 6 3 #8 1 17 37 7 7 6 6 4 #1 5 4 5 #2 1 #2 65 3 #7 8 1 #4 #3 2 3 2 3 1 2 5 1 3 5 6 #1 6 #1 #3 1 #4 7 17 6 7 6 7 7 6 0 #1 4 #3 2 37 6 6 7 6 7 4 1 #4 4 1 75 5 25 1 2 1 1 5 8

,Q([HUFLVHVDQG WKHFROXPQVRI Q ZHUHREWDLQHGE\ DSSO\LQJWKH*UDP²6FKPLGWSURFHVVWRWKHFROXPQVRI A )LQGDQ XSSHUWULDQJXODUPDWUL[ R VXFKWKDW A D QR &KHFN\RXUZRUN 2 3 2 3 5 9 5=6 #1=6 6 1 6 1=6 77 5=6 7 7 6 7 A D 6 4 #3 #5 5 Q D 4 #3=6 1=6 5 1 5 1=6 3=6 2 3 2 3 #2 3 #2=7 5=7 6 5 7 6 77 2=7 7 6 5=7 7 A D 6 4 2 #2 5 Q D 4 2=7 #4=7 5 4 6 4=7 2=7 )LQGD45 IDFWRUL]DWLRQRIWKHPDWUL[LQ([HUFLVH

)LQGD45 IDFWRUL]DWLRQRIWKHPDWUL[LQ([HUFLVH ,Q([HUFLVHVDQG DOOYHFWRUVDQGVXEVSDFHVDUHLQ Rn 0DUN HDFKVWDWHPHQW7UXHRU)DOVH -XVWLI\HDFKDQVZHU D ,I fY1 ; Y2 ; Y3 g LVDQRUWKRJRQDOEDVLVIRU W WKHQPXO WLSO\LQJ Y3 E\DVFDODU c JLYHVDQHZRUWKRJRQDOEDVLV fY 1 ; Y 2 ; c Y 3 g

E 7KH*UDP²6FKPLGWSURFHVVSURGXFHVIURPDOLQHDUO\LQ GHSHQGHQWVHW f[1 ; : : : ; [p g DQRUWKRJRQDOVHW fY1 ; : : : ; Yp g ZLWKWKHSURSHUW\WKDWIRUHDFK k WKHYHFWRUV Y1 ; : : : ; Yk VSDQWKHVDPHVXEVSDFHDVWKDWVSDQQHGE\ [1 ; : : : ; [k F ,I A D QR ZKHUH Q KDVRUWKRQRUPDOFROXPQV WKHQ R D QTA

D ,I W D 6SDQ f[1 ; [2 ; [3 g ZLWK f[1 ; [2 ; [3 g OLQHDUO\LQGH SHQGHQW DQGLI fY1 ; Y2 ; Y3 g LVDQRUWKRJRQDOVHWLQ W WKHQ fY1 ; Y2 ; Y3 g LVDEDVLVIRU W E ,I [ LVQRWLQDVXEVSDFH W WKHQ [ # SURMW [ LVQRW]HUR

F ,QD45 IDFWRUL]DWLRQ VD\ A D QR ZKHQ A KDVOLQ HDUO\LQGHSHQGHQWFROXPQV WKHFROXPQVRI Q IRUPDQ RUWKRQRUPDOEDVLVIRUWKHFROXPQVSDFHRI A

6XSSRVH A D QR ZKHUH Q LV m ! n DQG R LV n ! n 6KRZ WKDWLIWKHFROXPQVRI A DUHOLQHDUO\LQGHSHQGHQW WKHQ R PXVW EHLQYHUWLEOH >+LQW 6WXG\WKHHTXDWLRQ R[ D DQGXVHWKH IDFWWKDW A D QR@

6XSSRVH A D QR ZKHUH R LVDQLQYHUWLEOHPDWUL[ 6KRZ WKDW A DQG Q KDYHWKHVDPHFROXPQVSDFH >+LQW *LYHQ \ LQ &RO A VKRZWKDW \ D Q[ IRUVRPH [ $OVR JLYHQ \ LQ &RO Q VKRZWKDW \ D A[ IRUVRPH [@

*LYHQ A D QR DVLQ7KHRUHP GHVFULEHKRZWRÀQGDQ RUWKRJRQDO m ! m VTXDUH PDWUL[ Q1 DQGDQLQYHUWLEOH n ! n XSSHUWULDQJXODUPDWUL[ R VXFKWKDW ! " R A D Q1 0

7KH0$7/$% qr FRPPDQGVXSSOLHVWKLVÍXOOµ45 IDFWRU L]DWLRQZKHQ UDQN A D n

/HW X1 ; : : : ; Xp EHDQRUWKRJRQDOEDVLVIRUDVXEVSDFH W RI Rn DQGOHW T W Rn ! Rn EHGHÀQHGE\ T .[/ D SURMW [ 6KRZWKDW T LVDOLQHDUWUDQVIRUPDWLRQ

6XSSRVH A D QR LVD45 IDFWRUL]DWLRQRIDQ m ! n PD WUL[ A ZLWKOLQHDUO\LQGHSHQGHQWFROXPQV 3DUWLWLRQ A DV ŒA1 A2 ! ZKHUH A1 KDV p FROXPQV 6KRZKRZWRREWDLQD 45 IDFWRUL]DWLRQRI A1 DQGH[SODLQZK\\RXUIDFWRUL]DWLRQ KDVWKHDSSURSULDWHSURSHUWLHV >0@ 8VH WKH *UDP²6FKPLGW SURFHVV DV LQ ([DPSOH WR SURGXFHDQRUWKRJRQDOEDVLVIRUWKHFROXPQVSDFHRI 2 3 #10 13 7 #11 6 2 1 #5 37 6 7 7 #6 3 13 #3 AD6 6 7 4 16 #16 #2 55 2 1 #5 #7

>0@ 8VHWKHPHWKRGLQWKLVVHFWLRQWRSURGXFHD45 IDFWRU L]DWLRQRIWKHPDWUL[LQ([HUFLVH >0@ )RUDPDWUL[SURJUDP WKH*UDP²6FKPLGWSURFHVVZRUNV EHWWHUZLWKRUWKRQRUPDOYHFWRUV 6WDUWLQJZLWK [1 ; : : : ; [p DV LQ7KHRUHP OHW A D Œ [1 " " " x p ! 6XSSRVH Q LVDQ n ! k PDWUL[ZKRVHFROXPQVIRUPDQRUWKRQRUPDOEDVLVIRU WKHVXEVSDFH Wk VSDQQHGE\WKHÀUVW k FROXPQVRI A 7KHQ IRU [ LQ Rn QQT [ LVWKHRUWKRJRQDOSURMHFWLRQRI [ RQWR Wk 7KHRUHP LQ6HFWLRQ ,I [kC1 LVWKHQH[WFROXPQRI A WKHQHTXDWLRQ LQWKHSURRIRI7KHRUHP EHFRPHV YkC1 D [kC1 # Q.QT [kC1 / 7KH SDUHQWKHVHV DERYH UHGXFH WKH QXPEHU RI DULWKPHWLF RSHUDWLRQV /HW XkC1 D YkC1 =kYkC1 k 7KHQHZ Q IRUWKH QH[WVWHSLV Œ Q XkC1 ! 8VHWKLVSURFHGXUHWRFRPSXWHWKH 45 IDFWRUL]DWLRQRIWKHPDWUL[LQ([HUFLVH :ULWHWKH NH\VWURNHVRUFRPPDQGV\RXXVH WEB

$)"15&3


40-65*0/ 50 13"$5*$& 130#-&. 2 3 1 [2 ! Y1 /HW Y1 D [1 D 4 1 5 DQG Y2 D [2 # Y1 D [2 # 0Y1 D [2 6R f[1 ; [2 g LVDOUHDG\ Y1 ! Y1 1 RUWKRJRQDO $OOWKDWLVQHHGHGLVWRQRUPDOL]HWKHYHFWRUV /HW 2 3 2 p 3 1= 3 1 1 1 4 5 4 p 5 1 D 1=p3 X1 D Y1 D p kY1 k 3 1 1= 3 ,QVWHDGRIQRUPDOL]LQJ Y2 GLUHFWO\ QRUPDOL]H Y02 D 3Y2 LQVWHDG p 3 2 3 2 1=p6 1 1 1 4 1 5 D 4 1= 6 5 X2 D 0 Y02 D p p 2 kY2 k 1 C 12 C .#2/2 #2 #2= 6 7KHQ fX1 ; X2 g LVDQRUWKRQRUPDOEDVLVIRU W

-&"45426"3&4 130#-&.4 7KHFKDSWHU·VLQWURGXFWRU\H[DPSOHGHVFULEHGDPDVVLYHSUREOHP A[ D E WKDWKDGQR VROXWLRQ ,QFRQVLVWHQWV\VWHPVDULVHRIWHQLQDSSOLFDWLRQV WKRXJKXVXDOO\QRWZLWKVXFK DQHQRUPRXVFRHIÀFLHQWPDWUL[ :KHQDVROXWLRQLVGHPDQGHGDQGQRQHH[LVWV WKHEHVW RQHFDQGRLVWRÀQGDQ [ WKDWPDNHV A[ DVFORVHDVSRVVLEOHWR E 7KLQNRI A[ DVDQ DSSUR[LPDWLRQ WR E 7KHVPDOOHUWKHGLVWDQFHEHWZHHQ E DQG A[ JLYHQE\ kE # A[k WKHEHWWHUWKHDSSUR[LPDWLRQ 7KH JHQHUDOOHDVWVTXDUHVSUREOHP LVWRÀQGDQ [ WKDWPDNHV kE # A[k DVVPDOODVSRVVLEOH 7KHDGMHFWLYHÓHDVWVTXDUHVµ DULVHVIURPWKHIDFWWKDW kE # A[k LVWKHVTXDUHURRWRIDVXPRIVTXDUHV

%&'*/*5*0/

,I A LV m ! n DQG E LVLQ Rm D OHDVWVTXDUHVVROXWLRQ RI A[ D E LVDQ [O LQ Rn VXFKWKDW kE # AO[k % kE # A[k IRUDOO [ LQ Rn

7KHPRVWLPSRUWDQWDVSHFWRIWKHOHDVWVTXDUHVSUREOHPLVWKDWQRPDWWHUZKDW [ ZH VHOHFW WKHYHFWRU A[ ZLOOQHFHVVDULO\EHLQWKHFROXPQVSDFH &RO A 6RZHVHHNDQ [ WKDWPDNHV A[ WKHFORVHVWSRLQWLQ &RO A WR E 6HH)LJ 2IFRXUVH LI E KDSSHQVWREH LQ &RO A WKHQ E LV A[ IRUVRPH [ DQGVXFKDQ [ LVDÓHDVWVTXDUHVVROXWLRQµ b

Axˆ

0 Col A

Ax

Ax

'*(63& 7KHYHFWRU E LVFORVHUWR AO[ WKDQ

WR A[ IRURWKHU [

/HDVW6TXDUHV3UREOHPV

6ROXWLRQRIWKH*HQHUDO/HDVW6TXDUHV3UREOHP *LYHQ A DQG E DVDERYH DSSO\WKH%HVW$SSUR[LPDWLRQ7KHRUHPLQ6HFWLRQ WRWKH VXEVSDFH &RO A /HW EO D SURM&RO A E

%HFDXVH EO LVLQWKHFROXPQVSDFHRI A WKHHTXDWLRQ A[ D EO LV FRQVLVWHQW DQGWKHUHLV DQ [O LQ Rn VXFKWKDW AO[ D EO

6LQFH EO LVWKHFORVHVWSRLQWLQ &RO A WR E DYHFWRU [O LVDOHDVWVTXDUHVVROXWLRQRI A[ D E LIDQGRQO\LI [O VDWLVÀHV 6XFKDQ [O LQ Rn LVDOLVWRIZHLJKWVWKDWZLOOEXLOG EO RXWRI WKHFROXPQVRI A 6HH)LJ >7KHUHDUHPDQ\VROXWLRQVRI LIWKHHTXDWLRQKDVIUHH YDULDEOHV@ b

b – Axˆ

ˆ = Axˆ b

0

subspace of !m

Col A xˆ

A

!n '*(63& 7KHOHDVWVTXDUHVVROXWLRQ [O LVLQ Rn

O %\WKH2UWKRJRQDO'HFRPSRVLWLRQ7KHRUHPLQ6HF 6XSSRVH [O VDWLVÀHV AO[ D E WLRQ WKHSURMHFWLRQ EO KDVWKHSURSHUW\WKDW E # EO LVRUWKRJRQDOWR &RO A VR E # AO[ LVRUWKRJRQDOWRHDFKFROXPQRI A ,I Dj LVDQ\FROXPQRI A WKHQ Dj !.E # AO[/ D 0 DQG DTj .E # AO[/ D 0 6LQFHHDFK DTj LVDURZRI AT AT .E # AO[/ D

7KLVHTXDWLRQDOVRIROORZVIURP7KHRUHPLQ6HFWLRQ 7KXV

AT E # ATAO[ D ATAO[ D AT E 7KHVHFDOFXODWLRQVVKRZWKDWHDFKOHDVWVTXDUHVVROXWLRQRI A[ D E VDWLVÀHVWKHHTXDWLRQ

ATA[ D AT E

7KHPDWUL[HTXDWLRQ UHSUHVHQWVDV\VWHPRIHTXDWLRQVFDOOHGWKH QRUPDOHTXDWLRQV IRU A[ D E $ VROXWLRQRI LVRIWHQGHQRWHGE\ [O

5)&03&.

7KHVHWRIOHDVWVTXDUHVVROXWLRQVRI A[ D E FRLQFLGHVZLWKWKHQRQHPSW\VHWRI VROXWLRQVRIWKHQRUPDOHTXDWLRQV ATA[ D AT E

1300' $VVKRZQDERYH WKHVHWRIOHDVWVTXDUHVVROXWLRQVLVQRQHPSW\DQGHDFK OHDVWVTXDUHVVROXWLRQ [O VDWLVÀHVWKHQRUPDOHTXDWLRQV &RQYHUVHO\ VXSSRVH [O VDWLVÀHV ATAO[ D AT E 7KHQ [O VDWLVÀHV DERYH ZKLFKVKRZVWKDW E # AO[ LVRUWKRJRQDOWRWKH

$)"15&3


URZVRI AT DQGKHQFHLVRUWKRJRQDOWRWKHFROXPQVRI A 6LQFHWKHFROXPQVRI A VSDQ &RO A WKHYHFWRU E # AO[ LVRUWKRJRQDOWRDOORI &RO A +HQFHWKHHTXDWLRQ E D AO[ C .E # AO[/ LVDGHFRPSRVLWLRQRI E LQWRWKHVXPRIDYHFWRULQ &RO A DQGDYHFWRURUWKRJRQDOWR &RO A %\WKHXQLTXHQHVVRIWKHRUWKRJRQDOGHFRPSRVLWLRQ AO[ PXVWEHWKHRUWKRJRQDO O DQG [O LVDOHDVWVTXDUHVVROXWLRQ SURMHFWLRQRI E RQWR &RO A 7KDWLV AO[ D E

&9".1-& )LQGDOHDVWVTXDUHVVROXWLRQRIWKHLQFRQVLVWHQWV\VWHP A[ D E IRU 2

4 A D 40 1

3 0 2 5; 1

2

3 2 E D 4 05 11

40-65*0/ 7RXVHQRUPDOHTXDWLRQV FRPSXWH 3 2 ! " 4 ! 0 4 0 1 4 17 0 25 D ATA D 0 2 1 1 1 1 2 3 ! " 2 ! " 4 0 1 4 5 19 0 D AT E D 0 2 1 11 11

1 5

"

7KHQWKHHTXDWLRQ ATA[ D AT E EHFRPHV ! "! " ! " 17 1 x1 19 D 1 5 x2 11

5RZRSHUDWLRQVFDQEHXVHGWRVROYHWKLVV\VWHP EXWVLQFH ATA LVLQYHUWLEOHDQG 2 ! 2 LWLVSUREDEO\IDVWHUWRFRPSXWH ! " 1 5 #1 T !1 .A A/ D 84 #1 17 DQGWKHQWRVROYH ATA[ D AT E DV [O D .ATA/!1 AT E ! "! " ! " ! " 1 1 5 #1 19 84 1 D D D 2 84 #1 17 11 84 168 ,QPDQ\FDOFXODWLRQV ATA LVLQYHUWLEOH EXWWKLVLVQRWDOZD\VWKHFDVH 7KHQH[W H[DPSOHLQYROYHVDPDWUL[RIWKHVRUWWKDWDSSHDUVLQZKDWDUHFDOOHG DQDO\VLVRIYDULDQFH SUREOHPVLQVWDWLVWLFV

&9".1-& )LQGDOHDVWVTXDUHVVROXWLRQRI A[ D E IRU 2

1 61 6 61 AD6 61 6 41 1

1 1 0 0 0 0

0 0 1 1 0 0

3 0 07 7 07 7; 07 7 15 1

2

3 #3 6 #1 7 6 7 6 07 6 7 ED6 7 6 27 4 55 1


40-65*0/ &RPSXWH 2

1 61 T AAD6 40 0 2

1 61 A ED6 40 0 T

1 1 0 0

1 0 1 0

1 0 1 0

1 0 0 1

1 1 0 0

1 0 1 0

1 0 1 0

1 0 0 1

2 3 1 0 0 3 1 2 1 6 6 1 0 07 61 7 61 7 62 07 0 1 0 7D6 76 42 0 56 0 1 07 61 7 4 5 1 2 1 0 0 1 1 0 0 1 2 3 3 #3 2 3 7 1 6 4 6 #1 7 6 7 6 7 07 76 0 7 D 6 #4 7 6 7 5 4 0 6 27 25 4 5 1 6 5 1

7KHDXJPHQWHGPDWUL[IRU ATA[ D AT E LV 2 3 2 6 2 2 2 4 1 62 7 60 2 0 0 #4 6 7&6 42 0 2 0 25 40 2 0 0 2 6 0

0 1 0 0

2 2 0 0

3 2 07 7 05 2

2 0 2 0

3 0 1 3 0 #1 #5 7 7 1 #1 #2 5 0 0 0

7KHJHQHUDOVROXWLRQLV x1 D 3 # x4 x2 D #5 C x4 x3 D #2 C x4 DQG x4 LVIUHH 6R WKHJHQHUDOOHDVWVTXDUHVVROXWLRQRI A[ D E KDVWKHIRUP 2 3 2 3 3 #1 6 #5 7 6 17 7 6 7 [O D 6 4 #2 5 C x4 4 1 5 0 1

7KHQH[WWKHRUHPJLYHVXVHIXOFULWHULDIRUGHWHUPLQLQJZKHQWKHUHLVRQO\RQHOHDVW VTXDUHVVROXWLRQRI A[ D E 2IFRXUVH WKHRUWKRJRQDOSURMHFWLRQ EO LVDOZD\VXQLTXH

5)&03&.

/HW A EHDQ m ! n PDWUL[ 7KHIROORZLQJVWDWHPHQWVDUHORJLFDOO\HTXLYDOHQW

D 7KHHTXDWLRQ A[ D E KDVDXQLTXHOHDVWVTXDUHVVROXWLRQIRUHDFK E LQ Rm E 7KHFROXPQVRI A DUHOLQHDUO\LQGSHQGHQW F 7KHPDWUL[ ATA LVLQYHUWLEOH :KHQWKHVHVWDWHPHQWVDUHWUXH WKHOHDVWVTXDUHVVROXWLRQ [O LVJLYHQE\ [O D .ATA/!1 AT E

7KHPDLQHOHPHQWVRIDSURRIRI7KHRUHP DUHRXWOLQHGLQ([HUFLVHV² ZKLFK DOVRUHYLHZFRQFHSWVIURP&KDSWHU )RUPXOD IRU [O LVXVHIXOPDLQO\IRUWKHRUHWLFDO SXUSRVHVDQGIRUKDQGFDOFXODWLRQVZKHQ ATA LVD 2 ! 2 LQYHUWLEOHPDWUL[ :KHQDOHDVWVTXDUHVVROXWLRQ [O LVXVHGWRSURGXFH AO[ DVDQDSSUR[LPDWLRQWR E WKHGLVWDQFHIURP E WR AO[ LVFDOOHGWKH OHDVWVTXDUHVHUURU RIWKLVDSSUR[LPDWLRQ

&9".1-& *LYHQ A DQG E DVLQ([DPSOH GHWHUPLQHWKHOHDVWVTXDUHVHUURULQ WKHOHDVWVTXDUHVVROXWLRQRI A[ D E

$)"15&3


40-65*0/ )URP([DPSOH 2 3 2 E D 4 0 5 DQG 11

x3 (2, 0, 11) b !84

+HQFH ^ Ax

(0, 2, 1)

0

(4, 0, 1)

x1

4 AO[ D 4 0 1

3 2 3 0 ! " 4 1 25 D 445 2 1 3

3 2 3 2 3 2 4 #2 E # AO[ D 4 0 5 # 4 4 5 D 4 #4 5 11 3 8

p p kE # AO[k D .#2/2 C .#4/2 C 82 D 84 p 7KHOHDVWVTXDUHVHUURULV 84 )RUDQ\ [ LQ R2 WKHGLVWDQFHEHWZHHQ E DQGWKHYHFWRU p A[ LVDWOHDVW 84 6HH)LJ 1RWHWKDWWKHOHDVWVTXDUHVVROXWLRQ [O LWVHOIGRHVQRW DSSHDULQWKHÀJXUH

DQG

Col A

2

2

'*(63&

$OWHUQDWLYH&DOFXODWLRQVRI/HDVW6TXDUHV6ROXWLRQV 7KHQH[WH[DPSOHVKRZVKRZWRÀQGDOHDVWVTXDUHVVROXWLRQRI A[ D E ZKHQWKH FROXPQVRI A DUHRUWKRJRQDO 6XFKPDWULFHVRIWHQDSSHDULQOLQHDUUHJUHVVLRQSUREOHPV GLVFXVVHGLQWKHQH[WVHFWLRQ

&9".1-& )LQGDOHDVWVTXDUHVVROXWLRQRI A[ D E IRU 2

3 1 #6 6 1 #2 7 7; AD6 41 15 1 7

2

3 #1 6 27 7 ED6 4 15 6

40-65*0/ %HFDXVH WKH FROXPQV D1 DQG D2 RI A DUH RUWKRJRQDO WKH RUWKRJRQDO SURMHFWLRQRI E RQWR &RO A LVJLYHQE\ 8 45 E ! D1 E ! D2 EO D D1 C D2 D D1 C D2 D1 ! D1 D2 ! D2 4 90 2 3 2 3 2 3 2 #3 #1 6 2 7 6 #1 7 6 1 7 7 6 7 6 7 D6 4 2 5 C 4 1=2 5 D 4 5=2 5 2 7=2 11=2

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ñ ñ 7KH45 PHWKRGLVFRPSDUHGZLWKWKHVWDQGDUGQRUPDOHTXDWLRQPHWKRGLQ**ROXEDQG&9DQ/RDQ 0DWUL[&RPSXWDWLRQV UGHG %DOWLPRUH -RKQV+RSNLQV3UHVV SS²


5)&03&.

*LYHQDQ m ! n PDWUL[ A ZLWKOLQHDUO\LQGHSHQGHQWFROXPQV OHW A D 45 EHD 45 IDFWRUL]DWLRQRI A DVLQ7KHRUHP 7KHQ IRUHDFK E LQ Rm WKHHTXDWLRQ A[ D E KDVDXQLTXHOHDVWVTXDUHVVROXWLRQ JLYHQE\ [O D R!1 QT E

1300' /HW [O D R!1 QT E 7KHQ

AO[ D 45[O D 45R!1 QT E D QQT E

%\7KHRUHP WKHFROXPQVRI Q IRUPDQRUWKRQRUPDOEDVLVIRU &RO A +HQFH E\ O 7KHRUHP QQT E LVWKHRUWKRJRQDOSURMHFWLRQ EO RI E RQWR &RO A 7KHQ AO[ D E ZKLFKVKRZVWKDW [O LVDOHDVWVTXDUHVVROXWLRQRI A[ D E 7KHXQLTXHQHVVRI [O IROORZV IURP7KHRUHP

/6.&3*$"- /05& 6LQFH R LQ7KHRUHP LVXSSHUWULDQJXODU [O VKRXOGEHFDOFXODWHGDVWKHH[DFW VROXWLRQRIWKHHTXDWLRQ R[ D QT E ,WLVPXFKIDVWHUWRVROYH E\EDFNVXEVWLWXWLRQRUURZRSHUDWLRQVWKDQWR FRPSXWH R!1 DQGXVH

&9".1-& )LQGWKHOHDVWVTXDUHVVROXWLRQRI A[ D E IRU 2

1 61 AD6 41 1

3 1 1 3

3 5 07 7; 25 3

2

3 3 6 57 7 ED6 4 75 #3

40-65*0/ 7KH45 IDFWRUL]DWLRQRI A FDQEHREWDLQHGDVLQ6HFWLRQ 2 3 3 1=2 1=2 1=2 2 4 5 6 1=2 #1=2 #1=2 7 2 74 0 2 35 A D 45 D 6 4 1=2 #1=2 1=2 5 0 0 2 1=2 1=2 #1=2 7KHQ

3 2 3 3 1=2 1=2 1=2 1=2 6 6 7 5 7 4 5 QT E D 4 1=2 #1=2 #1=2 1=2 56 4 7 5 D #6 1=2 #1=2 1=2 #1=2 4 #3 2

2

3

7KHOHDVWVTXDUHVVROXWLRQ [O VDWLVÀHV R[ D QT E WKDWLV 2 32 3 2 3 2 4 5 x1 6 40 2 3 54 x2 5 D 4 #6 5 x3 0 0 2 4 2 3 10 7KLVHTXDWLRQLVVROYHGHDVLO\DQG\LHOGV [O D 4 #6 5 2

$)"15&3


13"$5*$& 130#-&.4 2 3 2 3 1 #3 #3 5 1 5 DQG E D 4 #3 5 )LQGDOHDVWVTXDUHVVROXWLRQRI A[ D E /HW A D 4 1 5 1 7 2 #5 DQGFRPSXWHWKHDVVRFLDWHGOHDVWVTXDUHVHUURU :KDWFDQ\RXVD\DERXWWKHOHDVWVTXDUHVVROXWLRQRI A[ D E ZKHQ E LVRUWKRJRQDO WRWKHFROXPQVRI A"

&9&3$*4&4 ,Q ([HUFLVHV ² ÀQG D OHDVWVTXDUHV VROXWLRQ RI A[ D E E\ D FRQVWUXFWLQJWKHQRUPDOHTXDWLRQVIRU [O DQGE VROYLQJIRU [O 2 3 2 3 #1 2 4 A D 4 2 #3 5 E D 4 1 5 #1 3 2 2 3 2 3 2 1 #5 0 5 E D 4 8 5 A D 4 #2 2 3 1 2 3 2 3 1 #2 3 6 #1 6 7 27 7 E D 6 1 7 A D 6 4 0 4 #4 5 35 2 5 2 2 3 2 3 1 3 5 A D 4 1 #1 5 E D 4 1 5 1 1 0 ,Q([HUFLVHVDQG HTXDWLRQ A[ D E 2 1 1 61 1 A D 6 41 0 1 0 2 1 1 61 1 6 61 1 A D 6 61 0 6 41 0 1 0

GHVFULEHDOOOHDVWVTXDUHVVROXWLRQVRIWKH 3 2 3 0 1 637 07 7 E D 6 7 485 15 1 2 3 2 3 0 7 627 07 7 6 7 6 7 07 7 E D 6 3 7 7 667 17 6 7 455 15 1 4

&RPSXWHWKHOHDVWVTXDUHVHUURUDVVRFLDWHGZLWKWKHOHDVW VTXDUHVVROXWLRQIRXQGLQ([HUFLVH &RPSXWHWKHOHDVWVTXDUHVHUURUDVVRFLDWHGZLWKWKHOHDVW VTXDUHVVROXWLRQIRXQGLQ([HUFLVH

,Q([HUFLVHV² ÀQGD WKHRUWKRJRQDOSURMHFWLRQRI E RQWR &RO A DQGE DOHDVWVTXDUHVVROXWLRQRI A[ D E 2 3 2 3 1 5 4 1 5 E D 4 #2 5 A D 4 3 #2 4 #3

2

3 2 3 1 2 3 4 5 E D 4 #1 5 A D 4 #1 1 2 5 2 3 2 3 4 0 1 9 6 1 #5 7 607 1 7 E D 6 7 AD6 46 405 1 05 1 #1 #5 0 2 3 2 3 1 1 0 2 6 1 657 0 #1 7 6 7 6 AD4 ED4 7 0 1 15 65 #1 1 #1 6 2 3 2 3 ! " 3 4 11 5 1 5 E D 4 #9 5 X D /HW A D 4 #2 DQG Y D #1 3 4 5 ! " 5 &RPSXWH AX DQG AY DQGFRPSDUHWKHPZLWK E #2 &RXOG X SRVVLEO\EHDOHDVWVTXDUHVVROXWLRQRI A[ D E" $QVZHUWKLVZLWKRXWFRPSXWLQJDOHDVWVTXDUHVVROXWLRQ 2 3 2 3 ! " 2 1 5 4 /HW A D 4 #3 #4 5 E D 4 4 5 X D DQG Y D #5 4 3 2 ! " 6 &RPSXWH AX DQG AY DQGFRPSDUHWKHPZLWK E ,V #5 LWSRVVLEOHWKDWDWOHDVWRQHRI X RU Y FRXOGEHDOHDVWVTXDUHV VROXWLRQRI A[ D E" $QVZHUWKLVZLWKRXWFRPSXWLQJDOHDVW VTXDUHVVROXWLRQ

,Q([HUFLVHVDQG XVHWKHIDFWRUL]DWLRQ A D QR WRÀQGWKH OHDVWVTXDUHVVROXWLRQRI A[ D E 2 3 2 3 2 3 " 2 3 2=3 #1=3 ! 7 3 5 4 5 D 4 2=3 2=3 5 A D 4 2 E D 435 0 1 1 1 1=3 #2=3 1 2 3 2 3 2 3 1 #1 1=2 #1=2 ! #1 " 61 6 6 67 3 47 1=2 7 7 6 1=2 7 2 7 A D 6 ;E D 6 4 1 #1 5 D 4 1=2 #1=2 5 0 4 55 5 1 4 1=2 1=2 7 ,Q([HUFLVHVDQG A LVDQ m ! n PDWUL[DQG E LVLQ Rm 0DUN HDFKVWDWHPHQW7UXHRU)DOVH -XVWLI\HDFKDQVZHU

D 7KHJHQHUDOOHDVWVTXDUHVSUREOHPLVWRÀQGDQ [ WKDW PDNHV A[ DVFORVHDVSRVVLEOHWR E

E $ OHDVWVTXDUHVVROXWLRQRI A[ D E LVDYHFWRU [O WKDW O ZKHUH EO LVWKHRUWKRJRQDOSURMHFWLRQRI VDWLVÀHV AO[ D E E RQWR &RO A F $ OHDVWVTXDUHVVROXWLRQRI A[ D E LVDYHFWRU [O VXFKWKDW kE # A[k % kE # AO[k IRUDOO [ LQ Rn

G $Q\VROXWLRQRI ATA[ D AT E LVDOHDVWVTXDUHVVROXWLRQ RI A[ D E

H ,IWKHFROXPQVRI A DUHOLQHDUO\LQGHSHQGHQW WKHQWKH HTXDWLRQ A[ D E KDVH[DFWO\RQHOHDVWVTXDUHVVROXWLRQ

D ,I E LVLQWKHFROXPQVSDFHRI A WKHQHYHU\VROXWLRQRI A[ D E LVDOHDVWVTXDUHVVROXWLRQ E 7KHOHDVWVTXDUHVVROXWLRQRI A[ D E LVWKHSRLQWLQWKH FROXPQVSDFHRI A FORVHVWWR E

F $ OHDVWVTXDUHVVROXWLRQRI A[ D E LVDOLVWRIZHLJKWV WKDW ZKHQDSSOLHGWRWKHFROXPQVRI A SURGXFHVWKH RUWKRJRQDOSURMHFWLRQRI E RQWR &RO A G ,I [O LV D OHDVWVTXDUHV VROXWLRQ RI A[ D E WKHQ [O D .ATA/!1 AT E

'HVFULEHDOOOHDVWVTXDUHVVROXWLRQVRIWKHV\VWHP

xCy D2 xCy D4 >0@ ([DPSOH LQ6HFWLRQ GLVSOD\HGDORZSDVVOLQHDU ÀOWHUWKDWFKDQJHGDVLJQDO fyk g LQWR fykC1 g DQGFKDQJHGD KLJKHUIUHTXHQF\ VLJQDO fwk g LQWR WKH ]HUR VLJQDO ZKHUH yk D FRV."k=4/ DQG wk D FRV.3"k=4/ 7KHIROORZLQJFDO FXODWLRQVZLOOGHVLJQDÀOWHUZLWKDSSUR[LPDWHO\WKRVHSURS HUWLHV 7KHÀOWHUHTXDWLRQLV

a0 ykC2 C a1 ykC1 C a2 yk D ´k

ykC2 ykC1 2 kD0 #:7 0 kD1 6 6 :: 6 #1 #:7 : 6 6 #:7 #1 6 6 0 #:7 6 6 :7 0 6 4 1 :7 kD7 :7 1

/HW A EHDQ m ! n PDWUL[ 8VHWKHVWHSVEHORZWRVKRZWKDWD YHFWRU [ LQ Rn VDWLVÀHV A[ D LIDQGRQO\LI ATA[ D 7KLV ZLOOVKRZWKDW 1XO A D 1XO ATA D 6KRZWKDWLI A[ D WKHQ ATA[ D

2

E 6XSSRVH ATA[ D ([SODLQZK\ [TATA[ D DQGXVH WKLVWRVKRZWKDW A[ D

F 'HWHUPLQHWKHUDQNRI A

6XSSRVH A LV m ! n ZLWKOLQHDUO\LQGHSHQGHQWFROXPQVDQG E LVLQ Rm 8VHWKHQRUPDOHTXDWLRQVWRSURGXFHDIRUPXOD O WKHSURMHFWLRQRI E RQWR &RO A >+LQW )LQG [O ÀUVW 7KH IRU E IRUPXODGRHVQRWUHTXLUHDQRUWKRJRQDOEDVLVIRU &RO A@

wkC2 wkC1

kD0 :7 kD1 6 6 :: 6 #1 : 6 6 :7 6 6 0 6 6 #:7 6 4 1 kD7 #:7

/HW A EHDQ m ! n PDWUL[VXFKWKDW ATA LVLQYHUWLEOH 6KRZ WKDWWKHFROXPQVRI A DUHOLQHDUO\LQGHSHQGHQW >&DUHIXO
8VH([HUFLVH WRVKRZWKDW UDQN ATA D UDQN A >+LQW +RZPDQ\FROXPQVGRHV ATA KDYH" +RZLVWKLVFRQQHFWHG ZLWKWKHUDQNRI ATA"@

IRUDOO k

.8/

%HFDXVHWKHVLJQDOVDUHSHULRGLF ZLWKSHULRG LWVXIÀFHV WRVWXG\HTXDWLRQ IRU k D 0; : : : ; 7 7KHDFWLRQRQWKH WZRVLJQDOVGHVFULEHGDERYHWUDQVODWHVLQWRWZRVHWVRIHLJKW HTXDWLRQV VKRZQEHORZ

I ,I A KDVD45 IDFWRUL]DWLRQ VD\ A D QR WKHQWKHEHVW ZD\WRÀQGWKHOHDVWVTXDUHVVROXWLRQRI A[ D E LVWR FRPSXWH [O D R!1 QT E

E ([SODLQ ZK\ A PXVW KDYH DW OHDVW DV PDQ\ URZV DV FROXPQV

)LQGDIRUPXODIRUWKHOHDVWVTXDUHVVROXWLRQRI A[ D E ZKHQ WKHFROXPQVRI A DUHRUWKRQRUPDO

H 7KHQRUPDOHTXDWLRQVDOZD\VSURYLGHDUHOLDEOHPHWKRG IRUFRPSXWLQJOHDVWVTXDUHVVROXWLRQV

/HW A EHDQ m ! n PDWUL[ZKRVHFROXPQVDUHOLQHDUO\LQGH SHQGHQW >&DUHIXO A QHHGQRWEHVTXDUH@ D 8VH([HUFLVH WRVKRZWKDW ATA LVDQLQYHUWLEOHPDWUL[


# 0 :7 #1 :7 0 #:7 1

ykC1 3 2 3 6 07 :7 7 7 6 7 2 3 6 #:7 7 07 7 a0 6 7 6 7 #:7 7 74 a1 5 D 6 #1 7 6 #:7 7 #1 7 7 a2 6 7 6 07 #:7 7 7 6 7 4 :7 5 05 :7 1 yk

wk 3 2 3 0 607 #:7 7 7 6 7 2 3 607 07 7 a0 6 7 6 7 :7 7 74 a1 5 D 6 0 7 7 607 #1 7 6 7 a2 607 :7 7 7 6 7 405 05 #:7 0

:ULWH DQ HTXDWLRQ A[ D E ZKHUH A LV D 16 ! 3 PDWUL[ IRUPHGIURPWKHWZRFRHIÀFLHQWPDWULFHVDERYHDQGZKHUH E LQ R16 LVIRUPHGIURPWKHWZRULJKWVLGHVRIWKHHTXDWLRQV )LQG a0 a1 DQG a2 JLYHQE\WKHOHDVWVTXDUHVVROXWLRQRI A[ D E 7KHLQWKHGDWDDERYHZDVXVHGDVDQDSSUR[ p LPDWLRQIRU 2=2 WRLOOXVWUDWHKRZDW\SLFDOFRPSXWDWLRQ LQDQDSSOLHGSUREOHPPLJKWSURFHHG ,IZHUHXVHG LQVWHDG WKHUHVXOWLQJÀOWHUFRHIÀFLHQWVZRXOGDJUHHWRDWOHDVW p p VHYHQGHFLPDOSODFHVZLWK 2=4; 1=2 DQG 2=4 WKHYDOXHV SURGXFHGE\H[DFWDULWKPHWLFFDOFXODWLRQV WEB

$)"15&3


40-65*0/4 50 13"$5*$& 130#-&.4 )LUVW FRPSXWH

2

1 ATA D 4 #3 #3 2 1 AT E D 4 #3 #3

1 5 1 1 5 1

32 3 2 1 1 #3 #3 3 7 54 1 5 1 5 D 4 9 2 1 7 2 0 32 3 2 3 1 5 #3 7 54 #3 5 D 4 #65 5 2 #5 #28

9 83 28

3 0 28 5 14

1H[W URZUHGXFHWKHDXJPHQWHGPDWUL[IRUWKHQRUPDOHTXDWLRQV ATA[ D AT E 2 3 2 3 2 3 3 9 0 #3 1 3 0 #1 1 0 #3=2 2 4 9 83 28 #65 5 & 4 0 56 28 #56 5 & " " " & 4 0 1 1=2 #1 5 0 28 14 #28 0 28 14 #28 0 0 0 0

7KHJHQHUDOOHDVWVTXDUHVVROXWLRQLV x1 D 2 C 32 x3 x2 D #1 # 12 x3 ZLWK x3 IUHH )RURQHVSHFLÀFVROXWLRQ WDNH x3 D 0 IRUH[DPSOH DQGJHW 2 3 2 [O D 4 #1 5 0 7RÀQGWKHOHDVWVTXDUHVHUURU FRPSXWH 2 32 3 2 3 1 #3 #3 2 5 1 54 #1 5 D 4 #3 5 EO D AO[ D 4 1 5 1 7 2 0 #5

,WWXUQVRXWWKDW EO D E VR kE # EO k D 0 7KHOHDVWVTXDUHVHUURULV]HUREHFDXVH E KDSSHQVWREHLQ &RO A ,I E LVRUWKRJRQDOWRWKHFROXPQVRI A WKHQWKHSURMHFWLRQRI E RQWRWKHFROXPQVSDFH RI A LV ,QWKLVFDVH DOHDVWVTXDUHVVROXWLRQ [O RI A[ D E VDWLVÀHV AO[ D

"11-*$"5*0/4 50 -*/&"3 .0%&-4 $ FRPPRQWDVNLQVFLHQFHDQGHQJLQHHULQJLVWRDQDO\]HDQGXQGHUVWDQGUHODWLRQVKLSV DPRQJVHYHUDOTXDQWLWLHVWKDWYDU\ 7KLVVHFWLRQGHVFULEHVDYDULHW\RIVLWXDWLRQVLQ ZKLFKGDWDDUHXVHGWREXLOGRUYHULI\DIRUPXODWKDWSUHGLFWVWKHYDOXHRIRQHYDULDEOH DVDIXQFWLRQRIRWKHUYDULDEOHV ,QHDFKFDVH WKHSUREOHPZLOODPRXQWWRVROYLQJD OHDVWVTXDUHVSUREOHP )RUHDV\DSSOLFDWLRQRIWKHGLVFXVVLRQWRUHDOSUREOHPVWKDW\RXPD\HQFRXQWHUODWHU LQ\RXUFDUHHU ZHFKRRVHQRWDWLRQWKDWLVFRPPRQO\XVHGLQWKHVWDWLVWLFDODQDO\VLVRI VFLHQWLÀFDQGHQJLQHHULQJGDWD ,QVWHDGRI A[ D E ZHZULWH X ˇ D \ DQGUHIHUWR X DV WKH GHVLJQPDWUL[ ˇ DVWKH SDUDPHWHUYHFWRU DQG \ DVWKH REVHUYDWLRQYHFWRU

/HDVW6TXDUHV/LQHV 7KH VLPSOHVW UHODWLRQ EHWZHHQ WZR YDULDEOHV x DQG y LV WKH OLQHDU HTXDWLRQ y D ˇ0 C ˇ1 x ñ ([SHULPHQWDOGDWDRIWHQSURGXFHSRLQWV .x1 ; y1 /; : : : ; .xn ; yn / WKDW ñ 7KLVQRWDWLRQLVFRPPRQO\XVHGIRUOHDVWVTXDUHVOLQHVLQVWHDGRI y D mx C b

$SSOLFDWLRQVWR/LQHDU0RGHOV

ZKHQJUDSKHG VHHPWROLHFORVHWRDOLQH :HZDQWWRGHWHUPLQHWKHSDUDPHWHUV ˇ0 DQG ˇ1 WKDWPDNHWKHOLQHDV´FORVHµWRWKHSRLQWVDVSRVVLEOH 6XSSRVH ˇ0 DQG ˇ1 DUH À[HG DQG FRQVLGHU WKH OLQH y D ˇ0 C ˇ1 x LQ )LJ &RUUHVSRQGLQJWRHDFKGDWDSRLQW .xj ; yj / WKHUHLVDSRLQW .xj ; ˇ0 C ˇ1 xj / RQWKHOLQH ZLWKWKHVDPH x FRRUGLQDWH :HFDOO yj WKH REVHUYHG YDOXHRI y DQG ˇ0 C ˇ1 xj WKH SUHGLFWHG y YDOXHGHWHUPLQHGE\WKHOLQH 7KHGLIIHUHQFHEHWZHHQDQREVHUYHG y YDOXHDQGDSUHGLFWHG y YDOXHLVFDOOHGD UHVLGXDO y

Data point

(xj , yj ) (xj , !0 + !1xj )

Point on line

Residual

Residual

y = !0 + !1x x1

xj

xn

x

'*(63& )LWWLQJDOLQHWRH[SHULPHQWDOGDWD

7KHUHDUHVHYHUDOZD\VWRPHDVXUHKRZ´FORVHµWKHOLQHLVWRWKHGDWD 7KHXVXDO FKRLFHSULPDULO\EHFDXVHWKHPDWKHPDWLFDOFDOFXODWLRQVDUHVLPSOH LVWRDGGWKHVTXDUHV RIWKHUHVLGXDOV 7KH OHDVWVTXDUHVOLQH LVWKHOLQH y D ˇ0 C ˇ1 x WKDWPLQLPL]HVWKH VXPRIWKHVTXDUHVRIWKHUHVLGXDOV 7KLVOLQHLVDOVRFDOOHGD OLQHRIUHJUHVVLRQRI \ RQ [ EHFDXVHDQ\HUURUVLQWKHGDWDDUHDVVXPHGWREHRQO\LQWKH y FRRUGLQDWHV 7KH FRHIÀFLHQWV ˇ0 ˇ1 RIWKHOLQHDUHFDOOHGOLQHDU UHJUHVVLRQFRHIÀFLHQWVò ,IWKHGDWDSRLQWVZHUHRQWKHOLQH WKHSDUDPHWHUV ˇ0 DQG ˇ1 ZRXOGVDWLVI\WKH HTXDWLRQV

:HFDQZULWHWKLVV\VWHPDV

X ˇ D \;

3UHGLFWHG y YDOXH

2EVHUYHG y YDOXH

ˇ0 C ˇ1 x1

y1

ˇ0 C ˇ1 x2 :: :

y2 :: :

ˇ0 C ˇ1 xn

yn

2

1 61 6 ZKHUH X D 6 :: 4: 1

3 x1 x2 7 7 :: 7 ; : 5

xn

ˇD

!

" ˇ0 ; ˇ1

2

3 y1 6 y2 7 6 7 \ D 6 :: 7 4 : 5

yn

2IFRXUVH LIWKHGDWDSRLQWVGRQ·WOLHRQDOLQH WKHQWKHUHDUHQRSDUDPHWHUV ˇ0 ˇ1 IRU ZKLFKWKHSUHGLFWHG y YDOXHVLQ X ˇ HTXDOWKHREVHUYHG y YDOXHVLQ \ DQG X ˇ D \ KDV QRVROXWLRQ 7KLVLVDOHDVWVTXDUHVSUREOHP A[ D E ZLWKGLIIHUHQWQRWDWLRQ 7KHVTXDUHRIWKHGLVWDQFHEHWZHHQWKHYHFWRUV X ˇ DQG \ LVSUHFLVHO\WKHVXPRI WKHVTXDUHVRIWKHUHVLGXDOV 7KH ˇ WKDWPLQLPL]HVWKLVVXPDOVRPLQLPL]HVWKHGLVWDQFH EHWZHHQ X ˇ DQG \ &RPSXWLQJWKHOHDVWVTXDUHVVROXWLRQRI X ˇ D \ LVHTXLYDOHQWWR ÀQGLQJWKH ˇ WKDWGHWHUPLQHVWKHOHDVWVTXDUHVOLQHLQ)LJ ò ,IWKHPHDVXUHPHQWHUURUVDUHLQ x LQVWHDGRI y VLPSO\LQWHUFKDQJHWKHFRRUGLQDWHVRIWKHGDWD .xj ; yj / EHIRUHSORWWLQJWKHSRLQWVDQGFRPSXWLQJWKHUHJUHVVLRQOLQH ,IERWKFRRUGLQDWHVDUHVXEMHFWWRSRVVLEOHHUURU WKHQ\RXPLJKWFKRRVHWKHOLQHWKDWPLQLPL]HVWKHVXPRIWKHVTXDUHVRIWKH RUWKRJRQDO SHUSHQGLFXODU GLVWDQFHVIURPWKHSRLQWVWRWKHOLQH 6HHWKH3UDFWLFH3UREOHPVIRU6HFWLRQ

$)"15&3


&9".1-& )LQGWKHHTXDWLRQ y D ˇ0 C ˇ1 x RIWKHOHDVWVTXDUHVOLQHWKDWEHVWÀWV WKHGDWDSRLQWV .2; 1/ .5; 2/ .7; 3/ DQG .8; 3/

40-65*0/ 8VHWKH x FRRUGLQDWHVRIWKHGDWDWREXLOGWKHGHVLJQPDWUL[ X LQ DQG WKH y FRRUGLQDWHVWREXLOGWKHREVHUYDWLRQYHFWRU \ 2 3 2 3 1 2 1 61 7 627 5 7; \ D 6 7 X D6 41 435 75 1 8 3

)RUWKHOHDVWVTXDUHVVROXWLRQRI X ˇ D \ REWDLQWKHQRUPDOHTXDWLRQVZLWKWKHQHZ QRWDWLRQ X TX ˇ D X T\ 7KDWLV FRPSXWH

X TX D

!

1 2

1 5

1 7

X T\ D

!

1 2

1 5

1 7

7KHQRUPDOHTXDWLRQVDUH

+HQFH ! " ! ˇ0 4 D ˇ1 22

22 142

!

"!1 !

4 22

9 57

"

3 2 2 " 1 ! 1 6 4 57 7D 61 8 41 75 22 1 8 2 3 " 1 ! " 7 1 6 9 627 D 8 435 57 3

22 142

"!

ˇ0 ˇ1

"

D

!

9 57

22 142

"

"

! "! " ! " ! " 1 142 #22 1 24 9 2=7 D D D 4 57 5=14 84 #22 84 30

7KXVWKHOHDVWVTXDUHVOLQHKDVWKHHTXDWLRQ

yD

5 2 C x 7 14

6HH)LJ y 3 2 1 1

2

3

4

5

6

7

8

9

x

'*(63& 7KHOHDVWVTXDUHVOLQH

yD

2 7

C

5 x 14

$ FRPPRQSUDFWLFHEHIRUHFRPSXWLQJDOHDVWVTXDUHVOLQHLVWRFRPSXWHWKHDYHUDJH x RIWKHRULJLQDO x YDOXHVDQGIRUPDQHZYDULDEOH x " D x # x 7KHQHZ x GDWDDUHVDLG WREHLQ PHDQGHYLDWLRQIRUP ,QWKLVFDVH WKHWZRFROXPQVRIWKHGHVLJQPDWUL[ZLOO EHRUWKRJRQDO 6ROXWLRQRIWKHQRUPDOHTXDWLRQVLVVLPSOLÀHG MXVWDVLQ([DPSOH LQ 6HFWLRQ 6HH([HUFLVHV DQG


7KH*HQHUDO/LQHDU0RGHO ,QVRPHDSSOLFDWLRQV LWLVQHFHVVDU\WRÀWGDWDSRLQWVZLWKVRPHWKLQJRWKHUWKDQDVWUDLJKW OLQH ,QWKHH[DPSOHVWKDWIROORZ WKHPDWUL[HTXDWLRQLVVWLOO X ˇ D \ EXWWKHVSHFLÀF IRUPRI X FKDQJHVIURPRQHSUREOHPWRWKHQH[W 6WDWLVWLFLDQVXVXDOO\LQWURGXFHD UHVLGXDOYHFWRU ! GHÀQHGE\ ! D \ # Xˇ DQGZULWH \ D Xˇ C !

$Q\HTXDWLRQRIWKLVIRUPLVUHIHUUHGWRDVD OLQHDUPRGHO 2QFH X DQG \ DUHGHWHUPLQHG WKHJRDOLVWRPLQLPL]HWKHOHQJWKRI ! ZKLFKDPRXQWVWRÀQGLQJDOHDVWVTXDUHVVROXWLRQ RI X ˇ D \ ,QHDFKFDVH WKHOHDVWVTXDUHVVROXWLRQ ˇO LVDVROXWLRQRIWKHQRUPDO HTXDWLRQV X TX ˇ D X T\

/HDVW6TXDUHV)LWWLQJRI2WKHU&XUYHV :KHQGDWDSRLQWV .x1 ; y1 /; : : : ; .xn ; yn / RQDVFDWWHUSORWGRQRWOLHFORVHWRDQ\OLQH LW PD\EHDSSURSULDWHWRSRVWXODWHVRPHRWKHUIXQFWLRQDOUHODWLRQVKLSEHWZHHQ x DQG y 7KHQH[WWZRH[DPSOHVVKRZKRZWRÀWGDWDE\FXUYHVWKDWKDYHWKHJHQHUDOIRUP

y D ˇ0 f0 .x/ C ˇ1 f1 .x/ C " " " C ˇk fk .x/

ZKHUH f0 ; : : : ; fk DUHNQRZQIXQFWLRQVDQG ˇ0 ; : : : ; ˇk DUHSDUDPHWHUVWKDWPXVWEH GHWHUPLQHG $VZHZLOOVHH HTXDWLRQ GHVFULEHVDOLQHDUPRGHOEHFDXVHLWLVOLQHDULQ WKHXQNQRZQSDUDPHWHUV )RUDSDUWLFXODUYDOXHRI x JLYHVDSUHGLFWHG RU´ÀWWHGµ YDOXHRI y 7KH GLIIHUHQFHEHWZHHQWKHREVHUYHGYDOXHDQGWKHSUHGLFWHGYDOXHLVWKHUHVLGXDO 7KH SDUDPHWHUV ˇ0 ; : : : ; ˇk PXVWEHGHWHUPLQHGVRDVWRPLQLPL]HWKHVXPRIWKHVTXDUHV RIWKHUHVLGXDOV

Average cost per unit

y

Units produced

x

'*(63&

$YHUDJHFRVWFXUYH

&9".1-& 6XSSRVHGDWDSRLQWV .x1 ; y1 /; : : : ; .xn ; yn / DSSHDUWROLHDORQJVRPH VRUWRISDUDERODLQVWHDGRIDVWUDLJKWOLQH )RULQVWDQFH LIWKH x FRRUGLQDWHGHQRWHVWKH SURGXFWLRQOHYHOIRUDFRPSDQ\ DQG y GHQRWHVWKHDYHUDJHFRVWSHUXQLWRIRSHUDWLQJDW DOHYHORI x XQLWVSHUGD\ WKHQDW\SLFDODYHUDJHFRVWFXUYHORRNVOLNHDSDUDERODWKDW RSHQVXSZDUG)LJ ,QHFRORJ\ DSDUDEROLFFXUYHWKDWRSHQVGRZQZDUGLVXVHGWR PRGHOWKHQHWSULPDU\SURGXFWLRQRIQXWULHQWVLQDSODQW DVDIXQFWLRQRIWKHVXUIDFH DUHDRIWKHIROLDJH)LJ 6XSSRVHZHZLVKWRDSSUR[LPDWHWKHGDWDE\DQHTXDWLRQRI WKHIRUP y D ˇ0 C ˇ1 x C ˇ2 x 2

'HVFULEHWKHOLQHDUPRGHOWKDWSURGXFHVDÓHDVWVTXDUHVÀWµRIWKHGDWDE\HTXDWLRQ

40-65*0/ (TXDWLRQ GHVFULEHVWKHLGHDOUHODWLRQVKLS 6XSSRVHWKHDFWXDOYDOXHVRI WKHSDUDPHWHUVDUH ˇ0 ˇ1 ˇ2 7KHQWKHFRRUGLQDWHVRIWKHÀUVWGDWDSRLQW .x1 ; y1 / VDWLVI\ DQHTXDWLRQRIWKHIRUP

Net primary production

y

y1 D ˇ0 C ˇ1 x1 C ˇ2 x12 C #1

ZKHUH #1 LVWKHUHVLGXDOHUURUEHWZHHQWKHREVHUYHGYDOXH y1 DQGWKHSUHGLFWHG y YDOXH ˇ0 C ˇ1 x1 C ˇ2 x12 (DFKGDWDSRLQWGHWHUPLQHVDVLPLODUHTXDWLRQ Surface area of foliage

'*(63&

3URGXFWLRQRIQXWULHQWV

x

y1 D ˇ0 C ˇ1 x1 C ˇ2 x12 C #1 y2 D ˇ0 C ˇ1 x2 C ˇ2 x22 C #2 :: :: : : yn D ˇ0 C ˇ1 xn C ˇ2 xn2 C #n

$)"15&3


,WLVDVLPSOHPDWWHUWRZULWHWKLVV\VWHPRIHTXDWLRQVLQWKHIRUP \ D X ˇ C ! 7RÀQG X LQVSHFWWKHÀUVWIHZURZVRIWKHV\VWHPDQGORRNIRUWKHSDWWHUQ 3 2 3 2 3 21 x1 x12 2 3 #1 y1 6 7 2 7 ˇ0 6 y2 7 6 1 x2 x2 76 7 6 #2 7 6 7 6 6 7 6 :: 7 D 6 : : :: 7 6 74 ˇ1 5 C 6 :: 7 :: 4 : 5 4 :: 4 : 5 : 5 ˇ2 yn 2 x 1 x # n

\

D

n

n

X

C

ˇ

!

&9".1-& ,IGDWDSRLQWVWHQGWRIROORZDSDWWHUQVXFKDVLQ)LJ WKHQDQ

y

DSSURSULDWHPRGHOPLJKWEHDQHTXDWLRQRIWKHIRUP

y D ˇ0 C ˇ1 x C ˇ2 x 2 C ˇ3 x 3 6XFKGDWD IRULQVWDQFH FRXOGFRPHIURPDFRPSDQ\·VWRWDOFRVWV DVDIXQFWLRQRIWKH OHYHORISURGXFWLRQ 'HVFULEHWKHOLQHDUPRGHOWKDWJLYHVDOHDVWVTXDUHVÀWRIWKLVW\SH WRGDWD .x1 ; y1 /; : : : ; .xn ; yn /

40-65*0/ %\DQDQDO\VLVVLPLODUWRWKDWLQ([DPSOH ZHREWDLQ x '*(63&

'DWDSRLQWVDORQJDFXELFFXUYH

2EVHUYDWLRQ YHFWRU

2

3 y1 6 7 6 y2 7 7 \D6 6 :: 7 ; 4 : 5

yn

2

1 6 61 X D6 6 :: 4: 1

'HVLJQ PDWUL[

x1 x2 :: : xn

x12 x22 :: : xn2

3 x13 7 x23 7 :: 7 7; : 5

xn3

3DUDPHWHU YHFWRU

2

3 ˇ0 6 7 6 ˇ1 7 7 ˇD6 6 ˇ 7; 4 25 ˇ3

5HVLGXDO YHFWRU

2

3 #1 6 7 6 #2 7 7 !D6 6 :: 7 4 : 5

#n

0XOWLSOH5HJUHVVLRQ 6XSSRVHDQH[SHULPHQWLQYROYHVWZRLQGHSHQGHQWYDULDEOHV³VD\ u DQG v ³DQGRQH GHSHQGHQWYDULDEOH y $ VLPSOHHTXDWLRQIRUSUHGLFWLQJ y IURP u DQG v KDVWKHIRUP

y D ˇ0 C ˇ1 u C ˇ2 v

$ PRUHJHQHUDOSUHGLFWLRQHTXDWLRQPLJKWKDYHWKHIRUP

y D ˇ0 C ˇ1 u C ˇ2 v C ˇ3 u2 C ˇ4 uv C ˇ5 v 2

7KLVHTXDWLRQLVXVHGLQJHRORJ\ IRULQVWDQFH WRPRGHOHURVLRQVXUIDFHV JODFLDOFLUTXHV VRLOS+DQGRWKHUTXDQWLWLHV ,QVXFKFDVHV WKHOHDVWVTXDUHVÀWLVFDOOHGD WUHQGVXUIDFH (TXDWLRQV DQG ERWKOHDGWRDOLQHDUPRGHOEHFDXVHWKH\DUHOLQHDULQWKH XQNQRZQSDUDPHWHUVHYHQWKRXJK u DQG v DUHPXOWLSOLHG ,QJHQHUDO DOLQHDUPRGHO ZLOODULVHZKHQHYHU y LVWREHSUHGLFWHGE\DQHTXDWLRQRIWKHIRUP

y D ˇ0 f0 .u; v/ C ˇ1 f1 .u; v/ C " " " C ˇk fk .u; v/ ZLWK f0 ; : : : ; fk DQ\VRUWRINQRZQIXQFWLRQVDQG ˇ0 ; : : : ; ˇk XQNQRZQZHLJKWV

&9".1-& ,Q JHRJUDSK\ ORFDO PRGHOV RI WHUUDLQ DUH FRQVWUXFWHG IURP GDWD .u1 ; v1 ; y1 /; : : : ; .un ; vn ; yn / ZKHUH uj vj DQG yj DUHODWLWXGH ORQJLWXGH DQGDOWLWXGH UHVSHFWLYHO\ 'HVFULEHWKHOLQHDUPRGHOEDVHGRQ WKDWJLYHVDOHDVWVTXDUHVÀWWRVXFK GDWD 7KHVROXWLRQLVFDOOHGWKH OHDVWVTXDUHVSODQH 6HH)LJ


'*(63& $ OHDVWVTXDUHVSODQH

40-65*0/ :HH[SHFWWKHGDWDWRVDWLVI\WKHIROORZLQJHTXDWLRQV y1 D ˇ0 C ˇ1 u1 C ˇ2 v1 C #1 y2 D ˇ0 C ˇ1 u2 C ˇ2 v2 C #2 :: :: : : yn D ˇ0 C ˇ1 un C ˇ2 vn C #n

7KLVV\VWHPKDVWKHPDWUL[IRUP \ D X ˇ C ! ZKHUH 2EVHUYDWLRQ YHFWRU

2

3 y1 6 y2 7 6 7 \ D 6 : 7; 4 :: 5

yn

SG

5IF (FPNFUSZ PG B -JOFBS .PEFM m

'HVLJQ PDWUL[

2

1 61 6 X D6: 4 ::

1

u1 u2 :: : un

3 v1 v2 7 7 :: 7 ; : 5

vn

3DUDPHWHU YHFWRU

2

3

ˇ0 ˇ D 4 ˇ1 5; ˇ2

5HVLGXDO YHFWRU

2

3 #1 6 #2 7 6 7 !D6 : 7 4 :: 5

#n

([DPSOH VKRZVWKDWWKHOLQHDUPRGHOIRUPXOWLSOHUHJUHVVLRQKDVWKHVDPHDEVWUDFW IRUPDVWKHPRGHOIRUWKHVLPSOHUHJUHVVLRQLQWKHHDUOLHUH[DPSOHV /LQHDUDOJHEUDJLYHV XVWKHSRZHUWRXQGHUVWDQGWKHJHQHUDOSULQFLSOHEHKLQGDOOWKHOLQHDUPRGHOV 2QFH X LVGHÀQHGSURSHUO\ WKHQRUPDOHTXDWLRQVIRU ˇ KDYHWKHVDPHPDWUL[IRUP QRPDWWHU KRZPDQ\YDULDEOHVDUHLQYROYHG 7KXV IRUDQ\OLQHDUPRGHOZKHUH X TX LVLQYHUWLEOH WKHOHDVWVTXDUHV ˇO LVJLYHQE\ .X TX/!1 X T\

)XUWKHU5HDGLQJ )HUJXVRQ - ,QWURGXFWLRQWR/LQHDU$OJHEUDLQ*HRORJ\ 1HZ
13"$5*$& 130#-&. :KHQWKHPRQWKO\VDOHVRIDSURGXFWDUHVXEMHFWWRVHDVRQDOÁXFWXDWLRQV DFXUYHWKDW DSSUR[LPDWHVWKHVDOHVGDWDPLJKWKDYHWKHIRUP

y D ˇ0 C ˇ1 x C ˇ2 VLQ .2"x=12/

ZKHUH x LVWKHWLPHLQPRQWKV 7KHWHUP ˇ0 C ˇ1 x JLYHVWKHEDVLFVDOHVWUHQG DQG WKHVLQHWHUPUHÁHFWVWKHVHDVRQDOFKDQJHVLQVDOHV *LYHWKHGHVLJQPDWUL[DQGWKH SDUDPHWHUYHFWRUIRUWKHOLQHDUPRGHOWKDWOHDGVWRDOHDVWVTXDUHVÀWRIWKHHTXDWLRQ DERYH $VVXPHWKHGDWDDUH .x1 ; y1 /; : : : ; .xn ; yn /

$)"15&3


&9&3$*4&4 ,Q([HUFLVHV² ÀQGWKHHTXDWLRQ y D ˇ0 C ˇ1 x RIWKHOHDVW VTXDUHVOLQHWKDWEHVWÀWVWKHJLYHQGDWDSRLQWV .0; 1/ .1; 1/ .2; 2/ .3; 2/ .1; 0/ .2; 1/ .4; 2/ .5; 3/ .#1; 0/ .0; 1/ .1; 2/ .2; 4/ .2; 3/ .3; 2/ .5; 1/ .6; 0/ /HW X EHWKHGHVLJQPDWUL[XVHGWRÀQGWKHOHDVWVTXDUHVOLQH WRÀWGDWD .x1 ; y1 /; : : : ; .xn ; yn / 8VHDWKHRUHPLQ6HFWLRQ WRVKRZWKDWWKHQRUPDOHTXDWLRQVKDYHDXQLTXHVROXWLRQ LIDQGRQO\LIWKHGDWDLQFOXGHDWOHDVWWZRGDWDSRLQWVZLWK GLIIHUHQW x FRRUGLQDWHV

6XSSRVH WKH LQLWLDO DPRXQWV M$ DQG M% DUH XQNQRZQ EXW D VFLHQWLVW LV DEOH WR PHDVXUH WKH WRWDO DPRXQWV SUHVHQWDWVHYHUDOWLPHVDQG UHFRUGVWKHIROORZLQJSRLQWV .ti ; yi / .10; 21:34/ .11; 20:68/ .12; 20:05/ .14; 18:87/ DQG .15; 18:30/ D 'HVFULEHDOLQHDUPRGHOWKDWFDQEHXVHGWRHVWLPDWH M$ DQG M% E >0@ )LQGWKHOHDVWVTXDUHVFXUYHEDVHGRQ

/HW X EHWKHGHVLJQPDWUL[LQ([DPSOH FRUUHVSRQGLQJWR DOHDVWVTXDUHVÀWRIDSDUDERODWRGDWD .x1 ; y1 /; : : : ; .xn ; yn / 6XSSRVH x1 x2 DQG x3 DUHGLVWLQFW ([SODLQZK\WKHUHLVRQO\ RQHSDUDERODWKDWÀWVWKHGDWDEHVW LQDOHDVWVTXDUHVVHQVH 6HH([HUFLVH $ FHUWDLQ H[SHULPHQW SURGXFHV WKH GDWD .1; 1:8/ .2; 2:7/ .3; 3:4/ .4; 3:8/ .5; 3:9/ 'HVFULEHWKHPRGHOWKDWSURGXFHV DOHDVWVTXDUHVÀWRIWKHVHSRLQWVE\DIXQFWLRQRIWKHIRUP

y D ˇ1 x C ˇ2 x 2 6XFKDIXQFWLRQPLJKWDULVH IRUH[DPSOH DVWKHUHYHQXHIURP WKHVDOHRI x XQLWVRIDSURGXFW ZKHQWKHDPRXQWRIIHUHGIRU VDOHDIIHFWVWKHSULFHWREHVHWIRUWKHSURGXFW D *LYHWKHGHVLJQPDWUL[ WKHREVHUYDWLRQYHFWRU DQGWKH XQNQRZQSDUDPHWHUYHFWRU E >0@ )LQGWKHDVVRFLDWHGOHDVWVTXDUHVFXUYHIRUWKHGDWD $ VLPSOHFXUYHWKDWRIWHQPDNHVDJRRGPRGHOIRUWKHYDUL DEOHFRVWVRIDFRPSDQ\ DVDIXQFWLRQRIWKHVDOHVOHYHO x KDVWKHIRUP y D ˇ1 x C ˇ2 x 2 C ˇ3 x 3 7KHUHLVQRFRQVWDQW WHUPEHFDXVHÀ[HGFRVWVDUHQRWLQFOXGHG D *LYHWKHGHVLJQPDWUL[DQGWKHSDUDPHWHUYHFWRUIRUWKH OLQHDUPRGHOWKDWOHDGVWRDOHDVWVTXDUHVÀWRIWKHHTXD WLRQDERYH ZLWKGDWD .x1 ; y1 /; : : : ; .xn ; yn / E >0@ )LQGWKHOHDVWVTXDUHVFXUYHRIWKHIRUPDERYHWRÀW WKHGDWD .4; 1:58/ .6; 2:08/ .8; 2:5/ .10; 2:8/ .12; 3:1/ .14; 3:4/ .16; 3:8/ DQG .18; 4:32/ ZLWKYDOXHVLQWKRX VDQGV ,ISRVVLEOH SURGXFHDJUDSKWKDWVKRZVWKHGDWD SRLQWVDQGWKHJUDSKRIWKHFXELFDSSUR[LPDWLRQ $ FHUWDLQH[SHULPHQWSURGXFHVWKHGDWD .1; 7:9/ .2; 5:4/ DQG .3; #:9/ 'HVFULEHWKHPRGHOWKDWSURGXFHVDOHDVWVTXDUHVÀW RIWKHVHSRLQWVE\DIXQFWLRQRIWKHIRUP

y D A FRV x C B VLQ x 6XSSRVHUDGLRDFWLYHVXEVWDQFHV$ DQG% KDYHGHFD\FRQ VWDQWVRIDQG UHVSHFWLYHO\ ,IDPL[WXUHRIWKHVHWZR VXEVWDQFHVDWWLPH t D 0 FRQWDLQV M$ JUDPVRI$ DQG M% JUDPVRI%WKHQDPRGHOIRUWKHWRWDODPRXQW y RIWKHPL[WXUH SUHVHQWDWWLPH t LV

y D M$ e !:02t C M% e !:07t

.6/

+DOOH\·V&RPHWODVWDSSHDUHGLQDQGZLOOUHDSSHDULQ >0@ $FFRUGLQJWR.HSOHU·VÀUVWODZ DFRPHWVKRXOGKDYH DQHOOLSWLF SDUDEROLF RUK\SHUEROLFRUELWZLWKJUDYLWDWLRQDO DWWUDFWLRQVIURPWKHSODQHWVLJQRUHG ,QVXLWDEOHSRODUFRRU GLQDWHV WKHSRVLWLRQ .r; #/ RIDFRPHWVDWLVÀHVDQHTXDWLRQRI WKHIRUP

r D ˇ C e.r " FRV #/ ZKHUH ˇ LVDFRQVWDQWDQG e LVWKH HFFHQWULFLW\ RIWKHRUELW ZLWK 0 % e < 1 IRUDQHOOLSVH e D 1 IRUDSDUDEROD DQG e > 1 IRUDK\SHUEROD 6XSSRVHREVHUYDWLRQVRIDQHZO\GLVFRYHUHG FRPHWSURYLGHWKHGDWDEHORZ 'HWHUPLQHWKHW\SHRIRUELW DQGSUHGLFWZKHUHWKHFRPHWZLOOEHZKHQ # D 4:6 UDGLDQV

#

r

>0@ $ KHDOWK\FKLOG·VV\VWROLFEORRGSUHVVXUH p LQPLOOLPH WHUVRIPHUFXU\ DQGZHLJKW w LQSRXQGV DUHDSSUR[LPDWHO\ UHODWHGE\WKHHTXDWLRQ

ˇ0 C ˇ1 OQ w D p 8VHWKHIROORZLQJH[SHULPHQWDOGDWDWRHVWLPDWHWKHV\VWROLF EORRGSUHVVXUHRIDKHDOWK\FKLOGZHLJKLQJSRXQGV 3 7KHEDVLFLGHDRIOHDVWVTXDUHVÀWWLQJRIGDWDLVGXHWR. ) *DXVV DQG LQGHSHQGHQWO\ WR$/HJHQGUH ZKRVHLQLWLDOULVHWRIDPHRFFXUUHG LQZKHQKHXVHGWKHPHWKRGWRGHWHUPLQHWKHSDWKRIWKHDVWHURLG &HUHV )RUW\GD\VDIWHUWKHDVWHURLGZDVGLVFRYHUHG LWGLVDSSHDUHGEHKLQG WKHVXQ *DXVVSUHGLFWHGLWZRXOGDSSHDUWHQPRQWKVODWHUDQGJDYHLWV ORFDWLRQ 7KHDFFXUDF\RIWKHSUHGLFWLRQDVWRQLVKHGWKH(XURSHDQVFLHQWLÀF FRPPXQLW\

w

OQ w

p


D 5HZULWHWKHGDWDLQ([DPSOH ZLWKQHZ x FRRUGLQDWHV LQPHDQGHYLDWLRQIRUP /HW X EHWKHDVVRFLDWHGGHVLJQ PDWUL[ :K\DUHWKHFROXPQVRI X RUWKRJRQDO" E :ULWHWKHQRUPDOHTXDWLRQVIRUWKHGDWDLQSDUWD DQG VROYHWKHPWRÀQGWKHOHDVWVTXDUHVOLQH y D ˇ0 C ˇ1 x " ZKHUH x " D x # 5:5

>0@ 7RPHDVXUHWKHWDNHRIISHUIRUPDQFHRIDQDLUSODQH WKH KRUL]RQWDOSRVLWLRQRIWKHSODQHZDVPHDVXUHGHYHU\VHFRQG IURP t D 0 WR t D 12 7KHSRVLWLRQVLQIHHW ZHUH DQG D )LQG WKH OHDVWVTXDUHV FXELF FXUYH y D ˇ0 C ˇ1 t C ˇ2 t 2 C ˇ3 t 3 IRUWKHVHGDWD

6XSSRVHWKH x FRRUGLQDWHVRIWKHGDWD P .x1 ; y1 /; : : : ; .xn ; yn / DUHLQPHDQGHYLDWLRQIRUP VRWKDW xi D 0 6KRZWKDWLI X LVWKHGHVLJQPDWUL[IRUWKHOHDVWVTXDUHVOLQHLQWKLVFDVH WKHQ X TX LVDGLDJRQDOPDWUL[

E 8VHWKHUHVXOWRISDUWD WRHVWLPDWHWKHYHORFLW\RIWKH SODQHZKHQ t D 4:5 VHFRQGV 1 1 /HW x D .x1 C " " " C xn / DQG y D .y1 C " " " C yn / n n 6KRZ WKDW WKH OHDVWVTXDUHV OLQH IRU WKH GDWD .x1 ; y1 /; : : : ; .xn ; yn / PXVWSDVVWKURXJK .x; y/ 7KDWLV VKRZ WKDW x DQG y VDWLVI\WKHOLQHDUHTXDWLRQ y D ˇO0 C ˇO1 x >+LQW 'HULYHWKLVHTXDWLRQIURPWKHYHFWRUHTXDWLRQ \ D X ˇO C ! 'HQRWHWKHÀUVWFROXPQRI X E\ 8VHWKHIDFWWKDWWKH UHVLGXDOYHFWRU ! LVRUWKRJRQDOWRWKHFROXPQVSDFHRI X DQG KHQFHLVRUWKRJRQDOWR @

([HUFLVHVDQGLQYROYHDGHVLJQPDWUL[ X ZLWKWZRRUPRUH FROXPQVDQGDOHDVWVTXDUHVVROXWLRQ ˇO RI \ D Xˇ &RQVLGHUWKH IROORZLQJQXPEHUV

O 2 ³WKH VXPRIWKH VTXDUHVRIWKHÚHJUHVVLRQWHUPµ L kX ˇk 'HQRWHWKLVQXPEHUE\ 665 O 2 ³WKH VXPRIWKH VTXDUHVIRU HUURUWHUP 'HQRWH LL k\ # X ˇk WKLVQXPEHUE\66( LLL k\k2 ³WKH´WRWDOµ VXPRIWKH VTXDUHVRIWKH y YDOXHV 'HQRWH WKLVQXPEHUE\667

*LYHQGDWDIRUDOHDVWVTXDUHVSUREOHP .x1 ; y1 /; : : : ; .xn ; yn / WKH IROORZLQJDEEUHYLDWLRQVDUHKHOSIXO P P P 2 Pn x D niD1 xi ; x D iD1 xi2 ; P Pn P P y D i D1 yi ; xy D niD1 xi yi

(YHU\VWDWLVWLFVWH[WWKDWGLVFXVVHVUHJUHVVLRQDQGWKHOLQHDUPRGHO \ D Xˇ C ! LQWURGXFHVWKHVHQXPEHUV WKRXJKWHUPLQRORJ\DQG QRWDWLRQYDU\VRPHZKDW 7RVLPSOLI\PDWWHUV DVVXPHWKDWWKH PHDQRIWKH y YDOXHVLV]HUR ,QWKLVFDVH 667 LVSURSRUWLRQDO WRZKDWLVFDOOHGWKH YDULDQFH RIWKHVHWRI y YDOXHV

7KHQRUPDOHTXDWLRQVIRUDOHDVWVTXDUHVOLQH y D ˇO0 C ˇO1 x PD\ EHZULWWHQLQWKHIRUP P P nˇO0 C ˇO1 x D y .7/ P P P ˇO0 x C ˇO1 x 2 D xy

-XVWLI\WKHHTXDWLRQ 667 D 665 C 66( >+LQW 8VHD WKHRUHP DQGH[SODLQZK\WKHK\SRWKHVHVRIWKHWKHRUHPDUH VDWLVÀHG@ 7KLVHTXDWLRQLVH[WUHPHO\LPSRUWDQWLQVWDWLVWLFV ERWKLQUHJUHVVLRQWKHRU\DQGLQWKHDQDO\VLVRIYDULDQFH

O 2 ˇO T X T\ >+LQW 5HZULWHWKHOHIWVLGH 6KRZWKDW kX ˇk DQGXVHWKHIDFWWKDW ˇO VDWLVÀHVWKHQRUPDOHTXDWLRQV@ 7KLV IRUPXODIRU665 LVXVHGLQVWDWLVWLFV )URPWKLVDQGIURP ([HUFLVH REWDLQWKHVWDQGDUGIRUPXODIRU66(

'HULYHWKHQRUPDOHTXDWLRQV IURPWKHPDWUL[IRUPJLYHQ LQWKLVVHFWLRQ 8VHDPDWUL[LQYHUVHWRVROYHWKHV\VWHPRIHTXDWLRQVLQ DQGWKHUHE\REWDLQIRUPXODVIRU ˇO0 DQG ˇO1 WKDWDSSHDULQPDQ\ VWDWLVWLFVWH[WV

T 66( D \T \ # ˇO X T \

40-65*0/ 50 13"$5*$& 130#-&. &RQVWUXFW X DQG ˇ VRWKDWWKH k WKURZRI Xˇ LVWKHSUHGLFWHG y YDOXHWKDWFRUUHVSRQGV WRWKHGDWDSRLQW .xk ; yk / QDPHO\

y

ˇ0 C ˇ1 xk C ˇ2 VLQ.2"xk =12/ ,WVKRXOGEHFOHDUWKDW x

6DOHVWUHQGZLWKVHDVRQDO ÁXFWXDWLRQV

2

1 6 :: X D4:

1

x1 :: : xn

3 VLQ.2"x1 =12/ 7 :: 5; :

VLQ.2"xn =12/

2

3 ˇ0 ˇ D 4 ˇ1 5 ˇ2

$)"15&3


*//&3 130%6$5 41"$&4 1RWLRQV RI OHQJWK GLVWDQFH DQG RUWKRJRQDOLW\ DUH RIWHQ LPSRUWDQW LQ DSSOLFDWLRQV LQYROYLQJDYHFWRUVSDFH )RU Rn WKHVHFRQFHSWVZHUHEDVHGRQWKHSURSHUWLHVRIWKH LQQHUSURGXFWOLVWHGLQ7KHRUHP RI6HFWLRQ )RURWKHUVSDFHV ZHQHHGDQDORJXHVRI WKHLQQHUSURGXFWZLWKWKHVDPHSURSHUWLHV 7KHFRQFOXVLRQVRI7KHRUHP QRZEHFRPH D[LRPV LQWKHIROORZLQJGHÀQLWLRQ

%&'*/*5*0/

$Q LQQHUSURGXFW RQDYHFWRUVSDFH V LVDIXQFWLRQWKDW WRHDFKSDLURIYHFWRUV X DQG Y LQ V DVVRFLDWHVDUHDOQXPEHU hX; Yi DQGVDWLVÀHVWKHIROORZLQJD[LRPV IRUDOO X Y Z LQ V DQGDOOVFDODUV c

hX; Yi D hY; Xi hX C Y; Zi D hX; Zi C hY; Zi hc X; Yi D chX; Yi hX; Xi $ 0 DQG hX; Xi D 0 LIDQGRQO\LI X D

$ YHFWRUVSDFHZLWKDQLQQHUSURGXFWLVFDOOHGDQ LQQHUSURGXFWVSDFH 7KHYHFWRUVSDFH Rn ZLWKWKHVWDQGDUGLQQHUSURGXFWLVDQLQQHUSURGXFWVSDFH DQG QHDUO\HYHU\WKLQJGLVFXVVHGLQWKLVFKDSWHUIRU Rn FDUULHVRYHUWRLQQHUSURGXFWVSDFHV 7KHH[DPSOHVLQWKLVVHFWLRQDQGWKHQH[WOD\WKHIRXQGDWLRQIRUDYDULHW\RIDSSOLFDWLRQV WUHDWHGLQFRXUVHVLQHQJLQHHULQJ SK\VLFV PDWKHPDWLFV DQGVWDWLVWLFV

&9".1-& )L[ DQ\ WZR SRVLWLYH QXPEHUV³VD\ DQG ³DQG IRU YHFWRUV X D .u1 ; u2 / DQG Y D .v1 ; v2 / LQ R2 VHW

hX; Yi D 4u1 v1 C 5u2 v2

6KRZWKDWHTXDWLRQ GHÀQHVDQLQQHUSURGXFW

40-65*0/ &HUWDLQO\ $[LRP LV VDWLVÀHG EHFDXVH hX; Yi D 4u1 v1 C 5u2 v2 D 4v1 u1 C 5v2 u2 D hY; Xi ,I Z D .w1 ; w2 / WKHQ hX C Y; Zi D 4.u1 C v1 /w1 C 5.u2 C v2 /w2 D 4u1 w1 C 5u2 w2 C 4v1 w1 C 5v2 w2 D hX ; Z i C hY ; Z i

7KLVYHULÀHV$[LRP )RU$[LRP FRPSXWH

hc X; Yi D 4.cu1 /v1 C 5.cu2 /v2 D c.4u1 v1 C 5u2 v2 / D chX; Yi

)RU$[LRP QRWHWKDW hX; Xi D 4u21 C 5u22 $ 0 DQG 4u21 C 5u22 D 0 RQO\LI u1 D u2 D 0 WKDWLV LI X D $OVR h; i D 0 6R GHÀQHVDQLQQHUSURGXFWRQ R2 ,QQHU SURGXFWV VLPLODU WR FDQ EH GHÀQHG RQ Rn 7KH\ DULVH QDWXUDOO\ LQ FRQQHFWLRQZLWK´ZHLJKWHGOHDVWVTXDUHVµSUREOHPV LQZKLFKZHLJKWVDUHDVVLJQHGWR WKHYDULRXVHQWULHVLQWKHVXPIRUWKHLQQHUSURGXFWLQVXFKDZD\WKDWPRUHLPSRUWDQFH LVJLYHQWRWKHPRUHUHOLDEOHPHDVXUHPHQWV )URPQRZRQ ZKHQDQLQQHUSURGXFWVSDFHLQYROYHVSRO\QRPLDOVRURWKHUIXQFWLRQV ZHZLOOZULWHWKHIXQFWLRQVLQWKHIDPLOLDUZD\ UDWKHUWKDQXVHWKHEROGIDFHW\SHIRU YHFWRUV 1HYHUWKHOHVV LWLVLPSRUWDQWWRUHPHPEHUWKDWHDFKIXQFWLRQ LV DYHFWRUZKHQ LWLVWUHDWHGDVDQHOHPHQWRIDYHFWRUVSDFH

,QQHU3URGXFW6SDFHV

&9".1-& /HW t0 ; : : : ; tn EHGLVWLQFWUHDOQXPEHUV )RU p DQG q LQ Pn GHÀQH hp; qi D p.t0 /q.t0 / C p.t1 /q.t1 / C " " " C p.tn /q.tn /

,QQHUSURGXFW$[LRPV ²DUHUHDGLO\FKHFNHG )RU$[LRP QRWHWKDW

hp; pi D Œp.t0 /!2 C Œp.t1 /!2 C " " " C Œp.tn /!2 $ 0

$OVR h; i D 0 7KHEROGIDFH]HURKHUHGHQRWHVWKH]HURSRO\QRPLDO WKH]HURYHFWRU LQ Pn ,I hp; pi D 0 WKHQ p PXVWYDQLVKDW n C 1 SRLQWV t0 ; : : : ; tn 7KLVLVSRVVLEOH RQO\LI p LVWKH]HURSRO\QRPLDO EHFDXVHWKHGHJUHHRI p LVOHVVWKDQ n C 1 7KXV GHÀQHVDQLQQHUSURGXFWRQ Pn

&9".1-& /HW V EH P2 ZLWKWKHLQQHUSURGXFWIURP([DPSOH ZKHUH t0 D 0 t1 D 12 DQG t2 D 1 /HW p.t / D 12t 2 DQG q.t / D 2t # 1 &RPSXWH hp; qi DQG hq; qi 40-65*0/

# $ # $ hp; qi D p.0/q.0/ C p 12 q 12 C p.1/q.1/ D .0/.#1/ C .3/.0/ C .12/.1/ D 12 # $ hq; qi D Œq.0/!2 C Œq 12 !2 C Œq.1/!2

D .#1/2 C .0/2 C .1/2 D 2

/HQJWKV 'LVWDQFHV DQG2UWKRJRQDOLW\ /HW V EHDQLQQHUSURGXFWVSDFH ZLWKWKHLQQHUSURGXFWGHQRWHGE\ hX; Yi -XVWDVLQ Rn ZHGHÀQHWKH OHQJWK RU QRUP RIDYHFWRU Y WREHWKHVFDODU p kYk D hY; Yi

(TXLYDOHQWO\ kYk2 D hY; Yi 7KLVGHÀQLWLRQPDNHVVHQVHEHFDXVH hY; Yi $ 0 EXWWKH GHÀQLWLRQ GRHVQRW VD\WKDW hY; Yi LVD´VXPRIVTXDUHVµ EHFDXVH Y QHHGQRWEHDQHOHPHQW RI Rn $ XQLWYHFWRU LVRQHZKRVHOHQJWKLV 7KH GLVWDQFHEHWZHHQ X DQG Y LV kX # Yk 9HFWRUV X DQG Y DUH RUWKRJRQDO LI hX; Yi D 0

&9".1-& /HW P2 KDYHWKHLQQHUSURGXFW RI([DPSOH &RPSXWHWKHOHQJWKV RIWKHYHFWRUV p.t / D 12t 2 DQG q.t / D 2t # 1

40-65*0/

% # $&2 C Œp.1/!2 kpk2 D hp; pi D Œp.0/!2 C p 12

D 0 C Œ3!2 C Œ12!2 D 153 p kpk D 153 p )URP([DPSOH hq; qi D 2 +HQFH kqk D 2

7KH*UDP±6FKPLGW3URFHVV 7KHH[LVWHQFHRIRUWKRJRQDOEDVHVIRUÀQLWHGLPHQVLRQDOVXEVSDFHVRIDQLQQHUSURGXFW VSDFHFDQEHHVWDEOLVKHGE\WKH*UDP²6FKPLGWSURFHVV MXVWDVLQ Rn &HUWDLQRUWKRJR QDOEDVHVWKDWDULVHIUHTXHQWO\LQDSSOLFDWLRQVFDQEHFRQVWUXFWHGE\WKLVSURFHVV 7KHRUWKRJRQDOSURMHFWLRQRIDYHFWRURQWRDVXEVSDFH W ZLWKDQRUWKRJRQDOEDVLV FDQEHFRQVWUXFWHGDVXVXDO 7KHSURMHFWLRQGRHVQRWGHSHQGRQWKHFKRLFHRIRUWKRJRQDO EDVLV DQGLWKDVWKHSURSHUWLHVGHVFULEHGLQWKH2UWKRJRQDO'HFRPSRVLWLRQ7KHRUHPDQG WKH%HVW$SSUR[LPDWLRQ7KHRUHP

$)"15&3


&9".1-& /HW V EH P4 ZLWKWKHLQQHUSURGXFWLQ([DPSOH LQYROYLQJHYDOXDWLRQ RISRO\QRPLDOVDW #2 #1 DQG DQGYLHZ P2 DVDVXEVSDFHRI V 3URGXFHDQ RUWKRJRQDOEDVLVIRU P2 E\DSSO\LQJWKH*UDP²6FKPLGWSURFHVVWRWKHSRO\QRPLDOV t DQG t 2

40-65*0/ 7KHLQQHUSURGXFWGHSHQGVRQO\RQWKHYDOXHVRIDSRO\QRPLDODW #2; : : : ; 2 VRZHOLVWWKHYDOXHVRIHDFKSRO\QRPLDODVDYHFWRULQ R5 XQGHUQHDWKWKHQDPHRIWKH SRO\QRPLDOñ 3RO\QRPLDO

2 3 1 617 6 7 7 9HFWRURIYDOXHV 6 6 1 7; 415 1

2

t

t2

3

2 3 4 617 6 7 607 6 7 415 4

#2 6 #1 7 6 7 6 0 7; 6 7 4 15 2

7KHLQQHUSURGXFWRIWZRSRO\QRPLDOVLQ V HTXDOVWKHVWDQGDUG LQQHUSURGXFWRIWKHLU FRUUHVSRQGLQJYHFWRUVLQ R5 2EVHUYHWKDW t LVRUWKRJRQDOWRWKHFRQVWDQWIXQFWLRQ 6R WDNH p0 .t / D 1 DQG p1 .t / D t )RU p2 XVHWKHYHFWRUVLQ R5 WRFRPSXWHWKHSURMHFWLRQ RI t 2 RQWR 6SDQ fp0 ; p1 g

ht 2 ; p0 i D ht 2 ; 1i D 4 C 1 C 0 C 1 C 4 D 10 hp0 ; p0 i D 5

ht 2 ; p1 i D ht 2 ; t i D #8 C .#1/ C 0 C 1 C 8 D 0

7KHRUWKRJRQDOSURMHFWLRQRI t 2 RQWR 6SDQ f1; t g LV

10 p 5 0

2

2

C 0p1 7KXV

p2 .t/ D t # 2p0 .t / D t # 2

$QRUWKRJRQDOEDVLVIRUWKHVXEVSDFH P2 RI V LV 3RO\QRPLDO

p0

2 3 1 617 6 7 7 9HFWRURIYDOXHV 6 6 1 7; 415 1

2

p1

3 #2 6 #1 7 6 7 6 0 7; 6 7 4 15 2

2

p2

3 2 6 #1 7 6 7 6 #2 7 6 7 4 #1 5 2

%HVW$SSUR[LPDWLRQLQ,QQHU3URGXFW6SDFHV $ FRPPRQSUREOHPLQDSSOLHGPDWKHPDWLFVLQYROYHVDYHFWRUVSDFH V ZKRVHHOHPHQWV DUHIXQFWLRQV 7KHSUREOHPLVWRDSSUR[LPDWHDIXQFWLRQ f LQ V E\DIXQFWLRQ g IURPD VSHFLÀHGVXEVSDFH W RI V 7KH´FORVHQHVVµRIWKHDSSUR[LPDWLRQRI f GHSHQGVRQWKH ZD\ kf # gk LVGHÀQHG :HZLOOFRQVLGHURQO\WKHFDVHLQZKLFKWKHGLVWDQFHEHWZHHQ f DQG g LVGHWHUPLQHGE\DQLQQHUSURGXFW ,QWKLVFDVH WKH EHVWDSSUR[LPDWLRQWR f E\ IXQFWLRQVLQ W LVWKHRUWKRJRQDOSURMHFWLRQRI f RQWRWKHVXEVSDFH W

&9".1-& /HW V EH P4 ZLWKWKHLQQHUSURGXFWLQ([DPSOH DQGOHW p0 p1 DQG p2 EHWKHRUWKRJRQDOEDVLVIRXQGLQ([DPSOH IRUWKHVXEVSDFH P2 )LQGWKHEHVW DSSUR[LPDWLRQWR p.t / D 5 # 12 t 4 E\SRO\QRPLDOVLQ P2 ñ (DFKSRO\QRPLDOLQ P4 LVXQLTXHO\GHWHUPLQHGE\LWVYDOXHDWWKHÀYHQXPEHUV !2; : : : ; 2 ,QIDFW WKH FRUUHVSRQGHQFHEHWZHHQ p DQGLWVYHFWRURIYDOXHVLVDQLVRPRUSKLVP WKDWLV DRQHWRRQHPDSSLQJRQWR R5 WKDWSUHVHUYHVOLQHDUFRPELQDWLRQV

,QQHU3URGXFW6SDFHV

40-65*0/ 7KHYDOXHVRI p0 ; p1 DQG p2 DWWKHQXPEHUV #2 #1 DQGDUHOLVWHG LQ R5 YHFWRUVLQ DERYH 7KHFRUUHVSRQGLQJYDOXHVIRU p DUH #3 DQG #3 &RPSXWH hp; p0 i D 8; hp0 ; p0 i D 5;

hp; p1 i D 0;

hp; p2 i D #31 hp2 ; p2 i D 14

7KHQWKHEHVWDSSUR[LPDWLRQLQ V WR p E\SRO\QRPLDOVLQ P2 LV

pO D SURMP2 p D

hp; p0 i hp; p1 i hp; p2 i p0 C p1 C p2 hp0 ; p0 i hp1 ; p1 i hp2 ; p2 i

D 85 p0 C

!31 p 14 2

D

8 5

#

31 2 .t 14

# 2/:

7KLVSRO\QRPLDOLVWKHFORVHVWWR p RIDOOSRO\QRPLDOVLQ P2 ZKHQWKHGLVWDQFHEHWZHHQ SRO\QRPLDOVLVPHDVXUHGRQO\DW #2 #1 DQG 6HH)LJ y

2 t

2 ˆ p(t) p(t) '*(63&

7KHSRO\QRPLDOV p0 p1 DQG p2 LQ([DPSOHV DQGEHORQJWRDFODVVRISRO\QRPL DOVWKDWDUHUHIHUUHGWRLQVWDWLVWLFVDV RUWKRJRQDOSRO\QRPLDOVò 7KHRUWKRJRQDOLW\UHIHUV WRWKHW\SHRILQQHUSURGXFWGHVFULEHGLQ([DPSOH

||v||

0 W

||projW v||

v

7ZR,QHTXDOLWLHV

||v – proj W v||

*LYHQDYHFWRU Y LQDQLQQHUSURGXFWVSDFH V DQGJLYHQDÀQLWHGLPHQVLRQDOVXEVSDFH W ZHPD\DSSO\WKH3\WKDJRUHDQ7KHRUHPWRWKHRUWKRJRQDOGHFRPSRVLWLRQRI Y ZLWK UHVSHFWWR W DQGREWDLQ

projW v

'*(63&

7KHK\SRWHQXVHLVWKHORQJHVWVLGH

5)&03&.

kYk2 D k SURMW Yk2 C kY # SURMW Yk2 6HH)LJ ,QSDUWLFXODU WKLVVKRZVWKDWWKHQRUPRIWKHSURMHFWLRQRI Y RQWR W GRHVQRW H[FHHGWKHQRUPRI Y LWVHOI 7KLVVLPSOHREVHUYDWLRQOHDGVWRWKHIROORZLQJLPSRUWDQW LQHTXDOLW\ 5IF $BVDIZ4DIXBS[ *OFRVBMJUZ )RUDOO X Y LQ V jhX; Yij % kXk kYk

ò 6HH 6WDWLVWLFVDQG([SHULPHQWDO'HVLJQLQ(QJLQHHULQJDQGWKH3K\VLFDO6FLHQFHV QGHG E\1RUPDQ / -RKQVRQDQG)UHG& /HRQH1HZ

$)"15&3


1300' ,I X D WKHQERWKVLGHVRI DUH]HUR DQGKHQFHWKHLQHTXDOLW\LVWUXHLQWKLV FDVH 6HH3UDFWLFH3UREOHP ,I X ¤ OHW W EHWKHVXEVSDFHVSDQQHGE\ X 5HFDOO WKDW kc Xk D jcj kXk IRUDQ\VFDODU c 7KXV + + + hY; Xi + jhY; Xij jhY; Xij jhX; Yij + kXk D k SURMW Yk D + X+ D kXk D hX; Xi + jhX; Xij kXk2 kXk

6LQFH k SURMW Yk % kYk ZHKDYH

jhX; Yij % kYk ZKLFKJLYHV kX k

7KH&DXFK\²6FKZDU]LQHTXDOLW\LVXVHIXOLQPDQ\EUDQFKHVRIPDWKHPDWLFV $ IHZ VLPSOHDSSOLFDWLRQVDUHSUHVHQWHGLQWKHH[HUFLVHV 2XUPDLQQHHGIRUWKLVLQHTXDOLW\KHUH LVWRSURYHDQRWKHUIXQGDPHQWDOLQHTXDOLW\LQYROYLQJQRUPVRIYHFWRUV 6HH)LJ

5)&03&.

kX C Yk % kXk C kYk

u+v

v ||u + v||

||v|| 0

5IF 5SJBOHMF *OFRVBMJUZ )RUDOO X; Y LQ V

||u||

1300'

kX C Yk2 D hX C Y; X C Yi D hX; Xi C 2hX; Yi C hY; Yi % kXk2 C 2jhX; Yij C kYk2 % kXk2 C 2kXk kYk C kYk2 &DXFK\²6FKZDU] 2 D .kXk C kYk/

u

'*(63&

7KHOHQJWKVRIWKHVLGHVRID WULDQJOH

7KHWULDQJOHLQHTXDOLW\IROORZVLPPHGLDWHO\E\WDNLQJVTXDUHURRWVRIERWKVLGHV

$Q,QQHU3URGXFWIRU C Œa; b! &DOFXOXVUHTXLUHG 3UREDEO\WKHPRVWZLGHO\XVHGLQQHUSURGXFWVSDFHIRUDSSOLFDWLRQVLVWKHYHFWRUVSDFH C Œa; b! RIDOOFRQWLQXRXVIXQFWLRQVRQDQLQWHUYDO a % t % b ZLWKDQLQQHUSURGXFWWKDW ZHZLOOGHVFULEH :HEHJLQE\FRQVLGHULQJDSRO\QRPLDO p DQGDQ\LQWHJHU n ODUJHUWKDQRUHTXDO WRWKHGHJUHHRI p 7KHQ p LVLQ Pn DQGZHPD\FRPSXWHDÓHQJWKµIRU p XVLQJWKH LQQHUSURGXFWRI([DPSOH LQYROYLQJHYDOXDWLRQDW n C 1 SRLQWVLQ Œa; b! +RZHYHU WKLVOHQJWKRI p FDSWXUHVWKHEHKDYLRUDWRQO\WKRVH n C 1 SRLQWV 6LQFH p LVLQ Pn IRU DOOODUJH n ZHFRXOGXVHDPXFKODUJHU n ZLWKPDQ\PRUHSRLQWVIRUWKH´HYDOXDWLRQµ LQQHUSURGXFW 6HH)LJ

p(t)

a

b

p(t) t

a

b

'*(63& 8VLQJGLIIHUHQWQXPEHUVRIHYDOXDWLRQSRLQWVLQ Œa; b! WRFRPSXWH

kpk2

t

,QQHU3URGXFW6SDFHV

/HWXVSDUWLWLRQ Œa; b! LQWR n C 1 VXELQWHUYDOVRIOHQJWK $t D .b # a/=.n C 1/ DQG OHW t0 ; : : : ; tn EHDUELWUDU\SRLQWVLQWKHVHVXELQWHUYDOV ∆t a

t0

tn

tj

b

,I n LVODUJH WKHLQQHUSURGXFWRQ Pn GHWHUPLQHGE\ t0 ; : : : ; tn ZLOOWHQGWRJLYHDODUJH YDOXHWR hp; pi VRZHVFDOHLWGRZQDQGGLYLGHE\ n C 1 2EVHUYHWKDW 1=.n C 1/ D $t =.b # a/ DQGGHÀQH 2 3 n n 1 X 1 4X hp; qi D p.tj /q.tj / D p.tj /q.tj /$t 5 n C 1 j D0 b # a j D0

1RZ OHW n LQFUHDVHZLWKRXWERXQG 6LQFHSRO\QRPLDOV p DQG q DUHFRQWLQXRXVIXQFWLRQV WKHH[SUHVVLRQLQEUDFNHWVLVD5LHPDQQVXPWKDWDSSURDFKHVDGHÀQLWHLQWHJUDO DQGZH DUHOHGWRFRQVLGHUWKH DYHUDJHYDOXHRI p.t /q.t / RQWKHLQWHUYDO Œa; b!

1 b#a

Z

b

p.t /q.t / dt a

7KLVTXDQWLW\LVGHÀQHGIRUSRO\QRPLDOVRIDQ\GHJUHHLQIDFW IRUDOOFRQWLQXRXV IXQFWLRQV DQGLWKDVDOOWKHSURSHUWLHVRIDQLQQHUSURGXFW DVWKHQH[WH[DPSOHVKRZV 7KHVFDOHIDFWRU 1=.b # a/ LVLQHVVHQWLDODQGLVRIWHQRPLWWHGIRUVLPSOLFLW\

&9".1-& )RU f g LQ C Œa; b! VHW hf; gi D

Z

b

f .t /g.t / dt

a

6KRZWKDW GHÀQHVDQLQQHUSURGXFWRQ C Œa; b!

40-65*0/ ,QQHUSURGXFW$[LRPV ²IROORZIURPHOHPHQWDU\SURSHUWLHVRIGHÀQLWH LQWHJUDOV )RU$[LRP REVHUYHWKDW hf; f i D

Z

a

b

Œf .t /!2 dt $ 0

7KHIXQFWLRQ Œf .t /!2 LVFRQWLQXRXVDQGQRQQHJDWLYHRQ Œa; b! ,IWKHGHÀQLWHLQWHJUDORI Œf .t /!2 LV]HUR WKHQ Œf .t /!2 PXVWEHLGHQWLFDOO\]HURRQ Œa; b! E\DWKHRUHPLQDGYDQFHG FDOFXOXV LQZKLFKFDVH f LVWKH]HURIXQFWLRQ 7KXV hf; f i D 0 LPSOLHVWKDW f LVWKH ]HURIXQFWLRQRQ Œa; b! 6R GHÀQHVDQLQQHUSURGXFWRQ C Œa; b!

&9".1-& /HW V EHWKHVSDFH C Œ0; 1! ZLWKWKHLQQHUSURGXFWRI([DPSOH DQG OHW W EHWKHVXEVSDFHVSDQQHGE\WKHSRO\QRPLDOV p1 .t/ D 1 p2 .t / D 2t # 1 DQG p3 .t / D 12t 2 8VHWKH*UDP²6FKPLGWSURFHVVWRÀQGDQRUWKRJRQDOEDVLVIRU W 40-65*0/ /HW q1 D p1 DQGFRPSXWH hp2 ; q1 i D

Z

1

ˇ1 ˇ .2t # 1/.1/ dt D .t # t /ˇˇ D 0 2

0

0

$)"15&3


6R p2 LVDOUHDG\RUWKRJRQDOWR q1 DQGZHFDQWDNH q2 D p2 )RUWKHSURMHFWLRQRI p3 RQWR W2 D 6SDQ fq1 ; q2 g FRPSXWH ˇ1 Z 1 ˇ 2 3ˇ hp3 ; q1 i D 12t " 1 dt D 4t ˇ D 4

hq1 ; q1 i D

hp3 ; q2 i D hq2 ; q2 i D

Z

Z

0

0 1

Z

DQG

ˇ1 ˇ 1 " 1 dt D t ˇˇ D 1 0

12t 2 .2t # 1/ dt D

0 1

Z

0

1

.24t 3 # 12t 2 / dt D 2

ˇ1 ˇ 1 1 3ˇ .2t # 1/ dt D .2t # 1/ ˇ D 6 3 0 2

0

7KHQ SURMW2 p3 D

0

1

hp3 ; q1 i hp3 ; q2 i 4 2 q1 C q2 D q1 C q2 D 4q1 C 6q2 hq1 ; q1 i hq2 ; q2 i 1 1=3 q3 D p3 # SURMW2 p3 D p3 # 4q1 # 6q2

$VDIXQFWLRQ q3 .t / D 12t 2 # 4 # 6.2t # 1/ D 12t 2 # 12t C 2 7KHRUWKRJRQDOEDVLV IRUWKHVXEVSDFH W LV fq1 ; q2 ; q3 g

13"$5*$& 130#-&.4 8VHWKHLQQHUSURGXFWD[LRPVWRYHULI\WKHIROORZLQJVWDWHPHQWV hY; i D h; Yi D 0 hX; Y C Zi D hX; Yi C hX; Zi

&9&3$*4&4 /HW R2 KDYH WKH LQQHU SURGXFW RI ([DPSOH DQG OHW [ D .1; 1/ DQG \ D .5; #1/ D )LQG k[k k\k DQG jh[; \ij2

/HW P3 KDYHWKHLQQHUSURGXFWJLYHQE\HYDOXDWLRQDW #3 #1 DQG /HW p0 .t/ D 1 p1 .t/ D t DQG p2 .t/ D t 2 D &RPSXWHWKHRUWKRJRQDOSURMHFWLRQRI p2 RQWRWKHVXE VSDFHVSDQQHGE\ p0 DQG p1

/HW R2 KDYHWKHLQQHUSURGXFWRI([DPSOH 6KRZWKDW WKH&DXFK\²6FKZDU]LQHTXDOLW\KROGVIRU [ D .3; #2/ DQG \ D .#2; 1/ >6XJJHVWLRQ 6WXG\ jh[; \ij2 @

E )LQG D SRO\QRPLDO q WKDW LV RUWKRJRQDO WR p0 DQG p1 VXFK WKDW fp0 ; p1 ; qg LV DQ RUWKRJRQDO EDVLV IRU 6SDQ fp0 ; p1 ; p2 g 6FDOH WKH SRO\QRPLDO q VR WKDW LWV YHFWRURIYDOXHVDW .#3; #1; 1; 3/ LV .1; #1; #1; 1/

E 'HVFULEHDOOYHFWRUV .´1 ; ´2 / WKDWDUHRUWKRJRQDOWR \

([HUFLVHV²UHIHUWR P2 ZLWKWKHLQQHUSURGXFWJLYHQE\HYDOXD WLRQDW #1 DQG 6HH([DPSOH &RPSXWH hp; qi ZKHUH p.t / D 4 C t q.t / D 5 # 4t 2

&RPSXWH hp; qi ZKHUH p.t / D 3t # t 2 q.t / D 3 C 2t 2

&RPSXWH kpk DQG kqk IRU p DQG q LQ([HUFLVH

&RPSXWH kpk DQG kqk IRU p DQG q LQ([HUFLVH &RPSXWHWKHRUWKRJRQDOSURMHFWLRQRI q RQWRWKHVXEVSDFH VSDQQHGE\ p IRU p DQG q LQ([HUFLVH &RPSXWHWKHRUWKRJRQDOSURMHFWLRQRI q RQWRWKHVXEVSDFH VSDQQHGE\ p IRU p DQG q LQ([HUFLVH

/HW P3 KDYHWKHLQQHUSURGXFWDVLQ([HUFLVH ZLWK p0 ; p1 DQG q WKHSRO\QRPLDOVGHVFULEHGWKHUH )LQGWKHEHVWDSSUR[ LPDWLRQWR p.t/ D t 3 E\SRO\QRPLDOVLQ 6SDQ fp0 ; p1 ; qg /HW p0 p1 DQG p2 EHWKHRUWKRJRQDOSRO\QRPLDOVGHVFULEHG LQ([DPSOH ZKHUHWKHLQQHUSURGXFWRQ P4 LVJLYHQE\ HYDOXDWLRQ DW #2 #1 DQG )LQG WKH RUWKRJRQDO SURMHFWLRQRI t 3 RQWR 6SDQ fp0 ; p1 ; p2 g )LQG D SRO\QRPLDO p3 VXFK WKDW fp0 ; p1 ; p2 ; p3 g VHH ([ HUFLVH LV DQ RUWKRJRQDO EDVLV IRU WKH VXEVSDFH P3 RI P4 6FDOHWKHSRO\QRPLDO p3 VRWKDWLWVYHFWRURIYDOXHVLV .#1; 2; 0; #2; 1/

$SSOLFDWLRQVRI,QQHU3URGXFW6SDFHV

/HW A EHDQ\LQYHUWLEOH n ! n PDWUL[ 6KRZWKDWIRU X Y LQ Rn WKHIRUPXOD hX; Yi D .AX/! .AY/ D .AX/T .AY/ GHÀQHV DQLQQHUSURGXFWRQ Rn

([HUFLVHV ² UHIHU WR V D C Œ0; 1! ZLWK WKH LQQHU SURGXFW JLYHQE\DQLQWHJUDO DVLQ([DPSOH

/HW T EHDRQHWRRQHOLQHDUWUDQVIRUPDWLRQIURPDYHFWRU VSDFH V LQWR Rn 6KRZWKDWIRU X Y LQ V WKHIRUPXOD hX; Yi D T .X/!T .Y/ GHÀQHVDQLQQHUSURGXFWRQ V

&RPSXWH hf; gi ZKHUH f .t/ D 5t # 3 DQG g.t/ D t 3 # t 2

8VHWKHLQQHUSURGXFWD[LRPVDQGRWKHUUHVXOWVRIWKLVVHFWLRQWR YHULI\WKHVWDWHPHQWVLQ([HUFLVHV² hX; c Yi D chX; Yi IRUDOOVFDODUV c ,I fX; Yg LVDQRUWKRQRUPDOVHWLQ V WKHQ kX # Yk D

p 2

hX; Yi D 14 kX C Yk2 # 14 kX # Yk2 kX C Yk2 C kX # Yk2 D 2kXk2 C 2kYk2 !p " !p " a b DQG Y D p *LYHQ a $ 0 DQG b $ 0 OHW X D p b a 8VHWKH&DXFK\²6FKZDU]LQHTXDOLW\WRFRPSDUHWKHJHRPHW p ULFPHDQ ab ZLWKWKHDULWKPHWLFPHDQ .a C b/=2 ! " ! " a 1 /HW X D DQG Y D 8VH WKH &DXFK\²6FKZDU] b 1 LQHTXDOLW\WRVKRZWKDW '

aCb 2

(2

%

a2 C b 2 2

&RPSXWH hf; gi ZKHUH f .t/ D 1 # 3t 2 DQG g.t/ D t # t 3 &RPSXWH kf k IRU f LQ([HUFLVH &RPSXWH kgk IRU g LQ([HUFLVH

/HW V EHWKHVSDFH C Œ#1; 1! ZLWKWKHLQQHUSURGXFWRI([DP SOH )LQGDQRUWKRJRQDOEDVLVIRUWKHVXEVSDFHVSDQQHGE\ WKHSRO\QRPLDOV 1 t DQG t 2 7KHSRO\QRPLDOVLQWKLVEDVLV DUHFDOOHG /HJHQGUHSRO\QRPLDOV /HW V EHWKHVSDFH C Œ#2; 2! ZLWKWKHLQQHUSURGXFWRI([DP SOH )LQGDQRUWKRJRQDOEDVLVIRUWKHVXEVSDFHVSDQQHGE\ WKHSRO\QRPLDOV 1 t DQG t 2 >0@ /HW P4 KDYHWKHLQQHUSURGXFWDVLQ([DPSOH DQGOHW p0 p1 p2 EHWKHRUWKRJRQDOSRO\QRPLDOVIURPWKDWH[DP SOH 8VLQJ\RXUPDWUL[SURJUDP DSSO\WKH*UDP²6FKPLGW SURFHVVWRWKHVHW fp0 ; p1 ; p2 ; t 3 ; t 4 g WRFUHDWHDQRUWKRJRQDO EDVLVIRU P4 >0@ /HW V EH WKH VSDFH C Œ0; 2"! ZLWK WKH LQQHU SURG XFW RI ([DPSOH 8VH WKH *UDP²6FKPLGW SURFHVV WR FUHDWH DQ RUWKRJRQDO EDVLV IRU WKH VXEVSDFH VSDQQHG E\ f1; FRV t; FRV2 t; FRV3 t g 8VHDPDWUL[SURJUDPRUFRPSXWD WLRQDOSURJUDPWRFRPSXWHWKHDSSURSULDWHGHÀQLWHLQWHJUDOV

40-65*0/4 50 13"$5*$& 130#-&.4 %\ $[LRP hY; i D h; Yi 7KHQ h; Yi D h0Y; Yi D 0hY; Yi E\ $[LRP VR h; Yi D 0

%\$[LRPV DQGWKHQDJDLQ hX; Y C Zi D hY C Z; Xi D hY; Xi C hZ; Xi D hX; Yi C hX; Zi

"11-*$"5*0/4 0' *//&3 130%6$5 41"$&4 7KHH[DPSOHVLQWKLVVHFWLRQVXJJHVWKRZWKHLQQHUSURGXFWVSDFHVGHÀQHGLQ6HFWLRQ DULVHLQSUDFWLFDOSUREOHPV 7KHÀUVWH[DPSOHLVFRQQHFWHGZLWKWKHPDVVLYHOHDVW VTXDUHVSUREOHPRIXSGDWLQJWKH1RUWK$PHULFDQ'DWXP GHVFULEHGLQWKHFKDSWHU·V LQWURGXFWRU\H[DPSOH

:HLJKWHG/HDVW6TXDUHV /HW \ EHDYHFWRURI n REVHUYDWLRQV y1 ; : : : ; yn DQGVXSSRVHZHZLVKWRDSSUR[LPDWH \ E\ DYHFWRU \O WKDWEHORQJVWRVRPHVSHFLÀHGVXEVSDFHRI Rn ,Q6HFWLRQ \O ZDVZULWWHQ DV A[ VRWKDW \O ZDVLQWKHFROXPQVSDFHRI A 'HQRWHWKHHQWULHVLQ \O E\ yO1 ; : : : ; yOn 7KHQWKH VXPRIWKHVTXDUHVIRUHUURU RU 66( LQDSSUR[LPDWLQJ \ E\ \O LV 66( D .y1 # yO1 /2 C " " " C .yn # yOn /2 7KLVLVVLPSO\ k\ # \O k2 XVLQJWKHVWDQGDUGOHQJWKLQ Rn

$)"15&3


1RZVXSSRVHWKHPHDVXUHPHQWVWKDWSURGXFHGWKHHQWULHVLQ \ DUHQRWHTXDOO\ UHOLDEOH 7KLVZDVWKHFDVHIRUWKH1RUWK$PHULFDQ'DWXP VLQFHPHDVXUHPHQWVZHUH PDGHRYHUDSHULRGRI\HDUV $VDQRWKHUH[DPSOH WKHHQWULHVLQ \ PLJKWEH FRPSXWHGIURPYDULRXVVDPSOHVRIPHDVXUHPHQWV ZLWKXQHTXDOVDPSOHVL]HV 7KHQ LWEHFRPHVDSSURSULDWHWRZHLJKWWKHVTXDUHGHUURUVLQ LQVXFKDZD\WKDWPRUH LPSRUWDQFHLVDVVLJQHGWRWKHPRUHUHOLDEOHPHDVXUHPHQWVñ ,IWKHZHLJKWVDUHGHQRWHG E\ w12 ; : : : ; wn2 WKHQWKHZHLJKWHGVXPRIWKHVTXDUHVIRUHUURULV :HLJKWHG66( D w12 .y1 # yO1 /2 C " " " C wn2 .yn # yOn /2

7KLVLVWKHVTXDUHRIWKHOHQJWKRI \ # \O ZKHUHWKHOHQJWKLVGHULYHGIURPDQLQQHU SURGXFWDQDORJRXVWRWKDWLQ([DPSOH LQ6HFWLRQ QDPHO\

h[; \i D w12 x1 y1 C " " " C wn2 xn yn ,WLVVRPHWLPHVFRQYHQLHQWWRWUDQVIRUPDZHLJKWHGOHDVWVTXDUHVSUREOHPLQWRDQ HTXLYDOHQWRUGLQDU\OHDVWVTXDUHVSUREOHP /HW W EHWKHGLDJRQDOPDWUL[ZLWKSRVLWLYH w1 ; : : : ; wn RQLWVGLDJRQDO VRWKDW 2 32 3 2 3 w1 0 """ 0 y1 w1 y1 6 0 7 6 y2 7 6 w2 y2 7 w2 6 76 7 6 7 W\D6 : 6 : 7D6 : 7 : : :: :: 7 4 :: 5 4 :: 5 4 :: 5 0 """ wn yn wn yn ZLWKDVLPLODUH[SUHVVLRQIRU W \O 2EVHUYHWKDWWKH j WKWHUPLQ FDQEHZULWWHQDV

wj2 .yj # yOj /2 D .wj yj # wj yOj /2 ,WIROORZVWKDWWKHZHLJKWHG 66( LQ LVWKHVTXDUHRIWKHRUGLQDU\OHQJWKLQ Rn RI W \ # W \O ZKLFKZHZULWHDV kW \ # W \O k2 1RZVXSSRVHWKHDSSUR[LPDWLQJYHFWRU \O LVWREHFRQVWUXFWHGIURPWKHFROXPQVRI DPDWUL[ A 7KHQZHVHHNDQ [O WKDWPDNHV AO[ D \O DVFORVHWR \ DVSRVVLEOH +RZHYHU WKHPHDVXUHRIFORVHQHVVLVWKHZHLJKWHGHUURU

kW \ # W \O k2 D kW \ # WAO[k2 7KXV [O LVWKHRUGLQDU\ OHDVWVTXDUHVVROXWLRQRIWKHHTXDWLRQ

WA[ D W \ 7KHQRUPDOHTXDWLRQIRUWKHOHDVWVTXDUHVVROXWLRQLV

.WA/T WA[ D .WA/T W \

&9".1-& )LQG WKH OHDVWVTXDUHV OLQH y D ˇ0 C ˇ1 x WKDW EHVW ÀWV WKH GDWD

.#2; 3/ .#1; 5/ .0; 5/ .1; 4/ DQG .2; 3/ 6XSSRVHWKHHUURUVLQPHDVXULQJWKH y YDOXHV RIWKHODVWWZRGDWDSRLQWVDUHJUHDWHUWKDQIRUWKHRWKHUSRLQWV :HLJKWWKHVHGDWDKDOI DVPXFKDVWKHUHVWRIWKHGDWD ñ 1RWHIRUUHDGHUVZLWKDEDFNJURXQGLQVWDWLVWLFV 6XSSRVHWKHHUURUVLQPHDVXULQJWKH yi DUHLQGHSHQGHQW UDQGRPYDULDEOHVZLWKPHDQVHTXDOWR]HURDQGYDULDQFHVRI %12 ; : : : ; %n2 7KHQWKHDSSURSULDWHZHLJKWVLQ DUH wi2 D 1=%i2 7KHODUJHUWKHYDULDQFHRIWKHHUURU WKHVPDOOHUWKHZHLJKW


40-65*0/ $VLQ6HFWLRQ ZULWH X IRUWKHPDWUL[ A DQG ˇ IRUWKHYHFWRU [ DQG REWDLQ 2 3 2 3 1 #2 3 6 1 #1 7 657 ! " 6 7 6 7 ˇ0 6 7 07 X D6 61 7; ˇ D ˇ1 ; \ D 6 5 7 41 5 445 1 1 2 3

)RUDZHLJKWLQJPDWUL[ FKRRVH W ZLWKGLDJRQDOHQWULHV DQG /HIW PXOWLSOLFDWLRQE\ W VFDOHVWKHURZVRI X DQG \ 2 3 2 3 2 #4 6 6 2 #2 7 6 10 7 6 7 6 7 6 7 07 WX D 6 62 7; W \ D 6 10 7 41 4 45 15 1 2 3 )RUWKHQRUPDOHTXDWLRQ FRPSXWH ! 14 T .WX / WX D #9

y y = 4.3 + .2x

2

DQGVROYH

y = 4 – .1x

–2

2

x

!

14 #9

#9 25

"

"!

DQG

ˇ0 ˇ1

"

D

T

.WX / W \ D !

59 #34

!

59 #34

"

"

7KHVROXWLRQRIWKHQRUPDOHTXDWLRQLVWRWZRVLJQLÀFDQWGLJLWV ˇ0 D 4:3 DQG ˇ1 D :20 7KHGHVLUHGOLQHLV y D 4:3 C :20x ,QFRQWUDVW WKHRUGLQDU\OHDVWVTXDUHVOLQHIRUWKHVHGDWDLV

'*(63&

:HLJKWHGDQGRUGLQDU\ OHDVWVTXDUHVOLQHV

#9 25

y D 4:0 # :10x %RWKOLQHVDUHGLVSOD\HGLQ)LJ

7UHQG$QDO\VLVRI'DWD /HW f UHSUHVHQWDQXQNQRZQIXQFWLRQZKRVHYDOXHVDUHNQRZQSHUKDSVRQO\DSSUR[ LPDWHO\ DW t0 ; : : : ; tn ,IWKHUHLVDÓLQHDUWUHQGµLQWKHGDWD f .t0 /; : : : ; f .tn / WKHQ ZHPLJKWH[SHFWWRDSSUR[LPDWHWKHYDOXHVRI f E\DIXQFWLRQRIWKHIRUP ˇ0 C ˇ1 t ,IWKHUHLVD´TXDGUDWLFWUHQGµWRWKHGDWD WKHQZHZRXOGWU\DIXQFWLRQRIWKHIRUP ˇ0 C ˇ1 t C ˇ2 t 2 7KLVZDVGLVFXVVHGLQ6HFWLRQ IURPDGLIIHUHQWSRLQWRIYLHZ ,QVRPHVWDWLVWLFDOSUREOHPV LWLVLPSRUWDQWWREHDEOHWRVHSDUDWHWKHOLQHDUWUHQG IURPWKHTXDGUDWLFWUHQGDQGSRVVLEO\FXELFRUKLJKHURUGHUWUHQGV )RULQVWDQFH VXSSRVHHQJLQHHUVDUHDQDO\]LQJWKHSHUIRUPDQFHRIDQHZFDU DQG f .t / UHSUHVHQWV WKHGLVWDQFHEHWZHHQWKHFDUDWWLPH t DQGVRPHUHIHUHQFHSRLQW ,IWKHFDULVWUDYHOLQJ DWFRQVWDQWYHORFLW\ WKHQWKHJUDSKRI f .t / VKRXOGEHDVWUDLJKWOLQHZKRVHVORSHLVWKH FDU·VYHORFLW\ ,IWKHJDVSHGDOLVVXGGHQO\SUHVVHGWRWKHÁRRU WKHJUDSKRI f .t/ ZLOO FKDQJHWRLQFOXGHDTXDGUDWLFWHUPDQGSRVVLEO\DFXELFWHUPGXHWRWKHDFFHOHUDWLRQ 7RDQDO\]HWKHDELOLW\RIWKHFDUWRSDVVDQRWKHUFDU IRUH[DPSOH HQJLQHHUVPD\ZDQW WRVHSDUDWHWKHTXDGUDWLFDQGFXELFFRPSRQHQWVIURPWKHOLQHDUWHUP ,IWKHIXQFWLRQLVDSSUR[LPDWHGE\DFXUYHRIWKHIRUP y D ˇ0 C ˇ1 t C ˇ2 t 2 WKH FRHIÀFLHQW ˇ2 PD\QRWJLYHWKHGHVLUHGLQIRUPDWLRQDERXWWKHTXDGUDWLFWUHQGLQWKHGDWD EHFDXVHLWPD\QRWEH´LQGHSHQGHQWµLQDVWDWLVWLFDOVHQVHIURPWKHRWKHU ˇi 7RPDNH

$)"15&3


ZKDWLVNQRZQDVD WUHQGDQDO\VLV RIWKHGDWD ZHLQWURGXFHDQLQQHUSURGXFWRQWKH VSDFH Pn DQDORJRXVWRWKDWJLYHQLQ([DPSOH LQ6HFWLRQ )RU p q LQ Pn GHÀQH

hp; qi D p.t0 /q.t0 / C " " " C p.tn /q.tn / ,QSUDFWLFH VWDWLVWLFLDQVVHOGRPQHHGWRFRQVLGHUWUHQGVLQGDWDRIGHJUHHKLJKHUWKDQ FXELFRUTXDUWLF 6ROHW p0 p1 p2 p3 GHQRWHDQRUWKRJRQDOEDVLVRIWKHVXEVSDFH P3 RI Pn REWDLQHGE\DSSO\LQJWKH*UDP²6FKPLGWSURFHVVWRWKHSRO\QRPLDOV t t 2 DQG t 3 %\6XSSOHPHQWDU\([HUFLVH LQ&KDSWHU WKHUHLVDSRO\QRPLDO g LQ Pn ZKRVHYDOXHV DW t0 ; : : : ; tn FRLQFLGHZLWKWKRVHRIWKHXQNQRZQIXQFWLRQ f /HW gO EHWKHRUWKRJRQDO SURMHFWLRQZLWKUHVSHFWWRWKHJLYHQLQQHUSURGXFW RI g RQWR P3 VD\

gO D c0 p0 C c1 p1 C c2 p2 C c3 p3 7KHQ gO LVFDOOHGDFXELF WUHQGIXQFWLRQ DQG c0 ; : : : ; c3 DUHWKH WUHQGFRHIÀFLHQWV RI WKHGDWD 7KHFRHIÀFLHQW c1 PHDVXUHVWKHOLQHDUWUHQG c2 WKHTXDGUDWLFWUHQG DQG c3 WKH FXELFWUHQG ,WWXUQVRXWWKDWLIWKHGDWDKDYHFHUWDLQSURSHUWLHV WKHVHFRHIÀFLHQWVDUH VWDWLVWLFDOO\LQGHSHQGHQW 6LQFH p0 ; : : : ; p3 DUHRUWKRJRQDO WKHWUHQGFRHIÀFLHQWVPD\EHFRPSXWHGRQHDW DWLPH LQGHSHQGHQWO\RIRQHDQRWKHU 5HFDOOWKDW ci D hg; pi i=hpi ; pi i :HFDQ LJQRUH p3 DQG c3 LIZHZDQWRQO\WKHTXDGUDWLFWUHQG $QGLI IRUH[DPSOH ZHQHHGHG WRGHWHUPLQHWKHTXDUWLFWUHQG ZHZRXOGKDYHWRÀQGYLD*UDP²6FKPLGW RQO\D SRO\QRPLDO p4 LQ P4 WKDWLVRUWKRJRQDOWR P3 DQGFRPSXWH hg; p4 i=hp4 ; p4 i

&9".1-& 7KHVLPSOHVWDQGPRVWFRPPRQXVHRIWUHQGDQDO\VLVRFFXUVZKHQWKH SRLQWV t0 ; : : : ; tn FDQEHDGMXVWHGVRWKDWWKH\DUHHYHQO\VSDFHGDQGVXPWR]HUR )LWD TXDGUDWLFWUHQGIXQFWLRQWRWKHGDWD .#2; 3/ .#1; 5/ .0; 5/ .1; 4/ DQG .2; 3/

40-65*0/ 7KH t FRRUGLQDWHVDUHVXLWDEO\VFDOHGWRXVHWKHRUWKRJRQDOSRO\QRPLDOV IRXQGLQ([DPSOH RI6HFWLRQ 3RO\QRPLDO

p0

2 3 1 617 6 7 7 9HFWRURIYDOXHV 6 6 1 7; 415 1

y = p(t)

pO D 2

D 2

'*(63&

$SSUR[LPDWLRQE\DTXDGUDWLF WUHQGIXQFWLRQ

p1

3

#2 6 #1 7 6 7 6 0 7; 6 7 4 15 2

2

p2

3

2 6 #1 7 6 7 6 #2 7; 6 7 4 #1 5 2

'DWD g

2 3 3 657 6 7 657 6 7 445 3

7KHFDOFXODWLRQVLQYROYHRQO\WKHVHYHFWRUV QRWWKHVSHFLÀFIRUPXODVIRUWKHRUWKRJRQDO SRO\QRPLDOV 7KHEHVWDSSUR[LPDWLRQWRWKHGDWDE\SRO\QRPLDOVLQ P2 LVWKHRUWKRJRQDO SURMHFWLRQJLYHQE\

y

–2

2

x

hg; p0 i hg; p1 i hg; p2 i p0 C p1 C p2 hp0 ; p0 i hp1 ; p1 i hp2 ; p2 i 20 p 5 0

#

1 p 10 1

#

7 p 14 2

DQG

p.t O / D 4 # :1t # :5.t 2 # 2/

6LQFHWKHFRHIÀFLHQWRI p2 LVQRWH[WUHPHO\VPDOO LWZRXOGEHUHDVRQDEOHWRFRQFOXGH WKDWWKHWUHQGLVDWOHDVWTXDGUDWLF 7KLVLVFRQÀUPHGE\WKHJUDSKLQ)LJ


)RXULHU6HULHV&DOFXOXVUHTXLUHG &RQWLQXRXVIXQFWLRQVDUHRIWHQDSSUR[LPDWHGE\OLQHDUFRPELQDWLRQVRIVLQHDQGFRVLQH IXQFWLRQV )RULQVWDQFH DFRQWLQXRXVIXQFWLRQPLJKWUHSUHVHQWDVRXQGZDYH DQHOHFWULF VLJQDORIVRPHW\SH RUWKHPRYHPHQWRIDYLEUDWLQJPHFKDQLFDOV\VWHP )RUVLPSOLFLW\ ZHFRQVLGHUIXQFWLRQVRQ 0 % t % 2" ,WWXUQVRXWWKDWDQ\IXQFWLRQ LQ C Œ0; 2"! FDQEHDSSUR[LPDWHGDVFORVHO\DVGHVLUHGE\DIXQFWLRQRIWKHIRUP a0 C a1 FRV t C " " " C an FRV nt C b1 VLQ t C " " " C bn VLQ nt 2 IRUDVXIÀFLHQWO\ODUJHYDOXHRI n 7KHIXQFWLRQ LVFDOOHGD WULJRQRPHWULFSRO\ QRPLDO ,I an DQG bn DUHQRWERWK]HUR WKHSRO\QRPLDOLVVDLGWREHRI RUGHU Q 7KH FRQQHFWLRQEHWZHHQWULJRQRPHWULFSRO\QRPLDOVDQGRWKHUIXQFWLRQVLQ C Œ0; 2"! GHSHQGV RQWKHIDFWWKDWIRUDQ\ n $ 1 WKHVHW

f1; FRV t; FRV 2t; : : : ; FRV nt; VLQ t; VLQ 2t; : : : ; VLQ nt g

LVRUWKRJRQDOZLWKUHVSHFWWRWKHLQQHUSURGXFW Z 2" hf; gi D f .t /g.t / dt

0

7KLVRUWKRJRQDOLW\LVYHULÀHGDVLQWKHIROORZLQJH[DPSOHDQGLQ([HUFLVHV DQG

&9".1-& /HW C Œ0; 2"! KDYHWKHLQQHUSURGXFW DQGOHW m DQG n EHXQHTXDO SRVLWLYHLQWHJHUV 6KRZWKDW FRV mt DQG FRV nt DUHRUWKRJRQDO

40-65*0/ 8VHDWULJRQRPHWULFLGHQWLW\ :KHQ m ¤ n Z 2" hFRV mt; FRV nt i D FRV mt FRV nt dt 0 Z 1 2" D ŒFRV.mt C nt / C FRV.mt # nt /! dt 2 0 ! "ˇ 1 VLQ.mt C nt / VLQ.mt # nt / ˇˇ2" D C ˇ D0 2 mCn m#n 0

/HW W EHWKHVXEVSDFHRI C Œ0; 2"! VSDQQHGE\WKHIXQFWLRQVLQ *LYHQ f LQ C Œ0; 2"! WKHEHVWDSSUR[LPDWLRQWR f E\IXQFWLRQVLQ W LVFDOOHGWKH QWKRUGHU )RXULHUDSSUR[LPDWLRQ WR f RQ Œ0; 2"! 6LQFHWKHIXQFWLRQVLQ DUHRUWKRJRQDO WKHEHVWDSSUR[LPDWLRQLVJLYHQE\WKHRUWKRJRQDOSURMHFWLRQRQWR W ,QWKLVFDVH WKH FRHIÀFLHQWV ak DQG bk LQ DUHFDOOHGWKH )RXULHUFRHIÀFLHQWV RI f 7KHVWDQGDUG IRUPXODIRUDQRUWKRJRQDOSURMHFWLRQVKRZVWKDW

ak D

hf; FRV kt i ; hFRV kt; FRV kt i

bk D

hf; VLQ kt i ; hVLQ k t; VLQ k t i

k$1

([HUFLVH DVNV\RXWRVKRZWKDW hFRV kt; FRV kt i D " DQG hVLQ k t; VLQ k t i D " 7KXV Z Z 1 2" 1 2" ak D f .t / FRV k t dt; bk D f .t / VLQ kt dt " 0 " 0

7KHFRHIÀFLHQWRIWKHFRQVWDQW IXQFWLRQ 1 LQWKHRUWKRJRQDOSURMHFWLRQLV ! Z " Z 2" 1 1 1 2" a0 hf; 1i D f .t /!1 dt D f .t / FRV.0!t / dt D h1; 1i 2" 0 2 " 0 2

ZKHUH a0 LVGHÀQHGE\ IRU k D 0 7KLVH[SODLQVZK\WKHFRQVWDQWWHUPLQ LV ZULWWHQDV a0 =2

$)"15&3


&9".1-& )LQGWKH nWKRUGHU)RXULHUDSSUR[LPDWLRQWRWKHIXQFWLRQ f .t / D t RQ WKHLQWHUYDO Œ0; 2"! 40-65*0/ &RPSXWH 1 1 a0 D ! 2 2 "

Z

2" 0

1 t dt D 2"

"

ˇ # 1 2 ˇˇ2" t D" 2 ˇ0

DQGIRU k > 0 XVLQJLQWHJUDWLRQE\SDUWV ! "2" Z 1 2" 1 1 t ak D t FRV kt dt D FRV k t C VLQ k t D0 " 0 " k2 k 0 ! "2" Z 1 2" 1 1 t 2 bk D t VLQ k t dt D VLQ k t # FRV kt D# " 0 " k2 k k 0 7KXVWKH nWKRUGHU)RXULHUDSSUR[LPDWLRQRI f .t / D t LV 2 2 " # 2 VLQ t # VLQ 2t # VLQ 3t # " " " # VLQ nt 3 n )LJXUH VKRZVWKHWKLUGDQGIRXUWKRUGHU)RXULHUDSSUR[LPDWLRQVRI f y

y 2!

2!

y=t

y=t

!

!

! (a) Third order

2!

t

!

2!

t

(b) Fourth order

'*(63& )RXULHUDSSUR[LPDWLRQVRIWKHIXQFWLRQ f .t/ D t

7KHQRUPRIWKHGLIIHUHQFHEHWZHHQ f DQGD)RXULHUDSSUR[LPDWLRQLVFDOOHGWKH PHDQVTXDUHHUURU LQWKHDSSUR[LPDWLRQ 7KHWHUP PHDQ UHIHUVWRWKHIDFWWKDW WKHQRUPLVGHWHUPLQHGE\DQLQWHJUDO ,WFDQEHVKRZQWKDWWKHPHDQVTXDUHHUURU DSSURDFKHV]HURDVWKHRUGHURIWKH)RXULHUDSSUR[LPDWLRQLQFUHDVHV )RUWKLVUHDVRQ LW LVFRPPRQWRZULWH 1 X a0 C f .t / D .am FRV mt C bm VLQ mt / 2 mD1 7KLVH[SUHVVLRQIRU f .t / LVFDOOHGWKH )RXULHUVHULHV IRU f RQ Œ0; 2"! 7KHWHUP am FRV mt IRUH[DPSOH LVWKHSURMHFWLRQRI f RQWRWKHRQHGLPHQVLRQDOVXEVSDFH VSDQQHGE\ FRV mt

13"$5*$& 130#-&.4 /HW q1 .t / D 1 q2 .t / D t DQG q3 .t / D 3t 2 # 4 9HULI\WKDW fq1 ; q2 ; q3 g LVDQRUWKRJ RQDOVHWLQ C Œ#2; 2! ZLWKWKHLQQHUSURGXFWRI([DPSOH LQ6HFWLRQ LQWHJUDWLRQ IURP #2 WR )LQGWKHÀUVWRUGHUDQGWKLUGRUGHU)RXULHUDSSUR[LPDWLRQVWR

f .t / D 3 # 2 VLQ t C 5 VLQ 2t # 6 FRV 2t


&9&3$*4&4 )LQGWKHOHDVWVTXDUHVOLQH y D ˇ0 C ˇ1 x WKDWEHVWÀWVWKH GDWD .#2; 0/ .#1; 0/ .0; 2/ .1; 4/ DQG .2; 4/ DVVXPLQJWKDW WKHÀUVWDQGODVWGDWDSRLQWVDUHOHVVUHOLDEOH :HLJKWWKHP KDOIDVPXFKDVWKHWKUHHLQWHULRUSRLQWV 6XSSRVHRXWRIGDWDSRLQWVLQDZHLJKWHGOHDVWVTXDUHV SUREOHPKDYHD y PHDVXUHPHQWWKDWLVOHVVUHOLDEOHWKDQWKH RWKHUV DQGWKH\DUHWREHZHLJKWHGKDOIDVPXFKDVWKHRWKHU SRLQWV 2QHPHWKRGLVWRZHLJKWWKHSRLQWVE\DIDFWRU RIDQGWKHRWKHUE\DIDFWRURI 12 $ VHFRQGPHWKRGLV WRZHLJKWWKHSRLQWVE\DIDFWRURIDQGWKHRWKHUE\D IDFWRURI 'RWKHWZRPHWKRGVSURGXFHGLIIHUHQWUHVXOWV" ([SODLQ )LWDFXELFWUHQGIXQFWLRQWRWKHGDWDLQ([DPSOH 7KH RUWKRJRQDOFXELFSRO\QRPLDOLV p3 .t / D 56 t 3 # 176 t

7RPDNHDWUHQGDQDO\VLVRIVL[HYHQO\VSDFHGGDWDSRLQWV RQH FDQXVHRUWKRJRQDOSRO\QRPLDOVZLWKUHVSHFWWRHYDOXDWLRQDW WKHSRLQWV t D #5; #3; #1; 1; 3 DQG D 6KRZWKDWWKHÀUVWWKUHHRUWKRJRQDOSRO\QRPLDOVDUH

p0 .t/ D 1;

p1 .t/ D t;

DQG

p2 .t/ D 38 t 2 #

35 8

7KHSRO\QRPLDO p2 KDVEHHQVFDOHGVRWKDWLWVYDOXHVDW WKHHYDOXDWLRQSRLQWVDUHVPDOOLQWHJHUV E )LWDTXDGUDWLFWUHQGIXQFWLRQWRWKHGDWD

.#5; 1/; .#3; 1/; .#1; 4/; .1; 4/; .3; 6/; .5; 8/ ,Q([HUFLVHV² WKHVSDFHLV C Œ0; 2"! ZLWKWKHLQQHUSURGXFW 6KRZWKDW VLQ mt DQG VLQ nt DUHRUWKRJRQDOZKHQ m ¤ n

6KRZWKDW VLQ mt DQG FRV nt DUHRUWKRJRQDOIRUDOOSRVLWLYH LQWHJHUV m DQG n 6KRZWKDW k FRV k tk2 D " DQG k VLQ kt k2 D " IRU k > 0

)LQGWKHWKLUGRUGHU)RXULHUDSSUR[LPDWLRQWR f .t/ D t # 1

)LQG WKH WKLUGRUGHU )RXULHU DSSUR[LPDWLRQ WR f .t / D 2" # t )LQG WKH WKLUGRUGHU )RXULHU DSSUR[LPDWLRQ WR WKH VTXDUH ZDYHIXQFWLRQ f .t/ D 1 IRU 0 % t < " DQG f .t / D #1 IRU " % t < 2" )LQGWKHWKLUGRUGHU)RXULHUDSSUR[LPDWLRQWR VLQ2 t ZLWKRXW SHUIRUPLQJDQ\LQWHJUDWLRQFDOFXODWLRQV

)LQGWKHWKLUGRUGHU)RXULHUDSSUR[LPDWLRQWR FRV3 t ZLWKRXW SHUIRUPLQJDQ\LQWHJUDWLRQFDOFXODWLRQV ([SODLQZK\D)RXULHUFRHIÀFLHQWRIWKHVXPRIWZRIXQFWLRQV LVWKHVXPRIWKHFRUUHVSRQGLQJ)RXULHUFRHIÀFLHQWVRIWKH WZRIXQFWLRQV 6XSSRVHWKHÀUVWIHZ)RXULHUFRHIÀFLHQWVRIVRPHIXQFWLRQ f LQ C Œ0; 2"! DUH a0 a1 a2 DQG b1 b2 b3 :KLFKRIWKH IROORZLQJWULJRQRPHWULFSRO\QRPLDOVLVFORVHUWR f " 'HIHQG \RXUDQVZHU a0 C a1 FRV t C a2 FRV 2t C b1 VLQ t g.t/ D 2 a0 h.t/ D C a1 FRV t C a2 FRV 2t C b1 VLQ t C b2 VLQ 2t 2 >0@ 5HIHUWRWKHGDWDLQ([HUFLVH LQ6HFWLRQ FRQ FHUQLQJWKHWDNHRIISHUIRUPDQFHRIDQDLUSODQH 6XSSRVHWKH SRVVLEOHPHDVXUHPHQWHUURUVEHFRPHJUHDWHUDVWKHVSHHGRI WKHDLUSODQHLQFUHDVHV DQGOHW W EHWKHGLDJRQDOZHLJKWLQJ PDWUL[ZKRVHGLDJRQDOHQWULHVDUH DQG )LQGWKHFXELFFXUYHWKDWÀWVWKHGDWDZLWK PLQLPXPZHLJKWHGOHDVWVTXDUHVHUURU DQGXVHLWWRHVWLPDWH WKHYHORFLW\RIWKHSODQHZKHQ t D 4:5 VHFRQGV >0@ /HW f4 DQG f5 EHWKHIRXUWKRUGHUDQGÀIWKRUGHU)RXULHU DSSUR[LPDWLRQVLQ C Œ0; 2"! WRWKHVTXDUHZDYHIXQFWLRQLQ ([HUFLVH 3URGXFHVHSDUDWHJUDSKVRI f4 DQG f5 RQWKH LQWHUYDO Œ0; 2"! DQGSURGXFHDJUDSKRI f5 RQ Œ#2"; 2"! SG

5IF -JOFBSJUZ PG BO 0SUIPHPOBM 1SPKFDUJPO m

40-65*0/4 50 13"$5*$& 130#-&.4 &RPSXWH

ˇ2 1 2 ˇˇ hq1 ; q2 i D 1!t dt D t ˇ D 0 2 !2 !2 ˇ2 Z 2 ˇ hq1 ; q3 i D 1! .3t 2 # 4/ dt D .t 3 # 4t /ˇˇ D 0 Z

2

!2

hq2 ; q3 i D

Z

2

!2

t ! .3t 2 # 4/ dt D

'

!2

(ˇ2 ˇ 3 4 t # 2t 2 ˇˇ D 0 4 !2

$)"15&3


y y = 3 – 2 sin t

7KHWKLUGRUGHU)RXULHUDSSUR[LPDWLRQWR f LVWKHEHVWDSSUR[LPDWLRQLQ C Œ0; 2"! WR f E\IXQFWLRQVYHFWRUV LQWKHVXEVSDFHVSDQQHGE\ FRV t FRV 2t FRV 3t VLQ t VLQ 2t DQG VLQ 3t %XW f LVREYLRXVO\ LQ WKLVVXEVSDFH VR f LVLWVRZQEHVW DSSUR[LPDWLRQ f .t / D 3 # 2 VLQ t C 5 VLQ 2t # 6 FRV 2t

y = f (t)

9 3

π 2π

–3

t

)RUWKHÀUVWRUGHUDSSUR[LPDWLRQ WKHFORVHVWIXQFWLRQWR f LQWKHVXEVSDFH W D 6SDQf1; FRV t; VLQ t g LV 3 # 2 VLQ t 7KHRWKHUWZRWHUPVLQWKHIRUPXODIRU f .t/ DUH RUWKRJRQDOWRWKHIXQFWLRQVLQ W VRWKH\FRQWULEXWHQRWKLQJWRWKHLQWHJUDOVWKDW JLYHWKH)RXULHUFRHIÀFLHQWVIRUDÀUVWRUGHUDSSUR[LPDWLRQ

)LUVWDQGWKLUGRUGHU DSSUR[LPDWLRQVWR f .t /

$)"15&3 4611-&.&/5"3: &9&3$*4&4 7KHIROORZLQJVWDWHPHQWVUHIHUWRYHFWRUVLQ Rn RU Rm / ZLWK WKHVWDQGDUGLQQHUSURGXFW 0DUNHDFKVWDWHPHQW7UXHRU )DOVH -XVWLI\HDFKDQVZHU D 7KHOHQJWKRIHYHU\YHFWRULVDSRVLWLYHQXPEHU E $ YHFWRU Y DQGLWVQHJDWLYH #Y KDYHHTXDOOHQJWKV F

7KHGLVWDQFHEHWZHHQ X DQG Y LV kX # Yk

G ,I r LVDQ\VFDODU WKHQ kr Yk D rkYk H

,IWZRYHFWRUVDUHRUWKRJRQDO WKH\DUHOLQHDUO\LQGHSHQ GHQW

I

,I [ LV RUWKRJRQDO WR ERWK X DQG Y WKHQ [ PXVW EH RUWKRJRQDOWR X # Y

J ,I kX C Yk2 D kXk2 C kYk2 WKHQ X DQG Y DUHRUWKRJRQDO

K ,I kX # Yk2 D kXk2 C kYk2 WKHQ X DQG Y DUHRUWKRJRQDO L

7KHRUWKRJRQDOSURMHFWLRQRI \ RQWR X LVDVFDODUPXOWLSOH RI \

M

,IDYHFWRU \ FRLQFLGHVZLWKLWVRUWKRJRQDOSURMHFWLRQRQWR DVXEVSDFH W WKHQ \ LVLQ W

N 7KHVHWRIDOOYHFWRUVLQ Rn RUWKRJRQDOWRRQHÀ[HGYHFWRU LVDVXEVSDFHRI Rn

U

$ OHDVWVTXDUHVVROXWLRQRI A[ D E LVWKHYHFWRU AO[ LQ &RO A FORVHVWWR E VRWKDW kE # AO[ k % kE # A[k IRU DOO [

V

7KH QRUPDO HTXDWLRQV IRU D OHDVWVTXDUHV VROXWLRQ RI A[ D E DUHJLYHQE\ [O D .ATA/!1 AT E

/HW fY1 ; : : : ; Yp g EHDQRUWKRQRUPDOVHW 9HULI\WKHIROORZLQJ HTXDOLW\E\LQGXFWLRQ EHJLQQLQJZLWK p D 2 ,I [ D c1 Y1 C " " " C cp Yp WKHQ

k[k2 D jc1 j2 C " " " C jcp j2 /HW fY1 ; : : : ; Yp g EHDQRUWKRQRUPDOVHWLQ Rn 9HULI\WKH IROORZLQJLQHTXDOLW\ FDOOHG %HVVHO·VLQHTXDOLW\ ZKLFKLVWUXH IRUHDFK [ LQ Rn

k[k2 $ j[ ! Y1 j2 C j[ ! Y2 j2 C " " " C j[ ! Yp j2 /HW U EH DQ n ! n RUWKRJRQDO PDWUL[ 6KRZ WKDW LI fY1 ; : : : ; Yn g LV DQ RUWKRQRUPDO EDVLV IRU Rn WKHQ VR LV fU Y1 ; : : : ; U Yn g

,I W LVDVXEVSDFHRI Rn WKHQ W DQG W ? KDYHQR YHFWRUVLQFRPPRQ

6KRZWKDWLIDQ n ! n PDWUL[ U VDWLVÀHV .U [/! .U \/ D [ ! \ IRUDOO [ DQG \ LQ Rn WKHQ U LVDQRUWKRJRQDOPDWUL[

P ,I fY1 ; Y2 ; Y3 g LVDQRUWKRJRQDOVHWDQGLI c1 c2 DQG c3 DUH VFDODUV WKHQ fc1 Y1 ; c2 Y2 ; c3 Y3 g LVDQRUWKRJRQDOVHW

6KRZWKDWLI U LVDQRUWKRJRQDOPDWUL[ WKHQDQ\UHDOHLJHQ YDOXHRI U PXVWEH ˙1

O

Q ,IDPDWUL[ U KDVRUWKRQRUPDOFROXPQV WKHQ U U T D I

R $ VTXDUHPDWUL[ZLWKRUWKRJRQDOFROXPQVLVDQRUWKRJR QDOPDWUL[ S ,IDVTXDUHPDWUL[KDVRUWKRQRUPDOFROXPQV WKHQLWDOVR KDVRUWKRQRUPDOURZV T ,I W LVDVXEVSDFH WKHQ k SURMW Yk2 C kY # SURMW Yk2 D kY k 2

$ +RXVHKROGHUPDWUL[ RUDQ HOHPHQWDU\UHÁHFWRU KDVWKH IRUP Q D I # 2XXT ZKHUH X LVDXQLWYHFWRU 6HH([HU FLVH LQWKH6XSSOHPHQWDU\([HUFLVHVIRU&KDSWHU 6KRZ WKDW Q LVDQRUWKRJRQDOPDWUL[ (OHPHQWDU\UHÁHFWRUVDUHRI WHQXVHGLQFRPSXWHUSURJUDPVWRSURGXFHD45 IDFWRUL]DWLRQ RIDPDWUL[ A ,I A KDVOLQHDUO\LQGHSHQGHQWFROXPQV WKHQ OHIWPXOWLSOLFDWLRQE\DVHTXHQFHRIHOHPHQWDU\UHÁHFWRUVFDQ SURGXFHDQXSSHUWULDQJXODUPDWUL[


/HW T W Rn ! Rn EHDOLQHDUWUDQVIRUPDWLRQWKDWSUHVHUYHV OHQJWKV WKDWLV kT .[/k D k[k IRUDOO [ LQ Rn D 6KRZ WKDW T DOVR SUHVHUYHV RUWKRJRQDOLW\ WKDW LV T .[/!T .\/ D 0 ZKHQHYHU [ ! \ D 0

([HUFLVHVDQGFRQFHUQWKHUHDO 6FKXUIDFWRUL]DWLRQ RIDQ n ! n PDWUL[ A LQWKHIRUP A D URU T ZKHUH U LVDQRUWKRJRQDO PDWUL[DQG R LVDQ n ! n XSSHUWULDQJXODUPDWUL[

/HW X DQG Y EHOLQHDUO\LQGHSHQGHQWYHFWRUVLQ Rn WKDWDUH QRW RUWKRJRQDO 'HVFULEHKRZWRÀQGWKHEHVWDSSUR[LPDWLRQ WR ] LQ Rn E\YHFWRUVRIWKHIRUP x1 X C x2 Y ZLWKRXWÀUVW FRQVWUXFWLQJDQRUWKRJRQDOEDVLVIRU 6SDQ fX; Yg

/HW A EHDQ n ! n PDWUL[ZLWK n UHDOHLJHQYDOXHV FRXQWLQJ PXOWLSOLFLWLHV GHQRWHGE\ &1 ; : : : ; &n ,WFDQEHVKRZQWKDW A DGPLWVDUHDO 6FKXUIDFWRUL]DWLRQ 3DUWVD DQGE VKRZ WKHNH\LGHDVLQWKHSURRI 7KHUHVWRIWKHSURRIDPRXQWVWR UHSHDWLQJD DQGE IRUVXFFHVVLYHO\VPDOOHUPDWULFHV DQG WKHQSLHFLQJWRJHWKHUWKHUHVXOWV D /HW X1 EH D XQLW HLJHQYHFWRU FRUUHVSRQGLQJ WR &1 OHW X2 ; : : : ; Xn EHDQ\RWKHUYHFWRUVVXFKWKDW fX1 ; : : : ; Xn g LV DQ RUWKRQRUPDO EDVLV IRU Rn DQG WKHQ OHW U D Œ X1 X2 " " " Xn ! 6KRZ WKDW WKH ÀUVW FROXPQ RI U T AU LV &1 H1 ZKHUH H1 LVWKHÀUVWFROXPQRIWKH n ! n LGHQWLW\PDWUL[

E 6KRZ WKDW WKH VWDQGDUG PDWUL[ RI T LV DQ RUWKRJRQDO PDWUL[

6XSSRVHWKHFROXPQVRI A DUHOLQHDUO\LQGHSHQGHQW 'HWHU PLQHZKDWKDSSHQVWRWKHOHDVWVTXDUHVVROXWLRQ [O RI A[ D E ZKHQ E LVUHSODFHGE\ c E IRUVRPHQRQ]HURVFDODU c ,I a b DQG c DUH GLVWLQFW QXPEHUV WKHQ WKH IROORZLQJ V\VWHPLVLQFRQVLVWHQWEHFDXVHWKHJUDSKVRIWKHHTXDWLRQV DUHSDUDOOHOSODQHV 6KRZWKDWWKHVHWRIDOOOHDVWVTXDUHV VROXWLRQVRIWKHV\VWHPLVSUHFLVHO\WKHSODQHZKRVHHTXDWLRQ LV x # 2y C 5´ D .a C b C c/=3

6KRZ WKDW LI A DGPLWV D UHDO 6FKXU IDFWRUL]DWLRQ A D URU T WKHQ A KDV n UHDOHLJHQYDOXHV FRXQWLQJPXOWLSOLF LWLHV

E 3DUWD LPSOLHVWKDW U TAU KDVWKHIRUPVKRZQEHORZ ([SODLQZK\WKHHLJHQYDOXHVRI A1 DUH &2 ; : : : ; &n >+LQW 6HHWKH6XSSOHPHQWDU\([HUFLVHVIRU&KDSWHU @

x # 2y C 5´ D a x # 2y C 5´ D b

2

&1 6 0 6 U TAU D 6 :: 4 :

x # 2y C 5´ D c

&RQVLGHUWKHSUREOHPRIÀQGLQJDQHLJHQYDOXHRIDQ n ! n PDWUL[ A ZKHQ DQ DSSUR[LPDWH HLJHQYHFWRU Y LV NQRZQ 6LQFH Y LVQRWH[DFWO\FRUUHFW WKHHTXDWLRQ

AY D &Y

.1/

ZLOO SUREDEO\ QRW KDYH D VROXWLRQ +RZHYHU & FDQ EH HVWLPDWHGE\DOHDVWVTXDUHVVROXWLRQZKHQ LVYLHZHG SURSHUO\ 7KLQNRI Y DVDQ n ! 1 PDWUL[ V WKLQNRI & DV DYHFWRULQ R1 DQGGHQRWHWKHYHFWRU AY E\WKHV\PERO E 7KHQ EHFRPHV E D &V ZKLFKPD\DOVREHZULWWHQDV V & D E )LQGWKHOHDVWVTXDUHVVROXWLRQRIWKLVV\VWHPRI n HTXDWLRQVLQWKHRQHXQNQRZQ & DQGZULWHWKLVVROXWLRQXVLQJ WKHRULJLQDOV\PEROV 7KHUHVXOWLQJHVWLPDWHIRU & LVFDOOHGD 5D\OHLJKTXRWLHQW 6HH([HUFLVHVDQGLQ6HFWLRQ 8VHWKHVWHSVEHORZWRSURYHWKHIROORZLQJUHODWLRQVDPRQJ WKH IRXU IXQGDPHQWDO VXEVSDFHV GHWHUPLQHG E\ DQ m ! n PDWUL[ A 5RZ A D .1XO A/? ;

&RO A D .1XO AT /?

D 6KRZWKDW 5RZ A LVFRQWDLQHGLQ .1XO A/? 6KRZWKDWLI [ LVLQ 5RZ A WKHQ [ LVRUWKRJRQDOWRHYHU\ X LQ 1XO A E 6XSSRVH UDQN A D r )LQG GLP 1XO A DQG GLP .1XO A/? DQGWKHQGHGXFHIURPSDUWD WKDW 5RZ A D .1XO A/? >+LQW 6WXG\WKHH[HUFLVHVIRU6HFWLRQ @ F ([SODLQZK\ &RO A D .1XO AT /? ([SODLQZK\DQHTXDWLRQ A[ D E KDVDVROXWLRQLIDQGRQO\ LI E LVRUWKRJRQDOWRDOOVROXWLRQVRIWKHHTXDWLRQ AT[ D

0

'

' A1

'

'

3 7 7 7 5

>0@ :KHQ WKH ULJKW VLGH RI DQ HTXDWLRQ A[ D E LV FKDQJHG VOLJKWO\³VD\ WR A[ D E C $E IRUVRPHYHFWRU $E³WKHVROXWLRQ FKDQJHVIURP [ WR [ C $[ ZKHUH $[ VDWLVÀHV A.$[/ D $E 7KHTXRWLHQW k$Ek=kEk LVFDOOHGWKH UHODWLYHFKDQJH LQ E RU WKH UHODWLYHHUURU LQ E ZKHQ $E UHSUHVHQWVSRVVLEOHHUURULQWKH HQWULHVRI E/ 7KHUHODWLYHFKDQJHLQWKHVROXWLRQLV k$[k=k[k :KHQ A LVLQYHUWLEOH WKH FRQGLWLRQQXPEHU RI A ZULWWHQDV FRQG.A/ SURGXFHVDERXQGRQKRZODUJHWKHUHODWLYHFKDQJHLQ [ FDQEH

k$Ek k$[k % FRQG.A/! k[ k kE k

.2/

,Q([HUFLVHV² VROYH A[ D E DQG A.$[/ D $E DQGVKRZ WKDWWKHLQHTXDOLW\ KROGVLQHDFKFDVH 6HHWKHGLVFXVVLRQRI LOOFRQGLWLRQHG PDWULFHVLQ([HUFLVHV²LQ6HFWLRQ ! " ! " ! " 4:5 3:1 19:249 :001 ED $E D A D 1:6 1:1 6:843 #:003 ! " ! " ! " 4:5 3:1 :500 :001 A D ED $E D 1:6 1:1 #1:407 #:003 ,IFRPSOH[QXPEHUVDUHDOORZHG HYHU\ n # n PDWUL[ A DGPLWVD FRPSOH[ 6FKXUIDFWRUL]DWLRQ A D URU !1 ZKHUH R LVXSSHUWULDQJXODU DQG U !1 LVWKH FRQMXJDWH WUDQVSRVHRI U 7KLVYHU\XVHIXOIDFWLV GLVFXVVHGLQ 0DWUL[$QDO\VLV E\5RJHU$ +RUQDQG&KDUOHV5 -RKQVRQ &DPEULGJH &DPEULGJH8QLYHUVLW\3UHVV SS ² 1

$)"15&3 2


3 2 3 #6 #4 1 :100 6 7 1 0 #2 7 7 E D 6 2:888 7 4 #1:404 5 11 7 #3 5 9 7 1 1:462 2 3 :49 6 7 #1:28 7 $E D 10!4 6 4 5:78 5 8:04

7 6 #5 6 A D 4 10 19

2

3 2 3 #6 #4 1 4:230 6 7 1 0 #2 7 7 E D 6 #11:043 7 4 49:991 5 11 7 #3 5 9 7 1 69:536 2 3 :27 6 7 7:76 7 $E D 10!4 6 4 #3:77 5 3:93

7 6 #5 6 A D 4 10 19

6\PPHWULF0DWULFHV DQG4XDGUDWLF)RUPV

*/530%6$503: &9".1-&

0XOWLFKDQQHO,PDJH3URFHVVLQJ $URXQGWKHZRUOGLQOLWWOHPRUH WKDQ PLQXWHV WKHWZR /DQGVDWVDWHOOLWHVVWUHDNVLOHQWO\DFURVVWKHVN\LQQHDU SRODURUELWV UHFRUGLQJLPDJHVRIWHUUDLQDQGFRDVWOLQH LQ VZDWKV NLORPHWHUVZLGH (YHU\ GD\V HDFKVDWHOOLWH SDVVHVRYHUDOPRVWHYHU\VTXDUHNLORPHWHURIWKHHDUWK·V VXUIDFH VRDQ\ORFDWLRQFDQEHPRQLWRUHGHYHU\GD\V 7KH/DQGVDWLPDJHVDUHXVHIXOIRUPDQ\SXUSRVHV 'HYHORSHUVDQGXUEDQSODQQHUVXVHWKHPWRVWXG\WKHUDWH DQGGLUHFWLRQRIXUEDQJURZWK LQGXVWULDOGHYHORSPHQW DQG RWKHUFKDQJHVLQODQGXVDJH 5XUDOFRXQWULHVFDQDQDO\]H VRLOPRLVWXUH FODVVLI\WKHYHJHWDWLRQLQUHPRWHUHJLRQV DQG ORFDWHLQODQGODNHVDQGVWUHDPV *RYHUQPHQWVFDQGHWHFW DQGDVVHVVGDPDJHIURPQDWXUDOGLVDVWHUV VXFKDVIRUHVW ÀUHV ODYDÁRZV ÁRRGV DQGKXUULFDQHV (QYLURQPHQWDO DJHQFLHVFDQLGHQWLI\SROOXWLRQIURPVPRNHVWDFNVDQG PHDVXUHZDWHUWHPSHUDWXUHVLQODNHVDQGULYHUVQHDUSRZHU SODQWV 6HQVRUVDERDUGWKHVDWHOOLWHDFTXLUHVHYHQVLPXO WDQHRXVLPDJHVRIDQ\UHJLRQRQHDUWKWREHVWXGLHG 7KH VHQVRUVUHFRUGHQHUJ\IURPVHSDUDWHZDYHOHQJWKEDQGV³ WKUHHLQWKHYLVLEOHOLJKWVSHFWUXPDQGIRXULQLQIUDUHGDQG WKHUPDOEDQGV (DFKLPDJHLVGLJLWL]HGDQGVWRUHGDVD UHFWDQJXODUDUUD\RIQXPEHUV HDFKQXPEHULQGLFDWLQJWKH VLJQDOLQWHQVLW\DWDFRUUHVSRQGLQJVPDOOSRLQWRU SL[HO

RQWKHLPDJH (DFKRIWKHVHYHQLPDJHVLVRQHFKDQQHORI D PXOWLFKDQQHO RU PXOWLVSHFWUDOLPDJH 7KHVHYHQ/DQGVDWLPDJHVRIRQHÀ[HGUHJLRQW\SLFDOO\ FRQWDLQPXFKUHGXQGDQWLQIRUPDWLRQ VLQFHVRPHIHDWXUHV ZLOODSSHDULQVHYHUDOLPDJHV

$)"15&3

6\PPHWULF0DWULFHVDQG4XDGUDWLF)RUPV

3ULQFLSDOFRPSRQHQWDQDO\VLVLVLOOXVWUDWHGLQWKH SKRWRVEHORZ WDNHQRYHU5DLOURDG9DOOH\ 1HYDGD ,PDJHV IURPWKUHH/DQGVDWVSHFWUDOEDQGVDUHVKRZQLQD ²F 7KHWRWDOLQIRUPDWLRQLQWKHWKUHHEDQGVLV UHDUUDQJHG LQ WKHWKUHHSULQFLSDOFRPSRQHQWLPDJHVLQ G ²I 7KHÀUVW FRPSRQHQWG GLVSOD\VRU´H[SODLQVµ RIWKHVFHQH YDULDQFHSUHVHQWLQWKHLQLWLDOGDWD ,QWKLVZD\ WKHWKUHH FKDQQHOLQLWLDOGDWDKDYHEHHQUHGXFHGWRRQHFKDQQHO

GDWD ZLWKDORVVLQVRPHVHQVHRIRQO\RIWKHVFHQH YDULDQFH (DUWK6DWHOOLWH&RUSRUDWLRQRI5RFNYLOOH 0DU\ODQG ZKLFKNLQGO\VXSSOLHGWKHSKRWRVVKRZQKHUH LV H[SHULPHQWLQJZLWKLPDJHVIURPVHSDUDWHVSHFWUDO EDQGV 3ULQFLSDOFRPSRQHQWDQDO\VLV HVVHQWLDOIRUVXFK PDVVLYHGDWDVHWV W\SLFDOO\UHGXFHVWKHGDWDWRDERXW XVDEOHSULQFLSDOFRPSRQHQWV WEB

-

6\PPHWULFPDWULFHVDULVHPRUHRIWHQLQDSSOLFDWLRQV LQRQHZD\RUDQRWKHU WKDQDQ\ RWKHUPDMRUFODVVRIPDWULFHV 7KHWKHRU\LVULFKDQGEHDXWLIXO GHSHQGLQJLQDQHVVHQWLDO ZD\RQERWKGLDJRQDOL]DWLRQIURP&KDSWHUDQGRUWKRJRQDOLW\IURP&KDSWHU 7KH GLDJRQDOL]DWLRQRIDV\PPHWULFPDWUL[ GHVFULEHGLQ6HFWLRQ LVWKHIRXQGDWLRQIRU WKHGLVFXVVLRQLQ6HFWLRQVDQGFRQFHUQLQJTXDGUDWLFIRUPV 6HFWLRQ LQWXUQ LVQHHGHGIRUWKHÀQDOWZRVHFWLRQVRQWKHVLQJXODUYDOXHGHFRPSRVLWLRQDQGRQWKHLPDJH SURFHVVLQJGHVFULEHGLQWKHLQWURGXFWRU\H[DPSOH 7KURXJKRXWWKHFKDSWHU DOOYHFWRUV DQGPDWULFHVKDYHUHDOHQWULHV

'LDJRQDOL]DWLRQRI6\PPHWULF0DWULFHV

%*"(0/"-*;"5*0/ 0' 4:..&53*$ ."53*$&4 $ V\PPHWULF PDWUL[LVDPDWUL[ A VXFKWKDW AT A 6XFKDPDWUL[LVQHFHVVDULO\VTXDUH ,WVPDLQGLDJRQDOHQWULHVDUHDUELWUDU\ EXWLWVRWKHUHQWULHVRFFXULQSDLUV³RQRSSRVLWH VLGHVRIWKHPDLQGLDJRQDO

&9".1-& 2IWKHIROORZLQJPDWULFHV RQO\WKHÀUVWWKUHHDUHV\PPHWULF 6\PPHWULF

!

"

1 0 ; 0 !3

1RQV\PPHWULF

!

" 1 !3 ; 3 0

2

3 0 !1 0 4 !1 5 8 5; 0 8 !7 2 3 1 !4 0 4 !6 1 !4 5; 0 !6 1

2

a 4b c 2 5 44 3

b d e 4 3 2

3 c e5 f

3 2 1

3 2 15 0

7REHJLQWKHVWXG\RIV\PPHWULFPDWULFHV LWLVKHOSIXOWRUHYLHZWKHGLDJRQDOL]DWLRQ SURFHVVRI6HFWLRQ 2

3 6 !2 !1 &9".1-& ,ISRVVLEOH GLDJRQDOL]HWKHPDWUL[ A D 4 !2 6 !1 5 !1 !1 5

40-65*0/ 7KHFKDUDFWHULVWLFHTXDWLRQRI A LV

0 D !!3 C 17!2 ! 90! C 144 D !.! ! 8/.! ! 6/.! ! 3/ 6WDQGDUGFDOFXODWLRQVSURGXFHDEDVLVIRUHDFKHLJHQVSDFH 2 3 2 3 !1 !1 ! D 8 W Y1 D 4 1 5I ! D 6 W Y2 D 4 !1 5I 0 2

2 3 1 ! D 3 W Y3 D 4 1 5 1

7KHVHWKUHHYHFWRUVIRUPDEDVLVIRU R3 ,QIDFW LWLVHDV\WRFKHFNWKDW fY1 ; Y2 ; Y3 g LV DQ RUWKRJRQDO EDVLVIRU R3 ([SHULHQFHIURP&KDSWHU VXJJHVWVWKDWDQ RUWKRQRUPDO EDVLVPLJKWEHXVHIXOIRUFDOFXODWLRQV VRKHUHDUHWKHQRUPDOL]HGXQLW HLJHQYHFWRUV 2 2 p 3 p 3 p 3 2 !1=p6 1= 3 !1=p2 6 7 6 p 7 X1 D 4 1= 2 5 ; X2 D 4 !1= 6 5; X3 D 4 1= 3 5 p p 0 2= 6 1= 3 /HW

2

p p !1=p2 !1=p6 6 P D 4 1= 2 !1= 6 p 0 2= 6

p 3 1=p3 7 1=p3 5 ; 1= 3

2

8 D D 40 0

0 6 0

3 0 05 3

7KHQ A D 3'3!1 DVXVXDO %XWWKLVWLPH VLQFH P LVVTXDUHDQGKDVRUWKRQRUPDO FROXPQV P LVDQ RUWKRJRQDO PDWUL[ DQG P !1 LVVLPSO\ P T 6HH6HFWLRQ 7KHRUHPH[SODLQVZK\WKHHLJHQYHFWRUVLQ([DPSOHDUHRUWKRJRQDO³WKH\FRU UHVSRQGWRGLVWLQFWHLJHQYDOXHV

5)&03&.

,I A LVV\PPHWULF WKHQDQ\WZRHLJHQYHFWRUVIURPGLIIHUHQWHLJHQVSDFHVDUH RUWKRJRQDO

$)"15&3


1300' /HW Y1 DQG Y2 EHHLJHQYHFWRUVWKDWFRUUHVSRQGWRGLVWLQFWHLJHQYDOXHV VD\ !1 DQG !2 7RVKRZWKDW Y1 ! Y2 D 0 FRPSXWH !1 Y1 ! Y2 D .!1 Y1 /T Y2 D .AY1 /T Y2 D D D

.YT1 AT /Y2 YT1 .!2 Y2 / !2 YT1 Y2 D

D

YT1 .AY2 /

6LQFH Y1 LVDQHLJHQYHFWRU 6LQFH AT D A 6LQFH Y2 LVDQHLJHQYHFWRU

!2 Y1 ! Y2

+HQFH .!1 ! !2 /Y1 ! Y2 D 0 %XW !1 ! !2 ¤ 0 VR Y1 ! Y2 D 0 7KHVSHFLDOW\SHRIGLDJRQDOL]DWLRQLQ([DPSOHLVFUXFLDOIRUWKHWKHRU\RIV\P PHWULFPDWULFHV $Q n " n PDWUL[ A LVVDLGWREH RUWKRJRQDOO\GLDJRQDOL]DEOH LIWKHUH DUHDQRUWKRJRQDOPDWUL[ P ZLWK P !1 D P T DQGDGLDJRQDOPDWUL[ D VXFKWKDW

A D 3'3T D 3'3!1

6XFKDGLDJRQDOL]DWLRQUHTXLUHV n OLQHDUO\LQGHSHQGHQWDQGRUWKRQRUPDOHLJHQYHF WRUV :KHQLVWKLVSRVVLEOH" ,I A LVRUWKRJRQDOO\GLDJRQDOL]DEOHDVLQ WKHQ

AT D .3'3T /T D P T T D T P T D 3'3T D A 7KXV A LVV\PPHWULF 7KHRUHPEHORZVKRZVWKDW FRQYHUVHO\ HYHU\V\PPHWULFPDWUL[ LVRUWKRJRQDOO\GLDJRQDOL]DEOH 7KHSURRILVPXFKKDUGHUDQGLVRPLWWHG WKHPDLQLGHD IRUDSURRIZLOOEHJLYHQDIWHU7KHRUHP

5)&03&.

$Q n " n PDWUL[ A LVRUWKRJRQDOO\GLDJRQDOL]DEOHLIDQGRQO\LI A LVDV\PPHWULF PDWUL[ 7KLVWKHRUHPLVUDWKHUDPD]LQJ EHFDXVHWKHZRUNLQ&KDSWHUZRXOGVXJJHVWWKDW LWLVXVXDOO\LPSRVVLEOHWRWHOOZKHQDPDWUL[LVGLDJRQDOL]DEOH %XWWKLVLVQRWWKHFDVH IRUV\PPHWULFPDWULFHV 7KHQH[WH[DPSOHWUHDWVDPDWUL[ZKRVHHLJHQYDOXHVDUHQRWDOOGLVWLQFW 2

3 !2 &9".1-& 2UWKRJRQDOO\GLDJRQDOL]HWKHPDWUL[ A D 4 !2 6 4 2 FKDUDFWHULVWLFHTXDWLRQLV

3 4 2 5 ZKRVH 3

0 D !!3 C 12!2 ! 21! ! 98 D !.! ! 7/2 .! C 2/ 40-65*0/ 7KHXVXDOFDOFXODWLRQVSURGXFHEDVHVIRUWKHHLJHQVSDFHV 2 3 2 3 2 3 1 !1=2 !1 ! D 7 W Y1 D 4 0 5; Y2 D 4 1 5 I ! D !2 W Y3 D 4 !1=2 5 1 0 1

$OWKRXJK Y1 DQG Y2 DUHOLQHDUO\LQGHSHQGHQW WKH\DUHQRWRUWKRJRQDO 5HFDOOIURP Y2 ! Y1 6HFWLRQWKDWWKHSURMHFWLRQRI Y2 RQWR Y1 LV Y1 DQGWKHFRPSRQHQWRI Y2 Y1 ! Y1 RUWKRJRQDOWR Y1 LV 2 3 2 3 2 3 !1=2 1 !1=4 !1=2 Y2 ! Y1 405 D 4 1 5 Y1 D 4 1 5 ! ] 2 D Y2 ! Y1 ! Y1 2 0 1 1=4


7KHQ fY1 ; ]2 g LVDQRUWKRJRQDOVHWLQWKHHLJHQVSDFHIRU ! D 7 1RWHWKDW ]2 LVDOLQHDU FRPELQDWLRQRIWKHHLJHQYHFWRUV Y1 DQG Y2 VR ]2 LVLQWKHHLJHQVSDFH 7KLVFRQVWUXFWLRQ RI ]2 LVMXVWWKH*UDP²6FKPLGWSURFHVVRI6HFWLRQ 6LQFHWKHHLJHQVSDFHLVWZR GLPHQVLRQDOZLWKEDVLV Y1 ; Y2 / WKHRUWKRJRQDOVHW fY1 ; ]2 g LVDQ RUWKRJRQDOEDVLV IRUWKH HLJHQVSDFH E\WKH%DVLV7KHRUHP 6HH6HFWLRQRU 1RUPDOL]H Y1 DQG ]2 WRREWDLQWKHIROORZLQJRUWKRQRUPDOEDVLVIRUWKHHLJHQVSDFH IRU ! D 7 2 p 3 2 p 3 !1=p18 1= 2 6 7 X1 D 4 0p 5 ; X2 D 4 4= 18 5 p 1= 2 1= 18 $QRUWKRQRUPDOEDVLVIRUWKHHLJHQVSDFHIRU ! D !2 LV

2 3 2 3 !2 !2=3 1 14 !1 5 D 4 !1=3 5 2Y3 D X3 D k2Y3 k 3 2 2=3

%\7KHRUHP X3 LVRUWKRJRQDOWRWKHRWKHUHLJHQYHFWRUV X1 DQG X2 +HQFH fX1 ; X2 ; X3 g LVDQRUWKRQRUPDOVHW /HW 2 p 3 p 2 3 1= 2 !1= 18 !2=3 7 0 0 p 6 7 7 05 P D Œ X1 X2 X3 " D 4 0 4= 18 !1=3 5 ; D D 4 0 p p 0 0 !2 1= 18 2=3 1= 2 7KHQ P RUWKRJRQDOO\GLDJRQDOL]HV A DQG A D 3'3!1

,Q([DPSOH WKHHLJHQYDOXHKDVPXOWLSOLFLW\WZRDQGWKHHLJHQVSDFHLVWZR GLPHQVLRQDO 7KLVIDFWLVQRWDFFLGHQWDO DVWKHQH[WWKHRUHPVKRZV

7KH6SHFWUDO7KHRUHP 7KHVHWRIHLJHQYDOXHVRIDPDWUL[ A LVVRPHWLPHVFDOOHGWKH VSHFWUXP RI A DQGWKH IROORZLQJGHVFULSWLRQRIWKHHLJHQYDOXHVLVFDOOHGD VSHFWUDOWKHRUHP

5)&03&.

5IF 4QFDUSBM 5IFPSFN GPS 4ZNNFUSJD .BUSJDFT $Q n " n V\PPHWULFPDWUL[ A KDVWKHIROORZLQJSURSHUWLHV D A KDV n UHDOHLJHQYDOXHV FRXQWLQJPXOWLSOLFLWLHV E 7KHGLPHQVLRQRIWKHHLJHQVSDFHIRUHDFKHLJHQYDOXH ! HTXDOVWKHPXOWLSOLFLW\ RI ! DVDURRWRIWKHFKDUDFWHULVWLFHTXDWLRQ F 7KH HLJHQVSDFHV DUH PXWXDOO\ RUWKRJRQDO LQ WKH VHQVH WKDW HLJHQYHFWRUV FRUUHVSRQGLQJWRGLIIHUHQWHLJHQYDOXHVDUHRUWKRJRQDO G A LVRUWKRJRQDOO\GLDJRQDOL]DEOH

3DUWD IROORZVIURP([HUFLVHLQ6HFWLRQ 3DUWE IROORZVHDVLO\IURPSDUW G 6HH([HUFLVH 3DUWF LV7KHRUHP %HFDXVHRID DSURRIRIG FDQEHJLYHQ XVLQJ([HUFLVHDQGWKH6FKXUIDFWRUL]DWLRQGLVFXVVHGLQ6XSSOHPHQWDU\([HUFLVH LQ&KDSWHU 7KHGHWDLOVDUHRPLWWHG

$)"15&3


6SHFWUDO'HFRPSRVLWLRQ 6XSSRVH A D 3'3!1 ZKHUHWKHFROXPQVRI P DUHRUWKRQRUPDOHLJHQYHFWRUV X1 ; : : : ; Xn RI A DQGWKHFRUUHVSRQGLQJHLJHQYDOXHV !1 ; : : : ; !n DUHLQWKHGLDJRQDOPDWUL[ D 7KHQ VLQFH P !1 D P T # A D 3'3T D X1 # D !1 X1

###

###

2

$6 Xn 4

2

XT1

!1

3

::

:

0

$6 : 7 !n Xn 4 :: 5 XTn

32

3 XT1 7 6 :: 7 54 : 5 !n XTn

0

8VLQJWKHFROXPQ²URZH[SDQVLRQRIDSURGXFW7KHRUHPLQ6HFWLRQ ZHFDQ ZULWH

A D !1 X1 XT1 C !2 X2 XT2 C # # # C !n Xn XTn

7KLVUHSUHVHQWDWLRQRI A LVFDOOHGD VSHFWUDOGHFRPSRVLWLRQ RI A EHFDXVHLWEUHDNV XS A LQWRSLHFHVGHWHUPLQHGE\WKHVSHFWUXPHLJHQYDOXHV RI A (DFKWHUPLQ LV DQ n " n PDWUL[RIUDQN )RUH[DPSOH HYHU\FROXPQRI !1 X1 XT1 LVDPXOWLSOHRI X1 )XUWKHUPRUH HDFKPDWUL[ Xj XTj LVD SURMHFWLRQPDWUL[ LQWKHVHQVHWKDWIRUHDFK [ LQ Rn WKHYHFWRU .Xj XTj /[ LVWKHRUWKRJRQDOSURMHFWLRQRI [ RQWRWKHVXEVSDFHVSDQQHGE\ Xj 6HH([HUFLVH

&9".1-& &RQVWUXFWDVSHFWUDOGHFRPSRVLWLRQRIWKHPDWUL[ A WKDWKDVWKHRU WKRJRQDOGLDJRQDOL]DWLRQ AD

!

7 2

2 4

"

D

!

p p "! 2=p5 !1=p5 8 1= 5 2= 5 0

0 3

"!

p " 1=p5 2= 5

p 2=p5 !1= 5

40-65*0/ 'HQRWHWKHFROXPQVRI P E\ X1 DQG X2 7KHQ A D 8X1 XT1 C 3X2 XT2 7RYHULI\WKLVGHFRPSRVLWLRQRI A FRPSXWH

p " ! " p $ 2=p5 # p 4=5 2=5 D 2= 5 1= 5 2=5 1=5 1= 5 p " ! ! " p p $ !1=p5 # 1=5 !2=5 T X2 X2 D !1= 5 2= 5 D !2=5 4=5 2= 5 X1 XT1 D

!

DQG

8X1 XT1

C

3X2 XT2

D

!

32=5 16=5

" ! 16=5 3=5 C 8=5 !6=5

!6=5 12=5

"

D

!

7 2

2 4

"

DA


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

13"$5*$& 130#-&.4 6KRZWKDWLI A LVDV\PPHWULFPDWUL[ WKHQ A2 LVV\PPHWULF 6KRZWKDWLI A LVRUWKRJRQDOO\GLDJRQDOL]DEOH WKHQVRLV A2

&9&3$*4&4 'HWHUPLQHZKLFKRIWKHPDWULFHVLQ([HUFLVHV²DUHV\PPHWULF ! " ! " 3 5 !3 5 5 !7 !5 3 2 3 ! " 0 8 3 2 2 0 !2 5 4 8 4 4 3 !2 0 2 3 2 3 !6 2 0 1 2 1 2 25 1 2 15 4 0 !6 4 2 0 0 !6 1 2 1 2 'HWHUPLQHZKLFKRIWKHPDWULFHVLQ([HUFLVHV²DUHRUWKRJR QDO ,IRUWKRJRQDO ÀQGWKHLQYHUVH " p p # ! " :6 :8 1=p2 !1=p2 :8 !:6 1= 2 1= 2 2 3 ! " !1 2 2 !5 2 25 4 2 !1 2 5 2 2 !1 2 3 2=3 2=3 1=3 p p 1=p 5 !2=p 5 5 4 p 0 5=3 !4= 45 !2= 45 2 3 :5 :5 !:5 !:5 6 !:5 :5 !:5 :5 7 7 6 4 :5 :5 :5 :5 5 !:5 :5 :5 !:5 2UWKRJRQDOO\GLDJRQDOL]HWKHPDWULFHVLQ([HUFLVHV² JLYLQJ DQRUWKRJRQDOPDWUL[ P DQGDGLDJRQDOPDWUL[ D 7RVDYH\RX

WLPH WKHHLJHQYDOXHVLQ([HUFLVHV²DUH !2 !50 ! ! " ! " 3 1 1 5 1 3 5 1 ! " ! " 16 !4 !7 24 !4 1 24 7 2 3 2 3 1 1 3 !2 !36 0 3 15 05 4 1 4 !36 !23 3 1 1 0 0 3 2 3 2 3 3 !2 4 7 !4 4 6 25 5 05 4 !2 4 !4 4 2 3 4 0 9 2 3 2 3 4 1 3 1 2 0 0 0 61 60 4 1 37 1 0 17 7 7 6 6 43 5 4 1 4 1 0 0 2 05 1 3 1 4 0 1 0 1 2 3 2 3 3 1 1 1 3 1 5 DQG Y D 4 1 5 9HULI\WKDWLVDQ /HW A D 4 1 1 1 3 1 HLJHQYDOXHRI A DQG Y LVDQHLJHQYHFWRU 7KHQRUWKRJRQDOO\ GLDJRQDOL]H A 2 3 2 3 2 3 5 !4 !2 !2 1 5 2 5 Y1 D 4 2 5 DQG Y2 D 4 1 5 /HW A D 4 !4 !2 2 2 1 0 9HULI\WKDW Y1 DQG Y2 DUHHLJHQYHFWRUVRI A 7KHQRUWKRJR QDOO\GLDJRQDOL]H A

$)"15&3


,Q([HUFLVHVDQG PDUNHDFKVWDWHPHQW7UXHRU)DOVH -XVWLI\ HDFKDQVZHU D $Q n " n PDWUL[WKDWLVRUWKRJRQDOO\GLDJRQDOL]DEOHPXVW EHV\PPHWULF E ,I AT D A DQGLIYHFWRUV X DQG Y VDWLVI\ AX D 3X DQG AY D 4Y WKHQ X ! Y D 0

F $Q n " n V\PPHWULFPDWUL[KDV n GLVWLQFWUHDOHLJHQYDO XHV G )RUDQRQ]HUR Y LQ Rn WKHPDWUL[ YYT LVFDOOHGDSURMHF WLRQPDWUL[ D (YHU\V\PPHWULFPDWUL[LVRUWKRJRQDOO\GLDJRQDOL]DEOH E ,I B D 3'3T ZKHUH P T D P !1 DQG D LVDGLDJRQDO PDWUL[ WKHQ B LVDV\PPHWULFPDWUL[ F $QRUWKRJRQDOPDWUL[LVRUWKRJRQDOO\GLDJRQDOL]DEOH G 7KHGLPHQVLRQRIDQHLJHQVSDFHRIDV\PPHWULFPDWUL[ HTXDOVWKHPXOWLSOLFLW\RIWKHFRUUHVSRQGLQJHLJHQYDOXH 6XSSRVH A LVDV\PPHWULF n " n PDWUL[DQG B LVDQ\ n " m PDWUL[ 6KRZWKDW B TAB B TB DQG BB T DUHV\PPHWULF PDWULFHV 6KRZWKDWLI A LVDQ n " n V\PPHWULFPDWUL[ WKHQ A[/! \ D [ !.A\/ IRUDOO [; \ LQ Rn

6XSSRVH A LV LQYHUWLEOH DQG RUWKRJRQDOO\ GLDJRQDOL]DEOH ([SODLQZK\ A!1 LVDOVRRUWKRJRQDOO\GLDJRQDOL]DEOH 6XSSRVH A DQG B DUHERWKRUWKRJRQDOO\GLDJRQDOL]DEOHDQG AB D BA ([SODLQZK\ AB LVDOVRRUWKRJRQDOO\GLDJRQDO L]DEOH /HW A D 3'3!1 ZKHUH P LVRUWKRJRQDODQG D LVGLDJRQDO DQGOHW ! EHDQHLJHQYDOXHRI A RIPXOWLSOLFLW\ k 7KHQ ! DSSHDUV k WLPHVRQWKHGLDJRQDORI D ([SODLQZK\WKH GLPHQVLRQRIWKHHLJHQVSDFHIRU ! LV k 6XSSRVH A D PRP !1 ZKHUH P LVRUWKRJRQDODQG R LVXSSHU WULDQJXODU 6KRZWKDWLI A LVV\PPHWULF WKHQ R LVV\PPHWULF DQGKHQFHLVDFWXDOO\DGLDJRQDOPDWUL[ &RQVWUXFWDVSHFWUDOGHFRPSRVLWLRQRI A IURP([DPSOH &RQVWUXFWDVSHFWUDOGHFRPSRVLWLRQRI A IURP([DPSOH /HW X EHDXQLWYHFWRULQ Rn DQGOHW B D XXT

D *LYHQDQ\ [ LQ Rn FRPSXWH B [ DQGVKRZWKDW B [ LVWKH RUWKRJRQDOSURMHFWLRQRI [ RQWR X DVGHVFULEHGLQ6HFWLRQ E 6KRZWKDW B LVDV\PPHWULFPDWUL[DQG B 2 D B

F 6KRZWKDW X LVDQHLJHQYHFWRURI B :KDWLVWKHFRUUH VSRQGLQJHLJHQYDOXH"

/HW B EHDQ n " n V\PPHWULFPDWUL[VXFKWKDW B 2 B $Q\ VXFKPDWUL[LVFDOOHGD SURMHFWLRQPDWUL[ RUDQ RUWKRJRQDO SURMHFWLRQPDWUL[ *LYHQDQ\ \ LQ Rn OHW \O D B \ DQG ] D \ ! \O D 6KRZWKDW ] LVRUWKRJRQDOWR \O E /HW W EHWKHFROXPQVSDFHRI B 6KRZWKDW \ LVWKHVXP RIDYHFWRULQ W DQGDYHFWRULQ W ? :K\GRHVWKLVSURYH WKDW B \ LVWKHRUWKRJRQDOSURMHFWLRQRI \ RQWRWKHFROXPQ VSDFHRI B "

>0@ 2UWKRJRQDOO\GLDJRQDOL]HWKHPDWULFHVLQ([HUFLVHV² 7RSUDFWLFHWKHPHWKRGVRIWKLVVHFWLRQ GRQRWXVHDQHLJHQYHFWRU URXWLQHIURP\RXUPDWUL[SURJUDP ,QVWHDG XVHWKHSURJUDPWR ÀQGWKHHLJHQYDOXHV DQG IRUHDFKHLJHQYDOXH ! ÀQGDQRUWKRQRU PDOEDVLVIRU 1XO.A ! !I / DVLQ([DPSOHVDQG 2 3 5 2 9 !6 6 2 5 !6 97 7 6 4 9 !6 5 25 !6 9 2 5 2 3 :38 !:18 !:06 !:04 6 !:18 :59 !:04 :12 7 7 6 4 !:06 !:04 :47 !:12 5 !:04 :12 !:12 :41 2 3 :31 :58 :08 :44 6 :58 !:56 :44 !:58 7 7 6 4 :08 :44 :19 !:08 5 :44 !:58 !:08 :31 2 3 10 2 2 !6 9 6 2 10 2 !6 97 6 7 2 2 10 !6 97 6 6 7 4 !6 !6 !6 26 95 9 9 9 9 !19

40-65*0/4 50 13"$5*$& 130#-&.4 .A2 /T D .AA/T D ATAT E\DSURSHUW\RIWUDQVSRVHV %\K\SRWKHVLV AT D A 6R .A2 /T D AA D A2 ZKLFKVKRZVWKDW A2 LVV\PPHWULF ,I A LVRUWKRJRQDOO\GLDJRQDOL]DEOH WKHQ A LVV\PPHWULF E\7KHRUHP %\3UDFWLFH 3UREOHP A2 LVV\PPHWULFDQGKHQFHLVRUWKRJRQDOO\GLDJRQDOL]DEOH7KHRUHP

4XDGUDWLF)RUPV

26"%3"5*$ '03.4 8QWLOQRZ RXUDWWHQWLRQLQWKLVWH[WKDVIRFXVHGRQOLQHDUHTXDWLRQV H[FHSWIRUWKHVXPV RIVTXDUHVHQFRXQWHUHGLQ&KDSWHUZKHQFRPSXWLQJ [T[ 6XFKVXPVDQGPRUHJHQHUDO H[SUHVVLRQV FDOOHG TXDGUDWLFIRUPV RFFXUIUHTXHQWO\LQDSSOLFDWLRQVRIOLQHDUDOJHEUD WRHQJLQHHULQJLQGHVLJQFULWHULDDQGRSWLPL]DWLRQ DQGVLJQDOSURFHVVLQJDVRXWSXW QRLVHSRZHU 7KH\DOVRDULVH IRUH[DPSOH LQSK\VLFVDVSRWHQWLDODQGNLQHWLFHQHUJ\ GLIIHUHQWLDOJHRPHWU\DVQRUPDOFXUYDWXUHRIVXUIDFHV HFRQRPLFVDVXWLOLW\IXQFWLRQV DQGVWDWLVWLFVLQFRQÀGHQFHHOOLSVRLGV 6RPHRIWKHPDWKHPDWLFDOEDFNJURXQGIRUVXFK DSSOLFDWLRQVÁRZVHDVLO\IURPRXUZRUNRQV\PPHWULFPDWULFHV $ TXDGUDWLFIRUP RQ Rn LVDIXQFWLRQ Q GHÀQHGRQ Rn ZKRVHYDOXHDWDYHFWRU [ n LQ R FDQEHFRPSXWHGE\DQH[SUHVVLRQRIWKHIRUP Q.[/ D [TA[ ZKHUH A LVDQ n " n V\PPHWULFPDWUL[ 7KHPDWUL[ A LVFDOOHGWKH PDWUL[RIWKHTXDGUDWLFIRUP 7KHVLPSOHVWH[DPSOHRIDQRQ]HURTXDGUDWLFIRUPLV Q.[/ D [TI [ D k[k2 ([ DPSOHVDQGVKRZWKHFRQQHFWLRQEHWZHHQDQ\V\PPHWULFPDWUL[ A DQGWKHTXDGUDWLF IRUP [TA[

&9".1-& /HW [ D D A D

!

4 0

0 3

"

!

" x1 &RPSXWH [TA[ IRUWKHIROORZLQJPDWULFHV x2 ! " 3 !2 E A D !2 7

40-65*0/

! "! " ! " 4 0 x1 4x1 D [ A[ D Œ x1 x2 " D Œ x1 x2 " D 4x12 C 3x22 3x2 0 3 x2 E 7KHUHDUHWZR !2 HQWULHVLQ A :DWFKKRZWKH\HQWHUWKHFDOFXODWLRQV 7KH .1; 2/ HQWU\LQ A LVLQEROGIDFHW\SH T

T

[ A[ D Œ x1

! "! " ! " 3 ! x1 3x1 ! x2 x2 " D Œ x1 x2 " !2 7 x2 !2x1 C 7x2

D x1 .3x1 ! x2 / C x2 .!2x1 C 7x2 / D 3x12 ! x1 x2 ! 2x2 x1 C 7x22

D 3x12 ! 4x1 x2 C 7x22

7KHSUHVHQFHRI !4x1 x2 LQWKHTXDGUDWLFIRUPLQ([DPSOHE LVGXHWRWKH !2 HQWULHVRIIWKHGLDJRQDOLQWKHPDWUL[ A ,QFRQWUDVW WKHTXDGUDWLFIRUPDVVRFLDWHGZLWK WKHGLDJRQDOPDWUL[ A LQ([DPSOHD KDVQR x1 x2 FURVVSURGXFW WHUP

&9".1-& )RU [ LQ R3 OHW Q.[/ D 5x12 C 3x22 C 2x32 ! x1 x2 C 8x2 x3 :ULWH

WKLVTXDGUDWLFIRUPDV [TA[

40-65*0/ 7KHFRHIÀFLHQWVRI x12 x22 x32 JRRQWKHGLDJRQDORI A 7RPDNH A V\P PHWULF WKHFRHIÀFLHQWRI xi xj IRU i ¤ j PXVWEHVSOLWHYHQO\EHWZHHQWKH .i; j /DQG .j; i /HQWULHVLQ A 7KHFRHIÀFLHQWRI x1 x3 LV ,WLVUHDGLO\FKHFNHGWKDW

Q.[/ D [TA[ D Œ x1 x2

2

5 !1=2 x3 " 4 !1=2 3 0 4

32 3 0 x1 4 5 4 x2 5 x3 2

$)"15&3


&9".1-& /HW Q.[/ D x12 ! 8x1 x2 ! 5x22 &RPSXWHWKHYDOXHRI Q.[/ IRU [ D !

" ! " ! " !3 2 1 DQG 1 !2 !3

40-65*0/ Q.!3; 1/ D .!3/2 ! 8.!3/.1/ ! 5.1/2 D 28

Q.2; !2/ D .2/2 ! 8.2/.!2/ ! 5.!2/2 D 16

Q.1; !3/ D .1/2 ! 8.1/.!3/ ! 5.!3/2 D !20

,QVRPHFDVHV TXDGUDWLFIRUPVDUHHDVLHUWRXVHZKHQWKH\KDYHQRFURVVSURGXFW WHUPV³WKDWLV ZKHQWKHPDWUL[RIWKHTXDGUDWLFIRUPLVDGLDJRQDOPDWUL[ )RUWXQDWHO\ WKHFURVVSURGXFWWHUPFDQEHHOLPLQDWHGE\PDNLQJDVXLWDEOHFKDQJHRIYDULDEOH

&KDQJHRI9DULDEOHLQD4XDGUDWLF)RUP ,I [ UHSUHVHQWVDYDULDEOHYHFWRULQ Rn WKHQD FKDQJHRIYDULDEOH LVDQHTXDWLRQRIWKH IRUP [ D P \; RUHTXLYDOHQWO\; \ D P !1 [ ZKHUH P LVDQLQYHUWLEOHPDWUL[DQG \ LVDQHZYDULDEOHYHFWRULQ Rn +HUH \ LVWKH FRRUGLQDWHYHFWRURI [ UHODWLYHWRWKHEDVLVRI Rn GHWHUPLQHGE\WKHFROXPQVRI P 6HH 6HFWLRQ ,IWKHFKDQJHRIYDULDEOH LVPDGHLQDTXDGUDWLFIRUP [TA[ WKHQ [TA[ D .P \/TA.P \/ D \TP TAP \ D \T.P TAP /\

DQGWKHQHZPDWUL[RIWKHTXDGUDWLFIRUPLV P TAP 6LQFH A LVV\PPHWULF 7KHRUHP JXDUDQWHHVWKDWWKHUHLVDQ RUWKRJRQDO PDWUL[ P VXFKWKDW P TAP LVDGLDJRQDOPDWUL[ D DQGWKHTXDGUDWLFIRUPLQ EHFRPHV \TD \ 7KLVLVWKHVWUDWHJ\RIWKHQH[WH[DPSOH

&9".1-& 0DNHDFKDQJHRIYDULDEOHWKDWWUDQVIRUPVWKHTXDGUDWLFIRUPLQ([ DPSOH LQWRDTXDGUDWLFIRUPZLWKQRFURVVSURGXFWWHUP

40-65*0/ 7KHPDWUL[RIWKHTXDGUDWLFIRUPLQ([DPSOHLV ! " 1 !4 AD !4 !5 7KHÀUVWVWHSLVWRRUWKRJRQDOO\GLDJRQDOL]H A ,WVHLJHQYDOXHVWXUQRXWWREH ! D 3 DQG ! D !7 $VVRFLDWHGXQLWHLJHQYHFWRUVDUH " " p # p # 2=p5 1=p5 ! D 3W I ! D !7 W !1= 5 2= 5 7KHVHYHFWRUVDUHDXWRPDWLFDOO\RUWKRJRQDOEHFDXVHWKH\FRUUHVSRQGWRGLVWLQFWHLJHQ YDOXHV DQGVRSURYLGHDQRUWKRQRUPDOEDVLVIRU R2 /HW " p p # ! " 3 0 2=p5 1=p5 P D ; DD 0 !7 !1= 5 2= 5 7KHQ A D 3'3!1 DQG D D P !1 $3 D P TAP DVSRLQWHGRXWHDUOLHU $ VXLWDEOHFKDQJH RIYDULDEOHLV ! " ! " x1 y1 [ D P \; ZKHUH [ D DQG \ D x2 y2

4XDGUDWLF)RUPV

7KHQ

x12 ! 8x1 x2 ! 5x22 D [TA[ D .P \/TA.P \/ D \TP T$3\ D \T D \

D 3y12 ! 7y22

7RLOOXVWUDWHWKHPHDQLQJRIWKHHTXDOLW\RITXDGUDWLFIRUPVLQ([DPSOH ZHFDQ FRPSXWH Q.[/ IRU [ D .2; !2/ XVLQJWKHQHZTXDGUDWLFIRUP )LUVW VLQFH [ D P \ VR \D +HQFH

"

\ D P !1 [ D P T [

p p #! p # " " 2 2=p5 !1=p5 6=p5 D !2 1= 5 2= 5 !2= 5

p p 3y12 ! 7y22 D 3.6= 5/2 ! 7.!2= 5/2 D 3.36=5/ ! 7.4=5/ D 80=5 D 16

7KLVLVWKHYDOXHRI Q.[/ LQ([DPSOHZKHQ [ D .2; !2/ 6HH)LJ x !2

xTAx

Multiplication by P

0

16

!

yTDy !2

y

'*(63& &KDQJHRIYDULDEOHLQ [TA[

([DPSOHLOOXVWUDWHVWKHIROORZLQJWKHRUHP 7KHSURRIRIWKHWKHRUHPZDVHVVHQ WLDOO\JLYHQEHIRUH([DPSOH

5)&03&.

5IF 1SJODJQBM "YFT 5IFPSFN /HW A EHDQ n " n V\PPHWULFPDWUL[ 7KHQWKHUHLVDQRUWKRJRQDOFKDQJHRI YDULDEOH [ D P \ WKDWWUDQVIRUPVWKHTXDGUDWLFIRUP [TA[ LQWRDTXDGUDWLFIRUP \TD \ ZLWKQRFURVVSURGXFWWHUP 7KHFROXPQVRI P LQWKHWKHRUHPDUHFDOOHGWKH SULQFLSDOD[HV RIWKHTXDGUDWLF IRUP [TA[ 7KHYHFWRU \ LVWKHFRRUGLQDWHYHFWRURI [ UHODWLYHWRWKHRUWKRQRUPDOEDVLV RI Rn JLYHQE\WKHVHSULQFLSDOD[HV

$ *HRPHWULF9LHZRI3ULQFLSDO$[HV 6XSSRVH Q.[/ D [TA[ ZKHUH A LVDQLQYHUWLEOH 2 " 2 V\PPHWULFPDWUL[ DQGOHW c EHD FRQVWDQW ,WFDQEHVKRZQWKDWWKHVHWRIDOO [ LQ R2 WKDWVDWLVI\ [TA[ D c

$)"15&3


HLWKHUFRUUHVSRQGVWRDQHOOLSVHRUFLUFOH DK\SHUEROD WZRLQWHUVHFWLQJOLQHV RUDVLQJOH SRLQW RUFRQWDLQVQRSRLQWVDWDOO ,I A LVDGLDJRQDOPDWUL[ WKHJUDSKLVLQ VWDQGDUG SRVLWLRQ VXFKDVLQ)LJ ,I A LVQRWDGLDJRQDOPDWUL[ WKHJUDSKRIHTXDWLRQ LV x2

x2

b b a

x1

x1

a

x 21 x 22 — + — = 1, a > b > 0 a2 b2

x 21 x 22 — – — = 1, a > b > 0 a2 b2 hyperbola

ellipse '*(63& $QHOOLSVHDQGDK\SHUERODLQVWDQGDUGSRVLWLRQ

URWDWHGRXWRIVWDQGDUGSRVLWLRQ DVLQ)LJ )LQGLQJWKH SULQFLSDOD[HV GHWHUPLQHG E\WKHHLJHQYHFWRUVRI A DPRXQWVWRÀQGLQJDQHZFRRUGLQDWHV\VWHPZLWKUHVSHFWWR ZKLFKWKHJUDSKLVLQVWDQGDUGSRVLWLRQ x2 y2

x2

y2

y1

1 1 1

1

x1

x1 y1

(a) 5x 21 – 4x1x 2 + 5x 22 = 48

(b) x 12 – 8x1x2 – 5x 22 = 16

'*(63& $QHOOLSVHDQGDK\SHUEROD QRW LQVWDQGDUGSRVLWLRQ

7KHK\SHUERODLQ)LJ E LVWKHJUDSKRIWKHHTXDWLRQ [TA[ D 16 ZKHUH A LVWKH PDWUL[LQ([DPSOH 7KHSRVLWLYH y1 D[LVLQ)LJ E LVLQWKHGLUHFWLRQRIWKHÀUVW FROXPQRIWKHPDWUL[ P LQ([DPSOH DQGWKHSRVLWLYH y2 D[LVLVLQWKHGLUHFWLRQRIWKH VHFRQGFROXPQRI P

&9".1-& 7KHHOOLSVHLQ)LJ D LVWKHJUDSKRIWKHHTXDWLRQ 5x12 ! 4x1 x2 C

5x22 D 48 )LQGDFKDQJHRIYDULDEOHWKDWUHPRYHVWKHFURVVSURGXFWWHUPIURPWKH HTXDWLRQ ! " 5 !2 40-65*0/ 7KHPDWUL[RIWKHTXDGUDWLFIRUPLV A D 7KHHLJHQYDOXHVRI !2 5 A WXUQRXWWREHDQG ZLWKFRUUHVSRQGLQJXQLWHLJHQYHFWRUV " p # " p # 1=p2 !1=p2 X1 D ; X2 D 1= 2 1= 2

4XDGUDWLF)RUPV

"

p p # 1=p2 !1=p2 /HW P D Œ X1 X2 " D 7KHQ P RUWKRJRQDOO\GLDJRQDOL]HV A VRWKH 1= 2 1= 2 FKDQJHRIYDULDEOH [ D P \ SURGXFHVWKHTXDGUDWLFIRUP \T D \ D 3y12 C 7y22 7KHQHZ D[HVIRUWKLVFKDQJHRIYDULDEOHDUHVKRZQLQ)LJ D

&ODVVLI\LQJ4XDGUDWLF)RUPV :KHQ A LVDQ n " n PDWUL[ WKHTXDGUDWLFIRUP Q.[/ D [TA[ LVDUHDOYDOXHGIXQFWLRQ ZLWKGRPDLQ Rn )LJXUH GLVSOD\VWKHJUDSKVRIIRXUTXDGUDWLFIRUPVZLWKGRPDLQ R2 )RUHDFKSRLQW [ D .x1 ; x2 / LQWKHGRPDLQRIDTXDGUDWLFIRUP Q WKHJUDSKGLVSOD\VWKH SRLQW .x1 ; x2 ; ´/ ZKHUH ´ D Q.[/ 1RWLFHWKDWH[FHSWDW [ D WKHYDOXHVRI Q.[/ DUH DOOSRVLWLYHLQ)LJ D DQGDOOQHJDWLYHLQ)LJ G 7KHKRUL]RQWDOFURVVVHFWLRQVRI WKHJUDSKVDUHHOOLSVHVLQ)LJV D DQGG DQGK\SHUERODVLQ)LJ F z

z

z

z x1

x1 x1

x2

x1

x2

x2

x2

(a) z = 3x 21 + 7x 22

(b) z = 3x 12

(c) z = 3x 21 – 7x 22

(d) z = – 3x 21 – 7x 22

'*(63& *UDSKVRITXDGUDWLFIRUPV

7KHVLPSOH 2 " 2 H[DPSOHVLQ)LJ LOOXVWUDWHWKHIROORZLQJGHÀQLWLRQV

%&'*/*5*0/

$ TXDGUDWLFIRUP Q LV D SRVLWLYHGHÀQLWH LI Q.[/ > 0 IRUDOO [ ¤ E QHJDWLYHGHÀQLWH LI Q.[/ < 0 IRUDOO [ ¤ F LQGHÀQLWH LI Q.[/ DVVXPHVERWKSRVLWLYHDQGQHJDWLYHYDOXHV

$OVR Q LVVDLGWREH SRVLWLYHVHPLGHÀQLWH LI Q.[/ $ 0 IRUDOO [ DQGWREH QHJDWLYH VHPLGHÀQLWH LI Q.[/ % 0 IRUDOO [ 7KHTXDGUDWLFIRUPVLQSDUWVD DQGE RI)LJDUH ERWKSRVLWLYHVHPLGHÀQLWH EXWWKHIRUPLQD LVEHWWHUGHVFULEHGDVSRVLWLYHGHÀQLWH 7KHRUHPFKDUDFWHUL]HVVRPHTXDGUDWLFIRUPVLQWHUPVRIHLJHQYDOXHV

5)&03&.

2VBESBUJD 'PSNT BOE &JHFOWBMVFT /HW A EHDQ n " n V\PPHWULFPDWUL[ 7KHQDTXDGUDWLFIRUP [TA[ LV D SRVLWLYHGHÀQLWHLIDQGRQO\LIWKHHLJHQYDOXHVRI A DUHDOOSRVLWLYH E QHJDWLYHGHÀQLWHLIDQGRQO\LIWKHHLJHQYDOXHVRI A DUHDOOQHJDWLYH RU F LQGHÀQLWHLIDQGRQO\LI A KDVERWKSRVLWLYHDQGQHJDWLYHHLJHQYDOXHV

$)"15&3


z

1300' %\WKH3ULQFLSDO$[HV7KHRUHP WKHUHH[LVWVDQRUWKRJRQDOFKDQJHRIYDULDEOH [ D P \ VXFKWKDW Q.[/ D [TA[ D \T D \ D !1 y12 C !2 y22 C # # # C !n yn2

x1 Positive definite

ZKHUH !1 ; : : : ; !n DUHWKHHLJHQYDOXHVRI A 6LQFH P LVLQYHUWLEOH WKHUHLVDRQHWR RQHFRUUHVSRQGHQFHEHWZHHQDOOQRQ]HUR [ DQGDOOQRQ]HUR \ 7KXVWKHYDOXHVRI Q.[/ IRU [ ¤ FRLQFLGHZLWKWKHYDOXHVRIWKHH[SUHVVLRQRQWKHULJKWVLGHRI ZKLFK LVREYLRXVO\FRQWUROOHGE\WKHVLJQVRIWKHHLJHQYDOXHV !1 ; : : : ; !n LQWKHWKUHHZD\V GHVFULEHGLQWKHWKHRUHP

x2

z x1

x2

&9".1-& ,V Q.[/ D 3x12 C 2x22 C x32 C 4x1 x2 C 4x2 x3 SRVLWLYHGHÀQLWH" 40-65*0/ %HFDXVHRIDOOWKHSOXVVLJQV WKLVIRUPÓRRNVµSRVLWLYHGHÀQLWH %XWWKH PDWUL[RIWKHIRUPLV 2 3 3 2 0 A D 42 2 25 0 2 1

Negative definite z x1

x2

DQGWKHHLJHQYDOXHVRI A WXUQRXWWREH DQG !1 6R Q LVDQLQGHÀQLWHTXDGUDWLF IRUP QRWSRVLWLYHGHÀQLWH

Indefinite

WEB

7KHFODVVLÀFDWLRQRIDTXDGUDWLFIRUPLVRIWHQFDUULHGRYHUWRWKHPDWUL[RIWKHIRUP 7KXVD SRVLWLYHGHÀQLWHPDWUL[ A LVD V\PPHWULF PDWUL[IRUZKLFKWKHTXDGUDWLFIRUP [TA[ LVSRVLWLYHGHÀQLWH 2WKHUWHUPV VXFKDV SRVLWLYHVHPLGHÀQLWHPDWUL[ DUHGHÀQHG DQDORJRXVO\

/6.&3*$"- /05& $ IDVWZD\WRGHWHUPLQHZKHWKHUDV\PPHWULFPDWUL[ A LVSRVLWLYHGHÀQLWHLV WRDWWHPSWWRIDFWRU A LQWKHIRUP A D RTR ZKHUH R LVXSSHUWULDQJXODUZLWK SRVLWLYHGLDJRQDOHQWULHV $ VOLJKWO\PRGLÀHGDOJRULWKPIRUDQ/8 IDFWRUL]DWLRQ LVRQHDSSURDFK 6XFKD &KROHVN\IDFWRUL]DWLRQ LVSRVVLEOHLIDQGRQO\LI A LV SRVLWLYHGHÀQLWH 6HH6XSSOHPHQWDU\([HUFLVHDWWKHHQGRI&KDSWHU

13"$5*$& 130#-&. 'HVFULEHDSRVLWLYHVHPLGHÀQLWHPDWUL[ A LQWHUPVRILWVHLJHQYDOXHV WEB

&9&3$*4&4 &RPSXWHWKHTXDGUDWLFIRUP [TA[ ZKHQ A D DQG D [ D

!

x1 x2

"

E [ D

! " 6 1

!

F [ D 2

4 &RPSXWHWKHTXDGUDWLFIRUP [TA[ IRU A D 4 3 0 DQG

5 1=3

! " 1 3

3 2 1

1=3 1

"

3

0 15 1

2

3 x1 D [ D 4 x2 5 x3

2

3 2 E [ D 4 !1 5 5

2

p 3 1=p3 6 7 F [ D 4 1= 3 5 p 1= 3

)LQGWKHPDWUL[RIWKHTXDGUDWLFIRUP $VVXPH [ LVLQ R2 E 5x12 C 3x1 x2 D 10x12 ! 6x1 x2 ! 3x22 )LQGWKHPDWUL[RIWKHTXDGUDWLFIRUP $VVXPH [ LVLQ R2 E x1 x2 D 20x12 C 15x1 x2 ! 10x22

)LQGWKHPDWUL[RIWKHTXDGUDWLFIRUP $VVXPH [ LVLQ R3 D 8x12 C 7x22 ! 3x32 ! 6x1 x2 C 4x1 x3 ! 2x2 x3 E 4x1 x2 C 6x1 x3 ! 8x2 x3

)LQGWKHPDWUL[RIWKHTXDGUDWLFIRUP $VVXPH [ LVLQ R3 D 5x12 ! x22 C 7x32 C 5x1 x2 ! 3x1 x3 E x32 ! 4x1 x2 C 4x2 x3

0DNH D FKDQJH RI YDULDEOH [ D P \ WKDW WUDQVIRUPV WKH TXDGUDWLFIRUP x12 C 10x1 x2 C x22 LQWRDTXDGUDWLFIRUPZLWK QRFURVVSURGXFWWHUP *LYH P DQGWKHQHZTXDGUDWLFIRUP /HW A EHWKHPDWUL[RIWKHTXDGUDWLFIRUP

9x12 C 7x22 C 11x32 ! 8x1 x2 C 8x1 x3 ,WFDQEHVKRZQWKDWWKHHLJHQYDOXHVRI A DUH DQG )LQGDQRUWKRJRQDOPDWUL[ P VXFKWKDWWKHFKDQJHRIYDULDEOH [ D P \ WUDQVIRUPV [TA[ LQWRDTXDGUDWLFIRUPZLWKQRFURVV SURGXFWWHUP *LYH P DQGWKHQHZTXDGUDWLFIRUP &ODVVLI\WKHTXDGUDWLFIRUPVLQ([HUFLVHV² 7KHQPDNHD FKDQJHRIYDULDEOH [ D P \ WKDWWUDQVIRUPVWKHTXDGUDWLFIRUP LQWRRQHZLWKQRFURVVSURGXFWWHUP :ULWHWKHQHZTXDGUDWLF IRUP &RQVWUXFW P XVLQJWKHPHWKRGVRI6HFWLRQ 3x12 ! 4x1 x2 C 6x22

2x12 C 10x1 x2 C 2x22

x12 ! 6x1 x2 C 9x22

9x12 ! 8x1 x2 C 3x22

!5x12 C 4x1 x2 ! 2x22 8x12 C 6x1 x2

>0@ !2x12 ! 6x22 ! 9x32 ! 9x42 C 4x1 x2 C 4x1 x3 C 4x1 x4 C 6x3 x4 >0@ 4x12 C 4x22 C 4x32 C 4x42 C 3x1 x2 C 3x3 x4 ! 4x1 x4 C 4x2 x3 >0@ x12 C x22 C x32 C x42 C 9x1 x2 ! 12x1 x4 C 12x2 x3 C 9x3 x4 >0@ 11x12 ! x22 ! 12x1 x2 ! 12x1 x3 ! 12x1 x4 ! 2x3 x4

:KDW LV WKH ODUJHVW SRVVLEOH YDOXH RI WKH TXDGUDWLF IRUP 5x12 C 8x22 LI [ D .x1 ; x2 / DQG [T[ D 1 WKDW LV LI x12 C x22 D 1" 7U\VRPHH[DPSOHVRI [ :KDWLVWKHODUJHVWYDOXHRIWKHTXDGUDWLFIRUP 5x12 ! 3x22 LI [ T [ D 1"

,Q([HUFLVHVDQG PDWULFHVDUH n " n DQGYHFWRUVDUHLQ Rn 0DUNHDFKVWDWHPHQW7UXHRU)DOVH -XVWLI\HDFKDQVZHU D 7KHPDWUL[RIDTXDGUDWLFIRUPLVDV\PPHWULFPDWUL[ E $ TXDGUDWLFIRUPKDVQRFURVVSURGXFWWHUPVLIDQGRQO\ LIWKHPDWUL[RIWKHTXDGUDWLFIRUPLVDGLDJRQDOPDWUL[ F 7KHSULQFLSDOD[HVRIDTXDGUDWLFIRUP [TA[ DUHHLJHQYHF WRUVRI A G $ SRVLWLYHGHÀQLWHTXDGUDWLFIRUP Q VDWLVÀHV Q.[/ > 0 IRUDOO [ LQ Rn

4XDGUDWLF)RUPV

H ,IWKHHLJHQYDOXHVRIDV\PPHWULFPDWUL[ A DUHDOOSRVL WLYH WKHQWKHTXDGUDWLFIRUP [TA[ LVSRVLWLYHGHÀQLWH I $ &KROHVN\IDFWRUL]DWLRQRIDV\PPHWULFPDWUL[ A KDV WKHIRUP A D RTR IRUDQXSSHUWULDQJXODUPDWUL[ R ZLWK SRVLWLYHGLDJRQDOHQWULHV D 7KHH[SUHVVLRQ k[k2 LVDTXDGUDWLFIRUP

E ,I A LVV\PPHWULFDQG P LVDQRUWKRJRQDOPDWUL[ WKHQ WKHFKDQJHRIYDULDEOH [ D P \ WUDQVIRUPV [TA[ LQWRD TXDGUDWLFIRUPZLWKQRFURVVSURGXFWWHUP F ,I A LVD 2 " 2 V\PPHWULFPDWUL[ WKHQWKHVHWRI [ VXFK WKDW [TA[ D c IRUDFRQVWDQW c FRUUHVSRQGVWRHLWKHUD FLUFOH DQHOOLSVH RUDK\SHUEROD

G $QLQGHÀQLWHTXDGUDWLFIRUPLVHLWKHUSRVLWLYHVHPLGHÀ QLWHRUQHJDWLYHVHPLGHÀQLWH H ,I A LVV\PPHWULFDQGWKHTXDGUDWLFIRUP [TA[ KDVRQO\ QHJDWLYHYDOXHVIRU [ ¤ WKHQWKHHLJHQYDOXHVRI A DUH DOOQHJDWLYH ([HUFLVHV DQG VKRZ ! KRZ WR " FODVVLI\ D TXDGUDWLF IRUP a b T Q.[/ D [ A[ ZKHQ A D DQG GHW A ¤ 0 ZLWKRXWÀQG b d LQJWKHHLJHQYDOXHVRI A ,I !1 DQG !2 DUHWKHHLJHQYDOXHVRI A WKHQWKHFKDUDFWHULVWLF SRO\QRPLDORI A FDQEHZULWWHQLQWZRZD\V GHW.A ! !I / DQG .! ! !1 /.! ! !2 / 8VHWKLVIDFWWRVKRZWKDW !1 C !2 D a C d WKHGLDJRQDOHQWULHVRI A DQG !1 !2 D GHW A 9HULI\WKHIROORZLQJVWDWHPHQWV D Q LVSRVLWLYHGHÀQLWHLI GHW A > 0 DQG a > 0 E Q LVQHJDWLYHGHÀQLWHLI GHW A > 0 DQG a < 0 F Q LVLQGHÀQLWHLI GHW A < 0 6KRZWKDWLI B LV m " n WKHQ B TB LVSRVLWLYHVHPLGHÀQLWH DQGLI B LV n " n DQGLQYHUWLEOH WKHQ B TB LVSRVLWLYHGHÀQLWH 6KRZWKDWLIDQ n " n PDWUL[ A LVSRVLWLYHGHÀQLWH WKHQWKHUH H[LVWVDSRVLWLYHGHÀQLWHPDWUL[ B VXFKWKDW A D B TB >+LQW :ULWH A D 3'3T ZLWK P T D P !1 3URGXFHDGLDJRQDO PDWUL[ C VXFKWKDW D D C TC DQGOHW B D P CP T 6KRZ WKDW B ZRUNV@ /HW A DQG B EHV\PPHWULF n " n PDWULFHVZKRVHHLJHQYDOXHV DUHDOOSRVLWLYH 6KRZWKDWWKHHLJHQYDOXHVRI A C B DUHDOO SRVLWLYH >+LQW &RQVLGHUTXDGUDWLFIRUPV@ /HW A EHDQ n " n LQYHUWLEOHV\PPHWULFPDWUL[ 6KRZWKDW LIWKHTXDGUDWLFIRUP [TA[ LVSRVLWLYHGHÀQLWH WKHQVRLVWKH TXDGUDWLFIRUP [TA!1 [ >+LQW &RQVLGHUHLJHQYDOXHV@ SG

.BTUFSJOH %JBHPOBMJ[BUJPO BOE 2VBESBUJD 'PSNT m

$)"15&3


40-65*0/ 50 13"$5*$& 130#-&. 0DNHDQRUWKRJRQDOFKDQJHRIYDULDEOH [ D P \ DQGZULWH

z

x1

[TA[ D \T D \ D !1 y12 C !2 y22 C # # # C !n yn2

x2 Positive semidefinite

DVLQHTXDWLRQ ,IDQHLJHQYDOXH³VD\ !i ³ZHUHQHJDWLYH WKHQ [TA[ ZRXOGEH QHJDWLYHIRUWKH [ FRUUHVSRQGLQJWR \ D Hi WKH i WKFROXPQRI In 6RWKHHLJHQYDOXHV RIDSRVLWLYHVHPLGHÀQLWHTXDGUDWLFIRUPPXVWDOOEHQRQQHJDWLYH &RQYHUVHO\ LIWKH HLJHQYDOXHVDUHQRQQHJDWLYH WKHH[SDQVLRQDERYHVKRZVWKDW [TA[ PXVWEHSRVLWLYH VHPLGHÀQLWH

$0/453"*/&% 015*.*;"5*0/ (QJLQHHUV HFRQRPLVWV VFLHQWLVWV DQGPDWKHPDWLFLDQVRIWHQQHHGWRÀQGWKHPD[LPXP RUPLQLPXPYDOXHRIDTXDGUDWLFIRUP Q.[/ IRU [ LQVRPHVSHFLÀHGVHW 7\SLFDOO\ WKH SUREOHPFDQEHDUUDQJHGVRWKDW [ YDULHVRYHUWKHVHWRIXQLWYHFWRUV 7KLV FRQVWUDLQHG RSWLPL]DWLRQSUREOHP KDVDQLQWHUHVWLQJDQGHOHJDQWVROXWLRQ ([DPSOHEHORZDQGWKH GLVFXVVLRQLQ6HFWLRQZLOOLOOXVWUDWHKRZVXFKSUREOHPVDULVHLQSUDFWLFH 7KHUHTXLUHPHQWWKDWDYHFWRU [ LQ Rn EHDXQLWYHFWRUFDQEHVWDWHGLQVHYHUDO HTXLYDOHQWZD\V k[k D 1; k[k2 D 1; [T [ D 1 DQG

x12 C x22 C # # # C xn2 D 1

7KHH[SDQGHGYHUVLRQ RI [T[ D 1 LVFRPPRQO\XVHGLQDSSOLFDWLRQV :KHQDTXDGUDWLFIRUP Q KDVQRFURVVSURGXFWWHUPV LWLVHDV\WRÀQGWKHPD[LPXP DQGPLQLPXPRI Q.[/ IRU [T[ D 1

&9".1-& )LQGWKHPD[LPXPDQGPLQLPXPYDOXHVRI Q.[/ D 9x12 C 4x22 C 3x32 VXEMHFWWRWKHFRQVWUDLQW [T[ D 1

40-65*0/ 6LQFH x22 DQG x32 DUHQRQQHJDWLYH QRWHWKDW 4x22 % 9x22

DQG

3x32 % 9x32

DQGKHQFH

Q.[/ D 9x12 C 4x22 C 3x32 % 9x12 C 9x22 C 9x32 D 9.x12 C x22 C x32 / D9

ZKHQHYHU x12 C x22 C x32 D 1 6RWKHPD[LPXPYDOXHRI Q.[/ FDQQRWH[FHHGZKHQ [ LVDXQLWYHFWRU )XUWKHUPRUH Q.[/ D 9 ZKHQ [ D .1; 0; 0/ 7KXVLVWKHPD[LPXP YDOXHRI Q.[/ IRU [T[ D 1 7RÀQGWKHPLQLPXPYDOXHRI Q.[/ REVHUYHWKDW

9x12 $ 3x12 ;

4x22 $ 3x22

DQGKHQFH

Q.[/ $ 3x12 C 3x22 C 3x32 D 3.x12 C x22 C x32 / D 3

ZKHQHYHU x12 C x22 C x32 D 1 $OVR Q.[/ D 3 ZKHQ x1 D 0 x2 D 0 DQG x3 D 1 6R LVWKHPLQLPXPYDOXHRI Q.[/ ZKHQ [T[ D 1

&RQVWUDLQHG2SWLPL]DWLRQ

,WLVHDV\WRVHHLQ([DPSOHWKDWWKHPDWUL[RIWKHTXDGUDWLFIRUP Q KDVHLJHQ YDOXHV DQGDQGWKDWWKHJUHDWHVWDQGOHDVWHLJHQYDOXHVHTXDO UHVSHFWLYHO\ WKH FRQVWUDLQHG PD[LPXPDQGPLQLPXPRI Q.[/ 7KHVDPHKROGVWUXHIRUDQ\TXDGUDWLF IRUP DVZHVKDOOVHH !

" 3 0 &9".1-& /HW A D DQGOHW Q.[/ D [TA[ IRU [ LQ R2 )LJXUHGLV 0 7 SOD\VWKHJUDSKRI Q )LJXUHVKRZVRQO\WKHSRUWLRQRIWKHJUDSKLQVLGHDF\OLQGHU WKHLQWHUVHFWLRQRIWKHF\OLQGHUZLWKWKHVXUIDFHLVWKHVHWRISRLQWV .x1 ; x2 ; ´/ VXFKWKDW ´ D Q.x1 ; x2 / DQG x12 C x22 D 1 7KH´KHLJKWVµRIWKHVHSRLQWVDUHWKHFRQVWUDLQHG YDOXHVRI Q.[/ *HRPHWULFDOO\ WKHFRQVWUDLQHGRSWLPL]DWLRQSUREOHPLVWRORFDWHWKH KLJKHVWDQGORZHVWSRLQWVRQWKHLQWHUVHFWLRQFXUYH 7KHWZRKLJKHVWSRLQWVRQWKHFXUYHDUHXQLWVDERYHWKH x1 x2 SODQH RFFXUULQJ ZKHUH x1 D 0 DQG x2 D ˙1 7KHVHSRLQWVFRUUHVSRQGWRWKHHLJHQYDOXHRI A DQG WKHHLJHQYHFWRUV [ D .0; 1/ DQG ![ D .0; !1/ 6LPLODUO\ WKHWZRORZHVWSRLQWVRQWKH FXUYHDUHXQLWVDERYHWKH x1 x2 SODQH 7KH\FRUUHVSRQGWRWKHHLJHQYDOXHDQGWKH HLJHQYHFWRUV .1; 0/ DQG .!1; 0/

z

x1

z

x2

'*(63& ´ D 3x12 C 7x22

x1

x2

'*(63& 7KHLQWHUVHFWLRQRI ´ D

3x12 C 7x22 DQGWKHF\OLQGHU x12 C x22 D 1

(YHU\SRLQWRQWKHLQWHUVHFWLRQFXUYHLQ)LJKDVD ´FRRUGLQDWHEHWZHHQDQG DQGIRUDQ\QXPEHU t EHWZHHQDQG WKHUHLVDXQLWYHFWRU [ VXFKWKDW Q.[/ D t ,Q RWKHUZRUGV WKHVHWRIDOOSRVVLEOHYDOXHVRI [TA[ IRU k[k D 1 LVWKHFORVHGLQWHUYDO 3 % t % 7 ,WFDQEHVKRZQWKDWIRUDQ\V\PPHWULFPDWUL[ A WKHVHWRIDOOSRVVLEOHYDOXHVRI [TA[ IRU k[k D 1 LVDFORVHGLQWHUYDORQWKHUHDOD[LV 6HH([HUFLVH 'HQRWHWKH OHIWDQGULJKWHQGSRLQWVRIWKLVLQWHUYDOE\ m DQG M UHVSHFWLYHO\ 7KDWLV OHW

m D PLQ f[TA[ W k[k D 1g;

M D PD[ f[TA[ W k[k D 1g

([HUFLVHDVNV\RXWRSURYHWKDWLI ! LVDQHLJHQYDOXHRI A WKHQ m % ! % M 7KH QH[WWKHRUHPVD\VWKDW m DQG M DUHWKHPVHOYHVHLJHQYDOXHVRI A MXVWDVLQ([DPSOH ñ

5)&03&.

/HW A EHDV\PPHWULFPDWUL[ DQGGHÀQH m DQG M DVLQ 7KHQ M LVWKHJUHDWHVW HLJHQYDOXH !1 RI A DQG m LVWKHOHDVWHLJHQYDOXHRI A 7KHYDOXHRI [TA[ LV M ZKHQ [ LVDXQLWHLJHQYHFWRU X1 FRUUHVSRQGLQJWR M 7KHYDOXHRI [TA[ LV m ZKHQ [ LVDXQLWHLJHQYHFWRUFRUUHVSRQGLQJWR m

ñ 7KHXVHRI PLQLPXP DQG PD[LPXP LQ DQG OHDVW DQG JUHDWHVW LQWKHWKHRUHP UHIHUVWRWKHQDWXUDO RUGHULQJRIWKHUHDOQXPEHUV QRWWRPDJQLWXGHV

$)"15&3


1300' 2UWKRJRQDOO\GLDJRQDOL]H A DV 3'3!1 :HNQRZWKDW $OVR

[TA[ D \TD \

ZKHQ [ D P \

k[k D kP \k D k\k

IRUDOO \

EHFDXVH P TP D I DQG kP \k2 D .P \/T .P \/ D \TP TP \ D \T\ D k\k2 ,QSDUWLFXODU k\k D 1 LIDQGRQO\LI k[k D 1 7KXV [TA[ DQG \TD \ DVVXPHWKHVDPHVHWRIYDOXHVDV [ DQG \ UDQJHRYHUWKHVHWRIDOOXQLWYHFWRUV 7RVLPSOLI\QRWDWLRQ VXSSRVHWKDW A LVD 3 " 3 PDWUL[ZLWKHLJHQYDOXHV a $ b $ c $UUDQJHWKHHLJHQYHFWRU FROXPQVRI P VRWKDW P D Œ X1 X2 X3 " DQG 2 3 a 0 0 05 D D40 b 0 0 c *LYHQDQ\XQLWYHFWRU \ LQ R3 ZLWKFRRUGLQDWHV y1 y2 y3 REVHUYHWKDW

ay12 D ay12

by22 % ay22 DQGREWDLQWKHVHLQHTXDOLWLHV

cy32 % ay32

\TD \ D ay12 C by22 C cy32

% ay12 C ay22 C ay32 D a.y12 C y22 C y32 / D ak\k2 D a

7KXV M % a E\GHÀQLWLRQRI M +RZHYHU \TD \ D a ZKHQ \ D H1 D .1; 0; 0/ VRLQ IDFW M D a %\ WKH [ WKDWFRUUHVSRQGVWR \ D H1 LVWKHHLJHQYHFWRU X1 RI A EHFDXVH 2 3 # $ 1 X2 X 3 4 0 5 D X1 [ D P H1 D X 1 0

7KXV M D a D HT1 D H1 D XT1AX1 ZKLFKSURYHVWKHVWDWHPHQWDERXW M $ VLPLODU DUJXPHQWVKRZVWKDW m LVWKHOHDVWHLJHQYDOXH c DQGWKLVYDOXHRI [TA[ LVDWWDLQHG ZKHQ [ D P H3 D X3 2

3 3 2 1 &9".1-& /HW A D 4 2 3 1 5 )LQGWKHPD[LPXPYDOXHRIWKHTXDGUDWLF 1 1 4 IRUP [TA[ VXEMHFWWRWKHFRQVWUDLQW [T[ D 1 DQGÀQGDXQLWYHFWRUDWZKLFKWKLVPD[L PXPYDOXHLVDWWDLQHG

40-65*0/ %\7KHRUHP WKHGHVLUHGPD[LPXPYDOXHLVWKHJUHDWHVWHLJHQYDOXHRI A 7KHFKDUDFWHULVWLFHTXDWLRQWXUQVRXWWREH 0 D !!3 C 10!2 ! 27! C 18 D !.! ! 6/.! ! 3/.! ! 1/

7KHJUHDWHVWHLJHQYDOXHLV 7KHFRQVWUDLQHGPD[LPXPRI [TA[ LVDWWDLQHGZKHQ [ LVDXQLWHLJHQYHFWRUIRU 2 p 3 2 3 1 1= 3 6 p 7 ! D 6 6ROYH .A ! 6I /[ D 0 DQGÀQGDQHLJHQYHFWRU 4 1 5 6HW X1 D 4 1= 3 5 p 1 1= 3


,Q7KHRUHPDQGLQODWHUDSSOLFDWLRQV WKHYDOXHVRI [TA[ DUHFRPSXWHGZLWKDGGL WLRQDOFRQVWUDLQWVRQWKHXQLWYHFWRU [

5)&03&.

/HW A; !1 DQG X1 EHDVLQ7KHRUHP 7KHQWKHPD[LPXPYDOXHRI [TA[ VXEMHFW WRWKHFRQVWUDLQWV [T[ D 1; [TX1 D 0 LVWKHVHFRQGJUHDWHVWHLJHQYDOXH !2 DQGWKLVPD[LPXPLVDWWDLQHGZKHQ [ LVDQ HLJHQYHFWRU X2 FRUUHVSRQGLQJWR !2

7KHRUHPFDQEHSURYHGE\DQDUJXPHQWVLPLODUWRWKHRQHDERYHLQZKLFKWKH WKHRUHPLVUHGXFHGWRWKHFDVHZKHUHWKHPDWUL[RIWKHTXDGUDWLFIRUPLVGLDJRQDO 7KH QH[WH[DPSOHJLYHVDQLGHDRIWKHSURRIIRUWKHFDVHRIDGLDJRQDOPDWUL[

&9".1-& )LQGWKHPD[LPXPYDOXHRI 9x12 C 4x22 C 3x32 VXEMHFWWRWKHFRQ

VWUDLQWV [T[ D 1 DQG [TX1 D 0 ZKHUH X1 D .1; 0; 0/ 1RWHWKDW X1 LVDXQLWHLJHQYHFWRU FRUUHVSRQGLQJWRWKHJUHDWHVWHLJHQYDOXH ! D 9 RIWKHPDWUL[RIWKHTXDGUDWLFIRUP

40-65*0/ ,IWKHFRRUGLQDWHVRI [ DUH x1 x2 x3 WKHQWKHFRQVWUDLQW [TX1 D 0 PHDQV VLPSO\WKDW x1 D 0 )RUVXFKDXQLWYHFWRU x22 C x32 D 1 DQG 9x12 C 4x22 C 3x32 D 4x22 C 3x32 % 4x22 C 4x32

D 4.x22 C x32 / D4 7KXVWKHFRQVWUDLQHGPD[LPXPRIWKHTXDGUDWLFIRUPGRHVQRWH[FHHG $QGWKLVYDOXH LVDWWDLQHGIRU [ D .0; 1; 0/ ZKLFKLVDQHLJHQYHFWRUIRUWKHVHFRQGJUHDWHVWHLJHQYDOXH RIWKHPDWUL[RIWKHTXDGUDWLFIRUP

&9".1-& /HW A EHWKHPDWUL[LQ([DPSOHDQGOHW X1 EHDXQLWHLJHQYHFWRU FRUUHVSRQGLQJWRWKHJUHDWHVWHLJHQYDOXHRI A )LQGWKHPD[LPXPYDOXHRI [TA[ VXEMHFW WRWKHFRQGLWLRQV [T[ D 1;

[TX1 D 0

40-65*0/ )URP([DPSOH WKHVHFRQGJUHDWHVWHLJHQYDOXHRI A LV ! D 3 6ROYH .A ! 3I /[ D WRÀQGDQHLJHQYHFWRU DQGQRUPDOL]HLWWRREWDLQ 2

p 3 1=p6 6 7 X2 D 4 1= 6 5 p !2= 6

7KHYHFWRU X2 LVDXWRPDWLFDOO\RUWKRJRQDOWR X1 EHFDXVHWKHYHFWRUVFRUUHVSRQGWRGLI IHUHQWHLJHQYDOXHV 7KXVWKHPD[LPXPRI [TA[ VXEMHFWWRWKHFRQVWUDLQWVLQ LV DWWDLQHGZKHQ [ D X2 7KHQH[WWKHRUHPJHQHUDOL]HV7KHRUHPDQG WRJHWKHUZLWK7KHRUHP JLYHVD XVHIXOFKDUDFWHUL]DWLRQRI DOO WKHHLJHQYDOXHVRI A 7KHSURRILVRPLWWHG

$)"15&3


5)&03&.

/HW A EH D V\PPHWULF n " n PDWUL[ ZLWK DQ RUWKRJRQDO GLDJRQDOL]DWLRQ A D 3'3!1 ZKHUH WKH HQWULHV RQ WKH GLDJRQDO RI D DUH DUUDQJHG VR WKDW !1 $ !2 $ # # # $ !n DQGZKHUHWKHFROXPQVRI P DUHFRUUHVSRQGLQJXQLWHLJHQ YHFWRUV X1 ; : : : ; Xn 7KHQIRU k D 2; : : : ; n WKHPD[LPXPYDOXHRI [TA[ VXEMHFW WRWKHFRQVWUDLQWV [T[ D 1;

[TX1 D 0;

:::;

[TXk !1 D 0

LVWKHHLJHQYDOXH !k DQGWKLVPD[LPXPLVDWWDLQHGDW [ D Xk

7KHRUHPZLOOEHKHOSIXOLQ6HFWLRQVDQG 7KHIROORZLQJDSSOLFDWLRQ UHTXLUHVRQO\7KHRUHP

&9".1-& 'XULQJWKHQH[W\HDU DFRXQW\JRYHUQPHQWLVSODQQLQJWRUHSDLU x KXQGUHGPLOHVRISXEOLFURDGVDQGEULGJHVDQGWRLPSURYH y KXQGUHGDFUHVRISDUNV DQGUHFUHDWLRQDUHDV 7KHFRXQW\PXVWGHFLGHKRZWRDOORFDWHLWVUHVRXUFHVIXQGV HTXLSPHQW ODERU HWF EHWZHHQWKHVHWZRSURMHFWV ,ILWLVPRUHFRVWHIIHFWLYHWRZRUN VLPXOWDQHRXVO\RQERWKSURMHFWVUDWKHUWKDQRQRQO\RQH WKHQ x DQG y PLJKWVDWLVI\D FRQVWUDLQW VXFKDV 4x 2 C 9y 2 % 36 6HH)LJ (DFKSRLQW .x; y/ LQWKHVKDGHG IHDVLEOHVHW UHSUHVHQWVDSRVVLEOHSXEOLF ZRUNVVFKHGXOHIRUWKH\HDU 7KHSRLQWVRQWKHFRQVWUDLQWFXUYH 4x 2 C 9y 2 D 36 XVH WKHPD[LPXPDPRXQWVRIUHVRXUFHVDYDLODEOH y Parks and recreation 2

4x 2 + 9y 2 = 36 Feasible set 3 Road and bridge repair

x

'*(63& 3XEOLFZRUNVVFKHGXOHV

,QFKRRVLQJLWVSXEOLFZRUNVVFKHGXOH WKHFRXQW\ZDQWVWRFRQVLGHUWKHRSLQLRQVRI WKHFRXQW\UHVLGHQWV 7RPHDVXUHWKHYDOXH RU XWLOLW\ WKDWWKHUHVLGHQWVZRXOGDVVLJQWR WKHYDULRXVZRUNVFKHGXOHV .x; y/ HFRQRPLVWVVRPHWLPHVXVHDIXQFWLRQVXFKDV

q.x; y/ D xy 7KHVHWRISRLQWV .x; y/ DWZKLFK q.x; y/ LVDFRQVWDQWLVFDOOHGDQ LQGLIIHUHQFHFXUYH 7KUHHVXFKFXUYHVDUHVKRZQLQ)LJ 3RLQWVDORQJDQLQGLIIHUHQFHFXUYHFRUUHVSRQG WRDOWHUQDWLYHVWKDWFRXQW\UHVLGHQWVDVDJURXSZRXOGÀQGHTXDOO\YDOXDEOHò )LQGWKH SXEOLFZRUNVVFKHGXOHWKDWPD[LPL]HVWKHXWLOLW\IXQFWLRQ q

40-65*0/ 7KHFRQVWUDLQWHTXDWLRQ 4x 2 C 9y 2 D 36 GRHVQRWGHVFULEHDVHWRIXQLW YHFWRUV EXWDFKDQJHRIYDULDEOHFDQÀ[WKDWSUREOHP 5HZULWHWKHFRQVWUDLQWLQWKH IRUP % x &2 % y &2 C D1 3 2 ò ,QGLIIHUHQFHFXUYHVDUHGLVFXVVHGLQ0LFKDHO',QWULOLJDWRU 5RQDOG*%RGNLQ DQG&KHQJ+VLDR (FRQRPHWULF0RGHOV 7HFKQLTXHV DQG$SSOLFDWLRQV 8SSHU6DGGOH5LYHU 1-3UHQWLFH+DOO


y Parks and recreation 1.4

4x 2 + 9y 2 = 36 (indifference curves) q(x, y) = 4 q(x, y) = 3 x

q(x, y) = 2 2.1 Road and bridge repair '*(63& 7KHRSWLPXPSXEOLFZRUNVVFKHGXOH

LV .2:1; 1:4/

DQGGHÀQH

x y ; x2 D ; WKDWLV; 3 2 7KHQWKHFRQVWUDLQWHTXDWLRQEHFRPHV x1 D

x D 3x1

DQG

y D 2x2

x12 C x22 D 1

!

" x1 DQGWKHXWLOLW\IXQFWLRQEHFRPHV q.3x1 ; 2x2 / D .3x1 /.2x2 / D 6x1 x2 /HW [ D x2 T 7KHQWKHSUREOHPLVWRPD[LPL]H Q.[/ D 6x1 x2 VXEMHFWWR [ [ D 1 1RWHWKDW Q.[/ D [TA[ ZKHUH ! " 0 3 AD 3 0 " p # " p # 1=p2 !1=p2 7KHHLJHQYDOXHVRI A DUH ˙3 ZLWKHLJHQYHFWRUV IRU ! D 3 DQG IRU 1= 2 1= 2 p ! D !3 7KXVWKHPD[LPXPYDOXHRI Q.[/ D q.x1 ; x2 / LV DWWDLQHGZKHQ x1 D 1= 2 p DQG x2 D 1= 2 x D 3x1 D p p,QWHUPVRIWKHRULJLQDOYDULDEOHV WKHRSWLPXPSXEOLFZRUNVVFKHGXOHLV 3= 2 & 2:1 KXQGUHGPLOHVRIURDGVDQGEULGJHVDQG y D 2x2 D 2 & 1:4 KXQGUHG DFUHVRISDUNVDQGUHFUHDWLRQDODUHDV 7KHRSWLPXPSXEOLFZRUNVVFKHGXOHLVWKHSRLQW ZKHUHWKHFRQVWUDLQWFXUYHDQGWKHLQGLIIHUHQFHFXUYH q.x; y/ D 3 MXVWPHHW 3RLQWV .x; y/ ZLWKDKLJKHUXWLOLW\OLHRQLQGLIIHUHQFHFXUYHVWKDWGRQRWWRXFKWKHFRQVWUDLQW FXUYH 6HH)LJ

13"$5*$& 130#-&.4 /HW Q.[/ D 3x12 C 3x22 C 2x1 x2 )LQGDFKDQJHRIYDULDEOHWKDWWUDQVIRUPV Q LQWR DTXDGUDWLFIRUPZLWKQRFURVVSURGXFWWHUP DQGJLYHWKHQHZTXDGUDWLFIRUP :LWK Q DVLQ3UREOHP ÀQGWKHPD[LPXPYDOXHRI Q.[/ VXEMHFWWRWKHFRQVWUDLQW [T[ D 1 DQGÀQGDXQLWYHFWRUDWZKLFKWKHPD[LPXPLVDWWDLQHG

&9&3$*4&4 ,Q([HUFLVHVDQG ÀQGWKHFKDQJHRIYDULDEOH [ D P \ WKDW WUDQVIRUPVWKHTXDGUDWLFIRUP [TA[ LQWR \TD \ DVVKRZQ 5x12 C 6x22 C 7x32 C 4x1 x2 ! 4x2 x3 D 9y12 C 6y22 C 3y32

3x12 C 2x22 C 2x32 C 2x1 x2 C 2x1 x3 C 4x2 x3 D 5y12 C 2y22 >+LQW [ DQG \ PXVWKDYHWKHVDPHQXPEHURIFRRUGLQDWHV VRWKHTXDGUDWLFIRUPVKRZQKHUHPXVWKDYHDFRHIÀFLHQWRI ]HURIRU y32 @

,Q([HUFLVHV² ÀQGD WKHPD[LPXPYDOXHRI Q.[/ VXEMHFWWR WKHFRQVWUDLQW [T[ D 1 E DXQLWYHFWRU X ZKHUHWKLVPD[LPXPLV DWWDLQHG DQGF WKHPD[LPXPRI Q.[/ VXEMHFWWRWKHFRQVWUDLQWV [T[ D 1 DQG [TX D 0 Q.[/ D 5x12 C 6x22 C 7x32 C 4x1 x2 ! 4x2 x3 6HH([HUFLVH

$)"15&3


Q.[/ D 3x12 C 2x22 C 2x32 C 2x1 x2 C 2x1 x3 C 4x2 x3 6HH ([HUFLVH Q.[/ D 5x12 C 5x22 ! 4x1 x2

Q.[/ D 7x12 C 3x22 C 3x1 x2

/HW Q.[/ D !2x12 ! x22 C 4x1 x2 C 4x2 x3 )LQGDXQLWYHFWRU [ LQ R3 DWZKLFK Q.[/ LVPD[LPL]HG VXEMHFWWR [T[ D 1 >+LQW 7KHHLJHQYDOXHVRIWKHPDWUL[RIWKHTXDGUDWLFIRUP Q DUH !1 DQG !4@

/HW Q.[/ D 7x12 C x22 C 7x32 ! 8x1 x2 ! 4x1 x3 ! 8x2 x3 )LQGDXQLWYHFWRU [ LQ R3 DWZKLFK Q.[/ LVPD[LPL]HG VXEMHFWWR [T[ D 1 >+LQW 7KHHLJHQYDOXHVRIWKHPDWUL[RI WKHTXDGUDWLFIRUP Q DUHDQG !3@

)LQG WKH PD[LPXP YDOXH RI Q.[/ D 7x12 C 3x22 ! 2x1 x2 VXEMHFWWRWKHFRQVWUDLQW x12 C x22 D 1 'RQRWJRRQWRÀQG DYHFWRUZKHUHWKHPD[LPXPLVDWWDLQHG

/HW ! EHDQ\HLJHQYDOXHRIDV\PPHWULFPDWUL[ A -XVWLI\ WKHVWDWHPHQWPDGHLQWKLVVHFWLRQWKDW m % ! % M ZKHUH m DQG M DUHGHÀQHGDVLQ >+LQW )LQGDQ [ VXFKWKDW ! D [TA[@

/HW A EHDQ n " n V\PPHWULFPDWUL[ OHW M DQG m GHQRWH WKHPD[LPXPDQGPLQLPXPYDOXHVRIWKHTXDGUDWLFIRUP [TA[ DQGGHQRWHFRUUHVSRQGLQJXQLWHLJHQYHFWRUVE\ X1 DQG Xn 7KHIROORZLQJFDOFXODWLRQVVKRZWKDWJLYHQDQ\QXPEHU t EHWZHHQ M DQG m WKHUHLVDXQLWYHFWRU [ VXFKWKDW t D [TA[ C ˛M IRUVRPHQXPEHU ˛ EHWZHHQ 9HULI\WKDW t D .1 ! ˛/mp p DQG 7KHQOHW [ D 1 ! ˛ Xn C ˛ X1 DQGVKRZWKDW [T[ D 1 DQG [TA[ D t >0@ ,Q([HUFLVHV² IROORZWKHLQVWUXFWLRQVJLYHQIRU([HU FLVHV² x1 x2 C 3x1 x3 C 30x1 x4 C 30x2 x3 C 3x2 x4 C x3 x4

)LQGWKHPD[LPXPYDOXHRI Q.[/ D !3x12 C 5x22 ! 2x1 x2 VXEMHFWWRWKHFRQVWUDLQW x12 C x22 D 1 'RQRWJRRQWRÀQG DYHFWRUZKHUHWKHPD[LPXPLVDWWDLQHG

3x1 x2 C 5x1 x3 C 7x1 x4 C 7x2 x3 C 5x2 x4 C 3x3 x4

6XSSRVH [ LVDXQLWHLJHQYHFWRURIDPDWUL[ A FRUUHVSRQGLQJ WRDQHLJHQYDOXH :KDWLVWKHYDOXHRI [TA["

!6x12 ! 10x22 ! 13x32 ! 13x42 ! 4x1 x2 ! 4x1 x3 ! 4x1 x4 C 6x3 x4

z

       4     x2   

x

x1

7KHPD[LPXPYDOXHRI Q.[/ VXEMHFWWR [T [ D 1 LV

4x12 ! 6x1 x2 ! 10x1 x3 ! 10x1 x4 ! 6x2 x3 ! 6x2 x4 ! 2x3 x4

40-65*0/4 50 13"$5*$& 130#-&.4 ! 3 7KHPDWUL[RIWKHTXDGUDWLFIRUPLV A D 1

" 1 ,WLVHDV\WRÀQGWKHHLJHQYDOXHV 3 " p # " p # 1=p2 !1=p2 DQG DQGFRUUHVSRQGLQJXQLWHLJHQYHFWRUV DQG 6RWKH 1= 2 1= 2 " p p # 1=p2 !1=p2 GHVLUHGFKDQJHRIYDULDEOHLV [ D P \ ZKHUH P D $ FRPPRQ 1= 2 1= 2

HUURUKHUHLVWRIRUJHWWRQRUPDOL]HWKHHLJHQYHFWRUV 7KHQHZTXDGUDWLFIRUPLV \TD \ D 4y12 C 2y22

7KHPD[LPXPRI Q.[/ IRU [ DXQLWYHFWRULV DQGWKHPD[LPXPLVDWWDLQHGDW ! p " ! " 1=p2 1 WKHXQLWHLJHQYHFWRU >$ FRPPRQLQFRUUHFWDQVZHULV 7KLVYHFWRU 0 1= 2 PD[LPL]HVWKHTXDGUDWLFIRUP \TD \ LQVWHDGRI Q.[/@

5)& 4*/(6-"3 7"-6& %&$0.104*5*0/ 7KHGLDJRQDOL]DWLRQWKHRUHPVLQ6HFWLRQVDQGSOD\DSDUWLQPDQ\LQWHUHVWLQJDS SOLFDWLRQV 8QIRUWXQDWHO\ DVZHNQRZ QRWDOOPDWULFHVFDQEHIDFWRUHGDV A D PDP !1 ZLWK D GLDJRQDO +RZHYHU DIDFWRUL]DWLRQ A D QDP !1 LV SRVVLEOHIRUDQ\ m " n PDWUL[ A $ VSHFLDOIDFWRUL]DWLRQRIWKLVW\SH FDOOHGWKH VLQJXODUYDOXHGHFRPSRVLWLRQ LVRQHRIWKHPRVWXVHIXOPDWUL[IDFWRUL]DWLRQVLQDSSOLHGOLQHDUDOJHEUD 7KHVLQJXODUYDOXHGHFRPSRVLWLRQLVEDVHGRQWKHIROORZLQJSURSHUW\RIWKHRUGLQDU\ GLDJRQDOL]DWLRQWKDWFDQEHLPLWDWHGIRUUHFWDQJXODUPDWULFHV 7KHDEVROXWHYDOXHVRIWKH HLJHQYDOXHVRIDV\PPHWULFPDWUL[ A PHDVXUHWKHDPRXQWVWKDW A VWUHWFKHVRUVKULQNV

7KH6LQJXODU9DOXH'HFRPSRVLWLRQ

FHUWDLQYHFWRUVWKHHLJHQYHFWRUV ,I A[ D ![ DQG k[k D 1 WKHQ

kA[k D k![k D j!j k[k D j!j

,I !1 LVWKHHLJHQYDOXHZLWKWKHJUHDWHVWPDJQLWXGH WKHQDFRUUHVSRQGLQJXQLWHLJHQYHF WRU Y1 LGHQWLÀHVDGLUHFWLRQLQZKLFKWKHVWUHWFKLQJHIIHFWRI A LVJUHDWHVW 7KDWLV WKH OHQJWKRI A[ LVPD[LPL]HGZKHQ [ D Y1 DQG kAY1 k D j!1 j E\ 7KLVGHVFULSWLRQRI Y1 DQG j!1 j KDVDQDQDORJXHIRUUHFWDQJXODUPDWULFHVWKDWZLOOOHDGWRWKHVLQJXODUYDOXH GHFRPSRVLWLRQ ! " 4 11 14 &9".1-& ,I A D WKHQWKHOLQHDUWUDQVIRUPDWLRQ [ 7! A[ PDSV 8 7 !2 WKHXQLWVSKHUH f[ W k[k D 1g LQ R3 RQWRDQHOOLSVHLQ R2 VKRZQLQ)LJ )LQGDXQLW YHFWRU [ DWZKLFKWKHOHQJWK kA[k LVPD[LPL]HG DQGFRPSXWHWKLVPD[LPXPOHQJWK x3

Multiplication by A

x2

(18, 6) x1

x2 x1 (3, – 9) '*(63& $ WUDQVIRUPDWLRQIURP R3 WR R2

40-65*0/ 7KHTXDQWLW\ kA[k2 LVPD[LPL]HGDWWKHVDPH [ WKDWPD[LPL]HV kA[k DQG kA[k2 LVHDVLHUWRVWXG\ 2EVHUYHWKDW kA[k2 D .A[/T .A[/ D [TATA[ D [T.ATA/[

$OVR ATA LVDV\PPHWULFPDWUL[ VLQFH .ATA/T D ATAT T D ATA 6RWKHSUREOHPQRZ LVWRPD[LPL]HWKHTXDGUDWLFIRUP [T .ATA/[ VXEMHFWWRWKHFRQVWUDLQW k[k D 1 %\ 7KHRUHPLQ6HFWLRQ WKHPD[LPXPYDOXHLVWKHJUHDWHVWHLJHQYDOXH !1 RI ATA $OVR WKHPD[LPXPYDOXHLVDWWDLQHGDWDXQLWHLJHQYHFWRURI ATA FRUUHVSRQGLQJWR !1 )RUWKHPDWUL[ A LQWKLVH[DPSOH 2 3 2 3 " 4 8 ! 80 100 40 4 11 14 ATA D 4 11 7 5 D 4 100 170 140 5 8 7 !2 14 !2 40 140 200

7KHHLJHQYDOXHVRI ATA DUH !1 D 360 !2 D 90 DQG !3 D 0 &RUUHVSRQGLQJXQLWHLJHQ YHFWRUVDUH UHVSHFWLYHO\ 2 3 2 3 2 3 1=3 !2=3 2=3 Y1 D 4 2=3 5; Y2 D 4 !1=3 5; Y3 D 4 !2=3 5 2=3 2=3 1=3 7KHPD[LPXPYDOXHRI kA[k2 LV DWWDLQHGZKHQ [ LVWKHXQLWYHFWRU Y1 7KHYHFWRU AY1 LVDSRLQWRQWKHHOOLSVHLQ)LJIDUWKHVWIURPWKHRULJLQ QDPHO\ 3 2 ! " 1=3 ! " 4 11 14 4 18 5 2=3 D AY1 D 8 7 !2 6 2=3 p p )RU k[k D 1 WKHPD[LPXPYDOXHRI kA[k LV kAY1 k D 360 D 6 10

([DPSOHVXJJHVWVWKDWWKHHIIHFWRI A RQWKHXQLWVSKHUHLQ R3 LVUHODWHGWRWKH TXDGUDWLFIRUP [T .ATA/[ ,QIDFW WKHHQWLUHJHRPHWULFEHKDYLRURIWKHWUDQVIRUPDWLRQ [ 7! A[ LVFDSWXUHGE\WKLVTXDGUDWLFIRUP DVZHVKDOOVHH

$)"15&3


7KH6LQJXODU9DOXHVRIDQ P " Q 0DWUL[

/HW A EHDQ m " n PDWUL[ 7KHQ ATA LVV\PPHWULFDQGFDQEHRUWKRJRQDOO\GLDJRQDOL]HG /HW fY1 ; : : : ; Yn g EHDQRUWKRQRUPDOEDVLVIRU Rn FRQVLVWLQJRIHLJHQYHFWRUVRI ATA DQG OHW !1 ; : : : ; !n EHWKHDVVRFLDWHGHLJHQYDOXHVRI ATA 7KHQ IRU 1 % i % n

kAYi k2 D .AYi /TAYi D YTi ATAYi D YTi .!i Yi / D !i

6LQFH Yi LVDQHLJHQYHFWRURI ATA

6LQFH Yi LVDXQLWYHFWRU

6RWKHHLJHQYDOXHVRI ATA DUHDOOQRQQHJDWLYH %\UHQXPEHULQJ LIQHFHVVDU\ ZHPD\ DVVXPHWKDWWKHHLJHQYDOXHVDUHDUUDQJHGVRWKDW

!1 $ !2 $ # # # $ !n $ 0 T 7KH VLQJXODUYDOXHV RI A DUHWKHVTXDUHURRWVRIWKHHLJHQYDOXHVRI p A A GHQRWHGE\ #1 ; : : : ; #n DQGWKH\DUHDUUDQJHGLQGHFUHDVLQJRUGHU 7KDWLV #i D !i IRU 1 % i % n %\HTXDWLRQ WKHVLQJXODUYDOXHVRI$ DUHWKHOHQJWKVRIWKHYHFWRUV AY1 ; : : : ; AYn

&9".1-& /HW A EHWKHPDWUL[LQ([DPSOH 6LQFHWKHHLJHQYDOXHVRI ATA DUH DQG WKHVLQJXODUYDOXHVRI A DUH p p p p #1 D 360 D 6 10; #2 D 90 D 3 10;

x2

Av1 x1 Av2 '*(63&

#3 D 0

)URP([DPSOH WKHÀUVWVLQJXODUYDOXHRI A LVWKHPD[LPXPRI kA[k RYHUDOOXQLW YHFWRUV DQGWKHPD[LPXPLVDWWDLQHGDWWKHXQLWHLJHQYHFWRU Y1 7KHRUHPLQ6HFWLRQ VKRZVWKDWWKHVHFRQGVLQJXODUYDOXHRI A LVWKHPD[LPXPRI kA[k RYHUDOOXQLW YHFWRUVWKDWDUH RUWKRJRQDOWR Y1 DQGWKLVPD[LPXPLVDWWDLQHGDWWKHVHFRQGXQLW HLJHQYHFWRU Y2 ([HUFLVH )RUWKH Y2 LQ([DPSOH 3 2 ! " !2=3 ! " 4 11 14 4 3 !1=3 5 D AY2 D 8 7 !2 !9 2=3 7KLVSRLQWLVRQWKHPLQRUD[LVRIWKHHOOLSVHLQ)LJ MXVWDV AY1 LVRQWKHPDMRUD[LV 6HH)LJ 7KHÀUVWWZRVLQJXODUYDOXHVRI A DUHWKHOHQJWKVRIWKHPDMRUDQGPLQRU VHPLD[HVRIWKHHOOLSVH 7KHIDFWWKDW AY1 DQG AY2 DUHRUWKRJRQDOLQ)LJ LVQRDFFLGHQW DVWKHQH[WWKHRUHP VKRZV

5)&03&.

6XSSRVH fY1 ; : : : ; Yn g LVDQRUWKRQRUPDOEDVLVRI Rn FRQVLVWLQJRIHLJHQYHFWRUVRI ATA DUUDQJHGVRWKDWWKHFRUUHVSRQGLQJHLJHQYDOXHVRI ATA VDWLVI\ !1 $ # # # $ !n DQG VXSSRVH A KDV r QRQ]HUR VLQJXODU YDOXHV 7KHQ fAY1 ; : : : ; AYr g LV DQ RUWKRJRQDOEDVLVIRU &RO A DQG UDQN A D r

1300' %HFDXVH Yi DQG !j Yj DUHRUWKRJRQDOIRU i ¤ j .AYi /T .AYj / D YTi ATAYj D YTi .!j Yj / D 0 7KXV fAY1 ; : : : ; AYn g LVDQRUWKRJRQDOVHW )XUWKHUPRUH VLQFHWKHOHQJWKVRIWKHYHF WRUV AY1 ; : : : ; AYn DUHWKHVLQJXODUYDOXHVRI A DQGVLQFHWKHUHDUH r QRQ]HURVLQJXODU YDOXHV AYi ¤ LIDQGRQO\LI 1 % i % r 6R AY1 ; : : : ; AYr DUHOLQHDUO\LQGHSHQGHQW


YHFWRUV DQGWKH\DUHLQ &RO A )LQDOO\ IRUDQ\ \ LQ &RO A³VD\ \ D A[³ZHFDQZULWH [ D c1 Y1 C # # # C cn Yn DQG \ D A[ D c1 AY1 C # # # C cr AYr C cr C1 AYr C1 C # # # C cn AYn D c1 AY1 C # # # C cr AYr C 0 C # # # C 0

7KXV \ LVLQ 6SDQ fAY1 ; : : : ; AYr g ZKLFKVKRZVWKDW fAY1 ; : : : ; AYr g LVDQRUWKRJRQDO EDVLVIRU &RO A +HQFH UDQN A D GLP &RO A D r

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

7KH6LQJXODU9DOXH'HFRPSRVLWLRQ 7KHGHFRPSRVLWLRQRI A LQYROYHVDQ m " n ´GLDJRQDOµPDWUL[ † RIWKHIRUP ! " D 0 †D 0 0 ! m ! r URZV

" n ! r FROXPQV

ZKHUH D LVDQ r " r GLDJRQDOPDWUL[IRUVRPH r QRWH[FHHGLQJWKHVPDOOHURI m DQG n ,I r HTXDOV m RU n RUERWK VRPHRUDOORIWKH]HURPDWULFHVGRQRWDSSHDU

5)&03&.

5IF 4JOHVMBS 7BMVF %FDPNQPTJUJPO /HW A EHDQ m " n PDWUL[ZLWKUDQN r 7KHQWKHUHH[LVWVDQ m " n PDWUL[ † DV LQ IRUZKLFKWKHGLDJRQDOHQWULHVLQ D DUHWKHÀUVW r VLQJXODUYDOXHVRI A #1 $ #2 $ # # # $ #r > 0 DQGWKHUHH[LVWDQ m " m RUWKRJRQDOPDWUL[ U DQGDQ n " n RUWKRJRQDOPDWUL[ V VXFKWKDW

A D U †V T

$Q\IDFWRUL]DWLRQ A D U †V T ZLWK U DQG V RUWKRJRQDO † DVLQ DQGSRVLWLYH GLDJRQDOHQWULHVLQ D LVFDOOHGD VLQJXODUYDOXHGHFRPSRVLWLRQ RU 69' RI A 7KH PDWULFHV U DQG V DUHQRWXQLTXHO\GHWHUPLQHGE\ A EXWWKHGLDJRQDOHQWULHVRI † DUH QHFHVVDULO\WKHVLQJXODUYDOXHVRI A 6HH([HUFLVH 7KHFROXPQVRI U LQVXFKD GHFRPSRVLWLRQDUHFDOOHG OHIWVLQJXODUYHFWRUV RI A DQGWKHFROXPQVRI V DUHFDOOHG ULJKWVLQJXODUYHFWRUV RI A ñ ,QJHQHUDO UDQNHVWLPDWLRQLVQRWDVLPSOHSUREOHP )RUDGLVFXVVLRQRIWKHVXEWOHLVVXHVLQYROYHG VHH 3KLOLS(*LOO :DOWHU0XUUD\ DQG0DUJDUHW+:ULJKW 1XPHULFDO/LQHDU$OJHEUDDQG2SWLPL]DWLRQ YRO 5HGZRRG&LW\ &$$GGLVRQ:HVOH\ 6HF

$)"15&3


1300' /HW !i DQG Yi EHDVLQ7KHRUHP VRWKDW fAY1 ; : : : ; AYr g LVDQRUWKRJRQDOEDVLV IRU &RO A 1RUPDOL]HHDFK AYi WRREWDLQDQRUWKRQRUPDOEDVLV fX1 ; : : : ; Xr g ZKHUH Xi D DQG

1 1 AYi D AYi kAYi k #i

AYi D #i Xi

.1 % i % r/

1RZH[WHQG fX1 ; : : : ; Xr g WRDQRUWKRQRUPDOEDVLV fX1 ; : : : ; Xm g RI Rm DQGOHW

U D Œ X1 X2 # # # X m "

DQG

V D Œ Y1 Y 2 # # # Y n "

%\FRQVWUXFWLRQ U DQG V DUHRUWKRJRQDOPDWULFHV $OVR IURP

AV D Œ AY1 # # # AYr

# # # " D Œ #1 X1 # # # #r Xr

### "

/HW D EHWKHGLDJRQDOPDWUL[ZLWKGLDJRQDOHQWULHV #1 ; : : : ; #r DQGOHW † EHDVLQ DERYH 7KHQ 2 3 #1 0 6 07 #2 6 7 6 7 : :: U † D Œ X 1 X2 # # # Xm " 6 7 6 7 4 0 5 #r 0 0

D Œ #1 X1 # # # #r Xr D AV

### "

6LQFH V LVDQRUWKRJRQDOPDWUL[ U †V T D AV V T D A 7KHQH[WWZRH[DPSOHVIRFXVDWWHQWLRQRQWKHLQWHUQDOVWUXFWXUHRIDVLQJXODUYDOXH GHFRPSRVLWLRQ $QHIÀFLHQWDQGQXPHULFDOO\VWDEOHDOJRULWKPIRUWKLVGHFRPSRVLWLRQ ZRXOGXVHDGLIIHUHQWDSSURDFK 6HHWKH1XPHULFDO1RWHDWWKHHQGRIWKHVHFWLRQ

&9".1-& 8VHWKHUHVXOWVRI([DPSOHVDQGWRFRQVWUXFWDVLQJXODUYDOXH ! " GHFRPSRVLWLRQRI A D

4 8

11 7

14 !2

40-65*0/ $ FRQVWUXFWLRQFDQEHGLYLGHGLQWRWKUHHVWHSV SG

$PNQVUJOH BO 47% m

6WHS )LQGDQRUWKRJRQDOGLDJRQDOL]DWLRQRI A TA 7KDWLV ÀQGWKHHLJHQYDOXHVRI ATA DQGDFRUUHVSRQGLQJRUWKRQRUPDOVHWRIHLJHQYHFWRUV ,I A KDGRQO\WZRFROXPQV WKHFDOFXODWLRQVFRXOGEHGRQHE\KDQG /DUJHUPDWULFHVXVXDOO\UHTXLUHDPDWUL[SUR JUDPò +RZHYHU IRUWKHPDWUL[ A KHUH WKHHLJHQGDWDIRU ATA DUHSURYLGHGLQ([DPSOH 6WHS 6HWXS9 DQG † $UUDQJHWKHHLJHQYDOXHVRI ATA LQGHFUHDVLQJRUGHU ,Q ([DPSOH WKHHLJHQYDOXHVDUHDOUHDG\OLVWHGLQGHFUHDVLQJRUGHU DQG 7KH FRUUHVSRQGLQJXQLWHLJHQYHFWRUV Y1 Y2 DQG Y3 DUHWKHULJKWVLQJXODUYHFWRUVRI A 8VLQJ ([DPSOH FRQVWUXFW 2 3 1=3 !2=3 2=3 V D Œ Y1 Y2 Y3 " D 4 2=3 !1=3 !2=3 5 2=3 2=3 1=3 ò 6HHWKH 6WXG\*XLGH IRUVRIWZDUHDQGJUDSKLQJFDOFXODWRUFRPPDQGV 0$7/$%IRULQVWDQFH FDQSURGXFH ERWKWKHHLJHQYDOXHVDQGWKHHLJHQYHFWRUVZLWKRQHFRPPDQG HLJ


7KHVTXDUHURRWVRIWKHHLJHQYDOXHVDUHWKHVLQJXODUYDOXHV p p #1 D 6 10; #2 D 3 10; #3 D 0

7KHQRQ]HURVLQJXODUYDOXHVDUHWKHGLDJRQDOHQWULHVRI D 7KHPDWUL[ † LVWKHVDPH VL]HDV A ZLWK D LQLWVXSSHUOHIWFRUQHUDQGZLWK·VHOVHZKHUH " p # " p # 6 10 0 6 10 0 0 p p DD ; † D ŒD 0" D 0 3 10 0 3 10 0 6WHS &RQVWUXFW8 :KHQ A KDVUDQN r WKHÀUVW r FROXPQVRI U DUHWKHQRUPDOL]HG YHFWRUVREWDLQHGIURP AY1 ; : : : ; AYr ,QWKLVH[DPSOH A KDVWZRQRQ]HURVLQJXODU YDOXHV VR UDQN A D 2 5HFDOOIURPHTXDWLRQ DQGWKHSDUDJUDSKEHIRUH([DPSOH WKDW kAY1 k D #1 DQG kAY2 k D #2 7KXV ! " " p # 1 1 18 3=p10 X1 D AY1 D p D 1= 10 #1 6 10 6 p # ! " " 1 1 3 1=p10 X2 D AY2 D p D !3= 10 #2 3 10 !9 1RWHWKDW fX1 ; X2 g LVDOUHDG\DEDVLVIRU R2 7KXVQRDGGLWLRQDOYHFWRUVDUHQHHGHGIRU U DQG U D Œ X1 X2 " 7KHVLQJXODUYDOXHGHFRPSRVLWLRQRI A LV 3 " p # 2 1=3 p # " p 2=3 2=3 3=p10 1=p10 6 10 4 !2=3 !1=3 p 0 0 2=3 5 A D 1= 10 !3= 10 0 3 10 0 2=3 !2=3 1=3 " U

" †

" VT

2

3 1 !1 &9".1-& )LQGDVLQJXODUYDOXHGHFRPSRVLWLRQRI A D 4 !2 2 5 2 !2 ! " 9 !9 40-65*0/ )LUVW FRPSXWH ATA D 7KHHLJHQYDOXHVRI ATA DUHDQG !9 9 ZLWKFRUUHVSRQGLQJXQLWHLJHQYHFWRUV " " p # p # 1=p2 1=p2 Y1 D ; Y2 D !1= 2 1= 2

7KHVHXQLWYHFWRUVIRUPWKHFROXPQVRI V "

V D Œ Y1 Y2 " D

p 1=p2 !1= 2

p # 1=p2 1= 2

p p 7KHVLQJXODUYDOXHVDUH #1 D 18 D 3 2 DQG #2 D 0 6LQFHWKHUHLVRQO\RQHQRQ]HUR p VLQJXODUYDOXH WKH´PDWUL[µ D PD\EHZULWWHQDVDVLQJOHQXPEHU 7KDWLV D D 3 2 7KHPDWUL[ † LVWKHVDPHVL]HDV A ZLWK D LQLWVXSSHUOHIWFRUQHU 2 3 2 p 3 D 0 3 2 0 † D 4 0 05 D 4 0 05 0 0 0 0 7RFRQVWUXFW U ÀUVWFRQVWUXFW AY1 DQG AY2 2 p 3 2=p2 6 7 AY1 D 4 !4= 2 5; p 4= 2

2 3 0 AY 2 D 4 0 5 0

$)"15&3


x2

1 v1

x1

x3

Av1

u1 u3

x1

'*(63&

u2

x2

p $VDFKHFNRQWKHFDOFXODWLRQV YHULI\WKDW kAY1 k D #1 D 3 2 2IFRXUVH AY2 D EHFDXVH kAY2 k D #2 D 0 7KHRQO\FROXPQIRXQGIRU U VRIDULV 2 3 1=3 1 X1 D p AY1 D 4 !2=3 5 3 2 2=3

7KHRWKHUFROXPQVRI U DUHIRXQGE\H[WHQGLQJWKHVHW fX1 g WRDQRUWKRQRUPDOEDVLVIRU R3 ,QWKLVFDVH ZHQHHGWZRRUWKRJRQDOXQLWYHFWRUV X2 DQG X3 WKDWDUHRUWKRJRQDOWR X1 6HH)LJ (DFKYHFWRUPXVWVDWLVI\ XT1 [ D 0 ZKLFKLVHTXLYDOHQWWRWKHHTXDWLRQ x1 ! 2x2 C 2x3 D 0 $ EDVLVIRUWKHVROXWLRQVHWRIWKLVHTXDWLRQLV 2 3 2 3 2 !2 Z 1 D 4 1 5; Z 2 D 4 0 5 0 1 &KHFNWKDW Z1 DQG Z2 DUHHDFKRUWKRJRQDOWR X1 $SSO\WKH*UDP²6FKPLGWSURFHVV ZLWKQRUPDOL]DWLRQV WR fZ1 ; Z2 g DQGREWDLQ 2 p 3 2 p 3 !2=p45 2=p5 6 7 X2 D 4 1= 5 5 ; X3 D 4 4= 45 5 p 0 5= 45

)LQDOO\ VHW U D Œ X1 X2 X3 " WDNH † DQG V T IURPDERYH DQGZULWH p p 32 p 2 3 2 3" p p # 1 !1 1=3 2=p5 !2=p45 3 2 0 1= 2 !1= 6 7 p2 p A D 4 !2 2 5 D 4 !2=3 4=p45 54 0 05 1= 5 2 1= 2 1= 2 !2 0 0 2=3 0 5= 45

$SSOLFDWLRQVRIWKH6LQJXODU9DOXH'HFRPSRVLWLRQ 7KH69' LVRIWHQXVHGWRHVWLPDWHWKHUDQNRIDPDWUL[ DVQRWHGDERYH 6HYHUDORWKHUQX PHULFDODSSOLFDWLRQVDUHGHVFULEHGEULHÁ\EHORZ DQGDQDSSOLFDWLRQWRLPDJHSURFHVVLQJ LVSUHVHQWHGLQ6HFWLRQ

&9".1-& 5IF $POEJUJPO /VNCFS 0RVWQXPHULFDOFDOFXODWLRQVLQYROYLQJDQ HTXDWLRQ A[ D E DUHDVUHOLDEOHDVSRVVLEOHZKHQWKH69' RI A LVXVHG 7KHWZR RUWKRJRQDOPDWULFHV U DQG V GRQRWDIIHFWOHQJWKVRIYHFWRUVRUDQJOHVEHWZHHQYHFWRUV 7KHRUHPLQ6HFWLRQ $Q\SRVVLEOHLQVWDELOLWLHVLQQXPHULFDOFDOFXODWLRQVDUH LGHQWLÀHGLQ † ,IWKHVLQJXODUYDOXHVRI A DUHH[WUHPHO\ODUJHRUVPDOO URXQGRIIHUURUV DUHDOPRVWLQHYLWDEOH EXWDQHUURUDQDO\VLVLVDLGHGE\NQRZLQJWKHHQWULHVLQ † DQG V ,I A LVDQLQYHUWLEOH n " n PDWUL[ WKHQWKHUDWLR #1 =#n RIWKHODUJHVWDQGVPDOOHVW VLQJXODUYDOXHVJLYHVWKH FRQGLWLRQQXPEHU RI A ([HUFLVHV²LQ6HFWLRQ VKRZHGKRZWKHFRQGLWLRQQXPEHUDIIHFWVWKHVHQVLWLYLW\RIDVROXWLRQRI A[ D E WR FKDQJHVRUHUURUV LQWKHHQWULHVRI A $FWXDOO\ D´FRQGLWLRQQXPEHUµRI A FDQEH FRPSXWHGLQVHYHUDOZD\V EXWWKHGHÀQLWLRQJLYHQKHUHLVZLGHO\XVHGIRUVWXG\LQJ A[ D E &9".1-& #BTFT GPS 'VOEBNFOUBM 4VCTQBDFT *LYHQDQ69' IRUDQ m " n

PDWUL[ A OHW X1 ; : : : ; Xm EHWKHOHIWVLQJXODUYHFWRUV Y1 ; : : : ; Yn WKHULJKWVLQJXODUYHFWRUV DQG #1 ; : : : ; #n WKHVLQJXODUYDOXHV DQGOHW r EHWKHUDQNRI A %\7KHRUHP LVDQRUWKRQRUPDOEDVLVIRU &RO A

fX1 ; : : : ; Xr g


ul A

5HFDOOIURP7KHRUHPLQ6HFWLRQWKDW .&RO A/? D 1XO AT +HQFH

N

fXr C1 ; : : : ; Xm g

LVDQRUWKRQRUPDOEDVLVIRU 1XO AT 6LQFH kAYi k D #i IRU 1 % i % n DQG #i LVLIDQGRQO\LI i > r WKHYHFWRUV Yr C1 ; : : : ; Yn VSDQDVXEVSDFHRI 1XO A RIGLPHQVLRQ n ! r %\WKH5DQN7KHRUHP GLP 1XO A D n ! UDQN A ,WIROORZVWKDW

v1 w Ro A

fYr C1 ; : : : ; Yn g

LVDQRUWKRQRUPDOEDVLVIRU 1XO A E\WKH%DVLV7KHRUHPLQ6HFWLRQ )URP DQG WKHRUWKRJRQDOFRPSOHPHQWRI 1XO AT LV &RO A ,QWHUFKDQJLQJ A DQG AT QRWHWKDW .1XO A/? D &RO AT D 5RZ A +HQFH IURP

x3

fY1 ; : : : ; Y r g u1 u3

x1 !

)

lA (Co

x2

u2

Co

lA

LVDQRUWKRQRUPDOEDVLVIRU 5RZ A )LJXUHVXPPDUL]HV ² EXWVKRZVWKHRUWKRJRQDOEDVLV f#1 X1 ; : : : ; #r Xr g IRU &RO A LQVWHDGRIWKHQRUPDOL]HGEDVLV WRUHPLQG\RXWKDW AYi D #i Xi IRU 1 % i % r ([SOLFLWRUWKRQRUPDOEDVHVIRUWKHIRXUIXQGDPHQWDOVXEVSDFHVGHWHUPLQHGE\ A DUH XVHIXOLQVRPHFDOFXODWLRQV SDUWLFXODUO\LQFRQVWUDLQHGRSWLPL]DWLRQSUREOHPV Multiplication by A

7KHIXQGDPHQWDOVXEVSDFHVLQ ([DPSOH Row A

v2

v1

...

Col A = Row AT

vr

σr ur

vr + 1

ur + 1

0

...

...

0

.. .

σ1u1 σ2u2

...

Av1

Nul A

vn – 1 vn

um

Nul AT

'*(63& 7KHIRXUIXQGDPHQWDOVXEVSDFHVDQGWKH DFWLRQRI A

7KHIRXUIXQGDPHQWDOVXEVSDFHVDQGWKHFRQFHSWRIVLQJXODUYDOXHVSURYLGHWKHÀQDO VWDWHPHQWVRIWKH,QYHUWLEOH0DWUL[7KHRUHP 5HFDOOWKDWVWDWHPHQWVDERXW AT KDYH EHHQRPLWWHGIURPWKHWKHRUHP WRDYRLGQHDUO\GRXEOLQJWKHQXPEHURIVWDWHPHQWV 7KHRWKHUVWDWHPHQWVZHUHJLYHQLQ6HFWLRQV DQG

5)&03&.

5IF *OWFSUJCMF .BUSJY 5IFPSFN DPODMVEFE

/HW A EHDQ n " n PDWUL[ 7KHQWKHIROORZLQJVWDWHPHQWVDUHHDFKHTXLYDOHQWWR WKHVWDWHPHQWWKDW A LVDQLQYHUWLEOHPDWUL[ X Y Z [

.&RO A/? D fg .1XO A/? D Rn 5RZ A D Rn A KDV n QRQ]HURVLQJXODUYDOXHV

$)"15&3


&9".1-& 3FEVDFE 47% BOE UIF 1TFVEPJOWFSTF PG " :KHQ † FRQWDLQVURZVRU FROXPQVRI]HURV DPRUHFRPSDFWGHFRPSRVLWLRQRI A LVSRVVLEOH 8VLQJWKHQRWDWLRQ HVWDEOLVKHGDERYH OHW r D UDQN A DQGSDUWLWLRQ U DQG V LQWRVXEPDWULFHVZKRVHÀUVW EORFNVFRQWDLQ r FROXPQV

U D Œ Ur V D Œ Vr

Um!r "; Vn!r ";

ZKHUH Ur D Œ X1 # # # Xr " ZKHUH Vr D Œ Y1 # # # Yr "

7KHQ Ur LV m " r DQG Vr LV n " r 7RVLPSOLI\QRWDWLRQ ZHFRQVLGHU Um!r RU Vn!r HYHQWKRXJKRQHRIWKHPPD\KDYHQRFROXPQV 7KHQSDUWLWLRQHGPDWUL[PXOWLSOLFDWLRQ VKRZVWKDW " #" # D 0 VrT A D Œ Ur Um!r " D Ur DVrT VnT!r 0 0 7KLVIDFWRUL]DWLRQRI A LVFDOOHGD UHGXFHGVLQJXODUYDOXHGHFRPSRVLWLRQ RI A 6LQFH WKHGLDJRQDOHQWULHVLQ D DUHQRQ]HUR D LVLQYHUWLEOH 7KHIROORZLQJPDWUL[LVFDOOHG WKH SVHXGRLQYHUVH DOVR WKH 0RRUH²3HQURVHLQYHUVH RI A

AC D Vr D !1 UrT

6XSSOHPHQWDU\([HUFLVHV²DWWKHHQGRIWKHFKDSWHUH[SORUHVRPHRIWKHSURSHUWLHV RIWKHUHGXFHGVLQJXODUYDOXHGHFRPSRVLWLRQDQGWKHSVHXGRLQYHUVH

&9".1-& -FBTU4RVBSFT 4PMVUJPO *LYHQWKHHTXDWLRQ A[ D E XVHWKHSVHX GRLQYHUVHRI A LQ WRGHÀQH

[O D AC E D Vr D !1 UrT E 7KHQ IURPWKH69' LQ

AO[ D .Ur DVrT /.Vr D !1 UrT E/ D Ur ''!1 UrT E D Ur UrT E

%HFDXVH VrT Vr D Ir

,WIROORZVIURP WKDW Ur UrT E LVWKHRUWKRJRQDOSURMHFWLRQ EO RI E RQWR &RO A 6HH 7KHRUHPLQ6HFWLRQ 7KXV [O LVDOHDVWVTXDUHVVROXWLRQRI A[ D E ,QIDFW WKLV [O KDVWKHVPDOOHVWOHQJWKDPRQJDOOOHDVWVTXDUHVVROXWLRQVRI A[ D E 6HH6XSSOHPHQWDU\ ([HUFLVH

/6.&3*$"- /05& ([DPSOHV²DQGWKHH[HUFLVHVLOOXVWUDWHWKHFRQFHSWRIVLQJXODUYDOXHVDQG VXJJHVWKRZWRSHUIRUPFDOFXODWLRQVE\KDQG ,QSUDFWLFH WKHFRPSXWDWLRQRI ATA VKRXOGEHDYRLGHG VLQFHDQ\HUURUVLQWKHHQWULHVRI A DUHVTXDUHGLQWKH HQWULHVRI ATA 7KHUHH[LVWIDVWLWHUDWLYHPHWKRGVWKDWSURGXFHWKHVLQJXODUYDOXHV DQGVLQJXODUYHFWRUVRI A DFFXUDWHO\WRPDQ\GHFLPDOSODFHV

)XUWKHU5HDGLQJ +RUQ 5RJHU$ DQG&KDUOHV5-RKQVRQ 0DWUL[$QDO\VLV &DPEULGJH &DPEULGJH 8QLYHUVLW\3UHVV /RQJ &OLII ´9LVXDOL]DWLRQRI0DWUL[6LQJXODU9DOXH'HFRPSRVLWLRQµ 0DWKHPDWLFV 0DJD]LQH SS ²


0ROHU &% DQG'0RUULVRQ ´6LQJXODU9DOXH$QDO\VLVRI&U\SWRJUDPVµ $PHU0DWK 0RQWKO\ SS ² 6WUDQJ *LOEHUW /LQHDU$OJHEUDDQG,WV$SSOLFDWLRQV WKHG %HOPRQW &$%URRNV &ROH :DWNLQV 'DYLG6 )XQGDPHQWDOVRI0DWUL[&RPSXWDWLRQV 1HZ
13"$5*$& 130#-&. *LYHQDVLQJXODUYDOXHGHFRPSRVLWLRQ A D U †V T ÀQGDQ69' RI AT +RZDUHWKH VLQJXODUYDOXHVRI A DQG AT UHODWHG"

WEB

&9&3$*4&4 )LQGWKHVLQJXODUYDOXHVRIWKHPDWULFHVLQ([HUFLVHV² ! " ! " 1 0 !5 0 0 !3 0 0

!p

6 0

p1 6

"

!p

3 0

p2 3

"

)LQG DQ 69' RI HDFK PDWUL[ LQ ([HUFLVHV ² >+LQW ,Q 2 3 !1=3 2=3 2=3 2=3 5 ,Q ([HUFLVH RQHFKRLFHIRU U LV 4 2=3 !1=3 2=3 2=3 !1=3 2 p 3 1=p6 6 7 ([HUFLVH RQHFROXPQRI U FDQEH 4 !2= 6 5@ p 1= 6

! !

2

!3 0

0 0

2 !1 2 2

"

7 4 0 5 2

"

3

1 05 5

3 !3 1 4 6 !2 5 6 !2

! !

!2 0 0 !1 2 0

3 2

2

"

"

3

4 !2 4 2 !1 5 0 0 2

1 4 0 !1

)LQGWKH69' RI A D

!

3 2

3 1 15 1

" 2 2 >+LQW :RUNZLWK AT @ 3 !2

,Q([HUFLVH ÀQGDXQLWYHFWRU [ DWZKLFK A[ KDVPD[LPXP OHQJWK 6XSSRVHWKHIDFWRUL]DWLRQEHORZLVDQ69' RIDPDWUL[ A ZLWKWKHHQWULHVLQ U DQG V URXQGHGWRWZRGHFLPDOSODFHV

2

:40 A D 4 :37 !:84 2 :30 " 4 :76 :58

32 !:78 :47 7:10 !:33 !:87 5 4 0 !:52 !:16 0 3 !:51 !:81 :64 !:12 5 !:58 :58

0 3:10 0

3 0 05 0

D :KDWLVWKHUDQNRI A"

E 8VHWKLVGHFRPSRVLWLRQRI A ZLWKQRFDOFXODWLRQV WR ZULWHDEDVLVIRU &RO A DQGDEDVLVIRU 1XO A >+LQW )LUVW ZULWHWKHFROXPQVRI V @ 5HSHDW([HUFLVHIRUWKHIROORZLQJ69' RID 3 " 4 PDWUL[ A 2 32 3 !:86 !:11 !:50 12:48 0 0 0 6:34 0 05 A D 4 :31 :68 !:67 5 4 0 :41 !:73 !:55 0 0 0 0 2 3 :66 !:03 !:35 :66 6 !:13 !:90 !:39 !:13 7 7 "6 4 :65 :08 !:16 !:73 5 !:34 :42 !:84 !:08

,Q([HUFLVHV² A LVDQ m " n PDWUL[ZLWKDVLQJXODUYDOXH GHFRPSRVLWLRQ A D U †V T ZKHUH U LVDQ m " m RUWKRJRQDO PDWUL[ † LVDQ m " n ´GLDJRQDOµPDWUL[ZLWK r SRVLWLYHHQWULHV DQGQRQHJDWLYHHQWULHV DQG V LVDQ n " n RUWKRJRQDOPDWUL[ -XVWLI\HDFKDQVZHU 6XSSRVH A LVVTXDUHDQGLQYHUWLEOH )LQGDVLQJXODUYDOXH GHFRPSRVLWLRQRI A!1 6KRZWKDWLI A LVVTXDUH WKHQ j GHW Aj LVWKHSURGXFWRIWKH VLQJXODUYDOXHVRI A 6KRZWKDWWKHFROXPQVRI V DUHHLJHQYHFWRUVRI ATA WKH FROXPQVRI U DUHHLJHQYHFWRUVRI AAT DQGWKHGLDJRQDO HQWULHVRI † DUHWKHVLQJXODUYDOXHVRI A >+LQW 8VHWKH 69' WRFRPSXWH ATA DQG AAT @ 6KRZWKDWLI A LVDQ n " n SRVLWLYHGHÀQLWHPDWUL[ WKHQDQ RUWKRJRQDOGLDJRQDOL]DWLRQ A D 3'3T LVDVLQJXODUYDOXH GHFRPSRVLWLRQRI A

$)"15&3


6KRZWKDWLI P LVDQRUWKRJRQDO m " m PDWUL[ WKHQ PA KDV WKHVDPHVLQJXODUYDOXHVDV A -XVWLI\WKHVWDWHPHQWLQ([DPSOHWKDWWKHVHFRQGVLQJXODU YDOXHRIDPDWUL[ A LVWKHPD[LPXPRI kA[k DV [ YDULHV RYHUDOOXQLWYHFWRUVRUWKRJRQDOWR Y1 ZLWK Y1 DULJKWVLQJXODU YHFWRUFRUUHVSRQGLQJWRWKHÀUVWVLQJXODUYDOXHRI A >+LQW 8VH7KHRUHPLQ6HFWLRQ@ /HW U D Œ X1 # # # Xm " DQG V D Œ Y1 # # # Xi DQG Yi DUHDVLQ7KHRUHP 6KRZWKDW

Yn " ZKHUHWKH

A D #1 X1 YT1 C #2 X2 YT2 C # # # C #r Xr YTr : 8VLQJWKHQRWDWLRQRI([HUFLVH VKRZWKDW AT Xj D #j Yj IRU 1 % j % r D UDQN A

/HW T W Rn ! Rm EHDOLQHDUWUDQVIRUPDWLRQ 'HVFULEHKRZ WRÀQGDEDVLV B IRU Rn DQGDEDVLV C IRU Rm VXFKWKDWWKH PDWUL[IRU T UHODWLYHWR B DQG C LVDQ m " n ´GLDJRQDOµ PDWUL[

>0@ &RPSXWHDQ69' RIHDFKPDWUL[LQ([HUFLVHVDQG 5HSRUWWKHÀQDOPDWUL[HQWULHVDFFXUDWHWRWZRGHFLPDOSODFHV 8VH WKHPHWKRGRI([DPSOHVDQG 2 3 !18 13 !4 4 6 2 19 !4 12 7 7 A D 6 4 !14 11 !12 85 !2 21 4 8 2 3 6 !8 !4 5 !4 6 2 7 !5 !6 47 7 A D 6 4 0 !1 !8 2 25 !1 !2 4 4 !8 >0@ &RPSXWH WKH VLQJXODU YDOXHV RI WKH 4 " 4 PDWUL[ LQ ([HUFLVHLQ6HFWLRQ DQGFRPSXWHWKHFRQGLWLRQQXPEHU #1 =#4 >0@ &RPSXWHWKHVLQJXODUYDOXHVRIWKH 5 " 5 PDWUL[LQ([ HUFLVHLQ6HFWLRQ DQGFRPSXWHWKHFRQGLWLRQQXPEHU #1 =#5

40-65*0/ 50 13"$5*$& 130#-&. ,I A D U †V T ZKHUH † LV m " n WKHQ AT D .V T /T †T U T D V †T U T 7KLVLVDQ 69' RI AT EHFDXVH V DQG U DUHRUWKRJRQDOPDWULFHVDQG †T LVDQ n " m ´GLDJRQDOµ PDWUL[ 6LQFH † DQG †T KDYHWKHVDPHQRQ]HURGLDJRQDOHQWULHV A DQG AT KDYHWKH VDPHQRQ]HURVLQJXODUYDOXHV >1RWH ,I A LV 2 " n WKHQ AAT LVRQO\ 2 " 2 DQGLWV HLJHQYDOXHVPD\EHHDVLHUWRFRPSXWHE\KDQG WKDQWKHHLJHQYDOXHVRI ATA@

"11-*$"5*0/4 50 *."(& 130$&44*/( "/% 45"5*45*$4 7KHVDWHOOLWHSKRWRJUDSKVLQWKLVFKDSWHU·VLQWURGXFWLRQSURYLGHDQH[DPSOHRIPXOWLGL PHQVLRQDO RU PXOWLYDULDWH GDWD³LQIRUPDWLRQRUJDQL]HGVRWKDWHDFKGDWXPLQWKHGDWD VHWLVLGHQWLÀHGZLWKDSRLQWYHFWRU LQ Rn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

&9".1-& $QH[DPSOHRIWZRGLPHQVLRQDOGDWDLVJLYHQE\DVHWRIZHLJKWVDQG KHLJKWVRI N FROOHJHVWXGHQWV /HW ;j GHQRWHWKH REVHUYDWLRQYHFWRU LQ R2 WKDWOLVWVWKH ZHLJKWDQGKHLJKWRIWKH j WKVWXGHQW ,I w GHQRWHVZHLJKWDQG h KHLJKW WKHQWKHPDWUL[

$SSOLFDWLRQVWR,PDJH3URFHVVLQJDQG6WDWLVWLFV

RIREVHUYDWLRQVKDVWKHIRUP

!

w1 h1 " ;1

w2 h2 " ;2

### ###

wN hN "

"

;N

7KHVHWRIREVHUYDWLRQYHFWRUVFDQEHYLVXDOL]HGDVDWZRGLPHQVLRQDO VFDWWHUSORW 6HH )LJ h

w '*(63& $ VFDWWHUSORWRIREVHUYDWLRQ YHFWRUV ;1 ; : : : ; ;N x3

x2

x1 '*(63&

$ VFDWWHUSORWRIVSHFWUDOGDWDIRUD VDWHOOLWHLPDJH

&9".1-& 7KHÀUVWWKUHHSKRWRJUDSKVRI5DLOURDG9DOOH\ 1HYDGD VKRZQLQWKH FKDSWHULQWURGXFWLRQ FDQEHYLHZHGDV RQH LPDJHRIWKHUHJLRQ ZLWK WKUHHVSHFWUDO FRPSRQHQWV EHFDXVHVLPXOWDQHRXVPHDVXUHPHQWVRIWKHUHJLRQZHUHPDGHDWWKUHH VHSDUDWHZDYHOHQJWKV (DFKSKRWRJUDSKJLYHVGLIIHUHQWLQIRUPDWLRQDERXWWKHVDPH SK\VLFDOUHJLRQ )RULQVWDQFH WKHÀUVWSL[HOLQWKHXSSHUOHIWFRUQHURIHDFKSKRWRJUDSK FRUUHVSRQGVWRWKHVDPHSODFHRQWKHJURXQGDERXWPHWHUVE\PHWHUV 7RHDFK SL[HOWKHUHFRUUHVSRQGVDQREVHUYDWLRQYHFWRULQ R3 WKDWOLVWVWKHVLJQDOLQWHQVLWLHVIRU WKDWSL[HOLQWKHWKUHHVSHFWUDOEDQGV 7\SLFDOO\ WKHLPDJHLV 2000 " 2000 SL[HOV VRWKHUHDUHPLOOLRQSL[HOVLQWKH LPDJH 7KHGDWDIRUWKHLPDJHIRUPDPDWUL[ZLWKURZVDQGPLOOLRQFROXPQV ZLWKFROXPQVDUUDQJHGLQDQ\FRQYHQLHQWRUGHU ,QWKLVFDVH WKH´PXOWLGLPHQVLRQDOµ FKDUDFWHURIWKHGDWDUHIHUVWRWKHWKUHH VSHFWUDO GLPHQVLRQVUDWKHUWKDQWKHWZR VSDWLDO GLPHQVLRQVWKDWQDWXUDOO\EHORQJWRDQ\SKRWRJUDSK 7KHGDWDFDQEHYLVXDOL]HGDVD FOXVWHURIPLOOLRQSRLQWVLQ R3 SHUKDSVDVLQ)LJ

0HDQDQG&RYDULDQFH

hˆ

w ˆ

7RSUHSDUHIRUSULQFLSDOFRPSRQHQWDQDO\VLV OHW Œ ;1 # # # ;N " EHD p " N PDWUL[RI REVHUYDWLRQV VXFKDVGHVFULEHGDERYH 7KH VDPSOHPHDQ 0 RIWKHREVHUYDWLRQYHFWRUV ;1 ; : : : ; ;N LVJLYHQE\ 1 0 D .;1 C # # # C ;N / N )RUWKHGDWDLQ)LJ WKHVDPSOHPHDQLVWKHSRLQWLQWKH´FHQWHUµRIWKHVFDWWHUSORW )RU k D 1; : : : ; N OHW O k D ;k ! 0 ; 7KHFROXPQVRIWKH p " N PDWUL[

ON " O1 ; O2 ### ; B D Œ;

'*(63&

:HLJKW²KHLJKWGDWDLQ PHDQGHYLDWLRQIRUP

KDYHD]HURVDPSOHPHDQ DQG B LVVDLGWREHLQ PHDQGHYLDWLRQIRUP :KHQWKH VDPSOHPHDQLVVXEWUDFWHGIURPWKHGDWDLQ)LJ WKHUHVXOWLQJVFDWWHUSORWKDVWKHIRUP LQ)LJ

$)"15&3


7KHVDPSOH FRYDULDQFHPDWUL[ LVWKH p " p PDWUL[ S GHÀQHGE\

SD

1 BB T N !1

6LQFHDQ\PDWUL[RIWKHIRUP BB T LVSRVLWLYHVHPLGHÀQLWH VRLV S 6HH([HUFLVHLQ 6HFWLRQZLWK B DQG B T LQWHUFKDQJHG

&9".1-& 7KUHHPHDVXUHPHQWVDUHPDGHRQHDFKRIIRXULQGLYLGXDOVLQDUDQGRP VDPSOHIURPDSRSXODWLRQ 7KHREVHUYDWLRQYHFWRUVDUH 2 3 2 3 2 3 1 4 7 ;1 D 4 2 5; ;2 D 4 2 5; ;3 D 4 8 5; 1 13 1

&RPSXWHWKHVDPSOHPHDQDQGWKHFRYDULDQFHPDWUL[

2 3 8 ;4 D 4 4 5 5

40-65*0/ 7KHVDPSOHPHDQLV 02 3 2 3 2 3 2 31 2 3 2 3 1 4 7 8 20 5 1 @4 5 4 5 4 5 4 5A 14 5 4 5 2 C 2 C 8 C 4 16 D 4 D 0D 4 4 20 1 13 1 5 5 6XEWUDFWWKHVDPSOHPHDQIURP ;1 ; : : : ; ;4 WRREWDLQ 2 3 2 3 2 3 !4 !1 2 O 2 D 4 !2 5; ; O 3 D 4 4 5; O 1 D 4 !2 5; ; ; !4 8 !4 2

DQG

!4 !1 2 B D 4 !2 !2 4 !4 8 !4

7KHVDPSOHFRYDULDQFHPDWUL[LV

3 3 05 0

2 3 3 O4 D 405 ; 0

3 2 3 !4 !2 !4 !4 !1 2 3 6 1 !1 !2 8 7 7 S D 4 !2 !2 4 0 56 4 2 4 !4 5 3 !4 8 !4 0 3 0 0 2 3 2 3 30 18 0 10 6 0 14 18 24 !24 5 D 4 6 8 !8 5 D 3 0 !24 96 0 !8 32 2

7RGLVFXVVWKHHQWULHVLQ S D Œsij " OHW ; UHSUHVHQWDYHFWRUWKDWYDULHVRYHUWKHVHW RIREVHUYDWLRQYHFWRUVDQGGHQRWHWKHFRRUGLQDWHVRI ; E\ x1 ; : : : ; xp 7KHQ x1 IRU H[DPSOH LVDVFDODUWKDWYDULHVRYHUWKHVHWRIÀUVWFRRUGLQDWHVRI ;1 ; : : : ; ;N )RU j D 1; : : : ; p WKHGLDJRQDOHQWU\ sjj LQ S LVFDOOHGWKH YDULDQFH RI xj 7KHYDULDQFHRI xj PHDVXUHVWKHVSUHDGRIWKHYDOXHVRI xj 6HH([HUFLVH ,Q ([DPSOH WKHYDULDQFHRI x1 LVDQGWKHYDULDQFHRI x3 LV 7KHIDFWWKDWLVPRUH WKDQLQGLFDWHVWKDWWKHVHWRIWKLUGHQWULHVLQWKHUHVSRQVHYHFWRUVFRQWDLQVDZLGHU VSUHDGRIYDOXHVWKDQWKHVHWRIÀUVWHQWULHV 7KH WRWDOYDULDQFH RIWKHGDWDLVWKHVXPRIWKHYDULDQFHVRQWKHGLDJRQDORI S ,Q JHQHUDO WKHVXPRIWKHGLDJRQDOHQWULHVRIDVTXDUHPDWUL[ S LVFDOOHGWKH WUDFH RIWKH PDWUL[ ZULWWHQ WU.S / 7KXV

fWRWDOYDULDQFHg D WU.S/


7KHHQWU\ sij LQ S IRU i ¤ j LVFDOOHGWKH FRYDULDQFH RI xi DQG xj 2EVHUYHWKDW LQ([DPSOH WKHFRYDULDQFHEHWZHHQ x1 DQG x3 LVEHFDXVHWKH .1; 3/HQWU\LQ S LV 6WDWLVWLFLDQVVD\WKDW x1 DQG x3 DUH XQFRUUHODWHG $QDO\VLVRIWKHPXOWLYDULDWHGDWD LQ ;1 ; : : : ; ;N LVJUHDWO\VLPSOLÀHGZKHQPRVWRUDOORIWKHYDULDEOHV x1 ; : : : ; xp DUH XQFRUUHODWHG WKDWLV ZKHQWKHFRYDULDQFHPDWUL[RI ;1 ; : : : ; ;N LVGLDJRQDORUQHDUO\ GLDJRQDO

3ULQFLSDO&RPSRQHQW$QDO\VLV )RUVLPSOLFLW\ DVVXPHWKDWWKHPDWUL[ Œ ;1 # # # ;N " LVDOUHDG\LQPHDQGHYLDWLRQ IRUP 7KHJRDORISULQFLSDOFRPSRQHQWDQDO\VLVLVWRÀQGDQRUWKRJRQDO p " p PDWUL[ P D Œ X1 # # # Xp " WKDWGHWHUPLQHVDFKDQJHRIYDULDEOH ; D P < RU 2 3 2 3 x1 y1 6 y2 7 6 x2 7 # $ 6 7 6 7 6 :: 7 D X1 X2 # # # Xp 6 :: 7 4 : 5 4 : 5 xp yp

ZLWKWKHSURSHUW\WKDWWKHQHZYDULDEOHV y1 ; : : : ; yp DUHXQFRUUHODWHGDQGDUHDUUDQJHGLQ RUGHURIGHFUHDVLQJYDULDQFH 7KHRUWKRJRQDOFKDQJHRIYDULDEOH ; D P < PHDQVWKDWHDFKREVHUYDWLRQYHFWRU ;k UHFHLYHVD´QHZQDPHµ
y1 D XT1 ; D c1 x1 C c2 x2 C # # # C cp xp

7KXV y1 LVDOLQHDUFRPELQDWLRQRIWKHRULJLQDOYDULDEOHV x1 ; : : : ; xp XVLQJWKHHQWULHVLQ WKHHLJHQYHFWRU X1 DVZHLJKWV ,QDVLPLODUIDVKLRQ X2 GHWHUPLQHVWKHYDULDEOH y2 DQG VRRQ

&9".1-& 7KHLQLWLDOGDWDIRUWKHPXOWLVSHFWUDOLPDJHRI5DLOURDG9DOOH\([ DPSOH FRQVLVWHGRIPLOOLRQYHFWRUVLQ R3 7KHDVVRFLDWHGFRYDULDQFHPDWUL[LVñ 2 3 2382:78 2611:84 2136:20 S D 4 2611:84 3106:47 2553:90 5 2136:20 2553:90 2650:71 ñ 'DWDIRU([DPSOHDQG([HUFLVHVDQGZHUHSURYLGHGE\(DUWK6DWHOOLWH&RUSRUDWLRQ 5RFNYLOOH 0DU\ODQG

$)"15&3


)LQGWKHSULQFLSDOFRPSRQHQWVRIWKHGDWD DQGOLVWWKHQHZYDULDEOHGHWHUPLQHGE\WKH ÀUVWSULQFLSDOFRPSRQHQW

40-65*0/ 7KHHLJHQYDOXHVRI S DQGWKHDVVRFLDWHGSULQFLSDOFRPSRQHQWVWKHXQLW HLJHQYHFWRUV DUH !1 D 7614:23 2 3 :5417 X1 D 4 :6295 5 :5570

!2 D 427:63 2 3 !:4894 X2 D 4 !:3026 5 :8179

!3 D 98:10 2 3 :6834 X3 D 4 !:7157 5 :1441

8VLQJWZRGHFLPDOSODFHVIRUVLPSOLFLW\ WKHYDULDEOHIRUWKHÀUVWSULQFLSDOFRPSRQHQW LV y1 D :54x1 C :63x2 C :56x3 7KLVHTXDWLRQZDVXVHGWRFUHDWHSKRWRJUDSKG LQWKHFKDSWHULQWURGXFWLRQ 7KH YDULDEOHV x1 x2 x3 DUHWKHVLJQDOLQWHQVLWLHVLQWKHWKUHHVSHFWUDOEDQGV 7KHYDOXHVRI x1 FRQYHUWHGWRDJUD\VFDOHEHWZHHQEODFNDQGZKLWH SURGXFHGSKRWRJUDSKD 6LPLODUO\ WKHYDOXHVRI x2 DQG x3 SURGXFHGSKRWRJUDSKVE DQGF UHVSHFWLYHO\ $WHDFKSL[HOLQ SKRWRJUDSKG WKHJUD\VFDOHYDOXHLVFRPSXWHGIURP y1 DZHLJKWHGOLQHDUFRPELQDWLRQ RI x1 ; x2 ; x3 ,QWKLVVHQVH SKRWRJUDSKG ´GLVSOD\VµWKHÀUVWSULQFLSDOFRPSRQHQWRI WKHGDWD ,Q([DPSOH WKHFRYDULDQFHPDWUL[IRUWKHWUDQVIRUPHGGDWD XVLQJYDULDEOHV y1 y2 y3 LV 2 3 7614:23 0 0 0 427:63 0 5 DD4 0 0 98:10

$OWKRXJK D LVREYLRXVO\VLPSOHUWKDQWKHRULJLQDOFRYDULDQFHPDWUL[ S WKHPHULWRI FRQVWUXFWLQJWKHQHZYDULDEOHVLVQRW\HWDSSDUHQW +RZHYHU WKHYDULDQFHVRIWKH YDULDEOHV y1 y2 y3 DSSHDURQWKHGLDJRQDORI D DQGREYLRXVO\WKHÀUVWYDULDQFHLQ D LVPXFKODUJHUWKDQWKHRWKHUWZR $VZHVKDOOVHH WKLVIDFWZLOOSHUPLWXVWRYLHZ WKHGDWDDVHVVHQWLDOO\RQHGLPHQVLRQDOUDWKHUWKDQWKUHHGLPHQVLRQDO

5HGXFLQJWKH'LPHQVLRQRI0XOWLYDULDWH'DWD 3ULQFLSDOFRPSRQHQWDQDO\VLVLVSRWHQWLDOO\YDOXDEOHIRUDSSOLFDWLRQVLQZKLFKPRVWRI WKHYDULDWLRQ RUG\QDPLFUDQJH LQWKHGDWDLVGXHWRYDULDWLRQVLQ RQO\DIHZ RIWKHQHZ YDULDEOHV y1 ; : : : ; yp ,WFDQEHVKRZQWKDWDQRUWKRJRQDOFKDQJHRIYDULDEOHV ; D P < GRHVQRWFKDQJH WKHWRWDOYDULDQFHRIWKHGDWD 5RXJKO\VSHDNLQJ WKLVLVWUXHEHFDXVHOHIWPXOWLSOLFDWLRQ E\ P GRHVQRWFKDQJHWKHOHQJWKVRIYHFWRUVRUWKHDQJOHVEHWZHHQWKHP 6HH([HUFLVH 7KLVPHDQVWKDWLI S D 3'3T WKHQ ' ( ' ( WRWDOYDULDQFH WRWDOYDULDQFH D D WU.D/ D !1 C # # # C !p RI x1 ; : : : ; xp RI y1 ; : : : ; yp 7KHYDULDQFHRI yj LV !j DQGWKHTXRWLHQW !j = WU.S/ PHDVXUHVWKHIUDFWLRQRIWKHWRWDO YDULDQFHWKDWLV´H[SODLQHGµRU´FDSWXUHGµE\ yj

&9".1-& &RPSXWHWKHYDULRXVSHUFHQWDJHVRIYDULDQFHRIWKH5DLOURDG9DOOH\ PXOWLVSHFWUDOGDWDWKDWDUHGLVSOD\HGLQWKHSULQFLSDOFRPSRQHQWSKRWRJUDSKV G ²I VKRZQLQWKHFKDSWHULQWURGXFWLRQ


40-65*0/ 7KHWRWDOYDULDQFHRIWKHGDWDLV WU.D/ D 7614:23 C 427:63 C 98:10 D 8139:96 >9HULI\WKDWWKLVQXPEHUDOVRHTXDOV WU.S/@ 7KHSHUFHQWDJHVRIWKHWRWDOYDULDQFH H[SODLQHGE\WKHSULQFLSDOFRPSRQHQWVDUH )LUVWFRPSRQHQW

6HFRQGFRPSRQHQW

7KLUGFRPSRQHQW

7614:23 D 93:5 8139:96

427:63 D 5:3 8139:96

98:10 D 1:2 8139:96

,QDVHQVH RIWKHLQIRUPDWLRQFROOHFWHGE\/DQGVDWIRUWKH5DLOURDG9DOOH\UHJLRQ LVGLVSOD\HGLQSKRWRJUDSKG ZLWKLQH DQGRQO\UHPDLQLQJIRUI 7KHFDOFXODWLRQVLQ([DPSOHVKRZWKDWWKHGDWDKDYHSUDFWLFDOO\QRYDULDQFHLQ WKHWKLUGQHZ FRRUGLQDWH 7KHYDOXHVRI y3 DUHDOOFORVHWR]HUR *HRPHWULFDOO\ WKH GDWDSRLQWVOLHQHDUO\LQWKHSODQH y3 D 0 DQGWKHLUORFDWLRQVFDQEHGHWHUPLQHGIDLUO\ DFFXUDWHO\E\NQRZLQJRQO\WKHYDOXHVRI y1 DQG y2 ,QIDFW y2 DOVRKDVUHODWLYHO\VPDOO YDULDQFH ZKLFKPHDQVWKDWWKHSRLQWVOLHDSSUR[LPDWHO\DORQJDOLQH DQGWKHGDWDDUH HVVHQWLDOO\RQHGLPHQVLRQDO 6HH)LJ LQZKLFKWKHGDWDUHVHPEOHDSRSVLFOHVWLFN

&KDUDFWHUL]DWLRQVRI3ULQFLSDO&RPSRQHQW9DULDEOHV ,I y1 ; : : : ; yp DULVHIURPDSULQFLSDOFRPSRQHQWDQDO\VLVRID p " N PDWUL[RIREVHU YDWLRQV WKHQWKHYDULDQFHRI y1 LVDVODUJHDVSRVVLEOHLQWKHIROORZLQJVHQVH ,I X LV DQ\XQLWYHFWRUDQGLI y D XT ; WKHQWKHYDULDQFHRIWKHYDOXHVRI y DV ; YDULHVRYHU WKHRULJLQDOGDWD ;1 ; : : : ; ;N WXUQVRXWWREH XT S X %\7KHRUHPLQ6HFWLRQ WKH PD[LPXPYDOXHRI XT S X RYHUDOOXQLWYHFWRUV X LVWKHODUJHVWHLJHQYDOXH !1 RI S DQG WKLVYDULDQFHLVDWWDLQHGZKHQ X LVWKHFRUUHVSRQGLQJHLJHQYHFWRU X1 ,QWKHVDPHZD\ 7KHRUHPVKRZVWKDW y2 KDVPD[LPXPSRVVLEOHYDULDQFHDPRQJDOOYDULDEOHV y D XT ; WKDWDUH XQFRUUHODWHG ZLWK y1 /LNHZLVH y3 KDVPD[LPXPSRVVLEOHYDULDQFHDPRQJDOO YDULDEOHVXQFRUUHODWHGZLWKERWK y1 DQG y2 DQGVRRQ

/6.&3*$"- /05& 7KHVLQJXODUYDOXHGHFRPSRVLWLRQLVWKHPDLQWRROIRUSHUIRUPLQJSULQFLSDOFRP SRQHQWDQDO\VLVLQSUDFWLFDODSSOLFDWLRQV ) p,I B LVD * p " N PDWUL[RIREVHUYDWLRQV LQPHDQGHYLDWLRQIRUP DQGLI A D 1= N ! 1 B T WKHQ ATA LVWKHFRYDULDQFH PDWUL[ S 7KHVTXDUHVRIWKHVLQJXODUYDOXHVRI A DUHWKH p HLJHQYDOXHVRI S DQGWKHULJKWVLQJXODUYHFWRUVRI A DUHWKHSULQFLSDOFRPSRQHQWVRIWKHGDWD $VPHQWLRQHGLQ6HFWLRQ LWHUDWLYHFDOFXODWLRQRIWKH69' RI A LVIDVWHU DQGPRUHDFFXUDWHWKDQDQHLJHQYDOXHGHFRPSRVLWLRQRI S 7KLVLVSDUWLFXODUO\ WUXH IRULQVWDQFH LQWKHK\SHUVSHFWUDOLPDJHSURFHVVLQJZLWK p D 224 PHQ WLRQHGLQWKHFKDSWHULQWURGXFWLRQ 3ULQFLSDOFRPSRQHQWDQDO\VLVLVFRPSOHWHGLQ VHFRQGVRQVSHFLDOL]HGZRUNVWDWLRQV

)XUWKHU5HDGLQJ /LOOHVDQG 7KRPDV0 DQG5DOSK:.LHIHU 5HPRWH6HQVLQJDQG,PDJH,QWHUSUHWDWLRQ WKHG 1HZ

$)"15&3


13"$5*$& 130#-&.4 7KHIROORZLQJWDEOHOLVWVWKHZHLJKWVDQGKHLJKWVRIÀYHER\V %R\

:HLJKWOE

+HLJKWLQ

)LQGWKHFRYDULDQFHPDWUL[IRUWKHGDWD 0DNHDSULQFLSDOFRPSRQHQWDQDO\VLVRIWKHGDWDWRÀQGDVLQJOH VL]HLQGH[ WKDW H[SODLQVPRVWRIWKHYDULDWLRQLQWKHGDWD

&9&3$*4&4 ,Q([HUFLVHVDQG FRQYHUWWKHPDWUL[RIREVHUYDWLRQVWRPHDQ GHYLDWLRQIRUP DQGFRQVWUXFWWKHVDPSOHFRYDULDQFHPDWUL[ ! " 19 22 6 3 2 20 12 6 9 15 13 5 ! " 1 5 2 6 7 3 3 11 6 8 15 11 )LQGWKHSULQFLSDOFRPSRQHQWVRIWKHGDWDIRU([HUFLVH )LQGWKHSULQFLSDOFRPSRQHQWVRIWKHGDWDIRU([HUFLVH >0@ $ /DQGVDWLPDJHZLWKWKUHHVSHFWUDOFRPSRQHQWVZDV PDGH RI +RPHVWHDG $LU )RUFH %DVH LQ )ORULGD DIWHU WKH EDVHZDVKLWE\KXUULFDQH$QGUHZLQ 7KHFRYDULDQFH PDWUL[RIWKHGDWDLVVKRZQEHORZ )LQGWKHÀUVWSULQFLSDO FRPSRQHQWRIWKHGDWD DQGFRPSXWHWKHSHUFHQWDJHRIWKH WRWDOYDULDQFHWKDWLVFRQWDLQHGLQWKLVFRPSRQHQW 2 3 164:12 32:73 81:04 539:44 249:13 5 S D 4 32:73 81:04 249:13 189:11

>0@ 7KH FRYDULDQFH PDWUL[ EHORZ ZDV REWDLQHG IURP D /DQGVDWLPDJHRIWKH&ROXPELD5LYHULQ:DVKLQJWRQ XVLQJ GDWDIURPWKUHHVSHFWUDOEDQGV /HW x1 x2 x3 GHQRWHWKH VSHFWUDO FRPSRQHQWV RI HDFK SL[HO LQ WKH LPDJH )LQG D QHZYDULDEOHRIWKHIRUP y1 D c1 x1 C c2 x2 C c3 x3 WKDWKDV PD[LPXPSRVVLEOHYDULDQFH VXEMHFWWRWKHFRQVWUDLQWWKDW c12 C c22 C c32 D 1 :KDWSHUFHQWDJHRIWKHWRWDOYDULDQFHLQ WKHGDWDLVH[SODLQHGE\ y1 " 2 3 29:64 18:38 5:00 20:82 14:06 5 S D 4 18:38 5:00 14:06 29:21 /HW x1 ; x2 GHQRWH WKH YDULDEOHV IRU WKH WZRGLPHQVLRQDO GDWDLQ ([HUFLVH )LQGDQHZ YDULDEOH y1 RIWKH IRUP y1 D c1 x1 C c2 x2 ZLWK c12 C c22 D 1 VXFKWKDW y1 KDVPD[ LPXPSRVVLEOHYDULDQFHRYHUWKHJLYHQGDWD +RZPXFKRI WKHYDULDQFHLQWKHGDWDLVH[SODLQHGE\ y1 " 5HSHDW([HUFLVHIRUWKHGDWDLQ([HUFLVH

6XSSRVHWKUHHWHVWVDUHDGPLQLVWHUHGWRDUDQGRPVDPSOH RIFROOHJHVWXGHQWV /HW ;1 ; : : : ; ;N EHREVHUYDWLRQYHFWRUV LQ R3 WKDW OLVW WKH WKUHH VFRUHV RI HDFK VWXGHQW DQG IRU j D 1; 2; 3 OHW xj GHQRWHDVWXGHQW·VVFRUHRQWKH j WKH[DP 6XSSRVHWKHFRYDULDQFHPDWUL[RIWKHGDWDLV 2 3 5 2 0 6 25 S D 42 0 2 7

/HW y EH DQ ´LQGH[µ RI VWXGHQW SHUIRUPDQFH ZLWK y D c1 x1 C c2 x2 C c3 x3 DQG c12 C c22 C c32 D 1 &KRRVH c1 ; c2 ; c3 VRWKDWWKHYDULDQFHRI y RYHUWKHGDWDVHWLVDVODUJHDV SRVVLEOH >+LQW 7KHHLJHQYDOXHVRIWKHVDPSOHFRYDULDQFH PDWUL[DUH ! D 3; 6 DQG@ 2 3 5 4 2 11 4 5 >0@ 5HSHDW([HUFLVHZLWK S D 4 4 2 4 5 *LYHQ PXOWLYDULDWH GDWD ;1 ; : : : ; ;N LQ Rp / LQ PHDQ GHYLDWLRQ IRUP OHW P EH D p " p PDWUL[ DQG GHÀQH +LQW /HW Z EHWKHYHFWRULQ RN ZLWKDLQHDFKHQWU\ 7KHQ Œ ;1 # # # ;N " Z D WKH]HURYHFWRULQ Rp /@ E 6KRZWKDWLIWKHFRYDULDQFHPDWUL[RI ;1 ; : : : ; ;N LV S WKHQWKHFRYDULDQFHPDWUL[RI <1 ; : : : ;
/HW ; GHQRWHDYHFWRUWKDWYDULHVRYHUWKHFROXPQVRID p " N PDWUL[RIREVHUYDWLRQV DQGOHW P EHD p " p RUWKRJRQDO PDWUL[ 6KRZWKDWWKHFKDQJHRIYDULDEOH ; D P < GRHVQRW FKDQJHWKHWRWDOYDULDQFHRIWKHGDWD >+LQW %\([HUFLVH LWVXIÀFHVWRVKRZWKDW WU .P T SP / D WU .S/ 8VHDSURSHUW\ RIWKHWUDFHPHQWLRQHGLQ([HUFLVHLQ6HFWLRQ@ 7KHVDPSOHFRYDULDQFHPDWUL[LVDJHQHUDOL]DWLRQRIDIRUPXOD IRUWKHYDULDQFHRIDVDPSOHRI N VFDODUPHDVXUHPHQWV VD\ t1 ; : : : ; tN ,I m LVWKHDYHUDJHRI t1 ; : : : ; tN WKHQWKH VDPSOH YDULDQFH LVJLYHQE\ n 1 X .tk ! m/2 N ! 1 kD1

.1/


1=.N ! 1/ WLPHVWKHVXPRI N PDWULFHVRIVL]H p " p )RU O k @ 1 % k % N ZULWH ;k ! 0 LQSODFHRI ;

6KRZKRZWKHVDPSOHFRYDULDQFHPDWUL[ S GHÀQHGSULRU WR ([DPSOH PD\ EH ZULWWHQ LQ D IRUP VLPLODU WR >+LQW 8VHSDUWLWLRQHGPDWUL[PXOWLSOLFDWLRQWRZULWH S DV

40-65*0/4 50 13"$5*$& 130#-&.4 )LUVWDUUDQJHWKHGDWDLQPHDQGHYLDWLRQIRUP 7KHVDPSOHPHDQYHFWRULVHDVLO\ ! " 130 VHHQWREH 0 D 6XEWUDFW 0 IURPWKHREVHUYDWLRQYHFWRUVWKHFROXPQVLQ 65 WKHWDEOH DQGREWDLQ ! " !10 !5 !5 5 15 BD !4 !5 !1 3 7 7KHQWKHVDPSOHFRYDULDQFHPDWUL[LV

2

3 !10 !4 ! "6 !5 !5 7 7 1 !10 !5 !5 5 15 6 6 !5 !1 7 SD 6 5 ! 1 !4 !5 !1 3 7 4 5 3 7 5 15 7 ! " ! " 1 400 190 100:0 47:5 D D 47:5 25:0 4 190 100

7KHHLJHQYDOXHVRI S DUHWRWZRGHFLPDOSODFHV

!1 D 123:02

!2 D 1:98 ! " :900 7KHXQLWHLJHQYHFWRUFRUUHVSRQGLQJWR !1 LV X D 6LQFH S LV 2 " 2 WKH :436 FRPSXWDWLRQVFDQEHGRQHE\KDQGLIDPDWUL[SURJUDPLVQRWDYDLODEOH )RUWKH VL]H LQGH[ VHW y D :900wO C :436hO DQG

ZKHUH wO DQG hO DUHZHLJKWDQGKHLJKW UHVSHFWLYHO\ LQPHDQGHYLDWLRQIRUP 7KH YDULDQFHRIWKLVLQGH[RYHUWKHGDWDVHWLV %HFDXVHWKHWRWDOYDULDQFHLV WU.S/ D 100 C 25 D 125 WKHVL]HLQGH[DFFRXQWVIRUSUDFWLFDOO\DOO RIWKH YDULDQFHRIWKHGDWD 7KHRULJLQDOGDWDIRU3UDFWLFH3UREOHPDQGWKHOLQHGHWHUPLQHGE\WKHÀUVW SULQFLSDOFRPSRQHQW X DUHVKRZQLQ)LJ ,QSDUDPHWULFYHFWRUIRUP WKHOLQHLV [ D 0 C t X ,WFDQEHVKRZQWKDWWKHOLQHLVWKHEHVWDSSUR[LPDWLRQWRWKHGDWD h 75 70 Inches

65 60 55 w 120

130

140

150

Pounds '*(63& $QRUWKRJRQDOUHJUHVVLRQOLQHGHWHUPLQHGE\WKH

ÀUVWSULQFLSDOFRPSRQHQWRIWKHGDWD

$)"15&3


LQWKHVHQVHWKDWWKHVXPRIWKHVTXDUHVRIWKH RUWKRJRQDO GLVWDQFHVWRWKHOLQHLV PLQLPL]HG ,QIDFW SULQFLSDOFRPSRQHQWDQDO\VLVLVHTXLYDOHQWWRZKDWLVWHUPHG RUWKRJRQDOUHJUHVVLRQ EXWWKDWLVDVWRU\IRUDQRWKHUGD\ 3HUKDSVZH·OOPHHWDJDLQ

$)"15&3 4611-&.&/5"3: &9&3$*4&4 0DUNHDFKVWDWHPHQW7UXHRU)DOVH -XVWLI\HDFKDQVZHU ,Q HDFKSDUW A UHSUHVHQWVDQ n " n PDWUL[ D ,I A LVRUWKRJRQDOO\GLDJRQDOL]DEOH WKHQ A LVV\PPHWULF E ,I A LVDQRUWKRJRQDOPDWUL[ WKHQ A LVV\PPHWULF F ,I A LVDQRUWKRJRQDOPDWUL[ WKHQ kA[k D k[k IRUDOO [ LQ Rn G 7KHSULQFLSDOD[HVRIDTXDGUDWLFIRUP [TA[ FDQEHWKH FROXPQVRIDQ\PDWUL[ P WKDWGLDJRQDOL]HV A H ,I P LVDQ n " n PDWUL[ZLWKRUWKRJRQDOFROXPQV WKHQ P T D P !1

I ,IHYHU\FRHIÀFLHQWLQDTXDGUDWLFIRUPLVSRVLWLYH WKHQ WKHTXDGUDWLFIRUPLVSRVLWLYHGHÀQLWH J ,I [TA[ > 0 IRUVRPH [ WKHQWKHTXDGUDWLFIRUP [TA[ LV SRVLWLYHGHÀQLWH K %\DVXLWDEOHFKDQJHRIYDULDEOH DQ\TXDGUDWLFIRUPFDQ EHFKDQJHGLQWRRQHZLWKQRFURVVSURGXFWWHUP L 7KHODUJHVWYDOXHRIDTXDGUDWLFIRUP [ A[ IRU k[k D 1 LVWKHODUJHVWHQWU\RQWKHGLDJRQDORI A T

M 7KHPD[LPXPYDOXHRIDSRVLWLYHGHÀQLWHTXDGUDWLFIRUP [TA[ LVWKHJUHDWHVWHLJHQYDOXHRI A N $ SRVLWLYHGHÀQLWHTXDGUDWLFIRUPFDQEHFKDQJHGLQWR DQHJDWLYHGHÀQLWHIRUPE\DVXLWDEOHFKDQJHRIYDULDEOH [ D P X IRUVRPHRUWKRJRQDOPDWUL[ P

O $QLQGHÀQLWHTXDGUDWLFIRUPLVRQHZKRVHHLJHQYDOXHV DUHQRWGHÀQLWH

P ,I P LVDQ n " n RUWKRJRQDOPDWUL[ WKHQWKHFKDQJHRI YDULDEOH [ D P X WUDQVIRUPV [TA[ LQWRDTXDGUDWLFIRUP ZKRVHPDWUL[LV P !1 $3 Q ,I U LV m " n ZLWKRUWKRJRQDOFROXPQV WKHQ U U T [ LV WKHRUWKRJRQDOSURMHFWLRQRI [ RQWR &RO U R ,I B LV m " n DQG [ LVDXQLWYHFWRULQ Rn WKHQ kB [k % #1 ZKHUH #1 LVWKHÀUVWVLQJXODUYDOXHRI B S $ VLQJXODUYDOXHGHFRPSRVLWLRQRIDQ m " n PDWUL[ B FDQ EH ZULWWHQ DV B D P †Q ZKHUH P LV DQ m " m RUWKRJRQDOPDWUL[ Q LVDQ n " n RUWKRJRQDOPDWUL[ DQG † LVDQ m " n ´GLDJRQDOµPDWUL[ T ,I A LV n " n WKHQ A DQG ATA KDYHWKHVDPHVLQJXODU YDOXHV

/HW fX1 ; : : : ; Xn g EHDQRUWKRQRUPDOEDVLVIRU Rn DQGOHW !1 ; : : : ; !n EHDQ\UHDOVFDODUV 'HÀQH

A D !1 X1 XT1 C # # # C !n Xn XTn D 6KRZWKDW A LVV\PPHWULF

E 6KRZWKDW !1 ; : : : ; !n DUHWKHHLJHQYDOXHVRI A /HW A EHDQ n " n V\PPHWULFPDWUL[RIUDQN r ([SODLQZK\ WKHVSHFWUDOGHFRPSRVLWLRQRI A UHSUHVHQWV A DVWKHVXPRI r UDQN 1 PDWULFHV /HW A EHDQ n " n V\PPHWULFPDWUL[ D 6KRZWKDW .&RO A/? D 1XO A >+LQW 6HH6HFWLRQ@

E 6KRZWKDWHDFK \ LQ Rn FDQEHZULWWHQLQWKHIRUP \ D \O C ] ZLWK \O LQ &RO A DQG ] LQ 1XO A

6KRZWKDWLI Y LVDQHLJHQYHFWRURIDQ n " n PDWUL[ A DQG Y FRUUHVSRQGVWRDQRQ]HURHLJHQYDOXHRI A WKHQ Y LVLQ &RO A >+LQW 8VHWKHGHÀQLWLRQRIDQHLJHQYHFWRU@ /HW A EHDQ n " n V\PPHWULFPDWUL[ 8VH([HUFLVHDQG DQHLJHQYHFWRUEDVLVIRU Rn WRJLYHDVHFRQGSURRIRIWKH GHFRPSRVLWLRQLQ([HUFLVHE 3URYHWKDWDQ n " n PDWUL[ A LVSRVLWLYHGHÀQLWHLIDQGRQO\ LI A DGPLWVD &KROHVN\IDFWRUL]DWLRQ QDPHO\ A D RTR IRU VRPHLQYHUWLEOHXSSHUWULDQJXODUPDWUL[ R ZKRVHGLDJRQDO HQWULHVDUHDOOSRVLWLYH >+LQW 8VHD45 IDFWRUL]DWLRQDQG ([HUFLVHLQ6HFWLRQ@ 8VH([HUFLVHWRVKRZWKDWLI A LVSRVLWLYHGHÀQLWH WKHQ A KDVDQ/8 IDFWRUL]DWLRQ A D LU ZKHUH U KDVSRVLWLYH SLYRWVRQLWVGLDJRQDO 7KHFRQYHUVHLVWUXH WRR ,I A LV m " n WKHQWKHPDWUL[ G D ATA LVFDOOHGWKH *UDPPDWUL[ RI A ,QWKLVFDVH WKHHQWULHVRI G DUHWKHLQQHUSURGXFWVRIWKH FROXPQVRI A 6HH([HUFLVHVDQG 6KRZ WKDW WKH *UDP PDWUL[ RI DQ\ PDWUL[ A LV SRVLWLYH VHPLGHÀQLWH ZLWKWKHVDPHUDQNDV A 6HHWKH([HUFLVHV LQ6HFWLRQ 6KRZWKDWLIDQ n " n PDWUL[ G LVSRVLWLYHVHPLGHÀQLWHDQG KDV UDQN r WKHQ G LVWKH*UDPPDWUL[RIVRPH r " n PDWUL[ A 7KLVLVFDOOHGD UDQNUHYHDOLQJIDFWRUL]DWLRQ RI G >+LQW &RQVLGHUWKHVSHFWUDOGHFRPSRVLWLRQRI G DQGÀUVWZULWH G DV BB T IRUDQ n " r PDWUL[ B @ 3URYHWKDWDQ\ n " n PDWUL[ A DGPLWVD SRODUGHFRPSRVLWLRQ RIWKHIRUP A D PQ ZKHUH P LVDQ n " n SRVLWLYHVHPLGHÀ QLWHPDWUL[ZLWKWKHVDPHUDQNDV A DQGZKHUH Q LVDQ n " n RUWKRJRQDOPDWUL[ >+LQW 8VHDVLQJXODUYDOXHGHFRPSRVL WLRQ A D U †V T DQGREVHUYHWKDW A D .U †U T /.UV T /@ 7KLVGHFRPSRVLWLRQLVXVHG IRULQVWDQFH LQPHFKDQLFDOHQ JLQHHULQJWRPRGHOWKHGHIRUPDWLRQRIDPDWHULDO 7KHPDWUL[ P GHVFULEHVWKHVWUHWFKLQJRUFRPSUHVVLRQRIWKHPDWHULDOLQ WKHGLUHFWLRQVRIWKHHLJHQYHFWRUVRI P DQG Q GHVFULEHVWKH URWDWLRQRIWKHPDWHULDOLQVSDFH


([HUFLVHV²FRQFHUQDQ m " n PDWUL[ A ZLWKDUHGXFHGVLQ JXODUYDOXHGHFRPSRVLWLRQ A D Ur DVrT DQGWKHSVHXGRLQYHUVH AC D Vr D !1 UrT

9HULI\WKHSURSHUWLHVRI AC D )RUHDFK \ LQ Rm AAC \ LVWKHRUWKRJRQDOSURMHFWLRQRI \ RQWR &RO A E )RUHDFK [ LQ Rn AC A[ LVWKHRUWKRJRQDOSURMHFWLRQRI [ RQWR 5RZ A F AAC A D A DQG AC AAC D AC

6XSSRVH WKH HTXDWLRQ A[ D E LV FRQVLVWHQW DQG OHW [C D AC E %\([HUFLVHLQ6HFWLRQ WKHUHLVH[DFWO\ RQHYHFWRU S LQ 5RZ A VXFKWKDW AS D E 7KHIROORZLQJ VWHSV SURYH WKDW [C D S DQG [C LV WKH PLQLPXP OHQJWK VROXWLRQ RI A[ D E D 6KRZWKDW [C LVLQ 5RZ A >+LQW :ULWH E DV A[ IRUVRPH [ DQGXVH([HUFLVH@ E 6KRZWKDW [C LVDVROXWLRQRI A[ D E

F 6KRZ WKDW LI X LV DQ\ VROXWLRQ RI A[ D E WKHQ k[C k % kXk ZLWKHTXDOLW\RQO\LI X D [C

*LYHQDQ\ E LQ Rm DGDSW([HUFLVHWRVKRZWKDW AC E LVWKH OHDVWVTXDUHVVROXWLRQRIPLQLPXPOHQJWK >+LQW &RQVLGHU O ZKHUH EO LVWKHRUWKRJRQDOSURMHFWLRQRI WKHHTXDWLRQ A[ D E E RQWR &RO A@ >0@ ,Q([HUFLVHVDQG FRQVWUXFWWKHSVHXGRLQYHUVHRI A %H JLQE\XVLQJDPDWUL[SURJUDPWRSURGXFHWKH69' RI A RU LIWKDW LVQRWDYDLODEOH EHJLQZLWKDQRUWKRJRQDOGLDJRQDOL]DWLRQRI ATA 8VHWKHSVHXGRLQYHUVHWRVROYH A[ D E IRU E D .6; !1; !4; 6/ DQGOHW [O EHWKHVROXWLRQ 0DNHDFDOFXODWLRQWRYHULI\WKDW [O LVLQ 5RZ A )LQGDQRQ]HURYHFWRU X LQ 1XO A DQGYHULI\WKDW kO[k < kO[ C Xk ZKLFKPXVWEHWUXHE\([HUFLVH F 2 3 !3 !3 !6 6 1 6 !1 !1 !1 1 !2 7 7 A D 6 4 0 0 !1 1 !1 5 0 0 !1 1 !1 2

4 6 !5 A D 6 4 2 6

0 !1 !2 0 3 5 0 !1 !2 0 !3 !6

3 0 07 7 05 0


7KH*HRPHWU\RI 9HFWRU6SDFHV

*/530%6$503: &9".1-&

7KH3ODWRQLF6ROLGV ,QWKHFLW\RI$WKHQVLQ %& WKH*UHHNSKLORVRSKHU 3ODWRIRXQGHGDQ$FDGHP\ VRPHWLPHVUHIHUUHGWRDVWKH ZRUOG·VÀUVWXQLYHUVLW\ :KLOHWKHFXUULFXOXPLQFOXGHG DVWURQRP\ ELRORJ\ SROLWLFDOWKHRU\ DQGSKLORVRSK\ WKH VXEMHFWFORVHVWWRKLVKHDUWZDVJHRPHWU\ ,QGHHG LQVFULEHG RYHUWKHGRRUVRIKLVDFDGHP\ZHUHWKHVHZRUGV ´/HWQR RQHGHVWLWXWHRIJHRPHWU\HQWHUP\GRRUVµ 7KH*UHHNVZHUHJUHDWO\LPSUHVVHGE\JHRPHWULF SDWWHUQVVXFKDVWKHUHJXODUVROLGV $ SRO\KHGURQLVFDOOHG UHJXODULILWVIDFHVDUHFRQJUXHQWUHJXODUSRO\JRQVDQGDOO WKHDQJOHVDWWKHYHUWLFHVDUHHTXDO $VHDUO\DV\HDUV EHIRUH(XFOLG WKH3\WKDJRUHDQVNQHZDWOHDVWWKUHHRIWKH UHJXODUVROLGV WKHWHWUDKHGURQWULDQJXODUIDFHV WKHFXEH VTXDUHIDFHV DQGWKHRFWDKHGURQWULDQJXODUIDFHV 6HH)LJ 7KHVHVKDSHVRFFXUQDWXUDOO\DVFU\VWDOVRI FRPPRQPLQHUDOV 7KHUHDUHRQO\ÀYHVXFKUHJXODUVROLGV WKHUHPDLQLQJWZREHLQJWKHGRGHFDKHGURQSHQWDJRQDO IDFHV DQGWKHLFRVDKHGURQWULDQJXODUIDFHV 3ODWRGLVFXVVHGWKHEDVLFWKHRU\RIWKHVHÀYHVROLGV LQ%RRN;,,, RIKLV (OHPHQWV DQGVLQFHWKHQWKH\KDYH FDUULHGKLVQDPH WKH3ODWRQLFVROLGV )RUFHQWXULHVWKHUHZDVQRQHHGWRHQYLVLRQJHRPHWULF REMHFWVLQPRUHWKDQWKUHHGLPHQVLRQV %XWQRZDGD\V PDWKHPDWLFLDQVUHJXODUO\GHDOZLWKREMHFWVLQYHFWRUVSDFHV

KDYLQJIRXU ÀYH RUHYHQKXQGUHGVRIGLPHQVLRQV ,WLVQRW QHFHVVDULO\FOHDUZKDWJHRPHWULFDOSURSHUWLHVRQHPLJKW DVFULEHWRWKHVHREMHFWVLQKLJKHUGLPHQVLRQV )RUH[DPSOH ZKDWSURSHUWLHVGROLQHVKDYHLQ VSDFHDQGSODQHVKDYHLQVSDFHWKDWZRXOGEHXVHIXO LQKLJKHUGLPHQVLRQV" +RZFDQRQHFKDUDFWHUL]HVXFK REMHFWV" 6HFWLRQVDQGSURYLGHVRPHDQVZHUV 7KHK\SHUSODQHVRI6HFWLRQZLOOEHLPSRUWDQWIRU XQGHUVWDQGLQJWKHPXOWLGLPHQVLRQDOQDWXUHRIWKHOLQHDU SURJUDPPLQJSUREOHPVLQ&KDSWHU :KDWZRXOGWKHDQDORJXHRIDSRO\KHGURQÓRRN OLNHµLQPRUHWKDQWKUHHGLPHQVLRQV" $ SDUWLDODQVZHU LVSURYLGHGE\WZRGLPHQVLRQDOSURMHFWLRQVRIWKHIRXU GLPHQVLRQDOREMHFW FUHDWHGLQDPDQQHUDQDORJRXVWRWZR GLPHQVLRQDOSURMHFWLRQVRIDWKUHHGLPHQVLRQDOREMHFW 6HFWLRQLOOXVWUDWHVWKLVLGHDIRUWKHIRXUGLPHQVLRQDO ´FXEHµDQGWKHIRXUGLPHQVLRQDO´VLPSOH[µ 7KHVWXG\RIJHRPHWU\LQKLJKHUGLPHQVLRQVQRW RQO\SURYLGHVQHZZD\VRIYLVXDOL]LQJDEVWUDFWDOJHEUDLF FRQFHSWV EXWDOVRFUHDWHVWRROVWKDWPD\EHDSSOLHGLQ R3 )RULQVWDQFH 6HFWLRQVDQGLQFOXGHDSSOLFDWLRQVWR FRPSXWHUJUDSKLFV DQG6HFWLRQRXWOLQHVDSURRILQ ([HUFLVH WKDWWKHUHDUHRQO\ÀYHUHJXODUSRO\KHGUDLQ R3

$)"15&3

7KH*HRPHWU\RI9HFWRU6SDFHV

'*(63& 7KHÀYH3ODWRQLFVROLGV

0RVWDSSOLFDWLRQVLQHDUOLHUFKDSWHUVLQYROYHGDOJHEUDLFFDOFXODWLRQVZLWKVXEVSDFHV DQGOLQHDUFRPELQDWLRQVRIYHFWRUV 7KLVFKDSWHUVWXGLHVVHWVRIYHFWRUVWKDWFDQEH YLVXDOL]HGDVJHRPHWULFREMHFWVVXFKDVOLQHVHJPHQWV SRO\JRQV DQGVROLGREMHFWV ,QGLYLGXDOYHFWRUVDUHYLHZHGDVSRLQWV 7KHFRQFHSWVLQWURGXFHGKHUHDUHXVHGLQ FRPSXWHUJUDSKLFV OLQHDUSURJUDPPLQJLQ&KDSWHU DQGRWKHUDUHDVRIPDWKHPDWLFVñ 7KURXJKRXWWKHFKDSWHU VHWVRIYHFWRUVDUHGHVFULEHGE\OLQHDUFRPELQDWLRQV EXW ZLWKYDULRXVUHVWULFWLRQVRQWKHZHLJKWVXVHGLQWKHFRPELQDWLRQV )RULQVWDQFH LQ 6HFWLRQ WKHVXPRIWKHZHLJKWVLV ZKLOHLQ6HFWLRQ WKHZHLJKWVDUHSRVLWLYH DQGVXPWR 7KHYLVXDOL]DWLRQVDUHLQ R2 RU R3 RIFRXUVH EXWWKHFRQFHSWVDOVRDSSO\ WR Rn DQGRWKHUYHFWRUVSDFHV

"''*/& $0.#*/"5*0/4 $QDIÀQHFRPELQDWLRQRIYHFWRUVLVDVSHFLDONLQGRIOLQHDUFRPELQDWLRQ *LYHQYHFWRUV RU´SRLQWVµ Y1 ; Y2 ; : : : ; Yp LQ Rn DQGVFDODUV c1 ; : : : ; cp DQ DIÀQHFRPELQDWLRQ RI Y1 ; Y2 ; : : : ; Yp LVDOLQHDUFRPELQDWLRQ

c1 Y1 C ! ! ! C cp Yp VXFKWKDWWKHZHLJKWVVDWLVI\ c1 C ! ! ! C cp D 1 ñ 6HH)ROH\ YDQ'DP )HLQHU DQG+XJKHV &RPSXWHU*UDSKLFV³3ULQFLSOHVDQG3UDFWLFH QGHGLWLRQ %RVWRQ $GGLVRQ:HVOH\ SS² 7KDWPDWHULDODOVRGLVFXVVHVFRRUGLQDWHIUHH´DIÀQH VSDFHVµ

%&'*/*5*0/

$IILQH&RPELQDWLRQV

7KHVHWRIDOODIÀQHFRPELQDWLRQVRISRLQWVLQDVHW S LVFDOOHGWKH DIÀQHKXOO RU DIÀQHVSDQ RI S GHQRWHGE\ DII S

7KHDIÀQHKXOORIDVLQJOHSRLQW Y1 LVMXVWWKHVHW fY1 g VLQFHLWKDVWKHIRUP c1 Y1 ZKHUH c1 D 1 7KHDIÀQHKXOORIWZRGLVWLQFWSRLQWVLVRIWHQZULWWHQLQDVSHFLDOZD\ 6XSSRVH \ D c1 Y1 C c2 Y2 ZLWK c1 C c2 D 1 :ULWH t LQSODFHRI c2 VRWKDW c1 D 1 " c2 D 1 " t 7KHQWKHDIÀQHKXOORI fY1 ; Y2 g LVWKHVHW \ D .1 " t /Y1 C t Y2 ;

ZLWK t LQ R

7KLVVHWRISRLQWVLQFOXGHV Y1 ZKHQ t D 0 DQG Y2 ZKHQ t D 1 ,I Y2 D Y1 WKHQ DJDLQGHVFULEHVMXVWRQHSRLQW 2WKHUZLVH GHVFULEHVWKH OLQH WKURXJK Y1 DQG Y2 7R VHHWKLV UHZULWH LQWKHIRUP \ D Y 1 C t .Y 2 " Y 1 / D S C t X ;

ZLWK t LQ R

ZKHUH S LV Y1 DQG X LV Y2 " Y1 7KHVHWRIDOOPXOWLSOHVRI X LV 6SDQ fXg WKHOLQHWKURXJK X DQGWKHRULJLQ $GGLQJ S WRHDFKSRLQWRQWKLVOLQHWUDQVODWHV 6SDQ fXg LQWRWKHOLQH WKURXJK S SDUDOOHOWRWKHOLQHWKURXJK X DQGWKHRULJLQ 6HH)LJ &RPSDUHWKLVÀJXUH ZLWK)LJ LQ6HFWLRQ

p + tu p

tu u

'*(63&

)LJXUHXVHVWKHRULJLQDOSRLQWV Y1 DQG Y2 DQGGLVSOD\V DII fY1 ; Y2 g DVWKHOLQH WKURXJK Y1 DQG Y2

y = v 1 + t(v 2 – v 1) aff{v 1 , v 2}

v1

v2

t(v 2 – v 1)

v2 – v1

'*(63&

1RWLFHWKDWZKLOHWKHSRLQW \ LQ)LJ LVDQDIÀQHFRPELQDWLRQRI Y1 DQG Y2 WKHSRLQW \ " Y1 HTXDOV t.Y2 " Y1 / ZKLFKLVDOLQHDUFRPELQDWLRQLQIDFW DPXOWLSOH RI Y2 " Y1 7KLVUHODWLRQEHWZHHQ \ DQG \ " Y1 KROGVIRUDQ\DIÀQHFRPELQDWLRQRISRLQWV DVWKH IROORZLQJWKHRUHPVKRZV

5)&03&.

$ SRLQW \ LQ Rn LVDQDIÀQHFRPELQDWLRQRI Y1 ; : : : ; Yp LQ Rn LIDQGRQO\LI \ " Y1 LVDOLQHDUFRPELQDWLRQRIWKHWUDQVODWHGSRLQWV Y2 " Y1 ; : : : ; Yp " Y1 :

$)"15&3


1300' ,I \ " Y1 LVDOLQHDUFRPELQDWLRQRI Y2 " Y1 ; : : : ; Yp " Y1 ; WKHUHH[LVWZHLJKWV c2 ; : : : ; cp VXFKWKDW \ " Y1 D c2 .Y2 " Y1 / C ! ! ! C cp .Yp " Y1 /

\ D .1 " c2 " ! ! ! " cp /Y1 C c2 Y2 C ! ! ! C cp Yp

7KHQ DQGWKHZHLJKWVLQWKLVOLQHDUFRPELQDWLRQVXPWR 6R \ LVDQDIÀQHFRPELQDWLRQRI Y1 ; : : : ; Yp &RQYHUVHO\ VXSSRVH \ D c1 Y1 C c2 Y2 C ! ! ! C cp Yp

ZKHUH c1 C ! ! ! C cp D 1 6LQFH c1 D 1 " c2 " ! ! ! " cp HTXDWLRQ PD\EHZULWWHQ DVLQ DQGWKLVOHDGVWR ZKLFKVKRZVWKDW \ " Y1 LVDOLQHDUFRPELQDWLRQRI Y 2 " Y1 ; : : : ; Y p " Y 1 : ,QWKHVWDWHPHQWRI7KHRUHP WKHSRLQW Y1 FRXOGEHUHSODFHGE\DQ\RIWKHRWKHU SRLQWVLQWKHOLVW Y1 ; : : : ; Yp : 2QO\WKHQRWDWLRQLQWKHSURRIZRXOGFKDQJH ! " ! " ! " ! " ! " 1 2 1 "2 4 &9".1-& /HW Y1 D Y2 D Y3 D Y4 D DQG \ D 2 5 3 2 1 ,ISRVVLEOH ZULWH \ DVDQDIÀQHFRPELQDWLRQRI Y1 ; Y2 ; Y3 DQG Y4

40-65*0/ &RPSXWHWKHWUDQVODWHGSRLQWV ! " ! " ! " 1 0 "3 Y2 " Y 1 D ; Y3 " Y1 D ; Y 4 " Y1 D ; 3 1 0

\ " Y1 D

!

3 "1

"

7RÀQGVFDODUV c2 c3 DQG c4 VXFKWKDW

c2 .Y2 " Y1 / C c3 .Y3 " Y1 / C c4 .Y4 " Y1 / D \ " Y1

URZUHGXFHWKHDXJPHQWHGPDWUL[KDYLQJWKHVHSRLQWVDVFROXPQV ! " ! " 1 0 "3 3 1 0 "3 3 # 3 1 0 "1 0 1 9 "10 7KLVVKRZVWKDWHTXDWLRQ LVFRQVLVWHQW DQGWKHJHQHUDOVROXWLRQLV c2 D 3c4 C 3 c3 D "9c4 " 10 ZLWK c4 IUHH :KHQ c4 D 0 \ " Y1 D 3.Y2 " Y1 / " 10.Y3 " Y1 / C 0.Y4 " Y1 / DQG \ D 8Y1 C 3Y2 " 10Y3 $VDQRWKHUH[DPSOH WDNH c4 D 1 7KHQ c2 D 6 DQG c3 D "19 VR \ " Y1 D 6.Y2 " Y1 / " 19.Y3 " Y1 / C 1.Y4 " Y1 / DQG \ D 13Y1 C 6Y2 " 19Y3 C Y4 :KLOHWKHSURFHGXUHLQ([DPSOHZRUNVIRUDUELWUDU\SRLQWV Y1 ; Y2 ; : : : ; Yp LQ Rn WKHTXHVWLRQFDQEHDQVZHUHGPRUHGLUHFWO\LIWKHFKRVHQSRLQWV Yi DUHDEDVLVIRU Rn )RUH[DPSOH OHW B D fE1 ; : : : ; En g EHVXFKDEDVLV 7KHQDQ\ \ LQ Rn LVDXQLTXH OLQHDU FRPELQDWLRQRI E1 ; : : : ; En 7KLVFRPELQDWLRQLVDQDIÀQHFRPELQDWLRQRIWKH E·VLIDQG RQO\LIWKHZHLJKWVVXPWR 7KHVHZHLJKWVDUHMXVWWKH B FRRUGLQDWHVRI \ DVLQ6HFWLRQ

$IILQH&RPELQDWLRQV

2 3 2 3 2 3 2 3 2 3 4 0 5 2 1 &9".1-& /HW E1 D 4 0 5 E2 D 4 4 5 E3 D 4 2 5 S1 D 4 0 5 DQG S2 D 4 2 5 3 2 4 0 2 7KHVHW B D fE1 ; E2 ; E3 g LVDEDVLVIRU R3 'HWHUPLQHZKHWKHUWKHSRLQWV S1 DQG S2 DUH DIÀQHFRPELQDWLRQVRIWKHSRLQWVLQ B

40-65*0/ )LQGWKH B FRRUGLQDWHVRI S1 DQG S2 ELQHGE\URZUHGXFLQJWKHPDWUL[ Œ E1 E2 E3 S1 2 2 3 1 4 0 5 2 1 6 6 40 5 4 2 0 2 # 60 4 3 2 4 0 2 0

7KHVHWZRFDOFXODWLRQVFDQEHFRP S2 ! ZLWKWZRDXJPHQWHGFROXPQV 3 2 0 0 "2 3 7 2 7 7 1 0 "1 3 5 0 1 2 " 13

5HDGFROXPQWREXLOG S1 DQGUHDGFROXPQWREXLOG S2 S1 D "2E1 " E2 C 2E3

DQG

S2 D 23 E1 C 23 E2 " 13 E3

7KHVXPRIWKHZHLJKWVLQWKHOLQHDUFRPELQDWLRQIRU S1 LV "1 QRW 1 VR S1 LV QRW DQ DIÀQHFRPELQDWLRQRIWKH E·V +RZHYHU S2 LV DQDIÀQHFRPELQDWLRQRIWKH E·V EHFDXVH WKHVXPRIWKHZHLJKWVIRU S2 LV 1

%&'*/*5*0/

$ VHW S LV DIÀQH LI S; T 2 S LPSOLHVWKDW .1 " t/S C t T 2 S IRUHDFKUHDOQXPEHU t *HRPHWULFDOO\ DVHWLVDIÀQHLIZKHQHYHUWZRSRLQWVDUHLQWKHVHW WKHHQWLUHOLQH WKURXJKWKHVHSRLQWVLVLQWKHVHW ,I S FRQWDLQVRQO\RQHSRLQW S WKHQWKHOLQHWKURXJK S DQG S LVMXVWDSRLQW D´GHJHQHUDWHµOLQH $OJHEUDLFDOO\ IRUDVHW S WREHDIÀQH WKHGHÀQLWLRQUHTXLUHVWKDWHYHU\DIÀQHFRPELQDWLRQRIWZRSRLQWVRI S EHORQJWR S 5HPDUNDEO\ WKLVLVHTXLYDOHQWWRUHTXLULQJWKDW S FRQWDLQHYHU\DIÀQHFRPELQDWLRQRI DQDUELWUDU\QXPEHURISRLQWVRI S

5)&03&.

$ VHW S LVDIÀQHLIDQGRQO\LIHYHU\DIÀQHFRPELQDWLRQRISRLQWVRI S OLHVLQ S 7KDWLV S LVDIÀQHLIDQGRQO\LI S D DII S

1300' 6XSSRVHWKDW S LVDIÀQHDQGXVHLQGXFWLRQRQWKHQXPEHU m RISRLQWVRI S RFFXUULQJLQDQDIÀQHFRPELQDWLRQ :KHQ m LVRU DQDIÀQHFRPELQDWLRQRI m SRLQWV RI S OLHVLQ S E\WKHGHÀQLWLRQRIDQDIÀQHVHW 1RZ DVVXPHWKDWHYHU\DIÀQHFRPELQD WLRQRI k RUIHZHUSRLQWVRI S \LHOGVDSRLQWLQ S DQGFRQVLGHUDFRPELQDWLRQRI k C 1 SRLQWV 7DNH Yi LQ S IRU i D 1; : : : ; k C 1 DQGOHW \ D c1 Y1 C ! ! ! C ck Yk C ck C1 Yk C1 ZKHUH c1 C ! ! ! C ck C1 D 1 6LQFHWKH ci ·VVXPWR DWOHDVWRQHRIWKHPPXVWQRWEH HTXDOWR %\UHLQGH[LQJWKH Yi DQG ci LIQHFHVVDU\ ZHPD\DVVXPHWKDW ck C1 ¤ 1 /HW t D c1 C ! ! ! C ck 7KHQ t D 1 " ck C1 ¤ 0 DQG #c ck $ 1 \ D .1 " ck C1 / Y1 C ! ! ! C Yk C ck C1 Yk C1 t t %\WKHLQGXFWLRQK\SRWKHVLV WKHSRLQW ] D .c1 =t /Y1 C ! ! ! C .ck =t /Yk LVLQ S VLQFHWKH FRHIÀFLHQWVVXPWR 7KXV GLVSOD\V \ DVDQDIÀQHFRPELQDWLRQRIWZRSRLQWVLQ S DQGVR \ 2 S %\WKHSULQFLSOHRILQGXFWLRQ HYHU\DIÀQHFRPELQDWLRQRIVXFKSRLQWVOLHV LQ S 7KDWLV DII S $ S %XWWKHUHYHUVHLQFOXVLRQ S $ DII S DOZD\VDSSOLHV 7KXV ZKHQ S LVDIÀQH S D DII S &RQYHUVHO\ LI S D DII S WKHQDIÀQHFRPELQDWLRQVRIWZR RUPRUH SRLQWVRI S OLHLQ S VR S LVDIÀQH

$)"15&3


7KHQH[WGHÀQLWLRQSURYLGHVWHUPLQRORJ\IRUDIÀQHVHWVWKDWHPSKDVL]HVWKHLUFORVH FRQQHFWLRQZLWKVXEVSDFHVRI Rn

%&'*/*5*0/

$ WUDQVODWHRIDVHW S LQ Rn E\DYHFWRU S LVWKHVHW S C S D fV C S W V 2 S g $ ÁDW LQ Rn LVDWUDQVODWHRIDVXEVSDFHRI Rn 7ZRÁDWVDUH SDUDOOHO LIRQHLVDWUDQVODWH RIWKHRWKHU 7KH GLPHQVLRQRIDÁDW LVWKHGLPHQVLRQRIWKHFRUUHVSRQGLQJSDUDOOHO VXEVSDFH 7KH GLPHQVLRQRIDVHW S ZULWWHQDVGLP S LVWKHGLPHQVLRQRIWKH VPDOOHVWÁDWFRQWDLQLQJ S $ OLQH LQ Rn LVDÁDWRIGLPHQVLRQ $ K\SHUSODQH LQ Rn LVDÁDWRIGLPHQVLRQ n " 1 ,Q R3 WKHSURSHUVXEVSDFHVó FRQVLVWRIWKHRULJLQ WKHVHWRIDOOOLQHVWKURXJK DQGWKHVHWRIDOOSODQHVWKURXJK 7KXVWKHSURSHUÁDWVLQ R3 DUHSRLQWV]HUR GLPHQVLRQDO OLQHVRQHGLPHQVLRQDO DQGSODQHVWZRGLPHQVLRQDO ZKLFKPD\RU PD\QRWSDVVWKURXJKWKHRULJLQ 7KHQH[WWKHRUHPVKRZVWKDWWKHVHJHRPHWULFGHVFULSWLRQVRIOLQHVDQGSODQHVLQ R3 DVWUDQVODWHVRIVXEVSDFHV DFWXDOO\FRLQFLGHZLWKWKHLUHDUOLHUDOJHEUDLFGHVFULSWLRQVDV VHWVRIDOODIÀQHFRPELQDWLRQVRIWZRRUWKUHHSRLQWV UHVSHFWLYHO\

5)&03&.

$ QRQHPSW\VHW S LVDIÀQHLIDQGRQO\LILWLVDÁDW

1300' 6XSSRVHWKDW S LVDIÀQH /HW S EHDQ\À[HGSRLQWLQ S DQGOHW W D S C ."S/ VRWKDW S D W C S 7RVKRZWKDW S LVDÁDW LWVXIÀFHVWRVKRZWKDW W LVDVXEVSDFHRI Rn 6LQFH S LVLQ S WKH]HURYHFWRULVLQ W 7RVKRZWKDW W LVFORVHGXQGHUVXPVDQG VFDODUPXOWLSOHV LWVXIÀFHVWRVKRZWKDWLI X1 DQG X2 DUHHOHPHQWVRI W WKHQ X1 C t X2 LVLQ W IRUHYHU\UHDO t 6LQFH X1 DQG X2 DUHLQ W WKHUHH[LVW V1 DQG V2 LQ S VXFKWKDW X1 D V1 " S DQG X2 D V2 " S 6R IRUHDFKUHDO t X1 C t X2 D .V1 " S/ C t .V2 " S/ D .1 " t/V1 C t .V1 C V2 " S/ " S /HW \ D V1 C V2 " S 7KHQ \ LVDQDIÀQHFRPELQDWLRQRISRLQWVLQ S 6LQFH S LVDIÀQH \ LVLQ S E\7KHRUHP %XWWKHQ .1 " t /V1 C t \ LVDOVRLQ S 6R X1 C t X2 LVLQ "S C S D W 7KLVVKRZVWKDW W LVDVXEVSDFHRI Rn 7KXV S LVDÁDW EHFDXVH S D W C S &RQYHUVHO\ VXSSRVH S LVDÁDW 7KDWLV S D W C S IRUVRPH S 2 Rn DQGVRPH VXEVSDFH W 7RVKRZWKDW S LVDIÀQH LWVXIÀFHVWRVKRZWKDWIRUDQ\SDLU V1 DQG V2 RI SRLQWVLQ S WKHOLQHWKURXJK V1 DQG V2 OLHVLQ S %\GHÀQLWLRQRI W WKHUHH[LVW X1 DQG X2 LQ W VXFKWKDW V1 D X1 C S DQG V2 D X2 C S 6R IRUHDFKUHDO t

.1 " t /V1 C t V2 D .1 " t /.X1 C S/ C t .X2 C S/ D .1 " t /X1 C t X2 C S 6LQFH W LVDVXEVSDFH .1 " t/X1 C t X2 2 W DQGVR .1 " t/V1 C t V2 2 W C S D S 7KXV S LVDIÀQH ò ,I S D WKHQWKHWUDQVODWHLVMXVW S LWVHOI 6HH)LJ LQ6HFWLRQ

ó $ VXEVHW A RIDVHW B LVFDOOHGD SURSHU VXEVHWRI B LI A 6D B 7KHVDPHFRQGLWLRQDSSOLHVWRSURSHU VXEVSDFHVDQGSURSHUÁDWVLQ Rn WKH\DUHQRWHTXDOWR Rn

x3 5 b3

b2 p2 0

b1

p1 5

x1

'*(63&

$IILQH&RPELQDWLRQV

7KHRUHPSURYLGHVDJHRPHWULFZD\WRYLHZWKHDIÀQHKXOORIDVHW LWLVWKHÁDWWKDW FRQVLVWVRIDOOWKHDIÀQHFRPELQDWLRQVRISRLQWVLQWKHVHW )RULQVWDQFH )LJ VKRZV WKHSRLQWVVWXGLHGLQ([DPSOH $OWKRXJKWKHVHWRIDOO OLQHDU FRPELQDWLRQVRI E1 E2 DQG E3 LVDOORI R3 WKHVHWRIDOO DIÀQH FRPELQDWLRQVLVRQO\WKHSODQHWKURXJK E1 E2 DQG E3 1RWHWKDW S2 IURP([DPSOH LVLQWKHSODQHWKURXJK E1 E2 DQG E3 ZKLOH S1 LVQRWLQWKDWSODQH $OVR VHH([HUFLVH 7KHQH[WH[DPSOHWDNHVDIUHVKORRNDWDIDPLOLDUVHW³WKHVHWRIDOOVROXWLRQVRID V\VWHP A[ D E

&9".1-& 6XSSRVHWKDWWKHVROXWLRQVRIDQHTXDWLRQ A[ D E DUHDOORIWKHIRUP 2 3 2 3

2 4 [ D x3 X C S ZKHUH X D 4 "3 5 DQG S D 4 0 5 5HFDOOIURP6HFWLRQ WKDWWKLV 1 "3 VHWLVSDUDOOHOWRWKHVROXWLRQVHWRI A[ D ZKLFKFRQVLVWVRIDOOSRLQWVRIWKHIRUP x3 X )LQGSRLQWV Y1 DQG Y2 VXFKWKDWWKHVROXWLRQVHWRI A[ D E LV DII fY1 ; Y2 g

40-65*0/ 7KHVROXWLRQVHWLVDOLQHWKURXJK S LQWKHGLUHFWLRQRI X DVLQ)LJ 6LQFH DII fY1 ; Y2 g LVDOLQHWKURXJK Y1 DQG Y2 LGHQWLI\WZRSRLQWVRQWKHOLQH [ D x3 X C S 7ZR VLPSOHFKRLFHVDSSHDUZKHQ x3 D 0 DQG x3 D 1 7KDWLV WDNH Y1 D S DQG Y2 D X C S VRWKDW 2 3 2 3 2 3 2 4 6 Y2 D X C S D 4 "3 5 C 4 0 5 D 4 "3 5: 1 "3 "2

,QWKLVFDVH WKHVROXWLRQVHWLVGHVFULEHGDVWKHVHWRIDOODIÀQHFRPELQDWLRQVRIWKHIRUP 2 3 2 3 4 6 [ D .1 " x3 /4 0 5 C x3 4 "3 5: "3 "2

(DUOLHU 7KHRUHPGLVSOD\HGDQLPSRUWDQWFRQQHFWLRQEHWZHHQDIÀQHFRPELQDWLRQV DQGOLQHDUFRPELQDWLRQV 7KHQH[WWKHRUHPSURYLGHVDQRWKHUYLHZRIDIÀQHFRPELQD WLRQV ZKLFKIRU R2 DQG R3 LVFORVHO\FRQQHFWHGWRDSSOLFDWLRQVLQFRPSXWHUJUDSKLFV GLVFXVVHGLQWKHQH[WVHFWLRQDQGLQ6HFWLRQ

%&'*/*5*0/

5)&03&.

! " Y )RU Y LQ R WKHVWDQGDUG KRPRJHQHRXVIRUP RI Y LVWKHSRLQW YQ D LQ RnC1 1 n

$ SRLQW \ LQ Rn LVDQDIÀQHFRPELQDWLRQRI Y1 ; : : : ; Yp LQ Rn LIDQGRQO\LIWKH KRPRJHQHRXVIRUPRI \ LVLQ 6SDQ fQY1 ; : : : ; YQ p g ,QIDFW \ D c1 Y1 C ! ! ! C cp Yp ZLWK c1 C ! ! ! C cp D 1 LIDQGRQO\LI \Q D c1 YQ 1 C ! ! ! C cp YQ p

1300' $ SRLQW \ LVLQ DII fY1 ; : : : ; Yp g LIDQGRQO\LIWKHUHH[LVWZHLJKWV c1 ; : : : ; cp VXFK WKDW ! " ! " ! " ! " \ Y1 Y2 Y D c1 C c2 C ! ! ! C cp p 1 1 1 1 7KLVKDSSHQVLIDQGRQO\LI \Q LVLQ 6SDQ fQY1 ; YQ 2 ; : : : ; YQ p g

2 3 2 3 2 3 2 3 3 1 1 4 &9".1-& /HW Y1 D 4 1 5 Y2 D 4 2 5 Y3 D 4 7 5 DQG S D 4 3 5 8VH7KH 1 2 1 0 RUHPWRZULWH S DVDQDIÀQHFRPELQDWLRQRI Y1 Y2 DQG Y3 LISRVVLEOH

$)"15&3


40-65*0/ 5RZUHGXFHWKHDXJPHQWHGPDWUL[IRUWKHHTXDWLRQ x1 YQ 1 C x2 YQ 2 C x3 YQ 3 D SQ 7RVLPSOLI\WKHDULWKPHWLF PRYHWKHIRXUWKURZRI 1·VWRWKHWRSHTXLYDOHQWWRWKUHH URZLQWHUFKDQJHV $IWHUWKLV WKHQXPEHURIDULWKPHWLFRSHUDWLRQVKHUHLVEDVLFDOO\WKH VDPHDVWKHQXPEHUQHHGHGIRUWKHPHWKRGXVLQJ7KHRUHP 2 3 2 3 1 1 1 1 1 1 1 1 63 6 1 1 47 17 7 # 6 0 "2 "2 7 Œ YQ 1 YQ 2 YQ 3 SQ ! # 6 41 5 4 2 7 3 0 1 6 25 1 2 1 0 0 1 0 "1 2 3 1 0 0 1:5 60 1 0 "1 7 7 # !!! # 6 40 0 1 :5 5 0 0 0 0 %\7KHRUHP 1:5Y1 " Y2 C :5Y3 D S 6HH)LJ ZKLFKVKRZVWKHSODQHWKDWFRQWDLQV Y1 Y2 Y3 DQG S WRJHWKHUZLWKSRLQWVRQWKHFRRUGLQDWHD[HV x3 3 v2

v1

v3

5 p

x1

15

x2

'*(63&

13"$5*$& 130#-&. ! " ! " ! " ! " 1 "1 3 4 3ORWWKHSRLQWV Y1 D Y2 D Y3 D DQG S D RQJUDSKSDSHU DQG 0 2 1 3 H[SODLQZK\ S PXVW EHDQDIÀQHFRPELQDWLRQRI Y1 Y2 DQG Y3 7KHQÀQGWKHDIÀQH FRPELQDWLRQIRU S >+LQW :KDWLVWKHGLPHQVLRQRI DII fY1 Y2 Y3 g‹!

&9&3$*4&4 ,Q([HUFLVHV² ZULWH \ DVDQDIÀQHFRPELQDWLRQRIWKHRWKHU SRLQWVOLVWHG LISRVVLEOH ! " ! " ! " ! " ! " 1 "2 0 3 5 Y1 D Y2 D Y3 D Y4 D \D 2 2 4 7 3 ! " ! " ! " ! " 1 "1 3 5 Y1 D Y2 D Y3 D \D 1 2 2 7 2

3 2 3 2 3 2 3 "3 0 4 17 Y1 D 4 1 5 Y2 D 4 4 5 Y3 D 4 "2 5 \ D 4 1 5 1 "2 6 5

2 3 2 3 2 3 2 3 1 2 4 "3 Y1 D 4 2 5 Y2 D 4 "6 5 Y3 D 4 3 5 \ D 4 4 5 0 7 1 "4

2 3 2 3 2 3 2 1 2 ,Q([HUFLVHVDQG OHW E1 D 4 1 5 E2 D 4 0 5 E3 D 4 "5 5 1 "2 1 DQG S D fE1 ; E2 ; E3 g 1RWHWKDW S LVDQRUWKRJRQDOEDVLVIRU R3 :ULWHHDFKRIWKHJLYHQSRLQWVDVDQDIÀQHFRPELQDWLRQRIWKH SRLQWVLQWKHVHW S LISRVVLEOH >+LQW 8VH7KHRUHPLQ6HFWLRQ LQVWHDGRIURZUHGXFWLRQWRÀQGWKHZHLJKWV@

2 3 2 3 2 3 3 6 0 D S1 D 4 8 5 E S2 D 4 "3 5 F S3 D 4 "1 5 4 3 "5 2 3 2 3 2 3 0 1:5 5 D S1 D 4 "19 5 E S2 D 4 "1:3 5 F S3 D 4 "4 5 "5 ":5 0

/HW

2 3 1 607 6 ; Y1 D 4 7 35 0 2 3 5 6 "3 7 7 S1 D 6 4 5 5; 3

2

3

2

3

"1 6 27 6 7; Y3 D 4 15 1 2 3 4 627 7 S3 D 6 4 8 5; 5

2 6 "1 7 6 7; Y2 D 4 05 4 2 3 "9 6 10 7 7 S2 D 6 4 9 5; "13

DQG S D fY1 ; Y2 ; Y3 g ,W FDQ EH VKRZQ WKDW S LV OLQHDUO\ LQGHSHQGHQW D ,V S1 LQ 6SDQ S " ,V S1 LQ DII S " E ,V S2 LQ 6SDQ S " ,V S2 LQ DII S "

2

3 3 6 07 7 Y3 D 6 4 12 5; "6 2

DQG

E ,I fE1 ; : : : ; Ek g LVDOLQHDUO\LQGHSHQGHQWVXEVHWRI Rn DQG LI S LVDOLQHDUFRPELQDWLRQRI E1 ; : : : ; Ek WKHQ S LVDQ DIÀQHFRPELQDWLRQRI E1 ; : : : ; Ek F 7KHDIÀQHKXOORIWZRGLVWLQFWSRLQWVLVFDOOHGDOLQH G $ ÁDWLVDVXEVSDFH H $ SODQHLQ R3 LVDK\SHUSODQH D ,I S D f[g WKHQ DII S LVWKHHPSW\VHW

E $ VHWLVDIÀQHLIDQGRQO\LILWFRQWDLQVLWVDIÀQHKXOO F $ ÁDWRIGLPHQVLRQLVFDOOHGDOLQH G $ ÁDWRIGLPHQVLRQLVFDOOHGDK\SHUSODQH H $ ÁDWWKURXJKWKHRULJLQLVDVXEVSDFH

6KRZ WKDW 6XSSRVH fY1 ; Y2 ; Y3 g LV D EDVLV IRU R3 6SDQ fY2 " Y1 ; Y3 " Y1 g LVDSODQHLQ R3 >+LQW :KDWFDQ \RXVD\DERXW X DQG Y ZKHQ 6SDQ fX; Yg LVDSODQH"@

6KRZWKDWLI fY1 ; Y2 ; Y3 g LVDEDVLVIRU R3 WKHQ DII fY1 ; Y2 ; Y3 g LVWKHSODQHWKURXJK Y1 Y2 DQG Y3

/HW A EHDQ m % n PDWUL[DQG JLYHQ E LQ Rm VKRZWKDWWKH VHW S RIDOOVROXWLRQVRI A[ D E LVDQDIÀQHVXEVHWRI Rn

/HW Y 2 Rn DQGOHW k 2 R 3URYHWKDW S D f[ 2 Rn W [ ! Y D kg LVDQDIÀQHVXEVHWRI Rn

F ,V S3 LQ 6SDQ S " ,V S3 LQ DII S " 5HSHDW([HUFLVHZKHQ 2 3 2 3 1 2 6 07 6 17 7 6 7 Y1 D 6 4 3 5; Y2 D 4 6 5; "2 "5 2 3 2 3 4 "5 6 "1 7 6 37 7 6 7 S1 D 6 4 15 5; S2 D 4 "8 5; "7 6

$IILQH&RPELQDWLRQV

3 1 6 67 7 S3 D 6 4 "6 5: "8

6XSSRVHWKDWWKHVROXWLRQVRIDQHTXDWLRQ ! "A[ D E DUHDOORI ! " 4 "3 WKHIRUP [ D x3 X C S ZKHUH X D DQG S D "2 0 )LQGSRLQWV Y1 DQG Y2 VXFKWKDWWKHVROXWLRQVHWRI A[ D E LV DII fY1 ; Y2 g 6XSSRVHWKDWWKHVROXWLRQVRIDQHTXDWLRQ 2 3 A[ D E DUHDOORI 2 3 5 1 WKHIRUP [ D x3 X C S ZKHUH X D 4 1 5 DQG S D 4 "3 5 "2 4 )LQGSRLQWV Y1 DQG Y2 VXFKWKDWWKHVROXWLRQVHWRI A[ D E LV DII fY1 ; Y2 g ,Q([HUFLVHVDQG PDUNHDFKVWDWHPHQW7UXHRU)DOVH -XVWLI\ HDFKDQVZHU D 7KHVHWRIDOODIÀQHFRPELQDWLRQVRISRLQWVLQDVHW S LV FDOOHGWKHDIÀQHKXOORI S

&KRRVHDVHW S RIWKUHHSRLQWVVXFKWKDW DII S LVWKHSODQHLQ R3 ZKRVHHTXDWLRQLV x3 D 5 -XVWLI\\RXUZRUN &KRRVHDVHW S RIIRXUGLVWLQFWSRLQWVLQ R3 VXFKWKDW DII S LV WKHSODQH 2x1 C x2 " 3x3 D 12 -XVWLI\\RXUZRUN

/HW S EHDQDIÀQHVXEVHWRI Rn VXSSRVH f W Rn ! Rm LVD OLQHDUWUDQVIRUPDWLRQ DQGOHW f .S/ GHQRWHWKHVHWRILPDJHV ff .[/ W [ 2 S g 3URYHWKDW f .S/ LVDQDIÀQHVXEVHWRI Rm /HW f W Rn ! Rm EHDOLQHDUWUDQVIRUPDWLRQ OHW T EHDQ DIÀQHVXEVHWRI Rm DQGOHW S D f[ 2 Rn W f .[/ 2 T g 6KRZ WKDW S LVDQDIÀQHVXEVHWRI Rn

,Q([HUFLVHV² SURYHWKHJLYHQVWDWHPHQWDERXWVXEVHWV A DQG B RI Rn RUSURYLGHWKHUHTXLUHGH[DPSOHLQ R2 $ SURRI IRUDQH[HUFLVHPD\XVHUHVXOWVIURPHDUOLHUH[HUFLVHVDVZHOODV WKHRUHPVDOUHDG\DYDLODEOHLQWKHWH[W ,I A $ B DQG B LVDIÀQH WKHQ DII A $ B

,I A $ B WKHQ DII A $ DII B

Œ.DII A/ [ .DII B/! $ DII .A [ B/ >+LQW 7R VKRZ WKDW D [ E $ F VKRZWKDW D $ F DQG E $ F @

)LQGDQH[DPSOHLQ R2 WRVKRZWKDWHTXDOLW\QHHGQRWKROGLQ WKHVWDWHPHQWRI([HUFLVH >+LQW &RQVLGHUVHWV A DQG B HDFKRIZKLFKFRQWDLQVRQO\RQHRUWZRSRLQWV@ DII .A \ B/ $ .DII A \ DII B/

)LQGDQH[DPSOHLQ R2 WRVKRZWKDWHTXDOLW\QHHGQRWKROGLQ WKHVWDWHPHQWRI([HUFLVH

$)"15&3


40-65*0/ 50 13"$5*$& 130#-&. x2 p v2 v1

v3

x1

6LQFH WKH SRLQWV Y1 Y2 DQG Y3 DUH QRW FROOLQHDU WKDW LV QRW RQ D VLQJOH OLQH DII fY1 ; Y2 ; Y3 g FDQQRWEHRQHGLPHQVLRQDO 7KXV DII fY1 ; Y2 ; Y3 g PXVWHTXDO R2 7R ÀQGWKHDFWXDOZHLJKWVXVHGWRH[SUHVV S DVDQDIÀQHFRPELQDWLRQRI Y1 Y2 DQG Y3 ÀUVW FRPSXWH ! " ! " ! " "2 2 3 Y2 " Y1 D ; Y3 " Y1 D ; DQG S " Y1 D 2 1 3

7RZULWH S " Y1 DVDOLQHDUFRPELQDWLRQRI Y2 " Y1 DQG Y3 " Y1 URZUHGXFHWKHPDWUL[ KDYLQJWKHVHSRLQWVDVFROXPQV # ! " " 1 1 0 "2 2 3 2 # 2 1 3 0 1 2 7KXV S " Y1 D 12 .Y2 " Y1 / C 2.Y3 " Y1 / ZKLFKVKRZVWKDW % & S D 1 " 12 " 2 Y1 C 12 Y2 C 2Y3 D " 32 Y1 C 12 Y2 C 2Y3

7KLVH[SUHVVHV S DVDQDIÀQHFRPELQDWLRQRI Y1 Y2 DQG Y3 EHFDXVHWKHFRHIÀFLHQWVVXP WR $OWHUQDWLYHO\ XVHWKHPHWKRGRI([DPSOHDQGURZUHGXFH 3 2 3 2 1 0 0 " 32 ! " 1 1 1 1 Y1 Y2 Y3 S 6 1 7 # 4 1 "1 3 4 5 # 4 0 1 0 2 5 1 1 1 1 0 2 1 3 0 0 1 2 7KLVVKRZVWKDW S D " 32 Y1 C 12 Y2 C 2Y3

"''*/& */%&1&/%&/$& 7KLVVHFWLRQFRQWLQXHVWRH[SORUHWKHUHODWLRQEHWZHHQOLQHDUFRQFHSWVDQGDIÀQHFRQ FHSWV &RQVLGHUÀUVWDVHWRIWKUHHYHFWRUVLQ R3 VD\ S D fY1 ; Y2 ; Y3 g ,I S LVOLQHDUO\ GHSHQGHQW WKHQRQHRIWKHYHFWRUVLVDOLQHDUFRPELQDWLRQRIWKHRWKHUWZRYHFWRUV :KDW KDSSHQVZKHQRQHRIWKHYHFWRUVLVDQ DIÀQH FRPELQDWLRQRIWKHRWKHUV" )RULQVWDQFH VXSSRVHWKDW Y3 D .1 " t/Y1 C t Y2 ; IRUVRPH t LQ R 7KHQ

.1 " t/Y1 C t Y2 " Y3 D :

7KLVLVDOLQHDUGHSHQGHQFHUHODWLRQEHFDXVHQRWDOOWKHZHLJKWVDUH]HUR %XWPRUHLV WUXH³WKHZHLJKWVLQWKHGHSHQGHQFHUHODWLRQVXPWR

.1 " t/ C t C ."1/ D 0:

7KLVLVWKHDGGLWLRQDOSURSHUW\QHHGHGWRGHÀQH DIÀQHGHSHQGHQFH

%&'*/*5*0/

$QLQGH[HGVHWRISRLQWV fY1 ; : : : ; Yp g LQ Rn LV DIÀQHO\GHSHQGHQW LIWKHUHH[LVW UHDOQXPEHUV c1 ; : : : ; cp QRWDOO]HUR VXFKWKDW

c1 C ! ! ! C cp D 0

DQG

c1 Y1 C ! ! ! C cp Yp D

2WKHUZLVH WKHVHWLV DIÀQHO\LQGHSHQGHQW

$IILQH,QGHSHQGHQFH

$QDIÀQHFRPELQDWLRQLVDVSHFLDOW\SHRIOLQHDUFRPELQDWLRQ DQGDIÀQHGHSHQ GHQFHLVDUHVWULFWHGW\SHRIOLQHDUGHSHQGHQFH 7KXV HDFKDIÀQHO\GHSHQGHQWVHWLV DXWRPDWLFDOO\OLQHDUO\GHSHQGHQW $ VHW fY1 g RIRQO\RQHSRLQWHYHQWKH]HURYHFWRU PXVWEHDIÀQHO\LQGHSHQGHQW EHFDXVHWKHUHTXLUHGSURSHUWLHVRIWKHFRHIÀFLHQWV ci FDQQRWEHVDWLVÀHGZKHQWKHUHLV RQO\RQHFRHIÀFLHQW )RU fY1 g WKHÀUVWHTXDWLRQLQ LVMXVW c1 D 0 DQG\HWDWOHDVWRQH WKHRQO\RQH FRHIÀFLHQWPXVWEHQRQ]HUR ([HUFLVHDVNV\RXWRVKRZWKDWDQLQGH[HGVHW fY1 ; Y2 g LVDIÀQHO\GHSHQGHQWLI DQGRQO\LI Y1 D Y2 7KHIROORZLQJWKHRUHPKDQGOHVWKHJHQHUDOFDVHDQGVKRZVKRZ WKHFRQFHSWRIDIÀQHGHSHQGHQFHLVDQDORJRXVWRWKDWRIOLQHDUGHSHQGHQFH 3DUWVF DQGG JLYHXVHIXOPHWKRGVIRUGHWHUPLQLQJZKHWKHUDVHWLVDIÀQHO\GHSHQGHQW 5HFDOO IURP6HFWLRQWKDWLI Y LVLQ Rn WKHQWKHYHFWRU YQ LQ RnC1 GHQRWHVWKHKRPRJHQHRXV IRUPRI Y

5)&03&.

*LYHQDQLQGH[HGVHW S D fY1 ; : : : ; Yp g LQ Rn ZLWK p & 2 WKHIROORZLQJVWDWH PHQWVDUHORJLFDOO\HTXLYDOHQW 7KDWLV HLWKHUWKH\DUHDOOWUXHVWDWHPHQWVRUWKH\ DUHDOOIDOVH D E F G

S LVDIÀQHO\GHSHQGHQW 2QHRIWKHSRLQWVLQ S LVDQDIÀQHFRPELQDWLRQRIWKHRWKHUSRLQWVLQ S 7KHVHW fY2 " Y1 ; : : : ; Yp " Y1 g LQ Rn LVOLQHDUO\GHSHQGHQW 7KHVHW fQY1 ; : : : ; YQ p g RIKRPRJHQHRXVIRUPVLQ RnC1 LVOLQHDUO\GHSHQGHQW

1300' 6XSSRVHVWDWHPHQWD LVWUXH DQGOHW c1 ; : : : ; cp VDWLVI\ %\UHQDPLQJWKH SRLQWVLIQHFHVVDU\ RQHPD\DVVXPHWKDW c1 ¤ 0 DQGGLYLGHERWKHTXDWLRQVLQ E\ c1 VRWKDW 1 C .c2 =c1 / C ! ! ! C .cp =c1 / D 0 DQG Y1 D ."c2 =c1 /Y2 C ! ! ! C ."cp =c1 /Yp

1RWHWKDWWKHFRHIÀFLHQWVRQWKHULJKWVLGHRI VXPWR 7KXVD LPSOLHVE 1RZ VXSSRVHWKDWE LVWUXH %\UHQDPLQJWKHSRLQWVLIQHFHVVDU\ RQHPD\DVVXPHWKDW Y1 D c2 Y2 C ! ! ! C cp Yp ZKHUH c2 C ! ! ! C cp D 1 7KHQ

.c2 C ! ! ! C cp /Y1 D c2 Y2 C ! ! ! C cp Yp

c2 .Y2 " Y1 / C ! ! ! C cp .Yp " Y1 / D

DQG 1RWDOORI c2 ; : : : ; cp FDQEH]HUREHFDXVHWKH\VXPWR 6RE LPSOLHVF 1H[W LIF LVWUXH WKHQWKHUHH[LVWZHLJKWV c2 ; : : : ; cp QRWDOO]HUR VXFKWKDW KROGV 5HZULWH DV DQGVHW c1 D ".c2 C ! ! ! C cp / 7KHQ c1 C ! ! ! C cp D 0 7KXV VKRZVWKDW LVWUXH 6RF LPSOLHVD ZKLFKSURYHVWKDWD E DQGF DUH ORJLFDOO\HTXLYDOHQW )LQDOO\ G LVHTXLYDOHQWWRD EHFDXVHWKHWZRHTXDWLRQVLQ DUHHTXLYDOHQWWRWKHIROORZLQJHTXDWLRQLQYROYLQJWKHKRPRJHQHRXVIRUPVRIWKHSRLQWV LQ S ! " ! " ! " Y Y c1 1 C ! ! ! C cp p D 1 1 0 ,QVWDWHPHQWF RI7KHRUHP Y1 FRXOGEHUHSODFHGE\DQ\RIWKHRWKHUSRLQWVLQ WKHOLVW Y1 ; : : : ; Yp 2QO\WKHQRWDWLRQLQWKHSURRIZRXOGFKDQJH 6R WRWHVWZKHWKHUD VHWLVDIÀQHO\GHSHQGHQW VXEWUDFWRQHSRLQWLQWKHVHWIURPWKHRWKHUSRLQWV DQGFKHFN ZKHWKHUWKHWUDQVODWHGVHWRI p " 1 SRLQWVLVOLQHDUO\GHSHQGHQW

$)"15&3


&9".1-& 7KHDIÀQHKXOORIWZRGLVWLQFWSRLQWV S DQG T LVDOLQH ,IDWKLUGSRLQW U LVRQWKHOLQH WKHQ fS; T; Ug LVDQDIÀQHO\GHSHQGHQWVHW ,IDSRLQW V LVQRWRQWKH OLQHWKURXJK S DQG T WKHQWKHVHWKUHHSRLQWVDUHQRWFROOLQHDUDQG fS; T; Vg LVDQDIÀQHO\ LQGHSHQGHQWVHW 6HH)LJ aff{p, q}

q p

r

s

'*(63& fS; T; Ug LVDIÀQHO\GHSHQGHQW

2 3 2 3 2 3 1 2 0 &9".1-& /HW Y1 D 4 3 5 Y2 D 4 7 5 Y3 D 4 4 5 DQG S D fY1 ; Y2 ; Y3 g 7 6:5 7 'HWHUPLQHZKHWKHU S LVDIÀQHO\LQGHSHQGHQW 3 2 3 2 1 "1 40-65*0/ &RPSXWH Y2 " Y1 D 4 4 5 DQG Y3 " Y1 D 4 1 5 7KHVHWZRSRLQWV ":5 0 DUHQRWPXOWLSOHVDQGKHQFHIRUPDOLQHDUO\LQGHSHQGHQWVHW S 0 6RDOOVWDWHPHQWVLQ 7KHRUHPDUHIDOVH DQG S LVDIÀQHO\LQGHSHQGHQW )LJXUHVKRZV S DQGWKHWUDQVODWHG VHW S 0 1RWLFHWKDW 6SDQ S 0 LVDSODQHWKURXJKWKHRULJLQDQG DII S LVDSDUDOOHOSODQH WKURXJK Y1 Y2 DQG Y3 2QO\DSRUWLRQRIHDFKSODQHLVVKRZQKHUH RIFRXUVH x3 v3 v1 v2

v3 – v1 v2 – v1 x1

aff{v1 , v2 , v3} x2 Span{v2 – v1 , v3 – v1}

'*(63& $QDIÀQHO\LQGHSHQGHQWVHW fY1 ; Y2 ; Y3 g

2 3 2 3 2 3 2 3 1 2 0 0 &9".1-& /HW Y1 D 4 3 5 Y2 D 4 7 5 Y3 D 4 4 5 DQG Y4 D 4 14 5 DQGOHW 7 6:5 7 6 S D fY1 ; : : : ; Y4 g ,V S DIÀQHO\GHSHQGHQW" 2 3 2 3 2 3 1 "1 "1 40-65*0/ &RPSXWH Y2 " Y1 D 4 4 5 Y3 " Y1 D 4 1 5 DQG Y4 " Y1 D 4 11 5 ":5 0 "1 DQGURZUHGXFHWKHPDWUL[ 2 3 2 3 2 3 1 "1 "1 1 "1 "1 1 "1 "1 4 4 1 11 5 # 4 0 5 15 5 # 4 0 5 15 5 ":5 0 "1 0 ":5 "1:5 0 0 0

5HFDOOIURP6HFWLRQRU6HFWLRQ WKDWWKHFROXPQVDUHOLQHDUO\GHSHQGHQWEH FDXVHQRWHYHU\FROXPQLVDSLYRWFROXPQ VR Y2 " Y1 ; Y3 " Y1 DQG Y4 " Y1 DUHOLQHDUO\

$IILQH,QGHSHQGHQFH

GHSHQGHQW %\VWDWHPHQWF LQ7KHRUHP fY1 ; Y2 ; Y3 ; Y4 g LVDIÀQHO\GHSHQGHQW 7KLV GHSHQGHQFHFDQDOVREHHVWDEOLVKHGXVLQJG LQ7KHRUHPLQVWHDGRIF 7KHFDOFXODWLRQVLQ([DPSOHVKRZWKDW Y4 " Y1 LVDOLQHDUFRPELQDWLRQRI Y2 " Y1 DQG Y3 " Y1 ZKLFKPHDQVWKDW Y4 " Y1 LVLQ 6SDQ fY2 " Y1 ; Y3 " Y1 g %\7KHRUHPLQ 6HFWLRQ Y4 LVLQ DII fY1 ; Y2 ; Y3 g ,QIDFW FRPSOHWHURZUHGXFWLRQRIWKHPDWUL[LQ ([DPSOHZRXOGVKRZWKDW Y4 " Y1 D 2.Y2 " Y1 / C 3.Y3 " Y1 / Y4 D "4Y1 C 2Y2 C 3Y3

6HH)LJ x3 v3

v4

v1 v2

v3 – v1

aff{v1 , v2 , v3} v4 – v1

x2

v2 – v1 x1 '*(63& Y4 LVLQWKHSODQH DII fY1 ; Y2 ; Y3 g

)LJXUHVKRZVJULGVRQERWK 6SDQfY2 " Y1 ; Y3 " Y1 g DQG DII fY1 ; Y2 ; Y3 g 7KHJULG RQ DII fY1 ; Y2 ; Y3 g LVEDVHGRQ $QRWKHU´FRRUGLQDWHV\VWHPµFDQEHEDVHGRQ LQ ZKLFKWKHFRHIÀFLHQWV "4 DQGDUHFDOOHG DIÀQH RU EDU\FHQWULF FRRUGLQDWHVRI Y4

%DU\FHQWULF&RRUGLQDWHV 7KHGHÀQLWLRQRIEDU\FHQWULFFRRUGLQDWHVGHSHQGVRQWKHIROORZLQJDIÀQHYHUVLRQRIWKH 8QLTXH5HSUHVHQWDWLRQ7KHRUHPLQ6HFWLRQ 6HH([HUFLVHLQWKLVVHFWLRQIRUWKH SURRI

5)&03&.

/HW S D fY1 ; : : : ; Yk g EHDQDIÀQHO\LQGHSHQGHQWVHWLQ Rn 7KHQHDFK S LQ DII S KDVDXQLTXHUHSUHVHQWDWLRQDVDQDIÀQHFRPELQDWLRQRI Y1 ; : : : ; Yk 7KDWLV IRU HDFK S WKHUHH[LVWVDXQLTXHVHWRIVFDODUV c1 ; : : : ; ck VXFKWKDW S D c1 Y1 C ! ! ! C ck Yk

%&'*/*5*0/

DQG

c1 C ! ! ! C ck D 1

/HW S D fY1 ; : : : ; Yk g EHDQDIÀQHO\LQGHSHQGHQWVHW 7KHQIRUHDFKSRLQW S LQ DII S WKHFRHIÀFLHQWV c1 ; : : : ; cp LQWKHXQLTXHUHSUHVHQWDWLRQ RI S DUHFDOOHG WKH EDU\FHQWULF RU VRPHWLPHV DIÀQH FRRUGLQDWHV RI S 2EVHUYHWKDW LVHTXLYDOHQWWRWKHVLQJOHHTXDWLRQ ! " ! " ! " S Y1 Y D c1 C ! ! ! C ck k 1 1 1

LQYROYLQJWKHKRPRJHQHRXVIRUPVRIWKHSRLQWV 5RZUHGXFWLRQRIWKHDXJPHQWHGPDWUL[ ' ( YQ 1 ! ! ! YQ k SQ IRU SURGXFHVWKHEDU\FHQWULFFRRUGLQDWHVRI S

$)"15&3


! " ! " ! " ! " 1 3 9 5 E D F D DQG S D )LQGWKHEDU\FHQ 7 0 3 3 WULFFRRUGLQDWHVRI S GHWHUPLQHGE\WKHDIÀQHO\LQGHSHQGHQWVHW fD; E; Fg 40-65*0/ 5RZUHGXFHWKHDXJPHQWHGPDWUL[RISRLQWVLQKRPRJHQHRXVIRUP PRYLQJ WKHODVWURZRIRQHVWRWKHWRSWRVLPSOLI\WKHDULWKPHWLF 2 3 2 3 1 3 9 5 1 1 1 1 ' ( 3 35 # 41 3 9 55 DQ EQ FQ SQ D 4 7 0 1 1 1 1 7 0 3 3 2 3 1 1 0 0 4 6 7 1 7 #6 1 0 40 3 5 5 0 0 1 12

&9".1-& /HW D D

7KHFRRUGLQDWHVDUH 14 13 DQG

5 12

VR S D 14 D C 13 E C

5 F 12

%DU\FHQWULFFRRUGLQDWHVKDYHERWKSK\VLFDODQGJHRPHWULFLQWHUSUHWDWLRQV 7KH\ ZHUHRULJLQDOO\GHÀQHGE\$ ) 0RHELXVLQIRUDSRLQW S LQVLGHDWULDQJXODU UHJLRQZLWKYHUWLFHV D E DQG F +HZURWHWKDWWKHEDU\FHQWULFFRRUGLQDWHVRI S DUH WKUHHQRQQHJDWLYHQXPEHUV mD ; mE DQG mF VXFKWKDW S LVWKHFHQWHURIPDVVRIDV\VWHP FRQVLVWLQJRIWKHWULDQJOHZLWKQRPDVV DQGPDVVHV mD mE DQG mF DWWKHFRUUHVSRQGLQJ YHUWLFHV 7KHPDVVHVDUHXQLTXHO\GHWHUPLQHGE\UHTXLULQJWKDWWKHLUVXPEH 7KLV YLHZLVVWLOOXVHIXOLQSK\VLFVWRGD\ñ )LJXUHJLYHVDJHRPHWULFLQWHUSUHWDWLRQWRWKHEDU\FHQWULFFRRUGLQDWHVLQ([DPSOH VKRZLQJWKHWULDQJOH "DEF DQGWKUHHVPDOOWULDQJOHV "SEF "DSF DQG "DES 7KH DUHDVRIWKHVPDOOWULDQJOHVDUHSURSRUWLRQDOWRWKHEDU\FHQWULFFRRUGLQDWHVRI S ,QIDFW 1 DUHD."SEF/ D ! DUHD."DEF/ 4 DUHD."DSF/ D

1 ! DUHD."DEF/ 3

DUHD."DES/ D

5 ! DUHD."DEF/ 12

a area = s · area(∆abc)

p

area = t · area(∆abc)

b

c

area = r · area(∆abc)

'*(63& S D r D C s E C t F +HUH r D 14

s D 13 t D

5 12

7KHIRUPXODVLQ)LJ DUHYHULÀHGLQ([HUFLVHV² $QDORJRXVHTXDOLWLHVIRU YROXPHVRIWHWUDKHGURQVKROGIRUWKHFDVHZKHQ S LVDSRLQWLQVLGHDWHWUDKHGURQLQ R3 ZLWKYHUWLFHV D E F DQG G ñ 6HH([HUFLVHLQ6HFWLRQ ,QDVWURQRP\ KRZHYHU ÉDU\FHQWULFFRRUGLQDWHVµXVXDOO\UHIHUWRRUGLQDU\ R3 FRRUGLQDWHVRISRLQWVLQZKDWLVQRZFDOOHGWKH ,QWHUQDWLRQDO&HOHVWLDO5HIHUHQFH6\VWHP D&DUWHVLDQ FRRUGLQDWHV\VWHPIRURXWHUVSDFH ZLWKWKHRULJLQDWWKHFHQWHURIPDVVWKHEDU\FHQWHU RIWKHVRODUV\VWHP

$IILQH,QGHSHQGHQFH

:KHQDSRLQWLVQRWLQVLGHWKHWULDQJOHRUWHWUDKHGURQ VRPHRUDOORIWKHEDU\FHQWULF FRRUGLQDWHVZLOOEHQHJDWLYH 7KHFDVHRIDWULDQJOHLVLOOXVWUDWHGLQ)LJ IRUYHUWLFHV D E F DQGFRRUGLQDWHYDOXHV r; s; t DVDERYH 7KHSRLQWVRQWKHOLQHWKURXJK E DQG F IRU LQVWDQFH KDYH r D 0 EHFDXVHWKH\DUHDIÀQHFRPELQDWLRQVRIRQO\ E DQG F 7KHSDUDOOHO OLQHWKURXJK D LGHQWLÀHVSRLQWVZLWK r D 1 r=

1

a

r= c

p b

s=

0 s=

0

1

'*(63& %DU\FHQWULFFRRUGLQDWHV

IRUSRLQWVLQ DII fD; E; Fg

%DU\FHQWULF&RRUGLQDWHVLQ&RPSXWHU*UDSKLFV :KHQZRUNLQJZLWKJHRPHWULFREMHFWVLQDFRPSXWHUJUDSKLFVSURJUDP DGHVLJQHUPD\ XVHD´ZLUHIUDPHµDSSUR[LPDWLRQWRDQREMHFWDWFHUWDLQNH\SRLQWVLQWKHSURFHVV RIFUHDWLQJDUHDOLVWLFÀQDOLPDJHò )RULQVWDQFH LIWKHVXUIDFHRISDUWRIDQREMHFW FRQVLVWVRIVPDOOÁDWWULDQJXODUVXUIDFHV WKHQDJUDSKLFVSURJUDPFDQHDVLO\DGGFRORU OLJKWLQJ DQGVKDGLQJWRHDFKVPDOOVXUIDFHZKHQWKDWLQIRUPDWLRQLVNQRZQRQO\DWWKH YHUWLFHV %DU\FHQWULFFRRUGLQDWHVSURYLGHWKHWRROIRUVPRRWKO\LQWHUSRODWLQJWKHYHUWH[ LQIRUPDWLRQRYHUWKHLQWHULRURIDWULDQJOH 7KHLQWHUSRODWLRQDWDSRLQWLVVLPSO\WKH OLQHDUFRPELQDWLRQRIWKHYHUWH[YDOXHVXVLQJWKHEDU\FHQWULFFRRUGLQDWHVDVZHLJKWV &RORUVRQDFRPSXWHUVFUHHQDUHRIWHQGHVFULEHGE\5*% FRRUGLQDWHV $ WULSOH .r; g; b/ LQGLFDWHVWKHDPRXQWRIHDFKFRORU³UHG JUHHQ DQGEOXH³ZLWKWKHSDUDPHWHUV YDU\LQJIURP 0 WR 1 )RUH[DPSOH SXUHUHGLV .1; 0; 0/ ZKLWHLV .1; 1; 1/ DQGEODFNLV .0; 0; 0/ 2 3 2 3 2 3 2 3 3 4 1 3 &9".1-& /HW Y1 D 4 1 5 Y2 D 4 3 5 Y3 D 4 5 5 DQG S D 4 3 5 7KHFRO 5 4 1 3:5 RUVDWWKHYHUWLFHV Y1 Y2 DQG Y3 RIDWULDQJOHDUHPDJHQWD .1; 0; 1/ OLJKWPDJHQWD .1; :4; 1/ DQGSXUSOH .:6; 0; 1/ UHVSHFWLYHO\ )LQGWKHLQWHUSRODWHGFRORUDW S 6HH)LJ

v1 v3 v2 '*(63& ,QWHUSRODWHGFRORUV

ò 7KH,QWURGXFWRU\([DPSOHIRU&KDSWHUVKRZVDZLUHIUDPHPRGHORID%RHLQJDLUSODQH XVHGWR YLVXDOL]HWKHÁRZRIDLURYHUWKHVXUIDFHRIWKHSODQH

$)"15&3


40-65*0/ )LUVW ÀQGWKHEDU\FHQWULFFRRUGLQDWHVRI S +HUHLVWKHFDOFXODWLRQXVLQJ KRPRJHQHRXVIRUPVRIWKHSRLQWV ZLWKWKHÀUVWVWHSPRYLQJURZWRURZ 2 3 2 3 1 1 1 1 1 0 0 :25 6 ' ( 63 4 1 3 7 1 0 :50 7 7 # 60 7 YQ 1 YQ 2 YQ 3 SQ # 6 41 3 5 3 5 4 0 0 1 :25 5 5 4 1 3:5 0 0 0 0

6R S D :25Y1 C :5Y2 C :25Y3 8VHWKHEDU\FHQWULFFRRUGLQDWHVRI S WRPDNHDOLQHDU FRPELQDWLRQRIWKHFRORUGDWD 7KH5*% YDOXHVIRU S DUH 2 3 2 3 2 3 2 3 1 1 :6 :9 UHG :254 0 5 C :504 :4 5 C :254 0 5 D 4 :2 5 JUHHQ 1 1 1 1 EOXH

2QHRIWKHODVWVWHSVLQSUHSDULQJDJUDSKLFVVFHQHIRUGLVSOD\RQDFRPSXWHUVFUHHQ LVWRUHPRYH´KLGGHQVXUIDFHVµWKDWVKRXOGQRWEHYLVLEOHRQWKHVFUHHQ ,PDJLQHWKH YLHZLQJVFUHHQDVFRQVLVWLQJRI VD\ DPLOOLRQSL[HOV DQGFRQVLGHUDUD\RUÓLQHRI VLJKWµIURPWKHYLHZHU·VH\HWKURXJKDSL[HODQGLQWRWKHFROOHFWLRQRIREMHFWVWKDWPDNH XSWKH' GLVSOD\ 7KHFRORUDQGRWKHULQIRUPDWLRQGLVSOD\HGLQWKHSL[HORQWKHVFUHHQ VKRXOGFRPHIURPWKHREMHFWWKDWWKHUD\ÀUVWLQWHUVHFWV 6HH)LJ :KHQWKHREMHFWVLQ WKHJUDSKLFVVFHQHDUHDSSUR[LPDWHGE\ZLUHIUDPHVZLWKWULDQJXODUSDWFKHV WKHKLGGHQ VXUIDFHSUREOHPFDQEHVROYHGXVLQJEDU\FHQWULFFRRUGLQDWHV

'*(63& $ UD\IURPWKHH\HWKURXJKWKHVFUHHQWR

WKHQHDUHVWREMHFW

7KHPDWKHPDWLFVIRUÀQGLQJWKHUD\WULDQJOHLQWHUVHFWLRQVFDQDOVREHXVHGWRSHU IRUPH[WUHPHO\UHDOLVWLFVKDGLQJRIREMHFWV &XUUHQWO\ WKLV UD\WUDFLQJ PHWKRGLVWRR VORZIRUUHDOWLPHUHQGHULQJ EXWUHFHQWDGYDQFHVLQKDUGZDUHLPSOHPHQWDWLRQPD\ FKDQJHWKDWLQWKHIXWXUHó

&9".1-& /HW 2

3 1 Y1 D 4 1 5; "6

2

3 8 Y2 D 4 1 5; "4

2

3 5 Y3 D 4 11 5; "2

2

3 0 D D 4 0 5; 10

2

3 :7 E D 4 :4 5; "3

DQG [.t/ D D C t E IRU t & 0 )LQGWKHSRLQWZKHUHWKHUD\ [.t / LQWHUVHFWVWKHSODQHWKDW FRQWDLQVWKHWULDQJOHZLWKYHUWLFHV Y1 Y2 DQG Y3 ,VWKLVSRLQWLQVLGHWKHWULDQJOH" ó 6HH-RVKXD)HQGHUDQG-RQDWKDQ5RVH ´$ +LJK6SHHG5D\7UDFLQJ(QJLQH%XLOWRQD)LHOG3URJUDPPDEOH 6\VWHPµ LQ 3URF ,QW &RQIRQ)LHOG3URJUDPPDEOH7HFKQRORJ\ ,((( $ VLQJOHSURFHVVRUFDQ FDOFXODWHPLOOLRQUD\WULDQJOHLQWHUVHFWLRQVSHUVHFRQG

$IILQH,QGHSHQGHQFH

40-65*0/ 7KHSODQHLV DII fY1 ; Y2 ; Y3 g $ W\SLFDOSRLQWLQWKLVSODQHPD\EHZULWWHQ DV .1 " c2 " c3 /Y1 C c2 Y2 C c3 Y3 IRUVRPH c2 DQG c3 7KHZHLJKWVLQWKLVFRPELQDWLRQ VXPWR 7KHUD\ [.t / LQWHUVHFWVWKHSODQHZKHQ c2 c3 DQG t VDWLVI\ .1 " c2 " c3 /Y1 C c2 Y2 C c3 Y3 D D C t E 5HDUUDQJHWKLVDV c2 .Y2 " Y1 / C c3 .Y3 " Y1 / C t ."E/ D D " Y1 ,QPDWUL[IRUP '

Y2 " Y1

Y3 " Y 1

3 c2 "E 4 c3 5 D D " Y1 t (

2

)RUWKHVSHFLÀFSRLQWVJLYHQKHUH 2 3 7 Y2 " Y1 D 4 0 5; 2

2

3 4 Y3 " Y1 D 4 10 5; 4

2

3 "1 D " Y1 D 4 "1 5 16

5RZUHGXFWLRQRIWKHDXJPHQWHGPDWUL[DERYHSURGXFHV 2

7 40 2

4 10 4

3 2 "1 1 "1 5 # 4 0 16 0

":7 ":4 3

0 1 0

0 0 1

3 :3 :1 5 5

7KXV c2 D :3 c3 D :1 DQG t D 5 7KHUHIRUH WKHLQWHUVHFWLRQSRLQWLV 2

3 2 3 2 3 0 :7 3:5 [.5/ D D C 5E D 4 0 5 C 54 :4 5 D 4 2:0 5 10 "3 "5:0

$OVR [.5/ D .1 " :3 " :1/Y1 C :3Y2 C :1Y3 2 3 2 3 2 3 2 3 1 8 5 3:5 D :64 1 5 C :34 1 5 C :14 11 5 D 4 2:0 5 "6 "4 "2 "5:0 7KHLQWHUVHFWLRQSRLQWLVLQVLGHWKHWULDQJOHEHFDXVHWKHEDU\FHQWULFZHLJKWVIRU [.5/ DUH DOOSRVLWLYH

13"$5*$& 130#-&.4 'HVFULEHDIDVWZD\WRGHWHUPLQHZKHQWKUHHSRLQWVDUHFROOLQHDU ! " ! " ! " ! " 4 1 5 1 7KHSRLQWV Y1 D Y2 D Y3 D DQG Y4 D IRUPDQDIÀQHO\GH 1 0 4 2 SHQGHQWVHW )LQGZHLJKWV c1 ; : : : ; c4 WKDWSURGXFH DQ DIÀQHGHSHQGHQFHUHODWLRQ c1 Y1 C ! ! ! C c4 Y4 D ZKHUH c1 C ! ! ! C c4 D 0 DQGQRWDOO ci DUH]HUR >+LQW 6HH WKHHQGRIWKHSURRIRI7KHRUHP@

$)"15&3


&9&3$*4&4 ,Q([HUFLVHV² GHWHUPLQHLIWKHVHWRISRLQWVLVDIÀQHO\GH SHQGHQW 6HH3UDFWLFH3UREOHP ,IVR FRQVWUXFWDQDIÀQH GHSHQGHQFHUHODWLRQIRUWKHSRLQWV ! " ! " ! " ! " ! " ! " 3 0 2 2 5 "3 "3 6 0 1 4 "2 2 3 2 3 2 3 2 3 1 "2 2 0 4 2 5 4 "4 5 4 "1 5 4 15 5 "1 8 11 "9 2 3 2 3 2 3 2 3 "2 0 1 "2 4 5 5 4 "3 5 4 "2 5 4 7 5 3 7 "6 "3 2 3 2 3 2 3 2 3 1 0 "1 0 4 0 5 4 1 5 4 5 5 4 5 5 "2 1 1 "3 2 3 2 3 2 3 2 3 1 0 2 3 4 3 5 4 "1 5 4 5 5 4 5 5 1 "2 2 0

,Q([HUFLVHVDQG ÀQGWKHEDU\FHQWULFFRRUGLQDWHVRI S ZLWK UHVSHFWWRWKHDIÀQHO\LQGHSHQGHQWVHWRISRLQWVWKDWSUHFHGHVLW 2 3 2 3 2 3 2 3 1 2 1 5 6 "1 7 6 1 7 6 2 7 6 47 7 6 7 6 7 6 7 6 4 2 5 4 0 5 4 "2 5 S 4 "2 5 1 1 0 2 2 3 2 3 2 3 2 3 0 1 1 "1 6 17 617 6 47 6 17 7 6 7 6 7 6 7 6 4 "2 5 4 0 5 4 "6 5 S 4 "4 5 1 2 5 0

,Q([HUFLVHVDQG PDUNHDFKVWDWHPHQW7UXHRU)DOVH -XVWLI\ HDFKDQVZHU D ,I Y1 ; : : : ; Yp DUHLQ Rn DQGLIWKHVHW fY1 " Y2 ; Y3 " Y2 ; : : : ; Yp " Y2 g LV OLQHDUO\ GHSHQGHQW WKHQ fY1 ; : : : ; Yp g LV DIÀQHO\GHSHQGHQW 5HDGWKLVFDUHIXOO\ E ,I Y1 ; : : : Yp DUHLQ Rn DQGLIWKHVHWRIKRPRJHQHRXV IRUPV fQY1 ; : : : ; YQ p g LQ RnC1 LVOLQHDUO\LQGHSHQGHQW WKHQ fY1 ; : : : ; Yp g LVDIÀQHO\GHSHQGHQW

F $ ÀQLWHVHWRISRLQWV fY1 ; : : : ; Yk g LVDIÀQHO\GHSHQGHQWLI WKHUHH[LVWUHDOQXPEHUV c1 ; : : : ; ck QRWDOO]HUR VXFKWKDW c1 C ! ! ! C ck D 1 DQG c1 Y1 C ! ! ! C ck Yk D

G ,I S D fY1 ; : : : ; Yp g LVDQDIÀQHO\LQGHSHQGHQWVHWLQ Rn DQG LI S LQ Rn KDV D QHJDWLYH EDU\FHQWULF FRRUGLQDWH GHWHUPLQHGE\ S WKHQ S LVQRWLQ DII S H ,I Y1 ; Y2 ; Y3 ; D DQG E DUHLQ R3 DQGLIDUD\ D C t E IRU t & 0 LQWHUVHFWVWKHWULDQJOHZLWKYHUWLFHV Y1 Y2 DQG Y3 WKHQWKHEDU\FHQWULFFRRUGLQDWHVRIWKHLQWHUVHFWLRQSRLQW DUHDOOQRQQHJDWLYH D ,I fY1 ; : : : Yp g LVDQDIÀQHO\GHSHQGHQWVHWLQ Rn WKHQWKH VHW fQY1 ; : : : ; YQ p g LQ RnC1 RIKRPRJHQHRXVIRUPVPD\EH OLQHDUO\LQGHSHQGHQW

E ,I Y1 Y2 Y3 DQG Y4 DUH LQ R3 DQG LI WKH VHW fY2 " Y1 ; Y3 " Y1 ; Y4 " Y1 g LVOLQHDUO\LQGHSHQGHQW WKHQ fY1 ; : : : ; Y4 g LVDIÀQHO\LQGHSHQGHQW

F *LYHQ S D fE1 ; : : : ; Ek g LQ Rn HDFK S LQ DII S KDV D XQLTXH UHSUHVHQWDWLRQ DV DQ DIÀQH FRPELQDWLRQ RI E 1 ; : : : ; Ek

G :KHQFRORULQIRUPDWLRQLVVSHFLÀHGDWHDFKYHUWH[ Y1 Y2 Y3 RIDWULDQJOHLQ R3 WKHQWKHFRORUPD\EHLQWHUSRODWHG DWDSRLQW S LQ DII fY1 ; Y2 ; Y3 g XVLQJWKHEDU\FHQWULFFRRU GLQDWHVRI S H ,I T LVDWULDQJOHLQ R2 DQGLIDSRLQW S LVRQDQHGJHRI WKHWULDQJOH WKHQWKHEDU\FHQWULFFRRUGLQDWHVRI S IRUWKLV WULDQJOH DUHQRWDOOSRVLWLYH ([SODLQZK\DQ\VHWRIÀYHRUPRUHSRLQWVLQ R3 PXVWEH DIÀQHO\GHSHQGHQW 6KRZWKDWDVHW fY1 ; : : : ; Yp g LQ Rn LVDIÀQHO\GHSHQGHQWZKHQ p & n C 2 8VHRQO\WKHGHÀQLWLRQRIDIÀQHGHSHQGHQFHWRVKRZWKDWDQ LQGH[HGVHW fY1 ; Y2 g LQ Rn LVDIÀQHO\GHSHQGHQWLIDQGRQO\LI Y1 D Y2 7KHFRQGLWLRQVIRUDIÀQHGHSHQGHQFHDUHVWURQJHUWKDQWKRVH IRUOLQHDUGHSHQGHQFH VRDQDIÀQHO\GHSHQGHQWVHWLVDXWR PDWLFDOO\OLQHDUO\GHSHQGHQW $OVR DOLQHDUO\LQGHSHQGHQWVHW FDQQRWEHDIÀQHO\GHSHQGHQWDQGWKHUHIRUHPXVWEHDIÀQHO\ LQGHSHQGHQW &RQVWUXFWWZROLQHDUO\GHSHQGHQWLQGH[HGVHWV S1 DQG S2 LQ R2 VXFKWKDW S1 LVDIÀQHO\GHSHQGHQWDQG S2 LVDIÀQHO\LQGHSHQGHQW ,QHDFKFDVH WKHVHWVKRXOGFRQWDLQ HLWKHURQH WZR RUWKUHHQRQ]HURSRLQWV ! " ! " ! " "1 0 2 Y2 D Y3 D DQG OHW S D /HW Y1 D 2 4 0 fY 1 ; Y 2 ; Y 3 g D 6KRZWKDWWKHVHW S LVDIÀ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ÀUVW FRRUGLQDWH LV QHJDWLYHEHFDXVH S1 LVRSSRVLWHWKH Y1 VLGHRIWKHOLQH WKURXJK Y2 DQG Y3 3RLQW S2 LVRQWKH Y2 Y3 HGJHRI T ,WV FRRUGLQDWHVDUH .0; C; C/ :LWKRXWFDOFXODWLQJWKHDFWXDO YDOXHV GHWHUPLQHWKHVLJQVRIWKHEDU\FHQWULFFRRUGLQDWHV RISRLQWV S6 S7 DQG S8 DVVKRZQLQ)LJ

y

6XSSRVHWKDW fS1 ; S2 ; S3 g LVDQDIÀQHO\LQGHSHQGHQWVHWLQ Rn DQG T LVDQDUELWUDU\SRLQWLQ Rn 6KRZWKDWWKHWUDQVODWHGVHW fS1 C T; S2 C T; S3 C Tg LVDOVRDIÀQHO\LQGHSHQGHQW

v2 p1

v1 p3

/HW fS1 ; S2 ; S3 g EHDQDIÀQHO\GHSHQGHQWVHWRISRLQWVLQ Rn DQGOHW f W Rn ! Rm EHDOLQHDUWUDQVIRUPDWLRQ 6KRZWKDW ff .S1 /; f .S2 /; f .S3 /g LVDIÀQHO\GHSHQGHQWLQ Rm

p7 p8

$IILQH,QGHSHQGHQFH

p2 p5

x

v3 p4 p6

'*(63&

! " ! " ! " ! " 0 1 4 3 Y1 D Y2 D Y3 D S1 D 1 5 3 5 ! " ! " ! " ! " 5 2 "1 0 S3 D S4 D S5 D S2 D 1 3 0 4 ! " ! " 1 6 S7 D DQG S D fY1 ; Y2 ; Y3 g S6 D 2 4 D 6KRZWKDWWKHVHW S LVDIÀQHO\LQGHSHQGHQW

/HW

E )LQGWKHEDU\FHQWULFFRRUGLQDWHVRI S1 S2 DQG S3 ZLWK UHVSHFWWR S F 2QJUDSKSDSHU VNHWFKWKHWULDQJOH T ZLWKYHUWLFHV Y1 Y2 DQG Y3 H[WHQGWKHVLGHVDVLQ)LJ DQGSORWWKH SRLQWV S4 S5 S6 DQG S7 :LWKRXWFDOFXODWLQJWKHDFWXDO YDOXHV GHWHUPLQHWKHVLJQVRIWKHEDU\FHQWULFFRRUGLQDWHV RISRLQWV S4 S5 S6 DQG S7 3URYH 7KHRUHP IRU DQ DIÀQHO\ LQGHSHQGHQW VHW S D fY1 ; : : : ; Yk g LQ Rn >+LQW 2QHPHWKRGLVWRPLPLF WKHSURRIRI7KHRUHPLQ6HFWLRQ@ /HW T EHDWHWUDKHGURQLQ´VWDQGDUGµSRVLWLRQ ZLWKWKUHH HGJHVDORQJWKHWKUHHSRVLWLYHFRRUGLQDWHD[HVLQ R3 DQG VXSSRVH WKH YHUWLFHV DUH aH1 b H2 c H3 DQG ZKHUH Œ H1 H2 H3 ! D I3 )LQGIRUPXODVIRUWKHEDU\FHQWULFFR RUGLQDWHVRIDQDUELWUDU\SRLQW S LQ R3

,Q([HUFLVHV² D E DQG F DUHQRQFROOLQHDUSRLQWVLQ R2 DQG S LVDQ\RWKHUSRLQWLQ R2 /HW "DEF GHQRWHWKHFORVHGWULDQJXODU UHJLRQGHWHUPLQHGE\ D; E DQG F DQGOHW "SEF EHWKHUHJLRQ GHWHUPLQHGE\ S E DQG F )RUFRQYHQLHQFH DVVXPHWKDW D E Q DQG F DUHDUUDQJHGVRWKDW GHW Œ DQ EQ FQ ! LVSRVLWLYH ZKHUH DQ E DQG FQ DUHWKHVWDQGDUGKRPRJHQHRXVIRUPVIRUWKHSRLQWV 6KRZ WKDWWKHDUHDRI "DEF LV GHW Œ DQ EQ FQ !=2 >+LQW &RQVXOW6HFWLRQVDQG LQFOXGLQJWKH([HUFLVHV@

/HW S EHDSRLQWRQWKHOLQHWKURXJK D DQG E 6KRZWKDW GHW Œ DQ EQ SQ ! D 0 /HW S EHDQ\SRLQWLQWKHLQWHULRURI "DEF ZLWKEDU\FHQWULF FRRUGLQDWHV .r; s; t/ VRWKDW 2 3 ' ( r DQ EQ FQ 4 s 5 D SQ t

8VH([HUFLVHDQGDIDFWDERXWGHWHUPLQDQWV&KDSWHU WR VKRZWKDW

r D DUHDRI "SEF/=DUHDRI "DEF/ s D DUHDRI "DSF/=DUHDRI "DEF/

t D DUHDRI "DES/=DUHDRI "DEF/

7DNH T RQWKHOLQHVHJPHQWIURP E WR F DQGFRQVLGHUWKHOLQH WKURXJK T DQG D ZKLFKPD\EHZULWWHQDV S D .1 " x/T C x D IRU DOO UHDO x 6KRZ WKDW IRU HDFK x GHW Œ SQ EQ FQ ! D x ! GHW Œ DQ EQ FQ ! )URP WKLV DQG HDUOLHU ZRUN FRQFOXGH WKDWWKHSDUDPHWHU x LVWKHÀUVWEDU\FHQWULFFRRUGLQDWHRI S +RZHYHU E\FRQVWUXFWLRQ WKHSDUDPHWHU x DOVRGHWHUPLQHV WKHUHODWLYHGLVWDQFHEHWZHHQ S DQG T DORQJWKHVHJPHQWIURP T WR D :KHQ x D 1 S D D :KHQWKLVIDFWLVDSSOLHGWR ([DPSOH LWVKRZVWKDWWKHFRORUVDWYHUWH[ D DQGWKHSRLQW T DUHVPRRWKO\LQWHUSRODWHGDV S PRYHVDORQJWKHOLQHEHWZHHQ D DQG T

40-65*0/4 50 13"$5*$& 130#-&.4 )URP([DPSOH WKHSUREOHPLVWRGHWHUPLQHLIWKHSRLQWVDUHDIÀQHO\GHSHQGHQW 8VHWKHPHWKRGRI([DPSOHDQGVXEWUDFWRQHSRLQWIURPWKHRWKHUWZR ,IRQHRI WKHVHWZRQHZSRLQWVLVDPXOWLSOHRIWKHRWKHU WKHRULJLQDOWKUHHSRLQWVOLHRQDOLQH 7KHSURRIRI7KHRUHPHVVHQWLDOO\SRLQWVRXWWKDWDQDIÀQHGHSHQGHQFHUHODWLRQ DPRQJSRLQWVFRUUHVSRQGVWRDOLQHDUGHSHQGHQFHUHODWLRQDPRQJWKHKRPRJHQHRXV

$)"15&3


IRUPVRIWKHSRLQWV XVLQJWKH VDPH ZHLJKWV 2 4 1 ' ( YQ 1 YQ 2 YQ 3 YQ 4 D 4 1 0 1 1 2 1 0 # 40 1 0 0

6R URZUHGXFH 3 2 5 1 1 4 25 # 44 1 1 1 3 0 "1 0 1:25 5 1 :75

1 1 0

1 5 4

3 1 15 2

9LHZWKLVPDWUL[DVWKHFRHIÀFLHQWPDWUL[IRU A[ D ZLWKIRXUYDULDEOHV 7KHQ x4 LVIUHH x1 D x4 x2 D "1:25x4 DQG x3 D ":75x4 2QHVROXWLRQLV x1 D x4 D 4 x2 D "5 DQG x3 D "3 $ OLQHDUGHSHQGHQFHDPRQJWKHKRPRJHQHRXVIRUPVLV 4QY1 " 5QY2 " 3QY3 C 4QY4 D 6R 4Y1 " 5Y2 " 3Y3 C 4Y4 D $QRWKHUVROXWLRQPHWKRGLVWRWUDQVODWHWKHSUREOHPWRWKHRULJLQE\VXEWUDFWLQJ Y1 IURPWKHRWKHUSRLQWV ÀQGDOLQHDUGHSHQGHQFHUHODWLRQDPRQJWKHWUDQVODWHG SRLQWV DQGWKHQUHDUUDQJHWKHWHUPV 7KHDPRXQWRIDULWKPHWLFLQYROYHGLVDERXW WKHVDPHDVLQWKHDSSURDFKVKRZQDERYH

$0/7&9 $0.#*/"5*0/4 6HFWLRQFRQVLGHUHGVSHFLDOOLQHDUFRPELQDWLRQVRIWKHIRUP

c1 Y1 C c2 Y2 C ! ! ! C ck Yk ;

ZKHUH c1 C c2 C ! ! ! C ck D 1

7KLVVHFWLRQIXUWKHUUHVWULFWVWKHZHLJKWVWREHQRQQHJDWLYH

%&'*/*5*0/

$ FRQYH[FRPELQDWLRQ RISRLQWV Y1 ; Y2 ; : : : ; Yk LQ Rn LVDOLQHDUFRPELQDWLRQRI WKHIRUP c1 Y1 C c2 Y2 C ! ! ! C ck Yk

VXFKWKDW c1 C c2 C ! ! ! C ck D 1 DQG ci & 0 IRUDOO i 7KHVHWRIDOOFRQYH[ FRPELQDWLRQVRISRLQWVLQDVHW S LVFDOOHGWKH FRQYH[KXOO RI S GHQRWHGE\ FRQY S

7KHFRQYH[KXOORIDVLQJOHSRLQW Y1 LVMXVWWKHVHW fY1 g WKHVDPHDVWKHDIÀQHKXOO ,QRWKHUFDVHV WKHFRQYH[KXOOLVSURSHUO\FRQWDLQHGLQWKHDIÀQHKXOO 5HFDOOWKDWWKH DIÀQHKXOORIGLVWLQFWSRLQWV Y1 DQG Y2 LVWKHOLQH \ D .1 " t /Y1 C t Y2 ;

ZLWK t LQ R

%HFDXVHWKHZHLJKWVLQDFRQYH[FRPELQDWLRQDUHQRQQHJDWLYH WKHSRLQWVLQ FRQY fY1 ; Y2 g PD\EHZULWWHQDV \ D .1 " t/Y1 C t Y2 ; ZLWK 0 ' t ' 1 ZKLFKLVWKH OLQHVHJPHQW EHWZHHQ Y1 DQG Y2 KHUHDIWHUGHQRWHGE\ Y1 Y2 ,IDVHW S LVDIÀQHO\LQGHSHQGHQWDQGLI S 2 DII S WKHQ S 2 FRQY S LIDQGRQO\LIWKH EDU\FHQWULFFRRUGLQDWHVRI S DUHQRQQHJDWLYH ([DPSOHVKRZVDVSHFLDOVLWXDWLRQLQ ZKLFK S LVPXFKPRUHWKDQMXVWDIÀQHO\LQGHSHQGHQW

&9".1-& /HW 2

3 3 6 07 7 Y1 D 6 4 6 5; "3

2

3 "6 6 37 7 Y2 D 6 4 3 5; 0

2 3 3 667 7 Y3 D 6 4 0 5; 3

2 3 0 637 7 S1 D 6 4 3 5; 0

2

3 "10 6 57 7 S2 D 6 4 11 5; "4

&RQYH[&RPELQDWLRQV

DQG S D fY1 ; Y2 ; Y3 g 1RWHWKDW S LVDQRUWKRJRQDOVHW 'HWHUPLQHZKHWKHU S1 LVLQ 6SDQ S DII S DQG FRQY S 7KHQGRWKHVDPHIRU S2

40-65*0/ ,I S1 LVDWOHDVWD OLQHDU FRPELQDWLRQRIWKHSRLQWVLQ S WKHQWKHZHLJKWV DUHHDVLO\IRXQG EHFDXVH S LVDQRUWKRJRQDOVHW /HW W EHWKHVXEVSDFHVSDQQHGE\ S $ FDOFXODWLRQDVLQ6HFWLRQVKRZVWKDWWKHRUWKRJRQDOSURMHFWLRQRI S1 RQWR W LV S1 LWVHOI S ! Y1 S ! Y2 S ! Y3 SURMW S1 D 1 Y1 C 1 Y2 C 1 Y3 Y1 ! Y 1 Y2 ! Y2 Y3 ! Y3 18 18 18 Y1 C Y2 C Y3 54 54 54 2 3 2 3 3 "6 6 7 16 0 7 7 C 16 3 7 C D 6 4 5 4 6 35 3 3 "3 0 D

2 3 2 3 3 0 7 637 16 6 6 7 D 6 7 D S1 34 0 5 4 3 5 3 0

7KLVVKRZVWKDW S1 LV LQ 6SDQ S $OVR VLQFHWKHFRHIÀFLHQWVVXPWR S1 LVLQ DII S ,Q IDFW S1 LVLQ FRQY S EHFDXVHWKHFRHIÀFLHQWVDUHDOVRQRQQHJDWLYH )RU S2 DVLPLODUFDOFXODWLRQVKRZVWKDW SURMW S2 ¤ S2 6LQFH SURMW S2 LVWKHFORVHVW SRLQWLQ 6SDQ S WR S2 WKHSRLQW S2 LVQRWLQ 6SDQ S ,QSDUWLFXODU S2 FDQQRWEHLQ DII S RU FRQY S 5HFDOOWKDWDVHW S LVDIÀQHLILWFRQWDLQVDOOOLQHVGHWHUPLQHGE\SDLUVRISRLQWVLQ S :KHQDWWHQWLRQLVUHVWULFWHGWRFRQYH[FRPELQDWLRQV WKHDSSURSULDWHFRQGLWLRQLQYROYHV OLQHVHJPHQWVUDWKHUWKDQOLQHV

%&'*/*5*0/

$ VHW S LV FRQYH[ LIIRUHDFK S; T 2 S WKHOLQHVHJPHQW ST LVFRQWDLQHGLQ S ,QWXLWLYHO\ DVHW S LVFRQYH[LIHYHU\WZRSRLQWVLQWKHVHWFDQ´VHHµHDFKRWKHU ZLWKRXWWKHOLQHRIVLJKWOHDYLQJWKHVHW )LJXUHLOOXVWUDWHVWKLVLGHD

Convex

Convex

Not convex

'*(63&

7KHQH[WUHVXOWLVDQDORJRXVWR7KHRUHPIRUDIÀQHVHWV

5)&03&.

$ VHW S LVFRQYH[LIDQGRQO\LIHYHU\FRQYH[FRPELQDWLRQRISRLQWVRI S OLHVLQ S 7KDWLV S LVFRQYH[LIDQGRQO\LI S D FRQY S

1300' 7KHDUJXPHQWLVVLPLODUWRWKHSURRIRI7KHRUHP 7KHRQO\GLIIHUHQFHLV LQWKHLQGXFWLRQVWHS :KHQWDNLQJDFRQYH[FRPELQDWLRQRI k C 1 SRLQWV FRQVLGHU \ D c1 Y1 C ! ! ! C ck Yk C ck C1 Yk C1 ZKHUH c1 C ! ! ! C ck C1 D 1 DQG 0 ' ci ' 1 IRU

$)"15&3


DOO i ,I ck C1 D 1 WKHQ \ D Yk C1 ZKLFKEHORQJVWR S DQGWKHUHLVQRWKLQJIXUWKHU WRSURYH ,I ck C1 < 1 OHW t D c1 C ! ! ! C ck 7KHQ t D 1 " ck C1 > 0 DQG #c ck $ 1 \ D .1 " ck C1 / Y1 C ! ! ! C Yk C ck C1 Yk C1 t t %\WKHLQGXFWLRQK\SRWKHVLV WKHSRLQW ] D .c1 =t /Y1 C ! ! ! C .ck =t /Yk LVLQ S VLQFHWKH QRQQHJDWLYHFRHIÀFLHQWVVXPWR 7KXVHTXDWLRQ GLVSOD\V \ DVDFRQYH[FRPELQDWLRQ RIWZRSRLQWVLQ S %\WKHSULQFLSOHRILQGXFWLRQ HYHU\FRQYH[FRPELQDWLRQRIVXFK SRLQWVOLHVLQ S 7KHRUHPEHORZSURYLGHVDPRUHJHRPHWULFFKDUDFWHUL]DWLRQRIWKHFRQYH[KXOO RIDVHW ,WUHTXLUHVDSUHOLPLQDU\UHVXOWRQLQWHUVHFWLRQVRIVHWV 5HFDOOIURP6HFWLRQ ([HUFLVH WKDWWKHLQWHUVHFWLRQRIWZRVXEVSDFHVLVLWVHOIDVXEVSDFH ,QIDFW WKH LQWHUVHFWLRQRIDQ\FROOHFWLRQRIVXEVSDFHVLVLWVHOIDVXEVSDFH $ VLPLODUUHVXOWKROGV IRUDIÀQHVHWVDQGFRQYH[VHWV

5)&03&.

/HW fS˛ W ˛ 2 Ag EHDQ\FROOHFWLRQRIFRQYH[VHWV 7KHQ \˛2A S˛ LVFRQYH[ ,I fTˇ W ˇ 2 B g LVDQ\FROOHFWLRQRIDIÀQHVHWV WKHQ \ˇ2B Tˇ LVDIÀQH

1300' ,I S DQG T DUHLQ \S˛ WKHQ S DQG T DUHLQHDFK S˛ 6LQFHHDFK S˛ LVFRQYH[ WKHOLQHVHJPHQWEHWZHHQ S DQG T LVLQ S˛ IRUDOO ˛ DQGKHQFHWKDWVHJPHQWLVFRQWDLQHG LQ \S˛ 7KHSURRIRIWKHDIÀQHFDVHLVVLPLODU

5)&03&.

)RUDQ\VHW S WKHFRQYH[KXOORI S LVWKHLQWHUVHFWLRQRIDOOWKHFRQYH[VHWVWKDW FRQWDLQ S

1300' /HW T GHQRWHWKHLQWHUVHFWLRQRIDOOWKHFRQYH[VHWVFRQWDLQLQJ S 6LQFH FRQY S LVDFRQYH[VHWFRQWDLQLQJ S LWIROORZVWKDW T $ FRQY S 2QWKHRWKHUKDQG OHW C EH DQ\FRQYH[VHWFRQWDLQLQJ S 7KHQ C FRQWDLQVHYHU\FRQYH[FRPELQDWLRQRISRLQWV RI C 7KHRUHP DQGKHQFHDOVRFRQWDLQVHYHU\FRQYH[FRPELQDWLRQRISRLQWVRIWKH VXEVHW S 7KDWLV FRQY S $ C 6LQFHWKLVLVWUXHIRUHYHU\FRQYH[VHW C FRQWDLQLQJ S LWLVDOVRWUXHIRUWKHLQWHUVHFWLRQRIWKHPDOO 7KDWLV FRQY S $ T 7KHRUHPVKRZVWKDW FRQY S LVLQDQDWXUDOVHQVHWKH´VPDOOHVWµFRQYH[VHWFRQ WDLQLQJ S )RUH[DPSOH FRQVLGHUDVHW S WKDWOLHVLQVLGHVRPHODUJHUHFWDQJOHLQ R2 DQG LPDJLQHVWUHWFKLQJDUXEEHUEDQGDURXQGWKHRXWVLGHRI S $VWKHUXEEHUEDQGFRQWUDFWV DURXQG S LWRXWOLQHVWKHERXQGDU\RIWKHFRQYH[KXOORI S 2UWRXVHDQRWKHUDQDORJ\ WKHFRQYH[KXOORI S ÀOOV LQ DOOWKHKROHVLQWKHLQVLGHRI S DQGÀOOV RXW DOOWKHGHQWVLQ WKHERXQGDU\RI S

&9".1-& D 7KHFRQYH[KXOOVRIVHWV S DQG T LQ R2 DUHVKRZQEHORZ

S

conv S

T

conv T

&RQYH[&RPELQDWLRQV

E /HW S EHWKHVHWFRQVLVWLQJRIWKHVWDQGDUGEDVLVIRU R3 ; S D fH1 ; H2 ; H3 g 7KHQ FRQY S LVDWULDQJXODUVXUIDFHLQ R3 ZLWKYHUWLFHV H1 H2 DQG H3 6HH)LJ

x3 e3

)! " * x W x & 0 DQG y D x 2 6KRZWKDWWKHFRQYH[KXOORI y )! " * x 2 S LVWKHXQLRQRIWKHRULJLQDQG W x > 0 DQG y & x 6HH)LJ y

x2

&9".1-& /HW S D e2

40-65*0/ (YHU\SRLQWLQ FRQY S PXVWOLHRQDOLQHVHJPHQWWKDWFRQQHFWVWZRSRLQWV RI S 7KHGDVKHGOLQHLQ)LJ LQGLFDWHVWKDW H[FHSWIRUWKHRULJLQ WKHSRVLWLYH y D[LVLVQRWLQ FRQY S EHFDXVHWKHRULJLQLVWKHRQO\SRLQWRI S RQWKH y D[LV ,WPD\ VHHPUHDVRQDEOHWKDW)LJ GRHVVKRZ FRQY S EXWKRZFDQ\RXEHVXUHWKDWWKHSRLQW .10!2 ; 104 / IRUH[DPSOH LVRQDOLQHVHJPHQWIURPWKHRULJLQWRDSRLQWRQWKHFXUYHLQ S" &RQVLGHUDQ\SRLQW S LQWKHVKDGHGUHJLRQRI)LJ VD\ ! " a SD ; ZLWK a > 0 DQG b & a2 b

0 e1

x1

'*(63&

y y = x2

x

'*(63&

5)&03&.

7KHOLQHWKURXJK DQG S KDVWKHHTXDWLRQ y D .b=a/t IRU t UHDO 7KDWOLQHLQWHUVHFWV S ZKHUH t!VDWLVÀHV".b=a/t D t 2 WKDWLV ZKHQ t D b=a 7KXV S LVRQWKHOLQHVHJPHQW b=a IURP WR ZKLFKVKRZVWKDW)LJ LVFRUUHFW b 2 =a2 7KHIROORZLQJWKHRUHPLVEDVLFLQWKHVWXG\RIFRQYH[VHWV ,WZDVÀUVWSURYHGE\ &RQVWDQWLQ&DUDWKHRGRU\LQ ,I S LVLQWKHFRQYH[KXOORI S WKHQ E\GHÀQLWLRQ S PXVWEHDFRQYH[FRPELQDWLRQRISRLQWVRI S %XWWKHGHÀQLWLRQPDNHVQRVWLSXODWLRQ DVWRKRZPDQ\SRLQWVRI S DUHUHTXLUHGWRPDNHWKHFRPELQDWLRQ &DUDWKHRGRU\·V UHPDUNDEOHWKHRUHPVD\VWKDWLQDQ nGLPHQVLRQDOVSDFH WKHQXPEHURISRLQWVRI S LQ WKHFRQYH[FRPELQDWLRQQHYHUKDVWREHPRUHWKDQ n C 1 $BSBUIFPEPSZ ,I S LVDQRQHPSW\VXEVHWRI Rn WKHQHYHU\SRLQWLQ FRQY S FDQ EHH[SUHVVHGDVDFRQYH[FRPELQDWLRQRI n C 1 RUIHZHUSRLQWVRI S

1300' *LYHQ S LQ FRQY S RQHPD\ZULWH S D c1 Y1 C ! ! ! C ck Yk ZKHUH Yi 2 S; c1 C ! ! ! C ck D 1 DQG ci & 0 IRUVRPH k DQG i D 1; : : : ; k 7KHJRDOLVWRVKRZWKDW VXFKDQH[SUHVVLRQH[LVWVIRU S ZLWK k ' n C 1 ,I k > n C 1 WKHQ fY1 ; : : : ; Yk g LVDIÀQHO\GHSHQGHQW E\([HUFLVHLQ6HFWLRQ 7KXVWKHUHH[LVWVFDODUV d1 ; : : : ; dk QRWDOO]HUR VXFKWKDW k X i D1

di Yi D

DQG

k X i D1

di D 0

&RQVLGHUWKHWZRHTXDWLRQV

c1 Y1 C c2 Y2 C ! ! ! C ck Yk D S DQG

d1 Y1 C d2 Y2 C ! ! ! C dk Yk D %\VXEWUDFWLQJDQDSSURSULDWHPXOWLSOHRIWKHVHFRQGHTXDWLRQIURPWKHÀUVW ZHQRZ HOLPLQDWHRQHRIWKH Yi WHUPVDQGREWDLQDFRQYH[FRPELQDWLRQRIIHZHUWKDQ k HOHPHQWV RI S WKDWLVHTXDOWR S

$)"15&3


6LQFHQRWDOORIWKH di FRHIÀFLHQWVDUH]HUR ZHPD\DVVXPHE\UHRUGHULQJVXE VFULSWVLIQHFHVVDU\ WKDW dk > 0 DQGWKDW ck =dk ' ci =di IRUDOOWKRVH i IRUZKLFK di > 0 )RU i D 1; : : : ; k OHW bi D ci " .ck =dk /di 7KHQ bk D 0 DQG k X

k X

k ck X bi D ci " di D 1 " 0 D 1 dk i D1 i D1 i D1

)XUWKHUPRUH HDFK bi & 0 ,QGHHG LI di ' 0 WKHQ bi & ci & 0 ,I di > 0 WKHQ bi D di .ci =di " ck =dk / & 0 %\FRQVWUXFWLRQ k !1 X i D1

bi Yi D D

k X i D1

k X i D1

bi Yi D ci Y i "

, k + X ck ci " di Yi dk i D1

k k X ck X di Yi D ci Yi D S dk i D1 i D1

7KXV S LVQRZDFRQYH[FRPELQDWLRQRI k " 1 RIWKHSRLQWV Y1 ; : : : ; Yk 7KLVSURFHVV PD\EHUHSHDWHGXQWLO S LVH[SUHVVHGDVDFRQYH[FRPELQDWLRQRIDWPRVW n C 1 RIWKH SRLQWVRI S 7KHIROORZLQJH[DPSOHLOOXVWUDWHVWKHFDOFXODWLRQVLQWKHSURRIDERYH

&9".1-& /HW ! " 1 Y1 D ; 0

! " 2 Y2 D ; 3

! " 5 Y3 D ; 4

! " 3 Y4 D ; 0

SD

"

10 3 5 2

#

;

DQG S D fY1 ; Y2 ; Y3 ; Y4 g 7KHQ 1 Y 4 1

C 16 Y2 C 12 Y3 C

1 Y 12 4

DS

8VHWKHSURFHGXUHLQWKHSURRIRI&DUDWKHRGRU\·V7KHRUHPWRH[SUHVV S DVDFRQYH[ FRPELQDWLRQRIWKUHHSRLQWVRI S

40-65*0/ 7KHVHW S LVDIÀQHO\GHSHQGHQW 8VHWKHWHFKQLTXHRI6HFWLRQWRREWDLQ DQDIÀQHGHSHQGHQFHUHODWLRQ "5Y1 C 4Y2 " 3Y3 C 4Y4 D

1H[W FKRRVHWKHSRLQWV Y2 DQG Y4 LQ ZKRVHFRHIÀFLHQWVDUHSRVLWLYH )RUHDFK SRLQW FRPSXWHWKHUDWLRRIWKHTXRWLHQWVLQHTXDWLRQV DQG 7KHUDWLRIRU Y2 LV 1 1 1 1 1 ( 4 D 24 DQGWKDWIRU Y4 LV 12 ( 4 D 48 7KHUDWLRIRU Y4 LVVPDOOHU VRVXEWUDFW 48 6 WLPHVHTXDWLRQ IURPHTXDWLRQ WRHOLPLQDWH Y4 %1 4

C

5 48

& % Y1 C 16 "

4 48

17 Y 48 1

&

Y2 C

C

%1

4 Y 48 2

2

C

C

3 48

& %1 Y3 C 12 "

27 Y 48 3

DS

4 48

& Y4 D S

7KLVUHVXOWFDQQRW LQJHQHUDO EHLPSURYHGE\GHFUHDVLQJWKHUHTXLUHGQXPEHURI SRLQWV ,QGHHG JLYHQDQ\WKUHHQRQFROOLQHDUSRLQWVLQ R2 WKHFHQWURLGRIWKHWULDQJOH IRUPHGE\WKHPLVLQWKHFRQYH[KXOORIDOOWKUHH EXWLVQRWLQWKHFRQYH[KXOORIDQ\ WZR

&RQYH[&RPELQDWLRQV

13"$5*$& 130#-&.4 2 3 2 3 2 3 2 3 2 3 6 7 "2 1 3 /HW Y1 D 4 2 5 Y2 D 4 1 5 Y3 D 4 4 5 S1 D 4 3 5 DQG S2 D 4 2 5 DQGOHW 2 5 "1 1 1 S D fY1 ; Y2 ; Y3 g 'HWHUPLQHZKHWKHU S1 DQG S2 DUHLQ FRQY S

/HW S EHWKHVHWRISRLQWVRQWKHFXUYH y D 1=x IRU x > 0 ([SODLQJHRPHWULFDOO\ ZK\ FRQY S FRQVLVWVRIDOOSRLQWVRQDQGDERYHWKHFXUYH S

&9&3$*4&4

" * )! "* S 2 0 W0'y<1 'HVFULEHRU y 0 VNHWFK WKHFRQYH[KXOORI S ! " x 'HVFULEHWKHFRQYH[KXOORIWKHVHW S RISRLQWV LQ R2 WKDW y VDWLVI\WKHJLYHQFRQGLWLRQV -XVWLI\\RXUDQVZHUV 6KRZ WKDWDQDUELWUDU\SRLQW S LQ S EHORQJVWR FRQY S D y D 1=x DQG x & 1=2

,Q R2 OHW S D

)!

E y D VLQ x

F y D x 1=2 DQG x & 0 &RQVLGHUWKHSRLQWVLQ([HUFLVHLQ6HFWLRQ :KLFKRI S1 S2 DQG S3 DUHLQ FRQY S " &RQVLGHUWKHSRLQWVLQ([HUFLVHLQ6HFWLRQ :KLFKRI S1 S2 DQG S3 DUHLQ FRQY S " /HW

([HUFLVHV²XVHWKHWHUPLQRORJ\IURP6HFWLRQ )! " ! " ! "* "1 2 4 D /HW T D ; ; DQGOHW 0 3 1 ! " ! " ! " ! " 2 3 2 0 S1 D ; S2 D ; S3 D ; DQG S4 D : 1 2 0 2 )LQGWKHEDU\FHQWULFFRRUGLQDWHVRI S1 S2 S3 DQG S4 ZLWK UHVSHFWWR T E 8VH\RXUDQVZHUVLQSDUWD WRGHWHUPLQHZKHWKHUHDFK RI S1 ; : : : ; S4 LQSDUWD LVLQVLGH RXWVLGH RURQWKHHGJH RI FRQY T DWULDQJXODUUHJLRQ )! " ! " ! "* 2 0 "1 ; ; DQG 5HSHDW([HUFLVHIRU T D 0 5 1 " # ! " ! " ! " 1 2 1 1 S1 D ; S2 D ; S3 D 1 ; DQG S4 D : 1 1 0 3

2

3 2 3 2 3 2 3 "1 0 1 1 Y1 D 4 "3 5; Y2 D 4 "3 5; Y3 D 4 "1 5; Y4 D 4 1 5; 4 1 4 "2 2 3 2 3 1 0 S1 D 4 "1 5; S2 D 4 "2 5; 2 2

DQG S D fY1 ; Y2 ; Y3 ; Y4 g 'HWHUPLQHZKHWKHU S1 DQG S2 DUHLQ FRQY S 2 3 2 3 2 3 2 3 "1 2 0 "2 6 27 6 07 6 "2 7 6 17 6 7 7 6 7 6 7 /HW Y1 D 6 7 4 "1 5 Y2 D 4 2 5 Y3 D 4 0 5 S1 D 6 4 " 32 5 2 1 2 5 2 2 13 2 3 2 3 "2 6 "1 6 07 6 "4 7 6 "2 7 6 7 6 7 6 7 DQGOHW S EH S2 D 6 1 7 S3 D 6 7 DQG S4 D 4 05 4 45 4 15 4 7 "1 4

WKHRUWKRJRQDOVHW fY1 ; Y2 ; Y3 g 'HWHUPLQHZKHWKHUHDFK Si LV LQ 6SDQ S DII S RU FRQY S E S2 F S3 G S4 D S1

/HW S D fY1 ; Y2 ; Y3 ; Y4 g EHDQDIÀ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c1 Y1 C c2 Y2 C c3 Y3 DQG c1 C c2 C c3 D 1 WKHQ \ LVDFRQYH[FRPELQDWLRQRI Y1 ; Y2 DQG Y3 E ,I S LVDQRQHPSW\VHW WKHQ FRQY S FRQWDLQVVRPHSRLQWV WKDWDUHQRWLQ S F ,I S DQG T DUHFRQYH[VHWV WKHQ S [ T LVDOVRFRQYH[

D $ VHWLVFRQYH[LI [ \ 2 S LPSOLHVWKDWWKHOLQHVHJPHQW EHWZHHQ [ DQG \ LVFRQWDLQHGLQ S E ,I S DQG T DUHFRQYH[VHWV WKHQ S \ T LVDOVRFRQYH[

$)"15&3


F ,I S LVDQRQHPSW\VXEVHWRI R5 DQG \ 2 FRQY S WKHQ WKHUHH[LVWGLVWLQFWSRLQWV Y1 ; : : : ; Y6 LQ S VXFKWKDW \ LV DFRQYH[FRPELQDWLRQRI Y1 ; : : : ; Y6 /HW S EH D FRQYH[ VXEVHW RI Rn DQG VXSSRVH WKDW f W Rn ! Rm LVDOLQHDUWUDQVIRUPDWLRQ 3URYHWKDWWKHVHW f .S/ D ff .[/ W [ 2 S g LVDFRQYH[VXEVHWRI Rm

/HW f W Rn ! Rm EH D OLQHDU WUDQVIRUPDWLRQ DQG OHW T EH D FRQYH[ VXEVHW RI Rm 3URYH WKDW WKH VHW S D f[ 2 Rn W f .[/ 2 T g LVDFRQYH[VXEVHWRI Rn ! " ! " ! " ! " 1 1 4 4 Y2 D Y3 D Y4 D DQG /HW Y1 D 0 2 2 0 ! " 2 &RQÀUPWKDW SD 1 S D 13 Y1 C 13 Y2 C 16 Y3 C 16 Y4 DQG Y1 " Y2 C Y3 " Y4 D : 8VHWKHSURFHGXUHLQWKHSURRIRI&DUDWKHRGRU\·V7KHRUHPWR H[SUHVV S DVDFRQYH[FRPELQDWLRQRIWKUHHRIWKH Yi ·V 'R WKLVLQ WZR ZD\V ! " ! " "1 0 5HSHDW ([HUFLVH IRU SRLQWV Y1 D Y2 D 0 3 ! " ! " ! " 3 1 1 Y4 D DQG S D JLYHQWKDW Y3 D 1 "1 2 SD

1 Y 121 1

C

72 Y 121 2

C

37 Y 121 3

C

1 Y 11 4

DQG

10Y1 " 6Y2 C 7Y3 " 11Y4 D : ,Q([HUFLVHV² SURYHWKHJLYHQVWDWHPHQWDERXWVXEVHWV A DQG B RI Rn $ SURRIIRUDQH[HUFLVHPD\XVHUHVXOWVRIHDUOLHU H[HUFLVHV ,I A $ B DQG B LVFRQYH[ WKHQ FRQY A $ B

,I A $ B WKHQ FRQY A $ FRQY B

D Œ.FRQY A/ [ .FRQY B/! $ FRQY .A [ B/

E )LQGDQH[DPSOHLQ R2 WRVKRZWKDWHTXDOLW\QHHGQRWKROG LQSDUWD D FRQY .A \ B/ $ Œ.FRQY A/ \ .FRQY B/!

E )LQGDQH[DPSOHLQ R2 WRVKRZWKDWHTXDOLW\QHHGQRWKROG LQSDUWD

/HW S0 S1 DQG S2 EH SRLQWV LQ Rn DQG GHÀQH I1 .t/ D .1 " t/S1 C t S2 DQG I0 .t/ D .1 " t/S0 C t S1 J.t/ D .1 " t/I0 .t/ C t I1 .t/ IRU 0 ' t ' 1 %)RUWKHSRLQWV & % & DVVKRZQEHORZ GUDZDSLFWXUHWKDWVKRZV I 0 12 I 1 12 DQG %1& J 2 p1

p0

5HSHDW([HUFLVHIRU I 0

p2

%3& 4

I1

%3& % & DQG J 34 4

/HW J.t/ EHGHÀQHGDVLQ([HUFLVH ,WVJUDSKLVFDOOHG D TXDGUDWLF%p]LHUFXUYH DQGLWLVXVHGLQVRPHFRPSXWHU JUDSKLFVGHVLJQV 7KHSRLQWV S0 S1 DQG S2 DUHFDOOHGWKH FRQWUROSRLQWV IRUWKHFXUYH &RPSXWHDIRUPXODIRU J.t / WKDWLQYROYHVRQO\ S0 S1 DQG S2 7KHQVKRZWKDW J.t/ LV LQ FRQY fS0 ; S1 ; S2 g IRU 0 ' t ' 1

*LYHQ FRQWURO SRLQWV S0 S1 S2 DQG S3 LQ Rn OHW J1 .t / IRU 0 ' t ' 1 EHWKHTXDGUDWLF%p]LHUFXUYHIURP([HUFLVH GHWHUPLQHG E\ S0 S1 DQG S2 DQG OHW J2 .t/ EH GH ÀQHGVLPLODUO\IRU S1 S2 DQG S3 )RU 0 ' t ' 1 GHÀQH K.t/ D .1 " t /J1 .t/ C t J2 .t/ 6KRZWKDWWKHJUDSKRI K.t / OLHV LQ WKH FRQYH[ KXOO RI WKH IRXU FRQWURO SRLQWV 7KLV FXUYHLVFDOOHGD FXELF%p]LHUFXUYH DQGLWVGHÀQLWLRQKHUH LVRQHVWHSLQDQDOJRULWKPIRUFRQVWUXFWLQJ%p]LHUFXUYHV GLVFXVVHGODWHULQ6HFWLRQ $ %p]LHUFXUYHRIGHJUHH k LVGHWHUPLQHGE\ k C 1 FRQWUROSRLQWV DQGLWVJUDSKOLHVLQ WKHFRQYH[KXOORIWKHVHFRQWUROSRLQWV

40-65*0/4 50 13"$5*$& 130#-&.4 7KHSRLQWV Y1 Y2 DQG Y3 DUHQRWRUWKRJRQDO VRFRPSXWH 2 3 2 3 2 3 2 3 1 "8 "5 "3 Y2 " Y1 D 4 "1 5; Y3 " Y1 D 4 2 5; S1 " Y1 D 4 1 5; DQG S2 " Y1 D 4 0 5 3 "3 "1 "1

$XJPHQWWKHPDWUL[ Œ Y2 " Y1 Y3 " Y1 ! ZLWKERWK S1 " Y1 DQG S2 " Y1 DQGURZ UHGXFH 2 3 2 3 1 1 0 1 1 "8 "5 "3 3 6 7 1 7 2 4 "1 2 1 05 # 6 1 40 3 2 5 3 "3 "1 "1 0 0 0 " 52

7KHWKLUGFROXPQVKRZVWKDW S1 " Y1 D 13 .Y2 " Y1 / C 23 .Y3 " Y1 / ZKLFKOHDGVWR S1 D 0Y1 C 13 Y2 C 23 Y3 7KXV S1 LVLQ FRQY S ,QIDFW S1 LVLQ FRQY fY2 ; Y3 g

+\SHUSODQHV

7KHODVWFROXPQRIWKHPDWUL[VKRZVWKDW S2 " Y1 LVQRWDOLQHDUFRPELQDWLRQRI Y2 " Y1 DQG Y3 " Y1 7KXV S2 LVQRWDQDIÀQHFRPELQDWLRQRI Y1 Y2 DQG Y3 VR S2 FDQQRWSRVVLEO\EHLQ FRQY S $QDOWHUQDWLYHPHWKRGRIVROXWLRQLVWRURZUHGXFHWKHDXJPHQWHGPDWUL[RI KRPRJHQHRXVIRUPV 2 3 1 0 0 0 0 6 7 1 ' ( 60 07 1 0 3 6 7 YQ 1 YQ 2 YQ 3 SQ 1 SQ 2 # 6 7 2 0 0 1 40 5 3

0

0

0

0

1

,I S LVDSRLQWDERYH S WKHQWKHOLQHWKURXJK S ZLWKVORSH "1 ZLOOLQWHUVHFW S DWWZR SRLQWVEHIRUHLWUHDFKHVWKHSRVLWLYH x DQG y D[HV

):1&31-"/&4 +\SHUSODQHVSOD\DVSHFLDOUROHLQWKHJHRPHWU\RI Rn EHFDXVHWKH\GLYLGHWKHVSDFHLQWR WZRGLVMRLQWSLHFHV MXVWDVDSODQHVHSDUDWHV R3 LQWRWZRSDUWVDQGDOLQHFXWVWKURXJK R2 7KHNH\WRZRUNLQJZLWKK\SHUSODQHVLVWRXVHVLPSOH LPSOLFLW GHVFULSWLRQV UDWKHU WKDQWKH H[SOLFLW RUSDUDPHWULFUHSUHVHQWDWLRQVRIOLQHVDQGSODQHVXVHGLQWKHHDUOLHU ZRUNZLWKDIÀQHVHWVñ $QLPSOLFLWHTXDWLRQRIDOLQHLQ R2 KDVWKHIRUP ax C by D d $QLPSOLFLWHTXD WLRQRIDSODQHLQ R3 KDVWKHIRUP ax C by C c´ D d %RWKHTXDWLRQVGHVFULEHWKH OLQHRUSODQHDVWKHVHWRIDOOSRLQWVDWZKLFKDOLQHDUH[SUHVVLRQDOVRFDOOHGD OLQHDU IXQFWLRQDO KDVDÀ[HGYDOXH d

%&'*/*5*0/

$ OLQHDUIXQFWLRQDO RQ Rn LVDOLQHDUWUDQVIRUPDWLRQ f IURP Rn LQWR R )RUHDFK VFDODU d LQ R WKHV\PERO Œf d ! GHQRWHVWKHVHWRIDOO [ LQ Rn DWZKLFKWKHYDOXH RI f LV d 7KDWLV

Œf d !

LVWKHVHW

f[ 2 Rn W f .[/ D d g

7KH ]HUR IXQFWLRQDO LVWKHWUDQVIRUPDWLRQVXFKWKDW f .[/ D 0 IRUDOO [ LQ Rn $OO RWKHUOLQHDUIXQFWLRQDOVRQ Rn DUHVDLGWREH QRQ]HUR

&9".1-& ,Q R2 WKHOLQH x " 4y D 13 LVDK\SHUSODQHLQ R2 DQGLWLVWKHVHWRI

SRLQWVDWZKLFKWKHOLQHDUIXQFWLRQDO f .x; y/ D x " 4y KDVWKHYDOXH 7KDWLV WKH OLQHLVWKHVHW Œf 13!

&9".1-& ,Q R3 WKHSODQH 5x " 2y C 3´ D 21 LVDK\SHUSODQH WKHVHWRISRLQWV

DWZKLFKWKHOLQHDUIXQFWLRQDO g.x; y; ´/ D 5x " 2y C 3´ KDVWKHYDOXH 7KLVK\SHU SODQHLVWKHVHW Œg 21! ,I f LVDOLQHDUIXQFWLRQDORQ Rn WKHQWKHVWDQGDUGPDWUL[RIWKLVOLQHDUWUDQVIRUPD WLRQ f LVD 1 % n PDWUL[ A VD\ A D Œ a1 a2 ! ! ! an ! 6R

Œf 0!

LVWKHVDPHDV

f[ 2 Rn W A[ D 0g D 1XO A

ñ 3DUDPHWULFUHSUHVHQWDWLRQVZHUHLQWURGXFHGLQ6HFWLRQ

$)"15&3


,I f LVDQRQ]HURIXQFWLRQDO WKHQ UDQN A D 1 DQGGLP 1XO A D n " 1 E\WKH5DQN 7KHRUHPò 7KXV WKHVXEVSDFH Œf W 0! KDVGLPHQVLRQ n " 1 DQGVRLVDK\SHUSODQH $OVR LI d LVDQ\QXPEHULQ R WKHQ

Œf d !

LVWKHVDPHDV

f[ 2 Rn W A[ D d g

5HFDOOIURP7KHRUHPLQ6HFWLRQWKDWWKHVHWRIVROXWLRQVRI A[ D E LVREWDLQHG E\WUDQVODWLQJWKHVROXWLRQVHWRI A[ D XVLQJDQ\SDUWLFXODUVROXWLRQ S RI A[ D E :KHQ A LVWKHVWDQGDUGPDWUL[RIWKHWUDQVIRUPDWLRQ f WKLVWKHRUHPVD\VWKDW

Œf d ! D Œf W 0! C S

IRUDQ\ S LQ Œf W d !

7KXVWKHVHWV Œf W d ! DUHK\SHUSODQHVSDUDOOHOWR Œf W 0! 6HH)LJ

[ f :d ] [ f :0]

p

'*(63& 3DUDOOHOK\SHUSODQHV

ZLWK f . S/ D d

:KHQ A LVD 1 % n PDWUL[ WKHHTXDWLRQ A[ D d PD\EHZULWWHQZLWKDQLQQHU SURGXFW Q ! [ XVLQJ Q LQ Rn ZLWKWKHVDPHHQWULHVDV A 7KXV IURP

Œf d !

LVWKHVDPHDV

f[ 2 Rn W Q ! [ D d g

n

7KHQ Œf 0! D f[ 2 R W Q ! [ D 0g ZKLFKVKRZVWKDW Œf 0! LVWKHRUWKRJRQDOFRPSOH PHQWRIWKHVXEVSDFHVSDQQHGE\ Q ,QWKHWHUPLQRORJ\RIFDOFXOXVDQGJHRPHWU\IRU R3 Q LVFDOOHGD QRUPDO YHFWRUWR Œf 0! $ ´QRUPDOµYHFWRULQWKLVVHQVHQHHGQRW KDYHXQLWOHQJWK $OVR Q LVVDLGWREH QRUPDO WRHDFKSDUDOOHOK\SHUSODQH Œf d ! HYHQ WKRXJK Q ! [ LVQRW]HURZKHQ d ¤ 0 $QRWKHUQDPHIRU Œf d ! LVD OHYHOVHW RI f DQG Q LVVRPHWLPHVFDOOHGWKH JUDGLHQW RI f ZKHQ f .[/ D Q ! [ IRUHDFK [ ! " ! " 3 1 DQG Y D DQGOHW H D f[ W Q ! [ D 12g VR H D 4 "6 Œf 12! ZKHUH f .x; y/ D 3x C 4y 7KXV H LVWKHOLQH 3x C 4y D 12 )LQGDQLPSOLFLW GHVFULSWLRQRIWKHSDUDOOHOK\SHUSODQHOLQH H1 D H C Y

&9".1-& /HW Q D

40-65*0/ )LUVW ÀQGDSRLQW S LQ H1 7RGRWKLV ÀQGDSRLQWLQ H DQGDGG Y WRLW ! " ! " ! " ! " 0 1 0 1 )RULQVWDQFH LVLQ H VR S D C D LVLQ H1 1RZ FRPSXWH 3 "6 3 "3 Q ! S D "9 7KLVVKRZVWKDW H1 D Œf " 9! 6HH)LJ ZKLFKDOVRVKRZVWKHVXEVSDFH H0 D f[ W Q ! [ D 0g 7KHQH[WWKUHHH[DPSOHVVKRZFRQQHFWLRQVEHWZHHQLPSOLFLWDQGH[SOLFLWGHVFULS WLRQVRIK\SHUSODQHV ([DPSOHEHJLQVZLWKDQLPSOLFLWIRUP ò 6HH7KHRUHPLQ6HFWLRQRU7KHRUHPLQ6HFWLRQ

+\SHUSODQHV

y 4

n

v x

4

–4 v

H = [ f : 12] H0 = [ f : 0]

–4

H1 = [ f : –9]

v '*(63&

&9".1-& ,Q R2 JLYHDQH[SOLFLWGHVFULSWLRQRIWKHOLQH x " 4y D 13 LQSDUD PHWULFYHFWRUIRUP

40-65*0/ 7KLVDPRXQWVWRVROYLQJDQRQKRPRJHQHRXVHTXDWLRQ A[ D E ZKHUH A D Œ 1 "4 ! DQG E LVWKHQXPEHULQ R :ULWH x D 13 C 4y ZKHUH y LVDIUHHYDULDEOH ,QSDUDPHWULFIRUP WKHVROXWLRQLV ! " ! " ! " ! " x 13 C 4y 13 4 [D D D Cy D S C y T; y 2 R y y 0 1 &RQYHUWLQJDQH[SOLFLWGHVFULSWLRQRIDOLQHLQWRLPSOLFLWIRUPLVPRUHLQYROYHG 7KH EDVLFLGHDLVWRFRQVWUXFW Œf 0! DQGWKHQÀQG d IRU Œf d ! ! " ! " 1 6 &9".1-& /HW Y1 D DQG Y2 D DQGOHW L1 EHWKHOLQHWKURXJK Y1 DQG 2 0 Y2 )LQGDOLQHDUIXQFWLRQDO f DQGDFRQVWDQW d VXFKWKDW L1 D Œf d !

40-65*0/ 7KHOLQH L1 LVSDUDOOHOWRWKHWUDQVODWHGOLQH L0 WKURXJK Y2 " Y1 DQGWKH RULJLQ 7KHGHÀQLQJHTXDWLRQIRU L0 KDVWKHIRUP ! " ! " x a Œa b! D 0 RU Q ! [ D 0; ZKHUH Q D y b

6LQFH Q LVRUWKRJRQDOWRWKHVXEVSDFH L0 ZKLFKFRQWDLQV Y2 " Y1 FRPSXWH ! " ! " ! " 6 1 5 " D Y2 " Y1 D 0 2 "2 DQGVROYH

'

a

b

!

(

5 "2

"

D0

%\LQVSHFWLRQ DVROXWLRQLV Œ a b ! D Œ 2 5 ! /HW f .x; y/ D 2x C 5y )URP L0 D Œf 0! DQG L1 D Œf d ! IRUVRPH d 6LQFH Y1 LVRQOLQH L1 d D f .Y1 / D 2.1/ C 5.2/ D 12 7KXV WKHHTXDWLRQIRU L1 LV 2x C 5y D 12 $VDFKHFN QRWHWKDW f .Y2 / D f .6; 0/ D 2.6/ C 5.0/ D 12 VR Y2 LVRQ L1 WRR 2 3 2 3 2 3 1 2 3 &9".1-& /HW Y1 D 4 1 5 Y2 D 4 "1 5 DQG Y3 D 4 1 5 )LQGDQLPSOLFLWGH 1 4 2 VFULSWLRQ Œf d ! RIWKHSODQH H1 WKDWSDVVHVWKURXJK Y1 Y2 DQG Y3

$)"15&3


40-65*0/ H1 LVSDUDOOHOWRDSODQH H0 WKURXJKWKHRULJLQWKDWFRQWDLQVWKHWUDQVODWHG SRLQWV 2 3 2 3 1 2 Y2 " Y1 D 4 "2 5 DQG Y3 " Y1 D 4 0 5 3 1

6LQFHWKHVHWZRSRLQWVDUHOLQHDUO\LQGHSHQGHQW H0 D 6SDQ fY2 " Y1 ; Y3 " Y1 g /HW 2 3 a Q D 4 b 5 EHWKHQRUPDOWR H0 7KHQ Y2 " Y1 DQG Y3 " Y1 DUHHDFKRUWKRJRQDOWR Q c 7KDWLV .Y2 " Y1 /! Q D 0 DQG .Y3 " Y1 /! Q D 0 7KHVHWZRHTXDWLRQVIRUPDV\VWHP ZKRVHDXJPHQWHGPDWUL[FDQEHURZUHGXFHG 2 3 2 3 ! " ' ( a ' ( a 1 "2 3 0 5 5 4 4 1 "2 3 2 0 1 b D 0; b D 0; 2 0 1 0 c c 5RZRSHUDWLRQV\LHOG a D ." 24 /c b D . 54 /c ZLWK c IUHH 6HW c D 4 IRULQVWDQFH 7KHQ 2 3 "2 Q D 4 5 5 DQG H0 D Œf 0! ZKHUH f .[/ D "2x1 C 5x2 C 4x3 4 7KHSDUDOOHOK\SHUSODQH H1 LV Œf d ! 7RÀQG d XVHWKHIDFWWKDW Y1 LVLQ H1 DQG FRPSXWH d D f .Y1 / D f .1; 1; 1/ D "2.1/ C 5.1/ C 4.1/ D 7 $VDFKHFN FRPSXWH f .Y2 / D f .2; "1; 4/ D "2.2/ C 5."1/ C 4.4/ D 16 " 9 D 7

7KHSURFHGXUHLQ([DPSOHJHQHUDOL]HVWRKLJKHUGLPHQVLRQV +RZHYHU IRUWKH VSHFLDOFDVHRI R3 RQHFDQDOVRXVHWKH FURVVSURGXFW IRUPXODWRFRPSXWH Q XVLQJD V\PEROLFGHWHUPLQDQWDVDPQHPRQLFGHYLFH Q D .Y2 " Y1 / % .Y3 " Y1 / ˇ ˇ ˇ 1 ˇ ˇ 2 L ˇˇ ˇˇ ˇ ˇ1 "2 0 ˇˇ ˇ M ˇˇ D ˇˇ D ˇˇ "2 0 L " ˇ ˇ3 3 1 ˇ 3 1 Nˇ 2 3 "2 D "2L C 5M C 4N D 4 5 5 4

ˇ ˇ ˇ 1 2 ˇˇ ˇ M C ˇ ˇ "2 1

ˇ 2 ˇˇ N 0ˇ

,IRQO\WKHIRUPXODIRU f LVQHHGHG WKHFURVVSURGXFWFDOFXODWLRQPD\EHZULWWHQ DVDQRUGLQDU\GHWHUPLQDQW ˇ ˇ ˇ 1 ˇ ˇ ˇ ˇ ˇ 2 x1 ˇˇ ˇˇ ˇ ˇ1 ˇ 1 "2 0 ˇˇ 2 ˇˇ 2 ˇˇ ˇ ˇ ˇ ˇ ˇ f .x1 ; x2 ; x3 / D ˇ "2 0 x2 ˇ D ˇ x " x C x 3 1 ˇ 1 ˇ 3 1 ˇ 2 ˇ "2 0 ˇ 3 ˇ 3 1 x3 ˇ

D "2x1 C 5x2 C 4x3

6RIDU HYHU\K\SHUSODQHH[DPLQHGKDVEHHQGHVFULEHGDV Œf d ! IRUVRPHOLQHDU IXQFWLRQDO f DQGVRPH d LQ R RUHTXLYDOHQWO\DV f[ 2 Rn W Q ! [ D d g IRUVRPH Q LQ Rn 7KHIROORZLQJWKHRUHPVKRZVWKDW HYHU\ K\SHUSODQHKDVWKHVHHTXLYDOHQWGHVFULSWLRQV

5)&03&.

$ VXEVHW H RI Rn LVDK\SHUSODQHLIDQGRQO\LI H D Œf d ! IRUVRPHQRQ]HUR OLQHDUIXQFWLRQDO f DQGVRPHVFDODU d LQ R 7KXV LI H LVDK\SHUSODQH WKHUH H[LVWDQRQ]HURYHFWRU Q DQGDUHDOQXPEHU d VXFKWKDW H D f[ W Q ! [ D d g

+\SHUSODQHV

1300' 6XSSRVHWKDW H LVDK\SHUSODQH WDNH S 2 H DQGOHW H0 D H " S 7KHQ H0 LVDQ .n " 1/GLPHQVLRQDOVXEVSDFH 1H[W WDNHDQ\SRLQW \ WKDWLVQRWLQ H0 %\WKH 2UWKRJRQDO'HFRPSRVLWLRQ7KHRUHPLQ6HFWLRQ \ D \1 C Q ZKHUH \1 LVDYHFWRULQ H0 DQG Q LVRUWKRJRQDOWRHYHU\YHFWRULQ H0 7KHIXQFWLRQ f GHÀQHGE\ f .[/ D Q ! [ IRU [ 2 Rn LVDOLQHDUIXQFWLRQDO E\SURSHUWLHVRIWKHLQQHUSURGXFW 1RZ Œf 0! LVDK\SHUSODQH WKDWFRQWDLQV H0 E\FRQVWUXFWLRQRI Q ,WIROORZVWKDW

H0 D Œf 0! >$UJXPHQW H0 FRQWDLQVDEDVLV S RI n " 1 YHFWRUV DQGVLQFH S LVLQWKH .n " 1/ GLPHQVLRQDOVXEVSDFH Œf 0! S PXVWDOVREHDEDVLVIRU Œf 0! E\WKH%DVLV7KHRUHP@ )LQDOO\ OHW d D f .S/ D Q ! S 7KHQ DVLQ VKRZQHDUOLHU

Œf d ! D Œf 0! C S D H0 C S D H 7KHFRQYHUVHVWDWHPHQWWKDW Œf d ! LVDK\SHUSODQHIROORZVIURP DQG DERYH 0DQ\LPSRUWDQWDSSOLFDWLRQVRIK\SHUSODQHVGHSHQGRQWKHSRVVLELOLW\RI´VHSDUDW LQJµWZRVHWVE\DK\SHUSODQH ,QWXLWLYHO\ WKLVPHDQVWKDWRQHRIWKHVHWVLVRQRQHVLGH RIWKHK\SHUSODQHDQGWKHRWKHUVHWLVRQWKHRWKHUVLGH 7KHIROORZLQJWHUPLQRORJ\DQG QRWDWLRQZLOOKHOSWRPDNHWKLVLGHDPRUHSUHFLVH 5010-0(: */ Rn 5&3.4 "/% '"$54 )RUDQ\SRLQW S LQ Rn DQGDQ\UHDO ı > 0 WKH RSHQEDOO B.S; ı/ ZLWKFHQWHU S DQG UDGLXV ı LVJLYHQE\ B.S; ı/ D f[ W k[ " Sk < ıg

*LYHQDVHW S LQ Rn DSRLQW S LVDQ LQWHULRUSRLQW RI S LIWKHUHH[LVWVD ı > 0 VXFKWKDW B.S; ı/ $ S ,IHYHU\RSHQEDOOFHQWHUHGDW S LQWHUVHFWVERWK S DQGWKH FRPSOHPHQWRI S WKHQ S LVFDOOHGD ERXQGDU\SRLQW RI S $ VHWLV RSHQ LILW FRQWDLQVQRQHRILWVERXQGDU\SRLQWV 7KLVLVHTXLYDOHQWWRVD\LQJWKDWDOORILWV SRLQWVDUHLQWHULRUSRLQWV $ VHWLV FORVHG LILWFRQWDLQVDOORILWVERXQGDU\SRLQWV ,I S FRQWDLQVVRPHEXWQRWDOORILWVERXQGDU\SRLQWV WKHQ S LVQHLWKHURSHQQRU FORVHG $ VHW S LV ERXQGHG LIWKHUHH[LVWVD ı > 0 VXFKWKDW S $ B.; ı/ $ VHW LQ Rn LV FRPSDFW LILWLVFORVHGDQGERXQGHG 7KHRUHP 7KHFRQYH[KXOORIDQRSHQVHWLVRSHQ DQGWKHFRQYH[KXOORID FRPSDFWVHWLVFRPSDFW 7KHFRQYH[KXOORIDFORVHGVHWQHHGQRWEHFORVHG 6HH ([HUFLVH

y S p2 p1

x B(0, 3)

'*(63&

7KHVHW S LVFORVHGDQGERXQGHG

&9".1-& /HW )!

" ! " ! " ! "* "2 "2 2 2 ; ; ; ; 2 "2 "2 2

!

" ! " "1 2 S D FRQY S1 D ; DQG S2 D ; 0 1 % & DVVKRZQLQ)LJ 7KHQ S1 LVDQLQWHULRUSRLQWVLQFH B S; 34 $ S 7KHSRLQW S2 LVDERXQGDU\SRLQWVLQFHHYHU\RSHQEDOOFHQWHUHGDW S2 LQWHUVHFWVERWK S DQGWKH FRPSOHPHQWRI S 7KHVHW S LVFORVHGVLQFHLWFRQWDLQVDOOLWVERXQGDU\SRLQWV 7KH VHW S LVERXQGHGVLQFH S $ B.; 3/ 7KXV S LVDOVRFRPSDFW

$)"15&3


1RWDWLRQ ,I f LVDOLQHDUIXQFWLRQDO WKHQ f .A/ ' d PHDQV f .[/ ' d IRUHDFK [ 2 A &RUUHVSRQGLQJQRWDWLRQVZLOOEHXVHGZKHQWKHLQHTXDOLWLHVDUHUHYHUVHGRUZKHQWKH\ DUHVWULFW

%&'*/*5*0/

7KHK\SHUSODQH H D Œf d ! VHSDUDWHV WZRVHWV A DQG B LIRQHRIWKHIROORZLQJ KROGV L f .A/ ' d DQG f .B/ & d LL f .A/ & d DQG f .B/ ' d

RU

,ILQWKHFRQGLWLRQVDERYHDOOWKHZHDNLQHTXDOLWLHVDUHUHSODFHGE\VWULFWLQHTXDO LWLHV WKHQ H LVVDLGWR VWULFWO\VHSDUDWH A DQG B 1RWLFHWKDWVWULFWVHSDUDWLRQUHTXLUHVWKDWWKHWZRVHWVEHGLVMRLQW ZKLOHPHUHVHS DUDWLRQGRHVQRW ,QGHHG LIWZRFLUFOHVLQWKHSODQHDUHH[WHUQDOO\WDQJHQW WKHQWKHLU FRPPRQWDQJHQWOLQHVHSDUDWHVWKHPEXWGRHVQRWVHSDUDWHWKHPVWULFWO\ $OWKRXJKLWLVQHFHVVDU\WKDWWZRVHWVEHGLVMRLQWLQRUGHUWRVWULFWO\VHSDUDWHWKHP WKLVFRQGLWLRQLVQRWVXIÀFLHQW HYHQIRUFORVHGFRQYH[VHWV )RUH[DPSOH OHW )! " * )! " * 1 1 x x AD W x & DQG ' y ' 2 DQG B D W x & 0 DQG y D 0 y y 2 x

7KHQ A DQG B DUHGLVMRLQWFORVHGFRQYH[VHWV EXWWKH\FDQQRWEHVWULFWO\VHSDUDWHG E\DK\SHUSODQHOLQHLQ R2 6HH)LJ 7KXVWKHSUREOHPRIVHSDUDWLQJRUVWULFWO\ VHSDUDWLQJ WZRVHWVE\DK\SHUSODQHLVPRUHFRPSOH[WKDQLWPLJKWDWÀUVWDSSHDU y

2

x 2

4

'*(63& 'LVMRLQWFORVHGFRQYH[VHWV

7KHUHDUHPDQ\LQWHUHVWLQJFRQGLWLRQVRQWKHVHWV A DQG B WKDWLPSO\WKHH[LVWHQFH RIDVHSDUDWLQJK\SHUSODQH EXWWKHIROORZLQJWZRWKHRUHPVDUHVXIÀFLHQWIRUWKLVVHFWLRQ 7KHSURRIRIWKHÀUVWWKHRUHPUHTXLUHVTXLWHDELWRISUHOLPLQDU\PDWHULDOó EXWWKHVHFRQG WKHRUHPIROORZVHDVLO\IURPWKHÀUVW

5)&03&.

6XSSRVH A DQG B DUHQRQHPSW\FRQYH[VHWVVXFKWKDW A LVFRPSDFWDQG B LV FORVHG 7KHQWKHUHH[LVWVDK\SHUSODQH H WKDWVWULFWO\VHSDUDWHV A DQG B LIDQG RQO\LI A \ B D ¿

5)&03&.

6XSSRVH A DQG B DUHQRQHPSW\FRPSDFWVHWV 7KHQWKHUHH[LVWVDK\SHUSODQHWKDW VWULFWO\VHSDUDWHV A DQG B LIDQGRQO\LI .FRQY A/ \ .FRQY B/ D ¿ ó $ SURRIRI7KHRUHPLVJLYHQLQ6WHYHQ5/D\ &RQYH[6HWVDQG7KHLU$SSOLFDWLRQV 1HZ

+\SHUSODQHV

1300' 6XSSRVHWKDW .FRQY A/ \ .FRQY B/ D ¿ 6LQFHWKHFRQYH[KXOORIDFRPSDFW VHWLVFRPSDFW 7KHRUHPHQVXUHVWKDWWKHUHLVDK\SHUSODQH H WKDWVWULFWO\VHSDUDWHV FRQY A DQG FRQY B &OHDUO\ H DOVRVWULFWO\VHSDUDWHVWKHVPDOOHUVHWV A DQG B &RQYHUVHO\ VXSSRVHWKHK\SHUSODQH H D Œf d ! VWULFWO\VHSDUDWHV A DQG B :LWK RXWORVVRIJHQHUDOLW\ DVVXPHWKDW f .A/ < d DQG f .B/ > d /HW [ D c1 [1 C ! ! ! C ck [k EHDQ\FRQYH[FRPELQDWLRQRIHOHPHQWVRI A 7KHQ f .[/ D c1 f .[1 / C ! ! ! C ck f .[k / < c1 d C ! ! ! C ck d D d

VLQFH c1 C ! ! ! C ck D 1 7KXV f .FRQY A/ < d /LNHZLVH f .FRQY B/ > d VR H D Œf d ! VWULFWO\VHSDUDWHV FRQY A DQG FRQY B %\7KHRUHP FRQY A DQG FRQY B PXVW EHGLVMRLQW

&9".1-& /HW

2 3 2 3 2 3 2 3 2 "3 3 1 D1 D 4 1 5; D2 D 4 2 5; D3 D 4 4 5; E1 D 4 0 5; 1 1 0 2

2

3 2 DQG E2 D 4 "1 5; 5

DQGOHW A D fD1 ; D2 ; D3 g DQG B D fE1 ; E2 g 6KRZWKDWWKHK\SHUSODQH H D Œf 5! ZKHUH f .x1 ; x2 ; x3 / D 2x1 " 3x2 C x3 GRHVQRWVHSDUDWH A DQG B ,VWKHUHDK\SHUSODQH SDUDOOHOWR H WKDWGRHVVHSDUDWH A DQG B " 'RWKHFRQYH[KXOOVRI A DQG B LQWHUVHFW"

40-65*0/ (YDOXDWHWKHOLQHDUIXQFWLRQDO f DWHDFKRIWKHSRLQWVLQ A DQG B f .D1 / D 2;

f .D2 / D "11;

f .D3 / D "6;

f .E1 / D 4;

DQG

f .E2 / D 12

6LQFH f .E1 / D 4 LVOHVVWKDQDQG f .E2 / D 12 LVJUHDWHUWKDQ SRLQWVRI B OLHRQERWK VLGHVRI H D Œf 5! DQGVR H GRHVQRWVHSDUDWH A DQG B 6LQFH f .A/ < 3 DQG f .B/ > 3 WKHSDUDOOHOK\SHUSODQH Œf 3! VWULFWO\VHSDUDWHV A DQG B %\7KHRUHP .FRQY A/ \ .FRQY B/ D ¿ &DXWLRQ ,IWKHUHZHUHQRK\SHUSODQHSDUDOOHOWR H WKDWVWULFWO\VHSDUDWHG A DQG B WKLVZRXOG QRW QHFHVVDULO\LPSO\WKDWWKHLUFRQYH[KXOOVLQWHUVHFW ,WPLJKWEHWKDWVRPH RWKHUK\SHUSODQHQRWSDUDOOHOWR H ZRXOGVWULFWO\VHSDUDWHWKHP

13"$5*$& 130#-&. 2 3 2 3 2 3 2 3 1 "1 1 "2 /HW S1 D 4 0 5 S2 D 4 2 5 Q1 D 4 1 5 Q2 D 4 1 5 OHW H1 EHWKHK\SHUSODQH 2 1 "2 3 3 SODQH LQ R SDVVLQJWKURXJKWKHSRLQW S1 DQGKDYLQJQRUPDOYHFWRU Q1 DQGOHW H2 EHWKHK\SHUSODQHSDVVLQJWKURXJKWKHSRLQW S2 DQGKDYLQJQRUPDOYHFWRU Q2 *LYHDQ H[SOLFLWGHVFULSWLRQRI H1 \ H2 E\DIRUPXODWKDWVKRZVKRZWRJHQHUDWHDOOSRLQWVLQ H1 \ H2

&9&3$*4&4

" ! " "1 3 DQG 4 1 )LQGDOLQHDUIXQFWLRQDO f DQGDUHDOQXPEHU d VXFKWKDW L D Œf d !

/HW L EHWKHOLQHLQ R2 WKURXJKWKHSRLQWV

!

! " ! " 1 "2 DQG 4 "1 )LQGDOLQHDUIXQFWLRQDO f DQGDUHDOQXPEHU d VXFKWKDW L D Œf d !

/HW L EHWKHOLQHLQ R2 WKURXJKWKHSRLQWV

$)"15&3


,Q([HUFLVHVDQG GHWHUPLQHZKHWKHUHDFKVHWLVRSHQRUFORVHG RUQHLWKHURSHQQRUFORVHG D f.x; y/ W y > 0g

E f.x; y/ W x D 2 DQG 1 ' y ' 3g F f.x; y/ W x D 2 DQG 1 < y < 3g G f.x; y/ W xy D 1 DQG x > 0g H f.x; y/ W xy & 1 DQG x > 0g

D f.x; y/ W x 2 C y 2 D 1g E f.x; y/ W x 2 C y 2 > 1g

F f.x; y/ W x 2 C y 2 ' 1 DQG y > 0g G f.x; y/ W y & x 2 g H f.x; y/ W y < x 2 g ,Q([HUFLVHVDQG GHWHUPLQHZKHWKHURUQRWHDFKVHWLVFRPSDFW DQGZKHWKHURUQRWLWLVFRQYH[ 8VHWKHVHWVIURP([HUFLVH 8VHWKHVHWVIURP([HUFLVH ,Q([HUFLVHV² OHW H EHWKHK\SHUSODQHWKURXJKWKHOLVWHG SRLQWV D )LQGDYHFWRU Q WKDWLVQRUPDOWRWKHK\SHUSODQH E )LQGDOLQHDUIXQFWLRQDO f DQGDUHDOQXPEHU d VXFKWKDW H D Œf d ! 2 3 2 3 2 3 2 3 2 3 2 3 1 2 "1 1 4 7 4 "2 5 4 "2 5 4 "4 5 4 1 5 4 4 5 4 "2 5 3 1 5 1 3 4 2 3 2 3 2 3 2 3 1 2 1 1 607 637 627 617 7 6 7 6 7 6 7 6 4 1 5 4 1 5 4 2 5 4 1 5 0 0 0 1 2 3 2 3 2 3 2 3 1 2 1 3 627 6 27 637 6 27 7 6 7 6 7 6 7 6 4 0 5 4 "1 5 4 2 5 4 "1 5 0 "3 7 "1 2 3 2 3 2 3 2 3 1 2 0 "2 6 "3 7 6 17 617 6 07 7 6 7 6 7 6 7 /HW S D 6 4 1 5 Q D 4 5 5 Y1 D 4 1 5 Y2 D 4 1 5 2 "1 1 3 2 3 1 647 4 7 DQG Y3 D 6 4 0 5 DQGOHW H EHWKHK\SHUSODQHLQ R ZLWK 4 QRUPDO Q DQGSDVVLQJWKURXJK S :KLFKRIWKHSRLQWV Y1 Y2 DQG Y3 DUHRQWKHVDPHVLGHRI H DVWKHRULJLQ DQGZKLFKDUH QRW" 2 3 2 3 2 3 2 3 2 3 "1 0 /HW D1 D 4 "1 5 D2 D 4 1 5 D3 D 4 6 5 E1 D 4 5 5 5 3 0 "1 2 3 2 3 2 3 1 2 3 E2 D 4 "3 5 E3 D 4 2 5 DQG Q D 4 1 5 DQG OHW "2 1 "2 A D fD1 ; D2 ; D3 g DQG B D fE1 ; E2 ; E3 g )LQGDK\SHUSODQH H

ZLWKQRUPDO Q WKDWVHSDUDWHV A DQG B ,VWKHUHDK\SHUSODQH SDUDOOHOWR H WKDWVWULFWO\VHSDUDWHV A DQG B " 2 3 2 3 2 3 2 1 1 6 "3 7 6 27 627 6 7 6 7 6 /HW S1 D 4 S2 D 4 Q1 D 4 7 DQG 15 "1 5 45 2 3 2 2 3 2 637 4 7 Q2 D 6 4 1 5 OHW H1 EHWKHK\SHUSODQHLQ R WKURXJK S1 ZLWK 5 QRUPDO Q1 DQGOHW H2 EHWKHK\SHUSODQHWKURXJK S2 ZLWK QRUPDO Q2 *LYHDQH[SOLFLWGHVFULSWLRQRI H1 \ H2 >+LQW )LQGDSRLQW S LQ H1 \ H2 DQGWZROLQHDUO\LQGHSHQGHQW YHFWRUV Y1 DQG Y2 WKDWVSDQDVXEVSDFHSDUDOOHOWRWKH GLPHQVLRQDOÁDW H1 \ H2 @

/HW F1 DQG F2 EHGLPHQVLRQDOÁDWVLQ R6 DQGVXSSRVHWKDW F1 \ F2 ¤ ¿ :KDWDUHWKHSRVVLEOHGLPHQVLRQVRI F1 \ F2 "

,Q([HUFLVHV² ZULWHDIRUPXODIRUDOLQHDUIXQFWLRQDO f DQG VSHFLI\DQXPEHU d VRWKDW Œf d ! LVWKHK\SHUSODQH H GHVFULEHG LQWKHH[HUFLVH ' ( /HW A EHWKH 1 % 4 PDWUL[ 1 "3 4 "2 DQGOHW b D 5 /HW H D f[ LQ R4 W A[ D bg ' ( /HW A EHWKH 1 % 5 PDWUL[ 2 5 "3 0 6 1RWHWKDW 1XO A LVLQ R5 /HW H D 1XO A

/HW H EH WKH SODQH LQ R3 VSDQQHG E\ WKH URZV RI B D ! " 1 3 5 7KDWLV H D 5RZ B >+LQW +RZLV H 0 2 4 UHODWHGWR 1XO B " 6HH6HFWLRQ @ /HW H EH WKH SODQH LQ R3 VSDQQHG E\ WKH URZV RI B D ! " 1 4 "5 7KDWLV H D 5RZ B 0 "2 8 2 3 1 0 2 5 /HW H EHWKHFROXPQVSDFHRIWKHPDWUL[ B D 4 4 "7 "6

7KDWLV H D &RO B >+LQW +RZLV &RO B UHODWHGWR 1XO B T " 6HH6HFWLRQ @ 2 3 1 0 2 5 /HW H EHWKHFROXPQVSDFHRIWKHPDWUL[ B D 4 5 "4 "4 7KDWLV H D &RO B

,Q([HUFLVHVDQG PDUNHDFKVWDWHPHQW7UXHRU)DOVH -XVWLI\ HDFKDQVZHU D $ OLQHDUWUDQVIRUPDWLRQIURP R WR Rn LVFDOOHGDOLQHDU IXQFWLRQDO E ,I f LVDOLQHDUIXQFWLRQDOGHÀQHGRQ Rn WKHQWKHUHH[LVWV DUHDOQXPEHU k VXFKWKDW f .[/ D k [ IRUDOO [ LQ Rn F ,I D K\SHUSODQH VWULFWO\ VHSDUDWHV VHWV A DQG B WKHQ A \ B D ¿

G ,I A DQG B DUHFORVHGFRQYH[VHWVDQG A \ B D ¿ WKHQ WKHUHH[LVWVDK\SHUSODQHWKDWVWULFWO\VHSDUDWHV A DQG B

D ,I d LVDUHDOQXPEHUDQG f LVDQRQ]HUROLQHDUIXQFWLRQDO GHÀQHGRQ Rn WKHQ Œf d ! LVDK\SHUSODQHLQ Rn E *LYHQ DQ\ YHFWRU Q DQG DQ\ UHDO QXPEHU d WKH VHW f[ W Q ! [ D d g LVDK\SHUSODQH

F ,I A DQG B DUHQRQHPSW\GLVMRLQWVHWVVXFKWKDW A LV FRPSDFWDQG B LVFORVHG WKHQWKHUHH[LVWVDK\SHUSODQH WKDWVWULFWO\VHSDUDWHV A DQG B

G ,I WKHUH H[LVWV DK\SHUSODQH H VXFK WKDW H GRHV QRW VWULFWO\ VHSDUDWH WZR VHWV A DQG B WKHQ .FRQY A/ \ .FRQY B/ ¤ ¿ ! " ! " ! " ! " 1 3 5 4 Y2 D Y3 D DQG S D )LQG /HW Y1 D 1 0 3 1 DK\SHUSODQH Œf d ! LQWKLVFDVH DOLQH WKDWVWULFWO\VHSDUDWHV S IURP FRQY fY1 ; Y2 ; Y3 g ! " ! " ! " 1 5 4 Y2 D Y3 D 5HSHDW([HUFLVHIRU Y1 D 2 1 4 ! " 2 DQG S D 3

3RO\WRSHV

! " 4 )LQGDK\SHUSODQH Œf d ! WKDWVWULFWO\VHSD 1 UDWHV B.; 3/ DQG B.S; 1/ >+LQW $IWHUÀQGLQJ f VKRZWKDW WKHSRLQW Y D .1 " :75/ C :75S LVQHLWKHULQ B.; 3/ QRULQ B.S; 1/:! ! " ! " 2 6 /HW T D DQG S D )LQGDK\SHUSODQH Œf d ! WKDW 3 1 VWULFWO\VHSDUDWHV B.T; 3/ DQG B.S; 1/

/HW S D

*LYHDQH[DPSOHRIDFORVHGVXEVHW S RI R2 VXFKWKDW FRQY S LVQRWFORVHG *LYHDQH[DPSOHRIDFRPSDFWVHW A DQGDFORVHGVHW B LQ R2 VXFKWKDW .FRQY A/ \ .FRQY B/ D ¿ EXW A DQG B FDQQRWEH VWULFWO\VHSDUDWHGE\DK\SHUSODQH 3URYHWKDWWKHRSHQEDOO B.S; ı/ D f[ W k[ " Sk < ıg LVD FRQYH[VHW >+LQW 8VHWKH7ULDQJOH,QHTXDOLW\@ 3URYHWKDWWKHFRQYH[KXOORIDERXQGHGVHWLVERXQGHG

40-65*0/ 50 13"$5*$& 130#-&. )LUVW FRPSXWH Q1 ! S1 D "3 DQG Q2 ! S2 D 7 7KHK\SHUSODQH H1 LVWKHVROXWLRQVHWRIWKH HTXDWLRQ x1 C x2 " 2x3 D "3 DQG H2 LVWKHVROXWLRQVHWRIWKHHTXDWLRQ "2x1 C x2 C 3x3 D 7 7KHQ

H1 \ H2 D f[ W x1 C x2 " 2x3 D "3 DQG "2x1 C x2 C 3x3 D 7g 7KLVLVDQLPSOLFLWGHVFULSWLRQRI H1 \ H2 7RÀQGDQH[SOLFLWGHVFULSWLRQ VROYHWKH V\VWHPRIHTXDWLRQVE\URZUHGXFWLRQ !

1 "2

1 "2 "3 1 3 7

"

#

"

1 0

0 " 53 " 10 3 1 " 13

2

6 C 13 x3 x3 D x3 /HW S D 6 4

1 3

" 10 3

#

3

2 3 5

7 637 5 1 1 7 DQG Y D 6 1 7 7KH 7KXV x1 D " 10 C x x D 2 3 3 3 3 435 3 5 0 1 JHQHUDOVROXWLRQFDQEHZULWWHQDV [ D S C x3 Y 7KXV H1 \ H2 LVWKHOLQHWKURXJK S LQ WKHGLUHFWLRQRI Y 1RWHWKDW Y LVRUWKRJRQDOWRERWK Q1 DQG Q2

10-:501&4 7KLVVHFWLRQVWXGLHVJHRPHWULFSURSHUWLHVRIDQLPSRUWDQWFODVVRIFRPSDFWFRQYH[VHWV FDOOHGSRO\WRSHV 7KHVHVHWVDULVHLQDOOVRUWVRIDSSOLFDWLRQV LQFOXGLQJJDPHWKHRU\ 6HFWLRQ OLQHDUSURJUDPPLQJ6HFWLRQVWR DQGPRUHJHQHUDORSWLPL]DWLRQ SUREOHPV VXFKDVWKHGHVLJQRIIHHGEDFNFRQWUROVIRUHQJLQHHULQJV\VWHPV

$)"15&3


$ SRO\WRSH LQ Rn LVWKHFRQYH[KXOORIDÀQLWHVHWRISRLQWV ,Q R2 DSRO\WRSH LVVLPSO\DSRO\JRQ ,Q R3 DSRO\WRSHLVFDOOHGDSRO\KHGURQ ,PSRUWDQWIHDWXUHVRI DSRO\KHGURQDUHLWVIDFHV HGJHV DQGYHUWLFHV )RUH[DPSOH WKHFXEHKDVVTXDUH IDFHV HGJHV DQGYHUWLFHV 7KHIROORZLQJGHÀQLWLRQVSURYLGHWHUPLQRORJ\IRU KLJKHUGLPHQVLRQVDVZHOODV R2 DQG R3 5HFDOOWKDWWKHGLPHQVLRQRIDVHWLQ Rn LV WKHGLPHQVLRQRIWKHVPDOOHVWÁDWWKDWFRQWDLQVLW $OVR QRWHWKDWDSRO\WRSHLVDVSHFLDO W\SHRIFRPSDFWFRQYH[VHW EHFDXVHDÀQLWHVHWLQ Rn LVFRPSDFWDQGWKHFRQYH[KXOORI WKLVVHWLVFRPSDFW E\WKHWKHRUHPLQWKHWRSRORJ\WHUPVDQGIDFWVER[LQ6HFWLRQ

%&'*/*5*0/

/HW S EHDFRPSDFWFRQYH[VXEVHWRI Rn $ QRQHPSW\VXEVHW F RI S LVFDOOHG DSURSHU IDFH RI S LI F ¤ S DQGWKHUHH[LVWVDK\SHUSODQH H D Œf d ! VXFK WKDW F D S \ H DQGHLWKHU f .S / ' d RU f .S / & d 7KHK\SHUSODQH H LVFDOOHG D VXSSRUWLQJK\SHUSODQH WR S ,IWKHGLPHQVLRQRI F LV k WKHQ F LVFDOOHGD NIDFH RI S ,I P LVDSRO\WRSHRIGLPHQVLRQ k WKHQ P LVFDOOHGD NSRO\WRSH $ IDFH RI P LVFDOOHGD YHUWH[ SOXUDO YHUWLFHV DIDFHLVDQ HGJH DQGD .k " 1/ GLPHQVLRQDOIDFHLVD IDFHW RI S

&9".1-& 6XSSRVH S LVDFXEHLQ R3 :KHQDSODQH H LVWUDQVODWHGWKURXJK R3 XQWLOLWMXVWWRXFKHVVXSSRUWV WKHFXEHEXWGRHVQRWFXWWKURXJKWKHLQWHULRURIWKH FXEH WKHUHDUHWKUHHSRVVLELOLWLHVIRU H \ S GHSHQGLQJRQWKHRULHQWDWLRQRI H 6HH )LJXUH H \ S PD\EHDGLPHQVLRQDOVTXDUHIDFHIDFHW RIWKHFXEH H \ S PD\EHDGLPHQVLRQDOHGJHRIWKHFXEH H \ S PD\EHDGLPHQVLRQDOYHUWH[RIWKHFXEH H H

S

S

S

H H ! S is 2-dimensional.

H ! S is 1-dimensional.

H ! S is 0-dimensional.

'*(63&

0RVWDSSOLFDWLRQVRISRO\WRSHVLQYROYHWKHYHUWLFHVLQVRPHZD\ EHFDXVHWKH\KDYH DVSHFLDOSURSHUW\WKDWLVLGHQWLÀHGLQWKHIROORZLQJGHÀQLWLRQ

%&'*/*5*0/

/HW S EHDFRQYH[VHW $ SRLQW S LQ S LVFDOOHGDQ H[WUHPHSRLQW RI S LI S LV QRWLQWKHLQWHULRURIDQ\OLQHVHJPHQWWKDWOLHVLQ S 0RUHSUHFLVHO\ LI [; \ 2 S DQG S 2 xy WKHQ S D [ RU S D \ 7KHVHWRIDOOH[WUHPHSRLQWVRI S LVFDOOHGWKH SURÀOH RI S

3RO\WRSHV

$ YHUWH[RIDQ\FRPSDFWFRQYH[VHW S LVDXWRPDWLFDOO\DQH[WUHPHSRLQWRI S 7KLV IDFWLVSURYHGGXULQJWKHSURRIRI7KHRUHP EHORZ ,QZRUNLQJZLWKDSRO\WRSH VD\ P D FRQY fY1 ; : : : ; Yk g IRU Y1 ; : : : ; Yk LQ Rn LWLVXVXDOO\KHOSIXOWRNQRZWKDW Y1 ; : : : ; Yk DUHWKHH[WUHPHSRLQWVRI P +RZHYHU VXFKDOLVWPLJKWFRQWDLQH[WUDQHRXVSRLQWV )RU H[DPSOH VRPHYHFWRU Yi FRXOGEHWKHPLGSRLQWRIDQHGJHRIWKHSRO\WRSH 2IFRXUVH LQWKLVFDVH Yi LVQRWUHDOO\QHHGHGWRJHQHUDWHWKHFRQYH[KXOO 7KHIROORZLQJGHÀQLWLRQ GHVFULEHVWKHSURSHUW\RIWKHYHUWLFHVWKDWZLOOPDNHWKHPDOOH[WUHPHSRLQWV

%&'*/*5*0/

7KHVHW fY1 ; : : : ; Yk g LVD PLQLPDOUHSUHVHQWDWLRQ RIWKHSRO\WRSH P LI P D FRQY fY1 ; : : : ; Yk g DQGIRUHDFK i D 1; : : : ; k; Yi 62 FRQY fYj W j ¤ i g (YHU\SRO\WRSHKDVDPLQLPDOUHSUHVHQWDWLRQ )RULI P D FRQY fY1 ; : : : ; Yk g DQGLI VRPH Yi LVDFRQYH[FRPELQDWLRQRIWKHRWKHUSRLQWV WKHQ Yi PD\EHGHOHWHGIURPWKH VHWRISRLQWVZLWKRXWFKDQJLQJWKHFRQYH[KXOO 7KLVSURFHVVPD\EHUHSHDWHGXQWLOWKH PLQLPDOUHSUHVHQWDWLRQLVOHIW ,WFDQEHVKRZQWKDWWKHPLQLPDOUHSUHVHQWDWLRQLVXQLTXH

5)&03&.

6XSSRVH M D fY1 ; : : : ; Yk g LVWKHPLQLPDOUHSUHVHQWDWLRQRIWKHSRO\WRSH P 7KHQ WKHIROORZLQJWKUHHVWDWHPHQWVDUHHTXLYDOHQW D S 2 M E S LVDYHUWH[RI P F S LVDQH[WUHPHSRLQWRI P

H

p

'*(63&

H'

Q

1300' D ) E 6XSSRVH S 2 M DQGOHW Q D FRQY fY W Y 2 M DQG Y ¤ Sg ,WIRO ORZVIURPWKHGHÀQLWLRQRI M WKDW S 62 Q DQGVLQFH Q LVFRPSDFW 7KHRUHPLPSOLHV WKHH[LVWHQFHRIDK\SHUSODQH H 0 WKDWVWULFWO\VHSDUDWHV fSg DQG Q /HW H EHWKH K\SHUSODQHWKURXJK S SDUDOOHOWR H 0 6HH)LJ 7KHQ Q OLHVLQRQHRIWKHFORVHGKDOIVSDFHV H C ERXQGHGE\ H DQGVR P ) H C 7KXV H VXSSRUWV P DW S )XUWKHUPRUH S LVWKHRQO\SRLQWRI P WKDWFDQOLHRQ H VR H \ P D fSg DQG S LVDYHUWH[RI P E ) F /HW S EHDYHUWH[RI P 7KHQWKHUHH[LVWVDK\SHUSODQH H D Œf d ! VXFK WKDW H \ P D fSg DQG f .P / & d ,I S ZHUHQRWDQH[WUHPHSRLQW WKHQWKHUHZRXOG H[LVWSRLQWV [ DQG \ LQ P VXFKWKDW S D .1 " c/[ C c \ ZLWK 0 < c < 1 7KDWLV + , + , 1 1 " 1 .[/ c \ D S " .1 " c/[ DQG \ D .S/ " c c + , 1 1 " 1 f .[/ %XW f .S/ D d DQG f .[/ & d VR ,WIROORZVWKDW f .\/ D f .S/ " c c + , + , 1 1 " 1 .d / D d f .\/ ' .d / " c c 2QWKHRWKHUKDQG \ 2 P VR f .\/ & d ,WIROORZVWKDW f .\/ D d DQGWKDW \ 2 H \ P 7KLVFRQWUDGLFWVWKHIDFWWKDW S LVDYHUWH[ 6R S PXVWEHDQH[WUHPHSRLQW 1RWHWKDW WKLVSDUWRIWKHSURRIGRHVQRWGHSHQGRQ P EHLQJDSRO\WRSH ,WKROGVIRUDQ\FRPSDFW FRQYH[VHW F ) D ,WLVFOHDUWKDWDQ\H[WUHPHSRLQWRI P PXVWEHDPHPEHURI M

$)"15&3


&9".1-& 5HFDOOWKDWWKHSURÀOHRIDVHW S LVWKHVHWRIH[WUHPHSRLQWVRI S 7KHRUHPVKRZVWKDWWKHSURÀOHRIDSRO\JRQLQ R2 LVWKHVHWRIYHUWLFHV 6HH)LJ 7KHSURÀOHRIDFORVHGEDOOLVLWVERXQGDU\ $QRSHQVHWKDVQRH[WUHPHSRLQWV VRLWV SURÀOHLVHPSW\ $ FORVHGKDOIVSDFHKDVQRH[WUHPHSRLQWV VRLWVSURÀOHLVHPSW\

'*(63&

([HUFLVHDVNV\RXWRVKRZWKDWDSRLQW S LQDFRQYH[VHW S LVDQH[WUHPHSRLQW RI S LIDQGRQO\LI ZKHQ S LVUHPRYHGIURP S WKHUHPDLQLQJSRLQWVVWLOOIRUPDFRQYH[ VHW ,WIROORZVWKDWLI S " LVDQ\VXEVHWRI S VXFKWKDW FRQY S " LVHTXDOWR S WKHQ S " PXVW FRQWDLQWKHSURÀOHRI S 7KHVHWVLQ([DPSOHVKRZWKDWLQJHQHUDO S " PD\KDYHWREH ODUJHUWKDQWKHSURÀOHRI S ,WLVWUXH KRZHYHU WKDWZKHQ S LVFRPSDFWZHPD\DFWXDOO\ WDNH S " WREHWKHSURÀOHRI S DV7KHRUHPZLOOVKRZ 7KXVHYHU\QRQHPSW\FRPSDFW VHW S KDVDQH[WUHPHSRLQW DQGWKHVHWRIDOOH[WUHPHSRLQWVLVWKHVPDOOHVWVXEVHWRI S ZKRVHFRQYH[KXOOLVHTXDOWR S

5)&03&.

/HW S EHDQRQHPSW\FRPSDFWFRQYH[VHW 7KHQ S LVWKHFRQYH[KXOORILWVSURÀOH WKHVHWRIH[WUHPHSRLQWVRI S

1300' 7KHSURRILVE\LQGXFWLRQRQWKHGLPHQVLRQRIWKHVHW S ñ 2QHLPSRUWDQWDSSOLFDWLRQRI7KHRUHPLVWKHIROORZLQJWKHRUHP ,WLVRQHRIWKH NH\WKHRUHWLFDOUHVXOWVLQWKHGHYHORSPHQWRIOLQHDUSURJUDPPLQJ /LQHDUIXQFWLRQDOV DUHFRQWLQXRXV DQGFRQWLQXRXVIXQFWLRQVDOZD\VDWWDLQWKHLUPD[LPXPDQGPLQLPXP RQDFRPSDFWVHW 7KHVLJQLÀFDQFHRI7KHRUHPLVWKDWIRUFRPSDFWFRQYH[VHWV WKH PD[LPXPDQGPLQLPXP LVDFWXDOO\DWWDLQHGDWDQH[WUHPHSRLQWRI S

5)&03&.

/HW f EHDOLQHDUIXQFWLRQDOGHÀQHGRQDQRQHPSW\FRPSDFWFRQYH[VHW S 7KHQ O RI S VXFKWKDW WKHUHH[LVWH[WUHPHSRLQWV YO DQG Z

f .OY/ D PD[ f .Y/ Y 2S

DQG

O / D PLQ f .Y/ f .Z Y2S

1300' $VVXPHWKDW f DWWDLQVLWVPD[LPXP m RQ S DWVRPHSRLQW Y0 LQ S 7KDWLV f .Y0 / D m :HZLVKWRVKRZWKDWWKHUHH[LVWVDQH[WUHPHSRLQWLQ S ZLWKWKHVDPH SURSHUW\ %\7KHRUHP Y0 LVDFRQYH[FRPELQDWLRQRIWKHH[WUHPHSRLQWVRI S 7KDW LV WKHUHH[LVWH[WUHPHSRLQWV Y1 ; : : : ; Yk RI S DQGQRQQHJDWLYH c1 ; : : : ; ck VXFKWKDW Y0 D c1 Y1 C ! ! ! C ck Yk

ZLWK c1 C ! ! ! C ck D 1

,IQRQHRIWKHH[WUHPHSRLQWVRI S VDWLVÀHV f .Y/ D m WKHQ

f .Y i / < m

IRU i D 1; : : : ; k

ñ 7KHGHWDLOVPD\EHIRXQGLQ6WHYHQ5/D\ &RQYH[6HWVDQG7KHLU$SSOLFDWLRQV 1HZ

3RO\WRSHV

VLQFH m LVWKHPD[LPXPRI f RQ S %XWWKHQ EHFDXVH f LVOLQHDU

m D f .Y0 / D f .c1 Y1 C ! ! ! C ck Yk / D c1 f .Y1 / C ! ! ! C ck f .Yk / < c1 m C ! ! ! C ck m D m.c1 C ! ! ! C ck / D m 7KLVFRQWUDGLFWLRQLPSOLHVWKDWVRPHH[WUHPHSRLQW YO RI S PXVWVDWLVI\ f .OY/ D m O LVVLPLODU 7KHSURRIIRU Z !

" ! " ! " "1 3 1 &9".1-& *LYHQSRLQWV S1 D S2 D DQG S3 D LQ R2 OHW S D 0 1 2 FRQY fS1 ; S2 ; S3 g )RUHDFKOLQHDUIXQFWLRQDO f ÀQGWKHPD[LPXPYDOXH m RI f RQWKH VHW S DQGÀQGDOOSRLQWV [ LQ S DWZKLFK f .[/ D m

D f1 .x1 ; x2 / D x1 C x2

E f2 .x1 ; x2 / D "3x1 C x2

F f3 .x1 ; x2 / D x1 C 2x2

40-65*0/ %\7KHRUHP WKHPD[LPXPYDOXHLVDWWDLQHGDWRQHRIWKHH[WUHPHSRLQWV RI S 6RWRÀQG m HYDOXDWH f DWHDFKH[WUHPHSRLQWDQGVHOHFWWKHODUJHVWYDOXH D f1 .S1 / D "1 f1 .S2 / D 4 DQG f1 .S3 / D 3 VR m1 D 4 *UDSKWKHOLQH f1 .x1 ; x2 / D m1 WKDWLV x1 C x2 D 4 DQGQRWHWKDW [ D S2 LVWKHRQO\SRLQWLQ S DWZKLFK f1 .[/ D 4 6HH)LJ D E f2 .S1 / D 3 f2 .S2 / D "8 DQG f2 .S3 / D "1 VR m1 D 3 *UDSKWKHOLQH f2 .x1 ; x2 / D m2 WKDWLV "3x1 C x2 D 3 DQGQRWHWKDW [ D S1 LVWKHRQO\SRLQWLQ S DWZKLFK f2 .[/ D 3 6HH)LJ E F f3 .S1 / D "1 f3 .S2 / D 5 DQG f3 .S3 / D 5 VR m1 D 5 *UDSKWKHOLQH f3 .x1 ; x2 / D m3 WKDWLV x1 C 2x2 D 5 +HUH f3 DWWDLQVLWVPD[LPXPYDOXHDW S2 DW S3 DQGDW HYHU\SRLQWLQWKHFRQYH[KXOORI S2 DQG S3 6HH)LJ F x2

x2

4

2

4 p3

2 p2

S –2

p1

x2

2

p3

2 p2

S 4

(a) x1 + x2 = 4

4

x1

–2

p1

2

(b) –3x1 + x2 = 3

p3 p2

S 4

x1

–2

p1

2

4

x1

(c) x1 + 2x2 = 5

'*(63&

7KHVLWXDWLRQLOOXVWUDWHGLQ([DPSOH IRU R2 DOVRDSSOLHVLQKLJKHUGLPHQVLRQV 7KH PD[LPXPYDOXHRIDOLQHDUIXQFWLRQDO f RQDSRO\WRSH P RFFXUVDWWKHLQWHUVHFWLRQRI DVXSSRUWLQJK\SHUSODQHDQG P 7KLVLQWHUVHFWLRQLVHLWKHUDVLQJOHH[WUHPHSRLQWRI P RUWKHFRQYH[KXOORIRUPRUHH[WUHPHSRLQWVRI P ,QHLWKHUFDVH WKHLQWHUVHFWLRQLV DSRO\WRSH DQGLWVH[WUHPHSRLQWVIRUPDVXEVHWRIWKHH[WUHPHSRLQWVRI P %\GHÀQLWLRQ DSRO\WRSHLVWKHFRQYH[KXOORIDÀQLWHVHWRISRLQWV 7KLVLVDQ H[SOLFLWUHSUHVHQWDWLRQRIWKHSRO\WRSHVLQFHLWLGHQWLÀHVSRLQWVLQWKHVHW $ SRO\WRSH PD\DOVREHUHSUHVHQWHGLPSOLFLWO\DVWKHLQWHUVHFWLRQRIDÀQLWHQXPEHURIFORVHGKDOI VSDFHV ([DPSOH LOOXVWUDWHVWKLVLQ R2

$)"15&3


&9".1-& /HW

! " 0 S1 D ; 1

! " 1 S2 D ; 0

! " 3 S3 D 2

DQG

LQ R2 DQGOHW S D FRQY fS1 ; S2 ; S3 g 6LPSOHDOJHEUDVKRZVWKDWWKHOLQHWKURXJK S1 DQG S2 LVJLYHQE\ x1 C x2 D 1 DQG S LVRQWKHVLGHRIWKLVOLQHZKHUH

x1 C x2 & 1

" x1 " x2 ' "1:

RU HTXLYDOHQWO\

6LPLODUO\ WKHOLQHWKURXJK S2 DQG S3 LV x1 " x2 D 1 DQG S LVRQWKHVLGHZKHUH

x1 " x2 ' 1 $OVR WKHOLQHWKURXJK S3 DQG S1 LV "x1 C 3x2 D 3 DQG S LVRQWKHVLGHZKHUH

"x1 C 3x2 ' 3: 6HH)LJXUH ,WIROORZVWKDW S FDQEHGHVFULEHGDVWKHVROXWLRQVHWRIWKHV\VWHPRI OLQHDULQHTXDOLWLHV "x1 " x2 ' "1 x1 " x2 ' 1 "x1 C 3x2 ' 3 7KLVV\VWHPPD\EHZULWWHQDV A[ ' E ZKHUH 2 3 ! " "1 "1 x1 A D 4 1 "1 5; [ D ; x2 "1 3

DQG

2

3 "1 E D 4 1 5: 3

1RWHWKDWDQLQHTXDOLW\EHWZHHQWZRYHFWRUV VXFKDV A[ DQG E DSSOLHVWRHDFKRIWKH FRUUHVSRQGLQJFRRUGLQDWHVLQWKRVHYHFWRUV x2

x1 – x2 = 1

4

2 p1

p3 S p2 2

–2

–x1 + 3x2 = 3

4

x1

x1 + x2 = 1 '*(63&

,Q&KDSWHU LWZLOOEHQHFHVVDU\WRUHSODFHDQLPSOLFLWGHVFULSWLRQRIDSRO\WRSHE\ DPLQLPDOUHSUHVHQWDWLRQRIWKHSRO\WRSH OLVWLQJDOOWKHH[WUHPHSRLQWVRIWKHSRO\WRSH ,QVLPSOHFDVHV DJUDSKLFDOVROXWLRQLVIHDVLEOH 7KHIROORZLQJH[DPSOHVKRZVKRZWR KDQGOHWKHVLWXDWLRQZKHQVHYHUDOSRLQWVRILQWHUHVWDUHWRRFORVHWRLGHQWLI\HDVLO\RQD JUDSK

&9".1-& /HW P EHWKHVHWRISRLQWVLQ R2 WKDWVDWLVI\ A[ ' E ZKHUH 2

1 A D 41 3

3 3 15 2

DQG

DQG [ & )LQGWKHPLQLPDOUHSUHVHQWDWLRQRI P

2

3 18 E D 4 85 21

3RO\WRSHV

40-65*0/ 7KHFRQGLWLRQ [ & SODFHV P LQWKHÀUVWTXDGUDQWRI R2 DW\SLFDOFRQGL WLRQLQOLQHDUSURJUDPPLQJSUREOHPV 7KHWKUHHLQHTXDOLWLHVLQ A[ ' E LQYROYHWKUHH ERXQGDU\OLQHV .1/ x1 C 3x2 D 18

.2/ x1 C x2 D 8

.3/ 3x1 C 2x2 D 21

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ÀHVDOOLQHTXDOLWLHVH[FHSWSRVVLEO\WKHWKLUGLQHTXDOLW\ 7HVWWKLV

3.3/ C 2.5/ D 19 < 21 7KLVLQWHUVHFWLRQSRLQWVDWLVÀHVWKHLQHTXDOLW\IRU VR S12 LVLQWKHSRO\WRSH 7KHLQWHUVHFWLRQRI DQG LV S23 D .5; 3/ 7KLVVDWLVÀHVDOOLQHTXDOLWLHVH[FHSW SRVVLEO\WKHLQHTXDOLW\IRU 7HVWWKLV

1.5/ C 3.3/ D 14 < 18 7KLVVKRZVWKDW S23 LVLQWKHSRO\WRSH % & )LQDOO\ WKHLQWHUVHFWLRQRI DQG LV S13 D 27 ; 33 7HVWWKLVLQWKHLQHTXDOLW\ 7 7 IRU % & % & 1 27 C 1 33 D 60 * 8:6 > 8 7 7 7

7KXV S13 GRHV QRW VDWLVI\WKHVHFRQGLQHTXDOLW\ ZKLFKVKRZVWKDW S13 LV QRW LQ P ,Q FRQFOXVLRQ WKHPLQLPDOUHSUHVHQWDWLRQRIWKHSRO\WRSH P LV )! " ! " ! " ! " ! "* 0 7 3 5 0 ; ; ; ; : 0 0 5 3 6

7KHUHPDLQGHURIWKLVVHFWLRQGLVFXVVHVWKHFRQVWUXFWLRQRIWZREDVLFSRO\WRSHV LQ R3 DQGKLJKHUGLPHQVLRQV 7KHÀUVWDSSHDUVLQOLQHDUSURJUDPPLQJSUREOHPV WKHVXEMHFWRI&KDSWHU %RWKSRO\WRSHVSURYLGHRSSRUWXQLWLHVWRYLVXDOL]H R4 LQD UHPDUNDEOHZD\

6LPSOH[ $ VLPSOH[ LVWKHFRQYH[KXOORIDQDIÀQHO\LQGHSHQGHQWÀQLWHVHWRIYHFWRUV 7RFRQVWUXFW D k GLPHQVLRQDOVLPSOH[RU k VLPSOH[ SURFHHGDVIROORZV VLPSOH[ S 0 VLPSOH[ S 1 VLPSOH[ S 2 :: :

DVLQJOHSRLQW fY1 g FRQY.S 0 [ fY2 g/ ZLWK Y2 QRWLQ DII S 0 FRQY.S 1 [ fY3 g/ ZLWK Y3 QRWLQ DII S 1

k VLPSOH[ S k

FRQY.S k !1 [ fYk C1 g/; ZLWK Yk C1 QRWLQ DII S k !1

7KHVLPSOH[ S 1 LVDOLQHVHJPHQW 7KHWULDQJOH S 2 FRPHVIURPFKRRVLQJDSRLQW Y3 WKDWLVQRWLQWKHOLQHFRQWDLQLQJ S1 DQGWKHQIRUPLQJWKHFRQYH[KXOOZLWK S2 6HH

$)"15&3

7KH*HRPHWU\RI9HFWRU6SDFHV v1

v1

v1

v1

v2

S0

v2

v4

v3

S1

v2

S2

v3 S3

'*(63&

)LJ 7KHWHWUDKHGURQ S3 LVSURGXFHGE\FKRRVLQJDSRLQW Y4 QRWLQWKHSODQHRI S 2 DQGWKHQIRUPLQJWKHFRQYH[KXOOZLWK S 2 %HIRUHFRQWLQXLQJ FRQVLGHUVRPHRIWKHSDWWHUQVWKDWDUHDSSHDULQJ 7KHWULDQJOH S 2 KDVWKUHHHGJHV (DFKRIWKHVHHGJHVLVDOLQHVHJPHQWOLNH S 1 :KHUHGRWKHVH WKUHHOLQHVHJPHQWVFRPHIURP" 2QHRIWKHPLV S 1 2QHRIWKHPFRPHVE\MRLQLQJWKH HQGSRLQW Y2 WRWKHQHZSRLQW Y3 7KHWKLUGFRPHVIURPMRLQLQJWKHRWKHUHQGSRLQW Y1 WR Y3
v1

v4

v2

v3

v5

v5

v1

v4 v1

v2

v3

v4

v2

'*(63& 7KH 4GLPHQVLRQDOVLPSOH[ S 4 SURMHFWHGRQWR R2 ZLWKWZR

WHWUDKHGUDOIDFHWVHPSKDVL]HG

v3

3RO\WRSHV

VXFKIDFHWV )LJXUHLGHQWLÀHVDOOWHQHGJHVRI S 4 DQGWKHVHFDQEHXVHGWRYLVXDOL]H WKHWHQWULDQJXODUIDFHV )LJXUHVKRZVDQRWKHUUHSUHVHQWDWLRQRIWKHGLPHQVLRQDOVLPSOH[ S 4 7KLVWLPH WKHÀIWKYHUWH[DSSHDUV´LQVLGHµWKHWHWUDKHGURQ S 3 7KHKLJKOLJKWHGWHWUDKHGUDOIDFHWV DOVRDSSHDUWREH´LQVLGHµ S 3 v4

v4

v5

v2

v1

v3

v1

v3

v4

v5

v1

v2

v4

v5

v2

v3

v1

v2

v3

'*(63& 7KHÀIWKYHUWH[RI S 4 LV´LQVLGHµ S 3

+\SHUFXEH /HW Ii D Hi EHWKHOLQHVHJPHQWIURPWKHRULJLQ WRWKHVWDQGDUGEDVLVYHFWRU Hi LQ Rn 7KHQIRU k VXFKWKDW 1 ' k ' n WKHYHFWRUVXPò

C k D I1 C I2 C ! ! ! C Ik

LVFDOOHGD k GLPHQVLRQDO K\SHUFXEH 7RYLVXDOL]HWKHFRQVWUXFWLRQRI C k VWDUWZLWKWKHVLPSOHFDVHV 7KHK\SHUFXEH C 1 LVWKHOLQHVHJPHQW I1 ,I C 1 LVWUDQVODWHGE\ H2 WKHFRQYH[KXOORILWVLQLWLDODQGÀ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ò 7KHYHFWRUVXPRIWZRVHWV A DQG B LVGHÀQHGE\ A C B D fF W F D D C E IRUVRPH D 2 A DQG E 2 B g

$)"15&3


C1

C2

C3

3

'*(63& &RQVWUXFWLQJWKHFXEH C

'*(63& C 4 SURMHFWHGRQWR R2

(a)

(b)

(c)

'*(63& 7KUHHRIWKHFXELFIDFHWVRI C 4

)LJXUHVKRZVDQRWKHUUHSUHVHQWDWLRQRI C 4 LQZKLFKWKHWUDQVODWHGFXEHLVSODFHG ´LQVLGHµ C 3 7KLVPDNHVLWHDVLHUWRYLVXDOL]HWKHFXELFIDFHWVRI C 4 VLQFHWKHUHLVOHVV GLVWRUWLRQ

'*(63& 7KHWUDQVODWHGLPDJHRI

C 3 LVSODFHG´LQVLGHµ C 3 WRREWDLQ C 4

$OWRJHWKHU WKHGLPHQVLRQDOFXEH C 4 KDVHLJKWFXELFIDFHV 7ZRFRPHIURPWKH RULJLQDODQGWUDQVODWHGLPDJHVRI C 3 DQGVL[FRPHIURPWKHVTXDUHIDFHVRI C 3 WKDWDUH VWUHWFKHGLQWRFXEHV 7KHVTXDUHGLPHQVLRQDOIDFHVRI C 4 FRPHIURPWKHVTXDUHIDFHV

3RO\WRSHV

RI C 3 DQGLWVWUDQVODWH DQGWKHHGJHVRI C 3 WKDWDUHVWUHWFKHGLQWRVTXDUHV 7KXVWKHUH DUH 2 % 6 C 12 D 24 VTXDUHIDFHV 7RFRXQWWKHHGJHV WDNHWLPHVWKHQXPEHURIHGJHV LQ C 3 DQGDGGWKHQXPEHURIYHUWLFHVLQ C 3 7KLVPDNHV 2 % 12 C 8 D 32 HGJHVLQ C 4 7KHYHUWLFHVLQ C 4 DOOFRPHIURP C 3 DQGLWVWUDQVODWH VRWKHUHDUH 2 % 8 D 16 YHUWLFHV 2QHRIWKHWUXO\UHPDUNDEOHUHVXOWVLQWKHVWXG\RISRO\WRSHVLVWKHIROORZLQJIRU PXOD ÀUVWSURYHGE\/HRQDUG(XOHU² ,WHVWDEOLVKHVDVLPSOHUHODWLRQVKLS EHWZHHQWKHQXPEHURIIDFHVRIGLIIHUHQWGLPHQVLRQVLQDSRO\WRSH 7RVLPSOLI\WKH VWDWHPHQWRIWKHIRUPXOD OHW fk .P / GHQRWHWKHQXPEHURI k GLPHQVLRQDOIDFHVRIDQ nGLPHQVLRQDOSRO\WRSH P ó (XOHU·VIRUPXOD

n!1 X ."1/k fk .P / D 1 C ."1/n!1 k D0

,QSDUWLFXODU ZKHQ n D 3; v " e C f D 2 ZKHUH v e DQG f GHQRWHWKHQXPEHURI YHUWLFHV HGJHV DQGIDFHWVUHVSHFWLYHO\ RI P

13"$5*$& 130#-&. )LQGWKHPLQLPDOUHSUHVHQWDWLRQRIWKHSRO\WRSH 2 3 2 P3GHÀQHGE\WKHLQHTXDOLWLHV A[ ' E 1 3 12 DQG [ & ZKHQ A D 4 1 2 5 DQG E D 4 9 5 2 1 12

&9&3$*4&4

! " ! " ! " 1 2 "1 S2 D DQG S3 D LQ R2 0 3 2 OHW S D FRQY fS1 ; S2 ; S3 g )RUHDFKOLQHDUIXQFWLRQDO f ÀQG WKHPD[LPXPYDOXH m RI f RQWKHVHW S DQGÀQGDOOSRLQWV [ LQ S DWZKLFK f .[/ D m E f .x1 ; x2 / D x1 C x2 D f .x1 ; x2 / D x1 " x2

*LYHQSRLQWV S1 D

F f .x1 ; x2 / D "3x1 C x2 ! " ! " ! " 0 2 1 *LYHQSRLQWV S1 D S2 D DQG S3 D LQ R2 "1 1 2 OHW S D FRQY fS1 ; S2 ; S3 g )RUHDFKOLQHDUIXQFWLRQDO f ÀQG WKHPD[LPXPYDOXH m RI f RQWKHVHW S DQGÀQGDOOSRLQWV [ LQ S DWZKLFK f .[/ D m E f .x1 ; x2 / D x1 " x2 D f .x1 ; x2 / D x1 C x2 F f .x1 ; x2 / D "2x1 C x2

5HSHDW([HUFLVHZKHUH m LVWKH PLQLPXP YDOXHRI f RQ S LQVWHDGRIWKHPD[LPXPYDOXH 5HSHDW([HUFLVHZKHUH m LVWKH PLQLPXP YDOXHRI f RQ S LQVWHDGRIWKHPD[LPXPYDOXH ,Q([HUFLVHV² ÀQGWKHPLQLPDOUHSUHVHQWDWLRQRIWKHSRO\WRSH GHÀQHGE\WKHLQHTXDOLWLHV A[ ' E DQG [ & ! " ! " 1 2 10 A D ED 3 1 15

A D

!

2

2 4

1 A D 4 1 4 2 2 A D 4 1 1

" ! " 3 18 ED 1 16 3 2 3 3 18 1 5 E D 4 10 5 1 28 3 2 3 1 8 1 5 E D 4 6 5 2 7

/HW S D f.x; y/ W x 2 C .y " 1/2 ' 1g [ f.3; 0/g ,VWKHRUL JLQDQH[WUHPHSRLQWRI FRQY S " ,VWKHRULJLQDYHUWH[RI FRQY S " )LQGDQH[DPSOHRIDFORVHGFRQYH[VHW S LQ R2 VXFKWKDWLWV SURÀOH P LVQRQHPSW\EXW FRQY P ¤ S )LQGDQH[DPSOHRIDERXQGHGFRQYH[VHW S LQ R2 VXFKWKDW LWVSURÀOH P LVQRQHPSW\EXW FRQY P ¤ S D 'HWHUPLQHWKHQXPEHURI k IDFHVRIWKHGLPHQVLRQDO VLPSOH[ S 5 IRU k D 0; 1; : : : ; 4 9HULI\WKDW\RXUDQVZHU VDWLVÀHV(XOHU·VIRUPXOD E 0DNHDFKDUWRIWKHYDOXHVRI fk .S n / IRU n D 1; : : : ; 5 DQG k D 0; 1; : : : ; 4 &DQ\RXVHHDSDWWHUQ" *XHVVDJHQHUDO IRUPXODIRU fk .S n /

ó $ SURRILVSUHVHQWHGLQ6WHYHQ5/D\ &RQYH[6HWVDQG7KHLU$SSOLFDWLRQV 1HZ

$)"15&3


D 'HWHUPLQHWKHQXPEHURI k IDFHVRIWKHGLPHQVLRQDO K\SHUFXEH C 5 IRU k D 0; 1; : : : ; 4 9HULI\WKDW\RXUDQ VZHUVDWLVÀHV(XOHU·VIRUPXOD

D $ FXEHLQ R3 KDVÀYHIDFHWV E $ SRLQW S LVDQH[WUHPHSRLQWRIDSRO\WRSH P LIDQGRQO\ LI S LVDYHUWH[RI P

E 0DNHDFKDUWRIWKHYDOXHVRI fk .C n / IRU n D 1; : : : ; 5 DQG k D 0; 1; : : : ; 4 &DQ\RXVHHDSDWWHUQ" *XHVVDJHQHUDO IRUPXODIRU fk .C n /

F ,I S LV D QRQHPSW\ FRPSDFW FRQYH[ VHW DQG D OLQHDU IXQFWLRQDODWWDLQVLWVPD[LPXPDWDSRLQW S WKHQ S LVDQ H[WUHPHSRLQWRI S

6XSSRVH Y1 ; : : : ; Yk DUH OLQHDUO\ LQGHSHQGHQW YHFWRUV LQ Rn .1 ' k ' n/ 7KHQWKHVHW X k D FRQY f˙Y1 ; : : : ; ˙Yk g LVFDOOHGD NFURVVSRO\WRSH D 6NHWFK X 1 DQG X 2

G $ GLPHQVLRQDOSRO\WRSHDOZD\VKDVWKHVDPHQXPEHU RIYHUWLFHVDQGHGJHV

E 'HWHUPLQHWKHQXPEHURI k IDFHVRIWKHGLPHQVLRQDO FURVVSRO\WRSH X 3 IRU k D 0; 1; 2 :KDWLVDQRWKHUQDPH IRU X 3 " F 'HWHUPLQHWKHQXPEHURI k IDFHVRIWKHGLPHQVLRQDO FURVVSRO\WRSH X 4 IRU k D 0; 1; 2; 3 9HULI\ WKDW \RXU DQVZHUVDWLVÀHV(XOHU·VIRUPXOD G )LQGDIRUPXODIRU fk .X n / WKHQXPEHURI k IDFHVRI X n IRU 0 ' k ' n " 1 $ NS\UDPLG P k LVWKHFRQYH[KXOORID .k " 1/SRO\WRSH Q DQGDSRLQW [ 62 DII Q )LQGDIRUPXODIRUHDFKRIWKH IROORZLQJLQWHUPVRI fj .Q/; j D 0; : : : ; n " 1 D 7KHQXPEHURIYHUWLFHVRI P n f0 .P n / E 7KHQXPEHURI k IDFHVRI P n fk .P n / IRU 1 ' k ' n " 2

F 7KH QXPEHU RI .n " 1/GLPHQVLRQDO IDFHWV RI P n fn!1 .P n /

,Q([HUFLVHVDQG PDUNHDFKVWDWHPHQW7UXHRU)DOVH -XVWLI\ HDFKDQVZHU D $ SRO\WRSHLVWKHFRQYH[KXOORIDÀQLWHVHWRISRLQWV E /HW S EHDQH[WUHPHSRLQWRIDFRQYH[VHW S ,I X; Y 2 S S 2 XY DQG S ¤ X WKHQ S D Y

F ,I S LVDQRQHPSW\FRQYH[VXEVHWRI Rn WKHQ S LVWKH FRQYH[KXOORILWVSURÀOH

G 7KHGLPHQVLRQDOVLPSOH[ S 4 KDVH[DFWO\ÀYHIDFHWV HDFKRIZKLFKLVDGLPHQVLRQDOWHWUDKHGURQ

/HW Y EHDQHOHPHQWRIWKHFRQYH[VHW S 3URYHWKDW Y LVDQ H[WUHPHSRLQWRI S LIDQGRQO\LIWKHVHW f[ 2 S W [ ¤ Yg LV FRQYH[ ,I c 2 R DQG S LVDVHW GHÀQH cS D fc [ W [ 2 S g /HW S EHDFRQYH[VHWDQGVXSSRVH c > 0 DQG d > 0 3URYHWKDW cS C dS D .c C d /S )LQGDQH[DPSOHWRVKRZWKDWWKHFRQYH[LW\RI S LVQHFHVVDU\ LQ([HUFLVH

,I A DQG B DUHFRQYH[VHWV SURYHWKDW A C B LVFRQYH[

$ SRO\KHGURQSRO\WRSH LVFDOOHG UHJXODU LIDOOLWVIDFHWV DUH FRQJUXHQW UHJXODU SRO\JRQV DQG DOO WKH DQJOHV DW WKH YHUWLFHVDUHHTXDO 6XSSO\WKHGHWDLOVLQWKHIROORZLQJSURRI WKDWWKHUHDUHRQO\ÀYHUHJXODUSRO\KHGUD D 6XSSRVHWKDWDUHJXODUSRO\KHGURQKDV r IDFHWV HDFKRI ZKLFK LV D k VLGHG UHJXODU SRO\JRQ DQG WKDW s HGJHV PHHWDWHDFKYHUWH[ /HWWLQJ v DQG e GHQRWHWKHQXPEHUV RIYHUWLFHVDQGHGJHVLQWKHSRO\KHGURQ H[SODLQZK\ kr D 2e DQG sv D 2e 1 1 1 1 E 8VH(XOHU·VIRUPXODWRVKRZWKDW C D C s k 2 e F )LQG DOO WKH LQWHJUDO VROXWLRQV RI WKH HTXDWLRQ LQ SDUW E WKDWVDWLVI\WKHJHRPHWULFFRQVWUDLQWVRIWKHSUREOHP +RZVPDOOFDQ k DQG s EH" )RU\RXULQIRUPDWLRQ WKHÀYHUHJXODUSRO\KHGUDDUHWKHWHWUD KHGURQ WKHFXEH WKHRFWDKHGURQ WKHGRGHFDKHGURQ DQGWKHLFRVDKHGURQ 7KHQXPEHUVLQSDUHQWKHVHVLQGLFDWHWKHQXPEHUVRI YHUWLFHV HGJHV DQGIDFHV UHVSHFWLYHO\

40-65*0/ 50 13"$5*$& 130#-&. 7KHPDWUL[LQHTXDOLW\ A[ ' E \LHOGVWKHIROORZLQJV\VWHPRILQHTXDOLWLHV D x1 C 3x2 ' 12 E x1 C 2x2 ' 9 F 2x1 C x2 ' 12 7KHFRQGLWLRQ [ & SODFHVWKHSRO\WRSHLQWKHÀUVWTXDGUDQWRIWKHSODQH 2QH YHUWH[LV .0; 0/ 7KH x1 LQWHUFHSWVRIWKHWKUHHOLQHVZKHQ x2 D 0 DUH DQG VR .6; 0/ LVDYHUWH[ 7KH x2 LQWHUFHSWVRIWKHWKUHHOLQHVZKHQ x1 D 0 DUH DQG VR .0; 4/ LVDYHUWH[

&XUYHVDQG6XUIDFHV

+RZGRWKHWKUHHERXQGDU\OLQHVLQWHUVHFWIRUSRVLWLYHYDOXHVRI x1 DQG x2 " 7KH LQWHUVHFWLRQRID DQGE LVDW SDE D .3; 3/ 7HVWLQJ SDE LQF JLYHV 2.3/ C 1.3/ D 9 < 12 VR SDE LVLQ P 7KHLQWHUVHFWLRQRIE DQGF LVDW SEF D .5; 2/ 7HVWLQJ SEF LQD JLYHV 1.5/ C 3.2/ D 11 < 12 VR SEF LVLQ P 7KHLQWHUVHFWLRQRID DQGF LVDW SDF D .4:8; 2:4/ 7HVWLQJ SDF LQE JLYHV 1.4:8/ C 2.2:4/ D 9:6 > 9 6R SDF LV QRWLQ P )LQDOO\ WKHWKUHHYHUWLFHVH[WUHPHSRLQWV RIWKHSRO\WRSHVDUH .0; 0/ .6; 0/ .5; 2/ .3; 3/ DQG .0; 4/ 7KHVHSRLQWVIRUPWKHPLQLPDOUHSUHVHQWDWLRQRI P 7KLVLV GLVSOD\HGJUDSKLFDOO\LQ)LJ x2 12

8

(c)

4 (a)

P 4

8

(b) 12

x1

'*(63&

$637&4 "/% 463'"$&4 )RUWKRXVDQGVRI\HDUV EXLOGHUVXVHGORQJWKLQVWULSVRIZRRGWRFUHDWHWKHKXOORIDERDW ,QPRUHUHFHQWWLPHV GHVLJQHUVXVHGORQJ ÁH[LEOHPHWDOVWULSVWROD\RXWWKHVXUIDFHVRI FDUVDQGDLUSODQHV :HLJKWVDQGSHJVVKDSHGWKHVWULSVLQWRVPRRWKFXUYHVFDOOHG QDWXUDO FXELFVSOLQHV 7KHFXUYHEHWZHHQWZRVXFFHVVLYHFRQWUROSRLQWVSHJVRUZHLJKWV KDV DSDUDPHWULFUHSUHVHQWDWLRQXVLQJFXELFSRO\QRPLDOV 8QIRUWXQDWHO\ VXFKFXUYHVKDYH WKHSURSHUW\WKDWPRYLQJRQHFRQWUROSRLQWDIIHFWVWKHVKDSHRIWKHHQWLUHFXUYH EHFDXVH RISK\VLFDOIRUFHVWKDWWKHSHJVDQGZHLJKWVH[HUWRQWKHVWULS 'HVLJQHQJLQHHUVKDG ORQJZDQWHGORFDOFRQWURORIWKHFXUYH³LQZKLFKPRYHPHQWRIRQHFRQWUROSRLQWZRXOG DIIHFWRQO\DVPDOOSRUWLRQRIWKHFXUYH ,Q D)UHQFKDXWRPRWLYHHQJLQHHU 3LHUUH %p]LHU VROYHGWKLVSUREOHPE\DGGLQJH[WUDFRQWUROSRLQWVDQGXVLQJDFODVVRIFXUYHV QRZFDOOHGE\KLVQDPH

%p]LHU&XUYHV 7KHFXUYHVGHVFULEHGEHORZSOD\DQLPSRUWDQWUROHLQFRPSXWHUJUDSKLFVDVZHOODV HQJLQHHULQJ )RUH[DPSOH WKH\DUHXVHGLQ$GREH,OOXVWUDWRUDQG0DFURPHGLD)UHHKDQG DQGLQDSSOLFDWLRQSURJUDPPLQJODQJXDJHVVXFKDV2SHQ*/7KHVHFXUYHVSHUPLWD SURJUDPWRVWRUHH[DFWLQIRUPDWLRQDERXWFXUYHGVHJPHQWVDQGVXUIDFHVLQDUHODWLYHO\ VPDOOQXPEHURIFRQWUROSRLQWV $OOJUDSKLFVFRPPDQGVIRUWKHVHJPHQWVDQGVXUIDFHV KDYHRQO\WREHFRPSXWHGIRUWKHFRQWUROSRLQWV 7KHVSHFLDOVWUXFWXUHRIWKHVHFXUYHV DOVRVSHHGVXSRWKHUFDOFXODWLRQVLQWKH´JUDSKLFVSLSHOLQHµWKDWFUHDWHVWKHÀQDOGLVSOD\ RQWKHYLHZLQJVFUHHQ ([HUFLVHVLQ6HFWLRQLQWURGXFHGTXDGUDWLF%p]LHUFXUYHVDQGVKRZHGRQHPHWKRG IRUFRQVWUXFWLQJ%p]LHUFXUYHVRIKLJKHUGHJUHH 7KHGLVFXVVLRQKHUHIRFXVHVRQTXDGUDWLF DQGFXELF%p]LHUFXUYHV ZKLFKDUHGHWHUPLQHGE\WKUHHRUIRXUFRQWUROSRLQWV GHQRWHG

$)"15&3


E\ S0 S1 S2 DQG S3 7KHVHSRLQWVFDQEHLQ R2 RU R3 RUWKH\FDQEHUHSUHVHQWHGE\ KRPRJHQHRXVIRUPVLQ R3 RU R4 7KHVWDQGDUGSDUDPHWULFGHVFULSWLRQVRIWKHVHFXUYHV IRU 0 ' t ' 1 DUH Z.t/ D .1 " t/2 S0 C 2t .1 " t/S1 C t 2 S2 3

2

2

3

[.t/ D .1 " t/ S0 C 3t.1 " t/ S1 C 3t .1 " t/S2 C t S3

)LJXUHVKRZVWZRW\SLFDOFXUYHV 8VXDOO\ WKHFXUYHVSDVVWKURXJKRQO\WKHLQLWLDODQG WHUPLQDOFRQWUROSRLQWV EXWD%p]LHUFXUYHLVDOZD\VLQWKHFRQYH[KXOORILWVFRQWURO SRLQWV 6HH([HUFLVHV²LQ6HFWLRQ p1

p0

p2

p1

p0

p2

p3

'*(63& 4XDGUDWLFDQGFXELF%p]LHUFXUYHV

%p]LHUFXUYHVDUHXVHIXOLQFRPSXWHUJUDSKLFVEHFDXVHWKHLUHVVHQWLDOSURSHUWLHVDUH SUHVHUYHGXQGHUWKHDFWLRQRIOLQHDUWUDQVIRUPDWLRQVDQGWUDQVODWLRQV )RULQVWDQFH LI A LVDPDWUL[RIDSSURSULDWHVL]H WKHQIURPWKHOLQHDULW\RIPDWUL[PXOWLSOLFDWLRQ IRU 0 ' t ' 1

A[.t / D AŒ.1 " t/3 S0 C 3t.1 " t/2 S1 C 3t 2 .1 " t/S2 C t 3 S3 !

D .1 " t/3 AS0 C 3t.1 " t/2 AS1 C 3t 2 .1 " t /AS2 C t 3 AS3

7KHQHZFRQWUROSRLQWVDUH AS0 ; : : : ; AS3 7UDQVODWLRQVRI%p]LHUFXUYHVDUHFRQVLGHUHG LQ([HUFLVH 7KHFXUYHVLQ)LJ VXJJHVWWKDWWKHFRQWUROSRLQWVGHWHUPLQHWKHWDQJHQWOLQHVWR WKHFXUYHVDWWKHLQLWLDODQGWHUPLQDOFRQWUROSRLQWV 5HFDOOIURPFDOFXOXVWKDWIRUDQ\ SDUDPHWULFFXUYH VD\ \.t/ WKHGLUHFWLRQRIWKHWDQJHQWOLQHWRWKHFXUYHDWDSRLQW \.t/ LVJLYHQE\WKHGHULYDWLYH \0 .t / FDOOHGWKH WDQJHQWYHFWRU RIWKHFXUYH 7KLVGHULYDWLYH LVFRPSXWHGHQWU\E\HQWU\

&9".1-& 'HWHUPLQHKRZWKHWDQJHQWYHFWRURIWKHTXDGUDWLF%p]LHUFXUYH Z.t/ LVUHODWHGWRWKHFRQWUROSRLQWVRIWKHFXUYH DW t D 0 DQG t D 1

40-65*0/ :ULWHWKHZHLJKWVLQHTXDWLRQ DVVLPSOHSRO\QRPLDOV Z.t/ D .1 " 2t C t 2 /S0 C .2t " 2t 2 /S1 C t 2 S2 7KHQ EHFDXVHGLIIHUHQWLDWLRQLVDOLQHDUWUDQVIRUPDWLRQRQIXQFWLRQV Z0 .t / D ."2 C 2t /S0 C .2 " 4t /S1 C 2t S2 6R Z0 .0/ D "2S0 C 2S1 D 2.S1 " S0 / Z0 .1/ D "2S1 C 2S2 D 2.S2 " S1 / 7KHWDQJHQWYHFWRUDW S0 IRULQVWDQFH SRLQWVIURP S0 WR S1 EXWLWLVWZLFHDVORQJ DVWKHVHJPHQWIURP S0 WR S1 1RWLFHWKDW Z0 .0/ D ZKHQ S1 D S0 ,QWKLVFDVH Z.t/ D .1 " t 2 /S1 C t 2 S2 DQG WKH JUDSK RI Z.t/ LV WKH OLQH VHJPHQW IURP S1 WR S2

&XUYHVDQG6XUIDFHV

&RQQHFWLQJ7ZR%p]LHU&XUYHV 7ZREDVLF%p]LHUFXUYHVFDQEHMRLQHGHQGWRHQG ZLWKWKHWHUPLQDOSRLQWRIWKHÀUVW FXUYH [.t/ EHLQJWKHLQLWLDOSRLQW S2 RIWKHVHFRQGFXUYH \.t / 7KHFRPELQHGFXUYHLV VDLGWRKDYH G 0 JHRPHWULFFRQWLQXLW\ DW S2 EHFDXVHWKHWZRVHJPHQWVMRLQDW S2 ,IWKH WDQJHQWOLQHWRFXUYH 1 DW S2 KDVDGLIIHUHQWGLUHFWLRQWKDQWKHWDQJHQWOLQHWRFXUYH 2 WKHQD´FRUQHUµ RUDEUXSWFKDQJHRIGLUHFWLRQ PD\EHDSSDUHQWDW S2 6HH)LJ p3

p1

p4

p2

p0 '*(63& G 0 FRQWLQXLW\DW S2

7RDYRLGDVKDUSEHQG LWXVXDOO\VXIÀFHVWRDGMXVWWKHFXUYHVWRKDYHZKDWLVFDOOHG G 1 JHRPHWULFFRQWLQXLW\ ZKHUHERWKWDQJHQWYHFWRUVDW S2 SRLQWLQWKHVDPHGLUHFWLRQ 7KDWLV WKHGHULYDWLYHV [0 .1/ DQG \0 .0/ SRLQWLQWKHVDPHGLUHFWLRQ HYHQWKRXJKWKHLU PDJQLWXGHVPD\EHGLIIHUHQW :KHQWKHWDQJHQWYHFWRUVDUHDFWXDOO\HTXDODW S2 WKH WDQJHQWYHFWRULVFRQWLQXRXVDW S2 DQGWKHFRPELQHGFXUYHLVVDLGWRKDYH C 1 FRQWLQXLW\ RU C 1 SDUDPHWULFFRQWLQXLW\ )LJXUHVKRZV G 1 FRQWLQXLW\LQD DQG C 1 FRQWLQXLW\ LQ E

p1

2

p3

p2

p0 0

p1

p3

p2

p4

p0 p4

0

4

2

6

8

10

(a)

12

14

(b)

'*(63& D G 1 FRQWLQXLW\DQGE C 1 FRQWLQXLW\

&9".1-& /HW [.t/ DQG \.t / GHWHUPLQHWZRTXDGUDWLF%p]LHUFXUYHV ZLWKFRQWURO SRLQWV fS0 ; S1 ; S2 g DQG fS2 ; S3 ; S4 g UHVSHFWLYHO\ 7KHFXUYHVDUHMRLQHGDW S2 D [.1/ D \.0/ D 6XSSRVHWKHFRPELQHGFXUYHKDV G 1 FRQWLQXLW\DW S2 :KDWDOJHEUDLFUHVWULFWLRQ GRHVWKLVFRQGLWLRQLPSRVHRQWKHFRQWUROSRLQWV" ([SUHVVWKLVUHVWULFWLRQLQJHRPHW ULFODQJXDJH E 5HSHDWSDUWD IRU C 1 FRQWLQXLW\ 40-65*0/ D )URP([DPSOH [0 .1/ D 2.S2 " S1 / $OVR XVLQJWKHFRQWUROSRLQWVIRU \.t/ LQ SODFHRI Z.t/ ([DPSOHVKRZVWKDW \0 .0/ D 2.S3 " S2 / G 1 FRQWLQXLW\PHDQVWKDW \0 .0/ D k [0 .1/ IRUVRPHSRVLWLYHFRQVWDQW k (TXLYDOHQWO\ S3 " S2 D k.S2 " S1 /;

ZLWK k > 0

$)"15&3


*HRPHWULFDOO\ LPSOLHVWKDW S2 OLHVRQWKHOLQHVHJPHQWIURP S1 WR S3 7R SURYHWKLV OHW t D .k C 1/!1 DQGQRWHWKDW 0 < t < 1 6ROYHIRU k WRREWDLQ k D .1 " t/=t :KHQWKLVH[SUHVVLRQLVXVHGIRU k LQ DUHDUUDQJHPHQWVKRZV WKDW S2 D .1 " t/S1 C t S3 ZKLFKYHULÀHVWKHDVVHUWLRQDERXW S2 E C 1 FRQWLQXLW\ PHDQV WKDW \0 .0/ D [0 .1/ 7KXV 2.S3 " S2 / D 2.S2 " S1 / VR S3 " S2 D S2 " S1 DQG S2 D .S1 C S3 /=2 *HRPHWULFDOO\ S2 LVWKHPLGSRLQWRI WKHOLQHVHJPHQWIURP S1 WR S3 6HH)LJ )LJXUHVKRZV C 1 FRQWLQXLW\IRUWZRFXELF%p]LHUFXUYHV 1RWLFHKRZWKHSRLQW MRLQLQJWKHWZRVHJPHQWVOLHVLQWKHPLGGOHRIWKHOLQHVHJPHQWEHWZHHQWKHDGMDFHQW FRQWUROSRLQWV

p4

p3

p0

p2 x(t)

y(t)

p5

p1 p6 '*(63& 7ZRFXELF%p]LHUFXUYHV

7ZRFXUYHVKDYH C 2 SDUDPHWULF FRQWLQXLW\ZKHQWKH\KDYH C 1 FRQWLQXLW\DQGWKH VHFRQG GHULYDWLYHV [00 .1/ DQG \00 .0/ DUHHTXDO 7KLVLVSRVVLEOHIRUFXELF%p]LHUFXUYHV EXWLWVHYHUHO\OLPLWVWKHSRVLWLRQVRIWKHFRQWUROSRLQWV $QRWKHUFODVVRIFXELFFXUYHV FDOOHG B VSOLQHV DOZD\VKDYH C 2 FRQWLQXLW\EHFDXVHHDFKSDLURIFXUYHVVKDUHWKUHH FRQWUROSRLQWVUDWKHUWKDQRQH *UDSKLFVÀJXUHVXVLQJ%VSOLQHVKDYHPRUHFRQWUROSRLQWV DQGFRQVHTXHQWO\UHTXLUHPRUHFRPSXWDWLRQV 6RPHH[HUFLVHVIRUWKLVVHFWLRQH[DPLQH WKHVHFXUYHV 6XUSULVLQJO\ LI [.t / DQG \.t/ MRLQDW S3 WKHDSSDUHQWVPRRWKQHVVRIWKHFXUYHDW S3 LVXVXDOO\WKHVDPHIRUERWK G 1 FRQWLQXLW\DQG C 1 FRQWLQXLW\ 7KLVLVEHFDXVHWKH PDJQLWXGHRI [0 .t/ LVQRWUHODWHGWRWKHSK\VLFDOVKDSHRIWKHFXUYH 7KHPDJQLWXGH UHÁHFWVRQO\WKHPDWKHPDWLFDOSDUDPHWHUL]DWLRQRIWKHFXUYH )RULQVWDQFH LIDQHZ YHFWRUIXQFWLRQ ].t / HTXDOV [.2t / WKHQWKHSRLQW ].t / WUDYHUVHVWKHFXUYHIURP S0 WR S3 WZLFHDVIDVWDVWKHRULJLQDOYHUVLRQ EHFDXVH 2t UHDFKHV 1 ZKHQ t LV :5 %XW E\WKHFKDLQ UXOHRIFDOFXOXV ]0 .t / D 2 ! [0 .2t / VRWKHWDQJHQWYHFWRUWR ].t / DW S3 LVWZLFHWKHWDQJHQW YHFWRUWR [.t / DW S3 ,QSUDFWLFH PDQ\VLPSOH%p]LHUFXUYHVDUHMRLQHGWRFUHDWHJUDSKLFVREMHFWV 7\SH VHWWLQJSURJUDPVSURYLGHRQHLPSRUWDQWDSSOLFDWLRQ EHFDXVHPDQ\OHWWHUVLQDW\SHIRQW LQYROYHFXUYHGVHJPHQWV (DFKOHWWHULQD3RVW6FULSW IRQW IRUH[DPSOH LVVWRUHGDVD VHWRIFRQWUROSRLQWV DORQJZLWKLQIRUPDWLRQRQKRZWRFRQVWUXFWWKH´RXWOLQHµRIWKH OHWWHUXVLQJOLQHVHJPHQWVDQG%p]LHUFXUYHV (QODUJLQJVXFKDOHWWHUEDVLFDOO\UHTXLUHV PXOWLSO\LQJWKHFRRUGLQDWHVRIHDFKFRQWUROSRLQWE\RQHFRQVWDQWVFDOHIDFWRU 2QFHWKH RXWOLQHRIWKHOHWWHUKDVEHHQFRPSXWHG WKHDSSURSULDWHVROLGSDUWVRIWKHOHWWHUDUHÀOOHG LQ )LJXUHLOOXVWUDWHVWKLVIRUDFKDUDFWHULQD3RVW6FULSWIRQW 1RWHWKHFRQWUROSRLQWV

&XUYHVDQG6XUIDFHV

Q '*(63& $ 3RVW6FULSWFKDUDFWHU

0DWUL[(TXDWLRQVIRU%p]LHU&XUYHV 6LQFHD%p]LHUFXUYHLVDOLQHDUFRPELQDWLRQRIFRQWUROSRLQWVXVLQJSRO\QRPLDOVDV ZHLJKWV WKHIRUPXODIRU [.t / PD\EHZULWWHQDV ' [.t / D S0 S1 S2 ' D S0 S 1 S2 ' D S0 S 1 S2

2

3 .1 " t/3 ( 6 3t .1 " t /2 7 7 S3 6 4 3t 2 .1 " t / 5 t3 2 3 1 " 3t C 3t 2 " t 3 ( 6 3t " 6t 2 C 3t 3 7 7 S3 6 4 5 3t 2 " 3t 3 t3 2 32 3 1 "3 3 "1 1 76 t 7 (60 3 "6 3 76 7 S3 6 40 0 3 "3 5 4 t 2 5 t3 0 0 0 1

7KHPDWUL[ZKRVHFROXPQVDUHWKHIRXUFRQWUROSRLQWVLVFDOOHGD JHRPHWU\PDWUL[ G 7KH 4 % 4 PDWUL[RISRO\QRPLDOFRHIÀFLHQWVLVWKH %p]LHUEDVLVPDWUL[ MB ,I X.t/ LV WKHFROXPQYHFWRURISRZHUVRI t WKHQWKH%p]LHUFXUYHLVJLYHQE\ [.t/ D GMB X.t/

2WKHUSDUDPHWULFFXELFFXUYHVLQFRPSXWHUJUDSKLFVDUHZULWWHQLQWKLVIRUP WRR )RULQVWDQFH LIWKHHQWULHVLQWKHPDWUL[ MB DUHFKDQJHGDSSURSULDWHO\ WKHUHVXOWLQJ FXUYHVDUH%VSOLQHV 7KH\DUH´VPRRWKHUµWKDQ%p]LHUFXUYHV EXWWKH\GRQRWSDVV WKURXJKDQ\RIWKHFRQWUROSRLQWV $ +HUPLWH FXELFFXUYHDULVHVZKHQWKHPDWUL[ MB LVUHSODFHGE\D+HUPLWHEDVLVPDWUL[ ,QWKLVFDVH WKHFROXPQVRIWKHJHRPHWU\PDWUL[ FRQVLVWRIWKHVWDUWLQJDQGHQGLQJSRLQWVRIWKHFXUYHVDQGWKHWDQJHQWYHFWRUVWRWKH FXUYHVDWWKRVHSRLQWVñ 7KH%p]LHUFXUYHLQHTXDWLRQ FDQDOVREHÍDFWRUHGµLQDQRWKHUZD\ WREHXVHG LQWKHGLVFXVVLRQRI%p]LHUVXUIDFHV )RUFRQYHQLHQFHODWHU WKHSDUDPHWHU t LVUHSODFHG

ñ 7KHWHUP EDVLVPDWUL[ FRPHVIURPWKHURZVRIWKHPDWUL[WKDWOLVWWKHFRHIÀFLHQWVRIWKH EOHQGLQJ SRO\ QRPLDOVXVHGWRGHÀQHWKHFXUYH )RUDFXELF%p]LHUFXUYH WKHIRXUSRO\QRPLDOVDUH .1 ! t /3 3t.1 ! t/2 3t 2 .1 ! t/ DQG t 3 7KH\IRUPDEDVLVIRUWKHVSDFH P3 RISRO\QRPLDOVRIGHJUHH 3 RUOHVV (DFKHQWU\LQWKH YHFWRU [.t/ LVDOLQHDUFRPELQDWLRQRIWKHVHSRO\QRPLDOV 7KHZHLJKWVFRPHIURPWKHURZVRIWKHJHRPHWU\ PDWUL[ G LQ

$)"15&3


E\DSDUDPHWHU s

2 3 S0 1 0 0 6 6 7 ' ( S "3 3 0 2 3 6 17 [.s/ D X.s/T MBT 6 4 S2 5 D 1 s s s 4 3 "6 3 S3 "1 3 "3 2 3 S0 6 ' ( S1 7 7 D .1 " s/3 3s.1 " s/2 3s 2 .1 " s/ s 3 6 4 S2 5 S3 2

32 3 0 S0 6 S1 7 07 76 7 0 54 S 2 5 S3 1

7KLVIRUPXODLVQRWTXLWHWKHVDPHDVWKHWUDQVSRVHRIWKHSURGXFWRQWKHULJKWRI EHFDXVH [.s/ DQGWKHFRQWUROSRLQWVDSSHDULQ ZLWKRXWWUDQVSRVHV\PEROV 7KH PDWUL[RIFRQWUROSRLQWVLQ LVFDOOHGD JHRPHWU\YHFWRU 7KLVVKRXOGEHYLHZHGDVD 4 % 1 EORFNSDUWLWLRQHG PDWUL[ZKRVHHQWULHVDUHFROXPQYHFWRUV 7KHPDWUL[WRWKHOHIW RIWKHJHRPHWU\YHFWRU LQWKHVHFRQGSDUWRI FDQEHYLHZHGDVDEORFNPDWUL[ WRR ZLWKDVFDODULQHDFKEORFN 7KHSDUWLWLRQHGPDWUL[PXOWLSOLFDWLRQPDNHVVHQVH EHFDXVH HDFKYHFWRU HQWU\LQWKHJHRPHWU\YHFWRUFDQEHOHIWPXOWLSOLHGE\DVFDODUDVZHOODV E\DPDWUL[ 7KXV WKHFROXPQYHFWRU [.s/ LVUHSUHVHQWHGE\

%p]LHU6XUIDFHV $ ' ELFXELFVXUIDFHSDWFKFDQEHFRQVWUXFWHGIURPDVHWRIIRXU%p]LHUFXUYHV &RQVLGHU WKHIRXUJHRPHWU\PDWULFHV ' ( S11 S12 S13 S14 ( ' S21 S22 S23 S24 ( ' S31 S32 S33 S34 ( ' S41 S42 S43 S44

DQGUHFDOOIURPHTXDWLRQ WKDWD%p]LHUFXUYHLVSURGXFHGZKHQDQ\RQHRIWKHVH PDWULFHVLVPXOWLSOLHGRQWKHULJKWE\WKHIROORZLQJYHFWRURIZHLJKWV 2 3 .1 " t/3 6 3t .1 " t /2 7 7 MB X.t / D 6 4 3t 2 .1 " t / 5 t3 /HW G EHWKHEORFNSDUWLWLRQHG 4 % 4 PDWUL[ZKRVHHQWULHVDUHWKHFRQWUROSRLQWV Sij GLVSOD\HGDERYH 7KHQWKHIROORZLQJSURGXFWLVDEORFN 4 % 1 PDWUL[ DQGHDFKHQWU\LV D%p]LHUFXUYH 2 32 3 S11 S12 S13 S14 .1 " t/3 6 6 S21 S22 S23 S24 7 " t/2 7 7 6 3t .1 7 GMB X.t / D 6 2 4 4 S31 5 S32 S33 S34 3t .1 " t / 5 S41 S42 S43 S44 t3 ,QIDFW

2

3 .1 " t /3 S11 C 3t .1 " t /2 S12 C 3t 2 .1 " t /S13 C t 3 S14 6 .1 " t /3 S21 C 3t .1 " t /2 S22 C 3t 2 .1 " t /S23 C t 3 S24 7 7 GMB X.t / D 6 4 .1 " t /3 S31 C 3t .1 " t /2 S32 C 3t 2 .1 " t /S33 C t 3 S34 5 .1 " t /3 S41 C 3t .1 " t /2 S42 C 3t 2 .1 " t /S43 C t 3 S44

&XUYHVDQG6XUIDFHV

1RZÀ[ t 7KHQ GMB X.t / LVDFROXPQYHFWRUWKDWFDQEHXVHGDVDJHRPHWU\YHFWRU LQHTXDWLRQ IRUD%p]LHUFXUYHLQDQRWKHUYDULDEOH s 7KLVREVHUYDWLRQSURGXFHVWKH %p]LHUELFXELFVXUIDFH [.s; t / D X.s/T MBT GMB X.t/;

ZKHUH 0 ' s; t ' 1

7KHIRUPXODIRU [.s; t / LVDOLQHDUFRPELQDWLRQRIWKHVL[WHHQFRQWUROSRLQWV ,IRQH LPDJLQHVWKDWWKHVHFRQWUROSRLQWVDUHDUUDQJHGLQDIDLUO\XQLIRUPUHFWDQJXODUDUUD\ DV LQ)LJ WKHQWKH%p]LHUVXUIDFHLVFRQWUROOHGE\DZHERIHLJKW%p]LHUFXUYHV IRXU LQWKH´s GLUHFWLRQµDQGIRXULQWKH´t GLUHFWLRQµ 7KHVXUIDFHDFWXDOO\SDVVHVWKURXJK WKHIRXUFRQWUROSRLQWVDWLWV´FRUQHUVµ :KHQLWLVLQWKHPLGGOHRIDODUJHUVXUIDFH WKH VL[WHHQSRLQWVXUIDFHVKDUHVLWVWZHOYHERXQGDU\FRQWUROSRLQWVZLWKLWVQHLJKERUV p 21

p 11 p 22

p 31 p 32 p 41

p 12 p 13

p 23 p 33

p 42

p 24

p 14

p 34

p 43

p 44 '*(63& 6L[WHHQFRQWUROSRLQWVIRUD%p]LHU ELFXELFVXUIDFHSDWFK

$SSUR[LPDWLRQVWR&XUYHVDQG6XUIDFHV ,Q&$' SURJUDPVDQGLQSURJUDPVXVHGWRFUHDWHUHDOLVWLFFRPSXWHUJDPHV WKHGHVLJQHU RIWHQZRUNVDWDJUDSKLFVZRUNVWDWLRQWRFRPSRVHD´VFHQHµLQYROYLQJYDULRXVJHRPHWULF VWUXFWXUHV 7KLVSURFHVVUHTXLUHVLQWHUDFWLRQEHWZHHQWKHGHVLJQHUDQGWKHJHRPHWULF REMHFWV (DFKVOLJKWUHSRVLWLRQLQJRIDQREMHFWUHTXLUHVQHZPDWKHPDWLFDOFRPSXWDWLRQV E\WKHJUDSKLFVSURJUDP %p]LHUFXUYHVDQGVXUIDFHVFDQEHXVHIXOLQWKLVSURFHVV EHFDXVHWKH\LQYROYHIHZHUFRQWUROSRLQWVWKDQREMHFWVDSSUR[LPDWHGE\PDQ\SRO\JRQV 7KLVGUDPDWLFDOO\UHGXFHVWKHFRPSXWDWLRQWLPHDQGVSHHGVXSWKHGHVLJQHU·VZRUN $IWHUWKHVFHQHFRPSRVLWLRQ KRZHYHU WKHÀQDOLPDJHSUHSDUDWLRQKDVGLIIHUHQW FRPSXWDWLRQDOGHPDQGVWKDWDUHPRUHHDVLO\PHWE\REMHFWVFRQVLVWLQJRIÁDWVXUIDFHV DQGVWUDLJKWHGJHV VXFKDVSRO\KHGUD 7KHGHVLJQHUQHHGVWR UHQGHU WKHVFHQH E\LQ WURGXFLQJOLJKWVRXUFHV DGGLQJFRORUDQGWH[WXUHWRVXUIDFHV DQGVLPXODWLQJUHÁHFWLRQV IURPWKHVXUIDFHV &RPSXWLQJWKHGLUHFWLRQRIDUHÁHFWHGOLJKWDWDSRLQW S RQDVXUIDFH IRULQVWDQFH UHTXLUHVNQRZLQJWKHGLUHFWLRQVRIERWKWKHLQFRPLQJOLJKWDQGWKH VXUIDFHQRUPDO³WKH YHFWRUSHUSHQGLFXODUWRWKHWDQJHQWSODQHDW S &RPSXWLQJVXFKQRUPDOYHFWRUVLVPXFK HDVLHURQDVXUIDFHFRPSRVHGRI VD\ WLQ\ÁDWSRO\JRQVWKDQRQDFXUYHGVXUIDFHZKRVH QRUPDOYHFWRUFKDQJHVFRQWLQXRXVO\DV S PRYHV ,I S1 S2 DQG S3 DUHDGMDFHQWYHU WLFHVRIDÁDWSRO\JRQ WKHQWKHVXUIDFHQRUPDOLVMXVWSOXVRUPLQXVWKHFURVVSURGXFW .S2 " S1 / % .S2 " S3 / :KHQWKHSRO\JRQLVVPDOO RQO\RQHQRUPDOYHFWRULVQHHGHGIRU UHQGHULQJWKHHQWLUHSRO\JRQ $OVR WZRZLGHO\XVHGVKDGLQJURXWLQHV *RXUDXGVKDGLQJ DQG3KRQJVKDGLQJ ERWKUHTXLUHDVXUIDFHWREHGHÀQHGE\SRO\JRQV $VDUHVXOWRIWKHVHQHHGVIRUÁDWVXUIDFHV WKH%p]LHUFXUYHVDQGVXUIDFHVIURPWKH VFHQHFRPSRVLWLRQVWDJHQRZDUHXVXDOO\DSSUR[LPDWHGE\VWUDLJKWOLQHVHJPHQWVDQG

$)"15&3


SRO\KHGUDOVXUIDFHV 7KHEDVLFLGHDIRUDSSUR[LPDWLQJD%p]LHUFXUYHRUVXUIDFHLVWR GLYLGHWKHFXUYHRUVXUIDFHLQWRVPDOOHUSLHFHV ZLWKPRUHDQGPRUHFRQWUROSRLQWV

5HFXUVLYH6XEGLYLVLRQRI%p]LHU&XUYHVDQG6XUIDFHV )LJXUHVKRZVWKHIRXUFRQWUROSRLQWV S0 ; : : : ; S3 IRUD%p]LHUFXUYH DORQJZLWKFRQWURO SRLQWVIRUWZRQHZFXUYHV HDFKFRLQFLGLQJZLWKKDOIRIWKHRULJLQDOFXUYH 7KHÓHIWµ FXUYHEHJLQVDW T0 D S0 DQGHQGVDW T3 DWWKHPLGSRLQWRIWKHRULJLQDOFXUYH 7KHÚLJKWµ FXUYHEHJLQVDW U0 D T3 DQGHQGVDW U3 D S3 p1

q2

r1

p2

q3 = r0

q1

r2

p0 = q0

p3 = r3

'*(63& 6XEGLYLVLRQRID%p]LHUFXUYH

)LJXUHVKRZVKRZWKHQHZFRQWUROSRLQWVHQFORVHUHJLRQVWKDWDUH´WKLQQHUµWKDQ WKHUHJLRQHQFORVHGE\WKHRULJLQDOFRQWUROSRLQWV $VWKHGLVWDQFHVEHWZHHQWKHFRQWURO SRLQWVGHFUHDVH WKHFRQWUROSRLQWVRIHDFKFXUYHVHJPHQWDOVRPRYHFORVHUWRDOLQH VHJPHQW 7KLV YDULDWLRQGLPLQLVKLQJSURSHUW\ RI%p]LHUFXUYHVGHSHQGVRQWKHIDFWWKDW D%p]LHUFXUYHDOZD\VOLHVLQWKHFRQYH[KXOORIWKHFRQWUROSRLQWV p1

q2

q1

r1

p2

q3 = r0

r2

p0 = q0

p3 = r3

'*(63& &RQYH[KXOOVRIWKHFRQWUROSRLQWV

7KHQHZFRQWUROSRLQWVDUHUHODWHGWRWKHRULJLQDOFRQWUROSRLQWVE\VLPSOHIRUPXODV 2IFRXUVH T0 D S0 DQG U3 D S3 7KHPLGSRLQWRIWKHRULJLQDOFXUYH [.t / RFFXUVDW [.:5/ ZKHQ [.t/ KDVWKHVWDQGDUGSDUDPHWHUL]DWLRQ [.t / D .1 " 3t C 3t 2 " t 3 /S0 C .3t " 6t 2 C 3t 3 /S1 C .3t 2 " 3t 3 /S2 C t 3 S3

IRU 0 ' t ' 1 7KXV WKHQHZFRQWUROSRLQWV T3 DQG U0 DUHJLYHQE\ T3 D U0 D [.:5/ D 18 .S0 C 3S1 C 3S2 C S3 /

7KHIRUPXODVIRUWKHUHPDLQLQJ´LQWHULRUµFRQWUROSRLQWVDUHDOVRVLPSOH EXWWKHGHULYD WLRQRIWKHIRUPXODVUHTXLUHVVRPHZRUNLQYROYLQJWKHWDQJHQWYHFWRUVRIWKHFXUYHV %\ GHÀQLWLRQ WKHWDQJHQWYHFWRUWRDSDUDPHWHUL]HGFXUYH [.t/ LVWKHGHULYDWLYH [0 .t/ 7KLV YHFWRUVKRZVWKHGLUHFWLRQRIWKHOLQHWDQJHQWWRWKHFXUYHDW [.t / )RUWKH%p]LHUFXUYH LQ [0 .t/ D ."3 C 6t " 3t 2 /S0 C .3 " 12t C 9t 2 /S1 C .6t " 9t 2 /S2 C 3t 2 S3

IRU 0 ' t ' 1 ,QSDUWLFXODU

[0 .0/ D 3.S1 " S0 /

DQG

[0 .1/ D 3.S3 " S2 /

&XUYHVDQG6XUIDFHV

*HRPHWULFDOO\ S1 LVRQWKHOLQHWDQJHQWWRWKHFXUYHDW S0 DQG S2 LVRQWKHOLQHWDQJHQW WRWKHFXUYHDW S3 6HH)LJ $OVR IURP [0 .t / FRPSXWH [0 .:5/ D 34 ."S0 " S1 C S2 C S3 /

/HW \.t / EHWKH%p]LHUFXUYHGHWHUPLQHGE\ T0 ; : : : ; T3 DQGOHW ].t / EHWKH%p]LHUFXUYH GHWHUPLQHGE\ U0 ; : : : ; U3 6LQFH \.t / WUDYHUVHVWKHVDPHSDWKDV [.t / EXWRQO\JHWVWR [.:5/ DV t JRHVIURP 0 WR 1 \.t / D [.:5t / IRU 0 ' t ' 1 6LPLODUO\ VLQFH ].t/ VWDUWVDW [.:5/ ZKHQ t D 0 ].t/ D [.:5 C :5t/ IRU 0 ' t ' 1 %\WKHFKDLQUXOHIRUGHULYDWLYHV \0 .t / D :5[0 .:5t /

DQG

0

0

]0 .t/ D :5[0 .:5 C :5t /

IRU 0 ' t ' 1

)URP ZLWK \ .0/ LQSODFHRI [ .0/ IURP ZLWK t D 0 DQGIURP WKHFRQWURO SRLQWVIRU \.t / VDWLVI\

3.T1 " T0 / D \0 .0/ D :5[0 .0/ D 32 .S1 " S0 /

3.T3 " T2 / D \0 .1/ D :5[0 .:5/ D 38 ."S0 " S1 C S2 C S3 /

0

0

)URP ZLWK \ .1/ LQSODFHRI [ .1/ IURP ZLWK t D 1 DQGIURP (TXDWLRQV DQG FDQEHVROYHGWRSURGXFHWKHIRUPXODVIRU T0 ; : : : ; T3 VKRZQLQ([HUFLVH *HRPHWULFDOO\ WKHIRUPXODVDUHGLVSOD\HGLQ)LJ 7KH LQWHULRUFRQWUROSRLQWV T1 DQG U2 DUHWKHPLGSRLQWV UHVSHFWLYHO\ RIWKHVHJPHQWIURP S0 WR S1 DQGWKHVHJPHQWIURP S2 WR S3 :KHQWKHPLGSRLQWRIWKHVHJPHQWIURP S1 WR S2 LVFRQQHFWHGWR T1 WKHUHVXOWLQJOLQHVHJPHQWKDV T2 LQWKHPLGGOH p1

q1

q0 = p0

1 p + p 2) 2( 1 q2

r1 q3 = r0

p2

r2

p3 = r3

'*(63& *HRPHWULFVWUXFWXUHRIQHZFRQWUROSRLQWV

7KLVFRPSOHWHVRQHVWHSRIWKHVXEGLYLVLRQSURFHVV 7KHÚHFXUVLRQµEHJLQV DQG ERWKQHZFXUYHVDUHVXEGLYLGHG 7KHUHFXUVLRQFRQWLQXHVWRDGHSWKDWZKLFKDOOFXUYHV DUHVXIÀFLHQWO\VWUDLJKW $OWHUQDWLYHO\ DWHDFKVWHSWKHUHFXUVLRQFDQEH´DGDSWLYHµDQG QRWVXEGLYLGHRQHRIWKHWZRQHZFXUYHVLIWKDWFXUYHLVVXIÀFLHQWO\VWUDLJKW 2QFHWKH VXEGLYLVLRQFRPSOHWHO\VWRSV WKHHQGSRLQWVRIHDFKFXUYHDUHMRLQHGE\OLQHVHJPHQWV DQGWKHVFHQHLVUHDG\IRUWKHQH[WVWHSLQWKHÀQDOLPDJHSUHSDUDWLRQ $ %p]LHUELFXELFVXUIDFHKDVWKHVDPHYDULDWLRQGLPLQLVKLQJSURSHUW\DVWKH%p]LHU FXUYHVWKDWPDNHXSHDFKFURVVVHFWLRQRIWKHVXUIDFH VRWKHSURFHVVGHVFULEHGDERYH FDQEHDSSOLHGLQHDFKFURVVVHFWLRQ :LWKWKHGHWDLOVRPLWWHG KHUHLVWKHEDVLFVWUDWHJ\ &RQVLGHUWKHIRXU´SDUDOOHOµ%p]LHUFXUYHVZKRVHSDUDPHWHULV s DQGDSSO\WKHVXEGLYL VLRQSURFHVVWRHDFKRIWKHP 7KLVSURGXFHVIRXUVHWVRIHLJKWFRQWUROSRLQWV HDFKVHW GHWHUPLQHVDFXUYHDV s YDULHVIURPWR $V t YDULHV KRZHYHU WKHUHDUHHLJKWFXUYHV HDFKZLWKIRXUFRQWUROSRLQWV $SSO\WKHVXEGLYLVLRQSURFHVVWRHDFKRIWKHVHVHWVRI IRXUSRLQWV FUHDWLQJDWRWDORIFRQWUROSRLQWV $GDSWLYHUHFXUVLRQLVSRVVLEOHLQWKLV VHWWLQJ WRR EXWWKHUHDUHVRPHVXEWOHWLHVLQYROYHGò ò 6HH)ROH\ YDQ'DP )HLQHU DQG+XJKHV &RPSXWHU*UDSKLFV³3ULQFLSOHVDQG3UDFWLFH QG(G %RVWRQ $GGLVRQ:HVOH\ SS ²

$)"15&3


13"$5*$& 130#-&.4 $ VSOLQH XVXDOO\UHIHUVWRDFXUYHWKDWSDVVHVWKURXJKVSHFLÀHGSRLQWV $ %VSOLQH KRZHYHU XVXDOO\GRHVQRWSDVVWKURXJKLWVFRQWUROSRLQWV $ VLQJOHVHJPHQWKDVWKH SDUDPHWULFIRUP ' [.t/ D 16 .1 " t /3 S0 C .3t 3 " 6t 2 C 4/S1 ( C ."3t 3 C 3t 2 C 3t C 1/S2 C t 3 S3 IRU 0 ' t ' 1 ZKHUH S0 S1 S2 DQG S3 DUHWKHFRQWUROSRLQWV :KHQ t YDULHVIURPWR [.t/ FUHDWHVDVKRUWFXUYHWKDWOLHVFORVHWR S1 S2 %DVLFDOJHEUDVKRZVWKDWWKH%VSOLQH IRUPXODFDQDOVREHZULWWHQDV ' [.t/ D 16 .1 " t /3 S0 C .3t .1 " t/2 " 3t C 4/S1 ( C .3t 2 .1 " t/ C 3t C 1/S2 C t 3 S3

7KLVVKRZVWKHVLPLODULW\ZLWKWKH%p]LHUFXUYH ([FHSWIRUWKH 1=6 IDFWRUDWWKHIURQW WKH S0 DQG S3 WHUPVDUHWKHVDPH 7KH S1 FRPSRQHQWKDVEHHQLQFUHDVHGE\ "3t C 4 DQGWKH S2 FRPSRQHQWKDVEHHQLQFUHDVHGE\ 3t C 1 7KHVHFRPSRQHQWVPRYHWKHFXUYH FORVHUWR S1 S2 WKDQWKH%p]LHUFXUYH 7KH 1=6 IDFWRULVQHFHVVDU\WRNHHSWKHVXPRIWKH FRHIÀFLHQWVHTXDOWR )LJXUH FRPSDUHVD%VSOLQHZLWKD%p]LHUFXUYHWKDWKDVWKH VDPHFRQWUROSRLQWV

'*(63& $ %VSOLQHVHJPHQWDQGD%p]LHUFXUYH

6KRZWKDWWKH%VSOLQHGRHVQRWEHJLQDW S0 EXW [.0/ LVLQ FRQY fS0 ; S1 ; S2 g $VVXP LQJWKDW S0 S1 DQG S2 DUHDIÀQHO\LQGHSHQGHQW ÀQGWKHDIÀQHFRRUGLQDWHVRI [.0/ ZLWKUHVSHFWWR fS0 ; S1 ; S2 g 6KRZWKDWWKH%VSOLQHGRHVQRWHQGDW S3 EXW [.1/ LVLQ FRQY fS1 ; S2 ; S3 g $VVXPLQJ WKDW S1 S2 DQG S3 DUHDIÀQHO\LQGHSHQGHQW ÀQGWKHDIÀQHFRRUGLQDWHVRI [.1/ ZLWK UHVSHFWWR fS1 ; S2 ; S3 g

&9&3$*4&4 6XSSRVHD%p]LHUFXUYHLVWUDQVODWHGWR [.t / C E 7KDWLV IRU 0 ' t ' 1 WKHQHZFXUYHLV [.t/ D .1 " t/3 S0 C 3t.1 " t/2 S1 C 3t 2 .1 " t/S2 C t 3 S3 C E 6KRZWKDWWKLVQHZFXUYHLVDJDLQD%p]LHUFXUYH >+LQW :KHUHDUHWKHQHZFRQWUROSRLQWV"@ 7KHSDUDPHWULFYHFWRUIRUPRID%VSOLQHFXUYHZDVGHÀQHG LQWKH3UDFWLFH3UREOHPVDV ' [.t/ D 16 .1 " t/3 S0 C .3t .1 " t/ " 3t C 4/S1 ( C.3t 2 .1 " t/ C 3t C 1/S2 C t 3 S3 IRU 0 ' t ' 1 ZKHUH S0 S1 S2 DQG S3 DUHWKHFRQWUROSRLQWV

D 6KRZWKDWIRU 0 ' t ' 1 [.t/ LVLQWKHFRQYH[KXOORIWKH FRQWUROSRLQWV E 6XSSRVH WKDW D %VSOLQH FXUYH [.t/ LV WUDQVODWHG WR [.t/ C E DVLQ([HUFLVH 6KRZWKDWWKLVQHZFXUYH LVDJDLQD%VSOLQH /HW [.t/ EHDFXELF%p]LHUFXUYHGHWHUPLQHGE\SRLQWV S0 S1 S2 DQG S3 D &RPSXWHWKH WDQJHQW YHFWRU [0 .t/ 'HWHUPLQHKRZ [0 .0/ DQG [0 .1/ DUH UHODWHG WR WKH FRQWURO SRLQWV DQG JLYH JHRPHWULFGHVFULSWLRQVRIWKH GLUHFWLRQV RIWKHVHWDQJHQW YHFWRUV ,VLWSRVVLEOHWRKDYH [0 .1/ D " E &RPSXWHWKHVHFRQGGHULYDWLYH [00 .t/ DQGGHWHUPLQHKRZ [00 .0/ DQG [00 .1/ DUHUHODWHGWRWKHFRQWUROSRLQWV 'UDZ

DÀJXUHEDVHGRQ)LJ DQGFRQVWUXFWDOLQHVHJPHQW WKDWSRLQWVLQWKHGLUHFWLRQRI [00 .0/ >+LQW 8VH S1 DVWKH RULJLQRIWKHFRRUGLQDWHV\VWHP@ /HW [.t/ EHWKH%VSOLQHLQ([HUFLVH ZLWKFRQWUROSRLQWV S0 S1 S2 DQG S3 D &RPSXWH WKH WDQJHQW YHFWRU [0 .t / DQG GHWHUPLQH KRZ WKHGHULYDWLYHV [0 .0/ DQG [0 .1/ DUHUHODWHGWRWKHFRQWURO SRLQWV *LYHJHRPHWULFGHVFULSWLRQVRIWKH GLUHFWLRQV RI WKHVHWDQJHQWYHFWRUV ([SORUHZKDWKDSSHQVZKHQERWK [0 .0/ DQG [0 .1/ HTXDO -XVWLI\\RXUDVVHUWLRQV E &RPSXWHWKHVHFRQGGHULYDWLYH [00 .t / DQGGHWHUPLQHKRZ [00 .0/ DQG [00 .1/ DUHUHODWHGWRWKHFRQWUROSRLQWV 'UDZ DÀJXUHEDVHGRQ)LJ DQGFRQVWUXFWDOLQHVHJPHQW WKDWSRLQWVLQWKHGLUHFWLRQRI [00 .1/ >+LQW 8VH S2 DVWKH RULJLQRIWKHFRRUGLQDWHV\VWHP@ /HW [.t/ DQG \.t / EHFXELF%p]LHUFXUYHVZLWKFRQWUROSRLQWV fS0 ; S1 ; S2 ; S3 g DQG fS3 ; S4 ; S5 ; S6 g UHVSHFWLYHO\ VRWKDW [.t/ DQG \.t/ DUHMRLQHGDW S3 7KHIROORZLQJTXHVWLRQVUHIHUWR WKHFXUYHFRQVLVWLQJRI [.t / IROORZHGE\ \.t / )RUVLPSOLFLW\ DVVXPHWKDWWKHFXUYHLVLQ R2 D :KDWFRQGLWLRQRQWKHFRQWUROSRLQWVZLOOJXDUDQWHHWKDW WKHFXUYHKDV C 1 FRQWLQXLW\DW S3 " -XVWLI\\RXUDQVZHU E :KDWKDSSHQVZKHQ [0 .1/ DQG \0 .0/ DUHERWKWKH]HUR YHFWRU" $ %VSOLQHLVEXLOWRXWRI%VSOLQHVHJPHQWV GHVFULEHGLQ ([HUFLVH /HW S0 ; : : : ; S4 EHFRQWUROSRLQWV )RU 0 ' t ' 1 OHW [.t/ DQG \.t/ EHGHWHUPLQHGE\WKHJHRPHWU\PDWULFHV Œ S0 S1 S2 S3 ! DQG Œ S1 S2 S3 S4 ! UHVSHFWLYHO\ 1RWLFH KRZ WKH WZR VHJPHQWV VKDUH WKUHH FRQWURO SRLQWV 7KHWZRVHJPHQWVGRQRWRYHUODS KRZHYHU³WKH\MRLQDWD FRPPRQHQGSRLQW FORVHWR S2 D 6KRZWKDWWKHFRPELQHGFXUYHKDV G 0 FRQWLQXLW\³WKDWLV [.1/ D \.0/

E 6KRZWKDWWKHFXUYHKDV C 1 FRQWLQXLW\DWWKHMRLQSRLQW [.1/ 7KDWLV VKRZWKDW [0 .1/ D \0 .0/

/HW [.t/ DQG \.t / EH%p]LHUFXUYHVIURP([HUFLVH DQGVXS SRVHWKHFRPELQHGFXUYHKDV C 2 FRQWLQXLW\ZKLFKLQFOXGHV C 1 FRQWLQXLW\ DW S3 6HW [00 .1/ D \00 .0/ DQGVKRZWKDW S5 LV FRPSOHWHO\GHWHUPLQHGE\ S1 S2 DQG S3 7KXV WKHSRLQWV S0 ; : : : ; S3 DQGWKH C 2 FRQGLWLRQGHWHUPLQHDOOEXWRQHRIWKH FRQWUROSRLQWVIRU \.t/ /HW [.t/ DQG \.t / EHVHJPHQWVRID%VSOLQHDVLQ([HUFLVH 6KRZWKDWWKHFXUYHKDV C 2 FRQWLQXLW\DVZHOODV C 1 FRQWLQXLW\ DW [.1/ 7KDWLV VKRZWKDW [00 .1/ D \00 .0/ 7KLV KLJKHURUGHUFRQWLQXLW\LVGHVLUDEOHLQ&$' DSSOLFDWLRQVVXFK DVDXWRPRWLYHERG\GHVLJQ VLQFHWKHFXUYHVDQGVXUIDFHV DSSHDUPXFKVPRRWKHU +RZHYHU %VSOLQHVUHTXLUHWKUHH WLPHVWKHFRPSXWDWLRQRI%p]LHUFXUYHV IRUFXUYHVRIFRP SDUDEOHOHQJWK )RUVXUIDFHV %VSOLQHVUHTXLUHQLQHWLPHVWKH FRPSXWDWLRQRI%p]LHUVXUIDFHV 3URJUDPPHUVRIWHQFKRRVH %p]LHUVXUIDFHVIRUDSSOLFDWLRQVVXFKDVDQDLUSODQHFRFNSLW VLPXODWRU WKDWUHTXLUHUHDOWLPHUHQGHULQJ

&XUYHVDQG6XUIDFHV

$ TXDUWLF%p]LHUFXUYHLVGHWHUPLQHGE\ÀYHFRQWUROSRLQWV S0 S1 S2 ; S3 DQG S4 [.t/ D .1 " t/4 S0 C 4t.1 " t/3 S1 C 6t 2 .1 " t/2 S2

C 4t 3 .1 " t /S3 C t 4 S4

IRU 0 ' t ' 1

&RQVWUXFWWKHTXDUWLFEDVLVPDWUL[ MB IRU [.t/ 7KH´%µ LQ%VSOLQHUHIHUVWRWKHIDFWWKDWDVHJPHQW [.t/ PD\ EHZULWWHQLQWHUPVRIDEDVLVPDWUL[ MS LQDIRUPVLPLODU WRD%p]LHUFXUYH 7KDWLV [.t/ D GMS X.t/

IRU 0 ' t ' 1

ZKHUH G LVWKHJHRPHWU\PDWUL[ Œ S0 S1 S2 S3 ! DQG X.t / LVWKHFROXPQYHFWRU .1; t; t 2 ; t 3 / ,QD XQLIRUP %VSOLQH HDFK VHJPHQWXVHVWKHVDPHEDVLVPDWUL[ EXWWKHJHRPHWU\PDWUL[ FKDQJHV &RQVWUXFWWKHEDVLVPDWUL[ MS IRU [.t/ ,Q([HUFLVHVDQG PDUNHDFKVWDWHPHQW7UXHRU)DOVH -XVWLI\ HDFKDQVZHU D 7KHFXELF%p]LHUFXUYHLVEDVHGRQIRXUFRQWUROSRLQWV E *LYHQDTXDGUDWLF%p]LHUFXUYH [.t/ ZLWKFRQWUROSRLQWV S0 S1 DQG S2 WKHGLUHFWHGOLQHVHJPHQW S1 " S0 IURP S0 WR S1 LVWKHWDQJHQWYHFWRUWRWKHFXUYHDW S0 F :KHQWZRTXDGUDWLF%p]LHUFXUYHVZLWKFRQWUROSRLQWV fS0 ; S1 ; S2 g DQG fS2 ; S3 ; S4 g DUHMRLQHGDW S2 WKHFRPELQHG %p]LHUFXUYHZLOOKDYH C 1 FRQWLQXLW\DW S2 LI S2 LVWKH PLGSRLQWRIWKHOLQHVHJPHQWEHWZHHQ S1 DQG S3 D 7KHHVVHQWLDOSURSHUWLHVRI%p]LHUFXUYHVDUHSUHVHUYHG XQGHUWKHDFWLRQRIOLQHDUWUDQVIRUPDWLRQV EXWQRWWUDQV ODWLRQV E :KHQWZR%p]LHUFXUYHV [.t/ DQG \.t/ DUHMRLQHGDWWKH SRLQWZKHUH [.1/ D \.0/ WKHFRPELQHGFXUYHKDV G 0 FRQWLQXLW\DWWKDWSRLQW F 7KH%p]LHUEDVLVPDWUL[LVDPDWUL[ZKRVHFROXPQVDUH WKHFRQWUROSRLQWVRIWKHFXUYH ([HUFLVHV²FRQFHUQWKHVXEGLYLVLRQRID%p]LHUFXUYHVKRZQ LQ )LJ /HW [.t/ EH WKH %p]LHU FXUYH ZLWK FRQWURO SRLQWV S0 ; : : : ; S3 DQGOHW \.t/ DQG ].t/ EHWKHVXEGLYLGLQJ%p]LHUFXUYHV DVLQWKHWH[W ZLWKFRQWUROSRLQWV T0 ; : : : ; T3 DQG U0 ; : : : ; U3 UHVSHF WLYHO\ D 8VHHTXDWLRQ WRVKRZWKDW T1 LVWKHPLGSRLQWRIWKH VHJPHQWIURP S0 WR S1 E 8VHHTXDWLRQ WRVKRZWKDW

8T2 D 8T3 C S0 C S1 " S2 " S3 : F 8VH SDUW E HTXDWLRQ DQG SDUW D WR VKRZ WKDW T2 LV WKH PLGSRLQW RI WKH VHJPHQW IURP T1 WR WKH PLGSRLQW RI WKH VHJPHQW IURP S1 WR S2 7KDW LV T2 D 12 ŒT1 C 12 .S1 C S2 /! D -XVWLI\HDFKHTXDOVVLJQ

3.U3 " U2 / D ]0 .1/ D :5[0 .1/ D 32 .S3 " S2 /:

$)"15&3


E 6KRZWKDW U2 LVWKHPLGSRLQWRIWKHVHJPHQWIURP S2 WR S3 F -XVWLI\HDFKHTXDOVVLJQ 3.U1 " U0 / D ]0 .0/ D :5[0 .:5/

G 8VH SDUW F WR VKRZ WKDW 8U1 D "S0 " S1 C S2 C S3 C 8U0

H 8VHSDUWG HTXDWLRQ DQGSDUWD WRVKRZWKDW U1 LV WKHPLGSRLQWRIWKHVHJPHQWIURP U2 WRWKHPLGSRLQWRIWKH VHJPHQWIURP S1 WR S2 7KDWLV U1 D 12 ŒU2 C 12 .S1 C S2 /!

6RPHWLPHVRQO\RQHKDOIRID%p]LHUFXUYHQHHGVIXUWKHU VXEGLYLGLQJ )RUH[DPSOH VXEGLYLVLRQRIWKHÓHIWµVLGH LVDFFRPSOLVKHGZLWKSDUWVD DQGF RI([HUFLVHDQG HTXDWLRQ :KHQERWKKDOYHVRIWKHFXUYH [.t/ DUHGLYLGHG LWLVSRVVLEOHWRRUJDQL]HFDOFXODWLRQVHIÀFLHQWO\WRFDOFXODWH ERWKOHIWDQGULJKWFRQWUROSRLQWVFRQFXUUHQWO\ ZLWKRXWXVLQJ HTXDWLRQ GLUHFWO\ D 6KRZWKDWWKHWDQJHQWYHFWRUV \0 .1/ DQG ]0 .0/ DUHHTXDO E 8VHSDUWD WRVKRZWKDW T3 ZKLFKHTXDOV U0 / LVWKH PLGSRLQWRIWKHVHJPHQWIURP T2 WR U1 F 8VLQJSDUWE DQGWKHUHVXOWVRI([HUFLVHVDQG ZULWH DQDOJRULWKPWKDWFRPSXWHVWKHFRQWUROSRLQWVIRUERWK \.t/ DQG ].t/ LQDQHIÀFLHQWPDQQHU 7KHRQO\RSHUDWLRQV QHHGHGDUHVXPVDQGGLYLVLRQE\ ([SODLQZK\DFXELF%p]LHUFXUYHLVFRPSOHWHO\GHWHUPLQHG E\ [.0/ [0 .0/ [.1/ DQG [0 .1/

7UXH7\SH IRQWV FUHDWHGE\$SSOH&RPSXWHUDQG$GREH 6\VWHPV XVH TXDGUDWLF %p]LHU FXUYHV ZKLOH 3RVW6FULSW IRQWV FUHDWHGE\0LFURVRIW XVHFXELF%p]LHUFXUYHV 7KH FXELFFXUYHVSURYLGHPRUHÁH[LELOLW\IRUW\SHIDFHGHVLJQ EXWLWLVLPSRUWDQWWR0LFURVRIWWKDWHYHU\W\SHIDFHXVLQJ TXDGUDWLFFXUYHVFDQEHWUDQVIRUPHGLQWRRQHWKDWXVHVFXELF FXUYHV 6XSSRVHWKDW Z.t/ LVDTXDGUDWLFFXUYH ZLWKFRQWURO SRLQWV S0 S1 DQG S2 D )LQGFRQWUROSRLQWV U0 U1 U2 DQG U3 VXFKWKDWWKHFXELF %p]LHUFXUYH [.t/ ZLWKWKHVHFRQWUROSRLQWVKDVWKHSURS HUW\WKDW [.t/ DQG Z.t/ KDYHWKHVDPHLQLWLDODQGWHUPLQDO SRLQWVDQGWKHVDPHWDQJHQWYHFWRUVDW t D 0 DQG t D 1 6HH([HUFLVH E 6KRZ WKDW LI [.t/ LV FRQVWUXFWHG DV LQ SDUW D WKHQ [.t/ D Z.t/ IRU 0 ' t ' 1

8VHSDUWLWLRQHGPDWUL[PXOWLSOLFDWLRQWRFRPSXWHWKHIROORZ LQJPDWUL[SURGXFW ZKLFKDSSHDUVLQWKHDOWHUQDWLYHIRUPXOD IRUD%p]LHUFXUYH 2 32 3 1 0 0 0 S0 6 "3 6 7 3 0 07 6 76 S1 7 4 3 "6 3 0 54 S2 5 S3 "1 3 "3 1

40-65*0/4 50 13"$5*$& 130#-&.4 )URPHTXDWLRQ ZLWK t D 0 [.0/ 6D S0 EHFDXVH

[.0/ D 16 ŒS0 C 4S1 C S2 ! D 16 S0 C 23 S1 C 16 S2 :

7KHFRHIÀFLHQWVDUHQRQQHJDWLYHDQGVXPWR VR [.0/ % LVLQ&FRQY fS0 ; S1 ; S2 g DQG WKHDIÀQHFRRUGLQDWHVZLWKUHVSHFWWR fS0 ; S1 ; S2 g DUH 16 ; 23 ; 16 )URPHTXDWLRQ ZLWK t D 1 [.1/ 6D S3 EHFDXVH [.1/ D 16 ŒS1 C 4S2 C S3 ! D 16 S1 C 23 S2 C 16 S3 :

7KHFRHIÀFLHQWVDUHQRQQHJDWLYHDQGVXPWR VR [.1/ % LVLQ&FRQY fS1 ; S2 ; S3 g DQG WKHDIÀQHFRRUGLQDWHVZLWKUHVSHFWWR fS1 ; S2 ; S3 g DUH 16 ; 23 ; 16

"11&/%*9

"

8QLTXHQHVVRIWKH5HGXFHG (FKHORQ)RUP 5)&03&.

6OJRVFOFTT PG UIF 3FEVDFE &DIFMPO 'PSN (DFK m ! n PDWUL[ A LVURZHTXLYDOHQWWRDXQLTXHUHGXFHGHFKHORQPDWUL[ U

1300' 7KHSURRIXVHVWKHLGHDIURP6HFWLRQ WKDWWKHFROXPQVRIURZHTXLYDOHQW PDWULFHVKDYHH[DFWO\WKHVDPHOLQHDUGHSHQGHQFHUHODWLRQV 7KHURZUHGXFWLRQDOJRULWKPVKRZVWKDWWKHUHH[LVWVDWOHDVWRQHVXFKPDWUL[ U 6XSSRVHWKDW A LVURZHTXLYDOHQWWRPDWULFHV U DQG V LQUHGXFHGHFKHORQIRUP 7KH OHIWPRVWQRQ]HURHQWU\LQDURZRI U LVDÓHDGLQJOµ &DOOWKHORFDWLRQRIVXFKDOHDGLQJ DSLYRWSRVLWLRQ DQGFDOOWKHFROXPQWKDWFRQWDLQVLWDSLYRWFROXPQ 7KLVGHÀQLWLRQ XVHVRQO\WKHHFKHORQQDWXUHRI U DQG V DQGGRHVQRWDVVXPHWKHXQLTXHQHVVRIWKH UHGXFHGHFKHORQIRUP 7KHSLYRWFROXPQVRI U DQG V DUHSUHFLVHO\WKHQRQ]HURFROXPQVWKDWDUH QRW OLQHDUO\GHSHQGHQWRQWKHFROXPQVWRWKHLUOHIW 7KLVFRQGLWLRQLVVDWLVÀHGDXWRPDWLFDOO\ E\D ÀUVW FROXPQLILWLVQRQ]HUR 6LQFH U DQG V DUHURZHTXLYDOHQWERWKEHLQJURZ HTXLYDOHQWWR A WKHLUFROXPQVKDYHWKHVDPHOLQHDUGHSHQGHQFHUHODWLRQV +HQFH WKH SLYRWFROXPQVRI U DQG V DSSHDULQWKHVDPHORFDWLRQV ,IWKHUHDUH r VXFKFROXPQV WKHQVLQFH U DQG V DUHLQUHGXFHGHFKHORQIRUP WKHLUSLYRWFROXPQVDUHWKHÀUVW r FROXPQVRIWKH m ! m LGHQWLW\PDWUL[ 7KXV FRUUHVSRQGLQJSLYRWFROXPQVRI U DQG V DUHHTXDO )LQDOO\ FRQVLGHUDQ\QRQSLYRWFROXPQRI U VD\FROXPQ j 7KLVFROXPQLVHLWKHU ]HURRUDOLQHDUFRPELQDWLRQRIWKHSLYRWFROXPQVWRLWVOHIWEHFDXVHWKRVHSLYRWFROXPQV DUHDEDVLVIRUWKHVSDFHVSDQQHGE\WKHFROXPQVWRWKHOHIWRIFROXPQ j (LWKHUFDVH FDQEHH[SUHVVHGE\ZULWLQJ U [ D IRUVRPH [ ZKRVH j WKHQWU\LV 7KHQ V [ D WRR ZKLFKVD\VWKDWFROXPQ j RI V LVHLWKHU]HURRUWKH VDPH OLQHDUFRPELQDWLRQRIWKH SLYRWFROXPQVRI V WR LWV OHIW 6LQFHFRUUHVSRQGLQJSLYRWFROXPQVRI U DQG V DUHHTXDO FROXPQV j RI U DQG V DUHDOVRHTXDO 7KLVKROGVIRUDOOQRQSLYRWFROXPQV VR V D U ZKLFKSURYHVWKDW U LVXQLTXH

"

"11&/%*9

#

&RPSOH[1XPEHUV

$ FRPSOH[QXPEHU LVDQXPEHUZULWWHQLQWKHIRUP

´ D a C bi ZKHUH a DQG b DUHUHDOQXPEHUVDQG i LVDIRUPDOV\PEROVDWLVI\LQJWKHUHODWLRQ i 2 D !1 7KHQXPEHU a LVWKH UHDOSDUW RI ´ GHQRWHGE\ 5H ´ DQG b LVWKH LPDJLQDU\SDUW RI ´ GHQRWHGE\ ,P ´ 7ZRFRPSOH[QXPEHUVDUHFRQVLGHUHGHTXDOLIDQGRQO\LIWKHLU UHDODQGLPDJLQDU\SDUWVDUHHTXDO )RUH[DPSOH LI ´ D 5 C .!2/i WKHQ 5H ´ D 5 DQG ,P ´ D !2 )RUVLPSOLFLW\ ZHZULWH ´ D 5 ! 2i $ UHDOQXPEHU a LVFRQVLGHUHGDVDVSHFLDOW\SHRIFRPSOH[QXPEHU E\LGHQWLI\LQJ a ZLWK a C 0i )XUWKHUPRUH DULWKPHWLFRSHUDWLRQVRQUHDOQXPEHUVFDQEHH[WHQGHGWR WKHVHWRIFRPSOH[QXPEHUV 7KH FRPSOH[QXPEHUV\VWHP GHQRWHGE\ C LVWKHVHWRIDOOFRPSOH[QXPEHUV WRJHWKHUZLWKWKHIROORZLQJRSHUDWLRQVRIDGGLWLRQDQGPXOWLSOLFDWLRQ

.a C bi / C .c C d i/ D .a C c/ C .b C d /i

.a C bi /.c C d i/ D .ac ! bd / C .ad C bc/i

7KHVHUXOHVUHGXFHWRRUGLQDU\DGGLWLRQDQGPXOWLSOLFDWLRQRIUHDOQXPEHUVZKHQ b DQG d DUH]HURLQ DQG ,WLVUHDGLO\FKHFNHGWKDWWKHXVXDOODZVRIDULWKPHWLF IRU R DOVRKROGIRU C )RUWKLVUHDVRQ PXOWLSOLFDWLRQLVXVXDOO\FRPSXWHGE\DOJHEUDLF H[SDQVLRQ DVLQWKHIROORZLQJH[DPSOH

&9".1-&

.5 ! 2i /.3 C 4i / D 15 C 20i ! 6i ! 8i 2 D 15 C 14i ! 8.!1/ D 23 C 14i

7KDWLV PXOWLSO\HDFKWHUPRI 5 ! 2i E\HDFKWHUPRI 3 C 4i XVH i 2 D !1 DQGZULWH WKHUHVXOWLQWKHIRUP a C bi 6XEWUDFWLRQRIFRPSOH[QXPEHUV ´1 DQG ´2 LVGHÀQHGE\

´1 ! ´2 D ´1 C .!1/´2 ,QSDUWLFXODU ZHZULWH !´ LQSODFHRI .!1/´ "

"11&/%*9 #

E\

&RPSOH[1XPEHUV

"

7KH FRQMXJDWH RI ´ D a C bi LVWKHFRPSOH[QXPEHU ´ UHDGDV´´ EDUµ GHÀQHG

´ D a ! bi 2EWDLQ ´ IURP ´ E\UHYHUVLQJWKHVLJQRIWKHLPDJLQDU\SDUW

&9".1-& 7KHFRQMXJDWHRI !3 C 4i LV !3 ! 4i ZULWH !3 C 4i D !3 ! 4i 2EVHUYHWKDWLI ´ D a C bi WKHQ

´´ D .a C bi /.a ! bi / D a2 ! abi C bai ! b 2 i 2 D a2 C b 2

6LQFH ´´ LVUHDODQGQRQQHJDWLYH LWKDVDVTXDUHURRW 7KH DEVROXWHYDOXH RU PRGXOXV RI ´ LVWKHUHDOQXPEHU j´j GHÀQHGE\

j´j D

p

´´ D

p a2 C b 2

p ,I ´ LVDUHDOQXPEHU WKHQ ´ D a C 0i DQG j´j D a2 ZKLFKHTXDOVWKHRUGLQDU\ DEVROXWHYDOXHRI a 6RPHXVHIXOSURSHUWLHVRIFRQMXJDWHVDQGDEVROXWHYDOXHDUHOLVWHGEHORZ w DQG ´ GHQRWHFRPSOH[QXPEHUV

´ D ´ LIDQGRQO\LI ´ LVDUHDOQXPEHU w C ´ D w C ´ w´ D w ´ LQSDUWLFXODU r´ D r´ LI r LVDUHDOQXPEHU ´´ D j´j2 " 0 jw´j D jwjj´j jw C ´j # jwj C j´j

,I ´ ¤ 0 WKHQ j´j > 0 DQG ´ KDVDPXOWLSOLFDWLYHLQYHUVH GHQRWHGE\ 1=´ RU ´!1 DQGJLYHQE\

1 ´ D ´!1 D ´ j´j2 2IFRXUVH DTXRWLHQW w=´ VLPSO\PHDQV w $ .1=´/

&9".1-& /HW w D 3 C 4i DQG ´ D 5 ! 2i &RPSXWH ´´ j´j DQG w=´ 40-65*0/ )URPHTXDWLRQ ´´ D 52 C .!2/2 D 25 C 4 D 29

p p )RUWKHDEVROXWHYDOXH j´j D ´´ D 29 7RFRPSXWH w=´ ÀUVWPXOWLSO\ERWKWKH QXPHUDWRUDQGWKHGHQRPLQDWRUE\ ´ WKHFRQMXJDWHRIWKHGHQRPLQDWRU %HFDXVHRI

"

"11&/%*9 #

&RPSOH[1XPEHUV

WKLVHOLPLQDWHVWKH i LQWKHGHQRPLQDWRU

3 C 4i w D ´ 5 ! 2i D D

3 C 4i 5 C 2i $ 5 ! 2i 5 C 2i

15 C 6i C 20i ! 8 52 C .!2/2

7 C 26i 29 26 7 C i D 29 29 D

*HRPHWULF,QWHUSUHWDWLRQ (DFKFRPSOH[QXPEHU ´ D a C bi FRUUHVSRQGVWRDSRLQW .a; b/ LQWKHSODQH R2 DV LQ)LJ 7KHKRUL]RQWDOD[LVLVFDOOHGWKH UHDOD[LV EHFDXVHWKHSRLQWVa; 0 RQLW FRUUHVSRQGWRWKHUHDOQXPEHUV 7KHYHUWLFDOD[LVLVWKH LPDJLQDU\D[LV EHFDXVHWKH SRLQWV .0; b/ RQLWFRUUHVSRQGWRWKH SXUHLPDJLQDU\QXPEHUV RIWKHIRUP 0 C bi RU VLPSO\ bi 7KHFRQMXJDWHRI ´ LVWKHPLUURULPDJHRI ´ LQWKHUHDOD[LV 7KHDEVROXWH YDOXHRI ´ LVWKHGLVWDQFHIURP .a; b/ WRWKHRULJLQ Imaginary axis z = a + bi

b

Real axis

a

z = a – bi '*(63& 7KHFRPSOH[FRQMXJDWHLVDPLUURULPDJH

$GGLWLRQRIFRPSOH[QXPEHUV ´ D a C bi DQG w D c C d i FRUUHVSRQGVWRYHFWRU DGGLWLRQRI .a; b/ DQG .c; d / LQ R2 DVLQ)LJ Im z w+z w

z Re z

'*(63& $GGLWLRQRIFRPSOH[QXPEHUV

"11&/%*9 #

&RPSOH[1XPEHUV

"

7RJLYHDJUDSKLFDOUHSUHVHQWDWLRQRIFRPSOH[PXOWLSOLFDWLRQ ZHXVH SRODUFRRUGL QDWHV LQ R2 *LYHQDQRQ]HURFRPSOH[QXPEHU ´ D a C bi OHW ' EHWKHDQJOHEHWZHHQ WKHSRVLWLYHUHDOD[LVDQGWKHSRLQW .a; b/ DVLQ)LJ ZKHUH !! < ' # ! 7KHDQJOH ' LVFDOOHGWKH DUJXPHQW RI ´ ZHZULWH ' D DUJ ´ )URPWULJRQRPHWU\

a D j´j FRV ';

b D j´j VLQ '

DQGVR

´ D a C bi D j´j.FRV ' C i VLQ '/ Im z z |z|

|z | sin ϕ

ϕ

Re z

| z| cos ϕ

'*(63& 3RODUFRRUGLQDWHVRI ´

,I w LVDQRWKHUQRQ]HURFRPSOH[QXPEHU VD\

w D jwj .FRV # C i VLQ #/ WKHQ XVLQJVWDQGDUGWULJRQRPHWULFLGHQWLWLHVIRUWKHVLQHDQGFRVLQHRIWKHVXPRIWZR DQJOHV RQHFDQYHULI\WKDW

w´ D jwj j´j ŒFRV.# C '/ C i VLQ.# C '/"

6HH)LJ $ VLPLODUIRUPXODPD\EHZULWWHQIRUTXRWLHQWVLQSRODUIRUP 7KHIRUPXODV IRUSURGXFWVDQGTXRWLHQWVFDQEHVWDWHGLQZRUGVDVIROORZV Im z wz !+ϕ

w

z |z| ! ϕ

Re z

'*(63& 0XOWLSOLFDWLRQZLWKSRODU

FRRUGLQDWHV

7KHSURGXFWRIWZRQRQ]HURFRPSOH[QXPEHUVLVJLYHQLQSRODUIRUPE\WKH SURGXFWRIWKHLUDEVROXWHYDOXHVDQGWKHVXPRIWKHLUDUJXPHQWV 7KHTXRWLHQW RIWZRQRQ]HURFRPSOH[QXPEHUVLVJLYHQE\WKHTXRWLHQWRIWKHLUDEVROXWHYDOXHV DQGWKHGLIIHUHQFHRIWKHLUDUJXPHQWV

Im z iz

ϕ ! π2 z=3+i

i π 2

ϕ

0XOWLSOLFDWLRQE\ i

Re z

&9".1-& D ,I w KDVDEVROXWHYDOXH WKHQ w D FRV # C i VLQ # ZKHUH # LVWKHDUJXPHQWRI w 0XOWLSOLFDWLRQRIDQ\QRQ]HURQXPEHU ´ E\ w VLPSO\URWDWHV ´ WKURXJKWKHDQJOH # E 7KHDUJXPHQWRI i LWVHOILV !=2 UDGLDQV VRPXOWLSOLFDWLRQRI ´ E\ i URWDWHV ´ WKURXJK DQDQJOHRI !=2 UDGLDQV )RUH[DPSOH 3 C i LVURWDWHGLQWR .3 C i/i D !1 C 3i

"

"11&/%*9 #

&RPSOH[1XPEHUV

3RZHUVRID&RPSOH[1XPEHU )RUPXOD DSSOLHVZKHQ ´ D w D r.FRV ' C i VLQ '/ ,QWKLVFDVH

´2 D r 2 .FRV 2' C i VLQ 2'/ DQG

´3 D ´ $ ´2

D r.FRV ' C i VLQ '/ $ r 2 .FRV 2' C i VLQ 2'/

D r 3 .FRV 3' C i VLQ 3'/ ,QJHQHUDO IRUDQ\SRVLWLYHLQWHJHU k

´k D r k .FRV k' C i VLQ k'/ 7KLVIDFWLVNQRZQDV 'H0RLYUH·V7KHRUHP

&RPSOH[1XPEHUVDQG R2 $OWKRXJKWKHHOHPHQWVRI R2 DQG C DUHLQRQHWRRQHFRUUHVSRQGHQFH DQGWKHRSHUDWLRQV RIDGGLWLRQDUHHVVHQWLDOO\WKHVDPH WKHUHLVDORJLFDOGLVWLQFWLRQEHWZHHQ R2 DQG C ,Q R2 ZHFDQRQO\PXOWLSO\DYHFWRUE\DUHDOVFDODU ZKHUHDVLQ C ZHFDQPXOWLSO\DQ\ WZRFRPSOH[QXPEHUVWRREWDLQDWKLUGFRPSOH[QXPEHU 7KHGRWSURGXFWLQ R2 GRHVQ·W FRXQW EHFDXVHLWSURGXFHVDVFDODU QRWDQHOHPHQWRI R2 :/ :HXVHVFDODUQRWDWLRQIRU HOHPHQWVLQ C WRHPSKDVL]HWKLVGLVWLQFWLRQ x2

Im z (2, 4)

2 + 4i –1 + 2i

(–1, 2) (4, 0)

(–3, –1)

–3 – i (3, –2)

7KHUHDOSODQH R2

4 + 0i

x1

3 – 2i

7KHFRPSOH[SODQH C

Re z

*ORVVDU\ " DGMXJDWH RU FODVVLFDODGMRLQW 7KHPDWUL[ DGM A IRUPHGIURP DVTXDUHPDWUL[ A E\UHSODFLQJWKH .i; j /HQWU\RI A E\WKH .i; j /FRIDFWRU IRUDOO i DQG j DQGWKHQWUDQVSRVLQJWKH UHVXOWLQJPDWUL[ DIÀQHFRPELQDWLRQ $ OLQHDUFRPELQDWLRQRIYHFWRUVSRLQWVLQ Rn LQZKLFKWKHVXPRIWKHZHLJKWVLQYROYHGLV DIÀQHGHSHQGHQFHUHODWLRQ $QHTXDWLRQRIWKHIRUP c1 Y1 C ! ! ! C cp Yp D ZKHUH WKH ZHLJKWV c1 ; : : : ; cp DUH QRW DOO ]HUR DQG c1 C ! ! ! C cp D 0 DIÀQHKXOO RU DIÀQHVSDQ RIDVHW S 7KHVHWRIDOODIÀQH FRPELQDWLRQVRISRLQWVLQ S GHQRWHGE\ DII S DIÀQHO\GHSHQGHQWVHW $ VHW fY1 ; : : : ; Yp g LQ Rn VXFKWKDWWKHUH DUHUHDOQXPEHUV c1 ; : : : ; cp QRWDOO]HUR VXFKWKDW c1 C ! ! ! C cp D 0 DQG c1 Y1 C ! ! ! C cp Yp D DIÀQHO\LQGHSHQGHQWVHW $ VHW fY1 ; : : : ; Yp g LQ Rn WKDWLVQRW DIÀQHO\GHSHQGHQW DIÀQHVHW RU DIÀQHVXEVHW $ VHW S RISRLQWVVXFKWKDWLI S DQG T DUHLQ S WKHQ .1 " t/S C t T 2 S IRUHDFKUHDOQXPEHU t DIÀQHWUDQVIRUPDWLRQ $ PDSSLQJ T W Rn ! Rm RIWKHIRUP T .[/ D A[ C E ZLWK A DQ m # n PDWUL[DQG E LQ Rm DOJHEUDLFPXOWLSOLFLW\ 7KHPXOWLSOLFLW\RIDQHLJHQYDOXHDVD URRWRIWKHFKDUDFWHULVWLFHTXDWLRQ DQJOH EHWZHHQQRQ]HURYHFWRUV X DQG Y LQ R2 RU R3 / 7KHDQJOH # EHWZHHQWKHWZRGLUHFWHGOLQHVHJPHQWVIURPWKHRULJLQWR WKHSRLQWV X DQG Y 5HODWHGWRWKHVFDODUSURGXFWE\ X ! Y D kXk kYk FRV # DVVRFLDWLYHODZRIPXOWLSOLFDWLRQ A.%&/ D .AB/C IRUDOO A B C DWWUDFWRU RIDG\QDPLFDOV\VWHPLQ R2 / 7KHRULJLQZKHQDOO WUDMHFWRULHVWHQGWRZDUG DXJPHQWHGPDWUL[ $ PDWUL[PDGHXSRIDFRHIÀFLHQWPDWUL[ IRUDOLQHDUV\VWHPDQGRQHRUPRUHFROXPQVWRWKHULJKW (DFKH[WUDFROXPQFRQWDLQVWKHFRQVWDQWVIURPWKHULJKWVLGH RIDV\VWHPZLWKWKHJLYHQFRHIÀFLHQWPDWUL[ DX[LOLDU\ HTXDWLRQ $ SRO\QRPLDO HTXDWLRQ LQ D YDULDEOH r FUHDWHGIURPWKHFRHIÀFLHQWVRIDKRPRJHQHRXVGLIIHUHQFH HTXDWLRQ

# EDFNVXEVWLWXWLRQ ZLWKPDWUL[QRWDWLRQ 7KHEDFNZDUGSKDVH RIURZUHGXFWLRQRIDQDXJPHQWHGPDWUL[WKDWWUDQVIRUPVDQ HFKHORQPDWUL[LQWRDUHGXFHGHFKHORQPDWUL[ XVHGWRÀQG WKHVROXWLRQV RIDV\VWHPRIOLQHDUHTXDWLRQV

EDFNZDUGSKDVH RIURZUHGXFWLRQ 7KHODVWSDUWRIWKHDO JRULWKPWKDWUHGXFHVDPDWUL[LQHFKHORQIRUPWRDUHGXFHG HFKHORQIRUP EDQGPDWUL[ $ PDWUL[ZKRVHQRQ]HURHQWULHVOLHZLWKLQDEDQG DORQJWKHPDLQGLDJRQDO EDU\FHQWULFFRRUGLQDWHV RIDSRLQW S ZLWKUHVSHFWWRDQDIÀQHO\ LQGHSHQGHQW VHW S D fY1 : : : ; Yk g 7KH XQLTXH VHW RI ZHLJKWV c1 ; : : : ; ck VXFKWKDW S D c1 Y1 C ! ! ! C ck Yk DQG c1 C ! ! ! C ck D 1 6RPHWLPHVDOVRFDOOHGWKH DIÀQHFRRUGLQDWHV RI S ZLWKUHVSHFWWR S EDVLFYDULDEOH $ YDULDEOHLQDOLQHDUV\VWHPWKDWFRUUHVSRQGV WRDSLYRWFROXPQLQWKHFRHIÀFLHQWPDWUL[ EDVLV IRUDQRQWULYLDOVXEVSDFH H RIDYHFWRUVSDFH V / $Q LQGH[HG VHW B D fY1 ; : : : ; Yp g LQ V VXFK WKDW L B LVD OLQHDUO\LQGHSHQGHQWVHWDQGLL WKHVXEVSDFHVSDQQHGE\ B FRLQFLGHVZLWK H WKDWLV H D 6SDQ fY1 ; : : : ; Yp g BFRRUGLQDWHVRI [ B

EHVWDSSUR[LPDWLRQ JLYHQYHFWRU

6HH FRRUGLQDWHVRI [ UHODWLYHWRWKHEDVLV 7KHFORVHVWSRLQWLQDJLYHQVXEVSDFHWRD

ELGLDJRQDOPDWUL[ $ PDWUL[ZKRVHQRQ]HURHQWULHVOLHRQWKH PDLQGLDJRQDODQGRQRQHGLDJRQDODGMDFHQWWRWKHPDLQ GLDJRQDO EORFNGLDJRQDO PDWUL[ $ SDUWLWLRQHGPDWUL[ A D ŒAij ! VXFK WKDWHDFKEORFN Aij LVD]HURPDWUL[IRU i ¤ j

EORFNPDWUL[

6HH SDUWLWLRQHGPDWUL[

EORFNPDWUL[PXOWLSOLFDWLRQ 7KHURZ²FROXPQPXOWLSOLFDWLRQ RISDUWLWLRQHGPDWULFHVDVLIWKHEORFNHQWULHVZHUHVFDODUV EORFN XSSHU WULDQJXODU PDWUL[ $ SDUWLWLRQHG PDWUL[ A D ŒAij ! VXFK WKDW HDFK EORFN Aij LV D ]HUR PDWUL[ IRU i > j ERXQGDU\SRLQW RIDVHW S LQ Rn $ SRLQW S VXFKWKDWHYHU\RSHQ EDOOLQ Rn FHQWHUHGDW S LQWHUVHFWVERWK S DQGWKHFRPSOHPHQW RI S ERXQGHGVHW LQ Rn $ VHWWKDWLVFRQWDLQHGLQDQRSHQEDOO B.; ı/ IRUVRPH ı > 0 BPDWUL[ IRU T $ PDWUL[ ŒT !B IRUDOLQHDUWUDQVIRUPDWLRQ T W V ! V UHODWLYHWRDEDVLV B IRU V ZLWKWKHSURSHUW\ WKDW ŒT .[/!B D ŒT !B Œ[!B IRUDOO [ LQ 9

$ &DXFK\²6FKZDU]LQHTXDOLW\ FKDQJHRIEDVLV

jhX; Yij $ kuk!kvk IRUDOO X Y

6HH FKDQJHRIFRRUGLQDWHVPDWUL[

"

"

(MPTTBSZ

FKDQJHRIFRRUGLQDWHVPDWUL[ IURPDEDVLV B WRDEDVLV C $ PDWUL[ C P B WKDWWUDQVIRUPV BFRRUGLQDWHYHFWRUVLQWR C FRRUGLQDWHYHFWRUV Œ[!C D P Œ[!B ,I C LVWKHVWDQGDUG EDVLVIRU Rn WKHQ

C

C

B

P LVVRPHWLPHVZULWWHQDV PB B

FKDUDFWHULVWLFHTXDWLRQ RI A GHW.A " "I / D 0 FKDUDFWHULVWLF SRO\QRPLDO RI A GHW.A " "I / RU LQ VRPH WH[WV GHW."I " A/ &KROHVN\IDFWRUL]DWLRQ $ IDFWRUL]DWLRQ A D RTR ZKHUH R LV DQLQYHUWLEOHXSSHUWULDQJXODUPDWUL[ZKRVHGLDJRQDOHQWULHV DUHDOOSRVLWLYH FORVHGEDOO LQ Rn $ VHW f[ W k[ " Sk < ıg LQ Rn ZKHUH S LV LQ Rn DQG ı > 0 FORVHGVHW LQ Rn $ VHWWKDWFRQWDLQVDOORILWVERXQGDU\SRLQWV FRGRPDLQ RIDWUDQVIRUPDWLRQ T W Rn ! Rm / 7KHVHW Rm WKDW FRQWDLQVWKHUDQJHRI T ,QJHQHUDO LI T PDSVDYHFWRUVSDFH V LQWRDYHFWRUVSDFH W WKHQ W LVFDOOHGWKHFRGRPDLQRI T FRHIÀFLHQWPDWUL[ $ PDWUL[ZKRVHHQWULHVDUHWKHFRHIÀFLHQWV RIDV\VWHPRIOLQHDUHTXDWLRQV FRIDFWRU $ QXPEHU Cij D ."1/iCj GHW Aij FDOOHGWKH .i; j / FRIDFWRURI A ZKHUH Aij LVWKHVXEPDWUL[IRUPHGE\GHOHWLQJ WKH i WKURZDQGWKH j WKFROXPQRI A FRIDFWRUH[SDQVLRQ $ IRUPXODIRU GHW A XVLQJFRIDFWRUVDVVR FLDWHGZLWKRQHURZRURQHFROXPQ VXFKDVIRUURZ GHW A D a11 C11 C ! ! ! C a1n C1n FROXPQ²URZ H[SDQVLRQ 7KH H[SUHVVLRQ RI D SURGXFW AB DV D VXP RI RXWHU SURGXFWV FRO1 .A/ URZ1 .B/ C ! ! ! C FROn .A/ URZn .B/ ZKHUH n LVWKHQXPEHURIFROXPQVRI A FROXPQVSDFH RIDQ m # n PDWUL[ A 7KHVHW &RO A RIDOO OLQHDUFRPELQDWLRQVRIWKHFROXPQVRI A ,I A D ŒD1 ! ! ! Dn ! WKHQ &RO A D 6SDQ fD1 ; : : : ; Dn g (TXLYDOHQWO\ &RO A D f\ W \ D A[ IRUVRPH [ LQ Rn g FROXPQVXP 7KHVXPRIWKHHQWULHVLQDFROXPQRIDPDWUL[ FROXPQYHFWRU $ PDWUL[ZLWKRQO\RQHFROXPQ RUDVLQJOH FROXPQRIDPDWUL[WKDWKDVVHYHUDOFROXPQV FRPPXWLQJ PDWULFHV 7ZR PDWULFHV A DQG B VXFK WKDW AB D BA FRPSDFW VHW LQ Rn $ VHW LQ Rn WKDW LV ERWK FORVHG DQG ERXQGHG FRPSDQLRQPDWUL[ $ VSHFLDOIRUPRIPDWUL[ZKRVHFKDUDF WHULVWLFSRO\QRPLDOLV ."1/n p."/ ZKHQ p."/ LVDVSHFLÀHG SRO\QRPLDOZKRVHOHDGLQJWHUPLV "n FRPSOH[HLJHQYDOXH $ QRQUHDOURRWRIWKHFKDUDFWHULVWLFHTXD WLRQRIDQ n # n PDWUL[ FRPSOH[ HLJHQYHFWRU $ QRQ]HUR YHFWRU [ LQ C n VXFK WKDW A[ D "[ ZKHUH A LVDQ n # n PDWUL[DQG " LVDFRPSOH[ HLJHQYDOXH FRPSRQHQW RI \ RUWKRJRQDO WR X IRU X ¤ 7KH YHFWRU \!X \" X X!X

FRPSRVLWLRQRIOLQHDUWUDQVIRUPDWLRQV $ PDSSLQJSURGXFHG E\DSSO\LQJWZRRUPRUHOLQHDUWUDQVIRUPDWLRQVLQVXFFHV VLRQ ,IWKHWUDQVIRUPDWLRQVDUHPDWUL[WUDQVIRUPDWLRQV VD\ OHIWPXOWLSOLFDWLRQE\ B IROORZHGE\OHIWPXOWLSOLFDWLRQE\ A WKHQWKHFRPSRVLWLRQLVWKHPDSSLQJ [ 7! A.B [/ FRQGLWLRQQXPEHU RI A 7KHTXRWLHQW #1 =#n ZKHUH #1 LVWKH ODUJHVWVLQJXODUYDOXHRI A DQG #n LVWKHVPDOOHVWVLQJXODU YDOXH 7KHFRQGLWLRQQXPEHULV C1 ZKHQ #n LV]HUR FRQIRUPDEOHIRUEORFNPXOWLSOLFDWLRQ 7ZRSDUWLWLRQHGPDWUL FHV A DQG B VXFKWKDWWKHEORFNSURGXFW AB LVGHÀQHG 7KH FROXPQSDUWLWLRQRI A PXVWPDWFKWKHURZSDUWLWLRQRI % FRQVLVWHQWOLQHDUV\VWHP $ OLQHDUV\VWHPZLWKDWOHDVWRQH VROXWLRQ FRQVWUDLQHGRSWLPL]DWLRQ 7KHSUREOHPRIPD[LPL]LQJDTXDQ WLW\VXFKDV [TA[ RU kA[k ZKHQ [ LVVXEMHFWWRRQHRUPRUH FRQVWUDLQWV VXFKDV [T[ D 1 RU [TY D 0 FRQVXPSWLRQPDWUL[ $ PDWUL[LQWKH/HRQWLHILQSXW²RXWSXW PRGHOZKRVHFROXPQVDUHWKHXQLWFRQVXPSWLRQYHFWRUVIRU WKHYDULRXVVHFWRUVRIDQHFRQRP\ FRQWUDFWLRQ $ PDSSLQJ [ 7! r [ IRU VRPH VFDODU r ZLWK 0 $ r $ 1 FRQWUROODEOH SDLURIPDWULFHV $ PDWUL[SDLU .A; B/ ZKHUH A LV n # n B KDV n URZV DQG UDQN Œ B

AB

A2 B

!!!

An!1 B ! D n

5HODWHGWRDVWDWHVSDFHPRGHORIDFRQWUROV\VWHPDQGWKH GLIIHUHQFHHTXDWLRQ [kC1 D A[k C B Xk .k D 0; 1; : : :/ FRQYHUJHQW VHTXHQFHRIYHFWRUV $ VHTXHQFH f[k g VXFKWKDW WKHHQWULHVLQ [k FDQEHPDGHDVFORVHDVGHVLUHGWRWKHHQWULHV LQVRPHÀ[HGYHFWRUIRUDOO k VXIÀFLHQWO\ODUJH FRQYH[ FRPELQDWLRQ RI SRLQWV Y1 ; : : : ; Yk LQ Rn $ OLQHDU FRPELQDWLRQRIYHFWRUVSRLQWV LQZKLFKWKHZHLJKWVLQWKH FRPELQDWLRQDUHQRQQHJDWLYHDQGWKHVXPRIWKHZHLJKWV LV FRQYH[KXOO RIDVHW S 7KHVHWRIDOOFRQYH[FRPELQDWLRQVRI SRLQWVLQ S GHQRWHGE\ FRQY S FRQYH[VHW $ VHW S ZLWKWKHSURSHUW\WKDWIRUHDFK S DQG T LQ S WKHOLQHVHJPHQW ST LVFRQWDLQHGLQ S FRRUGLQDWHPDSSLQJ GHWHUPLQHGE\DQRUGHUHGEDVLV B LQD YHFWRUVSDFH V $ PDSSLQJWKDWDVVRFLDWHVWRHDFK [ LQ V LWVFRRUGLQDWHYHFWRU Œ[!B FRRUGLQDWHVRI [ UHODWLYHWRWKHEDVLV B D fE1 ; : : : ; En g 7KH ZHLJKWV c1 ; : : : ; cn LQWKHHTXDWLRQ [ D c1 E1 C ! ! ! C cn En FRRUGLQDWHYHFWRURI [ UHODWLYHWR B 7KHYHFWRU Œ[!B ZKRVH HQWULHVDUHWKHFRRUGLQDWHVRI [ UHODWLYHWRWKHEDVLV B FRYDULDQFH RIYDULDEOHV xi DQG xj IRU i ¤ j 7KHHQWU\ sij LQ WKHFRYDULDQFHPDWUL[ S IRUDPDWUL[RIREVHUYDWLRQV ZKHUH xi DQG xj YDU\RYHUWKH i WKDQG j WKFRRUGLQDWHV UHVSHFWLYHO\ RIWKHREVHUYDWLRQYHFWRUV FRYDULDQFHPDWUL[ RU VDPSOHFRYDULDQFHPDWUL[ 7KH p # p PDWUL[ S GHÀQHG E\ S D .N " 1/!1 BB T ZKHUH B LV D p # N PDWUL[RIREVHUYDWLRQVLQPHDQGHYLDWLRQIRUP

(MPTTBSZ

"

&UDPHU·VUXOH $ IRUPXODIRUHDFKHQWU\LQWKHVROXWLRQ [ RI WKHHTXDWLRQ A[ D E ZKHQ A LVDQLQYHUWLEOHPDWUL[ FURVVSURGXFWWHUP $ WHUP cxi xj LQDTXDGUDWLFIRUP ZLWK i ¤ j FXEH $ WKUHHGLPHQVLRQDOVROLGREMHFWERXQGHGE\VL[VTXDUH IDFHV ZLWKWKUHHIDFHVPHWWLQJDWHDFKYHUWH[

GRPDLQ RIDWUDQVIRUPDWLRQ T 7KHVHWRIDOOYHFWRUV [ IRU ZKLFK T .[/ LVGHÀQHG GRWSURGXFW 6HH LQQHUSURGXFW G\QDPLFDOV\VWHP 6HH GLVFUHWHOLQHDUG\QDPLFDOV\VWHP

%

HFKHORQIRUP RU URZHFKHORQIRUP RIDPDWUL[ $QHFKHORQ PDWUL[WKDWLVURZHTXLYDOHQWWRWKHJLYHQPDWUL[ HFKHORQPDWUL[ RU URZHFKHORQPDWUL[ $ UHFWDQJXODUPDWUL[ WKDWKDVWKUHHSURSHUWLHV $OOQRQ]HURURZVDUHDERYH DQ\URZRIDOO]HURV (DFKOHDGLQJHQWU\RIDURZLVLQ DFROXPQWRWKHULJKWRIWKHOHDGLQJHQWU\RIWKHURZDERYH LW $OOHQWULHVLQDFROXPQEHORZDOHDGLQJHQWU\DUH]HUR HLJHQIXQFWLRQV RI D GLIIHUHQWLDO HTXDWLRQ [0 .t/ D A[.t/ $ IXQFWLRQ [.t/ D Ye "t ZKHUH Y LVDQHLJHQYHFWRURI A DQG " LVWKHFRUUHVSRQGLQJHLJHQYDOXH HLJHQVSDFH RI A FRUUHVSRQGLQJWR " 7KHVHWRI DOO VROXWLRQV RI A[ D "[ ZKHUH " LVDQHLJHQYDOXHRI A &RQVLVWVRIWKH ]HURYHFWRUDQGDOOHLJHQYHFWRUVFRUUHVSRQGLQJWR " HLJHQYDOXH RI A $ VFDODU " VXFKWKDWWKHHTXDWLRQ A[ D "[ KDVDVROXWLRQIRUVRPHQRQ]HURYHFWRU [ HLJHQYHFWRU RI A $ QRQ]HUR YHFWRU [ VXFKWKDW A[ D "[ IRU VRPHVFDODU " HLJHQYHFWRUEDVLV $ EDVLVFRQVLVWLQJHQWLUHO\RIHLJHQYHFWRUV RIDJLYHQPDWUL[ HLJHQYHFWRUGHFRPSRVLWLRQ RI [ $QHTXDWLRQ [ D c1 Y1 C ! ! ! C cn Yn H[SUHVVLQJ [ DVDOLQHDUFRPELQDWLRQRIHLJHQ YHFWRUVRIDPDWUL[ HOHPHQWDU\PDWUL[ $QLQYHUWLEOHPDWUL[WKDWUHVXOWVE\SHU IRUPLQJRQHHOHPHQWDU\URZRSHUDWLRQRQDQLGHQWLW\PDWUL[ HOHPHQWDU\URZRSHUDWLRQV 5HSODFHPHQW 5HSODFHRQH URZE\WKHVXPRILWVHOIDQGDPXOWLSOHRIDQRWKHUURZ ,QWHUFKDQJHWZRURZV 6FDOLQJ 0XOWLSO\DOOHQWULHVLQD URZE\DQRQ]HURFRQVWDQW HTXDOYHFWRUV 9HFWRUVLQ Rn ZKRVHFRUUHVSRQGLQJHQWULHVDUH WKHVDPH HTXLOLEULXPSULFHV $ VHWRISULFHVIRUWKHWRWDORXWSXWRIWKH YDULRXVVHFWRUVLQDQHFRQRP\ VXFKWKDWWKHLQFRPHRIHDFK VHFWRUH[DFWO\EDODQFHVLWVH[SHQVHV HTXLOLEULXPYHFWRU 6HH VWHDG\VWDWHYHFWRU HTXLYDOHQW OLQHDU V\VWHPV /LQHDU V\VWHPV ZLWK WKH VDPH VROXWLRQVHW H[FKDQJHPRGHO 6HH /HRQWLHIH[FKDQJHPRGHO H[LVWHQFHTXHVWLRQ $VNV ´'RHVDVROXWLRQWRWKHV\VWHPH[ LVW"µ 7KDWLV ´,VWKHV\VWHPFRQVLVWHQW"µ $OVR ´'RHVD VROXWLRQRI A[ D E H[LVWIRU DOO SRVVLEOH E"µ H[SDQVLRQE\FRIDFWRUV 6HH FRIDFWRUH[SDQVLRQ H[SOLFLWGHVFULSWLRQ RIDVXEVSDFH W RI Rn $ SDUDPHWULF UHSUHVHQWDWLRQRI W DVWKHVHWRIDOOOLQHDUFRPELQDWLRQVRI DVHWRIVSHFLÀHGYHFWRUV H[WUHPHSRLQW RIDFRQYH[VHW S $ SRLQW S LQ S VXFKWKDW S LV QRWLQWKHLQWHULRURIDQ\OLQHVHJPHQWWKDWOLHVLQ S 7KDWLV

GHFRXSOHGV\VWHP $ GLIIHUHQFHHTXDWLRQ \kC1 D A\k RUD GLIIHUHQWLDOHTXDWLRQ \0 .t / D A\.t/ LQZKLFK A LVDGLDJRQDO PDWUL[ 7KHGLVFUHWHHYROXWLRQRIHDFKHQWU\LQ \k DVD IXQFWLRQRI k RUWKHFRQWLQXRXVHYROXWLRQRIHDFKHQWU\ LQWKHYHFWRUYDOXHGIXQFWLRQ \.t / LVXQDIIHFWHGE\ZKDW KDSSHQVWRWKHRWKHUHQWULHVDV k ! 1 RU t ! 1 GHVLJQPDWUL[ 7KHPDWUL[ X LQWKHOLQHDUPRGHO \ D ;ˇ C ! ZKHUHWKHFROXPQVRI X DUHGHWHUPLQHGLQVRPHZD\E\WKH REVHUYHGYDOXHVRIVRPHLQGHSHQGHQWYDULDEOHV GHWHUPLQDQW RIDVTXDUHPDWUL[ A 7KHQXPEHU GHW A GHÀQHG LQGXFWLYHO\E\DFRIDFWRUH[SDQVLRQDORQJWKHÀUVWURZRI A $OVR ."1/r WLPHVWKHSURGXFWRIWKHGLDJRQDOHQWULHVLQDQ\ HFKHORQIRUP U REWDLQHGIURP A E\URZUHSODFHPHQWVDQG r URZLQWHUFKDQJHVEXWQRVFDOLQJRSHUDWLRQV GLDJRQDOHQWULHV LQDPDWUL[ (QWULHVKDYLQJHTXDOURZDQG FROXPQLQGLFHV GLDJRQDOL]DEOH PDWUL[ $ PDWUL[WKDWFDQEHZULWWHQLQIDF WRUHGIRUPDV PDP !1 ZKHUH D LVDGLDJRQDOPDWUL[DQG P LVDQLQYHUWLEOHPDWUL[ GLDJRQDOPDWUL[ $ VTXDUHPDWUL[ZKRVHHQWULHV QRW RQWKH PDLQGLDJRQDODUHDOO]HUR GLIIHUHQFHHTXDWLRQ RU OLQHDUUHFXUUHQFHUHODWLRQ $QHTXD WLRQRIWKHIRUP [kC1 D A[k k D 0; 1; 2; : : : ZKRVHVROX WLRQLVDVHTXHQFHRIYHFWRUV [0 ; [1 ; : : : : GLODWLRQ $ PDSSLQJ [ 7! r [ IRUVRPHVFDODU r ZLWK 1 < r GLPHQVLRQ RIDÁDW S 7KHGLPHQVLRQRIWKHFRUUHVSRQGLQJSDUDOOHO VXEVSDFH RIDVHW S 7KHGLPHQVLRQRIWKHVPDOOHVWÁDWFRQWDLQLQJ S RIDVXEVSDFH S 7KHQXPEHURIYHFWRUVLQDEDVLVIRU S ZULWWHQDV GLP S RIDYHFWRUVSDFH V 7KHQXPEHURIYHFWRUVLQDEDVLVIRU V ZULWWHQDV GLP V 7KHGLPHQVLRQRIWKH]HURVSDFHLV GLVFUHWHOLQHDUG\QDPLFDOV\VWHP $ GLIIHUHQFHHTXDWLRQRIWKH IRUP [kC1 D A[k WKDWGHVFULEHVWKHFKDQJHVLQDV\VWHP XVXDOO\DSK\VLFDOV\VWHP DVWLPHSDVVHV 7KHSK\VLFDO V\VWHPLVPHDVXUHGDWGLVFUHWHWLPHV ZKHQ k D 0; 1; 2; : : : ; DQGWKH VWDWH RIWKHV\VWHPDWWLPH k LVDYHFWRU [k ZKRVH HQWULHVSURYLGHFHUWDLQIDFWVRILQWHUHVWDERXWWKHV\VWHP GLVWDQFHEHWZHHQ X DQG Y 7KHOHQJWKRIWKHYHFWRU X " Y GHQRWHGE\ GLVW .X; Y/ GLVWDQFHWRDVXEVSDFH 7KHGLVWDQFHIURPDJLYHQSRLQWYHF WRU Y WRWKHQHDUHVWSRLQWLQWKHVXEVSDFH GLVWULEXWLYHODZV OHIW A.B C C / D AB C AC DQGULJKW .B C C /A D BA C CA IRUDOO A B C

&

"

(MPTTBSZ LI [ \ DUHLQ S DQG S LVRQWKHOLQHVHJPHQW [\ WKHQ S D [ RU S D \

' IDFWRUL]DWLRQ RI A $QHTXDWLRQWKDWH[SUHVVHV A DVDSURGXFW RIWZRRUPRUHPDWULFHV ÀQDOGHPDQGYHFWRU RU ELOORIÀQDOGHPDQGV 7KHYHFWRU G LQWKH/HRQWLHILQSXW²RXWSXWPRGHOWKDWOLVWVWKHGROODUYDOXHV RIWKHJRRGVDQGVHUYLFHVGHPDQGHGIURPWKHYDULRXVVHFWRUV E\WKHQRQSURGXFWLYHSDUWRIWKHHFRQRP\ 7KHYHFWRU G FDQUHSUHVHQWFRQVXPHUGHPDQG JRYHUQPHQWFRQVXPSWLRQ VXUSOXVSURGXFWLRQ H[SRUWV RURWKHUH[WHUQDOGHPDQG ÀQLWHGLPHQVLRQDO YHFWRU VSDFH $ YHFWRU VSDFH WKDW LV VSDQQHGE\DÀQLWHVHWRIYHFWRUV ÁDW LQ Rn

$ WUDQVODWHRIDVXEVSDFHRI Rn

ÁH[LELOLW\PDWUL[ $ PDWUL[ZKRVH j WKFROXPQJLYHVWKHGH ÁHFWLRQVRIDQHODVWLFEHDPDWVSHFLÀHGSRLQWVZKHQDXQLW IRUFHLVDSSOLHGDWWKH j WKSRLQWRQWKHEHDP ÁRDWLQJ SRLQW DULWKPHWLF $ULWKPHWLF ZLWK QXPEHUV UHSUH VHQWHGDVGHFLPDOV ˙ :d1 ! ! ! dp # 10r ZKHUH r LVDQLQWHJHU DQGWKHQXPEHU p RIGLJLWVWRWKHULJKWRIWKHGHFLPDOSRLQW LVXVXDOO\EHWZHHQDQG ÁRS

2QHDULWKPHWLFRSHUDWLRQ .C; "; %; =/ RQWZRUHDOÁRDWLQJ SRLQWQXPEHUV

IRUZDUGSKDVH RIURZUHGXFWLRQ 7KHÀUVWSDUWRIWKHDOJR ULWKPWKDWUHGXFHVDPDWUL[WRHFKHORQIRUP )RXULHUDSSUR[LPDWLRQ RIRUGHU n 7KHFORVHVWSRLQWLQWKH VXEVSDFHRI nWKRUGHUWULJRQRPHWULFSRO\QRPLDOVWRDJLYHQ IXQFWLRQLQ C Œ0; 2$! )RXULHUFRHIÀFLHQWV 7KHZHLJKWVXVHGWRPDNHDWULJRQRPHWULF SRO\QRPLDODVD)RXULHUDSSUR[LPDWLRQWRDIXQFWLRQ

WKHIUHHYDULDEOHVWKHSDUDPHWHUV LIDQ\ $IWHU6HFWLRQ WKHSDUDPHWULFGHVFULSWLRQLVZULWWHQLQYHFWRUIRUP *LYHQVURWDWLRQ $ OLQHDUWUDQVIRUPDWLRQIURP Rn WR Rn XVHGLQ FRPSXWHUSURJUDPVWRFUHDWH]HURHQWULHVLQDYHFWRUXVXDOO\ DFROXPQRIDPDWUL[ *UDPPDWUL[ RI A

7KHPDWUL[ ATA

*UDP²6FKPLGWSURFHVV $QDOJRULWKPIRUSURGXFLQJDQRU WKRJRQDORURUWKRQRUPDOEDVLVIRUDVXEVSDFHWKDWLVVSDQQHG E\DJLYHQVHWRIYHFWRUV

) KRPRJHQHRXV FRRUGLQDWHV ,Q R3 WKH UHSUHVHQWDWLRQ RI .x; y; ´/ DV .X; Y; Z; H / IRUDQ\ H ¤ 0 ZKHUH x D X=H y D Y =H DQG ´ D Z=H ,Q R2 H LVXVXDOO\WDNHQDV DQGWKHKRPRJHQHRXVFRRUGLQDWHVRI .x; y/ DUHZULWWHQDV .x; y; 1/ KRPRJHQHRXVHTXDWLRQ $QHTXDWLRQRIWKHIRUP A[ D SRV VLEO\ZULWWHQDVDYHFWRUHTXDWLRQRUDVDV\VWHPRIOLQHDU HTXDWLRQV ! " Y KRPRJHQHRXVIRUP RIDYHFWRU Y LQ Rn 7KHSRLQW YQ D 1 LQ RnC1 +RXVHKROGHU UHÁHFWLRQ $ WUDQVIRUPDWLRQ [ 7! Q[ ZKHUH Q D I " 2XXT DQG X LVDXQLWYHFWRU .XTX D 1/

K\SHUSODQH LQ Rn $ ÁDWLQ Rn RIGLPHQVLRQ n " 1 $OVR D WUDQVODWHRIDVXEVSDFHRIGLPHQVLRQ n " 1

* LGHQWLW\PDWUL[ GHQRWHGE\ I RU In $ VTXDUHPDWUL[ZLWKRQHV RQWKHGLDJRQDODQG]HURVHOVHZKHUH

)RXULHUVHULHV $QLQÀQLWHVHULHVWKDWFRQYHUJHVWRDIXQFWLRQ LQWKHLQQHUSURGXFWVSDFH C Œ0; 2$! ZLWKWKHLQQHUSURGXFW JLYHQE\DGHÀQLWHLQWHJUDO

LOOFRQGLWLRQHGPDWUL[ $ VTXDUHPDWUL[ZLWKDODUJHRUSRV VLEO\LQÀQLWH FRQGLWLRQQXPEHU DPDWUL[WKDWLVVLQJXODURU FDQEHFRPHVLQJXODULIVRPHRILWVHQWULHVDUHFKDQJHGHYHU VRVOLJKWO\

IUHHYDULDEOH YDULDEOH

LPDJH RIDYHFWRU [ XQGHUDWUDQVIRUPDWLRQ T DVVLJQHGWR [ E\ T

$Q\YDULDEOHLQDOLQHDUV\VWHPWKDWLVQRWDEDVLF

IXOOUDQN PDWUL[ RI m DQG n

$Q m # n PDWUL[ZKRVHUDQNLVWKHVPDOOHU

IXQGDPHQWDOVHWRIVROXWLRQV $ EDVLVIRUWKHVHWRIDOOVROXWLRQV RIDKRPRJHQHRXVOLQHDUGLIIHUHQFHRUGLIIHUHQWLDOHTXDWLRQ IXQGDPHQWDOVXEVSDFHV GHWHUPLQHGE\ A 7KHQXOOVSDFHDQG FROXPQVSDFHRI A DQGWKHQXOOVSDFHDQGFROXPQVSDFHRI AT ZLWK &RO AT FRPPRQO\FDOOHGWKHURZVSDFHRI A

( *DXVVLDQHOLPLQDWLRQ

6HH URZUHGXFWLRQDOJRULWKP

JHQHUDO OHDVWVTXDUHV SUREOHP *LYHQ DQ m # n PDWUL[ A DQG D YHFWRU E LQ Rm ÀQG [O LQ Rn VXFK WKDW kE " AO[k $ kE " A[k IRUDOO [ LQ Rn

JHQHUDOVROXWLRQ RIDOLQHDUV\VWHP $ SDUDPHWULFGHVFULSWLRQ RIDVROXWLRQVHWWKDWH[SUHVVHVWKHEDVLFYDULDEOHVLQWHUPVRI

7KHYHFWRU T .[/

LPSOLFLWGHVFULSWLRQ RIDVXEVSDFH W RI Rn $ VHWRIRQH RUPRUHKRPRJHQHRXVHTXDWLRQVWKDWFKDUDFWHUL]HWKHSRLQWV RI W ,P[ 7KHYHFWRULQ Rn IRUPHGIURPWKHLPDJLQDU\SDUWVRIWKH HQWULHVRIDYHFWRU [ LQ C n LQFRQVLVWHQWOLQHDUV\VWHP

$ OLQHDUV\VWHPZLWKQRVROXWLRQ

LQGHÀQLWHPDWUL[ $ V\PPHWULFPDWUL[ A VXFKWKDW [TA[ DV VXPHVERWKSRVLWLYHDQGQHJDWLYHYDOXHV LQGHÀQLWHTXDGUDWLFIRUP $ TXDGUDWLFIRUP Q VXFKWKDW Q.[/ DVVXPHVERWKSRVLWLYHDQGQHJDWLYHYDOXHV LQÀQLWHGLPHQVLRQDO YHFWRUVSDFH WKDWKDVQRÀQLWHEDVLV

$ QRQ]HURYHFWRUVSDFH V

LQQHUSURGXFW 7KHVFDODU XTY XVXDOO\ZULWWHQDV X ! Y ZKHUH X DQG Y DUHYHFWRUVLQ Rn YLHZHGDV n # 1 PDWULFHV $OVR FDOOHGWKH GRWSURGXFW RI X DQG Y ,QJHQHUDO DIXQFWLRQRQ

(MPTTBSZ DYHFWRUVSDFHWKDWDVVLJQVWRHDFKSDLURIYHFWRUV X DQG Y D QXPEHU hX; Yi VXEMHFWWRFHUWDLQD[LRPV 6HH 6HFWLRQ LQQHUSURGXFWVSDFH $ YHFWRUVSDFHRQZKLFKLVGHÀQHGDQ LQQHUSURGXFW LQSXW²RXWSXWPDWUL[ 6HH FRQVXPSWLRQPDWUL[ LQSXW²RXWSXWPRGHO 6HH /HRQWLHILQSXW²RXWSXWPRGHO LQWHULRUSRLQW RIDVHW S LQ Rn $ SRLQW S LQ S VXFKWKDWIRU VRPH ı > 0 WKHRSHQEDOO %.S; ı/ FHQWHUHGDW S LVFRQWDLQHG LQ S LQWHUPHGLDWHGHPDQGV 'HPDQGVIRUJRRGVRUVHUYLFHVWKDW ZLOOEHFRQVXPHGLQWKHSURFHVVRISURGXFLQJRWKHUJRRGV DQGVHUYLFHVIRUFRQVXPHUV ,I [ LVWKHSURGXFWLRQOHYHODQG C LVWKHFRQVXPSWLRQPDWUL[ WKHQ C [ OLVWVWKHLQWHUPHGLDWH GHPDQGV LQWHUSRODWLQJSRO\QRPLDO $ SRO\QRPLDOZKRVHJUDSKSDVVHV WKURXJKHYHU\SRLQWLQDVHWRIGDWDSRLQWVLQ R2 LQYDULDQWVXEVSDFH IRU A $ VXEVSDFH H VXFKWKDW A[ LVLQ H ZKHQHYHU [ LVLQ H LQYHUVH RIDQ n # n PDWUL[ A $Q n # n PDWUL[ A!1 VXFKWKDW AA!1 D A!1 A D In LQYHUVHSRZHUPHWKRG $QDOJRULWKPIRUHVWLPDWLQJDQHLJHQ YDOXH " RIDVTXDUHPDWUL[ ZKHQDJRRGLQLWLDOHVWLPDWHRI " LVDYDLODEOH LQYHUWLEOH OLQHDU WUDQVIRUPDWLRQ $ OLQHDU WUDQVIRUPDWLRQ T W Rn ! Rn VXFKWKDWWKHUHH[LVWVDIXQFWLRQ S W Rn ! Rn VDWLVI\LQJERWK T .S.[// D [ DQG S.T .[// D [ IRUDOO [ LQ Rn LQYHUWLEOHPDWUL[ $ VTXDUHPDWUL[WKDWSRVVHVVHVDQLQYHUVH LVRPRUSKLFYHFWRUVSDFHV 7ZRYHFWRUVSDFHV V DQG W IRU ZKLFKWKHUHLVDRQHWRRQHOLQHDUWUDQVIRUPDWLRQ T WKDWPDSV V RQWR W LVRPRUSKLVP $ RQHWRRQHOLQHDUPDSSLQJIURPRQHYHFWRU VSDFHRQWRDQRWKHU

, NHUQHO RIDOLQHDUWUDQVIRUPDWLRQ T W V ! W / 7KHVHWRI [ LQ V VXFKWKDW T .[/ D .LUFKKRII·VODZV YROWDJHODZ 7KHDOJHEUDLFVXPRIWKH RI YROWDJHGURSVLQRQHGLUHFWLRQDURXQGDORRSHTXDOVWKH DOJHEUDLFVXPRIWKHYROWDJHVRXUFHVLQWKHVDPHGLUHFWLRQ DURXQGWKHORRS FXUUHQWODZ 7KHFXUUHQWLQDEUDQFK LVWKHDOJHEUDLFVXPRIWKHORRSFXUUHQWVÁRZLQJWKURXJKWKDW EUDQFK

ODGGHUQHWZRUN $QHOHFWULFDOQHWZRUNDVVHPEOHGE\FRQQHFW LQJLQVHULHVWZRRUPRUHHOHFWULFDOFLUFXLWV OHDGLQJHQWU\ 7KHOHIWPRVWQRQ]HURHQWU\LQDURZRIDPDWUL[ OHDVWVTXDUHV HUURU 7KH GLVWDQFH kE " AO[k IURP E WR AO[ ZKHQ [O LVDOHDVWVTXDUHVVROXWLRQRI A[ D E OHDVWVTXDUHVOLQH 7KHOLQH y D ˇO0 C ˇO1 x WKDWPLQLPL]HVWKH OHDVWVTXDUHVHUURULQWKHHTXDWLRQ \ D Xˇ C !

"

OHDVWVTXDUHV VROXWLRQ RI A[ D E $ YHFWRU [O VXFK WKDW kE " AO[k $ kE " A[k IRUDOO [ LQ Rn OHIW LQYHUVH RI A $Q\ UHFWDQJXODU PDWUL[ C VXFK WKDW CA D I OHIWPXOWLSOLFDWLRQ E\ A 0XOWLSOLFDWLRQRIDYHFWRURUPDWUL[ RQWKHOHIWE\ A OHIWVLQJXODUYHFWRUV RI A 7KHFROXPQVRI U LQWKHVLQJXODU YDOXHGHFRPSRVLWLRQ A D U †V T p p OHQJWK RU QRUP RI Y 7KHVFDODU kYk D Y ! Y D hY; Yi /HRQWLHIH[FKDQJH RU FORVHG PRGHO $ PRGHORIDQHFRQRP\ ZKHUHLQSXWVDQGRXWSXWVDUHÀ[HG DQGZKHUHDVHWRISULFHV IRUWKHRXWSXWVRIWKHVHFWRUVLVVRXJKWVXFKWKDWWKHLQFRPH RIHDFKVHFWRUHTXDOVLWVH[SHQGLWXUHV 7KLV´HTXLOLEULXPµ FRQGLWLRQLVH[SUHVVHGDVDV\VWHPRIOLQHDUHTXDWLRQV ZLWK WKHSULFHVDVWKHXQNQRZQV /HRQWLHILQSXW²RXWSXWPRGHO RU /HRQWLHISURGXFWLRQHTXD WLRQ 7KHHTXDWLRQ [ D C [ C G ZKHUH [ LVSURGXFWLRQ G LVÀQDOGHPDQG DQG C LVWKHFRQVXPSWLRQRULQSXW²RXWSXW PDWUL[ 7KH j WKFROXPQRI C OLVWVWKHLQSXWVWKDWVHFWRU j FRQVXPHVSHUXQLWRIRXWSXW OHYHOVHW RU JUDGLHQW RIDOLQHDUIXQFWLRQDO f RQ Rn $ VHW Œf d ! D f[ 2 Rn W f .[/ D d g OLQHDUFRPELQDWLRQ $ VXPRIVFDODUPXOWLSOHVRIYHFWRUV 7KH VFDODUVDUHFDOOHGWKH ZHLJKWV OLQHDUGHSHQGHQFHUHODWLRQ $ KRPRJHQHRXVYHFWRUHTXDWLRQ ZKHUHWKHZHLJKWVDUHDOOVSHFLÀHGDQGDWOHDVWRQHZHLJKWLV QRQ]HUR OLQHDUHTXDWLRQ LQWKHYDULDEOHV x1 ; : : : ; xn / $QHTXDWLRQWKDW FDQEHZULWWHQLQWKHIRUP a1 x1 C a2 x2 C ! ! ! C an xn D b ZKHUH b DQGWKHFRHIÀFLHQWV a1 ; : : : ; an DUHUHDORUFRPSOH[ QXPEHUV OLQHDU ÀOWHU $ OLQHDUGLIIHUHQFHHTXDWLRQXVHGWRWUDQVIRUP GLVFUHWHWLPHVLJQDOV OLQHDUIXQFWLRQDO RQ Rn $ OLQHDUWUDQVIRUPDWLRQ f IURP Rn LQWR R OLQHDUO\GHSHQGHQW YHFWRUV $QLQGH[HGVHW fY1 ; : : : ; Yp g ZLWK WKHSURSHUW\WKDWWKHUHH[LVWZHLJKWV c1 ; : : : ; cp QRWDOO]HUR VXFKWKDW c1 Y1 C ! ! ! C cp Yp D 7KDWLV WKHYHFWRUHTXD WLRQ c1 Y1 C c2 Y2 C ! ! ! C cp Yp D KDVD QRQWULYLDO VROXWLRQ OLQHDUO\LQGHSHQGHQW YHFWRUV $QLQGH[HGVHW fY1 ; : : : ; Yp g ZLWK WKH SURSHUW\ WKDW WKH YHFWRU HTXDWLRQ c1 Y1 C c2 Y2 C ! ! ! C cp Yp D KDV RQO\ WKH WULYLDO VROXWLRQ c 1 D ! ! ! D c p D 0 OLQHDU PRGHO LQ VWDWLVWLFV $Q\ HTXDWLRQ RI WKH IRUP \ D Xˇ C ! ZKHUH X DQG \ DUHNQRZQDQG ˇ LVWREHFKRVHQ WRPLQLPL]HWKHOHQJWKRIWKH UHVLGXDOYHFWRU ! OLQHDUV\VWHP $ FROOHFWLRQRIRQHRUPRUHOLQHDUHTXDWLRQV LQYROYLQJWKHVDPHYDULDEOHV VD\ x1 ; : : : ; xn OLQHDU WUDQVIRUPDWLRQ 7 IURPDYHFWRUVSDFH V LQWRDYHF WRU VSDFH W $ UXOH T WKDW DVVLJQV WR HDFK YHFWRU [ LQ V D XQLTXH YHFWRU T .[/ LQ W VXFK WKDW L T .X C Y/ D T .X/ C T .Y/ IRU DOO X; Y LQ V DQG LL T .c X/ D cT .X/ IRUDOO X LQ V DQGDOOVFDODUV c 1RWDWLRQ

"

(MPTTBSZ

T W V ! W DOVR [ 7! A[ ZKHQ T W Rn ! Rm DQG A LVWKH VWDQGDUGPDWUL[IRU T OLQHWKURXJK S SDUDOOHOWR Y

7KHVHW fS C t Y W t LQ Rg

ORRSFXUUHQW 7KHDPRXQWRIHOHFWULFFXUUHQWÁRZLQJWKURXJKD ORRSWKDWPDNHVWKHDOJHEUDLFVXPRIWKH RI YROWDJHGURSV DURXQGWKHORRSHTXDOWRWKHDOJHEUDLFVXPRIWKHYROWDJH VRXUFHVLQWKHORRS ORZHUWULDQJXODUPDWUL[ GLDJRQDO

$ PDWUL[ZLWK]HURVDERYHWKHPDLQ

ORZHUWULDQJXODUSDUW RI A $ ORZHUWULDQJXODUPDWUL[ZKRVH HQWULHVRQWKHPDLQGLDJRQDODQGEHORZDJUHHZLWKWKRVH LQ A /8 IDFWRUL]DWLRQ 7KHUHSUHVHQWDWLRQRIDPDWUL[ A LQWKHIRUP A D LU ZKHUH L LVDVTXDUHORZHUWULDQJXODUPDWUL[ZLWK RQHVRQWKHGLDJRQDODXQLWORZHUWULDQJXODUPDWUL[ DQG U LVDQHFKHORQIRUPRI A

PDSSLQJ

7KHHQWULHVZLWKHTXDOURZDQG

6HH WUDQVIRUPDWLRQ

0DUNRY FKDLQ $ VHTXHQFH RI SUREDELOLW\ YHFWRUV [0 [1 [2 ; : : : ; WRJHWKHU ZLWK D VWRFKDVWLF PDWUL[ P VXFK WKDW [kC1 D P [k IRU k D 0; 1; 2; : : : : PDWUL[

$ UHFWDQJXODUDUUD\RIQXPEHUV

PDWUL[HTXDWLRQ $QHTXDWLRQWKDWLQYROYHVDWOHDVWRQHPDWUL[ IRULQVWDQFH A[ D E PDWUL[ IRU 7 UHODWLYH WR EDVHV B DQG C $ PDWUL[ M IRU DOLQHDUWUDQVIRUPDWLRQ T W V ! W ZLWKWKHSURSHUW\WKDW ŒT .[/!C D M Œ[!B IRUDOO [ LQ V ZKHUH B LVDEDVLVIRU V DQG C LVDEDVLVIRU W :KHQ W D V DQG C D B WKHPDWUL[ M LVFDOOHGWKH BPDWUL[IRU T DQGLVGHQRWHGE\ ŒT !B

PDWUL[RIREVHUYDWLRQV $ p # N PDWUL[ZKRVHFROXPQVDUH REVHUYDWLRQYHFWRUV HDFKFROXPQOLVWLQJ p PHDVXUHPHQWV PDGHRQDQLQGLYLGXDORUREMHFWLQDVSHFLÀHGSRSXODWLRQ RUVHW

PDWUL[WUDQVIRUPDWLRQ $ PDSSLQJ [ 7! A[ ZKHUH A LVDQ m # n PDWUL[DQG [ UHSUHVHQWVDQ\YHFWRULQ Rn

PD[LPDOOLQHDUO\LQGHSHQGHQWVHW LQ V $ OLQHDUO\LQGHSHQ GHQWVHW B LQ V VXFKWKDWLIDYHFWRU Y LQ V EXWQRWLQ B LV DGGHGWR B WKHQWKHQHZVHWLVOLQHDUO\GHSHQGHQW

PHDQGHYLDWLRQIRUP RIDPDWUL[RIREVHUYDWLRQV $ PDWUL[ ZKRVHURZYHFWRUVDUHLQPHDQGHYLDWLRQIRUP )RUHDFK URZ WKHHQWULHVVXPWR]HUR PHDQGHYLDWLRQIRUP RIDYHFWRU WR]HUR

m " n PDWUL[

$ PDWUL[ZLWK m URZVDQG n FROXPQV

0RRUH²3HQURVHLQYHUVH

6HH SVHXGRLQYHUVH

PXOWLSOHUHJUHVVLRQ $ OLQHDUPRGHOLQYROYLQJVHYHUDOLQGHSHQ GHQWYDULDEOHVDQGRQHGHSHQGHQWYDULDEOH

/ QHDUO\VLQJXODUPDWUL[

$QLOOFRQGLWLRQHGPDWUL[

QHJDWLYH GHÀQLWH PDWUL[ $ V\PPHWULF PDWUL[ A VXFK WKDW [TA[ < 0 IRUDOO [ ¤

QHJDWLYHVHPLGHÀQLWHPDWUL[ [TA[ $ 0 IRUDOO [

6HH QRUP

PDLQGLDJRQDO RIDPDWUL[ FROXPQLQGLFHV

PLQLPDOVSDQQLQJVHW IRUDVXEVSDFH H $ VHW B WKDWVSDQV H DQGKDVWKHSURSHUW\WKDWLIRQHRIWKHHOHPHQWVRI B LV UHPRYHGIURP B WKHQWKHQHZVHWGRHVQRWVSDQ H

QHJDWLYHGHÀQLWHTXDGUDWLFIRUP WKDW Q.[/ < 0 IRUDOO [ ¤

. PDJQLWXGH RIDYHFWRU

PLJUDWLRQPDWUL[ $ PDWUL[WKDWJLYHVWKHSHUFHQWDJHPRYH PHQWEHWZHHQGLIIHUHQWORFDWLRQV IURPRQHSHULRGWRWKH QH[W

$ YHFWRUZKRVHHQWULHVVXP

PHDQVTXDUHHUURU 7KHHUURURIDQDSSUR[LPDWLRQLQDQLQQHU SURGXFWVSDFH ZKHUHWKHLQQHUSURGXFWLVGHÀQHGE\DGHÀ QLWHLQWHJUDO

$ TXDGUDWLFIRUP Q VXFK

$ V\PPHWULFPDWUL[ A VXFKWKDW

QHJDWLYHVHPLGHÀQLWHTXDGUDWLFIRUP VXFKWKDW Q.[/ $ 0 IRUDOO [

$ TXDGUDWLFIRUP Q

QRQKRPRJHQHRXVHTXDWLRQ $QHTXDWLRQRIWKHIRUP A[ D E ZLWK E ¤ SRVVLEO\ZULWWHQDVDYHFWRUHTXDWLRQRUDVD V\VWHPRIOLQHDUHTXDWLRQV

QRQVLQJXODU PDWUL[

$QLQYHUWLEOHPDWUL[

QRQWULYLDO VROXWLRQ $ QRQ]HUR VROXWLRQ RI D KRPRJHQHRXV HTXDWLRQRUV\VWHPRIKRPRJHQHRXVHTXDWLRQV QRQ]HUR PDWUL[RUYHFWRU $ PDWUL[ZLWKSRVVLEO\RQO\RQH URZRUFROXPQ WKDWFRQWDLQVDWOHDVWRQHQRQ]HURHQWU\ p p QRUP RU OHQJWK RI Y 7KHVFDODU kYk D Y ! Y D hY; Yi

QRUPDO HTXDWLRQV 7KH V\VWHP RI HTXDWLRQV UHSUHVHQWHG E\ ATA[ D AT E ZKRVH VROXWLRQ \LHOGV DOO OHDVWVTXDUHV VR OXWLRQV RI A[ D E ,Q VWDWLVWLFV D FRPPRQ QRWDWLRQ LV X TXˇ D X T\

QRUPDOL]LQJ DQRQ]HURYHFWRU Y 7KHSURFHVVRIFUHDWLQJDXQLW YHFWRU X WKDWLVDSRVLWLYHPXOWLSOHRI Y QRUPDOYHFWRU WRDVXEVSDFH V RI Rn WKDW Q ! [ D 0 IRUDOO [ LQ V

$ YHFWRU Q LQ Rn VXFK

QXOOVSDFH RIDQ m # n PDWUL[ A 7KHVHW 1XO A RIDOOVROXWLRQV WRWKHKRPRJHQHRXVHTXDWLRQ A[ D 1XO A D f[ W [ LVLQ Rn DQG A[ D g

0 REVHUYDWLRQ YHFWRU 7KH YHFWRU \ LQ WKH OLQHDU PRGHO \ D Xˇ C ! ZKHUHWKHHQWULHVLQ \ DUHWKHREVHUYHGYDOXHV RIDGHSHQGHQWYDULDEOH RQHWRRQH PDSSLQJ $ PDSSLQJ T W Rn ! Rm VXFK WKDW HDFK E LQ Rm LVWKHLPDJHRI DWPRVW RQH [ LQ Rn RQWR PDSSLQJ $ PDSSLQJ T W Rn ! Rm VXFKWKDWHDFK E LQ Rm LVWKHLPDJHRI DWOHDVW RQH [ LQ Rn

(MPTTBSZ

"

7KHVHW f[ W k[ " Sk < ıg LQ Rn ZKHUH

SHUPXWHGORZHUWULDQJXODUPDWUL[ $ PDWUL[VXFKWKDWDSHU PXWDWLRQRILWVURZVZLOOIRUPDORZHUWULDQJXODUPDWUL[

RSHQVHW S LQ Rn $ VHWWKDWFRQWDLQVQRQHRILWVERXQGDU\ SRLQWV (TXLYDOHQWO\ S LVRSHQLIHYHU\SRLQWRI S LVDQ LQWHULRUSRLQW

SHUPXWHG/8 IDFWRUL]DWLRQ 7KHUHSUHVHQWDWLRQRIDPDWUL[ A LQWKHIRUP A D LU ZKHUH L LVDVTXDUHPDWUL[VXFKWKDW DSHUPXWDWLRQRILWVURZVZLOOIRUPDXQLWORZHUWULDQJXODU PDWUL[ DQG U LVDQHFKHORQIRUPRI A

RSHQEDOO %.S; ı/ LQ Rn ı > 0

RULJLQ

7KH]HURYHFWRU

RUWKRJRQDOEDVLV

$ EDVLVWKDWLVDOVRDQRUWKRJRQDOVHW

RUWKRJRQDOFRPSOHPHQW RI W / RUWKRJRQDOWR W

7KHVHW W

?

RIDOOYHFWRUV

RUWKRJRQDOGHFRPSRVLWLRQ 7KHUHSUHVHQWDWLRQRIDYHFWRU \ DV WKH VXP RI WZR YHFWRUV RQH LQ D VSHFLÀHG VXEVSDFH W DQG WKH RWKHU LQ W ? ,Q JHQHUDO D GHFRPSRVLWLRQ \ D c1 X1 C ! ! ! C cp Xp ZKHUH fX1 ; : : : ; Xp g LVDQRUWKRJRQDO EDVLVIRUDVXEVSDFHWKDWFRQWDLQV \ RUWKRJRQDOO\GLDJRQDOL]DEOH PDWUL[ $ PDWUL[ A WKDWDGPLWV DIDFWRUL]DWLRQ A D PDP !1 ZLWK P DQRUWKRJRQDOPDWUL[ .P !1 D P T / DQG D GLDJRQDO RUWKRJRQDOPDWUL[ U !1 D U T

$ VTXDUHLQYHUWLEOHPDWUL[ U VXFK WKDW

SLYRW $ QRQ]HURQXPEHUWKDWHLWKHULVXVHGLQDSLYRWSRVLWLRQ WRFUHDWH]HURVWKURXJKURZRSHUDWLRQVRULVFKDQJHGLQWRD OHDGLQJ ZKLFKLQWXUQLVXVHGWRFUHDWH]HURV SLYRWFROXPQ

$ FROXPQWKDWFRQWDLQVDSLYRWSRVLWLRQ

SLYRWSRVLWLRQ $ SRVLWLRQLQDPDWUL[ A WKDWFRUUHVSRQGVWRD OHDGLQJHQWU\LQDQHFKHORQIRUPRI A SODQHWKURXJKX Y DQGWKHRULJLQ $ VHWZKRVHSDUDPHWULF HTXDWLRQLV [ D s X C t Y s t LQ R/ ZLWK X DQG Y OLQHDUO\ LQGHSHQGHQW SRODUGHFRPSRVLWLRQ RI A $ IDFWRUL]DWLRQ A D PQ ZKHUH P LVDQ n # n SRVLWLYHVHPLGHÀQLWHPDWUL[ZLWKWKHVDPHUDQN DV A DQG Q LVDQ n # n RUWKRJRQDOPDWUL[

SRO\JRQ

$ SRO\WRSHLQ R2

RUWKRJRQDOSURMHFWLRQRI \ RQWR X RURQWRWKHOLQHWKURXJK X DQG \!X WKHRULJLQ IRU X ¤ 7KHYHFWRU \O GHÀQHGE\ \O D X X!X RUWKRJRQDOSURMHFWLRQRI\RQWR : 7KHXQLTXHYHFWRU \O LQ W VXFKWKDW \ " \O LVRUWKRJRQDOWR W 1RWDWLRQ \O D SURMW \

SRO\WRSH 7KHFRQYH[KXOORIDÀQLWHVHWRISRLQWVLQ Rn D VSHFLDOW\SHRIFRPSDFWFRQYH[VHW

RUWKRJRQDOWR :

SRVLWLYH GHÀQLWH PDWUL[ $ V\PPHWULF PDWUL[ A VXFK WKDW [TA[ > 0 IRUDOO [ ¤

RUWKRJRQDOVHW $ VHW S RIYHFWRUVVXFKWKDW X ! Y D 0 IRUHDFK GLVWLQFWSDLU X; Y LQ S 2UWKRJRQDOWRHYHU\YHFWRULQ W

RUWKRQRUPDOEDVLV YHFWRUV RUWKRQRUPDOVHW

$ EDVLVWKDWLVDQRUWKRJRQDOVHWRIXQLW

$QRUWKRJRQDOVHWRIXQLWYHFWRUV T

RXWHUSURGXFW $ PDWUL[SURGXFW XY ZKHUH X DQG Y DUHYHFWRUV LQ Rn YLHZHGDV n # 1 PDWULFHV 7KHWUDQVSRVHV\PEROLVRQ WKH´RXWVLGHµRIWKHV\PEROV X DQG Y RYHUGHWHUPLQHG V\VWHP $ V\VWHP RI HTXDWLRQV ZLWK PRUH HTXDWLRQVWKDQXQNQRZQV

$ SRO\WRSHLQ R3

SRVLWLYHFRPELQDWLRQ RISRLQWV Y1 ; : : : ; Ym LQ Rn $ OLQHDU FRPELQDWLRQ c1 Y1 C ! ! ! C cm Ym ZKHUHDOO ci & 0

SRVLWLYHGHÀQLWHTXDGUDWLFIRUP WKDW Q.[/ > 0 IRUDOO [ ¤

$ TXDGUDWLFIRUP Q VXFK

SRVLWLYHKXOO RIDVHW S 7KHVHWRIDOOSRVLWLYHFRPELQDWLRQV RISRLQWVLQ S GHQRWHGE\ SRV S

SRVLWLYHVHPLGHÀQLWHPDWUL[ [TA[ & 0 IRUDOO [

$ V\PPHWULFPDWUL[ A VXFKWKDW

SRVLWLYH VHPLGHÀQLWH TXDGUDWLF IRUP VXFKWKDW Q.[/ & 0 IRUDOO [

$ TXDGUDWLF IRUP Q

SRZHUPHWKRG $QDOJRULWKPIRUHVWLPDWLQJDVWULFWO\GRPLQDQW HLJHQYDOXHRIDVTXDUHPDWUL[

1 SDUDOOHOÁDWV 7ZRRUPRUHÁDWVVXFKWKDWHDFKÁDWLVDWUDQVODWH RIWKHRWKHUÁDWV SDUDOOHORJUDPUXOHIRUDGGLWLRQ $ JHRPHWULFLQWHUSUHWDWLRQRI WKHVXPRIWZRYHFWRUV X Y DVWKHGLDJRQDORIWKHSDUDOOHOR JUDPGHWHUPLQHGE\ X Y DQG SDUDPHWHUYHFWRU \ D Xˇ C !

SRO\KHGURQ

7KHXQNQRZQYHFWRU ˇ LQWKHOLQHDUPRGHO

SULQFLSDOD[HV RIDTXDGUDWLFIRUP [TA[ 7KHRUWKRQRUPDO FROXPQVRIDQRUWKRJRQDOPDWUL[ P VXFKWKDW P !1 AP LV GLDJRQDO 7KHVHFROXPQVDUHXQLWHLJHQYHFWRUVRI A 8VX DOO\WKHFROXPQVRI P DUHRUGHUHGLQVXFKDZD\WKDWWKH FRUUHVSRQGLQJHLJHQYDOXHVRI A DUHDUUDQJHGLQGHFUHDVLQJ RUGHURIPDJQLWXGH

SDUDPHWULF HTXDWLRQ RI D SODQH $Q HTXDWLRQ RI WKH IRUP [ D S C s X C t Y s t LQ R ZLWK X DQG Y OLQHDUO\ LQGHSHQGHQW

SULQFLSDO FRPSRQHQWV RI WKH GDWD LQ D PDWUL[ B RI REVHUYDWLRQV 7KH XQLW HLJHQYHFWRUV RI D VDPSOH FR YDULDQFHPDWUL[ S IRU B ZLWKWKHHLJHQYHFWRUVDUUDQJHG VR WKDW WKH FRUUHVSRQGLQJ HLJHQYDOXHV RI S GHFUHDVH LQ PDJQLWXGH ,I B LV LQ PHDQGHYLDWLRQ IRUP WKHQ WKH SULQFLSDO FRPSRQHQWV DUH WKH ULJKW VLQJXODU YHFWRUV LQ D VLQJXODUYDOXHGHFRPSRVLWLRQRI B T

SDUWLWLRQHGPDWUL[ RU EORFNPDWUL[ $ PDWUL[ZKRVHHQWULHV DUHWKHPVHOYHVPDWULFHVRIDSSURSULDWHVL]HV

SUREDELOLW\YHFWRU $ YHFWRULQ Rn ZKRVHHQWULHVDUHQRQQHJD WLYHDQGVXPWRRQH

SDUDPHWULF HTXDWLRQ RI D OLQH [ D S C t Y t LQ R

$Q HTXDWLRQ RI WKH IRUP

"

(MPTTBSZ

SURGXFW $[ 7KHOLQHDUFRPELQDWLRQRIWKHFROXPQVRI A XVLQJ WKHFRUUHVSRQGLQJHQWULHVLQ [ DVZHLJKWV SURGXFWLRQ YHFWRU 7KH YHFWRU LQ WKH /HRQWLHI LQSXW²RXWSXW PRGHOWKDWOLVWVWKHDPRXQWVWKDWDUHWREHSURGXFHGE\WKH YDULRXVVHFWRUVRIDQHFRQRP\ SURÀOH RIDVHW S LQ Rn

7KHVHWRIH[WUHPHSRLQWVRI S

SURMHFWLRQPDWUL[ RU RUWKRJRQDOSURMHFWLRQPDWUL[ $ V\P PHWULFPDWUL[ B VXFKWKDW B 2 D B $ VLPSOHH[DPSOHLV B D YYT ZKHUH Y LVDXQLWYHFWRU SURSHUVXEVHWRIDVHW 6 LWVHOI SURSHUVXEVSDFH V LWVHOI

$ VXEVHWRI S WKDWGRHVQRWHTXDO S

$Q\VXEVSDFHRIDYHFWRUVSDFH V RWKHUWKDQ

SVHXGRLQYHUVH RI A 7KHPDWUL[ VD !1 U T ZKHQ UDV T LVD UHGXFHGVLQJXODUYDOXHGHFRPSRVLWLRQRI A

2 45 IDFWRUL]DWLRQ $ IDFWRUL]DWLRQRIDQ m # n PDWUL[ A ZLWK OLQHDUO\ LQGHSHQGHQW FROXPQV A D QR ZKHUH Q LV DQ m # n PDWUL[ZKRVHFROXPQVIRUPDQRUWKRQRUPDOEDVLVIRU &RO A DQG R LVDQ n # n XSSHUWULDQJXODULQYHUWLEOHPDWUL[ ZLWKSRVLWLYHHQWULHVRQLWVGLDJRQDO TXDGUDWLFH%p]LHUFXUYH $ FXUYHZKRVHGHVFULSWLRQPD\EH ZULWWHQLQWKHIRUP J.t/ D .1 " t/I 0 .t/ C t I 1 .t / IRU 0 $ t $ 1 ZKHUH I 0 .t/ D .1 " t/S0 C t S1 DQG I 1 .t / D .1 " t/S1 C t S2 7KHSRLQWV S0 S1 S2 DUHFDOOHGWKH FRQWUROSRLQWV IRU WKHFXUYH TXDGUDWLFIRUP $ IXQFWLRQ Q GHÀQHGIRU [ LQ Rn E\ Q.[/ D [TA[ ZKHUH A LVDQ n # n V\PPHWULFPDWUL[FDOOHGWKH PDWUL[RIWKHTXDGUDWLFIRUP

3 UDQJH RIDOLQHDUWUDQVIRUPDWLRQ T 7KHVHWRIDOOYHFWRUVRI WKHIRUP T .[/ IRUVRPH [ LQWKHGRPDLQRI T UDQN RIDPDWUL[ A 7KHGLPHQVLRQRIWKHFROXPQVSDFHRI A GHQRWHGE\ UDQN A 5D\OHLJKTXRWLHQW R.[/ D .[TA[/=.[T[/ $QHVWLPDWHRIDQ HLJHQYDOXHRI A XVXDOO\DV\PPHWULFPDWUL[ UHFXUUHQFHUHODWLRQ

6HH GLIIHUHQFHHTXDWLRQ

UHGXFHG HFKHORQ IRUP RU UHGXFHG URZ HFKHORQ IRUP $ UHGXFHGHFKHORQPDWUL[WKDWLVURZHTXLYDOHQWWRDJLYHQ PDWUL[ UHGXFHGHFKHORQPDWUL[ $ UHFWDQJXODUPDWUL[LQHFKHORQIRUP WKDWKDVWKHVHDGGLWLRQDOSURSHUWLHV 7KHOHDGLQJHQWU\LQ HDFKQRQ]HURURZLV DQGHDFKOHDGLQJLVWKHRQO\QRQ]HUR HQWU\LQLWVFROXPQ UHGXFHG VLQJXODU YDOXH GHFRPSRVLWLRQ $ IDFWRUL]DWLRQ A D UDV T IRUDQ m # n PDWUL[ A RI UDQN r ZKHUH U LV m # r ZLWKRUWKRQRUPDOFROXPQV D LVDQ r # r GLDJRQDO PDWUL[ ZLWK WKH r QRQ]HUR VLQJXODU YDOXHV RI A RQ LWV GLDJRQDO DQG V LV n # r ZLWKRUWKRQRUPDOFROXPQV

UHJUHVVLRQFRHIÀFLHQWV 7KHFRHIÀFLHQWV ˇ0 DQG ˇ1 LQWKHOHDVW VTXDUHVOLQH y D ˇ0 C ˇ1 x

UHJXODUVROLG 2QHRIWKHÀYHSRVVLEOHUHJXODUSRO\KHGURQVLQ R3 WKHWHWUDKHGURQHTXDOWULDQJXODUIDFHV WKHFXEH VTXDUHIDFHV WKHRFWDKHGURQHTXDOWULDQJXODUIDFHV WKH GRGHFDKHGURQHTXDOSHQWDJRQDOIDFHV DQGWKHLFRVDKH GURQHTXDOWULDQJXODUIDFHV UHJXODUVWRFKDVWLFPDWUL[ $ VWRFKDVWLFPDWUL[ P VXFKWKDW VRPHPDWUL[SRZHU P k FRQWDLQVRQO\VWULFWO\SRVLWLYHHQWULHV

UHODWLYH FKDQJH RU UHODWLYH HUURU LQ E/ k%Ek=kEk ZKHQ E LVFKDQJHGWR E C %E

7KH TXDQWLW\

UHSHOORU RIDG\QDPLFDOV\VWHPLQ R2 / 7KHRULJLQZKHQDOO WUDMHFWRULHVH[FHSWWKHFRQVWDQW]HURVHTXHQFHRUIXQFWLRQ WHQGDZD\IURP

UHVLGXDO YHFWRU 7KH TXDQWLW\ ! WKDW DSSHDUV LQ WKH JHQHUDO OLQHDUPRGHO \ D Xˇ C ! WKDWLV ! D \ " Xˇ WKHGLIIHU HQFHEHWZHHQWKHREVHUYHGYDOXHVDQGWKHSUHGLFWHGYDOXHV RI y 5H[ 7KHYHFWRULQ Rn IRUPHGIURPWKHUHDOSDUWVRIWKHHQWULHV RIDYHFWRU [ LQ C n ULJKW LQYHUVH RI A AC D I

$Q\ UHFWDQJXODU PDWUL[ C VXFK WKDW

ULJKWPXOWLSOLFDWLRQ E\ A ULJKWE\ A

0XOWLSOLFDWLRQRIDPDWUL[RQWKH

ULJKWVLQJXODUYHFWRUV RI A 7KHFROXPQVRI V LQWKHVLQJXODU YDOXHGHFRPSRVLWLRQ A D U †V T

URXQGRIIHUURU (UURULQÁRDWLQJSRLQWDULWKPHWLFFDXVHGZKHQ WKHUHVXOWRIDFDOFXODWLRQLVURXQGHGRUWUXQFDWHG WRWKH QXPEHURIÁRDWLQJSRLQWGLJLWVVWRUHG $OVR WKHHUURUWKDW UHVXOWVZKHQWKHGHFLPDOUHSUHVHQWDWLRQRIDQXPEHUVXFKDV LVDSSUR[LPDWHGE\DÁRDWLQJSRLQWQXPEHUZLWKDÀQLWH QXPEHURIGLJLWV URZ²FROXPQUXOH 7KHUXOHIRUFRPSXWLQJDSURGXFW AB LQ ZKLFKWKH .i; j /HQWU\RI AB LVWKHVXPRIWKHSURGXFWVRI FRUUHVSRQGLQJHQWULHVIURPURZ i RI A DQGFROXPQ j RI B

URZHTXLYDOHQW PDWULFHV 7ZRPDWULFHVIRUZKLFKWKHUHH[LVWV DÀQLWH VHTXHQFHRIURZRSHUDWLRQVWKDWWUDQVIRUPVRQH PDWUL[LQWRWKHRWKHU URZUHGXFWLRQDOJRULWKP $ V\VWHPDWLFPHWKRGXVLQJHOHPHQ WDU\URZRSHUDWLRQVWKDWUHGXFHVDPDWUL[WRHFKHORQIRUPRU UHGXFHGHFKHORQIRUP URZUHSODFHPHQW $QHOHPHQWDU\URZRSHUDWLRQWKDWUHSODFHV RQHURZRIDPDWUL[E\WKHVXPRIWKHURZDQGDPXOWLSOHRI DQRWKHUURZ URZVSDFH RIDPDWUL[ A 7KHVHW 5RZ A RIDOOOLQHDUFRPELQD WLRQVRIWKHYHFWRUVIRUPHGIURPWKHURZVRI A DOVRGHQRWHG E\ &RO AT URZVXP

7KHVXPRIWKHHQWULHVLQDURZRIDPDWUL[

URZYHFWRU $ PDWUL[ZLWKRQO\RQHURZ RUDVLQJOHURZRID PDWUL[WKDWKDVVHYHUDOURZV URZ²YHFWRUUXOHIRUFRPSXWLQJ $[ 7KHUXOHIRUFRPSXWLQJD SURGXFW A[ LQZKLFKWKH i WKHQWU\RI A[ LVWKHVXPRIWKH

(MPTTBSZ SURGXFWVRIFRUUHVSRQGLQJHQWULHVIURPURZ i RI A DQGIURP WKHYHFWRU [

4 VDGGOHSRLQW RIDG\QDPLFDOV\VWHPLQ R2 7KHRULJLQZKHQ VRPHWUDMHFWRULHVDUHDWWUDFWHGWR DQGRWKHUWUDMHFWRULHVDUH UHSHOOHGIURP VDPH GLUHFWLRQ DV D YHFWRU Y PXOWLSOHRI Y

$ YHFWRU WKDW LV D SRVLWLYH

VDPSOHPHDQ 7KHDYHUDJH M RIDVHWRIYHFWRUV ;1 ; : : : ; ;N JLYHQE\ M D .1=N /.;1 C ! ! ! C ;N /

VFDODU $ UHDO QXPEHUXVHGWRPXOWLSO\HLWKHUDYHFWRURUD PDWUL[ VFDODUPXOWLSOHRI X E\ F 7KHYHFWRU c X REWDLQHGE\PXOWLSO\ LQJHDFKHQWU\LQ X E\ c

VFDOH DYHFWRU 0XOWLSO\DYHFWRURUDURZRUFROXPQRID PDWUL[ E\DQRQ]HURVFDODU 6FKXUFRPSOHPHQW $ FHUWDLQPDWUL[IRUPHGIURPWKHEORFNV RID 2 # 2 SDUWLWLRQHGPDWUL[ A D ŒAij ! ,I A11 LVLQYHUW LEOH LWV6FKXUFRPSOHPHQWLVJLYHQE\ A22 " A21 A!1 11 A12 ,I A22 LV LQYHUWLEOH LWV 6FKXU FRPSOHPHQW LV JLYHQ E\ A11 " A12 A!1 22 A21 6FKXUIDFWRUL]DWLRQ RI A IRUUHDOVFDODUV $ IDFWRUL]DWLRQ A D URU T RIDQ n # n PDWUL[ A KDYLQJ n UHDOHLJHQYDOXHV ZKHUH U LVDQ n # n RUWKRJRQDOPDWUL[DQG R LVDQXSSHU WULDQJXODUPDWUL[

VHWVSDQQHGE\ fY1 ; : : : ; Yp g

7KHVHW 6SDQ fY1 ; : : : ; Yp g

VLJQDO RU GLVFUHWHWLPHVLJQDO $ GRXEO\LQÀQLWHVHTXHQFHRI QXPEHUV fyk g DIXQFWLRQGHÀQHGRQWKHLQWHJHUV EHORQJVWR WKHYHFWRUVSDFH S

VLPLODU PDWULFHV 0DWULFHV A DQG B VXFKWKDW P !1 AP D B RUHTXLYDOHQWO\ A D PBP !1 IRUVRPHLQYHUWLEOHPDWUL[ P

VLPLODULW\WUDQVIRUPDWLRQ LQWR P !1 AP

$ WUDQVIRUPDWLRQWKDWFKDQJHV A

VLPSOH[ 7KHFRQYH[KXOORIDQDIÀQHO\LQGHSHQGHQWÀQLWHVHW RIYHFWRUVLQ Rn VLQJXODU PDWUL[

$ VTXDUHPDWUL[WKDWKDVQRLQYHUVH

VLQJXODUYDOXHGHFRPSRVLWLRQ RIDQ m # n PDWUL[ A A D U †V T ZKHUH U LVDQ m # m RUWKRJRQDOPDWUL[ V LVDQ n # n RUWKRJRQDOPDWUL[ DQG † LVDQ m # n PDWUL[ZLWKQRQ QHJDWLYHHQWULHVRQWKHPDLQGLDJRQDODUUDQJHGLQGHFUHDV LQJRUGHURIPDJQLWXGH DQG]HURVHOVHZKHUH ,I UDQN A D r WKHQ † KDVH[DFWO\ r SRVLWLYHHQWULHVWKHQRQ]HURVLQJXODU YDOXHVRI A RQWKHGLDJRQDO VLQJXODUYDOXHV RI A 7KHSRVLWLYH VTXDUHURRWVRIWKHHLJHQ YDOXHVRI ATA DUUDQJHGLQGHFUHDVLQJRUGHURIPDJQLWXGH VL]H RIDPDWUL[ 7ZRQXPEHUV ZULWWHQLQWKHIRUP m # n WKDWVSHFLI\WKHQXPEHURIURZVm DQGFROXPQVn LQWKH PDWUL[ VROXWLRQ RIDOLQHDUV\VWHPLQYROYLQJYDULDEOHV x1 ; : : : ; xn $ OLVW .s1 ; s2 ; : : : ; sn / RIQXPEHUVWKDWPDNHVHDFKHTXDWLRQLQ

"

WKHV\VWHPDWUXHVWDWHPHQWZKHQWKHYDOXHV s1 ; : : : ; sn DUH VXEVWLWXWHGIRU x1 ; : : : ; xn UHVSHFWLYHO\ VROXWLRQVHW 7KHVHWRIDOOSRVVLEOHVROXWLRQVRIDOLQHDUV\VWHP 7KHVROXWLRQVHWLVHPSW\ZKHQWKHOLQHDUV\VWHPLVLQFRQVLV WHQW 6SDQ fY1 ; : : : ; Yp g 7KH VHW RI DOO OLQHDU FRPELQDWLRQV RI Y1 ; : : : ; Yp $OVR WKH VXEVSDFHVSDQQHG RU JHQHUDWHG E\ Y1 ; : : : ; Yp VSDQQLQJVHW IRUDVXEVSDFH H / $Q\VHW fY1 ; : : : ; Yp g LQ H VXFKWKDW H D 6SDQ fY1 ; : : : ; Yp g VSHFWUDOGHFRPSRVLWLRQ RI A $ UHSUHVHQWDWLRQ

A D "1 X1 XT1 C ! ! ! C "n Xn XTn ZKHUH fX1 ; : : : ; Xn g LVDQRUWKRQRUPDOEDVLVRIHLJHQYHFWRUV RI A DQG "1 ; : : : ; "n DUHWKHFRUUHVSRQGLQJHLJHQYDOXHVRI A VSLUDOSRLQW RIDG\QDPLFDOV\VWHPLQ R2 7KHRULJLQZKHQ WKHWUDMHFWRULHVVSLUDODERXW VWDJHPDWUL[PRGHO $ GLIIHUHQFHHTXDWLRQ [kC1 D A[k ZKHUH [k OLVWVWKHQXPEHURIIHPDOHVLQDSRSXODWLRQDWWLPH k ZLWKWKHIHPDOHVFODVVLÀHGE\YDULRXVVWDJHVRIGHYHORSPHQW VXFKDVMXYHQLOH VXEDGXOW DQGDGXOW VWDQGDUGEDVLV 7KHEDVLV E D fH1 ; : : : ; Hn g IRU Rn FRQVLVWLQJ RIWKHFROXPQVRIWKH n # n LGHQWLW\PDWUL[ RUWKHEDVLV f1; t; : : : ; t n g IRU Pn VWDQGDUGPDWUL[ IRUDOLQHDUWUDQVIRUPDWLRQ T / 7KHPDWUL[ A VXFKWKDW T .[/ D A[ IRUDOO [ LQWKHGRPDLQRI T VWDQGDUGSRVLWLRQ 7KHSRVLWLRQRIWKHJUDSKRIDQHTXDWLRQ [TA[ D c ZKHQ A LVDGLDJRQDOPDWUL[ VWDWHYHFWRU $ SUREDELOLW\YHFWRU ,QJHQHUDO DYHFWRUWKDWGH VFULEHVWKH´VWDWHµRIDSK\VLFDOV\VWHP RIWHQLQFRQQHFWLRQ ZLWKDGLIIHUHQFHHTXDWLRQ [kC1 D A[k VWHDG\VWDWHYHFWRU IRUDVWRFKDVWLFPDWUL[ P $ SUREDELOLW\ YHFWRU T VXFKWKDW P T D T VWLIIQHVVPDWUL[ 7KHLQYHUVHRIDÁH[LELOLW\PDWUL[ 7KH j WK FROXPQRIDVWLIIQHVVPDWUL[JLYHVWKHORDGVWKDWPXVWEH DSSOLHGDWVSHFLÀHGSRLQWVRQDQHODVWLFEHDPLQRUGHUWR SURGXFHDXQLWGHÁHFWLRQDWWKH j WKSRLQWRQWKHEHDP VWRFKDVWLFPDWUL[ $ VTXDUHPDWUL[ZKRVHFROXPQVDUHSURED ELOLW\YHFWRUV VWULFWO\GRPLQDQWHLJHQYDOXH $QHLJHQYDOXH "1 RIDPDWUL[ A ZLWKWKHSURSHUW\WKDW j"1 j > j"k j IRUDOORWKHUHLJHQYDOXHV "k RI A VXEPDWUL[ RI A $Q\PDWUL[REWDLQHGE\GHOHWLQJVRPHURZV DQGRUFROXPQVRI A DOVR A LWVHOI VXEVSDFH $ VXEVHW H RIVRPHYHFWRUVSDFH V VXFKWKDW H KDV WKHVHSURSHUWLHV WKH]HURYHFWRURI V LVLQ H H LVFORVHGXQGHUYHFWRUDGGLWLRQ DQG H LVFORVHGXQGHU PXOWLSOLFDWLRQE\VFDODUV VXSSRUWLQJK\SHUSODQH WRDFRPSDFWFRQYH[VHW S LQ Rn $ K\SHUSODQH H D Œf d ! VXFKWKDW H \ S 6D ¿ DQGHLWKHU f .x/ $ d IRUDOO x LQ S RU f .x/ & d IRUDOO x LQ S V\PPHWULFPDWUL[ $ PDWUL[ A VXFKWKDW AT A

"

(MPTTBSZ

V\VWHPRIOLQHDUHTXDWLRQV RUD OLQHDUV\VWHP $ FROOHFWLRQ RIRQHRUPRUHOLQHDUHTXDWLRQVLQYROYLQJWKHVDPHVHWRI YDULDEOHV VD\ x1 ; : : : ; xn

XQLWFRQVXPSWLRQ YHFWRU $ FROXPQ YHFWRU LQ WKH /HRQWLHI LQSXW²RXWSXWPRGHOWKDWOLVWVWKHLQSXWVDVHFWRUQHHGVIRU HDFKXQLWRILWVRXWSXW DFROXPQRIWKHFRQVXPSWLRQPDWUL[

5

XQLWORZHUWULDQJXODUPDWUL[ $ VTXDUHORZHUWULDQJXODUPD WUL[ZLWKRQHVRQWKHPDLQGLDJRQDO

WHWUDKHGURQ $ WKUHHGLPHQVLRQDOVROLGREMHFWERXQGHGE\IRXU HTXDO WULDQJXODU IDFHV ZLWK WKUHH IDFHV PHHWLQJ DW HDFK YHUWH[ WRWDOYDULDQFH 7KHWUDFHRIWKHFRYDULDQFHPDWUL[ S RIDPDWUL[ RIREVHUYDWLRQV WUDFH RIDVTXDUHPDWUL[ A 7KHVXPRIWKHGLDJRQDOHQWULHVLQ A GHQRWHGE\ WU A WUDMHFWRU\ 7KHJUDSKRIDVROXWLRQ f[0 ; [1 ; [2 ; : : :g RIDG\QDP LFDOV\VWHP [kC1 D A[k RIWHQFRQQHFWHGE\DWKLQFXUYHWR PDNHWKHWUDMHFWRU\HDVLHUWRVHH $OVR WKHJUDSKRI [.t/ IRU t & 0 ZKHQ [.t/ LVDVROXWLRQRIDGLIIHUHQWLDOHTXDWLRQ [0 .t/ D A[.t/ WUDQVIHUPDWUL[ $ PDWUL[ A DVVRFLDWHGZLWKDQHOHFWULFDOFLU FXLWKDYLQJLQSXWDQGRXWSXWWHUPLQDOV VXFKWKDWWKHRXWSXW YHFWRULV A WLPHVWKHLQSXWYHFWRU WUDQVIRUPDWLRQ RU IXQFWLRQ RU PDSSLQJ 7 IURP Rn WR Rm $ UXOHWKDWDVVLJQVWRHDFKYHFWRU [ LQ Rn DXQLTXH YHFWRU T .[/ LQ Rm 1RWDWLRQ T W Rn ! Rm $OVR T W V ! W GHQRWHVDUXOHWKDWDVVLJQVWRHDFK [ LQ V D XQLTXHYHFWRU T .[/ LQ W WUDQVODWLRQ E\DYHFWRU S/ 7KHRSHUDWLRQRIDGGLQJ S WRD YHFWRURUWRHDFKYHFWRULQDJLYHQVHW WUDQVSRVH RI A $Q n # m PDWUL[ AT ZKRVHFROXPQVDUHWKH FRUUHVSRQGLQJURZVRIWKH m # n PDWUL[ A WUHQGDQDO\VLV 7KHXVHRIRUWKRJRQDOSRO\QRPLDOVWRÀWGDWD ZLWKWKHLQQHUSURGXFWJLYHQE\HYDOXDWLRQDWDÀQLWHVHWRI SRLQWV WULDQJOHLQHTXDOLW\ kX C Yk $ kXk C kYk IRUDOO X Y WULDQJXODUPDWUL[ $ PDWUL[ A ZLWKHLWKHU]HURVDERYHRU]HURV EHORZWKHGLDJRQDOHQWULHV WULJRQRPHWULFSRO\QRPLDO $ OLQHDUFRPELQDWLRQRIWKHFRQ VWDQWIXQFWLRQDQGVLQHDQGFRVLQHIXQFWLRQVVXFKDV FRV nt DQG VLQ nt WULYLDOVROXWLRQ 7KHVROXWLRQ [ D RIDKRPRJHQHRXVHTXDWLRQ A[ D

6 XQFRUUHODWHGYDULDEOHV $Q\WZRYDULDEOHV xi DQG xj ZLWK i ¤ j WKDWUDQJHRYHUWKH i WKDQG j WKFRRUGLQDWHVRIWKH REVHUYDWLRQYHFWRUVLQDQREVHUYDWLRQPDWUL[ VXFKWKDWWKH FRYDULDQFH sij LV]HUR XQGHUGHWHUPLQHGV\VWHP $ V\VWHPRIHTXDWLRQVZLWKIHZHU HTXDWLRQVWKDQXQNQRZQV XQLTXHQHVVTXHVWLRQ $VNV ´,IDVROXWLRQRIDV\VWHPH[LVWV LV LWXQLTXH³WKDWLV LVLWWKHRQO\RQH"µ

XQLWYHFWRU

$ YHFWRU Y VXFKWKDW kYk D 1

XSSHUWULDQJXODUPDWUL[ $ PDWUL[ U QRWQHFHVVDULO\VTXDUH ZLWK]HURVEHORZWKHGLDJRQDOHQWULHV u11 ; u22 ; : : : :

7 9DQGHUPRQGHPDWUL[ $Q n # n PDWUL[ V RULWVWUDQVSRVH ZKHQ V KDVWKHIRUP 2 3 1 x1 x12 !!! x1n!1 6 7 61 x2 x22 !!! x2n!1 7 6 7 V D6: :: :: :: 7 6 :: : : : 7 4 5

1

xn2

xn

!!!

xnn!1

YDULDQFH RIDYDULDEOH xj 7KHGLDJRQDOHQWU\ sjj LQWKHFRYDUL DQFHPDWUL[ S IRUDPDWUL[RIREVHUYDWLRQV ZKHUH xj YDULHV RYHUWKH j WKFRRUGLQDWHVRIWKHREVHUYDWLRQYHFWRUV

YHFWRU $ OLVWRIQXPEHUV DPDWUL[ZLWKRQO\RQHFROXPQ ,Q JHQHUDO DQ\HOHPHQWRIDYHFWRUVSDFH YHFWRU DGGLWLRQ HQWULHV

$GGLQJ YHFWRUV E\ DGGLQJ FRUUHVSRQGLQJ

YHFWRUHTXDWLRQ $QHTXDWLRQLQYROYLQJDOLQHDUFRPELQDWLRQ RIYHFWRUVZLWKXQGHWHUPLQHGZHLJKWV YHFWRUVSDFH $ VHWRIREMHFWV FDOOHGYHFWRUV RQZKLFKWZR RSHUDWLRQVDUHGHÀQHG FDOOHGDGGLWLRQDQGPXOWLSOLFDWLRQE\ VFDODUV 7HQD[LRPVPXVWEHVDWLVÀHG 6HHWKHÀUVWGHÀQLWLRQ LQ6HFWLRQ YHFWRUVXEWUDFWLRQ VXOWDV X " Y

&RPSXWLQJ X C ."1/Y DQGZULWLQJWKHUH

8 ZHLJKWHG OHDVW VTXDUHV /HDVWVTXDUHV SUREOHPV ZLWK D ZHLJKWHGLQQHUSURGXFWVXFKDV

h[; \i D w12 x1 y1 C ! ! ! C wn2 xn yn : ZHLJKWV

7KHVFDODUVXVHGLQDOLQHDUFRPELQDWLRQ

; ]HURVXEVSDFH YHFWRU

7KHVXEVSDFH fg FRQVLVWLQJRIRQO\WKH]HUR

]HUR YHFWRU 7KH XQLTXH YHFWRU GHQRWHG E\ VXFK WKDW X C D X IRUDOO X ,Q Rn LVWKHYHFWRUZKRVHHQWULHV DUHDOO]HURV

$QVZHUVWR2GG1XPEHUHG ([HUFLVHV 2

$IBQUFS 6HFWLRQ SDJH 7KHVROXWLRQLV .x1 ; x2 / D .!8; 3/ RUVLPSO\ .!8; 3/

.2; 1/

5HSODFH5RZE\LWVVXPZLWKWLPHV5RZ DQGWKHQ UHSODFH5RZE\LWVVXPZLWK 3 WLPHV5RZ 7KHVROXWLRQVHWLVHPSW\ .16; 21; 14; 4/

,QFRQVLVWHQW

.5; 3; !1/

,QFRQVLVWHQW

h ¤ 2

$OO h

&DOFXODWLRQVVKRZWKDWWKHV\VWHPLVLQFRQVLVWHQW VRWKH WKUHHOLQHVKDYHQRSRLQWLQFRPPRQ 0DUNDVWDWHPHQW7UXHRQO\LIWKHVWDWHPHQWLV DOZD\V WUXH *LYLQJ\RXWKHDQVZHUVKHUHZRXOGGHIHDWWKHSXUSRVHRI WKHWUXH²IDOVHTXHVWLRQV ZKLFKLVWRKHOS\RXOHDUQWRUHDG WKHWH[WFDUHIXOO\ 7KH 6WXG\*XLGH ZLOOWHOO\RXZKHUHWR ORRNIRUWKHDQVZHUV EXW\RXVKRXOGQRWFRQVXOWLWXQWLO\RX KDYHPDGHDQKRQHVWDWWHPSWWRÀQGWKHDQVZHUV\RXUVHOI k ! 2g C h D 0

7KHURZUHGXFWLRQRI ! " a b f WR c d g

b. ac /

3 2 4 8 3LYRWFROVDQG 4 2 4 8 5 3 6 9 12 ! " ! " ! " " " 0 0 0 0 0 0 8 8
D &RQVLVWHQW ZLWKPDQ\VROXWLRQV E &RQVLVWHQW ZLWKPDQ\VROXWLRQV $OO h

!

a 0

d!

b

f

b. ac /

f . ac /

g!

"

VKRZVWKDW d ! PXVWEHQRQ]HUR VLQFH f DQG g DUH DUELWUDU\ 2WKHUZLVH IRUVRPHFKRLFHVRI f DQG g WKH VHFRQGURZFRXOGFRUUHVSRQGWRDQHTXDWLRQRIWKHIRUP 0 D q ZKHUH q LVQRQ]HUR 7KXV ad ¤ bc

6ZDS5RZDQG5RZ VZDS5RZDQG5RZ

5HSODFH5RZE\5RZ C !4 5RZ UHSODFH5RZE\ 5RZ C 5RZ

5HYLHZ3UDFWLFH3UREOHPDQGWKHQ ZULWH DVROXWLRQ 7KH 6WXG\*XLGH KDVDVROXWLRQ

6HFWLRQ SDJH 5HGXFHGHFKHORQIRUP DDQGE (FKHORQIRUP G 1RWLQ HFKHORQIRUP F 2 3 2 0 !8 4 0 4 5 0 0 0 0 0

D ,QFRQVLVWHQWZKHQ h D 2 DQG k ¤ 8 E 8QLTXHVROXWLRQZKHQ h ¤ 2

F 0DQ\VROXWLRQVZKHQ h D 2 DQG k D 8 5HDGWKHWH[WFDUHIXOO\ DQGZULWH\RXUDQVZHUVEHIRUH\RX FRQVXOWWKH 6WXG\*XLGH 5HPHPEHU DVWDWHPHQWLVWUXH RQO\LILWLVWUXHLQDOOFDVHV 6LQFHWKHUHDUHIRXUSLYRWVRQHLQHDFKFROXPQRIWKH FRHIÀFLHQWPDWUL[ WKHDXJPHQWHGPDWUL[PXVWUHGXFHWR WKHIRUP 2 3 1 0 0 0 a 60 1 0 0 b7 6 7 40 1 0 c5 0 0 0 0 1 d DQGVR

Da

x1 x2 x3

D b D c

x4 D d

"

"

"OTXFST UP 0EE/VNCFSFE &YFSDJTFT 1RPDWWHUZKDWWKHYDOXHVRI D E F DQG G WKHVROXWLRQ H[LVWVDQGLVXQLTXH

,IWKHFRHIÀFLHQWPDWUL[KDVDSLYRWSRVLWLRQLQHYHU\URZ WKHQWKHUHLVDSLYRWSRVLWLRQLQWKHERWWRPURZ DQGWKHUHLV QRURRPIRUDSLYRWLQWKHDXJPHQWHGFROXPQ 6R WKH V\VWHPLVFRQVLVWHQW E\7KHRUHP ,IDOLQHDUV\VWHPLVFRQVLVWHQW WKHQWKHVROXWLRQLVXQLTXHLI DQGRQO\LI HYHU\FROXPQLQWKHFRHIÀFLHQWPDWUL[LVDSLYRW FROXPQ RWKHUZLVH WKHUHDUHLQÀQLWHO\PDQ\VROXWLRQV $QXQGHUGHWHUPLQHGV\VWHPDOZD\VKDVPRUHYDULDEOHVWKDQ HTXDWLRQV 7KHUHFDQQRWEHPRUHEDVLFYDULDEOHVWKDQWKHUH DUHHTXDWLRQV VRWKHUHPXVWEHDWOHDVWRQHIUHHYDULDEOH 6XFKDYDULDEOHPD\EHDVVLJQHGLQÀQLWHO\PDQ\GLIIHUHQW YDOXHV ,IWKHV\VWHPLVFRQVLVWHQW HDFKGLIIHUHQWYDOXHRID IUHHYDULDEOHZLOOSURGXFHDGLIIHUHQWVROXWLRQ
x1 C x2 D 2

x1 ! x2 D 0

3x1 C 2x2 D 5

6HFWLRQ SDJH

!

" ! " !4 5 1 4

1R E LV QRW DOLQHDUFRPELQDWLRQRI D1 D2 DQG D3 1R E LV QRW DOLQHDUFRPELQDWLRQRIWKHFROXPQVRI A h D 3 1RQLQWHJHUZHLJKWVDUHDFFHSWDEOH RIFRXUVH EXWVRPH VLPSOHFKRLFHVDUH 0 # Y1 C 0 # Y2 D DQG 2 3 2 3 3 !4 1 # Y 1 C 0 # Y 2 D 4 1 5; 0 # Y1 C 1 # Y2 D 4 0 5 2 1 2 3 2 3 !1 7 1 # Y 1 C 1 # Y 2 D 4 1 5; 1 # Y 1 ! 1 # Y 2 D 4 1 5 3 1

6SDQ fY1 ; Y2 g LVWKHVHWRISRLQWVRQWKHOLQHWKURXJK Y1 DQG EHFDXVH Y2 LVDPXOWLSOHRI Y1 ! " 2 2 h LVFRQVLVWHQWIRUDOO h DQG +LQW 6KRZWKDW !1 1 k k ([SODLQZKDWWKLVFDOFXODWLRQVKRZVDERXW 6SDQ fX; Yg

D 1R WKUHH x2

u – 2v

u

D 5Y1 LVWKHRXWSXWRIGD\VRIRSHUDWLRQRIPLQH E 7KHWRWDORXWSXWLV x1 Y1!C x2 Y2" VR x1 DQG x2 VKRXOG 240 VDWLVI\ x1 Y1 C x2 Y2 D 2824 F >0@ GD\VIRUPLQHDQGGD\VIRUPLQH

– 2v

u+v –v

x1

v

2

3 2 3 2 3 3 5 2 x1 4 !2 5 C x2 4 0 5 D 4 !3 5 8 !9 8 2 3 2 3 2 3 3x1 5x2 2 4 !2x1 5 C 4 0 5 D 4 !3 5 8 8x1 !9x 2 3 22 3 3x1 C 5x2 2 4 !2x1 5 D 4 !3 5 8 8x1 ! 9x2

3x1 C 5x2 D

E
F D1 D 1 # D1 C 0 # D2 C 0 # D3 u–v

!2x1

3 2 3 2 3 2 3 0 1 5 0 x1 4 4 5 C x2 4 6 5 C x3 4 !1 5 D 4 0 5 !1 3 !8 0

%HIRUH\RXFRQVXOW\RXU 6WXG\*XLGH UHDGWKHHQWLUH VHFWLRQFDUHIXOO\ 3D\VSHFLDODWWHQWLRQWRGHÀQLWLRQVDQG WKHRUHPVWDWHPHQWV DQGQRWHDQ\UHPDUNVWKDWSUHFHGHRU IROORZWKHP

p.t/ D 1 C 3t C 2t 2

2

2

D !3

8x1 ! 9x2 D 8 8VXDOO\WKHLQWHUPHGLDWHVWHSVDUHQRWGLVSOD\HG

D D X ! 2Y E D 2X ! 2Y F D 2X ! 3:5Y G D 3X ! 4Y
.17=14; !34=14; 16=14/ D .17=14; !17=7; 8=7/ ! " 10=3 D 2 E $GGJDW .0; 1/ DGGJDW .8; 1/ DQGDGGJDW .2; 4/ 5HYLHZ3UDFWLFH3UREOHPDQGWKHQ ZULWH DVROXWLRQ 7KH 6WXG\*XLGH KDVDVROXWLRQ

6HFWLRQ SDJH 7KHSURGXFWLVQRWGHÀQHGEHFDXVHWKHQXPEHURIFROXPQV LQWKH 3 $ 2 PDWUL[GRHVQRWPDWFKWKHQXPEHURIHQWULHV LQWKHYHFWRU 2 3 2 3 2 3 " 1 2 ! 1 2 !2 15 D !2 # 4 !3 5 C 3 # 4 1 5 D A[ D 4 !3 3 6 2 1 3 62 3 2 3 1 !2 6 4 D 4 65C4 35 D 4 95 !2 18 16

4FDUJPO 2

3 2 3 " 1 2 ! 1 # .!2/ C 2 # .3/ !2 15 E A[ D 4 !3 D 4 .!3/ # .!2/ C 1 # .3/ 5 3 1 6 1 # .!2/ C 6 # .3/ 2 3 4 D 4 9 5: 16 6KRZ\RXUZRUNKHUHDQGIRU([HUFLVHV² EXW WKHUHDIWHUSHUIRUPWKHFDOFXODWLRQVPHQWDOO\ ! " ! " ! " ! " ! " 1 2 !3 1 !4 2# !1# C1# !1# D !2 !3 1 !1 1 2 3 2 3 4 !5 7 2 3 6 6 !1 7 x1 6 !8 7 3 !8 6 74 x 5 D 6 7 4 7 !5 4 05 05 2 x3 !4 1 2 !7 ! " ! " ! " ! " 5 1 !3 8 x1 C x2 C x3 D DQG 0 2 4 0 2 3 ! " x ! " 5 1 !3 4 1 5 8 x2 D 0 2 4 0 x3 2 3 2 3 2 3 1 3 !4 !2 x1 !11 4 1 5 2 4 5 [ D 4 x2 5 D 4 3 5 !3 !7 6 12 0 x3

7KHHTXDWLRQ A[ D E LVQRWFRQVLVWHQWZKHQ 3b1 C b2 LV QRQ]HUR 6KRZ\RXUZRUN 7KHVHWRI E IRUZKLFKWKH HTXDWLRQ LV FRQVLVWHQWLVDOLQHWKURXJKWKHRULJLQ³WKHVHWRI DOOSRLQWV .b1 ; b2 / VDWLVI\LQJ b2 D !3b1 2QO\WKUHHURZVFRQWDLQDSLYRWSRVLWLRQ 7KHHTXDWLRQ A[ D E GRHV QRW KDYHDVROXWLRQIRUHDFK E LQ R4 E\ 7KHRUHP 7KHZRUNLQ([HUFLVHVKRZVWKDWVWDWHPHQWG LQ 7KHRUHPLVIDOVH 6RDOOIRXUVWDWHPHQWVLQ7KHRUHPDUH IDOVH 7KXV QRWDOOYHFWRUVLQ R4 FDQEHZULWWHQDVDOLQHDU FRPELQDWLRQRIWKHFROXPQVRI A $OVR WKHFROXPQVRI A GR QRW VSDQ R4 7KHPDWUL[ ŒY1 Y2 Y3 ! GRHVQRWKDYHDSLYRWLQHDFKURZ VRWKHFROXPQVRIWKHPDWUL[GRQRWVSDQ R4 E\7KHRUHP 7KDWLV fY1 ; Y2 ; Y3 g GRHVQRWVSDQ R4 5HDGWKHWH[WFDUHIXOO\DQGWU\WRPDUNHDFKH[HUFLVH VWDWHPHQW7UXHRU)DOVHEHIRUH\RXFRQVXOWWKH 6WXG\*XLGH 6HYHUDOSDUWVRI([HUFLVHVDQGDUH LPSOLFDWLRQV RIWKH IRUP ´,I hVWDWHPHQWi WKHQ hVWDWHPHQWiµ RUHTXLYDOHQWO\

´hVWDWHPHQWi LI hVWDWHPHQWiµ

0DUNVXFKDQLPSOLFDWLRQDV7UXHLI hVWDWHPHQWi LVWUXHLQ DOOFDVHVZKHQ hVWDWHPHQWi LVWUXH

c1 D !3 c2 D !1 c3 D 2

7KHPDWUL[HTXDWLRQFDQEHZULWWHQDV c1 Y1 C c2 Y2 C c3 Y3 C c4 Y4 C c5 Y5 D Y6 ZKHUH c1 D !3

"

c2 D 1 c3 D 2 c4 D !1 c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

6HFWLRQ SDJH 7KHV\VWHPKDVDQRQWULYLDOVROXWLRQEHFDXVHWKHUHLVDIUHH YDULDEOH x3 7KHV\VWHPKDVDQRQWULYLDOVROXWLRQEHFDXVHWKHUHLVDIUHH YDULDEOH x3 2 3 2 3 x1 !1 [ D 4 x2 5 D x3 4 !1 5 1 x3 2 3 2 3 2 3 x1 !9 8 6 x2 7 6 47 6 !5 7 7 6 7 6 7 [ D 6 4 x3 5 D x3 4 1 5 C x4 4 0 5 0 1 x4 2 3 2 [ D x2 4 1 5 0

+LQW 7KHV\VWHPGHULYHGIURPWKH UHGXFHG HFKHORQIRUPLV

x1 ! 4x2

x3

C 5x6 D 0 ! x6 D 0

x5 ! 4x6 D 0 0D0

7KHEDVLFYDULDEOHVDUH x1 x3 DQG x5 7KHUHPDLQLQJ YDULDEOHVDUHIUHH 7KH 6WXG\*XLGH GLVFXVVHVWZRPLVWDNHV WKDWDUHRIWHQPDGHRQWKLVW\SHRISUREOHP 2 3 2 3 5 4 [ D 4 !2 5 C x3 4 !7 5 D S C x3 T: *HRPHWULFDOO\ WKH 0 1 2 3 2 3 5 4 VROXWLRQVHWLVWKHOLQHWKURXJK 4 !2 5 SDUDOOHOWR 4 !7 5 0 1

"

"OTXFST UP 0EE/VNCFSFE &YFSDJTFT 2

3 2 3 2 3 !5 3 !2 /HW X D 4 1 5; Y D 4 0 5; S D 4 0 5 7KHVROXWLRQRI 0 1 0 WKHKRPRJHQHRXVHTXDWLRQLV [ D x2 X C x3 Y WKHSODQH WKURXJKWKHRULJLQVSDQQHGE\ X DQG Y 7KHVROXWLRQVHWRI WKHQRQKRPRJHQHRXVV\VWHPLV [ D S C x2 X C x3 Y WKH SODQHWKURXJK S SDUDOOHOWRWKHVROXWLRQVHWRIWKH KRPRJHQHRXVHTXDWLRQ 2 3 2 3 2 3 x1 8 !1 [ D 4 x2 5 D 4 !4 5 C x3 4 !1 5 7KHVROXWLRQVHWLVWKH x3 0 1 2 3 8 OLQHWKURXJK 4 !4 5 SDUDOOHOWRWKHOLQHWKDWLVWKHVROXWLRQ 0 VHWRIWKHKRPRJHQHRXVV\VWHPLQ([HUFLVH

[ D D C t E ZKHUH t UHSUHVHQWVDSDUDPHWHU RU ( ! " ! " ! " x1 D !2 ! 5t x1 !2 !5 [D D Ct RU x2 0 3 x2 D 3t ! " ! " 3 1 [ D S C t .T ! S/ D Ct !3 4

,WLVLPSRUWDQWWRUHDGWKHWH[WFDUHIXOO\DQGZULWH\RXU DQVZHUV $IWHUWKDW FKHFNWKH 6WXG\*XLGH LIQHFHVVDU\ D AZ D A.S C Yh / D AS C AYh D E C D E E AYh D A.Z ! S/ D AZ ! AS D E ! E D

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

D :KHQ A LVD 4 $ 4 PDWUL[ZLWKWKUHHSLYRWSRVLWLRQV WKH HTXDWLRQ A[ D KDVDIUHHYDULDEOHDQGKHQFHKDV QRQWULYLDOVROXWLRQV E :LWKWKUHHSLYRWSRVLWLRQV A GRHVQRWKDYHDSLYRW SRVLWLRQLQHDFKRILWVIRXUURZV %\7KHRUHPLQ 6HFWLRQ WKHHTXDWLRQ A[ D E GRHVQRWKDYHD VROXWLRQIRUHYHU\SRVVLEOH E 7KHZRUG´SRVVLEOHµLQ WKHH[HUFLVHPHDQVWKDWWKHRQO\YHFWRUVFRQVLGHUHGLQ WKLVFDVHDUHWKRVHLQ R4 EHFDXVH A KDVIRXUURZV D :KHQ A LVD 3 $ 2 PDWUL[ZLWKWZRSLYRWSRVLWLRQV HDFK FROXPQLVDSLYRWFROXPQ 6RWKHHTXDWLRQ A[ D KDV QRIUHHYDULDEOHVDQGKHQFHQRQRQWULYLDOVROXWLRQ E :LWKWZRSLYRWSRVLWLRQVDQGWKUHHURZV A FDQQRWKDYH DSLYRWLQHYHU\URZ 6RWKHHTXDWLRQ A[ D E FDQQRW KDYHDVROXWLRQIRUHYHU\SRVVLEOH E LQ R3 E\ 7KHRUHPLQ6HFWLRQ

" 1 !4 7KH 6WXG\*XLGH VKRZV 1 !4 KRZWRDQDO\]HWKHSUREOHPLQRUGHUWRFRQVWUXFW A ,I E LV DQ\YHFWRU QRW DPXOWLSOHRIWKHÀUVWFROXPQRI A WKHQWKH VROXWLRQVHWRI A[ D E LVHPSW\DQGWKXVFDQQRWEHIRUPHG E\WUDQVODWLQJWKHVROXWLRQVHWRI A[ D 7KLVGRHVQRW FRQWUDGLFW7KHRUHP EHFDXVHWKDWWKHRUHPDSSOLHVZKHQ WKHHTXDWLRQ A[ D E KDVDQRQHPSW\VROXWLRQVHW

2QHDQVZHULV A D

!

6XSSRVH AY D DQG AZ D 7KHQ VLQFH A.Y C Z/ D AY C AZ E\7KHRUHPD LQ6HFWLRQ A.Y C Z/ D AY C AZ D C D : 1RZ OHW c DQG d EH VFDODUV 8VLQJERWKSDUWVRI7KHRUHP A.c Y C d Z/ D A.c Y/ C A.d Z/ D cAY C dAZ D c C d D :

6HFWLRQ SDJH 7KHJHQHUDOVROXWLRQLV p* D :875p6 ; ZLWK p6 IUHH 2QH HTXLOLEULXPVROXWLRQLV p6 D 1000 DQG p* D 875 8VLQJ IUDFWLRQV WKHJHQHUDOVROXWLRQFRXOGEHZULWWHQ p* D .7=8/p6 DQGDQDWXUDOFKRLFHRISULFHVPLJKWEH p6 D 80 DQG p* D 70 2QO\WKH UDWLR RIWKHSULFHVLV LPSRUWDQW 7KHHFRQRPLFHTXLOLEULXPLVXQDIIHFWHGE\D SURSRUWLRQDOFKDQJHLQSULFHV D

'LVWULEXWLRQRI 2XWSXWIURP ) 3 0DQ 6HU

2XWSXW ! 2 :9 !:1 :9 E 4 !:8 !:1 !:8

!

! ,QSXW " " "

!:2 !:4 :6

3 0 05 0

3XUFKDVHGE\ ) 3 0DQ 6HU

F >0@ p) 3 % 30 p0 % 71 p6 D 100

D 'LVWULEXWLRQRI2XWSXWIURP $J 0DQ 6HU 7UDQVS 2XWSXW

!

!

!

! ,QSXW " " " "

3XUFKDVHGE\ $J 0DQ 6HU 7UDQVS

E 2QHVROXWLRQLV p$ D 7:99 p0 D 8:36 p6 D 14:65 DQG p7 D 10:00

4FDUJPO F

'LVWULEXWLRQRI2XWSXWIURP $J 0DQ 6HU 7UDQVS

! ! ! ! ,QSXW 3XUFKDVHGE\ " $J " 0DQ " 6HU " 7UDQVS G 2QHVROXWLRQLV p$ D 7:81 p0 D 7:67 p6 D 15:62 DQG p7 D 10:00 7KHFDPSDLJQKDVEHQHÀWHG6HUYLFHVWKHPRVW 2XWSXW

31D+&23 C +3 &6 +5 27 ! 1D3 &6 +5 27 C 3+2 2 C 3&22

%2 63 C 6+2 2 ! 2+3 %23 C 3+2 6

>0@ 160Q6 C 13$V2 &U10 235 C 374+2 624

! 16+0Q24 C 26$V+3 C 130&U63 212 C 327+2 2

8 ˆ x1 D x3 ! 40 ˆ ˆ 8 ˆ ˆ ˆ ˆ x2 D 50 x 2 D x3 C 10 ˆ ˆ ˆ ˆ ˆ ˆ
6HFWLRQ SDJH -XVWLI\\RXUDQVZHUVWR([HUFLVHV² /LQ LQGHS

/LQ GHSHQ

/LQ LQGHS

/LQ GHSHQ

D 1R h h D !4

E $OO h $OO h

/LQ GHSHQ /LQ GHSHQ /LQ LQGHS ,I\RXFRQVXOW\RXU 6WXG\*XLGH EHIRUH\RXPDNHDJRRG HIIRUWWRDQVZHUWKHWUXHIDOVHTXHVWLRQV \RXZLOOGHVWUR\ PRVWRIWKHLUYDOXH ! " ! " ! " " 0 0 0 0 0 0 0 0 0 2 3 2 3 " 0 60 7 6 07 7 DQG 6 0 7 6 40 40 05 05 0 0 0 0 $OOIRXUFROXPQVRIWKH 6 $ 4 PDWUL[ A PXVWEHSLYRW FROXPQV 2WKHUZLVH WKHHTXDWLRQ A[ D ZRXOGKDYHD IUHHYDULDEOH LQZKLFKFDVHWKHFROXPQVRI A ZRXOGEH OLQHDUO\GHSHQGHQW

"

A $Q\ 3 $ 2 PDWUL[ZLWKWKHVHFRQGFROXPQDPXOWLSOHRI WKHÀUVWZLOOKDYHWKHGHVLUHGSURSHUW\ B $Q\ 3 $ 2 PDWUL[ZLWKWZRQRQ]HURFROXPQVVXFKWKDW QHLWKHUFROXPQLVDPXOWLSOHRIWKHRWKHUZLOOZRUN ,QWKLV FDVH WKHFROXPQVIRUPDOLQHDUO\LQGHSHQGHQWVHW DQGVR WKHHTXDWLRQ B [ D KDVRQO\WKHWULYLDOVROXWLRQ 2 3 1 [ D 4 1 5 !1 7UXH E\7KHRUHP 7KH 6WXG\*XLGH DGGVDQRWKHU MXVWLÀFDWLRQ

7UXH E\7KHRUHP 7UXH $ OLQHDUGHSHQGHQFHUHODWLRQDPRQJ Y1 Y2 DQG Y3 PD\EHH[WHQGHGWRDOLQHDUGHSHQGHQFHUHODWLRQDPRQJ Y1 Y2 Y3 DQG Y4 E\SODFLQJD]HURZHLJKWRQ Y4 0@ 8VLQJWKHSLYRWFROXPQVRI A; 2 3 3 !4 7 6 !5 !3 !11 7 7 BD6 4 4 3 25 8 !7 4 2WKHUFKRLFHVDUHSRVVLEOH

>0@ (DFKFROXPQRI A WKDWLVQRWDFROXPQRI B LVLQWKH VHWVSDQQHGE\WKHFROXPQVRI B

6HFWLRQ SDJH !

" ! " 2 2a !6 2b 2 3 3 [ D 4 1 5 QRWXQLTXH 0 2 3 4 637 6 [ D x3 4 7 15 0

2 3 7 [ D 4 6 5 XQLTXHVROXWLRQ 3

a D 5; b D 6

v

u x1 T(u) T(v)

$ UHÁHFWLRQWKURXJKWKHRULJLQ

"

"OTXFST UP 0EE/VNCFSFE &YFSDJTFT

x2

T(u)

v

x2 T (u) u

T (u + v) u

u+v v

T (v)

x1

x1 x2

T(v) cT (u)

$ UHÁHFWLRQWKURXJKWKHOLQH x1 D x2 : ! " ! " ! " 8 !3 5 ; 2 9 11

!

" ! " 13 2x1 ! x2 ; 5x1 C 6x2 7

u

x1

+LQW 6LQFH fY1 ; Y2 ; Y3 g LVOLQHDUO\GHSHQGHQW \RXFDQZULWH DFHUWDLQHTXDWLRQDQGZRUNZLWKLW 2QHSRVVLELOLW\LVWRVKRZWKDW T GRHVQRWPDSWKH]HUR YHFWRULQWRWKH]HURYHFWRU VRPHWKLQJWKDWHYHU\OLQHDU WUDQVIRUPDWLRQ GRHV GR T .0; 0/ D .0; !3; 0/

7DNH X DQG Y LQ R3 DQGOHW c DQG d EHVFDODUV 7KHQ

´hVWDWHPHQWi LIDQGRQO\LI hVWDWHPHQWiµ

c X C d Y D .cu1 C dv1 ; cu2 C dv2 ; cu3 C dv3 /

0DUNVXFKDVHQWHQFHDV7UXHLI hVWDWHPHQWi LVWUXH ZKHQHYHU hVWDWHPHQWi LVWUXH DQG DOVR hVWDWHPHQWi LVWUXH ZKHQHYHU hVWDWHPHQWi LVWUXH

7KHWUDQVIRUPDWLRQ T LVOLQHDUEHFDXVH T .c X C d Y/ D .cu1 C dv1 ; 0; cu3 C dv3 / D .cu1 ; 0; cu3 / C .dv1 ; 0; dv3 / D c.u1 ; 0; u3 / C d.v1 ; 0; v3 / D cT .X/ C d T .Y/

D :KHQ b D 0; f .x/ D mx ,QWKLVFDVH IRUDOO x; y LQ R DQGDOOVFDODUV c DQG d

>0@ $OOPXOWLSOHVRI .!1; !1; 1; 0/

f .cx C dy/ D m.cx C dy/ D mcx C mdy D c.mx/ C d.my/ D c # f .x/ C d # f .y/

>0@
6HFWLRQ SDJH 2

7KLVVKRZVWKDW f LVOLQHDU E :KHQ f .x/ D mx C b ZLWK b QRQ]HUR f .0/ D m.0/ C b D b ¤ 0 F ,QFDOFXOXV f LVFDOOHGDÓLQHDUIXQFWLRQµEHFDXVHWKH JUDSKRI f LVDOLQH +LQW 6KRZWKDWWKHLPDJHRIDOLQHWKDWLV WKHVHWRI LPDJHVRIDOOSRLQWVRQDOLQH FDQEHUHSUHVHQWHGE\WKH SDUDPHWULFHTXDWLRQRIDOLQH $Q\SRLQW [ RQWKHSODQH 3 VDWLVÀHVWKHSDUDPHWULFHTXDWLRQ [ D s X C t Y IRUVRPHYDOXHVRI s DQG t %\OLQHDULW\ WKH LPDJH T .[/ VDWLVÀHVWKHSDUDPHWULFHTXDWLRQ

.s; t LQ R/

cu

T (u)

5HDGWKHWH[WFDUHIXOO\DQGZULWH\RXUDQVZHUVEHIRUH\RX FKHFNWKH 6WXG\*XLGH 1RWLFHWKDW([HUFLVHH LVD VHQWHQFHRIWKHIRUP

T .[/ D sT .X/ C tT .Y/

T (cu)

3 3 !5 ! 61 1 27 7 6 43 05 !3 1 0 p p " ! !1=p2 1=p2 1= 2 1= 2

" !

!

0 !1 !1 0

0 1 "

!1 0

"

7KHGHVFULEHGWUDQVIRUPDWLRQ T PDSV H1 LQWR !H1 DQGPDSV H2 LQWR !H2 $ URWDWLRQWKURXJK " UDGLDQVDOVRPDSV H1 LQWR !H1 DQGPDSV H2 LQWR !H2 6LQFHDOLQHDUWUDQVIRUPDWLRQLV FRPSOHWHO\GHWHUPLQHGE\ZKDWLWGRHVWRWKHFROXPQVRI WKHLGHQWLW\PDWUL[ WKHURWDWLRQWUDQVIRUPDWLRQKDVWKHVDPH HIIHFWDV T RQHYHU\YHFWRULQ R2 x2

"

7KHVHWRILPDJHVLVMXVW 6SDQ fT .X/ T .Y/g ,I T .X/ DQG T .Y/ DUHOLQHDUO\LQGHSHQGHQW 6SDQ fT .X/ T .Y/g LVDSODQH WKURXJK T .X/ T .Y/ DQG ,I T .X/ DQG T .Y/ DUHOLQHDUO\ GHSHQGHQWDQGQRWERWK]HUR WKHQ 6SDQ fT .X/; T .Y/g LVD OLQHWKURXJK ,I T .X/ D T .Y/ D WKHQ 6SDQ fT .X/; T .Y/g LV fg

0 1

T(2, 1) T(e2) T(e1) x1

$IBQUFS 4VQQMFNFOUBSZ &YFSDJTFT " 2

3 2 !4 0 0 !1 5 4 1 0 !1 3

!

1 0

!5 4 1 !6

"

2

1 60 6 4 0 0 [ D

!

2 0 2 1 7 !4

3 0 0 0 07 7 0 15 0 !1

"

$QVZHUWKHTXHVWLRQVEHIRUHFKHFNLQJWKH 6WXG\*XLGH -XVWLI\\RXUDQVZHUVWR([HUFLVHV² 4

1RWRQHWRRQHDQGGRHVQRWPDS R RQWR R

4

1RWRQHWRRQHEXWPDSV R3 RQWR R2 2 3 " " 60 "7 7 6 40 5 0 0 0 0

n ([SODLQZK\ DQGWKHQFKHFNWKH 6WXG\*XLGH +LQW ,I Hj LVWKH j WKFROXPQRI In WKHQ B Hj LVWKH j WK FROXPQRI B +LQW ,VLWSRVVLEOHWKDW m > n" :KDWDERXW m < n" >0@ 1R ([SODLQZK\ >0@ 1R ([SODLQZK\

6HFWLRQ SDJH 2

3 2 3 2 3 110 130 295 6 47 6 37 6 97 7 6 7 6 7 D x1 6 4 20 5 C x2 4 18 5 D 4 48 5 ZKHUH x1 LVWKH 2 5 8 QXPEHURIVHUYLQJVRI&KHHULRVDQG x2 LVWKHQXPEHURI VHUYLQJVRI1DWXUDO&HUHDO 2 3 2 3 110 130 ! " 295 6 4 6 7 37 7 x1 D 6 9 7 0L[VHUYLQJVRI E 6 4 20 4 48 5 18 5 x2 2 5 8 &KHHULRVWRJHWKHUZLWKVHUYLQJRI1DWXUDO &HUHDO

D 6KHVKRXOGPL[VHUYLQJRI0DFDQG&KHHVH VHUYLQJVRIEURFFROL DQGVHUYLQJRIFKLFNHQWRJHW KHUGHVLUHGQXWULWLRQDOFRQWHQW E 6KHVKRXOGPL[VHUYLQJVRIVKHOOVDQGZKLWH FKHGGDU VHUYLQJRIEURFFROL DQGVHUYLQJVRI FKLFNHQWRJHWKHUGHVLUHGQXWULWLRQDOFRQWHQW 1RWLFH WKDWWKLVPL[FRQWDLQVVLJQLÀFDQWO\OHVVEURFFROL VRVKH VKRXOGOLNHLWEHWWHU 2 32 3 2 3 11 !5 0 0 I1 50 6 !5 7 6 7 6 10 !1 0 76 I2 7 6 !40 7 7 D RL D Y 6 4 0 !1 9 !2 54 I3 5 4 30 5 0 0 !2 10 !30 I4 2 3 2 3 I1 3:68 6 I2 7 6 !1:90 7 7 6 7 >0@ L D 6 4 I3 5 D 4 2:57 5 I4 !2:49

2

32 3 2 3 12 !7 0 !4 I1 40 6 !7 6 7 6 7 15 !6 07 6 76 I2 7 D 6 30 7 RL D Y 4 0 !6 14 !5 54 I3 5 4 20 5 I4 !4 0 !5 13 !10 2 3 2 3 I1 11:43 6 I2 7 6 10:55 7 7 6 7 >0@ L D 6 4 I3 5 D 4 8:04 5 I4 5:84

[kC1 D M [k IRU k D 0; 1; 2; : : : ; ZKHUH ! " ! " :93 :05 800;000 M D DQG [0 D : :07 :95 500;000 ! " 741;720 7KHSRSXODWLRQLQIRU k D 2 LV [2 D 558;280 ! " :98363 :00167 D M D :01637 :99833 ! " 30;754;500 E >0@ [6 D 229;449;000

>0@ D 7KHSRSXODWLRQRIWKHFLW\GHFUHDVHV $IWHU\HDUV WKH SRSXODWLRQVDUHDERXWHTXDO EXWWKHFLW\SRSXODWLRQ FRQWLQXHVWRGHFOLQH $IWHU\HDUV WKHUHDUHRQO\ SHUVRQVLQWKHFLW\URXQGHGRII +RZHYHU WKHFKDQJHVLQSRSXODWLRQVHHPWRJURZ VPDOOHUHDFK\HDU E 7KHFLW\SRSXODWLRQLVLQFUHDVLQJVORZO\ DQGWKH VXEXUEDQSRSXODWLRQLVGHFUHDVLQJ $IWHU\HDUV WKH FLW\SRSXODWLRQKDVJURZQIURPWRDERXW

&KDSWHU6XSSOHPHQWDU\([HUFLVHV SDJH D J P V \

) ) 7 ) 7

E K Q W ]

) ) 7 7 )

F L R X

7 7 7 )

G M S Y

) ) 7 )

H N T Z

7 7 ) )

I O U [

7 ) 7 7

D $Q\FRQVLVWHQWOLQHDUV\VWHPZKRVHHFKHORQIRUPLV 2 3 2 3 " " " " " " 40 " " 5 RU 4 0 "5 0 0 2 0 0 0 0 0 0 0 3 0 " " "5 0 RU 4 0 0 0 0 0

E $Q\FRQVLVWHQWOLQHDUV\VWHPZKRVHUHGXFHGHFKHORQ IRUPLV I3 F $Q\LQFRQVLVWHQWOLQHDUV\VWHPRIWKUHHHTXDWLRQVLQ WKUHHYDULDEOHV

D 7KHVROXWLRQVHW L LVHPSW\LI h D 12 DQG k ¤ 2 LL FRQWDLQVDXQLTXHVROWXWLRQLI h ¤ 12 LLL FRQWDLQV LQÀQLWHO\PDQ\VROXWLRQVLI h D 12 DQG k D 2 E 7KHVROXWLRQVHWLVHPSW\LI k C 3h D 0 RWKHUZLVH WKH VROXWLRQVHWFRQWDLQVDXQLTXHVROXWLRQ

"


3 2 3 2 3 2 !4 !2 D 6HW Y1 D 4 !5 5 Y2 D 4 1 5 Y3 D 4 1 5 DQG 7 !5 !3 2 3 b1 E D 4 b2 5 ´'HWHUPLQHLI Y1 Y2 Y3 VSDQ R3 µ b3 6ROXWLRQ21R 3 2 !4 !2 1 1 5 ´'HWHUPLQHLIWKHFROXPQV E 6HW A D 4 !5 7 !5 !3 RI A VSDQ R3 µ F 'HÀQH T .[/ D A[ ´'HWHUPLQHLI T PDSV R3 RQWR R3 µ ! " ! " ! " ! " ! " ! " 4 2 7 1 5 5 8=3 7=3 D C RU D C 6 6 4=3 14=3 3 1 3 2

+LQW &RQVWUXFWD´JULGµRQWKH x1 x2 SODQHGHWHUPLQHGE\ D1 DQG D2

$ VROXWLRQVHWLVDOLQHZKHQWKHV\VWHPKDVRQHIUHH YDULDEOH ,IWKHFRHIÀFLHQWPDWUL[LV 2 $ 3 WKHQWZRRIWKH FROXPQVVKRXOGEHSLYRWFROXPQV )RULQVWDQFH WDNH ! " 1 2 " 3XWDQ\WKLQJLQFROXPQ 7KHUHVXOWLQJ 0 3 " PDWUL[ZLOOEHLQHFKHORQIRUP 0DNHRQHURZUHSODFHPHQW RSHUDWLRQRQWKHVHFRQGURZWRFUHDWHDPDWUL[ QRW LQ " ! " ! 1 2 1 1 2 1 HFKHORQIRUP VXFKDV ' 0 3 1 1 5 2

+LQW +RZPDQ\IUHHYDULDEOHVDUHLQWKHHTXDWLRQ A[ D " 2 3 1 0 !3 1 25 E D 4 0 0 0 0 D ,IWKHWKUHHYHFWRUVDUHOLQHDUO\LQGHSHQGHQW WKHQ a; c; DQG f PXVWDOOEHQRQ]HUR E 7KHQXPEHUV a; : : : ; f FDQKDYHDQ\YDOXHV +LQW /LVWWKHFROXPQVIURPULJKWWROHIWDV Y1 ; : : : ; Y4 +LQW 8VH7KHRUHP /HW M EHWKHOLQHWKURXJKWKHRULJLQWKDWLVSDUDOOHOWRWKH OLQHWKURXJK Y1 Y2 DQG Y3 7KHQ Y2 ! Y1 DQG Y3 ! Y1 DUH ERWKRQ M 6RRQHRIWKHVHWZRYHFWRUVLVDPXOWLSOHRIWKH RWKHU VD\ Y2 ! Y1 D k.Y3 ! Y1 7KLVHTXDWLRQSURGXFHVD OLQHDUGHSHQGHQFHUHODWLRQ .k ! 1/Y1 C Y2 ! k Y3 D $ VHFRQGVROXWLRQ $ SDUDPHWULFHTXDWLRQRIWKHOLQHLV [ D Y1 C t .Y2 ! Y1 /: 6LQFH Y3 LVRQWKHOLQH WKHUHLVVRPH t0 VXFKWKDW Y3 D Y1 C t0 .Y2 ! Y1 / D .1 ! t0 /Y1 C t0 Y2 6R Y3 LVDOLQHDUFRPELQDWLRQRI Y1 DQG Y2 DQG fY1 Y2 Y3 g LV OLQHDUO\GHSHQGHQW 2 3 1 0 0 05 a D 4=5 b D !3=5 4 0 !1 0 0 1 D 7KHYHFWRUOLVWVWKHQXPEHUVRIWKUHH WZR DQG RQHEHGURRPDSDUWPHQWVSURYLGHGZKHQ x1 ÁRRUVRI SODQ$ DUHFRQVWUXFWHG 2 3 2 3 2 3 3 4 5 E x1 4 7 5 C x2 4 4 5 C x3 4 3 5 8 8 9

F >0@ 8VHÁRRUVRISODQ$ DQGÁRRUVRISODQ%2U XVHÁRRUVRISODQ$ÁRRUVRISODQ%DQGÁRRUVRI SODQ&7KHVHDUHWKHRQO\IHDVLEOHVROXWLRQV 7KHUHDUH RWKHUPDWKHPDWLFDOVROXWLRQV EXWWKH\UHTXLUHD QHJDWLYHQXPEHURUDIUDFWLRQDOQXPEHU RIÁRRUVRI RQHRUWZRRIWKHSODQV ZKLFKPDNHVQRSK\VLFDOVHQVH

$IBQUFS 6HFWLRQ SDJH !

" ! " 0 2 3 !5 3 QRWGHÀQHG 10 !4 !7 6 !7 " 13 !6 " ! " 5 6 !15 5 9 !6 2 3 2 3 !10 11 D AE1 D 4 0 5; AE2 D 4 8 5; 2 26 3 !19 !10 11 85 AB D 4 0 26 !19 2 3 !1 # 4 C 3.!2/ !1.!2/ C 3 # 3 2.!2/ C 4 # 3 5 E AB D 4 2 # 4 C 4.!2/ 5 # 4 ! 3.!2/ 5.!2/ ! 3 # 3 2 3 !10 11 85 D4 0 26 !19

!4 !8 ! 1 !7 ! 1 !3

3 $ 7

k D !2 3 2 3 5 6 6 5 10 15 AD D 4 10 12 10 5 DA D 4 6 12 15 5 15 15 12 6 10 12 5LJKWPXOWLSOLFDWLRQWKDWLV PXOWLSOLFDWLRQRQWKHULJKW E\ D PXOWLSOLHVHDFK FROXPQ RI A E\WKHFRUUHVSRQGLQJ GLDJRQDOHQWU\RI D /HIWPXOWLSOLFDWLRQE\ D PXOWLSOLHV HDFK URZ RI A E\WKHFRUUHVSRQGLQJGLDJRQDOHQWU\RI D 7KH 6WXG\*XLGH WHOOVKRZWRPDNH AB D BA EXW\RX VKRXOGWU\WKLV\RXUVHOIEHIRUHORRNLQJWKHUH 2

+LQW 2QHRIWKHWZRPDWULFHVLV Q $QVZHUWKHTXHVWLRQVEHIRUHORRNLQJLQWKH 6WXG\*XLGH ! " ! " 3 1 E2 D E1 D 2 4 7KHWKLUGFROXPQRI AB LVWKHVXPRIWKHÀUVWWZRFROXPQV RI AB +HUH·VZK\ :ULWH B D Œ E1 E2 E3 ! %\ GHÀQLWLRQ WKHWKLUGFROXPQRI AB LV AE3 ,I E3 D E1 C E2 WKHQ AE3 D A.E1 C E2 / D AE1 C AE2 E\DSURSHUW\RI PDWUL[YHFWRUPXOWLSOLFDWLRQ 7KHFROXPQVRI A DUHOLQHDUO\GHSHQGHQW :K\" +LQW 6XSSRVH [ VDWLVÀHV A[ D DQGVKRZWKDW [ PXVWEH

4FDUJPO +LQW 8VHWKHUHVXOWVRI([HUFLVHVDQG DQGDSSO\WKH DVVRFLDWLYHODZRIPXOWLSOLFDWLRQWRWKHSURGXFW CAD XT Y D YT X D !3a C 2b ! 5c 2 3 !3a !3b !3c 2b 2c 5; XYT D4 2a !5a !5b !5c 2 3 !3a 2a !5a 2b !5b 5 YXT D4 !3b !3c 2c !5c

+LQW )RU7KHRUHPE VKRZWKDWWKH .i; j /HQWU\RI A.B C C / HTXDOVWKH .i; j /HQWU\RI AB C AC +LQW 8VHWKHGHÀQLWLRQRIWKHSURGXFW Im A DQGWKHIDFWWKDW Im [ D [ IRU [ LQ Rm

+LQW )LUVWZULWHWKH .i; j /HQWU\RI .AB/T ZKLFKLVWKH .j; i/HQWU\RI AB 7KHQ WRFRPSXWHWKH .i; j /HQWU\LQ B TAT XVHWKHIDFWVWKDWWKHHQWULHVLQURZ i RI B T DUH b1i ; : : : ; bni EHFDXVHWKH\FRPHIURPFROXPQ i RI B DQG WKHHQWULHVLQFROXPQ j RI AT DUH aj1 ; : : : ; aj n EHFDXVHWKH\ FRPHIURPURZ j RI A >0@ 7KHDQVZHUKHUHGHSHQGVRQWKHFKRLFHRIPDWUL[ SURJUDP )RU0$7/$%XVHWKH KHOS FRPPDQGWRUHDG DERXW ]HURV RQHV H\H DQG GLDJ )RUWKH7, VWXG\WKH GLP ILOO DQG LGHQ LQVWUXFWLRQV 7KH7, GRHVQRWKDYHD´GLDJRQDOµFRPPDQG >0@ 'LVSOD\\RXUUHVXOWVDQGUHSRUW\RXUFRQFOXVLRQV >0@ 'LVSOD\\RXUUHVXOWVDQGUHSRUW\RXUFRQFOXVLRQV 7KHPDWULFHVDSSHDUWRDSSURDFKWKHPDWUL[ 2 3 1=3 1=3 1=3 4 1=3 1=3 1=3 5 1=3 1=3 1=3

6HFWLRQ SDJH

!

2 !3 !5=2 4

"

! " ! " 1 !3 !3 1 1 ! RU 6 7 !2 !7=3 3

x1 D 7 DQG x2 D !9 ! " ! " ! " ! " !9 11 6 13 D DQG E DQG 4 !5 !2 !5

:ULWHRXW\RXUDQVZHUVEHIRUHFKHFNLQJWKH 6WXG\*XLGH 7KHSURRIFDQEHPRGHOHGDIWHUWKHSURRIRI7KHRUHP AB D AC ) A!1 AB D A!1 AC ) IB D IC ) B D C 1R LQJHQHUDO B DQG C FDQEHGLIIHUHQWZKHQ A LVQRW LQYHUWLEOH 6HH([HUFLVHLQ6HFWLRQ +LQW $SSO\WKHHOHPHQWDU\PDWULFHVXVHGWRURZUHGXFH A WR I; WRWKHPDWUL[ ŒA; B!: D D C !1 B !1 A!1 6KRZWKDW D ZRUNV

$IWHU\RXÀQG X D CB ! A VKRZWKDW X LVDVROXWLRQ

+LQW &RQVLGHUWKHHTXDWLRQ A[ D

"

+LQW ,I A[ D KDVRQO\WKHWULYLDOVROXWLRQ WKHQWKHUHDUH QRIUHHYDULDEOHVLQWKHHTXDWLRQ A[ D DQGHDFKFROXPQ RI A LVDSLYRWFROXPQ +LQW &RQVLGHUWKHFDVH a D b D 0 7KHQFRQVLGHUWKH ! " !b DQGXVHWKHIDFWWKDW ad ! bc D 0 YHFWRU a +LQW )RUSDUWD LQWHUFKDQJH A DQG B LQWKHER[ IROORZLQJ([DPSOHLQ6HFWLRQ DQGWKHQUHSODFH B E\ WKHLGHQWLW\PDWUL[ )RUSDUWVE DQGF EHJLQE\ZULWLQJ 2 3 URZ1 .A/ A D 4 URZ2 .A/ 5 URZ3 .A/ 2 3 ! " 8 3 1 !9 3 4 1 5 13 4 10 !4 1 7=2 3=2 1=2 7KHJHQHUDOIRUPRI A!1 LV 2 1 0 0 6 !1 1 0 6 6 0 !1 !1 1 A DBD6 : 6 : 4 :

0

0

###

### ::

:

!1

3 0 07 7 7 : :: 7 7 :5

1

+LQW )RU j D 1; : : : ; n OHW Dj Ej DQG Hj GHQRWHWKH j WK FROXPQVRI A; B DQG I UHVSHFWLYHO\ 8VHWKHIDFWVWKDW Dj ! Dj C1 D Hj DQG Ej D Hj ! Hj C1 IRU j D 1; : : : ; n ! 1 DQG Dn D En D Hn 2 3 3 # $ 4 0 5 )LQGWKLVE\URZUHGXFLQJ A H3 !1 C D

!

1 !1

1 !1 1 0

"

>0@ 7KHGHÁHFWLRQVDUH DQGLQFKHV UHVSHFWLYHO\ >0@ DQGQHZWRQV UHVSHFWLYHO\

6HFWLRQ SDJH 7KHDEEUHYLDWLRQ,07 KHUHDQGLQWKH 6WXG\*XLGH GHQRWHVWKH ,QYHUWLEOH0DWUL[7KHRUHP7KHRUHP ,QYHUWLEOH E\WKH,071HLWKHUFROXPQRIWKHPDWUL[LVD PXOWLSOHRIWKHRWKHUFROXPQ VRWKH\DUHOLQHDUO\ LQGHSHQGHQW $OVR WKHPDWUL[LVLQYHUWLEOHE\7KHRUHPLQ 6HFWLRQEHFDXVHWKHGHWHUPLQDQWLVQRQ]HUR 1RWLFHWKDW AT KDVDSLYRWLQHYHU\FROXPQ VRE\,07 AT LVLQYHUWLEOH +HQFHE\,07 A LVDOVRLQYHUWLEOH 1RWLQYHUWLEOH E\WKH,076LQFHWKLVPDWUL[KDVDFROXPQ RI]HURV LWVFROXPQVIRUPDOLQHDUO\GHSHQGHQWVHWDQG KHQFHWKHPDWUL[LVQRWLQYHUWLEOH

"


,QYHUWLEOH E\WKH,077KHPDWUL[URZUHGXFHVWR 2 3 !1 !3 0 1 6 7 6 0 !4 8 07 6 7 6 0 3 07 0 4 5 1 0 0 0 DQGKDVIRXUSLYRWSRVLWLRQV

>0@ 7KH 4 $ 4 PDWUL[KDVIRXUSLYRWSRVLWLRQV VRLWLV LQYHUWLEOHE\WKH,07 7KH 6WXG\*XLGH ZLOOKHOS EXWÀUVWWU\WRDQVZHUWKH TXHVWLRQVEDVHGRQ\RXUFDUHIXOUHDGLQJRIWKHWH[W $ VTXDUHXSSHUWULDQJXODUPDWUL[LVLQYHUWLEOHLIDQGRQO\LI DOOWKHHQWULHVRQWKHGLDJRQDODUHQRQ]HUR :K\" 1RWH 7KHDQVZHUVEHORZIRU([HUFLVHV²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n 6WDWHPHQWJ RIWKH,07 LVIDOVHIRU F VRVWDWHPHQWG LV IDOVH WRR 7KDWLV WKHHTXDWLRQ F [ D KDVDQRQWULYLDO VROXWLRQ +LQW 8VHWKH,07 ÀUVW /HW W EHWKHLQYHUVHRI AB 7KHQ ABW D I DQG A.BW / D I 8QIRUWXQDWHO\ WKLVHTXDWLRQE\LWVHOIGRHV QRWSURYHWKDW A LVLQYHUWLEOH :K\QRW" )LQLVKWKHSURRI EHIRUH\RXFKHFNWKH 6WXG\*XLGH 6LQFHWKHWUDQVIRUPDWLRQ [ 7! A[ LVRQHWRRQH VWDWHPHQW I RIWKH,07 LVWUXH 7KHQVWDWHPHQWL LVDOVRWUXHDQGWKH WUDQVIRUPDWLRQ [ 7! A[ PDSV Rn RQWR Rn $OVR A LV LQYHUWLEOH ZKLFKLPSOLHVWKDWWKHWUDQVIRUPDWLRQ [ 7! A[ LV LQYHUWLEOH E\7KHRUHP +LQW ,IWKHHTXDWLRQ A[ D E KDVDVROXWLRQIRUHDFK E WKHQ A KDVDSLYRWLQHDFKURZ7KHRUHPLQ6HFWLRQ &RXOGWKHUHEHIUHHYDULDEOHVLQDQHTXDWLRQ A[ D E" +LQW )LUVWVKRZWKDWWKHVWDQGDUGPDWUL[RI T LVLQYHUWLEOH 7KHQXVHDWKHRUHPRUWKHRUHPVWRVKRZWKDW ! " 7 9 T !1 .[/ D B [ ZKHUH B D 4 5

+LQW 7RVKRZWKDW T LVRQHWRRQH VXSSRVHWKDW T .X/ D T .Y/ IRUVRPHYHFWRUV X DQG Y LQ Rn 'HGXFHWKDW X D Y 7RVKRZWKDW T LVRQWR VXSSRVH \ UHSUHVHQWVDQ DUELWUDU\YHFWRULQ Rn DQGXVHWKHLQYHUVH S WRSURGXFHDQ [ VXFKWKDW T .[/ D \ $ VHFRQGSURRIFDQEHJLYHQXVLQJ 7KHRUHPWRJHWKHUZLWKDWKHRUHPIURP6HFWLRQ +LQW &RQVLGHUWKHVWDQGDUGPDWULFHVRI T DQG U ,I T PDSV Rn RQWR Rn WKHQWKHFROXPQVRILWVVWDQGDUG PDWUL[ A VSDQ Rn E\7KHRUHPLQ6HFWLRQ %\WKH ,07 A LVLQYHUWLEOH +HQFH E\7KHRUHP T LVLQYHUWLEOH DQG A!1 LVWKHVWDQGDUGPDWUL[RI T !1 6LQFH A!1 LVDOVR LQYHUWLEOH E\WKH,07LWVFROXPQVDUHOLQHDUO\LQGHSHQGHQW DQGVSDQ Rn $SSO\LQJ7KHRUHPLQ6HFWLRQWRWKH WUDQVIRUPDWLRQ T !1 VKRZVWKDW T !1 LVDRQHWRRQH PDSSLQJRI Rn RQWR Rn >0@ D 7KHH[DFWVROXWLRQRI LV x1 D 3:94 DQG x2 D :49 7KH H[DFWVROXWLRQRIUHYLHZ LV x1 D 2:90 DQG x2 D 2:00 E :KHQWKHVROXWLRQRI LVXVHGDVDQDSSUR[LPDWLRQIRU WKHVROXWLRQLQ WKHHUURULQXVLQJWKHYDOXHRI IRU x1 LVDERXW DQGWKHHUURULQXVLQJIRU x2 LV DERXW F 7KHFRQGLWLRQQXPEHURIWKHFRHIÀ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ÁRDWLQJSRLQWDULWKPHWLF 7KHSURGXFW AA!1 VKRXOG KDYHVHYHUDORIIGLDJRQDOHQWULHVWKDWDUHIDUIURPEHLQJ ]HUR ,IQRW WU\DODUJHUPDWUL[

6HFWLRQ SDJH

!

A EA C C

B EB C D

"

!

C A

D B

"

Y D B !1 H[SODLQZK\ X D !B !1 A Z D C X D A!1 ZK\" Y D !BA!1 Z D 0 ZK\"

!1 !1 A21 D !B21 B11 A31 D !B31 B11 !1 C22 D B22 ! B21 B11 B12

4FDUJPO

C DUHLQYHUWLEOH ([SODLQZK\ &RQYHUVHO\ VXSSRVH B DQG C DUHLQYHUWLEOH 7RSURYHWKDW A LVLQYHUWLEOH JXHVV ZKDW A!1 PXVWEHDQGFKHFNWKDWLWZRUNV " ! # $ XkT D Xk XkT C [kC1 [TkC1 GkC1 D Xk [kC1 [TkC1 D Gk C [kC1 [TkC1 2QO\WKHRXWHUSURGXFWPDWUL[ [kC1 [TkC1 QHHGVWREH FRPSXWHGDQGWKHQDGGHGWR Gk ! " ! " ! " I 0 I 0 I Y LV 6LPLODUO\ 7KHLQYHUVHRI X I !X I 0 I KDVDQLQYHUVH )URPHTXDWLRQ RQHREWDLQV ! "! "! " ! " I 0 A11 A12 I !Y A11 0 D " !X I A21 A22 0 I 0 S

,I A LVLQYHUWLEOH WKHQWKHPDWUL[RQWKHULJKWVLGHRI" LVD SURGXFWRILQYHUWLEOHPDWULFHVDQGKHQFHLVLQYHUWLEOH %\ ([HUFLVH A11 DQG S PXVWEHLQYHUWLEOH

W .s/ D Im ! C.A ! sIn /!1 B 7KLVLVWKH6FKXU FRPSOHPHQWRI A ! sIn LQWKHV\VWHPPDWUL[ ! "! " 1 0 1 0 D A2 D 2 !1 2 !1 ! " ! " 1C0 0C0 1 0 D D 2!2 0 C .!1/2 0 1 ! "! " A 0 A 0 E M 2 D I !A I !A ! 2 " ! " A C0 0C0 I 0 D D 2 A!A 0 C .!A/ 0 I

,I A1 DQG B1 DUH .k C 1/ $ .k C 1/ DQGORZHUWULDQJXODU ! " ! " a T b T WKHQZULWH A1 D DQG B1 D ZKHUH Y A Z B A DQG B DUH k $ k DQGORZHUWULDQJXODU Y DQG Z DUHLQ Rk DQG a DQG b DUHVXLWDEOHVFDODUV $VVXPHWKDWWKHSURGXFWRI k $ k ORZHUWULDQJXODUPDWULFHVLVORZHUWULDQJXODU DQG FRPSXWHWKHSURGXFW A1 B1 :KDWGR\RXFRQFOXGH" 8VH([HUFLVHWRÀQGWKHLQYHUVHRIDPDWUL[RIWKHIRUP ! " B11 0 BD ZKHUH B11 LV p $ p B22 LV q $ q DQG 0 B22 B LVLQYHUWLEOH 3DUWLWLRQWKHPDWUL[ A DQGDSSO\\RXUUHVXOW WZLFHWRÀQGWKDW 2 3 !5 2 0 0 0 6 3 !1 0 0 07 6 7 0 0 1=2 0 07 A!1 D 6 6 7 4 0 0 0 3 !4 5 0 0 0 !5=2 7=2

D E 7KHFRPPDQGVWREHXVHGLQWKHVHH[HUFLVHVZLOO GHSHQGRQWKHPDWUL[SURJUDP F 7KHDOJHEUDQHHGHGFRPHVIURPWKHEORFNPDWUL[ HTXDWLRQ ! "! " ! " A11 0 [1 E1 D A22 [2 E2 A21

ZKHUH [1 DQG E1 DUHLQ R20 DQG [2 DQG E2 DUHLQ R30 7KHQ A11 [1 D E1 ZKLFKFDQEHVROYHGWRSURGXFH [1

"

7KHHTXDWLRQ A21 [1 C A22 [2 D E2 \LHOGV A22 [2 D E2 ! A21 [1 ZKLFKFDQEHVROYHGIRU [2 E\URZ UHGXFLQJWKHPDWUL[ ŒA22 F! ZKHUH F D E2 ! A21 [1

6HFWLRQ SDJH

2

3 2 3 !7 3 L\ D E ) \ D 4 !2 5 U [ D \ ) [ D 4 4 5 6 !6 2 3 2 3 2 3 2 3 1 38 6 !5 6 37 6 16 7 7 6 7 \ D 4 12 5 [ D 4 !4 5 \ D 6 4 1 5 [ D 4 8:5 5 0 0 !4 !4 ! "! " 1 0 2 5 LU D !3=2 1 0 7=2 2 32 3 1 0 0 3 1 2 4 !3 1 0 54 0 3 25 3 2 1 0 0 4 2 32 3 1 0 0 3 7 2 4 2 1 0 54 0 5 05 !1 1 1 0 0 5 2 32 3 1 0 0 0 1 3 !5 !3 6 !1 6 1 0 07 3 17 6 76 0 !2 7 4 4 5 1 0 54 0 0 0 05 !2 !1 0 1 0 0 0 0 2 32 3 1 0 0 2 0 5 2 4 !3 1 0 54 0 3 2 35 2 3 1 0 0 0 4 2 3 1=2 !3=4 !2 U !1 D 4 0 !1=4 !1 5 0 0 !1=2 2 3 1 0 0 !1 1 0 5 L D4 2 !2 !1 1 2 3 3 5=4 !2 3=4 !1 5 A!1 D 4 3=2 1 1=2 !1=2

+LQW 7KLQNDERXWURZUHGXFLQJ Œ A

I !

+LQW 5HSUHVHQWWKHURZRSHUDWLRQVE\DVHTXHQFHRI HOHPHQWDU\PDWULFHV D 'HQRWHWKHURZVRI D DVWUDQVSRVHVRIFROXPQYHFWRUV 7KHQSDUWLWLRQHGPDWUL[PXOWLSOLFDWLRQ\LHOGV 2 T3 Y1 # $6 : 7 ### F4 4 :: 5 A D CD D F1 YT4

D

F1 YT1

C ### C

F4 YT4

E A KDVHQWULHV 6LQFH C KDVHQWULHVDQG D KDVHQWULHV WRJHWKHUWKH\RFFXS\RQO\RIWKH PHPRU\QHHGHGWRVWRUH A

"


([SODLQZK\ U D DQG V T DUHLQYHUWLEOH 7KHQXVHD WKHRUHPRQWKHLQYHUVHRIDSURGXFWRILQYHUWLEOHPDWULFHV i1

D

i2 1/2 ohm

v1

i2 6 ohms

v1

9/2 ohms

i2

v3

3/4 ohm

v3

" 1 C R3 =R2 !R1 ! R3 ! .R1 R3 /=R2 !1=R2 1 C R1 =R2 ! " 3 !12 E A D !1=3 5=3 ! "! "! " 1 !6 1 0 1 !2 D 0 1 !1=3 1 0 1 6HW R1 D 2 RKPV R2 D 3 RKPV DQG R3 D 6 RKPV

D

!

>0@

2

D L D

1 0 0 0 0 0 0 6 !:25 1 0 0 0 0 0 6 6 1 0 0 0 0 6 !:25 !:0667 6 6 0 !:2667 !:2857 1 0 0 0 6 6 0 0 !:2679 !:0833 1 0 0 6 6 6 0 0 0 !:2917 !:2921 1 0 6 4 0 0 0 0 !:2697 !:0861 1 0 0 0 0 0 !:2948 !:2931 2

U D

E [ D

!1 3:75 0 0 0 0 0 0

4 60 6 6 60 6 60 6 60 6 6 60 6 40 0

3 0 07 7 7 07 7 07 7 07 7 7 07 7 05

1

3 !1 0 0 0 0 0 !:25 !1 0 0 0 0 7 7 7 3:7333 !1:0667 !1 0 0 0 7 7 0 3:4286 !:2857 !1 0 0 7 7 0 0 3:7083 !1:0833 !1 0 7 7 7 0 0 0 3:3919 !:2921 !1 7 7 0 0 0 0 3:7052 !1:0861 5 0 0 0 0 0 3:3868

.3:9569; 6:5885; 4:2392; 7:3971; 5:6029; 8:7608; 9:4115; 12:0431/ 2

F A!1 D

:2953 6 :0866 6 6 6 :0945 6 6 :0509 6 6 :0318 6 6 6 :0227 6 4 :0100 :0082

:0866 :2953 :0509 :0945 :0227 :0318 :0082 :0100

:0945 :0509 :3271 :1093 :1045 :0591 :0318 :0227

:0509 :0945 :1093 :3271 :0591 :1045 :0227 :0318

:0318 :0227 :1045 :0591 :3271 :1093 :0945 :0509

!1

:0227 :0318 :0591 :1045 :1093 :3271 :0509 :0945

:0100 :0082 :0318 :0227 :0945 :0509 :2953 :0866

3 :0082 :0100 7 7 7 :0227 7 7 :0318 7 7 :0509 7 7 7 :0945 7 7 :0866 5 :2953

2EWDLQ A GLUHFWO\DQGWKHQFRPSXWH A ! U L!1 WRFRPSDUHWKHWZRPHWKRGVIRULQYHUWLQJDPDWUL[

6HFWLRQ SDJH 2

:10 :60 C D 4 :30 :20 :30 :10 2 3 44:44 [ D 4 16:67 5 16:67

!1

!1

3 2 3 & :60 % 60 LQWHUPHGLDWH 0 5 D 4 20 5 GHPDQG :10 10

[ D

!

110 120

"

1:6 1:2

"

E

!

111:6 121:2

"

+LQW 8VHSURSHUWLHVRIWUDQVSRVHVWRREWDLQ ST D ST C C YT VRWKDW ST [ D .ST C C YT /[ D ST C [ C YT [ 1RZFRPSXWH ST [ IURPWKHSURGXFWLRQHTXDWLRQ

i3

v2

!

F +LQW )LQGDIRUPXODLQYROYLQJ .I ! C /!1 6HHWKH 6WXG\*XLGH 2 3 82:8 [ D 4 131:0 5 110:3

i3

v2

i1

E

i2

D

>0@ [ D .99576; 97703; 51231; 131570; 49488; 329554; 13835/ 7KHHQWULHVLQ [ VXJJHVWPRUHSUHFLVLRQLQWKHDQVZHUWKDQ LVZDUUDQWHGE\WKHHQWULHVLQ G ZKLFKDSSHDUWREHDFFXUDWH RQO\WRSHUKDSVWKHQHDUHVWWKRXVDQG 6RDPRUHUHDOLVWLF DQVZHUIRU [ PLJKWEH [ D 1000 $ .100; 98; 51; 132; 49; 330; 14/ >0@ [.12/ LVWKHÀUVWYHFWRUZKRVHHQWULHVDUHDFFXUDWHWRWKH QHDUHVWWKRXVDQG 7KHFDOFXODWLRQRI [.12/ WDNHVDERXW ÁRSV ZKLOHURZUHGXFWLRQRI Œ.I ! C / G! WDNHVRQO\ DERXW ÁRSV ,I C LVODUJHUWKDQ 20 $ 20 WKHQIHZHU ÁRSVDUHQHHGHGWRFRPSXWH [.12/ E\LWHUDWLRQWKDQWR FRPSXWHWKHHTXLOLEULXPYHFWRU [ E\URZUHGXFWLRQ $VWKH VL]HRI C JURZV WKHDGYDQWDJHRIWKHLWHUDWLYHPHWKRG LQFUHDVHV $OVR EHFDXVH C EHFRPHVPRUHVSDUVHIRUODUJHU PRGHOVRIWKHHFRQRP\ IHZHULWHUDWLRQVDUHQHHGHGIRU UHDVRQDEOHDFFXUDF\

6HFWLRQ SDJH 2

3 2 3 1 :25 0 0 !1 !1 1 0 5 4 1 0 25 4 0 0 0 1 0 0 1 p 2 p 3 1=p2 1= p2 0 4 1= 2 !1= 2 05 0 0 1 p p 3 2 3 C 4p 3 p1=2 ! 3=2 4 3=2 1=2 4!3 35 0 0 1 6HHWKH3UDFWLFH3UREOHP

A.BD/ UHTXLUHVPXOWLSOLFDWLRQV .AB/D UHTXLUHV PXOWLSOLFDWLRQV 7KHÀUVWPHWKRGXVHVDERXWWZLFHDVPDQ\ PXOWLSOLFDWLRQV ,I D KDGFROXPQV WKHFRXQWVZRXOG EHDQG UHVSHFWLYHO\ 8VHWKHIDFWWKDW

1 VLQ2 ' ! D FRV ' FRV ' FRV ' ! " ! "! " A A S I S D )LUVWDSSO\WKHOLQHDU T 1 T 1 1 T WUDQVIRUPDWLRQ A DQGWKHQWUDQVODWHE\ S VHF ' ! WDQ ' VLQ ' D

4FDUJPO

.12; !6; !3/

2

1 60 6 4 0 0

0 p0 3=2 1=2 ! p 3=2 1=2 0 0

3 0 07 7 05 1

7KHWULDQJOHZLWKYHUWLFHVDW .7; 2; 0/ .7:5; 5; 0/ DQG .5; 5; 0/ 2 32 3 2 3 2:2586 !1:0395 !:3473 X R 2:3441 :0696 54 Y 5 D 4 G 5 >0@ 4 !1:3495 :0910 !:3046 1:2777 Z B

6HFWLRQ SDJH

"

2

3 2 3 2 3 1 4 !3 6 !1 7 6 2 7 6 3 7 7 6 7 6 7 %DVLVIRU &RO A 6 4 !2 5 4 2 5 4 5 5 3 6 !5 2 3 2 3 2 !7 6 !2:5 7 6 :5 7 6 7 6 7 6 7 17 %DVLVIRU 1XO A 6 6 7 6 0 7 4 5 4 0 !4 5 0 1

&RQVWUXFWDQRQ]HUR 3 $ 3 PDWUL[ A DQGFRQVWUXFW E WREH DOPRVWDQ\FRQYHQLHQWOLQHDUFRPELQDWLRQRIWKHFROXPQVRI A

7KHVHWLVFORVHGXQGHUVXPVEXWQRWXQGHUPXOWLSOLFDWLRQ E\DQHJDWLYHVFDODU 6NHWFKDQH[DPSOH

+LQW
7KHVHWLVQRWFORVHGXQGHUVXPVRUVFDODUPXOWLSOHV # $
,I &RO F ¤ R5 WKHQWKHFROXPQVRI F GRQRWVSDQ R5 6LQFH F LVVTXDUH WKH,07 VKRZVWKDW F LVQRWLQYHUWLEOH DQGWKHHTXDWLRQ F [ D KDVDQRQWULYLDOVROXWLRQ 7KDWLV 1XO F FRQWDLQVDQRQ]HURYHFWRU $QRWKHUZD\WRGHVFULEH WKLVLVWRZULWH 1XO F ¤ fg

D 7KHWKUHHYHFWRUV Y1 ; Y2 DQG Y3 E ,QÀQLWHO\PDQ\YHFWRUV F
,I 1XO C D fg WKHQWKHHTXDWLRQ C [ D KDVRQO\WKH WULYLDOVROXWLRQ 6LQFH C LVVTXDUH WKH,07 VKRZVWKDW C LV LQYHUWLEOHDQGWKHHTXDWLRQ C [ D E KDVDVROXWLRQIRUHDFK E LQ R6 $OVR HDFKVROXWLRQLVXQLTXH E\7KHRUHPLQ 6HFWLRQ

,I 1XO B FRQWDLQVQRQ]HURYHFWRUV WKHQWKHHTXDWLRQ B [ D KDVQRQWULYLDOVROXWLRQV 6LQFH B LVVTXDUH WKH,07 VKRZVWKDW B LVQRWLQYHUWLEOHDQGWKHFROXPQVRI B GRQRW VSDQ R5 6R &RO B LVDVXEVSDFHRI R5 EXW &RO B ¤ R5 >0@ 'LVSOD\WKHUHGXFHGHFKHORQIRUPRI A DQGVHOHFWWKH SLYRWFROXPQVRI A DVDEDVLVIRU &RO A )RU 1XO A ZULWH WKHVROXWLRQRI A[ D LQSDUDPHWULFYHFWRUIRUP 2 3 2 3 3 !5 6 !7 7 6 9 7 7 6 7 %DVLVIRU &RO A 6 4 !5 5; 4 7 5 3 !7 2 3 2 3 2 3 !2:5 4:5 !3:5 6 !1:5 7 6 2:5 7 6 !1:5 7 6 7 6 7 6 7 6 7 6 7 17 %DVLVIRU 1XO A 6 6 7; 6 0 7 ; 6 0 7 4 05 4 1 5 4 0 5 0 0 1

6HFWLRQ SDJH

! " ! " ! " 1 2 7 [ D 3E1 C 2E2 D 3 C2 D 1 !1 1 ! " ! " ! " ! " 1 4 2 1:5 ŒZ!B D Œ [ !B D 2 1 !1 :5 2 3 2 3 2 3 1 2 !6 637 6 17 6 57 7 6 7 6 7 %DVLVIRU &RO A 6 4 2 5 4 !1 5 4 9 5 GLP &RO A D 3 5 0 14

"


3 !3 6 17 7 %DVLVIRU 1XO A 6 4 0 5 GLP 1XO A D 1 0 2

3 2

P )

3

7KHYHFWRUV Y1 Y3 DQG Y4 IRUPDEDVLVIRUWKHJLYHQ VXEVSDFH H 6R GLPHQVLRQ H D 3

&RO A D R4 EHFDXVH A KDVDSLYRWLQHDFKURZ DQGVRWKH FROXPQVRI A VSDQ R4 1XO A FDQQRW HTXDO R2 EHFDXVH 1XO A LVDVXEVSDFHRI R6 ,WLVWUXH KRZHYHU WKDW 1XO A LV WZRGLPHQVLRQDO 5HDVRQ 7KHHTXDWLRQ A[ D KDVWZR IUHHYDULDEOHV EHFDXVH A KDVVL[FROXPQVDQGRQO\IRXURI WKHPDUHSLYRWFROXPQV

6HHWKH 6WXG\*XLGH DIWHU\RXZULWH\RXUMXVWLÀFDWLRQV 7KHIDFWWKDWWKHVROXWLRQVSDFHRI A[ D KDVDEDVLVRI WKUHHYHFWRUVPHDQVWKDW GLP 1XO A D 3 6LQFHD 5 $ 7 PDWUL[ A KDVVHYHQFROXPQV WKH5DQN7KHRUHPVKRZVWKDW UDQN A D 7 ! GLP 1XO A D 4 6HHWKH 6WXG\*XLGH IRUD MXVWLÀFDWLRQWKDWGRHVQRWH[SOLFLWO\PHQWLRQWKH5DQN 7KHRUHP $ 9 $ 8 PDWUL[KDVHLJKWFROXPQV %\WKH5DQN7KHRUHP GLP 1XO A D 8 ! UDQN A 6LQFHWKHUDQNLV GLP 1XO A D 1 7KDWLV WKHGLPHQVLRQRIWKHVROXWLRQVSDFH RI A[ D LV

&UHDWHD 3 $ 5 PDWUL[ A ZLWKWZRSLYRWFROXPQV 7KH UHPDLQLQJWKUHHFROXPQVZLOOFRUUHVSRQGWRIUHHYDULDEOHV LQWKHHTXDWLRQ A[ D 6RWKHGHVLUHGFRQVWUXFWLRQLV SRVVLEOH 7KH p FROXPQVRI A VSDQ &RO A E\GHÀQLWLRQ ,I GLP &RO A D p WKHQWKHVSDQQLQJVHWRI p FROXPQVLV DXWRPDWLFDOO\DEDVLVIRU &RO A E\WKH%DVLV7KHRUHP ,Q SDUWLFXODU WKHFROXPQVDUHOLQHDUO\LQGHSHQGHQW

D +LQW 7KHFROXPQVRI B VSDQ W DQGHDFKYHFWRU Dj LV LQ W 7KHYHFWRU Fj LVLQ Rp EHFDXVH B KDV p FROXPQV E +LQW :KDWLVWKHVL]HRI C " F +LQW +RZDUH B DQG C UHODWHGWR A" >0@
&KDSWHU6XSSOHPHQWDU\([HUFLVHV SDJH 7 7

E K

) 7

F L

7 7

G M

) )

H N

7

R

)

S

7

I

2 !5 6 3 7 6 !8 7 7 6 7 %DVLVIRU &RO A 6 4 0 5 4 9 5 GLP &RO A D 2 !3 !7 2 3 2 32 3 !2 !1 !1 6 17 6 076 07 6 7 6 76 7 7 6 76 7 %DVLVIRU 1XO A 6 6 0 7 6 0 7 6 !1 7 GLP 1XO A D 3 4 05 4 154 05 0 0 1

D J

Q

) 7

I O

) )

A2 D 2A ! I 0XOWLSO\E\ A A3 D 2A2 ! A 6XEVWLWXWH A2 D 2A ! I A3 D 2.2A ! I / ! A D 3A ! 2I 0XOWLSO\E\ A DJDLQ A4 D A.3A ! 2I / D 3A2 ! 2A 6XEVWLWXWHWKHLGHQWLW\ A2 D 2A ! I DJDLQ A4 D 3.2A ! I / ! 2A D 4A ! 3I 2 3 ! " 10 !1 !3 13 10 5 4 9 !8 27 !5 !3

D p.xi / D c0 C c1 xi C # # # C cn!1 xin!1 2 3 c0 6 7 D URZi .V / # 4 ::: 5 D URZi .V F/ D yi cn!1 E 6XSSRVH x1 ; : : : ; xn DUHGLVWLQFW DQGVXSSRVH V F D IRU VRPHYHFWRU F 7KHQWKHHQWULHVLQ F DUHWKHFRHIÀFLHQWV RIDSRO\QRPLDOZKRVHYDOXHLV]HURDWWKHGLVWLQFW SRLQWV x1 ; : : : ; xn +RZHYHU DQRQ]HURSRO\QRPLDORI GHJUHH n ! 1 FDQQRWKDYH n ]HURV VRWKHSRO\QRPLDO PXVWEHLGHQWLFDOO\]HUR 7KDWLV WKHHQWULHVLQ F PXVW DOOEH]HUR 7KLVVKRZVWKDWWKHFROXPQVRI V DUH OLQHDUO\LQGHSHQGHQW F +LQW :KHQ x1 ; : : : ; xn DUHGLVWLQFW WKHUHLVDYHFWRU F VXFKWKDW V F D \ :K\" D P 2 D .XXT /.XXT / D X.XT X/XT D X.1/XT D P E P T D .XXT /T D XT T XT D XXT D P

F Q2 D .I ! 2P /.I ! 2P / D I ! I.2P / ! 2PI C 2P .2P / D I ! 4P C 4P 2 D I; EHFDXVHRISDUWD /HIWPXOWLSOLFDWLRQE\DQHOHPHQWDU\PDWUL[SURGXFHVDQ HOHPHQWDU\URZRSHUDWLRQ

B ' E1 B ' E2 E1 B ' E3 E2 E1 B D C

6R B LVURZHTXLYDOHQWWR C 6LQFHURZRSHUDWLRQVDUH UHYHUVLEOH C LVURZHTXLYDOHQWWR B $OWHUQDWLYHO\ VKRZ C EHLQJFKDQJHGLQWR B E\URZRSHUDWLRQVXVLQJWKH LQYHUVHVRIWKH Ei 6LQFH B LV 4 $ 6 ZLWKPRUHFROXPQVWKDQURZV LWVVL[ FROXPQVDUHOLQHDUO\GHSHQGHQWDQGWKHUHLVDQRQ]HUR [ VXFKWKDW B [ D 7KXV AB [ D A D ZKLFKVKRZVWKDW WKHPDWUL[ AB LVQRWLQYHUWLEOH E\WKH,QYHUWLEOH0DWUL[ 7KHRUHP >0@ 7RIRXUGHFLPDOSODFHV DV k LQFUHDVHV 2 3 :2857 :2857 :2857 Ak ! 4 :4286 :4286 :4286 5 DQG :2857 :2857 :2857 2 3 :2022 :2022 :2022 B k ! 4 :3708 :3708 :3708 5 :4270 :4270 :4270

4FDUJPO

6HFWLRQ SDJH

RU LQUDWLRQDOIRUPDW 2 3 2=7 2=7 2=7 3=7 3=7 5 DQG Ak ! 4 3=7 2=7 2=7 2=7 2 3 18=89 18=89 18=89 33=89 33=89 5 B k ! 4 33=89 38=89 38=89 38=89

,QWHUFKDQJLQJWZRURZVUHYHUVHVWKHVLJQRIWKH GHWHUPLQDQW $ URZUHSODFHPHQWRSHUDWLRQGRHVQRWFKDQJHWKH GHWHUPLQDQW

$IBQUFS 6HFWLRQ SDJH

!5

!23

6WDUWZLWKURZ

6WDUWZLWKURZRUFROXPQ !5

ad ! bc cb ! da ,QWHUFKDQJLQJWZRURZVFKDQJHVWKH VLJQRIWKHGHWHUPLQDQW !2 .18 C 12k/ ! .20 C 12k/ D !2 $ URZUHSODFHPHQW GRHVQRWFKDQJHWKHYDOXHRIDGHWHUPLQDQW !5 k.4/ ! k.2/ C k.!7/ D !5k 6FDOLQJDURZE\D FRQVWDQW k PXOWLSOLHVWKHGHWHUPLQDQWE\ k

k

!7

,QYHUWLEOH

1RWLQYHUWLEOH

/LQHDUO\ LQGHSHQGHQW !32

6HHWKH 6WXG\*XLGH

+LQW 6KRZWKDW .GHW A/.GHW A!1 / D 1 +LQW 8VH7KHRUHP

!12 6WDUWZLWKFROXPQRUURZ

"

!1

7KHPDWUL[LVXSSHURUORZHUWULDQJXODU ZLWKRQO\·VRQ WKHGLDJRQDO 7KHGHWHUPLQDQWLV WKHSURGXFWRIWKH GLDJRQDOHQWULHV ! " c d D cb ! ad D .!1/.ad ! bc/ GHW EA D GHW a b

D .GHW E/.GHW A/ ! " a C kc b C kd GHW EA D GHW c d D .a C kc/d ! .b C kd /c D ad C kcd ! bc ! kdc D .C1/.ad ! bc/ D .GHW E/.GHW A/ ! " 15 5 5A D QR 20 10 +LQWVDUHLQWKH 6WXG\*XLGH

7KHDUHDRIWKHSDUDOOHORJUDPDQGWKHGHWHUPLQDQWRI ! " x Œ X Y ! ERWKHTXDO ,I Y D IRUDQ\ x WKHDUHDLVVWLOO 2 ,QHDFKFDVHWKHEDVHRIWKHSDUDOOHORJUDPLVXQFKDQJHG DQGWKHDOWLWXGHUHPDLQVEHFDXVHWKHVHFRQGFRRUGLQDWHRI Y LVDOZD\V >0@ ,QJHQHUDO GHW.A C B/ LVQRWHTXDOWR GHW A C GHW B >0@
+LQW 8VH7KHRUHPDQGDQRWKHUWKHRUHP ! " 6 0 GHW AB D GHW D 24 17 4 .GHW A/.GHW B/ D 3 # 8 D 24 D !12

F !3

E

G

1 4

H

GHW A D .a C e/d ! .b C f /c D ad C ed ! bc ! f c D .ad ! bc/ C .ed ! f c/ D GHW B C GHW C

+LQW &RPSXWH GHW A E\DFRIDFWRUH[SDQVLRQGRZQFROXPQ >0@ 6HHWKH 6WXG\*XLGH DIWHU\RXKDYHPDGHDFRQMHFWXUH DERXW ATA DQG AAT

6HFWLRQ SDJH

!

5=6 !1=6

"

!

4 5=2

"

2

3 3=2 4 4 5 !7=2

p 5s C 4 !4s ! 15 s ¤ ˙ 3 x1 D x2 D 6.s 2 ! 3/ 4.s 2 ! 3/ 1 4s C 3 s ¤ 0 !1 x1 D x2 D 3.s C 1/ 6s.s C 1/ 2 3 2 3 0 1 0 0 1 0 1 DGM A D 4 !3 !1 !3 5 A!1 D 4 !3 !1 !3 5 3 3 2 6 3 2 6 2 3 2 3 !1 !1 5 !1 !1 5 1 !1 1 5 A D 4 1 !5 15 DGM A D 4 1 !5 6 1 7 !5 1 7 !5 2 3 2 3 2 0 0 2 0 0 1 6 0 5 A!1 D 4 2 6 05 DGM A D 4 2 6 !1 !9 !1 !9 3 3 ! " a b ,I A D WKHQ C11 D d C12 D !c C21 D !b c d C22 D a 7KHDGMXJDWHPDWUL[LVWKHWUDQVSRVHRIFRIDFWRUV ! " d !b DGM A D !c a

"

"OTXFST UP 0EE/VNCFSFE &YFSDJTFT )ROORZLQJ7KHRUHP ZHGLYLGHE\ GHW A WKLVSURGXFHVWKH IRUPXODIURP6HFWLRQ

$ 3 $ 3 PDWUL[ A LVQRWLQYHUWLEOHLIDQGRQO\LILWVFROXPQV DUHOLQHDUO\GHSHQGHQWE\WKH,QYHUWLEOH0DWUL[7KHRUHP 7KLVKDSSHQVLIDQGRQO\LIRQHRIWKHFROXPQVLVLQWKH SODQHVSDQQHGE\WKHRWKHUWZRFROXPQV ZKLFKLV HTXLYDOHQWWRWKHFRQGLWLRQWKDWWKHSDUDOOHOHSLSHG GHWHUPLQHGE\WKHVHFROXPQVKDV]HURYROXPH ZKLFKLQ WXUQLVHTXLYDOHQWWRWKHFRQGLWLRQWKDW GHW A D 0

1 j GHW Œ 2

D 6HH([DPSOH

Y1

Y2 !j

E 4"abc=3

>0@ ,Q0$7/$%WKHHQWULHVLQ B ! LQY.A/ DUH DSSUR[LPDWHO\ 10!15 RUVPDOOHU 6HHWKH 6WXG\*XLGH IRU VXJJHVWLRQVWKDWPD\VDYH\RXNH\VWURNHVDV\RXZRUN >0@ 0$7/$% 6WXGHQW9HUVLRQXVHVÁRSVIRU LQY.A/ DQGÁRSVIRUWKHLQYHUVHIRUPXOD 7KH LQY$ FRPPDQGUHTXLUHVRQO\DERXWRIWKH RSHUDWLRQVIRUWKHLQYHUVHIRUPXOD 7KH 6WXG\*XLGH VKRZV KRZWRXVHWKH IORSV FRPPDQG

&KDSWHU6XSSOHPHQWDU\([HUFLVHV SDJH D 7 J 7 P )

E K Q

7 7 7

F L R

) ) )

G M S

) ) 7

H N

) 7

I O

) )

7KHVROXWLRQIRU([HUFLVHLVEDVHGRQWKHIDFWWKDWLIDPDWUL[ FRQWDLQVWZRURZVRUWZRFROXPQV WKDWDUHPXOWLSOHVRIHDFK RWKHU WKHQWKHGHWHUPLQDQWRIWKHPDWUL[LV]HUR E\7KHRUHP EHFDXVHWKHPDWUL[FDQQRWEHLQYHUWLEOH 0DNHWZRURZUHSODFHPHQWRSHUDWLRQV DQGWKHQIDFWRURXWD FRPPRQPXOWLSOHLQURZDQGDFRPPRQPXOWLSOHLQURZ ˇ ˇ ˇ ˇ ˇ1 a b C c ˇˇ ˇˇ 1 a b C c ˇˇ ˇ ˇ1 b a C c ˇˇ D ˇˇ 0 b!a a ! b ˇˇ ˇ ˇ1 c aCbˇ ˇ0 c!a a!c ˇ ˇ ˇ ˇ1 a b C c ˇˇ ˇ 1 !1 ˇˇ D .b ! a/.c ! a/ˇˇ 0 ˇ0 1 !1 ˇ

D0

!12 :KHQWKHGHWHUPLQDQWLVH[SDQGHGE\FRIDFWRUVRIWKHÀUVW URZ WKHHTXDWLRQKDVWKHIRUP ax C by C c D 0 ZKHUHDW OHDVWRQHRI a DQG b LVQRW]HUR 7KLVLVWKHHTXDWLRQRID OLQH ,WLVFOHDUWKDW .x1 ; y1 / DQG .x2 ; y2 / DUHRQWKHOLQH EHFDXVHZKHQWKHFRRUGLQDWHVRIRQHRIWKHSRLQWVDUH VXEVWLWXWHGIRU x DQG y WZRURZVRIWKHPDWUL[DUHHTXDO DQGVRWKHGHWHUPLQDQWLV]HUR

2

3 a2 b 2 ! a 2 5 c 2 ! a2 2 1 GHW T D .b ! a/.c ! a/ GHW 4 0 0 2 1 D .b ! a/.c ! a/ GHW 4 0 0

1 T ' 4 0 0

a b!a c!a

7KXV E\7KHRUHP

a 1 1 a 1 0

D .b ! a/.c ! a/.c ! b/

3 a2 bCa5 cCa 3 a2 bCa5 c!b

$UHD D 12 ,IRQHYHUWH[LVVXEWUDFWHGIURPDOOIRXU YHUWLFHV DQGLIWKHQHZYHUWLFHVDUH Y1 Y2 DQG Y3 WKHQ WKHWUDQVODWHGÀJXUHDQGKHQFHWKHRULJLQDOÀJXUH ZLOOEHD SDUDOOHORJUDPLIDQGRQO\LIRQHRI Y1 Y2 DQG Y3 LVWKHVXP RIWKHRWKHUWZRYHFWRUV

1 A D A!1 A D I GHW A %\WKH,QYHUWLEOH0DWUL[7KHRUHP DGM A LVLQYHUWLEOHDQG 1 A .DGM A/!1 D GHW A

%\WKH,QYHUVH)RUPXOD .DGM A/ #

D X D CA!1 Y D D ! CA!1 B 1RZXVH([HUFLVH F E )URPSDUWD DQGWKHPXOWLSOLFDWLYHSURSHUW\RI GHWHUPLQDQWV ! " A B D GHW Œ A.D ! CA!1 B/ ! GHW C D D GHW Œ AD ! ACA!1 B !

D GHW Œ AD ! CAA!1 B ! D GHW Œ AD ! CB !

ZKHUHWKHHTXDOLW\ AC D CA ZDVXVHGLQWKHWKLUGVWHS )LUVWFRQVLGHUWKHFDVH n D 2 DQGSURYHWKDWWKHUHVXOW KROGVE\GLUHFWO\FRPSXWLQJWKHGHWHUPLQDQWVRI B DQG C 1RZDVVXPHWKDWWKHIRUPXODKROGVIRUDOO .k ! 1/ $ .k ! 1/ PDWULFHV DQGOHW A B DQG C EH k $ k PDWULFHV 8VHDFRIDFWRUH[SDQVLRQDORQJWKHÀUVWFROXPQ DQGWKHLQGXFWLYHK\SRWKHVLVWRÀQG GHW B 8VHURZ UHSODFHPHQWRSHUDWLRQVRQ C WRFUHDWH]HURVEHORZWKHÀUVW SLYRWDQGSURGXFHDWULDQJXODUPDWUL[ )LQGWKHGHWHUPLQDQW RIWKLVPDWUL[DQGDGGLWWR GHW B WRJHWWKHUHVXOW >0@ &RPSXWH ˇ ˇ1 ˇ ˇ1 ˇ ˇ1 ˇ ˇ1 ˇ ˇ1 ˇ ˇ1 ˇ ˇ1 ˇ ˇ1

1 2 2 1 2 2 2 2

ˇ 1 ˇˇ 2 ˇˇ D 1; 3ˇ

1 2 3 3 3

1 2 3 4 4

ˇ ˇ1 1 ˇ ˇ1 2 ˇ ˇ1 2 ˇ ˇ1 2 ˇ 1 ˇˇ 2 ˇˇ 3 ˇˇ D 1 4 ˇˇ 5ˇ

1 2 3 3

ˇ 1 ˇˇ 2 ˇˇ D 1; 3 ˇˇ ˇ 4

4FDUJPO &RQMHFWXUH ˇ ˇ1 1 1 ˇ ˇ1 2 2 ˇ ˇ1 2 3 ˇ: ˇ: ˇ: ˇ ˇ1 2 3

:::

::

:

:::

ˇ 1 ˇˇ 2 ˇˇ 3 ˇˇ D 1 :: ˇ : ˇˇ nˇ

7RFRQÀUPWKHFRQMHFWXUH XVHURZUHSODFHPHQWRSHUDWLRQV WRFUHDWH]HURVEHORZWKHÀUVWSLYRW WKHQWKHVHFRQGSLYRW DQGVRRQ 7KHUHVXOWLQJPDWUL[LV ˇ ˇ ˇ1 1 1 ::: 1 ˇˇ ˇ ˇ0 1 1 1 ˇˇ ˇ ˇ0 0 1 1 ˇˇ ˇ: :: ˇ :: ˇ: ˇ: : : ˇˇ ˇ ˇ0 0 0 ::: 1ˇ ZKLFKLVDQXSSHUWULDQJXODUPDWUL[ZLWKGHWHUPLQDQW

$IBQUFS 6HFWLRQ SDJH D X C Y LVLQ V EHFDXVHLWVHQWULHVZLOOERWKEH QRQQHJDWLYH ! " 2 DQG c D !1 WKHQ X LVLQ V EXW E ([DPSOH ,I X D 2 c X LVQRWLQ V ! " :5 DQG c D 4 WKHQ X LVLQ H EXW c X LV ([DPSOH ,I X D :5 QRWLQ H
1R WKHVHWLVQRWFORVHGXQGHUPXOWLSOLFDWLRQE\VFDODUVWKDW DUHQRWLQWHJHUV 2 3 !2 H D 6SDQ fYg ZKHUH Y D 4 5 5 %\7KHRUHP H LVD 3 VXEVSDFHRI R3 2 3 2 3 2 3 W D 6SDQ fX; Yg ZKHUH X D 4 !1 5 Y D 4 0 5 %\ 0 2 7KHRUHP W LVDVXEVSDFHRI R3 D 7KHUHDUHRQO\WKUHHYHFWRUVLQ fY1 ; Y2 ; Y3 g DQG Z LVQRW RQHRIWKHP E 7KHUHDUHLQÀQLWHO\PDQ\YHFWRUVLQ 6SDQ fY1 ; Y2 ; Y3 g F Z LVLQ 6SDQ fY1 ; Y2 ; Y3 g EHFDXVH Z D Y1 C Y2

W LVQRWDYHFWRUVSDFHEHFDXVHWKH]HURYHFWRULVQRWLQ W 82 3 2 3 2 39 2 !1 0 > ˆ > ˆ <6 6 3 7 6 !1 7= 07 6 7 6 7 6 7 ; ; S D 4 !1 5 4 0 5 4 3 5> ˆ > ˆ ; : 0 3 0 +LQW 8VH7KHRUHP

"

:DUQLQJ $OWKRXJKWKH 6WXG\*XLGH KDVFRPSOHWHVROXWLRQVIRU HYHU\ RGGQXPEHUHGH[HUFLVHZKRVHDQVZHUKHUHLVRQO\D ´+LQWµ \RX PXVW UHDOO\WU\WRZRUNWKHVROXWLRQ\RXUVHOI 2WKHUZLVH \RXZLOOQRWEHQHÀWIURPWKHH[HUFLVH
D

E

X C .!1/X D 1X C .!1/X

D Œ1 C .!1/!X

F

G

$[LRP $[LRP

D 0X D ([HUFLVH )URP([HUFLVH LWIROORZVWKDW .!1/X D !X $Q\VXEVSDFH H WKDWFRQWDLQV X DQG Y PXVWDOVRFRQWDLQDOO VFDODUPXOWLSOHVRI X DQG Y DQGKHQFHPXVWFRQWDLQDOOVXPV RIVFDODUPXOWLSOHVRI X DQG Y 7KXV H PXVWFRQWDLQ 6SDQ fX; Yg +LQW )RUSDUWRIWKHVROXWLRQ FRQVLGHU Z1 DQG Z2 LQ H C K DQGZULWH Z1 DQG Z2 LQWKHIRUP Z1 D X1 C Y1 DQG Z2 D X2 C Y2 ZKHUH X1 DQG X2 DUHLQ H DQG Y1 DQG Y2 DUH LQ K >0@ 7KHUHGXFHGHFKHORQIRUPRI ŒY1 ; Y2 ; Y3 ; Z! VKRZV WKDW Z D Y1 ! 2Y2 C Y3 +HQFH Z LVLQWKHVXEVSDFH VSDQQHGE\ Y1 Y2 DQG Y3 : >0@ 7KHIXQFWLRQVDUH FRV 4t DQG FRV 6t 6HH([HUFLVHLQ 6HFWLRQ

6HFWLRQ SDJH 2

32 3 2 3 3 !5 !3 1 0 0 54 3 5 D 4 0 5 4 6 !2 !8 4 1 !4 0 VR Z LVLQ 1XO A 2 3 2 3 2 3 2 3 4 !2 2 !4 617 6 07 6 7 6 7 6 !3 7 6 2 7 6 0 7 6 5 7 7 6 7 6 6 7 6 7 4 15 4 05 405 4 15 0 1 0 0

W LVQRWDVXEVSDFHRI R3 EHFDXVHWKH]HURYHFWRU .0; 0; 0/ LVQRWLQ W W LVDVXEVSDFHRI R4 EHFDXVH W LVWKHVHWRIVROXWLRQVRI WKHV\VWHP

p ! 3q ! 4s

2p

D0

! s ! 5r D 0

W LVQRWDVXEVSDFHEHFDXVH LVQRWLQ W -XVWLÀFDWLRQ ,I DW\SLFDOHOHPHQW .s ! 2t; 3 C 3s; 3s C t; 2s/ ZHUH]HUR WKHQ 3 C 3s D 0 DQG 2s D 0 ZKLFKLVLPSRVVLEOH

"


3 1 !6 1 5 VR W LVDYHFWRUVSDFHE\ W D &RO A IRU A D 4 0 1 0 7KHRUHP 2 3 0 2 1 6 1 !1 27 7 6 43 1 05 2 !1 !1

D

E

D

E

2

3

6 ! " 6 !3 7 2 6 7 LVLQ 7KHYHFWRU LVLQ 1XO A DQGWKHYHFWRU 4 3 !9 5 9 &RO A 2WKHUDQVZHUVDUHSRVVLEOH Z LVLQERWK 1XO A DQG &RO A 6HHWKH 6WXG\*XLGH %\QRZ\RXVKRXOGNQRZKRZWRXVH LWSURSHUO\ 2 3 2 3 3 1 !3 !3 4 2 5 7KHQ [ LVLQ /HW [ D 4 2 5 DQG A D 4 !2 !1 !1 5 7 1XO A 6LQFH 1XO A LVDVXEVSDFHRI R3 10[ LVLQ 1XO A D A D VRWKH]HURYHFWRULVLQ &RO A E %\DSURSHUW\RIPDWUL[PXOWLSOLFDWLRQ A[ C AZ D A.[ C Z/ ZKLFKVKRZVWKDW A[ C AZ LVD OLQHDUFRPELQDWLRQRIWKHFROXPQVRI A DQGKHQFHLVLQ &RO A F c.A[/ D A.c [/ ZKLFKVKRZVWKDW c.A[/ LVLQ &RO A IRU DOOVFDODUV c D )RUDUELWUDU\SRO\QRPLDOV S T LQ P2 DQGDQ\VFDODU c ! " ! " .S C T/.0/ S.0/ C T.0/ T .S C T / D D S.1/ C T.1/ .S C T/.1/ ! " ! " S.0/ T.0/ D C D T .S / C T . T / S.1/ T.1/ ! " ! " c S.0/ S.0/ Dc D cT .S/ T .c S/ D S.1/ c S.1/ 6R T LVDOLQHDUWUDQVIRUPDWLRQIURP P2 LQWR P2 E $Q\TXDGUDWLFSRO\QRPLDOWKDWYDQLVKHVDWDQGPXVW EHDPXOWLSOHRI S.t/ D t .t ! 1/ 7KHUDQJHRI T LV R2 D )RU A B LQ M2"2 DQGDQ\VFDODU c

T .A C B/ D .A C B/ C .A C B/T D A C B C AT C B T 7UDQVSRVHSURSHUW\ D .A C AT / C .B C B T / D T .A/ C T .B/ T .cA/ D .cA/ C .cA/T D cA C cAT D c.A C AT / D cT .A/ 6R T LVDOLQHDUWUDQVIRUPDWLRQIURP M2"2 LQWR M2"2 E ,I B LVDQ\HOHPHQWLQ M2"2 ZLWKWKHSURSHUW\WKDW B T D B DQGLI A D 12 B WKHQ ' (T T .A/ D 12 B C 12 B D 12 B C 12 B D B

F 3DUWE VKRZHGWKDWWKHUDQJHRI T FRQWDLQVDOO B VXFK WKDW B T D B 6RLWVXIÀFHVWRVKRZWKDWDQ\ B LQWKH UDQJHRI T KDVWKLVSURSHUW\ ,I B D T .A/ WKHQE\ SURSHUWLHVRIWUDQVSRVHV

B T D .A C AT /T D AT C AT T D AT C A D B %! " & 0 b G 7KHNHUQHORI T LV W b UHDO !b 0

+LQW &KHFNWKHWKUHHFRQGLWLRQVIRUDVXEVSDFH 7\SLFDO HOHPHQWVRI T .U / KDYHWKHIRUP T .X1 / DQG T .X2 / ZKHUH X1 X2 DUHLQ U >0@ Z LVLQ &RO A EXWQRWLQ 1XO A ([SODLQZK\ >0@ 7KHUHGXFHGHFKHORQIRUPRI A LV 2 3 1 0 1=3 0 10=3 60 1 1=3 0 !26=3 7 6 7 40 0 0 1 !4 5 0 0 0 0 0

6HFWLRQ SDJH

2

3 1 1 1 1 1 5 KDVWKUHHSLYRW 7KH 3 $ 3 PDWUL[ A D 4 0 0 0 1 SRVLWLRQV %\WKH,QYHUWLEOH0DWUL[7KHRUHP A LVLQYHUWLEOH DQGLWVFROXPQVIRUPDEDVLVIRU R3 6HH([DPSOH

7KLVVHWGRHVQRWIRUPDEDVLVIRU R3 7KHVHWLVOLQHDUO\ GHSHQGHQWDQGGRHVQRWVSDQ R3 7KLVVHWGRHVQRWIRUPDEDVLVIRU R3 7KHVHWLVOLQHDUO\ GHSHQGHQWEHFDXVHWKH]HURYHFWRULVLQWKHVHW +RZHYHU 2 3 2 3 3 !3 0 0 1 !1 0 0 4 !3 7 0 !3 5 ' 4 0 4 0 !3 5 0 0 0 5 0 0 0 5

7KHPDWUL[KDVDSLYRWLQHDFKURZDQGKHQFHLWVFROXPQV VSDQ R3

7KLVVHWGRHVQRWIRUPDEDVLVIRU R3 7KHVHWLVOLQHDUO\ LQGHSHQGHQWEHFDXVHRQHYHFWRULVQRWDPXOWLSOHRIWKH 3 RWKHU +RZHYHU 2 3 WKHYHFWRUVGRQRWVSDQ R 7KHPDWUL[ !2 6 4 3 !1 5 FDQKDYHDWPRVWWZRSLYRWVVLQFHLWKDVRQO\ 0 5 WZRFROXPQV 6RWKHUHZLOOQRWEHDSLYRWLQHDFKURZ 2 3 2 3 2 3 2 3 !2 6 !1 7 7 6 4 1 5 4 0 5 4 15 0 1 0 2 3 2 3 !6 !5 6 !5=2 7 6 !3=2 7 7 6 7 %DVLVIRU 1XO A 6 4 1 5 4 0 5 0 1 2 3 2 3 !2 4 %DVLVIRU &RO A 4 2 5 4 !6 5 !3 8

4FDUJPO fY1 ; Y2 ; Y4 ; Y5 g

>0@ fY1 ; Y2 ; Y3 ; Y5 g

/LQHDUO\LQGHSHQGHQW -XVWLI\DQVZHUVWR([HUFLVHV²

7KHWKUHHVLPSOHVWDQVZHUVDUH fY1 ; Y2 g fY1 ; Y3 g DQG fY2 ; Y3 g 2WKHUDQVZHUVDUHSRVVLEOH 6HHWKH 6WXG\*XLGH IRUKLQWV

+LQW 8VHWKH,QYHUWLEOH0DWUL[7KHRUHP 1R :K\LVWKHVHWQRWDEDVLVIRU H " fFRV !t; VLQ !t g

/HW A EHWKH n $ k PDWUL[ Œ Y1 # # # Yk ! 6LQFH A KDV IHZHUFROXPQVWKDQURZV WKHUHFDQQRWEHDSLYRWSRVLWLRQLQ HDFKURZRI A %\7KHRUHPLQ6HFWLRQ WKHFROXPQV RI A GRQRWVSDQ Rn DQGKHQFHDUHQRWDEDVLVIRU Rn

+LQW ,I fY1 ; : : : ; Yp g LVOLQHDUO\GHSHQGHQW WKHQWKHUHH[LVW c1 ; : : : ; cp QRWDOO]HUR VXFKWKDW c1 Y1 C # # # C cp Yp D 8VHWKLVHTXDWLRQ 1HLWKHUSRO\QRPLDOLVDPXOWLSOHRIWKHRWKHUSRO\QRPLDO VR fS1 ; S2 g LVDOLQHDUO\LQGHSHQGHQWVHWLQ P3 /HW fY1 ; Y3 g EHDQ\OLQHDUO\LQGHSHQGHQWVHWLQWKHYHFWRU VSDFH V DQGOHW Y2 DQG Y4 EHOLQHDUFRPELQDWLRQVRI Y1 DQG Y3 7KHQ fY1 ; Y3 g LVDEDVLVIRU6SDQfY1 ; Y2 ; Y3 ; Y4 g

>0@
6HFWLRQ SDJH

!

!

3 !7

2

"

1 2 !3 !5

"

3 !7 4 4 5 3

! " 5 1

!

3 !1 4 !1 5 3 2 3 2 4 6 5 !1

2 !1

"

"

2

7KH 6WXG\*XLGH KDVKLQWV ! " 1 D 5Y1 ! 2Y2 D 10Y1 ! 3Y2 C Y3 D !Y2 ! Y3 1 LQÀQLWHO\PDQ\DQVZHUV

+LQW %\K\SRWKHVLV WKH]HURYHFWRUKDVDXQLTXH UHSUHVHQWDWLRQDVDOLQHDUFRPELQDWLRQRIHOHPHQWVRI S ! " 9 2 4 1

+LQW 6XSSRVHWKDW ŒX!B D ŒZ!B IRUVRPH X DQG Z LQ V DQG GHQRWHWKHHQWULHVLQ ŒX!B E\ c1 ; : : : ; cn 8VHWKHGHÀQLWLRQ RI ŒX!B 2QHSRVVLEOHDSSURDFK )LUVW VKRZWKDWLI X1 ; : : : ; Xp DUH OLQHDUO\ GHSHQGHQW WKHQ ŒX1 !B ; : : : ; ŒXp !B DUHOLQHDUO\ GHSHQGHQW 6HFRQG VKRZWKDWLI ŒX1 !B ; : : : ; ŒXp !B DUH OLQHDUO\GHSHQGHQW WKHQ X1 ; : : : ; Xp DUHOLQHDUO\ GHSHQGHQW 8VHWKHWZRHTXDWLRQVGLVSOD\HGLQWKHH[HUFLVH $ VOLJKWO\ GLIIHUHQWSURRILVJLYHQLQWKH 6WXG\*XLGH

/LQHDUO\GHSHQGHQW

2

3 2 3 2 3 1 !3 !4 D 7KHFRRUGLQDWHYHFWRUV 4 !3 5 4 5 5 4 5 5 5 !7 !6 2 3 1 4 0 5 GRQRWVSDQ R3 %HFDXVHRIWKHLVRPRUSKLVP !1 EHWZHHQ R3 DQG P2 WKHFRUUHVSRQGLQJSRO\QRPLDOVGR QRWVSDQ P2 2 3 2 3 2 3 2 3 0 1 !3 2 E 7KHFRRUGLQDWHYHFWRUV 4 5 5 4 !8 5 4 4 5 4 !3 5 1 !2 2 0 VSDQ R3 %HFDXVHRIWKHLVRPRUSKLVPEHWZHHQ R3 DQG P2 WKHFRUUHVSRQGLQJSRO\QRPLDOVVSDQ P2 2 3 2 3 2 3 3 5 0 677 6 17 6 17 7 6 7 6 7 >0@ 7KHFRRUGLQDWHYHFWRUV 6 4 0 5 4 0 5 4 !2 5 0 !2 0 2 3 1 6 16 7 4 6 7 4 !6 5 DUHDOLQHDUO\GHSHQGHQWVXEVHWRI R %HFDXVHRI 2 WKHLVRPRUSKLVPEHWZHHQ R4 DQG P3 WKHFRUUHVSRQGLQJ SRO\QRPLDOVIRUPDOLQHDUO\GHSHQGHQWVXEVHWRI P3 DQG WKXVFDQQRWEHDEDVLVIRU P3 : 2 3 ! " 1:3 !5=3 >0@ Œ[!B D >0@ 4 0 5 8=3 0:8

6HFWLRQ SDJH

2 3 2 3 1 !2 4 1 5 4 1 5 GLP LV 0 3 2 3 2 3 2 3 0 0 2 6 1 7 6 !1 7 6 0 7 7 6 7 6 7 6 4 0 5 4 1 5 4 !3 5 GLP LV 1 2 0 2 3 2 3 2 3 1 !2 0 6 27 6 07 657 6 7 6 7 6 7 GLP LV 4 0 5 4 !2 5 4 2 5 !3 0 6

1REDVLV GLP LV

6HHWKH 6WXG\*XLGH +LQW
+LQW 6XSSRVH S GRHVVSDQ V DQGXVHWKH6SDQQLQJ6HW 7KHRUHP 7KLVOHDGVWRDFRQWUDGLFWLRQ ZKLFKVKRZVWKDW WKHVSDQQLQJK\SRWKHVLVLVIDOVH

"


+LQW 8VHWKHIDFWWKDWHDFK Pn LVDVXEVSDFHRI P

1R ([SODLQZK\

-XVWLI\HDFKDQVZHU D 7UXH E 7UXH

5RZ A DQG 1XO A DUHLQ Rn &RO A DQG 1XO AT DUHLQ Rm 7KHUHDUHRQO\IRXUGLVWLQFWVXEVSDFHVEHFDXVH 5RZ AT D &RO A DQG &RO AT D 5RZ A

F 7UXH

+LQW 6LQFH H LVDQRQ]HURVXEVSDFHRIDÀQLWHGLPHQVLRQDO VSDFH H LVÀQLWHGLPHQVLRQDODQGKDVDEDVLV VD\ Y1 ; : : : ; Yp )LUVWVKRZWKDW fT .Y1 /; : : : ; T .Yp /g VSDQV T .H / >0@ D 2QHEDVLVLV fY1 ; Y2 ; Y3 ; H2 ; H3 g ,QIDFW DQ\WZRRI WKHYHFWRUV H2 ; : : : ; H5 ZLOOH[WHQG fY1 ; Y2 ; Y3 g WRDEDVLVRI R5

6HFWLRQ SDJH UDQN A D 2 GLP 1XO A D 2 2 3 2 3 1 !4 %DVLVIRU &RO A 4 !1 5 4 2 5 5 !6

%DVLVIRU 5RZ A .1; 0; !1; 5/ .0; !2; 5; !6/ 2 3 2 3 1 !5 6 5=2 7 6 !3 7 7 6 7 %DVLVIRU 1XO A 6 4 1 5 4 0 5 0 1

UDQN A D 3 GLP 1XO A D 3 2 3 2 3 2 3 2 6 3 6 !2 7 6 !3 7 6 0 7 7 6 7 6 7 %DVLVIRU &RO A 6 4 4 5 4 9 5 4 3 5 !2 3 3

%DVLVIRU 5RZ A .2; 6; !6; 6; 3; 6/ .0; 3; 0; 3; 3; 0/ .0; 0; 0; 0; 3; 0/ 2 3 2 3 2 3 3 0 !3 6 0 7 6 !1 7 6 0 7 6 7 6 7 6 7 617 6 07 6 07 7 6 7 6 7 %DVLVIRU 1XO A 6 607 6 17 6 07 6 7 6 7 6 7 405 4 05 4 05 0 0 1

5HFDOOWKDW GLP &RO A D m SUHFLVHO\ZKHQ &RO A D Rm RU HTXLYDOHQWO\ ZKHQWKHHTXDWLRQ A[ D E LVFRQVLVWHQWIRUDOO E %\([HUFLVH E GLP &RO A D m SUHFLVHO\ZKHQ GLP 1XO AT D 0 RUHTXLYDOHQWO\ ZKHQWKHHTXDWLRQ AT [ D KDVRQO\WKHWULYLDOVROXWLRQ 2 3 2a 2b 2c XYT D 4 !3a !3b !3c 5 7KHFROXPQVDUHDOO 5a 5b 5c PXOWLSOHVRI X VR &RO XYT LVRQHGLPHQVLRQDO XQOHVV a D b D c D 0: +LQW /HW A D Œ X &RO A :K\"

>0@ +LQW 6HH([HUFLVHDQGWKHUHPDUNVEHIRUH([DPSOH >0@ 7KHPDWULFHV C DQG R JLYHQIRU([HUFLVH ZRUN KHUH DQG A D CR

6HFWLRQ SDJH D LL

!

6 9 !2 !4

1R ([SODLQZK\
"

!

E

0 !2

"

2

3 2 3 4 !1 0 8 1 15 E 4 2 5 D 4 !1 0 1 !2 2 ! " ! " !3 1 !2 1 C P B D BPC D !5 2 !5 3 ! " ! " 2 3 1 3 C P B D B P C D 12 0 !1 0 !2

6HHWKH 6WXG\*XLGH 2 1 3 C P B D 4 !2 !5 1 4 D E F G

B LVDEDVLVIRU V

3 0 2 5 3

2

3 5 Œ!1 C 2t!B D 4 !2 5 1

7KHFRRUGLQDWHPDSSLQJLVDOLQHDUWUDQVIRUPDWLRQ 7KHSURGXFWRIDPDWUL[DQGDYHFWRU 7KHFRRUGLQDWHYHFWRURI Y UHODWLYHWR B

D >0@

6HHWKH 6WXG\*XLGH

X3 ! ,I X ¤ WKHQ X LVDEDVLVIRU

X2

P !1

2

6 6 6 16 6 D 32 6 6 6 4

32

0 32

16 0 16

0 24 0 8

12 0 16 0 4

0 20 0 10 0 2

3 10 07 7 15 7 7 07 7 67 7 05 1

"

4FDUJPO E FRV2 t FRV3 t FRV4 t FRV5 t FRV6 t

D .1=2/Œ1 C FRV 2t ! D .1=4/Œ3 FRV t C FRV 3t! D .1=8/Œ3 C 4 FRV 2t C FRV 4t! D .1=16/Œ10 FRV t C 5 FRV 3t C FRV 5t ! D .1=32/Œ10 C 15 FRV 2t C 6 FRV 4t C FRV 6t !

>0@ +LQW /HW C EHWKHEDVLV fY1 ; Y2 ; Y3 g 7KHQWKHFROXPQV RI P DUH ŒX1 !C ; ŒX2 !C DQG ŒX3 !C 8VHWKHGHÀQLWLRQRI C FRRUGLQDWHYHFWRUVDQGPDWUL[DOJHEUDWRFRPSXWH X1 X2 X3 7KHVROXWLRQPHWKRGLVGLVFXVVHGLQWKH 6WXG\*XLGH +HUHDUHWKHQXPHULFDODQVZHUV 2 3 2 3 2 3 !6 !6 !5 D X1 D 4 !5 5 X2 D 4 !9 5 X3 D 4 0 5 21 32 3 2 3 2 3 2 3 28 38 21 E Z1 D 4 !9 5 Z2 D 4 !13 5 Z3 D 4 !7 5 !3 2 3

6HFWLRQ SDJH ,I yk D 2 WKHQ ykC1 D 2 DQG ykC2 D 2 6XEVWLWXWLQJWKHVHIRUPXODVLQWRWKHOHIWVLGHRIWKHHTXDWLRQ JLYHV k

kC1

kC2

ykC2 C 2ykC1 ! 8yk D 2kC2 C 2 # 2kC1 ! 8 # 2k D 2k .22 C 2 # 2 ! 8/ D 2k .0/ D 0

IRUDOO k

6LQFHWKHGLIIHUHQFHHTXDWLRQKROGVIRUDOO k 2k LVD VROXWLRQ $ VLPLODUFDOFXODWLRQZRUNVIRU yk D .!4/k

7KHVLJQDOV 2k DQG .!4/k DUHOLQHDUO\LQGHSHQGHQWEHFDXVH QHLWKHULVDPXOWLSOHRIWKHRWKHU )RULQVWDQFH WKHUHLVQR VFDODU c VXFKWKDW 2k D c.!4/k IRUDOON %\7KHRUHP WKHVROXWLRQVHW H RIWKHGLIIHUHQFHHTXDWLRQLQ([HUFLVH LVWZRGLPHQVLRQDO %\WKH%DVLV7KHRUHPLQ6HFWLRQ WKHWZROLQHDUO\LQGHSHQGHQWVLJQDOV 2k DQG .!4/k IRUPD EDVLVIRU H ,I yk D .!2/k WKHQ

ykC2 C 4ykC1 C 4yk D .!2/kC2 C 4.!2/kC1 C 4.!2/k D .!2/k Œ.!2/2 C 4.!2/ C 4! D .!2/k .0/ D 0

IRUDOO k

6LPLODUO\ LI yk D k.!2/k WKHQ

ykC2 C 4ykC1 C 4yk

D .k C 2/.!2/kC2 C 4.k C 1/.!2/kC1 C 4k.!2/k D .!2/k Œ.k C 2/.!2/2 C 4.k C 1/.!2/ C 4k! D .!2/k Œ4k C 8 ! 8k ! 8 C 4k!

D .!2/k .0/

LQGHSHQGHQW 6LQFH GLP H D 2 WKHVLJQDOVIRUPDEDVLVIRU H E\WKH%DVLV7KHRUHP
1R WZRVLJQDOVFDQQRWVSDQWKHWKUHHGLPHQVLRQDOVROXWLRQ VSDFH ' 1 (k ' 2 (k . 12 /k .! 23 /k 3 3

Yk D c1 .:8/k C c2 .:5/k C 10 ! 10 DV k ! 1 p p yk D c1 .!2 C 3/k C c2 .!2 ! 3/k VHHÀJXUH = original data = smoothed data

10 8 6 4 2 0

2

4

6

8

10

12

14

k

D ykC1 ! 1:01yk D !450 y0 D 10;000 k 2 C c1 # .!4/k C c2

!2 C k C c1 # 2k C c2 # .!2/k

[kC1 D A[k ZKHUH 2 3 0 1 0 0 60 0 1 07 7; AD6 40 0 0 15 2 !6 8 !3

2

3 yk 6 ykC1 7 7 [D6 4 ykC2 5 ykC3

7KHHTXDWLRQKROGVIRUDOO k VRLWKROGVZLWK k UHSODFHGE\ k ! 1 ZKLFKWUDQVIRUPVWKHHTXDWLRQLQWR

ykC2 C 5ykC1 C 6yk D 0

IRUDOO k

7KHHTXDWLRQLVRIRUGHU )RUDOO k WKH&DVRUDWLPDWUL[ C.k/ LVQRWLQYHUWLEOH ,QWKLV FDVH WKH&DVRUDWLPDWUL[JLYHVQRLQIRUPDWLRQDERXWWKH OLQHDULQGHSHQGHQFHGHSHQGHQFHRIWKHVHWRIVLJQDOV ,Q IDFW QHLWKHUVLJQDOLVDPXOWLSOHRIWKHRWKHU VRWKH\DUH OLQHDUO\LQGHSHQGHQW +LQW 9HULI\WKHWZRSURSHUWLHVWKDWGHÀQHDOLQHDU WUDQVIRUPDWLRQ )RU fyk g DQG f´k g LQ S VWXG\ T .fyk g C f´k g/ 1RWHWKDWLI r LVDQ\VFDODU WKHQWKH k WK WHUPRI rfyk g LV ryk VR T .rfyk g/ LVWKHVHTXHQFH fwk g JLYHQE\

wk D rykC2 C a.rykC1 / C b.ryk /

+LQW )LQG TD.y0 ; y1 ; y2 ; : : : / DQG DT .y0 ; y1 ; y2 ; : : : /

6HFWLRQ SDJH D

IRUDOO k

7KXVERWK .!2/ DQG k.!2/k DUHLQWKHVROXWLRQVSDFH H RI WKHGLIIHUHQFHHTXDWLRQ $OVR WKHUHLVQRVFDODU c VXFKWKDW k.!2/k D c.!2/k IRUDOO k EHFDXVH c PXVWEHFKRVHQ LQGHSHQGHQWO\RI k /LNHZLVH WKHUHLVQRVFDODU c VXFKWKDW .!2/k D ck.!2/k IRUDOO k 6RWKHWZRVLJQDOVDUHOLQHDUO\

k

D

)URP 1 0" ! :7 :6 :3 :4

)URP + ," ! :95 :45 :05 :55

E 7R 1HZV 0XVLF

! " 1 0

F

E 7R +HDOWK\ ,OO

"

"OTXFST UP 0EE/VNCFSFE &YFSDJTFT ! " 1 : 0 2 3 1=4 4 1=2 5 1=4

F XVH [0 D

!

5=14 9=14

"

>0@ $ERXWRIWKH8QLWHG6WDWHVSRSXODWLRQ D 7KHHQWULHVLQDFROXPQRI P VXPWR $ FROXPQLQWKH PDWUL[ P ! I KDVWKHVDPHHQWULHVDVLQ P H[FHSWWKDW RQHRIWKHHQWULHVLVGHFUHDVHGE\ +HQFHHDFKFROXPQ VXPLV E %\SDUWD WKHERWWRPURZRI P ! I LVWKHQHJDWLYHRI WKHVXPRIWKHRWKHUURZV F %\SDUWE DQGWKH6SDQQLQJ6HW7KHRUHP WKHERWWRP URZRI P ! I FDQEHUHPRYHGDQGWKHUHPDLQLQJ .n ! 1/ URZVZLOOVWLOOVSDQWKHURZVSDFH $OWHUQDWLYHO\ XVHSDUWD DQGWKHIDFWWKDWURZ RSHUDWLRQVGRQRWFKDQJHWKHURZVSDFH /HW A EHWKH PDWUL[REWDLQHGIURP P ! I E\DGGLQJWRWKHERWWRP URZDOOWKHRWKHUURZV %\SDUWD WKHURZVSDFHLV VSDQQHGE\WKHÀUVW .n ! 1/ URZVRI A G %\WKH5DQN7KHRUHPDQGSDUWF WKHGLPHQVLRQRIWKH FROXPQVSDFHRI P ! I LVOHVVWKDQ n DQGKHQFHWKHQXOO VSDFHLVQRQWULYLDO ,QVWHDGRIWKH5DQN7KHRUHP \RX PD\XVHWKH,QYHUWLEOH0DWUL[7KHRUHP VLQFH P ! I LV DVTXDUHPDWUL[ D 7KHSURGXFW S [ HTXDOVWKHVXPRIWKHHQWULHVLQ [ )RUD SUREDELOLW\YHFWRU WKLVVXPPXVWEH E P D Œ S1 S2 # # # Sn ! ZKHUHWKH Si DUHSUREDELOLW\ YHFWRUV %\PDWUL[PXOWLSOLFDWLRQDQGSDUWD # $ # $ SP D S S1 S S2 # # # S Sn D 1 1 # # # 1 D S F %\SDUWE S.P [/ D .SP /[ D S [ D 1 $OVR WKH HQWULHVLQ P [ DUHQRQQHJDWLYHEHFDXVH P DQG [ KDYH QRQQHJDWLYHHQWULHV +HQFH E\D P [ LVDSUREDELOLW\ YHFWRU

&KDSWHU6XSSOHPHQWDU\([HUFLVHV SDJH D J P V

7 ) 7 7

E K Q W

7 ) ) )

F L R

) 7 7

G M S

) ) 7

H N T

7 ) )

I O U

7 ) 7

7KHVHWRIDOO .b1 ; b2 ; b3 / VDWLVI\LQJ b1 C 2b2 C b3 D 0

7KHYHFWRU S1 LVQRW]HURDQG S2 LVQRWDPXOWLSOHRI S1 VR NHHSERWKRIWKHVHYHFWRUV 6LQFH S3 D 2S1 C 2S2 GLVFDUG S3 6LQFH S4 KDVD t 2 WHUP LWFDQQRWEHDOLQHDUFRPELQDWLRQ RI S1 DQG S2 VRNHHS S4 )LQDOO\ S5 D S1 C S4 VRGLVFDUG S5 7KHUHVXOWLQJEDVLVLV fS1 ; S2 ; S4 g

/HW A EHWKHVWDQGDUG m $ n PDWUL[RIWKHWUDQVIRUPDWLRQ T D ,I T LVRQHWRRQH WKHQWKHFROXPQVRI A DUHOLQHDUO\ LQGHSHQGHQW7KHRUHPLQ6HFWLRQ VR GLP 1XO A D 0 %\WKH5DQN7KHRUHP GLP &RO A D UDQN A D n 6LQFHWKHUDQJHRI T LV &RO A WKHGLPHQVLRQRIWKHUDQJHRI T LV n E ,I T LVRQWR WKHQWKHFROXPQVRI A VSDQ Rm 7KHRUHP LQ6HFWLRQ VR GLP &RO A D m %\WKH5DQN 7KHRUHP GLP 1XO A D n ! GLP &RO A D n ! m 6LQFH WKHNHUQHORI T LV 1XO A WKHGLPHQVLRQRIWKHNHUQHORI T LV n ! m ,I S LVDÀQLWHVSDQQLQJVHWIRU V WKHQDVXEVHWRI S ³VD\ S 0 ³LVDEDVLVIRU V 6LQFH S 0 PXVWVSDQ V S 0 FDQQRWEHD SURSHUVXEVHWRI S EHFDXVHRIWKHPLQLPDOLW\RI S 7KXV S 0 D S ZKLFKSURYHVWKDW S LVDEDVLVIRU V

D +LQW $Q\ \ LQ &RO AB KDVWKHIRUP \ D AB [ IRUVRPH [ %\([HUFLVH UDQN PA ( UDQN A DQG UDQN A D UDQN P !1 PA ( UDQN PA 7KXV UDQN PA D UDQN A

7KHHTXDWLRQ AB D 0 VKRZVWKDWHDFKFROXPQRI B LVLQ 1XO A 6LQFH 1XO A LVDVXEVSDFH DOOOLQHDUFRPELQDWLRQVRI WKHFROXPQVRI B DUHLQ 1XO A VR &RO B LVDVXEVSDFHRI 1XO A %\7KHRUHPLQ6HFWLRQ GLP &RO B ( GLP 1XO A $SSO\LQJWKH5DQN7KHRUHP ZH ÀQGWKDW

n D UDQN A C GLP 1XO A & UDQN A C UDQN B D /HW A1 FRQVLVWRIWKH r SLYRWFROXPQVLQ A 7KH FROXPQVRI A1 DUHOLQHDUO\LQGHSHQGHQW 6R A1 LVDQ m $ r VXEPDWUL[ZLWK UDQN r E %\WKH5DQN7KHRUHPDSSOLHGWR A1 WKHGLPHQVLRQRI 5RZ A LV r VR A1 KDV r OLQHDUO\LQGHSHQGHQWURZV 8VH WKHPWRIRUP A2 7KHQ A2 LV r $ r ZLWKOLQHDUO\ LQGHSHQGHQWURZV %\WKH,QYHUWLEOH0DWUL[7KHRUHP A2 LVLQYHUWLEOH 2 3 0 1 0 # $ 2 B AB A B D 4 1 !:9 :81 5 1 :5 :25 2 3 1 !:9 :81 1 05 ' 40 0 0 !:56 7KLVPDWUL[KDV UDQN 3 VRWKHSDLU .A; B/ LVFRQWUROODEOH # $ >0@ UDQN B AB A2 B A3 B D 3 7KHSDLU .A; B/ LV QRWFRQWUROODEOH

4FDUJPO

$IBQUFS 6HFWLRQ SDJH

2

3 1
2 3 0 617 6 7 7 >0@ # D !4 6 6 2 7 # D !8 405 1 2 3 2 3 0 2 6 07 617 6 7 6 7 7 6 7 # D 12 6 6 !1 7 6 2 7 4 15 405 0 1

"

2 3 2 3 6 0 637 607 6 7 6 7 6 3 7 6 !1 7 6 7 6 7 425 405 0 1

6HFWLRQ SDJH #2 ! 4# ! 45 !5

#2 ! 16# C 48 4 12

#2 ! 3# ! 40 !5

#2 ! 9# C 32 QRUHDOHLJHQYDOXHV !#3 C 10#2 ! 33# C 36

!#3 C 18#2 ! 95# C 150

!#3 C 8#2 ! 19# C 12

-XVWLI\\RXUDQVZHU

+LQW 7KHHTXDWLRQJLYHQKROGVIRUDOO #

6HHWKH 6WXG\*XLGH DIWHU\RXKDYHZULWWHQ\RXUDQVZHUV

7KH 6WXG\*XLGH KDVKLQWV

+LQW 8VH7KHRUHP

+LQW )LQGDQLQYHUWLEOHPDWUL[ P VXFKWKDW RQ D P !1 AP ! " !1 D fY1 ; Y2 g ZKHUH Y2 D LVDQHLJHQYHFWRUIRU # D :3 1

+LQW 8VHWKHHTXDWLRQ A[ D #[ WRÀQGDQHTXDWLRQ LQYROYLQJ A!1 +LQW )RUDQ\ # .A ! #I /T D AT ! #I %\DWKHRUHP ZKLFKRQH" AT ! #I LVLQYHUWLEOHLIDQGRQO\LI A ! #I LV LQYHUWLEOH /HW Y EHWKHYHFWRULQ Rn ZKRVHHQWULHVDUHDOO 1·V 7KHQ AY D s Y +LQW ,I A LVWKHVWDQGDUGPDWUL[RI T ORRNIRUDQRQ]HUR YHFWRU Y DSRLQWLQWKHSODQH VXFKWKDW AY D Y D [kC1 D c1 #kC1 X C c2 $kC1 Y E A[k D A.c1 #k X C c2 $k Y/ D c1 #k AX C c2 $k AY D c1 #k #X C c2 $k $Y D [kC1

x2

/LQHDULW\ X DQG Y DUHHLJHQYHFWRUV

[0 D c1 Y1 C c2 Y2 C c3 Y3

6LQFH [0 DQG Y1 DUHSUREDELOLW\YHFWRUVDQGVLQFHWKH HQWULHVLQ Y2 DQGLQ Y3 HDFKVXPWR " VKRZVWKDW 1 D c1 F %\SDUWE

w T(u)

2

D AY1 D Y1 AY2 D :5Y2 AY3 D :2Y3 7KLVDOVRVKRZV WKDWWKHHLJHQYDOXHVRI A DUH DQG E fY1 ; Y2 ; Y3 g LVOLQHDUO\LQGHSHQGHQWEHFDXVHWKH HLJHQYHFWRUVFRUUHVSRQGWRGLVWLQFWHLJHQYDOXHV 7KHRUHP 6LQFHWKHUHDUHYHFWRUVLQWKHVHW WKHVHW LVDEDVLVIRU R3 6RWKHUHH[LVWXQLTXH FRQVWDQWVVXFK WKDW

Z[0 D c1 ZTY1 C c2 ZTY2 C c3 ZTY3

T(v)

u

1 Y 14 2 1 .:3/Y2 ; [2 D Y1 ! 141 .:3/2 Y2 DQG 14 1 .:3/k Y2 $V k ! 1 .:3/k ! 0 DQG 14

7KHQ

T(w)

v

E [0 D Y1 ! F [1 D Y1 ! [k D Y1 ! [k ! Y1

3

!1 >0@ # D 5 4 !1 5 2

[0 D Y1 C c2 Y2 C c3 Y3

x1

2

3

!3 # D 10 4 2 5 1

2 3 2 # D 15 4 2 5 1

8VLQJSDUWD [k D Ak [0 D Ak Y1 C c2 Ak Y2 C c3 Ak Y3 D Y1 C c2 .:5/k Y2 C c3 .:2/k Y3 ! Y1 DV k ! 1

."/

"


>0@ 5HSRUW\RXUUHVXOWVDQGFRQFOXVLRQV
6HFWLRQ SDJH !

" 226 !525 90 !209 2 3 !1 # D 2 4 1 5 !1

!

ak 0 bk 2.ak ! b k / 2 3 2 3 1 0 # D 3 4 !1 5 4 !1 5 0 1

"

:KHQDQDQVZHULQYROYHVDGLDJRQDOL]DWLRQ A D 3'3!1 WKH IDFWRUV P DQG D DUHQRWXQLTXH VR\RXUDQVZHUPD\GLIIHUIURP WKDWJLYHQKHUH ! " ! " 1 0 1 0 P D DD 3 1 0 !1 1RWGLDJRQDOL]DEOH 2 3 2 3 1 !1 !1 5 0 0 1 0 5 D D 4 0 !1 05 P D 4 2 3 0 1 0 0 !1 2 3 2 3 !1 2 1 5 0 0 0 5 D D 4 0 1 05 P D 4 !1 !1 1 0 1 0 0 1 2 3 2 3 !1 !1 !1 0 0 0 1 0 5 D D 4 0 1 05 P D 4 1 !1 0 1 0 0 1 1RWGLDJRQDOL]DEOH 2 3 2 1 3 !1 !1 5 60 60 2 !1 27 6 7 6 P D 4 DD4 0 0 1 05 0 0 0 0 1 0 6HHWKH 6WXG\*XLGH

0 3 0 0

2

1 !3 0 0 1 0 1 0 0 2

2 61 6 >0@ P D 6 60 40 0 2 7 0 60 7 6 0 0 DD6 6 40 0 0 0

0 0 0 0 7 0 0 !14 0 0

3 1 0 1 07 7 0 !1 7 7 1 05 0 1 3 0 07 7 07 7 05 !14

6HFWLRQ SDJH

!

3 !1 !5 6

0 4

"

D T .H1 / D E3 T .H2 / D !E1 ! 2E2 T .H3 / D 2E1 C 3E3 2 3 2 3 0 !1 E Œ T .H1 / !B D 4 0 5 Œ T .H2 / !B D 4 !2 5 1 0 2 3 2 Œ T . H 3 / !B D 4 0 5 3 2 3 0 !1 2 05 F 4 0 !2 1 0 3 D 9 ! 3t C t 2 C t 3 E )RUDQ\ S T LQ P2 DQGDQ\VFDODU c

0 0 2 0

T ŒS.t/ C T.t/! D .t C 3/ŒS.t/ C T.t/! D .t C 3/S.t/ C .t C 3/T.t/ D T ŒS.t/! C T ŒT.t/! T Œc # S.t/! D .t C 3/Œc # S.t/! D c # .t C 3/S.t/ D c # T ŒS.t/!

3 0 07 7 05 2

2

1R A PXVWEHGLDJRQDOL]DEOH ([SODLQZK\ +LQW :ULWH A D 3'3!1 6LQFH A LVLQYHUWLEOH LVQRWDQ HLJHQYDOXHRI A VR D KDVQRQ]HURHQWULHVRQLWVGLDJRQDO ! " 1 1 ZKRVHFROXPQVDUH 2QHDQVZHULV P1 D !2 !1 HLJHQYHFWRUVFRUUHVSRQGLQJWRWKHHLJHQYDOXHVLQ D1 +LQW &RQVWUXFWDVXLWDEOH 2 $ 2 WULDQJXODUPDWUL[ 2 3 2 0 !1 1 6 7 !1 0 !4 7 7 >0@ P D 6 47 0 2 05 0 1 0 3 2 3 !12 0 0 0 6 0 !12 0 07 7 DD6 4 0 0 13 05 0 0 0 13

3 3 0 0 61 3 07 7 F 6 40 1 35 0 0 1 2 3 3 0 0 4 5 !2 05 0 4 1 2 3 2 D 4 5 5 8 E +LQW &RPSXWH T .S C T/ DQG T .c # S/ IRUDUELWUDU\ S T LQ P2 DQGDQDUELWUDU\VFDODU c 2 3 1 !1 1 0 05 F 4 1 1 1 1 ! " ! " ! " !2 !2 1 1 E1 D E2 D 0 !1 1 3 ! " ! " 2 !1 E1 D E2 D 1 3

4FDUJPO D AE1 D 3E1 VR E1 LVDQHLJHQYHFWRURI A +RZHYHU A KDVRQO\RQHHLJHQYDOXH # D 3 DQGWKHHLJHQVSDFHLV RQO\RQHGLPHQVLRQDO VR A LVQRWGLDJRQDOL]DEOH ! " 3 1 E 0 3

%\GHÀQLWLRQ LI A LVVLPLODUWR B WKHUHH[LVWVDQLQYHUWLEOH PDWUL[ P VXFKWKDW P !1 AP D B 6HH6HFWLRQ 7KHQ B LVLQYHUWLEOHEHFDXVHLWLVWKHSURGXFWRILQYHUWLEOH PDWULFHV 7RVKRZWKDW A!1 LVVLPLODUWR B !1 XVHWKH HTXDWLRQ P !1 AP D B 6HHWKH 6WXG\*XLGH +LQW 5HYLHZ3UDFWLFH3UREOHP +LQW &RPSXWH B.P !1 [/ +LQW :ULWH A D 3%3!1 D .PB/P !1 DQGXVHWKHWUDFH SURSHUW\ )RUHDFK j I.Ej / D Ej 6LQFHWKHVWDQGDUGFRRUGLQDWH YHFWRURIDQ\YHFWRULQ Rn LVMXVWWKHYHFWRULWVHOI ŒI.Ej /!E D Ej 7KXVWKHPDWUL[IRU I UHODWLYHWR B DQGWKH VWDQGDUGEDVLV E LVVLPSO\ Œ E1 E2 # # # En ! 7KLVPDWUL[ LVSUHFLVHO\WKH FKDQJHRIFRRUGLQDWHV PDWUL[ PB GHÀQHGLQ 6HFWLRQ 7KH BPDWUL[IRUWKHLGHQWLW\WUDQVIRUPDWLRQLV In EHFDXVH WKH BFRRUGLQDWHYHFWRURIWKH j WKEDVLVYHFWRU Ej LVWKH j WK FROXPQRI In 2 3 !7 !2 !6 >0@ 4 0 !4 !6 5 0 0 !1

6HFWLRQ SDJH !

"

!

"

!1 C i !1 ! i # D 2 ! i 1 1 ! " ! " !1 ! i !1 C i # D 3 C 2i # D 3 ! 2i 4 4 ! " ! " 1!i 1Ci # D 4 C i # D 4 ! i 2 2 p # D 3 ˙ i ' D "=6 UDGLDQ r D 2 # D 2 C i

# D ˙2i ' D !"=2 UDGLDQV r D 2 p # D ! 3 ˙ i ' D !5"=6 UDGLDQ r D 2

,Q([HUFLVHV² RWKHUDQVZHUVDUHSRVVLEOH $Q\ P WKDW PDNHV P !1 AP HTXDOWRWKHJLYHQ C RUWR C T LVDVDWLVIDFWRU\ DQVZHU )LUVWÀQG P WKHQFRPSXWH P !1 AP ! " ! " !1 !1 2 !1 P D C D 1 0 1 2 ! " ! " 3 1 1 3 P D C D 0 2 !3 1 ! " ! " 1 2 !3 4 P D C D 0 !5 !4 !3 ! " ! " 2 !1 :96 !:28 P D C D 2 0 :28 :96

\ D

!

2 !1 C 2i

"

D

!1 C 2i 5

!

!2 ! 4i 5

"

"

D 3URSHUWLHVRIFRQMXJDWHVDQGWKHIDFWWKDW [T D [T E A[ D A[ DQG A LVUHDO F EHFDXVH [T A[ LVDVFDODUDQG KHQFHPD\EHYLHZHGDVD 1 $ 1 PDWUL[ G SURSHUWLHV RIWUDQVSRVHV H AT D A GHÀQLWLRQRI q +LQW )LUVWZULWH [ D 5H [ C i.,P [/ 2 3 !1 1 !1 !1 6 0 !1 0 27 7 >0@ P D 6 4 1 0 0 !2 5 0 0 2 0 2 3 !2 5 0 0 6 !5 !2 0 07 7 C D6 4 0 0 !4 10 5 0 0 !10 !4 2WKHUFKRLFHVDUHSRVVLEOH EXW C PXVWHTXDO P !1 AP

6HFWLRQ SDJH D +LQW )LQG c1 c2 VXFKWKDW [0 D c1 Y1 C c2 Y2 8VHWKLV UHSUHVHQWDWLRQDQGWKHIDFWWKDW Y1 DQG ! Y2 DUH" 49=3 HLJHQYHFWRUVRI A WRFRPSXWH [1 D 41=3 E ,QJHQHUDO [k D 5.3/k Y1 ! 4. 13 /k Y2

IRU k & 0

:KHQ p D :2 WKHHLJHQYDOXHVRI A DUHDQG DQG ! " ! " 1 2 [k D c1 .:9/k C c2 .:7/k ! DV k ! 1 1 1 7KHKLJKHUSUHGDWLRQUDWHFXWVGRZQWKHRZOV·IRRGVXSSO\ DQGHYHQWXDOO\ERWKSUHGDWRUDQGSUH\SRSXODWLRQVSHULVK ,I p D :325 WKHHLJHQYDOXHVDUHDQG 6LQFH 1:05 > 1 ERWKSRSXODWLRQVZLOOJURZDWSHU\HDU $Q HLJHQYHFWRUIRULV .6; 13/ VRHYHQWXDOO\WKHUHZLOOEH DSSUR[LPDWHO\VSRWWHGRZOVWRHYHU\WKRXVDQG Á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

"


3 2 3 2 !1 [k D Y1 C :1.:5/k 4 !3 5 C :3.:2/k 4 0 5 ! Y1 DV 1 1 k!1 ! " 0 1:6 D A D :3 :8 E 7KHSRSXODWLRQLVJURZLQJEHFDXVHWKHODUJHVW HLJHQYDOXHRI A LV ZKLFKLVODUJHUWKDQLQ PDJQLWXGH 7KHHYHQWXDOJURZWKUDWHLV ZKLFKLV SHU\HDU 7KHHLJHQYHFWRU .4; 3/ IRU #1 D 1:2 VKRZVWKDWWKHUHZLOOEHMXYHQLOHVIRUHYHU\DGXOWV F >0@ 7KHMXYHQLOH²DGXOWUDWLRVHHPVWRVWDELOL]HDIWHU DERXWRU\HDUV 7KH 6WXG\*XLGH GHVFULEHVKRZWR FRQVWUXFWDPDWUL[SURJUDPWRJHQHUDWHDGDWDPDWUL[ ZKRVHFROXPQVOLVWWKHQXPEHUVRIMXYHQLOHVDQGDGXOWV HDFK\HDU *UDSKLQJWKHGDWDLVDOVRGLVFXVVHG

6HFWLRQ SDJH

! " ! " 5 !3 4t 3 !1 2t e ! e 1 1 2 2 ! " ! " 5 !3 t 9 !1 !t ! e C e 7KHRULJLQLVDVDGGOHSRLQW 1 1 2 2 7KHGLUHFWLRQRIJUHDWHVWDWWUDFWLRQLVWKHOLQHWKURXJK .!1; 1/ DQGWKHRULJLQ 7KHGLUHFWLRQRIJUHDWHVWUHSXOVLRQLV WKHOLQHWKURXJK .!3; 1/ DQGWKHRULJLQ ! " ! " 1 1 4t 7 1 6t ! e C e 7KHRULJLQLVDUHSHOOHU 7KH 2 3 2 1 GLUHFWLRQRIJUHDWHVWUHSXOVLRQLVWKHOLQHWKURXJK .1; 1/ DQG WKHRULJLQ ! " ! " 1 1 4 0 6HW P D DQG D D 7KHQ 3 1 0 6 A D PDP !1 6XEVWLWXWLQJ [ D P \ LQWR [0 D A[ ZHKDYH

[.t/ D

d .P \/ D A.P \/ dt P \0 D PDP !1 .P \/ D PD \

/HIWPXOWLSO\LQJE\ P !1 JLYHV " # " y10 .t / 4 0 \ D D \; RU D 0 y2 .t / 0

0 6

#"

y1 .t / y2 .t /

#

FRPSOH[VROXWLRQ ! " ! " 1!i 1Ci c1 e .!2Ci /t C c2 e .!2!i/t 1 1 UHDOVROXWLRQ ! " ! " FRV t C VLQ t VLQ t ! FRV t c1 e !2t C c2 e !2t FRV t VLQ t 7KHWUDMHFWRULHVVSLUDOLQWRZDUGWKHRULJLQ ! " ! " !3 C 3i !3 ! 3i FRPSOH[ c1 e 3it C c2 e !3it 2 2 UHDO ! " ! " !3 FRV 3t ! 3 VLQ 3t !3 VLQ 3t C 3 FRV 3t c1 C c2 2 FRV 3t 2 VLQ 3t 7KHWUDMHFWRULHVDUHHOOLSVHVDERXWWKHRULJLQ

! " ! " 1Ci 1!i FRPSOH[ c1 e .1C3i/t C c2 e .1!3i/t 2 2 ! " ! " FRV 3t ! VLQ 3t VLQ 3t C FRV 3t UHDO c1 e t C c2 et 2 FRV 3t 2 VLQ 3t 7KHWUDMHFWRULHVVSLUDORXW DZD\IURPWKHRULJLQ 2 3 2 3 2 3 !1 !6 !4 !2t !t C c2 4 1 5e C c3 4 1 5e t >0@ [.t/ D c1 4 0 5e 1 5 4 7KHRULJLQLVDVDGGOHSRLQW $ VROXWLRQZLWK c3 D 0 LV DWWUDFWHGWRWKHRULJLQ $ VROXWLRQZLWK c1 D c2 D 0 LV UHSHOOHG

>0@ FRPSOH[ 2 3 2 3 !3 23 ! 34i t c1 4 1 5e C c2 4 !9 C 14i 5 e .5C2i /t C 1 3 2 3 23 C 34i c3 4 !9 ! 14i 5 e .5!2i /t 3 2 3 2 3 !3 23 FRV 2t C 34 VLQ 2t UHDO c1 4 1 5e t C c2 4 !9 FRV 2t ! 14 VLQ 2t 5 e 5t C 1 3 FRV 2t 2 3 23 VLQ 2t ! 34 FRV 2t c3 4 !9 VLQ 2t C 14 FRV 2t 5 e 5t 3 VLQ 2t 7KHRULJLQLVDUHSHOOHU 7KHWUDMHFWRULHVVSLUDORXWZDUG DZD\IURPWKHRULJLQ ! " !2 3=4 >0@ A D 1 !1 ! " ! " ! " 5 1 !:5t 1 !3 !2:5t v1 .t/ D e ! e v2 .t/ 2 2 2 2 ! " !1 !8 >0@ A D 5 !5 ! " ! " iL .t/ !20 VLQ 6t D e !3t vC .t/ 15 FRV 6t ! 5 VLQ 6t

6HFWLRQ SDJH (LJHQYHFWRU [4 D

# % 4:9978

(LJHQYHFWRU [4 D

! !

" ! " 1 4:9978 RU A[4 D :3326 1:6652

" ! " :5188 :4594 RU A[4 D 1 :9075

# % :9075 ! " ! " !:7999 4:0015 [ D A[ D 1 !5:0020 HVWLPDWHG # D !5:0020

>0@ ! " ! " ! " ! " ! " :75 1 :9932 1 :9998 ; ; ; ; [k W 1 :9565 1 :9990 1

$k W 11:5;

12:78;

12:96;

12:9948; 12:9990

>0@ $5 D 8:4233 $6 D 8:4246 DFWXDOYDOXH DFFXUDWHWRSODFHV

$IBQUFS 4VQQMFNFOUBSZ &YFSDJTFT "

$k W 5:8000; 5:9655; 5:9942; 5:9990 .k D 1; 2; 3; 4/I R.[k /W 5:9655; 5:9990; 5:99997; 5:9999993

>0@ %0 D 3:3384 %1 D 3:32119 DFFXUDWHWRSODFHVZLWK URXQGLQJ %2 D 3:3212209 $FWXDOYDOXH DFFXUDWHWRSODFHV >0@ D $6 D 30:2887 D $7 WRIRXUGHFLPDOSODFHV 7RVL[ SODFHV WKHODUJHVWHLJHQYDOXHLV ZLWK HLJHQYHFWRU .:957629; :688937; 1; :943782/ E 7KHLQYHUVHSRZHUPHWKRGZLWK ˛ D 0/ SURGXFHV !1 $!1 1 D :010141 $2 D :010150 7RVHYHQSODFHV WKH VPDOOHVWHLJHQYDOXHLV ZLWKHLJHQYHFWRU .!:603972; 1; !:251135; :148953/ 7KHUHDVRQIRUWKH UDSLGFRQYHUJHQFHLVWKDWWKHQH[WWRVPDOOHVW HLJHQYDOXHLVQHDU D ,IWKHHLJHQYDOXHVRI A DUHDOOOHVVWKDQLQPDJQLWXGH DQGLI [ ¤ WKHQ Ak [ LVDSSUR[LPDWHO\DQHLJHQYHFWRU IRUODUJH k E ,IWKHVWULFWO\GRPLQDQWHLJHQYDOXHLV DQGLI [ KDVD FRPSRQHQWLQWKHGLUHFWLRQRIWKHFRUUHVSRQGLQJ HLJHQYHFWRU WKHQ fAk [g ZLOOFRQYHUJHWRDPXOWLSOHRI WKDWHLJHQYHFWRU F ,IWKHHLJHQYDOXHVRI A DUHDOOJUHDWHUWKDQLQ PDJQLWXGH DQGLI [ LVQRWDQHLJHQYHFWRU WKHQWKH GLVWDQFHIURP Ak [ WRWKHQHDUHVWHLJHQYHFWRUZLOO LQFUHDVH DV k ! 1


7 ) ) )

E K Q W

) 7 7 7

F L R X

7 ) ) 7

G M S Y

) 7 7 7

H N T Z

7 ) ) )

I O U [

7 ) 7 7

D 6XSSRVH A[ D #[ ZLWK [ ¤ 7KHQ

.5I ! A/[ D 5[ ! A[ D 5[ ! #[ D .5 ! #/[:

7KHHLJHQYDOXHLV 5 ! # E .5I ! 3A C A2 /[ D 5[ ! 3A[ C A.A[/ D 5[ ! 3#[ C #2 [ D .5 ! 3# C #2 /[: 7KHHLJHQYDOXHLV 5 ! 3# C #2

6XSSRVH A[ D #[ ZLWK [ ¤ 7KHQ

p.A/[ D .c0 I C c1 A C c2 A2 C # # # C cn An /[ D c0 [ C c1 A[ C c2 A2 [ C # # # C cn An [ D c0 [ C c1 #[ C c2 #2 [ C # # # C cn #n [ D p.#/[ 6R p.#/ LVDQHLJHQYDOXHRIWKHPDWUL[ p.A/

,I A D PDP !1 WKHQ p.A/ D Pp.D/P !1 DVVKRZQLQ ([HUFLVH ,IWKH .j; j / HQWU\LQ D LV # WKHQWKH .j; j / HQWU\LQ D k LV #k DQGVRWKH .j; j / HQWU\LQ p.D/ LV p.#/ ,I p LVWKHFKDUDFWHULVWLFSRO\QRPLDORI A WKHQ p.#/ D 0

IRUHDFKGLDJRQDOHQWU\RI D EHFDXVHWKHVHHQWULHVLQ D DUH WKHHLJHQYDOXHVRI A 7KXV p.D/ LVWKH]HURPDWUL[ 7KXV p.A/ D P # 0 # P !1 D 0

,I I ! A ZHUHQRWLQYHUWLEOH WKHQWKHHTXDWLRQ .I ! A/[ D ZRXOGKDYHDQRQWULYLDOVROXWLRQ [ 7KHQ [ ! A[ D DQG A[ D 1 # [ ZKLFKVKRZVWKDW A ZRXOG KDYHDVDQHLJHQYDOXH 7KLVFDQQRWKDSSHQLIDOOWKH HLJHQYDOXHVDUHOHVVWKDQLQPDJQLWXGH 6R I ! A PXVWEH LQYHUWLEOH D 7DNH [ LQ H 7KHQ [ D c X IRUVRPHVFDODU c 6R A[ D A.c X/ D c.AX/ D c.#X/ D .c#/X ZKLFKVKRZV WKDW A[ LVLQ H E /HW [ EHDQRQ]HURYHFWRULQ K 6LQFH K LV RQHGLPHQVLRQDO K PXVWEHWKHVHWRIDOOVFDODU PXOWLSOHVRI [ ,I K LVLQYDULDQWXQGHU A WKHQ A[ LVLQ K DQGKHQFH A[ LVDPXOWLSOHRI [ 7KXV [ LVDQ HLJHQYHFWRURI A 5HSODFH a E\ a ! # LQWKHGHWHUPLQDQWIRUPXODIURP ([HUFLVHLQ&KDSWHU6XSSOHPHQWDU\([HUFLVHV GHW.A ! #I / D .a ! b ! #/n!1 Œa ! # C .n ! 1/b! 7KLVGHWHUPLQDQWLV]HURRQO\LI a ! b ! # D 0 RU a ! # C .n ! 1/b D 0 7KXV # LVDQHLJHQYDOXHRI A LIDQG RQO\LI # D a ! b RU # D a C .n ! 1/b )URPWKHIRUPXOD IRU GHW.A ! #I / DERYH WKHDOJHEUDLFPXOWLSOLFLW\LV n ! 1 IRU a ! b DQGIRU a C .n ! 1/b

GHW.A ! #I / D .a11 ! #/.a22 ! #/ ! a12 a21 D #2 ! .a11 C a22 /# C .a11 a22 ! a12 a21 / D #2 ! .WU A/# C GHW A 8VHWKHTXDGUDWLFIRUPXODWRVROYH WKHFKDUDFWHULVWLFHTXDWLRQ p WU A ˙ .WU A/2 ! 4 GHW A #D 2

7KHHLJHQYDOXHVDUHERWKUHDOLIDQGRQO\LIWKHGLVFULPLQDQW LVQRQQHJDWLYH WKDWLV .WU A/2 ! 4 GHW A & 0 7KLVLQHTXDOLW\ ) * WU A 2 VLPSOLÀHVWR .WU A/2 & 4 GHW A DQG & GHW A: 2 ! " 0 1 Cp D GHW.Cp ! #I / D 6 ! 5# C #2 D p.#/ !6 5 ,I p LVDSRO\QRPLDORIRUGHU WKHQDFDOFXODWLRQVXFKDVLQ ([HUFLVHVKRZVWKDWWKHFKDUDFWHULVWLFSRO\QRPLDORI Cp LV p.#/ D .!1/2 p.#/ VRWKHUHVXOWLVWUXHIRU n D 2 6XSSRVHWKHUHVXOWLVWUXHIRU n D k IRUVRPH k & 2 DQG FRQVLGHUDSRO\QRPLDO p RIGHJUHH k C 1 7KHQ H[SDQGLQJ GHW.Cp ! #I / E\FRIDFWRUVGRZQWKHÀUVWFROXPQ WKH GHWHUPLQDQWRI Cp ! #I HTXDOV 2 3 !# 1 ### 0 :: 7 6 :: 6 : 7 .!#/ GHW 6 : 7 C .!1/kC1 a0 4 0 1 5 # # # !ak ! # !a1 !a2

"

"OTXFST UP 0EE/VNCFSFE &YFSDJTFT 7KH k $ k PDWUL[VKRZQLV Cq ! #I ZKHUH q.t/ D a1 C a2 t C # # # C ak t k!1 C t k %\WKHLQGXFWLRQ DVVXPSWLRQ WKHGHWHUPLQDQWRI Cq ! #I LV .!1/k q.#/ 7KXV GHW.Cp ! #I / D .!1/kC1 a0 C .!#/.!1/k q.#/ D .!1/kC1 Œa0 C #.a1 C # # # C ak #k!1 C #k /! D .!1/kC1 p.#/ 6RWKHIRUPXODKROGVIRU n D k C 1 ZKHQLWKROGVIRU n D k %\WKHSULQFLSOHRILQGXFWLRQ WKHIRUPXODIRU GHW.Cp ! #I / LVWUXHIRUDOO n & 2

)URP([HUFLVH WKHFROXPQVRIWKH9DQGHUPRQGHPDWUL[ V DUHHLJHQYHFWRUVRI Cp FRUUHVSRQGLQJWRWKHHLJHQYDOXHV #1 #2 #3 WKHURRWVRIWKHSRO\QRPLDO p 6LQFHWKHVH HLJHQYDOXHVDUHGLVWLQFW WKHHLJHQYHFWRUVIRUPDOLQHDUO\ LQGHSHQGHQWVHW E\7KHRUHP LQ6HFWLRQ 7KXV V KDV OLQHDUO\LQGHSHQGHQWFROXPQVDQGKHQFHLVLQYHUWLEOH E\WKH ,QYHUWLEOH0DWUL[7KHRUHP )LQDOO\ VLQFHWKHFROXPQVRI V DUHHLJHQYHFWRUVRI Cp WKH'LDJRQDOL]DWLRQ7KHRUHP 7KHRUHPLQ6HFWLRQ VKRZVWKDW V !1 Cp V LVGLDJRQDO >0@ ,I\RXUPDWUL[SURJUDPFRPSXWHVHLJHQYDOXHVDQG HLJHQYHFWRUVE\LWHUDWLYHPHWKRGVUDWKHUWKDQV\PEROLF FDOFXODWLRQV \RXPD\KDYHVRPHGLIÀFXOWLHV

p

35

p 5 5

6HFWLRQ SDJH 1RWRUWKRJRQDO

1RWRUWKRJRQDO

2UWKRJRQDO

6KRZ X1 ! X2 D 0 PHQWLRQ7KHRUHP DQGREVHUYHWKDWWZR OLQHDUO\LQGHSHQGHQWYHFWRUVLQ R2 IRUPDEDVLV 7KHQ REWDLQ ! " ! " ! " ! " 2 2 26 6 1 6 C D 3 C [ D 39 13 !3 52 4 2 4 !3 6KRZ X1 ! X2 D 0 X1 ! X3 D 0 DQG X2 ! X3 D 0 0HQWLRQ 7KHRUHP DQGREVHUYHWKDWWKUHHOLQHDUO\LQGHSHQGHQW YHFWRUVLQ R3 IRUPDEDVLV 7KHQREWDLQ X C 189 X3 D 52 X1 ! 32 X2 C 2X3 [ D 52 X1 ! 27 18 2 ! " ! " ! " !2 !4=5 14=5 \ D C 1 7=5 8=5 ! " :6 \ ! \O D GLVWDQFHLV !:8 p 3 2 p 3 2 1=p3 !1= 2 0p 5 4 1=p3 5 4 1= 2 1= 3 2UWKRQRUPDO

2UWKRQRUPDO

6HHWKH 6WXG\*XLGH

+LQW
6HFWLRQ SDJH

>0@ 6WDWH\RXUFRQMHFWXUHDQGYHULI\LWDOJHEUDLFDOO\

+LQW kU [k2 D .U [/T .U [/ $OVR SDUWVD DQGF IROORZ IURPE

$IBQUFS

8 5

+LQW ,I [ LVLQ W ? WKHQ [ LVRUWKRJRQDOWRHYHU\YHFWRULQ W

2

3 3=35 4 !1=35 5 !1=7

!

!:6 :8

"

!

8=13 12=13

p 3 7=p69 4 2=p69 5 4= 69

1RWRUWKRJRQDO

2

"

2UWKRJRQDO

5HIHUWRWKH 6WXG\*XLGH DIWHU\RXKDYHZULWWHQ\RXU DQVZHUV +LQW 8VH7KHRUHPVDQGIURP6HFWLRQ X ! Y D 0 kXk2 D 30 kYk2 D 101 kX C Yk2 D .!5/2 C .!9/2 C 52 D 131 D 30 C 101 ! " !b 7KHVHWRIDOOPXOWLSOHVRI ZKHQ Y ¤ / a +LQW 8VHWKHGHÀQLWLRQRIRUWKRJRQDOLW\

+LQW &RQVLGHUDW\SLFDOYHFWRU Z D c1 Y1 C # # # C cp Yp LQ W

+LQW ,I\RXKDYHDFDQGLGDWHIRUDQLQYHUVH \RXFDQFKHFN WRVHHZKHWKHUWKHFDQGLGDWHZRUNV \!X X 5HSODFH X E\ c X ZLWK c ¤ 0 WKHQ 6XSSRVH \O D X!X \ !.c X/ c.\ ! X/ .c X/ D 2 .c/X D \O .c X/! .c X/ c X!X

/HW L D 6SDQ fXg ZKHUH X LVQRQ]HUR DQGOHW T .[/ D SURML [ %\GHÀQLWLRQ

T .[ / D

[!X X D .[ ! X/.X ! X/!1 X X!X

)RU [ DQG \ LQ Rn DQGDQ\VFDODUV c DQG d SURSHUWLHVRIWKH LQQHUSURGXFW7KHRUHP VKRZWKDW

T .c [ C d \/ D Œ.c [ C d \/! X!.X ! X/!1 X D Œc.[ ! X/ C d.\ ! X/!.X ! X/!1 X D c.[ ! X/.X ! X/!1 X C d.\ ! X/.X ! X/!1 X D cT .[/ C d T .\/ 7KXV T LVOLQHDU

4FDUJPO

6HFWLRQ SDJH [ D ! 89 X1 ! 29 X2 C 23 X3 C 2X4 2

3 !1 4 4 5 0 2

2

3

3 !1 4 2 5 D \ 6 2

3

10=3 !7=3 \ D 4 2=3 5 C 4 7=3 5 8=3 7=3 2

3 3 6 !1 7 7 6 4 15 !1

2

3 !1 6 !3 7 7 6 4 !2 5 3

2

3 2 3 0 10 6 !2 7 6 !6 7 7 6 7 [D6 4 4 5 C 4 !2 5 !2 2

2 3 2 3 2 2 6 4 7 6 !1 7 7 6 7 \ D 6 405C4 35 0 !1

p

40

3 8=9 !2=9 2=9 1 0 5=9 4=9 5 D U TU D U U T D 4 !2=9 0 1 2=9 4=9 5=9 2 3 2 3 2 2 E SURMW \ D 6X1 C 3X2 D 4 4 5 .U U T /\ D 4 4 5 5 5 !

"

2

2

3

2 3 0 0 $Q\PXOWLSOHRI 4 2=5 5 VXFKDV 4 2 5 1=5 1

:ULWH\RXUDQVZHUVEHIRUHFKHFNLQJWKH 6WXG\*XLGH +LQW 8VH7KHRUHPDQGWKH2UWKRJRQDO'HFRPSRVLWLRQ 7KHRUHP )RUWKHXQLTXHQHVV VXSSRVH AS D E DQG AS1 D E DQGFRQVLGHUWKHHTXDWLRQV S D S1 C .S ! S1 / DQG S D S C

6HFWLRQ SDJH 2

3 2 3 3 !1 4 0 5 4 5 5 !1 !3 2

3 2 3 1 5 6 !4 7 6 1 7 7 6 7 6 4 0 5 4 !4 5 1 !1 2

2

3 2 3 2 3 4 !5 5 4 3=2 5 1 3=2 2

p 3 2 p 3 2=p30 2=p6 4 !5=p30 5 4 1=p6 5 1= 30 1= 6

3 2 3 2 3 3 1 !3 6 17 6 37 6 17 7 6 7 6 7 6 4 !1 5 4 3 5 4 1 5 3 !1 3

R D

!

6 0

12 6

"

2

3 2 3 2 3 1 3 2 6 !1 7 6 0 7 6 0 7 6 7 6 7 6 7 7 6 7 6 7 6 6 !1 7 6 3 7 6 2 7 4 1 5 4 !3 5 4 2 5 1 3 !2

"

2

p 3 1=p5 1=2 1=2 6 !1= 5 0 0 7 6 7 p 6 Q D 6 !1=p5 1=2 1=2 7 7 4 1= 5 !1=2 5 1=2 p 1=2 !1=2 1= 5 p p 3 2p 5 ! 5 4 5 RD4 0 6 !2 5 0 0 4 6HHWKH 6WXG\*XLGH

6XSSRVH [ VDWLVÀHV R[ D WKHQ QR[ D Q D DQG A[ D 6LQFHWKHFROXPQVRI A DUHOLQHDUO\LQGHSHQGHQW [ PXVWEH]HUR 7KLVIDFW LQWXUQ VKRZVWKDWWKHFROXPQVRI R DUHOLQHDUO\LQGHSHQGHQW 6LQFH R LVVTXDUH LWLV LQYHUWLEOH E\WKH,QYHUWLEOH0DWUL[7KHRUHP 'HQRWHWKHFROXPQVRI Q E\ T1 ; : : : ; Tn 1RWHWKDW n ( m EHFDXVH A LV m $ n DQGKDVOLQHDUO\LQGHSHQGHQWFROXPQV 8VHWKHIDFWWKDWWKHFROXPQVRI Q FDQEHH[WHQGHGWRDQ RUWKRQRUPDOEDVLVIRU Rm VD\ fT1 ; : : : ; Tm g 7KH 6WXG\ *XLGH GHVFULEHVRQHPHWKRG /HW Q0 D Œ TnC1 # # # Tm ! DQG Q1 D Œ Q Q0 ! 7KHQ XVLQJSDUWLWLRQHGPDWUL[ ! " R D QR D A PXOWLSOLFDWLRQ Q1 0 +LQW 3DUWLWLRQ R DVD 2 $ 2 EORFNPDWUL[

>0@ 7KHGLDJRQDOHQWULHVRI R DUH DQG WRIRXUGHFLPDOSODFHV

6HFWLRQ SDJH

"! " ! " ! " 6 !11 x1 !4 3 D E [O D x2 !11 22 11 2 ! "! " ! " ! " 6 6 x1 6 4=3 D D E [O D x2 6 42 !6 !1=3 2 3 2 3 5 !1 p [O D 4 !3 5 C x34 1 5 2 5 0 1 2 3 ! " 1 2=7 D EO D 4 1 5 E [O D 1=7 0 2 3 2 3 3 2=3 6 17 7 D EO D 6 E [O D 4 0 5 4 45 1=3 !1 2 3 2 3 11 7 AX D 4 !11 5; AY D 4 !12 5 11 7 2 3 2 3 0 4 E ! AX D 4 2 5; E ! AY D 4 3 5 1R X FRXOGQRW !6 !2 SRVVLEO\EHDOHDVWVTXDUHVVROXWLRQRI A[ D E :K\" ! " 4 [O D 6HHWKH 6WXG\*XLGH !1

D

!

"


D ,I A[ D WKHQ ATA[ D AT D 7KLVVKRZVWKDW 1XO A LVFRQWDLQHGLQ 1XO ATA E ,I ATA[ D WKHQ [TATA[ D [T D 0 6R A[/T.A[/ D 0 ZKLFKPHDQVWKDW kA[k2 D 0/ DQG KHQFH A[ D 7KLVVKRZVWKDW 1XO ATA LVFRQWDLQHGLQ 1XO A +LQW )RUSDUWD XVHDQLPSRUWDQWWKHRUHPIURP&KDSWHU %\7KHRUHP EO D AO[ D A.ATA/!1 AT E 7KHPDWUL[ A.ATA/!1 AT RFFXUVIUHTXHQWO\LQVWDWLVWLFV ZKHUHLWLV VRPHWLPHVFDOOHGWKH KDWPDWUL[ ! "! " ! " 2 2 x 6 D ZKRVH 7KHQRUPDOHTXDWLRQVDUH 2 2 y 6 VROXWLRQLVWKHVHWRI .x; y/ VXFKWKDW x C y D 3 7KH VROXWLRQVFRUUHVSRQGWRSRLQWVRQWKHOLQHPLGZD\EHWZHHQ WKHOLQHV x C y D 2 DQG x C y D 4

9 4

yD

C

5 # x 14

D

9 4

C

5 .x 14

! 5:5/

+LQW 7KHHTXDWLRQKDVDQLFHJHRPHWULFLQWHUSUHWDWLRQ

6HFWLRQ SDJH D

p

! " 1 105 E $OOPXOWLSOHVRI 4 p p 56 5 2 3 3 25 C 14 t 25

D &RQVWDQWSRO\QRPLDO p.t/ D 5 E t 2 ! 5 LVRUWKRJRQDOWR p0 DQG p1 YDOXHV .4; !4; !4; 4/ DQVZHU q.t/ D 14 .t 2 ! 5/

17 t 5

9HULI\HDFKRIWKHIRXUD[LRPV )RULQVWDQFH

6HFWLRQ SDJH y D :9 C :4x

DQG .2:5; 3/ 7KHFROXPQVRI X DUHRUWKRJRQDOEHFDXVH WKHHQWULHVLQWKHVHFRQGFROXPQVXPWR ! "! " ! " 4 0 ˇ0 9 D E ˇ1 0 21 7:5

1:

y D 1:1 C 1:3x

,IWZRGDWDSRLQWVKDYHGLIIHUHQW x FRRUGLQDWHV WKHQWKHWZR FROXPQVRIWKHGHVLJQPDWUL[ X FDQQRWEHPXOWLSOHVRIHDFK RWKHUDQGKHQFHDUHOLQHDUO\LQGHSHQGHQW %\7KHRUHP LQ6HFWLRQ WKHQRUPDOHTXDWLRQVKDYHDXQLTXHVROXWLRQ 2 3 2 3 1:8 1 1 6 2:7 7 62 47 6 7 6 7 7 X D 6 3 3:4 97 D \ D Xˇ C ! ZKHUH \ D 6 6 7 6 7 4 3:8 5 44 16 5 3:9 5 25 2 3 &1 6 &2 7 ! " 6 7 ˇ1 7 ˇD !D6 6 &3 7 ˇ2 4 &4 5 &5 E >0@ y D 1:76x ! :20x 2 2

3 2 7:9 FRV 1 \ D Xˇ C ! ZKHUH \ D 4 5:4 5 X D 4 FRV 2 !:9 FRV 3 2 3 ! " &1 A ˇD ! D 4 &2 5 B &3

3 VLQ 1 VLQ 2 5 VLQ 3

>0@ ˇ D 1:45 DQG e D :811 WKHRUELWLVDQHOOLSVH 7KH HTXDWLRQ r D ˇ=.1 ! e # FRV #/ SURGXFHV r D 1:33 ZKHQ # D 4:6

>0@ D y D !:8558 C 4:7025t C 5:5554t 2 ! :0274t 3 E 7KHYHORFLW\IXQFWLRQLV v.t/ D 4:7025 C 11:1108t ! :0822t 2 DQG v.4:5/ D IWVHF

+LQW :ULWH X DQG \ DVLQHTXDWLRQ DQGFRPSXWH X TX DQG X T\ D 7KHPHDQRIWKH x GDWDLV xN D 5:5 7KHGDWDLQ PHDQGHYLDWLRQIRUPDUH .!3:5; 1/ .!:5; 2/ .1:5; 3/

hX; Yi D .AX/!.AY/ D .AY/!.AX/ D hY ; X i

'HÀQLWLRQ 3URSHUW\RIWKHGRWSURGXFW 'HÀQLWLRQ

hX; c Yi D hc Y; Xi

$[LRP

D chX; Yi

$[LRP

D chY; Xi

$[LRP

+LQW &RPSXWHWLPHVWKHULJKWKDQGVLGH p p p p p hX; Yi D pa b Cp b a D 2 ab kXk2 D . a/2 C . p b/2 D a C b 6LQFH a DQG bpDUH kYk D b C a QRQQHJDWLYH kXk D apC b 6LPLODUO\ p p %\ &DXFK\²6FKZDU] 2 ab ( a C b b C a D a C b p aCb +HQFH ab ( 2 p t 3t 2 ! 1 2= 5

>0@ 7KHQHZRUWKRJRQDOSRO\QRPLDOVDUHPXOWLSOHVRI !17t C 5t 3 DQG 72 ! 155t 2 C 35t 4 6FDOHWKHVH SRO\QRPLDOVVRWKHLUYDOXHVDW !2 !1 DQGDUHVPDOO LQWHJHUV

6HFWLRQ SDJH y D 2 C 32 t

p.t/ D 4p0 ! :1p1 ! :5p2 C :2p3 ' ( D 4 ! :1t ! :5.t 2 ! 2/ C :2 56 t 3 ! 176 t 7KLVSRO\QRPLDOKDSSHQVWRÀWWKHGDWDH[DFWO\

8VHWKHLGHQWLW\

VLQ mt VLQ nt D 12 ŒFRV.mt ! nt / ! FRV.mt C nt/!

1 C FRV 2kt 2 " C 2 VLQ t C VLQ 2t C 23 VLQ 3t >+LQW 6DYHWLPHE\XVLQJ WKHUHVXOWVIURP([DPSOH @ 8VHWKHLGHQWLW\ FRV2 kt D

1 2

! 12 FRV 2t :K\"

4FDUJPO +LQW 7DNHIXQFWLRQV f DQG g LQ C Œ0; 2"! DQGÀ[DQ LQWHJHU m & 0 :ULWHWKH)RXULHUFRHIÀFLHQWRI f C g WKDW LQYROYHV FRV mt DQGZULWHWKH)RXULHUFRHIÀFLHQWWKDW LQYROYHV VLQ mt .m > 0/ >0@ 7KHFXELFFXUYHLVWKHJUDSKRI g.t/ D !:2685 C 3:6095t C 5:8576t 2 ! :0477t 3 7KH YHORFLW\DW t D 4:5 VHFRQGV LV g 0 .4:5/ D 53:4 IW=VHF 7KLVLV DERXWIDVWHUWKDQWKHHVWLPDWHREWDLQHGLQ([HUFLVH LQ6HFWLRQ


) 7 7 )

E K Q

7 7 )

F L R

7 ) )

G M S

) 7 7

H N T

) 7 7

I O U

7 ) )

+LQW ,I fY1 ; Y2 g LVDQRUWKRQRUPDOVHWDQG [ D c1 Y1 C c2 Y2 WKHQWKHYHFWRUV c1 Y1 DQG c2 Y2 DUHRUWKRJRQDO DQG

k[k2 D kc1 Y1 C c2 Y2 k2 D kc1 Y1 k2 C kc2 Y2 k2

D .jc1 jkY1 k/2 C .jc2 jkY2 k/2 D jc1 j2 C jc2 j2

([SODLQZK\ 6RWKHVWDWHGHTXDOLW\KROGVIRU p D 2 6XSSRVHWKDWWKHHTXDOLW\KROGVIRU p D k ZLWK k & 2 OHW fY1 ; : : : ; YkC1 g EHDQRUWKRQRUPDOVHW DQGFRQVLGHU [ D c1 Y1 C # # # C ck Yk C ckC1 YkC1 D Xk C ckC1 YkC1 ZKHUH Xk D c1 Y1 C # # # C ck Yk

*LYHQ [ DQGDQRUWKRQRUPDOVHW fY1 ; : : : ; Yp g LQ Rn OHW [O EH WKHRUWKRJRQDOSURMHFWLRQRI [ RQWRWKHVXEVSDFHVSDQQHGE\ Y1 ; : : : ; Yp %\7KHRUHP LQ6HFWLRQ [O D .[ ! Y1 /Y1 C # # # C .[ ! Yp /Yp

%\([HUFLVH kO[k2 D j[ ! Y1 j2 C # # # C j[ ! Yp j2 %HVVHO·V LQHTXDOLW\IROORZVIURPWKHIDFWWKDW kO[k2 ( k[k2 QRWHG EHIRUHWKHVWDWHPHQWRIWKH&DXFK\²6FKZDU]LQHTXDOLW\ LQ 6HFWLRQ 6XSSRVH .U [/! .U \/ D [ ! \ IRUDOO [ \ LQ Rn DQGOHW H1 ; : : : ; Hn EHWKHVWDQGDUGEDVLVIRU Rn )RU j D 1; : : : ; n; U Hj LVWKH j WKFROXPQRI U 6LQFH kU Hj k2 D .U Hj /! .U Hj / D Hj ! Hj D 1 WKHFROXPQVRI U DUHXQLWYHFWRUV VLQFH .U Hj /! .U Hk / D Hj ! Hk D 0 IRU j ¤ k WKHFROXPQVDUHSDLUZLVHRUWKRJRQDO +LQW &RPSXWH QT Q XVLQJWKHIDFWWKDW .XXT /T D XT T XT D XXT

/HW W D 6SDQ fX; Yg *LYHQ ] LQ Rn OHW ]O D SURMW ] 7KHQ ]O LVLQ &RO A ZKHUH A D Œ X Y ! VD\ ]O D AO[ IRUVRPH [O LQ R2 6R [O LVDOHDVWVTXDUHVVROXWLRQRI A[ D ] 7KH QRUPDOHTXDWLRQVFDQEHVROYHGWRSURGXFH [O DQGWKHQ ]O LV IRXQGE\FRPSXWLQJ AO[ 2 3 2 3 2 3 x a 1 +LQW /HW [ D 4 y 5 E D 4 b 5 Y D 4 !2 5 DQG ´ c 5 2 T3 2 3 Y 1 !2 5 5 5 7KHJLYHQVHWRI A D 4 YT 5 D 4 1 !2 T 1 !2 5 Y

"

HTXDWLRQVLV A[ D E DQGWKHVHWRIDOOOHDVWVTXDUHV VROXWLRQVFRLQFLGHVZLWKWKHVHWRIVROXWLRQVRI ATA[ D AT E 7KHRUHP LQ6HFWLRQ 6WXG\WKLV HTXDWLRQ DQGXVHWKHIDFWWKDW .YYT /[ D Y.YT [/ D .YT [/Y EHFDXVH YT [ LVDVFDODU D 7KHURZ²FROXPQFDOFXODWLRQRI AX VKRZVWKDWHDFKURZ RI A LVRUWKRJRQDOWRHYHU\ X LQ 1XO A 6RHDFKURZRI A LVLQ .1XO A/? 6LQFH .1XO A/? LVDVXEVSDFH LWPXVW FRQWDLQDOOOLQHDUFRPELQDWLRQVRIWKHURZVRI A KHQFH .1XO A/? FRQWDLQV 5RZ A E ,I UDQN A D r WKHQ GLP 1XO A D n ! r E\WKH5DQN 7KHRUHP %\([HUFLVH F LQ6HFWLRQ GLP 1XO A C GLP.1XO A/? D n

6R GLP.1XO A/? PXVWEH r %XW 5RZ A LVDQ r GLPHQVLRQDOVXEVSDFHRI .1XO A/? E\WKH5DQN 7KHRUHPDQGSDUWD 7KHUHIRUH 5RZ A PXVWFRLQFLGH ZLWK .1XO A/? F 5HSODFH A E\ AT LQSDUWE DQGFRQFOXGHWKDW 5RZ AT FRLQFLGHVZLWK .1XO AT /? 6LQFH 5RZ AT D &RO A WKLV SURYHVF ,I A D URU T ZLWK U RUWKRJRQDO WKHQ A LVVLPLODUWR R EHFDXVH U LVLQYHUWLEOHDQG U T D U !1 / DQGVR A KDVWKH VDPHHLJHQYDOXHVDV R E\7KHRUHPLQ6HFWLRQ QDPHO\ WKH n UHDOQXPEHUVRQWKHGLDJRQDORI R

k'[k D :4618 k[k k'Ek D 3363 $ .1:548 $ 10!4 / D :5206 FRQG.A/ $ kEk 2EVHUYHWKDW k'[k=k[k DOPRVWHTXDOV FRQG.A/ WLPHV k'Ek=kEk

>0@

k'[k k'Ek D 7:178 $ 10!8 D 2:832 $ 10!4 k[k kE k 2EVHUYHWKDWWKHUHODWLYHFKDQJHLQ [ LV PXFK VPDOOHUWKDQ WKHUHODWLYHFKDQJHLQ E ,QIDFW VLQFH

>0@

FRQG.A/ $

k'Ek D 23;683 $ .2:832 $ 10!4 / D 6:707 kEk

WKHWKHRUHWLFDOERXQGRQWKHUHODWLYHFKDQJHLQ [ LVWR IRXUVLJQLÀFDQWÀJXUHV 7KLVH[HUFLVHVKRZVWKDWHYHQ ZKHQDFRQGLWLRQQXPEHULVODUJH WKHUHODWLYHHUURULQD VROXWLRQQHHGQRWEHDVODUJHDV\RXPLJKWH[SHFW

$IBQUFS 6HFWLRQ SDJH 6\PPHWULF

1RWV\PPHWULF 1RWV\PPHWULF " :6 :8 2UWKRJRQDO 1RWRUWKRJRQDO :8 !:6 2 3 p 2=3 0p 5=3 p 7 6 2UWKRJRQDO 4 2=3 1=p5 !4=p45 5 1=3 !2= 5 !2= 45 !

"


P D

"

P D

"

p p # ! 4 1=p2 !1=p2 DD 0 1= 2 1= 2 p !4=p17 1= 17

2

0 2

p # ! 17 1=p17 DD 0 4= 17

p p p 3 1=p3 1=p6 !1= 2 6 7 P D 4 1= 3 !2= 6 0p 5 p p 1= 6 1= 2 1= 3 2 3 5 0 0 2 05 D D 40 0 0 !2 2

p p !1=p5 4=p45 P D 4 2= 5 2=p45 0 5= 45 2 3 7 0 0 7 05 D D 40 0 0 !2 2

p

"

0 0

"

3 !2=3 !1=3 5 2=3

3

:5 !:5 !1= 2 0p 6 :5 27 :5 0 !1= 7 p P D 6 4 :5 !:5 0p 5 1= 2 :5 :5 0 1= 2 2 3 9 0 0 0 60 5 0 07 7 DD6 40 0 1 05 0 0 0 1 2

p p p 3 1=p3 !1=p2 !1=p6 6 7 P D 4 1= 3 1= 2 !1=p6 5 p 0 2= 6 1= 3 2 3 5 0 0 2 05 D D 40 0 0 2 6HHWKH 6WXG\*XLGH .B TAB/T D B TATB T T

D B TAB

3URGXFWRIWUDQVSRVHVLQ UHYHUVHRUGHU %HFDXVH A LVV\PPHWULF

7KHUHVXOWDERXW B TB LVDVSHFLDOFDVHZKHQ A D I .BB T /T D B T TB T D BB T VR BB T LVV\PPHWULF +LQW 8VHDQRUWKRJRQDOGLDJRQDOL]DWLRQRI A RUDSSHDOWR 7KHRUHP 7KH'LDJRQDOL]DWLRQ7KHRUHPLQ6HFWLRQVD\VWKDWWKH FROXPQVRI P DUHOLQHDUO\LQGHSHQGHQW HLJHQYHFWRUV FRUUHVSRQGLQJWRWKHHLJHQYDOXHVRI A OLVWHGRQWKHGLDJRQDO RI D 6R P KDVH[DFWO\ k FROXPQVRIHLJHQYHFWRUV FRUUHVSRQGLQJWR # 7KHVH k FROXPQVIRUPDEDVLVIRUWKH HLJHQVSDFH

A D 8X1 XT1 C 6X2 XT2 C 3X3 XT3 2 3 1=2 !1=2 0 1=2 05 D 8 4 !1=2 0 0 0 2 3 1=6 1=6 !2=6 1=6 !2=6 5 C 64 1=6 !2=6 !2=6 4=6 2 3 1=3 1=3 1=3 1=3 1=3 5 C 34 1=3 1=3 1=3 1=3

+LQW .XXT /[ D X.XT[/ D .XT[/X EHFDXVH XT[ LVDVFDODU

6HFWLRQ SDJH D 5x12 C ! 10 D !3 2 8 D 4 !3 2

2 xx 3 1 2

!3 !3

"

C x22

3 !3 2 7 !1 5 !1 !3

E

E

! 1 1 [ D P \ ZKHUH P D p 2 1

E F ! " 5 3=2 3=2 0 2 3 0 2 3 42 0 !4 5 3 !4 0 " !1 \T D \ D 6y12 ! 4y22 1

,Q([HUFLVHV² RWKHUDQVZHUVFKDQJHRIYDULDEOHVDQGQHZ TXDGUDWLFIRUP DUHSRVVLEOH 3RVLWLYHGHÀQLWH HLJHQYDOXHVDUHDQG

! 1 !1 &KDQJHRIYDULDEOH [ D P \ ZLWK P D p 2 5 1HZTXDGUDWLFIRUP 7y12 C 2y22

"

2 1

,QGHÀQLWH HLJHQYDOXHVDUHDQG !3

! " 1 1 !1 &KDQJHRIYDULDEOH [ D P \ ZLWK P D p 1 2 1 1HZTXDGUDWLFIRUP 7y12 ! 3y22

3RVLWLYHVHPLGHÀQLWH HLJHQYDOXHVDUHDQG! 1 1 &KDQJHRIYDULDEOH [ D P \ ZLWK P D p 10 !3 1HZTXDGUDWLFIRUP 10y12

3 1

"

>0@ 1HJDWLYHVHPLGHÀQLWH HLJHQYDOXHVDUH !6 !8 !12 &KDQJHRIYDULDEOH [ D P \ 2 p 3 3=p12 0p !1=2 0 6 7 1=2 0p 7 6 1= 12 !2=p6 P D6 p 7 4 1= 12 1=p6 1=2 !1=p2 5 p 1= 6 1=2 1= 2 1= 12 1HZTXDGUDWLFIRUP !6y22 ! 8y32 ! 12y42

>0@ ,QGHÀQLWH HLJHQYDOXHVDUHDQG !6:5 &KDQJHRIYDULDEOH [ D P \ 2 3 3 !4 3 4 1 65 0 !5 07 7 P D p 6 3 4 !3 5 50 4 4 0 5 0 5

4FDUJPO 1HZTXDGUDWLFIRUP 8:5y12 C 8:5y22 ! 6:5y32 ! 6:5y42

6HHWKH 6WXG\*XLGH

:ULWHWKHFKDUDFWHULVWLFSRO\QRPLDOLQWZRZD\V ! " a!# b GHW.A ! #I / D GHW b d !#

D #2 ! .a C d /# C ad ! b 2

DQG

.# ! #1 /.# ! #2 / D #2 ! .#1 C #2 /# C #1 #2 (TXDWH FRHIÀFLHQWVWRREWDLQ #1 C #2 D a C d DQG #1 #2 D ad ! b 2 D GHW A ([HUFLVHLQ6HFWLRQVKRZHGWKDW B TB LVV\PPHWULF $OVR [TB TB [ D .B [/TB [ D kB [k2 & 0 VRWKHTXDGUDWLF IRUPLVSRVLWLYHVHPLGHÀQLWH DQGZHVD\WKDWWKHPDWUL[ B TB LVSRVLWLYHVHPLGHÀQLWH +LQW 7RVKRZWKDW B TB LV SRVLWLYHGHÀQLWHZKHQ B LVVTXDUHDQGLQYHUWLEOH VXSSRVH WKDW [TB TB [ D 0 DQGGHGXFHWKDW [ D

+LQW 6KRZWKDW A C B LVV\PPHWULFDQGWKHTXDGUDWLF IRUP [T.A C B/[ LVSRVLWLYHGHÀQLWH

6HFWLRQ SDJH

2

3 1=3 2=3 !2=3 1=3 2=3 5 [ D P \ ZKHUH P D 4 2=3 !2=3 2=3 1=3 2 3 1=3 D E ˙4 2=3 5 F !2=3 " p # !1=p2 F D E ˙ 1= 2 2 3 1=3 p ˙4 2=3 5 5 C 5 2=3

+LQW ,I m D M WDNH ˛ D 0 LQWKHIRUPXODIRU [ 7KDWLV OHW [ D Xn DQGYHULI\WKDW [TA[ D m ,I m < M DQGLI t LV DQXPEHUEHWZHHQ m DQG M WKHQ 0 ( t ! m ( M ! m DQG 0 ( .t ! m/=.M ! m/ ( 1 6ROHW ˛ D .t ! m/=.M ! m/ 6ROYHWKHH[SUHVVLRQIRU ˛ WRVHHWKDW t D .1 ! ˛/m C ˛M $V ˛ JRHVIURPWR t JRHVIURP m WR M &RQVWUXFW [ DV LQWKHVWDWHPHQWRIWKHH[HUFLVH DQGYHULI\LWVSURSHUWLHV 2 3 :5 6 :5 7 6 F !:5 >0@ D E 4 7 :5 5 :5 2 p 3 !3=p12 6 7 6 1=p12 7 >0@ D !4 E 6 F !10 7 4 1= 12 5 p 1= 12

"

6HFWLRQ SDJH

7KHDQVZHUVLQ([HUFLVHV²DUHQRWWKHRQO\SRVVLELOLWLHV ! " ! "! "! " !3 0 !1 0 3 0 1 0 D 0 0 0 1 0 0 0 1 " p p #! p p # "" 3 0 1=p5 !2=p5 2=p5 1=p5 0 2 1= 5 2= 5 2= 5 !1= 5 p 2 p 32 p 3 1= 2 !1= 2 0 3 10 p0 54 0 4 0 0 1 10 5 p p 0 0 1= " 2 p 1= 2 p 0# 2=p5 1=p5 $ 2= 5 !1= 5 2 32 p 3 !1=3 2=3 2=3 3 10 0 2=3 5 4 0 4 2=3 !1=3 05 2=3 2=3 !1=3 0 0 " p p # 3=p10 !1=p10 $ 3= 10 1= 10 ! " 3 2 2 2 3 !2 " p p #! " 5 0 0 1=p2 !1=p2 D 0 3 0 1= 2 1= 2 p p 2 3 1=p 2 1=p 2 0 p $ 4 !1= 18 1= 18 !4= 18 5 !2=3 2=3 1=3 D UDQN A D 2

2

3 2 3 :40 !:78 E %DVLVIRU &RO A 4 :37 5; 4 !:33 5 !:84 !:52 2 3 :58 %DVLVIRU 1XO A 4 !:58 5 :58

5HPHPEHUWKDW V T DSSHDUVLQWKH69'

/HW A D U †V T D U †V !1 6LQFH A LVVTXDUHDQG LQYHUWLEOH UDQN A D n DQGDOOWKHHQWULHVRQWKHGLDJRQDORI † PXVWEHQRQ]HUR 6R A!1 D .U †V !1 /!1 D V †!1 U !1 D V †!1 U T +LQW 6LQFH U DQG V DUHRUWKRJRQDO

ATA D .U †V T /T U †V T D V †T U T U †V T D V .†T †/V !1 7KXV V GLDJRQDOL]HV ATA :KDWGRHVWKLVWHOO\RXDERXW V " /HW A D U †V T 7KHPDWUL[ 38 LVRUWKRJRQDO EHFDXVH P DQG U DUHERWKRUWKRJRQDO 6HH([HUFLVH LQ 6HFWLRQ 6RWKHHTXDWLRQ 3$ D .38 /†V T KDVWKHIRUP UHTXLUHGIRUDVLQJXODUYDOXHGHFRPSRVLWLRQ %\([HUFLVH WKHGLDJRQDOHQWULHVLQ † DUHWKHVLQJXODUYDOXHVRI PA +LQW 8VHDFROXPQ²URZH[SDQVLRQRI .U †/V T

"


+LQW &RQVLGHUWKH69' IRUWKHVWDQGDUGPDWUL[RI T ³VD\ A D U †V T D U †V !1 /HW B D fY1 ; : : : ; Yn g DQG C D fX1 ; : : : ; Xm g EHEDVHVFRQVWUXFWHGIURPWKHFROXPQVRI V DQG U UHVSHFWLYHO\ &RPSXWHWKHPDWUL[IRU T UHODWLYH WR B DQG C DVLQ6HFWLRQ 7RGRWKLV \RXPXVWVKRZ WKDW V !1 Yj D Hj WKH j WKFROXPQRI In 2 3 !:57 !:65 !:42 :27 6 :63 !:24 !:68 !:29 7 7 >0@ 6 4 :07 !:63 :53 !:56 5 !:51 :34 !:29 !:73 2 3 16:46 0 0 0 0 6 0 12:16 0 0 07 7 $6 4 0 0 4:87 0 05 0 0 0 4:31 0 2 3 !:10 :61 !:21 !:52 :55 6 !:39 :29 :84 !:14 !:19 7 6 7 7 $6 6 !:74 !:27 !:07 :38 :49 7 4 :41 !:50 :45 !:23 :58 5 !:36 !:48 !:19 !:72 !:29

SD

D

1 1 # O1 BB T D ; N !1 N !1

###

2

O T1 ; 6 $ : On 6 ; 6 :: 4 O TN ;

3 7 7 7 5

N N 1 X O OT 1 X ;k ;k D .;k ! 0/.;k ! 0/T N !1 1 N !1 1

&KDSWHU6XSSOHPHQWDU\([HUFLVHV SDJH D 7 J ) P 7

E K Q

) 7 )

F L R

7 ) 7

G M S

) ) 7

H N T

) ) )

I O

) )

,I UDQN A D r WKHQ GLP 1XO A D n ! r E\WKH5DQN 7KHRUHP 6RLVDQHLJHQYDOXHRIPXOWLSOLFLW\ n ! r +HQFH RIWKH n WHUPVLQWKHVSHFWUDOGHFRPSRVLWLRQRI A H[DFWO\ n ! r DUH]HUR 7KHUHPDLQLQJ r WHUPV FRUUHVSRQGLQJWRWKHQRQ]HURHLJHQYDOXHV DUHDOOUDQN PDWULFHV DVPHQWLRQHGLQWKHGLVFXVVLRQRIWKHVSHFWUDO GHFRPSRVLWLRQ

>0@ (1 =(5 D 68;622

,I AY D #Y IRUVRPHQRQ]HUR # WKHQ Y D #!1 AY D A.#!1 Y/ ZKLFKVKRZVWKDW Y LVDOLQHDU FRPELQDWLRQRIWKHFROXPQVRI A

6HFWLRQ SDJH

+LQW ,I A D RTR ZKHUH R LVLQYHUWLEOH WKHQ A LVSRVLWLYH GHÀQLWH E\([HUFLVH LQ6HFWLRQ &RQYHUVHO\ VXSSRVHWKDW A LVSRVLWLYHGHÀQLWH 7KHQE\([HUFLVH LQ 6HFWLRQ A D B TB IRUVRPHSRVLWLYHGHÀQLWHPDWUL[ B ([SODLQZK\ B DGPLWVD45 IDFWRUL]DWLRQ DQGXVHLWWR FUHDWHWKH&KROHVN\IDFWRUL]DWLRQRI A

" ! 12 7 B D 10 2 ! " 86 !27 SD !27 16 ! " :95 IRU # D 95:2 !:32

M D

!

" 10 !6 !9 !10 8 !4 !1 5 3 !5 !

:32 :95

"

,I A LV m $ n DQG [ LVLQ Rn WKHQ [TATA[ D .A[/T .A[/ D kA[k2 & 0 7KXV ATA LVSRVLWLYHVHPLGHÀQLWH %\ ([HUFLVH LQ6HFWLRQ UDQN ATA D UDQN A

IRU # D 6:8

>0@ RIWKHYDULDQFH y1 D :95x1 ! :32x2 y1 H[SODLQVRIWKHYDULDQFH

c1 D 1=3 c2 D 2=3 c3 D 2=3 WKHYDULDQFHRI y LV

D ,I Z LVWKHYHFWRULQ RN ZLWKDLQHDFKSRVLWLRQ WKHQ # $ ;1 # # # ;N Z D ;1 C # # # C ;N D EHFDXVHWKH ;k DUHLQPHDQGHYLDWLRQIRUP 7KHQ # $ <1 # # #
7KDWLV <1 C # # # C
###

;N

$#

;1

###

;N

$T

&RPSXWHWKHFRYDULDQFHPDWUL[RIWKH
+LQW :ULWHDQ69' RI A LQWKHIRUP A D U †V T D 34 ZKHUH P D U †U T DQG Q D 89 T 6KRZWKDW P LV V\PPHWULFDQGKDVWKHVDPHHLJHQYDOXHVDV † ([SODLQZK\ Q LVDQRUWKRJRQDOPDWUL[ D ,I E D A[ WKHQ [C D AC E D AC A[ %\ ([HUFLVH D [C LVWKHRUWKRJRQDOSURMHFWLRQRI [ RQWR 5RZ A E )URPD DQGWKHQ([HUFLVHF A[C D A.AC A[/ D .AAC A/[ D A[ D E F 6LQFH [C LVWKHRUWKRJRQDOSURMHFWLRQRQWR 5RZ A WKH 3\WKDJRUHDQ7KHRUHPVKRZVWKDW kXk2 D k[C k2 C kX ! [C k2 3DUWF IROORZV LPPHGLDWHO\ 2 3 2 3 !2 !14 13 13 :7 6 !2 !14 6 :7 7 13 13 7 7 6 7 1 6 7 [O D 6 !:8 7 !2 6 !7 !7 #6 >0@ AC D 6 7 6 7 40 4 2 !6 4 :8 5 7 75 4 !12 !6 !6 :6 ! " A 7KHUHGXFHGHFKHORQIRUPRI LVWKHVDPHDVWKH [T UHGXFHGHFKHORQIRUPRI A H[FHSWIRUDQH[WUDURZRI

4FDUJPO ]HURV 6RDGGLQJVFDODUPXOWLSOHVRIWKHURZVRI A WR [T FDQ SURGXFHWKH]HURYHFWRU ZKLFKVKRZVWKDW [T LVLQ 5RZ A 2 3 2 3 !1 0 6 17 607 6 7 6 7 7 6 7 %DVLVIRU 1XO A 6 6 0 7 6 1 7 4 05 415 0 0

$IBQUFS

6HFWLRQ SDJH 6RPHSRVVLEOHDQVZHUV \ D 2Y1 ! 1:5Y2 C :5Y3 \ D 2Y1 ! 2Y3 C Y4 \ D 2Y1 C 3Y2 ! 7Y3 C 3Y4

\ D !3Y1 C 2Y2 C 2Y3 7KHZHLJKWVVXPWR VRWKLVLVDQ DIÀQHVXP D S1 D 3E1 ! E2 ! E3 2 DII S VLQFHWKHFRHIÀFLHQWVVXP WR E S2 D 2E1 C 0E2 C E3 … DII S VLQFHWKHFRHIÀFLHQWVGR QRWVXPWR F S3 D !E1 C 2E2 C 0E3 2 DII S VLQFHWKHFRHIÀFLHQWV VXPWR D S1 2 6SDQ S EXW S1 … DII S E S2 2 6SDQ S DQG S2 2 DII S F S3 … 6SDQ S VR S3 … DII S ! " ! " !3 1 Y1 D DQG Y2 D 2WKHUDQVZHUVDUHSRVVLEOH 0 !2

6HHWKH 6WXG\*XLGH

6SDQ fY2 ! Y1 ; Y3 ! Y1 g LVDSODQHLIDQGRQO\LI fY2 ! Y1 ; Y3 ! Y1 g LVOLQHDUO\LQGHSHQGHQW 6XSSRVH c2 DQG c3 VDWLVI\ c2 .Y2 ! Y1 / C c3 .Y3 ! Y1 / D 6KRZWKDWWKLV LPSOLHV c2 D c3 D 0

/HW S D f[ W A[ D Eg 7RVKRZWKDW S LVDIÀQH LWVXIÀFHV WRVKRZWKDW S LVDÁDW E\7KHRUHP /HW W D f[ W A[ D g 7KHQ W LVDVXEVSDFHRI Rn E\ 7KHRUHPLQ6HFWLRQRU7KHRUHPLQ6HFWLRQ 6LQFH S D W C S ZKHUH S VDWLVÀHV AS D E E\7KHRUHP LQ6HFWLRQ S LVDWUDQVODWHRI W DQGKHQFH S LVDÁDW

$ VXLWDEOHVHWFRQVLVWVRIDQ\WKUHHYHFWRUVWKDWDUHQRW FROOLQHDUDQGKDYHDVWKHLUWKLUGHQWU\ ,ILVWKHLUWKLUG HQWU\ WKH\OLHLQWKHSODQH ´ D 5 ,IWKHYHFWRUVDUHQRW FROOLQHDU WKHLUDIÀQHKXOOFDQQRWEHDOLQH VRLWPXVWEHWKH SODQH ,I S; T 2 f .S / WKHQWKHUHH[LVW U; V 2 S VXFKWKDW f .U/ D S DQG f .V/ D T *LYHQDQ\ t 2 R ZHPXVWVKRZWKDW ] D .1 ! t /S C t T LVLQ f .S / 1RZXVHGHÀQLWLRQVRI S DQG T DQGWKHIDFWWKDW f LVOLQHDU 7KHFRPSOHWHSURRILV SUHVHQWHGLQWKH 6WXG\*XLGH 6LQFH B LVDIÀQH 7KHRUHPLPSOLHVWKDW B FRQWDLQVDOO DIÀQHFRPELQDWLRQVRISRLQWVRI B +HQFH B FRQWDLQVDOO DIÀQHFRPELQDWLRQVRISRLQWVRI A 7KDWLV DII A ) B 6LQFH A ) .A [ B/ LWIROORZVIURP([HUFLVHWKDW DII A ) DII .A [ B/ 6LPLODUO\ DII B ) DII .A [ B/ VR ŒDII A [ DII B! ) DII .A [ B/

"

7RVKRZWKDW D ) E \ F VKRZWKDW D ) E DQG D ) F 7KHFRPSOHWHSURRILVSUHVHQWHGLQWKH 6WXG\*XLGH

6HFWLRQ SDJH $IÀQHO\GHSHQGHQWDQG 2Y1 C Y2 ! 3Y3 D 7KHVHWLVDIÀQHO\LQGHSHQGHQW ,IWKHSRLQWVDUHFDOOHG Y1 Y2 Y3 DQG Y4 WKHQ fY1 ; Y2 ; Y3 g LVDEDVLVIRU R3 DQG Y4 D 16Y1 C 5Y2 ! 3Y3 EXWWKHZHLJKWVLQWKHOLQHDU FRPELQDWLRQGRQRWVXPWR !4Y1 C 5Y2 ! 4Y3 C 3Y4 D

7KHEDU\FHQWULFFRRUGLQDWHVDUH .!2; 4; !1/

6HHWKH 6WXG\*XLGH :KHQDVHWRIÀYHSRLQWVLVWUDQVODWHGE\VXEWUDFWLQJ VD\ WKHÀUVWSRLQW WKHQHZVHWRIIRXUSRLQWVPXVWEHOLQHDUO\ GHSHQGHQW E\7KHRUHPLQ6HFWLRQ EHFDXVHWKHIRXU SRLQWVDUHLQ R3 %\7KHRUHP WKHRULJLQDOVHWRIÀYH SRLQWVLVDIÀQHO\GHSHQGHQW ,I fY1 ; Y2 g LVDIÀQHO\GHSHQGHQW WKHQWKHUHH[LVW c1 DQG c2 QRWERWK]HUR VXFKWKDW c1 C c2 D 0 DQG c1 Y1 C c2 Y2 D 6KRZWKDWWKLVLPSOLHV Y1 D Y2 )RUWKHFRQYHUVH VXSSRVH Y1 D Y2 DQGVHOHFWVSHFLÀF c1 DQG c2 WKDWVKRZWKHLUDIÀQH GHSHQGHQFH 7KHGHWDLOVDUHLQWKH 6WXG\*XLGH ! " ! " 1 3 DQG Y3 ! Y1 D DUH D 7KHYHFWRUV Y2 ! Y1 D 2 !2 QRWPXOWLSOHVDQGKHQFHDUHOLQHDUO\LQGHSHQGHQW %\ 7KHRUHP S LVDIÀQHO\LQGHSHQGHQW ' ( ' ( ' ( E S1 $ ! 68 ; 98 ; 58 S2 $ 0; 12 ; 12 S3 $ 148 ; ! 58 ; ! 18 '6 5 7( '1 1 5( S4 $ 8 ; ! 8 ; 8 S 5 $ 4 ; 8 ; 8 F S6 LV .!; !; C/ S7 LV .0; C; !/ DQG S8 LV .C; C; !/

6XSSRVH S D fE1 ; : : : ; Ek g LVDQDIÀQHO\LQGHSHQGHQWVHW 7KHQHTXDWLRQ KDVDVROXWLRQ EHFDXVH S LVLQ DII S +HQFHHTXDWLRQ KDVDVROXWLRQ %\7KHRUHP WKH KRPRJHQHRXVIRUPVRIWKHSRLQWVLQ S DUHOLQHDUO\ LQGHSHQGHQW 7KXV KDVDXQLTXHVROXWLRQ 7KHQ DOVR KDVDXQLTXHVROXWLRQ EHFDXVH HQFRGHVERWKHTXDWLRQV WKDWDSSHDULQ 7KHIROORZLQJDUJXPHQWPLPLFVWKHSURRIRI7KHRUHPLQ 6HFWLRQ ,I S D fE1 ; : : : ; Ek g LVDQDIÀQHO\LQGHSHQGHQW VHW WKHQVFDODUV c1 ; : : : ; ck H[LVWWKDWVDWLVI\ E\ GHÀQLWLRQRI DII S 6XSSRVH [ DOVRKDVWKHUHSUHVHQWDWLRQ [ D d 1 E 1 C # # # C dk E k

DQG

d1 C # # # C dk D 1

D

IRUVFDODUV d1 ; : : : ; dk 7KHQVXEWUDFWLRQSURGXFHVWKH HTXDWLRQ D [ ! [ D .c1 ! d1 /E1 C # # # C .ck ! dk /Ek

E

7KHZHLJKWVLQE VXPWREHFDXVHWKH c ·VDQGWKH d ·V VHSDUDWHO\VXPWR 7KLVLVLPSRVVLEOH XQOHVVHDFKZHLJKW LQ LV EHFDXVH S LVDQDIÀQHO\LQGHSHQGHQWVHW 7KLV SURYHVWKDW ci D di IRU i D 1; : : : ; k

"


,I fS1 ; S2 ; S3 g LVDQDIÀQHO\GHSHQGHQWVHW WKHQWKHUHH[LVW VFDODUV c1 c2 DQG c3 QRWDOO]HUR VXFKWKDW c1 S1 C c2 S2 C c3 S3 D DQG c1 C c2 C c3 D 0 1RZXVH WKHOLQHDULW\RI f ! " ! " ! " a1 b1 c /HW D D ED DQG F D 1 7KHQ a2 b2 c 2 3 2 a1 b1 c1 GHW Œ DQ EQ FQ ! D GHW 4 a2 b2 c2 5 D 1 1 1 2 3 a1 a2 1 GHW 4 b1 b2 1 5 E\WKHWUDQVSRVHSURSHUW\RIWKH c1 c2 1 GHWHUPLQDQW7KHRUHPLQ6HFWLRQ %\([HUFLVHLQ 6HFWLRQ WKLVGHWHUPLQDQWHTXDOVWLPHVWKHDUHDRIWKH WULDQJOHZLWKYHUWLFHVDW D E DQG F 2 3 r Q WKHQ&UDPHU·VUXOHJLYHV ,I Œ DQ EQ FQ !4 s 5 D S t r D GHW Œ SQ EQ FQ != GHW Œ DQ EQ FQ ! %\([HUFLVH WKH QXPHUDWRURIWKLVTXRWLHQWLVWZLFHWKHDUHDRI 4SEF DQG WKHGHQRPLQDWRULVWZLFHWKHDUHDRI 4DEF 7KLVSURYHVWKH IRUPXODIRU r 7KHRWKHUIRUPXODVDUHSURYHGXVLQJ &UDPHU·VUXOHIRU s DQG t

6HFWLRQ SDJH 6HHWKH 6WXG\*XLGH 1RQHDUHLQFRQY S S1 D ! 16 Y1 C 13 Y2 C 23 Y3 C 16 Y4 VR S1 … FRQY S S2 D 13 Y1 C 13 Y2 C 16 Y3 C 16 Y4 VR S2 2 FRQY S

D 7KHEDU\FHQWULFFRRUGLQDWHVRI S1 S2 S3 DQG S4 DUH '1 1 1( ' 1 1( '1 1 3( ; ; 0; 2 ; 2 2 ; ! 4 ; 4 DQG 3 6 2 'UHVSHFWLYHO\ ( 1 3 ; ; ! 14 2 4 E S3 DQG S4 DUHRXWVLGH FRQY T S1 LVLQVLGH FRQY T S2 LVRQWKHHGJH Y2 Y3 RI FRQY T S1 DQG S3 DUHRXWVLGHWKHWHWUDKHGURQ FRQY S S2 LVRQWKH IDFHFRQWDLQLQJWKHYHUWLFHV Y2 Y3 DQG Y4 S4 LVLQVLGH FRQY S S5 LVRQWKHHGJHEHWZHHQ Y1 DQG Y3 6HHWKH 6WXG\*XLGH ,I S T 2 f .S/ WKHQWKHUHH[LVW U V 2 S VXFKWKDW f .U/ D S DQG f .V/ D T 7KHJRDOLVWRVKRZWKDWWKHOLQHVHJPHQW \ D .1 ! t /S C t T IRU 0 ( t ( 1 LVLQ f .S / 8VHWKH OLQHDULW\RI f DQGWKHFRQYH[LW\RI S WRVKRZWKDW \ D f .Z/ IRUVRPH Z LQ S 7KLVZLOOVKRZWKDW \ LVLQ f .S/ DQGWKDW f .S/ LVFRQYH[ S D 16 Y1 C 12 Y2 C 13 Y4 DQG S D 12 Y1 C 16 Y2 C 13 Y3

6XSSRVH A ) B ZKHUH B LVFRQYH[ 7KHQ VLQFH B LV FRQYH[ 7KHRUHPLPSOLHVWKDW B FRQWDLQVDOOFRQYH[ FRPELQDWLRQVRISRLQWVRI B +HQFH B FRQWDLQVDOOFRQYH[ FRPELQDWLRQVRISRLQWVRI A 7KDWLV FRQY A ) B

D 8VH([HUFLVHWRVKRZWKDW FRQY A DQG FRQY B DUH ERWKVXEVHWVRI FRQY .A [ B/ 7KLVZLOOLPSO\WKDWWKHLU XQLRQLVDOVRDVXEVHWRI FRQY .A [ B/

E 2QHSRVVLELOLW\LVWROHW A EHWZRDGMDFHQWFRUQHUVRID VTXDUHDQGOHW B EHWKHRWKHUWZRFRUQHUV 7KHQZKDWLV .FRQY A/ [ .FRQY B/ DQGZKDWLV FRQY .A [ B/"

()

f1

p1 f0

() 1 2

g

1 2

p2

() 1 2

p0

J.t/ D .1 ! t/I 0 .t/ C t I 1 .t/

D .1 ! t/Œ.1 ! t/S0 C t S1 ! C tŒ.1 ! t/S1 C t S2 !

D .1 ! t/2 S0 C 2t.1 ! t/S1 C t 2 S2 : 7KHVXPRIWKHZHLJKWVLQWKHOLQHDUFRPELQDWLRQIRU J LV .1 ! t/2 C 2t.1 ! t/ C t 2 ZKLFKHTXDOV .1 ! 2t C t 2 / C .2t ! 2t 2 / C t 2 D 1 7KHZHLJKWVDUHHDFK EHWZHHQDQGZKHQ 0 ( t ( 1 VR J.t/ LVLQ FRQY fS0 ; S1 ; S2 g

6HFWLRQ SDJH f .x1 ; x2 / D 3x1 C 4x2 DQG d D 13 D 2SHQ G &ORVHG D E F G H D E D E

E &ORVHG H &ORVHG

F 1HLWKHU

1RWFRPSDFW FRQYH[ &RPSDFW FRQYH[ 1RWFRPSDFW FRQYH[ 1RWFRPSDFW QRWFRQYH[ 1RWFRPSDFW FRQYH[ 2 3 0 Q D 4 2 5 RUDPXOWLSOH 3 f .[/ D 2x2 C 3x3 d D 11 2 3 3 6 !1 7 7 QD6 4 2 5 RUDPXOWLSOH 1 f .[/ D 3x1 ! x2 C 2x3 C x4 d D 5

Y2 LVRQWKHVDPHVLGHDV Y1 LVRQWKHRWKHUVLGH DQG Y3 LV LQ H 2 3 2 3 32 10 6 !14 7 6 !7 7 7 6 7 2QHSRVVLELOLW\LV S D 6 4 0 5 Y1 D 4 1 5 0 0 2 3 !4 6 17 7 Y2 D 6 4 0 5 1 f .x1 ; x2 ; x3 / D x1 ! 3x2 C 4x3 ! 2x4 DQG d D 5

f .x1 ; x2 ; x3 / D x1 ! 2x2 C x3 DQG d D 0

f .x1 ; x2 ; x3 / D !5x1 C 3x2 C x3 DQG d D 0

4FDUJPO 6HHWKH 6WXG\*XLGH

E

f .x1 ; x2 / D 3x1 ! 2x2 ZLWK d VDWLVI\LQJ 9 < d < 10 LVRQH SRVVLELOLW\ f .x; y/ D 4x C 1 $ QDWXUDOFKRLFHIRU d LV ZKLFK HTXDOV f .3; :75/ 7KHSRLQW .3; :75/ LVWKUHHIRXUWKVRIWKH GLVWDQFHEHWZHHQWKHFHQWHURI B.; 3/ DQGWKHFHQWHURI B.S; 1/ ([HUFLVHD LQ6HFWLRQJLYHVRQHSRVVLELOLW\ 2UOHW S D f.x; y/ W x 2 y 2 D 1 DQG y > 0g 7KHQ FRQY S LVWKH XSSHURSHQ KDOISODQH /HW [ \ 2 B.S; ı/ DQGVXSSRVH ] D .1 ! t/[ C t \ ZKHUH 0 ( t ( 1 7KHQVKRZWKDW

k] ! Sk D kŒ.1 ! t/[ C t \! ! Sk

D k.1 ! t/.[ ! S/ C t.\ ! S/k < ı:

6HFWLRQ SDJH D m D 1 DWWKHSRLQW S1 F m D 5 DWWKHSRLQW S3

E m D 5 DWWKHSRLQW S2

D m D !3 DWWKHSRLQW S3

E m D 1 RQWKHVHW FRQY fS1 ; S3 g

F m D !3 RQWKHVHW FRQY fS1 ; S2 g %! " ! " ! " ! "& 0 5 4 0 ; ; ; 0 0 3 5 %! " ! " ! " ! "& 0 7 6 0 ; ; ; 0 0 4 6 7KHRULJLQLVDQH[WUHPHSRLQW EXWLWLVQRWDYHUWH[ ([SODLQZK\

2QHSRVVLELOLW\LVWROHW S EHDVTXDUHWKDWLQFOXGHVSDUWRI WKHERXQGDU\EXWQRWDOORILW )RUH[DPSOH LQFOXGHMXVWWZR DGMDFHQWHGJHV 7KHFRQYH[KXOORIWKHSURÀOH P LVD WULDQJXODUUHJLRQ

S

"

conv P =

D f0 .C 5 / D 32 f1 .C 5 / D 80 f2 .C 5 / D 80 f3 .C 5 / D 40 f4 .C 5 / D 10 DQG 32 ! 80 C 80 ! 40 C 10 D 2

f0

f1

f2

f3

S1

S2

S3

4

S5

S

f4

)RUDJHQHUDOIRUPXOD VHHWKH 6WXG\*XLGH D f0 .P n / D f0 .Q/ C 1 E fk .P n / D fk .Q/ C fk!1 .Q/ F fn!1 .P n / D fn!2 .Q/ C 1

6HHWKH 6WXG\*XLGH

/HW S EHFRQYH[DQGOHW [ 2 cS C dS ZKHUH c > 0 DQG d > 0 7KHQWKHUHH[LVW V1 DQG V2 LQ S VXFKWKDW [ D c V1 C d V2 %XWWKHQ ) * d c [ D c V1 C d V2 D .c C d / V1 C V2 : cCd cCd

1RZVKRZWKDWWKHH[SUHVVLRQRQWKHULJKWVLGHLVDPHPEHU RI .c C d /S )RUWKHFRQYHUVH SLFNDW\SLFDOSRLQWLQ .c C d /S DQG VKRZLWLVLQ cS C dS

+LQW 6XSSRVH A DQG B DUHFRQYH[ /HW [ \ 2 A C B 7KHQWKHUHH[LVW D F 2 A DQG E G 2 B VXFKWKDW [ D D C E DQG \ D F C G )RUDQ\ t VXFKWKDW 0 ( t ( 1 VKRZWKDW Z D .1 ! t/[ C t \ D .1 ! t/.D C E/ C t.F C G/ UHSUHVHQWVDSRLQWLQ A C B

6HFWLRQ SDJH 7KHFRQWUROSRLQWVIRU [.t/ C E VKRXOGEH S0 C E S1 C E DQG S3 C E :ULWHWKH%p]LHUFXUYHWKURXJKWKHVHSRLQWV DQGVKRZDOJHEUDLFDOO\WKDWWKLVFXUYHLV [.t/ C E 6HHWKH 6WXG\*XLGH D [0 .t/ D .!3 C 6t ! 3W2 /S0 C .3 ! 12t C 9t 2 /S1 C .6t ! 9t 2 /S2 C 3t 2 S3 VR [0 .0/ D !3S0 C 3S1 D 3.S1 ! S0 / DQG [0 .1/ D !3S2 C 3S3 D 3.S3 ! S2 / 7KLVVKRZVWKDWWKH WDQJHQWYHFWRU [0 .0/ SRLQWVLQWKHGLUHFWLRQIURP S0 WR S1 DQGLVWKUHHWLPHVWKHOHQJWKRI S1 ! S0 /LNHZLVH [0 .1/ SRLQWVLQWKHGLUHFWLRQIURP S2 WR S3 DQGLVWKUHHWLPHV WKHOHQJWKRI S3 ! S2 ,QSDUWLFXODU [0 .1/ D LIDQG RQO\LI S3 D S2 E [00 .t/ D .6 ! 6t /S0 C .!12 C 18t/S1

C.6 ! 18t/S2 C 6t S3 ; VRWKDW [ .0/ D 6S0 ! 12S1 C 6S2 D 6.S0 ! S1 / C 6.S2 ! S1 / DQG [00 .1/ D 6S1 ! 12S2 C 6S3 D 6.S1 ! S2 / C 6.S3 ! S2 / )RUDSLFWXUHRI [00 .0/ FRQVWUXFWDFRRUGLQDWHV\VWHP ZLWKWKHRULJLQDW S1 WHPSRUDULO\ ODEHO S0 DV S0 ! S1 DQGODEHO S2 DV S2 ! S1 )LQDOO\ FRQVWUXFWDOLQHIURP 00

"

"OTXFST UP 0EE/VNCFSFE &YFSDJTFT WKLVQHZRULJLQWKURXJKWKHVXPRI S0 ! S1 DQG S2 ! S1 H[WHQGHGRXWDELW 7KDWOLQHSRLQWVLQWKHGLUHFWLRQRI [00 .0/ 0 = p1

p0 – p1

p2 – p1

w

w = (p0 – p1) + (p2 – p1) = 1 x"(0) 6

D )URP([HUFLVHD RUHTXDWLRQ LQWKHWH[W [0 .1/ D 3.S3 ! S2 / 8VHWKHIRUPXODIRU [0 .0/ ZLWKWKHFRQWUROSRLQWVIURP \.t/ DQGREWDLQ \0 .0/ D 3S3 C 3S4 D 3.S4 ! S3 / )RU C 1 FRQWLQXLW\ 3.S3 ! S2 / D 3.S4 ! S3 / VR S3 D .S4 C S2 /=2 DQG S3 LVWKHPLGSRLQWRIWKHOLQH VHJPHQWIURP S2 WR S4 E ,I [0 .1/ D \0 .0/ D WKHQ S2 D S3 DQG S3 D S4 7KXV WKHÓLQHVHJPHQWµIURP S2 WR S4 LVMXVWWKHSRLQW S3 >1RWH ,QWKLVFDVH WKHFRPELQHGFXUYHLVVWLOO C 1 FRQWLQXRXV E\GHÀQLWLRQ +RZHYHU VRPHFKRLFHVRI WKHRWKHU´FRQWUROµSRLQWV S0 S1 S5 DQG S6 FDQ SURGXFHDFXUYHZLWKDYLVLEOHFRUQHUDW S3 LQZKLFK FDVHWKHFXUYHLVQRW G 1 FRQWLQXRXVDW S3 @ +LQW 8VH [00 .t/ IURP([HUFLVHDQGDGDSWWKLVIRUWKH VHFRQGFXUYHWRVHHWKDW \00 .t/ D 6.1 ! t /S3 C 6.!2 C 3t /S4 C 6.1 ! 3t/S5 C 6t S6 7KHQVHW [00 .1/ D \00 .0/ 6LQFHWKHFXUYHLV C 1 FRQWLQXRXV DW S3 ([HUFLVHD VD\VWKDWWKHSRLQW S3 LVWKHPLGSRLQWRI WKHVHJPHQWIURP S2 WR S4 7KLVLPSOLHVWKDW S4 ! S3 D S3 ! S2 8VHWKLVVXEVWLWXWLRQWRVKRZWKDW S4 DQG S5 DUHXQLTXHO\GHWHUPLQHGE\ S1 S2 DQG S3 2QO\ S6 FDQEHFKRVHQDUELWUDULO\ :ULWHDYHFWRURIWKHSRO\QRPLDOZHLJKWVIRU [.t/ H[SDQG WKHSRO\QRPLDOZHLJKWV DQGIDFWRUWKHYHFWRUDV MB X.t/

2

3 1 ! 4t C 6t 2 ! 4t 3 C t 4 6 4t ! 12t 2 C 12t 3 ! 4t 4 7 6 7 6 7 6t 2 ! 12t 3 C 6t 4 6 7 4 5 4t 3 ! 4t 4 4 t 2 32 3 1 !4 6 !4 1 1 60 76 t 7 4 !12 12 !4 6 76 2 7 6 7 0 6 !12 67 D6 60 76t 7; 40 5 0 0 4 !4 4 t 3 5 t4 0 0 0 0 1 2 3 1 !4 6 !4 1 60 7 4 !12 12 !4 6 7 6 0 6 !12 67 MB D 6 0 7 40 0 0 4 !4 5 0 0 0 0 1

6HHWKH 6WXG\*XLGH

D +LQW 8VHWKHIDFWWKDW T0 D S0 E 0XOWLSO\WKHÀUVWDQGODVWSDUWVRIHTXDWLRQ E\ 83 DQGVROYHIRU 8T2 F 8VHHTXDWLRQ WRVXEVWLWXWHIRU 8T3 DQGWKHQDSSO\ SDUWD D )URPHTXDWLRQ \0 .1/ D :5[0 .:5/ D ]0 .0/ E 2EVHUYHWKDW \0 .1/ D 3.T3 ! T2 / 7KLVIROORZVIURP HTXDWLRQ ZLWK \.t/ DQGLWVFRQWUROSRLQWVLQSODFHRI [.t/ DQGLWVFRQWUROSRLQWV 6LPLODUO\ IRU ].t/ DQGLWV FRQWUROSRLQWV ]0 .0/ D 3.U1 ! U0 / %\SDUWD 3.T3 ! T2 / D 3.U1 ! U0 / 5HSODFH U0 E\ T3 DQGREWDLQ T3 ! T2 D U1 ! T3 DQGKHQFH T3 D .T2 C U1 /=2 F 6HW T0 D S0 DQG U3 D S3 &RPSXWH T1 D .S0 C S1 /=2 DQG U2 D .S2 C S3 /=2 &RPSXWH P D .S1 C S2 /=2 &RPSXWH T2 D .T1 C P/=2 DQG U1 D .P C U2 /=2 &RPSXWH T3 D .T2 C U1 /=2 DQGVHW U0 D T3 2S C S 2 S C 2S1 D U0 D S0 U1 D 0 U2 D 1 U3 D S2 3 3 E +LQW :ULWHWKHVWDQGDUGIRUPXOD LQWKLVVHFWLRQ ZLWK Ui LQSODFHRI Si IRU i D 0; : : : ; 3 DQGWKHQUHSODFH U0 DQG U3 E\ S0 DQG S2 UHVSHFWLYHO\ [.t/ D .1 ! 3t C 3t 2 ! t 3 /S0 C .3t ! 6t 2 C 3t 3 /U1 C .3t 2 ! 3t 3 /U2 C t 3 S2

LLL

8VHWKHIRUPXODVIRU U1 DQG U2 IURPSDUWD WRH[DPLQH WKHVHFRQGDQGWKLUGWHUPVLQWKLVH[SUHVVLRQIRU [.t/

,QGH[ $FFHOHUDWRUPXOWLSOLHUPRGHO Q $GMRLQW FODVVLFDO $GMXJDWH $GREH,OOXVWUDWRU $IÀQHFRPELQDWLRQV ² GHÀQLWLRQRI RISRLQWV ² ² $IÀQHFRRUGLQDWHV ² $IÀQHGHSHQGHQFH GHÀQLWLRQRI OLQHDUGHSHQGHQFHDQG ² $IÀQHKXOODIÀQHVSDQ JHRPHWULFYLHZRI RIWZRSRLQWV $IÀQHLQGHSHQGHQFH ² EDU\FHQWULFFRRUGLQDWHV ² GHÀQLWLRQRI $IÀQHVHW ² GLPHQVLRQRI LQWHUVHFWLRQRI $IÀQHWUDQVIRUPDWLRQ $LUFUDIWGHVLJQ $OJHEUDLFPXOWLSOLFLW\RIDQHLJHQYDOXH $OJHEUDLFSURSHUWLHVRI Rn $OJRULWKPV EDVHVIRU &RO A 5RZ A 1XO A ² FRPSXWHD%PDWUL[ GHFRXSOHDV\VWHP GLDJRQDOL]DWLRQ ² ÀQGLQJ$ï ² ÀQGLQJFKDQJHRIFRRUGLQDWHVPDWUL[ *UDP6FKPLGWSURFHVV ² LQYHUVHSRZHUPHWKRG ² -DFREL·VPHWKRG /8 IDFWRUL]DWLRQ ² 45 DOJRULWKP UHGXFWLRQWRÀUVWRUGHUV\VWHP URZ²FROXPQUXOHIRUFRPSXWLQJ AB URZUHGXFWLRQ ² URZ²YHFWRUUXOHIRUFRPSXWLQJ A[ VLQJXODUYDOXHGHFRPSRVLWLRQ ² VROYLQJDOLQHDUV\VWHP VWHDG\VWDWHYHFWRU ²

ZULWLQJVROXWLRQVHWLQSDUDPHWULF YHFWRUIRUP $PSV $QDO\VLVRIGDWD 6HHDOVR 0DWUL[IDFWRUL]DWLRQ GHFRPSRVLWLRQ $QDO\VLVRIYDULDQFH ² $QJOHVLQ R2 DQG R3 $QWLFRPPXWDWLYLW\ $SSUR[LPDWLRQ $UHD DSSUR[LPDWLQJ GHWHUPLQDQWVDV ² HOOLSVH SDUDOOHORJUDP ² WULDQJOH $UJXPHQWRIFRPSOH[QXPEHU $ $VVRFLDWLYHODZPXOWLSOLFDWLRQ $VVRFLDWLYHSURSHUW\DGGLWLRQ $VWURQRP\ EDU\FHQWULFFRRUGLQDWHVLQ Q $WWUDFWRU ÀJ $XJPHQWHGPDWUL[ $X[LOLDU\HTXDWLRQ $YHUDJHYDOXH $[LRPV LQQHUSURGXFWVSDFH YHFWRUVSDFH %FRRUGLQDWHYHFWRU ² %FRRUGLQDWHV %PDWUL[ %DFNVXEVWLWXWLRQ ² %DFNZDUGSKDVH %DODQFLQJFKHPLFDOHTXDWLRQV %DQGPDWUL[ %DU\FHQWULFFRRUGLQDWHV ² LQFRPSXWHUJUDSKLFV ² GHÀQLWLRQRI SK\VLFDODQGJHRPHWULFLQWHUSUHWDWLRQV RI ² %DVLFYDULDEOH %DVLV ² FKDQJHRI ² FKDQJHRI LQ Rn ² FROXPQVSDFH ² ² ² FRRUGLQDWHV\VWHPV ²

HLJHQVSDFH HLJHQYHFWRUV IXQGDPHQWDOVHWRIVROXWLRQV IXQGDPHQWDOVXEVSDFHV ² QXOOVSDFH ² ² RUWKRJRQDO ² ² ² RUWKRQRUPDO ² URZVSDFH ² VROXWLRQVSDFH VSDQQLQJVHW VWDQGDUG VXEVSDFH ² WZRYLHZV ² %DVLVPDWUL[ Q %DVLV7KHRUHP %HDPPRGHO %HVVHO·VLQHTXDOLW\ %HVWDSSUR[LPDWLRQ C Œa; b! )RXULHU P 4 ² WR \ E\HOHPHQWVRI W %HVW$SSUR[LPDWLRQ7KHRUHP %p]LHUEDVLVPDWUL[ %p]LHUFXUYHV ² DSSUR[LPDWLRQVWR ² LQ&$' SURJUDPV LQFRPSXWHUJUDSKLFV FRQQHFWLQJWZR ² FRQWUROSRLQWVLQ ² FXELF ² JHRPHWU\PDWUL[ PDWUL[HTXDWLRQVIRU ² TXDGUDWLF ² UHFXUVLYHVXEGLYLVLRQRI ² WDQJHQWYHFWRUVDQGFRQWLQXLW\ YDULDWLRQGLPLQLVKLQJSURSHUW\RI %p]LHU 3LHUUH %p]LHUVXUIDFHV ² DSSUR[LPDWLRQVWR ² ELFXELF UHFXUVLYHVXEGLYLVLRQRI ² YDULDWLRQGLPLQLVKLQJSURSHUW\RI %LGLDJRQDOPDWUL[ %LOORIÀQDOGHPDQGV %OHQGLQJSRO\QRPLDOV Q *

*

*OEFY

%ORFNPDWUL[ GLDJRQDO PXOWLSOLFDWLRQ XSSHUWULDQJXODU %RXQGDU\FRQGLWLRQ %RXQGDU\SRLQW %RXQGHGVHW %UDQFKFXUUHQW %UDQFKHVLQQHWZRUN %VSOLQHV XQLIRUP %XGJHWFRQVWUDLQW ² & ODQJXDJH C Œa; b! ² C n &$' SURJUDPV &DPEULGJH'LHW ² &DUDWKHRGRU\ &RQVWDQWLQ &DUDWKHRGRU\7KHRUHP ² &DVRUDWLPDWUL[ ² &DXFK\²6FKZDU]LQHTXDOLW\ ² &D\OH\²+DPLOWRQ7KHRUHP &HQWHURIJUDYLW\PDVV &HQWHURISURMHFWLRQ &)' 6HH &RPSXWDWLRQDOÁXLGG\QDPLFV &KDQJHRIEDVLV ² LQ Rn ² &KDQJHRIYDULDEOH IRUFRPSOH[HLJHQYDOXH LQGLIIHUHQWLDOHTXDWLRQ LQG\QDPLFDOV\VWHP ² LQSULQFLSDOFRPSRQHQWDQDO\VLV LQDTXDGUDWLFIRUP ² &KDQJHRIFRRUGLQDWHVPDWUL[ ² &KDUDFWHULVWLFHTXDWLRQRIPDWUL[ ² &KDUDFWHULVWLFSRO\QRPLDO &KDUDFWHUL]DWLRQRI/LQHDUO\'HSHQGHQW 6HWV7KHRUHP &KHPLFDOHTXDWLRQV &KROHVN\IDFWRUL]DWLRQ &ODVVLFDODGMRLQW &ODVVLÀFDWLRQRI6WDWHVDQG3HULRGLFLW\ &ORVHGVHW &RGRPDLQ &RHIÀFLHQW FRUUHODWLRQ ÀOWHU )RXULHU RIOLQHDUHTXDWLRQ PDWUL[ UHJUHVVLRQ WUHQG

&RIDFWRUH[SDQVLRQ ² &ROXPQVSDFH ² EDVLVIRU ² ² ² GLPHQVLRQRI OHDVWVTXDUHVSUREOHP ² DQGQXOOVSDFH ² VXEVSDFH ² 6HHDOVR )XQGDPHQWDOVXEVSDFHV &ROXPQURZH[SDQVLRQ &ROXPQV DXJPHQWHG GHWHUPLQDQWV RSHUDWLRQV RUWKRJRQDO RUWKRQRUPDO ² SLYRW $ VSDQ Rm VXP YHFWRU &RPHW RUELWRI &RPPXQLFDWLRQ&ODVVHV &RPPXWDWLYLW\ &RPSDFWVHW &RPSDQLRQPDWUL[ &RPSOHPHQW RUWKRJRQDO ² &RPSOH[HLJHQYDOXHV ² &RPSOH[QXPEHU $²$ DEVROXWHYDOXHRI $ DUJXPHQWRI $ FRQMXJDWH $ JHRPHWULFLQWHUSUHWDWLRQRI $²$ SRODUFRRUGLQDWHV $ SRZHUVRI $ DQG R2 $ UHDODQGLPDJLQDU\D[HV $ UHDODQGLPDJLQDU\SDUWV $ &RPSOH[URRW 6HHDOVR $X[LOLDU\HTXDWLRQ (LJHQYDOXH &RPSOH[YHFWRU Q UHDODQGLPDJLQDU\SDUWV ² &RPSOH[YHFWRUVSDFH Q &RPSRQHQWRI \ RUWKRJRQDOWR X &RPSRVLWLRQRIOLQHDUWUDQVIRUPDWLRQV &RPSRVLWLRQRIPDSSLQJV &RPSXWDWLRQDOÁXLGG\QDPLFV&)' &RPSXWHUJUDSKLFV EDU\FHQWULFFRRUGLQDWHVLQ ² %p]LHUFXUYHVLQ FHQWHURISURMHFWLRQ FRPSRVLWHWUDQVIRUPDWLRQV KRPRJHQHRXVFRRUGLQDWHV ² SHUVSHFWLYHSURMHFWLRQV ² VKHDUWUDQVIRUPDWLRQV

'² &RQGLWLRQQXPEHU VLQJXODUYDOXHGHFRPSRVLWLRQ &RQIRUPDEOHSDUWLWLRQ &RQMXJDWHSDLU $ &RQVLVWHQWV\VWHP ² PDWUL[HTXDWLRQ &RQVWDQWRIDGMXVWPHQW SRVLWLYH &RQVWUDLQHGRSWLPL]DWLRQ ² HLJHQYDOXHV ² ² IHDVLEOHVHW LQGLIIHUHQFHFXUYH ² 6HHDOVR 4XDGUDWLFIRUP &RQVXPSWLRQPDWUL[ &RQWLQXLW\RITXDGUDWLFFXELF%p]LHU FXUYHV JHRPHWULF .G 0 ; G 1 / FRQWLQXLW\ SDUDPHWULF .C 0 ; C 1 ; C 2 / FRQWLQXLW\ &RQWLQXRXVG\QDPLFDOV\VWHPV ² &RQWLQXRXVIXQFWLRQV ² ² &RQWUDFWLRQWUDQVIRUPDWLRQ &RQWUDVWEHWZHHQ 1XO A DQG &RO A ² &RQWUROSRLQWV LQ%p]LHUFXUYHV ² &RQWUROV\VWHP ² FRQWUROVHTXHQFH FRQWUROODEOHSDLU 6FKXUFRPSOHPHQW VSDFHVKXWWOH ² VWDWHYHFWRU VWDWHVSDFHPRGHO VWHDG\VWDWHUHVSRQVH V\VWHPPDWUL[ WUDQVIHUIXQFWLRQ &RQWUROODELOLW\PDWUL[ &RQYHUJHQFH ² 6HHDOVR ,WHUDWLYHPHWKRGV &RQYH[FRPELQDWLRQV ² FRQYH[VHWV ² ² ² GHÀQLWLRQRI ZHLJKWVLQ ² &RQYH[KXOO RI%p]LHUFXUYHFRQWUROSRLQWV ÀJ RIFORVHGVHW RIFRPSDFWVHW JHRPHWULFFKDUDFWHUL]DWLRQRI ² RIRSHQVHW &RQYH[VHWV ² GLVMRLQWFORVHG ÀJ

*OEFY H[WUHPHSRLQWRI ² K\SHUSODQHVHSDUDWLQJ ² LQWHUVHFWLRQRI SURÀOHRI 6HHDOVR 3RO\WRSHV &RRUGLQDWHPDSSLQJ ² ² &RRUGLQDWHV\VWHPV ² ² FKDQJHRIEDVLV ² JUDSKLFDO ² LVRPRUSKLVP ² SRODU $ Rn ² &RRUGLQDWHYHFWRU ² &RUUHODWLRQFRHIÀFLHQW &RVWYHFWRU &RXQWHUH[DPSOH &RYDULDQFH PDWUL[ ² &UDPHU·VUXOH ² &URVVSURGXFW &URVVSURGXFWWHUP &U\VWDOORJUDSK\ ² &XEH IRXUGLPHQVLRQDO &XELFFXUYH %p]LHU ² ² +HUPLWH &XELFVSOLQHV QDWXUDO &XUUHQWÁRZ &XUUHQWODZ &XUYHÀWWLQJ ² ² &XUYHV 6HH %p]LHUFXUYHV 'H0RLYUH·V7KHRUHP $ 'HFRPSRVLWLRQ HLJHQYHFWRU IRUFH RUWKRJRQDO ² SRODU VLQJXODUYDOXH ² 6HHDOVR )DFWRUL]DWLRQ 'HFRXSOHGV\VWHP 'HJHQHUDWHOLQH 'HVLJQPDWUL[ 'HWHUPLQDQW ² ² DGMXJDWH DUHDDQGYROXPH ² &DVRUDWLDQ FKDUDFWHULVWLFHTXDWLRQ ² FRIDFWRUH[SDQVLRQ ² FROXPQRSHUDWLRQV &UDPHU·VUXOH ² HFKHORQIRUP HLJHQYDOXHV

HOHPHQWDU\PDWUL[ ² JHRPHWULFLQWHUSUHWDWLRQ ÀJ DQGLQYHUVH ² OLQHDULW\SURSHUW\ PXOWLSOLFDWLYHSURSHUW\ n ! n PDWUL[ SURGXFWRISLYRWV SURSHUWLHVRI UHFXUVLYHGHÀQLWLRQ URZRSHUDWLRQV ² V\PEROLF 3 ! 3 PDWUL[ WUDQVIRUPDWLRQV ² WULDQJXODUPDWUL[ YROXPH ² 6HHDOVR 0DWUL[ 'LDJRQDOHQWULHV 'LDJRQDOPDWUL[ ² ² 'LDJRQDO0DWUL[5HSUHVHQWDWLRQ 7KHRUHP 'LDJRQDOL]DEOHPDWUL[ GLVWLQFWHLJHQYDOXHV ² QRQGLVWLQFWHLJHQYDOXHV ² RUWKRJRQDOO\ 'LDJRQDOL]DWLRQ7KHRUHP 'LIIHUHQFHHTXDWLRQ ² ² GLPHQVLRQRIVROXWLRQVSDFH HLJHQYHFWRUV ÀUVWRUGHU KRPRJHQHRXV ² OLQHDU ² QRQKRPRJHQHRXV ² SRSXODWLRQPRGHO ² UHFXUUHQFHUHODWLRQ UHGXFWLRQWRÀUVWRUGHU VLJQDOSURFHVVLQJ VROXWLRQVHWVRI ² ÀJ VWDJHPDWUL[PRGHO ² VWDWHVSDFHPRGHO 6HHDOVR '\QDPLFDOV\VWHP 0DUNRY FKDLQ 'LIIHUHQWLDOHTXDWLRQ ² ² FLUFXLWSUREOHP ² ² GHFRXSOHGV\VWHP HLJHQIXQFWLRQV LQLWLDOYDOXHSUREOHP VROXWLRQVRI 6HHDOVR /DSODFHWUDQVIRUP 'LIIHUHQWLDWLRQ 'LJLWDOVLJQDOSURFHVVLQJ 6HH 6LJQDOSURFHVVLQJ 'LODWLRQWUDQVIRUPDWLRQ

'LPHQVLRQRIDÁDWRUDVHW 'LPHQVLRQYHFWRUVSDFH ² ² FODVVLÀFDWLRQRIVXEVSDFHV ² FROXPQVSDFH QXOOVSDFH URZVSDFH ² VXEVSDFH ² 'LUHFWHGOLQHVHJPHQW 'LUHFWLRQ RIJUHDWHVWDWWUDFWLRQ RIJUHDWHVWUHSXOVLRQ 'LVFUHWHG\QDPLFDOV\VWHPV ² 'LVFUHWHOLQHDUG\QDPLFDOV\VWHP 6HH '\QDPLFDOV\VWHP 'LVFUHWHWLPHVLJQDO 6HH 6LJQDOV 'LVMRLQWFORVHGFRQYH[VHWV ÀJ 'LVWDQFH EHWZHHQYHFWRUDQGVXEVSDFH ² EHWZHHQYHFWRUV ² 'LVWRUWLRQ 'LVWULEXWLYHODZV 'RGHFDKHGURQ 'RPDLQ 'RWSURGXFW 'XDOLW\ '\QDPLFDOV\VWHP ² DWWUDFWRU FKDQJHRIYDULDEOH ² GHFRXSOLQJ GLVFUHWH ² HLJHQYDOXHVDQGHLJHQYHFWRUV ² ² ² HYROXWLRQRI JUDSKLFDOVROXWLRQV ² RZOSRSXODWLRQPRGHO ² ² SUHGDWRUSUH\PRGHO ² UHSHOOHU VDGGOHSRLQW ÀJ VSLUDOSRLQW VWDJHPDWUL[PRGHO ² ² 6HHDOVR 'LIIHUHQFHHTXDWLRQ 0DWKHPDWLFDOPRGHO (DUWK6DWHOOLWH&RUSRUDWLRQ (FFHQWULFLW\RIRUELW (FKHORQIRUP EDVLVIRUURZVSDFH ² FRQVLVWHQWV\VWHP GHWHUPLQDQW ÁRSV /8 IDFWRUL]DWLRQ ² SLYRWSRVLWLRQV ²

*

*

*OEFY

(GJHVRISRO\KHGURQ (IIHFWLYHUDQN (LJHQIXQFWLRQV ² (LJHQVSDFH ² GLPHQVLRQRI RUWKRJRQDOEDVLVIRU (LJHQYDOXH ² FKDUDFWHULVWLFHTXDWLRQ ² FRPSOH[ ² ² FRQVWUDLQHGRSWLPL]DWLRQ GHWHUPLQDQWV ² GLDJRQDOL]DWLRQ ² ² GLIIHUHQWLDOHTXDWLRQV ² GLVWLQFW ² G\QDPLFDOV\VWHPV ² LQYDULDQWSODQH ,QYHUWLEOH0DWUL[7KHRUHP LWHUDWLYHHVWLPDWHV ² PXOWLSOLFLW\RI QRQGLVWLQFW ² DQGTXDGUDWLFIRUPV DQGURWDWLRQ ² ÀJ ÀJ URZRSHUDWLRQV VLPLODULW\ VWULFWO\GRPLQDQW WULDQJXODUPDWUL[ 6HHDOVR '\QDPLFDOV\VWHP (LJHQYHFWRU ² EDVLV FRPSOH[ GHFRPSRVLWLRQ GLDJRQDOL]DWLRQ ² ² GLIIHUHQFHHTXDWLRQV G\QDPLFDOV\VWHP ² ² ² OLQHDUWUDQVIRUPDWLRQVDQG ² OLQHDUO\LQGHSHQGHQW 0DUNRYFKDLQ SULQFLSDOFRPSRQHQWV URZRSHUDWLRQV (OHFWULFDOQHWZRUNPRGHO ² FLUFXLWSUREOHP ² PDWUL[IDFWRUL]DWLRQ ² PLQLPDOUHDOL]DWLRQ (OHPHQWDU\PDWUL[ ² GHWHUPLQDQW ² LQWHUFKDQJH UHÁHFWRU URZUHSODFHPHQW VFDOH (OHPHQWDU\UHÁHFWRU (OHPHQWDU\URZRSHUDWLRQ (OHPHQWV3ODWR (OOLSVH DUHD

VLQJXODUYDOXHV ² (TXDOPDWULFHV (TXDWLRQ DX[LOLDU\ FKDUDFWHULVWLF ² GLIIHUHQFH ² ² GLIIHUHQWLDO ² ² LOOFRQGLWLRQHG RIDOLQH OLQHDU ² ² QRUPDO ² SDUDPHWULF ² SULFH SURGXFWLRQ WKUHHPRPHQW YHFWRU ² (TXLOLEULXPSULFHV ² (TXLOLEULXP XQVWDEOH (TXLOLEULXPYHFWRU ² (TXLYDOHQFHUHODWLRQ (TXLYDOHQWOLQHDUV\VWHPV (XOHU /HRQDUG ([LVWHQFHDQG8QLTXHQHVV7KHRUHP ([LVWHQFHRIVROXWLRQ ([LVWHQFHTXHVWLRQV ² ² ² ([SOLFLWGHVFULSWLRQ ² ([WUHPHSRLQW ² )DFHVRISRO\KHGURQ )DFHWRISRO\WRSH )DFWRUL]DWLRQ DQDO\VLVRIDG\QDPLFDOV\VWHP RIEORFNPDWULFHV FRPSOH[HLJHQYDOXH GLDJRQDO IRUDG\QDPLFDOV\VWHP LQHOHFWULFDOHQJLQHHULQJ ² 6HHDOVR 0DWUL[IDFWRUL]DWLRQ GHFRPSRVLWLRQ 6LQJXODUYDOXH GHFRPSRVLWLRQ69' )HDVLEOHVHW )H\QPDQ 5LFKDUG )LOWHUFRHIÀFLHQWV )LOWHU OLQHDU ORZSDVV PRYLQJDYHUDJH )LQDOGHPDQGYHFWRU )LQLWHVHW )LQLWHGLPHQVLRQDOYHFWRUVSDFH VXEVSDFH ² )LUVWSULQFLSDOFRPSRQHQW )LUVWRUGHUGLIIHUHQFHHTXDWLRQ 6HH 'LIIHUHQFHHTXDWLRQ

)ODWLQ Rn )OH[LELOLW\PDWUL[ )OLJKWFRQWUROV\VWHP ² )ORDWLQJSRLQWDULWKPHWLF )ORDWLQJSRLQWRSHUDWLRQÁRS )ORZLQQHWZRUN ² ² )RUFH GHFRPSRVLWLRQ )RUWUDQ )RUZDUGSKDVH )RXULHUDSSUR[LPDWLRQ )RXULHUFRHIÀFLHQWV )RXULHUVHULHV ² )UHHYDULDEOH )XOOUDQN )XQFWLRQ FRQWLQXRXV ² ² HLJHQIXQFWLRQ WUDQVIHU WUHQG XWLOLW\ 7KH)XQGDPHQWDO0DWUL[ )XQGDPHQWDOVROXWLRQVHW )XQGDPHQWDOVXEVSDFHV ÀJ ÀJ ² *DXVV &DUO)ULHGULFK Q Q *DXVVLDQHOLPLQDWLRQ Q *HQHUDOOHDVWVTXDUHVSUREOHP *HQHUDOOLQHDUPRGHO *HQHUDOVROXWLRQ ² *HRPHWULFFRQWLQXLW\ *HRPHWULFGHVFULSWLRQV RI R2 ² RI 6SDQ fX; Yg ² RI 6SDQ fYg ² *HRPHWULFSRLQW *HRPHWU\PDWUL[RID%p]LHUFXUYH *HRPHWU\RIYHFWRUVSDFHV ² DIÀQHFRPELQDWLRQV ² DIÀQHLQGHSHQGHQFH ² FRQYH[FRPELQDWLRQV ² FXUYHVDQGVXUIDFHV ² K\SHUSODQHV ² SRO\WRSHV ² *HRPHWU\YHFWRU *LYHQVURWDWLRQ *OREDO3RVLWLRQLQJ6\VWHP*36 ² *RXUDXGVKDGLQJ *UDGLHQW *UDPPDWUL[ *UDP6FKPLGWSURFHVV ² ² LQLQQHUSURGXFWVSDFHV ² /HJHQGUHSRO\QRPLDOV LQ P 4

*OEFY LQ Rn ² *UDP6FKPLGW3URFHVV7KHRUHP *UDSKLFV FRPSXWHU 6HH &RPSXWHUJUDSKLFV +HDWFRQGXFWLRQ +HUPLWHFXELFFXUYH +HUPLWHSRO\QRPLDOV +LGGHQVXUIDFHV +LOEHUWPDWUL[ +RPRJHQHRXVFRRUGLQDWHV ² ² +RPRJHQHRXVIRUPVDQGDIÀQH LQGHSHQGHQFH +RPRJHQHRXVIRUPRI Y LQ Rn ² +RPRJHQHRXVV\VWHP ² GLIIHUHQFHHTXDWLRQV LQHFRQRPLFV ² VXEVSDFH +RRNH·VODZ +RXVHKROGHUPDWUL[ UHÁHFWLRQ +RZDUG $ODQ+ +XOO DIÀQH JHRPHWULFYLHZRI +\SHUEROD +\SHUFXEH ² FRQVWUXFWLRQRI ² +\SHUSODQHV ² GHÀQLWLRQRI H[SOLFLWGHVFULSWLRQVRI ² LPSOLFLWGHVFULSWLRQVRI ² SDUDOOHO ² VHSDUDWLQJVHWVRI ² VXSSRUWLQJ +\SHUVSHFWUDOLPDJHSURFHVVLQJ ,FRVDKHGURQ ,GHQWLW\PDWUL[ ,OOFRQGLWLRQHGPDWUL[ ,OOFRQGLWLRQHGQRUPDOHTXDWLRQ ,PDJHSURFHVVLQJ PXOWLFKDQQHO ² ² ,PDJH YHFWRU ,PDJLQDU\D[LV $ ,PDJLQDU\QXPEHUV SXUH $ ,PDJLQDU\SDUW FRPSOH[QXPEHU $ FRPSOH[YHFWRU ² ,PSOLFLWGHÀQLWLRQRI 1XO A ,PSOLFLWGHVFULSWLRQ ,QFRQVLVWHQWV\VWHP 6HHDOVR /LQHDUV\VWHP ,QGH[HGVHW ,QGLIIHUHQFHFXUYH ²

,QHTXDOLW\ %HVVHO·V &DXFK\6FKZDU] ² WULDQJOH ,QÀQLWHGLPHQVLRQDOVSDFH ,QÀQLWHVHW Q ,QLWLDOYDOXHSUREOHP ,QQHUSURGXFW ² DQJOHV D[LRPV RQ C Œa; b! ² HYDOXDWLRQ OHQJWKQRUP RQ P n SURSHUWLHV ,QQHUSURGXFWVSDFH ² EHVWDSSUR[LPDWLRQLQ ² &DXFK\²6FKZDU]LQHTXDOLW\LQ ² GHÀQLWLRQRI GLVWDQFHVLQ LQ)RXULHUVHULHV ² *UDP²6FKPLGWSURFHVVLQ ² OHQJWKVQRUPV LQ RUWKRJRQDOLW\LQ IRUWUHQGDQDO\VLVRIGDWD ² WULDQJOHLQHTXDOLW\LQ ZHLJKWHGOHDVWVTXDUHV ² ,QSXWVHTXHQFH 6HHDOVR &RQWUROV\VWHP ,QWHUFKDQJHPDWUL[ ,QWHULRUSRLQW ,QWHUPHGLDWHGHPDQG ,QWHUQDWLRQDO&HOHVWLDO5HIHUHQFH6\VWHP Q ,QWHUSRODWHGFRORUV ² ,QWHUSRODWLQJSRO\QRPLDO ,QWURGXFWLRQDQG([DPSOHV ,QYDULDQWSODQH ,QYHUVH DOJRULWKPIRU ² DXJPHQWHGFROXPQV FRQGLWLRQQXPEHU GHWHUPLQDQW HOHPHQWDU\PDWUL[ ² ÁH[LELOLW\PDWUL[ IRUPXOD LOOFRQGLWLRQHGPDWUL[ OLQHDUWUDQVIRUPDWLRQ 0RRUH²3HQURVH SDUWLWLRQHGPDWUL[ SURGXFW VWLIIQHVVPDWUL[ ² WUDQVSRVH ,QYHUVHSRZHUPHWKRG ² ,QYHUWLEOH

*

OLQHDUWUDQVIRUPDWLRQ PDWUL[ ² ,QYHUWLEOH0DWUL[7KHRUHP ² ,VRPRUSKLFYHFWRUVSDFHV ,VRPRUSKLVP ² Q ,WHUDWLYHPHWKRGV HLJHQVSDFH ² HLJHQYDOXHV ² IRUPXODIRU .I " C /!1 ² LQYHUVHSRZHUPHWKRG ² -DFREL·VPHWKRG SRZHUPHWKRG ² 45 DOJRULWKP -DFRELDQPDWUL[ Q -DFREL·VPHWKRG -RUGDQIRUP -RUGDQ :LOKHOP Q -XQFWLRQV

k FURVVSRO\WRSH .HUQHO ² k IDFH .LUFKKRII·VODZV k S\UDPLG /DGGHUQHWZRUN ² ² /DJXHUUHSRO\QRPLDO /DPEHUVRQ 5 /DQGVDWLPDJH ² /$3$&. /DSODFHWUDQVIRUP /DZRIFRVLQHV /HDGLQJHQWU\ ² /HDGLQJYDULDEOH Q /HDVWVTXDUHVÀW FXELFWUHQG ÀJ OLQHDUWUHQG ² TXDGUDWLFWUHQG ² VFDWWHUSORW VHDVRQDOWUHQG ÀJ WUHQGVXUIDFH /HDVWVTXDUHVSUREOHP ² FROXPQVSDFH ² FXUYHÀWWLQJ ² HUURU ² OLQHV ² PHDQGHYLDWLRQIRUP PXOWLSOHUHJUHVVLRQ ² QRUPDOHTXDWLRQV ² RUWKRJRQDOFROXPQV SODQH ² 45 IDFWRUL]DWLRQ ² UHVLGXDOV VLQJXODUYDOXHGHFRPSRVLWLRQ

*

*OEFY

/HDVWVTXDUHVSUREOHPFRQWLQXHG VXPRIWKHVTXDUHVIRUHUURU ² ZHLJKWHG ² 6HHDOVR ,QQHUSURGXFWVSDFH /HDVWVTXDUHVVROXWLRQ DOWHUQDWLYHFDOFXODWLRQ ² PLQLPXPOHQJWK 45 IDFWRUL]DWLRQ ² /HIWGLVWULEXWLYHODZ /HIWVLQJXODUYHFWRU /HIWPXOWLSOLFDWLRQ /HJHQGUHSRO\QRPLDO /HQJWKRIYHFWRU ² VLQJXODUYDOXHV /HRQWLHI :DVLO\ Q H[FKDQJHPRGHO LQSXW²RXWSXWPRGHO ² SURGXFWLRQHTXDWLRQ /HYHOVHW /LQHV GHJHQHUDWH HTXDWLRQRI H[SOLFLWGHVFULSWLRQRI DVÁDW JHRPHWULFGHVFULSWLRQVRI LPSOLFLWHTXDWLRQRI SDUDPHWULFYHFWRUHTXDWLRQ RIUHJUHVVLRQ 6SDQ fYg WUDQVODWLRQRI /LQHVHJPHQW /LQHVHJPHQW GLUHFWHG /LQHDUFRPELQDWLRQ ² DIÀQHFRPELQDWLRQ 6HH $IÀQH FRPELQDWLRQV LQDSSOLFDWLRQV ZHLJKWV /LQHDUGHSHQGHQFH ² ÀJ DIÀQHGHSHQGHQFHDQG ² FROXPQVSDFH ² URZHTXLYDOHQWPDWULFHV $ URZRSHUDWLRQV /LQHDUGLIIHUHQFHHTXDWLRQ 6HH 'LIIHUHQFHHTXDWLRQ /LQHDUHTXDWLRQ ² 6HHDOVR /LQHDUV\VWHP /LQHDUÀOWHU /LQHDUIXQFWLRQDOV PD[LPXPYDOXHRI /LQHDULQGHSHQGHQFH ² HLJHQYHFWRUV PDWUL[FROXPQV LQ P 3

LQ Rn VHWV ² VLJQDOV ² ]HURYHFWRU /LQHDUPRGHO 6HH 0DWKHPDWLFDOPRGHO /LQHDUSURJUDPPLQJ SDUWLWLRQHGPDWUL[ /LQHDU3URJUDPPLQJ*HRPHWULF0HWKRG /LQHDU3URJUDPPLQJ6LPSOH[0HWKRG /LQHDUUHFXUUHQFHUHODWLRQ 6HH 'LIIHUHQFHHTXDWLRQ /LQHDUV\VWHP ² ² EDVLFVWUDWHJ\IRUVROYLQJ ² FRHIÀFLHQWPDWUL[ FRQVLVWHQWLQFRQVLVWHQW ² HTXLYDOHQW H[LVWHQFHRIVROXWLRQV ² ² JHQHUDOVROXWLRQ KRPRJHQHRXV ² ² OLQHDULQGHSHQGHQFH ² DQGPDWUL[HTXDWLRQ ² PDWUL[QRWDWLRQ QRQKRPRJHQHRXV ² RYHUXQGHUGHWHUPLQHG SDUDPHWULFVROXWLRQ ² VROXWLRQVHWV ² ² DQGYHFWRUHTXDWLRQV 6HHDOVR /LQHDUWUDQVIRUPDWLRQ 5RZRSHUDWLRQ /LQHDUWUDQVIRUPDWLRQ ² ² ² %PDWUL[ FRPSRVLWH FRPSRVLWLRQRI FRQWUDFWLRQGLODWLRQ RIGDWD ² GHWHUPLQDQWV ² GLDJRQDOPDWUL[UHSUHVHQWDWLRQ GLIIHUHQWLDWLRQ GRPDLQFRGRPDLQ JHRPHWULF ² *LYHQVURWDWLRQ +RXVHKROGHUUHÁHFWLRQ LQYHUWLEOH ² LVRPRUSKLVP ² NHUQHO ² PDWUL[RI ² ² QXOOVSDFH ² RQHWRRQHRQWR ² SURMHFWLRQ SURSHUWLHV RQ Rn ² UDQJH ² UHÁHFWLRQ ²

URWDWLRQ ÀJ VKHDU VLPLODULW\ ² VWDQGDUGPDWUL[ ² YHFWRUVSDFH ² ² 6HHDOVR ,VRPRUSKLVP 6XSHUSRVLWLRQ SULQFLSOH /LQHDUWUHQG /LQHDULW\SURSHUW\RIGHWHUPLQDQW IXQFWLRQ /LQHDUO\GHSHQGHQWVHW /LQHDUO\LQGHSHQGHQWHLJHQYHFWRUV /LQHDUO\LQGHSHQGHQWVHW ² ² 6HHDOVR %DVLV /RQJWHUPEHKDYLRU RIDG\QDPLFDOV\VWHP RID0DUNRYFKDLQ /RRSFXUUHQW /RZHUWULDQJXODUPDWUL[ ² /RZSDVVÀOWHU /8 IDFWRUL]DWLRQ ²

Mm"n 0DFURPHGLD)UHHKDQG 0DLQGLDJRQDO 0DSOH 0DSSLQJ FRPSRVLWLRQRI FRRUGLQDWH ² ² HLJHQYHFWRUV ² PDWUL[IDFWRUL]DWLRQV ² RQHWRRQH ² RQWR Rm VLJQDOSURFHVVLQJ 6HHDOVR /LQHDUWUDQVIRUPDWLRQ 0DUJLQDOSURSHQVLW\WRFRQVXPH 0DUN,, FRPSXWHU 0DUNRYFKDLQ ² FRQYHUJHQFH HLJHQYHFWRUV SUHGLFWLRQV ² SUREDELOLW\YHFWRU VWDWHYHFWRU VWHDG\VWDWHYHFWRU ² VWRFKDVWLFPDWUL[ 0DUNRY&KDLQVDQG%DVHEDOO6WDWLVWLFV 0DVV²VSULQJV\VWHP 0DWKHPDWLFD 0DWKHPDWLFDOHFRORJLVWV 0DWKHPDWLFDOPRGHO ² DLUFUDIW EHDP

*OEFY HOHFWULFDOQHWZRUN OLQHDU ² QXWULWLRQ ² SRSXODWLRQ ² ² SUHGDWRU²SUH\ ² VSRWWHGRZO ² VWDJHPDWUL[ ² ² 6HHDOVR 0DUNRYFKDLQ 0$7/$% 0DWUL[ ² DGMRLQWDGMXJDWH DQWLFRPPXWLQJ DXJPHQWHG EDQG ELGLDJRQDO EORFN &DVRUDWL ² FKDQJHRIFRRUGLQDWHV ² FKDUDFWHULVWLFHTXDWLRQ ² FRHIÀFLHQW RIFRIDFWRUV FROXPQVSDFH ² FROXPQVXP FROXPQYHFWRU FRPPXWDWLYLW\ FRPSDQLRQ FRQVXPSWLRQ FRQWUROODELOLW\ FRYDULDQFH ² GHVLJQ GLDJRQDO GLDJRQDOL]DEOH HFKHORQ HOHPHQWDU\ ² ² ÁH[LELOLW\ JHRPHWU\ *UDP +LOEHUW +RXVHKROGHU LGHQWLW\ LOOFRQGLWLRQHG LQWHUFKDQJH LQYHUVH LQYHUWLEOH ² -DFRELDQ Q OHDGLQJHQWU\ ² RIDOLQHDUWUDQVIRUPDWLRQ ² ² PLJUDWLRQ m ! n PXOWLSOLFDWLRQ ² ² QRQ]HURURZFROXPQ QRWDWLRQ QXOOVSDFH ² ² RIREVHUYDWLRQV

RUWKRJRQDO RUWKRQRUPDO Q RUWKRQRUPDOFROXPQV ² SDUWLWLRQHG ² 3DXOLVSLQ SRVLWLYHGHÀQLWHVHPLGHÀQLWH SRZHUVRI ² SURGXFWV ² ² SURMHFWLRQ SVHXGRLQYHUVH RITXDGUDWLFIRUP UDQNRI ² UHGXFHGHFKHORQ UHJXODUVWRFKDVWLF URZHTXLYDOHQW Q $ URZVSDFH ² URZ²FROXPQUXOH VFDODUPXOWLSOH ² VFDOH 6FKXUFRPSOHPHQW VLQJXODUQRQVLQJXODU VL]HRI VTXDUH VWDQGDUG ² VWLIIQHVV ² VWRFKDVWLF ² VXEPDWUL[RI VXP ² V\PPHWULF ² V\VWHP WUDFHRI WUDQVIHU ² WUDQVSRVHRI ² WULGLDJRQDO XQLWFRVW XQLWORZHUWULDQJXODU 9DQGHUPRQGH ]HUR 6HHDOVR 'HWHUPLQDQW 'LDJRQDOL]DEOH PDWUL[ ,QYHUVH 0DWUL[ IDFWRUL]DWLRQGHFRPSRVLWLRQ 5RZRSHUDWLRQ 7ULDQJXODUPDWUL[ 0DWUL[HTXDWLRQ ² 0DWUL[IDFWRUL]DWLRQGHFRPSRVLWLRQ ² &KROHVN\ FRPSOH[HLJHQYDOXH ² GLDJRQDO ² ² LQHOHFWULFDOHQJLQHHULQJ ² IXOO45 OLQHDUWUDQVIRUPDWLRQV ² /8² SHUPXWHG/8 SRODU 45 ² ² UDQN

*

UDQNUHYHDOLQJ UHGXFHG/8 UHGXFHG69' 6FKXU VLPLODULW\ ² VLQJXODUYDOXHGHFRPSRVLWLRQ ² VSHFWUDO ² 0DWUL[*DPHV 0DWUL[LQYHUVLRQ ² 0DWUL[PXOWLSOLFDWLRQ ² EORFN FROXPQ²URZH[SDQVLRQ DQGGHWHUPLQDQWV ² SURSHUWLHV ² URZ²FROXPQUXOH 6HHDOVR &RPSRVLWLRQRIOLQHDU WUDQVIRUPDWLRQV 0DWUL[QRWDWLRQ 6HH %DFNVXEVWLWXWLRQ 0DWUL[RIFRHIÀFLHQWV 0DWUL[RIREVHUYDWLRQV 0DWUL[SURJUDP 0DWUL[WUDQVIRUPDWLRQ ² 6HHDOVR /LQHDUWUDQVIRUPDWLRQ 0DWUL[²YHFWRUSURGXFW ² SURSHUWLHV UXOHIRUFRPSXWLQJ 0D[LPXPRITXDGUDWLFIRUP ² 0HDQ VDPSOH 0HDQVTXDUHHUURU 0HDQGHYLDWLRQIRUP 0LFURFKLS 0LJUDWLRQPDWUL[ 0LQLPDOUHDOL]DWLRQ 0LQLPDOUHSUHVHQWDWLRQRISRO\WRSH ² ² 0LQLPXPOHQJWKVROXWLRQ 0LQLPXPRITXDGUDWLFIRUP ² 0RGHO PDWKHPDWLFDO 6HH 0DWKHPDWLFDO PRGHO 0RGXOXV $ 0RHELXV $) 0ROHFXODUPRGHOLQJ ² 0RRUH3HQURVHLQYHUVH 0RYLQJDYHUDJH 0XLU 7KRPDV 0XOWLFKDQQHOLPDJH 6HH ,PDJHSURFHVVLQJ PXOWLFKDQQHO 0XOWLSOHUHJUHVVLRQ ² 0XOWLSOLFDWLYHSURSHUW\RIGHW 0XOWLSOLFLW\RIHLJHQYDOXH 0XOWLYDULDWHGDWD ² 1$' 1RUWK$PHULFDQ'DWXP 1DWLRQDO*HRGHWLF6XUYH\ 1DWXUDOFXELFVSOLQHV

*

*OEFY

1HJDWLYHGHÀQLWHTXDGUDWLFIRUP 1HJDWLYHÁRZ LQDQHWZRUNEUDQFK 1HJDWLYHRIDYHFWRU 1HJDWLYHVHPLGHÀQLWHIRUP 1HWZRUN ² EUDQFK EUDQFKFXUUHQW HOHFWULFDO ² ² ² ÁRZ ² ² ORRSFXUUHQWV ² 1RGHV 1RLVH UDQGRP 1RQKRPRJHQHRXVV\VWHP ² GLIIHUHQFHHTXDWLRQV ² 1RQOLQHDUG\QDPLFDOV\VWHP Q 1RQVLQJXODUPDWUL[ 1RQWULYLDOVROXWLRQ 1RQ]HURFROXPQ 1RQ]HURURZ 1RQ]HURVLQJXODUYDOXHV ² 1RUPRIYHFWRU ² 1RUPDOHTXDWLRQ ² LOOFRQGLWLRQHG 1RUPDOYHFWRU 1RUWK$PHULFDQ'DWXP1$' 1XOOVSDFH ² ² EDVLV ² ² DQGFROXPQVSDFH ² GLPHQVLRQRI ² HLJHQVSDFH H[SOLFLWGHVFULSWLRQRI ² OLQHDUWUDQVIRUPDWLRQ ² 6HHDOVR )XQGDPHQWDOVXEVSDFHV .HUQHO 1XOOLW\ 1XWULWLRQPRGHO ² 2EVHUYDWLRQYHFWRU ² 2FWDKHGURQ 2KP·VODZ 2LOH[SORUDWLRQ 2QHWRRQHOLQHDUWUDQVIRUPDWLRQ 6HHDOVR ,VRPRUSKLVP 2QHWRRQHPDSSLQJ ² 2QWRPDSSLQJ 2SHQEDOO 2SHQVHW 2SHQ*/ 2SWLPL]DWLRQ FRQVWUDLQHG 6HH &RQVWUDLQHGRSWLPL]DWLRQ 2UELWRIDFRPHW 2UGHUHG nWXSOH 2UGHUHGSDLU 2UWKRJRQDO HLJHQYHFWRUV

PDWUL[ SRO\QRPLDOV UHJUHVVLRQ VHW ² YHFWRUV ² 2UWKRJRQDOEDVLV ² ² IRUIXQGDPHQWDOVXEVSDFHV ² *UDP²6FKPLGWSURFHVV ² 2UWKRJRQDOFRPSOHPHQW ² 2UWKRJRQDO'HFRPSRVLWLRQ7KHRUHP 2UWKRJRQDOGLDJRQDOL]DWLRQ SULQFLSDOFRPSRQHQWDQDO\VLV TXDGUDWLFIRUP ² VSHFWUDOGHFRPSRVLWLRQ ² 2UWKRJRQDOSURMHFWLRQ ² ² JHRPHWULFLQWHUSUHWDWLRQ PDWUL[ SURSHUWLHVRI ² RQWRDVXEVSDFH ² VXPRI ÀJ 2UWKRJRQDOLW\ ² 2UWKRJRQDOO\GLDJRQDOL]DEOH 2UWKRQRUPDO EDVLV ² FROXPQV ² PDWUL[ Q URZV VHW ² 2XWHUSURGXFW 2YHUGHWHUPLQHGV\VWHP 2ZOSRSXODWLRQPRGHO ² ²

P Pn ² ² GLPHQVLRQ LQQHUSURGXFW VWDQGDUGEDVLV WUHQGDQDO\VLV 3DUDEROD 3DUDOOHO OLQH SURFHVVLQJ VROXWLRQVHWV ÀJ ÀJ 3DUDOOHOÁDWV 3DUDOOHOK\SHUSODQHV ² 3DUDOOHOHSLSHG 3DUDOOHORJUDP DUHDRI ² ODZ IRUYHFWRUV UHJLRQLQVLGH UXOHIRUDGGLWLRQ 3DUDPHWHUYHFWRU

3DUDPHWULF FRQWLQXLW\ GHVFULSWLRQ ² HTXDWLRQRIDOLQH HTXDWLRQRIDSODQH YHFWRUHTXDWLRQ ² YHFWRUIRUP 3DUWLDOSLYRWLQJ 3DUWLWLRQHGPDWUL[ ² DGGLWLRQDQGPXOWLSOLFDWLRQ ² DOJRULWKPV EORFNGLDJRQDO EORFNXSSHUWULDQJXODU FROXPQ²URZH[SDQVLRQ FRQIRUPDEOH LQYHUVHRI ² RXWHUSURGXFW 6FKXUFRPSOHPHQW VXEPDWULFHV 3DUWLWLRQV 3DWKV UDQGRP 3DXOLVSLQPDWUL[ 3HQWDWRSH ² 3HUPXWHG/8 IDFWRUL]DWLRQ 3HUVSHFWLYHSURMHFWLRQ ² 3KDVH EDFNZDUG IRUZDUG 3K\VLFV EDU\FHQWULFFRRUGLQDWHVLQ 3KRQJVKDGLQJ 3LYRW FROXPQ ² $ SRVLWLRQV ² SURGXFW 3L[HO 3ODQHV JHRPHWULFGHVFULSWLRQVRI LPSOLFLWHTXDWLRQRI 3ODWR 3ODWRQLFVROLGV ² 3RLQWV DIÀQHFRPELQDWLRQVRI ² ² ERXQGDU\ H[WUHPH ² LQWHULRU 3RLQWPDVVHV 3RODUFRRUGLQDWHV $ 3RODUGHFRPSRVLWLRQ 3RO\JRQ ² 3RO\KHGURQ UHJXODU 3RO\QRPLDOV EOHQGLQJ Q FKDUDFWHULVWLF GHJUHHRI

*OEFY +HUPLWH LQWHUSRODWLQJ /DJXHUUH /HJHQGUH RUWKRJRQDO LQ Pn ² ² VHW WULJRQRPHWULF ]HUR 3RO\WRSHV ² GHÀQLWLRQV 6HH ² H[SOLFLWUHSUHVHQWDWLRQRI K\SHUFXEH ² LPSOLFLWUHSUHVHQWDWLRQRI ² k FURVVSRO\WRSH k S\UDPLG PLQLPDOUHSUHVHQWDWLRQRI ² ² VLPSOH[ ² 3RSXODWLRQPRGHO ² ² ² ² ² 3RVLWLYHGHÀQLWHPDWUL[ 3RVLWLYHGHÀQLWHTXDGUDWLFIRUP 3RVLWLYHVHPLGHÀQLWHPDWUL[ 3RVW6FULSW IRQWV ² 3RZHUPHWKRG ² 3RZHUVRIDFRPSOH[QXPEHU $ 3RZHUVRIDPDWUL[ ² 3UHGDWRU²SUH\PRGHO ² 3UHGLFWHG y YDOXH 3UHSURFHVVLQJ 3ULFHHTXDWLRQ 3ULFHYHFWRU 3ULFHV HTXLOLEULXP ² 3ULQFLSDO$[HV7KHRUHP 3ULQFLSDOFRPSRQHQWDQDO\VLV ² ² FRYDULDQFHPDWUL[ ÀUVWSULQFLSDOFRPSRQHQW PDWUL[RIREVHUYDWLRQV PXOWLYDULDWHGDWD ² VLQJXODUYDOXHGHFRPSRVLWLRQ 3UREDELOLW\YHFWRU 3URFHVVFRQWUROGDWD 3URGXFW RIFRPSOH[QXPEHUV $ GRW RIHOHPHQWDU\PDWULFHV LQQHU ² RIPDWULFHV ² ² RIPDWUL[LQYHUVHV RIPDWUL[WUDQVSRVHV ² PDWUL[²YHFWRU RXWHU VFDODU

6HHDOVR &ROXPQ²URZH[SDQVLRQ ,QQHUSURGXFW 3URGXFWLRQHTXDWLRQ 3URGXFWLRQYHFWRU 3URÀOH 3URMHFWLRQ PDWUL[ SHUVSHFWLYH ² WUDQVIRUPDWLRQV 6HHDOVR 2UWKRJRQDOSURMHFWLRQ 3URSHUVXEVHW Q 3URSHUWLHV GHWHUPLQDQWV ² LQQHUSURGXFW OLQHDUWUDQVIRUPDWLRQ ² PDWUL[DGGLWLRQ ² PDWUL[LQYHUVLRQ PDWUL[PXOWLSOLFDWLRQ ² PDWUL[²YHFWRUSURGXFW $[ ² RUWKRJRQDOSURMHFWLRQV ² RI Rn UDQN WUDQVSRVH ² 6HHDOVR ,QYHUWLEOH0DWUL[7KHRUHP 3URSHUWLHVRI'HWHUPLQDQWV7KHRUHP 3VHXGRLQYHUVH 3XEOLFZRUNVFKHGXOHV ² IHDVLEOHVHW LQGLIIHUHQFHFXUYH ² XWLOLW\ 3XUHLPDJLQDU\QXPEHU $ 3\WKDJRUHDQ7KHRUHP 3\WKDJRUHDQV 45 DOJRULWKP 45 IDFWRUL]DWLRQ ² &KROHVN\IDFWRUL]DWLRQ IXOO45 IDFWRUL]DWLRQ OHDVWVTXDUHV ² 45 )DFWRUL]DWLRQ7KHRUHP 4XDGUDWLF%p]LHUFXUYH ² 4XDGUDWLFIRUP ² FKDQJHRIYDULDEOH ² FODVVLI\LQJ ² FURVVSURGXFWWHUP LQGHÀQLWH PD[LPXPDQGPLQLPXP ² RUWKRJRQDOGLDJRQDOL]DWLRQ ² SRVLWLYHGHÀQLWH SULQFLSDOD[HVRI JHRPHWULFYLHZRI ² 6HHDOVR &RQVWUDLQHGRSWLPL]DWLRQ 6\PPHWULFPDWUL[ 4XDGUDWLF)RUPVDQG(LJHQYDOXHV 7KHRUHP ²

*

Rn DOJHEUDLFSURSHUWLHVRI FKDQJHRIEDVLV ² GLPHQVLRQ LQQHUSURGXFW ² OHQJWKQRUP ² TXDGUDWLFIRUP VWDQGDUGEDVLV VXEVSDFH ² WRSRORJ\LQ R2 DQG R3 ² 5DQGRPSDWKV 5DQJHRIWUDQVIRUPDWLRQ ² 5DQN ² ² LQFRQWUROV\VWHPV HIIHFWLYH HVWLPDWLRQ Q IDFWRUL]DWLRQ ² IXOO ,QYHUWLEOH0DWUL[7KHRUHP ² SURSHUWLHVRI 6HHDOVR 2XWHUSURGXFW 5DQN7KHRUHP ² 5DQNUHYHDOLQJIDFWRUL]DWLRQ 5D\OHLJKTXRWLHQW 5D\WUDFLQJPHWKRG ² 5D\WULDQJOHLQWHUVHFWLRQV ² 5HDOD[LV $ 5HDOSDUW FRPSOH[QXPEHU $ FRPSOH[YHFWRU ² 5HDOYHFWRUVSDFH 5HFWDQJXODUFRRUGLQDWHV\VWHP 5HFXUUHQFHUHODWLRQ 6HH 'LIIHUHQFH HTXDWLRQ 5HFXUVLYHVXEGLYLVLRQRI%p]LHUFXUYHV VXUIDFHV ² 5HGXFHGHFKHORQIRUP EDVLVIRUQXOOVSDFH ² VROXWLRQRIV\VWHP XQLTXHQHVVRI $ 5HGXFHG/8 IDFWRUL]DWLRQ 5HGXFHGVLQJXODUYDOXHGHFRPSRVLWLRQ 5HGXFWLRQWRÀUVWRUGHUHTXDWLRQ 5HÁHFWLRQ ² +RXVHKROGHU 5HÁHFWRUPDWUL[ 5HJUHVVLRQ FRHIÀFLHQWV OLQH PXOWLSOH ² RUWKRJRQDO 5HJXODUSRO\KHGUD

*

*OEFY

5HJXODUSRO\KHGURQ 5HJXODUVROLGV 5HODWLYHFKDQJH 5HODWLYHHUURU 6HHDOVR &RQGLWLRQQXPEHU 5HQGHULQJJUDSKLFV 5HSHOOHU 5HVLGXDO 5HVLVWDQFH 5*% FRRUGLQDWHV ² 5LHPDQQVXP 5LJKWVLQJXODUYHFWRU 5LJKWGLVWULEXWLYHODZ 5LJKWPXOWLSOLFDWLRQ 5/& FLUFXLW ² 5RWDWLRQGXHWRDFRPSOH[HLJHQYDOXH ² ÀJ 5RWDWLRQWUDQVIRUPDWLRQ ÀJ ² 5RXQGDERXW 5RXQGRIIHUURU 5RZ²FROXPQUXOH 5RZHTXLYDOHQWPDWULFHV $ QRWDWLRQ Q 5RZRSHUDWLRQ ² EDFNVXEVWLWXWLRQ ² EDVLFIUHHYDULDEOH GHWHUPLQDQWV ² HFKHORQIRUP HLJHQYDOXHV HOHPHQWDU\ H[LVWHQFHXQLTXHQHVV ² LQYHUVH OLQHDUGHSHQGHQFHUHODWLRQV SLYRWSRVLWLRQV ² UDQN 6HHDOVR /LQHDUV\VWHP 5RZUHGXFWLRQDOJRULWKP ² EDFNZDUGSKDVH IRUZDUGSKDVH 6HHDOVR 5RZRSHUDWLRQ 5RZUHSODFHPHQWPDWUL[ 5RZVSDFH ² EDVLV ² GLPHQVLRQRI ,QYHUWLEOH0DWUL[7KHRUHP 6HHDOVR )XQGDPHQWDOVXEVSDFHV 5RZYHFWRU 5RZ²YHFWRUUXOH

S ² 6DGGOHSRLQW ÀJ ÀJ 6DPSOHFRYDULDQFHPDWUL[

6DPSOHPHDQ 6DPSOHYDULDQFH ² 6DPXHOVRQ 3$ Q 6FDODU 6FDODUPXOWLSOH ÀJ ² 6FDODUSURGXFW 6HH ,QQHUSURGXFW 6FDOHDQRQ]HURYHFWRU 6FDOHPDWUL[ 6FDWWHUSORW 6FHQHYDULDQFH ² 6FKXUFRPSOHPHQW 6FKXUIDFWRUL]DWLRQ 6HFRQGSULQFLSDOFRPSRQHQW 6HULHVFLUFXLW 6HWV DIÀQH ² ERXQGHG FORVHG FRPSDFW FRQYH[ ² OHYHO RSHQ YHFWRU 6HH 9HFWRUVHW 6KHDUWUDQVIRUPDWLRQ 6KHDUDQGVFDOHWUDQVIRUPDWLRQ 6KXQWFLUFXLW 6LJQDOSURFHVVLQJ DX[LOLDU\HTXDWLRQ ÀOWHUFRHIÀFLHQWV IXQGDPHQWDOVROXWLRQVHW OLQHDUGLIIHUHQFHHTXDWLRQ ² OLQHDUÀOWHU ORZSDVVÀOWHU PRYLQJDYHUDJH UHGXFWLRQWRÀUVWRUGHU 6HHDOVR '\QDPLFDOV\VWHP 6LJQDOV FRQWUROV\VWHPV GLVFUHWHWLPH ² ² IXQFWLRQ ² QRLVH VDPSOHG YHFWRUVSDFH 6 6LPLODUPDWULFHV ² 6HHDOVR 'LDJRQDOL]DEOHPDWUL[ 6LPLODULW\WUDQVIRUPDWLRQ 6LPSOH[ ² FRQVWUXFWLRQRI ² IRXUGLPHQVLRQDO 6LQJXODUPDWUL[ 6LQJXODUYDOXHGHFRPSRVLWLRQ69' ² FRQGLWLRQQXPEHU HVWLPDWLQJPDWUL[UDQN IXQGDPHQWDOVXEVSDFHV ²

OHDVWVTXDUHVVROXWLRQ m ! n PDWUL[ ² SULQFLSDOFRPSRQHQWDQDO\VLV SVHXGRLQYHUVH UDQNRIPDWUL[ UHGXFHG VLQJXODUYHFWRUV 6LQJXODU9DOXH'HFRPSRVLWLRQ7KHRUHP 6LQNRIG\QDPLFDOV\VWHP 6L]HRIDPDWUL[ 6ROLGV 3ODWRQLF ² 6ROXWLRQVHW ² RI$[ E GLIIHUHQFHHTXDWLRQV ² GLIIHUHQWLDOHTXDWLRQV H[SOLFLWGHVFULSWLRQRI IXQGDPHQWDO JHQHUDO ² ² ² JHRPHWULFYLVXDOL]DWLRQ ÀJ ÀJ ÀJ KRPRJHQHRXVV\VWHP ² PLQLPXPOHQJWK QRQKRPRJHQHRXVV\VWHP ² ² QXOOVSDFH SDUDPHWULF ² URZHTXLYDOHQWPDWULFHV VXEVSDFH ² VXSHUSRVLWLRQ WULYLDãQRQWULYLDO XQLTXH ² 6HHDOVR /HDVWVTXDUHVVROXWLRQ 6RXUFHRIG\QDPLFDOV\VWHP 6SDFHVKXWWOH ² 6SDQ ² DIÀQH OLQHDULQGHSHQGHQFH RUWKRJRQDOSURMHFWLRQ VXEVSDFH 6SDQQLQJVHW 6SDQQLQJ6HW7KHRUHP ² 6SDQ fX; Yg DVDSODQH ÀJ 6SDQ fYg DVDOLQH ÀJ 6SDQ fY1 ; : : : ; Yp g 6SDUVHPDWUL[ 6SDWLDOGLPHQVLRQ 6SHFWUDOFRPSRQHQWV 6SHFWUDOGHFRPSRVLWLRQ ² 6SHFWUDOGLPHQVLRQ 6SHFWUDOIDFWRUL]DWLRQ 6SHFWUDO7KHRUHP ² 6SLUDOSRLQW

*OEFY 6SOLQHV %² QDWXUDOFXELF 6SRWWHGRZO ² ² ² 6TXDUHPDWUL[ 6WDJHPDWUL[PRGHO ² ² 6WDQGDUGEDVLV 6WDQGDUGPDWUL[ ² 6WDQGDUGSRVLWLRQ 6WDWHYHFWRU 6WDWHVSDFHPRGHO 6WHDG\VWDWH KHDWÁRZ UHVSRQVH WHPSHUDWXUH YHFWRU ² ² 7KH6WHDG\6WDWH9HFWRUDQG*RRJOH·V 3DJH5DQN 6WLIIQHVVPDWUL[ ² 6WRFKDVWLFPDWUL[ ² ² UHJXODU 6WULFWO\GRPLQDQWHLJHQYDOXH 6WULFWO\VHSDUDWHK\SHUSODQHV 6XEPDWUL[ 6XEVHW SURSHU Q 6XEVSDFH ² EDVLVIRU ² FROXPQVSDFH ² GLPHQVLRQRI ² ² HLJHQVSDFH IXQGDPHQWDO ÀJ ² KRPRJHQHRXVV\VWHP LQWHUVHFWLRQRI OLQHDUWUDQVIRUPDWLRQ ÀJ QXOOVSDFH ² VSDQQHGE\DVHW VXP ]HUR 6HH DOVR9HFWRUVSDFH 6XPRIVTXDUHVIRUHUURU ² 6XSHUSRVLWLRQSULQFLSOH 6XSSRUWLQJK\SHUSODQH 6XUIDFHQRUPDO 6XUIDFHUHQGHULQJ 69' 6HH 6LQJXODUYDOXHGHFRPSRVLWLRQ 69' 6\PEROLFGHWHUPLQDQW 6\PPHWULFPDWUL[ ² GLDJRQDOL]DWLRQRI ² SRVLWLYHGHÀQLWHVHPLGHÀQLWH VSHFWUDOWKHRUHPIRU ² 6HH DOVR4XDGUDWLFIRUP 6\QWKHVLVRIGDWD 6\VWHP OLQHDU 6HH /LQHDUV\VWHP

6\VWHPPDWUL[ 7DQJHQWYHFWRU ² ² 7HWUDKHGURQ 7KHRUHP DIÀQHFRPELQDWLRQRISRLQWV ² %DVLV %HVW$SSUR[LPDWLRQ &DUDWKHRGRU\ ² &DXFK\²6FKZDU],QHTXDOLW\ &D\OH\²+DPLOWRQ &KDUDFWHUL]DWLRQRI/LQHDUO\ 'HSHQGHQW6HWV &ROXPQ²5RZ([SDQVLRQRI$% &UDPHU·V5XOH 'H0RLYUH·V $ 'LDJRQDO0DWUL[5HSUHVHQWDWLRQ 'LDJRQDOL]DWLRQ ([LVWHQFHDQG8QLTXHQHVV *UDP²6FKPLGW3URFHVV ,QYHUVH)RUPXOD ,QYHUWLEOH0DWUL[ ² ² 0XOWLSOLFDWLYH3URSHUW\RIGHW 2UWKRJRQDO'HFRPSRVLWLRQ 3ULQFLSDO$[HV 3\WKDJRUHDQ 45 )DFWRUL]DWLRQ 4XDGUDWLF)RUPVDQG(LJHQYDOXHV ² 5DQN ² 5RZ2SHUDWLRQV 6LQJXODU9DOXH'HFRPSRVLWLRQ 6SDQQLQJ6HW ² 6SHFWUDO ² 7ULDQJOH,QHTXDOLW\ 8QLTXH5HSUHVHQWDWLRQ 8QLTXHQHVVRIWKH5HGXFHG(FKHORQ )RUP $ 7KUHHPRPHQWHTXDWLRQ 7RWDOYDULDQFH IUDFWLRQH[SODLQHG 7UDFHRIDPDWUL[ 7UDMHFWRU\ 7UDQVIHUIXQFWLRQ 7UDQVIHUPDWUL[ 7UDQVIRUPDWLRQ DIÀQH FRGRPDLQ GHÀQLWLRQRI GRPDLQRI LGHQWLW\ LPDJHRIDYHFWRU[XQGHU UDQJHRI 6HHDOVR /LQHDUWUDQVIRUPDWLRQ 7UDQVODWLRQ YHFWRU

*

LQKRPRJHQHRXVFRRUGLQDWHV ² 7UDQVSRVH ² FRQMXJDWH Q RILQYHUVH RIPDWUL[RIFRIDFWRUV RISURGXFW SURSHUWLHVRI ² 7UHQGDQDO\VLV ² 7UHQGVXUIDFH 7ULDQJOH DUHDRI 7ULDQJOHLQHTXDOLW\ 7ULDQJXODUPDWUL[ GHWHUPLQDQWV HLJHQYDOXHV ORZHU ² XSSHU ² 7ULGLDJRQDOPDWUL[ 7ULJRQRPHWULFSRO\QRPLDO 7ULYLDOVROXWLRQ 7UXH7\SH IRQWV 8QFRUUHODWHGYDULDEOH 8QGHUGHWHUPLQHGV\VWHP 8QLIRUP%VSOLQH 8QLTXH5HSUHVHQWDWLRQ7KHRUHP 8QLTXHYHFWRU 8QLTXHQHVVTXHVWLRQ ² ² 8QLWFHOO ² 8QLWFRQVXPSWLRQYHFWRU 8QLWFRVWPDWUL[ 8QLWORZHUWULDQJXODUPDWUL[ 8QLWVTXDUH 8QLWYHFWRU 8QVWDEOHHTXLOLEULXP 8SSHUWULDQJXODUPDWUL[ ² 8WLOLW\IXQFWLRQ 9DOXHDGGHGYHFWRU 9DQGHUPRQGHPDWUL[ 9DULDEOH EDVLFIUHH OHDGLQJ Q XQFRUUHODWHG 6HHDOVR &KDQJHRIYDULDEOH 9DULDQFH ² Q VDPSOH ² VFHQH ² WRWDO 9DULDWLRQGLPLQLVKLQJSURSHUW\RI%p]LHU FXUYHVDQGVXUIDFHV 9HFWRUV DGGLWLRQVXEWUDFWLRQ DQJOHVEHWZHHQ ² DVDUURZV ÀJ FROXPQ

*

*OEFY

9HFWRUV FRQWLQXHG FRPSOH[ Q FRRUGLQDWH ² FRVW GHFRPSRVLQJ GLVWDQFHEHWZHHQ ² HTXDO HTXLOLEULXP ² ÀQDOGHPDQG JHRPHWU\ LPDJH OHIWVLQJXODU OHQJWKQRUP ² OLQHDUFRPELQDWLRQV ² OLQHDUO\GHSHQGHQWLQGHSHQGHQW ² QHJDWLYH QRUPDO QRUPDOL]LQJ REVHUYDWLRQ ² RUWKRJRQDO ² SDUDPHWHU DVDSRLQW ÀJ SULFH SUREDELOLW\ SURGXFWLRQ LQ R2 ² LQ R3 LQ Rn UHÁHFWLRQ ² UHVLGXDO VLQJXODU VWDWH

VWHDG\VWDWH ² ² VXP WDQJHQW ² WUDQVODWLRQV XQLTXH XQLW YDOXHDGGHG ZHLJKWV ]HUR 6HHDOVR (LJHQYHFWRU 9HFWRUDGGLWLRQ DVWUDQVODWLRQ 9HFWRUHTXDWLRQ OLQHDUGHSHQGHQFHUHODWLRQ ² SDUDPHWULF 9HFWRUVHW ² ² LQGH[HG OLQHDULQGHSHQGHQFH ² ² RUWKRJRQDO ² RUWKRQRUPDO ² SRO\QRPLDO 9HFWRUVSDFH ² RIDUURZV D[LRPV FRPSOH[ Q DQGGLIIHUHQFHHTXDWLRQV ² DQGGLIIHUHQWLDOHTXDWLRQV ² RIGLVFUHWHWLPHVLJQDOV ² ÀQLWHGLPHQVLRQDO ² RIIXQFWLRQV LQÀQLWHGLPHQVLRQDO

RISRO\QRPLDOV UHDO Q 6HHDOVR *HRPHWU\RIYHFWRUVSDFHV ,QQHUSURGXFWVSDFH 6XEVSDFH 9HFWRUVXEWUDFWLRQ 9HFWRUVXP 9HUWH[YHUWLFHV RISRO\KHGURQ ² 9LEUDWLRQRIDZHLJKWHGVSULQJ 9LHZLQJSODQH 9LUWXDOUHDOLW\ 9ROW 9ROXPH GHWHUPLQDQWVDV ² HOOLSVRLG SDUDOOHOHSLSHG ² WHWUDKHGURQ :HLJKWHGOHDVWVTXDUHV ² :HLJKWV DVIUHHYDULDEOHV :LUHIUDPHDSSUR[LPDWLRQ :LUHIUDPHPRGHOV =HURIXQFWLRQDO =HURPDWUL[ =HURSRO\QRPLDO =HURVROXWLRQ =HURVXEVSDFH =HURYHFWRU RUWKRJRQDO VXEVSDFH XQLTXH

3KRWR&UHGLWV 3DJH 0DUN,, FRPSXWHU %HWWPDQQ&RUELV :DVVLO\/HRQWLHI +XOWRQ$UFKLYH*HWW\ ,PDJHV 3DJH (OHFWULFS\ORQVDQGZLUHV +DDN6KXWWHUVWRFN /ODQEHULVODNHUDLOZD\LQ /ODQEHULVVWDWLRQLQ6QRZGRQLD *Z\QHGG 1RUWK:DOHV ':,PDJHV:DOHV$ODP\ 0DFKLQHIRUJULQGLQJVWHHO 0LUFHD%H]HUJKHDQX6KXWWHUVWRFN 3DJH *RRGV
1

[David C Lay] Linear Algebra and Its Applications(BookZZ.org)

Recommend Documents