MATCH
Communications in Mathematical
and in Computer Chemistry
MATCH Commun. Math. Comput. Chem. 65 (2011) 123-130
ISSN 0340 - 6253
6KDUS%RXQGVRQWKH3,DQG9HUWH[3,(QHUJ\
RI*UDSKV
0-1DGMDIL±$UDQL
'HSDUWPHQWRI0DWKHPDWLFV8QLYHUVLW\RI.DVKDQ
.DVKDQ,5,UDQ
5HFHLYHG0D\
$EVWUDFW
,Q WKLV SDSHU WKH QRWLRQV RI 3, DQG YHUWH[ 3, HQHUJ\ RI D JUDSK * DUH
LQWURGXFHG 7KH\ DUH GHILQHG DV WKH VXP RI WKH DEVROXWH YDOXHV RI 3, DQG
YHUWH[ 3, HLJHQYDOXHV RI * UHVSHFWLYHO\ ,Q WKLV SDSHU IRUPXODV IRU WKH 3,
DQG YHUWH[ 3, HQHUJ\ RI WKH &DUWHVLDQ SURGXFW RI JUDSKV DUH JLYHQ 6RPH
ERXQGVIRUWKHVHTXDQWLWLHVDUHDOVRSURYHG
,QWURGXFWLRQ
/HW * EH D JUDSK ZLWK YHUWH[ DQG HGJH VHWV 9* DQG (* UHVSHFWLYHO\ $V XVXDO WKH
GLVWDQFHEHWZHHQWKHYHUWLFHVXDQGYRI*LVGHQRWHGE\G*XYGXYIRUVKRUWDQGLWLV
GHILQHGDVWKHQXPEHURIHGJHVLQDVKRUWHVWSDWKFRQQHFWLQJWKHYHUWLFHVXDQGY
6XSSRVH *UDSK GHQRWHV WKH FODVV RI DOO JUDSKV $ PDS 7RS IURP *UDSKV LQWR UHDO
QXPEHUVLVFDOOHGDWRSRORJLFDOLQGH[LI*؆+ LPSOLHVWKDW7RS* 7RS+7KH:LHQHU
LQGH[ ZDV WKH ILUVW UHSRUWHG WRSRORJLFDO LQGH[ EDVHG RQ JUDSK GLVWDQFHV >@7KLV LQGH[ LV
GHILQHG DV WKH VXP RI DOO GLVWDQFHV EHWZHHQ YHUWLFHV RI WKH JUDSK XQGHU FRQVLGHUDWLRQ :H
HQFRXUDJH WKH UHDGHUV WR FRQVXOW SDSHUV >@ IRU EDFNJURXQG PDWHULDO DV ZHOO DV EDVLF
FRPSXWDWLRQDOWHFKQLTXHVUHODWHGWRWKH:LHQHULQGH[
-124-
/HWH XYEHDQHGJHRIWKHJUDSK*7KHQXPEHURIYHUWLFHVRI*ZKRVHGLVWDQFHWR
WKHYHUWH[XLVVPDOOHUWKDQWKHGLVWDQFHWRWKHYHUWH[YLVGHQRWHGE\QXH$QDORJRXVO\QYH
LVWKHQXPEHURIYHUWLFHVRI*ZKRVHGLVWDQFHWRWKHYHUWH[YLVVPDOOHUWKDQWKHGLVWDQFHWR
WKH YHUWH[ X 7KH HGJH YDULDQWV RI QXH DQG QYH DUH GHQRWHG E\ PXY DQG PYH
UHVSHFWLYHO\:H QRZ GHILQH IRXU WRSRORJLFDO LQGLFHV RI WKH 3, YHUWH[ 3, 6]HJHG DQG HGJH
6]HJHGLQGLFHVRIWKHJUDSK*DVIROORZV
ܲ‫ܫ‬ሺ‫ܩ‬ሻ ൌ ෍
௘ୀ௨௩
ܲ‫ܫ‬௩ ሺ‫ܩ‬ሻ ൌ ෍
ሾ݉௨ ሺ݁ሻ ൅ ݉௩ ሺ݁ሻሿ
௘ୀ௨௩
ܵ‫ݖ‬ሺ‫ܩ‬ሻ ൌ ෍
ܵ‫ݖ‬௘ሺீሻ ൌ ෍
ሾ݊௨ ሺ݁ሻ ൅ ݊௩ ሺ݁ሻሿ
௘ୀ௨௩
௘ୀ௨௩
݊௨ ሺ݁ሻǤ ݊௩ ሺ݁ሻ
݉௨ ሺ݁ሻǤ ݉௩ ሺ݁ሻ
7KLV WRSRORJLFDO LQGLFHV DWWUDFWHG UHFHQWO\ PXFK DWWHQWLRQ >@ 7KH
YHUWH[3,PDWUL[93,0*RI*LVGHILQHGVRWKDWLWVLMHQWU\DLMLVHTXDOWR
ƒ୧୨ ൌ ൜
݊௩೔ ሺ݁ሻ ൅ ݊௩ೕ ሺ݁ሻ ݁ ൌ ‫ݒ‬௜ ‫ݒ‬௝ Ͳ
‫݁ݏ݅ݓݎ݄݁ݐ݋‬
7KH 3,PDWUL[ 3,0* RI * LV GHILQHG DQDORJRXVO\ ,Q WKLV SDSHU ZH GLVFXVV DERXW
YHUWH[ 3, PDWUL[ 6LPLODUO\ WKHVH FRQFOXVLRQV DUH HVWDEOLVKHG IRU WKH 3, PDWUL[ 7KH
HLJHQYDOXHV RI 93,0* DUH VDLG WR EH WKH YHUWH[ 3,HLJHQYDOXHV RI * DQG IRUP WKH
93,0*VSHFWUXPRI*GHQRWHGE\93,6SHF*
6LQFHWKHYHUWH[3,PDWUL[LVV\PPHWULFDOORILWVHLJHQYDOXHVɁLL QDUH
UHDO DQG FDQ EH ODEHOHG VR WKDW į • į •  • įQ ,I įL ! įL !  ! įLN DUH GLVWLQFW HLJHQYDOXHVWKHQWKH93,VSHFWUXPFDQEHZULWWHQDV
ߜ
െ ’‡…ሺ
ሻ ൌ ൬ ௜ଵ
݉ଵ
ߜ௜ଶ
݉ଶ
ǥ ߜ௜௞
൰ǡ
ǥ ݉௞
ZKHUHPMLQGLFDWHVWKHDOJHEUDLFPXOWLSOLFLW\RIWKHHLJHQYDOXHįLM&OHDUO\PPÂÂÂPN
Q7KHRUGLQDU\VSHFWUXPRI*ZKLFKLVWKHVSHFWUXPRIWKHDGMDFHQF\PDWUL[RI*LVZHOO
VWXGLHGDQGPDQ\SURSHUWLHVRIJUDSKVLQFRQQHFWLRQZLWKWKHVSHFWUXPDUHUHYHDOHGGXULQJ
WKHSDVW\HDUV7KHQRWLRQRIWKHHQHUJ\RIDJUDSKZDVLQWURGXFHGE\,YDQ*XWPDQ>@,W
LV GHILQHG DV ‫ܧ‬ሺ‫ܩ‬ሻ ൌ σ௡௜ୀଵȁߣ௜ ȁ ,Q D VLPLODU ZD\ WKH YHUWH[ HQHUJ\ RI D JUDSK *
ܸܲ‫ܧܫ‬ሺ‫ܩ‬ሻ ൌ σ௡௜ୀଵȁߜ௜ ȁLVGHILQHG
,Q WKLV SDSHU VRPH ERXQGV IRU WKH 3, DQG YHUWH[ 3, HQHUJ\ RI JUDSKV DQG LWV
UHODWLRQVKLSZLWKVRPHWRSRORJLFDOLQGLFHVDUHREWDLQHG
-125-
0DLQ5HVXOWV
,QWKLVVHFWLRQVRPHERXQGVIRUWKHYHUWH[3,HQHUJ\RIJUDSKVDUHREWDLQHG
7KHRUHP 6XSSRVH * * « *V DUH FRQQHFWHG JUDSKV ZLWK Q Q « QV YHUWLFHV
UHVSHFWLYHO\7KH93,HLJHQYDOXHVRI* *î*î«î*VDUHDVIROORZV
ߜ௜భ ǡ௜మ ǡǥǡ௜ೞ ൌ ൬ෑ
௦
ȁ‫ܩ‬௧ ȁ൰ ෍
௧ୀଵ
௦
௝ୀଵ
ߜ௜ೕ ሺ‫ܩ‬௝ ሻ
ห‫ܩ‬௝ ห
ǡ
ͳ ൑ ݅௝ ൑ ݊௝ ǡ ͳ ൑ ݆ ൑ ‫ݏ‬Ǥ
,QSDUWLFXODULI+ * * « *VWKHQߜ௜భ ǡ௜మ ǡǥǡ௜ೞ ൌ ȁ‫ܪ‬ȁ௦ିଵ σ௦௝ୀଵ ߜ௜ೕ ሺ‫ܪ‬ሻDQG]HURLVDQ
HLJHQYDOXHRI‫ ܪ‬௡ IRUQ _+_
3URRI:HILUVWVXSSRVHV * *DQG+ *,Iሺܽǡ ܾሻሺ‫ݔ‬ǡ ‫ݕ‬ሻLVDQHGJHRI‫ ܩ‬ൈ ‫ܪ‬WKHQE\
GHILQLWLRQܽ ൌ ‫ݔ‬DQGܾ‫ܧ א ݕ‬ሺ‫ܪ‬ሻRUܽ‫ܧ א ݔ‬ሺ‫ܩ‬ሻDQGܾ ൌ ‫ݕ‬7KHUHIRUH݊ሺ௔ǡ௕ሻ ሺሺܽǡ „ሻሺܽǡ …ሻሻ ൌ
ȁ‫ܩ‬ȁ݊௕ ሺܾܿሻ
DQG
݊ሺ௔ǡ௕ሻ ሺሺܽǡ „ሻሺ‫ݔ‬ǡ „ሻሻ ൌ ȁ‫ܪ‬ȁ݊௔ ሺܽ‫ݔ‬ሻ
6R
݊ሺ௔ǡ௕ሻ ൫ሺܽǡ „ሻሺܽǡ …ሻ൯ ൅
݊ሺ௔ǡ௖ሻ ሺሺܽǡ „ሻሺܽǡ …ሻሻ ൌ ȁ‫ܩ‬ȁሺ݊௕ ሺܾܿሻ ൅ ݊௖ ሺܾܿሻሻ 2Q WKH RWKHU KDQG ሺ
ൈ ሻ ൌ
ȁୌȁ ۪ȁȁǤ ሺ
ሻ ൅ ȁ
ȁǤ ܸܲ‫ܯܫ‬ሺ‫ܪ‬ሻ۪‫ܫ‬ȁீȁ 7KLV
LPSOLHV
WKDW
ܸܲ‫ ܫ‬െ ܵ‫ܿ݁݌‬ሺ‫ ܩ‬ൈ ‫ܪ‬ሻ ൌ
ሼȁ‫ܪ‬ȁߜ௜ ሺ‫ܩ‬ሻ ൅ ȁ‫ܩ‬ȁߜ௝ ሺ‫ܪ‬ሻȁͳ ൑ ݅ ൑ ȁ‫ܩ‬ȁǡ ͳ ൑ ݆ ൑ ȁ‫ܪ‬ȁሽ7RGHGXFHWKHUHVXOWLWLVHQRXJKWRQRWLFH
WKDW ‫ܩ‬ଵ ൈ ‫ܩ‬ଶ ൈ ǥ ൈ ‫ܩ‬௦ ൌ ሺ‫ܩ‬ଵ ൈ ‫ܩ‬ଶ ൈ ǥ ൈ ‫ܩ‬௦ିଵ ሻ ൈ ‫ܩ‬௦ DQG DSSO\ DQ LQGXFWLYH DUJXPHQW 7R
SURYHWKHVHFRQGSDUWZHQRWHWKDWߜଵǡଶǡǥǡ௡ ൌ ͲZKLFKSURYHVRXUWKHRUHPᇝ
&RUROODU\:LWKQRWDWLRQRI7KHRUHPZHKDYH
ୱ
ሺ
ሻ ൑ ෑሺȁ
୲ ȁሻଶ ෍
୲ୀଵ
ୱ
୨ୀଵ
ሺ
୨ ሻ
ห
୨ ห
ଶ
ǡ ͳ ൑ Œ ൑ •
ZLWKHTXDOLW\LIDQGRQO\LIDWPRVWRQHRIWKHJUDSKV*«*VDUHQRQHPSW\
3URRI$SSO\7KHRUHPᇝ
/HPPD/HW*EHQYHUWH[JUDSKDQG െ ’‡…ሺ
ሻ ൌ ሼߜଵ ǡ ߜଶ ǡ ǥ ǡ ߜ௡ ሽ7KHQ
௡
෍
ߜ௜ ଶ ൑ ‫݊݅ܯ‬ሼʹܲ‫ܫ‬௩ ሺ‫ܩ‬ሻଶ ǡ ʹ݊ܲ‫ܫ‬௩ ሺ‫ܩ‬ሻǡ Ͷሺ݊ െ ͳሻଶ ݉ ൅ Ͷܵ‫ݖ‬ሺ‫ܩ‬ሻǡ ʹሺ݊ െ ͳሻܲ‫ܫ‬௩ ሺ‫ܩ‬ሻ ൅ Ͷܵ‫ݖ‬ሺ‫ܩ‬ሻሽǤ
௜ୀଵ
3URRI6XSSRVHܵ ൌ σ௡௜ୀଵ ߜ௜ ଶ 7RSURYHWKHOHPPDLWLVHQRXJKWRVKRZWKDWܵ ൑ ʹܲ‫ܫ‬௩ ሺ‫ܩ‬ሻଶ
ܵ ൑ ʹሺ݊ െ ͳሻܲ‫ܫ‬௩ ሺ‫ܩ‬ሻDQGܵ ൑ Ͷሺ݊ െ ͳሻଶ ݉ ൅ ͺܵ‫ݖ‬ሺ‫ܩ‬ሻǤ,WLVFOHDUWKDW
-126-
ܵ ൌ ሺσ௡௜ୀଵ ߜ௜ ሻଶ െ ʹ σ௜ழ௝ ߜ௜ ߜ௝ ൌ െʹ σ௜ழ௝ ߜ௜ ߜ௝ DQGE\>3URSRVLWLRQ@ܿଶ ൌ σ௜ழ௝ ߜ௜ ߜ௝ 7KHUHIRUH
ܵ ൌ ʹ σ௜ழ௝ሺ݊௩ ௜ ൅ ݊௩ ௝ ሻଶ ൑ ʹܲ‫ܫ‬௩ ሺ‫ܩ‬ሻଶ 2QWKHRWKHUKDQG
ܵ ൌ ʹ෍
ܵ ൌ ʹ෍
ܵൌ෍
௜ழ௝
௜ழ௝
௜ழ௝
ሺ݊௩ ௜ ൅ ݊௩ ௝ ሻଶ ൑ ʹሺ݊ െ ͳሻ ෍
ሺ݊௩ ௜ ൅ ݊௩ ௝ ሻଶ ൌ ʹ ෍
ሺ݊௩ ௜ ൅ ݊௩ ௝ ሻଶ ൌ ʹ ෍
௜ழ௝
௜ழ௝
௜ழ௝
ሺ݊௩ ௜ ൅ ݊௩ ௝ ሻ ൑ ʹሺ݊ െ ͳሻܲ‫ܫ‬௩ ሺ‫ܩ‬ሻǡ
ሺ݊௩ ௜ ଶ ൅ ʹ݊௩ ௜ ݊௩ ൅ ݊௩ೕ ଶ ሻ ൑ ʹሺ݊ െ ͳሻଶ ݉ ൅ Ͷܵ‫ݖ‬ሺ‫ܩ‬ሻǡ
௝
ሺ݊௩ ௜ ଶ ൅ ʹ݊௩ ௜ ݊௩ ൅ ݊௩ ௝ ଶ ሻ ൑ ʹሺ݊ െ ͳሻܲ‫ܫ‬௩ ሺ‫ܩ‬ሻ ൅ Ͷܵ‫ݖ‬ሺ‫ܩ‬ሻǤ
௝
7KLVFRPSOHWHVRXUSURRIᇝ
/HPPD/HW െ ’‡…ሺ
ሻ ൌ ሼߜଵ ǡ ߜଶ ǡ ǥ ǡ ߜ௡ ሽǡZKHUHߜଵ ൑ ߜଶ ൑ ‫ ڮ‬൑ ߜ௡ 7KHQ
ଶሺ௡ିଵሻ
ଶሺ௡ିଵሻ
ܲ‫ܫ‬௩ ሺ‫ܩ‬ሻǡ ට ௡ ඥሾሺ݊
௡
ߜ௡ ൑ ‫݊݅ܯ‬ሼට
ሺ௡ିଵሻ
െ ͳሻܲ‫ܫ‬௩ ሺ‫ܩ‬ሻ ൅ ʹܵ‫ݖ‬ሺ‫ܩ‬ሻሿǡ ට
௡
ඥሾͶ݉ሺ݊ െ ͳሻଶ ൅ Ͷܵ‫ݖ‬ሺ‫ܩ‬ሻሽ
௡ିଵ
௡ିଵ ଶ
3URRI 6LQFH σ௡௜ୀଵ ߜ௜ ൌ Ͳ ߜ௡ ൌ σ௡ିଵ
௜ୀଵ ߜ௜ 7KHUHIRUH ȁߜ௡ ȁ ൑ σ௜ୀଵ ȁߜ௜ ȁ ൑ ξ݊ െ ͳටσ௜ୀଵ ߜ௜ ଶ
ଶ
ଶ
ଶ
௡
DQGVRߜ௡ ଶ ൑ ሺ݊ െ ͳሻ σ௡ିଵ
௜ୀଵ ߜ௜ ൌ ሺ݊ െ ͳሻൣσ௜ୀଵሾߜ௜ െ ߜ௡ ൧Ǥ7KHUHIRUHߜ௡ ൑
ሺ௡ିଵሻ
௡
σ௡௜ୀଵ ߜ௜ ଶ Ǥ
1RZLWLVHQRXJKWRDSSO\/HPPDWRFRPSOHWHWKHOHPPDᇝ
7KHRUHP /HW * EH D FRQQHFWHG JUDSK 7KHQሺ
ሻ ൑ ඥʹ݊ σ௘ୀ௨௩ሺ݊௨ ൅ ݊௩ ሻଶ 8SSHU
ERXQGLVDFKLHYHGLIDQGRQO\LI*LVDWULYLDOJUDSK0RUHRYHUሺ
ሻ ൑ ξ݊ߙLQZKLFK
ߙ ൌ ‫݊݅ܯ‬ሼʹܲ‫ܫ‬௩ ሺ‫ܩ‬ሻଶ ǡ ʹሺ݊ െ ͳሻܲ‫ܫ‬௩ ሺ‫ܩ‬ሻǡ Ͷሺ݊ െ ͳሻଶ௠ ൅ Ͷܵ‫ݖ‬ሺ‫ܩ‬ሻǡ ʹሺ݊ െ ͳሻܲ‫ܫ‬௩ ሺ‫ܩ‬ሻ ൅ Ͷܵ‫ݖ‬ሺ‫ܩ‬ሻሽǤ
3URRI%\GHILQLWLRQDQG/HPPD
9ܲ‫ܧܫ‬ሺ‫ܩ‬ሻ ൌ σ௡௜ୀଵȁߜ௜ ȁ ൑ ට݊ σ௡௜ୀଵ ߜ௜ ଶ ൌ ටʹ݊ σ௘ୀ௨௩ఢாሺீሻሾ݊௨ ሺ݁ሻ ൅ ݊௩ ሺ݁ሻሿଶ ǡ
ܸܲ‫ܧܫ‬ሺ‫ܩ‬ሻ ൑ ξ݊ඥʹܲ‫ܫ‬௩ ሺ‫ܩ‬ሻଶ ൌ ξʹ݊ܲ‫ܫ‬௩ ሺ‫ܩ‬ሻ
ܸܲ‫ܧܫ‬ሺ‫ܩ‬ሻ ൑ ξ݊ඥʹሺ݊ െ ͳሻܲ‫ܫ‬௩ ሺ‫ܩ‬ሻ ൌ ඥʹ݊ሺ݊ െ ͳሻܲ‫ܫ‬௩ ሺ‫ܩ‬ሻǡ
ܸܲ‫ܧܫ‬ሺ‫ܩ‬ሻ ൑ ξ݊ඥͶሺ݊ െ ͳሻଶ ݉ ൅ Ͷܵ‫ݖ‬ሺ‫ܩ‬ሻ ൌ ʹξ݊ඥʹሺ݊ െ ͳሻଶ ݉ ൅ ܵ‫ݖ‬ሺ‫ܩ‬ሻǡ
ZKLFKFRPSOHWHVRXUWKHRUHPᇝ
/HPPD6XSSRVH*LVELSDUWLWHJUDSKZLWKHLJHQYDOXHV ߣଵ ൑ ߣଶ ൑ ‫ ڮ‬൑ ߣ௡ DQGYHUWH[
HLJHQYDOXHVRIߜଵ ൑ ‫ ڮ‬൑ ߜ௡ 7KHQܸܲ‫ܯܫ‬ሺ‫ܩ‬ሻ ൌ ݊‫ܣ‬ሺ‫ܩ‬ሻDQGߜ௜ ൌ ݊ߣ௜ ǡ ͳ ൑ ݅ ൑ ݊
-127-
3URRI 6XSSRVH * LV ELSDUWLWH 6LQFH * GRHV QRW KDYH RGG F\FOHV ୳ ሺ‡ሻ ൅ ୴ ሺ‡ሻ ൌ  IRU
HGJHH XY7KHUHIRUHܸܲ‫ܯܫ‬ሺ‫ܩ‬ሻ ൌ ݊‫ܣ‬ሺ‫ܩ‬ሻ
&RUROODU\ ,I * LV ELSDUWLWH WKHQ †‡–ሺሺ
ሻሻ ൌ ݊௡ †‡–ሺ‫ܣ‬ሺ‫ܩ‬ሻሻ DQG ݊ଶ ೙ඥ†‡–ሺ‫ܣ‬ሻ ൑
ܸܲ‫ܧܫ‬ሺ‫ܩ‬ሻ ൑ ݊ξʹ݉݊
/HW*EHDJUDSKZLWK9* ^YYYQ`DQGYHUWH[PDWUL[37KHQWKHYHUWH[
GHJUHHRIYLGHQRWHGE\3LLVJLYHQE\ܲ௜ ൌ σ௡௝ୀଵ ܽ௜௝ /HWWKHYHUWH[GHJUHHVHTXHQFHEH
^33 3Q` 7KHQ WKH VHFRQG YHUWH[ GHJUHH RI YL GHQRWHG E\ 7L LV JLYHQ E\ ܶ௜ ൌ
σ௡௝ୀଵ ܽ௜௝ ܲ௝ ,I^333Q`LVWKHGHJUHHVHTXHQFHWKHQ*LVNYHUWH[UHJXODULI3L NIRUDOOL,I*KDVWKHYHUWH[GHJUHHVHTXHQFH^333Q`DQGVHFRQGYHUWH[GHJUHH
்
GHJUHHVHTXHQFH^777Q`WKHQ*LVSVHXGRNYHUWH[GHJUHHUHJXODULI ௉೔ ൌ ݇IRU
೔
DOOL
7KHRUHP>@6XSSRVH*LVDJUDSKG GLDP*DQGW W*LVWKHQXPEHURIGLVWLQFW
HLJHQYDOXHV RI * 7KHQ ‫ ݐ‬൑ ݀ ൅ ͳ ,Q SDUWLFXODU W* LI DQG RQO\ LI * LV &RPSOHWH
0RUHRYHULI*KDVWKHYHUWH[GHJUHHVHTXHQFHሼଵ ǡ ଶ ǡ ǥ ǡ ୬ ሽDQGVHFRQGYHUWH[GHJUHH
VHTXHQFHሼଵ ǡ ଶ ǡ ǥ ǡ ୬ ሽ7KHQ
Ɂଵ ൒ ඨ
ଵ ଶ ൅ ଶ ଶ ൅ ǥ ൅ ୬ ଶ
ଵ ଶ ൅ ଶ ଶ ൅ ǥ ൅ ୬ ଶ
Ǥ
(TXDOLW\KROGVLIDQGRQO\LI*LVSVHXGRYHUWH[UHJXODU
7KHRUHP:LWKWKHQRWDWLRQVGHVFULEHGDERYH
ଶ
ଶ
σ௡ ܶ௜
σ௡ ܶ௜
ܸܲ‫ܧܫ‬ሺ‫ܩ‬ሻ ൑ ඨ ௡௜ୀଵ ଶ ൅ ሺ݊ െ ͳሻඨܵ െ ௡௜ୀଵ ଶ σ௜ୀଵ ܲ௜
σ௜ୀଵ ܲ௜
ZKHUH 6 LV WKH VXP RI WKH VTXDUHV RI HQWULHV LQ WKH YHUWH[ PDWUL[(TXDOLW\KROGVLIDQG
RQO\LIHLWKHU*LVDFRPSOHWHJUDSKRUDSVHXGRNYHUWH[UHJXODUJUDSKZLWKWKUHHGLVWLQFW
ௌି௞ మ
ௌି௞ మ
HLJHQYDOXHሺ݇ǡ ට ௣ିଵ ǡ െට ௣ିଵ ሻ
3URRI/HW^ߜ ߜ įQ`EHWKHHLJHQYDOXHVRI*7KHQ
෍
௡
ߜ௜ ൌ Ͳǡ ෍
௜ୀଵ
௡
ߜ௜ ଶ ൌ ܵ ൌ ෍
௜ୀଵ
௡
ሺܽ௜௝ ሻଶ
௜ǡ௝ୀଵ
-128-
1RZDSSO\LQJWKH&DXFK\±6FKZDUW]LQHTXDOLW\WRWKHWZRQíYHFWRUVDQG
_į__į__įQ_ZHJHW
ሺܸܲ‫ܧܫ‬ሺ‫ܩ‬ሻ െ ߜଵ ሻଶ ൑ ሺ݊ െ ͳሻሺܵ െ ߜଵ ଶ ሻ
7KXV
ܸܲ‫ܧܫ‬ሺ‫ܩ‬ሻ ൑ ߜଵ ൅ ටሺ݊ െ ͳሻሺܵ െ ߜଵ ଶ ሻ
ଶ௉ூ
'HILQH D IXQFWLRQ ݂ሺ‫ݔ‬ሻ ൌ ‫ ݔ‬൅ ඥሺ݊ െ ͳሻሺܵ െ ‫ ݔ‬ଶ ሻ IRU ೡ ൑ ‫ ݔ‬൑ ξܵ 7KHQ E\ DSSO\LQJ WKH
௡
ௌ
PD[±PLQWHFKQLTXHVRIFDOFXOXVZHFDQVHHWKDWI[LVPRQRWRQLFDOO\GHFUHDVLQJLQ‫ ݔ‬൒ ට௡
1RZE\&DXFK\±6FKZDUW]LQHTXDOLW\ZHKDYH
ܲ௜ ଶ ൌ ሺ෍
7KHQ
෍
௡
ܲ௜ ଶ ൑ ෍
௜ୀଵ
௡
௡
௝ୀଵ
݊෍
௜ୀଵ
ܽ௜௝ ሻଶ ൑ ݊ ෍
௡
௝ୀଵ
ܽ௜௝ ଶ ൌ ݊ ෍
௡
௝ୀଵ
௡
௜ୀଵ
ܽ௜௝ ଶ ෍
௡
௝ୀଵ
ܽ௜௝ ଶ ൌ ݊ܵ
$OVR ܶ௜ ൌ σ௡௝ୀଵ ܽ௜௝ ܲ௝ ൒ σ௡௝ୀଵ ܽ௜௝ ଶ DQG σ௡௜ୀଵ ܶ௜ଶ ൒ σ௡௜ୀଵሺσ௡௝ୀଵ ܽ௜௝ ଶ ሻଶ ൒ ܵ ଶ +HQFH ߜ௜ ൒
σ೙
்೔మ
మ
೔సభ ௉೔
ටσ೙೔సభ
ௌమ
σ೙ ் మ
ௌ
൒ ට௡ௌ ൌ ට௡Ǥ 7KHUHIRUH ܸܲ‫ܧܫ‬ሺ‫ܩ‬ሻ ൑ ݂ሺߜଵ ሻ ൑ ݂ሺටσ೙೔సభ ௉೔మ ሻ DQG WKXV WKH WKHRUHP LV
೔సభ ೔
SURYHG:HQRZVXSSRVHWKHHTXDOLW\KROGV7KHQ
σ௡ ܶ௜ଶ
ߜଵ ൌ ඨ ௜ୀଵ
σ௡௜ୀଵ ܲ௜ଶ
DQG E\ 7KHRUHP ZH KDYH * LV SVHXGR NYHUWH[ UHJXODU $OVR HTXDOLW\ KROGV LQ WKH
&DXFK\±6FKZDUW] LQHTXDOLW\ +HQFH ȁߜଶ ȁ ൌ ȁߜଷ ȁ ൌ ‫ ڮ‬ൌ ȁߜ௡ ȁ DQG VR ሺσ௡௜ୀଶȁߜ௜ ȁሻଶ ൌ
ௌିఋభ మ
ሺ݊ െ ͳሻሺܵ െ ߜଵ ଶ ሻ +HQFHȁߜ௜ ȁ ൌ ට
௡ିଵ
ௌିఋభ మ
ǡ݅ ൌ ʹǡ͵ǡ ǥ ǡ ݊Ǥ 6LQFH ȁߜ௜ ȁ ൌ ට
௡ିଵ
ǡ ߜ௜ FDQ KDYH DW
PRVWWZRGLVWLQFWYDOXHVDQGZHDUULYHDWWKHIROORZLQJ
‡ * KDV H[DFWO\ RQH HLJHQYDOXH 7KHQ DOO HLJHQYDOXHV DUH ]HUR DV WKH VXP RI
HLJHQYDOXHVLVWKHWUDFHRI3DQGDV*LVFRQQHFWHG6R* .
‡*KDVH[DFWO\WZRGLVWLQFWHLJHQYDOXHV7KHQE\7KHRUHP*LVFRPSOHWH
-129-
σ೙
்మ
‡ * KDV H[DFWO\ WKUHH GLVWLQFW HLJHQYDOXHV 7KHQ ߜଵ ൌ ටσ೙೔సభ ௉೔మ DQG ȁߜ௜ ȁ ൌ
೔సభ ೔
మ
ටௌିఋభ ǡ݅ ൌ ʹǡ͵ǡ ǥ ǡ ݊ $OVR ்೔ ൌ ݇ IRU DOO L 7KHQ ZH JHW * DV D JUDSK ZLWK WKUHH GLVWLQFW
௡ିଵ
௉೔
ௌି௞ మ
ௌି௞ మ
HLJHQYDOXHVሺ݇ǡ ට
ǡ െට
ሻSURYLQJWKHUHVXOWᇝ
௣ିଵ
௣ିଵ
5HIHUHQFHV
$ 5 $VKUDIL 0 -DODOL 0 *KRUEDQL 0 9 'LXGHD &RPSXWLQJ 3, DQG RPHJD
SRO\QRPLDOV RI DQ LQILQLWH IDPLO\ RI IXOOHUHQHV 0$7&+ &RPPXQ 0DWK &RPSXW
&KHP±
$ 5 $VKUDIL 0 0LU]DUJDU 7KH HGJH 6]HJHG SRO\QRPLDO RI JUDSKV 0$7&+
&RPPXQ0DWK&RPSXW&KHP±
' 0 &YHWNRYLü 0 'RRE + 6DFKV 6SHFWUD RI *UDSKV $FDGHPLF 3UHVV 1HZ
<RUN
+ 'HQJ 2Q WKH 3, LQGH[ RI D JUDSK 0$7&+ &RPPXQ 0DWK &RPSXW&KHP ±
$ $ 'REU\QLQ 5 (QWULQJHU , *XWPDQ :LHQHU LQGH[ RI WUHHV 7KHRU\ DQG
DSSOLFDWLRQV$FWD$SSO0DWK±
$ $ 'REU\QLQ , *XWPDQ 6 .ODYåDU 3 äLJHUW :LHQHU LQGH[ RI KH[DJRQDO
V\VWHPV$FWD$SSO0DWK±
* + )DWK±7DEDU 0 - 1DGMDIL±$UDQL 0 0RJKDUUDE $ 5 $VKUDIL 6RPH
LQHTXDOLWLHVIRU6]HJHG±OLNHWRSRORJLFDOLQGLFHVRIJUDSKV0$7&+&RPPXQ0DWK
&RPSXW&KHP±
, *XWPDQ $ 5 $VKUDIL 2Q WKH 3, LQGH[ RI SKHQ\OHQHV DQG WKHLU KH[DJRQDO
VTXHH]HV0$7&+&RPPXQ0DWK&RPSXW&KHP±
, *XWPDQ 7KH HQHUJ\ RI JUDSKV %HU 0DWK6WDWLVW 6HFW )RUVFKXQJV] *UD] ±
- +DR 6RPH ERXQGV IRU 3, LQGLFHV 0$7&+ &RPPXQ 0DWK &RPSXW &KHP ±
70DQVRXU06FKRUN7KH3,LQGH[RIEULGJHDQGFKDLQJUDSKV0$7&+&RPPXQ
0DWK&RPSXW&KHP±
0 0LU]DJDU 3, 6]HJHG DQG HGJH 6]HJHG SRO\QRPLDOV RI D GHQGULPHU QDQRVWDU
0$7&+&RPPXQ0DWK&RPSXW&KHP±
-130-
$5$VKUDIL0-1DGMDIL±$UDQL*+)DWK±7DEDU00LU]DUJDU6KDUSERXQGV
RQWKH3,VSHFWUDOUDGLXVVXEPLWWHG
0-1DGMDIL±$UDQL*+)DWK±7DEDU$5$VKUDIL([WUHPDOJUDSKVZLWKUHVSHFW
WRWKHYHUWH[3,LQGH[$SSO0DWK/HWW±
+:LHQHU6WUXFWXUDOGHWHUPLQDWLRQRIWKHSDUDIILQERLOLQJSRLQWV-$P&KHP6RF
±