• No results found

download full version

N/A
N/A
Protected

Academic year: 2020

Share "download full version"

Copied!
11
0
0

Loading.... (view fulltext now)

Full text

(1)

8NU-3K\V2SW9 <X9DV\ONLY<X1DVWLVKLQDQG59ORNK

,QVWLWXWHRI3K\VLFDO2SWLFV'UDJRPDQRY6W/YLY8NUDLQH HPDLOYORNK#LIROYLYXD

5HFHLYHG

$EVWUDFW

:H KDYH SUHVHQWHG FRPSXWHU VLPXODWLRQV RI FRQRVFRSLF SDWWHUQV RFFXUULQJ LQ RSWLFDOO\ XQLD[LDOJ\URWURSLFFU\VWDOVLQWKHYLFLQLW\RILVRWURSLFSRLQWIRUDQXPEHURIVHWVRIFU\VWDO SDUDPHWHUV DQG RULHQWDWLRQV 7KH DSSHDUDQFH RI VSHFLDO GLUHFWLRQV FKDUDFWHUL]HG E\ HTXDOL]DWLRQRIOLQHDUDQGFLUFXODUELUHIULQJHQFHVKDVEHHQUHYHDOHG

.H\ZRUGVFRQRVFRSLFSDWWHUQVJ\URWURSLFFU\VWDOVLVRWURSLFSRLQW

3$&6E/F+]

,QWURGXFWLRQ

7KH WHUP LVRWURSLF SRLQW FRQFHUQV WKH FRQGLWLRQVXQGHUZKLFKRSWLFDOELUHIULQJHQFHRI D QRQFXELF RSWLFDOO\ XQLD[LDO FU\VWDO EHFRPHV ]HUR6XFKWKHFRQGLWLRQVDUHXVXDOO\UHDOL]HGDW DFHUWDLQZDYHOHQJWKlLRIRSWLFDOUDGLDWLRQDQG WKH\ LPSO\ LQ IDFW D UHYHUVDO RI WKH OLQHDU ELUHIULQJHQFH VLJQ 2QH RI WKH IHDWXUHV RI WKH LVRWURSLF SRLQW LV WKDW WKH FU\VWDO UHPDLQV DQLVRWURSLF DW lL WKXV RIIHULQJ D XQLTXH SRVVLELOLW\ IRU REVHUYDWLRQ RI DQLVRWURS\ RI WKH RWKHU RSWLFDO HIIHFWV IRU LQVWDQFH WKH RSWLFDO DFWLYLW\ VHH HJ >±@ ,W LV ZHOO NQRZQ WKDW WKH LQGLFDWLYH VXUIDFH RI WKH RSWLFDO DFWLYLW\ GHVFULELQJWKHVHFRQGUDQND[LDO WHQVRUJLM FDQ EH SUHVHQWHG DV D SVHXGR VFDODU J\UDWLRQ SDUDPHWHU

LM L M

* J O O=

ZKHUH O=VLQ FRVQ j O=VLQ VLQQ j DQG

FRV

O = Q GHILQH WUDQVIRUPDWLRQ UHODWLRQV EHWZHHQ WKH SRODU DQG &DUWHVLDQ FRRUGLQDWH V\VWHPVDQGWKHXQLWYHFWRUUrFRLQFLGHVZLWKWKH ZDYHYHFWRUVHH)LJ>@$FFRUGLQJWRRQHRI

WKH SURSHUWLHV RI WKH LQGLFDWLYH VXUIDFHV RI JLM WHQVRUWKHVHVXUIDFHVFRUUHVSRQGWRWKHVXUIDFHV RIFLUFXODUELUHIULQJHQFH

F U O *

Q Q Q

Q rl

p

D = - = =

ZKHUH QU DQG QO DUH WKH UHIUDFWLYH LQGLFHV RI ULJKWDQGOHIWFLUFXODUO\SRODUL]HGZDYHVrWKH VSHFLILF URWDWRU\ SRZHU DQG Q WKH PHDQ UHIUDFWLYH LQGH[ RI OLQHDUO\ SRODUL]HG ZDYHV SHFXOLDU IRU WKH JLYHQ GLUHFWLRQ RI OLJKW SURSDJDWLRQ 7KH ELUHIULQJHQFH LQ J\URWURSLF FU\VWDOV ZLWK WKH LVRWURSLF SRLQW GRHV QRW

(2)

FRPSOHWHO\GLVDSSHDUDWlLEHFDXVHWKHFLUFXODU ELUHIULQJHQFH UHPDLQV WR H[LVW 0RUHRYHU WKH FLUFXODUELUHIULQJHQFHVKRXOGEHFKDUDFWHUL]HGE\ VRPH VSDWLDO GLVWULEXWLRQ VLQFH WKH FU\VWDO UHPDLQV DQLVRWURSLF DW lL ,Q SDUWLFXODU WKH GLVWULEXWLRQRIWKHFLUFXODUELUHIULQJHQFHVKRXOG OHDG WR DSSHDUDQFH RI FRQRVFRSLF IULQJHV >@ $VZHKDYHHDUOLHUVKRZQ>±@WKHFRQRVFRSLF IULQJHV FDXVHG E\ VSDWLDO GLVWULEXWLRQ RI WKH FLUFXODU ELUHIULQJHQFH GLIIHU IURP WKRVH FDXVHG E\ WKH OLQHDU ELUHIULQJHQFH 7KH REVHUYDWLRQ RI ³FLUFXODU FRQRVFRSLF IULQJHV´ UHSUHVHQWV UDWKHU GLIILFXOW H[SHULPHQWDO WDVN EHFDXVH WKH FLUFXODU ELUHIULQJHQFH HIIHFW LVXVXDOO\ VHYHUDO RUGHUV RI PDJQLWXGH VPDOOHU WKDQ WKH OLQHDU RQH 5HDOO\ WKH XVXDO YDOXH RI WKH FLUFXODU ELUHIULQJHQFH LV a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

Q Q Q

D = D + D

ZKHUHDQOLVWKH OLQHDU ELUHIULQJHQFH ,W ZRXOG EHQDWXUDOWRDVVXPHWKDWWKHFRQRVFRSLFIULQJHV LQ WKLV VSHFWUDO UDQJH DUH GHWHUPLQHG E\ VSDWLDO GLVWULEXWLRQ RI WKH WRWDO ELUHIULQJHQFH RU WKH ELUHIULQJHQFH IRU HOOLSWLFDOO\ SRODUL]HG HLJHQZDYHV DQGWKXVWKH VKDSH RI WKH IULQJHV VKRXOGGLIIHUIURPWKRVHFDXVHGE\WKHSXUHRU TXDVLSXUHOLQHDURUFLUFXODUELUHIULQJHQFHV

7KHDLPRIWKHSUHVHQWSDSHULVWRDQDO\]H WKH VKDSH RI FRQRVFRSLF IULQJHV IRU RSWLFDOO\ XQLD[LDOJ\URWURSLFFU\VWDOVLQWKHFORVHVSHFWUDO UDQJHRIWKHLVRWURSLFSRLQW

6LPXODWLRQ$OJRULWKP

7RKHOSLQXQGHUVWDQGLQJRIWKHUHVXOWVSUHVHQWHG EHORZOHWXVUHFDOOLQEULHIWKHNQRZQDOJRULWKP >@IRUWKHFDOFXODWLRQVRI FRQRVFRSLF SDWWHUQV LQ WKH FDVH RI ELUHIULQJHQW J\URWURSLF FU\VWDOV 7KH RSWLFDO WUDQVPLWWDQFH WKH OLJKW LQWHQVLW\ QRUPDOL]HGWRLWVYDOXHDWWKHHQWUDQFHLQWHUIDFH RI D QRQDEVRUELQJ J\URWURSLF ELUHIULQJHQW FU\VWDO SODWH SODFHG EHWZHHQ FURVVHG SRODUL]HUV WDNHVWKHIROORZLQJIRUP>@

FRV VLQ VLQ VLQ , = c- b b c- d

ZKHUH

p QG d

l

= D

LV WKH SKDVH GLIIHUHQFH ZLWK DQ EHLQJ GHWHUPLQHGE\ (Tc UHSUHVHQWV WKH DQJOH GHILQLQJ WKH RIIRULHQWDWLRQ RI SRODUL]HU DQG DQDO\]HU DQG b WKH DQJOH EHWZHHQ RQH RI FU\VWDOORSK\VLFDO GLUHFWLRQV O\LQJ LQ WKH SODQH SHUSHQGLFXODU WR WKH RSWLFDO EHDP DQG WKH WUDQVPLWWDQFH GLUHFWLRQ RI SRODUL]HU )RU RXU JHRPHWU\ c p= DQG b p= ZH FDQ UHZULWH(TDVIROORZV

VLQ O *

, G Q

Q p

l

= D +

/HW XV UHPLQG WKDW (T LV ZULWWHQ RQO\ IRU WKH FHQWUDO EHDP RI WKH GLYHUJHQW UD\ $FFRXQWLQJIRUVSDWLDOGLVWULEXWLRQRIWKHOLQHDU DQG FLUFXODU ELUHIULQJHQFHV PRGLILHV (T UHVXOWLQJLQ

VLQ VLQ LM L M

P

J O O

, G Q

Q p

l

= D Q +

ZKHUH DQP GHQRWHV WKH PD[LPXP YDOXH RI WKH OLQHDU ELUHIULQJHQFH LQ XQLD[LDO FU\VWDOV IRU WKH SURSDJDWLRQGLUHFWLRQSHUSHQGLFXODUWRWKHRSWLF D[LV )LQDOO\ DIWHU FRQVLGHULQJ WKH DQJXODU GHSHQGHQFHRIGZHREWDLQWKHUHODWLRQ

(

)

VLQ

VLQ LM L M P

,

J O O

G Q Q p l h x = í Ä Ô

í + + ÅÅD Q + ÕÕ

Æ Ö

(3)

8NU-3K\V2SW9 ZKHUHhDQGx DUHWKHFRRUGLQDWHVRIWKHH[LW

LQWHUIDFH RI WKH FU\VWDO SODWH WKH RULJLQ RI WKH FRRUGLQDWH V\VWHP h x FRLQFLGHV ZLWK WKH RXWSXW SRLQW IRU WKH FHQWUDO EHDP RI WKH UD\ FRQH± VHH )LJ 3ULRU WR GHVFULSWLRQ RI WKH REWDLQHG UHVXOWV LW ZRXOG EH UHDVRQDEOH WR SUHVHQWDIHZGHILQLWLRQVXVHGODWHU

LWKH RSWLF D[LV IRU OLQHDUO\ SRODUL]HG ZDYHV LV D GLUHFWLRQ LQ FU\VWDO LQ ZKLFK WKH YHORFLWLHV RU WKH FRUUHVSRQGLQJ UHIUDFWLYH LQGLFHV RI RUWKRJRQDO OLQHDUO\ SRODUL]HG HLJHQZDYHVDUHWKHVDPH

LLWKH RSWLF D[LV IRU FLUFXODUO\ SRODUL]HG ZDYHVLVDGLUHFWLRQLQDQLVRWURSLFFU\VWDODWWKH ZDYHOHQJWK RI WKH LVRWURSLF SRLQW LQ ZKLFK WKH YHORFLWLHV RU WKH UHIUDFWLYH LQGLFHV RI OHIW DQG ULJKW FLUFXODUO\ SRODUL]HG HLJHQZDYHV DUH WKH VDPHZHKDYH*=IRUWKHVHGLUHFWLRQV

5HVXOWVDQGGLVFXVVLRQ

D3URSDJDWLRQRIOLJKWDORQJ]D[LVLQFU\VWDOV EHORQJLQJWRWKHSRLQWV\PPHWU\JURXSV DQG 7KHUHVXOWVRIFRPSXWHUVLPXODWLRQVRIWKH FRQRVFRSLFSDWWHUQVIRUWKHFDVHRISURSDJDWLRQ RIOLJKWDORQJ]GLUHFWLRQLQFU\VWDOVWKDWEHORQJ WR WKH SRLQW JURXSV RI V\PPHWU\ DQGDUHSUHVHQWHGLQ)LJXUHVWR7KH J\UDWLRQWHQVRUIRUWKHVHV\PPHWU\JURXSVWDNHV WKHIRUP LM J J J J = RU LM J J J J -= -

+HQFH WKH SVHXGR VFDODU J\UDWLRQ SDUDPHWHUPD\EHUHSUHVHQWHGDV VLQ FRV

VLQ VLQ FRV VLQ FRV * J J J J J j j

= Q +

+ Q + Q =

= Q + Q

RU VLQ FRV VLQ VLQ FRV

FRV VLQ * J J J J J j j

= - Q

-- Q + Q =

= Q - Q

:H GHILQH WKH DSHUWXUH DQJOH RI WKH GLYHUJHQWRSWLFDOUD\WREHHTXDOWRŠ/HWWKH FU\VWDO SODWH KDYH WKH WKLFNQHVV G =P VXFK DUWLILFLDOO\ODUJHYDOXHLVXVHGIRULQFUHDVLQJWKH FLUFXODU UHWDUGDWLRQ DQG WKH J\UDWLRQ WHQVRU FRPSRQHQWV DUH í - :H ZLOO FKDQJH WKH

OLQHDUELUHIULQJHQFHLQWKHUDQJHRI - -$V

D ILUVW VWHS ZH GHPRQVWUDWH WKH FRQRVFRSLF IULQJHV IRU WKH VLPSOHVW FDVH ZKHQ

J =J =J þ DQG D =QO VHH )LJ +HUH WKH DSSHDUDQFH RI LQWHUIHUHQFH SDWWHUQV LV FDXVHG PHUHO\ E\ DQJXODU GHSHQGHQFH RI WKH UHWDUGDWLRQ RQ WKH VDPSOH WKLFNQHVV :KHQ

J =J =J = DQG D =QO WKH GDUN ILHOG ZRXOGEHREVHUYHGRQWKHVFUHHQ

,Q WKH QH[W WZR FDVHV WKH J\UDWLRQ WHQVRU VXUIDFHV GR QRW H[KLELW D VLJQ UHYHUVDO DQG VR WKHUH LV QR GLUHFWLRQ LQ FU\VWDO ZKHUH WKH YHORFLWLHV RI WKH ULJKW DQG OHIW FLUFXODUO\ SRODUL]HGZDYHVDUHHTXDO,QFUHDVHLQWKHOLQHDU ELUHIULQJHQFH LQ WKRVH FDVHV VHH )LJ DQG OHDGV RQO\ WR LQFUHDVLQJ GHQVLW\ RI ULQJOLNH FRQRVFRSLFIULQJHV

*UDGXDO FKDQJHV LQ WKH FRQRVFRSLF IULQJHV IRU WKH FU\VWDOV ZLWK VLJQUHYHUVHG J\UDWLRQ VXUIDFH RFFXUULQJ XQGHU LQFUHDVLQJ OLQHDU ELUHIULQJHQFH H[KLELW SHFXOLDULWLHV FRQVLVWLQJ LQ DSHULRGLFLW\LHWKHDSSHDUDQFHRIWKLFN

)LJ &RQRVFRSLF SDWWHUQV IRU J\URWURSLF FU\VWDOV LQ FDVH RI J=J=Jþ DQG

O Q

(4)

D E F G

(5)

8NU-3K\V2SW9 LQWHUIHUHQFHULQJWKDW PRYHVWRZDUGVWKH FHQWUH

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

,QFDVHRIWKHJ\UDWLRQVXUIDFHVKRZLQJQR VLJQ UHYHUVDO WKH FRQRVFRSLF IULQJHV DUH WKH VDPH DV WKH XVXDO FRQRVFRSLF IULQJHV FDXVHG RQO\ E\ WKH OLQHDU ELUHIULQJHQFH LQ RSWLFDOO\ XQLD[LDOFU\VWDOVDQGREVHUYHGLQWKHGLUHFWLRQRI OLJKWSURSDJDWLRQSHUSHQGLFXODUWRWKHRSWLFD[LV VHH)LJ

+RZHYHU WKH FKDQJHV LQ WKH FRQRVFRSLF SDWWHUQV DUH HVVHQWLDOO\ GLIIHUHQW LQ WKH RWKHU FDVHV)RUH[DPSOHDZLGHLQWHUIHUHQFHEDQGLV REVHUYHG XQGHU WKH FRQGLWLRQV J J J í DQGDQO )LJD 7KLV EDQG LV SHUSHQGLFXODUWR]D[LV,WLVQRWHZRUWK\WKDWWKH [\SODQHLVWKHQWKHSODQHRIDEVHQFHRIFLUFXODU ELUHIULQJHQFHRULQVRPHRWKHUWHUPVWKHSODQH RIRSWLFD[LVIRUWKHFLUFXODUO\SRODUL]HGZDYHV :LWKLQFUHDVLQJOLQHDUELUHIULQJHQFHWKLVEDQGLV WUDQVIRUPHGWRRYDOVHH)LJFDQGGZKLFKLV SHFXOLDU IRU WKH OLJKW SURSDJDWLRQ DORQJ WKH ELVHFWRURIDFXWHDQJOHEHWZHHQWKHRSWLFD[HVLQ RSWLFDOO\ ELD[LDO FU\VWDOV 7KH DQJOH EHWZHHQ WKRVH ³RSWLF D[HV´ LQFUHDVHV ZLWK VXEVHTXHQW LQFUHDVH LQ WKH OLQHDU ELUHIULQJHQFH )LJH 7KHGLUHFWLRQVRIWKH³RSWLFD[HV´IRUPWKHFRQH ZKRVH DQJOH GHFUHDVHV ZLWK LQFUHDVLQJ OLQHDU ELUHIULQJHQFH7KHVHFWLRQRIWKLVFRQHE\WKH[\ SODQHUHSUHVHQWVDWKLFNLQWHUIHUHQFHULQJVHHQLQ )LJGDQGH)LQDOO\WKHFRQHDQJOHKHUHWKH PDWWHU FRQFHUQV WKH FRQH RI VSHFLDO GLUHFWLRQV FKDUDFWHUL]HG E\ H[WUHPH YDOXHV RI WKH WRWDO

)LJ&RQRVFRSLFSDWWHUQVIRUWKHSURSDJDWLRQGLUHFWLRQDORQJ]D[LVLQJ\URWURSLFFU\VWDOVDWGLIIHUHQW YDOXHV RI OLQHDU ELUHIULQJHQFH DQG WKH FRUUHVSRQGLQJ VXUIDFHV RI WRWDO HOOLSWLFDO ELUHIULQJHQFH DD =QO ED =QO -FD = íQO -GD =QO -HD = íQO - DQG ID = íQO -

J = í -

(6)

HOOLSWLFDO ELUHIULQJHQFH DQG HTXDO YDOXHV RI WKH OLQHDU DQG FLUFXODU ELUHIULQJHQFHV WHQGV WR WKH FULWLFDO DQJOH ZKRVH PDJQLWXGH LV GHWHUPLQHG

E\WKHUDWLRRIOLQHDUDQGFLUFXODUELUHIULQJHQFHV )LJI

+HUHZHDUHWRUHPLQGWKDWWKHHLJHQZDYHV )LJ&RQRVFRSLFSDWWHUQVIRUWKHSURSDJDWLRQGLUHFWLRQSHUSHQGLFXODUWRWKHRSWLFD[LVLQJ\URWURSLF FU\VWDOV DW GLIIHUHQW YDOXHV RI OLQHDU ELUHIULQJHQFH D D =QO E D =QO - DQG

F

O

Q

-D =

J = í -

J = l=QPDQGG=P

(7)

8NU-3K\V2SW9 GHDOW ZLWK DUH HOOLSWLFDOO\ SRODUL]HG ZKHQHYHU

WKH FLUFXODU DQG OLQHDU ELUHIULQJHQFHV VLPXOWDQHRXVO\H[LVW7KXVRQHVKRXOGFRQVLGHU WKH UHIUDFWLYH LQGLFHV DQG WKH ELUHIULQJHQFH IRU

WKRVH HOOLSWLFDOO\ SRODUL]HG ZDYHV 7KH FKDQJHV LQWKHFRQRVFRSLFSDWWHUQVSUHVHQWHGLQ)LJD±G FRUUHVSRQG WR WKH FRQGLWLRQV XQGHU ZKLFK WKH GLUHFWLRQRIHTXDOLW\RIWKHUHIUDFWLYHLQGLFHVIRU )LJ &RQRVFRSLF SDWWHUQV IRU WKH SURSDJDWLRQ GLUHFWLRQ FKDUDFWHUL]HG ZLWK WKH FRQGLWLRQV

F

*= D =Q Š ZLWK UHVSHFW RI ] D[LV LQ WKH ][ SODQH DQG SHUSHQGLFXODU WR WKH RSWLF D[LV LQ J\URWURSLFFU\VWDOVDWGLIIHUHQWYDOXHVRIOLQHDUELUHIULQJHQFHDQGWKHFRUUHVSRQGLQJVXUIDFHVRIWRWDO HOOLSWLFDOELUHIULQJHQFHDD =QO E

O

Q

-D = F

O

Q

-D = G

O

Q

-D = í DQGH

O

Q

-D =

J = í -

J = - í - l=QPDQGG =P

(8)

WKHULJKWDQGOHIWFLUFXODUO\SRODUL]HGZDYHVLWLV LQ WKH [\ SODQH± VHH )LJD WUDQVIRUPV JUDGXDOO\ WR WKH FRQH RI VSHFLDO GLUHFWLRQV ZKLFKDUHFKDUDFWHUL]HGE\DQRPDORXVYDOXHVRI WKHWRWDOHOOLSWLFDOELUHIULQJHQFHVHH)LJFDQG G )XUWKHU LQFUHDVH LQ WKH OLQHDU ELUHIULQJHQFH JLYHV ULVH WR GHFUHDVLQJ FRQH DQJOH 6LPLODU EHKDYLRXU LV REVHUYHG LQ FDVH ZKHQ WKH OLJKW SURSDJDWHV DORQJ WKH GLUHFWLRQ RI ]HUR J\UDWLRQ SDUDPHWHULHD]HURFLUFXODUELUHIULQJHQFHLQ FU\VWDOVZLWKWKHVLJQUHYHUVHGJ\UDWLRQVXUIDFH F 3URSDJDWLRQ RI OLJKW LQ WKH GLUHFWLRQ ZKHUH

F

*= D =Q LQ WKH XQLD[LDO FU\VWDOV RI SRLQW V\PPHWU\JURXSVDQG

,Q )LJ RQH FDQ VHH KRZ WKH VSHFLDO GLUHFWLRQV DSSHDU ZKHQ WKH OLJKW SURSDJDWHV LQ ][SODQHDORQJWKHŠGLUHFWLRQZLWKUHVSHFWWR ]D[LV:LWKLQFUHDVLQJOLQHDUELUHIULQJHQFHWKH DQJOHRIWKHFRQHZKHUHWKHVHVSHFLDOGLUHFWLRQV EHORQJ QHDUV WR WKH VPDOOHVW YDOXH GHWHUPLQHG E\ WKH UDWLR RI WKH FLUFXODU DQG OLQHDU

ELUHIULQJHQFHVVHHHJ>@)LJF±I

G3URSDJDWLRQ RI OLJKW DORQJ WKH RSWLF D[LV LQ WKH XQLD[LDO FU\VWDOV RI SRLQW V\PPHWU\ JURXS

P

7KHJ\UDWLRQWHQVRU IRU WKH SRLQW JURXSRI

V\PPHWU\PKDVWKHIROORZLQJIRUP

LM J

J = -J

7KHQWKHVFDODUJ\UDWLRQSDUDPHWHUPD\EH UHSUHVHQWHGDV

VLQ FRV VLQ VLQ

VLQ FRV VLQ

* J J

J

j j

j j

= Q - Q =

= Q -

7KH FRQRVFRSLF SDWWHUQV W\SLFDO IRU SURSDJDWLRQRIOLJKWDORQJ]D[LVDUHGHSLFWHGLQ )LJ $V VHHQ IURP )LJ WKH SDWWHUQV IRU WKH FDVH RI DQOpDQF UHYHDO TXLWH XQXVXDO VKDSH WKH\ DUH LQYDULDQW ZLWK UHVSHFW WR WKH V\PPHWU\RSHUDWLRQRIIRXUIROGD[LV0RUHRYHU WKH LQWHUIHUHQFH FXUYHV EHFRPH FORVHG DQG )LJ&RQRVFRSLFSDWWHUQVIRUWKHSURSDJDWLRQGLUHFWLRQDORQJ[RU\GLUHFWLRQVLQJ\URWURSLFFU\VWDOV RISRLQWJURXSPDWGLIIHUHQWYDOXHVRIOLQHDUELUHIULQJHQFHDQGWKHFRUUHVSRQGLQJVXUIDFHVRIWRWDO HOOLSWLFDO ELUHIULQJHQFH D D =QO ED =QO - FD =QO í - GD = íQO -

H

O

Q

-D = í DQGI O

Q

-D =

(9)

8NU-3K\V2SW9 DSHULRGLF LI

O Q

-D f 2QH FDQ FRQFOXGH WKDW WKH UHJLRQ RI UDUHILFDWLRQ RI WKH LQWHUIHUHQFH FXUYHV FRUUHVSRQGV WR DSSHDUDQFH RI WKH PHQWLRQHG VSHFLDO GLUHFWLRQV ZKLFK OLH RQ D UDWKHU FRPSOLFDWHG VXUIDFH WKH RQH LQYDULDQW ZLWK UHVSHFW WR WKH IRXUIROG V\PPHWU\ D[LV :LWKIXUWKHULQFUHDVLQJOLQHDUELUHIULQJHQFHWKH VXUIDFH FRQYROYHV LQWR D VLQJOH OLQH RULHQWHG DORQJ]D[LVZKLFKLVLQIDFWWKHRUGLQDU\RSWLF D[LVLQXQLD[LDOFU\VWDOV

H3URSDJDWLRQRIOLJKWDORQJ[RU\D[HVLQWKH XQLD[LDOFU\VWDOVRISRLQWV\PPHWU\JURXSP

7KH FRQRVFRSLF IULQJHV SHFXOLDU IRU WKH SURSDJDWLRQRIOLJKWDORQJ[DQG\GLUHFWLRQVLQ FU\VWDOVEHORQJLQJWRWKHSRLQWJURXSRIV\PPHW U\PDUHWKHVDPH:KHQWKHOLJKWSURSDJDWHV DORQJ[RU\ GLUHFWLRQV XQGHU WKH FRQGLWLRQV RI

O Q

D = WKH LQWHUIHUHQFH IULQJHV DUH HOOLSVHV 7KRVH HOOLSVHV WUDQVIRUP WR K\SHUEROHV ZLWK LQFUHDVLQJ OLQHDU ELUHIULQJHQFH WKURXJK WKH LQWHUPHGLDWH FLUFXODU DQG K\SHUEROLFHOOLSWLFDO

VWDWHVVHH)LJ1RWLFHWKDWWKHHOOLSWLFDORU FLUFXODU VKDSHV RI WKH LQWHUIHUHQFH SDWWHUQV RFFXUULQJ LQ WKLV FDVH PHDQ WKDW WKH VXUIDFH RI WKH WRWDO ELUHIULQJHQFH SRVVHVVHV WKH VDPH V\PPHWU\DVWKHFRQRVFRSLFIULQJHV

I3URSDJDWLRQ RI OLJKW LQ WKH GLUHFWLRQ ZKHUH

F

*= D =Q Š ZLWK UHVSHFW WR [ D[LV LQ [\ SODQHLQWKHXQLD[LDOFU\VWDOVRISRLQWV\PPHWU\ JURXSP

:KHQ WKH OLJKW SURSDJDWHV GLUHFWO\ DORQJ WKH VRFDOOHG RSWLF D[LV IRU FLUFXODUO\ SRODUL]HG ZDYHVLHDORQJWKHGLUHFWLRQZKHUHD =QF RQHFDQREVHUYHDWUDQVIRUPDWLRQRILQWHUIHUHQFH IULQJHVIURPDQHOOLSWLFILJXUHWRWKHRWKHURQH WKURXJKDVWUDLJKWLQWHUIHUHQFHEDQGRULHQWHGSD UDOOHOWR]D[LV)LJD:LGHQLQJRIWKLVEDQG DQG VLPXOWDQHRXV PRYHPHQW RI WKH VSHFLDO GL UHFWLRQV WRZDUGV WKH ] GLUHFWLRQ RFFXU ZLWK LQFUHDVLQJOLQHDUELUHIULQJHQFHVHH)LJE±H 7KLVFRQFOXVLRQLVLQDJUHHPHQWZLWKWKHUHVXOWV VKRZQ LQ )LJE±I 2QFH WKH OLQHDU ELUHIULQ )LJ &RQRVFRSLF SDWWHUQV IRU WKH SURSDJDWLRQ GLUHFWLRQ FKDUDFWHUL]HG ZLWK WKH FRQGLWLRQ

F

*= D =Q ŠZLWKUHVSHFWWR[D[LVLQWKH[\SODQHLQJ\URWURSLFFU\VWDOVRISRLQWJURXSPDW GLIIHUHQWYDOXHVRIOLQHDUELUHIULQJHQFHDQGWKHFRUUHVSRQGLQJVXUIDFHVRIWRWDOHOOLSWLFDOELUHIULQJHQFH

D D =QO E

O Q

-D = F

O

Q

-D = í G

O Q

-D = H

O

Q

-D = í DQG I

O Q -D =

(10)

JHQFHEHFRPHVODUJHUWKDQWKHFLUFXODURQHWKH XVXDOFRQRVFRSLFILJXUHLVREVHUYHG)LJI

'LVFXVVLRQ

/HWXVDQDO\]HVRPHSHFXOLDULWLHVRIWKHVSHFLDO SURSDJDWLRQ GLUHFWLRQV UHYHDOHG E\ XV :KHQ LQYHVWLJDWLQJWKHIXQFWLRQ

VLQ LM L M P

J O O Q Q

Q

D = D Q +

IRU WKH H[LVWHQFH RI H[WUHPD QDPHO\ WKH PLQLPDGHSHQGLQJRQWKHDQJOHVQDQGjRQH FDQ HDVLO\ ILQG WKDW WKH H[WUHPH YDOXHV FRUUHVSRQG MXVW WR WKH UHYHDOHG VSHFLDO GLUHFWLRQV0RUHRYHULWLVVHHQIURPWKHILJXUHV RI WRWDO ELUHIULQJHQFH WKDW WKH PHQWLRQHG GLUHFWLRQVDUHDOVRFKDUDFWHUL]HGE\LQWHUVHFWLRQ RI WKH VXUIDFHV RI FLUFXODU DQG OLQHDU ELUHIULQJHQFHV DQG WKH HTXDOLW\ RI WKHVH ELUHIULQJHQFHV

VLQ LM L M P

J O O Q

Q

D Q = ‹

7KHQ WKH HOOLSWLFLW\ RI WKH HLJHQZDYHV SURSDJDWHG DORQJ WKRVH GLUHFWLRQV XQGHU WKH FRQGLWLRQ RI D = DQF QO ZLOO DFTXLUH D VSHFLILF YDOXH

( )

* Q N * % % % % Q = = - - - +

ZKHUH%DQG%DUHWKHFRPSRQHQWVRIRSWLFDO LPSHUPHDELOLW\ WHQVRU DFWXDO IRU WKH FURVV VHFWLRQRIWKHRSWLFDOLQGLFDWUL[SHUSHQGLFXODUWR WKH GLUHFWLRQ RI OLJKW SURSDJDWLRQ ,W LV LQWHUHVWLQJWRUHFDOOLQWKLVUHODWLRQWKDWWKHJLYHQ HLJHQZDYH HOOLSWLFLW\ KDV EHHQ UHYHDOHG LQ WKH VWXG\ >@ EXW XQIRUWXQDWHO\ WKH DXWKRUV >@ KDYH QRW QRWLFHG WKDW WKH YDOXH

N = VKRXOGEHSHFXOLDUIRUWKHFRQGLWLRQ

F O

Q Q

D = D 1RWLFH DOVR WKDW WKH YDOXH RI WKH DERYHHLJHQZDYHHOOLSWLFLW\FDQEHDFKLHYHGQRW RQO\ LQ WKH YLFLQLW\ RI WKH LVRWURSLF SRLQW EXW DOVR IRU WKH SURSDJDWLRQ RI OLJKW DORQJ DQ\ GLUHFWLRQ ZKHUH WKH FRQGLWLRQ D = DQF QO LV

VDWLVILHG LH VRPHZKHUH IDU IURP WKH ZDYHOHQJWKRIWKHLVRWURSLFSRLQW)RUH[DPSOH WKH FLUFXODU DQG OLQHDU ELUHIULQJHQFHV LQ TXDUW] FU\VWDOV DUH WKH VDPH LI WKH OLJKW SURSDJDWHV LQ WKH GLUHFWLRQ GHILQHG E\ WKH DQJOH êŠ ZLWK UHVSHFWWRWKHRSWLFD[LV>@DQGZHKDYH

N = WRR

&RQFOXVLRQ

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

7KH UHVXOWV RI IXUWKHU VWXGLHV RI WKLV SKHQRPHQD ZLOO EH GHVFULEHG LQ D IRUWKFRPLQJ SDSHU

$FNQRZOHGJHPHQW

(11)

8NU-3K\V2SW9

5HIHUHQFHV

9ORNK 2* DQG 7VDU\N $9 8NU - 3K\V LQ5XVVLDQ

9ORNK2*+RORYH\0,DQG7VDU\N$9 8NU-3K\VLQ5XVVLDQ 9ORNK 2* 7VDU\N $9 DQG 1HNUDVRYD

,0 8NU - 3K\V LQ 5XVVLDQ

+REGHQ 09 $FWD&U\VW$ +REGHQ09$FWD&U\VW$

+REGHQ 09 1DWXUH

+REGHQ091DWXUH 9ORNK 2* 6SDWLDO GLVSHUVLRQ SKHQRPHQD

LQSDUDPHWULFFU\VWDORSWLFV/YLY9\VKFKD 6KNRODLQ5XVVLDQ

9ORNK58NU-3K\V2SW

9ORNK 5 3\DWDN < DQG 6NDE ,

)HUURHOHFWULFV

9ORNK 52 3\DWDN <X$ DQG 6NDE ,3 6RY3K\V6RO6WDWH 1DVWLVKLQ <X$ 9ORNK 2* DQG 'RYJ\L

2%8NU-3K\V2SW

%RUQ 0 :ROI ( 3ULQFLSOHV RI RSWLFV 2[IRUG/RQGRQ(GLQEXUJK1HZ<RUN 3DULV)UDQNIXUW³3HUJDPRQ3UHVV´ 6LURWLQ <X, DQG 6KDVNROVND\D 03

)XQGDPHQWDOV RI FU\VWDO SK\VLFV 0RVFRZ ³1DXND´LQ5XVVLDQ

9ORNK 2* DQG .RE\O\DQVN\ 9% 8NU -3K\VLQ8NUDLQLDQ

References

Related documents

It will also describe the StoneFly Backup Advantage data protection and data management solution and its role in enabling the most effective use of disk for backup, mirroring

§ Sequences of words, tags, morphemes, phonemes ( n-grams, FSAs, FSTs ; regex compilation, best-paths , forward-backward , collocations ).. § Vectors (

The research participants comprise of programme coordinators of Bachelor’s Degree of Social work, final year students within the programme and teachers teaching courses relevant

The current study attempted to address this know- ledge gap by using qualitative methods to explore: 1) common ecological stressors faced by adolescents in two predominantly

A major reason for this is the lack of adequate training algorithms for deep SNNs, since spike signals (i.e., discrete events produced by a spiking neuron whenever its internal

It is envisioned that the strategy and protocol will address the development of supervision plans that reflect the juvenile offender’s needs and activities that must be

Our purpose-built labs are suitable for food related work including food demonstrations, food product development, shelf life testing and food safety management.. They are fully

All these economies have experienced rapid growth of their physical capital stock and very high rate of investment in human capital.. On one hand, the supporters of the