Qrqolqhdu Frqwuro ri d Frqwlqxrxvo| Yduldeoh Wudqvplvvlrq

+FYW, iru K|eulg Yhklfoh Srzhuwudlqv

S1 Vhwoxu



/ M1 U1Zdjqhu

2

/ G1 P1 Gdzvrq

2

dqg E1 Vdpxhov



Dxwrprwlyh Uhvhdufk Oderudwru|/ Ioxru Gdqlho Hqjlqhhulqj Lqqrydwlrq Exloglqj/

Fohpvrq Xqlyhuvlw|/ Fohpvrq/ Vrxwk Fdurolqd 5<9671 hpdlo= mzdjqhuCfohpvrq1hgx

Devwudfw=

Dxwrprwlyh hqjlqhhuv duh frqwlqxrxvo| h{sorulqj

ydulrxv hqjlqh/ wudqvplvvlrq/ dqg fkdvvlv whfkqrorjlhv wr lq0

fuhdvh ryhudoo yhklfoh shuirupdqfh/ ixho hfrqrp|/ dqg vdihw|1

Rqh surplvlqj srzhuwudlq frqfhsw lv wkh frqwlqxrxvo| yduldeoh

wudqvplvvlrq +FYW, zklfk rhuv d frqwlqxxp ri lqqlwho| ydul0

deoh jhdu udwlrv ehwzhhq hvwdeolvkhg plqlpxp dqg pd{lpxp olp0

lwv1 Wklv frqwlqxrxv jhdu udwlr vshfwuxp fdq lqfuhdvh wkh ryhudoo

srzhuwudlq h!flhqf| dqg holplqdwh wkh xqzdqwhg mhunv dvvrfl0

dwhg zlwk pdqxdo dqg dxwrpdwlf wudqvplvvlrqv1 Dowkrxjk edvlf

FYW ghvljqv pd| kdyh gl!fxow| zlwk kljk wrutxh2orz vshhg uh0

txluhphqwv/ d k|eulg srzhu vsolw frqwlqxrxvo| yduldeoh wudqvplv0

vlrq frqjxudwlrq rhuv erwk {hg jhdulqj dqg dgmxvwdeoh sxooh|v

wr vdwlvi| gulylqj ghpdqgv1 Wkh hhfwlyh frqwuro ri wkh yduldeoh

udglxv sxooh|v doorzv wkh ghvljqdwlrq ri hqjlqh wrutxh2vshhg wr

lpsuryh ryhudoo v|vwhp shuirupdqfh iru d jlyhq rshudwlqj frq0

glwlrq1 Lq wklv sdshu/ wkh ixqgdphqwdo frpsrqhqwv/ frqjxud0

wlrq/ dqg nlqhpdwlfv ri d srzhu vsolw FYW zloo eh glvfxvvhg1 D

vxlwh ri pdwkhpdwlfdo prghov zloo eh suhvhqwhg zklfk lqfoxghv

wkh lqwhuqdo frpexvwlrq vsdun ljqlwlrq hqjlqh/ foxwfk/ wudqvplv0

vlrq glhuhqwldo/ dqg fkdvvlv g|qdplfv1 Wkh sureohp ri zkhho

vshhg +l1h1/ fuxlvh, frqwuro ri d FYW htxlsshg yhklfoh zloo eh

frqvlghuhg1 Dq lqqrydwlyh dgdswlyh qrqolqhdu frqwuroohu zloo eh

ghvljqhg wr hqvxuh dv|pswrwlf wudfnlqj ri wkh ghvluhg zkhho vshhg1

Qrphqfodwxuh Olvw

8 U

dlu0wr0ixho udwlr lq xhqfh



_

dhurg|qdplf gudj frh!flhqw

S

6

pdqlirog frh!flhqw



f

hqjlqh frh!flhqw



o

uroolqj uhvlvwdqfh frh!flhqw

S

A

wrutxh frqvwdqw +Q0p0v2nj,

e

zkhho vshhg wudfnlqj huuru +udg2v,

}

dffhohudwlrq gxh wr judylw| +p2v

5

,



FJ khljkw +p,

a

e

hqjlqh lqhuwld +nj0p

5

,

a

|

ryhudoo wudqvplvvlrq lqhuwld +nj0p

5

,

a



zkhho lqhuwld +nj0p

5

,

g

e

srvlwlyh frqwuro jdlq +huuru,

g

6

srvlwlyh frqwuro jdlq +wkurwwoh,

g

#

srvlwlyh frqwuro jdlq +jhdu udwlr,



yhklfoh pdvv +nj,

6

@

pdqlirog dlu pdvv +nj,

6

@_

ghvluhg pdqlirog dlu pdvv +nj,

6

@

pdqlirog dlu pdvv rz udwh+nj2v,

6

s

ixho rz udwh +nj2v,

A 

wkurwwoh fkdudfwhulvwlf



FYW frqwuro lqsxw +v

04

,

k

wkurwwoh shufhqw



S

rxwsxw wr lqsxw sxooh| udwlr



Uhvhdufk Dvvlvwdqwv lq wkh Ghsduwphqwv ri Hohfwulfdo ) Frp0

sxwhu Hqjlqhhulqj dqg Phfkdqlfdo Hqjlqhhulqj

2

Idfxow| phpehuv lq wkh Ghsduwphqwv ri Hohfwulfdo ) Frp0

sxwhu Hqjlqhhulqj dqg Phfkdqlfdo Hqjlqhhulqj

K

sdudphwhu dgdswdwlrq jdlqv

w

judgh dqjoh +udgldqv,

b

dlu0wr0ixho udwlr

#

6

pdqlirog dlu pdvv huuru +nj,

#



yroxphwulf h!flhqf|



e

hqjlqh wrutxh +Q0p,



?

c 

T Ace

FYW lqsxw wrutxh +Q0p,



J|

c 

T Ac

FYW rxwsxw wrutxh +Q0p,





zkhho ordg wrutxh +Q0p,

/

e

hqjlqh vshhg +udg2v,

/

?

FYW lqsxw vshhg +udg2v,

/

J|

FYW rxwsxw vshhg +udg2v,

/



zkhho vshhg +udg2v,

/

_

ghvluhg zkhho vshhg +udg2v,

D

ryhudoo wudqvplvvlrq udwlr

4 Lqwurgxfwlrq

K|eulg hohfwulf yhklfohv krog wkh surplvh ri lqfuhdvhg rshu0

dwlqj h!flhqf| wkurxjk wkh xvh ri pxowlsoh srzhu vrxufhv

dqg d frqwlqxrxvo| yduldeoh wudqvplvvlrq +FYW,1 D FYW lv

w|slfdoo| frpsrvhg ri wzr k|gudxolfdoo|/ ru vsulqj/ dfwxdwhg

yduldeoh udgll sxooh|v dqg d fkdlq/ ru phwdo sxvklqj ehow/ dv

vkrzq lq Ilj1 41 Wr odxqfk wkh yhklfoh iurp uhvw/ wkh lq0

sxw sxooh| udglxv zloo eh vpdoohu wkdq wkh rxwsxw sxooh|

udglxv uhvxowlqj lq d vshhg uhgxfwlrq dqg wrutxh pxowlsol0

fdwlrq wudqvplwwhg wr wkh gulyh vkdiw1 Iru lqfuhdvhg rxwsxw

vkdiw vshhg/ wkh sxooh| udgll duh lqyhuvho| pdqlsxodwhg vl0

pxowdqhrxvo| +l1h1/ lqsxw sxooh| udglxv lqfuhdvhv dv wkh rxw0

sxw sxooh| udglxv ghfuhdvhv,1 Qrwh wkdw wkh sxooh| udgll

duh vlpxowdqhrxvo| pdqlsxodwhg dv wkh ehow*v ohqjwk pxvw

uhpdlq frqvwdqw wr pdlqwdlq wkh ehow whqvlrq1 Rqh gudz0

edfn ri wklv FYW ghvljq lv wkh uhtxluhphqw iru wkh ehow wr

wudqvplw doo wkh ghpdqghg wrutxh zklfk pd| eh gl!fxow

lq phglxp wr khdy|0gxw| yhklfoh dssolfdwlrqv1 Krzhyhu/ lq

sdvvhqjhu2oljkw0gxw| yhklfohv/ edvlf FYW ghvljqv wkdw rqo|

xwlol}h phwdo sxvklqj ehowv wr phhw wkh wrutxh uhtxluhphqwv

pd| eh dffhswdeoh1

Iljxuh 4=

Sdudooho k|eulg hohfwulf yhklfoh ihdwxulqj d frq0

wlqxrxvo| yduldeoh wudqvplvvlrq1

Wr ryhufrph wkh wrutxh olplwdwlrqv ri edvlf FYWv/ dowhu0

qdwlyh ghvljqv ihdwxulqj {hg jhdu wudlqv kdyh ehhq frqvlg0

huhg +h1j1/ ^4`0^7`,1 Iru lqvwdqfh/ d srzhu vsolw FYW +FYW0

SV, ihdwxuhv d FYW zlwk dq lqwhjudwhg sodqhwdu| jhdu wudlq

+SJW,1 D SJW frqvlvwv ri irxu frpsrqhqwv= vxq jhdu/ ulqj

jhdu/ sodqhw jhduv/ dqg fduulhu1 Wkh v|vwhp*v srzhu rz/

vkrzq lq Ilj1 5/ uhyhdov wkdw wkh lqsxw srzhu/ 

?

/

?

/ lv

vsolw ehwzhhq 

U

/

U

dqg 

U2

/

U2

1 Wklv srzhu vsolw rf0

fxuv ehfdxvh wkhuh duh wzr lqsxw sdwkv wr wkh SJW1 Iluvw/

wkh vxq jhdu lv gluhfwo| frxsohg wr wkh FYW lqsxw sxooh|/

dqg vhfrqg/ wkh ulqj jhdu lv frxsohg wr wkh FYW rxwsxw sxo0

oh| wkurxjk d {hg jhdu v|vwhp1 Lq rwkhu zrugv/ d sruwlrq

ri wkh srzhu wudyhov wkurxjk wkh FYW lqsxw sxooh| gluhfwo|

wr wkh vxq jhdu ri wkh SJW zkloh wkh uhpdlqlqj srzhu lv

wudqvplwwhg iurp wkh lqsxw sxooh| wr wkh ehow0gulyhq rxw0

sxw sxooh| wkhq wr wkh SJW ulqj jhdu1 Wkhvh wzr srzhu

lqsxwv duh wudqvplwwhg wr wkh fduulhu +l1h1/ v|vwhp rxwsxw



J|

/

J|

, wkurxjk wkh sodqhw jhduv1 Wkh FYW0SV frqjx0

udwlrq doorzv prvw ri wkh srzhu wr eh wudqvplwwhg gluhfwo|

wr wkh vxq jhdu e|sdvvlqj wkh FYW rxwsxw sxooh| gxulqj

kljk wrutxh2orz vshhg vlwxdwlrqv1 Wkh rssrvlwh rffxuv lq

orz wrutxh2kljk vshhg vlwxdwlrqv zlwk srzhu suhgrplqdwho|

wudyholqj wkurxjk wkh FYW rxwsxw sxooh|1 Wklv srzhu vsolw

skhqrphqrq lv ghshqghqw xsrq wkh rxwsxw0wr0lqsxw sxooh|

udwlr/ 

S

1 Zkhq 

S

lv odujh/ wkh lqsxw sxooh| lv urwdwlqj

idvwhu wkdq wkh rxwsxw sxooh| dqg 

U

/

U

: 

U2

/

U2

>

khqfh wklv prgh ri rshudwlrq doorzv wkh pdmrulw| ri srzhu

wr eh wudqvplwwhg gluhfwo| wr wkh vxq jhdu1 Zkhq 

S

lv

vpdoo/ wkh rssrvlwh rffxuv dqg prvw ri wkh srzhu lv wudqv0

plwwhg wkurxjk wkh FYW wr wkh ulqj jhdu1 Wklv srzhu vsolw

vkrxog doorz wkh wudqvplvvlrq wr eh xwlol}hg lq phglxp wr

khdy|0gxw| dxwrprwlyh rshudwlrqv1

Iljxuh 5=

Srzhu rz wkurxjk d srzhu vsolw FYW zlwk

h{sdqghg sodqhwdu| jhdu wudlq1

Wkh lpsohphqwdwlrq ri d FYW rhuv wkh rssruwxqlw| iru

hqkdqfhg hqjlqh dqg zkhho +yhklfoh, vshhgv e| wkh hqjlqh

+HFX, dqg wudqvplvvlrq +WFX, frqwuro xqlwv lq d kdupr0

qlrxv pdqqhu iru ydulrxv gulylqj frqglwlrqv dv vkrzq lq

Ilj1 6 ^8`1 Iru lqvwdqfh/ d FYW pd| doorz wkh hqjlqh wr

rshudwh dw d vshflf vshhg zkloh fkdqjlqj wkh sxooh|v* udgll

wr dfklhyh wrutxh pxowlsolfdwlrq/ dffhohudwlrq/ dqg vshhg dv

shu wkh yhklfoh*v yhorflw|/ ordg uhtxluhphqwv/ hqjlqh srzhu/

dqg jhdu udwlrv1 D vhfrqg frqwuro vfhqdulr lv wr doorz wkh

hqjlqh wr rshudwh zlwklq d vshflhg vshhg udqjh dqg vlpxo0

wdqhrxvo| dgmxvw wkh hqjlqh wkurwwoh dqg FYW jhdu udwlr

iru hqjlqh wrutxh2vshhg sdwkv zlwk dffhswdeoh h!flhqf|1

Wkhvh sdwkv pd| eh fkrvhq wr rhu lqfuhdvhg shuirupdqfh

ryhu frqyhqwlrqdo yhklfohv/ dqg doorz wkh hqjlqh wr dvvlvw wkh

FYW lq jhqhudwlqj wkh qhfhvvdu| vshhg2wrutxh wr ryhufrph

jhdu udwlr uhvwulfwlrqv1 Wkhvh wzr rshudwlqj surohv surylgh

wkh uhvhdufk prwlydwlrq iru FYW g|qdplfv dqg qrqolqhdu

frqwuro dojrulwkpv1 Lq wklv sdshu/ zh frqvlghu wkh ghvljq

ri d qrqolqhdu/ dgdswlyh/ edfnvwhsslqj frqwuroohu wr hqvxuh

dv|pswrwlf zkhho vshhg dqg jhdu udwlr wudfnlqj1

Iljxuh 6=

Srzhuwudlq +l1h1/ hqjlqh dqg wudqvplvvlrq, frq0

wuro v|vwhpv1

Wklv sdshu lv rujdql}hg dv iroorzv1 Wkh sureohp vwdwhphqw

lv suhvhqwhg lq Vhfwlrq 51 D qrqolqhdu dgdswlyh prgho0edvhg

FYW frqwuroohu/ vxssruwhg e| d vwdelolw| surri/ lv ghvljqhg

edvhg rq d vlpsolhg yhklfoh prgho lq Vhfwlrq 61 D vxp0

pdu| ri wkh sdshu lv suhvhqwhg lq Vhfwlrq 71

5 Sureohp Vwdwhphqw

Wr lpsuryh wkh shuirupdqfh ri d FYW0edvhg k|eulg yhkl0

foh srzhuwudlq iru ydulrxv gulylqj vfhqdulrv/ wkh hqjlqh dqg

wudqvplvvlrq ixqfwlrqv pxvw eh v|qfkurqrxvo| frqwuroohg1 D

qrqolqhdu frqwuro dojrulwkp/ zklfk vlpxowdqhrxvo| dgmxvwv

wkh wudqvplvvlrq jhdu udwlr dqg hqjlqh wkurwwoh dqjoh +l1h1/

hohfwurqlf wkurwwoh frqwuro dfwxdwru, zkloh prqlwrulqj wkh

hqjlqh dqg zkhho vshhgv wr vdwlvi| wdujhw vshhgv zloo eh gh0

vljqhg iru d vsdun ljqlwlrq +VL, hqjlqh dqg srzhu vsolw FYW1

Wkh frqwuroohu pxvw irufh wkh zkhho vshhg/ /



E|/ wr wudfn

d ghvluhg wudmhfwru|/ /

_

E|1

6 Qrqolqhdu Prgho0edvhg FYW Frqwuroohu

Wkh HFX dqg WFX/ lqwhjudo sduwv ri wkh hqjlqh dqg wudqv0

plvvlrq vxev|vwhpv/ dv vkrzq lq Ilj1 6/ duh ixqfwlrqdoo|

lqghshqghqw xqlwv zklfk frruglqdwh dqg vkduh v|vwhp lq0

irupdwlrq wr uhjxodwh wkh srzhuwudlq rshudwlrq1 Wkh HFX

prqlwruv d qxpehu ri vhqvru vljqdov lqfoxglqj wkh hqjlqh

vshhg/ pdqlirog devroxwh suhvvxuh/ dqg yhklfoh vshhg wr sur0

gxfh d frqwuro vljqdo wkdw jryhuqv wkh wkurwwoh srvlwlrq/ ixho

lqmhfwruv/ vsdun gholyhu|/ dqg d krvw ri rwkhu ixqfwlrqv1 Wkh

WFX uhfhlyhv lqirupdwlrq frqfhuqlqj wkh ghvluhg jhdu udwlr

suroh/ zklfk lv d ghqhg lqsxw/ dqg wkh vshhgv rq hlwkhu

vlgh ri wkh FYW/ wr surylgh wkh uhtxluhg frqwuro lqsxw wr

wkh FYW/ wr hqvxuh wkdw wkh jhdu udwlr iroorzv wkh ghvluhg

suroh1

Wr lqlwldwh wkh prgho0edvhg frqwuroohu ghvljq surfhvv/ d g|0

qdplf prgho pxvw eh hvwdeolvkhg zklfk shuplwv uhdo0wlph

dssolfdwlrqv1 Wkh vlpsolhg yhklfoh prgho zloo uhsuhvhqw

wkh FYW zlwk d uvw0rughu lqwhjudwru dv zhoo dv dffrpsd0

q|lqj g|qdplfv iru wkh hqjlqh dqg wkh zkhho1 Wkh dgdswlyh

qrqolqhdu frqwuro dojrulwkp zloo eh suhvhqwhg iru wkh dv|ps0

wrwlf wudfnlqj ri wkh ghvluhg zkhho vshhg dv zhoo dv hqvxulqj

wkdw wkh FYW udwlr wudfnv d ghvluhg wudmhfwru|1 D ghwdlohg

vwdelolw| dqdo|vlv zloo eh suhvhqwhg wr ydolgdwh wkh frqwuroohu

ghvljq1

614 Vlpsolhg G|qdplf Prgho

Wr ghyhors d vlpsolhg yhklfoh prgho zlwk lqwhjudwhg FYW/

wkh g|qdplfv duh glylghg lqwr wkuhh sduwv dv vkrzq lq Ilj1

71 Wkh dqdo|vlv ehjlqv zlwk wkh hqjlqh vshhg g|qdplfv/

prghohg dv d vlqjoh lqhuwld v|vwhp/ jlyhq dv

/

e

'



a

e

d

e

3 

T Ace

o

+4,

zkhuh 

e

E| lv wkh wrutxh jhqhudwhg e| wkh hqjlqh dqg



T Ace

E| ghqrwhv wkh wrutxh dssolhg wr wkh FYW e| wkh

hqjlqh1 Wkh FYW lv prghohg dv vlqjoh lqwhjudwru zlwk d

oxpshg lqhuwld sodfhg rq wkh hqjlqh vlgh ri wkh FYW ^48`1

Wkh wudqvplvvlrq g|qdplfv duh jlyhq e|



T Ac

' d

T Ace

3 a

|

/

e

o D

+5d,

D ' 

+5e,

zkhuh D E| lv wkh hhfwlyh jhdu udwlr/ dqg  E| lv wkh FYW

frqwuro lqsxw1 Wkh nlqhpdwlf uhodwlrqvkls ri wkh vkdiw

vshhgv rq hlwkhu vlgh ri wkh FYW lv jlyhq e|

/

e

' D/



+6,

zkhuh /



E| lv wkh zkhho vshhg1 Ilqdoo|/ wkh zkhho vlgh ri

wkh yhklfoh zdv prghohg dv d vlqjoh lqhuwld zlwk

/



'



a







FY W>z

3 





+7,

zkhuh 

FY W>z

E| ghqrwhv wkh wrutxh dssolhg wr wkh zkhhov

e| wkh FYW1 Wkh ordg wrutxh dw wkh zkhho/ 



E|/ lv prg0

hohg dv





' 

_

/

2



n 6} t? w

+8,

Iljxuh 7=

Vlpsolhg yhklfoh prgho lqfrusrudwlqj d FYW1

Wkhvh yhklfoh g|qdplfv pd| eh frqghqvhg lqwr wzr glhu0

hqwldo htxdwlrqv e| wdnlqj wkh wlph ghulydwlyh ri htxdwlrq

+6,/ dqg xvlqj htxdwlrqv +4,/ +5d,/ dqg +7, wr |lhog

/



'



}ED



3


2

}EDc /



3 



n D

e



+9d,

D ' 

+9e,

zkhuh }ED lv d srvlwlyh vfdodu ixqfwlrq ghqhg dv iroorzv

}ED ] a



n Ea

e

n a

|

 D

2

+:,

dqg }EDc  ghqrwhv wkh wlph ghulydwlyh ri }ED zklfk fdq

fdofxodwhg e| xvlqj +9e, dqg +:, dv iroorzv

}EDc  ' 2 Ea

e

n a

|

 D

+;,

D vsdun ljqlwlrq hqjlqh prgho lv qrz suhvhqwhg wr surgxfh

wkh hqjlqh wrutxh 

e

E| lq +9d, ^49`1 Vshflfdoo|/ wkh lqwdnh

pdvv dlu rz g|qdplfv

6

duh dvvxphg wr eh ri wkh irup

6

@

' 3

f

6

@

/

e

n 



A Ek

+<,

zkhuh wkh pdvv dlu rz lv uhodwhg wr wkh hqjlqh wrutxh df0

fruglqj wr



e

' 

f

6

@

zlwk 

f

' S

A

S

6

#

T

8 U Eb

+43,

E| xvlqj +<, dqg +43,/ htxdwlrqv +9d, dqg +9e, pd| eh

uhzulwwhq dv iroorzv

}ED



f

/



' 3



2

f

}EDc /



3







f

n D6

@

+44d,

D ' 

+44e,

Li wkh dlu0wr0ixho udwlr/ b/ fdq eh phdvxuhg ru hvwlpdwhg/

wkhq wkh fruuhvsrqglqj pdvv ixho rz uhodwlrqvkls ehfrphv

6

s

' 

b

6

@

+45,

zklfk pd| eh xvhg dv d fulwhulrq iru hqjlqh0wudqvplvvlrq

h!flhqf| lq ixwxuh uhvhdufk1

615 Frqwuro V|qwkhvlv

Dv phqwlrqhg lq Vhfwlrq 5/ wkh frqwuro remhfwlyh lv wr

irufh wkh zkhho vshhg/ /



E|/ wr wudfn d ghvluhg wudmhfwru|/

/

_

E|/ dv vlploduo| grqh lq ^4:`1 Lw lv dvvxphg wkdw wkh gh0

vluhg zkhho vshhg dqg lwv uvw wzr wlph ghulydwlyhv uhpdlq

erxqghg iru doo wlph +l1h1/ /

_

E|/ /

_

E|/ /

_

E| M n

"

,1

Lw lv dovr dvvxphg wkdw wkh vljqdov /



E|/ /

e

E|/ 6

@

E|/ dqg

w E| duh dydlodeoh iru phdvxuhphqw dqg wkdw w E| lv vorzo|

wlph ydu|lqj +l1h1/ w E| E f,1

Wr txdqwli| wkh frqwuro remhfwlyh/ wkh wudfnlqj huuru vljqdo

lv ghqhg dv

e ' /

_

3 /



+46,

Wkh rshq0orrs wudfnlqj huuru g|qdplfv pd| eh fdofxodwhg

e| wdnlqj wkh wlph ghulydwlyh ri htxdwlrq +46,/ pxowlso|lqj
e|

}ED



3

/ xwlol}lqj htxdwlrq +44d,/ dqg vxevwlwxwlqj +46, edfn

lqwr wkh uhvxowlqj htxdwlrq wr rewdlq

}ED



f

e ' 3 

2

f

}EDc e n }ED



f

/

_

+47,

n 

2

f

}EDc /

_

n 





f

3 D6

@

6

Lw lv dvvxphg wkdw wkh hqjlqh vshhg xfwxdwlrqv duh pdlq0

wdlqhg zlwklq 8( ri wkh jlyhq wdujhw vshhg ^49`> khqfh/ wkh prgho

sdudphwhuv/ 

f

/ 



dqg 

f

lq +<, dqg +43, duh dvvxphg wr eh xq0

nqrzq frqvwdqwv iru wkh frqwuroohu ghvljq1

Wr idflolwdwh wkh edfnvwhsslqj0edvhg frqwuro ghvljq v|qwkh0

vlv/ dq dx{loldu| wudfnlqj huuru vljqdo/ ghqrwhg e| #

6

E|/ lv

ghqhg dv iroorzv

#

6

' 6

@_

3 6

@

+48,

zkhuh 6

@_

E| ghqrwhv wkh vxevhtxhqwo| ghvljqhg ghvluhg

pdqlirog dlu pdvv vljqdo1

Wkh rshq0orrs wudfnlqj hu0

uru g|qdplfv lq htxdwlrq +47, duh qrz uhzulwwhq lqwr

d pruh dgydqwdjhrxv irup e| xvlqj htxdwlrq +48, dqg

dgglqj2vxewudfwlqj wkh whup D6

@_

E| dv

}ED



f

e ' 3



2

f

}EDc e n

}ED



f

/

_

+49,

n 

2

f

}EDc /

_

n 





f

3 D6

@_

n D#

6

Diwhu vxevwlwxwlqj iru 



E| iurp +8,/ wkh rshq0orrs wudfn0

lqj huuru g|qdplfv ri +49, fdq eh uhzulwwhq dv

}ED



f

e ' 3 

2

f

}EDc e

+4:,

n`



EDc c /



c | 



3 D6

@_

n D#

6

zkhuh `



EDc c /



c | M ^

fe

uhsuhvhqwv wkh nqrzq uhjuhv0

vlrq pdwul{/ dqg 



M ^

ef

frqwdlqv xqnqrzq frqvwdqw

sdudphwhuv +Vhh Dsshqgl{ D iru wkh h{solflw ghqlwlrqv ri

`



Eu dqg 



,1

Edvhg rq wkh vxevhtxhqw vwdelolw| dqdo|vlv/ wkh ghvluhg pdq0

lirog dlu pdvv vljqdo lv ghvljqhg dv iroorzv

6

@_

' 

D



`







n g

e

e



+4;,

zkhuh 



E| M ^

ef

ghqrwhv d vxevhtxhqwo| ghvljqhg sdud0

phwhu hvwlpdwh yhfwru iru 



1 Diwhu vxevwlwxwlqj +4;, lqwr

+4:,/ wkh iroorzlqj forvhg0orrs wudfnlqj huuru g|qdplfv fdq

eh rewdlqhg

}ED



f

e ' 3 

2

f

}EDc e n `







3 g

e

e n D#

6

+4<,

zkhuh wkh sdudphwhu hvwlpdwlrq huuru/ ghqrwhg e| 



E| M

^

ef

/ lv ghqhg dv





' 



3 



+53,

Edvhg rq wkh vxevhtxhqw vwdelolw| dqdo|vlv dqg wkh forvhg0

orrs huuru v|vwhpv jlyhq lq +4<, dqg +58,/ wkh sdudphwhu

xsgdwh odz iru 



E| lv ghvljqhg dv

u





' K



`

A



e

+54,

zkhuh K



M ^

efe

lv d frqvwdqw/ srvlwlyh gldjrqdo jdlq pd0

wul{1

Frqwlqxlqj zlwk wkh edfnvwhsslqj frqwuro ghvljq/ wkh wlph

ghulydwlyh ri +48, lv fdofxodwhg dqg wkh uhvxowlqj htxdwlrq
lv pxowlsolhg rq erwk vlghv e|

}ED



r

1 Diwhu dgglqj dqg vxe0

wudfwlqj wkh whup



2

3

}EDc #

6

wr wkh uljkw0kdqg vlgh/ dqg

wkhq xvlqj htxdwlrq +<,/ wkh iroorzlqj rshq0orrs g|qdplf

h{suhvvlrq iru #

6

E| fdq eh frqvwuxfwhg

}ED



f

#

6

' 3 

2

f

}EDc #

6

n }ED



f

6

@_

+55,

n 

f



f

}ED

e

6

@

n 

2

f

}EDc #

6

3







f

}EDA Ek

Diwhu wdnlqj wkh wlph ghulydwlyh ri +4;, dqg vxevwlwxwlqj

wkh uhvxowlqj h{suhvvlrq lqwr +55, iru 6

@_

E|/ wkh rshq0orrs

g|qdplfv iru #

6

E| fdq eh uhzulwwhq lqwr wkh iroorzlqj dg0

ydqwdjhrxv irup

}ED



f

#

6

' 3 

2

f

}EDc #

6

n l



+56,

n`

2



2

3 `







A  Ek

zkhuh `

2

Eu M ^

fS

/ `



Eu M ^

f2

uhsuhvhqw nqrzq uh0

juhvvlrq pdwulfhv frqvlvwlqj ri phdvxudeoh vljqdov/ l



E| M

^



ghqrwhv dq dx{loldu| phdvxudeoh vljqdo/ dqg 

2

M ^

Sf

/





M ^

2f

ghqrwh wkh frqvwdqw/ xqnqrzq sdudphwhu yhf0

wruv +Vhh Dsshqgl{ D iru wkh h{solflw ghqlwlrqv ri `

2

Eu/

`



Eu/ l



E|/ 

2

/ dqg 



,1 Edvhg rq wkh vxevhtxhqw vwdelo0

lw| dqdo|vlv/ wkh wkurwwoh dqjoh k E| lv ghvljqhg dv

k ' A 

3

+



`









l



n `

2



2

n g

6

#

6

n De

,

+57,

zkhuh 

2

E| M ^

Sf

dqg 



E| M ^

2f

ghqrwh vxevhtxhqwo|

ghvljqhg sdudphwhu hvwlpdwh yhfwruv iru 

2

dqg 



/ uhvshf0

wlyho|1 Diwhu vxevwlwxwlqj +57, lqwr +56,/ wkh forvhg orrs

huuru g|qdplfv iru #

6

E| fdq eh zulwwhq dv

}ED



f

#

6

' 3



2

f

}EDc #

6

+58,

3g

6

#

6

3 De n `

2



2

3

`







`







q

`

2



2

n g

6

#

6

n De

r

zkhuh wkh sdudphwhu hvwlpdwlrq huuru whupv/ ghqrwhg e|



2

E| M ^

Sf

dqg 



E| M ^

2f

/ duh ghqhg dv iroorzv



2

' 

2

3 

2

+59,





' 



3 



+5:,

Edvhg rq wkh vxevhtxhqw vwdelolw| dqdo|vlv dqg wkh forvhg0

orrs huuru v|vwhpv jlyhq lq +4<, dqg +58,/ wkh sdudphwhu

xsgdwh odzv iru 

2

E| dqg 



E| duh ghvljqhg dv iroorzv

u



2

' K

2

`

A

2

#

6

+5;d,

u





' h tl

2



+5;e,

zkhuh l

2

' 3K



`

A



#

6

`







q

`

2



2

n g

6

#

6

n De

r

zkhuh K

2

M ^

SfS

dqg K



M ^

2f2

duh frqvwdqw srvlwlyh gldj0

rqdo dgdswdwlrq jdlq pdwulfhv/ dqg h tu ghqrwhv wkh sur0

mhfwlrq dojrulwkp

7

ghqhg lq ^4;`1 Wkh surmhfwlrq dojrulwkp

lv uhtxluhg wr dyrlg srwhqwldo vlqjxodulwlhv lq wkh frqwuro odz

wkdw duh lqwurgxfhg e| glylvlrq ri wkh whup `







lq +57,1

Wkh surmhfwlrq dojrulwkp h{sorlwv wkdw idfw wkdw `







: f

+Wklv idfw lv reylrxv iurp dq lqvshfwlrq ri wkh ghqlwlrq ri

`



Eu dqg 



jlyhq lq Dsshqgl{ D,1

Lq dgglwlrq wr wkh zkhho vshhg wudfnlqj remhfwlyh/ dq dx{lo0

ldu| frqwuro remhfwlyh lv wkh wudfnlqj remhfwlyh iru wkh FYW

jhdu udwlr1 Wr txdqwli| wklv wudfnlqj remhfwlyh/ wkh FYW

wudfnlqj huuru vljqdo/ ghqrwhg e| # E| M ^



/ lv ghqhg dv

# ' D

_

3 D

+5<,

7

Iru pruh ghwdlov rq wkh xvh ri wkh surmhfwlrq dojrulwkp/ wkh

uhdghu lv uhihuuhg wr sdjhv 8450847 ri ^4;` 1

zkhuh D

_

E| M ^



lv wkh ghvluhg jhdu udwlr suroh1 Lw lv

dvvxphg wkdw wkh ghvluhg jhdu udwlr suroh dqg lwv uvw

wlph ghulydwlyh uhpdlq erxqghg iru doo wlph +l1h1/ D

_

E|/

D

_

E| M n

"

,1 Wr dyrlg vlqjxodulwlhv lq wkh deryh zkhho

vshhg frqwuroohu/ zh zloo dvvxph wkdw D

_

Ef D Ef dqg

D

_

E| : f1 Vshflfdoo|/ wkhvh frqglwlrqv hqvxuh wkdw d sr0

whqwldo vlqjxodulw| +lqwurgxfhg e| glylvlrq ri wkh whup D E|

lq +4;,, lv dyrlghg1

Diwhu wdnlqj wkh wlph ghulydwlyh ri htxdwlrq +5<,/ dqg wkhq

vxevwlwxwlqj +44e,/ wkh rshq0orrs g|qdplfv iru # E| pd| eh

h{suhvvhg dv

# ' D

_

3 

+63,

Edvhg rq wkh vxevhtxhqw vwdelolw| dqdo|vlv/ wkh FYW frqwuro

lqsxw  E| lv ghvljqhg dv

 ' D

_

n g

#

#

+64,

Diwhu vxevwlwxwlqj htxdwlrqv +64, lqwr +63,/ wkh forvhg0orrs

h{suhvvlrq iru # E| lv dv

# ' 3g

#

#

+65,

616 Vwdelolw| Dqdo|vlv

Wkh vwdelolw| ri wkh dgdswlyh frqwuroohu pd| eh h{dplqhg

edvhg rq wkh forvhg0orrs huuru v|vwhpv jlyhq lq htxdwlrqv

+4<,/ +58, dqg +65,1

Wkhruhp 4 Wkh sursrvhg frqwuroohu jlyhq lq +4;,/ +57,/

dqg +64, hqvxuhv dv|pswrwlf yhklfoh zkhho vshhg dqg jhdu

udwlr wudfnlqj lq wkh vhqvh wkdw

*4

|<"

e E| / # E| ' f

+66,

zkhuh e E| dqg # E| duh ghqhg lq +46, dqg +5<,/ uhvshf0

wlyho|1

Surri1 Wkh wkhruhp surri lv h{hfxwhg lq wzr vwhsv1 Iluvw/

lw pd| eh qrwhg wkdw vwdqgdug olqhdu frqwuro dujxphqwv fdq

eh dssolhg wr htxdwlrq +65, wr vkrz wkdw

# E| ' # Ef i TE3g

#

|

+67,

Wkh vhfrqg wudfnlqj remhfwlyh jlyhq lq htxdwlrq +66, iro0

orzv1

Diwhu vxevwlwxwlqj htxdwlrq +5<, lqwr +67, dqg wkhq uhdu0

udqjlqj/ wkh iroorzlqj h{suhvvlrq/ lv rewdlqhg

D E| ' D

_

E| 3 ED

_

Ef 3 D Ef i TE3g

#

|

+68,

Li D

_

Ef D Ef dqg D

_

E| : f/ wkhq lw pd| eh revhuyhg

iurp htxdwlrq +68, wkdw

D E| : f | D f

+69,

Vhfrqg/ frqvlghu d qrq0qhjdwlyh vfdodu ixqfwlrq T

2

E| M ^



ghqhg dv

T

2

'



2

f

} ED e

2

n 

2

f

} ED #

2

6

+6:,

n

2



A


K

3







n 

2



A
2

K

3

2



2

n 

2



A


K

3







Diwhu wdnlqj wkh wlph ghulydwlyh ri htxdwlrq +6:,/ dqg xvlqj

htxdwlrqv +4<, dqg +58,/ T

2

E| fdq eh zulwwhq dv

T

2

' 3g

e

e

2

3 g

6

#

2

6

+6;,

ne`







n #

6

`

2



2

3#

6

`







`







q

`

2



2

n g

6

#

6

n De

r

3

A


K

3



u





3

A
2

K

3

2

u



2

3

A


K

3



u





zkhuh wkh htxdolwlhv







' 3

u





c





2

' 3

u



2

c







' 3

u





+6<,

kdyh ehhq xwlol}hg1 Diwhu vxevwlwxwlqj wkh sdudphwhu hvwl0

pdwh xsgdwh odzv iurp htxdwlrqv +54,/ +5;d,/ dqg +5;e,/ lw

lv hdv| wr vkrz wkdw T

2

E| lq htxdwlrq +6;, pd| eh erxqghg

deryh dv

T

2

$ 3g

e

e

2

3 g

6

#

2

6

+73,

Lq wklv fdofxodwlrq/ wkh idfw wkdw 

A


K

3



h tl

2

 $ f kdv

ehhq xwlol}hg +Uhihu wr ^4;` iru ghwdlov,1

Diwhu xwlol}lqj htxdwlrqv +6:, dqg +73,/ lw pd| eh vkrzq

wkdw e E|/ #

6

E|/ 



E|/ 

2

E|/ dqg 



E| M n

"

dqg wkdw

e E|/ #

6

E| M n

2

1 Lw lv fohdu iurp wkh idfwv wkdw /

_

E|/

/

_

E|/ /

_

E| M n

"

dqg htxdwlrqv +46,/ +53,/ +59,/ +5:,/

+54,/ +5;d, dqg +5;e,/ wkdw /



E|/ 



E|/ 

2

E|/ 



E|/

u





E|/ dqg

u



2

E| M n

"

1 Vlqfh #E|/ D

_

E| M n

"

/ lw lv

hdv| wr vhh iurp htxdwlrq +5<, wkdw D E| M n

"

1 Diwhu xvlqj

htxdwlrq +64,/ lw fdq eh vkrzq wkdw  E| M n

"

1 Khqfh/

iurp htxdwlrq +63,/ # E| M n

"

1 Iurp wkh ghqlwlrq jlyhq

lq htxdwlrq +:,/ lw lv fohdu wkdw } ED/ } EDc  M n

"

1 Dovr/

iurp wkh h{solflw h{suhvvlrqv iru `



Eu dqg `



Eu jlyhq lq

Dsshqgl{ D dqg iurp htxdwlrq +69,/ lw lv fohdu wkdw `



Eu/

`



Eu M n

"

1 Lw pd| qrz eh vwdwhg iurp htxdwlrq +4;,

wkdw 6

@_

E| M n

"

1 Qrz/ iurp htxdwlrq +4:,/ e E| M n

"

1

Iurp wkh h{suhvvlrqv jlyhq lq Dsshqgl{ D/ lw fdq qrz eh

vkrzq wkdw l



E|/  E|/ `

2

Eu M n

"

1 Xvlqj vwdqgdug vlj0

qdo fkdvlqj dujxphqwv/ lw fdq eh vkrzq wkdw doo vljqdov uh0

pdlq erxqghg1 Ilqdoo|/ dvvxplqj wkdw wkh wkurwwoh fkdu0

dfwhulvwlf A  Eu lv lqyhuwleoh/ zh fdq xvh htxdwlrq +57, wr

vwdwh wkdw k E| M n

"

1 Lw fdq qrz eh vwdwhg wkdw e E| lv

xqlirupo| frqwlqxrxv iurp wkh idfw wkdw e E| M n

"

1 Vlqfh

e E| M n

"

K n

2

dqg e E| lv xqlirupo| frqwlqxrxv/ d fruro0

odu| wr Eduedodw*v Ohppd ^4<` pd| eh hpsor|hg wr suryh

wkh uvw wudfnlqj remhfwlyh vwdwhg lq htxdwlrq +66,1

7 Frqfoxvlrq

Lq wklv sdshu/ d vlpsolhg prgho iru d yhklfoh zlwk d vsdun

ljqlwlrq hqjlqh dqg d frqwlqxrxvo| yduldeoh wudqvplvvlrq lq

d srzhu vsolw frqjxudwlrq kdv ehhq suhvhqwhg1 Ixuwkhu/

wkh urdg judgh/ dqg dhurg|qdplf gudj hhfwv kdyh ehhq

h{soruhg1 D qrqolqhdu frqwuro dojrulwkp/ lpsohphqwhg e|

wkh HFX/ zdv ghyhorshg wr irufh wkh yhklfoh wr wudfn d suh0

vfulehg vshhg wudmhfwru|1 Wkh WFX dojrulwkp zdv dovr gh0

yhorshg wr hqvxuh wkdw wkh FYW jhdu udwlr wudfnv d ghvluhg

suroh zklfk vdwlvhv erwk yhklfoh dqg hqjlqh vshhg uhtxluh0

phqwv1 Wkh frqwuroohu frpshqvdwhg iru xqnqrzq yhklfoh

prgholqj sdudphwhuv xvlqj g|qdplf sdudphwhu hvwlpdwhv1

D frpsohwh vwdelolw| surri zdv dovr suhvhqwhg wr jxdudqwhh

dv|pswrwlf yhklfoh vshhg wudfnlqj1 Ixwxuh zrun lqfoxghv

wkh h{shulphqwdo whvwlqj ri wkh sursrvhg frqwuro vwudwhj|

rq d k|eulg yhklfoh srzhuwudlq1

Dfnqrzohgjhphqw

Wkh dxwkruv judwhixoo| dfnqrzohgjh wkh vxssruw ri wklv zrun

e| wkh X1V1 Dup| Wdqn Dxwrprwlyh Frppdqg wkurxjk wkh

Dxwrprwlyh Uhvhdufk Fhqwhu/ Xqlyhuvlw| ri Plfkljdq1

Uhihuhqfhv

^4`

Kdqdfkl/ V1/ D Vwxg| ri wkh G|qdplfv ri d Vsolw0

Wrutxh/ Jhduhg0Qhxwudo Wudqvplvvlrq Phfkdqlvp/ Mrxu0

qdo ri Phfkdqlfdo Ghvljq/ Yro1 445/ ss1 5940 5:3/ Vhsw1 4<<31

^5`

Ydkde}dghk/ K1/ dqg Olq}hoo/ V1 P1/ Prgholqj/ Vlp0

xodwlrq/ dqg Frqwuro Lpsohphqwdwlrq iru d Vsolw0Wrutxh/

Jhduhg Qhxwudo/ Lqqlwho| Yduldeoh Wudqvplvvlrq/ VDH Sd0

shu Qr1 <4373</ 4<<41

^6`

Pxflqr/ Y1 K1/ Vplwk/ M1 H1/ Frzdq/ E1/ dqg Npl0

flnlhzlf}/ P1/ Sdudphwulf Prgholqj dqg Dqdo|vlv ri d

Sodqhwdu| Jhdu0FYW Phfkdqlvp/ VDH Sdshu Qr1 <7384</

4<<71

^7`

Pxflqr/ Y1 K1/ Vplwk/ M1 H1/ Vkdupd/ S1 N1/ Frzdq/

E1/ dqg Nplflnlhzlf}/ P1/ D Grxeoh Sodqhwdu| Jhdu Wudlq0

FYW Wudqvplvvlrq zlwk Pxowlsoh Dssolfdwlrqv VDH Sdshu

Qr1 <833<7/ 4<<81

^8`

Urggd/ Z1 M1/ Ndvwxud/ M1 O1/ dqg Pdkdq/ V1 W1/

Dxwrprwlyh Hohfwurqlf Lpsolfdwlrqv ri REG LL/ VDH Sur0

fhhglqjv/ Qr1 S0593/ ss1 7407:/ Rfw1 4<<51

^9`

Prvnzd/ M1 M1/ dqg Khgulfn/ M1 N1/ Prgholqj dqg

Ydolgdwlrq ri Dxwrprwlyh Hqjlqhv iru Frqwuro Dojrulwkp

Ghyhorsphqw/ Wudqvdfwlrqv ri wkh DVPH Mrxuqdo ri G|0

qdplf V|vwhpv/ Phdvxuhphqw/ dqg Frqwuro/ Yro1 447/ Qr1

5/ ss1 5:;05;8/ Mxqh 4<<51

^:`

Fkr/ G1/ dqg Khgulfn/ M1 N1/ D Qrqolqhdu Frq0

wuroohu Ghvljq Phwkrg iru Ixho0Lqmhfwhg Dxwrprwlyh Hq0

jlqhv/ Wudqvdfwlrqv ri wkh DVPH Mrxuqdo ri Hqjlqhhulqj

iru Jdv Wxuelqhv dqg Srzhu/ Yro1 443/ Qr1 6/ ss1 6460653/

Mxo| 4<;;1

^;`

Nrwzlfnl/ D1 M1/ G|qdplf Prghov iru Wrutxh Frq0

yhuwhu Htxlsshg Yhklfohv/ VDH Sdshu Qr1 ;536<6/ 4<;51

^<`

Jloohvslh/ W1 G1/ Ixqgdphqwdov ri Yhklfoh G|qdp0

lfv/ VDH/ Zduuhqgdoh/ SD/ 4<<51

^43` Pxflqr/ Y1 K1/ Vplwk/ M1 H1/ Frzdq/ E1/ dqg Npl0

flnlhzlf}/ P1/ D Frqwlqxrxvo| Yduldeoh Srzhu Vsolw Wudqv0

plvvlrq iru Dxwrprwlyh Dssolfdwlrqv/ VDH Sdshu Qr1

<:39;:/ 4<<:1

^44` Ox/ ]1/ Dffhohudwlrq Vlpxodwlrq ri d Yhklfoh zlwk

d Frqwlqxrxvo| Yduldeoh Srzhu Vsolw Wudqvplvvlrq/ P1 V1

Wkhvlv/ Ghsduwphqw ri Phfkdqlfdo Hqjlqhhulqj/ Zhvw Ylu0

jlqld Xqlyhuvlw|/ Mxo| 4<<;1

^45` Vdpxhov/ E1/ Vhwoxu/ S1/ Gdzvrq/ G1 P1/ Kdtxh/

L1/ dqg Zdjqhu/ M1 U1/ Frqwlqxrxvo| Yduldeoh Wudqvplv0

vlrq Prgholqj dqg dqdo|vlv iru K|eulg Yhklfohv xvlqj d

Srzhu Vsolw Frqjxudwlrq Whfkqlfdo Uhsruw/ Dxwrprwlyh

Uhvhdufk Fhqwhu/ Ghsdudwphqw ri Phfkdqlfdo Hqjlqhhulqj/

Fohpvrq/ VF/ Pdufk/ 53341

^46` Sdxvfk/ P1/ dqg Sihlhu/ I1/ Vlpxodwlrq ri d Fkdlq

Gulyh FYW dv d Phfkdwurqlf V|vwhp/ Frqwuro ri Rvflo0

odwlrqv dqg Fkdrv Surfhhglqjv/ Yro1 6/ ss1 6<406<7/ Dxj1

4<<:1

^47` Fkr/ G1/ dqg Khgulfn/ M1 N1/ Dxwrprwlyh Srzhuwudlq

Prgholqj iru Frqwuro/ Wudqvdfwlrqv ri wkh DVPH Mrxuqdo

ri G|qdplf V|vwhpv/ Phdvxuhphqw/ dqg Frqwuro/ Yro1 444/

ss1 89;08:9/ Ghfhpehu 4<;<1

^48` Jx}}hood/ O1/ dqg Vfkplg/ D1 P1/ Ihhgedfn Olq0

hdul}dwlrq ri Vsdun0Ljqlwlrq Hqjlqhv zlwk Frqwlqxrxvo|

Yduldeoh Wudqvplvvlrqv/ LHHH Wudqvdfwlrqv rq Frqwuro

V|vwhpv Whfkqrorj|/ Yro1 6/ Qr1 4/ ss1 8708;/ Pdu1 4<<81

^49` Zdjqhu/ M1 U1/ Gdzvrq/ G1 P1/ dqg ]h|x/ O1/ Qrq0

olqhdu Dlu0Wr0Ixho Udwlr dqg Hqjlqh Vshhg Frqwuro iru

Vsdun Ljqlwlrq Hqjlqhv/ surfhhglqj ri wkh DVPH LPHFH/

G|qdplf V|vwhpv dqg Frqwurov Glylvlrq/ GVF0Yro1 9<04/

ss1 9660973/ 53331

^4:` Prvnzd/ M1 M1 dqg Khgulfn/ M1 N1/ Qrqolqhdu Do0

jrulwkpv iru Dxwrprwlyh Hqjlqh Frqwuro/ LHHH Frqwuro

V|vwhpv Pdjd}lqh/ ss1 ;;0<6/ Dsulo 4<<31

^4;` Nuvwlf/ P1/ Ndqhoodnrsrxorv/ L1/ dqg Nrnrwrylf/

S1/Qrqolqhdu dqg Dgdswlyh Frqwuro Ghvljq/ Mrkq Zloh|

) Vrqv/ Lqf1/ 4<<81

^4<` Vorwlqh/ M1 M1 dqg Ol/ Z1 Dssolhg Qrqolqhdu Frq0

wuro/ Hqjohzrrg Fol/ QM= Suhqwlfh Kdoo/ Lqf1/ 4<<41

Dsshqgl{ D lv dydlodeoh xsrq uhtxhvw1