Some computational aspects of iterated structures

Vrph frpsxwdwlrqdo dvshfwv ri
lwhudwhg vwuxfwxuhv
4
Mrkdq E|vwuùp/ 5 Mrkdq Khovlqj dqg 6 Dqqhwwh Phlghoo
4
5
Ghsw1 ri Pdwk1/ Oxohè Xqlyhuvlw| ri Whfkqrorj|/ Vzhghq
Ghsw1 ri Vrolg Phfk1/ Ur|do Lqvwlwxwh ri Whfkqrorj|/ Vzhghq
6
Qduyln Xqlyhuvlw| Froohjh/ Qruzd|
Devwudfw
Zh frqvlghu vrph frpsxwdwlrqdo dvshfwv ri hhfwlyh surshuwlhv iru
vrph pxowlvfdoh vwuxfwxuhv1 Lq sduwlfxodu zh glvfxvv lwhudwhg vtxduh
krqh|frpev dqg dqrwkhu w|sh ri vtxduh krqh|frpev frqwdlqlqj xs wr
7333 vpdoo glvfv udqgrpo| glvwulexwhg lqvlgh hdfk vtxduh1 Zh suhvhqw
vrph qxphulfdo phwkrgv iru hvwlpdwlqj wkh hhfwlyh frqgxfwlylw| zlwk
jrrg frqwuro ri wkh dffxudf|1
4
Lqwurgxfwlrq
E| krqh|frpev rqh xvxdoo| phdqv wzr0glphqvlrqdo wzr0frpsrqhqw shulrglf
vwuxfwxuhv zkhuh wkh lqwhulru sduw frqvlvwv ri sro|jrqv +w|slfdoo| kh{djrqv ru
uhfwdqjohv,1 Lwhudwhg krqh|frpev duh/ urxjko| vshdnlqj/ krqh|frpev lq krq0
h|frpev1 Vhyhudo w|shv ri vxfk vwuxfwxuhv +dqg dovr vrph pxowlglphqvlrqdo
jhqhudol}dwlrqv, zdv vwxglhg wkhruhwlfdoo| e| G1 Oxnndvvhq zlwk uhvshfw wr
olqhdu hodvwlf dqg wkhupdo hhfwlyh surshuwlhv lq ^4:/ 4;/ 4</ 53`1 Kh irxqg
vrph vlpsoh irupxodh iru xsshu dqg orzhu erxqgv ri wkh fruuhvsrqglqj hi0
ihfwlyh sdudphwhuv1 Wkhvh erxqgv kdsshq wr frqyhujh wr wkh vdph olplw dv
wkh qxpehu ri ohqjwk vfdohv wxuqv wr lqqlw|/ dqg/ lw wxuqv rxw wkdw wkhvh
olplwv duh rswlpdo1 Frpsrvlwh vwuxfwxuhv ri wklv w|sh duh xvxdoo| uhihuuhg wr
dv rswlpdo1 Rwkhu zhoo nqrzq h{dpsohv duh wkh Kdvklq hoolsvrlgdo vwuxfwxuh
dqg wkh pxowludqn odplqdwh vwuxfwxuh +vhh h1j1 wkh errn ^47`,1 Krzhyhu/ lw
lv lpsruwdqw wr qrwh wkdw vwuxfwxuhv zklfk kdyh wkh vdph hhfwlyh ehkdylru
4
riwhq duh glhuhqw zlwk uhvshfw wr surshuwlhv vxfk dv orfdo exfnolqj uhvlv0
wdqfh/ orfdo dqg joredo vwuhqjwk/ vwuxfwxudo froodsvh hwf1 Iurp wklv srlqw
ri ylhz lwhudwhg krqh|frpev vhhp wr eh yhu| lqwhuhvwlqj/ hvshfldoo| ehfdxvh
wkh| srvvhvv fhuwdlq vxlwdeoh v|pphwu| surshuwlhv +vhh ^53`,1
Lq wklv sdshu zh frqvlghu vrph frpsxwdwlrqdo dvshfwv ri pxowlvfdoh vwuxf0
wxuhv ri wklv w|sh1 Lq Vhfwlrq 5 dqg Vhfwlrq 6 zh glvfxvv lwhudwhg vtxduh
krqh|frpev1 Lq sduwlfxodu zh suhvhqw d qxphulfdo phwkrg iru hvwlpdwlqj
wkh hhfwlyh frqgxfwlylw| zlwk jrrg frqwuro ri wkh dffxudf|1 Wklv phwkrg
surylghv d uholdeoh wrro iru frpsdulqj glhuhqw w|shv ri lwhudwhg vtxduh krq0
h|frpev1 Xs wr qrz rqo| wkh fdvh ri zlgho| vhsdudwhg ohqjwk vfdohv kdv
ehhq frqvlghuhg1 Iurp d pdqxidfwxulqj srlqw ri ylhz lw lv lqwhuhvwlqj wr lq0
yhvwljdwh dqg frpsduh lqwhuphgldwh fdvhv1 Zh wkhuhiruh dovr frpsduh vrph
frqfuhwh lwhudwhg krqh|frpev1
Lq Vhfwlrq 7 zh frqvlghu dqrwkhu w|sh ri vtxduh krqh|frpev frqwdlqlqj xs
wr 7333 vpdoo glvfv udqgrpo| glvwulexwhg lqvlgh hdfk vtxduh1 Qrw yhu| orqj
wlph djr frpsxwdwlrqdo sureohpv ri wklv w|sh zhuh frqvlghuhg lpsrvvleoh
wr vroyh1 Lw pd| wkhuhiruh frph dv d vxusulvh wkdw zh duh deoh wr frpsxwh
wkh fruuhvsrqglqj hhfwlyh frqgxfwlylw| ri vxfk sureohpv/ dqg hyhq zlwk
h{wuhpho| jrrg dffxudf|1 Wkh edvlv iru rxu qxphulfdo hvwlpdwhv lv wkh zrun
ri Khovlqj ^45/ 46` zkr kdv ghyhorshg d idvw pxowlsroh0dffhohudwhg lwhudwlyh
vfkhph iru frpsxwdwlrqv ri sureohpv zlwk lqfoxvlrqv ri duelwudu| vkdsh1
Vhfwlrq 8 lv ohiw iru vrph qdo frpphqwv1
5
Lwhudwhg vtxduh krqh|frpev
Zh vwduw e| ghqlqj wkh lwhudwhg vtxduh krqh|frpe0vwuxfwxuh e| d fkdu0
dfwhulvwlf ixqfwlrq 6c&c +lqglfdwru ixqfwlrq,1 Ohw f / ohw &c c 6
eh
l lqwhjhuv dqg ohw eh wkh fkdudfwhulvwlf ixqfwlrq ri wkh lqwhuydo
k srvlwlyh
*E26
Zh ghqh wkh fkdudfwhulvwlf ixqfwlrq 6c&c rq -2 dv
fc 6c&c E% ' E6 E% E6 E%2 c
En
zkhuh E6 lv irxqg lwhudwlyho| e| ohwwlqj
eh wkh dfc o0shulrglf h{whqvlrq
E
*E26
*E26
ri wkh ixqfwlrq E E& *
*2c Ef ' 

E| xvlqj wklv fkdudfwhulvwlf ixqfwlrq zh fdq ghqh wkh frqgxfwlylw|b6c&c E%
dv iroorzv=
b6c&c E% ' blq 6c&c E% n bfrq  6c&c E% 5
Iljxuh 4= Lwhudwhg vtxduh krqh|frpev1
Khuh/ blq dqg bfrq duh wkh frqgxfwlylwlhv ri wkh lqwhulru dqg wkh frqqhfwhg
sduw ri wkh lwhudwhg vtxduh krqh|frpe0vwuxfwxuh/ uhvshfwlyho|1
Lq rughu wr eh deoh wr vshdn derxw hhfwlyh frqgxfwlylw| iru wkh deryh
vwuxfwxuh zh frqvlghu wkh iroorzlqj wzr olplw0fdvhv=
{ 6 dqg & dqg ohw dssurdfk lqqlw|1 Wkh fruuhvsrqglqj hhfwlyh
frqgxfwlylw| lv lq wklv fdvh ghqrwhg b6c& { 6 dqg ohw ' & dssurdfk lqqlw|1 Wkh fruuhvsrqglqj hhfwlyh
frqgxfwlylw| lv lq wklv fdvh ghqrwhg b6c" Wkh uvw ri wkhvh fdvhv duh wuhdwhg e| vwdqgdug shulrglf krprjhql}dwlrq/
l1h1 +vhh h1j1 wkh errn ^47`,
<
;
$2
$2
#
#
@
?  ]
Y
Y
b6c& ' 4?
c
n

b
n
b
_%
6c&c
 E%
6c&c
 E%
>
Y%
Y%2
`shc2u Et = mt m t
M
+4,
c2
zkhuh t ' dfc o2 c `shu
Et lv wkh vsdfh ri t 0shulrglf ixqfwlrqv ri wkh xvxdo
Vrerohy vsdfh ` c2 Et Pruhryhu/ xvlqj wkh zhoo nqrzq surshuw| ri frpsoh0
phqwdu| hqhuj| lq wzr glphqvlrqv/ vhh h1j1 ^47/ s1 6<`/ zh jhw wkh iroorzlqj
6
Iljxuh 5= Lwhudwhg vtxduh krqh|frpev iru ' &
htxlydohqw yduldwlrqdo irupxod=
<
;
#
#
$2
$2
@
?  ]
Y
Y


' 4?
n

_%
n
>
b6c& `shc2u Et = mt m t b6c&c E% Y%
b6c&c E% Y%2

M
+5,
Lq wkh vhfrqg fdvh wkh hhfwlyh frqgxfwlylw| b6c" ' ^6 c zkhuh ^6 lv
ghwhuplqhg e| wkh iroorzlqj lwhudwlyh yduldwlrqdo irupxod=
^
<
;
$2
$2
#
#
@
?  ]
Y
Y
c
n

K
E%
n
K
E%
_%
'
4?
>
Y%
Y%2
`shc2u Et = mt m t
M
7
+6,
K E% ' ^3 n bfrq E / ' c c 6c
^f ' blq Vlploduo| dv lq +5, zh rewdlq wkh iroorzlqj htxlydohqw yduldwlrqdo irupxod=
;
?  ]


' 4?

c
2
=
^ M`s h u Et mt m t K E%
#
$2
Y
n
Y%

n
K E%
#
Y
Y%2
$2
<
@
_%> +7,
Uhpdun 4 Wkh lwhudwlyh surfhgxuh iroorzv e| wkh Lwhudwhg Krprjhql}dwlrq
Wkhruhp ri Ehqvrxvvdq/ Olrqv dqg Sdsdqlfrodrx ^8`1 Wklv wkhruhp lv ydolg
iru fdvhv zkhq wkh frqgxfwlylw| b6cc lv ri wkh irup
b6cc E% ' bE%c 2 %c c 6 %c
zkhuh b lv shulrglf lq hdfk yduldeoh +zklfk lv wkh fdvh iru wkh krqh|frpev
frqvlghuhg lq ^4:/ 53`,1 Lq rxu fdvh wklv dvvxpswlrq lv voljkwo| prglhg1
Krzhyhu/ e| plplfnlqj wkh surri jlyhq h1j1 lq ^49` lw lv qrw kdug wr yhuli| wkdw
wkh lwhudwlyh surfhvv uhpdlqv ydolg1 Pruhryhu/ e| xvlqj wkh vdph dujxphqwv
dv lq wkdw surri lw lv dovr srvvleoh wr yhuli| wkdw
b6c& $ b6c"
dv & $ 4
Iru lwhudwhg vtxduh krqh|frpev wkh dssur{lpdwlrq irupxod ri Oxnndvvhq
+vhh ^4:/ 53`, uhgxfhv wr



o6  '
n
c
b6c" bfrq
blq bfrq  n 26 bfrq +8,
zkhuh *E26 o6  +lq idfw/ wklv irupxod lv dovr ydolg iru lwhudwhg kh{dj0
rqdo krqh|frpev/ vhh ^4;`,1 Qrwh wkdw +8, |lhogv wkh frqyhujhqfh
b6c" $ b"c"



' bfrq n
n
blq bfrq 2bfrq 3

c
+9,
dv 6 $ 4 / zklfk lv suhflvho| wkh Kdvklq0Vkwulnpdq xsshu erxqg iru
blq bfrq dqg wkh Kdvklq0Vkwulnpdq orzhu erxqg li blq : bfrq 1 Zh uh0
fdoo wkdw wkh Kdvklq0Vkwulnpdq erxqgv duh wkh ehvw srvvleoh zlwklq wkh fodvv
ri pdfurvfrslfdoo| lvrwurslf vwuxfwxuhv +iru wkh ruljlqdo surri vhh ^44`,1
8
Iljxuh 6= H{dpsoh ri xqlw fhoov iru vrph w|shv ri lwhudwhg vtxduh krqh|frpev
lq wkh fdvh ' fD2
6
Qxphulfdo frpsxwdwlrq ri hhfwlyh surs0
huwlhv iru lwhudwhg vtxduh krqh|frpev
Lq wklv vhfwlrq zh vwxg| qxphulfdo frpsxwdwlrqv ri wkh hhfwlyh frqgxfwly0
lwlhv b6c& dqg b6c" 1 Wkh phwkrg iru rxu qxphulfdo hvwlpdwlrqv lv edvhg rq
c2
c2
uhsodflqj `shu
Et zlwk d qlwh glphqvlrqdo vxevhw ri `shu
Et lq +4,/ +5,
dqg +6,/ +7,/ uhvshfwlyho|1 Lq wklv zd| zh rewdlq xsshu dqg orzhu qxphulfdo
hvwlpdwhv bn
c iru b6c& Lq sduwlfxodu/ zh rewdlq xsshu dqg orzhu
dqg b3
6c&
6c&
n
c iru b6c" zklfk duh irxqg lwhudwlyho| e|
qxphulfdo hvwlpdwhv b6c" dqg b3
6c"
n
3
uhsodflqj ^ E% zlwk ^ E% dqg ^ E% lq +6, dqg +7,c uhvshfwlyho|1 Zh qrwh
wkdw wkh jhrphwu| ri wkh sureohp lv wkh vdph rq hdfk vwhs1 Edvhg rq wklv
jhrphwu| zh fdq fuhdwh d {hg IHP0phvk +zlwk qhvw phvk0uhqhphqw lq
wkh fruqhuv, zklfk zh fdq xvh rq doo wkh 26 vwhsv1 Dv dq dssur{lpdwlrq iru
h
b6c& zh xvh wkh qxpehu b
6c&
n
3
h
c
b
6c& ' *2 b6c& n b6c&
dqg ohw o6c& ghqrwh wkh fruuhvsrqglqj pd{lpxp srvvleoh uhodwlyh qxphulfdo
9
huuru hvwlpdwh +lq (, l1h1
o6c& ' ff
3
bn
6c& b6c&
3 bn
6c& n b6c&
Pruhryhu/ ohw _6c& ghqrwh wkh ghyldwlrq ehwzhhq b6c& dqg b"c" +lq (, l1h1
_6c& ' ff
b"c" b6c&
b6c&
+:,
h
dqg ohw _6c& ghqrwh wkh dssur{lpdwh ghyldwlrq ehwzhhq b
6c& dqg b"c" +lq
(, l1h1
h
b"c" b
6c&
_6c& ' ff
h
b6c&
+zlwk pd{lpxp srvvleoh qxphulfdo huuru ohvv wkdq o6c& Eb"c" *bn
6c& ,1 Li wkh
pd{lpxp huuru hvwlpdwh lv wrr odujh zh kdyh wr uhqh wkh phvk hyhq pruh
dqg vwduw doo ryhu djdlq zlwk shuiruplqj dqrwkhu 26 frpsxwdwlrqv1
Lq wkh fdvh & 4 zh rqo| kdyh wr shuirup wzr frpsxwdwlrqv lqvwhdg ri
3
261 Khuh bn
6c& dqg b6c& duh irxqg gluhfwo| iurp +4, dqg +5,1 Krzhyhu/ zkhq
& lv odujh wkh sureohp ehfrphv vwurqjo| qrq0krprjhqhrxv dqg lq rughu wr
rewdlq dq dffxudwh dssur{lpdwlrq zh zloo qhhg dq hqruprxv dprxqw ri qlwh
hohphqwv1 Zh kdyh shuiruphg d qxpehu ri qxphulfdo fdofxodwlrqv1 Lq Iljxuh
7 +vhh dovr Dsshqgl{, zh suhvhqw vrph ri wkh h{dfw sorwv ri _6c& +dfwxdoo|
ri _6c& exw wkh glhuhqfh lv lqvljqlfdqw, dv d ixqfwlrq ri wkh duhd0iudfwlrq iru vrph ydoxhv ri 6 dqg & Hdfk sorw lv iru wkh h{wuhph fdvh blq *bfrq f
+iru zklfk _6c& lv dvvxphg wr eh odujhvw srvvleoh/ f1i1 ^54` dqg ^56`,1
E| hydoxdwlqj b6c" iurp wkh irupxod ri Oxnndvvhq +8, zlwk o6 '  dqg
uhsodflqj b6c& lq +:, zlwk wklv ydoxh zh reylrxvo| rewdlq dq xsshu erxqg iru
_6c& Lq rughu wr frpsduh zlwk rxu qxphulfdo fdofxodwlrqv zh kdyh sorwwhg
wklv xsshu erxqg iru vrph ydoxhv ri 6 lq Iljxuh 81
Ohw xv jlyh d ihz frpphqwv wr wkh qxphulfdo uhvxowv1 Lw vhhpv wkdw iru d
{hg ydoxh ri 6 wkh frqgxfwlylw| lqfuhdvhv zlwk &1 Krzhyhu/ wkh frqyhujhqfh/
dv & jrhv wr 4/ lv vorzhu wkdq zh h{shfwhg +vhh Iljxuh 7,1 Iurp d pdqxidf0
wxulqj srlqw ri ylhz lw lv reylrxv wkdw wkh qxpehu ri zklwh vtxduhv &2E63
lq hdfk xqlw fhoo lv dq lpsruwdqw sdudphwhu +lw lv hdvlhu wr pdqxidfwxuh d
vwuxfwxuh zlwk ihz odujh vtxduhv wkdq pdq| vpdoo rqhv,1 Zh revhuyh iurp
Iljxuh 7 wkdw wkh frqgxfwlylw| pd| qrw qhfhvvdulo| lqfuhdvh zlwk wkh qxpehu
&2E63 +frpsduh h1j1 wkh fdvh 6 ' 2c & ' H zlwk wkh fdvh 6 ' c & ' ,1
:
Iljxuh 7= Glhuhqfh ehwzhhq wkh hhfwlyh frqgxfwlylw| dqg wkh Kdvklq0
Vkwulnpdq xsshu erxqg1
Iljxuh 8= Xsshu erxqg iru _6c" dffruglqj wr wkh dssur{lpdwlrq irupxod ri
Oxnndvvhq
;
Krzhyhu/ gxh wr wkh huuru hvwlpdwhv +vhh wkh odvw wdeoh lq wkh Dsshqgl{, zh
fdq qrw gudz dq| up frqfoxvlrqv rq wklv pdwwhu iru wkh frpsxwhg fdvhv1
Iru odujh ydoxhv ri & wklv idfw lv reylrxv iurp wkh & ' 40fxuyhv lq Iljxuh 7/
dqg wkh frqyhujhqfh uhvxow ^9`1
7
Vtxduh krqh|frpev zlwk kljko| lqkrprjh0
qhrxv vtxduhv
E| xvlqj vwdqgdug qlwh hohphqw phwkrg surjudpv wkh dffxudf| ri wkh frp0
sxwdwlrqv zkhq & ' 4 lv vx!flhqw iru wkh sxusrvh ri frpsdulvrq1 Wkh
frpsxwdwlrqv iru qlwh ydoxhv ri & duh ohvv dffxudwh +vhh wkh wdeoh ydoxhv lq
wkh Dsshqgl{,1 Zkhq wkh xqlw fhoo frqvlvwv ri pruh wkdq durxqg ff remhfwv
+lq wklv fdvh vtxduhv, lw wxuqv rxw wr eh doprvw lpsrvvleoh wr rewdlq xvhixo
uhvxowv e| vwdqgdug qlwh hohphqw phwkrg surjudpv1
Qrz/ frqvlghu wkh fdvh zkhq wkh froohfwlrq ri vtxduhv lv uhsodfhg zlwk
d pxfk odujhu froohfwlrq ri udqgrpo| glvwulexwhg glvnv1 Wkhuh duh uhfhqwo|
ghyhorshg phwkrgv ^</ 46` wkdw duh fdsdeoh ri udslgo| vroylqj yhu| frpsoh{
fhoo vwuxfwxuhv wr kljk dffxudf|1 Lq rughu wr looxvwudwh/ zh zloo khuh xvh dq
lqwhjudo htxdwlrq edvhg dgdswlyh vfkhph ghyhorshg e| Khovlqj ^45/ 46` dqg
dffhohudwhg zlwk wkh idvw pxowlsroh phwkrg ^43`1 Zh zloo xvh wklv vfkhph wr
frpsxwh wkh hhfwlyh surshuwlhv ri svhxgrudqgrp vwuxfwxuhv zlwk f2e dqg
efbS lqfoxvlrqv lq wkh fruuhvsrqglqj fhoov1
Wkh svhxgrudqgrp glvn frqjxudwlrqv zhuh jhqhudwhg e| d Prqwh Fduor
surfhgxuh ri Phwursrolv w|sh ^57`1 Zh ehjlq e| sodflqj doo glvnv dw wkh
odwwlfh srlqwv ri d uhjxodu vtxduh duud|1 Hdfk glvn lv wkhq jlyhq d vpdoo
whqwdwlyh glvsodfhphqw lq d udqgrp gluhfwlrq1 Wkh glvsodfhphqw lv dffhswhg
ru uhmhfwhg dffruglqj wr zkhwkhu ru qrw wkh qhz srvlwlrq zloo fdxvh wkh glvn
wr ryhuods lwv qhljkeruv1 Rqh lwhudwlrq vwhs frqvlvwv ri wu|lqj wr pryh hdfk
glvn rqfh vr wkdw wkh udwlr ri pryhg glvnv wr wkh wrwdo qxpehu ri glvnv lv
derxw rqh kdoi1 Wklv surfhvv lv lwhudwhg vhyhudo wlphv xqwlo htxloleulxp lv
dfklhyhg1 Wkh udqgrp frqjxudwlrqv zh xvh lq rxu h{dpsohv zhuh jhqhudwhg
xvlqj wkrxvdqgv ri lwhudwlrq vwhsv1
Lq rxu h{dpsoh wkh jhrphwulhv frqvlvw ri hlwkhu f2e ru efbS udqgrpo|
sodfhg glvnv zlwk fhqwhuv lq d vtxduh ri vl}h fH fH1 Doo glvnv duh ri wkh
vdph vl}h dqg fkrvhq vxfk wkdw wkh wrwdo duhd iudfwlrq ri wkh glvnv lv fS1 Wkh
glvnv kdyh frqgxfwlylw| fff dqg wkh vxuurxqglqj pdwul{ kdv frqgxfwlylw|
<
1 Ilqdoo| wklv vtxduh lv wkhq lwvhoi sodfhg lq d vtxduh ri vl}h f f/ vhh
Iljxuh 91
Wkh fhoo sureohp lv wkhq vroyhg rq wkhvh xqlw fhoov zlwk shulrglf erxqgdu|
frqglwlrqv dqg wkh krprjhql}hg frqgxfwlylw| lv frpsxwhg1 Wklv surfhgxuh
lv h{hfxwhg iru yh glhuhqw uhdol}dwlrqv ri wkh fhoo vwuxfwxuh1 Irxu vwuxfwxuhv
frqwdlq f2e glvnv dqg rqh vwuxfwxuh frqwdlqv efbS glvnv1 Wkh uhodwlyh huuru
iru wkh frqgxfwlylwlhv lq wkh f2e glvn frpsxwdwlrqv lv hvwlpdwhg wr eh f3b
dqg wkh uhodwlyh huuru iru wkh frqgxfwlylw| lq wkh efbS glvn frpsxwdwlrq
lv hvwlpdwhg wr eh f3. 1 Wkh phdq ydoxh dqg pd{lpxp ghyldwlrq iru doo
yh frqjxudwlrqv duh wkhq frpsxwhg wr jlyh wkh udqjh ri wkh krprjhql}hg
frqgxfwlylw|1
Zh frpsduh wkh frqgxfwlylw| ri wkh wzr0vfdoh fhoo vwuxfwxuh zlwk wkdw ri d
fhoo zkhuh wkh vtxduh zlwk glvnv lv uhsodfhg zlwk d vtxduh zlwk d krprjhqhrxv
pdwhuldo zlwk wkh vdph hhfwlyh surshuwlhv dv wkh vtxduh zlwk glvnv1 Wkh
vtxduhv zlwk glvn lqfoxvlrqv kdyh frqgxfwlylwlhv htxdo wr
D  ff. c
+;,
2D ff2 +<,
dqg wkh fruuhvsrqglqj wzr0vfdoh vwuxfwxuhv zlwk wkhvh vtxduhv lqfoxghg lq
wkh odujhu vtxduh kdyh frqgxfwlylwlhv ri
E| lqvwhdg uhsodflqj wkh lqfoxghg vtxduh frqvlvwlqj ri glvnv zlwk d vtxduh
frqvlvwlqj ri d krprjhqhrxv pdwhuldo zlwk hhfwlyh frqgxfwlylw| htxdo wr
D / zh rewdlq d frqgxfwlylw| htxdo wr 2 Dc zklfk lv zlwklq wkh hvwlpdwhv
jlyhq lq +<,1 Wklv djuhhphqw vxjjhvwv wkdw lw lv dgplvvleoh wr uhsodfh wkh
xqghuo|lqj khwhurjhqhrxv vfdoh e| d krprjhqhrxv rqh/ zklfk zdv h{shfwhg1
Rqh pljkw wklqn wkdw wkh ydoxh lq +;, lq wkh udqgrp fdvh lv forvh wr
wkh hhfwlyh frqgxfwlylw| ri d shulrglf glvwulexwlrq ri vxfk glvnv1 Krzhyhu/
lw wxuqv rxw wkdw wkh frqgxfwlylwlhv duh D 22 I orzhu lq wkh shulrglf
fdvh/ ehlqj e22 iru vtxduh sdfnlqj dqg ef iru kh{djrqdo sdfnlqj1 Wklv
vkrzv wkdw shufrodwlrq hhfwv lq wkh udqgrp fdvh frqvlghudeo| dhfwv wkh
frqgxfwlylwlhv iru wklv yroxph iudfwlrq ri fS
8
Vrph qdo frpphqwv
Wkh wkhruhwlfdo zrun rq lwhudwhg krqh|frpev shuiruphg lq ^4:/ 4;/ 4</ 53` zdv
odwhu iroorzhg e| vwxglhv ri vrph qrq0olqhdu surshuwlhv +vhh ^9/ 4:/ 5;/ 5<`,1
43
Iljxuh 9= Xqlw fhoo ri vtxduh krqh|frpe frqwdlqlqj 73<9 glvnv1 Wkh hhfwlyh
frqgxfwlylw| lq wkh %0gluhfwlrq lv hvwlpdwhg wr 5186<59861
Lq sduwlfxodu/ lq ^9` lw zdv revhuyhg wkdw wkh pdfurvfrslf ehkdylru ri uh0
lwhudwhg krqh|frpev pd| eh pruh h{wuhph wkdq wkdw ri wkh ehvw srvvleoh
odplqdwhv ri udqn 2 Wklv lv dq lqwhuhvwlqj revhuydwlrq vlqfh iru olqhdu sure0
ohpv udqn 20odplqdwhv kdyh suryhq wr eh ehvw srvvleoh zlwklq wkh fodvv ri
wzr0frpsrqhqw vwuxfwxuhv1 Lq doo wkhvh vwxglhv qrqolqhdu yhuvlrqv ri Wkh Lw0
hudwhg Krprjhql}dwlrq Wkhruhp kdv sod|hg dq hvvhqwldo uroh1 Iru wkh prvw
jhqhudo qrqolqhdu yhuvlrqv zh uhihu wr ^9/ 48/ 49`1 Dq dqlvrwurslf yhuvlrq ri
wkh irupxod ri Oxnndvvhq fruuhvsrqglqj wr dq lqqlwh qxpehu ri lwhudwlrq
ohyhov zdv frpsduhg zlwk wkh hhfwlyh surshuwlhv ri uhfwdqjxodu krqh|frpev
lq ^55` dqg ^56`1 Khuh/ lw zdv revhuyhg d vxusulvlqjo| jrrg djuhhphqw ehwzhhq
wkh qxphulfdo fdofxodwlrqv dqg wklv irupxod1
Frqfhuqlqj wkh Lwhudwhg Krprjhql}dwlrq Wkhruhp iru olqhdu sureohpv
lqfoxglqj hodvwlflw| zh uhihu wr ^5`1 Vhh dovr ^4` iru d glhuhqw dssurdfk e| xvh
ri vr0fdoohg pxowlvfdoh frqyhujhqfh1 Wkhuh duh pdq| rwkhu h{dpsohv lq wkh
olwhudwxuh zkhuh lwhudwhg krprjhql}dwlrq duh xvhg1 Iru lwhudwhg odplqdwhv zh
uhihu wr Iudqfiruw dqg Pxudw ^;`/ Plowrq ^58/ 59`/ Dyhoodqhgd hw do1 ^6/ 7` dqg
wkh uhihuhqfhv jlyhq wkhuh1 Wkhuh lv dovr d qhz fodvv ri rswlpdo plfurvwuxfwxuh
44
irxqg uhfhqwo|/ vhh Vljpxqg ^5:`1
Wr jhw vdwlvidfwru| hhfwlyh frqgxfwlylwlhv iru fhuwdlq w|shv ri udqgrp
phgld/ rqh kdv wr vroyh fhoo sureohpv zlwk wkrxvdqgv ri udqgrp lqfoxvlrqv1
Wklv lv d wdvn wkdw lv h{wuhpho| kdug zlwk vwdqgdug w|sh vriwzduh iru qx0
phulfdo vroxwlrq ri sduwldo glhuhqwldo htxdwlrqv/ vxfk dv wkh qlwh hohphqw
phwkrg1 Wkh fdofxodwlrqv shuiruphg lq Vhfwlrq 7 lqglfdwhv wkdw wkh phwkrg
ghyhorshg lq ^46/ 45` lv d srzhuixo wrro1
Uhlwhudwhg krprjhql}dwlrq fdq eh xvhg wr vroyh pdq| sudfwlfdo sureohpv1
Zh khuh uhihu wr ^:` zkhuh uhlwhudwhg krprjhql}dwlrq zdv xvhg wr frpsxwh
wkh hodvwlf surshuwlhv ri zryhq frpsrvlwhv1
Uhihuhqfhv
^4` J1 Doodluh dqg P1 Euldqh/ Pxowlvfdoh frqyhujhqfh dqg uhlwhudwhg krprjh0
ql}dwlrq1 Surfhhglqjv ri wkh Ur|do Vrflhw| ri Hglqexujk/ 459D/ 5<:0675/
4<<91
^5` P1 Dyhoodqhgd/ Lwhudwhg krprjhql}dwlrq/ glhuhqwldo hhfwlyh phglxp
wkhru|/ dqg dssolfdwlrqv1 Frpp1 Sxuh Dsso1 Pdwk1/ 73/ ;36;7:/ 4<;:1
^6` P1 Dyhoodqhgd dqg D1 Y1 Fkhundhy dqg N1 D1 Oxulh dqg J1 Z1 Plo0
wrq/ Rq wkh hhfwlyh frqgxfwlylw| ri sro|fu|vwdov dqg d wkuhh0glphqvlrqdo
skdvh0lqwhufkdqjh lqhtxdolw|1 M1 Dsso1 Sk|v1/ 96/ 43/ sdjhv 7<;<8336/
4<;;1
^7` P1 Dyhoodqhgd dqg D1 Y1 Fkhundhy dqg O1 Y1 Jleldqvn| dqg J1 Z1
Plowrq dqg P1 Uxghovrq/ D frpsohwh fkdudfwhul}dwlrq ri wkh srvvleoh
exon dqg vkhdu prgxol ri sodqdu sro|fu|vwdov1 M1 Phfk1 Sk|v1Vrolgv/ 77/
44:<454;/ 4<<91
^8` D1 Ehqvrxvvdq/ M1O1 Olrqv/ dqg J1F1 Sdsdqlfrodrx/ Dv|pswrwlf Dqdo|vlv
iru Shulrglf Vwuxfwxuhv1 Qruwk Kroodqg/ Dpvwhugdp/ 4<:;1
^9` D1 Eudlghv dqg G1 Oxnndvvhq/ Uhlwhudwhg krprjhql}dwlrq ri lqwhjudo
ixqfwlrqdov1 Pdwk1 Prg1 Phwk1 Dsso1 Vfl/ 43/ 4/ 7::4/ 53331
^:` M1 E|vwuùp/ Q1 Mhndevrqv dqg M1 Yduqd/ Dq hydoxdwlrq ri glhuhqw prg0
hov iru suhglfwlrq ri hodvwlf surshuwlhv ri zryhq frpsrvlwhv1 Frpsrvlwhv
Hqjlqhhulqj/ Sduw E/ 64/ 4/ :053/ 53331
45
^;` J1D1 Iudqfiruw dqg I1 Pxudw/ Krprjhql}dwlrq dqg rswlpdo erxqgv lq0
olqhdu hodvwlflw|1 Dufk1 Udw1 Phfk1 Dqdo1/ <7/ 4944::/ 4<;91
^<` K1 Fkhqj dqg O1 Juhhqjdug/ Rq wkh qxphulfdo hydoxdwlrq ri hohfwurvwdwlf
hogv lq ghqvh udqgrp glvshuvlrq ri f|olqghuv1 M1 Frpsxw1 Sk|v1/ 469/
95<096</ 4<<:1
^43` O1 Juhhqjdug dqg Y1 Urnkolq/ D idvw dojrulwkp iru sduwlfoh vlpxodwlrqv1
M1 Frpsxw1 Sk|v1 :6/ 658/ 4<;:1
^44` ]1 Kdvklq dqg V1 Vkwulnpdq/ D yduldwlrqdo dssurdfk wr wkh wkhru| ri
hhfwlyh pdjqhwlf shuphdelolw| ri pxowl skdvh pdwhuldov1 M1 Dsso1 Sk|v
66/ 645806464/ 4<951
^45` M1 Khovlqj/ Wklq eulgjhv lq lvrwurslf hohfwurvwdwlfv1 M1 Frps1 Sk|v1 45:/
4750484/ 4<<91
^46` M1 Khovlqj/ D kljk0rughu dffxudwh dojrulwkp iru hohfwurvwdwlfv ri ryhuods0
slqj glvnv1 M1 Vwdw1 Sk|v1 <3 / 4794047:6/ 4<<;1
^47` Y1Y1 Mlnry/ V1P1 Nr}ory dqg R1D1 Rohlqln/ Krprjhql}dwlrq ri glhuhq0
wldo rshudwruv dqg lqwhjudo ixqfwlrqdov1 Vsulqjhu0Yhuodj/ Ehuolq/ 4<<71
^48` M1O1 Olrqv/ G1 Oxnndvvhq/ O1H1 Shuvvrq dqg S1 Zdoo/ Uhlwhudwhg krprjh0
ql}dwlrq ri prqrwrqh rshudwruv1 F1 U1 Dfdg1 Vfl/ Vhulhv L/ 663+;,=9:809;3/
53331
^49` M1O1 Olrqv/ G1 Oxnndvvhq/ O1H1 Shuvvrq dqg S1 Zdoo/ Uhlwhudwhg krprj0
hql}dwlrq ri qrqolqhdu prqrwrqh rshudwruv1 Wr dsshdu lq= Fklq1 Dqq1
Pdwk1/ Vhu1 E / 55/ 4/ 53341
^4:` G1 Oxnndvvhq/ Irupxodh dqg erxqgv frqqhfwhg wr rswlpdo ghvljq dqg
krprjhql}dwlrq ri sduwldo glhuhqwldo rshudwruv dqg lqwhjudo ixqfwlrqdov1
Sk1G1 wkhvlv/ Ghsw1 ri Pdwk1/ Wurpvù Xqlyhuvlw|/ Qruzd|/ 4<<91
^4;` G1 Oxnndvvhq= Erxqgv dqg krprjhql}dwlrq ri rswlpdo uhlwhudwhg krqh|0
frpev1 Lq= Frpsxwhu Dlghg Rswlpxp Ghvljq ri Vwuxfwxuhv Y/ Hgv1 V1
Khuqäqgh}/ F1 D1 Euhggld/ 59:05:9/ Frpsxwdwlrqdo Phfkdqlfv Sxeolfd0
wlrqv/ Vrxwkdpswrq/ 4<<:1
46
^4<` G1 Oxnndvvhq/ Vrph dvshfwv ri uhlwhudwhg krqh|frpev1 Lq= Phfkdqlfv
ri Frpsrvlwh Pdwhuldov dqg Vwuxfwxuhv/ hgv1 F1D1 Prwd Vrduhv/ F1P1
Prwd Vrduhv dqg P1M1P1 Iuhlwdv/ Yro1 LLL / 6<<073</ QDWR DVL/ Wurld/
Sruwxjdo/ 4<<;1
^53` G1 Oxnndvvhq/ D qhz uhlwhudwhg vwuxfwxuh zlwk rswlpdo pdfurvfrslf eh0
kdylrxu1 VLDP M1 Dsso1 Pdwk1/ 8<+8,/ 4;584;75/ 4<<<1
^54` D1 Phlghoo/ Wkh0Rxw0ri0sodqh vkhdu prgxoxv ri wzr0frpsrqhqw uhjxodu
krqh|frpev zlwk duelwudu| wklfnqhvv1 Lq= Phfkdqlfv ri Frpsrvlwh Pd0
whuldov dqg Vwuxfwxuhv/ hgv1 F1D1 Prwd Vrduhv/ F1P1 Prwd Vrduhv dqg
P1M1P1 Iuhlwdv/ QDWR DVL/ Wurld/ Sruwxjdo/ Yro1 LLL/ 69:06:</ 4<<;1
^55` D1 Phlghoo dqg O1H1 Shuvvrq/ Rswlpdo ghvljq ri uhfwdqjxodu krqh|frpev
lq uhfwdqjxodu vdqgzlfk sodwhv/ Surfhhglqjv ri 9 Lqwhuqdwlrqdo Frqihu0
hqfh rq Frpsrvlwhv Hqjlqhhulqj/ 89:089;/ Ruodqgr/ 4<<<1
^56` D1 Phlghoo dqg S1 Zdoo1 Krprjhql}dwlrq dqg ghvljq ri vwuxfwxuhv zlwk
rswlpdo pdfurvfrslf ehkdylrxu1 Lq= Frpsxwhu Dlghg Rswlpxp Ghvljq ri
Vwuxfwxuhv Y/ Hgv1 V1 Khuqäqgh}/ F1 D1 Euhggld/ 6<607351 Frpsxwdwlrqdo
Phfkdqlfv Sxeolfdwlrqv/ Vrxwkdpswrq/ 4<<:1
^57` Phwursrolv/ D1Z1 Urvhqeoxwk/ P1Q1 Urvhqeoxwk/ D1K1 Whoohu/ dqg H1
Whoohu/ Htxdwlrq ri Vwdwh Fdofxodwlrqv e| Idvw Frpsxwlqj Pdfklqhv1 M1
Fkhp1 Sk|v1/ 54/ 43;:043<5/ 4<861
^58` J1 Z1 Plowrq/ Prgholqj wkh surshuwlhv ri frpsrvlwhv e| odplqdwhv1 Lq
Krprjhql}dwlrq dqg Hhfwlyh Prgxol ri Pdwhuldov dqg Phgld/ hg1 e|
M1O1 Hulfnvhq/ G1 Nlqghuohkuhu/ U1 Nrkq/ dqg M10O1 Olrqv1/ 4/ ss1 483
4:7/ Wkh LPD yroxphv lq pdwkhpdwlfv dqg lwv dssolfdwlrqv/ Vsulqjhu0
Yhuodj/ 4<;91
^59` J1 Z1 Plowrq/ Rq fkdudfwhul}lqj wkh vhw ri srvvleoh hhfwlyh whqvruv
ri frpsrvlwhv= Wkh Yduldwlrqdo Phwkrg dqg wkh Wudqvodwlrq Phwkrg1
Frpp1 rq Sxuh dqg Dsso1 Pdwk1/ Yro1 [OLLL/ 960458/ 4<<31
^5:` Roh Vljpxqg/ D qhz fodvv ri h{wuhpdo frpsrvlwhv1 M1 Phfk1 Sk|v1 Vrolgv/
7; / qr1 5/ 6<:75;/ 53331
47
^5;` S1 Zdoo/ Krprjhql}dwlrq ri vrph sduwldo glhuhqwldo rshudwruv dqg lq0
whjudo ixqfwlrqdov1 Sk1G1 wkhvlv/ Ghsw1 ri Pdwk1/ Oxohè Xqlyhuvlw| ri
Whfkqrorj|/ Oxohè/ Vzhghq/ 4<<;1
^5<` S1 Zdoo/ Rswlpdo erxqgv rq wkh hhfwlyh vkhdu prgxol iru vrph qrqolqhdu
dqg uhlwhudwhg sureohpv1 Dfwd1 Vfl1 Pdwk1/ 98=8860899/ 53331
48
Dsshqgl{
Zh khuh suhvhqw wkh qxphulfdo ydoxhv iru wkh frpsxwdwlrqv ri wkh vrolg olqhv
lq Iljxuh 71
4
p@4=
fb
fH
f.
fS
fD
fe
f
f2D
f2
f
ffD
h
b
6c&
ffD2fSefe
ffH.H.
f.DfH
f2efDeb.
fHbHff
fefbefS
fDS.
fD..Df
fSeeb.SH
fHfefS2
fHbSebe
pd{ huuru4
fb2f f3D
f.D f3e
f2 f3e
f2b f3e
f.2 f3e
feHS f3e
fDff f3e
fSff f3e
fHf f3e
fS2H f3e
fSDe f3e
o6c& dIo
fH f3
fS f3
f2 f3
f2 f3
f2 f3
f2 f3
fb. f32
ff f3
f f3
f.H f32
f. f32
_6c& dIo
fb
2e

b
eDf
eSH
e
b2
S
He
fbe
p@5=
fb
fH
f.
fS
fD
fe
f
f2D
f2
f
ffD
h
b
6c&
ffD2Sb
fff2H2
f.ef2e2
f2eDSe2D
f2SD.b
feb2.
fD2Se.f
fDH.fD2S
fSDe
fHf.fDH
fHb.DeDb
pd{ huuru
ff f3e
f2eD f3e
f2S f3e
f22. f3e
f2HD f3e
f f3e
feH f3e
f2b. f3e
fefH f3e
f2e f3e
f2f f3e
o6c& dIo
f2D f3
f22 f3
f2 f3
fb2 f32
fH. f32
fHf f32
fSS f32
fD f32
fS2 f32
ff f32
f2 f32
_6c& dIo
fD2
fbH
e
..
2f.
22D
22H
22f
2fe
H
fHf
@5
.
p>n
p>n
49
p@6=
fb
fH
f.
fS
fD
fe
f
f2D
f2
f
ffD
h
b
6c&
ffD2eD.b
ffeeH
f.ebD
f2e.2D2D
f2bfHDe
fe22S2.D
fDfHe2
fDb.fb
fSD.beDS
fHfe2f
fHbb2S
pd{ huuru
f f3e
f2DD f3e
f2e f3e
f2DH f3e
f2. f3e
f2H f3e
ff f3e
f2S f3e
f2b2 f3e
f2fS f3e
f2D f3e
o6c& dIo
f2 f3
f2 f3
fe f3
f f3
fH f32
fSS f32
fD. f32
fee f32
fee f32
f2D f32
fe f32
_6c& dIo
f
fS
fHb

2b
e
ee
ef

fb.
fS2
Lq wkh wdeoh ehorz zh suhvhqw wkh qxphulfdo ydoxhv iru wkh frpsxwdwlrqv
ri wkh srlqwv lq Iljxuh 71
6
Yroxph0 2
iudfwlrq
2
EfH2 ' 2
fSDS
2
2
2
2
2
2
2
2
e
5
&

2
e
D
S
.
H
b
f
4
2
4
2
QRV5

e
b
S
2D
S
eb
Se
H
ff
4
S
H
4
Se
h
b
6c&
f2ffSDf
f2fDe
f2f2eD
f2f2DH2
f2f2Hb2
f2fe.
f2f
f2feH
f2fDH
f2fS.
f2feeS
f2f22He
f2f22
f2fDSD
f2f2beD
Qxpehu ri vtxduhv lq wkh xqlw fhoo1
4:
o6c& dIo
e f3D
S f3e
S f3e
. f3e
2 f3e
2S f3e
SS f3e
D. f3e
S f3e
Sf f3e
2b f3D
. f3e
S f3e
2f f3D
S f3e
_6c&
ebb
fHf
2.2SD
2Dfeb
2 eH
22bH
22.H
2 fD2S
2ff2e
bD.e
D.D
2SDDb
2H
fbb2b
2 2S