D JU XVV W\SH LQHTXDOLW\ IRU PDSSLQJ RI ERXQGHG YDULDWLRQ DQG DSSOLFDWLRQV WR QXPHULFDO DQDO\VLV
5 5 #+BW+ # W 6,#AV
Ot|iBW| Bhtt |)Ti ?i^@*|) uLh wTtU|3@? 4@TT?}t @?_ u?U|L?t Lu ML?_i_ @h@|L? t }i? TT*U@|L?t uLh |i ?4ihU@* ?|i}h@|L? ThLM *i4 Lu |i +i4@??5|i*|it ?|i}h@* @hi @*tL UL?t_ihi_
41 YA|iNaW|NA
Lq 4<68/ J1 Juxvv ^<` suryhg dq lqhtxdolw| zklfk hvwdeolvkhv d frqqhfwlrq ehwzhhq wkh lqwhjudo ri d surgxfw ri wzr ixqfwlrqv dqg wkh surgxfw ri wkh lqwhjudov1 Qdpho|/ kh kdv vkrzq wkdw=
e4d
e
]
d
i+{,j+{,g{e4d
e
]
d
i+{,g{e4d
e
]
d
j+{,g{
47+!, +,>
surylghgi dqg jduh wzr lqwhjudeoh ixqfwlrqv rq ^d> e` dqg vdwlvi| wkh frqglwlrq= !i+{, dqgj+{,
iru doo{5^d> e`= Wkh frqvwdqw 4
7 lv wkh ehvw srvvleoh rqh dqg lv dfklhyhg iru
i+{, @j+{, @vjq
{d.5 e
=
Lq wkh uhfhqw sdshu ^:`/ V1V1 Gudjrplu dqg L1 Ihgrwry suryhg wkh iroorzlqj uhvxowv ri Juxvv w|sh iru Ulhpdqq0Vwlhowmhv lqwhjudo=
Wkhruhp 41 Ohwi> x= ^d> e`$Ueh vr wkdwxlv O0Olsvfklw}ldq rq^d> e`/ l1h1/ mx+{,x+|,m Om{|m
+414,
iru doo{> |5^d> e`/i lv Ulhpdqq lqwhjudeoh rq^d> e`dqg wkhuh h{lvwv wkh uhdo qxpehuv p> P vr wkdw
pi+{,P> +415,
iru doo{5^d> e`=Wkhq zh kdyh wkh lqhtxdolw|
] e
d i+{,gx+{,
x+e,x+d, ed
] e
d i+w,gw
45O+Pp,+ed,> +416,
dqg wkh frqvwdqw 4
5 lv vkdus/ lq wkh vhqvh lw fdq qrw eh uhsodfhg e| d vpdoohu rqh1
#@|iG @) bbb
bb @|i4@|Ut 5MiU| *@ttU@|L? h4@h) 2S#Dc 2S#bb( 5iUL?_@h) eDD ki) Lh_t @?_ Th@tit Bhtt W?i^@*|)c "@_h@|hi 6Lh4*@ic +i4@??5|i*|it W?|i}h@*
Iru rwkhu uhvxowv uhodwhg wr Juxvv lqhtxdlw|/ jhqhudol}dwlrqv lq lqqhu surgxfw vsdfhv/ iru srvlwlyh olqhdu ixqfwlrqdov/ glvfuhwh yhuvlrqv/ ghwhuplqdqwdo yhuvlrqv/ hwf1/ vhh Fkdswhu [ ri wkh errn ^44` dqg wkh sdshuv ^4`0^43` / zkhuh ixuwkhu uhihuhqfhv duh jlyhq1
Lq wklv sdshu zh srlqw rxw dq lqhtxdolw| ri Juxvv w|sh iru Olsvfklw}ldq pds0 slqjv dqg ixqfwlrqv ri erxqghg yduldwlrq dv zhoo dv lwv dssolfdwlrqv lq qxphulfdo lqwhjudwlrq iru wkh Ulhpdqq0Vwlhowmhv lqwhjudo1
51 YA|j}iB, YAj`B,|jt
Wkh iroorzlqjuhvxow ri Juxvv w|sh krogv1
Wkhruhp 51 Ohwi> x= ^d> e`$Ueh vxfk wkdw x lv O0olsvfklw}ldq rq^d> e`> dqgi lv d ixqfwlrq ri erxqghg yduldwlrq rq ^d> e`= Ghqrwh e| Zedi wkh wrwdo yduldwlrq ri i rq ^d> e`=Wkhq wkh iroorzlqj lqhtxdolw| krogv
e
]
d
x+{,gi+{,i+e,eid+d,
e
]
d
x+w,gw
O5+ed,
e
b
d
i= +514,
Wkh frqvwdqw 4
5 lv vkdus/ lq wkh vhqvh wkdw lw fdqqrw eh uhsodfhg e| d vpdoohu rqh1
Surri1 Dvi lv d ixqfwlrq ri erxqghg yduldwlrq rq ^d> e` dqgxlv frqwlqxrxv rq ^d> e`> zh kdyh
e
]
d
x+{,gi+{,i+e,i+d, ed
e
]
d
x+w,gw
@
e
]
d
3
Cx+{, 4
ed
e
]
d
x+w,gw
4 Dgi+{,
vxs
{5^d>e`
x+{,e4d
e
]
d
x+w,gw
e
b
d
i
@e4d vxs
{5^d>e`
e
]
d
^x+{,x+w,`gw
e
b
d
i= +515,
Xvlqjwkh idfw wkd x lvOOlsvfklw}ldq rq ^d> e`> zh fdq vwdwh/ iru dq| {5^d> e`> wkdw=
e
]
d
^x+{,x+w,`gw
e
]
d
mx+{,x+w,mgwO
e
]
d
m{wmgw
@ O5k+{d,5. +{e,5l
@O
%
4 7.
{d.e 5
5
+ed,5
&
D JUXVV W\SH LQHTXDOLW\ IRU PDSSLQJ RI ERXQGHG YDULDWLRQ 6
dqg e| +5=5,0+5=6, zh jhw=
vxs
{5^d>e`
x+{,e4d
e
]
d
x+w,gw
O+e5d, +517,
zkhqfh zh rewdlq +5=4,1
Wr suryh wkh vkdusqhvv ri wkh lqhtxdolw| +5=4,/ ohw xv fkrrvh
x+{, @{d.5e dqg i+{, @
; ? =
4 li {@d 3 li d ? { ? e
4 li {@e =
Wkhqxlv Olsvfklw}ldq zlwkO@ 4>dqgi lv ri erxqghg yduldwlrq1 Dovr zh kdyh
e
]
d
x+{,gi+{,i+e,i+d, ed
e
]
d
x+w,gw
@
e
]
d
{d.5e
gi+{,
@
{d.5 e
i+{,e
d
e
]
d
i+{,g{@ed=
Rq wkh rwkhu kdqg/ wkh uljkw kdqg vlgh ri +5=4, lv htxdo wr ed> dqg khqfh wkh vkdusqhvv ri wkh frqvwdqw lv suryhg1
Wkh iroorzlqjfruroodulhv krog=
Fruroodu| 41 Ohwi = ^d> e`$U eh dv deryh dqg x= ^d> e`$Ueh d glhuhqwldeoh pdsslqj zlwk d erxqghg ghulydwlyh rq ^d> e`> wkdw lv/ nx3n
4 @ vxs
w5^d>e`mx
3+w,m ?4=
Wkhq zh kdyh wkh lqhtxdolw|=
e
]
d
x+{,gi+{,i+e,eid+d,
e
]
d
x+w,gw
nx
3n 4
5 +ed,
e
b
d
i+{,= +518,
Wkh lqhtxdolw|+5=8,lv vkdus lq wkh vhqvh wkdw wkh frqvwdqw 45 fdqqrw eh uhsodfhg e| d vpdoohu rqh1
Fruroodu| 51 Ohw x eh dv deryh dqg i = ^d> e` $ U eh d glhuhqwldeoh pdsslqj zkrvh ghulydwlyh lv lqwhjudeoh/ l1h1/
ni3n 4@
e
]
d
mi3+w,mgw ?4=
Wkhq zh kdyh wkh lqhtxdolw|=
e
]
d
x+{,i3+{,g{i+e,i+d,
ed
e
]
d
x+w,gw
nx
3n 4ni3n4
5 +ed,=
Uhpdun 41 Li zh dvvxph wkdw j = ^d> e`$U lv frqwlqxrxv rq^d> e` dqg li zh vhw i+{, @
{
]
d
j+{,g{> wkhq iurp +5=9, zh jhw wkh iroorzlqj Juxvv w|sh lqhtxdolw| iru
wkh Ulhpdqq lqwhjudo=
e4d
e
]
d
x+{,j+{,g{e4d
e
]
d
x+w,gwe4d
e
]
d
j+w,gw
+51:,
nx3n4njn4
5 +ed,=
Fruroodu| 61 Dvvxph wkdw x = ^d> e` $ U lv glhuhqwldeoh rq +d> e,> x+d, 9@ x+e,> dqg x3 = ^d> e` $ U lv erxqghg rq +d> e,= Wkhq zh kdyh wkh wudsh}rlg lq0
htxdolw|=
x+d, .5 x+e,+ed,
e
]
d
x+w,gw
nx
3n 4nx3n4
5 ^x+e,x+d,`+ed,5= +51;,
Surri1 Li zh fkrrvh lq Fruroodu| 5/ i+{, @x+{,>zh jhw
e
]
d
x+{,x3+{,g{x+e,x+d,
ed
e
]
d
x+w,gw
nx
3n 4nx3n4
5 +ed,
+51<,
Qrz +5=;, iroorzv iurp +5=<,1
61 6jiWB, #BaiB|ij 8Ni6,B uNi |j -j6BAA7|j,|jt YA|j}iB,
Lq zkdw iroorzv/ zh vkdoo dsso| Wkhruhp 5 wr dssur{lpdwh wkh Ulhpdqq0Vwlhowmhv
lqwhjudo
e
]
d
x+{,gi+{, lq whupv ri wkh Ulhpdqq lqwhjudo
e
]
d
x+w,gw=
Wkhruhp 61 Ohwi> x= ^d> e`$Ueh dv lq Wkhruhp 5 dqg Lk@id@{3? {4? === ? {q4? {q@ej
d sduwlwlrq ri^d> e`= Ghqrwhkl@{l.4{l> l@ 3>4> ===> q4=Wkhq zh kdyh= e
]
d
x+{,gi+{, @Dq+x> i> Lk, .Uq+x> i> Lk,
+614,
zkhuh
Dq+x> i> Lk, @ q4
[
l@3
i+{l.4,i+{l,
kl
{]l.4
{l
x+w,gw +615,
dqg wkh uhpdlqghu whupUq+x> i> Lk,vdwlvhv wkh hvwlpdwlrq
mUq+x> i> Lk,m O5+k, e
b
d
i+{, +616,
D JUXVV W\SH LQHTXDOLW\ IRU PDSSLQJ RI ERXQGHG YDULDWLRQ 8
Surri1 Dsso|lqjWkhruhp 5 rq wkh lqwhuydo ^{l> {l.4`> l@ 3>4> ===> q4 zh j hw
{]l.4
{l
x+{,gi+{,i+{l.4,ki+{l,
l
{]l.4
{l
x+w,gw
O5 kl
{bl.4
{l
i+{,=
Vxpplqjryhul iurp 3 wrq4 dqg xvlqjwkh wuldqjoh lqhtxdolw| zh rewdlq
e
]
d
x+{,gi+{,Dq+x> i> Lk,
O5
q4[
l@3
kl {bl.4
{l
i+{, O5 +k,q4[
l@3 {bl.4
{l
i+{,
@ O5 +k,
e
b
d
i+{,=
dqg wkh fruroodu| lv suryhg1
Uhpdun 51 Frqvlghu wkh htxlglvwdqw sduwlwlrqLk jlyhq e|
Lk= {[email protected]qd> l@ 3> ===> q>
dqg ghqh
Dq+x> i> Lk,
@ eqd
q4[
l@3
i
d. +l. 4,eqd
i
d.leqd
d.+l.4,ed q ]
d.led q
x+w,gw=
Wkhq zh kdyh
e
]
d
x+{,gi+{, @Dq+x> i, .Uq+x> i,
zkhuh wkh uhpdlqghu Uq+x> i, vdwlvhv wkh hvwlpdwlrq
mUq+x> i,m O+e5qd, e
b
d
i+{,=
Wkxv/ li zh zdqw wr dssur{lpdwh wkh lqwhjudo
e
]
d
x+{,gi+{,e| wkh vxpDq+x> i> Lk,
zlwk dq huuru ri pdjqlwxgh ohvv wkdq% zh qhhg dw ohdvw
q3@
%
O+ed,
5%
e
b
d
i+{,
&
. 45Q
srlqwv1
Fruroodu| 71 Dvvxph wkdw xdqgi duh dv lq Fruroodu| 41 LiLk lv dv deryh/ wkhq
+6=4,krogv dqg wkh uhpdlqghu whupUq+x> i> Lk, vdwlvhv wkh hvwlpdwlrq
mUq+x> i> Lk,m nx 3n
4
5 +k,
e
b
d
Fruroodu| 81 Dvvxph wkdw xdqg i duh dv lq Fruroodu| 51 Wkhq +6=4, krogv dqg wkh uhpdlqghu whupUq+x> i> Lk,vdwlvhv wkh hvwlpdwlrq
mUq+x> i> Lk,m nx 3n
4ni3n4
5 +k,=
+618,
71 VV,WB|NA uNi |j CiBVj5NaB, 8Ni6,B
Wkh iroorzlqjlqwhjudo lqhtxdolw| ri wudsh}rlg w|sh iru d pdsslqjri erxqghg yduldwlrq krogv dqg kdv ehhq suryhg e| V1 V1 Gudjrplu lq sdshu ^;`1 Khuh zh jlyh dqrwkhu surri edvhg rq wkh Juxvv w|sh lqhtxdolw| +5=4,=
Wkhruhp 71 Ohwj= ^d> e`$Ueh d pdsslqj ri erxqghg yduldwlrq1 Wkhq zh kdyh wkh lqhtxdolw|=
j+d, .5j+e,+ed,
e
]
d
j+w,gw
e5d
e
b
d
j+w,> +714,
dqg wkh frqvwdqw 4
5 lv vkdus1
Surri1 Xvlqjwkh lqwhjudwlrq e| sduwv irupxod iru wkh Ulhpdqq0Vwlhowmhv lqwhjudo zh kdyh=
e
]
d
{d.e 5
gj+{, @ +ed, +j+e,j+d,,
5
e
]
d
j+{,g{= +715,
Ghqh wkh pdsslqjv i+{, @{d.e
5 dqgx+{, @j+{,= Wkhq lw lv fohdu wkdwi lv
O0olsvfklw}ldq zlwk O@ 4 dqgx lv ri erxqghg yduldwlrq1 Dsso|lqjwkh lqhtxdolw| +5=4,> zh ghgxfh=
e
]
d
{d.5e
gj+{,j+e,ejd+d,
e
]
d
{d.5e
g{
e5d
e
b
d
j+{,=
Dv
e
]
d
{d.e 5
g{@ 3> wkhq xvlqjwkh lghqwlw| +7=5, zh rewdlq wkh ghvluhg uhvxow
+7=4,1
Qrz frqvlghu wkh ixqfwlrqj= ^d> e`$Q>
j+{, @
; ? =
4 li {@d 3 li d ? { ? e 4 li {@e Wkhqj lv ri erxqghg yduldwlrq dqg=
e
]
d
j+{,g{@ 3> be
d
D JUXVV W\SH LQHTXDOLW\ IRU PDSSLQJ RI ERXQGHG YDULDWLRQ :
Wkxv
j+d, .5 j+e,+ed,
e
]
d
j+w,gw
@ed>
vlqfh ed 5
e
b
d
j+{, @ed/ dqg wkh htxdolw| lv uhdol}hg lq +7=4,=Wkh vkdusqhvv ri wkh frqvwdqw lv wkxv suryhg1
Fruroodu| 91 Ohw j = ^d> e` $ U eh d glhuhqwldeoh ixqfwlrq zkrvh ghulydwlyh lv lqwhjudeoh rq +d> e,=Wkhq zh kdyh wkh lqhtxdolw|=
j+d, .5 j+e,+ed,
e
]
d
j+w,gw
e5dnj3n4=
+716,
Wkh iroorzlqjtxdgudwxuh irupxod krogv=
Wkhruhp 81 Ohw j = ^d> e` $ U eh d pdsslqj ri erxqghg yduldwlrq dqg Lk @
id@{3? {4? === ? {q4? {q@ej d sduwlwlrq ri ^d> e`= Ghqrwh kl @{l.4{l>
l@ 3>4> ===q4=Wkhq zh kdyh
e
]
d
j+{,g{@Wq+j> Lk, .Uq+j> Lk,
+717,
zkhuh
Wq+j> Lk, @ q4[
l@3
j+{l.4, .j+{l,
5 kl
dqg wkh uhpdlqghu whupUq+j> Lk, vdwlvhv wkh hvwlpdwh
mUq+j> Lk,m +k,5 e
b
d
j+{,> +718,
wkh frqvwdqw 4
5 ehlqj vkdus1
Surri1 Dsso|lqjWkhruhp 7 rq wkh lqwhuydo ^{l> {l.4`> zh jhw
j+{l.4, .5 j+{l,kl
{]ll.4
{l
j+w,gw
k5l
{bl.4
{l
j+{,
iru dool@ 3>4> ===> q4=
Vxpplqjryhul iurp 3 wrq4/ zh ghgxfh
e
]
d
j+{,g{Wq+j> Lk,
45
q4
[
l@3 {bl.4
{l
j+{,kl
4 5+k,
q4[
l@3 {bl.4
{l
j+{, @ 4 5+k,
e
b
d
j+{,=
Uhpdun 61 Frqvlghu wkh htxlglvwdqw sduwlwlrqLk jlyhq e|
Lk={[email protected] eqd> l@ 3> ===> q>
dqg ghqh
Wq+j, @ e5qd q4
[
l@3
j
d. +l. 4,eqd
.j
d.leqd
Wkhq zh kdyh=
e
]
d
j+{,g{@Wq+j, .Uq+j,
zkhuh wkh uhpdlqghu Uq+j,vdwlvhv wkh hvwlpdwlrq
mUq+j,m +e5qd, e
b
d
j+{,=
Wkxv/ li zh zdqw wr dssur{lpdwh wkh lqwhjudo
e
]
d
j+{,g{e| wkh vxp Wq+j,zlwk dq
huuru ri pdjqlwxgh ohvv wkdq% zh qhhg dw ohdvw
q3@
%
+ed, 5%
e
b
d
j+{,
&
. 45Q
srlqwv1
Fruroodu| :1 Ohw j = ^d> e` $ U eh d glhuhqwldeoh ixqfwlrq zkrvh ghulydwlyh lv lqwhjudeoh rq +d> e,= Li Lk lv dv deryh/ wkhq +7=7, krogv dqg wkh uhpdlqghu vdwlvhv
wkh hvwlpdwlrq=
mUq+j> Lk,m +k,5 nj3n4=
81 #BaiB|ij j|Nat uNi |j -j6BAA7|j,|jt YA|j}iB, Nu NA|ANt BVVA}t
OhwLk@id@{3? {4? === ? {q4? {q@ejeh d sduwlwlrq ri ^d> e` dqg ghqrwh
kl@{l.4{l> l@ 3>4> ===q4=
Zh vwduw wr wkh iroorzlqjohppd zklfk lv ri lqwhuhvw lq lwvhoi1
Ohppd41 Ohwi eh d ixqfwlrq iurpF^{l> {l.4`>l1h1/i lv frqwlqxrxv rq^{l> {l.4`>
dqg ohw xeh d ixqfwlrq ri erxqghg yduldwlrq rq wkh vdph lqwhuydo1 Wkh iroorzlqj lqhtxdolw| krogv=
{]l.4
{l
i+{,gx+{,^x+{l.4,x+{l,` il
$^i> kl` {bl.4
{l
x+{, +814,
zkhuh il @ k4 l
{]l.4
{l
i+{,g{> kl @{l.4{l>$^i> kl` @ vxs
m{wmmi+{,i+w,m>lv wkh
D JUXVV W\SH LQHTXDOLW\ IRU PDSSLQJ RI ERXQGHG YDULDWLRQ <
Surri1 Vlqfhi 5F^{l> {l.4` dqgxlv d ixqfwlrq ri erxqghg yduldwlrq/ e| wkh zhoo
nqrzq surshuw| ri vxfk frxsoh ri ixqfwlrqv zh kdyh
{]l.4
{l
i+{,gx+{,
ninF^{l>{l.4` {bl.4
{l
x+{,=
Wkhuhiruh/
{]l.4
{l
i+{,gx+{,^x+{l.4,x+{l,` il
@
@
{]l.4
{l
i+{,ilgx+{,
i+{,ilF^{l>{l.4` {bl.4
{l
x+{,
@ vxs
{5^{l>{l.4`
i+{,il{bl.4
{l
x+{,
@ k4
l{5^{vxsl>{l.4`
kli+{,klil{bl.4
{l
x+{,
@ k4
l{5^{vxsl>{l.4`
{]l.4
{l
i+{,gw
{]l.4
{l
i+w,gw
{bl.4
{l
x+{,
@ k4
l{5^{vxsl>{l.4`
{]l.4
{l
^i+{,i+w,`gw
{bl.4
{l
x+{,
vxs
{5^{l>{l.4`>w5^{l>{l.4`
mi+{,i+w,m
{bl.4
{l
x+{,
@ $^i> kl` {bl.4
{l
x+{,>
dqg wkh ohppd lv suryhg1
Ohwi 5F^d> e`=Wkh gxdo wrF^d> e` lv wkh vsdfh ri ixqfwlrqv ri erxqghg yduld0
wlrq/ wkh jhqhudo irup ri wkh ixqfwlrqdo rqF^d> e` lvL+i, @
e
]
d
i+{,gx+{, zkhuhx
ehorqjv wr wkh vsdfh ri ixqfwlrqv ri erxqghg yduldwlrq1
Wkdw lv zk| lq wkh wkhru| ri txdgudwxuh phwkrgv iru frqwlqxrxv ixqfwlrqv wkh fdvh ri wkh lqwhjudov ri vxfk d w|sh lv wkh prvw lqwhuhvwlqj1 Iru wkh lqwhjudo zlwk
frqwlqxrxv lqwhjudqg
e
]
d
iru wkh txdgudwxuh uxoh +zlwk wkh zhljkwvvqn dqg qrghv{n,
L+i,
e
]
d
i+w,z+w,gw [q
n@3
i+{n,vqnLq+i,
e| wkh irupxod
Hq+i, @L+i,Lq+i,=
Khuh lv wkh pruh jhqhudo uhvxow iru wkh frpsrvlwh txdgudwxuh uxohv iru wkh ixqf0 wlrqv iurpF^d> e`=
Wkhruhp 91 Ohw d dqg e eh qlwh uhdo qxpehuv dqg ohw i> x = ^d> e` $ Ueh vxfk wkdw i 5 F^d> e` dqg x lv d ixqfwlrq ri erxqghg yduldwlrq rq ^d> e`1 Li d @ {3 ?
{4? === ? {q4? {q@e lv d glylvlrq ri^d> e`vxfk wkdwmklm? >;l@ 3>4> ===> q4
zkhuhkl@{l.4{l> wkhq wkh iroorzlqj hvwlpdwlrq iru wkh Huuru ixqfwlrqdo ri wkh
Ulhpdqq0Vwlhowmhv txdgudwxuh uxoh lv wuxh
mHfrps
q +i,m $^i> ` e
b
d
x+{,> +815,
zkhuh
Hfrps q +i, @
e
]
d
i+{,gx+{,q4[
l@3
x+{l.4,x+{l,
kl
{]l.4
{l
i+{,g{>
dqg $^i> `lv wkh prgxoxv ri frqwlqxlw| ri i zlwk uhvshfw wr= Surri1 Iru d jlyhq glylvlrq ri ^d> e` dv deryh/ zh kdyh
e
]
d
i+{,gx+{, @
q4
[
l@3 {]l.4
{l
i+{,gx+{,=
Wkhq e| Ohppd 4/ zh fdq zulwh vxffhvvlyho|=
mHfrps
q +i,m @
e
]
d
i+{,gx+{,
q4
[
l@3
x+{l.4,x+{l,
kl
{]l.4
{l
i+{,g{
@
q4[
l@3 {]l.4
{l
i+{,gx+{,q4[
l@3
x+{l.4,x+{l,
kl
{]l.4
{l
i+{,g{
q4[
l@3
{]l.4
{l
i+{,gx+{,x+{l.4,kx+{l,
l
{]l.4
{l
i+{,g{
q4[
l@3
$^i> kl` {bl.4
{l
x+{,$^i> `q4[
l@3 {bl.4
{l
x+{,
@ $^i> `be
d
x+{,>
D JUXVV W\SH LQHTXDOLW\ IRU PDSSLQJ RI ERXQGHG YDULDWLRQ 44
-jujijAWjt
do 55 #h@}L4hc Bhtt ?i^@*|) ? ??ih ThL_U| tT@Uitc Ai t|h@*@? @| 5LU B@3i||ic ? Thitt
d2o 55 #h@}L4hc }i?ih@*3@|L? Lu Bhtt< ?i^@*|) ? ??ih ThL_U| tT@Uit @?_ @TT*U@|L?tc a @| ?@* TT*c @UUiT|i_
do 55 #h@}L4hc Bhtt |)Ti ?|i}h@* ?i^@*|) uLh 4@TT?}t Lu hOL*_ih<t |)Ti @?_ @TT*U@ |L?t uLh |h@Ti3L_ uLh4*@c A@4! @?} a Lu @|c @UUiT|i_
deo 55 #h@}L4hc 5L4i _tUhi|i ?i^@*|it Lu Bhtt |)Ti @?_ @TT*U@|L?t ? }itt?} |iLh)c tM4||i_
dDo 55 #h@}L4hc 5L4i ?|i}h@* ?i^@*|it Lu Bhtt |)Tic tM4||i_
dSo 55 #h@}L4h @?_ Bw LL|c ? @ BhttwT@t |)Ti ?i^@*|) @?_ |t @TT*U@|L?t uLh |i it|4@|L? Lu T4L4i?|t Lu }itt?} 4@TT?}tc tM4||i_
d.o 55 #h@}L4h @?_ W 6i_L|Lc ? ?i^@*|) Lu Bhtt< |)Ti uLh +i4@??5|i*|it ?|i}h@* @?_ @TT*U@|L?t uLh tTiU@* 4i@?tc A@4!@?} a Lu @|c 2bEeEbbHc 2HS2b2
dHo 55 #h@}L4hc ? |i |h@Ti3L_ uLh4*@ uLh 4@TT?}t Lu ML?_i_ @h@|L?c tM4||i_ dbo B Bhttc NMih _@t @ 44 _it @MtL*|i? i|h@}it L?
K3@ UK
@sE%}E%_% 3
EK3@5
UK @sE%_%
UK
@}E%_% c @| ~ c bEbDc 2D22S dfo 6?!c |hi@|ti L? Bhtt< ?i^@*|)c tM4||i_
do #5 |h?LUc a, iU@hU @?_ 6?!c *@ttU@* @?_ i W?i^@*|it ? ?@*)ttc k*ih U@_i4U M*tihtc #Lh_hiU|c bb
E5 5 #h@}L4h 7WNN, Nu N66AWB|NAt BAa YAuNi6B|Wte XW|NiB PAjit|+ Nu CjWAN,N}+e N! gg4Ie j,ONiAj |+e Ihh t|iB,B
,4@* @__hittG pffe72>z'[>f[>
EW 6i_L|L $jVBi|6jA| Nu B|j6B|Wte Cj PAjit|+ Nu CiBAt"je P6|B|B Fhhe 7N| uiWB