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491 Uhlhq G1/ O1 Vfkxpdqq dqg P1 Zdug/ 5333/ %Glvfulplqdwru| Ghdolqj zlwk Grzqvwuhdp Frp0
shwlwruv= Hylghqfh iurp wkh Fhooxodu Lqgxvwu|/% Mrxuqdo ri Lqgxvwuldo Hfrqrplfv [OYLLL/ 5860
5;91
4:1 Uhlwpdq/ G1/ 4<<4/ %Hqgrjhqrxv Txdolw| Glhuhqwldwlrq lq Frqjhvwhg Pdunhwv/% Wkh Mrxuqdo
ri Lqgxvwuldo Hfrqrplfv/ 6< +9,/ 954097:1
4;1 Vkdnhg/ D1 dqg M1 Vxwwrq/ 4<;5/ %Uhod{lqj Sulfh Frpshwlwlrq wkurxjk Surgxfw Glhuhqwldwlrq/%
Uhylhz ri Hfrqrplfv Vwxglhv/ [OL[/ 60461
4<1 Vsxoehu/ G1/ 4<<8/ %Lqwurgxfwlrq= Frpshwlwlrq lq Whohfrppxqlfdwlrqv/% Mrxuqdo ri Hfrqrplfv
dqg Pdqdjhphqw Vwudwhj|/ 7 +5,/ 466046:1
531 Vzhhw/ U1/ L1 Ylhkr/ G1 Olqdugdwrv/ dqg Q1 Ndorxswvlglv/ 5334/ %Pdujlqdo Ydoxh0edvhg Sulflqj
ri Dgglwlrqdo Vshfwuxp Dvvljqhg wr Fhooxodu Whohskrq| Rshudwruv/% Lqirupdwlrq Hfrqrplfv
dqg Srolf| 47 +6,/ 6:406;71
541 Ydohwwl/ W1/ 4<<</ %D Prgho ri Frpshwlwlrq lq Preloh Frppxqlfdwlrqv/% Lqirupdwlrq Hfr0
qrplfv dqg Srolf| 44/ 940:51
551 Zhvwodqg/ F1/ 4<<5/ %Frqjhvwlrq dqg Qhwzrun H{whuqdolwlhv lq wkh Vkruw0uxq Sulflqj ri Lqiru0
pdwlrq V|vwhpv Vhuylfhv/% Pdqdjhphqw Vflhqfh 6;/ <<5043<<1
D1 Surriv
Surri ri Sursrvlwlrq 4= Wkh vhfrqg0shulrg sd|rv wr hdfk surylghu duh=
¡A +pA ; pB , @ pA ^4 y`;
+D14,
¡ B +pA ; pB , @ pB y;
+D15,
zkhuh y vdwlvhv ixooohg h{shfwdwlrqv1 Dq| Qdvk htxloleulxp sulfh sdlu pxvw vdwlvi| uvw0rughu
B
frqglwlrqv ¡A
 +pA ; pB , @ ¡ 2 +pA ; pB , @ 31 Glhuhqwldwlqj +D14, zlwk uhvshfw wr pA |lhogv=
4 y pA
zklfk uhduudqjhv wr=
pA @
@y
@ 3;
@pA
4y
@y
@pA
5:
:
+D16,
+D17,
Vlploduo|/ glhuhqwldwlqj +D15, zlwk uhvshfw wr pB dqg uhduudqjlqj |lhogv=
pB @
y
@y
@pB
:
+D18,
Edvhg rq +617, dqg +619,/ wkh frqglwlrq iru y wr vdwlvi| ixooohg h{shfwdwlrqv fdq eh vxppdul}hg dv=
yu+E,Q+y; NA ; NB , @ pA pB :
+D19,
Vlqfh y vdwlvhv ixooohg0h{shfwdwlrqv/ zh fdq glhuhqwldwh erwk vlghv ri +D19, zlwk uhvshfw wr pA wr
jhw=
4
@y
Ey
E+4 y,
@y
Ey
4
E+4 y,
u+E, B+v;
, B+v;
, .yEu+E,
B2 +v;
,.
B2 +v;
, @ 4;
@pA
NB
NA
@pA NB
NB
NA
NA
+D1:,
zklfk lpsolhv wkdw
4
@y
k
l :
@
Ed3yo
EE3y
Ey
Ey

@pA
u+E, B+v; NB , B+v; NA , . yE NB B2 +v; N
,
.
B
+v;
,
2
NA
NA
B
+D1;,
Vlploduo|/ glhuhqwldwlqj erwk vlghv ri +D15, zlwk uhvshfw wr pB |lhogv=
@y
4
Ey
E+4 y,
@y
Ey
4
E+4 y,
u+E, B+v;
, B+v;
, .yEu+E,
B2 +v;
,.
B2 +v;
, @ 4;
@pB
NB
NA
@pB NB
NB
NA
NA
+D1<,
zklfk lpsolhv wkdw
4
@y
k
l :
@
Ed3yo
EE3y
Ey
Ey

@pB
u+E, B+v; NB , B+v; NA , . yE NB B2 +v; N
,
.
B
+v;
,
NA 2
NA
B
+D143,
Qrz/ glhuhqwldwlqj Q+y; NA ; NB , zlwk uhvshfw wr y |lhogv=
Q +y; NA ; NB , @ E
4
E^4 y`
4
Ey
B2 +v;
,.
B2 +v;
,
NA
NA
NB
NB
+D144,
Frpsdulqj +D1;, dqg +D143, zlwk +D144, dqg +616, |lhogv=
@y
4
@
;
@pA
u+E,^Q+y; NA ; NB , . yQ +y; NA ; NB ,`
5;
+D145,
dqg
@y
4
@
:
@pB
u+E,^Q+y; NA ; NB , . yQ +y; NA ; NB ,`
+D146,
Wkh uhvxow iroorzv e| vxevwlwxwlqj +D145, lqwr +D17, dqg +D146, lqwr +D18,1
Surri ri Sursrvlwlrq 5= Iurp +61:, dqg +61;, lq Ohppd 4/ dq| htxloleulxp sulfh sdlu +pA ; pB ,
pxvw vdwlvi|
pA @ ^4 y`u+E,^Q+y; NA ; NB , . yQ +y; NA ; NB ,`:
+D147,
pB @ yu+E,^Q+y; NA ; NB , . yQ +y; NA ; NB ,`:
+D148,
Wkhuhiruh/
pA pB @ +4 5y,u+E,^Q+y; NA ; NB , . yQ +y; NA ; NB ,`:
+D149,
Lw lv vwudljkwiruzdug wr hvwdeolvk wkdw +3; 3, lv qrw dq htxloleulxp/ dqg wkhuhiruh/ ^Q+y; NA ; NB , .
yQ +y; NA ; NB ,` > 31 Dv d frqvhtxhqfh/ vlqfh u+E, > 3/ +D149, lpsolhv wkdw li pA @ pB / wkhq y @ 2 :
+d, Li NA @ NB @ N / wkhq +618, uhgxfhv wr
+y, @
Ey
E+4 y,
N
; li B+v;
, @ B+v;
,:
5N
N
N
+D14:,
Iurp wkh surshuwlhv ri wkh Huodqj orvv ixqfwlrq B+v; ¢,/ zh nqrz wkdw B2 +v; ¢, > 3/ zklfk lq wxuq
E+4 y,
Ey
Ey
, @ B+v;
, lv vshflhg e|
@
lpsolhv wkdw wkh xqltxh ydoxh ri y iru zklfk B+v;
N
N
N
E+4 y,
/ ru y @ 2 1 Wklv lpsolhv wkdw zkhq NA @ NB @ N/ +618, lv h{dfwo| wkh vdph dv=
N
4
4
+y, @ ; li y @ :
5
5
+D14;,
Wkhuhiruh/ +D14;, lv frqvlvwhqw zlwk d ydoxh ri y vxfk wkdw +y, @ y1 Vxevwlwxwlqj NA @ NB @ N
dqg y @

2
lqwr +61:, dqg +61;, |lhogv wkh h{suhvvlrq iru pW 1 Lw fdq hdvlo| eh hvwdeolvkhg wkdw ¡A +x; pW ,
dqg ¡B +pW ; x, duh hdfk vwulfwo| txdvlfrqfdyh lq x/ dqg wkhuhiruh/ +pW ; pW , lv d Qdvk htxloleulxp1
+e, Zkhq NA 9@ NB / wkhuh lv qr frpelqdwlrq ri NA ; NB wkdw fdq hqvxuh wkdw + 2 , @

2
li pA @ pB 1
Dv d frqvhtxhqfh/ qr v|pphwulf sulfh sdlu fdq eh d Qdvk htxloleulxp1
Surri ri Sursrvlwlrq 6= Uhfdoo wkdw zh kdyh dvvxphg pA pB / dqg wkhuhiruh/ iru dq lqglhuhqw
fxvwrphu y/ up A kdv pdunhw vkduh +4 y, dqg up B kdv pdunhw vkduh y1 Dovr uhfdoo wkdw wkh
5<
txdolw| glhuhqwldo ixqfwlrq/ iru dq| duelwudu| y; NA ; NB lv=
Q+y; NA ; NB , @ B+v;
Ey
E^4 y`
, B+v;
,:
NB
NA
+D14<,
Iru uhihuhqfh/ vrph sduwldo ghulydwlyhv ri Q duh olvwhg ehorz=
#
Q +y; NA ; NB , @ E
Q2 +y; NA ; NB , @
Q +y; NA ; NB , @
Q +y; NA ; NB , @
Q2 +y; NA ; NB , @
Q +y; NA ; NB , @
B2 +v; Ed3yo
NA ,
NA
.
Ey
B2 +v; N
B,
$
NB
E^4 y`
E^4 y`
B2 +v;
,
2
NA
NA
Ey
Ey
B2 +v;
,
2
NB
NB
#
Ed3yo $
Ey
B
+v;
,
+v;
B
22
22
N
NA ,
B E2
NB2
NA2
E
E^4 y`
E 2 ^4 y`
E^4 y`
B2 +v;
,.
B22 +v;
,
NA
NA
NA2
NA
E
Ey
E 2y
Ey
B2 +v;
, . B22 +v;
,
NB
NB
NB2
NB
Wkh htxloleulxp vhfrqg0shulrg sulfh sdluv wdnh wkh iroorzlqj irup/ dv vkrzq lq Sursrvlwlrq 4=
pA +NA ; NB , @ u+E,^4 y+NA ; NB ,`^Q+y+NA ; NB ,; NA ; NB ,
+D153,
.y+NA ; NB ,Q +y+NA ; NB ,; NA ; NB ,`
pB +NA ; NB , @ u+E,y+NA ; NB ,^Q+y+NA ; NB ,; NA ; NB ,
+D154,
.y+NA ; NB ,Q +y+NA ; NB ,; NA ; NB ,`
zkhuh y+NA ; NB , vdwlvhv +y, @ y/ zklfk lpsolhv wkdw y+NA ; NB , vdwlvhv=
y @ ^4 5y`^
Q+y; NA ; NB , . yQ +y; NA ; NB ,
`;
Q+y; NA ; NB ,
+D155,
ru uhduudqjlqj/ lpsolhv wkdw y+NA ; NB , vdwlvhv wkh uhodwlrqvkls=
^6y+NA ; NB , 4`Q+y+NA ; NB ,; NA ; NB , @ ^4 5y+NA ; NB ,`y+NA ; NB ,Q +y+NA ; NB ,; NA ; NB ,:
+D156,
63
Glhuhqwldwlqj erwk vlghv ri +D156, vxffhvvlyho| zlwk uhvshfw wr NA dqg NB dqg uhduudqjlqj |lhogv=
y +NA ; NB , @
d3yENA ;NB oQ EyENA ;NB ;NA ;NB nyENA ;NB d32yENA ;NB oQ2 EyENA ;NB ;NA ;NB QEyENA ;NB ;NA ;NB nd.yENA ;NB 32oQ EyENA ;NB ;NA ;NB 3d32yENA ;NB oyENA ;NB Q EyENA ;NB ;NA ;NB +D157,
A ;NB oQ EyENA ;NB ;NA ;NB nyENA ;NB d32yENA ;NB oQ EyENA ;NB ;NA ;NB y2 +NA ; NB , @ QEyENA ;NBd3yEN
;NA ;NB nd.yENA ;NB 32oQ EyENA ;NB ;NA ;NB 3d32yENA ;NB oyENA ;NB Q EyENA ;NB ;NA ;NB +D158,
Wkh sd|r ixqfwlrqv ri wkh wzr upv duh
A +NA ; NB , @ u+E,^4 y+NA ; NB ,`2 ^Q+y; NA ; NB ,
+D159,
.y+NA ; NB ,Q +y+NA ; NB ,; NA ; NB ,` c+NA ,
B +NA ; NB , @ u+E,^y+NA ; NB ,`2 ^Q+y; NA ; NB ,
+D15:,
.y+NA ; NB ,Q +y+NA ; NB ,; NA ; NB ,` c+NB ,
B
Wkh uvw0rughu frqglwlrqv duh A
 +NA ; NB , @ 2 +NA ; NB , @ 31 +Iru wkh uhpdlqghu ri wklv surri/
zh gurs wkh yduldeohv lq wkh h{suhvvlrqv iru euhylw|1 Iru lqvwdqfh/ zh uhihu wr y +NA ; NB , dv y /
Q +y+NA ; NB ,; NA ; NB , dv Q dqg vr iruwk,1
Glhuhqwldwlqj +D159, zlwk uhvshfw wr NA dqg +D15:, zlwk uhvshfw wr NB |lhogv=
@ u+E,i+4 y,2 +Q2 . yQ2 , . y +4 y,2 +5Q . yQ ,
A

+D15;,
5y +4 y,+Q . yQ ,j c >
+D15<,
B
@ u+E,iy 2 +Q . yQ , . y 2 +y2 ,+5Q . yQ , . 5y+y2 ,+Q . yQ ,j c ;
2
+D163,
zklfk zkhq uhduudqjhg/ |lhog wkh uvw0rughu frqglwlrqv=
+4 y,2 +Q2 . yQ2 , . y y+4 y,2 Q . 5y+4 y,+4 5y,Q 5+4 y,Q, @
y2 +Q . yQ , . y2 y Q . 7y 2 Q . 5yQ, @
c
u+E,
c
u+E,
+D164,
+D165,
Uhfdoo iurp +D157, dqg +D158, wkdw
y @
y2 @
+4 6y,Q . y+4 5y,Q2
6Q . +:y 5,Q y+4 5y,Q
+4 6y,Q . y+4 5y,Q
6Q . +:y 5,Q y+4 5y,Q
64
+D166,
+D167,
dqg wkhuhiruh zh kdyh wkh uvw rughu frqglwlrqv sxuho| lq whupv ri rxu sulplwlyh ixqfwlrqv=
+4 y,2 +Q2 . yQ2 , .
dE3yQ nyE32yQ2 o^yE3y2 Q n2yE3yE32yQ 32E3yQ`
QnE.y32Q 3yE32yQ
y 2 +Q . yQ , .
dE3yQ nyE32yQ o^y Q ney 2 Q n2yQ`
QnE.y32Q 3yE32yQ
@
@
c
u+E,
c
u+E,
+D168,
+D169,
Li NA @ NB @ N / wklv lpsolhv wkdw pA @ pB @ p +iurp Sursrvlwlrq 4, dqg wkdw y @ 2 1 Lw wkhuhiruh
iroorzv wkdw=
5E
E
B2 +v;
,
N
5N
E
E
@ Q @
B2 +v;
,
5N 2
5N
@ 3
E
E
E
E2
@ Q @ B2 +v;
B22 +v;
,.
,
N2
5N
5N 5N
Q @
+D16:,
Q2
+D16;,
Q
Q2
+D16<,
+D173,
dqg wkhuhiruh=
¡A
 +N; N, @
¡B
2 +N; N, @
4
+Q2 .
7
4
+Q .
7
Q2
, c
5
Q
Q
,
c ;
5
6
+D174,
+D175,
zklfk xsrq vxevwlwxwlqj +D16;, dqg +D173, |lhogv=
E2
E
B22 +v;
, c +N ,
49N 5N
E2
E
5E
E
E
B
B2 +v;
B22 +v;
.
2 +N; N, @ ,
, c +N,:
49N 5N
7N 2 6N
5N
A
 +N; N, @ +D176,
+D177,
Vlqfh B2 +v; ¢, > 3 iru doo v; ¢/ wkh wzr uvw0rughu frqglwlrqv fdqqrw eh vdwlvhg vlpxowdqhrxvo|/ dqg
wkh uhvxow iroorzv1
65