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wWZ,/ zklfk ixuwkhu lpsolhv wkdw plq%�ZE%�% @ wWZ1

Zh qrz suryh wkdw Ohppd 6 dqg ; duh Fruroodulhv ri Ohppd 431

Surri ri Ohppd 6= Ohw h|f +uhvshfwlyho| h|�, ghqrwh wkh udqgrp yduldeoh wdnlqj ydoxhv

lq \ @ i�|> |j dqg vxfk wkdw Suih|f @ �|j @ s +uhvshfwlyho| Suih|� @ �|j @ t,1 Zh gh�qh wkh

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dvvxph wkdw �o A oW +rwkhuzlvh wkh uhvxow lv wulyldoo| vdwlv�hg,1 Iru dq| vhtxhqfh ri vljqdov

gulylqj sod|hu 5*v olnholkrrg udwlr xs wr �o dw gdwh w/ wkhuh h{lvwv d odvw gdwh w� wf vxfk wkdw

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+59,/ dqg V&? dqg pZ dv lq Ohppd 431 Zh fdq fkrrvh uf vpdoo hqrxjk vr wkdw iru dq| u � uf

dqg � � 5�W/ p�*� ? 3 +e| gh�qlwlrq ri �W dqg ehfdxvh j lv forvh wr 3 zkhq u lv forvh wr 3,1

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zklfk jlyhv xv wkh ghvluhg uhvxow1

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Dsshqgl{/ zh frqvlghu d shuwxuedwlrq � ri wkh jdph1 Wkh Ohppd lq wkh pdlq wh{w zloo eh

suryhg lq wkh frqwh{w ri wkh shuwxuehg jdph1 +Qrwh wkdw wkh shuwxuedwlrq iru zklfk �|� @ 3

iru doo l/w lv d srvvleoh shuwxuedwlrq/ dqg fruuhvsrqgv wr wkh xqshuwxuehg jdph1,

Zh �uvw lqwurgxfh vrph qrwdwlrq wkdw zloo doorz xv wr ghdo zlwk wkh shuwxuehg jdph1 D

uhdol}dwlrq lv qrz dq lq�qlwh sdwk ri dfwlrqv/ vljqdov dqg w|shv iru hdfk sod|hu1 Zh ghqrwh

e| wl� 5 Q ^ i.4j wkh gdwh dw zklfk sod|hu l ehfrphv d w|sh wkdw dozd|v ghihfwv1 Qrwh

wkdw zh kdyh=

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�� lv qrz gh�qhg e|

%|� � Suj�'j� cj� clihw� @ w m hw� � w> wl� � wj=

D klvwru| k|� qrz frqvlvwv ri d klvwru| ri dfwlrqv/ vljqdov +xs wr gdwh w� 4, dqg w|sh +xs wr

gdwh w,1 Wkh suredelolw| s|+k|�, zloo qrz uhihu wr

s|+k|�, � Su@d�|�ocj�cl

id|� @ F m k|�j>

D sulydwh klvwru| hk|� uhihuv wr d klvwru| iru zklfk sod|hu l kdv qrw vzlwfkhg wr ghihfw ehiruh

w dqg lv vwloo d udwlrqdo w|sh dw gdwh w1 Wkh gh�qlwlrqv ri o|�^k|�`/ y

|�+�> �� m k|�, dqg �y|�+�> ��,

vwloo dsso| +zh rplw khuh wkh uhihuhqfh wr wkh shuwxuedwlrq �,1 Ilqdoo|/ zh qrz gh�qh wkh

frqwlqxdwlrq ydoxh �y|�+��, dv=

�y|�+��, � Hj�'j� cj� ^4� h3oE4�?th|� c|l� �3|� m hw� A w> wl� A w`1 +64,

Qrwh wkdw wkhvh gh�qlwlrqv frlqflgh zlwk wkh rqhv jlyhq lq wkh pdlq wh{w zkhq � @ 3/

vlqfh wkhq zh kdyh wl� @ .41 Dovr qrwh wkdw wkh gh�qlwlrqv ri �|+�, dqg �|cA +=, duh nhsw

xqfkdqjhg1 Lq sduwlfxodu/ wkh surri ri Sursrvlwlrq 6 dssolhv wr dq| shuwxuedwlrq12�

Surri ri Ohppd 41+H{lvwhqfh ri wkuhvkrogv,1

2�, UiT| uLh |�i _ih��@|�L? Lu @ *L�ih ML�?_ L? 4|cA � Etii wi44@ �2 Mi*L���

63

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�y�+��> �(, lv erxqghg ehorz e| �e+4 � �,/ dqg wkdw �y|�+��> ��, � �y|�+�

(> ��, lv erxqghg

deryh e| f1 Vr �o|� 5 ^3> �O`1 Frqvlghu d klvwru| hk|�1 Li o|�^hk|�` A �o|� / wkhq e| frqvwuxfwlrq/ iru doo

� 9@ �(/

y|�+�> �� m hk|�, @ s|+hk|�,�y|�+�> ��, . +4� s|+hk|�,,y|�+�> �(, ? s|+hk|�,�y|�+�(> ��,dqg lw lv wkxv rswlpdo iru sod|hu l wr vzlwfk wr ghihfw1

Li o|�^hk|�` ? �o|�/ wkhq zh pxvw kdyh �o|� A 3 vlqfh o|�^

hk|�` lv qrq0qhjdwlyh1 Vr e| gh�qlwlrq ri

�o|� / wkhuh pxvw h{lvw d vwudwhj| � 9@ �( vxfk wkdw y|�+�> �� m hk|�, A y|�+�(> �� m hk|�,1 Lw iroorzv

wkdw diwhu hk|�/ rswlpdo ehkdylru grhv qrw frqvlvwv lq ghihfwlqj iru hyhu1 Vlqfh �� lv rswlpdo

dqg vlqfh lw lv d wuljjhu vwudwhj| e| dvvxpswlrq/ lw pxvw rswlpdo iru sod|hu l wr frqwlqxh wr

frrshudwh diwhu wkh klvwru| hk|�1Surri ri Ohppd 7 +ydoxh iurp ghihfwlqj,1

Fkrrvh � A 4/ Wq @ � m oq u m dqg gh�qh �%|� @ pd{r$Aq %|3r� 1 Dvvxph wkdw �%|� � �%|21 Vlqfh

o|3r2 � %|3r� iru doo v/ Htxdwlrq +9, lpsolhv

o|2 � �%|�^�+|,`Aq

Li sduw| 4 ghihfwv iurp gdwh w xqwlo gdwh w. W / wkhq iurp Ohppd 6/ lw pxvw eh wkdw

o|nA2 � o|2 h{sWp(

5

zlwk suredelolw| dw ohdvw htxdo wr 4� ue1 E| fkrrvlqj W @ gf m oq u�%|� m dqg gf @2

6(^5 . � m

oq�+|, m`/ zh hqvxuh wkdw o|nA2 lv odujhu wkdq u32 +khqfh odujhu wkdq �O vlqfh �O lv frpsdudeoh

wr u3�,1 Sod|hu 5 wkxv vzlwfkhv wr ghihfw zlwk suredelolw| 4� ue ehiruh gdwh w. W / zklfk

lpsolhv

�y|�+�(> �2, � uW .R++uW ,2,

dv ghvluhg1

Vlqfh sod|hu 4 dfwxdoo| vzlwfkhv dw vrph gdwh w � vf/ vf � Wq zlwk suredelolw| �%|� +e|

gh�qlwlrq ri �%|�,/ dqg vlqfh wklv wuljjhuv d uhvsrqvh e| sod|hu 5 ehiruh gdwh w . W zlwk

suredelolw| 4� u2/ wkhuh pxvw h{lvw d gdwh w� 5 iw� vf> ==> w. Wj vxfk wkdw

%|�2 � �%|�Wq . W

= +65,

Li w� � w/ wkhq �%|2 � %|�2 / dqg zh pd| dsso| wkh dqdo|vlv deryh wr wkh fdvh zkhuh sod|hu 5

vwduwv ghihfwlqj dw w/ dqg rewdlq wkdw sod|hu 4 vzlwfkhv wr ghihfw ehiruh gdwh w . W � +zlwk

W � @ gf m oq u�%|2 m, zlwk suredelolw| 4� ue1 Li w� � w/ wkhq zh pd| dsso| wkh dqdo|vlv deryh

64

wr wkh fdvh zkhuh sod|hu 5 vwduwv ghihfwlqj dw w�/ dqg rewdlq wkdw sod|hu 4 vzlwfkhv wr ghihfw

ehiruh gdwh w�.W � zlwk suredelolw| 4� ue1 Lq erwk fdvhv/ sod|hu 4 vzlwfkhv wr ghihfw ehiruh

gdwh w. W . W �1 Vlqfh +65, lpsolhv W � � 5W +ehfdxvh oqW ?? W ,/ zh �qdoo| rewdlq=

�y|2+�(> �2, � 6uW .R++uW ,2,

Ehiruh surylqj Ohppd 8 dqg 9/ zh vwduw zlwk wkh iroorzlqj suholplqdu| uhvxow=

Ohppd 44 Fkrrvh � dv lq Ohppd 6 dqg ohw �W @ �*? o�o 1 Dovr ohw ew @ plqihw2> wl�j1 Wkhuh

h{lvw d> uf> vxfk wkdw ;u � uf/ ;��/ ;W � �W / zh kdyh=

d, Su#Wqcj�cliew � hw� � ew. � m oq u m m ew � Wj � 4� u2

e, Su#Wqcj�clihw� � W m ew � Wj � 4� u2

Surri1 Lw zloo eh frqyhqlhqw wr ohw Wf @ � m oq u m1 Zkhq ew @ w dqg hw� ? w/ wkh glvwulexwlrq

ryhu klvwrulhv jhqhudwhg xs wr gdwh hw� e| wkh wulsohw +�Wq> �� > �, frlqflghv zlwk wkdw jhqhudwhg

e| +�� > ��,1 Wkhuhiruh/ iru dq| w� ? w/ zh kdyh +dsso|lqj Ohppd 6,=

Su#Wqcj� clihw� @ w� m ew @ wj @ Suj� cj�i

|�\r'|�3Af

�+|r�,j � 4j � ue>

zklfk ixuwkhu lpsolhv wkdw iru doo w/

Su#Wqcj� clihw� � w m ew @ wj � wue= +66,

Vlploduo|/ li ew @ w wkhq hlwkhu wl� � w. Wf/ lq zklfk fdvh hw� � wl� � w. Wf ru wl� A w. Wf/

khqfh hw2 @ w e| gh�qlwlrq ri ew1 Dw gdwh w. Wf . 4/ xqghu wkh hyhqw iew @ w> wl� A w. Wfj/ wkhglvwulexwlrq ryhu vljqdov uhfhlyhg e| sod|hu 4 vlqfh w frlqflghv zlwk wkdw jhqhudwhg +�� > �(,1

Dsso|lqj Ohppd 6/ zh rewdlq=

Su#Wqcj�clihw� A w. Wf m ew @ w> wl� A w. Wfj � Suj� cj(i

|nAf\r'|

�+|r�, � 4j � ue= +67,

Frpelqlqj lqhtxdolwlhv +66, dqg +67,/ dqg revhuylqj wkdw ue �W � u2 |lhogv wkh ghvluhg

uhvxow1

Surri ri Ohppd 8= +orzhu erxqg rq frqwlqxdwlrq ydoxhv, Fkrrvh l @ 4/ dqg ohwew @ plqihw2> wl�j dqg Wf @ � m oq u m1 Dvvxph sod|huv iroorz wkh sur�oh +�Wq> �2, dqg frqvlghu

wkh hyhqwv D � ihw� 5 iew> ===> ew . Wfj> ew � �Wj dqg E � ihw� � �W> ew � �Wj1 Xqghu hyhqw D ^E/

wkh glvwulexwlrq ryhu ew frlqflghv zlwk wkdw jhqhudwhg e| +�� > �2> �,/ dqg sod|hu 4*v h{shfwhg

sd|r� lv htxdo wr Hj� cj2cl ^4� h3oe| m D ^E` .R+u m oq u m,1 Vlqfh

Su#Wqcj2cliD ^Ej � 4� 5u2

65

e| Ohppd 44/ zh jhw wkh ghvluhg lqhtxdolwlhv1

Surri ri Ohppd 9= Frqvlghu d wuljjhu htxloleulxp � 5 H+u, dqg dvvxph wkdw � 9@+�(> �(,1 +Rwkhuzlvh wkh uhvxow lv lpphgldwh,1 Ohw ew @ plqihw2> wl�j1

Vwhs 4= Zh �uvw suryh wkdw

y�+�, � Hj� cj2cl ^4� h3oe| m wl� A 3` .R+u m oq u m,

Ohw � ghqrwh wkh vhw ri wuljjhu vwudwhjlhv iru sod|hu 41 Qrz frqvlghu wkh qhz jdph zkhuh

dw hdfk gdwh/ sod|hu 4 zrxog eh lqiruphg derxw zkhwkhu sod|hu 5 vzlwfkhv wr ghihfw dw wkdw

gdwh1 Ohw �� ghqrwh wkh vhw ri wuljjhu vwudwhjlhv iru sod|hu 4 lq wkdw qhz jdph1 Vlqfh

� � ��/ dqg vlqfh �� 5 �/ zh pxvw kdyh=

y�+��> �2, � pd{7j�M7P�

y�+���> �2,

Qrz revhuyh wkdw zkhq sod|hu 4 lv wrog wkdw sod|hu 5 vzlwfkhv/ lw lv rswlpdo iru sod|hu 4 wr

vzlwfk wr ghihfw wrr1 Li sod|hu 4 lv wrog wkdw klv rssrqhqw lv vwloo jrlqj wr frrshudwh/ wkhq zh

vkrz qrz wkdw lw lv rswlpdo iru klp wr frqwlqxh wr frrshudwh wrr1 Vxssrvh e| frqwudglfwlrq

wkdw iru doo � 5 �/ �y|�+�> �2, ? �y|�+�

(> �2,1 Wkhq e| htxdwlrq +7,/ y|�+�> �( m hk|�, zrxog eh

vwulfwo| qhjdwlyh iru doo klvwrulhv hk|�1 Wklv zrxog lpso| wkdw +lq wkh ruljlqdo jdph,/ iurp

gdwh w rq/ sod|hu 4 ghihfwv diwhu dq| klvwru| hk|�1 Vr lq wkh htxloleulxp � ri wkh ruljlqdo jdph/

diwhu dq| w�klvwru|/ erwk sod|huv zrxog ghihfw iru hyhu1 E| edfnzdug lqgxfwlrq zh zrxog

jhw wkdw erwk sod|huv ghihfw dw doo gdwhv/ frqwudglfwlqj wkh dvvxpswlrq wkdw � 9@ +�(> �(,1

Lw iroorzv wkdw zkhq sod|hu 4 lv doorzhg wr fkrrvh d wuljjhu vwudwhj| lq ��/ lw lv rswlpdo

iru klp wr vzlwfk wr ghihfw h{dfwo| dw wkh vdph gdwh dw sod|hu 5 grhv1 Ohw ��W� ghqrwh wkdw

vwudwhj|1 Zh zlvk wr vkrz wkdw

y�+��W�> �2, � Hj� cj2cl ^4� h3oe| m wl� A 3` .R+u m oq u m,=

Ohw Vl� @ iw> �|� � u� ru w � �Wj/ dqg frqvlghu wkh hyhqw D @ iwl� 5 Vl

� > wl� ? hw2j1 Xqghu

hyhqw D/ sod|hu 4 vzlwfkhv wr ghihfw ehiruh sod|hu 5 grhv/ dqg kh pd| wkhuhiruh h{shfw wr

jhw dq h{shfwhg jdlq vwulfwo| odujhu wkdq 4� h3oe|1 Krzhyhu/ wkh h{wud jdlq fdqqrw eh odujhu

wkdq R+u m oq u m, = zkhq wl� � �W / wkh glvfrxqwhg ydoxh ri wkh sd|r�v uhfhlyhg diwhu �W lv

frpsdudeoh wr u +vlqfh h3o7A @ u,/ dqg zkhq �|� � u�/ sod|hu 4 rewdlqv d frqwlqxdwlrq sd|r�

erxqghg e| 7gu m oq u m e| Ohppd 71 Vlqfh Su7jW� cj2cliDj � 4� u� �W e| frqvwuxfwlrq/ zh jhw

wkh ghvluhg erxqg rq y�+��W�> �2,/ khqfh rq y�+�,1

Vwhs 5= y�+�, @ +4� %f2,�yf� .R+u m oq u m,

Iluvw revhuyh wkdw y�+�, � y�+�Wq> �2,/ dqg Ohppd 8 wkxv lpsolhv y�+�, � +4�%f2,�yf��gu m

oq u m1

66

Dovr/ e| gh�qlwlrq ri �yf�/ zh kdyh Hj� cj2cl ^4�h3oe| m wl� A 3` @ Suj� cj2clihw2 A 3j�yf�1 Vlqfhwkh suredelolw| wkdw sod|hu 5 vwduwv e| ghihfwlqj ghshqgv rqo| rq +�2> �,/ zh dovr kdyh

Suj� cj2clihw2 A 3j @ Sujclihw2 A 3j @ 4� %f21

Dqg vwhs 4 shuplwv wr frqfoxgh1

Vwhs 6= w @ plqihw�> hw2j= Zh vkrz

y�+�, @ Hjcl ^4� h3o| m w A 3` .R+u m oq u m,=

Ohw V @ iw><l> %|� � u�j dqg frqvlghu wkh hyhqwv D @ iw 5 Vj dqg E @ iw � �Wj1 Xqghu

hlwkhu hyhqw D ru hyhqw E/ sod|hu 4 grhv qrw rewdlq d sd|r� wkdw gl�huv iurp 4 � h3o| e|

pruh wkdq R+u m oq u m,12e Vlqfh SujcliD^Ej � 4� u� �W e| frqvwuxfwlrq/ zh jhw wkh ghvluhg

uhvxow1

Ilqdo vwhs= Wkh dujxphqw ri vwhs 6 pd| eh dssolhg wr ghulyh y2+�,/ zklfk wkhuhiruh

fdqqrw gl�hu y�+�, e| pruh wkdq R+u m oq u m,1 Frpelqlqj wklv revhuydwlrq zlwk vwhs 5 |lhogv

wkh ghvluhg uhvxow1

Surri ri Ohppd :1

Wkh vwudwhj| �"cq frlqflghv zlwk �Wq / dqg wkdw wkh odwwhu vwudwhj| jlyhv sduw| 4 d sd|r�

forvh wr y�+�, +iurp Ohppd 8 dqg 9,1 Zh zloo qrz frpsxwh d orzhu erxqg rq

y�+��cq > �2,� y�+�

"cq> �2,

Ohw wl @ plqiwl�> wl2j/ dqg Wf @ � m oq u m1 Frqvlghu wkh hyhqwv D @ i;l> wl � hw� � wl . Wf

dqg wl � Wj dqg E @ i;l> hw� � wl dqg wl � Wj1 Xqghu wkh dvvxpswlrq ri wkh Ohppd/ dqg

wkdqnv wr Ohppd 44/ zh kdyh=

Suj�cqcj2cliD ^Ej � 4� u�*S � 5u2=

Lq dgglwlrq/ wkh glvwulexwlrq ryhu wl lv lqghshqghqw iurp wkh vwudwhjlhv iroorzhg e| wkh

udwlrqdo sod|huv1 Xqghu hyhqw D^E/ hlwkhu erwk sod|huv vzlwfk wr ghihfw dssur{lpdwho| dw

wkh vdph gdwh wl ru wkh| vzlwfk wr ghihfw diwhu W 1 Lq hlwkhu fdvh/ wkh dgglwlrqdo sd|r� wkdw

sod|hu 4 rewdlqv gxulqj wkh W W@5 �uvw shulrgv iurp iroorzlqj ��cq udwkhu wkdq �"cq lv dw

ohdvw htxdo wr

+f� 4,Suliwl � W W@5 m D ^Ej4� h3oAW*2

�= +68,

2e�L|i |�@| wi44@ e �4T*�it |�@| ��i? | ' h|�c T*@)ih � U@??L| }i| @ UL?|�?�@|�L? T@)Lg *@h}ih |�@?

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69