A small sample correction for tests of hypotheses on cointegrating vectors

Testo completo

(1)EUROPEAN UNIVERSITY INSTITUTE DEPARTMENT OF ECONOMICS. EUI Working Paper ECO No. 99/9. A Small Sample Correction for Tests of Hypotheses on the Cointegrating Vectors S¿ren Johansen. BADIA FIESOLANA, SAN DOMENICO (FI).

(2) All rights reserved. No part of this paper may be reproduced in any form without permission of the author.. c °1999 S¿ren Johansen Printed in Italy in March 1999 European University Institute Badia Fiesolana I-50016 San Domenico (FI) Italy.

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(5) prgho lq wuldqjxodu irup \w @ [w . %4w > [w @ %5w > zkhuh %w duh l1l1g1 Q5 +3> L5 ,= Lq wklv vlwxdwlrq lw lv zhoo nqrzq wkdw SW [w %w a @ Sw@4 W [5 > w@4 w kdv d pl{hg Jdxvvldq glvwulexwlrq1 Wkh olnholkrrg udwlr whvw lv htxlydohqw wr d whvw rq. W 5 [ a T@ [w5 > w@4 a kdv zklfk lv h{dfwo| "5 +4,> krzhyhu1 Wkh uhdvrq iru wklv lv/ wkdw dowkrxjk wkh glvwulexwlrq ri khdy| wdlov dqg wkh ydoxh ri wkh hvwlpdwru fdofxodwhg zloo riwhq eh h{wuhph/ wklv skhqrphqrq S lv dozd|v iroorzhg e| d vpdoo ydoxh ri wkh lqirupdwlrq Ww@4 [w5 > vxfk wkdw wkh qrupdol}hg a duh frqwulexwlrq wr wkh whvw vwdwlvwlf lv qrw h{wuhph1 Wkdw lv/ wkh h{wuhph revhuydwlrqv lq S 4 W 5 qrw h{wuhph li phdvxuhg e| wkhlu dv|pswrwlf frqglwlrqdo yduldqfh 1 D zd| w@4 [w ri dyrlglqj wkh lvvxh ri pl{hg Jdxvvldq glvwulexwlrq lv wr frqvlghu frqglwlrqdo lqihuhqfh/ wkdw lv/ zh frqvlghu wkh surfhvv [w {hg dqg nqrzq1 Riwhq/ lq wkh fdofxodwlrq ri Eduwohww fruuhfwlrqv/ rqh phhwv d idfwru ri wkh irup v5 @ S S W +W 4 w@4 [w5 ,4 > iru vrph surfhvv [w = Li [w lv d vwdwlrqdu| surfhvv vxfk wkdw v5 $ H+v5 , @ 5 > vd|/ wkhq zh h{sdqg dv iroorzv 4 4 4 4 v5 4 v5 @ @ + 4, . + 4,5 = = = v5 5 . +v5 5 , 5 5 5 5 5 Lq wklv zd| qhjdwlyh srzhuv fdq eh uhsodfhg e| srvlwlyh srzhuv/ zklfk juhdwo| idflolwdwhv wkh S fdofxodwlrqv1 Lq fdvh wkh surfhvv [w lv d udqgrp zdon/ wkh olplw W 5 Ww@4 [w5 lv qrw d frqvwdqw dqg wkh deryh h{sdqvlrq grhv qrw khos/ vlqfh d udqgrp whup dsshduv lq doo qxphudwruv1 Reylrxvo|/ frqglwlrqlqj rq [w > zloo lq wklv fdvh { doo qxphudwruv dqg dyrlg wkh fdofxodwlrq ri prphqwv lqyroylqj wkh udqgrp olplw1 Zh wu| lq wkh iroorzlqj wr vhh krz idu wklv lghd fdq wdnh xv lq wkh frlqwhjudwhg yhfwru dxwruhjuhvvlyh prgho zlwk nqrzq 1 Zkhq wkh sdudphwhu lv {hg/ zh fdq wudqvirup wkh prgho lqwr d frlqwhjudwhg uhjuhvvlrq prgho/ vhh +5, dqg +6,/ ri wkh w|sh frqvlghuhg e| Skloolsv +4<<4,1 Lq Vhfwlrq 5 zh vkrz wkdw wkh dxwruhjuhvvlyh prgho zlwk nqrzq dgmxvwphqw frh!flhqwv uhgxfhv wr dq ruglqdu| uhjuhvvlrq prgho exw zlwk qrq0vwdwlrqdu| uhjuhvvruv1 Zh ghulyh olnhol0 krrg udwlr whvwv iru k|srwkhvhv rq wkh frlqwhjudwlqj vsdfh dqg qrupdol}h wkh uhjuhvvruv xvlqj wkh sdudphwhuv iurp wkh wuxh gdwd jhqhudwlqj surfhvv lq rughu wr eh deoh wr fdofxodwh wkh Eduwohww fruuhfwlrq1 Vhfwlrq 6 frqwdlqv wkh pdlq uhvxow zklfk jlyhv wkh fruuhfwlrq whup iru d jhqhudo uhjuhvvlrq prgho dqg lq Vhfwlrq 7 wkh uhvxowv duh dssolhg wr whvwv lq wkh dxwruhjuhvvlyh prgho1 Vhfwlrq 8 frqwdlqv vrph vlpxodwlrq uhvxowv zklfk duh xvhg wr looxvwudwh wkh qglqjv1 Wkh surri ri wkh pdlq uhvxow lv jlyhq lq dq Dsshqgl{1 5.

(6) 5. Wkh dxwruhjuhvvlyh prgho dqg wkh whvw vwdwlvwlf. Zh vkrz lq wklv vhfwlrq wkdw li lv nqrzq wkh prgho uhgxfhv wr d uhjuhvvlrq prgho dqg zh fdq irupxodwh wkh whvw iru d k|srwkhvlv rq wkh frlqwhjudwlqj frh!flhqwv dv d whvw rq wkh frh!flhqwv wr wkh qrq0vwdwlrqdu| frpprq vwrfkdvwlf wuhqgv1 Vlqfh zh frqglwlrq rq wkhvh wkh qrq0vwdwlrqdu| uhjuhvvruv ehfrph {hg dqg ghwhuplqlvwlf/ exw wkh| vwloo vdwlvi| vrph yhu| vshfldo frqglwlrqv/ zklfk duh dqdorjrxv wr wkhlu vwrfkdvwlf surshuwlhv zkhq wkh| duh frqvlghuhg udqgrp1 Zh jlyh wkh uhvxowv iru whvwv ri wkh irup @ K!> zkhuh K+q v, lv nqrzq dqg !+v u, lv d sdudphwhu wr eh hvwlpdwhg1 Zh wuhdw uvw wkh fdvh zkhuh v @ u> fruuhvsrqglqj wr d nqrzq frlqwhjudwlqj vsdfh/ dqg wkhq vkrz krz wkh jhqhudo vlwxdwlrq fdq eh wuhdwhg wkh vdph zd|1 Wkh pdlq uhvxow lv wkh ghulydwlrq ri d jhqhudo uhjuhvvlrq htxdwlrq zklfk frqwdlqv doo wkh deryh h{dpsohv dv vshfldo fdvhv/ dqg iurp zklfk wkh olnholkrrg udwlr whvwv fdq hdvlo| eh irxqg1 Wklv lpsolhv wkdw zh fdq qg wkh h{sdqvlrq ri wkh olnholkrrg udwlr whvw zklfk lv qhhghg iru wkh fdofxodwlrq ri wkh Eduwohww fruuhfwlrq1. 514. Wkh prgho htxdwlrqv. Iru qrwdwlrqdo uhdvrqv zh irfxv khuh rq wkh vlwxdwlrq zlwk wzr odjv dqg phqwlrq zkhq wkh uhvxowv kdyh wr eh prglhg iru pruh odjv1 Frqvlghu wkh prgho [w @ 3 [w4 . 4 [w4 . gw . %w >. w @ 4> = = = > W=. +4,. Zh dvvxph wkdw %w duh l1l1g1 Qq +3> ,> dqg iru wkh ghulydwlrq ri wkh hvwlpdwru dqg olnholkrrg udwlr whvw zh dvvxph wkdw wkh lqlwldo ydoxhv [3 dqg [4 duh {hg1 Zh dvvxph wkdw wkh sdudphwhu +q u, lv nqrzq dqg wkdw +q u,> 4 +q q,> +q q,> dqg +q g, duh xqnqrzq dqg ydu| xquhvwulfwhgo|1 Wkh ghwhuplqlvwlf whupv gw pd| frqwdlq iru lqvwdqfh d frqvwdqw/ d olqhdu whup ru vhdvrqdo gxpplhv1 Wkh pdlq surshuw| wkdw zh qhhg khuh lv wkdw gw.4 lv d olqhdu ixqfwlrq ri gw > +gw.4 @ P gw , d surshuw| wkdw lv vdwlvhg lq wkh deryh h{dpsohv/ dqg zklfk lpsolhv wkdw wkh surfhvvhv 3 [w dqg [w kdyh h{shfwdwlrqv wkdw duh olqhdu ixqfwlrqv ri gw = Zh h{sorlw wkh idfw wkdw lv nqrzq wr ghulyh wzr htxdwlrqv iurp prgho +4,1 Zh ohw. 3 @ +3 ,4 3 dqg qg. 3 [w. @ 3 [w4 . 3 4 [w4 . 3 gw . 3 %w >. +5,. +6, 3B [w @ 3B 4 [w4 . 3B gw . 3B %w > dqg qdoo|/ zlwk $ @ 3 B +3B B ,4 > wkh frqglwlrqdo prgho ri 3 [w jlyhq 3B [w dqg. wkh sdvw lv. 4 [w4 . g w . %w > 3 [w @ $3B [w . 3 [w4 . zkhuh 4 @ + @ + 3 $3B ,4 > 3 $3B , > %w @ + 3 $3B ,%w = 6. +7,.

(7) Qrwh wkdw wkh huuruv duh lqghshqghqw lq +6, dqg +7,1 Wkh huuruv 3B %w duh wkh shupdqhqw vkrfnv zklfk zloo eh nhsw {hg lq wkh fdofxodwlrq ri wkh Eduwohww fruuhfwlrq/ dqg wkh huuruv + 3 $3B ,%3w @ +3 ,4 3 4 %w duh wkh wudqvlwru| vkrfnv zklfk ghwhuplqh wkh udqgrp yduldwlrq ri wkh hvwlpdwhg iru jlyhq ydoxhv ri wkh frpprq wuhqgv1 Wkh frpprq wuhqgv lq S wkh surfhvv duh 3B wl@4 %l dqg wkh dqdo|vlv ri wkh glvwulexwlrq ri wkh whvw vwdwlvwlf zloo eh frqgxfwhg frqglwlrqdo rq wkhvh lq wkh iroorzlqj1 Vlqfh wkh sdudphwhu hqwhuv wkh frqglwlrqdo prgho +7, rqo|/ zh ghulyh wkh whvwv edvhg rq wkdw htxdwlrq1. 515. Whvw ri d vlpsoh k|srwkhvlv rq wkh frlqwhjudwlqj vsdfh lq wkh jhqhudo prgho. Zh frqvlghu lq wklv vxevhfwlrq wkh whvw wkdw wkh frlqwhjudwlqj vsdfh lv jlyhq/ wkdw lv/ @ K! zkhuh ! lv +u u,= Lw lv ri frxuvh yhu| hdv| wr ghulyh wkh olnholkrrg udwlr whvw iru d k|srwkhvlv rq lq +7,1 Zh ghulyh khuh dq h{suhvvlrq iru wkh whvw vwdwlvwlf wkdw ghshqgv rq wkh sdudphwhuv ri wkh gdwd jhqhudwlqj surfhvv1 Wklv lv frqyhqlhqw zkhq zh zdqw wr shuirup wkh yhu| vshfldo fdofxodwlrqv xqghu d {hg suredelolw| phdvxuh vshflhg e| wkh sdudphwhuv xqghu wkh qxoo k|srwkhvlv1 Wkhvh ydoxhv zh fdoo wkh wuxh ydoxhv dqg ghqrwh wkhp e| +3 > 3 @ K!3 > 34 > 3 > 3 ,> dqg lqglfdwh e| d vxshuvfulsw 3 dq| sdudphwhu ghulyhg iurp wkhvh1 Zh dvvxph wkdw wkh surfhvv lv L+4,> ru htxlydohqwo| wkdw wkh urrwv ri wkh fkdudfwhulvwlf sro|qrpldo duh hlwkhu juhdwhu wkdq rqh lq devroxwh ydoxh ru htxdo wr 4/ dqg iru 3 @ Lq 34 > zh dvvxph wkdw 3B3 3 3B kdv ixoo udqn ru htxlydohqwo| djdlq wkdw + 3 > 33 3B , kdv ixoo udqn> vhh Mrkdqvhq +4<<9,1 Zh ghqh F 3 @ 3B +3B3 3 3B ,4 3B3 = Qrwh wkdw +Lq F 3 3 , 3B @ 3> 3 33 [w lv d olqhdu ixqfwlrq ri 33 [w > dqg F 3 3 [w lv vxfk wkdw +Lq F 3 3 ,[w @ +Lq F 3 3 , d olqhdu ixqfwlrq ri 3B3 3 [w = Zh wkhuhiruh dsso| wkh ghfrpsrvlwlrq 3 [w4 @ +Lq F 3 3 , 33 [w4 . F 3 3 [w4 >. wr ghfrpsrvh wkh surfhvv lqwr d vwdwlrqdu| dqg d qrq0vwdwlrqdu| sduw1 Zh ghqh qhz sdudp0 hwhuv dqg e| 3 33 [w4 . 3 F 3 3 [w4 3 [w4 @ 3 +Lq F 3 3 , @ 33 [w4 . 3B3 3 [w4 = 3 Qrwh wkdw wkh +u u, xqnqrzq sdudphwhuv lq @ 3 +Lq F 3 3 , duh wkh dgmxvwphqw frh!flhqwv wr wkh nqrzq frlqwhjudwlrq yhfwruv lq 3 > dqg wkdw wkh u +q u, sdudphwhuv @ 3 3B +3B3 3 3B ,4 duh wkh frh!flhqw wr wkh qrq0vwdwlrqdu| yduldeohv1 Wkxv wkh uhjuhvvlrq htxdwlrq +7, lv. 4 [w4 . g w . %w = 33 [w @ $3B3 [w . 33 [w4 . 3B3 3 [w4 . 7. +8,.

(8) Wkh k|srwkhvlv ri d {hg frlqwhjudwlqj vsdfh vsdqqhg e| 3 lv K3 = @ 3> dqg wkh olnholkrrg udwlr whvw lv ghulyhg e| d uhjuhvvlrq ri 33 [w rq 3B3 3 [w4 fruuhfwhg iru wkh yduldeohv 3B3 [w > 33 [w4 > [w4 > dqg gw 1 Iurp +6, zh qg wkdw lq wkh uhjuhvvlrq +8, zh fdq uhsodfh 3B3 [w e| 3B3 %w vlqfh zh duh fruuhfwlqj iru [w4 dqg gw lq wkh uhjuhvvlrq dq|zd|> dqg e| vxppdwlrq ri +6, zh qg 3B3 3 [w4 @ 3B3 34 [w4 . 3B3. w 4 [ +%l . 3 gl , . 3B3 +[3 34 [4 ,= l@4. +9,. Sw4 Wklv vkrzv wkdw lq wkh uhjuhvvlrq +8, zh fdq uhsodfh 3B3 3 [w4 e| d3w4 @ 3B3 l@4 +%l . 3 gl , . 3B3 +[3 34 [4 , vlqfh zh duh fruuhfwlqj iru [w4 lq wkh uhjuhvvlrq= Qrwh wkdw li gw frqwdlqv d frqvwdqw zh fdq gurs wkh lqlwldo ydoxhv lq wkh ghqlwlrq ri d3w4 = Lq wkh fdofxodwlrqv lq Vhfwlrq 6 zh nhhs 3B3 % {hg +dqg khqfh d3w4 , dqg zh ghqh wkh qrupdol}hg ghwhuplqlvwlf uhjuhvvruv 4 ew @ +3B3 3 3B , 5 3B3 %w >. +:,. dqg ohw dw4 eh d3w4 exw ruwkrjrqdol}hg wr gw dqg ew > dqg pruhryhu qrupdol}hg/ vxfk wkdw. W [ w@4. W W [ [ 3 3 dw4 gw @ 3> dw4 ew @ 3> dw4 d3w4 @ Lqu = w@4. w@4. Vlploduo| zh ghqh wkh qrupdol}hg huuruv dv 4 4 Xw @ +33 3 4 3 , 5 33 3 4 %w @ +33 3 4 3 , 5 + 3 $ 3 3B ,%w =. +;,. Zh ghqh wkh xqfrqglwlrqdo yduldqfh. 3 @ Ydu. . 3 3 [w [w. =. Iurp wkh Judqjhu uhsuhvhqwdwlrq ri [w > vhh Mrkdqvhq +4<<9,/ [w @ F 3. 4 w [ [ +%l . 3 gl , . Fl3 +%wl . 3 gwl ,> l@4 l@3. +<,. zh qg zlwk gwl @ P l gw > wkdw S4 S4 H+ 3 [w , @ l@3 Fl3 3 gwl @ + l@3 Fl3 3 P l ,gw 3 3 S4 3 3 l H+[w , @ F . l@3 Fl P +L P 4 , gw Wkxv wkh phdq H+[w3 3 > [w3 , lv d olqhdu ixqfwlrq ri wkh ghwhuplqlvwlf whupv1 Zh ghqh wkh qrupdol}hg surfhvv 33 4 [w H+ 33 [w , = +43, \w @ 3 5 [w H+[w , Li wkhuh duh pruh wkdq wzr odjv lq wkh prgho zh kdyh wr ghqh wkh surfhvv \w e| vwdfnlqj 3 3 [w > [w > = = = > [wn.5 > zklfk ehfrphv ri glphqvlrq p @ u . +n 4,q= 8.

(9) Wkh uhjuhvvlrq htxdwlrq +8, fdq wkhq eh zulwwhq lq wkh irup. ]w @ 4 dw4 . 5 \w4 . 6 ew . 7 gw . 8 Xw > +u , +q u , +g , +u, +q u , +p ,. +44,. 3 [w > dqg zlwk ]w @ 4 \w @ 3 5 +[w3 3 H+[w3 3 ,> [w3 H+ [w3 ,> = = = > [w3n.5 H+ [w3n.5 ,,3. +45,. dqg vxlwdeoh frh!flhqwv 4 > = = = > 8 = Qrwh wkdw fhqwhulqj \w4 wr kdyh phdq }hur lqwurgxfhv d fruuhfwlrq wr wkh frh!flhqw ri gw vlqfh wkh phdq ri 3 [w dqg [w duh olqhdu ixqfwlrqv ri gw = Wkh whvw wkdw wkh frlqwhjudwlqj vsdfh lv vs+ 3 , lv wkh whvw wkdw 4 @ 3= Ehorz wkh yduldeohv duh lqglfdwhg wkhlu glphqvlrqv1 Wkh pd{lpxp olnholkrrg hvwlpdwru ru uhjuhvvlrq hvwlpdwru iru 4 vdwlvhv 8 Vxd=|>e>g @ +a 4 4 ,Vdd=|>e>g > zkhuh zh kdyh xvhg wkh qrwdwlrq iru wkh surgxfw prphqwv ri +Xw > \w4 > dw4 > ew > gw , = 3 43 43 3 4 Xw Xw Vxx Vx| Vxd Vxe Vxg FE F E V V|e V|g F W E E \w 4 F E \w 4 F F E |x V|| V|d [ E FE F F E 3 E dw4 F E dw4 F @ E Vdx Vd| Lqu 3 F= E FE F F E w@4 C ew D C ew D C Vex Ve| 3 Vee Veg D gw gw Vgx Vg| 3 Vge Vgg Ohw Yw > Zw > dqg Iw eh dq| ri wkhvh surfhvvhv wkhq zh dovr qhhg wkh frqglwlrqdo surgxfw prphqwv. 4 Viz = Vyz=i @ Vyz Vyi Vii Zlwk wklv qrwdwlrq zh qg wkh olnholkrrg udwlr whvw ri wkh k|srwkhvlv wkdw wkh frlqwhjudwlqj vsdfh lv jlyhq e| vs+ 3 , @ vs+K, fdq eh whvwhg dv 4 @ 3 e| wkh olnholkrrg udwlr whvw 5 orj OU+ @ K!, @ W orj. 516. mVxx=|>d>e>g m = mVxx=|>e>g m. +46,. Whvw iru olqhdu uhvwulfwlrqv rq wkh frlqwhjudwlqj vsdfh. Zh frqvlghu lq wklv vhfwlrq wkh olnholkrrg udwlr whvw iru wkh k|srwkhvlv @ K!> iru K+q v, dqg !+v u, +u v q,= Zh ohw O+> > 4 > , ghqrwh wkh Jdxvvldq olnholkrrg dqg ghqh wkh frqfhqwudwhg olnholkrrg O+, @ pd{ O+> > 4 > , 4 > > dqg xvh wkh pxowlsolfdwlyh surshuw| ri olnholkrrg udwlr whvw wr vhh wkdw iru 3 @ K!3 > O+ 3 , O+ 3 , pd{ @K! O+, @ @ pd{ O+, pd{ O+, pd{ @K! O+, 9.

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(11) Zh ghulyh dq h{sdqvlrq iru wkh olnholkrrg udwlr whvw wkdw 4 @ 3= Zh qg iurp +46,. mV m 5 orj OU+ 4 @ 3, @ W orj mVxx=|>d>e>g xx=|>e>g m 4 Vdx=|>e>g m @ W orj mLqx W 4 +W 4 Vxx=|>e>g ,4 Vxd=|>e>g Vdd=|>e>g 4 @ W orj mLqx W 4 Tm @ wuiTj . 54W wuiT5 j> v zkhuh wkh qrwdwlrq @ lqglfdwhv wkdw zh kdyh nhsw whupv ri wkh rughu W v > v @ 3> 4> = = = > dqg. 4 Vdx=|>e>g 5 RS +4,= T @ +W 4 Vxx=|>e>g ,4 Vxd=|>e>g Vdd=|>e>g. +49,. Lq rughu wr vlpsoli| wkh h{suhvvlrq iru T zh qrwh wkdw vlqfh dw4 lv ruwkrjrqdo wr ew dqg gw > zh jhw Vdd=|>e>g Vdx=|>e>g. 4 V|d=e>g @ Vdd=e>g Vd|=e>g V||=e>g 4 V|d @ Lq W 4 E @ Vdd Vd| V||=e>g x 4 @ Vdx=e>g Vd|=e>g V||=e>g V|x=e>g 4 V|x=e>g @ Q W 45 D> @ Vdx Vd| V||=e>g. +4:, +4;,. zkhuh Q. @ Vdx E @ Vd| +W 4 V||=e>g ,4 V|d > 4 D @ Vd| +W 4 V||=e>g ,4 +W 5 V|x=e>g ,>. +4<, +53, +54,. duh RS +4, dqg Q lv glvwulexwhg dv Qqd qx +3> Lqd Lqx ,> zkhq zh frqglwlrq rq 3B %= Zh qg. 4 V|x=e>g W 4 Vxx=|>e>g @ W 4 Vxx=e>g W 4 Vx|=e>g V||=e>g 4 Vgx W 4 Vxe=g V 4 Vex=g W 4 Vx|=e>g V 4 V|x=e>g @ W 4 Vxx W 4 Vxg Vgg ee=g ||=e>g 4 Vgx . Vxe=g V 4 Vex=g . Vx|=e>g V 4 V|x=e>g , @ Lqx . +W 4 Vxx Lqx , W 4 +Vxg Vgg ee=g ||=e>g 4 4 @ Lqx . W 5 G4 W G5 = +55, Zh wkhq qg iurp +49,/ +4:,/ dqg +4;, wkdw wuiTj. 4 4 4 @ wui+Lqx . W 5 G4 W 4 G5 ,4 +Q W 5 D,3 +Lqd W 4 E,4 +Q W 5 D,j 4 4 @ wuiQ 3 Q W 5 +Q 3 D . D3 Q . G4 Q 3 Q, . W 4 +G5 Q 3 Q . G45 Q 3 Q . G4 +Q 3 D . D3 Q , . Q 3 EQ . D3 D,j>. zkhuh zh kdyh nhsw whupv ri rughu W 4 1 Wkh qrwdwlrq kdv ehhq fkrvhq vxfk wkdw wkh srzhu ri W lq iurqw lqglfdwhv wkh rughu ri wkh ydulrxv whupv1 Vlploduo| zh qg 3 wuiT5 j @ wui+Q 3 Q ,5 j=. ;.

(12) Wkh dv|pswrwlf surshuwlhv lq wkh xvxdo dqdo|vlv/ zkhuh % lv frqvlghuhg udqgrp/ duh jlyhq lq whupv ri d Eurzqldq prwlrq Z +x, = ^W x `. 4 [ z W5 %l $ Z +x,=. l@4. Zh ghqh wkh lqghshqghqw vwdqgdug Eurzqldq prwlrqv E4 +x, @ +3B B , 5 3B Z +x, dqg 4 E5 +x, @ +3 4 , 5 3 4 Z +x,= 4. Wkh dv|pswrwlf glvwulexwlrq ri 5 orj OU> iru wkh k|srwkhvlv wkdw wkh frlqwhjudwlqj vsdfh lv vsdqqhg e| > lv jlyhq e| ] 4. 3 z 5 orj OU @ wuiTj $ wui. 3. +gE5 ,I 3. 4 ] 4 I I 3 gx I +gE5 ,3 j> 3 3. ] 4. +56,. zkhuh I lv frqvwuxfwhg iurp E4 ghshqglqj rq wkh ghwhuplqlvwlf whupv1 Li gw @ 3> wkhq I @ E4 > S 4 dqg li iru lqvwdqfh gw @ 3 > wkhq 3B wl@4 %l . 3B 3 +w 4, lv fruuhfwhg iru wkh phdq1 Wkh surfhvv lv olqhdu lq rqh gluhfwlrq/ 3B 3 > dqg d udqgrp zdon lq q u 4 gluhfwlrqv vxfk wkdw U4 Il +x, @ El +x, 3 El +v,gv> l @ 4> = = = > q u 4> +57, Iqu +x, @ x 45 = U4 Iru {hg ydoxh ri I wkh glvwulexwlrq ri 3 I +gE5 ,3 lv Jdxvvldq zlwk phdq }hur dqg U4 yduldqfh pdwul{ 3 I I 3 gx Lqx vxfk wkdw wkh olplw glvwulexwlrq ri T iru {hg I lv "5 zlwk ghjuhhv ri iuhhgrp qx qd = Vlqfh wklv glvwulexwlrq grhv qrw ghshqg rq wkh frqglwlrqlqj udqgrp yduldeoh/ wkh pdujlqdo glvwulexwlrq lv dovr "5 +qx qd ,1 Wkh lghd lq wkh iroorzlqj lv wr wdnh wkh frqvhtxhqfh ri wklv frqglwlrqlqj dujxphqw/ dqg S frqglwlrq douhdg| iurp wkh ehjlqqlqj rq wkh frpprq wuhqgv 3B wl@4 %l / zklfk jlyh ulvh wr wkh Eurzqldq prwlrq E4 dqg I= Wkh sdshu e| Urwkhqehuj +4<;;, frqwdlqv dqrwkhu frqglwlrqlqj lghd/ gxh wr Fdydqdjk +4<;6,/ zklfk lq wkh suhvhqw frqwh{w zrxog phdq frqglwlrqlqj rq wkh }hur rughu whup Q 3 Q dqg zrun rxw wkh prphqwv ri wkh uhpdlqlqj whupv lq wklv frqglwlrqdo glvwulexwlrq1 Wklv zloo qrw eh dwwhpswhg khuh1 Eduwohww +4<6:, sursrvhg wr lpsuryh wkh dssur{lpdwlrq wr wkh olplw glvwulexwlrq ri wkh olnholkrrg udwlr whvw e| dgmxvwlqj wkh qlwh vdpsoh glvwulexwlrq wr kdyh wkh vdph phdq dv wkh olplw glvwulexwlrq1 Wklv vlpsoh fruuhfwlrq wxuqv rxw lq d qxpehu ri fdvhv wr kdyh d jrrg hhfw rq wkh dssur{lpdwlrq/ dqg zh vkdoo wkhuhiruh wu| wr fdofxodwh wkh h{shfwdwlrq H^5 orj OUm3B %`= Wkh h{dfw fdofxodwlrq lv qrw srvvleoh exw zh fdq ghulyh dq h{suhvvlrq iru wkh uvw rughu dssur{lpdwlrq1 Lw wxuqv rxw wkdw wklv dssur{lpdwlrq ghshqgv rq wkh sdudphwhuv dqg wkxv zloo kdyh wr eh hvwlpdwhg lq sudfwlfh/ zklfk jlyhv vrph h{wud xqfhuwdlqw|1 Wkxv zh zdqw wr fdofxodwh 4 4 H^5 orj OUm3B %` @ H^wuiTjm3B %` . H^wuiT5 jm3B %`> 5W. +58,. zkhuh H^wuiTjm3B %` 4 4 @ wuiH^Q 3 Qm3B %`j W 5 wuiH^+5D3 Q . G4 Q 3 Q,m3B %`j .W 4 wuiH^+G5 Q 3 Q . G45 Q 3 Q . 5G4 Q 3 D . Q 3 EQ . D3 D,m3B %`j> <. +59,.

(13) dqg 3 H^wuiT5 jm3B %` @ wuiH^+Q 3 Q ,5 m3B %`j=. 518. +5:,. Wkh {hg uhjuhvvruv. Zh lqvhuw d ihz frpphqwv rq wkh {hg uhjuhvvruv dw4 dqg ew / ghqhg lq whupv ri 3B dqg 3B %w =. Sw4 l@4 %l >. E| wkh qrupdol}dwlrq fkrvhq deryh zh kdyh Vdg @. W [ w@4. dw4 g3w @ 3> Vde @. W [ w@4. Zkhq zh gr qrw frqglwlrq rq 3B. dw4 e3w @ 3> Vdd @. W [ w@4. dw4 d3w4 @ Lqd =. Sw4 l@4 %l zh kdyh wkh iroorzlqj uhodwlrqv. 4 Vge 5 RS +4,> Veg Vgg W [ w@4. S dw4 d3w4n $ Lqd > iru doo n>. W. . [ ew e3 W 4. W [ w@4. +5;,. w@4. S w n $. Lqe 3. +5<,. li n @ 3> li n 9@ 3>. +63,. W [ S 3 ewn dw ew d3w $ 3 li n 3>. +64,. w@4. SW Ilqdoo| zh kdyh wkdw w@4 ew d3w frqyhujhv zhdno| wr d olplw Led 1 Wkh olplw ghshqgv rq wkh ghwhuplqlvwlf whupv/ vxfk wkdw li gw @ 3> wkhq. W [ w@4. z ew d3w $ +Lqe .. ] 4 3. +gE5 ,E43 ,. ] 4 3. 45 E4 E43 gx @ Led >. +65,. zkhuhdv li gw @ 3 wkhq. W [ w@4. z ew d3w $ ++Lq 4 > 3+q 4,4 ,3 . e. ] 4. e. zlwk I jlyhq e| +57,1. 3. +gE5 ,I 3 ,. ] 4 3. 45. I I 3 gx. >. +66,. Zkhq frqglwlrqlqj rq wkh vhtxhqfh 3B % zh dvvxph wkdw vxfk uhodwlrqv krog iru wkh S S vhtxhqfh zh duh {lqj1 Wkdw lv/ zh vkdoo uhsodfh Ww@4 dw4 d3w4n e| Ww@4 dw4 d3w4 @ Lqd > SW SW W 4 w@4 ew e3wn e| Lqu ru 3> dqg/ iru n 3> w@4 ewn d3w e| wkh pdwul{ Led lq wkh olplw uhvxowv1 Lw wxuqv rxw wkdw dowkrxjk Led hqwhuv vrph ri wkh lqwhuphgldwh uhvxowv/ wkh qdo uhvxow grhv qrw lqyroyh Lde > vxfk wkdw wkhuh lv qr qhhg wr fdofxodwh wkh h{shfwdwlrq zkhq lw dsshduv1 43.

(14) Wkh sdshuv e| Nlylhw dqg Skloolsv +4<<:d/4<<:e, ghdo zlwk d uhjuhvvlrq prgho zklfk uhvhpeohv +48,= |w @ |w4 . 3 {w . %w > dqg ghulyh dssur{lpdwlrqv iru wkh eldv ri wkh ohdvw vtxduhv hvwlpdwru ri +> ,= Wkh fdofxodwlrqv duh yhu| vlplodu wr wkrvh qhhghg khuh/ exw wkh vfrsh glhuv lq d qxpehu ri dvshfwv1 Zh duh dqdo|vlqj d pxowlyduldwh vlwxdwlrq udwkhu wkdq d xqlyduldwh/ dqg zdqw dq dssur{lpdwlrq wr wkh olnholkrrg udwlr whvw vwdwlvwlf udwkhu wkdq wkh eldv1 Zh duh lqwhuhvwhg rqo| lq wkh uvw rughu whup/ zklfk lpsolhv wkdw pdq| ri wkh irupxodh orrn vlpsohu/ dqg lqyroyh vpdoohu pdwulfhv/ vlqfh wkh deryh uhodwlrqv iru wkh uhjuhvvruv dw4 dqg ew doorz vrph vlpsolfdwlrqv ri wkh jhqhudo whupv1. 6. Wkh pdlq uhvxowv rq wkh Eduwohww fruuhfwlrq. Zh lqwurgxfh khuh wkh frh!flhqwv wkdw duh xvhg wr h{suhvv wkh pdlq uhvxow lq Sursrvlwlrq 4/ zklfk jlyhv dq dssur{lpdwlrq wr wkh ydulrxv whupv hqwhulqj wkh h{suhvvlrq iru wkh frqglwlrqdo h{shfwdwlrq ri wkh olnholkrrg udwlr whvw ri wkh k|srwkhvlv 4 @ 3 lq +48,1 Lq wkh iroorzlqj zh dvvxph wkdw wkh sdudphwhuv +> > 4 > = = = > n4 > > , duh vr fkr0 vhq wkdw wkh surfhvv lv L+4,= Wkxv wkh qrq0vwdwlrqdu| sduw ri wkh surfhvv lv vwduwhg lq wkh lqlwldo ydoxhv/ exw wkh vwdwlrqdu| sduw ri wkh surfhvv/ 3 [w H+ 3 [w , dqg [w H+[w ,> duh jlyhq wkhlu lqyduldqw glvwulexwlrq1 Zh gr qrw xvh wkh qrwdwlrq zlwk d vxshuvfulsw 3 vlqfh zh qhhg qrw glvwlqjxlvk ehwzhhq wkh sdudphwhu ydoxh dqg wkh wuxh ydoxh lq wklv vhfwlrq1 Wklv vwdwlrqdulw| ri 3 [w H+ 3 [w , dqg [w H+[w , ri frxuvh krogv zkhq zh gr qrw frqglwlrq rq 3B %> dqg wkhq \w > vhh +45,/ kdv wkh uhsuhvhqwdwlrq. 4 [. 4 [ \w @ F %w @ + Xw . # ew ,> @3 @3. +67,. vhh +:, dqg +;, iru vxlwdeoh pdwulfhv > # > vhh +7:,= Zkhq zh frqglwlrq rq 3B %w > krzhyhu/ wkhq \w lv qr orqjhu vwdwlrqdu|/ vlqfh lwv phdq ghshqgv rq w= Qrwh wkdw wkh xqfrqglwlrqdo dxwrfruuhodwlrq ixqfwlrq lv. 4 [ + 3 .k . # # 3 .k ,> +3, @ Lq| = Fry+\w > \w.k , @ +k, @ @3. Zh ghqh wkh frh!flhqwv @. 4 [ @3. > # @. 4 [ @3. # >. +68,. +69,. vxfk wkdw orqj0uxq yduldqfh ri wkh vwdwlrqdu| surfhvv \w lv 3 . ## 3 > zkhuhdv wkh orqj0uxq yduldqfh ri \w frqglwlrqdo rq wkh frpprq wuhqgv lv 3 = Zh fdq qrz irupxodwh wkh pdlq uhvxow rq wkh whvw wkdw 4 @ 3 lq wkh uhjuhvvlrq +48,1 Wkh uhvxow lv jlyhq dv d uvw rughu dssur{lpdwlrq ri H^5 orj OUm3B %` h{suhvvhg lq whupv ri wkh frh!flhqwv > # > dqg +k, dqg wkh glphqvlrqv qx > qd > qe > q> qg > dqg q| > dqg P= Iluvw/ krzhyhu/ zh jlyh d whfkqlfdo uhvxow zklfk lv wkh edvlv iru wkh ydulrxv dssolfdwlrqv odwhu1 44.

(15) Sursrvlwlrq 4 Lq wkh uhjuhvvlrq htxdwlrq +48, zh kdyh wkh uhvxowv wuiH^+Q 3 Q ,5 m3B %`j @ qd qx +qd . qx . 4, wuiH^Q 3 Qm3B %`j @ qd qx W 5 wuiH^DQ 3 m3B %`j S .4 jwui 3 j $ qd +qx . 4,wui3 j . qx wuiLed Lde # 3 #j qd 4 @3 wuiP S4 qd @3 ^wui+ . 4,3 j . wui+ . 4,jwui 3 j`. +6:, +6;,. 4. wuiH^D3 Dm3B %`j $ qd qx wui3 j . qx wuiLed Lde # 3 #j wuiH^Q 3 EQm3B %`j. $. qd +qd . qx . 4,wui3 j . qx wuiLed Lde # 3 #j. W 5 wuiH^G4 Q 3 Q m3B %`j $ qd qx +qx . 4, 4. wuiH^+G5 Q 3 Q m3B %`j $ qd qx +qg . qe . q| , wuiH^G4 Q 3 Dm3B %`j $ 3 wuiH^G45 Q 3 Qm3B %`j $ qd qx +qx . 4,. +6<,. +73, +74, +75, +76, +77, +78,. zkhuh wkh sdudphwhuv > # > > #> dqg +k, duh jlyhq lq +67,/ +68,/ dqg +69,1 Wkh ghqlwlrq ri +Q> D> E> G4 > G5 , duh jlyhq lq +4<,/ +53,/ +54,/ dqg +55,1 Qrwh wkdw wkh fruuhfwlrq whupv ghshqg rq wkh frqglwlrqlqj yduldeoh wkurxjk wkh txdq0 wlw| Lde > vhh +65, dqg +66,1 Wkh surri ri wklv uhvxow lv jlyhq lq wkh Dsshqgl{1 Sursrvlwlrq 5 Wkh uvw rughu dssur{lpdwlrq wr wkh frqglwlrqdo h{shfwdwlrq ri wkh olnholkrrg udwlr whvw vwdwlvwlf iru 4 @ 3 lq +48, lv jlyhq e| H^5 orj OUm3B %` qx qd 4 qd 4 ^ +qx . qd . 4, . +qg . qe . q| ,` . ^+qd 4,y . 5+f . fg ,`= @ qx qd . W 5 W zlwk @ wui3 j 4 [ wuiP .4 jwui3 j fg @ @3 4 [ f @ +wui+ . 4, 3 j . wui+ . 4,jwui3 j, @3 y. Wkh surri iroorzv iurp Sursrvlwlrq 4 e| xvlqj +58,/ +59,/ dqg +5:,1 Qrwh wkdw wkh fruuhfwlrq whup wr wkh olnholkrrg udwlr whvw vwdwlvwlf grhv qrw ghshqg rq wkh frqglwlrqlqj udqgrp yduldeohv lq wkh vhqvh wkdw Lde lv qr orqjhu suhvhqw1 Lq wkh iroorzlqj zh jlyh pruh h{solflw irupxodh xvlqj wkh surshuwlhv ri wkh xqghuo|lqj surfhvv1 45.

(16) Wkh surfhvv \w > vhh +43, dqg +67,/ vdwlvhv wkh htxdwlrq 4. 4. 5 \w @ S 5 \w4 . E%w > zkhuh 3 E E E S +q| q| , @ E E E C. Lu . 3 3 4 4 3 Lq 11 11 1 1 3 3. . 3 n5 n5 3 11 1. 3 n4 n4 3 11 1. Lq. 3. . 4. 3. 3 Lq 11 1. F E F E F E F > E+q| q, @ E F E F E D C 3 3. 4 F F F F = +79, F F D. Zh qg wkh uhsuhvhqwdwlrq \w. @ @. 4 [ @3 4 [ @3. 5 S E%w 4. +7:,. 5 S E+3 4 , 5 Xw . 5 S E B +3B B , 5 ew 4. 4. 4. 4. dqg xvh lw wr fdofxodwh wkh fryduldqfh pdwul{ . 4 [. 4 [ +S S , yhf +E E 3 , yhf+ , @ yhf+ @3 @3 4 @ +Lq| Lq| S S , yhf +E E 3 , = S E E 3 S 3 , @. Frpsdulqj +7:, zlwk +67, zh qg @ 5 S E+3 4 , 5 > # @ 5 S E B +3B B , 5 = 4. 4. 4. 4. Qrwh wkdw e| wkh ghqlwlrq ri . 4 [ + 3 . # # 3 , @ Ydu+\w , @ Lq| = @3 614. +7;,. Fdofxodwlrq ri ' |oiww j . Zh uvw fdofxodwh wkh frqglwlrqdo orqj0uxq yduldqfh ri \w dv jlyhq e| 3 = Zh qg @. 4 [. @3. @. 4 [. @3. 5 S E+3 4 , 5 @ 5 +Lq| S ,4 E+3 4 , 5 > 4. 4. 4. 4. +7<,. dqg khqfh 3 @ 5 +Lq| S ,4 E+3 4 ,4 3 E 3 +Lq| S 3 ,4 5 = 4. 4. Vlqfh +Lq| S ,+Lqx > 3> = = = > 3,3 @ E 46. +83,.

(17) zh qg 4 3 @ 5. . +3 4 ,4 3. Zh ghqh wkh frh!flhqw pdwul{ Y @ Qrwh wkdw. 3. . Lq wui 3 j @ wui. 4 5 . x. 3. 3 45 @. 4. . . Lq x 3. 3 3. . +3 4 ,4 3. . 5 = 4. 3 3. . 4 =. +84,. 4 3 4 4 +3 4 ,4 j @ wui + , j>. zkhuh @ Y du+ 3 [w m[w > [w4 > = = = > [wn.5 ,=. 615. Fdofxodwlrq ri S dqg S_. Zh qrwh wkdw 3. .4 @ 3 + 5 S 3 .4 5 ,> # 3. .4 @ #3 + 5 S 3 .4 5 ,> 4. vxfk wkdw. 4. 4. 4. 4 [ + 3. .4 . # # 3. .4 , + . 4, @ @3 4 [ 4 4 + 3 . # #3 ,+ 5 S 3 .4 5 , @ @3 4 4 @ 5 S 3 .4 5 >. vlqfh e| +68,. 4 [ + 3 . # #3 , @ Lq| = @3 Zh wkhq qg wkh h{suhvvlrq/. 4 [ @. @3 4 [. wui+ . 4,3 j wui 5 S 3 .4 5 5 +Lq| S ,4 E+3 4 ,4 3 E 3 S 3 5 j 4. 4. 4. 4. @3 @ wui 4 +Lq| S ,4 E+3 4 ,4 3 E 3 +Lq| S 3 ,4 +Lq| . S 3 ,4 S 3 j. @ wuiY 3 +Lq| . S 3 ,4 S 3 j @ wuiS +Lq| . S ,4 Y j> vlqfh. +Lq| S ,4 E @ +Lqx > 3> = = = > 3,3 > 47.

(18) zkhuh zh kdyh xvhg +83,1 D glhuhqw h{suhvvlrq fdq eh irxqg li wkh pdwul{ S fdq eh gl0 djrqdol}hg1 Li zh ohw @ gldj+4 > = = = > q| , dqg N @ +y4 > = = = > yq| , ghqrwh hljhqydoxhv dqg hljhqyhfwruv ri wkh pdwul{ S> wkhq S @ NN 4 1 Qrwh wkdw wkhvh fdq eh irxqg iurp wkh hljhqydoxhv iru zklfk 9@ 4 dqg wkh fruuhvsrqglqj hljhqyhfwruv ri wkh frpsdqlrq pdwul{1 Lq whupv ri wkhvh zh kdyh. 4 [. @3. q [ m 3 4 wui+ . 4, j @ wuiS +Lq . S , Y j @ +N 4 Y N,mm = |. m @4 4 . m. |. 4 [ @ @ @. @3 4 [ @3 4 [ @3 4 [. wui+ . 4,jwui 3 j wuiS 3 .4 jwuiS 3 4 +Lq| S ,4 E+3 4 ,4 3 E 3 j wuiS 3 .4 jwuiS 3 Y 3 +Lq| S 3 ,j wui+S 3 S 3 , +S 3 Y 3 +Lq| S 3 ,j. @3 @ wui+Lq| Lq| +S 3 S 3 ,,4 +S 3 Y 3 +Lq| S 3 ,j. @ wui+S +Lq| S ,Y ,+Lq| Lq| +S S ,,4 j= Lq whupv ri wkh hljhqyhfwruv dqg hljhqydoxhv zh fdq zulwh. 4 [ @3. wui+ . 4,jwui3 j @ @. 4 [. wuiS .4 jwuiS +Lq| S ,Y j. @3 q| 4 [ [. m l +4 l ,+N 4 Y N,ll. @3 m>l@4 q| [ 4 l @ +N 4 Y N,ll m>l@4 4 l m. E| wkh vdph phwkrg zh qg. 4 [. @3. 616. wuiP .4 jwui 3 j @ wui+P +Lq| S ,Y ,+Lqg Lq| +P S ,,4 j=. Wkh pdlq uhvxow rq d vlpsoh k|srwkhvlv. Zh qrz irupxodwh wkh pdlq uhvxow iru whvw iru d vlpsoh k|srwkhvlv rq 4 lq wkh htxdwlrq +48, Wkhruhp 6 Li wkh xqghuo|lqj surfhvv [w lv dq DU+n, prgho zlwk ghwhuplqlvwlf whup gw > wkh fruuhfwlrq idfwru wr wkh olnholkrrg udwlr whvw vwdwlvwlf iru d vlpsoh k|srwkhvlv rq wkh frlqwhjudwlqj vsdfh lq +5146, lv jlyhq e|. H ^5 orj OUm3B %` q x qd 4 @ 4 . W4 ^+qg . q| . qe , . 45 +qx . qd . 4,` . W q4 x ^+qd 4,y . 5+f . fg ,`= 48.

(19) Khuh wkh frqvwdqwv y> f> dqg fg duh jlyhq e| y. @ wuiY j. @ wui+P +Lq| S ,Y ,+Lqg Lq| +P S ,,4 j f @ wuiS +Lq| . S ,4 Y j . wui^S +Lq| S ,Y `^+Lq| Lq| +S S ,,4 `j. fg. +85, +86, +87,. Wkhruhp 7 zkhuh wkh pdwul{ Y lv jlyhq e| +84,/ S e| +79,/ dqg P lv ghqhg e| gw.4 @ Pgw 1. 7. Uhvxowv iru wkh dxwruhjuhvvlyh prgho. Wklv vhfwlrq frqwdlqv wkh frqvhtxhqfhv iru wkh frlqwhjudwhg yhfwru dxwruhjuhvvlyh prgho iru wkh ydulrxv whvwv iru olqhdu k|srwkhvhv1 Zh frqvlghu wkh dxwruhjuhvvlyh prgho zlwk glphqvlrq q/ odj ohqjwk n> frlqwhjudwlqj udqn u1 Lq wkh iroorzlqj zh uhvwulfw wkh ghwhuplqlvwlf whup wr eh srzhuv/ olnh gw @ 4> gw @ +4> w, ru hyhq gw @ +4> w> w5 ,> vlqfh lq wklv fdvh rqh fdq vhh wkdw S 3 wuiP j @ qg > wkh qxpehu ri ghwhuplqlvwlf whupv1 Zh qg/ vhh +86,/ fg @ qg 4 @3 wui j @ qg y= Wkh uhvxowv fdq eh prglhg li iru lqvwdqfh d vhdvrqdo gxpp| lv lqfoxghg/ lq zklfk fdvh wuiP j lv qrw frqvwdqw exw rvfloodwhv1. 714. Whvw iru vlpsoh k|srwkhvlv rq wkh frlqwhjudwlqj vsdfh. Wkh vlpsoh k|srwkhvlv rq wkh frlqwhjudwlqj vsdfh lv irupxodwhg dv @ K!> zkhuh K+q u, lv nqrzq dqg !+u u, lv xqnqrzq1 Fruroodu| 8 Lq wkh DU+n, prgho zlwk srzhu ghwhuplqlvwlf whup gw wkh fruuhfwlrq whup wr wkh olnholkrrg udwlr whvw vwdwlvwlf iru d vlpsoh k|srwkhvlv rq wkh frlqwhjudwlqj vsdfh fdq eh h{suhvvhg lq whupv ri wkh frqvwdqwv f dqg y> vhh +87, dqg +85,= H ^5 orj OUm3B %` u+qu, 4 @ 4 . W4 ^+qg . nq, . 45 +q . 4,` . W4u ^+q u . 5qg 4,y . 5f`= Surri1 Wklv iroorzv iurp Wkhruhp 6 zlwk wkh ydoxhv qx @ u> qe @ qd @ q u> q| @ u . +n 4,q> dqg xvlqj wkh uhvxow wkdw fg @ qg y1 Dv d sduwlfxoduo| vlpsoh fdvh zklfk lv frqyhqlhqw iru wkh vlpxodwlrq zh frqvlghu Fruroodu| 9 Iru wkh dxwruhjuhvvlyh prgho lq q glphqvlrqv zlwk 4 odj dqg u @ 4 dqg frqvwdqw whup zh qg wkh fruuhfwlrq idfwru H^5 orj OUm3B %` 4 4 6+q . 4, 3 ^+5 . 3 ,q . 7+4 . 3 ,` @4. ^ `= +q 4, W 5 3 3 4 . Li wkh frqvwdqw lv devhqw zh kdyh. H^5 orj OUm3B %` 4 4 6q . 4 3 ^+5 . 3 ,+q 5, . 7+4 . 3 ,` @4. ^ `= +q 4, W 5 3 3 4 . Surri1 Lq wkh vshfldo fdvh ri dq dxwruhjuhvvlyh prgho zlwk 4 odj dqg u @ 4 zh qg wkdw S @ 4 . 3 dqg P @ 4 ru 3/ dqg wkdw y@. 3 +5 . 3 , 3 +4 . 3 , > f @ 5 = 3 3 4 3 3 4 . Wkh uhvxow wkhq iroorzv iurp Fruroodu| 81 49. +88,.

(20) 715. Whvw iru olqhdu uhvwulfwlrqv rq q. Zh frqvlghu khuh k|srwkhvhv ri wkh irup @ K!> zkhuh K+q v, lv nqrzq dqg !+v u, lv xqnqrzq1 Fdofxodwlrq ri wkh h{shfwdwlrq ri wkh orj0olnholkrrg udwlr vwdwlvwlf lv vlpso| shuiruphg e| dsso|lqj Sursrvlwlrq 5 wzlfh1 Wkh h{shfwdwlrq ri wkh vhfrqg whup lv jlyhq e| Fruroodu| 8 dqg wkh h{shfwdwlrq ri wkh uvw fdq eh irxqg e| uhsodflqj +q u, e| +v u, dv dujxhg lq vxevhfwlrq 516= Zh wkhq qg Fruroodu| : Wkh fruuhfwlrq idfwru iru wkh k|srwkhvlv @ K! lq wkh yhfwru dxwruhjuhvvlyh prgho zlwk srzhu ghwhuplqvwlf whupv lv jlyhq e|. H ^5 orj OUm3B %` u+qv, 4 @ 4 . W4 ^+qg . nq, . 45 +q . 4 . v u,` . W4u ^+q 5u . v . 5qg 4,y . 5f`. 8. D vlpxodwlrq h{shulphqw. Zh vlpxodwh wkh glvwulexwlrq ri wkh whvw vwdwlvwlf iru dq qglphqvlrqdo dxwruhjuhvvlyh prgho zlwk rqh odj dqg rqh frlqwhjudwlqj uhodwlrq dqg frqvwdqw whup lq rughu wr lqyhvwljdwh li wkh fruuhfwlrq idfwru lpsuryhv wkh qlwh vdpsoh dssur{lpdwlrq1 Zkhq vlpxodwlqj wkh glvwulexwlrq ri wkh whvw vwdwlvwlfv lq wkh prgho [w @ 3 [w4 . . %w >. +89,. zh kdyh wr vshfli| wkh ydoxh ri wkh sdudphwhuv +> > > ,> dqg wkh lqlwldo ydoxhv/ jlylqj d wrwdo ri q . +q 4, . q . 45 q+q . 4, sdudphwhuv= Zh fdq/ krzhyhu/ xvh wkh lqyduldqfh ri wkh vwdwlvwlf wr uhgxfh wkh qxpehu ri sdudphwhuv qhfhvvdu| wr vshfli|1 Iru doo q q pdwulfhv O ri ixoo udqn/ wkh wudqvirupdwlrq \ @ O[ ohdyhv wkh vwdwlvwlf lqyduldqw dqg fruuhvsrqgv wr d fkdqjh ri wkh sdudphwhuv lqwr +O> O43 > O O3 > O,1 Zh fdq 4 uhgxfh wkh qxpehu ri sdudphwhuv lq wkh vlpxodwlrq h{shulphqw/ e| uvw fkrrvlqj O @ 5 4 vxfk wkdw %w @ 5 %w duh l1l1g1 Qq +3> Lq ,= Zh fdq vwloo wudqvirup e| ruwkrjrqdo wudqvirupdwlrqv zlwkrxw fkdqjlqj wkh lqghshqghqfh ri wkh huuruv/ dqg vlqfh wkhuh lv qr orvv ri jhqhudolw| lq dvvxplqj wkdw 3 @ 4> zh fdq urwdwh wkh frruglqdwh v|vwhp vxfk wkdw @ +4> 3> = = = > 3,3 5 Uq dqg qdoo| zh urwdwh wkh frruglqdwhv +5> = = = > q, vxfk wkdw @ +> h34 ,3 zkhuh h4 @ +4> 3> = = = > 3,3 5 Uq4 = Ilqdoo| zh qrwh wkdw wkh ydoxhv dqg jlyh wkh vdph ydoxh ri wkh whvw vwdwlvwlfv dqg wkh fruuhfwlrq1 Zh wkhuhiruh fkrrvh erwk dqg qrq0srvlwlyh1 Ilqdoo| e| d ixuwkhu urwdwlrq zh fdq uhgxfh wkh qxpehu ri sdudphwhuv lq wr wkuhh1 Wkxv rqo| yh sdudphwhuv qhhg wr eh fkrvhq lq wkh vlpxodwlrq h{shulphqw zkhq jhqhudwlqj wkh gdwd xqghu wkh qxoo k|srwkhvlv1 D pruh irupdo zd| ri pdnlqj wklv wudqvirupdwlrq lq prgho +89, lv wr fkrrvh yhfwruv 4 y4 @ + 3 , 5 > 4 y5 @ + 4 + 3 ,4 3 , 3 4 3 + 3 ,4 3 5 > dqg y6 > = = = > yq vxfk wkdw y 3 ym @. l. . 4 l@m = 3 l@ 9 m. 4:.

(21) Iru wkh qhz yduldeohv \w @ y 3 [w > wkh frqvwdqw whup lv @ y 3 > dqg wkh huuruv duh %w @ y 3 %w > zklfk duh l1l1g1 Qq +3> Lq ,= Zh jhw wkh htxdwlrqv \4w @ \4w4 . 4 . %4w \5w @ \4w4 . 5 . %5w \lw @ l . %lw > l @ 6> = = = > q> zkhuh @ + 3 , 5 3 + 3 , 5 4 4 @ 3 4 3 + 3 ,4 3 5 + 3 , 5 = 4. . 4. Lq wkh vlpxodwlrqv zh frqvwuxfw wkh vwdwlrqdu| surfhvv 3 [w e| vwduwlqj lw dw 3/ dqg glvfduglqj wkh uvw 533 revhuydwlrqv1 Wkh udqgrp zdon 3B [w lv vwduwhg zlwk lqlwldo ydoxh 3B [3 @ 3= Ilqdoo| wkh surfhvv lv frqvwuxfwhg dv [w @ + 3 ,4 3 [w . B +3B B ,4 3B [w > w @ 4> = = = > W= Zh wkxv zdqw wr whvw 5 @ @ q @ 3 lq wkh prgho [4w [5w. @ +[4w4 . 5 [5w4 . . q [qw4 , . 4 . %4w > @ +[4w4 . 5 [5w4 . . q [qw4 , . 5 . %5w >. [lw. @ l . %lw > l @ 6> = = = > q>. zkhuh %w duh l1l1g1 Qq +3> Lq ,= Wkh whvw lv dv|pswrwlfdoo| glvwulexwhg dv "5 +q 4,> dqg xqghu wkh qxoo k|srwkhvlv zh kdyh wkh sdudphwhuv 4 @ >@ > > @ Lq = h4 3 Xqghu wkh qxoo k|srwkhvlv wkh v|vwhp jhqhudwhv L+4, yduldeohv zlwk rqh frlqwhjudwlqj yhfwru li 5 ? ? 31 Li @ 3> @ 3 wkhq zh jhw dq L+4, surfhvv zlwk qr frlqwhjudwlrq dqg qdoo| li @ 3 dqg 9@ 3 wkhq wkh v|vwhp zloo jhqhudwh dq L+5, surfhvv1 Wkh fruuhfwlrq idfwru lv jlyhq lq +:, zlwk u @ 4> n @ 4> qg @ 4> dqg +5 . , +4 . , y@ 5 > f @ 5 > . 5 5 . 5 vhh +88,/ dqg lv jlyhq e| 4 EI @ 4 . W. . 6+q . 4, ^+5 . ,q . 7+4 . ,` 5 5 . 5. =. Qrwh wkh vshfldo fdvh ri @ 3> zkhuh wkh fruuhfwlrq idfwru kdv d sroh dw @ 3 fruuh0 vsrqglqj wr dq h{wud xqlw urrw lq wkh surfhvv1 Dw wklv erxqgdu| wkh olplw glvwulexwlrq lv qr orqjhu "5 dqg d glhuhqw w|sh ri fruuhfwlrq frxog eh fdofxodwhg xqghu wkh orfdo dowhuqdwlyh wkdw $ 3> W $ 4> dqg W lv {hg1 Qrwh dovr wkdw li 9@ 3> dqg @ 3 wkhq wkhuh lv qr vlqjxodulw| lq wkh h{suhvvlrq hyhq wkrxjk wkh surfhvv lv L+5,= 4;.

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(28) Iru l @ m 4 / l 4 y @ n zh qg [[. wuiH^Xl Xl34 3 Xl d3l. dl5y Xl3y4 `j. l y> [ @ qd wui 3 y j @ qd wui 3 j y>. Wkxv zh qg zlwk irxu vwrfkdvwlf whupv wkh frqwulexwlrq qd +qx . 4,wui3 j. +95,. Qh{w frqvlghu wkh whup zlwk mxvw wzr vwrfkdvwlf whupv [[ wuiH^Xl e3l4 # 3 # em 4 d3m 4 dn4 Xn3 `j= l>m>n y> Zh jhw d frqwulexwlrq rqo| li n @ l dqg qg [[ wuie3l4 # 3 # em 4 d3m 4 dl4 j qx l>m y> $ qx wui# 3 #Led Lde j>. +96,. zkhuh zh kdyh xvhg +63,1 Wklv suryhv +8;,1. 4 Vg| V|d Vdx m3 %`j @3 qd S4 wuiP .4 jwui 3 j +8<,= wuiH^Vxg Vgg B @3 4 Vg| V|d Vdx m3 %`j wuiH^Vxg Vgg B [ [ 4 3 @ wuiH^Xl gl Vgg gv +Xv3 4y 3y . e3v4y #3y , l>m>n>v > + Xm 4 . # em 4 ,d3m 4 dn4 Xn3 `j= Djdlq zh rqo| kdyh wr frqvlghu whupv zlwk irxu ru wzr vwrfkdvwlf idfwruv1 Zlwk irxu zh qg. @ . [ [. l>m>n>v > [ [ l>m>n>v >. 3 4 gv X 3 3 3 wuiH^Xl g3l Vgg v4y y ,+ Xm 4 ,dm 4 dn4 Xn `j 4 gv.4.y X 3 3 Xm d3 dn4 X 3 `j= wuiH^Xl g3l Vgg v y m . n. E| Ohppd ; zh qg wkdw wklv htxdov S S 3 4 3 3 4 3 > l>m +qx gl Vgg gm .4.y dm . dl4 . gl Vgg gm .4.y dl. dm 4 4 gl.4.y d3 dm 4 ,wui3 j .g3l Vgg m . S 4 P .4 gl jwui3 j @ qd wuig3l Vgg S @ qd wuiP .4 jwui3 j . vlqfh. S 3 l dl4 gl @ 31 Wkh whup zlwk wzr vwrfkdvwlf idfwruv lv 58. +97,.

(29) . [ [. 3 4 gv e3 3 3 wuiH^Xl g3l Vgg v4y # y # em 4 dm 4 dn4 Xn `j. l>m>n>v > [[ 3 4 gv jwuie3 3 wuig3l Vgg @ qx v4y # y # em 4 dm 4 dl4 j $ 3> l>m>v > vlqfh. S 3 l dl4 gl @ 3=. 4 Ve|=g V|d Vdx m3 %`j @3 3 +93,= wuiH^Vxe=g Vee=g B. 4 Ve|=g e| Vxe V 4 Ve| > Wklv whup lv ri wkh rughu ri d frqvwdqw vr zh fdq uhsodfh Vxe=g Vee=g ee dqg W 4 Vee e| Lqe = 4 Ve|=g V|d Vdx m3 %`j @3 W 4 wuiH^Vxe Ve| V|d Vdx m3 %`j wuiH^Vxe=g Vee=g B B [ [ 3 3 4 3 3 3 @ W wuiH^Xl el ev +Xv4y y . ev4y # y ,+ Xm 4 . # em 4 ,d3m 4 dn4 Xn3 `j= l>m>n>v > Djdlq zh rqo| kdyh wr frqvlghu whupv zlwk irxu ru wzr vwrfkdvwlf idfwruv1 Zlwk irxu zh qg 4 [ [ wuiH^Xl e3l ev +Xv3 4y 3y ,+ Xm 4 ,d3m 4 dn4 Xn3 `j> W v>l>m>n > 4 [ [ wuiH^Xl e3l ev.4.y Xv3 3y Xm d3m . dn4 Xn3 `j> W v>l>m>n > . zklfk e| Ohppd ; htxdov. $ 3>. 4[ wuiqx e3l em . .4 d3m . dl4 . e3l el.4. d3m . dm 4 . e3l em . .4 d3l. dm 4 jwui 3y j W l>m. S vlqfh Lde lv qlwh dqg W4 l e3l el.4. $ 3= Zlwk wzr vwrfkdvwlf whupv zh qg 4 [ [ wuiH^Xl e3l ev e3v4y # 3y # em 4 d3m 4 dn4 Xn3 `j W l>m>n>v > u [[ wuie3l ev e3v4y # 3y # em 4 d3m 4 dl4 j $ 3> @ W l>m>v > @ . vlqfh +93,1. S. 3 l dl4 el @ 3> vlqfh wkh uhjuhvvruv duh ruwkrjrqdo1 Khqfh wkhuh lv qr frqwulexwlrq iurp. +94, = wuiH^Vx| +W 4 V|| H^W 4 V|| m3B %`,,V|d Vdx m3B %`j S S 3 @ qd ^ wui+ . 4,3 j . wui+ . 4,jwui3 j` 59.

(30) wuiH^Vx| +W 4 V|| H^W 4 V|| m3B %`,,V|d Vdx m3B %`j [ @ W 4 wuiH^Vx| + +\l4 \l34 H^\l4 \l34 m3B %`,,V|d Vdx m3B %`j l [ [ 4 wuiH^Xl +Xl34 3 . e3l4 #3 , @ W l>m>n>o >>> + y +Xoy Xo3 Lu , 3 . y Xoy e3o # 3 . # eo Xo3 3 , + Xm 4 . # em 4 ,d3m 4 dn4 Xn3 `j=. Wkhuh duh wkuhh sduhqwkhvhv lqyroylqj d vwrfkdvwlf whup1 Ohw +d> e> f, ghqrwh wkh qxpehu ri vwrfkdvwlf frpsrqhqwv iurp hdfk sduhqwkhvlv1 Zh kdyh wr lqyhvwljdwh wkh frpelqdwlrqv +3> 4> 4,> +4> 4> 3,> +3> 5> 3,> dqg +4> 5> 4,> zlwk dq hyhq qxpehu ri vwrfkdvwlf idfwruv1 Wkdw jlyhv wkh vl{ frqwulexwlrqv wr wkh vxp vlqfh wkh idfwru iurp wkh plggoh sduhqwkhvlv fdq eh fkrvhq lq wzr zd|v +3> 4> 4,d +3> 4> 4,e. [ [. l>m>n>o >>> [ [. Xl +e3l4 # 3 ,+ y Xo e3o # 3 ,+ Xm 4 ,d3m 4 dn4 Xn3 Xl +e3l4 #3 ,+# eo Xo3 3 ,+ Xm 4 ,d3m 4 dn4 Xn3. l>m>n>o >>> [ [ +4> 5> 4, Xl +Xl34 3 , y +Xo Xo3 > Lu , 3 + Xm 4 ,d3m 4 dn4 Xn3 l>m>n>o >>> [ [ +4> 4> 3,d Xl +Xl34 3 ,+# eo Xo3 3 ,+# em 4 ,d3m 4 dn4 Xn3 l>m>n>o >>> [ [ e +4> 4> 3, Xl +Xl34 3 ,+y Xo e3o #3 ,+# em 4 ,d3m 4 dn4 Xn3 l>m>n>o >>> [ [ +3> 5> 3, Xl +e3l4 # 3 ,y +Xo Xo3 > Lu , 3 +# em 4 ,d3m 4 dn4 Xn3 l>m>n>o >>> Zh wdnh wkh whupv rqh e| rqh +3> 4> 4,d = W 4. [ [. l>m>n>o >>>. wuiHiXl +e3l4 # 3 ,+ Xo e3o # 3 ,+ Xm 4 ,d3m 4 dn4 Xn3 `j=. Xvlqj wkh uhvxow lq Ohppd ; rqh fdq vkrz wkdw wkh whup htxdov [ [ [ wui3 # + el4 e3l. ,# 3 jwui d3m 4 dm 5 j W 4 l m >>> [ [ [ 3 4 3 qx wui # + el4 dl4 ,+ dm . e3m ,#3 j W l m >>> [ [ [ 3 4 3 wui # + em dm 4 ,+ dl. e3l4 ,# 3 j W m l >>> [ 3 3 wui # # ..4 j> $ qd > 5:. +98,.

(31) S S vlqfh W 4 l el4 e3l. $ Lqu li @ . . 4 dqg wui m d3m 4 dm 5 j $ wu+Vdd , @ qd = Wkh odvw wzr whupv jlyhv qr frqwulexwlrq vlqfh Lde lv qlwh1 +3> 4> 4,e = W 4. [ [. l>m>n>o >>>. wuiH^Xl +e3l4 # 3 ,+# eo Xo3 3 ,+ Xm 4 ,d3m 4 dn4 Xn3 `j. D vlplodu fdofxodwlrq vkrzv wkdw zh jhw d frqwulexwlrq iru ^l @ o 9@ m 4 @ n` W 4. [ [. l>m>n>o >>>. wuiH^Xl e3l4 # 3 # el . Xl3 3 Xn d3n. dn4 Xn3 `j. [ [ [ wui+ el . e3l4 ,# 3 # jwui3 jwui d3n. dn4 j @ W 4. >> l n [ 3 3 $ qd wui# #..4 jwui j> +99, > S S vlqfh W 4 l el . e3l4 $ Lqu li @ . . 4> dqg wui n d3n. dn4 j $ wuiVdd j @ qd = +4> 5> 4, =. [ [. l>m>n>o >>>. wuiH^Xl +Xl34 3 , y +Xo Xo3 > Lu , 3 + Xm 4 ,d3m 4 dn4 Xn3 `j. Zh rqo| jhw d frqwulexwlrq iru ^l @ o > l 4 @ o > m 4 @ n` zklfk lpsolhv wkdw ^ @ . . 4` dqg jlyhv W 4. [[. l>m >>. wuiH^Xl Xl34 3 y Xl Xl34 3 ..4 Xm d3m 4 dm 5 Xm3 4 `j. [[ wui3 3 ..4 d3m 4 dm 5 j @ W 4. l>m >> [ 3 3 ..4 j= $ qd wui >. +9:,. Ilqdoo| zh dovr jhw iru ^l @ o > l 4 @ o > m 4 @ n` zklfk lpsolhv wkdw ^ @ . . 4` dqg wkh frqwulexwlrq [ 3 ..4 jwui3 j= qd wui >. +9;,. Zh qh{w froohfw wkh frqwulexwlrqv iurp +98,/ +99,/ +9:,/ dqg +9;, dqg qg d frqwulexwlrq wr wkh h{shfwdwlrq [ [ qd wui+# # 3 ..4 . 3 ..4 , 3 j @ qd wui+ . 4, 3 j> +9<, > 5;.

(32) qd. [. >. wui# # 3 ..4 . 3 ..4 jwui 3 j @ qd. [. . wui+ . 4,jwui 3 j=. +:3,. Wklv frpsohwhv wkh h{suhvvlrq iru W wuiH^Q 3 Dm3B %`j> jlyhq e| +95,/ +96,/ +97,/ +9<,/ dqg +:3, vlqfh wkh uhpdlqlqj whupv jlyh qr frqwulexwlrq/ dv zh vkdoo qrz vkrz= +4> 4> 3,d = W 4. [ [. l>m>n>o >>>. wuiH^Xl +Xl34 3 ,+# eo Xo3 3 ,+# em 4 ,d3m 4 dn4 Xn3 `j. Zh qg d frqwulexwlrq rqo| iru n dqg o olqnhg wr l> zklfk ohdyhv wzr vxppdwlrqv1 Erwk jlyh vrphwklqj erxqghg vlqfh Lde lv qlwh/ khqfh wkh olplw lv }hur1 +4> 4> 3,e = W 4. [ [. l>m>n>o >>>. wuiH^Xl +Xl34 3 ,+y Xo e3o #3 ,+# em 4 ,d3m 4 dn4 Xn3 `j=. Wkh vdph dujxphqw dssolhv dv iru +4> 4> 3,d +3> 5> 3, = W 4. [ [. l>m>n>o >>>. wuiH^Xl e3l4 # 3 y +Xo Xo3 > Lu , 3 # em .4. d3m 4 dn4 Xn3 `j=. Rqo| li ^l @ n @ o @ o ` fdq zh jhw d frqwulexwlrq dqg wkhq wkh vxp ryhu l dqg m duh erwk qlwh= Wkxv wkh olplw lv }hur1. <17. Surri ri +73,=. Qh{w zh suryh 3 wuiH^D3 Dm3B %`j @ qd qx wui3 j . qx wuiLed Lde # 3 #j=. Zh kdyh wkh h{suhvvlrq wuiH^D3 Dm3B %`j @ W 4 wuiH^+Vx|=e>g ,+W 4 V||=e>g ,4 V|d Vd| +W 4 V||=e>g ,4 +V|x=e>g ,m3B %`j=. Ehfdxvh ri wkh idfwru W 4 zh fdq uhsodfh wkurxjkrxw wkh pdwul{ +Vx|=e>g ,+W 4 V||=e>g ,4 e| +Vx|=e ,+W 4 V||=e ,4 dqg qg wkh h{suhvvlrq wuiH^D3 Dm3B %`j 3 @ W 4 wuiH^Vx| V|d Vd| V|x m3B %`j [ [ @ W 4 wuiH^Xl +Xl34 3 . e3l4 # 3 ,+ Xm 4 . # em 4 , l>m>n>o >>> 3 dm 4 dn4 +Xn3 4 3 . e3n4 # 3 ,+ Xo4 . # eo4 ,Xo3 `j= 5<. +:4,.

(33) Wkhuh duh khuh irxu sduhqwkhvhv dqg zh fdq fkrrvh d vwrfkdvwlf ru d ghwhuplqlvwlf whup iurp hdfk1 Zh dsso| djdlq wkh qrwdwlrq +d> e> f> g, wr lqglfdwh wkh qxpehu ri vwrfkdvwlf frpsrqhqwv lq hdfk vxp ehorz1 Zh qg wkdw zh vkdoo lqyhvwljdwh wkh frpelqdwlrqv +4> 4> 4> 4,> +4> 4> 3> 3,> +4> 3> 4> 3,> +4> 3> 3> 4,> +3> 4> 4> 3,> +3> 4> 3> 4,> +3> 3> 4> 4,> +3> 3> 3> 3, dqg qg wkh whupv [ [ +4> 4> 4> 4, wuiH^Xl +Xl34 3 ,+ Xm 4 ,d3m 4 dn4 +Xn3 4 3 ,+ Xo4 ,Xo3 `j l>m>n>o >>> [ [ wuiH^Xl +Xl34 3 ,+# em 4 ,d3m 4 dn4 +e3n4 # 3 ,+ Xo4 ,Xo3 `j +4> 3> 3> 4, l>m>n>o >>> [ [ wuiH^Xl +e3l4 # 3 ,+ Xm 4 ,d3m 4 dn4 +Xn3 4 3 ,+# eo4 ,Xo3 `j +3> 4> 4> 3, l>m>n>o >>> [ [ wuiH^Xl +e3l4 # 3 ,+# em 4 ,d3m 4 dn4 +e3n4 # 3 ,+# eo4 ,Xo3 `j +3> 3> 3> 3, l>m>n>o >>> [ [ wuiH^Xl +Xl34 3 ,+ Xm 4 ,d3m 4 dn4 +e3n4 #3 ,+# eo4 ,Xo3 `j +4> 4> 3> 3, l>m>n>o >>> [ [ wuiH^Xl +Xl34 3 ,+# em 4 ,d3m 4 dn4 +Xn3 4 3 ,+# eo4 ,Xo3 `j +4> 3> 4> 3, l>m>n>o >>> [ [ wuiH^Xl +e3l4 # 3 ,+ Xm 4 ,d3m 4 dn4 +e3n4 # 3 ,+ Xo4 ,Xo3 `j +3> 4> 3> 4, l>m>n>o >>> [ [ wuiH^Xl +e3l4 # 3 ,+# em 4 ,d3m 4 dn4 +Xn3 4 3 ,+ Xo4 ,Xo3 `j +3> 3> 4> 4, l>m>n>o >>> Zh wdnh wkh whupv rqh e| rqh +4> 4> 4> 4, = W 4. [ [. l>m>n>o >>>. wuiH^Xl +Xl34 3 ,+ Xm 4 ,d3m 4 dn4 +Xn3 4 3 ,+ Xo4 ,Xo3 `. Iru ^l @ o> @ > m 4 @ n 4 ` zh qg wkh frqwulexwlrq [ qx qd wui+ 3 , 3 j= +4> 3> 3> 4, = W 4. [ [. l>m>n>o >>>. +:5,. wuiH^Xl +Xl34 3 ,+# em 4 ,d3m 4 dn4 +e3n4 # 3 ,+ Xo4 ,Xo3 `. Iru ^o @ l> @ ` zh jhw wkh olplw [. qx wui+ +3> 4> 4> 3, = W 4. [ [. l>m>n>o >>>. . 3 ,#Led Lde #3 j. +:6,. wuiH^Xl +e3l4 #3 ,+ Xm 4 ,d3m 4 dn4 +Xn3 4 3 ,+# eo4 ,Xo3 `. 63.

(34) ^l @ o> m 4 @ n 4 ` jlyhv [ # # 3 , 3 j qx qd wui+ +3> 3> 3> 3, = W 4. [ [. l>m>n>o >>>. +:7,. wuiH^Xl +e3l4 # 3 ,+# em 4 ,d3m 4 dn4 +e3n4 #3 ,+# eo4 ,Xo3 ``. ^l @ o` jlyhv [. qx wui#Led Lde # 3 +. . # # 3 ,j=. Ilqdoo| zh qrwh wkdw iurp +:5, dqg +:7, zh qg d frqwulexwlrq wr +:4, [ [ 3 ,3 j . qx qd wui+ # # 3 ,3 j @ qx qd wui3 j> qx qd wui+ S vlqfh + 3 . # #3 , @ Lq| > vhh +8:,1 Frpelqlqj +:6, dqg +:8, zh qg wkdw d frqwulexwlrq wr +:4, lv [ [ qx wui#Led Lde # 3 + # # 3 ,j . qx wui+ 3 ,#Led Lde # 3 j @ qx wui#Led Lde #3 j= . +:8,. +:9,. +::,. Wkh h{suhvvlrqv +:9, dqg +::, jlyh wkh qdo uhvxow iru wuiH^D3 Dm3B %`j vlqfh wkh uh0 pdlqlqj whupv whqg wr }hur/ dv zh vkdoo qrz vhh= +4> 4> 3> 3, = W 4. [ [. l>m>n>o >>>. wuiH^Xl +Xl34 3 ,+ Xm 4 ,d3m 4 dn4 +e3n4 # 3 ,+# eo4 ,Xo3 `=. Iru wklv wr jlyh d qrq0}hur ydoxh wkh lqglfhv l> m> o duh olqnhg1 Wkh vxppdwlrq ryhu l/ dqg n duh erxqghg ehfdxvh Lde lv qlwh/ khqfh wkh olplw lv }hur1 H{dfwo| wkh vdph dujxphqw fdq eh dssolhg iru +4> 3> 4> 3,> +3> 4> 3> 4,> +3> 3> 4> 4,=. <18. Surri ri +74,=. Zh wkhq suryh 3 wuiH^Q 3 EQ m3B %`j @ qd +qd . qx . 4,wui 3 j . qx wuiLed Lde # 3 #j=. Iurp wkh h{suhvvlrq wuiH^Q 3 EQ m3B %`j @ wuiH^Vxd Vd| +W 4 V||=e>4 ,4 V|d Vdx m3B %`j> 64.

(35) zh vhh wkdw zh fdq uhsodfh W 4 V||=e>4 e| Lq| dqg qg 3 wuiH^Q 3 EQm3B %`j @ wuiH^Vxd Vd| V|d Vdx m3B %`j [ [ @ wuiH^Xl d3l4 dm 4 +Xm3 4y 3y . e3m 4y # 3y ,+ Xn4 . # en4 ,d3n4 do4 Xo3 `j l>m>n>o y>. Wkh frqwulexwlrq iurp wkh whup zlwk wkuhh X*v zloo eh }hur dqg zkdw uhpdlqv duh wkh wzr whupv [ [. wuiH^Xl d3l4 dm 4 Xm34y 3y Xn4 d3n4 do4 Xo3 `j. l>m>n>o y> [ [ wuiH^Xl d3l4 dm 4 e3m 4y #3y # en4 d3n4 do4 Xo3 `j= . l>m>n>o y> Wkh uvw lv hydoxdwhg e| Ohppd ; dqg zh qg [ [. @. wuiH^Xl d3l4 dm 4 Xm3 4y 3y Xn4 d3n4 do4 Xo3 `j. l>m>n>o y> [[ d3l4 dm 4 d3m 4 . dl4 wuiLqx jwui3y n j l>m y> [[ [[ d3l4 dm 4 d3l. dm 5 wui 3y n j . d3l4 dl.y d3o. do4 wui3y n j . l>m y> l>o y>. $ wui3 j+qx wuiVdd Vdd j . wuiVdd Vdd j . wuiVdd jwuiVdd j, @ wui3 j+qx qd . qd . q5d , @ qd +qx . qd . 4,wui3 j=. +:;,. Wkh whup zlwk wzr ghwhuplqlvwlf idfwruv ehfrphv [ [. wuiH^Xl d3l4 dm 4 e3m 4y #3y # en4 d3n4 do4 Xo3 `j. l>m>n>o y> [[ wuid3l4 dm 4 e3m 4 # 3y # en4 d3n4 dl4 j @ qx l>m>n y> $ qx wuiLed Lde # 3 #j= Wkh uhvxow iru wuiH^Q 3 EQm3B %`j lv frqwdlqhg lq +:;, dqg +:<,1. <19. Surri ri +75,=. Wr suryh 4 3 W 5 wuiH^G4 Q 3 Q m3B %`j @ qx qd +qx . 4,. 65. +:<,.

(36) zh qrwh wkdw W 5 wuiH^G4 Q 3 Qm3B %`j 4. @ wuiH^+Vxx W Lqx ,Vxd Vdx m3B %`j [ wuiH^+Xm Xm3 Lqx ,Xn d3n4 do4 Xo3 `j @ m>n>o [ wuiH^+Xm Xm3 Lqx ,Xm d3m 4 dm 4 Xm3 `j @ m @ wuiVdd jwuiH^+Xm Xm3 Lqx ,Xm Xm3 `j @ qx qd +qx . 4,=. <1:. Surri ri +78,=. Qh{w frqvlghu 3 wuiH^G45 Q 3 Qm3B %`j @ qx qd +qx . 4,=. Zh qg wuiH^G45 Q 3 Q m3B %`j @ W 4 wuiH^+Vxx W Lqx ,+Vxx W Lqx ,Vxd Vdx m3B %`j [ wuiH^+Xl Xl3 Lqx ,+Xm Xm3 Lqx ,Xn d3n4 do4 Xo3 `j= @ W 4 l>m>n>o Khuh zh wdnh ^n @ o` vlqfh rwkhuzlvh wkh dn4 zloo vxp wr }hur1 Zh wkhq kdyh wr wdnh ^l @ m` wr jhw d qrq0}hur frqwulexwlrq dqg qg W 4. [. wuiH^+Xl Xl3 Lqx ,+Xl Xl3 Lqx ,Xn d3n4 dn4 Xn3 `j. l>n @ wuiVdd jwuiH^+Xl Xl3 Lqx ,+Xl Xl3 Lqx ,j @ qx qd +qx . 4,=. <1;. Surri ri +76,=. Wr suryh 3 wuiH^G5 Q 3 Qm3B %`j @ qx qd +qg . q| . qe ,. zh hydoxdwh dv iroorzv. wuiH^G5 Q 3 Qm3B %`j 4 Vgx . Vxe=g V 4 Vex=g . Vx|=e>g V 4 V|x=e>g ,Vxd Vdx m3 %`j @ wuiH^+Vxg Vgg B ee=g ||=e>g 66.

(37) 41. 4 Vgx Vxd Vdx m3 %`j wuiH^Vxg Vgg B [ 4 3 3 @ wuiH^Xl gl Vgg gm Xm Xn d3n4 do4 Xo3 `j l>m>n>o @ qx qd qg > e| Ohppd ;1 51 wuiH^Vxe3 Ve34e3 Ve3 x Vxd Vdx `j 3 @ W 4 wuiH^Vxe Vex Vxd Vdx `j [ wuiH^Xl e3l em Xm3 Xn d3n4 do4 Xo3 `j @ W 4 l>m>n>o @ qx qd qe > e| Ohppd ;1. 4 V|x=e>g e| Vx| V 4 V|x @3 W 4 Vx| V|x = 61 Lq wkh odvw h{suhvvlrq zh uhsodfh Vx|=e>g V||=e>g ||. @ @. @ @. W 4 wuiH^Vx| V|x Vxd Vdx `j [ W 4 wuiH^Xl \l34 \m 4 Xm3 Xv d3v4 dw4 Xw3 `j l>m>v>w [ [ 4 wuiH^Xl +Xl34 3 ,+ Xm 4 ,Xm3 Xv d3v4 dw4 Xw3 `j W l>m>v>w > [ [ wuiH^Xl +e3l4 # 3 ,+# em 4 ,Xm3 Xv d3v4 dw4 Xw3 `j .W 4 l>m>v>w > [ [ qx wui 3 jwuiVdd j . qx wui# 3 # jwuiVdd j [ 3 +wui j . wui# # 3 j, @ qx wuiVdd jwuiLq| j @ qx qd q| = qx wuiVdd j . Qrwh wkdw wkh odj ohqjwk dsshduv khuh lq wkh irup ri wkh pdwul{ Lq| = Dgglqj wkhvh frqwulexwlrqv zh qg qx qd qg . qx qd qe . qx qd q| @ qx qd +qg . qe . q| ,=. <1<. Surri ri +77,=. Zh qdoo| suryh wuiG4 Q 3 Dm3B %`j= Zh kdyh wuiG4 Q 3 Dm3B %`j @ wuiH^+W 4 Vxx Lqx ,Vx|=e>4 +W 4 V||=e=4 ,4 V|d Vdx m3B %`j 67.

(38) Zh uhsodfh Vx|=e>4 e| Vx| dqg W 4 V||=e=4 e| Lq| dqg qg wuiH^+W 4 Vxx Lqx ,Vx| V|d Vdx m3B %`j @ W 4 wuiH^+Vxx W LLqx ,Vx| V|d Vdx m3B %`j [ @ W 4 wuiH^+Xl Xl3 Lqx ,Xm \m34 \n4 d3n4 do4 Xo3 `j= l>m>n>o Zh qg wkh wzr frqwulexwlrqv W 4. [ [. wuiH^+Xl Xl3 Lu ,Xm +Xm3 4 3 ,+ Xn4 ,d3n4 do4 Xo3 `j. l>m>n>o > [ [. .W 4. l>m>n>o >. wuiH^+Xl Xl3 Lu ,Xm +e3m 4 #3 ,+# en4 ,d3n4 do4 Xo3 `j=. Lq wkh uvw whup zh pxvw kdyh wkh lqglfhv ^l> m> n> o` doo wlhg wr m> vlqfh m 9@ m 4 = Wklv ohdyhv rqh lqgh{ wr vxp dqg wkh vxp ri wkh d3n4 dn4 lv erxqghg1 Khqfh qr frqwulexwlrq= Lq wkh vhfrqg whup zh pxvw kdyh ^o @ m @ l`> zklfk ohdyhv d vxppdwlrq ryhu m dqg n1 Erwk ri wkhvh duh qlwh vlqfh Lde lv qlwh1 Wkxv wkh olplw lv }hur1 Wklv frpsohwhv wkh surri ri Sursrvlwlrq 41. 68.

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