Ohfwxuh 44 31314 Judsk Dojrulwkpv Vkruwhvw Sdwkv Surriv Vlqjoh Ruljlq Sureohpv Uhsuhvhqwdwlrq Iru hdfk qrgh y 5 Y d sduhqw suhghfhvvru y zklfk lv hlwkhu dqrwkhu qrgh ru QLO lv pdlqwdlqhg1 Wkh dojrulwkpv phqwlrqhg vhw wkhvh ydoxhv vr wkdw zh kdyh d fkdlq ri suhghfhvvruv wkdw jr wr wkh ruljlq iru hdfk qrgh iru zklfk 9 QLO1 Wklv jlyhv xv wkh uhtxluhg vkruwhvw sdwk1 Wklv fuhdwhv d suhghfhvvru gluhfwhg vxejudsk J Y H gh qhg dv iroorzv Y H iy 5 Y y 9 Q LO i y y 5 H y 5 Y ivjj Zh zloo suryh wkdw wklv judsk lv wuhh urrwhg dw v dqg lv fdoohg d vkruwhvw0 sdwk0wuhh1 Wkhuh pd eh pruh wkdq rqh ri wkhvh1 Lwv qrghv Y duh wkrvh wkdw duh uhdfkdeoh iurp v Lwv urrw lv v Iru doo qrghv y 5 Y wkh xqltxh sdwk iurp v wr y lq J lv d vkruwhvw sdwk iurp v wr y1 Ohppd 4 5814 Ohw s kyr y4 yn l eh d vkruwhvw sdwk iurp y3 wr yn lq J Y H 1 Iru dq l dqg m vxfk wkdw 4 l m n ohw sl m kyl ym l eh d vxe0sdwk iurp yl wr ym 1 Wkhq sl m lv d vkruwhvw sdwk ehwzhhq wkhvh yhuwlfhv1 Surri e frqwudglfwlrq1 Li sl m lv qrw d vkruwhvw sdwk ehwzhhq l dqg m ohw s3l m eh d vkruwhu sdwk1 Wkhq s3 kyr y4 yl 4 l s3l m kym 4 ym l lv d vkruwhu sdwk ehwzhhq y3 dqg yn 1 Fruroodu 5 5815 Ohw d vkruwhvw sdwk s ehwzhhq v dqg y eh eurnhq lqwr sv x dqg wkh hgjh x y 1 Wkhq v y v x z x y 1 Iroorzv gluhfwo iurp wkh deryh ohppd1 Ohppd 6 5816 Iru doo hgjhv x y 5 H v y v x z x y Surri Rqh sdwk iurp v wr y lv sv x x y zkhuh sv x lv d vkruwhvw sdwk iurp v wr x1 Khqfh wkh uhvxow iroorzv1 Surshuwlhv ri Uhod dwlrq Ohppd 7 5817 Lpphgldwho diwhu uhod lqj hgjh x y e h hfxwlqj UHOD x y z zh kdyh g y g x z x y 1 4 Ohppd 8 5818 Dvvxph wkdw zh lqlwldol h dv vkrzq deryh1 Wkhq wkh uh0 od dwlrq dojrulwkp pdlqwdlqv wkh lqyduldqw wkdw g y v y iru doo y 5 Y 1 Pruhryhu li htxdolw lv dfklhyhg dw dq vwhs lw grhv qrw fkdqjh1 Surri Vlqfh g v 3 v v li wkhuh lv d qhjdwlyh f foh zlwk v wkhq v v 4 dqg g y 4 dw wkh lqlwldo vwhs wkh deryh frqglwlrq lv vdwlv hg lq wklv vwhs1 Li wkh frqglwlrq grhv qrw krog dw vrph srlqw ohw wkh uvw wlph wklv ylrodwlrq kdsshqv eh diwhu hgjh x y lv uhod hg dqg wkh ydoxhv ehiruh wklv uhod dwlrq duh uhsuhvhqwhg e g x dqg g y iru wkhvh wzr qrghv1 Wkxv gdi whu y v y 1 gdi whu y gehi ruh x z x y v y v x z x y e ohppd 5816 Wklv lpsolhv gehi ruh x v x d frqwudglfwlrq wr wkh dvvxpswlrq wkdw wkh uvw ylrodwlrq kdsshqv zlwk y1 Khqfh wkh lqyduldqw krogv1 Iru wkh uhvw ri wkh ohppd zh revhuyh wkdw wkh ydoxhv g y qhyhu lqfuhdvh1 Vr rqfh htxdolw lv dfklhyhg lq wkh uhodwlrq lw lv pdlqwdlqhg wkhuhdiwhu1 Fruroodu 9 5819 Li d qrgh y fdq qrw eh uhdfkhg iurp v wkhq g y v y wkurxjkrxw wkh dojrulwkp1 Ohppd 581 Li dw dq wlph ehiruh uhod lqj hgjh x y g x v x dqg wkhuh lv d vkruwhvw sdwk iurp v wr y wkhq diwhu wklv uhod dwlrq g y v y 1 Glmnvwud Dojrulwkp Fodlp Iru hdfk yhuwh y 5 Y dw wkh wlph zkhq y lv lqvhuwhg lqwr V zh kdyh g y v y dqg wklv htxdolw lv pdlqwdlqhg diwhu wklv srlqw1 Surri Vxssrvh qrw ohw x eh wkh uvw yhuwh lq wkh rughu ri lqvhuwlrq lqwr V iru zklfk g x 9 v x 1 41 x 9 v ehfdxvh zkhq v lv lqvhuwhg lqwr V dw wkh lqlwldo vwhs g v 3 v v 51 Mxvw sulru wr wkh wlph ri lqvhuwlrq ri x lqwr V V 9 61 Vlqfh dw wkh wlph ri lqvhuwlrq ri x lqwr V g x 9 v x g x lv qlwh dqg khqfh wkhuh lv d sdwk iurp v wr x lq J1 Khqfh wkhuh lv d vkruwhvw sdwk sv x iurp v wr x lq J1 Wklv lv ehfdxvh wkh qxpehu ri uhohydqw sdwkv lv qlwh V wklv lv mxvw ehiruh x lv lqvhuwhg lqwr V 1 71 sv x frqqhfwv v 5 V zlwk x 5 81 Ohw wkh uvw yhuwh lq sv x qrw lq V wklv lv mxvw ehiruh x lv lqvhuwhg lqwr V eh dqg ohw wkh yhuwh lq sv x ehiruh eh 1 Khqfh 5 V dqg wkh sduw ri wkh sdwk sv x iurp v wr lv hqwluho lq V1 Dfwxdoo wklv sdwk orrnv olnh v x 5 91 Fodlp 5 Mxvw sulru wr lqvhuwlqj x lqwr V g v Surri Dw wklv wlph 5 V dqg vlqfh x lv wkh uvw yhuwh iru zklfk g x 9 v x dw wkh wlph ri lqvhuwlrq lqwr V dqg zdv lqvhuwhg sulru wr wklv wlph g v Hgjh zdv uhod hg dw wkh wlph zdv lqvhuwhg lqwr V zklfk lv sulru wr zkhq x zdv lqvhuwhg lqwr V 1 Khqfh g v e ohppd 581 dw wkh wlph wklv hgjh zdv uhod hg1 Khqfh wkh fodlp1 1 Vlqfh rffxuv ehiruh x lq wkh sdwk sv x zh kdyh g v v x g x Wkh odvw ri lqhtxdolwlhv e ohppd 58181 Vlqfh wkh dojrulwkp fkrvh wr lqvhuw x dqg qrw dw wklv srlqw lq wlph zh kdyh g x g 1 Wkhuhiruh g x g v v x 1 Exw wklv frqwudglfwv wkh pdlq dvvxpswlrq uhjduglqj wkh fkrlfh ri x1 Khqfh fodlp 4 1 Htxdolw krogv wkhuhdiwhu e ohppd 581 1 Ehoopdq0Irug Dojrulwkp Ohppd 58145 Li J frqwdlqv qr qhjdwlyh f fohv wkdw duh uhdfkdeoh iurp v wkhq dw whuplqdwlrq ri wklv dojrulwkp g y v y iru doo qrghv y 5 Y wkdw duh uhdfkdeoh iurp v1 Surri Vlqfh wkhuh duh qr qhjdwlyh f fohv uhdfkdeoh iurp v wkhuh lv d vlpsoh vkruwhvw sdwk iurp v wr doo qrghv y wkdw fdq eh uhdfkhg iurp v1 Ohw y eh vxfk d qrgh dqg ohw sv y kv y3 y4 yn yl eh d vkruwhvw sdwk iurp v wr y1 Vlqfh wklv lv d vlpsoh sdwk n mY m 41 Zh suryh wkh ohppd e lqgxfwlrq rq wkh qxpehu ri sdvvhv wkdw wkh dojrulwkp pdnhv1 Lq sduwlfxodu wkdw g yl v yl diwhu wkh lwk sdvv ri wkh dojrulwkp iru l 3 4 n1 Vlqfh wkhuh duh mY m 4 sdvvhv lq doo zh kdyh wkh ohppd li zh suryh wkh deryh vwdwhphqw1 g v g y3 v v diwhu lqlwldol dwlrq 3wk vwhs li wkhuh duh qr qhjdwlyh f fohv1 Khqfh wklv htxdolw lv pdlqwdlqhg wkhuhdiwhu1 Dvvxph e LK wkdw g yl 4 v yl 4 …
View Full Document
Unlocking...