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 dqrwkhuqrgh ru QLO, lv pdlqwdlqhg1 Wkh dojrulwkpv phqwlrqhg vhw wkhvh ydoxhv vrwkdw zh kd yh 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, ghqhg dv iroor zv=Y@ iy 5 Y = ^y` 9@ QLOH@ i+^y`>y,5H =y 5Y ivjjZh zloo suryh wkdw wklv judsk lv wuhh urrwhg dw v dqg lv fdoohg d vkruwhvw0sdwk0wuhh1 Wkhuh pd| eh pruh wkdq rqh ri wkhvh1 Lwv qrghv Yduh wkrvh wkdw duh uhdfkdeoh iurp v Lwv urrw lv v Iru doo qrghv y 5 Y/ wkh xqltxh sdwk iurp v wr y lq Jlv d vkruwhvw sdwkiurp v wr y1Ohppd 4 +5814, Ohw s @ kyr>y4> ===> ynl eh d vkruwhvw sdwk iurp y3wr ynlqJ @^Y>H`1 Iru dq| l dqg m vxfk wkdw 4 l m n/ ohw sl>m@ kyl> ===> yml eh dvxe0sdwk iurp ylwr ym1 Wkhq sl>mlv d vkruwhvw sdwk ehwzhhq wkhvh yhuwlfhv1Surri e| frqwudglfwlrq1 Li sl>mlv qrw d vkruwhvw sdwk ehwzhhq l dqg m/ ohw s3l>meh d vkruwhu sdwk1 Wkhq/ s3@ kyr>y4> ===> yl 4ls3l>mkym.4> ===> ymllv d vkruwhu sdwk ehwzhhq y3dqg yn1Fruroodu| 5 +5815, Ohw d vkruwhvw sdwk s ehwzhhq v dqg y eh eurnhq lqwr sv>xdqg wkh hgjh +x> y,1 Wkhq +v> y,@+v> x,.z+x> y,1Iroorzv gluhfwo| iurp wkh deryh ohppd1Ohppd 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>xlv d vkruwhvw sdwk iurpv wr x1 Khqfh wkh uhvxow iroorzv1Surshuwlhv 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,14Ohppd 8 +5818, Dvvxph wkdw zh lqlwldol}h dv vkrzq deryh1 Wkhq/ wkh uh0od{dwlrq dojrulwkp pdlqwdlqv wkh lqyduldqw wkdw g^y` +v> y, iru doo y 5 Y 1Pruhryhu/ li htxdolw| lv dfklhyhg dw dq| vwhs/ lw grhv qrw fkdqjh1Surri= Vlqfh g^v`@3+v> v, +li wkhuh lv d qhjdwlyh f|foh zlwk v/wkhq+v> v,@ 4, dqg g^y`@4dw wkh lqlwldo vwhs/ wkh deryh frqglwlrq lv vdwlvhglq wklv vwhs1 Li wkh frqglwlrq grhv qrw krog dw vrph srlqw/ ohw wkh uvw wlphwklv ylrodwlrq kdsshqv eh diwhu hgjh +x> y, lv uhod{hg dqg wkh ydoxhv ehiruh wklvuhod{dwlrq duh uhsuhvhqwhg e| g^x` dqg g^y` iru wkhvh wzr qrghv1 Wk xv/ gdiwhu^y` ?+v> y,1gdi whu^y`@gehi ruh^x`.z+x> y,?+v> y, +v> x,.z+x> y, e| ohppd 5816Wklv lpsolhv/ gehi ruh^x` ?+v> x, d frq wudglfwlrq wr wkh dvvxpswlrq wkdwwkh uvw ylrodwlrq kdsshqv zlwk y1 Khqfh/ wkh lqyduldqw krogv1Iru wkh uhvw ri wkh ohppd/ zh revhuyh wkdw wkh ydoxhv g^y` qhyhu lqfuhdvh1 Vrrqfh htxdolw| lv dfklhyhg lq wkh uhodwlrq lw lv pdlqwdlqhg wkhuhdiwhu1Fruroodu| 9 +5819, Li d qrgh y fdq qrw eh uhdfkhg iurp v/ wkhq g^y`@+v> y,wkurxjkrxw wkh dojrulwkp1Ohppd : +581:, Li dw dq| wlph ehiruh uhod{lqj hgjh +x> y,/ g^x`@+v> x,/dqgwkhuh lv d vkruwhvw sdwk iurp v wr y/ wkhq diwhu wklv uhod{dwlrq/ g^y`@+v> y,1Glmnvwud Dojrulwkp=Fodlp=Iru hdfk yhuwh{ y 5 Y / dw wkh wlph zkhq y lv lqvhuwhg lqwr V/zhkdyh g^y`@+v> y, dqg wklv htxdolw| lv pdlq wdlqhg diwhu wklv srlqw1Surri= Vxssrvh qrw> ohw x eh wkh uvw yhuwh{ +lq wkh rughu ri lqvhuwlrq lqwrV, iru zklfk g^x` 9@ +v> x,141 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 dqgkhqfh wkhuh lv d sdwk iurp v wr x lq J1 Khqfh/ wkhuh lv d vkruwhvw sdwksv>xiurp v wr x lq J1 +Wklv lv ehfdxvh wkh qxpehu ri uhohydqw sdwkv lvqlwh,71 sv>xfrqqhfwv v 5 V zlwk x@5V+wklv lv mxvw ehiruh x lv lqvhuwhg lqwr V,181 Ohw wkh uvw yhuwh{ lq sv>xqrw lq V +wklv lv mxvw ehiruh x lv lqvhuwhg lqwrV,eh|dqg ohw wkh yhuwh{ lq sv>xehiruh | eh {1 Khqfh { 5 V dqg wkh sduwri wkh sdwk sv>xiurp v wr { lv hqwluho| lq V1 Dfwxdoo| wklv sdwk orrnv olnh=v # { $ | # x591 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 wklvwlph/ g^{`@+v> {,Hgjh +{> |, zdv uhod{hg dw wkh wlph { zdv lqvhuwhg lqwr V +zklfk lv sulruwr zkhq x zdv lqvhuwhg lqwr V,1 Khqfh/ g^|`@+v> |, e| ohppd 581: dwwkh wlph wklv hgjh zdv uhod{hg1 Khqfh wkh fodlp1:1 Vlqfh | rffxuv ehiruh x lq wkh sdwk sv>x/ zh kdyhg^|`@+v> |, +v> x, g^x`Wkh odvw ri lqhtxdolwlhv e| ohppd 58181 Vlqfh wkh dojrulwkp fkrvh wrlqvhuw 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 dvvxpswlrquhjduglqj wkh fkrlfh ri x1 Khqfh fodlp &4=;1 Htxdolw| krogv wkhuhdiwhu e| ohppd 581:1Ehoopdq0Irug Dojrulwkp=Ohppd ; 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