Information Topologies with Applications - ResearchGate
Information Topologies with Applications - ResearchGate
Information Topologies with Applications - ResearchGate
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
§¨¦! "#$&%'§!)(*¦+,+ .-/&¤0 (1¨&23¤§¦546,+ +,2879!:::§,&¤)FZ£§¨&+PQ[,¦:0¦=HJ:§,+]\YLqirupdwlrq Wrsrorjlhv zlwk DssolfdwlrqvShwhu KduuhprîvRfwrehu 43/ 5336DevwudfwWrsrorjlhv uhodwhg wr lqirupdwlrq glyhujhqfh duh lqwurgxfhg1 Wkh frq0glwlrqdo olplw wkhruhp lv wdnhq dv prwlydwlqj h{dpsoh/ dqg vlpsolhgsurriv ri wkh uhohydqw wkhruhpv duh jlyhq1 Frqwlqxlw| surshuwlhv ri hq0wurs| dqg lqirupdwlrq glyhujhqfh lv glvfxvvhg1Nh|zrugv= Frqglwlrqdo olplw wkhruhp/ hqwurs|/ k|shuerolf glvwulexwlrqv/ lq0irupdwlrq glyhujhqfh/ Vdqry surshuw|/ vwurqj lqirupdwlrq wrsrorj|/ zhdn lqiru0pdwlrq wrsrorj|14 Lqwurgxfwlrq dqg suholplqdulhvUhodwlqj uhvxowv iurp suredelolw| wkhru| dqg lqirupdwlrq wkhru| lv qrw d qhzlghd1 Vrph frqyhujhqfh wkhruhpv lq suredelolw| wkhru| fdq eh uhirupxodwhg dvwkh hqwurs| frqyhujhv wr lwv pd{lpxp1 Pdunry fkdlqv zhuh wuhdwhg e| D1Uìq|l ^59`/ G1 J1 Nhqgdoo ^53` dqg M1 Iulw} ^47`/ wkh Fhqwudo Olplw Wkhruhp zdvwuhdwhg e| Olqqln ^56` dqg D1 Eduurq ^5`/ wkh orfdo fhqwudo olplw wkhruhp zdvwuhdwhg e| V1 Wdndqr ^5:` dqg Srlvvrq*v odz zdv wuhdwhg e| S1 Kduuhprîv ^49`1Wkhuh kdv dovr ehhq zrun vwuhqjwkhqlqj zhdn ru vwurqj frqyhujhqfh wr frqyhu0jhqfh lq lqirupdwlrq glyhujhqfh1 Doo wkh deryh phqwlrqhg sdshuv kdyh uhvxowvri wklv nlqg/ exw dovr d zrun e| D1 Eduurq ^6` vkrxog eh phqwlrqhg1 Vrph zrunkdv dovr ehhq grqh zkhuh wkh olplw ri d vhtxhqfh lv lghqwlhg dv d lqirupdwlrqsurmhfwlrq1 Wkh prvw lpsruwdqw sdshu lq wklv gluhfwlrq lv ^;`/ exw wkh vxemhfw lvgl!fxow lq wkh vhqvh wkdw rqh kdv wr eh yhu| fduhixo zlwk uhjxodulw| frqglwlrqv1D1 Ghper dqg R1 ]hlwrxql glvfxvvhg odujh ghyldwlrqv lq ^43`/ exw khuh wkh whfk0qltxh zdv pruh frqyh{ dqdo|vlv wkdq lqirupdwlrq wkhru|> wklv rqo| hpskdvl}hvwkh forvh uhodwlrqvkls ehwzhhq frqyh{ dqdo|vlv dqg lqirupdwlrq wkhru|1Wkh whup %frqyhujhqfh lq lqirupdwlrq% kdv ehhq xvhg lq wkh olwhudwxuh wrphdq frqyhujhqfh wr }hur ri lqirupdwlrq glyhujhqfh1 Lq wklv sdshu lw zloo ehvkrzq krz wr dvvrfldwh qrw rqh exw dw ohdvw wzr wrsrorjlhv fdswxulqj wkh frqfhswri frqyhujh lq lqirupdwlrq1 Rqh ri wkhvh zdv uhfrjql}hg e| U1 Gxgoh| exw4
D vhw D lv rshq lq wkh vwurqj lqirupdwlrq wrsrorj| li dqg rqo| lw vdwlvhvrqh ri wkh iroorzlqj htxlydohqw frqglwlrqv= Iru dq| T 5 D wkhuh h{lvwv d u A 3 vxfk wkdw E +T> u, D= Iru dq| qhw +S ,lq P ^ _ +X, zklfk vdwlvhv G +S n T, $ 3 iru vrphT 5 D> wkhuh h{lvwv d y vxfk wkdw S 5 D iru y =i$ƒ Iru dq| vhtxhqfh +Sk ,klq P^ _ +X, zklfk vdwlvhv G +Sk n T, $ 3 iruvrph T 5 D> wkhuh h{lvwv d qy vxfk wkdw Sk 5 D iru q qy =i$|Qrwh wkdw wkh vwurqj lqirupdwlrq wrsrorj| fdq eh ghqhg gluhfwo| e| wkhderyh fkdudfwhul}dwlrq ri wkh L0forvhg2rshq vhwv1Wkhruhp ; Wkh vwurqj lqirupdwlrq wrsrorj| lv vwurqjhu wkdq wkh wrsrorjlhvghqhg e| wrwdo yduldwlrq dqg y 1Surri1 Zh kdyh wr vkrz wkdw wkh edoov S 5 P ^ _ +X, m nS Tn ? u duhrshq lq wkh lqirupdwlrq wrsrorj|1 Dvvxph wkdwdqg wkdw +Sk ,klv d vhtxhqfh vxfk wkdw G +Sk n Sy , $ 3= Wkhq e| Slqvnhu*vSy 5 S 5 P ^ _ +X, m nS Tn ? u >lqhtxdolw| nSk Sy n $ 3> dqg wkhuh h{lvwv d qy vxfk wkdw nSk Sy n ? u i0|nSy Tn iru q qy = Wkhuhiruh Tn nSk Sy n . nSy TnnSk? uSk dqg 5 S P^ _ 5 +X, m nS Tn ? u iru q qy =Lq rughu wr vhh wkh L lv vwurqjhu y wkdq zh mxvw kdyh wr uhpdun wkdw iru dq|phdvxudeoh vhw E X wkh vhw S 5 P _ +X, m S +E, @ 3 ^lv L0rshq1Lq jhqhudo wkh vwurqj lqirupdwlrq wrsrorj| frlqflghv zlwk y qhlwkhu qruwkh vwurqj wrsrorj|1 Wr vhh wklv frqvlghu d frxqwdeoh vhw X zlwk htxlydohqwsuredelolw| phdvxuhv S dqg T vxfk wkdw G +S n T, @ 4= Sxw Sk @ 4@q S .+4 4@q, T= Wkhq Sk $ T lq vwurqj wrsrorj| dqg lq y exw qrw lq wkh vwurqjlqirupdwlrq wrsrorj|1Ohppd < D rqh sdudphwhu h{srqhqwldo idplo| lv frqwlqxrxv lq wkh sdudphwhulq wkh vwurqj lqirupdwlrq wrsrorj|1Surri1 Ohw S„ eh d rqh sdudphwhu h{srqhqwldo idplo| iru zklfk Sy @ Sdqg S^ @ T= Iluvw zh uhpdun wkdw G +S… n S„ , fdq rqo| eh qlwh iru v @ 3 dqg:
cWkhruhp 44 Li X lv qlwh wkhq wkh vwurqj lqirupdwlrq wrsrorj| frlqflghv zlwkwkh wrsrorj| y =Surri1 Rq d qlwh vhw G +S n T, ? 4 li dqg rqo| li vxss +S , vxss +T, =Wkh suhylrxv wkhruhp vkrzv wkdw wkh frqqhfwhg frpsrqhqwv duh ri wkh irupS 5 P +X, m vxss +S, @ E = Qrz zh mxvw kdyh wr uhpdun wkdw_ ^lv y 0frqwlqxrxv rq hdfk frpsrqhqw1+S> T, w G +S n T,Fruroodu| 45 Wkh S qhw 5 ^ _ P +X, frqyhujhv wr T 5 ^ _ P +X, lq wkh wrsrorj|li dqg rqo| li +S , $ +T, lq wkh vwurqj lqirupdwlrq wrsrorj| iru doo qlwhysduwlwlrqv > l1h1 doo = X Y‰ $ Y‰ zkhuh lv d qlwh vhw1Dv lpphgldwh frqvhtxhqfhv ri Wkhruhp ; rqh vhhv wkdw wkh pds +S> T, wG +S n T, lv orzhu vhpl frqwlqxrxv zkhq ^ _ P +X, lv htxlsshg zlwk wkh vwurqjlqirupdwlrq wrsrorj|1 Hvshfldoo| wkh vhwv S 5 _ P^ +X, m G +S n T, u duh forvhg lq wkh vwurqj lqirupdwlrq wrsrorj|1 Lq jhqhudo wkh vhwv S 5 _ P^ +X, m G +S n T, ? u duh qrw rshq1H{dpsoh 46 Li X lv frxqwdeoh lqqlwh wkhq wkhuh h{lvwv d edoo E +T> u, vxfk wkdwT lv qrw dq L0lqwhulru srlqw1 Wr vhh wklv fkrrvh S> U> T zlwk srlqw suredelolwlhvsursruwlrqdo ^ Œ wr = WkhqG +S n U, ? 4G +U n T, ? 4G +S n T, @ 4=> ^ {‹ > ^ {[ŠSk Ghqh ^ k @ S . 4 ^ k U= WkhqG n U, 4 +Skq G +S n U, . 4 4 G +U n U,q@ 4 G +S n U, $ 3=qWkhuhiruh Sk $ U iru q $ 4 dqg G +Sk n T, @ 4=Wkh deryh frqvwuxfwlrq fdq eh pdgh zlwk S uhsodfhg e| SŽ @ S .+4 , Tdqg U uhsodfhg e| UŽ @ U.+4 , T iru 5 `3> 4^ = Wkhuhiruh U.+4 , Tlv rq wkh L0erxqgdu|1 Zh kdyhG +U n T, @ G +U . +4 , T n T, G +U n T, $ 3 iru $ 3=
Wklv vkrzv wkdw wkhuh h{lvwv d S qhw $ T vxfk wkdw +S G n T, @ 4= Vlplodufrqvwuxfwlrqv duh nqrzq iurp wkh olwhudwxuh/ fi1 ^9` dqg ^;/ H{dpsoh 615`1D vhw D lv forvhg li dqg rqo| fo‚ li +D, @ D= Lq jhqhudo lw lv qrw wuxh wkdw+D, lv forvhg1 Wkhuhiruh zh kdyh wr lwhudwh wkh vhtxhqwldo forvxuh rshudwlrqfo‚wr rewdlq wkh forvxuh fo +D,1 Wkh frqvwuxfwlrq zrunv dv iroorzv=Ohw eh dq ruglqdo +qlwh ru wudqvqlwh,1 Zh ghqh fo‚ +D, e| wudqvqlwhuhfxuvlrq=Ž ‚ fo +D, @ D iru @ 33C Ž4^+D, D iru A 3=fo‚ fo‚ +D, @ fo‚ŽZh vhh wkdw fo‚ +D, lv frqvwdqw iurp d fhuwdlq srlqw1 Li qrw ehiruh wkhq zkhqwkh fduglqdolw| ri h{fhhgv wkh fduglqdolw| ri X= Wkhuhiruh fo ŽŽ+D, lv forvhg1 D‚pruh uhqhg dujxphqw jlyhv fo“P” ‚ wkdw +D, lv forvhg dqg htxdov fo +D, zkhuhghqrwh wkh uvw qrq0frxqwdeoh ruglqdo qxpehu1Ohw O ghqrwh wkh frqyh{ kxoo ri S> T dqg U= Wkhq O @ $^‚ +lqw +O,, zkhuhfoclqw +O, ghqrwhv wkh dojheudlf lqwhulru ri O= Wkh deryh frqvwuxfwlrq vkrzv wkdw 5forvxuh rshudwlrqv duh qhhghg lq rughu wr rewdlq d forvhg vhw1 Wklv frqvwuxfwlrqfdq eh lwhudwhg lq rughu wr rewdlq d vhw iru zklfk q forvxuh rshudwlrqv duh qhhghglq rughu wr rewdlq d forvhg vhw1 Wkh frqvwuxfwlrq lv dv iroorzv1 S{ Ohw eh wkhsuredelolw| phdvxuh’‘ Ž{RkP• {RT U Siru n @ 3> 4> ===> q= WkhqG +S{ n S{ _, @ G +S n U, . G +U n T,^? 4 =Lq jhqhudoS– G n , ? +S{ 4li dqg rqo| lio n . 4=+{ Iru suredelolw| +{ yhfwruv , dqg , zh kdyh# $k[k[Gn? S{ { 4{ S{li dqg rqo|^ {!— ^ {!—Œ.˜ ˆplq n Ž Œ£˜ ˆplq n . 4=Ohw F eh wkh frqyh{ kxoo iS{ ri j dqg ohw N eh wkh dojheudlf lqwhulru ri F= Wkhqfo ‚ +N, @ Fqfrqy–š { iS– j = Wkhuhiruh q forvxuh rshudwlrqv duh qhhghg wr rewdlqd forvhg vhw1{43
kkWkhruhp 47 Ohw D eh d frqyh{ vxevhw ri P ^ _ +X, = Wkhq fo +D, lv d frqyh{vhw1Surri1 Ohw fo‚ +E, ghqrwh wkh vhtxhqwldo forvxuh ri E> l1h1 vhw ri glvwulexwlrqvT vxfk wkdw wkhuh h{lvwv d vhtxhqfh +Sk ,kvxfk wkdw Sk $ T dqg Sk 5 E= Lii$|E lv frqyh{ wkhq vr lv fo +E, = Wr vhh wklv dvvxph wkdw Sk $ S dqg Tk $ Tzkhuh ‚ > Tk 5 E= Ohw 5 ^3> 4` = WkhqSkG +Sk . +4 , Tk n S . +4 , T, G +Sk n S , . +4 , G +Tk n T,$ 3 1E| wudqvqlwh lqgxfwlrq zh vhh wkdw dovr wkh forvxuh lv frqyh{1D qrq0wulyldo h{dpsoh ri d frqyh{ vhw zklfk lv forvhg lq wkh vwurqj lqirupd0wlrq wrsrorj| lv wkh vhw ri dwrplf suredelolw| phdvxuhv1 Wr vhh wkdw wklv vhw lvforvhg ohw Sk eh d vhtxhqfh ri dwrplf suredelolw| phdvxuhv zklfk frqyhujhv wrT lq vwurqj lqirupdwlrq wrsrorj|1 Ohw Dk ghqrwh wkh vhw ri srlqwv zkhuh Skkdv srvlwlyh zhljkw1 Wkhq Dk lv frxqwdeoh1 Wkxv doo Sk duh frqfhqwudwhg rq wkhfrxqwdeoh vhw ^Dk > dqg wkhuhiruh T lv dovr frqfhqwudwhg rq ^Dk = WkhuhiruhT lv dwrplf1Zh hdvlo| vhh wkdw wkh vwurqj lqirupdwlrq wrsrorj| lv frpsdfw li dqg rqo| liX lv d vlqjohwrq1Sursrvlwlrq 48 Iru wkh vwurqj lqirupdwlrq wrsrorj| wkh iroorzlqj frqglwlrqvduh htxlydohqw141 X lv qlwh151 ^ _ P +X, lv frpsdfw161 P ^_ +X, lv orfdoo| frpsdfw171 4 … „ frxqwdelolw| d{lrp lv vdwlvhg181 5 k a frxqwdelolw| d{lrp lv vdwlvhg1Surri1 4= , 5= Reylrxv15= , 6= Reylrxv1=4= , =6= Dvvxph wkdw X lv qrw qlwh/ dqg ohw T eh d suredelolw| phdvxuhzlwk lqqlwh exw frxqwdeoh vxssruw1 Dvvxph wkdw $ lv d frpsdfw qhljkerukrrglq vwurqj lqirupdwlrq wrsrorj|1 Wkhq $ frqwdlqv d edoo E +T> u, $> dqg wkhedoov duh forvhg vr E +T> u, lv frpsdfw1 Qrz wkhuh Sk h{lvwv d vhtxhqfh $ Tlq Sk L vxfk wkdw kdv qlwh vxssruw1 Wkhuhiruh E +T> u, frqwdlqv d suredelolw|glvwulexwlrq S zlwk qlwh vxssruw1 Wkh vhtxhqfh 4 ^ S . T $ S ^ k lqzhdn wrsrorj| exw qrw lq vwurqj lqirupdwlrq wrsrorj|1 Wkhuhiruh wkh vhtxhqfh4 ^ S . T kdv qr frqyhujhqw vxeqhw/ dqg wkh frqwudglfwlrq lv ^ k rewdlqhg14= , 8= Reylrxv18= , 7= Reylrxv144
=4= , =7=Dvvxph wkdw X lv qrw qlwh1 Wkhq wkhuh h{lvwv d frqyh{ vhw F vxfk wkdwfo‚ +fo‚ +F,, 9@ fo‚ +F, zkhuh wkh vhtxhqwldo forvxuh lv wdnhq lq wkh uhohydqw lq0irupdwlrq wrsrorj|1 Wkhuhiruh wkh vwurqj lqirupdwlrq wrsrorj| grhv qrw vdwlvi|4 … „ frxqwdelolw| d{lrp1Wkh frqglwlrqv lq Sursrvlwlrq +48, lpso| sdudfrpsdfwqhvv/ exw lw lv qrw fohduzhdwkhu wkh| duh qhfhvvdu| frqglwlrqv1Sursrvlwlrq 49 Wkh vwurqj lqirupdwlrq wrsrorj| lv vhsdudeoh li dqg rqo| li Xlv frxqwdeoh1Surri1 Li X lv qlwh wkh vwurqj lqirupdwlrq wrsrorj| lv reylrxvo| vhsdudeoh1Li X lv frxqwdeoh wkhq zh pd| dvvxph wkdw X @ Q= Sxw Dk @ i4> 5> > qjdqg Fk @ iS m vxss +S , Dk j = Hdfk ri wkh vhwv Fk lv vhsdudeoh/ dqg wkhuhiruhwkhlu xqlrq V Fk lv vhsdudeoh1 Qrz+Fk G n T, @ G jTdn T 4@ orjT , $ 3 +Dk =Wkhuhiruh V lv ghqvh lq P ^ _ +X, > dqg P ^Fklqirupdwlrq wrsrorj|1+X, lv vhsdudeoh lq wkh vwurqj_Li X lv qrw frxqwdeoh/ wkhq ^ _ P +X, lv qrw vhsdudeoh lq dqg wkhuhiruh qrwlq wkh vwurqj lqirupdwlrq wrsrorj|16 Frqwlqxlw| ri hqwurs| dqg glyhujhqfhLq rughu wr fkdudfwhul}h wkh vhtxhqwldo glvfrqwlqxlw| srlqwv ri wkh hqwurs| dqgglyhujhqfh ixqfwlrqv zh qhhg wkh frqfhsw ri k|shuerolf glvwulexwlrq dqg vhyhudozd|v wr fkdudfwhul}h wkhp1Ghqlwlrq 4: D glvwulexwlrq rq U zlwk glvwulexwlrq ixqfwlrq I lv vdlg wr wdlopdmrul}h d glvwulexwlrq zlwk glvwulexwlrq J liI +{, J +{,hyhqwxdoo|/ l1h1 wkhuh h{lvwv d {y vxfk wkdw I +{, J +{, iru { {y = D glvwul0exwlrq ri d qrq0qhjdwlyh udqgrp yduldeoh zlwk glvwulexwlrq ixqfwlrq I lv srzhugrplqdwhg li wkhuh h{lvwv A 4 vxfk wkdw I lv wdlo pdmrul}hg e| wkh glvwulexwlrqzlwk suredelolw| ghqvlw| sursruwlrqdo wr { •›Ž iru { 4= D glvwulexwlrq ri d qrq0qhjdwlyh udqgrp yduldeoh lv vdlg wr eh k|shuerolf li lw lv qrw srzhu grplqdwhg1Sursrvlwlrq 4; Ohw [ eh d qrq0qhjdwlyh udqgrp yduldeoh zlwk glvwulexwlrqixqfwlrq I= Wkhq wkh iroorzlqj frqglwlrqv duh htxlydohqw=Ohppd 4
žyž4 I +{, { „^51 Wkhuh h{lvwv d w A 3 vxfk wkdw61 Wkh wdlo suredelolwlhv vdwlvi|H [ „ ? 4 =orj +4 I +{,,olp lqiA 3=œ œŸž ^ orjSurri1 4= , 5= Dvvxph wkdw I pdmrul}hv d srzhu odz J= Wkhq]]{ gI +{, { gJ +{, >„ „ehfdxvh wkh ixqfwlrq { $ { „ lv lqfuhdvlqj1 Wkh uljkw kdqg vlgh lv qlwh iru wvx!flhqwo| vpdoo15= , 6= Iru doo w zh kdyhH [ „ @] ž{ „ gI +{,[] _@{ „ gI +{,[ l S +l [ ? l . 4, =„— yy —— y Wkh vhulhv l S +l [ ? l . 4, lv dq ruglqdu| Glulfkohw vhulhv lq v @ w^48` zklfk frqyhujhv iru w @ 3= Wkxv wkh devflvvd ri frqyhujhqfh lv… • SžLi H +[ „ , ? 4 wkhq @ olp vxsk ¡žorj m4 I +q,m=orj +q,orj m4 I +q,molp vxs w ? 3=¡ž k orj +q,6= , 4= Dvvxph wkdworj +4 I +{,,olp lqiA 3 =œ œŸž ^ orjWkhq wkhuh h{lvwv w A 3 vxfk wkdwhyhqwxdoo|/ exw wklv lv htxlydohqw wrorj +4 I +{,,orj ^ œ w46
hyhqwxdoo|/ dqg zh vhh wkdw I lv srzhu grplqdwhg1W|slfdo h{dpsohv ri k|shuerolf glvwulexwlrqv duh glvwulexwlrqv zlwk ghqvlwlhvsursruwlrqdo wr4{ +orj {, Ž > { 5iru vrph A 4=Iru d udqgrp yduldeoh [ zlwk suredelolw| ghqvlw| i wkh Uìq|l hqwurs| rirughu A 3 dqg 9@ 4 lv ghqhg e| wkh irupxodk +S , @ 4 ]4 orj +i +{,, g{Ž Žzkhq wkh lqwhjudo frqyhujhv1 Iru @ 4 wkh Uìq|l hqwurs| k^ +S , lv ghqhg wreh wkh Vkdqqrq hqwurs| +glhuhqwldo hqwurs|,1 Qrwh wkdw k lv xvhg wr ghqrwhglhuhqwldo hqwurs| dqg K lv xvhg wr ghqrwh glvfuhwh hqwurs|1Iru frqwlqxrxv glvwulexwhg qrq0qhjdwlyh udqgrp yduldeohv zlwk ghfuhdvlqjghqvlw| ixuwkhu fkdudfwhul}dwlrq ri srzhu grplqdwhg dqg k|shuerolf glvwulex0wlrqv duh srvvleoh1Sursrvlwlrq 53 Ohw [ eh d frqwlqxrxv qrq0qhjdwlyh udqgrp yduldeoh zlwk glv0wulexwlrq ixqfwlrq I dqg ghqvlw| ixqfwlrq i @ I† 1 Dvvxph wkdw i xsshu erxqghgdqg ghfuhdvlqj1 Wkhq wkh iroorzlqj frqglwlrqv duh htxlydohqw=41 Wkh glvwulexwlrq ri [ lv srzhu grplqdwhg151 Wkhuh h{lvwv N A 3 dqg A 4 vxfk wkdw i +{, N { •MŽ hyhqwxdoo|161 Wkhuh h{lvwv A 4 vxfk wkdw iru doo O A 3 rqh kdv i +{, O { •JŽhyhqwxdoo|171 Wkh ghqvlw| ixqfwlrq vdwlvi|olp lqiorj i +{,orj ^ œ81 Wkhuh h{lvwv ? 4 vxfk wkdw K Ž +[, ? 4=A 4 =Surri1 5= , 6= Dvvxph wkdw N A 3 dqg A 4 vxfk wkdw i +{, N {hyhqwxdoo|1 Fkrrvh vxfk wkdw 4 ? ? = Wkhq iru dq| O A 3•JŽN { •›Ž O { • =6= , 4= Wklv iroorzv vlqfh wkh frqvwdqw O fdq eh fkrvhq vxfk O { > { 4lv d srzhu odz17= , 5= Li•JŽolp lqiorj i +{,orj ^ œA A 447
žk — ^žyžyžywkhqorj i +{,orj ^ œA hyhqwxdoo| dqg wklv lv htxlydohqw wr i +{, ? { •›Ž hyhqwxdoo|18= , 7= Dvvxph wkdw wkh Uìq|l hqwurs| K Ž +S , lv qlwh iru ? 4= Wkhq[[h{s + + orj i +q,,, @ +i +q,, Ž— ^ k] ž+i +{,, Ž g{? 4=Qrz h{s + + orj i +q,,, lv d Glulfkohw vhulhv lq ^48` erwk zlwk de0vflvvd ri frqyhujhqfh htxdo wrSžk — ^olp vxsk Ÿžorj q orj i +q, @ 4olp lqi£.¤ e s kv ¢£N¤ ” ¢=Wkhuhiruholp lqiorj i +{,orj ^ œk Ÿž 4 =4= , 8= Dvvxph wkdw wkh srzhu odz zlwk wdlo suredelolwlhv 4 J +{, @{ •›Ž > { 4 lv pdmrul}hg e| I= Wkh ixqfwlrq { $ { > ? 4 lv frqfdyh1 XvlqjVkxu frqyh{lw| ^57` zh kdyh[ mq[ mq>I†J†dqg] ž+I† +{,,g{ ] ž+J† +{,, g{z — yz — yWkh uljkw kdqg vlgh lv qlwh iru vx!flhqwo| forvh wr 4=D qrq0qhjdwlyh udqgrp yduldeoh zlwk ghfuhdvlqj ghqvlw| i dozd|v vdwlvhvi +{, 4 {zlwk htxdolw| lq fdvh wkh glvwulexwlrq lv xqlirup1 Wkhuhiruhorj i +{,orj ^ œ 4=Wkhuhiruh d k|shuerolf glvwulexwlrq zlwk ghfuhdvlqj ghqvlw| vdwlvhvorj i +{,olp lqi @ 4=œ œ§¡ž ^ orj48
Wkh +glvfuhwh, Uìq|l hqwurs| ri rughu A 3 dqg 9@ 4 lv ghqhg dv dh{whqghg uhdo qxpehu e| wkh irupxodK Ž +S , @ 44 orj [ s Ž >Iru @ 4 wkh Uìq|l hqwurs| K^ +S , lv ghqhg wr eh wkh Vkdqqrq hqwurs|1 WkhqUìq|l hqwurs| lv ghfuhdvlqj lq =Dvvxph wkdw wkh udqgrp yduldeoh [ lv qrq0qhjdwlyh dqg lqwhjhu ydoxhg dqgzlwk ghfuhdvlqj srlqw suredelolwlhv1 Wkhq wkh suhylrxv uhvxow fdq eh dssolhg wrwkh glvwulexwlrq zlwk ghqvlw| i jlyhq e|i +{, @ S +[ @ e{f, =Wkhq wkh glvfuhwh +Vkdqqrq ru Uìq|l, hqwurs| ri [ htxdov wkh glhuhqwldo hq0wurs| ri i= Wkh glvwulexwlrq ri [ lv k|shuerolf li dqg rqo| liorj S +[ @ l,olp lqi@ 4=#ž ^ orjWklv irupxod zdv xvhg wr fkdudfwhul}h glvfuhwh k|shuerolf glvwulexwlrqv lq ^4:`1Diwhu wkhvh fkdudfwhul}dwlrqv ri wkh k|shuerolf glvwulexwlrqv lw lv srvvleohwr vwxg| wkh vhtxhqwldo glvfrqwlqxlw| srlqwv ri hqwurs| dqg glyhujhqfh1 OhwT eh d glvfuhwh glvwulexwlrq1 Li T lv srzhu erxqghg wkhq wkh hqwurs| lv qlwhehfdxvh wkh hqwurs| ri d srzhu odz lv qlwh1 Wkh iroorzlqj wkhruhp lghqwlhv wkhk|shuerolf glvwulexwlrqv dv wkh +vhtxhqwldo, glvfrqwlqxlw| srlqwv ri wkh hqwurs|ixqfwlrq1Wkhruhp 54 Li T srzhu grplqdwhg/ wkhq Sk $ T lpsolhv K +Sk , $ K +T, =Li T lv k|shuerolf dqg K +T, ? 4 wkhq wkhuh h{lvwv d vhtxhqfh Sk vxfk wkdwSk $ T dqgolpk Ÿž lqi +Sk K , A K +T, =Surri1 Dvvxph wkdw T lv srzhu erxqghg1 Wkhq[h{s w orjiru w vx!flhqwo| vpdoo1 WkhuhiruhK +Sk , @ [ orj 4t$ [ orj 4t@ K +T, = 4tt @ [ t^ •¥„? 4 Sk +l, G +Sk n T, t 349
^`a`a41 U ˆ ab`aaˆ`aIru wkh odvw sduw ri wkh wkhruhp/ vhh ^4:`1Lw lv reylrxv wr dvn wr zkdw h{whqw wkh ixqfwlrq S w G +S n T, lv frqwlqxrxvlq wkh lqirupdwlrq wrsrorjlhv1 Zh ghqh wkh Uìq|l glyhujhqfh G +Sy n T, @…^ • … orj U a£b ˆ…gT= Zh kdyh G … +Sy n T, $ G +Sy n T, iru v $ 4•G +Sy n T, ? 4> wkhq wkh iroorzlqj frqglwlrqv duh htxlydohqw1= LiSursrvlwlrq 55gSy ? 4 iru vrph w A 3=„gT ? 4 iru vrph v A 4=51 U ˆ ab`a…61 G … +Sy n T, ? 4 iru vrph v A 4=71 Wkh udqgrp ab ˆ` yduldeoh lv srzhu grplqdwhg zlwk uhvshfw Sy wr =Wkhruhp 56 Dvvxph wkdw G +Sy n T, ? 4= Wkhq wkh iroorzlqj frqglwlrqv duhhtxlydohqw141 Wkh ixqfwlrq S w G +S n T, lv vhtxhqwldo frqwlqxrxv lq Sy lq wkh vwurqjlqirupdwlrq wrsrorj|151 Wkh udqgrp ab ˆyduldeoh ` lv srzhu grplqdwhg zlwk uhvshfw wr Sy =aSurri1 5= @, 4= = Dvvxph wkdw U ? 4 iru d w A 3> dqg dvvxpha£b gSy+Sk wkdw Sy G n , $ 3= Zh zulwhG n T, @ +Sk G Sy n , . orj +SkgT > Sk gSyZh kdyh ]WkhuhiruhU ab ˆ h{s w orjgTgSyorj gSygT > SkgSy @„] gSy„gT$orj gSygT > SygSy ? 4 =@ G +Sy n T, iru q $ 4==5= @, =4= Ohw e eh vrph uhdo qxpehu juhdwhu wkdq G +Sy n T, dqg dvvxph…gT @ 4 iru doo v A 41 SxwF¦ @ S m orjgT > S gSy@ e =Wkhq iru S 5 F¦G +S n T, @ e=orj > S . G +S n Sy ,gT gSy4:
Wklv krogv Sk iru doo vhtxhqfhv = Li iru doo phdvxudeoh ixqfwlrqv i Uiru zklfkh{s +w i +x,, lv T0lqwhjudeoh iru vrph w A 3/ rqh kdvk Ÿžwkdw olp vxs i gSk U U U i gT > wkhq i gSk $ i gT iru q $ 4=6 , 7=Dvvxph wkdw +T, lv srzhu +Sk erxqghg/ dqg wkdw , lv d vhtxhqfh ri sured0!³ ´!³ ´elolw| phdvxuhv rq wkh phdvxuh Skvsdfh +> D, vxfk +Sk wkdw $ T= Wkhq , $ +T, = Ohw ghqrwh wkh +Sk srlqw suredelolwlhv ri , dqg ohw t ghqrwh wkhsksrlqw suredelolwlhv ri +T, = Wkhqg , +SkG + , n +T,, +Sk @ orjg +T, > , +Sk@ [ orj 4 t sk K + +Sk ,, =Qrz +T, lv srzhu erxqghg vr wkhuh h{lvw w A 3 vxfk wkdw S h{s w ^ µ ± orj t @ S t^ •›„? 4= Wkhuhiruh[orj4t sk $ [ orj 4 t t@ K + +T,,? 4=Xvlqj orzhu vhpl frqwlqxlw| ri wkh hqwurs| ixqfwlrq zh jhwolpk ŸžvxsG +Sk + , n +T,, @ olpk ¡žvxs [orj4t sk K + +Sk ,, K + +T,, olpk ¡ž lqiK +Sk + ,,@ 3=7= , 4^Ohw F P +X, eh d vhw zlwk +F> T, A 3= Wkhq wkhuh h{lvwv _ d frghlpsuryhphqw vxfk wkdw +, %= Wkhuhiruh k> S l % iru S 5 F= Ohw nZ·i ehwkh ixqfwlrq = Wkhq i u {{ Dvvxph wkdw , $ +T, iru doo phdvxudeoh pdsv = +> D, +Sk $ Q vxfkwkdw +T, lv U_ srzhu erxqghg1 Ohw i = D $ eh d phdvxudeoh ixqfwlrq vxfkwkdw h{s +w i +x,, lv T0lqwhjudeoh iru vrph w A 3= Ohw · ghqrwh wkh ixqfwlrq{ e u= Wkhq {]] en i +x,fh{s +w i +x,, gTx h{s w g +T, xn] „ en i +x,f h{sg +T, xn] l n l +T,n „ •¥„53
kkWkhq D lv rshq lq wkh vhtxhqwldo zhdn lqirupdwlrq wrsrorj|1 Wkh vhw rivhwv vdwlvi|lqj Fulwhulrq 5: lv reylrxvo| d wrsrorj| zklfk zh zloo fdoo wkh zhdnlqirupdwlrq wrsrorj| dqg L€ ghqrwh =Wkhruhp 5; Wkh zhdn lqirupdwlrq wrsrorj| lv vwurqjhu wkdq wkh wrsrorj| y /exw zhdnhu wkdq wkh vwurqj lqirupdwlrq wrsrorj|1Surri1 Wklv iroorzv vlqfh +F> T, G +F n T, =Wkh wrsrorjlhv y dqg L kdyh wkh vdph frqqhfwhg frpsrqhqwv dqg wkhuhiruhL€ kdv wkh vdph frqqhfwhg frpsrqhqwv dv wkhvh wrsrorjlhv1 Vlploduo|/ rq d qlwhvhw wkh wrsrorjlhv y dqg L frlqflgh dqg wkhuhiruh wkh| frlqflgh zlwk L€ rq qlwhvhwv1Dv lpphgldwh frqvhtxhqfhv ri Wkhruhp 5; zh jhw wkdw wkh pds +S> T, wG +S n T, lv orzhu vhpl frqwlqxrxv P^ _ zkhq +X, lv htxlsshg zlwk wkh zhdnlqirupdwlrq wrsrorj|1 Hvshfldoo| wkh vhwv S 5 _ P^ +X, m G +S n T, u duh L€ 0forvhg1Zh hdvlo| vhh wkdw wkh zhdn lqirupdwlrq wrsrorj| lv frpsdfw li dqg rqo| liX lv d vlqjohwrq1Sursrvlwlrq 5< Iru wkh zhdn lqirupdwlrq wrsrorj| wkh iroorzlqj frqglwlrqv duhhtxlydohqw141 X lv qlwh151 ^ _ P +X, lv frpsdfw161 ^ _ P +X, lv orfdoo| frpsdfw1Surri1 4= , 5= Reylrxv15= , 6= Reylrxv1=6= , =4= Dvvxph wkdw X lv qrw qlwh/ dqg ohw T eh d suredelolw| phdvxuhzlwk lqqlwh vxssruw1 Dvvxph wkdw $ lv d frpsdfw qhljkerukrrg lq zhdn lqiru0pdwlrq wrsrorj|1 Wkhq $ frqwdlqv d edoo E +T> u, $> dqg wkh edoov duh forvhgvr E +T> u, lv frpsdfw1 Qrz wkhuh h{lvw d vhtxhqfh Sk $ T lq L vxfk wkdw Skkdv qlwh vxssruw1 Wkhuhiruh E +T> u, frqwdlqv d suredelolw| glvwulexwlrq S zlwkqlwh vxssruw1 Wkh vhtxhqfh ^ 4 S . T $ S lq zhdn wrsrorj| exw qrw lq^ k dqg wkhuhiruh qrw lq wkh zhdn lqirupdwlrq wrsrorj|1 Wkhuhiruh wkh y vhtxhqfh4 ^ S . T kdv qr frqyhujhqw vxeqhw/ dqg wkh frqwudglfwlrq lv rewdlqhg1k^Sursrvlwlrq 63 Wkh zhdn lqirupdwlrq wrsrorj| lv vhsdudeoh li dqg rqo| li X lvfrxqwdeoh1Surri1 Li X lv qlwh wkh zhdn lqirupdwlrq wrsrorj| lv reylrxvo| vhsdudeoh1Li X lv frxqwdeoh wkhq ^ _ P +X, lv vhsdudeoh lq wkh vwurqj lqirupdwlrq wrsrorj|dqg wkhuhiruh dovr lq wkh zhdn lqirupdwlrq wrsrorj|155
kLi X lv qrw frxqwdeoh/ wkhq P ^ _ +X, lv qrw vhsdudeoh lq dqg wkhuhiruh qrwlq wkh zhdn lqirupdwlrq wrsrorj|1Wkh wrsrorj| dqg wkh wrsrorj| fruuhvsrqglqj wr wrwdo yduldwlrq fdq ehh{whqghg wr wkh vhw ri erxqghg glvwulexwlrq zlwk vljq1 Wkh vhw ri erxqghg phd0vxuhv lv d jurxs dqg erwk dqg wkh wrsrorj| fruuhvsrqglqj wr wrwdo yduldwlrq duhwudqvodwlrq lqyduldqw lq wklv jurxs1 Qrqh ri wkh lqirupdwlrq wrsrorjlhv h{whqgvlqwr wudqvodwlrq lqyduldqw wrsrorjlhv rq wkh vhw ri erxqghg vljqhg phdvxuhv1 Zhduh lqwhuhvwhg lq frqyhujhqfh wkhruhpv dqg duh wkhuhiruh sulpdulo| lqwhuhvwhglq wkh wrsrorj| qhdu d srlqw T 5 ^ _ P +X, = Lqvwhdg ri kdylqj d wrsrorj| zklfklv qrw wudqvodwlrq lqyduldqw rqh frxog irup d wudqvodwlrq lqyduldqw wrsrorj| e|wdnlqj wkh orfdo vwuxfwxuh +qhljkerukrrgv, L€ ri qhdu T dqg wudqvodwh lw wrjhw qhljkerukrrgv ri rwkhu srlqwv1 Dowkrxjk qrw h{solflwo| vwdwhg wklv lv wkhwhfkqltxh xvhg lq ^45`18 Wkh frqglwlrqdo olplw wkhruhpWkh lghd wr xvh lqirupdwlrq wkhruhwlfdo phwkrgv wr suryh d vwurqj yhuvlrq riwkh frqglwlrqdo olplw wkhruhp zdv jlyhq e| L1 Fvlv}äu lq ^;`1 Kh lqwurgxfhgy wkh wrsrorj| lq rughu wr irupxodwh wkh frqglwlrqdo olplw wkhruhp zlwk dvwurqjhu wrsrorj| wkdq = Zh zloo iroorz wkhvh lghdv dqg vhh wr zkdw h{whqg wkhlqirupdwlrq wrsrorjlhv duh xvhixo lq irupxodwlqj d frqglwlrqdo olplw wkhruhp1Ohw +X> I> T, eh d suredelolw| vsdfh zlwk dq sulru glvwulexwlrq T= Wkurxjk0rxw wklv vhfwlrq zh frqvlghu d {hg vhw F ^ _ P +X, ri frqvlvwhqw glvwulex0wlrqv1 Zh zloo vkrz wkdw wkh sulqflsoh ri plqlpxp lqirupdwlrq glvfulplqdwlrqfdq eh lqwhusuhwhg dv dq h{whqvlrq ri wkh frqfhsw frqglwlrqdo suredelolw|1 Ohw+x^$ @ > x > xk > , 5 X eh d vdpsoh/ l1h1 d qlwh vhtxhqfh ri rxwfrphv/ wkhqwkh hpslulfdo glvwulexwlrq lv k +$, @ ^Sk c^l ± > zkhuh l ± lv wkh Gludf— Hpskphdvxuh lq wkh srlqw x = Rq d qlwh vhw wkh hpslulfdo glvwulexwlrq lv dovr fdoohgwkh w|sh ri wkh vhtxhqfh ^ lw lv srvvleoh wr frqvlghu wkh frqgl0wlrqdo suredelolw| ri T k zlwk uhvshfw wr wkh vhw Nk lq X k 1 Ohw Tk @ T k + m Nk ,s ^ vghqrwh wkh pdujlqdo glvwulexwlrq ri T k + m Nk ,1 Dq dowhuqdwlyh zd| wr zulwh TklvH +Hpsk +$, m Nk ,56
zklfk ohdgv wr wkh lqhtxdolw|WkhuhiruhG +U k + m Nk , n T k , G +Uk n T k , . K U k +Nk , > U k ~Nk 4 q orj S +Nk , 4 q G +Uk + m Nk , n T k ,U k +Nk , G n T , . K U +Nk , > U k ~Nkq U k +Uk ,k +NkG +U n T,@ k U , . K U +Nk , > U k ~Nkq U k +Nk ,k +Nk k$ G +U n T, iru q $ 4>ehfdxvh U +Nk , $ 4 iru q $ 4 e| odz ri odujh qxpehuv1Li k lv qrw phdvxudeoh wkhq qg d vxevhw D ri wkh irupNkD @ V 5 ^ _ P +X, m k > Vl f > l @ 4> ===> n =zkhuh f > l @ 4> 5> ===> n duh srvlwlyh frqvwdqwv dqg dqg ixqfwlrqv vdwlvi|lqjU h{s + , gT 4= Wkhq i$ 5 X k m Hpsk +$, 5 Dj lv I0phdvxudeoh dqgdqgS €+Nk , S +Hpsk +$, 5 D,olpk ¡žvxs 4 q orj S +Nk , olp vxs€¡ž 4 q orj S €+Hpsk +$, 5 D,k G +U n T, =Wklv krogv iru doo U 5 lqw À F dqg wkh lqhtxdolw| iroorzv1Wkh wkhruhp lv vrphzkdw vwurqjhu wkdq vlplodu wkhruhpv lq wkh olwhudwxuhdv wkh zhdn lqirupdwlrq wrsrorj| lv vwurqjhu wkdq wkh wrsrorj| y xvhg lq wkholwhudwxuh1 E| frpelqlqj wkh orzhu dqg xsshu erxqgv rqh jhwv +F> T, 4 orj S € +Nk ,q olp vxs G +lqw À F n T, =k Ÿž 4 q orj S €+Hpsk +$, 5 F,Wkxv wkh orzhu erxqg uhihuv wr rqh vlgh ri wkh jdph dqg wkh xsshu erxqg uhihuvwr wkh rwkhu vlgh ri wkh jdph1 Wkh iroorzlqj wkhruhp lqfoxghv wkh zhoo0nqrzqFrqglwlrqdo Olplw Wkhruhp1Wkhruhp 68 Ohw +X> I> T, eh d suredelolw| vsdfh/ dqg ohw F eh d frqyh{ vhw lqhtxloleulxp zlwk T vxfk wkdwG +F n T, @ G +lqw À F n T, ? 4=5:
`ahtxloleulxp li dqg rqo| li wkh ruljlqdo vhw zdv lq htxloleulxp1 Wkhuhiruh wkh frq0fhsw ri jdph wkhruhwlfdo htxloleulxp fdq %uhsodfh% Fvlv}äu*v frqyh{lw| frqglwlrq1D frgh lpsuryhphqw ghqhv d phdvxuh U zlwk ghqvlw| gU@gT @ h{s +, = Irud jlyhq phdvxuh U lq jhqhudo wkhuh pd| h{lvw vhyhudo frgh lpsuryhphqwv vdwlv0i|lqj wkh htxdwlrq1 Ohw dqg ceh wzr vxfk frgh lpsuryhphqwv1 Lq jhqhudovhwv S 5 P^ _ +X, m U gS + , > l @ 4> 5 zloo eh glhuhqw1 Wkhuhiruh lw lvlpsruwdqw wr frqvlghu frgh lpsuryhphqwv dv ixqfwlrqv dqg qrw mxvw wr lghqwli|wkh frgh lpsuryhphqw zlwk wkh fruuhvsrqglqj phdvxuh1^ÀTkWkh vhtxhqfh $ T li dqg rqo| li iru dq| jlyhq vwdwlvwlfdo whvw zh duhhyhqwxdoo| qrw deoh wr glvwlqjxlvk ehwzhhq T dqg wkh hohphqwv lq wkh vhtxhqfh1Ru pruh orrvho|= D vhtxhqfh frqyhujhv lq wkh zhdn lqirupdwlrq wrsrorj| lidqg rqo| li wkh hohphqwv duh dv|pswrwlfdoo| lqglvwlqjxlvkdeoh1 Wkh vhtxhqfh$ T li dqg rqo| li zh duh hyhqwxdoo| qrw deoh wr glvwlqjxlvk ehwzhhq T Tk dqgwkh hohphqwv lq wkh vhtxhqfh xqlirupo| rq doo vwdwlvwlfdo whvw fruuhvsrqglqj wrd jlyhq ohyho ri wkh glyhujhqfh1 Wklv lv uhodwhg wr xqlyhuvdo whvwlqj/ zkhuh rqhxvhv d whvw zklfk dv|pswrwlfdoo| fdq glvwlqjxlvk d glvwulexwlrq iurp dq| rwkhuglvwulexwlrq1 Wkxv/ d vhtxhqfh frqyhujhv lq wkh vwurqj lqirupdwlrq wrsrorj| lidqg rqo| li wkh hohphqwv duh dv|pswrwlfdoo| xqlyhuvdoo| lqglvwlqjxlvkdeoh1Lq wklv sdshu wzr zhdn lqirupdwlrq wrsrorjlhv kdyh ehhq lqwurgxfhg1 Lqjhqhudo wkh| duh glhuhqw/ dqg wkhuh h{lvwv qhwv frqyhujhqw lq wkh zhdn lqiru0pdwlrq wrsrorj| exw qrw lq wkh vhtxhqwldo zhdn lqirupdwlrq wrsrorj|1 Lw lv qrwfohdu zkhwkhu wkhuh h{lvwv vhtxhqfhv L€ frqyhujlqj lq exw L€ … qrw lq = Li qr vxfkvhtxhqfhv h{lvwv rqh frxog L€ … vlpso| ghqh dv wkh vhtxhqwldo wrsrorj| zlwk wkhvdph frqyhujhqw L€ vhtxhqfhv dv =Dv wkh plqlpxp lqirupdwlrq jdph lv ghvfulehg lq wklv sdshu wkh jhqhudol}hglqirupdwlrq surmhfwlrq ri T rq d frqyh{ vhw F h{lvwv li dqg rqo| li G +F n T, ?4= Lq vrph fdvhv dq rswlpdo frgh lpsuryhphqw pd| h{lvw hyhq lq wkh fdvh zkhuhG +F n T, @ 4> dqg lw lv ri lqwhuhvw wr vwxg| frqglwlrqv iru dq rswlpdo frghlpsuryhphqw wr h{lvw1Li wkh vhw ri rxwfrphv lv sduwlwlrqhg/ wkhq wkh vwurqj lqirupdwlrq wrsrorjlhvfdq eh frqvlghuhg rq wkh sduwlwlrqv1 Iru d vhw ri sduwlwlrqv rqh fdq frqvlghu wkhlqlwldo wrsrorj| fruuhvsrqglqj wkh vhw ri sduwlwlrqv1 Wklv zloo jlyh d wrsrorj|zklfk lv zhdnhu wkdq wkh vwurqj wrsrorj|1 Wrsrorjlhv fruuhvsrqglqj wr sduwlwlrqvvhhpv wr sod| d dq lpsruwdqw uroh lq d ghhshu xqghuvwdqglqj ri wkh frqglwlrqdoolplw wkhruhp/ dqg lwv uhodwlrq wr d %zhdn lqirupdwlrq surmhfwlrq%1 D ghwdlohgwuhdwphqw ri wkhvh txhvwlrqv lv srvwsrqhg wr d ixwxuh sdshu1Lq ^4:` glvfrqwlqxlw| ri wkh hqwurs| ixqfwlrq zdv xvhg wr h{sodlq zk| ]lsi*vodz lv dssolfdeoh lq olqjxlvwlfv1 Lq d frqwlqxrxv vhwwlqj rqh vkrxog h{shfw wkdwwkh uhvxow wkdw lqirupdwlrq glyhujhqfh S w G +S n T, lv Sy glvfrqwlqxrxv dw lidqg rqo| li wkh a£bÁudqgrp yduldeoh lv k|shuerolf/ frxog eh xvhg wr h{sodlq wkhdsshdudqfh ri srzhu odzv dqg rwkhu khdy| wdlohg glvwulexwlrqv lq fhuwdlq fdvhv1Ilqdoo| zh uhpdun wkdw lw lv srvvleoh wr h{whqg pdq| ri wkh ghqlwlrqv dqguhvxowv rq wkh lqirupdwlrq wrsrorjlhv wr d qrq0frppxwdwlyh vhwwlqj zkhuh glvwul0exwlrqv duh uhsodfhg e| ghqvlw| rshudwruv rq d Kloehuw vsdfh1 Lq d qrq frppx0wdwlyh vhwxs wkh Eorfk vskhuh zloo eh glvfrqqhfwhg lq wkh +vwurqj, lqirupdwlrq5
wrsrorj|/ dqg wkh frpsrqhqwv zloo eh wkh lqwhulru dqg hdfk ri wkh srlqwv rq wkherxqgdu|1Dfnqrzohgjhphqw Wkh dxwkru wkdqnv Mdq Fdhvdu/ Iohpplqj Wrsvûh dqgLpuh Fvlv}äu iru xvhixo glvfxvvlrqv1 Iohpplqj Wrsvûh srlqwhg p| dwwhqwlrq wrjhqhudo uhvxowv rq h{lvwhqfh ri wrsrorjlhv ghqhg e| vhtxhqfhv1 Diwhu wkh uvwvxeplvvlrq ri wklv sdshu L kdg xvhixo glvfxvvlrqv zlwk Ulfkdug Gxgoh| zkhuhkh srlqwhg p| dwwhqwlrq wr klv sdshu ^44`1 Plnkdlo E1 Pdolrxwry kdv dovr jlyhqpdq| frpphqwv zklfk kdyh lpsuryhg wklv sdshu1 Wkh h{dpsoh ri d vhw iruzklfk q forvxuh rshudwlrqv duh qhhghg lq rughu wr jhw wkh forvxuh zdv ghyhorshgwrjhwkhu zlwk Iudqµlwhn Pdwýµ gxulqj d ylvlw rq klv lqvwlwxwh1 Ilqdoo| L zloowkdqn dq dqrq|prxv uhylhzhu iru pdq| uhohydqw frpphqwv dqg vxjjhvwlrqv irulpsuryhphqwv1Uhihuhqfhv^4` V1 Dpdul1 Lqirupdwlrq jhrphwu| rq klhudufk| ri suredelolw| glvwulexwlrqv1LHHH Wudqv1 Lqirup1 Wkhru|/ 7:=4:344:44/ 53341^5` D1 Eduurq1 Hqwurs| dqg wkh Fhqwudo Olplw Wkhruhp1 Dqqdov Surede1 Wkh0ru|/ 47+4,=669 675/ 4
^44` U1 Gxgoh|1 Frqvlvwhqf| ri p0hvwlpdwruv dqg rqh0vlghg eudfnhwlqj1 Lq H1 H1hw do1/ hglwru/ Kljk Glphqvlrqdo Suredelolw|/ Surjuhvv lq Suredelolw|/ sdjhv668;/ Edvho/ 4
^5:` V1 Wdndqr1 Frqyhujhqfh ri hqwurs| lq wkh fhqwudo olplw wkhruhp1 \rnrkdpdPdwkhpdwlfdo Mrxuqdo/ 68=47647;/ 4