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NA|jA|t 41 Zkdw duh phdvxuhv dqg zk| _phdvxudeoh% vhwv 4141 Wkh sureohp zlwk Ohehvjxh _phdvxuh% 51 Olplwv/ vxpv/ dqg rwkhu edvlfv 5141 Vhw Rshudwlrqv 5151 Olplwv/ Olpvxsv/ dqg Olplqiv1 5161 Vxpv ri srvlwlyh ixqfwlrqv 5171 Vxpv ri frpsoh{ ixqfwlrqv 5181 Lwhudwhg vxpv 5191 Frpsdfw Vhwv lq Uq 51:1 Edvlf phwulf vsdfh qrwlrqv 61 Wrsrorjlhv/ Dojheudv dqg ~ Dojheudv 71 Phdvxuhv 7141 Uhjxodulw| Uhvxowv 81 Frqvwuxfwlqj h{dpsohv ri phdvxuhv 8141 Frqvwuxfwlqj Phdvxuhv iurp Suhphdvxuhv 8151 Surri ri Wkhruhp 81: 8161 H{dpsohv ri qlwho| dgglwlyh phdvxuhv 8171 _Udgrq% phdvxuhv rq +U> E, 91 Dsshqgl{= Frqvwuxfwlrq ri Phdvxuhv 9141 Rxwhu Phdvxuhv 9151 Fdudwkrgru|*v Frqvwuxfwlrq Wkhruhp h 9161 Uhjxodulw| uhvxowv uhylvlwhg 9171 Frqvwuxfwlrq ri phdvxuhv rq d vlpsoh surgxfw vsdfh1 :1 Frqwlqxrxv dqg phdvxudeoh ixqfwlrqv :141 Uhodwlyh Wrsrorjlhv dqg ~ Dojheudv :151 Surgxfw Vsdfhv :161 Phdvxudelolw| rq Frpsohwh Phdvxuh Vsdfhv :171 Dsshqgl{= Vshfldo fdvhv ri wkh Surgxfw Wkhruhpv ;1 Lqwhjudwlrq wkhru| ;141 Lqwhjudo ri Vlpsoh ixqfwlrqv 5d|i q viRwih P Pfff TBiqRidthi!|i2 ! _ et hitidh "dt dh|dBBA tvh|i wA ST fhdc| 5S Ph! 5idh|Ric| vu d|iRd|t fP xciht|A vu %dBuvhcd Sdc 5i}v wd vBBd % PffP ! R 5 6 8 8 9 ; 43 47 49 4: 4< 58 5: 65 65 67 69 6; 74 74 76 79 7: 7< 85 86 8: 8< 93 93 S ipr xI qvr| ;151 Lqwhjudov ri srvlwlyh ixqfwlrqv ;161 Frpsdulvrq wr wkh Ulhpdqq Lqwhjudo <1 Prghv ri Frqyhujhqfh 431 Ghqvlw| Wkhruhpv 441 Ixelql*v Wkhruhp 44141 Surgxfw Phdvxuh 451 Ohehvjxh phdvxuh rq Ug 45141 Srodu Frruglqdwhv dqg Vxuidfh Phdvxuh 461 Vljqhg Phdvxuhv 46141 Kdkq Ghfrpsrvlwlrq Wkhruhp 46151 Mrugdq Ghfrpsrvlwlrq 471 Udgrq0Qlnrg|p Wkhruhp 481 Frpsoh{ Phdvxuhv 48141 Devroxwh Frqwlqxlw| rq dq Dojheud 491 Phdvxuh Glhuhqwldwlrq Wkhruhpv rq Uq 49141 D Fryhulqj Ohppd dqg Dyhudjlqj Rshudwruv 49151 Pd{lpdo Ixqfwlrqv 49161 Ohehvtxh Vhw 4:1 Wkh Ixqgdphqwdo Wkhruhp ri Fdofxoxv 4;1 Dsshqgl{= Frpsdfwqhvv rq phwulf vsdfhv 4<1 Dsshqgl{= G|qnlq*v ~ Wkhruhp 531 Dsshqgl{= Pxowlsolfdwlyh V|vwhp Wkhruhp 95 :5 :7 :: ;3 ;3 ;: <6 <9 <: << 434 439 43< 43< 443 444 446 448 456 458 459 41 b B| Bij 6jBt ijt BAa + T6jBt iBO,j tj|t Ghqlwlrq 414 +Suholplqdu|,1 Vxssrvh wkdw [ lv d vhw dqg S +[ , ghqrwhv wkh froohfwlrq ri doo vxevhwv ri [= D phdvxuh rq [ lv d ixqfwlrq = S +[ , $ ^3> 4` vxfk wkdw +4, +>, @ 3 Q +5, Li iDl jldR lv d qlwh +Q ? 4, ru frxqwdeoh +Q @ 4, froohfwlrq ri vxevhwv ri [ zklfk duh sdlu0zlvh glvmrlqw +l1h1 Dl _ Dm @ > li l 9@ m , wkhq Q [ +Dl ,= +^Q Dl , @ ldR ldR H{dpsoh 4151 Vxssrvh wkdw [ lv dq| vhw dqg { 5 [ lv d srlqw1 Iru D [> ohw 4 li {5D { +D, @ 3 rwkhuzlvh1 Wkhq @ { lv d phdvxuh rq [ fdoohg wkh dw {= H{dpsoh 4161 Vxssrvh wkdw lv d phdvxuh rq [ dqg A 3> wkhq lv dovr d S phdvxuh rq [= Pruhryhu/ li i = 5 M j duh doo phdvxuhv rq [> wkhq @ SM > l1h1 [ +D, iru doo D [ +D, @ SM lv d phdvxuh rq [= +Vhh Vhfwlrq 5 iru wkh phdqlqj ri wklv vxp1, Zh pxvw vkrz wkdw R lv frxqwdeo| dgglwlyh1 Vxssrvh wkdw iDl jldR lv d froohfwlrq ri sdlu0zlvh glvmrlqw hu SUQh yrpr ra rhr ur vxevhwv ri [> wkhq +^R Dl , @ ldR R [ +Dl , @ T R [[ +Dl , ldR SM [ @ +Dl , @ +^R Dl , ldR SM ldR SM @ +^R Dl , ldR ldR R [[ zkhuh lq wkh wklug htxdolw| zh xvhg Wkhruhp 5153 ehorz dqg lq wkh irxuwk zh xvhg wkdw idfw wkdw lv d phdvxuh1 H{dpsoh 4171 Vxssrvh wkdw [ lv d vhw = [ $ ^3> 4` lv d ixqfwlrq1 Wkhq [ +{, { =@ {S [ lv d phdvxuh/ h{solflwo| +D, @ [ {S D +{, iru doo D [= 4141 Wkh sureohp zlwk Ohehvjxh _phdvxuh%1 Txhvwlrq 41 Grhv wkhuh h{lvw d phdvxuh = S +U, $^3> 4` vxfk wkdw +4, +^d> e,, @ +e d, iru doo d ? e dqg +5, +D . {, @ +D, iru doo { 5 UB Wkh xqiruwxqdwh dqvzhu lv qr zklfk zh ghprqvwudwh qrz1 Lq idfw wkh dqvzhu lv qr hyhq li zh uhsodfh 41 e| wkh frqglwlrq wkdw 3 ? ++3> 4`, ? 4= Ohw xv lghqwli| ^3> 4, zlwk wkh xqlw flufoh V =@ i} 5 F = m} m @ 4j e| wkh pds !+w, @ hlSw 5 V iru w 5 ^3> 4,= Xvlqj wklv lghqwlfdwlrq zh pd| xvh wr ghqh d ixqfwlrq rq S +V , e| +!+D,, @ +D, iru doo D ^3> 4,= Wklv qhz ixqfwlrq lv d phdvxuh rq V zlwk wkh surshuw| wkdw 3 ? ++3> 4`, ? 4= Iru } 5 V dqg Q V ohw +414, }Q =@ i}q 5 V = q 5 Q j> wkdw lv wr vd| hl Q lv Q urwdwhg frxqwhu forfnzlvh e| dqjoh = Zh qrz fodlp wkdw lv lqyduldqw xqghu wkhvh urwdwlrqv/ l1h1 +415, +}Q , @ +Q , iru doo } 5 V dqg Q V= Wr yhuli| wklv/ zulwh Q @ !+D, dqg } @ !+w, iru vrph w 5 ^3> 4, dqg D ^3> 4,= Wkhq !+w,!+D, @ !+w . D prg 4, zkhuh Iru Q ^3> 4, dqg 5 ^3> 4,> ohw w . D prg 4 @ id . w prg 4 5 ^3> 4, = d 5 Q j @ +d . D _ id ? 4 wj, ^ ++w 4, . D _ id 4 wj, = ipr xI qvr| U Wkxv +!+w,!+D,, @ +w . D prg 4, @ ++d . D _ id ? 4 wj, ^ ++w @ ++d . D _ id ? 4 wj,, . +++w 4, . D _ id 4 wj,, 4, . D _ id 4 @ +D _ id ? 4 wj, . +D _ id 4 wj, @ ++D _ id ? 4 wj, ^ +D _ id 4 wj,, wj,, @ +D, @ +!+D,,= Wkhuhiruh lw vxfhv wr suryh wkdw qr qlwh phdvxuh rq V vxfk wkdw Ht1 +415, krogv1 Wr gr wklv zh zloo _frqvwuxfw% d qrq0phdvxudeoh vhw Q @ !+ D, iru vrph D ^^3> 4,= Wr gr wklv ohw U eh wkh frxqwdeoh vhw U =@ i} @ hlSw = w 5 ^3> 4, _ Tj= Dv deryh U dfwv rq V e| urwdwlrqv dqg glylghv V xs lqwr htxlydohqfh fodvvhv/ zkhuh }> z 5 V duh htxlydohqw li } @ uz iru vrph u 5 U= Fkrrvh +xvlqj wkh d{lrp ri fkrlfh, rqh uhsuhvhqwdwlyh srlqw q iurp hdfk ri wkhvh htxlydohqfh fodvvhv dqg ohw Q V eh wkh vhw ri wkhvh uhsuhvhqwdwlyh srlqwv1 Wkhq hyhu| srlqw } 5 V pd| eh xqltxho| zulwwhq dv } @ qu zlwk q 5 Q dqg u 5 U= Wkdw lv wr vd| d +uQ , +416, V@ u SU c zkhuh D lv xvhg wr ghqrwh wkh xqlrq ri sdlu0zlvh glvmrlqw vhwv iD j = E| Htv1 +415, dqg +416, zh qg wkdw [ [ +uQ , @ +Q ,= +V , @ u SU u SU Wkh uljkw phpehu iurp wklv htxdwlrq lv hlwkhu 3 ru 4/ 3 li +Q , @ 3 dqg 4 li +Q , A 3= Lq hlwkhu fdvh lw lv qrw htxdo +V , 5 +3> 4,= Wkxv zh kdyh uhdfkhg wkh ghvluhg frqwudglfwlrq1 Surri1 +Vhfrqg surri, Iru Q ^3> 4, dqg 5 ^3> 4,> ohw Q @ Q . prg 4 @ id . prg 4 5 ^3> 4, = d 5 Q j @ + . Q _ id ? 4 j, ^ ++ 4, . Q _ id 4 j, = Li lv d phdvxuh vdwlvi|lqj wkh surshuwlhv ri wkh Txhvwlrq zh zrxog kdyh +Q , @ + . Q _ id ? 4 j, . ++ 4, . Q _ id 4 j, @ +Q _ id ? 4 @ +Q _ id ? 4 +417, j, . +Q _ id 4 j ^ +Q _ id 4 j, j,, @ +Q ,= Zh zloo qrz frqvwuxfw d edg vhw Q zklfk frxsohg zlwk Ht1 +417, zloo ohdg wr d frqwudglfwlrq1 Vhw T{ i{ . u 5 U = u5 Tj @{ . T= hu SUQh yrpr ra rhr ur V Qrwlfh wkdw T{ _ T| 9@ > lpsolhv wkdw T{ @ T| = Ohw R @ iT{ = { 5 Uj ~ wkh ruelw vsdfh ri wkh T dfwlrq1 Iru doo D 5 R fkrrvh i +D, 5 ^3> 4@6, _ D=R Ghqh Q @ i +R,= Wkhq revhuyh= +4, i +D, @ i +E , lpsolhv wkdw D _ E 9@ > zklfk lpsolhv wkdw D @ E vr wkdw i lv lqmhfwlyh1 +5, R @ iTq = q 5 Q j= Ohw U eh wkh frxqwdeoh vhw/ U T _ ^3> 4,= Zh qrz fodlp wkdw Q u _ Q v @ > li u 9@ v dqg ^3> 4, @ ^uSU Q u = +418, +419, Lqghhg/ li { 5 Q u _ Q v 9@ > wkhq { @ u . q prg 4 dqg { @ v . qQ prg 4> wkhq q q3 5 T> l1h1 Tq @ TqQ 1 Wkdw lv wr vd|/ q @ i +Tq , @ i +TqQ , @ qQ dqg khqfh wkdw v @ u prg 4> exw v> u 5 ^3> 4, lpsolhv wkdw v @ u= Ixuwkhupruh/ li { 5 ^3> 4, dqg q =@ i +T{ ,> wkhq { q @ u 5 T dqg { 5 Q u prg R = Qrz wkdw zh kdyh frqvwuxfwhg Q> zh duh uhdg| iru wkh frqwudglfwlrq1 E| Htxd0 wlrqv +417~419, zh qg [ [ +Q u , @ +Q , 4 @ +^3> 4,, @ @ u SU uSU 4 li +Q , A 3 = 3 li +Q , @ 3 zklfk lv fhuwdlqo| lqfrqvlvwhqw1 Lqflghqwdoo| zh kdyh mxvw surgxfhg dq h{dpsoh ri vr fdoohg _qrq ~ phdvxudeoh% vhw1 Ehfdxvh ri wklv h{dpsoh dqg rxu ghvluh wr kdyh d phdvxuh rq U vdwlvi|lqj wkh surshuwlhv lq Txhvwlrq 4/ zh qhhg wr prgli| rxu ghqlwlrq ri d phdvxuh1 Zh zloo jlyh xs rq wu|lqj wr phdvxuh doo vxevhwv D U> l1h1 zh zloo rqo| wu| wr ghqh rq d vpdoohu froohfwlrq ri _phdvxudeoh% vhwv1 Vxfk froohfwlrq zloo eh fdoohg ~ dojheudv1 Wkhvh qrwlrq zloo eh lqwurgxfhg lq wkh qhvw vhfwlrq1 51 w6|te t 6te BAa N| ji OBtWt 5141 Vhw Rshudwlrqv1 Vxssrvh wkdw [ lv d vhw1 Iru D [ ohw @ Df @ [ q D @ i{ 5 [ = { 5 Dj dqg pruh jhqhudoo| li D> E [ ohw E q D @ i{ 5 E = { 5 Dj= @ Zh dovr ghqh wkh v|pphwulf glhuhqfh ri D dqg E e| D7E @ +E q D, ^ +D q E , = Dv xvxdo li iD jSL lv dq lqgh{hg froohfwlrq ri vxevhwv ri [ zh ghqh wkh xqlrq dqg wkh lqwhuvhfwlrq ri wklv froohfwlrq e| ^SL D =@ i{ 5 [ = < 5 L 6 { 5 D j dqg _SL D =@ i{ 5 [ = { 5 D ; 5 L j= Ri di ti |i 2vR vu vi ihi !i! T M5 i u df Bo( 9 ipr xI qvr| W Qrwlfh wkdw ^ lv forvho| uhodwhg wr < dqg _ lv forvho| uhodwhg wr ;= Iru h{dpsoh ohw R iDq jqdR eh d vhtxhqfh ri vxevhwv iurp [ dqg ghqh iDq l1r1j =@ i{ 5 [ = & iq = { 5 Dq j @ 4j dqg iDq d1d1j =@ i{ 5 [ = { 5 Dq iru doo q vxflhqwo| odujhj1 Wkhq { 5 iDq l1r1j l ;Q 5 Q <q Q 6 { 5 Dq zklfk pd| eh zulwwhq dv iDq l1r1j @ _RdR ^q$Q Dq Q dqg vlploduo|/ { 5 iDq d1d1j l <Q 5 Q 6 ;q Q> { 5 Dq zklfk pd| eh zulwwhq dv iDq d1d1j @ ^RdR _q$Q Dq = Q 5151 Olplwv/ Olpvxsv/ dqg Olplqiv1 Qrwdwlrq 5141 Wkh lv wkh vhw U =@ U^ i 4j > l1h1 lw lv U zlwk wzr qhz srlqwv fdoohg 4 dqg 4= Zh xvh wkh iroorzlqj frqyhqwlrqv/ 4 3 @ 3> 4 . d @ 4 iru dq| d 5 U> 4 . 4 @ 4 dqg 4 4 @ 4 zkloh 4 4 lv qrw ghqhg1 Li U zh zloo ohw vxs dqg lqi ghqrwh wkh ohdvw xsshu erxqg dqg juhdwhvw orzhu erxqg ri uhvshfwlyho|1 Zh zloo dovr xvh wkh iroorzlqj frqyhqwlrq/ li @ >> wkhq vxs > @ 4 dqg lqi > @ .4= R Qrwdwlrq 5151 Vxssrvh wkdw i{q jqdR U lv d vhtxhqfh ri qxpehuv1 Wkhq olp lqi {q @ olp lqi i{n = n qj @ vxs lqi i{n = n qj +514, q6R q6R q dqg +515, olp vxs {q @ olp vxsi{n = n qj @ lqi vxsi{n = n qj= q6R q q6R Zh zloo dovr zulwh olp lqi iru olp dqg olp vxs iru olp= Uhpdun 5161 Qrwlfh wkdw li dn =@ lqi i{n = n qj dqg en =@ vxsi{n = n qj>wkhq idn j lv dq lqfuhdvlqj vhtxhqfh zkloh ien j lv d ghfuhdvlqj vhtxhqfh1 Wkhuhiruh wkh olplwv lq Ht1 +514, dqg Ht1 +515, dozd|v h{lvw1 Wkh iroorzlqj sursrvlwlrq frqwdlqv vrph edvlf surshuwlhv ri olplqiv dqg olpvxsv1 Sursrvlwlrq 5171 Ohw idq jR dqg ieq jR eh wzr vhtxhqfhv ri uhdo qxpehuv1 qdR qdR Wkhq +4, Vkrz olp lqi q6R dq olp vxsq6R dq dqg wkh olpq6R dq h{lvwv lq U l olp lqi q6R dq @ olp vxsq6R dq 5 U= +5, Vxssrvh wkdw olp vxsq6R dq @ P 5 U> vkrz wkdw wkhuh lv d vxevhtxhqfh idqn jR ri idq jR vxfk wkdw olpn6R dqn @ P= qdR ndR +6, Vxssrvh wkdw olp vxsq6R dq ? 4 dqg olp vxsq6R eq A 4> wkhq suryh wkdw +516, olp vxs +dq . eq , olp vxs dq . olp vxs eq = q6R q6R q6R Lq rwkhu zrugv/ Ht1 +516, krogv surylghg wkh uljkw vlgh ri wkh htxdwlrq lv zhoo ghqhg1 +7, Vxssrvh wkdw dq 3 dqg eq 3 iru doo q 5 Q= Vkrz +517, olp vxs +dq eq , olp vxs dq olp vxs eq > q6R q6R q6R surylghg wkh uljkw kdqg vlgh ri +517, lv qrw ri wkh irup 3 4 ru 4 3= hu SUQh yrpr ra rhr ur X Surri1 Zh zloo rqo| suryh sduw 41 dqg ohdyh wkh uhvw dv dq h{huflvh wr wkh uhdghu1 Zh ehjlq e| qrwlflqj wkdw lqi idn = n qj vxsidn = n qj ; q vr wkdw olp lqi dq olp vxs dq = q6R q6R Qrz vxssrvh wkdw olp lqi q6R dq @ olp vxsq6R dq @ d 5 U= Wkhq iru doo A 3> wkhuh lv dq lqwhjhu Q vxfk wkdw d lqi idn = n Q j vxsidn = n Q j d . > l1h1 zh kdyh d dn d . iru doo n Q= Khqfh e| wkh ghqlwlrq ri wkh olplw/ olpn6R dn @ d= Li olp lqi q6R dq @ 4> wkhq zh nqrz iru doo P 5 +3> 4, wkhuh lv dq lqwhjhu Q vxfk wkdw P lqi idn = n Q j dqg khqfh olpq6R dq @ 4= Wkh fdvh zkhuh olp vxsq6R dq @ 4 lv kdqgohg vlploduo|1 Frqyhuvho|/ vxssrvh wkdw olpq6R dq @ D 5 U h{lvwv1 Li D 5 U> wkhq iru hyhu| A 3 wkhuh h{lvwv Q +, 5 Q vxfk wkdw mD dq m iru doo q Q +,> l1h1 D dq D . iru doo q Q +,= Iurp wklv zh ohduq wkdw D olp lqi dq olp vxs dq D . = q6R q6R Vlqfh A 3 lv duelwudu|/ lw iroorzv wkdw D olp lqi dq olp vxs dq D> q6R q6R l1h1 wkdw D @ olp lqi q6R dq @ olp vxsq6R dq = Li D @ 4> wkhq iru doo P A 3 wkhuh h{lvw Q +P , vxfk wkdw dq P iru doo q Q +P ,= Wklv vkrzv wkdw olp lqi dq P q6R dqg vlqfh P lv duelwudu| lw iroorzv wkdw 4 olp lqi dq olp vxs dq = q6R q6R Wkh surri lv vlplodu li D @ 4 dv zhoo1 Wkh iroorzlqj hohphqwdu| ohppd zloo riwhq eh xvhixo1 Ohppd 5181 Ohw [ dqg \ eh vhwv/ U [ \ dqg vxssrvh wkdw d = U $ U lv d ixqfwlrq1 Ohw { U =@ i| 5 \ = +{> | , 5 Uj dqg U| =@ i{ 5 [ = +{> |, 5 Uj = Wkhq vxs d+{> | , @ vxs vxs d+{> | , @ vxs vxs d+{> | , dqg {S[ |S{ U |S\ {SU| lqi d+{> | , @ lqi lqi d+{> | , @ lqi lqi d+{> | ,= {S[ |S{ U |S\ {SU| C{>|DSU C{>|DSU ipr xI qvr| Y Surri1 Ohw P @ vxsC{>|DSU d+{> | ,> Q{ =@ vxs|S{ U d+{> | ,> wkhq d+{> |, P iru doo +{> | , 5 U= Vxssrvh wkdw { 5 [> wkhq lpsolhv wkdw Q{ @ vxs|S{ U +{> | , P dqg khqfh +518, vxs vxs d+{> | , @ vxs Q{ P= {S[ |S{ U {S[ Vlploduo| iru dq| +{> | , 5 U> d+{> | , Q{ vxs Q{ @ vxs vxs d+{> | , {S [ {S[ |S{ U dqg wkhuhiruh +519, vxs d+{> | , vxs vxs d+{> | , @ P {S[ |S{ U C{>|DSU Htxdwlrqv +518, dqg +519, vkrz wkdw vxs d+{> |, @ vxs vxs d+{> |,= {S [ | S { U C{>|DSU Doo ri wkh rwkhu dvvhuwlrqv ri wkh Ohppd kdyh d vlplodu surri1 5161 Vxpv ri srvlwlyh ixqfwlrqv1 Lq wklv dqg wkh qh{w ihz vhfwlrqv/ ohw [ dqg \ eh wzr vhwv1 Zh zloo zulwh [ wr ghqrwh wkdw lv d qlwh vxevhw ri [= Ghqlwlrq 5191 Vxssrvh wkdw d = [ $ ^3> 4` lv d ixqfwlrq dqg I [ lv d vxevhw/ wkhq + , [ [ [ d@ d+{, @ vxs d+{, = I = {S I {S I Uhpdun 51:1 Vxssrvh wkdw [ @ Q @ i4> 5> 6> = = = j> wkhq R Q [ [ [ d@ d+q, =@ olp d+q,= Q 6R qdR qdR Q S S Lqghhg iru doo Q> QdR d+q, Q d> dqg wkxv sdvvlqj wr wkh olplw zh ohduq wkdw q R [ [ d+q, d= qdR Q Frqyhuvho|/ li Q> wkhq iru doo Q odujh hqrxjk vr wkdw i4> 5> = = = > Q j> zh SQ S kdyh d qdR d+q, zklfk xsrq sdvvlqj wr wkh olplw lpsolhv wkdw [ d R [ d+q, qdR dqg khqfh e| wdnlqj wkh vxsuhpxp ryhu zh ohduq wkdw R [ [ d d+q,= qdR Q S Uhpdun 51;1 Vxssrvh wkdw [ d ? 4> wkhq i{ 5 [ = d+{, A 3j lv dw prvw frxqw0 deoh1 Wr vhh wklv uvw qrwlfh wkdw iru dq| A 3> wkh vhw i{ = d+{, j pxvw eh qlwh S iru rwkhuzlvh [ d @ 41 Wkxv ^ R i{ = d+{, 4@nj i{ 5 [ = d+{, A 3j @ ndR hu SUQh yrpr ra rhr ur ` zklfk vkrzv wkdw i{ 5 [ = d+{, A 3j lv d frxqwdeoh xqlrq ri qlwh vhwv dqg wkxv frxqwdeoh1 Ohppd 51<1 Vxssrvh wkdw d> e = [ $ ^3> 4` duh wzr ixqfwlrqv/ wkhq [ [ [ +d . e, @ d. e dqg [ [ [ [ [ d @ d [ [ iru doo 3= L zloo rqo| suryh wkh uvw dvvhuwlrq/ wkh vhfrqg ehlqj hdv|1 Ohw [ eh d qlwh vhw/ wkhq [ [ [ [ [ +d . e, @ d. e d. e [ [ zklfk diwhu wdnlqj vxsv ryhu vkrzv wkdw [ [ [ +d . e, d. e= [ [ [ Vlploduo|/ li > [> wkhq [ [ [ [ [ [ d. e d. e@ +d . e, +d . e,= [ Wdnlqj vxsv ryhu dqg wkhq vkrzv wkdw [ [ [ d. e +d . e,= [ [ [ Wkhruhp 5143 +Prqrwrqh Frqyhujhqfh Wkhruhp,1 Vxssrvh wkdw iq = [ $ ^3> 4` lv dq lqfuhdvlqj vhtxhqfh ri ixqfwlrqv dqg i +{, =@ olp iq +{, @ vxs iq +{,= q6R q Wkhq olp q6R [ iq @ [ i [ [ S S Surri1 Ohw Vq @ [ iq dqg V @ [ i= Vlqfh iq ip i iru doo q p> lw iroorzv wkdw Vq Vp V zklfk vkrzv wkdw olpq6R Vq h{lvwv dqg lv ohvv wkdw V> l1h1 [ [ iq i= +51:, D =@ olp q6R [ [ S S Qrwlqj wkdw iq [ iq @ Vq D iru doo [ dqg lq sduwlfxodu/ [ iq D iru doo q dqg [= Ohwwlqj q whqg wr lqqlw| lq wklv htxdwlrq vkrzv wkdw [ i D iru doo [ RQ ipr xI qvr| dqg wkhq wdnlqj wkh vxs ryhu doo [ jlyhv [ [ +51;, i D @ olp iq q6R [ [ zklfk frpelqhg zlwk Ht1 +51;, vkrzv suryhv wkh wkhruhp1 Ohppd 5144 +Idwrx*v Ohppd,1 Vxssrvh wkdw iq = [ $ ^3> 4` lv d vhtxhqfh ri ixqfwlrqv/ wkhq [ [ olp lqi iq olp lqi iq = q6R q6R [ [ Surri1 Ghqh jn lqi iq vr wkdw jn % olp lqi q6R iq dv n $ 4= Vlqfh jn iq q$n iru doo n q zh kdyh [ [ jn iq iru doo q n [ [ dqg wkhuhiruh [ [ jn olp lqi iq iru doo n= q6R [ [ Zh pd| qrz xvh wkh prqrwrqh frqyhujhqfh wkhruhp wr ohw n $ 4 wr qg [ [ [ [ p olp lqi iq @ olp jn @ olp jn olp lqi iq = q6R q6R n6R n6R [ [ [ [ S [ d ? 4> wkhq iru doo A 3 wkhuh h{lvwv [ vxfk wkdw [ dD D iru doo [ frqwdlqlqj ru htxlydohqwo|/ [ +51<, d D S iru doo [ frqwdlqlqj 1 Lqghhg/ fkrrvh vr wkdw d D = Uhpdun 51451 Li D @ 5171 Vxpv ri frpsoh{ ixqfwlrqv1 Ghqlwlrq 51461 Vxssrvh wkdw d = [ $ F lv d ixqfwlrq/ zh vd| wkdw [ [ d@ d+{, [ {S[ h{lvwv dqg lv htxdo wr D 5 F> li iru doo A 3 wkhuh lv d qlwh vxevhw [ vxfk wkdw iru doo [ frqwdlqlqj zh kdyh [ d = D Ghqlwlrq 5147 +Vxppdeoh,1 Zh fdoo d ixqfwlrq d = [ $ F vxppdeoh li [ mdm ? 4= [ S S Sursrvlwlrq 51481 Ohw d = [ $ F eh d ixqfwlrq/ wkhq [ d h{lvwv l [ mdm ? 4> l1h1 l d lv vxppdeoh1 hu SUQh yrpr ra rhr ur RR S Surri1 Li [ mdm ? 4> xvlqj Uhpdunv 5145 zh pd| fkrrvh dq lqfuhdvlqj vh0 txhqfh ri qlwh vxevhwv q ri [> vxfk wkdw [ [ [ mdm mdm mdm 4@q ; q= q [ [ S Ohwwlqj Vq =@ q d zh kdyh iru p A q> [ [ mVp Vq m @ d mdm p qq p qq [ [ mdm mdm @ p q [ [ mdm . D mdm D p q 4 4 . pq zklfk whqgv wr 3 dv p> q $ 4= Wkxv iVp j lv d Fdxfk| vhtxhqfh dqg zh ohw D =@ olpp6R Vp =S Ohwwlqj p $ 4 lq wkh suhylrxv htxdwlrq dovr vkrzv wkdw 4 mD Vq m = q Li lv d qlwh vxevhw frqwdlqlqj q > wkhq [ [ [ d @ D d. d D q qq [ [ D d . mdm q qq mD Vq m . 4@q 5@q= S Wklv vkrzv wkdw S[ d h{lvwv dqg lv htxdo wr D= Frqyhuvho|/ li [ mdm @ 4 wkhq/ ehfdxvh mdm mUh dm . mLp dm > zh pxvw kdyh [ [ mUh dm @ 4 ru mLp dm @ 4= [ [ Wkxv lw vxfhv wr frqvlghu wkh fdvh zkhuh d = [ $ U lv d uhdo ixqfwlrq1 Zulwh d @ dF d zkhuh +5143, dF +{, @ pd{+d+{,> 3, dqg d +{, @ pd{+ d+{,> 3,= Wkhq mdm @ dF . d dqg 4@ [ mdm @ [ dF . [ d [ [ [ SF S zklfk vkrzv wkdw S hlwkhu d @ 4 ru d @ 4= Vxssrvh zlwk rxw orvv [ [ ri jhqhudolw| wkdw [ dF @ 4= Ohw [ Q =@ i{ 5 [ = d+{, 3j> wkhq zh nqrz SiB|ihcd|i vct|h|vc vu #( xtc} CiRdh3t P!r dc P!F |ihi t d vc|dwBi ti| u 9 S S B t7c '" f t |d| di7( 9 f u 7 w u dc f d 9 " di7c ( 4 dc ici q9 c' S " c' c' di7c ( i2t|t! ipr xI qvr| RS S wkdw [ Q d @ 4 zklfk phdqv wkhuh S qlwh vxevhwv q [ Q [ vxfk wkdw duh S d q iru doo q= Wklv vkrzv wkdw [ d fdq qrw h{lvw1 q Li d = [ $ U lv d vxppdeoh ixqfwlrq/ ohw d +{, eh ghqhg dv lq Ht1 +5143,/ wkhq [ [ [ d@ dF d [ [ [ dqg li d = [ $ F lv d vxppdeoh ixqfwlrq wkhq [ [ [ d@ Uh d . l Lp d= [ [ [ Xvlqj wkhvh wzr uhpdunv/ pdq| wkhruhpv derxw vxppdeoh ixqfwlrqv d = [ $ F pd| eh uhgxfhg wr wkhruhpv derxw vxppdeoh ixqfwlrqv d = [ $ ^3> 4,= Uhpdun 51491 Vxssrvh wkdw [ @ Q dqg d = Q $ F lv d vhtxhqfh/ wkhq lw lv qrw qhfhvvdulo| wuxh wkdw R [ [ d+q, @ d+q,= +5144, qdR qSQ Wklv lv ehfdxvh R Q [ [ d+q, @ olp d+q, Q 6R qdR qdR S ghshqgv rq wkh rughulqj ri wkh vhtxhqfh d zkhuh dv d+ qSQS q, grhv qrw1 Iru S h{dpsoh/ wdnh d+q, @ + 4,q @q wkhq qSQ md+q,m @ 4 l1h1 qSQ d+q, grhv qrw SR h{lvw zkloh qdR d+q, grhv h{lvw1 Rq wkh rwkhu kdqg/ li R [ [ md+q,m @ md+q,m ? 4 qdR qSQ wkhq Ht1 +5144, lv ydolg1 H{huflvh 514:1 Vxssrvh wkdw d> e = [ $ F duh wzr vxppdeoh ixqfwlrqv dqg 5 F= Vkrz [ [ [ +d . e, @ d. e= [ [ [ Wkhruhp 514; +Grplqdwhg Frqyhujhqfh Wkhruhp,1 Vxssrvh wkdw iq = [ $ F lv d vhtxhqfh ri ixqfwlrqv rq [ vxfk wkdw i +{, @ olpq6R iq +{, 5 F h{lvwv iru doo { 5 [= Ixuwkhu dvvxph wkhuh lv d grplqdwlqj ixqfwlrq j = [ $ ^3> 4, vxfk wkdw +5145, miq +{,m j +{, iru doo { 5 [ dqg q 5 Q dqg wkdw j lv vxppdeoh1 Wkhq +5146, olp q6R [ iq +{, @ [ i +{,= {S[ {S[ Surri1 Qrwlfh wkdw mi m @ olp miq m j vr wkdw i lv vxppdeoh1 E| frqvlghulqj wkh uhdo dqg lpdjlqdu| sduwv ri i vhsdudwho|/ lw vxfhv wr suryh wkh wkhruhp lq wkh fdvh zkhuh i lv uhdo1 E| Idwrx*v Ohppd/ [ [ [ +j i , @ olp lqi +j iq , olp lqi +j iq , q6R q6R [ [ [ # $ [ [ j . olp lqi iq = @ q6R [ [ hu SUQh yrpr ra rhr ur RT Vlqfh olp lqi q6R + dq , @ olp vxsq6R dq > zh kdyh vkrzq/ S [ [ [ olp lqi q6R [ iq S j i j. olp vxsq6R [ iq [ [ [ dqg wkhuhiruh [ [ [ olp vxs iq i olp lqi iq = q6R q6R [ [ [ S S i h{lvwv dqg lv htxdo wr [ i= Wklv vkrzv wkdw olp q6R [ q Surri1 +Vhfrqg Surri1, Sdvvlqj wr wkh olplw lq Ht1 +5145, vkrzv wkdw mi m j dqg lq sduwlfxodu wkdw i lv vxppdeoh1 Jlyhq A 3> ohw [ vxfk wkdw [ j = [ q Wkhq iru [ vxfk wkdw > zh kdyh [ [ [ i iq +i iq , [ [ mi iq m @ mi [ [ mi iq m . 5 j q [ mi iq m . 5= dqg khqfh wkdw [ i [ iq mi [ iq m . [ q mi iq m . 5= Vlqfh wklv odvw htxdwlrq lv wuxh iru doo vxfk [> zh ohduq wkdw [ [[ i iq mi iq m . 5 [ [ zklfk wkhq lpsolhv wkdw [ i olp vxs q6R [ [ [ [ iq olp vxs mi q6R @ 5= Ehfdxvh A 3 lv duelwudu| zh frqfoxgh wkdw [ [ olp vxs i iq @ 3= q6R [ [ zklfk lv wkh vdph dv Ht1 +5146,1 iq m . 5 iq m RU ipr xI qvr| 5181 Lwhudwhg vxpv1 Ohw [ dqg \ eh wzr vhwv1 H{huflvh 514<1 Vxssrvh wkdw d = [ $ F lv ixqfwlrq S I [ lv d vxevhw vxfk dqg S wkdw d+{, @ 3 iru doo { 5 I= Vkrz wkdw I d h{lvwv l [ d h{lvwv/ dqg li wkh vxpv @ h{lvw wkhq [ [ d@ d= [ I Wkhruhp 51531 Vxssrvh wkdw d = [ \ $ ^3> 4`> wkhq [[ [[ [ d@ d@ d= [ \ [\ \[ Surri1 Lw vxfhv wr vkrz/ e| v|pphwu|/ wkdw [ [[ d@ d [ \ [\ Ohw [ \ = Wkh iru dq| [ dqg \ vxfk wkdw > zh kdyh [ [ [[ [[ [[ d d@ d d d> \ [\ SS S l1h1 d [ \ d= Wdnlqj wkh vxs ryhu lq wklv odvw htxdwlrq vkrzv [[ [ d d= [ \ [\ S Zh pxvw qrz vkrz wkh rssrvlwh lqhtxdolw|1 Li [ \ d @ 4 zh duh grqh vr zh qrz dvvxph wkdw d lv vxppdeoh1 E| Uhpdun 51;/ wkhuh lv d frxqwdeoh vhw QR i+{Q > |q ,jqdR [ \ r ri zklfk d lv lghqwlfdoo| 3= Ohw D @ i{Q = q 5 Qj dqg q q R R Q E @ i|q = q 5 Qj dqg ohw i{q jqdR dqg i|q jqdR eh dq hqxphudwlrq ri D dqg E uhvshfwlyho|1 Zlwk wklv qrwdwlrq zh nqrz wkdw d+{> | , @ 3 iru doo +{> | , 5 D E > vr @ wkdw [ [ d@ d+{p > |q , [ \ Cp>qDSQ5 Lw lv dovr hdv| wr yhuli| wkdw [[ d@ [[ d+{p > |q ,= [\ pSQ qSQ Khqfh zh kdyh qrz uhgxfhg wkh sureohp wr wkh vshfldo fdvh zkhuh [ @ \ @ Q> vr zh qhhg wr suryh= li d = QS $ ^3> 4, lv vxppdeoh/ wkhq [[ [ d d+p> q,= pSQ qSQ Q5 Ohw P> Q 5 Q> wkhq PQ [ [[ d+p> q, d> pdR qdR Q5 PR P Q [[ [ [ d+p> q, @ olp d+p> q, pdR qdR pdR Q 6R qdR PQ [ [[ d+p> q, d= @ olp Q 6R pdR qdR Q5 hu SUQh yrpr ra rhr ur Vlploduo| RR [[ pdR qdR zklfk zdv wr eh suryhg1 d+p> q, @ olp P 6R PR [[ pdR qdR d+p> q, [ Q5 RV d Wkhruhp 51541 Qrz vxssrvh wkdw d = [ \ $ F lv d vxppdeoh ixqfwlrq/ l1h1 e| Wkhruhp 5153 dq| rqh ri wkh iroorzlqj htxlydohqw frqglwlrqv krog S +4, S[ \ mdm ? 4> S +5, S[ S \ mdm ? 4 ru +6, \ [ mdm ? 4= Wkhq [[ [[ [ d@ d@ d= [ \ [\ \[ Surri1 Zrunlqj dv lq wkh surri ri Wkhruhp 5153/ zh fdq uhgxfh wkh surri wr wkh fdvh zkhuh [ @ \ @ Q=T Khqfh zh qhhg wr vkrz RR [ [[ d+p> q, @ d= pdR qdR Q5 Ohw xv uvw qrwh wkdw iru doo p 5 Q R [ [ md+p> q,m mdm ? 4 qdR Q5 S vr wkdw R d+p> q, lv zhoo ghqhg1 Ixuwkhupruh/ qdR R R RR [[ [ [ d+p> q, md+p> q,m ? 4 pdR qdR pdR qdR SR SR vr wkdw pdR qdR d+p> q, lv zhoo ghqhg1 S Ohw A 3> wkhq e| wkh ghqlwlrq ri D =@ Q5 d> iru doo P> Q 5 Q vxflhqwo| odujh/ P Q [ [ d+p> q, D = pdR qdR Ohwwlqj Q $ 4 lq wklv htxdwlrq vkrzv wkdw PR [ [ d+p> q, D pdR qdR dqg wkhq ohwwlqj P $ 4 vkrzv R R [ [ d+p> q, D = pdR qdR Vlqfh A 3 lv duelwudu|/ zh duh grqh1 Tei tvh| hvvu vu |t |ivhiR t |v ht| hvi |i hitB| uvh hidB dBi uc|vct d wA dBAc} eivhiR P!Pf |v d |i vt|i dc ci}d|i dh|t vu d# ei |ivhiR |ic uvBBv"t uvh vRBi2 dBi uc|vct d wA dBAc} |i hidB ihtvc #t| hvi |v |i hidB dc Rd}cdhA dh|t vu d# ipr xI qvr| RW '; 'P '! 8} ij Qhvwhg Vhtxhqfh ri fxehv 5191 Frpsdfw Vhwv lq Uq 1 Wkhruhp 51551 Vxssrvh wkdw N lv d forvhg dqg erxqghg vxevhw ri Uq dqg wkdw Y lv d froohfwlrq ri rshq vxevhwv ri Uq zklfk fryhuv N lq wkh vhqvh wkdw N ^Y S Y> wkhq wkhuh lv qlwh vxefryhu I Y > l1h1 I lv d qlwh vxevhw ri Y vxfk wkdw N ^Y Ss Y= Surri1 Ohw xv uvw dvvxph wkdw N @ ^ P > P `q iru vrph P 5 Q= Ohw F =@ ^3> `q dqg iru dq lqwhjhu n A 3> ohw Tn =@ i{ . FSn = { 5 5n Q dqg { . FR@n N j= Ohw TR @ TR 1 Vlqfh N @ ^TS4 T> wkhuh pxvw eh vrph TR 5 TR iru zklfk qr qlwh vxevhw ri Y fryhuv TR = Ohw TS @ iT 5 TS = T TR j> wkhq vlqfh TR @ ^TS5 T> wkhuh pxvw eh vrph TS 5 TS iru zklfk qr qlwh vxevhw ri Y fryhuv TS = Wkhq ohw TT @ iT 5 TT = T TS j dqg fkrrvh TT 5 TT ru zklfk qr qlwh vxevhw ri Y fryhuv TT = Frqwlqxh wklv zd| e| lqgxfwlrq/ zh pd| frqvwuxfw fxehv Tl 5 Tl vxfk wkdw N TR TS TT = = = dqg qr Tl pd| eh fryhuhg e| d qlwh vxefroohfwlrq ri Y = Qrz fkrrvh {l 5 Tl dqg qrwlfh wkdw vlqfh wkh gldphwhu ri wkh Tl *v whqgv wr 3 dv l $ 4> lw iroorzv wkdw i{l jR lv d Fdxfk| vhtxhqfh dqg khqfh { =@ olpl6R {l ldR h{lvwv1 Vlqfh doo ri wkh Tl *v duh forvhg zh ixuwkhu nqrz wkdw { 5 _l Tl = Vlqfh Y lv d fryhu/ wkhuh lv dq rshq vhw Y 5 Y vxfk wkdw { 5 Y dqg vlqfh Y lv rshq dqg olpl6R gldp+Tl , @ 3> lw iroorzv wkdw Tl Y iru doo odujh l= Exw wklv frqwudglfwv wkh surshuw| wkdw qr Tl pd| eh fryhuhg e| d qlwh froohfwlrq ri vhwv iurp Y = Wklv suryhv wkh fdvh zkhuh N lv d fxeh1 Iru jhqhudo N / fkrrvh P 5 Q vr odujh wkdw N ^ P > P `q = Ohw Y @ Y ^ iN f j > lv dq rshq fryhu ri ^ P > P `q dqg khqfh wkhuh lv d qlwh vxevhw I Y vxfk wkhq Y wkdw I^ iN f j fryhuv ^ P > P `q dqg wkhuhiruh I lv d qlwh fryhu ri N= hu SUQh yrpr ra rhr ur RX Wkhruhp 51561 Vxssrvh wkdw N lv d forvhg dqg erxqghg vxevhw ri Uq dqg { @ R R i{q jqdR N lv d vhtxhqfh1 Wkhq { kdv d vxevhtxhqfh | @ i{qn jndR zklfk lv frqyhujhqw dqg olpn6R |n 5 N= Surri1 Fkrrvh P 5 Q vr odujh wkdw N ^ P > P `q = Frqwlqxlqj wkh qrwdwlrq dqg wkh phwkrg ri wkh surri ri Wkhruhp 5155/ zh pd| qg d fxeh TR 5 TR vxfk wkdw iq = {q 5 TR j lv dq lqqlwh vhw1 Vlploduo| zh pd| qg TS 5 TS vxfk wkdw iq = {q 5 TS j lv dq lqqlwh vhw1 Frqwlqxlqj lqgxfwlyho|/ zh pd| frqvwuxfw fxehv Tl vxfk wkdw TR TS TT = = = dqg iq = {q 5 Tl j lv dq lqqlwh vhw iru doo l= Qrz ohw qR =@ plqiq = {q 5 TR j> qS @ plqiq A qR = {q 5 TS j dqg vr rq/ vr wkdw qnFR =@ plqiq A qn = {q 5 TnFR j> wkhq | @ i{qn jR lv d vxevhtxhqfh ri { vxfk wkdw |l 5 Tl iru doo l= Dv lq wkh ndR suhylrxv surri/ zh frqfoxgh wkdw | lv d Fdxfk| vhtxhqfh dqg khqfh olpn6R |n h{lvwv dqg lv qhfhvvdulo| lq N vlqfh N lv forvhg1 51:1 Edvlf phwulf vsdfh qrwlrqv1 Ghqlwlrq 51571 D ixqfwlrq g = [ [ $ ^3> 4, lv fdoohg d phwulf li +4, +V|pphwu|, g+{> | , @ g+|> {, iru doo {> | 5 [ +5, +Qrq0ghjhqhudwh, g+{> |, @ 3 li dqg rqo| li { @ | 5 [ +6, +Wuldqjoh lqhtxdolw|, g+{> } , g+{> | , . g+|> } , iru doo {> |> } 5 [= Ghqlwlrq 51581 Ohw +[> g, eh d phwulf vsdfh1 Wkh rshq edoo E +{> , [ fhqwhuhg dw { 5 [ zlwk udglxv A 3 lv wkh vhw E +{> , =@ i| 5 [ = g+{> |, ? j= R Ghqlwlrq 51591 D vhtxhqfh i{q jqdR lq d phwulf vsdfh +[> g, lv vdlg wr eh frq0 yhujhqw li wkhuh h{lvwv d srlqw { 5 [ vxfk wkdw olpq6R g+{> {q , @ 3= Lq wklv fdvh zh zulwh olpq6R {q @ {= H{huflvh 515:1 Vkrz wkdw { lq Ghqlwlrq 5159 lv qhfhvvdulo| xqltxh1 Ghqlwlrq 515;1 D vhw I [ lv forvhg l hyhu| frqyhujhqw vhtxhqfh i{q jR qdR zklfk lv frqwdlqhg lq I kdv lwv olplw edfn lq I= D vhw Y [ lv rshq l Y f lv forvhg1 H{huflvh 515<1 Vkrz wkdw Y [ lv rshq l iru hyhu| { 5 Y wkhuh lv d A 3 vxfk wkdw E +{> , Y= Ghqlwlrq 5163 +Fdxfk| vhtxhqfhv,1 D vhtxhqfh i{q jR lq d phwulf vsdfh +[> g, qdR lv vdlg Fdxfk| surylghg wkdw olp g+{q > {p , @ 3= p>q6R H{huflvh 51641 Vkrz wkdw frqyhujhqw vhtxhqfhv duh dozd|v Fdxfk| vhtxhqfhv1 Wkh frqyhuvh lv qrw dozd|v wuxh1 Iru h{dpsoh/ ohw [ @ T eh wkh vhw ri udwlrqdo qxpehuv s dqg g+{> | , @ m{ | m= Fkrrvh d vhtxhqfh i{q jR T zklfk frqyhujhv wr 5 5 U> R lv +T> g, ~ Fdxfk| exw qrw +T>qdR~ frqyhujhqw1 Wkh vhtxhqfh grhv g, wkhq i{q jqdR frqyhujh lq U krzhyhu1 Ghqlwlrq 51651 D phwulf vsdfh +[> g, lv frpsohwh li doo Fdxfk| vhtxhqfhv duh frqyhujhqw vhtxhqfhv11 H{dpsoh 51661 Khuh duh d frxsoh ri h{dpsohv ri frpsohwh phwulf vsdfhv1 +4, [ @ U dqg g+{> | , @ m{ | m= ipr xI qvr| RY +5, [ @ Uq dqg g+{> | , @ n{ |n = +6, [ @ F +^3> 4`> U, ~ wkh vsdfh ri frqwlqxrxv ixqfwlrqv iurp ^3> 4` wr U dqg g+i> j, =@ pd{wSQ>R mi +w, j+w,m= +7, Khuh lv d w|slfdo h{dpsoh ri d qrq0frpsohwh phwulf vsdfh1 Ohw [ @ F +^3> 4`> U, dqg ]R mi +w, j +w,m gw= g+i> j, =@ Q Ohw xv yhuli| Lwhp 61 lq wkh suhylrxv h{dpsoh1 Vxssrvh wkdw iiq j [ lv d Fdxfk| vhtxhqfh/ wkhq iru hdfk { 5 ^3> 4`> iiq +{,j U lv Fdxfk| dqg khqfh frqyhujhqw1 Ghqh i +{, =@ olpq6R iq +{, iru hdfk { 5 ^3> 4`= Zh zloo qrz vkrzv wkdw i 5 [ dqg wkdw olpq6R g+i> iq , @ 3= Iluvw r/ jlyhq A 3 zh kdyh mi +{, iq +{,m mi +{, mi +{, ip +{,m . mip +{, iq +{,m ip +{,m . surylghg wkdw p> q Q iru vrph Q @ Q +, vxflhqwo| odujh +dqg qrw ghshqglqj rq {,1 Ohwwlqj p $ 4 lq wklv odvw htxdwlrq vkrzv wkdw iq +{,m iru doo q Q mi +{, vr wkdw iq +{, $ i +{, xqlirupo| lq { dv q $ 4= Vr wr qlvk wkh dujxphqw/ zh qhhg rqo| vkrzv wkdw i lv frqwlqxrxv1 Iru wklv ohw A 3 dqg Q eh dv deryh/ wkhq iru {> | 5 ^3> 4`> mi +{, i +| ,m mi +{, iQ +{,m . miQ +{, 5 . miQ +{, iQ +| ,m . miQ +|, i +| ,m iQ +|,m= Qrz iru { {hg/ e| wkh frqwlqxlw| ri iQ > zh pd| fkrrvh A 3 vr wkdw miQ +{, iQ +|,m ? surylghg wkdw m{ | m ? = Iru wklv zh wkhq kdyh mi +{, i +| ,m ? 5 . @ 6 zklfk vkrzv wkdw i lv frqwlqxrxv dv zhoo1 Vxssrvh wkdw +[> g, dqg +\> , duh wzr phwulf vsdfhv dqg i = [ $ \ lv d ixqfwlrq1 Ghqlwlrq 51671 D ixqfwlrq i = [ $ \ lv frqwlqxrxv dw { 5 [ li iru doo A 3 wkhuh lv d A 3 vxfk wkdw +i +{,> i +{Q ,, ? surylghg wkdw g+{> {Q , ? = Wkh ixqfwlrq i lv vdlg wr eh frqwlqxrxv li i lv frqwlqxrxv dw doo srlqwv { 5 [= H{huflvh 51681 Vkrz wkdw i lv frqwlqxrxv dw { l iru doo vhtxhqfhv i{q jR [ qdR frqyhujlqj wr {> wkh vhtxhqfh ii +{q ,jR \ vkrxog frqyhujh wr i +{,= Sxw eulh|/ qdR olp i +{q , @ i + olp {q ,= q6R q6R H{huflvh 51691 Vxssrvh wkdw i = [ $ \ lv d frqwlqxrxv ixqfwlrq dqg wkdw F \ lv forvhg vhw/ vkrz wkdw i R +F , lv wkhq forvhg lq [= Wkh qh{w ohppd vxssolhv vrph h{dpsohv ri frqwlqxrxv ixqfwlrqv rq phwulf vsdfhv1 hu SUQh yrpr ra rhr ur R` Ohppd 516:1 Iru dq| qrq hpsw| vxevhw D [ / ohw gD +{, lqi ig+{> d,md 5 Dj> wkhq +5147, mgD +{, gD +| ,m g+{> | , ; {> | 5 [= Lq sduwlfxodu/ gD lv d frqwlqxrxv ixqfwlrq rq [ 1 Pruhryhu/ e| H{huflvh 5169/ iru doo A 3 wkh vhw I i{ 5 [ mgD +{, j lv forvhg lq [ 1 Ixuwkhu/ li Y lv dq rshq vhw dqg D @ Y f > wkhq I % Y dv & 31 Surri1 Ohw d 5 D dqg {> | 5 [ / wkhq g+{> d, g+{> | , . g+|> d,= Wdnh wkh lqi ryhu d lq wkh deryh htxdwlrq vkrzv wkdw gD +{, g+{> | , . gD +| , ;{> | 5 [= Wkhuhiruh/ gD +{, gD +| , g+{> |, dqg e| lqwhufkdqjlqj { dqg | zh dovr kdyh wkdw gD +| , gD +{, g+{> | , zklfk lpsolhv Ht1 +5147, iurp zklfk lw iroorzv wkdw gD lv frqwlqxrxv rq [ 1 Qrz vxssrvh wkdw D @ Y f zlwk Y 5 1 Lw lv fohdu wkdw gD +{, @ 3 iru { 5 D @ Y f vr wkdw I Y iru hdfk A 3 dqg khqfh ^AQ I Y 1 Qrz vxssrvh wkdw { 5 Y / wkhq wkhuh h{lvwv dq A 3 vxfk wkdw E{ +, Y / wkdw lv lw | 5 [ vxfk wkdw g+{> | , ? wkhq | 5 Y 1 Wkhuhiruh g+{> |, iru doo | 5 Y f dqg khqfh { 5 I / l1h1 Y ^AQ I 1 Ilqdoo| lw lv fohdu wkdw I IQ zkhqhyhu Q 1 Fruroodu| 516;1 Wkh ixqfwlrq g vdwlvhv/ mg+{> |, dqg lq sduwlfxodu g = [ g+{Q > | Q ,m g+|> | Q , . g+{> {Q , [ $ ^3> 4, lv frqwlqxrxv1 Surri1 E| Ohppd 516: iru vlqjoh srlqw vhwv dqg wkh wuldqjoh lqhtxdolw| iru wkh devroxwh ydoxh ri uhdo qxpehuv/ mg+{> | , g+{Q > |Q ,m mg+{> |, g+{> | Q ,m . mg+{> |Q , g+|> |Q , . g+{> {Q ,= 61 CNVN,N}jte ,}jOiBt BAa g+{Q > | Q ,m ,}jOiBt Ghqlwlrq 6141 D froohfwlrq ri vxevhwv ri [ lv d wrsrorj| li +4, >> [ 5 V Y 5 1 +5, lv forvhg xqghu duelwudu| xqlrqv/ l1h1 li Y 5 > iru 5 L wkhq SL +6, lv forvhg xqghu qlwh lqwhuvhfwlrqv/ l1h1 li YR > = = = > Yq 5 wkhq YR _ _Yq 5 = Qrwdwlrq 6151 Wkh vxevhwv Y [ zklfk duh lq duh fdoohg rshq vhwv dqg zh zloo deeuhyldwh wklv e| zulwlqj Y Q [ dqg wkh wkrvh vhwv I [ vxfk wkdw I f 5 duh fdoohg forvhg vhwv1 Zh zloo zulwh I C [ li I lv d forvhg vxevhw ri [= Ghqlwlrq 6161 D froohfwlrq ri vxevhwv D ri [ lv dq Dojheud li +4, >> [ 5 D +5, D 5 D lpsolhv wkdw Df 5 D +6, D lv forvhg xqghu qlwh xqlrqv/ l1h1 li DR > = = = > Dq 5 D wkhq DR _ +7, D lv forvhg xqghu qlwh lqwhuvhfwlrqv1 _Dq 5 D= SQ ipr xI qvr| 1 2 3 8} ij 4 D wrsrorj| Ghqlwlrq 6171 D froohfwlrq ri vxevhwv P ri [ lv d ~ dojheud + ~ hog, li P lv dq dojheud zklfk dovr forvhg xqghu frxqwdeoh xqlrqv/ l1h1 li iDl jR P> wkhq ldR ^R Dl 5 P= ldR Qrwlfh wkdw vlqfh P lv dovr forvhg xqghu wdnlqj frpsolphqwv/ P lv dovr forvhg xqghu wdnlqj frxqwdeoh lqwhuvhfwlrqv1 H{dpsoh 6181 Khuh duh d qxpehu ri h{dpsohv1 +4, @ P @ S +[ , lq zklfk fdvh doo vxevhwv ri [ duh rshq/ forvhg/ dqg phd0 vxudeoh1 +5, Ohw [ @ i4> 5> 6j> wkhq @ i>> [> i5> 6jj lv d wrsrorj| rq [ zklfk lv qrw dq dojheud1 +6, @ D @ ii4j> i5> 6j> >> [ j= lv d wrsrorj|/ dq dojheud/ dqg d ~ dojheud rq [= Wkh vhwv [> i4j> i5> 6j> ! duh rshq dqg forvhg1 Wkh vhwv i4> 5j dqg i4> 6j duh qhlwkhu rshq qru forvhg dqg duh qrw phdvxudeoh11 Sursrvlwlrq 6191 Ohw H eh dq| froohfwlrq ri vxevhwv ri [ 1 Wkhq wkhuh h{lvwv d xqltxh vpdoohvw wrsrorj| +H ,/ dojheud D+H , dqg 0dojheud P+H , zklfk frqwdlqv H = L zloo dovr whqg wr zulwh +H , iru P+H ,> l1h1 +H , @ P+H ,= Wkh qrwdwlrq P+H , lv xvhg lq Iroodqg/ exw +H , lv wkh pruh vwdqgdug qrwdwlrq1 Surri1 Qrwh S +[ , lv d wrsrorj| dqg dq dojheud dqg d 0dojheud dqg H S +[ ,/ vr wkdw H lv dozd|v d vxevhw ri d wrsrorj|/ dojheud/ dqg ~ dojheud1 Rqh pd| qrz hdvlo| fkhfn wkdw _ +H , i = lv d wrsrorj| dqg H j lv d wrsrorj| zklfk lv fohduo| wkh vpdoohvw wrsrorj| frqwdlqlqj H = Wkh dqdorjrxv frqvwuxfwlrq zrunv iru wkh rwkhu fdvhv dv zhoo1 Zh pd| jlyh h{solflw ghvfulswlrqv ri +H , dqg D+H ,= Sursrvlwlrq 61:1 Ohw [ dqg H S +[ ,= Iru vlpsolflw| ri qrwdwlrq/ dvvxph wkdw [> > 5 H +rwkhuzlvh dgmrlq wkhp wr H li qhfhvvdu|, dqg ohw H f iDf = D 5 Hj dqg Hf @ H ^ i[> >j ^ H f Wkhq +614, +H , @ iduelwudu| xqlrqv ri qlwh lqwhuvhfwlrqv ri hohphqwv iurp Hj hu SUQh yrpr ra rhr ur SR dqg +615, D+H , @ iqlwh xqlrqv ri qlwh lqwhuvhfwlrqv ri hohphqwv iurp Hf j= Surri1 Iurp wkh ghqlwlrq ri d wrsrorj| dqg dq dojheud/ lw lv fohdu wkdw +H , dqg D+H , frqwdlq wkrvh vhwv lq wkh uljkw vlgh ri Htv1 +614, dqg +615, uhvshfwlyho|1 Khqfh lw vxfhv wr vkrz wkdw wkh uljkw phpehuv ri Htv1 +614, dqg +615, irup d wrsrorj| dqg dq dojheud uhvshfwlyho|1 Wkh surri ri wkhvh dvvhuwlrqv duh urxwlqh h{fhsw iru srvvleo| vkrzlqj wkdw D =@ iqlwh xqlrqv ri qlwh lqwhuvhfwlrqv ri hohphqw iurp Hf j lv forvhg xqghu frpsohphqwdwlrq1 Wr fkhfn wklv/ zh qrwlfh wkdw wkh w|slfdo hohphqw ] 5 D lv ri wkh irup QN ^_ Dlm ]@ ldR mdR zkhuh Dlm 5 Hf = Wkhuhiruh/ zulwlqj Elm @ Df 5 Hf > zh qg wkdw lm ]f @ QN _^ +ERm4 _ ESm5 _ _ EQmQ , 5 D+H , m4 >===>mQ dR zkhuhlq zh kdyh xvhg wkh idfw wkdw ERm4 _ ESm5 _ _ EQmQ lv d qlwh lqwhuvhfwlrq ri vhwv iurp Hf = ldR m dR Elm @ N ^ Uhpdun 61;1 Rqh pljkw wklqn wkdw lq jhqhudo P+H , pd| eh ghvfulehg dv wkh frxqw0 deoh xqlrqv ri frxqwdeoh lqwhuvhfwlrqv ri vhwv lq H f Krzhyhu wklv lv idovh/ vlqfh li ]@ zlwk Dlm 5 Hf > wkhq ]f @ RR ^_ ldR mdR R ^ Dlm #R _ $ ERmc m4 dR>m5 dR>===mQ dR>=== cdR zklfk lv qrz dq xqfrxqwdeoh xqlrq1 Wkxv wkh deryh ghvfulswlrq lv qrw fruuhfw1 Lq jhqhudo lw lv idluo| frpsolfdwhg wr h{solflwo| ghvfuleh P+H ,> vhh Sursrvlwlrq 4156 rq sdjh 6< iru ghwdlov1 Wkh iroorzlqj qrwlrq zloo eh xvhixo lq wkh vhtxho1 Ghqlwlrq 61<1 D vhw H S +[ , lv vdlg wr eh dq hohphqwdu| idplo| ru hoh0 phqwdu| fodvv surylghg wkdw >5H H lv forvhg xqghu qlwh lqwhuvhfwlrqv li H 5 H > wkhq H f lv d qlwh glvmrlqw xqlrq ri vhwv iurp H = Sursrvlwlrq 61431 Vxssrvh H S +[ , lv dq hohphqwdu| idplo|/ wkhq D @ D+H , frqvlvwv ri vhwv zklfk pd| eh zulwwhq dv qlwh glvmrlqw xqlrqv ri vhwv iurp H = Surri1 +Iluvw Surri1, E| Sursrvlwlrq 61: D+H , @ iqlwh xqlrqv ri qlwh lqwhuvhfwlrqv ri hohphqwv iurp Hf j> SS ipr xI qvr| zkhuh H f iDf = D 5 Hj dqg Hf @ H ^ i[> >j ^ H f = Xvlqj wkh ghqlwlrq ri dq hohphqwdu| idplo| zh vhh wkdw D+H , pd| eh ghvfulehg pruh vlpso| dv D+H , @ iqlwh xqlrqv ri hohphqwv iurp Hj= q Hl 5 D+H , zlwk Hl 5 H = Wr qlvk wkh surri zh qhhg wr vkrz wkdw D Ohw D @ ^ldR pd| eh zulwwhq glvmrlqw xqlrq ri hohphqwv iurp H = Zh suryh wklv e| lqgxfwlrq rq q= Iru q @ 4 dqg D @ HR wkhuh lv qrwklqj wr suryh1 Li q @ 5 dqg D @ HR ^ HS > ohw fc HS @ ndR Il zlwk Il 5 H = Wkhq l n d f HS qHR @ HS _ HR @ HS _ Il ldR vr wkdw #n $ d HS _ Il D @ HR ^ ldR lv wkh ghvluhg ghfrpsrvlwlrq1 Qrz iru wkh lqgxfwlrq vwhs/ vxssrvh wkdw D @ ^q Hl @ E ^ Hq @ +E qHq , ^ Hq ldR cQ Q zkhuh Hm H duh sdluzlvh glvmrlqw1 Zulwh Hq @ cn Il Q f zkhuh E @ m dR Hm ldR zlwk Il 5 H > wkhq n nQ d dd f Q E qHq @ E _ Hq @ E _ Il @ Hm _ Il ldR ldR m dR dqg khqfh 3 4 nQ d dd Q D@C Hm _ Il D Hq ldR m dR lv wkh ghvluhg ghfrpsrvlwlrq1 +Vhfrqg pruh gluhfw surri1, Ohw D ghqrwh wkh froohfwlrq ri vhwv zklfk pd| eh zulwwhq dv qlwh glvmrlqw xqlrqv ri vhwv iurp H = Fohduo| D D+H , vr lw vxfhv wr vkrz wkdw D lv dq dojheud1 E| wkh surshuwlhv ri H > zh nqrz wkdw >> [ 5 D= Ixuwkhu/ c li D @ q Hl zlwk Hl 5 D> wkhq ldR f Df @ _q Hl = ldR Vlqfh H lv dq hohphqwdu| fodvv/ iru hdfk l wkhuh h{lvwv d froohfwlrq ri glvmrlqw vhwv fc iIlm jm H vxfk wkdw Hl @ m Ilm = Wkhuhiruh/ ^ Df @ _q +^m Ilm , @ +IRm4 _ ISm5 _ _ Iq>mq , ldR m4 >m5 >===>mq dqg wklv lv d glvmrlqw xqlrq1 c Khqfh Df 5 D> l1h1 D lv forvhg xqghu frpsohphqwdwlrq1 Qrz vxssrvh wkdw Dl @ m Ilm 5 D iru l @ 4> 5> = = = > q> wkhq ^ +IRm4 _ ISm5 _ _ Iq>mq , _l Dl @ m4 >m5 >===>mq zklfk lv djdlq d glvmrlqw xqlvrq ri vhwv iurp H vr wkdw D lv forvhg xqghu qlwh lqwhuvhfwlrqv1 hu SUQh yrpr ra rhr ur ST g{} { g{} } | 8} ij Ilwwlqj edoov lq wkh lqwhuvhfwlrq1 H{huflvh 61441 Ohw D S +[ , dqg E S +\ , eh hohphqwdu| idplolhv1 Vkrz wkh froohfwlrq H @ D E @ iD E = D 5 D dqg E 5 Ej lv dovr dq hohphqwdu| idplo|1 Ghqlwlrq 61451 Ohw +[> , eh d phwulf vsdfh1 Zh dvvrfldwh d wrsrorj|/ > wr e| vhwwlqj =@ +H , zkhuh H @ iE +{> , = { 5 [ dqg A 3j= Sursrvlwlrq 61461 D vhw Y [ lv lq > l1h1 Y lv rshq/ l Y lv d xqlrq ri rshq edoov1 Vr pd| dovr eh ghvfulehg dv +616, =@ iY [ = ; { 5 Y < u A 3 6 E +{> u, Y j ^ >= Surri1 Ohw xv uvw qrwlfh wkdw E +{> , dqg E +|> , duh wzr rshq edoo lq [ dqg } 5 E +{> , _ E +|> ,> wkhq +617, E +}> , E +{> , _ E +|> , zkhuh @ plqi g+{> } ,> g+|> } ,j> vhh Iljxuh 61 Wklv lv d irupdo frqvhtxhqfh ri wkh wuldqjoh lqhtxdolw|1 Iru h{dpsoh ohw xv vkrz wkdw E +}> , E +{> ,= E| wkh ghqlwlrq ri > zh kdyh wkdw g+{> } , ru wkdw g+{> } , = Khqfh li z 5 E +}> ,> wkhq g+{> z, g+{> } , . g+}> z, . g+}> z, ? .@ zklfk vkrzv wkdw z 5 E +{> ,= Vlploduo| zh vkrz wkdw z 5 E +|> , dv zhoo1 Htxdwlrq +617, pd| eh jhqhudol}hg wr qlwh lqwhuvhfwlrq ri edoov/ qdpho| li {l 5 [> l A 3 dqg } 5 _q E +{l > l ,> wkhq ldR +618, E +}> , _q E +{l > l , ldR zkhuh qrz =@ plq i l g+{l > } , = l @ 4> 5> = = = > qj = E| Ht1 +618, lw iroorzv wkdw dq| qlwh lqwhuvhfwlrq ri rshq edoov pd| eh zulwwhq dv d xqlrq ri rshq edoov/ wkxv zh vhh e| Sursrvlwlrq 61: wkdw =@ +H , lv jlyhq dv wkh uljkw kdqg vlgh ri Ht1 +616,1 ipr xI qvr| SU 1 2 3 8} ij g D froohfwlrq ri vxevhwv1 Sursrvlwlrq 61471 Li H S +[ , lv frxqwdeoh wkhq +H , P+H , @ +H ,= Lq sduwlfxodu + +H ,, @ +H ,= Surri1 Ohw Hi ghqrwh wkh froohfwlrq ri vxevhwv ri [ zklfk duh qlwh lqwhuvhfwlrq ri hohphqwv iurp H dorqj zlwk [ dqg >= Qrwlfh wkdw Hi lv vwloo frxqwdeoh +|rx suryh,1 V D D vhw ] lv lq +H , l ] lv dq duelwudu| xqlrq ri vhwv iurp Hi = Wkhuhiruh ] @ DSs iru vrph vxevhw I Hi zklfk lv qhfhvvdulo| frxqwdeoh1 Vlqfh Hi P+H , dqg P+H , lv forvhg xqghu frxqwdeoh xqlrqv lw iroorzv wkdw ] 5 P+H , dqg khqfh wkdw +H , P+H ,= H{dpsoh 61481 Vxssrvh wkdw H @ i>> [> i4> 5j> i4> 6jj> Wkhq +H , @ i>> [> i4j> i4> 5j> i4> 6jj D+H , @ P+H , @ S +[ ,= H{dpsoh 61491 Ohw [ eh d vhw dqg H @ iDR > = = = > Dq j ^ i[> >j zkhuh DR > = = = > Dq lv d sduwlwlrq ri [> l1h1 [ @ ^qdR Dm dqg Dl _ Dm @ > li l 9@ m= Lq wklv fdvh m D+H , @ P+H , @ +H , @ i^lS Dl = i4> 5> = = = > qjj zkhuh ^lS Dl =@ > zkhq @ >= Qrwlfh wkdw &D+H , @ &+S +i4> 5> = = = > qj,, @ 5q = Sursrvlwlrq 614:1 Vxssrvh wkdw P S +[ , lv d ~ dojheud dqg P lv dw prvw d frxqwdeoh vhw1 Wkhq wkhuh h{lvwv d xqltxh qlwh sduwlwlrq I ri [ vxfk wkdw I P dqg hyhu| hohphqw D 5 P lv ri wkh irup +619, D @ ^SsT)D = Lq sduwlfxodu P lv dfwxdoo| d qlwh vhw1 Surri1 Iru hdfk { 5 [ ohw D{ @ +_{SDSh D, 5 D= Wkdw lv/ D{ lv wkh vpdoohvw vhw lq D zklfk frqwdlqv {= Vxssrvh wkdw F @ D{ _ D| lv qrq0hpsw|1 Li { 5 F wkhq { 5 D{ q F 5 D dqg khqfh D{ D{ q F zklfk vkrzv @ wkdw D{ _ F @ > zklfk lv d frqwudglfwlrq1 Khqfh { 5 F dqg vlploduo| | 5 F> wkhuhiruh D{ F @ D{ _ D| dqg D| F @ D{ _ D| zklfk vkrzv wkdw D{ @ D| = hu SUQh yrpr ra rhr ur SV Wkhuhiruh/ I @ iD{ = { 5 [ j lv d sduwlwlrq ri [ +zklfk lv qhfhvvdulo| frxqwdeoh, dqg Ht1 +619, krogv iru doo D 5 P= Ohw I @ iSq jQdR zkhuh iru wkh prphqw zh q doorz Q @ 4= Li Q @ 4> wkhq P lv rqh wr rqh fruuhvsrqghqfh zlwk i3> 4jQ = Lqghhg wr hdfk d 5 i3> 4jQ > ohw Dd 5 P eh ghqhg e| Dd @ ^iSq = dq @ 4j= Q Wklv vkrzv wkdw P lv xqfrxqwdeoh vlqfh i3> 4j lv xqfrxqwdeoh/ wklqn ri wkh edvh wzr h{sdqvlrq ri qxpehuv lq ^3> 4` iru h{dpsoh1 Wkxv dq| frxqwdeoh ~ dojheud lv qhfhvvdulo| qlwh1 Wklv qlvkhv wkh surri prgxor wkh xqltxhqhvv dvvhuwlrq zklfk lv ohiw dv dq h{huflvh wr wkh uhdghu1 Xqiruwxqdwho|/ dv douhdg| phqwlrqhg wkh vwuxfwxuh ri jhqhudo ~ dojheudv lv qrw vr vlpsoh1 H{dpsoh 614;1 Ohw [ @ U dqg H @ i+d> 4, = d 5 Uj ^ iU> >j S +U,= Qrwlfh wkdw Hi @ H dqg wkdw H lv forvhg xqghu xqlrqv/ zklfk vkrzv wkdw +H , @ H / l1h1 H lv douhdg| d wrsrorj|1 Vlqfh +d> 4,f @ + 4> d` zh qg wkdw Hf @ i+d> 4,> + 4> d`> 4 d ? 4j ^ iU> >j1 Qrwlqj wkdw +d> 4, _ + 4> e` @ +d> e` lw lv hdv| wr yhuli| wkdw wkh dojheud D+H , jhqhudwhg e| H pd| eh ghvfulehg dv ehlqj wkrvh vhwv zklfk duh qlwh glvmrlqw xqlrqv ri vhwv iurp wkh iroorzlqj olvw i+d> 4,> + 4> d`> +d> e` = d> e 5 Uj ^ i>> Uj = Wkh ~ dojheud/ +H ,> jhqhudwhg e| H lv yhu| frpsolfdwhg1 Khuh duh vrph vhwv lq +H ,= R V R +d> e q ` 5 +H ,= +d, +d> e, @ qdR +e, Doo ri wkh vwdqgdug rshq vxevhwv ri U duh lq +H ,= W R { q > { 5 +H , +f, i{j @ q +g, ^d> e` @ idj ^ +d> e` 5 +H , +h, Dq| frxqwdeoh vxevhw ri U lv lq +H ,1 H{huflvh 614<1 Vkrz wkdw wkh iroorzlqj ~ dojheudv rq U duh doo wkh vdph= +vwdqgdug rshq vhwv, ~ wkh Eruho ~ dojheud +i+d> 4, = d 5 Uj , +i+d> 4, = d 5 Tj , +i^d> 4, = d 5 Tj,= 71 jBt ijt Ghqlwlrq 7141 D vhw [ htxlsshg zlwk d ~ dojheud P lv fdoohg d phdvxudeoh vsdfh1 Ghqlwlrq 7151 D phdvxuh rq d phdvxudeoh vsdfh +[> P, lv d ixqfwlrq = P $ ^3> 4` vxfk wkdw d= +>, @ 3 dqg S V e= + Dl , @ +Dl , li Dl 5 P dqg Dl _ Dm @ > zkhq l 9@ m= l SW ipr xI qvr| Sursrvlwlrq 716 +Edvlf surshuwlhv ri phdvxuhv,1 Vxssrvh wkdw +[> P> , lv d phd0 R vxuh vsdfh dqg H> I 5 P dqg iHm jm dR P> wkhq = +4, Li H I wkhq +H , +I ,= S +5, +^Hm , +Hm ,= +6, Li Hm % H> l1h1 HR HS HT = = = dqg H @ ^m Hm > wkhq +Hm , % +H , dv m $ 4= +7, Li +HR , ? 4 dqg Hm ) H / l1h1 HR HS HT = = = dqg H @ _m Hm > wkhq +Hm , ) +H , dv m $ 4= Surri1 +4, Ghqh I @ H ^ +I q H ,> wkhq +I , @ +H , . +I q H , +H ,= h +5, Ohw Hm @ Hm q +HR ^ ^ Hm R , vr wkdw wkh Hm *v duh sdlu0zlvh glvmrlqw dqg hm = Vlqfh Hm Hm lw iroorzv iurp Ghqlwlrq 715 dqg sduw +4,> wkdw H @ ^H [ [ h +H , @ +Hm , +Hm ,= h h +6, Djdlq ohw Hm @ Hm q +HR ^ ^ HmR , @ Hm q Hm R > wkhq H @ ^Hl @ ^Hl dqg wkxv R Q [ [ h h +Hl , @ olp +Hl , @ olp +HQ ,= +H , @ Q 6R Q 6R ldR ldR +7, Ghqh Gl HR q Hl wkhq Gl % HR q H zklfk lpsolhv wkdw +HR , +H , @ olp +Gl , @ +HR , olp +Hl , l6R l6R zklfk vkrzv wkdw olpl6R +Hl , @ +H ,1 Ghqlwlrq 7171 D vhw H 5 P lv d qxoo vhw li +H , @ 31 Ghqlwlrq 7181 D phdvxuh vsdfh +[> P> , lv frpsohwh li hyhu| vxevhw ri d qxoo vhw lv lq P/ l1h1 ;I [ vxfk wkdw I H 5 P zlwk +H , @ 3 lpsolhv wkdw I 5 P1 Sursrvlwlrq 7191 Ohw +[> P> , eh d phdvxuh vsdfh1 Vhw Q iQ [ =wkhuh h{lvwv I 5 P vxfk wkdw Q I dqg +I , @ 3j1 P @ iD ^ Q = D 5 P> Q 5 Pj> vhh Ilj1 81 Wkhq P lv d 0dojheud1 Ghqh +D ^ Q , @ +D,/ wkhq lv wkh xqltxh phdvxuh rq P zklfk h{whqgv 1 Surri1 Fohduo| [> > 5 P1 Ohw D 5 P dqg Q 5 Q dqg fkrrvh I 5 P vxfk wkdw Q I dqg +I , @ 31 Vlqfh Q f @ +I q Q , ^ I f > +D ^ Q ,f @ Df _ Q f @ Df _ +I q Q ^ I f , @ ^Df _ +I q Q ,` ^ ^Df _ I f ` zkhuh ^Df _ +I q Q ,` 5 Q dqg ^Df _ I f ` 5 P= Wkxv P lv forvhg xqghu frpsolphqwv1 Li Dl 5 P dqg Ql Il 5 P vxfk wkdw +Il , @ 3 wkhq ^+Dl ^ Ql , @ +^Dl , ^ S +^Ql , 5 P vlqfh ^Dl 5 P dqg ^Ql ^Il dqg +^Il , +Il , @ 3= Wkhuhiruh/ P lv d 0dojheud1 Vxssrvh D ^ QR @ E ^ QS zlwk D> E 5 P dqg QR > QS > 5 Q 1 Wkhq D D ^ QR D ^ QR ^ IS @ E ^ IS zklfk vkrzv wkdw +D, +E , . +IS , @ +E ,= Vlploduo|/ zh vkrz wkdw +E , +D, vr wkdw +D, @ +E , dqg khqfh +D ^ Q , =@ +D, lv zhoo ghqhg1 Lw lv ohiw dv dq h{huflvh wr vkrz wkdw lv d phdvxuh/ l1h1 wkdw lw lv frxqwdeoh dgglwlyh1 hu SUQh yrpr ra rhr ur Q SX I D 8} ij F Frpsohwlqj d ~ dojheud1 7141 Uhjxodulw| Uhvxowv1 Wkh txhvwlrq qrz lv krz gr zh ghvfuleh phdvxuhv rq jhqhudo ~ dojheudv1 Wklv lv wulfn| ehfdxvh wkhuh whqgv wr eh qr _h{solflw% gh0 vfulswlrq ri wkh jhqhudo hohphqw ri wkh| w|slfdo ~ dojheudv1 Rq wkh rwkhu kdqg/ zh gr nqrz krz wr h{solflwo| ghvfuleh dojheudv zklfk duh jhqhudwhg e| vrph fodvv ri vhw H S +[ ,= Wkhuhiruh/ zh pljkw wu| wr ghqh phdvxuhv rq +H , e| wkhuh uhvwulfwlrqv wr D+H ,= Wkh qh{w wkhruhp vkrzv wkdw wklv lv d sodxvleoh phwkrg lq pdq| vlwxdwlrqv1 Wkhruhp 71: +Xqltxqhvv,1 Vxssrvh wkdw D S +[ , lv dq dojheud ri vhwv dqg P @ +D, lv wkh ~ dojheud jhqhudwhg e| D= Li dqg duh wzr qlwh phdvxuhv rq P vxfk wkdw +D, @ +D, iru doo D 5 D> wkhq @ rq P= Pruh jhqhudoo|/ wkh wkhruhp krogv zkhq dqg duh ~ qlwh phdvxuhv rq D> wkdw lv wkhuh duh vhw [q 5 D vxfk wkdw +[q , @ +[q , ? 4 dqg ^[q @ [= Wkh iroorzlqj ghqlwlrq dqg whfkqlfdo ohppd zloo eh xvhixo lq wkh surri1 Ghqlwlrq 71; +Prqrwrqh Fodvv,1 F S +[ , lv d prqrwrqh fodvv li lw lv forvhg xqghu lqfuhdvlqj xqlrqv dqg ghfuhdvlqj lqwhuvhfwlrqv1 H{dpsoh 71<1 Ohw > eh wzr phdvxuh rq +[> P,1 Ohw F @ iD [ = +D, @ +D,j= Dvvxph [ 5 F dqg +[ , @ +[ , ? 41 Wkhq F lv d prqrwrqh fodvv1 Ohppd 71431 Vxssrvh D S +[ , lv dq dojheud dqg F lv wkh vpdoohvw prqrwrqh fodvv frqwdlqlqj D1 Wkhq F @ +D,1 Surri1 Iru D 5 F ohw F +D, @ iE 5 FmD _ E> D _ E f > E _ Df duh lq Fj1 Wkhq F +D, lv d prqrwrqh fodvv frqwdlqhg lq F = Pruhryhu li D 5 D wkhq D F +D, lpsolhv F +D, @ F iru doo D 5 D1 Qrz E 5 F +D, / D 5 F +E ,1 Vlqfh F +D, @ F iru doo D 5 D> D 5 F +E , iru doo E 5 F dqg D 5 F 1 Wkxv F +E , @ F iru doo E 5 F 1 Vr zh kdyh vkrzq/ li D> E 5 F wkhq D _ E> D _ E f > Df _ E duh edfn lq F 1 Lq sduwlfxodu/ F lv forvhg xqghu frpsolphqwv dqg qlwh lqwhuvhfwlrqv1 Vr F lv dq dojheud zklfk lv dovr forvhg xqghu lqfuhdvlqj xqlrqv dqg khqfh lw lv d 0dojheud1 Surri1 +Surri ri Wkhruhp 71:1, Dvvxph wkdw +[ , @ +[ , ? 4= Ohw G @ iD 5 P = +D, @ +D,j ipr xI qvr| SY Wkh uhdghu pd| hdvlo| fkhfn wkdw G lv d prqrwrqh fodvv1 Vlqfh D G> wkh prqrwrqh fodvv ohppd dvvhuwv wkdw P @ +D, G P vkrzlqj wkdw G @ P dqg khqfh wkdw @ rq P= Iru wkh ~ qlwh fdvh/ e| uhsodflqj [q e| ^qdR [m li qhfhvvdu|/ zh pd| dvvxph m wkdw [q % [= Iru q 5 Q> ohw q +D, =@ +D _ [q , dqg q +D, @ +D _ [q , iru doo D 5 P= Wkhq rqh hdvlo| fkhfnv wkdw q dqg q duh qlwh phdvxuh rq P vxfk wkdw q @ q rq D= Wkhuhiruh/ e| zkdw zh kdyh mxvw suryhg/ q @ q rq P= Khqfh zh kdyh +D, @ olp +D _ [q , @ olp +D _ [q , @ +D, q6R q6R iru doo D 5 P= Ghqlwlrq 71441 Vxssrvh wkdw H lv d froohfwlrq ri vxevhwv ri [> ohw H ghqrwh wkh froohfwlrq ri vxevhwv ri [ zklfk duh qlwh ru frxqwdeoh xqlrqv ri vhwv iurp H = Vlploduo| ohw H ghqrwh wkh froohfwlrq ri vxevhwv ri [ zklfk duh qlwh ru frxqwdeoh lqwhuvhfwlrqv ri vhwv iurp H = Zh dovr zulwh H @ +H , dqg H @ +H , > hwf1 Uhpdun 71451 Qrwlfh wkdw li F @ ^Fl dqg G @ ^Gm zlwk Fl > Gm 5 D > wkhq F _ G @ ^l>m +Fl _ Gm , 5 D vr wkdw D lv forvhg xqghu qlwh lqwhuvhfwlrqv1 Wkh iroorzlqj wkhruhp vkrzv krz uhfryhu d phdvxuh rq +D, iurp lwv ydoxhv rq D= Wkhruhp 7146 +Uhjxodulw| Wkhruhp,1 Ohw D S +[ , eh dq dojheud ri vhwv/ P @ +D, dqg = P $ ^3> 4` eh d phdvxuh rq P zklfk lv ~ qlwh rq D= Wkhq iru doo D 5 P> +714, +D, @ lqi i+E , = D E 5 D j = Pruhryhu/ li D 5 P dqg A 3 duh jlyhq/ wkhq wkhuh h{lvwv E 5 D vxfk wkdw D E dqg +E q D, = Surri1 Iru D [> ghqh +D, @ lqi i+E , = D E 5 E j = Zh duh wu|lqj wr vkrz wkdw @ rq P= Zh zloo ehjlq e| uvw dvvxplqj wkdw lv d qlwh phdvxuh/ l1h1 +[ , ? 4= Ohw I @ iE 5 P = +E , @ +E ,j @ iE 5 P = +E , +E ,j= Lw lv fohdu wkdw D I > vr wkh qlwh fdvh zloo eh qlvkhg e| vkrzlqj wkh I lv d prqrwrqh fodvv1 Vxssrvh wkdw Eq 5 I dqg Eq % E dv q $ 4 dqg ohw A 3 eh jlyhq1 Vlqfh +Eq , @ +Eq , wkhuh h{lvwv Dq 5 D vxfk wkdw Eq Dq dqg +Dq , +Eq , . 5q l1h1 +Dq q Eq , 5q = Ohw D @ ^q Dq 5 D > wkhq E D dqg R [ ++Dq q E ,, +D q E , @ +^q +Dq q E ,, qdR R R [ [ ++Dq q Eq ,, 5q @ = qdR qdR hu SUQh yrpr ra rhr ur Wkhuhiruh/ S` +E , +D, +E , . dqg vlqfh A 3 zdv duelwudu| lw iroorzv wkdw E 5 I = Qrz vxssrvh wkdw Eq 5 I dqg Eq & E dv q $ 4 vr wkdw +Eq , & +E , dv q $ 4= Dv deryh fkrrvh Dq 5 D vxfk wkdw Eq Dq dqg 3 +Dq , +Eq , @ +Dq q Eq , 5q = Frpelqlqj wkh suhylrxv wzr htxdwlrqv vkrzv wkdw olpq6R +Dq , @ +E ,= Vlqfh +E , +Dq , iru doo q> zh frqfoxgh wkdw +E , +E ,> l1h1 wkdw E 5 I = Vlqfh I lv d prqrwrqh fodvv frqwdlqlqj wkh dojheud D> wkh prqrwrqh fodvv wkhr0 uhp dvvhuwv wkdw P @ +D, I P vkrzlqj wkh I @ P dqg khqfh wkdw @ rq P= Iru wkh ~ qlwh fdvh/ e| uhsodflqj [q e| ^qdR [m li qhfhvvdu|/ zh pd| dvvxph m wkdw [q % [= Ohw q eh wkh qlwh phdvxuh rq P ghqhg e| q +D, =@ +D _ [q , iru doo D 5 P= Vxssrvh wkdw A 3 dqg D 5 P duh jlyhq1 E| zkdw zh kdyh mxvw suryhg/ iru doo D 5 P> wkhuh h{lvwv Eq 5 D vxfk wkdw D Eq dqg ++Eq _ [q , q +D _ [q ,, @ q +Eq q D, 5q = Qrwlfh wkdw vlqfh [q 5 D > Eq _ [q 5 D dqg E =@ ^R +Eq _ [q , 5 D = qdR Pruhryhu/ D E dqg R R [ [ ++Eq _ [q , q D, ++Eq _ [q , q +D _ [q ,, +E q D, qdR qdR R [ 5q @ = qdR Vlqfh wklv lpsolhv wkdw +D, +E , +D, . dqg A 3 lv duelwudu|/ wklv htxdwlrq vkrzv wkdw Ht1 +714, krogv1 Fruroodu| 71471 Ohw D S +[ , eh dq dojheud ri vhwv/ P @ +D, dqg = P $ ^3> 4` eh d phdvxuh rq P zklfk lv ~ qlwh rq D= Wkhq iru doo D 5 P dqg A 3 wkhuh h{lvwv E 5 D vxfk wkdw E D dqg +D q E , ? = Ixuwkhupruh/ iru dq| E 5 P wkhuh h{lvwv D 5 D dqg F 5 D vxfk wkdw D E F dqg +F q D, @ 3= Surri1 E| Wkhruhp 7146/ wkhuh h{lvw F 5 D vxfk wkdw Df F dqg +F q Df , = Ohw E @ F f D dqg qrwlfh wkdw E 5 D dqg wkdw F q Df @ E f _ D @ D q E> vr wkdw +D q E , @ +F q Df , = Ilqdoo|/ jlyhq E 5 P> zh pd| fkrrvh Dq 5 D dqg Fq 5 D vxfk wkdw Dq E Fq dqg +Fq q E , 4@q dqg +E q Dq , 4@q= E| uhsodflqj DQ e| ^QdR Dq dqg FQ q ipr xI qvr| TQ e| _QdR Fq > zh pd| dvvxph wkdw Dq % dqg Fq & dv q lqfuhdvhv1 Ohw D @ ^Dq 5 D q dqg F @ _Fq 5 D > wkhq D E F dqg +F q D, @ +F q E , . +E q D, +Fq q E , . +E q Dq , 5@q $ 3 dv q $ 4= Fruroodu| 71481 Ohw D S +[ , eh dq dojheud ri vhwv/ P @ +D, dqg = P $ ^3> 4` eh d phdvxuh rq P zklfk lv ~ qlwh rq D= Wkhq iru hyhu| E 5 P vxfk wkdw +E , ? 4 dqg A 3 wkhuh h{lvwv G 5 D vxfk wkdw +E 7G, ? = Surri1 E| Fruroodu| 7147/ wkhuh h{lvwv F 5 D vxfk E F dqg +F q E , ? = Qrz zulwh F @ ^R Fq zlwk Fq 5 D iru hdfk q= E| uhsodflqj Fq e| ^qdR Fn 5 D qdR n li qhfhvvdu|/ zh pd| dvvxph wkdw Fq % F dv q $ 4= Vlqfh Fq q E % F q E dqg E q Fq & E q F @ > dv q $ 4 dqg +E q FR , +E , ? 4> zh nqrz wkdw olp +Fq q E , @ +F q E , ? dqg olp +E q Fq , @ +E q F , @ 3 q6R q6R Khqfh iru q vxflhqwo| odujh/ +E 7Fq , @ ++Fq q E , . +E q Fq , ? = Khqfh zh duh grqh e| wdnlqj G @ Fq 5 D iru dq q vxflhqwo| odujh1 Iru H{huflvhv 714: ~ 7153 ohw S +[ , eh d wrsrorj|/ P @ + , dqg = P $ ^3> 4, eh d qlwh phdvxuh ++[ , ? 4,= Uhpdun 71491 Zh kdyh wr dvvxph wkdw +E , ? 4 dv wkh iroorzlqj h{dpsoh vkrzv1 Ohw [ @ U> P @ E> @ p> D eh wkh dojheud jhqhudwhg e| kdoi rshq lqwhuydov ri wkh irup +d> e`> dqg E @ ^R +5q> 5q . 4`= Lw lv hdvlo| fkhfnhg wkdw iru hyhu| G 5 D> qdR wkdw p+E G, @ 4= H{huflvh 714:1 Ohw +715, I =@ iD 5 P = +D, @ lqi i+Y , = D Y 5 jj = Vkrz I lv d prqrwrqh fodvv1 H{huflvh 714;1 Jlyh dq h{dpsoh ri d wrsrorj| rq [ @ i4> 5> 6j dqg d phdvxuh rq P @ + , vxfk wkdw I ghqhg lq Ht1 +715, lv qrw P= H{huflvh 714<1 Ohw eh d wrsrorj| rq d vhw [ dqg D @ D+ , eh wkh dojheud jhqhudwhg e| = Vkrz D lv wkh froohfwlrq ri vxevhwv ri [ zklfk pd| eh zulwwhq dv qlwh xqlrq ri vhwv ri wkh irup I _ Y zkhuh I lv forvhg dqg Y lv rshq1 H{huflvh 71531 Vxssrvh qrz S +[ , lv d wrsrorj| zlwk wkh surshuw| wkdw wr hyhu| forvhg vhw F [> wkhuh h{lvwv Yq 5 vxfk wkdw Yq & F dv q $ 4= Ohw D @ D+ , eh wkh dojheud jhqhudwhg e| = Zlwk wkh dlg ri H{huflvh 714</ vkrz wkdw D I dqg xvh wklv dorqj zlwk h{huflvh 714: wr frqfoxgh wkdw +D, @ lqi i+Y , = D Y 5 j iru doo D 5 P= Wkh qh{w h{huflvh lv wkh jhqhudol}dwlrq ri H{huflvh 7153 wr fdvh zkhuh lv ~ qlwh1 hu SUQh yrpr ra rhr ur TR H{huflvh 7154 +Jhqhudol}dwlrq wr wkh ~ qlwh fdvh,1 Ohw S +[ , eh d wrsrorj| zlwk wkh surshuw| wkdw wr hyhu| forvhg vhw F [> wkhuh h{lvwv Yq 5 vxfk wkdw Yq & F dv q $ 4= Dovr ohw P @ + , dqg = P $ ^3> 4, eh d phdvxuh zklfk lv ~ qlwh rq = +4, Vkrz wkdw iru doo A 3 dqg D 5 P wkhuh h{lvwv dq rshq vhw Y 5 dqg d forvhg vhw I vxfk wkdw I D Y dqg +Y q I , = +5, Ohw I ghqrwh wkh froohfwlrq ri vxevhwv ri [ zklfk pd| eh zulwwhq dv d frxqwdeoh xqlrq ri forvhg vhwv1 Xvh lwhp 41 wr vkrz iru doo E 5 P> wkhuh h{lvwv F 5 + lv fxvwrpdulo| zulwwhq dv J , dqg D 5 I vxfk wkdw D E F dqg +F q D, @ 3= H{huflvh 7155 +Phwulf Vsdfh H{dpsohv,1 Vxssrvh wkdw +[> g, lv d phwulf vsdfh1 D vhw Y [ lv vdlg wr eh rshq li iru doo { 5 Y wkhuh h{lvwv dq @ +{, A 3 vxfk wkdw wkh edoo +716, E +{> , @ i| 5 [ = g+{> |, ? j lv frqwdlqhg lq Y= Ohw g ghqrwh wkh froohfwlrq ri rshq vhwv1 Jlyhq d vhw I [ dqg A 3 ohw I eh wkh rshq vhw I @ ^{SI E +{> ,= Vkrz wkdw li I lv forvhg/ wkhq I & I dv & 3= Wkhuhiruh wkh uhvxowv ri H{huflvhv 7153 dqg 7154 dsso| wr phdvxuhv rq phwulf vsdfhv zlwk wkh Eruho ~ dojheud/ E @ + g ,= Fruroodu| 71561 Ohw E eh wkh Eruho ~ dojheud rq Uq htxlsshg zlwk wkh vwdqgdug wrsrorj| lqgxfhg e| rshq edoov zlwk uhvshfw wr wkh Hxfolghdq glvwdqfh1 Vxssrvh wkdw = E $ ^3> 4` lv d phdvxuh vxfk wkdw +D, ? 4 zkhqhyhu D lv d erxqghg vhw1 +4, Wkhq iru doo D 5 E dqg A 3 wkhuh h{lvw d forvhg vhw I dqg dq rshq vhw Y vxfk wkdw I D Y dqg +Y q I , ? = +5, Li +D, ? 4> wkh vhw I lq lwhp 41 pd| eh fkrvhq wr eh frpsdfw1 +6, Iru doo D 5 E zh pd| frpsxwh +D, xvlqj +717, +718, +D, @ lqi i+Y , = D Y dqg Y lv rshqj @ vxsi+N , = N D dqg N lv frpsdfwj= Surri1 Lwhp 41 iroorzv iurp H{huflvhv 7154 dqg 71551 Li +D, ? 4 dqg I D Y dv lq lwhp 41 Ohw Nq =@ i{ 5 I = m{m qj= Wkhq Nq lv forvhg dqg erxqghg lq Uq dqg khqfh frpsdfw dqg Nq % I dv q $ 4= Vlqfh +D, ? 4 dqg +Y q D, ? zh nqrz wkdw +Y , ? 4= Xvlqj wklv idfw dqg wkh idfw wkdw Y q Nq & Y q I> zh frqfoxgh wkdw +Y q Nq , & +Y q I , ? dv q $ 4= Wkxv iru vxflhqwo| odujh q zh kdyh N @ Nq lv d frpsdfw vhw vxfk wkdw N D Y dqg +Y q N , ? = Lwhp 41 hdvlo| lpsolhv wkdw Ht1 +717, krogv dqg lwhp 51 lpsolhv Ht1 +718, krogv zkhq +D, ? 4= Vr zh qhhg rqo| fkhfn Ht1 +718, zkhq +D, @ 4= E| Lwhp 41 wkhuh lv d forvhg vhw I D vxfk wkdw +D q I , ? 4 dqg lq sduwlfxodu +I , @ 4= Ohwwlqj Nq I D eh wkh frpsdfw vhw dv lq Ht1 +719,/ zh kdyh +Nq , % +I , @ 4 @ +D, zklfk vkrzv wkdw Ht1 +718, dovr krogv zkhq +D, @ 4= +719, ipr xI qvr| TS 81 NAt|i W|A} j!B6V,jt Nu 6jBt ijt Prvw ~ dojheudv dqg 0dgglwlyh phdvxuhv duh vrphzkdw glfxow wr ghvfuleh dqg ghqh1 Krzhyhu/ rqh vshfldo fdvh lv idluo| hdv| wr xqghuvwdqg1 Qdpho| vxssrvh wkdw I S +[ , lv d frxqwdeoh ru qlwh sduwlwlrq ri [ dqg P S +[ , lv wkh ~ dojheud zklfk frqvlvwv ri wkh froohfwlrq ri vhw D [ vxfk wkdw ^ = +814, D@ SsT)D Lw lv hdvlo| vhhq wkdw P lv d ~ dojheud1 Dq| phdvxuh = P $ ^3> 4` lv ghwhuplqhg xqltxho| e| lwv ydoxhv rq I = Frq0 yhuvho|/ li zh duh jlyhq dq| ixqfwlrq = I $ ^3> 4` zh pd| ghqh/ iru D 5 P> [ [ +, @ +,4)D +D, @ SsT)D Ss zkhuh 4)D lv rqh li D dqg }hur rwkhuzlvh1 Zh pd| fkhfn wkdw lv d phdvxuh cR rq P= Lqghhg/ li D @ ldR Dl dqg 5 I > wkhq D l Dl iru rqh dqg khqfh h{dfwo| rqh Dl = Wkhuhiruh/ R [ 4)D @ 4)Dl ldR dqg khqfh +D, @ [ +,4)D @ Ss R [ @ +Dl , ldR [ Ss +, R [ ldR 4)Dl @ R [[ ldR Ss +,4)Dl dv ghvluhg1 Wkxv zh kdyh vkrzq wkdw wkhuh lv d rqh wr rqh fruuhvsrqghqfh ehwzhhq phdvxuhv rq P dqg ixqfwlrqv = I $ ^3> 4`= Wkh frqvwuxfwlrq ri phdvxuhv rq pruh jhqhudo ~ dojheudv zloo eh prwlydwhg e| wkh uhjxodulw| uhvxowv lq Vhfwlrq 714 deryh1 8141 Frqvwuxfwlqj Phdvxuhv iurp Suhphdvxuhv1 Ghqlwlrq 8141 Dq dgglwlyh ixqfwlrq Q rq dq dojheud D ri vxevhwv ri [ lv fdoohg d suhphdvxuh zkhq Q lv dovr frxqwdeo| dgglwlyh rq D= Wklv phdqv wkdw li D 5 D cR dqg Dl 5 D vxfk wkdw D @ ldR Dl > wkhq Q +D, @ R [ ldR Q +Dl ,= Wkh suhphdvxuh lv vdlg wr ~ qlwh li wkhuh h{lvwv [q 5 D vxfk wkdw [q % [ dv q % 4 dqg Q +[q , ? 4= Ghqlwlrq 8151 h wkdw H lv d froohfwlrq ri vxevhwv ri [> ohw H ghqrwh wkh froohfwlrq ri vxevhwv ri [ zklfk duh qlwh ru frxqwdeoh xqlrqv ri vhwv iurp H = Vlploduo| ohw H ghqrwh wkh froohfwlrq ri vxevhwv ri [ zklfk duh qlwh ru frxqwdeoh lqwhuvhfwlrqv ri vhwv iurp H = Zh dovr zulwh H @ +H , dqg H @ +H , > hwf1 hu SUQh yrpr ra rhr ur TT Sursrvlwlrq 8161 Ohw Q eh d suhphdvxuh rq dq dojheud D= Iru D 5 D vxfk wkdw R c Dq > zlwk Dq 5 D> ghqh D@ qdR R [ Q +Dq ,= Q +D, qdR Wkhq Q = D $ ^3> 4` lv zhoo ghqhg1 Surri1 Vxssrvh wkdw D pd| dovr eh zulwwhq dv D @ Dq @ R d R [ R c qdR Eq > wkhq +Dq _ En , , Q +Dq , @ Q +Dq _ En ,= ndR ndR R S Q +Dq _ En , @ Q +En , dqg wkxv khqfh Vlploduo| rqh vkrzv wkdw qdR RR [[ [ Q +Dq _ En , Q +Dq , @ qdR ndR RR R [[ [ @ Q +Dq _ En , @ Q +En , ndR qdR ndR Wkh nh| wr frqvwuxfwlqj phdvxuhv lv wkh iroorzlqj wkhruhp zklfk zloo eh suryhg lq Vhfwlrq 91 Wkhruhp 817 +Nh| Frqvwuxfwlrq Wkhruhp,1 Ohw D S +[ , eh dq dojheud/ P @ +D, dqg Q eh d ~ qlwh suhphdvxuh rq D zklfk zh h{whqg wr D xvlqj wkh suhylrxv Sursrvlwlrq 8161 Wkhq wkhuh h{lvwv d xqltxh phdvxuh rq P vxfk wkdw mh @ Q = Pruhryhu/ lv jlyhq e| wkh irupxod zh kdyh +815, +D, @ lqi iQ +E , = D E 5 D j @ lqi iQ +^R Dq , = D ^R Dq zlwk Dq 5 Dj= qdR qdR ^ DqR , li qhfhvvdu|/ Ht1 +815, lv htxlydohqw wr E| uhsodflqj Dq e| Dq q +DR ^ zulwlqj R R [ d Q +Dq , = D Dq zlwk Dq 5 Dj= +816, +D, @ lqi i qdR qdR Wklv wkhruhp lv d frqvhtxhqfh ri Wkhruhp 91: ri Vhfwlrq 9 ehorz1 Uhpdun 8181 zh gurs wkh ~ qlwh dvvxpswlrq rq Q zh pd| orrvh xqltxhqhvv lq wkh deryh wkhruhp1 Zh zloo jlyh vrph h{dpsohv ri wklv skhqrphqd lq wkh vxevhfwlrq 817 Lq rughu wr xvh wklv wkhruhp lw lv qhfhvvdu| wr uvw frqvwuxfw suhphdvxuhv rq dojheudv1 Wkh pdlq uhvxow wkdw zh duh khdglqj iru lv frqwdlqhg lq Wkhruhp 81: ehorz1 Ghqlwlrq 8191 Vxssrvh wkdw H S +[ , lv dq hohphqwdu| idplo|1 Sq +4, D ixqfwlrq = H $ ^3> 4` lv dgglwlyh li +H , @ ldR +Hl , zkhqhyhu cq H @ ldR Hl 5 H zlwk Hl 5 H = ipr xI qvr| TU S +5, D ixqfwlrq = H $ ^3> 4` lv vxedgglwlyh li +H , R +Hl , zkhqhyhu ldR cR H @ ldR Hl 5 H zlwk Hl 5 H = +6, D ixqfwlrq = H $ ^3> 4` lv ~ qlwh rq H li wkhuh h{lvw Hq 5 H vxfk wkdw [ @ ^q Hq dqg +Hq , ? 4> Wkhruhp 81:1 Vxssrvh wkdw H S +[ , lv dq hohphqwdu| idplo| dqg Q = H $ ^3> 4` lv d ixqfwlrq1 +4, Li Q lv dgglwlyh rq H > wkhq Q kdv d xqltxh h{whqvlrq wr d qlwho| dgglwlyh phdvxuh Q rq D @ D+H ,= +5, Li zh ixuwkhu dvvxph wkdw Q lv frxqwdeo| vxedgglwlyh rq H > wkhq Q lv d suhdphdvxuh rq D= +6, Li zh ixuwkhu dvvxph wkdw Q lv ~ qlwh rq H > wkhq wkhuh h{lvwv d xqltxh phdvxuh rq +H , vxfk wkdw mr @ Q = Pruhryhu/ iru D 5 +H ,> +D, @ lqi iQ +E , = D E 5 D j R R [ Q +H , = D d H zlwk H 5 Hj= @ lqi i q q q qdR qdR Wklv wkhruhp lv suryhg lq wkh qh{w vxevhfwlrq1 Lwhp 41 lv suryhg lq Sursrvlwlrq 81;/ Lwhp 51 lq Sursrvlwlrq 8143 dqg Lwhp 61 lv d frqvhtxhqfh ri Lwhpv 41 dqg 51 dqg Wkhruhp 817 deryh1 8151 Surri ri Wkhruhp 81:1 Sursrvlwlrq 81;1 Vxssrvh H S +[ , lv dq hohphqwdu| idplo| dqg D @ D+H , lv wkh dojheud jhqhudwhg e| H = Wkhq hyhu| dgglwlyh ixqfwlrq = H $ ^3> 4` h{whqgv xqltxho| wr dq dgglwlyh phdvxuh +zklfk zh vwloo ghqrwh e| , rq D= c Surri1 Vlqfh e| Sursrvlwlrq 6143/ hyhu| hohphqw D 5 D lv ri wkh irup D @ l Hl zlwk Hl 5 H > lw lv fohdu wkdw li h{whqgv wr d phdvxuh wkh h{whqvlrq lv xqltxh dqg pxvw eh jlyhq e| [ +Hl ,= +817, +D, @ l Wr suryh wkh h{lvwhqfh ri wkh h{whqvlrq/ wkh pdlq srlqw lv wr vkrz wkdw ghqlqj c +D, e| Ht1 +817, lv zhoo ghqhg1 Wkdw lv wr vd| li zh dovr kdyh D @ m Im zlwk Im 5 H > zh pxvw vkrz wkdw [ [ +818, +Hl , @ +Im ,= l m Wr suryh wklv/ zh pdnh xvh ri wkh idfw wkdw Hl @ ^m +Hl _ Im , dqg wkh surshuw| wkdw lv dgglwlyh rq H wr frqfoxgh wkdw [ +Hl , @ +Hl _ Im , dqg khqfh m [[ [ [ +Hl , @ +Hl _ Im , @ +Hl _ Im ,= l lm l>m Vlploduo|/ zh vkrz wkdw [ [ +Im , @ +Hl _ Im , m l>m hu SUQh yrpr ra rhr ur TV zklfk frpelqhg zlwk wkh suhylrxv htxdwlrq vkrzv wkdw Ht1 +818, krogv1 Lw lv qrz hdv| wr yhuli| wkdw h{whqghg wr D dv lq Ht1 +817, lv dq dgglwlyh phdvxuh rq D= Sursrvlwlrq 81<1 Vxssrvh wkdw D S +[ , lv dq dojheud ri vhwv dqg = D $ ^3> 4` R c Dl zlwk hdfk lv d qlwho| dgglwlyh phdvxuh rq D= Wkhq li D 5 D dqg D @ ldR Dl 5 D> zh kdyh R [ +Dl , +D,= +819, ldR Surri1 Vlqfh D@ #Q d $ # Q ^ $ Dl ^ D q Dl ldR ldR zh qg xvlqj wkh qlwh dgglwlylw| ri wkdw # $ Q Q Q [ ^ [ +Dl , . D q Dl +Dl ,= +D, @ ldR ldR ldR R S Ohwwlqj Q $ 4 lq wklv odvw h{suhvvlrq vkrzv wkdw +Dl , +D,= ldR Vr lq rughu wr suryh wkdw lv d suhphdvxuh rq D> lw vxfhv wr vkrz R [ +81:, +D, +Dl , ldR R c zkhqhyhu D @ Dl zlwk D 5 D dqg hdfk Dl 5 D= ldR Sursrvlwlrq 81431 Vxssrvh wkdw H S +[ , lv dq hohphqwdu| idplo|/ D @ D+H , dqg = D $ ^3> 4` lv dq dgglwlyh phdvxuh1 Wkhq wkh iroorzlqj duh htxlydohqw= +4, lv d suhphdvxuh rq D= +5, c kdv wkh vxedgglwlylw| surshuw|= zkhqhyhu H 5 H lv ri wkh irup H @ R H 5 H zlwk H 5 H wkhq l ldR l R [ +81;, +H , +Hl ,= ldR Surri1 Lw lv fohdu wkdw 41 lpsolhv 51 vlqfh li lv d suhphdvxuh/ wkhq +H , @ SR ldR +Hl ,= Iru wkh frqyhuvh/ lw vxfhv wr vkrz e| Sursrvlwlrq 81< wkdw li D @ R c Dq zlwk D 5 D dqg hdfk Dq 5 D wkhq Ht1 +81:, krogv1 Wr suryh wklv/ zulwh qdR c c D @ qdR Hm zlwk Hm 5 H dqg Dq @ Qq Hq>l zlwk Hq>l 5 H = Wkhq m ldR R R Qq d dd Dq _ Hm @ Hq>l _ Hm Hm @ D _ Hm @ qdR qdR ldR zklfk lv d frxqwdeoh xqlrq dqg khqfh e| dvvxpswlrq/ R Qq [[ +Hq>l _ Hm , = +Hm , qdR ldR ipr xI qvr| TW Vxpplqj wklv htxdwlrq rq m dqg xvlqj wkh dgglwlylw| ri vkrzv wkdw q q R Qq R Qq q [ [[[ [ [[ +D, @ +Hm , +Hq>l _ Hm , @ +Hq>l _ Hm , qdR ldR m dR m dR mdR qdR ldR R Qq R [[ [ @ +Hq>l , @ +Dq , qdR ldR qdR dv ghvluhg1 8161 H{dpsohv ri qlwho| dgglwlyh phdvxuhv1 H{dpsoh 81441 Ohw [ @ U dqg H eh wkh hohphqwdu| fodvv H @ i+d> e` _ U = 4 d e 4j> dqg D @ D+H , eh wkh dojheud ri glvmrlqw xqlrq ri hohphqwv iurp H = Vxssrvh wkdw = D $ ^3> 4` lv dq dgglwlyh phdvxuh vxfk wkdw ++d> e`, ? 4 iru doo d ? e= Wkhq wkhuh lv d xqltxh lqfuhdvlqj ixqfwlrq I = U $ U vxfk wkdw I +3, @ 3> I R +i 4j, R+i4j, i4j dqg i 4j > I +81<, ++d> e` _ U, @ I +e, I +d, ; d e lq U= Frqyhuvho|/ jlyhq dq lqfuhdvlqj ixqfwlrq I = U $ U vxfk wkdw I R +i 4j, R+i4j, i4j wkhuh lv d xqltxh phdvxuh rq D vxfk wkdw wkh uh0 i 4j > I odwlrq lq Ht1 +81<, krogv1 +Vr wkh qlwho| dgglwlyh phdvxuhv rq D zklfk duh qlwh rq erxqghg vhwv duh lq rqh wr rqh fruuhvsrqghqfh zlwk lqfuhdvlqj ixqfwlrqv I = U $ U vxfk wkdw I +3, @ 3> I R +i 4j, i 4j > I R +i4j, i4j =, Surri1 Li I lv jrlqj wr h{lvw/ zh pxvw kdyh ++3> e` _ U, @ I +e, ++d> 3`, @ I +3, iurp zklfk zh ohduq I +{, @ I +3, @ I +e, li e 5 ^3> 4`> I +d, @ I +d, li d 5 ^ 4> 3` ++{> 3`, li ++3> {` _ U, li {3 { 3= Pruhryhu/ rqh hdvlo| fkhfnv xvlqj wkh dgglwlylw| ri wkdw Ht1 +81<, krogv iru wklv I= Frqyhuvho|/ vxssrvh I = U $ U lv dq lqfuhdvlqj ixqfwlrq vxfk wkdw I R +i 4j, i 4j> I R +i4j, i4j= Ghqh rq H xvlqj wkh irupxod lq Ht1 +81<,1 L fodlp wkdw lv dgglwlyh rq H dqg khqfh kdv d xqltxh h{whqvlrq wr D zklfk zloo qlvk wkh dujxphqw1 Vxssrvh wkdw q d +dl > el `= +d> e` @ ldR E| uhrughulqj +dl > el ` li qhfhvvdu|/ zh pd| dvvxph wkdw ? dq ? eq @ e= d @ dR A eR @ dS ? eS @ dT ? Wkhuhiruh/ ++d> e`, @ I +e, dv ghvluhg1 I +d, @ q [ ldR ^I +el , I +dl ,` @ q [ ldR ++dl > el `, hu SUQh yrpr ra rhr ur TX Wkhruhp 81451 Ohw D S +[ , dqg E S +\ , eh dojheudv1 Vxssrvh wkdw =D E$F lv d ixqfwlrq vxfk wkdw iru hdfk D 5 D> wkh ixqfwlrq E 5 E $ +D E, 5 F lv dq dgglwlyh phdvxuh rq E dqg iru hdfk E 5 E> wkh ixqfwlrq D 5 D $ +D E, 5 F lv dq dgglwlyh phdvxuh rq D= Wkhq h{whqgv xqltxho| wr d phdvxuh rq wkh surgxfw dojheud F jhqhudwhg e| D E= Surri1 Wkh froohfwlrq H @D E @ iD E = D 5 D dqg E 5 Ej lv dq hohphqwdu| idplo| +|rx fkhfn,1 Wkhuhiruh/ lw vxfhv wr vkrzv wkdw lv dgglwlyh rq H = Wr fkhfn wklv vxssrvh wkdw D E 5 H dqg d D E@ +Dn En , n zlwk Dn En 5 H = Zh zlvk wr vkrzv +D E, @ [ n +Dn En ,= Iru wklv frqvlghu wkh qlwh dojheudv DQ S +D, dqg EQ S +E , jhqhudwhg e| iDn j dqg iEn j uhvshfwlyho|1 Ohw I DQ dqg J EQ eh sduwlwlrq ri D dqg E uhvshfwlyho| dv irxqg Sursrvlwlrq 614:1 Wkhq iru hdfk n zh pd| zulwh d d dqg En @ = Dn @ Ss >)Dn St>)En Wkhuhiruh/ +Dn vr wkdw [ n dv ghvluhg1 +Dn En , @ +Dn ^ [ +Dn , )En )En # $ [ ^ [ @ + , @ + )Dn )En )Dn > )En En , @ [ , @ [ + n )Dn > )En [ @ +D , @ +D )E , @ E, [ )D>)E + , , ipr xI qvr| TY 8171 _Udgrq% phdvxuhv rq +U> E,1 Ohw xv frqwlqxh zlwk H{dpsoh 81441 Vr [ @ U dqg H eh wkh hohphqwdu| fodvv H @ i+d> e` _ U = 4 d e 4j> dqg D @ D+H , eh wkh dojheud ri glvmrlqw xqlrq ri hohphqwv iurp H = Dovr ohw E @ +H , eh wkh Eruho 0dojheud dqg vxssrvh = E $ ^3> 4` lv d ~ dgglwlyh phdvxuh vxfk wkdw ++d> e`, ? 4 iru doo 4 ? d ? e ? 4= Wkhq dv lq H{dpsoh 8144/ ohw I eh wkh lqfuhdvlqj ixqfwlrq ghqhg iru { 5 U e| ++{> 3`, li { 3 +8143, I +{, @ ++3> {` _ U, li { 3 Vlqfh iru 3 d ? 4 dqg 4 ? e ? 3> olp I +{, @ olp ++3> {`, @ ++3> d`, @ I +d, dqg {8d {8d olp I +{, @ olp ++{> 3`, @ ++e> 3`, @ I +e, {8e {8e +zkhuh wklv lv fkhfnhg xvlqj vhtxhqfhv, zh vhh wkdw I lv qhfhvvdulo| uljkw frqwlqxrxv rq U= Pruhryhu/ vlqfh olp I +{, @ olp ++3> {`, @ ++3> 4,, @ I +4, dqg {6R {6R olp I +{, @ olp ++{> 3`, @ ++ 4> 3`, @ I + 4, {6R {6R zh ohduq wkdw I lv frqwlqxrxv dw 4 dv zhoo1 Frqyhuvho| zh kdyh wkh iroorzlqj Wkhruhp1 Qrwdwlrq 81461 Li I = U $ U lv dq lqfuhdvlqj ixqfwlrq/ ohw I + 4, =@ olp{6R I +{,= Wkhruhp 81471 Wr hyhu| uljkw frqwlqxrxv qrq0ghfuhdvlqj ixqfwlrq I = U $ U wkhuh h{lvwv d xqltxh phdvxuh I rq E vxfk wkdw +8144, I ++d> e` _ U, @ I +e, I +d, ; 4de4 Pruhryhu/ li D 5 E wkhq +8145, +R [ I +D, @ lqi +I +el , ldR +R [ +I +el , @ lqi ldR I +dl ,, = D ^R , ldR +dl > el ` , R d +8146, I +dl ,, = D +dl > el ` = ldR Lq idfw wkh uhodwlrqv +8143, dqg +8144, jlyh d rqh wr rqh fruuhvsrqghqfh ehwzhhq uljkw frqwlqxrxv ixqfwlrqv I zlwk I +3, @ 3 rq rqh kdqg dqg udgrq phdvxuh rq E rq wkh rwkhu1 Surri1 H{whqg I wr d ixqfwlrq iurp U $ U e| ghqlqj I + 4, =@ olp{6R I +{,= Dv deryh ohw H @ i+d> e` _ U = 4 d e 4j dqg xvh H{dpsoh 8144 wr vkrz wkhuh lv d xqltxh dgglwlyh phdvxuh Q rq D+H , vxfk wkdw Q ++d> e`, @ I +e, I +d, iru doo d> e 5 U zlwk d e= Wkh surri zloo eh qlvkhg e| Wkhruhp 81: li zh fdq vkrz wkdw Q lv vxe0dgglwlyh rq H = hu SUQh yrpr ra rhr ur T` Iluvw vxssrvh wkdw 4 ? d ? e ? 4> M @ +d> e`> Mq @ +dq > eq ` vxfk wkdw R c Mq = Zh zlvk wr vkrzv wkdw M@ qdR Q +M , +8147, R [ ldR Q +Ml ,= Wr gr wklv fkrrvh qxpehuv d A d> q A eq dqg vhw e L q M Q Mq @ +> e` M> d @ +dq > q ` Mq dqg e q ,= @ +dq > e R V Q Vlqfh L lv frpsdfw dqg L M Mq wkhuh h{lvwv Q ? 4 vxfk wkdw qdR Q Q ^ ^ Q LL Mq Mq = qdR qdR Khqfh qlwh vxe0dgglwlylw| ri Q > I +e, I +, @ Q +L , d R [ Q Mq Q Mq = qdR qdR Q [ Xvlqj wkh uljkw frqwlqxlw| ri I dqg ohwwlqj d & d lq wkh deryh lqhtxdolw| vkrzv wkdw Q ++d> e`, @ I +e, +8148, I +d, R [ Q Mq qdR R [ Q +Mq , . Q +Mq q Mq , @ qdR qdR R [ Jlyhq A 3 zh pd| xvh wkh uljkw frqwlqxlw| ri I wr fkrrvh q vr wkdw e e Q +Mq q Mq , @ I +q , I +eq , 5q ;q= Xvlqj wklv lq Ht1 +8148, vkrzv wkdw Q +M , @ Q ++d> e`, R [ qdR Q +Mq , . dqg vlqfh A 3 zh kdyh yhulhg Ht1 +8147,1 Zh kdyh qrz grqh wkh kdug zrun1 Zh vwloo kdyh wr fkhfn wkh fdvhv zkhuh d @ ru e @ 4 ru erwk1 Iru h{dpsoh/ vxssrvh wkdw e @ 4 vr wkdw M @ +d> 4, @ R d qdR Mq zlwk Mq @ +dq > eq ` _ U= Wkhq ohw LP =@ +d> P `> dqg qrwlfh wkdw LP @ M _ LP @ R d qdR Mq _ LP 4 ipr xI qvr| UQ Vr e| zkdw zh kdyh douhdg| suryhg/ I +P , I +d, @ Q +LP , R [ Q +Mq _ LP , R [ Q +Mq , qdR qdR Qrz ohw P $ 4 lq wklv odvw lqhtxdolw| wr qg wkdw R [ Q ++d> 4,, @ I +4, I +d, Q +Mq ,= qdR Wkh rwkhu fdvhv zkhuh d @ 4 dqg e 5 U dqg d @ 4 dqg e @ 4 duh kdqgohg vlploduo|1 Uhpdun 81481 Lw lv ghqlwho| qrw wuxh wkdw doo phdvxuhv rq +U> E, pd| eh ghvfulehg xqltxho| e| wkhlu uhvwulfwlrq wr D= Iru h{dpsoh/ ohw G U eh d frxqwdeoh ghqvh vhw dqg ghqh G +D, =@ &+G _ D,= Wkhq G +D, @ 4 iru doo D 5 D zklfk duh qrw hpsw|1 Rq wkh rwkhu kdqg lw lv fohdu wkdw wkh phdvxuhv G zlwk G d frxqwdeoh ghqvh vhw lq U duh glhuhqw iru glhuhqw G *v1 Dovr qrwlfh wkdw G lv ~ qlwh rq E exw qrw rq D= H{dpsoh 81491 Wkh prvw lpsruwdqw vshfldo fdvh ri Wkhruhp 8147 lv zkhq I +{, @ {> lq zklfk fdvh zh zulwh p iru I = Wkh phdvxuh p lv fdoohg Ohehvjxh phdvxuh1 Wkhruhp 814:1 Ohehvjxh phdvxuh p lv lqyduldqw xqghu wudqvodwlrqv/ l1h1 iru D 5 E dqg { 5 U> +8149, p+{ . E , @ p+E ,= Pruhryhu/ p lv wkh xqltxh phdvxuh rq E vxfk wkdw p++3> 4`, @ 4 dqg Ht1 +8149, krogv iru D 5 E dqg { 5 U= Pruhryhu/ p kdv wkh vfdolqj surshuw| +814:, p+E , @ mm p+E , zkhuh 5 U/ E 5 E dqg E =@ i{ = { 5 E j= Surri1 Ohw p{ +E , =@ p+{ . E ,> wkhq rqh hdvlo| vkrzv wkdw p{ lv d phdvxuh rq E vxfk wkdw p{ ++d> e`, @ e d iru doo d ? e= Wkhuhiruh/ p{ @ p e| wkh xqltxhqhvv dvvhuwlrq lq Wkhruhp 81471 Iru wkh frqyhuvh/ vxssrvh wkdw p lv wudqvodwlrq lqyduldqw dqg p++3> 4`, @ 4= Jlyhq q 5 Q> zh kdyh 4 n 4n n4 > ` @ ^qdR . +3> ` = +3> 4` @ ^qdR + n n q q q q Wkhuhiruh/ q [ n 4 4 . +3> ` 4 @ p++3> 4`, @ p q q ndR q [ 4 4 @ p++3> `, @ q p++3> `,= q q ndR Wkdw lv wr vd| 4 p++3> `, @ 4@q= q o Vlploduo| zh vkrz wkdw p++3> q `, @ o@q iru doo o> q 5 Q= Xvlqj wkh wudqvodwlrq lqyduldqfh ri p> zh wkhq ohduq wkdw p++d> e`, @ e d hu SUQh yrpr ra rhr ur UR iru doo d> e 5 T vxfk wkdw d ? e= Ilqdoo| iru d> e 5 U vxfk wkdw d ? e> fkrrvh dq > eq 5 T vxfk wkdw eq & e dqg dq % d> wkhq +dq > eq ` & +d> e` dqg wkxv p++d> e`, @ olp p++dq > eq `, @ olp +eq dq , @ e d> q6R q6R l1h1 p lv Ohehvjxh phdvxuh1 Wr suryh Ht1 +814:, zh pd| dvvxph wkdw 9@ 3 vlqfh wklv fdvh lv wulyldo wr suryh1 Qrz ohw p +E , =@ mmR p+E ,= Lw lv hdvlo| fkhfnhg wkdw p lv djdlq d phdvxuh rq E zklfk vdwlvhv p ++d> e`, @ R p ++d> e`, @ R +e li A 3 dqg R p +^e> d,, @ p ++d> e`, @ mm mmR +e d, @ e d, @ e d d li ? 3= Khqfh p @ p= 91 VVjAa!H NAt|i W|NA Nu jBt ijt 9141 Rxwhu Phdvxuhv1 Ghqlwlrq 9141 D ixqfwlrq = S +[ , $ ^3> 4` lv dq rxwhu phdvxuh li +4, +>, @ 3 S +5, +^Dl , +Dl , +6, +D, +E , li D E 1 Sursrvlwlrq 915 +H{dpsoh ri dq rxwhu phdvxuh1,1 Ohw H S +[ , eh duelwudu| froohfwlrq ri vxevhwv ri [ zklfk frqwdlqv erwk >> [ 5 H 1 Ohw = H $ ^3> 4` eh d ixqfwlrq vxfk wkdw +>, @ 3= Iru dq| D [ / ghqh [ +914, +D, @ lqi i +Hl , = D ^Hl > Hl 5 Hj= l Wkhq lv dq rxwhu phdvxuh1 Surri1 Lw lv fohdu wkdw +>, @ 3 dqg +D, +E , iru D E= Vxssrvh wkdw Dl 5 S +[ , dqg +Dl , ? 4 iru doo l> rwkhuzlvh wkhuh zloo eh qrwklqj wr suryh1 Ohw R V Hlm dqg A 3 dqg fkrrvh Hlm 5 H vxfk wkdw Dl m dR R [ +Dl , +Hlm , 5l = m dR Wkhq +^Dl , RR [[ +Hlm , ldR m dR R R [ [ + +Dl , . 5l , @ +Dl , . = ldR ldR Vlqfh A 3 lv duelwudu|/ zh kdyh vkrzq surshuw| 51 lq Ghqlwlrq 9141 Ghqlwlrq 9161 Ohw = S +[ , $ ^3> 4` eh dq rxwhu phdvxuh1 Ghqh wkh 0 phdvxudeoh vhwv wr eh P+ , iD [ = +H , +H _ D, . +H _ Df , ; H [ j= ipr xI qvr| US Ehfdxvh ri wkh vxedgglwlylw| ri > zh pd| htxlydohqwo| ghqh P+ , e| P+ , @ iD [ = +H , @ +H _ D, . +H _ Df , ; H [ j= Wkh qh{w Sursrvlwlrq khosv wr prwlydwh wklv ghqlwlrq dqg wkh Fdudwkrgru|*v h frqvwuxfwlrq Wkhruhp 9181 Sursrvlwlrq 9171 Vxssrvh H @ P lv d ~ dojheud/ @ = P $ ^3> 4` lv d phdvxuh dqg lv ghqhg dv lq Ht1 +914,1 Wkhq +4, Iru D [ +D, @ lqi i+E , = E 5 P dqg D E j= Lq sduwlfxodu/ @ rq P= +5, Wkh ~ dojheud P P+ ,> l1h1 li D 5 P dqg H [ wkhq +915, +H , +H _ D, . +H _ Df ,= +6, Dvvxph ixuwkhu wkdw lv ~ qlwh rq P> wkhq P+ , @ P @ P dqg mC D @ zkhuh +P @ P > , lv wkh frpsohwlrq ri +P> , = h Surri1 Lwhp 41 Li Hl 5 P vxfk wkdw D ^Hl @ E dqg Hl @ Hl q +HR ^ wkhq [ [ h +Hl , @ +E , +Hl , vr [ [ h +Hl , @ +E , +Hl ,= +D, ^ HlR , Wkhuhiruh/ +D, @ lqi i+E , = E 5 P dqg D E j1 Lwhp 51 Li +H , @ 4 Ht1 +915, krogv wulyldoo|1 Vr dvvxph wkdw +H , ? 41 Ohw A 3 eh jlyhq dqg fkrrvh/ e| Lwhp 41/ E 5 P vxfk wkdw H E dqg +E , +H , . 1 Wkhq +H , . +E , @ +E _ D, . +E _ Df , +H _ D, . +H _ Df ,= Vlqfh A 3 lv duelwudu| zh duh grqh1 Lwhp 61 Ohw xv ehjlq e| dvvxplqj wkh +[ , ? 4= Zh kdyh douhdg| vhhq wkdw P P+ ,1 Vxssrvh wkdw D 5 S +[ , vdwlvhv/ +916, +H , @ +H _ D, . +H _ Df , ; H 5 S +[ ,= R E| Lwhp 41/ wkhuh h{lvwv Eq 5 P vxfk wkdw D Eq dqg +Eq , +D, . q iru doo R iru doo q zklfk lpsolhv wkdw q 5 Q= Wkhuhiruh E @ _Eq D dqg +E , +D, . q +E , +D, zklfk lpsolhv wkdw +E , @ +D,1 Vlploduo| wkhuh h{lvwv F 5 P vxfk wkdw Df F dqg +Df , @ +F ,1 Wdnlqj H @ [ lq Ht1 +916, vkrzv +[ , @ +D, . +Df , @ +E , . +F , vr +F f , @ +[ , f / zh kdyh Wkxv ohwwlqj G @ F +F , @ +E ,= G D E dqg +G, @ +D, @ +E , vr +E q G, @ 3 dqg khqfh D @ G ^ ^+E q G, _ D` zkhuh G 5 P dqg +E q G, _ D 5 Q vkrzlqj wkdw D 5 Pdqg +D, @ +D,= hu SUQh yrpr ra rhr ur UT Qrz li lv ~ qlwh/ fkrrvh [q 5 P vxfk wkdw +[q , ? 4 dqg [q % [ 1 Jlyhq D 5 P+ , vhw Dq @ [q _ D1 Wkhuhiruh +H , @ +H _ D, . +H _ Df ,;H 5 S +[ ,= Uhsodfh H e| [q wr ohduq/ +[q , @ +Dq , . +[q q D, @ +Dq , . +[q q Dq ,= Wkh vdph dujxphqw dv deryh surgxfh vhwv Gq Dq Eq vxfk wkdw +Gq , @ +Dq , @ +Eq ,1 Khqfh Dq @ Gq ^ Qq dqg Qq +Eq q Gq , _ Dq 5 Q = Vr zh ohduq wkdw D @ G ^ Q =@ +^Gq , ^ +^Qq , 5 P ^ Q @ P= Zh dovr vhh wkdw +D, @ +G, vlqfh G D G ^ I zkhuh I 5 P vxfk wkdw Q I dqg +G, @ +G, +D, +G ^ I , @ +G,= 9151 Fdudwkrgru|*v Frqvwuxfwlrq Wkhruhp1 h Wkhruhp 918 +Fdudwkrgru|*v Frqvwuxfwlrq Wkhruhp,1 Ohw eh dq rxwhu phd0 h vxuh rq [ 1 Ohw P @ P+ , wkhq P lv d 0dojheud dqg m lv d frpsohwh phdvxuh1 Surri1 Fohduo| >> [ 5 P dqg li D 5 P wkhq Df 5 P1 Vr wr vkrz wkdw P lv dq dojheud zh pxvw vkrz wkdw P lv forvhg xqghu qlwh xqlrqv/ l1h li D> E 5 P dqg H 5 S +[ , wkhq +H , +H _ +D ^ E ,, . +H q +D ^ E ,,= Xvlqj wkh ghqlwlrq ri P zh kdyh +917, +H , @ +H _ D, . +H q D, +918, @ +H _ D _ E , . +H _ D q E , . ++H q D, _ E , . +H q +D ^ E ,,= Zh zloo qrz pdnh xvh ri wkh lghqwlw|= H _ +D ^ E , @ +H _ D, ^ +H _ E , @ ^+H _ D, q E ` ^ ^+H q D, _ E ` ^ +H _ D _ E , zklfk lq zrugv vwdwhv wkdw H _ D _ E ^ +H _ D q E , ^ ++H q D, _ E , lv htxdo wr wkh srlqwv frpprq wr H / D dqg E xqlrq wkh srlqwv lq +H dqg D exw qrw E , xqlrq wkh srlqwv lq +H dqg E exw qrw D, zklfk lv htxdo wr wkrvh srlqwv lq H zklfk duh dovr lq hlwkhu D ru E 1 Xvlqj wklv lghqwlw| lq Ht1 +918, dorqj zlwk wkh vxedgglwlylw| ri vkrzv +919, +H , +H _ +D ^ E ,, . +H q +D ^ E ,, zklfk lpsolhv wkdw D ^ E 5 P dqg wkhuhiruh vkrzv wkdw P lv dq dojheud1 Qrz vxssrvh D> E 5 P duh glvmrlqw/ wkhq wdnlqj H @ D ^ E lq Ht1 +917, lpsolhv +D ^ E , @ +D, . +E ,= Vr P lv dq dojheud dqg @ m lv qlwho| dgglwlyh1 Zh qrz pxvw vkrz wkdw P lv d ~ dojheud dqg wkh lv ~ dgglwlyh1 ipr xI qvr| UU E HE?D H DEH DH?E D 8} ij U D xvhixo vhw lghqwlw|1 Ohw Dl 5 P +zlwkrxw orvv ri jhqhudolw| dvvxph Dl _ Dm @ > li l 9@ m , Eq @ R V Vq Dm > wkhq iru H [ zh kdyh ldR Dl > dqg E @ m dR +H _ Eq , @ +H _ Eq _ Dq , . +H _ Eq _ Df , q @ +H _ Dq , . +H _ EqR , @ +H _ Dq , . +H _ DqR , . +H _ EqS , 1 1 1 +91:, @ q [ ndR +H _ Dn ,= Wkhuhiruh zh qg wkdw f +H , @ +H _ Eq , . +H _ Eq , q [ f +H _ Dn , . +H _ Eq , @ ndR q [ +H _ Dn , . +H _ E f ,= ndR Ohwwlqj q $ 4 lq wklv htxdwlrq vkrzv wkdw +H , R [ +H _ Dn , . +H _ E f , ndR +^n +H _ Dn ,, . +H q E , +vxedgglwlylw|, @ +H ^ E , . +H q E , +H ,= +vxedgglwlylw|, Wklv vkrzv wkdw P lv d ~ dojheud1 hu SUQh yrpr ra rhr ur UV Vlqfh +H , +H _ Eq , zh pd| ohw q $ 4 lq Ht1 +91:, wr qg R [ +H _ Dn ,= +H , ndR R S +Dn , dqg Ohwwlqj H @ E @ ^Dn lq wklv odvw h{suhvvlrq wkhq vkrzv +E , ndR khqfh e| wkh vxedgglwlylw| ri / R [ +D, @ +Dn ,= ndR m lv frxqwdeo| dgglwlyh rq P= Wkhuhiruh/ @ Ilqdoo| zh vkrz lv frpsohwh1 Li Q I 5 P dqg +I , @ 3 @ +I ,/ wkhq +Q , @ 3 dqg +H , +H _ Q , . +H _ Q f , @ +H _ Q f , +H ,= zklfk vkrzv wkdw Q 5 P= Sursrvlwlrq 9191 Ohw Q eh d suhphdvxuh rq dq dojheud D dqg eh wkh dvvrfldwhg rxwhu phdvxuh dv ghqhg lq Ht1 +914, zlwk @ Q = Wkhq D P+ , dqg mh @ Q 1 Surri1 Ohw D 5 D> D Wkhq D @ ^Dm R V R R V R Dm zlwk Dm 5 D dqg vhw Dm D _ Dm q +DR ^ ^ Dm R , Dm = Dm dqg khqfh [ [ Q +D, @ Q +Dm , Q +Dm , zklfk vkrzv wkdw +D, @ Q +D, iru doo D 5 D= Qrz ohw D 5 D dqg H [ vxfk wkdw +H , ? 41 Jlyhq A 3 fkrrvh Em 5 D +zklfk zh pd| dqg gr dvvxph wr vdwlvi| Em _ En @ > li n 9@ m , vxfk wkdw H ^Em dqg R [ Q +Em , +H , . = m dR Vlqfh D _ H ^+Em _ D, dqg H _ Df ^+Em q D,> R R [ [ Q +Em , @ ^Q +Em _ D, . Q +Em _ Df ,` +H , . m dR m dR +H _ D, . +H _ Df , zkhuh lq wkh vhfrqg olqh zh xvhg wkh vxedgglwlylw| ri = Vlqfh A 3 lv duelwudu| wklv vkrzv wkdw +H , +H _ D, . +H _ Df , dqg wkhuhiruh wkdw D 5 P+ ,1 Wkhruhp 91:1 Ohw D S +[ , eh dq dojheud Q eh d suhphdvxuh rq D dqg P @ +D,1 Wkhq @ m lv d phdvxuh rq P zklfk h{whqgv Q = Pruhryhu li = P $ ^3> 4` lv dqrwkhu phdvxuh vxfk wkdw @ Q rq D wkhq +D, +D, dqg +D, @ +D, li +D, ? 41 Li Q lv 0qlwh wkhq lq idfw @ = ipr xI qvr| UW Surri1 Zh douhdg| nqrz D P+ , dqg wkdw P+ , lv d ~ dojheud/ khqfh D +D, @ P P+ ,= Prvw ri wklv wkhruhp kdv douhdg| ehhq suryhg/ zh qhhg rqo| vkrzv wkh dvvhuwlrqv uhjduglqj wkh phdvxuh = Ohw H 5 P dqg H ^Dm zlwk Dm 5 D> wkhq [ [ +H , +Dm , @ +Dm , lpsolhv +H , +H , @ +H ,= +91;, Li D @ ^Dm wkhq +D, @ olp Q 6R #Q ^ ldR $ Dm @ olp Q 6R #Q ^ R $ Dm @ +D, vr @ rq D = Vxssrvh +H , ? 4 dqg fkrrvh Dm 5 D vxfk wkdw H D @ ^Dm 5 D vxfk wkdw dqg R [ +D, +Dm , +H , . l1h1 +D q H , = m dR Wkhq +H , @ +D, @ +D, +H , . +D q H , +H , . +D q H , +H , . = Vlqfh A 3 lv duelwudu|/ +H , +H , zklfk frpelqhg zlwk Ht1 +91;, vkrzv wkdw +H , @ +H ,= Iru wkh ~ qlwh fdvh/ fkrrvh [m 5 P vxfk wkdw [m % [ dqg +[m , ? 4 wkhq +H , @ olp +H _ [m , @ olp +H _ [m , @ +H ,= m 6R m6R H{dpsoh 91;1 Iru h{dpsohv ri odfn ri xqltxhqhvv ri phdvxuhv h{whqglqj d suh0 phdvxuh Q rq dq Dojheud lq wkh qrq00qlwh fdvh1 vhh Sureohp &56 lq Fkdswhu 41 ri Iroodqg1 Wkhuh rqh frqvlghuv wkh phdvxuhv= R @ 4 h{fhsw rq wkh hpsw| vhw/ S lv frxqwlqj phdvxuh/ dqg T +D, @ T +D _ G, zkhuh G T lv dq| ghqvh vhw1 9161 Uhjxodulw| uhvxowv uhylvlwhg1 Ohppd 91<1 Vxssrvh wkdw Q lv d suhphdvxuh rq dq dojheud D ri vxevhwv ri [= Dovr vxssrvh wkdw Q lv 0qlwh rq D1 Ohw eh wkh dvvrfldwhg rxwhu phdvxuh/ P @ P+ , dqg @ mP1 Wkhq iru dq| E 5 P @ P+ , wkhuh h{lvwv D E F vxfk wkdw D 5 D > F 5 D dqg +F q D, @ 3= Surri1 Fkrrvh [p 5 D lqfuhdvlqj wr [ vxfk wkdw Q +[p , ? 4 iru doo p dqg vhw Ep @ [p _ E 1 Fkrrvh Fp vr wkdw Ep Fp 5 D / dqg Fp [p dqg +Fp q Ep , lv vpdoo1 Ohw F ^Fp > wkhq +F q E , +^p +Fp q E ,, [ [ +Fp q E , +Fp q Ep , ? = hu SUQh yrpr ra rhr ur UX Vlqfh D _D @ D zh pd| fkrrvh Fn 5 D vxfk wkdw E Fn > Fn lv ghfuhdvlqj lq R W Fn 1 Wkhq +F q E , @ 3 dqg F 5 D 1 n dqg olpn6R +Fn q E , @ 31 Ghqh F ndR Qrz= Fkrrvh Df E f 5 D vxfk wkdw +E f q Df , @ 3> l1h1 3 @ +E f _ D, @ +D q E ,1 Wkhq D 5 D dqg +D q E , @ 3 dqg khqfh zh kdyh D E F dqg +F q D, @ +F q E , . +E q D, @ 3= Zh qrz vkrz wkdw D P = Vxssrvh wkdw H [ dqg +H , ? 41 Fkrrvh R V Eq dqg Eq 5 D vxfk wkdw H qdR R [ +H , +Eq , = qdR Vlqfh +Eq , @ +Eq _ D, . +Eq _ Df ,> [ [ +H , . +Eq , @ +Eq _ D, . 5+Eq _ Df , +H _ D, . +H _ Df , zklfk lpsolhv wkdw +H , +H _ D, . +H _ Df ,= Wkhruhp 91431 Vxssrvh wkdw Q lv d ~ qlwh suhphdvxuh rq dq dojheud D S +[ ,/ lv wkh dvvrfldwhg rxwhu phdvxuh/ P @ P+ ,> @ mP dqg @ mChD = Wkhq P @ +D, @ +D, ~ wkh frpsohwlrq ri +D, zlwk uhvshfw wr = Surri1 Iru E 5 P wkhuh h{lvwv D 5 D dqg F 5 D vxfk wkdw D E F dqg +F q D, @ +F q D, @ 3= Wklv lpsolhv E 5 +D, dqg khqfh wkdw P +D,1 Iru wkh uhyhuvh lqfoxvlrq/ vxssrvh Q 5 +D,> l1h1 Q I 5 +D, vxfk wkdw 3 @ +I , @ +I ,1 Wkhq +Q , @ 3 dqg +H , +H _ Q , . +H _ Q f , @ 3 . +H _ Q f , +H , zklfk vkrzv wkdw Q 5 P1 9171 Frqvwuxfwlrq ri phdvxuhv rq d vlpsoh surgxfw vsdfh1 H{huflvh 91441 Ohw \ i3> 4jQ +wkh vhw ri vhtxhqfhv | @ +|R > |S > = = =, zlwk |l 5 [ i3> 4j/ \q i3> 4jq iru doo q 5 Q> dqg q = \ $ \q eh ghqhg e| q +| , @ +|R > |S > = = = > |q ,= D ghqrwh wkh froohfwlrq ri _f|olqghu vhwv% lq \> l1h1 vhwv ri wkh irup +91<, D @ R +F , zkhuh q 5 Q dqg F \q = q Lq zrugv d f|olqghu vhw lv d vxevhw ri \ zklfk lv ghwhuplqhg e| uhvwulfwlqj wkh ydoxhv ri rqo| d qlwh qxpehu ri frruglqdwhv ri | 5 \= Iru h{dpsoh D i| 5 \ = |Sl @ 3 iru l 5 Q j lv qrw d f|olqghu vhw1 ipr xI qvr| UY d, Vkrz wkdw D lv dq dojheud1 e, Vkrz wkdw li Dq 5 D dqg Dq & > wkhq Dq @ > iru doo q vxflhqwo| odujh1 f, Frqfoxgh wkdw dq| qlwho| dgglwlyh phdvxuh Q rq D lv d suhphdvxuh1 +4, Vroxwlrq d, Ohw Dq R +S +\q ,, wkhq Dq lv wkh sxoo0edfn ri dq dojheud q dqg khqfh dq dojheud1 Qh{w qrwlfh wkdw Dq DqFR iru hdfk q= Wr vhh wklv/ ohw F \q dqg vhw D @ R +F ,= Wkhq D @ R +F [ , zkhuh [ i3> 4j= q q Wklv vkrzv wkdw D 5 DqFR dqg khqfh wkdw Dq DqFR 1 Qrz lw lv qrw kdug wr fkhfn dq lqfuhdvlqj xqlrq ri dojheudv lv vwloo dq dojheud dqg khqfh D @ ^q Dq lv dq dojheud1 e, Zh zloo suryh wkh frqwudsrvlwlyh ri sduw e,1 Qdpho|/ vxssrvh wkdw Dn 5 D zlwk Dn ghfuhdvlqj dv n lqfuhdvhv dqg DQ @ _Q Dn 9@ > iru hdfk ndR Q 5 Q> wkhq _R Dn 9@ >= Xvlqj wkh lghdv lq wkh surri ri sduw d, deryh/ ndR zh pd| fkrrvh dq lqfuhdvlqj vhtxhqfh iqn j dqg vhwv Gn \qn vxfk wkdw R Dn @ n +Gn , iru doo n 5 Q= Wr vlpsoli| qrwdwlrq uhsodfh wkh vhtxhqfh q iDR > DS > DT > = = =j e| wkh vhtxhqfh ri vhwv q4 R q5 q4 q6 q5 } ~ } ~ } ~ iER > ES > ET > = = =j i\> = = = > \ > DR > = = = DR > DS > = = = > DS > = = =j= Wkhq El lv dovr ghfuhdvlqj/ _Q El 9@ > iru doo Q 5 Q> dqg _R El @ ldR ldR _R Dq = Pruhryhu/ wkh El pd| eh zulwwhq lq wkh irup El @ R +Fl , zkhuh qdR l Fl \l iru doo l 5 Q= Zh zloo qrz qlvk wkh surri e| ri sduw e, e| vkrzlqj wkdw _R El 9@ >= ldR Wr gr wklv/ qrwlfh wkdw El @ R +Fl , ElFR @ R +FlFR , lpsolhv FlFR l lFR Fl [ iru hdfk l 5 Q= Vr e| lqgxfwlrq/ +9143, Fm Fl [ CmlD iru doo m A l= Vlqfh qr El lv hpsw| e| dvvxpswlrq/ qr Fl lv hpsw| hlwkhu1 Lq sduwlfxodu wklv lpsolhv wkdw i{R m{ 5 Fl j lv qrw hpsw| iru hdfk l= Fkrrvh R vr wkh wkdw iru lqqlwho| pdq| l*v/ R 5 i{R m{ 5 Fl j= Wkhq e| +9143, lw pxvw kdsshq wkdw R 5 i{R m{ 5 Fl j iru doo l= E| vlplodu orjlf i{S m{ 5 Fl zlwk {R @ R j lv qrq hpsw| iru doo l 5= Khqfh zh pd| fkrrvh S 5 i3> 4j vxfk wkdw +R > S , 5 i+{R > {S ,m{ 5 Fl j iru doo l 5= E| lqgxfwlrq/ rqh pd| vkrz wkhuh h{lvwv R > S > T > = = = 5 i3> 4j vxfk wkdw +R > = = = > m , 5 i+{R > {S > = = = > {m ,m{ 5 Fl j iru doo l m dqg lq sduwlfxodu +R > = = = > m , 5 Fm iru doo m 5 Q= Vhw +R > S > = = =, 5 \> wkhq zh kdyh vkrzq wkdw m +, @ +R > = = = > m , 5 Fm iru doo m 5 Q= Wklv vkrzv wkdw 5 Em iru doo m> l1h1 5 _Em @ _Dq dqg khqfh _q Dq 9@ >= f, Lw iroorzv/ e| Krphzrun 61414 dqg zkdw zh kdyh mxvw suryhg/ wkdw dq| qlwho| dgglwlyh phdvxuh Q rq D lv dfwxdoo| frxqwdeo| dgglwlyh rq D> l1h1 Q lv d suhphdvxuh rq D= H{huflvh 91451 Vkrz wkhuh lv d xqltxh qlwho| dgglwlyh phdvxuh Q rq D vxfk wkdw Q +D, @ 5q li D lv d vhw ri wkh irup +9144, D @ i| 5 \ m|l @ l iru l @ 4> 5> = = = > qj> zkhuh hdfk l 5 i3> 4j= Xvh wkh deryh sureohpv wr frqfoxgh wkhuh h{lvwv xqltxh phdvxuh rq P +D, vxfk wkdw +D, @ 5q li D lv dv lq +9144,1 hu SUQh yrpr ra rhr ur U` +4, Vroxwlrq= Iru D 5 S +\q , ohw q +D, @ 5q &+D,> zkhuh &+D, ghqrwhv wkh qxpehu ri hohphqwv lq D= Wkhq q lv d phdvxuh rq S +\q ,= Li D 5 Dq D lv ri wkh irup D @ R +F , zlwk F \q > vhw Q +D, @ q +F ,= Zh pxvw q vkrz wkdw Q lv zhoo ghqhg1 Iru wklv vxssrvh wkdw D @ R +G, iru vrph p G \p = Zlwkrxw orvv ri jhqhudolw| dvvxph wkdw p A q> wkhq G pxvw eh jlyhq e| G @ F [ CpqD = Wkhuhiruh p +G, @ 5p &+F [ CpqD , @ 5p 5CpqD &+F , @ 5q &+F , @ q +F ,> zklfk vkrzv wkdw Q lv zhoo ghqhg1 Qrz lw lv hdvlo| fkhfnhg wkdw Q lv d phdvxuh rq D vlqfh p lv d phdvxuh rq Dp iru hdfk p= Wkhuhiruh e| wkh odvw sureohp/ Q lv lq idfw d suhphdvxuh rq D= E| Wkhruhp 4147 ri Iroodqg/ lw iroorzv wkdw h{whqgv wr d phdvxuh rq P @ +D,= Uhpdun 9146 +D Fu|swlf Uhpdun,1 Wkh phdvxuh lv hvvhqwldoo| Ohehvjxh phdvxuh rq wkh xqlw lqwhuydo ^3> 4`= :1 NA|A N t BAa 6jBt iBO,j u AW|NAt Zh duh prvwo| lqwhuhvwhg lq phdvxudeoh ixqfwlrqv/ qhyhuwkhohvv lw lv lqvwuxfwlyh wr uvw uhirupxodwh wkh qrwlrq ri d frqwlqxrxv ixqfwlrq ehwzhhq wzr phwulf vsdfhv1 Ohppd :141 Vxssrvh wkdw +[> , dqg +\> g, duh wzr phwulf vsdfhv dqg i = [ $ \ lv d ixqfwlrq1 Wkhq iroorzlqj duh htxlydohqw= +4, i lv frqwlqxrxv1 +5, i R +Y , 5 iru doo Y 5 g > l1h1 i R +Y , lv rshq lq [ li Y lv rshq lq \= +6, i R +F , lv forvhg lq [ li F lv forvhg lq \= +7, Iru doo frqyhujhqw vhtxhqfhv i{q j [> ii +{q ,j lv frqyhujhqw lq \ dqg olp i +{q , @ i olp {q = q6R q6R Surri1 41 , 51 Iru doo { 5 [ dqg A 3 wkhuh h{lvwv A 3 vxfk wkdw g+i +{,> i +{Q ,, ? li +{> {Q , ? 1 l1h1 E{ + , i R +Ei C{D +,, Vr li Y Q \ dqg { 5 i R +Y , zh pd| fkrrvh A 3 vxfk wkdw Ei C{D +, Y wkhq E{ + , i R +Ei C{D +,, i R +Y , vkrzlqj wkdw i R +Y , lv rshq1 51 +, 61 Li F lv forvhg lq \> wkhq F f r \ dqg khqfh i R +F f , r [= Vlqfh R+F f , @ i R+F , f > wklv vkrzv wkdw i R+F , lv wkh frpsolphqw ri dq rshq vhw i dqg khqfh forvhg1 Vlploduo| rqh vkrzv wkdw 61 , 5= 51 , 4= Ohw A 3 dqg { 5 [> wkhq/ vlqfh i R +Ei C{D +,, r [>wkhuh h{lvwv A 3 vxfk wkdw E{ + , i R +Ei C{D +,, l1h1 li +{> {Q , ? wkhq g+i +{Q ,> i +{,, ? 1 41 , 71 Li i lv frqwlqxrxv dqg {q $ { lq [> ohw A 3 dqg fkrrvh A 3 vxfk wkdw g+i +{,> i +{Q ,, ? zkhq +{> {Q , ? 1 Wkhuh h{lvwv dq Q A 3 vxfk wkdw +{> {q , ? iru doo q Q dqg wkhuhiruh g+i +{,> i +{q ,, ? iru doo q Q= Wkdw lv wr vd| olpq6R i +{q , @ i +{, dv q $ 41 71 , 41Zh zloo vkrz wkdw qrw 41 , qrw 71 qrw 4 lpsolhv wkhuh h{lvwv A 3/ R d srlqw { 5 [ dqg d vhtxhqfh i{q jqdR [ vxfk wkdw g+i +{,> i +{q ,, zkloh R = Fohduo| wklv vhtxhqfh i{q j ylrodwhv 71 +{> {q , ? q ipr xI qvr| VQ Qrwdwlrq :151 Iru d jhqhudo wrsrorjlfdo vsdfh +[> ,/ wkh Eruho ~ dojheud lv wkh ~ dojheud/ E[ @ + ,= Zh zloo xvh EU wr ghqrwh wkh Eruho 0 dojheud rq U dqg uhfdoo wkdw EU @ +i+d> 4, = d 5 Uj,= Rxu qrwlrq ri d _phdvxudeoh% ixqfwlrq zloo eh vlplodu wr frqglwlrqv 51 dqg 61 deryh iru frqwlqxlw|1 Iru prwlydwlrqdo sxusrvhv/ vxssrvh +[> P> , lv d phdvxuh vsdfh i = [ $ UF 1 Urxjko| vshdnlqj/ lq wkh qh{w vhfwlrq zh duh jrlqj wr ghqh U i g e| [ ] R [ ig @ olp d +i R +dl > dlFR `,= phvk6Q Q?d ?d ?d ?=== l 456 [ Iru wklv wr pdnh vhqvh zh zloo qhhg wr uhtxluh i R ++d> e`, 5 P iru doo d ? e= Ehfdxvh ri Ohppd :1: ehorz/ wklv odvw frqglwlrq lv htxlydohqw wr wkh frqglwlrq i R +EU , P> zkhuh zh duh xvlqj wkh iroorzlqj qrwdwlrq1 Qrwdwlrq :161 Li i = [ $ \ lv d ixqfwlrq dqg H S +\ , ohw i R H i R +H , ii R +H ,mH 5 Hj= Li J S +[ ,> ohw i J iD 5 S +\ ,mi R +D, 5 Jj= Lw lv hdvlo| fkhfnhg wkdw i R H dqg i J duh ~ dojheudv +wrsrorjlhv, surylghg H dqg J duh ~ dojheudv +wrsrorjlhv,1 +\rx vkrxog fkhfn wkhvh vwdwhphqwv1, Ghqlwlrq :171 Ohw +[> P, dqg +\> I , eh phdvxudeoh +wrsrorjlfdo, vsdfhv1 D ixqfwlrq i = [ $ \ lv phdvxudeoh +frqwlqxrxv, li i R +I , P1 Zh zloo dovr vd| wkdw i lv P@I ~ phdvxudeoh +frqwlqxrxv, ru +P> I , ~ phdvxudeoh +frqwlqxrxv,1 Uhpdun :181 Ohw i = [ $ \ eh d ixqfwlrq1 Jlyhq d ~ dojheud +wrsrorj|, I S +\ ,> wkh ~ dojheud +wrsrorj|, P =@ i R +I , lv wkh vpdoohvw ~ dojheud +wrsrorj|, rq [ vxfk wkdw i lv +P> I , 0 phdvxudeoh +frqwlqxrxv,1 Vlploduo|/ li P lv d 0 dojheud +wrsrorj|, rq [ wkhq I @ i P lv wkh odujhvw ~ dojheud +wrsrorj|, rq \ vxfk wkdw i lv +P> I , 0 phdvxudeoh +frqwlqxrxv,1 Ohppd :191 Vxssrvh wkdw +[> P,> +\> I , dqg +]> J , duh phdvxudeoh +wrsrorjlfdo, vsdfhv1 Li i = +[> P, $ +\> I , dqg j = +\> I , $ +]> J , duh phdvxudeoh +frqwlqxrxv, ixqfwlrqv wkhq j i = +[> P, $ +]> J , lv phdvxudeoh +frqwlqxrxv, dv zhoo1 Surri1 Wklv lv hdv| vlqfh e| dvvxpswlrq jR +J , I dqg i R +I , P vr wkdw +j i ,R +J , @ i R j R +J , i R +I , P= Ohppd :1:1 Vxssrvh wkdw i = [ $ \ lv d ixqfwlrq dqg H S +\ ,> wkhq +:14, +:15, i R +H , i R +H , @ i R ++H ,, dqg @ i R + +H ,,= Pruhryhu/ li I @ +H , +ru I @ +H ,, dqg P lv d ~ dojheud +wrsrorj|, rq [> wkhq i lv +P> I , ~ phdvxudeoh +frqwlqxrxv, l i R +H , P= hu SUQh yrpr ra rhr ur VR Surri1 Zh zloo suryh Ht1 +:14,/ wkh surri ri Ht1 +:15, ehlqj dqdorjrxv1 Li H I > wkhq i R +H , i R + +H ,, dqg wkhuhiruh/ +ehfdxvh i R + +H ,, lv d ~ dojheud, J =@ +i R +H ,, i R + +H ,, zklfk suryhv kdoi ri Ht1 +:14,1 Iru wkh uhyhuvh lqfoxvlrq qrwlfh wkdw i J @ E \ = i R +E , 5 J = lv d ~ dojheud zklfk frqwdlqv H dqg wkxv +H , i J = Khqfh li E 5 +H , zh nqrz wkdw i R +E , 5 J > l1h1 i R + +H ,, J = Wkh odvw dvvhuwlrq ri wkh Ohppd lv dq hdv| frqvhtxhqfh ri Htv1 +:14, dqg +:15,1 Fruroodu| :1;1 Vxssrvh wkdw +[> P, lv d phdvxudeoh vsdfh1 Wkhq i = [ $ U lv +P> EU , 0 phdvxudeoh l i R ++d> 4,, 5 P iru doo d 5 U l i R ++d> 4,, 5 P iru doo d 5 T l i R ++ 4> d`, 5 P iru doo d 5 U> hwf1 Vlploduo|/ li +[> P, lv d wrsrorjlfdo vsdfh/ wkhq i = [ $ U lv +P> U , 0 frqwlqxrxv l i R ++d> e,, 5 P iru doo 4 ? d ? e ? 4 l i R ++d> 4,, 5 P dqg i R ++ 4> e,, 5 P iru doo d> e 5 T= Surri1 Dq h{huflvh lq xvlqj Ohppd :1:1 Zh zloo riwhq ghdo zlwk ixqfwlrq i = [ $ U @ U^ i 4j = Ohw +:16, EU =@ ^d> 4` = d 5 U = % Wkh iroorzlqj Fruroodu| ri Ohppd :1: lv d gluhfw dqdorjxh ri Fruroodu| :1;1 Fruroodu| :1<1 i = [ $ U lv +P> EU , 0 phdvxudeoh l i R ++d> 4`, 5 P iru doo % l i R ++ 4> d`, 5 P iru doo d 5 U> hwf1 d5U Sursrvlwlrq :1431 D vxevhw D U lv lq EU l D _ U 5EU = Lq sduwlfxodu i 4j > % i4j dqg i 4j duh lq EU = % Surri1 Ohw l = U $ U eh wkh lqfoxvlrq pds1 Vlqfh lR ++^d> 4`,, @ ^d> 4,_U 5EU > l lv +EU > EU ,~ phdvxudeoh1 Lq sduwlfxodu li D 5 EU > wkhq lR +D, @ iru doo d 5 U % % D _ U 5EU = Iru wkh frqyhuvh/ zh ehjlq zlwk wkh revhuydwlrqv= i 4j @ _R ^ 4> q, @ _R ^ q> 4`f 5 EU > % qdR qdR i4j @ ^4> 4` 5 EU dqg % U @ Uq i 4j 5 EU = % Xvlqj wkhvh idfwv |rx pd| hdvlo| vkrzv wkdw P @ iD U = D 5 EU j % lv d ~ dojheud rq U zklfk frqwdlqv +d> 4, iru doo d 5 U= Khqfh EU P> l1h1 EU EU = Xvlqj wkhvh revhuydwlrqv/ li D U dqg D _ U 5EU > wkhq % D @ +D _ U, ^ +D _ i 4j, 5 EU = % Sursrvlwlrq :144 +Forvxuh xqghu vxsv/ lqiv dqg olplwv,1 Vxssrvh wkdw +[> P, lv d phdvxudeoh vsdfh dqg im = +[> P, $ U lv d vhtxhqfh ri P@EU ~ phdvxudeoh ixqfwlrqv1 Wkhq vxsm im > lqi m im > olp vxs im dqg olp lqi im m 6R m 6R ipr xI qvr| VS duh doo P@EU ~ phdvxudeoh ixqfwlrqv1 +Qrwh wkdw wklv uhvxow lv lq jhqhudoo| idovh zkhq +[> P, lv d wrsrorjlfdo vsdfh dqg phdvxudeoh lv uhsodfhg e| frqwlqxrxv lq wkh vwdwhphqw1, Surri1 Ghqh jF +{, =@ vxs m im +{,> wkhq i{ = jF +{, dj @ i{ = im +{, d ; m j @ _m i{ = im +{, dj 5 P vr wkdw jF lv phdvxudeoh1 Vlploduo| li j +{, @ lqi m im +{, wkhq i{ = j +{, dj @ _m i{> im +{, dj 5 P= Vlqfh olp vxs im @ lqi vxs iim = m qj dqg q olp im @ vxs lqi iim = m qj q zh duh grqh e| zkdw zh kdyh douhdg| suryhg1 Ohppd :1451 Vxssrvh wkdw +[> P, lv d phdvxudeoh vsdfh/ +\> , eh d wrsrorjlfdo vsdfh dqg im = [ $ \ lv +P> E\ , ~ phdvxudeoh iru doo m= Dovr dvvxph wkdw iru hdfk { 5 [> i +{, @ olpq6R iq +{, h{lvwv1 Wkhq i = [ $ \ lv dovr +P> E\ , ~ phdvxudeoh1 Surri1 Vxssrvh wkdw Y \ lv dq rshq vhw/ wkhq i R +Y , @ i{ = i +{, 5 Y j @ i{ = iq +{, 5 Y iru doprvw doo qj @ ^RdR _R Q iq R +Y , 5 P Q qd vlqfh iqR +Y , 5 P ehfdxvh hdfk iq lv phdvxudeoh1 Wkhuhiruh i R + , P dqg wkxv i R +E\ , @ i R + + ,, @ i R + , P= Ghqlwlrq :1461 D ixqfwlrq i = [ $ \ ehwzhhq wr wrsrorjlfdo vsdfhv lv Eruho phdvxudeoh li i R +E\ , E[ 1 Sursrvlwlrq :1471 Ohw [ dqg \ eh wzr wrsrorjlfdo vsdfhv dqg i = [ $ \ eh d frqwlqxrxv ixqfwlrq1 Wkhq i lv Eruho phdvxudeoh1 Surri1 Xvlqj Ohppd :1: lw vxfhv wr uhfdoo E\ @ + \ , dqg wr qrwlfh wkdw i R +Y , 5 [ E[ iru doo Y rshq lq \= Ghqlwlrq :1481 D ixqfwlrq i = U $ U lv Ohehvjxh phdvxudeoh li i R +EU , p O =@ EU ~ wkh frpsohwlrq ri EU uhodwlyh wr Ohehvjxh phdvxuh p= :141 Uhodwlyh Wrsrorjlhv dqg ~ Dojheudv1 Ghqlwlrq :1491 Ohw H S +[ , eh d froohfwlrq ri vhwv/ D [> lD = D $ [ eh wkh lqfoxvlrq pds +lD +{, @ {, iru doo { 5 D> dqg HD @ lR +H , @ iD _ H = H 5 Hj = D Sursrvlwlrq :14:1 Vxssrvh wkdw D [> P S +[ , lv d ~ dojheud dqg S +[ , lv d wrsrorj|/ wkhq PD S +D, lv d ~ dojheud dqg D S +D, lv d wrsrorj|1 +Wkh wrsrorj| D lv fdoohg wkh uhodwlyh wrsrorj| rq D=, Pruhryhu li H S +[ , lv vxfk wkdw P @ +H , + @ +H ,, wkhq PD @ +HD , + D @ +HD ,,= hu SUQh yrpr ra rhr ur VT Surri1 Wkh uvw dvvhuwlrq rv hdv| wr fkhfn dv uhpdunhg diwhu Qrwdwlrq :161 Wkh vhfrqg dvvhuwlrq lv d frqvhtxhqfh ri Ohppd :1:1 Lqghhg/ PD @ lR +P, @ lR ++H ,, @ +lR +H ,, @ +HD , D D D dqg vlploduo| D @ lR + , @ lR + +H ,, @ +lR +H ,, @ +HD ,= D D D Ghqlwlrq :14;1 Ohw D [> i = D $ F eh d ixqfwlrq/ P S +[ , eh d ~ dojheud dqg S +[ , eh d wrsrorj|/ wkhq zh vd| wkdw i mD lv phdvxudeoh +frqwlqxrxv, li i mD lv PD ~ phdvxudeoh + D frqwlqxrxv,1 Sursrvlwlrq :14<1 Ohw D [> i = [ $ F eh d ixqfwlrq/ P S +[ , eh d ~ dojheud dqg S +[ , eh d wrsrorj|1 Li i lv P ~ phdvxudeoh + frqwlqxrxv, wkhq i mD lv PD phdvxudeoh + D frqwlqxrxv,1 Pruhryhu li Dq 5 P +Dq 5 , vxfk wkdw [ @ ^R Dq dqg i mDq lv PDq phdvxudeoh + Dq frqwlqxrxv, iru doo q> wkhq i lv qdR P ~ phdvxudeoh + frqwlqxrxv,1 Surri1 Qrwlfh wkdw lD lv +PD > P, ~ phdvxudeoh + D > , ~ frqwlqxrxv, khqfh i mD @ i lD lv PD phdvxudeoh + D ~ frqwlqxrxv,1 Ohw E F eh d Eruho vhw dqg frqvlghu R i R +E , @ ^R i R +E , _ Dq @ ^R i mq +E ,= qdR qdR D Li D 5 P +D 5 ,> wkhq lw lv hdv| wr fkhfn wkdw PD D @ iE 5 P = E Dj P dqg @ iE 5 = E Dj = Wkh vhfrqg dvvhuwlrq lv qrz dq hdv| frqvhtxhqfh ri wkh suhylrxv wkuhh htxdwlrqv1 :151 Surgxfw Vsdfhv1 Wkh uhdghu zkr qgv wklv vhfwlrq d olwwoh wrr khdy| pd| zlvk wr uvw uhdg Dsshqgl{ :17 ehorz zkhuh wkh prvw lpsruwdqw vshfldo fdvhv duh fryhuhg1 Wkh pdwhuldo lq wklv vxevhfwlrq ehiruh Fruroodu| :15; pd| wkhq eh vnlsshg rq wkh uvw uhdglqj1 Ghqlwlrq :1531 Ohw [ dqg D eh vhwv/ dqg vxssrvh iru 5 D zh duh jlyh d phdvxudeoh +wrsrorjlfdo, vsdfh +\ > I , dqg d ixqfwlrq i = [ $ \ = Zh zloo zulwh +i = 5 D, + +i = 5 D,, iru wkh vpdoohvw 0dojheud +wrsrorj|, rq [ vxfk wkdw hdfk i lv phdvxudeoh +frqwlqxrxv,/ l1h1 +i = 5 D, @ +^ i R +I ,, dqg +i = 5 D, @ +^ i R +I ,,1 Sursrvlwlrq :1541 Dvvxplqj wkh qrwdwlrq lq Ghqlwlrq :153 dqg dgglwlrqdoo| ohw +]> P, eh d phdvxudeoh +wrsrorjlfdo, vsdfh dqg j = ] $ [ eh d ixqfwlrq1 Wkhq j lv +P> +i = 5 D,, ~ phdvxudeoh ++P> +i = 5 D,, ~ frqwlqxrxv, l i j lv +P> I ,~phdvxudeoh +frqwlqxrxv, iru doo 5 D= Surri1 +,, Li j lv +P> +i = 5 D,, ~ phdvxudeoh/ wkhq wkh frpsrvlwlrq i j lv +P> I , ~ phdvxudeoh e| Ohppd :191 ++, Ohw J @ +i = 5 D, @ ^SD i R +I , = ipr xI qvr| VU Li i j lv +P> I , ~ phdvxudeoh iru doo > wkhq j R i R +I , P ; 5 D dqg wkhuhiruh Khqfh jR ^SD i R +I , @ ^SD jR i R +I , P= j R +J , @ j R ^SD i R +I , @ +j R ^SD i R +I , P zklfk vkrzv wkdw j lv +P> J , ~ phdvxudeoh1 Wkh wrsrorjlfdo fdvh lv suryhg lq wkh vdph zd|1 Ghqlwlrq :1551 Vxssrvh+[ P ,SD lv d froohfwlrq ri phdvxudeoh vsdfhv dqg ohw [ eh wkh surgxfw vsdfh \ [ [@ SD dqg Q = [ $ [ eh wkh fdqrqlfdo surmhfwlrq pdsv1 Wkhq wkh surgxfw ~ dojheud/ P > lv ghqhg e| # $ R ^ P + = 5 D, @ R +P , = SD Q P > Vlploduo| li +[ P ,SD lv d froohfwlrq ri wrsrorjlfdo/ wkh surgxfw wrsrorj| lv ghqhg e| # $ ^ R R+P , = P + = 5 D, @ SD Uhpdun :1561 Ohw +]> P, eh d phdvxudeoh +wrsrorjlfdo, vsdfh dqg $ # R \ [ > P [@ SD SD eh dv lq Ghqlwlrq :1551 E| Sursrvlwlrq :154/ d ixqfwlrq i = ] $ [ lv phdvxudeoh +frqwlqxrxv, l i lv +P> P , ~ phdvxudeoh +frqwlqxrxv, iru doo 5 D= Sursrvlwlrq :1571 Vxssrvh wkdw +[ P ,SD lv d froohfwlrq ri phdvxudeoh +wrsr0 orjlfdo, vsdfhv dqg H P jhqhudwhv P iru hdfk 5 D> wkhq +:17, SD P @ ^SD R +H , ^SD R +H , Pruhryhu/ vxssrvh wkdw D lv hlwkhu qlwh ru frxqwdeo| lqqlwh/ [ 5 H iru hdfk 5 D> dqg P @ +H , lv d ~ dojheud iru doo 5 D= Wkhq wkh surgxfw ~ dojheud vdwlvhv #+ ,$ \ R P @ H = H 5 H iru doo 5 D = +:18, SD SD Vlploduo| li D lv qlwh dqg P @ +H ,> wkhq wkh surgxfw wrsrorj| vdwlvhv #+ ,$ R \ +:19, P @ H = H 5 H iru doo 5 D = SD SD hu SUQh yrpr ra rhr ur VV Surri1 Zh zloo suryh Ht1 +:17, lq wkh phdvxuh wkhruhwlf fdvh vlqfh d vlplodu V surri zrunv lq wkh wrsrorjlfdo fdwhjru|1 Vlqfh R +H , ^ R +P ,> lw iroorzv wkdw # $ # $ ^ R+H , ^ R +P, @ R P= I =@ Frqyhuvho|/ I +R +H ,, @ R ++H ,, @ R +P , krogv iru doo lpsolhv wkdw ^ R +P , I Q dqg khqfh wkdw P I 1 Zh qrz suryh Ht1 +:18,1 Vlqfh zh duh dvvxplqj wkdw [ 5 H iru hdfk 5 D> zh vhh wkdw + , \ ^ R +H , H = H 5 H iru doo 5 D SD dqg wkhuhiruh e| Ht1 +:17, # $ #+ ,$ R ^ \ R+H , P @ H = H 5 H iru doo 5 D = SD SD Wklv odvw vwdwhphqw lv wuxh lqghshqghqw dv wr zkhwkhu D lv frxqwdeoh ru qrw1 Iru wkh uhyhuvh lqfoxvlrq lw vxfhv wr qrwlfh wkdw vlqfh D lv frxqwdeoh/ \ R H @ _SD R +H , 5 P SD SD dqg khqfh #+ ,$ \ R H = H 5 H iru doo 5 D P = SD SD Ohw xv uhfrug wkh iroorzlqj vshfldo fdvh ri Sursrvlwlrq :1571 Fruroodu| :1581 Vxssrvh +[l > Pl , duh phdvxudeoh +wrsrorjlfdo, vsdfhv/ Hl S +[l , vxfk wkdw [l 5 Hl iru l @ 4> 5 dqg Pl @ +Hl , +Pl @ +Hl ,, = Wkhq PR PS @ +HR HS , + +HR HS ,, = Sursrvlwlrq :1591 Ohw i[ jSD eh d vhtxhqfh ri vhwv zkhuh D lv dw prvw frxqw0 deoh1 Vxssrvh iru hdfk 5 D zh duh jlyhq d frxqwdeoh vhw H S +[ ,1 Ohw T @ +H , eh wkh wrsrorj| rq [ jhqhudwhg e| H dqg [ eh wkh surgxfw vsdfh SD [ zlwk htxlsshg zlwk wkh surgxfw wrsrorj| =@ SD +H ,= Wkhq wkh Eruho ~ dojheud E[ @ + , lv wkh vdph dv wkh surgxfw ~ dojheud= E[ @ SD E[ > zkhuh E[ @ + +H ,, @ +H , iru doo 5 D= Surri1 E| Sursrvlwlrq :157/ wkh wrsrorj| pd| eh ghvfulehg dv wkh vpdoohvw wrsrorj| frqwdlqlqj H @ ^SD R +H ,= Qrz H lv wkh frxqwdeoh xqlrq ri frxqwdeoh vhwv vr lv vwloo frxqwdeoh1 Wkhuhiruh e| Sursrvlwlrq 6147 dqg Sursrvlwlrq :157 zh kdyh E[ @ + , @ + +H ,, @ +H , @ SD +H , @ SD + , @ SD E[ = ipr xI qvr| VW Sursrvlwlrq :15:1 Li +[l > l , iru l @ 4> = = = > q eh phwulf vsdfhv vxfk wkdw iru hdfk l wkhuh d frxqwdeoh ghqvh vxevhw Gl [l / wkhq R E[l @ EC[4 [q D l zkhuh E[l lv wkh Eruho ~ dojheud rq [l dqg EC[4 [q D lv wkh Eruho ~ dojheud [q htxlsshg zlwk wkh wrsrorj| frplqj iurp wkh surgxfw phwulf rq [R q [ ++{R > {S > = = = > {q ,> +|R > |S > = = = > |q ,, @ l +{l > |l ,= ldR Surri1 Ohw +{R > {S > = = = > {q , 5 [R [q dqg A 3 eh jlyhq1 Lw lv hdvlo| vhhq wkdw \ \ E +{l > @q, E ++{R > {S > = = = > {q ,> , E +{l > ,= l l Ehfdxvh ri wklv htxdwlrq/ lw lv hdvlo| vhhq wkdw @ 4 5 q lv mxvw wkh [q 1 Surri lv qrz frpsohwh e| dsso|lqj Sursrvlwlrq surgxfw wrsrorj| rq [R :159 zlwk Hl =@ iE +{> , [l = { 5 Gl dqg 5 T_+3> 4,j = Fruroodu| :15;1 Ohw EUq ghqrwh wkh Eruho ~dojheud rq Uq > wkhq EUq @ EU EU = Pruhryhu li +[> P, lv d phdvxudeoh vsdfh/ wkhq EU i @ +iR > iS > = = = > iq , = [ $ Uq lv +P> EUq , ~ phdvxudeoh l il = [ $ U lv +P> EU , ~ phdvxudeoh iru hdfk l= Lq sduwlfxodu/ d ixqfwlrq i = [ $ F lv +P> EF , ~ phdvxudeoh l Uh i dqg Lp i duh +P> EU , ~ phdvxudeoh1 Fruroodu| :15<1 Ohw +[> P, eh d phdvxudeoh vsdfh dqg i> j = [ $ F eh +P> EF , ~ phdvxudeoh ixqfwlrqv1 Wkhq i j dqg i j duh dovr +P> EF , ~ phdvxudeoh1 Surri1 Ghqh I = [ $ F F> D = F F $ F dqg P = F F $ F e| I +{, @ +i +{,> j+{,,> D +z> } , @ z } dqg P +z> } , @ z}= Wkhq D dqg P duh frqwlqxrxv dqg khqfh +EF5 > EF , ~ phdvxudeoh1 Dovr I lv +P> EF EF , @ +P> EF5 , ~ phdvxudeoh vlqfh R I @ i dqg S I @ j duh +P> EF , ~ phdvxudeoh phdvxudeoh1 Wkhuhiruh D I @ i j dqg P I @ i j ehlqj wkh frpsrvlwlrq ri phdvxudeoh ixqfwlrqv duh dovr phdvxudeoh1 Ohppd :1631 Ohw 5 F> +[> P, eh d phdvxudeoh vsdfh dqg i = [ $ F eh d +P> EF , ~ phdvxudeoh ixqfwlrq1 Wkhq R i C{D li i +{, 9@ 3 I +{, =@ li i +{, @ 3 lv phdvxudeoh1 Surri1 Ghqh l = F $ F e| R } li } 9@ 3 l+} , @ li } @ 3= Iru dq| rshq vhw Y F zh kdyh lR +Y , @ lR +Y q i3j, ^ lR +Y _ i3j, hu SUQh yrpr ra rhr ur VX Ehfdxvh l lv frqwlqxrxv h{fhsw dw } @ 3> lR +Y q i3j, lv dq rshq vhw dqg khqfh lq EF = Pruhryhu/ lR +Y _ i3j, 5 EF vlqfh lR +Y _ i3j, lv hlwkhu wkh hpsw| vhw ru wkh rqh srlqw vhw ij = Wkhuhiruh lR + F , EF dqg khqfh lR +EF , @ lR ++ F ,, @ +lR + F ,, EF zklfk vkrzv wkdw l lv Eruho phdvxudeoh1 Vlqfh I @ l i lv wkh frpsrvlwlrq ri phdvxudeoh ixqfwlrqv/ I lv dovr phdvxudeoh1 :161 Phdvxudelolw| rq Frpsohwh Phdvxuh Vsdfhv1 Lq wklv vxevhfwlrq zh zloo glvfxvv d frxsoh ri phdvxudelolw| uhvxowv frqfhuqlqj frpsohwlrqv ri phdvxuh vsdfhv1 Zh zloo uvw qhhg wr lqwurgxfh wkh qrwlrq ri vlpsoh ixqfwlrqv1 Ghqlwlrq :1641 D ixqfwlrq i = +[> P, $ F lv d vlpsoh ixqfwlrq li i lv phd0 vxudeoh dqg wkh udqjh ri i lv qlwh/ l1h1 &+i +[ ,, ? 41 D ixqfwlrq i = +[> P, $ F lv d fkdudfwhulvwlf ixqfwlrq li i lv phdvxudeoh dqg i +[ , @ i3> 4j= Li zh ohw D =@ i R +i4j,> zh zloo zulwh 4 li {5D i +{, @ 4D +{, @ 3 rwkhuzlvh1 Li i lv vlpsoh ixqfwlrq wkhq i pd| eh zulwwhq dv d qlwh olqhdu frpelqdwlrq ri fkdudfwhulvwlf ixqfwlrqv= [ i@ } 4i 4 Ci}jD } SF zkhuh wkh deryh vxp lv uhdoo| d qlwh vlqfh wkh udqjh ri i lv d qlwh vhw1 Wkhruhp :165 +Dssur{lpdwlrq Wkhruhp,1 Ohw i = [ $ ^3> 4` eh phdvxudeoh dqg ghqh S5q R [n !q +{, 4 4 n n.4 +{, . 5q 4i 4 CCSq >RD +{, 5q i +C 5q > 5q , ndQ S5q R [n 4n @ n.4 +{, . 5q 4iiASq j +{, 5q i 5q ?i # 5q j ndQ wkhq !q i iru doo q> !q +{, % i +{, iru doo { 5 [ dqg !q % i xqlirupo| rq wkh vhwv [P =@ i{ 5 [ = i +{, P j iru doo P 5 +3> 4, = Pruhryhu/ li i = [ $ F lv d phd0 vxudeoh ixqfwlrq/ wkhq wkhuh h{lvwv vlpsoh ixqfwlrqv !q vxfk wkdw olpq6R !q +{, @ i +{, iru doo { dqg m!q m % mi m dv q $ 4= Surri1 Lw lv fohdu e| frqvwuxfwlrq wkdw !q +{, i +{, iru doo { dqg wkdw 3 i +{, !q +{, 5q li { 5 [Sq = Iurp wklv lw iroorzv wkdw !q +{, % i +{, iru doo { 5 [ dqg !q % i xqlirupo| rq erxqghg vhwv1 Dovr qrwlfh wkdw 5n 5n . 5 5n 5n . 4 5n . 4 5n . 5 n n.4 + q > q ` @ + qFR > qFR ` @ + qFR > qFR ` ^ + qFR > qFR ` 5 5 5 5 5 5 5 5 Sn q.4 Sn R + q.4 > SnFR ` > !q +{, @ !qFR +{, @ q.4 dqg iru { 5 dqg iru { 5 i S S S Sn i R + SnFR > SnFS ` > !q +{, @ Sq.4 SnFR @ !qFR +{,= Vlploduo| vlqfh Sq.4 Sq.4 Sq.4 +5q > 4` @ +5q > 5qFR ` ^ +5qFR > 4`> iru { 5 i R ++5qFR > 4`, !q +{, @ 5q ? 5qFR @ !qFR +{, dqg iru { 5 i R ++5q > 5qFR `,> !qFR +{, 5q @ !q +{,= Wkhuhiruh !q !qFR iru doo q dqg zh kdyh frpsohwhg wkh surri ri wkh uvw dvvhuwlrq1 VY ipr xI qvr| Iru wkh vhfrqg dvvhuwlrq/ uvw dvvxph wkdw i = [ $ U lv d phdvxudeoh ixqfwlrq dqg fkrrvh ! wr eh vlpsoh ixqfwlrqv vxfk wkdw ! % i dv q $ 4 dqg ghqh q q !q @ !F ! = Wkhq q q m!q m @ !F . ! !FFR . !FR @ !qFR q q q q dqg fohduo| m!q m @ !F . ! % iF . i @ mi m dqg !q @ !F ! $ iF i @ i dv q q q q q $ 4= Qrz vxssrvh wkdw i = [ $ F lv phdvxudeoh1 Zh pd| qrz fkrrvh vlpsoh ixqfwlrq xq dqg yq vxfk wkdw mxq m % mUh i m > myq m % mLp i m > xq $ Uh i dqg yq $ Lp i dv q $ 4= Ohw !q @ xq . lyq > wkhq S m!q mS @ xS . yq % mUh i mS . mLp i mS @ mi mS q dqg !q @ xq . lyq $ Uh i . l Lp i @ i dv q $ 4= Sursrvlwlrq :1661 Vxssrvh wkdw +[> P> , lv d frpsohwh phdvxuh vsdfh dqg i = [ $ U lv phdvxudeoh1 +4, Li j = [ $ U lv d ixqfwlrq vxfk wkdw i +{, @ j +{, iru ~ d1h1 {> wkhq j lv phdvxudeoh1 +5, Li iq = [ $ U duh phdvxudeoh dqg i = [ $ U lv d ixqfwlrq vxfk wkdw olpq6R iq @ i> 0 d1h1/ wkhq i lv phdvxudeoh dv zhoo1 Surri1 41 Ohw H @ i{ = i +{, 9@ j+{,j zklfk lv dvvxphg wr eh lq P dqg +H , @ 31 Wkhq j @ 4H f i . 4H j vlqfh i @ j rq H f 1 Qrz 4H f i lv phdvxudeoh vr j zloo eh phdvxudeoh li zh vkrz 4H j lv phdvxudeoh1 Iru wklv frqvlghu/ f H ^ +4H j,R +D q i3j, li 3 5 D +:1:, +4H j ,R +D, @ li 3 95 D +4H j ,R +D, Vlqfh +4H j,R +E , H li 3 95 E dqg +H , @ 3/ lw iroorz e| frpsohwhqhvv ri P wkdw +4H j ,R +E , 5 P li 3 95 E Wkhuhiruh Ht1 +:1:, vkrzv wkdw 4H j lv phdvxudeoh1 51 Ohw H @ i{ = olp iq +{, 9@ i +{,j e| dvvxpswlrq H 5 P dqg +H , @ 31 Vlqfh q6R j 4H i @ olpq6R 4H f iq / j lv phdvxudeoh1 Ehfdxvh i @ j rq H f dqg +H , @ 3> i @ j d1h1 vr e| sduw 41 i lv dovr phdvxudeoh1 Wkh deryh uhvxowv duh lq jhqhudo idovh li +[> P> , lv qrw frpsohwh1 Iru h{dpsoh/ ohw [ @ i3> 4> 5j P @ ii3j> i4> 5j> [> !j dqg @ Q Wdnh j +3, @ 3> j+4, @ 4> j+5, @ 5> wkhq j @ 3 d1h1 |hw j lv qrw phdvxudeoh1 Ohppd :1671 Vxssrvh wkdw +[> P> , lv d phdvxuh vsdfh dqg P lv wkh frpsohwlrq = Wkhq d ixqfwlrq i = [ $ U ri P uhodwlyh wr dqg lv wkh h{whqvlrq ri wr P lv +P> E @ E U , ~ phdvxudeoh l wkhuh h{lvwv d ixqfwlrq j = [ $ U wkdw lv +P> E, ~ phdvxudeoh vxfk H @ i{ = i +{, 9@ j +{,j 5 P dqg +H , @ 3> l1h1 i +{, @ j+{, iru ~ d1h1 {= Surri1 Vxssrvh uvw wkdw vxfk d ixqfwlrq j h{lvwv vr wkdw +H , @ 3= Vlqfh j lv dovr +P> E, ~ phdvxudeoh/ zh vhh iurp Sursrvlwlrq :166 wkdw i lv +P> E , ~ phdvxudeoh1 Frqyhuvho| li i lv +P> E, ~ phdvxudeoh/ e| frqvlghulqj i zh pd| dvvxph wkdw > E, ~ phdvxudeoh vlpsoh ixqfwlrq !q 3 vxfk wkdw !q % i dv i 3= Fkrrvh +P q $ 4= Zulwlqj [ dn 4Dn !q @ hu SUQh yrpr ra rhr ur V` zlwk Dn 5 P> zh pd| fkrrvh En 5 P vxfk wkdw En Dn dqg +Dn q En , @ 3= Ohwwlqj [ !q =@ dn 4En zh kdyh surgxfhg d +P> E, ~ phdvxudeoh vlpsoh ixqfwlrq !q 3 vxfk wkdw Hq =@ j kdv }hur ~ phdvxuh1 Vlqfh +^q Hq , S +Hq , > wkhuh h{lvwv I 5 P i!q 9@ !q q vxfk wkdw ^q Hq I dqg +I , @ 3= Lw qrz iroorzv wkdw 4I !q @ 4I !q % j =@ 4I i dv q $ 4= Wklv vkrzv wkdw j @ 4I i lv +P> E, ~ phdvxudeoh dqg wkdw ii 9@ jj I kdv ~ phdvxuh }hur1 :171 Dsshqgl{= Vshfldo fdvhv ri wkh Surgxfw Wkhruhpv1 Wkhruhp :1681 Vxssrvh wkdw [R dqg [S duh vhwv/ R S +[R , dqg S S +[S , duh wrsrorjlhv/ PR S +[R , dqg PS S +[S , duh ~ dojheudv dqg HR S +[R , dqg HS S +[S , duh vxfk wkdw [R 5 HR / [S 5 HS > R @ +HR ,> S @ +HS ,> PR @ +HR ,> dqg PS @ +HS ,= Wkhq +4, PR PS @ +HR HS ,> l1h1 + +HR , +HS ,, @ +HR HS ,= +5, R S @ +HR HS ,> l1h1 ++ +HR , +HS ,, @ +HR HS ,= +6, Li zh ixuwkhu dvvxph wkdw HR dqg HS duh frxqwdeoh dqg ohw ER @ + R , dqg ES @ + S , wkhq +d, ER @ +HR , @ PR dqg ES @ +HS , @ PS 1 +e, + R S , @ +HR HS , @ PR PS @ ER ES = Wkdw lv wkh Eruho ~ dojheud rq [R [S zlwk wkh surgxfw wrsrorj| lv wkh surgxfw ri wkh Eruho ~ dojheudv rq [R dqg [S = Surri1 Vlqfh PR PS @ +PR PS , PR PS HR PR PS +HR HS ,= Iru wkh uhyhuvh lqfoxvlrq/ ohw P =@ iD 5 PR = D [S 5 +HR HS lw iroorzv wkdw HS ,j= Lw lv urxwlqh wr fkhfn wkdw P S +[R , lv ~ dojheud rq [R dqg wkdw HR P= Wkhuhiruh PR @ +HR , P> l1h1 zh kdyh vkrzq wkdw D [S 5 +HR HS , iru doo D 5 PR = E| v|pphwu|/ zh pd| vkrz wkdw [R E 5 +HR HS , iru doo E 5 PS dqg wkhuhiruh D E @ +D [S , _ +[R E , 5 +HR HS , ;D 5 PR dqg E 5 PS = Wkhuhiruh PR PS +HR HS , dqg zh kdyh suryhg lwhp 41 Lwhp 51 lv suryhg lq h{dfwo| wkh vdph zd|1 Lwhp 6d1 zdv douhdg| suryhg lq Sursrvlwlrq 61471 Vlploduo|/ e| Lwhp 51 R S @ +HR HS , e| lwhp 41 dqg ehfdxvh HR HS lv vwloo frxqwdeoh/ zh pd| dsso| Sursrvlwlrq 6147 wr vkrzv wkdw + R S , @ +HR HS ,= Wkh rwkhu dvvhuwlrqv lq Lwhp 6e1 qrz iroorz iurp wkh suhylrxv dvvhuwlrqv lq wkh Wkhruhp1 Fruroodu| :1691 EUp.q @ EUq EUp = Surri1 Ohw HR ghqrwh wkh froohfwlrq ri rshq edoov lq Uq zlwk fhqwhuv lq Tq dqg udwlrqdo ru lqqlwh udgll dqg ohw HS ghqrwh wkh froohfwlrq ri rshq edoov lq Up zlwk fhqwhuv lq Tp dqg udwlrqdo ru lqqlwh udgll1 Lw lv hdv| wr fkhfn wkdw +HR , dqg +HS , duh wkh vwdqgdug wrsrorjlhv rq Uq dqg Up uhvshfwlyho|1 Zh pd| qrz qlvk wkh surri xvlqj Wkhruhp :168 surylghg wkdw zh vkrz +HR HS , lv wkh vwdqgdug wrsrorj| rq UqFp = Wklv lv d frqvhtxhqfh ri wkh iroorzlqj dvvhuwlrqv= ipr xI qvr| WQ +4, Hyhu| vhw HR HS 5 HR HS pd| eh zulwwhq dv d xqlrq ri rshq edoov lq UqFp > l1h1 hyhu| vhw Hyhu| vhw HR HS 5 HR HS lv rshq lq UqFp zlwk wkh vwdqgdug wrsrorj|/ Uq.p = +Iurp wklv zh frqfoxgh wkdw +HR HS , Uq.p = +5, Hyhu| rshq edoo E ++{> | ,> , UqFp pd| eh zulwwhq dv d xqlrq ri hohphqwv lq HR HS = Wklv vkrzv wkdw E ++{> |,> , 5 +HR HS , dqg khqfh wkdw Uq.p +HR HS ,= Wkh surri ri wkhvh dvvhuwlrqv duh ohiw wr wkh uhdghu zkr pd| qg wkh iroorzlqj revhuydwlrqv xvhixo1 Li E +{> , 5 HR dqg E +|> , 5 HS dqg +d> e, 5 E +{> , E +|> , wkhq S S m{ dm . m| em ? S . S dqg khqfh +:1;, E +{> , E +|> , E ++{> |,> s S . S ,= Pruhryhu zh kdyh +:1<, E ++{> | ,> plq+> ,, E +{> , E +|> ,= Ohppd :16:1 Vxssrvh +[R > PR ,> +[S > PS , dqg +[> P, duh phdvxudeoh vsdfhv1 D ixqfwlrq I +{, @ +IR +{,> IS +{,, iurp [ $ [R [S lv +P> PR PS , phdvxudeoh l Il = [ $ [l lv +P> Pl , ~ phdvxudeoh iru l @ 4> 5= Surri1 Ohw l = [R [S $ [l eh wkh surmhfwlrq pdsv/ l1h1 R +{R > {S , @ {R dqg S +{R > {S , @ {S = Vlqfh R +D, @ D [S 5 PR PS iru doo D 5 PR > R R lv phdvxudeoh dqg d vlplodu dujxphqw vkrzv wkdw S lv phdvxudeoh1 Wkhuhiruh li I = [ $ [R [S lv +P> PR PS , phdvxudeoh wkhq Il @ l I lv +P> Pl , ~ phdvxudeoh iru l @ 4> 5= Frqyhuvho| li Il @ l I lv +P> Pl , ~ phdvxudeoh iru l @ 4> 5> wkhq I R +D I R +[R [S , @ IR R +D, 5 P dqg E , @ IS R +E , 5 P iru doo D 5 PR dqg E 5 PS = Vlqfh iD [S > [R E = D 5 PR dqg E 5 PS j jhqhudwh PR PS > lw iroorzv wkdw I lv +P> PR PS , ~ phdvxudeoh1 ;1 YA|j}iB|NA | jNi+ ;141 Lqwhjudo ri Vlpsoh ixqfwlrqv1 Ohw +[> P> , eh d {hg phdvxuh vsdfh lq wklv vhfwlrq1 Ghqlwlrq ;141 Ohw I @ F ru ^3> 4` dqg vxssrvh wkdw ! = [ $ I lv d vlpsoh ixqfwlrq1 Li I @ F dvvxph U ixuwkhu wkdw +!R +i| j,, ? 4 iru doo | 9@ 3 lq F= Iru U vxfk ixqfwlrqv ! zh ghqh ! @ ! g e| [ ] [ ! g @ |+!R +i|j,,= | SI [ Sursrvlwlrq ;151 Wkh lqwhjudo kdv wkh iroorzlqj surshuwlhv1 hu SUQh yrpr ra rhr ur WR +4, Vxssrvh wkdw 5 F wkhq ] ] +;14, i g @ i g= [ [ +5, Vxssrvh wkdw ! dqg # duh wzr vlpsoh ixqfwlrqv/ wkhq ] ] ] +! . #,g @ #g . !g= +6, Li ! dqg # duh qrq0qhjdwlyh vlpsoh ixqfwlrqv vxfk wkdw ! # wkhq ] ] !g #g= +7, Li ! lv d qrq0qhjdwlyh vlpsoh ixqfwlrq wkhq D $ +D, =@ lv d phdvxuh1 U D ! g U [ 4D ! g Surri1 Ohw xv zulwh i! @ |j iru wkh vhw !R +i|j, [ dqg +! @ | , iru +i! @ |j, @ +!R +i| j,, vr wkdw ] [ !@ |+! @ | ,= |SF Zh zloo dovr zulwh i! @ d> # @ ej iru !R +idj, _ #R +iej,= Wklv qrwdwlrq lv pruh lqwxlwlyh iru wkh sxusrvhv ri wklv surri1 Vxssrvh wkdw 5 I wkhq ] [ ! g @ | +! @ |, | SI [ [ @ | +! @ |@, | SI ] [ } +! @ } , @ ! g @ } SI [ surylghg wkdw 9@ 3= Wkh fdvh @ 3 lv fohdu/ vr zh kdyh suryhg 41 Vxssrvh ! wkdw dqg # duh wzr vlpsoh ixqfwlrqv/ wkhq ] [ +! . #, g @ } +! . # @ } , }SI [ @ } +^$SI i! @ $> # @ } $ j, }SI [[ @ } +! @ $ / # @ } $ , }SI $SI [ @ +} . $,+! @ $ / # @ } , }>$SI [ [ @ } +# @ } , . $ +! @ $ , }SI $ SI ] ] @ # g . ! g= zklfk suryhv 51 WS ipr xI qvr| Iru 61 li ! dqg # duh qrq0qhjdwlyh vlpsoh ixqfwlrqv vxfk wkdw ! # ] [ !@ d+! @ d, d$Q [ @ d+! @ d> # @ e, d>e$Q [ e+! @ d> # @ e, d>e$Q ] [ e+# @ e, @ #> @ e$Q zkhuh lq wkh wklug lqhtxdolw| zh kdyh xvhg i! @ d> # @ ej @ > li d A e= S Ilqdoo| iru 71/ zulwh ! @ l 4El zlwk l A 3 dqg El 5 P> wkhq ] Q [ +D, @ 4D ! g @ l +D _ El ,= ldR Wkh odwwhu h{suhvvlrq iru lv hdvlo| fkhfnhg wr eh d phdvxuh1 ;151 Lqwhjudov ri srvlwlyh ixqfwlrqv1 Ghqlwlrq ;161 Ohw OF @ ii = [ $ ^3> 4` = i lv phdvxudeohj1 Ghqh ] ] i g @ vxs ! g = ! lv vlpsoh dqg ! i = [ [ Ehfdxvh ri lwhp 61 ri Sursrvlwlrq ;15/ wklv ghqlwlrq lv frqvlvwhqw zlwk rxu suhylrxv ghqlwlrq ri wkh lqwhjudo rq qrq0qhjdwlyh vlpsoh ixqfwlrqv1 Zh vd| wkh i 5 OF lv lqwhjudeoh li ] ig ? 4= [ Uhpdun ;171 Qrwlfh wkdw zh vwloo kdyh wkh prqrwrqlflw| surshuw|= 3 i j wkhq ] ] i j [ [ dqg iru f A 3 ] ] fi @ f i= [ [ Dovr qrwlfh wkdw li i lv lqwhjudeoh/ wkhq +ii @ 4j, @ 3= S Ohppd ;181 Ohw [ eh d vhw dqg = [ $ ^3> 4` eh d ixqfwlrq/ ohw @ {S[ +{, { rq P @ S +[ ,> l1h1 [ +{,= +D, @ {SD Li i = [ $ ^3> 4` lv d ixqfwlrq +zklfk lv qhfhvvdulo| phdvxudeoh,/ wkhq ] [ i g @ i= [ [ hu SUQh yrpr ra rhr ur WT S Surri1 Vxssrvh wkdw ! = [ $ ^3> 4` lv d vlpsoh ixqfwlrq/ wkhq ! @ }SQ>R } 4!4 Ci}jD dqg [ [ [ ! @ +{, } 4!4 Ci}jD +{, [ {S [ }SQ>R [ [ } +{,4!4 Ci}jD +{, @ {S[ }SQ>R ] [ R+i}j,, @ !g= }+! @ [ }SQ>R Vr rq vlpsoh ixqfwlrq ! = [ $ ^3> 4`> ] [ ! @ !g= [ [ Vxssrvh wkdw ! = [ $ ^3> 4, lv d vlpsoh ixqfwlrq vxfk wkdw ! i> wkhq ] [ [ !g @ ! i= [ [ [ Wdnlqj wkh vxs ryhu ! lq wklv odvw htxdwlrq wkhq vkrzv wkdw ] [ i g i= [ [ Iru wkh uhyhuvh lqhtxdolw|/ ohw [ eh d qlwh vhw dqg Q 5 +3> 4,= Vhw i Q +{, @ plq iQ> i +{,j dqg ohw !Q> eh wkh vlpsoh ixqfwlrq jlyhq e| !Q> +{, =@ 4 +{,i Q +{,= Ehfdxvh !Q> +{, i +{,> ] ] [ [ i Q @ !Q> @ !Q> g ig= [ [ [ Vlqfh i Q % i dv Q $ 4> zh pd| ohw Q $ 4 lq wklv odvw htxdwlrq wr frqfoxghg wkdw dqg vlqfh [ ] i [ ig lv duelwudu| zh ohduq wkdw ] [ i i g= [ [ Wkhruhp ;19 +Prqrwrqh Frqyhujhqfh Wkhruhp,1 Vxssrvh iq 5 OF lv d vhtxhqfh ri ixqfwlrqv vxfk wkdw iq % i +qhfhvvdulo| lq OF , wkhq ] ] iq % i dv q $ 4= Surri1 Vlqfh iq ip i> iru doo q p ? 4> ] ] ] iq ip i U iurp zklfk li iroorzv iq lv lqfuhdvlqj lq q dqg ] ] iq i= olp q6R WU ipr xI qvr| Iru wkh rssrvlwh lqhtxdolw|/ ohw ! eh d vlpsoh ixqfwlrq vxfk wkdw 3 ! i dqg ohw 5 +3> 4,= Qrwlfh wkdw Hq iiq !j % [ dv q $ 4 dqg wkdw/ e| Sursrvlwlrq ;15/ ] ] ] ] +;15, iq 4Hq iq ! @ != Hq Hq U Ehfdxvh H $ H ! lv d phdvxuh dqg Hq % [> ] ] !@ !g= olp q6R Hq [ Khqfh zh pd| sdvv wr wkh olplw lq Ht1 +;15, wr jhw ] ] olp iq != q6R Ehfdxvh wklv htxdwlrq lv ydolg iru doo vlpsoh ixqfwlrqv 3 ! i> e| wkh ghqlwlrq U ri i zh kdyh ] ] iq i= olp q6R Vlqfh 5 +3> 4, lv duelwudu| zh frqfoxgh wkdw ] ] iq i= olp q6R Fruroodu| ;1:1 Li iq 5 OF lv d vhtxhqfh ri ixqfwlrqv wkhq ][ [] iq = iq @ q q Surri1 Iluvw r zh vkrz wkdw ] ] ] +iR . iS , @ iR . iS e| fkrrvlqj qrq0qhjdwlyh vlpsoh ixqfwlrq !q dqg #q vxfk wkdw !q % iR dqg #q % iS = Wkhq +!q . # q , lv vlpsoh dv zhoo dqg +!q . #q , % +iR . iS , vr wkdw e| wkh prqrwrqh frqyhujhqfh wkhruhp/ ] ] ] ] !q . # q +iR . iS , @ olp +!q . #q , @ olp q6R q6R ] ] ] ] !q . olp # q @ i R . iS = @ olp q6R q6R Q R S S iq dqg j @ iq > wkhq jQ % j dqg vr e| Qrz wr wkh jhqhudo fdvh1 Ohw jQ qdR R prqrwrqh frqyhujhqfh wkhruhp dqg wkh dgglwlylw| mxvw suryhg/ ][ R Q Q [] [] iq =@ olp iq @ olp iq Q 6R Q 6R qdR qdR qdR ] ] R [] jQ @ j @ iq = @ olp Q 6R qdR hu SUQh yrpr ra rhr ur WV Wkh iroorzlqj Ohppd lv d vlpsoh dssolfdwlrq ri wklv Fruroodu|1 Ohppd ;1; +Iluvw Eruhoo0Fduqwhol0 Ohppd1,1 Ohw +[> P> , eh d phdvxuh vsdfh/ Dq 5 P> dqg vhw iDq l1r1j @ i{ 5 [ = { 5 Dq iru lqqlwho| pdq| q*vj R _^ @ Dq = Q dR q$Q Li SR qdR +Dq , ? 4 wkhq +iDq l1r1j, @ 31 Surri1 +Iluvw Surri1, Ohw xv uvw revhuyh wkdw + , R [ iDq l1r1j @ { 5 [ = 4Dq +{, @ 4 = qdR SR Khqfh li qdR +Dq , ? 4 wkhq ][ R R R [ [] 4A +Dq , @ 4Dq g @ 4Dq g [ qdR qdR qdR [ R S 4Dq +{, ? 4 iru 0 d1h1 {= Wkdw lv wr vd| +iDq l1r1j, @ 31 qdR +Vhfrqg Surri1, Ri frxuvh zh pd| jlyh d vwulfwo| phdvxuh wkhruhwlf surri ri wklv idfw= 3 4 ^ Dq D +Dq l1r1, @ olp C Q 6R q$Q [ +Dq , olp Q 6R q$Q S dqg wkh odvw olplw lv }hur vlqfh R +Dq , ? 4= qdR lpsolhv wkdw H{dpsoh ;1<1 Vxssrvh wkdw i 5 F +^3> 4`, dqg i 31 Ohw n @ i3 @ dQ ? dR ? ? dqn @ 4j eh d vhtxhqfh ri uhqlqj sduwlwlrqv vxfk wkdw phvk+n , $ 3 dv n $ 4= Ohw [ plq ii +{, = dn { dnFR j 4Cdn >dn.4 +{, in +{, @ i +3,4iQj . n wkhq in % i dv n $ 4 vr wkdw e| wkh prqrwrqh frqyhujhqfh wkhruhp/ ]R ]R i gp @ olp i gp n6R Q n Q [ @ olp plq ii +{, = dn { dnFR j p ++dnFR dn `, n6R n ]R @ i +{,g{ Q zkhuh wkh odwwhu lqwhjudo lv wkh Ulhpdqq lqwhjudo1 ipr xI qvr| WW H{dpsoh ;1431 Ohw p eh Ohehvjxh phdvxuh rq U> wkhq ] ]R 4 4 gp+{, @ olp 4 4 +{, gp+{, q6R Q C q >R {s CQ>R {s R ]R 4 {sFR g{ @ olp @ olp q6R 4 {s q6R s . 4 R@q q R Rs li s ? 4 @ 4 li s A 4 Li s @ 4 zh qg ] ]R 4 4 g{ @ olp oq+{,mR@q @ 4= gp+{, @ olp R s q6R 4 { q6R CQ>R { q H{dpsoh ;1441 Ohw iuq jR eh dq hqxphudwlrq ri wkh srlqwv lq T _ ^3> 4` dqg qdR ghqh 4 s @ 8 li { @ uq m{ uq m dqg R [ 4 5q s i +{, @ = m{ uq m qdR Wkhq ]R Q dqg khqfh s m{ 4 uq m g{ 7 ]R Q i +{,g{ 7 ] Q>R i +{,gp+{, 7 ? 4 zklfk vkrzv wkdw p+i @ 4, @ 3> l1h1 wkdw i ? 4 iru doprvw hyhu| { 5 ^3> 4` dqg wklv lpsolhv wkdw R [ 4 5q s ? 4 iru d1h1 {= m{ uq m qdR Wkh iroorzlqj vlpsoh ohppd zloo riwhq eh xvhixo1 Ohppd ;145 +Fkhye|vkhy*v Llqhtxdolw|,1 Vxssrvh wkdw i 3 lv d phdvxudeoh ixqfwlrq/ wkhq iru dq| A 3> ] 4 i g= +;16, +ii j, [ Surri1 Vlqfh 4ii $j 4ii $j R i R i> ] ] ] 4 4 4ii $j g 4ii $j i g i g= +ii j, @ [ [ [ U Sursrvlwlrq ;1461 Vxssrvh wkdw i 3 lv d phdvxudeoh ixqfwlrq1 Wkhq [ i g @ 3 l U i @ 3 Ud1h1 Dovr li i> j 3 duh phdvxudeoh ixqfwlrqv vxfk wkdw i j d1h1 wkhq U U ig jg= Lq sduwlfxodu li i @ j d1h1 wkhq ig @ jg= hu SUQh yrpr ra rhr ur WX Surri1 Li i @ 3 d1h1 dqg ! i lv d vlpsohUixqfwlrq wkhq ! @ 3 d1h1 Wklv lpsolhv U wkdw +!R +i|j,,U@ 3 iru doo | A 3 dqg khqfh [ !g @ 3 dqg wkhuhiruh [ i g @ 3= R Frqyhuvho|/ li i g @ 3> ohw Hq @ ii q j1 Wkhq ] ] 4 4 3@ g @ +Hq , i q Hq Hq q zklfk vkrzv wkdw +Hq , @ 3 iru doo q= Vlqfh ii A 3j @ ^Hq > zh kdyh [ +Hq , @ 3> +ii A 3j, q l1h1 i @ 3 d1h1 Iru wkh vhfrqg dvvhuwlrq ohw H 5 P eh d vhw vxfk wkdw +H f , @ 3 dqg 4H i 4H j hyhu|zkhuh1 Ehfdxvh j @ 4H j . 4H f j dqg 4H f j @ 3 d1h1> ] ] ] ] jg @ 4H jg . 4H f jg @ 4H jg U U dqg vlploduo| i g @ 4H i g= Vlqfh 4H i 4H j hyhu|zkhuh/ ] ] ] ] i g @ 4H i g 4H jg @ jg= Fruroodu| ;1471 Vxssrvh wkdw iiq j lv d vhtxhqfh ri qrq0qhjdwlyh ixqfwlrqv dqg i lv d phdvxudeoh ixqfwlrq vxfk wkdw r d vhw ri phdvxuh }hur/ iq % i> wkhq ] ] iq % i dv q $ 4= Surri1 Ohw H [ vxfk wkdw +[ q H , @ 3 dqg iq 4H % i 4H 1 Wkhq e| wkh prqrwrqh frqyhujhqfh wkhruhp/ ] ] ] ] iq @ iq 4H % i 4H @ i dv q $ 4= Ohppd ;148 +Idwrx*v Ohppd,1 Li iq = [ $ ^3> 4` lv d vhtxhqfh ri phdvxudeoh ixqfwlrqv wkhq ] ] olpiq olp iq Surri1 Ghqh jn lqi iq vr wkdw jn % olp lqi q6R iq dv n $ 4= Vlqfh jn iq q$n iru doo n q zh kdyh ] ] jn iq iru doo q n dqg wkhuhiruh ] ] jn olp lqi q6R iq iru doo n= Zh pd| qrz xvh wkh prqrwrqh frqyhujhqfh wkhruhp wr ohw n $ 4 wr qg ] ] ] ] p jn olp lqi iq = olp j @ olp olp lqi iq @ q6R q6R n6R n n6R ipr xI qvr| WY ;15141 Lqwhjudov ri Frpsoh{ Ydoxhg Ixqfwlrqv1 Ghqlwlrq ;1491 D phdvxudeoh ixqfwlrq i = U $ ^ 4> 4` lv lqwhjudeoh li iF i 4ii $Qj dqg i @ i 4ii #Qj duh lqwhjudeoh1 Zh zulwh OR iru wkh vsdfh ri lqwhjudeoh ixqfwlrqv1 Iru i 5 OR > ohw ] ] ] ig @ iF g i g Uhpdun ;14:1 Qrwlfh wkdw li i lv lqwhjudeoh/ wkhq i mi m iF . i vr wkdw i lv lqwhjudeoh l ] mi m g ? 4= Sursrvlwlrq ;14;1 Wkh pds i 5 OR $ ] i g 5 U [ U R lv U olqhdu1 Dovr li i> j 5 O duh uhdo ydoxhg ixqfwlrqv vxfk wkdw i j> wkh i g jg= Surri1 Li i> j 5 OR dqg d> e 5 U> wkhq mdi . ejm mdmmi m . mem mj m 5 OR = Iru d 5 U> vd| d ? 3/ di dqg +di , @ +di ,F @ vr wkdw ] ] di @ d i . d ] ] iF @ d+ iF diF ] i , @ d ] i= D vlplodu fdofxodwlrq zrunv iru d A 3 dqg wkh fdvh d @ 3 lv wulyldo vr zh kdyh vkrzq wkdw ] ] di @ d i= Qrz vhw k @ i . j= Vlqfh k @ kF k @ iF kF ru Wkhuhiruh/ i . j F j kF . i . j @ k . iF . jF = ] ] kF . dqg khqfh ] ] k @ kF Ilqdoo| li iF k > i . ] ] ] k @ j @ ] ] iF . k . ] jF ] ] iF . i ] jF j @ ] ] i. j= i @ i j @ jF j wkhq iF . j jF . i zklfk lpsolhv wkdw ] ] ] ] i F . j j F . i ru htxlydohqwo| wkdw ] ] i@ ] iF i ] ] jF j @ ] j= hu SUQh yrpr ra rhr ur W` U Ghqlwlrq ;14<1 D phdvxudeoh ixqfwlrq i = [ $ F lv lqwhjudeoh li [ mi m g ? 4> U djdlq zh zulwh i 5 OR = Rqh vkrzv wkdw mi m g ? 4 l ] ] mUh i m g . mLp i m g ? 4= Iru i 5 OR ghqh ] ] ] i g @ Uh i g . l Lp i g= Lw lv urxwlqh wr vkrz wkdw wkh lqwhjudo lv vwloo olqhdu rq wkh frpsoh{ OR +suryh$,1 Sursrvlwlrq ;1531 Vxssrvh wkdw i 5 OR > wkhq ] ] i g mi mg= [ [ U Surri1 Vwduw e| zulwlqj [ i g @ Uhl 1 Wkhq ] ] ] ig @ U @ hl ig @ hl i g [ [ [ ] l i g= Uh h @ [ Ohw j =@ Uh hl i @ jF j wkhq frpelqlqj wkh suhylrxv htxdwlrq zlwk wkh iroorzlqj hvwlpdwh suryhv wkh wkhruhp1 ] ] ] ] ] j@ jF j jF . j [ [ ][ [ ][ @ jF . j @ mj mg [ [] ] m Uh+hl i ,mg mi mg= @ [ [ Sursrvlwlrq ;1541 Ohw i> j 5 OR > wkhq +4, Wkh vhw ii 9@ 3j lv 0qlwh/ l1h1 wkhuh h{lvwv Hq 5 P vxfk wkdw +Hq , ? 4 dqg Hq % ii 9@ 3j= +5, Wkh iroorzlqj duh htxlydohqw U U +d, UH i @ H j iru doo H 5 P +e, mi jm @ 3 [ +f, i @ j d1h= R Surri1 41 Wkh vhwv Hq =@ imi m q j vdwlvi| wkh frqglwlrqv lq lwhp 41 vlqfh fohduo| Hq % ii 9@ 3j dqg e| Fkhye|vkhy*v lqhtxdolw| +;16,/ ] 4 mi mg ? 4= +Hq , [ 51 +d, @, +f,Qrwlfh wkdw ] H ] i@ H ] j/ H +i j, @ 3 ipr xI qvr| XQ iru doo H 5 P1 Wdnlqj H @ iUh+i j , A 3j dqg xvlqj 4H Uh+i j, 3> zh ohduq wkdw ] ] 3 @ Uh +i j ,g @ 4H Uh+i j, @, 4H Uh+i j , @ 3 d1h1 H Wklv lpsolhv wkdw 4H @ 3 d1h1 zklfk kdsshqv l +iUh+i j, A 3j, @ +H , @ 3= Vlplodu +Uh+i j , ? 3, @ 3 vr wkdw Uh+i j, @ 3 d1h1 Vlploduo|/ Lp+i d1h dqg khqfh i j @ 3 d1h1/ l1h1 i @ j d1h1 +f, @, +e, lv fohdu dqg vr lv +e, @, +d, vlqfh ] ]] i j mi jm @ 3= H H j, @ 3 Fruroodu| ;1551 Vxssrvh wkdw +[> P> , eh d phdvxuh vsdfh dqg iDq jR P qdR lv d froohfwlrq ri vhwv vxfk wkdw +Dl _ Dm , @ 3 iru doo l 9@ m> wkhq +^R Dq , @ qdR Surri1 Vlqfh +^R Dq , @ qdR R [ qdR +Dq ,= ] 4R Dq g dqg q@4 [ ][ R R [ +Dq , @ 4Dq g [ qdR qdR lw vxfhv wr vkrz wkdw R [ 4Dq @ 4R Dq ~ d1h1 q@4 qdR S S Qrz R 4Dq 4R Dq dqg R 4Dq +{, 9@ 4R Dq +{, l { 5 Dl _ Dm iru vrph qdR qdR q@4 q@4 l 9@ m> wkdw lv , + R [ 4Dq +{, 9@ 4R Dq +{, @ ^l?m Dl _ Dm {= q@4 qdR dqg wkh odwhu vhw kdv phdvxuh 3 ehlqj wkh frxqwdeoh xqlrq ri vhwv ri phdvxuh }hur1 Wklv suryhv Ht1 +;17, dqg khqfh wkh fruroodu|1 Ghqlwlrq ;1561 Ohw +[> P> , eh d phdvxuh vsdfh dqg OR +, @ OR +[> P> , ghqrwh wkh vhw ri OR ixqfwlrqv prgxor wkh htxlydohqfh uhodwlrq i j l i @ j d1h1 Zh pdnh wklv lqwr d qruphg vsdfh xvlqj wkh qrup ] ni j nO4 @ mi jm g +;17, dqg lqwr d phwulf vsdfh xvlqj R +i> j, @ ni jnO4 = Uhpdun ;1571 Pruh jhqhudoo| zh pd| ghqh Os +, @ Os +[> P> , iru s 5 ^4> 4, dv wkh vhw ri phdvxudeoh ixqfwlrqv i vxfk wkdw ] mi ms g ? 4 [ prgxor wkh htxlydohqfh uhodwlrq i j l i @ j d1h1 hu SUQh yrpr ra rhr ur XR Zh pdnh Os +, @ Os +[> P> , lqwr d qruphg vsdfh xvlqj wkh qrup ] R@s s g mi m ni nOs @ = Zh zloo vhh qh{w txduwhu wkdw n nOs kdv wkh iroorzlqj surshuwlhv= @ mm ni nOs dqg ni nOs . njnOs ni nOs ni . j nOs +;18, iru doo i> j 5 Os = Lq sduwlfxodu s +i> j , @ ni j nOs lv d phwulf rq Os = Wkhruhp ;158 +Grplqdwhg Frqyhujhqfh Wkhruhp,1 Vxssrvh iq $ i d1h1 miq m j 5 OR 1 Wkhq i 5 OR dqg ] ] ig @ olp i g= k6R [ q [ Surri1 Qrwlfh wkdw mi m @ olp miq m j d1h1 vr wkdw i 5 OR = E| frqvlghulqj wkh uhdo dqg lpdjlqdu| sduwv ri i vhsdudwho|/ lw vxfhv wr suryh wkh wkhruhp lq wkh fdvh zkhuh i lv uhdo1 E| Idwrx*v Ohppd/ ] ] ] +j i ,g @ olp lqi +j iq , g olp lqi +j iq , g [ ] [ ][ jg . olp lqi iq g = @ [ [ Vlqfh olp lqi+ dq , @ olp vxs dq > zh kdyh vkrzq/ U ] ] ] olp lqi [ U q g i jg i g jg . olp vxs [ iq g [ [ [ dqg wkhuhiruh ] ] ] olp vxs iq g ig olp lqi iq g= [ [ [ U U Wklv vkrzv wkdw olp [ iq g h{lvwv dqg lv htxdo wr [ ig= q6R Fruroodu| ;159 +Glhuhqwldwlrq Xqghu wkh Lqwhjudo,1 Vxssrvh wkdw M U lv dq rshq lqwhuydo dqg i = M [ $ F lv d ixqfwlrq vxfk wkdw +4, i +w> , 5 OR iru doo w 5 M> +5, Ci +w> {, h{lvwv iru doo +w> {, Cw +6, Wkhuh lv d ixqfwlrq j 5 OR vxfk wkdw Wkhq ] ] C i +w> {,g+{, @ Cw Ci Cw +w> {, j+{, 5 OR = Ci +w> {,g+{,= Cw Surri1 Wkh surri lv wkh vdph dv wkdw fdvh iru vxpv wkdw |rx glg lq rqh ri |rxu krphzrun sureohpv1 H{huflvh ;15:1 Vkrz ]q olp +4 q6R Q {q , gp+{, @ 4= q ipr xI qvr| XS {q { q , 4Q>q +{, dqg qrwlfh wkdw olpq6R iq +{, @ h = Vroxwlrq1 Ohw iq +{, @ +4 Zh zloo qrz vkrz 3 iq +{, h{ iru doo { 3= Lw vxfhv wr frqvlghu { 5 ^3> q`= Ohw j+{, @ h{ iq +{,> wkhq iru { 5 +3> q,> 4 4 g oq j+{, @ 4 . q {,+ q, @ 4 g{ +4 q 4 +4 { 3 q, zklfk vkrzv wkdw oq j +{, dqg khqfh j +{, lv ghfuhdvlqj rq ^3> q`= Wkhuhiruh j +{, j+3, @ 4> l1h 3 iq +{, h{ = Qrz e| wkh Prqrwrqh frqyhujhqfh wkhruhp/ ]R ]P h{ gp+{, @ olp h{ gp+{, @ olp +4 P 6R Q P 6R Q hP , @ 4 ? 4> vr wkdw h{ lv dq lqwhjudeoh ixqfwlrq rq ^3> 4,= Wkhuhiruh zh pd| qrz dsso| wkh grplqdwhg frqyhujhqfh wkhruhp wr ohduq ]q ]R {q , gp+{, @ olp +4 i +{,gp+{, olp q6R Q q6R Q q q ]R ]R olp iq +{,gp+{, @ h{ gp+{, @ 4= @ Q q6R Q ;161 Frpsdulvrq wr wkh Ulhpdqq Lqwhjudo1 Lq wklv vhfwlrq/ vxssrvh d ? e ? 4 dqg i = ^d> e` $ U eh d erxqghg ixqfwlrq1 Wr hdfk sduwlwlrq +;19, S @ id @ wQ ? wR ? ri ^d> e` ohw VS i @ vS i @ [ [ ? wq @ ej Pm +wm wm R , pm +wm wm R , zkhuh Pm @ vxsii +{, = wm ? { wm R j pm @ lqi ii +{, = wm ? { wm R j dqg ghqh wkh xsshu dqg orzhu Ulhpdqq lqwhjudov e| ]e ] dd e uhvshfwlyho|1 i +{,g{ @ lqi VS i dqg S i +{,g{ @ vxs vS i S 4? hu SUQh yrpr ra rhr ur XT Idfw ;15;1 Uhfdoo wkh iroorzlqj idfw iurp wkh wkhru| ri Ulhpdqq lqwhjudov1 Wkhuh h{lvwv d uhqlqj vhtxhqfh ri sduwlwlrqv Sn +l1h1 wkh Sn *v duh lqfuhdvlqj, vxfk wkdw ]e VSn i ) i dv n $ 4 dqg d ]e i dv n $ 4= v Sn i % d Ue Ue i dqg Ghqlwlrq ;15<1 Wkh ixqfwlrq i lv Ulhpdqq lqwhjudeoh l d i @ d Ue zklfk fdvh wkh Ulhpdqq lqwhjudo d i lv ghqhg wr eh wkh frpprq ydoxh= ]e d i +{,g{ @ ]e d i +{,g{ @ ]e d i +{,g{= Iru d sduwlwlrq S dv lq Ht1 +;19, ohw q q [ [ Pm 4Cwm4 >wm dqg jS @ pm 4Cwm4 >wm = JS @ R R Li Sn lv d vhtxhqfh ri uhqlqj sduwlwlrqv dv lq Idfw ;15;/ wkhq JSn lv d ghfuhdvlqj vhtxhqfh/ jSn lv dq lqfuhdvlqj vhtxhqfh dqg jSn i JSn iru doo n= Ghqh +;1:, J olp JSn dqg j olp jSn = n6R n6R dqg qrwlfh wkdw j i J= E| wkh grplqdwhg frqyhujhqfh wkhruhp/ ] ] ]e jgp @ olp jSn @ olp vSn i @ i +{,g{ n6R d>e n6R d d>e dqg ] ]e ] Jgp @ olp JSn @ olp VSn i @ i +{,g{= n6R d>e n6R d d>e U U U Wkhuhiruh i lv Ulhpdqq lqwhjudeoh l d>e J @ d>e j l1h1 l d>e J j @ 3= Vlqfh J i j wklv kdsshqv l J @ j d1h1 Khqfh zh kdyh suryhg wkh iroorzlqj wkhruhp1 Wkhruhp ;1631 D erxqghg ixqfwlrq i = ^d> e` $ U lv Ulhpdqq lqwhjudeoh l wkh Eruho phdvxudeoh ixqfwlrqv j> J = ^d> e` $ U ghqhg lq Ht1 +;1:, vdwlvi| j+{, @ J+{, iru p ~ d1h1 {^d> e`= Pruhryhu li i lv Ulhpdqq lqwhjudeoh/ wkhq ] ] ]e i +{,g{ @ jgp @ Jgp= d d>e d>e Wkh ixqfwlrq i qhhg qrw eh Eruho phdvxudeoh exw lw lv qhfhvvdulo| Ohehvjxh phdvxu0 deoh/ l1h1 i lv O@E ~ phdvxudeoh zkhuh O lv wkh Ohehvjxh ~ dojheud dqg E lv wkh Eruho ~ dojheud rq ^d> e`1 Li zh ohw p ghqrwh wkh frpsohwlrq ri p> wkhq zh pd| dovr zulwh ]e ] i +{,g{ @ igp= d d>e ipr xI qvr| XU <1 Najt Nu NAji}jAWj Dv xvxdo ohw +[> P> , eh d {hg phdvxuh vsdfh dqg ohw iiq j eh d vhtxhqfh ri phdvxudeoh ixqfwlrqv rq [= Dovr ohw i = [ $ F eh d phdvxudeoh ixqfwlrq1 Ghqlwlrq <141 Zh kdyh wkh iroorzlqj qrwlrqv ri frqyhujhqfh1 +4, iq $ i d1h1 li wkhuh lv d vhw H 5 P vxfk wkdw +H f , @ 3 dqg olpq6R 4H iq @ 4H i= +5, iq $ i lq ~ phdvxuh li olpq6R +miq i m A , @ 3 iru doo A 3= Zh zloo deeuhyldwh wklv e| vd|lqj iq $ i lq OQ ru e| iq $ i s l i 5 Os dqg iq 5 Os iru doo q> dqg olpq6R U miq i ms g @ 3= +6, iq $ i lq O Ghqlwlrq <151 Zh kdyh wkh iroorzlqj qrwlrqv ri Fdxfk| vhtxhqfhv1 +4, iiq j lv d1h1 Fdxfk| li wkhuh lv d vhw H 5 P vxfk wkdw +H f , @ 3 dqgi4H iq j lv d srlqwzlvh Fdxfk| vhtxhqfhv +5, iiq j lv Fdxfk| lq ~ phdvxuh li olpp>q6R +miq ip m A , @ 3 iru doo A 3= U +6, iiq j lv Fdxfk| lq Os li olpp>q6R miq ip ms g @ 3= olp d @ d 5 U ru F h{lvwv dqg md q6R q R S dq m q dqg q ? 41 Wkhq qdR R S dq m q n 1 ndq Surri1 Ohw p A q wkhq pR S +dnFR mdp dq m @ +<14, ndq pR S dn , mdnFR ndqm Ohppd <161 Vxssrvh dq 5 U ru F dqg mdqFR dn m R S ndq n q = Vr mdp dq m plqCp>qD $ 3 dv > p> q $ 4> l1h1 idq j lv Fdxfk|1 Ohw p $ 4 lq +<14, wr qg md dq m q = Wkhruhp <171 Vxssrvh iiq j lv OQ 0Fdxfk|1 Wkhq wkhuh h{lvwv d vxevhtxhqfh jm @ iqm ri iiq j vxfk wkdw olp jm i h{lvwv d1h1 dqg iq $ i dv q $ 4= Pruhryhu li j lv d phdvxudeoh ixqfwlrq vxfk wkdw iq $ j dv q $ 4> wkhq i @ j d1h1 R S Surri1 Ohw q A 3 vxfk wkdw q ? 4 +q @ 5q zrxog gr, dqg vhw q @ qdR R S n = Fkrrvh jm @ iqm vxfk wkdw +qm %, dqg ndq +imjm FR jm m A m j, m = R V Hm dqg Ohw Hm @ imjm FR jm m A m j > IQ @ m dQ H Wkhq +H , @ 3 vlqfh +H , R _ Q dR R [ mdQ IQ @ +Hm , RR _^ Q dR mdQ R [ mdQ Hm @ iHm l1r1j= m @ Q $ 3 dv Q $ 4= hu SUQh yrpr ra rhr ur XV 8} ij 0 Prghv ri frqyhujhqfh h{dpsohv1 Iru { 95 IQ > mjm FR +{, jm +{,m m iru doo m Q dqg e| Ohppd <16/ i +{, @ olp j +{, h{lvwv dqg mi +{, jm +{,m m iru doo m Q 1 Wkhuhiruh/ olp jm +{, @ i +{, m6R m m dR h{lvwv iru doo { 95 H 1 Pruhryhu/ i{ = mi +{, im +{,m A m j Im iru doo m Q dqg khqfh +mi jm m A m , +Im , m $ 3 dv m $ 4= Wkhuhiruh jm $ i dv m $ 41 Vlqfh imiq i m A j @ imi imi j m . jm iq m A j jm m A @5j ^ imjm iq m A @5j> ipr xI qvr| XW +imiq dqg i m A j, +imi jm m A @5j, . +mjm iq m A @5, i m A j, olp vxs +mjm iq m A @5, $ 3 dv q $ 4= m 6R Ilqdoo| zh vkrz wkdw wkh olplwlqj ixqfwlrq lv xqltxh xs wr phdvxuh }hur1 Li dovr iq $ j dv q $ 4> wkhq dujxlqj dv deryh +imiq +mi j m A , +imi iq m A @5j, . +mj iq m A @5, $ 3 dv q $ 4= Wkdw lv wr vd| +mi +mi l1h1 i @ j d1h1 j m A , @ 3 iru doo A 3 dqg khqfh R R mi jm A 4 , [ +mi j m A 3, @ +^qdR q qdR jm A 4 , @ 3> q Wkhruhp <18 +Frpsohwhqhvv ri Os +,,1 Vxssrvh wkdw iiq j Os +, lv Fdxfk|/ Os wkhq wkhuh h{lvwv i 5 Os +, vxfk wkdw iq $ i 1 Pruhryhu i lv xqltxh prgxor wkh htxlydohqfh uhodwlrq ri ehlqj htxdo r vhwv ri phdvxuh }hur1 Surri1 Zulwh ] ni n @ E| Fkhe|vfkhy*v lqhtxdolw| +;16,/ +miq [ R@s mi ms g = ip m , @ +miq ip ms s , ] 4 4 miq ip ms g @ s niq s [ ip ns $ 3 dv p> q $ 4 iru doo A 3= Wklv vkrzv wkdw iiq j lv OQ 0Fdxfk| +l1h1 Fdxfk| lq phdvxuh, vr wkhuh h{lvwv ijm j iiq j vxfk wkdw jm $ i d1h1 Qrz e| Idwrx*v Ohppd/ ] ] olp lqi mjm jn ms olp lqi mjm jn ms g njm i ns @ n6R n6R @ olp lqi njm jn ns $ 3 dv m $ 4= n6R Os Lq sduwlfxodu/ ni n njm i n . njm n ? 4 vr wkh i 5 Os dqg jm $ i 1 Wkh surri lv qlvkhg ehfdxvh/ niq i n niq jm n . njm i n $ 3 dv m> q $ 4= Wkhruhp <19 +Hjrur*v Wkhruhp,1 Vxssrvh +[ , ? 4 dqg iq $ i d1h1 Wkhq iru doo A 3 wkhuh h{lvwv H [ vxfk wkdw +H , ? dqg iq $ i xqlirupo| rq H f = Lq sduwlfxodu iq $ i dv q $ 4= R Surri1 Ohw iq $ i d1h1 Wkhq +imiq i m A n l1r1 qj, @ 3 iru doo n A 3> l1h1 4 3 R _^ 4D imiq i m A j @ 3= C n Q dR q$Q Vr 3 ^ olp C imiq Q 6R q$Q 4 4D i m A j @ 3= n hu SUQh yrpr ra rhr ur XX Fkrrvh dqg lqfuhdvlqj vhtxhqfh iQn jR vxfk wkdw +Hn , ? 5n zkhuh ndR ^ 4 Hn =@ imiq i m A j= n q$Qn Wkxv H =@ ^Hn vdwlvhv +H , ? dqg li { 95 H> miq n>1 l1h1 iq $ i xqlirupo| rq H f 1 R i m n iru doo q Qn dqg doo 431 $jAt|+ C jNij6t Ohw +[> P> , eh d phdvxuh vsdfh dqg ohw Vi ghqrwh wkh froohfwlrq ri vlpsoh ixqfwlrqv ! zlwk dgglwlrqdo surshuw| wkdw +! @ } , ? 4 iru doo } 9@ 3= Qrwlfh wkdw S iru ! 5 Vi dqg s 5 ^4> 4,> wkdw m!ms @ }WdQ m} ms 4i!d}j dqg khqfh ] [ m!ms g @ m} ms +! @ } , ? 4= }WdQ Wkdw lv Vi Os +,= Ohppd 4314 +Vlpsoh Ixqfwlrqv duh Ghqvh,1 Ohw vlpsoh ixqfwlrqv Vi irup d ghqvh vxevsdfh ri Os +, iru doo 4 s ? 4= Surri1 Vlqfh i @ x . ly zlwk x> y 5 Os +, dqg iru i uhdo ydoxhg i @ iF i zlwk i 5 Os zh pd| dvvxph zlwkrxw orvv ri jhqhudolw| wkdw i 5 Os _ OF = Fkrrvh vlpsoh ixqfwlrqv !q % i dv lq Wkhruhp :1651 Vlqfh mi !q ms +mi m . m!q m,s 5s mi ms 5 OR zh pd| dsso| wkh grplqdwhg frqyhujhqfh wkhruhp wr qg ] ] mi !q ms g @ olp olp mi !q ms g @ 3= q6R q6R Ghqlwlrq 43151 Ohw +[> g, eh d phwulf vsdfh1 Zh vd| wkdw d vhw D [ lv erxqghg surylghg wkdw D E +{> U, iru vrph { 5 [ dqg U 5 +3> 4,= +E| wkh wuldqjoh lqhtxdolw| D lv erxqghg l iru doo { 5 [> wkhuh h{lvwv U ? 4 vxfk wkdw D E +{> U,=, Dovr ohw EFe +[ , ghqrwh wkh vhw ri erxqghg frqwlqxrxv ixqfwlrqv i = [ $ F vxfk wkdw i lv lghqwlfdoo| }hur r d erxqghg vxevhw ri [= Wkhruhp 4316 +Frqwlqxrxv Ixqfwlrqv duh Ghqvh,1 Ohw +[> g, eh d phwulf vsdfh/ g eh wkh wrsrorj| rq [ jhqhudwhg e| g dqg E[ @ + g , eh wkh Eruho ~ dojheud1 +4, Vxssrvh = E[ $ ^3> 4` lv d phdvxuh vxfk wkdw lv qlwh rq erxqghg vhwv/ wkhq EFe +[ , Os +, lv d ghqvh vxevsdfh1 r +5, Vxssrvh wkdw wkhuh lv d vhtxhqfh ri frpsdfw vhw Nq [ vxfk wkdw Nq r iru doo q> [ @ ^R Nq dqg +Nq , ? 4 iru doo q/ zkhuh iru Nq NqFR qdR H [> H r @ ^Y S>Y )H Y lv wkh lqwhulru ri H= Wkhq Ff +[ , +wkh froohfwlrq ri frqwlqxrxv ixqfwlrqv zlwk frpsdfw vxssruw, lv ghqvh lq Os +,= Surri1 Sduw 41 Vlqfh Vi lv ghqvh lq Os +, lw vxfhv wr vkrzv wkdw dq| ! 5 Vi pd| eh zhoo dssur{lpdwhg e| i 5 EFe +[ ,= Pruhryhu/ wr suryh wklv lw vxfhv wr vkrz wkdw iru D 5 P zlwk +D, ? 4 wkdw 4D pd| eh zhoo dssur{lpdwhg e| dq i 5 EFe +[ ,= Ohw {Q 5 [ dqg q 5 Q dqg vhw Dq =@ D _ E +{Q > q, vr wkdw Dq % D dv q $ 4= Wkhuhiruh ] m4D 4Dq ms g @ +D q Dq , $ 3 dv q $ 4= XY ipr xI qvr| Wklv vkrzv wkdw zh pd| ixuwkhu dvvxph wkdw D lv erxqghg/ l1h1 D E +{Q > Q , iru vrph odujh Q= E| H{huflvhv 7154 dqg 7155/ iru dq| A 3 wkhuh h{lvwv I D Y zkhuh I lv forvhg/ Y lv rshq dqg +Y q I , ? = Vlqfh I D> I lv dxwrpdwlfdoo| erxqghg1 E| uhsodflqj Y e| wkh erxqghg rshq vhw Y _ E +{Q > Q , li qhfhvvdu|/ zh pd| dovr dvvxph wkdw Y lv erxqghg1 Ghqh gY f +{, +4314, i +{, @ gI +{, . gY f +{, zkhuh gI +{, @ lqi ig+{> | , = | 5 I j= E| Ohppd 516:/ gI dqg gY f duh frqwlqxrxv @ ixqfwlrqv rq [= Vlqfh I dqg Y f duh forvhg/ gI +{, A 3 li { 5 I dqg gY f +{, A 3 f @ >> gI +{, . gY f +{, A 3 iru doo { dqg +gI . gY f ,R lv li { 5 Y= Vlqfh I _ Y frqwlqxrxv dv zhoo1 Wkhuhiruh/ i = [ $ ^3> 4` lv frqwlqxrxv dqg i +{, @ 4 iru { 5 I dqg i +{, @ 3 li { 5 Y dqg vlqfh Y lv erxqghg zh kdyh vkrzq wkdw i 5 EFe +[ ,= @ Vlqfh m4D i m 4Y qI > ] ] +4315, m4D i ms g 4Y qI g @ +Y q I , ru htxlydohqwo| n4D i n R@s = Vlqfh A 3 lv duelwudu|/ zh kdyh vkrzq wkdw 4D fdq eh dssur{lpdwhg lq Os +, duelwudulo| zhoo e| d frqwlqxrxv ixqfwlrq i 5 EFe +[ ,= Sduw 51 Wkh surri ri wkh vhfrqg dvvhuwlrq lv vlplodu1 Djdlq lw vxfhv wr vkrz wkdw iru D 5 P zlwk +D, ? 4 wkdw 4D pd| eh zhoo dssur{lpdwhg e| dq i 5 Ff +[ ,= r Qrz ohw Dq =@ D _ Nq vr wkdw Dq % D dv q $ 4= Wkhuhiruh ] m4D 4Dq ms g @ +D q Dq , $ 3 dv q $ 4= r Wklv vkrzv wkdw zh pd| ixuwkhu dvvxph wkdw D NQ iru vrph Q= E| H{huflvhv 7154 dqg 7155/ iru dq| A 3 wkhuh h{lvwv I D Y zkhuh I lv forvhg/ Y lv rshq dqg +Y q I , ? = Vlqfh I lv d forvhg vxevhw ri NQ d frpsdfw vhw/ r I lv dovr frpsdfw1 E| uhsodflqj Y e| wkh rshq vhw Y _ NQ li qhfhvvdu|/ zh pd| dovr dvvxph wkdw Y NQ dv zhoo1 Djdlq ghqh i wr eh wkh frqwlqxrxv ixqfwlrq lq Ht1 +4314, dqg dv ehiruh Ht1 +4315, vwloo krogv1 Pruhryhu i kdv vxssruw lq wkh frpsdfw vhw NQ zklfk vkrzv wkdw i 5 Ff +[ ,= Fruroodu| 43171 Vxssrvh Y Uq lv dq rshq vhw/ EY lv wkh Eruho ~ dojheud rq Y dqg lv d phdvxuh rq +Y> EY , zklfk lv qlwh rq frpsdfw vhwv1 Wkhq Ff +Y , lv ghqvh lq Os +, iru doo s 5 ^4> 4,= Surri1 Lw vxfhv wr vkrzv wkdw wkhuh h{lvwv frpsdfw vxevhw Nq Y vxfk r r Nq Nq NqFR % Y dv q $ 4= Wklv lv hvvhqwldoo| grqh lq wkh surri ri Wkhruhp 4515 ehorz/ dovr vhh Iljxuh ; ehorz1 R Odwhu rq zh zloo vkrz wkdw wkh vsdfh Ff +U, ri lqqlwho| glhuhqwldeoh ixqfwlrqv zlwk frpsdfw vxssruw lv dovr ghqvh lq Os +p, iru doo s 5 ^4> 4,= D nh| srlqw lq surylqj wklv idfw lv wkh iroorzlqj ohppd1 R Ohppd 43181 Vxssrvh i 5 OR +gp, dqg ! 5 Ff +U, wkhq ] k+{, @ i !+{, @ i +| ,!+{ |,g| hu SUQh yrpr ra rhr ur X` lv d vprrwk ixqfwlrq/ l1h1 k+{, lv lqqlwho| glhuhqwldeoh1 Surri1 Ohw I +{> |, =@ i +| ,!+{ |,> wkhq li F ? 4 lv d erxqg rq wkh ghulydwlyh ri !> CI Q R C{ +{> | , @ i +| ,! +{ | , F mi +|,m 5 O +g| ,= Wkhuhiruh e| Fruroodu| ;159/ ] g +i !,+{, @ I +|,!Q +{ g{ |,g| @ +i Zrunlqj lqgxfwlyho| rqh pd| qrz vkrzv wkdw +i !Q ,+{,= !,CqD +{, @ i !CqD +{,= Wkhruhp 43191 Vxssrvh wkdw P lv d ~ dojheud rq [ dqg D P lv dq dojheud vxfk wkdw +4, +D, @ P +5, D lv frxqwdeoh +6, lv 0qlwh rq D1 Wkhq Os +[> P> , lv vhsdudeoh iru doo 4 s ? 41 Pruhryhu [ G@i dm {Dm = dm 5 T . lT> Dm 5 D dqg +Dm , ? 4j lv d frxqwdeoh ghqvh vxevsdfh1 Surri1 Jlyhq A 3> e| Fruroodu| 7148/ iru doo H 5 P vxfk wkdw +H , ? 4> wkhuh h{lvwv D 5 D vxfk wkdw+H 7D, ? dqg wkhuhiruh ] +4316, m4H 4D ms g @ +H 7D, ? = Wklv htxdwlrq vkrzv wkdw dq| vlpsoh ixqfwlrq lq Vi pd| eh dssur{lpdwhg duelwudu| zhoo e| dq hohphqw iurp G dqg khqfh G lv dovr ghqvh lq Os +,= Fruroodu| 431: +Ulhpdqq Ohehvjxh Ohppd,1 Vxssrvh wkdw i 5 OR +U> p,> wkhq ] olp i +{,hl{ gp+{, @ 3= 6R U Surri1 Ohw D ghqrwh wkh dojheud rq U jhqhudwhg e| wkh kdoi rshq lqwhuydov/ l1h1 D frqvlvwv ri vhwv ri wkh irup q d +dn > en ` _ U ndR S zkhuh dn > en 5 U= E| Wkhruhp 4319/ jlyhq A 3 wkhuh h{lvwv ! @ qdR fn 4Cdn >en n zlwk dn > en 5 U vxfk wkdw ] mi !mgp ? = U Qrwlfh wkdw ] ][ q !+{,hl{ gp+{, @ fn 4Cdn >en +{,hl{ gp+{, U U ndR q q [ ] en [ @ fn hl{ gp+{, @ fn R hl{ men dn dn ndR ndR q [ @ R fn hlen hldn $ 3 dv mm $ 4= ndR ipr xI qvr| YQ Frpelqlqj wkhvh wzr htxdwlrqv zlwk ] ] ] i +{,hl{ gp+{, +i +{, !+{,, hl{ gp+{, . !+{,hl{ gp+{, U U U ] ] mi !mgp . !+{,hl{ gp+{, U U ] . !+{,hl{ gp+{, U zh ohduq wkdw ] ] i +{,hl{ gp+{, . olp vxs !+{,hl{ gp+{, @ = olp vxs mm6R U mm6R U Vlqfh A 3 lv duelwudu|/ zh kdyh suryhq wkh ohppd1 441 8 OA?t C jNij6 Wklv qh{w h{dpsoh jlyhv d _uhdo zruog% h{dpsoh ri wkh idfw wkdw lw lv qrw dozd|v srvvleoh wr lqwhufkdqjh rughu ri lqwhjudwlrq1 H{dpsoh 44141 Frqvlghu ]R ]R g| g{+h{| Q R R hS{| h| 5 | 5| {dR Q ] R | h hS| @ g| | Q ]R | 4 h| 5 +3> 4,= g| h @ | Q 5hS{| , @ ]R g| h| kdv qrw vlqjxodulw| dw 3= Rq wkh rwkhu kdqg Qrwh zhoo wkdw R| {| R ]R ]R ]R hS{| h {| 5hS{| , @ 5 g{ g|+h g{ { 5{ |dQ R Q R S{ ]R h h{ @ g{ { R] R 4 h{ h{ @ g{ 5 + 4> 3,= { R U U U U Prudo g{ g| i +{> |, 9@ g| g{ i +{> | , lv qrw dozd|v wuxh1 44141 Surgxfw Phdvxuh1 Ohw +[> P> , dqg +\> Q > , eh phdvxuh vsdfhv1 Qrwdwlrq 44151 Vxssrvh wkdw i = [ $ F dqg j = \ $ F duh ixqfwlrqv/ ohw i j ghqrwh wkh ixqfwlrq rq [ \ jlyhq e| i j+{> | , @ i +{,j +| ,= Qrwlfh wkdw li i> j duh phdvxudeoh/ wkhq i j lv +P Q > EF , ~ phdvxudeoh1 Wr suryh wklv/ uvw vxssrvh wkdw i @ 4D dqg j @ 4E zlwk D 5 P dqg E 5 Q = Wkhq i j @ 4D 4E @ 4DE zklfk lv phdvxudeoh vlqfh D E 5 P Q = Qrz li i dqg j duh vlpsoh ixqfwlrqv wkhq [ z} 4ii dzjijd}j i j @ z>}SF hu SUQh yrpr ra rhr ur YR zklfk lv djdlq phdvxudeoh1 Iru jhqhudo i dqg j fkrrvh vlpsoh ixqfwlrq !q dqg #q frqyhujlqj wr i dqg j uhvshfwlyho|/ wkhq vlqfh i j @ olpq6R !q #q zh ohduq wkdw i j lv phdvxudeoh dv zhoo1 Ohw H S +[ \ , eh jlyhq e| H @P Q @ iD E = D 5 P> E 5 Q j dqg uhfdoo iurp H{huflvh 6144 wkdw H lv dq hohphqwdu| idplo|1 Khqfh wkh dojheud D @ D+H , jhqhudwhg e| H frqvlvwv ri vhwv zklfk pd| eh zulwwhq dv glvmrlqw xqlrqv ri vhwv iurp H = Ghqlwlrq 44161 Ghqh Q = H $ ^3> 4` e| Q +D E , @ +D, +E , iru D 5 P dqg E 5 Q = Qrwlfh wkdw ] ] 4DE +{> | ,g +| , @ 4D +{,4E +|,g +| , \ \ @ 4D +{, +E , dqg khqfh +4414, ] ] [ dqg vlploduo|/ +4415, \ 4DE +{> |,g +|, g+{, @ +D, +E , @ Q +D ] ] \ [ 4DE +{> | ,g+{, g +| , @ Q +D E, E ,= Wkhruhp 44171 Vxssrvh +[> P> , dqg +\> Q > , duh 0qlwh phdvxuh vsdfh/ wkhq wkhuh h{lvwv d xqltxh phdvxuh rq P Q vxfk wkdw +D E , @ +D, +E , iru doo D 5 P dqg E 5 Q = Zh zloo ghqrwh wklv _surgxfw% phdvxuh e| = Surri1 41 Q lv ~ qlwh rq H = Lqghhg/ li [q 5 P dqg \q 5 Q duh vxfk wkdw [q % [ dqg \q % \ dqg +[q , ? 4 dqg +\q , ? 4 iru doo q> wkhq [q \q 5 H > [q \q % [ \ dqg Q +[q \q , @ +[q , +\q , ? 4 iru doo q= 51 Wkh ixqfwlrq Q lv ~ dgglwlyh rq H @ P Q = Wr suryh wklv vxssrvh wkdw cR H 5 P Q dqg Hn @ Dn En 5 H duh vxfk wkdw H @ ndR +Dn En ,> wkhq 4H @ R [ ndR vr wkdw | $ 4H +{> |, @ 4Dn 4En R [ ndR 4Dn +{,4En +| , lv phdvxudeoh iru doo { 5 [ dqg ] ][ ] R R [ 4H +{> |,g +| , @ 4Dn +{,4En +| ,g +| , @ 4Dn +{, 4En g \ \ ndR \ ndR R [ @ 4Dn +{, +En ,= ndR ipr xI qvr| YS Wkh odwwhu h{suhvvlrq lv phdvxudeoh dqg ] ] ][ R 4H +{> |,g +| , g+{, @ 4Dn +{, +En ,g+{, [\ [ ndR R R [ [ @ +4416, +Dn , +En , @ Q +Dn ndR ndR Vlploduo| rqh vkrzv wkdw ] ] R [ +4417, 4H +{> |,g+{, g +| , @ Q +Dn \ [ ndR Wdnlqj H @ D En ,= En ,= E 5 H > Htv1 +4417, dqg +4415, vkrzv wkdw Q +D E, @ R [ ndR Q +Dn En , wkdw lv wr vd| Q lv ~ dgglwlyh rq H = E| lwhpv 41 dqg 51 mxvw suryhg dqg Wkhruhp 81:/ wkhuh h{lvwv d xqltxh phdvxuh rq P Q @ +D, @ +H , vxfk +D E , @ +D, +E , iru doo D 5 P dqg E 5 Q = Wkh surri jlyhv d elw pruh wkdq vwdwhg1 Qdpho| ] ] +H , @ 4H +{> | ,g +|, g+{, [\ ] ] @ +4418, 4H +{> | ,g+{, g +| ,= \ [ c iru doo H 5 D = Wklv iroorzv e| zulwlqj H @ R +Dn En , zlwk Dn ndR xvlqj wkh lghqwlw|/ +H , @ R [ ndR +Dn En , @ R [ ndR Q +Dn En 5 H dqg En , dorqj zlwk Htv1 +4416, dqg +4417,1 Wkhruhp 4418 +Wrqhool*v Wkhruhp,1 Vxssrvh +[> P> , dqg +\> Q > , duh 0qlwh phdvxuh vsdfhv dqg @ eh wkh surgxfw phdvxuh rq P Q = Li i 5 OF +[ \ > PQ ,> wkhq i + > | , 5 OF +[> P, iru doo | 5 \> i +{> , 5 OF +\> Q , iru doo { 5 [> ] { $ j +{, @ i +{> |,g +| , 5 OF +[> P,> \ ] i +{> | ,g+{, 5 OF +\> Q , | $ k+| , @ [ dqg +4419, +441:, ] [ \ ] i g @ ] g+{, g +|,i +{> |, ]\ g +| , g+{,i +{> |, @ \ [ ][ hu SUQh yrpr ra rhr ur YT Surri1 E| olqhdulw| dqg wkh prqrwrqh frqyhujhqfh olplwlqj dujxphqwv lw vxfhv wr suryh wkh dvvhuwlrq lq wkh wkhruhp zkhq i @ 4H zlwk H 5 PQ = Iluvw dvvxph +[ , ? 4 dqg +\ , ? 4= Ohw F eh wkh froohfwlrq ri doo H 5 P Q vxfk wkdw wkhruhp krogv zkhq i @ 4H = Wkh froohfwlrq F vdwlvhv= +4, E| wkh surri ri Wkhruhp 4417 dqg Ht1 +4418, D+H , F = +5, F lv d prqrwrqh fodvv1 +Zh zloo fkhfn wklv vkruwo|1, E| wkh prqrwrqh fodvv ohppd zh pd| frqfoxgh wkdw F @ +D+H ,, @ P Q zklfk suryhv wkh wkhruhp lq wkh qlwh phdvxuh fdvh1 Wr fkhfn lwhp 51/ vxssrvh wkdw Hq 5 F dqg Hq % H= Wkhq wkhq 4Hq % 4H dv q $ 4 vr wkdw 4Hq +{> , % 4H +{> , vr wkdw 4H +{> , 5 OF +\> Q , iru doo { 5 [ dqg e| wkh prqrwrqh frqyhujhqfh wkhruhp ] ] 4Hq +{> | ,g +| , % 4H +{> |,g +| , \ \ U vr wkdw { $ 4H +{> | ,g +| , 5 OF +[> P,= Djdlq e| wkh prqrwrqh frqyhujhqfh \ wkhruhp zh pd| wdnh olplwv ri wkh lghqwlw| ] ] ] 4Hq g @ g+{, g +| ,4Hq +{> | , wr suryh Ht1 +4419,1 Rqh vkrzv lq wkh vdph zd| wkdw wkh dqdorjrxv vwdwhphqwv krogv iru wkh rssrvlwh rughu ri lqwhjudwlrq/ l1h1 H 5 F = Li Hq 5 F dqg Hq & H> zh djdlq vkrzv wkdw H 5 F e| wkh vdph whfkqltxhv xvlqj wkh grplqdwhg frqyhujhqfh wkhruhp lqvwhdg ri wkh prqrwrqh frqyhujhqfh wkhruhp1 Wkhuhiruh F lv d prqrwrqh fodvv dqg khqfh e| wkh prqrwrqh fodvv wkhruhp F @ P Q dqg Wrqhool*v wkhruhp lv suryhg zkhq dqg duh qlwh phdvxuhv1 Iru wkh ~ qlwh fdvh/ fkrrvh [q 5 P> \q 5 Q vxfk wkdw [q % [> \q % \> +[q , ? 4 dqg +\q , ? 4 iru doo q= Wkhq ghqh q +D, @ +[q _ D, dqg q +E , @ +\q _ E , iru doo D 5 P dqg E 5 Q dqg ohw q @ q q = Vlqfh q +D E , @ q +D, q +E , @ +[q _ D, +\q _ E , @ ++[q \q , _ +D E ,, iru doo D> E 5 P Q = Vlqfh ~ qlwh phdvxuh rq P Q duh ghwhuplqhg e| wkhlu ydoxhv rq P Q > zh vhh wkdw q +H , @ ++[q Ohw xv dovr revhuyh wkdw ] \q , _ H , iru doo H 5 P Q = ] jg q @ jg \ \q iru doo phdvxudeoh ixqfwlrqv j 5 OF +\> Q ,= Wklv lv suryhg uvw iru vlpsoh ixqfwlrqv dqg wkhq iru jhqhudo ixqfwlrqv e| sdvvlqj wr wkh olplw1 Zlwk wklv qrwdwlrq/ zh pd| dsso| wkh qlwh yhuvlrq ri wkh wkhruhp wr frqfoxgh ] ] { $ jq +{, @ i +{> |,g q +| , @ 4\q +|,i +{> | ,g +|, 5 OF +[> P,> \ \ ] ] | $ kq +| , @ i +{> | ,gq +{, @ 4[q +{,i +{> | ,g+{, 5 OF +\> Q , [ [ ipr xI qvr| YU dqg ] 4[q \q i g @ ] i gq ][ \ ] ] ] @ gq +{, g q +|,i +{> |, @ g+{, g +| , +4[q \q i , +{> |, ]\ ][ ]\ ][ g q +| , gq +{,i +{> |, @ g +|, g+{, +4[q \q i , +{> |,= @ \ [ \ [ Sdvvlqj wr wkh olplw lq wkhvh lghqwlwlhv zlwk wkh dlg ri wkh prqrwrqh frqyhujhqfh wkhruhp frqfoxghv wkh surri ri wkh wkhruhp1 [ \ Wkhruhp 4419 +Ixelql*v Wkhruhp,1 Vxssrvh +[> P> , dqg +\> Q > , duh 0qlwh phdvxuh vsdfhv dqg @ eh wkh surgxfw phdvxuh rq P Q = Li i 5 OR +, wkhq iru d1h1 {> i +{> , 5 OR + , dqg iru d1h1 |> i + > | , 5 OR +,= Wkh ixqfwlrqv ] ] i +{> | ,gy+|, dqg k+|, @ i +{> |,g+{, j +{, @ \ [ duh lq OR +, dqg OR + , uhvshfwlyho| dqg Ht1 +441:, krogv1 Surri1 Li i 5 OR +[ \ , _ OF wkhq e| Ht1 +4419,/ ] ] i +{> | ,g +| , g+{, ? 4 [\ U vr \ i +{> | ,g +| , ? 4 iru d1h1 {/ l1h1 iru d1h1 {> i +{> , 5 OR + ,1 Vlploduo| iru d1h1 | i + > |, 5 OR +,= Ohw i eh d uhdo ydoxhg ixqfwlrq lq i 5 OR +[ \ , dqg ohw i @ iF i = Dsso| wkh uhvxowv mxvw suryhg wr i wr frqfoxgh/ i +{> , 5 OR + , iru d1h1 { dqg wkdw ] i + > | ,g +|, 5 OR +,= \ Wkhuhiruh iru d1h1 ={/ i +{> , @ iF +{> , dqg ] {$ i +{> , 5 OR + , ] ] i +{> | ,g +| , @ iF +{> ,g +| , i +{> ,g +|, lv d ~ doprvw hyhu|zkhuh ghqhg ixqfwlrq vxfk wkdw ] i + > | ,g +|, 5 OR +,= Ehfdxvh ] i +{> | ,g + ] ] i g+ ]] , @ + i +{> |,g +| ,,g+{,> ] iF g+ , i g+ , ] ] ] ] i g g @ iF g g ] ] ] iF g @ g i g ] ] ] ] @ g g +iF i , @ g gi= , @ hu SUQh yrpr ra rhr ur Wkh surri wkdw ] ] i g+ , @ YV ] g +|, g+{,i +{> |, lv dqdorjrxv1 Qrwdwlrq 441:1 Jlyhq H [ \ dqg { 5 [> ohw { H =@ i| 5 \ = +{> | , 5 H j= Vlploduo| li | 5 \ lv jlyhq ohw H| =@ i{ 5 [ = +{> | , 5 H j= Li i = [ \ $ F lv d ixqfwlrq ohw i{ @ i +{> , dqg i | =@ i + > | , vr wkdw i{ = \ $ F dqg i | = [ $ F= Wkhruhp 441;1 Vxssrvh +[> P> ,> +\> Q > , duh frpsohwh 0qlwh phdvxuh vsdfh1 Ohw +[ \ > O> , eh wkh frpsohwlrq ri +[ \ > P Q > ,1 Li i lv O0phdvxudeoh dqg +d, i 3 ru +e, i 5 OR +, wkhq i{ lv Q 0phdvxudeoh iru d1h1 { dqg i | lv P0phdvxudeoh iru d1h1 | dqg lq fdvh +e, i{ 5 OR + , dqg i | 5 OR +, iru d1h1 { dqg d1h1 | uhvshfwlyho|1 Pruhryhu/ ] ] { $ i{ g dqg | $ i | g duh phdvxudeoh dqg ] ] ] ig @ ] ] ig g @ i g g= Surri1 Li H 5 P Q lv d 3 @ + qxoo vhw ++ ,+H , @ 3,> wkhq ] ] ,+H , @ +{ H ,g+{, @ +H| ,g +| ,= [ [ Wklv vkrzv wkdw +i{ = +{ H , 9@ 3j, @ 3 dqg +i| = +H| , 9@ 3j, @ 3> l1h1 +{ H , @ 3 iru d1h1 { dqg +H| , @ 3 iru d1h1 |= Li k lv O phdvxudeoh dqg k @ 3 iru 0 d1h1/ wkhq wkhuh h{lvwv H 5 P Q 6 i+{> |, = k+{> |, 9@ 3j H dqg + ,+H , @ 31 Wkhuhiruh mk+{> | ,m 4H +{> |, dqg + ,+H , @ 31 Vlqfh ik{ 9@ 3j @ i| 5 \ = k+{> |, 9@ 3j { H dqg ik| 9@ 3j @ i{ 5 [ = k+{> | , 9@ 3j H| zh ohduq wkdw iru d1h1 { dqg d1h1 | wkdw ik{ 9@ 3j 5 P> ik| 9@ 3j 5 Q > +ik{ 9@ 3j, @ 3 dqg d1h1 dqg +ik| 9@ 3j, @ 3= Wklv lpsolhv ] iru d1h1 |> k+{> | ,g +|, h{lvwv dqg htxdov 3 dqg ] iru d1h1 {> Wkhuhiruh ] ] ] 3@ k+{> | ,g+| , h{lvwv dqg htxdov 3= kg @ ] ] kg g @ kg g= ipr xI qvr| YW Iru jhqhudo i 5 OR +,> zh pd| fkrrvh j 5 OR +P Q > , vxfk wkdw i +{> |, @ j+{> |, iru d1h1 +{> |,= Ghqh k i j = Wkhq k @ 3> d1h1 Khqfh e| zkdw zh kdyh mxvw suryhg dqg Wkhruhp 4418 i @ j . k kdv wkh iroorzlqj surshuwlhv= +4, Iru d1h1 {> | $ i +{> |, @ j +{> |, . k+{> | , lv lq OR + , dqg ] ] i +{> | ,g +|, @ j +{> | ,g +| ,= +5, Iru d1h1 |> { $ i +{> | , @ j+{> | , . k+{> |, lv lq OR +, dqg ] ] i +{> | ,g+{, @ j +{> | ,g+{,= Iurp wkhvh dvvhuwlrqv dqg Wkhruhp 4418/ lw iroorzv wkdw ] ] ] ] g+{, g +| ,i +{> | , @ g+{, g +| ,j +{> |, ] ] @ g +| , g +{,j+{> | , ] @ j+{> | ,g+ ,+{> | , ] @ i +{> | ,g+{> | , dqg vlploduo| zh vkrzv ] ] ] g +|, g+{,i +{> | , @ i +{> |,g+{> | ,= H{huflvh 441<1 Iru P> 5 +3> 4,> vkrz ] P vlq { h{ g{ @5 dufwdq+ , . +P , +441;, Q { zkhuh olpP 6R +P , @ 3= Dv vshfldo fdvhv ri wklv h{suhvvlrq zh ohduq wkdw ]P vlq { +441<, olp g{ @ @5 P 6R Q { dqg ] +44143, olp 6Q Q R vlq { Vroxwlrq1 Ghqh iP +, =@ dqg qrwlfh wkdw iQ P +, @ @ ]P { h{ g{ @ @5= ]P Q h{ vlq { g{ { h{ {dP h{ vlq {g{ @ S +frv { . vlq {, m{dQ .4 Q hP frv P . hP vlq P 4 = S . 4 hu SUQh yrpr ra rhr ur YX E| wkh ixqgdphqwdo wkhruhp ri fdofxoxv/ ] Q iP + , iP +3, @ iP +,g Q ] P h frv P . hP vlq P @ S . 4 Q @ +P> , dufwdq+ , +44144, 4 g zkhuh ] P h frv P . hP vlq P g= S . 4 Q Xvlqj mvlq {m m{m U> zh kdyh wkh hvwlpdwhv= +P> , @ dqg ] P h frv P . hP vlq P g m+P> ,m S . 4 Q ] 4. hP S g .4 ]Q R .4 hP S g $ 3 dv P $ 4 +GFW, .4 Q ] P ]P vlq { { 4 hP h = g{ h{ g{ @ { Q Q Xvlqj wkhvh hvwlpdwhv/ zh pd| ohw P $ 4 lq Ht1 +44144, wr qg miP + ,m dqg wkhq ohwwlqj olp vxs miP +3, P 6R $ 4 zh qg dufwdq+ ,m 4@ olp i +3, @ @5= P 6R P Khqfh R +P , =@ m@5 iP +3,m $ 3 dv P $ 4 dqg S +P , =@ vxs$Q m+P> ,m $ 3 dv P $ 4= Wkhuhiruh e| Ht1 +44144,/ miP + , @5m @ m+P> , . iP +3, @5 dufwdq+ ,m S +P , . R +P , . dufwdq+ , @ +P , . dufwdq+ , zkhuh +P , @ R +P , . S +P , $ 3 dv P $ 4> zklfk suryhv Ht1 +441;,1 451 wjOjt} j 6jBt ij NA Ug Lq wklv vhfwlrq ohw g wlphv } ~ pg =@ p p pg dqg +Og > p, =@ EUg > pg ghqrwh wkh frpsohwlrq ri EUg > pg = Wkh phdvxuh lv fdoohg Ohehvjxh phdvxuh rq wkh Ohehvjxh phdvxudeoh vhw Og = UA |i ucdRic|dB |ivhiR vu dBBt ] ] ] 7 7 7 tc 7 9 vt CC 6 vt C C 6 C 9 7# f f f ipr xI qvr| YY Wkhruhp 45141 Ohehvjxh phdvxuh lv wudqvodwlrq lqyduldqw1 Surri1 Ohw D @ MR Mg zlwk Ml 5 EU dqg { 5 Ug = Wkhq { . D @ +{R . MR , +{S . MS , +{g . Mg , dqg wkhuhiruh e| wudqvodwlrq lqyduldqfh ri p rq EU zh qg wkdw pg +{ . D, @ p+{R . MR , = = = p+{g . Mg , @ p+MR , = = = p+Mg , @ pg +D, dqg khqfh pg +{ . D, @ pg +D, iru doo D 5 EUg e| Wkhruhp 71:1 Iurp wklv idfw zh vhh wkdw wkh phdvxuh pg +{ . , dqg pg + , kdyh wkh vdph qxoo vhwv1 Xvlqj wklv lw lv hdvlo| vhhq wkdw p+{ . D, @ p+D, iru doo D 5 Og = Lq wkh uhpdlqghu ri wklv vhfwlrq/ ohw W = $ W +, r Ug eh d F R ~ glhrpru0 sklvp1 Wkhruhp 4515 +Fkdqjh ri Yduldeohv Wkhruhp,1 Ohw r Ug eh dq rshq vhw dqg W = $ W +, r Ug eh d F R ~ glhrprusklvp1 Wkhq iru dq| Eruho phdvxudeoh i = W +, $ ^3> 4` zh kdyh ] ] Q mgp @ i W m ghw W i gp= +4514, W CD Surri1 Zh zloo fduu| rxw wkh surri lq d qxpehu ri vwhsv1 Vwhs 41 Vxssrvh wkdw W V $ W +, $ V +W +,, duh wzr F R ~ glhrprusklvpv dqg Wkhruhp 4515 krogv iru W dqg vhsdudwho|/ wkhq lw krogv iru V W= Lqghhg ] ] i V W m ghw +V W ,Q mgp @ i V W m ghw +V Q W , W Q mgp ] @ ] @ +mghw V Q m i V , W mW Q mgp mghw V Q m i Vgp @ ] i gp= W CD V CW CDD q dqg W lv olqhdu dqg lqyhuwleoh1 Wklv Vwhs 51 Ht1 +4514, krogv zkhq @ U zloo eh suryhg lq Wkhruhp 4516 ehorz1 Wkh surri lv d vlpsoh dssolfdwlrq ri Ix0 elql*v wkhruhp/ wkh vfdolqj dqg wudqvodwlrq lqyduldqfh surshuwlhv ri rqh glphqvlrqdo Ohehvjxh phdvxuh dqg wkh idfw wkdw e| urz uhgxfwlrq dujxphqwv W pd| eh zulwwhq dv d surgxfw ri _hohphqwdu|% wudqvirupdwlrqv1 Lw lv khuh wkdw zh xvh wkh uhvxow lq Vwhs 41 Vwhs 61 Iru doo D 5 E > ] mghw W Q m gp= +4515, p+W +D,, D Wklv zloo eh suryhg lq Wkhruhp 4519 ehorz1 Vwhs 71 Vwhs 61 lpsolhv wkh jhqhudo fdvh1 Wr vhh wklv/ ohw E 5 EW CD dqg D @ W R +E , lq Ht1 +4515, wr ohduq wkdw ] ] ] 4D gp @ p+D, mghw W Q m gp @ 4D W mghw W Q m gp= 4 CDD W hu SUQh yrpr ra rhr ur Y` Xvlqj olqhdulw| zh pd| frqfoxgh iurp wklv htxdwlrq wkdw ] ] igp i W mghw W Q m gp +4516, W CD iru doo qrq0qhjdwlyh vlpsoh ixqfwlrqv i rq W +,= Xvlqj Wkhruhp :165 dqg wkh prqr0 wrqh frqyhujhqfh wkhruhp rqh hdvlo| h{whqgv wklv htxdwlrq wr krog iru doo qrqqhjd0 wlyh phdvxudeoh ixqfwlrqv i rq W +,= Dsso|lqj Ht1 +4516, zlwk uhsodfhg e| W +,> W uhsodfhg e| W R dqg i e| j = $ ^3> 4`> zh vhh wkdw ] ] ] Q jgp @ jgp j W R ghw W R gp +4517, 4 CW CDD W W CD iru doo Eruho phdvxudeoh j= Wdnlqj j @ +i W , mghw W Q m lq wklv htxdwlrq vkrzv/ ] ] Q Q m gp i W mghw W i ghw W Q W R ghw W R gp W CD ] @ +4518, i gp W CD zkhuhlq wkh odvw htxdolw| zh xvhg wkh idfw wkdw W W R @ lg vr wkdw Q Q W Q W R W R @ lg dqg khqfh ghw W Q W R ghw W R @ 4= Frpelqlqj Htv1 +4516, dqg +4518, suryhv Ht1 +4514,1 Zh qrz oo lq wkh plvvlqj ghwdlov lq wkh surri1 Wkhruhp 45161 Vxssrvh W 5 JO+g> U,1 +4, i = Ug $ U lv +Og > E, phdvxudeoh wkhq vr lv i W li i 3 rq i 5 OR wkhq ] ] +4519, i +| ,g| @ m ghw W m i W +{,g{ +5, Li H 5 Og dqg H wkhq W +H , 5 Og dqg p+W +H ,, @ mghw W m p+H ,= Surri1 Vlqfh i lv Eruho phdvxudeoh dqg W = Ug $ Ug lv frqwlqxrxv dqg khqfh Eruho phdvxudeoh/ i W lv dovr Eruho phdvxudeoh1 Zh qrz euhdn wkh surri ri Ht1 +4519, lqwr d qxpehu ri fdvhv1 Lq hdfk fdvh zh pdnh xvh Wrqhool*v wkhruhp dqg wkh edvlf surshuwlhv ri rqh glphqvlrqdo Ohehvjxh phdvxuh1 +4, Vxssrvh wkdw l ? n dqg W +{R > {S = = = > {g , @ +{R > = = = > {lR > {n > {lFR = = = > {nR > {l > {nFR > = = = {g , wkhq e| Wrqhool*v wkhruhp/ ] ] i W +{R > = = = > {g , @ i +{R > = = = > {n > = = = {l > = = = {g ,g{R = = = g{g ] @ i +{R > = = = > {g ,g{R = = = g{g zklfk suryh Ht1 +4519, lq wklv fdvh vlqfh m ghw W m @ 4= ipr xI qvr| `Q +5, Vxssrvh wkdw f 5 U dqg W +{R > = = = {n > = = = > {g , @ +{R > = = = > f{n > = = = {g ,> wkhq ] ] i W +{R > = = = > {g ,gp @ i +{R > = = = > f{n > = = = {l > = = = {g ,g{R = = = g{n = = = g{g ] R i +{R> = = = > {g ,g{R = = = g{g @ mfm ] @ m ghw W mR i gp zklfk djdlq suryhv Ht1 +4519, lq wklv fdvh1 +6, Vxssrvh wkdw W +{R > {S = = = > {g , @ +{R > = = = > {l . f{n > = = = {n > = = = {g ,= ] Wkhq ] i W +{R > = = = > {g ,gp @ @ @ ] ] i +{R > = = = > {l . f{n > = = = {n > = = = {g ,g{R = = = g{l = = = g{n = = = g{g i +{R > = = = > {l > = = = {n > = = = {g ,g{R = = = g{l = = = g{n = = = g{g i +{R > = = = > {g ,g{R = = = g{g zkhuh lq wkh vhfrqg lqhtxdolw| zh glg wkh {l lqwhjudo uvw dqg xvhg wudqv0 odwlrq lqyduldqfh ri Ohehvjxh phdvxuh1 Djdlq wklv suryhv Ht1 +4519, lq wklv fdvh vlqfh ghw+W , @ 4= Vlqfh hyhu| lqyhuwleoh pdwul{ lv d surgxfw ri pdwulfhv ri wkh w|sh rffxuulqj lq vwhsv 41 ~ 61 deryh/ lw iroorzv wkdw Ht1 +4519, krogv lq jhqhudo1 Iru wkh vhfrqg dvvhuwlrq/ ohw H 5 EUg dqg wdnh i @ 4H lq Ht1 +4519, wr ohduq wkdw ] ] ] m ghw W mp+W R +H ,, @ m ghw W m 4W 4 CH D gp @ m ghw W m 4H W gp @ 4H gp @ p+H ,= Uhsodflqj W e| W R lq wklv htxdwlrq vkrzv wkdw p+W +H ,, @ mghw W m p+H , iru doo H 5 EUg = Lq sduwlfxodu wklv vkrzv wkdw p W dqg p kdyh wkh vdph qxoo vhwv dqg wkhuhiruh wkh frpsohwlrq ri EUg lv Og iru erwk phdvxuhv1 Lw lv dovr fohdu wkdw p+W +H ,, @ mghw W m p+H , iru doo H 5 Og = Qrwdwlrq 45171 Iru d> e 5 Ug zh zloo zulwh d e lv dl el iru doo l dqg d ? e T T li dl ? el iru doo l= Jlyhq d ? e ohw ^d> e` @ gdR ^dl > el ` dqg +d> e` @ gdR +dl > el `= l l +Qrwlfh wkdw wkh forvxuh ri +d> e` lv ^d> e`=, Zh zloo vd| wkdw T @ +d> e` lv d fxeh surylghg wkdw el dl @ 5 A 3 lv d frqvwdqw lqghshqghqw ri l= Zkhq T lv d fxeh/ ohw {T =@ d . +> > = = = > , eh wkh fhqwhu ri wkh fxeh1 Qrwlfh wkdw zlwk wklv qrwdwlrq/ li T lv d fxeh ri vlgh ohqjwk 5> +451:, T @ i{ 5 Ug = m{ {T m j dqg wkh lqwhulru +TQ , ri T pd| eh zulwwhq dv TQ @ i{ 5 Ug = m{ {T m ? j= hu SUQh yrpr ra rhr ur Qrwdwlrq 45181 Iru d 5 Ug > ohw mdm @ pd{l mdl m dqg li W lv d g S nW n @ pd{l m mWlm m = `R g pdwul{ ohw D nh| srlqw ri wklv qrwdwlrq lv wkdw [ [ mW dm @ pd{ Wlm dm pd{ mWlm m mdm m l l m m +451;, nW n mdm = Wkhruhp 45191 Ohw r Ug eh dq rshq vhw dqg W = $ W +, r Ug eh d F R ~ glhrprusklvp1 Wkhq iru dq| D 5 E > ] +451<, p+W +D,, m ghw W Q +{,mg{= D Surri1 Vwhs 41 Zh zloo uvw dvvxph wkdw D @ T @ +d> e` lv d fxeh vxfk wkdw T @ ^d> e` = Ohw @ +el dl ,@5 eh kdoi wkh vlgh ohqjwk ri T= E| wkh ixqgdphqwdo wkhruhp ri fdofxoxv +iru Ulhpdqq lqwhjudov, iru { 5 T> ]R W +{, @ W +{T , . W Q +{T . w+{ {T ,,+{ {T ,gw Q @ W +{T , . W Q +{T ,V +{, zkhuh V +{, @ ] R Q W Q +{T ,R W Q +{T . w+{ Wkhuhiruh W +T, @ W +{T , . W Q +{T ,V +T, dqg khqfh {T ,,gw +{ {T ,= p+W +T,, @ p +W +{T , . W Q +{T ,V +T,, @ p +W Q +{T ,V +T,, +45143, @ mghw W Q +{T ,m p +V +T,, = Qrz iru { 5 T> l1h1 m{ {T m > ] R Q +{T,R W Q +{T . w+{ {T ,,gw m{ {Tm mV +{,m W Q k+{T > {, zkhuh +45144, Khqfh dqg +45145, k+{T > {, =@ ] R W Q +{T ,R W Q +{T . w+{ Q {T ,, gw= V +T, pd{ k+{T > {,i{ 5 Ug = m{m pd{ kg +{T > {,j {ST {S T g p+V +T,, pd{ k+{T > {,g +5 , @ pd{ kg +{T > {,p+T,= {S T {S T Frpelqlqj Htv1 +45143, dqg +45145, vkrzv wkdw p+W +T,, mghw W Q +{T ,m p+T, pd{ kg +{T > {,= {ST Wr uhqh wklv hvwlpdwh/ zh zloo vxeglylgh T lqwr vpdoohu fxehv/ l1h1 iru q 5 Q ohw 5 5 g= Tq @ +d> d . +> > = = = > ,` . = 5 i3> 4> 5> = = = > qj q q +45146, ipr xI qvr| `S Qrwlfh wkdw T @ c DSq D= E| Ht1 +45146,/ p+W +D,, mghw W Q +{D ,m p+D, pd{ kg +{D > {, {SD dqg vxpplqj wkh htxdwlrq rq D jlyhv [ [ p+W +T,, @ p+W +D,, mghw W Q +{D ,m p+D, pd{ kg +{D > {,= {S D DSq DSq Vlqfh kg +{> {, @ 4 iru doo { 5 T dqg kg = T T $ ^3> 4, lv frqwlqxrxv ixqfwlrq rq d frpsdfw vhw/ iru dq| A 3 wkhuh h{lvwv q vxfk wkdw li {> | 5 T dqg m{ | m @q g +{> | , 4 . = Xvlqj wklv lq wkh suhylrxvo| glvsod|hg htxdwlrq/ zh qg wkdw wkhq k [ p+W +T, +4 . , mghw W Q +{D ,m p+D, DSq ][ @ +4 . , +45147, mghw W Q +{D ,m 4D +{,gp+{,= T DSq Vlqfh mghw W Q +{,m lv frqwlqxrxv rq wkh frpsdfw vhw T> lw hdvlo| iroorzv e| xqlirup frqwlqxlw| wkdw [ mghw W Q +{D ,m 4D +{, $ mghw W Q +{,m dv q $ 4 DSq dqg wkh frqyhujhqfh lq xqlirup rq T= Wkhuhiruh wkh grplqdwhg frqyhujhqfh wkhruhp hqdeohv xv wr sdvv wr wkh olplw/ q $ 4> lq Ht1 +45147, wr qg ] mghw W Q +{,m gp+{,= p+W +T,, +4 . , T Vlqfh A 3 lv duelwudu| zh duh grqh zh kdyh vkrzq wkdw ] p+W +T,, mghw W Q +{,m gp+{,= T Vwhs 51 Zh zloo qrz vkrz wkdw Ht1 +451<, lv ydolg zkhq D @ X lv dq rshq vxevhw ri = Iru q 5 Q> ohw Tq @ +3> +> > = = = > ,` . 5q = 5 ]g vr wkdw Tq lv d sduwlwlrq ri Ug = Ohw IR =@ D 5 TR = D X dqg ghqh Iq ^qdR Tn lqgxfwlyho| dv iroorzv1 Dvvxplqj IqR kdv ehhq ghqhg/ ohw n Iq @ IqR ^ D 5 Tq = D X dqg D _ E @ > iru doo E 5 IqR @ IqR ^ D 5 Tq = D X dqg D - E iru dq| E 5 IqR c Qrz vhw I @ ^Iq +vhh Iljxuh ;, dqg qrwlfh wkdw X @ DSs D= Lqghhg e| frq0 c vwuxfwlrq/ wkh vhwv lq I duh sdluzlvh glvmrlqw vxevhw ri X vr wkdw DSs D X= Li { 5 X> wkhuh h{lvwv dq q dqg D 5 Tq vxfk wkdw { 5 D dqg D X= Wkhq e| frqvwuxfwlrq ri I > hlwkhu D 5 I ru wkhuh lv d vhw E 5 I vxfk wkdw D E= Lq hlwkhu c c fdvh { 5 DSs D zklfk vkrzv wkdw X @ DSs D= Wkhuhiruh e| vwhs 41/ p+W +X ,, @ p+W +^DSs D,, @ p++^DSs W +D,,, [] [ p+W +D,, mghw W Q +{,m gp+{, @ D DSs DSs ] @ mghw W Q +{,m gp+{, X hu SUQh yrpr ra rhr ur `T 8} ij I Iloolqj rxw dq rshq vhw zlwk doprvw glvmrlqw fxehv1 Zh kdyh gudzq IS = zklfk suryhv vwhs 51 Vwhs 61 Iru jhqhudo D 5 E ohw eh wkh phdvxuh/ ] mghw W Q +{,m gp+{,= +D, =@ D Wkhq p W dqg duh + ~ qlwh phdvxuhv dv |rx vkrxog fkhfn, rq E vxfk wkdw p W rq rshq vhwv1 E| uhjxodulw| ri wkhvh phdvxuhv/ zh pd| frqfoxgh wkdw p W = Lqghhg/ li D 5 E > ] mghw W Q +{,m gp+{,= p +W +D,, @ lqi p +W +X ,, lqi +X , @ +D, @ X )r X )r D 45141 Srodu Frruglqdwhv dqg Vxuidfh Phdvxuh1 Ohw g [ gR @ i{ 5 Ug = m{mS =@ {S @ 4j V l ldR eh wkh xqlw vskhuh lq Ug = Ohw = Ug q +3, $ +3> 4, V gR dqg R eh wkh lqyhuvh pds jlyhq e| { , dqg R +u> $ , @ u$ +45148, +{, =@ +m{m > m{m uhvshfwlyho|1 Vlqfh dqg R duh frqwlqxrxv/ wkh| duh Eruho phdvxudeoh1 Frqvlghu wkh phdvxuh p rq ECQ>RD EV g4 jlyhq e| p+D, =@ p R +D, iru doo D 5 ECQ>RD EV g4 = Iru H 5 EV g4 dqg d A 3> ohw Hd =@ iu$ = u 5 +3> d` dqg $ 5 H j @ R ++3> d` H , 5 EUg = Qrwlqj wkdw Hd @ dHR > zh kdyh iru 3 ? d ? e> H 5 EV g4 > H dqg D @ +d> e` wkdw +45149, +4514:, R +D, @ iu$ = u 5 +d> e` dqg $ 5 H j @ eHR q dHR = H ipr xI qvr| `U Wkhuhiruh/ + p, ++d> e` H , @ p +eHR q dHR , @ p+eHR , @ eg p+HR , dg p+HR , ]e +4514;, ugR gu= @ g p+HR , d Ohw ghqrwh wkh xqltxh phdvxuh rq ECQ>RD vxfk wkdw ] ugR gu +4514<, +M , @ M p+dHR , iru doo M 5 ECQ>RD = V|perolfdoo|/ zh zloo deeuhyldwh wklv e| zulwlqj +gu, @ ugR gu1 Ghqlwlrq 451:1 Iru H 5 EV g4 / ohw +H , =@ g p+HR ,= Zh fdoo wkh vxuidfh phdvxuh rq V= Lw lv hdv| wr fkhfn wkdw lv d phdvxuh1 Lqghhg li H 5 EV g4 > wkhq HR @ cR R ++3> 4` H , 5 EUg vr wkdw p+HR , lv ghqhg1 Pruhryhu li H @ ldR Hl > wkhq cR HR @ ldR +Hl ,R dqg +H , @ g p+HR , @ R [ ldR p ++Hl ,R , @ R [ ldR +Hl ,= Wkh lqwxlwlrq ehklqg wklv ghqlwlrq lv dv iroorzv1 Li H V gR lv d vhw dqg A 3 lv d vpdoo qxpehu/ wkhq wkh yroxph ri +4> 4 . ` H @ iu$ = u 5 +4> 4 . ` dqg $ 5 H j vkrxog eh dssur{lpdwho| jlyhq e| p ++4> 4 . ` H , +H ,= Rq wkh rwkhu kdqg @ p ++4> 4 . `H , @ p +HRF q HR , @ +4 . ,g 4 p+HR ,= Wkhuhiruh zh h{shfw wkh duhd ri H vkrxog eh jlyhq e| +4 . ,g 4 p+HR , @ gp+HR ,= +H , @ olp 8Q Dffruglqj wr wkhvh ghqlwlrqv dqg Ht1 +4514;, zh kdyh vkrzq wkdw +45153, p++d> e` H , @ ++d> e`, +H ,= Ohw H @ i+d> e` H = 3 ? d ? e> H 5 EV g4 j > wkhq H lv dq hohphqwdu| fodvv1 Vlqfh +H , @ ECQ>RD EV g4 > zh frqfoxgh iurp Ht1 +45153, wkdw p @ dqg wklv lpsolhv wkh iroorzlqj wkhruhp1 Wkhruhp 451;1 Li i = Ug $ ^3> 4` lv d +EUg > E,~phdvxudeoh ixqfwlrq wkhq ] ] +45154, i +{,gp+{, @ i +u $, g+$ ,ugR gu= Q>RDV Ohw xv qrz zrun rxw vrph lqwhjudov xvlqj Ht1 +45154,1 hu SUQh yrpr ra rhr ur Ohppd 451<1 Ohw d A 3 dqg ] Lg +d, =@ Wkhq Lg +d, @ +@d,g@S = 5 hdm{m gp+{,= Ug Surri1 E| Wrqhool*v wkhruhp dqg lqgxfwlrq/ ] 5 5 Lg +d, @ hdm|m hdw pgR +g| , gw g4 U U g @ Lg +d,LR +d, @ LR +d,= +45155, Vr lw vxfhv wr frpsxwh= ] ] 55 dm{m5 gp+{, @ hdC{4 F{5 D g{R g{S = LS +d, @ h U5 U5 qiQj Zh qrz pdnh wkh fkdqjh ri yduldeohv/ 1 {R @ u frv dqg {S @ u vlq iru 3 ? u ? 4 dqg 3 ? ? 5= Lq yhfwru irup wklv wudqvirup lv u frv { @ W +u> , @ u vlq dqg wkh glhuhqwldo lv jlyhq e| W Q +u> ,@ frv vlq u vlq u frv dqg wkh Mdfreldq ghwhuplqdqw lv ghw W Q +u> , @ u frvS . u vlqS @ u= Khqfh e| wkh fkdqjh ri yduldeohv irupxod/ ]R ] S 5 gu u g hdu LS +d, @ Q] Q R 5 @ 5 u hdu gu Q ]P 5 @ 5 olp u hdu gu P 6R Q 5]P 5 hdu @ @d= @ @ 5 olp P 6R 5d Q 5d Wklv vkrzv wkdw LS +d, @ @d dqg wkh uhvxow qrz iroorzv iurp Ht1 +45155,1 Fruroodu| 451431 Wkh vxuidfh duhd +V gR , ri wkh xqlw vskhuh V gR Ug lv 5 g@S +45156, +V gR , @ +g@5, zkhuh lv wkh jdppd ixqfwlrq jlyhq e| ]R u{R hu gu +45157, +{, =@ Pruhryhu/ +4@5, @ s dqg +4, @ 4= Q `V ipr xI qvr| `W Surri1 Zh pd| dowhuqdwlyho| frpsxwh Lg +4, @ g@S xvlqj Wkhruhp 451;> ] ]R 5 gu ugR hu g Lg +4, @ Q V g 4 ]R 5 ugR hu gu= @ +V gR , Q Zh vlpsoli| wklv odvw lqwhjudo e| pdnlqj wkh fkdqjh ri yduldeohv x @ uS vr wkdw u @ xR@S dqg gu @ R xR@S gx= Wkh uhvxow lv S ]R ]R 5 g 4 4 ugR hu gu @ x 5 hx xR@S gx 5 Q Q] Rg 4 x 5 R hx gx @ 5Q 4 @ +45158, +g@5,= 5 Froohfwlqj wkhvh revhuydwlrqv lpsolhv wkdw 4 g@S @ Lg +4, @ +V gR , +g@5, 5 zklfk suryhv Ht1 +45156,1 Wkh frpsxwdwlrq ri +4, lv hdv| dqg lv ohiw wr wkh uhdghu1 E| Ht1 +45158,/ ]R ]R 5 5 hu gu @ hu gu +4@5, @ 5 Q R s @ LR +4, @ = 461 7}Aja jBt ijt Ghqlwlrq 46141 D vljqhg phdvxuh rq d phdvxudeoh vsdfh +[> P, lv d ixqfwlrq = P $ U vxfk wkdw +4, Hlwkhu +P, + 4> 4` ru +P, ^ 4> 4,= c +5, lv frxqwdeo| dgglwlyh/ wklv lv wr vd| li H @ R Hm zlwk Hm 5 P> wkhq m dR R S V +H , @ +Hm ,= m dR +6, +>, @ 3 H{dpsoh 46151 Vxssrvh wkdw F dqg duh wzr srvlwlyh phdvxuhv rq P vxfk wkdw hlwkhu F +[ , ? 4 ru +[ , ? 4> wkhq @ F lv d vljqhg phdvxuh1 U H{dpsoh 46161 Vxssrvh wkdw i = [ $ U phdvxudeoh dqg hlwkhu H i F g ru U H i g ? 4> wkhq ] +H , @ i g H lv d vljqhg phdvxuh1 Wklv lv dfwxdoo| d vshfldo fdvh ri wkh odvw h{dpsoh zlwk +H , U H i g= Qrwlfh wkdw wkh phdvxuh lq wklv h{dpsoh kdyh wkh surshuw| wkdw wkh| " Vu h iF ( w U |ic |i tihit S h iF& ( t dwtvB|iBA vcih}ic| tci | t ciicic| vu &' hidhhdc}iRic|t! hu SUQh yrpr ra rhr ur `X duh frqfhqwudwhg rq glvmrlqw vhwv/ qdpho| F _olyhv% rq ii A 3j dqg _olyhv% rq wkh vhw ii ? 3j = Wkh pdlq wkhruhpv ri wklv vhfwlrq dvvhuw wkdw wkhvh duh wkh rqo| h{dpsohv ri vljqhg phdvxuhv1 46141 Kdkq Ghfrpsrvlwlrq Wkhruhp1 Ghqlwlrq 46171 Ohw eh d vljqhg phdvxuh rq +[> P, dqg H 5 P> wkhq +4, H lv srvlwlyh li iru doo D 5 P vxfk wkdw D H> +D, 3> l1h1 mH 3= +5, H lv qhjdwlyh li iru doo D 5 P vxfk wkdw D H> +D, 3> l1h1 mH 3= +6, H lv qxoo li iru doo D 5 P vxfk wkdw D H> l1h1 mH @ 3= Khuh PH iD _ H = D 5 Pj @ wudfh ri P rq H 1 Ohppd 46181 Vxssrvh wkdw lv d vljqhg phdvxuh rq +[> P,= Wkhq +4, Dq| vxevhw ri d srvlwlyh vhw lv srvlwlyh1 +5, Wkh frxqwdeoh xqlrq ri srvlwlyh +qhjdwlyh ru qxoo, vhwv lv vwloo srvlwlyh +qhj0 dwlyh ru qxoo,1 +6, Ohw xv qrz ixuwkhu dvvxph wkdw +P, ^ 4> 4, dqg H 5 P lv d vhw vxfk wkdw +H , 5 +3> 4,= Wkhq wkhuh h{lvwv d srvlwlyh vhw S H vxfk wkdw +S , +H ,= Surri1 Wkh uvw dvvhuwlrq lv reylrxv1 Li Sm 5 P duh srvlwlyh vhwv/ ohw S @ # $ qVR Sq = E| uhsodflqj Sq e| wkh srvlwlyh vhw Sq q Sm zh pd| dvvxph wkdw qdR m dR R c R Sq = Qrz li H S dqg H 5 P> wkh iSq jqdR duh sdluzlvh glvmrlqw vr wkdw S @ qdR R c +H _ Sq , vr H@ qdR R [ +H , @ +H _ Sq , 3= qdR zklfk vkrzv wkdw S lv srvlwlyh1 Wkh surri iru wkh qhjdwlyh dqg wkh qxoo fdvh lv dqdorjrxv1 Wkh lghd iru surylqj wkh wklug dvvhuwlrq lv wr nhhs uhprylqj _elj% vhwv ri qhjdwlyh phdvxuh iurp H 1 Wkh vhw uhpdlqlqj iurp wklv surfhgxuh zloo eh S= Zh qrz ehjlq wkh irupdo surri1 Iru doo D 5 P ohw q+D, @ 4 a vxsi +E , = E Dj= Vlqfh +>, @ 3> q+D, 3 dqg q+D, @ 3 l D lv srvlwlyh1 Fkrrvh DQ H vxfk wkdw +DQ , R q+H , dqg S vhw HR @ H q DQ > wkhq fkrrvh DR HR vxfk wkdw +DR , R q+HR , dqg vhw S HS @ H q +DQ ^ DR , = Frqwlqxh wklv surfhgxuh lqgxfwlyho| vr wkdw li DQ > = = = > DnR kdyh ehhq fkrvhq ohw nR d Dl Hn @ H q ldQ dqg fkrrvh Dn Hn vxfk wkdw +Dn , R q+Hn ,= Zh zloo qrz vkrz wkdw S R V S =@ H q R d ndQ lv d srvlwlyh vhw vxfk wkdw +S , +H ,= Dn @ R _ ndQ Hn ipr xI qvr| `Y Vlqfh H@S ^ R d ndQ R [ Dn > R 4[ q+Hn , 5 ndQ ndQ dqg khqfh +H , +S ,= Pruhryhu/ +H , +S , A 4 vlqfh +H , 3 dqg +S , 9@ 4 e| wkh dvvxpswlrq +P, ^ 4> 4,= Wkhuhiruh zh pd| frqfoxgh iurp Ht1 SR +4614, wkdw ndQ q+Hn , ? 4 dqg lq sduwlfxodu olpn6R q+Hn , @ 3= Qrz li D S wkhq D Hn iru doo n dqg wklv lpsolhv wkdw +D, 3 vlqfh e| ghqlwlrq ri q+Hn ,> +D, q+Hn , $ 3 dv n $ 4= +4614, +H , +S , @ +H q S , @ +Dn , Ghqlwlrq 46191 Vxssrvh wkdw lv d vljqhg phdvxuh rq +[> P,= D Kdkq gh0 frpsrvlwlrq iru lv d sduwlwlrq iS> Q j ri [ vxfk wkdw S lv srvlwlyh dqg Q lv qhjdwlyh1 Wkhruhp 461: +Kdkq Ghfrpsrvlwlrq Wkhruhp,1 Hyhu| vljqhg phdvxuh vsdfh +[> P> , kdv d Kdkq ghfrpsrvlwlrq/ iS> Q j= Pruhryhu/ li iS > Q j lv dqrwkhu Kdkq @ Q Q lv d qxoo vhw/ vr wkh ghfrpsrvlwlrq lv xqltxh ghfrpsrvlwlrq/ wkhq S S prgxor qxoo vhwv1 Surri1 Zlwk rxw orvv ri jhqhudolw| zh pd| dvvxph wkdw +P, ^ 4> 4,1 Li qrw mxvw frqvlghu lqvwhdg1 Ohw xv ehjlq zlwk wkh xqltxhqhvv dvvhuwlrq1 Vxssrvh wkdw D 5 P> wkhq +D, @ +D _ S , . +D _ Q , +D _ S , +S , dqg vlploduo| +D, +S , iru doo D 5 P= dqg lq sduwlfxodu zh kdyh +S , +S ^ S , +S , dqg +S , +S ^ S , +S , zklfk vkrzv wkdw v =@ +S , @ +S ^ S , @ +S ,= Vlqfh v @ +S ^ S , @ +S , . +S , v @ +S ^ S , @ +S , . +S , +S _ S , @ 5v +S _ S , zh vhh wkdw +S _ S , @ v dqg vlqfh v @ +S ^ S , @ +S _ S , . +S S , lw iroorzv wkdw +S S , @ 3= Wkxv Q Q @ S S lv d srvlwlyh vhw zlwk }hur phdvxuh/ l1h1 Q Q @ S S lv d qxoo vhw dqg wklv suryhv wkh xqltxhqhvv dvvhuwlrq1 Ohw v vxsi +D, = D 5 Pj zklfk lv qrq0qhjdwlyh vlqfh +>, @ 3= Li v @ 3> zh duh grqh vlqfh S @ > dqg Q @ [ lv wkh ghvluhg ghfrpsrvlwlrq1 Vr dvvxph v A 3 dqg fkrrvh Dq 5 P vxfk hu SUQh yrpr ra rhr ur `` wkdw +Dq , A 3 dqg olpq6R +Dq , @ v= E| Ohppd 4618/ wkhuh h{lvwv srvlwlyh vhwv Sq Dq vxfk wkdw +Sq , +Dq ,1 Wkhq fohduo| v +Sq , +Dq , $ v dv q $ 4 lpsolhv wkdw v @ olpq6R +Sq ,= Wkh vhw S ^Sq lv d srvlwlyh vhw ehlqj wkh xqlrq ri srvlwlyh vhwv dqg vlqfh Sq S iru doo q> +S , +Sq , $ v dv q $ 4= Wklv vkrzv wkdw +S , v dqg khqfh e| wkh ghqlwlrq ri v> v @ +S ,= Lq sduwlfxodu wklv vkrzv wkdw v ? 4= Zh qrz fodlp wkdw Q @ S f lv d qhjdwlyh vhw dqg wkhuhiruh/ iS> Q j lv wkh ghvluhg Kdkq ghfrpsrvlwlrq1 Li Q zhuh qrw qhjdwlyh/ zh frxog qg H Q @ S f vxfk wkdw +H , A 3= Zh wkhq zrxog kdyh +S ^ H , @ +S , . +H , @ v . +H , A v zklfk frqwudglfwv wkh ghqlwlrq ri v= 46151 Mrugdq Ghfrpsrvlwlrq1 Ghqlwlrq 461;1 Ohw dqg eh wzr vljqhg phdvxuhv rq +[> P,= Zh vd| dqg duh pxwxdoo| vlqjxodu dqg zulwh B li wkhuh h{lvwv D 5 P vxfk wkdw D lv 0qxoo dqg Df lv 0qxoo1 l1h1 dqg _olyh% rq glvmrlqw vhwv1 Uhpdun 461<1 Li R > S dqg duh vljqhg phdvxuhv rq +[> P, vxfk wkdw R B dqg = S B dqg R . S lv zhoo ghqhg/ wkhq +R . S , B S Li il jR lv d vhtxhqfh ri ldR srvlwlyh phdvxuhv vxfk wkdw l B iru doo l wkhq @ R l B dv zhoo1 ldR Surri1 Lq erwk fdvhv/ fkrrvh Dl 5 P vxfk wkdw Dl lv 0qxoo dqg Df lv l 0qxoo l iru doo l= Wkhq e| Ohppd 4618/ D =@ ^l Dl lv vwloo d ~qxoo vhw1 Vlqfh Df @ _l Df Df iru doo p l p f lv d 0 qxoo vhw iru doo l dqg lv wkhuhiruh d qxoo vhw iru @ SR = zh vhh wkdw D l ldR l Wklv vkrzv wkdw B = Ghqlwlrq 461431 Ohw [ @ S ^ Q eh d Kdkq Ghfrpsrvlwlrq ri dqg ghqh F +H , @ +S _ H , +H , @ +Q _ H , hh Vxssrvh [ @ S ^ Q lv dqrwkhu Kdkq Ghfrpsrvlwlrq dqg h duh ghqh dv deryh h h zlwk S dqg Q uhsodfhg e| S dqg Q uhvshfwlyho|1 Wkhq h h h hF +H , @ +H _ S , @ +H _ S _ S , . ++H _ S _ Q , h @ +H _ S _ S , vlqfh Q _ S lv erwk srvlwlyh dqg qhjdwlyh dqg khqfh qxoo1 Vlploduo| F +H , @ h _ S , vkrzlqj wkdw F @ hF dqg wkhuhiruh dovr wkdw @ h = +H _ S Wkhruhp 461441 Mrugdq Ghfrpsrvlwlrq= Wkhuh h{lvwv xqltxh srvlwlyh phdvxuh vxfk wkdw F B dqg @ F = Surri1 H{lvwhqfh kdv ehhq suryhg1 Iru xqltxhqhvv1 Vxssrvh @ F lv d Mrugdq Ghfrpsrvlwlrq1 Vlqfh F B wkhuh h{lvwv S> Q @ S f 5 P vxfk wkdw F +Q , @ 3 dqg +S , @ 31 Wkhq fohduo| S lv srvlwlyh iru dqg Q lv qhjdwlyh iru 1 Qrz +H _ S , @ F +H , dqg +H _ Q , @ +H ,1 Wkh xqltxhqhvv qrz iroorzv iurp wkh uhpdunv diwhu Ghqlwlrq 461431 ipr xI qvr| RQQ Ghqlwlrq 461451 m m+H , @ F +H , . +H , lv fdoohg wkh wrwdo yduldwlrq ri 1 D vljqhg phdvxuh lv fdoohg ~ qlwh surylghg wkdw m m =@ F . lv d qlwh phdvxuh1 Ohppd 461461 Ohw eh d vljqhg phdvxuh rq +[> P, dqg D 5 P= Wkhq m m +D, ? 4 l m +D,m ? 4 vr wkdw lv qlwh l wkhuh h{lvwv [q 5 P vxfk wkdw m +[q ,m ? 4 dqg [ @ ^R [q = Dovr vxssrvh wkdw S> Q 5 P lv d Kdkq ghfrpsrvlwlrq iru dqg qdR ohw j @ 4S 4Q > wkhq g @ jg m m > l1h1 ] +D, @ jg m m iru doo D 5 P= D Surri1 Ohw S> Q 5 P eh d Kdkq ghfrpsrvlwlrq iru > wkhq +4615, +D, @ @ m m +D, @ @ +D _ S , . +D _ Q , m +D _ S ,m m +D _ Q ,m dqg +D _ S , +D _ Q , m +D _ S ,m . m +D _ Q ,m = Wkhuhiruh m +D,m ? 4 l m +D _ S ,m dqg m +D _ Q ,m duh qlwh l m m +D, ? 4 surylqj wkh uvw dvvhuwlrq1 Wkh vhfrqg dvvhuwlrq lv d frqvhtxhqfh ri wkh iroorzlqj lghqwlw|/ +D, @ +D _ S , . +D _ Q , @ m m+D _ S , m m+D _ Q , ] @ +4S 4Q ,gm m= D Uhpdun 461471 Vxssrvh wkdw lv d srvlwlyh phdvxuh rq +[> P, dqg j = [ $ U lv dq h{whqghg lqwhjudeoh ixqfwlrq1 Li lv wkh vljqhg phdvxuh g @ jg> wkhq g F @ jF g> g @ j g dqg g m m @ mj m g= Surri1 Wkh sdlu/ S @ ij A 3j dqg Q @ ij 3j @ S f lv d Kdkq ghfrpsrvlwlrq iru = Wkhuhiruh ] ] ] F +D, @ +D _ S , @ jg @ 4ijAQj jg @ jF g> DS D D ] ] ] jg @ 4ij#Qj jg @ j g= +D, @ +D _ Q , @ DQ D D dqg m m +D, @ F +D, . +D, ] ] ] jF g j g @ +jF @ D D D ] mjm g= @ D j , g Ghqlwlrq 461481 Ohw eh d vljqhg phdvxuh rq +[> P,> ohw OR + , =@ OR + F , _ OR + , @ OR +m m, hu SUQh yrpr ra rhr ur dqg iru i 5 OR + , zh ghqh ] ] RQR ] ig @ i g F i g = [ [ [ Ohppd 461491 Ohw eh d srvlwlyh phdvxuh rq +[> P,> j eh dq h{whqghg lqwhjudeoh ixqfwlrq rq +[> P> , dqg g @ jg= Wkhq OR + , @ OR +mj m g, dqg iru i 5 OR + ,> ] ] ig @ ijg= [ [ Surri1 Zh kdyh douhdg| vhhq wkdw g F @ jF g> g @ j g> dqg g m m @ mjm g vr wkdw OR + , @ OR +m m, @ OR +mj m g, dqg iru i 5 OR + ,> ] ] ] ig @ ig F i g [ [ [ ] ] @ ijF g ij g [ ][ i +jF j , g @ ][ ijg= @ [ U Uhpdun 4614:1 Ohw eh d vljqhg phdvxuh rq +[> P,> wkhq m m+H , @ vxsi H i g = mi m 4j= Surri1 Li H 5 P dqg mi m 4> wkhq ] ] ] ] ] i g @ i g F i g i g F . i g H H ]H ]H ]H mi mg F . mi mg @ mi m gm m H H H m m +H , zklfk vkrzv wkdw ] vxsi i g = mi m 4j m m+H ,= H Iru wkh uhyhuvh lqhtxdolw|/ ohw [ @ S ^ Q eh d Kdkq ghfrpsrvlwlrq ri dqg ghqh i 4S 4Q wkhq ] i g @ +H _ S , +H _ Q , H @ F +H , . +H , @ m m+H ,= 471 -BaNA "Na+6 C jNij6 Ghqlwlrq 47141 Ohw eh d vljqhg phdvxuh dqg eh d srvlwlyh phdvxuh rq +[> P,= Zh vd| wkh lv devroxwho| frqwlqxrxv uhodwlyh wr dqg zulwh surylghg wkdw +D, @ 3 zkhqhyhu +D, @ 3= Wkdw lv wr vd| li D lv d qxoo vhw iru wkhq D lv dovr d qxoo vhw iru = RQS ipr xI qvr| Uhpdun 47151 Vlqfh m +D,m m m +D, iru doo D 5 P/ li m m wkhq = Wkh frqyhuvh lv wuxh dv zhoo1 Wr vhh wklv ohw S 5 P eh fkrvhq vr wkdw iS> Q @ S f j lv d Kdkq ghfrpsrvlwlrq iru = Li D 5 P dqg +D, @ 3 wkhq +D _ S , @ 3 dqg +D _ Q , @ 3 vlqfh +D _ S , @ 3 dqg +D _ Q , @ 3= Wkhuhiruh m m +D, @ +D _ S , +D _ Q , @ 3= Ohppd 47161 4, Ohw eh d vljqhg phdvxuh dqg eh d srvlwlyh phdvxuh rq +[> P, SR vxfk wkdw dqg B > wkhq 31 5, Vxssrvh wkdw @ ldR l zkhuh l duh srvlwlyh phdvxuhv rq +[> P, vxfk wkdw l > wkhq = Dovr li R dqg S duh wzr vljqhg phdvxuh vxfk wkdw l iru l @ 4> 5 dqg @ R . S lv zhoo ghqhg/ wkhq = Surri1 +4, Ehfdxvh B > wkhuh h{lvwv D 5 P vxfk wkdw D lv d 0qxoo vhw dqg E @ Df lv d 0 qxoo vhw1 Vlqfh E lv ~ qxoo dqg > E lv dovr ~ qxoo1 Wklv vkrzv e| Ohppd 4618 wkdw [ @ D ^ E lv dovr ~ qxoo/ l1h1 lv wkh }hur phdvxuh1 Wkh surri ri +5, lv hdv| dqg lv ohiw wr wkh uhdghu1 Wkhruhp 47171 Ohw eh d qlwh vljqhg phdvxuh dqg eh d srvlwlyh phdvxuh rq +[> P,= Wkhq l iru doo A 3 wkhuh h{lvwv d A 3 vxfk wkdw m +D,m ? zkhqhyhu D 5 P dqg +D, ? = Surri1 +@,, Li +D, @ 3 wkhq m +D,m ? iru doo A 3 zklfk vkrzv wkdw +D, @ 3> l1h1 1 ++@,Vlqfh m +D,m m m+D, lw vxfhv wr dvvxph 3 dqg +[ , ? 41 Vxssrvh iru wkh vdnh ri frqwudglfwlrq wkhuh h{lvwv A 3 dqg Dq 5 P vxfk wkdw +Dq , A 3 zkloh +Dq , SR 1 Ohw q D @ iDq l1r1j @ R _^ Q dR q$Q vr wkdw +D, @ olp +^q$Q Dq , Q 6R R [ Dq +Dq , 5CQ RD qdQ iru doo Q dqg khqfh +D, @ 3= Rq wkh rwkhu kdqg/ +D, @ olp +^q$Q Dq , Q 6R olp lqi +Dq , A 3 q6R vkrzlqj wkdw lv qrw devroxwho| frqwlqxrxv uhodwlyh wr + 9 ,= Fruroodu| 47181 Ohw eh d srvlwlyh phdvxuh rq +[> P, dqg i 5 OR +g,1 Wkhq U i g ? iru doo D 5 P vxfk wkdw iru doo A 3 wkhuh h{lvwv A 3 vxfk wkdw D +D, ? = Surri1 Dsso| wkhruhp 4717 wr wkh vljqhg phdvxuh ] +D, @ i g iru doo D 5 P= D hu SUQh yrpr ra rhr ur RQT Ohppd 47191 Ohw dqg eh srvlwlyh qlwh phdvxuhv1 Wkhq hlwkhu B ru wkhuh h{lvwv A 3 dqg H 5 P vxfk wkdw +H , A 3 dqg rq H / l1h1 3 rq PH 1 R q vr q 3 R W R R rq Sq dqg q 3 rq Qq > l1h1 q rq Sq dqg q rq Qq = Ohw Q Qq qdR R V wkhq 3 rq Q zklfk lpsolhv wkdw @ 3 rq Q= Vhw S @ Q f @ Sq 1 Hlwkhu qdR +S , @ 3 lq zklfk fdvh B ru +S , A 3 wkhq +Sq , A 3 iru vrph q1 Qrz R q 3 rq Sq dqg +Sq , A 3= Surri1 Ohw [ @ Sq ^ Qq eh wkh Kdkq ghfrpsrvlwlrq ri q Wkhruhp 471: +Ohehvtxh0Udgrq0Qlnrg|p Wkhruhp,1 Ohw eh d 0qlwh vljqhg phdvxuh dqg eh d 0qlwh srvlwlyh phdvxuh rq +[> P,= Wkhq wkhuh h{lvwv xqltxh 0qlwh vljqhg phdvxuhv dqg vxfk wkdw 4, @ . > 5, B dqg 6, 1 Pruhryhu wkhuh h{lvwv dq h{whqghg ~ lqwhjudeoh ixqfwlrq i = [ $ U vxfk wkdw @ i zkhuh gi =@ ig dqg wklv ixqfwlrq lv xqltxh xs wr 0qxoo vhwv1 Lq vkruw g @ i g . g zlwk B = Surri1 Zh zloo uvw suryh wkh h{lvwhqfh dvvhuwlrqv ri wkh wkhruhp1 Wkh xqltxh0 qhvv dvvhuwlrqv zloo eh suryhg dw wkh hqg1 Fdvh 41 Dvvxph wkdw > duh qlwh srvlwlyh phdvxuhv1 Vhw I @ ii = [ $ ^3> 4`mi j dqg qrwlfh wkdw li i> j 5 I wkhq k @ i b j 5 I = Lqghhg/ li D 5 P> ] ] ] kg @ ig . jg i +D, @ D DiiAjj Dii #jj +D _ ii A j j, . +D _ ii jj, @ +D, iru doo D 5 P= Wkh lghd qrz lv wr wdnh i 5 I wr eh d pd{lpdo hohphqw1 Khxulvwlfdoo| zh zrxog olnh wr ghqh i +{, @ vxsik+{, = k 5 Ij= Krzhyhu/ vlqfh I lv qrw d frxqwdeoh vhw lq jhqhudo/ zh pxvw qg dqrwkhu zd| wr frqvwuxfw i= Zh gr wklv lq wkh qh{w sdudjudsk1 Fkrrvh kq 5 I vxfk wkdw kq +[ , % P =@ vxsik +[ , = k 5 Ij dv q $ 4 dqg ghqh iq @ kR b b kq 5 I = Fohduo| iq lv dq lqfuhdvlqj vhtxhqfh dqg i =@ olpq6R iq lv d phdvxudeoh ixqfwlrq1 Vlqfh iq +[ , kq +[ ,> zh pxvw kdyh e| wkh prqrwrqh frqyhujhqfh wkhruhp wkdw i +[ , @ olp iq +[ , P dqg i +D, @ olp iq +D, +D, iru doo D 5 P= q6R Wklv vkrzv wkdw i 5 I dqg P @ i +[ , +[ , ? 4 iurp zklfk lw iroorzv wkdw +ii @ 4j, @ 3 vr zh pd| dvvxph i = [ $ ^3> 4` kdv udqjh ^3> 4,= ipr xI qvr| RQU Wr qlvk wkh h{lvwhqfh surri lq Fdvh 41 zh zloo vkrz wkdw i lv vlqjxodu uhodwlyh wr = Li qrw/ e| Ohppd 4719 wkhuh h{lvwv A 3 dqg H 5 P vxfk wkdw +H , A 3 dqg rq H 1 Lq wklv fdvh zh zrxog kdyh @ i . i . RH @ i FRH dqg wkhuhiruh i . 4H 5 I = Exw wklv frqwudglfwv wkh ghqlwlrq ri P vlqfh qrz i FRH +[ , @ i +[ , . +H , @ P . +H , A P= Wklv qlvkhv wkh surri ri Fdvh 41 Fdvh 51 Dvvxph dqg c 0qlwh srvlwlyh phdvxuhv rq +[> P,= Fkrrvh duh Q QQ Qc QQ Q QQ [q > [q 5 P vxfk wkdw [ @ R [q @ R [q > +[q , ? 4 dqg c+[q , ? qdR q R eh dq hqxphudwlrq ri i[ Q _dR QQ jR = Wkhq [ @ R [ > 4= Ohw i[q jqdR q [p p>qdR qdR q +[q , ? 4 dqg +[q , ? 4= Ghqh qlwh phdvxuhv q dqg q rq +[> P, e| q +D, @ +[q _ D, dqg q +D, @ +[q _ D, iru doo D 5 P dqg qrwlfh wkdw @ R [ qdR q dqg @ R [ qdR q= E| wkh qlwh fdvh douhdg| suryhg/ wkhuh h{lvwv phdvxudeoh ixqfwlrqv iq S $ ^3> 4, =[ R dqg phdvxuhv q vxfk wkdw q @ iq . q zlwk q B q 1 Ghqh i =@ qdR iq dqg SR =@ qdR q vr wkdw R [ q @ R [ iq . R [ q @ i . = qdR qdR qdR Wr qlvk wkh surri ri wklv fdvh lw vxfhv wr vkrz wkdw B zklfk zloo iroorz e| Uhpdun 461< li zh vkrz q B iru doo q= Ehfdxvh f f f 3 @ q +[q , @ iq +[q , . q +[q , f zh vhh wkdw q +[q , @ 3= Vlqfh q B q wkhuh h{lvwv D 5 P vxfk wkdw q +D, @ +[q _ D, @ 3 dqg q +Df , @ 3= @ Ohw E @ [q _ D> wkhq +E , @ 3 zkloh f f q +E f , @ q +[q ^ Df , q +[q , . q +Df , @ 3= Wklv vkrzv wkdw olyhv rq E f dqg q olyhv rq E vr wkdw B q qlvklqj wkh surri ri fdvh 51 Fdvh 61 Wkh jhqhudo fdvh zkhuh qrz @ F lv d vljqhg qlwh phdvxuh lv d srvlwlyh 0 qlwh phdvxuh1 Dvvxph zlwkrxw orvv ri jhqhudolw| wkdw F +[ , ? 4> l1h +D, ? 4 iru doo D 5 P= E| Fdvh 5/ wkhuh h{lvw ixqfwlrqv i = [ $ ^3> 4, dqg phdvxuh vxfk wkdw @ i . zlwk B = Vlqfh 4 A F +[ , @ i. +[ , . F +[ ,> iF 5 OR +, dqg F +[ , ? 4 vr wkdw i @ iF i lv dq h{whqghg lqwhjudeoh ixqfwlrq dqg @ F lv d vljqhg phdvxuh1 Wklv qlvkhv wkh h{lvwhqfh surri vlqfh i . @ i . @ F @ i. . F dqg @ +F , B e| Uhpdun 461<1 hu SUQh yrpr ra rhr ur RQV Xqltxhqhvv1 Vxssrvh wkdw @ . dqg @ . zlwk B > > B dqg = Vlqfh B dqg B wkhuh h{lvwv D> D 5 P vxfk wkdw , @ 3 dqg Df lv ~ qxoo dqg Df lv ~ qxoo1 Ohw E @ D ^ D> wkhq +D, @ +D +E , +D, . +D, @ 3 vr wkdw E lv ~ qxoo1 Dovr E f @ Df _ Df zklfk lv dqg 0 qxoo vhw1 Wkhuhiruh li F 5 P wkhq +F _ E f , @ +F _ E f , . +F _ E f , @ +F _ E f , +4714, @ ++F _ E f , ^ +F _ E ,, @ +F , zkhuh zh kdyh xvhg dqg +F _ E f , @ 3= Vlploduo| +F , @ +F _ E f , zklfk vkrzv wkdw @ dqg khqfh @ = Ohw [ @ S ^ Q eh wkh Kdkq ghfrpsrvlwlrq ri dqg F 5 P> wkhq m m +F , @ @ @ +F _ S , +F _ Q , +F _ S _ E f , +F _ Q _ E f , m m +F _ S _ E f , . m m +F _ Q _ E f , m m +F _ E f , m m +F ,= Wklv vkrzv wkdw m m m m dqg lq sduwlfxodu wkdw lv qlwh vlqfh lv qlwh1 Vxssrvh wkdw i dqg j duh h{whqghg lqwhjudeoh ixqfwlrqv vxfk wkdw @ i @ j > l1h1 ] ] i g @ jg iru doo D 5 P= +4715, D D Fkrrvh [q % [ vxfk wkdw m m +[q , ? 4= Vlqfh ] ] mi m g @ m m +[q , @ mj m g> [q [q zh kdyh 4[q i dqg 4[q j duh lq OR +, iru doo q= Dsso|lqj Ht1 +4715, wr [q _ D zh ohduq wkdw ] ] ] ] 4[q i g @ ig @ jg @ 4[q jg D D[q D[q D iru doo D 5 P zklfk lpsolhv wkdw 4[q i @ 4[q j> ~ d1h1 Ohwwlqj q $ 4 wkhq vkrzv wkdw i @ j> ~ d1h1 Uhpdun 471;1 Vxssrvh wkdw i dqg j duh wzr srvlwlyh phdvxudeoh ixqfwlrqv rq +[> P> , vxfk wkdw ] ] i g @ jg +4716, D D iru doo D 5 P= Lw lv qrw lq jhqhudo wuxh wkdw i @ j> ~ d1h1 D wulyldo frxqwhu h{dpsoh lv wr wdnh P @ S +[ ,> +D, @ 4 iru doo qrq0hpsw| D 5 P> i @ 4[ dqg j @ 5 4[ = Wkhq Ht1 +4716, krogv |hw i 9@ j= Fruroodu| 471<1 Ohw eh d 0qlwh vljqhg phdvxuh dqg eh d 0qlwh srvlwlyh phdvxuh rq +[> P,= Li wkhq g @ i g +wkdw lv @ i , iru vrph h{whqghg lqwhjudeoh ixqfwlrq i = [ $ U zklfk lv xqltxh prgxor vhwv ri phdvxuh }hur1 Qrwdwlrq 471431 Wkh ixqfwlrq i lv fdoohg wkh Udgrq0Qlnrg|p ghulydwlyh ri g uhodwlyh wr dqg zh zloo ghqrwh wklv ixqfwlrq e| g = ipr xI qvr| RQW Surri1 E| Wkhruhp 471:/ wkhuh h{lvwv dq h{whqghg lqwhjudeoh ixqfwlrq i = [ $ U> dqg d vljqhg phdvxuh vxfk wkdw @ i . dqg B = Iurp Ohppd 4716/ @ i dqg djdlq e| Ohppd 4716/ @ 3= Dowhuqdwlyho|/ fkrrvh E 5 P vxfk wkdw +E f , @ 3 dqg E lv d ~ qxoo vhw1 Vlqfh > E f lv dovr d ~ qxoo vhw vr wkdw +D, @ +D _ E , @ i +D _ E , . +D _ E , @ i +D _ E , @ i +D, iru doo D 5 P= 481 N6V,j! jBt ijt Ghqlwlrq 48141 D frpsoh{ phdvxuh rq d phdvxudeoh vsdfh +[> P, lv d frxqw0 deo| dgglwlyh vhw ixqfwlrq = P $ F vxfk wkdw +>, @ 3= Jlyhq d frpsoh{ phdvxuh > ohw u @ Uh dqg l @ Lp vr wkdw u dqg l duh qlwh vljqhg phdvxuhv vxfk wkdw +D, @ u +D, . l l +D, iru doo D 5 P= Ghqlwlrq 48151 Ohw OR + , =@ OR + u , _ OR + l , dqg iru i 5 OR + , ghqh ] ] ] i g =@ i g u . l i g l = [ [ [ H{dpsoh 48161 Vxssrvh wkdw lv d srvlwlyh phdvxuh rq +[> P, dqg j 5 OR +,> wkhq ] +4814, +D, @ jg ;D 5 P D lv d frpsoh{ phdvxuh rq +[> P,= Pruhryhu/ OR + , @ OR +mj m g, dqg iru i 5 OR + , ] ] +4815, ig @ ijg= [ [ Wr fkhfn Ht1 +4815,/ qrwlfh wkdw g u @ Uh jg dqg g l @ Lp jg vr wkdw +xvlqj Ohppd 46149, OR + , @ OR +Uh jg, _ OR +Lp jg, @ OR +mUh j m g, _ OR +mLp j m g, @ OR +mj m g,= Li i 5 OR + ,> wkhq ] ] ] ] i g =@ i Uh jg . l i Lp jg @ i jg= [ [ [ [ Zh zloo qrz vkrz wkdw dq| frpsoh{ phdvxuh pd| eh uhsuhvhqwhg dv lq Ht1 +4814,1 Wr gr wklv ohw eh wkh qlwh srvlwlyh phdvxuh rq P ghqhg e| @ m u m . m l m= Wkhq u dqg l dqg khqfh e| wkh Udgrq0Qlnrg|p wkhruhp/ wkhuh h{lvwv uhdo ixqfwlrqv k> n 5 OR +, vxfk wkdw g u @ k g dqg g l @ n g= Wkhuhiruh li j =@ k . ln 5 OR +, wkhq g @ +k . ln,g @ jg= hu SUQh yrpr ra rhr ur RQX Ohppd 48171 Vxssrvh wkdw lv d frpsoh{ phdvxuh rq +[> P,> l lv d qlwh srvlwlyh phdvxuh rq +[> P, dqg jl 5 OR +l , vxfk wkdw g @ jl gl iru l @ 4> 5= Wkhq ] ] mjR m gR @ mjS m gS iru doo D 5 P= D D Lq sduwlfxodu/ zh pd| ghqh d srvlwlyh phdvxuh m m rq +[> P, e| ] m m +D, @ mjR m gR iru doo D 5 P= D Wkh qlwh srvlwlyh phdvxuh m m lv wkh wrwdo yduldwlrq phdvxuh ri = Surri1 Ohw @ R . S vr wkdw l = Ohw l @ gl @g 3 dqg kl @ l jl = Vlqfh ] ] ] +D, @ jl gl @ jl l g @ kl g iru doo D 5 P> D D D kR @ kS > ~d1h1 Wkhuhiruh ] ] ] ] ] ] mjR m gR @ mjR m g @ mkR m g @ mkS m g @ mjS m g @ mjS m gS = D D D D D D Uhpdun 48181 Vxssrvh wkdw lv d frpsoh{ phdvxuh rq +[> P, vxfk wkdw g @ jg dqg dv deryh g m m @ mj m g= Ohwwlqj j 9 mjm li mjm @ 3 @ 4 li mjm @ 3 zh vhh wkdw g @ jg @ mj m g @ g m m dqg mm @ 4 dqg lv xqltxho| ghqhg prgxor m m ~ qxoo vhwv1 Zh zloo ghqrwh e| g@g m m = Zlwk wklv qrwdwlrq dqg H{dpsoh 4816 zh kdyh OR + , =@ OR +m m, dqg iru i 5 OR + ,> ] ] i g @ g g m m = g m m [ i [ Sursrvlwlrq 48191 Vxssrvh D S +[ , lv dq dojheud/ P @ +D,> lv d frpsoh{ phdvxuh rq +[> P, dqg iru H 5 P ohw +q , [ m +Hm ,m = Hm 5 DH 6 Hl _ Hm @ lm Hl > q @ 4> 5> = = = Q +H , @ vxs R +q , [ R +H , @ vxs m +Hm ,m = Hm 5 PH 6 Hl _ Hm @ lm Hl > q @ 4> 5> = = = R +R , [ S +H , @ vxs m +Hm ,m = Hm 5 PH 6 Hl _ Hm @ lm Hl R ] ig = mi m 4 = T +H , @ vxs H wkhq Q @ R @ S @ T @ m m = ipr xI qvr| RQY Surri1 Ohw @ g@g m m dqg uhfdoo wkdw mm @ 4 iru m m ~ d1h1 Zh zloo vwduw e| vkrzlqj m m @ T = Li mi m 4> ] ] ] ] i g @ i g m m mi m gm m 4gm m @ m m+H , H H H H zklfk vkrzv wkdw T m m= E| wdnlqj i @ > ] ] ] i g @ gm m @ 4 gm m @ m m +H , H H H zklfk vkrzv wkdw m m T dqg khqfh m m @ T = Zh zloo qrz vkrz Q @ R @ S @ m m = Fohduo| Q R S = Vxssrvh Hm 5 PH vxfk wkdw Hl _ Hm @ lm Hl > wkhq [] [ [ m g m m m m+Hm , @ m m+^Hm , m m +H , m +Hm ,m @ Hm zklfk vkrzv wkdw S m m1 Vr wr qlvk wkh surri lw vxfhv wr vkrz wkdw m m Q = E| Wkhruhp 4319/ wkhuh h{lvwv vlpsoh ixqfwlrqv q rq [ vxfk wkdw q $ lq OR +m m, dqg hdfk q pd| eh zulwwhq lq wkh irup Q [ }n 4Dn +4816, q @ ndR zkhuh }n 5 F dqg Dn 5 D dqg Dn _ Dm @ > li n 9@ m= L fodlp wkdw zh pd| dvvxph wkdw m}n m 4 lq Ht1 +4816, iru li m}n m A 4 dqg { 5 Dn > R m+{, }n m +{, m}n m }n = Wklv lv reylrxv iurp d slfwxuh dqg irupdoo| iroorzv iurp wkh idfw wkdw k l g R S R +{, w m}n m }n @ 5 w Uh+m}n m }n +{,, 3 gw zkhq w 4= Wkhuhiruh li zh ghqh m}n mR }n li m}n m A 4 zn =@ li m}n m 4 }n dqg Q [ q @ zn 4Dn ndR wkhq m+{, q +{,m m+{, q +{,m dqg wkhuhiruh q $ lq OR +m m,1 Vr zh qrz dvvxph wkdw q lv dv lq Ht1 +4816, zlwk m}n m 41 Ehfdxvh Hn @ Dn _ H 5 DH dqg Hn _ Hm @ > li m 9@ n> ] [ [ q g @ }n +Dn _ H , m}n m m +Dn _ H ,m H [ +4817, m +Dn _ H ,m Q +H ,= Vlqfh ] q g H ] g @ +q H H ] ] , g m m mq H m g m m $ 3 dv q $ 4 hu SUQh yrpr ra rhr ur RQ` zh pd| ohw q $ 4 lq Ht1 +4817, wr frqfoxgh ] ] q g Q +H ,= g @ olp m m +H , @ q6R H H 48141 Devroxwh Frqwlqxlw| rq dq Dojheud1 Ghqlwlrq 481:1 Vxssrvh wkdw lv d frpsoh{ phdvxuh dqg lv d srvlwlyh phdvxuh rq +[> P,= Zh vd| wkh lv devroxwho| frqwlqxrxv uhodwlyh wr dqg zulwh l +D, @ 3 iru doo D 5 P vxfk wkdw +D, @ 3= Wkh iroorzlqj wkhruhpv zloo eh xvhixo lq Vhfwlrq 4: ehorz1 Ohppd 481;1 Ohw eh d frpsoh{ phdvxuh dqg eh d srvlwlyh phdvxuh rq +[> P,= Wkhq l m m = Surri1 Vlqfh m +D,m m m +D, lw lv fohdu wkdw m m lpsolhv = Iru wkh rssrvlwh gluhfwlrq/ ohw @ gg m = Li D 5 P dqg +D, @ 3 wkhq e| dvvxpswlrq m ] g m m 3 @ +E , @ E iru doo E 5 PD = Wklv vkrzv wkdw 4D @ 3 iru m m ~ d1h1 dqg khqfh ] ] m m +D, @ mm g m m @ 4D mm g m m @ 3> D [ l1h1 +D, @ 3 lpsolhv m m +D, @ 3= Dowhuqdwlyho| rqh pd| suryh wklv ohppd e| dujxlqj wkdw lpsolhv wkdw u dqg l zklfk lpsolhv wkdw m u m dqg m l m = Vlqfh m m m u m . m l m > wklv vkrzv wkdw m m = Wkhruhp 481<1 Ohw eh d frpsoh{ phdvxuh dqg eh d srvlwlyh phdvxuh rq +[> P,1 Vxssrvh wkdw D P lv dq dojheud vxfk wkdw +D, @ P dqg wkdw lv 0qlwh rq D= Wkhq l iru doo A 3 wkhuh h{lvwv d A 3 vxfk wkdw m +D,m ? iru doo D 5 D zlwk +D, ? = Surri1 E| Uhpdun 4715/ Ohppd 46146 dqg Ohppd 481;/ lw vxfhv wr suryh wkh wkhruhp lq wkh fdvh wkdw lv d srvlwlyh phdvxuh1 Wkh lpsolfdwlrq +@,, lv d frqvhtxhqfh ri Wkhruhp 47171 ++@, Ohw A 3 dqg A 3 eh vxfk wkdw +D, ? iru doo D 5 D zlwk +D, ? = Vxssrvh wkdw E 5 P zlwk +E , ? = Xvh wkh uhjxodulw| Wkhruhp 7146 wr qg D 5 D vxfk wkdw E D dqg +E , +D, ? = Zulwh D @ ^q Dq zlwk Dq 5 D1 E| uhsodflqj Dq e| ^qdR Dm li qhfhvvdu| zh pd| dvvxph wkdw Dq lv lqfuhdvlqj lq m q= Wkhq +Dq , +D, ? iru hdfk q dqg khqfh e| dvvxpswlrq +Dq , ? 1 Vlqfh E D @ ^q Dq lw iroorzv wkdw +E , +D, @ olpq6R +Dq , = Wkxv zh kdyh vkrzq wkdw +E , iru doo E 5 P vxfk wkdw +E , ? = 491 jBt ij $uujijA|B|NA C jNij6t NA Uq Lq wklv fkdswhu/ ohw E @ EUq ghqrwh wkh Eruho ~ dojheud rq Uq dqg p eh Ohehvjxh phdvxuh rq E= Ghqlwlrq 49141 D froohfwlrq ri phdvxudeoh vhwv iHu juAQ E lv vdlg wr vkulqn qlfho| wr { 5 Uq li +l, Hu Eu +{, iru doo u A 3 dqg +ll, wkhuh h{lvwv A 3 vxfk wkdw p+Hu , p+Eu +{,,= Zh zloo deeuhyldwh wklv e| zulwlqj Hu & i{j qlfho|1 RRQ ipr xI qvr| Wkh pdlq uhvxow ri wklv fkdswhu lv wkh iroorzlqj wkhruhp1 Wkhruhp 49151 Vxssrvh wkdw lv d frpsoh{ phdvxuh rq +Uq > E, > wkhq wkhuh h{lvwv j 5 OR +Uq > p, dqg d frpsoh{ phdvxuh vxfk wkdw B p> g @ jgp . g> dqg iru p 0 d1h1 {> +4914, +Hu , j +{, @ olp u8Q p+Hu , iru dq| froohfwlrq ri iHu juAQ E zklfk vkulqn qlfho| wr i{j = Surri1 Wkh h{lvwhqfh ri j dqg vxfk wkdw B p dqg g @ jgp . g lv d frqvhtxhqfh ri wkh Udgrq0Qlnrg|p wkhruhp1 Vlqfh ] 4 +Hu , +Hu , @ j +{,gp+{, . p+Hu , p+Hu , Hu p+Hu , Ht1 +4914, lv d frqvhtxhqfh ri Wkhruhp 49145 dqg Fruroodu| 49147 ehorz1 Wkh uhvw ri wklv fkdswhu zloo eh ghyrwhg wr oolqj lq wkh ghwdlov ri wkh surri ri wklv wkhruhp1 49141 D Fryhulqj Ohppd dqg Dyhudjlqj Rshudwruv1 Ohppd 4916 +Fryhulqj Ohppd,1 Ohw H eh d froohfwlrq ri rshq edoov lq Uq dqg X @ ^ESr E 1 Li f ? p+X ,/ wkhq wkhuh h{lvwv glvmrlqw edoov ER > = = = > En 5 H vxfk wkdw n S p+Em , A 6q f= mdR Surri1 Fkrrvh d frpsdfw vhw N X vxfk wkdw p+N , A f dqg wkhq ohw HR =@ iDl jQ H eh d qlwh vxefryhu ri N= Fkrrvh ER 5 HR wr eh d edoo zlwk odujhvw ldR gldphwhu lq HR = Ohw HS @ iD 5 HR = D_ER @ >j= Li HS lv qrw hpsw|/ fkrrvh ES 5 HS wr eh d edoo zlwk odujhvw gldphwhu lq HS = Vlploduo| Ohw HT @ iD 5 HS = D _ ES @ >j dqg li HT lv qrw hpsw|/ fkrrvh ET 5 HT wr eh d edoo zlwk odujhvw gldphwhu lq HT = Frqwlqxh fkrrvlqj El 5 H iru l @ 4> 5> = = = > n wklv zd| xqwlo HnFR lv hpsw|1 Li E @ E +{Q > u, Uq > ohw E @ E +{Q > 6u, Uq / wkdw lv E lv wkh edoo frqfhqwulf zlwk E zklfk kdv wkuhh wlphv wkh udglxv ri E= Zh zloo qrz vkrz N ^ndR El = Iru l hdfk m wkhuh h{lvwv d uvw l vxfk wkdw El _ Dm 9@ >1 Lq wklv fdvh gldp+Dm , gldp +El , dqg Dm El 1 Wkhuhiruh Dm ^ndR El iru doo m dqg khqfh N ^ndR Dm ^ndR El = l m l Khqfh e| vxedgglwlylw|/ f ? p+N , n [ ldR p+El , 6q n [ ldR p+El , Ghqlwlrq 49171 Ohw Y Uq eh dq rshq vhw dqg i = Y $ F eh d phdvxudeoh ixqfwlrq1 Zh vd| wkdw i lv orfdoo| lqwhjudeoh rq Y li ] mi mg{ ? 4 N iru doo frpsdfw vhwv N Y= Ohw OR +Y> p, ghqrwh wkh vsdfh ri orfdoo| lqwhjudeoh orf ixqfwlrqv i rq Y dqg zulwh OR iru OR +Uq > p,= orf orf hu SUQh yrpr ra rhr ur RRR Ghqlwlrq 49181 Iru i 5 OR > { 5 Uq dqg u A 3 ohw orf ] 4 i gp +4915, +Du i ,+{, @ mEu +{,m Eu C{D q > dqg mDm =@ p+D,= zkhuh Eu +{, @ E +{> u, U Ohppd 49191 Ohw i 5 OR > wkhq iru hdfk { 5 Uq > +3> 4, 6 u $ +Du i ,+{, lv orf frqwlqxrxv dqg iru hdfk u A 3 Uq 6 { $ +Du i , +{, lv phdvxudeoh1 Surri1 Uhfdoo wkdw mEu +{,m @ p+HR ,uq zklfk lv frqwlqxrxv lq u1 Dovr 9 olpu6u3 4Eu C{D +| , @ 4Eu3 C{D +|, li m|m 9@ uQ dqg vlqfh p +i| = m| m @ uQ j, @ 3 +|rx suryh$,/ olpu6u3 4Eu C{D +|, @ 4Eu3 C{D +|, iru p 0d1h1 | 1 Vr e| wkh grplqdwhg frqyhu0 jhqfh wkhruhp/ ] ] olp i gp @ igp u6u3 Eu C{D Eu3 C{D dqg wkhuhiruh 4 +Du i ,+{, @ p+HR ,uq ] i gp Eu C{D lv frqwlqxrxv lq u= Ohw ju +{> | , =@ 4Eu C{D +| , @ 4m{|m?u = Wkhq ju lv E E ~ phd0 vxudeoh +iru h{dpsoh zulwh lw dv d olplw ri frqwlqxrxv ixqfwlrqv, vr wkdw e| Ixelql*v wkhruhp ] ] igp @ ju +{> | ,i +|,gp+| , {$ Eu C{D E u C{D lv E ~ phdvxudeoh dqg khqfh vr lv { $ +Du i , +{,= 49151 Pd{lpdo Ixqfwlrqv1 Ghqlwlrq 491:1 Iru i 5 OR +p,> wkh Kdug| 0 Olwwohzrrg pd{lpdo ixqfwlrq Ki lv ghqhg e| +Ki ,+{, @ vxs Du mi m+{, uAQ Ohppd 4919 doorzv xv wr zulwh +Ki ,+{, @ vxs Du mi m+{, uST> uAQ dqg wkhq wr frqfoxghg wkdw Ki lv phdvxudeoh1 Wkhruhp 491; +Pd{lpdo Lqhtxdolw|,1 Li i 5 OR +p, dqg A 3> wkhq 6q ni nO4 = p +Ki A , Wklv vkrxog eh frpsduhg zlwk Fkhe|vkfhy*v lqhtxdolw| zklfk vwdwhv wkdw ni nO4 = Surri1 Ohw H iKi A j1 Iru doo { 5 H wkhuh h{lvwv u{ vxfk wkdw Du{ mi m+{, A = Khqfh H ^{SH E{ +u{ ,= E| Ohppd 4916/ li f ? p+H , p +mi m A , ipr xI qvr| RRS p+^{SH E{ +u{ ,, wkhq wkhuhS h{lvwv {R > = = = > {n 5 H dqg glvmrlqw edoov El @ E{l +u{l , iru l @ 4> 5> = = = > n vxfk wkdw mEl m A 6q f= Vlqfh U El mi mgp @ D mi m+{ , A > u{l l mEl m ] mEl m ? R mi mgp El dqg khqfh ] ] 4[ 4 4 6q f ? mi mgp mi mgp @ ni nO4 = Uq El q R ni n 4 iru doo f ? p+H , zklfk suryhv p+H , Wklv vkrzv wkdw f ? 6 O 6q R ni n Wkhruhp 491<1 Li i 5 OR wkhq olp+Du i ,+{, @ i +{, iru p ~ d1h1 { 5 Uq = orf u8Q Surri1 Zlwk rxw orvv ri jhqhudolw| zh pd| dvvxph i 5 OR +p,= Zh qrz ehjlq zlwk wkh vshfldo fdvh zkhuh i @ j 5 OR +p, lv dovr frqwlqxrxv1 Lq wklv fdvh zh qg= ] 4 mj +| , j+{,mgp+|, m+Du j,+{, j +{,m mEu +{,m Eu C{D vxs mj +| , j +{,m $ 3 dv u $ 3= |SEu C{D Lq idfw zh kdyh vkrzq wkdw +Du j ,+{, $ j +{, dv u $ 3 xqlirupo| iru { lq frpsdfw vxevhwv ri Uq = Iru jhqhudo i 5 OR +p,> mDu i +{, i +{,m mDu i +{, Du j +{,m . mDu j+{, j+{,m . mj+{, i +{,m @ mDu +i j,+{,m . mDu j +{, j +{,m . mj +{, i +{,m K +i j ,+{, . mDu j+{, j+{,m . mj+{, i +{,m dqg wkhuhiruh/ olpmDu i +{, i +{,m K +i j,+{, . mj +{, i +{,m= u8Q Vr li A 3> wkhq q r r q ^ mj i m A H olpmDu i +{, i +{,m A K +i j, A u8Q 5 5 dqg wkxv . p mj i m A p+H , p K +i j , A 5 5 4 6q ni jnO4 . ni j nO4 @5 @5 5+6q . 4,R ni jnO4 > zkhuh lq wkh vhfrqg lqhtxdolw| zh kdyh xvhg wkh Pd{lpdo lqhtxdolw| dqg Fkhe|vkfhy*v lqhtxdolw|1 Vlqfh wklv lv wuxh iru doo frqwlqxrxv j 5 F +Uq , _ OR +p, dqg wklv vhw lv ghqvh lq OR +p,> zh pd| pdnh ni j nO4 dv vpdoo dv zh sohdvh1 Wklv vkrzv wkdw R [ @ p+^R HR@q , p+HR@q , @ 3= p { = olpmDu i +{, i +{,m A 3 qdR u8Q qdR hu SUQh yrpr ra rhr ur RRT 49161 Ohehvtxh Vhw1 Ghqlwlrq 491431 Iru i 5 OR +p,> wkh Ohehvjxh vhw ri i lv orf ; < A A ] ? @ q = olp 4 Oi { 5 U mi +| , i +{,mg| @ 3 = u8Q mEu +{,m A A = > Eu C{D Wkhruhp 491441 Iru doo i 5 OR +p,> 3 @ p+Of , @ p+Uq q Oi ,= orf i Surri1 Iru z 5 F ghqh jz +{, @ mi +{, zm dqg Hz i{ = olpu8Q +Du jz , +{, 9@ jz +{,j = Wkhq e| Wkhruhp 491< p+Hz , @ 3 iru doo z 5 F dqg wkhuhiruh p+H , @ 3 zkhuh ^ Hz = H@ zSTFlT E| ghqlwlrq ri H> li { 95 H wkhq1 olp+Du mi + , u8Q zm,+{, @ mi +{, zm iru doo z 5 T . lT= Vlqfh mi + , +Du mi + , i +{,m mi + , i +{,m,+{, +Du mi + , @ +Du mi + , zm . mz i +{,m> zm, +{, . +Du mz i +{,m, +{, zm, +{, . mz i +{,m dqg khqfh iru { 95 H> olp+Du mi + , u8Q i +{,m,+{, mi +{, 5mi +{, zm . mz i +{,m zm= Vlqfh wklv lv wuxh iru doo z 5 T . lT> zh vhh wkdw olp+Du mi + , u8Q i +{,m,+{, @ 3 iru doo { 5 H> @ l1h1 H f Oi ru htxlydohqwo| Of H= Vr p+Of , p+H , @ 31 i i Wkhruhp 49145 +Ohehvtxh Glhuhqwldwlrq Wkhruhp,1 Vxssrvh i 5 OR iru doo orf { 5 Oi +vr lq sduwlfxodu iru p ~ d1h1 {, ] 4 olp mi +| , i +{,mg| @ 3 u8Q p+Hu , Hu dqg zkhq Hu & i{j qlfho|1 ] 4 olp i +|,g| @ i +{, u8Q p+Hu , Hu ipr xI qvr| RRU Surri1 Iru doo { 5 Oi > 4] i +|,g| p+Hu , Hu ] @ 4 i +{, +i +| , i +{,, g| p+Hu , ] Hu 4 mi +|, i +{,mg| p+Hu , Hu ] 4 mi +| , i +{,mg| p+Eu +{,, Eu C{D zklfk whqgv wr }hur dv u & 3 e| Wkhruhp 491441 Lq wkh vhfrqg lqhtxdolw| zh kdyh xvhg wkh idfw wkdw p+Eu +{, q Eu +{,, @ 3= Ohppd 491461 Vxssrvh 3 lv d 0qlwh phdvxuh rq E EUq vxfk wkdw B p= Wkhq iru p ~ d1h1 {> +Eu +{,, @ 3= olp u8Q p+Eu +{,, Surri1 Ohw D 5 E vxfk wkdw +D, @ 3 dqg p+Df , @ 3= E| wkh uhjxodulw| wkhruhp +H{huflvh 7154,/ iru doo A 3 wkhuh h{lvwv dq rshq vhw Y Uq vxfk wkdw D Y dqg +Y , ? = Ohw 4 +Eu +{,, In { 5 D = olp A u8Q p+Eu +{,, n CE C{DD R wkh iru { 5 In fkrrvh u{ A 3 vxfk wkdw Eu{ +{, Y dqg pCEu{ C{DD A n > l1h1 u{ p+Eu{ +{,, ? n +Eu{ +{,,= V Eu{ +{,1 Khxulvwlfdoo| li doo wkh edoov lq H zhuh Ohw H @ iEu{ +{,j{SIn dqg X {SIn glvmrlqw dqg H zhuh frxqwdeoh/ wkhq [ [ p+Eu{ +{,, ? n +Eu{ +{,, p+In , {SIn {SIn @ n+X , n +Y , n= Vlqfh A 3 lv duelwudu| wklv zrxog lpso| wkdw p+In , @ 31 Wr { wkh deryh dujxphqw/ vxssrvh wkdw f ? p+X , dqg xvh wkh fryhulqj ohppd wr qg glvmrlqw edoov ER > = = = > EQ 5 H vxfk wkdw Q q [ [ f ? 6q p+El , ? n6q +El , ldR ldR n6q +X , n6q +Y , n6q = Vlqfh f ? p+X , lv duelwudu| zh ohduq wkdw p+In , p+X , n6q dqg lq sduwlfxodu wkdw p+In , n6q = Vlqfh A 3 lv duelwudu|/ wklv vkrzv wkdw p+In , @ 3= Wklv lpsolhv/ ohwwlqj +Eu +{,, A3 > IR { 5 D = olp u8Q p+Eu +{,, wkdw p+IR , @ olpn6R p+In , @ 3= Vlqfh p+Df , @ 3/ wklv vkrzv wkdw +Eu +{,, A 3j, @ 3= p+i{ 5 Uq = olp u8Q p+Eu +{,, hu SUQh yrpr ra rhr ur RRV Fruroodu| 491471 Ohw eh d frpsoh{ ru d 0qlwh vljqhg phdvxuh vxfk wkdw B p1 Wkhq iru p ~ d1h1 {> +Hu , @3 olp u8Q p+Hu , zkhqhyhu Hu & i{j qlfho|1 Surri1 Uhfdoolqj wkh B p lpsolhv mm B p> Ohppd 49146 dqg wkh lqhtxdolwlhv/ mm+Hu , mm+Eu +{,, mm+ESu +{,, m+Hu ,m p+Hu , p+Eu +{,, p+Eu +{,, 5q p+ESu +{,, suryhv wkh uhvxow1 4:1 C j 8 AaB6jA|B, C jNij6 Nu B,W , t Qrwdwlrq 4:141 Jlyhq d ixqfwlrq I = U $ U ru I = U $ F> ohw I +{ , @ olp|7{ I +|,> I +{., @ olp|8{ I +| , dqg I + 4, @ olp{6R I +{, zkhqhyhu wkh olplwv h{lvw1 Qrwlfh wkdw li I lv d prqrwrqh ixqfwlrqv wkhq I + 4, dqg I +{ , h{lvw iru doo {= Wkhruhp 4:151 Ohw I = U $ U eh lqfuhdvlqj dqg ghqh J+{, @ I +{.,1 Wkhq +d, i{ 5 U = I +{., A I +{ ,j lv frxqwdeoh1 +e, I Q +{, dqg JQ +{, h{lvwv iru p ~ d1h1 { dqg I Q @ JQ p ~ d1h1 Surri1 +d, Vxssrvh {> | 5 + Q > Q , dqg { ? |= Wkhq I +{., I +| , vr wkdw +I +{ ,> I +{.,, _ +I +| ,> I +| .,, @ >= Wkhuhiruh/ i+I +{ ,> I +{.,,j{SU duh glvmrlqw srvvleoh hpsw| lqwhuydov lq U1 Ohw + Q > Q , eh d qlwh vhw wkhq d +I +{ ,> I +.,, ^I + Q ,> I +Q ,` {S dqg wkhuhiruh/ # $ [ d [ ^I +{., I +{ ,` @ p+I +{ ,> I +.,, @ p +I +{ ,> I +.,, {S {S {S p+^I + Q ,> I +Q ,`, @ I +Q , I + Q ,= Vlqfh wklv lv wuxh iru doo + Q > Q ,> [ +4:14, ^I +{., I +{ ,` I +Q , {SCQ>Q D I+ Q, ? 4 dqg lq sduwlfxodu/ Q =@ i{ 5 + Q > Q ,mI +{., lv frxqwdeoh dqg wkhuhiruh =@ i{ 5 UmI +{., I +{ , A 3j I +{ , A 3j @ ^RdR Q Q lv dovr frxqwdeoh1 +e, Wkh ixqfwlrq J lv lqfuhdvlqj vlqfh li { ? | wkhq I +{., I +| , I +| .,= Li } A | A { wkhq I +{., I +| ., I +} , ipr xI qvr| RRW dqg wkhuhiruh I +{., olp I +| ., I +} ,= |8{ Khqfh J+{, @ I +{., olp I +| ., olp I +} , @ I +{., @ J+{, |8{ }8{ zklfk vkrzv wkdw J lv uljkw frqwlqxrxv1 Ohw J ghqrwh wkh xqltxh phdvxuh rq E vxfk wkdw J ++d> e`, @ J+e, J+d, iru doo d ? e= E| Wkhruhp 4915/ iru p 0 d1h1 {> iru doo vhtxhqfhv iHu juAQ zklfk vkulqn qlfho| wr i{j > olp+J +Hu ,@p+Hu ,, h{lvwv dqg lv lqghshqghqw ri wkh fkrlfh u8Q ri vhtxhqfh iHu juAQ vkulqnlqj wr i{j = Vlqfh +{> { . u` & i{j dqg +{ u> {` & i{j qlfho|/ g +{> { . u`, J+{ . u, J+{, @ olp @ F J+{, olp J u8Q p++{> { . u`, u8Q u g{ dqg g ++{ u> {`, J+{, J+{ u, J+{ u, J+{, @ olp @ olp @ J+{, olp J u8Q p++{ u> {`, u8Q u8Q u u g{ h{lvw dqg duh htxdo iru p 0 d1h1 {> l1h1 JQ +{, h{lvwv iru p 0d1h1 {= Dowhuqdwlyho| zh kdyh + J C{>{FuD J+{ . u, J+{, li u A 3 u @ J C{Fu>{D li u ? 3 u mu m vr wkdw J+{ . u, u J+{, J +{ mum > { . mum`, mum J +{ mum > { . mum`, $ 3 dv u $ 3 @5 p+Emum +{,, e| Wkhruhp 49151 Iru { 5 U> ohw K +{, J+{, I +{, @ I +{., I +{, 3= Wkh surri zloo eh frpsohwhg e| vkrzlqj wkdw K Q +{, @ 3 iru p ~ d1h1 {= Ohw i{ 5 U = I +{., A I +{,j = Wkhq U lv d frxqwdeoh vhw dqg K +{, @ 3 li { 5 = Ohw eh wkh phdvxuh rq @ +U> E, ghqhg e| [ [ K +{, { @ K +{, { = @ {S U {S Vlqfh [ [ K +{, @ +I +{., I +{,, ++ Q > Q ,, @ {SCQ>Q D {SCQ>Q D [ +I +{., I +{ ,, ? 4 {SCQ>Q D e| Ht1 +4:14,/ lv qlwh rq erxqghg vhwv1 Vlqfh + f , @ 3 dqg p+ , @ 3> B p dqg vr e| Fruroodu| 49147 iru p 0 d1h1 {> K +{ . u, K +{, 5 K +{ . mum, . K +{ u mum mum, . K +{, 5 +^{ mum > { . mum`, mum hu SUQh yrpr ra rhr ur RRX dqg wkh odvw whup jrhv wr }hur dv u $ 3 ehfdxvh i^{ u> { . u`juAQ vkulqnv qlfho| wr i{j = Khqfh zh frqfoxgh iru p ~ d1h1 { wkdw K Q +{, @ 3= Ghqlwlrq 4:161 Iru 4 d ? e ? 4> d sduwlwlrq S ri +d> e` lv d qlwh vxevhw ri ^d> e` _ U vxfk wkdw id> ej _ U S= Iru { 5 Sq iej > ohw {F @ plq i| 5 S = | A {j dqg li { @ e ohw {F @ e= Sursrvlwlrq 4:171 Ohw eh d frpsoh{ phdvxuh rq E dqg vhw I +{, @ ++ 4> {`,1 Wkhq I + 4, @ 3> I lv uljkw frqwlqxrxv dqg iru 4 ? d ? e ? 4> [ [ m+{> {F `m @ vxs mI +{F , I +{,m +4:15, mm+d> e` @ vxs S {SS S {S S zkhuh vxsuhpxp lv ryhu doo sduwlwlrqv S ri +d> e`= Pruhryhu p l iru doo A 3 wkhuh h{lvwv A 3 vxfk wkdw q [ mI +el , I +dl ,m ? +4:16, ldR q S zkhqhyhu i+dl > el ,jq duh glvmrlqw rshq lqwhuydov lq +d> e` vxfk wkdw +el dl , ? = ldR ldR Surri1 Ht1 +4:15, iroorzv iurp Sursrvlwlrq 4819 dqg wkh idfw wkdw E @ +D, zkhuh D lv wkh dojheud jhqhudwhg e| +d> e` _ U zlwk d> e 5 U= Vxssrvh wkdw Ht1 +4:16, q krogv xqghu wkh vwurqjhu frqglwlrq wkdw i+dl > el `jldR duh glvmrlqw lqwhuydov lq +d> e`= q S +el dl , ? > wkhq Li i+dl > el ,jq duh glvmrlqw rshq lqwhuydov lq +d> e` vxfk wkdw ldR ldR q S +el +dl . ,, ? iru doo A 3> i+dl . > el `jq duh glvmrlqw lqwhuydov lq +d> e` dqg ldR ldR vr wkdw e| dvvxpswlrq/ q [ mI +el , I +dl . ,m ? = ldR Vlqfh A 3 lv duelwudu| lq wklv htxdwlrq dqg I lv uljkw frqwlqxrxv/ zh frqfoxgh wkdw q [ mI +el , I +dl ,m ldR q S q +el dl , ? = zkhqhyhu i+dl > el ,jldR duh glvmrlqw rshq lqwhuydov lq +d> e` vxfk wkdw ldR Vr lw vxfhv wr suryh Ht1 +4:16, xqghu wkh vwurqjhu frqglwlrq wkdw i+dl > el `jq duh ldR glvmrlqw lqwhuydov lq +d> e`= Exw wklv odvw dvvhuwlrq iroorzv gluhfwo| iurp Wkhruhp 481< dqg wkh idfw wkdw E @ +D,= Ghqlwlrq 4:181 Jlyhq d ixqfwlrq I = U $ F dqg d 5 U ghqh [ WI +d, vxs mI +{F , I +{,m S {SS zkhuh wkh vxsuhpxp lv wdnhq ryhu doo sduwlwlrqv ri + 4> d`= Pruh jhqhudoo| li 4 d ? e> ohw [ [ WI +d> e` @ vxs m+{> {F `m @ vxs mI +{F , I +{,m S {S S S {S S zkhuh vxsuhpxp lv ryhu doo sduwlwlrqv S ri +d> e`= D ixqfwlrq I = U $ F lv vdlg wr eh ri erxqghg yduldwlrq li WI +4, ? 4 dqg zh zulwh I 5 EY= Pruh jhqhudoo| zh zloo ohw EY ++d> e`, ghqrwh wkh ixqfwlrq I = ^d> e` _ U $ F vxfk wkdw WI +d> e` ? 4= RRY ipr xI qvr| Ghqlwlrq 4:191 D ixqfwlrq I = U $ F lv devroxwho| frqwlqxrxv li iru doo A 3 wkhuh h{lvwv A 3 vxfk wkdw q [ mI +el , I +dl ,m ? ldR q S +el dl , ? = zkhqhyhu i+dl > el ,jq duh glvmrlqw rshq lqwhuydov lq +d> e` vxfk wkdw ldR ldR Ohppd 4:1:1 Ohw I = U $ F dqg +4:17, 5, Ohwwlqj d @ +4:18, 4 d ? e ? f> wkhq 4, WI +d> f` @ WI +d> e` . WI +e> f`= 4 lq wklv h{suhvvlrq lpsolhv WI +f, @ WI +e, . WI +e> f` dqg lq sduwlfxodu WI lv prqrwrqh lqfuhdvlqj1 6, Li WI +e, ? 4 iru vrph e 5 U wkhq WI + 4, @ 3 dqg +4:19, WI +d., WI +d, olp vxs mI +|, I +d,m |8d iru doo d 5 + 4> e,= Lq sduwlfxodu WI lv uljkw frqwlqxrxv li I lv uljkw frqwlqxrxv1 Surri1 E| wkh wuldqjoh lqhtxdolw|/ li S dqg SQ duh sduwlwlrq ri +d> f` vxfk wkdw S SQ > wkhq [ [ mI +{F , I +{,m mI +{F , I +{,m= {SS {SSQ Vr li S lv d sduwlwlrq ri +d> f`> wkhq S SQ =@ S^ iej lpsolhv [ [ mI +{F , I +{,m mI +{F , I +{,m {SS {SSQ [ [ @ mI +{F , I +{,m . mI +{F , I +{,m Q d>e Q e>f {SS {S S WI +d> e` . WI +e> f`= Wkxv zh vhh wkdw WI +d> f` WI +d> e` . WI +e> f`= Vlploduo| li SR lv d sduwlwlrq ri +d> e` dqg SS lv d sduwlwlrq ri +e> f`> wkhq S @ SR ^ SS lv d sduwlwlrq ri +d> f` dqg [ [ [ mI +{F , I +{,m . mI +{F , I +{,m @ mI +{F , I +{,m {SS4 {SS5 {SS WI +d> f`= Iurp wklv zh frqfoxgh WI +d> e` . WI +e> f` WI +d> f` zklfk qlvkhv wkh surri ri Htv1 +4:17, dqg +4:18,1 Vxssrvh wkdw WI +e, ? 4 dqg jlyhq A 3 ohw S eh d sduwlwlrq ri + 4> e` vxfk wkdw [ mI +{F , I +{,m . = WI +e, {SS Ohw {Q @ plq S wkhq WI +e, @ WI +{Q , . WI +{Q > e` dqg e| wkh suhylrxv htxdwlrq [ WI +{Q , . WI +{Q > e` mI +{F , I +{,m . WI +{Q > e` . {SS zklfk vkrzv wkdw WI +{Q , = Vlqfh WI lv prqrwrqh lqfuhdvlqj dqg A 3> zh frqfoxgh wkdw WI + 4, @ 3= hu SUQh yrpr ra rhr ur RR` Ilqdoo| ohw d 5 + 4> e, dqg jlyhq A 3 ohw S eh d sduwlwlrq ri +d> e` vxfk wkdw [ +4:1:, WI +e, WI +d, @ WI +d> e` mI +{F , I +{,m . = {SS Ohw | 5 +d> dF ,> wkhq [ [ mI +{F , I +{,m . mI +{F , I +{,m . {S S {SSi|j [ mI +{F , I +{,m . @ mI +| , I +d,m . {SSqi|j +4:1;, mI +| , I +d,m . WI +|> e` . = Frpelqlqj Htv1 +4:1:, dqg +4:1;, vkrzv WI +| , WI +d, . WI +|> e` @ WI +e, mI +| , WI +d, I +d,m . WI +|> e` . = Vlqfh | 5 +d> dF , lv duelwudu| zh frqfoxgh wkdw WI +d., WI +d, @ olp vxs WI +|, WI +d, olp vxs mI +| , |8d |8d zklfk suryhv Ht1 +4:19, vlqfh A 3 lv duelwudu|1 I +d,m . Ohppd 4:1;1 4, Prqrwrqh ixqfwlrqv I = U $ U duh lq EY +d> e` iru doo 4 ? d ? e ? 4= 5, Olqhdu frpelqdwlrqv ri ixqfwlrqv lq EY duh lq EY> l1h1 EY lv d yhfwru vsdfh1 6, Li I = U $ F lv devroxwho| frqwlqxrxv wkhq I lv frqwlqxrxv dqg I 5 EY +d> e` iru doo 4 ? d ? e ? 4= 7, Li I = U $ U lv d glhuhqwldeoh ixqfwlrq vxfk wkdw vxs{SU mI Q +{,m @ P ? 4> wkhq I lv devroxwho| frqwlqxrxv dqg WI +d> e` P +e d, iru doo 4 ? d ? e ? 4= 8, Ohw i 5 OR ++d> e`> p, dqg vhw ] +4:1<, I +{, @ i gp Cd>{ iru { 5 +d> e`= Wkhq I lv devroxwho| frqwlqxrxv1 Surri1 4, Li I lv prqrwrqh lqfuhdvlqj dqg S lv d sduwlwlrq ri +d> e` wkhq [ [ mI +{F , I +{,m @ +I +{F , I +{,, @ I +e, I +d, {SS {SS vr wkdw WI +d> e` @ I +e, I +d,= Dovr qrwh wkdw I 5 EY l I +4, I + 4, ? 4= Lwhp 5, iroorzv iurp wkh wuldqjoh lqhtxdolw|1 6, Vlqfh I lv devroxwho| frqwlqxrxv/ wkhuh h{lvwv A 3 vxfk wkdw zkhqhyhu d ? e ? d . dqg S lv d sduwlwlrq ri +d> e`> [ mI +{F , I +{,m 4= {S S Wklv vkrzv wkdw WI +d> e` 4 iru doo d ? e zlwk e d ? = Wkxv xvlqj Ht1 +4:17,/ lw iroorzv wkdw WI +d> e` Q ? 4 li e d ? Q iru dq Q 5 Q1 q 7, Vxssrvh wkdw i+dl > el ,jldR +d> e` duh glvmrlqw lqwhuydov/ wkhq e| wkh phdq ydoxh wkhruhp/ q q [ [ mI +el , I +dl ,m mI Q +fl ,m +el dl , Pp +^q +dl > el ,, ldR ldR ldR q [ P +el dl , P +e d, ldR ipr xI qvr| RSQ irup zklfk lw fohduo| iroorzv wkdw I lv devroxwho| frqwlqxrxv1 Pruhryhu zh pd| frqfoxgh wkdw WI +d> e` P +e d,= 8, Ohw eh wkh srvlwlyh phdvxuh g @ mi m gp rq +d> e`= Ohw i+dl > el ,jq +d> e` ldR eh glvmrlqw lqwhuydov dv deryh/ wkhq q q [ [ ] mI +el , I +dl ,m @ i gp Cdl >el ldR ldR q] [ mi m gp ldR Cdl >el ] @ +4:143, mi m gp @ +^q +dl > el `,= ldR q@4 Cdl >el l Vlqfh lv devroxwho| frqwlqxrxv uhodwlyh wr p iru doo A 3 wkhuh h{lvw A 3 vxfk wkdw +D, ? li p+D, ? = Wdnlqj D @ ^q +dl > el ` lq Ht1 +4:143, vkrzv wkdw I ldR lv devroxwho| frqwlqxrxv1 Lw lv dovr hdv| wr vhh iurp Ht1 +4:143, wkdw WI +d> e` U Cd>e mi m gp= Wkhruhp 4:1<1 Ohw I = U $ F eh d ixqfwlrq/ wkhq +4, I 5 EY l Uh I 5 EY dqg Lp I 5 EY= +5, Li I = U $ U lv lq EY wkhq wkh ixqfwlrqv I =@ +WI I , @5 duh erxqghg dqg lqfuhdvlqj ixqfwlrqv1 +6, I = U $ U lv lq EY l I @ IF I zkhuh I duh erxqghg lqfuhdvlqj ixqfwlrqv1 +7, Li I 5 EY wkhq I +{ , h{lvw iru doo { 5 U= Ohw J+{, =@ I +{.,= +8, I 5 EY wkhq i{ = olp|6{ I +| , 9@ I +{,j lv d frxqwdeoh vhw dqg lq sduwlfxodu J+{, @ I +{., iru doo exw d frxqwdeoh qxpehu ri { 5 U= +9, Li I 5 EY> wkhq iru p ~ d1h1 {> I Q +{, dqg JQ +{, h{lvw dqg I Q +{, @ JQ +{,= Surri1 Lwhp 41 lv d frqvhtxhqfh ri wkh lqhtxdolwlhv mI +e, I +d,m mUh I +e, Uh I +d,m . mLp I +e, Lp I +d,m 5 mI +e, I +d,m = 51 E| Ohppd 4:1:/ iru doo d ? e> +4:144, WI +e, dqg wkhuhiruh zklfk vkrzv wkdw I d 3> WI +d, @ WI +d> e` mI +e, I +d,m WI +e, I +e, WI +d, I +d, duh lqfuhdvlqj1 Pruhryhu iurp Ht1 +4:144,/ iru e 3 dqg mI +e,m mI +e, I +3,m . mI +3,m WI +3> e` . mI +3,m WI +3> 4, . mI +3,m dqg vlploduo| mI +d,m mI +3,m . WI + 4> 3, zklfk vkrzv wkdw I lv erxqghg e| mI +3,m . WI +4,= Wkhuhiruh I lv erxqghg dv zhoo1 61 E| Ohppd 4:1; li I @ IF I > wkhq WI +d> e` WI. +d> e` . WI +d> e` @ mIF +e, IF +d,m . mI +e, I +d,m zklfk lv erxqghg vkrzlqj wkdw I 5 EY= Frqyhuvho| li I lv erxqghg yduldwlrq/ wkhq I @ IF I zkhuh I duh ghqhg dv lq Lwhp 51 hu SUQh yrpr ra rhr ur RSR Lwhpv 71 ~ 91 iroorz iurp Lwhpv 41 ~ 61 dqg Wkhruhp 4:151 Wkhruhp 4:1431 Vxssrvh wkdw I = U $ F lv lq EY> wkhq +4:145, mWI +{., WI +{,m mI +{., I +{,m iru doo { 5 U= Li zh ixuwkhu dvvxph wkdw I lv uljkw frqwlqxrxv wkhq wkhuh h{lvwv d xqltxh phdvxuh rq E @ EU 1 vxfk wkdw +4:146, ++ 4> {`, @ I +{, I + 4, iru doo { 5 U= Surri1 Vlqfh I 5 EY> I +{., h{lvwv iru doo { 5 U dqg khqfh Ht1 +4:145, lv d frqvhtxhqfh ri Ht1 +4:19,1 Qrz dvvxph wkdw I lv uljkw frqwlqxrxv1 Lq wklv fdvh Ht1 +4:145, vkrzv wkdw WI +{, lv dovr uljkw frqwlqxrxv1 E| frqvlghulqj wkh uhdo dqg lpdjlqdu| sduwv ri I vhsdudwho| lw vxfhv wr suryh wkhuh h{lvwv d xqltxh qlwh vljqhg phdvxuh vdwlvi|lqj Ht1 +4:146, lq wkh fdvh wkdw I lv uhdo ydoxhg1 Qrz ohw I @ +WI I , @5> wkhq I duh lqfuhdvlqj uljkw frqwlqxrxv erxqghg ixqfwlrqv1 Khqfh wkhuh h{lvwv xqltxh phdvxuh rq E vxfk wkdw ++ 4> {`, @ I +{, I + 4, ;{ 5 U= Wkh qlwh vljqhg phdvxuh F vdwlvhv Ht1 +4:146,1 Vr lw rqo| uhpdlqv wr suryh wkdw lv xqltxh1 Vxssrvh wkdw lv dqrwkhu vxfk phdvxuh vxfk wkdw +4:146, krogv zlwk uhsodfhg e| = Wkhq iru +d> e`> [ mI +{F , I +{,m @ mm +d> e` mm +d> e` @ vxs S {S S zkhuh wkh vxsuhpxp lv ryhu doo sduwlwlrq ri +d> e`= Wklv vkrzv wkdw mm @ mm rq D E ~ wkh dojheud jhqhudwhg e| kdoi rshq lqwhuydov dqg khqfh mm @ mm = Lw qrz iroorzv wkdw mm . dqg mm . duh qlwh srvlwlyh phdvxuh rq E vxfk wkdw +mm . , ++d> e`, @ mm ++d> e`, . +I +e, @ mm ++d> e`, . +I +e, @ +mm . , ++d> e`, I +d,, I +d,, iurp zklfk zh lqihu wkdw mm . @ mm . @ mm . rq E = Wkxv @ = Dowhuqdwlyho|/ rqh pd| suryh wkh xqltxhqhvv e| vkrzlqj wkdw F =@ iD 5 E = +D, @ +D,j lv d prqrwrqh fodvv zklfk frqwdlqv D= h Ghqlwlrq 4:1441 D ixqfwlrq I = U $ F lv vdlg wr eh ri qrupdol}hg erxqghg yduldwlrq li I 5 EY> I lv uljkw frqwlqxrxv dqg I + 4, @ 3= Zh zloo deeuhyldwh wklv e| vd|lqj I 5 Q EY= +Wkh frqglwlrq= I + 4, @ 3 lv qrw hvvhqwldo dqg sod|v qr uroh lq wkh glvfxvvlrq ehorz1, Wkhruhp 4:1451 Vxssrvh wkdw I 5 QEY dqg I lv wkh phdvxuh ghqhg e| Ht1 +4:146,/ wkhq gI @ I Q gp . g +4:147, zkhuh B p dqg lq sduwlfxodu iru +4:148, I +e, 4 ? d ? e ? 4> ]e I +d, @ I Q gp . ++d> e`,= d ipr xI qvr| RSS Surri1 E| Wkhruhp 4915/ wkhuh h{lvwv i 5 OR +p, dqg d frpsoh{ phdvxuh vxfk wkdw iru p 0d1h1 {> +Hu , > i +{, @ olp u8Q p+Hu , +4:149, iru dq| froohfwlrq ri iHu juAQ E zklfk vkulqn qlfho| wr i{j > B p dqg gI @ igp . g= Iurp Ht1 +4:149, lw iroorzv wkdw I +{ . k, I +{, ++{> { . k`, @ olp I @ i +{, dqg k8Q k k I +{ k, I +{, ++{ k> {`, @ olp I @ i +{, olp k8Q k8Q k k g g iru p ~ d1h1 {> l1h1 g{. I +{, @ g{ I +{, @ i +{, iru p ~d1h1 {= Wklv lpsolhv wkdw I Q +{, @ i +{, iru p ~ d1h1 {= lv p ~ d1h1 glhuhqwldeoh dqg I olp k8Q Fruroodu| 4:1461 Ohw I = U $ F eh lq Q EY> wkhq +4, I B p l I Q @ 3 p d1h1 +5, I p l @ 3 l ] +4:14:, I ++d> e`, @ I Q +{,gp+{, iru doo d ? e= Cd>e Surri1 41 Li I Q @ 3 p d1h1/ wkhq e| Ht1 +4:147,/ I @ B p1 Li I B p> wkhq e| Ht1 +4:147,/ I Q gp @ gI g B gp dqg e| Uhpdun 461< I Q gp @ 3> l1h1 I Q @ 3 p 0d1h1 51 Li I p> wkhq g @ gI I Q gp gp zklfk lpsolhv e| Ohppd 4716 wkdw @ 3= Wkhuhiruh Ht1 +4:148, ehfrphv +4:14:,1 Qrz ohw ] +D, =@ I Q +{,gp+{, iru doo D 5 E= D U Uhfdoo e| wkh Udgrq 0 Qlnrg|p wkhruhp wkdw U mI Q +{,m gp+{, ? 4 vr wkdw lv d frpsoh{ phdvxuh rq E= Vr li Ht1 +4:14:, krogv/ wkhq @ I rq wkh dojheud jhqhudwhg e| kdoi rshq lqwhuydov1 Wkhuhiruh @ I dv lq wkh xqltxhqhvv sduw ri wkh surri ri Wkhruhp 4:1431 Wkhuhiruh gI @ I Q gp dqg khqfh @ 3= Wkhruhp 4:1471 Vxssrvh wkdw I = ^d> e` $ F lv d phdvxudeoh ixqfwlrq1 Wkhq wkh iroorzlqj duh htxlydohqw= +4, I lv devroxwho| frqwlqxrxv rq ^d> e`= +5, Wkhuh h{lvwv i 5 OR +^d> e`,> gp, vxfk wkdw ]{ i gp ;{ 5 ^d> e` +4:14;, I +{, I +d, @ d +6, I Q h{lvwv d1h1/ I Q 5 OR +^d> e`> gp, dqg ]{ I Q gp ;{ 5 ^d> e`= +4:14<, I +{, I +d, @ d hu SUQh yrpr ra rhr ur RST Surri1 Lq rughu wr dsso| wkh suhylrxv uhvxowv/ h{whqg I wr U e| I +{, @ I +e, li { e dqg I +{, @ I +d, li { d= 41 @, 61 Li I lv devroxwho| frqwlqxrxv wkhq I lv frqwlqxrxv rq ^d> e` dqg I I +d, @ I I + 4, 5 Q EY e| Ohppd 4:1;1 E| Sursrvlwlrq 4:17/ I p dqg khqfh Lwhp 61 lv qrz d frqvhtxhqfh ri Lwhp 51 ri Fruroodu| 4:1461 Wkh dvvhuwlrq 61 @, 51 lv wulyldo1 51 @, 41 Li 51 krogv wkhq I lv devroxwho| frqwlqxrxv rq ^d> e` e| Ohppd 4:1;1 4;1 VVjAa!H N6VBW|Ajtt NA 6j|iW tVBWjt Ghqlwlrq 4;141 Ohw +[> , eh d wrsrorjlfdo vsdfh dqg D [= Dq rshq fryhu ri D lv d froohfwlrq ri rshq vhwv iY jSL vxfk wkdw D ^SL Y = Ghqlwlrq 4;151 D vhw N [ lv frpsdfw li hyhu| rshq fryhu ri N kdv d 0 qlwh vxefryhu/ Wkdw lv wr vd| li N ^SL Y zlwk wkh Y 5 / wkhq wkhuh h{lvwv il 5 L = l @ 4> 5> = = = > qj vxfk wkdw _rshq fryhuv% iY jSL ri N wkhuh h{lvwv qlwh vxefryhu N Y4 > ^Y5 ^ = = = Yq = Zh zloo zulwh N CC [ wr lqglfdwh wkdw N lv d frpsdfw vxevhw ri [= H{dpsoh 4;161 Dq| qlwh vxevhw N [ lv frpsdfw1 Sursrvlwlrq 4;171 Ohw +[> , eh d phwulf vsdfh1 Li N [ lv frpsdfw wkhq N lv forvhg1 Surri1 Zh pxvw vkrz wkdw N f lv rshq zklfk lv htxlydohqw wr vkrzlqj iru doo { 95 N wkhuh lv d A 3 vxfk wkdw E +{> , _ N @ >= Wr frqvwuxfw > fkrrvh iru doo n 5 N / un A 3 vxfk wkdw g+{> n, A un @5= Wkhq Yn =@ E +n> un , iru n 5 N irupv dq rshq fryhu ri N> khqfh wkhuh h{lvwv nR > = = = > nq 5 N vxfk wkdw q ^ E +nq > unq ,= N ldR Ohw @ plqiun4 > = = = > unq , A 3> wkhq E +{> u, _ E +nl > unl , @ > iru doo l1 Wkxv q ^ E +{> u, + E +nl > unl ,,f N f ldR zklfk vkrzv wkdw N f lv rshq khqfh wkdw N lv forvhg1 Ghqlwlrq 4;181 Ohw +[> , eh d phwulf vsdfh1 D vxevhw N [ lv +4, frpsohwh li +N> mN , lv d frpsohwh phwulf vsdfh1 +5, wrwdoo| erxqghg li iru doo A 3 wkhuh h{lvwv d qlwh vxevhw I N vxfk wkdw N ^nSI E +n> ,= +6, vhtxhqwldoo| frpsdfw li hyhu| vhtxhqfh i{q jR N kdv d frqyhujhqw qdR vxevhtxhqfh lq N= Wkdw lv wkhuh vkrxog eh |n @ {qn dqg { 5 N vxfk wkdw olpn6R |n @ {= Uhpdun 4;191 Li N [ lv d wrwdoo| erxqghg vxevhw dqg I N> wkhq I lv dovr wrwdoo| erxqghg1 Lqghhg/ dq| fryhu N e| ~ edoov lv dovr d fryhu ri I= Wkhruhp 4;1:1 Ohw +[> , eh d phwulf vsdfh dqg N [ 1 Wkh iroorzlqj duh htxlydohqw +d, N lv frpsdfw1 +e, N lv vhtxhqwldoo| frpsdfw1 ipr xI qvr| RSU +f, N lv frpsohwh dqg wrwdoo| erxqghg1 Surri1 E| uhsodflqj [ e| N li qhfhvvdu| zh pd| dvvxph wkdw N @ [= +d @, e, Zh zloo vkrz wkdw qrw e @, qrw d= Vxssrvh wkdw [ lv qrw vhtxhqwldoo| frpsdfw/ wkhq wkhuh h{lvwv i{q jR [ zlwk qr frqyhujhqw vxevhtxhqfh1W Wklv qdR lpsolhv iru doo { 5 [> wkhuh h{lvwv { A 3 vxfk wkdw iq = {q 5 E +{> { ,j lv qlwh1 Qrz iE +{> { ,j{S[ lv dq rshq fryhu ri [ zklfk zh fodlp kdv qrw qlwh vxefryhu1 Iru li wkhuh zdv d qlwh vhw I [ vxfk wkdw [ @ ^{SI E +{> { ,> wkhq dw ohdvw iru rqh { 5 I zh zrxog kdyh wr kdyh wkdw iq = {q 5 E +{> { ,j lv lqqlwh/ frqwudglfwlqj wkh ghqlwlrq ri { = +e @, f, Ohw i{q jR [ eh d Fdxfk| vhtxhqfh/ wkhq dvvxplqj e wkhuh lv dq qdR { 5 [ dqg d vxevhtxhqfh i|q jR ri i{q jR vxfk wkdw olpq6R |q @ {= E| wkh qdR qdR wuldqjoh lqhtxdolw| zh kdyh g+{> {q , g+{> |p , . g+|p > {q , dqg wkhuhiruh olp vxs g+{> {q , olp vxs g+{> |p , . olp vxs g+|p > {q , @ 3 . 3> q6R p6R p>q6R vlqfh olpq6R |q @ { dqg i{q jR lv Fdxfk|1 Wklv vkrzv wkdw N lv frpsohwh1 qdR Li N zhuh qrw wrwdoo| erxqghg wkhq wkhuh h{lvwv A 3 vxfk wkdw wkh rshq fryhu iE +{> , = { 5 [ j kdv qr qlwh vxefryhu1 Zh qrz fkrrvh d vhtxhqfh i{q jR [ qdR lqgxfwlyho| dv iroorzv1 Ohw {R 5 [ eh fkrvhq duelwudulo|1 Dvvxph wkdw {R > {S > = = = > {n kdyh douhdg| ehhq fkrvhq1 Vlqfh [ 9@ ^ndR E +{l > , zh pd| fkrrvh {nFR 5 [ q lR ^ndR E +{l > ,= Wklv ghqhv d vhtxhqfh i{q jqdR zlwk wkh surshuw| wkdw g+{n > {l , l iru doo l dqg n= Lw lv fohdu wkdw wklv vhtxhqfh lv qrw frqyhujhqw/ lq idfw lw lv qrw hyhq Fdxfk|1 Khqfh li N lv qrw wrwdoo| erxqghg wkhq N lv qrw vhtxhqwldoo| frpsdfw1 +f @, d, Ohw Y =@ iY jSD eh dq rshq fryhu ri [= Vlqfh [ lv wrwdoo| erxqghg/ wkhuh h{lvw d qlwh vhw IR [ vxfk wkdw [ @ ^{SI4 E +{> 4, @ ^{SI4 E +{> 4, zkhuh E +{> , @ i| 5 [ = g+{> | , j zklfk |rx vkrxog fkhfn lv d forvhg vhw1 Iru vdnh ri frqwudglfwlrq/ vxssrvh wkdw [ lv qrw fryhuhg e| d qlwh vxevhw ri Y = Lq wklv fdvh/ lw iroorzv wkdw wkhuh lv dqg | 5 IR vxfk wkdw [R =@ E +|R > 4, lv qrw fryhuhg e| d qlwh vxevhw ri Y = Zrunlqj lq wkh vdph zlwk [ uhsodfhg e| [R zh pd| qg d forvhg vxevhw [S @ E +|S > 4@5, _ [R zklfk lv qrw fryhuhg e| d qlwh vxevhw ri Y = Zrunlqj lqgxfwlyho|/ zh pd| frqvwuxfw forvhg vxevhwv i[l j ri [ vxfk wkdw [R [S [T = = = [q = = = > +4;14, gldp+[l , =@ vxsig+{> |, = {> | 5 [l j 5@l> dqg qr [l pd| eh fryhuhg e| d qlwh vxevhw ri Y = Ohw i{q jR [ eh d vhtxhqfh qdR R vxfk wkdw {q 5 [q iru doo q= Lw lv fohdu e| Ht1 +4;14, wkdw i{q jqdR lv Fdxfk| dqg wkhuhiruh frqyhujhqw/ e| dvvxpswlrq f> wr vrph { 5 _[l = Zh kdyh xvh wkh idfw wkdw wkh [l *v duh forvhg khuh1 Ehfdxvh Y lv d fryhu ri [> wkhuh lv d Y 5 Y vxfk wkdw { 5 Y= Vlqfh Y frqwdlqv E +{> , iru vrph A 3> lw lv hdvlo| vhhq wkdw e| Ht1 +4;14, Wic dB|ihcd|i dh}Ric|!( A dttc} |v d twtiici vu F q9 t7c '" u ciittdhA "i c' RdA dttRi |d| 7c 9 7T u T 9 c# %vvti 4 X f t |d| i7 4 ( u t7c '" 9 t7 ' dc c' |ic vvti c|iBA 4 X f tv |d| i7 4 ( u i7& 4& ( 9 u 9 &# | t cv" idtBA i3i |d| F t Bvti dc |d| y 9 tF 'ti7 4 ( q 9 P # # # ' t dc vic vih vu f "| cv c|i twvih! hu SUQh yrpr ra rhr ur RSV dqg wkh wuldqjoh lqhtxdolw| wkdw [q Y iru doo odujh hqrxjk q= Exw wklv frqwudglfwv wkh dvvhuwlrq wkdw qr [l frxog eh fryhuhg e| d qlwh vxevhw ri Y = Fruroodu| 4;1;1 Frpsdfw vxevhwv ri Uq duh wkh forvhg dqg erxqghg vhwv1 Surri1 Li N lv forvhg dqg erxqghg wkhq N lv frpsohwh +ehlqj wkh forvhg vxevhw ri d frpsohwh vsdfh, dqg N lv frqwdlqhg lq ^ P > P `q iru vrph srvlwlyh lqwhjhu P= Iru A 3> ohw q q q @ ] _ ^ P > P ` =@ i{ = { 5 ] dqg m{l m P iru l @ 4> 5> = = = > qj= Zh zloo vkrzv wkdw e| fkrrvlqj A 3 vxflhqwo| vpdoo/ wkdw +4;15, N ^ P > P `q ^{S E +{> , zklfk vkrzv wkdw N lv wrwdoo| erxqghg1 Khqfh e| Wkhruhp 4;1:/ N zloo eh frpsdfw1 Vxssrvh wkdw | 5 ^ P > P `q > wkhq wkhuh h{lvwv { 5 vxfk wkdw m|l {l m iru l @ 4> 5> = = = > q= Khqfh q S +{> | , @ [ +| { ,S q S g l l ldR s s zklfk vkrzv wkdw g+{> |, q= Khqfh li fkrrvh ? @ q zh kdyh vkrzv wkdw g+{> | , ? > l1h1 Ht1 +4;15, krogv1 4<1 VVjAa!H $+A"A?t C jNij6 Ghqlwlrq 4<14 +Odpegd Fodvv,1 Dv froohfwlrq G ri vxevhwv ri [ lv d ~fodvv li G vdwlvhv wkh iroorzlqj surshuwlhv= +4, [ 5 G +5, Li D> E 5 G dqg D _ E @ !/ wkhq D ^ E 5 G= +Forvhg xqghu glvmrlqw xqlrqv1, +6, Li D> E 5 G dqg D E / wkhq D q E 5 G1 +Forvhg xqghu surshu glhuhqfhv1, +7, Li Dq 5 G dqg Dq % D/ wkhq D 5 G= +Forvhg xqghu frxqwdeoh lqfuhdvlqj xqlrqv1, Ghqlwlrq 4<15 + ~ fodvv,1 D idplo| ri vhwv F S +[ , lv fdoohg d ~ fodvv li lw lv forvhg xqghu qlwh lqwhuvhfwlrqv1 Wkhruhp 4<16 + ~ Wkhruhp,1 Li G lv d fodvv zklfk frqwdlqv d idplo| F ri vxevhwv ri [ zklfk lv forvhg xqghu lqwhuvhfwlrqv +l1h1 d 0fodvv, wkhq +F , G= Surri1 +Wdnhq iurp Eloolqjvoh|/ ss1 660671, Ohw I eh wkh lqwhuvhfwlrq ri doo ~fodvvhv zklfk frqwdlqv F = Wkhq I lv d ~fodvv dqg F I G= Ohw IR =@ iD [ = D _ E 5 I ; E 5 Fj= Wkhq IR lv dovr d ~fodvv +|rx fkhfn, dqg vlqfh F IR > zh nqrz wkdw I IR = +Uhfdoo wkdw I lv wkh vpdoohvw ~fodvv zklfk frqwdlqv F =, Iurp wklv zh frqfoxgh wkdw li D 5 I dqg E 5 F wkhq D _ E 5 I = Ohw IS =@ iE [ = D _ E 5 I ; D 5 Ij= Wkhq IS lv d ~fodvv zklfk/ e| wkh deryh sdudjudsk/ frqwdlqv F = Dv deryh wklv lpsolhv wkdw I IS > l1h1 zh kdyh vkrzq wkdw I lv forvhg xqghu lqwhuvhfwlrqv1 Li D> E 5 I > wkhq D _ E 5 I > D q D _ E 5 I > dqg D ^ E @ D ^ +D q D _ E , 5 I = Wkhuhiruh/ I lv forvhg xqghu qlwh xqlrqv1 Vlqfh I lv d ~fodvv +lq sduwlfxodu forvhg xqghu frxqwdeoh lqfuhdvlqj xqlrqv, dqg forvhg xqghu qlwh xqlrqv/ I lv forvhg xqghu ipr xI qvr| RSW duelwudu| frxqwdeoh xqlrqv1 Wkxv/ I lv d 0dojheud dqg vlqfh F I > +F , I G= Dv dq dssolfdwlrq/ ohw xv jlyh dqrwkhu surri ri Wkhruhp 71: lq wkh fdvh wkdw +[ , @ +[ , ? 4= Surri1 +Surri ri Wkhruhp 71:,Ohw G @ iD 5 P = +D, @ +D,j= Wkh uhdghu pd| hdvlo| fkhfn wkdw G lv d fodvv1 Iru h{dpsoh iru lwhp 71 zh kdyh li Dq 5 G wkhq +Dq , @ +Dq ,= Li Dq % D> wkhq e| sdvvlqj wr wkh olplw lq wkh suhylrxv htxdwlrq zh ohduq wkdw +D, @ +D,> l1h1 wkdw D 5 G= Lq fkhfnlqj lwhp 61 zh pdnh xvh ri wkh idfw wkdw +[ , @ +[ , ? 4=, Wkh surri ri wkh qlwh fdvh qrz iroorzv iurp G|qnlq*v ~ Wkhruhp 4<161 531 VVjAa!H ,|V,WB|j 7+t|j6 C jNij6 Ghqlwlrq 5314 +Erxqghg Frqyhujhqfh,1 Ohw eh d vhw1 Zh vd| wkdw d vh0 txhqfh ri ixqfwlrqv [q iurp wr U ru F frqyhujhv erxqghgo| wr d ixqfwlrq [ li olpq6R [q +$, @ [ +$, iru doo $ 5 dqg vxsim[q +$ ,m = $ 5 dqg q @ 4> 5> = = =j ? 4= Ghqlwlrq 5315 +Pxowlsolfdwlyh V|vwhp,1 D froohfwlrq ri uhdo ydoxhg ixqfwlrqv +T, rq lv d pxowlsolfdwlyh v|vwhp surylghg ij 5 T zkhqhyhu i> j 5 T= Wkhruhp 5316 +G|qnlq*v Pxowlsolfdwlyh V|vwhp Wkhruhp,1 Ohw K eh d olqhdu vsdfh ri ixqfwlrqv zklfk frqwdlqv wkh frqvwdqw ixqfwlrqv dqg lv forvhg xqghu erxqghg frqyhujhqfh1 Li T K lv pxowlsolfdwlyh idplo| ri erxqghg ixqfwlrqv/ wkhq K frqwdlqv doo erxqghg +T,0phdvxudeoh ixqfwlrqv1 Wkhruhp 5317 +Frpsoh{ Pxowlsolfdwlyh V|vwhp Wkhruhp,1 Ohw K eh d frpsoh{ yhfwru vsdfh ri frpsoh{ ixqfwlrqv rq vxfk wkdw= 4 5 K> K lv forvhg xqghu frpsoh{ frqmxjdwlrq/ dqg K lv forvhg xqghu erxqghg frqyhujhqfh1 Vxssrvh T lv d pxowl0 solfdwlyh idplo| ri erxqghg ixqfwlrqv frqwdlqhg lq K zklfk lv dovr forvhg xqghu frqmxjdwlrq1 Wkhq K frqwdlqv doo ri wkh erxqghg +T, phdvxudeoh ixqfwlrqv1 Wkh surri ri wkhvh Wkhruhpv lv edvhg rq G|qnlq*v Wkhruhp 4<161 Surri1 +ri wkh Pxowlsolfdwlyh V|vwhp Wkhruhpv1, Sxw D 5 G li 4D 5 K dqg ghqrwh e| F wkh idplo| ri doo vhwv ri wkh irup= +5314, E =@ i$ 5 = [R +$ , 5 UR > = = = > [p +$, 5 Up j zkhuh p @ 4> 5> = = = > [n 5 T dqg Un duh rshq lqwhuydov lq wkh uhdo fdvh dqg Un duh uhfwdqjohv lq F lq wkh frpsoh{ fdvh1 Rqh pd| fkhfn wkdw G lv d ~v|vwhp/ F lv forvhg xqghu lqwhuvhfwlrqv/ dqg wkdw +T, @ +F ,= Wkh ~ Wkhruhp 4<16 zloo lpso| wkdw +T, @ +F , G rqfh zh vkrz wkdw F G= Wklv lv rxu qh{w wdvn1 Lw lv hdv| wr frqvwuxfw/ iru hdfk n d xqlirupo| erxqghg vhtxhqfh ri frqwlqx0 n rxv ixqfwlrqv iq frqyhujlqj wr wkh fkdudfwhulvwlf ixqfwlrq 4Un = E| Zhlhuvwudvv* n wkhruhp/ wkhuh h{lvwv sro|qrpldov sn +{, vxfk wkdw sn +{, iq +{, 4@q iru p q n +}> } , lq } dqg } m{m n[n nR lq wkh uhdo fdvh dqg sro|qrpldov sro|qrpldov sp hu SUQh yrpr ra rhr ur vxfk wkdw sn +}> } , q ixqfwlrqv RSX n iq +} , 4@q iru m} m n[n nR lq wkh frpsoh{ fdvh1 Wkh Iq =@sR +[R ,sS +[S , = = = sp +[p , +uhdo fdvh, q q q Iq =@sR +[R [R ,sS +[S > [S , = = = sp +[p > [p , +frpsoh{ fdvh, q q q rq duh xqlirupo| erxqghg/ ehorqj wr K dqg frqyhujh srlqwzlvh wr 4E dv q $ 4/ zkhuh E lv wkh vhw lq Ht1 +5314,1 Khqfh wklv vhw lv dq hohphqw ri G dqg wkhuhiruh F G= Vlqfh +T, G> li [ lv dq duelwudu| erxqghg +T,0phdvxudeoh ixqfwlrq wkhq n i q ? [ nFR j 5 +T, G lq wkh uhdo fdvh dqg q n . 4 nQ nQ . 4 n ? Uh[ > ? Lp[ 5 +T, G q q q q lq wkh frpsoh{ fdvh1 Wkhuhiruh R [n 4 n ?[ # n.4 5 K +uhdo fdvh, [q @ q q ndR q dqg [ [ n . lnQ 4q n ?h[ # n.4 > nQ ?vp[ # nQ .4 r [q 5 K= +frpsoh{ fdvh, [q @ q q qq q Q nn Ehfdxvh [q 5 K frqyhujhv wr [ erxqghgo| dqg K lv forvhg xqghu erxqghg frq0 yhujhqfh/ [ lv lq K= ... View Full Document

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