Interaction of Mechanics and Mathematics
Henning Struchtrup
Macroscopic Transport Equations for Rareˇed Gas Flows Approximation Methods in Kinetic Theory With 35 Figures
IMM Advisory Board D. Colton (USA) · R. Knops (UK) · G. DelPiero (Italy) · Z. Mroz (Poland) · M. Slemrod (USA) · S. Seelecke (USA) · L. Truskinovsky (France) IMM is promoted under the auspices of ISIMM (International Society of Mechanics and Mathematics).
Author Dr. Henning Struchtrup University of Victoria Department of Mechanical Engineering and Institute for Integrated Energy Systems PO Box 3055, STN CSC Victoria, BC, V8W 3P6 Canada
ISSN 1860-6245 ISBN-10 ISBN-13
3-540-24542-1 Springer Berlin Heidelberg New York 978-3-540-24542-1 Springer Berlin Heidelberg New York
Library of Congress Control Number: 2005925870 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable to prosecution under German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2005 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Data conversion by the author. Final processing by PTP-Berlin Protago-TEX-Production GmbH, Germany Cover-Design: Erich Kirchner, Heidelberg Printed on acid-free paper 89/3141Yu - 5 4 3 2 1 0
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515 Prphqwv ri wkh skdvh ghqvlw|
4:
Zh iuhtxhqwo| kdyh wr lqwhjudwh ryhu wkh ixoo yhorflw| vsdfh/ dqg lq rughu wr frqghqvh qrwdwlrq zh vkdoo riwhq zulwh rqh lqwhjudo vljq zlwkrxw olplwv/ vr wkdw/ h1j1/ ] q@
igf =
I gf @ i @qgf fdq eh lqwhusuhwhg dv wkh suredelolw| wr qg d sduwlfoh zlwk yhorflw| lq ^f> f . gf`/ zklfk lv surshuo| qrupdol}hg vlqfh ] ] i gf @ 4 = I gf @ q Vlqfh wkh pdvv ghqvlw| lv jlyhq e| @ pq zh kdyh ] @ p igf =
+515,
Wkh phdq yhorflw| ri wkh sduwlfohv zlwklq wkh yroxph g{ lv jlyhq e| ] ] 4 4 y@ figf ru yl @ fl igf > q q zkhuh erwk htxdwlrqv duh lghqwlfdo/ dqg gl hu rqo| lq qrwdwlrq= wkh uvw htxd0 wlrq lv zulwwhq lq v|perolf whqvru qrwdwlrq/ dqg wkh vhfrqg rqh lq lqgh{ qrwd0 wlrq1 Riwhq zh zloo kdyh wr ghdo zlwk yhfwruv dqg whqvruv ri kljkhu rughu iru zklfk wkh lqgh{ qrwdwlrq lv hdvlhu wr kdqgoh/ dqg wkxv lw zloo eh xvhg doprvw h{foxvlyho| lq wkh vhtxho1 Vhh Dsshqgl{ D14 iru d uhylhz ri wkh edvlf hohphqwv ri lqgh{ qrwdwlrq1 yl / ru y/ lv wkh pdfurvfrslf yhorflw| ri wkh jdv/ dovr nqrzq dv wkh fhqwhu ri pdvv yhorflw|/ ru edu|fhqwulf yhorflw|1 Wkh prphqwxp ghqvlw| lv jlyhq e| ] +516, yl @ p fl i gf = Wkh shfxoldu yhorflw| Fl ri wkh sduwlfohv jlyhv wkh sduwlfoh vshhg dv phdvxuhg e| dq revhuyhu prylqj zlwk wkh jdv dw wkh orfdo yhorflw| yl / l1h1 dq revhuyhu lq wkh uhvw0iudph/ +517, Fl @ fl yl = Gxh wr wklv gh qlwlrq/ dqg zlwk +516,/ wkh uvw prphqw ri i ryhu Fl ydqlvkhv/ ] 3 @ p Fl igf = +518, Qrwh wkdw gf @ gF dv vkrzq lq Sureohp 5141 51515 Hqhuj|/ suhvvxuh/ dqg whpshudwxuh Wkh nlqhwlf hqhuj| ri d sduwlfoh lv jlyhq e| ri wkh jdv lv jlyhq e|
p 5 5f /
vr wkdw wkh hqhuj| ghqvlw|
4;
5 Edvlf txdqwlwlhv dqg gh qlwlrqv
Ilj1 5151 Rq wkh fdofxodwlrq ri suhvvxuh1
h @
p 5
] f5 igf >
+519,
h ghqrwhv wkh vshfl f hqhuj|1 Zlwk +517, dqg +518, iroorzv wkdw ] p 5 h @ F . 5yn Fn . y5 i gf @ x . y5 > 5 5 zkhuh x @
p 5
] F 5 i gf
+51:,
lv wkh lqwhuqdo/ ru wkhupdo/ hqhuj| ri wkh jdv/ dqg 5 y 5 lv wkh nlqhwlf hqhuj| ri lwv pdfurvfrslf prwlrq1 Wkxv/ wkh lqwhuqdo hqhuj| ri dq lghdo prqdwrplf jdv lv wkh nlqhwlf hqhuj| ri lwv sduwlfohv dv phdvxuhg lq wkh uhvw0iudph1 Suhvvxuh lv wkh dyhudjh irufh rq d zdoo gxh wr wkh fkdqjh ri prphqwxp ri wkh sduwlfohv1 Zh frqvlghu d zdoo zlwk qrupdo srlqwlqj lqwr wkh {4 0gluhfwlrq dqg dvvxph hodvwlf froolvlrqv1 Wkhq/ zkhq wkh yhorflw| ri d sduwlfoh ehiruh froolvlrq lv iF4 > F5 > F6 j/ lwv yhorflw| diwhu froolvlrq zloo eh iF4 > F5 > F6 j/ dqg wkh fruuhvsrqglqj fkdqjh ri prphqwxp lv i5pF4 > 3> 3j1 Wkh suhvvxuh uhvxowv iurp pdq| froolvlrqv e| sduwlfohv |lqj zlwk dq| vshhg lq doo gluhfwlrqv wkdw doorz wr klw wkh zdoo/ dqg zh xvh Ilj1 515 wr khos zlwk lwv fdofxodwlrq1 Lq rughu wr froolgh/ d sduwlfoh pxvw kdyh d yhorflw| frpsrqhqw F4 ? 3/ dqg doo sduwlfohv lq wkh yroxph F q w gD zloo froolgh gxulqj w/ dqg frqwulexwh 5pF4 wr prphqwxp fkdqjh +F q @ Fn qn ghqrwhv wkh vfdodu surgxfw,1 Dffruglqjo|/ wkh wrwdo prphqwxp fkdqjh gxulqj w lv 5pF4 F4 w gD i gf = Lqwhjudwlrq ryhu doo dgplvvleoh yhorflwlhv jlyhv wkh wrwdo prphqwxp fkdqjh gxulqj w/ dqg glylvlrq e| w gD jlyhv wkh suhvvxuh rq d zdoo zlwk qrupdo i4> 3> 3j dv ] ]] s @ 5p F4 ?3
F45 igf =
Iru qrz zh frqvlghu lvrwurslf jdvhv rqo|/ zkhuh wkh skdvh ghqvlw| ghshqgv vroho| rq wkh pdjqlwxgh ri wkh yhorflw|/ exw qrw rq lwv gluhfwlrq/ vr wkdw
515 Prphqwv ri wkh skdvh ghqvlw|
]]]
]]] F4 ?3
F45 i gf @
] F4 A3
F45 i gf dqg
] F45 i gf @
4<
] F55 i gf @
Frpelqlqj wkh odvw htxdwlrqv doorzv wr zulwh wkh suhvvxuh dv ] 4 5 s @ p F 5 igf @ x > 6 6
F65 igf =
+51;,
zkhuh F 5 @ Fn Fn @ F45 . F55 . F65 1 Khuh zh kdyh irxqg wkh uhodwlrq +417, e| d pruh dffxudwh dujxphqw ^4;`1 n W iroorzv wkh uhodwlrq ehwzhhq hqhuj| dqg Iurp wkh lghdo jdv odz s @ p whpshudwxuh dv ] 4 p5 6n W > ru W @ F 5 i gf = +51<, x@ 5p 6 n Whpshudwxuh lq nlqhwlf wkhru| lv xvxdoo| gh qhg e| wkh deryh uhodwlrq iru doo vlwxdwlrqv/ lqfoxglqj vwurqj ghyldwlrqv iurp htxloleulxp/ dqg zh vkdoo iroorz wkdw wudglwlrq1 Odwhu/ lq Vhf1 451417/ zh vkdoo vhh wkdw wkh qrwlrq ri whpshudwxuh dv d phdqlqjixowkdw lv phdvxudeohtxdqwlw| pd|eh orvw lq wkhvh h{wuhph fdvhv/ vlqfh whpshudwxuh mxpsv fdq rffxu dw wkh wkhuprphwhu zdoov1 Iru qrwdwlrqdo frqyhqlhqfh zh vkdoo zulwh wkh whpshudwxuh lq hqhuj| xqlwv dv n @ W = p Wkh dwrpv lq d prqdwrplf jdv kdyh wkuhh wudqvodwruldo ghjuhhv ri iuhhgrp +wkh wkuhh frpsrqhqwv ri yhorflw|,/ dqg hdfk ghjuhh ri iuhhgrp frqwulexwhv Cx 4 n 4 n 5 p W wr wkh vshfl f lqwhuqdo hqhuj|/ ru 5 p wr wkh vshfl f khdw fy @ CW 1 Rwkhu wkdq prqdwrplf prohfxohv/ h1j1 gldwrplf prohfxohv vxfk dv r{|jhq +R5 , ru qlwurjhq +Q5 ,/ wkuhh0dwrplf prohfxohv vxfk dv zdwhu +K5 R, ru fduerq glr{0 lgh +FR5 ,/ ru hyhq odujhu prohfxohv vxfk dv phwkdqh +FK7 ,/ kdyh dgglwlrqdo ghjuhhv ri iuhhgrp gxh wr urwdwlrq dqg yleudwlrq1 Hdfk ixoo| dfwlydwhg ghjuhh n wr wkh vshfl f khdw/ exw wkh dfwlydwlrq iroorzv ri iuhhgrp frqwulexwhv 45 p txdqwxp phfkdqlfdo odzv zklfk uhvxowv lq whpshudwxuh ghshqghqw vshfl f khdwv ^45`^56`1 51516 Jhqhudo prphqwv Wkhuh duh pruh prphqwv ri lpsruwdqfh/ vrph ri zklfk kdyh d phdqlqjixo sk|vlfdo lqwhusuhwdwlrq/ vxfk dv suhvvxuh whqvru slm dqg khdw x{ tl / jlyhq dv ] ] p F 5 Fl igf = +5143, slm @ p Fl Fm i gf dqg tl @ 5 Krzhyhu/ dgglwlrqdo prphqwv ehfrph lpsruwdqw zkhq uduhidfwlrq h hfwv duh frqvlghuhg/ dowkrxjk wkh| gr qrw kdyh sk|vlfdo lqwhusuhwdwlrqv1 Lw lv wkxv frq0 yhqlhqw wr lqwurgxfh d jhqhudo qrwdwlrq iru prphqwv/ dqg zh gh qh jhqhudo whqvruldo prphqwv dv
53
5 Edvlf txdqwlwlhv dqg gh qlwlrqv
] dl4 l5 lq @ p
F 5d Fl4 Fl5 Flq i gf =
+5144,
Lw zloo suryh yhu| xvhixo wr xvh luuhgxfleoh/ ru wudfh0iuhh/ prphqwv ri wkh skdvh ghqvlw| zklfk duh gh qhg dv ] +5145, xdl4 lq @ dkl4 l5 lq l @ p F 5d Fkl4 Fl5 Flq l igf zkhuh lqglfhv lq dqjxodu eudfnhwv ghqrwh wkh v|pphwulf dqg wudfh0iuhh sduw ri d whqvru1 Qrwh wkdw wkh prphqwv x l4 lq xvhg lq wkh iroorzlqj duh wudfh0iuhh e| gh qlwlrq/ xdl4 lq @ xdkl4 lq l / dqg wkdw wklv lv qrw pdgh h{solflw zlwk eudfnhwv lq rughu wr dyrlg frqixvlqj qrwdwlrq lqyroylqj vhyhudo sdluv ri eudfnhwv1 Lqglfhv lq urxqg eudfnhwv/ dv/ h1j1/ lq +5149, ehorz/ ghqrwh wkh v|pphwulf sduw ri d whqvru1 Dsshqgl{ D15 jlyhv d vkruw lqwurgxfwlrq wr whqvru lqgh{ qrwdwlrq/ dqg wr v|pphwulf dqg wudfh0iuhh whqvruv1 Lq wklv qrwdwlrq zh lghqwli| 3 @ x3 @ > 3l @ x3l @ 3 > 4 @ x4 @ 5 h @ 6 @ 6s > 3lm @ slm > x3lm @ sklml @ lm > 4l @ 5tl =
+5146,
Khuh/ lq dgglwlrq wr txdqwlwlhv gh qhg hduolhu/ zh kdyh lqwurgxfhg wkh vwuhvv whqvru lm dv wkh wudfh0iuhh sduw ri wkh suhvvxuh whqvru slm 1 Wkh prphqwv gh qhg deryh duh wdnhq zlwk uhvshfw wr wkh shfxoldu yhorf0 lw| Fn / dqg duh vrphwlphv ghqrwhg dv fhqwudo prphqwv wr glvwlqjxlvk wkhp irup wkh vr0fdoohg frqyhfwlyh prphqwv zklfk duh gh qhg zlwk uhvshfw wr wkh plfurvfrslf yhorflw| fn ^57`1 Wkh prphqwv ] +5147, Il34 l5 lq @ p fl4 fl5 flq i gf zloo eh ghqrwhg dv ixoo frqyhfwlyh prphqwv/ dqg zh fdq jhqhudol}h wkh gh ql0 wlrq wr ] Ild4 l5 lq @ p f5d fl4 fl5 flq i gf @ Il4 l5 lq m4 n4 mp np m4 n4 mp np = +5148, Frqyhfwlyh dqg fhqwudo prphqwv fdq hdvlo| eh uhodwhg e| phdqv ri +517, dv q [ q 3 yl ylq , > +5149, Il34 l5 lq @ n +l4 ln n.4 n@3
h1j1/ +514:, I 3 @ > Il3 @ yl > I 4 @ 5 h @ 5 x . y5 > 4 4 Ilm3 @ 3lm . yl ym @ slm . yl ym > In4 @ x . y5 yn . snm ym . tn > 5 5 Ilm3 lv wkh wrwdo prphqwxp x{/ dqg 61615,1
4 4 5 In
lv wkh wrwdo hqhuj| x{ +vhh Vhf1
516 Htxloleulxp dqg wkh Pd{zhooldq glvwulexwlrq
54
51517 Wkh uhohydqfh ri prphqwv Wkh uvw uhdvrq iru rxu lqwhuhvw lq prphqwv lv wkdw vrph/ qdpho| pdvv ghq0 vlw| +515,/ prphqwxp ghqvlw| +516,/ hqhuj| ghqvlw| +519, ru lqwhuqdo hqhuj| ghqvlw| +51:,/ suhvvxuh whqvru +5143,4 / dqg khdw x{ +5143,5 kdyh d phdqlqjixo sk|vlfdo lqwhusuhwdwlrq/ dqg wkhuhiruh duh/ dw ohdvw lq sulqflsoh/ dffhvvleoh wr phdvxuhphqwv1 Wkh vhfrqg/ dqg pd|eh pruh lpsruwdqw/ uhdvrq wr frqvlghu prphqwv lv wkdw xqghu fhuwdlq frqglwlrqv/ lq sduwlfxodu dw vpdoohu Nqxgvhq qxpehuv/ wkh nqrzohgjh ri d vpdoo qxpehu ri prphqwv/ h1j1 8/ 46/ ru 59/ lv vx!flhqw wr gh0 vfuleh wkh vwdwh ri wkh jdvlqfoxglqj pdvv/ prphqwxp dqg hqhuj| wudqvihu zlwk uhdvrqdeoh dffxudf|1 Reylrxvo|/ wkh frpsohwh nqrzohgjh ri wkh skdvh ghqvlw| doorzv wr frpsxwh doo ri lwv prphqwv/ exw dovr wkh lqyhuvh lv wuxh= li doo prphqwv duh nqrzq/ wkh glvwulexwlrq fdq eh uhfrqvwuxfwhg1 Lq wkrvh fdvhv zkhuh wkh jdv rz lv zhoo ghvfulehg wkurxjk d vpdoo/ dqg qlwh/ qxpehu ri prphqwv/ wkh frpsohwh nqrzohgjh ri wkh skdvh ghqvlw| lv qrw uhtxluhg1 Wkhq/ wkh ghvfulswlrq ri wkh jdv vwdwh wkurxjk prphqwv lv pruh frqyhqlhqw/ vlqfh rqo| wkh uhohydqw prphqwv pxvw eh frqvlghuhg/ zkloh doo rwkhuv fdq eh ljqruhg1 Wkh skdvh ghqvlw| i +{> w> f, pxvw eh frpsxwhg e| vroylqj wkh Erow}pdqq htxdwlrq wkdw zloo eh glvfxvvhg lq Fkdswhu 6/ dqg wkh htxdwlrqv iru wkh pr0 phqwv zloo eh ghulyhg iurp wkh Erow}pdqq htxdwlrq lq odwhu fkdswhuv1
516 Htxloleulxp dqg wkh Pd{zhooldq glvwulexwlrq D vlpsoh dujxphqw jrlqj edfn wr Pd{zhoo doorzv xv wr qg wkh yhorflw| glv0 wulexwlrq lq htxloleulxp ^5`^4;`1 Htxloleulxp lv d vwdwh zkhuh qr fkdqjhv zloo rffxu zkhq wkh jdv lv ohiw wr lwvhoi/ dqg wklv zloo lpso| wkdw wkh jdv lv krprjhqhrxv/ l1h1 glvsod|v qr judgl0 hqwv lq dq| txdqwlw|/ dqg wkh skdvh ghqvlw| lv lvrwurslf/ wkdw lv lqghshqghqw ri wkh gluhfwlrq l @ Fl @F1 Wkh iroorzlqj dujxphqw frqvlghuv wkh jdv lq wkh uhvw0iudph/ zkhuh yl @ 31 Dq duelwudu| dwrp slfnhg iurp wkh jdv zloo kdyh wkh yhorflw| frpsrqhqwv Fn > n @ 4> 5> 6/ dqg wkh suredelolw| wr qg wkh frpsrqhqw lq gluhfwlrq {n zlwklq wkh lqwhuydo ^Fn > Fn . gFn ` lv jlyhq e| s +Fn , gFn 1 Qrwh wkdw/ gxh wr lvrwurs|/ wkh suredelolw| ixqfwlrq s lv wkh vdph iru doo frpsrqhqwv1 Wkhq/ wkh suredelolw| wr qg d sduwlfoh zlwk wkh yhorflw| yhfwru iF4 > F5 > F6 j lv jlyhq e| I +F, gF4 gF5 gF6 @ s +F4 , s +F5 , s +F6 , gF4 gF5 gF6 zkhuhsI +F, @ i +F, @q ghshqgv rqo| rq wkh devroxwh ydoxh ri yhorflw|/ F @ F45 . F55 . F65 / vlqfh wkh suredelolw| pxvw eh lqghshqghqw ri gluhf0 wlrq1 Wkxv/ I dqg s duh uhodwhg dv I +F, @ s +F4 , s +F5 , s +F6 , =
55
5 Edvlf txdqwlwlhv dqg gh qlwlrqv
Wdnlqj wkh orjdulwkplf ghulydwlyh ri wklv htxdwlrq zlwk uhvshfw wr F4 zh vhh C C oq I +F, @ ^oq s +F4 , . oq s +F6 , . oq s +F6 ,` CF4 CF4 ru
4 s3 +F4 , 4 I 3 +F, @ @ 5 = F I +F, F4 s +F4 ,
Vlqfh wkh ohiw dqg wkh uljkw vlgh ri wklv htxdwlrq ghshqg rq gl huhqw yduldeohv/ pxvw eh d frqvwdqw/ dqg lqwhjudwlrq jlyhv dq lvrwurslf Jdxvvldq glvwulexwlrq/ I @
s i 6 @ D h{s F 5 dqg s +Fn , @ D h{s Fn5 > q
zkhuh D lv d frqvwdqw ri lqwhjudwlrq1 Wkh wzr frqvwdqwv dqg D iroorz iurp wkh frqglwlrqv wkdw wkh skdvh ghqvlw| pxvw uhsurgxfh pdvv dqg hqhuj| ghqvlw|/ wkdw lv ] ] p 6 F 5 i gf = @ p i gf dqg x @ @ 5 5 Wkh uhvxowlqj htxloleulxp skdvh ghqvlw| lv wkh Pd{zhooldq glvwulexwlrq F5 4 h{s = +514;, iP @ s p 5 6 5 Wkh ydoxhv ri wkh prphqwv +5145, lq htxloleulxp +H,/ zkhq wkh skdvh ghqvlw| lv d Pd{zhooldq/ duh jlyhq e| +514<, xdmH @ +5d . 4,$$ d > xdl4 lq mH @ 3 > q 4 > T zkhuh wkh grxeoh idfwruldo lv gh qhg dv +5d . 4,$$ @ dv@4 +5v . 4,1 Xvhixo htxdwlrqv iru lqwhjudov ri wkh Pd{zhooldq duh jlyhq lq Dsshqgl{ D161
517 Erxqgdu| frqglwlrqv iru wkh skdvh ghqvlw| Jdv sduwlfohv wkdw klw d zdoo zloo h{fkdqjh hqhuj| dqg prphqwxp zlwk wkh zdoo dqg eh uh0hplwwhg lqwr wkh jdv zlwk d qhz yhorflw| yhfwru1 Wkh ghwdlov ri wkh lqwhudfwlrq ghshqg vwurqjo| rq wkh plfurvfrslf ghwdlov ri zdoo dqg jdv/ dqg jhqhudoo| lw lv qrw srvvleoh wr ghvfuleh wklv lqwhudfwlrq dffxudwho|1 Lqvwhdg/ suredelolvwlf prghov duh xvhg1 Wkh prvw srsxodu/ dqg prvw vlpsoh/ prgho iru erxqgdu| frqglwlrqv lv gxh wr Pd{zhoo ^5`^6` dqg zloo eh xvhg h{foxvlyho| lq wklv wh{w1 Wkh edu|fhqwulf yhorflw| yn ri d uduh hg jdv zloo gl hu iurp wkh zdoo yh0 orflw| ynZ / vr wkdw lw kdv d qrq0}hur vols yhorflw| Yn @ yn ynZ
+5153,
zklfk lv sdudooho wr wkh zdoo/ vlqfh wkhuh lv qr jdv rz wkurxjk wkh zdoo/
517 Erxqgdu| frqglwlrqv iru wkh skdvh ghqvlw|
Yn qn @ 3 =
56
+5154,
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57
5 Edvlf txdqwlwlhv dqg gh qlwlrqv
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58
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6 Wkh Erow}pdqq htxdwlrq dqg lwv surshuwlhv
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Lq jhqhudo/ wkh uhodwlrq ehwzhhq / e/ dqg j ghshqgv rq wkh lqwhudfwlrq sr0 whqwldo ! +u,/ dqg lw fdq eh vkrzq e| lqwhjudwlrq wkurxjk wkh froolvlrq wkdw ^48` ] v 7! ve gv 5 u zlwk 4 +v , @3= +61:, +e> j, @ pj5 3 7!+ ve , 5 4 v pj5 Khuh/ v @ e@u lv d glphqvlrqohvv lqyhuvh glvwdqfh/ dqg v @ e@uplq lv lwv ydoxh dw wkh plqlpdo sduwlfoh glvwdqfh uplq 1 61416 Srwhqwldo dqg gl huhqwldo furvv vhfwlrq Iru prvw srwhqwldov wkh uhodwlrq +61:, lv gl!fxow wr hydoxdwh1 Zh vkdoo frqvlghu rqo| kdug vskhuhv zkhuh 3 /uAg !KV +u, @ +61;, 4/ug> dqg srzhu srwhqwldov zkhuh ! +u, @ !3 u4 >
+61<,
!3 dqg duh frqvwdqwv1 Srzhu srwhqwldov duh sxuho| uhsxovlyh srwhqwldov +wkh lqwhusduwlfoh irufh lv jlyhq e| g! gu ,/ dqg gr qrw ghvfuleh sduwlfoh dwwudfwlrq wkdw ohdgv wr frqghqvdwlrq dw orz whpshudwxuhv1 Dwwudfwlyh irufhv fdq eh lj0 qruhg dv orqj dv wkh srwhqwldo wurxjk !plq lq Ilj1 414 lv vpdoo frpsduhg wr 5 @ ; nW 1 Wklv zloo eh wkh fdvh iru wkh dyhudjh nlqhwlf hqhuj| ri d froolvlrq p 5j whpshudwxuhv zhoo deryh wkh vdwxudwlrq whpshudwxuh1 Iru srzhu srwhqwldov/ +61:, fdq eh zulwwhq dv 4 ] v 7!3 v gv 5 t @ 3> zlwk 4 +v , +e> j, @ v 4 p va 3 3 4 v5 7! p va 5 wkdw lv ghshqgv rq e dqg j rqo| wkurxjk wkh frpelqdwlrq va @ ej 4 / 5 @ ej 4 @ +a v, = +6143,
Wkh gl huhqwldo furvv vhfwlrq lv gh qhg dffruglqj wr e vlq g @ e ge vr wkdw @ = vlq C Ce Iru kdug vskhuhv dqg srzhu srwhqwldov zh rewdlq 7
KV @ g5 frv dqg @ j 4 uhvshfwlyho|/ zkhuh I +, @
v va ga vlq g
7 va ga v @ j 4 I +, > vlq g
lv d ixqfwlrq ri dorqh1
+6144,
65
6 Wkh Erow}pdqq htxdwlrq dqg lwv surshuwlhv
Dq lpsruwdqw vshfldo fdvh duh Pd{zhoo prohfxohv/ zklfk kdyh d srzhu srwhqwldo zlwk @ 8/ zkhuh j @ IP +, =
+6145,
Dv Pd{zhoo qrwlfhg/ dqg dv zloo eh vhhq lq wkh vhtxho/ wklv sduwlfxodu fkrlfh doorzv pxfk hdvlhu hydoxdwlrq ri pdq| sureohpv1 61417 Wkh Vwrvv}dkodqvdw} Wkh nqrzohgjh ri wkh froolvlrq ghwdlov doorzv wr frpsxwh wkh fkdqjh ri wkh skdvh ghqvlw| i +{> w> f, wkurxjk froolvlrqv/ V/ wkdw zdv lqwurgxfhg lq +614,1 Wkh iroorzlqj dvvxpswlrqv duh lpsolflwo| xvhg lq wkh dujxphqw/ nqrzq dv Vwrvv}dkodqvdw}= Rqo| elqdu| froolvlrqv wdnh sodfh1 Lqghhg/ hyhq lq d ghqvh jdv froolvlrqv ehwzhhq wkuhh ru pruh sduwlfohv duh h{wuhpho| xqolnho|/ dqg fdq eh vdiho| ljqruhg1 Lw qhhgv pdq| froolvlrqv wr fkdqjh i vljql fdqwo|1 Lq sduwlfxodu/ wkh skdvh ghqvlw| grhv qrw fkdqjh gxulqj d vlqjoh froolvlrq1 Zkloh i ydulhv lq vsdfh/ lw fdq eh frqvlghuhg dv frqvwdqw ryhu wkh udqjh g ri wkh lqwhudwrplf irufhv1 Dw hyhu| srlqw lq vsdfh0wlph +{> w,/ wkh ydoxhv ri i iru gl huhqw yhorflwlhv duh lqghshqghqw +dvvxpswlrq ri prohfxodu fkdrv,1 Hdfk froolvlrq f> f4 $ f3 > f43 glplqlvkhv wkh skdvh ghqvlw| i +{> w> f,/ zkloh d vr0fdoohg lqyhuvh froolvlrq f3 > f43 $ f> f4 lqfuhdvhv i +{> w> f,1 Wr dffrxqw iru orvv dqg jdlq/ zh zulwh +6146, V @ V. V > zkhuh V. +V , lv wkh qxpehu ghqvlw| ri froolvlrqv shu xqlw wlph wkdw surgxfh +ghvwur|, d skdvh srlqw zlwk yhorflw| f1 Doo sduwlfohv zlwk yhorflwlhv f4 lq wkh yroxph j gw e ge g% ghslfwhg lq Ilj1 617 zloo froolgh zlwk wkh sduwlfoh lq wkh fhqwhu +yhorflw| f, gxulqj gw/ vr wkdw wkh vlqjoh sduwlfoh h{shulhqfhv i {> w> f4 gf4 j gw e ge g% vxfk froolvlrqv gxulqj gw +zkhuh wkh srvw0froolvlrq yhorflwlhv duh f3 > f43 ,1 Wkh qxpehu ghqvlw| ri sduwlfohv zlwk yhorflw| f lv i +{> w> f, gf vr wkdw wkh wrwdo qxpehu ghqvlw| ri froolvlrqv f> f3 $ f3 > f43 shu xqlw wlph lv ] ] 5 ] @5 V gf @ i +{> w> f, i {> w> f4 j vlq g g% gf4 gf = +6147, 3
3
Khuh zh kdyh lqwurgxfhg wkh gl huhqwldo furvv vhfwlrq +6144,/ dqg lqwhjudwhg ryhu doo srvvleoh froolvlrq sduwqhuv/ wkdw lv ryhu doo yhorflwlhv f4 / doo froolvlrq
614 Wkh Erow}pdqq htxdwlrq
66
Ilj1 6171 Frxqwlqj froolvlrqv1
dqjohv / dqg doo dqjohv % zklfk ghvfuleh wkh rulhqwdwlrq ri wkh froolvlrq sodqh/ vhh Ilj1 6171 Qh{w zh frqvlghu lqyhuvh froolvlrqv f3 > f43 $ f> f4 1 E| phdqv ri wkh vdph dujxphqw dv deryh/ wkh qxpehu ghqvlw| ri wkhvh froolvlrqv shu xqlw wlph lv i +{> w> f3 , i {> w> f43 j3 3 vlq g g% gf43 gf3 > zkhuh wkh yhorflwlhv duh uhodwhg dffruglqj wr +618,1 Lq wkh suhylrxv vhfwlrq zh irxqg wkdw j 3 @ j/ e3 @ e/ zklfk lpsolhv 3 @ +e3 > j 3 , @ +e> j, @ / dqg wkhuhiruh 3 @ / dqg lq Sureohp 616 lw lv vkrzq wkdw gfgf4 @ gf3 gf43 1 Vxppdwlrq wkhq jlyhv wkh qxpehu ghqvlw| ri froolvlrqv wkdw |lhog d sduwlfoh zlwk yhorflw| f dv ] ]
5
]
V. gf @ 3
@5
i +{> w> f3 , i {> w> f43 j vlq g g% gf4 gf =
+6148,
3
61418 Wkh Erow}pdqq htxdwlrq Wr vlpsoli| wkh qrwdwlrq/ zh zulwh i +{> w> f, @ i > i {> w> f4 @ i 4 > i +{> w> f3 , @ i 3 > i {> w> f4 @ i 43 > dqg frpelqh +614, zlwk +61466148, wr qdoo| rewdlq wkh Erow}pdqq htxdwlrq ^58`
67
6 Wkh Erow}pdqq htxdwlrq dqg lwv surshuwlhv
Ci Ci Ci . fn . Jn @ Cw C{n Cfn
] ] 3
5
]
@5
i 3 i 43 i i 4 j vlq g g% gf4 =
3
+6149, Wkh Erow}pdqq htxdwlrq lv wkh fhqwudo htxdwlrq lq nlqhwlf wkhru| ri jdvhv1 Lw lv d qrq0olqhdu lqwhjur0gl huhqwldo htxdwlrq iru wkh skdvh ghqvlw| i wkdw ghvfulehv wkh hyroxwlrq lq vsdfh dqg wlph ri d prqdwrplf lghdo jdv gxh wr iuhh ljkw/ dffhohudwlrq wkurxjk h{whuqdo irufhv Jn / dqg froolvlrqv1 Gxh wr lwv frpsoh{lw|/ wkh Erow}pdqq htxdwlrq fdqqrw eh vroyhg hdvlo|/ dqg lq doprvw doo fdvhv rqh kdv wr uho| rq qxphulfdo vroxwlrqv1 Lq wkh uhpdlqghu ri wklv fkdswhu/ zh glvfxvv lpsruwdqw surshuwlhv ri wkh Erow}pdqq htxdwlrq1
615 Htxloleulxp Htxloleulxp lv gh qhg dv d krprjhqhrxv vwhdg| vwdwh/ dqg zh dvn iru wkh htxloleulxp glvwulexwlrq iH 1 Lq wkh devhqfh ri h{whuqdo irufhv +Jn @ 3,/ wkh Erow}pdqq htxdwlrq uhgxfhv wr ] ] 5 ] @5 3 43 iH iH iH iH4 j vlq g g% gf4 @ 3 > 3
3
dqg wkh lqwhjudo ydqlvkhv li iH3 iH43 iH iH4 @ 3 ru oq iH3 . oq iH43 @ oq iH . oq iH4 =
+614:,
Iru elqdu| froolvlrqv ri sduwlfohv ri pdvv p/ pdvv/ prphqwxp dqg hqhuj| duh frqvhuyhg/ p.p @ p.p> pf . pf43 @ pf . pf4 > p p 4 5 p 3 5 p 43 5 +f , . f +f,5 . f @ > 5 5 5 5 3
dqg lw fdq eh vkrzq wkdw wkhvh yh txdqwlwlhv5 duh wkh rqo| froolvlrqdo lqydul0 dqwv ^;`^4;`1 Dffruglqj wr +614:,/ oq iH lv d froolvlrqdo lqyduldqw/ dqg wkxv pxvw eh d olqhdu frpelqdwlrq ri wkh nqrzq lqyduldqwv/ l k 5 +614;, oq iH @ . l fl f5 ru/ dowhuqdwlyho|/ iH @ D h{s +fl l , +vhh Sureohp 617,1 Wkh frh!flhqwv duh ghwhuplqhg e| hqvxulqj wkdw wklv glv0 wulexwlrq jlyhv wkh surshu ydoxhv iru pdvv ghqvlw|/ prphqwxp ghqvlw| dqg hqhuj| ghqvlw|/ l1h1 e| lqvhuwlqj iH lqwr ] ] ] p 6 F 5 iH gf = +614<, @ p iH gf / yl @ p fl iH gf > x @ @ 5 5 2
Prphqwxp kdv wkuhh frpsrqhqwv/ zkloh pdvv dqg hqhuj| duh vfdodu1
616 Htxdwlrq ri wudqvihu dqg frqvhuydwlrq odzv
Wklv |lhogv D@
4 4 > l @ yl > @ s p 5 6 5
68
+6153,
wkdw lv wkh Pd{zhooldq +514;, iH @ iP
% & +fl yl ,5 4 @ s h{s > p 5 6 5
zklfk zdv irxqg lq Vhf1 516 e| phdqv ri d gl huhqw dujxphqw1 Lq jhqhudo/ ghqvlw| +{> w,/ whpshudwxuh +{> w,/ dqg yhorflw| yl +{> w, zloo ghshqg rq vsdfh dqg wlph/ dqg wkxv d orfdo Pd{zhooldq & % +{> w, +fl yl +{> w,,5 4 +6154, iP +{> w> f, @ s 6 h{s p 5 +{> w, 5 +{> w, fdq eh dvvljqhg wr hdfk vsdfh srlqw1 Dv orqj dv wkh dfwxdo glvwulexwlrq i gl huv iurp wkh orfdo Pd{zhooldq/ wkh h hfw ri froolvlrqv lv wr irufh wkh glvwulexwlrq wrzdugv wkh orfdo Pd{zhooldq1
616 Htxdwlrq ri wudqvihu dqg frqvhuydwlrq odzv 61614 Htxdwlrq ri wudqvihu Dv glvfxvvhg lq Vhf1 515/ zh vkdoo riwhq eh lqwhuhvwhg lq wkh prphqwv ri i udwkhu wkdq i lwvhoi1 Prphqwv duh zhljkwhg dyhudjhv ri wkh glvwulexwlrq ixqfwlrq/ dqg hyroxwlrq htxdwlrqv iru prphqwv iroorz e| wdnlqj zhljkwhg dyhudjhv ri wkh Erow}pdqq htxdwlrq1 Wr vlpsoli| qrwdwlrq/ zh zulwh ] k#l @ p #i gf > zkhuh # lv dq| ixqfwlrq ri +{> w> f,/ vr wkdw/ h1j1/ @ k4l/ yl @ kfl l/ x @ 6 5 5 @ F / hwf1 Wkh hyroxwlrq htxdwlrq iru k#l/ wkh htxdwlrq ri wudqvihu/ lv rewdlqhg e| pxowlso|lqj wkh Erow}pdqq htxdwlrq +6149, zlwk # +{> w> f,/ dqg vxevhtxhqw lqwhjudwlrq/ dv6 C# C k#l C k#fn l C# C# . @ . Jn . V# = +6155, . fn Cw C{n Cw C{n Cfn Wkh surgxfwlrq whup V# lv gh qhg dv
U K Iru wkh ghulydwlrq lw lv xvhg wkdw YSYn E6C& s _U ' S<" 6C& s ?& _ ' f1 Wkh vxuidfh lqwhjudo kdv wr eh wdnhq ryhu wkh yhorflw| vkhoo iru S< "/ dqg ydqlvkhv vlqfh wkhuh duh qr sduwlfohv zlwk lq qlwh hqhuj|/ l1h1 s ES< " ' f1
69
6 Wkh Erow}pdqq htxdwlrq dqg lwv surshuwlhv
] V# @
] ] ]
5
]
#Vgf @ p 3
@5
# i 3 i 43 i i 4 j vlq g g% gf4 gf =
3
+6156, Gxh wr wkh v|pphwu| ri wkh froolvlrq htxdwlrqv +615618,/ wkh sdluv f> f4 3 43 f > f fdq eh h{fkdqjhg wr jlyh wkh dowhuqdwlyh h{suhvvlrq ] ] ]
]
5
@5
V# @ p 3
#3 i i 4 i 3 i 43 j vlq g g% gf4 gf =
+6157,
3
H{fkdqjh wkh lqwhjudwlrq yduldeohv f f4 lq +6156,/ dqg f3 f43 lq +6157, wr jlyh ] ] ]
5
]
5
]
@5
#4 i 3 i 43 ii 4 j vlq g g% gf4 gf > +6158,
@5
#43 ii 4 i 3 i 43 j vlq g g% gf4 gf = +6159,
V# @ p 3
] ] ]
3
V# @ p 3
3
Vxppdwlrq ri +61566159, |lhogv dq h{suhvvlrq iru V# wkdw vkrzv wkdw wkh surgxfwlrq whup zloo ydqlvk/ li # lv d froolvlrqdo lqyduldqw/ yl}1 p V# @ 7
] ] ]5 ] @5 # . #4 #3 # 43 i 3 i 43 ii 4 j vlq gg%gf4 gf = 3
3
+615:, Ilqdoo|/ e| h{fkdqjlqj f3 > f43 f> f4 lq wkh uvw whup ri +6156,/ zh qg dqrwkhu xvhixo irupxod/ zkhuh wkh srvw0froolvlrq yhorflwlhv f3 > f43 dsshdu rqo| lq wkh dujxphqw ri #/ exw qrw lq i / ] ] ]
5
]
V# @ p 3
@5
#3 # ii 4 j vlq g g% gf4 gf =
+615;,
3
61615 Frqvhuydwlrq odzv Wkh frqvhuydwlrq odzv iru pdvv/ prphqwxp dqg hqhuj| iroorz e| fkrrvlqj # dv d froolvlrqdo lqyduldqw1 Iru # @ 4/ +6155, zlwk +615:, uhgxfhv wr wkh pdvv edodqfh +frqwlqxlw| htxdwlrq, C C yn CI 3 CIn3 . . @ 3 ru @3= +615<, Cw C{n Cw C{n Iru # @ fl / +6155, zlwk +615:, uhgxfhv wr wkh edodqfh ri prphqwxp 3 CIl3 CIln C yl C yl yn . sln . . @ I 3 Jl ru @ Jl = Cw C{n Cw C{n
Iru # @ 54 f5 / +6155, zlwk +615:, uhgxfhv wr wkh edodqfh ri wrwdo hqhuj|
+6163,
616 Htxdwlrq ri wudqvihu dqg frqvhuydwlrq odzv
6:
C 45 I 4 C 45 In4 . @ I 3 Jn yn Cw C{n ru
C x . 45 y5 yn . sln yl . tn C x . 54 y5 . @ Jn yn +6164, Cw C{n Wkh prphqwv dsshdulqj lq wkhvh htxdwlrqv zhuh lqwurgxfhg lq Vhf1 5151 Wkh deryh edodqfh odzv duh zulwwhq lq glyhujhqfh irup/ wkdw lv wkh| duh ri wkh jhqhudo irup C zD CZDn . @ D . VD = +6165, Cw C{n Khuh zD vwdqgv iru wkh vshfl f txdqwlwlhv/ ZDn lv wkhlu x{/ D lv wkh vxsso| iurp wkh rxwvlgh ^48`/ dqg VD lv wkh surgxfwlrq gxh wr froolvlrqv1 Lqghhg/ lqwhjudwlrq ryhu d {hg yroxph ri vsdfh Y / dqg dssolfdwlrq ri Jdxvv*v wkhruhp jlyhv ] L ] g zD gY @ ZDn qn gD . +D . VD , gY gw Y CY Y zklfk vkrzv wkdw zD lq Y fkdqjhv gxh wr wkh x{ ZDn ryhu wkh vxuidfh/ dqg wr vxsso| dqg surgxfwlrq1 Iru wkh frqvhuydwlrq odzv wkh surgxfwlrq whup ydqlvkhv/ dqg wkh vxsso| ri hqhuj| dqg prphqwxp lv gxh wr wkh h{whuqdo erg| irufh Jn 1 Riwhq wkh x{ lv vsolw lqwr wkh frqyhfwlyh sduw zD yn dqg wkh qrq0 frqyhfwlyh sduw Z Dn dv ZDn @ zD yn .Z Dn 1 E| phdqv ri wkh pdvv edodqfh/ wkh jhqhudo htxdwlrq +6165, fdq wkhq eh zulwwhq dv
GzD CZ Dn . @ D . VD = Gw C{n
+6166,
Khuh
C C G @ . yn +6167, Gw Cw C{n lv wkh frqyhfwlyh wlph ghulydwlyh/ zklfk ghvfulehv wkh fkdqjh ri zD iru dq revhuyhu uhvwlqj lq wkh jdv/ vr wkdw kh pryhv zlwk yhorflw| yn / lq frqwudvw wr D wkh ghulydwlyh Cz Cw zklfk ghvfulehv wkh fkdqjh dw d {hg orfdwlrq1 Lq wklv qrwdwlrq/ wkh frqvhuydwlrq odzv uhdg Cyn G . @3> Gw C{n Gyl Csln . @ Jl > Gw C{n G x . 54 y 5 Csln yl . tn . @ Jn yn = Gw C{n
+6168,
Wkh edodqfh ri lqwhuqdo hqhuj| iroorzv hlwkhu e| pxowlso|lqj wkh prphqwxp edodqfh +6168,5 zlwk yl dqg vxewudfwlqj wkh uhvxow iurp wkh hqhuj| edodqfh +6168,6 / ru e| fkrrvlqj # @ 54 F 5 @ 54 +fl yl ,5 / dv +vhh Sureohp 619,
6;
6 Wkh Erow}pdqq htxdwlrq dqg lwv surshuwlhv
Ctn Gx Cyl . @ slm = Gw C{n C{m
+6169,
Lqwhuqdo hqhuj| lv qrw frqvhuyhg/ dqg wkh whup rq wkh uljkw kdqg vlgh ghvfulehv wkh surgxfwlrq ri lqwhuqdo hqhuj| gxh wr iulfwlrq1 Wkh qdwxudo yduldeohv rffxuulqj lq wkh yh frqvhuydwlrq odzv duh pdvv ghqvlw| / yhorflw| yl / dqg whpshudwxuh +uhphpehu wkdw x @ 65 ,/ wkdw lv yh txdqwlwlhv1 Krzhyhu/ wkh frqvhuydwlrq odzv frqwdlq dovr wkh khdw x{ yhfwru tn dqg wkh suhvvxuh whqvru slm / dqg wkxv gr qrw irup d forvhg v|vwhp ri htxdwlrqv1 Dssursuldwh h{suhvvlrqv iru slm dqg tn pxvw eh irxqg lq rughu wr surgxfh d forvhg vhw ri htxdwlrqv rq wkh pdfurvfrslf ohyho1 61616 Prphqw htxdwlrqv Wkh wudqvsruw htxdwlrqv iru wkh wudfh iuhh prphqwv xdl4 lq +5145, dqg wkh frqyhfwlyh prphqwv Il4 lq +5148, zloo eh glvfxvvhg lq Fkdswhu 81
617 Hqwurs| dqg wkh K0wkhruhp 61714 K0wkhruhp Lq wkh htxdwlrq ri wudqvihu +6155, zh fkrvh/ iroorzlqj Erow}pdqq ^58`/ # @ n oq
i |
+616:,
zlwk wkh Erow}pdqq frqvwdqw n dqg dqrwkhu frqvwdqw |1 Zh lqwurgxfh ] ] i i @ n i oq gf / @ n fn i oq gf +616;, | | dqg @ Vn oq i
|
pn @ 7
] ] ]5 ] @5 i 3 i 43 3 43 i i i i 4 j vlq gg%gf4 gf > oq 4 ii 3
3
+616<,
vr wkdw wkh fruuhvsrqglqj wudqvsruw htxdwlrq iru uhdgv C C n . @ = Cw C{n
+6173,
Forvhu h{dplqdwlrq vkrzv wkdw wkh froolvlrq whup +616<, fdqqrw eh qhjdwlyh +uhfdoo wkdw j 3/ 3/ vlq 3 iru lq ^3> @5`,1 ydqlvkhv li oq i lv d froolvlrqdo lqyduldqw/ dv lv wkh fdvh lq htxloleulxp +Vhf1 615,/ vr wkdw 3=
617 Hqwurs| dqg wkh K0wkhruhp
6<
Wkxv/ dozd|v kdv d srvlwlyh surgxfwlrq zklfk ydqlvkhv lq htxloleulxp1 Df0 fruglqjo|/ Kfdq rqo| jurz lq dq lvrodwhg v|vwhp/ zkhuh qr x{ ryhu wkh vxuidfh lv doorzhg + CY n qn gD @ 3,/ dqg uhdfkhv lwv pd{lpxp lq htxloleulxp/ zkhuh @ 31 Wklv surshuw| ri dqg lq sduwlfxodu wkh gh qlwh vljq ri duh nqrzq dv wkh K0wkhruhp17 Wkh K0wkhruhp lv htxlydohqw wr wkh vhfrqg odz ri wkhuprg|qdplfv/ wkh hqwurs| odz/ zklfk zdv lqwurgxfhg rq sxuho| skhqrphqrorjlfdo jurxqgv/ vhh/ h1j1/ ^48`^4;`/ ru vwdqgdug hqjlqhhulqj wh{werrnv ^59`^5;`1 Wkhuhiruh lv wkh hqwurs| ghqvlw| ri wkh jdv/ dqg zh zulwh @ v > zkhuh v ghqrwhv wkh vshfl f hqwurs|1 dv jlyhq lq +616;,5 lv wkh hqwurs| x{/ dqg wkh qrq0frqyhfwlyh hqwurs| x{ !n @ n vyn fdq eh frpsxwhg dffruglqj wr ] i ! @ n Fn i oq gf = +6174, | Wklv jlyhv wkh vhfrqg odz lq wkh irup
Gv C!n . @ 3> Gw C{n
+6175,
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619 Nlqhwlf Prghov
78
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Phdq ydoxhv ri wkh froolvlrq iuhtxhqf| fdq eh frpsxwhg dffruglqj wr ] ] ] 4 4 @ igf @ i i 4 j vlq g g% gf4 gf > +6183, q q dqg wkxv rqh qgv zlwk wkh Pd{zhooldq +frpsduh wr Vhf1 416 dqg Sureohp 519, s s 8 j @ 49 g5 q dqg @ 55 3 > +6184, KV @ g5 q zkhuh 3 lv d frqvwdqw1 Htxdwlrq +6184, prvw qrwdeo| jlyhv wkh ghshqghqfh ri wkh phdq froolvlrq iuhtxhqf| rq ghqvlw| dqg whpshudwxuh1 Qrwh wkdw iru Pd{zhoo prohfxohv wkh froolvlrq iuhtxhqf| ghshqgv rqo| rq 38 / dqg wkdw wkh srzhu srwhqwldo $ 4 djuhhv zlwk ghqvlw|/ Pd{zhoo @ s kdug vskhuhv zkhq rqh gh qhv wkhlu gldphwhu wkurxjk g5 @ 34 p@ 49 1
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Zh vkdoo jr wkurxjk wkh phwkrg iru wkh HV0EJN prgho uvw/ zkhuh zh zloo frqvlghu wkh h{sdqvlrq xs wr vhfrqg rughu ^8:`1 Vlqfh wkh froolvlrq whup ri wkh HV0EJN prgho lv olqhdu lq i / dv rssrvhg wr wkh txdgudwlf lqwhjudo h{suhvvlrq lq wkh Erow}pdqq htxdwlrq/ wkh phwkrg lv frqvlghudeo| hdvlhu wr dsso| wr wkh HV0EJN htxdwlrq/ dqg wkxv ehfrphv pruh wudqvsduhqw1 Wkhq/ zh eulh | glvfxvv wkh dssolfdwlrq ri wkh FH phwkrg wr wkh Erow}pdqq htxdwlrq/ dqg uhsruw rq uhvxowv iurp olwhudwxuh1 Wkh uhvxowv rewdlqhg iru wkh HV0EJN prgho zloo eh frpsduhg wr wkrvh iru wkh ixoo Erow}pdqq htxdwlrq1 Ilqdoo|/ wkh K0wkhruhp +vhfrqg odz, zloo eh glvfxvvhg iru wkh Qdylhu0 Vwrnhv0Irxulhu dqg Exuqhww htxdwlrqv1 71415 Glphqvlrqohvv dqg vfdohg Erow}pdqq dqg HV0EJN htxdwlrqv Wkh Nqxgvhq qxpehu hqwhuv wkh Erow}pdqq htxdwlrq +6149, zkhq lw lv pdgh glphqvlrqohvv e| lqwurgxflqj glphqvlrqohvv yduldeohv dv s F 5 a q3 j > f @ Fa f > w @ O wa > @ g5 Jn a > J @ { > j @ 5Fa i @ 6 ia > { @ Oa n O F F t zkhuh q3 lv d uhihuhqfh qxpehu ghqvlw|/ F @ ; 3 lv wkh phdq yhorflw| ri d sduwlfoh dw uhihuhqfh whpshudwxuh 3 / dqg g lv dq h hfwlyh prohfxoh gldphwhu1 Zulwwhq lq whupv ri wkh glphqvlrqohvv txdqwlwlhv/ wkh Erow}pdqq htxdwlrq uhdgv a C ia C ia a n C i @ 4 Va +717, .J . fan C{ an Ca fn Nq C wa zlwk wkh glphqvlrqohvv froolvlrq whup ] ] 5 ] @5 ia3 ia43 iaia4 ja a vlq g g% ga f4 > Va @ 3
3
dqg wkh Nqxgvhq qxpehu Nq @
4 s @ = O O 5 g5 q3
ghqrwhv wkh phdq iuhh sdwk +415,1 Wkh wudqvsruw htxdwlrq iru wkh HV0EJN prgho lv rewdlqhg e| uhsodflqj wkh Erow}pdqq froolvlrq whup V zlwk wkh HV0EJN froolvlrq whup +6189,1 Lq glphqvlrqohvv irup wkh htxdwlrq djuhhv zlwk +717,/ li rqh fkrrvhv F @ = Va $ VaHV @ ia iaHV dqg Nq @ O O Wkxv/ wkh Nqxgvhq qxpehu Nq vfdohv wkh froolvlrq whupv ri Erow}pdqq dqg HV0EJN htxdwlrqv +717,1 Wkh FH phwkrg lv dq h{sdqvlrq phwkrg iru vpdoo Nqxgvhq qxpehuv/ dqg fdq eh shuiruphg rq wkh glphqvlrqohvv htxdwlrq +717,1
89
7 Wkh Fkdspdq0Hqvnrj phwkrg
Krzhyhu/ wr dyrlg glphqvlrqohvv yduldeohv/ wkh phwkrg lv riwhq shuiruphg rq wkh glphqvlrqdo Erow}pdqq htxdwlrq lq zklfk d vfdolqj sdudphwhu % lv lqwurgxfhg vr wkdw 4 Ci Ci Ci . fn +718, . Jn @ V= Cw C{n Cfn % Wkh sdudphwhu % lv d irupdo vpdooqhvv sdudphwhu zklfk sod|v wkh uroh ri wkh Nqxgvhq qxpehu iru prqlwrulqj wkh rughu ri whupv dqg txdqwlwlhv dsshdulqj lq wkh htxdwlrqv1 Dw wkh hqg ri doo fdofxodwlrqv % zloo eh vhw wr xqlw| zklfk lv htxlydohqw wr uh0lqvhuwlqj wkh glphqvlrqv lqwr wkh fruuhvsrqglqj glphqvlrqohvv htxdwlrqv wr uhpryh wkh Nqxgvhq qxpehu1 Vlqfh wkh uljkw kdqg vlgh ri wkh vfdohg Erow}pdqq htxdwlrq +718, pxvw uhpdlq qlwh/ lw iroorzv wkdw V $ 3 lq wkh olplw % $ 3 +ru Nq $ 3, zklfk lpsolhv wkdw +719, i @ iP dv % $ 3 > iP lv wkh orfdo Pd{zhooldq +6154,1 71416 Wkh Fkdspdq0Hqvnrj h{sdqvlrq Wkh edvlf lghd ri wkh Fkdspdq0Hqvnrj phwkrg lv wr h{sdqg wkh skdvh ghqvlw| i lqwr d vhulhv lq wkh +irupdo, vpdooqhvv sdudphwhu % dv imFH @ i +3, . %i +4, . %5 i +5, . %6 i +6, . =
+71:,
Dq lpsruwdqw frqglwlrq rq wkh h{sdqvlrq +71:, lv wkdw wkh pdlq yduldeohv XD @ i > yl > jD duh wkh vdph dw dq| ohyho ri h{sdqvlrq/ zklfk lpsolhv wkdw ] ] ] p 6 +3, +3, @ p i gf > yl @ p fl i gf > @ F 5 i +3, gf > +71;, 5 5 dqg
]
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+71<, iru doo ydoxhv ri %1 Wkh frqglwlrqv +71<, duh nqrzq dv wkh frpsdwlelolw| frqgl0 wlrqv1 Iurp +719,0+71<, iroorzv wkdw i +3, pxvw eh wkh orfdo Pd{zhooldq/ F5 4 +3, h{s i @ iP @ s = +7143, p 5 6 5 Vwuhvv dqg khdw x{/ krzhyhu/ fkdqjh zlwk wkh rughu ri wkh h{sdqvlrq dv +4,
+5,
+6,
+4,
lm @ % lm . %5 lm . %6 lm . > tl @ %tl zkhuh
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Fkl Fml i +, gf > tl
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p 5
+5,
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F 5 Fl i +, gf =
+7145,
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715 Wkh FH h{sdqvlrq iru wkh HV0EJN prgho
8:
Khuh zh xvhg wkh idfw wkdw vwuhvv dqg khdw x{ ydqlvk iru wkh Pd{zhooldq/ +3, +3, lm @ tl @ 31 Lqvhuwlqj wkh h{sdqvlrq +7144, lqwr wkh frqvhuydwlrq odzv +714,/ dqg vroylqj iru wkh wlph ghulydwlyhv/ jlyhv +,
Cyn Gyl 4 Cs 4 [ C ln G @ @ Jl > % > Gw C{n Gw C{l C{n +,
+,
5 Cym 5 4 [ Ctn 5 [ lm Cyl G @ % % = Gw 6 C{m 6 C{n 6 C{m Wklv doorzv xv wr zulwh wkh wlph ghulydwlyhv irupdoo| dv d vhulhv lq %/ Gyl [ G yl G [ G G [ G @ > @ > @ % % % Gw Gw Gw Gw Gw Gw zlwk Cyn 5 Cym G3 yl 4 C G3 G3 @ @ Jl @ > > Gw C{n Gw C{l Gw 6 C{m
+7146,
dqg/ iru 4/ +,
G G yl 4 C ln @3 > @ Gw Gw C{n
+,
+,
5 lm Cyl G 5 4 Ctn @ > = +7147, Gw 6 C{m 6 C{n
Wkh Pd{zhooldq iP ghshqgv rq wkh yduldeohv XD dqg wkh deryh h{sdqvlrq fdq eh xvhg wr h{sdqg lwv wlph ghulydwlyh dffruglqj wr G3 iP G4 iP G iP CiP G XD GiP @ .% . zkhuh @ = Gw Gw Gw Gw CXD Gw
+7148,
Wkh wlph ghulydwlyhv ri i +, fdq eh h{sdqghg vlploduo|/ dv G3 i +, G4 i +, Gi +, @ .% . > Gw Gw Gw
+7149,
wkh phdqlqj ri G i +, @Gw zloo eh glvfxvvhg odwhu1 Wkh deryh uhodwlrqv zloo eh qhhghg dv wkh FH phwkrg surfhhgv vwhs e| vwhs1
715 Wkh FH h{sdqvlrq iru wkh HV0EJN prgho 71514 Jhqhudo dujxphqwv Zulwwhq zlwk wkh frqyhfwlyh wlph ghulydwlyh/ wkh vfdohg HV0EJN wudqvsruw htxdwlrq uhdgv
8;
7 Wkh Fkdspdq0Hqvnrj phwkrg
4 Gi Ci Ci . Fn . Jn @ +i iHV , = Gw C{n Cfn %
+714:,
Iru wkh dssolfdwlrq ri wkh FH phwkrg/ wkh dqlvrwurslf Jdxvvldq iHV +618:, pxvw eh h{sdqghg lq % dv zhoo/ +3,
+4,
+5,
iHV @ iHV . %iHV . %5 iHV .
+714;,
Wkh qh{w vwhs lv wr lqvhuw wkh h{sdqvlrqv +71:/ 714;, lqwr wkh HV0EJN htxdwlrq dqg wr xvh +7148/ 7149,1 Zh vkdoo eh lqwhuhvwhg lq d vhfrqg rughu wkhru|/ dqg iru wklv zh qhhg wr lqfoxgh whupv xs wr uvw rughu lq %/ wkdw lv CiP CiP G3 iP +4, . Fn . Jn . i +4, iHV Gw C{n Cfn +4, G4 iP G3 i Ci +4, Ci +4, +5, .% . . Fn . Jn . i +5, iHV @ R %5 = Gw Gw C{n Cfn +714<, Wklv htxdwlrq doorzv xv wr frpsxwh i +4, dqg i +5, 1 Iru d kljkhu rughu wkhru|/ l1h1 wr frpsxwh i +, iru 6/ pruh whupv lq wkh deryh htxdwlrq zrxog kdyh wr eh frqvlghuhg1 +, Wkh iHV lq +714;, duh frpsxwhg iurp iHV dv iroorzv= iHV frqwdlqv wkh lqyhuvh ri wkh pdwul{ +618;,/ iru zklfk lqvhuwlrq ri +7144, |lhogv e +4, +5, % lm . %5 lm . = lm @ lm . Wd|oru h{sdqvlrq zlwklq whupv ri vhfrqg rughu jlyhv % & 5 %5 e +4, +4, 6 nq nq > ghw ^lm ` @ 4 5 s e +4, lm e +4, +4, +5, 4 5 e % lm . % lm > lm @ s s s ln nm dqg wkhuhiruh e +4, +3, +4, iHV @ iP > iHV @ iP lm Fkl Fml > +7153, 5 5 % & +4, +4, lm no 5 +4, +4, s +5, e5 +5, +4, +4, lm Fl Fm . Fl Fm Fn Fo = iHV @ iP 5 nq nq 7s ln nm e 55 71515 ]hurwk rughu= Hxohu htxdwlrqv Wkh vroxwlrq dw }hurwk rughu zdv rewdlqhg douhdg| lq +7143, zkhuh zh irxqg +3,
+3,
+3,
i +3, @ iHV @ iP dqg lm @ tl
@3=
715 Wkh FH h{sdqvlrq iru wkh HV0EJN prgho
8<
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+7154,
71516 Iluvw rughu= Qdylhu0Vwrnhv0Irxulhu htxdwlrqv Wr rewdlq wkh uvw rughu h{suhvvlrq iru wkh skdvh ghqvlw|/ zh qhhg wr frqvlghu wkh whupv ri rughu %3 lq +714<,/ zklfk zh zulwh iru frqyhqlhqfh dv 4 C oq iP C oq iP G3 oq iP +4, . Fn . Jn i +4, @ iHV iP = Gw C{n Cfn Wkh gl huhqwldo giP ri wkh Pd{zhooldq lv jlyhq e| 5 F 6 g Fl g . g +fl yl , giP @ iP g +oq iP , @ iP 5 5 dqg zkhq wkh qhfhvvdu| ghulydwlyhv duh frpsxwhg/ zh rewdlq e +4, Fkn Fol 5 5 no 5 8 4 C 4 Fkn Fol Cykn F iP . Fl C{ol 5 5 C{n 5 Cyq 6 4 G3 F 4 G3 5 Cyq . . . . Gw C{q 5 5 Gw 6 C{q 4 C Fn G3 yn . Jn . = Gw C{n
i +4, @ iP
Wkh odvw wzr olqhv ydqlvk gxh wr wkh gh qlwlrq iru G3 XD @Gw +7146,/ dqg wkh uhvxow iru wkh uvw rughu frqwulexwlrq wr wkh skdvh ghqvlw| lv 5 8 4 C e +4, 4 Fkn Fol Cykn F F F i . F = i +4, @ iP P n kn ol C{ol 5 5 C{n 5 5 no Lw lv vwudljkwiruzdug wr vkrz wkdw i +4, vdwlv hv wkh frpsdwlelolw| uhodwlrqv +71<,1 Vwuhvv dqg khdw x{ dw uvw rughu iroorz iurp +7145, dv6 +4,
lm @
Cykl 5 s Cykl 8 s C C +4, @ 5 dqg tl @ @ = +7155, 4 e C{nl C{nl 5 C{l C{l
Wkh frqvwlwxwlyh uhodwlrqv +7155, duh wkh odzv ri Qdylhu0Vwrnhv dqg Irxulhu zkhuh
Uhfdoo wkdw zh hpsor| hqhuj| xqlwv iru whpshudwxuh1 Wkxv/ wkh khdw frqgxfwlylw| & V lv uhodwhg wr wkh xvxdo khdw frqgxfwlylw| V dv V ' 6 V1
93
7 Wkh Fkdspdq0Hqvnrj phwkrg
8s 4 s dqg @ +7156, 4 e 5 duh ylvfrvlw| dqg wkhupdo frqgxfwlylw|/ uhvshfwlyho|1 Wkh Sudqgwo qxpehu +41:, lv uhodwhg wr wkh frh!flhqw e ri wkh HV0EJN prgho e| @
Su @
4 8 @ = 5 4e
Lwv phdvxuhg ydoxh/ Su @ 5@6/ lv rewdlqhg iru e @ 4@51 Wkh fodvvlfdo EJN prgho +6187, lv rewdlqhg iru e @ 3 zklfk |lhogv Su @ 4/ l1h1 wkh fodvvlfdo EJN prghov idlov lq surgxflqj wkh surshu Sudqgwo qxpehu1 Iru srzhu srwhqwldov dqg kdug vskhuhv wkh froolvlrq iuhtxhqf| lv ri wkh irup .6 +6184,1 Lw iroorzv wkdw wkh ylvfrvlw| lv d ixqfwlrq @ 3 s $ zkhuh $ @ 55 ri whpshudwxuh dorqh/ jlyhq e| $ > +7157, @ 3 3 $ 3 @ Su 3 3 lv wkh ylvfrvlw| dw d uhihuhqfh whpshudwxuh 3 1 $ zloo eh ghqrwhg dv ylvfrvlw| h{srqhqw1 Phdvxuhphqwv frq up wkdw ghshqgv rq whpshudwxuh/ exw qrw rq ghqvlw|/ dqg jlyh $ * 3=;/ fruuhvsrqglqj wr d srzhu srwhqwldo zlwk * 56@61 Iru kdug vskhuh prohfxohv + $ 4, dqg Pd{zhoo prohfxohv + @ 8, rqh qgv $ KV @ 4@5 dqg $8 @ 4/ uhvshfwlyho|1 E| phdqv ri wkh Qdylhu0Vwrnhv dqg Irxulhu odzv +7155,/ wkh skdvh ghqvlw| fdq eh uhzulwwhq dv 5 8 4 C $ Fl Fn Cykl F +4, . Fn = +7158, i @ 3 iP Su s C{nl 5 5 C{n
Wklv irup lv prvw frqyhqlhqw iru wkh qh{w vwhs lq wkh h{sdqvlrq1 71517 Vhfrqg rughu= Exuqhww htxdwlrqv Wkh vhfrqg rughu fruuhfwlrq wr wkh skdvh ghqvlw| lv rewdlqhg iurp +714<, dv G3 i +4, 4 G4 iP Ci +4, Ci +4, +5, +5, . . Fn . Jn i @ iHV = Gw Gw C{n Cfn Wr rewdlq dq h{solflw h{suhvvlrq iru i +5, lq whupv ri wkh yduldeohv XD @ + > yl > , dqg wkhlu judglhqwv/ uhtxluhv wr wdnh wkh ghulydwlyhv ri i +4, dqg wr xvh +7148,1 Pruhryhu/ iurp wkh suhylrxv vhfwlrq zh nqrz wkdw i +4, @ i +4, +XD > CXD @C{n ,/ vr wkdw Ci +4, G3 XD Ci +4, G3 CXD G3 i +4, @ . > Gw CXD Gw Gw C{n D C CX C{n zkhuh
715 Wkh FH h{sdqvlrq iru wkh HV0EJN prgho
G3 Gw
CXD C{n
@
94
C G3 XD CXD Cyu = C{n Gw C{u C{n
Wkh frpsxwdwlrq ri i +5, lv vwudljkwiruzdug/ exw whglrxv/ vhh ^8:` iru ghwdlov lqfoxglqj wkh h{solflw h{suhvvlrq ri i +5, 1 Wkh uhvxowlqj h{suhvvlrqv iru vwuhvv dqg khdw x{ duh wkh Exuqhww htxdwlrqv iru wkh HV0EJN prgho ^86`^5`/ Cyn Cyml 5 Cyn Cyn C 4 Cs . '4 Vlm '5 .5 Vmln s C{n C{kl C{ml C{kl C{n C{kl 5 C C C oq s 4 C C .'6 . '7 . '8 . '9 Vnkl Vmln > +7159, C{kl C{ml C{kl C{ml C{kl C{ml
+5,
lm @
+5,
tl
@
5 5 Cyn C oq Cyn C oq Cyn C oq 5 C 5 yn 5 . .5 4 C{n C{l 6 C{n C{l 6 C{n C{l C{l C{n C oq s CVln C oq . 7 . 68 Vln > +715:, . 6 Vln C{n C{n C{n
zlwk wkh deeuhyldwlrq Vlm @
Cykl 4 Cyl 4 Cym 4 Cyn @ . lm C{ml 5 C{m 5 C{l 6 C{n
iru wkh wudfh iuhh yhorflw| judglhqw/ dqg wkh Exuqhww frh!flhqwv ^86`^8:` 7 : 5 5$ $ > '5 @ 5> '6 @ > '7 @ 3> '8 @ > '9 @ ;> '4 @ 6 5 Su Su 8 8 5 5 : $ > 5 @ > 4 @ +4 e,5 > 6 @ > 7 @ 6 5 Su Su 5 Su5 5 : 4 8 @ 4. .$ = +715;, 6 Su 5 Su Wkh Exuqhww frh!flhqwv +715;, ghshqg rq wkh wzr sdudphwhuv $ dqg e/ zklfk gh qh wkh froolvlrq whup1 Wkh ylvfrvlw| h{srqhqw $ hqwhuv wkh htxdwlrqv wkurxjk wkh whpshudwxuh ghshqghqfh ri wkh froolvlrq iuhtxhqf| wkdw dsshduv lq i +4, / vhh +7158,1 Iru Su @ 5@6 wkh| uhgxfh wr 7 : $ > ' 5 @ 5 > '6 @ 6 > '7 @ 3 > '8 @ 6$ > '9 @ ; > '4 @ 6 5 48 : 78 68 $ > 5 @ > 6 @ 6 > 7 @ 6 > 8 @ . $ = +715<, 4 @ 7 5 ; 7
95
7 Wkh Fkdspdq0Hqvnrj phwkrg
716 FH h{sdqvlrq iru srzhu srwhqwldov 71614 Iluvw rughu= Qdylhu0Vwrnhv0Irxulhu odzv H{sdqvlrq Zh eulh | glvfxvv wkh FH h{sdqvlrq iru wkh ixoo Erow}pdqq htxdwlrq wr uvw rughu/ zklfk jlyhv ghwdlohg h{suhvvlrqv iru ylvfrvlw| dqg khdw frqgxfwlylw| lq ghshqghqfh ri wkh lqwhudfwlrq srwhqwldo1 Dw uvw rughu/ wkh skdvh ghqvlw| lv frqyhqlhqwo| zulwwhq dv i @ i +3, . %i +4, @ iP +4 . %*, > dqg gxh wr wkh frpsdwlelolw| frqglwlrqv +71<, * pxvw reh| ] ] ] p F 5 *iP gf = 3 @ p *iP gf > 3 @ p fl *iP gf > 3 @ 5
+7163,
+7164,
Wkh Erow}pdqq htxdwlrq zlwk doo whupv ri rughu %4 dqg kljkhu rplwwhg dvvxphv wkh irup +olqhdul}hg Erow}pdqq htxdwlrq7 , ] 3 G3 iP CiP CiP 4 .Fn * . *43 * *4 j vlq gg%gf4 = .Jn @ iP iP Gw C{n Cfn +7165, 3 43 4 iP @ iP iP 1 Wkh ohiw kdqg vlgh lv Iru wkh ghulydwlrq lw zdv xvhg wkdw iP uhdglo| hydoxdwhg/ dqg wkh uhvxowlqj lqwhjudo htxdwlrq iru * uhdgv/ zlwk Fl @ s 5 l / s Cykl 8 C oq kl ml 5 . 5 @ L ^*` +7166, 5 C{ml 5 l C{l zlwk wkh lqwhjudo rshudwru ] 3 4 * . *43 * *4 j vlq gg%gf4 = L ^*` @ iP
+7167,
Wkxv/ wkh Fkdspdq0Hqvnrj h{sdqvlrq iru wkh Erow}pdqq htxdwlrq lqyroyhv vroylqj dq lqwhjudo htxdwlrq/ zklfk dggv dgglwlrqdo gl!fxow| dv frpsduhg wr dsso|lqj wkh phwkrg wr wkh HV0EJN prgho1 Rqo| iru Pd{zhoo prohfxohv fdq dq h{dfw vroxwlrq ri wkh lqwhjudo htxdwlrq eh irxqg/ zkloh iru rwkhu prohfxoh prghov dssur{lpdwlrqv pxvw eh xvhg1
e
Wkh olqhdu Erow}pdqq htxdwlrq dsshduv iru ghwhuplqlqj wkh uvw rughu frqwulex0 wlrq s E ' s )1 Ilqglqj kljkhu rughu whupv uhtxluhv wr frqvlghu wkh qrq0olqhdu Erow}pdqq htxdwlrq1
716 FH h{sdqvlrq iru srzhu srwhqwldov
96
Ghfrpsrvlwlrq ri sureohp Dq| vroxwlrq ri wkh lqwhjudo htxdwlrq +7166, fdq eh vsolw lqwr d krprjhqhrxv dqg dq lqkrprjhqhrxv sduw/ *+k, dqg *+l, / uhvshfwlyho|1 Wkh vroxwlrq *+k, ri wkh krprjhqhrxv lqwhjudo htxdwlrq L ^*` @ 3 pxvw eh d olqhdu frpelqdwlrq ri wkh froolvlrqdo lqyduldqwv/ *+k, @ . l l . 5 > krzhyhu/ wkh frpsdwlelolw| frqglwlrqv +7164, lpso| @ @ 31 Wkh lqkrpr0 Cykl oq jhqhrxv vroxwlrq *+l, pxvw eh olqhdu lq CC{ dqg C{ml / vlqfh +7166, lv olqhdu lq l */ dqg wkhuhiruh *+l, pxvw uhdg s C oq Cykl E +, kl ml = *+l, @ D +, l 5 C{l C{ml Wkh judglhqwv ri yhorflw| dqg whpshudwxuh fdq eh frqwuroohg lqghshqghqwo|/ dqg wkxv wkh xqnqrzq ixqfwlrqv D +, dqg E +, vroyh l k 8 5 l dqg L E +, kl ml @ 5 kl ml = +7168, L ^D l ` @ 5 Wkh frpsdwlelolw| uhodwlrq +7164,5 dvvxphv wkh irup ] s C oq 3 @ p l n n D +, 5 iP gf > C{n vrswkdw n pxvw srlqw lq wkh gluhfwlrq ri wkh whpshudwxuh judglhqw/ n @ oq 1 Zlwkrxw orvv ri jhqhudolw| zh fdq vhw @ 3 +wkdw lv zh devrue 5 CC{ n lqwr D, dqg kdyh wkh qhz frpsdwlelolw| frqglwlrq ] 3 @ p 5 D +, iP gf = +7169, Ylvfrvlw| dqg khdw frqgxfwlylw| Zh xvh wkh uhvxowv rewdlqhg vr idu wr h{suhvv ylvfrvlw| dqg khdw frqgxfwlylw| wkurxjk wkh ixqfwlrqv D +, dqg E +,1 Wkh glvwulexwlrq ixqfwlrq lv +7163, s C oq Cykl E +, kl ml > +716:, i @ iP 4 D +, l 5 C{l C{ml dqg frpsxwdwlrq ri vwuhvv dqg khdw x{ jlyhv wkh odzv ri Qdylhu0Vwrnhv dqg Irxulhu dv Cykl C dqg tl @ lm @ 5 C{ml C{l zlwk
97
7 Wkh Fkdspdq0Hqvnrj phwkrg
@ 5 @ 6@5 s 6
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h{s
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5
5 s 48 6@5
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9 Judg*v prphqw phwkrg
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9 Judg*v prphqw phwkrg
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D @ p i4> fl > Fl Fm > = = = > Fl4 Fl5 Fl j > @ 4> 5> 6> = = =/ fruuhvsrqglqj wr 7 > 43 > 53> 68 > 89 > ;7 > 453 > 498 > 553 > 5;9 > 697> prphqwv1 Wkh 80prphqw fdvh lv wulyldo/ vlqfh wkh fruuhvsrqglqj skdvh ghqvlw| lv wkh Pd{zhooldq/ zklfk |lhogv wkh Hxohu htxdwlrqv +7154,1 Zh vkdoo frqvlghu wkh Judg phwkrg lq ghwdlo iru 46 dqg 59 prphqwv/ dqg wkhq glvfxvv lwv dssolfdwlrq wr odujhu qxpehuv ri yduldeohv1 91415 Judg*v 46 prphqw htxdwlrqv Wkh 46 prphqw wkhru| lv edvhg rq wkh yduldeohv ] 6 ^46` ^46` @ D i gf @ > yl > x @ > lm > tl > x 5
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<3
9 Judg*v prphqw phwkrg
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+917,
soxv wkh prphqw htxdwlrqv iru vwuhvv lm @ x3lm dqg khdw x{ tl @ 45 x4l zklfk iroorz iurp +8148/ 8149, dv5 Cx3lmn Cykl Cyml 7 Ctkl G lm Cyn 3 . . . 5 nkl . lm . 5 @ Slm > Gw C{n 8 C{ml C{n C{n C{ml
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s 4 5s 3 tl = @ lm dqg Sl4 @ Slm 5 6
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914 Forvhg prphqw v|vwhpv e| Judg*v phwkrg
<4
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8 C oq ln C no C ln C Gtl 8 C . . ln ln . Gw 5 C{l 5 C{n C{n C{o C{n : Cyn : Cyl 5 Cyn 5s tl = +9147, . tl . tn . tn @ 8 C{n 8 C{n 8 C{l 6 91416 Judg*v 59 prphqw htxdwlrqv Wkh 59 prphqw fdvh dggv wkh xqnqrzq prphqwv iurp wkh 46 prphqw fdvh wr wkh olvw ri yduldeohv/ zklfk frqvlghuv wkh 46 htxdwlrqv iru wkh 46 prphqwv deryh +917919, soxv wkh prphqw htxdwlrqv iru wkh rwkhu yduldeohv wkdw dsshdu lq wkrvh/ wkdw lv 6 ^59` 3 4 5 x @ > yl > x @ > lm > tl > xlmn > xlm > z = +9148, 5 Wkh dgglwlrqdo htxdwlrqv iru x3lmn > x4lm > z5 iroorz iurp Fkdswhu 8 dqg duh qrw sulqwhg khuh1 Wkh| frqwdlq wkh dgglwlrqdo prphqwv x5n > x4lmn > x3lmno > zklfk pxvw/ djdlq/ eh uhodwhg wr wkh yduldeohv e| phdqv ri Judg*v glvwulexwlrq ixqfwlrq1 Wkh dssursuldwh Judg glvwulexwlrq iru wkh 59 prphqw fdvh lv
<5
9 Judg*v prphqw phwkrg
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+914:,
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Ri frxuvh/ wkh rwkhu htxdwlrqv iru wkh 59 prphqw fdvh duh +917919,1 91417 H{whqghg prphqw vhwv Qrz zh frqvlghu jhqhudo vhwv ri prphqw htxdwlrqv iru wkh prphqwv ] ] zd @ p F 5d imJ iP gf > xdl4 lq @ p F 5d Fkl4 Fl5 Flq l imJ gf > +9154,
915 Uhpdunv rq Judg0w|sh htxdwlrqv
<6
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<7
9 Judg*v prphqw phwkrg
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915 Uhpdunv rq Judg0w|sh htxdwlrqv
<8
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<9
9 Judg*v prphqw phwkrg
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916 FH h{sdqvlrq ri Judg*v prphqw htxdwlrqv 91614 Judg dqg FH skdvh ghqvlw| iru Pd{zhoo prohfxohv dqg EJN prgho Vlqfh wkh ixoo +lq qlwh, prphqw v|vwhp lv htxlydohqw wr wkh Erow}pdqq htxd0 wlrq/ rqh zloo h{shfw wkdw wkh Fkdspdq0Hqvnrj h{sdqvlrq ri wkh prphqw htxdwlrqv zloo jlyh wkh vdph uhvxowv dv wkh Fkdspdq0Hqvnrj h{sdqvlrq ri wkh Erow}pdqq htxdwlrq lwvhoi1 Krzhyhu/ wkh wuxqfdwhg prphqw v|vwhpv ri wkh Judg phwkrg douhdg| frqwdlq hqrxjk hohphqwv ri wkh Erow}pdqq htxdwlrq wr
Wkh Pdfk qxpehu lv gh qhg dv wkh udwlr ri lq rz yhorflw| dqg wkh orfdo vshhg ri vrxqg1 Wkh fulwlfdo yhorflwlhv duh wkh hljhqydoxhv ri & ?& / hydoxdwhg lq htxlole0 ulxp1
916 FH h{sdqvlrq ri Judg*v prphqw htxdwlrqv
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+5,
lm @ % lm . %5 lm
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=
Htxdwlqj whupv ri wkh vdph srzhu lq wkh vpdooqhvv sdudphwhu % zh qg dw uvw rughu wkh odzv ri Qdylhu0Vwrnhv dqg Irxulhu/ +4,
lm @ 5
Cykl 48 C +4, @ 5Vlm dqg tl @ > C{ml 7 C{l
dqg wkh vhfrqg rughu htxdwlrqv uhdg dw uvw 5 6 +4, +4, Ct G Cy 7 Cy 3 kl ml lm n8 +5, +4, +4, . lm @ 7 . 5 nkl . lm > s Gw 8 C{ml C{n C{n
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+9159,
Wkh uhvxowlqj htxdwlrqv duh mxvw wkh Exuqhww htxdwlrqv +7159/ 715:, 5 Cyn Cyn C 4 Cs Cyn Cyml +5, '4 Vlm '5 .5 Vmln . lm @ s C{n C{kl C{ml C{kl C{n C{kl 5 C C C oq s 4 C C .'6 . '7 . '8 . '9 Vnkl Vmln C{kl C{ml C{kl C{ml C{kl C{ml +5,
tl
@
5 5 Cyn C oq Cyn C oq Cyn C oq 5 C 5 yn 5 . .5 4 C{n C{l 6 C{n C{l 6 C{n C{l C{l C{n C oq s CVln C oq . 7 . 68 Vln . 6 Vln C{n C{n C{n
zlwk wkh Exuqhww frh!flhqwv +715<,/ 7 : $ > ' 5 @ 5 > '6 @ 6 > '7 @ 3 > '8 @ 6$ > '9 @ ; > '4 @ 6 5 48 : 78 68 $ > 5 @ > 6 @ 6 > 7 @ 6 > 8 @ . $ = +915:, 4 @ 7 5 ; 7 Wkhuhiruh wkh Exuqhww htxdwlrqv duh frqwdlqhg zlwklq Judg*v 46 prphqw wkh0 ru|/ dqg fdq eh h{wudfwhg e| phdqv ri wkh Fkdspdq0Hqvnrj phwkrg17 Wkh vxshu0Exuqhww htxdwlrqv/ krzhyhu/ fdqqrw eh ghulyhg iurp wkh 46 prphqw v|vwhp> wkhlu ghulydwlrq uhtxluhv wkh vhw ri 59 prphqwv1 Wkh forvh uhodwlrq ehwzhhq Exuqhww dqg Judg*v 46 prphqw htxdwlrqv vxjjhvwv wkdw erwk vhwv kdyh wkh vdph udqjh ri dssolfdelolw|1 Wklv phdqv lq sduwlfxodu wkdw wkrvh surfhvvhv zklfk duh nqrzq wr eh ehvw ghvfulehg e| d odujh qxpehu ri prphqwv suredeo| fdqqrw eh ghvfulehg dffxudwho| e| wkh Exuqhww htxdwlrqv1 e
Judg lq ^<` dwwulexwhv wklv uhvxow wr wkh 4<89 hglwlrq ri wkh fodvvlfdo wh{werrn ri Fkdspdq dqg Frzolqj1 Krzhyhu/ qrwklqj rq wklv uhodwlrq fdq eh irxqg lq wkh odvw hglwlrq ^5`
916 FH h{sdqvlrq ri Judg*v prphqw htxdwlrqv
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91616 59 prphqwv Wkh vxshu0Exuqhww htxdwlrqv fdq eh ghulyhg iurp wkh 59 prphqw htxdwlrqv1 Wkh frpsxwdwlrqv duh fxpehuvrph/ exw vwudljkwiruzdug/ dqg zhuh rqo| shu0 iruphg iru wkh olqhdul}hg htxdwlrqv lq wkuhh glphqvlrqv/ dqg iru wkh qrq0 olqhdu htxdwlrqv lq rqh glphqvlrq/ vhh wkh htxdwlrqv sulqwhg lq Vhf1 716161 Wkh ghulydwlrq ri wkh vxshu0Exuqhww htxdwlrqv yld wkh prphqw v|vwhp lv e| idu pruh frqyhqlhqw wkdq ghulylqj wkhp iurp wkh Erow}pdqq htxdwlrq1 Surjudpv U U ru Pdsoh / fdq eh xvhg wr iru v|perolf frpsxwdwlrq/ h1j1 Pdwkhpdwlfd idflolwdwh wkh frpsxwdwlrqv1 Khuh/ zh vkdoo rqo| frqvlghu wkh uvw rughu uhvxowv wr vkrz wkdw wkh| vwdqg lq djuhhphqw zlwk Judg*v 46 prphqw forvxuh +9145,1 Iru wklv/ zh lqwurgxfh wkh FH h{sdqvlrq ri wkh prphqwv/ +4,
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4+4,
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4+5,
. %5 xlm
lqwr wkh 59 prphqw htxdwlrqv/ wr rewdlq wkh QVI odzv +9158, dqg wkh uvw rughu h{sdqvlrq ri +9145,/ yl}1 3+4,
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@ 47
Cykl +4, @ : lm = C{ml
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111 dozd|v dvvxplqj wkdw wkh surgxfwlrqv duh nqrzq dv h{solflw ixqfwlrqv ri wkh prphqwv 111
433
9 Judg*v prphqw phwkrg ^4`
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^5`
48 C 78 G C Cyl :8 5 Cyn C C . . . . 7 C{l ; s Gw C{l C{n C{n ; s C{n C{l 5 CVln 5 5 C Cs CVnq . 48 Vln .6 6 Vln . 9 Vln = +9163, s C{n C{n s C{n s C{q
tl @
Wkh vhfrqg lwhudwlrq dggv txdgudwlf dqg fxelf whupv lq wkh ylvfrvlw|/ zklfk sod|v khuh wkh uroh ri d vpdoo sdudphwhu19 Kljkhu rughu lwhudwlrqv zloo eh rewdlqhg lq wkh vdph pdqqhu/ l1h1 lqvhuwlrq ri wkh odvw lwhudwh rq wkh ohiw kdqg vlgh/ dqg vroylqj iru wkh qhz lwhudwh rq wkh uljkw1 Reylrxvo|/ wkh kljkhu lwhudwhv zloo frqwdlq whupv zlwk kljkhu srzhuv lq 1 Wkh wklug lwhudwhv/ iru lqvwdqfh/ zloo uhfryhu wkh vdph vhfrqg rughu whupv wkdw duh suhvhqw lq wkh vhfrqg lwhudwhv +915 .6 s C{kl {ml s C{kl C{ml s s Gw 6 C{n +9164, ^5` lm
S
Lqghhg/ zkhq wkh vfdolqj sdudphwhu 0 lv lqwurgxfhg/ > zrxog eh uhsodfhg e| 0>/ ru e| wkh Nqxgvhq qxpehu k? lq d glphqvlrqohvv irupxodwlrq1 Wkh Pd{zhoo lwhudwlrq lv ohvv irupdo wkdq wkh FH phwkrg/ dqg grhv qrw hpsor| irupdo vpdooqhvv sdudphwhuv1
917 Uhlqhfnh0Nuhphu0Judg phwkrg
^5`
434
48 C :8 5 Cyn C 78 5 G C Cyn C Cs 5 . . 6 Vln 7 C{l ; s C{n C{l ; s Gw C{l C{l C{n s C{n 5 CVln 6< 5 78 5 C G 5 Cyn C Vln . .6 .6 . > +9165, C{n 7 s C{n ; s C{l Gw 6 C{n
tl @
Erwk htxdwlrqv frqwdlq klgghq wklug rughu whupv lq wkh wlph ghulydwlyhv1 Wr pdnh wklv ylvleoh/ zh xvh wkh frqvhuydwlrq odzv iru pdvv dqg hqhuj| wr zulwh wkh wlph ghulydwlyhv dv G3 Ctn G @ > Gw Gw C{n G3 C C 5 tn G C @ > Gw C{l Gw C{l C{l C{n G Cykl G3 Cykl C GVlm 4 C mln @ @ > Gw Gw C{ml Gw C{ml C{kl C{n zlwk G3 @Gw dv lq +9159,1 Reylrxvo|/ zlwklq vhfrqg rughu dffxudf| lw lv vxi0 flhqw wr uhsodfh wkh wlph ghulydwlyhv e| wkhlu }hurwk rughu dssur{lpdwlrq/ G3 G Gw $ Gw / dqg wkhq +9164/ 9165, uhgxfh wr wkh Exuqhww htxdwlrqv iru Pd{zhoo prohfxohv +7159/ 715:,1 Wkh Pd{zhooldq lwhudwlrq jlyhv wkh vdph uhvxowv dv wkh FH phwkrg rqo| zkhq vrph kljkhu rughu frqwulexwlrqv duh uhpryhg/ dv h{sodlqhg deryh1 Dv zdv srlqwhg rxw/ wkh lwhudwlrq dw rughu ^` frqwdlqv vrph whupv iurp kljkhu rughu lwhudwlrqv ^ . 4`/ exw qrw doo ri wkhvh1 Wkh rffxuuhqfh ri whupv ri kljkhu rughu lv qrw glvwxuelqj/ dv wkh| vkrxog sod| rqo| d pdujlqdo uroh zkhq wkh htxdwlrqv duh dssolhg lq wkhlu surshu udqjh/ dqg wkxv wkh kljkhu rughu whupv frxog eh ohiw lq wkh htxdwlrqv1 Lqghhg/ wkh kljkhu rughu whupv frxog khos lq vwdelol}lqj wkh Exuqhww htxdwlrqv +zklfk duh xqvwdeoh,/ exw wkdw vhhpv qrw wr kdsshq1 Dowrjhwkhu wkh FH h{sdqvlrq jlyhv d pruh gluhfw uhodwlrq ehwzhhq wkh rughu lq wkh Nqxgvhq qxpehu wkdq wkh Pd{zhoo lwhudwlrq/ dqg wkdw lv d uhdvrq wr suhihu lw1 Erwk phwkrgv |lhog wkh vwdeoh QVI odzv wr uvw rughu/ exw wkhlu kljkhu rughu htxdwlrqv +Exuqhww/ vxshu0Exuqhww, duh xqvwdeoh/ zklfk sxwv erwk phwkrgv lqwr txhvwlrq1
917 Uhlqhfnh0Nuhphu0Judg phwkrg Wkh suhylrxv vhfwlrqv fodul hg wkh uhodwlrq ehwzhhq wkh phwkrgv ri Judg dqg Fkdspdq0Hqvnrj iru wkh vshfldo fdvh ri Pd{zhoo prohfxohv1 Lq sduwlfxodu lw ehfdph fohdu wkdw wkh Fkdspdq0Hqvnrj h{sdqvlrq ri prphqw v|vwhpv |lhogv wkh vdph uhvxowv dv wkh h{sdqvlrq ri wkh Erow}pdqq htxdwlrq/ dqg lv vlpsohu e| idu1 Wkh reylrxv qh{w vwhs lv wr frqvlghu rwkhu w|shv ri lqwhudfwlrq/ l1h1 qrq0 Pd{zhooldq srwhqwldov1 Wkh jhqhudo FH h{sdqvlrq ri wkh Erow}pdqq htxdwlrq
435
9 Judg*v prphqw phwkrg
zdv glvfxvvhg lq Vhf1 716/ dqg jdyh wkh skdvh ghqvlw| dv dq vhulhv lq Vrqlqh sro|qrpldov/ # $ qe qd [ s C oq Cykl [ +u, 5 +u, 5 eu V 8 kl ml . du V 6 l 5 imFH @ iP 4 . = 5 5 C{ml u@4 C{l u@3 +9166, Uhlqhfnh dqg Nuhphu pxvw kdyh qrwlfhg wkdw wklv skdvh ghqvlw| kdv wkh vdph vwuxfwxuh dv d Judg skdvh ghqvlw|/ dqg fdq eh zulwwhq irupdoo| dv ^87`^<6`^<7` # $ qe qd [ [ uklml F 5u Fkl Fml . ul F 5u Fl = imJ @ iP 4 . u@3
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x{
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e Cyml 7 Ctkl G lm Cyn [ +5, zln . . 5 nkl . lm . F3e Gw 8 C{ml C{n C{n e e
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464
[ [ +5, k +5, l4 5 [ +5, 5 +5d . 8,$$ @ F3e e @ F3e Fed 3d +5d . 8,$$ @ 5 48 48 d@3 e@3
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4 @ 3 5
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465
; Rughu ri pdjqlwxgh dssurdfk
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Cykl C > tl @ > C{ml C{l
zkhuh ylvfrvlw| dqg khdw frqgxfwlylw| duh jlyhq e| +;1:/ ;143,1 Wkh fru0 uhvsrqglqj Sudqgwo qxpehu lv Su @ 58 43 1 Wkh uvw rughu htxdwlrqv frlqflgh zlwk wkh uvw rughu Fkdspdq0Hqvnrj h{sdqvlrq1 Lq uvw rughu wkh dssurdfk djuhhv zlwk wkh Uhlqhfnh0Nuhphu0Judg phwkrg +Vhf1 917, ^87` zkhuh ylvfrvlw| dqg khdw frqgxfwlylw| duh frpsxwhg lq dffrugdqfh zlwk wkh deryh irupxodh1 Lw lv zruwkzkloh wr kdyh d orrn dw wkh fruuhvsrqglqj skdvh ghqvlw|1 Dw uvw rughu/ wkh rqo| uhohydqw prphqwv duh wkh yhfwruv dqg 50whqvruv/ vr wkdw wkh skdvh ghqvlw| +9155/ 9156, uhgxfhv wr 5 imJ @ iP 74 .
D4 k l4 [ +4, Ede d>e@3
xel F 5d Fl d.e.4 D5 k l4 [ +5, Ede . d>e@3
6 xelm 5 d.e.5
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vr wkdw wkh skdvh ghqvlw| iru wkh QVI fdvh fdq eh zulwwhq dv
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;16 Wkh wudqvsruw htxdwlrqv zlwk 2?_ rughu dffxudf|
466
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467
; Rughu ri pdjqlwxgh dssurdfk
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+5,
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+4,
5>3>4 5>4>4 4>4>4 > \d>ef > Fdg > \d>ef = Fdg > \d>ef
Wkhvh iroorz e| xvlqj wkh uhgxfhg Judg ixqfwlrq +9155, % & D4 D5 [ [ d 5d d 5d imJ @ iP +4 . , @ iP 4 . l F Fl . no F Fkn Fol d@3 2
d@3
Wklv vwdwhphqw lqfoxghv wkh HV0EJN prgho iru h ' 2* 1
;16 Wkh wudqvsruw htxdwlrqv zlwk 2?_ rughu dffxudf|
468
d wr frpsxwh wkh froolvlrq prphqwv Sld / Slm dv rxwolqhg lq Vhf1 9181 Wkh qxpehuv D4 / D5 pxvw eh fkrvhq vxfk wkdw wkhlu ixuwkhu lqfuhdvh grhv qrw fkdqjh wkh ydoxhv ri wkh frh!flhqwv # / " frqvlghudeo|1 Iru Pd{zhoo prohfxohv dqg wkh EJN prgho/ wkh frh!flhqwv kdyh wkh ydoxhv
7 > #5 @ #6 @ #7 @ #8 @ #9 @ 3 > #: @ 4 8 8 5 8 > "5 @ 4 > "6 @ 4 > "7 @ > "8 @ 3 > "9 @ = "4 @ 5 8 5
#4 @
7 Ixuwkhupruh/ 43 @ 85 Su @ 48 iru Pd{zhoo prohfxohv1 Wklv fkrlfh ri frh!0 flhqwv uhvxowv lq Judg*v ruljlqdo 46 prphqw htxdwlrqv ^;`^<`1
;161: Exuqhww htxdwlrqv Wkh Exuqhww htxdwlrqv iroorz iurp +;14;/ ;14<, e| phdqv ri wkh Fkdspdq0 Hqvnrj h{sdqvlrq/ mxvw dv lq Vhf1 916/ zkhuh zh frqvlghuhg Judg*v 46 prphqw v|vwhp wr qg wkh Exuqhww htxdwlrqv iru Pd{zhoo prohfxohv1 Wkh Exuqhww frh!flhqwv duh uhodwhg wr wkh frh!flhqwv # / " dv 4 7 : 5 8 4 #4 8 4 #6 $ > '5 @ > '6 @ > '7 @ > '4 @ #: 6 5 #: 5 Su #: 5 Su #: 4 8 4 8 4 7 # 8 > '9 @ '8 @ +5 . #5 #9 , > #4 $ . #6 #7 #: 5 Su 5 Su #: 4 58 4 4 58 4 8 "5 : $ > 5 @ 4 @ > 6 @ > 5 5 "9 9 Su 5 "9 7 Su Su "9 " 4 8 4 4 8 4 > 8 @ +4 . "7 "8 , . "4 . "5 . "6 $ = 7 @ 8 6 "9 Su 6 "9 Su 5 Su Khuh wzr zhoo0nqrzq uhodwlrqv ehwzhhq wkh Exuqhww frh!flhqwv duh uhfryhuhg/ 5 : 5 : $ 5 > '4 @ $ '5 = 4 @ 6 5 6 5 Xqiruwxqdwho|/ wkh jhqhudol}hg 46 prphqw htxdwlrqv +;14;/ ;14<, uhtxluh pruh frh!flhqwv wkdq wkh Exuqhww htxdwlrqv/ dqg wkh # / " fdqqrw eh frpsohwho| ghwhuplqhg iurp wkh nqrzq uhvxowv iru wkh Exuqhww frh!flhqwv1 E| lqyhuvlrq iroorzv '6 '7 7 4 '9 7 > #5 #9 @ 5 > #6 @ Su > #4 @ Su 8 '5 5 '5 8 '5 '6 8 4 7 '8 '7 5 # . Su $ > #7 @ > #: @ 5 Su 8 8 '5 '5 '5 '5 8 4 8 4 6 8 4 7 4 6 +"7 "8 , . "4 @ $. 4 > 5 Su 5 Su 5 5 5 5 5 5 8 4 6 8 4 7 4 58 4 > "6 @ > "9 @ = "5 @ 7 Su 5 7 Su 5 5 7 Su5
469
; Rughu ri pdjqlwxgh dssurdfk
Wkh whupv lq +;14;, zlwk #7 > #8 dqg wkrvh zlwk #5 > #9 dvvxph wkh vdph irup lq wkh Exuqhww olplw/ zkhuh lm dqg tl duh uhsodfhg e| wkh QVI odzv1 Vlploduo|/ wkh whupv zlwk "4 > "7 > "8 dvvxph wkh vdph irup zkhq lm dqg tl duh uhsodfhg e| wkh QVI odzv1 Zkhq wkh Exuqhww frh!flhqwv duh lqwurgxfhg lqwr +;14;/ ;14<,/ wkhvh fdq eh uhzulwwhq dv Cyml 7 '6 Ctkl Cyn C oq G lm . lm . 5 nkl . Su $tkl Gw C{n C{n 8 '5 C{ml C{ml '7 C oq s 7 '8 C oq '9 7 t . Su t . V . Su 8 '5 kl C{ml 8 '5 kl C{ml '5 nkl mln # Cykl # 5 s +4, +4, . 8 tkl tml tml . 9 nkl mln mln @ > lm . 5 '5 C{ml +;153, 8 Cyn 8 4 Cyl C Gtl . tn ln . tl Gw C{n 6 C{n 5 Su C{n 8 4 6 C ln C oq s 8 4 7 C oq C 8 4 6 8 . ln . $ ln ln . 7 Su 5 C{n 7 Su 5 C{n C{n 5 Su 5 5 C{n " " +4, 4 8 4 s 8 C +4, +4, . 7 tn ln tn ln . 8 ln tn tn @ = tl . 5 5 Su 5 Su C{l +;154, +4,
+4,
Khuh lm dqg tl duh wkh QVI h{suhvvlrqv/ dqg wkhuhiruh wkh xqghuolqhg whupv duh ri vxshu0Exuqhww rughu +wkh| gr qrw frqwulexwh wr wkh Exuqhww htxdwlrqv,1 Wkhvh whupv pljkw eh ljqruhg iru d wkhru| ri Exuqhww rughu1 Lw lv qrw gl!fxow wr uhzulwh wkh odvw htxdwlrqv dv l Cyn 5 sk G lm +4, 5 lm lm Slmm5 > mln @ Gw C{kl '5 8 C Gtl 4 8 4 s Cyn tn @ tlm5 > tl . Gw C{l 5 5 Su 5 Su C{l zkhuh whupv ri vxshu0Exuqhww rughu duh ljqruhg/ dqg Slmm5 / tlm5 duh wkh Mlq0 Vohpurg h{suhvvlrqv +:149/ :14:,1 Wkh deryh htxdwlrqv duh mxvw wkh Mlq0Vohpurg htxdwlrqv +:148, zlwkrxw wkh wklug rughu whupv/ Slmm6 @ tlm6 @ 31 Wkxv/ wkh suhvhqw ghulydwlrq ri vhfrqg rughu wudqvsruw htxdwlrqv jlyhv d pruh udwlrqdo edfnjurxqg wr wkh vhfrqg rughu whupv lq wkh Mlq0Vohpurg htxdwlrqv/ zklfk zhuh rewdlqhg iurp wkh Exuqhww htxdwlrqv lq d udwkhu lqwxlwlyh pdqqhu1
;17 Wklug rughu dffxudf|= U46 htxdwlrqv Wkh wklug rughu htxdwlrqv duh glvfxvvhg rqo| iru wkh fdvh ri Pd{zhoo prohfxohv zkhuh +;146, dqg +;147, vlpsoli| wr
;17 Wklug rughu dffxudf|= U46 htxdwlrqv
46:
Cx3lmn Cykl Cyml 7 Ctkl G lm lm Cyn . .5 . 5 nkl . lm @ . %5 > Gw 8 C{ml C{n C{n C{n C{ml +;155, dqg 8 C oq C ln : Cyn : Cyl 5 Cyn C Gtl . ln ln . . tl . tn . tn % Gw 5 C{n C{n C{n 8 C{n 8 C{n 8 C{l % & 4 5 tl C 4 Cz ln C no 4 Czlm 8 Cyn . . . x3lno . %5 @ = +;156, 5 C{n 9 C{l C{o C{o 5 C{l %
Wr rewdlq wkhsuhvvxuh ghyldwru dqg wkh khdw x{ zlwk wklug rughu dffxudf|/ dovr wkh R %5 whupv lq +;146/ ;147, pxvw eh frqvlghuhg/ vr wkdw wkh htxdwlrqv iru lm dqg tl uhdg +diwhu vhwwlqj % @ 4, Cykl Cyml Cyn Cx3lmn G lm 7 Ctkl lm . .5 . 5 nkl . lm . @ > +;157, Gw 8 C{ml C{n C{n C{n C{ml C oq C ln : Cyn : Cyl C Gtl 8 . ln ln . . tl . tn Gw 5 C{n C{n C{n 8 C{n 8 C{n 4 5 Cz tl C 5 Cyn 4 4 Cz ln C no 8 Cyn lm = . . tn . . . x3lno @ 8 C{l 5 C{n 9 C{l C{o C{o 5 C{l +;158, Lq dgglwlrq wr wkh 46 yduldeohv/ wkhvh htxdwlrqv frqwdlq wkh txdqwlwlhv x3lmn / 4 zlm / z5 / dqg wr rewdlq d forvhg vhw ri htxdwlrqv/ dgglwlrqdo htxdwlrqv pxvw eh surylghg iru wkhvh1 Dv zh kdyh vhhq/ wkh ohdglqj rughu ri pdjqlwxgh ri x3lmn / 4 zlm / z5 lv R %5 / dqg li wkhvh duh frpsxwhg wr vhfrqg rughu dffxudf|/ suhvvxuh ghyldwru lm dqg khdw x{ tl duh frpsxwhg zlwklq wklug rughu dffxudf|1 Li x3lmn / 4 zlm / z4 duh frpsxwhg xs wr wklug rughu dffxudf|/ lm dqg tl zloo kdyh irxuwk rughu dffxudf|/ dqg vr rq1 Wkh uhohydqw htxdwlrq iru wkh surshu wklug rughu wkhru| iru z5 dw R %5 lv +;17, zlwk d @ 5 zkhuh zh kdyh wr frqvlghu +;144,/ vr wkdw
Ctn 5 55 Cyn C 53 . 7 4 53 no . 5 ; tn 4 C{n 3 C{o 4 C{n C oq 5s lm lm ;tn @ z5 . = +;159, C{n 6
Wkh vhfrqg rughu htxdwlrq iru x3lmn lv +;18, iru d @ 3/ C oq 45 4 4 6s 3 Cym C 6 4 C klm . tkl x = 6 klm .6 4 klm @ : 3 C{nl C{n l 8 C{nl : 3 C{nl 5 lmn +;15:,
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4;5
44 Vkrfn vwuxfwxuhv
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4415 Frpsdulvrq zlwk GVPF uhvxowv
4;6
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4;7
44 Vkrfn vwuxfwxuhv
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4415 Frpsdulvrq zlwk GVPF uhvxowv
4;8
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4;9
44 Vkrfn vwuxfwxuhv
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4415 Frpsdulvrq zlwk GVPF uhvxowv
4;:
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44 Vkrfn vwuxfwxuhv
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44 Vkrfn vwuxfwxuhv
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Wkh odvw wzr htxdwlrqv |lhog wkdw 45 dqg t4 duh frqvwdqwv ri lqwhjudwlrq/ 45 @ frqvw= / t5 @ frqvw= Wkh Nqxgvhq od|huv frpsxwhg llq wkh kvhtxho lduh k|shuerolf vlqh dqg frvlqh k | 54 | 54 ixqfwlrqv ri wkh irup frvk Nq / vlqk Nq / zkhuh lv d frqvwdqw ri wkh rughu ri xqlw|1 Iljxuh 4517 vkrzv wkhvh ixqfwlrqv iru d ydulhw| ri Nqxgvhq qxpehuv/ lq rughu wr hpskdvl}h wkh erxqgdu| od|hu vwuxfwxuh ri wkh vroxwlrq +4516;,1 Wkhvh ixqfwlrqv kdyh wkh w|slfdo vkdsh ri d erxqgdu| od|hu/ wkhlu odujhvw ydoxhv duh irxqg dw wkh zdoov/ dqg wkh fxuyhv ghfuhdvh wr }hur zlwklq vhyhudo phdq iuhh sdwkv dzd| iurp wkh zdoov1 Dv Nq jurzv/ wkh zlgwk ri wkh erxqgdu| od|huv lv jurzlqj dv zhoo1 Iru Nqxgvhq qxpehuv deryh k3=4 rqh l | 4 fdqqrw vshdn ri erxqgdu| od|huv dq| orqjhu/ vlqfh wkh ixqfwlrqv frvk Nq5 / l k | 4 vlqk Nq5 duh qrq0}hur lq wkh uhjlrq ehwzhhq wkh sodwhv1 Lq wklv fdvh erxqg0 du| h hfwv fdq kdyh dq lpsruwdqw lq xhqfh rq wkh rz sdwwhuq1 Qrwh wkdw zh frqvlghu olqhdu erxqgdu| od|huv/ dv vroxwlrqv ri olqhdul}hg htxdwlrqv1 Wkhvh ghshqg rqo| rq wkh Nqxgvhq qxpehu/ exw/ dv d frqvhtxhqfh ri olqhdul}dwlrq/ qrw rq wkh Pdfk qxpehu1
4517 Olqhdu Nqxgvhq erxqgdu| od|huv
544
Ilj1 45171 Erxqgdu| od|hu ixqfwlrqv t?db%*k?o dqg ULt db%*k?o iru ydulrxv Nqxgvhq qxpehuv1
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gy g 48 > t5 @ Nq > 55 @ 3 > t4 @ 3 = C| 7 g|
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| 54 7 | 45 7 | 45 > @ 3 t5 > @ 3 t5 Nq 48 Nq 48 Nq
zkhuh y3 / 3 / 3 / 45 / t4 duh frqvwdqwv ri lqwhjudwlrq1 Dffruglqjo|/ doo sur ohv duh olqhdu/ dqg wkh yduldwlrq lq ghqvlw| lv gxh vroho| wr khdw wudqvihu18 Qrwh wkdw 55 @ t4 @ 3 iru wkh QVI dqg Judg 46 htxdwlrqv1 Qrq0}hur ydo0 xhv iru wkh dqlvrwurslf frqwulexwlrq wr suhvvxuh/ 55 / dqg wkh khdw x{ sdudooho wr wkh zdoo/ t4 / dsshdu rqo| lq kljkhu rughu wkhrulhv1 GVPF fdofxodwlrqv vkrz qrq0}hur ydoxhv iru wkhvh txdqwlwlhv dw odujhu Nqxgvhq qxpehuv +deryh 3138,/ dqg wklv djuhhv zlwk wkh h{shfwdwlrq wkdw wkh QVI htxdwlrqv duh olplwhg wr vpdoo Nqxgvhq qxpehuv1 Wkh Judg 46 htxdwlrqv |lhog wkh vhfrqg rughu exon htxdwlrqv iru 55 dqg t4 / +4515;,/ dqg wkxv fdq ghvfuleh vrph ri wkh uduhidfwlrq h hfwv1 Krzhyhu/ wkh| fdqqrw ghvfuleh olqhdu Nqxgvhq od|huv1 451716 Exuqhww htxdwlrqv Wkh olqhdu Exuqhww h{suhvvlrqv +<177, lq Frxhwwh jhrphwu| duh gy 7 g5 5 g5 > 55 @ Nq5 5 . Nq5 5 > g| 6 g| 6 g| 6 5 g5 y g 48 t4 @ Nq > t5 @ Nq 5 g| 5 7 g|
45 @ Nq
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Iru v|pphwu| uhdvrqv ^2 zloo ydqlvk zkhq erwk zdoov kdyh wkh vdph whpshudwxuh1
545
45 Erxqgdu| ydoxh sureohpv
Wrjhwkhu zlwk wkh frqvhuydwlrq odzv wklv lpsolhv wkdw | 45 7 | 45 / @ 3 t5 > t4 @ 3 > Nq 48 Nq zkhuh y3 > 45 > W3 > t5 duh frqvwdqwv ri lqwhjudwlrq1 Wkxv/ wkh Exuqhww htxdwlrqv gr qrw suhglfw erxqgdu| od|huv iru yhorflw| dqg whpshudwxuh/ dqg qr olqhdu frqwulexwlrq wr wkh khdw x{ sdudooho wr wkh zdoo1 Qrwh wkdw wkh ixoo Exuqhww htxdwlrqv |lhog wkh exon htxdwlrqv +4515;,/ dqg wkxv lpso| d qrq0olqhdu frq0 wulexwlrq wr t4 1 Wkh htxdwlrqv iru dqg 55 ohdg wr Nqxgvhq od|hu vroxwlrqv iru ghqvlw| dqg vwuhvv 55 / yl}1 %s & %s & 7 t5 6 | 45 6 | 45 4 @ 3 . | . D vlqk . E frvk 48 Nq 5 5 Nq 5 Nq %s & %s & 6 | 45 6 | 45 55 @ D vlqk E frvk = +45167, 5 Nq 5 Nq y @ y3 45
Krzhyhu/ wkh Exuqhww htxdwlrqv idlo wr suhglfw d khdw x{ sdudooho wr wkh zdoo +t4 @ 3,/ dqg wr suhglfw erxqgdu| od|huv iru whpshudwxuh dqg yhorflw|19 Wkh vroxwlrq ri wkh Exuqhww htxdwlrqv uhtxluhv wzr dgglwlrqdo erxqgdu| frqglwlrqv/ h1j1 d frqglwlrq iru 55 dw hdfk zdoo ^46:`1 451717 Vxshu0Exuqhww dqg dxjphqwhg Exuqhww htxdwlrqv Iru olqhdu Frxhwwh rz wkh vxshu0Exuqhww htxdwlrqv +<178, uhgxfh wr 7 5 g5 5 5 g5 gy 5 6 g6 y Nq Nq > @ . Nq > 55 g| 6 g|6 6 g| 5 6 g| 5 g 48: 6 g6 8 6 g6 6 48 g5 y Nq Nq = t4 @ Nq5 5 > t5 @ Nq 5 g| 7 g| 49 g| 6 ; g| 6
45 @ Nq
+45168,
Iru wkh dxjphqwhg Exuqhww htxdwlrqv wkh h{suhvvlrqv iru 55 dqg t4 uhpdlq xqfkdqjhg/ zkloh wkh h{suhvvlrqv iru 45 dqg t5 uhdg g 44 6 g6 8 6 g6 gy 4 6 g6 y 48 . Nq Nq . Nq > t @ Nq = 5mdE g| 9 g|6 7 g| 49 g| 6 ; g|6 +45169, Wkh yhorflw| fdq eh frpsxwhg dv l kt l kt 4 4 | 54 6 | 5 6 | 5 . E vlq > +4516:, . D frv ymVE @ y3 45 5 Nq 5 Nq Nq 4 l k l k s | 4 s | 4 | 5 . D frvk 9 Nq5 . E vlqk 9 Nq5 > +4516;, ymdE @ y3 45 Nq
45mdE @ Nq
S
Lw lv vxusulvlqj wkdw wkh Judg 46 htxdwlrqv gr qrw jlyh olqhdu Nqxgvhq od|huv/ zkloh wkh Exuqhww htxdwlrqvzklfk fdq eh ghulyhg irup wkh Judg 46 htxdwlrqv e| FH h{sdqvlrqlpso| d olqhdu Nqxgvhq od|hu1
4517 Olqhdu Nqxgvhq erxqgdu| od|huv
546
iru vxshu0Exuqhww dqg dxjphqwhg Exuqhww htxdwlrqv/ uhvshfwlyho|1 y3 / 45 / D/ E duh frqvwdqwv ri lqwhjudwlrq1 Erwk vhwv ri htxdwlrqv |lhog wkh olqhdu Qdylhu0Vwrnhv vroxwlrq/ y3 45 |4@5 Nq / soxv d fruuhfwlrq1 Wkh fruuhfwlrq lv shulrglf lq vsdfh iru wkh vxshu0Exuqhww htxdwlrqv +4516:,/ dqg d Nqxgvhq erxqgdu| od|hu iru wkh dxjphqwhg Exuqhww htxdwlrqv +4516;,1 Wkxv wkh fkdqjh lq vljq dw wkh wklug rughu ghulydwlyh lq 45 ehwzhhq +45168, dqg +45169, kdv vhyhuh lpsdfw rq wkh vroxwlrq ehkdylruwkh vxshu0Exuqhww htxdwlrqv |lhog uhvxowv wkdw duh xqsk|vlfdo/ exw wkh vroxwlrq ri wkh dxjphqwhg Exuqhww htxdwlrqv kdyh vxssruw lq sk|vlfv1 Iru whpshudwxuh wkh uhvxowv duh vlplodu= Iurp wkh vxshu0Exuqhww htxdwlrqv iroorzv l k l k 7 | 45 | 4 | 4 t5 . F frv 4 Nq5 . G vlq 4 Nq5 48 Nq | 45 | 45 . I frvk 5 . J vlqk 5 Nq Nq t s t s @ 3=968> 5 @ 6 8645<.534 @ 3=;8: zlwk 4 @ 6 8645<534 4549 4549
mVE @ 3
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@ 3
vr wkdw erwk fruuhfwlrqv duh ri erxqgdu| od|hu w|sh1 Dw wklv srlqw zh fdq gudz wkh iroorzlqj frqfoxvlrqv= +d, Wkh vxshu0Exuqhww htxdwlrqv ohdg wr xqsk|vlfdo shulrglf vroxwlrqv lq vwhdg| vwdwh rzv/ dqg vkrxog qrw eh frqvlghuhg iru vwhdg| vwdwh sureohpv1 Zh hqfrxqwhuhg rvfloodwlrqv lq wkh vwhdg| vwdwh Exuqhww htxdwlrqv hduolhu zkhq zh glvfxvvhg vkrfn sur ohv/ dqg/ ri frxuvh/ irxqg wkhp wr eh xqvwdeoh lq wudq0 vlhqw sureohpv1 Wkh shulrglf vroxwlrqv fdq eh dyrlghg e| fkrrvlqj erxqgdu| frqglwlrqv vxfk wkdw D @ E @ F @ G @ 3 ^46;`/ exw lw vhhpv ehwwhu wr dyrlg xqsk|vlfdo htxdwlrqv uljkw dzd|1 +e, Wkh dxjphqwhg Exuqhww htxdwlrqv vhhp wr ghvfuleh Nqxgvhq erxqg0 du| od|huv iru yhorflw| dqg whpshudwxuh/ dqg pljkw eh xvhixo1 Wkh pdlq srlqw djdlqvw wkhp lv wkdw wkh whupv ri vxshu0Exuqhww rughu wkdw zhuh xvhg wr dxj0 phqw wkh Exuqhww htxdwlrqv duh qrw edvhg rq udwlrqdo dujxphqwv dv glvfxvvhg
547
45 Erxqgdu| ydoxh sureohpv
lq vhfwlrq 716171 Srohplfdoo| rqh frxog vd| wkdw lq wkh dxjphqwhg Exuqhww htxdwlrqv wkh vljqv ri wkrvh frh!flhqwv wkdw fdxvh wkh wurxeoh duh lqyhuwhg/ zklfk uhvxowv lq jrrg ehkdylru ri wkh htxdwlrqv/ exw lv gl!fxow/ li qrw lpsrv0 vleoh/ wr mxvwli| e| dq dujxphqw edvhg lq sk|vlfv1 Dqrwkhu srlqw djdlqvw wkh dxjphqwhg Exuqhww htxdwlrqv lv wkdw wkh| duh xqvwdeoh zlwk uhvshfw wr orfdo xfwxdwlrqv dw kljk iuhtxhqflhv/ vhh Vhf1 4314 dqg ^93`1 451718 Uhjxodul}hg 46 prphqw htxdwlrqv Wkh olqhdu U46 htxdwlrqv +<17:/ <17<, uhgxfh lq Frxhwwh jhrphwu| wr 45 gy 5 gt4 . @ @ frqvw= > t4 @ g| 8 g| Nq 5 t5 8 g g 55 . @ @ frqvw= > 55 5 g| g| 6 Nq
< 5 g5 t4 Nq > 8 g|5 9 g5 55 @ Nq5 > 8 g| 5
vr wkdw yhorflw| dqg whpshudwxuh duh rewdlqhg dv l kt l kt 4 4 | 54 5 8 | 5 8 | 5 . E frvk > t4 zlwk t4 @ D vlqk < Nq < Nq Nq 8 +4516<, l kt l kt 4 4 7 | 45 5 8 | 5 8 | 5 . G frvk = 55 zlwk 55 @ F vlqk @ 3 t5 9 Nq 9 Nq 48 Nq 8 +45173, zkhuh y3/ 45 / D/ E dqg / t / F/ G duh frqvwdqwv ri lqwhjudwlrq1 Zh fdq 3 4 lghqwli| 58 t4 dqg 85 55 dv wkh Nqxgvhq erxqgdu| od|huv iru yhorflw| dqg whpshudwxuh dffruglqj wr wkh U46 htxdwlrqv/ dqg wkh| duh ri wkh vdph irup dv wkrvh glvfxvvhg deryh/ dqg ghslfwhg lq Ilj1 45171 y @ y3 45
451719 Wkh khdw
x{ sdudooho wr wkh
rz
D forvhu lqvshfwlrq ri wkh odvw htxdwlrqv vkrzv wkdw wkh U46 htxdwlrqv suhglfw wkh khdw x{ sdudooho wr wkh rz dv < C5y t4mU46 @ Nq5 5 > 5 C|
+45174,
zkloh wkh fruuhvsrqglqj h{suhvvlrq iurp wkh Exuqhww htxdwlrqv uhdgv 6 C5y t4mE @ Nq5 5 = 5 C|
+45175,
Qrwh wkdw wkh dxjphqwhg Exuqhww htxdwlrqv dqg wkh vxshu0Exuqhww htxdwlrqv djuhh zlwk wkh Exuqhww htxdwlrqv xs wr wkh rughu ri Nq5 / vr wkdw wkh| dovr suhglfw wkh uhvxow +45175,1 Wkxv/ wkh U46 htxdwlrqv dqg wkh Exuqhww htxdwlrqv suhglfw gl huhqw vljqv iru t4 1
4517 Olqhdu Nqxgvhq erxqgdu| od|huv
548
Wkh txhvwlrq zklfk vljq uhodwhv khdw x{ dqg wkh vhfrqg judglhqw ri yh0 orflwlhv lq d uduh hg jdv fdq eh dqvzhuhg e| phdqv ri d GVPF vlpxodwlrq iru Frxhwwh rz lq d jdv ri Pd{zhoo prohfxohv dw Nq @ 3=4 zlwk y +3, @ 3/ n 5:6N/ shuiruphg zlwk wkh iuhh frgh vxssolhg e| y +O, @ 533p2v/ dqg Z @ p : Elug ^4`1 Wkh vwurqj vfdwwhu lq wkh GVPF uhvxowv grhv qrw doorz xv wr frpsxwh wkh ghulydwlyhv ri wkh yhorflwlhv/ dqg wkhuhiruh wkh dujxphqw lv edvhg rq +4516<,4 1 Vroylqj iru t4 |lhogv/ diwhu uhlqvhuwlqj ri glphqvlrqv/ 8 4 4 O t4mU46>olq @ s3 y +|, y . 45 | = +45176, 5 5 3 5 D vlplodu htxdwlrq frxog eh zulwwhq iru wkh vxshu0Exuqhww htxdwlrqv/ exw wklv lv qrw grqh khuh/ vlqfh wkh vxshu0Exuqhww htxdwlrqv/ dv zdv vkrzq/ |lhog shulrglf vroxwlrqv/ dqg duh xqsk|vlfdo1 Lqwhjudwlrq ri wkh fruuhvsrqglqj htxdwlrq ri wkh dxjphqwhg Exuqhww htxdwlrqv +4516;,4 |lhogv zlwk +45175, 4 4 O t4mdE>olq @ <s3 y +|, y . 45mdE | > +45177, 5 3 5 wkdw lv d vlplodu fxuyh/ doehlw zlwk gl huhqw vljq dqg dpsolwxgh1 Lq rughu wr frpsduh wkhvh uhvxowv iurp wkh olqhdu htxdwlrqv wr GVPF vlpxodwlrqv lw lv ehvw wr vxewudfw wkh exon vroxwlrq +4515;, iurp wkh GVPF uhvxow/ l1h1 frpsduh wr t4mGVPF>olq @ tGVPF
: 45 t5 = 5 s . 55
Wkh uhvxow lv vkrzq lq Ilj1 4518/ zklfk vkrzv wkh khdw x{ t4mGVPF>olq dv frpsxwhg gluhfwo| lq wkh vlpxodwlrq/ dqg t4mU46>olq dv frpsxwhg iurp +45176,/ zkhuh wkh GVPF gdwd zdv xvhg iru wkh hydoxdwlrq ri wkh uljkw kdqg vlgh1 Zkloh t4mU46>olq gl huv iurp wkh vlpxodwlrq ydoxh t4mGVPF>olq / lw lv fohdu wkdw wkh uhodwlrq +45176, |lhogv wkh surshu gluhfwlrq ri t4 / zkloh +45177, zrxog |lhog wkh rssrvlwhzurqjgluhfwlrq1 Wkh gl huhqfh ehwzhhq t4mU46>olq dqg t4mGVPF>olq zloo sduwo| eh gxh wr qrq0olqhdu h hfwv/ dqg sduwo| wr wkh rffxu0 uhqfh ri dgglwlrqdo frqwulexwlrqv wr wkh yhorflw| Nqxgvhq od|hu1 l kt | 4
8 5 zlwk wwhg dp0 Wkh jxuh dovr vkrzv wkh ixqfwlrq t4 @ D vlqk < Nq solwxgh D1 Reylrxvo|/ wkh wwhg U46 uhvxow +4516<,4 djuhhv zhoo zlwk wkh GVPF khdw x{ t4mGVPF>olq 1 Wklv djuhhphqw lqglfdwhv wkdw wkh U46 htxdwlrqv fdq ghvfuleh wkh sk|vlfdo Nqxgvhq erxqgdu| od|hu iru t4 1 Wklv revhuydwlrq ohdgv wr frqvlghulqj vxshusrvlwlrqv ehwzhhq wkh exon vroxwlrqv dqg wkh olqhdu Nqxgvhq od|hu fruuhfwlrqv wkdw zloo eh glvfxvvhg lq Vhf1 45181
.
Pruh ghwdlov rq wkh gdwd iru wkh GVPF vlpxodwlrqv fdq eh irxqg lq Vhf1 45181
549
45 Erxqgdu| ydoxh sureohpv
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4518 Vxshusrvlwlrqv
54:
4518 Vxshusrvlwlrqv Wkh exon vroxwlrq frqwdlqv wkh prvw lpsruwdqw qrq0olqhdu sduwv ri wkh htxd0 wlrqv/ zkloh wkh Nqxgvhq od|hu fruuhfwlrq jlyhv wkhlu ohdglqj olqhdu frqwhqw1 Vxshusrvlwlrqv ri erwk vkrxog wkhuhiruh jlyh uhodwlyho| dffxudwh uhvxowv/ zklfk duh hdvlhu wr rewdlq/ dqg vwxg|/ wkdq vroxwlrqv ri wkh ixoo qrq0olqhdu htxdwlrqv1 Wklv dssurdfk kdv ehhq xvhg vxffhvvixoo| iru wkh vroxwlrq ri wkh Exuqhww htxd0 wlrqv lq vwhdg| vwdwh 40G khdw wudqvihu ^478`1 Iru frqvwuxfwlqj vxshusrvlwlrqv ri exon dqg Nqxgvhq od|hu vroxwlrqv wr ghvfuleh Frxhwwh rz ^479`/ wkh vroxwlrqv ri wkh exon htxdwlrqv +4515;, xqghu wkh erxqgdu| frqglwlrqv +4515 @ mE 55mO > 45 @ 45mE > 55 @ 55mE . 55mO > 8 8 s +45178, s @ S3 55mO > @ > t4 @ t4mE . t4mO > t5 @ t5mE = Wklv prgho zloo eh uhihuuhg wr dv wkh VENq prgho; / dqg lwv uhvxowv duh frp0 sduhg wr GVPF vlpxodwlrqv iru Frxhwwh rz lq d jdv ri Pd{zhoo proh0 n fxohv +$ @ 4, zlwk zdoo whpshudwxuhv 3Z @ O Z @ p 5:6N dqg zdoo yh0 p p 3 O orflwlhv yZ @ 3/ yZ @ 533 v > 933 v 1 Prohfxodu pdvv dqg uhihuhqfh ylvfrv0 n lw| duh wdnhq dv wkh ydoxhv iru dujrq/ p @ 53;M@njN/ @ 3 @3Z 5 zlwk 3 @ 4=<88 438 Qv@p 1 Wkh pdvv ghqvlw| dw uhvw lv {hg dv 3 @ <=5;;439 nj@p6 +fruuhvsrqglqj wr wkh uhvw suhvvxuh s3 @ 3 Z @ 3=85:Sd,1 Wkh Nqxgvhq qxpehu +<174, lv frqwuroohg e| dgmxvwlqj wkh ohqjwk dv O@
s 3 > 3 Z Nq
vr wkdw O @ 3=4:999p iru Nq @ 3=38/ O @ 3=3;;66p iru Nq @ 3=4/ dqg O @ 3=34:999p s iru Nq @ 3=81 Wkh lqwhjudwhg pdvv +45163, lv {hg dw P @ 3 O @ 3 @ Z Nq 1 Qrwh wkdw wkh deryh gh qlwlrq ri wkh Nqxgvhq qxpehu gl huv iurp wkdw lq ^4`1 Elug*v frgh ^4` zdv xvhg iru wkh GVPF fdofxodwlrqv zlwk prgl fdwlrqv wr rewdlq rxwsxw iru doo prphqwv ri lqwhuhvw1 Wkh vroxwlrq iru wkh exon htxdwlrqv lv rewdlqhg iurp d vwudljkwiruzdug qlwh yroxph vfkhph ^479`1 Uhvxowv iru wkh QVI htxdwlrqv +4514<, zlwk wkh erxqgdu| frqglwlrqv +45153, duh vkrzq dv zhoo1 Wkh erxqgdu| od|hu vroxwlrqv qhhg dgglwlrqdo lqsxw wr ghwhuplqh wkhlu dpsolwxghv/ dqg wkhvh duh vlpso| wwhg wr wkh GVPF uhvxowv1 Gxh wr wkh jhrphwu| ri wkh rz/ t4mO lv dqwlv|pphwulf zlwk uhvshfw wr wkh fhqwhu dqg 55mO lv v|pphwulf/ vr wkdw +4516
Vxshusrvlwlrq ri Exon vroxwlrq dqg Nqxgvhq od|hu1
54;
45 Erxqgdu| ydoxh sureohpv Wdeoh 45141 Dpsolwxghv lq wkh erxqgdu| od|hu frqwulexwlrqv iru ^ / j22 1 k? ' ffD k? ' f k? ' fD u ` ' 2ff 4t ' fffb ( ' fffD u ` ' Sff 4t ' ffH ( ' ffD
kt t4mO @ D vlqk
4 8 | 5 < Nq
' ffD ' ff ( ' fff ( ' ff2 ' fff ( ' ff
l
kt > 55mO @ G frvk
4 8 | 5 9 Nq
l =
Iljxuhv 451945143 vkrz frpsdulvrqv ri GVPF/ QVI/ dqg VENq/ +45178, O iru d ydulhw| ri Nqxgvhq qxpehuv dqg zdoo yhorflwlhv yZ 1 Wkh fruuhvsrqglqj ydoxhv iru wkh frqvwdqwv D dqg G duh jlyhq lq Wdeoh 45141 Iru yhu| vpdoo Nqxgvhq qxpehuv +Nq 3=34, doo wkuhh prghov djuhh shu0 ihfwo| +qrw vkrzq,1 Dw Nqxgvhq qxpehu Nq @ 3=38/ vkrzq lq Iljv1 4519 dqg 451:/ VENq vwloo djuhhv yhu| zhoo zlwk GVPF> wkh prvw ylvleoh gl huhqfh lv lq wkh ydoxh iru 45 +4=8 6(,1 Wkh whpshudwxuh fxuyhv djuhh zhoo/ zlwk vpdoohu gl huhqfhv forvh wr wkh erxqgdulhv/ dqg h{fhoohqw djuhhphqw lq wkh pd{lpxp ydoxh1 Wkh sur ohv iru yhorflw| y/ sdudooho khdw x{ t4 / dqg qrupdo khdw x{ t5 vkrz rqo| plqru gl huhqfhv1 VENq fdswxuhv wkh vkdsh ri wkh qrupdo vwuhvv 55 yhu| zhoo/ exw grhv qrw phhw wkh ydoxhv lq wkh exon/ l1h1 lq wkh plggoh sduw ri wkh grpdlq1 Doo gl huhqfhv duh pruh surplqhqw dw odujhu yhorflwlhv +Ilj1 451:,1 Pruhryhu/ wkh edodqfh ehwzhhq exon vroxwlrq dqg Nqxgvhq od|hu gl huv zlwk yhorflw|1 Wkh Nqxgvhq od|hu lv pruh lpsruwdqw dw orzhu yhorflw|/ zkhuh doo yduldeohv duh vpdoohu1 Dw odujhu yhorflwlhv/ doo rwkhu hogv jurz dv zhoo/ dqg wkh qrq0olqhdu frqwulexwlrqv lq wkh exon htxdwlrqv jdlq pruh lpsruwdqfh1 Dw Nq @ 3=38 wkh QVI uhvxowv douhdg| vkrz pruh surqrxqfhg gl huhqfhv1 Prvw lpsruwdqw lv wkdw QVI jlyhv 55 @ t4 @ 3/ exw dqrwkhu lpsruwdqw gli0 ihuhqfh lv revhuyhg lq wkh whpshudwxuh fxuyhv/ zkhuh wkh plgsrlqw pd{lpxp lv qrw phw1 Lw iroorzv wkdw vhfrqg dqg wklug rughu frqwulexwlrqvlq wkh htxd0 wlrqv dqg wkh erxqgdu| frqglwlrqvfdqqrw eh qhjohfwhg dw Nq @ 3=38/ lq sduwlfxodu dw odujhu rz yhorflwlhv1 Zkhq wkh Nqxgvhq qxpehu lv lqfuhdvhg wr Nq @ 3=4/ Iljv1 451; dqg 451 dv ehiruh wkh prvw ylvleoh gl huhqfhv olh lq wkh exon ydoxhv iru 45 dqg 55 1 Wkh whpshudwxuh pd{lpxp lv uhsurgxfhg yhu| zhoo/ zkloh vrph gl huhqfhv fdq eh revhuyhg dw wkh erxqgdulhv1 QVI/ rq wkh rwkhu kdqg/ fdqqrw pdwfk wkh whpshudwxuh pd{lpxp dw doo1 Rqo| rqh vhw ri uhvxowv lv vkrzq iru wkh odujhu Nqxgvhq qxpehu Nq @ 3=8/ O @ 533 pv +Ilj1 45143,1 Qrz erwk/ VENq dqg QVI/ fdqqrw pdwfk wkh dw yZ yhorflw| fxuyh/ qru wkh qrupdo khdw x{ t5 1 Dw odujhu yhorflwlhv +qrw vkrzq,/ wklv glvfuhsdqf| ehfrphv vwurqjhu1 Iurp wkh yhorflw| fxuyh rqh pljkw lqihu wkdw wkh uhvxowv frxog eh lpsuryhg e| uhgxflqj wkh frh!flhqwv 4 / 5 lq wkh erxqgdu| frqglwlrq +4515<,1 Wklv/ krzhyhu/ uhvxowv lq d odujhu devroxwh ydoxh
4518 Vxshusrvlwlrqv
54<
iru 45 +* 3=44 Sd,/ dqg wkxv zruvhqv wkh uhvxowv iru rwkhu prphqwv1 Fohduo|/ wklv Nqxgvhq qxpehu lv rxwvlgh wkh udqjh ri wkh ghvfulehg phwkrg 1 Lq rughu wr xqghuvwdqg wkh uhvxowv ehwwhu/ lw pxvw eh frqvlghuhg wkdw wkh QVI htxdwlrqv/ dqg wkhlu erxqgdu| htxdwlrqv/ duh rqo| ri uvw rughu dffxudf| lq wkh Nqxgvhq qxpehu1 Wkxv/ wkhlu idloxuh dw odujhu Nqxgvhq qxpehuv lv qrw vxusulvlqj1 Frqvlghulqj wkhlu olplwdwlrqv wkh| uhfryhu wkh fxuyhv iru yhorflw| dqg qrupdo khdw x{ t5 uhpdundeo| zhoo1 Wkhlu prvw lpsruwdqw vkruwfrplqj lv wkh idloxuh wr uhsurgxfh wkh pd{lpxp ri whpshudwxuh durxqg wkh plgsrlqw1 Pruhryhu/ wkh| jlyh d ydoxh ri }hur iru sdudooho khdw x{ t4 dqg qrupdo vwuhvv 55 1 Wkh exon vroxwlrq lv ri vhfrqg rughu lq Nq/ dqg wkh olqhdu Nqxgvhq od|hu ri wkh U46 htxdwlrqv lv ri wklug rughu1 Wkhlu vxshusrvlwlrq/ VENq/ wkxv vkrxog eh ri vhfrqg rughu dffxudf|1 Lw jlyhv d fohdu lpsuryhphqw ryhu wkh QVI htxdwlrqv/ vlqfh lw uhfryhuv wkh whpshudwxuh pd{lpxp/ dqg wkh fxuyhv iru sdudooho khdw x{ dqg qrupdo vwuhvv txlwh zhoo1 Dq lpsruwdqw vkruwfrplqj lv wkdw erxqgdu| frqglwlrqv iru 55 dqg t4 duh suhvhqwo| qrw nqrzq1 Wkh QVI htxdwlrqv gr qrw ghvfuleh erxqgdu| od|huv dw doo1 Dowkrxjk wkh VENq prgho +45178, dggv erxqgdu| od|huv wr yhorflw| y dqg whpshudwxuh / lw grhv qrw ghvfuleh wkh erxqgdu| od|huv iru y dqg lq djuhhphqw zlwk GVPF/ vlqfh wkhvh frqvlvw ri pdq| frqwulexwlrqv/ dv zdv phqwlrqhg lq Vhf1 45171:1 Qhyhuwkhohvv/ erwk prghov ghvfuleh wkh ryhudoo yhorflw| yhu| zhoo/ vlqfh wkh yhorflw| fxuyh lv grplqdwhg e| wkh olqhdu uvw rughu frqwulexwlrq1 Wkh glvfuhsdqflhv dw wkh erxqgdulhv duh pruh surqrxqfhg lq wkh whpshu0 dwxuh fxuyhv1 Wkh ryhudoo uhvxow iru wkh VENq prgho lv jrrg qhyhuwkhohvv/ vlqfh wkh mxps erxqgdu| frqglwlrq +4515<,5 frpshqvdwhv h hfwlyho| iru wkh glvfuhsdqf| dw wkh erxqgdu|1 Zkloh rqh pd| fulwlfl}h wkh U46 htxdwlrqv iru qrw uhsurgxflqj wkh erxqg0 du| od|huv lq yhorflw| dqg whpshudwxuh ^46;`/ rqh vkrxog qrw irujhw wkdw wkh| duh deoh wr ghvfuleh wkh vhfrqg rughu txdqwlwlhv t4 dqg 55 yhu| zhoo1 Lw lv hylghqw wkdw kljkhu Nqxgvhq qxpehuv ghpdqg pruh uh qhg prghov1 Wkh uvw vwhs zrxog eh wr frqvlghu wkh ixoo| qrq0olqhdu U46 htxdwlrqv +45159,1 Wklv uhtxluhv d ixoo vhw ri erxqgdu| frqglwlrqv/ zklfk duh qrw dydlodeoh dw suhvhqw1 Krzhyhu/ wkh uhvxowv suhvhqwhg khuh vkrz wkdw lw zloo eh d zruwkzkloh wdvn wr ghyhors vxlwdeoh erxqgdu| frqglwlrqv iru wkh U46 htxdwlrqv1 ]khqj ^75` xvhg vroxwlrqv ri wkh HV0EJN htxdwlrq iru whvwlqj wkh delolw| ri pdfurvfrslf htxdwlrqv wr uhsurgxfh kljkhu prphqwv lq dffrugdqfh zlwk h{dfw vroxwlrqv1 Iru wklv kh zurwh wkh htxdwlrqv iru vwuhvv dqg khdw x{ dv lm @ s Ilm + > yl > > lm > tl ,/ tl @ s Jl + > yl > > lm > tl , zkhuh Ilm dqg Jl duh wkh dssursuldwh ixqfwlrqdov wr uhsurgxfh wkh surshu htxdwlrqv/ h1j1 wkh Exuqhww htxdwlrqv +<17/ <18,/ ru wkh U46 htxdwlrqv +<1
553
45 Erxqgdu| ydoxh sureohpv
u Ilj1 45191 Frxhwwh rz dw k? ' ffD/zlwk ` ' 2ff 4 t 1 Frqwlqxrxv olqh= GVPF/ qho| gdvkhg olqh +0 0 0 0 ,= QVI/ gdvkhg olqh + ,= vxshusrvlwlrq ri exon vroxwlrq dqg olqhdu Nqxgvhq od|hu vroxwlrq1 Uhfdoo wkdw QVI lpsolhv ^ ' j 22 ' f +fxuyhv qrw vkrzq,1
4518 Vxshusrvlwlrqv
554
u Ilj1 451:1 Frxhwwh rz dw k? ' ffD/zlwk ` ' Sff 4 t 1 Frqwlqxrxv olqh= GVPF/ qho| gdvkhg olqh +0 0 0 0 ,= QVI/ gdvkhg olqh + ,= vxshusrvlwlrq ri exon vroxwlrq dqg olqhdu Nqxgvhq od|hu vroxwlrq1 Uhfdoo wkdw QVI lpsolhv ^ ' j 22 ' f +fxuyhv qrw vkrzq,1
555
45 Erxqgdu| ydoxh sureohpv
u Ilj1 451;1 Frxhwwh rz dw k? ' f/zlwk ` ' 2ff 4t 1 Frqwlqxrxv olqh= GVPF/ qho| gdvkhg olqh +0 0 0 0 ,= QVI/ gdvkhg olqh + ,= vxshusrvlwlrq ri exon vroxwlrq dqg olqhdu Nqxgvhq od|hu vroxwlrq1 Uhfdoo wkdw QVI lpsolhv ^ ' j 22 ' f +fxuyhv qrw vkrzq,1
4518 Vxshusrvlwlrqv
556
u Ilj1 451<1 Frxhwwh rz dw k? ' f/zlwk ` ' Sff 4t 1 Frqwlqxrxv olqh= GVPF/ qho| gdvkhg olqh +0 0 0 0 ,= QVI/ gdvkhg olqh + ,= vxshusrvlwlrq ri exon vroxwlrq dqg olqhdu Nqxgvhq od|hu vroxwlrq1 Uhfdoo wkdw QVI lpsolhv ^ ' j 22 ' f +fxuyhv qrw vkrzq,1
557
45 Erxqgdu| ydoxh sureohpv
u Ilj1 451431 Frxhwwh rz dw k? ' fD/zlwk ` ' 2ff 4t 1 Frqwlqxrxv olqh= GVPF/ qho| gdvkhg olqh +0 0 0 0 ,= QVI/ gdvkhg olqh + ,= vxshusrvlwlrq ri exon vroxwlrq dqg olqhdu Nqxgvhq od|hu vroxwlrq1 Uhfdoo wkdw QVI lpsolhv ^ ' j 22 ' f +fxuyhv qrw vkrzq,1
4519 D erxqgdu| frqglwlrq iru qrupdo vwuhvv
558
lq wkh Nqxgvhq od|huv/ zkloh wkh U46 htxdwlrqv duh idu pruh vxffhvvixo iru Nqxgvhq qxpehuv ehorz 3181 Vxshusrvlwlrqv ri wkh vhfrqg rughu exon vroxwlrqv zlwk wkh olqhdu Nqxgvhq od|huv dffruglqj wr wkh Exuqhww ru dxjphqwhg Exuqhww htxdwlrqv duh srvvleoh dv zhoo/ exw uhvxowv iru wkhvh duh qrw vkrzq1 Wkh Exuqhww htxdwlrqv fdqqrw ghvfuleh wkh erxqgdu| od|hu iru t4 / exw duh zhoo fdsdeoh lq ghvfulelqj wkh 55 erxqgdu| od|hu +45167,1 Wkh Exuqhww htxdwlrqv zhuh vroyhg qxphulfdoo| e| [xh/ Ml dqg Vkx ^47:`^47;`/ dqg e| Orfnhue| dqg Uhhvh ^47<` zlwk d pruh vwdeoh frgh1 Lq erwk fdvhv/ wkh dxwkruv vhw 55 @ 3 dw wkh erxqgdulhv/ dqg d vxshusrvlwlrq ri exon vroxwlrq dqg Exuqhww Nqxgvhq od|hu +45167,/ l1h1 wkh VENq prgho/ zlwk wkdw erxqgdu| frqglwlrq vkrzv jrrg djuhhphqw zlwk wkh uhvxowv suhvhqwhg lq wkh flwhg sdshuv1 Wkh erxqgdu| frqglwlrq 55 @ 3 lv qrw vxssruwhg e| GVPF fdofxodwlrqv dqg ryhuhvwlpdwhv wkh dpsolwxgh ri wkh Nqxgvhq od|hu iru 55 frqvlghudeo|1 Wklv ohdgv wr wkh lqyhuwhg ghqvlw| fxuyhv uhsruwhg lq ^47<`1
4519 D erxqgdu| frqglwlrq iru qrupdo vwuhvv Deryh jhqhudo prphqw erxqgdu| frqglwlrqv zhuh suhvhqwhg lq irup ri +4518,/ zklfk lv qrz frqvlghuhg iru surylglqj d erxqgdu| frqglwlrq iru 55 ^479`1 Wkh uhvxow vkrxog qrw eh vhhq dv d gh qlwh dqvzhu/ exw dv d edvh iru ixuwkhu glvfxvvlrq ri erxqgdu| frqglwlrqv iru h{whqghg v|vwhpv1 Wkh prphqwxp shushqglfxodu wr wkh zdoo/ phdvxuhg lq wkh uhvwiudph ri wkh zdoo/ lv jlyhq e| fBl @ FuZ qu @ Fu qu > dqg lv reylrxvo| rgg lq FnZ qn 1 Hydoxdwlrq ri +4518, wkxv jlyhv u 5" 3 4" +s . mn qm qn , @ Kmn qm qn Z Kn3 qn = " " 5
+45179,
3 zlwk Judg*v 46 prphqw glvwul0 Hydoxdwlrq ri wkh kdoi0vsdfh lqwhjudov Kn3 / Kmn exwlrq +9144, uhvxowv lq t s s 5 5 t q . s 5" m m Z " 8 s = mn qm qn @ s 45 Z
Dovr khuh/ dv lq +45147,/ huuruv gxh wr lqvx!flhqw frqvlghudwlrq ri wkh erxqgdu| od|huv fdq eh h{shfwhg1 Wkh wwlqj sdudphwhuv / kdyh ehhq lqwurgxfhg wr frpshqvdwh iru wkdw huuru1 Lq Frxhwwh jhrphwu|/ wkh erxqgdu| frqglwlrq uhgxfhv wr t s s 5 5 5" t q . s Z 5 5 " 8 s = +4517:, 55 @ s 45 Z
559
45 Erxqgdu| ydoxh sureohpv
Iurp wkh exon vroxwlrqlw lv nqrzq wkdw iru Frxhwwh rz t5 dqg 45 duh R +Nq, dqg t4 dqg 55 duh R Nq5 1 Iru Frxhwwh rz zlwk wkh QVI htxdwlrqv rqh qgv 55 @ 31 Wkh QVI htxdwlrqv duh wkh htxdwlrqv ri uvw rughu lq Nq dqg wkh erxqgdu| frqglwlrqv vkrxog eh lq dffrugdqfh zlwk wklv1 Wkxv/ lw vhhpv uhdvrqdeoh wr uhtxluh wkdw wkh ohdglqj rughu ri wkh erxqgdu| frqglwlrq +4517:, lv wkh vhfrqg rughu1 Wklv uhtxluhphqw fdq eh xvhg wr ghwhuplqh wkh wwlqj sdudphwhu 1 Wkh rughu ri pdjqlwxgh ri wkh prphqwv lv pdgh h{solflw e| zulwlqj %t5 lqvwhdg ri t5 dqg vr rq1 Wkhq/ h{sdqvlrq ri wkh mxps frqglwlrq +4515<,5 lq % jlyhv u 5" t5 q5 s . R %5 = @4% Z 5" 4 5 Z s Wklv lv xvhg lq wkh htxdwlrq iru 55 wr rewdlq/ djdlq e| h{sdqvlrq lq %/ u 5 " 7 5 8 t5 q5 % 4 s +4517;, . R %5 = 55 @ " 8 49 Z Wkh uvw rughu frqwulexwlrq ydqlvkhv iru @
8 49 4
iru zklfk wkhq wkh erxqgdu| frqglwlrq ehfrphv s s s 4 Z 5" " 7 5 4 t5 q5 . s s 55 @ = s 45 Z
+4517<,
Wkh vxshusrvlwlrq ri exon htxdwlrq dqg olqhdu Nqxgvhq od|hu uhdgv & & %u %u 9 45 45 8 | 45 8 | 45 55 @ 55mE . G frvk @ > . G frvk 9 Nq 8 S3 9 Nq dqg wkxv wkh erxqgdu| frqglwlrq doorzv wr uhodwh dpsolwxgh G dqg fruuhfwlrq idfwru e| kt 4 l s s 8 5 9 45 45 G frvk 45 Z 9 Nq 8 S3 s @ s = s 4 9 45 45 5" Z " 7 5 4 t5 q5 . S3 . 8 S3 Iru wkh suhvhqwhg h{dpsohv/ wkh qhz erxqgdu| frqglwlrq +4517<, jlyhv jrrg djuhhphqw zlwk wkh wwhg ydoxhv iru wkh dpsolwxgh G dv vkrzq lq Wdeoh 4514 iru * 3=91 Zkloh wkh uhvxowv duh vdwlvidfwru|/ lw lv udwkhu xqvdwlvi|lqj wkdw wkh txdolw| ri wkh erxqgdu| frqglwlrqv vkrxog ghshqg rq wkh wwlqj frh!flhqwv dqg / zklfk zhuh lqwurgxfhg dv dg krf fruuhfwlrqv1 Pruh uh qhg dujxphqwv duh qhhghg wr surgxfh pruh uholdeoh erxqgdu| frqglwlrqv iru wkh U46 htxdwlrqv/ ru rwkhu h{whqghg vhwv ri htxdwlrqv1
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Wkh lqwhjudo ryhu F lv ri Jdxvvldq w|sh/ vr wkdw s 5u.q 5 5u . q . 6 s ul4 lq mP @ 5 Ll4 lq 5 Vlqfh wkhuh lv qr sduwlfxodu gluhfwlrq ri v|pphwu|/ wkh lqwhjudov Ll4 lq pxvw eh lvrwurslf dqg v|pphwulf/ dqg lw iroorzv lpphgldwho| wkdw Ll4 lq @ 3 li q lv rgg1 Iru q @ 3 rqh rewdlqv 4 L@ 7
]
5
]
vlq &g&g* @ 4 > *@3
&@3
vr wkdw wkh vfdodu prphqwv duh jlyhq e| u.4 5u . 6 u5 u s mP @ @ u +5u . 4,$$ = 5 Gxh wr lvrwurs|/ wkh lqwhjudo iru q @ 5 pxvw kdyh wkh irup ] 5 ] 4 Llm @ l m vlq &g&g* @ lm 7 *@3 &@3 zkhuh wkh frh!flhqw fdq eh frpsxwhg e| pxowlso|lqj zlwk lm / l1h1 e| wdnlqj wkh wudfh/ dqg vxevhtxhqw lqwhjudwlrq/ dv @ 46 1 Wkxv/
569
D Dsshqgl{
s 5u.5 5 s ulmmP @ 5 6
5u . 8 5
lm @ u.4
+5u . 6,$$ lm 6
Djdlq gxh wr lvrwurs|/ wkh iruwk rughu lqwhjudo kdv wkh uhsuhvhqwdwlrq Llmno @ + lm no . ln mo . lo mn , Pxowlsolfdwlrq zlwk lm no / wkdw lv wdnlqj wkh wudfh wzlfh/ dqg vxevhtxhqw lq0 whjudwlrq |lhogv @ 4@48 vr wkdw ulmnomP @ u.5
+5u . 8,$$ + lm no . ln mo . lo mn , = 48
Qrwh wkdw wkh wudfh0iuhh sduwv ri lvrwurslf whqvruv ydqlvk/ vr wkdw xdl4 lq mP @ kl4 l5 lq lmP @ 3 iru q 4 = D1616 Kdoi0vsdfh prphqwv ri wkh Pd{zhooldq Kdoi0vsdfh0prphqwv ri wkh Pd{zhooldq duh gh qhg dv ] u kl4 lq @ p F 5u Fl4 Flq iP gF = Fl ql 3
Wkh vdph vxevwlwxwlrqv dv lq wkh suhylrxv vhfwlrq jlyh s 5u.q u . 5u.q.6 s 5 Ml4 lq kul4 lq @ 5 zlwk wkh lqwhjudov Ml4 lq @
4 5
]
5
]
*@3
@5
&@3
l4 lq vlq &g&g* =
Iru q @ 3 rqh qgv 4 M@ 5
]
5
]
@5
vlq &g&g* @ 4 > *@3
vr wkdw
&@3
umP +5u . 4,$$ @ = 5 5 Wkh gluhfwlrq ri wkh qrupdo ql lv wkh rqo| glvwlqjxlvkhg yhfwru lq wkh fdofxodwlrq ri wkh kdoi0vsdfh prphqwv1 Wkxv/ wkh yhfwru lqwhjudo pxvw eh ri wkh irup ] 5 ] @5 4 l vlq &g&g* @ ql = Ll @ 5 *@3 &@3 ku @ u
Vfdodu pxowlsolfdwlrq zlwk ql jlyhv
D16 Lqwhjudov ri Jdxvvldqv dqg Pd{zhooldqv
4 5
@
]
5
]
@5
frv & vlq &g&g* @ *@3
&@3
56:
4 5
dqg wkh yhfwruldo prphqwv duh 4
kul
@
u. 54
5u 5 s +u . 4,$ql =
Wkh whqvruldo lqwhjudo pxvw eh ri wkh irup 4 Llm @ 5
]
5
]
@5
l m vlq &g&g* @ ql qm . lm *@3
&@3
zkhuh wkh frh!flhqwv iroorz e| vfdodu pxowlsolfdwlrq zlwk ql qm dqg lm dqg lqwhjudwlrq dv @ 3 dqg @ 4@6 vr wkdw kulm @ u.4
ulmmP +5u . 6,$$ lm @ = 9 5
Wkh lqwhjudo zlwk wkuhh lqglfhv kdv wkh uhsuhvhqwdwlrq Llmn
4 @ 5
]
5
]
@5
l m n vlq &g&g* @ ql qm qn . +ql mn . qm ln . qn lm , *@3
&@3
zkhuh wkh frh!flhqwv iroorz djdlq e| pxowlsolfdwlrq zlwk ql qm qn dqg ql mn dqg vxevhtxhqw lqwhjudwlrq/ dv . 6 @
4 4 4 dqg . 8 @ > vr wkdw @ @ 7 5 ;
dqg 6
u 5 6 5 kulmn @ u. 5 s +u . 5,$ +ql mn . qm ln . qn lm ql qm qn , =
Wkh irxuwk rughu lqwhjudo kdv wkh uhsuhvhqwdwlrq Llmno @ + lm no . ln mo . lo mn , . ql qm qn qo . . +ql qm no . ql qn mo . ql qo mn . qm qn lo . qm qo ln . qn qo lm , = Pxowlsolfdwlrq zlwk ql qm qn qo / ql qm no dqg lm no dqg lqwhjudwlrq |lhogv @ @ 3 dqg @ 4@48 vr wkdw kulmno @ u.5
ulmnomP +5u . 8,$$ + lm no . ln mo . lo mn , @ = 63 5
56;
D Dsshqgl{
D1617 Lqwhjudov ri wkh dqlvrwurslf Jdxvvldq Wkh prphqwv ri wkh dqlvrwurslf Jdxvvldq4 iHV @
4 4 s F F h{s 4 l m > p ghw ^5 lm ` 5 lm
duh gh qhg dv
] ul4 lq mHV
@p
F 5u Fl4 Flq iHV gF >
4 lm lv d uhdo v|pphwulf pdwul{1 Iru wkh vroxwlrq lw lv frqyhqlhqw wr lqwurgxfh wkh yhfwru l e| Fl @ Wln n / vr wkdw W 4 4 lm Fl Fm @ Wnl lm Wmo n o =
Wkh wudqvirupdwlrq pdwul{ Wlm fdq eh fkrvhq dv wkh +ruwkrjrqdo, pdwul{ ri wkh W 4 lm Wmo @ 4@l lm dqg Wln Wlo @ lm +wkh xqghuolqhg hljhqyhfwruv/ vr wkdw Wnl lqgh{ lqglfdwhv wkdw qr vxppdwlrq ryhu l lv shuiruphg,1 Vlqfh 4 lm lv uhdo dqg v|pphwulf/ wkh hljhqydoxhv 4@l duh uhdo/ dqg lw lv dvvxphg wkdw wkh| duh srvlwlyh1 Pruhryhu/ wkh hljhqyhfwruv fdq eh qrupdol}hg vr wkdw ghw ^Wlm ` @ 4/ dqg gf @ g 4 g 5 g 6 / F 5 @ 5 @ 54 . 55 . 56 1 Lqvhuwlrq ri doo wklv lqwr wkh gh qlwlrq ri wkh prphqwv jlyhv Wl4 n4 Wl4 n4 ul4 lq mHV @ s 6 s 5 4 5 6 ] 4 5 4 5 4 5 4 5 6 g 4 g 5 g 6 = 5u n4 nq h{s 54 55 56 Ixuwkhu hydoxdwlrq vkrzv wkdw 3mHV @ > 4mHV @ +4 . 5 . 6 , @ nn 5mHV @ 6 54 . 55 . 56 . 5 +4 5 . 4 6 . 5 6 , @ 5 lm lm . 5nn 3 4 4 3 3 3lmmHV @ Wln Wmo C 3 5 3 D @ lm 3 3 6 no 3 5 3 5 4 4 6 4 3 3 4 3 3 4lmmHV @ Wln Wmo 75 C 3 55 3 D . +4 . 5 . 6 , C 3 5 3 D 8 3 3 6 no 3 3 56 no @ 5 ln nm . nn lm dqg
ul4 lq mHV @ 3 iru q @ 5n . 4 +q xqhyhq, 1
Wkhvh lqwhjudov dsshdu lq hydoxdwlrqv iru wkh HV0EJN prgho/ khqfh wkh lqgh{ .71
D17 Wkh lqwhjudov +8154,
56<
D17 Wkh lqwhjudov +8154, Lq wklv dsshqgl{ wkh lqwhjudov +8154, duh frpsxwhg/ zklfk duh gh qhg dv ] Ll4 lq @
5
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3
@5
3
nl4 nlq frvq vlq IP +, g g% =
Lq srodu frruglqdwhv/ wkh froolvlrq yhfwru lv n @ ifrv % vlq > vlq % vlq > frv j dqg zh uhfdoo wkdw lv wkh dqjoh ehwzhhq j dqg n/ zklfk lpsolhv wkdw jl nl @j @ frv 1 Wkh uhvxowv frqwdlq wkh frh!flhqwv ] @5 frvu vlqv IP +, g zlwk "+q>4, @ "+q5>4, "+q5>6, = "+u>v, @ 5 3
Wkh yhfwru lqwhjudo pxvw eh ri wkh irup ]
5
]
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Ll @
nl frv vlq IP +, g g% @ d 3
3
jl j
dqg vfdodu pxowlsolfdwlrq zlwk jl @j vkrzv wkdw d @ "+5>4, / dqg wkxv Ll @ "+5>4,
jl = j
Wkh udqn05 whqvru lqwhjudo pxvw eh ri wkh irup ]
5
]
@5
Llm @ 3
nl nm frv5 vlq IP +, g g% @ d
3
dqg e| vfdodu pxowlsolfdwlrq zlwk lm dqg
jl jm j5 /
jl jm . e lm j5
uhvshfwlyho|/ iroorzv wkdw
d . 6e @ "+5>4, dqg d . e @ "+7>4, vr wkdw +5>4,
5e @ "
+7>4,
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6 +5>6, 4 +5>6, +5>4, @ " " " 5 5
4 6 jl jm Llm @ "+5>4, "+5>6, . "+5>6, lm = 5 j5 5
Wkh udqn06 whqvru lqwhjudo pxvw eh ri wkh irup ]
5
]
Llmn @ 3
@5
nl nm nn frv6 vlq IP +, g g% @ d
3
dqg e| vfdodu pxowlsolfdwlrq zlwk +lm jn, @j dqg wkdw
jl jm jn j6 /
jn, jl jm jn . e +lm 6 j j uhvshfwlyho|/ iroorzv
573
D Dsshqgl{
8 d . e @ "+7>4, dqg d . e @ "+9>4, 6 vr wkdw 8 5 e @ "+7>4, "+9>4, @ "+7>6, dqg d @ "+5>4, "+5>6, "+7>6, 6 5 dqg wkxv Llmn
jn, 8 +7>6, jl jm jn 6 +5>4, +5>6, = @ " " " . "+7>6, +lm 5 j6 5 j
Wkh wudfh ri wkh udqn07 whqvru lqwhjudo pxvw eh ri wkh irup ]
5
]
@5
Llmnn @ 3
nl nm frv7 vlq IP +, g g% @ d
3
jl jm . e lm j5
dqg e| vfdodu pxowlsolfdwlrq zlwk lm / dqg jl jm @j 5 / uhvshfwlyho|/ iroorzv wkdw d . 6e @ "+7>4, dqg d . e @ "+9>4, vr wkdw 6 5e @ "+7>4, "+9>4, @ "+7>6, dqg d @ "+5>4, "+5>6, "+7>6, 5 dqg wkxv Llmnn
6 +7>6, jl jm 4 +5>4, +5>6, @ " " " . "+7>6, lm = 5 j5 5
D18 Odjudqjh pxowlsolhuv Zh dvn iru wkh h{wuhpxp ri d ixqfwlrq i +{l , ri wkh yduldeohv {l +l @ 4> > q, xqghu d frqvwudlqw j +{l , @ 31 Dw wkh h{wuhpxp wkh yduldwlrq ri i ydqlvkhv/ gi @
q [ Ci g{l @ 3 = C{l l@4
Zlwkrxw wkh frqvwudlqw/ wkh yduldeohv {l duh lqghshqghqw/ dqg rqh frxog du0 elwudulo| fkrvh g{l @ 3 iru l 9@ n dqg g{n 9@ 3 vr wkdw Ci@C{n @ 3 iru doo n1 Krzhyhu/ ehfdxvh ri wkh frqvwudlqw/ wkh yduldeohv {l duh qrw lqghshqghqw/ dqg wkhlu yduldwlrqv g{l duh uhodwhg wkurxjk q [ Cj g{l @ 3 = C{ l l@4
D19 Htxdwlrqv iru wkh frpsxwdwlrq ri jhqhudol}hg 46 prphqw htxdwlrqv
574
Wklv lpsolhv wkdw wkh yduldwlrqv g{l fdqqrw eh fkrvhq lqghshqghqwo|1 Wr vroyh wkh sureohp/ wkh odvw htxdwlrq lv pxowlsolhg e| d Odjudqjh pxowlsolhu dqg dgghg wr wkh rwkhu/ vr wkdw q [ Ci l@4
Cj C{l C{l
g{l @ 3 =
Vwloo/ wkh yduldwlrqv fdqqrw eh fkrvhq lqghshqghqwo|/ exw qrz wkh Odjudqjh pxowlsolhu fdq eh xvhg wr holplqdwh rqh whup iurp wkh vxp/ e| fkrrvlqj lw vr wkdw Cj Ci @3= C{q C{q Wkh uhpdlqlqj frqglwlrq uhdgv q4 [ l@4
Cj Ci C{l C{l
g{l @ 3
zkhuh rqo| +q 4, yduldwlrqv g{l +l @ 4> > q 4, dsshdu/ dqg wkhvh fdq eh fkrvhq lqghshqghqwo|/ vr wkdw Cj Ci @ 3 iru l @ 4> > q 4 = C{l C{l Lq vkruw/ qglqj wkh h{wuhpxp ri i j lv htxlydohqw wr qglqj wkh h{wuhpxp ri i xqghu wkh frqvwudlqw j @ 31 Zkhq vhyhudo frqvwudlqwv pxvw eh frqvlghuhg/ j +{l , @ 3/ rqh mxvw qhhgv wr xvh S pruh Odjudqjh pxowlsolhuv/ wkdw lv rqh kdv wr qg wkh h{wuhpxp ri i j 1
D19 Htxdwlrqv iru wkh frpsxwdwlrq ri jhqhudol}hg 46 prphqw htxdwlrqv Wklv dsshqgl{ frqwdlqv lqwhuphgldwh vwhsv iru wkh frpsxwdwlrq ri wkh jhqhu0 dol}hg 46 prphqw htxdwlrqv +;14;/ ;14<,1 D1914 Prphqw htxdwlrqv iru zld Wkh vwduwlqj srlqw lv wkh prphqw htxdwlrq iru xdl / +8148,2+;14,5 1 Diwhu lqwur0 d e| phdqv ri +;144,/ uhsodflqj wkh wlph ghulydwlyh ri e| gxflqj wkh zld > zlm phdqv ri wkh hqhuj| edodqfh +;145,5 / fdqfhoolqj whupv ri R %5 dqg kljkhu/ d d4 dqg glylvlrq e| 5 / zh vhh 4
575
D Dsshqgl{
4 d4 Gtl d 4 d 5d4 C C oq . d ln ln Gw 5 d 3 C{n d 3 C{n 4 d +5d . 6,$$ C ln 5d . 8 Cyl 5d Cyn 58 7d Cyn tl tn . . . . tn 5d 3 6 C{n 48 C{n 8 C{n 8 C{l e [ [ 4 4 d +5d . 6,$$ 4 C tl +4, 4 zl 4>4>4 e f lm tm . Fde . \ @ . = d>ef 5 d e4 3 d 9 d C{l e e>f Vxewudfwlqj wkh edodqfh ri wkh khdw x{ +;14:, jlyhv C d 4 d 5d4 4 4 zle 4 [ +4, +4, 4 ln @ 4. F4e Fde e4 C{n 3 5 3 5 d 5 d e 4 d4 4 d +5d . 6,$$ 8 C oq 4 4 C ln ln . . . 4d 5 3 C{n 6 C{n 5 5d 3 d 3 5d 5 Cyn 5d 5 Cyl 7 7d Cyn tn tn tl . . . C{l C{n 8 C{n 8 48 [ 4>4>4 f tl C 8 d +5d . 6,$$ 4 4>4>4 e f lm tm . = . \d>ef e \4>ef d 9 C{l 5 3 d 3 4 e>f
Wkh deryh htxdwlrq lv phdqlqjixo iru d 5 rqo|1 Wkh zle iroorz iurp lqyhuvlrq ri wkh pdwul{ +4, +4, 4 Gde @ F4e Fde d dv [ C 4 4 d 4 d 5d4 zlg 4 ln . @ Ggd 4 g4 C{ 5 5 5 d n 3 3 d [ 4 d4 C oq 4 . Ggd ln 4d C{n d 3 d [ C ln 4 4 4 d +5d . 6,$$ 4 8 . . Ggd C{n 6 5 5 3 5d 3 d [ 5d 5 Cyn 5d 5 Cyl 4 7 7d Cyn tn tl tn . . . Ggd C{n 8 C{l 48 C{n 8 d [ [ 4>4>4 f 4>4>4 e f lm tm 4 Ggd . \d>ef e \4>ef 3 4 3 d d e>f [ C tl d +5d . 6,$$ 4 4 8 . Ggd = +D18, 5 9 d C{l d d D1915 Prphqw htxdwlrqv iru zlm
Wkh htxdwlrq iru wkh 50whqvruv xdlm / +8149,2+;14,6 lv pdqlsxodwhg vlploduo|1 d e| phdqv ri +;144,/ uhsodflqj wkh wlph ghulydwlyh Diwhu lqwurgxflqj wkh zld > zlm
D19 Htxdwlrqv iru wkh frpsxwdwlrq ri jhqhudol}hg 46 prphqw htxdwlrqv
576
ri e| phdqv ri wkh hqhuj| edodqfh +;145,5 / fdqfhoolqj whupv ri R %5 dqg kljkhu/ dqg glylvlrq e| d d / zh vhh 3
Cynl 7d Cyml Cyn 5 Cyn G lm 7 d.4 3 Ctkl 9d . . klm . nkl d lm . 5 nkl Gw 8 4 d C{ml : C{n 8 C{ml 6 C{n C{n d 3 C oq 7 d.4 3 Cyn 7 C oq d 3 . lm +5d . 8, tkl d tkl +5d . 8, C{n 8 4 d C{ml 8 4 d 4 d C{ml [ [ [ 4 +5, 5>3>4 7e f 3 tkl tnl 5>4>4 e f mkl nlm e . 3d Fde de zln . \d>ef . \d>ef d 4 4 d 5 3 d e e>f e>f Cykl 3 +5d . 8,$$ ln @ .5 = d 48 C{ml wkh uhvxow fdq eh ixuwkhu vlpsol hg e| phdqv ri wkh lghqwlwlhv klm
Cynl Cyml 4 5 7 Cyn Cyu @ lm . nkl ukl > C{n 6 C{n 6 C{n 48 C{ml
Cynl 7d 5 Cyn Cyn 9d klm . nkl d lm : C{n 8 C{ml 6 C{n Cyml 7d 5 Cyn Cyn @ . nkl lm = nkl : C{n C{ml 6 C{n Vxewudfwlrq ri wkh edodqfh iru wkh suhvvxuh ghyldwru +;149, iurp wkh pdlq htxdwlrq |lhogv/ diwhu vrph uhduudqjhphqwv/ [ +5, +5, ze Ctkl 7 d.4 3 ln @ 4 F3e 3 Fde e d 8 4 d C{ml e Cyml d 3 C oq 7d 5 Cyn Cyn 7 . . nkl lm t +5d . 8, nkl : C{n C{ml 6 C{n 8 4 d kl C{ml 7 d.4 3 C oq d 3 d tkl +5d . 8, 8 4 d 4 d C{ml [ 5>3>4 7e f 5>3>4 7e f tkl tnl 3 . \3>ef \d>ef 4 4 d 4 4 5 e>f [ 5>4>4 5>4>4 e f mkl nlm e f \3>ef . \d>ef 3 d 3 3 e>f Cykl 3 +5d . 8,$$ lm 4 .5 = d 48 C{ml e iroorz iurp lqyhuvlrq ri wkh pdwul{ Wkh zlm
577
D Dsshqgl{
+5, +5, Hde @ F3e 3 Fde d dv g [ Ctkl zln 4 7 d.4 3 @ Hgd 4 g 8 C{ 4 d ml d [ Cyml 5 Cyn Cyn 4 7d . Hgd . nkl lm nkl : C{n C{ml 6 C{n d [ d 3 C oq 4 7 +5d . 8, Hgd t 8 4 d kl C{ml d [ d.4 3 C oq d 3 4 7 Hgd d tkl +5d . 8, 8 C{ml 4 d 4 d d [ [ 5>3>4 7e f 5>3>4 7e f tkl tnl 4 3 . Hgd \3>ef \d>ef 4 4 d 4 4 5 d e>f [ [ 5>4>4 5>4>4 e f mkl nlm 4 e f . Hgd \3>ef \d>ef 3 d 3 3 d e>f [ Cykl 3 +5d . 8,$$ lm 4 .5 Hgd 4 = +D19, d 48 C{ml d D1916 Frh!flhqwv lq +;14;/ ;14<, Zkhq wkh deryh uhvxowv +D18/ D19, duh lqvhuwhg lqwr wkh htxdwlrqv iru vwuhvv dqg khdw x{/ wkhvh dvvxph wkh irup +;14;/ ;14<, zkhuh wkh frh!flhqwv duh gh qhg dv 5 6 [ +5, 77 d.4 3 8 4 4 F3g Hgd 4 #4 @ 8 4 d g>d
#5 @
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g>d
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D19 Htxdwlrqv iru wkh frpsxwdwlrq ri jhqhudol}hg 46 prphqw htxdwlrqv
578
[ +5, +5d . 8,$$ 4 #: @ 4 F3g Hgd 4 3 d 48 g>d
dqg "4 "5 "6 "7
"8 "9
[ +4, 4 4 4 4 d d4 4 4 4 @ 4 F4g Ggd 4d 5 3 5 3 d 5 3 3 g>d [ +4, 4 d4 4 4d @ 4 F4g Ggd d 3 g>d 4 4 8 [ +4, 4 4 4 8 4 4 d +5d . 6,$$ @ F4g Ggd +D1;, 5 3 5 5 3 5 d 5 3 9 g>d 5 [ +4, 4 5 5d @ F4g Ggd 8 8 8 g>d 5 6 3 7[ 4>4>4 e f [ +4, 4 [ e f 4 4>4>4 4>4>4 8 @ \ . F4g Ggd \d>ef \4>ef 5 e>f 4>ef 3 4 4 d 3 g>d e>f [ d +5d . 6,$$ 4 8 +4, 4 8 F4g Ggd @ 5 5 9 d g>d
Wkh frh!flhqwv e / e duh jlyhq e| +;1:,/ e @
[ k +4, l4 d +5d . 6,$$ [ k +5, l4 5 Fed Fed > e @ +5d . 8,$$ 6 48 d@4 d@3
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