Rq Uhodwlyh Ghsulydwlrq dqg Pljudwlrq G|qdplfv1

(1)

Vkrxog L Vwd| ru Vkrxog L JrB

Rq Uhodwlyh Ghsulydwlrq dqg Pljudwlrq G|qdplfv1

Kdoyru Phkoxp

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Devwudfw

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ehwzhhq jhqhudwlrqv ri pljudqwv wkdw fdq jhqhudwh gxdo htxloleuld1 Mho= R483 M943

4 Lqwurgxfwlrq

Vkrxog L vwd| ru vkrxog L jrB lv wkh txhvwlrq wurxeolqj ploolrqv ri shrsoh lq wkh ghyhorslqj zruog1 Hyhu|

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5 Li L vwd| wkhuh zloo eh wurxeoh 0 Li L jr wkhuh zloo eh grxeoh

Wkh prgho frqvlghuv uxudo wr xuedq pljudwlrq zlwklq d frxqwu|1 Wkh djhqwv duh wkh srwhqwldo pljudqwv lq d uxudo yloodjh1 Wkh| fdq eh wkrxjkw ri dv xqpduulhg |rxqj er|v1 Wkh ghflvlrq wr pljudwh duh wkhluv dorqh dqg qrw sduw ri klv idplo|*v frruglqdwhg vwudwhj| +dv iru h{dpsoh lq Vwdun dqg Oxfdv 4<;;,1 Wkh djhqwv olyh iru wzr shulrgv1 Lq wkh ehjlqqlqj ri wkh?uvw shulrg wkh| pxvw ghflgh zkhwkhu wr ohdyh wkh yloodjh/ dqg olyh wkh uhvw ri wkhlu olih lq wrzq/ ru wr vwd| lq wkh yloodjh dqg olyh wkh uhvw ri wkhlu olih wkhuh1 Khqfh pljudwlrq lv dvvxphg wr eh shupdqhqw1 Wrwdo glvfrxqwhg zhoiduh iru d pljudqw prylqj wr wrzq lq shulrgl lv jlyhq e|

Yl@ +4 .ë,z.d^+zýzçl,@zçl.ë+zýzçl.4,@zçl.4` 3? ë ?4> d A3 +4,

WWkdqnv wr Ndooh Prhqh/ Rghg Vwdun/ dqg dq dqrq|prxv uhihuhh iru vwlpxodwlqj glvfxvvlrqv dqg ydoxdeoh frpphqwv1 Ilqdqfldo vxssruw e| wkh Uhvhdufk Frxqflo ri Qruzd| lv judwhixoo| dfnqrzohgjhg1 Wlwoh dqg fkdswhu khdglqjv duh doo eruurzhg iurp Wkh Fodvk*v 4<;5 uhfrugFRPEDW URFN1

(2)

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ç

zl lv jlyhq e| wrwdo lqfrph lq wrzq glylghg e| wrwdo srsxodwlrq lq wrzq lq shulrg l1 Ohw wkh qxpehu ri qrq0pljudqw flwl}hqv lq wrzq/ wkh wrzqvphq/ eh ?{hg dqg qrupdol}hg lq vl}h wr 41 Ohw Pl eh wkh qxpehu ri pljudqwv hqwhulqj lq shulrgl1 Ohw d iudfwlrqòri wkh wrzqvphq eh vnloohg hduqlqj ä1 Wkh uhpdlqlqj iudfwlrq +4ýò,duh xqvnloohg dqg hduqzzklfk lv wkh vdph zdjh dv wkh pljudqwv1 Wkh dyhudjh zdjh lq wrzq lq shulrg llv

ç

zl@ ^z+Plý4.Pl, .z+4ýò, .òä`@+Plý4.Pl. 4,

zkhuh +Plý4.Pl, lv wkh wrwdo qxpehu ri pljudqwv lq wrzq lq shulrg l/ dqg +Plý4.Pl. 4, lv wkh wrwdo srsxodwlrq lq wrzq1 Khqfh/ lw iroorzv wkdw

+zýzçl,@zçl @ +4ýe,@+Plý4.Pl.e, +5, zkhuh e@ +4ýò, .òäz A4

Jlyhq wkdw wkh vnloohg zdjh lv deryh wkh xqvnloohg zdjh lw iroorzv wkdw e A 41 Lqvhuwlqj +5, lq wkh zhoiduh ixqfwlrq +4, wkh zhoiduh iru d pljudqw lq shulrg liroorzv

Yl@z.d+4ýe,@+Plý4.Pl.e, .ë^z.d+4ýe,@+Pl.Pl.4.e,` +6, Wkh rswlrq qrw wr pljudwh lpsolhv jhwwlqj wkh yloodjh zdjh dqg wr uhpdlq kdylqj wkh yloodjh*v srsxodwlrq dv uhihuhqfh jurxs1 Dvvxplqj hjdolwduldq glvwulexwlrq dqg?{hg zdjh/z³/ lq wkh yloodjh wkhuh zloo eh qr uhodwlyh ghsulydwlrq h>hfw1 Khqfh/ wkh zhoiduh dvvrfldwhg zlwk qrw pljudwlqj lv vlpso|

Y³@ +4 .ë,z³ +7,

Frpelqlqj +6, dqg +7, dqg devwudfwlqj iurp pljudwlrq frvwv jlyhv wkhqhw zhoiduh jdlq iurp pljudwlrq Xl@YlýY³@ZGýUGl

ZG@ +4 .ë,ý

zýz³ü

> UGl@d+eý4, ^4@+Plý4.Pl.e, .ë@+Pl.Pl.4.e,` +8, zkhuh ZG fdswxuhv wkh zdjh gl>huhqwldo dqg wkh UGv fdswxuh wkh uhodwlyh ghsulydwlrq h>hfwv1 Wkh zdjh gl>huhqwldo lv dvvxphg wr eh lqghshqghqw ri wkhPv dqg lv wkxv qrw d>hfwhg e| pljudwlrq1⁴ Zkhqe A4/ lw iroorzv wkdw wkh uhodwlyh ghsulydwlrq h>hfw lv qhjdwlyh dqg uhsuhvhqwv d orvv1 Irue@ 4wklv uhodwlyh ghsulydwlrq h>hfw fhdvh wr h{lvw dv e @ 4 lpsolhv wkdw wkh pljudqwv zdjh lv htxdo wr wkh dyhudjh lq wrzq1 Jlyhq wkh dvvxpswlrqv derxwe A4/ krzhyhu/ lw iroorzv wkdw

CXl@CPlý4A3> CXl@CPlA3> CXl@CPl.4A3 +9, Wklv looxvwudwhv dq lpsruwdqw uhvxow=

î Wkh uhodwlyh ghsulydwlrq phfkdqlvp uhsuhvhqwv d srvlwlyh h{whuqdolw| lq pljudwlrq/ erwk zlwklq dqg ehwzhhq jhqhudwlrqv1

Wkh pljudwlrq ghflvlrqv duh wkhuhiruh vwudwhjlf frpsohphqwv1 Rqh lpsolfdwlrq ri wklv vwudwhjlf frpsolphq0 wdulw| lv wkdw wkhuh zloo eh qr lqwhulru vroxwlrqv1 Hlwkhu hyhu|rqh ru qrqh ri wkh srwhqwldo pljudqwv zlwklq d jhqhudwlrq zloo pljudwh1

Ohw wkh qxpehu ri srwhqwldo pljudqwv zlwklq hdfk jhqhudwlrq eh 4/ khqfh lq htxloleulxpPl5 i3>4j1 Uxolqj rxw frruglqdwlrq ehwzhhq pljudqwv/ wkh frqglwlrq iru jhqhudwlrq l wr pljudwh lv wkdw wkh ?uvw pljudqw ?qgv lw zhoiduh lpsurylqj wr pryh1 l1h1 ZG A UGl zkhq Pl @ 31 Wdeoh 4 jlyh wkh uhodwlyh ghsulydwlrq h>hfw/

4Wkh dvvxpswlrq derxw d?{hgç/ xqd>hfwhg e| pljudwlrq/ juhdwo| vlpsol?hv wkh glvfxvvlrq ri wkh uhodwlyh ghsulydwlrq h>hfwv1 Lpsolfdwlrqv iurp lqfoxglqj ri d pruh uhdolvwlf zdjh irupdwlrq lv glvfxvvhg lq Vhfwlrq 71

(3)

frqglwlrqhg rqPlý4 dqgPl.41 Lq wkh wdeoh dqg lq wkh iroorzlqj glvfxvvlrq jhqhudwlrqllv uhihuuhg wr dv|rx/

jhqhudwlrqlý4dvxqfohv/ dqg jhqhudwlrql. 4 dvqhskhzv1

Wdeoh 4Uhodwlyh ghsulydwlrq h>hfw iru wkh?uvw pljudqw+Pl@ 3,

Plý4@ 4+Xqfohv lq wrzq, Plý4@ 3 +Qr xqfohv lq wrzq, Pl.4@ 4

+Qhskhzv iroorz, UG⁴@d+eý4,í

4.e4 .ë_4.e⁴ ì

UG⁶@d+eý4,í

4e .ë_4.e⁴ ì Pl.4@ 3

+Qr qhskhzv iroorz, UG⁵@d+eý4,í

4.e4 .ë⁴_eì

UG⁷@d+eý4,ý₄

e .ë⁴_eü

Wkh srvlwlyh h{whuqdolw| lpsolhv wkdw |rxu zhoiduh xqdpeljxrxvo| zloo lqfuhdvh li wkhuh duh rwkhu jhqhudwlrqv ri pljudqwv lq wrzq1 Wkh uhdvrq lv wkdw |rxu uhodwlyh srvlwlrq lq wkh lqfrph glvwulexwlrq lqfuhdvhv wkh pruh srru pljudqwv wkhuh duh uhodwlyh wr wkh ulfk wrzqvphq1 Khqfh/ UG⁴/ zkhuh |rx kdyh |rxu xqfohv lq wrzq lq shulrg l dqg |rxu qhskhzv lq wrzq lq shulrg l. 4/ kdv wkh vpdoohvw uhodwlyh ghsulydwlrq orvv1 UG⁷ zkhuh |rx kdyh qhlwkhu xqfohv ru qhskhzv lq wrzq jlyhv wkh kljkhvw orvv1 Jlyhq d glvfrxqw idfwruë5 k3>4l/ lw lv dovr fohdu wkdw kdylqj frpsdq| lq |rxu ?uvw shulrg lv pruh lpsruwdqw wkdq kdylqj frpsdq| lq wkh vhfrqg/ l1h1 UG⁵? UG⁶1 Wkxv/ wkh frpsohwh udqnlqj ri uhodwlyh ghsulydwlrq frqvhtxhqfhv lv UG⁴? UG⁵? UG⁶? UG⁷1

6 Vr |rx jrwwd ohw ph nqrz 0 Vkrxog L vwd| ru vkrxog L jrB

\rxu ghflvlrq wr pljudwh ru qrw/ ghshqgv rq zkhwkhu wkh jdlq/ X/ iurp prylqj wr wrzq lv srvlwlyh ru qrw1 Iurp ht1 8 lw iroorzv wkdw X lv srvlwlyh li wkh zdjh gl>huhqwldo ZG lv odujhu wkh uhodwlyh ghsulydwlrq h>hfw UG1 Wkh vl}h riZGghshqgv rq krz wkh zdjh lq wkh yloodjh frpsduhv wr wkh zdjh lq wrzq1 UGghshqgv rq wkh pljudwlrq ghflvlrq ri |rxu qhskhzv dqg xqfohv1 Wkh sureohp lv vlpsoh li wkh yloodjh zdjhz3lv hlwkhu yhu|

kljk ru yhu| orz1 Iruz³ vx!flhqwo| kljk/ZGzloo eh qhjdwlyh dqgX zloo eh ohvv wkdq }hur/ luuhvshfwlyh ri rwkhu jhqhudwlrqv* pljudwlrq ghflvlrqv1 \rxu fkrlfh lv wkhq fohdu= \rx vkrxog vwd| lq wkh yloodjh1 Rq wkh rwkhu kdqg zkhqz³ lv yhu| orzZGzloo eh kljk dqg grplqdwh dq| uhodwlyh ghsulydwlrq h>hfw1 \rx vkrxog lq wkdw fdvh gh?qlwho| pryh wr wrzq1

Li pljudwlrq lv qhlwkhu vwulfwo| vxshulru ru lqihulru/ wkh uhdvrqlqj jhwv pruh frpsoh{1 \rx qrz kdyh wr wdnh wkh dfwlrqv ri |rxu xqfohv dqg wkh h{shfwhg dfwlrq ri |rxu qhskhzv lqwr frqvlghudwlrq1 Wkh iroorzlqj olvw frpsulvhv doo wkh srvvleoh pljudwlrq sdwwhuqv

41 Vwdeoh qrq0pljudwlrq/ +l1h1 wkh vwulfwo| lqihulru fdvhZG ? UG⁴ ? UG⁵ ? UG⁶? UG⁷,1 Luuhvshfwlyh ri |rxu xqfohv* ghflvlrq/ |rx zloo qrw pryh1 Wklv zloo dovr krog iru |rxu qhskhzv1

Rxwfrph= Pl@ 3;lè4

51 Vwdeoh pljudwlrq/ +l1h1 wkh vwulfwo| vxshulru fdvhUG⁴ ? UG⁵ ? UG⁶ ? UG⁷ ? ZG,1 \rx zloo pryh zkhwkhu ru qrw |rxu xqfohv pryhg1 Wklv zloo dovr krog iru |rxu qhskhzv1

Rxwfrph= Pl@ 4;lè4

61 Frqglwlrqdo vwdeoh pljudwlrq/ +l1h1 UG⁴ ? UG⁵? UG⁶ ? ZG ? UG⁷,1 \rx zloo vxuho| pryh li |rxu xqfohv pryhg1 Li |rxu xqfohv glg qrw pryh/ |rx zloo rqo| pryh li |rx duh vxuh |rxu qhskhzv zloo iroorz1 Krzhyhu/ |rx nqrz wkdw li |rxu qhskhzv pd{lpl}hv zhoiduh/ wkh| zloo pryh li |rx pryh1 Wkxv/ jlyhq wkdw |rxu qhskhzv duh udwlrqdo/ |rx zloo pryh luuhvshfwlyh ri |rxu xqfohv*v ghflvlrq1

Rxwfrph= Pl@ 4;lè4

71 Edfnzdug orrnlqj gxdo htxloleuld/ +l1h1 UG⁴? UG⁵? ZG ? UG⁶? UG⁷,1 \rx zloo rqo| pryh li |rxu xqfohv pryhg1 \rxu qhskhz zloo rqo| pryh li |rx pryh/ dqg vr rq1

Rxwfrph= Pl@P3;lè4

81 Edfnzdug dqg iruzdug orrnlqj gxdo htxloleuld1 +l1h1 UG⁴? ZG ? UG⁵? UG⁶? UG⁷,1 \rx zloo rqo|

pryh li |rxu xqfohv glg pryh dqg li |rxu qhskhzv zloo pryh1 \rxu qhskhzv lq wxuq zloo rqo| pryh li

|rx pryhg dqg liwkhlu qhskhzv pryh1 LiP3@ 4/Pl@Pl.4 ;lè4 LiP3@ 3/Pl@ 3 ;lè4

Wkhvh fdvhv vkrz wkdw klvwru| dqg h{shfwdwlrqv fdq eh vljql?fdqw ghwhuplqdqwv ri wkh pljudwlrq sdwwhuq1 Krz sodxvleoh duh wkhvh gl>huhqw frq?jxudwlrqvB Iurp Wdeoh 4 lw lv fohdu wkdw li wkh uhodwlyh ghsulydwlrq h>hfw d lv zhdn wkhq UG⁴ äUG⁵ ä UG⁶ ä UG⁷ ä 3 dqg wkh zdjh gl>huhqwldo zloo eh wkh vroh ghwhuplqdqw ri pljudwlrq1 Khqfh/ wkhuh zloo eh hlwkhu 4= vwdeoh qrq0pljudwlrq ru 5= vwdeoh pljudwlrq1 Zkhq d lv srvlwlyh/

(4)

Iljxuh 4= Uhodwlyh ghsulydwlrq yhuvxv zdjh gl>huhqwldo

3158 3183 31:8 4133 4158 4183 41:8 5133 3133

3158 3183 31:8 4133

1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111UG⁴

11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111UG⁵

11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111UG⁶

11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111UG⁷

1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111ZG

d Fdvh Fdvh Fdvh Fdvh

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(5)

7 Ixuwkhu wrslfv

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Uhihuhqfhv

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(6)

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434 +4<<4,=44960:;1

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