Çekirdek üreten uzay yönteminin matematiksel temelleri ve bazı uygulamaları / Mathematical foundations and some applications of the reproducing kernel method

T.C.

FIRAT ÜNİVERSİTESİ

FEN BİLİMLERİ ENSTİTÜSÜ

ÇEKİRDEK ÜRETEN UZAY YÖNTEMİNİN MATEMATİKSEL TEMELLERİ VE BAZI

UYGULAMALARI

DOKTORA TEZİ Ali AKGÜL

Anabilim Dalı: Matematik Programı: Uygulamalı Matematik Danışmanı: Doç. Dr. Mustafa İNÇ

T.C.

FIRAT ÜNİVERSİTESİ

FEN BİLİMLERİ ENSTİTÜSÜ

ÇEKİRDEK ÜRETEN UZAY YÖNTEMİNİN MATEMATİKSEL

TEMELLERİ VE BAZI UYGULAMALARI

DOKTORA TEZİ

Ali AKGÜL

Anabilim Dalı: Matematik

Programı: Uygulamalı Matematik

Danışmanı: Doç. Dr. Mustafa İNÇ

Tezin Enstitüye Verildiği Tarih: 12 Şubat 2014

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phwrgxq êù}ÿpÿ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 7: Vhnlo 8141 *qq iduno ghjhuohul lêlq |dnodvn êù}ÿpohulq judjl1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 95 Vhnlo 9141 Lnl sdudoho sodnd dudvqgdnl gxudjdq vnvdq hnvhqho vlphwuln

elu dnvndq ^;3`11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 97 Vhnlo 9151 p @ Uh @ 4 lêlq UNKVP/ RKDP yh UN7

êù}ÿpohulqlq nduvodvwuopdv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 :: Vhnlo 9161 p @ Uh @ 4 lêlq pxwodn kdwd +PK, yh edjo kdwd +EK,1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 :; Vhnlo 9171 p @ 6 yh Uh @ 4 lêlq UNKVP/ RKDP yh UN7 êù}ÿpohulqlq nduvodvwuo0 pdv11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 :; Vhnlo 9181 p @ ; yh Uh @ 4 lêlq pxwodn kdwd +PK, yh edjo kdwd +EK,1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 :< Vhnlo 9191 p @ 4 yh Uh @ 7 lêlq pxwodn kdwd +PK, yh edjo kdwd +EK,11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 :< Vhnlo 91:1 p @ 4 yh Uh @ 43 lêlq pxwodn kdwd +PK, yh edjo kdwd +EK,1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ;3 Vhnlo 91;1 Vdelw elu Uh|qrogv vd|v Uh @ 4 yh duwdq pdq|hwln dodq hwnlvl p @ 4> 6> ;> 53 lêlq UNKVP êù}ÿpÿqÿq nduvodvwuopdv1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ;3 Vhnlo 91<1 Vdelw elu pdq|hwln dodq hwnlvl p @ 4 yh duwdq Uh|qrogv vd|v Uh @ 4> 7> 43 lêlq UNKVP êù}ÿpÿqÿq nduvodvwuopdv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ;4

Vhnlo :141 Ùuqhn :14 lêlq dqdolwln êù}ÿp +DÊ, yh UNKVP loh hogh hglohq êù}ÿpÿq nduvodvwuopdv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ;<

Vhnlo :151 Ùuqhn :14 lêlq YLP/ KSP yh UNKVP loh hogh hglohq pxwodn

kdwdoduq nduvodvwuopdv1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 <3 Vhnlo :161 Ùuqhn :15 lêlq dqdolwln êù}ÿpÿ yh UNKVP êù}ÿpÿqÿ jùvwhuhq judn111<5 Vhnlo :171 Ùuqhn :15 lêlq DGP/ YSP/ KDP yh UNKVP loh hogh hglohq

pxwodn kdwdoduq nduvodvwuopdv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 <5 Vhnlo :181 Ùuqhn :16 lêlq dqdolwln êù}ÿp +DÊ, yh UNKVP êù}ÿpÿqÿq

nduvodvwuopdv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 <8 Vhnlo :191 Ùuqhn :16 lêlq DGP/ KDP yh UNKVP loh hogh hglohq pxwodn kdwdoduq nduvodvwuopdv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 <8

WDEORODU O LVWHV L

Vd|id Qr

Wdeor 7141 Ùuqhn 714 lêlq hogh hglohq vd|vdo vrqxêodu11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6; Wdeor 7151 Ùuqhn 714 lêlq w @ 4 gxuxpxqgd hogh hglohq

vd|vdo vrqxêodu 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6; Wdeor 7161 Ùuqhn 714 lêlq pxwodn yh edjo kdwdoduq

nduvodvwuopdv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6< Wdeor 7171 Ùuqhn 715 lêlq hogh hglohq PK yh EK1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 74 Wdeor 7181 Ùuqhn 715 lêlq w @ 4 gxuxpxqgd hogh hglohq

vd|vdo vrqxêodu 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 74 Wdeor 7191 Ùuqhn 715 lêlq pxwodn yh edjo kdwdoduq

nduvodvwuopdv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 75 Wdeor 71:1 Ùuqhn 715 lêlq { @ 5=8 gxuxpxqgd hogh hglohq vd|vdo vrqxêodu11 1 1 1 1 1 1 175 Wdeor 71;1 Ùuqhn 715 lêlq { @ 8=3 gxuxpxqgd hogh hglohq vd|vdo vrqxêodu 1 1 1 1 1 1 1 1 75 Wdeor 71<1 Ùuqhn 715 lêlq { @ :=8 gxuxpxqgd hogh hglohq vd|vdo vrqxêodu 1 1 1 1 1 1 1 1 76 Wdeor 71431 Ùuqhn 715 lêlq { @ 43=3 gxuxpxqgd hogh hglohq vd|vdo vrqxêodu 1 1 1 1 1 1 76 Wdeor 71441 Ùuqhn 715 lêlq { @ 5=8 +elulqfl vud, yh { @ 8=3 +lnlqfl vud, lnhq pxwodn kdwdoduq nduvodvwuopdv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 77 Wdeor 71451 Ùuqhn 715 lêlq { @ :=8 +elulqfl vud, yh { @ 43=3 +lnlqfl vud,

lnhq pxwodn kdwdoduq nduvodvwuopdv1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 178 Wdeor 71461 Ùuqhn 715 lêlq { @ 3=39 gxuxpxqgd hogh hglohq

vd|vdo vrqxêodu1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 78 Wdeor 71471 Ùuqhn 715 lêlq { @ 3=3; gxuxpxqgd hogh hglohq vd|vdo vrqxêodu1 1 1 1 1 1 78 Wdeor 71481 Ùuqhn 715 lêlq { @ 3=4 gxuxpxqgd hogh hglohq vd|vdo vrqxêodu1 1 1 1 1 1 1 79 Wdeor 71491 Ùuqhn 715 lêlq pxwodn kdwdoduq nduvodvwuopdv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 79 Wdeor 8141  @ 4 gxuxpxqgd hogh hglohq vd|vdo vrqxêodu111 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 18< Wdeor 8151  @ 5 gxuxpxqgd hogh hglohq vd|vdo vrqxêodu 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 8< Wdeor 8161  @ 6=84 gxuxpxqgd hogh hglohq vd|vdo vrqxêodu 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 93 Wdeor 8171  @ 4 gxuxpxqgd Dgrpldq d|uvp phwrgx/ Odsodfh d|uvp phwrgx/ E0vsolqh phwrgx yh êhnlughn ÿuhwhq x}d| phwrgx loh hogh hglohq pxwodn kdwdodu11 1 1 1 1 1 193 Wdeor 8181  @ 5 gxuxpxqgd Odsodfh d|uvp phwrgx/ Dgrpldq d|uvp phwrgx/ E0vsolqh phwrgx yh êhnlughn ÿuhwhq x}d| phwrgx loh hogh hglohq pxwodn kdwdodu 1 1 1 1 1 1 1 94

Wdeor 8191  @ 6=84 gxuxpxqgd E0vsolqh phwrgx/ êrn whulpol ropd|dq hjul phwrgx/ Olh0juxs êhnlp phwrgx yh êhnlughn ÿuhwhq x}d| phwrgx loh hogh hglohq pxwodn kdwdodu 94 Wdeor 9141 p @ 4 yh Uh @ 4 lêlq hogh hglohq vd|vdo vrqxêodu 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 :7 Wdeor 9151 p @ 6 yh Uh @ 4 lêlq hogh hglohq vd|vdo vrqxêodu1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 :8 Wdeor 9161 p @ ; yh Uh @ 4 lêlq hogh hglohq vd|vdo vrqxêodu1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 :8 Wdeor 9171 p @ 53 yh Uh @ 4 lêlq hogh hglohq vd|vdo vrqxêodu1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 :9 Wdeor 9181 p @ 4 yh Uh @ 7 lêlq hogh hglohq vd|vdo vrqxêodu1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 :9 Wdeor 9191 p @ 4 yh Uh @ 43 lêlq hogh hglohq vd|vdo vrqxêodu1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 :: Wdeor :141 Ùuqhn :14 lêlq hogh hglohq vd|vdo vrqxêodu1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ;; Wdeor :151 Ùuqhn :14 lêlq KSP/ YLP yh UNKVP loh hogh hglohq pxwodn kdwdoduq nduvodvwuopdv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ;< Wdeor :161 Ùuqhn :15 lêlq hogh hglohq vd|vdo vrqxêodu11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 <4 Wdeor :171 Ùuqhn :15 lêlq YSP/ DGP/ KDP yh UNKVP loh hogh hglohq pxwodn kdwdoduq nduvodvwuopdv1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 <4 Wdeor :181 Ùuqhn :16 lêlq hogh hglohq vd|vdo vrqxêodu11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 <7 Wdeor :191 Ùuqhn :16 lêlq DGP/ KDP yh UNKVP loh hogh hglohq pxwodn kdwdoduq nduvodvwuopdv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 <7

VHPEROOHU O LVWHV L O= Olqhhu Rshudwùu O3_{= Lqwhjudo Rshudwùuÿ} S_{= Wrsodp Vhperoÿ} = Doskd = Ehwd = Ghowd  = Vljpd  = Ukr %= Hsvlorq = Hwd = Odpegd

U_{= Uhho Vd|odu Fÿpohvl}

NLVDOWPDODU

KDP= Krprwrsl dqdol} phwrgx NgY= Nruwhzhj0gh Yulhv ghqnohpl KSP= Krprwrsl shuwÿuedv|rq phwrgx GWP= Glihudqvl|ho gùqÿvÿp phwrgx

UNKVP= Êhnlughn ÿuhwhq Kloehuw x}d| phwrgx DGP= Dgrpldq d|uvp |ùqwhpl

YLP= Ydu|dv|rqho lwhudv|rq phwrgx YSP= Ydu|dv|rqho sdudphwuhohu phwrgx

PKSP= Jhqlvohwloplv krprwrsl shuwÿuedv|rq phwrgx PGP= Jhqlvohwloplv d|uvp phwrgx

OJP= Olh juxs phwrgx EVP= E0vsolqh phwrgx DÊ= Dqdolwln êù}ÿp \Ê= \dnodvn êù}ÿp PK= Pxwodn kdwd EK= Edjo kdwd [LL

41 EÙOßP J LU LV

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Êhnlughn ÿuhwhq x}d| ù}ho elu Kloehuw x}d|gu1 Êhnlughn ÿuhwhq x}d|q whrulvlqlq roxv0 wxuxopdv 4<3;*ohuh gd|dqpdnwdgu1 \dnq }dpdqodugd eluêrn glihudqvl|ho ghqnohp/ nvpl glihudqvl|ho ghqnohpohu/ lqwhjudo ghqnohpohu yh vqu ghjhu sureohpohul ex phwrgod lqfhohq0 plvwlu1 Glihudqvl|ho ghqnohplq êù}ÿp vduwoduqq êhvlwol wdqpodpdoduqd jùuh exqd nduvon jhohq êhnlughn ÿuhwhq x}d|odu pdqwno elu vhnlogh lqvd hglohelolu1 Ex x}d|odu wdqpodqgn0 wdq vrqud ex x}d|odugd ÿuhwlohq êhnlughn irqnvl|rqodu nrod|onod exoxqdelolu1 \dnq wdulkwh olqhhu ropd|dq sureohpohulq ex phwrgod êù}ÿpÿ loh lojlol shn êrn êdovpd |dsog1 Dvoqgd ÿuhwlohq êhnlughjlq whrulvlqlq jholvlp hyuhvlqlq ùqfhvlqgh êdovpdoduq eluêrjx Ehujpdq êhnlughnohul loh lojlol êdovpdodugu ^4  :`= 4<63*odugd yh 4<73*odugd wduwvodq êhnlughnohulq êrjx Ehujpdq êhnlughnohulglu ^;`1 ßuhwlohq êhnlughjlq whrulvl kdnnqgd gdkd vrqud Phufhu ^<` êdovpdodu |dspvwu1 Phufhu sr}lwli wdqpo lqwhjudo ghqnohplqlq vÿuhnol êhnlughjlqlq

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i+|, @ ki+{,> n+{> |,l

vhnolqgh lvsdwodpvwu ^;`1 ßuhwlohq êhnlughnohulq whrulvl ^43`/ 531 |ÿ}|oq edvoduqgd ]duhped wdudiqgdq kduprqln yh elkduprqln irqnvl|rqodu lêlq vqu ghjhu sureohpohul ÿ}0 hulqh |dswj êdovpdgd nxoodqopvwu1 \dnq wdulkohugh ex phwrg elolpvho eluêrn prgho lêlq ùqhpol elu lojl dodq ropxvwxu1 Ex phwrgod x|jxodpdo elolpohugh vhul êù}ÿpohul wdp

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51 EÙOßP

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lêlq d, k{> |l @ k|> {l> +Nrpsohnv hvohqln, e, k{> |l @  k{> |l .  k{> |l > f, k{> {l  3> g, k{> {l @ 3 / { @ 3> vduwodu vdjodquvd k=> =l = H  H $ F>

gùqÿvÿpÿ H gh elu lê êdusp wdqpodu1 Elu yhnwùu x}d|qd lê êdusp loh eluolnwh lê êdusp x}d| ghqlu ^6:`1

Wdqp 5161 F^d> e` x}d|qgd ^d> e` dudoj ÿ}hulqgh wÿp vÿuhnol nrpsohnv ghjhuol irqnvl|rqodu lêlq lê êdusp ki+{,> j+{,l @ ] e d i+{,j+{,g{> vhnolqgh wdqpodqu ^6:`1

Wdqp 5171 Wdp rodq lê êdusp x}d|qd Kloehuw x}d| ghqlu ^6:`1

Whruhp 5141 +Ulhv} jùvwhulpl, i irqnvl|rqhol KKloehuw x}d|qgd vquo olqhhu elu irqnvl|rqho rovxq1 Wÿp { 5 K lêlq/

i+{, @ k{> {fl

rodfdn vhnlogh elu {f5 K yh nin @ n{fn hvlwoljl phyfxwwxu ^6:`1

Wdqp 5181 +Fdxfk| gl}lvl, Khu % A 3 lêlq hjhu khu p> q A P rogxjxqgd n{p {qn ? %>

rodfdn vhnlogh elu P vd|v ydu lvh ex wdnwlugh qrupox elu x}d|gd yhnwùuohulq elu gl}lvl rodq +{q, gl}lvlqh Fdxfk| gl}lvl ghqlu ^6:`1

Wdqp 5191 +Qrup, { $ n{n >

H yhnwùu x}d|qgdq U*|h elu gùqÿvÿp ropdn ÿ}huh hjhu

d, n{n @ 3 / { @ 3>

e, n{n @ mm n{n > ;{ 5 H yh  5 I/ f, n{ . |n  n{n . n|n > ;{> | 5 H/

vduwodu vdjodquvd ex gùqÿvÿp qrup rodudn dgodqguou ^6:`1

Wdqp 51:1 D|q I vndohu flvpl ÿ}hulqgh wdqpodqpv X yh Y yhnwùu x}d|oduq jù} ùqÿqh dodop1 Elu D = X $ Y irqnvl|rqx/

+l, ;x> x25 X> D+x. x2, @ D+x, . D+x2, +Wrsodqdeloluoln,>

+ll, ;x 5 X yh ; 5 I> D+x, @ D+x, +Krprjhqoln,>

nrvxooduq vdjo|ruvd elu olqhhu gùqÿvÿp |d gd olqhhu rshudwùu dgq dou ^6;`1

Whruhp 5151 X yh Y qrupox x}d|odu yh D = X $ Y elu olqhhu rshudwùu rovxq1 Khu x 5 X lêlq

nDxnY  N nxnX

rodfdn vhnlogh elu N A 3 vdelwl yduvd D rshdwùuÿqh vquogu ghqlu ^6;`1

Whruhp 5161 X yh Y qrupox x}d|odu yh D = X $ Y elu olqhhu rshudwùu rovxq1 D rshudwùuÿ dqfdn yh dqfdn vquo lvh vÿuhnolglu ^6;`1

Whruhp 5171 K/ elu lê êdusp x}d| rovxq1 Khu x 5 K yhnwùuÿ lêlq nxn @skx> xl>

irqnvl|rqx K ÿ}hulqgh elu grjdo qrup roxu ^6;`1

Whruhp 5181 K/ elu Kloehuw x}d| yh V @ ilj >

vd|odelolu elu ruwrqrupdo dow nÿph rovxq1 idlj vndohu gl}lvl loh roxvwxuxodqS"l'dllvhulvl

dqfdn yh dqfdn " [ l' d2_l ? 4> 8

nrvxox vdjodquvd |dnqvdnwu ^6;`1 Wdqp 51;1 K @ ; ? =

im i uhho yh|d nrpsohnv ghjhuol elu irqnvl|rq/ { 5 [> [ khukdqjl elu fÿpoh/

< @ > rovxq1

ki> jlK +i> j 5 K,>

lê êdusp|od elu Kloehuw x}d| wdqpodu1 Hjhu U| irqnvl|rqx khu elu | 5 [ lêlq phyfxwvd/

ex wdnwlugh U|5 K roxu yh khu elu i 5 K lêlq

ki> U|l_K @ i+|,>

vdjodq|ruvd U|irqnvl|rqxqd K*q ÿuhwlohq êhnlughn irqnvl|rqx yh KKloehuw x}d|qd gd

êhnlughn ÿuhwhq Kloehuw x}d| ghqlu ^;`1

Wdqp 51<1 [ elu lê êdusp x}d| rovxq1 Ex wdnwlugh wÿp {> | 5 [ lêlq mk{> |lm2 k{> {l k|> |l

hvlwvl}oljl phyfxwwxu yh ex hvlwvl}oljh Fdxfk|0Vfkzdu} hvlwvl}oljl ghqlu ^6<`1

Wdqp 51431 H fÿpohvl wrsrormln x}d|qq elu dow fÿpohvl rovxq1 Hjhu H @ rox|ruvd ex wdnwlugh H fÿpohvlqh x}d|qgd |rjxqgxu ghqlu ^73`1

Wdqp 51441 [ x}d| D = G+D,  [ $ [ loh eluolnwh G+D, fÿpohvl [ x}d|qgd |rjxq rodfdn vhnlogh elu Kloehuw x}d| rovxq1 Ex wdnwlugh D*qq dgmrlqwl DW_{loh eholuwlolu yh}

G+DW, @ ; ? =

| 5 [m wÿp { 5 G+D, lêlq kD{> |l @ k{> zl hvlwoljlql vdjod|dfdn z 5 [ phyfxwwxu

< @ > yh

DW| @ z

vhnolqgh lidgh hglolu ^74`1

Wdqp 51451 O+[, vquo rshudwùuohu x}d| ropdn ÿ}huh D 5 O+[, rovxq1 L elulp rshudwùu ropdn ÿ}huh hjhu ED @ L yh DE @ L rodfdn vhnlogh elu E rshudwùuÿ yduvd ex wdnwlugh ex E rshudwùuÿqh D rshudwùuÿqÿq whuvl ghqlu yh E @ D3_{vhnoolqgh jùvwhulolu ^75`1}

Kdwuodwpd 5141 KKloehuw x}d| êhnlughn ÿuhwhq Kloehuw x}d| yh U| irqnvl|rqx

gd [ ÿ}hulqgh ÿuhwlohq êhnlughn irqnvl|rq rovxq1 Ex wdnwlugh wÿp {> | 5 [ lêlq/ +l, U|+|,  3>

+ll, U|+{, @ U{+|,>

+lll, mU|+{,m2 U{+{,U|+|,> +Fdxfk|0Vfkzdu} hvlwvl}oljl,>

+ly, {f5 [ rovxq1 Ex wdnwlugh dvdjgdnl ÿê gxuxp eluelulqh hvghjhuglu1

+d, U{f+{f, @ 3>

+e, Wÿp | 5 [ lêlq U|+{f, @ 3>

+f, Wÿp i 5 K lêlq i+{f, @ 3>

|d}odelolu ^76`1

Ùuqhn 5141 q er|xwox Uq _{Ùnolg x}d| êhnlughn ÿuhwhq elu x}d|gu1}

Uq_{@ idm d @ +d> d2> ===> dq,> dl5 Uj >} ropdn ÿ}huh hp+l, @ ; ? = 3> l 9@ p> 4> l @ p> +3  l> p  q, êhnlughjl Uq _{qlq ÿuhwlohq êhnlughjlglu ^;`1}

Ùuqhn 5151 K> q0er|xwox elu Kloehuw x}d|/ ihljql' qrupdo ruwrjrqdo ed} rovxq1 \dql

khl+w,> hm+w,l_K@ ; ? = 3> l 9@ m> 4> l @ m> ropdn ÿ}huh Uv+w, @ q [ l' hl+w,hl+v,>

K*q ÿuhwlohq elu êhnlughjl roxu1 Jhuêhnwhq khu i+w, 5 K lêlq i+w, @ q [ l' dlhl+w,/ +dlnrpsohnv vd|odu,> ropdn ÿ}huh ki+w,> Uv+w,lK@ - _q [ l' dlhl+w,> q [ m' hm+w,hm+v, . K @ q [ l' dlhl+v, @ i +v,> :

hvlwoljl vdjodqu ^;`1

Wdqp 51461 x elu glihudqvl|ho ghqnohplq dqdolwln êù}ÿpÿ/ xq lvh ex ghqnohplq

|dnodvn êù}ÿpÿ rovxq1 mx  xqm idunqd |dnodvpq pxwodn kdwdv yh ƒx3x_ƒxƒqƒ rudqqd lvh

|dnodvpq edjo kdwdv dg yhulolu ^77`1

61 EÙOßP

ÊHN LUGHN ßUHWHQ X]D\ODU YH ßUHW LOHQ ÊHN LUGHNOHU LQ Ù]HOO LNOHU L

6141 ßuhwlohq Êhnlughjlq Whpho Ù}hoolnohul

Ù}hooln 6141 Hjhu K> êhnlughn ÿuhwhq elu x}d| lvh ex wdnwlugh K*wdnl U| ÿuhwlohq

êhnlughjl hvohqln vlphwuln roxu1 \dql

U|+{, @ U{+|,>

hvlwoljl vdjodqu ^;`1

Ù}hooln 6151 Hjhu K> êhnlughn ÿuhwhq elu x}d| lvh ex wdnwlugh K*wdnl U| ÿuhwlohq

êhnlughn irqnvl|rqx whnwlu ^;`1

Ù}hooln 6161 Hjhu U| irqnvl|rqx> K*wd ÿuhwlohq elu êhnlughn lvh ex wdnwlugh khuelu

{ 5 [ lêlq U{+{,  3 roxu1 U{+{, @ 3 ropdv lêlq jhuhn yh |hwhu nrvxo K @ i3j ropdvgu

^;`1

Ù}hooln 6171 U| ÿuhwlohq êhnlughjl sr}lwli wdqpo elu êhnlughnwlu/ |dql l> m +l> m @

4> ===> q, khukdqjl nrpsohnv vd|odu ropdn ÿ}huh khu {l5 [ lêlq q

[

l>m'

U{l+{m,lm  3> hvlwvl}oljl phyfxwwxu ^;`1

Ù}hooln 6181 K0 Kloehuw irqnvl|rq x}d|qq êhnlughn ÿuhwhq elu x}d| ropdv lêlq jhuhn yh |hwhu nrvxo khukdqjl elu vdelw { 5 [ lêlq L+i, @ i+{, olqhhu irqnvl|rqholqlq vquo ropdvgu ^;`1

6151 Zp

2 ^d> e` Êhnlughn ßuhwhq X}d| yh Rqxq ßuhwlohq Êhnlughn Irqnvl|rqx

Wdqp 6141 ^d> e` dudoj ÿ}hulqgh elu i irqnvl|rqx yhulovlq1 i+dn> en,jqn' fÿpohvl

lnlvhu lnlvhu d|un dên +dn> en,  ^d> e` dudonoduq elu fÿpohvl rovxq1 Hjhu khu % A 3 lêlq q

[

l'

rogxjxqgd

q

[

l'

mi+el,  i+dl,m ? %>

rodfdn vhnlogh q*h edjo ropd|dq hq d} elu  phyfxwvd/ ex wdnwlugh i irqnvl|rqxqd ^d> e` dudoj ÿ}hulqgh pxwodn vÿuhnolglu ghqlu ^;`1

Ùqhuph 6141 Elu i irqnvl|rqx/ ^d> e` dudoj ÿ}hulqgh pxwodn vÿuhnol lvh d|q }dpdqgd vÿuhnolglu ^;`1

Ùqhuph 6151 ^d> e` dudoj ÿ}hulqgh yhulohq elu i irqnvl|rqx lêlq hjhu { 5 ^d> e` ropdn ÿ}huh



i_+{,_{  P>}

hvlwvl}oljl phyfxwvd ex wdnwlugh i irqnvl|rqx pxwodn vÿuhnol roxu ^;`1

Vrqxê 6141 Hjhu i _{irqnvl|rqx ^d> e` dudoj ÿ}hulqgh vÿuhnol lvh ex wdnwlugh i irqnvl|0}

rqx ex dudon ÿ}hulqgh pxwodn vÿuhnol roxu ^;`1

Ùqhuph 6161 Hjhu i irqnvl|rqx ^d> e` dudoj ÿ}hulqgh lqwhjudoohqhelolu elu irqnvl|rq lvh ex wdnwlugh/

] { d

i+w,gw>

irqnvl|rqx ex dudon ÿ}hulqgh pxwodn vÿuhnol roxu ^;`1

6161 Zp 2 ^d> e` Irqnvl|rq X}d| Zp 2 ^d> e` irqnvl|rq x}d| Z2p^d> e` @ q

i m iEp3 pxwodn vÿuhnol/ iEp

5 O2^d> e`> { 5 ^d> e`r rodudn wdqpodqu1 Zp

2 ^d> e` irqnvl|rq x}d|qgd lê êdusp yh qrup khukdqjl i> j 5 Z2p^d> e`

irqnvl|rqodu lêlq/ vudv|od/ ki> jlZp 2 @ p3_[ l'f iEl+d,jEl+d, . ] e d iEp+{,jEp+{,g{ +614, yh nin_{Z p} 2 @tki> il_{Z p} 2 > +615, 43

vhnolqgh wdqpodqu ^;`1

Whruhp 6141 Zp

2 ^d> e` irqnvl|rq x}d| elu Kloehuw X}d|gu ^;`1

Whruhp 6151 Zp

2 ^d> e` irqnvl|rq x}d| êhnlughn ÿuhwhq elu x}d|gu ^;`1

Zp

2 ^d> e` x}d|qgd U|ÿuhwlohq êhnlughn irqnvl|rqxqxq qdvo hogh hglogljlql lqfhoh|holp1

U|*qlq Z2p^d> e` x}d|qq ÿuhwlohq êhnlughn irqnvl|rqx rogxjxqx idu} hgholp1 Ex wdnwlugh

khukdqjl vdelw elu | 5 ^d> e` yh khukdqjl i 5 Zp

2 ^d> e` lêlq U| irqnvl|rqx/

ki> U|l_Zp

2 @ i+|,> +616,

hvlwoljlql vdjodpdogu1 +616, hvlwoljlqh êhnlughn ÿuhwph ù}hooljl dg yhulolu1 Ex ù}hooln +614,*gh nxoodqouvd/ ki> U|l_Zp 2 @ p3_[ l'f iEl+d,C l_U |+d, C{l . ] e d iEp+{,C p_U |+{, C{p g{ @ i+|,> +617,

exoxqxu yh nvpl lqwhjudv|rq |dugp|od/ ] e d iEp+{,C p_U |+{, C{p g{ @ p3_[ l'f +4,liEp3l3+{,C pnl_U |+{, C{pnl      e {'d +618, .+4,p ] e d i+{,C 2p_U |+{, C{2p g{>

hogh hglolu1 Ghjlvnhq ghjlvwlulophvl|oh

p3_[ l'f +4,liEp3l3+{,C pnl_U |+{, C{pnl @ p3_[ l'f +4,Ep3l3iEl+{,C 2p3l3_U |+{, C{2p3l3 > +619,

|d}odelolu1 +618, yh +619, ghqnohpohul +617, ghqnohplqgh |hulqh |d}ouvd

ki> U|l_Zp 2 @ p3_[ l'f iEl+d,  Cl_U |+d, C{l  +4, p3l3C2p3l3U|+d, C{2p3l3  . p3_[ l'f +4,p3l3iEl+e,C 2p3l3_U |+e, C{2p3l3 .+4,p ] e d i+{,C 2p_U |+{, C{2p g{ @ i +|,> 44

exoxqxu1 Exqgdq grod| U| irqnvl|rqx ; A A A A A A A A A A A ? A A A A A A A A A A A = +4,pC 2p_U |+{, C{2p @ +{  |,> Cl_U |+d, C{l +4, p3l3C2p3l3U|+d, C{2p3l3 @ 3> C2p3l3_U |+e, C{2p3l3 @ 3> l @ 3> 4> ===> p  4> +61:,

jhqhoohvwluloplv glihudqvl|ho ghqnohplqlq êù}ÿpÿgÿu1 Eù|ohfh/

+4,pC 2p_U |+{, C{2p @ 3> +61;, ghqnohpl/ 2p@ 3>

ndudnwhulvwln ghqnohplqh yh 5p ndwo/

 @ 3>

ù} ghjhulqh vdklswlu1 { 9@ | lnhq elol|rux} nl U| irqnvl|rqx 5p phuwhehol vdelw ndwvd|o

krprjhq +61;, glihudqvl|ho ghqnohplqlq/ Cl_U |+d, C{l  +4, p3l3C2p3l3U|+d, C{2p3l3 @ 3> l @ 3> 4> ===> p  4> +61<, C2p3l3_U |+e, C{2p3l3 @ 3> l @ 3> 4> ===> p  4> +6143,

vqu vduwoduq vdjod|dq elu êù}ÿpÿgÿu1 Ex qhghqoh +61:,*qlq jhqho êù}ÿpÿ/

U|+{, @ ; A A A ? A A A = S2p l'fl+|,{l3> {  |> S2p l'gl+|,{l3> { A |> +6144, roxu1 +4,pC 2p_U |+{, C{2p @ +{  |,> rogxjxqgdq/ Cl_U |+|3, C{l @ Cl_U |+|n, C{l > l @ 3> 4> ===> 5p  5> +6145, 45

+4,p  C2p3_U |+|n, C{2p3  C2p3_U |+|3, C{2p3  @ 4> +6146, |d}odelolu1 fl+|, yh gl+|, +l @ 4> 5>1==> 5p, ndwvd|odu +61<,/ +6143,/ +6145, yh +6146,

ghqnohpohulqghq hogh hglolu ^;`1

Ùqhuph 6171 U|irqnvl|rqx> Z2p^d> e` x}d|qq ÿuhwlohq êhnlughjl rovxq1 Ex wdnwlugh {

yh |*|h jùuh/ ClnmU|+{, C{l_C|m 5 O 2<sub>^d> e`></sub> l . m @ 5p  4> roxu ^;`1 Vrqxê 6151 ClnmU|+{, C{l_C|m > 3  l . m  5p  5>

irqnvl|rqx { yh |*|h jùuh ^d> e` dudojqgd pxwodn vÿuhnol elu irqnvl|rqgxu ^;`1

Whruhp 6161 U| irqnvl|rqx/ Z2p^d> e` x}d|qq ÿuhwlohq êhnlughjl rovxq1 Ex wdnwlugh

{ yh |*|h jùuh ClnmU|+{, C{l_C|m 5 Z p 2 ^d> e`> l . m @ p  4> roxu ^;`1 Whruhp 6171 Zp

2 ^d> e` êhnlughn ÿuhwhq elu x}d| yh

iq> i 5 Z2p^d> e`> q @ 4> 5> ===

rovxq1 Hjhu iq> i*h n=nZp

2 qrupxqgd |dnqv|ruvd ex wdnwlugh/ 3  n  p  4>

ropdn ÿ}huh iqEn> iEn*|d gÿ}jÿq |dnqvdu ^;`1

6171 Z₂Ep>q+G, Lnlol +Elqdu|, Êhnlughn ßuhwhq X}d|

G/ ^d> e`  ^f> g`  U2vhnolqgh wdqpo elu glngùuwjhqvho eùojh rovxq1 Wdqp 6151 G eùojhvlqgh yhulohq elu i+{> |, irqnvl|rqx lêlq/

iLn@ ^dn> en`  ^fn> gn`jqn'>

fÿpohvl lnlvhu lnlvhu d|un Ln G eùojhohulqlq elu fÿpohvl rovxq1 Hjhu/

i+Ll, @ i+el> gl,  i+el> fl,  i+dl> gl, . i +dl> fl,>

ropdn ÿ}huh ;% A 3 lêlq q [ l' mLlm ? > rogxjxqgd/ q [ l' mi+Ll,m ? %>

rodfdn vhnlogh  A 3 yduvd yh i+{> f,> j+d> |, irqnvl|rqodu/ vudv|od/ { yh |*|h jùuh pxwodn vÿuhnol irqnvl|rqodu lvh ex wdnwlugh i+{> |, irqnvl|rqxqd G eùojhvlqgh wdpdphq vÿuhnolglu ghqlu ^;`1

Ùqhuph 6181 Hjhu i+{> |, irqnvl|rqx G eùojhvlqgh wdpdphq vÿuhnol elu irqnvl|rq lvh ex wdnwlugh khu elu vdelw | 5 ^f> g` lêlq {*lq elu irqnvl|rqx rodudn i+{> |, irqnvl|rqx ^d=e` dudoj ÿ}hulqgh pxwodn vÿuhnol elu irqnvl|rq roxu1 Ehq}hu vhnlogh khu elu vdelw { 5 ^d> e` lêlq |*qlq elu irqnvl|rqx rodudn i+{> |, irqnvl|rqx ^f=g` dudoj ÿ}hulqgh pxwodn vÿuhnol elu irqnvl|rq roxu ^;`1

Ùqhuph 6191 Hjhu i+{> |, irqnvl|rqx G eùojhvl ÿ}hulqgh wdpdphq vÿuhnol elu irqnvl|rq lvh ex wdnwlugh i+{> |, irqnvl|rqx ex eùojh ÿ}hulqgh d|q }dpdqgd vÿuhnol elu irqnvl|rqgxu ^;`1

Ùqhuph 61:1 Hjhu i+{> |, irqnvl|rqx G eùojhvl ÿ}hulqgh lqwhjudoohqhelolu lvh ex wdnwlugh/ ] { d ] | f i+w> v,gwgv> +{> |, 5 G>

irqnvl|rqx G eùojhvlqgh wdpdphq vÿuhnol elu irqnvl|rq roxu ^;`1

Ùqhuph 61;1 Hjhu i+{, irqnvl|rqx ^d> e` dudojqgd pxwodn vÿuhnol elu irqnvl|rq lvh ex wdnwlugh ex irqnvl|rq ex eùojhgh wdpdphq vÿuhnol elu irqnvl|rqgxu ^;`1

Ùqhuph 61<1 Hjhu i+{, yh j+|, irqnvl|rqodu vudv|od ^d> e` yh ^f> g` dudonoduqgd pxwodn vÿuhnol irqnvl|rqodu lvh ex wdnwlugh i+{,j+|, irqnvl|rqx G eùojhvlqgh wdpdphq vÿuhnolglu ^;`1

Wdqp 6161 Lnlol irqnvl|rq x}d| Z2Ep>q+G, @

; ? =

i+{> |,m C_C{pnq32p3_C|iE{>|q3 irqnvl|rqx G eùojhvlqgh wdpdphq vÿuhnol/

Cpnq_iE{>|

C{p_C|q 5 O2+G,>

< @ > vhnolqgh wdqpodqu1 Ex x}d|gdnl lê êdusp yh qrup/ vudv|od/

ki+{> |,> j+{> |,l_ZEp>q 2 EG@ [p3 l'f g ] f k Cq C|q C l C{li+d> |,C q C|q C l C{lj+d> |, l g| . q3 [ m'f G Cm C|mi+=> f,> C m C|mj+=> f, H Zp 2 .U U G Cp C{p C q C|qi+{> |,C p C{p C q C|qj+{> |,g{g| yh nin_ZEp>q 2 @ t ki> il_ZEp>q 2 > i 5 Z Ep>q 2 +G,>

edjqwodu loh yhulolu ^;`1

Whruhp 6181 Z₂Ep>q+G, irqnvl|rq x}d| elu Kloehuw x}d|gu ^;`1 Whruhp 6191 i+{> |,> j+{,j2+|, 5 Z Ep>q 2 +G,> rovxq/ ex wdnwlugh/ ki+{> |,> j+{,j2+|,l<sub>Z</sub>Ep>q 2 @ G ki+{> |,> j+{,lZp 2 > j2+|, H Zq 2 > hvlwoljl vdjodqu ^;`1 Vrqxê 6161 i+{,i2+|,> j+{,j2+|, 5 Z Ep>q 2 +G,> rovxq1 Ex wdnwlugh/ ki+{,i2+|,> j+{,j2+|,l_ZEp>q 2 @ ki+{,> j+{,jlZ p 2 ki2+|,> j2+|,lZ2q> roxu ^;`1 48

Vrqxê 6171 Hjhu i 5 Zp

2 ^d> e` lvh ex wdnwlugh q 5 Q lêlq/

ninZp

2 @ ninZ2Ep>q> |d}odelolu ^;`1

Whruhp 61:1 Z2Ep>q+G, x}d| êhnlughn ÿuhwhq elu x}d| roxs ex x}d|q ÿuhwlohq êhnlu0

ghjl NE>> U> Têhnlughnohul vudv|od Z2p^d> e`> Z2q^f> g` x}d|oduqq ÿuhwlohq êhnlughnohul

ropdn ÿ}huh/

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hogh hglolu1 Ex gd lvsdw wdpdpodu1 

Wdqp 71:1 Lnlol êhnlughn ÿuhwhq Z + , x}d|

Z + , @ ; A A A A ? A A A A = x m C e_y C{2_Cw2> @ ^3> 4`  ^3> W ` eùojhvlqgh wdpdphq vÿuhnol/ CSy C{_Cw 5 O2+ , > x+{> 3, @ xw+{> 3, @ x+3> w, @ x+4> w, @ 3> < A A A A @ A A A A >

rodudn wdqpodqu1 Z + , x}d|qgdnl lê êdusp yh qrup x> j 5 Z + , ropdn ÿ}huh/ vudv|od/ kx> jlZ @ 2 [ l'f ] W f  C Cw Cl C{lx+3> w, C Cw Cl C{lj+3> w,  gw . 2 [ m'f  Cm Cwmx+=> 3,> Cm Cwmj+=> 3, S 2 . ] W f ]  f  C C{ C Cwx+{> w, C C{ C Cwj+{> w,  g{gw yh nxnZ @ t kx> xlZ> x 5 Z + , > 5;

vhnolqghglu ^;`1

Ohppd 7171 Z + , x}d| êhnlughn ÿuhwhq elu x}d| roxs ex x}d|q ÿuhwlohq êhnlughjl> E|> uv ÿuhwlohq êhnlughn irqnvl|rqodu/ vudv|od/ S2^3> 4` yh I2^3> W ` x}d|oduqq ÿuhwlohq

êhnlughn irqnvl|rqodu ropdn ÿ}huh/

NE|>v@ E|uv> |d}odelolu1 Khukdqjl x 5 Z + , lêlq/ x+|> v, @ x> NE|>v  Z> yh NE|>v+{> w, @ NE{>w+|> v, > hvlwolnohul phyfxwwxu ^;`1

Wdqp 71;1 Lnlol êhnlughn ÿuhwhq fZ + , x}d| f Z + , @ ; A ? A = x m x> @ ^3> 4`  ^3> W ` eùojhvlqgh wdpdphq vÿuhnol/ C2x C{Cw5 O 2_{+ , >} < A @ A > rovxq1 fZ + , x}d|qgdnl lê êdusp yh qrup x> j 5 fZ + , ropdn ÿ}huh/ vudv|od/

kx> jlfZ @ ] W f  C Cwx+3> w, C Cwj+3> w,  gw . kx+=> 3,> j+=> 3lJ 2 . ] W f ]  f  C C{ C Cwx+{> w, C C{ C Cwj+{> w,  g{gw yh nxnfZ @ t kx> xlfZ> x 5 fZ + , > vhnolqghglu ^;`1 5<

Ohppd 7181 fZ + , x}d| êhnlughn ÿuhwhq elu x}d| roxs ex x}d|q ÿuhwlohq êhnlughjl> T|> tvêhnlughnohul vudv|od J2^3> 4` yh K2^3> W ` x}d|oduqq ÿuhwlohq êhnlughn irqnvl|rqodu

roxs

JE|>v@ T|tv>

glu ^;`1

716151 Z + , X}d|qgd Êù}ÿpÿq Vxqxpx Vhêlohq rshudwùu olqhhu yh vquo ropdv jhuhnwljlqghq/

@ ^3> 4`  ^3> W `

ropdn ÿ}huh O olqhhu rshudwùuÿ/

O = Z + , $ fZ + ,>

vhnolqgh roxs/ +718, ghqnohplqghq

Oy @ yww y{{> |d}odelolu1 P+{> w, @ +{  4, i+{,k  +w, k+3,  {k  2+w,  5i_+{,k +w, k+3,  +{  4, i_+{,k +w, k+3, .5i+{, . {i+{, . w  j+{, .5j+3,i _+{, k+3, . +{  4, i+{,j+3, k+3,   vlq 5 7 y+{> w, 

E{3i E{kEw

kEf . {k2+w, . {i+{,  {k2+3, .wj+{, .E{3i E{jEf_kEf  {j+4,

6 8 > rodudn doquvd k+3, 9@ 3 ropdn ÿ}huh +718,+71:, ghqnohpohul

; ? =

Oy+{> w, @ P+{> w,> +{> w, 5 ^3> 4`  ^3> W `  U2_>

y+{> 3, @ yw+{> 3, @ y+3> w, @ y+4> w, @ 3>

+71<,

kdolql dou1 Exudgd edvodqjê yh vqu vduwoduqq krprjhqohvwlulophvl lêlq

y+{> w, @ x+{> w, .+{  4, i+{,k+w, k+3,  {k2+w,  {i+{, . {k2 +3, w  j+{, .+{  4, i+{,j+3, k+3,  {j+4,  > 63

gùqÿvÿpÿ nxoodqopvwu1 Qrwdv|rq nduvnojq ùqohphn ednpqgdq +71<, gd y+{> w, |hulqh x|jxq ropdv lêlq whnudu x+{> w, |d}dop1

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mOx+{> w,m  d nxnZ>    _C{C Ox+{> w,      f nxnZ>    _CwC Ox+{> w,      e nxnZ>    _C{C _CwC Ox+{> w,      g nxnZ>

hvlwvl}olnohul |d}odelolu1 Exqgdq grod|

nOxn2Zf @ ] W f  C CwOx+3> w, 2 gw . kOx+=> 3,> Ox+=> 3,lJ 2 . ]  f ] W f  C C{ C CwOx+{> w, 2 gwg{ @ ] W f  C CwOx+3> w, 2 gw . ^Ox+3> 3,`2 . ]  f  C C{Ox+{> 3, 2 g{ . ]  f ] W f  C C{ C CwOx+{> w, 2 gwg{  d2. f2. We2. g2nxn2Z>

hogh glolu1 Ex lvh lvsdw wdpdpodu1 

Kdwuodwpd 7141 @ ^3> 4`  ^3> W ` fÿpohvlqlq vd|odelolu |rjxq i+{> w, > +{2> w2, > ===j

dow fÿpohvlql vhêholp1 OW _{rshudwùuÿ O rshudwùuÿqÿq dgmrlqw rshudwùuÿ ropdn ÿ}huh/}

*l+{> w, @ JE{l>wl+{> w, >

l @ OW*l irqnvl|rqoduq wdqpod|dop1 Z + , x}d|qq q e l r" l' ruwrqrupdo vlvwhpl/

il+{> w,j"l' lidghvlqlq Judp0Vfkplgw ruwrjrqdoohvwluph vÿuhflqghq |duduodqouvd

e l@ l [ n' _lnn>

rodudn exoxqxu ^8:`1 Exudgd ln*odu ruwrjrqdo ndwvd|odugu1

Whruhp 715 i+{l> wl,j"l'> *gd |rjxq rovxq1 Ex wdnwlugh ilj"l'> Z + , x}d|qgd

elu wdp vlvwhp roxu yh l+{> w, @ ONE{l>wl+{> w,> lidghvl vdjodqu1 Lvsdw= +{> w, 5 lêlq l+{> w, @ +OW*l, +{> w, @ OW*l> NE{>w  Z @ *l> ONE{>w  f Z @ ONE{>w> *l  f Z @ ONE{>w> JE{l>wl  f Z @ ONE{>w+{l> wl, @ ONE{l>wl+{> w, >

|d}odelogljlqghq l+{> w, 5 Z + , rogxjx dvlndugu1 Khuelu vdelw x 5 Z + , lêlq hjhu/

kx> llZ @ 3> l @ 4> 5> === roxuvd ex wdnwlugh/ kx> OW*llZ @ kOx> *llZf@ Ox> JE{l>wl  f Z @ Ox +{l> wl, @ 3> l @ 4> 5> ===>

exoxqxu1 i+{l> wl,j"l'> gl}lvl gd |rjxq rogxjxqgdq/

Ox +{> w, @ 3>

|d}odelolu1 O3 _{lq yduojqgdq/}

x @ 3>

roxu1 Ex lvh lvsdw wdpdpodu1 

Whruhp 7161 Hjhu i+{l> wl,j"l'> *gd |rjxq lvh ex wdnwlugh +71<,*xq êù}ÿpÿ/

x @ " [ l' l [ n' lnP+{n> wn, el> +7143, 65

edjqwvq vdjodu1

Lvsdw= x irqnvl|rqx +71<,*xq êù}ÿpÿ rovxq1 Whruhp 715*ghq qel+{> w,

r" l'lidghvl

Z + ,*gd elu wdp vlvwhpglu1 Eù|ohfh

x @ " [ l' G x> el H Z e l@ " [ l' l [ n' lnkx> nlZel @ " [ l' l [ n' lnkx> OW*nlZ el @ " [ l' l [ n' lnkOx> *nlZfel @ " [ l' l [ n' ln Ox> JE{n>wn  f Z el @ " [ l' l [ n' lnOx +{n> wn, el @ " [ l' l [ n' _lnP+{n> wn, el> |d}odelolu1 Ex lvh lvsdw wdpdpodu1 

xq|dnodvn êù}ÿpÿ x dqdolwln êù}ÿpÿqÿq q whulplqlq jù} ùqÿqh doqpdv|od hogh hglolu1

\dql/ |dnodvn êù}ÿp xq@ q [ l' l [ n' lnP+{n> wn, el>

rodudn exoxqxu1 Eù|ohfh/ q $ 4 lnhq nxq xn $ 3 roxu1

716161 Phwrgxq X|jxodqpdv

Hjhu P olqhhu lvh ex wdnwlugh +71<,*xq dqdolwln êù}ÿpÿ +7143, loh exoxqdelolu1 Hjhu P olqhhu ghjlovh ex wdnwlugh +71<,*xq dqdolwln êù}ÿpÿ +7143, |d gd

; A A ? A A = Khukdqjl vdelw xf5 Z + ,> xq @ q [ l' Dlel> +7144,

edjqwv|od exoxqdelolu/ exudgd ; A A A A A A A A A A A A A ? A A A A A A A A A A A A A = D@ P+{n> wn,> D2@ 2 [ n' 2nP+{n> wn,> === Dq@ q [ n' qnP +{n> wn,> +7145,

glu1 Vlpgl +7145, loh yhulohq xq |dnodvn êù}ÿpÿqÿq dqdolwln êù}ÿph |dnqvdgjq jùvwhuh0

olp1

Ohppd 71:1 Hjhu/ xq

8=8

$ ex> nxqn vquo/ +{q> wq, $ +|> v, yh P+{> w, vÿuhnol/

vduwodu vdjodquvd ex wdnwlugh/

P+{q> wq, $ P+|> v,>

|d}odelolu1

Lvsdw= ßuhwlohq êhnlughjlq whpho ù}hooljl yh Fdxfk|0Vfkzdu} hvlwvl}oljlqghq mx+{> w,m @  x+|> v,> NE{>w+|> v,  Z     nx+|> v,nZ  NE{>w+|> v,   Z @ Qnx+|> v,nZ> |d}odelolu1 Eù|ohfh    Cx+{> w,_C{     @      x+|> v,>CNE{>w+|> v, C{ _Z      nx+|> v,nZ     CNE{>w+|> v, C{     Z @ Q2nx+|> v,nZ yh    Cx+{> w,_Cw     @      x+|> v,>CNE{>w+|> v, Cw _Z      nx+|> v,nZ     CNE{>w+|> v, Cw     Z @ Qnx+|> v,n_Z>

hvlwvl}olnohul hogh hglolu1 Elu wdudiwdq

mxq3+|> v,  ex+|> v,m @    xq3+{> w,  ex+{> w,> NE|>v+{> w,  Z     nxq3+{> w,  ex+{> w,n_Z  NE{>w+|> v,   Z @ Qenxq3+{> w,  ex+{> w,n_Z>

phyfxw lnhq gljhu wdudiwdq

mxq3+{q> wq,  ex+|> v,m @ mxq3+{q> wq,  xq3+|> v, . xq3+|> v,  ex+|> v,m

 muxq3+> ,m m+{q> wq,  +|> v,m . mxq3+|> v,  ex+|> v,m >

phyfxwwxu1 \xndugdnl hvlwvl}olnohughq yh

xq 8=8

$ ex

lidghvlqghq

mxq3+|> v,  ex+|> v,m $ 3> muxq3+> ,m 

t f2

.f22nxnZ>

|d}odelolu1 q $ 4 lnhq/ nxqn vquo rodfdjqgdq

mxq3+{q> wq,  ex+|> v,m $ 3>

roxu1 P+{> w,*qlq vÿuhnololjlqghq q $ 4 lnhq/ P+{_q> w_q, $ P+|> v,>

exoxqxu1 Ex lvh lvsdw wdpdpodu1 

Whruhp 7171 +7144,*ghnl nxqn*qlq vquo rogxjxqx yh +71<,*xq whn elu êù}ÿph vdkls

rogxjxqx ndexo hgholp1 Hjhu i+{l> wl,j"l'> *gd |rjxq lvh ex wdnwlugh xq |dnodvn êù}ÿpÿ

x dqdolwln êù}ÿpÿqh |dnqvdu yh x @ " [ l' Dlel> |d}odelolu1

Lvsdw= Ùqfholnoh xq*qlq |dnqvdn rogxjxqx jùvwhuholp1 Exqxq lêlq +7144,*ghq/

xqn@ xq.Dqneqn> exoxqxu1 qel r" l' qlq ruwrqrupdooljlqghq/ nxqnn2@ nxqn2. D2qn@ qn [ l' D2l> +7146, roxu1 +7146,*whq lvh/ nxqnn A nxqn >

hogh hglolu1 Ex lvh nxqn*qlq prqrwrq duwdq rogxjxqx jùvwhulu1 nxqn vquo gd rogxjxqgdq

nxqn |dnqvdnwu yh "

[

l'

D2l @ g>

rodfdn vhnlogh vrqox elu g vd|v phyfxwwxu1 Eù|ohfh/

iDlj"l'5 o2>

rogxjx jùuÿoÿu1 Hjhu p A q lvh ex wdnwlugh/ nxp xqn2 @ nxp xp3. xp3 xp32. === . xqn xqn2  nxp xp3n2. nxp3 xp32n2. === . nxqn xqn2> |d}odelolu1 nxp xp3n2@ D2p> rogxjxqgdq vrqxê rodudn q> p $ 4 lnhq/ nxp xqn2@ p [ o'qn D2o $ 3>

exoxqxu1 Z + ,*qq wdpojqgdq q $ 4 lnhq xq$ ex roxu1 Vlpglex *qxq +71<,*xq êù}ÿpÿ

rogxjxqx jùvwhuholp1 +7144, lidghvlqgh khu lnl wdudiq olplwl doquvd/

e x @ " [ l' Dlel> exoxqxu1 Eù|ohfh/ Ox @e " [ l' DlO el> yh Ox+{e n> wn, @ " [ l' DlO el+{n> wn, @ " [ l' Dl G O el+{> w,> *n+{> w, H f Z @ " [ l' Dl G e l+{> w,> OW*n+{> w, H Z @ " [ l' Dl G e l+{> w,> n+{> w, H Z> rogxjxqgdq l [ n' lnOex+{n> wn, @ " [ l' Dl -e l+{> w,> l [ n' lnn+{> w, . Z @ " [ l' Dl G e l+{> w,> en+{> w, H Z @ Dn> hogh hglolu1 +7145,*ghq/ Ox+{e _n> wn, @ P+{n> wn,>

|d}odelolu1 i+{l> wl,j"l'> gd |rjxq rogxjxqgdq khuelu +|> v, 5 lêlq m $ 4 lnhq/

 {qm> wqm



$ +|> v,>

rodfdn vhnlogh elu{qm> wqm "

m' dow gl}lvl phyfxwwxu1 Eù|ohfh/

Oxe{qm> wqm 

@ P+{qm> wqm,>

exoxqxu1 m $ 4 lnhq P*qlq vÿuhnololjlqghq yh Ohppd 71:*ghq Oex+|> v, @ P+|> v,>

|d}odelolu1 Ex gd ex*qxq +71<,*x vdjodgjq jùvwhulu1 Eù|ohfh lvsdw wdpdpodqpv roxu1 

Kdwuodwpd 7151

i+{> w, > +{2> w2, > ===j 5 @ ^3> 4`  ^3> W `>

irupxqgdnl |rjxq fÿpoh|l jù} ùqÿqh dodop1

*_l+{> w, @ JE{l>wl+{> w,> l+{> w, @ O W_* l+{> w,> e l+{> w, @ l [ n' _lnn+{> w,>

ropdn ÿ}huh ln ruwrjrqdo ndwvd|odu

 @ 4 nn> ll@ 4 t nln2 l3 [ n' f2 ln > lm @  l3 [ n'm flnnm t nln2 l3 [ n' f2 ln > fln@ G l> en H >

edjqwodu |dugp|od exoxqxu ^;`1

716171 Vd|vdo Vrqxêodu

Ex eùoÿpgh êhnlughn ÿuhwhq phwrg |dugp|od lnl ùuqhn êù}ÿogÿ1 Exoxqdq vd|vdo ghjhuohu wdeorodu yh judnohu kdolqgh jùvwhulogl1

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x+{> 3, @ vlq+{,> Cx

Cw+{> 3, @ 3>