Diferensiyel operatörler için yarı ters problemler / Half-inverse problems for differential operators

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FIRAT ÜNĐVERSĐTESĐ FEN BĐLĐMLERĐ ENSTĐTÜSÜ

DĐFERENSĐYEL OPERATÖRLER ĐÇĐN YARI TERS PROBLEMLER

Yüksek Lisans Tezi Tüba YALÇIN

Anabilim Dalı: Matematik Programı: Uygulamalı Matematik Tez Danışmanı: Prof. Dr. Etibar PENAHLI

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glu1 Exudgd 3 vlpjhvl vrqvx} êduspgd n ' q whulplqlq phyfxw ropdgjq jùvwhulu1 Whruhp := s E{ yh t E{ > df> o dudojqgd wdqpo uhho irqnvl|rqodu yh n0qf phu0 whehghq wÿuhyohul O2_{Ef>  gh ropdn ÿ}huh i}

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dvlpswrwln irupÿoohulqlq vdjodqpdv jhuhn yh |hwhuglu1

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\du whuv sureohpohu nrqxvx lon rodudn K1 Krfkvwdgw/ E1 Olhehupdq dbo gdkd vrqud O1 Vdnkqrylfk dDDo/ dDSo H1 V1 Shqdkry/ K1 Nr|xqedndq d2.o/ d2Ho yh eluêrn Êlqol pdwhp0 dwlnêl wdudiqgdq lqfhohqplvwlu1

Lvrvshnwudo sureohpoh lojlol rodudn Wuxerzlw} dS> 2f> 2o/ gdkd vrqud Pd{ Mrghlw yh Ohylwdq d2eo lqwhjudo ghqnohpl yh gùqÿvÿp rshudwùuÿqh gd|do |hql elu |ùqwhpoh lvrvshnwudo sureohpl lqfhohplvwlu1

Ex êdovpdgd vrqox dudonwd Vwxup0Olrxylooh ghqnohpl lêlq |du whuv sureohpohu/ Vwxup0Olrxylooh ghqnohpl yh Gludf vlvwhpl lêlq lvrvshnwudo sureohp lqfhohqplvwlu1

51 WHPHO WDQLP YH WHRUHPOHU

5141 Glihuhqvl|ho Rshudwùuohulq Vshnwudo Whrulvlqgh Nxoodqodq Ùqhpol Ndyudpodu

514141 Wdqp + Lê Êdusp X}d|/ Kloehuw X}d| ,= N/ U yh|d F ropdn ÿ}huh [ elu yhnwùu x}d| rovxq1 ?> AG [  [ $ N gùqÿvÿpÿ dvdjgdnl ù}hoolnohuh vdklsvh ?> A |h [ ÿ}hulqgh elu lê êdusp/ E[> ?> A lnlolvlqh gh lê êdusp x}d| ghqlu1

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phwuljlqh jùuh wdp lê êdusp x}d|qd lvh Kloehuw x}d| ghqlu do1 514151 Wdqp +w2_{dd> eo X}d|,= d  {  e ropdn ÿ}huh} w2dd> eo ' ; ? =i E{ G e ] d di E{o2_{g{ ? 4} < @ >1 Ex x}d|gd lê êdusp ? i E{ > j E{ A' e ] d i E{ j E{g{ vhnolqgh wdqpogu do1

514161 Wdqp +Vqu vduwodu,= Vquo elu G eùojhvlqgh/ }lnvho elu rod|q pdwh0 pdwlnvho gdyudqv nrqwuro hwphn lvwhql|ruvd/ G eùojhvlqlq vquqgd/ jhqhoolnoh edjpo ghjlvnhqh edjo ed} vduwodu |ÿnohqphvl jhuhnlu1 Yhulohq ex vqu ghjhuohulqh glihuhqvl|ho ghqnohp lêlq vqu vduwodu ghqlu1 Vqu vduwodu 6*h d|uou dSo1

4, Glulfkohw vqu vduw= Ex gxuxpgd x irqnvl|rqx vquo elu G eùojhvlqlq vqu ÿ}hulqgh wdqpogu1 G ' Ef> O eùojhvlqgh Glulfkohw vqu vduwodu> x Ef ' > x EO '  vhnolqgh wdqpodqu1 Exudgd >  vdelwwlu1 Elu glngùuwjhqvho eùojhgh f ? { ? O yh

f ? | ? O2 ropdn ÿ}huh Glulfkohw vqu vduwodu> x Ef> | > x EO> | > x E{> f yh x E{> O2

rodudn yhulolu1 Hjhu edjpo ghjlvnhq x/ vquq khukdqjl elu qrnwdvqgd viu lvh ex vqu vduwqd krprmhq vqu vduw/ dnvl wdnwlugh krprmhq ropd|dq vqu vduw ghqlu1

5, Qhxpdqq vqu vduw= Elu O x}xqoxjx lêlq Qhxpdqq vqu vduwodu x{Ef> w ' > x{EO> w ' 

rodudn wdqpodqu1

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gq qrupdo wÿuhyl vqu ÿ}hulqgh wdqpogu1 Hjhu x Ef> | n x{Ef> | yh|d x EO> | n x{EO> | jlel vduwodu yduvd ex vqu vduwodu nduvn+Urelq, vqu vduwodugu1

51417 Wdqp +Edvodqjê vduwodu,= Gliÿ}|rq ghqnohpl yh gdojd ghqnohpl jlel nvpl glihuhqvl|ho ghqnohpohulq êrjx }dpdqd edjogu1 Edvodqjê }dpdq rodq w ' f dqqgd edjpo ghjlvnhqlq edvodqjê ghjhuohul yhulopholglu1 Elulqfl phuwhehghq elu glihu0 hqvl|ho ghqnohpgh edvodqjê vduw x E{> f/ lnlqfl phuwhehghq elu glihuhqvl|ho ghqnohpgh lvh x E{> f yh xwE{> f vhnolqghglu dSo1

51418 Wdqp +Rshudwùu,= Wdqp yh ghjhu fÿpohvl yhnwùu x}d| rodq gùqÿvÿpohuh rshudwùu ghqlu deeo1

514191 Ùuqhn= F dd> eo ghq nhqgl lêlqh rodq yh W { Ew ' w ] d { E  g > w 5 dd> eo

vhnolqgh wdqpodqdq W gùqÿvÿpÿ elu rshudwùugÿu1 Ex rshudwùuh lqwhjudo rshudwùuÿ ghqlu do1

5141:1 Wdqp +Olqhhu rshudwùu,= H{ yh H| lnl uhho olqhhu wrsrormln x}d| rovxq1

Ghjhu eùojhvl H| gh exoxqdq yh H{ gh wdqpo | ' D{ rshudwùuÿqÿ jù}ùqÿqh doqvq1

D rshudwùuÿ lêlq

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vduwodu vdjodq|ruvd D rshudwùuÿqh olqhhu rshudwùu ghqlu d2So1 43

5141;1 Ùuqhn= N Ew> v > f  w> v   vÿuhnol elu irqnvl|rq/ { Ev 5 F df> o ropdn ÿ}huh | Ew '  ] f N Ew> v { Ev gv

hvlwoljl loh wdqpo | ' D{ rshudwùuÿ elu olqhhu rshudwùugÿu deeo1

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nO{n  f n{n

rodfdn vhnlogh elu f uhho vd|v yduvd O rshudwùuÿqh vquogu ghqlu1 Ex hvlwvl}oljl vdjod|dq hq nÿêÿn f vd|vqd lvh O rshudwùuÿqÿq qrupx ghqlu dD.o1

5141431 Wdqp +Dgmrlqw Rshudwùu,= K yh K2 lnl Kloehuw x}d| yh O G K $ K2

vquo olqhhu rshudwùu rovxq1 Hjhu OW _{G K}

2 $ K rshudwùuÿ

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rshudwùu ghqlu deDo1

5141441 Wdqp +Dqdolwln irqnvl|rq,= Elu iE} nrpsohnv irqnvl|rqx gÿ}ohplq nh| elu }f qrnwdvqq  nrpvxoxjxqxq wÿp qrnwdoduqgd glihuhqvl|hoohqhelol|ruvd iE}

irqnvl|rqxqd }f qrnwdvqgd dqdolwln ghqlu deSo1

5141451 Wdqp +Wdp Irqnvl|rq,= Elu iE} nrpsohnv irqnvl|rqx gÿ}ohplq wÿp qrnwdoduqgd dqdolwln lvh iE} *|h wdp irqnvl|rq ghqlu deSo1

5141461 Wdqp +Wdp Irqnvl|rqxq Phuwhehvl,= iE} wdp irqnvl|rq rodfdn elêlpgh U '*4 q<" s mdnm 3 +514, irupÿoÿ loh wdqpo U

iE} ' " [ n'f dn}n +515, 44

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P Eu ? i T Eu +516,

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5141471 Wdqp= iE} wdp irqnvl|rqxqxq phuwhehvl vrqox  Ef ?  ? 4 rodfdn elêlpgh/ u A U lêlq

P Eu ? i T Eu +518,

ropdn ÿ}huh  A f vd|v yduvd iE} vrqox wlsh vdklswlu ghqlu1

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5141491 Wdqp +Ù}ghjhu/ ù}irqnvl|rq,= O vquo olqhhu elu rshudwùu rovxq1 O rshudwùuÿqÿq wdqp nÿphvlqgh

O| ' |

rodfdn vhnlogh elu | 9' f irqnvl|rqx yduvd  vd|vqd O rshudwùuÿqÿq ù}ghjhul/ | irqnvl0 |rqxqd lvh  ù}ghjhulqh nduvon jhohq ù}irqnvl|rq ghqlu d.o1

51414:1 Wdqp +Vwxup0Olrxylooh Rshudwùuÿ,= s E{ > t E{ yh $ E{ irqnvl|rq0 odu dd> eo dudojqgd vÿuhnol irqnvl|rqodu ropdn ÿ}huh Vwxup0Olrxylooh ghqnohpl

_g{g  s E{ g| g{  n t E{ | ' $ E{ |> 45

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vqu vduwodu hogh hglolu1 Hjhu t E{ uhho ghjhuol yh vÿuhnol irqnvl|rq> K yh k vd|odu gd vrqox lvh E2=.  E2=f sureohplqh uhjÿohu Vwxup0Olrxylooh sureohpl/ ex vduwodugdq khukdqjl elul er}xogxjxqgd ex sureohph vlqjÿohu Vwxup0Olrxylooh sureohpl ghqlu d.o1

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o| ' |n t E{ | ' |> f ? { ?  +5144,

X E| ' |_{Ef  k| Ef ' f> Y E| ' |}E n K| E ' f

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* Ef>  ' > *Ef>  ' k

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ndsdo dow x}d|odu ropdn ÿ}huh H x}d|qq wdpdpqgd wdqpo/ H ghq H2 |h gùqÿvÿp

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gxu/ d.o1

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5141561 Wdqp +Gludf0Ghowd Irqnvl|rqx,= dEw  wf ' ; A A A ? A A A = f > _{f  w ? w}f d  2d > wf d  w ? wfn d f > _{w  w}fn d 48

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ghqnohpl yh

| Ef ' | E ' f +615,

vqu nrvxoodu jù}ùqÿqh doqvq1

| Ef>  ' f> |Ef>  '  +616,

nrvxooduq vdjod|dq E= ghqnohplqlq êù}ÿpÿ | E{>  rovxq1 | E{>  | E{>  ' t? s { s  n { ] f t?sw s  N E{> w gw +617,

vhnolqgh lidgh hglolu dSo1 Exudgd N E{> w

N E{> | '  2 E{n|@2_] E{3|@2 s Ew gw n E{n|@2_] E{3|@2 E{3|@2_] f s Ew n v N Ew n v> w  v gv gw +618, lqwhjudo ghqnohplql vdjod|dq vÿuhnol glihuhqvl|hoohqhelolu elu irqnvl|rqgxu1 Jhqhooljl er}pdgdq



]

rogxjx yduvd|odelolu1 Exudgdq

N E>  ' f +61:,

hogh hglolu1

| E>  irqnvl|rqx 4250lqfl phuwhehghq dqdolwln irqnvl|rqgxu1 E= > E=2 sureohp0 lqlq vshnwuxpx | E>  qq viuoduqq nÿphvl loh êdnvu1 Ex lvh | E>  irqnvl|rqxqxq ex vshnwuxpx wdqpodpdv dqodpqd jholu1 E=e irupÿoÿqghq

kfEw ' N E> w +61;,

irqnvl|rqx E= > E=2 sureohplqlq vshnwuxpx rodudn wdqpodqu1

Vwxup0Olrxylooh sureohplqlq vshnwuxpx yh f  {  

2 dudoj ÿ}hulqgh s E{

yhulogljlqgh/ s E{ dudojq wdpdpqgd Ef  {   wdqpodqdelolu1 E=e edjqwvqgd N E{> w irqnvl|rqx C2_N C{2  s E{ N ' C2_N Cw2 +61<, glihuhqvl|ho ghqnohplqlq N E{> { '  2 { ] f s Ev gv +6143, N E{> f ' f

nrvxooduq vdjod|dq elu êù}ÿpÿgÿu1 \hql  ' { n w>  ' {  w ghjlvnhqohul |dugp loh E=b ghqnohpl C2_D CC '   es   n  2  D +6144,

irupxqgd |d}odelolu1 Jhuêhnwhq/

N{{ Nww s E{ N ' f>  ' { n w>  ' {  w C E>  C E{> w '       _{ _w {w       '                 ' 2 9' f N{ ' N{n N{ 4;

N{{ ' EN{n N{ {n N{{n EN{n N{ {n N{{ , N{{ ' EN= n N=  n N=f n EN= n N=  n N=f , N{{ ' Nn 2Nn N Nw ' Nwn Nw Nww ' ENwn Nw wn Nwwn ENwn Nw wn Nww , Nww ' EN=  N=  n N=f n EN=  N= E n N=f , Nww ' N 2Nn N

Ex lidghohu E=b gd |hulqh |d}ouvd eN s E{ N ' f , N '



es E{ N hogh hglolu1 Exudgd

D E>  ' N   n  2 >    2  +6145, rodudn wdqpodquvd E= hogh hglolu1 E=f nrvxox dvdjgdnl irupd lqglujhqlu yh

D E> f '  2 @2 ] f s Ev gv +6146, D E>  ' f

glu1 Jhuêhnwhq E=2 yh E=f edjqwoduqgdq

D E> f ' N   n f 2 >   f 2  ' N   2>  2  '  2 @2 ] f s Ev gv> D E>  ' N   n  2 >    2  ' N E> f ' f> exoxqxu1 E= nrvxooduq nxoodqdudn E= ghqnohpl

D E>  ' _e  ]   ] f s  w n v 2  D Ew> v gvgw n 2 @2 ] @2 s Ew gw= +6147,

lqwhjudo irupxqgd |d}odelolu1 Jhuêhnwhq/ D '   es   n  2  D 4<

ghqnohplqgh  |d jùuh lqwhjudo doquvd DE>   DE> f '   e  ] f s  v n w 2  D Ew> v gv hogh hglolu1 Ex vrq ghqnohpgh  |h jùuh lqwhjudo doquvd

D E>   D E>   D E> f n D E> f ' _e

 ]   ] f s  w n v 2  D Ew> v gvgw hogh hglolu1 E= vduwqgdq

D E>  '  e  ]   ] f s  w n v 2 

D Ew> v gvgw n D E> f  D E> f

D E>  '  e  ]   ] f s  w n v 2  D Ew> v gvgw n 2 @2 ] f s Ew gw  2 @2 ] f s Ew gw D E>  ' _e  ]   ] f s  w n v 2  D Ew> v gvgw n 2 @2 ] f s Ew gw n 2 f ] @2 s Ew gw D E>  ' _e  ]   ] f s  w n v 2  D Ew> v gvgw n 2 @2 ] @2 s Ew gw hogh hglolu dDSo1

614141 Whruhp= kfEw Ef  w   yh S E{



f  {  2



irqnvl|rqodu dvdjgdnl nrvxoodu vdjodvq=

4, kfEw irqnvl|rqx vquo wÿuhyh vdklswlu yh kfEf ' f gu1

5, S E{ irqnvl|rqx f  {   2 dudojqgd vquogu1 6, tT f$w$m k  Ew m n_e tT f${$ 2 m S E{ m? ₂ +6148, gu1 Ex gxuxpgd s E{ ' S E{ > f  {  ₂ +6149,

ropdn ÿ}huh s E{ vquo irqnvl|rqx ydugu yh | E>  irqnvl|rqx | E>  ' t? s  s  n  ] f t?sw s  kfEw gw= +614:, 53

vhnolqghglu dDSo1

Lvsdw1 D E>  irqnvl|rqx E=e loh wdqpo ropdn ÿ}huh

k E ' D E> 2   >     2> +614;,

kE ' D E> f > f    > +614<,

irqnvl|rqodu wdqpodqvq1 Ex irqnvl|rqodu

k E ' kfE   >     2 +6153, kE '  2 @2 ] f S Ev gv> f     +6154,

edjqwodu loh kfEw yh S E{ irqnvl|rqoduqd edjogu1

jiEw> v '  Ci E> f C m'wnv  i Ew> v > +6155, W i '   ]   ] f ji Ew> v gvgw> +6156, xiE ' EW i _'23 '   ] 23 23 ] f jiEw> v gvgw= +6157,

hvlwolnohul vdjodqvq1 E=e ghqnohpl

D E>  ' * E>  n WD E>  = +6158,

rshudwùu irupxqgd |d}odelolu1 Exudgd * E>  ' ; ? = kE  kE > f     k E  kE n kE2   >     2 +6159, 54

Wi ' ; ? = W i _{> f    } W i  xiE >     2 +615:, vhnolqghglu1 Jhuêhnwhq f     lnhq E=2S yh E=2. ghq

* E>  n WD E>  ' kE  kE n W D

roxs E=2 yh E= ghq

* E>  n WD E>  ' D E> f  D E> f n W D

exoxqxu1 E= yh E=2 ghq * E>  n WD E>  '  2 @2 ] f s Ev gv  2 @2 ] f s Ev gv   ]   ] f EDE> f m'wnv D Ew> v gvgw= exoxqxu1 E= ghq D E> f '  2 @2 ] f s Ev gv glu1 Exudgdq DE> f '  es  w n v 2 

exoxqxu1 Exqd jùuh * E>  n WD E>  '  2 @2 ] @2 s Ev gv   e  ]   ] f s  w n v 2  D Ew> v gvgw roxs E=e ghq

* E>  n WD E>  ' D E> 

hogh hglolu1     2 lnhq E=2S yh E=2. ghq

* E>  n WD E>  ' k E  kE n kE2   n W D  xDE

glu1 E=H ghq k E ' D E> 2   roxs 2   '  doquvd  n  ' 2 hogh hglolu1 Exqd jùuh

* E>  n WD E>  ' D E>   kE n kE n EW D_'23 EW D_'23

, * E>  n WD E>  ' D E>  hogh hglolu1 CxiE C '  23 ] f jiE> v gv  23 ] f jiE2  > v gv n  ] 23 jiEw> 2   gw= +615;,

irupÿoÿ lohulgh nxoodqodfdnwu1 nin ' tT  4@ mi E> m > 4@     Ci E> f C      > f      2>  n   2 +615<, qrupx loh i E>  irqnvl|rqoduqq x}d| jù}ùqÿqh doqdfdnwu1 iq ' Wqif/ ir ' * E> 

yh pq ' niqn rovxq1 E=D nrvxox yh E=2S edjqwvqgdq

pf ?



2 +6163,

hogh hglolu1 Jhuêhnwhq

pf ' nifn ' n* E> n ' tT  4@ m* E> m > 4@     C* E> f C      1

f     rovxq1  n  ' 2 rogxjxqgdq  ' 2   gu1 Exudgdq     2 roxu1 f      2 rogxjxqgdq  '  '  hogh hglolu1 E=2S gdq

m* E> m '         2 @2 ] @2 S Ev gv        ' f> f     +6164, exoxqxu yh     C* E> f C     '       C C 5 7 2 @2 ] f S Ev gv 6 8       '  e_f$tT 2$  2    S   2     roxs E=D vduwqgdq

    C* E> f C     ?  2> f     +6165,

exoxqxu1     2 rovxq1  n  ' 2 gu1 Exudgdq  ' 2   roxs/ f     roxu1  ' 2 lvh  ' f roxu1

m* E> m ' mk Em ' mD E> m roxs E=e ghq m* E> m '        _e  ]   ] f s  w n v 2  D Ew> v gvgw n 2 @2 ] @2 s Ew gw        56

m* E> m   e  ]   ] f    s  w n v 2     mD Ew> vm mgvm mgwm n  2 @2 ] @2 ms Ewm mgwm roxs E=. yh E=f gdq

m* E> m   e_f$tTwnv 2 $  2    s  w n v 2     mD Ew> vm m  m m  fm n f m* E> m   e_f$tTwnv 2 $  2    s  w n v 2     mD Ew> vm m2  fm mf  fm m* E> m ' f>     2 +6166,

exoxqxu1 Exudgd E=2S gdq     2 lnhq     C* E> f C     '     C

Cdk E  kEf n kE2  o         C* E> f C     '       C C 5 7k E n  2 E23@2_] f S Ev gv 6 8       '    kE   eS  2   2         C* E> f C      tT_f$$m kE m n  e_f$tT23 2 $  2    S  2   2     ?  2 ,     C* E> f C     ?  2 +6167, exoxqxu1 Eù|ohfh pf ' nifn ' tT  4@ mf> fm > 4@      2>  2       ' 2 , pf ?  2 +6168,

roxs E=f hogh hglolu1 Gljhu wdudiwdq E=22  E=2H irupÿoohulqghq

pqn 2p2q +6169,

hvlwvl}oljl hogh hglolu1 Jhuêhnwhq pqn' niqnn '  Wqn  * E>    ' tT + 4@ Wqn  * E>    > 4@      CW_qn* E> f C      , q ' f lêlq p  2p2f rogxjxqx jùvwhuholp= p ' nin ' nW* E> n ' tT  4@ mW* E> m > 4@     C EW* E> f C      1 roxs f     lnhq mW* E> m ' mW * E> m '        ]   ] f  *_{E> f m}'wnv  * Ew> v gvgw       > 57

mW* E> m   ]   ] f  *_{E> f m}'wnv   m* Ew> vm mgvm mgwm exoxqxu1 f     lvh     2 roxu1 D|ufd

* E> f ' * E n > f ' kE n   kEf

roxs E=2 ghq kEf '

 2 f@2 ] f S Ev gv ' f roxu1 * E> f ' kE n  '  2 En@2_] f S Ev gv Exudgdq *E> f '  eS _ 2  ,*E> f    _e tT f$ 2$  2  S 2   roxu1 E=D ghq  *_E> f ? 

2 exoxqxu1 E= yh E=2 ghq

mW* E> m   eE   E  f f ' f , mW* E> m ' f +616:, hogh hglolu1 f     lnhq     C EW* E> f C     '   EW *E> f    '      x*E_'23'f  E> f     '     Ex*_E    roxs E=2H irupÿoÿqgh  ' f doquvd

    C EW* E> f C     '        ] f j*Ew> f gw       '        ] f *_wEw> f * Ew> f gw        ] f m*wEw> fm m* Ew> fm mgwm

roxu1 E= yh E=2 ghq     C EW* E> f C     ' f +616;, exoxqxu1     2 lnhq

mW* E> m ' mW *  x*Em ' mfm , mW* E> m ' f +616<,

yh     C EW* E> f C     '     Cf C     ' f ,     C EW* E> f C     ' f +6173, 58

glu1 Exudgdq p ' nin ' nW* E> n ' tT  4@ mW* E> m > 4@     C EW* E> f C      p ' tT i4@ mf> fm > 4@ mf> fmj ' f> f    2

exoxqxu1 E=D edjqwvqgdq pf ?  2 , 2p 2 f ?  2 , p  2p 2 f +6174,

hogh hglolu1 q '  lêlq p2  2p2 rogxjxqx jùvwhuholp=

p2 ' ni2n '  W2 * E>    ' nWEW* E> n p2 ' tT  4@ W2 * E>    > 4@     C EW2 * E> f C      roxs f     lnhq  W2 * E>    '        ]   ] f  EW *_{E> f m}'wnv  EW * Ew> v gvgw       glu1 E=. yh E=H ghq

 W2 * E>      ]   ] f   EW *E> f m'wnv    mW * Ew> vm mgvm mgwm  W2 * E>    ' f exoxqxu1 f     lnhq     C EW2 * E> f C     '   W2*E> f    '      xW *E_'23'f  E> f         C EW2 * E> f C     '     ExW *_E    roxs E=2H ghq     C EW2 * E> f C     '        ] f jW *Ew> f gw       '        ] f EW *_wEw> f EW * Ew> f gw           C EW2 * E> f C       ] f

mEW *wEw> fm mEW * Ew> fm mgwm ' f

hogh hglolu1     2 lnhq  W2 * E>    ' mWEW *  x*Em ' mWEfm ' f ,  W2 * E>    ' f yh     C EW2 * E> f C     '     Cf C     ' f ,     C EW2 * E> f C     ' f exoxqxu1 Exudgdq p2 ' ni2n '  W2 * E>    ' nWEW* E> n p2 ' tT  4@ W2 * E>    > 4@     C EW2 * E> f C      p2 ' tT i4@ mf> fm > 4@ mf> fmj ' f

hogh hglolu1 E=e ghq

p ' f , 2p2 ' f , p2 ' 2p2

yh eù|ohfh

p> pe> pD> === ' f , pqn 2p2q

exoxqxu1 Eù|ohfh

* n W* n W2* n === ' D E>  +6175,

vhulvl gÿ}jÿq |dnqvdnwu1 Jhuêhnwhq mNqE> m '      " [ n'qn W_q*      f n f n f n === ' f ?  glu1 Eù|ohfh hogh hglohq D E>  irqnvl|rqx vquogu=

mD E> m '* n W* n W2* n ===   mD E> m  m*m n mW*m n  W2 *   nW *   nWe *   nWD *   n === mD E> m  f n f n f n === , mD E> m  f> f     mD E> m '* n W* n W2* n ===   mD E> m  m*m n mW*m n  W2 *   nW *   nWe *   nWD *   n === 5:

mD E> m  f n f n f n === , mD E> m  f>     2 yh E=e2 ghqnohplql vdjodu=

* E>  n WD E>  ' * E>  n W



* n W* n W2* n ===



* E>  n WD E>  ' * E>  n W* n W2* n W* n === ' D E> 

CD E> f C irqnvl|rqx gd vquogu=     CD E> f C     '     C* E> f C n C EW* E> f C n C EW2 * E> f C n ===         CD E> f C          C* E> f C     n     C EW* E> f C     n     C EW2 * E> f C     n ===     CD E> f C       2n f n f n === ,     CD E> f C      f=D> f         CD E> f C     '     C* E> f C n C EW* E> f C n C EW2* E> f C n ===         CD E> f C          C* E> f C     n     C EW* E> f C     n     C EW2 * E> f C     n ===     CD E> f C       2n f n f n f n === ,     CD E> f C      f=D>     2 E=22  E=2. irupÿoohulqh jùuh

D E>  ' f> D E> f '  2 @2 ] f S Ev gv> f     +6176, D E> 2   ' k E >     2 +6177,

nrvxoodu vdjodqu1 Jhuêhnwhq

D E>  ' * E>  n WD E>  ' kE  kE ' f , D E>  ' f> f    

D E> f ' * E> f n WD E> f ' kE  kEf

glu1 kEf '  2 f@2 ] f S Ev gv rogxjxqgdq D E> f ' kE '  2 @2 ] f S Ev gv> f     5;

exoxqxu1 Grod|v|od E=e vdjodqu1 Gljhu wdudiwdq

D E> 2   ' * E> 2   n WD E> 2  

D E> 2   ' k E  kE2   n kE2   n EW D  xDE2   ' k E

roxs E=ee vdjodqu1

s   2  ' eCD E> f C > f    2 +6178,

rovxq1 s E{ srwdqvl|hol f  {   dudoj ÿ}hulqgh E=eD irupÿoÿ loh wdqpodqu1 s E{ ' S E{ > f  {  ₂

hvlwoljl jhêhuolglu1 Jhuêhnwhq E=e yh E=eD ghq CD E> f C '  eS   2  , eCD E> f_C ' S   2  , s   2  ' S   2  1 f    2 , f  2   roxs , s E{ ' S E{ > f  {  ₂ hogh hglolu1

Dvdjgdnl 614151 whruhpl E= > E=2 vqu ghjhu sureohplqlq q vshnwuxpx loh

kfEw ' N E> w irqnvl|rqx dudvqgdnl edjqw| yhulu dDDo1

614151 Whruhp= iqj"_q' " [ q'  q q22   ? 4

nrvxoxqx vdjod|dq vd|oduq elu gl}lvl rovxq= Ex wdnwlugh dvdjgdnl vduwodu vdjod|dq |doq} yh |doq} elu x E irqnvl|rqx ydugu=

4, q vd|odu/ x E qq edvlw viuodugu1

5, x E edvnd viuodud vdkls ghjloglu1

6, kfEw vÿuhnol wÿuhyh vdkls yh k Ef ' k E ' f ropdn ÿ}huh x E irqnvl|rqx

dvdjgdnl vhnloghglu dDDo1 x E ' t? s  s  n  ] f t?sw s  kfEw gw ' | E>  1 5<

Vwxup0Olrxylooh sureohplqlq vshnwuxpx yh s E{ srwdqvl|holqlq s E{  s E  {

sduêdv yhuloplvnhq s E{ E f  {   wdqpodqdelolu1

Ed} nrvxoodu dowqgd ex nvpgd 614141 Whruhplqlq êù}ÿpÿqÿq yduoj lvsdwodqg yh ex êù}ÿpÿ |hqlghq nxupdqq |ùqwhpl yhulogl1 Ex êdovpdgd kfEw qlq qrupodu

ÿ}hulqgh gdkd }d|i vquodpdodu yh s E{  s E  { loh 614141 Whruhpl whnudu jù}ùqÿqh doqdfdnwu1 Jhqhooljl er}pdgdq



]

f

s Ev gv ' f +6179,

doqdelolu1

614161 Whruhp= kfEw Ef  w   yh t E{ ' t E  { Ef  {   irqnvl|rqodu

dvdjgdnl nrvxoodu vdjodvq=

4, kfEw irqnvl|rqx vquo wÿuhyh vdklswlu yh kfEf ' kfE ' f gu1

5, t E{ irqnvl|rqx f  {   dudojqgd vquogu1 6,  2  tT f$w$m k  fEw m n tT f${$  e m t E{ m  ?  +617:, glu1 Ex gxuxpgd

s E{  s E  { ' t E{ +617;,

ropdn ÿ}huh s E{ Ef  {   vquo irqnvl|rqx ydugu yh | E>  irqnvl|rqx E=. loh wdqpogu dDSo1 Lvsdw= k E ' kfE   ' D E> 2   >     2 +617<, k2E ' D E> f n D E2  > f '  2 @2 ] f t Ev gv> f    2 +6183, 63

irqnvl|rqodu wdqpodqvq1 E=e ghqnohpl

D E>  ' [ E>  n W2D E>  +6184,

rshudwùu irupxqgd |d}odelolu1 Exudgd [ E>  ' ; ? = k2E3kE23 2  k2E3kE23 2 > f     k2EnkE 2  k2E3kE23 2 >     2 +6185, W2i ' ; ? = W i n  2dxiE2    xiE2  o > f     W i   2dxiE n xiE2  o >     2 +6186, gu1 f     lnhq

D E>  ' [ E>  n W2D E> 

glu1 Jhuêhnwhq f     lnhq [ E>  ' k2E  k E2   2  k2E  k E2   2 ' k2E  k2E 2 n k E2    k E2   2

k E2   ' D E2  >  gu1 2   '  rovxq1 Exudgdq  n  ' 2 roxu1 E=Df ghq k2E  k2E ' dD E> f n D E2  > fo  dD E> f n D E2  > fo

k2E  k2E ' dD E> f n D E> fo  dD E> f n D E> fo ' f

, [ E>  ' f n k E2    k E2  ₂ ' k E  k E 2 roxs E=eb gdq

[ E>  ' D E> 2    D E> 2  

2 '

D E>   D E>  2

gu1 Exudgdnl D E>  khvdsodquvd D E>  '  e  ]   ] f s  v n w 2  D Ev> w gwgv n 2 @2 ] @2 s Ew gw D E>  '  e  ]   ] f s  w n v 2  D Ew> v gvgw  ₂ @2 ] @2 s Ew gw 64

D E>  '  5 9 7_e  ]   ] f s  w n v 2  D Ew> v gvgw n  2 @2 ] @2 s Ew gw 6 : 8 , D E>  ' D E>  exoxqxu1 Ex lidgh |xndugdnl ghqnohpgh |hulqh |d}ouvd

[ E>  ' D E>   ED E> 

2 ' D E>  > f    

hogh hglolu1 E=D ghq W2D E>  ' W D n  2dxDE2    xDE2  o W2D E>  ' W D n  2EW D'23E23  2EW D'23E23 W2D E>  ' W D n  2EW D'   2EW D' W2D E>  '  2W D W2D E>  '   2  ]   ] f EDE> f m'wnv D Ew> v gvgw

hogh hglolu1 DE> f khvdsodquvd E= yh E=eS gdq

D E> f '  2 @2 ] f s Ev gv '  2  ] f s Ev gv ' f , DE> f ' f roxs W2D E>  ' f

, [ E>  n W2D E>  ' D E>  > f    

hogh hglolu1     2 lnhq

D E>  ' [ E>  n W2D E> 

glu1 Jhuêhnwhq     2 lnhq

[ E>  ' k2E n k E

2  k2E  k E2   2 ' k2E  k2E 2 n k E n k E2   2

roxs E=eb gdq k E ' D E> 2   gu1 Exudgdq 2   '  doquvd  n  ' 2 hogh hglolu1 E=Df ghq

k2E  k2E ' dD E> f n D E2  > fo  dD E> f n D E2  > fo

k2E  k2E ' dD E> f n D E> fo  dD E> f n D E> fo ' f

k E n k E2   ' k E n k E ' 2D E> 2   ' 2D E>  hogh hglolu1 Exudgdq

[ E>  ' f n D E>  ' D E>  >     2 W2D E>  ' W D   2dxDE n xDE2  o ' W D   2dxDE n xDEo W2D E>  ' W D  xDE ' EW D  EW D'23 ' f

[ E>  n W2D E>  ' D E>  >     2

hogh hglolu1 i E>  qq qrupx E=2b loh wdqpodqvq1 iq ' W2qif/ pq ' niqn yh

if ' [ E>  rovxq1 E=e. nrvxox yh E=D edjqwvqgdq pr ?  hogh hglolu1

pf ' nifn ' n[ E> n ' tT  4@ m[ E> m > 4@     C[ E> f C      roxs f     lnhq m[ E> m '     k2E  k2E 2 n D E>      '         e @2 ] @2 t Ev gv n D E>         m[ E> m   e_f$tT 2$  2    t   2     @2 ] @2 mgvmnmD E> m   e_f$tT 2$  2    t   2         2  nmD E> m m[ E> m  _e tT f$ 2$  2    t   2         2  n mD E> m ' f +6187, yh     C[ E> f C     '       C C 5 7 e @2 ] f t Ev gv n D E> f 6 8       '      Ht   2       H_f$tT 2$  2    t   2     ,     C[ E> f C     ? > f     +6188, hogh hglolu1     2 lnhq m[ E> m '     k2E  k2E 2 n D E>      '         e @2 ] @2 t Ev gv n D E>          f +6189, 66

yh     C[ E> f C     '       C C 5 7 e @2 ] f t Ev gv n D E> f 6 8       ?  +618:, hogh hglolu1 pf ' nifn ' n[ E> n ' tT  4@ m[ E> m > 4@     C[ E> f C      pf ' tT i4@ mf> fm > 4@ m> mj '  , pf ?  +618;,

roxs E=22  E=2S yh E=D edjqwoduqgdq

pqn p2q +618<,

hogh hglolu1 Jhuêhnwhq q ' f lêlq p  p2f rogxjxqx jùvwhuholp=

p ' nin ' nW2[ E> n ' tT  4@ mW2[ E> m > 4@     C EW2[ E> f C      glu1 f     lnhq mW2[ E> m '    W [ n  2 k EW [_{' EW ['}l  mW2[ E> m '  2mW [m '  2        ]   ] f E[E> f m'wnv [ Ew> v gvgw       mW2[ E> m   2  ]   ] f m[E> f m'wnvm m[ Ew> vm mgvm mgwm roxs E=De ghq mW2[ E> m ' f> f     +6193, exoxqxu1 f     lnhq     C EW2[ E> f C     '     C C   2W [  E> f         C EW2[ E> f C     '      2 Cx[E> f C     '        2  ] f j[Ew> f       67

    C EW2[ E> f C     '        2  ] f [wEw> f [ Ew> f gw        2  ] f m[wEw> fm m[ Ew> fm mgwm     C EW2[ E> f C       2  ] f m[wEw> fm m[ Ew> fm mgwm

exoxqxu1 E=De yh E=DD ghq     C EW2[ E> f C      f> f     +6194, hogh hglolu1     2 lnhq mW2[ E> m '    W [   2 k EW [_'23n EW [_'l_ glu1 Exudgd  n  ' 2 rogxjxqgdq

mW2[ E> m ' mW [  W [m ' f , mW2[ E> m ' f>     2 +6195,

hogh hglolu1 Exudgdq     2 lnhq     C EW2[ E> f C     '     Cf C     ' f ,     C EW2[ E> f C     ' f>     2 +6196, roxs p ' nin ' nW2[ E> n ' tT  4@ mW2[ E> m > 4@     C EW2[ E> f C      p ' tT i4@ mf> fm > 4@ mf> fmj ' f , p  p2f +6197,

exoxqxu1 q '  lêlq p2  p2 rogxjxqx jùvwhuholp=

p2 ' ni2n '  W2 2[ E>    ' tT  4@ W22[ E>    > 4@     C EW2 2[ E> f C      1 f     lnhq  W2 2[ E>    ' mW2EW2[ E> m '    W2   2W [      W2 2[ E>    '    W   2W [   ₂kx 2W [E2    x  2W [E2   l    W2 2[ E>    '      2W   2W [      W2 2[ E>    '        2  ]   ] f #  2W [   E> f m'wnv $   2W [  Ew> v gvgw       68

roxs E=Sf yh E=S ghq  W2 2[ E>    ' f> f     +6198, exoxqxu1 f     lnhq     C EW2 2[ E> f C     '     C C   2W   2W [  E> f         C EW2 2[ E> f C     '  2      Cx 2W [E> f C     '        2  ] f j 2 W[ Ew> f           C EW2 2[ E> f C     '        2  ] f   2W [  w Ew> f m'wnv   2EW [ Ew> f gw           C EW2 2[ E> f C     ' f> f     +6199, glu1 E=S2 ghq     2 lnhq  W2 2[ E>    ' mW2EW2[ E> m ' mW2Efm ' f ,  W2 2[ E>    ' f +619:, yh E=S ghq     2 lnhq     C EW2 2[ E> f C     '     CW2 C Ef     ' f ,     C EW2 2[ E> f C     ' f +619;, exoxqxu1 p2 ' ni2n '  W2 2[ E>    ' tT  4@ W2 2[ E>    > 4@     C EW2 2[ E> f C      roxs E=Se ghq p2 ' tT i4@ mf> fm > 4@ mf> fmj ' f , p2  p2 +619<,

hogh hglolu1 q ' 2 lêlq p  p22 rogxjxqx jùvwhuholp=

p ' nin '  W 2[ E>    ' tT  4@ W 2[ E>    > 4@     C EW 2[ E> f C      roxs f     lnhq  W 2[ E>    'W2  W₂2[ E>  '      2W  W₂2[ E>       W 2[ E>    '        2  ]   ] f  W₂2[_{E> f m}'wnv   W₂2[Ew> v gvgw       ' f 69

glu1 E=SD yh E=SS gdq  W 2[ E>    ' f> f     exoxqxu1 f     lnhq     C EW 2[ E> f C     '     C C   2W  W₂2[  E> f         C EW 2[ E> f C     '  2     CxW2 2[E> f C     '        2  ] f j W 2₂[Ew> f           C EW 2[ E> f C     '        2  ] f  W₂2[_{wEw> f m}'wnv   W₂2[Ew> f gw       ' f     C EW 2[ E> f C     ' f> f     yh E=S. ghq     2 lnhq  W 2[ E>    'W2  W₂2[ E> _{ ' mW}2Efm ' f ,  W 2[ E>    ' f>     2 glu1 E=SH ghq     2 lnhq     C EW 2[ E> f C     '     CW2 C Ef     ' f ,     C EW 2[ E> f C     ' f>     2 roxs p ' nin '  W 2[ E>    ' tT  4@ W 2[ E>    > 4@     C EW 2[ E> f C     

exoxqxu1 E=Sb jù}ùqÿqh doquvd

p ' tT i4@ mf> fm > 4@ mf> fmj ' f , p ' f , p  p22

hogh hglolu1 Eù|ohfh

pe ' pD ' pS ' === ' f , pqn p2q

exoxqxu1 Ex gxuxpgd

[ n W2[ n W22[ n === ' D E>  +61:3,

vhulvl gÿ}jÿq |dnqvdnwu1 Jhuêhnwhq mNqE> m '      " [ n'qn W₂q[      f n f n f n === ' f ? = 6:

Hogh hglohq D E>  irqnvl|rqx vquogu= mD E> m '[ n W2[ n W22[ n W2[ n ===   mD E> m  m[m n mW2[m n  W2 2[   nW 2[   n === mD E> m  f n f n f n === , mD E> m  f> f     mD E> m  m[m n mW2[m n  W2 2[   nW 2[   n === mD E> m  f n f n f n === , mD E> m  f>     2 glu yh E=.f ghqnohplql vdjodu=

[ E>  n W2D E>  ' [ E>  n W2



[ n W2[ n W22[ n ===



[ E>  n W2D E>  ' [ E>  n W2[ n W22[ n === , [ E>  n W2D E>  ' D E> 

CD E> f C irqnvl|rqx gd vquogu=     CD E> f C     '     C[ E> f C n C EW2[ E> f C n C EW2 2[ E> f C n ===         CD E> f C          C[ E> f C     n     C EW2[ E> f C     n     C EW2 2[ E> f C     n ===     CD E> f C       n  e n f n === ' =2D ,     CD E> f C      =2D> f     yh     CD E> f C     '     C[ E> f C n C EW2[ E> f C n C EW2 2[ E> f C n ===         CD E> f C          C[ E> f C     n     C EW2[ E> f C     n     C EW2 2[ E> f C     n ===     CD E> f C       n f n f n === '  ,     CD E> f C      >     2 exoxqxu1 E=D2 > E=D irupÿoohulqghq

D E>  ' f> f     +61:4,

D E> 2   ' k E >     2 +61:5,

nrvxoodu vdjodqu1 Jhuêhnwhq

D E>  ' [ E>  n W2D E>  ' k2E  k E2   2  k2E  k E2   2 ' f D E>  ' f> f    

roxs E=. vdjodqu1 Gljhu wdudiwdq

D E> 2   ' [ E> 2   n W2D E> 2  

D E> 2   ' k2E n k E₂  k2E2    k E₂ n W D ₂dxDE n xDEo

D E> 2   ' k2E n k E₂  k2E2    k E₂ =

Exudgd k2E ' D E> fnD E2  > f yh k2E2   ' D E2  > fnD E> f rogxjxqgdq

k2E ' k2E2   hogh hglolu1 Ex/ |xndugdnl vrq ghqnohpgh |hulqh |d}ouvd E=.2

hogh hglolu1

D E> f n D E2  > f ' k2E > f    2 +61:6,

glu1 Jhuêhnwhq f     lnhq

D E> f n D E2  > f ' [ E> f n W2D E> f n [ E2  > f n W2D E2  > f

D E> f ' k2E  k E2   2  k2Ef  k E2 2 n EW D E> f n n 2 k xDE2  '  xDE2'232'f l

D E2  > f ' k2E2    k E₂  k2Ef  k E2₂ n EW D E2  > f  ₂kxDE2  ' n xDE2

l

glu1 Exudgd

k E2 ' kfE ' f> k2Ef '

 2 f@2 ] f t Ev gv ' f rogxjxqgdq

D E> f n D E2  > f ' k2E  k E2  ₂ nk2E2    k E 2

D E> f n D E2  > f ' k2E  k E n k E2  

2

exoxqxu1 k E ' D E> 2   yh k E2   ' D E2  >  gu1 D E> 2   ' D E2  >  rogxjxqgdq k E ' k E2   gu1 Eù|ohfh

D E> f n D E2  > f ' k2E > f    

exoxqxu1     2 lnhq

D E> f n D E2  > f ' [ E> f n W2D E> f n [ E2  > f n W2D E2  > f

D E> f ' k2E n k E

2  k2Ef  k E2 2 n EW D E> f   2dxDE n xDE2o D E2  > f ' k2E2   n k E2  ₂ k2Ef  k E2₂ n EW D E2  > f ₂dxDE2   n xDE2o

D E> f n D E2  > f ' k2E n k E₂ nk2E2   n k E2   2

D E> f n D E2  > f ' k2E >     2

roxs exudgdq

D E> f n D E2  > f ' k2E > f    2

hogh hglolu1 s E{ srwdqvl|hol E=eD irupÿoÿ loh wdqpogu1 Whruhp lvsdwodqpvwu1

71 LVRVSHNWUDO SUREOHP

7141 Nodvln Vwxup0Olrxylooh Ghqnohplqlq Wdqpodqpdv Nodvln Vwxup0Olrxylooh sureohpl loh

|n t E{ | ' |> f  {   +714,

|_{Ef  k| Ef ' f} +715,

|E n K| E ' f +716,

vhnolqghnl lnl qrnwdo vqu ghjhu sureohpl ndvwhglolu1 Exudgd t E{ uhho vÿuhnol elu irqnvl|rq/ k yh K uhho vd|odu yh  ù}ghjhuglu1 Lvrvshnwudo sureohp/ d|q vshnwuxpd vdkls Ee=0Ee=20Ee= vhnolqghnl eÿwÿq sureohpohuglu1 Ex sureohp Wuxerzlw} wdudiq0 gdq ghwd|o rodudn êdovopvwu dSo > d2fo > d2o1

Ex êdovpdgdnl whpho vrqxêodugdq ed}odu eholuwlohq êdovpdoduod ghqn jholu/ idndw |ùqwhpohu wdpdphq idunogu1 Exudgdnl whpho dudêodu Jhoidqg0Ohylwdq lqwhjudo ghqn0 ohpl yh gùqÿvÿp rshudwùuohulglu1 dSo/d2fo > d2o > dDDo > dDSo pdndohohulqghnl vrqxêodu t E{ lq O2_{Ef>  x}d|qgd rogxjx ndexoÿ loh hogh hgloplvwlu1 Exqxq |hulqh t E{ lq elulqfl}

phuwhehghq wÿuhyohulqlq O2_{Ef>  gh rogxjxqx yduvd|dop1 Ex ndeÿo I E{> | irqnvl0}

|rqxqxq glihuhqvl|hoohqheloluoljlqlq lvsdwqgd nrod|on vdjodu1

Ex eùoÿpÿq dqd vrqxêoduqq ed} dvdpdoduq dênodpdn lêlq nodvln Vwxup0Olrxylooh sureohpl ÿ}hulqgh |ùqwhplq dqd dvdpdoduqq wdvodjq |dspdn id|gdo rodfdnwu1

Whuv sureohplq prghuq whrulvl J1Eruj*xq whpho pdndohvl loh edvodpvwu deo = Eruj*xq pdndohvlqghnl lnl whpho vrqxê grjuxgdq grjux|d ex nrqx loh lojlolglu1 Eruj>

4, iqj lq elolqhq vshnwuxp ropdv gxuxpxqgd t E{ srwdqvl|holql yh Ek> K vd|

êliwlql lêhuhq eluêrn Et> k> K vrqvx} ÿêoÿvÿqÿq ydu rogxjxqx yh nduvon jhohq Ee=0 Ee=20Ee= sureohpohulqlq iqj vshnwuxpxqd vdkls rogxjxqx yh

5, Ee=0Ee=20Ee= sureohplqlq vshnwuxpxqxq f ?  ? === ? q ? ===ropdv yh

K 9' K ropdn ÿ}huh f ?  ? === ? q? === lq

|E n K| E ' f +716*,

vhnolqghnl Ee=_{yh Ee=2 vqu nrvxoox Ee= ghqnohplqh nduvon jhohq sureohplq vshn0}

wuxpx ropdv gxuxpxqgd/ iqj yh iqj vshnwuxpoduqq whn rodudn t E{ srwdqvl|holql

Nodvln whuv Vwxup0Olrxylooh sureohplqlq jholvphvlqgh elu vrqudnl ùqhpol dgp Q1Oh0 ylqvrq do wdudiqgdq hoh doqg1

714141 Wdqp= Ee=0Ee=20Ee= Vwxup0Olrxylooh sureohplqh t E{ ' t E  { yh k ' K vduwodu vdjodqdfdn elêlpgh *vlphwulnwlu* ghqlu d2eo1

714151 Whruhp +Ohylqvrq ^66`,= Vlphwuln Vwxup0Olrxylooh sureohpohu vqiqgd iqj/q  f vshnwuxpx/ t E{ srwdqvl|holql yh k vd|vq whn rodudn wdqpodu1

4<83*gh Y1Pdufkhqnr deo gùqÿvÿp rshudwùuohulql nxoodqdudn |du dudon ÿ}hulqghnl

|n tE{ | ' |  |n t2E{ | ' |

|Ef ' k| Ef |Ef ' k2| Ef

Vwxup0Olrxylooh sureohpohulqlq d|q vshnwudo irqnvl|rqodud vdkls ropdodu gxuxpxqgd t2E{  tE{ yh k2 ' k rogxjxqx lvsdwodpvwu1

* E{>  > df> 4 dudoj ÿ}hulqgh | Ef ' > |_{Ef ' k ropdn ÿ}huh}

|n t E{ | ' |

edvodqjê ghjhu sureohplqlq êù}ÿpÿ rovxq1 Dvdjgdnl whruhp K1Zh|o*lq êdovpdvqd gd|dqu1

i E{ gÿ}jÿq nrpsdnw ghvwhnohqhq elu irqnvl|rq yh I E '

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i E{ * E{>  g{

rovxq1 Ex gxuxpgd d}dopd|dq hq d} elu  E irqnvl|rqx ydugu nl i E{ $ I E gùqÿvÿpÿ ÿqlwhuglu1 Ex lvh " ] f i2_{E{ g{ '} " ] f I2_{E g E}

ghphnwlu1  E irqnvl|rqxqd vshnwudo irqnvl|rq ghqlu1

Eruj*xq whnoln whruhpl/ Pdufkhqnr*qxq whnoln whruhplqghq hogh hglohelolu1

J1Eruj/ Q1Ohylqvrq yh Y1Pdufkhqnr*qxq vrqxêodu ùqhpolglu yh rqodu gdkd id}od d0 udvwupd lêlq |ro jùvwhuplvwlu/ idndw rqodu vdghfh whnoln whruhpohulglu1 Vrqud hwnlq elu vhnlogh t E{ srwdqvl|holql yh vqu nrvxooduqgd k yh K vd|oduq nxupd sureohpl roxvwx1

Ex sureohp L1P1Jhoidqg yh E1 P1 Ohylwdq wdudiqgdq êù}ÿogÿ deo1 Rqoduq whpho gÿvÿqfhohul gùqÿvÿp rshudwùuohul yh vÿuhnol ruwrjrqdooln nul|gl1 Ex lnl gÿvÿqfh vshn0 wudo irqnvl|rq dêvqgdq whuv sureohplq wdp elu dudvwupdvq |dspd|d rodqdn vdjod|dq elu lqwhjudo ghqnohp yhulu1 Ex lqwhjudo ghqnohp ex eùoÿpgh hoh doqu1

Whuv Vwxup0Olrxylooh sureohpl lêlq edvnd elu |ùqwhp/ P1Nuhlq dfo wdudiqgdq jholvwlu0 loplvwlu1

7151 Jhoidqg0Ohylwdq Lqwhjudo Ghqnohplqlq Êù}ÿpÿqÿq Yduoj yh Whnoljl

Ex nrqxgd Jhoidqg0Ohylwdq lqwhjudo ghqnohplqlq +nodvln Vwxup0Olrxylooh sureohp0 lqlq |dsvqgd, nvphq |hql elu lvsdw yh gùqÿvÿp rshudwùuohulqlq yduojqq lvsdw yh0 ulogl1

tfE{ Ef  {   > uhho glihuhqvl|hoohqhelolu elu irqnvl|rq/ kf yh Kf vdelw uhho vd|odu

rovxq1

|n tfE{ | ' |> f  {   +717,

|_{Ef  k}f| Ef ' f +718,

|E n Kf| E ' f

nodvln Vwxup0Olrxylooh sureohpl jù}ùqÿqh doqvq1 *_fE{> / Ee=e ghqnohplqlq *_fEf>  ' > *_fEf>  ' kf

edvodqjê nrvxooduq vdjod|dq êù}ÿpÿ rovxq1 Eù|ohfh khu  lêlq *fE{> / Ee=D lon vqu

nrvxoxqx vdjodu1 Ee=e0Ee=D sureohplqlq f ?  ? 2 ? === ù}ghjhuohul

*fE>  n Kf*fE>  ' f +719,

ghqnohplqlq nùnohulglu yh q ohuh nduvon jhohq ù}irqnvl|rqodu lvh *fE{> q/ q  f gu1

Ex irqnvl|rqodu O2_{Ef>  gh qrupdoohvwlulophplvwlu1}

Dvdjgdnl jlel lqwhjudo ghqnohplqlq êhnlughjlql nxudop1 fq> q ' f> > 2> === q $ 4

lnhq êrn k}o elu vhnlogh viud |dnqvd|dq nh| uhho vd|odu rovxq1 I E{> | '

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[

q'f

fq*fE{> q *fE|> q +71:,

irqnvl|rqx vÿuhnol yh lnlqfl phuwhehghq vÿuhnol wÿuhyohqheloluglu1 Ex eùoÿpgh whuv sureohpgh ùqhpol elu |huh vdkls

N E{> | n I E{> | n

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]

f

N E{> w I Ew> | gw ' f> f  |  {   +71;, lqwhjudo ghqnohpl hoh doqu1 Ex ghqnohp Jhoidqg0Ohylwdq ghqnohpl rodudn elolqlu1 Ee=H lqwhjudo ghqnohplqlq êhnlughjl I Ew> | glu yh I E{> | | qlq elolqph|hq irqnvl|rqxgxu/ { sdudphwuhglu yh N E{> | +| qlq elolqph|hq irqnvl|rqx rodudn, elolqph|hq irqnvl|rqgxu1

715141 Whruhp= ;q  f lêlq 2

q>f A f ropdn ÿ}huh

 n fq2q>fA f +71<,

rovxq1 Ex wdnwlugh Ee=H lqwhjudo ghqnohpl ;{> f  {   lêlq elu whn êù}ÿph vdklswlu d2eo1 Lvsdw= Whruhplq lvsdw lêlq k{E| n { ] f I Ew> | kwE| gw ' f +7143,

krprjhq ghqnohplqlq vdghfh dvlndu êù}ÿph vdkls rogxjxqx jùvwhuphn |hwhuolglu1 Eù|oh0

fh k{E| Ee=f ghqnohplqlq dvlndu ropd|dq elu êù}ÿpÿgÿu1 Ee=f ghqnohplql k{E|

loh êdusouvd yh | |h jùuh f gdq { h lqwhjudoohqluvh

{ ] f k2_{E| g| n { ] f { ] f I Ew> | kwE| gwg| ' f

hogh hglolu1 Sduvhydo hvlwoljl nxoodqodudn

{ ] f k2_{E| g| ' " [ q'f  2 q>f 3 C { ] f k2_{E| * E|>  g| 4 D 2 hogh hglolu1 { ] f k2 {E| g| n { ] f { ] f I Ew> | kwE| gwg| ' ' " [ q'f  2 q>f 3 C { ] f k2_{E| * E|>  g| 4 D 2 n { ] f { ] f # _" [ q'f fq*fEw>  *fE|>  $ kwE| gwg| 77

{ ] f k2_{E| g|n { ] f { ] f I Ew> | kwE| gwg| ' " [ q'f  2 q>f   n fq2q>f  3 C { ] f k2_{E| * E|>  g| 4 D 2 ' f roxu1 Vrq ghqnohpghq yh Ee=b nrvxoxqgdq

{

]

f

k2_{E| * E|>  g| ' f> q ' f> > 2> 111

hogh hglolu1 i*qE{>  > q  fj vlvwhpl wdp rogxjxqgdq k{E| ' f roxu1 Lvsdw wdpdp0

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