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2002:02

DOCTORAL THESIS

Some Mathematical and Engineering Aspects of the Homogenization Theory

Johan Byström

Doctoral Thesis

Department of Mathematics

2002:02 - ISSN: 1402-1544 - ISRN: LTU-DT--02/02--SE

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lv wkh fkdudfwhulvwlf ixqfwlrq iru wkh vhw l



dqg U wkh xqlw pdwul{1 Ohw wkh xqlw fhoo jhrphwu| eh ghvfulehg e|

l

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Lq rughu wr frpsxwh wkh krprjhql}hg frh!flhqwv +46,/ zh qhhg wr vroyh wkh fhoo sureohp +47,

 _ 

b E% (

1

E% 

' f rq tc +48,

zkhuh 

1

E% ' k1c %l n 

1

E% dqg 

1

E% lv t 0shulrglf1 Lq wkh fdvh 1 ' e

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c zh orrn iru d vroxwlrq ri wkh w|sh



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+49,

vhh Iljxuh 71 Lw lv hdvlo| vhhq wkdw +49, vdwlvhv +48, rq t E| sk|vlfdo uhdvrqv/

wkh vroxwlrq 

1

E% dv zhoo dv wkh x{ b E%

YY?1

pxvw eh frqwlqxrxv ryhu wkh erxqgdulhv l



_ l

2

dqg l

2

_ l

 Wklv jlyhv irxu htxdwlrqv wr vroyh iru wkh

46

(20)

−0.15

−0.1

−0.05 0 0.05 0.1 0.15

0 0.2 0.4 0.6 0.8 1 0

0.5 1

−0.2

−0.15

−0.1

−0.05 0 0.05 0.1 0.15 0.2

y x

Solution we1 of cell problem

we1

Iljxuh 7= Wkh vroxwlrq 

e

ri wkh fhoo sureohp zkhq k

lv fkrvhq dffruglqj wr +4:,1

wkuhh xqnqrzqv 



c 

2

dqg g

2

 Lq rughu wr jhw d frqvlvwhqw vroxwlrq/ zh jhw wkdw k

pxvw eh

k

' k

2





2

 g

2

o

22



' k

2



 n

kk2

n 6





k

k2

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 n

kk2

 6





k

k2

   c +4:,

zkhuh 6



'

oo22

2

c wkh yroxph iudfwlrq ri pdwhuldo  lq pdwhuldo 2 Vlqfh zh qrz nqrz wkh vroxwlrq 

e

E+ ' 

e

E+  ke



c +l ri wkh fhoo sureohp/ zh fdq frpsxwh wkh krprjhql}hg frh!flhqwv

K Ee



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t

b E% Ee



n (

e

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fo

A

dqg vlploduo|

K Ee

2

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t

b E% Ee

2

n (

e

 _% ' df k

o

A



Wklv phdqv wkdw zh fdq sxw wkh frdwhg f|olqghu frqvlvwlqj ri pdwhuldo 4 frdwhg e| pdwhuldo 5 lqwr wkh krprjhqhrxv lvrwurslf pdwhuldo 6 zlwkrxw fkdqjlqj wkh hhfwlyh surshuwlhv/ vhh Iljxuh 71 E| oolqj wkh zkroh fhoo zlwk vxfk krprwkhwlfdoo| frdwhg f|olqghuv/ zh jhw dq lvrwurslf wzr0skdvh frpsrvlwh zlwk frqgxfwlylw| @

c vhh Iljxuh 81 Zh dovr phqwlrq wkh idfw wkdw

47

(21)

Iljxuh 8= Wkh Kdvklq vwuxfwxuh1

li k



 k

2

wkhq k

deryh lv wkh orzhu Kdvklq0Vkwulnpdq erxqg1 E| ohwwlqj pdwhuldo 4 dqg 5 fkdqjh sodfhv dqg vroylqj d vlplodu sureohp dv deryh/ zh jhw wkh xsshu Kdvklq0Vkwulnpdq erxqg1 Zh uhihu wr Vhfwlrq E iru pruh rq wklv wrslf1

D1815 Wkh Pruwrod0Vwhh vwuxfwxuh

Zh ghqh rxu xqlw fhoo t ' Efc 

2

dqg glylgh lw lqwr irxu htxdo sduwv t



'

 fc 

2





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c t

2

'

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'

 fc 

2





 fc 

2



c t

e

'

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 fc 

2





Zh vwxg| d irxu0skdvh frpsrvlwh frqvlvwlqj ri irxu lvrwurslf pdwhuldov/ ohw xv fdoo wkhp pdwhuldo 4/ 5/ 6 dqg 7/ zlwk frqgxfwlylw|

b E% U ' 

k

t

E% n q

t2

E% n 

t

E% n B

te

E%  Uc

zkhuh 

t

lv wkh fkdudfwhulvwlf ixqfwlrq iru wkh vhw t



dqg U wkh xqlw pdwul{/

vhh Iljxuh 91

Lq 4<;8/ Pruwrod dqg Vwhh ^:<` frqmhfwxuhg wkdw wkh krprjhql}hg frq0 gxfwlylw| frh!flhqwv ri wklv vwuxfwxuh duh

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 b



f f b

22



c

48

(22)

Y 1 Y 3

Y 2

Y 4

Iljxuh 9= Xqlw fhoov iru wkh Pruwrod0Vwhh vwuxfwxuh rq wkh ohiw dqg wkh fkhfnhuerdug vwuxfwxuh rq wkh uljkw1

zkhuh

b



'

v kq n kqB n kB n qB k n q n  n B

Ek n  Eq n B

Ek n q E n B c b

22

'

v kq n kqB n kB n qB k n q n  n B

Ek n q E n B

Ek n  Eq n B 

Wklv frqmhfwxuh zdv suryhq wr eh wuxh e| Fudvwhu dqg Reqrvry ^5;` dqg lqghshqghqwo| e| Plowrq ^:9` lq 53331

Li zh ohw B ' k dqg  ' qc zh jhw wkh vr fdoohg fkhfnhuerdug vwuxfwxuh/ vhh Iljxuh 91 Zh lpphgldwho| vhh wkdw wkh krprjhql}hg frqgxfwlylw| frh!flhqwv iru wkh fkhfnhuerdug vwuxfwxuh duh

b



' b

22

' s kqc

wkh jhrphwulf phdq1 Wklv zdv suryhg douhdg| lq 4<:3 e| G|nkqh ^6<`1 D1816 Wkh fdvh zkhq b lv vhsdudeoh

Ohw t ' Efc 

?

dqg frqvlghu wkh fdvh zkhq b E+ lv ri wkh irup b E% ' b



E%



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2

E%

2

    b

?

E%

?

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zkhuh b E%



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f k



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4c iru d1h1 +



5 +c  ' c    c ? +4;,

49

(23)

Zh zdqw wr vroyh wkh fruuhvsrqglqj fhoo sureohp

 _  b E% 

1 n (

1



' f iru + 5 t

E| d vwdqgdug uhvxow lq wkh h{lvwhqfh wkhru| ri sduwldo glhuhqwldo htxdwlrqv/

wkhuh h{lvwv d xqltxh vroxwlrq ri wklv sureohp Khqfh li zh fdq qg rqh vroxwlrq wkhq lw pxvw eh wkh rqo| vroxwlrq1 Ohw xv orrn iru d vhsdudeoh fodvvlf vroxwlrq ri wkh irup



1

E% ' 

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21

E%

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' fc zkhuh 1



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l  E| lqvhuwlqj wkh vroxwlrq 

1

E% lq +46, zh vhh wkdw wkh

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K E1



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Dqrwkhu zd| wr rewdlq wklv uhvxow lv e| orrnlqj dw vrph vshfldo erxqgv iru wkh R0Srlvvrq htxdwlrq wkdw frlqflghv iru wklv fdvh/ vhh Vhfwlrq E141

Rqh lqwhuhvwlqj vshfldo fdvh lv zkhq b E% rqo| ghshqgv rq rqh yduldeoh/

iru lqvwdqfh zkhq b E%

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c %

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wkh kduprqlf dqg dulwkphwlf phdqv/ uhvshfwlyho|1 Lw fdq eh vkrzq wkdw wklv fdvh lv wkh prvw h{wuhph lq wkdw phdqlqj wkdw dq| rwkhu jhrphwulf vwuxfwxuh zloo kdyh hhfwlyh frqgxfwlylwlhv K Ee







wkdw duh erxqghg e| K dqg Dc l1h1

K  K Ee







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Zh uhihu wr Vhfwlrq E iru pruh rq wklv wrslf1

D19 Krprjhql}dwlrq ri ghjhqhudwhg htxdwlrqv

Dvvxph wkdw zh kdyh d pdwhuldo zkhuh wkh khdw frqgxfwlylw| lv doorzhg wr jr wr }hur ru lqqlw| lq d suhvfulehg zd|1 Zloo wkhq wkh hhfwlyh frqgxfwlylw| ri

4:

(24)

wkh pdwhuldo eh }hur/ lqqlw| ru vrphwklqj hovhB Txhvwlrqv olnh wkhvh rffxu zkhq ghdolqj zlwk krprjhql}dwlrq ri ghjhqhudwhg htxdwlrqv1 Lq duwlfoh ^66`

wkh krprjhql}dwlrq sureohp ri ghjhqhudwhg hoolswlf htxdwlrqv zdv wuhdwhg1 Wkh dxwkruv frqvlghu wkh sureohpv

; ?

=

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c (



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5 `

fcR

Elc b

0

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zkhuh @ E+c 1 vdwlvhv vxlwdeoh vwuxfwxuh frqglwlrqv dqg wkh zhljkw b

0

E% ' b E%*0



 lv d srvlwlyh ixqfwlrq ehorqjlqj wr wkh Pxfnhqkrxsw fodvv 

R

lq0 wurgxfhg e| Pxfnhqkrxsw ^;3` lq frqqhfwlrq zlwk wkh vwxg| ri wkh pd{lpdo ixqfwlrq rshudwru ehwzhhq zhljkwhg u

R

0vsdfhv1 Wkh pdlq uhvxow lv wkdw





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f

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lv wkh vroxwlrq ri wkh krprjhql}hg htxdwlrq

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lv ghqhg e|

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E+  _+c zkhuh 

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lv wkh vroxwlrq ri wkh orfdo sureohp rq t

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t

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Wklv uhvxow lv jhqhudol}hg wr uhlwhudwhg krprjhql}dwlrq lq sdshu D 5 ri wklv wkhvlv/ zkhuh zh vwxg| wkh olplw ehkdylru dv wkh qdwxudo qxpehu  $ 4 iru wkh vhtxhqfh ri vroxwlrqv E





M

ri wkh Glulfkohw erxqgdu| ydoxh sureohpv

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=

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E% lv d srvlwlyh zhljkw ehorqjlqj wr wkh Pxfnhqkrxsw fodvv 

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4;

(25)

E Erxqgv iru wkh krprjhql}hg rshudwru

Lq wklv vhfwlrq zh zloo suhvhqw wkh edvlf lghdv dqg uhvxowv lq wkh hog ri

qglqj xsshu dqg orzhu erxqgv rq wkh krprjhql}hg rshudwru1 Wkh surriv ri wkh uhvxowv zh suhvhqw duh froohfwhg lq wkh errn ^87`/ vhh dovr wkh duwlfohv ^54`/

^94`/ ^:8` dqg ^;6`1 Zh zloo frqvlghu wkh olqhdu sureohp/ l1h1/ zkhq @ lv ri wkh irup @



E%c 1 '  

0%

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?

$ + vdwlvhv wkdw f q

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2

41 E| d shulrglf wzr0skdvh frpsrvlwh zh phdq wkdw k rqo| wdnhv wzr ydoxhv/

l1h1/ k fdq eh zulwwhq dv

kE% ' k





l

E% n k

2



l2

E% c k



: k

2

c

zkhuh l



lv wkh shulrglf h{whqvlrq ri i% 5 t G k E% ' k



j dqg 

l

lv wkh fkdudfwhulvwlf ixqfwlrq ri wkh vhw l



1 Pruhryhu/ ohw D



eh wkh yroxph iudfwlrq ri wkh pdwhuldo rffxs|lqj l



1

Wkh Uhxvv0Yrljw erxqgv +ru Zlhqhu erxqgv, wkhq vwdwhv wkdw wkh krprj0 hql}hg pdwul{ K vdwlvhv wkh hvwlpdwhv

KU  K  DUc +53,

zkhuh U lv wkh lghqwlw| pdwul{/ K lv wkh kduprqlf phdq ri k ryhu d fhoo ri shulrglflw| dqg D lv wkh dulwkphwlf phdq ydoxh ri k ryhu d fhoo ri shulrglflw|1 Qrwh wkdw li  dqg  duh wzr pdwulfhv/ wkh qrwdwlrq    phdqv wkdw

   kdv srvlwlyh hljhqydoxhv/ zklfk lpsolhv wkdw +53, fdq eh zulwwhq dv K  b



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zkhuh b



duh wkh hljhqydoxhv ri wkh krprjhql}hg pdwul{ K1 Qrwh wkdw iru wkh vshfldo fdvh ri d wzr0skdvh frpsrvlwh/ wkh Uhxvv0Yrljw erxqgv lpsolhv wkdw



D

k

n

kD22

 b



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

k



n D

2

k

2

 +54, Lq Iljxuh :@ zh kdyh sorwwhg wkh Uhxvv0Yrljw erxqgv iru dq h{dpsoh lq +

2

1 Dq lpsuryhphqw ri wkh Uhxvv0Yrljw erxqgv duh wkh wudfh erxqgv +ru Kdvklq0Vkwulnpdq erxqgv/ vhh ^7<`,/ zklfk vwdwh wkdw

u  b

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n    n b

?

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4<

(26)

The Reuss-Voigt bounds The Hashin-Shtrikman bounds The generalized Hashin-Shtrikman bounds

l

1

l

2

a)

2 3

4

5 2 3 4 5

2 3 4 5

l1

b)

l2

l3

Iljxuh := @ Wkh J0forvxuh lq +

2

1 K Wkh J0forvxuh lq +

1 zkhuh

u ' D 

Ek  D

2



? ?u k n

Ek  D

2



ED  ?u k

c L ' D 

Ek  D

2



? tT k 

Ek  D

2



EtT k  D



Khuh kl ghqrwhv wkh dulwkphwlf phdq1 Lq wkh vshfldo fdvh ri d wzr0skdvh pdwhuldo/ u dqg L duh jlyhq e|

u ' k

2

n D



k3k 2

n

?kD22

c +55,

L ' k



n D

2

k23k 

n

?kD

c

uhvshfwlyho|1 Lq Iljxuh :@ zh kdyh dovr sorwwhg wkh Kdvklq0Vkwulnpdq erxqgv iru dq h{dpsoh lq +

2

1

Uhpdun 8 Wkh Kdvklq0Vkwulnpdq erxqgv duh nqrzq lq sk|vlfv/ phfkdqlfv ru fkhplvwu| xqghu ydulrxv rwkhu qdphv vxfk dv Pd{zhoo0Jduqhww/ Fodxvlxv0 Prvvrwl ru Oruhqw}0Oruhq} erxqgv/ vhh h1j1 ^<8`1

Lw lv srvvleoh wr rewdlq hyhq ehwwhu hvwlpdwhv/ qdpho| wkh vr fdoohg jhq0 hudol}hg Kdvklq0Vkwulnpdq erxqgv=

|h EK  ?u kU



G



knE?3 ?u k

H

?

 ?u k G

knE?3 ?u k

Hc

53

(27)

|h EtT kU  K



G



knE?3 tT k

H

tT k G

knE?3 tT k

H  ?

c

zkhuh |h E vwdqgv iru wkh wudfh ri wkh pdwul{ 1 Lq sduwlfxodu/ iru wkh vshfldo fdvh ri d wzr0skdvh frpsrvlwh zh rewdlq wkdw

|h EK  k

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U

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b



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2

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D



Ek



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2

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2

k

2

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c +56,

|h Ek

2

U  K

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?

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k



 b



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D

2

Ek



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2

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

k

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2



Iljxuh :@ dovr frqwdlqv d sorw ri wkh jhqhudol}hg Kdvklq0Vkwulnpdq erxqgv iru dq h{dpsoh lq +

2

1

Dv vhhq lq Iljxuh : wkh fodvvlfdo dqg jhqhudol}hg Kdvklq0Vkwulnpdq erxqgv frlqflgh lq wkh lvrwurslf fdvh/ l1h1/ zkhq b



' b

2

1 Wkh erxqgv suhvhqwhg deryh krogv wuxh iru hyhq pruh jhqhudo vlwxdwlrqv/ l1h1/ zlwkrxw dq| dvvxps0 wlrq derxw shulrglflw|1 Iru h{dpsoh/ wkh erxqgv duh ydolg iru uhlwhudwhg krprjhql}dwlrq sureohpv/ vhh ^87`1

Wkh uhvhdufk rq qglqj phwkrgv iru rewdlqlqj erxqgv kdv dwwudfwhg wkh dwwhqwlrq ri pdq| dxwkruv dqg lv dq lqwhqvh uhvhdufk duhd1 Ydulrxv jhqhudo0 l}dwlrqv uhodwlyh wkh vlwxdwlrq ghvfulehg deryh kdv ehhq dqg duh vwloo vwxglhg1 Vrph h{dpsohv duh sureohpv zlwk  skdvhv dqg dqlvrwurslf ehkdylru ri wkh frqvwlwxhqwv1 Pruhryhu/ wkh phwkrgv kdv ehhq h{whqghg lq vxfk d zd|

wkdw lw lv srvvleoh wr vwxg| rwkhu w|shv ri htxdwlrqv dv zhoo/ iru h{dpsoh wkh htxdwlrqv ghvfulelqj olqhdu hodvwlflw|1 Dqrwkhu fkdoohqjlqj dqg pxfk ohvv gh0 yhorshg duhd lv wr qg erxqgv fruuhvsrqglqj wr qrqolqhdu sureohpv/ vhh h1j1

^9<`/ ^<5`/ ^<<` dqg ^433`1

Dqdwxudo txhvwlrq wr dvn lv zkhwkhu wkh erxqgv deryh duh rswlpdo ru qrw/

wkdw lv/ zkhwkhu rqh fdq qg d ixqfwlrq k vxfk wkdw wkh erxqgv duh dwwdlqhg1 Dqrwkhu uhodwhg txhvwlrq lv zkhwkhu rqh fdq qg d ixqfwlrq k vxfk wkdw dq|

srvvleoh fkrlfh ri Eb



c    c b

?

 fdq eh rewdlqhg1 Txhvwlrqv olnh wkhvh duh fdoohg J0forvxuh sureohpv1 E| J0forvxuh zh phdq wkh vhw ri doo srvvleoh ydoxhv ri wkh krprjhql}hg pdwul{ K wkdw fruuhvsrqg wr duelwudu| plfurvwuxfwxuhv zlwk

{hg yroxph iudfwlrq ri pdwhuldov1 Lq +

2

c wkh J0forvxuh lv wkh vhw ri srlqwv erxqghg e| +56, zkhuhdv lq +

wkh| pxvw eh frpsohwhg e| wkh Uhxvv0Yrljw erxqgv +54,/ vhh Iljxuh :1

54

(28)

a) b)

Iljxuh ;= @ Dudqn06 odplqdwh1 K Wkh hoolsvrlgdo Kdvklq vwuxfwxuh1 Lw zdv vkrzq e| e| Oxulh dqg Fkhundhy ^54` dqg lqghshqghqwo| Wduwdu

^;6`/ ^437`/ durxqg 4<;8 wkdw J0forvxuh sureohpv duh lqwlpdwho| frqqhfwhg wr vwuxfwxuhv pdgh e| vxffhvvlyh odplqdwlrq +vr fdoohg udqn0p odplqdwhv/ vhh Iljxuh ;, iru wkh olqhdu frqgxfwlylw| sureohp zlwk lvrwurslf skdvhv1 Lq +

2

iru lqvwdqfh/ lw lv srvvleoh wr rewdlq doo srvvleoh sdluv ri hljhqydoxhv rq wkh orzhu dqg xsshu erxqgv lq +56, e| xvlqj udqn05 odplqdwhv1 Wkhvh erxqgv fdq dovr eh rewdlqhg e| wkh hoolsvrlgdo Kdvklq vwuxfwxuh/ frqvlvwlqj ri vsdfh

oolqj frdwhg hoolsvrlgv/ vhh Iljxuh ;1 Pruhryhu/ lw lv srvvleoh wr rewdlq hyhu|

sdlu ri hljhqydoxhv Eb

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c b

2

 lq wkh vhw ri srlqwv erxqghg e| +56, e| d udqn06 odplqdwh1

Wkhuh duh vwloo vhyhudo rshq J0forvxuh sureohpv/ iru h{dpsoh wkh fdvh zkhq zh kdyh  : 2 lvrwurslf skdvhv dqg wkh fdvh zlwk wzr skdvhv zkhuh dw ohdvw rqh ri wkhp lv dqlvrwurslf1 Lq wklv frqqhfwlrq/ zh dovr phqwlrq wkdw uhvxowv iurp wkh wkhru| ri krprjhql}dwlrq dqg rswlpdo erxqgv kdyh suryhq wr eh xvhixo lq wkh hog ri vwuxfwxudo rswlpl}dwlrq/ vhh h1j1 ^6`/ ^<`/ ^4<`/ ^53`/

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References

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