Bivariate spline interpolation with optimal approximation order

(1)

Davydov, Oleg and Nurnberger, G. and Zeilfelder, F. (2001) Bivariate

spline interpolation with optimal approximation order. Constructive

Approximation, 17 (2). pp. 181-208. ,

http://dx.doi.org/10.1007/s003650010034

This version is available at

https://strathprints.strath.ac.uk/36669/

Strathprints

is designed to allow users to access the research output of the University of

Strathclyde. Unless otherwise explicitly stated on the manuscript, Copyright © and Moral Rights

for the papers on this site are retained by the individual authors and/or other copyright owners.

Please check the manuscript for details of any other licences that may have been applied. You

may not engage in further distribution of the material for any profitmaking activities or any

commercial gain. You may freely distribute both the url (

https://strathprints.strath.ac.uk/

) and the

content of this paper for research or private study, educational, or not-for-profit purposes without

prior permission or charge.

Any correspondence concerning this service should be sent to the Strathprints administrator:

[email protected]

The Strathprints institutional repository (

https://strathprints.strath.ac.uk

) is a digital archive of University of Strathclyde research

outputs. It has been developed to disseminate open access research outputs, expose data about those outputs, and enable the

(2)

(

)*+-,' . $/ 021

354768:92;<=9?>@BAC47DFE

G?HJILKNMOHQP+MRH

8

I

<TS4VU MOWYX[Z\MOX

<

MRH

]^`_.abdcfega

hji.kmlTniopkrqstoguwvQxfytozk\st{Qu{g|+}\{Q~pi{Qyvg{Qufogyw{Q~ogstuC ogufyti.km`

l wi uw{gkriNk\wi7}rOogi7{g|ogytyfnOsdozqstozkri{Qyuf{Q~stogyw}rRystuwi }+{g|}\~p{{gk\fuwi }\}L-ogufwi.vgqi.i

sk\q\i }ri kkr{l ¡ifi.gi y{gk\wi£¢Rq}rkV¤Ni.q~skripk[¥ipstu¥kri.q\{Qytozk\s{Qu¦}\wi ~i |§{gq

l¨

£©ª L«¬¨

w{Q}rimozffq\{s~ozk\s{Qui.q\q\{gq?st}Ln{QxfuRwi ozn{giVn®¯ r°²± ¨

wi.qi¯*st}³k\wi~odwst~ogyfstog~pi.kri.q{g|Bk\wikrqstogufvQyi }³su´l¨ oguR)k\fi{Quf}\k\oguJkp® {Qufywi.i ufR}{Quk\fi£}\~ogytyi }rk7ogufvQyi{g|k\wi£krqstogufvQxfytozk\s{QuoguRst}Nstufwi.i uffi uJk{g| uwi ozqrµ¶fi.vgi uwi.qozkrii wvgi }oguf&uwi ozqrµ¶}stuwvQxfytozqmgi.q\k\sti } ·{gq\i.{gi.q ¨`k\wip|§xRuffog~pi u¥k\ogy |¸xfufk\s{Quf}+{g|{Qxwq2}\wi ~i7ozq\i7~sufst~ogytyt-}\xwf{gq\kri poguf|¹{gq~ºo-y{.ogyty³ystuwi ozqy³stuµ wi.i ufwi u¥knRog}\s}+|¹{gqo}\xwi.q}\Rytstuwi}\xwnR}\Rogi{g|j

l »Bfst}}\f{

}Lk\fozk k\wiB{gRk\s¼µ ~ogy¥ozffq\{st~ozk\s{Qu{gqwi.q².ogu£ni2ogfsi.gi £n¥-xR}\stuwv7~stufst~ogyyN}\xff{gq\kri -}rRystuwi }.

½

xwq~pi.k\f{{g|`fq\{{g|²s}{Q~pRyi.kri yfs¼¾i.q\i u¥k?|¹q\{Q~¿k\wi-À¥xRog}\s¼µ¶stu¥kri.q\{Qytozk\s{Qukri wµ ufstÀxwi }2|¹{gq?k\wi}rk\xRwp{g|jk\wiozfRq\{ wst~ozk\s{Qu{

i.q{g|jnRstdozqsozkrim}rOytstuwi }2wi.gi y{gi stu&ÁÃÂ¥¨jÄ ÅzÆÇ

È ÉÊÊQÉËÌ.ÍtÎQÏÐVÌ Í¹ÑÒ\ÓÔ¥ÕJÖ×ÏYÒÍ-ØÙÚÌ.ÓYÓrÑÛ`ÒYÌ Í¹ÑÜ ÝÞ´ßàYáãâVäà[åæ¥Éç2èégê²æJÉç2ëìJí+îzïñð\òóJôQä àYåæJÉñçLõdégêjæ¥Éç7Éñégê ëdé öõégêgëdé öõd÷zø

ùVÏ\úûjÜ ü[ýÓÌ ÝýþQÎzüYÌÓÏrÓÞÿ`áäà[á¼äYï ÚáãóQïêYà[á¼ä ó¼ä[áãòóRê²áãó[ïàò¼ä[áãòóRê`äQàYòzáãâäYáÚòóòàôgïàê âVáãóQáÚâäãåòà[ïñôJä¶áêÚògðäãåãáÚóJïä àãå-áÚó¥ôgïïóJôQïóJä[áø

(3)

! "$#&%('*)+#&,- +%/.0#&1 +23,546 +,0.7%8:9;.0<,0#=> ?@)&A0%B =?C>?D2B +,0)+A0%B =@23#&,E#+FGIH JK0:L@"- =M<:#+F

28N= +?D2B =@C"$#&%8'O,0#+123 +%PL@"0%823,0L#+FQ.R<)&?D<TSU +,R.ELD1#O#+@K0,0<LDLWVYXW28@K?@<LD"ZM >#9[28L.0\-,R<.

'

]

_^

9`badcfe=ghji

^

W`Eakg ljh7m

Fn#&?W +%3%poqh9rGs t!uvVxwySGs

XWK0<?D

m

adczL@"0 +,Te|{-}B~=jak tRs/tRs $uvSOr

23Lb@K0:L@"- +MY#=F 2(N= +?@23 =>"Z#&%('O,0#&12B +%8L#+F/>#+> +%6.0)&?@:S*H

,Y>KR%32(>?> =@A0?@=4"$#&28,_/LD>L>K- = +.R12(AR,023_A0G +)&?> =,0)& +,0.U<?D12(> 23,>?@"$#&%B >23#+, 'xL@"0 +M<L ]

^

9`#+F$L@"R%323,R<L#+F$.0)&?@ST +,0.ULD1#O#+@K0,0<LDLVXG<?DM#&,0L>?@ARM>.Fn#+?QM<?D#&L@LDM<AR "- +?>28@23#&,RL!9423,"- +?D@23MA0%B +?xFn#+?9 ± =,0.y9 D "- +?D@28@23#&,0LE0 4*4 0 4+O4G+R4=R4 4G+=H L@A0%(>L#+, >K07 ="0"0?@#¢¡*231 =@23#&,#&[email protected]<?#+FT@K0<LD23,>?@"$#&%B >23#+,£1@K0#*.RL¤X?@)&2(N&,v23, ¥*4 0 *4 0| 4$ 0 4$+O4$&¦_4$ 4P+§H

,E@K0YM< +L@:#+F +,E 2(>?> =?D'¤@?@23 +,0)&A0%3 =>28#&,¨9¨4-LD> +,0.- =?@.\-,028@ <%31,><M©KR,023_A0LU«ª= "0?@#|NO23.RG b@#*#&%=>#WM<#&,RLD>?DA0MQ?@128@Q'O"Z23,>?@"$#&%B >23#+,IL@M©K01L/Fn#+? ]

_^

9`5XW28@K#+"R>281¤ +% +"0"0?D#¢¡R281¤ >23#+,U#&[email protected]?b¬

^

°²± `®4XWK0<?D

28L>K01¤ ¡R281¤ =%R.023 +1®>?#+F5@K0@?@23 +,0)&%8<L23,9¨H #|XGN&?<4Z +LWL@K0#|XW,E23,«&t=40>K028L1®>K0#O.#&,0%8'X#+?@¯_L2(F/SUv&V

0

H

° ,7>KR#=>K0?K0 +,0.64>KR ="0"0?@#¢¡*231 =@23#&,j"$#|X?Y#=FG@K0!L@"0%823,0ULD"- +M< ]

_^

9`Fn#+?TS± +VWyxX +LCL>A0.028<.²23,³¥R4 4$O4 0 =§H´X +LL@K0#|XW,j28,µ >K- Fn#&? !L@AR¶M23<,@%8'EL@1#*#=>K FnA0,0M®>23#+,±·/4

.023L

^

·Ps ] _^

9`D`u¸

°²± s

^

0 H0 `

XWK0<?DE¸ 28LU M<#&,0L© +,!>K0 =U.0"Z,0.0L#&,0%8'#&,£·4 VO4Sj +,0.¹>K0¨LD1¤ =%3%3LDx +,0)&%8¨º¢»23, 9¨H,²@K0:¢ +?D%328<?M<#+,0LD@?@A0M®>23#+,0L28,µ«R4P=@K0M<#&,0L© +,C¸¼23L

#&A0,0.R<.±AR,0.0<? +.0.R28>28#&,- +% +L@LDA01"R>28#&,µ>K- x>K0¨@?@2B =,0)&A0%3 =>28#&,K0 +LU,0#7,0¢ =? LD23,0)+A0%B +?N&?D>28M<L<Hµ½,0#+>KR<?U"0?D#*#+F#+F

^

0

H

0

`Xb +LT?D<M<,>%('jL@AR)&)&<L>.28,¾

0

=§HT´FSw¿+Vv*4Z>K0,7@K0x#&"*>231 +%/ +"0"R?@#¢¡*231 =>28#&, #&[email protected]<?bFÀ +28%3LFn#&?M<?© +28,@?@23 +,0)&A0%3 =>28#&,0L

^

L@Á+B`®H

JK07"0?@# %3<1 #+FYM<#&,0L>?DA0M@23,0) ¹?@128@23,>?@"$#&%B =@23#&,£#&"$<?> >#&?L@ =>28LDFÂ'O23,R)

^

0 H0 ` Xb =LT +%3?D¢ +.*'

?@#&A0)+K_CA0"23,µ¥§HC#|X®N&?<46>KRx_A0<L>23#+,28F@K0kM#&,0LD> +,:¸ÃFn#&?L@A0M©KÄ +, #&"$<?> >#&?M#&A0%3.

#&AR,0.0<.E23,R.0<"$<,0.R<,>%('¨#+F,0¢ +?

LD23,0)&AR%B +?D28>28<LG?@<1 +28,0<.#&"$<,6H ´¤28L@K0²"RA0?@"$#&L@²#+FC@K023L"- +"$<?!>#.0N+<%8#&" >KRE\-?DLD<?D12(>¨'O"Z²23,>?@"$#&%B >23#+, L@M©K01GFn#&? ]

^

9`®4&Sx+V-¤*4>K- ="Z#&LDL@L@L@L #&"R>281¤ =%_ +"0"0?D#|¡*281¤ =@23#&,#&?D.0<?28,Y@K0bL@,0L@ >K- C>KRkM<#+?@?@L@"$#&,0.028,0)M#&,0LD> +,¸Ã23,

^

0

H

0

`28LC28,0.0<"$<,R.0<,T#+F ,R¢ +?

.0)&<,0?> =@x<.R)&<L +,0.¹,0< +? L@28,0)&A0%3 +?YN+<?D@23M<L<HJK0.0> +23%8Lk#+F#&AR?xM<#&,RLD>?DA0M@23#&, +?D¨)&28N+<,¹28,Å*<M®>23#+,*4 XWK0<?D¢ +LI@K0U1 +23,j?D<LDA0%8T#=F>K0U"- ="Z?<46JK0#&?@1ÆRH

0

(4)

+,- +%('OL@23L#+F*>K0 FnA0,0.0 +1,© +%FnA0,0M®>28#&,0L#+F-#&AR?/LDM©K0<1G@K- =/Fn#&?@1q =L@23LFn#&?/ LDA0"$<?@LD"0%328,0 L@A L@"- +MY#=F

]

^ 9`®H

³I1"RK- +L@2<C@K- =#+A0?b +"R"0?@# =M©K¨23LbM#&1"R%3@<%('!.02Z<?D<,bFn?@#+1 @K- =b#=Fb

6 +,0.Ä

0

=H ,j¢ =M©K#+F>K0L@x"0 +"Z?@LT LD© %3Y%3#OM¢ +% +LD23LFn#&?I ¤LDA0"$<?@LD"0%328,0kLDA LD"- +M<U#=F ]

_^

9`WX +L M<#&,RLD>?DA0M@<.\-?DLD '±A0L@28,0)?@,0L>23, 23<?><M©KR,023_A0L<4 +,0.²>K0,j>K0 +L@28LFnA0,0M®>28#&,0L X?@EA0L@.£@#

A023%8.AR" _A- +LD2

28,><?D"Z#+%B =@23#&,µ#+"Z?> =@#&?x>K- 'O28<%3.R<.¹>K0E#&"R@231 +%G +"

"0?@#¢¡*231 =@23#&,¨#&?D.0<?HG,²M<#&,@?> +L@#@K023L40XY +?D)&A0.023?D<M@%8'XW28>KE,0#O.- +%6FnA0,RM>28#&,- +%8LW +,0. .0N+<%8#&" ,0®X +"R"0?@# =M©K@K- =!K- +LU2(>LU?D#*#+@LU23,µ@K023.0< 7#=F X< +¯¹23,>?@"$#&%B >23#+,±28, >?D#*.0ARM<<.v28,¿¥

0

C +,0. FnAR?D>KR<?.0®N&%3#&"$<.v28,¿¥&¦C +,0.f

0

H-A0?D@K0<?D1#+?@+4bXGj,0<. ,0X .R<L@M?@28"R>28#&, #+FTi L@1#*#=>K0,0L@L¤ +M<?D#&L@L<.0)&L23,¾>?@1L¤#+FI,0#*.0 +%FnA0,0M@23#&,0 +%3L

^

LD< <11¤ ¨RHd&` Hx½CL ?@<LDA0%8<46X1 +,- +)&Y>#®¡*"0%8#&28:%823,0< +?I.0"Z,0.0<,RM<23L XG<,µ.028?@<M >28#&,- +%G.0<?D28N= =>2(N&L#=FCLD1#O#+>KµFnAR,0M@23#&,0LU@#ÄLDA0"0"0?D<LDL>KR¨23,0L©

28%32('³#+F>KRE123,R231 +%3%(' L@A0"R"Z#&?>.jFnA0,0.- +1<,> +%FnA0,RM>28#&,0L23,±@K0x,023)&K #&?@K0#O#O.±#+F ¤,R¢ +? L@28,0)&A0%3 +?WN&<?> ¡54LD< <11¤ ²RHd =,0.³>KR¨.023LDM<A0LDL@23#+,¾ =k@K0<,R.#=FÅ*<M®>23#+,¹0H#+@¤>K0 =x28, 4 0 =G1xA0M©K 1#+?@kM#&1"0%328M¢ =@<.

=L@23LIFnAR,0M@23#&,0LXW28>K%3 +?@)+<?CLDA0"0"$#&?D@LIX?@!M<#+,0LD@?@A0M®><.Ä23,#[email protected]?C># +M©K028N&:L© 28%32('&H

½C1#&,0)#+>KR<? +.RN= +,© =)&<L:#+F#&A0?: +"0"R?@# +M©KÄ23LT@K- =T>KRLD>281¤ >23#+,j#+F>K0 +"0"0?D#¢¡R2 1¤ >23#+,Ä?@?D#&?

^

RHd&`I23LM<#&1"0%8>%8'±%3#OM¢ +%4(42(Y28,N&#&%(N&<LY#&,0%('j>K0%3#OM¢ =% Å*#

#&%3®N7,0#&?D1 #+F·7#&,>KR@?@2B =,0)&%3oq#&,XWK023M©KE>K0<?D?@#&?28LW1< +L@A0?D<.6H²#&?D<#|N&?<4->K0?@L@A0%(?@<1 +28,0L N= +%328.xFn#&? +,'!,0#&,0M#&,N&®¡¤"$#&%8'O)&#&,0 +%0.0#&1 +28,XW28>K!>KRM#&,0LD> +,b¸ 23,0.0"Z,0.0,_#+F$>K0 23"0LDM©K028TM<#&,0L© +,#+F/>K0

#&A0,0.- =?D'¨#+FQIH

½CL ' "R?@#O.0A0M#+F#&A0?WM#&,0LD@?@A0M®>28#&,640XG)+ U,0#*.0 +% +LD23LbFn#&?>KRLD"- +M#=FL@AR"Z? L@"0%823,0L

] z^

9`bac¿e=gYh ] _^

9`akgYhi ^

`Fn#&?W +%8%ZN&?D@23M<L

#+FU9rGs

XWK0<?D! UcV#" r°²±

$

+,R.¨SU+Vµ*HJK0 +L@28LbM<#+,0L@28LD>L#=F>K0CFnAR,0.- +1<,© =%ZFnA0,0M®>28#&,0L

g

±

s&%&%&%s g'#+Fb#+A0?:23,>?@"$#&%B =@23#&,LDM©K0<1+H¨Å*#&1"R?@#&"$<?>23L:#+F@K023L

+LD23L +?DL>A0.028<.23, Å*<M®>28#&,RH +1<%8'+428:28LLDK0#|XW,@K- =!e=g

±

s&%&%&%s g'*r23L)(+*,-.(/(01(2/354-.67(02/398:4<;=4>398:4>3=?C +,0. >K_A0L(+4-A@>?B@>C;D;=*A67?E48¥¦4)FG 4/@K0L@AR"0"Z#+?D>LU#=F>K0 +L@28LkFnA0,0M®>23#+,0LUg

}

+?D¨ +LYL@1 +%3%G +L "Z#+L@L@2

%3=4=XWK028M©KU23L/,0#+>K0bM< +L@GFn#&?@K0

=L@23L/FnA0,0M®>23#+,0LQM#&,0LD@?@A0M®>.28,

4

0

HH²#&?D<#|N&?<4 X:LDK0#|Xq>K- Te=g

±

s%&%&%®s g'*rY23LB@>?-DI(+4J2K9L8:*4M@N35*A?O,M*A3=?-.2/3P354-.67QR8:4S:4>354>6T-.?E448S:4M@®H ³,0#+>>K- =>KRIL@A0"R"Z#&?b#+F< +M©KFnA0,0.0 +1,© +%PFnA0,0M®>23#+,¨g

}

K- =L =G1#&Lb#&,RN&?D@®¡ #+F5@K0W>?@23 +,0)&AR%B =@23#&,U28,2(>L 23,>?@28#&?<4LD23123%B =?Q>#:>K0123,0281¤ +%8%8':LDA0"0"$#&?D@<.

=L@23L FnA0,0M®>28#&,0L Fn#&? 2(N+ =?@2B >LD"0%328,0<LU#+FC.0<)+?@<S +V¿M<#+,0LD@?@A0M®><.£23,z«R4 0 ¦O4 0 =§HVU +?D@23M<AR%B +?D%8'&4 >K0YL@A0"R"Z#&?>L#+Fg

±

s&%&%%<s g'M<#&28,0M<28.0:XW28>KE>K0YL@A0"R"Z#&?>LI#=F>K0 +LD23LL@"0%823,0LFn#&?

]

z^

9` )&28N+<,¹28,

0

=§H7#|X®N&<?4G@K0<?D +?@231"Z#&?© +,k.02Z<?D<,0M<L

XG<<,#&A0?

(5)

0 0

0 ¦O4 0 =/M¢ =,0,0#+ .028?@<M®>%('>?@ +,0LDFn#+?@1<.±28,># ¤?@128@28,><?D"Z#+%B =@23#&,¨LDM©K0<1+4Z28W.0#O<L ,0#+TLD<1Æ"Z#&LDL@2 %3k@#¨<1"0%3#|'²>K0#&LD +LD<L:23,7 ¨"0?D#*#+F#+F#&"*>231 +% +"0"0?D#¢¡R281¤ >23#+,j#&[email protected]<? 23,E>K0x"R?@<LD<,0MU#+F ,0< +?

L@23,R)&A0%B =?@28@23L<HWJK0x1 +23,².02Z<?D<,0M

XG<<,7#&AR?I23,>?@"$#&%B >23#+, #&"$<?> >#&? =,0.U>K0W#&,0"R?@<LD<,>.23,E«=-28L>K- XW.0#:,0#+ ¢N+#&23.!A0LD23,0):,0#&?D1¤ =%R.0?@28N= =@28N+<L >#T>KR.0)&<LG +L ,0#O.- +%0FnA0,RM>28#&,- +%8L<HJK0<?DFn#&?D+4_X +L@LD23)&,U@K0,R#*.- =%0FnA0,0M@23#&,0 +%3LQ@#Y®N&?D' N&?D> ¡Z4$.0)& +,0.>?D2B +,R)&%3I28,E !1#&?DTLD'O11@?@28MIFÀ +L@K028#&,6H

¼E<

2(N&,v ³?@)&A0%3 +?@?@2B =,0)&A0%3 =>28#&,94XGj.0,0#+> ' @K0j,_A01 <?#=FT>?@23 +,0)&%8<L4 ' >K0k,OAR1

?I#+F N&?D@23M<L<4

' +,0.µ>K0k,OAR1

?T#+F 23,@<?@28#&? +,0.

#&A0,0.- =?D'±N+<?>23M<L ?@L@"$<M@28N+<%8'+46±c4 '>K0C,OAR1 ?#+F.0)&L<40 =,0. 'W +,0.j>KRI,_A01 <? #+F23,>?@23#+? +,0. #+A0,0.- +?'¨<.R)&<LW?@L@"$<M®>28N+<%('&4QcH ´W23LXG<%3%5¯O,R#XW,E>K-

c !s

c Q! µWs

c "Q! ¹ %

^

*H0 `

,µ

0

=2(Xb =LLDK0#|XW,²@K- =

.0281 ] z^

9` c # °

%$

&#'# )(+* ,(±

$ ¾-#

,( °²± $'$

0

^

VW 0 `

^

=S. ¹. I¾V/ &`)

#

,( °²± $0

s

^

*Hd&`

XW28@K

0

23,0)>K0,_A01

<?#+FL@28,0)&A0%3 +?IN&?D@23M<Lx#+Fb94/XWK0<?D¨ @>2/3:S.C:(-.621.4>67?E4)3

23L N&?D> ¡XWK028M©K23LFn#&?@1<. 'XG#U%323,R<L XWK023M©K¨M?@#&LDL =

H´23LG¢ +L'@#xL@C>K- YN+<?D@®¡

23LL@23,R)&A0%B =?2(FP +,0.x#+,0%8'28F- =%8¢ +LQ>KR?@<.0)&L [email protected]<)+<,0?> =@W =

4+XWK0?@@K0b.0)&<,R<?> =M' #+FQ +,<.0)&23L.R\-,0.j +LbFn#+%3%3#|XWLH

46587:9<;>=';>?9A@CBEDÅ*A0"0"$#&LDF

±

48F

*

+?@b@K0?@M<#&,0LD<MAR>2(N&W<.0)+<L =D© +M©K0.@#: CN&?D> ¡

H

JK0:<.0)+GF

28LbL> +28.># 8:4S:4>354>6-.?<4:

XWK0,0N+<?W>K0T.0)&<LF

±

+,0.HF

*

+?@TM#&%3%823,0< +?<H ½C,7<.R)&IF =D© +M©K0.Ä@#

23LTL@ +23.@# 135*A398:4S:4>354>6T-.?E4U =

28F2(T23LT28>KR<?: #&AR,0.- +?' <.0)+#+?W +,²28,><?D23#&?<.0)&TXWK028M©KEFÀ =23%3LG>#

:.0)&<,R<?> >+H

,²>KR:\0,028@%31,_1@K0#*."028<MXW23LDY"$#&%('*,R#&123 +%Z@?@23 +%ZFnA0,RM>28#&,0L +?DYA0LDA- +%8%8'.R

>?@123,0. '>K023?N= +%3AR<L +,0.±.R<?@2(N= =>2(N&<L =L@#&1"Z#&28,>L<4PLD# M< +%3%8<.,0#O.- +%N= +%3A0L

^

(6)

0 0 0 0 0 L@"0%823,0ULD"- +M<L ] ±

*^

9` 4S± ¦*4/ +,0.³LDA0"$<?@LD"0%328,0<L ] ^

9`IXW28@K =V¨ +,0.Sj Y 0 4 ?@L@"$<M@28N+<%8'+H

³:L@®

i

^

9`bac¿e=·7hji

^

W`ax· l hi

^

oC`GFn#&?W +%8%Zo¿h9rGs ±cztRs 0 s&%%&%5s

+,0..0<,R#+>

' @K0.0<?D28N= =>2(N&#&"$<?@ =>#+?23,U>K0.028?@<M®>28#&,U#+F TA0,R28 N&M>#+?¤c

^

_s<` 23,¨>KR"R%B +,0=40L@#!>K0 =

|·adc ·! <·ks ·adc

·

{

s¢·jac

·

~

%

46587:9<;>=';>?9A@CB@

2(N&,±·hi °

^

9`®4Gs³tR4- =,_',OAR1 ?

·Ec!

#"

%$

^

· l ` ^'&

`5s

^

*H`

XWK0<?Do[hv9¨4

&

h£o4 +,0.(

±

s

+?@!L@#+1!A0,02(TN&M>#+?@Lk28,Ä@K0!"0%B =,0+4/23LTL@ +23.7># 35*8D-.( 1A-.(C 4T#+F·4Q +,0.³@K0¤%823,0< +?TFnA0,0M@23#&,0 +% a/i

°

^

9`*) [.0®\-,0.

'

^

*Hd&`:28Lk 35*8D-.(.K7C:35,>? 2*A39-.(4RXW2(>K,+

^

`badc-

<23,R)!@K0 8:4S.6 4M4:#+F H

P#+?bL@#&1ILD"ZM<2B =%5M©K0#&28M<L#+F

&

s

±

s

2(b23LM<#&,N&,023,@#UA0L@T@K0CFn#&%8%3#|XW23,0)YL@281"0%828\-. ,0#+> =>28#&,EXWKR23M©K)&#O<L

+M©¯>#7

0

ªH

0

`´F

23LC N+<?D@®¡j#+F9 +,0. Fx28LI +,j.0)&k =@> +M©K0<. >#

4RXGYLD

.

·

^ `adc!

^

· l ` ^

`5s µ 0 s

XWK0<?D/23L@K0TA0,02(bN&M@#&?23,@K0T.023?D<M®>23#+,#+F FY ¢X ¢'Fn?@#&1

40 +,R.o h9 28Lb#&,0T#=F>K0 >?D2B +,0)+%3<LXW28@KÄ.0)&%F&HxJK0!,0#=© =@23#&,28LIM<#&?D?@ML@28,0M<U28,7@K0UM¢ =L@UXWK0<,7>KR<?@ +?DxX# .02Z<?D<,W>?D2B +,0)+%3<LGo

±

sDo

=@> +M©K0.E@#IF&4$· l

"

+,0.±· l

$

M<#&28,0M<28.0: +%3#+,0)2F&4P +,0.K0,0M<

^

· l

"

`

^ `c!

^

· l

$

`

^ `5%

&`´F

23LW xN&?D@®¡²#+FQ9[ +,0.!F

±

s F

+?@IXG#M<#&,0LD<MAR>2(N&k.0)&LC =D© +M©KR<.>#

4RXL@®

.

"

.

$

·

^ `adc!

0"

%$

^

· l ` ^

`5s Gs1 0 s

XWK0<?DYofh9 28L>K0>?@23 +,0)&%8IXW28@K²N&?D@®¡

+,0.±)&.0L/F

±

s F

4Z +,0.,

}

23LW>K0A0,02(N+<M®>#&? 23,>K0.F

}

.028?@<M®>28#&,¤ ¢X ¢'Fn?D#&1

H` -#&?N+<?'<.R)&/FI#+F5>K0>?@23 +,0)&AR%B =@23#&,!9XGIM©K0#O#&L@ UA0,028N+<M®>#&?243

^

#&,RT#+FQXG#"$#&LDL@2 %3|`#+?D>KR#&)&#&,- =%Z># F +,0.EL@

.65

·

^'&

`bac!

5

·

^'& `5s

&

h F s µ

0

(7)

^'&

³:,0#|X¿ =L@L@#OM<23 =>TXW2(>K¨>K0:LDA0"Z?@LD"0%328,0:L@"- +M ] ^

9`®40XW28@KSx+VW£! =,0.

UczVW V

0

s

^

*H¥_`

UL@ #=FQ,0#O.- +%6FnAR,0M@23#&,- =%3L<4R +LFn#&%3%8#|XWL<H P#+?N&?D'¨N+<?D@®¡

#+F9¨4-%8o ±

s&%&%%<s@o '

: +%3%5@?@23 +,0)&%8<L =D© +M©KR<.>#

+,R.E,_A01

?@.¤M<#+A0,><?DM<%8#*M©¯XW23LD

^

LD© =?D>28,0):Fn?@#&1

#&A0,0.- =?D'U>?D2B +,R)&%32(F

28LG

#&A0,0.- =?D'UN&?D@®¡0`®H I<,0#=>

' F

}

>K0M<#&11#+,.0)&¤#=Fbo }(±

+,0.Äo

}

4QIc *s&%&%%<s ^

`®H´F

28Lk +,23,>?@28#&?

N&?D> ¡Z4 F

±

c F ' °²±

.R<,0#+@¨>K0¨M#&11#&,¹.0)&¨#+Fo

±

=,0.¹o '

H ° >KR<?DXW28L@=4 F

±

+,0.

F ' ¹°²±

+?@T>KR #&A0,R.- +?D'<.0)&L

^

=@> +M©K0.E@#

`#+F/o ±

=,0.Eo '

?D<L@"$<M®>2(N&<%('&H ³E.0\0,0

^ `k>#

>K0²L@®¤#+FI,0#*.0 +%bFnA0,0M®>23#+,- +%3L +LDL@28)&,023,R)Ä@#³N+<?'FnAR,0M@23#&, ·jhi

^

`±i

^

9`>K0TFn#&%8%3#|XW23,R)x,0#O.- +%5N= +%8A0<La

^

N 0 ` · ^

`GFn#&?W +%3%

^

s1`h

±

40XWKR<?@

±

adcfe

^

s1`Gh

a*µtRs tRs µu 0rGs

^

NR&` .

.

"

·

^

`GFn#+?W +%3%

^

GsQ`h

4RXWK0?@

acfe

^

Gs1`Gh

a*µuVs µuvVs µ C 0 rs

+,0.¨Fn#&?< +M©K± h³e 0 s&%&%&%®s ^

` rYL@A0M©KE>K- =F

}

28L,R#&,0.0)&<,0?> =@ =

4

^

N*` .

.

"

·

^

`GFn#+?W +%3%

^

GsQ`h

*

4RXWK0?@

*

adcfe

^

Gs1`Gh a µvVW

0

sj ³u+V

0

s T

0

rs

+,0.¨Fn#&?< +M©K± h³e 0 s&%&%&%®s ^

` rYL@A0M©KE>K- =F

}

28L.R<)&,0<?@ => =

4

^

NO_`

.

"

.

$

·

^

` +,0. .

"

.!"

·

^

` Fn#&? +%3%

^

s1`h

*

28F

28L

#+A0,0.- +?'N+<?>®¡Z4

+,0.

^

NR¦&`

.

"

.

$

·

^

` Fn#&?W +%3%

^

GsQ`h#

28F

(8)

A

1

2

3 (r,r)

ρ

β

α

; B @-BEDBJK0:L@®>L

±

4

=,0.

*

H

°

,EN+<?'<.0)& F#+F9¨40XW2(>K¨N&?D>28M<L

±

=,0.

40XG> +¯+:"$#&23,@L

&

}

.

ac

±

} °²±

^

±

`5s c 0 s%&%&%<s

s jctRs&%&%&%s>Vs

^

*Hd¦&`

XWKR<?@

adcS/ ¾=V. 0

^

V `R1#O.W:czS/ ¹=V/ 0 " \°²± (

$

s

^

*H`

+,0..0®\-,0

^

F`W@#

k@K0UL@®:#+F,0#O.- +%FnA0,0M@23#&,0 +%3LT =L@L@28)&,028,0)@#¨N&?D'jFnAR,0M@23#&,Ä·¾h i

^

W`>KR:Fn#+%3%3#|XW28,0)k,0#O.- +%6N= +%8A0<La

^

|`

.65

·

^'&

±

.

` s&%&%&%®s

.65

·

^'&

.

`GFn#&?W =%3% ²c tRs%&%&%®s>V*H

,µN+<?D'Ä>?D2B +,R)&%3Uo h9¨4QXW28@KN+<?D@23M<L

±

4

+,R.

*

4/X© +¯=¤A0,R28Fn#&?D1%('L@"- =M<<. "Z#+23,>L

&

}

l

adc

^

±

Ä

*

`Sbs $I cSs

^

*H `

+,0.7.0®\-,0

^

oC`># x>K0ULD#+F,0#O.- +%QFnAR,0M@23#&,- =%3LT +LDL@28)&,023,R)¤>#N+<?'FnA0,0M@23#&,Ä·h

i

^

W`>K0TFn#&%8%3#|XW23,0)Y,0#O.- +%6N= +%3AR<L<a

^

`E· ^'&

}

l

`Fn#&?W +%8%5@s´&s LDA0M©K²@K- =$C cS! +,R.EVUw£@s&s wyS. ¾=VOH

³:L@®

adc

^ `

.

^

F`

l

^

(9)

M¢ =?@. c.R231 ] ^

9`5%

^

*H`

'?C? B´28L¢ +L'>#L@T>K- =

M< +?@.

±

c # °

$

s M< +?@.

czM< +?@.

*

c # )( ñ°²±

$

s

M¢ =?@.

^

F`c

^

V

0

`

^

S. ¾=V/

0

`: " \°²± $

s M¢ =?@.

^

oC`c # )(+* ,(±

$

%

^

*Hª`

JKR<?@®Fn#&?@=4

M¢ +?D. c # ñ°

$

v # # )(+* )(±

$ £ #

,( °²± $ $

# )( ñ°²±

$

^@^ VW

0

`

^

S ¾+V/

0

`¾V/ O` # )( ñ°²±

$0

%

JK0:%311 x,0#|X Fn#&%8%3#|XWLFn?D#&1

^

*H0 `®4

^

OH&`b =,0.E LD231"0%3IM#&1"0AR© >23#+,6H

T T³% Y<

58?!'5 CB D#" 4>? VE 0%$

SE=VI -.398) Ec V " \°²± $'&)(

21.4>37·h i

^

W`

$

?*=4>6M4 4)3 2+@>?@B- C:3=2,+C=4 @<;9(2/354g.-h ]

^

9` @-.?2+@ K70A2/3:S ?/=4 K&*A(/(+*10H2/3:S32J4>654 2/?E4 ?07;=4 2/3=?<4>6;=*A(-.? 2*A3 ,M*A398.2/? 2*A3 @

g6-Cc · K&*A6J-.(/( hÃs

^

RH

0

`

0=4>6 4 2+@ 8:4/7 3548 -DI>* 1.4 &98

*A6 4M* 1.4>6 $ 2K·jhi3:

^

W`;=< h³e==Vs&%%&%<s>S 0 r?> -.398o¿hj9 $ ?/=4>3

@

^

·6 g6-` @5ACB

l

uz¸

: ( (

l

1¤ ¡ DFE

E

:

@

G: (

·

@IH

l

s

^

RHd&`

K&*A6 -.(/( s1v t $ ± uJ< $ 0=4>6M4

l

2+@)?/=4 8.2-'4 4>?E4>6 *EKIo $ -.398¸ 2+@ -,M*A3 @>?-.3=? 0:2,K8:4<;=4>398A@ *A3=(0 *A3¨Vs>S-.398 ?/=4B@I4 -.(/(+4M@>? -.3:S.(+4º¢» 2/3¨9

&

³XW28%3%="0?@#|N& JK0#&?@1¿RH

0

=>KR<,0.#+F_>K028LL@M>28#&,64 =FÂ@<?<L©

%323LDK023,R)bL@®N&<?@ +%_%311 +L<H ,j>K0k\0?@LDTXG#¨%311 +LWXUM#&,0LD23.0?T L@281"0%8@?@23 +,0)&A0%3 =>28#&,EM<#&,0LD23L>23,R)#+FXG#>?@2

+,0)&%8<L =,0.<LD> %823L@K¨LD#&1T?@<%3 =>28#&,0L ®X<,E,0#*.0 +%5N= +%3AR<Lb#+F/XG#"$#&%('*,R#&123 +%3L .0®\-,0. #&,7¢ +M©KÄ>?D2B +,0)+%3U +,R.MLD#+23,0.7@#&)&®>K0?:XW2(>KÄi

LD1#O#+@K0,0<LDL +M<?D#&L@LY ¨M<#&11#+,<.0)&!#+F >K0T@?@2B =,0)&%3L<H

®o

±

+,R.o

IXG#U>?D2B +,0)+%3<LGL@K- =?@23,R)! !M#&11#&,¤<.0)&GFCc

±

s

40 +,R.E%8F }ON

cF

>KR²#+@K0<?<.0)+²#+FCo

}

XW28@K <,R.0"Z#+23,

±

4bc

0

s *H I,0#+@

'64

±

4

(10)

± ±

}

+,0)&%8 XG<<, ± =,0.,

}

4-c 0 s *H

^

Å*<Q23)RH0RH0 Hd`

v

₂

T

₂

T

₁

v

1 θ

e

₁

2 θ

e

₂

e

1

; B CBEDB

PAR?D>KR<?@1#&?D+40%8g

: "R23<MXW28L@:"Z#+%8'O,0#&12B +%PFnA0,0M®>28#&,#&,¨o

±

Eo

LDA0M©K²@K- =

g l

c

}

hm

s c 0 s %

° A0?\-?DLD%8<11¤ WM©K- +?@ +M@<?D2+<Li

L@1#O#+>K0,R<L@LQ#+FZg +M<?D#&L@L F28,Y>?@1LQ#+FP2(>L,0#*.0 +%ON= +%8A0<LH

^

Å*11 !U23,Ä«=Fn#&?W UL@28128%B +?GM©K- +?> =M>?@2¢ =@23#&,¨A0LD23,0) =?D'OM<,_@?@28MYM#*#&?D.023,0 =>L<Hd`

5CB@ " 4>?VkuS

&

>KWº

±

£º N

c

$

?*=4>3gYhi

^

o

±

o

`J2KN-.398 *A3=(0 2K

L@28,

º

±

$ (

^

±

`c

D

^

0

` #

$ LD23,

(

^

º

±

£º

`RLD23, º

0"

(

±

^

±

`®s

^

RH`

K&*A6J-.(/(7c tRs&%%&%[email protected]%&%&%<s@S

&

>KWº

±

£º

c $ ?*=4>3gYhi

^

o

±

o

`J2KN-.398 *A3=(0 2K

%$

(

^

±

`c

^

0

`# #"

(

±

^

±

` s 7c tRs%&%&%®s>Vs Ec!s&%&%&%®s>S %

^

RH¥_`

'?C? BNO28.0<,>%('&4Pghi

^

o

±

o

`28F +,R.E#+,0%8'28FFn#&?bLD#&1TA0,028N&<M®>#&? Z,R#&,0M<#+%3%328,0¢ =? XW28@K 54

^%&

`c-

±

^%&

` s Fn#&?W =%3% s17tR4¨uvVO4- +,0.² +%8%

&

(11)

^

}

.

0

<_A02(N+ =%3<,@#!@K0:M<#&,R.028@23#&,

(

^

±

`

(

±

^

±

` s 7czt*s&%&%&%s>Vs c-Gs%&%&%<s@S %

³C,0#|X M©K0#O#&LDRc

H´Fº

±

¹º

c 4O>K0,

±

c

4- +,R.X:2311<.023 =>%8'U)&®

^

*H_` H ° >K0?DXW23LD+4-2(FQº

±

£º

N

c 4R@K0<,>KR:N+<M®>#&?DL 54

±

+,0.

LD> |'23,¨>K0:?D<%B >23#+,

WL@23,

^

º

±

£º

`c

±

L@23,Wº

±

s

XWK023M©KE231"0%823<L

L@28,

º

±

%$ (

±

^

±

` c

D

^

0

` #

$ L@28,

(

^

º

±

£º

`RL@28, º

0"

(

±

^

±

` s

+,0.¨>KR:\0?@LDLD> =>1,#=F@K0:%3<11 YFn#&%3%8#|XWL<H

JKOARL<4@K0j,0#O.- +%N= +%3A0L¤#+Fkgyh i

^

o

±

¾o

`LD> ¢'y23, ?@<%3 =>28#&,0L

^

RH` H¿JK0L> +1 ?@%B =@23#&,0L K0#+%3.!Fn#&?G®N&?D'L@AR¶M<28<,>%('L@1#*#+@KFnA0,RM>28#&,¨·/HHG'L@#&%(NO23,0)Y Y%823,0< +? LD'OLD@<1 X M¢ +,¨LD@231 =>ILD#&1I#+F>KRT,0#*.0 +%ZN= +%8A0<Lb#=F·2 µg: =

±

23,N+#&%8N+<.23,

^

RH`G23,@<?D1Lb#=F>K0 #+>KR<?@LH

5CB C;D;=*@4B?* -.?g

$

-A@N8:4/7 3548 -DI>* 1.4

$

2+@ 2/3i

^

o

±

ko

`

$

-.398 (+4>?6·7hi

^

o

±

o

`!K&*A6 @*14 4 h eG U 0 s%&%&%<s©=VOr

&

/KUº

±

qº N

c $ ?/=4>3 K&*A6 4 1.4>6>0 cÆ H +V 0 s&%&%%s ¹V/ 0%$

0"

(

^

·

±

`

^

±

`

u ¸ 1 ¡

DFE

E

(+* ,(R

0" (

^

·

±

`

^

±

`

L@28, (f

^

º

±

º

`

1¤ ¡

"$

(

^

·

}

`

^

±

`

s

^

RHd¦&`

0=4>6 4¸ 8:4<;=4>398A@ *A3=(0*A3¨V-.398xº

&

'?C? BÅ*23,0MY·hji ^

±

`®40XGYK0 |N+

L@28,

º

±

%$

(

·

^

±

`c

D

^

0 `

#

'$ LD23,

(

^

º

±

£º

`RLD23,

º

#"

(

·

^

±

`

(12)

^

c

D

^

0 ` #

$ ±<

s 7cq ¾=V. 0 s&%%&%s@Vs

^

RH`

XWK0<?D

±<

adcLD23, (

^

º

±

£º

`RLD23, º

#"

(

^

·

±

`

^

±

`5s

adcLD23,

(

^

º

±

£º

`RLD23,

º

±

$

(

^

·

`

^

±

`5%

#&,0LD23.0?

^

RH`b +L ULD'OLD@<1

{¨c

#+F&V. :£U%823,0< +?bOA0 =>28#&,0LW23,&V/ ¹ : xA0,0¯_,0#|XW,0L

}

s cq ¾=V. 0 s&%%&%s ¹V. 0 s c 0 s %

JK_A0L<4-XGYK- ¢N+

{c

^

±< (+* ,(± s&%%&%<s

±< (f)(±

s

(+* ,(± s&%&%&%s

(f,(± ` s

C23LW

^

&V/ ¾ :£&`

N&<M®>#&?XWK0#+L@M<#&1"Z#+,0<,>L +?D:L@#&1T%328,0¢ +?M<#&1

23,- =@23#&,RLb#+F

±<

s 7czt*s&%&%&%s ¾+V/ ¹*s +,0.

}

s c ¹Vs&%%&%<s>Vs c

0

s Os

+,0.

zc

i

s

XWK0<?D

ic

^

0 ` \°

+#

'

° }

'

( : °

$

:

}

²±

s XW2(>K 7adc ¹V. 0 s < adc =V. 0 s

!23LG +,M< <;28.0<,@28'1¤ >?@2¡Z423L +,M<M< <?@#U1 =>?D2(¡Z4O +,0.

28Lb kM<?© +28,M<M< 1¤ >?@2¡ZH Å*23,RM<I>KR.R>?@123,- =,_#+FQi 28LW !,R#&,<?@#M<#&,0L© +,1xA0%(>28"0%3#+F

.0 ±

: °

}

(

:

} ²±

N

(13)

@

{

@

u

@

(± @ @

@

s

XWK0<?D

@

(± @

28L #+A0,0.0. 'Ä EM#&,0L© +,.0<"$<,R.0<,x#+,0%8'#+,VOH UQ +?>23MA0%B =?@%8'+46Fn#&?: +%8% cq µ+V/

0

s%&%&%<s ¹V

0

4

±<

u¸

±

1¤ ¡ DFE

E

(+* )(R

±<

1¤ ¡ (f

E

±<

1¤ ¡ (f

E

s

XWK0<?Dk¸

±

.R<"$<,0.0L#&,0%('#&,VOH M¢ +%8%323,R)k>K0:.0®\-,02(>23#+,E#=F

} 40c

0

s O4RX:#

0"

(

^

·

±

`

^

±

`

c

±< L@23,

^

º

±

º

`RL@23, ( º

u¸

±

1 ¡ DFE

E

(+* ,(R

L@28, (

^

º

±

º

`RL@28, ( º

0" (

^

·

±

`

^

±

`

1¤ ¡ (f

E

LD23, (

^

º

±

º

`RL@23, ( º

0"

(

^

·

±

`

^

±

`

1¤ ¡ (f

E

L@28, (

^

º

±

º

`RL@28,

º

±

L@28, (

º

$

(

^

·I

`

^

±

`

s

+,0.

^

*H¦&`GFn#+%3%3#|XWLH

³ +%8L@#!,0<.²@K0TFn#&%3%8#XW28,0)xA0,R28N= +?@23 => X< +¯¨23,@<?@"$#&%3 =>28#&,k%8<11¤

^

M<#&1"- +?D 0 4 1¤ =?@¯¨¦232/ +,0.

0

*4-<11 !Â` H

5CBJ" 4>? Fq

I>4)-.3 2/3=?E4>6 1A-.( 0H2/?/ 4>398>;=*A2/3=?/@

±

s

$

-.398 (+4>? ¾h£etRs%&%&%<s@VOr -.398 <ÆheV "

\°²± ( $

s&%%&%s@S

0

Qr &

=4>3 K&*A6 -.3=0x·h³i :

^

F`

$

-.3=0

h³m )(

-.398 4 1.4>670¨c tRs&%&%%s < $

@

^

·

`

@

H

. u ¸

(

:

@

:

·

@

H

.

1 ¡

±

E

}

E

^

·I

`

^%&

}

.

`

1¤ ¡

"

$

"$

^

·I

`

^

}

`

s

^

RH

`

0=4>6 4

2+@B?*=4 (+4>3:S.?**EK F

$

8:4>35*A?<4M@ ?/=4 C 3=2/?:1.4M,>?E*A6 2/3?*=4N8.2/6 4M,>?2*A3P*EK F

$

&

}

.

-.398

-.6 4 8:4/7 35482/3 ; &

>-.398 ; &

> $ 6 4M@<;=4M,>?2 1.4>(0 $ -.398U¸ 2+@ - ,M*A3 @>?-.3=? 0:2,KP8:4<;=4>398A@ *A3=(0 *A3S

&

(14)

±

^

^ JK0<, c

^

0 s©t`®4 Cc&4

&

}

.

c

^

} °²±

s>t`®40c 0 s&%&%&%®s

H

Å*23,0MY·hji : ¥t*s

§4RXGK- ¢N&

@

^

·

`

@IH

D

u

: (

@

:

·

@IH

D

s Eczt*s&%&%&%s < s

^

RH`

XWK0<?D

28Lb>K0

^

AR,028N= +?D2B =@|`GJ ¢'*%8#&?"$#&%8'O,0#&12B =%§4

^

{Z`badc : (±

D

-

D

{ %

JK0<?DFn#&?D+4

@

^

·

`

@

H

D

u

@

^

·I

`

@

H

D

@

^

`

@

H

D

u

: (

@

:

·

@ H

D

@

^

`

@ H

D

s

+,0.¨XGY#&,R%8',R<<.E>#!<L>281¤ =@

@

^

`

@ H

D

H

®

adcVW " \°²± (

$

%

Å*23,RM<

^

0

`cS/

0

40@K0TFn#&%3%8#|XW23,0)k<?D12(>T28,_@<?D"Z#&%3 =>28#&,"0?D#

%8<1

^%&

}

.

`c

}

s Qc 0 s&%&%%<s

s

^

`c

s 7c tRs&%&%%s

sxUc 0 s *s

K- +L A0,023_A0!LD#&%3AR@23#&,

+1#&,0)A0,02(N+ =?@2B >x"$#&%8'O,0#+123 +%3L#+F.R<)&?D< =1#&L:S 46Fn#&? +,'¨)&2(N&<,E.- =>

}

4-c

0

s&%%&%<s

4P +,0.

447cztRs&%&%&%s

4&Uc

0

s OHJK0<,

^

`

^ `c

}

²±

^

`

^%& }

.

`

}

^ `

²±<

D

^

`

^

`

^

` s

h¾«tRs

§s

XWK0<?D

}

4-Qc

0

s&%%&%<s

4P +,0.

4c tRs&%&%%<s

4_!c

0

s *40.R<,0#+@@K0:FnA0,0.- +1<,> +% "Z#+%8'O,0#&12B +%8LQ#+FZ>K0

#|N&23,>?@"$#&%B =@23#&,k"0?@#

%3<1E4(4>K0®' +?@WA0,R28N= +?@23 =>"$#&%('*,R#&123 +%3L #+F.0<)&?D< =1#&LDWS/ 40A0,028_A0<%('¨.0@<?D128,0<. '>K0:M#&,0.02(>23#+,0L

}

^

°²±

`c }

s @skc

0

s&%%&%s

s

}

^

t`c-

} ^§

` c tRs 7c tRs&%%&%s

s Qc

0

s&%&%%<s

s

(15)

}

^

°²±

`c tRs xc 0 s&%&%&%s

s 7c tRs&%%&%s

s¤c 0 s *s

±<

^

`c

s s ctRs&%&%%<s

s

±<

^§ `c!

^

t`cztRs Gs c tRs%&%&%®s

s

?@L@"$<M@28N+<%8'+H G' !A0,023_A0,0<LDLC =?@)&A01<,<4-2828L¢ +L'¨>#!M©K0<M©¯¨@K- =

} ^

` c

(

}±

^

`®s

h¾¥t*s

§s

^

` c

(

±

^

`®s

h¾¥t*s

§s

+,0.64-M#&,0LD<_A0<,@%8'&4

@

} @IH

D

c

(

@

}± @IH

D

±

s

@

H

D

c

(

@

±

@

H

D

±

%

JK0<?DFn#&?D+4-X:K- ¢N+

@

^

`

@IH

D

u

}²±

^

`

^%& }

.

`

(

@

}± @IH

D

±

²±<

D

^

`

^

`

(

@

± @IH

D

±

s

Å*23,RM<

c

^

·6`

^

·

` 4

^

RH`231"0%823<L

^

`

^%& }

.

`

u

: @

:

·

@IH

D

^

·I

`

^%& }

.

`

s

^

`

^

`

u

: (

@

:

·

@ H

D

^

·6

`

^

`

s

+,0.>K0G%3<11 Fn#+%3%3#|XWL <M¢ =A0L@

@

}

± @ H

D

± +,0.

@

± @ H

D

±

+?@ #&AR,0.0<. 'Y M<#&,0L© +, .0<"$<,R.0<,#&,0%8'¤#&,ES*H

Å*23,0Mk#&AR?28,_@<?D"Z#&%3 =>28#&,¨L@M©KR<1x23L

+LD<.j#&,²,0#O.- +%N= +%3A0L28,N&#&%(N*28,0)"- +?D@2B +%.R<?@2(N=

>2(N&<L/23,N= +?@28#&A0L.023?D<M®>23#+,0L<4XG,0<.U @#*#+%&>#C?D<M< +LD>K0

^

XG¢ +¯*`28,><?D"Z#+%B =@23#&,TM#&,0.02(>23#+,0L 23,L@A0M©K :Fn#&?@1@K- =>K023?23,>?> +M®>23#+,

<M<#+1LG>?@ +M>

%8+H½Lb PM<#&11#+,A0,R28 XXW23%8% [email protected]<?@2(N= =>2(N&<L#+F>K0I'O"$

.

.65

^

·2 µg` HJK0T,0®¡OWXG#U%3<11 +L"0?D#|N*28.0T<L>281¤ =@23#&,RL #+F/>[email protected]<?D28N= =>2(N&L23,¨>?@1L#=FQ,0#O.- +%ZN+ =%3A0L#+F#&AR?L@M©K01=H

(16)

±

+,0.

H

^

#+>:>K- .FY1¤ ¢' k #&A0,0.- =?D'<.R)&Y#=F9¨H` I,0#+@ '!F ±<±

=,0. F ±<

X# #+>KR<?C.0)&LC#+Fo

±

4Z =@> +M©K0<.j@#

±

+,0.

4$?@L@"$<M@28N+<%8'+46 +,R. 'jº ±<

}

>K0k +,0)&%8 XG<, Fk +,R.!F

±<} 4-c

0

s *H

^

Å* Q28)0H0*H*Hd`

v

₁

v

₂

T

₁

e

_1,2

e

_1,1

1,1

θ

e

1,2

θ

; B CB@CB

5CB " 4>? gh ] z^

9` -.398!·¹h³i :

^

W` ;=< h£e=Vs%&%&%®s>S

0

r?> I>4!S.21.4>3 &

=4>3 K&*A6J-.(/(4²c tRs&%&%%s>V -.398EctRs&%&%%<s < $

@

.

. 5

^

·I g` @IH

. u ¸

(

:

1 ¡ DFE

E

@

: (

.

. 5

·

@IH

.

1¤ ¡

.65

^

· ¾g` ^%&

}

.

`

1¤ ¡

"

$

"$

°

.

" ^

· g` ^

}

`

s

^

RHª`

0=4>6 4

2+@!?*=4!(+4>3:S.?* *EK F

$

?*=4B@4>?/@ ±

*

-.6 4N8:4/7 3548 -DI>* 1.4

$

-.398Y¸ 8:4<;=4>398A@J*A3=(0 *A3 S-.398k128,$eº

±<± s©º

±<

r

&

'?C? BÅ*23,0M

^

Gs1`Gh#

±

*

tUu ¹uvVW " r°²± (

$

s ±cztRs&%%&%<s>Vs

±

@

.0^

· g` @ H

. u ¸

±

(

: @

:

.

·

@ H

.

1¤ ¡ DFE

}

E

^

·I g` ^%&

D

}

.

`

1¤ ¡

"

$

"$

.Z^ · g`

^

}

`

s Ec tRs%&%&%®s <s

(17)

^

K0#&%8.0L5Fn#&?tRs%&%&%<s 0 H½) +23, ':11 RH¥04 +"0"0%823<.I@#2

. 5

·7hji : (

^ F`6 +,0. c! . 5 g&4 X:)&®WFn#&?W +%3% Ec tRs%&%&%®s < Q4

@ . .65 ^ ·I g` @ H . u ¸ ± ( : ( @ : ( . .65 · @ H .

1¤ ¡ DFE } E . 5 ^ · g` ^%& } . `

1¤ ¡

" $ "$ . . 5 ^ · ¾g` ^ } ` % JK_A0L<4bXG±,0<. @#³LD>281¤ >, . . 5 ^

· ¿g` ^ } `!28,@<?D1L#+F . . " ^ · qg` ^ } `UXW2(>K ³u HJ/#!>K028L<,0.64-XG:A0L@:>K0:?D<%B >23#+,

±<} c M<#&LPº ±<} 3 LD23,º ±<} s c 0 s bs XWK0<?D ±<}

4 ! +,0. 3¨ +?@G>K0bAR,028N+<M®>#&?DL23,[email protected]?D<M®>23#+,0LQ#+FF ±<}

48FW +,0.2F 3?@L@"$<M®>28N+<%('&4 L@#!>K- . . " ^ · ¾g` ^ } ` c D # $ M#&L ( º ±<} L@28, º ±<} ° ( . .65 ^ ·6 ¹g` ^ } ` s ^

RH0 t`

=,0.²KR<,0M+4 . .65 ^ · g` ^ } ` u . . " ^ · g` ^ } ` ¸ 1 ¡ DFE E (± ° ( . . 5 ^ · ¾g` ^ } `

s c 0 s *s

XWK0<?DÄ¸ .0<"$<,0.RL²#&,R%8'y#+,f +,R.123,$eº ±<± s©º ±< r_H#-A0?>K0?@1#&?@=4 'y@K0723,0.0ARM>28#&, K'*"$#+@K0<LD23L<4 ° ( . . 5 ^ · ¾g` ^ } ` u ¸ ( ( : 1 ¡ DFE E @ : ( . . 5 · @IH .

1 ¡ .65 ^ · g` ^'& } . `

1¤ ¡

(18)

.0<,R#+> '±o

>K0x@?@23 +,0)&%8k23,79 >K- TL@K- =?@2FUXW28>Kjo

±

Hk®GF ±

+,0.F

xXG##+>KR<? <.0)+<L:#+F o

4 @© =M©K0<.>#

±

+,0.

4?D<LD"ZM>2(N&%8'&4/ +,R.Ä%8:º

}

x>KR =,0)&%3 ®X<, F +,0.HF

} 40c

0

s *H

^

ÅO<JQ23)RH0RHRHd`

v

₁

v

₂

e

_2,1

T

2 T

₁

e

_2,2

e

_1,2

e

_1,1

1,1

θ

e

2,1

θ

2,2

θ

1,2

θ

; B CBCB

PAR?D>KR<?@1#&?D+40%8

.0,0#+@:@K0:%3,0)+>K#+F:F&H

5CB " 4>?gh ] ^

9` -.398U·jhi:

^

W` ;=< h³e==Vs&%&%%s>S

0

r?> I>4BS.2 1.4>3

&

>KWº ±<

£º N

c

$

?*=4>3 K&*A6-.(/( ±cztRs&%%&%<s>V-.398c tRs&%&%%s <

$

@

.

. 5

^

·6 ¹g` @IH

. u¸

( (

:

1 ¡ DFE

E

@

:

(

.

. 5

·

@IH

.

1¤ ¡

. 5

^

· g` ^'&

}

.

`

1¤ ¡

"

$

°

.

""

^

· ¾g` ^

±

`

1¤ ¡

"

4

E

°

.

"$

^

· ¾g` ^

`

L@28, (f

^

º

±< £º

`

1 ¡

$

}²±<

°

.

.! $

^

· g` ^

`

s

^

RH 0 0

`

0=4>6 4¸ 8:4<;=4>398A@ *A3=(0*A3¨S-.398k128,$eº ±<±

s©º ±<

s©º

r

&

> K I>*A?/¹º ±<±

º ±

N

c -.398Äº ±<

fº

N

c $ ?/=4>3 K&*A6 -.(/( [c tRs&%&%%s>V -.398 c t*s&%&%&%s <

(19)

@ . . 5 ^ ·I g` @IH . u¸ ( ( : 1 ¡ DFE E @ : ( . . 5 · @IH .

1¤ ¡ .65 ^ · g` ^'& } . `

1¤ ¡

" "$ ° . . " ^ · g` ^ `

1 ¡ ²±<

LD23, (f ^ º ±< £º ` 1¤ ¡ $

}²±<

° . . ^ ·I g` ^ ` s ^

RH0 &`

'?C? B 0 `xJK0<LDL@<,@2B +%.025?@,0M< XG<<,

^

RHª`23Lb@K- =@K0T><?D1L

° . . "$ ^ ·I g` ^ ` s ^ GsQ`h *

s ³u s

23,¨>KR?D23)&KK0 +,[email protected]#=F ^ RHdª&`b +?@:L@A LD@28>A*><. ' LD23, (f ^ º ±< º ` ° . . $ ^ · g` ^ ` s ^ s1`Gh

s c 0 s %

´F ±cztR4>KR<,¤e

^

Gs1`Gh

*

a µu QrIc04+ =,0.

^

RH0 0 `623L/ L>?@ +23)&KDFn#&?DX [email protected]<#+,0L@OAR<,0M #+F

^

*Hdª` H!²#+?@<#|N+<?<428,±#&?D.0<?>#"$<?Fn#&?@1 23,R.0A0M@23#&,²#&, 4$XU#+,0%8',0<.Ä +,jLD@231 =>28#&, #+F/>K0TFn#&?D1 1 ¡ ° . . "$ ^ ·I g` ^ ` u¸ ± 1¤ ¡ " ° @ . .65 ^ ·6 ¹g` @IH . L@28, (f ^ º ±< £º ` 1 ¡ $

}²±<

° . .! $ ^ ·I g` ^ ` % ^

RH0 `

^

s1`h

*

40XW2(>K ³u Q4

. . "$ ^ · g` ^ ` u¸ 1 ¡ DFE E

° (+* ,(R

° ( . . "$ ^ · ¾g` ^ ` L@28, (f ^ º ±< º ` 1 ¡ ° (f E E } ²±< ° ( . .! $ ^ · ¾g` ^ ` % Å*23,RM< u ^

¨ `: ¾+V :c

u 0 s

(20)

^ 0 1¤ ¡ DFE E

° (+* ,(R

° ( . . "$ ^ ·6 g` ^ `

u 1 ¡ DFE E (± ° ( . . "$ ^ ·6 g` ^ ` u¸ * 1 ¡ DFE E (± ° ( . . 5 ^ · ¾g` ^ ` u¸ * 1¤ ¡ DFE E (± @ ° ( . . 5 ^ · ¾g` @ H . % -A0?D@K0<?D1#&?D+40LD23,0M ! ¹Vxu uvV:c ^ s `h s X:K- ¢N+ 1¤ ¡ ° (f E E

}²±<

° ( . . $ ^ ·6 ¹g` ^ `

u 1 ¡

$ "$ . . $ ^ ·I g` ^ ` s +,0. ^ *H 0 &`GFn#&%3%8#XWLH

&`/JK028L"- +?M< +, <LD> %823L@KR<. 'x ¡R +M@%8'>K0bL@ +1b +?D)&A01<,>L4+>K0#&,0%8'.025?@,0M<

23,0)U@K- =>KR:@<?D1L ° . . "" ^ ·I g` ^ ± ` s ^ GsQ`h *

s ³u s

,0#|X¿ +%8L@#!K- ¢N&:># :<LD@231 =>.6H

®Iohµ9; =,0.%3®

k N&?D@®¡7#+FQoYHCJK0<,oc¿o

}

^

`

^

LD<Å*M>28#&,Ä&` HJ! +?@k)&#&23,R)¤>#¨.0®\-,0UN= +?@28#&A0LILDA LD>L #+F

^

`I =,0. @K- =:XW23%8%

28,0LD@?@A01<,> +%23,>KR"0?D#*#=F#+FJK0<#+?@<1 *H

0

+,0.7>K0!¯=' <11¤ URHRH

³.0\0,0 l ^ ` ^ `>#

@K0YLD#+F,R#*.- =%6FnA0,0M@23#&,0 +%3LWM<#+?@?@L@"$#&,0.028,0)!>#!>K0 Fn#&%3%8#|XW23,0)x,R#*.- =%5N= +%3A0L<a

^ N_ 0 ` · ^ `GFn#&?W +%3% ^ s1`h ± 4 ^ N_ &` . . " · ^ `GFn#+?W +%3% ^ GsQ`h 28F:F }

28L,0#&,0.R<)&,0<?@ =>Y =

40#&? . " .! · ^ `$Fn#&?/ +%8% ^ Gs1`h 2(F F }

28L/.R<)&,0<?@ =>G =

4 AR F } (± 23L,0#&,0.R<)&,0<?@ => = 40#&? . $ .! " · ^ `Fn#&?I =%3% ^ GsQ`Ch

28F #+@K F

}

+,0. F

}

(±

[email protected])&<,R<?> >! =

4 AR

F

}

(R

23L,0#+,0.0<)+<,0?> =@k =

40#+? . " . $ · ^

` Fn#&?W +%3%

^

GsQ`h#

28F

28LW LD23,0)&AR%B +?N+<?>®¡Z4

(21)

^ .

"

.

$

^ ^

}°²±

#&?

.

"

.

$

·

^

`GFn#&?W +%8%

^

s1`Gh

*

2(F F }°²±

23L.R<)&,0<?@ =>Y =

40#&?

.

"

.

·

^

`GFn#+?W +%3%

^

GsQ`h

*

28F:F }°²±

28LW

#&AR,0.- +?'¨<.0)++4P +,0.

^

N_>_` .

.

"

·

^

`Fn#&? +%8%

^

GsQ`h

*

28FP28@K0<? F

}

23L.0)&<,R<?> >W =

#&?:F

}

23L #&AR,0.- +?' <.R)&+4-#&?

.

"

.!

·

^

`Fn#&?W =%3%

^

Gs1`Gh

2(F #+@K!F

}

+,R. F } (±

+?D,R#&,0.0)&<,0?> =@ =

4R#&?

.

$

.!

"

·

^

`Fn#&?G +%3%

^

s1`Gh

2(F F

}

28L,0#&,R.0<)&,0<?@ =>I =

4 F }(±

23L.R<)&,0<?@ => =

4- +,R. F }(R

23LW =) +23,¨,R#&,0.0)&<,0?> =@k

4-#+?

.

.!

$

·

^

`Fn#+?G =%3%

^

Gs1`h

2(FCF

}

23L,0#+,0.0<)+<,0?> =@I =

4

#+@K%F }(±

+,R.IF }(R +?DT.0<)&,0<?@ =>Y =

4P +,0.HF

}

(+*

23L,R#&,0.0)&<,0?> =@k

H

-A0?D@K0<?D1#&?D+40.R<,0#+@

'

l

^ `b

l

^

>K0ULD:#+FFnA0,0M®>28#&,- +%8LCM<#+?@?@L@"$#&,0.028,0)¨>#@K0U,0#O.- +%QN= +%8A0<LI%823L><.28,

^

N_

0

`®4

^

N_©&`®4

^

NO>` +,0.

^

N_@_`b?@<LD"ZM@28N&%8'+H ³ +%8L@#!.0\0,0

l

^ `

l

^

`

}

+,R.2F }°²±

23L/<2(>K0?Q.0)&<,0?> =@#&?Q%823L#&,Y>K0 #&A0,R.- +?D'+4 >K0,

l

^

`kadc 0428F F

}

°²±

23L +,Ä23,>?@23#+?I,0#&,0.R<)&,0<?@ =>¤ =

<.R)&+4 ARF

}

23L:,0#+<4@K0<,

l

^ `Wadc

l

^

`

l

* ^

`®4P28F 4-M#&,N&?@L@%8'+4CF

}

28L +,±28,_@<?D23#&?W,0#&,R.0<)&,0<?@ =>k =

<.0)++4

ARF }°²±

23L,0#=¢40>KR<,

l

^ `badc

l

^

`

l

^

`®4- +,R.640\-,- +%8%8'+4028F

#+>K!F

}

+,0.!F }°²±

=?@

23,>?@28#&?b,0#&,R.0<)&,0<?@ =>Y =

<.0)&L<4->KR<,

l

^ `adc

l

^

`

l

* ^

`

l

^ `®H P#+?N&?D'¨@?@2B =,0)&%3oqh9 XW2(>K¨N&?D@23M<L

±

s

*

=,0.².0)&LF

±

s F

*

40%8

l

adc

*

}²±

l

^

}

`

*

}²±

^

F

}

`

^

oC`5s

l

adc

*

}²±

l

^

}

(22)

Q23,- =%3%8'+4X:,0<.² xL@W#=F,R#*.- =%5FnA0,0M®>23#+,- +%3L #+F\-,028@

<%31,b'O"Z=HG®oqh79¨4 %3®

xN&?D@®¡²#+F/oY4P =,0.²%[email protected])&L.F

±

s F

HJKR<,

l

^ `23L .0\0,0<.E># T@K0:L@#+F,0#O.- +%5FnAR,0M@23#&,- =%3LM<#+?@?@L@"$#&,0.028,0)U>#!>K0:,0#O.- +%ZN+ =%3A0L

.

"

.

$

·

^

`5sbFn#&?W +%8%

^

s1`h

±

*

s

XWK0<?D

*

adce

^

Gs1`h

a

^

s Q`h

*

r %

-A0?D@K0<?D1#&?D+4Fn#&?N+<?D'<.R)&.FI#=F5ovXGC.0®\-,0

l

^

F`@#

@K0CLD#+F6,0#O.- +%-FnA0,RM>28#&,- +%8L +L@LD23)&,R23,0)x@#®N&?D'FnA0,0M®>23#+,±·jhi

^

W`>K0TFn#&%8%3#|XW23,0)k,0#*.0 +%5N= +%3AR<L<a

·

^%& ±

.

` s&%&%&%s

·

^%&

.

` Fn#&?W +%3% ±cztRs&%&%&%®s>VGs

XWK0<?D

&

}

.

+,0.

+?@.0\0,0<.23,

^

*Hd¦&` +,0.

^

*Hd&`?@L@"$<M®>28N+<%('&4- +,R. -28LG@K0CA0,R28N+<M®>#&? 23,¨>KR.R23?@M@23#&,Fn?D#&1 @K0:128.0.0%3C"Z#&28,W#+F F:>#U>KR:N+<?>®¡E#=Fo¿#+"0"Z#+L@28@I>#%F&H

P#+?N&?D'¨@?@2B =,0)&%3oqh9 XW2(>K¨N&?D@23M<L

±

s

*

=,0.².0)&LF

±

s F

*

4RXGYLD

l

adc

*

}²±

l

^

}

`

*

}²±

l

^

F

}

`

^

oC`5%

5CB 4 - 1.4

M¢ +?D.

l

czM¢ +?D.

l

c # °

$

%

^

RH0 _`

8

*A6 4M* 1.4>6

$

l

2+@Um

Q<C:3=2+@*A(1.4>3=?

$

FG

$

K&*A6 -.3=0P6

$

h

l

$

?*=4>6M4 4)3 2+@>?/@ -C 3=2,+C=4 ;=*A(0A35*14 2-.(

hm

@>C ,K ?* -.?

c

K&*A6 -.(/( h

l

&

'?C? B °

NO23#+A0L@%('&4

M¢ =?@.

l

cz M< +?@.

±

M¢ +?D.

£M¢ +?D.

*

£M¢ =?@.

^

F`M< +?@.

^

oC`>%

G'

^

*Hª`5 +,0.:LD#&1%31,_> +?D'CM<#&1"0AR> =>28#&,64¢XG #

l

c # r°

2$

H -A0?>K0?@1#&?@=4

M< +?@.

l

^ `c

²± M< +?@.

l

^

`czM< +?@.

±

M¢ +?D.

cM¢ +?D.

l

^ `>%

,0M<=4bM¢ +?D.

l

c M< +?@.

l

^

RH0 _`®HVU +?D@23M<AR%B +?D%8'&4 M< +?@.

l

cÆ.0231£m

H

<M< +A0LD#+F>KR23L<46@K0!L@M<#&,0.LD> =>1,_:#=FG@K0!%3<11 XW23%8%Fn#&%8%3#|Xf28F XL@KR#X >K- T>K0