Multiscale modelling of Arabidopsis thaliana phyllotaxis perturbation

HAL Id: tel-00859869

https://tel.archives-ouvertes.fr/tel-00859869

Submitted on 9 Sep 2013

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

Yassin Refahi

To cite this version:

Yassin Refahi. Multiscale modelling of Arabidopsis thaliana phyllotaxis perturbation. Modeling and

Simulation. Université Montpellier II - Sciences et Techniques du Languedoc, 2011. English. �NNT :

2011MON20108�. �tel-00859869�

. "$! $ $' 0$ 1!"23$ 43 56!73$4*

!"#$

!"# !$%&'(# )& *#+,& ,&

8* '$3% 4$ ,9 !#$%&"'( )*!'+$,,"$%

--(. ( )('& 0 1'2!#3+%(4"&

5 !)& -! %!#+)& 0 1'2!#3+%(!'.6 7%#" %"#&.6 78.%&3. 91:7;

<+#

:6&&"! ;<=>?@

%&'()*#+,*&- ./),*0( !$))$ '$# 2$3,/34+,*&-# '$

)+ 2!5))&,+6*$ '783+4*'&2#*# ,!+)*+-+ 7!". )+ ,(#& %(!' ,&0 A1%"&'*+1$ B*4"! C6! 0%66& <'"$!!$ =6% *' !"#$#%&'$(&$ )$ *+(%,$-./$ 01** !"# $ 23 4/5 !"#$%7 8 9:;< 7=/>#5?$('

23 @$A(8B& &'()* 8 C:;CD 7;AEE%/'$&/

2-$3 2A !A+,-'./-, 8 9:;< 7;AEE%/'$&/

23 F,$# 01213.,' 8 9C;DG 74HA-5(A'$&/ 5(,5'>

23 9!/5#'%E!$&'4,5 8 C:;CD 7G5/$ '$&/

23 @A(6 ##% 8 C:;D 79%8?5/$ '$&/

. "$! $ $' 0$ 1!"23$ 43 56!73$4*

!"#$#

!"#$%%&' ()* %+& '&,*&& )(

8* '*% *9 :1",*&*+1; <! =*>+3'$% . "$! $ "-?6&&"! @ABCD< %&'($# *'" +,-"''$./ ,0 12*3$-,4#$# (!*'$*.* 4!5'',(*6$# 4"2(&23*($,.# .'/$ )* 0 =1%"&'*+1$ E*4"! F6! 0%66& A'"$!!$ B6% *' !"#$# %"&"'%"% (')(*"+,"- ./

th

01.. ,"&(-" 2!" 34-5 (+7(#"%(&8 !"# $ 9: ;-$ !"#$%< = >)?@ <A-B#$%"'2 9: C"D'=E4 &'()* = F)?FG < ?D77(-2"4-9+": 9D !D+,-'./-, = >)?@ < ?D77(-2"4-9: H*"# 01213.,' = >F?GI <;JD+$'D2"4- $'*$2B 9: >!-$#2(7!"&'4,5 = F)?FG < I$-" 2"4-9: CD'6 ##% = F)?G < >(=%$-" 2"4-9: ;2$"''"712 '9 = F)?FG <;' D%-D'2

! "#$%&'(!)&'" !* +)!,'!) -'!$ !.+)',!) ,!" +-$" +)#/#*0" )!,!) '!,!*(" 2 ,#* 0')! (!$) 0!

(%3"!4 5%)'"(#+%! 6#0'*7 8- & 9(9 +#$) ,#' $* 0')! (!$) 0! (%3"! !. !+('#**!-7 :#$( !* ,!

-&'""&*( $*! ;)&*0! -'<!)(9 0&*",!" %#'.4 "!" #*"!'-" &='"9" !( )'('>$!" 0! ,#* ()&=&'- ,?#*(

<!&$ #$+&'092 #)'!*(!),#*()&=&'- 0&*"-!" <#**!"0')! ('#*"7

!)!,!) '!"'* 3)!,!*(,#* #@0')! (!$)0!(%3"!4 &*:)&&"7 !-!)!,!) '!+#$)-& #*A&* !

!( -&-'<!)(9>$?'- ,?& & #)09!"7

?&0)!""!,!" ='/")!,!) '!,!*("2,#*!* &0&*(0!(%3"!4B('!**! C&) #(7 ?&'!$ -& %&* !

0?&=#')$* !* &0)&*( 0#*( -!">$&-'(9"%$,&'*!"!( +)#/!""'#**!--!" "#*( !. !+('#**!--!"7 ! "$'"

=)&',!*(A!) 0?D()!"#*9($0'&*(7 B('!**!&(#$E#$)"+)'"0$(!,+"+#$),?!.+-'>$!)!(0'" $(!)7

F!) ' +#$) "&;)&*0! 0'"+#*'<'-'(94 "&;!*('--!""! !( "!" !* #$)&;!,!*(" +!),&*!*("7 ?!.+)',!

,&+)#/#*0! )! #**&'""&* !+#$) (#$(-!(!,+" >$?'- ,?& #*"& )97

F!" )!,!) '!,!*( =#*( 9;&-!,!*( 2 G&** 6$90#*7 H!" #*"!'-" !.+!)("4 !( "!" )!,&)>$!"

&='"9!" #*( -&);!,!*( #*()'<$9 2 &,9-'#)!) ,& (%3"!7 ! *! +!$. +&" #$<-'!) 0! "&-$!) "!"

>$&-'(9"%$,&'*!"4 "&;!*('--!""!4!( "&+&('!* !7

! =#$0)&'" )!,!) '!)B)' I'=&-" 0?&=#')& !+(9 0?D()! -! +)9"'0!*( 0$E$)J0! -&(%3"!7

! )!,!) '! 9;&-!,!*( !&*@K$ 6#$L9 !( F& %& M'N#-"N' +#$) &=#') & !+(9 0?D()! )&++#)@

(!$)" 0! ,& (%3"! !( 0?!* /&')! $* #,+(!@)!*0$7 K& 0'" $""'#* 0! ='=! =#'. !( -!" )!,&)>$!"

#*"()$ ('=!"0! !&*@K$ ,?#*( &'09 2&,9-'#)!) -!,&*$" )'(7

F!" )!,!) '!,!*(" "?&0)!""!*( 9;&-!,!*( 2 G=!" 5&)&;-'#4 +#$) &=#') !.&,'*9 -& (%3"! !(

+#$) -?'*(9)D( >$?'-& +#)(9 2-&(%3"!7

5!((! (%3"! 9(&*( +-$)'0' '+-'*&')! 2 -?'*(!)/& ! 0! -& <'#-#;'!4 0!" ,&(%9,&('>$!" !( 0!

-?'*/#),&('>$!4 E?&' !$ -& %&* ! 0! 0'&-#;$!) &=! -!" <'#-#;'"(!"7 ! =#$0)&'" (#$( +&)(' $-'3)!@

,!*( )!,!) '!) :!=& O!)*#$. !( C&<)' ! P!"*&)0 >$' ,?#*( /#$)*'!" -!" 0#**9!" <'#-#;'>$!"

$('-'"9!" 0&*" -& (%3"! !( >$' "! "#*( ,#*()9" 0'"+#*'<-!" +#$) (#$(!" ,!" >$!"('#*"7 5!-& ,?&

9(9 $*! #--&<#)&('#* ()3" !*)' %'""&*(! !( +&""'#**&*(!7 ?&0)!""!,!" )!,!) '!,!*( 2 O'* !*(

F')&<!(>$',?&/#$)*'-!""9>$!* !";9*9)9!"+&)"#*,#03-!0!+%J--#(&.'!7 !-!)!,!) '! +#$)

"# $!#%& ' (#)#( !#( +(,%-.!& /,( 0 1.2( , .!( 3!#% .242 1,($! !15 ' ).% .)!$5 6# $78&#9

1.2( &,:#%$!44#&&#9 #$ &#& .%&#!4&;

/#%6,%$ ), $78&#9 <=,! #2 4, 7,% # 6=#%&#!:%#( ' 4=2%! #(&!$5 6# >.%$1#44!#( ?; "# $!#%& '

(#)#( !#( /7!4!11# ",%&&#%9 @%%#A)2(!#4 @(!:.%9 #$ B7#(#&# C!3.2(#4 1.2( 4, .%D,% # E2=!4&

)=.%$ , .(65# #$ 4#2(6!&1.%!3!4!$5;

"#&.27,!$#(,!&(#)#( !#(4#&)#)3(#&6#4=5E2!1#F!($2,4/4,%$&G H7(!&$.17#/(,6,4#$+(#6A

#(! I.26.%J1.2(4#& %.)3(#2&#&6!& 2&&!.%&#$ 4#2(,!6#K9"5()#9 M(! 9N,%!#49 B7.),&J1.2(

4#2( ,!6##$ 4,:#%$!44#&&#K9#$ ,2&&!$.2&4#&,2$(#& )#)3(# 6#4=5E2!1#G /,& ,49 H7,OO(!$9N, !69

PQ0).%9 >! 7,#49 >!O9R.),!%9N,)!#%9 "#,%9 C5.9/!#((#9 >,$7!46#9@%%#AC,2(#9 ;;; ;

"# .26(,!& (#)#( !#( )#& 1,(#%$& 1.2( 4# &.2$!#% !% .%6!$!.%%#4 E2=!4 )=.%$ , .(65 ,!%&!

E2# 4#2(& #% .2(,:#)#%$& 1#(),%#%$& #$ 4#2( ,).2(; "# (#)#( !# 5:,4#)#%$ )#& &S2(& M4%,Q

#$ P.(.2(9 )#& 3#,2T U(8(#&9 V,)3!Q #$ B..(,< 1.2( 4#2( &.2$!#% &,%& .234!#( ).% 1#$!$ %# #2

/,(7,)1.2(4,<.!#E2=!4 )#1(. 2(#; W%:(,%6>MRHX'),)#( #!44#2&#&S2(P.(.2(E2!)=,

!"#$%%&'( )*+%%",-.&/ -, #&0$1$', / -%$/3 - /,-,$ "4 -1, 5

! "#$%%&'()*+(' ,( .&+ &/* + (%0 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 1

! ! 2*3&4( *45,30.+ (46 '#0 7&%604.('*& ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 1

! !8 "#$%%&'( '* /(''0.4+ ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 9

!8 "#$%%&'()*+(' '#0 ,0.*+'0,+ (%0: 6$4(,* (%,&60%+ ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ;

!8! <&60%+/.&/&+06 3$=&5(6$ (46 >&560.! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 8

!? <&%0 5%(. .075%('&.+ &@/#$%%&'()*+ ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! A

!?! B075%('*&4 &@'#0 04'.(% C&40 &@ ,0.*+'0, ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! A

!?!8 "(''0.4 @&.,('*&4*4 '#0 DE< *4F&%F*47GHD! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ;

!?!? IF0.F*0J&@ ,&60%%*47(//.&( #0+ ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 8

!?!1 B075%('*&4 &@/#$%%&'()*+3$#&.,&40+ ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 8?

!1 >&4 %5+*&4! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 89

6 7"8)9,-,&"'-% 8"#$%%&'( "4 ($'$,& -'# :&" *$8& -% '$,;"1</ 6=

8! K(+* 3*&%&7$ &@7040 .075%('*&4 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 8A

8!8 D/('*&L'0,/&.(% ,&60%%*47(46 +*,5%('*&4 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ?M

8!? >&,/5'('*&4(% ,&60%%*47 &@7040 .075%('&.$40'J&.N+: @&.,(%*+, .0F*0J ! ! ! ! ! ?8

8!?! O4'0.( '*&47.(/# ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ??

8!?!8 K&&%0(4 ,&60% ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ?1

8!?!? =*P0.04'*(% 0Q5('*&4+! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ?R

8!?!1 "*0 0J*+0L%*40(. 6*P0.04'*(% 0Q5('*&4+ ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 1M

8!1 D/('*(% /(''0.4 @&.,('*&4 J*'# .0( '*&46*P5+*&4 +$+'0,+! ! ! ! ! ! ! ! ! ! ! ! ! ! 1

8!1! >%(++* (%,&60%+ &@ .0( '*&46*P5+*&4 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 1

8!1!8 H+*47 0%%5%(. (5'&,('( @&. .0( '*&4L6*P5+*&4 +$+'0,+ ! ! ! ! ! ! ! ! ! ! ! 11

8!9 >&4 %5+*&4! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 19

5 !"#$%%&'( -'# -'-%+/&/ "4 )$1,91:-,&"' )-,,$1'/ &' /)&1-% )*+%%",-.&/ >?

!"# $ %&'()*'(+ ),)&+-.- " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " /0

!"! 1*)*.-*. )& 3'45&&.,6 " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " /!

!"7 8'39.,)*'(.)& 3'45& :'(3;&)*.', " " " " " " " " " " " " " " " " " " " " " " " " " " " /<

!"7"0 =5>,.*.',- " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " /<

!"7"# ?('%5(*.5- ':,@)43.--.9&5 -5A;5, 5- " " " " " " " " " " " " " " " " " " " " " /B

!"7"! C5'35*(. )& %('%5(*+':

n

@)43.--.9&5 -5A;5, 5- " " " " " " " " " " " " " " " <#

!"7"7 $,;35()*.', ':,@)43.--.9&5-5A;5, 5- " " " " " " " " " " " " " " " " " " " " <#

!"/ =5*5 *.,6 ,@)43.--.9.&.*+ ., ,'.-+-5A;5, 5- " " " " " " " " " " " " " " " " " " " " " </

!"/"0 ?('9&53- " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " </

!"/"# D--.6,35,* ': *E5'(5*. )&),6&5- *' 35)-;(54),6&5- " " " " " " " " " " " " <F

!"/"! D&6'(.*E3- " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " <B

!"< G5-;&*-" " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " F/

!"<"0 D,)&+-.-': ';( 4)*)-5* " " " " " " " " " " " " " " " " " " " " " " " " " " " " " F<

!"<"# 15,-.*.H.*+),)&+-.- " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " I#

!"<"! $-*.3)*.,6 *E5 ),',. )&4.H5(65, 5 ),6&5': )35)-;(54 -5A;5, 5 " " " " I#

!"F 1':*J)(5 .3%&535,*)*.', " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " I!

!"I =.- ;--.', " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " I/

!"B 8', &;-.'," " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " I<

!"#$$%&'()* '& +,-)*&,+* '$,/ '**,**+,0& %1 ' +%2,$ %1 3,-&4-5'&)%0 67

7"0 K'45&-%('%'-54 9+=';)4+ ),4 8';45(L (5 )&& " " " " " " " " " " " " " " " " " " " II

7"0"0 M,*('4; .,6,'.-5 ., *E5 -5 ',4 3'45& " " " " " " " " " " " " " " " " " " " " II

7"# N;),*.> )*.', ': -%.()&.,H5(-.',- " " " " " " " " " " " " " " " " " " " " " " " " " " " B0

7"! G5-;&*-" " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " B!

7"!"0 M,H5-*.6)*.,6-%.()&.,H5(-.',- ., J.&4@*+%5 ),4 )E%<3;*),* " " " " " " " " B!

7"!"# D,)&+-.-': *E5 -.3;&)*.',- " " " " " " " " " " " " " " " " " " " " " " " " " " BI

7"7 =.- ;--.', " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " 0O/

7"/ 8', &;-.'," " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " 0O<

8 9%2,$$)0: &", -,:4$'&)%0 %1 &", ,0&-'$ ;%0, %0' 2):)&);,2 < +,-)*&,+ =>?

/"0 K'45&-%('%'-54 9+PQ,--', ! "#$ " " " " " " " " " " " " " " " " " " " " " " " " " " 00O /"# 8',-*(; *.', ':) (5)&.-*. 7=35(.-*53 " " " " " " " " " " " " " " " " " " " " " " " 007 /"#"0 K5*E'4- %('%'-549+R5(,),45S ! "#$ " " " " " " " " " " " " " " " " " " " " 007 /"#"# 8',-*(; *.,6 7=4.6.*.S54 35(.-*53 " " " " " " " " " " " " " " " " " " " " " 00/ /"#"! D%%&. )*.', *' ) T'()&35(.-*53" " " " " " " " " " " " " " " " " " " " " " " " 00< /"#"7 U)(65* %)**5(, ', 7=4.6.*.S54 35(.-*53 " " " " " " " " " " " " " " " " " " " 00<

"#$ %&!'()*!+),-!.!)/ 01( 2(32(**+2 '+&(- # # # # # # # # # # # # # # # # # # # # # # # # 454

"#" 6(*7-0* ,)&&!* 7**!+) # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # 459

"#"#4 :!'7-,0!)/ 9;, 0!<,0+2,)& 2(32(**+2 '+&(-+) 01( *0,0! 0!**7( # # # # # 45$

"#"#5 =>: 3,00(2)+) , 2+? +@ (--* # # # # # # # # # # # # # # # # # # # # # # # # 45A

"#"#9 :!'7-,0!)/ 01( ,&!'()*!+),-!.(& '+&(-+) ,$;&!/!0!.(& '(2!*0(' # # # # 499

"#A B+) -7*!+)# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # 49C

!"#$%&'( )*+,(*(-$&$)"- ./0

%#4 D()( 2(/7-,0+2E)(0?+2F *!'7-,0+2G , 32+0+0E3( # # # # # # # # # # # # # # # # # # # 4$C

1 !"#$%&'( )*+,(*(-$&$)"- .2.

3 4(5-)$)"- "# 67)88(-9 :&')&;,(<"'8(' =&'>"? 7&)-A .2B

!"#$% &!"' "(")*+ +*!, -# "!(*%$ "!! "%&, $% */ 01("# "#2"#-("! !-/,3 40,'&+*5-2, /**2/*+

*$0,+ %&, -,%6 ("$,+-"! /*+ *#%$+1 $-*#6 0,(- "!% /*+ &0"+(" ,1$- "! -#21%$+'6 *7'8,# /*+ $0,

"$(*%&0,+,6#*$$*(,#$-*#$0,",%$0,$- "#2 1!$1+"!1%,%*/9*:,+%3 ;, "1%,$0,'"+, *#%$"#$!'

8+*:-#8 $0+*180*1$ $0,-+ !-/,6$0,-+ %0"&, -%*/ +1 -"! -(&*+$"# , /*+ $0,-+ 2,% +-&$-*#"#2 "#

<,,7$+,(,!' *(&!,73 =,%&-$,$0-% *(&!,7-$'6 $0,-+/*+("#2%$+1 $1+,-%0-80!'+,81!"+6"$$0,

!,5,! */ "7-% "#2 *+8"#%3 40-% +,81!"+-$' 0"% "$$+" $,2 $0, "$$,#$-*# */ % -,#$-%$% /*+ ,#$1+-,%

"#2 -%+,/,++,2$* "%&0'!!*$"7-%3

0'!!*$"7-%0"%<,,#%$12-,2/+*("8,*(,$+- "!"#22,% +-&$-5, &*-#$*/5-,:21+-#8$0,!"%$

,#$1+-,%3 >, 0"#-%$- "!!'6 &0'!!*$"7-% -% $0, +,%1!$ */ 2'#"(- "! &+* ,%%,% :0- 0 $"?, &!" , -#

" $-%%1, *(&*%,2 */ 1#2-@,+,#$-"$,2 ,!!% "!!,2 (,+-%$,(3 40, %0**$ "&- "! (,+-%$,( ABC>D

!* "$,2"$$0,,7$+,(-$'*/$0,%$,(8,#,+"$,%"!!"<*5,E8+*1#2,!,(,#$%*/&!"#$%%1 0"%!,"5,%6

"#2 9*:,+%3 40, BC> -% 2-5-2,2 -#$* /1# $-*#"! 2*("-#%6 $0, ,#$+"! F*#, "#2 $0, &,+-&0,+"!

F*#,6 -# &"+$- 1!"+3 40,%, 2*("-#%"+, ("+?,2 <'$0, ,7&+,%%-*# */ %&, -G 8,#,%3 C!$,+"$-*#%

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

"&&+*" 0,%3 H+*( *(&1$,+ % -,# , &*-#$ */ 5-,:6 $0,%, "&&+*" 0,% &+,%,#$ 0"!!,#8,%6 :0- 0

:-!! !,"21% $* 2,5,!*&*+-8-#"! (*2,!%"#2 "!8*+-$0(% -# $0, *1+%,*/ $0-% $0,%-%3

40*180 &0'!!*$"7-% -% " 0-80!' +,81!"+ &0,#*(,#*#6 " !*%,+ !**? -#2- "$,% $0"$ -$ "# <,

%1<), $ $* #*# #,8!-8-<!, &,+$1+<"$-*#%3 =1+-#8 $0, !*#8 0-%$*+' */ %$12'-#8 &0'!!*$"7-%6 *#!'

!"#$%&'%&&()#*#%+",(&#-'. /$.00+#*1%&/()#-)2*#%+" *#'%3()("# & *0(&-&%"5 +6/-#*#%+"*0

6+'(00%"57 8+ * $%(9( #$%&5+*0: ,( )(0.+" 2%+0+5% *0'*#* *# '%3()("# & *0(&7 8$( +)5*"%;*#%+"

+< #$( '%&&()#*#%+" *0&+ +))(&/+"'&#+ #$( *2+9( 6("#%+"(' #$)(( & *0(&7 8$( =)&# #,+ $*/#()&

%"#)+'- (#$(&#*#(+<*)#+</$.00+#*1%&*"'5("("(#,+)>&#$*#,%00$(0/-"'()&#*"'#$(6(#$+'&

*"' )(&-0#& +2#*%"('%" #$%&#$(&%&7

?%" ( #$( *'9("# +< 6+0( -0*) 2%+0+5.: * 0*&&% *0 ,*. #+ '( %/$() #$( )+0( +<5("(& %" /0*"#

'(9(0+/6("#%&#+ +"&%'()#$((3( #+<6-#*#%+"&7 @&%"5#$%&*//)+* $&+6(2%+0+5%&# +00(*5-(&

$*9(2(("0('#+ 5*#$()'*#*+"*6-#*"#+<#$( 6+'(0/0*"# !"#$%&'($()*"+$","7 !"/*)#% -0*):

#$(. +2&()9(' /()#-)2*#%+"& %" #$( 6(*&-)(''%9()5(" ( *"50(& A%7(7 #$( *"50( 2(#,((" #,+&- B

(&&%9(+)5*"&C%"2+#$,%0'B#./(*"'#$(6-#*"#7 !"D$*/#()E:,(&#-'.#$(&(/()#-)2*#%+"&*#

6* )+& +/% & *0(:-&%"5 '*#* /)+9%'(' 2.#$(&( +00(*5-(&7 8$(&( /()#-)2*#%+"&#-)" +-# #+ 2(

$%5$0. &#)- #-)('*"' <)(F-("#7 G( /)+/+&(#$*##$( /()#-)2*#%+"& *" 2( (1/0*%"('2./()6-B

#*#%+"& +< &- (&&%9( +)5*"& +" #$( &#(67 8+ &#-'. #$%& $./+#$(&%& ,( '(9(0+/ * +62%"*#+)%*0

6+'(0*"'*05+)%#$6&7

8$()(0*#%9(0. $%5$ <)(F-(" .+< +2&()9(' /()#-)2*#%+" /*##()"&%" D$*/#() E:&-55(&#& #$*#

*".6+'(0 +</$.00+#*1%&&$+-0' 2( *20(#+ )(/)+'- ( #$(67 !"D$*/#() H:-&%"5#$( *05+)%#$6&

*"' #++0& /)+/+&(' %" D$*/#() E: ,( *&&(&& #$( *2%0%#. +< * /()#-)2*#%+" 6+'(0 #+ )(5("()*#(

/()6-#*#%+"& +< +)5*"&7 G( ="' #$*# #+ &#-'. #$( &%6-0*#%+" '*#*: ,( "((' #+ *-56("# +-)

+)/-& +< 6(#$+'& #+ * +-"# <+) /()#-)2*#%+"& #$*# *" "+# 2( (1/0*%"(' 2. /()6-#*#%+"&7

8$%&*"*0.&%&+<#$( /()#-)2*#%+"6+'(0$%5$0%5$#&#$( )- %*0)+0( +<#$( ("#)*0 ;+"( &%;(%" #$(

)(5-0*#%+" +< /$.00+#*1%&7

8$(6*%"5("(&)(5-0*#%"5#$( ("#)*0;+"(+<#$(6()%&#(6*"'#$(%)%"#()* #%+"& *)(>"+,":

*"' '%3()("# 6+'(0& $*9( 2((" /)+/+&(' %" #$( 0%#()*#-)(7 !" D$*/#() I: ,( <-)#$() (0*2+)*#(

+" #$(&(6+'(0&:-&%"5 '%5%#%;(' 6()%&#(6& %" +)'()#+ #*>( %"#+ * +-"# #$(&/*#%*0 %"<+)6*#%+"

%" 6+'(00%"5 5("( "(#,+)>&7 8$( =)&# &%6-0*#%+"& +" * #$)(( '%6("&%+"*0 #%&&-( '%& *)'%"5 #$(

(00-0*) 5)+,#$ *"' '%9%&%+" 0(*' -& #+ #$%& $./+#$(&%& #$*# #$( (1/)(&&%+" ;+"( +< * 5("( *"

2( '(#()6%"(' 2. #$( 5(+6(#). +< 6()%&#(67 8+ <-)#$() (1/0+)( #$%& $./+#$(&%& ,( +"'- #

&%6-0*#%+"& +" *#(6/+)*0&()%(& +<#$)(( '%6("&%+"*0'%5%#%;(' 6()%&#(6&7

8+ &-6 -/: %" #$%& '%&&()#*#%+": ,( &#-'. /()#-)2*#%+" +< /$.00+#*1%& *# '%3()("# & *0(7 G(

&$+, $+, #$( /)+ (&&(& + -))%"5*# '%3()("# & *0(&*)( )(0*#('#+ (* $ +#$()7 J. %"9(&#%5*#%"5

#$(+)%5%"&+</$.00+#*1%&*#0+,()& *0(&:,(&$+,$+,%"#()* #%+"&*#*& *0( *")(&-0#%" +6/0(1

!"#$$%&' ()*$$!+,-%. ,+ "%/#0#&+

. ,$#.2 , .+,+# !3 ,0+

!"#$%& "($!

!" #$ %&" '#(% #!(*+ ,#,( *-%%".!( +! */-!%( +( %&" #.0-!+(-%+#! #$ /"-1"( -.#,!2 %&" (%"'3

4&" -..-!0"'"!% #$ /"-1"( -!2 56 "7%"!(+#! #$ #%&". 5#%-!+ -/ #.0-!( (, & -( 8#9".(: +( -//"2

*&6//#%-7+(3 4&"(%,!!+!0."0,/-.+%6#$*&6//#%- %+ *-%%".!(&-(2.-9!%&"-%%"!%+#!#$( +"!%+(%(

(+! " "!%,.+"( ;<2/". ! "#$:=>>? ℄3 4&" &+0&/6 ."0,/-. *-%%".!( -." #$%"! #'*-."2 %# .6(%-/(:

-!2 A.-1-+( 5.#%&".(."$".."2%# %&"' -(#%&%'( *+,!"#, ;A.-1-+( !"#$: =BC? ℄3

4&"#.0-!( #$ */-!%( -."-// 0"!".-%"2 56- (*" +D %+((,": %&" - *%,! -3 <*+ -/'".+(%"'(:

+! *-.%+ ,/-.: -."('-// *#*,/-%+#!( #$ ,!2+E"."!%+-%"2 "//(: /# -%"2 -% %&" "7%."'+%+"( #$ -7"(3

4&" "//( +! %&"(" %+((,"( #!(%-!%/6 0.#9 -!2 2+1+2" %&.#,0&#,% %&" /+$" #$ %&" */-!%: (%-.%+!0

%# 2+E"."!%+-%" -(%&"6-." (&+$%"2-9-6 56%&+( ,!+!%"..,*%"2 0.#9%&3 4&"6"1"!%,-//6 5" #'"

%&" */-!% #.0-!(: /"-2+!0 %# %&"*&6//#%- %+ *-%%".!(9"F,(%'"!%+#!"23

G&6//#%-7+( +(- 26!-'+ *.# "((3 4# 5"%%". ,!2".(%-!2 +%(26!-'+ *.#*".%+"(: %&" - %+1+%6

#$'".+(%"'(&-(%#5"$#//#9"2+! %+'":-!2+!*-.%+ ,/-. %&"- %+1+%6#$0"!"(%&-%-.""7*."(("2

-!2+!%".- %+!'".+(%"'-%+ %+((,"(&-1"%#5"$#//#9"2+!%+'"3 4#(&"2/+0&%#!0"!"(."0,/-%+!0

*&6//#%-7+(:'#2"/(#$*&6//#%-7+((&#,/2+! /,2"2"( .+*%+#!(#$%&"26!-'+ (#$0"!"%+ *.# "(("(

+! '".+(%"'(3

H! %&+( &-*%".: 9" *."("!% (#'" #$ %&" '-+! '#2"/( %&-% &-1" 5""! *.#*#("2 %# 2"( .+5"

-!2 -!-/6(" *&6//#%-7+(3 4&+( *."("!%-%+#! 9+// #!/6 5" *-.%+-/: 5" -,(" %&" /+%".-%,." #! %&"

(,5F" %+(&,0": 9+%&%&" "-./+"(%9#.I(0#+!0 5- I%# <! +"!%G".+#2: -!2 %&" 1".6." "!% %+'"(

!"#$%$&$ ($) (!$%(*)+$%(!"(%$$,$),-%(.$&$#"/(0-. (!$1*.1-%$-0(!+% (!$%+%23!+ ! "+,%"(

+,1.-#+/4-*.5/-3&$)4$-01&"/(,-.1!-4$/$%+%"()+6$.$/(% "&$%2.$&7+/4-/(!$(--&%1.-#+)$)

87 -,1*($.% +$/ $9

:!$ %(.* (*.$ -0 (!+% !"1($. +% 8"%$) -/ (!$ (!.$$ % "&$% (!"( -..$%1-/) (- (!$ (!.$$ ;.%(

1"."4."1!% "8-#$9 <+.%(2 3$ -/%+)$. 1!7&&-("=+% "( (!$ ," .-% -1+ % "&$9 >( (!+% % "&$ (!$

&+($."(*.$ ,-%(&7 1.-#+)$% *% 3+(! )$% .+1(+#$ ,-)$&%2 3!+ ! "($4-.+%$ (!$ 4$-,$(.7 -0 -.4"/

1-%+(+-/% -/(!$ %($,9 :!$/3$)$% .+8$(!$ ,"+/ &"%%$%-0 ,-)$&%(!"(!"#$ 8$$/ 1.-1-%$)

(-4$/$."($ 1!7&&-(" (+ 1"(($./%2"( (!$% "&$ -0 (!$3!-&$ ,$.+%($,2 3!$.$ (!$ %,"&&$%(4$-,$(.+

$&$,$/(% ".$ /$3&7 $,$.4+/4 -.4"/%2 -. !"#$!%"&9 <+/"&&72 3$ )+% *%% ,-)$&% (!"( ("5$

+/(-" -*/( (!$)7/",+ % -0 ,-&$ *&". 1.- $%%$%(!"( -/(.-& 1!7&&-("=+%9

! "#$%%&'()*+ (' ,( .&+ &/* + (%0

?!7&&-("=+%3"%)$% .+8$) "/)%(*)+$) 0.-,)+.$ ( -8%$.#"(+-/-0 8-("/+ "& -.4"/".."/4$,$/(%9

:!$4$-,$(.+ "&"/"&7%+%-0(!$-8%$.#$)1!7&&-(" (+ 1"(($./%"&&-3$)(- !"." ($.+@$1!7&&-(" (+

,-)$%9 A/(!$ 0-&&-3+/423$ )+% *%%1!7&&-(" (+ ,-)$%9

! ! "#$%&' # &)*$+,- '&. /0+ 1%2.+& ,'/#%

:!$,"(!$,"(+ +"/B$-/".)-<+8-/" +CDDEFGDH IJ-0?+%" -/%+)$.$)(!$4.-3(!-0"/+)$"&+@$)

1-1*&"(+-/-0."88+(%9 :!$1.-8&$,3"%(- -*/((!$/*,8$.-01"+.-0."88+(%3+(!+/"7$".4+#$/

-/$ 1"+. -0 ."88+(% C,"&$"/) 0$,"&$J 87 "%%*,+/4 (!"(" 8-./ 1"+. -0."88+(% 1.-)* $ "/-(!$.

1"+. $" !,-/(!2 "/) ."88+(%/$#$. )+$9 :- %-&#$ (!+%1.-8&$, !$+/(.-)* $) (!$ %$K*$/ $

(1, 1, 2, 3, 5, ..., F

k−1

, F

k

, F

k+1

, ...)

3!$.$

F

k+1

= F

k−1

+ F

k

2

F

0

= 1

2 "/)

F

1

= 1

L>)&$.'( &)*2DMME ℄9

O-!"//$% P$1&$. CDFEDGDQRIJ " S$.,"/ ,"(!$,"(+ +"/ "/) "%(.-/-,$. 3"% (!$ ;.%(

3!--8%$.#$) (!$ $=+%($/ $ -0 <+8-/" + /*,8$.% +/ 1&"/(%9 T$ 3"% %1$ +"&&7 +/($.$%($) +/ /*,8$.

;#$ L>)&$. '(&)*2 DMME 2 U!+1,"/'( &)2 HIIF ℄9 T$ -8%$.#$) (!"( (!$ ."(+- -0 (3- -/%$ *(+#$

$&$,$/(% -0<+8-/" + %$K*$/ $ -/#$.4$% "/) !$ -/ &*)$) (!"(V

lim

k→∞

F

k+1

F

k

= φ,

3!$.$

φ

(*./% -*( (- 8$ (!$ 4-&)$/ ."(+-2

φ = (

√

5 + 1)/2

_{≈ 1.61803399}

9 :!$ 4-&)$/ ."(+- -..$%1-/)%(- ")+#+%+-/-0"&+/$ +/(- (3-%$4,$/(%%* ! (!"((!$ ."(+- -0(!$ &$/4(!-0(!$ &+/$

!"

φ

=

AB

AC

=

AC

CB

#"

φ

=

AB

BD

=

BD

BC

"

φ

=

a+b

a

=

a

b

!"#$%&'(') *+,-./0&1 2& 4".1"2+ /"1&0"5"0&0 "14. 46.2&#7&1428 49& %+4". .: 49& /&1#49.: 49&

/"1& 4. 49& /&1#49.: 49& ;"##&%2&#7&14 "2 &<$+/4. 49& %+4". .:49& /&1#49 .:49& ;"##&% 2&#7&14

4.49& /&1#49.:49&27+//&%2&#7&14849& %+4"."2 +//&0#./0&1%+4".( *;,-./0&1%& 4+1#/&0%+61

:%.7 49& #./0&1 2& 4".1

ACB

8 49& %+4". .: 49& /&1#49 .: 49& /.1#&% 2"0& 4. 49& /&1#49 .: 49&

29.%4&%2"0&"2&<$+/ 4.#./0&1%+4".( =:6&0%+649&/+%#&24>.22";/&2<$+%&6"49"18 49&%&7+"1"1#

%& 4+1#/& "2 +/2. + #./0&1%& 4+1#/&( * , =:6& 0"5"0& 49& "% $7:&%&1 & .: + "% /& "14. 46. +% 28

2$ 9 49+4 49& %+4". .: 49& /&1#49 .: 49& /+%#&24 +% 4. 49& /&1#49 .: 49& 27+//&% +% "2 &<$+/ 4.

49& %+4". .: "% $7:&%&1 & 4. 49& /&1#49 .:49& /+%#&24+% 8 49& 27+//&%.: 49& 46. +1#/&2 %&+4&0

"2 +//&0 #./0&1+1#/&28

θ = 360

◦

₋

360

◦

φ

≈ 137.5

◦

(

! !" #$%&&'() (+ ,)((-./0

=1 >/+142 0"?&%&14 4@>&2 .: >9@//.4+ 4" >+44&%12 +1 ;& "0&14"A&0( B1& .: 49& 9+%+ 4&%"24" 2

.: 49& >9@//.4+ 4" >+44&%12 "2 49& 1$7;&% .: .%#+12 "1"4"+4"1# +4 49& 2+7& 1.0& 69" 9 "2 +//&0

!"# % .% &'()* +,-. CD&+1 ./ #*08'EEF 8 G7"49./ #*08HIIJ ℄( D$#+ @ "2 0&1.4&0 ;@

j

"1 49& :./L

/.6"1#( =: .1/@ .1& .%#+1 +>>&+%2 +4 + 4"7& *

j = 1

,8 49& >+44&%1 "2 +//&0 $1"M$#+4&( N1.49&%

<$+14"4+4"5& 9+%+ 4&%"24" .: >+44&%128 "2 49& 1,2.)".3 . #3"*.( =1 +2& .: $1"M$#+4& >9@//.4+O"2

49& 0"5&%#&1 & +1#/& "2 49& +1#/& ;&46&&1 46. 2$ &22"5& .%#+12 +10 "2+>>%.O"7+4&/@ .124+14(

=: 49& 0"5&%#&1 & +1#/& "2 &<$+/ 4.

180

◦

49& >+44&%1 "2 +//&0 1,+/, '(!+8 ": 49& 0"5&%#&1 & +1#/&

"2 0"?&%&14 :%.7

180

◦

8 49& >+44&%1 "2 +//&0 +4,)#*( G>"%+/ >+44&%12 +%& 49& 7.24 :%&<$&14 >+4L

4&%12.;2&%5&0"1>/+142CD&+1./ #*08 'EEF℄( G$1P.6&%84.;+ .8 +1049&7.0&/>/+145)#6,1(4+,+

/'#*,#3# 9+5& 2>"%+/ >9@//.4+O"2 CQ&"19+%04./ #*08HII' ℄8 2&& !"#$%& '(H ( =1 2>"%+/ >/+1428 /"1RL

"1# .%#+12 "1 49&"% .%0&% .: "1"4"+4".1 %&+4&2 + $%5& 49+4 "2 +//&0 ".3.)#/,2. +4,)#*( S9& 4"7&

;&46&&1 "1"4"+4".1 .: 46. 2$ &22"5& .%#+12 "2 +//&0 4*#+/( ')(3( T+ 9 .%#+1 9+21&"#9;.$%"1#

.%#+12 +//&0 "42 (3/# /+( U.11& 4"1# 49& .%#+12 4. 49& .14+ 42 %&+4&2 2& .10+%@ 2>"%+/2 *"1

!"#$%& '()* +,"%-. ,/0..12-3"4 "5 !"#$%&'($( )*"+$","6 2/& 7"8&%#&5 & -5#.& "4 -,,%13":-2&.0

137.5

◦

6 2/& #1.7&5-5#.&(

,-22&%546 4&& !"#$%& '(; ( </& 4,"%-. ,/0..12-3"4 "4 /-%- 2&%"=&7 >02/& 5$:>&% 1? ,-%-42" /"&4(

@5 2/& 42$70 1? ,/0..12- 2" ,-22&%546 $4$-..0 2A1 5&-%&42 152- 24 B.&-7"5# 21 2A1 ,-%-42" /"&4

5$:>&%4C -%& 154"7&%&76 4"5 &2A1 ,-%-42" /"&4 -%&&51$#/ 21 /-%- 2&%"=& 4,"%-. ,-22&%54(

!"#$$%&'(!')*+*,-!(!&') /)0&*,1*+)%23 4"#$$%&'(!')*+*,-!(!&') /)0&*,-)+0 *+03

!"#$%& '(;* B-C D15"$: ,.-52( ) 4,"%-.4 "5 1$52&% .1 EA"4& 1%"&52-2"15 -57 ; 4,"%-.4 "5

.1 EA"4& 1%"&52-2"15( B>C +"7& 8"&A 1? ,"5& 15&( F 1$52&% .1 EA"4& 4,"%-.4 -57 '; .1 EG

A"4& 4,"%-.4( B)6 ;C6 BF6 ';C -%& 154& $2"8& &.&:&524 1? 2/& !">15- " 4&H$&5 &( I" 2$%&4

?%1: JAAA(:-2/(4:"2/(&7$K,/0..1℄(

M" /-%74JM" /-%746 'NO' ℄42$7"&72/& %-7"-. 4,- "5#1?4$ &44"8&,%":1%7"- -.15#2/&#&5&%G

-2"8&4,"%-.( P&"52%17$ &7 ,.-421 /%15%-2"1A/" /"42/&%-2"11?2/&7"42-5 &1?15&,%":1%7"$:

?%1: 2/& &52&% 1? :&%"42&: 21 2/-2 1? 2/& ,%& &7"5# ,%":1%7"$: "5 2/& #&5&%-2"8& 4,"%-.6 4&&

!"#$%& '()* +&,%&-&./0/"1. 12 /3& -,"%04,0//&%. 0.5 ,40/1 3%1. %0/"1( 73&.$89&%- 1%%&-,1.5

/1 1%5&%121%#0."."/"0/"1.: ; 1%%&-,1.5-/1 /3& <1$.#&-/1%#0.( 73&0%%1=- ".5" 0/& 41 >="-&

0.5 1$./&% 41->="-& -,"%04-( 73&,40-/1 3%1. %0/"1 "- /3& %0/"1 125"-/0. &- 12 /=1 1.-& $/"?&

1%#0.-2%18/3& &./&%: 21%&@08,4&/3&5"-/0. &12/3&1%#0..$89&% AB2%18/3& &./&%5"?"5&5

9</3& 5"-/0. &12 /3& 1%#0..$89&% A) 2%18/3& &./&%( C" /$%& 2%18 DE"/ 3"-1. ! "#$:'F ℄(

H&/

P

i

0.5

P

i+1

9& /=1 1.-& $/"?& ,%"81%5"0 ". #&.&%0/"?& -,"%04 0.5

d

i

0.5

d

i+1

9& /3& 5"-/0. &- 12 /3& ,%"81%5"0

P

i

0.5

P

i+1

2%188&%"-/&8 &./&%: /3& ,40-/1 3%1. %0/"1 I

P R

J"-*

P R =

d

i+1

d

i

.

C3<441/0 /" ,0//&%.- 0. 9& 30%0 /&%"K&5 9<

P R

( L3&.

P R

"- 41-& /1 ': /3& ,40./ #&.M

&%0/&- 3"#3 ,0%0-/" 3"&- .$89&%-: 21% &@08,4& -$.N1=&% 0. #&.&%0/& $, /1 IBB: OFJ I 41 >M

="-&: 1$./&% 41 >="-&1% ?" &?&%-0J -,"%04- DP&0. !"#$: 'FF) :+&".30%5/ ! "#$: A;;'℄(

L3&%&0-/3& ,1/0/1 30-

P R = 1.3

0.5 #&.&%0/&- IA: QJ I 41 >="-&: 1$./&% 41 >="-& 1% ?" & ?&%-0J -,"M

%04- D+&".30%5/ !"#$: A;;' ℄( C0//&%.- ="/3 3"#3 .$89&%- 12 ,0%0-/" 3"&- 0%& 044&5 3"#3 1%5&%

,0//&%.-(

73&%& 0%& -&R$&. &- ". =3" 3 /&%8- 0%& /3& -$8 12 /=1 ,%& &5".# 1.&- 0- /3& !"91.0 "

-&R$&. &-: 9$//3& S%-/ /&%8-12 /3& -&R$&. &0%&5"T&%&./ 2%18/3& S%-//&%8-12 /3& !"91.0 "

-&R$&. &( U-&R$&. &9&#"..".#9<'0.5

p

".=3" 3&0 3/&%8 "-/3&-$812/=1 ,%&?"1$-

/&%8-"- 044&5

(p

− 2)

/3 " &&'() -&R$&. &DP&0. ! "#$:'FF) : V8"/3 ! "#$:A;;W ℄*

!"#$%&' ( #&&*%+&#,-#. #

(1, 3, 4, 7, 11, ...)

/& (0&* (00#1 2- (&&#,-#. #3

4&#,-#. # *.&'%- '#1/. '"# &(5# 6(+7 8#9/../.9 8+:(.1

2q + 1

7

q > 1

7 /& (00#1 !"# $

%"&' ;<#(. () %*7 =>>? 7@5/'" () %*7:AAB ℄D

(2, 2q + 1, 2q + 3, ..., a

k−1

, a

k

, a

k+1

, ...), a

k+1

= a

k−1

+ a

k

.

E.(11/'/*. '*1/&'/ "*-&(.1&F/%(0F"+00*'( '/ F(''#%.&7'"#%# (0&*#G/&'&6"*%0#1(.15-0H

'/I-9('#F(''#%.&3 E.6"*%0#1(.15-0'/I-9('# F(''#%.&&#J#%(0*%9(.&/./'/('# (' '"# &(5#.*1#3

!"/& 5#(.& '"(' '"# I-9( +7

j

7 K1#$.#1 (& '"# .-58#% *L *%9(.& (FF#(%/.9 (' '"# &(5# .*1#M

j

_{≥ 2}

3 E.6"*%0#1F"+00*'( '/ (%%(.9#5#.'&

j

*%9(.&(FF#(%(''"#&(5#6"*%0('(.90#&

360

◦

_/j

L%*5 #( " *'"#%3 !"#*%9(.& *L '"# .#G' 6"*%0/./'/('# /. '"# &F( #& (' '"# #.'%#*L '"# &F( #&

8#'6##.'"#*%9(.&*L'"#F%#J/*-&6"*%03 !"#1/J#%9#. #(.90#8#'6##.(.*%9(.*L(6"*%0(.1

'"# .#(%#&'*%9(. *L '"# .#G' 6"*%0/&

θ = 180

◦

_/j

3 !"# 1/J#%9#. # (.90#

θ

(. 8# &5(00#%'"(.

180

◦

/j

3 E.'"/& (&#7 (&F/%(0(%%(.9#5#.' (.8#*8&#%J#1(.1/& (00#15-0'/I-9('#3 N-0'/I-9('# F(''#%.& (%# "(%( '#%/O#1 8+ '"# .-58#% *L F(%(&'/ "/#& 6"/ " (%#5-0'/F0#& *L '"# .-58#% *L

F(%(&'/ "/#& *L -./I-9('# F(''#%.&7 &## P/9-%#=3Q 3 E.5-0'/I-9('# F(''#%.&7 '"# I-9( +K

j

M '-%.&

*-' '* 8# #,-(0 '* '"# 9%#('#&' *55*. 1/J/&*% *L '"# F(%(&'/ "+ .-58#%& (.1 /& /. L( ' '"#

.-58#% *L 9#.#%('/J# &F/%(0&3 @## P/9-%#=3RL*% & "#5(& *L '"# F"+00*'( '/ F(''#%.&3

P0*6#%& "(J# *L'#. 6"*%0#1 F"+00*'(G/&7 6/'" &#F(0& (.1 F#'(0&(%%(.9#1 /. 6"*%0&3 S/'"/.

'"# 6"*%0#1 F(''#%.&7 '"# .-58#% *L *%9(.& /. #( " 6"*%0 L-%'"#% "(%( '#%/O#& '"# F(''#%.&7

(& F(''#%.& 6/'" : *% T *%9(.& F#% 6"*%0 (%# (00#1 1# -&&('# (.1 '%/ -&&('# %#&F# '/J#0+7 &##

P/9-%# =3B3

!* &-5 -F7 '"# F"+00*'( '/ F(''#%.& (. 8# 1#& %/8#1 8+ '"# 1/J#%9#. # (.90# K

θ

M (.1 '"#

.-58#% *L *%9(.& /./'/('/.9&/5-0'(.#*-&0+ K

j

M(&L*00*6&D

j

U

1 :



1/&'/ "*-&

,

/L

θ = 180

◦

_,

&F/%(0

,

/L

θ

6= 180

◦

_,

j

_{≥ 2 :}

(

6"*%0#1

,

/L

θ = 180

◦

_/j,

5-0'/I-9('#

,

/L

θ

6= 180

◦

_/j.

!"# '%(.&/'/*. L%*5 1# -&&('# F"+00*'(G/& '* &F/%(0 *.# (. 8# *8&#%J#1 /. 5(.+ F0(.'&3

P*% #G(5F0#7 0#(J#& *L 4%(8/1*F&/& (' $%&' (%# (%%(.9#1 6/'" ( 1/J#%9#. # (.90# 0*&# '*

90

◦

9#.#%('/.9(1# -&&('#F(''#%.3 4%(8/1*F&/&9%(1-(00+9#.#%('#&&F/%(0F"+00*'(G/&(&/'1#J#0*F&

L%*5J#9#'('/J#9%*6'"'*V*6#%/.93 !"#&/0/,-#&K/3#34%(8/1*F&/&L%-/'&ML*%5(&F/%(0F"+00*'(G/&

6/'" ( 1/J#%9#. # (.90# 0*&# '*

137.5

◦

!"#$%& '()* +",$#-.& !"/01- " 3-..&%1

(j = 2)

( 45"6 #7810 -97 "$8 5-6 3-%-6." 57 1$8/&%6

(10, 16) = 2(5, 8)

:5" 5 -%& .:0 016& $.";& &9&8&1.6 0<

2

× (1, 2, 3, 5, 8, 11, ...).

=" .$%& <%08 >:::(8-.5(68".5(&?$@357990℄(

!"#$%&'(B* C50%9&?3-..&%1 D.%" $6-..&E( =" .$%& <%08 >:::(8-.5(68".5(&?$@357990℄(

!" #$%&&'()*+, )( ($- .-/+,(-. , )&-1 2%3).+ )& .'2-&,

45& 5"#5%&#$9-%".7 0< 357990.- ." 3-..&%165-6?%-:1 .5& -..&1."01 0<6 "&1."6.6 <%08 ?"F&%&1.

?08-"16( +"090#"6.6G 3576" "6.6G 8-.5&8-." "-16G -6 :&99 -6 083$.&% 6 "&1."6.6 5-;& 3%0306&?

80?&960<.5"635&108&101( H"F&%&1.8& 5-1"6865-;&/&&16$##&6.&?.0&I39-"150:.5&0%#-16

-%& <0%8&? -1? 50: .5&"% 306"."016 -%& ?&.&%8"1&? -. .5& J-1K 0< .5& 8&%"6.&8( H&63".& .5&

;-%"&.7 0< 8& 5-1"686G .5&7 -99 "1;09;& .5& "1J$&1 & 0< 3%&L&I"6."1# 0%#-16 01 .5& 306"."01 0<

<$.$%& 0%#-1"1"."-."016(

M1'NBN O0<8&"6.&%>O0<8&"6.&%G 'NBN ℄3%0306&? - %$9& ?&6 %"/"1# .5& "1"."-."01 0< - 1&: 0%L

#-1( O& 6$##&6.&? .5-. .5& 1&: 0%#-1 "1"."-.&6 3&%"0?" -997 D"(&( .5& 39-6.0 5%01 "6 %&#$9-%E

-. .5& 9-%#&6. -;-"9-/9& 63- & 9&<. /7 09?&% 0%#-16( P10: -1? P10: >P10: !"#$G ' Q' ℄ <$%.5&%

"1;&6."#-.&? .5"6 "?&- /7 6$%#" -997 "609-."1# .5& 6".& 0< -1 "1 "3"&1. 3%"80%?"$8( R1 "1 "3"&1.

3%"80%?"$8 D0% 80%& #&1&%-997 -1 "1 "3"&1. 0%#-1E "6 - 3%"80%?"$8 "1 ".6 S%6. 6.-#&6 0< ?&;&9L

038&1.E( 450$#5.5& 1&I. 3%"80%?"$8 :-6 <0%8&? -. .5& &I3& .&? 306"."01 :".5 -?";&%#&1 &

-1#9& 906& .0

137.5

◦

G .5& 6& 01? 1&I. 3%"80%?"$8 -33&-%&? 906&% .0 .5& "609-.&? 3%"80%?"$8

.5-1 $6$-9 D:".5- ?";&%#&1 & -1#9&-/0$.

165

◦

EG 6&& !"#$%&'(N( P10: -1? P10: $6&? .5& ?"6L

"#$%&' ()* + -'./012 3-45516/ 6# 3/66'&70( "1& '/ -3/66'&78 9%$/ 4

j

/7: :#;'&$'7 ' /7$5'

θ

!"#$%& '() * ,&-./ 01 2,& &34&%"-&52 01 *506 .57 *506' 8,& "5 "4"&52 4%"-0%7"$-

I

1

"/ "/09.2&7:;.5"5 "/"05' 8,&5&324%"-0%7"$-

I

2

"5"2".2&/6"2,.7"<&%#&5 &.5#9& 90/&20

137.5

◦

=

2,& /& 0575&324%"-0%7"$-

L

3

"5"2".2&/6"2, .7"<&%#&5 &.5#9&01

165

◦

90/&%2,.5 $/$.920

I

1

'

6"2, .-"5"-$-7"/2.5 &1%0-097&%0%#.5/' 8,"/-& ,.5"/-"/>$"2&/"-"9.%202,.201?01-&"/@

2&% &3 &422,& 4&%"07" "5"2".2"05"-40/&7:;2,& ?01-&"/2&% -& ,.5"/-= 6," , "/.:/&52 "5 2,&

4%040/"2"05 01*506.57 *506'

80&349."52,&4,;9902. 2" 4.22&%5/=02,&%-& ,.5"/-/ 6&%&.9/04%040/&7' A502,&%:"04,;/" .9

-& ,.5"/- "/ :./&7 05& 2,& 7"B&%&52".9 #%062, :&26&&5 2,& 0$2&% 9.;&%/ 01 2,& *AC .57 2,&

"52&%5.9 9.;&% 012,&*AC'D52,"/-& ,.5"/-2,& 0$2&%9.;&%/#%061./2&%2,.5 2,&"52&%5.9 9.;&%

6," , 9&.7/ :$ E9"5# .2 2,& /$%1. &=%&/$92"5# "5 0%#.510%-.2"05 FG%&&5= HHI =G%&&5= HHJ ℄'

D5 H L=* ,0$2&4%040/&7.2,&0%;207&2&%-"5&2,&49. &01"5"2".2"05010%#.5/' A 0%7"5#20

,"/2,&0%;= .4%"-0%7"$-4%07$ &/. ,&-" .9"5,":"20%2,.24%&<&52/2,&10%-.2"05015&60%#.5/

5&.%:;' 8,"/"7&."/E5065./M"5,":"20%; N&972,&0%;M' A 0%7"5#20 2,"/2,&0%;= 4%"-0%7"..57

2,& -&%"/2&- /$--"2 .%& /$%%0$57&7 :; "5,":"20%; N&97/ 2,.2 .5 :& #%.7"&52/ 01 5$2%"&52/ 0%

/"#5.99"5#-09& $9&/' 8,&N&97/#&5&%.2&7 :;097&% 4%"-0%7".4%&<&52/ "5"2".2"0501.5&60%#.5'

O0$.7;.57P0$7&% FO0$.7; ! "#$= HHJQ.R =O0$.7; ! "#$= HHJQ:R =O0$.7; !"#$= HHJQ R ℄

4%040/&72607;5.-" .9-07&9/6"2,"5,":"20%;N&97/.%0$574%"-0%7".' 8,&;$/&72,&/&-07&9/

204%0<"7&.5"5@7&42,/2$7;014,;9902. 2" 4.22&%5/' S02.:9;= 2,&;6&%&.:9&20%&4%07$ &-0/2

01 2,& E5065 /4"%.9 .57 6,0%9&7 4.22&%5/ 7&/ %":&7 "5 2,& 4%&<"0$/ /& 2"05= /,06"5# 2,.2 2,&

,;402,&/"/ 01 "5,":"20%; N&97/ 6./ . /.2"/1. 20%; &349.5.2"05 10% 2,& &-&%#&5 & 01 4,;9902.3"/

1%0-2,"/906&% / .9&7;5.-" .9/;/2&-' D5P,.42&%T6&/2$7;.-07"N&7<&%/"050105&012,&"%

!"! #$%&'( )*$)$(&% +, -$./%, /0% 1$.%&*

"# $ %&'(&% )* +,'&& -$-&'% ./)0$12 ! "#$3 4456$7 3 /)0$12 ! "#$3 445687 3

/)0$12 ! "#$3 4456 7 ℄/)0$12 $#1;)01&' '&<(%(+&1+,& (1&$% )*=)*>&(%+&'.=)*>&(%+&'3 ?5? ℄

$#1 @#)A $#1 @#)A .@#)A ! "#$3 4B! ℄ *)' +,& *)'>$+()# )* $ #&A )'C$#D E,&2 -')-)%&1 +A)

12#$>( $F >)1&F% +) '&-')10 & -,2FF)+$ +( -$++&'#%D E,& G'%+ )#& (% 8$%&1 )# +,& )#1(+()#%

)* (#(+($+()# )* $ #&A )'C$# -0+ *)'A$'1 82 =)*>&(%+&' $#1 +,& %& )#1 )#& (% 8$%&1 )# +,&

)#1(+()#% -')-)%&182@#)A $#1 @#)AD

!"#$ %&'()* "(+!#!$!,- $.( ./0&$.(#(# 0"&0&#('1/ 2&3%(!#$("

/)0$12$#1;)01&''&H&I$>(#&1+,&,2-)+,&%&%)*=)*>&(%+&'.=)*>&(%+&'3 ?5? ℄$#1'&-')10 &1

%-('$F $#1 1(%+( ,)0% -$++&'#% (# $ -,2%( % &I-&'(>&#+ $% A&FF $%(# #0>&'( $F %(>0F$+()#% )* $

12#$>( $F>)1&FD

J )'1(#C +) +,& ,2-)+,&%&% -')-)%&1 82=)*>&(%+&'3 $# )'C$# $--&$'% -&'()1( $FF23 A(+, $

GI&1 +(>& -&'()1

T

6(D&D +,& -F$%+) ,')#&(% '&C0F$'73 $+ +,& -&'(-,&'$F K)#& )* >&'(%+&> (# +,&

F$'C&%+ %-$ &F&*+ 82)F1&' )#&%D L& $0%& )*+,& C')A+,3 +,& )'C$#% >)<& $A$2'$1($FF2 *')>+,&

&#+'& )* +,& @JMA(+, $ <&F) (+2

V (r)

3

r

8&(#C +,& 1(%+$# &)* +,& -'(>)'1(0> *)'>+,& @JM

&#+&'D N# & $# )'C$# (% *)'>&13 (+% $#C0F$' -)%(+()# 1)&% #)+ ,$#C& )<&' +(>& )0+%(1& )* $

'&C()# )* '$1(0%

R

0

D E,&@JM (%%0--)%&1+) 8& $I(%2>>&+'( D

4.( (50("!%(,$

/)0$12 $#1 ;)01&' -&'*)'>&1 $ -,2%( % &I-&'(>&#+ (# A,( , +,& @JM A$% >)1&FF&1 $% $

1(%, GFF&1 A(+, %(F( )# )(F -F$ &1 )# $ <&'+( $F >$C#&+( G&F1D E,& (#(+($+()# )* +,& )'C$#% A$%

>)1&FF&1 82 1')-F&+% )* *&'')>$C#&+( F(O0(1 *$FF(#C $+ +,& &#+&' )* +,& 1(%, $+ '&C0F$' +(>&

(#+&'<$F%

T

D E,& 1')-F&+% A&'& -)F$'(K&1 (# +,& -'&%&# & )* +,& >$C#&+( G&F1 A,( , ,$1 $

'$1($F C'$1(&#+P >(#(>0> $+ +,& &#+&' $#1 >$I(>0> $+ +,& -&'(-,&'2D E,&-)F$'(K&11')-F&+%

'&-&FF&1 )#& $#)+,&'D E,&2A&'&$++'$ +&1+) +,& -&'(-,&'2)* +,&1(%, A,&'&+,& >$C#&+( G&F1

A$% >$I(>0>3 +,0% +,&2 >)<&1 $A$2 *')> +,& &#+'&D L2 +0#(#C

T

+,& &I-&'(>&#+ '&$+&1

1(%+( ,)0% -,2FF)+$ +( >)1& 6*)' F$'C& <$F0&% )*

T

7 $#1 %-('$F -,2FF)+$ +( >)1&% 6*)' %>$FF

<$F0&%)*

T

73%&&Q(C0'& D4 D /)0$12$#1;)01&''&-)'++,$+1')-F&+%(#+&'$ +&1A(+,&$ ,)+,&'

(#%(1& $ '&C()# )* '$1(0% )*

R

c

D N0+%(1& +,(% '&C()# +,& 1')-F&+% A&'& +)) *$' *')> &$ , )+,&' +) (#+&'$ + $#1 +,&2 )#F2 1'(*+&1 $A$2 '$1($FF2 *')> +,& &#+&'D E,& $0+,)'% %,)A&1 +,$+ +,&

-$++&'#%-')10 &1A&'& )#+')FF&182$# $1(>&#%()#$F-$'$>&+&'

G = V

0

T /R

c

A,&'&

V

0

A$%+,& $1<& +()# <&F) (+2 A,( , (% )#+')FF&1 82 >$C#&+( G&F1D E,&2 $F%) %,)A&1 +,$+

G = ln(P R)

3

!"#$%&$ ()*%+),- ."/0$1!2+),- " /0$1!2+),-)3($4(-1)1,-1

"#$%&' ()* +#,'&'-. /01223.4 .# /4..'&-6 37.4#-'8 71 .0' '9/'&#:'-.( ;4< +#6.# 03%6 :38'=

8#>'&$'- ' 4-$2' #6

180

◦

;7<?/#&42 :38' @#.0 /4&46.# 01 -%:7'&6 ; = A<( B0' $'-'&4.#>' 6/#&42

#6&'/&'6'-.'8 71.0'8460'82#-'( ; <?/#&42 /4..'&-3C0#$0'& 3&8'&@#.0/4&46.# 01-%:7'&6 ;D=

E<( F# .%&' C&3:G+3%481 ! "#$= ))H;4< ℄(

!"#$

J46'83-.0'473>':'-.#3-'801/3.0'6'6=+3%4814-8K3%8'&/&3/36'84 3:/%.4.#3-42:38'2(

L- .0#6 :38'2= .0' 616.': @46 C%&.0'& 6#:/2#M'8 71 6%//36#-$ .04. .0' ?NO @46 /24-4& 4-8

3&$4-6 @'&' #& 2'6 3C &48#%6

d

0

( B3 :4P' .0' 3&$4-6 4//'4& #- .0' 24&$'6. 6/4 ' 2'C. 71 328'& 3-'6= #. #6 466%:'8.04. 328'& 3&$4-6 '9'&.'84 &'/%26#>' C3& ' 3- #- #/#'-. 3&$4-6( B0#6 4- 7'

>#'@'8 4646#:/2' @41.3 :38'26/4 ' 3 %/4.#3- 3C 3&$4-6(

O3&' /&' #6'21= #- .0#6 :38'2 3&$4-6 4//'4&'8 3- 4 #& 2'

C

@#.0 &48#%6

R

0

@#.0 4 .#:' /'&#38# #.1 3C

T

3- 4 /24-4& :'&#6.':( B0'1 :3>'8 4@41 C&3: .0' '-.'& @#.0 >'23 #.1

V (r)

(

Q4 0 3&$4- &'4.'8 4 &'/%26#>' '-'&$1=

E(d)

= @0'&'

d

#6 .0' 8#6.4- ' 7'.@''- .0' /&#:3&8#%:

4-8 4 64:/2#-$ /3#-.

S

3- .0' #& 2'

C

= 6'' "#$%&' ( R ( B0' &'/%26#>' '-'&$1 &'/&'6'-.6 .0'

#-0#7#.3&1 M'286 /&'>#3%621 :'-.#3-'8( S'.

{P

1

, ..., P

l

}

7' .0' 6'. 3C '9#6.#-$ /&#:3&8#4= .0' 3:7#-'8&'/%26#>' '-'&$1

h

3-.0' /3#-.

S

#6 $#>'- 71*

h(S) =

l

X

i=1

E(d(P

i

, S)),

; ( < @0'&'

d(P

i

, S)

#6.0'8#6.4- '7'.@''-.0'/3#-.

S

4-8.0'/&#:3&8#4

P

i

=

1

≤ i ≤ l

( B0'C%- .#3-6

E(d)

.04.@'&'%6'8#- .0'6#:%24.#3-6@'&'

1/d

=

1/d

3

3&

exp(

−d/l)

( +3%4814-8K3%8'& &'/3&.

.04..0'6' C%- .#3-6$'-'&4.'8 T%42#.4.#>'21 .0' 64:'/4..'&-6(

B38'.'&:#-'.0' 23 4.#3-3C4-'@/&#:3&8#%:3- #& 2'

C

= .0'C%- .#3-

h

#6 42 %24.'8C3&4

8#6 &'.' 6'.3C /3#-.63- #& 2'

C

( N-3&$4-4//'4&64. '4 0/'&#38

T

4. .0'/3#-. C3& @0# 0 .0'

C%- .#3-

h

#6:#-#:%:( N6:'-.#3-'87'C3&'=.0' 6#:%24.#3-63C.0#6:38'2$'-'&4.'88#6.# 03%6

"#$%&' ( )* + -'./ 01 2-' 34/5/& .06'4 01 2-' +78( 9-' #5 #3#'52 0&$/5 01 5%.:'&

n

#;

&'3&';'52'6 #5 :4/ <( 9-' 30;#2#05 01 2-' 0&$/55%.:'&

n

#; / 1%5 2#05 01 30;#2#05; 01 3&'=#0%;

05';>

n

− 1

>

n

− 2

> ((( ( ?2 #; 6'@5'6 :A2-' 6#='&$'5 ' /5$4'

θ

> /56 2-' &/2#0 01 2-'#& 6#;2/5 ';

20 2-' '52'&> !"#$% '(%)* ("$+%( 9-' 0&$/5;.0='&/6#/44A 1&0. '52'& B#2- / ='40 #2A 01

V (r)

>

r

:'#5$ 2-' 6#;2/5 ' 1&0. 2-' '52'&( 9-' #5-#:#20&A @'46 4#5'; 01 302'52#/4

E(d)

/&' ;-0B5 /2 '/ - 0&$/5( 9-' &'6 6022'64#5' &'3&';'52; 2-' $'5'&/2#=' ;3#&/4(

! #$% &#%!'( )!*+,+-+$. -/! /01#-/!,!, 1)#1#,!% 20 $#3 4$% $#3

+50B/56 +50BC+50B*$ "!,> DEF ℄3&030;'6/.06#@ /2#05202-';'201-A302-';';3%210&B/&6

:AH01.'#;2'&( 9-'A;#.34A &'.0='6 2-' 2#.' 3'&#06# #2A01 #5#2#/2#05 01 0&$/5; /56 ;%$$';2'6

2-/2 2-' #5#2#/2#05 01 /5 0&$/5 2/<'; 34/ ' /2 2-' 3'&#3-'&/4 I05' 01 2-' +78 B-'&' /56 B-'5

2-'&' #; '50%$- ;3/ ' 10& #2; 10&./2#05( +#.#4/& 20 2-' -A302-';'; 01 H01.'#;2'&> 0&$/5; .0='6

/B/AB#2-/&/6#/4='40 #2A01

V (r)

( 9-'/5$%4/&30;#2#05;013&#.0&6#/6#6502 -/5$'0='&2#.'

0%2;#6'01 / &'$#0501 &/6#%;

R

0

(

J/;'605 2-#; 5'B;'201 -A302-';'; K0%/6A/56 L0%6'& 3&030;'6/ ;' 056.06'4#5

B-# -2-'A /;;%.'6 2-/22-' ;-/3' 012-' +78 #;3/&/:040#6/4 /56 3&#.0&6#/ -/=' /I#.%2-/44A '405M

$/2'6 ;-/3'( K0%/6A /56 L0%6'& 200< 2-';' /;;%.32#05; #520 / 0%52 :A / #52&06% #5$ /

;%334'.'52/&A 3/&/.'2'& /44'6 2-' 05# #2A( +#.#4/& 20 2-' @&;2 .06'4> '/ - 3&#.0&6#%. &'M

/2'6 /&'3%4;#=' '5'&$A

E(d)

/562-' 0.:#5'6&'3%4;#=' '5'&$A

h

05 2-' 30#52

S

$'5'&/2'6 :A

2-' 'N#;2#5$ 3&#.0&6#/

{P

1

, ..., P

l

}

#; $#='5 :A2-' OP%/2#05 ( (

90 @56 B-'&' / 5'B 3&#.0&6#%. /33'/&;> 2-'A /4 %4/2'6 052#5%0%;4A 2-' =/4%' 01 2-'

"#$%&' ( ) *&+,-'. +/ 01' &'2%-.#3' '4'&$5 /%4 0#+4

E(d)

7 89%:0#+4 ; (<=7 /+&

d

0

= 1

:4>

α

∈ {1, 3, 5, 7, 9}

( ?.

α

#4 &':.'. 01' %&3' @' +A'. .0''2'& ;

α = 1

#. 01' &#$10A+.0 %&3' :4>

α = 9

01' -'/0A+.0=(

/%4 0#+4

h

#.A#4#A%A B1'4'3'& 01' 3:-%'+/ /%4 0#+4

h

#.@'-+B: '&0:#401&'.1+->

E

S

( C1'/%4 0#+4

E(d)

%.'>#4 01' .' +4>A+>'- #.

E(d) =

−1 +



tanh α

_d

d

0



−1

−1 + (tanh α)

−1

,

; (<=

B1'&'

tanh

#. 01' 152'&@+-# 0:4$'40 :4>

α

&'2&'.'40. 01' .0#D4'.. +/ 01' #40'&: 0#+47 .'' "#$E

%&' ( ( C1' /%4 0#+4 #. : >' &':.#4$ /%4 0#+4 +/ >#.0:4 ' :4> 01' 1#$1'& #.

α

01' .0#D'& :&'

#40'&: 0#+4.(

C1':%01+&..1+B'>01:001'.0:0'+/01'.5.0'A :4@'>'. &#@'>@501'/+--+B#4$2:&:A'0'&)

Γ = d

0

/R

0

.

; (F=

G/ 01' +4# #05#. 0:H'4 #40+ : +%40

Γ = l

1

/R

0

,

B1'&'

l

1

= d

0

/N

1/2

#. 01' +&01+&:>#:- .#I' +/ 2&#A+&>#:7

1/N

&'2&'.'40#4$ 01' +4# #05( J''

"#$%&' ( <#4 B1# 101'@#/%& :0#+4>#:$&:A+/>#3'&$'4 ':4$-'.:.:/%4 0#+4+/

Γ

#.2&'.'40'>(

8: 1 2+#40 #4 01' >#:$&:A +&&'.2+4>. 0+ : .0':>53:-%' +/ >#3'&$'4 ' :4$-'. +@0:#4'> #4 .#AE

%-:0#+4.( ?. +4' :4 +@.'&3' #4 01' ,$%&'7 01' A+>'- &'2&+>% '. >#D'&'40 .0':>5 215--+0: 0#

2:00'&4. 1:&: 0'&#I'> @5

Γ

(

K#+-+$# :-&'-'3:4 '+/01'#41#@#0+&5,'->.#.4+0A'40#+4'>#401'A+>'-.2&+2+.'>@5L+%:>5

:4> M+%>'&( C1'5 +4-5 +4.#>'&'> :4 :@.0&: 0 /+&A +/ .% 1 ,'->.( C1'&'/+&'7 A% 1 &'.':& 1

"#$%&' ( )* +#,$&,- ./ 0#1'&$'2 ' ,2$4'

θ

,5 , /%2 6#.2 ./

Γ

( 78#&,4 8,66'&25 ,&' 4,9

:'44'0 :; 6<' 2%-:'& ./ 4. =>#5' ,20 .%26'& 4. =>#5' 8,&,56# <#'5

(m, n)

( ?<.&4'0 8,69

6'&25 ,&' 4,:'44'0 :; 6<' 2%-:'& .&$,25 ,6 ', < ><.&4

(j, j)

(

1/N = 3

@ ,20

α = 8

( "#$%&'

/&.- A+.%,0; ! "#$@ BB!C:D ℄(

7-#6< !"#$A7-#6< ! "#$@)FF! ℄8&.8.5'0,-.0'4#2><# <6<'58,6#,40#56&#:%6#.2,20,$'./

.40'&.&$,250'6'&-#2'6<'#2#6#,6#.284, './,2'>.&$,2( 7#-#4,&6.6<'5' .20-.0'48&.8.5'0

:;+.%,0; ,20G.%0'&@6<' 8&'9'H#56#2$.&$,25 &',6'0 &'8%45#1' '2'&$;-.0'44'0 :;,2#2<#:#9

6#.2/%2 6#.2( I2.&$,2,88',&'0><'&',20><'26<'5%-./#2<#:#6.&;#2J%'2 './8&'9'H#56#2$

.&$,25 >,5 :'4.>, 8&'0'K2'0 6<&'5<.40

E

s

( L. &'8&.0% ' ><.&4'0 ,20 -%46#M%$,6' 8,66'&25, 5#-%4,6#.2 -.0'4 >#6< 6>. #2<#:#6#.2 /%2 6#.25 >,5 8&.8.5'0( N.6< /%2 6#.25 &'8&'5'26'0 6<'

#2<#:#6.&; .26&#:%6#.2 ./ .40'& 8&#-.&0#, :;6,=#2$ #26. , .%26 6<'#& 58,6#,4 0#56&#:%6#.2 ,20

6<'#& ,$'( O2 ,00#6#.2@#2 .2' ./ 6<' /%2 6#.25@6<' #2<#:#6.&; 'P' 6 ./2'>4; #2#6#,6'0 .&$,25 #29

&',5'5$&,0%,44; .1'& 6#-'( L<#5,44.>'0 6. &',6' 8<;44.6, 6# 8,66'&25#2><# <5'1'&,4.&$,25

#2#6#,6' 5#-%46,2'.%54; #2 ,&.:%56 -,22'&(

Q.&'8&' #5'4;@ 4'6

{P

1

, ..., P

l

}

:' 6<' 5'6 ./ 'H#56#2$.&$,25 ,20

t

i

:' 6<' ,$' ./ .&$,2 C8&#9 -.&0#%-D

P

i

@

1

≤ i ≤ l

@ 6>. #2<#:#6#.2 /%2 6#.25,&' 0'K2'0 ,5*

h

1

(S) = (1.0

− e

−b

d

t

i

)

l

X

i=1

1

d(P

i

, S)

e

−b

1

t

i

_,

C (RD

h

2

(S) =

l

X

i=1

1

d(P

i

, S)

e

−b

2

t

i

_,

C (SD ><'&'

d(P

i

, S)

#5 6<' 0#56,2 ' :'6>''2 .&$,2

P

i

,20 , 5,-84#2$ 8.#26

S

.2 6<' ,8'H(

b

1

,20

"#$%&' ( )* + -'./ 01 2-' 230 #4-#5#2#04 1%4 2#046 &'/2#4$ / 2&# %66/2' 7/22'&4( 84-#5#2#4$

#49%'4 ' 01'/ -0&$/4#66-0345:;'<=<#4'6/2 '/ -0&$/4( >'=<#4'6 0&&'6704=20

h

2

1%4 2#04 /4= 5</ ? ;'<= <#4'6 0&&'6704= 20

h

1

1%4 2#04( 84-#5#2#04 1%4 2#046

h

1

/4=

h

2

/&' =';4'= #4 @A%/2#046 B (CD /4=B (E D &'67' 2#F'<:(

0&$/4 -/4$#4$ $&/=%/<<: 0F'& 2#.'(

G-'1%4 2#04

h

1

&'7&'6'426/7-/6'=#4-#5#2#4$ /H' 201/4'3<:10&.'=0&$/4/4=/<<0362-' 6#.%<2/4'0%6 /77'/&/4 ' 01 6'F'&/< 0&$/46 /6#4 3-0&<'=/4= .%<2#I%$/2' 7-:<<02/ 2# 7/22'&46(

G-'7&05<'.3#2-2-#6='</:.' -/4#6.#62-/22-'2-&'6-0<=

E

s

/45'&'/ -'=10&4'#$-50%&#4$ 6/.7<#4$ 70#4263#2- 6-0&27</620 -&04( 84 046'A%'4 'J6'F'&/< 0&$/46 /4/77'/& <06' 20

'/ -02-'&( G07&'F'422-#67&05<'.2-'#4-#5#2#04 1%4 2#04

h

2

#6/=='= 202-'6:62'.( K7&#.0&#=%. /77'/&6 3-'4/4= 3-'&' 502- 1%4 2#046

h

1

/4=

h

2

/&'502- 5'<03 2-' 7&'=';4'= 2-&'6-0<=

E

s

J 6'' "#$%&' ( ) ( G-#6 .0='< #6 /5<' 20 $'4'&/2' .0&' &05%62 3-0&<'= /4= .%<2#I%$/2' 7/22'&46

0.7/&'= 20 2-' .0='<7&0706'=5:L0%/=: /4= M0%='&(

M-/7./4 /4= "'&&: NM-/7./4 ! "#$J OP! ℄ 7&0706'= / 0.7<'.'42/&: .' -/4#6. 01 #4R

-#5#20&: ;'<=6( G-#6.' -/4#6. #4 <%='= /40&$/4 #4#2#/2#4$1/ 20&( G-' ='7<'2#04 012-' 1/ 20&

5: 7&'R'S#62#4$ 0&$/46 7&'F'42'= /77'/&/4 ' 01 / 4'3 0&$/4 #4 2-' F# #4#2: 2-' 0&$/46( G-#6

"#$%&##&%'"$()$ '+,)+,+,-$%)+,.$'"+%/0&"%1$,2' !"#$3 4555 ℄7 /0&"%1$,2' !"#$3 45589$: ℄;,+<

;+#&2 '1$' %&= +,-$%# 2&;(&'& $>?"%3 '1"# -&%&,$'&# $ +% &%',$'"+% -,$2"&%' +) $>?"% '1$' "#

&@>"A$(&%' '+ "%1"B"'+,C D&(2#7 E1& ,+(& +)$>?"% "% '1& ,&->($'"+% +) ;1C((+'$?"#"# 2"# >##&2 "%

'1& %&?' #& '"+%7

!" #$%& (%)* *&+(%),$*- $.

!"! #$%&'()*+, +- ).$ $,)0(' 1+,$ +- 2$0*3)$2

E1& $&,"$( '"##>&# +) ;($%'# $,& -&%&,$'&2 ),+. '1& #1++' $;" $( .&,"#'&. 9FGH:7 E1& FGH "#

2"A"2&2 "%'+ )>% '"+%$( 2+.$"%# '1$'$,& 1$,$ '&,"I&2 BC 2"J&,&%' ;$''&,%# +) -&%& &?;,&##"+%7

E1& &%',$(I+%&9KL:#"'>$'&2$''1&'";+)'1&.&,"#'&. +%#"#'"%-+)#'&. &((#"#'1&2+.$"%+)

&?;,&##"+%+) KMGNGEG8 9KMN8:-&%&7 F'&. &((#"% &%',$( I+%& 2"A"2&$%2 '1&2$>-1'&, &((#

&%'&, '1& ;&,";1&,C +) .&,"#'&.3 $((&2 ;&,";1&,$( I+%& 9OL:3 =1&,& (&$) $%2 P+=&, ;,".+,2"$

$,& )+,.&27 Q% $ ;,&2" '$B(& I+%& R>#' B&(+= '1& KL3 '1& +,-$%"I"%- &%'&, 9SK:3 '1& .$#'&,

,&->($'+,-&%&TUFKVWM9TUF:"#&?;,&##&23#&&X"->,& 7 Y 7 TUF>;,&->($'&#'1&&?;,&##"+%

+)KMN8=1"(&"%,&'>,%"% +%R>% '"+%="'1'1&,& &;'+,Z"%$#&KMGNGEG 9KMN :,&;,&##&#'1&

&?;,&##"+% +) TUF7 E1"# %&-$'"A& )&&2B$ Z %&'=+,Z "# B&("&A&2 '+ ,&->($'& '1& .$"%'&%$% & +)

'1& &%',$( I+%&$%2 "'#-&%&'" "2&%'"'C7 E1"# +%',+(#'1& .$"%'&%$% & +) '1&#'&. &(( "2&%'"'C

"%.&,"#'&.#/F$B(+=#Z"3 455[ 3F1$,.$ !"#$3 4558 ℄3#&&X"->,&# 7 \$%2 7 ] 7 Q%K1$;'&,\3=&

#'>2C '1& ,+(& +) %&-$'"A& )&&2B$ Z %&'=+,Z B&'=&&% TUF3 KMN 3 $%2 KMN8 "% ,&->($'"%- '1&

#"I&+) &%',$(I+%&7 Q%;$,'" >($,3=&#'>2C'1&I+%&+)&?;,&##"+%+)TUF+%$'1,&&<2".&%#"+%$(

.+2&(+)'1& FGH7E1&,&)+,&3"% '1& )+((+="%- =&;,&#&%'$ #'$'&+)$,'$B+>';$''&,%)+,.$'"+%

"%A+(A"%- TUF$%2 .+2&(# '1$'1$A& B&&% ;,+;+#&2'+ 2&# ,"B& '1&#& ;,+ &##&#7

O,".+,2"$ '1$' $,"#& +% '1& P$%Z# +) '1& FGH $% 2&A&(+; "%'+ P+=&,# $%2 "% '1"#

$#& $,& $((&2 P+,$( .&,"#'&.7 E1"# ;,+ &## "# +%',+((&2 BC -&%&# /M+1.$%% ! "#$34554 3

O$, C !"#$3 ^^! 3 O$, C ! "#$3455\ ℄7 _"J&,&%' #'$-&# $% B& "2&%'"D&2 2>,"%- 2&A&(+;.&%'

+) $ P+=&,7 F.C'1 ! "#$ /F.C'1 ! "#$3 ^^5 ℄ 2"A"2&2 '1& 2&A&(+;.&%' ;,+ &## "%'+ 4 #'$-&#7

F'$-&# $,&2&D%&2 "% '&,.#+) +B#&,A$B(&.+,;1+(+-" $( &A&%'#"% P+=&, 2&A&(+;.&%'7 E1&D,#'

#'$-& "#'1& $;;&$,$% & +)P+=&, ;,".+,2">.$#$B>'',&##+% '1&P$%Z+) '1&FGH7E1&%P+,$(

;,".+,2">. -,+=# +>' $%2B& +.&##&;$,$'& ),+.'1& FGH 9#'$-& '=+:7 F'$-&'1,&& "#.$,Z&2

BC '1& $;;&$,$% & +) #&;$(#7 Q% #'$-& )+>, #&;$(# 2&A&(+; '+ +A&,("& '1& P+,$( .&,"#'&.7 E1&

P+,$( .&,"#'&.2&A&(+;# >%'"( "'B& +.&#$ +.;(&'&(C2&A&(+;&2P+=&,7 Q% '1"#2"##&,'$'"+%3=&

)+ >#+%'1& FGH3;$,'" >($,(C "% K1$;'&,\3=& #'>2C '1&-&%& ,&->($'+,C%&'=+,Z#"% '1&D,#'

8#'$-&#7 E1&,&)+,&3=&2+%+'2&# ,"B&#'$-&#DA&'+'=&(A&/F.C'1 ! "#$3 ^^5 ℄3#&&X"->,& 7 [ 7

Q% $(( 2C%$." $( .+2&(# +) ;1C((+'$?"# .&%'"+%&2 B&)+,&3 '1& &%',$( I+%& ;($C# $ ,> "$(

"#$%&' ( )* "%+ -#.+/0 1.2/#+3 .4 35..- /6# /0 2'&#3-'2( 7-'2 '003 /+1 .&$/+#8#+$ '+-'&

/&' 2/&9'1 :; :0%' /+1 &'1 .0.&3 &'36' -#<'0;( => /+1 ?> #+1# /-' -5' '+-&/0 8.+' /+1

-5' 6'&#65'&/0 8.+'( @5' '003 #+ '+-&/0 8.+' 1#<#1' /+1 '+-'& -5' 6'&#65'&/0 8.+' A5'&' -5'

1#B'&'+-#/-' /+14.&26&#2.&1#/( @5'/&&.A3&'6&'3'+- -5' -&/+3#-#.+-. 1#B'&'+-#/-#.+( ?# -%&'

4&.2 C7/:0.A39#D EFFG℄(

"#$%&' ( I* J'$/-#<' 4''1:/ 9 +'-A.&9 :'-A''+ KL7D =MN /+1 =MNO( KL7 #+1% '3 :.-5

=MN /+1=MNO( =MN #+ .+P%+ -#.+ A#-5=MNO#+1% '3/3#$+/0#+1# /-'1:;Q-5/-&'6&'33'3

KL7(

&/-#. .4-5' 1#/2'-'&.46&#2.&1#/ -. -5' '+-&/0 8.+' 1#/2'-'&D/3A'2'+-#.+'1 -5#36/&/2'-'&

/+ 'R60/#+ /00-5' 65;00.-/ -# &'$#2'3 .4-5' 2.1'0 /3:#4%& /-#.+'<'+-3 A5'+#-#3 </&#'1D 3''

"#$%&' ( E ( S03.D:#.0.$# /0 'R6'&#2'+-35/<' 35.A+-5/-2.1#T /-#.+3 .4 -5#3&'$#.+D '#-5'&.4

$'+'-# .&#$#+.&:;1#&' -/0-'&/-#.+.4-5'2'&#3-'2D-;6# /00;5/1/2/P.&#26/ -.+65;00.-/R#3(

".& -5'3' &'/3.+3D 3'<'&/0 2.1'03 .4-5' $'+' &'$%0/-.&; +'-A.&9 -5/- .+-&.03 -5#3 &'$#.+ 5/<'

:''+ 6&.6.3'1 #+ -5' &' '+- ;'/&3 CU'#'&D !"#$D EFFV D U.&1.+ !"#$D EFF! D WX+33.+ ! "#$DEFFI D

Y.52 !"#$D EF F ℄( Z+ -5' 4.00.A#+$D A' T&3- 6&'3'+- -5' :#.0.$# /0 4/ -3 &'0/-'1 -. -5' '+-&/0

8.+' /+1 -5'+ A' 6&'3'+- -5'3'2.1'03(

!"!# $%&&'() *+(,%&-+) -) &.' /01 -)2+32-)4 56/