ΕΥκλείδη Γεωμετρία Tomos-3

“ ”

ÔÏÌÏÓ ÉÉÉ

Ç ÃÅÙÌÅÔÑÉÁ ÔÏÕ !ÙÑÏÕ

Âéâëßá !É, !ÉÉ, !ÉÉÉ

ÔÏÌÏÓ ÉÉÉ

Ç ÃÅÙÌÅÔÑÉÁ ÔÏÕ !ÙÑÏÕ

Âéâëßá !É, !ÉÉ, !ÉÉÉ

!!"#$%!!$!&""'!!(#'#$)$'!!"#!!!("#%!&'$' "!#!#!(#!!#$

!*+,&-'.!“'(/01&-)” !'2314/5.!)67'/*.!8&!&0*)393:

&6&;.3:*&0<!+)0!*1/,0)*87

#%)%'!!###! $!+!=)!#$#"!#%&!>=$%&

,0?,-)!>#%!>##%!>###

"@:5)!&''(

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

'&"#!A!'#!'!#%&!!$+%&

!60*(.8/50+7<!&6&2@*5/< +#'%,%!-.!-&7'94/<!+#!!;)41B+/< &6&2@*5/<!C43/* $/0)"12".!+#'%,%!-.!,)33C,.<!"(D0)14:*(/< !"#!!$"#%&%'!(%)#*!!+!,#-"%"$'.!$+/!0##!1$"$"23+/!%-"#4#! ,"!-&7'94/<!+#!!;)41B+/<.

'*334).0+C<!%8B'&< 1.!%8B')) &35!!$"#!!1$2(6!$#!,/)3&.!$&#!,!'#6#3&#!#$!%$!&!,#$!'!$+/!$72+/!(

• ,)33C,.<!"(D0)14:*(/<, !!"#$%&$'!(")*+*#,'!#$%!-".$!/%#*0"$%!&)*.1. • %.8:(40/<!!/5(/30B55.<, '/(2##$&"'!3")*0"#/#1."!4506$%7$'!#$%!

-"/("+8+/#$5!).%#/#$5#$% • #)*/2,)!#*03E5., (")*+,#&/"!'$%#$&$62)0/"'!!#!""($%%*'#

2.!%8B'))!&35!!$"#!!1$2(6!$#!,/)3&.!$&#!,!'#6#3&#!(!,#$!)!$+/!$72+/!( • "5B43*4/<!/&,/24.<, !!"#$%&$'!(")*+*#,'!#$%!!3- • >4:*(/<!'()(&4B<, '/(2##$&"'!3")*0"#/#1."!49$7/#:'!4506$%7$' • '(C.)5/<!)C(.<%!(")*+*#,'!'$%#$&$62)0/"'!!#!""($%%*' • #/4'B5.<!>40*(/'/2,/*, (")*+*#,'!'$%#$&$62)0/"'!!#!""($%%*'

3.!%8B'))!&1$2!6'-",!!,/)3&.!$+#!$72+!(( • +&E430/<!%.8B+/<, ;<##$&"'!#$%!-".$!/%#*0"$%!&)*.1. • ()5)30E(.<!,,B8/<, '/(2##$&"'!3")*0"#/#1. • +&E430/<! )6)4),:<, '/(2##$&"'! 3")*0"#/#1."! #")*+*#,'! '$%#$&$62)0/"'!

!#!""($%%*' •!!!!"5(E50/<!%/25)?.<, (")*+*#,'!'$%#$&$62)0/"'!!#!""($%%*'

4.!%8B'))!&1$2!6'-",!!,/)3&.!$+#!$72+!((( • %0/52*0/<!AB66)<, !!"#$%&$'!(")*+*#,'!#$%!-".$!/%#*0"$%!&)*.1. • !*40B+/<!!)86/2+/<, (")*+*#,'!'$%#$&$62)0/"'!!#!""($%%*' • %.8:(40/<!A0/*'B+.<, (")*+*#,'!'$%#$&$62)0/"'!!#!""($%%*' • +&E430/<!#)**76/*,/<, (")*+*#,'!'$%#$&$62)0/"'!!#!""($%%*'

!!!!!89-!! +29)#! !35!! $"#! ,:)$#! !/-:#"! %$#! $+! $2'2#! $+/! ()%+/! 1+/! ###-'()!$#$!1#)#19#&"! !*2&.!76!.!+$!+29)!.! !)%9"$",##! %$#!+67,6")+! $+! ()%+$!!35##! "/5#(.! "/###$'"!$.$! "/#!)%#"3!.! ,#$! ##$#66#%(.! #17*!&#! %$#! $"! )$#-27)'&""!$&#!,!$2(#&#$!,#-4.!,#$!%$#!$"#!$!6$,'!2+)''!1+/!1')!!$+!()%+"

%($"$",()-!&/(&3$&(#*!8#66$71"-+###!;#$<""$#/)$#+: $/(&3$&(#!'4%&5!6#*!=$,76#+.!,#)9#". $/(&3$&(#!'&$(("/"0%'%.:!8#3$"!%6!>+1+:6+/

#%)%'!#$##%'

$!+!=)!#$#"!#%&!>=$%&

,#,A#"!>#%!>##%!>###

($%A%+%'

-! 1)4$"! ###'+)9! "$"#! ##9%,"! %$#! #/$+$!6'! ##91$/>"! ,#$!

-!2!63&""! $".!.!&2!$)3#.! $+/!;4)+/!##9%!$#$!"!! !1+5(.!1)+%!#("$!)!.!

$".!"/%%)#''.!$&#!4#$/9$"8."!?$"#!$-$7/#$""$!$+/!069$&##!+(%,b-!/19)-5+/#! $($+$!.! ###'+)(."! ?$+! !)9'$+! #/$7$! "!! !6!:-!)"! #17)+""$! ###'(-)+#$#$!5#)#,$")$"$$,9!$#!!>'.*

"=$!72)$'!$*$"7$#"/!%#$!:#/!0$#2!"!:! #$!$!"!$($!0$#"6,#"0$!%$!<."! %#$&$:! !$%! !7<$.! $##$7$"! !$&/*$&2"! 98&"'! !&$*+$%0<.8'! ."!

0$7$#,%$%0$! #$!%#$&$:!%#*.!#")%!"%#:!$%%""! #$%#!=$!%8%#:!:08'!

$"."/! 0$#2! "!:! #*! ($5#$&*! ."! 0$7$#,%$%0$! #*.! &"%>*0<.*'! #&"#*!

(/2%#"%*#!(2#/! #<#$/$"!.$0"?8"!6&"%#$#"/!0$! #*.!"5>*%*!%#$%'!#5-6$%'!#"/!%$!#2)$!%9,0"!!$%!<9$/!62)$'# – &%#2! $"."/! %8%#2#!+08'"!48#&2#*"! .$0"?8! :#/! %#*.! $!/%#,0*!

"%#,!($.!<9$%.!+".$/!"#:0"!"."#"75@$/'# – 3!$&$"! ."! $>*+*)$"! #"/!"%#:#!A*$"7$#"/! %$! (5$!"*$&0<'#!-"-&:7$!!$%!*!<&$%."!%#/'!%!$%(<'!"%#<'!$"."/!#$!/1(*'"!($.!#"#"-6277$.#"/! 0$+27$'! !&$%!2)$/$'! +/"! .<$'! "."#"75@$/'"! $!$/(,!

"%#<'!$/!%!$%(<'!($!9""&$%.!$##"0*%*'!%$!#"0/2!!:7*#!&!"/#$"#"/!

7$/!:.!#2!$/$'!!$%!."!#")$(*+$"!#$%'!$&$%.*#<'"!"77/1'!!"%#$"!

($!)"!0!$&<%$%.!."!!&$8),%$%.! #/'! <&$%.<'! #$%'#!!"."/!:08'!

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

%#/'!%%.),#$'!!$%!#1&"!$!/#&"#$5."!"#:0"!#"/!".!!"&$%%/"%)$"!

#2!$/$'"! ($.! $"."/!(%."#:.! ."! $/%"#$%%)$"!"!:! #$%'! $&$%.*#<'!

!$%!"%9$7$5.#"/!0$!"%#2!#"!!&$67,0"#""!"*$5!$"."/!#$.:($>$/#!

+08'"!".!*!!:7*!$##/0,%$/!#/'!%!$%(<'!"%#<'!#"/!"."726$/!#*.!

$!$!#$""! #$%'"! #:#$! #"/! $/! $&$%.*#<'! )"! !&$)%0$!$/*)$5.! ."!

$/%"#$5%$%.!#$.!$(*+:!#"/!)"!+".$%.!<.#$.$'!"."?*#,%$/'"!!$%!

)"! "!$#"75@$%.! #/'! %9<%$/'! $/! $!$"$'! "*$&$5.! %$! "%#2! #"!

".#/#$"0$."#!&#:0"!#"/!#1&""!!"&2!#$!:#/!!$77$"!#"#"%%#$*"-.#$5.!#"/!%!$6")0"?$%.! #/'!%!$%(<'!"%#<'!#"/!$/! "(/$/!$/!$&$%-.*#<'!($.!%!$&"%!"?$.#"/!#*!9&*%/0:#*#2!#$%'"!!"&:7"!"%#2!#"!

$0!:(/""!$/!%!$%(<'!"%#<'!$%($#/0$5."!+/"#"!#<&!$%.!#"/!*%%/#2!

#2!$#$!$/!"."#"75@$/'!!$%!#1&"!+".$.#"/!)"!(*0$%/$!$/*)$5.$#!

-! ,+):'&""! 72&.! $".! ##91$/>".! $".!.!&2!$)3#.! $+/!;4)+/!"/-#(1!"!!2!!$"#!2!%#6+'//#!$+/!%)5$2')"!,#$!$$.!#>!1()#"$!.!2!-7)+/.!1+/ !'9)2+"!! "$+#! /1+6+%$"27! !2@#)4#! ,#$! 7%,&#$! 2!-7)+/.! 1+/! "/%-,)3#+#$#$!"!!#/"$")7$"$#!27#+!2!!$+!":%5)+#+!%1!$)+"$$,7!A+%$"27"

.$#! $"!2!6($"! $)$+$'$&#!"$!)!4#!"&29$&#!'"23<!$#$!,#$!+!0"-27,)$$+."!*1&.!"#'4.!###'()!$#$!#17! $+#!%)5$2')"! +"$"#! !$"#%&%'!

$".! $3$):($%$! $+/-$! +! 0"27,)$$+.! ###,96/*!! $"! "5(""! 2!$#>:! $&#!

7%,&#!1/)#23)#.!,#$!1)3"2#$+."! !#! 1)+@6'2#$#! 1+/! ###,:1$+/#! ,#$9! $"!2!$9@#""! #17! $+! !13-

1!)+!"$+!54)+!'#3#!$#$!##!'$##!##$$,!32!#+!!,$!#4#!"/<"$'"!&#!,#$9!

$"#!#)5#$7$"$#$!71&.!1)+,:1$!$!,#$!#17!$+!!>'.!5&)3+!$+/!06+:$#)5+/$!

$+!"5!$$,7!2!!$$.!###,#6:*!$.!$+/!0"27,)$$+/*! $&.! %$! <.".! #1.$"! $/! #%#7/#<'! #$0<'! !$%! 9"&2%%$.#"/!

!"&277*7"!!&$'! #*!62%*! #$%! $"."/! "%$'"!!1'! $"."/! #:#$!(%."#:.!$!

#1.$'! ."! (/"*<&$/! "!:! #$.! #57/.(&$(! ("/! "."! :%$! ".$&9:0"%#$!

!&$'!#*.!#$&%*,!#$%!#1.$%"!$/!#$0<'!"%#<'!:7$!#"/!0/#&"".$%."!($!

)"!!&<!$/!#:#$!*!$!/*2.$/"!#$%!#1.$%!."!*"".$#"/!#7/0"#8#,"!$.1!

*!"(/"!$"."/!7$""($ !#!@$@63#!;($!;((!,#$!;(((!$&#!4#$/9$"8.!###'()+#$#$!"!!"5'2#$#!

$+/!54)+/$!)$#1)#%2#$!:+#$#$!)"6#)'!#/$7!1+/!"'2!)#!6(2!!'(&4&/-8&(4-).

!#! $)3#!#/$9!@$@63#!6#2@9#+#$#$!&.!2$#! !#7$"$#$!,9$$!1+/! (5!$!

')"!"/2@!3!,#$!2!!$#!##$3"$+$5#!.%)$-2"$$,9.!@$@63#"!?:2'&##!2!!#/$'!

$"!-!4)"""$!+$!+)$"2+3!,#$! %$#! $#! $)3#!@$@63#! $3-!#$#$!"$"#!#)5'! $+/!

@$@63+/!;("!!+!1!)$!572!#+!!3##$!)$#)-)&2(#+!,#$9!$($+$+!$)71+$!4"$!!

,9-!! @$@63+! ##! "$+5!:!$! "!! "/%,!,)$2(##$! 2!-+)+6+%$,9! )$#,)$$9$!

#1+$!6("2#$#"!1$"$$!$+!@$@63+!;(!#'/!!#7.!6!$$+/)%!3!&.!"/66+%'!@#"$-,4#!1)+$9"!&#!1+/!-#!5)""$2+1+$"-+:#!"$#!!172!##!,#$!#'!$()+/!1#-)+/"$9<!$! $"!":%,)$""! $&#! 1#)#66"6!1$1()&#"!?$+! @$@63+!;((! %3#!$#$!

"/"$"2#$$,'! 5)'""! $".! $0$):($%! #*'! $>2.#7*%*'$! "$"! ":%,)$""! !13-1!)&#!,#$!"$!)!4#!"5"29$&#"!!(6+.$!"$+!@$@63+!;(((!$#!1(#$!!,##+#$,9!

"$!)!9!,#$#",!/9<+#$#$!,#$!!%%)9'+#$#$!"!!"'#3)#"! ?$#! $)3#! #/$9! @$@63#! /19)5+/#! %,! +)$"2+3! ,#$! 0(! 1)+$9"!$."!

*1&.!')"!(5+/2!!###'()!$!+$!+)$"2+3!1!)$(5+#$#$!76+$!"$+!@$@63+!;($!

!#4! +$! 1)+$9"!$.! ,#$##(2+#$#$!&.! !>'.*!B19)5+/#! &1! 1)+$9"!$.! "$+!

@$@63+!;(!,#$!#17!#,!1)+$9"!$.!"!!,#-(##!#17!$#!!172!##!@$@63#"

(!$#!>%)!""

#78/<!*/< 0)76+%+....................................................................................................9 !&.B,)0/!()!!$!+&98&(4-)!*(/!1E4/ 1. &$"#%&%' ..........................................................................................13 2. C)$"2+3 .............................................................................................14 3. 0#)#66"63#!,#$!,#-!$7$"$#!"$+!54)+ ............................................19 4. !)3!))!.!"$!)!(.!%&#3!.!,#$!"!5)'""!$+/.!"$"#!,#$#",!/'!

1#)#66"6!1$1()&# ...........................................................................39 5. ?576$#!19#&!"$$.!$)3!))!.!%&#3!. ....................................................56

("#!8)$$')$#!$"7$"$#.!$)$())&#!%&#$4# ..........................................56 ("%!8#$9!,+)/''#!$)3!))!.!%&#3!....................................................60

6. D($)"""!1#)#66"6!1$1()&#!,#$!$)$%&#$,4#!1)$"29$&#................63 7. ?576$#!"$"!2($)"""!$&#!1)$"29$&# ................................................84

7.1 !+!22"!"$+$5!$4)!.3!$".!-!&)3#.!$+/!7%,+/ .............................84 7.2 0#)#66"6!131!)#!,#$!,#$#",!/#"$$,9!1)+@6'2#$# ................86

!&.B,)0/!&)!!03+/0!+)0!.!8C@/'/<!(.<!&;B5(,.*.< 1. &$"#%&%'! .........................................................................................89 2. -!2(-+)+.!$".!!>9#$6""". ...............................................................90 3. ?:%,)$""!):+!,:,6&#.......................................................................92 4. ?576$#!"$+!-!4)"2#!":%,)$"".!,:,6&# ..........................................97 5. !)$%&#$,(.!1/)#23)!.!,#$!1)3"2#$#...............................................102 6. ?576$#!"$"!2($)"""!1/)#23)&#!,#$!1)$"29$&#.............................114

6.1. D($)"""!1/)#23)#. ................................................................114 6.2. D($)"""!1)3"2#$+. ................................................................117

7. ?:%,)$""!,/63#))&#!,#$!,4#&# .....................................................122 8. ?:%,)$""!"'#$)4# ..........................................................................135 !&.B,)0/!*)!!#)!+)5/50+B!*(&4&B 1. &$"#%&%'! .......................................................................................145 2. 0$#3)!""!"!!9,)+!,#$!2("+!67%+"!&'#)2+%(. ................................146 3. ("716!/)+!$)3%&#+!,#$!,##+#$,7!1!#$9%&#+"!

891+$!.!!'#)2+%(..........................................................................154 4. 8##+#$,7!$!$)9!))+!–!,#$#",!/'..................................................163

5. 8##+#$,7!+,$9!))+!–!,#$#",!/'................................................... 166 6. 8##+#$,7!!>9!))+!+*!,:@+.-!–!,#$#",!/'...................................... 168 7. 8##+#$,7!!$,+"9!))+!–!,#$#",!/'................................................ 169 8. 8##+#$,7!)&)!,9!))+!–!,#$#",!/' .............................................. 174 9. ?:%,)$""!$&#!16!/)4#!$&#!,##+#$,4#!"$!)!4#"!

?/21()#"2#.................................................................................... 178 10. .!#$,9!?576$# ................................................................................. 184

#4"#!8#$#",!/(.!2!!5)'""!%#96/"".!,#$!?:#-!"". ..................... 184 #4"#"#!8#$#",!/'!$+/!,:@+/!,#$9!$+#!0911+ ........................ 185 #4"#"%!8#$#",!/'!$+/!!$,+"#())+/!,#$9!$+#!0911+ ............. 185

10.2 0+6:!))#$!,##+#$,9!"$!)!9!,#$!"/22!$)3# .......................... 188 10.2.1 !$!!3##$!1+6:!))+ ........................................................ 188 10.2.2 C!$:1+.!$+/!Euler........................................................ 188 10.2.3 -2$,##+#$,9!"$!)!9.................................................... 190 10.2.4 -!"/22!$)3#!"$"!.!&2!$)3# ....................................... 191

'*5/6(0+7<!,0?,0/34).0+7<!/'.37<!30)!(.!'(&4&/8&(4-) .....196 ,0?,0/34).-) ..............................................................................197 !*4&(:40/....................................................................................199 +&50+:!,0?,0/34).-)..................................................................203

.

!!""##$% 1 !"!!""!XI

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

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

$#2%1$/34$01!5&$%!1#!#*$$%'6&$71!$,(#!%7'%-8,%$#!/6,3% #$!*,5$$!/6,$.$!*,$$-&%%.!&!/69,%!&#$!'%#*,#:/#$%0%$#%!$#!*,;'

$#,9%8-!&$$%9%(#! $$7!95,$7!8#%! $%.!/%$#20! $$7.!&96&%%.%!"%!*,$$-&%%.!

#7$6.!&7:8,$$$01!$$!4%;,3$%8+!7*+&#4,$!:%#!$3!/%$-&#&3!#*+!$$!%*('

*%'$!&$$!95,$% #$!'%0$%,$!/6,$.$!*,$$-&%%.!()!/69,%!(*$!/%'%$-!$%.!$,(%',%.!&$%'

,%6.!:;1(%.$!&%!#1$%&$$%9(#!*,$.!$#!$,(:;1#$!!8#%!$%.!%(#,/$:6.!$$7.!&%!

8-*$%%.!8#$#&8%76.% <$$!$,($$!/6,$.$!*,$$-&%%.! (+!/69,%! "#$!*#,$7&%-=%$#%!3!&714%-

$%8>!4%;,(#!$$7!+:8$7$!3!$*$(#!8#%!%(#,/+=%$#%!&$3!&0:8,%&3!*#,#''

'3'%*%*6';1%

)!$#8$%8>!*$7!#8$'$74%($#%!&$%.!#*$'%(2%%.!:%#!$#!'0$!*,5$#!/6-,3!$$7!8%(#'#($7!#7$$0$!%(1#%!3!#1#:;:>!8-4%!*,$&'>/#$$.!&%!%*(*%'#!

&9>/#$#!8#%!/%$-!3!9,>&3!'%'$/61;1!$3.!**%*%'$/%$,(#.!*$7!69$71!>'3!

8#$$97,;4%($!&%!$%'%8>!#1-'7&3!'3'#'>!3!9,>&3!$3.!%&+$3$#.!$;1!$,%-:51;1%!+7$+!*%4#15.!1#!#*$$%'%(!8#%!/%#!'%8#%$'$:(#!:%#!$31!#*$7&(#!

14 *?@!*",)!,<#"")*"+-

'%#$0*;&3.!8,%$3,(;1!%&+$3$#.!:%#!$#!&$%,%-!&9>/#$#!*$7!#1$%&$$%9$01!

&$#!$,(:;1#$!'3'#'>!$#!$,%:;1%8-!*,(&/#$#!8#%!$#!$%$,-%',#$!:%:$1+.!

*$7!69%%!61$$1#!&9$'%#&4%(!#*+!$$7.!&0:9,$1$7.!/%'%$3$6.. A%#! $6$$%#! *,$&*-4%%#! &>/%,#! *,$B*$46$%%! &7&$3/#$%8>! #1--

*$723! $;1! &96&%;1! %74%%51! 8#%! %*%*6';1! &$$! 95,$$! 8-$%! *$7! '%%8*%-,#%51%$#%!#*$'%8$-!&$%.!&0:9,$1%.!4%/%'%5&%%.%!

C#,#&-''$1$#.!$31!#1-*$723!$$7!&%&'($7!)"!/%!$#!&%&'(#!"!/69,%!

"V!$3.!**%*%'$/%$,(#.$!*#,#$3,$0/%!+$%!%'5!'%!:(1%$#%!*,$&*-4%%#!8#-4$,%&/$0!$$7!8#$-!*+&$!61#!#*$$6'%&/#!%2#,$-$#%!>!+9%!#*+!$3!4%;,(#!

$;1!#1#'$:%51%!<.!#7$-! $#!*'#(&%#!'%(*%%!8#%!3!/%'6$3! $;1! %'%$$>$;1!

$3.!&(#(,#.$!&%!#1$(4%&3!/%!$31!%*#,8>!#1-*$723!$;1!%'%$$>$;1!$$7!80-8'$7!&$31!**%*%'$/%$,(#%!"!Mueller!*#,#$3,%(!+$%!#7$+!*%4#15.!1#!$-(%('%$#%!&$31!6''%%D3!%*#,8$0.!4%/%'(;&3.!$;1!!!!!""#"$%"&'#()!!$;1!

&$%,%51! &93/-$;1! :%1%8-$! #($0! 3! &(#(,#! %(9%! %2#1$'3$%8-! 9,3&%/$'

*$%34%(!&%!#&$,$1$/%8-!46/#$#%! +*+!&0:9,$1$7.!/%'%$3$6.!%8(,-=%$#%!3!-*$D3!+$%!$$!*,5$$!/%-

,$.! $$7! &%&'($7!)"! #1#*$0&&%$#%!&%! #1$%&$$%9(#$! #8+/#! 8#%! 8#$-! $31!

#,(4/3&3$!!/%!$,%&/61%.!%1+$3$%.!$$7!&%&'($7!"!//69,%!$31!(#3!*,+$#&30%!

!6&#%#$!#7$>!3! !#1$%&$$%9(#!7*+8%%$#%!&%!&3/#1$%86.!'%#($,6.!;.!*,$.!

$$!*%,%%9+/%1$$!+*;.1!! !"#31! *'>,3! #*$7&(#! ,3$-! %8(,#&/61;1! #2%;/-$;1! :%#! $#!

&$$%9%(#!*$7!&7:8,$$$01!$$1!95,$% !"#3!/3!#1#($,-!&%!8,%$>,%#!%&+$3$#.% !"#3!/3!'%*$$/%,>!#1#($,-!$3.!*#,#''3'(#.!&$$!95,$!/%!$+&3!

'%*$$/6,%%#$!+&3!&$$!&%&'($!"% <9%$%8-!/%!$%.!#1$%&$$%9(%.!#,(4/3&3.$!*%$!*%&$-!$3,$01$#%!#7$6.!

$;1!*,$$-&%;1!&&$!&($!()$!((!8#%!("%

$#!.($''%)! 1. !!"""#!"$%#&!#'!#!()'!*+"&!,-.)/!.#&!(01!)/!.#&!213)/" 2. 4)!(*"#/!$!""")5!"$%#&!"(&%1%"&#" 3. &'3"$#! "$%#&! .13"!6! (")/! "($("')#! #!#%! $+6,#!$7"&! )"3*/!

89%$"/!,"!.13"!"'3"$#!!)'!"(&(*')'!()'!$'%#%!1"

)!-*EA*#F"+!<#"!)EF" 15

4. &($("')! "$%#&!.13"!)!(")/! "($("')#!#!#%! "'3"$"/! !)'! "%#/!"(&(*')'!()'!18)%!#&!.13"!#!(")/!!6%!.)&%-!!)'/!!),-#!"$$%#&!.13"!"/!.#&!(")/!!)!100)!"($("')"!

5. :0$$6! "'3"$#/! (")/! "($("')! "$%#&! 6! 89%$#! ()'! (""&*+"!#&!

#(#!!6%!"'3"$#!.#&!#(#!"."$%6!!6%!"'3"$#!()'!(").5(!"&!9/!";-/%!$'%'*"&! !)%!(#'#! !6/!.13"!6/!(")/! !)! "($("')#!()'!18"!#&! #(#! !)! *%#! (*"#/! !6/! &#"+&.1! '"'),*%6/'! "'3"$#/!.#&!!)!(*"#/!()'!'6,&)'"8"$!#&!9/!!),-!!6/!!,"!!)!"($("')"

6. :0$$6! "(&(*')'!(")/! "($("')! "$%#&!6! );"$#! 89%$#!6! )()$#!

(""&*+"!#&! #(#! "'3"$"/! ()'! 2"$$.)%!#&! (1%9! $"! .13"! *%#!#(#! !#! "($("'#! .#&! 18)%!#&! .13"!#! (")/! !6%! .)&%-! !),-!!9%!"(&(*'9%!.#&!$!)!$'&)!$6,"$)!!6/"!

7. ("58)/! "(&(*'9%! 3#! "$%#&! #,)&#! .".0&,*%)! #(9/! *%#!100)!

7"58)/#!#!#%!)&!#%!$$!)&+"/!.0$$"&/!"$%#&!,"!#;5!!)'/!$$"/" 8. <#"10060#!"($("'#!"$%#&!#'!1!!#!)()$#!'"!$'%#%!=%!#&" 9. ),)&#!$!"""1!$+-,#!#!"$%#&!#'!1!()'!(""&*+)%!#&!#(#!$$#!

$!)!(0-3)/!#,)&#!"($("'#!&$+-,#!#'" 10. *$#!.#&!#,)&#!$!"""1!$+-,#!#!"$%#&!#'!1!()'!(""&*+)%!#&!

#(#! $$#!$!)!(0-3)/!.#&!!)!,*8"3)/!#,)&#!"($("'#! &$+-,#$!#'"

11. !!"""1!89%$#!"$%#&!6!$58.0&$6!6!)()$#!'6,&)'"8"$!#&!#!#%!

$'%!"*+)'%! (""&$$#!"""/! #(#! '5)! 8"#,,*/#! )&! )()$"/! '"!2"$$.)%!#&!$!6%!$'&#!"(&%1%"&#"!+00&=/#!$!"""1!"$%#&!6!89$%$#!()'!(""&*+"!#&!$"!(""&$$#!"""/!#(#!'5)!"($("'"/!89%$$"/#!)&!)()$"/!'"!2"$$.)%!#&!$!)!$'&)!"($("')!.#&!$'%!"*+)'%!$!)!$'&)!$6,"$)"

12. <'"#,$'#!"$%#&!!)!$!"""#!$+-,#!()'!(""&*+"!#&!#(#!"($("$

'##! !#!)()$#! ;".&%)5%!#(#! *%#! "($("')!.#&!.#!#0-8)'%!$"!*%#!$6,"$)"

16 *?@!*",)!,<#"")*"+-

13. <"$$,#!"$%#&!!)!$!"""#!$+-,#!()'!(""&*+"!#&!#(#!"($("'#!$+-,#!##!".!!9%!)()$9%!!#!,"%!'5)!#(*%#%!&!"$%#&! $$##!#$,)&#! .#&! (#"10060##! "%=! !#! '(#0)&(#! "$%#&! (#"#0-060#8"#,,#"

14. !%#$"#!"$%#&!!)!$!"""#!$+-,#!()'!'6,&)'"8"$!#&!#!#%#!"%=!

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

15. ,;)%#/!!6/!$%#$"#/!"$%#&!6!"'3"$#!()'!(#"#,*%"&!#.$%6!6!

.#&!(""$!!6%!)()$#!$!"*%"!#&!!)!6,&.5.0&)" 16. 4)!.*%!")!!6/!$%#$"#/!"$%#&!!)!$'&)!,"!#'!#!!)'!6,&.'.0$)'" 17. -&1,"!")/!!6/!$%#$"#/!"$%#&!,&#!"'3"$#!()'!18"!#&!'&#!!)'!

.*%!")'!&!6/!$%#$"#/'!.#&!*+"&!!#!1."#!!6/!$!6%!"(&%1%"&#!!6/!$%#$"#/"

18. :=%)/!"$%#&!!)!$!"""#!$+-,#!()'!'6,&)'"8"$!#&!#!#%#!"%=!6!

,&#!.13"!6!(0"'"1!)"3)89%$)'!!"&8=%)'!(#"#,*%"&!#.$%6$!6#!!)!!"$89%)!(""&$!"*%"!#&!.#&!"(#%*"+"!#&!$!6!3*$6!#(#!#()'!1"+&$"!!6%!(""&$!")%-"

19. ,;)%#/!!)'!.=%)'!"$%#&!6!"'3"$#!()'!(#"#,*%"&!#.$%6!6!.#&!

(""$!!6%!)()$#!$!"*%"!#&!!)!!"$89%)" 20. .1$6! !)'!.=%)'! "$%#&! )!.5.0)/!()'!'&#8"1%"!#&!#(#! !6%!

(""&$!""%#,"%6!"'3"$#" 21. :50&%'")/! "$%#&! !)! $!"""#! $+-,#! ()'! '6,&)'"8"$!#&! #!#%#!

"%=! 6! ,&#! .13"!6! (0"'"1! "%#/! )"3)89%$)'! (#"#0060)-8"1,,)'! (#"#,*%"&! #.$%6!6#! !)! (#"#0060#8"#,,)! (""&$$!"*%"!#&!.#&! "(#%*"+"!#&!$!6!3*$6!#(#!#()'!1"+&$"! !6%!(""&$!")%-"

22. ,;)%#/!!)'!.'0$%'")'!"$%#&!6!"'3"$#!()'!(#"#,*%"&!#.$%6!6!

.#&!(""$!!6%!)()$#!$!"*%"!#&!!)!(#"#0060#8"#,,)"

)!-*EA*#F"+!<#"!)EF" 17

23. .1$"&/! !)'! .'0$%'")'! "$%#&! )&! .5.0)&! ()'! '6,&)'"8)5%!#&!#(#! !&/!'5)!(""&$!""%#,"%"/! !#(*%#%!&!(0"'"*/! !)'!)"3)$89%$)'!(#"#0060)8"1,,)'"

24. -5)!.=%)&!&-!.50&%'")&'!"$%#&!#,)&)&!#!#%#!!#$)!)&!'&1,"!")&!

!9%!21$"9%!#$)!.#&!)&!1;)%*/!!)'/!"$%#&!#%10)8)&" 25. :52)/!"$%#&!!)!$!"""#!$+-,#!()'!(""&*+"!#&!#(#!*;&!$$#!!"$

!"189%#" 26. >.!1"'")!"$%#&!!)!$!"""#!$+-,#!()'!(""&*+"!#&!#(#!).!=!

$$#!.#&!&$#(0"'"#!!"$89%#" 27. &&.)$1"'")!"$%#&!!)!$!"""#!$+-,#!()'!(""&*+"!#&!#(#!"$.)$

$&!$$#!.#&!&$#(0"'"#!!"$89%#"! 28. -9'".1"'")!"$%#&!!)!$!"""#!$+-,#!()'!(""&*+"!#&!#(#!'=$

'".#!$$##!&$#(0"'"#!.#&!&$)8=%&#!("%!189%#"

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

"%!'0$!*,5$$%!$,%&/$(!*,$&'%$,(=$71!$31!611$%#!$$7!&$%,%$0!&5-/#$$.! 8#$-! $,+*$! *,-:/#$%!#2%$4#0/#&$$! 8#%!&7/*#:>$! %1&;/#$51$-1$#.!*%,%:,#(6.!'%-&*#,$%.!8#%!#1%91%0&%/%.!#8+/#!8#%!&>/%,#!&%!8'#'

&%8-!8%(/%1#!$$7!C'-$;1#!8#%!$$7!+,%&$$$6'3%!.1#!:%:$1+.!*$7!#2(=%%!

1#!%*%&3/#14%(!%(1#%!3!#*$(7:>!9,>&3.!$3.!/#,%&$$$%'%8>.!*,$6'%7&3.0!

611$%#.! !!!*#"$#(!$! 8-$%! *$7! '%1! #8$'$74%($#%! #*+! $$7.! /%$#:%16&$%'

,$7.!/,;1#!8#%!06;1#!$$1!</7,1#($%!**%*'6$1$!3!9,>&3!$$7!+',$"") /&71+,$70! ;.! %*%*,+&4%$3.! *,$&'%$,%&$%8>.! %'%+$3$#.! $;1! %11$%51!-,$..#, %+!&*%%!$, #"%,%/$! 7*$8,0*$%%! /%#! *,$&*-4%%#! 2%*#:;:%8$02!8#4$,%&/$0!%'%$$>$;1!:%;/%$,%851!#1$%8%%/61;1!8#%!#(%$6,$7$!&0/(;'

1#!/%!+&#!69%%!%%&3:34%(!$!8#43:3$>.!<%!C%!1%,&+.$!/%#!&#(>!$$*$'$:%'

8>!9,$%-!&$31!#1$('3D3!$;1!:%;/%$,%851!$,%&/51%

"%!$,%&/$(! "!8#%! 3$!*$7!#1#(6,$1$#%!&$31!8#4%$+$3$#!%74%%51!8#%!

%*%*6';1! &$$! 95,$$! '%#$7*51$1$#%! /%! &9%'+1! &0:9,$13! $,$'$:(#%!<$$7.!

$,%&/$0.!4!8#%!*!%*%9%%,%($#%!:%1%8+$%,#!1#!*%,%:,#(%(!#7$+!*$7!&>/%,#!4#!

'6:#/%!!-$%0$"%1&%0$)"2$!"%+!+'!"1!!>!!-$%0$"!3""%+!+'!$%!%!-%#!$$!&8$*+!

#7$+!%%&-:%$#%!$!+,$.!!240#(!#!&$$1!$,%&/+! 4$!*%,%:,-(%$#%!3!'%#'%8#&(#!

18 *?@!*",)!,<#"")*"+-

*,$&$'>.!8#%!/6&;!#7$>.!#1-:%$#%!3!!240#(!"&%!:;1(%.%!"!$,%&/+.!5!$'3:%(!

&%!#7$+!*$7!&>/%,#!9,3&%/$*$%%($#%!;.!!!0%!,")"-$%0$!!8#%$!+*;.!#*6'%%2%!

$!Legendre$!%*%'69%$#%!9%%,%&/+!/%:64$7.!/+*;.!$%!:;1(%.0%

"%!$,%&/$(!#!8#%!&)!#($,$01!&$31!%&+$3$#!8#%!$/$%+$3$#!&$$!95'

,$!8#%!%(1#%!$%!*%$!&9$'%#&/61$%!8#%!#/(%&&3$>&%/$%%!)!*,5$3!61&$#&3!

*$7!4#!/*$,$0&%!1#!:(1%%!%(1#%!+$%$!#($0!3!%&+$3$#!$,(=%$#%!&$31!#,9>!

$$7!&%&'($7! "!/%! $3!'%#'%8#&(#! $3.! %1#*+4%&3.!8#%!&0/*$;&3.! /%1%#(#!

:%#!+'#!$#!&9>/#$#0$!3!%&+$3$#!&$$!95,$!%%'%8+$%,#!4#!*,6*%%!1#!*,$'

80*$%%%!,3'#'>$!+$%!$%!$,%&/$(!#!8#%!&)!4#!6*,%*%!1#!%(1#%!4%;,>/#$#%!

*2-''$7$!3!/$,(>!$3.!'%#$0*;&3.!#7$51!$;1!$,%&/51!#1-:%%!$$1!6'%:'

9$!$3.!$/$%+$3$#.!/>!%&+$3$#.0!$;1!&$%,%51!&$31!$/$%+$3$#!/>!%&+$3$#0!

%*(*%';1!&93/-$;1!8#%!6$&%!(#(1%$#%!1#!%*69$71!46&3!8,%$3,(;1%!C,-:'

/#$%$! #1! *%,%$,%&$$0/%! &%! 87,$-! *$'0%',#$! +*;.! #*6'%%2%! $!Cauchy, %(1#%!8,%$>,%#$!'3'#'>!61#!87,$+!*$'0%',$!8#4$,(=%$#%!*'>,;.!#*+!$%.!

6',%.!$$7%

"!$,%&/+.!&&!%%&-:%%!$31!611$%#!$3.!&$%,%-.!:;1(#.!/%!'0$!$,+-*$7.$!*$7!$!8-4%!61#.!#*+!#7$$0.!&3/#$$'$$%(!/%#!'%#($,%$%8>!-*$D3%!

C,5$#!9,3&%/$*$%%($#%!$!+,$.!!240#(!!*$7!*#,#*6/*%%!&$31!*%,%:,#(>!

/%#.!&$%,%-.!:;1(#.!;.!&7&$>/#$$.!'%6',;1!:;1%51!/%!8$%1>!8$,7(>%!

<$31!%1#''#8$%8>!'%#$0*;&3!9,3&%/$*$%%($#%!*'#$;1%8>!$,$'$:(#!/#*+!

$$1!!50.$!"!0$!+*;.!7*$'3'51%%!3!9,>&3!$3.!'623.!!#1%!#"$.'%$%!!8#%!

%11$%($#%!3!&$%,%-!:;1(#!;.!#*$$%'$0/%13!#*+!%*(*%'%.!:;1(%.%!<$$!8%(-/%1$!*$7!#8$'$74%(!*#,$7&%-=$1$#%!/+1$1!$,(%',%.!&$%,%6.!:;1(%.!8#%!

+*;.!/*$,%(!1#!#*$'%%94%($!%%'%8-!:%#!$%.!$,(%',%.$!$%!'0$!#*+D%%.!%(1#%!

%&$'01#/%.%

<$$7.!7*+'$%*$7.!$,%&/$0.!/&(!6;.!8#%!(+0!%%&-:$1$#%!8#%!*%,%:,--($1$#%!&7:8%8,%/61#!&$%,%-!&9>/#$#!*$7!4#!*#,$7&%#&4$01!&$$!87,(;.!

8%(/%1$$!/%!4#7/#&$>!$%8$1$/(#!&$3!*#,$7&(#&3%!##!&$%,%-!#7$-$!$/#'$'

*$%$01$#%!;.!%2>.1!*7,#/('#!–!*,(&/#$!&(#(,#!–!80'%1',$.–!851$.!8#%!$6''$.!$#!C'#$;1%8-!<$%,%-%!#$!%1'%#(6,$1!&$$%9%($!%'5!%(1#%!+$%!87,(;.!:%#!

$3!&(#(,#!/8#%!'%7$%,%7+1$;.!:%#!$$1!80'%1',$!8#%!$$1!851$0!%*%'6:%$#%!3!

*#,$7&(#&3!;.!&$%,%51!%8!*%,%&$,$(>.!/'3'#'>!'%#!$3.!:%16&%;.0!8#%!+9%!

3!*%,%:,#(>!/6&;!%'%+$3$#.$!+*;.!:(1%$#%!:%#!$$1!808'$!/1&3/%(#!*$7!#*6'

9$71!&$#4%,>!#*+&$#&3!#*+!&3/%($0%!*%'%8-!&$$7.!$,%&/$0.!*$7!#1#(6,$'

1$#%!&.!#7$-!$#!$,(#!&$%,%-!8#%!$#!&$$%9%(#!*$7!$#!&7:8,$$$01$!7*-,9%%!61#!

&#&%8+!/$$(&$$!$$!$*$($!8#$-!*%,(*$;&3!&7/*'3,51%$#%!/%!/%8,6.!*#,#''

'#:6.!:%#!1#!($-&$7/%!&$$!$%'%8+!8%(/%1$%

)!-*EA*#F"+!<#"!)EF" 19

%#!/#(###0#)#!1#$!1#2!&3&0&#!!'&%!*+(%

/(3&#'0!"!&4)#!"'

-"%! "$%#&!''%#!#%!*%#!,*")/!,&#/!"'3"$#/!8"#,,-/!%#!2"$$."!#&!(1%9!$"!*%#!"($("'#!!6/!.#&!*%#!100)!,*")/!!6/!%#!,"!#&9""$!#&. *53+!$60 .&$;!+$%!$$!/6,$.!+!!$3.!%74%(#.!+!-! !&,(&8%$#%!*-1;!&%!61#!

%*(*%'$!*$7!'%6,9%$#%!#*+!#7$+$!%15!$$!/6,$.!!-!'%1!#1>8%%!&.!#7$+!$$!

%*(*%'$% -%#!$$!/6,$.!+!!$3.!%74%(#.!4#!7*-,9%%!&71%9>.!*,$68$#&3!!,$!3!

$*$(#!4#!#1>8%%!&$$!%1!'+:;!%*(*%'$%!.$&%$!$$!/6,$.!+!!#*$$%'%(!8$%1+!

$/>/#!$;1!%74%%51!+!-!8#%!+!,%

67#.$""$

#+$%!+/;.$!/%!861$,$!$$!&3/%($!!!8#%!#8$(1#!$$!8$%1+!$/>/#!+!!

%(1#%! '71#$+1! 1#! :,-D$7/%!808'$$!&$$1!$*$($! '0$! '%-/%$,$%!*$7!&,('

&8$1$#%!*-1;!&$%.!%74%(%.!+!-!8#%!+!,!*,$&'%$,(=$71!'0$!-1%&#!$+2#%! +7$+!%(1#%!#'01#$$!8#%!&71%*5.!'%1!7*-,9%%!/6,$.!!-!$3.!%74%('

#.!+!-!*$7!&,(&8%$#%!%8$+.!%*%*6'$7!*$7!*%,%69%%!$$!+!% !."%'%

/(3&#'0!$!&4)#!$' &1%!'')! "'3"$"/! !*,%)%!#&#!."$%!#&! "($! "%#/! "(&(*')'!.#&!.13"!!"$89%)!."$!#&!"($!"%#/!"(&(*')'"

20 *?@!*",)!,<#"")*"+-

*53+!$60 .&$;!+!!8#%!-,!'0$!%74%(%.!*$7!$6/1$1$#%!&$$!&3/%($!*%!C#(,'

1$7/%!*-1;!&$%.!*-!8#%!*!!$79#(#!$#!&3/%(#!1!8#%!)!#1$(&$$%9#$!8#%!

(6,$7/%!$%.!1)!8#%!!-%!**(&3.!*-1;!&$3!!-!*#(,1$7/%!$#!$79#(#!&3'

/%(#!0!8#%!@$!%15!(6,$7/%!$%.!10!8#%!)@%!0#!#*$'%%94%(!*,5$#!+$%!$$!

$,(:;1$!*!-!8%($#%!%*(!%1+.!%*%*6'$7%

67#.$""%

*-1!&$$!$,(:;1$!*!-!$$!/6,$.!10-!>!$$!)!@!8%($#%!*-1;!&$$!('

'%$!%*(*%'$$!%15!$$!7*+'$%*$!*-1;!&%!-''$!%*(*%'$$!$+$%!4#!8%($#%!/6'

,$.!8#4%/%-.!$;1!%74%%51!*-!8#%!*!!*-1;!&$$!'$461!%*(*%'$$!%15!61#!

/6,$.!$$7.!*-1;!&%!-''$!%*(*%'$% **(&3.$!%-1!&$$!$,(:;1$!*!-!$$!/6,$.!1-!)!&,(&8%$#%!*-1;!&$$!

'$461!%*(*%'$$!%15!$$!7*+'$%*$!*-1;!&%!-''$$!4#!%(1#%!/6,$.!8#%!$;1!

%74%%51!*-!8#%!*!!*-1;!&$$!'$461!%*(*%'$$!%15!/6,$.!$$7.!4#!%(1#%!*--1;!&%!-''$!%*(*%'$$!*,-:/#!*$7!%(1#%!-$$*$!/*,+$#&3!&0%!

+*+!$#!*#,#*-1;!&71-:%$#%!+$%!$$!$,(:;1$!*-!!8%($#%!*-1;!&$$!

('%$!%*(*%'$%!<.!#7$+!$$!%*(*%'$!*$7!$,(=%$#%!#*+!$$!$,(:;1$!*!-$!&,(-&8$1$#%!$%!*!!8#%!*-!8#%!&71%*5.!$%!%74%(%.!+!!8#%!-,$!+*;.!8#%!8-4%!

$,(:;1$!$;1!%74%%51!+!!8#%!-,!&,(&8%$#%!*-1;!&$$!%*(*%'$!*$7!#7$6.!

$,(=$71% !."%'%

/(3&#'0!%!&4)#!%' &1%!'5)!"($("'#!!*,%)%!#&#!!#!"!6!.)&%-!!!),-!!)'/!"$%#&!"'3"$#" *53+!$60! .&$;!+$%!$#!'0$!%*(*%'#!/!,+0!8#%!/!,-0!$6/1$1$#%!8#%!3!8$%1>!

$$/>!$$7.!%(1#%!3!:,#//>!!,%!0#!#*$'%%94%(!+$%!3!!,!%(1#%!%74%(#%

)!-*EA*#F"+!<#"!)EF" 21

67#.$"&

.&$;! +$%! 3! !,! '%1! %(1#%! %74%(#%! 26,$7/%! *-1;! &$$! %*(*%'$!

/!,+0!$31!%74%(#!!*,!8#%!*-1;!&$$!%*(*%'$!/!,-0!$31!%74%(#!!1,%!"%!

'0$!%74%(%.!69$71!8$%1-!*6,#$#!8#%!&71%*5.!4#!*%,%69$71!/%#!%*%(-1%%#$!

*,-:/#!-$$*$$!#($0!$+$%!'%!4#!%(1#%!%74%(%.% .$&%!*,$80*$%%!+$%!'%1!7*-,9%%!8-*$%#!-''3!%74%(#!*$7!1#!'%6,9%'

$#%!#*+!$#!!!8#%!,$!*#,-!/+1$!3!8$%1>!$$/>!$;1!'0$!%*%*6';1$!*$7!%(1#%!

3!%74%(#!!,% !."%'%

<#"#!-"6$6

##!'$:%8-!&>/#$#!$3.!#*$'%%8$%8>.!'%#'%8#&(#.!#*$'('$1$#%!;.!%2>.1 .&$;! +$%! 3! :,#//>! ,!! %(1#%! 3! 8$%1>! $$/>! $;1! '0$! %*%*6';1%!

/!,+0!8#%!/!,-0% *-1!3!,!!'%1!%(1#%!%74%(#$!$+$%!$,(=%$#%!&$$!%*(*%'$! /!,+0!/%#!

%74%(#$!3!!*,!$!8#%!&$$!%*(*%'$!/!,-0!/(#!%74%(#$!3!!1,% .$&%!4#!69$7/%!'0$!%74%(%.!/%!'0$!8$%1-!&3/%(#$!*$7!$,(=$71!/%'

$#20!$$7.!/(#!%*%(-1%%#%!+7$+!%(1#%!-$$*$%

!"#$%!!"#%(!,)&#-"$%(!"–3

"%! $,%%.!*,5$%.!*,$$-&%%.!*$7!%*%'6:$1$#%!:%#!$31!#,9%8>!/%$--&#&3!&$$!95,$$!%*69$71!,+'$!#2%;/-$;1!&$%.!'%-($,%.!&0:9,$1%.!*#-,$7&%-&%%.!4%/%'(;&3.!$3.!<$%,%$/%$,(#.% <$%.!*,$$-&%%.!#7$6.$!3!1$3-$%8-!:1;&$>!#*+!$31!**%*%'$/%$,(#!!#(.%!$2#!!(0&3!$3.!%74%(#.$!9,3-

22 *?@!*",)!,<#"")*"+-

&%/$*$%%($#%!#,9%8-!:%#!1#!*,$&'%$,%&4%(!3!(7&%$:1;/(#!$$7!!%+!+'!"1! ;.!%*%(-1%%#.$!*$7!'%6,9%$#%!'%.!#7$>.%

"%!'0$!$%/1+/%1%.!%74%(%.!8#4$,(=$71!&$$!95,$!$3!46&3!%1+.!/$-1#'%8$0!%*%*6'$7$!+*;.!$#!'0$!&3/%(#!8#4$,(=$71!&$$!%*(*%'$!$3!46&3 /%#.!/$1#'%8>.! %74%(#.%!A%! $31! ('%#!#1$('3D3$! +*;.! $$!&3/%($! *%,1-!

&$$!2'%#%&43$%8+2!95,$!;.!$$/>!'0$!%74%%51$!6$&%!8#%!3!%74%(#!*#,$7'

&%-=%$#%!;.!$$/>!'0$!%*%*6';1%! -%1%8-$!$%!611$%%.!$$7!#(.%0"1#!$3.!%1&%0$)!8#%!$$7!%+!+'!"1"*%,-

1$01!&$$! 2'%#%&43$%8+2!95,$$!#($0!*,5$#!&,$71!$31!#*$8-'7D>!$$7.!

&$$!21$3$%8+2!$+*$!$;1!/$,(51!$$7.!//64%230% -%#!*$''$0.!#%51%.!$%!$,%%.!#7$6.!*,5$%.!*,$$-&%%.!#*$$6'%&#1!

/%#!%'#9%&$$*$%3/613!%*#,8>!8-'7D3!:%#!$31!#1-*$723!8,%$3,(;1!8#4%-$+$3$#.!%74%(#.!8#%!%*%*6'$7%

/(3&#'0!(!&4)#!(' &1%!,&#! "'3"$#! "$%#&!.13"!6!$"! !'5)! "'3"$"/!$!)!.)&%#!$6,"$)!!6/!!),-/!!)'/#!!#!"!"$%#&!.13"!6!.#&!(")/!!)!"($("')!!9%!"'$3"&=%!#'!=%" *53+!$60 ?D51$7/%!%74%(#!*1!8-4%$3!&$%.!%74%(%.!+!!8#%!-,!$!&$$!&3/%($!

$$/>.!$$7.!*%!C#(,1$7/%!*-1;!&$%.!%74%(%.!+!!8#%!-,!$#!(&#!$/>/#$#!

*+$!*!$!*-!8#%!*,%

67#.$""'

)!-*EA*#F"+!<#"!)EF" 23

,%#!$$7!*!(6,$7/%!$79#(#!%74%(#!*-1;!&$$!%*(*%'$!$;1!+!!8#%!

-,$!*$7!$6/1%%!!$#!&71%*(*%'#!$/>/#$#!+,!8#%!-!!&$#!&3/#(#!)!8#%!0!

#1$%&$$(9;.%!+,8%(!1#!#*$'%%94%(!+$%!3!%74%(#!*1!%(1#%!8-4%$3!&$31!)0!

/$,%&/+.!"0% *151$7/%!$$!$79#($!&3/%($!1!$3.!%74%(#.!*1!/%!$#!&3/%(#!+$!!$!

-$!,$!)!8#%!0%!**%%'>!+*!6!*!$!!-*!6!*,!8#%!∧

!"# = ∧

ΓΕΒ /;.!8#$#8$'

,7(>10! $#! $,(:;1#!+*,! 8#%! -*!! %(1#%! (&#%!.$&%!∧

#!" = ∧

"$% $! %15!∧

!"& = ∧

!"' ! !8#%!+*!6!*!%!?*-,9$71!'$%*+1!'0$!$,(:;1#!+*)!8#%!

!*0!/%!/(#!*'%7,-!(&3!8#%!$%.!*,$&8%(/%1%.!&.!#7$>1!:;1(%.!(&%.%!3,#!

%(1#%!(&#!8#%!6$&%!69$7/%!!)*!6!*0!8#%!!+)!6!!0% <$#!$,(:;1#!+*1!8#%!!*1!%(1#%!!+*!6!*!!!8#%!!1*!8$%1>!*'%7,-!

8#%!8-4%$3!&$31!!+!%!3,#!!1+!6!1!% "/$(;.!*,$80*$%%!+$%!!1-!6!1,!$!8#%!#($0!!+,!6!!-$!69$7/%!+$%!

$#!$,(:;1#!1+,!!1!-!%(1#%!(&#$!$*+$%!∧∧

ΖΒΓ=(!# .

**%%'>!+)!6!!0$!!1+!6!1!!8#%!∧

(!# = ∧

($' $!!$#!$,(:;1#!1+)!

8#%!1!0!%(1#%! (&#$!$*+$%!1)!6!10%!**(&3.$!#($0!)*!6!*0!8#%!3!1*!

%(1#%!8-4%$3!&$31!)0$!$%'%8-!4#!69$7/%!+$%!∧

("& = ∧

'"(6!$,4>%

3,#!8#4%/%-!#*+!$%.!:;1(%.!∧

&"( !8#%!∧

'"( !%(1#%!$,4>$!$*+$%!3!

1*!%(1#%!8-4%$3!&$31!'%#!$$7!!*!!$79#(#!%74%(#!)0%! !."%'%

/(3&#'0!)!&4)#!)' &1%!,&#!"'3"$#!"$%#&!.13"!6!$"!!""&/!"'3"$"/!()'!(""%)5%!#(#!!)!$'&)!$6,"$)!.#&!'&*"+"!#&!#(#!!)!$6,"$)!#'!##!!#!"!)&!!""&/!"'$3"$"/!2"$$.)%!#&!(1%9!$!)!$'&)!"($("')" *53+!$60 .&$;!+$%!3!%74%(#!+!!%(1#%!8-4%$3!&$%.!$,%%.!%74%(%.!!-$!!,!8#%!

!*!&$$!8$%1+!&3/%($!$$/>.!$$7.$!*$7!%(1#%!$$!!%!06'$7/%!1#!#*$'%(2$7'

/%!+$%!$%!%74%(%.!!-$!!,!8#%!!*!&,(&8$1$#%!&$$!('%$!%*(*%'$% ?*$46$$7/%!+$%!$%!$,%%.!%74%(%.!!-$!!,!8#%!!*!'%!&,(&8$1$#%!&$$!

('%$! %*(*%'$! 8#%!&7:8%8,%/61#! +$%!3!!-! &,(&8%$#%! %8$+.! $$7! %*%*6'$7!

$;1!!*!8#%!!,%!

24 *?@!*",)!,<#"")*"+-

.&$;!+$%!$$!%*(*%'$!$;1!+!!8#%!!-!$6/1%%!$$!%*(*%'$!$;1!!*!8#%!

!,$!8#$-!$31!%74%(#!!1%!#+$%!$%!%74%(%.!+!$!!-$!!1!%(1#%!$/$%*(*%'%.$!

+*;.!8#%!$%!%74%(%.!!1!$!!,!8#%!!*% **%%'>!3!+!!%(1#%!8-4%$3!&$%.!!,!8#%!!*$!%(1#%!8-4%$3!&$$!%*(-

*%'+!$$7.!8#%!&71%*5.!8#%!&$31!%74%(#!!1!*$7!#1>8%%!&.!#7$+%!

.$&%!3!:;1(#!∧

"!# !%('

1#%! $,4>$! +*;.! 8#%! 3! :;1(#!∧

ΑΒΓ /#*+! $31! 7*+4%&30%!

3,#!∧

"!# = ∧

ΑΒΓ $!%15!+!$!

!-!8#%!!1!%(1#%!$/$%*(*%'%.%!

+7$+!%(1#%!-$$*$%!3,#!3!!-!

&7/*(*$%%!/%!$3!!1!8#%!&71%'

*5.!&,(&8%$#%!*-1;!&$$!%*('

*6'$!$;1!!*!8#%!!,%!3,#!!3!

!-! '%1! /*$,%(! 1#! &,(&8%$#%!

%8$+.! $$7! %*%*6'$7! $;1! !,$!

!*%!.$&%$!$%!$,%%.!%74%(%.!!-$!

!,$!!-!&,(&8$1$#%!*-1;!&$$!('%$!!%*(*%'$% !."%'%

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

"%!*,$$-&%%.!3!8#%!4!#*$$%'$01!8,%$>,%#!*$7!$,%$4%$$01!$%.!'7-1#$+$3$%.!8#4%$+$3$#.!%74%(#.!8#%!%*%*6'$7!&$$!95,$$!&0/(;1#!/%!$$!

:1;&$+!$,%&/+%!.$&%$!&$#!*'#(&%#!$3.!8#4%$+$3$#.!%74%(#.!8#%!%*%*6'$7$!

3!&7/*%,%($,-!$;1!-*%%,;1!%74%%51!/%#.!

%%'%8>.! 8%1$,%8>.! '6&/3.! %74%%51! &$$!

95,$!8#4$,(=%$#%!&%!&96&3!/%!/(#!#8$(1#!

/%0!$3.!'6&/3.!/&'6*%!&9>/#!4%&0% 4$#1! $%! %74%(%.! $3.! '6&/3.! *'31!

$3.! /%0! %(1#%! &71%*(*%'%.! /*,+$#&3! 30$!

$+$%!3!8#4%$+$3$#!$3.!/%0!*,$.!$$!%*(*%'+!

$$7.! %2#&(#'(=%$#%! /%! $31! 8#4%$+$3$-!

$3.!*,$.!'0$!/+1$!%74%(%.!$3.!'6&/3.% )! *,+$#&3! 3! #*$$%'%(! 7*+'%%:/#!

%*#:;:%8$0! &7''$:%&/$0$! *$7! &#&(=%$#%!

&$31!2%1&%!"-%%#2!(0&3!$$7!%*%*6'$7%!+-1$(4%$#$!&$31!*,+$#&3!4!9,3&%/$*$%%($#%!

67#.$""(

67#.$"()$

)!-*EA*#F"+!<#"!)EF" 25

3! 611$%#! $3.!8#4%$+$3$#.! $3.! %74%(#.! /%0!*,$.! $%.!-*%%,%.!#8$(1%.! $3.!

'6&/3.!:%#!1#!%846&%%!$3!&71%*(*%'3! %'%+$3$#!$;1!%74%%51!$3.!'6&/3.%!

A%!#7$>!$31!611$%#!3!*,+$#&3!9#,#8$3,(=%$#%!;.!61#.! 2*#,#:;:%8+.2!!

&7''$:%&/+.% /(3&#'0!*!&4)#!*'

&1%!'5)!"'3"$"/!"$%#&!.13"!"/!$!)! $'&)!"($("')#!!#!"!)&!"'3"$"/!"$%#&!(#"10060"/" *53+!$60 .&$;!+$%!$%!%74%(%.!-,!8#%!+!!%(1#%!8-4%$%.!&$$!('%$!%*(*%'$$!/%!

!!8#%!,!$#!(913!$$7.%!0#!#*$'%%94%(!+$%!$%!'0$!%74%(%.!-,!8#%!+!!%(1#%!

*#,-''3'%.% *151$7/%! $#! &3/%(#! !!

8#%!,!$;1!'0$!%74%%51!8#%!(6-,$7/%! $31! ,*! *-1;! &$$! %*(-*%'$$!8-4%$3!&$3!!,!8#%!(&3!/%!

$31! +!%! **(&3.! (6,$7/%! $%.!

+*$!!!*!8#%!+,% **%%'>!3!+!!%(1#%!8-4%'

$3! &$$! %*(*%'$$! %(1#%! 8-4%$3!

8#%! &%! 8-4%! %74%(#! $$7! %*%*6'

'$7! *$7! *%,1-! #*+! $$! (91$.!

$3.%! 3,#! $%! :;1(%.!∧

!$# ! 8#%!∧

ΑΒΕ !%(1#%!$,46.%

"/$(;.$! %*%%'>! !3!-,! %(1#%!8-4%$3!&$$! %*(*%'$$!$%!:;1(%.!∧

ΓΔΒ

8#%!∧

ΓΔΕ !%(1#%!$,46.% **%%'>!!!,!!8$%1>!8#%!+!!6!,*$!$#!$,4$:51%#!$,(:;1#!!+!,!8#%!

!,*!%(1#%!(&#$!$*+$%!+,!6!!*% **(&3.!#($0!+,!6!!*$! !+!!6!,*! ! !8#%! !+*! !8$%1>$!$#!$,(:;1#!

+,*!8#%!+!*!%(1#%!(&#%

.$&%!$%!:;1(%.!∧

ΑΔΕ !8#%!∧

ΑΒΕ !%(1#%! (&%.!8#%!#($0!3!$%'%7$#(#!

%(1#%!$,4>$!%(1#%!$,4>!8#%!3!∧

ΑΔΕ %!3,#!3!!*,!!%(1#%!8-4%$3!&$31!,+!8#%!

/+*;.!:1;,(=$7/%!0!&$%.!,!!8#%!,-%

67#.$""*

26 *?@!*",)!,<#"")*"+-

<71%*5.!$%!%74%(%.!,-$!,!!8#%!,+!%(1#%!$/$%*(*%'%.$!8#45.!8#%!3!

+!$!#($0!$$!$,(:;1$!+!,!$,(=%%!%*(*%'$%!.$&%$!$%!+!!8#%!-,!%(1#%!$/$'

%*(*%'%.!8#%!#($0!$%!:;1(%.!∧

%#$ !8#%!∧

ΑΒΔ !%(1#%!$,46.$!$%!'0$!%74%(%.!

%(1#%!*#,-''3'%.%! !."%'%

/(3&#'0!+!&4)#!+' &1%! !'5)!"'3"$"/!"$%#&! !(#"10060"/!.#&!06%3)5%!(1%9!$"!.13"!,&#!#(#!#'!*/!!'+#$#!$6,"$##!6!"'3"$#!()'!$'%'*"&! !!#!$6,"$#!#'!1!."$!#&!$!)!$'&)!"($("')!,"!!&/!'5)!!(#"10060"/"! *53+!$60 .&$;!*)1!3!%74%(#!*$7!$,(=%$#%!#*+!$#!'0$!$79#(#!&3/%(#!*!8#%!

1!$;1!*#,-''3';1!%74%%51!

+!!8#%!-,!#1$(&$$%9#%!0#!

#*$'%%94%(! +$%! $#! &3/%(#!

$3.!*1!&,(&8$1$#%!&$$!%*(-*%'$! $;1! %74%%51! +!! 8#%!

-,% ?*$46$$7/%!+$%!3!%7-

4%(#!*)1!'%!&,(&8%$#%!*-'

1;!&$$!%*(*%'$!$;1!+!!8#%!

-,% #+$%$!#1!(6,$7/%!%*(*%'$!*$7!*%,1-!#*+!$31!*)1$!#7$+!4#!$6/1%%!

$$!%*(*%'$!$;1!+!!8#%!-,!8#$-!%74%(#!:,#//>$!$31!*1% .$&%!$%!%74%(%.!*)1!8#%!*1!4#!$,(=$71!/(#!%*%(-1%%#$!*,-:/#!-'

$$*$%!/,0$!%74%(%.!/%!'0$!8$%1-!&3/%(#$!$#7$(=$1$#%0% 3,#!3!%74%(#!*1!8%($#%!*-1;!&$$!%*(*%'$!$;1!+!!8#%!-,%!!!."%'% <#"#!-"6$6% A%!$31!*,+$#&3!*!:(1%$#%!(#1%,+!+$%!$%!*#,-''3'%.!%74%(%.!&$$!

95,$!4%;,$01$#%!;.!$/$%*(*%'%.%

67#.$""+

)!-*EA*#F"+!<#"!)EF" 27

/(3&#'0!,!&4)#!,' &1%!'5)!"'3"$"/!"$%#&!(#"10060"/!.#&!6!,&#!#(#!#'!*/!"$%#&!.1$3"!6!(")/!*%#!"($("')#!!#!"!.#&!6!1006!3#!"$%#&!.13"!6!(")/!#'$!#!!)!"($("')" *53+!$60 .&$;!+$%!$%!%74%(%.!+!!8#%!-,$!*$7!%(1#%!*#,-''3'%.$!$6/1$71!$$!

('%$!%*(*%'$!&$#!&3/%(#!!!8#%!,!#1$(&$$%9#!8#%!3!+!!%(1#%!8-4%$3!*,$.!

#7$+!$$!%*(*%'$%!0#!#*$'%%94%(!+$%!8#%!3!-,!%(1#%!8-4%$3!*,$.!$$!%*(-*%'$%

*151$7/%!$#!(913!!!8#%!,!$;1!'0$!%74%%51!/%!$31!%74%(#!!,!8#%!

(6,$7/%!*-1;!&$$!%*(*%'$!$31!,*!8-4%$3!&$3!!,!8#%! (&3!/%!$31!+!%!

**(&3.!(6,$7/%!$%.!+*$!!*!8#%!+,% **%%'>! 3! +!! %(1#%!

8-4%$3! &$$! %*(*%'$$! %(1#%!

8-4%$3!8#%!&%!8-4%! %74%(#!

$$7! %*%*6'$7! *$7! *%,1-!

#*+! $$! (91$.! $3.%! 3,#! $%!

:;1(%.!∧

+!, !8#%!∧

ΑΒΕ ! %(-1#%!$,46.%

**%%'>!$%!+!!8#%!-,!

%(1#%! *#,-''3'%.! 8#%! $6-/1$1$#%! #*+! $31! !,$! $%!

:;1(%.!∧

+!, ! 8#%!∧

ΓΔΒ 6'9$71! -4,$%&/#! '0$! $,46.!

8#%!#($0!3!∧

+!, !%(1#%!$,4>$!6*%$#%!+$%!8#%!3!∧

ΓΔΒ !%(1#%!$,4>%! 3,#!3!-,!%(1#%!8-4%$3!&$31!!,% **%%'>!$#!$,4$:51%#!$,(:;1#!+!,!8#%!!,*!69$71!8$%1>!*'%7,-!

$3!!!,!!!8#%!!!+!!6!,*$!!%(1#%!(&#$!$*+$%!8#%!!+,!6!!*% **%%'>!+!! 6!,*$!+,! 6!!*! 8#%!+*! 8$%1>! *'%7,-! &$#! $,(:;1#!

+,*!8#%!+!*$!#7$-!4#!%(1#%!(&#$!$*+$%!!∧

ABE = ∧

ΑΔΕ .

.$&%$!#($0!3!:;1(#!∧

ABE !%(1#%!$,4>$!%(1#%!$,4>!8#%!3!∧

ΑΔΕ . <71%*5.$!3!*,!%(1#%!8-4%$3!&$31!+,% **%%'>!3!*,!%(1#%!8-4%$3!&$31!+,!8#%!$31!,!$!%(1#%!8-4%$3!&$$!

%*(*%'$!$;1!*#,-''3';1!%74%%51!+!!8#%!-,% .$&%$!3!-,!%(1#%!8-4%$3!&$31!,*%

67#.$"",

28 *?@!*",)!,<#"")*"+-

<71%*5.$!3!-,!%(1#%!8-4%$3!8#%!&$31!!,!/+*;.!'%(9438%0$!8#45. 8#%!&$31!,*$!'3'#'>!%(1#%!8-4%$3!&$$!%*(*%'$$!#($0!%(1#%!8-4%$3!&%!'0$!

%74%(%.!$$7!%*%*6'$7!*$7!*%,1$01!#*+!$$!(91$.!$3.%!!!."%'%

/(3&#'0!-!&4)#-' &'3"$"/!(#"10060"/!(")/! !6%! $'&#! "'3"$#!.#&!)&!)()$"/!'"!2"$$$.)%!#&! $!)! $'&)!,"!#'!-%! "($("')#! "$%#&! !,"!#;5! !)'/!(#"10-060"/" *53+!$60 .&$;! +$%! $%! %74%(%.!+!!

8#%! -,! %(1#%! *#,-''3'%.! *,$.!

$31!*1!8#%!'%!&,(&8$1$#%!+'%.!

&$$!('%$!%*(*%'$%!0#!#*$'%%94%(!

+$%!$%!%74%(%.!+!!8#%!-,!%(1#%!

*#,-''3'%.% C-1;! &$$! %*(*%'$! $;1!

*#,#''>';1!*1!8#%!-,!(6,$7-/%!$31!%74%(#!)@!8-4%$3!&$31!

*1!&%!$79#($!&3/%($!$3.!)% **(&3.$!*-1;!&$$! %*(*%'

'$!$;1!*#,#''>';1!+!!8#%!*1!(6,$7/%!$31!%74%(#!)0!8-4%$3!&$31!*1!

&$$!&3/%($!$3.!)% **%%'>!3!*1!%(1#%!8-4%$3!*,$.!$%.!'0$!%74%(%.!)0!8#%!)@$!4#!%('

1#%!8-4%$3!8#%!&$$!%*(*%'$!*$7!#7$6.!$,(=$71% .$&%$!#($0!3!*1!%(1#%!8-4%$3!&$$!%*(*%'$!$;1!)0!8#%!)@$!%15!3!

+!!%(1#%!*#,-''3'3!/%!$31!*1$!6*%$#%!/&0/(;1#!/%!$31!*,+$#&3!+0!+$%!

3!+!!%(1#%!8-4%$3!&$$!%*(*%'$!$;1!)0!8#%!)@% "/$(;.$!#($0!3!*1!%(1#%!8-4%$3!&$$!%*(*%'$!$;1!)0!8#%!)@$!%15!

3!-,!%(1#%!*#,-''3'3!/%!$31!*1$!6*%$#%!+$%!3!-,!%(1#%!8-4%$3!&$$!%*('

*%'$!$;1!)0!8#%!)@% 3,#$!%(+&$1!$%!%74%(%.!+!!8#%!-,!%(1#%!8-4%$%.!&$$!%*(*%'$!$;1!

%74%%51! !)@! !8#%! !)0$!%(1#%!/&0/(;1#!/%!$31!*,+$#&3!50!/%$#20!$$7.!*#,-''3'%.%

!."%'%

67#.$"""-

)!-*EA*#F"+!<#"!)EF" 29

/(3&#'0!!".!&4)#!".' &1%! '5)! !",%#,"%"/! "'3"$"/! "$%#&! #%!$$!)&+#! (#"10060"/! (")/!'5)! !",%#,"%"/! "'3"$"/#! ()'! '"%! 2"$$.)%!#&! $!)! "($("'#! !)'/#!!#!"!!#!'5)!7"586!!9%!!",%),*%9%!"'3"&=%!)"$7)'%!$$"/!89%$"/" *53+!$60 .&$;!+$%!$%! $%/1+/%1%.! %74%(%.!!-!8#%!!+! %(1#%!#1$(&$$%9#!*#-

,-''3'%.!8#%!/3!&71%*(*%'%.! !/%!

$%.! $%/1+/%1%.! %74%(%.! *1! 8#%!

*,%! C#(,1$7/%! *-1;! &$#! '0$!

/3! &71%*(*%'#! =%0:3! $;1! $%-/1+/%1;1! %74%%51! $#! (&#! $/>-/#$#! !-! 6!!+! 6! *1! 6! *,! 8#%!

(6,$7/%! $%.! %74%(%.!!*$!+,! 8#%!

-1% 0#!#*$'%%94%(!+$%!

∧

ΓAB = ∧

ΔΕΖ . **%%'>!!-!(&3!!8#%!*#,-''

'3'3!/%!$31!*1$!6*%$#%!+$%!8#%!3!

-1! %(1#%! (&3! 8#%! *#,-''3'3! /%!

$31! !*%! "/$(;.$! #($0! !+! (&3!

8#%!*#,-''3'3!/%!$31!*,$!6*%$#%!

+$%!3!+,!%(1#%!(&3!8#%!*#,-''3'3!/%!$31!!*% .$&%$!#($0!$%!-1!8#%!+,!%(1#%!(&%.!8#%!*#,-''3'%.!/%!$31!!*!8#%!

'%!&,(&8$1$#%!&$$!('%$!%*(*%'$!/%!#7$>$!4#!%(1#%$!&0/(;1#!/%!$31!*,+'

$#&3!#$!/(&%.!8#%0!*#,-''3'%.!/%$#20!$$7.% ##!(&#!8#%!*#,-''3'#!$/>/#$#!+,!8#%!-1!&71'6$1$#%!/%!$#!$/>'

/#$#!+-!8#%!,1$!$*+$%!8#%!#7$-!%(1#%!/%$#20!$$7.!*#,-''3'#!8#%!(&#%!!

**$/61;.$!$#!$,(:;1#!+!-!8#%!,*1!69$71! !+!!6!,*$! !!-!6!*1! ! !8#%!!

+-!6!,1%!3,#!%(1#%!(&#$!$*+$%!8#%!3!:;1(#!∧

+!- !%(1#%!(&3!/%!$3!:;1(#!∧

ΔΕΖ . !."%'%

67#.$"$.

30 *?@!*",)!,<#"")*"+-

!"#$%!!"#%(!,)&#-"$%(!!%!- 10

A%!$%.!*#,#*-1;!*,$$-&%%.!%%&-:%$#%!3!611$%#!$3.!*#,#''3'(#.!

%74%%51!&$$!95,$$!'%#!/6&$7!$3.!>'3!4%/%'%;/613.!8#4%$+$3$#.!%74%(#.!

8#%!%*%*6'$7% )! *,+$#&3! 5! 4%;,%($#%! &#&%8>$! '%+$%! 8#$-! $31! #*$'%%8$%8>! $3.!

'%#'%8#&(#!%2#&(#'(=%$#%!+$%!'0$!8-4%$%.!%74%(%.!&%!%*(*%'$!%(1#%!*,5-$%&$#!$/$%*(*%'%.!8#%!/%$-!*#,-''3'%.%!#$!:%:$1+.!#7$+!%1%&90%$#%!/%!

$31!*,+$#&3!*%!.$&%$!$$!+$%!'0$!*#,-''3'%.!%74%(%.!%(1#%!$/$%*(*%'%.$!

'(1%%! $3! '71#$+$3$#! 3! 8#4%$+$3$#! $3.! /%#.! &%! 61#! %*(*%'$! 1#! &71%-*-:%$#%!8#%!$31!8#4%$+$3$#!$3.!-''3.!&%!#7$+!/*,+$#&3!+0%!A%!#7$+!$$1!

$,+*$!%%&-:%$#%!+9%!,3$-$!#''-!7*$1$$0/%1#!8#%!3!8#4%$+$3$#!'0$!%*%'

*6';1% )!*#,#''3'(#!'0$!%74%%51!&$$!95,$$!*$7!%(1#%!*#,-''3'%.!*,$.!

$,($3$!#*$'%%810%$#%!/%!$3!9,>&3!$$7!8-4%$$7!%*%*6'$7$!+*;.!8#%!3!*#'

,#''3'(#!'0$!%74%%51!*#,#''>';1!*,$.!$,($3!&$$!%*(*%'$!#*$'%%810%-$#%!/%!$3!9,>&3!$3.!8-4%$3.!%74%(#.!/*,+$#&3!#0%!

#6'$.$!3!*,+$#&3!&)!%1$-&&%$#%!&$31!%1+$3$#!$;1!*,$$-&%;1!*$7!

#1#(6,$1$#%! &$31! *#,#''3'(#! %74%%51! &$$! 95,$$! 8#%! *#,$7&%-=%%! $31!

%&+$3$#!'0$!:;1%51!/%!#1$(&$$%9#!*#,-''3'%.!*'%7,6.$!#''-!%%&-:%%!8#%!

$31!*#,#''3'(#!'0$!%*%*6';1!*$7!'%#*,#:/#$%0%$#%!3!*,+$#&3!&4%

/(3&#'0!""!&4)#!""' +(#!')3*%!$6,"$)!()'!2"$$."!#&! ".!#/! "%#/! "(&(*')'! %#!#+3"$!"'3"$#!8"#,,-!.13"!6!!(")/!!)!"($("')" *53+!$60 26,$7/%!$79#(#!%74%(#!

!-! *-1;! &$$! '$461! %*(*%'$!

8#%! #*+! $$! '$461! &3/%($! +!

$31!8-4%$3!%74%(#!+,!%*(!$3.!

!-%! /#$! &3/%($!+! 8#%! 3! %7-4%(#!!-!$,(=$71!%*(*%'$0%

*-1!3!+,!%(1#%!8-4%$3!

&$$! '$461! %*(*%'$$! $+$%! 69%%!

#94%(! #*+! $$! &3/%($! +! 8--4%$3!&$$!%*(*%'$%

67#.$""$$

)!-*EA*#F"+!<#"!)EF" 31

*-1!3!+,!'%1!%(1#%!8-4%$3!&$$!'$461!%*(*%'$$!(6,$7/%!*-1;!&$$!

%*(*%'$!$31!%74%(#!*,!8-4%$3!&$3!!-!&$$!&3/%($!,%!**(&3.$!*-1;!&$$!

%*(*%'$! $;1! $%/1+/%1;1! %74%%51!,*! 8#%!,+!(6,$7/%! $31!+1! 8-4%$3!

&$31!,*%!#6'$.$!*-1;!&$$!'$461!%*(*%'$!(6,$7/%!$31!%74%(#!)10!*#'

,-''3'3!*,$.!$31!!-% **%%'>!3!!-!%(1#%!8-4%$3!8#%!&$%.!'0$!%74%(%.!,*!8#%!,+$!4#!%(1#%!

8-4%$3!8#%!*,$.!$$!%*(*%'+!$$7.!8#%!#($0!3!)0!%(1#%!*#,-''3'3!/%!$31!

!-$!4#!%(1#%!8#%!#7$>!/&0/(;1#!/%!$31!*,+$#&3!#0!8-4%$3!&$$!%*(*%'$!

$;1!,*!8#%!,+% )!!%74%(#!)0!;.!8-4%$3!&$$!%*(*%'$!$;1!,*!8#%!,+!%(1#%!8-4%$3!

&%! 8-4%! %74%(#!#7$$0! $$7! %*%*6'$7! *$7! '%6,9%$#%!#*+! $$! (91$.! $3.!1%!

.$&%!3!)0!%(1#%!8-4%$3!8#%!&$3!1+% <71%*5.$!3!1+!%(1#%!8-4%$3!$+&$!&$31! !*,! !+&$!8#%!&$31! !)0$!

$*+$%!%(1#%!8-4%$3!8#%!&$$!%*(*%'+!$$7.$!*$7!%(1#%!$$!'$461!%*(*%'$%! !."%*%

/(3&#'0!"$!&4)#!"$'

+(#!$6,"$)!!)!)()$)!2"$$."!#&!$"!*%#!"($("')#!%#!'?93"$!.13"$!6!(")/!!)!"($("')" *53+!$60 0%;,$0/%! 61#! $79#($!

&3/%($! !! %8$+.! $$7! '$461$$.!

%*%*6'$7. +*+! $$! !! (6,$7/%! $31!

8-4%$3! %74%(#!!-! *,$.! $$! '$'

461! %*(*%'$! /&0/(;1#! /%! $31!

*,+$#&3!&&0% <$$!%*(*%'$!$$7!&3/%($7!

+!8#%!$3.!%74%(#.!!-!(6,$7/%!

$31!%74%(#!+,!*#,-''3'3!*,$.!

$31!!-% **%%'>! 3! +,! %(1#%! *#-

,-''3'3!*,$.!$31!!-$!8#%!3!%7-4%(#!!-!%(1#%!8-4%$3!&$$!'$461!%*(*%'$$!6*%$#%!+$%!8#%!3!+,!%(1#%!3!=3'

$$0/%13!8-4%$3!*,$.!$$!'$461!%*(*%'$!/*,+$#&3!+0%! !."%*%

67#.$""$%

32 *?@!*",)!,<#"")*"+-

/(3&#'0!"%!&4)#!"%' +(#! *%#! $6,"$)! "%#/! "(&(*')'! '"%! "$%#&! ''%#!#%! %#! '?93)5%!!'5)!.13"!"/!(")/!!)!"($("')!#'!##!)&!)()$"/!%#!2"$$.)%!#&!(")/!!)!$'&)!,*")/!!)'!"(&(*')'" *53+!$60 .&$;!+$%!%(1#%!'71#$+1!

#*+!$$!&3/%($!+!$$7!'$461$$.!

%*%*6'$7! 1#! 7D5&$7/%! '0$!

8-4%$%.!+!!8#%!+-!*,$.!#7-$+$! $%! $*$(%.! 1#! &,(&8$1$#%!

8#%! *,$.! $$! ('%$! /6,$.! $$7!

%*%*6'$7% #$!%*(*%'$!$;1!%74%%51!

+!!8#%!+-!4#!$6/1%%!$$!'$461!

%*(*%'$!8#$-!/(#!%74%(#$!6&$;!

$31!,+*% "%! %74%(%.!+!$!+-!8#%!

,+*!%(1#%!&71%*(*%'%.$!%15!3!

-+! 8#%! 3! !+! %(1#%! 8-4%$%.!

&$$!'$461!%*(*%'$%!.$&%!3!-+!8#%!3!!+!%(1#%!8-4%$%.!&%!8-4%!%74%(#!$$7!

'$461$$.!%*%*6'$7!*$7!'%6,9%$#%!#*+!$$!(91$.!$$7.!+$!8#%!&71%*5.!&$31!

,+*%

3,#!3!:;1(#!∧

!+* !%(1#%!$,4>$!+*;.!8#%!3!∧

ΓΑΕ $!%15!&,(&8$1$#%!

*-1;!&$$! ('%$! %*(*%'$%!+7$+! %(1#%!#'01#$$$!8#%!&71%*5.!$%!'0$! %74%(%.!

+!!8#%!+-!&7/*(*$$71$!$*+$%!'%1!%(1#%!'71#$+1!1#!7D;4$01!'0$!8-4%$%.%!!!."%'%

!"#$%!!"#%(!,)&#-"$%(!""!*/")%!"&

"%!$,%%.!*,$$-&%%.!8#'0*$$71!$3!'71#$+$3$#!8#$#&8%7>.!%74%(#.!

8-4%$3.!*,$.!61#!%*(*%'$!#*+!61#!&3/%($!%8$+.!%*%*6'$7!>!*-1;!&.!#7'

$+% )!*,+$#&3! &&!#*$$%'%(! $31!#74%1$%8>!/$,(>! $$7!:1;&$$0!4%;'

,>/#$$.!!"$%"",!8%"2$&'"$%!!!/&'6*%!)#!&$$!&9>/#!&"%&0$!$$!$*$($!&$%.!

&0:9,$1%.!*#,$7&%-&%%.!$3.!<$%,%$/%$,(#.!%/(#1(=%$#%!8#%!/%!$%.!/$,'

(6.!))%!8#%!)))#

67#.$"$&

)!-*EA*#F"+!<#"!)EF" 33

67#.$"$&)$

@)"%-!/" { ( ) ( ) ( )C1,C!,C A ⊆Δ⊆Γ∉ $!1,!⊥!!-$!+,!⊥!!-7!⇒!+1!⊥/C0 @)"%-!// { ( ) ( )C!,C A ⊆Γ∉ $!+1!⊥!/C0$!1,!⊥!!-7!!⇒!!!+,!!⊥ !!-

@)"%-!!/// { ( ) ( )C!,C A ⊆Γ∉ , +1!⊥!/C0$!+,!⊥!!-7!!⇒!!!1,!⊥!!-

/(3&#'0!"(!&4)#!"(' 4#!"($("'#!"($!!9%!)()$9%!6! $'&#!"'3"$#!"$%#&!.13"!6!"$%#&!,"$!#;5!!)'/!(#"10060#" *53+!$60 .&$;! +$%! 3! %74%(#!+!! %(-

1#%! 8-4%$3! *,$.! '0$! '$&/61#!!

%*(*%'#%! 0#! #*$'%%94%(! +$%! $#!

%*(*%'#!#7$-!%(1#%!*#,-''3'#%! *-1! $#! %*(*%'#! '%1! %(1#%!

*#,-''3'#$!$+$%! /*,$%8$%%1+/%1#0!

4#!$6/1$1$#%!8#$-!%74%(#!:,#//>$!

6&$;!$31!)0% 67#.$"""$'

34 *?@!*",)!,<#"")*"+-

C-1;!&$31!%74%(#!)0!*#(,1$7/%!$79#($!&3/%($!@!8#%!(6,$7/%!$%.!

%74%(%.!+@!8#%!!@!*$7!&,(&8$1$#%!#1$(&$$%9#!*-1;!&$#! '0$! '$461$#!

%*(*%'#% **%%'>!3!+!! %(1#%!8-4%$3!&$#!'0$! %*(*%'#$!4#! %(1#%!8-4%$3!8#%!

&$%.!%74%(%.!+@!8#%!!@!$;1!'0$!'$461$;1!%*%*6';1%

.$&%!&$$!$,(:;1$!+!@!$%!:;1(%.!∧

@+! !8#%!∧

ΚΒΑ !%(1#%!$,46.$!*$7!

%(1#%!#'01#$$%!3,#!$#!%*(*%'#!'%1!$6/1$1$#%!8#%!&71%*5.!%(1#%!*#,-'-'3'#%!!

!."%'%

/(3&#'0!")!&4)#!")' &1%!'5)!!",%#,"%"/!"'3"$"/!"$%#&!#%!$$!)&+#!(#"10060"/!(")/!'5)!!",%#,"%"/!"'3"$"/!.#&!'"!2"$$.)%!#&!(1%9!$!)!$'&)!,"!#'!*/!"($$("')#!!#!"!!#!"($("'#!!9%!"'3"&=%!#'!=%!"$%#&!(#"10060#" *53+!$60 .&$;! $%! $%/1+/%1%.! %74%(%.!

!+! 8#%! !-$! *$7! %(1#%! #1$(&$$%9#!

*#,-''3'%.! *,$.! $%.! $%/1+/%1%.!

%74%(%.! *,! 8#%! *1! 8#%! '%! &,(-&8$1$#%!&$$! ('%$! %*(*%'$!/%!#7$6.%!

0#! #*$'%%94%(! +$%! $#! %*(*%'#! *$7!

$,(=$1$#%! #*+! $%.!!-$!!+! 8#%!*1$!

*, #1$(&$$%9#$!%(1#%!*#,-''3'#%! +*+!$$!&3/%($!!!(6,$7/%!$31!

8-4%$3! %74%(#! !)! *,$.! $$! %*(*%'$!

$;1!$%/1$/61;1!%74%%51!*1!8#%!*,% +*+!$$!)!(6,$7/%!&$$!%*(*%'$!$;1!!*1!!8#%!!*,!!$%.!%74%(%.!!)@!

8#%!)0$!*#,-''3'%.!*,$.!$%.!*1!8#%!*,!#1$(&$$%9#% **%%'>!3!!)!%(1#%!8-4%$3!&$$!%*(*%'$!$;1!%74%%51!*1!8#%!*,$!%('

1#%!8-4%$3!8#%!&$%.!%74%(%.!)@!8#%!)0!#7$$0!$$7!%*%*6'$7$!*$7!'%6,9$'

1$#%!#*+!$$!(91$.!$3.%!.$&%$!$%!:;1(%.!∧

!)0 !8#%!∧

ΒΗΚ !%(1#%!$,46.% +($0!+/;.!$%!%74%(%.!!+!8#%!!-!%(1#%!*#,-''3'%.!*,$.!$%.!)0!

8#%!)@!#1$(&$$%9#$!$+&$!$%!:;1(%.!∧

!)0 !8#%!∧

)!+ $!+&$!8#%!$%!:;1(%.!∧

ΒΗΚ !8#%!∧

ΗΒΓ !69$71!-4,$%&/#!'0$!$,46.%

67#.$"""$(

)!-*EA*#F"+!<#"!)EF" 35

.$&%!$%!:;1(%.!∧

)!+ !8#%!∧

ΗΒΓ !%(1#%!$,46.$!$*+$%!3!!)!%(1#%!8-'

4%$3!&$%.!'0$!$%/1+/%1%.!%74%(%.!!+!8#%!!-%!3,#!3!!)!%(1#%!8-4%$3!8#%!

&$$!%*(*%'$!$;1!$%/1+/%1;1!%74%%51!!+!8#%!!-% <71%*5.$!/&0/(;1#!/%!$31!*,+$#&3!&30$!#($0!$#!'0$!%*(*%'#!$;1!

$%/1+/%1;1!%74%%51!!+$!!-!8#%!*,$!*1!%(1#%!8-4%$#!&$31! ('%#!%74%(#!

!)$!$#!'0$!#7$-!%*(*%'#!%(1#%!*#,-''3'#%! !."%'%

/(3&#'0!"*!&4)#!"*' &1%!'5)!(#"10060#!"($("'#!!*,%)%!#&!#(#!*%#!"($("')#!!#!"!)&!.)&%*/!!),*/!!9%!"(&(*'9%!#'!=%!"$%#&!"'3"$"/!(#"10060"/" *53+!$60 .&$;!+$%!$#!'0$!*#,-''3'#!%*(-

*%'#! $6/1$1$#%! #*+! $$! %*(*%'$!*10)!

8#$-! $%.! %74%(%.!*1! 8#%!)0%!0#! #*$'

'%%94%(! +$%! $%! *1! 8#%! )0! %(1#%! *#-,-''3'%.%!

*-1!$%!%74%(%.!*1!8#%!)0!'%1!%('

1#%! *#,-''3'%.$! $+$%! *,$%8$%%1+/%1%.!

%*(! $$7! %*%*6'$7! *10)! 4#! $6/1$1$#%!!

*,$.!$$!/6,$.!$;1!!1!!8#%!!0!!>!*,$.!$$!

/6,$.!$;1!*!8#%!)% .&$;! +$%! $6/1$1$#%! *,$.! $$!/6'

,$.!$;1!1!8#%!0$!&$$!&3/%($!@% **%%'>!$%!%74%(%.!*1@!8#%!)0@!

%(1#%! %74%(%.! *$7! #1>8$71! #1$(&$$%9#!

&$#!'0$!*#,-''3'#!%*(*%'#$!6*%$#%!+$%!

$#! '0$! %*(*%'#! 69$71!8$%1+!&3/%($! $$!

@%!+7$+!+/;.!%(1#%!-$$*$$!:%#$(!$#!'0$!%*(*%'#!%(1#%!*#,-''3'#% 3,#!$%!&71%*(*%'%.!%74%(%.!*1!8#%!)0!'%1!$6/1$1$#%!8#%!&71%*5.!

%(1#%!*#,-''3'%.%! !."%'%

67#.$"""$*

36 *?@!*",)!,<#"")*"+-

/(3&#'0!"+!&4)#!"+' &1%!'5)!"'3"$"/!!*,%)%!#&!#(#!(#"10060#!"($("'##!3#!!*,%)%!#&!$"!,*"6!#%10)8#" *53+!$60 .&$;!+$%!$%!%74%(%.!+!!8#%!-,!$6/1$1$#%!#*+!$#!*#,-''3'#!%*('

*%'#!!C$!F!!8#%!!<!!&$#!&3/%(#!!+$!*$!!!!8#%!!-$!1$!,!!#1$(&$$%9#%!0#!#*$'

'%%94%(!+$%!ΖΔΓ

=Z

*!

+* .

26,$7/%!$%.!%74%(%.!+-$!!,!8#%!+,$!8#%!6&$;!+$%!3!+,!$6/1%%!$$!

%*(*%'$!!F!!&$$!&3/%($!G%!26,$7/%!6$&%!$%.!!G1!!8#%!!G*% **%%'>! $#! *#,-''3'#! %*(-

*%'#! !F! !8#%! !<! ! $6/1$1$#%!#*+! $$!

%*(*%'$!G*!,$!$%!$$/6.!G*!8#%!,!!

%(1#%!*#,-''3'%.% "/$(;.$! %*%%'>! $#! *#,-'-

'3'#! %*(*%'#! F! 8#%! C! $6/1$1$#%!

#*+!$$!%*(*%'$!G1-+$!$%!$$/6.!G1!

8#%!+-!%(1#%!*#,-''3'%.% <$$! $,(:;1$! +,!! 3! %74%(#!

G*! %(1#%! *#,-''3'3! /%! $31! ,!$!

$*+$%1!ΞΔ

ΑΞ=

*!

+* .

**(&3.$! &$$! $,(:;1$!,+-! 3!

%74%(#!!1G!!%(1#%!*#,-''3'3!/%!$31!!-+$!!$*+$%1!ΖΔ

ΓΖ=

G,

+G .

.$&%!69$7/%!ΖΔ

ΓΖ=

*!

+*$!$*+$%!$%!'0$!%74%(%.!$6/1$1$#%!#*+!$#!

*#,-''3'#!%*(*%'#!&%!/6,3!#1-'$:#%!!!."%'%

!"#$%!!"#%(!,)&#-"$%(!"#-17

"%!*,$$-&%%.!#7$6.!%&$%-=$1$#%!&$31!*#,#''3'(#!%*%*6';1%!C#,-!

$$! :%:$1+.! +$%! /*$,%(! 1#! #*$'%%94%(! 3! 0*#,23! 8#%! /$1#'%8+$3$#! %*%-*6'$7!*#,-''3'$7!*,$.!'$461!%*(*%'$$!*$7!1#!'%6,9%$#%!#*+!&3/%($!%'

8$+.!#7$$0$!$$!#*$$6'%&/#!#7$+!'%!'%#$7*51%$#%!%746;.!;.!*,+$#&3%!

67#.$"""$+

)!-*EA*#F"+!<#"!)EF" 37

+*+! $%.! *#,#*-1;! $6&&%,%.! *,$$-&%%.! $%! '0$! *,5$%.! 9,3&%-/$*$%$01! &$31! #*+'%%2>! $$7.! $31! 8$%1>! 8-4%$$! '0$! *#,#''>';1! %*%-*6';1$!%15!$%!'0$!%*+/%1%.!%*%8%1$,51$1$#%!&$31!*#,#''3'(#!$;1!$$-/51!'0$!*#,#''>';1!%*%*6';1!#*+!61#!$,($$!%*(*%'$%

/(3&#'0!!",!&4)#!",' &1%!"'3"$#!"$%#&!.13"!6!(")/!*%#!"($("')#!!#!"!#0#!!#!"($("'#!!#!)()$#!'&*"+)%!#&!#(#!!6%!"'3"$#!#'!-!3#!"$%#&!"($$6/!.13"!#!(")/!#'!#!!)!"($("')" *53+!$60 .&$;!+$%!3! %74%(#!+!! %(-

1#%!8-4%$3!*,$.! 61#!'$461! %*(-*%'$!8#%!'%6,9%$#%!#*+!#7$>! $$!

$79#($! %*(*%'$! C%! 0#! #*$'%%-94%(! +$%! $$! %*(*%'$!C! %(1#%!8--4%$$!*,$.!$$!'$461!%*(*%'$%!

#$!C!$6/1%%!$$!'$461!%*('

*%'$!8#$-!$31!%74%(#!-!*% C-1;!&$31!-!*!*#(,1$7-

/%!61#!$79#($!&3/%($!1!8#%!(6'

,$7/%!*-1;!&$$! %*(*%'$!C! $31!

1)!8-4%$3!&$31!-*% **%%'>!3!+!!%(1#%!8-4%$3!&$$!'$461!%*(*%'$$!%(1#%!8-4%$3!8#%!&%!

8-4%! %74%(#! $$7! %*%*6'$7!*$7!'%6,9%$#%!#*+! $$! (91$.! $3.$!$*+$%!3!+!!

%(1#%!8-4%$3!&$3!-*%

.$&%!3!:;1(#!∧

+!1 !%(1#%!$,4>$!%15!%(1#%!$,4>!8#%!3!:;1(#!∧

ΗΖΒ , $*+$%!$%!%74%(%.!+!!8#%!)1!$$7!%*%*6'$7!C!%(1#%!*#,-''3'%.%

<71%*5.$!#($0!3!+!!%(1#%!8-4%$3!&$$!'$461!%*(*%'$$!4#!%(1#%!8-'

4%$3!&.!#7$+$!+*;.!8#%!3!*#,-''3'>!$3.!)1% **%%'>! '0$! %*(*%'#! %(1#%! 8-4%$#! +$#1! $%! 8-4%$%.! *$7! -:$1$#%$!

*,$.! $31!8$%1>! $$/>!8#%!*$7!#1>8$71!&$$!61#!#*+!#7$-!%(1#%!8-4%$%.!

8#%!&$$!-''$!/$,%&/+.!30$!6*%$#%!+$%!$$!%*(*%'$!C!%(1#%!8-4%$$!&$$!'$461!

%*(*%'$%!**%%'>!+/$%#!#*+'%%23!:(1%$#%!8#%!:%#!8-4%!-''$! /$79#($0!%*('

*%'$!*$7!'%6,9%$#%!#*+!$31!%74%(#!+!$!$$!=3$$0/%1$!#*$'%(9438%%!!!."%'%

67#.$""$,

38 *?@!*",)!,<#"")*"+-

/(3&#'0!"-!&4)#!"-' &1%!'5)!!",%#,"%#!"($("'#!"$%#&!.13"!#!$"!*%#!"($("')#!!#!"!!6!.)&%-!!),-!!)'/!3#!"$%#&!"($$6/!.13"!6!(")/!#'!#!!)!"($("')" *53+!$60

67#.$"$-

.&$;!+$%!$#!%*(*%'#!!+!8#%!!-!%(1#%!8-4%$#!*,$.!61#!'$461!%*('

*%'$!8#%!+$%!!,!%(1#%!3!8$%1>!$$/>!$$7.%!0#!#*$'%%94%(!+$%!3!!,!%(1#%!

8-4%$3!*,$.!$$!'$461!%*(*%'$% +1!3!!,!'%1!%(1#%!8-4%$3!&$$!'$461!%*(*%'$$!(6,$7/%!$3!,*!*-1;!

&$$!%*(*%'$!!+!8-4%$3!&$3!,+$!8#%!$31!,1!*-1;!&$$!%*(*%'$!!-!8-4%'

$3!&$3!,-$!$*+$%!3!,*!4#!%(1#%!'%#($,%$%8>!#*+!$3!,1% **%%'>!$$!%*(*%'$!!+!%(1#%!8-4%$$!&$$!'$461!8#%!(6,#/%!%74%(#!

,*!$$7!%*%*6'$7!!+!8-4%$3!&$31!$$/>!,+!$;1!'0$!%*%*6';1$!6*%$#%!+$%!

/$,%&/+.!30!3!,*!%(1#%!8-4%$3!&$$!'$461!%*(*%'$%! A%!+/$%$! $,+*$!*,$80*$%%!+$%!8#%!3!,1!%(1#%!8-4%$3!&$$!'$461!

%*(*%'$%! 3,#!#*+!$$! ('%$!&3/%($!,!$$7!'$461$$.!%*%*6'$7!7D51$1$#%!'0$!

8-4%$%.!*,$.!#7$+$!$%!,*!8#%!,1%! +7$+! %(1#%! -$$*$! /*,+$#&3! &"0$! $*+$%! 3! /$1#'%8>! 8-4%$$.! *$7!

/*$,%(!1#!7D;4%(!#*+!$$!&3/%($!,!*,$.!$$!%*(*%'$!%(1#%!3!,!$!*$7!#*$'

$%'%(!$31!8$%1>!$$/>!$;1!'0$!$%/1+/%1;1!%*%*6';1$!$#!$*$(#!%(1#%!8-4%'

$#!&$$!'$461!%*(*%'$%!!!!."%'%

)!-*EA*#F"+!<#"!)EF" 39

!"#$%!!"#%(!,)&#-"$%(!!"'!- 19

"%! '0$! *,$$-&%%.! #1#(6,$1$#%! &$31! 8#4%$+$3$#! %*%*6';1%!)! *,+-$#&3!&+!*#,$7&%-=%%!$31!8-4%$3!%74%(#!*,$.!61#!%*(*%'$!;.!($,6#!/%#.!'%-&/3.!8#46$;1!%*%*6';1!*,$.!#7$+$!%15!3!*,+$#&3!&#!#*$'%%810%%!+$%!8-4%!

=%0:$.!$%/1+/%1;1!%*%*6';1$!8#46$;1!&%!'$461!%*(*%'$$!$,(=%%!8#%!/(#!%7'

4%(#!8-4%$3!*,$.!$$!'$461!%*(*%'$$!*$7!%(1#%!3!$$/>!$$7.%

7*-'#!1#&#8%'-9!&&8!5(%&"'!&8!!"!– 19

+,9%8>!/%$-&#&3!&$$!95,$$!&9%$%86.!46&%%.!

%74%%51!8#%!%*%*6';1% %!5(%&"'!$9

@#4%$+$3$#!%74%%51!!8#%!%*%*6';1 $!5(%&"'!$9! C#,#''3'(#!%74%%51!&$$!95,$% )!5(%&"'!$9 @#4%$+$3$#!%74%(#.!&%!%*(*%'$!!!!!!!!!!!!!!!! #*+!&3/%($!%8$+.!>!%*(!$$7!%*%*6'$7%

%!5(%&"'!$9

C#,#''3'(#!%*%*6';1% (!5(%&"'!$9 @#4%$+$3$#!%*%*6';1 $!5(%&"'!$9

(#!:()!+(!9!'&!(!;9!,&8)!9!1#$!0!*(-'0!&%<9!

'&08!1#&#'1!<-!5#(###0#!5$5;+&8!

<0/(;1#!/%!$%.!#*#%$>&%%.!$;1!$,%&/51!#!8#%!&)!$$7!&%&'($7!)"$!

$#!&$%,%-!&9>/#$#!8#4$,(=$1$#%!*'>,;.!#*+!$#!%*(*%'#!&9>/#$#!*$7!$#!

*%,%&-''$71%!.$&%!&%!/%#!$,(%',3!&$%,%-!:;1(#!$%!6',%.!4%;,$01$#%!$,(:;'

1#!/%!8$,7(6.!$31!8$,7(>!$3.!$,(%',3.!8#%!3!&-&3!$3.!$,(%',3.!%(1#%!$,('

:;1$!/%!*'%7,6.! $%.!&-&%%.! $;1! %',51! $3.! $,(%',3.%!**(!*'6$1$!$%!#8/6.!

1$$01$#%!;.!/*%*%,#&/61#0!%740:,#//#!$/>/#$#%!-%#!1#!&71'%4%(!'$%*+1!3!

$,69$7&#!*,#8$%8>!/%!#7$6.!$%.!#*#%$>&%%.$!*,6*%%!*-1;!&$%.!//3!&71%*(*%'

'%.0!&71$,69$7&%.!3/%%74%(%.!Ox, Oy, Oz!1#!'#/&-1$1$#%!&3/%(#!+$!!$!-!5&$%! 1#! '3/%$7,:34%(!3! $,(%',3!"%+!-%!A%!#7$>! $31!$*$%8>$!3! $,(%',3!

%(1#%!$$!#1$(&$$%9$!$$7!$,%:51$7!&$$!95,$%!*(1#%!'$%*+1!%1$%'5.!(7&%8+$!1#!

#1#=3$34$01!&96&%%.!/%$#20!$;1!&$$%9%(;1!/%#.!$,(%',3.!8#%!1#!'%#$7*;'

4$01!8#$#&8%7#&$%8-!8,%$>,%#%!+7$+! :(1%$#%!&$%.!*,$$-&%%.! ("!/69,%! (5$!

+*$7! '%#$7*51$1$#%! $#!#*+'7$#!#*#,#($3$#!#*$$%'6&/#$#! :%#! $31!8#$#'

&8%7>!*#,#''3'%*%*6';1$!*$7!:(1%$#%!&$31!*,+$#&3!(+%!@-*$%%.!*,$%8$-'

&%%.!$;1!&7/*%,#&/-$;1$!+*;.!8#%!8-*$%%.!/%$6*%%$#!4%;,>&%%.$!*#,#$(4%'

1$#%!7*+!/$,(>!&9$'(;1.

40 !!"#!"#$!"!$%&"'!"%#

!"#!"#$!"#!$%$%!"#& !!!"#"!#$%$%%!&'!("!)%$#*+%$"#!"),!$$%#-!%)()%&%-!&'!(%-"!$,$%!.#!&/.!"),!"0$*-"!,)'-!1"#!!"!23'4./!"!*+.0!!%4$.##""!"%&"-2/$%$.!$3-!$$($3-# %&#&''($! &'((!)!(*+)**)!'()*),!-(.++!/)!01*2,3!%$!412!4)*56,77)(+5!+48!

(59! )4+4)*)9! -(.+)9!∧

!"# , $#"%∧

∧

%"! %! -+! +41*)5:;)+! 8(5! (1! ,&

;*15'/+!((.!*<1!+48!+2(=9!)+.+5!

4,.(+! /)-+7<()*1! +48! ().! ).+-41/=.12'+!(*+()%!

'! 4*8(+')! 4*1,+.>9! 5&

':<)5! 8(+.! 15! (*)59!+2(=9! -(.+)9!

)+.+5!/)(+?<!(129!+')9%! &'((! 71548.! 8(5! *).! )+.+5!

87)9!/)(+?<!(129!+')9!0+5!)!-(.++!∧

#"% )+.+5!)!451!/)-,7)!+48!+2&

(=9%!%8()$! +*0)+! .+! *)+?12/)! 8(5!∧∧∧

+< #"!%"!#"% %!'(*+9! 67,6)! ()9!

-).508()(+9$!/41*1</)!.+!241;='12/)!8(5!∧∧

> #"!#"% %!-)(*1</)!')&

/)+1! !#! ! ()9! !%x! !0+5! ).(89! ()9! -(.++9!∧

#"% ')/)+1! !!$! ! (=(151!>'()!∧∧

= !"##"! !0+5! !%!!(!%#%!%=719$!+48!(1!!!0+5!).(89!()9!-(.++9!∧

#"% ,-)(+5!)!)2;)++!!))*$!412!'2.+.(,!(59!)2;)+)9!%y!0+5!Az!'(+!')/)++!.!0+5!/!+.(+'(15:+%!

%+!(*+-(.+!!%#.!0+5!!%.!!)+.+5!!+'+$!!+,1<!=:12.!().!!%.!!015&

.3$!%#!(!%!!0+5!∧∧

ΒΑΕ="#$ !)+48!0+(+'0)23*%!!41/=.(9!15!)6,')59*!

#.!!0+5!!.!!)+.+5!+')9$!*)7+*3!!#.!(!.!%! %48!().!(*5-(.503!+.5'8()(+!'(1!(*+-(.1!#./!=:12/)!#.!+!#/!,!

,!./%!0/(9$!+,1<!!./!(!.!!+!!/!!0+5!!.!!(!.#$!;+!5':<)5!8(5!!#/!,!!/% %0171<;(9$! (+! (*+-(.+!%#/!0+5!%!/! =:12.!%#! (!%!$! ().!%/!

015.3!0+5!#/!,!!/%!!41/=.(9!;+!5':<)5!)!':=')!∧∧

ΕΑΓ>"#% .

!"#$!!!"#

$!/!@A!%B"%!$%&!'@B& 41

0/(9$!)!-(.++!∧

ΔΑΒ +41*)+:;)0)!+')!4*19!()!-(.++!∧

ΒΑΕ $!148()!∧∧∧∧∧

ΒΑΓ=ΕΑΓ+ΒΑΕ>ΔΑΓ+ΔΑΒ $!*)7+*3!∧∧∧

>+ #"%!"%!"# . A)!8/151! (*841!/41*1</)! .+!0,.12/)! ().!+48*)5?)!0+5! -5+! (59!

2487154)9!4)*54(>')59%! !."%*%

!"#!"#$!"'!$%$%!"'& 5%4%!#$%$%%!&'!("!)%$#*+%$"#!"),!%)()%&%-!&'!(%-"!$.!%4$.##""!$'!!.).('!!%(!"#!"#1$,$%$.!"),!$*##%$#-!.$4*-# %&#&''($ &'((! 8(5! )! (*+)**)! '()*),!

-(.++! /)! 01*2,3! (1! %! 4)*5=:)(+5!

+48! (59! )4+4)*)9! -(.+)9! $∧

ΒΑΓ ∧

ΓΑΔ 0+5!∧

ΔΑΒ $!8412!(+!')/)++!.$!

/!0+5!#!=:12.!7),;)+!)(2:+++*!4,.(!

'(59! +0/=9! 0+5! =:12.! +:;)+! 15! ./$!

/#!0+5!#.%!-+!+41*)+?12/)!8(5-

&'"% <ΔΑΒ+ΓΑΔ+∧∧∧

!1*;=9% $! '()*),! -(.++! /)! 01*2,3!

(1!.!4)*5=:)(+5!+48!(*)59!)4+4)*)9!

-(.+)9$!-5+!(59!141+)9!(1!,;*15'/+$!

*<1!1415(.*341()!+48!+2(=9!)+.+5!/)-+7<()*1!+48!().!(*+()%!!41/=.(9$!

;+! 5':<)5-!∧∧∧

ΓΒΔ>ΑΒΔ+ΓΒΑ %! /5+! (1.! +*51! 78-1! ;+! =:12/)! 8(5!∧∧∧

ΓΔΒ>ΑΔΒ+ΓΔΑ $!84(9!0+5!∧∧∧

ΒΓΔ>ΑΓΔ+ΑΒ% %!-=(1.(+9!71548. ∧∧∧∧∧∧∧

+++++= %#./#%%/#./%%.#/.%' $!=:12/)- ∧∧∧∧

++> /#../#/.#' . 0/(9!15!-(.+)9 ∧∧∧

ΒΓΔΒΔΓ $$%$" )(9!-(.+)9!

(12!(*5->.12!./#*!=:12.!,;*15'/+!*<1!1*;=9$!)41/=.(9 ∧

& > 2 1*;=9. &/1+(9$!0,;)!=.+!+48!(+!(*+-(.+!%./$!%/#$!%#.!=:)5!,;*15'/+!

-(.5>.!*<1!1*;=9$!)41/=.(9!5':<)5-

!"#$!!!"$

42 !!"#!"#$!"!$%&"'!"%#

=ΒΑΔ+ΔΒΑ+ΑΔΒ+ΓΑΔ+ΓΔΑ+ΑΓΔ+ΒΑΓ+ΑΓΒ+ΓΒΑ∧∧∧∧∧∧∧∧∧

(!.!1*;=9$!*)7+*3!!∧

& + ($#% =ΒΑΔ+ΓΑΔ+∧∧∧

1*;=9

0/(9$!+,1<!=:12/) ∧

& ,!/!1*;=9$!;+!4*=4)5!.+!)+.+5-

&$#% <ΒΑΔ+ΓΑΔ+∧∧∧

1*;=9%! !."%*% 6"$"$7$3#3 A41*)+!)4+')9!.+!!+41*)5:;)+!8(5!(1!,;*15'/+!((.!)4+4)*(.!-(.5>.!

()9!01*2,39!(2:+++9!'()-*),9! )0+5! 8:5! /8.1! (*+-)**)9*! -(.++9! *).!/41*)+!

.+!24)*6++.)5!(59!(='')*59!

1*;=9%!%2(8! -+.)(+5! 7+/-6,.1.(+9!248C)!(8'1!().!

+.5'1(503! ':=')! -5+! (59!

)4+4)*)9! -(.+)9! (*5=**12!

412! /8759! +41*)+:;)0)$!

8'1! 0+5! (1! +41(=7)'/+!

-5+!(1!,;*15'/+!((.!)'(-()*50>.! -(.5>.! 02*(1<!

4172->.12%!%9!')/)5(;)+!8(5!+2(8!(1!-).508!+41(=7)'/+$!+.!0+5!*).!*5+&

(24>.)(+5!412;).,!'(+!0!"%&"#'!0$!:*)'5/1415)+(+5!'(1!6567+1!'"""!)67=4)!

$2/4=*+'/+!/)(,!().!4*8(+')!'"""%12*% %&#&''($ D*,-/+(5$!-5+!()!'()*),!-(.++!&%!%./#$!8412!(+!')/)++!%$!.$!/$!

#$!%!%!%!)+.+5!'2.)4+4)*+$!')/)5>.12/)!! $!$)$) "#'!

∧∧∧

!(59!))4+4)*)9*!-(.+&

)9!()9!01*2,39$! *$$+ ,-

∧∧

ΒΑ *$$+ ,-

∧∧

ΓΒ !%!%!%!$!!!(59!-(.+)9!((.!(*5->.(.!&%.$!

&./$!%!%!%!!0+5!∧∧∧

ΓΒ$$# , %!%!%!(59!-(.+)9!(12!)02*(1<*!4172->.12!%./#!%!%!%!!)67=4)!':3/+!/1+*!0+5!;=(12/)-

'!(!! ...+Ο+Ο+Ο γ

∧

β

∧

α

∧

!"#$!!"$!

$!/!@A!%B"%!$%&!'@B& 43

$!(!∧∧∧∧

Α++Δ+Γ+Β ... = (2v –!3*!1*;=9

'6 = =Β+Α+∧∧∧

216& /!1*;=9$!'- = =Γ+Β+∧∧∧

,-#! /!1*;=9$!074% %48!(1!+.(+'(15:1!+41(=7)'/+!)4*8(+')!/1*!!-5+!(59!(*+)**)9!-(&

.+)9!.%%/&$!/%.#&$!%!%!%!!=:12/)!)4+')9!(59!+.5'1(50=9!':=')59-

,-,- $$

∧∧∧∧∧∧

Γ+Γ<ΓΒ+Β< , . . .

%)750,$!7+/6,.1.(+9!248C)!(+!451!4,.($!6*+'012/)!)/)!4*8';)&

')!0+(,!/=7)*-

)/.!–!3*!1*;=9!(!$!4!)∧∧

+ ,- "" ) + (∧∧

+ ,- '' ) + … =

= (∧∧

+ ,- '" ) + *%$+ ,-

∧∧

+ + … =

!(!)/!1*;=9!– '

∧

Ο *!+!)/!1*;=9!– #

∧

Ο ) + …!(!)/.*!1*;=9!–!'$!148()!4*10<4()5!8(5!'!4!3!1*;=9%

!"#!"#$!!""!$%$%!"") !!!&.4./!!$$%#-!%)()%&%-!&'!(%-"!&#"!$#-!.).(%-!$.!%4$.##""!&/.!!.).#'!&7).$%!"),!"0$*-!%(!"#!"%&"2/$%$.!$3-!$$($3-!1"#!)%!'!#$#-!)2%0$*-!$.0-!23'4./!! (#"!$"7""$""!$,$%!%(!"#!&0!"$,!!!"!1"$"#1%0"#4%(!$$(&'!.!).0!!"!*+%#!'-!)2%0$*-!$#-!8%#%#-!$'!!$$#9!!$##.#1%29!%!$$#&9!'!!$"!.).("!)$.1/)$.0!#

%&#&''($ %9!;)(*3'12/)!(*)59!)4+4)*)9!-(.+)9!/)!01*2,=9!.$!!$!-$!0+5!+9!

0+(+'0)2,'12/)!(+!)5'1'0)73*!(*+-(.+!!%./$!#!1!0+5!$-"!!/)!47)2&

*=9!.%!(!./(!!!#!(!!1!(!-$!(!-"$!!-5+!(+!141++!5':<)5-! ∧∧∧

ΗΘΚ>ΔΕΖ+#$% , ∧∧∧

ΑΒΓ>ΗΘΚ+ΔΕΖ , ∧∧∧

ΔΕΖ>ΑΒΓ+ΗΘΚ .

44 !!"#!"#$!"!$%&"'!"%#

!"#$!!!""

-+!+41*)5:;)+!8(5!(+!)2;<-*+//+!(/3/+(+!%/$!#1!0+5!$"!/41*)+!

.+!+41()7='12.!47)2*=9!(*5->.12$!8(5!*)7+*3!(1!,;*15'/+!*<1!)?!!+2&

(>.!)+.+5!/)-+7<()*1!(12!(*+(12%

%.!15!-(.+)9!∧∧∧

ΗΘΚΔΕΖ $$#$% !)+.+5!+')9$!(8()!)+.+5!,+.)*8!8(5!

0+5!(+!(/3/+(+!%/$!#1$!$"!)+.+5! +'+!0+5!+48!+2(,!0+(+'0)2,E)(+5!)5&

'847)2*1*!(*+-(.1%

&'((! 8(5! 15! -(.+)9! *).! )+.+5!

87)9! +')9! /)(+?<! (129! 0+5! 8(5! 0+(+-

'0)2,E)(+5! )! -(.++!∧

ΚΘΛ $! ),)?39!

()9!∧

ΗΘΚ 0+5! +')! 4*19! ()! -(.++!∧

ΑΒΓ %!%1!')/)+1!#!=:)5!7),;)+!=('5!

>'()!-"!(!-#%!2=*12/)!0+5!(59!)2&

;)+)9!"#$!$#%!%48!().! 5'8()(+!((.!

(*5->.(.! %./! 0+5! "-#! 4*10<4()5!

8(5!!%/!(!"#%

%,1<!∧∧∧

ΔΕΖ>ΗΘΚ+#$% !0+5! ,%./∧∧

ΚΘΛ= ;+!)+.+5!0+5 ∧

()$ >∧

ΔΕΖ !)67=4)!4+*+(3*)')!/)(,!(1!(=719!()9!+48*)5?)9*%!%+!(*+-(&

.+!71548.!$-#!0+5!#!1!=:12.!!-$!(!!#$!!-#!(!!1!!0+5!!∧∧

ΔΕΖ>()$ . !41/=.(9!;+!)+.+5!0+5!!$#!,!#1%!

0/(9$!'(1!(*+-(.1!$"#!)+.+5!!$"!+!"#!,!$#$!,*+!;+!)+.+5!0+5!

$"!+!"#!,!#1!%! 0/(9$!+,1<!"#!(!%/$!;+!)+.+5!0+5!$"!+!%/!,!#1%

!"#$!!!""!

$!/!@A!%B"%!$%&!'@B& 45

&/1+(9!/41*)+!.+!+41*)5:;)+!8(5!=:12/)!#1!+!$"!,!%/$!84(9!0+5!

8(5!%/!+!#1!,!$"% !41/=.(9$!)+.+5!*2.+(8.!.+!0+(+'0)2+';)+!=.+!(*+-(.1!/)!47)2&

*=9!+')9!4*19!(59!%/$!#1!0+5!$"!% !."%*%

!"#$!!""(

6"$"$7$3#3 A5+! 4+*8/15+! +48*)5?)! )45(2-:,.)(+5$! 241;=(1.(+9! 8(5! /5+! +48!

(59!-(.+)9$!4%:%!)!-(.++!∧

#$% $!)+.+5!)!/=-5'()!)67=4)!':3/+!//6*%

D*,-/+(5$!+,1<!∧∧

ΗΘΚ≥ΑΒΓ 0+5!!∧∧

ΔΕΖ≥ΒΓ" $!;+!=:12/)-! %/!≥!$"!0+5!%/!≥!#1%!%*0)+!71548.!.+!+41*)+?12/)!8(5!#1!+!$"!,!%/%

$(1!)'(()*508!()9!-(.++9!∧

ΑΒΓ !,=*)(+5!)!.#$!=('5!>'()!.+!5':<&

)5!∧∧

ΑΒΛ=ΗΘΚ $!8412!=:)5!7),;)+!.#!(!%.!(!./!0+5!=:12.!+:;)+!15!

%#!0+5!/#% %48!().!5'8()(+!((.!(*5->.(.!$-"!0+5!%.#$!4*10<4()5!8(5!

$"!(%#%!!4+')9$!/41*1</)!.+!4+*+()*3'12/)!8(5- ∧∧∧∧∧∧

ΔΕΖ<ΗΘΚ−ΑΒΓ=ΑΒΛ−ΑΒΓ=ΛΒΓ $!148()!+48!(+!(*+-(.+!#!1!

0+5!#./!=:12/)!#/!4!#1% A)!:*3')!(*5-(.5039!+.5'8()(+9!'(1!(*+-(.1!%#/$!4*10<4()5!8(5-

%/!4!%#!+!#/!(!$"!+!#/!4!$"!+!#1$! *)7+*3!8(5!A/!4!$"!+!#1$!148()!/41*)+!.+!0+(+'0)2+';)+!(*+-(.1!/)!

47)2*=9!%/$!$"$!#1%

46 !!"#!"#$!"!$%&"'!"%#

!"#!"#$!"(!$%$%!"(& :"!1"$"#1%0"#4%(!#$%$%%! &'!("!"),! $$%#-! %)()%&%-! &'!(%-"! &#"!$#-!.).(%-!$.!%4$.##""!&/.!%;!!"0$9!"!,)'-!1"#!!"!23'4./!"!%(!"#!)%!$"!"%&"2/$%$.!"),!$3!!$$($3#! !("'-!)$*)%#!$.!%4$.##""!1"#!

$'!!$$#9!!&'!#9!!!"!%(!"#!"#1$,$%$.!$'!!$%##%$'!!.$49!. %&#&''($

&'((!8(5!15!*1;)+')9!-(.+)9!)+.+5!15!∧∧∧

ΗΘΚΔΕΖ $$#$% !0+5!8(5!(1!

,;*15'/+! ((.! *<1! )?! +2(>.$! 84(9! 0+5! .+! 7),;1<.$! )+.+5! 4,.(1()! /)-+-

7<()*1!+48!().!(*+()$!0+5!)4547=1.!)+.+5! &"'% <ΗΘΚ+ΔΕΖ+∧∧∧

1*;=9%!

D*=4)5! 71548.! +48! (*)59! -(.+)9! +')9! 4*19! (59!∧∧∧

ΗΘΚΔΕΖ $$#$% ! .+!

0+(+'0)2+';)+!'()*),!-(.++%

!"#$!!!"%

&:1.(+9!3*)!7,6)5!(59!47)2*=9!%.$!./$!#!$!!1$!$-$!-"!+')9!0+5!

,=*1.(+9! (59!%/$! #1$!$"! )+.+5! ))41-/=.(9*! *2.+(8.! .+! 0+(+'0)2+';)+!

(*+-(.1$!412!.+!=:)5!47)2*=9!+')9!4*19!

(59!%/$!#1$!$"!)4*8(+')!//*% &'((!#AF! (1! (*+-(.1!412!0+&

(+'0)2,E)(+5$! 1<((9! >'()!%/! (!#A$!

#1!(!AF!0+5!$"!(!F#%!-)(*1</)!(1!

0=.(*1!&!(12!4)*5-)-*+//=.12!0<0712!

'(1!(*+-(.1!#AF!0+5!(59!+0(+.)9!&#$!

&A$!&F% "':2*5E8/+'()!8(5!%.!,!&#%!

!"#$!!!"%!

$!/!@A!%B"%!$%&!'@B& 47

%.!*).! 5':<)5!+2(8$!(8()!;+!=4*)4)!.+!'2/6++.)5!)+()!%.!(!&#$!

)+()!%.!4!&#% &'((!8(5!5':<)5!(1!4*>(1$!)*)7+*3!8(5!%.!(!&#*%!%8()!(+!(*+-(.+!

%./!0+5!&#A!=:12.!.%!(!&#$!./!(!&A!!0+5!+')9!6,')59!#A!(!%/$!,*+!

;+!)+.+5!0+5!∧∧

ΛΟΜ=#$% %!/5+!(1.!+*51!78-1!;+!)+.+5!∧∧

ΜΟΝ=ΔΕΖ !0+5!∧∧

ΝΟΛ=ΗΘΚ .

%8()!8/(9!=:12/)-

=ΗΘΚ+ΔΕΖ+∧∧∧

#$%

∧∧

ΜΟΝ+ΛΟΜ∧

ΝΟΛ+ !(!3!1*;=9% %2(8!)+.+5!,(141$!+,1<!+48!().!248;)')!5':<)5!

&#$% <ΗΘΚ+ΔΕΖ+∧∧∧

1*;=9%

!41/=.(9$!15!.%!0+5!&#!*).!/41*)+!.+!)+.+5!+')9% &'((!%.!4!#&%!%8()!/41*1</)!.+!4,*12/)!!&$!(!%.$!!&%!(!./!!

0+5!.+!,=*12/)!().!!$%%!%,1<!!./!(!.%$!!;+!)+.+5!0+5!!&$!(!&%$!#$!(!

(!A%$!148()!)! !$%! !;+!)+.+5!4+*,77)7)!4*19!().!#A!0+5!'2.)4>9!!(1!

(*+-(.1! #A&! ;+! )+.+5! 5'1->.51! 4*19! (1! (*+-(.1! $%&%! &('5! 5':<)5!

ΣΤ

ΟΣ=

#A

&#$! +48! 8412! 7+/6,.)(+5!

ΣΤ

ΛΜ=

&$

&#)-5*58()(+! ).+77,?*%!

0/(9! !#&!,!&$$!148()!;+!=:12/)!#A!,!$%%!%,1<!8/(9!)+.+5!#A!(!

(%/$!;+!)+.+5!0+5!%/!,!$%%!$(+!(*+-(.+!%./!0+5!&$%!)+.+5!.%!(!&$$!

./!(!&%!0+5!$%!4!%/%!!3*+$!;+!)+.+5!0+5!∧∧

ΣΟΤ>"'% =∧

$!% .

A)! 8/151! (*841! +41*)50.<)(+5! 8(5! ;+! )+.+5!∧∧

ΜΟΝ>"!* ! 0+5!∧∧

ΝΟΛ>()& .

!41/=.(9! &"'% =ΝΟΛ+ΜΟΝ+ΛΟΜ>ΗΘΚ+ΔΕΖ+∧∧∧∧∧∧

1*;=9%

%2(8!)+.+5!,(141$!+,1<! &"'% <ΗΘΚ+ΔΕΖ+∧∧∧

1*;=9%!3*+$!*).!

/41*)+!.+!)+.+5!%.!4!&#%!0/(9$!=:)5!+41*)5:;)+!8(5!*).!/41*)+!.+!)+.+5!

1<()!0+5!!%.!(!&#%!!41/=.(9!;+!)+.+5!!%.!,!#&%

!C>.)(+5!(>*+!(1!)2;<-*+//1!(/3/+!&B!0,;)(+!4*19!(1!)4+4)*1!

(12!0<0712$!'(1!')/)+1!&$!=('5!>'()!.+!5':<)5!

&B2!(!%.2 –!#&2

48 !!"#!"#$!"!$%&"'!"%#

0+5!,=*1.(+5!(+!!B#$!BA$!BF%! %,1<! )! &B! )+.+5! 0,;)()!

4*19! (1! )4+4)*1! (12! 0<0712$! ':)-/+(+E)5! 1*;=9! -(.+)9! /)! 0,;)! /5+!

+48! (59! )2;)+)9! !#&$!A&$!F&%!%+!

1*;1->.5+! (*+-(.+! !#&B! $!A&B! $!

F&B! =:12.! (59! 0,;)()9! 47)2*=9!

(129! +')9! /5+! 4*19! /5+$! ,*+! )+.+5!

+'+$!148()!!B#!(!BA!(!BF% %0171<;(9$!+,1<!(1!(*+-(.1!

#&B!)+.+5!1*;1->.51$!=:12/)-

#B2!(!&#2!+!&B2 = (!&#2!+!%.2 –!#&2!(!%.2.

&/1+(9!(1!%.!)+.+5!+'1!/)!0,;)!=.+!+48!(+!./$!#!$!!1$!$-$!-"$!0+5!(1!

B#! +'1!4*19!0,;)!=.+!+48!(+!BA$!BF%!!41/=.(9!0+5!0,;)!=.+!+48!(+!

%.$!./$!#!$!!1$!$-$!-"!)+.+5!+'1!/)!0,;)!=.+!+48!(+!B#$!BA$!BF% $(+!(*+-(.+!#BA!0+5!%./!15!47)2*=9!#B$!BA!)+.+5!+.(+'(15:+!

+')9!4*19!(59!47)2*=9!%.$!./$!84(9!0+5!)!6,')!#A!)+.+5! +')!4*19!()!

6,')!%/%!!41/=.(9$!;+!)+.+5!∧∧

ΑΒΓ=$'% %!/5+!(1.!+*51!78-1!;+!)+.+5!∧∧

ΔΕΖ=%'( !0+5!∧∧

= ()&$'. . 3*+!0+(+'0)2,';)0)!'()*),!-(.++!/)!01*2,3!(1!B!412!4)*5=:)-

(+5!+48! (59! )4+4)*)9! -(.+)9!∧∧∧

ΛΡΝ$%'($$'% $!15!141+)9! )+.+5!+.(+&

'(15:+!+')9!4*19!(59!)4+4)*)9!-(.+)9!∧∧∧

ΗΘΚ$"!*$#$% . 8%)%4%

)*++" <.! %04/&$""".! $"7""! =>"!").$%(!!"!23'4%(!*$##!9#$%!!"!##+/%#&

=>2!'!!)2 –!?=2.

%&#&''($ &:1.(+9!8(5!%.! ,!#&$!/)!*5,&

/)(*1! %.! -*,,)(+5! )/50<0751$! '(1!

141+1!/41*)+!.+!7),;)+!:1**3!%/!(!#&$!+,1<!)!#&!*).!)+.+5!/)-+7<()-

!"#$!!!"%(

!"#$!!!"%&)

$!/!@A!%B"%!$%&!'@B& 49

*)!()9!*5+/=(*12%!2=*12/)!0+5!().!!/.%

$!-(.++!∧

#%$ !)+.+5!)--)-*+//=.)!'(1!)/50<0751!%/.$!,*+!)+.+5!

1*;3!0+5!)41/=.(9!%.2!(!%/2!+!./2%!!3*+!;+!)+.+5!0+5!./2!(!%.2 –!%/2. %,1<!)+.+5!%/!(!#&$!;+!5':<)5!./2!(!%.2 –!#&2%!#+/6,.1.(+9!71548.!

&B!(!./$!=:12/)!&B2!(!%.2 –!#&2.

6"$"$7$3#3# )$!+48*)5?)!()9!4*8(+')9!/5!')!)5*50=9!4)*54(>')59*% $"& $*! "%!+,*"-%!.! "%/!0#-&)#)-!$$,*%/!+1+2%/!(-'%+#"!&!03*4!

%#!$&!!!05! "&6!02#/-,6! "%/! "-&)7*%/!89:'!0&"&!%";*!02#/-3!

9:'!"5"#!+!&!032&!<!!&%"1#&!$&!(!8.&

D*,-/+(5$!+9!241;='12/)!4*>(+!

8(5!/41*)+!.+!5':<)5!%.!(!#&%!%8()!;+!

=:12/)!AF! (! /#&%!%,1<! 8/(9! )+.+5!

AF(!#1!0+5!#&!(!%.!(!!#!(!!1$!'(1!

(*+-(.1!#!1!;+!5':<)5!!#1!(!#!!+!!1$!

412!)+.+5!+*<.+(1.%!#).!/41*)+!71548.!

.+!=:12/)!%.!(!#&% /5+! (129! +*5129! 78-129$!+.!'2/-

6++.)5!%.!4!#&$!;+!4*10<C)5!)!':=')!

#1!,!#!!+!!1$!412!)+.+5!0+5!4,75!+*<&

.+(1% 3*+$!4,.(1()!=:12/)-!%.!,!#&$!

148()! 2C>.1.(+9! ().!&B! 0,;)()! '(1!

)4+4)*1!(12!0<0712!0+5!=('5!>'()!.+!5':<)5!&B2!(!%.2 –!&#2$!,(,.12/)!

'().!'<.;)')!(12!4*1673/+(19%

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

%4"#-&+5!"%/!"-&)7*%/!89:'!"5"#!+!&!032&!<!!&%"1#&!$&!(!8.&

D*,-/+(5$!+9!241;='12/)!4*>(+!

8(5!/41*)+!.+!5':<)5!%.!(!#&%!%8()$!(+!

(*+-(.+!%./!0+5!#A&!)+.+5!+'+$!84(9!

0+5!(+!(*+-(.+!#!1!0+5!#F&% !41/=.(9!=:12/)-!

∧∧

ΜΟΛ=#$% 0+5!∧∧

ΛΟΝ="!* $!+48!

(59!141+)9!4*10<4()5!8(5!∧∧∧

ΜΟΝ=ΔΕΖ+#$% $!-(.++!)!141++!

!"#$!!!"%'

!"#$!!!"%#

50 !!"#!"#$!"!$%&"'!"%#

6=6+5+!)+.+5!+')!4*19!()!-(.++!∧

()& %!&('5!,(,.12/)!'()!5'8()(+!∧∧∧

ΗΘΚ=ΔΕΖ+#$% $!412!1*)-)+!')!!,(141%!!41/=.(9!*).!/41*)+!.+!

5':<)5!!%.!(!#&% %.!241;='12/)!8(5!5':<)5!!%.!4!#&$!(8()!/)!)45:)5*3/+(+!4+*8&

/15+!/)!+2(,!()9!02*+(9!+48*)5?)9!)*)7+*3!/)!:*3')!8/15(.!(*5->.(.*!

/41*1</)!.+!0+(+73?12/)!84(9!4*5.!')!,(141%

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

A5+! )'2.(1/8()*)*! 0+(+'0)23! (*5=**12$! 412! 05.)+(+5! '(+! +*5+!

47++'5+!/)!+2(,!()9!4*8(+')9!/5$!/41*)+!.+!-+.)5!(9!)?39-! A)!*)*1/=.)9!(59!)4+4)*)9!-(.+)9!+$!6$!-$!!-5+!(59!141+)9!5':<12.!15!

':=')59-!+!+!6!+!-!4!3!1*;=9$!6!4!+!+!-!0+5!241;=(1.(+9!8(5!)!6!)+.+5!)!

/=-5'()$!0+(+'0)2,E12/)!(59!-(.+)9!4*>(+!*5+*1:50=9!/)!01*2,3!&!0+5!

/)!()!')5*,!+$!6$!-!0+5!0+(845.!)4+0).(*)9%!#8-(!()9!'2.;30)9!++6+-4 43!1*;=9$!*).!)450+7<4()(+5!)!4)*5,=*)5+!)67=4)!':3/+!/5%1*%!

!"#$!!!"%&$

"+(+'0)2,E1.(+9!%1!0+5!#1!>'()!15!:1**=9!%%1!0+5!##1!.+!)+&

.+5!+.(+'(15:+!0,;)()9!'(59!+0(+.)9!&.!0+5!&/$!+&-"'.!/"!"&-&''01'. 8(5!15!%%1!0+5!##1!(=/.1.(+5!')!=.+!')/)+1!"!).(89!(12!0<0712$!*)7+*3!

&"!4!&.% !C>.1.(+9!0,;)()!"$!'(1!)4+4)*1$!=('5!>'()!"$2!(!&.2 –!&"2,

4*10<4()5! )! (*+)**)! &%! $./$! 412! /41*)+! .+! +41*)5:;)+! 8(5! )+.+5! )! E)-(1</).).

$!/!@A!%B"%!$%&!'@B& 51

!"#!"#$!!!")!$%$%!")& !!! *!"! %;%%&$.!)%$#*+%$"#!"%$";/!)"$%2232'!!"!%!&/.! %)#)*(&'!"!$,$%!.#!")*!"!$#!*&$%-!$.0!%(!"#!)"$"2232,&$""""!(#"# %&#&''($ %9!;)(*3'12/)!'()*)8! ))?,)**1*!412!4)*507)+)(+5!+48!(+!E)<-)!

4+*,77)7(.! )454=*(.! )B1! (!%./#$! B14! (!%4.4/4#4*$! )B2! (!%..4%4$! B24!(!/##4/4*!0+5!)B3!(!!.//4.4!$!!B34!(!%##4%4*%!-+!+41*)+?12/)!8(5!

15!+4=.+.(5!=**)9!(12!'()*)1<!)+.+5!4+*+77)78-*+//+$!/)(+?<!(129!+'+%

!"#$!!!"*

&5!(1/=9!((.!)454=*(.!B1$!B14!/)!(1!B2!;+!)+.+5!4+*,77)7)9!)4*8&

(+')1.*$!*)7+*3!%.!66!%4.4%!&/1+(9!!15!(1/=9!((.!!B3$!!B34!/)!(1!B2!;+!

)+.+5!4+*,77)7)9$!*)7+*3!%%466!..4%!!41/=.(9!(1!()(*,47)2*1!%..34!

)+.+5!4+*+77)78-*+//1% A)!(1.! +*51!+0*56>9!(*841!+41*)50.<)(+5!8(5!0+5!15!2487154)9!=&

**)9!(12!'()*)1<!)+.+5!4+*+77)78-*+//+%

!?,7712! 15! -(.+)9!∧∧

Δʹ′Αʹ′Βʹ′$$#" ! ! =:12.! 47)2*=9! 4+*,77)7)9$!

,*+!)+.+5!+')9!)4*8(+')!17*%!%+!(*+-(.+!71548.!%.#$!%4.4#4!)+.+5!+'+$!

+,1<! 5':<12.!15!':=')59!%.!(!%4.4! $!%#!(!%4#4! !0+5!∧∧

Δʹ′Αʹ′Βʹ′=$#" )4*8(+')!"%!3*%

0/(9! (1! 4+*+77)78-*+//1!%./#! )+.+5! *547,'51! (12! (*5->.12!

%.#! 0+5! (1! 4+*+77)78-*+//1! %4.4/4#4$! *547,'51! (12! (*5->.12!

%4.4#4%!3*+!0+5!(+!4+*+77)78-*+//+!%./#$!%4.4/4#4!!)+.+5!+'+. &/1+(9!+41*)50.<)(+5!8(5!0+5!15!2487154)9!=**)9!(12!'()*)1<!)+&

.+5!+')9%!! !."%*%

52 !!"#!"#$!"!$%&"'!"%#

6"$"$7$3#3! $(+!!"%&"#'!!(12!!207)+*)!)!*5+(<4(')!()9!4*8(+')9!/3!+.+,=-

*)(+5!')!(2:++1!'()*)8!0+5!8:5!')!)?,)**1%!D*1,+.>9$!0,(5!(=(151!*).!

/41*)+!.+!5':<)5!:(*+9!*5)20*5.+')59$!4%:%!')!4*+'/+!/)!6,')59!0+.1.50,!

)?,-(.+% !?,7712$!84(9!=-5.)!,+.)*8!+48! ().!+48*)5?)!412!4*1)-3;)0)$!!

/)!(1.!8*1!&%5";"!!"4*!!0,*!*"&!#'-7*!)*>!)..11</)!8:5!/8.1!5')/6+*5-0,!+77,!0+5!'2/4(>'5/+!':3/+(+%

!"#!"#$!!"*!$%$%!"*& !!! *!"! )"$"2232%)()%&.! $"3(4%(! "),! %)()%&.! )"$%2232.!)$.-! &/.! ")*!"!$#! *&$%-! $.0"!$,$%! $"! &/.! )"$"2232%)()%&"!#$"!.).("!&#"#$%($"#!$.!"$+#1,!%(!"#! "!%2.&"! $'!! 8%#%9!!$.0-"! ).0! .$(@.!$"#! )%!'! #%!1%).#"!%223!*&$"!$.0!"$+#1./# %&#&''($ -)(*1</)! /5+! (1/3! "#AF!

4+*,77)7)! '(59! 6,')59! %1.1/1#1, %41.41/41#41!).89!4+*+77)7)454=*12%!

%9! '2/617+'12/)! /)! V, V4! (+!

4+*+77)7)4+4)*+!"#AF!–!%1.1/1#1, 0+5!"#AF!–!%41.41/41#41!+.(+'(15:+%!%8()!+*0)+!.+!*)+?12/)!8(5-!

( )( )ΝΔʹ′Αʹ′Κ

ΝΔΚΑ=

ʹ′--

--

/

/

D,.( '(59 )/5)2;)+)9 "%, F# ;)(*1</) (+ 7 '(1 473;19 +'+ /)(+?< (129 )2;<-*+//+ (/3/+(+ "%1, %1%2, %2%3, … , %7-1 %7 0+5 F#1, #1#2, #2#3 , . . . , #7-1#7 +.(+'(15:+ !4+')9, 4,.( '(59 )/5)2;)+)9 "%4, F#4 (+ / +'+ (/3/+(+ "%4, %314, %14%24, ... , %4/-1 %4/ , 0+5 F#41, #41#42, #42 #43, . . . , #4/-1#4/ +.(+'(15:+.

!"#$!!!"+

$!/!@A!%B"%!$%&!'@B& 53

%+! 4+*+77)7)4+4)*+! /)! 4+*,47)2*)9! +0/=9! (+! (/3/+(+!"%1, %1%2$!%2%3$!8!$!%7-1!%7!!0+5!6,')59!+')9!/)!().!"#AF!!)+.+5!+'+!/)!(1!

V$! ! ).>! (+! 4+*+77)7)4+4)*+! /)! 4+*,47)2*)9! +0/=9! "%14$! %41%42, %24%34$!%!%!%!!$!%4/-1%4/!0+5!6,')59!+')9!/)!().!"#AF!)+.+5!+'+!/)!(1!V4%

D+*+()*1</)!+08/)!8(5!5':<12.!15!'2.)4+-(-=9- 7!⋅!)"%1#1F*!,!/!)"%14#14F*!⇒ ⇒!)"%7#7F*!,!)"%/4#/4F*!⇒ ⇒ "%7!,!"%/4!⇒!7!⋅ V!,!/!⋅ V4 )"*!!! &/1+(9! 7!⋅!)"%1#1!F*!4!/!)"%14#14F*!⇒ ⇒ 7!⋅ V!4!/!⋅ V4 )""*!!! 0+5! 7!⋅!)"%1#1F*!(!/!)"%14#14F* ⇒ ⇒ 7!⋅ V!(!/!⋅ V4!!!! )"""* &48()$!'</,(.+!/)!(1.!1*5'/8!5'8()(+9!78-(.!0+(,!(1.!!<*1?1!

(V%!1*5'/89!7*$!;+!=:12/)- ( )( )ΝΔʹ′ʹ′Κ

ΝΔΚΑ=

ʹ′ 11

11

AVV

!."%*%

6"$"$7$3#3 &5!'2.)4+-(-=9! "$! ""$! """! ()9!4*8(+-

')9! /7! )'"%/7*! 4*10<4(12.! +.! 4,.(! '(59!

)/5)2;)+)9! "%7 ,! F#7! 4,*12/)! (/3/+(+!

"%44/(! "%4/! $! F#44/(! F#4/! ! +.(+'(15:+$!

148() )"%44/#44/F*!(!)"%/4#/4F*!!!!0%74%

!"#!"#$!"+!$%$%!"+&

6%!'!#%!&.#"*!3! %04%("!1"#!"%!*!"!#3"%(.!$3-!'-!1.$0'7!!"!1"$"#1%0"#$%(!#$%$%%!&'!("! $$$(%&$3%"! (#3!"%!&.#"*!3!#$%$%%!&'!("# %&#&''($ #+.1.(+5!)!)/5)2;)++!%x!0+5!)! (*+)**)!%1.x1y1z1%!-)(*1</)! (2&

:++1!')/)+1!#1!()9!%1z1!0+5!().!4*16173!!(12!#41!'(1!)4+4)*1!x1A1y1.

!"#$!!!"+!

54 !!"#!"#$!"$%&"'!"%#

!"#$!!!"#

"&'&(())*+,)-)!./01&!∧

!"# !1(2!-)!'20!∧

$$$ #"! (&3!4*0/!(',!)41-

4)*,!x%y!'2!./01&!∧

!"% !1(2!-)!'20! ,#Ax 111

∧ʹ′ !5'(3!6(')!,3!%y$!%t!0&!5%

7,)0!/8!49,8!'20!Ax!'20!1*3&!:5(2!4,)!57,)0!,3!%1y1, A1#1+!/8!49,8!'20!

%1x1!&49;'&(2!I'!()*'!<*0/!('20!2-3)):)1&!%t!:)/9,=-)!',!(2-)1,!,+$!'5',3,!6(')!%#++!%1#+1!(&3!',!'->-&!#+#!+!#1+#1!!(*:)',!(',!)414)*,!

xAy!&49;'&(2!,(*'!-('/!%z!2!2-3)):)1&!-)!&97>!%!4,)!4)9357)3!',!#'!.&!*)1?,)-)!;'3!2!(')9)*!./01&!%'xyz!)10&3!1(2!-)!'20!%1.x1y1z1.

<*0/!('38!%x, A1x1!:)/9,=-)!'&!(2-)1&!/$!/1!&0'3(',17/8!5'(3!

6(')!%/!+!%1/1! '!<9,0&068!2! !#1#+1!)10&3!(*:)'2!('38! !#1+%1$!#1+/1 (&3!2!!##+!(*:)'2!('38!!#+%$!#+/'!

%&!'91./0&!%1/1#1+!(&3!%/#+!57,)0!%1/1! !+!%/$! !%1#1+!+!%#+!

(&3!∧∧

Δʹ′ΒΑ=Δʹ′Α $$$! '!19&!)10&3!1(&!&49;'&(2!"'!-*$!,4;')!

#1+/1!+!#+/' (1)

%&!,9:,.603&!'91./0&!@,34;0!#1#1+/1! $!##+/!57,)0!'38!(*:)')8!

4@))958!',)8! 1()8!-3&!49,8!-1&! !&/1#1++!/#+! $! !#1#1++!##+*'!19&!)10&3!

1(&$!,4;')! /1#1!+!/# (2)

&-,1/8! '&!,9:,.603&! '91./0&!%1#1#1+$!%##+! 57,)0! '38!(*:)')8!

4@))958!',)8! 1()8!-3&!49,8!-1&! &%1#1++!%#+! $!#1#1+!+!##+*'!19&!)10&3!

1(&$!,4;')! %1#1!+!%# (3)

$!2!AB!%C"%!$%&!'AC&! 55

%4;!'38!(75()38!&,*$!&(*$!&)*!49,(=4')3!2!3(;'2'&!'/0!'93.60/0!%1/1#1

(&3!%/#!&57,)0!'9)38!4@))958!1()8!-3&!49,8!-1&*'!19&!∧∧

ΒΑΔ=ΔΑ $$$& . !0')@68! &0*@,.&$! &0! @*D,)-)! )4;E2! -&8! ;'3! (&3! ,3! ./01)8!

'#"" $$$

∧ʹ′

∧

ΑΔʹ′ # !)10&3!1()8!&/8!&:9,1(-&'&!>!*3&0,958!1(/0!./0360*!(&3!

:)/9>(,)-)!4*0/!('38!%y$!%1y1!!'&!(2-)1&!2$!21$!5'(3!6(')!%2!+!%121,

-4,9,=-)!0&!*)1?,)-)!;'3.!∧∧

ΓΑ=ΔΑ "# $$$ . &3!'91)*9)8!@,34;0!4)9357,0'&3!&4;!)414)*)8!./01)8!1()8!(&3!,-,1%

/8!!()1-)0)8$!,4;')!:&!)10&3!1()8'! !."'*'

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

"#"*01!'!*#!%"!+,"1!""##$%"!+$,$--&-!+*+!"""

%&#&''($ #1*,0'&3!',!'->-&!%/!(&3!',!4&9&@@2@)414)*,

%1/121#1 –!%1+!/1+!21+!#1+' .5@,)-)! 0&! (&'&(())*(,)-)! 4&9&@@2@)414)*,!%/2#! –!%+/+2+#+!

;-,3,!(&3!,-,1/8!()1-)0,!49,8!',!*,(-50,'

!"#$!!!"$

$=-0/0&!-)!'20!49;'&(2!(/$!-4,9,=-)!0&!(&'&(())*(,)-)!'91)%

*92!./01&!%'/xy!1(2!-)!'20!%1'/1%1+#1$!!'5',3&!6(').

=∧

!$!∧

ʹ′$$$ ""& &&*!$!

∧∧ʹ′= $$$ "$"!"# &D*$!!

(&3!!!!∧∧

Β=ΔΒ #"" $$$ &.*'

56 !!"#!"#$!"$%&"'!"%#

<*0/!('20!%x!,91+,)-)!',!(2-)1,!%+$!5'(3!6(')

Αʹ′Α

ΑΒ=

ʹ′ΑΑ

Β

11

11A (1)

&49,0&068!',!%%+!@&-D*0)'&3!/8!')'*9'2!&0*@,.,8!'/0!)):=.9&--/0!

'-2-*'/0!%1/1$!%1%1+$!%/*' $'2!()057)3&!4*0/!('20!%y!,91+,)-)!,-,1/8!',!(2-)1,!#$!5'(3!6(')

ΑΔ

Αʹ′Α=

ΔΑ

ʹ′

$$

$$$" (2)

%4;!'38!(75()38!&,*$!&(*!-)!)0&9-,.>!'28!3*3;'2'&8!0"%#!&$'(0!&49;-'&(2! V'! ((*! >$! -)! (2-)930,=8! ;9,)8$! 4,@@&4@&(3*+,0'&8! (&'*! -5@2$!

49,(=4')3

ΑΔ

ΑΒ=

ΔΑ

Β

$$

$$" (3)

.)/9,=-)!',!4&9&@@2@)414)*,!-)!'9)38!()0'957,)()8!&(-58!%/$!

%%+$!%#$! !5('/!',!%/2#!–!%+/+2+#+'!%4;!'38!(75()38! &&*$! &,*!49,(=%4')3!2!,-,3;'2'&!'/0!4&9&@@2@,.9*--/0!%//1+$!%1/1/1+%1+'!&-,1/8!

&4;!'38!&D*! $!&(*!2!,-,3;'2'&!'/0!%##+%+! $!%1#1#1+%1+!(&3!&4;!'38!&.*$!

&)*!2!,-,3;'2'&!'/0!%/2#$!%1/121#1 . !4,-50/8!(&3!'&!+)=.2!'/0!1(/0!49,8!&)'58!&450&0'3!)*960!)10&3!

)41(28! ;-,3&! (&3! ,-,1/8! ()1-)0&! 4&9&@@2@;.9&--&'! %,! 4&9&@-@2@)414)*,!@,34;0!%/2#!–!%+/+2+#+! ! !)10&3!;-,3,!(&3!,-,1/8!()1-)0,!49,8!',!*,(-50,'!!

!."'*'

'%!)*#+'"!&,-.!#!'/!!"0'&"'/!1.-0'/ 5.1. 2"'!3"'"!'##!$!"/!!"'4&".-!1.-'5-.

A8! !'91)*92!(')9)*!./01&!:)/9>(&-)!',!0(7>-&0!',)!769,)!4,)!

49,(=4')3!&4;!'9)38!-2!()0)414)*)8!2-3)):)1)8!&&x$!&y$!&z*!(,30>8!&9%7>8!&'!"=93&!(',37)1&!'28!'91)*928!(')9)*8!./01&8!)10&3!,3!'9)38!)414)%

*)8! ./01)8! '28! ()*!'#*)'!*#+∧∧∧

! (&3! ,3! '9)38! *1)*9)8! ./01)8! '28!!

).,D,&(∧∧∧

.

$!2!AB!%C"%!$%&!'AC&! 57

(x - &z - y) ⎟⎠⎞

⎜⎝⎛Β↔

∧∧∧xOyD

(y - &x - z) ⎟⎠⎞

⎜⎝⎛Γ↔

∧∧∧yOz.

(z - &y - x) ⎟⎠⎞

⎜⎝⎛Α↔

∧∧∧xOz&

$'38! '91)*9)8! (')9)58! ./01)8!

-4,9)1! 0&!&4,*)37:)1!;'3!,3! )414)*)8!

./01)8!57,)0!'20!1*3&!()39*!-).5:,)8!-)!'38!&450&0'3!*1)*9)8!./01)8!(&3!

&0'3('9;0/8'!-'(3$!'&!(=93&!(',37)1&!'28!'91)*928!(&:,91+,0'&3!(&3!*3&-'*((,0'&3!0-)'93(*0!(',!769,!&0*!*=,'

B)'&?=!'/0!(2-)1/0!(&3!'/0!)):)360!(',!)414)*,!)4*97)3!-3&!&%

0'3(',172(2! ! 4,)! (>-)9&! (&'&0,)1'&3!/8! *)F(;'2'&'!%0!()! (*:)! 49;-'&(2!>!()!(*:)!(7>-&!)0&@@&.,=0!,3!@5?)38!%$)$%&'%!(&3!%%(*%&!%, %+%&&&!%%!(&3!%"%,-"%&!%%, %&,$.'.&!%%!(&3!%$(.",'.&!% $%%!('@4'$!49,(=4')3!-3&!05&!049;'&(20!>!50&!05,!0(7>-&0$!4,)!(&@)1'&3!"(/+#!>!"(/+0!',)!&973%

(,='!-'(3!4'7'!',!&?1/-&!%"1'!$)$%&!!$(.",'.&!%!$%!$%!!$'.!"%+#!%(*%&!% 57)3!/8!*)F(;!',!0&?1/-&0!%"1'!%(*%&%2!&,$.'.&!%!$%!,.!!$'.!"%+0!$)$%&&

'%!!(&3!&0'3('9;0/8' <&9;-,3&!(&'*!'20!-)'*D&(2!(',!769,!(&3!4*@3!,3!@5?)38!%$)$%&&

'%, %+'-('#%, %%(*13-!$$'!&$#$!%, %45%(-6%!(&3!%%4&4%"'2!37.&!%, &0'3-(',37,=0!-)!%%(*%&!%, %!+$#%, %%4&4%"'2!37.&!%, %,"-!%!(&3!!%"&%"-'2!37-.&!%!(@4'

-'(3$!-)'&?=!'91)*928!(')9)*8!./01&8!(&3!'93.60,)$!-)!79>(2!(&3!

'28!*)F(;'2'&8!*2-3,)9.)1'&3!!-3&!)00,3,@,.3(>!()(75'3(2'

67$(89:!%)(

67$*+7,!(89$%!:%%)(

4@))9*!%/

)414)*2!./01& ∧

$!!

)414)*2!./01&!∧

%2/ *1)*92!./01&!%!-!&2!-!/

!"#$!!!'()

58 !!"#!"#$!"$%&"'!"%#

"&'1!&)';!',0!'9;4,!-1&!(75(2! 3(;'2'&8!*=,!'935*9/0!4,)!(&%

:,91+)'&3!&4;!'&!&@@2@,)?&9'6-)0&!(=93&!(',37)1&!'/0!*=,!'935*9/0$!

-4,9)1!0&!()(7)'3(:)1!)00,3,@,.3(*!(&3!0&!0&0&(&@)0:)10!&4;!'&!(93'>%

93&! 3(;'2'&8!*=,!'93.60/0$!&9()1!0&!?)(&:&93(')1!4;(&!(&3!4,3&!(',3%

7)1&!&9(,=0!.3&!',0!4@>92!(&:,93(-;!-3&8!'91)*928'!

!"#$!!!'("

∗ 8!!"1'!&-&37.!!&'(!$"#$!&'2!'("!,"'(.!"1'!45%(-,2!$%!$,&-!!)"!+!%!)*!

$'.6"%2(

∗ (&.!%!0$'%!+!!''1! 5,11827

1218

812

=== !'40&%!∧∧∧∧

== ),*7 ϕ +!+!%!

∧∧= +* !!556!'%!37.&%2!"%.!,"'(.!&).!&"%!!"%6&!9)!,"%.!%&.!%!&$%2!%$&!!

4-'2!$&!%(

%(;-&$! 500,3)8! ;4/8! 4!-!45)-7$!&%+#! 37.&!! -4,9,=0! 0&! )4)%

('&:,=0!(',!769,$!,4;')!,3!)414)*)8!./01)8!-3&8!'91)*928!.10,0'&3!*1)%

*9)8!'28!4&9&4@29/-&'3(>8!'28'!%)@3(*$!&4;!'&!:)/9>-&'&!(&'&(())>8!

-3&8! '91)*928!&4;! '&!(',37)1&! '28$!49,(=4')3!;'3! )10&3!*)0&';0!-)! 5?2!

*,(-50&!(=93&!(',37)1&!2!'91)*92!4,)!(&'&(())*+)'&3!0&!-20!)10&3!-,%

0&*3(>'!%)';!*)0!4954)3!0&!0&0)1!4)91)9.,$!;'&0!&(;-&!(&3!2!(&'&(())>!

)0;8!'93.60,)!&4;!450')!*,(-50&!(',37)1&!-4,9)1!0&!-20!)10&3!-,0&*3%

(>$!&0!@20:,=0!)4;E2!&4&3'>()38!*3*'&?28!'/0!(',37)1/0!(&3!49,(&0&%

',@3(-,=!(',!)414)*,'!!!!

23&! '20! &4;*)3?2! '28! -2! -,0&*3(>8! '91)*928! &4;! '&! 5?3! (=93&!

(',37)1&!.10)'&3!)41(@2(2!'28!(&'*!(,9)0>!'91)*928'

$!2!AB!%C"%!$%&!'AC&! 59

&3!*=,!(&'*!(,9)0>!'91)*9)8!57,)0.!

'#*!!*#∧∧ʹ′ʹ′ʹ′= ')*##*)

∧∧ʹ′ʹ′=

∧∧ʹ′ʹ′= !*))*! !(&3!

x -!&y - z = x+-!&y+- z+ y - &z - x = y+- &z+- x+ z - &x - y = z+- &x+- y+

;-/8!-4,9)1!0&!)10&3!1()8!>!()--)'93(58' &B4,9)1!0&!&4,*)37:)1!;'3!.10,0'&3!1()8!&.!()-4'6(3-)8*!;'&0!*=,!

)414)*)8!!./01)8!',)8!)10&3!1()8.!3(,(()@>8!'91)*92!./01&*'

!"#$!!!'(-

z - :x - y ≠ z - :y – x, ∧∧

ʹ′ʹ′≠ yOxxOy

z - :x - y = z - :y – x, ∧∧ʹ′ʹ′= yOxxOy

%)@3(*$!'&!!(93'>93&!3(;'2'&8!*=,!'935*9/0!*3&-,9060,0'&3!/8!)?>8.

• !*0!*=,!'91)*9)8!./01)8!57,)0!-1&!5*9&!1(2!(&3!'38!49,(()1-)%

0)8!(1!&)'>0!*1)*9)8!&0*!-1&!1()8$!';')!&)'58!)10&3!1()8!>!()---)'93(58

• !*0!*=,!'91)*9)8!57,)0!*=,!)414)*)8!./01)8!!1()8!-1&!49,8!-1&!

(&3!'38!4)93)7;-)0)8!()!&)'58!*1)*9)8!&0'1(',37&!!1()8$!';')!&)%

'58!)10&3!1()8!>!()--)'93(58'

• !*0!*=,! '91)*9)8! ./01)8! ! 57,)0! '38! )414)*)8! ./01)8! ',)8! 1()8!

-1&!49,8!-1&$!';')!&)'58!)10&3!1()8!>!()--)'93(58'

60 !!"#!"#$!"$%&"'!"%#

• !*0!*=,!'91)*9)8!./01)8!57,)0!'38!*1)*9)8!./01)8!',)8!1()8!-1&!49,8!-1&$!';')!&)'58!)10&3! 1()8!>!()--)'93(58! &.;-/8!'91./0&!

-)!)!./01)8!1()8!)10&3!1(&!>!;-,3&*' %4;! '20! 4&9*:)(2! &4,')@)(-*'/0! 4,)! 49,2.>:2()!-4,9,=-)!

0&!&4,0&0:,=-)!;'3!*=,!'91)*9)8!(')9)58!./01)8!-)!'38!)414)*)8!./01)8!

',)8! 1()8! (&3! ,-,1/8! ()1-)0)8! )10&3! 1()8$! *2@&*>! #)#+%2#/#!1# ;4/8!&(93D68!)43'*((,)0!,3!,93(-,1!2!(&3!,3!',)!D3D@1,)!'"'

5.2 2"!,!;<"=-3-!!"0'&"'/!1.-0'/ "&!$! 500,3&! '28! 3(;'2'&8!*=,! '935*9/0!()-414')3! 4*3&3(:2'3(*5!

-)!'20!&0:964302!*)0&';'2'&!'&='3(>8!',)8' 34/8!49,(=4')3!&4;!'20!49;'&(2!(($!'9)38!*03()8!)414)*)8!./01)8!

4@29,=0!'38!49,G4,:5()38!(&'&(())>8!-3&8!'935*928!./01&8!;'&0. i) !!!(&:)-3*!)10&3!-3(9;')92!&4;!',!*:9,3(-&!'/0!*=,!*@@/0!(&3! ii*!!',!*:9,3(-*!',)8!)10&3!-3(9;')9,!&4;!-!,9:58' 3'&0! (&'&(())*(,)-)! '91)*92! ./01&!&'%/2! &4;! '9)38! *,(-50)8!

5*9)8!&)414)*)8!./01)8*!&%/$!&/2!(&3!&%2$!(&3!49,)(')10,)-)!'38!&(-58!

'28$!D91((,)-)!(&3!'20!(&'*!(,9)0>0!'28!'91)*92!./01&!&'%+/+2+!&$7>-&!

6'-*'!$'&! 0!&'%"%&!0!;-/8!2! '91)*92!(&:,91+)'&3!&)('29*!&4;! '38! 5*9)8!

4,)! '20! 4)93D*@@,)0$! ,3! ,4,1)8! )10&3! '91./0&'!<9,2.,=0'&3! *2@&*>! 50%

0,3)8$!;4/8!+!&6!+'-('#., 4!-!45)-7$!&%+,2!('@4'

!"#$!!!'(.

$!2!AB!%C"%!$%&!'AC&! 61

&3!*=,!'91)*9)8!@,34;0!4,)!49,5()E&0$!57,)0!1()8!5*9)8!(&3!1()8!

*1)*9)8$! &@@*! )10&3! *03()8$! -)! '20! 500,3&! ;'3! *)0! -4,9,=0! 0&! -)'&-(302:,=0!-5(/!-3&8! 0(')9)*80!(102(28!(&3!0&!.10,)0!()-4'6(3-)8'!.&!

)437)392:)1! -3&! )4,4'3(>! *3)9)=02(2! ',)! '9;4,)! (102(28! (&3! '28! (=%

-4'/(28'

>&!!4)3*>!,!4&9&'292'>8!D91(()'&3!(',!769,$!-4,9)1!0&!:)/9)1!

&(102'2!'20!'91)*92!&'%/2!(&3!0&!(30)1!'20!&'%+/+2+!.3&!0&!)@5.?)3!&0!

()-414')3!-)!'20!*,:)1(&!('&:)9>!'91)*92'!!4*97,)0!;-/8$!()!&0'1:)(2!

-)!'&!49,D@>-&'&!(',!)414)*,$!4,@@58!*)0&';'2')8!0&!.10)3!&)'>!2!(')%

9)*!(102(2!(',!769,'!

$'950,0'&8!4'7'!'2!./01&!∧

ʹ′ʹ′/%& !(&'*!,73,$!6(')!0&!()-45()3!

-)!'20!∧

ΟΑΒ $!2!&2+!4)90*!&4;!',0!2-3769,!4,)!D93((;'&0!/8!49,8!',!

)414)*,!%+/+&%/!(',0!2-3769,!4,)!D91(()'&3!2!&2!&$7'!6'-*' %,!&0!2!&2+!:&!()-45()3!>!;73!-)!'20!&2!)?&9'*'&3!&4;!',!&0!2!

7*9&?2!-3&8!2-3)):)1&8!&4;!',!&!4,)!0&!(72-&'1+)3!*,(-50)8!./01)8!-)!

'38!&%!(&3!&/!)10&3!-,0&*3(>!(',!769,' 3@)8! ,3! 2-3)):)1)8! 4,)!

(72-&'1+,)0!-)!'20!&%!./01&! 1(2!

-)! '20!∧

%82 ! D91((,0'&3! 4*0/!

('20! )430*0)3&! )0;8! (60,)! -)! *-?,0&!&%!&$7'!6'6*.

&-,1/8$! ,3! 2-3)):)1)8! 4,)!

(72-&'1+,)0!-)! '20!&/! ./01&! 1(2!

-)!'2!∧

/82 !D91((,0'&3!4*0/!('20!

)430*0)3&! )0;8! (60,)! -)! *?,0&!

&/' &3!*=,!(60,3!'5-0,0'&3!(&'*!

*=,!)):)1)8!!!&"1!!(&3!!&"2$!!&0,=! ∧

$*# + ∧∧

> "*&&*# .

-'(3$!&4;!'38!5*9)8!∧

$*# !(&3!∧

!*# ! (72-&'1+,0'&3! *=,! '91)*9)8$!

,3!&'%"1/!(&3!2!&'%"2/$!,4;')!2!

&(->! !&2+! !'28!(&'*!(,9)0>0! !'91)*928!*)0!()-414')3!-)!'20!&(->!&2!

'28!&973(>8'

!"#$!!!5.5

62 !!"#!"#$!"$%&"'!"%#

3&#!*0#& %,!)414)*,!4,)!*3597)'&3!&4;!'20!%/!(&3!'5-0)3!',)8!*=,!(60,)8!

(&'*!)@@)1E)38$!792(3-,4,3>:2()!.3&!)4,4'3(,=8!@;.,)8'!$!4@>928!-)%

@5'2! ',)! 49,D@>-&',8! &4&3')1! '2! 79>(2! -)'&(72-&'3(-60! (',! 769,$!

(*'3! 4,)! 792(3-,4,12()!()('2-&'3(*! ,!Euler!('2!-)@5'2! '28! (102(28!)0;8!(')9),=!(6-&',8'

1&!!$'2!()057)3&!:&!)4,*)37:)1!-3&!*3&*3(&(1&!4,)!)4)(')10)3!(',!

769,!'2!*3&*3(&(1&!.3&!'20!(=-4'/(2!(',!)414)*,!*=,!'93.60/0!-)!1(&!

+)=.2! ,-;@,./0! 4@))960! (&3! 4,)! *3))(9301+)3! '2! *)0&';'2'&! (=--4'/(28!*=,!(&'*!(,9)0>0!3(,(()@60!'935*9/0'!

459,)-)!'2!*37,';-,!xy!'/0!(&'*!(,9)0>!)*960!&%/!(&3!&%+/+!(&3!)414)*,!&<*!(*:)',!(',!)414)*,!'/0!)*960!&%/$!&%+/+$!4,)!0&!',!

'5-0)3!(&'*!'20!xy!&$7'!6'/*'!$!'91'2!&(->!&2!(&3!2!49,5('&(>!'28!D91-((,0'&3!()!*=,!*3&0,9)'3(,=8!2-3769,)8!&<*!-!/!(&3!&<*!-!/+!&0'1(',37&!&&0!2!&(->!2+&2!D93((;'&0!)41!',)!&<*$!2!'91)*92!:&!>'&0!3(,(()@>8*'!

459,)-)!'20!)):)1&!z!(*:)'2!(',! &<*! (',! (2-)1,! &'! %0! ('95-

E,)-)! '2! ./01&!∧

ʹ′ʹ′BAO ! (&'*! ,73, (',! 769,! .=9/! &4;! ',0! *?,0&! +z, 6(')!2!2-3)):)1&!&%+!0&!()-45()3!

-)! '20!&/! (&3! 2!&/+! -)! '20!&%$!

';')!2!('&:)9>!(&'*!:5(2!&(',!76%

9,*! &2+! :&! ?&0&D9):)1! (',0! 1*3,!

2-3769,!&<*!-!/+!4,)!>'&0!&973(*'!-'(3$! 2!&21+! 4,)! &4,')@)1! '2! 05&!

:5(2! '28!&2$! *)! ()-414')3! -)! '20!

&2$! &0,=! D91((,0'&3! ()! *=,! *3&%

0,9)'3(,=8!2-3769,)8' .&! 4&9&'29>(,)-)! ;'3! -3&!

&0*@,.2! *3&*3(&(1&! (',! )414)*,!

,*2.)1!4*0'&!('20!'&='3(2!*=,!'93%

.60/0!4,)!57,)0!'38!'9)38!!4@))958!',)8!&0'1(',37&!1()8!&7/918!0&!@&-%

D*0,0'&3!)4;E2!&4&3'>()38!49,(&0&',@3(-,=*'!!*6!;-/8!79)3*+)'&3!0&!

D.,=-)!5?/!&4;!',!)414)*,!.3&!0&!&0&4')7:)1!2!'&='3(2!(',0!4)93D*@-@,0'&!769,'

!"#$!!!'(#

!!"#!"##$$%!%#&!'!$& 63

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

!!&()*+,&-!'./!(01()+)22./!&3-24'./!&'5!(676(*5!(/'4&&('),!

&'5!85),*9!&3:2)"!!"!#"#$!–!#$#%!#"#$!–!%$&$&!'" %(2+./)!2(!)0':!'-/!*8,24*.&-#!9')/!*(5!&3:2)')!*(/!(7/),!&0$

26';&,2)#!265+(7!/)!(7/),!925,)!*),!9')/!*(/!(2676'50/!&',<!)6),':&(,<!

'-<!525,9'-')<#!265+5(/!/)!&0)*+,15(/!.<!6+5<!'-/!(6,+4/(,)!650!6(+,$

*8(750/%!!!5+19'-')!)0':<!'-<!(6,85):<!*,*),;/('),!)69!)0'9!650!&:2($

+)!(7/),!)/.&'9!.<!'5 "!"!"##"$$#%!"&%!'#()*%%!"!+!#,-%$!.!!"&/!#0*0#$&!&#(,#")*%", *),!650!)65*(731-*(!298,<!&'5!'=85<!'50!&'50!),;/)%!

!! &3(',*:! 1(.+7)! 1(2(8,;/('),! *),! )/)6'(&&('),! (6,'03;<! &')!

>,>87)!$!*),!$$!'./!1"!"##*(/%!!!(6=*')&-!*),!)/46'0?-!'50!)/'7&'5,350!

&3:2)'5<!&'5!3;+5#!&026(+,8)2>4/5/')<!'-/!=//5,)!!()#*, 6)+50&,4@(, )/06=+>8-'(<!*0&*587(<%!,2.<#!)/!6(+,5+,&'5(2(!&(!6)+)88-8(676(*)#!

)/)6'(&&('),!2,)!,*)/565,-',*:!1(.+7)#!&(!68:+-!)/)85)7)!2(!'5!(676(-*5%!%0'9!)*+,>;<!(7/),!*),!'5!)/',*(72(/5!'./!6+5'4&(./!()!2=3+,!*'%!

%(!)0'4!')!68)7&,)!-!,&9'-')!&'(+(;/#!9')/!*(/!(7/),!&(26'.&-#!(//5(7$

'),!.<!,&9'-')!9)*./#!*-8)*:!+,&5*0/)27)+%!-'&,!)65*(,*/(('),!9',!"2!

345!%! #"(/!456(/$!$.!!4,!"(/!0%)%33&3#0"07$(/!#*/%"!!!6.-!%!"!+!34%5!+!!,!"4"&"%%"! !,6+9')&-!**-#!96.<!*),!9',!"1#!$.!!*!%!#&*!(/!456(/$!

0%)%33&3#0*0#$%!!"!-8!#"%!#*/%"!%/""!")4'(%!%/83!5#%!!"(/!+9$/"!,6+9$

')&-!*.-%! "(!&-2(+,/:!5+585)7)!&/!)0':!'-!*,)'(6.&-!(268=*5/'),!')!2()=1-!

",:6!%", "#,-%$4/", "456!%"'!650!&3('7@5/'),!2(!(01()+)22)!'2:2)')#!(67$6(*)!*),!&'(+(4!&3:2)')!)/'7&'5,3)%!!#=85<#!6+=6(,!/)!&-2(,.1(7!-!)650-&7)! &026(+)&24'./! 650! )65'(85(/! &:2(+)! +42(&-+! )/;&-! &(! 6+5&()-)7&(,<!650!(6,*(/'+;/5/'),!&'-!2='+-&-#!96.<!6%3%!9',!&(!=/)!6)+)88-8($

676(*5!'5!),/92(/5!(2>)*5(!>4&-<!*),!(A50<!(7/),!&')1(+9% ",)!/)!)7/(,!*)')/5-':!-!*,)6+))24'(0&-!'./!(//5,;/!&',<!6+5-

'4&(,<! 650! )*585015(/! 1)! 6+=6(,! /)! *4/502(! 2,)! &(/'52-! )/)*+52:!

&'5/!'+965!3(,+,&25(!'50!(2>)*5(!'./!(01()+)22./!(676(*./!&3-24-'./%!&,!)6)+3=<!2,)<!&0/1(',*:<!)/46'0?-<! '-<!1(.+7)<! '50!#2>)*5(!

),)!')!(676(*)!&3:2)')!>+7&*5/'),!:*-!&'5!>,>875!$!'./!1"!"##*(/ (2('4!'-/!6+9')&-!*0-%!#*(7!&0)*+7/5/'),!6)+)88-89)+)22)!2(!'-/!7*,)!>4&-!

*),!7&)!(A-!*),!)*58501(7'),!,2(!&()3+5/-!5+585)7)-!-!(?:<!*,)*,*)&7)#!

9650!(/'9<!6)+(/1=&(.<!&-2(,;/5/'),!')!3.+7)!'50!(6,6=*50!650!065$

*-8;/50/!')!)/'7&'5,3)!)+422)')%

64 #BCD#$.!!1%#&$'#$%2

1#:,%!(!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!1#:,%!)

%'-/!6+;'-!6(+76'.&-!,%3%%-! E1!3!,%/".-#!!E2!3!,%/#0- #!3!,%/#.-#!!!!#2!3!,0.%-!!!!!!!!! #1!3!,#"/- E1!3!#1!4!#!!!! E2!3!!#2!4!##!

,2.<#!'+7)./5!,0.%-!3 3!'+7)./5!,#"/-# 569'(!#1!3!#2#!*),!='&,!!E1!3!E2

%'-!*(('(+-!6(+76'.&-!,%3%!/- E1!3!,%/".-#!!!#!3!,#.&-#!! #1!3!,#"/-#!!!!!!#13!,%&/-! #2!3!,0.%-#!!!!!!!E2 = (%/#0- E1!4!#!3!#1!4!#1#!! E2!4!#!3!#2!4!#1!

,2.<#!'+7)./5!,#"/-!3 3!'+7)./5!,0.%-#! 569'(!#1!3!#2#!*),!='&,!!E1!3!E2

#)!6,5!64/.!065*(,*/(50/!9',!'5!(2>)*9/!265+5(2(!/)!'5!3(,+,-

&15(2(!.<!2=)(15<#!9650" ,)-!!2&)!&3:2)')!=350/!7&)!(2>)*4%! ,>-! !#5!(2>)*9/!(7/),!1(',*9!*),!6+5&1(',*9!2(!'-/!(?:<!=//5,)"!

,')/! =/)!&3:2)! *,)2(+,&'(7! 6%"8!%0!$#6"4! ")40!'! '5! (2>)*9/!

'50!&3:2)'5<!6+5*(6'(,!.<!'5!41+5,&2)!'./!(2>)*;/!'./!&3--24'./!'-<!*,)2=+,&-<%

!!)+3:!,>-!8(,'5(+)-&(!&'-/!6(+76'.&-!650!298,<!6)+50&,4&'-*(!

2(!*(5!'+9650<"

,>1) *!!$"%#()*!",%!#()*%' ,>2) *!!!+,03&)$!",%!#()*%'

9650!'5!@-'5(2(/5!(2>)*9/!6+5-=*0A(! )69! 41+5,&2)! (2>)*;/!

7&./!&3-24'./!,%3%!%-

9650!'5!@-'5(2(/5!(2>)*9/!6+5=-*0A(! )+5(! '5! &3:2)! &0268-+;-1-*(#!;&'(!/)!6+5*(A(,!41+5,&2)!

7&./!&3-24'./!,%3%!/-

!!"#!"##$$%!%#&!'!$& 65

&,!*,)*,*)&7(<!,)-!*),!,>-!265+5(/!2(!',<!)/))*)7(<!!6)+)*53=<!/)!

(6(*')15(/!&(!,*0+'4-!6580)./,*4!3.+7)#!569'(!5*-)5(/!&(!658()./)!

,&5*,)3.+7&,2)! *),!&(! 658()./)! ,&5&0268-+;&,2)%!'+-&,2565,;/')<!

'+,)./565,:&(,<! !*),!'-/!)+3:!'50!%+3,2:*--#0*9?50#!265+(7!/)!)69-*(,31(7! 9',! ')! ,&5*,)3.+7&,2)! 3.+7)! (7/),! ,&5&0268-+;&,2)! *),! )/',$

&'+9+.<% !!2('4>)&-!&'-!2='+-&-!)6),'(7!'-/!*)')&*(0:!2,)<!!+/8)"&!&%!

#,-%$!.(!!!,*=)!),)!'-/!*)')&*(0:!'-<!>+7&*('),!&'5/!5+,&29!'50!(2>)$

*5(!6+;')!&')!'+7)./)!*),!*)'96,/!'-/!(6=*')&-!,6+5&1(',*4-!&')!6580$

)./,*4!3.+7)#!2=&.!'+,)./565,:&(./%!E+4)2)',#!)0'9!(6,'0)34/('),!6)$

+)'-+;/')<!9',!&(! =/)! '+7)./5! '5! ),/92(/5!2,)<!68(0+4<!*),! '50!)/'7$

&'5,350! &/! )0':/! (A50<! (7/),! )/(?4+'-'5! )69! '-! 1(.+5(2(/-! 68(0+4%!

#9'(!5+7@502(!'5!(2>)*9/!&')!'+7)./)!2=&.!'50!)/.&'5(!'(650"

(2>)*9/!3!21,>4&--!⋅!,(A5<-%

"65+5(2(!='&,!/)!06585)7&502(!'5!(2>)*9/!(/9<!6580);/50!.<!

'5!41+5,&2)!'./!(2>)*;/!'./!'+,);/./!2,)<!'+,)./5657-&-<#!)+5(!>=$

>),)!)65*(,31(7!9',!(7/),!)/(?4+'-'5!)69!'-/!'+,)./5657-&-% !! 1(.+7)! '50! (2>)*5(!&'5! (676(*5! 650!)/)6'(&&('),!2(!)0'9/!

'5/! '+965! (7/),!!"!"##"$$&%!2(! '-/!)*98501-! =//5,)"!.(5!6580)./,*4!

3.+7)!(7/),! ,&(2>)*,*4!)*+,>;<!'9'(#!)/!(7/),!,&5*,)3.+7&,2)!,4+)!*),!

,&5&0268-+;+,2)-%!%(!2,)!)68:#!(/'06.&,)*:!,)884!,&5*(/)2--!*,)'($

6.&-"!*!#,-%$"68!0!3+5(/"68!#()*%!##()*%!,#!"!!*$"!!#,-%$4/$!,0!)!./!

/%!$"%,#)"!'!./!!"!/!*$"!!%)"',4!*!(/!")"5$/(/( !!)/46'0?-!2,)<!)/'7&'5,3-<!1(.+7)<!2='+-&-<!&'5!3;+5!6)+50$

&,4@(,!*0&*587(<!*),!'(8,*4!)65*(731-*(!9',!*(/!265+(7!/)!064+?(,!&'5,$

3(,;*-<!1(.+7)!9)*50!&'5!3;+5%!",)!&(/5A-!'-<!98-<!*)'4&')&-<!=3(,!

.<!(?:<" ,&-!%'5!&0/1(',*9!2=+5<!'-<!6+5&=)),&-<#!)/',*)1,&';/')<!')!'+7-

)./)! 2(! '('+4(*+)! )65*(731-*(! ,'5! &'.*! )69! '5/! Sylder-! 9',! ,&5&0$268-+;&,2)!658((*+)!(7/),!*),!,&5*,)3.+7&,2)%

,(-!,&5/!)+5+4!&'-/!*)')&*(0:!2,)<!&0/4+'-&-<!9)*50#!)0'9!($

6,'0)34/('),!5+7@5/')<!'5/!9)*5!&'5!'('+4(*+5!.<!'5!31!'50!),/52=/50!

'50!(2>)*5(!2,)<!=*+)<!*),!'50!)/'7&'5,350!&/!)0':/!(A50<%!C)'96,/#!

(6(*'(7/('),!&')!658((*+)!2(!3+:&-!*,)2(+7&(./!!&(!'('+4(*+)% ,*-!%0':!-!*)')&*(0:!=3(,!.<!&0/=6(,)#!6)+)88-8(676(*)!,:!*),!

6+7&2)')-!2(! 7&50<!9)*50<!/)!(7/),! ,&5*,)3.+7&,2)! !,2.<#!96.<!)65-

66 #BCD#$.!!1%#&$'#$%2

*(,*/(('),!2(!3+:&-!)65'(8(&24'./!'50!M. Dehn#!=/)!*)/5/,*9!'('+4-(*+5!*(/!265+(7!/)!(7/),! ,&5*,)3.+7&,25!2(!=/)!*(>5!'50! 7*,50!9)*50%!

,/8=6(!*),!&398,5!2('4!'-/!6+9')&-!*'-% #652=/.<#!-!1(.+7)!2='+-&-<!&'5! 3;+5!650!)/)6'(&&('),!*)'/

)0'9! '5/! '+965! *(/! (7/),! &'5,3(,;*-<#! 6=+)/! './! 6+,&24'./%! ",)! 2,)!

&0&'-2)',*:!2(8='-!)6),'5(/'),!,7'!$!"!#;8/"3&!&%!:!3+:&-!'-<!%)#:%!

"!+!Cavallieri!,*-8)*:!)6(,+5&',*9<!85),&29<-% #5!)?,51)(2)&'5!=)*(,'),!&'5!)()5/9<!9',!'5!'$!>,>875!'./!1"!"-

##*(/!,650!*,)6+))2)'((('),!'-!&()*+,&-!'./!6+,&24'./-!&')2)'4!(*(7!

)*+,>;<!650!)6),'(7'),!-!)88)):!'./!2(19*./#!(/;!-!,7'!$!%!"&%!#;8%

/"3&!&%!(+)+29@('),!&'5!>,>875!'$$%

/#0!#"$##$%#&1&"#$%'

''!+$,$--"-%+.+%&#!#$"/%.!$+!!##!%+.+%&#!##!#+#.#!&%0,1%#$%!$+!! #%*! &%$(2'%%*! &3#! $+0'$'#%! %&,2'! ##4#! #!#%! &%$%,%.#$%! "%!&3#!."$!$%!%"#&3'$$$&!+,."$$#$" '%0%$'2$# 3(.+5(2(! '5! 6)+)88--

8(676(*5! %/".! –! %1/1"1.1!

*),!',<!*,));/,(<!%"#!%1"1!*(5!

>4&(;/!'50%!#)!6+7&2)')!%/"!

–! %1/1"1! *),! %."! –! %1.1"1!(7/),! 7&)#! )+5(! 5,! =*+(<! '50<!

(7/),!2,)!6+5<!2,)! 7&(<!*),!6)$

+488-8(<!:!')0',@92(/(<%

5$,$#6,"""!! &!+985<!!'-<!6+9')&-<!()!(7/),!1(2(8,;*-<!&(!9#',!)*58501(7% ,6.<!'5/7&1-*(!-!1(.+7)!'-<!&()*+,&-<!'./!6)+)88-8(6,6=*./!

*),! './!6+,&24'./!)/)6'(&&('),!&(!68:+-!)/',&'5,37)!2(! '-/!1(.+7)!

&()*+,&-<!6)+)88-85)+422./!*),!'+,);/./!)69!'5!6+;'5!>,>875#!2(!'5!

(6,68=5/!)()5/9<!'-<!&0&'-2)',*:<!3+:&-<!&3=&(./!525,9'-')<!*),!)$

/)85),;/!650!)0':!&0/(64)('),% %'-/!(,&)).),*:!6+9')&-!()#!=/)!6)+)88-8(676(*5!*,),+(7'),!&(!

*(5! '+,)./,*4! 6+7&2)')#! 96.<! =/)! 6)+)88-89)+)225! *,),+(7'),! ,)69!

27)!*,));/,5-!&(!*(5!,7&)-!'+7)./)%!,2.<#!')!6+7&2)')!650!6+5*(6'50/#!

1#:,%!!)*

!!"#!"##$$%!%#&!'!$& 67

9/')<!&022('+,*4#!$#/!,0!)#*!/%!#*/%"!!+,0"$!",%(!-'&,!'5!)65'=8(&2)!

650!6+5*(6'(,!5+(78(,!/)!(+2-/(01(7!&')!68)7&,)!650!1='(,!5!5+,&29<!&5!

*),! 2(! &-2(+,/5(<! 9+50<! /)! )65*51(7!(%! "!!$+/%,*%'! *-8)*:! "!4"&"%!

456(/(! &!'+965<!2(!'5/!56575!(+)+29@('),!'5!&026=+)&2)!(7/),!*)')80-

',*9<%!#6,'+=6(,!2,)!&0/1(',*:!6+5&=)),&-!'-<!1(.+7)<!'50!9)*50#!96.<!

=),/(!*),!2(!'5!(2>)*9/!*),!'(8,*4!'-/!)/',2(';6,&:!'50!.<!2()=150<% "(! '-/!6+9')&-! *.!585*8-+;/('),!-!1(.+7)!),)!')!6)+)88-8(-

676(*)#!)+5(!2(!3+:&-!'-<!265+(7!/)!)65*(,31(7!9',!&(!=/)!6)+)88-8(-676(*5!'5!),/92(/5!'-<!>4&-<!*),!'50!(A50<!650!)/',&'5,3(7!&/!)0':!(7$

/),!&')1(+9%

/#0!#"$#$(#&1&"#$(' ''!&3#!+$,$--"-%+.+%&$!01#4'!$%$!&7""!)#%'6#!##!.&%#!38#*!)$%!#%!$)$0*!+#4!&%0,1#'#$%!$+!!#%*!)#,4(0*!$4#6*!#"*!&7""*!)$-#$-6(#4'!"#%*!.&%%*!%4/%.%*#!#!#%!%.'$%!."$!$%!%"#&3'$$$&" '%0%$'2$ 3(.+5(2(!!!')!!!6)+)88--

8(676(*)!!%/".!-!%1/1"1.1!!*),!!%/".! -!%11/11"11.11#! 650! ,*)$/565,5(/! ',<! 6+5F651=&(,<! '50!

1(.+:2)'5<%! #9'(! )69! ')! 6)$

+)88-89)+)22)! %//1%1! *),!

%//11%11!=3502("! %1/1!3!%/!3!%11/11!,$%*.-#!569-'(!%1%11!3!/1/11!*),!)+5(!%%13!

3!//1#!%%11!3!//11!')!'+7)./)!

%%411!#!//1/11!1)!(7/),!7&)!,=350/!'+(,<!68(0+=<!7&(<!27)!6+5<!27)-% &257.<!')!'+7)./)!..1.11#!""1"11!(7/),!7&)#!*)1;<!*),!')!6)+)8$

8-89)+)22)!%..11%11#!%""11/11% #=85<#! ')! '('+468(0+)!%1.1.11%11! *),!/1"1"11/11! (7/),! 7&)! ,=$

350/!',<!68(0+=<!7&(<!27)!6+5<!27)#!*)1;<!*),!',<!)./7(<!'50<-% #)! 6+7&2)')! 85,69/!%%411! -!..1.11! *),!//1/11! -!""1"11! (7/),!

7&)%!%/! 6+5&'(1(7! &/! )0'4! '5! &'(+(9! 650! 6(+,*8(7('),! )69! ',<! =*+(<!

%/".#!%1/11"11.1#!%..1%1#!/""11/11#!1)!6+5*(A(,!-! ,&9'-')! ,,&5*0$

/)27)-!'./!)+3,*;/!6)+)88-8(6,6=*./%

1#:,%!!)+

68 #BCD#$.!!1%#&$'#$%2

/#0!#"$#)*#&1&")*' ''!&3#!+$,$--"-%+.+%&$!01#4'!)#%'6!&7""#!##! .&%#!38#*!)$%!#%!$)$0*!+#4!&%0,1#'#$%!$+!!#%*!)#,4(0*!$4#6*!#"*!&7""*!&%'!)$-#$-6(#4'!"#%*!.&%%*!%4/%.%*#!#!#%!%.'$%!."$!$%!%"#&3'$$$&" '%0%$'2$ 3(.+5(2(! ')! 6)-

+)88-8(676(*)! ! ! %/".- -%1/1"1.1! ! *),! !%/".! - -%11/11"11.11#! 650! ,*)-/565,5(/! ',<! 6+5F651=-&(,<! '50! 1(.+:2)'5<%!

-&'.!"1!'5!&-2(75!'52:<!

'./! .11"11! *),! /1"1#! /1 '5! &-2(75! '52:<! './!

%11/11!*),!/1"1!)/'7&'5,-3)%! &+7@5/'),! ')! &-2(7)!

%1!*),!.1#!569'(!6+5*(6'(,!'5!&'(+(9!%/".!–!%1/1"1.1. %(2+./)!2(!'-/!6+9')&-!('!=3502(!,%/".!-!%1/1"1.1-!3!

3!,%/".!-!%1/1"1.1-#!)+5(!5,!)*2=<!'50<!650!*,=+35/'),!)69!',<!*5+0$

+=<!'-<!>4&-<!%/".!*)')8:)50/!&',<!7*,(<!(01(7(<!%1.1#!/1"1!% #67&-<#! ,%/".! -!%11/11"11.11-! 3! ,%/".! -!%1/1"1.1-#! )+5(! 5,!

)*2=<!'50<!650!*,=+35/'),!)69!',<!*5+0+=<!'-<!>4&-<!%/".! !*)')8:$

)50/!&',<!7*,(<!(01(7(<!%11/11#!"11.11% 4+)!,%/".!-!%1/1"1.1-!3!,%/".!-!%11/11"11.11-%! !."%*%

/#0!#"$#)+#&1&"#)+' ''!&3#!+$,$--"-%+.+%&$! !01#4'! ."%*!&7"%%*!)$%!#$!38"!+#4!$''#%"##%1#3'!"(!$4#0*!."$#!#!#%!$4#7!%.'$%!."$!$%%"#&3'$$$&" '%0%$'2$# 3(.+5(2(! ')!6)+)88-8(676(*)!%/".! -!%1/1"1.1!*),!CD"G! -

C1D1"1G1!2(!>4&(,<!%/".!3!CD"G!*),!')!(6/!)0';/!(A-!7&)%!3=850$

2(!/)!)65*(7?502(!9',!')!6)+)88-8(676(*)!)0'4!(7/),!,&5*(/)2)%

#'#B651='502(!)+3,*4!9',!')!6)+)88-8(676(*)!(7/),!5+14#!*-8)*:!

%%1⊥!%/".!*),!CC1!⊥!CD"G!*),#!3.+7<!>84>-!'-<!)(/,*9'-')<#!9',!

1#:,%!!!,-

!!"#!"##$$%!%#&!'!$& 69

%/!66!CD%!-&'.!/#%$!6)+)88-89)+)225!2(!'5!&-2(75!#!&'-/!6+5=*')-

&-!'-<!-2,(01(7)<!/%!'='5,5#!;&'(!∧∧

ΚΝΜ=!!" !*),!!/#!3!CD%

#9'(!')!6)+)88-89)+)22)!/#%$!*),!CD"G!(7/),!7&)!*),!925,)%!

3(.+5(2(!'5!6)+)88-8(676(*5#!/#%$-/1#1%1$1%

%+5(!∧∧ʹ′=ʹ′ #$#!"! !,3!&!5+1:-!*),!//13!CC1!,.<!(A-!'./!)+$

3,*;/! 6)+)88-8(6,6=*./-#! ')! 5+15);/,)! 6)+)88-89)+)22)! /##1/1#!

CDD1C1!1)!(7/),!7&)!*),!925,)!,&026';&,2)-%! ",)!'5/! 7*,5!89)5!')!

//1$1$! #! GG1C1C! ! (7/),!

7&)!*),!925,)%! #)! 6)+)88-8(676(*)!

85,69/! /$%#! -! /1$1%1#1!

*),!CD"G!-!C1D1"1G1!(7-),!7&)!*),!925,)#!)+5(!98(<!

5,! =*+(<! '50<!(7/),! 7&(<!*),!

925,(<! *),! ')! (676(*)! '50<!

6)+488-8)! =/)! 6+5<! =/)!!()+5(!%/!66!CD!*),! %%1!66!CC1-%

-&'.!3!'5!&-2(75!'5-2:<! './!/"#!$%!*),! '5!6)$

+)88-89)+)225!/#!3%!%/!

0!'5!&-2(75!'52:<!'-<!".!2(!

'-/!#!#!'9'(!')!6)+)88-8(-676(*)! /3!#! -! /1351#1!

*),! /$%#! -! /1$1%1#1! (7/),!7&)!,6+9')&-!('-!#!569'(!1)!

(7/),!*),! ,/3!#!-!/13151#1-!3!,CD"G!-!C1D1"1G1- (i)

%884!(7/),!,/$%#-!3!,/3!#-!,6+9')&-!7%*0-#!4+)!1)!=3502(!*),!

,/3!#-!3!,%/".-%

,2.<!,&3((,!,6+9')&-!V.7) ( )( )

( )( )ΒΓΖΤ

ΒΘΗΤ=

ΒΓΖΤ

%/". (1)

#67&-<#!)+5(!'5!(676(*5!//1"1"!(7/),!6)+488-85!&',<!>4&(,<!'50!

6)+)88-8(676(*50!%%1.1.!-!##1010!#!&(2+./)!2(!'-/!6+9')&-!(0#!1)!(7/),"

1#:,%!!,.

70 #BCD#$.!!1%#&$'#$%2

( )( )

=ΒΓΖΤ

%/". ( )( )Τʹ′Ζʹ′Γʹ′Βʹ′−ΒΓΖΤ

ʹ′ʹ′ʹ′ʹ′ ."/%-%/". (2)

#/'(8;<! 925,)#! )+5(! '5! (676(*5!//1#1#! ! (7/),! 6)+488-85!&',<!

>4&(,<!'50!6)+)88-8(676(*50!!3!!131-!"001"1#!1)!=3502("!

( )( )

=ΒΓΖΤ

ΒΘΗΤ ( )( )Τʹ′Ζʹ′Γʹ′Βʹ′−ΒΓΖΤ

Τʹ′Ηʹ′Θʹ′Βʹ′−ΒΘΗΤ (3)

%69!',<!&3=&(,<!,&-#!,(-#!,*-!6+5*(6'(,!9',! ,%/".!-!%1/1"1.1-!3!,/3!#!-!/1351#1-#!(652=/.<!1)!=3502(!*), ,%/".!-!%1/1"1.1-!3!,CD"G!-!C1D1"1G1-#!89).!'-<!,i).

1#:,%!!,.!%

3'!%/! ')! 6)+)88-8(676(*)! *(/! :')/! *),! ')! *(5! 5+14#! *-8)*:"!

%%1!*(/!(7/),!*41('-!&'5!%/".!!:!!CC1!!*(/!(7/),!*41('-!&'5!CD"G#!

'9'(! 6+5>4885/')<! 5+14! ',<! =*+(<!C1D1"1G1! *),!%1/1"1.1! ! (67! './!

CD"G#! %/".! )/'7&'5,3)#! *-2,50+)5(2(! ')! 5+14! 6)+)88-8(676(*)!

C1D1"1G1 -!C1D1"1G1!*),!%1/1"1.1 -!%1/1"1.1% ,2.<#!,C1D1"1G1 -C1D1"1G1-!3!,CD"G!-!C1D1"1G1-!!*),

,%1/1"1.1 -!%1/1"1.1-!3!,%/".!-!%1/1"1.1-!,6+9')&-!*5-!*), ,C1D1"1G1 -!CD"G-!3!,%1/1"1.1 -!%/".-!!&(2+./)!2(!'-/!6+5-)5($2(/-!6(+76'.&-!'./!5+1;/!6)+)88-8(6,6=*./%

4+)!*),!648,!1)!=3502(!

,%/".!-!%1/1"1.1-!3!,CD"G!-!C1D1"1G1-%!!!."%*%

!!"#!"##$$%!%#&!'!$& 71

/#0!#"$#)$#&1&"#)$'# )3#! +$,$--"-%+.+%&$! $%! ."$! 38"! %.'$%! $'7-#($! #9'! &7"%2'!##4*"! '%0%$'2$ 3(.+5(2(! ')!6)+)88-8(676(*)!%/".! -!%1/1"1.1!*),!CD"G! -

C1D1"1G1!2(!7&)!(A-!650!)/',&'5,35(/!&',<!>4&(,<!%/".#!CD"G%!3=$

8502(!/)!*(7?502(!9',!(7/),!)/485))!'./!>4&(./!'50<!%/".#!CD"G%!

1#:,%!!,)

3(.+5(2(!'5!6)+)88-89)+)225!"D%#!!7&5!2(!'5!%/".#!9650!'5!

%! )/:*(,! &'-/! 6+5=*')&-! '-<! -2,(01(7)<!DC! ,6+9')&-! $%.0-#! *),! &0$

268-+;/502(! '5! &3:2)!;&'(! /)! 6+5*(A(,! 6)+)88-8(676(*5!"D%#! - "1D1%1#1! 7&5! ,,&5*(/)25-! !2(! '5!%/".! -!%1/1"1.1#!&(2+./)!2(! '-/!6+9')&-!*&!,=350/!,&5*(/)2(<!>4&(,<!*),!7&)!(A--%

%+5(! '5! (676(*5!"DD1"1!(7/),!6)+488-85!&',<!>4&(,<! '50!6)$

+)88-8(6,6=*50!CGG1C1!-!%##1%1#!1)!=3502("

( )( )Τʹ′Σʹ′Λʹ′Μʹ′−ΜΛΣΤ

ʹ′ʹ′ʹ′ʹ′ G"DC-CD"G = ( )( )ΜΛΣΤ

ΚΛΜΝ!!!,6+9')&-!(0-%

%884!=3502(!,"D%#-!3!,%/".-!!*),!

,"D%#!-!"1D1%1#1-!3!,%/".!-!%1/1"1.1-%

4+)!1)!,&3((,!( )( )ΑΒΓΔ

ΚΛΜΝ = ( )

( )Δʹ′Γʹ′Βʹ′Αʹ′−ΑΒΓΔ

ʹ′ʹ′ʹ′ʹ′ G"DC-CD"G

!."%*%

72 #BCD#$.!!1%#&$'#$%2

/#0!#"$#))#&1&"#))' :!-!(#*!&3#!#$#.9'!+$,$--"-%+%+0&9'! %"#3#$%!$%!##!-!(#!#9'!)3&9'!#9'!#$!-#(9'!$)$2'!##4*"! '%0%$'2$ 3(.+5(2(!')!925,)!6)+)88-8(676(*)!V!3!%/".!-!%1/1"1.1!*),!

V1 = A1B1"1.1 - A11/11"11.11#! 9650! ,%/#!%1/1-! #! ,%.#!%1.1-! #! ,%%1#!

%1%11-!!(7/),!')!@(()-!'./!52589)./!)*2;/!'50<%

3=8502(!/)!*(7?502(!9',"!( )

( )3

3

1 11VV

ΒΑ

ΑΒ=

1#:,%!!,,

E+5(*'(7/502(!',<!-2,(01(7(<!'./!)*2;/!/%#!/"#!//1!*)'4!!/C3!

3!%1/1! #!/D!3!%1.1#!/"!3!%1!%11#!569'(!*-2,50+)5(/'),!')!6)+)88-$

8(676(*)!/C$"!-!DC1$1"1!*),!%1/1"1.1 - A11/11"11.11#!')!5657)!(7/),!

7&)!*),!925,)!,&026';&,2)-#!&(2+./)!2(!'-/!6+9')&-!(.#!)+5(!')!@(()-!

,/CC1D#!A1/1"1.1-! #!,/D"1"! #!A1!%11.11.1-! !*),! !,/C$"#!%1/1/11%11-!!

')!5657)!*(7/'),!(67!(6,6=*./!650!(7/),!)/4!*(5!6)+488-8)#!)65'(85(/'),!

)69!7&)!6)+)88-89)+)22)% 3(.+5(2(!')!6)+)88-8(676(*)!/CC1D!-!/1C1C11D1!!*),!"/CG!-

"1/1C1G1 .

%+5(!,&3((,!ʹ′ΑΑ

Αʹ′Α=

ΔΑ

ΑΔ=

Β 111111%

%/!!,6+9')&-!VI%&-!*),!

!!"#!"##$$%!%#&!'!$& 73

/C!3!%1/1#!/D!3!%1!.1#!!/"!3!%1%11#!1)!=3502(!*),

ΒΜ

Αʹ′Α=

ΒΛ

ΑΔ=

ΒΚ

"!#!*-8)*:!!

ΒΜ

Βʹ′Β=

ΒΛ

ΒΓ=

/C

/% .

%884!!!( )( )ΓΒΚΝ

ΑΒΓΔ=

ΒΚ

ΒΑ , ( )( )ΚΚʹ′ΒΛ

ΓΒΚΝ=

ΒΛ

ΒΓ , ( )( )ΒΚΡΜ

Βʹ′ΒΚΚ=

ΒΜ

Βʹ′Β 1 .

#652=/.<!1)!,&3((,!!( )( )

( )( )ΚΚʹ′ΒΛ

ΓΒΚΝ=

ΓΒΚΝ

ΑΒΓΔ = ( )( )ΒΚΡΜ

Βʹ′ΒΚΚ 1 .

#6,68=5/!=3502(!!( )( )

( )( )11ΝΚΒʹ′Γʹ′−ΓΒΚΝ

Δʹ′Γʹ′Βʹ′Αʹ′−ΑΒΓΔ=

ΓΒΚΝ

ΑΒΓΔ ,

( )( )

( )#$ %%

%%

ΚΚʹ′ΛΒʹ′−ΚΚʹ′ΒΛ

ΝΚΒʹ′Γʹ′−ΓΒΚΝ=

ΚΚʹ′ΒΛ

ΓΒΚΝ ,

( )( ) ( )ΡΡʹ′ʹ′−ΚΚʹ′ΒΛ

ΚΚʹ′ΛΒʹ′−ΚΚʹ′ΒΛ=

ΒΚΡΜ

Βʹ′ΒΚΚ

&&

#$ %%%% .

%/!92.<!'=&&(+)!2()=1-!)#!>#!)#!*!!>+7&*5/'),!&(!&0/(3:!)/)85$

)7)#!*-8)*:! ,&3((,!δ

γ=

γ

β=

>

)! #!'9'(!5!89)5<!'50!6+;'50!6+5<!'5/!'=$

')+'5!,&5('),!2(!'5!89)5!'./!*(>./!'50!6+;'50!*),!'50!*(('(+50#!*-$

8)*:! 3

3

>

)

*

)= (V#!5+,&29<!&5-%!

8E+4)2)',#!)/!δ

γ=

γ

β=

>

)3!8#!!'9'(!)!3!8>!3!82)!3!83*!!⇒

⇒ '

''

'

%

#

%

#&

$

#=⎟⎟

⎠

⎞⎜⎜⎝

⎛== ].

4+)!!( )

( )( )

( )311

3."/%-%/".

ΝΚΒʹ′Γʹ′−ΓΒΚΝ

Δʹ′Γʹ′Βʹ′Αʹ′−ΑΒΓΔ=

ΡΡʹ′Μʹ′Μ−ΚΚʹ′ΒΛ

ʹ′ʹ′ʹ′ʹ′ =

= ( )( )

( )( )3

3

3

3

ΒΚ

ΑΒ=

ΓΒΚΝ

ΑΒΓΔ!#!!569'(!

( )( )3

11

3

1

%/

VV

ΒΑ= .!!

!."%*%

74 #BCD#$.!!1%#&$'#$%2

/#0!#"$#),#&1&. 34) ''!&3#!+$,$--"-%+.+%&$!%.'$%! ."$! $%%"#&3'$$$&#!#!#%!#%!&7"%%*!##4*!%.'$%!$'#%"#,!(9*!$'7-#(%*!#9'!482'!##4*" ''!#%!&7"%%*!&3#!+$,$--"-%+%+0&9'!%.'$%!$'#%"#,!(9*!$'7-#(%*!#9'!482'!##4*#!#!#%!#$!+$,$--"-%+.+%&$!%.'$%!."$!$%%"#&3'$$$&" '%0%$'2$ 3(.+5(2(!')!6)+)88-8(676(*)!!V1!3!%/".!-!%1/1"1.1!!*),!!

V2!3!CD"G!–!C2D2"2G2 2(!(A-! !01#!02#!650!)/',&'5,35(/!&',<!>4&(,<!

%/".#!CD"G%

(i) B651='502(!)+3,*4!9',!')!6)+)88-8(676(*)!(7/),!5+14#!*-8)$

*:!%%1 ⊥!%/".!*),!CC2 ⊥!CD"G%

#'#.(392)&'(!9',!!V1 = V2!!*),!1=8502(!/)!*(7?502(!9',"

1

2

0

0

CD"G

%/".=

%/!01!3!02#!'9'("!1

2

2

1

0

01VV

CD"G

%/".===

%/!01 ≠!02#!6%3%!!01!9!02#!'9'(!1)!064+3(,!2(')?(!'./!C#!C2!&-2(75!

C1!'='5,5!;&'(!CC13!01%!,%3%!*.-%

1#:,%!!,/

3(.+5(2(!'5!6)+)88-8(676(*5!V11!3!CD"G!-!C1D1"1G1%

%+5(!!V1 = V2#!!1)!=3502(!!1

2

1

1

VV

VV

ʹ′=

ʹ′ (1)

!!"#!"##$$%!%#&!'!$& 75

#67&-<#!)+5(!01!3!CC1!1)!=3502(!( )( )#$'(

"!%&

(

(

%

% =ʹ′

(2)

#?48850!( )( ) %

))))

% )

)

##

##

((##

((##

(

() =ʹ′

=ʹ′ʹ′

=ʹ′

(3)

%69!',<!&3=&(,<!,&-#!,(-#!,*-!6+5*(6'(,!9',!( )( ) %

)

)

)

#$'(

"!%&= .

3'#.(392)&'(!9',!( )( ) %

)

)

)

#$'(

"!%&= !!*),!1=8502(!/)!*(7?502(!9',"!

V1 = V2.

%/!01!3!02#!'9'(!%/".!3!CD"G#!569'(!V1 = V2.

%/!01 ≠!02! !6%3%!01!9!02#!'9'(!1(.+;/')<!'5!6+7&2)!V11#!96.<!')!

*)')&*(04&)2(!6+5-)502=/.<#!1)!=3502(!

( )( ) Κʹ′Κ

ΚΚ= )

#$'(

"!%& (4)

%884!! ( )( ) ʹ′

=1

1

V

VCD"G

%/". (5)

)+5(!=350/!7&)!(A-!,6+9')&-!*(-%! #67&-<#!,6+9')&-!(0-

KK

KK

V

V 2

1

2

ʹ′=

ʹ′ (6)

%69!',<!&3=&(,<!,.-#!,0-#!,:-!6+5*(6'(,!9',%

)

%

%

(

(

(

(

ʹ′=

ʹ′#!!569'(!V1 = V2.

(ii)! %<! 0651=&502(! ';+)! 9',! *(/! (7/),! *),! ')! *(5! 6)+)88-8(-

676(*)!5+1)#!*-8)*:!(7'(!9',!-!)*2:! !%%1! !*(/!(7/),!*41('-!&'-!>4&-!!

%/".#!!(7'(!9',!-!)*2:!CC2!!*(/!(7/),!*41('-!!&'-!>4&-!CD"G% 3(.+5(2(! '9'(! ',<! 5+1=<! 6+5>58=<! %1/1"1.1#! C1D1"1G1! './!

%1/1"1.1!!*),!C2D2"2G2!&',<!)6=/)/',!=*+(<!)/'7&'5,3)!,%3%!*.!)-%

76 #BCD#$.!!1%#&$'#$%2

1#:,%!!,/!%

-3502(!V1!3!%1/1"1.1-!%1/1"1.1!*),!V2!3!C1D1"1G1-!C2D2"2G2 ,6+5'4&(,<!('!*),!*5-%

#'#.(392)&'(!9',!V1 = V2!*),!1=8502(!/)!*(7?502(!9',"!

( )( ) %

)

)

)

#$'(

"!%&= #!!9650!!01!3!%1%1!!*),!!02!3!C2C1%

",)! ')! 5+14! 6+7&2)')! %1/1"1.1! -! %1/1"1.1! *),! C1D1"1G1- C2D2"2G2!!&(2+./)!2(!'-/!6+5-)5(2(/-!6(+76'.&-!,(01(-#!1)!=3502("!

( )( ) 1

2

1

2

2222

1111

0

0

%%

CC

G"DC

."/%=

ʹ′

ʹ′= #!!569'(!*),!

( )( ) %

)

)

)

#$'(

"!%&= .

3'#.(392)&'(!9',"!%

)

)

)

#$'(

"!%&= !*),!1=8502(!/)!*(7?502(!9',:

V1 = V2.

",)! ')! 5+14! 6+7&2)')! %1/1"1.1-! %1/1"1.1! ! *),! C1D1"1G1-

C2D2"2G2!!1)!=3502(!( )( ) %

)

)

)

('$#

&%!"=

ʹ′ʹ′ʹ′ʹ′

ʹ′ʹ′ʹ′ʹ′#!569'(!

,%1/1"1.1-!%1/1"1.1-!3!!,C1D1"1G1-!C2D2"2G2)

,&(2+./)!2(!'-/!6+5-)5(2(/-!6(+76'.&-#!,i-!>-% 4+)!*),!!V1 = V2. !."%*%

!!"#!"##$$%!%#&!'!$& 77

5$,$#",6"%%*!"#%*!+,##7"%%*!)*!- 34 "(!'-/!6+9')&-! *.!585*8-+;1-*(!-!&0/1(',*:!6)+50&7)&-!'-<!

1(.+7)<!'50!9)*50!),)!')!6)+)88-8(676(*)%!%+37@5/')<!2(!'-!1(2(8,;-*-! 6)+)':+-&-! '-<! 6+9')&-<! ()! ),)! '-/! *,)7+(&-! 6)+)88-8(6,6=*50!

)69!=/)!*,));/,5!(676(*9!'50!*),!&0/(37@5/')<!2(!(6,3(,+:2)')!*,)2=-+,&-<! *),! &0268:+.&-<! (/9<! 6)+)88-8(6,6=*50#! *)')8:)502(! &'-/!

6+9')&-!*.#!)69!'-/!5657)!2(!&()3+5/-!5+585)7)!265+(7!/)!)69*(,31(7!

9',"!1#! 7/%!0%)%33&3#0*0#$!! "!! 5"/4,#/!!,"%%!-8!&%!6%"! "!+!.9!+%!0!+!

%/""!"!"##*!!0!%+":!#*/%"!!"%'#)4( !! *,)*,*)&7)!*),!-!&(,+4!650!)*58501:1-*(! (7/),!4A5)-#!92.<!

*(/!6+5&+=+('),!&(!06585),&25(<%!&!Legendre!!(,&-):1-*(!'-!&()3+5/-!1(.+7)!'50!9)*50#!)884@5/')<!'-!&(,+4!'-<!6)+50&7)&-<!.<!(?:<"

%*58501;/')<!'5!)(/,*9!5+,&29!'-<!,&9'-')<!&'(+(;/!&3-24'./#!

265+(7! /)!)65*(,31(7!9',!*(5!6+7&2)')! ')!5657)! =350/! ',<! =*+(<! !2,)<!

29/5/!&'(+(4<!'50<!)./7)<!7&(<!*),!5257.<!*(72(/(<!(7/),!7&)!,&026';$

&,2)-#!569'(!5+14!,:!*585>4-!6+7&2)')!2(!7&(<!>4&(,<!*),!7&)!(A-!(7/),!

7&)% %*585(1.<#!=/)!684),5!6+7&2)!(7/),!,&5*(/)25!2(!=/)!5+19!6+7$

&2)!650!=3(,!.<!>4&-!2,)!*41('-!'52:!'50!6+7&2)'5<!*),!(A5<!2,)!6)$

+488-8-!)*2:!'50% '+-&,2565,;/')<!)0'9!)*+,>;<!'5!

)65'=8(&2)! 265+5(2(! /)! )65*(7?502(! 9',!

=/)! *,));/,5! (676(*5! *,),+(7! '5! 6)+)88--8(676(*5!&(!*(5!,&5*(/)2)!6+7&2)')%!E+4-)2)',#!*41(!=/)!)69!')!6+7&2)')!650!6+5$

*(6'50/!(7/),! ,&5*(/)25!2(!=/)!'+,)./,*9!

6+7&2)! 650! =3(,!.<! (A5<! '-/!6)+468(0+-!

)*2:! '50! 6)+)88-8(6,6=*50! *),!.<! >4&-!

=/)!)69!')!,7&)-!'+7)./)#!&')!5657)!-!*,)$

);/,5<! *,),+(7! '5! 6)+)88-89)+)225! '-<!

*41('-<! '52:<%! -'&,! +'4/502(! &'-! 1(2($

8,;*-!6+9')&-!()%

/#0!#"$#)-#&1&"#)-'# ''!&3#!%+.+%&%*!(9'.%*!%.'$%!."%*!)$%!$+!!#%*!)#,4(0*!##4*!$1/#3'!&3#! %4/%.%*! %)#!*! #9'! %+%+0&9'! ##4*!+#4!"1"$$#.;#4'!$%! #%*!$-'#."##%1%*!+-%4,0*!##4*! ."%*!(9'.%*#!)%!$'!+,#&-"/#3'!&3#!#41$.$!""$%.$!#9'!%4/%%2'!+#4!&,.")#'#$%!%)#!*!#9'!%+%+0&9'!"#$!%+.+%-

1#:,%!,/!-

78 #BCD#$.!!1%#&$'#$%2

&$!#9'!$,1%)2'!(9'%2'#!#!#%!#%!%4/%.%*!+#4!#,.;#'#$%!$+!!#%*!+,#'&#-0*! #9'! ""$%.9'! $4#2'! )$%! #%*! )#,4(0*! #9'! (9'%2'! "1"-$$#.;#4'!."%*!(9'.%*!$%!#%*!%4/%.%*!#%!#+#.%*!&%0,1#'#$%!$+!!#%*!)#',4(0*!##4*!)$%!)%.'#$%!%)#!*!#9'!%+%+0&9'!##4*" '%0%$'2$

3(.+5(2(! ',<! 7&(<! )./7(<!∧∧

%%% *+,-,+* ! 64/.! &')! (676(*)! ,$-#!

,$1-!)/'7&'5,3)#!*),!',<!(01(7(<!&;#!&1z1 (*'9<!'./!(6,6=*./!)0';/#!'=$

'5,(<!;&'(! .zOyyOz,zOxxOz 111111

∧∧∧∧

== 3(.+5(2(!(6,68=5/!')!&-2(7)!

.1#!"1!64/.!&',<!&z#!&1z1!)/'7&'5,3)!*),!',<!6+5>58=<!'50<!.1#!"11!64/.!

&')!,$-#!,$,-%!3=8502(!/)!*(7?502(!9',"!∧∧

Γʹ′Γ=ʹ′ %%%+&&* .

1#:,%!!,1

E4/.!&'-/!&z!1(.+5(2(!&-2(75!"#!'='5,5!;&'(!&"!3!&1"1!*),!'-/!

""1!66!..1% %+5(!!..1⊥!,$-#!!1)!(7/),!*),!!""1!⊥!,$-% %/!!"1%!⊥!&x!*),!"11%1 ⊥!&1x1#!'9'("!,&"1-

2!3!,&%-2!4!,%"1-2!*),!

,%"-2!!3!,%"1-2!4!,""1-2!#!569'(!,&"-2!3!,&"1-2!4!,""1-2 =

3!,&%-2!4!,%"1-2!4!,""1-2!3!,&%-2!4!,%"-2%!4+)!!∧

ΟΑΓ !3!&!5+1:%!

&257.<!>+7&*502(!∧

ΓΑΟ %%% !3!&!5+1:#!!569'(!∧

ΟΑΓ = ∧

ΓΑΟ %%% .

#)!'+7)./)!85,69/!&%"#!&1%1"1!=350/!&"!3!&1"1!*),!∧

ΑΟΓ =

= ∧

ΓΟΑ %% % , ∧

ΟΑΓ = ∧

ΓΑΟ %%% !,!3&!5+1:-%!!4+)!(7/),!7&)!#!569'(!!&%!3!&1%1. #/'(8;<!925,)#!)/!"1/!⊥!&B!#!"11/1 ⊥!&1B1!#!1)!=3502("!

!!"#!"##$$%!%#&!'!$& 79

∧

ΓΟ! = ∧

ΓΒΟ %%% !3!&!5+1:!*),!&/!3!&1/1%!#)!'+7)./)!85,69/!&%/! #!

&1%1/1!(7/),! 7&)#!)+5(!&%!3!&1%1#!&/!3!&1/1! !*),!∧∧

ΒΟΑ= %%%"*! .

4+)!%/3%1/1, ∧∧

ΑΟ= 111 B&%/ !*),!∧∧

ΑΒΟ= 111&/% !569'(!*), ∧∧

ΒΑΓʹ′=ΑΒΓʹ′ %%% , ∧∧

ΑΒΓʹ′=Γʹ′ %%%!. .<!&0268-+.2)',*=<!7&./!)./,;/% #)!'+7)./)!%/"1!*),!%1/1"11!=350/!68=5/!%/!3!%1/1 ,

∧∧

ΒΑΓʹ′=ΑΒΓʹ′ %%% , ∧∧

ΑΒΓʹ′=ΒΑΓʹ′ %%% %!4+)!(7/),! 7&)#!569'(!"1%!3!"11%1, "1/!3!"11/1!,6+9')&-!$%!(:-

#)!'+7)./)!&%"1!*),!&1%1"11!=350/!&%!3!&1%1#!!"1%!3!"11%1!*),!∧∧

Γʹ′ΑΟ=ʹ′ %%%%*" !!,3!&!5+1:-%!!4+)!(7/),!7&)#!569'(!&"13!&1"1. %884!!&"13!&1"1 ⇒!,&"1-2!3!,&1"1)2 ⇒

⇒ (&")2 - (""1)2 = (&1"1)2 - ("1"11)2 ⇒ (""1)2 = ("1"11)2 ⇒ ""1= "1"11 ()+5( &" = &1"1). #)!'+7)./)!85,69/!&""1!*),!&"1"11!=350/!&"!3!&1"1, &"13!&1"11!!*),!""13!"1"11%!3)!(7/),!(652=/.<!7&)#!569'(

∧∧

Γʹ′ΟΓ=ʹ′ %%%%%* .!!!."%*% 5$,$#6,""" !!)69*(,?-!1)!265+5(&(!/)!&0/'52(0'(7!),&1-'4!2(!'-!3+:&-!'50!

1(.+:2)'5<!'./!'+,;/!*)1='./%!#?48850#!2(!')!*+,':+,)! ,&9'-')<!5+$

15)./7./! '+,);/./! 265+5(2(! /)! )65+()502(! '-/! (6)/)8)2>)/92(/-!

3+:&-!'50!E01))5+(750!1(.+:2)'5<% #=85<#!*)'4!'-/!(*+;/-&-!5,!(01(7(<!6+=6(,!/)!(*8-+15(/!.<!-$

2,(01(7(<!650!*(7/'),!6+5<!'5!7*,5!2=+5<!'50!(6,6=*50!'./!)./,;/%

/0#'"-# ''!&3#!%+.+%&%*!(9'.%*!%.'$%!."%*!)$%!$1/#3'!$+!!#%*!)#,4(0*!##4*!%)#!*!#9'!%+%+0&9'!##4*!&3#!."$!+-7(%$!#$6$$#$#!+#4!"1"$$#.;#4'!."%*!(9'.%*!$%!#%*!+-%4,0*!#9'!$,1%)2'!(9'%2'!$'#."##%1$!#!#%!#$!7),$!$4#2'!#9'!+-$(.9'!#$"$7#9'!%"$+01#4'!$+!!#$!%+.+%&$!#9'!(9'%2'"

'%0%$'2$ E+4)2)',#!&'-/!6+5-)5(2(/-!6+9')&-#!9650!(73)2(!*),!&"!3!&1"1,

*(7?)2(!9',"!""13!"1"11%

80 #BCD#$.!!1%#&$'#$%2

/#0!#"$#)!#&1&"#)!' ''!#,.$!#$6$$#$!%.'$%!$'7-#($#!&"-$&6!"4'%"##3'!"4'%16!$'$--#(.$#! #!#%! ##!+$,$--"-%+.+%&#!$%!+-%4,0*!$4#7! #$! #,.$! #$6-$$#$! %.'$%! ."#! $%%"#&3'$$#&!$%! ##! %"!+-%4,#!+$,$--"-%+.+%&#!$%!+-%4,7!#"!$0""!$'7-#(#!)$%!%"#(2'%#!+,#*!##!+,2##" '%0%$'2$

3(.+5(2(!'+7)!'2:2)')!)#!>#!)!'='5,)!;&'(!!+

%

%

#= (1)

3=8502(!/)!*(7?502(!9',!*(5! ,&5);/,)!6)+)88-8(676(*)#!)69!')!

5657)!'5!6+;'5!=3(,!68(0+=<!)#!

>#! )! *),! '5! *(('(+5! (7/),! ,&9-68(0+5!2(!68(0+4!>#!(7/),! ,&5-*(/)2)%

%<! 0651=&502(! 9',! ')! 6)-+)88-8(676(*)!%/".!-!%1/1"1.1! *),!CD"G!-!C1D1"1G1!=350/!

%/! 3!)#!%.! 3! )#!%%13!>!*),!

CD!3!!CG!3!CC13!>#!(/; ∧∧

ΛΚΝ=!"& , ∧∧

Κʹ′ΛΚ=Αʹ′ΒΑ

*),!∧∧

Κʹ′ΝΚ=Αʹ′ΔΑ . %69! '-! &3=&-! ,&-! =350$

2("!ΑΔ

ΚΝ=

ΚΛ

ΑΒ#!*-8)*:!')!(0$

1()+)22)!'2:2)')!%/#!%.!(7/),!)/',&'+9+.<!)/485))!'./!CD#!CG% %+5(! ')! 6)+)88-89)+)22)! %/".! #! CD"G! =350/! ',<! )./7(<!

∧∧

ΛΚΝΒΑΔ - ! 7&(<!*),!',<!68(0+=<!'50<!)/',&'+9+.<!)/485)(<#!1)!(7/),!

,&5*(/)2)!,,&(2>)*,*4-!!,6+9')&-!VI. 14). C),!&(2+./)!2(!'5!6+5-)5(2(/5!69+,&2)#!)+5(!')!'2:2)')!%%1#!

CC1!&3-2)'7@50/!7&(<!)./7(<!2(!',<!68(0+=<!'./!)./,;/! -∧

ΒΑΔ∧

ΛΚΝ )$/'7&'5,3)#!5,!)65&'4&(,<!%1%1!#!C1C1!'./!&-2(7./!%1#!C1!)69!')!(676(*)!

%/".!*),!CD"G!1)!(7/),!7&(<% #)!6)+)88-8(676(*)!85,69/!%"1!*),!C"1!!=350/!,&5*(/)2(<!>4-

&(,<!*),!7&)!(A-%!4+)!(7/),!,&5*(/)2)%! !."%*%

1#:,%!!!,2

!!"#!"##$$%!%#&!'!$& 81

/#0!#"$#).#&1&"#).' ''!#0""%,$!#$6$$#$!%.'$%!$'7-#($#!#!#%!)$%!#$!%+(!$4#2'!!$#%$!)$%!#$#.9*!$'$(,$(!$%'$!+$,$--"-%+.+%&$!$&"-$&6!#$!+$,$-'-"-%+.+%&$!+#4!01#4'!#$!#$6$$#$!$4#7!9*!#$!-#(%*!$)$0*&!%.''$%!$'7-#($" ''!#$!%+.!#%""7,9'!#$"$7#9'!!$#%$!)$%!#$#.9*!$'$(,$(!$%'$!+$,$--"-%+.+%&$! %.'$%!$'7-#($#! #!#%!)$%! #$! #0""%,$! #$6$$#$!%.'$%!$'7-#($" '%0%$'2$ #'!-&'.! 9',! ')! '2:2)')!)#! >#! )#! *! (7/),!)/485))#! *-8)*:!&0/,$

&'5(/! '-/! )/)85)7)!δ

γ=

%

#%! 3(.+5(2(! ')! 925,)! 6)+)88-8(676(*)!

V1,V2,V3 ,V4! !2(!52985)(<!)*2=<!',<!)#!>#!)#!*!)/'7&'5,3)%!3=8502(!/)!

*(7?502(!9',!4

3

2

1

VV

VV

= .

1#:,%!!,3

E+4)2)',#!)+5(!')!V1, V2!(7/),!925,)#!1)!=3502(!

'

%

#

)

%

(

(⎟⎟⎠

⎞⎜⎜⎝

⎛= ,6+9$

')&-!**-%!#67&-<#!)+5(!')!V3 !*),!V4!!(7/),!925,)#!1)!=3502(!

'

$

+

(

(

/

' ⎟⎠

⎞⎜⎝

⎛= .

%884!!/

'

)

%

(

(

(

(

$

+

%

#

$

+

%

#''

=⇒⎟⎠

⎞⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⇒= .

82 #BCD#$.!!1%#&$'#$%2

3'!-&'.!9',!,&3((,!4

3

2

1

VV

VV

= %!#9'(!1)!)65*(7?502(!!9',!!$

+

%

#= .

-3502(!*),!648,!',<!&3=&(,<

'

%

#

(

(

)

%

⎟⎟⎠

⎞⎜⎜⎝

⎛= ,

'

$

+

(

(

/

' ⎟⎠

⎞⎜⎝

⎛= %!&69'(!!

4

3

2

1

VV

VV

= ⇒ *

)

>

)

*

)

>

)33

=⇒⎟⎠

⎞⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛ .

!."%*%

/#0!#"$#)%#&1&"#)%' ''!#%!+-%4,0*!&4#!$+0'$'#%!%&,2'!)3&#4!&%1###$"/#3'!)$%!$+!!#%*!##$0*!&%0-/#4'! %+.+%&$#!#!#%!"!##$6!#9'! %+%+0&9'!)$%!$.$!&%$(2'%#*!##4!)3&#4!&%1###$#3'#$%" '%0%$'2$ 3(.+5(2(! ')! 2=&)! C#D#"#G!!

'./! 68(0+;/! '-<! =*+)<!%/".! (/9<!

*(>50! *),! ')! )/'7&'5,3)! 2=&)!

C1#D1#"1#G1! ! './!68(0+;/! '-<!)6=-/)/',!=*+)<!'50!%1/1"1.1%!#)!(676(*)!

'./! 6)+)88-85)+422./! C""1C1!

*),!DGG1D1! '=2/5/'),!*)'4!'-/!(0-1(7)!331%!%/!!&!!'5!&-2(75!'52:<!'-<!

331!2(!'-!*,));/,5!.1/!'50!*(>50#!

1)!*(7?502(!9',!!&3!3!&31!*),!&/!3!

3!&.1% #)! '+7)./)! 3.G#! 3/D! =$

350/! .G! 3! /D#! G3! 3! 3D! *),!∧∧

ΘΛΒ=&(' #!.<!(/'9<!(/)884?!'./!6)+)88:8./!%.#!/"#!650!'=2/5-

/'),!)69!'-/!GD!,6+9')&-!$%('-%!4+)!(7/),!7&)! #!569'(!∧∧

ΒΘΛ=('& . #652=/.<"

∧∧∧

ΝΘΒ+ΔΘΝ=&'! = ∧∧∧

=ΝΘΒ+ G3D/3D !3!(5+1=<# 569'(!')!&-2(7)!.#!3#!/!(7/),!&0/(01(,)*4%!

&257.<!')!&-2(7)!.1#!31#!/1!(7/),!&0/(01(,)*4%

1#:,%!,*

!!"#!"##$$%!%#&!'!$& 83

%884!'5!%%1!(7/),! 7&5!*),!6)+488-85!6+5<!')!//1#!..1#! !569'(!

')!//1!*),!!..1!(7/),!7&)!*),!6)+488-8)!,6+9')&-!'-!*),!)69!)0'9!6+5$

*(6'(,!9',!')!./!!*),!.1/1!(7/),!7&)!*),!6)+488-8)!,6+9')&-!$%**-% #)! '+7)./)!85,69/!&/3!*),!&.131! =350/!/3! 3!.131! ,.<!2,&4!

7&./! '2-24'./-#!∧∧

Θʹ′Δʹ′Ο=&/3 ! ,.<! (/'9<! (/)884?! './! 6)+)88:8./!

./#!.1/1!650!'=2/5/'),!)69!'-/!/.1-!*),!∧∧

Δʹ′ΟΘʹ′=ΘΟΒ ,.<!*)'4!*5$+0+:/-#!,6+9')&-!$%&0-%

4+)! (7/),! 7&)#! 569'(! =350/! *),! ',<! 488(<! 68(0+=<! '50<! 7&(<!2,)!

6+5<!27)!,6+9')&-!$%(:-#!*-8)*:!&3!3!&31!*),!&/!3!&.1%! !."%*%

/#0!#"$#)(#&1&"#)(' ''!"%!&3#!#,%(9'%)7!+,."$$#$! !$%$!+$,7+-%4,"!0&,$!##4!+,2'##4!%.'$%!&%+-7"%$!$+!!#"!&7""!##4!&%3#%,#4#!)$%!"!$+!"#$""!#"*!$+0'$'#%!$)$6*!$+!!$4#6!#"'!+$,7+-%4,"!0&,$!##4!+,2##4!%.'$%! .""!$%!##!38#*!##4!&%3#%,#4#!#!#%!#$!+,."$$#$! %.'$%! ."$!$%%"#&3'$$$&" '%0%$'2$ 3(.+5(2(!')!'+,)./,*4!6+7&2)')!%/"!-!%1/1"1!*),!CD"!-!C1D1"1#

'='5,)!;&'(!-! =*+)!/""1/1! '50!6+;'50!/)! (7/),!*,684&,)!)69! '-!>4&-!

CD"!'50!*(('(+50!*),!'5!(A5<! !CC1!'50!*(('(+50!/)!(7/),! 7&5!2(!'-/!

)69&')&-!%%1!'-<!)*2:<!%%1!)69!'-/!=*+)!/""1/1!'50!6+;'50%!3)!)$

65*(7?502(!9',!')!'+,)./,*4!6+7&2)')!(7/),!,&5*(/)2)%

1#:,%!!39

84 #BCD#$.!!1%#&$'#$%2

%0268-+;/502(! ')! 6+7&2)')! )0'4! *-2,50+);/')<! ')! *,684&,4!

'50<!6)+)88-8(676(*)!%/"%!-!%1/1"1%1!*),!CD"#!-!C1D1"1#1!)/'7$

&'5,3)% %'5!*(('(+5!6)+)88-8(676(*5!-!>4&-!CD"#!(7/),!*,684&,)!)69!

'-!>4&-!CD"!'50!)/'7&'5,350!6+7&2)'5<#!4+)!7&-!2(!'-!>4&-!/""1/1!

'50!6+;'50!6)+)88-8(6,6=*50%!#)!6)+)88-8(676(*)!85,69/!=350/! ,&5$

*(/)2(<!>4&(,<!*),!7&)!(A-#!4+)!(7/),!,&5*(/)2)!,6+9')&-!*&-% #652=/.<!*),!')!)+3,*4!6+7&2)')!1)!(7/),!,&5*(/)2)!.<!2,&4!)0$

';/!'./!6)+)88-8(6,6=*./%!!!."%*% 5$,$#6,""" !!*,)'(6.&-!'-<!6+5-)5(2(/-<!6+9')&-<!,6+9')&-!*'-!&')!1"!"%

##*%!!)65*7*('),!.<!(?:<"!"(/!$.!!0)*!,%"%!#*/%"!"!!<9:!6%"!"!!7/%!7##"!

-8!&! 0%)%33&345)%,,!! #/$! "!! 833!! ")*5(/!'! 6%"! #*/%"! "!! 0%)%33&34%

5)%,,!!$"038!"!!%04!"!!")*5(/!'!"4"#!"%!0)*!,%"%!#*/%"!*!%"( ,2.<! =/)! 6+7&2)! 2(! >4&-! 6)+)88-89)+)225! (7/),! 6)+)88-8(-

676(*5%!-'&,!+)7/('),!/)!,&30+,@92)&'(!'5!(?:<&!"(/!7/%!0%)%33&3#0*0#%

$!!,"!'5!6+;'5!6+7&2)-!7##"!-8!&!$"038!"%!%04!"&!-8!&!"!+!$#."#)!+!")"%

5(/"6!.!0)*!,%"!%!6%"!"%!$.!!0)*!,%"%!#*/%"!"!!<9:'!"4"#!'%!#*/%"!*!%"( #7/),!92.<!6+5+)/=<!9',!=/)!'='5,5!6)+)88-8(676(*5!(7/),!*,684&,5!)69!

'5!*(('(+5!'+,)./,*9!6+7&2)% ",)! '-/! )65+0):! 2,)<! '='5,)<! 6)+)/9-&-<#! (6,8=31-*(! -! 6+5-$

)5(2(/-!*,)'(6.&-%

."#450&'##"!$#-"!#$"$#!()#%#'"-.!() ."+#67#"-$#"!7'5$',%$8"#!$8#9$(#:#8#!7;#0+*7;

%',<! 6+5'4&(,<! ()! 2=3+,! *'! )/)+(+1:*)2(! &3(*9/! )65*8(,&',*4!

&(!&'(+(4!650!(7/),!6)+)88-8(676(*)!*),!-!&()*+,&-!)0';/!'./!&3-24$

'./!=),/(!3+-&,2565,;/')<!*,)*,*)&7(<!*,)2=+,&-<!:!&0268:+.&-<#!2(!

)65'=8(&2)!/)!265+5(2(!/)!2,842(!),)!,&5*(/)2)!&3:2)')% #)! ,&5*(/)2)! &3:2)')! ,68-/! './! 6)+)88-8(6,6=*./-! &'5! 3;+5!

265+(7! /)! 2-/! (7/),! 64/')! ,&5*,)3.+7&,2)%! '+-&,2565,;/')<! '-! 1(-2(8,;*-! 6)+)':+-&-! '50! Cauchy#! 9',! *41(! *0+'9! 658((*+5! 265+(7! /)!*,),+(1(7!&(!'('+4(*+)!')!5657)!=350/!2,)!*5+0+:!*5,/:!*),!5,!06985,6(<!

!!"#!"##$$%!%#&!'!$& 85

*5+0+=<!(7/),!*5+0+=<!'50!)+3,*5(! !658((*+50#!*41(!658((*+5!*,)2(+7-@('),! &(! '('+4(*+)%! #)! ,&5*(/)2)! 658((*+)! 265+5(/! 92.<! /)! *,)2(-+,&15(/!&'5/!)0'9!)+,129! '('+)=*+./!650!265+5(/! /)!)/',&'5,3-15(/#!

;&'(!/)!*-2,50+)5(/'),!@(()-!'('+)=*+./!650!/)!=350/! ,&5*(/)2(<!>4-&(,<!*),!7&)!(A-<

%'5,3(,.*;<!265+(7! /)!)65*(,31(7!9',! ),)!*(5!658((*+)#!2(!*7($

*+(<!)./7(<!=)1#%! %! %#)n>#!=>1#! %! %! %! #!>n>!#!650!(7/),! ,&5&0268-+;&,2)! ,:!

,&5*,)3.+7&,2)-!064+350/!)*=+),5,!p1, . . . ,pn , q1, . . . ,qn, p, p1!#!q, q1!!'='5,5,#!;&'(!/)!,&3((,!-!&3=&-"

p1)1 + . . . + pn)n + (p + 2p1-!6!3!q1>1 + . . . + qm!>m + (q + 2q1-!6%

%/! 85,69/! 064+350/! 658((*+)! ,&5*(/)2)! ,7&./! 9)*./-! ),)! ')!

5657)!)0':!-!&3=&-!*(/!,&3((,#!)0'4!*(/!265+(7!/)!(7/),!,&5*,)3.+7&,2)!

,:!,&5&0268-+;&,2)-%!#652=/.<#!-!1(.+7)!'50!9)*50!&'5!3;+5!6)((,!/)!

(7/),!&'5,3(,;*-<%!%0'9!)*+,>;<!)65*(,*/((,!'5!)*985015!1(;+-2)"

($,#$-##(Dehn, 1902). *'$!)$'#'%)!!#%#,7%&,#!)$%!0'$*!%"#&3'$$#*!$."#4!!()#4&!+,#*!$4#!!)3&#*!&%'!$+#,%.!'$!%.'$%!#3#%!%"#&%$19,."%$$!#3#%!%"#"4'$+-",2"%$$" 4<)!7-$#"*'#+#.)$"$#!$8#'%0%$'2$8. %/!1!(7/),!-!*7(*+-!)./7)!'50!'('+)=*+50#!)+5(!-!)./7)!'50!*($

>50!(7/),!7&-!2(!66(#!-!)/))*)7)!&0/1:*-!)+4+('),"!

∑=

=ʹ′++0

%1

222,#3)$3-31( )6q2q

26q

12

1 ii∑

=

ʹ′++⎟⎠

⎞⎜⎝

⎛

%69!'-!&3=&-!)0':!!6+5*(6'(,!9',"!r1!3!s6!,9650!r,s!6+;'5,!2(')$

?(!'50<!)*=+),5,-%!,2.<#!&0/1!3!31!*),#!96.<!265+(7!/)!)65*(,31(7#!*(/!

064+350/!)*=+),5,!!?#!s!!'='5,5,!;&'(!!r!'5?&0/! ⎟⎠

⎞⎜⎝

⎛31 = s6%

86 #BCD#$.!!1%#&$'#$%2

."$##/###&&$&$%:%$%##*#'#*#!#"*$;#"!'*.#%#73&=-#!# !! 6+9')&-! **! )65'(8(7! ),)! '5!

3;+5! '-/!)/'7&'5,3-! '-<!6+9')&-<!VI. (*#! 9650! )65*(,*/(('),! 9',! ! +!! 345!%!

"(/! #,-%$$/! $.!! !,!*(/! 0%)%33&3!-5)8,,(/!#*/%"!*!!%!,#!"!!"#")85(/!!"!+!

345!+!!,!"4"&"%%". 1#'&,#!*)'4! '-!2('4>)&-!)69! '5!

(676(*5!&'5! 3;+5! *),!)69! '5! (2>)*9/!

&'5/!9)*5#!)/)>,>4@('),!-!*(/)2-!'50!

89)50!*)'4! =/)! ,)69! *(5!&(! '+7)-%!,$

2.<#! 96.<! '5/7&'-*(! *),! &'5! (,&)).-),*9! &-2(7.2)#! -! 98-! *)'4&')&-! &'5!

3;+5!(7/),!6(+,&&9'(+5!&(/1('-%! ",)!6(+76'.&-!650!&3('7@('),!2(!'-/!6+9')&-!*.!=3(,!.<!(?:<" %'5!>,>875!VI!*),!)2=&.<!2('4!'-/!6+9')&-!(*!'71('),!*),!8(/('),!

'5!(?:<!6+9>8-2)!,6+9')&-!VI. 25). "=%!6%"%!6#+%!'#*!#0*0#$!!#+'.5)%,,!!!#:,%!4,!"!!0)!%!680!"!!

$!'7/!6%"!"!#,-%$"64!0)!%!680!"!!833!". %'50<!E01))5+(750<! )65*7*('),! -! *,)67&'.&-! 9',#! )/! *51(7! =/)!

'('+4)./5!68(0+4<!)#!'9'(!(7/),!*0/)'9/!/)!*)')&*(0)&1(7!2(!*)/9/)!

*),!*,)>:'-!=/)!/=5! '('+4)./5!*,684&,50!(2>)*5(! ,&3%! @%()-%!D5),*4#!

(6(,*:!5,!E01))9+(,5,!)&3585(/')/!*),!2(!'-!2(8='-!&'(+(;/#!(7/),!6,$

1)/9! /)! =1(&)/! '5!)/485)5! 6+9>8-2)! *),! ),)! '5/!)3&#.#5! 6+9>8-2)!)0'9!>=>),)!(7/),!(,*,*:!6(+76'.&-!'-<!)6)7'-&-<"! +=%!6%"%!6#+%!'#*!

!"#)#4! 0%)%33&3#0*0#$!! 4,!"!! 0)!%! 680!"!! $!'7/! 6%"! *!!+! 456!+! 0)!%!680!"!!833!+#!650!1)!)65'(85(&(!*),!'-!+0&,585),*:!(6=*')&-!'-<!6+9$

')&-<!VI%!(0!&'5!3;+5% !!8(&-!)0'5(!'50!6+5>8:2)'5<!*(/!(7/),!42(&-#!96.<!'50!)/'7$

&'5,350!6+5>8:2)'5<!'-<!#6,6(*52('+7)<#!*),!5,!)88(6488-8(<!6+5&64$

1(,(<!*),!)65'037(<!5*:)-&)/!&'-!*-2,50+)7)!1+(8./!),)!'-/!),'7)!650!

*-2,5(+)-&(!)0'9!'5!6+9>8-2)!),)!'5/!*(>5% +D(&(,<+!),)!'5!*,68)&,)&29!'50!*(>50!*91-*)/!)69!65885(<!)(.$

2='+(<!'-<!)+3),9'-')<#!'50!"(&)7./)#!*),!'./!G=./!3+9/./%!C)'4!3+5$

/585),*:!&(,+4!)/)+=+5/'),!8(&(,<! "!'50! $665*+4'50<#!'50!%+3(')#!'50!

#0*9?50#!'50!"(/)73250#!'50!E84'./)#!'50!#+)'5&1=/50<#!'50!G,*52:$

*50<#! '50!%6588./750#! '50!.,5*8=50<#! '50!5+./5<#! '50!E46650#! ! '50 Descartes#!'50!Longchamps!*),!488./%

1#:,%!3()%

!!"#!"##$$%!%#&!'!$& 87

&!#0'9*,5<!!,:5<!2%'%!),;/)<-#!650!06:+?(!&358,)&':<!'./!=+)./!

'50!%+3,2:*-#!)/)+=+(,!)+*('=<!)69! ',<!8(&(,<! './!)+3)7./!#88:/./!

)(.2('+;/!),)!'5!6+9>8-2)!'50!*,68)&,)&25(!'50!*(>50%!%'-/!,&'5+,*:!

)0':! 6)+50&7)&-! './! 8(&(./#! 9')/! 5!#0'9*,5<!2,84! ),)! '-! 8(&-! '50!

#+)'5&1=/50<! ,(@:-&'0!6%'%-#!)/)+=+(,!9',!5!#+)'5&1=/-<!06=>)8(!'-!

8(&-! '50!&'5/!>)&,8,4! '-<!%,)(6'50!E'58(2)75! '5/! $$$! ,(.:!6%'%-#!2(!

(6,&'58:!&'-/!5657)!6)+50&7)&(!*),!',<!*,4+5+(<!(**53=<!'-<!)=//-&-<!

'50! 6+5>8:2)'5<%!C)'4! 27)! (**53:! 5!"7/.)<! (73(! *,)'4?(,! /)! *)')$

&*(0)&1(7!),)!'5!),5!'50! +"8)(*5+!'4+5<!&3:2)'5<!*(>50%!,')/!92.<!

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

*,68)&,)&1(7%!C)'4!488-!(**53:!=/)<!3+-&29<!(6=>)8(!&'50<!*)'57*50<!

'-<!/:&50!.:850!/)!*,68)&,4&50/!=/)!*0>,*9!>.29!'50!1(5(!%6988.$

/)#!(?!50!*),!'5!6+9>8-2)!6:+(!'-/!5/52)&7)!)6-%#!5,!&-"$$. %'-!&()3+5/-!2)1-2)',*:!*,)'(6.&-#!)0'9!650!&0/*=('),!2(!'5!

+:3"!!>)4-3&,%!(7/),!-!*,)67&'.&-!9',"!2!345!%!$.!!6.-(/!#*/%"!*!!%!,#!

"!/!6.-!!"!+!345!+!$.!!%6,$/!"!+%'!650!6+5*(6'(,!42(&)!)69!'-/!6+9$

')&-!'$%**% ,8(<!5,!8(&(,<!'50!++&3*!+!0)!-3:,%"!%+!!&'5!6=+)&2)!'50!3+9$

/50!50&,)&',*4!&'-+7@5/')/!&'-/!6+;'-!,*=)!650!=*.&(!5!$665*+4'-<!5!

'75<%!%(2+./)! 2(! '5/! $665*+4'-#! '5! 6+9>8-2)! '50! 06585),&25(! '-<!

68(0+4<!*(>50!650!1)!=3(,!*,684&,5!9)*5!)69!=/)!*51=/')!*(>5!68(0$

+4<! )#! )/4)('),! &'-/! ((+(&-! *),! *)')&*(0:! *(5! 2=&./! )/)89)./! &(!

&0/(3:!)/)85)7)#!2(')?(!'./!*(5!*51=/'./!2()(1;/!!)!!*),!!()% &,!8(&(,<!6+5&)/)'587&1-*)/!4885'(!&'-/! +)+,12-',*:+!:! ! +2-$

3)/,*:+! !*)')&*(0:!'./!*(5!2=&./!)/)89)./!x!*),!A!*),!4885'(!&'-/!+)/)80',*:+#!2(!'-!3+:&-!'./!'52;/!*(5!*)26(8./%

!!8(&-!92.<!650!(/)+25/7@(,!'-/!)689'-')!2(!'-!&()3+5/-!)/)80$

',*:!)/;&-!'./!*)26(8./!&3('7@('),!2(!'-!*,68:!8(&-!650!=*.&(!5!"=$

/),325<!,2)1-':<!'50!#0*9?50!*),!'50!E84'./)-%!&!#0'9*,5<!)/)+=+(,!9',!

5!"=/),325<!(6='03(!'-!8(&-!2(!'-!>5:1(,)!*(5!*,)+5+(',*;/!*)')&*(0;/#!

650!>)&7@5/'),!&',<!*./,*=<!'52=<!,6)+)>58:!*),!06(+>58:-%! "(!&()3+5/-!5+585)7)#!-!(?7&.&-!x2!3!()3!6+5*(6'(,#!)/!)69!'-!

&0/(3:! )/)85)7)!2)y

yx

x)

== ! 1(.+:&502(! ',<! ,&9'-'(<!yx

x)

= ! *),!!

)#

*

*

,= !*),!)7/(,!)6)85,+:!'50!y!2(')?(!'./!*(5!&3=&(./!)y = x2!!*),!

()x = y2 !650!6+5*(6'50/!)69!)0'=<%!!

88 #BCD#$.!!1%#&$'#$%2

-'&,#!)69!'-!25++:!)#

*

*

,

,

#== !'50!$665*+4'-!=3502(! ,&5*(/)2)!')

)*98501) *(5!&0&':2)')"

$%#)#,*

*#,

)

)

⎭⎬⎫

=

= !*), $)#

)#,*

#*,

)

)

⎭⎬⎫

=

=

#652=/.<#!!5,!*(5!2=&5,!)/485)5,!9+5,!B!*),!A#!2(')?(!'./!2()($

1;/! !)!*),! !()#!265+5(/!/)!6+5*(A50/!.<!+5,!&0/'('))2=/(<+!'50!&-$

2(750!'52:<#!(7'(!'./!*(5!6)+)>58;/!,&(&'-2)!,&--#!(7'(!'-<!6)+)>58:<!

*),!'-<!06(+>58:<!,&(&'-2)!,(--%

1#:,%!3()!-

,6.<! =3(,! )/)80',*4! )/)+(+1(7! *),! &358,)&1(7#! -! )(.2('+,*:!

*)')&*(0:!)0';/!'./!8(&(./!2(!*)/9/)!*),!*,)>:'-!(7/),!)*(/)'-%

.

!!""##$% 2 !"!!""!XI"

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

"#!!($'#&/&0

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

+1$#0'&%2$! #%! &%'%.0! '*3+&10'3%#%! #4-! $+$(4-! 56$*-! '*-$+#%.0!42!

&782% (16%.0$!'#9-!+1:#%'9!&$!!%+$$&%.-/&#%%!:#%!$/$!./.'$%!&(-%%!%-0-

'$;$%!#4-!#&#1%;,-4-!#4-!$%%35#14-!#$*2'!(2!'93&%4<&(!:#%$!!3&!'93&(

1%-$/2!:1$*2$! #$! &3&%$:-!./.'$*! '%3&0-&#%%!42!:1%$! #92!%.$'$*<(%2!!

( !! )$!+$*!$1(=$*-!#%!&3&%$0!#4-!&;;&;1%335-4-!'#$-!./.'$!.%-$-%-.,-!+$'*;,-4-!3&!&n +'&*152'

##9!'*-56&%%!%+$$&%.-/&#%%!:#%!$/$!+*1%3($&2!*#1%;4-%.52!%16%.0!

.%%!3&#0!+$'*;4-%.52)!3&!('%!/>9!&(-%%!%-0'$;&2!#4-!&0'&,-!#$*2$!;%%!

-%! .%#%'87$*3&! '#9-! +1:#%'9! +$! %+:! :+$*! +1$./+#&%! #$! <&3&'%,$&2!

'*3+51%'3%$!'/3)4-%!3&!#$!$+$($!.0<&!#1%;4-%.:!+1('3%!$%%%1&(#%%!'&!

#1&%2!('&2!*%'$$/-%3&2)!#1%;4-%.52!+*1%3($&2'!*#'%$!$!':;$2!$/$!:3$%4-!

#1%;4-%.,-!+*1%3($4-!%-0;&#%%!'&!':;$!+%1%''9'&+%+5$4-$! ;%%!-%!%(

+$$&%6<&(!#&'%.0!:#%! %'$/#%%!3&!#$-!./&$!#$*!':;$*!#4-!$3:'$;4-!%.-3,-!#$*2!*+1:#%'9!#)'

90 +?@!+",-!,#A"")+"(-

(.$'$*<&(! 9! '/;.1%'9! .,-$*! .%%! .*'(-$1$*!3&! ('&2! &0'&%2! .%%!

('%!/>9'!(16%.0!%+$$&%.-/&#%%!:#%$!%-!$/$!.,-$%!*8!./'%-$1$%)!56$*-!(-'%!/>9$!#:#&!&(-%%!%-0'$;$%!#4-!&0'&,-!#$*2!*+1:#%'9!"")'!(.$'$/<42!

%+$$&%.-/&#%%! :#%! $/$! :3$%$%! .,-$%! *8! ./'%-$1$%)! &(-%%! %-0'$;$%! #4-!

./&4-!#4-!$%%35#14-!#4-!&0'&,-!#$*2!*+1:#%'9!"&)'! B&!%-%)$10!'#9-! %':#9#%! #4-!':;4-!.%#0! #$-!+/$$7$$!%+:$&%-

.-/&#%%!:#%!$/$!./'%-$1$%! *8!.,-$%)!3&! ('&2!&0'&%2!&(-%%!%-0'$;$%!#4-!

*>,-! #$*2$!.%#%'8;$-#%2!'#$!:#%!$/$! ('$%!./'%-$1$%! *8!.,-$%)! ! 56$*-!

/>9!%-#%'#1:)42!%-0'$;%!#4-!&0'&,-!#$*2!%''0!.%%!%-#('#1$)%!*+1:(

#%'9!".)' B&!'93&1%-$/2!:1$*2$!$!:;.$2!V!&-:2!.*'(-$1$*!.%%!$!:;.$2!!V.$!

&-:2!.,-$*!'%3&0-$-#%%!42!:1%%!#4-!%.$'$*<%,-!* n2V )!.%%!* n2Vʹ′ )$!+$*!

$1(=$-#%%! %+:! #$*2! :;.$*2! #4-! &;;&;1%335-4-! +1%'30#4-! .%%! +$*1%-3($4-!%-#%'#$(642$!3&!&0'&%2!.%-$-%.0!+$'/;4-%!3&!&n!+'&*152'

A$!.&)0'%%$!#&'&%,-&%!3&!#9-!+1:#%'9!"#$!'#9-!$+$(%!%+$$&%.-/-&#%%!:#%!$/$!')%(1&2!&(-%%!%-0'$;&2!#4-!./&4-!#4-!$%%35#14-!#$*2'!A92!

%+:$&%792!%*#82!+1$9;&(#%%!9!%1.&#0!$/'.$'9!.%%!&-$%%)51$*'%!+1:#%(

'9!"+$!'/3)4-%!3&!#9-!$+$(%$!%-!$$<$/-!$/$!$3:.&-#1&2!')%(1&2$!#:#&!

*+016&%! $*-%#:#9#%! -%! &;;10>$*3&!'#9!3&;0'9!')%(1%!+$'/&$1$!+$*!

-%!39-!56&%!.$%-0!'93&(%!3&!#9!3%.18!')%(1%'

2. )!)*+%%%,!&(,!!-".&#('(, ##$!&%&'($!)"!$%!'*;.1('&%2!+%1%''9'&+%+5$4-!.%%!+1%'30#4-!5(

;%-%-!3&!&0'9! #9!'*3+'814'9!8! #9-!$%%351%'9! #4-!'6930#4-!%*#,-$!

,'#&!-%!+1$./>$*-!('%!'#&1&0'! ##$!&%&'($!)""!+%1$*'%0=&#%%!9!%-0;.9! -%!$1%'<&(! ! #$! &3&%$:-!

#$*!./.'$*!3&!&0'9! #$! &3&%$:-!+$'*;,-$*!8!$!:;.$2!.*'(-$1$*! *.%%!

.,-$*)!3&!&0'9!#$-!:;.$!+1('3%#$2!*+*1%3($%2)'!++('92!+%1$*'%0=&#%%!

9!%-0;.9!-%!'*;.1%<$/-!+*1%3($%!.%%!+1('3%!+$*!56$*-!('&2!&0'&%2!.%%!

('%!/>9'!#&!:'&2!%*#52!#%2!+&1%+#,'&%2!.%#%)&/;$*3&!'#9-!%7%$<%/3%(

'#9!35<$$$!#$*!+/$$7$*$!#9!!!"#$!$""%&"#'()"*%$%&!# "%!%16%($%!*''9-&2! &(6%-!'*-&%$9#0!&7$&&'('&%! #9-!5--$%%! #$*!

$1($*!.%%!#$*!%+&(1$*!%+:!#%2!%+$$&%.#%.52!#$*2!$%%$%.%'(&2'!(*#:!&(6&!

42! %+$#5'&'3%! -%! .%#%)&/;$*-! '&! %7%$<%/3%'#&2! 3&-$! %''0! %1.&#0!

$/2.%3+#&2! 3&<:$$*2$! +51%-! #92! &-#*+4'%%.82! +10;3%#%! $%%$%.%'(%2!

+$*!%.$'$*<$/'%-!;%%!-%!*+$>%%'#$/-!&.!#4-!+1$#514-!#9-!#%38!!!.0-

"/@""!@("!-!B+0","#!A-#!+C(DA!-#-# 91

+$%%2$!.%#0!.%-:-%! ;&43&#1%.82$!+$':#9#%2!($! #9-!$+$(%!+1$'+%<$/-'%-!-%!*+$'$;('$*-'

!0'9!%*#82!#92!+1$'+0<&%%2!+1$2!#9-!#&'%.8!%+:$&%79!#92!%':#9(

#%2!%+$#&'&(!9!!"#$!$&""%&"#'()"*%$%&$!+$*!&%'8;%;&!$!+/$$7$2$!.%%!&(

)013$'&!3&!&7%%1&#%.8!&+%#*6(%!$!(16%38$92$!%-!.%%!&-($#&!3&!.0+$%&2!

+%1%''%;52' -!35<$$$2$!3&!'93&1%-8!+1$'5;;%'9!56&%!.%#0!&0'9!42!&782% !1('.$*3&!3%%!%/7$*'%!%.$'$*<(%!*%-)!.%%!3%%!)<(-$*'%!%.$'$*<(%!

*&-)$!!5#'%!,'#&!-%!%'6/$*-!$%!'65'&%2%!!%- ≤!(!≤!&-!$!!%- ≤!!!≤!&- , ∀!-! ∗Ν∈ .%%

&-/" –!%-/" ≤ 21!*&- –!%-), ∀!-! ∗Ν∈ (1)

##$!&%&'($!)!#4-!+"$,-#./)!%+$$&%.-/&#%%$!3&!'93&1%-$/2!:1$*2!

$%%#*+435-9$!9!&782!+1:#%'9%!

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

1$!4- ≥ !

" # , ∀!-! ∗Ν∈ , (#2'"#3 tv+1 ≤ !

$ tv, ∀-! ∗Ν∈ ), %-%$ ∀!&!0!1!$!

∃1!∈!D!%4%%#%#!,&%$!!"!#&0$$#!!t1!2!&'

B&! &0'9! %*#8! #9-! +1:#%'9$! +1$)%-,2! ;-4'#8! '#$-!+/$$7$$! 9!

'65'9!*")!3%2!$$9;&(!'#9!$%%+('#4'9!:#%%!∀!&!0!1$! ∃1∈D*$!#5#$%$!,'#&!

-%!%'6/&%%!!&1 -!%1!2!&'!!!(''0$!

%- ≤!(!≤!&-!.%%!%- ≤!!!≤!&- ∀!-! ∗Ν∈ ⇒

⇒!!%- -!&- ≤!(!-!!!≤!&- -!%- , ∀!-! ∗Ν∈ ⇒

⇒ B-A ≤!&- -!%- , ∀!-! ∗Ν∈

(-!'$%+:-!$&6#$/3&!:#%!(!≠!!$!#:#&!;%%!&!3!2

B -A !0!1$!<%!*+01(

6&%! ! 1∈D*! ! #5#$%$$! ,'#&! B-A ≤! &1 –! %1 < 2

B -A $! $+:#&! %&' <

< 2

B -A $!$9'%$8!! B-A !2!1$!+$*!&(-%%!0#$+$'!11%!<%!56$*3&!(!3!!'

92 +?@!+",-!,#A"")+"(-

$#!12&'3$'(!%2%!'2'#/.! 2#%-!'#%!&+:3&-%!%-%)&1:3%'#&!'&! !0<1$%'3%$!$%%)$10!8!':;$!

$/$!'6930#4-!#1!$!#2!!*'*3&$'%.0!!#1!/!#2!$!!!#1 −!#2, !

!

$

!

)$!3&!'/;61$-9!

$1$'$;(%! &--$$/3&! 0<1$%'3%$! $%%)$10! 8! ':;$! #4-! %-#('#$%64-! &3&%-$,-$!%-! #%!'683%#%! &(-%%! &+(+&$%$!8! #4-!%-#('#$%64-!:;.4-$!%-! &(-%%!

'#&1&0!'683%#%'! (-0'$;9!+%1%#819'9!%'6/&%!.%%!;%%!#9-!%':#9#%!!#1!3!#2$!&-,!3&!!

-#1!!<%!&--$$/3&!#$!0<1$%'3%!--:14-$!*#1/#1/!'!'!'!/#1)$!;%%!.0<&!v∈IN*. /&-%.:#&1%$!3&!'⋅#1! $!'∈IR!&--$$/3&!#$!''*#1)$!:+$*!*#1)!&(-%%!#$!&3&%(

$:-!8!$!:;.$2!#$*!%-#%'#$(6$*!'683%#$2'! ##%!+'%('%%!%*#0!9!+1:#%'9!+$*!%.$'$*<&(!$)&('&%!-%!&--$9<&(!

42!&782%! !0"*12$&" "/)"#!3%!4)"!5$"$!$./)"6$*724)/)"#22#28%!!")/)"$#"

959*$7&" ,$$5"%,"!#""$"*12$""/)""#"8%24)/)""/)"!,%!""8/)""/)"959*/)"

%7"4)!#"

435&#'(!"!%6**#!"& '$%! -1%#"! 5%'$"/!"! $""$"."114!"! &$! &$&'%(+! $*!"#! "!6'%""!%/!!%$%."",!/!!%/!!##"14%./!!%/!!&$&'/!!"(%,!$ +75%!$-( (2! <&418'$*3&$! ;%%! +%10$&%;3%$! #%! &;;&;1%335-%! :3$%%! +&-#0(

;4-%!(!/,+$!(.!./.,.+.!'#$*2!./.'$*2!*"$1)!.%%!*".$1.)!%-#('#$%6%$

+-:!%"$

"/@""!@("!-!B+0","#!A-#!+C(DA!-#-# 93

3&!$3:'$;%!=&/;9!.$1*),-!*($!(.)$!*!$!!.)$!*/$!/.)$!*,$!,.)$!*+$!+.)!.%%!

#%2!%-#('#$%6&2!$%%35#1$*2!!!B$!!.B.'!0%!%+$$&%6<&(!:#%!

( )( )2

2

MBBM

ʹ′ʹ′= ("#$%!)

("&#&$&%&!&)

()$/!∧

#"! =∧

ʹ′ʹ′ʹ′ !"# !.%%!"#

"!= "&#&

"&!& (VI$!$1%'3:2!")$!#%!#1(;4-%!

(!+$!(.!.+.!<%!&(-%%!%'$;,-%%!*+1:#%'9!VI'!4)'!!11%!∧

"!# = ∧

ʹ′ʹ′ʹ′ #!" .

(''0!∧

"!# = ∧

""# !.%%!∧

ʹ′ʹ′ʹ′ #!" = ∧

ʹ′ʹ′ʹ′ #"" $!!$%:#%!&%(-$*-!'&!('%!

#:7%'!+(-%%!$&!.%%!∧

#"" = ∧

ʹ′ʹ′ʹ′ ""# (=90$)'!A%!#1(;4-%! !'$%+:-!(!B$!

(.!.B.!&(-%%!%'$;,-%%$!$+:#&!#"

#&"&= "#

"&#&!!*+1:#%'9!VI. 4).

2342! !#"

#&"&

⎛⎝⎜

⎞⎠⎟!

= (#")

(#&"&)

!

!!.%%!

"#

"&#&

⎛⎝⎜

⎞⎠⎟!

= ("#$%!)

("&#&$&%&!&)!*+1:#%(

'9!VI'!&1)'!!11%!( )( )2

2

MBBM

ʹ′ʹ′= ("#$%!)

("&#&$&%&!&).

(-0'$;9! %+:$&%79! 3+$1&(! -%! ;(-&%! ! :#%-! #%! +$'/;4-%! 56$*-! $(

+$%$$8+$#&!%1%<3:!+'&*1,-'!!! !."'$'

435&#'(!'!%6**#!'& '$%!&$&'%#!$*!"#!"!6'%"%#!%/!!%$%."",!/!!%/!!##"14%./!!%%(+$! +75%!$-(! 0&41$/3&!#$*2!./.'$*2!@$!@.!3&!%.#(-&2!1$!1.!.%%!.5-#1%!O, O.!

%-#('#$%6%'

(1.&(!-%!$&(7$*3&!:#%!#

#&=

!

$&

$⎟⎟⎠

⎞⎜⎜⎝

⎛ .

*'#4!#

#&≠

$

$&

⎛

⎝⎜

⎞

⎠⎟

!

'!!A:#&!$

$&

⎛

⎝⎜

⎞

⎠⎟

!

= #

*!!:+$*!!S ≠ K.'

94 +?@!+",-!,#A"")+"(-

+-:!%"%

#&!*'#4!%16%.0!S < K.'

(-!<&418'$*3&!#$-!%1%<3:$!&!3!@.− S!0!1!#:#&$!'/3)4-%!3&!#$!/&-%.:!!833%!+$*!%-%)51&#%%!'#$!#5'$2!#92!%+:$&%792$!<%!*+016&%!+$(

'/;4-$!+.-!!&;;&;1%335-$!'#$-!./.'$!@.!#5#$%$$!,'#&$!@.–!+.-!2!&'!11%!

@.–!+.-!2!@.– S$!!$+:#&!

+.- > S (1) ,

(-!+-! ! &(-%%! #$! %-#('#$%6$! .%-$-%.:! +$'/;4-$! #$! &;;&;1%335-$!

'#$-!./.'$!@$! #:#&$!':;4! #92!+1:#%'92! "$!<%! 56$*3&!

!

%

%ʹ′= $

$&

⎛

⎝⎜

⎞

⎠⎟

!

= #

*.

2342!%'6/&%!*

!%ʹ′= #

!%

!0!"$!!%)$/!@!0!+-'!11%!<%!56$*3&!!!

S > +.- (2)

"%!'65'&%2!*")$!*&)!$$9;$/-!'&!0#$+$$!$9'%$8!$&-!3+$1&(!-%!%'6/&%!S < K.'

8&!*'#4!#,1%!:#%!S > K.'!!A:#&!%+:!#9!'65'9!$

$&

⎛

⎝⎜

⎞

⎠⎟

!

= #*!!+%(1(

-$*3&!

2

1

1.⎟⎟⎠

⎞⎜⎜⎝

⎛ = *#

'!(-! %'6/&%!*

#= #&

&$! !#:#&!<%!56$*3&!.%%!

#

&= *

#ʹ′>1,

%)$/!S!0!@.'!11%!@!0!A!*&'5+&!.%%!#$!'833%!+$*!%.$'$*<&()'!++$35-42!

$

$&

⎛

⎝⎜

⎞

⎠⎟

!

=A

@.!!3&!A!2!@'

"/@""!@("!-!B+0","#!A-#!+C(DA!-#-# 95

A$!+1:&'93%!'$%+:-!%-0;&#%%!'#9-!+1$9;$/3&-9!+&1(+#4'9!*%)!

.%%!3&!#$*2! ($%$*2!:+42!+1%-!'*''$;%'3$/2!$$9;&(!'#%2!'65'&%2! !+-!0!A!

.%%!+-!2!A$!+$*!%-#%)0'.$*-'!++$35-42!$&-!3+$1&(!-%!%'6/&%!S!0!@.'!

A&'%.0$!%'6/&%!!S!3!@.$!$+:#&! ##&

= $

$&

⎛

⎝⎜

⎞

⎠⎟

!

.!!! !."'$'

90))#! !!! 4!"!0/.*%!S! ! $*!"#!1$""'$%$.%! $!-+!&$&'%(!7#!%-%$!%!'-"%+!%%(!S!5.%+!4!"!!6''%!&$&'%!7(!$*!"#!*&%+!1$!%%!'-"%!%%(!7!5.%+!4!"!0/.*%!8!1#&.-%$.%!%%(!&$&'%(!7($!'2'"#3#!"!! !S > K! !&"#!!

!

"

ʹ′=!

"#!%-%$!8!%!7($

E10;3%#%$!%+:!#9-!*

+ʹ′= #

&!+1$./+#&%!

ʹ′#

&= *

+!0!"$!%)$/!S > K.

11% T < K.. 9"."%3.2&2$ B&!3%%! %'$$/-%39!$%%#/+4'9!#$!'833%!%*#:!56&%!

42!&782%! !&)"S > K&""1"#"%89#.")%"!#.'$7!#"1","76(8-#,"-/8.$";"'"<'"""""$,$&"

4$"#""!

"

ʹ′=!

" ".

E10;3%#%$!56$*3&!*

+!0!"!!.%%!!

*

+!3!'!3!

'

@

@ʹ′ = ʹ′#

&$!:+$*!A!3!

ʹ′#

'.

11%!ʹ′#

&0"$! $+:#&! A%! 2!@.! .%%! %)$/!

*

+= ʹ′#

&$! <%! 56$*3&! .%%!

*

+ʹ′= #

&.

@%#0!#9-!%+:$&%79!#92!+1:#%'92!5!&%'%.:!1:'$!5+%%7&!9!%.:'$*-

<9!<&3&'%,$92!+%1%#819'9$!+$*!<%!%-%)51&#%%!'#%!&+:3&-%!42!)$!#&-!:311"$

96 +?@!+",-!,#A"")+"(-

,!.$'5!90))#(! !!!#%)$*!".#)1-+!$!&!'!&"#!&$&'%+!7#!%-%$!(56.0$#!$""$"."114!%!&"!%!#&-!!-"/!%!*!!#!,&%$!2!##"+%.6!%%(!"5-!%%!!&$&'%!!"!$*!"#!1#&.-%$.2!%%(!$#!#2'"#3!7!-!*!!%!$$ +75%!$-( 0&41$/3&!#$!&;;&;1%335-$!#
4-$!(!/,!.%%!#$!+&1%;&;1%3-

35-$!#
4-$!+3-0!'#$-!./.'$!@'!E1$)%-,2!#$!(!/,!&(-%%!('$!3&!

#$!3%':!#$*!+3-0$!01%!3&;%'/#&1$!%+:!#

!! '!()%%1,-#%2!'$%+:-!%+:!

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

3&;%'/#&19!%+:!#$!#

!.

(-!<&418'$*3&!#$!&;;&;1%3-5-$!.%-$-%.:!$.#0;4-$ (B!D/F,#(!.%%!#$!%)%%15'$*3&!

%+:!#$-!./.'$$!#:#&!%)%%1$/3&!$*-'%%'#%.0!%+:!#$!+1$9;$/3&-$!*+:-'$%+$$! *1 = K - E4$! #%! #1(;4-%!

(B!$!!D/$!/F,$!,#('! 2342$!'693%#(=$-#%2! #$! +%(

1%''9':;1%33$! ((.!.!! 3&! #9-!

+'&*10!(.!.! &)%+#:3&-9! ! #$*!./(

.'$*! '#$! B$! +%1%#91$/3&! :#%! #$!

#1(;4-$! &(-%%! ('$! 3&! #$! 3%':! #$*!

+%1%''9'$;1033$*!(!!.(.$! .%#0!

'*-5+&%%!3&;%'/#&1$!%+:!#$!.*.'%(

.:!#383%!(B!('! ()%%1,-#%2! '$%+:-! #%! #1(;4-%!B(!$!D!/$! F/,$! #,(! %+:! #$!

*+:'$%+$$!*1!$!%)%%1$/3&!+$':#9#%!3&;%'/#&19!%+:!#$!3%':!#$*!!*1!!.%%!

&1('.$*3&!*+:'$%+$! !*2!3!@!-!+8! '!-!$%%$%.%'(%!%*#8$!'/3)4-%!3&!#9-!+1:#%'9!X'"$!<%!3%2!$$9;8'&%!'&!*+:'$%+$! !*3!3!@! -!+2

3/"! !3%.1:#&1$!

%+:!#$-!%1%<3:!!&'

+-:!%""%#$

"/@""!@("!-!B+0","#!A-#!+C(DA!-#-# 97

)#!1:5#$#!'&%!+!;3()#!'2&'3$'(,!'2'#/.# ##%!+"$,-#.%!#$*!+*.'&($9!.%%!'#9-!+1:#%'9!&!+$*!+1$9;8<9.&$!

;%%! -%! %+$$&%6<&(! 9! '65'9!

2

1

1

KK

⎟⎟⎠

⎞⎜⎜⎝

⎛ʹ′

=ʹ′

$! %+$$&(6<9.&! :#%! $%! '65'&%2!

2

1

1

@

@⎟⎟⎠

⎞⎜⎜⎝

⎛ʹ′

<ʹ′

!!.%%!

2

1

1

@

@⎟⎟⎠

⎞⎜⎜⎝

⎛ʹ′

>ʹ′

!$$9;$/-!'&!0#$+$'

-!+$1&(%!:342! #92!%+:$&%792!+$*!%.$'$*<&(#%%!'#9-!+&1(+#4'9!!

$

$

#

#⎟⎟⎠

⎞⎜⎜⎝

⎛ʹ′

<ʹ′

! !.%%!&%'(=&#%%!'#$!!833%$!$&-!3+$1&(!-%!&)%13$'#&(!'#9-!

+&1(+#4'9!

!

$

$

#

#⎟⎟⎠

⎞⎜⎜⎝

⎛ʹ′

>ʹ′

.

/%%!#$!':;$!%*#:$!9!'65'9!

!

$

$

#

#⎟⎟⎠

⎞⎜⎜⎝

⎛ʹ′

>ʹ′

!;10)&#%%!42!

!

$

$

#

#

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ ʹ′<ʹ′

.%%!#:#&!&)%13:=&#%%!9!$%%$%.%'(%!+$*!%.$'$*<8<9.&!+0-4!'#$!./.'$!

@!%-#(!#$*!@.'!

-!&/'6939!%*#8!%+$)*;8!#92!&*<&(%2!%+:$&%792!;%%!#9-!+&1(+#4(

'9!

!

$

$

#

#⎟⎟⎠

⎞⎜⎜⎝

⎛ʹ′

>ʹ′

!+1$.0'&'&!$%0)$1%!'6:'%%$!%-%;.0=$-#%2!#$-!6789:!-%!

+1$#07&%!5-%!%.:3%!'833%$!+1$.&%35-$*!-%!&+%#/6&%!&*<&(%!%+:$&%79'!A$!

'833%! +$*! '&(+&%! ;(-&#%%! 03&'%! )%-&1:$! %-! %-%;-,'$*3&! +1$'&.#%.0!

#9-!+1$9;$/3&-9!+&1(+#4'9!

!

$

$

#

#⎟⎟⎠

⎞⎜⎜⎝

⎛ʹ′

<ʹ′

.

+.&(! <5'%3&!*

#

$

$!

=⎟⎟⎠

⎞⎜⎜⎝

⎛ʹ′

$! $+:#&[email protected]! S.! .%%! $%%+%'#,'%3&! :#%!

*+016&%!!-!∈!D!!,'#&!

@[email protected] S. (1)

$9'%$8!+-.0!S.'!

98 +?@!+",-!,#A"")+"(-

A,1%!56$*3&!

!

$

$

#

#⎟⎟⎠

⎞⎜⎜⎝

⎛ʹ′

>ʹ′

$!$+:#&!%-!<5'$*3&!*

#

$

$!

=⎟⎟⎠

⎞⎜⎜⎝

⎛ʹ′

$!<%!&((

-%%!S!0!@.'!/%%!-%!%+%'&%)<&(!:342!#$!@.!%1.&(!-%!&1&<&(!'65'9!%-0'$(;9!#92!*")$!+$*!<%!56&%!)*'%.0!#9!3$1)8!)!-!@.!2!S -!@.'!+(-%%!'$%+:-!&/'$;$$! #$!)! -%!+%1%'#0-&%! #$! &3&%$:-!.%-$-%.$/!+$'*;,-$*!+&1%;&(

;1%335-$*!'#$-!./.'$!@.' (-!'*3&$'('$*3&!3&!;- , E-.!#%!!&3&%$0!#4-!.%-$-%.,-!+$'*;,-4-!

+$*!&(-%%!+&1%;&;1%335-%!'#$*2!./.'$*2!@$!@.$!#:#&!#$!'833%!'#$!$+$($!<%!

.%#%)/;$*3&!.%%!#$!$+$($!+10;3%#%!5.%-&!$!Heath$!&(-%%!#$!&782%

!(,%"9(##"#"(")"76(8-#,")∈="4$"#"*) -"<"'"#&"+%)".$"$,-%"*)'-"<''"#,!#"

-!%+:$&%78!#$*!:342!&(-%%!+%$!$/'.$'9!%+:!#$!+1$9;$/3&-$!!833%'

+-:!%"%#%

(16%.0!+%1%#91$/3&!.%%!+0'%!:#%!21 E4 = E8 < K$!$+:#&!E4 –!@!2! 2

1 E4.

##9!'*-56&%%!56$*3&!;8 -!@!3!;4 -!@!- (E4 - E8) (1.&(!'$%+:-!-%!$&(7$*3&!:#%%!

E4 – E8 > 2

K+ 4 − (2)

E1$)%-,2 E4 - K = 4Ex , E8 - K = 8Ey .%% E4 - E8 = 4 ((!/) *&'5+&!'683%!&'&)'!(1.&(!'$%+:-!%-#(!#92!*&)!-%!$&(7$*3&!:#%%

<!*(!/)!0!&!+x!$!!8!!&!*(!/)!0!+x , 8!&!*(!/)!0!*(!/)!/!&+y

"/@""!@("!-!B+0","#!A-#!+C(DA!-#-# 99

8!*(!/)!0!&!+y$!!!8!!&*(B/)!0!&!+y$!!8!!*(B/)!0!+y.

E10;3%#%$!+1$)%-,2!56$*3&!/,!3!/B!2!(/!.%%!&+&%$8!#%!#1(;4(

-%!(B/$!,B/!56$*-!.$%-:!/>$2!%+:!#$!B!$!<%!%'6/&%! ( )( )

$""$

>ΔΓΑΓ

=ΔΜΓ

, $+:#&!*(B/)!0!*,B/)!0!+y.

+-#&',2!%-0'$;%$!56$*3&!!;16 -!@!3!;8 -!@!- (E8 - E16)!!.%%!#%2!'65'&%2%!

E8 - E16 > 2

KE8 −!!.''+'!-!%'6/2!#$*!!833%#$2!&(-%%!+'5$-!)%-&18'

-! %-#%3&#,+%'9! #92! +&1(+#4'92!

!

$

$

#

#⎟⎟⎠

⎞⎜⎜⎝

⎛ʹ′

>ʹ′

$! 641(2! %-%;4;8!

'#9-!+1$9;$/3&-9!+&1(+#4'9!

2

1

1

@

@⎟⎟⎠

⎞⎜⎜⎝

⎛ʹ′

<ʹ′

$!!3+$1&(!-%!;(-&%!42!&782%

*'#4!

!

$

$

#

#⎟⎟⎠

⎞⎜⎜⎝

⎛ʹ′

>ʹ′

= SK

'!A:#&!S > K.$!$+:#&!S - K.!0!1!'

/%%!&!3!S -!@.!!<%!*+016&%$!'/3)4-%!3&!#$!'833%$!!-!∈ N!#5#$%$!

,'#&%[email protected]!S - K.'!11%

E-. < S (i).

2342$!%)$/!!@!2!;-, 1+

@

+

SS@

1

1

+

+

--

2

-

- <=ʹ′

⇒=⎟⎟⎠

⎞⎜⎜⎝

⎛ʹ′

>ʹ′

,

++$35-42$!! S < E-. (ii).

"%!'65'&%2!*i) , (ii)!$$9;$/-!'&!0#$+$'

B%%!0''9!%+:$&%79!#92!+1:#%'92!&!5;%-&!%+:!#$-!Legendre'!-!&%('%.8! %$5%! #92!%+:$&%792!%*#82! *+016&%!'#%!+"$,-#.%$!&%&'($!XII$!.%#0 #9-!%+:$&%79!#$*!%-#('#$%6$*!<&4183%#$2!;%%!')%(1&2! *$9'%$8!#$*!:#%!

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

'$%+:-!%+$$&%.-/&#%%!#$!&782% !&)" !$#$5)" !5$"$!19#)"8#&"$)%.8#&&" "1"#" 76(8-#,"6$*5#!8$" #22#-

28%!!")$" $"%)" #'/"#8,9:" $)%.8%&" "$7" $6$.$7" $," "!8#&" !#)" "-$7)"

9$,)("$%!#.%"!#""%)"#$/"#8,9:"$)%.8%!# "!+*.'&($92!+%<%-,2!.%#5)*;&!'=!%*#8!#9-!&+%'$;8$!&+&%$8!%.1%-

&,2! 9! &)%13$;8! %-#('#$%64-! $%%$%.%'%,-! 3&! %*#52! #92! %+:$&%792! #92!

100 +?@!+",-!,#A"")+"(-

+1:#%'92!&$!<%!%+%%#$/'&!&;;1%)8! *'&!')%(1%)!.%-$-%.,-!+$'*5$14-'!

2342$!$&-!*+016&%!$*-%#:#9#%!&;;1%)82!%+&(14-!.%-$-%.,-!+$'*5$14-!

'&!')%(1%$!%)$/!#%!.%-$-%.0!+$'/&$1%!&(-%%!3:-$!+5-#&!*#&#10&$1$$!&-70&$1$$!$.#0&$1$$!$4$&.0&$1$!.%%!&%.$'0&$1$)'

"!Legendre!'$%+:-!%+5$&%7&!:#%!;%%!$/$!$3:.&-#1$*2!./.'$*2! %('6/&%!#$!&782%

!>6(8-#,"9%)$),91"6$*52/)$" #22#28%!!")$"$"$"!#2(*$"959*$"6$7"

38.$9#"%,"#9"1&""$7"!,98$5&"9%#4&"9%,"9%)$),91"6$*52/)$"6#8,2#-

28%!!")$"$"$"!,981"959*$"6$7"38.$9#"%,"#)"1&""$7"!#2(*$7!# -!.%-$-%.:#9#%!&5&%%%!#4-!+$'*;,-4-!&(-%%!&3)%-52!:#%!$&-!&((

-%%!%+%1%(#9#9'!(1.&(!-%!<&418'$*3&!%-#('#$%6%!3&!%*#0!:3$%%!+$'/(

;4-%! '#$! $&/#&1$! ./.'$! *@.)! .%#0! #$-! %+$.'&%'3:! #92! +&1(+#4'92!!

$

$

#

#⎟⎟⎠

⎞⎜⎜⎝

⎛ʹ′

≠ʹ′

.

-!%+:$&%79!#$*!Legendre!56&%!+'5$-!42!&782%!*'#4

KK

1

1

@

@2

ʹ′

ʹ′ʹ′=⎟⎟

⎠

⎞⎜⎜⎝

⎛ʹ′

>ʹ′

*5)!$!$+:#&!@!0!@..!'

+$,$!'&!%-#(<&'9!3&! #$-!+*.'&($9!$!Legndre!$56&#%%!:#%! #$!@..!!&(-%%!&3&%$:-!./.'$*$!&-,!$!+*.'&($92!#$!<&41&(!#*6%($!&3&%$:-!!S.

A$!%-!3+$1&(!-%!.%#%'.&*%'#&(!./.'$2!@..!$3:.&-#1$2!#$*!@!3&!

%.#(-%!!1..!2!1!!+$*!-%!%.%-$+$%&(!#9-!'65'9!*5)$!$&-!&(-%%!.%<:'$*!+1$(

)%-52'

+-:!%"%#.

"/@""!@("!-!B+0","#!A-#!+C(DA!-#-# 101

(-! $&6#$/3&! '$%+:-! #9-! /+%179! &-:2! #5#$%$*! ./.'$*!@..! &-#:2!

#$*!@$!3+$1$/3&!-%!<&418'$*3&!.%-$-%.:!+$'/;4-$!(1!(2!'!'!'!(-!!&;;&-;1%335-$!'#$-!@! &.#:2! #$*!@..! .%%! #$!:3$%:! #$*!.%-$-%.:!+$'/;4-$!!

(1.!(2.'!'!'!(-.!&;;&;1%335-$!'#$-!./.'$!@.'

+-:!%"%#/

(-!!+-$!+-.!#%!%-#('#$%6%!&3&%$0!#$*2$!<%!56$*3&%!

+

+

$

$

!

,!

%

%

ʹ′

ʹ′ʹ′=⎟⎟

⎠

⎞⎜⎜⎝

⎛ʹ′

=ʹ′

(4)

(''0!+-.2!@.!Κʹ′

>ʹ′

⇒1

+

1-

!.%%!%)$/!;-!0!@..$!<%!56$*3&%!

+

+, %

ʹ′

ʹ′ʹ′>

Εʹ′ν (5).

"%!'65'&%2!*<)$!*.)!$$9;$/-!'&!0#$+$'

*'#4!#,1%!+

+

$

$

#

#!

ʹ′

ʹ′ʹ′=⎟⎟

⎠

⎞⎜⎜⎝

⎛ʹ′

<ʹ′

'!A:#&!@..!0!@'

0&41$/3&!'$%+:-!./.'$!@..!$3:.&-#1$!#$*!@!3&!%.#(-%!1..'! A:#&!1..0!1'

0&41$/3&!.%-$-%.:!+$'/;4-$!+-!3!(1!(2!'!'!'!!(-!+&1%;&;1%335-$!

'#$-!@..!.%%!&.#:2!#$*!@!.%%!#$!:3$%:!#$*!.%-$-%.:!+$'/;4-$!+-.!3!

3!(1.!(2.'!'!'!(-.!!+&1%;&;1%335-$!'#$-!./.'$!@.'!

102 +?@!+",-!,#A"")+"(-

+-:!%"%#0

A:#&!<%!56$*3&!!+

+

$

$

!

!!

%

%

ʹ′

ʹ′ʹ′=⎟⎟

⎠

⎞⎜⎜⎝

⎛ʹ′

= (6)

(''0!!;.-!0!@!Κʹ′

<Εʹ′

⇒ν

11!.%%!;-!2!@..$!!$+:#&!!

+

+%

ʹ′

ʹ′ʹ′<

Εʹ′Ε

ν

(7).

"%!'65'&%2!*4)$!*+)!$$9;$/-!'&!0#$+$'!

A&'%.0!'$%+:-!<%!56$*3&!

!

$

$

#

#⎟⎟⎠

⎞⎜⎜⎝

⎛ʹ′

=ʹ′

.

*#!<3$&/.$'*,!7=3#)>%!,!'#$!73>')#&#

435&#'(!$!%6**#!$&! 76)$!%.#"/!#&3!5(."1*#"!##"#.$*%"#!&$!#$%!5(."1*#$+! *&$+!&"#!-1%#$+!5.%+!%2!!".0#&3!&"#!&$!#(%! *&"! ("#&%#$!"1")!5.*&1"%"$!8%!6).%#&1"!%/!!#$%!"(%,!!5.#&16%/!!(5$.;"*!$#!%%!1#&-!%2+!5(."1*#"+$

"/@""!@("!-!B+0","#!A-#!+C(DA!-#-# 103

+75%!$-( 0&41$/3&! #%! 35'%!

+$!3$!-$!0$!@$!!!#4-!%.3,-!

(!$! !/$! /($! ,($! ,!$! ,/!

#92!+*1%3($%2!,'!(!/' E1$)%-,2$!+0!>>!,!!

.%%! 0@! >>! (!! *+1:#%'9!VI'&)'!A$!0+!@! &(-%%! '$%-+:-! +%1%''9':;1%33$$! $-+:#&! *+1:#%'9! "'! 5<)!0@!3!

3!+!!3!+('!+(-%%!:342

0(!3!0,!.%%!∧

'"!=∧

%'# *+1:#%'9! "'&?)$! $+:#&! #%!

#1(;4-%! (+0$! 0@,! &(-%%!

('%'!/%%!#$-! ($%$!':;$!&(-%%!

('%!.%%!#%!#1(;4-%!(0-$!0,!'! !"%!;4-(&2!∧

('! , ∧

(%# $! !56$*-!+'&*152!

+%10''9'&2! 01%! &(-%%! ('&2! *+1:#%'9! XI'"1)'! +(-%%! :342! .%%! ,!! 3!0-$! ,@!3!0+$!$+:#&!#%!#1(;4-%!-0+$!!,@!&(-%%!('%!*+1:#%'9!"'!<)

/%%!#$-! ($%$!':;$!<%!&(-%%! ('%!.%%!#%!#1(;4-%!(-+$!0!@'!-!+*(

1%3($%! '$%+:-! 0'(-+! &(-%%! ('9! .%%! :3$%%! 3&! #9-! +*1%3($%! ,'0!@!

*+1:#%'9! )"'"1)'! "%! '#&1&52! ;4-(&2! #$*2! &(-%%! +1$)%-,2! ('&2$! %)$/!

'693%#(=$-#%%!%+:!+%10''9'%!&+(+&$%' ()$/!0@!>>!(!$!#%!#1(;4-%!,(!$!,0@!&(-%%!%'$;,-%%!*+1:#%'9!

"'!&?)$!01%!&(-%%!:3$%%!*VI$!!$1%'3:2!")' /%%!#$-!($%$!':;$!&(-%%!:3$%%!#%!#1(;4-%!,!/$!,!@$!.%<,2!.%%!#%!

,(/$!,0!'

++('92!$%!;4-(&2!∧

#"$ , ∧

+'( !!&(-%%!('&2$!%)$/!56$*-!+'&*152!+%(

10''9'&2'

+(-%%!:342!.%%!"#

"$= !'#

!'(= '#

'($!$+:#&!#%!#1(;4-%!(!/$!0@!!

&(-%%!:3$%%!!*+1:#%'9!VI. 6). -!+*1%3($%!'$%+:-!,'!(!/!&(-%%!:3$%%!+1$2!#9-!,'0!@$!01%!.%%!

+1$2! #9-!0'(-+! '!"%!'#&1&52! ;4-(&2! #$*2! &(-%%! +1$)%-,2! ('&2$! %)$/!

'693%#(=$-#%%!+%10''9'%!&+(+&$%' ()$/! !!3!3!3/$!#$!+%1%''9':;1%33$!+!3-!&(-%%!$%+'0'%$!%+:!

#$!#1(;4-$!-3/!*+1:#%'9!"'!<")

+-:!%".

104 +?@!+",-!,#A"")+"(-

(-!<&418'$*3&!.%%!#$!+%1%''9':;1%33$!0@!A$!#:#&!#$!+1('3%!

0-+'@3!!&(-%%!#$!3%':!#$*!+1('3%#$2!-+0A'3!@!!3!+!3-'0@!A$!

01%!('$!3&!#$!+1('3%!-/3!'!0@!!*+1:#%'9!XI'!5?)'!E1$)%-,2$!#$!+1(-'3%!0-+!'!@3!!&(-%%!3&;%'/#&1$!%+:!#9-!+*1%3($%!@'!+!3$!+$*!&(-%%!

('9!3&!#9-!+*1%3($%!0'!(-+$!%)$/!56$*-!#%2!5$1&2!#$*2! ('&2!3%%!+1$2!

3%%$!.%<,2!.%%!#%2!'#&1&52!;4-(&2!#$*2' .+#'%! '$%+:-! 9! +*1%3($%! ,'!(!/! $%%%15<9.&! '&! $/$! +*1%3($&2!

,'0!@$!0'(-+!('&2!.%%!:3$%&2!+1$2!#9-!%16%.8!.%%!'#%!$/$!%'$$/-%3%!

+1('3%#%!0-+'@3!$!-/3'0!@'! ()$/!.0<&!5-%!%+:!%*#0!#%!+1('3%#%!&(-%%!3&;%'/#&1$!%+:!.0<&!

+*1%3($%$! #%! $/$! +1('3%#%! 3%=(! *+&1&%(-$*-! #%2! $/$! +*1%3($&2$! 01%!

*+&1&%(-$*-!#$!3%':!#92!%16%.82!+*1%3($%2'!!!."'$'

90))#! !!!)$/.3&%(1$!&"#!#$$%$.2!5(."1*#"! #&%<=3! %2+!'$!,)#!%2!!'($!(,()(#!&"#!)$/.3&%(1$!%"!"!%*&%%#0"!14&"!%/!!"&1,!!%2+#!%-%$!%"!5.*&1"%"!7:-!-!.)/!&"#!7(:(-(-!.()(/(!$*!"#!"!6'%""!%/!!;6&$/!!!,)!&"#!!(,()($! +75%!$-(! ()$/!#$!&+(+&$$!@!0!$%-

6$#$3&(!#9-!,(!.%%!&(-%%!+%10'-'9'$!'#9!&0'9!(!/$!<%!$%6$#$(

3&(!.%%!#$!/>$2!,,1!#92!+*1%3((

$%2!,'(!/'!"3$(42$!#$!@.!.0.!

<%!$%6$#$3&(! #$!/>$2!,.,.1! #92!

+*1%3($%2!,.'(.!./.' A%! +1('3%#%! '$%+:-!

@!0-3/-!!.%%!@.!.0.-!3./.-.!<%! &(-%%! %'$G>8$! .%<,2! .%%! #%!

+%1%''9'&+(+&$%!@!A0-!3-+!!.%%! ! ! !@.!.A.0.-!!.3.-.+.'!A%!+%1%''9'&+(+&$%! '$%+:-! %*#0!

<%! &(-%%! %-0'$;%! #4-! &0'&4-!

@!A0!.%%!@.!.A.0.$!+$*!&(-%%!$%+'0'%&2! #4-!@!0!.%%!@.!.0.!%-#((

'#$%6%'!(''0!.%%!#%!+%1%''9'&+(+&$%!%*#0!&(-%%!$%+'0'%%!#4-!+1%'30(

#4-!@!0!-!3/-!.%%!@.!.0.-!3./.-.!%-#('#$%6%'

+-:!%"/#$

"/@""!@("!-!B+0","#!A-#!+C(DA!-#-# 105

11%!%'6/&%!!( )

( )( )

( )0!@2@!02

-/30!@23/--@!02

ʹ′ʹ′ʹ′=

ʹ′ʹ′ʹ′−ʹ′ʹ′ʹ′!$!!$+:#&!56$*3&

( )

( )( )

( )( )

( )=

ʹ′ʹ′ʹ′=

ʹ′ʹ′ʹ′=

ʹ′ʹ′ʹ′−ʹ′ʹ′ʹ′ -/3

3/-

0!@

@!0

-/30!@

3/--@!0

= ( )

( )

( )( )/!(

(!/

/!(41

(!/41

ʹ′ʹ′ʹ′=

ʹ′ʹ′ʹ′ .

435&#'(!)!%6**#!)& 8%!6).%#&1"!%/!!5.#&16%/!!&%"!%5%*"!##"#.4)2&$!2!5(."1*#"!'$!,)!!&"#!%%!6).%#&1"!%/!!5.#&16%/!!&%"!%5%*"!##"#.4)2&$!2!#&%<=3+!%2+!5(."1*#"!'($!!(,()(#!&$1+/!"!1$!%2!!5.-%"&2! *!&"#!%%!:311"!5%(!%2!!"&%'%$)2&$#! $*!"#!"!6'%""!%/!!;6&$,!!%%(+$ +75%!$-( 2+42!$&(7%3&$!#%!+1('3%#%!+-0!-!!3@!.%%!+.-.0.-!!.3.@.!&(-%%!

%-#%'#$(642!('%!3&!#%!+1('3%#%!@!0!-!3/-!!.%%!@.!.0.-!3./.-.!!*&'5(+&!'683%!<'")'

()$/!:342!%'6/&%!( )

( )( )

( )/!(

(!/

-/30!@

3/--@!0

ʹ′ʹ′ʹ′=

ʹ′ʹ′ʹ′−ʹ′ʹ′ʹ′$!<%!&(-%%!.%%!

( )( )

( )( )/!(

(!/

@3!0-+

!3@-+-0

ʹ′ʹ′ʹ′=

ʹ′ʹ′ʹ′−ʹ′ʹ′ʹ′.

++$35-42!56$*3&

( ) ( )( ) ( )

( )( )/!(

(!/

@3!0-+-/30!@

!3@+-03/--@!0

ʹ′ʹ′ʹ′=

ʹ′ʹ′ʹ′−ʹ′ʹ′ʹ′+ʹ′ʹ′ʹ′−ʹ′ʹ′ʹ′−+ .

106 +?@!+",-!,#A"")+"(-

435&#'(!*!%6**#!*& !!!#$%!%.#"/!#&4+!5(."1*#$+!40%(!!*&"!$=2#!%-%$!$*!"#!"!6'%"$+!%/!!;6&$,!!%%(+$ +75%!$-(! #*3&$'(=$*3&!3&!V, V.!#%2!%'$G>&(2!#1%;4-%.52!+*1%3($&2!,'!(!/!

.%%!,.'!(.!./.!.%%!3&!!b, b.!!#%2!&0'&%2!'#$*2!(!/$!!(.!./.!%-#('#$%6%'

05'$*3&!-%!$&(7$*3&!:#%%!-

-

.

.

ʹ′=ʹ′

.

+-:!%""0

*'#4!:#%!&(-%%!!bb

VV

ʹ′≠ʹ′

'!!A:#&!<%!%'6/&%!!!XV

bb

=ʹ′

!!3&!!!! V ) ʹ′≠ .

#&!(16%.0!*+$<5#$*3&!:#%%!!X < V.'!A:#&$!:+42!$&(7%3&$!%)%%1,(

-#%2!%+:!#9-!+*1%3($%!V.!$/$!+1('3%#%! ('%!3&!#$!-./.3.−0.!.@.$!%(

)%%1$/3&!$*'%%'#%.0!+$':#9#%!3&;%'/#&19!%+:! .2

Vʹ′!(-!!&)%13:'$*3&!

#9-! ($%%!$%%$%.%'(%!'#%2!+*1%3($&2!0.'(.-.+.!.%%! !,.'0.@.!.!+$*!%+5(

3&%-%-$! #:#&! <%! %)%%15'$*3&! %+:! %*#52! +$':#9#%! 3&;%'/#&19! %+:! #$!

3%':!#$*2' ++$35-42$!'/3)4-%!3&!#$!/&-%.:!!833%!*+$*!$%%#*+,<9.&!3&#0!

#9-!+1:#%'9!&)$!;%%!&!3!V.–!)!0!1!.%%!;%%!P1.!$!!P2.!$!'!'!'!$!P-. !#%!+1('3%(

#%!+$*!%)%%15'%3&!%+:!#9-!V.$!<%!*+017&%!.0+$%$!*+:'$%+$!

V. – (P1. + P2. + . . . + P-.) < &.

"/@""!@("!-!B+0","#!A-#!+C(DA!-#-# 107

,9'%$8!!V.!− (P1.!/!P2.!/!'!'!'!'!'/!P-.)!2!V.!− X!$!01%!!!

X < P1. + P2. + . . . . .+ P-. (1)

(-!F1! $!F2! $! '! '! '! $!Fv!&(-%%!#%!%-#('#$%6%!+1('3%#%!+$*!%)%%1$/3&!

%+:!#9-!V$!#:#&!#%!+1('3%#%!F.$!F..$!!!.!3"$!&!$!'!'!'$!-$!!+$*!&(-%%!#1%;4-%(

.0!.%%!56$*-!!&0'&%2!!:3$%&2!3&!#%2!!b, b.$!56$*-!('%!/>9$!$+:#&!<%!%'6/&%!

bbʹ′

=Ρʹ′Ρ

κ

κ !$!;%%!.!3!"$!&$!!'!'!'!$-'!!

*#'%$!#&'%.0!<%!56$*3&%!!bb

F

F

F

F

F

F

-

-

2

2

1

1

ʹ′=

ʹ′==

ʹ′=ʹ′

! '!11%!<%!&(-%%!.%%!!!

/

.

-

-

!$

!$ =ʹ′

=Ρʹ′++Ρʹ′+Ρʹ′Ρ++Ρ+Ρ

ν

ν

!

!

$!!$+:#&!!-21 FFF

Xʹ′++ʹ′+ʹ′ !

=

= -21 FFF

V+++ !

>1,

%)$/!V 0!F1!/!F2!/!'!'!'!/!F-. ++$35-42!<%!%'6/&%!9!'65'9!!

X!!0!F.1!/!F.2!/!'!'!'!!/!F.- (2)

"%!'65'&%2! *")$! *&)!:342!$$9;$/-!'&!0#$+$$!01%!$&-!3+$1&(!-%! %(

'6/&%!X < V.'

8&!*'#4!X > V.!'!A:#&!VX

bb

=ʹ′

'!!!(-!!A

VVX ʹ′

= $!!!#:#&!VX

A

Vʹ′

= > 1,

%)$/!!X > V.!'!!11%!!V > T.

++$35-42!Τ

ʹ′=ʹ′ V

bb

!!3&!!A!2!V'!A$!+1:&'93%!'$%+:-!%-0;&#%%!'#9-!

+1$9;$/3&-9!+&1(+#4'9!.%%!$$9;&(!'#%2!'65'&%2!!A!2!!F1!/!F2!/!'!'!'!!/!F- .%%!A!0!!F1/!F2!!/!'!'!'!!/!F-!$!$%!$+$(&2!%-#%)0'.$*-'!!++$35-42!$&-!3+$(

1&(!-%!%'6/&%!$/#&!9!'65'9!)!0!V.'

11%!56$*3&!!VV

bb

ʹ′=ʹ′

'!E15+&%!'$%+:-!-%!%'6/&%!!)!3!V.'

!."'$'

108 +?@!+",-!,#A"")+"(-

435&#'(!+!!%6**#!+& '$%!#&%<=$*+!5(."1*#$+!1$!;6&$#+!5%'$"/!"!$*!"#!"!6'%"$+!%/!!;6&$,!!%%(+$ +75%!$-( (2! <&418'$*3&$! ;%%! +%10$&%;3%$! #%2! %'$G>&(2! +*1%3($&2!

B'(!/,+!.%%!B.'(.!./.,.+.!!3&!&0'&%2!+&-#0;4-%'!0%!%+$$&%6<&(!:#%!

%'6/&%!9!'65'9%

( )( )+,/!(

(!/,+

+,/!(.B

(!/,+.B

ʹ′ʹ′ʹ′ʹ′ʹ′=ʹ′ʹ′ʹ′ʹ′ʹ′ʹ′

.

+-:!%""1

#*3&$'(=$*3&!!3&!E1!$!E2!$!E3!!#%2!#1%;4-%.52!+*1%3($&2!3&!.$1*(

)8!B!.%%!&0'&%2!!&1!3!(!/$!&2!3!(/,$!&3!3!(,+!!.%%!3&!!E1.!$!E2.$!E3.$!!

&1.$! &2.$! &3.! ! #%! %-#('#$%6%! '683%#%! +$*! %-%)51$-#%%! '#9-! +*1%3($%!

B.'(.!./.,.+.'!A:#&!*'/3)4-%!3&!#9-!+1:#%'9!.)!<%!56$*3&!2

1

2

1

&

&

E

E= ,

$9'%$8!!

2

21

2

21

&

&&

E

EE +=

+ (1)

"3$(42!<%!%'6/&%!

3

2

3

2

&

&

E

E= (2)

"/@""!@("!-!B+0","#!A-#!+C(DA!-#-# 109

(+:!#%2!'65'&%2!*")$!*&)!+1$./+#&%!:#%!!3

21

3

21

&

&&

E

EE +=

+!!*+1:#%-

'9!V'!&&)'!11%%!!

3

321

3

321

&

&&&

E

EEE ++=

++ (3)

"3$(42$!'#9-!+*1%3($%!B.'!(.!./.,.+.!56$*3&%

ʹ′

ʹ′+ʹ′+ʹ′=

ʹ′

ʹ′+ʹ′+ʹ′

0

0!$

0

0!$

)

)))

*

***$!$+:#&!

0!$

0

0!$

0

)

)

*

*

βʹ′+βʹ′+ʹ′

ʹ′=

Πʹ′+Πʹ′+ʹ′

ʹ′ (4)

(''0!$%!#1%;4-%.52!+*1%3($&2!E3!$!E3.!!!&(-%%!%'$G>&(2$!!$+:#&%!

ʹ′β

β=ʹ′Π

Π

0

0

0

0 (5)

(+:!#%2!'65'&%2!!*5)!$!*.)!*'/3)4-%!3&!#9-!+1:#%'9!V'&&)!+1$./+#&%!!!

0

0!$

0

0!$

)

)))

*

***

ʹ′++

=ʹ′

++ (6)

(+:!#%2!*<)!$!*4)!+1$./+#&%!:#%%!0!$

0!$

0!$

0!$

)))

)))

***

***

ʹ′+ʹ′+ʹ′++

=ʹ′+ʹ′+ʹ′

++ ,

$9'%$8!!!!!!!!!!!!!!!( )

( )!%$#"

"#$%!

!%$#"1"

"#$%!1"

ʹ′ʹ′ʹ′ʹ′ʹ′=ʹ′ʹ′ʹ′ʹ′ʹ′ʹ′

!!!."'$'

435&#'(!,!%6**#!,& 76)$! %.#"/!#&-! 5.*&1"! ##"#.$*%"#! &$! %.$#+! *&$+! ("#&%#$!"1$+)!5(."1*#$+$

110 +?@!+",-!,#A"")+"(-

+75%!$-( 0&41$/3&!#$!+1('3%!(!/!'!,+3!!451$*3&!#%2!+($!+/$!,/$!!$+:#&!

#$!+1('3%!$%%%1&(#%%!!'&!#1&%2!+*1%3($&2!+'(!/$!+'(/,$!+',/3'!0%!$&(-7$*3&!:#%!%*#52!&(-%%!%'$$/-%3&2$!$9'%$8!56$*-!('$*2!:;.$*2'

()$/!#%!'683%#%!(!+,$!(/3+!&(-%%!+%1%''9':;1%33%$!$%!$%%(

;,-%52! #$*2!(+$!/,! #%!$%%%1$/-!'&!$/$! ('%! #1(;4-%! ! *+1:#%'9! "'! 5<)$!

$9'%$8!(!+!3!(,+$!!(/,!3!3/,'!11%%

+'(!/!3!/'(!+!!3!/'(,+!3!+'(/,!3!+',/3!

()$/! $%! +*1%3($&2! +'(!/$! +'(/,$!

+',/3!&(-%%! ('&2$!.0<&!3(%!<%!&(-%%! ('9!3&!#$!

31!#$*!+1('3%#$2!(!/!–!,+3$!

+'6'!*,'(!/)!3!!31*(!/',+3)!!! !."'$'

453$')#!

76)$! %.#"/!#&3!5(."1*#"! $*!"#! *&2! ("#&%#$!"12)!1$! %%!#

$! %%(!

%.#"/!#&%$!5.*&1"%%+!5%(!40$#!%2!!*##"!!;6&2!&"#!%%!*##%!$=%+$

435&#'(!!-!%6**#!-& '$%!-1%#$+!%.#"/!#&4+!5(."1*#$+!$*!"#!"!6'%"$+!%/!!&$;/!!%/!!%1%'-"/!!5'$(.,!!%%(+$ +75%!$-(

#683%!!#

+-:!%""2

"/@""!@("!-!B+0","#!A-#!+C(DA!-#-# 111

0&41$/3&! $/$! :3$%&2! #1%;4-%.52!+*1%3($&2!('!/,!.%%!(.'!./.,.'!

0&41$/3&!&+('92!#%!+%1%''9'&+(+&$%!3&!&0'&%2!!/+,$!!./.+.,.!.%%!/>9!

('%!3&!#%!%-#('#$%6%!/>9!#4-!+*1%3($4-!%+:!#%2!.$1*)52!($!(.'!0%!%+$$&%(

6<&(!:#%!( )

( )

3

/!

!/

,/!.(

!/,('⎟⎠

⎞⎜⎝

⎛ʹ′ʹ′

=ʹ′ʹ′ʹ′ʹ′

.

A:#&! $93%$*1;$/-#%%! #%! =&/;9! #4-! $3$(4-! +%1%''9'$;10334-!

(!/@!∼!(.!./.@.$!(!,B!∼!(.!.,.B.!.%%!!/+,!∼!!./.+.,.$!%)$/!':;4!#92!$3$%:#9#%2!#4-!+*1%3($4-!#%!+%1%''9':;1%33%!%*#0!56$*-!

$/$!+'&*152!%-0'$;&2!.%%!#%2!+&1%&6:3&-&2!;4-(&2!('&2' /%%!#$-!($%$!':;$!.%%!#%!%+5-%-#%!#4-!(!/@$!(!,B$!!/+,!+%(

1%''9':;1%33%!&(-%%!:3$%%!+1$2!#%!+%1%''9':;1%33%!+$*!&1('.$-#%%!

%+5-%-#%!#4-!(.!.,.@.$!(.!.,.B.$!!./.+.,.!%-#('#$%6%$!%)$/!%-0!$/$!

&(-%%!('%' A%!'#&1&0!!,!-!@!!.%%!!.,.!-!@.!.!56$*-!#%2!'#&1&52!;4-(&2!#$*2!

('&2!.%%!+&1%56$-#%%!%+:!#$!($%$!+'8<$2!$3$(4-!+$'*;,-4-$!01%!<%!&(-%%!

:3$%%'!"+:#&$!'/3)4-%!3&!#9-!+1:#%'9!)"'!55$!<%!%'6/&%

( )( )

3

/!

!/

!@-,!

@!-!,⎟⎠

⎞⎜⎝

⎛ʹ′ʹ′


.

++%+'5$-$! 9! +*1%3($%! ('!/,! &(-%%! #$!31! #$*! +1('3%#$2!

!/,'(@B!.%%!#$!+1('3%!!/,'(@B!&(-%%!#$!21!#$*!+%1%''9'&+%+5$$*!

!,'@!'!A&'%.0!'$%+:-!<%!56$*3&!*('!/,) = 61!*!,'@!)$!8!!*!,'@!)!3!

3!4!*('!/,)!

++$35-42!<%!%'6/&%%!( )

( )

3

/!

!/

,/!.(6!/,('6

⎟⎠

⎞⎜⎝

⎛ʹ′ʹ′


$!$9'%$8

( )( )

3

/!

!/

,/!.(

!/,('⎟⎠

⎞⎜⎝

⎛ʹ′ʹ′


!!!."'$'

112 +?@!+",-!,#A"")+"(-

453$')#! '$%! -1%#$+! 5%'("/!#&4+! 5(."1*#$+! $*!"#! "!6'%"$+! %/!! &$;/!!#$%!%1-'%"/!!"&1,!!%%(+$ +75%!$-(!

+-:!%""3#$

(2!<&418'$*3&$!$/$!:3$%&2!+*1%3($&2!@'(!/,+!.%%!@.'(.!./.,.+.!

3&!&0'&%2!+&-#0;4-%' #*3&$'(=$*3&!3&!E1$!E2$!E3!#%2!+*1%3($&2!3&!&0'&%2! !&1!3!(!/$!

&2! 3!(/,$! !&3!3!(,+!.%%!3&!E1.$!E2.$!E3.$! !&1.$&2.$!&3.! !#%!%-#('#$%6%!

'683%#%!+$*!%-%)51$-#%%!'#9-!+*1%3($%!@.'!(.!./.,.+.'

#/3)4-%!3&!#9-!+1:#%'9! +!56$*3&! 3

3

3

2

2

1

1 'E

E

E

E

E

E=

ʹ′=

ʹ′=

ʹ′$!:+$*!'!

&(-%%!$!':;$2!$/$!$3$':;4-!+'&*1,-! *$9'%$8$! !=ʹ′ʹ′

=ʹ′ʹ′ (@

@(

!(

(! =' ).

++$35-42$!<%!%'6/&%!!! 3

321

321 'EEE

EEE=

ʹ′+ʹ′+ʹ′++

$!01%!<%!56$*3&%

( )( ) 3

333

)!(((AB)

!(

(!'

+,/!(.@

(!/,+@'

ʹ′ʹ′=⎟

⎠

⎞⎜⎝

⎛ʹ′ʹ′

==ʹ′ʹ′ʹ′ʹ′ʹ′ʹ′

.

435&#'(!.!%6**#!.& !!! #$%! %.#"/!#&4+! 5(."1*#$+! $*!"#! *&$+! ("#&%#$!"1$+)#! %-%$! %#!;6&$#+! %%(+! $*!"#! "!%#&%.-+/+! "!6'%"$+! %/!! (=,!! &"#! "!%*-

"/@""!@("!-!B+0","#!A-#!+C(DA!-#-# 113

&%.%+"#! "!! %#! ;6&$#+! #$%! %.#"/!#&,!! 5(."1*#/!! $*!"#! "!%#-&%.-+/+!"!6'%"$+!%/!!(=,!#!%-%$!%#!5(."1*#$+!$*!"#!*&$+!("#&%-#$!"1$+)$ +75%!$-( (2!<&418'$*3&!#%2!+*1%3($&2!('!/,!.%%!(.'!./.,.!3&!/>9!*$!*.$!

&0'&%2!!/,$!!./.,.!.%%!#%!%'$G>8!+1$2!%*#52!+%1%''9'&+(+&$%!!,-@!!

.%%!!!.,.-@.!.!%-#%'#$(642!*&'5+&!'683%!'#9-!+1:#%'9!#)' %#&!!*'#4!:#%!$%!+*1%3($&2!*('!/,)!$!*(.'!./.,.)!!&(-%%!%'$$/-%(

3&2$! $9'%$8! %'6/&%! *('!/,)! 3! *(.'!./.,.)'! 0%! %+$$&(7$*3&! :#%!

( )( ) *

*

,/!

!/, ʹ′=

ʹ′ʹ′ʹ′.

()$/!*!,!-!@!)!3!4*('!/,)!.%%!*!.,.-!@.!.)!3!4*(.'!./.,.)$!#%!+%1%''9'&+(+&$%! *!,-@!)! $! *!.,.-@.!.)! &(-%%! %'$$/-%3%$! $9'%$8! %('6/&%!*!,-@!)!3!*!.,.-@.!.)'!#/3)4-%!3&!#9-!+1:#%'9!)"'!5<$!;%%!#%2!&0'&%2!#4-!+%1%''9'&+%+5$4-!%*#,-! *+$*!&(-%%!+%1%''9':;1%33%)!<%!

56$*3&!( )

( ) +

+

!%$#

#$%! ʹ′=

ʹ′ʹ′ʹ′ʹ′$!$+:#&!

( )( ) +

+

%$#!

#$%! ʹ′=

ʹ′ʹ′ʹ′!*+1:#%'9!"'!5<)'!

A&'%.0!'$%+:-!%'6/&%!( )

( ) *

*

,/!

!/, ʹ′=

ʹ′ʹ′ʹ′.

%8&! !*'#4! #,1%!:#%!$%!&0'&%2! #4-!+*1%3($4-!&(-%%!%-#%'#1:)42!

%-0'$;&2! #4-! *>,-$!( )

( ) +

+

%$#

#$% ʹ′=

ʹ′ʹ′ʹ′'!0%! %+$$&(7$*3&! :#%! $%! +*1%3($&2

*('!/,)!$!*(.'!./.,.)!&(-%%!%'$$/-%3&2$!$9'%$8!*('!/,)!3!*(.'!./.,.)'

()$/!!( )

( ) +

+

%$#

#$% ʹ′=

ʹ′ʹ′ʹ′$!'/3)4-%!3&!#9-!+1:#%'9!"'!5<$!<%!56$*3&%!

( )( )

( )( )

( )( ) *

*

,/!

!/,

,/!2!/,2

,+/!

!/+, ʹ′=

ʹ′ʹ′ʹ′=

ʹ′ʹ′ʹ′=

ʹ′ʹ′ʹ′ʹ′.

++$35-42!*!,'-@!)!3!*!.,.-@.!.)!*+1:#%'9!XI'!5<)$!$9'%$8!!4*('!/,)!3!4*(.'!!./.,.)$!$+:#&!*('!!/,)!3!*(.'!./.,.)'

!."'$'

114 +?@!+",-!,#A"")+"(-

+#!1:5#$#!'&(!)*&3('(!7=3#)>%/.!'#$!73$')"&/. +#"#!?*&3('(!7=3#)>%#,!!

B+$1$/3&!-%!%+$$&(7$*3&!%+=!&*<&(%2!:#%!$/$!+*1%3($&2!3&! ('%!

/>9!.%%!%'&3&%$%.52!&0'&%2!56$*-!('$*2!:;.$*2$!3&!35<$$$!+$*!+'9'%0(

=&%!+&1%'':#&1$!'#9!'/;61$-9!$%%$%.%'(%!$'$.'814'92' E10;3%#%$!<&41$/3&!#%2!+*1%3($&2!V!3!#!-!(!/$!V.!3!#.-!(.!./.!

3&!/>$2!!*!!.%%!&0'&%2!&3&%$$/!&!*&'5+&!'683%!4'")'!(1.&(!-%!$&(7$*3&!

:#%!%+$.'&($-#%%!$%!'65'&%2!V > V.!.%%!V < V.'

+-:!%"1#$#$

)41(=$*3&!#9-!%.38!!#(!!'&!!@!!('%!#383%#%!3&!'93&(%!$%%(1&'92!

#%!'93&(%!(1$!(2,…$!(n-1!.%%!%+:!#%!'93&(%!%*#0!)51$*3&!#%!+%10''9'%!

'#9!&0'9!(!/!&+(+&$%'!"-$30=$*3&!!1!/1!$!!2!/2 ,…$!!n-1!/n-1!#%2!#$352!

#4-!&+%+5$4-!%*#,-!3&!#9-!5$1%!#!/!!.%%!!1.!/1.!$!!2.!/2.!$…$!!n-1.!/n-1.!

#%2!+%10''9'&2!+1$2!#9-!#(!+1$&$'52!#4-!!1!/1!$!!2!/2 ,…$!!n-1!/n-1!!&+(!

"/@""!@("!-!B+0","#!A-#!+C(DA!-#-# 115

#4-!&$1,-!(2!!2!/2$!(3!!3!/3 ,…$!(n-1 Bn-1!/n-1$!!(!/!%-#('#$%6%'! ##9-!+*1%3($%!#-(!/$!<&41$/3&!#%!+1('3%#%!E1$!E2 ,…$!En!3&!

&0'&%2!!&1!3!(1!1/1$!!&2!3!(2!2/2 ,…$!!&n-1 = An-1Bn-1/n-1$!&!3!(!/!%-#((

'#$%6%$!+$*!#$!.%<5-%!#$*2!56&%!/>$2!n*!!.%%!#%!%-#('#$%6%!%'$G>8!+1((

'3%#%!F1$!F2 , …$!Fn-1 !3&!&0'&%2!&1$!&2,…$!&n-1 . (-!<5'$*3&!Sn!3!E1!/!E2 + …!/!En!.%%!sn = P1 + P2 + … + Pn-1,

#:#&!+1$)%-,2!%'6/&%%!sn < V < Sn. (.$'$*<,-#%2!#9-! ($%%!$%%$%.%'(%!'#9-!+*1%3($%!#.-(.!./.!.%%!!

619'%3$+$%,-#%2!%-#('#$%6$*2!'*3&$'%'3$/2$!<%!56$*3&!sn.!2!V.!2!Sn.$!

:+$*!Sn.!3!E1./!E2./!…!/!En.$!!sn.!3!P1.!/!F2.!/!…!!/!F.n-1. 2342!Ei!3!Ei.$!;%%!.0<&!i ∈ {1, 2, …, nA!!.%%!Pi = Pi.!$!!;%%!.0<&

i ∈ {1, 2, …, (n-")A$!%)$/!56$*-! ('%!/>9!.%%!&0'&%2! %'&3&%$%.52$!&):(

'$-!

2i

ni

&

&⎟⎠

⎞⎜⎝

⎛= $!;%%!.0<&!!i ∈ {1, 2, …, (n-1)}.

11%!Sn = Sn.!.%%!!sn = sn.$!$+:#&!sn < V.!2!Sn. *'#4!:#%!%'6/&%!!V > V.'!A:#&$!$-$30=$-#%2!W!#$!+1('3%!(!/!-!#A?!

3&! />$2! ! *! ! .%%! &3&%$:-! &0'92! ! &! ! .%%! '%3&0-$-#%2! *+:>9! #%2!'65'&%2 Fi!3!Ei$!;%%!.0<&!i ∈ {1, 2, . . .(n-")A$!<%!56$*3&!

V < Sn, - V.2!- sn ⇒ V - V.!2!Sn - sn =

3!*E1!/!E2!/!'!'!'!/En) -!*F1!/!F2!/!'!'!'!/!Fn-1) =!En = nW (1).

#/3)4-%!:342!3&!#$!%7(43%!+*$:7$*!–!(16%38$9$!*+016&%!!n∈D!!

#5#$%$$!,'#&!n (V – V.)!0W$!$+:#&!V – V.!0!nW

!.%%!':;4!#92!*")$!

nW > V – V.!+$*!&(-%%!0#$+$'!++$35-42!$&-!3+$1&(!-%!%'6/&%!V>V.'

*'#4!#,1%!:#%!%'6/&%!!V < V.'!A:#&!$3$(42!<%!56$*3&!

V.- V < Sn - sn = En = nW (2),

.%%!%)$/!V.– V!0!1$!'/3)4-%!3&!#$!%7(43%!+*$:7$*!–!(16%38$9$

116 +?@!+",-!,#A"")+"(-

<%!*+016&%!n∈D!#5#$%$$!,'#&!n(V.- V) > W'!"+:#&!V.– V > nW

!.%%!':(

;4!#92!*&)$!!nW > V.– V$!+$*!&(-%%!0#$+$'!

++$35-42!$&-!3+$1&(!-%!%'6/&%!V < V.' A&'%.0!'$%+:-!<%!56$*3&!!V = V.' 9"."%3.2&2"

B&!&0'9!#%2!'65'&%2!

!

2

3

2

)

)⎟⎠

⎞⎜⎝

⎛= !$!;%%!.0<&!i ∈ {1, 2, . . . , (n-1)}

.%%!<&41,-#%2!;-4'#:!:#%!;%%!#$-!:;.$!#$*!+1('3%#$2!W!3&!&0'9!&!.%%!

/>$2!*!56$*3&!W!3!&.*!*&'5+&!.%%!'6:'%$!+$*!%.$'$*<&()$!3+$1$/3&!-%!

*+$'$;('$*3&!%+=!&*<&(%2!#%!:1%%!#4-!%.$'$*<%,-!*Sn), (sn)!.%%!-%!$&((

7$*3&!:#%!&(-%%!('%!3&!!3W

!42!&782%

*6$*3&!Sn!3!E1!/!E2!/!'!'!'!/!En!3!*&1!/!&2!/!'!'!'!/&n-1!/!&)!. 3

+ =

= n*

&&n

1-n...&n2

&n1 222

⋅⎥⎥⎦

⎤

⎢⎢⎣

⎡+⋅⎟

⎠

⎞⎜⎝

⎛++⎟

⎠

⎞⎜⎝

⎛+⎟

⎠

⎞⎜⎝

⎛ =

= ( ) ( ) ( )

3

+)

34

$!3$5353

3

+)

3

3$&3111!$!!

!!!!

⋅⋅++

=⋅⋅++++ =

= 4

+)

3

$!

3

$$

⋅⋅⎟

⎠

⎞⎜⎝

⎛+⎟

⎠

⎞⎜⎝

⎛+ !$!$+:#&!!lim Sn = 1 ⋅2 ⋅

0

6

0

+)

4

+)=

⋅=

⋅

++('92!!sn = P1 + P2 + . . . + Pn-1!3!*&1 /!&2!/!'!'!'!/!&n-1)

3

+ =

( ) ( ) ( )

4

+)

3

$!

3

$$

3

+)

34

$&!33$&3

3

+)

3

$&3111!$!!

!!!

⋅⋅⎟⎠

⎞⎜⎝

⎛−⎟

⎠

⎞⎜⎝

⎛−=

=⋅⋅⋅

=⋅⋅+++

=

"/@""!@("!-!B+0","#!A-#!+C(DA!-#-# 117

*6$*3&!'$%+:-! ! lim sn = 1 ⋅ 2 ⋅ 0

6

0

+)

4

+)=

⋅=

⋅'!A$!.$%-:!:1%$!

#4-!*Sn), (sn)!<&41$/3&!+'5$-!42!:;.$!#92!+*1%3($%2'

+#'!!?*&3('(!73>')#&%,! (-!<&418'$*3&!#$-!./&$$!#$*!$+$($*!9!%.38!&(-%%!3$-0$%!35#19(

'92!38.$*2!42!3$-0$%!35#19'92!:;.$*$!$9'%$8!3&!:;.$!Vo! 3! "$! #:#&!

&/.$'%!3+$1$/3&!-%!$&(7$*3&!:#%!$!:;.$2!+%1%''9'&+%+5$$*!3&!&0'9!&!

.%%!/>$2!!*!!&(-%%!('$2!3&!!V!3!&!⋅!*' @%#0!#9-!%+:$&%79!<%!<&418'$*3&!$&$$35-$!:#%!#$!&3&%$:-!$1(

<$;4-($*!+%1%''9'$;1033$*!&(-%%! ('$!3&!#$!;%-:3&-$!#4-!$%%'#0'&,-!

#$*$! #$!$+$($!+1$./+#&%!3&!'*''$;%'3$/2!%-#('#$%6$*2!3&!%*#$/2!+$*!

%.$'$*<$/-!.%%!%+:!#9-!%-0+#*79!+$*!+1$9;8<9.&!'#$!&%&'($!""'! E10;3%#%$!5'#4!$!./&$2!V!3!(!/,!-!(.!./.,.!*&'5+&!'683%!4'&)'

+-:!%"1#%

B+$1$/3&! -%! .%#%'.&*0'$*3&! $1<$;,-%$! +%1%''9':;1%33$!

@!/,!%'&3&%$%.:!3&!#$!(!/,!.%%!'#9!'*-56&%%!$1<$;,-%$!+%1%''9'&(

+(+&$$!V1!3!@!/,!-!@1!1/1,1!!3&!#9-!5$1%!@1!1/1,1!&+(!#$*!&+%+5$$*!P! 3! (.!./.,.'! A:#&$! '/3)4-%! 3&! #9-! +1:#%'9! )"'5&$! <%! 56$*3&%!

( )( )

1VV

1=

ΚΛΓΔ

ΑΒΓΔ= $!$+:#&!V = V1 .

05#$*3&!@!!3!x$!@,!3!y! !.%%!<&41$/3&!#$!$1<$;,-%$!+%1%''9('&+(+&$$!V.!!!3&!$%%'#0'&%2!!x, 1, 1.

118 +?@!+",-!,#A"")+"(-

#/3)4-%!+0'%!3&!#9-!+1:#%'9!)"'!5&!<%!56$*3&! +7$$

+7

.

.$ ⋅=⋅

⋅=

ʹ′,

x111x

VV

0=

⋅

⋅=

ʹ′'!11%! *yx

VV

0

1 ⋅⋅= $!$+:#&!

V = V1 = x y!*!. V0 = x y!*!3!*@!/,)!⋅!*!3!*(!/,)!*!3!&!⋅!*!'

@0<&!#1%;4-%.:!+1('3%!3&!&0'9!!&!!3+$1&(!-%!<&419<&(!42!#$!3%(

':!+%1%''9'&+%+5$$*!3&!#$!($%$!/>$2!*!.%%!&0'9!&&'!++$35-42$!$!:;.$2!

#$*!%'$/#%%!3&!V = 21!&&!⋅!*!3!&!⋅!*!'

A5'$2$!.0<&!+1('3%!3&!&0'9!!&!!.%%!/>$2!!*!!3+$1&(!-%!%-%'*<&(!

'&! #1%;4-%.0!+1('3%#%$!$+:#&!$!:;.$2! #$*!.%%!+0'%!<%!&1&<&(! ('$2!3&!! V!3!&!⋅!*'

+#$!!)!+3:0!&%=!!#8##$*3$ -!'/;.1%'9! #4-! &3&%$,-! $/$! &+(+&$4-!'6930#4-!8! #4-!:;.4-!

$/$!'#&1&,-!'6930#4-!3&!('%!/>9!3+$1&(!-%!;(-&%!&/.$'%!3&!&0'9!#9-!

%-%+:$&%.#9!3&-!%''0!3&!&+%#*6(%!&)%13$=:3&-9!".03!%%(!7";"'#4.## '#9-!$+$(%!.%%!<%!%-%)&1<$/3&!+%1%.0#4'!

+-:!%"1#.#$

"/@""!@("!-!B+0","#!A-#!+C(DA!-#-# 119

+!!!#$%!$5*5$#"!&031"%"! *')$! *'.)!%41!%!%"#!"5-!%#+!5".6''2-'$+!$()$*$+!*&0)$!*&1)$!*&2)$!*&3)!$!'!'!'!$*&)!&"%6!%"!"!%*&%%#0"!>$$"2!$()$-"."11/!!%1216%/!! !*(0!!0$!(0.!0.)$!!*(1!!1$!(1.!1.)$!!*(2!2$!(2.!2.)$!

*(3! !3$! (3.!3.)$! …$! ! *(!$! (.!.)! ! 4%&#! ,&%$"! ! *(0.!0.)! 3! '! *(0!0), *(1.!1.)! 3!'! *(1!1)$! ! *(2.!2.)! 3!'! *(2!!2)$! *(3 !3)! 3!'! *(3 !3), . . . , *(.!.)!3!'*(!)$!!%-%$!#&0$$#!!*'.)!3!'!⋅!*')B'!

B$%0=&%!!5#'%!'%-!-%!<&41$/-#%%!#%!'683%#%!!*')$!*'.)!42!'/-$'%!

&*</;1%334-!#3930#4-$!$9'%$8!'6930#4-!3&!39$&-%.:!&3&%$:-'

*#'%!'$%+:-!<%!56$*3&!!*')!3! ( )∑ ii!( !.%%!

*'.) = ( )∑ ʹ′ʹ′ ii!( = ( )∑ ii!( ' = ( )∑λ ii!( !3!'!⋅!*')'

@0#%! #5#$%$!&5&%%%! &3+&1%56&%! #$! ($%$!+%10$$7$!+$*!&3)%-(=&#%%!

'#%!+%10$$7%! #$*!38-4-%$!:+$*!3%%!.(-9'9!<&41&(#%%!42!5-%!'/-$'$!

%+:!%.%-9'(&2!8!5-%!&*</;1%33$!#383%!42!'/-$'$!'93&(4-$!$9'%$8!+$(

'$#8#4-!3&!39$&-%.:!38.$2' E%1:'9!#9-!+%1%$$7$'$;(%!#$*!)%%-$35-$*!9!%168!%*#8!$(-&%!$1(

<0!'*3+&10'3%#%$!.0#%!30'%'#%!#$!$+$($!$!Leibniz!&)013$'&!'&!+01%!+$''52!+&1%+#,'&%2!%+$$&6:3&-$2!#9-!%168!641(2!.%3(%!%3)%'&8#9'9'

E$'/!+1%-!%+:! #$-!@%&%'%51%$!$!(16%38$92!&(6&!619'%3$+$%8'&%!

+%1:3$%%!'*''$;%'#%.8!+1$.&%35-$*!-%!#&#1%;4-('&%!#$!+%1%&$'%.:!64-1($$! <&41,-#%2! &.&(! #$! 641($!42! '/-$'$! &*</;1%334-! #3930#4-'!A$!

<&,19'&!$9'%$8!42!3(%!&+(+&$9!+'0.%!3&!'*;.&.1%35-$!&01$2$!9!$+$(%!

%+%1#(=&#%%!%+:! &*</;1%33%! #383%#%$!+$*! #$!.%<5-%! #$*2!56&%!&+('92!

5-%!'*;.&.1%35-$!&01$2'!2342$!'&!%-#(<&'9!3&!#$-!Leibniz$!$!(16%38-$92!#$-(=&%!:#%!%*#8!&(-%%!%+',2!3%%!35<$$$2!;%%!-%!*+$>%%'#&(!#9!'65'9!

#$*!&3&%$$/!#$*!+%1%&$'%.$/!641($*!3&!#$!&3&%$:-!&-:2!#1%;,-$*!.%%!

$&-! %+$#&'&(! %+:$&%79'! /%%! -%! $,'&%! %*'#918! %+:$&%79! &;.',&%'&! #$!

&3&%$:-!#$*!+%1%&$'%.$/!641($*!3&#%7/!$/$!%<1$%'30#4-!+$*!%+$#&(

'$/-#%%!%+:!0+&%1%!#1(;4-%!#$!.%<5-%'

+-:!%"1#.#%

120 +?@!+",-!,#A"")+"(-

B(%!%+'8!&+%'8<&*'9!#92!%1682!#$*!@%&%'%51%!56$*3&!'#9-!+&1(-+#4'9! #4-! $/$! #1%;,-4-! '#%! $+$(%! 641(=&#%%! 5-%! +%1%''9':;1%33$!

(!/,!%+:!#9!$%%;,-%:!#$*!!,'! E10;3%#%$! #%! #1(;4-%!(!,$!!/,$! #%!$+$(%! 56$*-! ('%!/>9!,,.$!

!!.$!3+$1&(!-%!<&419<$/-!42!%+$#&'$/3&-%!%+:! ('%!&*</;1%33%!#38(

3%#%!(i!i .%%!!/i!,i!+%10''9'%!'#%2!&0'&%2!#$*2!(!$!/,!.%%!'&!('&2!%(

+$'#0'&%2!%+:!%*#52'! *#'%!'$%+:-$!%)$/!(iBi!3!/i,i$!<%!&(-%%!.%%!!#*(iBi) = #*/i!,i)$!$9(

'%$8!!*(!,)!3!*!/,)' (-!<&418'$*3&!&5&%%%!;-4'#8!#9-!5--$%%!#$*!$1%'35-$*!$'$.'9(

1,3%#$2$!#:#&!;%%!&3&%$0!'6930#4-!+$*!+&1%.'&($-#%%!3&#%7/!#4-!;1%)%(

.,-!+%1%'#0'&4-!$/$!'*-%1#8'&4-!%*#:!&(-%%!&/.$'%!%+$$&(7%3$' /%%! +%10$&%;3%$! %-! *'#$! '683%! 4'5'5)! <&418'$*3&! #%! &+(+&$%!

'683%#%!!!*'1)$!*'2)$!*'3)!+$*!$1(=$-#%%!%+:!#%!=&/;9!#4-!'*-%1#8'&4-%

⎭⎬⎫

+=

=

1-xx-y1-y

2 , ⎭⎬⎫

+=

=

x2xy3xy

2 , ⎭⎬⎫

+=

+=

62xy62xy

2 .

.%%!&(-%%!#$+$<&#935-%!3&#%7/!#4-!+%1%''8'4-!&*<&%,-!!x!3!1!!.%%!!x = 1 #:#&$!.0<&!&*<&(%!!x!3!1!!3&!!1!≤!1!≤!"!#53-&%!#%!'683%#%!'1$!'2$!'3!.%#0!#%!

#383%#%! ! (1! !1$!(2! !2$!(3! !3! ! .%%! %'6/&%! ! (2! !2! 3!(1! !1$!(3! !3 = !3!&(1 !1.

+-:!%"1#.#.

"/@""!@("!-!B+0","#!A-#!+C(DA!-#-# 121

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

*'1) = ( ) ( )∫ ∫ ∫=+−=−−−+−$

8

$

8

!

$

8

!! 9::&:9:::9:$$:: !!!.%%!

*'2) = ( ) ( )∫ ∫∫ ==+$

8

$

8

!

$

8

!! 9::&:9::&:9::!:&:0 ,

*'3) = ( ) 9:4:!&4:!

$

8

!∫ ++ = dx2x-x2 21

0∫ =

= ( ) dxx-x2 dxx-x2 21

0

21

0∫∫ =

-!%-#('#$%69!%8-:"<%3%*,"8,!'#$!6,1$!3+$1&(!-%!&)%13$'#&(!'&!

'#&1&0!'683%#%!3&!('%!/>9!42!&782%

+!!!#$%!&%$.$6 V, V.!%41!%!%"#!"5-!%"!5".6''2'"!$5*5$#"!*F0), *F1)$!*F2)$!*F3)$!'!'!'!$!*F)!&"%6!%"!"!%*&%%#0"!>$$"2!$5#54#/!!&0216%/!

*'0$!'0.)$!*'1, '1.)$!!*'2$!'2.)$!*'3$!'3.)$!'!'!'*'$!'.)!!4%&#!,&%$" *'0.)!3!'!*'0)$!*'1.)!3!'!*'1)$!*'2.)!3!'!*'2)$!*'3.)!3!'!*'3)!$!'!'!'!$!*'.)!3!'!*')!

%-%$!#&0$$#" (V.)!3!'!⋅ (V) ".

+-:!%""1#3.4

@%%!+0'%!'$%+:-!)%(-&#%%!-%!<&41$/-#%%!#%!'683%#%!V, V.!42!'/(

-$'%!&+(+&$4-!'6930#4-$!$9'%$8!'6930#4-!3&!39$&-%.:!:;.$'

122 +?@!+",-!,#A"")+"(-

*#'%!'$%+:-!*&'5+&!'683%!4'5'<)!<%!56$*3&%!*V)!3!# ( )2) !.%%!!

(V.)!3!#! ( )2)ʹ′ = ( )2)'∑ !3!'! ( )∑ 2) !3!!'!!*V).

,#!12&'3$'(!'=#>.%3/.!'#$!';./.! 435&#'(!!"/!%6**#"/&

76)$!&,!%+!$*!"#!*&%+!("#&%#$!"1%+)!1$!%%!31!&('*!#.%(!%!%5%*,

%+!40$#!!%2!!*##"!;6&2!&"#!*&%!$=%+$ +75%!$-(! 0&41$/3&! #$-!./'%-$1$! !V! !.%%! #$-!.,-$!V.! #$-! &;;&;1%335-$!

'#$-!./'%-$1$'!05'$*3&!-%!$&(7$*3&!:#%!V.3!!31 V!$!$9'%$8!V = 3V.'

+-:!%"$)

#&!*'#4!:#%!V > 3V.$!$+:#&!<%!56$*3&!V−3V.!3!&!0!1'!(+$.:+#$(

-#%2!%+:!#9!&0'9!.%-$-%.0!+$'/;4-%!+-!$!-!3!<$!#$!"4$!'!'!'!&;;&;1%335-%!

'#$-!./.'$$!%)%%1$/3&!%+:!#9!&0'9!.0<&!)$10$!+$':#9#%!3&;%'/#&19!

%+:!#$!3%':!#$*!+1$9;$/3&-$*!*+$'$(+$*$!:+42!%+$$&(6<9.&!'#$!/&-%.:!

!833%'! (+$.:+#$-#%2!%.$'$/<42!%+:!#$-!./'%-$1$!#%!+1('3%#%! %'$G>8!

+1$2!%*#:-!.%%!3&!&0'&%2!#%!+$'/;4-%!+-!$!-!3!<$!#$!"4$!'!'!'$!!%)%%1$/3&!

"/@""!@("!-!B+0","#!A-#!+C(DA!-#-# 123

%+:! #$-! ./'%-$1$! .0<&! )$10$! +$':#9#%! 3&;%'/#&19! %+:! #$! 3%':! #$*!

+1$9;$/3&-$*!*+$'$(+$*' #/3)4-%!.%%!+0'%!3&!#$!/&-%.:!!833%$!<%!+1$./>$*-!.0+$%%!'#%;(

38!42!*+:'$%+$!3&1%.0!.*'%-$1%.0!#383%#%$!+$*!#$!0<1$%'30!#$*2!x!!&(-%%!3%.1:#&1$!%+:!#$!!&!3!V – 3V.$!!$9'%$8!x < V - 3V.$!$+:#&!5V.!2!C!- x.

2342!#:#&!#$!'#&1&:!y = V − x! !&(-%%!#$!&;;&;1%335-$!'#$-!./('%-$1$!+1('3%$!$+:#&!<%!56$*3&!5V.!2!!y.

A$!+1('3%!:342!%*#:! 56&%!:;.$! #1%+'0'%$!%+:! #$-!:;.$!Vo! #92!

+*1%3($%2!3&!#9-!($%%!&0'9!.%%!('$!/>$2!!*+1:#%'9!+$!+:1%'3%)$!$9'%$8!y = 3 Vo!$!$+:#&!5V.!2!5Vo$!01%!!V.2!Vo.

2342! !Vo < V.$!%)$/!9!+*1%3($%!%*#8!&(-%%!&;;&;1%335-9!'#$-!

.,-$'!(*#:!:342!&(-%%!0#$+$$!&+$35-42!$&-!3+$1&(!-%!%'6/&%!V > 3V.'

8&!!*'#4!#,1%!:#%!!V < 3 V.'!A:#&!V.!0!31 V$!$+:#&!V.−

31 V!3!&!0!1'

(+$.:+#$-#%2!%+:! #$-!.,-$! #%2! %'$G>&(2!+1$2!%*#:-!+*1%3($&2!

3&!&0'&%2!#%!.%-$-%.0!+$'/;4-%!+-$!-!3!<$!#$!"4!'!'!'$!%)%%1$/3&!%+:!#$-!

.,-$!.0<&!)$10!+$':#9#%!3&;%'/#&19!%+:!#$!3%':!#$*!+1$9;$/3&-$*!

*+$'$(+$*' #/3)4-%!3&! #$!/&-%.:!!833%$!<%!+1$./>$*-!.0+$%%!'#%;38!42!

*+:'$%+$! 3&1%.0! .4-%.0! #383%#%! 3&! 0<1$%'3%! x$! 3%.1:#&1$! %+:! #$!

&!3!V.−31 V$!!$9'%$8!x < V.!–

31 V$!$+:#&!V.– x >

31 V'!2342!V.– x =

= Vo = 31 y$!:+$*!#,1%!Vo!&(-%%!9!&;;&;1%335-9!'#$-!.,-$!+*1%3($%!

.%%!y!#$!&;;&;1%335-$!'#$-!./'%-$1$!+1('3%'!11%! !31 y >

31 V$!$+:#&!

y > V'!(''0!y < V!$!%)$/!#$!+1('3%!&(-%%!&;;&;1%335-$!'#$-!./'%-$1$'!

(*#:!:342!&(-%%!0#$+$$!$+:#&!$&-!3+$1&(!-%!%'6/&%!V < 3V.' 11%!<%!%'6/&%!!V = 3V.'!!!."'$'

435&#'(!""!%6**#""& '$%!#&%<=$*+!&,!%#!3!&$'#!#.%#!$*!"#!"!6'%"%#!%/!!;6&$,!!%%(+$

124 !!"#!"#$!"$%&"'!"%#

!!""##$% %(! )&*+,'-./&! (0-! 1'-23&4(! 567-.(! V1 , V2! /&! 89'&1(! )-.(!

505:-.(!"1$!"2%!*;:-./&!7+!(&4<-./&!=)1!2

1

2

1

"

"

VV

= .

,')*!!2

1

2

1

VV

"

"≠ %!%=)&!!

SV

"

" 1

2

1 = !!/&!! 2VS ≠ .

$!!,')*!=)1!1'>0&1!!S < V2!%!%=)&!)+!;>-./&!V2 - S!&!&!'!(%

!"#$!!""

-?*(!(1+?1')6'+/&!')-!.&715=!#,//+$!+?-5=?)-7)+(!+?=!)@!89-'@!"2! )+! &AA&A++//;7+! 5+7-7159! ?-:0A*7+! ! !7! $! 7! &! )$! *$! +,$! %! %! %$!

+/+1+-0/&!+?=! )-7!505:-!"2 59)&!/-+9!?-'=)@)+!/&A+:0)&+@!+?=! )-!

/1'=!)-.!?+-@A-0/&7-.!.?-:-4?-.% &/-4*($!+?-5=?)-7)+(!+?=!)-7!567-!V2!)1(!?.++/4(&(!/&!89'&1(!

!7! $!7!&!)$!*$!+,$! %! %! %$! !5+1! 4(1-!03-(!/&!)-7!567-$!+/+1+-0/&!+?=!)-7!

567-! V2! 59)&! /-+9! ?-'=)@)+! /&A+:0)&+@! +?=! )-! /1'=! )-.! ?+-@-A-0/&7-.!.?-:-4?-.%

$0//*7+! 5+1! ?9:1! /&! )-! .&715=! #,//+! )+! ?+-503-.7! 59?-1+!

')1A/,!*(!.?=:-1?-$!)/,/+)+!)-.!567-.!/&!9)+-1'/+!X$!/15+=)&+-!)-.!

)&)15-0!++1)/-0!&!&!V2 - S$!!(@:+(,!X < V2 - S$!-?=)&!S < V2 - X. $!(1+/-+9!=/*(!V2 - X!?++1')97&1!)@7!?.++/4(+!!B2!?-.!+?-/;7&1!

5+1!&47+1!&AA&A++//;7@!')-7!567-!V2!$!(@:+(,!!!B2 = V2 - X. %7! )&*+,'-./&! )@7! +7)4')-1>@! ?.++/4(+!B1$! )@7! &AA&A++//;7@!

')-7!567-!V1$!)=)&!-1!89'&1(!)*7!?.++/4(*7!B1$!B2!&47+1!+79:-A&(!?+-(!

)+!)&)+9A*7+!)*7!(1+/;)+*7!)*7!505:*7!"1$!"2!-?+=)+'@!+.!$!9++!5+1!

&."&"!"%"!$!C!*&#&$!%$$!!D%E%#$$$$ 125

+79:-A&(!)*7!505:*7!"1$!"2!-?+=)+'@!/.$!(@:+(,!! SV

"

"

B

B 1

2

1

2

1 == .

0++! 1B

VB

S1

1

2>= $!+/-0!V1! '!B1%!!?-/;7*(!)+! 1'>0&1!S! '!B2 =

= V2 – X$! ! &76! ;>-./&!S < V2 – X%!%.)=!=/*(! &47+1!9)-?-%!,)'1!(&7!

/?-+&4!7+!1'>0&1!S < V2.

&!!,')*!)6++!=)1!1'>0&1!S > V2%!%=)&!11

2

VS

"

"= .!,')*!

SV

VS 2

1 ʹ′= ,

-?=)&!! 1S

VVS 2

1<=

ʹ′$!+/-0!V2 < S$!9++!S1!0!V1!!-8:;?&!:,//+!/&)9!)@7!

?+=)+'@!/.%!!?-/;7*(!!SV

"

" 2

1

2

ʹ′= !!!/&!!!S1!0!V1.

!/++/=F-7)+(!)6++!')-7!567-!V1!)@!(1+(15+'4+!?-.!+5-:-.),'+/&!

')@7! ?&+4?)*'@! -+.! ')-7! 567-! V2$! /&! ?++=/-1-.(! '.::-A1'/-0(! 5+-)+:,A-./&!'&!9)-?-$!-?=)&!(&7!/?-+-0/&!7+!;>-./&!S > V2.

%7!:-1?=7!1'>0&1!SV

"

" 1

2

1 = !)=)&$!+?=!)+!-+.!5+1!-8.!?+-50?)&1!=)1!

S = V2$!(@:+(,!!2

1

2

1

VV

"

"= .

%7!)&*+,'-./&!)-.(!+7)4')-1>-.(!5.:47(+-.(!V11$!V21$!)=)&!-+?=!

)@7!?+=)+'@!+(.!5+1!?9:1!)+!;>-./&!2

1

2

1

2

1

2

1

"

"

VV

V3V3

VV

===ʹ′

ʹ′.

!."%(%

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

"%$$1%+/(!%/(!0-&#'(!%"23" !!""##$% *&*+-0/&!(0-!=/-1-.(!567-.(!V, V1!!/&!5-+./;(!)+!'@/&4+!$$!$1!

5+1!)+!&AA&A++//;7+!')1(!89'&1(!)-.(!)&)+9A*7+!%2.#$!%121.1#1!/&!)1(!

+7)4')-1>&(! ?:&.+;(! )-.(! ?++9::@:&($! %2! 11! %121$…! %! *;:-./&! 7+!

(&4<-./&!=)12!( )( )3

3

#2

2#

VV

ʹ′ʹ′=

ʹ′.

126 !!"#!"#$!"$%&"'!"%#

!"#$!!"#

,')*!!( )( ) S

V#2

2#3

3

=ʹ′ʹ′

!!/&!!!S ≠ V1%!

$!!,')*!S < V1%!%=)&!V1- S!&!&!'!(%

%?-5=?)-7)+(!+?=!)-7!505:-!-&1.$!?-.!&47+1!89'@!)-.!567-.!V1!

)+!&AA&A++//;7+!5+7-7159!?-:0A*7+!!!7!$!7!&!)$!*$!+,$!%!%!%!$!+/+1+-0/&!

-'0//*7+! /&! )-7!.&715=!#,//+.! +?=! )-7! 505:-! +.)=! 59)&!/-+9! ?--'=)@)+!/&A+:0)&+@!+?=!)-!/1'=!)-.!?+-@A-0/&7-.!.?=:-1?-.%

%?-5=?)-7)+(!+5-:-0)*(!+?=!)-7!567-!V1!)1(!?.++/4(&(!/&!89-'&1(!)+!?-:0A*7+!!7$!7!&!)$!*$!+,$!%!%!%!!5+1!03-(!4(1-!/&!)-.!567-.$!+/+1-+-0/&!+?=!)-7!567-!+.)=!59)&!/-+9!?-'=)@)+!/&A+:0)&+@!+?=!)-!/1'=!

)-.!?+-@A-0/&7-.!.?=:-1?-.% $0//*7+! 5+1! ?9:1! /&! )-! .&715=!#,//+$! )+! ?+-503-.7! 59?-1+!

')1A/,!*(!.?=:-1?-!)/,/+)+!)-.!567-.!/&!9)+-1'/+!'$!/15+=)&+-!)-.!

)&)15-0!++1)/-0!&!&!V1- S$!(@:+(,!

X < V1– S, -?=)& S < V1– X. (1)

$!(1+/-+9!=/*(!V1– X!?++1')97&1!)@7!?.++/4(+!B1!?-.!+?-/;7&1!

5+1!@!-?-4+!&47+1!&AA&A++//;7@!')-7!567-!V1$!(@:+(,!B1 = V1– X. ,')*!B!@!+7)4')-1>@!?.++/4(+!@!&AA&A++//;7@!')-7!567-!V. !?-);)-./&! ->*+4(! 8:98@! )@(! A&715=)@)+(.$! =)1! @! '>;'@! -+.!

?+-50?)&1!?%>%!A1+!7!&!*$!(@:+(,!=)+7!?9+-./&!)+!/;'+!"$!#$!C$!E!!)*7!

&."&"!"%"!$!C!*&#&$!%$$!!D%E%#$$$$ 127

)=<*7! ΑΒ ΒΓ ΓΔ ΔΑ∩ ∩ ∩ ∩

! ! ! !!5+1!)+!+7)4')-1>+!/;'+!"1$!#1$!C1$!E1!!)*7!

)=<*7!! ʹ′ ʹ′ ʹ′ ʹ′ ʹ′ ʹ′ ʹ′ ʹ′∩ ∩ ∩ ∩

Α Β Β Γ Γ Δ Δ Α! ! ! . *+!(&4<-./&!=)1!)+!)+4A*7+!?-.!?&+15:&4-.7!5+1!5+)-+4F-.7!+.);(!

)1(!?.++/4(&(!&47+1!+7)1')-4>*(!=/-1+!;7+!?+-(!;7+%!.1+!?++9(&1A/+$!)+!

(&4<-./&!=)1!)+!)+4A*7+!!$"2$!!$1"121!&47+1!=/-1+% 3;+-./&!)1(!&"$!&1"1%!!?&1(,!-1!567-1!&47+1!=/-1-1$!+7!+!5+1!+1!

&47+1!-1!+5)47&(!)*7!89'&*7!)-.($!)+!;>-./&2

"&

&"

2&

&2

+

+

+2+2

#2

2#

$&

&$

ʹ′ʹ′=

ʹ′ʹ′=

ʹ′=

ʹ′=

ʹ′ʹ′=

ʹ′ʹ′.

%/-0!2&

&2

$&

&$

ʹ′ʹ′=ʹ′ʹ′

! !5+1! !∧∧

Σʹ′Οʹ′Βʹ′=ΒΟΣ !-&!+!-+),.$!'0//*7+!/&!

)@7! ?+=)+'@! VI%! +,$! )+! ;>-./&! =)1! )+! )+4A*7+! ΒΟΣ $! 21&1$1! &47+1!

=/-1+$!!-?=)&2!

+

+

2&

&2

2$

$2

ʹ′=

ʹ′ʹ′=

ʹ′ʹ′ (2)

!?4'@(!+/-0!!"&

&"

$&

&$


!!5+1!!∧∧

Σʹ′Οʹ′Κʹ′=ΚΟΣ !!-&!+!-+),.$!)+!)+4-

A*7+!"&$$!"1&1$1!!&47+1!=/-1+$!-?=)&2!

+

+

"&

&"

"$

$"

ʹ′=

ʹ′ʹ′=

ʹ′ʹ′ (3)

%;:-($!+/-0!! ⎟⎟⎠

⎞⎜⎜⎝

⎛ʹ′

=ʹ′ʹ′

=ʹ′ʹ′ +

+

"&

&"

2&

&2!5+1!

∧∧ʹ′Οʹ′ʹ′=Ο KBKB ( 4

81⋅= !-+);(.$!)+!)+4A*7+!2&"$!21&1"1!&47+1!

=/-1+$!-?=)&!!!!!!+

+

2&

&2

2"

"2

ʹ′=


(4)

%?=! )1(! '>;'&1(! -/.$! -3.$! -).! ?+-50?)&12!2"

"2

"$

$"

2$

$2

ʹ′ʹ′=

ʹ′ʹ′=

ʹ′ʹ′,

-?=)&!)+!)+4A*7+!$"2$!$1"121!)+!&47+1!5+1! 1'-A671+! ! -?+=)+'@!VI.5), (@:+(,!=/-1+!-?+=)+'@!VI$!-+1'/=(!+.%

!?-/;7*($!'0//*7+!/&!)-7!-+1'/=!'"%4$!-1!?.++/4(&(!$%&2"!5+1!

128 !!"#!"#$!"$%&"'!"%#

$1%&121"1! &47+1!=/-1&($!+/-0!?&+1;>-7)+1!+?=!=/-1+!5+1! 4'-.!?:,)-.(!

?-:0A*7+%!0++!!( )

( )( )( )

( )( )3

3

3

3

#2

2#

2&

&2

"2&$

&2"$

ʹ′ʹ′=

ʹ′ʹ′=

ʹ′ʹ′ʹ′⋅ʹ′⋅ .

%-!4(1-!'./8+47&1!'&!=:&(!)1(!?.++/4(&(!/&!5-+./;(!$$!$1!5+1!89-

'&1(!)+!1'-'5&:,!)+4A*7+!?-.!;>-.7!5-+./;(!)+!5;7)++!)*7!505:*7!&$!

&1! 5+1! 89'&1(! )1(! ?:&.+;(! )*7! -5)+A67*7! )*7! &AA&A++//;7*7! ')-.(!

505:-.(% !?&1(,!=/*(!'&! 4'+!5:9'/+)+!-!:=A-(!&7=(!)*7!@A-./;7*7!?+-(!

;7+7!+?=!)-.(!&?=/&7-.(!1'-0)+1!/&!)-!:=A-!=:*7!)*7!@A-./;7*7!?+-(!

=:-.(!)-.(!&?=/&7-.(!-?+=)+'@!V%+/.$!?+-50?)&12

( )( )

( )( )

( )( )3

3

#2

2#

"2&$

&2"$

E#C.#2"%$

%"2#.C#E$

ʹ′ʹ′=

ʹ′ʹ′ʹ′⋅ʹ′⋅

=ʹ′ʹ′ʹ′ʹ′ʹ′ʹ′ʹ′ʹ′⋅ʹ′

⋅

%::9!( )( )3

3

#2

2#

SV

ʹ′ʹ′= $! -?=)&! !

B

B

SV

ʹ′= %! !0++! 1

B

VB

S>=

ʹ′$!+/-0!@!

?.++/4(+!B!&47+1!&AA&A++//;7@!')-7!567-!V!5+1!1'>0&1!V!'!B% 0++!S!'!B1!&!V1– X%!-/*($!:=A*!)@(!'>;'@(!-+.$!;>-./&!S < V1– X,

)-!-?-4-!&47+1!9)-?-%!!?-/;7*(!(&7!/?-+&4!7+!1'>0&1!S < V1%

&!!,')*!)6++!=)1!&47+1!S > V1%!!?-/;7*(!;>-./&!( )( )3

3

2#

#2

VS ʹ′ʹ′

= !5+1!

+7! !SV

VS

ʹ′

ʹ′= $! !)=)&!-'0//*7+!/&!)-!?=+1'/+!)@(!?+=)+'@(!V%!5.$!)+!&47+1!

1VS

SV

>ʹ′

=ʹ′

$!!+/-0!S > V1%!0++!V > S1!-:,//+!/&)9!)@7!?+=)+'@!XII. 2).

!?-/;7*(!( )( ) S

V2#

#23

3

ʹ′

ʹ′=

ʹ′ʹ′! !/&!S1!0!V!5+1!)-!-?-4-!/&!'.::-A1'/-0(!?++=-

/-1-.(!/&!+.)-0(!)@(!?&+4?)*'@(!-+.!-(@A&4!'&!9)-?-%!!

%?-5:&4&)+1!:-1?=7!7+!1'>0&1!( )( )3

3

#2

2#

SV

ʹ′ʹ′= !/&!S ≠ V1%!

!?-/;7*(!1'>0&1!!( )( )3

3

#2

2#

VV

ʹ′ʹ′=

ʹ′.

&."&"!"%"!$!C!*&#&$!%$$!!D%E%#$$$$ 129

%7!!V1, V11!&47+1!-1!+7)4')-1>-1!50:17(+-1$!)=)&!!)+!;>-./&2!!V1 = 3 V, V11!&!3!V1$!-?+=)+'@!XII%+(.$!-?=)&2

( )( )3

3

1

1

#2

2#

VV

V33V

VV

ʹ′ʹ′=

ʹ′=

ʹ′=

ʹ′.

!."%(%

4$+$%)+5&5# $)+!!"%&"#'! 5+)+:,A-./&!')@!(1+?4')*'@!=)1! )+! )+4A*7+!$"2$!

$1"121!&47+1!=/-1+!/&!/1+!?-:.?:-5=)&+@!(1+(15+'4+$!&/++/=F-7)+(!)@7!

&$&(")"!! $&$! '%%*! ! -?+=)+'@! V%! //.$! >+@'1/-?-167)+(! 8;8+1+! )+! 4(1+!

F&0A@!-/-4*7!)+1A67*7% $.A5&5+1/;7+$!+?=!)1(!-/-1=)@)&(!)*7!)+1A67*7!2&$$!21&1$1!5+1!

2&"$!!21&1"1!!?+-50?)-.7!-1!'>;'&1(2!

2&

2$

&2

$2

ʹ′ʹ′

ʹ′ʹ′= -6.!!!!!!!!!5+1

"2

2&

2"

&2

ʹ′ʹ′

ʹ′ʹ′= (6)

+?=!)1(!-?-4&(!/&!&/++/-A,!)@(!&$&(")"!+!!$&$!'%%*!?+4+7-./&!!

"2

2"

2&

&2,2&

&2

2$

$2

ʹ′ʹ′=

ʹ′ʹ′ʹ′ʹ′=

ʹ′ʹ′!%!!0++!!

"2

2"

2$

$2

ʹ′ʹ′=

ʹ′ʹ′$!-?=)&

"2

2$

2"

$2

ʹ′ʹ′

ʹ′ʹ′= !,!!

2$

"2

$2

2"

ʹ′ʹ′

ʹ′ʹ′= (7)

$!'>;'@!-5.!)+!/?-+-0'&!7+!?+-503&1!9/&'+!/&!?-::+?:+'1+'/=!

)*7! -6.$! -,.! 5+)9! /;:@%! %;)-1+! 1(1=)@)+! =/*(! (&7! +7+/;+&)+1! ')+!

!"%&"#'!!)-.!!.5:&4(@%! %?=! )1(! -/-1=)@)&(! )*7! )+1A67*7! $&"$! $1&1"1! 5+1! 2&"$!

21&1"1!!?+-50?)-.7!-1!'>;'&1(2!"&

"$

&"

$"

ʹ′ʹ′

ʹ′ʹ′= !!5+1!!

2"

"&

"2

&"

ʹ′ʹ′

ʹ′ʹ′= !!+?=!)1(!

-?-4&(!/&!)-7!4(1-!=?*(!?+17!)+=?-!5+)+:,A-./&!')@!'>;'@!

2"

"$

"2

$"

ʹ′ʹ′

ʹ′ʹ′= (8)

%?=!)1(!'>;'&1(!-5.$!-*.!-/-4*(!8+4'5-./&!

130 !!"#!"#$!"$%&"'!"%#

2$

"$

$2

$"

ʹ′ʹ′

ʹ′ʹ′= (9)

-?*(!/+47&)+1!+?=!)1(!'>;'&1(!-*.!5+1!-4.!)+!)+4A*7+!!$"2$!$1"121!!

;>-.7!)1(!?:&.+;(!)-.(!+79:-A&($!9++!!&47+1!5+1!1'-A671+$!(@:+(,!=/-1+%!

'("%$&%"#'"%)''&"#'!

'(!&!*%("+"3!%$56#,!$#!#7,7#""!7$+-**5*"!&%%3!0-&#%3!%"2$!%#%#!

"%$%+#,%$%!&#!"!"!&2*,("+"23!7"2!#,($%!$(-*"."%!%/(!$8#(/(!%"23" !!""##$% *&*+-0/&!)-7!50:17(+-!/&!9<-7+!%2!5+1!+?=!'@/&4-!.!)-.!%2!

&?4?&(-!?++9::@:-!')1(!89'&1($!)-!-?-4-!(1+1+&4!)-7!++>15=!50:17(+-!'&!

(0-!5.:47(+-.(!V1 , V2!/&!9<-7&(!.%!&!+!$!.2!&!8%!!*;:-./&!7+!(&4<-./&!

=)1!8

+

VV

2

1 = .

!"#$!!"%

*&*+-0/&! &?4'@(! )-7! 50:17(+-!V11! /&! 9<-7+!.#! &! 5%+! 5+1! )-7!

50:17(+-!V21!/&!9<-7+!.!!&!:!⋅!8!=?-.!5$!:!/.'15-4!++1)/-4%!%=)&!V11!& &!!5⋅V1!!5+1!V21!&!:⋅V2

%7!!5+!'!:8$!!)=)&!!!V11!'!!V21$!!(@:+(,!!!5V1!!'!!:V2 %7!!5+!0!:8$!!)=)&!!!V11!0!V21$!!!(@:+(,!!!5V1!0!:!V2 %7!!5+!&!:8$!!)=)&!!!V11!&!V21$!!!(@:+(,!!!5!V1!&!:!V2 1%++$!'0//*7+!/&!)@!)&*+4+!?&+4! 1'=)@)+(!:=A*7!)-.!!0(-<-.$!

)+!?+;?&1!7+!1'>0&1!8

+

VV

2

1 = .

!."%(%

&."&"!"%"!$!C!*&#&$!%$$!!D%E%#$$$$ 131

4$+$%)+5&5! C&!/1+!9::@!(1+)0?*'@!@!+?=(&1<@!)+!/?-+-0'&!7+!A47&1!*(!&<,(2!

*;:-./&! 7+! (&4<-./&! =)1!8

+

VV

2

1 = %!$0//*7+! /&! )@! )&*+4+! ?&+4!

1'=)@)+(!:=A*7!)-.!!.(=<-.$!++5&4!7+!(&4<-./&!=)1!(&7!.?9+>&1!+@)=(!

')-!@/1+7-15)=!(19')@/+!/&!95++!8

+,VV

2

1 .

%7!2

1

VV

:

5

8

+<≤ $!)=)&!:+!!≤!!58!5+1!5V2!!0!!:V1.

-/*(!:+!≤!!58!⇒!.#!≤!!.!!!⇒ V11!≤ V21!⇒!!:!V1 ≤!!5!V2!!-9)-?-.

%7!2

1

VV

:

5

8

+>≥ $!)=)&!!:+!≥!58!!5+1!!5V2!'!:!V1 .

-/*(!!:+!≥!58!!⇒!.#!≥!.!!⇒ V11!≥ V21!⇒!!:!V1 ≥!5!V2!$!-9)-?-.%

%79:-A+!&+A+F=/+')&!+7!2

1

VV

:

5

8

+≤< !!,!!!

2

1

VV

8

+≥

λ

κ> .

'("%$&%"#("%)''&"#(!

!!"!&!*%("+"%!)!&'("%!$#!,&#3!0-&#%3!#,($%!$(-*"."%!%/(!29'(!%"23"! !!""##$%" *&*+-0/&!)-.(!5.:47(+-.(!V1 , V2!/&!4'&(!89'&1(!(1+/;)+*7!%2$!

.#!5+1!03@!$*!&!+!$!"#!&!8%!!*;:-./&!7+!(&4<-./&!=)12!8

+

VV

2

1 = .

!"#$!!"&

132 !!"#!"#$!"$%&"'!"%#

B+-&5)&47-./&!)@7!"#!5+)9!#E!&!+!!5+1!)&*+-0/&!)-7!50:17(+-!

V11!!/&!89'@!.#!03-(!#E% &1! 50:17(+-1! V1, V11! ;>-.7! 4'+! 03@$! -?=)&! &47+1! +79:-A-1! )*7!

89'&67!!)-.(!-?+=)+'@!++.%!%::9!-1!89'&1(!)*7!5.:47(+*7!+.)67!&47+1!

4'&($!-?=)&!V1 = V11!% !?&1(,! -! 50:17(+-(! /&! 89'@! .1#1! ?-.! &47+1! 4'@! /&! )@7! .#! 5+1!

03-(!!"E!&!+!7!8!$!;>&1!)/@)&4!/&!&?4?&(-!?++9::@:-!')1(!89'&1(!)-.$!)+!

;>-./&2!8

+

#"

#E

VV

2

1 ==ʹ′

! -?+=)+'@! +3.%!-/*(! ;>-./&! V1 = V11$! -?=)&!

1'>0&1!8

+

VV

2

1 = .

.1+!)-.(!+7)4')-1>-.(!567-.(!Q1, Q2!!/&!4'&(!89'&1(!(1+/;)+*7!

%2$!.#!5+1!03@!!$*!&!+$!!"#!&!8!!(&4<+/&!-?+=)+'@!+(.!=)1!!Q1 = 31 V1,

Q2 = 31 V2.

!?-/;7*(!)+!;>-./&!8

+

VV

V31

V31

QQ

2

1

2

1

2

1 === .!!!."%(%

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

%"23!#,($%!$(%%&%+#(/3!$(-*".#3!%/(!29'(!%"23"! '(! "%! 0-&#%3! "!"! &2*,("+/(! )! &'(/(! #,($%! $(%%&%+#(/3! $(--*".#3!%/(!29'(!%"23!%#%#!#,($%!,&"%!%#%&""!($$"%&"

!!""##$% (i) *&*+-0/&!?+6)+!(0-!4'-.(!-1'-(07+/-.(.!5.:47(+-.(!V1 , V2

/&!89'&1(!81$!82!5+1!03@!"1#1!$!"2#2%!!*;:-./&!7+!(&4<-./&!=)12

2

1

8

8 =11

22

ΛΚ

ΛΚ .

O!!""!!#"!$!#%&"'"$!%$$!%&#'%($$$$ 133

!"#$!!"#

!!"!#)!!!1(1!"!!2(2#!(*()!*++,!V1 = V2!!-*.!2

1

2

1

/

/

VV

= !$01*(*,2!

%%&#!3*!!45+67)!!/1!"!/2!#!+0*()!2

1

11

22

/

/

(!

(!= = 1.

!!"!#)!!!1(1!'!!2(2!#!(*()!)08!(+6!!2(2!0*81)+67)!(797*!!!2(1-"!

"!!1(1!-*.!*0*!(+!,27)8+!(1-!+41+67)!)080).+!0*1:;;2;+!01+<!(.<!/:(

,).<!(+6!V2. ")+7:=+67)!V1-!!(+)!-,;.).1+!7)!/:,2!/2!-*.!,>+<!!2(1-) #++,!V1 = V2! (*()! ,,7+?)*! 7)! (2)! 01*(*,2!V)*#! 3*! .,5,).! 2!

,54,2!

1

2

1

1

VV

VV

ʹ′=ʹ′

(1)

#;;:#!,,7+?)*!7)!(2)!01*(*,2!%%#!*++,!+.!-,;.).1+.!45+6)!8,*!

,>2!3*!.,5,).!2!,54,2!!!

2

1

1

1

/

/

VV

=ʹ′

(2)

%@:;;+6#!,,7+?)*!7)!(2)!01*(*,2!%+#!

11

22

12

22

1

2

(!

(!

(!

(!

VV

=ʹ′

=ʹ′

(3)

#0*!(.<!,54,).<!$%&#!$,&!#!$+&!01+-,0().!*(.!!11

22

2

1

(!

(!

/

/= .

134 %A!(%"'$!-$%""B%"#.

ii)!/,(?!(C1*!*(.!D.*!(+6<!-6;8).1+6<!V1!-*.!V2!7)!*)(8,(+.5)<!

/:,).<!/1!-*.!/2!!45+67)!11

22

2

1

(!

(!

/

/= )!&4;+67)!)*!.)8@+67)!*(.!!V1 = V2

!!"!!#)!!!1(1!"!!2(2#!!(*()!/1!"!/2!#!+0*()!!V1 = V2

!!"!#)!!1(1 '!!2(2#!(*()!3)?1C)(*<!(+)!-,;.).1+!V1-!$*0?<!*6(

(*<!-*(*,-)6:,(2-)!01+2D+674)?<!,(+!$i&!/&#!3*!45+67)/

12

22

2

1

(!

(!

/

/

ʹ′= (4)

#;;:#!,,7+?)*!7)!(2)!01*(*,2!%%#!*++,!+.!-,;.).1+.!45+6)!8,*!

,>2!3*!.,5,).!

1

1

2

1

VV

/

/

ʹ′= (5)

%@:;;+6#!,,7+?)*!7)!(2)!01*(*,2!%+#

1

2

12

22

VV

(!

(!

ʹ′=

ʹ′ (6)

#0*!(.<!,54,).<!$0&#!$1&#!$2&!01+-,0().!*(.1

2

1

1

VV

VV

ʹ′=

ʹ′#!+0*()!V1 = V2.

!"#$!!"#$"

$(+6<!-C)+6<!)+*17*=+)(*.!+.!8..+.!7)!(+6<!01+2D+,7))+6<!,6;-;+D.,7+,<#!*0?<!+*8))(*.!-*.!,(+!,597*!%1)%#!+0*()!*0+.).-),)(*.!-*.!

(+!*)(8,(+.5+!*0+(4;),7*)!!! !.").)

O!!""!!#"!$!#%&"'"$!%$$!%&#'%($$$$ 135

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

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

'$'*"! -"*$+.!"! )'&-*#/("! %#! 0($)"! -*1#"2! -*#/(3!! %4! #&%--$&%#!"!$"/!%)'(&$#("/!'$'*"/"!"4*%"1!$5$")"!3'$#!")!-*#/(52!

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

1+,7)! )63)8*!0+6! (47)).! (+!7)-D*;,()1+! -,-;+! ,(*! ,27)8*! #!

-*.! 0! -*.! (+! 7.-1*()1+! ,(*! !!

-*.!')!$!-:3)(2!,(2)!#0!,(+!'#!

(47)).!(+!7)D*;,()1+!-,-;+!,(*!

,27)8*! %! -*.! 1! -*.! 01++*)C<!

)+:0()(*.! (+6! 7.-1*()1+6! -,(

-;+6) '.5+(+7+,7)!01C(*! (+! (*-

@+!∩

ΑΒ #! -*.! (+! (*@+! 0+6! 01+-

-,0().!(+!,67/+;8=+67)!7)!∩

ΑΒ21 .

#-+;+,3?<!..5+(+7+,7)!(+!(*@+!∩

#021

#!+0*()!01+-,0().!(+!(*@+!∩

#041

!-*.!

,6))58=+67)!(2)!8..*!..*..-*,8*)!#0*!(2!..*..-*,8*!*6(9!-:0+()!3*!01+-,>).!

4)*!(*@+!∩

ΒΚ 7.-1*()1+!(+6!∩

ΒΕ !$01*(*,2!X. I). $!-:3)(+<!*0*!(+!,27)8+!!!,(2)!#0!(47)).!(2)!#0!,(+!,27)8+!$!

-*.!(+!7)D:;+!-,-;+!,(+!()!%*()!45+67)!∩∩

=0(0! !-*.!$01+(:,).<!III. 3

-*.!I. 4), ∩∩

< %01!0( #!+0*()!"$!3!"'#!.2;*.9!(+!,27)8+!$!)8)*.!)-(*<!

(+6!7.-1*()1+6!-,-;+6) #++,!+.!5+1.4<!!(#! !%1!)8)*.!0*1:;;2;)<!-*.!(+!$!)8)*.!)-(*<!

(+6!7.-1*()1+6!-,-;+6#!2!!(!.)!3*!45).!-+.):!,27)8*!7)!(+!7.-1*()1+!

-,-;+#!.2;*.9!3*!)8)*.!)-(*<!*6(+,)!!*(:!7)8=+)*!;*D+!-*.!+.!5+1.4<!

!0#!0(!3*!/18,-+)(*.!)-(*<!(+6!7.-1*()1+6!-,-;+6) E1++*)C<#!60:15+6)!*1(8+6!0;93+6<!..*.+5.-4<!5+1.4<!(+6!7)D*-

!"#$!!"%

136 %A!(%"'$!-$%""B%"#.

;,()1+6!-,-;+6! 8,)<!7)!(2)!0!#!+.!+0+8)<!/18,-+)(*.!)-(*<!(+6!7.-1*-()1+6! -,-;+6! -*.! *0+();+,)! 0;)614<! .,*0;)61+6! 0+;6DC)+6#! *1(.*-0;)61+6#!)DD)D1*774)+6!,(+!7)D*;,()1+!-,-;+)!!

!.")0)

+&,$!"(%&(%!-%%$%&(" !!!""#"$!!"$"!"%&'#!$(#2!'&%,(#2"!!%!#++(%&#,!'$4!%#+%*$$#(4!

'&%,(%!-"*$#"(""!$"/!"-","/!")!5"(#2!!%!%4!!56"/!!'")!0!'4%#,%!

%#!$4!%)'(&$#(4!'&%,(%! $%4!#&%-$&%#!"!$42!%)'(&$#(42!'&%,%

(%2&# &.,'('/( #<!(723+,)!+.!,+*81)<!*0*!)080).+!E#!0+6!..415)(*.!*0*!(+!-4)-

(1+!"!*6(C)) ".!(+74<!3*!)8)*.!74D.,(+.!-,-;+.#!*++,!-:3)!,+*81*!01+415)(*.!

*0*!,(1++9!27.-6-;8+6!D,1?!*0*!,(*3)19!..:7)(1*!(+6)!

!"#$!!"&

/(,.#!,)!+0+.*.90+()!34,2!-*.!*)!3)?19,+67)! (+!27.-,-;.+#! (+!

)080).+!0+6!+18=).!3*!,527*(8,).!0:)?!,(2)!)0.+:)).*!(2<!,+*81*<!-,(

-;+!-*.!7:;.,(*!74D.,(+#!*++,!2!..:7)(1+<! (2<!,+*81*<!0+6!)8)*.!-*.!

O!!""!!#"!$!#%&"'"$!%$$!%&#'%($$$$ 137

..:7)(1+<! (+6! 27.-6-;8+6#! )8)*.! 2!7)D*;,()12! *0*! (.<! 5+1.4<! (+6! -,--;+6!9!(2<!,+*81*<)!&)?1+,7)!.,+!-:3)()<!..*74(1+6<!#!#!0'!(+6!7)-D.,(+6!-,-;+6!(2<!7)D*;,()12<!,+*81*<!0+6!(47)+6)!(+)!*)(8,(+.5+!7)-D.,(+! -,-;+! (2<!7.-1*()12<!,+*81*<! -*(:! (.<! ..*74(1+6<!#-!-! #!0-'-!

*)(8,(+.5*) #<!)DD1*+)8!,(+!7)D*;,()1+!-,-;+!0+;,D?)+!.,*0;)61+!-*.!*1(.*(

0;)61+#!(+6!+0+8+6!+.!0;)614<!)*!)8)*.!)-(*<!(+6!7.-1*()1+6!-,-;+6. A0+34(+67)!0)5)!*(.!+.!0;)614<!(+6!0+6!/18,-+)(*.!,(+!()(*1(2(

7*1.+!!∩

ΑΒ !!)8)*.!+.!0!#!!(#!(##!##) &)?1+,7)! (2! ..:7)(1+!!'! -*.! (2)!-:3)(2!,(+! )080).+! (+6!-,(

-;+6!#0!'!,(+!"#!0+6!(47)).!(2!,+*81*!,(*! !&! #!&-)!%*!)080).*!$0'#!

"&&!-*.! $!'#"&&!,527*(8=+6)!0:)?!,(2)!)0.+:)).*!(2<!,+*81*<!74D.(

,(+6<!-,-;+6<)

&)?1+,7)!(*!27.-,-;.:!(+6<!∩

0&' -*.!∩

!&' . #++,!2!"&!)8)*.!-:3)(2!,(+!)080).+!#0!'#!-:3)!)080).+!..*!(2<!

"&!3*!)8)*.!-:3)(+!,(+!)080).+!#0!'!!$01*(*,2!B")%4&)!21*!(*!)080)(

.*!0&'!-*.!!&'!3*!)8)*.!-:3)(*!,(+!)080).+!#0!')

#++,!(*!27.-,-;.*!∩∩∩

K&&,0&',0#' !45+6)!8,)<!..*74(1+6<!$"""#!+(

1.,7*<!%&!3*!)8)*.!8,*#!+0*()!-*.!(*!()(*1(27*1.*!∩∩∩

!&,0&,0# 3*!)8)*.!8,*)

3,)<!0;)614<!(+6!0+;6DC)+6!60:15+6)!,(+!()(*1(27*1.+!∩

0# #!(*(

,)<!3*!60:15+6)!-*.!,(*!()(*1(27*1.*!∩

0& !-*.!!∩

!& !8,)<!01+<!(.<!0!#!!(#!

(##!##)!/,(?!+.!0&#!&E#!EF#!F&!-*.!!$#!$%#!%A#!A&!!*)(8,(+.5*) 441)+67)!(2!&$!-*.!*0*!(*!&#!$!(.<!-:3)()<!,(+!)080).+!#0!')!%*!

85)2!(+6<!&-#!$-!3*!04,+6)!,(.<!(+74<!0'#!!'!(?)!)0.04.?)!0&'#!!&'!

7)!(+!#0!'#!*++,!-*.!(*!.,+!*6(:!)080).*!)8)*.!-:3)(*!,(+!#0!') #++,!,(*!8,*!27.-,-;.*!091*7)!8,)<!5+1.4<!!0&#!!$!!-*.!+.!!&&-#!$$-!!

)8)*.!-:3)()<!,(+!)080).+!#0!'#!3*!)8)*.!!&&-!"!$$-!-*.!0&-!"!!$-!$01+(:(

,).<!""")!,*!-*.!")!,2&)! $#6(*!01+-,0().!*0*!(+!*(.!∩∩

= "!$"0& , *++,!(*!.,+,-);9!(18D?)*!"0&#!"!$!!45+6)!8,)<!(.<!D?)8)<!(?)!-+16+C)!(+6<&)

37?<!"0!"!"!!#!+0*()!"0!–!0&-"!"!!–!!$-#!.2;*.9!"&-"!"$-)!

21*! !"$

$!

"&

&0

ʹ′

ʹ′=

ʹ′

ʹ′#!+0*()!+.!&-$-!-*.! !0!!)8)*.!0*1:;;2;)<! ! $01*(*,2!

VI. 2). %0)..9!+.!&&-#!$$-!)8)*.!-:3)()<!,(+!)080).+!#0!'#!3*!)8)*.!0*-

138 %A!(%"'$!-$%""B%"#.

1:;;2;)<!$01*(*,2!XI)!2&!-*.!*++,#!*0?<!.)8@*7)#!&&-"!$$-#!3*!45+67)!*(.!+.!!&-$-#!&$!3*!)8)*.!8,)<!-*.!0*1:;;2;)<!$01*(*,2!")!++&!)!#;;:!+.!

&-$-!#!0!!)8)*.!0*1:;;2;)<#!+0*()!-*.!+.!&##!0!!3*!)8)*.!0*1:;;2;)<!

$01*(*,2! ")! +5&)!".! )63)8)<! ;+.0*)!0&#!!$!0+6!,6).4+6)!,27)8*! (?)!

.,+!*6(C)!0*1*;;9;?)#!3*!)8)*.!,6))080).)<) %0+74)?<#!(+!()(1:0;)61+!!0&$!)8)*.!)080).+! $(1*04=.+&! $01*(

(*,2!B")!*&)!!.*!(+)! 8..+!;*D+!(*!()(1:0;)61*!$&E%#!%EFA!)8)*.!)08(

0).*#!*0?<!-*.!(+!(18D?)+!AF&!)8)*.!)080).+!,597*) #)!))C,+67)!(*!,27)8*!&#!$#!E#!%#!F#!A#!&!!7)!(+!"!3*!,527*(.(

,()8!7)(*@,!(?)!(*@?)!∩

0& , ∩

!& !,()1)*!,597*#!0+;,).1+#!*0+();+,7)(

)+!*0*!061*78.)<!7)!/:,).<!(*!()(1:0;)61*!!0&$#!$&E%#!%EFY!!-*.!

(+!(18D?)+!AF&#!-+16+9!.)!(+!,27)8+!")! #)!-:)+67)!,)!-:3)!78*!*0*!(.<!0;)614<!!(#!(##!##!(.<! 8..)<!

-*(*,-)64<!0+6!-:)*7)!)08!(2<!0!#!-*3C<!-*.!,(.<!0;)614<!0+6!/18,-+(

)(*.!,(*!:;;*!(18*!()(*1(27*1.*!∩∩∩

ΓΒΔΓΔ ,,A !$-*.!,(+!:;;+!27.,+*81.+&#!

3*!,527*(.,()8!,597*!0+;,).1+!)DD)D1*774)+!,(2!7)D*;,()12!,+*81*#!

0+6!*0+();)8(*.!*0*!061*78.)<!7)!/:,).<!(*!()(1:0;)61*!-*.!(+!(18D?)+!

0+6!01+*)*+41*7)#!-*3C<!-*.!(*!+7+(*D9!01+<!*6(:!$+7+8?<!..*()(*D(

74)*&#!-*.!-+16+9!(+!,27)8+!") ",561.=*7*,()!*(.!+.!4.1)<!*6(+,!(+6!0+;64.1+6!/18,-+)(*.!)-(*<!

(2<! 7.-1*()12<! ,+*81*<! 0:)?! ,(2)! )0.+:)).*! (2<! +0+8*<! /18,-)(*.! +!

-,-;+<!#-0-!-'-) #<!*53)8!*0*!(+!"!2!-:3)(2!,(+!)080).+!(+6!()(1*0;),1+6!!0&$!

-*.!*<!,6)*)(9,).!(+!)080).+!*6(*!,(+!G) 441+67)! (.<! G0#! G!)! #++,! 2! "G! )8)*.! -:3)(2! ,(+! )080).+!

!0&$#!3*!)8)*.!-:3)(2!,)!*;)<!(.<!)63)8)<!(+6!0+6!..415+)(*.!*0*!(+)!

0*.*!(2<! $B"#!+1.,7*<!+&)!21*!2!"G!)8)*.!-:3)(2!,(.<!0G#!G!#!+0*()!

"02!"!G"2!6!G02! #!"!2!"!G"2!6!G!2#! !*++,! !"! !"#∧ ∧

= !"!%+139!

$01*(*,2!")!0*&)!#;;:!"0!"!"!#!+0*()!G"2!6!G02!"!G"2!6!G!2!)!21*!

G0!"!G!! )!$$!.,*(2(*!*6(9!01+-,0().!:7),*!-*.!*0*!(2)!.,*(2(*!(?)!

(1.DC)?)! !&G0#!"G!#! (*!+0+8*!45+6)!"0! "!"!#!G#!-+.)9!-*.!)8)*.!

+13+DC).*!,(+!G&)! "7+8?<!01+-,0().!*(.!();.-:!45+67)!G$!"!G&!"!G0!"!G!)!%0+-

74)?<!(+!()(1:0;)61+!!0&$!)8)*.!)DD1:>.7+!,)!-,-;+!7)!-4)(1+!G!-*.!

*-(8)*!G0)!$%+!*(.!)8)*.!)DD1:>.7+!01+-,0().!-*.!:7),*#!*++,!01*-).(

(*.!D.*!.,+,-);4<!(1*04=.+&)

O!!""!!#"!$!#%&"'"$!%$$!%&#'%($$$$ 139

#++,!!!0!3!&-$-!!-*.!!&-$-!"!&$#!3*!)8)*.!!0!3!&$)!#;;:!!0!"!

"!!$!!"!0&!3!&$#!+0*()!

!02!3!,G02 (1)

$E1:D7*(.!!+∩∩

+ &$!0 "!0!+134<#!+0*()!!0! >∩

!0 0!+134<!#!.2;*.9!∩

!0 !3%!+139)!-#1*!∧

!G0 !3!%!+139#!+0*()!!02!3!G02!6!G!2!"!,G02, *++,!2!!0!)8)*.!0;)61:!(1.DC)+6!*04)*)(.!,)!*7/;)8*!D?)8*&)

#0*!(+!!!+41+67)!(2)!-:3)(2!,(2!0'#!0+6!(2)!(47)).!,(+!,27)8+!$)!

%*()!'$! 3!'"#!+0*()! ,'$! 3!,'"! "!'0)!#;;:! 2

2

!$

!0

0$'$

0$'0

'$

'0=

⋅

⋅= ,

*++,!!∧

0!' "!%+139!$01*(*,2!""")!+%!-*.!01*(*,2!VI)!4#!0*1.,7*&)!%0+74()?<!3*!45+67)!

!02 '!,!!$2 (2)

#0*!(.<!,54,).<!!$%&!-*.!$,&!!01+-,0().!*(.!!$2!3!G02. 37?<!"02!"!"G2!6!G02!#!!"!2!"!!$2!6!"$2!!-*.!"0!"!"!#!+0*(

()!"G2!6!G02!"!!$2!6!"$2!!-*.!*++,!!$2!3!G02!!3*!)8)*.!"$2!'!"G2, .2;*.9!!"$!'!"G)!!*(:!7)8=+)*!;*D+!;+.0*)#!3*!)8)*.!"0-'!"G)!$'2(

;*.9!.))!/;:0()(*.!2!D)).-*(2(*!*)!60+34,+67)!*(.!"0-!'!"$)!#;;.C<!

3*!..0;*,.:=+67)!,6))5C<!(+!0;93+<!(?)!0;)61C)!(+6!)DD)D1*774)+6!

0+;6DC)+6!0!(##!)!)!)#!!7451.<!*(+6!0)(,5+67)!)*!.,5,).!"0-'!"$&) %8)*.!*7?<!2!"G!*0*,(*,2!(+6!-4)(1+6!!"!!(2<!,+*81*<!*0*!(2)!

4.1*!!0&$!(+6!0+;64.1+6#!))C!2!"0-!*-(8)*!(2<!7.-1*()12<!,+*81*<)!

$!4.1*!*6(9!;+.0*)!/18,-)(*.!)-(*<!(2<!7.-1*()12<!,+*81*<) %+! 8..+!*0+.).-),)(*.!-*.!D.*!(.<!60*;+.0)<!4.1)<!$&E%#!%EFA#!

AF&#!-*3C<!-*.!D.*!(.<!+7+(*D)8<!(+6<!$+7+8?<!..*()(*D74))<!01+<!*6(

(4<&#!*0?<!+*8))(*.!*0*! (2)!0*1*(912,2!0+6!*-+;+63)8!7)(:! (+!0*1.(

,7*)! !*(*,-)6:,(2-)! ;+.0*)! 4(,.! 0+;,).1+! )DD)D1*774)+!,(2!7)D*(

;,()12! ,+*81*#! (+6! +0+8+6! +.! 4.1)<! /18,-+)(*.! )-(*<! (2<! 7.-1*()12<!

,+*81*<)!!!.")0)

140 %A!(%"'$!-$%""B%"#.

+,&'"0! !!!'%)!'#!0**4!'&%,(%!#++(%&#,!-"*$#"("!&%")"!-("2!$"!-"*$#-"("!-"/!#++(07%%#!'$4!'&%,(%!!()*"!$%!"$"!-"*$#"(%!#%!#,!%)!

%!0*"+%!$.!!")%%5$(.!!$.!!"$"!'&%)(3!# &.,'('/( %*!0+;,).1*!E1#!E2!5?18=+)(*.!,)!*7+.)<!061*78.)<!7)!+7*;+D)<!

0;)614<!(.<!*-(8))<!11#!12!(?)!.,+!,+*.1C))!21*!!!( )( )3

2

31

32

31

2

1

2121

1

1

E

E== .

8%(%$1(4'4 !*(:!(2)!*0*.).@2!.,56-

1.,(9-*7)! *(.#! ..0;*,.:=+)(*<!

,6))5C<! (+! 0;93+<! (?)! 0;)6-1C)! (+6! )DD)D1*774)+6! 0+-;6DC)+6!0!(##! )! )! )! #! !70+-1+,7)! )*! 0)(,5+67)! )*! .,5,).!!

"0-!'!"$! )!#1-)8!D.7!*6(*!)*!

.)8@+67)! *(.! (+!0$! 70+1)8! )*!

D8)).! $7)! ,6))59! ..0;*,.*,7*!

(?)! 0;)61C)! (+6! 0+;6DC)+6&!

7.-1*()1+!*0*!+0+.+.90+()!)6-3,D1*77+! (797*#! :1*! -*.!*0*!

(+!"0!-!"0-)!%*()!2!,54,2!0$!'!"0!-!"0-#!,27*8)).!01:D7*(.!"0-!'!!"0!-!0$!"!"$)

!.*!)*!70+1)8!;+.0*)!(+!0$!)*!D8)).!7.-1*()1+!*0*!+0+.+.90+()!

)63,D1*77+!(797*#!*1-)8! $,,7+?)*!7)!(+!!)).-*!(977*&!)*!.)8@+67)!

*(.#!*)!3)?19,+67)!0+;,D?)+!7)!0;)614<!8,)<!7)!0!-"!!-!#!.2;*.9!7)!

..0;:,.+!0;93+<!0;)61C)#!(*()!(+!0$-!)8)*.!7.-1*()1+!*0*!(+!20$

#!9!

*(.!!$$-!3!20$ .

E1:D7*(.#!*++,!!-0!"!!-!#!3*!45+67)!"!-⊥!0!#!!"0!"!"!#!+0*()!

-*.!!"-$!"!"-0!"!!"

!.

#++,!""!'!"!-#!3*!!)8)*.!-*.!""-'!"$-#!+0*()!$"-'!$$-#!.2;*.9!!

20$

!'!$$-)

!"#$!!"&$"

O!!""!!#"!$!#%&"'"$!%$$!%&#'%($$$$ 141

#)! ,27)1.)+,<! *1+6<! /4/*.*! *)! 34,+67)! ν

∧

ω==)

3600"!

"

,

(*()!3*!45+67)!2?

900""90!0" )++ −=−=∧∧

)!!21*!0$!"!0!)!!

,6)!$85+! - $!

% &!"!0!!27! ⎟⎠

⎞⎜⎝

⎛⋅=⎟

⎠

⎞⎜⎝

⎛⋅⋅=⎟

⎠

⎞⎜⎝

⎛2?

27R22?

270"22? )2)) ,

*0+6!R!"!"0)

#++,!?) →!5!!-*.!!!27!$

!"

%⎛⎝⎜

⎞⎠⎟ → #!3*!45+67)!-*.!!lim!$0$&!"!5)

21*#!D.*!!)!!*1-+,)(?<!7)D:;+!3*!D8)).!(+!0$!7.-1*()1+!(+6! "0!-!"0-"!R - R-)!

",561.,(9-*7)! )08,2<! -*(:! (2)! *0*.).@2! *(.! -*.! +.! *0+,(:,).<!

(+6!"!*0*!(.<!4.1)<!$&E%#!%EFA#!AF&!)8)*.!7)D*;,()1)<!*0*!(2)!*-(8(

)*!R-"!OB-! (2<!7.-1*()12<!,+*81*<#!.2;*.9!/18,-+)(*.!)-(*<! (2<!7.(-1*()12<!,+*81*<)

%0)..9!"G2!"!"02 -!0G2 = = R2 -!0G2, !*1-)8!)*!.)8@+67)!*(.! +.! *-(8))<! (?)! 0)1.D)D1*7-74)?)! -,-;?)! ,(*! 0+;,D?)*!

!0&$#!$&E%#!%EFA#!AF&!/*8(

)+6)!);*((+,7)))<!9!(+6;:5.,(+)!

72!*6@*)*7)))<) &)?1+,7)! (+6<! 0)1.D)(

D1*774)+6<! -,-;+6<! ,(*! ()(

(1:0;)61*! !0&$#! $&E%! ! 7)!

*-(8))<!11!#!12!*)(8,(+.5*!$/;40)!,597*!%*)!,&) /5+67)!!0!"!0&!"!!$!"!*!3!&$!"!/!-*.!&E!"!$%!"!*!3!&$!"!/!3!!E%)

A0+34(+67)! *(.! 12 3! 11. /,(?

∧

!0"1 =∧

&0"1 = x$!"1 =∧

, y&"1 =Σ∧

! -*.!!∧

E&"2 = ∧

%$"2 !"!3#! ?$&"2 =∧

, &! ' #!

∧

= )!%*()!y < x!!-*.!+!'!?#!*++,!*!3!/!3!!E%)

/5+67)!*7?<!27!11 21

*

1

2*

2x

== #!27!(

!

$

!)!

= !-*.#!*++,!12!3!11#!3*!

!"#$!!"&$'

142 %A!(%"'$!-$%""B%"#.

)8)*.!!27(

!'!27!

2x#!!+0*()!!

(

!< #

! (< 90+&!!-*.!3!' x)!"7+8?<!?!'!y < x,

+0*()!-*.!+!'!?!'!y. 21*!,3!6!?!6!+!'!,x + x + y = 3x + y = 360o!#!))C!,3!6!?!6!+!"!+25+ ,

01:D7*!:(+0+)!%0+74)?<!!12 ≤!!11 .

+&,$!"(%&#%!-%%$&#"% 9)!'&%,(#2!#,!%)!%!0*"+#2!$.!!'$:.!!$.!!")%%5$(.!!$"/2# &.,'('/( &)?1+,7)!(.<!,+*81)<!V1 ,

V2!7)!*-(8))<!11#!12! !-*.!-4)(1*!

"1#!"2)!&4;+67)!)*!.)8@+67)!*(./

( )( )3

2

31

2

1

2121

VV

= #!9!!!

3

2

1

2

1

1

1

VV

⎟⎟⎠

⎞⎜⎜⎝

⎛= .

/,(?! ;+.0*)! *(.! .,5,).!3

2

1

2

1

1

1

VV

⎟⎟⎠

⎞⎜⎜⎝

⎛≠ )! &*! .)8@+67)! *(.!

*6(*!+.2D)8!,)!:(+0+)

E1:D7*(.#!(*()!3*!45+67)!VV

1

1 13

2

1 =⎟⎟⎠

⎞⎜⎜⎝

⎛#!!7)!!V ≠ V2!)!A0+34(+6(

7)!*(.!2!V!!)8)*.!7.*!,+*81*!+7*-))(12!(2<!V2!7)!*-(8)*!1) !"!/,(?!*(.!)8)*.!V < V2!)!%*()!3)?1+,7)!(+!0+;,).1+!E2!(+!)D(

D)D1*774)+!,(2)!V2!!7)!(.<!4.1)<!(+6!)-(*<!(2<!,+*81*<!V!$01*(*,2!%*&!

-*.!(+!*7+.+!01+<!*6(*!0+;,).1+!E1!(+!)DD)D1*774)+!,(2!,+*81*!V1 . %0+74)?<#!,,7+?)*!7)!(+!0*1.,7*!(2<!01*(*,2<!%*#!3*!45+67)/

( )( )

3

2

13

2

31

2

1

1

1

2121

E

E⎟⎟⎠

⎞⎜⎜⎝

⎛== .

21*!3*!.,5,).!2

1

2

1

VV

E

E= #!+0*()!45+67)!

1

1

2 E

VE

V= *++,!)8)*.!

V1!3!E1)!!%0+74)?<#!,,7+?)*!7)!(2)!01*(*,2!9)!%,#!!45+67)!V!3!E2.

!"#$!!"(

O!!""!!#"!$!#%&"'"$!%$$!%&#'%($$$$ 143

37?<!V!'!E2#!!*++,!2!,+*81*!V!)8)*.!,(+!),?()1.-*!(+6!E2)!!*(

(*;9@*7)!;+.0*)!,)!:(+0+)!%0+74)?<!.))!70+1)8!)*!)8)*.!V<V2.

!"%/,(?!(C1*!*(.!)8)*.!V > V2! )!/5+67)! !

3

1

2

1 1

1

VV

⎟⎟⎠

⎞⎜⎜⎝

⎛= ! $01*(*,2!

V)!*#!!0*1.,7*&)

#)!VV

VV 2

1 ʹ′= #! ! (*()! ! 1

VV

VV

2

1 >=ʹ′

,

*++,! !V > V2)! -#1*! !V1 > V-! $01*(*,2! ,#!

(977*&)

%0+74)?<!)

)

$

$

!

%

!⎛

⎝⎜

⎞

⎠⎟ =

ʹ′

&

!7)!!V-!'!V1.

#):D)(*.! ;+.0*)! ,(2)! 01+2D+,7))2! 0)18-0(?,2! -*.! 7)! 0*1*7+.+6<! *0?<! 01.)! ,6;-;+D.,7+,<! +.2D+,7*,()! ,)! :(+0+)! %0+74()?<#!.))!70+1)8!)*!.,5,).!V > V2#!:1*!)8)*.

V = V2#!!+0*()!

3

2

1

2

1

1

1

VV

⎟⎟⎠

⎞⎜⎜⎝

⎛= .

!.").)

8%(%$1(4'4 2@.+!01+,+59<!)8)*.!(+!D)D+)*<!*(.!0+63)):!,(*!!"%&"#'!!(+6!%6(

-;)8.2! .))! )7+*)8=)(*.! +*.)*7))+! 74(12,2<! ))*<! )080).+6! 9! ,()1)+,!

,597*(+<!*0?<!(2)!)))++,7)!,97)1*# .2;*.9!.))!60+;+D8=)(*.!)7/*.*)!

9!*D-+<!7)!/:,2!-:0+.*!,6D-)-1.74)2!7+):.*!74(12,2<)! %+!347*!*6(*!*)(.7)(C0.,)!..)@+..-:!+!#15.79.2<!+3:)+)(*<!74(

51.!(2!74(12,2!-,-;+6#!-6;8).1+6#!-C)+6!-*.!,+*81*<!74,?!(?)!)DD)(

D1*774)?)!0+;6DC)?)#!01.,7:(?)#!061*78.?)!*)(8,(+.5*!-*.!/18,-+(

)(*<! (.<!,54,).<!0+6! 45).!+!*D-+<! ))*<!-6;8).1+6!7)! (+6<!*D-+6<! (?)!

)DD)D1*774)?)!,7!*6(*)!-C)+6!-*.!,+*81*<) $(+!41D+!(+6!)()*+%,!-."/0$&1)!-*(:+)1)!)*!/1).!78*!,27*)(.-9!

01+,4DD.,2!(+6!*1.37+,!0#!(2)!+#%0%2#!74,?!(?)!;*D?)!2RE) #!)!"!2#!%,#!

,0#!04#!82#!*0+6!E)!2!0)187)(1+<!-*)+).-+,!)-DC)+6!)DD)D1*774)+6!,)!-,-;+!*-(8)*<!R.

!"#$! 18.1

.

!!""##$% 3 !"!!""!XI""

&#!'#(%($'"!$%!)!"

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

%-.!#%.%'!1&.(.$13.!$%','3.!(0+&(2!%()!%'%,&%(,()"!%()!145()"!%()!(-1%&%(,()"! %()! '$1($&%(,()! 1&$! %()! (-('1&%(,()#!))%6!&.%$''%-#/7(-.%&$!$%$8!#,(%6$'$8!$%!'%9,$!1&$!$&"!$%(!!$5+/(!XIII.

:&;%.&!&#*! %&!1&.(.$16!$%','6!1&%&$1')67'%&$!1&$!'&&,6*'%&$!

$'!(($'%.0!$*&/,&#!##$#+%(."!'<'%67'%&$!0!$9%$0!''%&<4!%08!#+'),68!

%()!$%','(4!1&$!%08!($&'%%,()!%08!#',$&'&,&''%.08!$*&/,&8#!=%+(8"!-8!

$)'#%,&$'&"!&.&*%,'%&$!*%$!('.!)#6,9().!6++&!1&.(.$16!$%','6"!'1%*8!

&#*!&)%6!%&!(#(/&!%9().!2(0!1&%&$1')&$;'/# )#*! %$8! )#*+($#'8! $'!$).(+$16! #,(%6$'$8! %()!1'*&+&/()! $%"! ($!

%<$! #,3%'8! &.&*%,(.%&$!$%0! ($&/,'$0! ');4&,&''()! %'2'&%(8!$'!61,(!

1&$!'%$(!+*&("!($!'#*''.'8!#%.%'!$%(!1&.(.$1*!#'.%6&-.(!1&$!0!(-(%(

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

#&,&$1')&$%$1%8!&$&!%$8!)#*+($#'8!#%.%'!#,(%6$'$8"!(0+&(2!&$&!%$8!#,((

%6$'$8!$%"!$)"!$*"!$+!1&$!$&#!>$%*$("!(,$$'%.'8!&#*!&)%%8!#&,()$$67().!

$($&/%',(! '.($&*%,(.#!?%0.! #,*%&$0! +! &#(('$1.4'%&$! *%$! &.! %.&! ');4-&,&''(!%'2'&!($&$,';'/!$'!61,(!1&$!'%$(!+*&("!%*%'!1&;%.&!&#*!%&!%'2-'&%&! %08! ($&/,'$08! '/.&$"! $'! $9%$0! ''! %(! ');4&,&''(! %'2'&"! 6,,0%(!

146 #@:!#"+,!,?=""A#")-

.6+(&(/!1&$!'/.&$!0!+'&*''.0!#("!")*#!?%0.!#,*%&$0!$$!&#(('$1.4'%&$!

*%$!&.!$'!141+(!''!,0%2!($6''%,(!'&&,&*'/!1&.(.$1*!#'.%6&-.("!%*%'!0!

#+'),6! %()! #'.%&&3.()! '/.&$! 6,,0%(8! .6+(&(8/! 1&$! '/.&$! 0! +'&*''.0!"+,$$&'#!?%0.!#,*%&$0!$'!%%+(8"!'1%/;'.%&$!$%(! /($(!$92'&!($!#+'),%8!

%-.!#%.%'!1&.(.$13.!$%','3.!1&$!$)&1,/.(.%&$!''%&<4!%()8#!

$#!'$#-)!$.!$!!"')%!'#$!/0$%!#1*%#!&"#)/%*02! ?%$8!#,(%6$'$8! $!'%9,$! +!($&%)#3.(.%&$!1&$!&#(('$1.4(.%&$!$9%-

$'$8! ''%&<4! %-.! %'0'6%-.! .2! 16#($-.! 1&%&$1')3./! #()! #,(14#%().!

&#*!%0!($&/,'$0!'.*8!');4&,&''()!%'2'&%(8!$'!61,(!1&$!'%$(!+*&(#!"$!

&.%/$%($9'8!''%,$1%8!$9%$'$8!&#(('$1.4(.%&$!''! ''5&(6!1&$!0!'%;(((8!

;&! '#(,(4$'! .&! '.%&9;'/! $%0! 9,2$0! %()! &.3'(.&"! *#-8! &)%2! ''+'-%2;01'!1&$!$9(+$6$%01'!$%(!!$5+/(!""#!

3)1%#$.!" (XIII.1) !!"!"#$%&'#(()!$(*(#!%+#+'"$",!&"!!-')!-#+!(.&)!/0&)"!$)!$"#$'!&1")! $)#! "#$%&'#(()#! $(*(#$)2!3)#!3')-%3$"+!12!!$')+-&(#!$)#!("&#/%$"')#!$(*(#$)2!$42!%+#,'"&42!-#+!$)#!(+&)%!$)#!#'5+-)%!$(*(#$)2"!","#+!3""$#3/!&+)!#30!$)!$"$'!&1")!$)#!(+#&)%!$)#!#'5+-)%!$(*(#$)2$ (41)!$5. =%'.()''! %(! %'2'&!)!!$'! 61,(!

1&$! '%$(! +*&(! $%(! $0''/(! -! ''! ''&&-+4%',(! %'2'&! %(! )-! 1&$! $%0.! #,(%-1%&$0! %08! -)! +&'56.()''! %'2'&! )+!

/$(!''!%(!'$$*!%()!)!#! .&!&#(('$9;'/!!*%$!!+-2!0!*)+2. ).&&,6*()''!%&!%'%,6&-.&!

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

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

(2!%(!)!!%9'$!($&$,';'/!$%(!-!$'!61,(!

1&$!'%$(!+*&(!%9()''1

)!·!-!0!)-2 !$*)#!"

=)!:)B"B":)!?=#C#)! 147

:&$!%(!)!·)-!$$(4%&$!''!%(!''5&(*.!%()!(,;(&-./()!-!#,"!'.3!%(!)-2!''!%(!''5&(*.!%()!%'%,&&3.()!D.E/#!

0,&! !''5! .-!#,/!0!''5! .D.E//#!##'$(2!!)!0!2)+"!!)!0!:)!

1&$!)+!0!)."!%#'%&$!*%$!:)!0!2).#

##('%.-8! ,)()(

ΑΘ

ΑΚ=

ΑΓΜΘεµβ

ΑΓΗΚεµβ6,&!''5!.)-,:/!0!2''5!.).E-/#!!

1%$$"!%(!''5&(*.!%()!&.3'(.&!!".)-/!$$(4%&$!''!%(!''5&(*.!%()!%'-%,&&3.()!)!#:"!(0+&(2!''!))+2! "!($*%$!)!!0!2)+#!0,&!(!&.3'(.&8! !".)-/!'/.&$!/$(8!''!))+2"!'#('%.-8!!+-2!0!*)+2.!!!

!."#(#

3)1%#$.!$!%6***#!$& !!"! $)! $"$'!&1")! "#$#&'!(()#! $(*(#$)2! ","#+! 3""$#3/!&+)!#30! $)! $"$'!&1")! ""02! $(*(#$)2! ##$)%! -#+! $)! %+3/!&+)! $)#!$(*(#$)2!%+#+'"$",!&"!!-')!-#+!(.&)!/0&)"!$0$"!$)!("&#/%$"')!$(*(#!$42!%+#,'"&42!","#+!$)!#30/)+3)!$)#!#'5+-)%!$(*(#$)2$ (41)!$5. 1$%-! *%$! %(! %'%,6&-.(! %()!

%'2'&%(8! )!! '/.&$! #'.%&#+6$$(!

%()! %'%,&&3.()! %()! %'2'&%(8!)-!

1&$!%$%-!-+!0!2)-#!.&!&#(('$9;'/!

*%$! &.! %(! %'2'&! -+! ($&$,';'/! $'!

61,(! 1&$! '%$(! +*&("! %*%'! %(! ''-&&+4%',(!&#*!%&!%'2'&%&!%08!($&/(

,'$08!'/.&$!%(!%'2'&!-!# ).&&,6*()''!%&!%'%,6&-.&!

)!/!!1&$!-+,:# ##'$(2!)!2!0!*)-2 %9()''!

''5!.)!/!/!0!*''5!.)-."/

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

%()! ''5&((4! %()! %'%,&&3.()! )-."#! 2'-8! %(! ''5&(*.! %()!

&.3'(.&! $$(4%&$! ''! %(! ''5&(*.!

%()!%'%,&&3.()!-+,:#! !$*)#! 2

148 #@:!#"+,!,?=""A#")-

##$#+%(.!+-!0!2-)"!!+-!0!-:!!!1&$!!!)-!0!-.!!.6,&!-:!0!2-./# ##('%.-8!''5!.-!#:/!0!2''5!.-!E./!1&$!''5!.D/E./!0!''5!.!+,#/# ##/$08"!%(!''5&(*.!%()!%'%,&&3.()!D/E.!$$(4%&$!''!!-2!1&$!%(!

''5&(*.!%()!(,;(&-./()!!+,#!$$(4%&$!''!!+·-+#! 0,&!!-2!0!-+·!+"!(0+&(2!%(!!!($&$,'/!%(!-+!$'!61,(!1&$!'%$(!

+*&(!''!''&&+4%',(!%'2'&!%(!!-#!!!."#(#

7,//# %'(!&!)($ (41)!$5. 1$%-!*%$!%9()''!2)-!0!!-#!=*%'!!-2!!0!))-2"!6,&!!-2!3!-)2 =

0!*)-2. 19()''!*'-8!*%$!!)2!0!*-)2!"!6,&!!)2!0!!-2!3!-)2"!%(!(#(/(!'/(

.&$!&(4.&%(# "'(/-8!&#(('$1.4'%&$!*%$!(4%'!0!$9%$0!2)-!4!!-!'/.&$!().&%*.!

.&!$$94'$"!($*%$!1&%6!''/7(.&!+*&(!;&!'/.&$!6%(#(#!##('%.-8!2)-!5!!-# !."#(#

3)1%#$.!'!%6***#!'& !!"!"#$%&'#(()!$(*(#!%+#+'"$",!&"!!-')!-#+!(.&)!/0&)"!$0$"!$)!$"$'!&1")!$)#!"#$%&'#(()#!$(*(#$)2!3)#!3')-%3$"+!12!!$')+-&(#!$)#!(+-'0$"')#!$(*(#$)2!$42!%+#,'"&42!-#+!$)#!(+&)%!$)#!("&#/%$"')#"!","#+!3""$#3/!&+)!#30!$)!$"$'!&1")!$)#!(+&)%!$)#!("&#/%$"')#!$(*(#$)2$ (41)!$5. 1$%-! )!! ');4&,&''(!

%'2'&"! %(! (#(/(! %9'$! ($&$,';'/!

$%(!$0''/(!-!$'!61,(!1&$!'%$(!

+*&(! ''! ''&&+4%',(! %'2'&! %(!

)-!1&$!%$%-!+!%(!'%$(!%()!)-#!

.&!&#(('$9;'/!*%$!+!2!0!*+-2. ).&&,6*()''! %(! %'%,6-

&-.(!)!#"#!##'$(2!)-! 0!2+-!

%9()''!)-2!0!)+-2"!(0+&(2!%(!

''5&(*.!%()!%'%,&&3.()!C"?"!

!$*)#! 3

=)!:)B"B":)!?=#C#)! 149

'/.&$!%'%,&#+6$$(!%()!''5&((4!%()!%'%,&&3.()!,:/"!# ##'$(2!)!·!-!0!)-2"!%(!''5&(*.!%()!(,;(&-./()!-!#?!$$(4%&$!

''!%(!''5&(*.!%()!%'%,&&3.()!C"?"#!2'-8"!$$94'$!*%$1!

''5!.C"?"/!0!)!''5!.,"/:/"!'#('%.-8!''5!.-!#?/!0!)!''5!.,"/:/#

##'$(2!)+!0!+-!'/.&$!.:!0!:/"!6,&!,:!0!:!"!(0+&(2!EB!0!B##! 1%$$!%9()''1

''5!."EB//!0!''5!./B#?/!!1&$!''5!./B#?/!0!''5!.+-",/#

##('%.-8! %(! ''5&(*.! %()! &.3'(.&!+,"/B!! $$(4%&$! ''! %(! ''-5&(*.!%()!(,;(&-./()!-!#?"!(0+&(2!''!%(!%'%,&#+6$$(!%()!''5&((4!%()!

%'%,&&3.()!,"/:#!0,&!%(!''5&(*.!%()!%'%,&&3.()!+!B:!'/.&$!#'.%&(

#+6$$(!%()!''5&((4!%()!%'%,&&3.()!,"/:"!(0+&(21!!+!2!0!*!+-2. !."#(#

3)1%#$.!(!%6***#!(& !!"!"#$%&'#(()!$(*(#!%+#+'"$",!&"!!-')!-#+!(.&)!/0&)"!$0$"!$)!!$')+&(#! $1"! $"$'#&6"1"! $)#! 0/)#! $(*(#$)2! -#+! $)#! (+-'0-$"')#!$(*(#$)2!$42!%+#,'"&42!","#+!$'+3/!&+)!#30!$)!$"$'!&1")!$)#!("&#/%$"')#!$(*(#$)2$ (41)!$5. 1$%-! *%$! %(! ');4&,&''(!

%'2'&!)!!%9'$!($&$,';'/!$%(!$0-''/(!-!$'!61,(!1&$!'%$(!+*&(!''!

''&&+4%',(! %'2'&! %(! )-#! .&!

&#(('$9;'/!*%$1!! )!2!3!!-2!0!%!-)2.

).&&,6*()''! %(! %'%,6&-(

.(!)!#+#! ##'$(2! %(! %'2'&!)!!

%9'$! ($&$,';'/! $%(! $0''/(! -! $'!

61,(! 1&$! '%$(! +*&("! %9()''1!!)!·!-!0!)-2

:&$!%(!''.!)!·!-!$$(4%&$!''!%(!''5&(*.!%()!(,;(&-./()!)!:.!%(!('!)-2!''!%(!''5&(*.!%()!%'%,&&3.()!./,+#! 0,&!!''5!.)!:./!0!''5!../,+/#

!$*)#! 4

150 #@:!#"+,!,?=""A#")-

##/$08!''5!.)!:./!0!''5!.-!#,/#!0,&!%(!''5&(*.!%()!&.3'(.&!

)./,#!!$).!%(!''5&(*.!%()!%'%,&&3.()!-!:/!$$(4%&$!''!%(!($#+6$$(!

%()!''5&((4!%()!%'%,&&3.()!./,+#!##('%.-81!

''5!.)!#+/!3!''5!.-!://!0!%''5!../,+/"!

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

3)1%#$.!)!%6***#!)& !!"! "#$%&'#(()! $(*(#! %+#+'"$",! &"! !-')! -#+! (.&)! /0&)! -#+!3')"-$#$",!-#$!!$(*(#!,&)!3')2!$)!("&#/%$"')!$(*(#!$42!%+#,-'"&42"!$0$"!$)!0/)!"#$%&'#(()!$(*(#!.5"+!%+#+'"$",!&"!!-')!-#+!(.&)!/0&)!-#+!$)!("&#/%$"')!$(*(#!$42!".#2!%+#,'"&42!","#+!$)!#'5+-0!"#$%&'#(()!$(*(#$

(41)!$5. 1$%-!*%$!%(!%'2'&!)!!

%9'$! ($&$,';'/! $%(! $0''/(! -!

$'!61,(!1&$!'%$(!+*&(!''!''-&&+4%',(!%'2'&!%(!)-#!F,((

'1%'/.()''! %(!!)! 1&%6! %'2(

'&!)+!0!)-# .&! &#(('$9;'/! *%$! %(!

%'2'&!+!!%9'$!%'0;'/!$'!61,(!

1&$! '%$(! +*&(! $%(! $0''/(!)!

''! ''&&+4%',(! %'2'&! %(!)!"!(0+&(2!!!+·+)!0!)!2.

).&&,6*()''! %(! %'%,6&-.(!)!#/#!##'$(2! %(!)!! %9'$!($&$,';'/!

$%(!-!$'!61,(!1&$!'%$(!+*&(!''!''&&+4%',(!%'2'&!%(!)-"!%9()''! )!·!!-!0!)-2. 0,&"!

''5!.-!#"/!0!''5!.)-,./#!)++6!''5!.-!#"/!0!''5!..:#//#!

##/$08!!''5!.)-,./!0!''5!.+).!/#! ##('%.-8!''5!.)!#//!0!''5!.!:!+/"!(0+&(2!!+·+)!0!)!2. !."#(#

!$*)#! 5

=)!:)B"B":)!?=#C#)! 151

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

"$! #%.%'! #,3%'8! #,(%6$'$8! #()! &#(('/9;01&.! #&,&#6.-! 1&$! ($!

(#(/'8!&*(,(4.!$%0!($&/,'$0!');4&,&''()!%'2'&%(8!$'!61,(!1&$!'%$(!

+*&("!'/.&$!5(0;0%$1%8!&$&!*$&!&1(+();(4.!1&$!1),/-8!$%0.!1&%&$1')2!

%()! 1&.(.$1(4! (-('1&%(,()! 1&$! %()! '$1($&%(,()#!##/$08"! &)%%8!'#(-,(4.!$2'',&!.&!&#(('$9;(4.!1&$!&+&'5,$16"!*#-8!;&!&/.'$!*&.',*!#&(

,&16%-#

1$%-!&"!6!');4&,&''&!%'2'&%&"!''! !6! !'$1,*%',(!%()! !&#!).!%(!

%'2'&!&!($&$,';'/!$'!61,(!1&$!'%$(!+*&("!%*%'!%(!6!'/.&$!%(!''&&+4%',(!

%'2'&!%08!($&/,'$08"!&.!1&$!'*.(.!&.!!$$94'$!0!$9%$01!62 0!&.&!– x), $($((4.&'&!!62!3!&6!0!&2.

).!+4$()''!&+&'5,$16!%0.!#&,&#6.-!'</$-$0!.&#(,,/#%(.%&8!%0.!

&,.0%$12!,/7&/"!5,/$1()''!!6!0!( )

215& −#!"$94().!'#/$08!($!$9%$'$81!

&!– x = ( )2

53& −!!!!!1&$!!!!!

251

x&

xx& +

=−

=

&/!,!#,*%&$0!$!1&$!0!&.%/$%,(*2!%08!#,*%&$0!2!#,(14#%().!6''(

$&!&#*!%0.!$$(().&'/& 2

2&x ⎟⎠

⎞⎜⎝

⎛+ = 5 ⇔⎟

⎠

⎞⎜⎝

⎛2

2& x2!3!&6!0!&2

5/!,!#,*%&$0!%!#,(14#%'$!&#*!%0.!$$(().&'/&

22

2x5

2x

& ⎟⎠

⎞⎜⎝

⎛=⎟

⎠

⎞⎜⎝

⎛− ⇔!!!&2 –!&6!0!62

&/!,!#,*%&$0!)!#,(14#%'$!&#*!%0.!$$(().&'/&1

&2!3!.&!– x)2 = 3x2 ⇔!!!&2 –!&6!0!62

(/!,!#,*%&$0!*!#,(14#%'$!-8!'<281!

152 #@:!#"+,!,?=""A#")-

).! $$94'$!x&

xx&

−= "!%*%'!&#*!%0.! $($*%0%&!%-.!&.&+(&$3.!#,(-

14#%'$!6''$&!*%$!!&

&xx& +

= .

3)1%#$.!*!%6***#!*&

!!"!'4$*1!"#$",#!$(4$",!&"!!-')!-#+!(.&)!/0&)"!-#$."#!#30!$#!

$(*(#$#!$42!%+#,'"&42!","#+!"#$",#!!''4$)2"!4!/"&0(""4!#3)$)#(*$ (41)!$5. 1$%-!)!!,0%*!');4&,&''(!%'2'&"!%(!(#(/(!($&$,(4''!$%(!$0''/(!

-!$'!61,(!1&$!'%$(!+*&(!''!''&&+4%',(!%'2'&!%(!)-#!.&!&#(('$9;'/!*%$!

16;'!%.&!&#*!%&!%'2'&%&!)-"!-!!'/.&$!6,,0%("!0!+'&*''.0!#("!")*.

!$*)#!#

F,('1%'/.()''! %(! %'2'&!!)!1&%6! %'2'&!)+! 0!2

AB . ##'$(2! %(!

%'2'&!)!! %9'$!($&$,';'/!$'!'%$(!1&$!61,(!+*&(!$%(!-!''!''&&+4%',(!

%'2'&!%(!)-!1&$!%(!%'2'&!)+!'/.&$!%(!'$$*!%()!)!"!%9()''1! +-2!0!*)+2 !.#,*%&$0!$/#

0,&! %(!+-2!#,(8! %(!)+2 %9'$!+*&(!&,$;'*!#,(8!&,$;'*"!6,&! %(!+-2!'/.&$!$4'''%,(!#,(8!%(!)+2!!.#,*%&$0!A#!+/

=(!)+2!'/.&$!,0%*"!($*%$!%(!%'2'&!)+!'/.&$!,0%*"!-8!%(!'$$*!%()!

,0%(4!%'2'&%(8!)!#!0,&!.$4'*-.&!''!%(.!(,$$'*!7!%()!5$5+/()!A/"!%(!

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

+)2!('.! %9'$!+*&(! %(.!(#(/(! %9'$! %'%,6&-.(8!&,$;'*8!#,(8! %'%,6&-.(!

&,$;'*"!%(!'21(8!-+!'/.&$!&$4'''%,(!#,(8!%(!'21(8!)+!.#,*%&$0!A#7/!

0,&!%(!)-!'/.&$!#("!")*!.#,*%&$0!A#&%/ ##'$(2!%(!-!($&$,'/!%(!)!!$'!61,(!1&$!'%$(!+*&(!''!''&&+4%',(!

%'2'&!%(!)-"!%9()''!

)-2!0!)!!⋅!-!

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

=)!:)B"B":)!?=#C#)! 153

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

!-!'/.&$!#,3%0 #("!")*!.#,*%&$0!A#!7&/#! !."#(#

!"#$%)!

?%0!($&%4#-$0!%08!#,*%&$08!+"!*#-8!1&$!$%$8!#,(%6$'$8!$$!1&$!$&!#()! &1(+();(4."! 9,0$$'(#($'/%&$! (,(+(&/&! #()!&.&#%49;01'! 1&$!$9(-+$6$%01'! '1%'.38! $%(!!$5+/(!A! #! ?).(G/7()''! $4.%('&! %&! &#(+4%-8!&#&,&/%0%&!&$&!%0.!&.6&.-$0!&)%3.!%-.!($&%)#3$'-.#

?4'*-.&!''!%()8!(,$$'(48!%()!!$5+/()!A"!&%(("$'#!("&.$4 +%-&(.%&$!'1'/.&!&$&! %&!(#(/&!)#6,9'$!1($.*!'%%,(!1&$!#&%(("$'#!'1'/.&!&$&!%&!(#(/&!('.!)#6,9'$!1($.*!'%%,(#

#);'/'8! .1! 8');4&,&''&!%'2'&%&8/!%#"!("+!&%(("$')+!+%&(.%&$!($ ');'/'8!%-.!(#(/-.!%&!%'%,6&-.&!'/.&$!$4'''%,&!1&$!%#"!("+!#&%(("$')+ &)%%8!%-.!(#(/-.!%&!%'%,6&-.&!'/.&$!&$4'''%,&#!)#(('$1.4'%&$!*%$"!&.!((;'/!'1!%-.!#,(%%,-.!'/&!');'/&"!)#6,9().!-8!#,(8!&)%2.!6#'$,'8!');'/('8!$4'''%,($!1&$!&$4'''%,($!'/%'!-8!#,(8!%(!'21(8!'/%'!1&$!().6''$#

,!');'/&!#()!;'-,'/%&$!'1!%-.!#,(%%,-.!1&+'/%&$!'4$*, ($!$4'''(%,'8!#,(8!&)%2!'/%'!'21'$!'/%'!'*.(!().6''$!1&+(4.%&$!'4$.2!1&$!($!.()(.6''$/!&$4'''%,'8!!''4$"2!.2!!/)&)+)#!=(!%'%,6&-.(!%08!;'-,(4''.08!');'/&8!1&+'/%&$!'4$0!1&$!%&!$4'''%,&!#,(8!&)%*!9-,/&!'4$!"!'.3!%&!&$4'''%,&!#,(8!&)%*!!/)&#.

?%0.!#,*%&$0!A#&%! %()! $9()!1'*&+&/()!&#(('$1.4'%&$!*%$!&.!&#*!

,0%2!');'/&!&*&$,';'/!,0%2!');'/&!$4'''%,0!''!%0.!&,9$12!().6''$!'*.(."!

%*%'!0!)#*+($#0!'/.&$!6+(&(8!1&$!1&+'/%&$!#3)$)(*. +0+&(2"!&.!6"!:!,0%6!%'2'&%&!''!6!5!:"!6"!:!&$4'''%,&!1&$!62, y2!$4'''%,&"!%*%'!%(!%'2'&!6!– y '/.&$!6+(&("!0!+'&*''.0!#3)$)(*#!.-$&!#&,6('$&'&!%(!%'2'&! 2 ,!–!,!"!*(#()!,!,0%*!%'2'&!'/.&$!&#(%('2/#!=(!%'2'&!:!(.('67'%&$!3')&#'(07)#&#.

).! %(! %'%,6&-.(! %08! *+08! )#',%9'$! %()! %'%,&&3.()! %08! #,(-$&,'*7()$&8!1&%6!%(!%'%,6&-.(!#+'),68!$)''%%,()!#,(8!%0.!*+0"!%*%'!

0!&#(%('2!1&+'/%&$!3'6$4!#3)$)(*#!+0+&(2"!&.!!62 = y2 + z2!!1&$!!z, x $4'''%,&"!%*%'!0!&#(%('2!!6!–!:!!(.('67'%&$!3'6$4!#3)$)(*.

).!%(!%'%,6&-.(!%08!*+08!)#',%9'$!%()!%'%,&&3.()!%08!#,($&,('*7()$&8!1&%6!%'%,6&-.(!#+'),68!().6''$!'*.(.!$)''%%,()!#,(8!%0.!*+0"!%*%'!0!&#(%('2!1&+'/%&$!$"$!'$4!#3)$)(*. +0+&(2"!&.!!62 = y2 + z2 1&$!!z, x!!&$4'''%,&"!%*%'!%(!6!-!:!(.('67'%&$!$"$!'$4!#3)$)(*.

154 #@:!#"+,!,?=""A#")-

?%0.! #,*%&$0!A#2$!&#(('$1.4'%&$! *%$! %(! (,;(&3.$(! #()! #',$%9'%&$!''%&<4!,0%3.!().6''$!'*.(.!$4'''%,-.!');'$3.!'/.&$!6+(&("!1&+'/%&$!!(.&) 1&$!0!#+'),6!%()!$$((4.&'()!%'%,&&3.()!'/.&$!6+(&(8!1&$!(.('67'%&$!(.&4.

?%0.!#,*%&$0!A#&+!&#(('$1.4'%&$!*%$!&.!&#*!');'/&!&*&$,';'/!')-;'/&"!0!(#(/&! '/.&$!#,(8! %0.!*+0!().6''$!&$4'''%,(8!1&$!$90'&%/7'$!''!&)%2.!%(!''.!6;,($$'&!%-.!%'%,&&3.-.!%()8!,0%*!%(!('!(,;(&3.$(!'%$("!%*%'!0!)#*+($#0!'/.&$!6+(&(8!1&$!(.('67'%&$!"/!&&1".

8#'#$*'4&4 ).!%(!,0%*!');4&,&''(!%'2'&!&!($&$,';'/!$'!61,(!1&$!'%$(!+*&("!

%*%'!%(!''&&+4%',(!%'2'&!x!%08!($&/,'$08!$$(4%&$!''

x = &2

15 −"!!(0+&(2!!!x = &

21–&

25 ,

%(!(#(/(!$4'*-.&!''!%(!$9*+$(!#()!#,(0&2;01'!'/.&$!#("!")*.

3. *$14#!8)%!%)-*+(%!'#$!'#(%($'1!4!(%"*+(%#! !"4%$!2!!"#)/%*02!

?%$8!#,(%6$'$8! &!'%9,$! $2! '1%/;'.%&$!($!#+'),%8! %()! $$*#+'),()!

%,$&3.()!1&$!%-.!1&.(.$13.!#'.%&&3.()"!'<&&3.()!1&$!('1&&3.()!'&(

&'&,&''%.-.!$%(.!/($(!141+(!1&$!1&;(,/7(.%&$!$'!$9%$0!''!%0.!&1%/.&!

%()!141+()#

3)1%#$.!+!%6***#!+& !!"!$'"+2!&1","2! +&03/"#')#!3""$#&6")#"!",$"!##$.2!","#+!%+#%)-

5+-.2!",$"!","#+!(4!%+#%)5+-.2"!","#+!,&"2"!$)!3""$!&1")!","#+!+&)&6"+)$! )#*('$<0 1$%-! *%$! %,'$8! ($&((9$1%8! &-./'8!!

∧

A , ∧

Β , ∧

Γ "!!%()!$$*#+'),()!#'.%&&3.()!)!-+#!'/.&$!/$'8#!.&!&#(('/<()''!*%$!%(!

#'.%6&-.(!'/.&$!$$(&3.$(#! 3%,()''! %$8! )-"! !#! 1&$! %0! /+"!

*#()!/!%(!$0''/(!%('28!%-.!)-"!!##! ##'$(2!($!#+'),%8!!-"!!)!'/.&$! /(

$'8! &.%/$%($9&! #,(8! %$8! !)"! )#! 1&$! 0!

&-./&!∧

ΓΒΑ ! '/.&$! /$0! #,(8! %0! &-./&!!!$*)#! 7

=)!:)B"B":)!?=#C#)! 155

EAB∧

"! %&! %,/&-.&!)!-"!)!#! '/.&$! /$&"!6,&!($!#+'),%8!)-"!!#! '/.&$!

/$'8"!0!&-./&! BA∧

Γ !'/.&$! /$0!#,(8!%0!&-./&! ΒΕΑ∧

!1&$!0!&-./&! EBA∧

'/.&$!/$0!#,(8!%0!&-./&! ΓΑΒ∧

#!##('%.-8!%(!%'2'&!)/!'/.&$!/$(!#,(8!%(!

!/!1&$!'#'$(2!&#(('/9%01'!*%$!%(!)-!'/.&$!/$(!#,(8!%(!!#"!%9()''!/-!0!

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

($*%$! %9().! 1($.2! %0.! #+'),6! /+! 1&$!

%$8! #+'),%8! /#"! #+! /$'8! #,(8! %$8! /-"!

-+!&.%/$%($9&#!0,&!0!&-./&! ΔΕΖ∧

!'/-

.&$! /$0! #,(8! %0! &-./&! ΔΓΖ∧

!)#(('/-

9;01'!*%$!0!&-./&! ΒΕΑ∧

!'/.&$!/$0!#,(8!

%0!&-./& ΒΓΑ∧

"!6,&!0!&-./&! ΔΕΑ∧

!'/-

.&$! /$0!#,(8!%0!&-./&! ! ΔΓΒ∧

#!"'(/-8!

&#(('$1.4'%&$!*%$!0! &-./&! ΕΔΓ∧

! '/.&$!

/$0!#,(8!%$8!&-./'8!∧

! , ∧

Β , ∧

Γ "! ! !6,&!%(!#'.%6&-.(!'/.&$!$$(&3.$(#

1$%-!%3,&!*%$!($!%,'$8!'0!($&((9$1%8!&-./'8!!∧

A , ∧

Β , ∧

Δ !!'/.&$!/(

$'8!.?9#!&&/#!!! 3%,()''! %$8!!+"!!##!##'$(2!($!#+'),%8!!)"!)#!'/.&$! /$'8!#,(8!

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

6,&!0!#+'),6!!#!'/.&$!/$0!#,(8!%0.!#+'),6!!+!1&$!0!&-./&! ΒΕΑ∧

!'/.&$!

/$0!#,(8!%0!&-./&!! ΓΔ∧

B .

=*%'!1&$!0!&-./&! ΒΕΔ∧

!'/.&$!/$0!#,(8!%0!&-./&!! ΕΔΒ∧

#!0,&!0!&-(

./&! ΔΕΑ∧

!'/.&$!/$0!#,(8!%0!&-./&!! ΓΔΕ∧

#!"'(/-8!&#(('$1.4()''!*%$!1&$!

0!&-./&! ΓΒΑ∧

!'/.&$!/$0!#,(8!%$8!&-./'8!∧

! , ∧

Β , ∧

Δ . 0,&!%(!#'.%6&-.(!'/.&$!$$(&3.$(#! !."#(# 94(",1&4 ?%&! '#*''.&! %.&! $$*#+'),(! 1&$! $$(&3.$(! #(+4&-.(! ;&! &.&*%-

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

!$*)#!$#

156 #@:!#"+,!,?=""A#")-

3)1%#$.!,!%6***#!,& 9"!-!$"!-#")"+-0!3""$!&1")!%%)!%+#%)5+-.2!%+#&6"+)+!$.(")"$#+!&"!!-')!-#+!(.&)!/0&)!-#+!$#!("&#/%$"'#!$(*(#$#!$42!%+#,'"&42!","#+!$)!-#$."#!,&)!("!$4"!3/"#'!!$)#!3""$#&6")#$ (41)!$5. 1$%-!)!-+#! 1&.(.$1*! #'.%6&-.(! 1&$!.! %(! $0''/(! %('28! %-.!

($&&-./-.!)-! 1&$!!##!.&! &#(('$9;'/! *%$! 1&;''$6! &#*! &)%%8! %%'.'%&$!

$%(!.!$'!61,(!1&$!'%$(!+*&(!1&$! %(!''&&+4%',(! %'2'&! %08!($&/,'$08!

$$(4%&$! ''! %0.! #+'),6! %()! 1&.(.$-1(4!#'.%&&3.()#

F',$&,6*()''!141+(!$%(!1&.(-.$1*!#'.%6&-.("!$4'*-.&!''!%0.!#,*(

%&$0! IV#$)#! ##'$(2! %&! %'2'&%&! #)"!

)!!'/.&$! /$&!#,(8!%&!)!"!!-!1&$!#'(

,$%9().! /$'8! &-./'8"! %&! %,/&-.&!)!#!

1&$!)!-!'/.&$!/$&"!6,&!!#!0!)-!1&$!($!

)#*+($#'8! &-./'8! '/.&$! /$'8#! 0,&! 0!

&-./&! Γ∧

AB ! $$(4%&$! ''! %0! &-./&!!

ΕΒΑ∧

"! 6,&!∧∧

ΒΑΘ= 2).# ! .#,*%&-

$0!"#!%2/#!##'$(2!%(!%*<(! ΓΔΕ∩

!'/.&$!

($#+6$$(! %()! %*<()!∩

ΒΓ ! 1&$! 0! &-./&! ΓΑΕ∧

! '/.&$! ($#+6$$&! %08! &-./&8!!

ΓΑΒ∧

"!!6,&!0!&-./&!! ΕΑΘ∧

!!'/.&$!/$0!#,(8!%0!&-./&!! ΕΘΑ∧

#!0,&!%(!%'2-'&!.#!'/.&$!/$(!#,(8!%(!%'2'&!#)"!(0+&(2!%(!)!#!:&$!'#'$(2!%(!%'2'&!!)!

'/.&$!/$(!#,(8!%(!#)"!1&$!0!&-./&!! ΕΒΑ∧

!!;&!$$(4%&$!''!%0!&-./&!! ΑΕΒ∧

.

)++6!1&$!0!&-./&!! ΕΒΑ∧

!!('/9;01'!/$0!#,(8!%0.!! .AB∧

"!6,&!;&!

'/.&$!1&$!0!! ΑΕΒ∧

!!/$0!#,(8!%0!! .AB∧

#!,!&-./&!! ΕΒΑ∧

!!'/.&$!1($.2!%-.!

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

!.

!)

!)

#!= !1&$!'#'$(2!!!)!0!#."!!%9()''!

.!

#.

#.

!#= .

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

%'2'&!.#!''&&+4%',(!%()!.!#!0,&!%(!%'2'&!!#!%9'$!%'0;'/!$'!61,(!

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

!$*)#! 8

=)!:)B"B":)!?=#C#)! 157

#,(8! %0.! #+'),6! %()! #'.%&&3.()#!E'! *'($(! %,*#(!&#(('$1.4()''!*%$!

1&$! %(!)-! ! %9'$! %'0;'/!$%(!.!$'!61,(!1&$!'%$(!+*&(!1&$!*%$!%(!''&&-+4%',(!%'2'&"!%(!-."!'/.&$!/$(!#,(8!%0.!#+'),6!%()!#'.%&&3.()#!!

!."#(#

3)1%#$.!-!%6***#!-& !!"!)+!3/"#'.2!$)#!-#")"+-)%!":#&6")#!-#+!%"-#&6")#!3)#!"&#&'!*)"$#+!&$)"!,%+)!-%-/)!&#"$"$)%"!&"!(+#!"#$",#"!$0$"!$)!0/)!$(*(#!%+#+'",$#+!&"!!-')!-#+!(.&)!/0&)!-#+!$)!("&#/%$"')!$(*#(#!$42!%+#,'"&42!","#+!4!3/"#'!!$)#!":#&6")#$ (41)!$5. 1$%-! (! 141+(8!)!-! ($&'%%,()!

)!"! 0! #+'),6! !-! %()! 1&.(.$1(4! ('-1&&3.()!1&$!0!#+'),6!-+!%()!1&.(.$-1(4! '<&&3.()! #()! '&&,6*(.%&$! $%(.!

#&,&#6.-! 141+(#!1$%-! '#/$08! *%$! %&!

%'2'&%&! !-"! -+! 1'/.%&$! '#;! ');'/&8#!

.&! &#(('/<()''! *%$! %(! ');4&,&''(!

%'2'&!!+!%9'$!%'0;'/!&#*!%(!-!$0''/(!!

$'!61,(!1&$!'%$(!+*&("!''!''&&+4%',(!

%'2'&!%(!-+# 1$%-! #! %(! 1%.%,(! %()! 141+()#!

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

#,('1%'/.()''!%0!!#!'%9,$!%(!)#!##'$(2!0!#+'),6!!-!'/.&$!#+'),6!1&(

.(.$1(4! ('1&&3.()"! %(! %*<(!∩

ΓΒ! ! '/.&$! #'.%&#+6$$(! %()! %*<()!∩

ΒΓ ,

6,&!%(!%*<(!!∩

ΑΓ !!'/.&$!%'%,&#+6$$(!%()!%*<()!∩

ΒΓ #!0,&!0!&-./&! ΓΕΑ∧

'/.&$!%'%,&#+6$$&!%08!! ΒΕΓ"

#!:&$!'#'$(2!0!&-./&!! ΓΒΕ∧

!!'/.&$!/$0!#,(8!

%0.! ! ΒΓΕ∧

, %9()''!∧∧

ΕΓΒ= 2)#- #! ##/$08! #-! 0! -+"! ($*%$! -+! '/.&$!

#+'),6!1&.(.$1(4!'<&&3.()!'&&'&,&''%.()!$%(.!141+(!)!-#

##('%.-8!∧∧

ΓΔΕ=-#+ "! 1&$! %%$$!∧∧

ΕΔΓ=ΕΓΒ 2 #! )++6! &#(('/-

9;01'!*%$!∧∧

ΕΓΒ=ΑΕΓ 2 "!6,&!∧∧

ΔΓ=ΑΕΓ E4 #!0,&!∧∧

ΒΕΓ=ΕΔΓ .

!$*)#! 9

158 #@:!#"+,!,?=""A#")-

,!&-./&! ! ΔΒΕ∧

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

!#+#!0,&!1&$!0!&-./&! ! ΔΕΒ∧

! !'/.&$! /$0!#,(8!%0.! ! ΒΓΕ∧

#!0,&!%&!%,/-

&-.&!#!+"!#!-!'/.&$!$$(&3.$&#!##('%.-8!ΒΓ

ΒΕ=

!#

!+!1&$!'#'$(2!!#!0!

0!-+! %#'%&$!*%$!ΓΒΔΓ

=ΔΓΒΔ

#!##/$08! '/.&$!!+! 5!+-!6,&!1&$!+-! 5!-!#!

1%$$!0!');'/&!!+!%9'$!%'0;'/!$'!61,(!1&$!'%$(!+*&(!1&$!%(!''&&+4%',(!

%'2'&!&)%28!%08!($&/,'$08!'/.&$!%(!+-#!!!."#(# 8#'#$*'4&4!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ).!x!'/.&$!0!#+'),6!%()!1&.(.$1(4!('1&&3.()!1&$!r!0!&1%/.&!%()!

#',$&'&,&''%.()!141+()"!%*%'!0!#+'),6!%()!('1&&3.()!'#(,'/!.&!)#((

+(&$$%'/!&+&'5,$16!&#*!%0!$9%$0!1!

(r + x) x = r2!"!!!(#*%'!;&!!%9()''1!!x = ( )152r

−

3)1%#$.!".!%6***#!".&

!!"!"&&'#*",!&"!-%-/)!-#")"+-0!3""$!&1")"! $0$"! $)! $"$'!&1")!!$42!3/"#'!2!$)#!3""$#&6")#!",-"#+! +&)%%"#()! 3')2! $)! !$')+-&(#!$1"!$"$'#&6"1"!3)#!.5)#"!12!3/"#'.2! $4"!3/"#'!! $)#!-#-")"+-)%!":#&6")#!-#+!$4"!3/"#-'!! $)#! -#")"+-)%! %"-#&6")#!3)#! "&&'!*)"$#+! &$)"! ,%+)! -%#-/)$ (41)!$5. 1$%-!)!-+#!1&.(.$1*!#'.%6&-.(!'&&'&,&''%.(!$'!141+(#! 1$%-!/!%(!1%.%,(!%()!141+()!1&$!),!($6''%,(8#!)#*!%(!/!6&()-

''!16;'%0!/.!#,(8!%0.!)!"!0!(#(/&!#,('1%'$.*''.0!%%'.'$!%(.!141+(!

$%(!$0''/(!:#!3%,()''!%$8!');'/'8!!:"!:)!1&$!&#*!%(!/!*%,()''!16;'-%0!/!!#,(8!%0.!:)"!0!(#(/&!%%'.'$!%(.!141+(!$%(!E!1&$!%0.!)!!$%(!B#

!$*)#! 10

=)!:)B"B":)!?=#C#)! 159

##'$(2!%(!%*<(!!∩

ΓΗ!# !!'/.&$!/$(!#,(8!%(!%*<(!∩

ΑΕΔΗ !1&$!%(!%*(

<(!∩

ΑΒΓ !'/.&$!/$(!#,(8!%(!%*<(!!∩

ΑΕΔ "!%#'%&$!*%$!%(!%*<(!!∩

ΓΗ !!'/.&$!/$(!

#,(8!%(!%*<(!!∩

ΗΔ #!0,&!%(!-,!'/.&$!#+'),6!1&.(.$1(4!('1&&3.()#!##/(

$08!/)!0!/!!1&$!/.!16;'%0!$%0.!)!"!6,&!∧∧

ΚΖΒ=)/: !.#,(%6$'$8!"#!*!

1&$!"#!2+/#!0,&!%(!%*<(!!∩

ΑΚ !!'/.&$!/$(!#,(8!%(!%*<(!!∩

ΚΒ "!1&%6!$).%#'$&!

%(! %*<(! !∩

ΑΒ ! ! '/.&$!($#+6$$(! %()! %*<()! !∩

ΒΚ ! !1&$! %(! %'2'&!):!'/.&$!

#+'),6!1&.(.$1(4!('1&&3.()#!-$&!%()8!/($()8!+*&()8!1&$!%(!%*<(!!∩

ΑΚ

'/.&$!($#+6$$(!%()!%*<()!∩

ΚΜ #!:&$!'#'$(2!%(!%*<(!!∩

ΑΒ !!'/.&$!($#+6$$(!

%()!%*<()!!∩

ΒΚ !!1&$!%(!%*<(!!∩

ΓΔ !!/$(!#,(8!%(!%*<(!!∩

ΑΒ "!!'/.&$!1&$!%(!

%*<(!!∩

ΓΔ !!($#+6$$(!%()!%*<()!!∩

ΒΚ . ##/$08!%(!%*<(!%()!!∩

ΓΔ !!;&!'/.&$!

($#+6$$(! %()! %*<()! !∩

ΓΗ #!0,&! %(! %*<(! ! !∩

ΓΗ ! ! '/.&$! /$(!#,(8! %(! %*<(!!∩

ΒΚ #!)++6!%(!%*<(!!∩

ΒΚ !!'/.&$!($#+6$$(!%()!∩

ΚΜ "!!'#'$(2!'/.&$!($#+6$$(!

1&$!%()!!∩

ΚΑ "!6,&!1&$!%(!!∩

ΓΗ '/.&$!($#+6$$(!%()!∩

ΚΜ #!)++6!1&$!%(!%*<(!!∩

ΓΒ !!'/.&$!($#+6$$(!%()!%*<()!!∩

ΒΚ "!($*%$!%(!%*<(!!∩

ΓΒ !!'/.&$!/$(!#,(8!%(!!∩

ΒΑ #!0,&!%(!%*<(!!∩

ΗΒ !!'/.&$!($#+6$$(!%()∩

ΒΜ "!6,&!1&$!0!&-./&!!∧

ΗΖΒ

'/.&$!($#+6$$&!%08!!∧

ΒΖΜ .

:&$!0!&-./&!!∧

ΗΖΒ !'/.&$!($#+6$$&!%08!!∧

ΖΑΒ "!($*%$!0!&-./&!!∧

ΖΑΒ

'/.&$!/$0!#,(8!%0!&-./&!!∧

ΑΒΖ #!0,&!1&$!0!&-./&!!∧

ΒΖΝ !!'/.&$!/$0!#,(8!

%0.! !∧

ΖΑΒ #!,! &-./&! !∧

ΑΒΖ ! ! '/.&$! 1($.2! %-.! (4(! %,$&3.-.!)!/! 1&$!

!/B#!0,&!%&!%,/&-.&!)!/!1&$!!/B!'/.&$!$$(&3.$&# "$94'$!'#('%.-81!!

ΒΝ

ΖΒ=

!/

)!"!!6,&!!!/2!0!)!!·!!B

##'$(2!)!! 0!!:"!!B! 1($.2! 16;'%0"! 6,&!:B! 0!)B"! '#('%.-8!∧∧

ΛΑΝ=!:B .

160 #@:!#"+,!,?=""A#")-

)++6!∧∧

ΚΒΝ=!)B "!6,&!∧∧

ΚΒΝ=ΛΚΝ . :&$!0!&-./&!$%(!)!'/.&$!1($.2!%-.!(4(!%,$&3.-.!):!!1&$!):B#!

0,&!1&$!0!&-./&!∧

ΚΒA '/.&$!/$0!#,(8!%0.!∧

ΚΝΑ .

##('%.-8"! $$94'$!0!&.&+(&/&!ΑΝ

ΚΑ=

ΑΚ

ΒΑ"!6,&!!)·)B! 0!:)2,

1&$!$).'#38!)!·!!B!3!!)·!)B!0!!/2!3!:)2. )#(('/9;01'!(0+&(2!*%$!!)2!0!!/2!3!):2, *#()!!0!!)!'/.&$!#+'),6!

%()!1&.(.$1(4!#'.%&&3.()"!0!!/!#+'),6!%()!1&.(.$1(4!'<&&3.()!1&$!0!):!

'/.&$!#+'),6!%()!1&.(.$1(4!('1&&3.()!'&&'&,&''%.0!$%(.!/($(!141+(#! !."#(#

3)1%#$.!""!%6***# 11) !!"!&"!-%-/)!("!'4$*!%+!("$')!"&&'#*",!-#")"+-0!3""$!&1")"!4!3/"#'!! $)#! 3""$#&6")#! ","#+! !/)&)2"! 4! )3),#! )")(!7"$#+! -#+!!"/!&&1"$ (41)!$5. #&&,6*()''!1&.(.$1*!#'.%6&-.(!)!-+#!$'!141+(!''!,0%2!($6(

''%,(#!.&!&#(('$9;'/!*%$!0!#+'),6!%()!#'.%&&3.()!'/.&$!6,,0%(8"!0!+'(&*''.0!"+,$$&'.

F&/,.()''!/!%(!1%.%,(!%()!141+()"!*%,()''!%$8!($&'%%,()8!!."!),"!%0!($&&3.$(!)-!1&$!$%(!%'2'&!/.!#&/,.()''!%'2'&!/:!/$(!''!%(!%.&!%%%&,%(!%08!&1%/.&8#

##'$(2!%(!%*<(!∩

ΓHA !'/.&$!/$(!#,(8!%(!!∩

ΑΔΗ !!1&$!%(!%*<(!!∩

ΑΒΓ

#,(8!%(!%*<(!!∩

ΑΕΔ "!%#'%&$!*%$!%&!%*<&!!∩

ΓΗ , ∩

ΗΔ !!'/.&$!/$&# #6.! *%,()''! %0.! )+"! ($! &-./'8! $%(! !! '/.&$! (,;%8! 1&$! -+! 0!

0! 2-!#!-$&! %()8! /($()8!+*&()8!($! &-./'8!$%(!E! '/.&$!(,;%8!1&$!)-! 0!

0!2-E#!=&!%,/&-.&!)E/"!)!-!'/.&$! $$(&3.$&"!($*%$!($!&-./'8!∧

ΑΜΖ , ∧

ΑΛΓ !'/.&$!/$'8!1&$!0!&-./&!!∧

ΛΑΓ !'/.&$!1($.2#!0,&!1&$!($!&-./'8!!∧

ΑΖΜ , ∧

ΑΓΛ !'/.&$!/$'8#

!!!"!#$#""!!%!#&#! 161

$'%!&!

(&

'!

)'= !*%+!

&!

2('!

2) Ζ=

Γ"!#,(+)-!

2&!

(&

&!

(&2= #!!./012(!

2&!

(&

'!

)'2= #! 3'%!

4&!

(&

2'!

)'2= #! )45%)-!

&"

(&

'(

*'= "!"%+! )+%! +167.-

+(89!"&

("

'(

'(*'=

+#!3'%!

( )2

2

2

2

"&

("

'(

'(*'=

+ .

"%+! (,(+)-#! :,%6! ,247(;! ,0!

,2-2%!!'!+(! 3*'0! *%+!2.+0! 5:<0!

,0!2(<%5=,('0!,2-2%!(;6%+!4!,5(1$

'3!,01!,(6,%<>601#!)45%)-!,0!*'#!

%,':,%+4!&'!*%+!!,0!,(,'3<860!,01!

2(<%5=,('01! ,2-2%,09! %1?42.601!

*%,3!,0!2++:!,01!:501!(;6%+!,(6,%$

,53++0! ,01! ,(,'%<>601! ,01! 2++0=!

,01!:501!%,':,%+4!('#!++/=(+!

%*'!)!'('2!*!+'(2

$'%!("2!*!+"&2. !0!"&2!(;6%+!'4,:#!)+:,+!4!)+32(,'09!(;6%+!'4,-#!(,02.689!,0!("2

(;6%+!'4,:!*%+!,0!("!(;6%+!'4,:" #,;+49!(;6%+!-&!*!,&"#!(,02.689!-"!*!+&"!*%+!-"2!*!-+!&"2 ,

3'%!7%!./012(!-"2!*!+!("2"!#,(+)-!,0!-"2!,'09!,0!("2!)(6!./(+!5:<0!

,06!0,0;0!./(+!,(,'3<8609!%'+72:9!,'09!,(,'3<860!%'+72:#!,0!-"!(;6%+!

%+=22(,'0!89!,'09!,0!2-*09!,'09!,0!(""!$'%! %+=2.86%!2(!,46!,':-,%+4!@"./'!,0!,2-2%!(-!(;6%+!%,0,02-!*%+!,0!("!(;6%+!,'0+%'2:A01-+%"!/%!)(;?012(!:,+!(;6%+!,.,%',4!%,0,02-"!

0+,8!-"2 -!"(2! *!#2#! )45%)-! ,0! ,(,'3<860! ,01!-"! 1,('./(+!

%,:!,0!,(,'3<860!,01!"(!*%,3!#2"!#,(+)-!,0!"&!(;6%+!+=22(,'0!,'09!

,0!-&#!*%+!,0!"-!*!"&!)!&-!(;6%+!+=22(,'0!,'09!,0!-&"!!553!,0!-"!

(;6%+!+=22(,'0!,'09!,0!-/#!3'%!,0!-"!(;6%+!+=22(,'0!,'09!,0!-/"

#,(+)-!15

"(

-"2

2

= #!)+0!%6,++,'0.-9!./012(!!45

#

-"2

2

= #!,0!0,0;0!

)(6!(;6%+!5:<09!,(,'3<8601!%'+720=!,'09!,(,'3<860!%'+72:#!3'%!,0!-"!

(;6%+!%+=22(,'0!*%,3!2-*09!,'09!,0!#"!$'%!,0!-"2!1,('./(+!,01!"(2

!"#$!! 11

162 #B")#"*1!1%!$"@#"!2

*%,3!,(,'3<860!,5(1'39!%+=22(,'49! %*%,3!2-*09'!,'09!,0!-"!*%+!,0!

-"!(;6%+!+=22(,'0!,'09!,0!'4,:!,2-2%!-/#!3'%!,0!(-!(;6%+!,(,3',4!

%,0,02-" 2289! ,0!0'70<>6+0!,01!,('+./(,%+!%,:!'4,:! ,2-2%!*%+! ,.,%',4!

%,0,02-!(;6%+!350<0"!#,;+49#!4!,5(1'3!,01!++0)=6%201!2(!%1,:!,(,'%-<>601!(;6%+!350<09#!*%+!06023A(,%+!!"%&##'(!%,':,%+4!X. 94)

#,;+49! ++/=(+!/-! ⋅!-(!*!!-2#!)+:,+!,0!,';<860!!-(!(;6%+! ++0$

<>6+0!,'09!,0!,';<860!!-/!*%+!++/=(+!ΒΜ

ΒΑ=

ΒΑ

ΘΒ .

$'%! %,0)(;/74*(! :,+! 4! ,5(1'3! ,01! ,(6,%<>601! )(6! (;6%+! '4,-#!

)45%)-!(;6%+!350<09!%3''4,09'#!,01!06023A(,%+!"%&##'(!%-5.,(!+/:5+0!

2(,3!,46!,':,%+4!3'"! !."")" !!"!"#"$#$!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !6!y! (;6%+!4!,5(1'3! ,01!*%606+*0=!,(6,%<>601!*%+!x!4!,5(1'3!

,01!*%606+*0=!)(*%<>601!(<<'%22.601!+(!*=*50!%*,;6%9! r#!,:,(#!+=2$.86%!2(!:+%!%,0)(;/74*%6!+,46!,':,%+4!((#!!7%!++/=(+!4!+/.+44!! y2 = x2 + r2 . 2289!%,:!,46!,%'%,-'4+4!2(,3!,46!,':,%+4!5#!./012(

x = )15(2r

− "!#,02.6894!

y2 = ( ) 22

r5264r

+− ⇔ y2 = ( )52104r 2

− ,

$'%! ! <+%! ,46! ,5(1'3! 6 ,01! *%606+*0=! ,(6,%<>601! (<<(<'%22.601! +(!

*=*50!%*,;6%9!r!7%!./012(! !y = 52102r

− "!!,0)(+*6=(,%+!:,+!,0! !y

89!)+%.0'3!(17=<'%2286!,2423,86!<'3.(,%+!

y = 5252r525

2r

−−+ #!,0!0,0;0!89!(17(;%!(;6%+!(53++86"

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

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

&!2!"*0!'(')*0$

!!!"!#$#""!!%!#&#! 163

%&#&''($ #<<'3.012(! ++:,5(1'0! ,';<860!

!-'! +(! *=*50! 2(! *.6,'0!*"!C'0(*,(;-6012(!,46!!*#!4!0,0;%!,.26(+!,06!*=*50!

+,0!#"!/%!%,0)(+/7(;!:,+!!-2!*!/*#2. #,(+)-!,0!,';<860!!-'!(;6%+!++:$

,5(1'0#!,0!,:?0!!∩

ΓBE !!(;6%+!,0!.6%!,';-,0! ,49! ,('+..'(+%9! ,01! *=*501#! 3'%! ,0!

,:?0!!∩

ΒΕ !!(;6%+!,0!.6%!.*,0!,49!,('+..-'(+%9!,01!*=*501!*%+!,0!,2-2%!-#!(;6%+!

;+0!2(!,46!%*,;6%!,01!*=*501! %,':,%+4!IV"(+#! ,:'++2%'"! #,(+)-!!#! *! -*##! .$

/012(!!#2!*!,!*#2#!)45%)-!!#2!*!,!-#2. "+/=(+!!-2!*!!#2 –!-#2#!3'%!

!-2!*!/!-#2#!)45%)-!!-2!*!/!*#2. !."")"

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

!"#!"#$!"'!$%$$$%"'( 4!!'!"!#'%0!#5%,!/0"!6,'!! %&!'!&*&-'.!"%""%%'"*'"!&!!%++"!#

&%,!#%!'*#67&$!#&!,"!!'!-!&!!!/*'%-85%,!."-!"*!"%""%+1&*!"$2!

'-!67""*0! "$2! #&!,"!2! %,&!-! "!! "",!! '%("%"!! "*0! "%""!+3&*0!

"$2!/)%0"%2!"$2!/0"!6,'!2$ %&#&''($ 0+,8! !-! 4! )+32(,'09! ,49!

+.%;'%9"! !.26012(! ,0! ,2-2%! !-!

+,0!+42(;0!'#!>+,(!,0!,2-2%!!'!6%!

(;6%+!)+,53++0!,01!'-!*%+!<'3.01-2(! 42+*=*5+0! )+%2.,'01! !-"! %,0!

+42(;0!'!..'012(!*37(,4!+,0!,2--2%!!-#!4!0,0;%!,.26(+!,0!42+*=*5+0!

+,0!+42(;0!*!*%+!..'012(!,0!,2-2%!

!*"

!"#$!! 12

!"#$!!"#

164 #B")#"*1!1%!$"@#"!2

9!"!#'%0#!"*0!"%""!7'"*0$ /(8'0=2(! ,06! *=*50! 2(! *.6,'0!/!

*%+!%*,;6%!*'!*%+!(<<'3.012(! ++:,5(1'0!

,';<860! &#1"! %,0! +42(;0! /! 1D>6012(!

*37(,4!+,0!(,;,()0!,01!*=*501!*%+!,%;'-6012(!,2-2%!/"!;+0!2(!!'!%%/"!(/%).

1! (17(;%!/"! 89! *37(,4! +,0! (,;-,()0!,01!*=*501#!(;6%+!*37(,4!+,+9!(17(;-(9!/##!/&#!/1"!

#,(+)-! ,0! ,2-2%!!'! (;6%+! ;+0! 2(!

,0!/"#!*%+!,0!'*! ;+0!,'09!,0!/#!*%+!($

,(+)-!,('+./016!0'7.9!<86;(9#!,0!!*!(;6%+! ;+0!2(!,0!"#!%,':,%+4!"",'"!

'+%!,06!;)+0!5:<0!*%7.6%!%,:!,%!,2-2%,%!"&#!"1!++0=,%+!2(!,0!*!"! $'%!"#!*!"&!*!"1" #,(+)-!,0!,2-2%!!'!(;6%+!)+,53++0!,01!'-#!,0!!-!(;6%+!,'+,53++0!

,01!-'" #,;+49#!:,89!%,0)(+*6=(,%+!+,0!5-22%!,01!%*05017(;#!++/=(+!4!

+/.+44! 2

2

*'

!*

-'

!-= .

$'%! !!*2 *!/!*'2"!#,;+49! ++/=(+ !&#2 *!/!#/2! ! %,':,%+4!(-'#!

*%+!(,(+)-!*'!*!#/#!!.,(,%+!!:,+4!!*!!*!#&" !553!,0!*!!)(;/74*(!;+0!2(!*%7.6%!%,:!,%!"##!"&#!"1!*%+!*%$

7.6%!%,:!,%!#&#!&1#!1#!(;6%+!;+0!2(!*%7.6%!%,:!,%!"##!"&#!"1"!$'%!

,%!,';<86%!#&1#!"1&#!"&1#!"#1!(;6%+!++:,5(1'%" #,02.689!*%,%+*(13+74*(!,1'%2;)%#!,01!%,0,(5(;,%+!%,:!,.+-

+('%!++:,5(1'%!,';<86%#!4!E3+4!,49!0,0;%9!(;6%+!,0!++:,5(1'0!,';<860!

#&1!*%+!*0'1.-!,49!,0!+42(;0!""! $++"!&#!#%!#&!,"!$ C'0(*,(;6012(! ,0! ,2-2%! "/! *%,3! ,2-2%! /)! *! '-"! #,(+)-!

'-

'*

'*

!'= ! %,':,%+4!VI#!&'!!'!*!"/#!'*!*!/#! !*%+!'-!*!/)!./012(!

/)

#/

/#

"/= !3'%!!"/!⋅!/)!*!#/2.

#,;+49#!0+!<86;(9! !∧

ΘΕK , ∧

ΕΘΛ !(;6%+!0'7.9"!#,02.689!,0!42+*=-*5+0!2(!)+32(,'0!")!7%!)+.57(+!%,:!,0!##!%)+:,+#!%6!..'012(!,0!#)#!4!<8$

6;%!!∧

ΛΕΚ !<;6(,%+!0'7-#!(.:+06!,0!,';<860!#)"!(;6%+!++0<>6+0!,'09!*%$

!"#$!!"#!

!!!"!#$#""!!%!#&#! 165

7.6%! %,:! ,%! ,';<86%! #)/#! #/"'"! #36! 50+,:6! ,('++,'.D012(! ,0! 42+-*=*5+0!)+%,4'>6,%9!+,%7('-!,46!")#!,:,(!%1,:!7%!)+.57(+!*%+!%,:!,%!+4$

2(;%!&#!1"!#,02.689!4!,1'%2;)%!(<<'3.(,%+!+,4!+.%;'%!)+%2.,'01!")" (!#&!,"!!%,&!-!$!'*#67&$$! 1!)+32(,'09!")!,49!+.%;'%9!(;6%+! ;+4!,'09!,4!)+32(,'0!!-!,49!

)0+2.649!+.%;'%9#!(,(+)-!,0!,2-2%!!'!(;6%+!;+0!,'09!,0!"/!*%+!,0!'-!

(;6%+!;+0!,'09!,0!/)" :*!"%""%+1&*!"$2!'-!67""*0!"$2!#&!,"!2!%,&!-!"!!"",!!'%("%"!!

"*0!"%""!+3&*0!"$2!/)%0"%2!"$2!/0"!6,'!2$!

#,(+)-!!'!*!-!'-!./012(!!-!*!/'-#!3'%!-!!*!!

"!!'"!#,;+49!

:,89!,'0*=,,(+!%,:!,46!%,:)(+?4!,01!5-22%,09!,01!%*05017(;#!!*2 =

*!-!! ⋅!'#! 0,:,(!!"

"#

!"

"$

"

"= "!#,02.689! 7%! ./012(4!-!2 = !

"!!*2,

:,01!!-! (;6%+!4!)+32(,'09! ,49!)0+2.649!+.%;'%9!*%+! ,0! ,2-2%!!*! +$

+0=,%+!2(!,46!,5(1'3!,49!,1'%2;)%9"! !."")"

0/11"!!

2

2

)*

+)

,*

+,=

%&#&''($ /(8'0=2(! %%/"! (/E'!,0!,'0-

4<0=2(60!42+*=*5+0!*%+!*%,%+*(1-3A012(!,0!,(,'3<860!!'&#!*%+!,0!

0'70<>6+0! -'&1"! #,(+)-! ,0! ,';$

<860!*!-! (;6%+! ++0<>6+0! ,'09! ,0!

,';<860!*!'!++/=(+: !'

*!

!*

-!= ,

3'%!-!!⋅!!'!*!!*2. 2289!++/=016!0+!+/.+(+94 ( )( )"!%&

!#'%

"!

!#= , (!-1#'!*!!-!⋅!!'!*!!*2!!*%+!!%-'&1'!*!-'⋅!'!*!*'2,

%,:!:,01!,'0*=,,(+!4!,'09!%,:)(+?4!+/.+4"!

!"#$!!!"#)

166 #B")#"*1!1%!$"@#"!2

!!"!"#"$#$ !6!r!(;6%+!4!%*,;6%!,49!,('+<(<'%22.649!+.%;'%9!*%+!x!4!%*2-!,49!

,1'%2;)%9! ,01! *%,%+*(13+74*(! %)45%)-! 4! ,5(1'3! ,01! *%606+*0=! ,($

,'3()'01'#!,:,(!:,89!%,0)(;/74*(#!++/=(+!4!+/.+44!

4r2 = 23 x2#!!3'%!!x = r6

32r

322 = .

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

!"#!"#$!"&!$%$$$%!"&( 4!!'!"!#'%0!#5%,!'!&*&-'.!*'"%%'"*!'!-!&!!%++"!&%,!#%!#&!,-"!"!./12!7+-&%!/"*$+*067&12!6%!"$&!/0"!6,'!"!'!-!&!!!/*'%--85%,!."-!"*!"%""%+1&*!"$2!'-!67""*0!"$2!#&!,"!2!%,&!-!'-/)%#-*!

!/.!"*!"%""%+1&*!"$2!/)%0"%2!"*0!*'"!7'"*0$ %&#&''($ 0+,8! !-! 4! )+32(,'09! ,49!

)0+2.649! +.%;'%9! *%+! '! ,0! 2.+0!

,49"!/(8'0=2(!,0!42+*=*5+0!)+%2.-,'01!!-! *%+! ,46! *37(,4! ,'09! ,46!

!-!+,0!+42(;0!'#!4!0,0;%!,.26(+!,0!

42+*=*5+0!+,0!+42(;0!*!*%+!..'01$

2(!,0!,2-2%!*-" 9!"!#'%0#!"*0!*'"!7'"*0$ "%,%+*(13A012(!,(,'3<860!#&1/!2(!,5(1'3!;+4!2(!*-!*%+!.+,8!

"! ,0!+42(;0! ,02-9! ,86!/&#!#1"!%,0!+42(;0!"!1D>6012(!*37(,4!+,0!

(,;,()0!,01!,(,'%<>601!*%+!(*%,.'87(6!,01!(,+,.)01!,%;'6012(!,2-2%$

,%!")#!"(!;+%!2(!,%!,2-2%,%!"##!"&#!"1#!"/"!3.'012(!,%!,2-2%,%!

)##!)&#!)1#!)/#!(##!(&#!(1#!(/!%%/"!(,%'"

#,(+)-!"#!*!"/!*%+!4!<86;%!∧

ΘEK !(;6%+!0'7-#!./012(!/#2!*!-!#"2..

#,(+)-! ++/=(+4!)"!*!"#!*%+!4!<86;%! !∧

ΛΚΕ ! !(;6%+!0'7-#!./012(!#)2 = *!-!#"2"!$'%!)#2!*!#/2#!!0,:,(!)#!*!#/"!'+%!,06!;)+0!5:<0!)/!*!/#"!!

$'%!,0!,';<860!)#/!(;6%+!++:,5(1'0"!((!,06!;)+0!,':,0!%,0)(+*6=012(!

:,+! *%7.6%! %,:! ,%! 1,:50+,%! ,';<86%#! ,86! 0,0;86! 0+! E3+(+9! (;6%+! 0+!

!"#$!! 14

!!!"!#$#""!!%!#&#! 167

,5(1'.9! ,01! ,(,'%<>601! #&1/! *%+!

*0'1..9! ,%! +42(;%!)#!(! (;6%+! ++:$

,5(1'%"! #,02.689! *%,%+*(13+,4*(!

0*,3()'0#! ,0! 0,0;0! ,('+./(,%+! +(! 0-*,>!++:,5(1'%!,';<86%.

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

%6-*(+!+,0!42+*=*5+0!)+%2.,'01!)("!

'+%!,06! ;)+0!5:<0#!%6!,('++,'%.(;!,0!

42+*=*5+0!*%+!(,%6.57(+!+,46!%'/+*-!

,01! 7.+4#! )+%,4'>6,%9! +,%7('-! ,4!

)(#!7%!)+.57(+!*%+!%,:!,%!+42(;%!&#!

1#!/"!$'%!,0!0*,3()'0!(<<'3.(,%+!+(!+.%;'%!)+%2.,'01!)(" (!#&!,"!!%,&!-!$!'*#67&$$!

#,(+)-!)#!*!#(!*%+!4!<86;%! !∧

ΛEM ! !(;6%+!0'7-#!./012(!)(2 = *!-!)#2"!#,;+49#!(,(+)-!!'!*!'-#!++/=(+!!-!*!-!-'"

#,+,5.06!++/=(+! 2

2

-*

!-

-'

!-= !%,':,%+4!VI"!&!*%+!E+E5;0!V!0'++2:9!5'"

#,02.689!!-2!*!-!-*2. !,0)(;/74*(!:289!:,+!!)(2!*!-!)#2!*%+!%*:2%!++/=(+!:,+!*-2 =

*!)#2#!)+:,+!#/!*!*-" $'%!7%!(;6%+!!-2!*!)(2#!)45%)-!!-!*!)("! #,02.689! 4! )+32(,'09!)(! ,49!+.%;'%9! ++0=,%+!2(! ,4! )+32(,'0!

,49!)0+2.649!+.%;'%9!!-!*%+!,%1,:/'06%!%,0)(;/74*(!:,+!,0!,(,'3<860!

,49!)+%2.,'01!,49!+.%;'%9!(;6%+!)+,53++0!%,:!,0!,(,'3<860!,49!,5(1'39!

,01!0*,%.)'01"!!!."")" !!"!"#"$#$ !6!r!(;6%+!4!%*,;6%!,49!,('+<(<'%22.649!+.%;'%9!*%+!x!4!%*2-!,01!

0*,%.)'01!,01!*%,%+*(13+74*(#!)45%)-! ,01!*%606+*0=!0*,%.)'01#!%$

,0)(;/74*(!:,+!++/=(+!4!+/.+44!,r2 = 2x2. #,02.689!7%!./012(!!!

x = r2 ⋅

!"#$!! "$!

168 #B")#"*1!1%!$"@#"!2

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

!"#!"#$!")!$%$$$%!")( 4!!'!"!#'%0!#5%,!'(;*2!'!-!&!!%++"!&%,!#%!#&!,"!"!./12!7+-&%!

/"*$+*067&12! '!-! 6%! "$&! /0"!6,'!"! '!-! &!! '%-85%,! ."-! "*! "%-""%+1&*! "$2! '-!67""*0! "$2! #&!,"!2! %,&!-! ""-/)%#-*! !/.! "*!

"%""%+1&*!"$2!/)%0"%2!"*0!'(;*0$ %&#&''($ 0+,8!!-!4! )+32(,'09! ,49! )0-

+2.649! +.%;'%9#! ,46! 0,0;%! ,.26012(!

+,0!+42(;0!'#!>+,(!!'! *! -!'-"!%,0!

+42(;0!'!..'012(!*37(,4!+,46!(17(;%!

!-#!4!0,0;%!,.26(+!,0!42+*=*5+0!)+%$

2.,'01!!-!+,0!+42(;0!!*" 9!"!#'%0#!"*0!'(;*0$ /(8'0=2(! %%/"! (+%'! ,(,'3-

<860!#&1/!2(!,5(1'3!;+4!,'09!,0!

,2-2%!*-"!%,%!+42(;%!##!&#!1#!/!

..'012(! *37(,(9! +,0! (,;,()0! ,01!

,(,'%<>601!*%+!,%;'6012(!,2-2%,%!

#"#! &)#! 1(#! /#! ;+%! ,'09! ,46!

,5(1'3!,01!,(,'%<>601!*%+!,'09!,0!

;)+0!2.'09!89!,'09! ,0! (,;,()0! ,01!

,(,'%<>601"!#,02.689!*%,%+*(13-+74*(!*=E09!0!0,0;09!,('+./(,%+!+(!

.?+!;+%!,(,'3<86%"! $++"!&#!#%!#&!,"!$! 3.'012(! ,%! ,2-2%,%!"1#!#1"!#,(+)-!4!(17(;%!"#!(;6%+!*37(,4!

+,0!(,;,()0!#&1/#!4!<86;%! HEK !(;6%+!0'7-#!3'%!,0!42+*=*5+0!)+%2.$

,'01!"1!7%!)+.57(+!%,:!,0!+42(;0!#"!#,;+49#!(,(+)-!4!(17(;%!1&!(;6%+!

*37(,4!+,+9!(17(;(9!#&#!&)!7%!(;6%+!*37(,4!*%+!+,46!(17(;%!"&#!3'%!,0!

42+*=*5+0!)+%2.,'01!"1!7%!)+.57(+!*%+!%,:!,0!&"!'+%!,06!;)+0!5:<0#!%6!

,0!42+*=*5+0!,('++,'%.(;!)+%,4'>6,%9!+,%7('-!,46!"1#!7%!)+.57(+!*%+!

%,:!,+9!1,:50+,(9!*0'1..9!,01!*=E01#!%.0=!(,%6.57(+!+,46!%'/+*-!,01!

7.+4"

!"#$!! 15

!"#$!! "%!

!!!"!#$#""!!%!#&#! 169

(!#&!,"!!%,&!-!$!'*#67&$$!

#,(+)-!1&!*!#&!*%+!4!<86;%!!∧

EZH !!(;6%+!0'7-!./012(!#12!*!-!#&2 .

#,;+49!#"!*!#&!*%+!4!<86;%! !∧

KEH ! !(;6%+!0'7-#!3'%!"12!*"#2!)!#12. #,02.689!./012(!"12!*!/!#&2.

#,;+49!(;6%+!!-!*!/-'!*%+! 2

2

*-

!-

-'

!-= #!!3'%!++/=(+!!-2!*!/!-*2.

!,0)(;/74*(!(,;+49!:,+!!1"2!*!/#&2!*%+!./012(!#&!*!*-#!(,02.$

689!"1!*!!-#!)45%)-!4!)+32(,'09!"1!++0=,%+!,'09!,4!)0+2.64!)+32($

,'0!!-"! !,0)(;/74*(!50+,:6!:,+! ,0! ,(,'3<860! ,49!)+%2.,'01! ,49!,('+<(-

<'%22.649! +.%;'%9! ,01! *=E01! (;6%+! ,'+,53++0! %,:! ,0! ,(,'3<860! ,49!

,5(1'39!%%*2-9'!,01!*=E01"! !."")" !!"!"#"$#$! !6!r!(;6%+!4!%*,;6%!,49!,('+<(<'%22.649!+.%;'%9!*%+!x!4!%*2-!,01!

*=E01#!,:,(#!:,89!%,0)(;/74*(#!++/=(+!4!+/.+44

4r2 = 3x2#!!!!0,:,(!7%!(;6%+!!!x = r332 .

,%!)"*+*',#!'',+#-'&"+!–!,"!"#,'./!

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

#&!,"!"!./12!7+-&%!'!-!6%!"!!/"*$+*(6%&!!#8#6!"!"!'!-!&!!!#

/*'%-85%,!."-!$!/)%0"%! !"*0!%-'*#%%'"*0!%,&!-!%""$"*2"!$!)%+.-6%&$!%)%##1&$

%&#&''($ (/1"!"+ $++"!&#! '!&*&-'*(! /%&"!-

+3&*0! #%! '(')*"! "*0! */*,*0! "*!

"%""%+1&*!"$2!!'",&!2!-#*("!-!6%!

"*!()*!"*0!"%""!+3&*0!"$2!'-!67#

""*0!"$2!#&!,"!2$

!"#$!! 16

170 #B")#"*1!1%!$"@#"!2

0+,8!!-!4!)+32(,'09!,49!)0+2.649! !+.%;'%9!*%+!.+,8!'!+42(;0!

,49! )+%2.,'01!!-! ,.,0+0!>+,(!A'! *! ,'B"!((!)+32(,'0! ,0! ,2-2%!!-!

<'3.012(!42+*=*5+0!*%+!+,0!+42(;0! !'!..'012(!*37(,4!+,46!!-#!4!0$

,0;%!,.26(+!,0!42+*=*5+0!+,0!+42(;0!*#!*%+!(6>6012(!,%!+42(;%!B#!*" /(8'0=2(!*=*50!*.6,'01!3!2(!%*,;6%!;+4!,'09!B*!*%+!*%606+*:!

,(6,3<860!#&1/"!(<<(<'%22.60!+,06!,%'%,368!*=*50"!*+/0,020=2(!

,%!,:?%!#&#!&1#!1/#!/"#!KE!+,%!+42(;%!)#!(#!##!F#!$!*%+!..'012(!,%!

,2-2%,%!)(#!(##!#F#!F$#!$)#!#$"!!0!,(6,3<860!)(#F$!(;6%+!(,;$

+49!*%606+*:!*%+!,0!(17=<'%220!,2-2%!!#$!(;6%+!,5(1'3!*%606+*0=!)($

*%<>601!(<<(<'%22.601!+,06!,%'%,368!*=*50"! %,%!+42(;%!##!&#!1#!/#!"!!1D>6012(!*37(,(9!+,0!(,;,()0!,01!*=$

*501#!,%;'6012(!,2-2%,%!#C#!&&#!1%#!/!#!"B! ;+%!2(!,46!%*,;6%!,01!

*=*501! *%+!..'012(! ,%! ,2-2%,%!C&#!&%#!%!#!!B#!BC#!C)#!)&#!&(#!

(%#!%##!#!#!!F#!FB#!B$#!$C"!

!"#$!!!"&! !!"#$!!!"&) :*!/%&"%+1&*!!<=:>!%,&!-!'!&*&-'.. C'3<2%,+#!%.0=!0+!(17(;(9!#C#!"B!(;6%+!*37(,(9!+,0!;)+0!(,;,()0#!

(;6%+! 2(,%?=! ,019! ,%'35545(9"!#,;+49! ,%! ,2-2%,%!#C#!"B! (;6%+! ;+%"!

#,02.689!,%!,2-2%,%!CB#!EK!(;6%+!;+%"!$'%!,0!,2-2%!CB!(;6%+!;+0!2(!

,46! ,5(1'3! ,01! *%606+*0=! ,(6,%<>601!#&1/""!'+%! ,06! ;)+0! 5:<0! ,%!

,2-2%,%!C&#!&%#!%!#!!B!(;6%+!;+%!2(!,46!,5(1'3!,01!*%606+*0=!,(6,%$

<>601!#&1/"!"!$'%!,0!,(6,3<860!C&%!B!(;6%+!*%606+*:" (/1"!#+ :!!"",+1&!!!?>!'!-!!@?!%,&!-!-#./)%0"!$ #,(+)-!C#!(;6%+!,5(1'3!*%606+*0=!(?%<>601#!E$!*%606+*0=!)(-

*%<>601! *%+! 4! <86;%!∧

ΠEO ! (;6%+! 0'7-#! ,0! ,2-2%!C$! ++0=,%+! 2(! ,46!

!!!!"!#$#""!!%!#&#!! "#"!

'($)*+!%,)!-&.,./-,0!'$.%&12.,)$!'/3%/!%,!%$%*+14.,!%56!'($)*+6!%,)!

'$.%&12.,)!$7.&/!7(,!8$!%,!%$%*+14.,!%56!'($)*+6!%,)!$9&12.,)!().!%,!

%$%*+14.,!%56!'($)*+6!%,)!'$-&12.,)$!

)/&!%,.!7'/,!(31,!%,!$:!/(,0%&/!8$!%5.!'($)*+!%,)!-&.,./-,0!'$%

.%&12.,)$!

!,!%8;8&!<:!=>$/!;'5!&','$/>?$7!7(,!8$!%5.!'($)*+!%,)!-&.,./%

-,0!'$.%&12.,)$!**&!%,!%*714.,!<$:!$7.&/!/(3'($)*,$!

)/&!%,.!7'/,!(31,!-&/!%&!%*714.&!<@&&!!&A%&!%#!&!'B:!$7.&/!/%

(3'($)*&$!

!"#!$%""!!A$! &)%3! %,.! %*3',! -&%&(-$)+(?5-&.! ;'5! ,/! '=-&!

(")*!&'3!%/6!$7-,(/!(+)*!='*$6!%,)!C5%,08$.,)!$/-,(&='*,)$!

#'7(56&!&+,0!-+?$!%8;8&!<@&!<$!!&','$7>?5-$!7(,!8$!%5.!'($)%

*+!%,)!-&.,./-,0!'$.%&12.,)!-&/!%,!%8;8&!@$!!$7.&/!'($)*+!%,)!-&.,%

./-,0!'$.%&12.,)&!='$%&/!3%/!%,!%*714.,!<@$!$7.&/!/(3'($)*,$!

)/&!%,.!7'/,!(31,!-&?=.&!&'3!%&!%*714.&!@&A&!A%#&!#!B&!B:$!!

$7.&/!/(3'($)*&$!

!

!!""!"#!!

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

%%,!-=.%*,!,!%,)!-0-(,)!',)!

?$4*;(&8$! (%,!-;8&! ",! )D2.,)8$!

-+?$%5!(%,! $'7'$',! %,)!-0-(,)!-&/!

'&7*.,)8$! '*,6! %,! &)%3! 8=*,6! '/&%

',>/-+! %8;8&%&! ,E! &! EF! ! 7(&! 8$!

%5.!'($)*+!%,)!-&.,./-,0!$9&12.,)!

-&/! %,)! '$-&12.,)! &.%7(%,/>&&! -&/!

'*,6!%,!+((,!8=*,6!%8;8&!,G! ! 7(,!

8$! %5.!'($)*+! %,)!-&.,./-,0!'$-&%

12.,)$!

#'$/';!,E&!<#!$7.&/!7(&!%8;%

8&%&!-&/!-+?$%&!(%,!$'7'$',!%,)!-0%

-(,)&! ='$%&/! 3%/! $7.&/! 7(&! -&/! '&*+((5(&$!**&! %,!<E! /(,0%&/!8$! %5.!

'($)*+!%,)!-&.,./-,0!$9&12.,)$!

-."$/$+%0."123""!$0#4"4"567!(/.$""

<*+18&%/&! $'$/';! <E! $7.&/! '($)*+! -&.,./-,0! $9&12.,)&! EF!

'($)*+!-&.,./-,0!'$-&12.,)!-&/!5!14.7&!!

"#, !$7.&/!,*?;&!='$%&/!3%/!

%,!%8;8&!<F!/(,0%&/!8$!%5.!'($)*+!%,)!-&.,./-,0!'$.%&12.,)!('*3%&%

!

!"#$!!!"#%!

!"#$! !!"#!"#$!%$%&"'!"%&!

&(!"'()!')(!)*+!,*)*!-./*!)*!)010(!

!2! )&*3)()!0+!)(+!4-+567!)*5!8(*

+*+)8*3! 4++)(/9+*5)! ,6(! )*! )6,*

/:+*! ;2!! +,+()! )&.4-+56*)! ')(!

)*+! ,*)*! -./*! 8(<=+(! (4.! )(! 54.*

-*)4(! )6,/:+(! 0+! >7&+)?! ;@+! @$+!

$%+!%!!8()!8*65-1!)*!2!!+,+()!+4,*

&(?!)&.4-+56*)!

&0*,:?+! )(! )6,/:+(! A#B+!

ABC+!ACD+!AD&+!A&#!!+,+()!8()!

(5)7!)&.4-+56()!!

!"#!$%""!!"()(&8+57&<(8(+!=)&)!*)!54.-*)4+?!*=8(!,"'(!=*6+?!)*5!

E()*30++*5!+)8*&(=*6*5)!,6(!8()(&8+57&)(8+!+)8*&7+*6*!4*5!4+6)8-+,+*

)()!(4.!+,8*&)!)&.4-+56(!)6,/:+()!!

!

!

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

#&&'$%(""!""%$$'$#!!

!4+)*1!.'!+,+()!4-+567!8(+*+)8*3!+F(/9+*5!8()!'2!4-+567!8(*

+*+)8*3!*+8(/9+*5+!)*!)010(!.2!=G+)!*)()6+<+,!&+!0=&*!8()!786*!-./*!

&)*!'!,46.)(&(!-(!8()!.'!+,+()!!)*!0+/(-3)+6*!)010(!)(?!*)(,6+&(?)!').!

(5).!)*!-./*! )&G3+)/!'2

.'

.'

2.! )! !!4,&(?! ! !.'!0!.!! !8()! !'2!0!.!+!!

76(!!.!

.!

.!

2.! )!

&)!/:+,+?! !"

#$% +!"

%$& !+,+()!*6<=?+!76(!(!/:+,(! !"

&%# ! !+,+()!

*6<1+!-./:!)(?!*0*).)()(?!):+!)6)/9+:+!!!2.!+!!!.A!,46.)(&(!12)!3()!

')(!)*+!,*)*!-./*+!(-*3!

'2

.'

.'

2.! +! ! ! !2.!0!A'!8()! !.'!0!';+! !=4+)()!.))!

'2

;'

';

A'! +!

*4.)+!(!/:+,(! !"

&'# ! !+,+()!*6<1! ,46.)(&(!12)! 3()!,6(!)*!(0)838-)*!

*)(0=)6*5!A2!*)=6G+)()!(4.!)*!&(0+,*!;)!!

!

"()!(+!)*!(0)838-)*!4+6)&)6(-+,!/36:!(4.!)*!2A!8()!+4(+=-<+)!

&)(+!,*)(!<=&(!(4.!)(+!*4*,(!F+8,+(&++!<(!*)=-<+)!(4.!)*!;!!8()!)(!54.*

-*)4(!&(0+,(!)*5!+)8*&(=*6*5)!

!

!"#$!!!"#"!

!!!"!#$#""!!%!#&#! 173

(!#&!,"!!%,&!-!$!'*#67&$$! 0+,8!!4!,0!2.+0!,01!,2-2%,09!3@"!!.0=!

,0!3G!./(+!)+%+'(7(;!+(!2.+0!*%+!3*'0!5:<0!+,0!

@! *%+! ! @G! (;6%+! ,0! 2+*':,('0! ,2-2%#! ./012(4!!

%G!4'2! *! +! %!4@'2 %,':,%+4! /'"!#,;+49! !GH! *! *!-!G!4!*%+!!3@!*!-!A4@#!!3'%!%GH'2!*!+!%3@'2 *%+!%.0=!!!'!*!,!'-#!!.,(,%+!:,+!!AB!*!+!-'"

"+/=(+!(,;+49!4!+/.+4! ! 2

2

-*

AB-'

!-= ! !%,'0-

,3+(+9!VI"!&!*%+!!!V"!5'"!!$'% (AB)2 = 5 (B*)2. #,;+49#!!B*!*%+!3@!(;6%+!;+%!2(!,46!%*,;$

6%! ,01! *=*501!#&1/"#! 0,:,(!!-! *!HG"!$'%!

HG!(;6%+!)+32(,'09!,49!)0+2.649!+.%;'%9" (/1"!)+ (!/)%0"%!"*0!%-'*#%%'"*0!%,&!-!%""$"$$!! #,(+)-!4!)+32(,'09!,49!+.%;'%9!(;6%+!'4,-!*%+!,0!,(,'3<86:!,49!

(;6%+!++0)=6%20!,'09!,0!,(6,%,53++0!,(,'3<860!,49!%*,;6%9!,01!*=*501!

(;6%+!'4,-!*%+!4!%*,;6%!,01!*=*501"!!$*!!+!,!-!%,&$"&*./!&.0!+1+%.0!"2'

(!,!*-&#( #36!)(!+(!*=*50!2(!'4,-!)+32(,'0!(<<'%.(;!*%606+*:!,(6,3<860!

4!,5(1'3!,01!,(6,%<>601!(;6%+!350<09!%3''4,4'#!*%+!5.<(,%+!"%&##'(. 1!,5(1'3! ,01!,(6,%<>601!#&1/"! (;6%+!4!,5(1'3! ,01! (+*0+%.$

)'01"!!!."")" !!"!"#"$#$ !6! r!(;6%+!4!%*,;6%!,49!,('+<(<'%22.649!+.%;'%9#!,:,(!4!%*,;6%!

,01!*=*501!%,:! ,06!0,0;0! <;6(,%+!4!*%,%+*(1-! ,01! (+*0+%.)'01! (;6%+!

r5

2p = .

1!%*,;6%!,01!(<<(<'%22.601!,(6,%<>601!(;6%+4

( )55105r5210

5r5210

2p

−=−=−

!"#$!!!"&"

174 #B")#"*1!1%!$"@#"!2

-%!)"*+*',#!&5&',-'&"+!-!,"!"#,'./

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

#&!,"!"!./12!7+-&%!'!-!6%!"!!/"*$+*(6%&!!#8#6!"!"!'!-!&!!!/*-'%-85%,!."-!$!/)%0"%!"*0!'1'%'!7'"*0!%,&!-!%""$"$"!$!)%+.6%&$!

!/*"*6#$ (/1"!"+ 9!"!#'%0#

!"#$!!!") !"#$!!!")!

/(8'0=2(!,06!*=E0!,01!*%,%+*(13+74*(!+,46!,':,%+4!(+"!C%;'-6012(!)=0!(,;,()%!*37(,%!2(,%?=!,019#!,%!!-'*#!'-#&!*%+!,.26012(!

,+9!,5(1'.9!!-#!-'#!'*#!*!#!#&#!#-#!&'!+,%!2.+%!1#!/#!"#!)#!(#!##!F"!

0+,8!$!,0!+42(;0!,02-9!,86!#F#!(/!*%+!C!,0!+42(;0!,02-9!,86!/)#!

1""!*+%+'0=2(!*%7.6%!%,:!,%!,2-2%,%!#$#!$F#!/C!+(!3*'0!*%+!2.+0!

5:<0!+,%!+42(;%!&#!%#!!#!2(!2(<%5=,('%!,2-2%,%!,86!)+%+'.+(86!,%!&$#!

$%#!!C" %,%!+42(;%!&#!%#!!!1D>6012(!*37(,(9!+,%!(,;,()%!,01!*=E01!*%+!

(?8,('+*3!,01!*=E01!,%;'6012(!,2-2%,%!&B#!%3#!!@!!!;+%!2(!&$#!$%#!

!C!*%+!..'012(!,%!,2-2%,%!B-#!-@#!@'#!'3#!3B" (/1"!#+ 6+!&'*!-&7'."+!8(%)*!'9*"'!'##&7'."+.!'&9&'&+!,"'!'#+:;*'+% :*!/%&"%+1&*!%,&!-!-#./)%0"*$!

!!!"!#$#""!!%!#&#! 175

$<012(!,%!,2-2%,%!&-#!%-#!3-"! #,(+)-!,0!,2-2%!#$!)+%+'(;,%+! ! !+,0!+42(;0!&!+(!3*'0!*%+!2.+0!

5:<0!*%+!,0!2(<%5=,('0!,2-2%!(;6%+!,0!&$!++/=(+4

$#2 )!#&2!*!/!$&2!!!!%,':,%+4!!,' #,;+49#!$#!*!#-!!*%+!$&!*!&B"

$'%!-#2!)!#&2!*!/!&B2. #,+,5.06!-#2!)!#&2!*!-&2!!%!,':,%+4!""!,.'

$'%!-&2 *!/&B2#!*%,3!+16.,(+%!!-&2!)!&B2!*!,!&B2!!!*%+!(,(+)- -&2!)!&B2!*!-B2#!!.,(,%+!:,+!!!-B2!*!,!&B2.

$'%!-B!*!-!&B" #,;+49!3B!*!-&B#!)+:,+!%&!*!-$&#!)45%)-!!!%&!*!-!&B"! %16(,>9!-B!*!B3" $20;89!%,0)(+*6=(,%+!:,+!*37(!,2-2%!%,:!,%!-@#!@'#!'3!(;6%+!

;+0!,'09!,%!-B#!B3" :*!/%&"%+1&*!%,&!-!%/,/%'*. %,0!+42(;0!$!1D>6012(!*37(,4!+,0!(,;,()0!,01!*=E01!*%+!(?8$

,('+*3!,01!*=E01!,%;'6012(!,2-2%!$H!;+0!2(!&B" 3.'012(!,%!,2-2%,%!!H/#!/@!*%+!7%!)(;?012(!:,+!4!H/@!(;6%+!(17(;%" #,(+)-!,0!,2-2%!/C!)+%+'(;,%+!+,0!!!+(!3*'0!*%+!2.+0!5:<0!*%+!

,0!2(<%5=,('0!,2-2%!(;6%+!,0!C!#!./012(!!/

C!

C!

/C= . #,;+49!/C!*!/$!

*%+!C!!*!!@!*!$H"!4!'%!!!!/

@!

$H

/$=

1! (17(;%!/$! (;6%+!,%'355454!+,46!!@#! (,(+)-!*%+!0+!)=0! (;6%+!*37(,(9!+,0!;)+0!(,;,()0!%,':,%+4!XI. 6).

1!(17(;%!!/!(;6%+!,%'355454!,'09!,46!(17(;%!$H#!(,(+)-!*%+!0+!)=0!(;6%+!*37(,(9!+,0!;)+0!(,;,()0"!"%+!%6!)=0!,';<86%#!:,89!,%!H$/#!/!@!,%!0,0;%!./016!)=0!,5(1'.9!%6350<(9#!,0,07(,470=6!2%A;!+(!2+%!<86;%!%./016!*0+6-!*0'1.-!,0!/'!*%+!0+!02:50<(9!,5(1'.9!(;6%+!,%'35$545(9#!,:,(!0+!355(9!)=0!,5(1'.9!7%!E';+*06,%+!+,46!;)+%!(17(;%!!%,':$,%+4!VI. 32).

$'%!,%!,2-2%,%!H/#!/@!E';+*06,%+! !+,46!;)+%!(17(;%"!"37(!(1$7(;%!E';+*(,%+!+(!.6%!(,;,()0!%,':,%+4!XI. 1).

$'%!+(!.6%!(,;,()0!7%!E';+*(,%+!*%+!,0!,(6,3<860!B-@'3"!

176 !!"#!"#$!"$%&"'!"%#

!"!#!$"%&'$"!!($#)!)$"&*$)"" !(&)'*! (+! (,*,)!-&! '))).&/())!*(+!*0,&/+!1!*&!,2*+!3))!43.+!

567+!,&!,&7)58(&.+!(,*,)!(+!1&$!(+!(,*,)!-$!'))).&/())!!*(+!&!*&!4%

3.+!3))!,2*+!567+!,&!,&7)58(&.+!(,*,)!(+!-&!&(.6()*0!'()! +.)!-$2!*!$&2!+!,!-&2!&(.6()*0!!-() !(/*09!-&!+!-,!!3))!&$!+!$-)!+.)!-$2!*!$-2!+!,!-,2$!&(+,2%

:;9!!-$2!*!$-

2!*!-,

2!+!-!-,

2. !(&)'*!$-

2!*!-,

2!+!$,

2!!!&(.6()*0!I)!-.($!2(&())!6()!,$2!*!$-2 =

+!-!-,2$!'05)'*!!!,-2!+!-!-,2. +.)!,-!+!/!,-)!!(/*09$!,.!+!/!,-$!4.)!,-!+!,.)! !(&)'*!,!!+!,'$!!-!+!.'!!3))!!,-!+!,.!()!(./7;:)!,!-$!,'.!

&/:))!/*)$!4.)!0!7;:/)!,!-!&/:))!/*0!,&!(0!7;:/)!,'.) &,+/;9! )(+'&)3:8+<,&! 6()! 0! 7;:/)!!-.! &/:))! /*0! ,&! (0! 7;:/)!

,'.)!+.)!+)!(.&)9!7;:/&9!,'.$!,!-$!!-.!&/:))!,&()=8!(+<9!/*&9)!!4:!

*&!)*6(5&<.+!(&:(47;:+!+)!(.&)9!7;:/&9!&/:))!/*&9!,&()=8!(+<9$!(+!(&:(4-7;:+!&/:))!)*+7>:)+!&(.6()*0!.()!+.)!(+!(&:(47;:+!,!-.'!&/:))!)*+%

7>:)+)!!(+,2:;9!(+!(&:(47;:+!,!-.'!&/:))!3):+:)36!&3))!0(4:;0!)(6!

(0:!(5&<.4!,.!(+<!38?+<() !4:!,&!34@&!,))!)(6!()9!'>'&3)!(5&<.29!(+<!38?+<!3)()*3&<4-

*+<,&!,&!(+:!/')+!(.6(+!(&:(47;:)$!@)!3)()*3&<)*@&/!*(&.&6!*A*,)$!(+!

+(+/+!(&.)2A&())!*&!'>'&3)!3):+:)34!(&:(47;:)$!3))!(+!+(+/+!3)5&/())!

!!!""#"!$%. !!""!"# %&&+#&,!$!!$&#(+# B.+&3(&/:+<,&!(0:!&<@&/)!C&!3))!2*(;!D!(+!*0,&/+!(+,*9!,&!(0!

'))7>:)+!(+<!38?+<) %+!*0,&/+!D! &/:))! (+!32:(.+! (09!*/)/.)9!(+<!(&.)5),?4:&)! (+:!

38?+!3))!(+!(,*,)!D&!&/:))!(+!,)*6!(09!(5&<.49!(+<!38?+<)!!(&)'*!(+!

(,*,)!-$!'))).&/())!*(+!&!*&!43.+!3))!,2*+!567+!3))!,&7)58(&.+!(,*%

,)!&/:))!(+!-&$!2A+<,&1!-$2!*!$&

2!+!,!-&

2!!!&(.6()*0!-(

!()(52+:!-$!+!CD$!-D!+!&D$!C&!+!&$!3))!&$!+!C!)!

+.)!DC2!*!C!

2!+!,!-&

2. !(&)'*!DC

2!*!C!

2!+!D!

2!!!!&(.6()*0!I)!-.(!2(&())!!D

2!+!,!-&

2. !(/*09$!(+!(&(.47;:+!(09!)3(/:)9!(09!*/)/.)9!(+<!(&.)5),?4:&)!

(+:!38?+$!&/:))!(.)(54*)+!)(6!(+!(&(.47;:+!(+<!,)*+8!(09!(5&<.49!(+<!

38?+<!!&(.6()*0!2'() "))!&(&)'*!(+!(,*,)!-&!&/:))!(+!,)*6!(09!(5&<.49!(+<!38?+<$!(+!

%%!"%-&-""%!$%!1!% 177

(,*,)!!D! &/:))! /*+!,&! (0:!)3(/:)! (09!*/)/.)9! (+<! (&.)5),?4:&)! (+:!

38?+)!+.)!(+!*0,&/+!!!):*3&)!*(0:!&()/4:&))!(09!*/)/.)9) !!""!## '!#-!.+%!"".!('(!/#0(+".!!($#)!%++1"1#!#.",!#".!"$"2%3!"#)!/#)!

##"""2," !(&)'*!(+!(,*,)!-&!2A&)!'))).&@&/!*(+!*0,&/+!1!*&!43.+!3))!,2*+!

567+!,&!,&7)58(&.+!(,*,)!(+!1&!3))!&(&)'*!(+!(,*,)!&E!2A&)!'))).&@&/!

*&!43.+!3))!,2*+!567+!*(+!*0,&/+!$!,&!,&7)58(&.+!(,*,)!(+!&$$!):!(+!

(,*,)!-E!'))).&@&/!*&!43.+!3))!,2*+!567+$!(6(&!(+!(,*,)!1$!@)!&/:))!(+!

,&7)58(&.+)

"*A8&)!-1

&1

&1

-&= $! ! 4.)!

2--

2112112--

= $! +(6(&!$E1-

1$

1$

-E

+=

&'))/.&*0!(09!(5&<.49!(+<!38?+<!*&!43.+!3))!,2*+!567+!(.6()*0!V.15). !(+,2:;9!-E!3!1$$!!4.)!!1$!3!1-!*!$E)

+.)!(+!(,*,)!-E!'))).&/())!*&!43.+!3))!,2*+!567+!,&!,&7)58(&%

.+!(,*,)!(+!!1$)!!(/*09!!1$!+!!-)!!(+,2:;9$!):!(+!(,*,)!!-E!!'))).&%

@&/!*&!43.+!3))!,2*+!567+$!(6(&!(+!,&7)58(&.+!(,*,)!(09!'))/.&*09!&/%

:))!(+!!-)!"))!&(&)'*!0!')4,&(.+9!(09!*/)/.)9!&/:))!.0(*!3))!(+!(&(.4%

7;:6!(09!&/:))!(.)(54*)+!)(6!(+!(&(.47;:+!(09!(5&<.49!(+<!38?+<$!2(&%

())!6()!3))!0!(5&<.4!(+<!38?+<!&/:))!.0(*) !4:!6,;9!2:)!.0(6!(,*,)!'))).&@&/!*&!43.+!3))!,2*+!567+$!(6(&!

3)@2:)!)(6!()!(,*,)()!!(09!'))/.&*09!&/:))!4..0(+!3))!527&())!#&%$%'(. +.)!0!(5&<.4!!-!(+<!';'&3)2'.+<!&/:))!4..0(+9!)(+(+,*)!!!.")')

$%&'#"" )"#$!1!#-!.+%!"".!/45".!()#)+!6!(!$!!%/+"!/#)!20$"!-7&"#!""!2!&#-4"!+"! "2,2#! "18! ()#(+!$18! !($#)!1!#-!.+%! "".! ('(!/#0$(+"." 9#+#",+1$1 %:!!!!!0!)3,*!(+<!38?+<!3))!!r!!0!)3(/:)!(09!*/)/.)9$!(6(&!0!(5&<-

.4!!)!!(+<!';'&3)2'.+<!(+<!3)()*3&<4*@03&!&/:))1!!

)!+! !2

15 −!!4.)!!!)!!+!! ( ) ( )315

3r5

3r

−=− !

178 !!"#!"#$!"$%&"'!"%#

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

$&%$"#,!%&!'4&&&$!%&)

:#! !/"!6"4$! ")! #-!.+08! "'$!#0$"!!$"!+!*$!$;12%"'$!#".!/#"#$/!.%$61/#$! /#)! $#! $.$&/+)6"4$!2!"#<4!"".8"

'/%(%'5, 0*(;!%,! 0! ')4,&(.+9! (09! '+-

*,2:09! */)/.)9! 3))!.$!#! *0,&/)! (09!

(2(+))$!>*(&!%.!+!.,!3))!%#!+!/#,) F&! ')4,&(.+!%,! 7.4/+<,&!0%

,)3835)+!3))!*()!*0,&/)!.$!#!/2.+<-,&! 34@&(&9! *(0:! %,! +)! +(+/&9! (2%

,:+<:!(+!0,)3835)+!*()!*0,&/)!!$!1!):(/*(+)A)) !(&)'*!%#!+!/#,$!2A+<,&!%,!+!,#,)!+.)!')4!):()*(.+/*9$!,%!+!

=23!%#)!!(/*09! 2

2

%#

,%

ΑΖ

ΒΑ= $!')6()!(+!(./7;:+!%1,!&/:))!)*+7>:)+!(.+9!

(+!(./7;:+!%1#!&(.6()*0!VI. 8).

+.)!!%,2 =

23!%1

2)!!2,;9$!(+!(&(.47;:+!(09!')),2(.+<!(09!*/)/%

.)9! &/:))! ()! (./)! '&8(&.)! (+<! (&(.)7>:+<! (09! (5&<.49! (09! (<.),/')9!

&(.6()*0!!2,($!4.)!(+!(,*,)!%1!)*+8())!,&!(0:!(5&<.4!(09!(<.),/')9)! !(/*09$!)(6!(0:!+,+)6(0()!(;:!(.)7>:;:!%,1$!,1#!2A+<,&1!!!

2

2

,1

%,

,#

AB= $!4.)!%,2!+!,,12.

2,;9$! !(+!(&(.47;:+!(09!')),2(.+<!(09!*/)/.)9!&/:))!!(.)(54*)+!

)(6!(+!!(&(.47;:+!(09!(5&<.49!(+<!38?+<!&(.6()*0!2'()!+.)!(+!(,*,)!

,1!&/:))!0!(5&<.4!(+<!38?+<) !(/*09$!&(&)'*!%.!+!.,$!2A+<,&!%,

2!+!/,!

2.!"))!(+!(&(.47;:+!(09!

')),2(.+<! (09!*/)/.)9! &/:))!')(54*)+!)(6! (+! (&(.47;:+! (09!(5&<.49! (+<!

+3()2'.+<!&(.6()*0!!2-()!+.)!(+!(,*,)!,!!&/:))!0!(5&<.4!(+<!+3()2'.+<) $(+!*0,&/+!%!/2.+<,&!34@&(0!*(0:!%,!3))!()/.:+<,&!(,*,)!%$!/*+!

,&!%,)!-2.+<,&!(0:!$.$!0!+(+/)!(2,:&)!(+!0,)3835)+!*(+!*0,&/+!3!3))!)(6!

(+!3!/2.+<,&!&<@&/)!34@&(0!*(0:!%,$!0!+(+/)!(2,:&)!(0:!%,!*(+!")!

)*('#!!"#

%%!"%-&-""%!$%!1!% 179

!(&)'*!%$!+!/%.!3))!".

3"

%.

%$= !2A+<,&!"3!+!/".)!

+.)!"32! +! -".

2! 3))! &(&)'*!.3

2! +!"3

2! *!".

2$! 2A+<,&!.3

2 = +!'".

2$!!4.)!,.

2!+!'."

2. !(&)'*!%,!+!/.,!3))!%#!+!/#,!2A+<,&!#,!+!/.#$!4.)!,.!+!,.#$!

3))!,.2!+!5.#

2)!")(4!*<:2(&))!."

2!3!.#

2$!!&(+,2:;9!2A+<,&!."!3!.#)

B)/.:+<,&!(,*,)!.#! /*+!(.+9!(+!."!3))!*(+!*0,&/+!#!/2.+<,&!

34@&(0!*(0:!%,$!0!+(+/)!(2,:&)!(+!0,)3835)+!*(+!*0,&/+!F)! !(&)'*!,.

2 = 5".2$!%,! +! /,.! 3))!"#! +! /".! 2A+<,&!%,

2 = +! '"#

2)!2,;9$! (+! (&(.47;:+! (09! ')),2(.+<! (09!*/)/.)9! &/:))! (&:()-

(54*)+! (+<! (&(.)7>:+<! (09! )3(/:)9! (+<! 3835+<! )(6! (+:! +(+/+! 3)()-*3&<4*@03&!(+!&)3+*4&'.+!&(.6()*0!26($!4.)!(+!"#!&/:))!0!)3(/:)!(+<!

3835+<!)(6!(+:!+(+/+!3)()*3&<4*@03&!(+!&)3+*4&'.+)!0(*)$!(+!"#! )-*+8())! ,&! (0:! (5&<.4! (+<! 3):+:)3+8! &=)7>:+<$! (+! +(+/+! &77.4/&())!

*(+:!3835+!(+<!(.+):)/2.),&!&(.6()*0!IV)2'()!"))!&(&)'*!0!')4,&(.+9!(09!*/)/.)9!)(+(&5&/())!)(6!,))!(5&<.4!3):+:)3+8! &=)7>:+<!3))! '8+!

(5&<.29! 3):+:)3+8! '&3)7>:+<! +)! +(+/&9! &77.4/+:())! *(+:! 3835+! (+<!

(.+):)/2.),&! &(.6()*0! 26(! 3))! &(&)'*! (+!%,! &/:))! 0! ')4,&(.+9! (09!

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

'&3)7>:+<)! "*A8&)! &(/*09!F#! +!3"$!')6()!)(2A+<:! (+! /')+!)(6! (+!32%

:(.+$!4.)!F#!+!"#)!%(6!(+!+.@+7>:)+!(./7;:+!F#,!2A+<,&!F,2 =

+!F#2!*!#,

2,!&(+,2:;9!(+!F,! )*+8())!,&!(0:!(5&<.4!(+<!3):+:)3+8!

(&:()7>:+<!&(.6()*0!27()!$!(5&<.4!6,;9!(+<!(&:()7>:+<!&/:))!(5&<.4!

(+<! &)3+*)2'.+<! &(.6()*0! 26($!4.)! (+! (,*,)!F,! &/:))!0!(5&<.4! (+<!

&)3+*)2'.+<) %2,:+<,&! (+! (,*,)!,1$! (+! +(+/+! &/:))!0! (5&<.4! (+<!38?+<$!*&!

43.+!3))!,2*+!567+!*(+!*0,&/+!-!,&!,&7)58(&.+!(,*,)!(+!-,)!+.)!(+!

(,*,)!-,!&/:))!0!(5&<.4!(+<!';'&3)2'.+<!&(.6()*0!2.$!(6.)*,)!() 0A&)!6,;9!)(+'&)A@&/!6()!(+!(&(.47;:+!(09!')),2(.+<!(09!*/)/.)9!

&/:))!()!(./)!'&8(&.)!(+<!(&(.)7>:+<!(09!(5&<.49!%1!(09!(<.),/')9$!')-(54*)+!(+<!(&(.)7>:+<!(09!(5&<.49!,!!(+<!+3()2'.+<!3))! !(.)(54*)+!

(+<!(&(.)7>:+<!(09!(5&<.49!1,!(+<!38?+<)!%:!(+!(&(.47;:+!(09!'))-,2(.+<!(09!*/)/.)9!&/:))! /*+!,&!2=)!(&(.47;:)$!(6(&!(+!(&(.47;:+!(09!

(5&<.49! (09! (<.),/')9! &/:))! /*+!,&! (2**&.)! (2(+))! (&(.47;:)$! (+! (&-(.47;:+!(09!(5&<.49!(+<!+3()2'.+<!&/:))!/*+!,&!(./)!(2(+))!(&(.47;:)!

3))! (+! (&(.47;:+! (09! (5&<.49! (+<! 38?+<! &/:))! /*+! ,&! '8+! (2(+))! (&-(.47;:))!

180 !!"#!"#$!"$%&"'!"%#

!(+,2:;9!(+!(&(.47;:+!(09!(5&<.49!(09!(<.),/')9!&/:))!()!(2*%

*&.)! (./()! (+<! (&(.)7>:+<! (09!(5&<.49! (+<!+3()2'.+<!3))! &(/*09! ')%

(54*)+! )(6! (+! (&(.47;:+! (09! (5&<.49! (+<! 38?+<)! %+! (&(.47;:+! (09!

(5&<.49!(+<!+3()2'.+<!&/:))!()!(./)!'&8(&.)!(+<!(&(.)7>:+<!(09!(5&<%

.49!(+<!38?+<)! &)!(5&<.29!(;:!(.)>:!*A0,4(;:!()9!+(+/&9!(.+):)/2.),&$!'05)'*!

(09!(<.),/')9$!(+<!+3()2'.+<!3))!(+<!38?+<!?./*3+:())!0!,/)!(.+9!(0:!

4550!*&!.0(6!567+$!&/:))!'05)'*!,&()=8!(+<9!.0(29)! &)!<(65+)(&9!'8+$!'05)'*!+)!(5&<.29!(+<!&)3+*)2'.+<!3))!(+<!';-

'&3)2'.+<$!'&:!&/:))!+8(&!,&()=8!(+<9!!.0(29!+8(&!.0(29!(.+9!()9!(.+0-7+8,&:&9!(5&<.29$!)/+8!&/:))!4..0(&9$!0!,))!;9!"+#%%!,!&(.6()*0!26(!

3))!0!4550!;9!#&%$%'(!&(.6()*0!2.()! $!(5&<.4!F,!(+<!&)3+*)2'.+<!&/:))!,&7)58(&.0!)(6!(0:!(5&<.4!

-,!(+<!';'&3)2'.+<)!B.47,)()$!&(&)'*!(+!(./7;:+!1#,!&/:))!)*+7>:)+!

(.+9! (+! (./7;:+!1%,$! )*A8&)! 0! ):)5+7/)!%,

,1

,1

#,= $! ! )(6! (0:! +(+/)!

(.+38((&)!6()! 2

2

,1

#,

%,

#,= )!!(+,2:;9!@)!)*A8&)! 2

2

#,

,1

,#

%,= .

"*A8&)!&(/*09!0!*A2*0!%,!+!,#,$!4.)!,12!+!,,#

2. !(/*09!2A+<,&!%#

2!+!-,#

2 . +.)!%#

2!3!,1

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

*&!43.+!3))!,2*+!567+!,&!,&7)58(&.+! (,*,)! (+!"#! &!(.6()*0! 5(!3))!

&(/*09!(+!-!'))).&/!(+!,1!*&!43.+!3))!,2*+!567+!,&!,&7)58(&.+!(,*,)!

(+!-,$!2A+<,&!"#!3!-,)!!(/*09!"#!+!#F$!4.)!#F!3!-,!3))!3)(4!

,&/G+:)!567+!F,!3!-,)!!!.")') 9#+#",+1$1 0*(;!r!0!)3(/:)!(09!*/)/.)9)!%6(&!+)!(5&<.29!(;:!(2:(&!*(&.&>:!(+<!

3)()*3&<4*(03):!'/:+:())$!;9!*<:4.(0*0!(09!)3(/:)9!r$!)(6!(+<9!(8(+<91

(5&<.4!(&(.)2'.+<!!!!)4 = r632

⋅

(5&<.4!+3()2'.+<!!!!!)8 = r2 ⋅

(5&<.4!38?+<!!!!!!!!!!!!)6 = r332

⋅

%%!"%-&-""%!$%!1!% 181

(5&<.4!&)3+*)2'.+<!!)20 = ( )55105r

−

(5&<.4!';'&3)2'.+<!)12 = ( )3153r

−

!(/*09!)(+'&/A@03&!6()!)*A8+<:!+)!*A2*&)91!

4r2 = 26

28

24 3)2))

23

== .

%)! (,*,)()! /8$! )4$! )6$! )8! &/:))! .0(4! ,&! (0:! 2::+))! (+<! !<35&/'0!

&?52(&!*A65)+!,&(4!(0:!(.6()*0!6($!&:>!()!(,*,)()!)20!$!)12!&/:))!4..0()) !=455+<$!6(;9!)(+'&/A@03&!3))!&83+5)!'))()*(>:&())!3))!)(6!(+:!

()+!(4:;!3)(45+7+$!)*A8&)!0!*A2*01!!)6!9!)8 < )4.

$!*A2*0!)20!3!)12!,(+.&/!:)!)(+'&)A@&/!)57&?.)34!;9!&=*91

)20!3!)12 ⇔ ( ) ( )222315

9r5510

25r

−>−

⇔ ( ) ( )5323159155

52

⋅−+>−

⇔ 5322

5522 −>− ⇔

352

552

< $!(+!+(+/+!)*A8&))

&,+/;9!,(+.&/!:)!)(+'&)A@&/!6()!)6!3!)20:

!)20!9!)6 ⇔ 26

220) α< ⇔ ( ) 2

2

r3945510

25r

⋅⋅<−⋅

⇔ ( ) ( ) ⇔<−⇔<−3255

51

3455

52

32

551 <−

⇔ ⇔<⇔<−5

131

55

321

51

91

< $!(+!+(+/+!)*A8&))!

!(+,2:;9!2A+<,&1!! )12!9!)20!9!)6!9!)8!9!)4.

!"#$%&! $(+!?)?5/+!XIII$!*(0:!23'+*0!(+<!Heiberg$!,&(4!(0:!(.6()*0!2:!

*8.&())! ,))! 7.),,*! 3))! (.+*(/@&())! 2:)! 3&/,&:+$! (+! +(+/+! ).A/G&)!;9!

&=*91

182 !!"#!"#$!"$%&"'!"%#

!!"#! $""! #%&! #'($! %$! )%(*$!+'! #!+%)! ,-"$'%'! .&! ,/,%'01",)%'&!'%)(.+!,-"$'!#)(&)-($)+.+!)#(! %,.#2)*(#+!%)!3'&! %,.0"#++#+!,,#+!-22"2.&4%!

%(6!(+55+89!*A+5))*(29! (+!3&/,&:+!)<(6!&35),?4:+:())!;9!*<%

,(2.)*,)!(+<!?)?5/+<!'""")!$&!*87A.+:0!)(6'+*0!(+!3&/,&:+!)<(6!3))!

6*)!(+!)3+5+<@+8:!2A+<:!;9!&=*91 =0&'! !#($18! 7")! /#$0$#!%--"!$;,2#! !/"78! "'$!#0$"!!#+"#$#-

&!+60$"'$!(!$!/#"#$/!.%3!"#)#! ""!"#"("! $#!#!+)0;!"#)!.#7! )$"#-!4$+'$!/#)!)$"&'$('$!$;12%"'$!($'$!2!"#<4!"".8"

'/%(%'5, #)6()!,))!*(&.&4!7;:/)!'&:!,(+.&/!:)!3)()*3&<)*(&/!)(6!'8+!(./%

7;:)!*!&(/(&')!&:!72:&)) F&!(./)!(./7;:)!3)()*3&<4G&())!0!*(&.&4!7;:/)!(09!(<.),/')9$!

,&!(2**&.)!(+<!+3()2'.+<!3))!,&!(2:(&!(+<!&)3+*)2'.+<) F&!2=)!(./7;:)!)*6(5&<.)!3))!)*+7>:))!(+<!*<:2.A+:())!*(+!/')+!

*0,&/+!'&:!<(4.A&)!*(&.&4!7;:/))!#)6()$!&(&)'*!0!7;:/)!(+<!)*6(5&<.+<!

(.)7>:+<!&/:))!'8+!(./()!(09!+.@*9$!@)!&/:))!3))!+)!2=)!/*&9!,&!&1@)!2A+<:!

4@.+)*,)(! (2**&.)9! +.@29$! (+! +(+/+! &/:))! )'8:)(+$! ')6()! 34@&! *(&.&4!

7;:/)!(&.)2A&())!*&!7;:/&9!,&!4@.+)*,)!,)3.6(&.+!(;:!(&**4.;:!+.@>:!

&(.6()*0!'")/2() .))! (+:! /')+!567+$!+8(&!)(6!(&.)**6(&.)! (;:! 2=)! &()(2';:! &/:))!

'<:)(6:!:)!3)()*3&<)*@&/!*(&.&4!7;:/)) %(6! (./)! (&(.47;:)! *A0,)(/G&())! 0! 7;:/)! (+<! 38?+<$! &:>! )(6!

(2**&.)!&/:))!)'8:)(+!:)!*A0,)()*(&/!*(&.&4!7;:/)) %(6!(./)!(&:(47;:)!3):+:)34!&)*+7>:))!3))!)*6(5&<.)($!*A0,)(/-

G&())!0!7;:/)!(+<!';'&3)2'.+<$!)554!)(6!(2**&.)!&/:))!)'8:)(+)!#)6()$!

&(&)'*! 0! 7;:/)! (+<! )*6(5&<.+<! (&:()7>:+<! &3):+:)3+8(! &/:))! ,))! 3))!

2:)!(2,((+!(09!+.@*9$!@)!&/:))!+)!(2**&.)9!7;:/&9!,&7)58(&.&9!(;:!(&*%

*4.;:!+.@>:$!(+!+(+/+!&/:))!)'8:)(+) 2,;9$!7))!(+:! /')+!567+$!+8(&!)(6!455)!(+587;:)!*A*,)()!&/:))!

'<:)(6:!:)!*A0,)()*(&/!*(&.&4!7;:/))! #&:! 3)()*3&<4G&())! &(+,2:;9! 455+! *(&.&6! *A*,)! (&.)&A6,&:+!

)(6!)*6(5&<.)!3))!)*+7>:))!*A*,)()$!&3(69!)(6!()!(2:(&!(+<!):)/2.)%

,&!()+!(4:;)

6!"""! '!&'$(#!"".!)$7#-!.+".!/#)!)$"&*$)".!#!$"#&*$".!!($#)!2(#!"+$6,!/#)!0$#!#02#""!"18!"+6,8!&'$(#8"

%%!"%-&-""%!$%!1!% 183

'/%(%'5,! 0*(;! %,.#!! )*6(5&<.+! 3))!

)*+7>:)+! (&:(47;:+! 3))! )9! (&.)-7.)/&/!*&!)<(6!3835+9!,&!32:(.+!1)!

%6(&!+)! &<@&/&9!%1$!,1$!.1$!#1!3))!

!1!')A+(+,+8:!()9!7;:/&9!(+<) !(&)'*!+)!7;:/&9!,&!3+.</*!(+!

1!&/:))! /*&9!3))!2A+<:!4@.+)*,)!(2*-*&.)9!+.@29$!0!3)@&,/)!)(6!)<(29$!4-

.)!3))!0!7;:/)! !∧

AZB $! ! &/:))! /*0!,&!,/)!+.@*!,&/+:!2:)!(2,((+!(09!+.@*9)!

+.)! *(+! (./7;:+! %1,! +)! 7;:/&9!!∧

ZAB !!3))!!∧

ABZ !2A+<:!4@.+)*,)!!,/)!

+.@*!3))!2:)!(2,((+!(09!+.@*9)!!(+,2:;9!!∧

ΓAB $!(+<!&/:))!!0!7;:/)!(+<!

(&:()7>:+<$!&/:))!,/)!+.@*!3))!2:)!(2,((+!(09!+.@*9) 9#+#",+1$1 $!().)(4:;!(.6()*0!7))!(0!,+:)')36(0()$!,&!(+:!(.6(+!(+<!2A&)!

'))(<(;@&/$! 6()! '05)'*! &3(69! (;:! (2:(&! (.+):)/&.@2:(;:! *(&.&>:!

*A0,4(;:! &(+5<2'.;:(! '&:! <(4.A+<:! 3))! 455)! (+<! :)! (&.)35&/+:())!

,&()=8! )*6(5&<.;:! 3))! )*+7>:);:! *A0,4(;:$! 2A&)! (.+3)52*&)! (+554!

*A65))!3))!*<G0(**&)9!6*+:!)/+.4!*(0:!+.@6(0()!(09!'))(8(;*09)!.))!

().4'&)7,)$! 2:)! &=4&'.+!(+<!)(+(&5&/())!)(6!'8+!3):+:)329! &(.)7;:)%

329(!(<.),/'&9!%,.#$!%-.!!,&!3+):*!?4*0!%,.!3))!3+.</29!(+<!3&/%

:())!&3)(2.;@&:!(09!?4*09!'/:&)!2:)!*A*,)!(+<!)3):+(+)&/!65&9!()9!)())%

(**&)9!(09!'))(8(;*09!(+<!*<,(&.4*,)(+9!3))!6,;9!'))/2.&)!&,/):>9!

)(6!()!(2:(&!*(&.&4!()!+(+/)!&3@2*),&) 3)!().)(0.**+<,&!6()!*(0:!).A*!(+<!?)?5/+<!'"!+./G+:())!()!(2-

:(&!*<73&3.),2:)!3):+:)34!*(&.&4$!'05)'*!(+!(&(.4&'.+$!(+!&=4&'.+$!+!

38?+9$!(+!+3(4&'.+!3))!(+!&)3+*4&'.+$!)554!'&:!'/:&())!(+<@&:4!7&:)369!

+.)*,69!&:69!03):+:)3+8!(+5<2'.+<0)!%(6!(0!'))(8(;*0!(09!(.6()*09!

*<:47&())!6()!0!03):+:)36(0()0!*<:'2&())!,&!()9!&=*9!'8+!*<:@*3&91 )(!&)!2'.&9!&/:))!3):+:)34!(+587;:)) ?(!&)!2'.&9!&/:))!,&()=8!(+<9!/*&9) %+!&.>(0,)!(+<!):)38((&)!&/:))!(+!&=*91!B+))!&()(52+:!*<:@*30!

(.2(&)!:)!(.+*(&@&/$!>*(&!()!(2:(&!(.+):)/&.@2:()!*A*,)()!:)!&/:))!()!

,+:)')34!(+<!)3):+(+)+8:!()9!*<:@*3&9!)<(29;!

)*('#!!"$

184 !!"#!"#$!"$%&"'!"%#

$()9! (.+(4*&)9! (+<! (.+07*@03):! (+<! $<,(&.4*,)(+9$! )(+'&/-A@03&!6()!!3)@2:)!)(6!()!().)(4:;!(2:(&!*A*,)()!&77.4/&())!*&!*/)/%

.))!$(+:!B54(;:)!&(/*09!):)/2.&())!6()!(+!(&(.4&'.+!&/:))!(+!)(5+8%

*(&.+!*(&.&6$! (+! +(+/+! A;./G&)! (0:! &()/4:&))! (09!*/)/.)9!*&! /*)! 3))!

6,+))!*A*,)()) $!*<:@*30!)<(*!(09!&77.)/*9!*&!*/)/.)!&/:))!(.47,)()!0!G0(+8%

,&:0$! '05)'*! ()! (+58&'.)! ,&! 2'.&9$! ,&()=8! (+<9! /*)$! 3):+:)34! (+-587;:)!()!+(+/)!&77.4/+:())!*&!*/)/.)!&/:))!)3.)?>9!()!(2:(&!(.+)%

:)/&.@2:()) $!*<:@*30!)<(*!,(+.&/!:)!):()3)()*()@&/!,&!).3&(29!)*+'8:),&9!

'))(<(>*&)9$!,&.)329!)(6!()9!+(+/&9!(&.)7.4/+:())!*(+!&(6,&:+)

)%2&,""!(Cromwell!&25..($!*&5)!..()! *$"'!P! /.+"7! #"-4!(+"#! ")! 0(+!8! "".! "#"(".! !($#)! ($#! 2!"#<4!

"".8!/#$"$)/%!#"-4&'$#"!!7"!!")!#/7-".6!8!$.$6,/!8!!($#)!)$"(4$#2!8% 1. !"!>!!($#)!!&&+%?)2"!$!!$&#(+#" 2. )-!8!")!((!(+!8!&'$(!8!"".!>!!($#)!2!"#<4!"".8!($!8" 3. )-!8!")!$"!+!08!&'$(!8!"".!>!!($#)!2!"#<4!"".8!($!8 !&%$.2#"*$)2!8'" 4. )-!8!")!/"+.&08!#!+)5%--"$"#)!##7!""$!(()"!#+)627!!(+*$"

%($!*%.'+7!(8%0'"

%($%!!9"$"#+%13:!"%!8&!#,!'.701#,:!+"'!().;%#,:! B).)(0.+8,&!6()$!):!3))!*&!65&9!()9!(&.)((>*&)9!(09!3)()*3&<*9!

(;:!3):+:)3>:!*(&.&>:!0!3)()*3&<*!(+<!7/:&())!)*+'<:),&/!,&!&77.)/*!

(+<!*(&.&+8!*&!,))!'+*,2:0!*/)/.)$!+!!<35&/'09!'&:!3)()*3&<4G&)!)%

(&<@&/)9!(+!*(&.&6!,2*)!*(0!*/)/.)$!)554!3)()*3&<4G&)!2:)!*(&.&6!*(+!

+(+/+!(&.)7.4/&())!,/)!*/)/.)!/*0!,&!(0!'+*,2:0)! &!B4((+9!'))(.)7,)(&<6,&:+9!()!/'))!(.+?5*,)()!3)()*3&<4G&)!

()!):(/*(+)A)!*(&.&4!,&!(0!206"("!"18!#$%-.$18!3))!(09!$4$6!$18$!,&%@6'+<9!()9!+(+/&9!&/A&!,&5&(**&)!3))!):)((8=&)! )'))/(&.))!&)!58*&)9!(+<!

().+<*)4G+<:!&:'))/2.+:$!):!3))!)())(+8:!34(+)&9! )')6(0(&9!(09!*/)/%

.)9$!+)!+(+/&9!'&!?./*3+:())!*()!)$%-*".#)!.))!567+<9!(50.6(0()9$!*(0!

*<:2A&))!().+<*)4G+:())!'8+!)(6!()9!3)()*3&<29!)<(29)

%%!"%-&-""%!$%!1!% 185

%($%$%!9"$"#+%1!!$#1!+)<#1!+"$7!$#.!$7//# +$%-.$1" 0*(;! 6()! (+!

(.6?50,)! 2A&)!5<@&/!3))!6()!+)!

3+.</29! (+<! 38?+<! &/:))! ()!

*0,&/)!%$!,$!.$!#$!!$!1$!$$!3!

&$A)! /7()! %)! &(/(&')! %,.#$!

!1$3!(2,:+<:!(0!*/)/.)!3)(4!

'8+! ().45505+<9! 3835+<9$! +)!

+(+/+)! &/:))! /*+)$! ')6()! ()! &7-7&7.),,2:)! (&(.47;:)! %,.#!

3))! !13$! &/:))! /*))!-2.+<,&!

(+! (,*,)! !.$! (+! +(+/+! &/:))!

')4,&(.+9!(09!*/)/.)9) 0A+<,&! !3

2! +! /!$

2 =

+!/3.2!3))!&(&)'*! ,90!3. ο

∧

= 2A+<,&!!.

2! +!!3

2! *!3.

2! 4.)!

!.2 = 23!3

2)!%554!(+!!.

2!&/:))!'+*,2:+$!4.)!3))!(+!!3

2!&/:))!'+*,2:+$!

&(+,2:;9!3))!+)!3835+)!!13$$!%,.#$!3)@>9!3))!()!&77&7.),,2:)!(&%

(.47;:)!&/:))!'+*,2:))

@4$6!$1"!-2.+<,&!'8+!().45505&9!3<35)329!(+,29!,&!/*&9!')),2%

(.+<9!'1!2(*)!>*(&1

'2 =

23' 2

1 , !6(+<!!'!!0!')4,&(.+9!(09!'+*,2:09!*/)/.)9)!

$&! 2:):! )(6! (+<9! '8+! 3835+<9! &77.4/+<,&! 2:)! (&(.47;:+$! )9!

(+8,&!(+!%,.#) $(+:!455+!3835+!7.4/+<,&!A+.'*!13!/*0!3))!().455050!(.+9!(0!

,.!3))!*(0!*<:2A&))!&77.4/+<,&!(+!(&(.47;:+!!13$)!&)!+3(>!3+.</29!

(+<!G0(+8,&:+<!3835+<!2A+<:!+.)*@&/)

%($%$)!!9"$"#+%1!!$#1!%'+##"3(&#1!+"$7!$#.!$7//# $!58*0!(+<!B4((+<!'))/2.&)!*0,):()34!)(6!)<(*:!(+<!!<35&/'0)!

!:>!+!!<35&/'09!A.0*),+(+)&/!'8+!3<35)329!(+,29!(09!*/)/.)9$!+!B4(-

)*('#!%&

186 !!"#!"#$!"$%&"'!"%#

(+9!?./*3&)!(2**&.)9!().45505&9!3<35)329!(+,29$!3)@&,/)!)(6!()9!+(+/&9!

')2.A&())! )(6! (.&)9! 3+.</29! (+<! &)3+*)2'.+<)!#8+! )(6! (+<9! 3835+<9!

&/:))!,)3.+/!3835+)$!(&.)7&7.),,2:+)!*&!'8+!):(/@&(&9!(.)7;:)329!2'.&9!

3))! +)! 455+)! '8+! 3835+)! ?./*3+:())! ,&()=8! )<(>:$! &/:))! ().45505+)!

(.+9!)<(+89!3))!/*+)!,&()=8!(+<9)

)*('#!!%"

+$%-.$1)!!(+@2(+<,&!6()!(+!(.6?50,)!2A&)!5<@&/!3))!%$!,$!.$!#$!

!$!-$!1$!$!"$!#$!F$!-!3+.</29! (+<! &)3+*)2'.+<! &$A)! /2()!!(&)'*! ()!

(,*,)()!,%$!,.$!,-$!,1$!,!!&/:))!&77&7.),,2:)!*(0!*/)/.)!3))! /*)!

,&()=8!(+<9$!()!*0,&/)!!%$!.$!-$!1$!!!&/:))!*<:&(/(&')) !(/*09!%.!+!.-!+!-1!+!1!!+!!%$!4.)!(+!(&:(47;:+!%.-1!!&/%

:))!3):+:)36)!&,+/;9!()!(&:(47;:)!"!,.#$!#$F1,!&/:))!3):+:)34) -2.+<,&!!-$!"$)!%6(&!!%.!<<!!-!&)(6!(+!(&:(47;:+!%.!1!(!3))!!

%.!<<!"F!&)(6!(+!(&:(47;:+!%"-$1($!4.)!!-!<<!"$)! !()(52+:$!"$!!<<!#F!&)(6!(+!(&:(47;:+!#"#$F() &,+/;9!,.$!!#$!1$$!#-!&/:))!().45505)!,&()=8!(+<9!3))!()!,%$!

-#$!1"$!F-!&/:))!().45505)!,&()=8!(+<9) %/+8!,.!+<<!#-!3))!,%!+<<!F-!+)!3835+)!%,.$!#F-!&/:))!/*+)!

3))!().45505+))!&,+/;9!+)!3835+)!#!-$!"1$!&/:))!/*+)!3))!().45505+))! $(+<9! /*+<9!3))!().45505+<9!3835+<9!#!-$!"1$!+)!A+.'29!!-$!

"$!&/:))!/*&9!3))!().45505&9)!+.)!-"!&/:))!')4,&(.+9!(09!*/)/.)9!3))!

0!7;:/)!-!"!&/:))!+.@*) $(+!(&:(47;:+!1!%.-$!):!0!!-!'))).&@&/!*&!43.+!3))!,2*+!567+$!

%%!"%-&-""%!$%!1!% 187

(+!,&7)58(&.+!(,*,)!&/:))! /*+!,&!%1)!+.)!10

6

%.

!-

λ

λ= )!!(/*09!!-

2 +

*%.2!+!!-

2*!!"

2!+!'

2$!6(+<!'!0!')4,&(.+9!(09!*/)/.)9)

0(*)!-"$!!-$!!.!&/:))!6(;9!+)!(5&<.29!3):+:)3+8!(&:()7>:+<$!

&=)7>:+<!3))!'&3)7>:+<!(+<!&77.4/+:())!*(+:!/')+!3835+) %554!0!')4,&(.+9!-"!(09!*/)/.)9!&/:))!'+*,2:0$!4.)!!-$!!.!&/%

:))!'+*,2:))!0(*)!+)!)3(/:&9!(;:!3835;:!!-#$!%.,!&/:))!'+*,2:&9!&):!

.$!.4!&/:))!)3(/:&9!(+<9$!(6(&! 22 !-31

. = , 22 %131).( =ʹ′ ).

!(+,2:;9!+)!3835+)!&/:))!'+*,2:+)!3))!(+!/')+!)*A8&)!3))!7))!(+<9!

3835+<9!"$1$!#F-$!+)!+(+/+)!&/:))!().45505+)!(.+9!)<(+89) @4$6!$1" %:!'!0!')4,&(.+9!(09!*/)/.)9$!/2.+<,&!()!(,*,)()!x, y

(2(+))!>*(&!'$!x, y!:)!&/:))!):45+7)!(.+9!()9!(5&<.29!3):+:)3+8!(&:()%7>:+<$!&=)7>:+<!3))!'&3)7>:+<!):(/*(+)A)!(+<!&77.4/+:())!*(+:! /')+!

3835+) -2.+<,&! &"( '8+!/*&9!3))!().45505&9!3<35)329!(+,29!*(0!*/)/.)!,&!)3(/%

:&9!.$!>*(&!.2= 2x

31

$!()9!#!-$!"1$!3))!

&""(!'8+! /*&9!3))!().45505&9!3<35)329!(+,29$!()9!%,.$!#F-$!,&!

)3(/:&9!.4!>*(&!! ( ) 22 y31

. =ʹ′ .

$(+<9! 3835+<9! &"(!/2.+<,&!!-$!"$!;9! (5&<.29! &77&7.),,2:;:!

)*6(5&<.;:!(.)7>:;:!().45505;:!,&()=8!(+<9!3))!*(+<9!3835+<9!&""() -2.+<,&!%.$!#F!;9!(5&<.29!&77&7.),,2:;:! )*6(5&<.;:!(.)7>%

:;:!().45505&9!(.+9!()9!!-$!"$)!$<,(50.>:+<,&!(+!*A*,))!$!+.@6(0%

()! (09! 3)()*3&<*9! )(+'&)3:8&())! 6(;9! *(0:! ):45<*0)! !(/*09! 2(&())!

&52&)!+!B4((+9(!6()!

&')4,&(.+9!(09!*/)/.)9(2!+!,&(5&<.4!(&:()7>:+<!*(+:!3835+!#!-(2.

#)6()!)(6!3)()*3&<*$!6

5

5

5

-!

"-= )!!(/*09!!)*A8&)!!

13

5

"-

6= ,

4.)!!13

5

"-

5= )!!(+,2:;9!!!"-

2 = 3 255 .

188 !!"#!"#$!"$%&"'!"%#

%($)!$#0)%(&"*!+".#.'+7!#$%&%7!+"'!#1""%$&="

10.2.1 >'!%=."'!/#0)%(&#$ $()!)$%-*".#$!*(0:!).A*!(+<!?)?5/+<!'"$!):)/2.+:())!+:+,)*()34!

,6:+!(2:(&!*(&.&4!*A*,)()!&(50:!(09!*/)/.)9$!(+<!3<5/:'.+<!3))!(+<!3>-:+<()!.))!(0:!)3./?&))!'&!'/:&())!+8(&!+.)*,69!(+<!(+5<2'.+<)!.))!()9!)%

:473&9!(+<!3&),2:+<!(+<!)3+5+<@&/!@)!'+@&/!2:)9!(&.)7.)/)369!+.)*,69!

(+<!(+5<2'.+<!*&!*87A.+:0!'))(8(;*0!&Cromwell$!&255.($!*&5)!/75() 9"-4!(+"! &/:))! 0! 2:;*0! (&(&.)*,2:+<! (5*@+<9! (+5<7>:;:! (2-

(+);:!>*(&1 1. "4@&!G&87+9!(+5<7>:;:!*<:):()2())!,6:+!3)(4!,/)!(5&<.4!*!

3+.</*!) 2. "4@&!(5&<.4!&:69!(+5<7>:+<! &)(+(&5&/!,))!)3,*!(+<!(+5<2-

'.+<(!3))!):*3&)!,6:+:!*&!2:)!)36,)!(+587;:+) 3. !/:))!'<:)(6:!:)!,&()?+8,&!)(6!(+!&*;(&.)36!&:69!(+5<7>:+<!

*(+!&*;(&.)36!&:69!455+<!(+5<7>:+<$!().),2:+:()9!*(+!(+%

58&'.+) 4. 0*(;!V!,))!3+.</*!3))!F1, F2 . . . Fn!()!!n!!*(+!(5*@+9!(+58%

7;:)!(+<!*<:):(>:())!*(0!3+.</*!V)!%6(&$!&/:))!'<:)(6:!:)!

,&()?+8,&!)(6! 2:)!(+587;:+! !Fi! !*&! 2:)!455+! !Fj! !A;./9!:)!

(&.4*+<,&!)(6!(+!!V.

10.2.2 >)/#:!$#1!EULER. F))!&:)55)3()3*!)(6'&)=0!(09!,+:)')36(0()9!(;:!(2:(&!3):+:)-

3>:!*(&.&>:!,(+.&/!:)!'+@&/!,&!(0!?+*@&))!(+<!(8(+<!(+<!Euler!7))!()!(+58&'.))

)%2&,""!&(+<!Euler). @!!/%6!!/.+"7!#"-4!(+"#!""!#-,6"8!/!"'$!/"+.&*$!$.$!""!#-,$

6"8! !!"'$! !(+*$! )$"4"#)!2!!""!#-,6"8!#!"'$!#/2*$!#.<120$"!/#"%!(4"#!(1-#(,!)$;4!)!"!"4#"8%!!/!(!!!)!#!(!*"

'/#(%=5%':!*'"!$"!+".#.'+7!#$%&%7!"%!8&!#,!$#1!;%-&!""$#:!

Euler: 0*(;! 6()! +)! 2'.&9! &:69! 3):+:)3+8! (+58&'.+<! &/:))! 3):+:)34! :-

7;:)!3))!*&!34@&!3+.</*!*<:(.2A+<:!!,!!)3,29.

%%!"%-&-""%!$%!1!% 189

%6(&!2A+<,&1

:&!+!/) (1)

')6()!34@&!)3,*!):*3&)!*&!'8+!2'.&9!3))

!,3!+!/)! (2)

')6()!34@&!)3,*!):*3&)!*&!'8+!*(&.&29!7;:/&9) &! (8(+9! 3! *! &! +! )! *! /! ! &(+<!Euler(! 567;! (;:! &2($! &/(! 7/:&())1!

22:&

&,

&:+=+ $!)(6!6(+<!(.+38((&)!6()

&!+!µν−µ+ν

µ

224

!()(52+:$!)*A8&)!!,!≥!,!!3))!!/:!*!/,!-!,:!3!7$!!4.)!!,!9!2-:

:2

%:!!:!+!,$!(6(&!,!9!6$!'05)'*!!,!+!,!!*!!,!+!-!!*!!,!+!'!!3))!):(/%

*(+)A)!!&!+!-!!*!!&!+!:!!*!!&!+!/7 #05)'*$!,&!2'.&9! )*6(5&<.)!(./7;:)!,(+.&/!:)!*A0,)()*@&/!3)%

:+:)36!(&(.4&'.+!*!+3(4&'.+!*!&)3+*4&'.+ %:!!:!+!-!$!(6(&!!,!9!-$!'05)'*!!,!+!,!!3))!!&!+!6)!+.)!,&!2'.&9!(&-

(.47;:)!,(+.&/!:)!*A0,)()*@&/!&=4&'.+!&38?+9()

%:!!:!+!'$!(6(&!!,!9!3

10$!'05)'*!!,!+!,!!3))!!&!+!2/)!+.)!,&!2'.&9!

3):+:)34!(&:(47;:)!,(+.&/!:)!*A0,)()*@&/!,6:+!';'&34&'.+)

%:! !:!≥!6$!(6(&!2-:

:2 - 3 = 02-:

:6≤

−!4.)! !,!9!,)!$<:&(>9!'&:!<-

(4.A&)!3):+:)36!(+58&'.+!,&!2'.&9!3):+:)36!:-7;:+!,&!!:!≥ 6. 0(*)!5+)(6:!*()!3):+:)34!*(&.&4!2A+<,&!(0:!)365+<@0!3)():+,*!

7))!()9!3+.</29$!()9!)3,29!3))!()9!2'.&9)

9#&1+3: '+"3: ,(&%: %&(.4&'.+ 4 6 4 "8?+9 8 12 6 &3(4&'.+ 6 12 8 #;'&34&'.+ 20 30 12 !)3+*4&'.+ 12 30 20

190 !!"#!"#$!"$%&"'!"%#

B).)(0.+8,&! (.+/):>9! 34-(+))!+,)'+(+/0*0$!)36,)!3))!*&!)<-(6:!(+:!(&.)+.)*,2:+!3)(45+7+)!0(*)!

2A+<,&1! (&(.4&'.+$! 38?+9! –! +3(4-&'.+$!&)3+*4&'.+!–!';'&34&'.+)!

%<(6! (+<! <(+3.8((&())! &/:))!

2:)9!'<H*,69$!(+<!7&;,&(.)34!,(+.&/!

:)! &=070@&/!;9! &=*91!"4@&! 2:)! )(6!

()! *(&.&4! &:69! G&87+<9! (.+38((&)!

)(6! (+!455+!*(&.&6! (+<! G&87+<9$!):!

*(0:!(&.)7&7.),,2:0!*/)/.)!3))!*()!

*0,&/)! (;:! 3+.</>:! /2.+<,&! ()! &%

/)((6,&:)! &(/(&')! 3))! @&;.**+<,&!

(+!*(&.&6!(+<!(.+38((&)!)(6!()9!(+-,29!)<(>:! (;:! &()(2';:)! &,52(&!*A)!

//$! 6(+<!().+<*)4G&())!+!38?+9!3))! (+!3):+:)36!+3(4&'.+()!F&!)<(*!

(0:!2::+))!(+!3):+:)36!(&(.4&'.+!&/:))!)<(+'<H36)

10.2.3 -"'+".#.'+7!($%&%7$! $(+!(2,((+!?)?5/+!(09!/#01'#$-"(2!)3,#4!4(2!+!B4((+9!)(+'/-

'&)!(0:!):)345<I0!'&3)(.)>:!(+5<2'.;:!*(+:!%.A),*'0)!$(0!*<:2A&))!

(&.)7.4/&)! ()! '&3)(./)!*A*,)()$! ()! ()=):+,&/!):45+7)!,&! (+:!).)@,6!

(;:! &'.>:!3))!):)/2.&)! (+! &/'+9! (;:! &'.>:!34@&!*(&.&+8)!%)!*(&.&4!

)<(4!**,&.)!+:+,4G+:())!%.A),*'&))!*!0,)3):+:)34!*(&.&4)!&)! 2'.&9!

34@&!0,)3):+:)3+8!*(&.&+8!&/:))!3):+:)34!(+587;:)$!)554!6A)!(+<!)'/+<!

(8(+<$!):(/@&()!)(6!6$()!*<,?)/:&)!*()!3):+:)34!*(&.&4)!3)!().)(0.*%

*+<,&! &(/*09! 6()! 3))! ()! 0,)3):+:)34! *(&.&4! &77.4/+:())! *&! */)/.)$!!

6(;9!)3.)?>9! !*<,?)/:&)!,&!()!3):+:)34!*(&.&4)!"4(+))!*(+)A&/)!/)/%

:+:())!*(+:!().)34(;!(/:)3)$!6(+<!(&.)7.4/&())!+!(8(+9!3))!(+!(5*@+9!

(;:!&'.>:!(+<!34@&!*(&.&+8)!$!&.7)*/)!(+<!%.A),*'0!(+<!'))(.)7,)%

(&8&())!()!0,)3):+:)34!*(&.&4!/)/:&())!:)!2A&)$!'<*(<A>9$!A)@&/)

)*('#!!%%

%%!"%-&-""%!$%!1!% 191

$0!;#:!

$-. .(&2.

>&'*2.-. >%$&"+

*2.-. $%.$"+

*2.-. .5"*2- .-.

?+$"+

*2.-.

/%+"*2- .-.

8 4 4 14 8 6 14 6 8 14 8 6 26 8 18 26 12 8 6 32 20 12 32 12 20 32 20 12 38 32 6 62 20 30 12 62 30 20 12 92 80 12

!!!!!3+5+?6!(&(.4&'.+!!!!!!!!!38?+-+3(4&'.+!!!!!!!!!!!!!!!!3+5+?6!&=4&'.+

5$.#!#&6!$#!1'-"#,%,-"#!%$"$"#!'"!$-2!%1'"$-,72!%,%'#%."2!$%32'

)*('#!%(

10.2.4 -!(1""%$&="!#$,!*%-"%$&="$!

$()9! *87A.+:&9! ().+<*)4*&)9! (09! .&;,&(./)9! ()! 3):+:)34! (+58-

7;:)!3))!(+58&'.)!()/G+<:!2:)!3&:(.)36!.65+!3))!A.0*),+(+)+8:())!;9!

<(+'&/7,)()!7))!(0:!&)*)7;7*!(;:!,&()*A0,)()*,>:!3))!(09!*<,,&(./)9)!

192 !!"#!"#$!"$%&"'!"%#

D9!,&()*A0,)()*,+/!:++8:())!#'&-'%,%%('#,$"2!"#-!"&.!)(&)3+:/-*&)9!(+<!)/*:+<:!2:)!*A*,)!):)55+/;(+!3))!(.+38((+<:!)(6!(&.)+.)%

*,+89!&*(+!*A*,)(!)<(+,+./)*,>:!(09!'+,*9!(+<!(&.)?455+:(+9!A>.+<)!

$!*<,,&(./)!(+<!*A*,)(+9!*A&(/G&())!,&!(+!*8:+5+!)<(>:!(;:!,&()*A0%

,)()*,>:$!(+<!2A+<:!'+,*!+,4')9)!!)')34!7))!(0:!.&;,&(./)!(+<!&()(2%

'+<!3))!(+<!A>.+<$!)(+'&)3:8&())!6()!0!?)*)3*!'+,*!(+<!(.2(&)!:)!)%

())(**+<,&!:)!'))(0.&/())!&/:))!0!)(6*()*0!'8+!*0,&/;:$!+(6(&!):)76%

,)*(&!*&!)*+,&(./&9)! ,)*)369!7))!(0!.&;,&(./)!(+<!&()(2'+<!&/:))!+!,&()*A0,)()*,69!

(+<!(.+38((&)!;9!):435)*0!;9!(.+9!4=+:)! &'05)'*!0!&8.&*0!0*<,,&-(.)3+80!&:69!*0,&/+<!;9!(.+9!,/)!&<@&/)($!)/+8!,(+.&/!:)!)(+'&)A@&/!6()!!

34@&! )*+,&(./)!*(+! &(/(&'+! &/:))!*8:@&*0! (.)>:! (+!(+58!):)354*&;:!

&&:!72:&)!;9!(.+9!'))/+.&()3+89!4=+:&9()!B).6,+))!'))(.)7,4(&<*0!*(+!

A>.+! 7/:&())! 7))! ()9! ):)354*&)9!;9! (.+9! &(/(&'+)!2,;9$! ?)*)36! .65+!

&'>!()/G+<:!+)!*(.+/29!;9!(.+9!4=+:)!*(+!A>.+)!!()527+:()9!,))!).A*!

*(+!A>.+!3))!&)*47+:()9!2:)!+.@+3):+:)36!*8*(0,)!*<:(&()7,2:;:$!0!

.&;,&(./)!(+<!&()(2'+<!&:(4**&())!*()!(5)/*))!(09!%:)5<()3*9!.&;,&%

(./)9!(+<!A>.+<)!0(*)$!?)*)36!.65+!*(0!,&52(0!(;:!*<,,&(.)>:!()/G&)!0!

+,4')!SO&,($!0!+(+/)!)(+(&5&/())!)(6!7.),,)329!)(&)3+:/*&)9!(+<!'))%(0.+8:!()9!)(+*(4*&)9!3))!(+:!(.+*):)(+5)*,6)

$4!)<(4!()!(5)/*))!@)!7/:&)!,))!*8:(+,0!().+<*/)*0!(;:!+,4';:!

*<,,&(.)>:!(;:!3):+:)3>:!(+5<7>:;:!3))!(+5<2'.;:)

?"7("!#1""%$&'2.!+".#.'+#)!/#01*2.#1$ 0*(;!!m ≥!,!!3))!!.m!!(+!3):+:)36!m-7;:+!,&!32:(.+!&!3))!3+.<%

/29!=v1, v2 . . . vm>)!F&!L!*<,?+5/G+<,&!(+!&<@87.),,+!(,*,)!(+<!=&3)%:4!)(6!(0:!3+.</*!v1$!')2.A&())!)(6!(+!32:(.+!&!3))!=):)*<:):(4!(+!

.m. %+! .m! 2A&)! +.)*,2:&9! (.+/):&/9! *<,,&(./&9)!0*(;!%! 0! *(.+/*!

78.;!)(6!(+!&!3)(4!7;:/)!/(<m!3))!,!+!3)(+((.)*,69! &):435)*0(!;9!

(.+9!L)!%6(&$!):!,&!2!*<,?+5/*+<,&!(+:!()<(+()36!,&()*A0,)()*,6$!2%

A+<,&!%m!+!,2!+!2!3))!,%,-1+!%-1. 0*(;!s!34(+))!*<,,&(./)!(+<!.m)!%6(&$!):!s(v1) = vi!3))!s(v2) = vj

,(+.&/!:)!)(+'&)A@&/!6()!@)!(.2(&)!:)!)*A8&)!3))! !j ≡ (i ± 1) modulo m. %)!!i, j$!5+)(6:!+./G+<:!(+!!s)!0(*)$!!s = Ai!!!,&!!!j ≡ (i ± 1)modulo m!!*!! s = Ai B!,&!!j ≡ (i – 1) modulo m.

%%!"%-&-""%!$%!1!% 193

F&! 455)! 567))$! 0! +,4')!*<,,&(.)>:! (+<!.m! '/:&())!,&!*A2*&)9!

7&::0(6.;:!;9!&=*91

Dm : Am = B2 = 1, BAB-1 = A-1

%0:!Dm!+:+,4G+<,&!!')&'.)3*!+,4')!?)@,+8!m!3))!&/:))!(4=&;9!/m.

)*('#!!%)#!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!)*('#!%)8

F(+.+8,&!:)!@&;.**+<,&!(+!(+587;:+!.m!;9!<(+*8:+5+!(+<!R3 (+<!(&.)2A&())!*(+!&(/(&'+!x&y!,&!(0:!).A*!(;:!)=6:;:!&!*(+!*0,&/+!

(0, 0, 0) . %6(&!(+!%!&/:))!*(.+/*!78.;!)(6!(+:!4=+:)!!z4z!!3))!(+!!,!!&/:))!

&)')3*!*(.+/*!78.;!)(6!(+!L)!0(*)!0!+,4')!Dm!)(+(&5&/())!)(6!7:**)&9!

&+./G+<*)9!2(!*(.+/29!3))!Dm ⊂ SO&,()!$0,&)>:+<,&!&(/*09!6()!2A+<,&!

3))!(0:!3<35)3*!+,4')!!1m = {A>$!(4=09!m$!<(++,4')!(09!SO(3). F(+.&/! :)! )(+'&)A@&/! 6()! 34@&! (&(&.)*,2:0! +,4')! )*+,&(.)>:!

*(+!&(/(&'+!&/:))!&/(&!3<35)3*!&/(&!')&'.)3*$!'05)'*!+,4')!*<,,&(.)>:!

&:69!3):+:)3+8!(+5<7>:+<)

?"7(%:!#1""%$&'2.!$-.!+".#.'+2.!#$%&%2. F&!/4 *<,?+5/G+<,&!(+!3):+:)36!(&(.4&'.+$!3):+:)36!(+58&'.+!

,&! -!3+.</29$! 6!)3,29!3))! -!2'.&9$!+)!+(+/&9!&/:))! )*6(5&<.)!(./7;:))!

0*(;!=vi>!+)!3+.</29$!eij!0!)3,*!(+<!*<:'2&)!!()9!3+.</29!vi!!3))!!vj!,&! i < j)!F&!1!*<,?+5/G+<,&!(0!*(.+/*!3)(4!7;:/)!!(!!78.;!)(6!(0:!&<@&/)!

(+<!&:>:&)!()!,2*)!(;:!e12!3))!e34)!$!!1!&:)554**&)!()!v1 v2!3))!()!v3 ,

194 !!"#!"#$!"$%&"'!"%#

v4)!F&!Q!*<,?+5/G+<,&!(0!*(.+/*!3)(4!7;:/)!(!78.;!)(6!(0:!&<@&/)!

(+<!*<:'2&)!()!,2*)!(;:!e13!3))!e24)!$!?!&:)554**&)!()!v1, v3!3))!()!v2, v4)!!()(52+:!@?!+!?@!3))!&/:))!0!*(.+/*!3)(4!7;:/)!(!78.;!)(6!(0:!!

&<@&/)!(+<!&:>:&)!()!,2*)!(;:!!e14!3))!!e23. 0*(;!%!0!*(.+/*!3)(4!7;:/)!

/(<,!78.;!)(6!(0:!&<@&/)!(+<!&:>:&)!

(0:!3+.</*!v1!3))!(+!32:(.+!(09!)(2%

:):()!2'.)9)!$!%!'))(0.&/!(0:!v1!3))!

)(&)3+:/G&)!()!v2 , v3, v4! !*()!v3, v4, v2 ):(/*(+)A))!"4@&! *<,,&(./)! (+<!/4 (.+2.A&())!)(6!*<:@2*&)9!(;:!%$!

1$!?)!0(*)!0!+,4')!*<,,&(.)>:! (+<!

/4 '/:&())!,&!*A2*&)9!7&::0(6.;:!;9!

&=*91!! %: %3 = P2 = Q2 = 1, PQ = Q P, APA-1 = Q, AQA-1 = PQ $! %! +:+,4G&())! $"$$#"!$-"(!

%'#!#$! 2A&)! (4=0! 2/! 3))! &/:))! <(+-+,4')!(09!SO(3).

F&!/8!*<,?+5/G+<,&! (+!3):+-

:)36!+3(4&'.+$!(+!+(+/+!2A&)!6!3+.<%

/29$! 2/!)3,29!3))! :!2'.&9$!(+<!&/:))!

)*6(5&<.)!(./7;:)) 0*(;!'!0!*(.+/*!3)(4!7;:/)!

(</!78.;!)(6! (0:!&<@&/)!(+<!&:>:&)!

()9! v1, v6! 3))! 1! +! '2! )! 0*(;! Q! 0!

*(.+/*! 3)(4! 7;:/)! (! 78.;!)(6! (0:!

&<@&/)!(+<!&:>:&)!()!v2!3))!v4)!!%6(&!

1?!+!?@!3))!&/:))!0!*(.+/*!3)(4!7;:/)!(!78.;!)(6!(0:!&<@&/)!(+<!&:>%

:&)!()!v3!!3))!v5!)!0*(;!%!0!*(.+/*!3)(4!7;:/)!/(<,!78.;!)(6!(0!&<@&/)!

(+<!&:>:&)!()!32:(.)!(;:!&'.>:!v1 v2 v3! !3))! ! !v4 v5 v6)!%6(&!%3 = 1, %1%-1!+!?! !3))!%?A-1 +!@?)!$!+,4')!*<,,&(.)3>:!(+<!/8!().47&())!

)(6!(+!*8:+5+!=P, Q, A, X>!3))!0!'!,(+.&/!:)!):()3)()*()@&/!)(6!2:):!455+!7&::*(+.)!R$!>*(&!0!+,4')!*<,,&(./)9!:)!'/:&())!)(61

)*('#!%*

)*('#!%+

%%!"%-&-""%!$%!1!% 195

⎪⎩

⎪⎨⎧

====

=====1-1-1-1-1-

-1-1223

QRQR,QPRPR,PQAQA,QAPAARAR,QPPQ,1QPA

O

H O!3)5&/())!%"$#"!$-"(!%'#!#,!2A&)!(4=0!/-$!(&.)2A&)!(0:!%!+!=%$!1$!Q>!;9!3)%

:+:)3*!<(++,4')!,&!'&/3(0!/!3))!&/:))!<(+%

+,4')!(09!SO(3).

F&!/20 *<,?+5/G+<,&!(+!3):+:)36!&)-3+*4&'.+$! (+! +(+/+! 2A&)! 2/! 3+.</29$! ,7!

)3,29! 3))! /7! 2'.&9! (+<! &/:))! )*6(5&<.)!

(./7;:)) $!+,4')!*<,,&(./)9! (+<!/20 '/:&())!

)(6 "!1!%3!+!,2 = C5 = ABC = 1

$! "! +:+,4G&())! "-"%%#"!$-"(!%'#!#, 2A&)!(4=0!67$!(&.)2A&)!(0:!%!3))!&/:))!<(++,4')!(09!SO(3).

%(+,2:+<:! '8+! 3):+:)34! (+58&'.)! (+! &=4&'.+! &38?+9(!!6! 3))! (+!

';'&34&'.+! !2,;9!+!38?+9!!6!3))!(+!+3(4'&.+!/8!&/:))!'<H34!&):!(+!

2:)!&77.)/&/!*&!*/)/.)$!(6(&!(+!455+!*A0,)(/G&())!)(6!()!&/)((6,&:)!&(/%

(&')!(09!*/)/.)9!*()9!3+.</29!(+<($!3)(4!*<:2(&))!2A+<:!()9! /')&9!+,4'&9!

*<,,&(./)9)!&,+/;9!! 3))!#20!&/:))!'<H34!3))!(+!/4 &/:))!)<(+'<H36)

)*('#!!%#!!!!!!!!!!!!!!!!!!!!!!!!!!!!!)*('#!%$!

2(;9! ,(+.&/! :)! )(+'&)A@&/$! 34@&! (&(&.)*,2:0! <(++,4')! (+<!

SO&,(! &/:))!,))!3<35)3*$!')&'.)3*$! (&(.)&'.)3*!*!&)3+*)&'.)3*!+,4'))!%:!'8+!(&(&.)*,2:&9!<(++,4'&9!(+< SO&,(!&/:))!)*6,+./&9$!(6(&!)<(29!&/:))!*<G<7&/9!*(0:!SO(3).

)*('#!!%-

196 !!"#!"#$!"$%&"'!"%#

(1.#/$'+%:!<'<0'#*&"+'+%:!#(,*%:! *'"!$,!($%&%#"%$&="

!3(69!)(6!(0!.&:)3*!,)?5)+7.)//)$!(+<!*<,(50.>:&)!(+!2.7+$!+<-

*))*()3*!*(0:!):47:;*0!(09!$(&.&+,&(./)9! !&/:))!0!,)?5)+7.)//)!(+<!

)3+5+<@&/)!$(0!*<:2A&))$!()!?)?5/)!(+<!A.0*),+(+)*@03):$!().+<*)4G+-:())!*8:(+,)!3)(4!@&,)()329!&:6(0(&9!!3))!&)')36(&.+!&:'))/2.+:)

"5)*)3*!().),2:&)!0!23'+*0!(;:!)$%-*".!,!)(6!(+:!BCDEC8F$! !(+<!

*()!!550:)34!2A&)!)(+'+@&/!)(6!(+:!!)!$(),4(0!&25'/()!$!)775)3*!23'+*0!

(;:!)$%-*".!,!)(6!(+:!GD8!HIJKLM!L)!BCLNI!!(&.)2A&)!(5+8*))!*A65))!)*(+-.)3+8!3))!,)@0,)()3+8!&:'))/2.+:(+9!3))$!().4!()!6(+))!3&:4$!().),2:&)!

,2A.)!**,&.)!)=&(2.)*(0)!$(+)A&/)!)(6!(0:!)*(+./)!(;:!).A)/;:!,)@0,)%

()3>:!(&.)2A+:())!*(0!,+:+7.)//)!(+<!O)!P)!QLR!SC8!TLC8SCR!&25'-($!*'0!

,&()/.)*,2:0!*()!!550:)34!)(6!(+:!.)!'.)*()):/'0!&/777()! F&! (0!5+7)3*!))()+5670*0!*&!).A)/)!,)@0,)()34!3&/,&:)!)*A+-

5&/())!(+!?)?5/+!(+<!UCNV!&2555($!&:>!!0!5+7)3*!)5505+<A/)!(;:!(.+(4%

*&;:!*()!)$%-*".#!().+<*)4G&())!')&=+')34!*(+!WXeYYC8!&25:2()!"4(+)&9!

().+<*)4*&)9! @&,4(;:! ,&! *87A.+:&9! (.+&3(4*&)9! &()A&).+8:())! *()!

A8NKLRR!&2555(!3))!ARFYDR!LRS!PLKECZ!&255'() F))! (.;(6(<(0! 23@&*0! @&,4(;:!$(&.&+,&(./)9$! ,&! )*(+.)329! ):)-

'.+,29$! &:'))/2.+<*&9! @&;.0()329! '))(.)7,)(&8*&)9! &()3&:(.;,2:&9! *()!

(+58&'.)!3))!,&!2,/)*0!*()!3):+:)34!*(&.&4!&/:))!(+![8JK\CYY!&255.() $!,2(.0*0!*(0!.&;,&(./)!3))!0!,2(.0*0!(;:!*(&.&>:!&)')36(&.)$!

)(+(&5&/!(4:()!2:)!&(/3)).+!@2,))!&)!&.7)*/&9!!(+<!%.A),*'0!+)!*A&()%

329!,&!(0!,2(.0*0!(;:!*A0,4(;:!2A+<:!().+<*))*@&/!*&!,&(4/.)*0!3))!

*A+5))*,6!(+<! !GD8!HIJKLM!L)!BCLNI)!$!*<:@&()3*!@&;./)$!*(+!(:&8,)!

(;:! )$%-*".!,$! 7))! (+! &,?)'6:! 3))! (+:! 673+! (2@03&! )(6! (+:! BDYEC8N!

&257/(!3))!23(+(&!)(+(&5&/!):)(6*()*(+!(,*,)!(;:!*87A.+:;:!().+<%

*)4*&;:!(09!!<35&/'&))9!.&;,&(./)9$!6(;9!&/:))!7))!().4'&)7,)!()!]QCM!

&25./($!]^^DKJQ!&256:($!_J8SC8!&25/.($!3))!WJDMC!&256,()! $!'70%!%2!$12!"9#,$+1%12!,&!()9!)*(+.)329!(09!!./G&9!3))!(0!*A2*0!

(09! ! ,&! (0! ,2(.0*0! 3))! ()9! @&;./&9! +5+35*.;*09$! ().+<*)4G&())! *(0!,+:+7.)//)!(+<!OJ`C8!&25-5(!3)@>9!3))!*(+!]S\L8SM!a8)!&25.5()!%.3&(4!5&((+,&.29!*(0!().+<*/)*0!(09!&=25)=09!(;:!7&;,&(.)3>:!,&@6';:!&/%:))!3))!(+![JJYDSFC!&25-7()

$!,)@0,)()3*!'))(.)7,4(&<*0!(09!7&;,&(.)3*9!*<,,&(./)9!3))!0!2:()=0!(;:!3):+:)3>:!*(&.&>:!*(+!(5)/*)+!)<(6!,(+.&/!:)!):)G0(0@&/ 3))!*()!bLYC!&256:($!cCCM!&25:,($!TJY^!&256,(!!3))!455))

!!"#!$%&"#'" 1. Anglin W. S. and Lambek J., "The Heritage of Thales", Springer, 1995 2. Artmann B., "Euclid – The Greation of Mathematics", Springer, 1999. 3. Boyer C., "The History of the Calualus and its Conceptual

Developments", Dover, 1949. 4. Coolidge J. L., "A History of Geometrical Methods", Oxford Univ.

Press, 1940 (Dover reprint 1998). 5. Cromwell P.R., "Polyhedra", Cambridge Univ. Press, 1997. 6. Edwards C. H. Jr., "The Historical Development of the Calculus",

Springer- Verlag, 1979. 7. Effimov N. V., "Higher Geometry", Mir Publishers, 1968. 8. Eves H., "A Survey of Geometry" (Revised Edition), Allyn and

Bacon, 1972. 9. Forder H. G., "The Foundations of Euclidean Geometry", Cambridge

Univ. Press, 1927 (Dover reprint). 10. Heath, T.L., "The Thirteen Books of Euclids Elements" (Second

Edition), Dover reprint, 1956. 11. Heath T.L., "The works of Archimedes", Dover reprint (no date) 12. Heiberg T. L. (Editor) Euclidis, "Elementa", Lipsia, 1885. 13. Hilbert D., "Foundations of Geometry", 10th ed., La Salle-Open

Court, 1971. 14. Moise E. E., "Elementary Geometry from an Advanced Stand-point",

Addisson-Wesley, 1963. 15. Mueller I., "Philosophy of Mathematics and deductive structure in

Euclid’s Elements", M.I.T., 1981. 16. Netz R., "The Shaping of Deduction in Greek Mathematics",

Cambridge Univ. Press, 1999. 17. Rees E. G., "Notes on Geometry", Springer 1983. 18. van der Waerden, B. L., !!""#!"#$$%"%%&"&"$$%'(%&!!"!!""##$!#$%!

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

19. Wolf J., "Spaces of Constant Curvature", Mc Graw Hill, 1963. 20. Yale P. B., "Geometry and Symmetry", Holden – Day, 1968 (Dover

reprint, 1988). 21. 3$%*)$$-!&(%!!&)*+',(-)".%'/'-('%/,)!%!4&*+!)*+&(!

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

150, 152, 156, 157, 177, 179, 186 '#".")!"%('!"%&)!"%&*!"%+, #/!#$$*!0"%&1+&$(&!"$"$"23("$!"

192 !4&1+!"'$(-4"$"-"#(2*+0(5!"&(!"

115 '4"/!)"$5)"")!&(!)!"%& '4"/!)"$"$"!$#&/(("$!"%& '4"/!)"$"$"!3/"$!"%& #6o##3/*")!"#& #6&*(""$*!-)"7".*"+-)!"%) !6"$"+0!"%'&!"%(*!"%(,!"%('!"%&%!"

177, 180 #(*"$"$,#5)!"%+ '((5$5"&$%&&!!"%(, '((5$")"!6"$"+0!"%++ !(20"$"$"8!9!#*,(*!"&&!"%%#!"%*)!"

121 #(2*+0(&*!"0"5+*!!/"/*!'""$&(&'!"

190 #(2*+0(5)!"%)!"#+!"$%!"%%$!"%',!"

190, 196 #(2:$!)!"#& !":++&$(!"+&.,%5!"%(,!"%&% !:4"$"!"!!"#"$%&!!"$% 9'"5"$"$"!3/"$!"%& %&/*!-"70++!!"$(!"$&!"%)&!"%*,!"

124, 126, 140 %#!:!")!"#+ ./3+"/!)!"%'&!"%'+!"%'$ .(!++0!"%+ .1/&!"(:""&6*6,(1/!"%+ .1/&!"&$%&&!)"!!*"&6*6,("$!"%+ (,"+5"&$%&*3/!"*( (,"+5"!'%&$1/"&6*6,(1/!",$ *0#*"";(-9#5+!!"#+ *5+-!(*$")!"%)

(*!&(&"5"&$%:.(!++"$"$+0+!$")""&"'!(""!!*"+,"""#-."!"%'(!"%'&!"151

(*'+&$(")"$5)"")!&(!)!"%& (*'"$!"5!"%+ (*&((*!0""+'(!!"%$( (&&((")".1/&!!"%#!"(+!"&) **"!#0)!"#& (*6#!"*!"+-)"$"$"!:9"$!"#&!"#+ ($<!'"6"#:&((!!"%$( ($<!0"6(-$!"5!"(+ ($<!-""20+!!"(+ ($/'+&*"!":++&$("*"&$%&&&)!"%(,!"

154 ($/'+&*"":++&$("*"&$%&&&)!"%(, (1(&!'&(("!"%+!"%'(!"%+&!"%++!"

181, 190, 195 &..&.(!++,/!"6(&"+!$!!"$) &*!""!&((*!0""+'(!!"%$( &*!""'&(("!"%+!"%'(!"%+$!"%#%!"%#*!"

183, 185, 186, 189, 190 &#'""1/!"%'&!"%('!"%&)!"%&*!"%+,!"

180 ,##&*=5!"&* &+9!(-/""-"*"-$5$!"-""+"*-$5$!!"&, &4'&(("!"(%!"%#,!"%$( &6!.1.*!-)""$##".*"+-)!"*' &6&6&(5".1/&!!"'%!"'*!"',!"'&!"()!"

57, 59, 77, 79 &6&6&("""20+!!"%)!"%#!",$!"&,!"%%#!"

119, 143 &6&6&("!"** '6*6&("+&$(&!!"*% &6*)'/&*!!"%'!"%+!"** '(!$""%,/5)!"#&!"#+ ':("4")!"(,!"#&!"#+!"$)!"$%!"%,) &$%&*".&/0)!"*' '$!#&&(5)!"(*!"$+!"$$!"%))!"%',!"

181, 184, 185 '$$-!*")!"#+ +&(9-)">-;-!"%+

!!"!#""#$

200 '?87',*-".>@A,B',#/"

C&:.")"$&+/-+&/1/"&6*6,(1/!",$ +0/1/!)!"%%$ -+*!!/"/*!-">$&(&-!"%$) 5+*!:!#*"!"%& .(1/!)!"%+!"#& /,1/!)""">+$(/!&")!"%+ %&3(5+!"$"$"Euler, 188 %&3(5+!"$1/"$(*3/"!!%,$1/!",*!"

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

189, 190, 193, 196 !!/"/*!-"$&$('&(("!"#(!"%&,!"%#$!"

193 !!$'"!"($)0/"*"""!&#&&)"$(&&((&)!"

62 !!$'"!"($)0/"$(&&((&)".1/&&)!"(#!"

60, 61, 62 !&/$(*!0"(,"+5"&$%&*3/!"*' !,/$(""$5)"")!&(!)!"%& !#&"5!"%+!"%#

!#&"5"&6*6,("$"6(")"&6&6&("!"%( !#&"5"&$%&&!)!"%( !"#"9-"&4'&(("!"%$% !"#"9-"$&$('&(("!"%$% !:9"-"!$'&(("!"%$% !:9")!"%+!"&&!"#*!"#&!"%%+!"%'(!"

168, 175, 176, 177, 178, 180, 182, 185, 190, 195 !$!#*!0""+'(!!"%$( 8:!#"$"D,$(5"*)! 143 !:#*/((")!"%&!"%+!"%#!"$)!"%**!"%*,!"

125, 130, 131, 132, 133, 134, 143, 188 !$($'"6"#$.1/*!'"21(&!!"&( !$($-"6"#:.1/"!"'* !$($-"6"#:&(("!"%#!"#' !1/*!0"$"+0!"#+ !3/")!"%&!"%+!"%#!"&%!"&*!"$)!"%**!"

123, 124, 125, 126, 127, 128, 131, 132, 134, 143, 188 D!%5+!$*!0">$/!.1.0! 190 +,.*"$")"!:!#")!"%,&!"%,+ +,%&45!"** +,%"(")"&4'/$#5"5)!"%)!"&&!"$%!"

196 +,%"(")"!/'#$"5)"!!*"":/%&"5)!"

184 !"#$!$%, 10 D,/!*2+")!"#&!"#+ +,"5"6#&$('!"%(' +,""""(%".3/*"!"%(' +&$!"25+!$*"+-)!"%$%!"%$* D&/1!)!"#+ E*!"+0(5)!"#& -.!")!"&,!"&+!"#& -.!1/"*"-$5$!!"&+ "!$!&((*!0""+'(!!"%$( "!$'&(("!"%+!"%'(!"%&+!"%+#!"%+$!"

180, 182, 183, 189, 190, 195 "+'(!""$++&$(&!)!"%$( "+'(!!"%$* "+"&6&6&(&)"&$%&&&)!"*&!",) -+"*!"6"#:.1/!!"$* -+"*!""$&(&'""20+!$!!"%(

'?F'@-F,A 201

-+"*"*"!:#*/(("*!"%+ -+"*"*"!3/"*!"%+ "+"*-$5$!!"%# "+-#".&)"!!+,)!"+* "(%-"6(&"+!!"+&!"++ ;'66")!"#&!"%#'!"%#(!"%#+!"%$) 6!(!9"#0!"#+!"## 6!(!9"#*!-"21(&"!"%%$ 6!('##5#!"&6&6&(!!"%( 6!(!##5#&6&6&("!",$!"(*!"((!"(&!"

63, 65, 66, 67, 68, 69, 70, 71, 72, 74, 75, 77, 80, 81, 84, 86, 89, 90, 113, 117 6!(!##5#&!"&$%&*3/""$""23("!",) 6!(!6#5(1+!$*!0".1/&!!"(# 6&6&(!"+,/5""+'(!!"%$( 6,(!)!"%+ 6#'.*""6(&"+!!"+7 ;#'$1/!)!"$!"%+!"#&!"#+!"%#' ;#!$1/*!'""$&(&'!"%# ;#":$!(2")!"%) ;"#*$&&!! 9 6"#$.1/*!0"6$(!+&(!!"%%* 6"#:&(("!"#(!"$)!"%,&!"%,#!"%,$!"

140, 142, 183, 184, 188, 192, 196 6(&"+!!"%&!"%#!"(*!"&&!"&+!"++!"#'!"

102, 104, 105, 107, 110, 114, 115, 116, 117, 123, 143 6(""!(+-C"$"!!"%(, 6(3$5"!6"$"+0!"%(, ;$"#&+!&")""",II, 87 ;$%!.-(&*""%&3(5+!!"+$ ;$%!.-(&*"*!"#& 6$(!+&(!"%(!"%#!"#$!"$)!"%)'!"%)(!"

107, 108, 109, 110, 113, 114, 115, 117, 123, 124, 126, 127, 128, 138, 140, 143, 163, 164, 165, 166, 168, 178, 179, 180, 182 (5$0"(*'+&$(")!"%&) (5$0"&$%&&!!"%(*!"%(, (5$-"$+0+!!"%++ (5$-)"#-.")!"%#) >$!+'$5)!"%$&

"$&(&'".1/&!!"%(!"%#!"')!"'%!"'*!"46, 48, 53, 103, 104, 182, 184, 189

"$&(&-"&!"6&(*"$(")0)!"%# "$&(&-""20+!!"%)!"%#!",$!"&,!"++!""

143, 176, 183 "$&(&-""3+!!"%'! 17, 84, 184 >$&(&"+&$(&!!"$!"%)!"*%!",*!"%$& "$"*2&*3(&)"$5)"%&1(&!)"$"$"&+9!1(":""$5/"&6&6&(5"%&1+&$(&!!"&,

":++&$(!"+&.,%5!"%(, ":++&$(&)"&$%&&&)!"%(' "$++&$(&!"$"$""20+!$")!"%$* "$++&$(*!,)"$(&&((&)".1/&&)!"&) "$+6$3"*+!""20+!$!!"(* "$+6$3"*+&)"$(&&((&), 60, 61 "$/'($5"5"&+9!(":!"&( "$/%&$*!0"%&1(&!"$"$"-.!"$!"%, ")!&(!!"%'!"%&!"%#!"$)!"$$!"%))!"

135, 136, 137, 138, 139, 140, 141, 142, 143, 145, 163, 164, 165, 166, 167, 168, 172, 173, 176, 177, 178, 179, 180, 184, 185, 187, 188

$!$$"$*!-)"+&$!"25+!$*"+-)! 192 $&$'($5"!6"$"+0!"%(,!"%&%!"%&* $&$(!&((*!0""+'(!!"%$'!"%$( $&$('&(("!"&(!"#'!"#(!"%'(!"%&'!"

183, 190 @&+!*"), 18 $"+0"(:""&6*6,(1/!"** $(*.1/*!0"!/*"-$5$!!"'( $(*.1/*!0"6$(!+&(!!"%)*!"%)&!"%%* $(*.1/*!-"6(&"+!!"&,!"#,!"#$!"%)$ $(&&((&)"!!$'"!"($)0/!"(8 $(&&((5""$&(&'".1/&!!"%,!"%#!",$!"

40, 42, 55, 56, 57, 58, 59, 60, 61 $(&&((5)".1/&!)"!!$!"!&$0!"() $:6")"$"$"Euler, 188 $6&(9"#0!"## )%&/"$"!"!!"#"$%&!!"$% B(*"$*!/&(5) %., 196 Anglin and Lambek, 196

202 '?87',*-".>@A,B',#/"

Artmann, 196 Boyer, 196 Cauchy, 18, 84 Coolidge, 196 Cromwell, 196 Dehn D., 66 Descartes, 86 Edwards Jr., 196 Effimov, 196 Euler, 62 Eves, 196 Forder, 196 Heath, 97, 98, 196

Heiberg, 181, 196 Hilbert, 196 Legendre, 18, 77, 99, 100 Leibniz, 119 Longchamps, 86 Moise, 196 Mueller, 196 Netz, 196 Rees, 196 van der Waerden B. L., 196 Wolf, 196 Yale, 196

!!"!!!#!"!$!!!!%!#!%!#!#!$!!!%!&!'!#!&

1. Aaboe A. "Episodes from the Early History of Mathematics", The

Mathematical Association of America, Washington D.C. 1964. 2. Alexander of Aphrodisias. "Commentary on Aristotle's Sophistici

Elenchi", Wallies, Berlin, p.76.23, 1898. 3. Allen R.E. (ed.) "Studies in Plato’s Metaphysics", London 1965. 4. Allman G.J. "Greek Geometry from Thales to Euclid", Logmans, Green

and Co, Dublin - London 1889. 5. Amir - Moez A. "Discussion of Difficulties in Euclid", Scripta

mathematica p.p. 276 – 303,1959. 6. Andrews G. E. Berndt, B. C and Ranklin, R.A. (Ed.) "Ramanujan

Revisited": Proceedings of the Centenary Conference, University of Ilinois at Urbana - Champaign, Boston, MA: Academic Press, 1988.

7. Annas J. "Die gegenstände der Mathematik dei Aristotles", Rhein Verlag Zürich 1950.

8. Apostle H. G. "Aristotle’s Philosophy of Mathematics", Chicago 1965. 9. Apostol T. M. "Introduction to analytic Number Theory", Springer, New

York 1976. 10. Archibald C.R., "Euclid's Book on Divisions of Figures with a

Restoration Based on Woepcke's Text and the Practica Geometriae of Leonardo Pisano", Cambridge 1915.

11. Arrighi G. "An Aderlardian version of Euclid", Rend. Acad. Naz. Sci, 16, No 1, 1992.

12. Artmann B. "Euclid, The Creation of Mathematics", Springer, N. Y. 1999.

13. Artmann B. "Uber Voreuklidishe Elemente", Archive for History of Exact Sciences, vol. 33, p.p.291-306, 1985.

14. Artmann B. "Ueber voreuklidische Elemente der Raumgeometrie' aus der Schule des Eudoxos", Arch. Hist. Sci.19, No.2, 121-135, 1988.

15. Artmann, B. "Euclid's elements and its prehistory", Edmonton, Apeiron 24, 1-47, 1992.

16. Ayoub R.G. "An Introduction to the Analytic Theory of Numbers", Amer. Math. Soc., Providence 1963.

17. Ball W. W.R "A short account of the history of Mathematics", Dover 1980.

18. Barbin E. "The Role of Problems in the History and Teaching Ma-thematics", Vita Mathematica, p.p. 17-25, 1966.

19. !"!""#$! #%&! '"""(!%)"! !!"#$!"$%&!! #$*+",%%-! ./$%%-$"! O&si""!Bruxelles 1996.

204 EY'0&'()!%123'4&'*&

20. Barnes J., Schofield M. and Sorabji R. (eds). "Articles on Aristotle", vol. I, II, Duckworth, London 1978.

21. Becker O. "Das Mathematische Denken der Antike", Vandenhoeck, Wis. Buch. Göttingen 1957.

22. Becker O. (ed.) "Zur Geschiche der griechischen Mathematik", Wis. Buch. Darmstadt, 1965.

23. Bekker I. 1%)$" (1%&5$"), (&""-6. 7+#",!'$!&&&"!*896"!"!:':'! 24. Bell E.T. "Men of Mathematics", Simon and Schuster, N. Y. 1965 25. Bellman R.E. "Anlytic Number Theory: An Introduction Reading", MA

Benjamin / Cummings, 1980. 26. Benoit P., Chemla K., Ritter J. "Histoire de fractions, fractions d'

histoire", 1992. 27. Berggren J. L. "Episodes in the Mathematics of Medieval Islam",

Springer, N.Y. 1986. 28. Berggren J. L. "History of Greek Mathematics:A survey of recent re-

search", Historia Mathematica, vol. 11, p.p. 394-410, 1984. 29. Bergstrasser G. "Pappus Kommentar zum Zehnten Buch von Euclid's

Elementen", Der Islam 21, 1933. 30. Boezio R. "Porfirio Isagoge", Wien - Leipzig 1906 (+;"6+/. G. Girgenti,

Milano 1995). 31. Boltianski V. "Tretii Proplemi Hilberta", Nauka, Moscow 1968. 32. Boltyanski G. V. "Equivalent and Equide-composable Figures" ($+#. +/

#%& !,%.), D.C. Heath and Company, Boston 1963. 33. Bolyai J. Appendix "The Theory of Space", Akademiai kiado, Budapest

1987. 34. Bonola R. "Non - Euclidean geometry", Dover, N. Y. 1955. 35. Borel E. and Stäckel P. "Die Element der Mathematik, II. Geometrie",

Teubner, Leipzig - Berlin 1909. 36. Borsuk K. – Smielew W. "Foudations of Geometry", North – Holland,

Amstesdam 1960. 37. Borwein J. M. And Borvein, P.B. Pi & the AGM: "A Study in Analytic

Number Theory and Comporational Complex", New York Wiley, 1987. 38. Boyer C. B. "A History of Mathematics", John Wiley & Sons, N.Y. 1989. 39. Brentjes S. "Textzeugen und Hypothesen zum arabischen Euklid in der

Ueberlieferung von al-Haggag b. Yusuf b. Matar", Arch. Hist. Exact Sci. 47, No.1, 1994.

40. Brentjes S. "Varianten einer Haggag – Version von Buch II der Elemente", Rodopi B. V., Amsterdam 1993.

41. Bretschneider K.A. "Die Geometrie ynd die Geometer vor Euklides", Leipzig 1870 (+;"6. Olms 1975)

42. Briggs H. "Elementorum Euclidis libri VI Priores", p.p.195-222, London 1620.

43. Brunschvicg L. "Le Role de Pythagorisme dans l’ evolution des idees", Blanchard, Paris 1937.

-&<'')!!'!0'3-=*.'* 205

44. Brunschvicg L. "Les Etapes de la Philosophie Mathematique", Blanchard, Paris 1991.

45. Burkert W. "Weissheit und Wissenschaft: Studien zu Pythagoras, Philolaos und Platon", Carl, Nürnberg 1962.

46. Burnet J. "Early Greek Philosophy", Mcmillan, London 1930. (+;"6+/. Penguin 1965)

47. Burton M. D. "The History of Mathematics", Dover, N. Y. 1985 48. Bury R. G., "Sextus Empiricus", vol. I-IV, Loeb Clas. Libr., London

1983 49. Busard H.L.L. "A thirteenth – century adaptation of Robert of Chester' s

version of Euclid's Elements" 2vols, Algorismus, Muenchen 1996. 50. Busard H.L.L. and Folkerts M. "Robert of Chester's redaction of Euclid's

Elements. The so called Adelard II version" Vol. I, Basel: Birkhaeuser 1992.

51. Busard H.L.L. "The Translation of the Elements of Euclid from the Arabic into Latin by Hermann of Carinthia" (?), Leiden 1968.

52. C. H. Edwards H. C., Jr. "The Historical Development of the Calculus", Springer 1979.

53. Cajori F. "A History of Mathematics", N.Y. 1893 (+;"6+/. Dover 1919) 54. Cantor M. "Vorlesungen über Geschichte der Mathematik", 3- ./$%%-,

Leipzig 1907. 55. Cardies J. L. "La proposotion 14 du livre V dans L' economies des

Elements d' Euclide", Rev. Hist. Sci. 44, No.3/4, 1991 56. Casey J. "A Sequest to the First Six Books of the Elements of Euclid", 6th

ed Dublin Hodges Figgis & Co., 1892. 57. Caveing M. "La figure et le nombre", PUF, Paris 1997. 58. Caveing M. "The debate between H. G. Zeuthen and H. Vogt on the

historical source of the knowledge of irrational quantities", Centaurus 38, No.2-3, 1996.

59. Caveing M. "These. Bernard Vitrac, De quelques questions touchant au traitement de la proportionnalite dans Les Elements d' Euclide", Rev. Hist. Sci 47, No.2, 1994

60. Clagett M. "Greek Sience in Antiguity", London 1957. 61. Clagett M. "The science of Mechanics in the middle Ages", Madison,

Wiskonsin 1954. 62. Clagett M. "The Medieval Latin Translations From the Arabic of the

Elements of Euclid, with special Emphasis on the version of Adelar of Bath", Isis 44, p.p. 16-42, 1953.

63. Colerus E. "Von Pythagoras bis Hilbert", Springer, Berlin, Wien, Leipzig 1937.

64. Colli G. "Zenone di Elea", Adelphi, Milano 1998. 65. Cousin V. "Procli comentarium in Platonis Parmenidem", G.Olms,

Hildesheim 1961. 66. Coxeter H. S. M. "Introduction to Geometry", J. Willey, N.Y. 1961

206 EY'0&'()!%123'4&'*&

67. Cromwell R. P. "Polyhedra", Cambridge 1997. 68. Curtze M. "Anaritii in decem libros priores Elementotum Euclidis

commentarii", Teubner, Leipzig 1899. 69. Curtze M. "Das algebrische werkes Eukleides über die Waage", Zeitchrift

für Mathematik und Physik, 19, 1874. 70. Curtze M. "Zwei Beiträge zur Geschichte der Physik im mittelalter",

Biblioteca Mathematica I, 3rd ser, 1900. 71. Davenport H. "The Higher Arithmatic: And Introduction to the Theory of

Numbers", 6th ed Cambridge University Press, 1992 72. De Morgan A. "Proportion", Penny Cyclopaedia XIX, p.p. 49-53

(0%6$*6% 1841). 73. De Young G. "The Arabic textual tradicions of Euclid' elements", History

of Science, p.p. 147-160, 1984. 74. Delacy A. E. "Euclid and Geometry", London 1965. 75. Dence J. B. and Dence, T. P. "Elements of the theory of numbers", CA

Harcourt / Academic Press, San Diego 1999. 76. Descartes R. "The Geometry", Dover, N. Y. 1954. 77. Dickson L. E. "History of the Theory of Numbers", 3 vols. Chelsea, N.Y.

1952. 78. Diedonne J. "Abrege d’ histoire des mathematiques", Hermann, Paris

1996. 79. Diels H. "Simplicius, In Aristotelis physicorum libros comentarius", Vol.

2 Teubner, Berlin 1882 – 1885. 80. Diels H. and Kranz W. "Die Fragmente der Vorsokratiker" Berlin 1936-

37 (+;"6./$%%- Weidmann, Berlin 1950,1951). 81. Dijksterhuis E. J. "De elemente van Euclides" (vol. 1-2), Noordhoff,

Groningen 1929-30. 82. Dixon R. "Mathographics", Dover p.p. 26-27, New York 1991. 83. Drake, S. "Euclid Book V from Eudoxus to Dedekind", Hist. Philos. Sci,

Nouv.Ser. 21, Toronto 1987. 84. Dudley U. "A. Budget of trisections", Springer N.Y. 1987. 85. Duhem P. "Les origines de la statique", Paris 1905. 86. Dunham W. "Journey through genius. The great theorems of

mathematics", Wiley Science Editions, New York 1990. 87. Effimov V. N. "Higher Geometry", Mir Publishers, Moscow 1968. 88. Engel F. - Stäckel P. "Die theorie der Parallelinien von Euclid bis auf

Gauss", Weidmann, Lepzig 1895. 89. Enriques F. and Amaldi U. "Elementi di geometria", Bologna 1904 (+;"6.

Feltrinelli), Roma 1990. 90. Eudemi, Phodii Peripatetici, "Fragmenta quae supersut" (Collegit L.

Spengel), Berolini 1866. 91. Eves H. "An Introduction to the History of mathematics", Reinhart, N.Y.

1953. 92. Eves H. "College Geometry", Allyn and Bacon, Boston 1995.

-&<'')!!'!0'3-=*.'* 207

93. Eves H. "Great moments in Mathematics before 1650", MAA, N.Y. 1985. 94. Eves H. and Newson C.V. "Foundations and Fundamental Concepts of

Mathematics", Dover, N.Y. 1997. 95. &>"!:#/%?!-'!/'!(#$!$#'#&"!"$"($)*+$"$%,(!#@$%?!*0"!1"++(!" (1991)

/"A!#@$%?!!0!*6#"&%-"!#)**+$!*896"'! 96. &>"!:#/%?! -'!/'! (%!""-.$" "'/"($)*+$"$%0/(! #@$%?!*0,! 2"!7"8-$"#A/#!

#,6! -"(&",6*,6! /"A! #,6! *!:"*,6! *A+&;#*,6! #)**.$! /"A! #@$%?! !0,! 2"!7"8-$"#A/#!#,6!'6$B6!/"A!#,6!'A6.C,6!#)***$!*896"'!

97. &>"!:#/%?! -'/'"! !&" $-1$.$" #22,'$" %"$".'$" "*3" +$)*+$"$%&3" !%45*3!# D!"/#A/#! ).%&!D"6+""96A%&!1&6+$!*%&!7"8-$"#A/9?!D"A$+*"?"!&'7'&'"!*896"!/000'

98. Exarchakos Th. "The method of mathematical induction and the Pell equation in ancient Greek Mathematics", Mentor, A Journal of Scientific and Educational Research, Hellenic Pedagogical Institute, vol 1, p.p. 4-16, Athens 1999.

99. Forder G. H., "The Foundations of Euclidean Geometry", Cambridge Univ. Press, 1927 (Dover reprint).

100. Fowler D.H. "The Mathematics of Plato’s Academy. A new re-construction", Oxford 1987.

101. Fowler, D. H. "Book II of Euclid's elements and a pre-Eudoxan theory of radio", Arch. Hist. Exact Sci. 22, 1980.

102. Frajese A. "Attraverso la Storia della Mathematica", Rusconi, Firenze 1977.

103. Frajese A. "La Mathematica nel mondo antico", Rusconi, Roma 1972. 104. Frajese A. "Platone e la Mathematica nel mondo antico", Roma 1963. 105. Frajese A. and Macioni L. "Gli Elementi di Euclide", La Monnier, Turin

1970. 106. Frank E. "Plato und die sogenanten Pythagoreer", Halle 1923 (+;"6+/.

Wissenschaftliche Buchgesellschaft, Darmstadt, 1962). 107. Frankland B. W. "The First Book of Euclid's Elements with a

Documentary Based Principally upon that of Proclus Diadochus", Cambridge 1905.

108. Frege G. "The Foundations of Arthmetic", Basic Blackwell, Oxford 1950. 109. Friedlein G. "Proclus, In Primum Euclidis Elementorum Commentari",

Teubner, Leipzig 1873 (+;"6. Olms G., Hildsheim 1965). 110. Gallop D. "Parmenides of Elea Framentas", Oxford 1984. 111. Gardies J.-L. "L' organisation du livre XII des Elements d' Euclide et ses

anomalies" (Organisation of Book XII of Euclid's Elements and its anomalies), Rev. Hist Sci. 47, No. 2, 1994.

112. Gauss C. F. "Disauitiones Arithmeticae", New Haven, CT: Yale University Press, 1966.

113. Gericke H. "Mathematik in Antike und Orient", Fourier, Wiesbaden 1992. 114. Gilbert 3. "Aristoteles Urtaile über die pythagorische Lehre", Archive

Gesch XXII (1953), p. 165.

208 EY'0&'()!%123'4&'*&

115. Görland A. "Aristoteles und die Mathematik", Marburg 1899. 116. Graser A. "Mathematics and Metaphysics in Aristotle", Bern 1987. 117. Grattan – Guiness, J. "Numbers, magnitudes, ratios, and proportions in

Euclid's it Elements: How did he handle them", Historia Mathematica 23, No.4, 1996

118. Greenberg J. M. "Euclidean and non - Euclidean Geometries", Springer, N.Y. 1993.

119. Gregory D. "#6%2(.'"6 "$ !'78+(/$", Euclidis quae supersunt omnia, Oxford 1703.

120. Gruber M. P. "History of Convexity, Handbook of Convex Geometry", vol. A, Elsevier, Amstrerdam 1993.

121. Guillaumin Y. J. "Mathematiques dans l’ Antiquite", IREM, Saint - Etienne 1992.

122. Guy R. K. "Unsolned Problmes in Number Theory" (2nd ed.) Springer - Verlag, New York, 1994.

123. Hankel H. "Beiträge. zur Geschichte der Mathematik im Altertum und Mitelalter", Leipzig 1874 (+;"6. G.Olms Hildesheim 1961).

124. Hara K. "Quelques ouvrages de geometrie more veterum de Roberval. I." Hist. Svi., II. Ser.2, No.1, 1992.

125. Hardy G. H. And Wright, E. A. "An Introduction to the Theory of Numbers", 5th ed Oxford, England: Clarendon Press, 1959.

126. Hartshorne R. "Teaching Geometry According to Euclid", Notices of AMS, vol. 47, (4) p.p. 460-465, 2000.

127. Hasse H. "Number Theory", Springer, Berlin 1980. 128. Hasse H. and Scholz H$"!&"%-.!$3""'/"$-10/""*3"#22*/$%&3"($)*+$"$%&3"

#9$!"&+*3!#!(+"#*%!#-?!&7&!)*1/' 129. Hasse, H. "Mathematicals Wissenschaft", Kunst und Macht, Mitt.

Deutsche Mathematiker vereinigung vol 4, p.p.28-38, 1947. 130. Hauser G. "Geometrie der Griechen von Thales bis Euclid", Atena,

Lucerne 1955. 131. Heath T. L. "A History of Greek mathematics", Oxford 1921 (+;"6. Dover

1975). 132. Heath T. L. "A manual of Greek Mathematics", Oxford 1931. 133. Heath T. L. "Mathematics in Aristotle", Oxford 1949 134. Heath T. L. "The Thirteen Books of the Elements", 2nd ed., vol. 1-3.

Dover, New York 1956. 135. Heiberg J. L. - Stamatis E. "Euclides Elementa" (vol. 1-2), Teubner,

Berlin 1969. 136. Heiberg J. L. "Euclide, Les Elements", ($+##,!"%- B. Vitrac $+ +A%"+.

#%& M. Caveing), PUF, Paris 1990. 137. Heiberg J. L. "Geschichte der Mathematik und Naturwissenschaften im

Altertum", Minerva, München 1925. 138. Heiberg J. L. and Menge H., "Euclidis opera omnia", 8 vols, Teubner,

Leipzig 1883-1916.

-&<'')!!'!0'3-=*.'* 209

139. Heiberg J. L., "Paralipomena zu Euklid", Hermes 38, 46-74, 161-201, 321-356, 1903.

140. Heiberg L. J., "Naturwissenschaflen Mathematik und Medizin im Klas-sischen Altertum" ($+)#+!- ./$%%-), Leipzig 1920.

141. Heinemann G. "Greek Mathematical work (I), Thales to Euclid", London 1939.

142. Hervagius J. "Euclidis de levi et Ponderoso Fragmentum" Euclidis Megarensis Mathematica Clarissimi Elementorum geometricorum libri xv, Basel 1537.

143. Herz - Fischler, R. "Theorem XIV, of the Elements", Arch. Int. Hist. Sci. 38, Nov. 120, 1988.

144. Hiller E. "Expositio rerum matematicorum", Teubner, Leipzig 1873 (+;"6. 1995)

145. Hofmann E. J. "Geschichte der Mathematik", Bd. 1,2, Sam. Gössen, Berlin 1953-1957.

146. Hoppe E. "Mathematik und Astronomie in Klassischen Altertum", Heidelberg 1911.

147. Hoyrup J. "The Four sides and the Area", Vita mathematica, N. Y. 1996. 148. Hultsch F. "Autolykos und Euclid", Berichte der Verhandlung der Kgl.

Sächsischen Gesellschaft der wissenschaft zu Leipzig, Phil, hist, classe. 38, 1886.

149. Hultsch F. "Euklides 8", Pauly-Wissowa II, Leipzig 1907. 150. Hultsch F. "Pappi Alexandrini collectionis quae supersunt" 3vols, Berlin

1876-1878 $+ $A" +"""A/9 $+##,!"%- ";@ #%6 Paul ver Eecke. 151. Husserl E. (tr. et intr. J. Derrida) "L’ origine de la Geometrie", PUF, Paris

1962. 152. Jones Al. "Pappus of Alexandria Book 7 of the Collection", Springer,

N.Y. 1986. 153. Joseph G. G. "The Crest of the Peacok" (Non - European Roots of

Mathematics), Norton, London 1991. 154. Kasir S. D. "The Algebra of Omar Khayyam", Chelsea, N.Y. 1931. 155. '-;%&!@?! 4'! !)-'/"3" *2(:$/'-4'3$" ;/8+$"$" +('+("-$%0/" 8-'/$"

+('+("-$%,!#!&7&"!*896"!)**+' 156. Kidd I. G. "Posidonius Fragmenta", Cambridge 1988. 157. Klee V. and Wagon S. "Old and New Unsolved Problems in Plane

Geometry and Number Theory", Math Assoc. Amer, Washington DC.1991.

158. Klein J. "Greek Mathematical Thought and the Origin of Algebra" (tr. Brann E.), Dover 1992.

159. Knorr W. R. "Rational diameters and the discovery of incom-mensurability", Am. Math. Mon. 105, No 5, 421-429, 1998.

160. Knorr W. R. "The ancient tradition of Geometric Problems", Dover, N. Y. 1986.

210 EY'0&'()!%123'4&'*&

161. Knorr W. R. "The practical element in ancient exact sciences", Synthese 81, No.3, 1989.

162. Knorr W. R. "The Pre-Euclidean Theory of Incommensurable Magnitudes", Harvard Un. Press, Cambridge Mass., 1973.

163. '%6#%+A#66-?!('!/"A!<#CA":!9%#%?!&$"!,$!$%43"-//"$(3""*3"+('+("-.$3!, *896"!)***'!

164. Kruoll W. "Proclus, In Platonis republican commentari", Leipzig 1901 (+;"6. Olms G., Hidelsheim 1972).

165. 0#/,6%?!!'!!<""$1(.$"+('+("-.$3!"!*896"!)2*/' 166. Landau E. "Elementary Number Theory", 2nd ed Springer New York

1994. 167. Lauenstein H. "Arithmetik und Geometrie in Raffaels Schule von Athen.

Die geheimnisvolle der Tafeln im fuer das Konzept harmonischer Komposition und der ungeahnte Bezug zum Athenatempel von Paestum. Peter" Lang, Frankfutr / Main 1998.

168. Lentz A. Paully - Wissowa. "Real - Encyslopädie der Classischen altertumwissenschaft", Stutgard 1894

169. Levet J. "P. Gerbert d’ Aurillac Traité de Géométrie", Irem 1997. 170. Liddell G. H. -! 34566! 7'!7.+"! "+>A/@! #-?! +""-6A/9?! +"B%%"?"!*896"!

1948. 171. Liddell H. G. Scott R., Jones H.S. "A Greek - English Lexicon", Oxford

1940. 172. Loomis E.S. "The Pythagorean Proposition", National Counsil of

Teachers of Mathematics, Wasington 1968. 173. Loria G. !%!""-.$" "'/"($)*+$"$%0/(! #7+##,!"%-!7'',("*%&$"!&7&"!

*896"!)*.)' 174. Lueneburg H. "Leonardi Pisani oder Lesevergnuegen eines

Mathematikers", Mannheim B.I. Wissenschaflverlag 1993. 175. Mackenzie J.D. "Dialogue and proof", History of mathematics, Clayton /

aust. 1980 176. Maddalena A. "I Pitagorici", Atena, Bari 1954. 177. Maeenpaeae P. von Plato, J. "The logic of Euclidean construction

procedures", Acta Philos. Fenn. 49, 1990. 178. Martin G. E. "Geometric constructions", Springer, New York, 1998. 179. Matvievskaya G. "The theory of quadratic irrationals in medieval

oriental mathematics", King, David A. (ed) et al., New York 1987. 180. Mau J. "Eukleides 3", %#% Die Klein Pauly, II %+". 416-419, Stuttgart

1967. 181. Meschkowski H. "Ways of Thought of Great Mathemaricians", Reidel,

Boston 1964. 182. Michel H. P. "De Pythagore a Euclide", Les Belles Lettres, Paris 1950. 183. Milhaud G. "Les Philosophes - geaometers de la Grece", Al can F. Paris

1900 (+;"6. Arno 1976).

-&<'')!!'!0'3-=*.'* 211

184. Moise E. E. "Elementary Geometry from an Advanced Stand-point", Addisson-Wesley, 1963.

185. Mol%dsci V. "Istorikomathematiceskie iseldovania" (!,%.) Nauka, Moscow - Leningrad 1942.

186. Morrow G. "Proclus, A Commentary of the first book of Euclid’s Elements", Princeton 1970.

187. Mueller I. "Euclid's Elements and the axiomatic method", British J. Philos. Sci., 20, 1969.

188. Mueller I. "Greek Mathematics and Greek Logic", in Corcoran J. "Ancient Logic and its modern Interpretation", Dordrecht, p.p. 35-70, 1974.

189. Mueller I. "On mathematics. Essays on ancient mathematics and its later development based on a conference", Oberwolfach, Germany, August 19-25, 1990, Apeiron 24, 1992.

190. Mueller I. "Philosophy of mathematics and deductive structure in Euclid's Elements", Cambridge, Massachussetts – London, MIT 1981.

191. Mugler C. "Dictionaire historique de la terminologie geometriques des Greecs", Les belles Letres, Paris 1958.

192. Mugler C. "Euclide, Extrait des Elements", Paris 1967. 193. Murata T. "A tentative reconstruction of the formation process of Book

XIII of Euclid's Elements" Comment. Math. Univ. St. Pauli 38, No.1 1989.

194. Netz R. "The Shaping of Deduction in Greek Mathematics", Cambridge, 1999.

195. Neuenschwander E. "Die ersten vier Bücher der Elemente Eucklids", Archive for History of Exact Sciences 9, p.p.325-380, 1973.

196. Niebel E. "Untezsuchungen uber die Bedeutung der geometrischen Konstruction in der Antike", Kant-Studien v.76, Cologne 1959.

197. Niven I. M., Zuckerman H.S. and Mongomery H. L. "An Introduction to the Theory of Numbers", Wiley N.Y. 1991.

198. O’ Meara D. J. "Pythagoras Revised", Oxford 1989. 199. Plato "Theaetetus" (tr Mc Dowell J.), Oxford 1973. 200. Porphyrius, "Fragmenta" (Ed. A. Smith), Stutgardiae et Lipsiae 1993. 201. Porphyry "On Aristotle’ s Categories" (Trans. S. K. Strange), N. Y. 1992. 202. Raven E. J. "Pythagoreans and Elatics", Ares, Chicago 1948. 203. Reed D., "Figures of thougth Mathematics and mathematical texts",

Routleds London 1995. 204. Reidemeister K. "Das exacte Denke der Griechen", Kroll, Hamburg 1949. 205. Reidemeister K. "Die Arithmetik der Griechen", de Gruyer, Berlin 1940. 206. Rey A. "Les Mathematiques en Grece au milieu de v’ siecle", Gauthier,

Paris 1935. 207. Russo L. "The definitions of fundamental geometrie entities contained in

book I of Euclid’s Elements", Arch. Hist. Exact. Sci. 52, No 3, 1998. 208. Sabra L. A. "Simpicius Proof of Euclid's Parallels Postulate", Journal of

the Warburg and Courtauld Institutes 32, p.p. 1-24, 1969.

212 EY'0&'()!%123'4&'*&

209. Sabra L. A. "Thabit ibn Qüra on Euclid's Parallels Postulaté", Journal of the Warburg and conrtau institutes, p.p. 19-32, 1968.

210. Saccheri G. "Euclides Vindicatus", G. B. Halssted, Chicago 1920, (+;"6. Chelsea 1956).

211. Sachs E. "Die fünf platonischen Körper", Berlin 1917. 212. Saito K. "Book II of Euclid’s Elemente in the light of the theory of Conic

Sections", Historia Scientiarum , 28, 1985. 213. Sarton G. "Introduction to the Theory of Science", Biblioteca Mathe-

matica I, 1900. 214. Sarton G., "Ancient science and Modern Civilization", Lincoln, Nebr.

1954. 215. Schmitz M. "Euklids Geometrie und ihre mathematiktheoretische

Grundlegung in der neuplatonischen Philosophie des Proklos", Epistemata, Wuerzburger Wiss. Schr., Wuerzburg 1991.

216. Simon M. "Euclid und die sechs planimetrischen Bücher", Teubner, Leipzig 1901.

217. Simplicius "Corollaries on Place and Time" (Trans J. U. Urmson), Durkworth, London 1992.

218. Simplicius "In Aristotelis physicorum libros comentarius" (Ed. H. Diels), Vol. 2, Berlin 1882 – 1885.

219. Simplicius "On Aristotle Physics 6" (Trans. D. Konstan), London 1989. 220. Smith A. "Porphyrius, Fragmenta", Teubner, Stutgardiae et Lipsiae 1993. 221. Smith E. D. "History of Mathematics", N.Y. 1923 – 25 (+;"6. Dover,

1958). 222. Sodan R. A. "Porfirio" .$28!"."3" $!""-.$!"! /+*$+6%! "!"(A/@! +""-6A/@"!

Milano 1997. 223. Spengel L. "Eudemi, Phodii Peripatetici, Fragmenta quae supersut",

Berolini 1866. 224. 1#"$##-!&'!(*9"22'/."6"='/$%,!#!!#@$%A!8"!2'&'&'!"!*896"!)*.+' 225. 1#"$##-!&'!!#6%2(.'*%">(-."'$$$-4!('/("!*896"!)*+.' 226. 1#"$##-!&'!!&"#22*/$%&"#9$!"&+*("!*896"!)*92' 227. 1#"$##-?!E. !#6%2(.'"6"<"(-("+("-.$!!3&(!!)*+/! 228. Stein H. "Eudoxos and Dedekind: On the ancient Greek theory of ratios

and its retation to modern mathematics", Synthese 84, No.2 1990. 229. Stevens H. F. and S. H. Hall. "Euclid’s Elements", Mc Millan, London

1894. 230. Suter H. "Des Kommentar des Pappus zum X Buche der Eucklides aus der

arabischen Übersetzung des Abü Othman al-Dimashki ins Deutsche übertragen", Abhandlungen zur Geschichte der Naturwissenschaften und der medizin 4, p.p. 9-78, 1922.

231. Suter Heinrich "Das Mathematiken – Verzeichniss im Fihrist", Abhandlungen sur Geschichte der Mathematik, p.p. 1-87, Leipzig 1892.

232. Szabo Arpad "Die Entfaltung der Griechischen Mathematik", Bibl. Institute, Mannheim 1994.

-&<'')!!'!0'3-=*.'* 213

233. Szabo Arpad "*9$-1$. "'/ #22*/$%0/ ($)*+$"$%0/", $+##,!"%- .. !"%A"+*%&, T.E.E. *896" 1973.

234. Taisbak C. M. "A table of half sums and differences. Ancient tricks with numbers", Centaurus 36, No 1 1993.

235. Taisbak C.M. "Coloured quadrangles A. guide to the tenth book of Euclid's Elements", Museum Tusculanum Press, Copenhagen 1982.

236. Taisbak C.M. "Elements of Euclid's data" Edmonto: Academic Printing and Publishing, Apeiron 1992.

237. Taisbak C.M. "Zeuthen and Euclid's Data 86. Algebra or a lemma about intersecting hyperbolas" Centaurus 38, No.2-3, 1996.

238. Tannery P. "La Geometrie Greque", Paris 1887 (+;"6. Olms G., Hidelsheim 1988)

239. Tannery P. "Mémoires Scientifiques" (pub. J-L. Heiberg & H.-G. Zenthen), vol. I Paris 1876, (+;"6. Gambay, J. 1995).

240. Tannery P. "Pour l' histoire des lignes et surfaces courbes dans l' antiquité", Bulletin des Sciences mathématiques, 1883-1884.

241. Tannery P. "Sur l' authenticité des axiomes d' Euclide", Bulletin des Sciences mathématiques, t VIII p.p. 162-175,1884.

242. Tannery P. "Sur la locution (:' $!"6", Revue des études grecques, t. X, p.p. 14-18,1897.

243. Taylor E. A. "Commentary on Plato's Timaeus", Oxford 1928. 244. Taylor T. "The Philosophical and mathematical Commedaries of Proclus

on the First Book of Euclid's Elements", Wissenschafthiche Buchgesellschaft, London 1788-1789 /"A 1791.

245. Thaer C. "Euclid, Die Elemente", Loeb Cl. Lib., Darmstadt 1989. 246. Thaer C. "Euclid, Die Data von Euclid", Springer, Berlin 1962. 247. Thomas – Stanford C. "Early editions of Euclid's Elements" (ed. Alan

Wofsy), Fine Arts, San Francisco 1977. 248. Thomas I. "Selections illustrating the history of Greek Mathematics",

vol.2, Loeb Classical Library, London 1980. 249. Thorup A. "A pre-Euclidean theory of proportions", Arch. Hist Exact.

Sci. 45, No 1, 1992. 250. Todhunter '. "The Elements of Euclid, for the use of Schools and

Colleges", London 1882. 251. Unguru S. "Greek mathematics and mathematical induction" Physis,

Nuova Der. 28, No 2 1991. 252. Unguru S. "On the Need to Rewrite the History of Greek mathematics",

Archive of Hist. Of Exact Scrienses, p.p. 67-113, 1975. 253. Van Der Waerden B. L. "Science Awakening", Noordhoff, Groningen

1954. 254. Van Der Waerden B.L. "Die Pythagoreer" Artemis, Zurich 1979. 255. Van der Waerden, B.L., "Die Postulate und Konstruktionen in der

fruehgrieschen Greometrie" Arch. Hist. Exact Sci. 18, 1978. 256. Van Pesch G.J. "De Procli fontibus", Leiden 1900

214 EY'0&'()!%123'4&'*&

257. Ver Eecke P. "Pappus Alexandrini. La collection mathematique", Bruges, Paris 1933.

258. Ver Eecke P. "Proclus de Lycie, Les Commentares sur le premier livre des Elements d'Euclide", Paris 1948.

259. Vinogradov I. M. "Elements of Number Theory", 5th rev. Ed, Dover, New York 1964.

260. Vitrac B. "La definition V.8 des Elements d' Euclide", Centaurus 38, No.2-3, 1996.

261. Vitruvius. ">(-."$-1$"(%""/$%&3(!(A("*"!)0"!&+6"#*""!/+%%""%6*/-!)*22. 262. Vogt H. "Die Entdeckungeschichte des irrationalen nach Plato usw",

Biblioteca Mathematica vol. 10, 3% #+):%?, %+"*$" 155 /"A 158, 1910. 263. Vogt H. "Die Lebenzeit Euclids", Biblioteca Mathematica 13, p.p. 193-

200, 1913. 264. Von Fritz K. "Grundprobleme der Geschichte der antiken Wissenschaft",

de Gruyter, Berlin 1971. 265. Wehrli F. (her). "Die Schule des Aristoteles" (Texte und Kommentar),

Basel 1953. 266. Weil A. "Number Theory An Approach Through History From

Hammurapi to Legendre", MA: Birkhauser, Boston 1984. 267. Woepcke F. "Essai d'une restitution de travaux Perdus d' Apollonius sur les

quantités irrationnelles" Memoires presentés par divers Savants a l' Academie des sciences, vol. 14, p.p. 658-720, 1856.

268. Woepcke F. "Notice sur des traductions arabes de deux auvrages perdus d' Euclide", Journal asiatique 18, ser.4th, %+"*$+? 233-247, 1851.

269. Woepke F. "Extrait du Fakhrist", Paris 1853 (+;"6. Olms G., Hildesheim 1982).

270. Young G. "The Arabic textnaltradicions of Euclid' elements", History of Science 1984, %+". 147-160.

271. Young J. W.A. "The Theory of Numbers" Ch. 7 in Monographs on Topics of Modern Mathematics to the Elementary Field, (W. A. Young) (Ed.) Dover, p.p. 306-349, New York 1955.

272. Zacharias M. "Das parallelen problem und seine Lösung", Sam Gössen, Leipzig und Berlin 1937.

273. Zajtsev E.A. "The notion of measure in early – medieval geometry", Istor. – Mat. Issled. 35 (1994) (Russian).

274. Zervos P.S. "On the development of Mathematical Intuition, on the genesis of Geometry, further remarks", Tensor 26, 1972.

275. Zeuthen H. G. "Histore des Mathematiques dans l' Antiquitè et le Moyen Age", Paris 1902.

ΕΥκλείδη Γεωμετρία Tomos-3

Documents

Transcript of ΕΥκλείδη Γεωμετρία Tomos-3