Phương pháp học máy với Support vector machine trong nhận dạng chữ viết tay trực tuyến

Luận văn thạc sỹ công nghệ thông tin - Chuyên ngành Khoa học máy tính Đề tài :
Tìm hiểu phương pháp SVM và ứng dụng trong nhận dạng chữ viết tay trực tuyến
(SVM - Support Vector Machine - phương pháp học máy)

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'-UDTU

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( &1 * )* ;- TN&+C= A(RG1(K h3iU = F B α C =& ) &- &* J(

α !(2& 1 2NG@.!* ?+I& → ∞ D =G>/ =G,G0& +.! E+.= E(2! E+>? E )

=&.1+& N+,/ G+ / +^

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T T !n UU

T T !n

T *?9XW N+,E (& L (C! E(S!*&'()! =+^/ ; N+,E+.= = 2 G&4 (ST *?9XW ]+Q' A+ = F

B α &= &9 )1 0 = = α 4 (& L (C! E(S C =&.E(& ]+Q' A+ N+,T +K! LD(AJ !(!L 1D3dbdU-

L@ - )1 L = 1 &= &80 90 A & )* + ( 1B B α )1 0 =

=& )* &1 α &8

(& E > / > !* l+ N+,=+CK Λ G>!* J@> J+2! E( /+! / K +Q = Q+ ?+0! ? C= E (&

L (C! !+,E > TE(S α∈Λ→ U4 +' ! > /+>/ L+>/ N+,* @> +.! ? (! ]+>/ EB! Λ /&Q+

α N+ +K! LJ+, (AJ !(!L 1 h3`ih3bi E+0=F? A+ ? C= 0&C 6 9& &1Λ %

! H ) 9 & &8 B &1 &.(

A !* =AG@,!* @.K ! +C! H+.!*D / > = (SE+\= F Eα L+>/ E ? C= / &>= !

+ +0' ](? ](>= ](? L α = +' E S =A !* Λ = F/ > R+ ! ()& &9&M( 9&

> α ]+Q' A+4 FF E α / > + L+Q!+O!* N+,$ +\/ 3 !()! 1 2L+Q!+O!* /&Q+ > α =

! +2= N+,3 J+,! (& ! +2= N+, : (C& 1 2L+Q!+O!* !+,' R+e!* E+ NG@.!*

NΛ E! E 3 E ! > K &.= & C/ J+,=+CKΛJ+,H+0' E!E 3 E

y(>= 1G.N ()! M&+! * G0+ F E J+, 4 +\= α ? ! α α 3 α

α ∈ Λ α ∈ {−1D3} ?3 4 > = (A = +2' 1 2L+Q!+O!* /&Q+ > α N+,R+e!* J@> 1 2L+Q!+O!* /&Q+ α 8&,!* ! +C! ](>= !+,'D =+ / > = (S* +Q = / F! / '> !* l+ /&Q+E+ NG@.!* NΛ E!E 3 E ! G1+&-

NΛ E!E 3 E &8)* &9 &M( 0 A 0&- )* ! )&C ! ) &2 &- &= +9

! 3 :&O( &8B α α & 0)A ) (Λ

56 - J@> ? ∈ ;#D ? T/ > + ?+I& ! UD α? & : ∈ Λ?

α ? & P: P TK G@!* =AF! EG@,!* = +g!*U

Λ ! " ! # $##$%

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> = (SL(2= N&+C! NΛ E!E 3 E ≤ 4

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!+,' / > ]+>/ 1&+2= N+,TE! E 3 E A&!* RF! /&Q+ E+ NG@.!* !+,' EG@./ * N+,(!=A K' /&Q+ =+CK +,? F E α J@> α ∈ ΛJ+,L'> (C& N+,$Λ

)), ,,((ln)

( 1 2

, , 2

z E N z z z l

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=A !* EB! N'> C =&.@Q?&./ #4#U N+,

=F? EG@./ +,? α J@> α∈Λ ?+,/ > = (SK +)! +./ EG@./ H+0' M&+! 1+>= =A()!

= +,! + N@>K GJ+C' =+CK +,? Dα =A()! N+,? C= =+CK +,? j> &- B& kD L )!*

! +C! R (2= EG@./ B)* > & 1( +2& &8D !(2& EG+ J+,K +)! N@>K c3 = F=+CK +,?

!+,' /&0!* / > = (S* +Q = / EG@./D !*G@./ N+ !(2& EG+ J+,K +)! N@>K 3 = F! > /&0!*

* +Q = / EG@./4 8 E > =+CK +,? !+,' L )!* = (S* +Q M&'(2= EG@./ J+2! E( ! +C! H+.!*4

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?*K 2D 6F2.$2 741061O ,'(1E5: ' 4,S 6

B! N'> (!=A K' / =+ = ()& / &+S! E(S R+Q E+Q? A+e!*

U T U

&' ! ()! EB! N'> !+,' L )!* / =+ R (2= = 2/ E C C =&.N+,! +! +' / +C?4

A !* = G./ !* (C? = 2/ E C C =&.N+,J+2! E( M&+! =A !* JF!(2& = 2/ E C C =&./ +C?

= F/+! K +Q / > N@>! E(SR+QE+Q? B α ≈ α +,N&>/ E > 1 2NG@.!* ?+I&

E!E 3 E N+,N@>!4 8 E > /+! K +Q ]+>/ EB! = ()? = ()& / &+S! E(S1G C =&./&Q+ ;N+,! +! 4 &>/ E > / w/+! ! QJ+,H E > =+CK ?+I& E!E 3 E /&0!* ! Q4

6Z=A()! =+ E+0L'> (C& $Λ ? H N! NΛ E!E 3 E J+,EG@./ * N+,H ) /&Q+

E E

6Z=A()! =+ E+0! > A+e!* Λ =0

∞

→

$ T U

N ? N+,E (& L (C! E(S B α C =&.J(

α 4 +,=+ /&0!* / > = ()? EB! N'> 1+& E+)

= F1G C =&./&Q+ !*&'()! =+^/ ; +8& & +8& )%( )1 &8) &%:)* &- &* /&Q+ /+>/ ?+I&4 +)' N+,? C= ?(C! E( M&+! =A !*4 (2& =+ 1GQH&.!* $Λ+' $Λ& = F/ &>!* K &.= & C/ J+,K +)! R2]+>/ 1&+2= ?+I& J+,H E > E(S= ! = +>!

EG@./ / &>!* N+,R+2= L+Q= 4 ,! Λ = FL )!* K &.= & C/ K +)! R2]+>/ 1&+2=D / w

K &.= & C/ J+,=+CKΛ !()! / > = (S= ! EG@./ Λ H(IH+,!* @!4

+O? 3dVb +K! LJ+, (AJ !(!L 1 E+0/ G>!* ? ! EG@./ = ! / +2= 1+& /E+ NG@.!* NΛ E!E 3 E - +\/ N+,R+e!* +\/ N+,RB* @> +.! R@Q -

EE

NΛT 1D444D U≤

=A !* E > J(2K +Q / wK &.= & C/ J+,? C= = +? 1 2TN+,= +? 1 2E+\/ =AG!* /

=+CKΛD EG@./ * N+,5R B ) J+,1(0EG@./ E( /+CK =A !* K +! 1+&U4 G,= ! / +2=

=A()! / > = (S1&' A+ = ! / +2= /&Q+ Λ

N ? JFΛ =+O!* = ( ?G>/N! N+,?G>/ =+O!* A+2= / +C? T/ +C? @! /+Q U4

Λ

N!

?*\ $,- 7'75 < /7<5'Q72' 61O ,4, +$,B < '$, ' /,0 '(

6ZK +! #4a E+0! > E(2! L+> ! (C? NΛ E! E 3 E !* l+ N+,1 2L+Q!+O!* E+>!

1 2"3 +' 3 / /+>/ K +! =GQ! 3 4 : +> ! (C? 5R B ) /&0!* HG.+ J+,

J (C/ E+>! 1 2/+>/ K +! =GQJ+,EG@./ K +>= R (S& !

G1+&-5R B ) 9& B)1 &1 &8B α α∈Λ &8) )* ;- &* &) )

)-A ).0+; &= +9! 3 B&8 - ( )- A 0+;.0&- )* G! & ! )

&* &9 +; (

&-> = (S/ > ! G0!* H+0' ! 3 ?+,J (C/ E+>! 1 2EG@./ =+2= /+Q K G@!* +>!

L )!* = G./ (C! EG@./4 &' ! ()! / w/+! = ! =+ ? C= H+0' ! 3 ?+,J (C/ E+>!

1 2= G./ (C! EG@./ = FE+0/ > = (SL(2= N&+C! 5R B ) N+,

G.L+>/ ! +& * G0+ NΛ E! E 3 E J+,5R B ) N+,NΛ E! E 3 E K &.= & C/J+,Λ N+I! H+0' E! E 3 E4 ,! 5R B ) = F/ wK &.= & C/ J+,Λ ?+,= ) D !()!E+ NG@.!* 5R B ) EG@./ H&,!* N+,? E+\/ =AG!* / =+CK Λ +& E+)' N+,J H&.J(

0 1

A@QN+ J@> L+> ! (C? ΛTNUD @Q=A()! =+ E+0! > !(2& ]+>/ EB! = F

N ? 4 GJ+C' 5R B ) ]+>/ EB! N+,E (& L (C! /&Q+ 1G C =&./&Q+

!*&'()! =+^/ ; N+,JG,+ ! +! JG,+ L )!* K &.= & C/ J+,K +)! R2]+>/ 1&+2= /&Q+

=+CK ?+I&4

?*] d'$U%&0 '(Q%0$2 741061O ,'(1E5: ' 4,S 6

A !* ?&./ #4# E+0E( /+CK E(2! 1G C =&.E(& = ( ]+>/ 1&+2

+,@Q?&./ #4%D #4[ E+0E( /+CK E(2! E (& L (C! E(S1G C =&.!+,' N+,JG,+ ! +!

JG,+ L )!* K &.= & C/ J+,K +)! R2]+>/ 1&+2= /&Q+ =+CK ?+I&4 ,! N+ J+2! E( N+,/ >

= (S! > *FJ( E+ NG@.!*ε =A !* R (S& = G>/ * @> +.! =A()!u

+K! LE+0E( ]&+2= A+e

!*-41

P+2= E+g!* = G>/T *?9)?W / G>!* = QA+e!* = +' JF=F? α E(S1+ 1 2= G./ α ! Q

TL > = G./ (C!UD =+ / > = (S=F?α E(S= S!* R()! J(2K +Q N+,! QD N&>/ E > / +^/ / +^! N+,

α /&0!* 1(0! Q4 2+.!* = G> + /&Q+ = S!* R()! K +Q EG@./ * N+,5R )

= ! EG@./ M&+ 5R B ) /&Q+ =+CKΛ 2+.!* = G>! +2= N+,1+ 1 2= G./ !* (C?

HB & 4 + * J(2K +Q /&Q+ R+2= E+g!* = G>/ =A()! N+,& )* ( ;- &./&Q+ α

P+2= E+g!* = G>/ !+,' = (S (C! ? C= 1 2= ! / +2=

1+&-3U (2& =+ =F? ? C= +,? ! +C! H+.!* =A !* ? C= =+CK +,? / > jL+Q!+O!*k M&+>N@>! TΛ / - N@>!U = F?+\/ H&,=+ / > EG@./ B α ! Q! G!* K +Q =A+Q* +>N+,5R ) N+ N@>!4

#U (2& =+ =F? +,? ! +C! H+.!* =A !* =+CK +,? / > L+Q!+O!* ! QT +,? =&'(2!

= ! / +g!* +.!U = F5R ) ! Q! G!* 1+ 1 2B α N+ N@>!4

A !* /+Q+ =AG@,!* @.K E(& L )!* R+QE+Q? α ! QH+I! E(2! +,? =F?EG@./ L )!* K +Q N+,+,? ! +C! H+.!* = 2=4

> N+,HG@> * >/ E C5R B ) /&Q+ =+CK +,? ! +C! H+.!*D / ,! HG@> * >/ E C/&Q+ 1 2NG@.!* ?+I& = F-

• (2& N@>!D 5R ) ! Q4 &>/ E > 1+ 1 2= G./ α 1(0*+! R+e!* J@> 1+ 1 2B α 8 E > K G@!* K +>K ; = / @.K L N@>!4

• ,! J@> ! QD B α ! QL )!* R+QE+Q? A+e!* α 1(0! Q=G@!* G>!*4+C' J@> ! QJ+2! E( /+! = G./ (C! N+,N+,? 1+ E(S]+>/ EB! =+CK Λ J+8α 1+/ B α J+,5R ) E !* = @, ! Q4 B α K &.= & C/ J+,? C= +,?

α /&.= (SD / ,! 5R B ) K &.= & C/ J+,= +,! RC=+CKΛ +)' N+,J+2! E( L >

= G./ (C! JF L )!* K +Q N+,? C= R (2! N ()! =&./ ?+,! > N+,? C= R (2! !*&'()!4 (S]GQN'> J+2! E( !+,' +K! LE( A+ K G@!* K +>K & J B E& ) ; 4

?*` $%&'(+$,- + 4316413,U 7Q> 7'7<7V,472' 9

+)' N+,K G@!* K +>K E(S=F? N@, * +Q L 1 2NG@.!* ?+I& N+,! Q4 G@!* K +>K

!+,' =F? α 1+ / B α J+,5R ) E !* = @, ! QE(S1+ 1 2* @> +.! N+,

! Q! +2=4 (SN+,? EG@./ E (& !+,' = F/+! N+,? / 5R B ) N+,E+ NG@.!* / >

= (SE (& L (S! EG@./4 G@!* K +>K !+,' = (2= N+CK ? C= /+2& =A&>/ N !* ! +& !

& )* ( ;- &.

?*a 5D 4U1, ' A5NUE-4$1E5D 4$20 64$2D '(>5:

'> = &'(2= / = 2!* L()=+CK =A&!* / &Q'(2& J+,J (C/ N+,? = (2!+,E(S/ > +,?

! +C! H+.!* = 2= ! +2= ?+,/ wHG.+ =A()! ? C= 1 2* @> +.! ?+I& M&+! 1+>= / > EG@./4 A !*

E > =+CK =A&!* / &Q'(2& J+,K G@!* K +>K ; 4

'> = &'(2= E+0EG+ A+ E (& L (C! E(S!*&'()! =+^/ ; = G./ = (C& M&+Q4 &+

E > E+0EG+ A+ ? C= L+> ! (C? ?@> N+,L+> ! (C? 5R B ) E+\/ =AG!* / ? C=

=+CK +,? T=AG@>/ E+)' = G@,!* 1GQH&.!* 1 2/ (& L )!* * +! N+,? E+\/ =AG!*U4 (S

!*&'()! =+^/ ; E&>!* = F5R B ) /&Q+ =+CK +,? =F? L (2? K +Q G0& +.!4

G@!* K +>K ; +.= E C!* = 2= / wL 1 2NG@.!* ?+I& N@>!4 '> = &'(2= /

= 2!* L()/&0!* EG+ A+ ? C= !*&'()! =+^/ 1&' H (I! ?@> E > N+,!*&'()! =+^/ ; 4G@!* K +>K !+,' N+,? J (C/ !*+' /+QL J@> 1 2NG@.!* ?+I& N+,! Q! G!* J+I! (C&M&+QD =F? A+ L(2= M&+Q/ > = (S/ +2K ! +C! EG@./4

*&'()! =+^/ ; J+,; N+,! G0!* !*&'()! =+^/ N'> = &'(2=4 G@!* K +>KN+,? C= ? )F! /&.= (S= G./ = /+>/ !*&'()! =+^/ !+,'4 (2K = ( 1+& 1(0=AF!R+,' J( K G@!* K +>K 4 G!* =AG@>/ L E J+,/ = (2=D =A !* / G@!* = (2K

= ( T/ G@!* aUD 1(0=AF! R+,' ? C= 1 2N'> = &'(2= J( = +>! = 2 G& M&' +./ K =&'(2!JF! > EG@./ 1GQH&.!* =A !* 4 EG@./ =AF! R+,' =A !* / G@!* %4

P+, = +>! = 2 G& EG@./ K +>= R (S& !

G1+&-R ) &8B &8∈ &8B = &- 0@ 9& U<B ∈ &) ) &8:

-&* ) ( +;8( ;.B )* <B +.0& < )- A & :&O( G

W+.& % ( ≤ ;- ?! 3 B

N * &8&8B )=& &8 &8&1 )= ;- ( &8&- &8B % &9

& <0 = 1 &= &80 90 A &8+ A 9& &8

U)= & ! 3 B &)

T

( ? J@> ? ! 3 B

≥ J@> ? ! 3 B

∇ ∇( &80&.) &8B 9& &8( &

> = (S* +Q = / '> !* l+ F! / /&Q+ E (& L (C! :: !

D

( ? ( ?

)-∇

∇ ! P !∇(! ! P ∇( ! ?

?(C! E( !+,' / G>!* = QA+e!* J(/= A ∇ ! / > = (SEG@./ K +)! = / ? C= )A (

* 6 /&Q+ + J(/= A ∇(! ! J+,∇( ! 4 (& !+,' / w/ > = (S]+Q' A+ L J(/= A ∇ ! = & C/ F! / >K =+.R@Q + J(/= A∇(! ! J+,∇( ! 4

A@QN+ J@> F! J(0D E 2 J@> E (S? !D / > = (S= +2' A+e!* J(/= A c∇ !+e? * G0++ J(/= A ∇(! J+,∇( !()! 1&' A+ ?(C! E( :: EG@./ = +Q?+0! =+.!4 (& E >/ G>!* = QA+e!* ! N+,/G./ = (S& /&Q+ =A !* ? (! 4

,! E 2 J@> E (S? = FJ(/= A c∇ L )!* !+e? * G0+ + J(/= A ∇( J+,∇(()! 1(0L )!* = ! =+ J+, E(S-

X*? "E-4$1E5D 4F2D 7'(,= 1T 1,U74EW

? C= R+, = +>! = 2 G& TR+, = +>! K +>= R (S& E+& / G@!*UD /+>/ ! +,= +>! /

=F? A+ ? C= R+, = +>! = 2 G& L+>/ N ()! M&+! / +\= / (0J@> R+, = +>! E >4 P+, = +>! = G>

! +2= EG@./ * N+,R+, = +>! * 2/ T B& ): BUD R+, = +>! = G> + EG@./ * N+,R+,

= +>! E 2 !*+I& T & ): BU +' / ,! EG@./ * N+,:&8 )&- 0)* (&/#&( & ( &h#ihVi4 8G@> ? C= J+, * +Q= (2= J( = ! N D R+, = +>! * 2/ J+,R+, = +>! E 2 !*+I& / >/&,!* L(2= M&+QT/G./ = (S& /&Q+ R+, = +>! * 2/ R+e!* J@> /G./ E+ /&Q+ R+, = +>! E 2 !*+I&U4

G,!* (C? /&Q+ R+, = +>! E 2 !*+I& 1&' A+ EG@./ !* (C? /&Q+ R+, = +>! * 2/ J+,!*G@./N+ 4

+2! E( E 2 !*+I& A+2= / > / 4 (& L R+, = +>! * 2/ M&+> K G>/ =+.K E(S=F?

!* (C?D !*G@, =+ / &'(S! M&+ * +Q R+, = +>! E 2 !*+I& = FE@! * +Q! @!D J+,=G,E >1&' !*G@./ A+ N+ !* (C? /&Q+ R+, = +>! * 2/4 G@!* K +>K / > H&,!* E(2!

-T !nUT

θ

+>/ EG@./ * N+,/+>/ (C1 2! +)! +*A+!* +!4

L@ - N * &8&- &8B )= &8 &8&1 )= < * > &9:&8 )&- ()* &8

* > &9:&8)&- 0)* (&/ &8:&O( & ( X& &8

$; +2& * &8( 1B 9& :&8 )&- ()* &8 &8( 1B 9& :&8 0)* (&/ < &8 % 1;- & > & 0 = 1Y& Y U 9& :&8)&- ()*

T

+)' N+,= ! / +2= M&+! =A !* JF! > * &>K =+ =F? EG@./ !* (C? /&Q+ R+, = +>! * 2/

= )!* M&+ !* (C? /&Q+ R+, = +>! E 2 !*+I& 1+& L E+0* +Q R+, = +>! E 2 !*+I&4

Y7!* l+ F! / /&Q+ R+, = +>! E 2 !*+I& =A !* ? 2 M&+! (CJ@> R+, = +>! * 2/EG@./ ? ! + M&+ J H&.1+& TR+, = +>! * 2/ E@! * +Q! / w* ? / > ? C= R+2= K G@!*

; 0A+,!* TE !DE U N+,E (S? / wEB! =A()! F! J(04

P+)' * @,/ ≥ D =+ K +Q ]+>/ EB! θ /&Q+ R+, = +>! E 2 !*+I& 5<θ ? !n{ P ( } !()! E(S=F? θ =+ H / &'(S! EG@,!* = +g!* P ( ? E P E!

? α 1+ / α R(> ! +2= ?+,J+I! R+QE+Q? EG@,!* = +g!* !+,' / > * + J@> +C'

θ N+,* +> =ABα L EG@,!* = +g!* = (2K =&'(2! J@> ! G=A !* F! aca4

8'$X9X 7'$$20 ,E-'($e,R,B 742,- ' F2D 7'(,= 1

+ E+0=F? EG@./ θ G>!* J@> ? C= ≥ !+,E >4 (S=F? θ N@>! ! +2= T?&./

= ()& R+, = +>! E 2 !*+I&U =+ ] +' EG@,!* = +g!* 1+ / ! > /+^= =A&./ =&!* =+ E (S? N@>!

! +2=4 ; 0A+,!* E (S? !+,' /&0!* / ! N+,E (S? TE !DE U /&Q+ R+, = +>! * 2/4 θ E+.=EG@./ N@>! ! +2= J@> (C1 2* >/ EG@,!* = +g!* N+,c 4

+C'θ ? E ? !* l+ N+,L(2= M&+Q/&Q+ R+, = +>! E 2 !*+I& R+e!* J@> L(2=M&+QR+, = +>! * 2/4

X*X 7,O 7R,B 742,- ' F2D 7'(,= 1R,[ '(+$%&'(+$,- +(7,O < F,0 2$,B <*

C= K G@!* K +>K * +Q R+, = +>! E 2 !*+I& E@! * +Q! N+,K G@!* K +>K * +Q? E+.+,? T & J ) U4 * +, A+ / ,! / > ? C= 1 2K G@>!* K +>K L+>/ ! GK G@!*

K +>K ?+\= K +g!* /+^=qhVi4

G@!* K +>K * +Q? E+ +,? HG.+ =A()! ! +C! ](>= 1+& E+)' J( E+ +,

?-? C= E (S? D !(2& E+ +,? ∇θT U x $ = FθT U /G./ E+ =+ D !(2& E+.+,? ∇θT U ≠ $ = FH / &'(S! H / = ( J(/= A ∇θT U -

P!? Pλ∇θT U J@> λ ≥ $ = / @.K

= F1(0/+Q = (C! EG@./ * +> =AB/&Q+θT UD N+,? =+O!* H+! * +> =AB/&Q+θT U J+,N+,?

* +Q? E+ +,? ∇θT U4 G>= (2! +,! * +Q? E+ +,? / E(2! L ∇θT U x $ = FN+,!* (C? /&Q+ R+, = +>!4

-T !nUT

θ

&>!* =+ / ,! / > = ()? E (& L (C! ≥ !()! =A !* M&+> =AF! H / &'(S! K +Q/ &> '> E(S=+2= /+QB = +,! K +! /&Q+ P! E(& L )!* +)? TH ∈ BU4 8 E > !(2& / >

x $ J+,∇θT U =G@!* G>!* |$ T1(0N+,? / P! |$ U = F=+ K +Q = +' ∇θT UR+e!* J(/= A ∇θ}T U ! G1+&-

00

+' JFH / &'(S! = ( ∇θT U = F1(0H / &'(S! = ( ∇θ}T U J+,E (& L (C!

$%&'(K

9

A !* / G@!* 3 E+0=AF! R+,' !*&'()! =+^/ ; 4 *&'()! =+^/ ; / 2*+^!*

= 2 = (S& E !* = @, + E+ NG@.!* HB & J+,5R R) E(S/ > EG@.// +\! =A()! R(> ! +2= /&Q+ 1+ 1 2= G./4 G@!* K +>K ! +C! H+.!* N+,? C= ? )F!

= G./ = !*&'()! =+^/ N'> = &'(2= !+,'4 h`ihbih3`ih3bi4

K*) 432'(43%&B '($&0 +4, +<,= 1+$,: ' $2,0 6$41E5D ' 4H'$F%&0 6

+)' N+,=AG@,!* @.K E@! * +Q! ! +2=4 A !* =AG@,!* @.K !+,'D =+CK ?+I& N+,? C= =+CK)> A 0+; &% & * 6 :&O( B)1 % &'( +, E =F? 1 ()&

K +g!* !+,'4

Q ! ! & 0)& &* 1 5J

+Q1GQ=+CK ?+I& / > EG@./ * ? K +! =GQN+,

8'$K9) 75: 1+$,; '(+$,: ' 6$7,$,74, +<,= 1

+Q1GQK G@!* =AF! 1 ()& K +g!* /+! =F? N+," P :x $ =A !* E > " N+,K +>KJ(/= A /&Q+ 1 ()& K +g!* "∈ 4 + / > + R+2= K G@!* =AF! 1+&-

" P : c3 J@> x c3

" P :≥ ~3 J@> x ~3L(2= @.K + R+2= K G@!* =AF! =A()!-

N&>/ E > ! G0!* 1&KK A= J(/= A = Q+ ?+0! K G@!* =AF! " ~:x c3 = F!+e?

=A()! 1 ()& K +g!* 3D K G@!* =AF! " ~: x 3 = F!+e? =A()! 1 ()& K +g!* #4

: +Q!* /+>/ / > H+2& 3=G,* 2/ = + E CE(2! 3N+,- 1 = 1T −:Uz "

: +Q!* /+>/ / > H+2& #=G,* 2/ = + E CE(2! #N+,- 2 = 1T −− :Uz "

&' A+ L +Q!* /+>/ K +)! +./ * G0+ 3J+,#N+,- = 1− 2 =2z "

8 E > E(S/ > N@>! ! +2= = F||"|| K +Q ! Q! +2= +' ! > /+>/ L+>/ K +Q E =F?/G./ = (S& /&Q+ 2

"

L +Q !* /+>/ K +) ! + /

2 / = + E C

# 3

: 4"4

P+ ]&!* M&+! /+>/ E (S? ?+I& N+,/+>/ F! /+& R+>! L! 3zD4 ()& K +g!*

=F? EG@./ K +Q L )!* EG@./ /+^= /+>/ F! /+& T! GJ+C' = F?@> / > ≥ #zDU4 (2&

L )!* / > A+,!* R& C/ !+,' = F=+CK 1 ()& K +g!* R+ * ? =+2= /+Q/+>/ 1 ()& K +g!* =A !*

L )!* * +! 4 &' ! ()! J@> A+,!* R& C/ !+,'D )* +;.( &- % &'( :@( ;- &

&.4 > !* l+ N+,5R B ) /&Q+ =+CK 1 ()& K +g!* * @> +.! 1(0! Q@! 5R

8 / > = ! N !()! E(S* +Q R+, = +>! =A()! =+ / > = (S/ &'(S! M&+ * +Q R+, = +>!

E 2 !*+I& =G@!* G>!* T/ G@!* a ?&./ #U4 P+, = +>! E 2 !*+I& 1(0N+,

T

1

2

12

T+.& % ≥ ?3D #D qD

0

=

+Q R+, = +>! !+,' =+ =F? EG@./ J+,=G,E > = ! EG@./ " = ( / )!* = G>/T *K9?W4(S= ! :D J+C! H&.!* E (& L (C! :: / R+, = +>! * 2/D =+ / > EG@./-

♦ , $ - N&>/ !+,' T" P :Uc 3x $ 1&' A+ !+e? =A()! 1 ()& K +g!* R ()! 3+' #4 EG@./ * N+,1&KK A= J(/= A J+,= +? * + J+,J (C/ = ! o4G@,!* = F1 2NG@.!* 1&KK A= J(/= A ! Q@! ! (& 1 J@> 1 2NG@.!* ?+I&4

8 E > E(S= ! R/ w/+! / ! ? C= ?+,/ > •$D N&>/ E

>-T" P :U " 3 x $ !()! : x T3z U " ? " T *K9KW

+C' =+ E+0= ! EG@./ " J+,:!()! ]+>/ EB! EG@./ 1 ()& K +g!* K +)! +./ 4

Q ! & 0)& &1 &.(

P+)' * @,* +Q1GQ/ > ? C= ?+I& HG0N (C& /+! ! +C! H+.!* 7 !+,E >D = FK +)! N@>K'{/&Q+ ]{T c3 +' 3U EG@./ ]+>/ EB! = )!* M&+ / )!* = G>/-

=+

=

TU

{T{U

T

&0!* * 2!* ! GK +! =AG@>/D !(2& /G./ = (S& ||"|| = F1(0N+,? / 5R B )

! QJ+,=G,E > 5R ) ! Q4 8 E > E(S= G./ = !*&'()! =+^/ ; T/G./ = (S&/&,!* N&>/ HB & J+,5R ) UD =+ / > = (S/G./ = (S& R (S& = G>/ 1+&-

A !* E > R N+,? C= +e!* 1 2HG@!* =G./ !4 (C/ / ! R = (S (C! !*&'()!

=+^/ ; 4 (2& / ! R N@>! 1(0N+,? / HB & ! Qc H =+CK =A&!* /G./ = (S&

ξ4 ,! !(2& R ! Q= FN+,? / 5R ) ! Qc H /G./ = (S& "4 T +>/ !+,'EG@./ * N+,c H HG.+ =A()! * +> =AB/&Q+ L +Q!* /+>/ N I D !* +, A+ / ,! ? C=

K G@!* +>! L+>/ N+,= 2 = (S& 1 2E (S? N I D EG@./ * N+,νc h3ViU4

5;- T" P :U c3 Pξ ≥ $ ?3DqD

ξ ≥ $P+, = +>! E 2 !*+I& /&Q+ R+, = +>! =A()! 1(0N+,-

DDT

ξ

ξUDD

&>/ E > * +> =AB/G./ = (S& /&Q+ # N+,

-UDT

θ