Nalejvárny

Příloha A
Nalejvárny
A.1 Nalejvárna z teorie míry
•
!"#$%&'( )*+,$+*
(X, X )'
X ⊆ P(X) ≡ {Y ; Y ⊆ X} ! σ
! "#$ %&!
("!
-.&/%0*.
4*567 ! ),45!"7 1&1 *$".1)(89
X 6= ∅
! )!*+,-"), .)$/%)0 0
! 1&2 *$".)$/%)' &$/ -)031' /!
X ∈ X'
:%%; A ∈ X ⇒ X \ A ∈ X '
S
:%%%; An ∈ X ' n = 1, 2, . . . ⇒
n An ∈ X <
=%>$#$5)? *+@)%( σ A05B!>!+ )0 X ! σ A05B!>+0 )0 X < C*!&%,5)D' ∀ B ⊆ P(X)
!E%4F7 ! )! .!)G1 σ A05B!>+0 )0 X $>4027 1&1 B ' (F!+, 4! )0-?#, σ A05B!>+0
B 0 $-)037 ! σ(B)<
• X
! H7)(&! f : X → [−∞, +∞] F0($#,' /!
:%;
/%'%*+1.'2
-3!"#$%&'2 45'67%
∀a ∈ R
8+59#' 3!"#$%&'(7: )*+,$+*;
f−1 [−∞, a] ∈ X .
Q (Xi , Xi )' i ∈ N ($)!3)? 484FK. .DL%F!5)?&2
*+$4F$+@< M$F$. *$5$/1.! X ≡
i∈N Xi 0 X ≡ σ A05B!>+0 B!)!+$#0), .DL%F!5A
Q
)?.% $>"K5)1(8
i∈N Ai ' Ai ∈ Xi < C$73%)$#$7 σ A05B!>+7 X >7"7 $-)03$#0F
Q
48.>$5!.
i∈N Xi <
•
)0 (X, X ) ! H7)(&! µ : X → [0, +∞] 4*567 1&1
:%; µ(∅) = 0'
:%%; An ∈ X ' n = 1, 2, . . . *$ "#$7 "%4 7)(F)1
•
< I7J
<%=2)+*'2 3>*.
µ
∞
[
n=1
An =
∞
X
µ(An ) .
n=1
M$".1)(0 :%%; 4! )0-?#, 4*$3!F), 0"%F%#%F0< N!-,*$+), .1+0
O
O
6+'%9'2
σ -6+'%9'2
S
'
X=
n
µ(X) < ∞'
!4F5%/! ∃ An ∈ X , n = 1, 2, . . .
!
!4F5%/!
'
An '
F0($#K' /! *50F1
µ(An ) < ∞
0
240
A Nalejvárny
!"#$% &$&'(&)*(+! "#$%&'(# µ(X) = 1)
Příklady.
1. *&#+#,%-.,/+ 01/2&34#+ "# "!,*-.*,/01 -+!" ,3 25,#6,7 ,#0.-84,7 +,59
(',: X ) ; %5+%5 4'$2.7%,/+ 01/034: $# '+0&'<'%,: 01#4052&-4- X =
P(X)! %#4= σ 93&>#?.5@ "# 05%#,6,/ +,5(',3 X ) A.'%+#%'<2- +/.3 032
23(47 054+,5(',: A ⊆ X 01'134/ 056#% "#"/<B 0.C2DE ν(A) = |A|)
2. F',G+ 8-2&34,/+ 01/2&34#+ "# 2.$(&!&3-%!(1 4.'.)56.&#" -+!" ) ; %5+%5
$05"'%7+ 01/034: "# X = R 3 X %8C) '&!.7&#)01 σ 8"75.'!"! %") σ 93&>#?.3
>#,#.5C3,- $=$%7+#+ 05&5@83C1#,G<B ',%#.C3&DE X = σ({(a, b] ; a < b}))
H5%5+ ,3 X #I'$%@"# "#4',- ,#8-05.,- +/.3 λ %325C-! (# +/.3 ',%#.C3&@
"# "#B5 47&23E ∀ a < b λ((a, b]) = b − a) J3%5 +/.3 $# ,38GC- K#?#$>@#9
5C3)
• 9&#(&)* )0&!& #:6$. C8B&#4#+ 2 ,#8-05.,7 +/1#) L@M f, g : X → R N@,2<#!
5?: X 9+:1'%#&,7! 3 µ +/.3 ,3 (X, X )) H32 4#O,@"#+#
f = g µ9$25.5 CP@4#
⇔
µ( {x ∈ X ; f (x) 6= g(x)} ) = 0 .
H1' 8,36#,/ $# 63$%5 05@(/C- 82.3%23E µ9$)C)
• H58,3+#,#"+#! (# C 4'$2.7%,/+ 01/034:! %") 24=( X "# 25,#6,- +,5(',3! "#
.5C,5$% N@,2</ f 3 g $25.5 CP@4# C8B&#4#+ 2 µ #2C'C3&#,%,/ 05(343C2@! (#
N@,2<# f 3 g $# .5C,3"/ 3&#$05Q ,3 ,5$'6' +/.= µ! %#4= ,3 {x ∈ X; µ({x}) > 0})
J5 3&# "'( ,#,/ 0.3C43 5?#<,:) J/+ +-+ ,3 +=$&'! (# .5C,5$% $25.5 CP@4# ,#&8#
C(4= <B3.32%#.'85C3% "325 .5C,5$% N@,2</ ,3 ,:"327 054+,5(',: R+,5(',= X S)
F',G+' $&5C=! +,5(',3 %8C) 0&,7 +/.=! ,3 ,/( $# N@,2<# f 3 g +3"/ .5C,3% $#
+D(# $ 4C5"'</ N@,2</ f 3 g +:,'% 3 ,#+@$/ #I'$%5C3% (-4,- T@,'C#.8-&,/U
+,5(',3 "325 C 4'$2.7%,/+ 01/034:) F325 01/2&34 ?=<B @C#4& "#4,5.58+:.,5@
K#?#$>@#5C@ +/.@) L@M f N@,2<# ,3 R '4#,%'<2= .5C,- ,@&# f ≡ 0! 3 0.5
23(47 a ∈ R! ?@M ga ',4'2-%5.5C- N@,2<# $',>&#%5,@ {a}E ga (x) = 1 0.5 x = a!
ga (x) = 0 0.5 x 6= a) H32 0.5 23(47 a ∈ R "# f = ga λ9$25.5 CP@4#! 3&#
+,5(',3 ?54D x 24# f (x) = ga (x) 0.5 CP#<B,3 a "# 0.-84,-)
• 4.'.)56.;# ,(*.5!17) L@M (X, X ) +:1'%#&,G 0.5$%5. 3 µ ,#8-05.,- +/.3 ,3
(X, X )) V3(47 X 9+:1'%#&,7 N@,2<' f : X → [0, +∞] 3 23(47 +,5(',: A ∈ X
25,$%.@2<#! 050$3,- ,301/2&34 C @6#?,'<' RW@4', XYYZS!
01'134/ 6/$&5
Z
A
f (x) 4µ(x) ∈ [0, +∞] ,
2%#.7 $# ,38GC- ,(*.5!17 8 N@,2<# f 01#$ +,5(',@ A 054&# +/.=
µ) [,%#>.-&
R
85?#<Q@"#
05"#+ C-(#,7B5 $5@6%@E 0.5 X 25,#6,5@ 0&3%/ A f (x) 4µ(x) =
P
x∈A f (x) · µ({x}))
• <(*.5!&#"*.7(1 =6(0/. "# X 9+:1'%#&,- N@,2<# f : X → R %325C-! (#
Z
|f (x)| 4µ(x) < ∞ .
X
241
A.1 Nalejvárna z teorie míry
R
R
!"#$%&! '()*! +!,!- * %-#!./0$1 X f + (x) ,µ(x)2 X f − (x) ,µ(x) +! 3(-!4-52
'63 "! 7)-38! f -6*590 !"#$%$&'()*+!"'(,&- 6 +!+:; %-#!./0$!; +! 95/6*
Z
Z
Z
+
f (x) ,µ(x) ≡
f (x) ,µ(x) −
f − (x) ,µ(x) .
X
X
X
</( f, g 396*%=%-#!./(96#!$->
f = g µ="?9?
⇔
∀A ∈ X
Z
f (x) ,µ(x) =
A
Z
g(x) ,µ(x) .
A
• ./0+,1'&2 03+4$'+0'5 @!8AB (X, X ) +! ;CD%#!$-5 '/("#(/2 µ, ν -!*0'(/-> ;:/E
-6 (X, X )? F!3-!;!2 &! ν +! 6G"($)#-C "'(+%#0 914% µ2 *0'%" ν ≪ µ2 +!"#$%&!
∀A ∈ X
µ(A) = 0 ⇒ ν(A) = 0 .
Fakt (Radonova-Nikodymova věta). H)I (X, X ) ;CD%#!$-5 '/("#(/2 µ σ =3(-!4-0
;:/6 -6 (X, X )2 ν 3(-!4-0 ;:/6 -6 (X, X ) 6 ν ≪ µ? <(#(; !J%"#)+! K-!*0=
'(/-0L X =;CD%#!$-0 7)-38! f -6*596-0 6"7+&+!+1%8$ +79:+!+1 7(*$!";2 ν
9*A$!,!; 3 µ2 #63(902 &!
Z
f (x) ,µ(x) .
∀ A ∈ X ν(A) =
A
M6#( 7)-38! +! )/4!-6 +!,-(*-64-C 9 /0;8% /(9-("#% µ="3(/( 9N),!? </(#( 36&=
,() +!,-(#$%9() 7)-38% "'$O)+:8: 95N! *;:-C-() /(9-("# G),) -6*596# !(*#2
ν
6"7+&+!9%8$ +79:+!9 7(*$!";( P "#6-,6/,-C "! (*-64)+! "E;G($!; µ
4% ,ν/,µ ?
Q$#!/-6#%9-: -0*!9 +! <10'+'" ν 9*A$!,!; 3 K,(;%-)+:8:L ;:D! µ?
• =(&0(&+!" &(*+!&+0'? R)-38! φ : I → R K3,! I ⊆ R +! %-#!/96$L +! +&!(>&2
'(3), ∀ α ∈ [0, 1]2 ∀ x, y ∈ I '$6#:
φ(α · x + (1 − α) · y) ≤ α · φ(x) + (1 − α) · φ(y) .
<(3),2 '/( 36&,> x 6= y 6 α ∈ (0, 1)2 +! -!/(9-("# ("#/02 '63 φ +! *9#(
+&!(>&2? </( '/69,C'(,(G-("#-: ;:/) µ -6 ;CD%#!$->; '/("#(/) (X, X )2
%-#!./(96#!$-() X =;CD%#!$-() 7)-38% f : X → I 6 95N! *;:-C-() 3(-9!J-:
7)-38% φ '63 =(&0(&+!" &(*+!&+0' D:30
Z
Z
φ( f (x) ,µ(x)) ≤
φ(f (x)) ,µ(x) .
X
X
S 'D:'6,C /E*! 3(-9!J-: 7)-38! φ /(9-("# -6"#090 '/09C #!A,E2 3,E& !J%"#)+!
3(-"#6-#6 k ∈ I 2 &! f (x) = k '/( µ="?9? x ∈ X ?
• ?+1@$& σ % +&(@&A;< :B*? H)I (Xi , Xi )2 i ∈ N ;CD%#!$-> '/("#(/E 6 µi 2 i ∈ N
+!,-6
σ =3(-!4-> ;:/E -6 (Xi , Xi )? <63 !J%"#)+! '/09C
Q
Q -!*0'(/-0 ;:/6 µ -6
"()4%-) ;CD%#!$-58A '/("#(/1 (X, X ) = ( i∈N Xi , i∈N Xi )2 #63(902 &! -6
;CD%#!$-58A (G,>$-:8:8A +! )/4!-6 -0"$!,)+:8: A(,-(#()T
Y
Y
µ(
Ai ) =
µi (Ai ) '/( Ai ∈ Xi , i ∈ N .
i∈N
i∈N
242
A Nalejvárny
!"# $%&! N
µ '( &#)*+, σ -.#*(/*01 *!23)0 4( !"#$%&' $+& µi ! #2*!/5'(
46$7#8($ i∈N µi 9 : ;<%;!=+ 4#5/>*5 =)#5 $+& µ ! ν ;%?($( µ ⊗ ν 9
@<%.8!=($ '( ()*&+!,'-+%. /&0& 1"&!(2 ')+2 9 : "#$"# ;<%;!=+ '(1
;&# .!,=A i ∈ N 1 Xi = R1 Xi 7#&(8#)4.0 σ-!8B(7&! *! R ! µi = λ '(=*#&#2$+&*0 C(7(4B5(#)! $%&!9
Příklad.
D!&"A24.3 4#5/>* i∈N Xi ) 5)(=(*A$ ;<%;!=+ Xi = R 75=5 #2*!/#)!" 46$7#8($ RN 1 ."(&3$,"# 2*!/(*%$ EFE> 2=G&!2*>"1 ,( $*#,>*5 ;&#$+**3EF N !;&>#&*+ *(;#)!,5'($( 2! 54;#<0=!*#59 H!#;!.1 ) ;<%;!=+1 ,(
'( I/(8*A $%" .#*(/*#5 >*=(J#)#5 $*#,>*5 N 8>*(0&*+ 54;#<0=!*#51 "'9 .=6,
'> )8!4"*+ >*"(&;&("5'($( '!.# I4(. E(8#/%4(8*A ?.086 N = [n] ≡ {1, 2, . . . , n}1
n ≥ 11 !*(7# ;&# "# 75=#5 '>*A =G)#=61 ;!. *!$%4"# 46$7#85 RN 75=5
;#5,%)!" 7+,*+'?% 2*!/(*% ;&# n-&#2$+*3 &(08*3 (5.8>=#)4.3 ;&#4"#& Rn 9
Poznámka.
Q
A.2 Nalejvárna ze svazů
K4;#<0=!*0 $*#,>*!1 *(7# 3! &4 (L, )1 E#, '( 2.&!".! 2 !*B8>E.AF# 32+4$2556
!+7&+&7 &41 '( *(;&02=*0 $*#,>*! L )67!)(*0 #. 4&#%8' " 3!9.7.%)' 1 "#
'(4" 7>*0&*% &(8!E% 1 ."(&0 '(
L>M +&:&;$(%) N
∀ x ∈ L x x1
L>>M 2%4$ 6'&4+$*<. N ∀ x, y ∈ L x y, y x ⇒ x = y 1
L>>>M 4+2% $4$(%) N
∀ x, y, z ∈ L x y, y z ⇒ x z 9
K4;#<0=0*% *!2)($( 4!4.5%) 1 '(4"8>,( *!)%E ∀ x, y ∈ L 75O x y *(7# y x9
P6$7#8 x ≺ y 2*!/% x y ! x 6= y9
• @#.5= x ≺ y ! *((J>4"5'( z ∈ L "!.#)A1 ,( x ≺ z ! z ≺ y 1 ;!. 4( x *!23)0
7!5%) !" &7 y 1 &(4;(.">)( y =!+%) !" &7 x9
• D#*(/*3 ;#4(" 4( 2#7&!25'( ;#$#E% >2 &!(2 7$21+2'"9 : Q!44(#)+ =>!B&!$5
'( .!,=3 ;&)(. .#*(/*AF# ;#4("5 2#7&!2(* .#8(/.($ L/> #)08($M9 R(-8> x F#&*%$ 4#54(=($ y1 ;!. 4( 5=+80 4;#'*>E( x ! y1 ;<>/($, x '( ;!. 5$%4"+*# *!=
y9
• R(4"8>,( M ⊆ L1 ;!. ;&)(. x ∈ M '( '$%$'.5%) 3+(&< M )2F8(=($ . 1
'(4"8>,( *((J>4"5'( z ∈ M "!.#)A1 ,( z ≺ x9 S*!8#B>E.61 y ∈ M '( '2;$'.5%)
3+(&< M )2F8(=($ . 1 ;#.5= *((J>4"5'( z ∈ M 4;8T5'%E% z ≻ x9
• ?"3+&'"' $*#,>*6 M ⊆ L ) ;#4("5 (L, )1 2*!/(*A sup M 1 ! &#)*+, *!23)!*A %&@'&%A) =!+%) ,.(!+2 M 1 '( ;&)(. y ∈ L "!.#)31 ,(
L>M z y ;&# .!,=A z ∈ M 1 !)?!.
L>>M y y′ ;&# .!,=A y′ ∈ L 4;8T5'%E% z y′ ;&# z ∈ M 9
@#.5= 45;&($5$ (J>4"5'(1 '( '(=*#2*!/*+ 5&/(*#9
?3!@&%) ;&).G x, y ∈ L1 #2*!/#)!*A x ∨ y '( 45;&($5$ $*#,>*6 {x} ∪ {y}9
• S*!8#B>E.61 $%B'"' $*#,>*6 M ⊆ L1 2*!/(*A inf M 1 ! *!23)!*A "!.A %&@C
(-4A) 7!5%) ,.(!+2 M '( ;&)(. x ∈ L "!.#)31 ,(
L>M x z ;&# .!,=A z ∈ M 1 !)?!.
•
243
A.2 Nalejvárna ze svazů
!!"
•
x′ x
#$% &'()*
x′ ∈ L
+#,-./010
x′ z
z ∈ M2
#$%
!"#$% #$3&4 x, y ∈ L5 %67'8%3'79 x ∧ y5 /: !7;<.< <7%(!7= {x} ∪ {y}2
&'()(*+ '(,(*-. /: #%+:> (L, ) >'&%395 (: #$% &'()* x, y ∈ L :?!+>./: +.#$:@
x∨y
L2
• &*-. /: #%+:> (L, ) >'&%395 (: #$% &'()* x, y ∈ L :?!+>./: /'& +.#$:<.< x∨y
>'& !7;<.< x ∧ y 3 L2 A3'6 /: /0#1!02310*45 /:+>,!(: #$% &'()* x, y, z ∈ L
<.<
3
x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) a x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) .
Příklad. B=#!1&9 #C0&,') )!+>$!D.>!370E% +3'6. /:
7:#$F6)7* <7%(!7=
N5
>/2 +%.D%$
&%7:87*" #$47!&= ' +/:)7%1:702
•
9',4+ #*-.
/: #%+:>
<.< ! !7;<.< 3
L2
(L, )
(%!36 '(/74(804
R ⊆ P(N ) ≡ {Y ; Y ⊆ N }
>'&%395 (: &'()F #%)<7%(!7'
&%7:87*
.6'3C:79 7'
M ⊆L
<F +.#$:@
Poznámka. G$% %3HC:705 (: #%+:> L /: I#,79 +3'6 +>'80 %3HC!>5 (: &'()F M ⊆ L
<F !7;<.<
7:D% ).F,7H5 (: &'()F #%)<7%(!7'
L
<F +.#$:<.<"2 J'()9
&%7:879 +3'6 /: #C0&,'):< I#,7*E% +3'6.2
•
:3,(*+7 '!*%$7
<7%(!7=
•
•
•
x0 z
3 L2
>'&%395 (:
I#,7*E% +3'6.
#$% &'()*
<$/4(1%(*+7 '!*%$7
#$% &'()*
G$3:&
x
z ∈ L2
G$3:&
x
/:
z ∈ L2
+: $%6.<0
I#,7*E% +3'6. /:
1$,4+ "5 /:+>,!(:
L
+: $%6.<0
4$)7$4;5 '!*$%
L5
>/2
x0 ∈ L
K:70 >% 7!1 /!7*E% 7:( +.#$:<.< #$F6)7*
4$)*=1;5 '!*$%
∨>4$!(.,(801$,4+
L5
>/2
y1 ∈ L
D.). C0&'>
>'&%3*5 (:
z y1
#3'!$737 4$!(.,(80>
x 6= sup {z ∈ L ; z ≺ x } .
∧>4$!(.,(801$,4+
>% /:+>
04?737 4$!(.,(801$,4+ "5 #%&.)
x 6= inf {z ∈ L ; x ≺ z } .
Poznámka. G$3:& &%7:87*E% +3'6. /: ∨@7:$%6,%(!>:,79 #$F3H >:E)=5 &)=( <F
#$F3H /:)7%E% )%,70E% +%.+:)'2 L7',%M!1&=5 #$3:& /:
>:E)=5 &)=( <F #$F3H /:)7%E% E%$70E% +%.+:)'2
•
∧@7:$%6,%(!>:,79
#$F3H
∨@7:$%6,%(!>:,791E #$3&4 3 &%7:87*< +3'6. (L, ) /: 7:/<:7O0 <7%@
(!7' M ⊆ L &>:$F /: #3'!$737>63#1@ 5 1%( 67'805 (: #$% &'()* x ∈ L :?!+>./:
M ′ ⊆ M >'&%3*5 (: x = sup M ′ 2
• L7',%M!1&=5 <7%(!7' ∧@7:$%6,%(!>:,791E #$3&4 3 L /: 7:/<:7O0 <7%(!7' M ⊆
L &>:$F /: 04?737>63#1@ 5 >/2 #$% &'()* y ∈ L :?!+>./: M ′ ⊆ M +#,-./010
y = inf M ′ 2
• A>'7)'$)70 #C0&,') ∨@7:$%6,%(!>:,7*E% #$3&. I#,7*E% +3'6. L /: /:E% -1(7 5
1%( /: E%$70 +%.+:) 7.,%3*E% #$3&. L2
• A(-1(7$7 L +: $%6.<0 )%,70 +%.+:) /:)7%>&%3*E% #$3&. L2
N7%(!7'
244
A Nalejvárny
!"#$ %&'( L %) #'($&* !"#$%&' + !,-./ 0#,12#' 3)4, '5,06 3) %.!7)0.08
4.%5* & L9 )-&2&'")#5#:+ !,-./ 3)/2#$02 ∨8#)7,(",125)"#$02 !7&-; 3%,. '5,0;<
• =&'( 3) &" !"#$(!$%&' + !,-./ 0#,12#' -,'5,06 3) 2#>0.084.%5* & L+ 53<
3)/2#? ∧87,(",125)"#? !7&-; 3%,. -,'5,0;<
• @.A (L1 , 1 )+ (L2 , 2 ) B!"#? %&'(;< )*!$$("#"+,(#-# .(/"0121*3 5:C45, !,8
%)56 %) D./) 7,(.0:5 (,D7'()#E φ : L1 → L2 5'-,&?+ 1)
•
∀ x, y ∈ L1
F,D7'()#E %) #'(&) (4
x 1 y ⇔ φ(y) 2 φ(x) .
5"4'# *!$$("#"+,(#-# +
∀ M ⊆ L1
3)%5"21) !"'5E
φ(sup M ) = inf { φ(y) ; z ∈ M }
' φ(inf M ) = sup { φ(y) ; z ∈ M } .
G&2/)#5#:+ -'1/$ '#522%,0,7>%0.% .%!,H*/*#E 3) 2 %&'(,&$ '#522%,0,7>%0.%<
A.3 Nalejvárna o maticích
@.A M, N #)!7*(/#? -,#)I#? 0#,12#;< J)*"#,. M × N 6# !$%3 %) D./) 7,8
(.0:5 7)*"#* K.#-C) #' M × N < F*!2% Σ = (σij )i∈M, j∈N <
=;0D," ΣA·B D./) ,(#'I,&'5 A×B 8!,/0'52C2 0'52C) Σ+ -/) A ⊆ M + B ⊆ N <
• =;0)572C-,. N × N 80'52C2 #'(&)0) /"($!$4*7 (-#$2-,*$!*3+ !,-./
x⊤ · Σ · x ≥ 0 !7, -'1/? x ∈ RN ,
7)%!)-52&) /"($!$4*7 2-,*$!*3+ !,-./
x⊤ · Σ · x > 0 !7, -'1/? 6= x ∈ RN .
L'52C) 3) !,%252&#: /)>#25#E+ !7*&: -/;1 3) 7)M."*7#E ' !,%252&#: %)02/)>#25#E<
N#&)7(#E 0'52C) - !,%252&#: /)>#25#E 3) 7,&#:1 !,%252&#: /)>#25#E<
• 8"9-%*7*". $*4-+53 N ×N 80'52C) Σ %) D./) 7,(.0:5 -'1/* 0'52C) Σ− 5'-,&*+
1) Σ · Σ− · Σ = Σ< O:-5)HE '.5,H2+ #'!HE-"'/ PQ#/:" RSTUV+ !,.1E&'3E #'0E%5,
5,4, 5)70E# /(-.2"$*4-+5*3 # !$%-< W'-,&* 0'52C) &1/; )X2%5.3)+ '") #)#E
.7I)#' 3)/#,(#'I#:< Y !HE!'/:+ 1) Σ 3) 7)M."*7#E+ !'- 3) 3)3E (,D)C#:#* 2#&)7()
3)/#,(#'I#: .7I)#' ' %4,/.3) %) % 2#&)7(#E 0'52CE Σ−1 <
• Z)%5"21) Σ 3) 7)'"#* N × N 0'52C)9 A, C ⊆ N 3%,. /2%3.#-5#E+ !'- :%;.+<4
2"/=*7& !,/0'52C) ΣC·C P& ΣAC·AC V 3) A × A80'52C) /)>#,&'#* &(5'4)0
•
ΣA|C = ΣA·A − ΣA·C · (ΣC·C )− · ΣC·A ,
!H2I)01 !"'5E -,#&)#C) ΣA|∅ ≡ ΣA·A < [7, !,%252&#: %)02/)>#25#E 0'52C) 5'5,
0'52C) #)(*&2%E #' &,"D: !%)./,2#&)7(#E 0'52C) (ΣC·C )− <
Fakt. Z)%5"21) 3) ΣC·C 7)M."'7#E+ !'- 3) ΣA|C 7)M."*7#E+ !7*&: -/;1 3) 7)M.8
"*7#E 0'52C) ΣAC·AC + ' & 5,05, !HE!'/: !"'5E
(ΣA|C )−1 = ((ΣAC·AC )−1 )A·A .
245
A.4 Nalejvárna o konvexních množinách
!"#$ %& Σ '!()*)+,- $&.,)*,/0 1&('&"*)+& '!()*)+,- (&2)$&.,)*,/0 '3" *45
%& '!()*)+,- $&.,)*,/0 1&('&"*)+& '!()*)+,- (&2)$&.,)*,/6 7!,&8,-0 '93*/
,3(9&$#%/:/ !"#$" %!&#'"%"("%); %(!#<9) A, B, C ⊆ N '! $+!# $)(%#,"*,/0 ΣBC·BC
) ΣC·C 1&=#9>1,/0 '3" '93*/
ΣA|C
ΣA|BC = (ΣAB|C )A|B .
A.4 Nalejvárna o konvexních množinách
•
•
? *!2*! !$$/9& %& @>2-1,- !@,38!+>, A1&>9,BC &#"9)$!+("B '1!(*!1 (D2E!9&2
Rn 0 "$& n ≥ 1 3 ,)"!9)+ RN '1! 2,!5),# '1!2-,,B:F N 0 :!5 %& ,&%83(*-%G/
!@,38&,/ &#"9)$!+("4F! '1!(*!1# + *-:F*! ("1)'*&:F A+)@ '!@,>2"3 ,3 (*13,HIH 8) J K6LC6 M! '1!*!0 ,/5& #+&$&,> N3"*3 %(!# N3"*):"D A35 ,3 %&$,# +B%)2"#C
+D#5)*3 + "!,*&O*# ('&:)>9,/F! '1!(*!1# RP(N ) 0 "$& P(N ) ≡ {A; A ⊆ N } %&
(D(*42 '!$2,!5), ,&'1>@$,4 "!,&8,4 2,!5),D '1!2-,,B:F N A+)@ J I6P6HC6
QR"3@D +-*G),D @$& #+&$&,B:F *+1@&,/ 9@& ,394@* ,3'S/"93$ + ",)@& AT1U,$(*&$
VWXPC6
Y,!5),3 K ⊆ Rn 0 n ≥ 1 (& ,3@B+> *+#(,-#. 0 %&(*9)5&
∀ x, y ∈ K
•
∀ α, β ≥ 0, α + β = 1
α·x+β·y ∈K.
7!,+&O,/ 2,!5),3 K %& /0&(1,#20 '!"#$ xn ∈ K, xn → x ∈ Rn ⇒ x ∈ K 6
3"4,#', A#@3+S&,4C "!,+&O,/ 2,!5),D K ⊆ Rn %& $&.,!+>,3 %3"! $)2&,(&
&5##.6+ +7&8/ K 0 *&$D 2,!5),D
n
n
x∈R ; x=
X
z∈J
αz · z
'1! "!,&8,!# J ⊆ K ,
X
z∈J
αz = 1 , αz ∈ R
o
,
:!5 %& A+5$DC '!(#,#*B 9),&>1,/ '!$'1!(*!10 *&$D 2,!5),3 {y + z ; z ∈ L}0
"$& L %& 9),&>1,/ '!$'1!(*!1 Rn 3 y ∈ Rn 6 1! ,&'1>@$,!# K %& $)2&,(& K
AZ $)2&,(& LC :&94 8/(9! 2&@) 0 3 n6
Příklady.
1.
793():"B2 'S/"93$&2 #@3+S&,4 "!,+&O,/ 2,!5),D %& +8)%+ 0 $&.,!+3,B
%3"! *+#(,-#. +7&8 ,-%3"4 "!,&8,4 2,!5),D B ⊆ Rn ;
!"# (B) ≡ { x ∈ Rn ; x =
2.
X
z∈B
αz · z
'1!
αz ≥ 0 ,
X
z∈B
αz = 1 } .
[E&:,-%G/2 'S/"93$&2 %& +8),9!0 ,-"$D *45 ,3@B+3,B 4#+6+'%:#0 $&.<
,!+3,B %3"! '1R,)" "!,&8,4F! '!8*# #@3+S&,B:F '!9!'1!(*!1R0 *%6 2,!<
5), *+31#
n
{x ∈ R ; hx, yi ≥ c},
K,=9):"B *&12/, %&
"$&
+8)6,9!+#6
n
y∈R ,c∈R
3
hx, yi ≡
n
X
i=1
x i · yi .
246
A Nalejvárny
•
!"#$!% K ⊆ R & n ≥ 1 '( !%)*+, !"#$%"&' ()$*$' & -('./$#(
0$1 ≡ [0, . . . , 0] ∈ K &
0$$1 ∀ x, y ∈ K ∀ α, β ≥ 0 α · x + β · y ∈ K 2
3+$4(!.!5 '( -(4!, " 6"!+(7!8 9!"#$!:2 ;!</$=6* .(>98! -( +!"#$% +!"$2
n
Příklady.
1.
?/%'$=6*9 @A86/%4(9 :)%+A(!BC" 6"!+(7!8C" 6:#(/( -( !",+ - !./* !(D
@>,)4!B 9!"#$!E ∅ =6 B ⊆ R F
X
!" (B) ≡ { x ∈ R ; x =
α ·z @>" 6"!(G!": ∅ =
6 C ⊆ B % α ≥ 0}.
n
n
z
z
z∈C
H @A8@%45 @>,)4!B 9!"#$!E B @A$-89,9( 6"!+(!=$ !" (∅) = { }2
2. I$!*9 @A86/%4(9 :)%+A(!BC" 6"!+(7!8C" 6:#(/( -( 0(1*"& ()$* @>" 9!"D
#$!: A ⊆ R F
n
A∗ = { x ∈ Rn ; ∀ y ∈ A hx, yi ≥ 0 } .
?:#(/ !%)+(9( 2$3*/" & %!(J" .B# 4!*5$06,+ - ()$*& -('./$#( (7$'.:-( 6"!(G!,
B ⊆ R .%6"+,& #( K = !" (B)& >('@(6.$+( 6/+,!"1*"& 2$3*/"& -('./$#( (7$'.:-(
6"!(G!, 9!"#$!% >%=$"!,/!8=C +(6.">K B ⊆ Q & #( K = !" (B)2
Poznámka. L>" 6"!(G!": B ⊆ R @/%.8 !" (B) = (B ) 2
Fakt.
!"#$!% K ⊆ R & n ≥ 1 -( >%=$"!,/!8 -(C/%! @>,+5 .(C4E& -('./$#(
K = A 64( A ⊆ Q -( 6"!(G!, 9!"#$!% >%=$"!,/!8=C +(6.">K2
• 789"!( :)%+A(!B 6"!+(7!8 9!"#$!E K ⊆ R '( J:4( >"):95. -(-8 6"!+(7!8
@"49!"#$!% F ⊆ K 6.(>, 9, !,'/(4:-8=8 +/%'.!"'.F @"6:4 @>" y, z ∈ K
".(+A(!, M'(G6% (y, z) ≡ {x ∈ R ; x = α · y + β · z α, β > 0, α + β = 1 }
@>".8!, F & .-2 @"6:4 F ∩ (y, z) 6= ∅& @%6 y, z ∈ F 0="# N%6.$=6E )!%G8& #( =(/,
M'(G6% /(#8 + F 12 ;!</$=6* .(>98! @>" '.5!: -( :/+$2
• 36+$+%/(!.!8 4(O!$=( '.5!E + @A8@%45 @"/E."@: K ⊆ R -( @>K'(6 K ' -(C"
,;!*(2&+& "/06!#,"!(& .-2 F = K ∩ {x ∈ R ; hx, yi = k}& 64( y ∈ R % k ∈ R
-'": .%6"+B& #( K ⊆ {x ∈ R ; hx, yi ≥ k}2
• 36+$+%/(!.!8 4(O!$=( !(@>,)4!B '.5!E + @A8@%45 :)%+A(!BC" 6"!+(7!8C" 6:D
#(/( K ⊆ R -( 9!"#$!% F ⊆ K '@/P:-8=8 !,'/(4:-8=8 .A$ @"498!6EF
0$1 ∈ F &
0$$1 x, y ∈ F α, β ≥ 0 ⇒ α · x + β · y ∈ F &
0$$$1 x, y ∈ K, x + y ∈ F ⇒ x, y ∈ F 2
Poznámka. Q.5!% 0>%=$"!,/!8C"1 -(C/%!: -( "@5. 0>%=$"!,/!81 -(C/%!2
• R(@>,)4!B '.5!E :)%+A(!B 0!(@>,)4!B1 6"!+(7!8 9!"#$!E ∅ =
6 K ⊆ R & ="#
-'": "@5. :)%+A(!B 6"!+(7!8 9!"#$!E& /)( 6/%'$O6"+%. @"4/( 4$9(!'(F
•
n
n
∗ ∗
n
n
∗
n
n
n
n
n
N
n
n
n
247
A.5 Nalejvárna z grafů
!"#$% &'()$!) 0* $)+,-' .)&$,/012,13 ($,4'$%* !) $56715.8 !"#$%&* 5$)+,
"34 '()!'*+%,- .$/& 95$:-';27 ")0(8$ /0, 10;<,- .) '!)'( $)+, '()!'*'
0$1,)=*
!"#$% &'()$!) 1 !) $56715.8 #!2,&* 5$:-';2% '/3'4*
!"#$% &'()$!) 2 1 ",(", ")>"? $5671@( !$ 1,,5 4)6,&*
!"#$% &'()$!) k − 1* 2&) k .) &'()$!) K !) $56715.8 724')&* 5$:-';2%
72"')4A
B5(,6C).(#* .)&'$@ !"#$5 &'()$!) k .) ;)-@ 2,$1)>$8 ($,4'$5 K A
• D651C)$7 2?4)- K ⊆ Rn .) 82$4)9',: * .)!"-'4) K ∩ (−K) = { }A
Fakt. D651C)$7 2?4)- K ⊆ Rn .) 65,!"C)$7* /0@1# 2&%4
∃y ∈ R
n
∀x ∈ K \ { }
hx, yi ≡
n
X
i=1
x i · yi > 0 .
• ;20!4'< :)$)0,15$7 $)$?-,17( 1)2",0)( 6= x ∈ Rn .) ($,4'$5
R = {α · x ; α ≥ 0 }A
• E)F-' K ⊆ Rn 65,!"C)$7 ?651C)$7 2,$1)>$8 2?4)-* /52 .)<, /5/0!)2 R ⊆ K
$561)() '()!'*+%,- 91 K =* .)!"-'4)
∀ x, y ∈ K
x + y ∈ R '(/-'2?.) x, y ∈ R .
E'$7(' !-,1%* R .) !"#$,? K * 2,$203"$# <05$,? K A G-5!"$#* 1 /C8/5&# 65F
,!"C)$3<, $)/0@6&$3<, /,-%)&0';23<, 2?4)-) .!,? /,.(% )>"0)(@-$8 /5/0!)2
5 <05$5 !%$,$%(5A
Fakt (důsledek Krein-Milmanovy věty). H54&7 65,!"C)$7 ?651C)$7 2,$1)>$8 2?4)-
K .) 2,$';27 ,+5- !17;< )>"0)(@-$8;< /5/0!2IA J,2,$;) K .) 2,$';27 ,+5.5232,-' !13 /,&($,4'$%* 1 $84 .) 254&7 /5/0!)2 65!",?/)$ 5-)!/,K .)&$8(
:)$)0?.8;8( 1)2",0)(A
Fakt. L5,!"C)$7 ?651C)$7 2?4)- .) 05;',$@-$8 .)<-5$ /0@1# ")<&%* 2&%4 (@
2,$)M$# ($,<, )>"0)(@-$8;< /5/0!2I* /C'M)(4 254&7 /5/0!)2 .) :)$)0,15$7
05;',$@-$8( 1)2",0)(A
Fakt. NC8&5 !"#$ 65,!"C)$3<, .)<-5$? "1,C8 5",(';27 !156A
9'= E)&$,"2,17 /01)2 !156? .) ;)-3 K A
9''= O?-,17 /01)2 !156? .) 2?4)- { }A
9'''= P",(% ",<,", !156? .!,? )>"0)(@-$8 /5/0!2% K A
A.5 Nalejvárna z grafů
• =&.!1/,-* 3!27'* !) +?&) 0,6?(#" "0,.';) G = (N, L, A), 2&)
9'= N .) $)/0@6&$@ 2,$)M$@ ($,4'$5 >8%?*
248
A Nalejvárny
!!"
!!!"
L #$ %&'(!&) &$'*!$&+',)&-./ /*)&0 &$1'2! !"#$0 +#3 4,'56*,7',-./ 6'48
%&'(!& N 9 L ⊆ {A ⊆ N ; |A| = 2}3
Zápis9 a
b , G ⇔ {a, b} ∈ L0
A #$ %&'(!&) '*!$&+',)&-./ /*)&0 &$1'2! %!&#$0 +#3 5:6';<4)&-./ 4,'#!.
*=>&-./ 5>2=9 A ⊆ N × N \ {(a, a) ; a ∈ N }3
Zápis9 a → b , G ⇔ (a, b) ∈ A0
!," 6'()45#$ :$
"#"'()*")(+ ,-." 9
∀ a, b ∈ N a 6= b
a→b
?$:+2!($
N
N3
L = ∅3
D@1*!4&A B*)C #$
".1
G ⇒ ¬{b → a
,
G} ∧ ¬{b
a
,
G} .
G0 6)7 ;A7<%$0 ($ G
#$:+2!($ A = ∅ ) )-!#"+)2."3 0
#$
/-.0
,
#$ %&'(!&) 5>2= /@1*!4&A/' B*)C5
"#)-!#"+)2."3 0
#$:+2!($
Poznámka. E6=:'1 >),$4$&A B*)C= ,2):+&F >)*5G5#$ )1:$&.! :%@G$73
•
H,'#!.$ 5>2=
!"
!!"
!!!"
a
b
a→b
b→a
[a, b]
,
,
,
G
G0
G3
#$
,
,-.".
G
#$:+2!($ &):+)&$ #$4&) >$ :!+5).A9
.'( #$ +'+I(0 .'
b
a
,
G"0
J'>&)%$&$#%$0 ($ C'*%<2&F #$ /*)&) >),$4$&) #)7' 5:6';<4)&< 4,'#!.$0 )2$
C)7+!.7@ #$ +' &$5:6';<4)&< 4,'#!.$3
•
G̃ = (Ñ , L̃, Ã) &)>,$%$ &)1/-.0#4 /@1*!4&A/' B*)C5
G = (N, L, A)0 #$:+2!($ Ñ ⊆ N 0 L̃ ⊆ L ) Ã ⊆ A3
• 5"16$)2."3 &)1/-.0 B*)C5 G 6*' &$6*<>4&'5 %&'(!&5 T ⊆ N #$ B*)C
D@1*!4&A B*)C
GT = (T, LT , AT ),
•
?$82!
a∈N
74$
LT = L ∩ P(T )
)
AT = L ∩ (T × T ) .
5>$2 /@1*!4&A/' B*)C50 6)7
!G (a) = {b ∈ N ; a b
"#G (a) = {b ∈ N ; b → a
$%G (a) = {b ∈ N ; a → b
,
G}
#$ %&'(!&)
()6(#17
,
G}
#$ %&'(!&)
-)1!87
,
G}
#$ %&'(!&)
19+:
5>25
5>25
5>25
•
4A+F K
a0
a3
E&)G$&A #$ %'+!,',<&' )&B2!.7'5 +$*%!&'2'B!A9 :'5:$4 K
&.-#"+0
a0
"#!/,*)6-0
*'4!G K
;,! 13
G :$ 154$ *'>5%F+ 6':2'56&':+ 5>2= B*)C5 ρ :
a1 , . . . , an 0 n ≥ 1 +)7',<0 ($ ∀ i = 1, . . . , n − 1 #$ [ai , ai+1 ] /*)&) , G3
=-.>": 5>2@ :2$45 ρ #:'5 a1 ) an 0 2"!+?": 5>2@
#$& , 6;A6)4F n ≥ 3" #:'5
a2 , . . . , an−1 3 ?$:+2!($ A, B ⊆ N #:'5 %&'(!&@ 5>2=0 a1 ∈ A ) an ∈ B 0 6)7
;$7&$%$0 ($ :2$4 2#1# @ A 1) B &F74@ +I( %$>! A ) B "3
L6'>'*M5#!0 ($ 4$N&!.$ :2$45 >)/*&5#$ ! 6;A6)4 n = 1O
• P2$4 :$ &)>-,< ;#(+.0 #$:+2!($ &),A. a1 , . . . , an #:'5 -7@"A 5>2@3
• P2$4 #$
< #1#4
Q
, /@1*!4&A% B*)C5
"#)-!#"+)2."3
#$:+2!($
∀ i = 1, . . . , n − 1
#$
ai
ai+1
,
G0
249
A.5 Nalejvárna z grafů
! "#$%&! "#$%& ∀ i = 1, . . . , n − 1 "'()*! +, -%. ai
ai+1 / G 0,-#
ai → ai+1 / G!
"'()"%* ! "#$%& ! 1,2)'3+, ∀ i = 1, . . . , n − 1 1, ai → ai+1 / G4
Neorientované grafy
• 5,2)'3+, G 1, 0,#63,0)#/(07 86(9 0(& N ! "($ :0#+30% A ⊆ N 0(;/,:, +$,%-#
/ G! 1,2)'3+,
∀ a, b ∈ N, a 6= b 1, a
b / G.
• .,()-# 86(9% G 2, -%&, 6#;%:<) :(=3:>'0* ?"'0> "#&:0#+30( N /, 2:@2'%
30$'%;,4 A(:#)07 0,#63,0)#/(07 86(9 G 0(& N 2, 0(;/, +$,%&! 1,2)'3+, N 1,
?"'0> :0#+30( / G4
• B,#63,0)#/(07 86(9 G 0(& N 0(;/,:, /($0'"("%1! 1,2)'3+, ,=32)%1, 6#;$'(& N =
A∪B ! A∩B = ∅ )($#/7! +, "#$%& a b / G! "($ |{a, b}∩A| = 1 = |{a, b}∩B|4
• 2!-'(!%"-30%& 45),# / C@-63&0*: 86(9% G 0(& N 1, "#2'#%"0#2) %;'D a1 , . . . , an+1 !
n ≥ 3! )($#/>! +,
E3F an+1 = a1 !
E33F a1 , . . . , an 12#% 6D;0G!
ai+1 / G4
E333F ∀ i = 1, . . . , n ai
H*2'# n 2, "($ 0(;7/> 67,)-# 45),#4 8*"(3-# I@$'% a1 , . . . , an+1 ! $&, n ≥ 4! 2,
6#;%:* '30$( ai
aj / G )($#/>! +, 1 ≤ i! j ≤ n ( 1 < j − i < n − 14
• J@-63&0* 86(9 G 0(& N 1, -#3( ,&! 1,2)'3+, ∀ a, b ∈ N ,=32)%1, / G 0,#63,0)#K
/(0> I,2)( ; a &# b4
• B,#63,0)#/(07 86(9 2, 0(;7/> "'-9 ! 1,2)'3+, 1, 2#%/32'7 ( 0,:> +>&07 0,#K
63,0)#/(07 I@$'%24 B,#63,0)#/(07 86(9! $),67 0,:> 0,#63,0)#/(07 I@$'%2! 2,
0(;7/> ,! 4
Fakt. B,#63,0)#/(07 86(9 G 0(& N 1, 2)6#: "6>/< ),C&@! 1,2)'3+, ∀ a, b ∈ N
/ G ,=32)%1, "6>/< 1,&0( I,2)( ; a &# b4
Orientované grafy
• :'(!%"-30%& 45),# / C@-63&0*: 86(9% G 0(& N 1, "#2'#%"0#2) %;'D a1 , . . . , an+1 !
n ≥ 3 )($#/>! +,
E3F an+1 = a1 !
E33F a1 , . . . , an 12#% 6D;0G!
E333F ∀ i = 1, . . . , n ai → ai+1 / G4
H*2'# n 2, 0(;7/> &G'$#% I@$'%4 L63,0)#/(07 86(9 2, 0(;7/> 045),(4)& ! 1,2)'3+,
/ 0<: 0,,=32)%1, #63,0)#/(07 I@$'%24
Fakt. L63,0)#/(07 86(9 G 0(& N 1, (I@$'3I$7! "6>/< $&@+ ,=32)%1, #M*2'#/>0*
a1 , . . . , an ! n = |N | 1,C# %;'D! $),6G 1, -#;,0 %7
∀ 1 ≤ i, j ≤ n
-'(!%"041 <($!)
ai → aj / G ⇒ i < j .
/ G! )14
250
A Nalejvárny
•
!"#!$!% &! '(!) a *! !"#"$ '()' b + ,-./01#2$ 3/45' G% /!67!"80+! &! b *!
%&%'"$ a + G% *!68)0&! + G !9068'*! 6!68'7#: 6)!1 ( a 1; b <!"+0+4)!#8#=% ( a
+!1! + G 6!68'7#> ?!684 1; b@A B#;&0#4 7;8;$"C '()' a + G 6! .'1! (#4D08
6-$.;)!$ !G (a)% $#;&0#4 7E!1"C $#;&0#- A ⊆ N 6-$.;)!$ "#G (A)A
F ;/0!#8;+4#G$ 3/45' G (#4$!#>% &! a *! 7E!1!" b% &! .'H a = b
#!.; + G +!1! ( a 1; b 68/0"8#= 6!68'7#: 6)!1% /!67!"80+! 68/0"8#= 6!68'7#>
?!684A I#4D!#2 *! $;80+;+>#; 4#3)0?";' 8!/$0#;);302J 7;8;$!" K #"()"*#+*&%
7E!1!" K +*)"(&%,A
Poznámka.
A.6 Nalejvárna o rozděleních
•
L'H P /;(1=)!#2 #=*4"G,; </!>)#G,;@ #>,;1#G,; +!"8;/'% 8!1- 7/4+1=7;M
1;.#;68#2 $2/4 #4 !'")01;+6"G$ 7/;68;/' RN % 7E0D!$& |N | = n ≥ 1A N!,;
(>")41#2 #'$!/0?";' ?,4/4"8!/0680";' *! -"$&%, (&!"#*.)/ /%#*%& e = [ei ]i∈N %
*!,;& 6);&"- *6;' 1!O#;+>#- #>6)!1;+#=J
Z
ei =
xi 1P (x)
7/;
i ∈ N,
RN
"1! xi ;(#4D'*! iM8;' 6);&"' +!"8;/' x ∈ RN A F 7/490 6! 8G$=E +:,/41#=
'+4&'*2 /;(1=)!#2% 7/; #=& ($2#=#G 0#8!3/>)- ei !9068'*2 4 *6;' ";#!D#GA
• P4)Q2 +:(#4$#;' #'$!/0?";' ?,4/4"8!/0680";' P *! *!,; $%-+,0+*1*. '+&0)"
Σ = (σij )i,j∈N % *!*2& 6);&"- *6;' 1!O#;+>#- #>6)!1'*2?2 5;/$')2J
Z
σij =
(xi − ei ) · (xj − ej ) 1P (x),
RN
(4 0$7)0?08#2,; 7E!17;")41'% &! P $> +!"8;/ 68E!1#2?, ,;1#;8A F 7/490 '+4M
&;+4#> /;(1=)!#2 $4*2 (7/4+01)4 ";#!D#G ,;1#;8- σij % #4(:+4#G $%-+,0+*)"A
• 2"3456,*. 3+4((%-($7 ,%8#95"*. *! +2?!/;($=/#G 67;*08G /;(1=)!#2A R/; n ∈ N
621)2 7E26)'Q#G nM/;($=/#G /;(1=)!#2 #4 <+:.=/;+G$@ 7/;68;/' RN % 7E0D!$&
|N | = nA N! 74/4$!8/0(;+>#; +!"8;/!$ e ∈ RN 4 7;6080+#= 1!O#08#2 N × N M
$480?2 ΣA S;/$')!
fe,Σ (x) = √
1
(2π)|N | · det (Σ)
(x − e)⊤ · Σ−1 · (x − e)
· exp −
2
7/; x ∈ RN % 1!O#'*! ,'68;8' 7E26)'Q#G </!3')>/#2 34'66;+6"G@ $2/- +(,)!1!$
" T!.!63'!;+= $2E! #4 RN A I#4D2 6! 6-$.;)!$ N (e, Σ)A F!"8;/!$ 68E!1#2?,
,;1#;8 *! 74" e 4 ";+4/04#D#2 $480?2 ΣA
• U;(1=)!#2 N (e, Σ) 6! (4+>12% ;+Q!$ ";$7)0";+4#=*Q2$ (7C6;.!$% 0 + 7E2741=
;.!?#=*Q2 7;6080+#= 6!$01!O#08#2 $480?! ΣA V)! "1-& $480?! Σ #!#2 7;6080+#=
1!O#08#2 <K /!3')>/#2@% 74" 1!O#;+4#> $2/4 *0& #!621)2 #4 ?!)G$ 7/;68;/' RN %
#:./& *! 6;'68E!1=#> #4 #=*4"G$ 4O##2$ <K 7;6'#'8G$ )0#!>/#2$@ 7;17/;M
68;/' RN A R;8;$ ,;+;E2$! ; (0*3456,*.' 3+4((%-($7' ,%8#95"*. A W0?$G#=
0 + 8;$8; 7E2741= 7)482% &! e *! +!"8;/!$ 68E!1#2?, ,;1#;8 4 Σ ";+4/04D#2
$480?2A
251
A.6 Nalejvárna o rozděleních
•
!"#$%&'$()* +&,-."/%0
! "#$!%&'()%*+ ,-./%+0*# %&',)1!*# 23 %&',)1!*# 4056
781!/ 9!0*&.0#:; .<!$-=1*) /&*0-*>!*9*#$? 05781!/@A B%&
r6%&'()%*+ %&',)1!*# *=.1!,8 #$# <5%5(!0%EF
Pr
2-@ θ1 , . . . , θr 05/&"+; G! θk > 0 5
k=1 θk = 1;
2--@ d ∈ NA
r ∈ N
(= <C#.18D*+
H#,1# *5 /&*!9*+ (*&G-*) 23 "I7)%&"+( <%&.0&%8@
[d1 , . . . , dr ] ; dk ∈ N,
J8.0&05 2"K9- 5%-0(!0-$/+ (#%)@
p([d1 , . . . , dr ]) =
L*59!*#
Pr
k=1
dk = d .
! 8%9!*5 "'&%$!(F
d!
· θd1 · . . . · θrdr .
d1 ! · . . . · dr ! 1
M(θ1 , . . . , θk , d)A
Poznámka.
M!.01-G!
ξ1 , . . . , ξ d
! *=?&,*I "I7)% 2"-' N!O*-$! PAQ@ ' %&',)1!*#
X = {1, . . . , r} . ?8.0&0&8 p(k) = θk ; k = 1, . . . , r; <5/ "!/0&% [ρ1 , . . . , ρr ];
/,! ρk = |{ℓ; ξℓ = k}|; (= "ID! 8"!,!*+ %&',)1!*#A R!*0& "!/0&% ! "15.0*)
*5
05781/5 9!0*&.0# .!.05"!*= *5 '=/15,) ,5057='! 20 A <&.1&8<*&.0- ?&,*&0 *=6
?&,*+?& "I7)%8@A
•
!"!#$%&'()( "(*+,%&-.
! "#$!%&'()%*+ .<& -0+ %&',)1!*# 23 %&',)1!*# *5 4,-.6
r ∈ N; r ≥ 2 (= <C#.18D*+ r6
α1 , . . . , αr 05/&"+; G! αk > 0A S<&'&%T8 -; G!
/%+0*#$? <%5",)<&,&7*&.0*#$? (#%=$?:@A B%&
%&'()%*+ %&',)1!*# <5%5(!0%E
(!'- <5%5(!0%E
α1 , . . . , αr
-&-. "5'75U N-%-$?1!0&"& %&',)1!*# .#,1# *5 (*&G-*)
23 "I7)%&"+( <%&.0&%8@
Θ=
[θ1 , . . . , θr ] ; θk > 0,
Pr
k=1 θk
=1 .
M! '5,=*& ?8.0&0&8 "K9- 40%5*.V&%(&"5*+: W!7!.>8!&") (#C!A BC!.*) -F 8"56
G8 (! %!.0%-/$-
(r − 1)6%&'()%*+
W!7!.>8!&"E (#%E *5
{ [θ1 , . . . , θr−1 ] ; θk > 0,
5 <&0+ 0%5*.V&%(5$-
r
[θk ]r−1
k=1 7→ [θk ]k=1 ;
/,!
Pr−1
k=1 θk
θr = 1 −
< 1}
Pr−1
k=1 θk A
J8.0&05 N-%-$?1!0&"5 %&',)1)*# 2"K9- '(#*)*+ ,&(-*8 #$# (#C!@
f ([θ1 , . . . , θr ]) =
Γ(α1 + . . . + αr )
· θα1 −1 · . . . · θrαr −1 ,
Γ(α1 ) · . . . · Γ(αr ) 1
R∞
Γ(α) = 0 y α−1 · e−y ,y ; α > 0
D(α1 , . . . , αr )A
/,!
! ?&,*&05 X5((5 V8*/$!A L*59!*#
Y!/0&% .0C!,*#$? ?&,*&0 5 /&"5%-5*9*# (50-$!
Z
$&"
(θk ) =
(θk , θl ) =
"5%
!
(θk ) =
αk
α ,
−αk ·αl
α2 ·(α+1) ,
(α−αk )·αk
α2 ·(α+1) .
/,!
.&8 8%9!*E 0)(-0& "'05?EF
α=
k 6= l,
Pr
k=1
αr ,
252
A Nalejvárny
Fakt (základní fakt z hlediska bayesovského přístupu k učení).
Pr
!"#$%&! '$(#)
*%+ d =
dk ∈ N, -.! k=1 dk = d, θ =
*%%+ h(d|θ) /0 /1$#%23/%4-5 637.8$!2) M(θ1 , . . . , θk , d),
*%%%+ π(θ) /0 9%6%4:$!#3;3 637.8$!2) D(α1 , . . . , αr ),
'(- π(θ|d) /0 9%6%4:$!#3;3 637.8$!2) D(α1 + d1 , . . . , αr + dr ) .
[dk ]rk=1 ,
[θk ]rk=1 ,
<"-1#-1,
π(θ|d) ∼ π(θ) · h(d|θ) ∼
r
Y
k=1
θkαk −1
·
r
Y
k=1
θkdk
=
r
Y
(αk +dk )−1
θk
,
k=1
-.! "=/>3$ ∼ 72(?) 63;23"# (& 2( /1$#%'$%-(#%;2) @(-#36A B)-0/!, &! 9%6%4:$!C
#3;3 637.8$!2) D! !"#$%!&'"( - /1$#%23/%4-5/1A
Literatura
!"#$%%&! ' ( )*"+,*! -( .#$/0*! )-
!""#$%& '( )*+ ,$-./0 +1230$4+(5+
!"#$%#"&%"#
54$66+6 7/- 5*$3( 8-$9*6: 2(;3-+5)+; 8-$9*6 $(; $5<5435 ;38-$9*6=
'()*#"+ (, -"-%.-%!.: !
!%: >!?!@A=
!"#$%%&! ' ( )*"+,*! -( .#$/0*! )-
!""#B%& C 5*$-$5)+-3D$)3/( /7 ,$-E
!"#$%%&! ' ( )*"+,*! -( .#$/0*! )-
A@@!%& C4)+-($)30+ ,$-./0 9-/9+-E
./0 +1230$4+(5+ 54$66+6 7/- $5<5435 ;38-$9*6=
)3+6 7/- 5*$3( 8-$9*6=
/##"+. (, -"-%.-%!.: "
!"#$%#"&%"# '()*#"+ (, -"-%.-%!. #
!%: HH?>F=
!"#$%%&! ' ( )*"+,*! -( .#$/0*! )-( 1+23*$"%&! 4'
,$-./0 J/;+46 3( J24)30$-3$)+ $($4<636= C 5*$9)+- 3(
.%6# (, 7891*%:1#-.3 "#$ )*&12 ":9+%#6
A%: F@F?FG!=
!"""%& I-$9*35$4
0)+-%&"*%"-1 /#"+2.%.3 415
K= I*/6* +;=%= ,$-5+4 L+..+-: M+N
O/-.= P*$9)+- >: !>#?AH@=
!"#$%%&! ' ( .#$/0*! )-
!""H%& Q$))35+ J/;+46 7/- 5/(;3)3/($4 3(;+9+(E
;+(5+ 3( $ J24)30$-3$)+ (/-J$4 ;36)-3B2)3/(=
/##"+. (, -"-%.-%!.: $
H%: !H!>?
!HF>=
!"5/ 6 !">F%& 0"-1:"-%!;< .-"-%.-%;"= KMRQ C47$%: S-$*$=
!"5/ 6 A@@#%& =<;+"$2 :"-1:"-%!;> .-"-%.-%;2= ,$)7<DS-+66: S-$*$=
7&829*#$: 11 !""F%& ?"21.%"# @1+%1, #1-A(*;. B ,*(: !(#.-*)!-%(# -( %#,1*1#!1 =
S*L )*+636: T(30+-63)< /7 T)-+5*)=
7$;!"%:#"
!">H%&
/# C#-*($)!-%(# -( D(#&18 E(+2-(91. = K9-3(8+-EU+-4$8: M+N
O/-.=
7&829*#$: 11( ':8"#!< )
(+66= W(&
!""F%& P*$3( 8-$9*6& 6+J$()356 $(; +V9-+6630+E
2:@(+%! "#$ F)"#-%-"-%&1 /99*("!G1. -( H1".(#%#6 "#$ I#!1*-"%#-2
P= X-/3;+0$2V: Y= Z/*4$6 +;6=%: Q+5)2-+ M/)+6 3( C-)3[53$4 W()+4438+(5+ "G\:
K9-3(8+-EU+-4$8: ,WKR' \#?#\= LUC MC]UO ZMW^O_
78!:+!# =
+%61#!1 KK
!""F%& P*$3( 8-$9*6 7/- 4+$-(3(8= W(&
I#!1*-"%#-2 %# /*-%J!%"+ C#-1+5
S= `+6($-;: K= ^$(.6 +;6=%: ,/-8$( Z$27J$((: ,WKR': G\?FG=
>&?#// 1@( -*?+" .( A*8$+:B#! 'A( 'C+#,#/3*/:#$ -6
!"""%&
E*(@"@%+%.-%!
L1-A(*;. "#$ 7891*- 2.-1:. = K9-3(8+-: M+N O/-.=
>&D -1( =#$08:3 E !""\%& 0)+-%&"*%"-1 4191#$1#!%1. B 0($1+.3 /#"+2.%. "#$
C#-1*9*1-"-%(#= ,/(/8-$9*6 /( K)$)36)356 $(; C9943+; S-/B$B343)< \#: P*$9J$(
$(; ^$44: Q/(;/(=
254
Literatura
!""#$% &'( )!*"+,-./ 0)( 01..2 34
6.)(765+2 8+0./1 9+) 7+26523.27: 6(;/.1<
!"#$%& '()*+, -./01 (20 /+34/52.() 524
!!"#$ %& '("()$()*$=
>%= ?@@A?>"<
!5+2 64
!"B"%& C+20565+2(/ 520.D.20.27. 52 16(651657(/ 6E.+):=
!5+2 64
@$$!%& F.D()+501& ( 8(6E.8(657(/ 9)(8.G+)* 9+) 7+20565+2(/ 520.D.24
0%1"# '("()$()*"# '%*)/(1= !" !%= !A>!<
0.27. (20 5))./.,(27.=
>>?A>B@<
+%,-!"# %& (./
!!"#$ %& 2"(./3"()*$ "!4 -()5*)"# 6!(/##)7/!*/ = #$ !HI%=
2. 7!81#9 ):( ;*.,. &<( :#"!= 0
8-%9"9)#)(1 )!(/-:"#$; " (%%# &%,!*/-(")! -/"$%!)!7; (/*.!)*"# -/<%-( =>?' 6@ABCDE < J25,.)156: +9 K)(2(0(<
2. 7!81#9 ):
!"">%&
!""L%& CE()(76.)5M(65+2 +9 0.7+8D+1(;/. 0.D.20.27: 8+0./1<
+%,-!"# %& -()5*)"# 6!(/##)7/!*/ 0/$/"-*. = %= @#"A>$$<
.819,." 64
!"LB%& JDD.) (20 /+G.) D)+;(;5/565.1 520N7.0 ;: ( 8N/65,(/N.0
!!"#$ %& 2"(./3"()*"# '("()$()*$= ""= >@?A>>"<
.819,." 64 !"B@%& C+,()5(27. 1./.765+2< F)%3/(-)*$= $ = !?BA!B?<
",#/ :( 0,*"8>.=9 ?( 0*==+@!/, 0 @$$"%& G/*(,-/$ %! #7/9-")* '("()$()*$<
8(DD523<
FD)523.)4O.)/(3= 'PFQR<
*A#+9 ( 4"!2. ;
!"##%&
8%$$)9)#)(1 H./%-1 I "! <<-%"*. (% ?%3<,(/-)J/4
8-%*/$$)!7 %& K!*/-(")!(1< S/.2N8 S).11= T.G U+)*<
<=#"./9 &B4( :#*$%!", :( C#=+/ &B: !""$%& >#/3/!($ %& F"1/$)"! '("()$@
()*$< '()7./ V.**.)= T.G U+)*<
<"D2./A."E : !""$%& QE. 7E(52 3)(DE '()*+, D)+D.)6:< '*"!4)!":)"! +%,-!"#
%& '("()$()*$= "& I%= >>>A>?><
F.+E." ( 4.!"= & !""$%& R2 6E. /+357 +9 7(N1(/ 8+0./1< P2& K!*/-(")!(1 )!
-()5*)"# 6!(/##)7/!*/ L W< V< FE(7E6.)= Q< F< X.G566= X< T< Y(2(/= Z< [< X.88.)
.01<%< T+)6E4\+//(20= ]816.)0(8= >A!I<
F##28!/ )6
!"B>%& QE. (2(/:151 +9 8N/65058.215+2(/ 7+26523.27: 6(;/.1 GE.2
1+8. ,()5(;/.1 (). D+16.)5+ 6+ +6E.)1 A ( 8+05-.0 D(6E (2(/:151 (DD)+(7E<
%3/(-)M"= '(= !B"4!"@<
;!A."8!/ 0& !"BI%& H./ !"#1$)$ %& N-/O,/!*1 ="("<
F)@
J25,.)156: +9 CE57(3+
S).11= 'PFQR<
;!*E%,#/ :6
9(85/:<
!"##%& R2 6E. 7E+57. +9 ( 8+0./ 6+ -6 0(6( 9)+8 (2 .^D+2.265(/
!!"#$ %& '("()$()*$= "'= >I@A>??<
;GH.I 4( ;!@"G/.I 3( &+"#*J.I C
!""@%& K!*/-(")! 6!&%-3"()%! 8-%*/$$)!7
)! >P</-( '1$(/3$< CWC S).11= _+7( W(6+2<
;.$I."8!/ !""?%& (,(%-)"# %! #/"-!)!7 F"1/$)"! !/(Q%-M$; (/*.!)*"# -/<%-(
2'0@H0@AR@DE< '57)+1+96 W.1.()7E= W.08+26<
;.$I."8!/ ( F.+E." ( 7%+$I."+/E :
!""?%& X.()2523 _(:.15(2 2.6G+)*1&
QE. 7+8;52(65+2 +9 *2+G/.03. (20 16(651657(/ 0(6(<
@I><
2"*.)!/ G/"-!)!7= $(= !"IA
;.88.$I. C( :#",#/ &( 0%+* 6( 0,*"8>.=9 ?( K+./!/2 L
.^(8D/.1 +2 1.8543)(DE+501<
@$$#%& QE).. 7+N26.)4
?%39)!"(%-)*$; 8-%9"9)#)(1 "!4 ?%3<,()!7 =
"&=
255
!"# $%&
!"#$% &'( )#*+,-!. /0 '(")*+, -./01.21 34567589& -., !"#$%&' $% ()! *+$%,
-$./!' "%# 0../$-"($1%' 12 3!-$'$1% 0%"/4'$'
':& ;& <=>573? @& A& B5CD19=.
139&+& EC75C1642 F12494=. G7=0H? B-EIJ? (#%K &
1+234,$2.5 6) '(""$+, ; C75.9L=785C4=.5M 2D5752C174N5C4=. =L 1O04P5M1.C Q5R1S
945. .1C>=7T 9C702C0719& -., 5%-!+("$%(4 $% 0+($6-$"/ 7%(!//$&!%-! 88 'U& Q19.573?
E& <5.T9 139&+& B=765. V50L85..? B-EIJ? )%#")&
1+234,$2.5 6) '
%"/ 12 ;"-)$%!
WW +, JHC485M 9C702C071 431.C4X25C4=. >4CD 67113R 91572D&
<!"+%$%& !'!"+-) ? ? $W%#$$*&
91:+,
/,.-,. 78 ' WW(+, ="4!'$"% %!(>1+?' "%# 3!-$'$1% @+".)' & EH74.617? Y1> Z=7T&
9#::,.;,$5 < '(""%+, A1:%#"($1%' 12 ;1#!+% *+1B"B$/$(4& EH74.617? Y1> Z=7T&
9#=,$>#.. ? '(""K+, J. 5 305M4N5C4=. =L 675HD425M G509945. 8=31M9& C-"%#$%",
D$"% 91:+%"/ 12 C("($'($-'
! '(+? (W$#((K&
9!-*,$ /@' '(""K+, B57T=P H7=H17C419 =L .=.7120794P1 25095M 8=31M9& 0%%"/' 12
C("($'($-' !"'$+?
(*)# (%%&
A#=$2*B,. CA '(") +, <!-(:+!' 1% E1%($%&!%-4 F"B/!'?
;5M\=76 U7199? B-EIJ&
.3 134C4=.& [.4P1794CR =L
A#=$2*B,. CA '(")"+, B4]13 675HD425M 599=245C4=. 8=31M9& C-"%#$%"D$"% 91:+%"/
12 C("($'($-'? #$'*+?
%!#!WK&
A#=$2*B,. CA '(""K+, @+".)$-"/ ;1#!/'& ^M571.3=. U7199? J]L=73&
A#=$2*B,. CA( 6#"2% 'D( A#$-,. EF( A,2>,$ G? '(""W+, -.31H1.31.21 H7=S
H17C419 =L 34712C13 B57T=P X1M39&
A#=$2*B,. CA( &23+#$%-!. @C '
C17H71C5C4=.9&
G!(>1+?'? !%'$+? *"(#$W$&
WW(+, ^D54. 675HD 8=31M9 5.3 CD147 25095M 4.S
(#!K(&
91:+%"/ 12 ()! 14"/ C("($'($-"/ C1-$!(4 = ? $"? !
A#=$2*B,. CA( CH2,5,:+#:*,$ 6/ '("))+, _=25M 2=8H0C5C4=.9 >4CD H7=\5\4M4S
C419 =. 675HD425M 9C702C0719 5.3 CD147 5HHM425C4=. C= 1]H17C 9R9C189&
()! 14"/ C("($'($-"/ C1-$!(4 =? &%' +? ($%# *&
91:+%"/ 12
A#=$2*B,. CA( I,$>=*+ F '(")*+, ;$H!# $%(!+"-($1% I1#!/'J +!'!"+-) +!.1+(
,KL,K& -.9C& AM12& ER9&? [.4P1794CR =L ;5M\=76? ;5M\=76&
A#=$2*B,. CA( I,$>=*+ F '(")"+, G75HD425M 8=31M9 L=7 599=245C4=.9 \1C>11.
P5745\M19? 9=81 =L >D42D 571 O05M4C5C4P1 5.3 9=81 O05.C4C5C4P1&
($'($-'? #''(+? !(#$%&
0%%"/' 12 C(",
A,2>,$ G? '(""!+, JHC485M 312=8H=94C4=. \R 2M4O01 91H575C=79& 3$'-+!(! ;"()!,
I"($-'? ##
? ""#( !&
A.J.2K4# &( )#*LM 7 '
WW%+, J. G509945. 2=.34C4=.5M 4.31H1.31.21 9C702C0S
719& M4B!+%!($?"? " '!+? ! %#!* &
A!NO, ) '("$$+, *+1B"B$/$(4 F)!1+4J A1:%#"($1%'J "%#1I *+1-!''!' & F& P5. Y=9S
C75.3? I=7=.C=&
)#:O,-*=*! 7) '(")!+, ID1=7R =L 75.3=8 =\917P5\M19 4. 71M5C4=.5M 35C5 \5919&
7%21+I"($1% C4'(!I'? ('*+?
)(# )"&
256
Literatura
!"#$%&'&( )
!""#$% & '()*'+ ,-./01 2324+/ ,-. 5)(0.3 6+7-/8-2)4)-( -,
6040502+ .+104)-(29 8.-505)1)43 6)24.)5'4)-(2 0(6 :.08;2<
!9 #!=>#<
!%%$* +,
!&-. )/ 0&'1$23
!&-. )
!&-. )
!&-. )
$$4 5
!"#$%&'(#! )*(+!*+, 9
!""?$% @.)A)((0B0 )(4+.8.+407)B0 21'A0B(37; C+1)A)(
/0+%1 -+$+2&3( !"#$%&*(( 9
"#
C .'D4)(E$< -$#.
!$9 !!#=!!?<
!"">$% F-(6)4)-(01 )(6+8+(6+(7+2 0/-(: ,-'. .0(6-/
C0.)051+2 G< 4#%/(!&'#$(*,5 -$#/&/(0('1 &!2 4#%67'(!8 9
$
H$9 #?"=#IJ<
!"">$% F-(6)4)-(01 )(6+8+(6+(7+2 0/-(: ,-'. .0(6-/ C0.)051+2 GG< 4#%.
/(!&'#$(*,5 -$#/&/(0('1 &!2 4#%67'(!8 9
$
H$9 HKI=H!I<
!"""$% F-(6)4)-(01 )(6+8+(6+(7+2 0/-(: ,-'. .0(6-/ C0.)051+2 GGG<9
L(01 7-(71'2)-(< 4#%/(!&'#$(*,5 -$#/&/(0('1 &!2 4#%67'(!8 9
%
M$9 #?"=#I?<
#KK#$% N+(:4;2 -, 2+/):.08;-)6 )(,+.+(7+2< 9!!&0, #" :&';+%&'(*, &!2
9$'(<*(&0 !'+00(8+!*+ 9
"
9 #JI=#"H<
!"">0$% F0'201 )(,+.+(7+ 0(6 70'201 +O810(04)-( P)4; 507Q:.-'(6 Q(-P1+6:+<
G(% =!*+$'&(!'1 (! 9$'(<*(&0 !'+00(8+!*+ >>
@< R+2(0.69 S< T0(Q2 +62<$< U-.:0(
V0',/0((9 UGSWX9 HKM=H!K<
$$4 5
!"">5$% S4.-(: 7-/81+4+(+22 0(6 ,0)4;,'1(+22 )( R03+2)0( (+4P-.Q2< G(%
=!*+$'&(!'1 (! 9$'(<*(&0
!'+00(8+!*+ >>
@< R+2(0.69 S< T0(Q2 +62<$< U-.:0(
V0',/0((9 UGSWX9 H!!=H!J<
$$4 5
$21$"%(2 6
(7&(2 +
('89!7& / :(";2 +<
('%%('7;% +
=$!>(";&!2 :6
=$!>(";&!2 :6
?$!7" +
?$!7" +/ ?!@ A
!""I$% ?$&6;(*&0 %#2+0,5 ,+0+*'(!8 *&7,&0 &!2 ,'&'(,'(*&0 %#2+0, < @;Y
4;+2)29 F0.(+:)+ U+1-( Z()C+.2)43<
!""I$%
!'$#27*'(#! '# :&';+%&'(*&0 @#8(*5 A'; +2('(#! < F;08/0(
0(6 T011[F\F9 UGSWX<
#KKI$% ?+#%+'$1 #" *#!2('(#!&0 (!2+6+!2+!*+ < @;69 Z()C+.2)43 -, F0]
1),-.()0 R+.Q+1+3<
!"JH$% & (-4+ -( 7-(6)4)-(01 )(6+8+(6+(7+ P)4; 240]
4)24)701 0881)704)-(2< )'&'(,'(*&9
$$
H$9 >>I=>JH<
!"IH$% ^)552 0(6 U0.Q-C 8.-8+.4)+2 -C+. '(6).+74+6 :.08;2< B#7$.
!&0 #" )'&'(,'(*&0 -;1,(*, 9
&'
!$9 !!=M!<
!""K$% -$#/&/(0(,'(* C+&,#!(!8 (! DE6+$' )1,'+%, < _-;( `)1+39
UGSWX<
!""H$% @+&$!(!8 F&1+,(&! G+'H#$I, < @+0.2-( @.+(4)7+ T0119 UGSWX<
!"JJ$% -$#/&/(0(,'(* C+&,#!(!8 (! !'+00(8+!' )1,'+%, J G+'H#$I, #" -0&7,(.
/0+ !"+$+!*+< U-.:0( V0',/0((9 UGSWX<
!"JI$% ^.08;-)629 :.08;]502+6 1-:)7 ,-. .+02-()(: 05-'4 .+1+]
C0(7+ .+104)-(2< G(% 92K&!*+, (! 9$'(<*(&0
!'+00(8+!*+
R< Y' R-'1039 Y<
T-::9 N< S4++12 +62<$< a-.4;]T-110(69 &/24+.60/9 M>I=M?M<
?$7$@ A
ε
?$7$@ A/ +;7('.$4 :
!"II$%
]06/)22)51+ 2)/81)L704)-(2 -, 4;+ 6+8+(6+(7+ 24.'74'.+ -, 0 2+4
-, .0(6-/ C0.)051+2< L1/+$!+'(I&9
&"
?$9 HM"=HH"<
!"J>$% F-(24.'74)(: 0( )(4+(2)-(01 +O8+.4 2324+/ GabS$<
G(% :+2(*&0 M+*(,(#! :&I(!8 < a-.4;]T-110(69 &/24+.60/9 MKI=M!><
257
!"#$%&'( )* +,#(- )* ./0'( 1* 234/# !""#$% !"#$%&& '()*+,-). /0)1)&
5(" 65 !"23$% '()#&% *+&+(,+(-&! .)/#%#)-# &)0 .+, 11!(-&+(2) & 4516 7-8+9. :+;
<50=> ?+@=A BC+=8)*. '()*+,-). /0)1) !"DE&
5,#$(!78"- 19
!""2$% ' B58965,-)8FG-,+ )8H50-G1, I50 *+(-*-6H J)0=5K +LM-F
K)8+6(+ 5I *-0+(G+* (9(8-( H0)B1-()8 ,5*+8@& N6%
3)-#%+&()+4 () %+(5-(&! .)+#!!(6
"#)-# 78 O& P50K-GQ. R& 4+6@+6 +*@&$& J50H)6 S)MI,)66. JNTUV. W2XYW2"&
5,#$(!78"- 19* )(:!,;&/- 9) X##X$% Z1)-6 H0)B1 ,5*+8@ )6* G1+-0 ()M@)8 -6F
G+0B0+G)G-56& 92:%)&! 2/ +;# <24&! *+&+(,+(-&! *2-(#+4 = . !. [X!Y[WE&
5,#$(!78"- 19* 90,!;/8 X##X$% '6(+@G0)8 H0)B1 J)0=5K ,5*+8@& ))&!, 2/
*+&+(,+(-,. "#. "2XY!#[#&
5"</!(;" = X##3$% ' M6-\+* )BB05)(1 G5 G1+ (1)0)(G+0-@)G-56 5I +LM-K)8+6(+
(8)@@+@ 5I ]'^@. (1)-6 H0)B1@ ;-G1 65 _)H@ )6* (1)-6 H0)B1@& *-&)0()&>(&)
92:%)&! 2/ *+&+(,+(-,. "$ X$. X"3Y[!X&
5"</!(;" =* 9;:7/-> ? X##2$% ' H0)B1-()8 0+B0+@+6G)G-56 5I +LM-K)8+6(+ (8)@F
@+@ 5I 'J/ (1)-6 H0)B1@& 92:%)&! 2/ ?&-;()# '#&%)()" <#,#&%-; . %. !#W3Y!#DE&
5:7,- @ X##[$% )&!@A& > %#B!)CD & E2D1!#F)GD 2$2%: & '()*+,-). /0)1)&
9(A,< B* @(C/#'( 9D !"EX$% TM`@+G *+B+6*+6(-+@ )6* (5,B8+G+6+@@ 0+@M8G I50
) @M`(8)@@ 5I +,`+**+* ,M8G-K)8M+* *+B+6*+6(-+@& 92:%)&! 2/
,,2-(&+(2) /2%
H2D1:+()" ?&-;()#%4. $& !$. !#[Y!!D&
9#$":;/ E !"!!$% '6)89G-( G0+)G,+6G 5I G1+ B589G5B+@ 0+HM8)089 *+0-K+* I05,
0+HM8)0 B589G5B+@& I#%;&)0#!()"#) 0#% J2)()E!(KE# E&0#D(# >&) L#+#),-;&16
1#) +# D,+#%0&D '' [$. [D#Y[E!&
9$(F/! G !"D2$% ?&+;#D&+(-&! M;#2%4 2/ N>(0#)-# & /0-6(+G56 a6-K+0-G9 /0+@@.
/0-6(+G56&
9$(F/! G !""2$% /05`)`-8-@G-( ObB+0G T9@G+,@& H=?*6O*P <#"(2)&! H2)/#%#)-#
*#%(#, () 11!(#0 ?&+;#D&+(-,. %& TN'J. /1-8)*+8B1-)&
9$(#$;/! 5H !"E2$% OK)8M)G-6H -6_M+6(+ *-)H0),@& Q1#%&+(2) <#,#&%-;. "!.
ED!YEEX&
9$(#$;/! 5H
!""E$% c)9+@F`)88. G1+ 0)G-56)8 B)@G-,+
I50 *+G+0,-6-6H -00+8+F
K)6(+ )6* 0+LM-@-G+ -6I50,)G-56 -6 `+8-+I 6+G;50=@ )6* -6_M+6(+ *-)H0),@$& N6%
3)-#%+&()+4 () %+(5-(&! .)+#!!("#)-# 7R
R& ^& Z55B+0. T& J50)8 +*@&$. J50H)6
S)MI,)66. JNTUV. WE#YWED&
9$/-"I
!""W$% Z56*-G-56)8 -6*+B+6*+6(+ -6 K)8M)G-56F`)@+* @9@G+,@& .)+#%6
)&+(2)&! 92:%)&! 2/ 11%2F(D&+# <#&,2)()" . '# [$. X#[YX[W&
94,;$ JK !"E"$% N6_M+6(+ *-)H0),@ I50 @G)G-@G-()8 ,5*+88-6H& ))&!, 2/ *+&+(,6
+(-,. '% X$. 23WY2DX&
90//7 1 !"D"$% ' 65G+ 56 6+)0+@GF6+-H1`5M0 ^-``@ )6* J)0=5K B05`)`-8-G-+@.
T)6=19¯
) ' W!. !EWY!"D& :OdVeaJNJ
90,/A/C$(C;/! HJ* )(:!,;&/- 9)
!""#$% T+LM+6G-)8 MB*)G-6H 5I (56*-G-56)8 B05F
`)`-8-G-+@ 56 *-0+(G+* H0)B1-()8 @G0M(GM0+@&
O#+S2%E,. $# 3$. 3D"Y2#3&
258
Literatura
!"#$%& '( )*+,-.# /( 01%"2%& 3
!""#$%
'()*+,-)./-)01,2 3-4 56)7&
!-12 4
!"89$% ':6;<1=:*; *+>-(-+>-+;-2 ;1?=10 *+>-(-+>-+;- 1+> =<*-0>1@*0*:A&
0&"*'!1 &2 )/%1&#&3/%-!1 4&5%- 2
!-12 4
!"#!$%&'()*+,%-$%&' !', .+!*-/ &
!$2 B#C""&
!"88$% D)>*+10 ;6+>*:*6+10 E?+;:*6+= C 1 >A+1F*; :<-6)A 6E -(*=:-F*;
!"#!$%&' %' 6+-%#%&'( 7+1%+2 /!'5+( !', .$!$%#$%-# 88
=:1:-=& G+%
H& I& J1).
(-)2 '& '7A)F= ->=&$& K0?4-)2 L6)>)-;<:2 !9MC!#N&
$#5.&6 3
O99N$%
UVW*;6&
$.7%28 9
9' #+3!*&%,#& P<L :<-=*=2 Q+*R-)=*>1> 31;*6+10 S?:6+TF1 >-
!"8"$% U?0:**+E6)F1:*6+ 1+> :<- ()6@0-F 6E ;<1)1;:-)*X1:*6+ 6E ;6+.
>*:*6+10 *+>-(-+>-+;- )-01:*6+=&
!"
!$2 #C!Y&
$.7%28 9
)*&:1+;# &2 &'$*&1 !', 8'2&*;!$%&' </+&*= 2
!""O$% Z6+>*:*6+10 *+>-(-+>-+;- )-01:*6+= <1R- +6 [+*:- ;6F(0-:-
8'2&*;!$%&' </+&*=( .$!$%#$%-!1 6+-%#%&' >"'-$%&'# !',
?!',&; )*&-+##+#( <*!'#!-$%&'# &2 @@$/ )*!5"+ &'2+*+'-+( A&1B 7 '& K?@\72
;<1)1;:-)*X1:*6+& G+%
]& ^& /\_-7 ->=&$& K0?4-)2 UG'`D2 #BBC#"Y&
8'$+*'!$%&'!1 0&"*'!1
$.7%28 9
!""B$% S )-;6R-)A 10,6)*:<F E6) ;<1*+ ,)1(<=&
$.7%28 9
!""81$% a1A-=*1+ +-:46)7= E)6F :<- (6*+: 6E R*-4 6E ;<1*+ ,)1(<=& G+%
&2 C33*&D%;!$+ ?+!#&'%'52 !# OC#$2 OYMCO"#&
E'-+*$!%'$= %' C*$%F-%!1 8'$+11%5+'-+ @G
b& c& Z66(-)2 '& U6)10 ->=&$2 U6),1+
K1?EF1++2 UG'`D2 N"YCM9#&
$.7%28 9
!""8@$% Z6F(0-W*:A 6E =:)?;:?)10 F6>-0=& G+%
)*!5"+ .$&-/!#$%-#HIJ
U& J?_76Rd2 P& I1;<6?:2 ]& ^& /\_-7 ->=&$2 Q+*6+ 6E ZX-;< U1:<-F1:*;*1+= 1+>
P<A=*;*=:=2 P)1,?-2 MO!CMO8&
$.7%28 9
O99N$% Z<1)1;:-)*X1:*6+ 6E -==-+:*10 ,)1(<= @A F-1+= 6E :<- 6(-.
)1:*6+ 6E 0-,10 F-),*+, 6E ;6F(6+-+:=&
8'$+*'!$%&'!1 0&"*'!1 &2 E'-+*$!%'$=(
>"KK%'+## !', L'&M1+,5+N7!#+, .=#$+;# 2 !$2 N#CYO&
$.7%28 9 O99M1$% )*&:!:%1%#$%- &',%$%&'!1 8',+3+',+'-+ .$*"-$"*+# & '()*+,-).
/-)01,2 I6+>6+&
$.7%28 9
O99M@$% Z<1)1;:-)*X1:*6+ 6E *+;0?=*6+ +-*,<@6)<66> *+ :-)F= 6E :<-
-==-+:*10 ,)1(<&
#9"&
8'$+*'!$%&'!1 0&"*'!1 &2 C33*&D%;!$+ ?+!#&'%'5 2 %" #$2 O8#C
O!$/+;!$%-!1 !#3+-$# &2 1+!*'%'5 7!=+#%!' '+$M&*P#Q 7!=+N
#%!' R"!1%$= -*%$+*%!& e-=-1);< )-(6): +& OO#N2 G+=:*:?:- 6E G+E6)F1:*6+ `<-6)A
$.7%28 9
O998$%
1+> S?:6F1:*6+2 P)1,?-&
$.7%28 9( :-;%< '
!""N$% ZG F6>-0= 1)*=*+, 1F6+, N )1+>6F R1)*1@0-=& G+%
S&*P#/&3 &' E'-+*$!%'$= )*&-+##%'5 TSE)U.V( .+3$+;:+* @@N@W( @IIG( <X+YZ(
K+-/ ?+3":1%-2 OY8CO8O&
$.7%28 9( :-.0<5%#$ 33
!""8$% D+ ;<1*+ ,)1(< F6>-0= E6) >-=;)*(:*6+ 6E
;6+>*:*6+10 *+>-(-+>-+;- =:)?;:?)-=&
$.7%28 9( =%>25#-?@ A
C''!1# &2 .$!$%#$%-#2 $& N$2 !N#NC!N"M&
!""8$% `<- F?0:**+E6)F1:*6+ E?+;:*6+ 1= 1 :660 E6)
259
!"#$%&'( #)*+,"#)&+ -!.!'-!'+!/ 0'1 !"#$%$& %$ '#"()%*"+ ,-.!+/ 23/ 0/ 4*%5
-"' !-/6/ 78$9!% :+"-! &+ ;$<8&#,!%#= >*#)*'/ ?@AB?CD/
!"#$%& '( )*+$,-!* .( /!012%*+2 / 2?EEC61 F9* *.!%")&*'# *G !%(&'( "'#.8&))&'( +* .*'!')# &' " +,"&' (%".,/ 012!#$!3%4"= != ?EDB?HD/
!"#$%& '( 3*45$5 6 2?EEC61 : %!+*'#)%$+)&*' "8(*%&), G*% ),! !##!')&"8 (%".,/
5$3!#$"3%-$"+ 6-7#$"+ -8 9((#-:%;"3! <!"/-$%$& = !"= IDJBHAI/
/74$8$9 : 2?EEK61 =!>?@%/+-/3$A ;-.!+1/ ;,/L/ -&#!%)"M'N .%O+!= 3PP Q7= ;%","/
/!012% 6 2ACDK61 B!-#%! (#"@.C(-.-2$-/3% / :+"-! &"= ;%","/
;-9$4",- .( <-=>74",- ;( $7 ;( ;-%-9- < 2?EAA61 9$ !:";(+! -8 ;%$%D
;"+ 3#%"$&7+"3%-$ E)-/! 7$%-$ %/ $-3 *-;(+!3! = #*$R%* O R*%!#.*'-!'+!/
;-,?-% )@( A-%%-9-97= ' 2ACDH61 S& .8! 8&'!"%5)& ! "8(*%&), # )* )!#) +,*%5
-"8&)T *G (%".,#= )!#) "+T+8&+&)T *G ,T.!%(%".,#= "'- #!8!+)&U!8T %!-$+! "+T+8&+
,T.!%(%".,#/ F59, 6-7#$"+ -$ G-;(73%$&= #$= J@@BJKC/
3-?#- B 2ACDC61 B)!-#1 -8 F3"3%/3%*"+ 5$8!#!$*! "$. 5$8-#;"3%-$ / 78$9!%= 30SFV/
+-% :"!!$% C( +-% >"11$% 6D 2ACDJ61 0'U"%&"'+! .%*.!%)&!# *G +*'-&)&*'"8 &'5
-!.!'-!'+! %!8")&*'/ 9$$"+/ -8 H#-2"2%+%31= #$2I6= CIHBCHJ/
3$?%-,*+2 6 2ACCC61 W*'-&)&*'"8 &'-!.!'-!'+! &' .*##&<&8&)T ),!*%T/ 0'1 H#-*!D
!.%$&/ -8 5F5HB9 II JK/3 5$3!#$"3%-$"+ F1;(-/%7; -$ 5;(#!*%/! H#-2"2%+%3%!/
"$. 3)!%# 9((+%*"3%-$/L 2X/ -! W** "'= P/ X/ W*Y "'= S/ 3*%"8= ;/ Z"88!T
!-#/6= IHIBIJA/
3$,4- ;( :$-,5 6 2ACCE61 W"$#"8 '!)9*%R#= #! ")&+# "'- ![.%!##&U!'!##/ 0'1 M$D
*!#3"%$31 %$ 9#3%N*%"+ 5$3!++%&!$*! O 2\/ S/ S,"+,)!%= F/ S/ ]!9&))= ]/ ^/ 7"'"8=
4/ P/ ]! !% !-#/6/ ^*%),5_*88"'-= 30SFV= @CBK@/
3$,4- ;( :$-,5 6 2ACCA61 `a$&U"8!'+! "'- #T'),!#&# *G +"$#"8 *-!8#/ 0'1 M$D
*!#3"%$31 %$ 9#3%N*%"+ 5$3!++%&!$*! P 2;/;/ >*'&##*'!= 3/ _!'%&*'= ]/^/ 7"'"8=
4/ P/ ]! !% !-#/6/ `8#!U&!%= 30SFV= ??EB??K/
3*5E '( !"#$%& ' 2ACCC61 : (%".,&+"8 +,"%"+)!%&Y")&*' *G ),! 8"%(!#) +,"&'
(%".,#/ 5$3!#$"3%-$"+ 6-7#$"+ -8 9((#-:%;"3! <!"/-$%$& = %"= ?ECB?I@/
F-!-%-G$ 2AC@E6 1 0'G*% ")&*' ),!*%!)&+"8 "'"8T#&# *G $8)&U"%&")! +*%%!8")&*'/
5Q, 6-7#$"+ -8 <!/!"#*) "$. R!@!+-(;!$3 = = @@BDA/
F$,4"!> H 2ACK@61 :'"8*(&!# <!)9!!' $8)&.8&+")&U! *-!8# G*% +*')&'(!'+T )"<8!#
"'- +*U"%&"'+! #!8!+)&*'/ Q%-;!3#%*/= $%= CJBAED/
F$,4"!> H 2ACDE61 ]&'!"% %!+$%#&U! !a$")&*'#= +*U"%&"'+! # !8!+)&*'= "'- ."),
"'"8T#&#/ 6-7#$"+ -8 3)! 9;!#%*"$ F3"3%/3%*"+ 9//-*%"3%-$ = &!2IK?6= C@IBCK?/
F>7!!-9$, 6 2ACCE61 '#"()%*"+ ,-.!+/ %$ 9((+%!. ,7+3%@"#%"3! F3"3%/3%*/ / 4*,'
Z&8!T= 30SFV/
F*%I <'( F-%I JF 2ACCH61 V' "[&* "Y")&*' *G .%*<"<&8&#)&+ +*'-&)&*'"8 &'5
-!.!'-!'+&!#/ 0'1 M$*!#3"%$31 %$ 9#3%N*%"+ 5$3!++%$&!$*! KS 2\/ ]/ 3"')"%"#= L/
;**8! !-#/6= 3*%("' 7"$G "''= 30SFV= JCABJCK/
F,7I>! 2ACAD61 V' ),! '")$%! *G #&Y! G"+)*%#/ '!$!3%*/= $= I@KBIKH/
260
Literatura
!"#$% &
!"#$%& '() *)+,-./) .012*-,34) 25 ()*)6.-7 ,36 )3/.*230)3- .3 6)-)*8
!"#$$%&'() "* +,$ -.+&"'./ 0#.%$12
0.3.39 -() 1.):,+6 1,--)*3 25 9;.3),81.9<=
"* 3#&$'#$)>
> ?#$@??#=
!"#$% &
4"5!'./ "* 0(!&#5/+5!./ 6$)$.!#,
!"
!"#$% &
0''./) "* 7.+,$1.+&#./ 3+.8
+&)+&#) #
'())(*(*"+ ,
3907
4"5!'./ "' 0/($:!.&# .'% ;&)#!$+$ 7$+,"%) !
-(./$ 01
<5==2 3$+) .'%
32)+$1) $
-23$(! 4
>&'$?!'@ !$A5!)&B'@ )2)+C12 . )+!5A+5!2 D"%1@'E'C '$=?B&)/")+&
!"#!%& A2**)+,-.23 ,36 4,;<,-.23=
>
> BBC@BDB=
!"?E%& '() 0)-(26 25 1,-( 42)F4.)3-<=
>
> !G!@#!B=
!"D!%& A201;-.39 -() 0.3.0;0 H++8.3 .< IJ84201+)-)=
>
> CC@C"=
!"CD%& K;LL7 <)-< ,< , :,<.< 52* , -()2*7 25 12<<.:.+.-7=
!%> ?@#D=
#$!$%&
M,N,+OP<NO 1*O4)> QKK RS> J*,(,=
=