Nalejvárny

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Nalejvárny
Příloha A
Nalejvárny
A.1 Nalejvárna z teorie míry
•
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(X, X )'
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S
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n An ∈ X <
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B 0 $-)037 ! σ(B)<
• X
! H7)(&! f : X → [−∞, +∞] F0($#,' /!
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Q
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i∈N Ai ' Ai ∈ Xi < C$73%)$#$7 σ A05B!>+7 X >7"7 $-)03$#0F
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i∈N Xi <
•
)0 (X, X ) ! H7)(&! µ : X → [0, +∞] 4*567 1&1
:%; µ(∅) = 0'
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•
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µ
∞
[
n=1
An =
∞
X
µ(An ) .
n=1
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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: $# '+0&'<'%,: 01#4052&-4- X =
P(X)! %#4= σ 93&>#[email protected] "# 05%#,6,/ +,5(',3 X ) A.'%+#%'<2- +/.3 032
23(47 054+,5(',: A ⊆ X 01'134/ 056#% "#"/<B 0.C2DE ν(A) = |A|)
2. F',G+ 8-2&34,/+ 01/2&34#+ "# 2.$(&!&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&[email protected]#,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#) [email protected] f, g : X → R [email protected],2<#!
5?: X 9+:1'%#&,7! 3 µ +/.3 ,3 (X, X )) H32 4#O,@"#+#
f = g µ9$25.5 [email protected]#
⇔
µ( {x ∈ X ; f (x) 6= g(x)} ) = 0 .
H1' 8,36#,/ $# 63$%5 [email protected](/C- 82.3%23E µ9$)C)
• H58,3+#,#"+#! (# C 4'$2.7%,/+ 01/034:! %") 24=( X "# 25,#6,- +,5(',3! "#
.5C,5$% [email protected],2</ f 3 g $25.5 [email protected]# C8B&#4#+ 2 µ #2C'C3&#,%,/ 05([email protected]! (#
[email protected],2<# f 3 g $# .5C,3"/ 3&#$05Q ,3 ,5$'6' +/.= µ! %#4= ,3 {x ∈ X; µ({x}) > 0})
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F',G+' $&5C=! +,5(',3 %8C) 0&,7 +/.=! ,3 ,/( $# [email protected],2<# f 3 g +3"/ .5C,3% $#
+D(# $ 4C5"'</ [email protected],2</ f 3 g +:,'% 3 ,#[email protected]$/ #I'$%5C3% (-4,- [email protected],'C#.8-&,/U
+,5(',3 "325 C 4'$2.7%,/+ 01/034:) F325 01/2&34 ?=<B @C#4& "#4,5.58+:.,[email protected]
K#?#$>@#[email protected] +/[email protected]) [email protected] f [email protected],2<# ,3 R '4#,%'<2= .5C,- ,@&# f ≡ 0! 3 0.5
23(47 a ∈ R! [email protected] ga ',4'2-%5.5C- [email protected],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$25.5 [email protected]#! 3&#
+,5(',3 ?54D x 24# f (x) = ga (x) 0.5 CP#<B,3 a "# 0.-84,-)
• 4.'.)56.;# ,(*.5!17) [email protected] (X, X ) +:1'%#&,G 0.5$%5. 3 µ ,#8-05.,- +/.3 ,3
(X, X )) V3(47 X 9+:1'%#&,7 [email protected],2<' f : X → [0, +∞] 3 23(47 +,5(',: A ∈ X
25,$%[email protected]<#! 050$3,- ,301/2&34 C @6#?,'<' [email protected]', XYYZS!
01'134/ 6/$&5
Z
A
f (x) 4µ(x) ∈ [0, +∞] ,
2%#.7 $# ,38GC- ,(*.5!17 8 [email protected],2<# f 01#$ +,5(',@ A 054&# +/.=
µ) [,%#>.-&
R
85?#<[email protected]"#
05"#+ C-(#,7B5 [email protected]%@E 0.5 X 25,#6,[email protected] 0&3%/ A f (x) 4µ(x) =
P
x∈A f (x) · µ({x}))
• <(*.5!&#"*.7(1 =6(0/. "# X 9+:1'%#&,- [email protected],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 ?
• [email protected]$& σ % +&(@&A;< :B*? H)I (Xi , Xi )2 i ∈ N ;CD%#!$-> '/("#(/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
!"# $%&! N
µ '( &#)*+, σ -.#*(/*01 *!23)0 4( !"#$%&' $+& µi ! #2*!/5'(
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@<%.8!=($ '( ()*&+!,'-+%. /&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
[email protected]&%) ;&).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
#$%
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&'()(*+ '(,(*-. /: #%+:> (L, ) >'&%395 (: #$% &'()* 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=
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/: #%+:>
<.< ! !7;<.< 3
L2
(L, )
(%!36 '(/74(804
R ⊆ P(N ) ≡ {Y ; Y ⊆ N }
>'&%395 (: &'()F #%)<7%(!7'
&%7:87*
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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
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I#,7*E% +3'6.
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#$% &'()*
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x
z ∈ L2
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x
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z ∈ L2
+: $%6.<0
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x0 ∈ L
K:70 >% 7!1 /!7*E% 7:( +.#$:<.< #$F6)7*
4$)*=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'!$737>63#[email protected] 5 1%( 67'805 (: #$% &'()* 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#[email protected] 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% #$3&. L2
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244
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!"#$ %&'( 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*! !$$/9& %& @>2-1,- [email protected],38!+>, A1&>9,BC &#"9)$!+("B '1!(*!1 (D2E!9&2
Rn 0 "$& n ≥ 1 3 ,)"!9)+ RN '1! 2,!5),# '1!2-,,B:F N 0 :!5 %& ,&%83(*-%G/
[email protected],38&,/ &#"9)$!+("4F! '1!(*!1# + *-:F*! ("1)'*&:F A+)@ '[email protected],>2"3 ,3 (*13,HIH 8) J K6LC6 M! '1!*!0 ,/5& #+&$&,> N3"*3 %(!# N3"*):"D A35 ,3 %&$,# +B%)2"#C
+D#5)*3 + "!,*&O*# ('&:)>9,/F! '1!(*!1# RP(N ) 0 "$& P(N ) ≡ {A; A ⊆ N } %&
(D(*42 '!$2,!5), ,&'1>@$,4 "!,&8,4 2,!5),D '1!2-,,B:F N A+)@ J I6P6HC6
QR"[email protected] +-*G),D @$& #+&$&,B:F *[email protected]&,/ [email protected]& ,[email protected]* ,3'S/"93$ + ",)@& AT1U,$(*&$
VWXPC6
Y,!5),3 K ⊆ Rn 0 n ≥ 1 (& ,[email protected]+> *+#(,-#. 0 %&(*9)5&
∀ x, y ∈ K
•
∀ α, β ≥ 0, α + β = 1
α·x+β·y ∈K.
7!,+&O,/ 2,!5),3 K %& /0&(1,#20 '!"#$ xn ∈ K, xn → x ∈ Rn ⇒ x ∈ K 6
3"4,#', A#@3+S&,4C "!,+&O,/ 2,!5),D K ⊆ Rn %& $&.,!+>,3 %3"! $)2&,(&
&5##.6+ +7&8/ K 0 *&$D 2,!5),D
n
n
x∈R ; x=
X
z∈J
αz · z
'1! "!,&8,!# J ⊆ K ,
X
z∈J
αz = 1 , αz ∈ R
o
,
:!5 %& A+5$DC '!(#,#*B 9),&>1,/ '!$'1!(*!10 *&$D 2,!5),3 {y + z ; z ∈ L}0
"$& L %& 9),&>1,/ '!$'1!(*!1 Rn 3 y ∈ Rn 6 1! ,&'1>@$,!# K %& $)2&,(& K
AZ $)2&,(& LC :&94 8/(9! 2&@) 0 3 n6
Příklady.
1.
793():"B2 'S/"93$&2 #@3+S&,4 "!,+&O,/ 2,!5),D %& +8)%+ 0 $&.,!+3,B
%3"! *+#(,-#. +7&8 ,-%3"4 "!,&8,4 2,!5),D B ⊆ Rn ;
!"# (B) ≡ { x ∈ Rn ; x =
2.
X
z∈B
αz · z
'1!
αz ≥ 0 ,
X
z∈B
αz = 1 } .
[E&:,-%G/2 'S/"93$&2 %& +8),9!0 ,-"$D *45 ,[email protected]+3,B 4#+6+'%:#0 $&.<
,!+3,B %3"! '1R,)" "!,&8,4F! '!8*# #@3+S&,B:F '!9!'1!(*!1R0 *%6 2,!<
5), *+31#
n
{x ∈ R ; hx, yi ≥ c},
K,=9):"B *&12/, %&
"$&
+8)6,9!+#6
n
y∈R ,c∈R
3
hx, yi ≡
n
X
i=1
x i · yi .
246
A Nalejvárny
•
!"#$!% K ⊆ R & n ≥ 1 '( !%)*+, !"#$%"&' ()$*$' & -('./$#(
0$1 ≡ [0, . . . , 0] ∈ K &
0$$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!"#$!E ∅ =6 B ⊆ R F
X
!" (B) ≡ { x ∈ R ; x =
α ·z @>" 6"!(G!": ∅ =
6 C ⊆ B % α ≥ 0}.
n
n
z
z
z∈C
H @[email protected]%45 @>,)4!B 9!"#$!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!$=( '.5!E + @[email protected]%45 @"/E."@: K ⊆ R -( @>K'(6 K ' -(C"
,;!*(2&+& "/06!#,"!(& .-2 F = K ∩ {x ∈ R ; hx, yi = k}& 64( y ∈ R % k ∈ R
-'": .%6"+B& #( K ⊆ {x ∈ R ; hx, yi ≥ k}2
• 36+$+%/(!.!8 4(O!$=( !(@>,)4!B '.5!E + @[email protected]%45 :)%+A(!BC" 6"!+(7!8C" 6:D
#(/( K ⊆ R -( 9!"#$!% 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 ",(", ")>"? [email protected]( !$ 1,,5 4)6,&*
!"#$% &'()$!) k − 1* 2&) k .) &'()$!) K !) $56715.8 724')&* 5$:-';2%
72"')4A
B5(,6C).(#* .)&'[email protected] !"#$5 &'()$!) k .) ;)[email protected] 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)$7* /[email protected]# 2&%4
∃y ∈ R
n
∀x ∈ K \ { }
hx, yi ≡
n
X
i=1
x i · yi > 0 .
• ;20!4'< :)$)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<, $)/[email protected]&$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;',[email protected]$8 .)<-5$ /[email protected]# ")<&%* 2&%4 (@
2,$)M$# ($,<, )>"0)(@-$8;< /5/0!2I* /C'M)(4 254&7 /5/0!)2 .) :)$)0,15$7
05;',[email protected]$8( 1)2",0)(A
Fakt. NC8&5 !"#$ 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 .) $)/[email protected]&[email protected] 2,$)[email protected] ($,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
[email protected]*!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)[email protected] #$ +' &$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
[email protected]*!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>[email protected] :2$45 ρ #:'5 a1 ) an 0 2"!+?": 5>[email protected]
#$& , 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$4 2#1# @ A 1) B &[email protected] +I( %$>! A ) B "3
L6'>'*M5#!0 ($ 4$N&!.$ :2$45 >)/*&5#$ ! 6;A6)4 n = 1O
• P2$4 :$ &)>-,< ;#(+.0 #$:+2!($ &),A. a1 , . . . , an #:'5 [email protected]"A 5>[email protected]
• 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),# / [email protected]&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-# [email protected]$'% a1 , . . . , an+1 ! $&, n ≥ 4! 2,
6#;%:* '30$( ai
aj / G )($#/>! +, 1 ≤ i! j ≤ n ( 1 < j − i < n − 14
• [email protected]&0* 86(9 G 0(& N 1, -#3( ,&! 1,2)'3+, ∀ a, b ∈ N ,=32)%1, / G 0,#63,0)#K
/(0> I,2)( ; a &# b4
• B,#63,0)#/(07 86(9 2, 0(;7/> "'-9 ! 1,2)'3+, 1, 2#%/32'7 ( 0,:> +>&07 0,#K
63,0)#/(07 [email protected]$'%24 B,#63,0)#/(07 86(9! $),67 0,:> 0,#63,0)#/(07 [email protected]$'%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),# / [email protected]&0*: 86(9% G 0(& N 1, "#2'#%"0#2) %;'D a1 , . . . , an+1 !
n ≥ 3 )($#/>! +,
E3F an+1 = a1 !
E33F a1 , . . . , an 12#% 6D;0G!
E333F ∀ i = 1, . . . , n ai → ai+1 / G4
H*2'# n 2, 0(;7/> &G'$#% [email protected]$'%4 L63,0)#/(07 86(9 2, 0(;7/> 045),(4)& ! 1,2)'3+,
/ 0<: 0,,=32)%1, #63,0)#/(07 [email protected]$'%24
Fakt. L63,0)#/(07 86(9 G 0(& N 1, ([email protected]$'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; [email protected] 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
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*!,;& 6);&"- *6;' 1!O#;+>#- #>6)!1;+#=J
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ei =
xi 1P (x)
7/;
i ∈ N,
RN
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Z
σij =
(xi − ei ) · (xj − ej ) 1P (x),
RN
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|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) = √
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(x − e)⊤ · Σ−1 · (x − e)
· exp −
2
7/; x ∈ RN % 1!O#'*! ,'68;8' 7E26)'Q#G </!3')>/#2 34'66;+6"[email protected] $2/- +(,)!1!$
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Pr
[email protected] θ1 , . . . , θr 05/&"+; G! θk > 0 5
k=1 θk = 1;
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r ∈ N
(= <C#.18D*+
H#,1# *5 /&*!9*+ (*&G-*) 23 "I7)%&"+( <%&.0&%[email protected]
[d1 , . . . , dr ] ; dk ∈ N,
J8.0&05 2"K9- 5%-0(!0-$/+ (#%)@
p([d1 , . . . , dr ]) =
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k=1
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· θd1 · . . . · θrdr .
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Poznámka.
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ξ1 , . . . , ξ d
! *=?&,*I "I7)% 2"-' N!O*-$! [email protected] ' %&',)1!*#
X = {1, . . . , r} . ?8.0&0&8 p(k) = θk ; k = 1, . . . , r; <5/ "!/0&% [ρ1 , . . . , ρr ];
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23 "I7)%&"+( <%&.0&%[email protected]
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[θ1 , . . . , θr ] ; θk > 0,
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=1 .
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· θα1 −1 · . . . · θrαr −1 ,
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R∞
Γ(α) = 0 y α−1 · e−y ,y ; α > 0
D(α1 , . . . , αr )A
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(θk , θl ) =
"5%
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αk
α ,
−αk ·αl
α2 ·(α+1) ,
(α−αk )·αk
α2 ·(α+1) .
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α=
k 6= l,
Pr
k=1
αr ,
252
A Nalejvárny
Fakt (základní fakt z hlediska bayesovského přístupu k učení).
Pr
!"#$%&! '$(#)
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dk ∈ N, -.! k=1 dk = d, θ =
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*%%%+ π(θ) /0 9%6%4:$!#3;3 637.8$!2) D(α1 , . . . , αr ),
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[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
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=
r
Y
(αk +dk )−1
θk
,
k=1
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#3;3 637.8$!2) D! !"#$%!&'"( - /1$#%23/%4-5/1A
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