f f v v v f v f I V V V F G I I(F ) = I(G)

Transcript

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2 M! NPOIQSRSTVUWTAXRZYS[]\A^R`_a^_ab c +*&+(+D($4&%'d&d.0*,13*,4&%'(e8&2?8-fD1921'#++&>%'%9 AI&+(+2>$4%9gg@> #+&+(@h>&v)i V,,,, j kl;>+-d($4&%' (!?2$m&%'(I4&1'>+S!#$=>gnJoIpeqI(DE%r;>+-D($4&%'s()>K8&2221'#+ >%'%9EBf&t F F u (DI1'+@>%9v&*,+ V F u D1 (D)&+:=>%'=>$>+fv G*#%9?wn qxe+s=(%9=($>+fe F F F F u D1 Fy A Fz 2 #+fe(i=(%9=($>+f) G&*,%' #V F (Fy Fz G ) (Fy Fz G ) (Fy Fz > ) (Fy Fz ) D?+?EI=>%'=>$>+fV) F { F}>~EL 9} t B? %r; #+ƒ=@a19v*:>+ +&#*,21'#+ƒ8*,221'>%'%'>$>+f8*#2(+f2 &=D=( (+ (p ( q)) r p q r 19%'1'*,+fE%'(I8&1' #1H=(ID19.0*,+f(wt G&8&1' G&8&&13,, c +S8-(d*,-221<21'%'1'2>)&+s+&,V*021'#+S8 #%'#+*#1'2 #:&+&w+&#*019?+s*#24 #(@A(+f2#5 p q r ˆ OEŠ ^armuwtaœ[sb c + =>Ž+&19a(v! #+@A21'#+w.#=(2192= f G f G f G f G f 2&w%r;>+2>$m4&%'md.0*#%9(&gm.#=>192= {v, f} #:2& {v, f} {v, f}.0*,%'>&v)&*,+- t {v, f} v f f f v f v f v v f f f f f v f v v v f v f f v f v v f f v v f v f v v f f f v E+m & & r W & ( J š# r L ( fœž Wœ0Ÿf ( & ( J <;+ 2 #D (+2>$4%9 xw&+&*,8&8%913@>*,21' #+ &*,+ I V V V {v, { F}>~EL 9})s.0*,%'*,21' 5 f} #+:D& t V = {p, q, r} I(p) = f, I(q) = I(r) = v E+e J L?œž œ0ÿ? ( r - <;&+&!?2$m&%9 l.0*,13*,4&%'(l8& #8?21921'#+&+&(%9%'( I F {v 1, v 2,...v n (DC%9l82?%9?+& #($ (+f } <;&+&s1'+f2>8&=av*021' #+ 2(% Bf& I {v 1, v 2,...v n } u D1 F = F 1 G I(F ) = f (I(F 1 )) u D1 F = Fy Fz 2D8C5 Fy Fz G Fy Fz G Fy Fz G ] I(F ) = f (I(Fy ), I(Fz )) (28C5 f (I(Fy ), I(Fz G )) f (I(Fy ), I(Fz G )) f (I(Fy ), I(Fz )) *ž.?(@d%'*.0*,%'*019?+ 8&2=@A=>+f2?G I I(p q r) = v d>k! #$&%'( > 2 #+f Ÿfª<! & J F L ( F«n q 21)A2>&%'>$(+? 21rG8 #2#&2 1'+?>8&=A*,21'#+ G F G I #+:* 5 I(F ) = I(G) { F}>~EL 9}h gt p q p q p (q r) (p q) r A22I>+&1' >l=(bf&1'.0*,%'>+@>e$ #+f2b?e%' x *#D@>1'*,219 nj<;&+8 #1'+f.F&/2=>$ *#+?1'Bf&žqhG#+a8 #2V*

3 !#$&% ')( *+, (2((GF2#19 G-w$0/>$s8 #&)%' G&$ *,13I8*#I8 # 5 p q r c +.#=(219Ž *#D1d%9 %'#13 6 #?*#+š>`%3* 13x21'4&19.F192= 8*#V*,8&8 #2*, GAS& 8*, V*,8&8 #2I*# 5 E+w!#$&%' { F}>~EL 9}h gt (DE&+&s & W ( J 21- r D1 G&8 #&I#s19+f2(282=>*019?+ G 5 F I I(F ) = v p p p (p q) q (p r) (q r) p q E+w!#$&%' (DE&+&a L ( J r 31 J 21 G&8-?&I2?2s1'+f2(28&=AV*019?+ G 5 F I I(F ) = f { F}>~EL 9}h gt p p p (p q) q 4 1 G&?+S&1HvBf& «J«5! 5 I(F ) = v I F E+/!?2$m&%' & L ( rª< W w<;&+ >+2>$m4&%9 /! #$&%'( 21f2#&219+f2(282=>*019?+ F x 6# «(ŸFª< f $6# «Ÿ7 Σ Bf&1 *01'DJ*,19 nj@#;x2 Σ &19sB?1C2*,213xJ*#1HI2?2)%'(!#$%9) 8&9;:=<;:>9;? t qi2*,213xj*#1he*, Σ F Σ = F { F}>~EL 9}h gt {p, p q} = q {p r, q r} = p q E+:(+D($4&%'sw!#$%9I(D)«J«5 J«@ r (;1'%<AK13xa*,S$?19+-I&+&s1'+?>8&=A*,21'#+SB?1C%9a*01'DJ*,19(5 xe*01'dj*,19e8*, {p q, q p, q} I(p) = f, I(q) = v E+ (+2>$4%9&! #$&%'(wx J «( J«5 & J«@ J?A6# L >œ0,b 6, - Jœ0 21/*#@A&+ 19+f>8&2=>*,21' #+ +& %' *0213DJ*,19(5 { F}>~EL 9}h gt {p, p} {p p} {p, p q, q} {p r, q r, p, q} 8&9;:=<;:>9;? t Σ = f Cwœ0 œ0 JŸf (Ÿ7t c +S* D1C>ED(&%9($>+fvD1 (DE@A #+f*?1'@a2?19#5 Σ = F Σ { F } D È XGFŠ bih R`XGFŠ ^a\abih E+ J J > Ÿ?œ0 & (D +&e!?2$m&%'bf&1f(d(g02#19 +&/.0*#213*,4&%'e82?8-fD1921'#+&+>%'%9?G2#19 +&/+=>?*,21'#+&/.0*,13*,4&%' 8& #8?21H19?+&+&(%9%'#5 { F}>~EL 9}h gt G G G 5 p q p q E+! #$&%' Bf&1exa&+ 1'Jx #+-@h21' #+ `@A #+;Jx #+@A21' #+s%919d=>v*,k x 1H D?K5 -œ7 W -œl7 & r B J«=M( $6, ( J - ANPOIQSRUT (p q) ( p q) 5 E+!#$&%' 13-Jx?+@h19?+a%919D=>V*,K x 1H D?K5 -œ7 W -œl7 & r 6# VM( $6, ( J - WNPOIQYX0Tž5 ( p q) (p q)) ZK[ ŸF œl\v7 t$]?2!#$&%' *#%'@>&%_^e2?8-fd1921'#+&+>%exm=(bf&1'.ž*#%9(+f2 +&! #$%92#-s!#$ +&?2$ *,%'w1'jx#+-@h21'.#?5 `baa~c9l? d:+e!<;:>9;? 8-*,)=(@A2>+@>g2&)%'s+& #$m4&gs.0*#213*,4&%'(?gf>#+d2@a21'.#sw%'* J*ihA?+S2&1'.ž*#+f2t 4?1H % ;(+2>$4%9aI.0*,1'*#4&%9I<;+&w!#$&%' 5 4?1H {p 1, p 2,..., p n } F H i = q 1... q n *ž.?(@

4 !#$&% ')( *+, - u q j = p j 21C&*,+-I%'*%'19?+& S*#4&%'(*,:s.?=>1H=#G i p j.0*, v u q j = p j 21 &*#+I%'* %'19?+& `*#4&%'(*, w.?=>1h=#g i p j.0*, f H xw=(bf&1'.0*,%'>+f2 i (DE.F*#19w&*,+-I%r;19+f>8&2=>*,21' #+:s%'* %'19?+& &+&13Bf&>$(+?5 i F %3* d #4>+F& >+ 82(+*,+fw%'*`1'Jx#+-@h21'#+ ( H i 8-?&g2?v%'( >%3gBf& i F (DE.F*#19 %'* %'19?+& 5 i Bf&1Cx).F*#19s21 p ((p q) r) 8&>++&>+fE%'(I.0*,%'>&V #? (DE=(Bf&1'.0*,%'>+f2 p, q, r v, v, v v, f, v v, f, f Bf&w%r; (p#+:8 >v21'$ 8%919Ž>v>+ q r) q r) (p q r) (p r) (p q r) 8&&13)(+ 5 (p r) (p q) v}>~c<le } e/%'* +&?+:&+&13@A192=aw%3* d 5 ds$b/($#g ZK[ ŸF œl\v7 t$]?2!#$&%' *#%'@>&%_^e2?8-fd1921'#+&+>%exm=(bf&1'.ž*#%9(+f2 +&! #$%92#-s!#$ +&?2$ *,%'w@a#+;jx#+@a21'.##5 `baa~c9l? d:+e!<;:>9;?st c +S8-(v8&>+&2w%3* d s%3*m+=>?*,21' #+SAE+&1'>)%'w2=D%HV*0(5 c +S8-(d*,-221C@A?+D13=(2(/%'( G i = q 1... q n *ž.#(@ u q j = p j 21 &*,+-I%'* %'19?+& S*#4&%9*,:s.#=(2192=?G i p j.ž*# v u q j = p j 21C&*,+I%3*%'19?+& S*#4&%9*,:s.#=(2192=?G i p j.ž*# f G (Dw=(Bf&1'.ž*#%9(+f2 i x).f*#19s*, &*,+-I%r;19+f>8&2=>*,21' #+:w%'*%'19?+& 5 i %'* F #4&2>+F&m>+782(+*,+fd%3*@>#+LJx?+@h19?+7( G i 8-?&d2#-v%'( 2>%3gB? i F (D)J*,-22 %3* %91' #+ 5 i p ((p q) r) Bf&1g(DJ*, &>+&+>+f`%'(.ž*#%9(&V # # #? (D p, q, r v, v, f f, v, v f, v, f f, f, v f, f, f =(Bf&1'.0*,%'>+f2 ( p q r) (p q r) (p q r) (p q r) (p q r) Bf&w%r; #+:8 >v21'$ 8%919Ž>v>+ ( q r) (p q) (p q) 8&&13)(+ 5 ( q r) p E?+:&+&13@A192=aw%3* 5 ^ #&e$ >D2E+&)!#$%9E2# d? G#?+ 8 >&/19%'1'2>e%'*a&=>$ *,V@V &)@A?+x2-@h21'.#)82=@A=(&>+f2?G #+S8 >d*#d1 t u 2($ 8%'*?@A> 8*,)%3*m! #$&%'w=(bf&1'.ž*#%9(+f2 p q (p q) (q p) u 2($ 8%'*?@A> 8*,)%3*m! #$&%'w=(bf&1'.ž*#%9(+f2 p q p q u J*,1'2w(@A(+w%9I+&=> f*021' #+-/8-*,I%9!#$&%'()&w67#?*,+ Gf%'w8&%' %r;19+f=>19(&ew%'*m!?2$m&%9 u *,8&8&%'13B?> %3* 1'D21'4&21'.F1H= 8*# *#8&8 #2 *# Ge # % ;1'+F.#(2#G JxBfC; #4>+&1' %3*ƒ!#$ +& #$ *,%'g@v &(@V =>#5 p ((p q) r) p ( p n d gq q r) p ( ( p q) r) p ((p q) r) (p q) (p r) q p (p r) ( q r) (p ( q r) n

5 ! #$ %& ' ( ) * + E+K6# J W«(DE&+&a&1'Jx#+@A21'#+:s%'1H22=>V*,&KL5 c +`2V*ž.ž*#19%'%'>V*$ *,1'+f2(+*,+f)&+&13Bf&>$>+fv*ž.?(@g()@>%'*#D>5 k * 8& #821'=A=w& (;>K8&21'$ss%'*mJ*ihA #+:2&19.0*#+?t ZK[ ŸF œl\v7 :t ]W#g!#$&%'w(DE=(Bf&1'.ž*#%9(+f2 &+&a@a?+ljx#+-@h21'#+:s@a%3*,-d>5 c +S+&,>V* +`(+D($4&%'as@A%3*,-D) #+f)%'*@>#+ljx?+@h19?+`xe=b?19.0*,%'>+f C(F ) 5 F C((p q) = { p q, p q} E+ 1'+f2(28&=AV*019?+ «( r«5 & D1s>7D(&%9($>+f7D1s(%9%' 2*,213xJ*#1H 5)l+ 8*#D1'@>&%'19( F (DE&+&a*#+f21'%9? #1'#5 x C(F ) C(F ) F M,.-)_a\Ab YSb/, OIH X\A[ U TAXaR 8&9;:=<;:>9;?& t 4 1 (Dw+ %'1H22=(*#% C;&+@>%'*#2 G<?+ +&#2(* %3*:@A%3*,2#+fg2#-w%9w%919D=>V*,K u C C \ u 2 #+fe@a>&k` 2*# 5 C u (p q r) \ q = p r k *@>%'*#D (D)&+&sœ0ŸF«(! & & L )@A%3*,2( C Cy > Cz >;19% AK1'D2w&+:%'1H22=(*#% 2>%WB? u u (DI+S%'19D2=(*#%C u Cy G u (DE&+:%919D=>V*,%C u Cz G u C = (Cy \ u) (Cz \ u) { F}>~EL 9}h gt u A #++&>+f)%'*=(2#%'.0*,+f2 p r q r p q u A #+&+>+f p q r r s t q p q s t u A #+&+>+f A p0 q r p q s q r q s p r p s D2(+f21' #+CG(%9%'()+w&#+&+&(+fE &«r s u A #++&>+f p q p q u > #+&+>+f)%'*@a%3*,-dw.f1'&s+&,=> 5 p p E+v œž F W - œ21ÿf«- r J E%3*a@A%3*,-D F 8-*,221'/E% ;>+2>$m4&%9dv@A%3*,-D a@>%'*#d x/&+&v2&192ež-+&19 C C 1, C 2,...C n 2>%'%'(EBf&?`419(+ C i C C i (DE&+&s=(2 #%'.0*,+f2sa(K C j 8&=(=>+f(G&> 5 C n = F 8&9;:=<;:>9;? t C F {p, p q, q r} r >+:8&>+*#+? 3

6 !#$&% ')( *+, - G C 1 = p (DE&+&s=(2 G&*,+I% ;(+2>$4%9as=(8*,2(G C 2 = p #%'.ž*#+f2w q C 3 = q C 1 A C 2 G C 4 = q (DE&+&s=(2 G &*,+I% ;>+2>$m4&%9sa&=>8*#DG r #%'.ž*#+f2w C 5 = r C 3 A C 4 5 E+sœžŸV5 W ( J <;&+:>+2>$m4&%9sa@>%'*#2( (D)&+&s8&>.#sw%'*@>%'*#Dw.F13 C 8*,221'E 5 C 8&9;:=<;:>9;? t C v}>~c<le }gt 1'$8&%'1'Bf& $ *#1' +C;1'$ 8%913Bf&s «9L?3:e2} F}A~v '}wt C F G C C(F ) C C(F ) C F C = {p, q} F = p q { F}>~EL 9}?} e!a : <L:+>9L?:t n q n #q n q>n 3 q p q r p q s% ;(+2>$4%9sa@>%'*#D n.#q p q n?q?+`?421'>+fv2@>@>(d1'.#($ (+f n?q n.?qanfq n,q q n #q>n fq p p n 3 q n?q nfqan #q r ^a\at>ỳ TVU O Ỳ b \A^ F -)_a\ab YSb, OIH X\A[ U TAXaR Cwœ0 œ0 JŸf (Ÿt0kW*v >?%9E=(2 #%'21'#+m(D «& J nj2#&+-qhg,@#;x2 1'2?22=D?%9.0*#+?ex@A?+D=Bf&>+@> 2=>$ *,+f213bf&s()@>%'*#2()?+?)(%9%'s(d)1'd#5 Xm œ0 J r &!œ0 tv]w #v@a%3*,2 =(&&1Hd<;&+ (+D($4&%' v@a%3*,-d8*,)=(2 #%'19?+ x@a#+-d=b?>+@> F C 2=>$ *,+f213bf&s 5 4 C 1 *,%'#V C F C = F Xm œ0 J r &!œ0 t v+s(+2>$4%9sa@>%'*#di*#$a2*#+?e+&w2=>!*,21' #+:(DE@A#+f*?1'@A2?19#5 4 1 *,%'#V C C = f D XmŠS\AOU [`YSb Ỳ b \A^ Š OUSX Ỳ b YSb, OIH X\A[ U TAXaR ZK[ ŸF œl\v7 St&k * $=A &s)=(2#%'21'#+`(d 6# 27 \f ( 8-?&I%'* =A!V*019?+CG@,;(DD 19#G21 >+-D($4&%'sa@A%3*,2(I@> #+f2v*#13@h#1'2?g*,%'#v/1'%c>kf13d2w&+&s=a!v*019?+s 5 (DI+ C C v}>~c<le }gt&l%'%9s+ ;xe8*?)@a?$ 8%9 >2w8-?&)%3* =(-@h21'#+ 5 {? eah ~cawt #+ * C F C = F C C( F ) C C( F ) = f v}>~c<le } t %3* $=A2 F&`x +& #+ =>2(2$1'+&1'D2?GW%'`2 &=( # >$19 Bf&`%'* 2=>!*,21'#+ >KF13D2#GW19%+&:19 8*?)@A #$$>+f)%3* 2#&.?>Gf+1CD1C%3* 2@V &>V@V &wdg>$ 1'+&?5 [Sb)\AŒ[`bIH H<U F ^wuwo)_at>bih!$#&%(')*,+-%(.0/%(#1+2*,+2354 c + +C;*#2?2132 %3*6)=(2 #%'19?+ BfC;>+f2 #+fs%r;&+& xw 9ž > :+>87ž}nJ@,;(DD 1'2 +C;*:*,@>&+ %'1H22=>V*,%<+=>?*,219q v}>~c<le })t:9 %<;*a*, $?19+e&+&g@A%3*,2)8-fD1921'.#d&*,+e&+ (+D($4&%'E@A?+f2V*#13@h #1'2?5n 4 1'+&?+ %r;>+2>$m4&%' 2>V*,19)*01'DJ*,19)8*,)% ;19+f2(282=>*019?+ #LJx?&J*#D?5q = e!9v2e>a:=atk *>)=D?%9&21' #+S8-fD1921'.#w(DE@A?$ 8%9 >2s8 #&)%3* 2=>!*,21' #+C5

7 !#$&% ')( *+, - { F}>~EL 9}?} e!a : <L:+>9L?,90 d> n :+>870}st q p q r s% ;(+2>$4%9sa@>%'*#D n.#q p q n?q p n 3 q r!$#&%(')*,+-%(..! *,+-3(4?+`?421'>+fv2@>@>(D1'.#($ (+f n #q n q/an?q q r n?q n+.#q/an?q q n,q n #q/an?q r n?q n 3 q/an,q c +7+C;*,&2 #13Da%3* E=(2 #%'21'#+ BfC;>+f2&>K7@A%3*,-Dd?+fd% ;&+(Dd+=>?*,21'.#nr@,;(DD 1'2a+ ;*`*,@>&+ %'1H22=>V*,%<8?219219q v}>~c<le }It 9 %,; *w*,$#1'+&+&e@>%'*#2+=>?*,21'.#)&*#+ &+ >+-D($4&%')@A#+f*?1'@A2?19#5n 4 19+&?+%r;>+2>$m4&%' 2>V*,19)*01'DJ*,19)8*,)% ;19+f2(282=>*019?+ #LJx?&I.FV*,1'#5 q = e!9v2e>a:=atk *>)=D?%9&21'#+S+&=> f*021'.#g(dv@a?$8&%9 >2w8 #&)%3*2=>!*,21'#+C5 { F}>~EL 9}?} e!a : <L:+>9L?Y?2a i<;:>870}st s% ;(+2>$4%9sa@>%'*#D n q p q r n.#q p q n?q p n 3 q r?+`?421'>+fv2@>@>(d1'.#($ (+f!$#&%(')*,+-%(. '+.! *!$#&%(' ) *,+-%(. '+.! +,4 / 4$.5*,! 4 n #q n q>n 3 q p q n?q n.#q>n?q n,q p nfqan fq E+ 8&>&.? 8*, E=(2 #%'21'#+ >? a<;> e2}m<;+&@a%3*,-d F 8*#D19aC;&+ C 2&1H C 0 G C 1 GL5'595'G C n 2(%9%'sBf& A)8 #I# G C 0 C i C i x)&+g=(2 #%'.0*,+f2g C i 1 >E<;+&s@A%3*,2?SC;&+ C C j 8&2=@A=>+f(5 k * 8&>.#w(Ds >?2ad<L> e}) <Le}? :+e!av}i21c#+s8&>+2?ljx#&vi&+&a@>%'*#2w& nj@a%3*,-ds<;>+f2=>žqh5 C { F}>~EL 9}?} e!a : <L:+>9L? >?2ad<L> n e}) <Le }?3:e!aV}dt q n #q n.#q>nfq p q r q s% ;(+2>$4%9sa@>%'*#D n.#q p q n?q?+`?421'>+fv2@>@>(d1'.#($ (+f n?q n #q>n q p r n,q n?q>nfq p r n 3 q n?q n,q>n 3 q r = e!9v2e>a:=a tik * )=(2 #%'19?+ >2 8-?&`%3* =A!&*019?+C5Ik * E=(2 #%'21'#+ %'19+=(*,1' 8*# >+f2=(g+ ;xe8*?)@a?$ 8%9 >2#5 9L?3:e2} F}A~v '}wt n q n #q n p q s% ;(+2>$4%9sa@>%'*#D n.#q qan+.#q p p q n?q #+S#419(+fE%'* =A!V*021'#+S%'19+&=*,1'2 n?q n #qan?q n,q q n fqan 3 q p q p n 3 q n?q n,qan #q p q c ++&E8 >I*ž.#?19E=A!&*019?+ %91'+&=*,1'2)8*#>+f2=>d@>*,%'(/@A%3*,2(e<;(+f2=>)#+f#(e&>K%919D=>V*,K AE+s8-(&.#(+?E?+&+&(E19(@A2($ E+b6? J & «( B -œž šxw&+&@a%3*,2mbf&1l*`*, 8&%9-d+ %'1H22=(*#% 8?219219x5Lk (w@a%3*,-dv+&=(?*,21'.#(g2#+f )@A%3*,2()v #+C5 = e!9v2e>a:=atk *>)=D?%9&21'#+S%91'+&=(*#19g8*,)(+?2=(w(Dv@A#$8&%'Āw8-?&)%'*=A!&*019?+S&()@A%3*,-D)E?2+C5 kcs@a%3*,2(s& 8&>$1'>s>KF($8&%9 =>*#19(+?as@A%3*,2(s&v #+CGC?+ =>*,19a #+-@ *#D&=%r;AK13x>+@> <;&+&d2=>!*,21' #+ %'19+&=*,1'2)8-*,e(+f2=>#5 >%'%9e&>K1' >$EAK($ 8%9d+C;=>*#19(+?/8 #(e&(@a%3*,2(l& E?2+ 5

8 !#$&% ')( *+, - 5* *! (+4 4 ' 4$. #:4 '4 #1)/ /% * 4?1H &+SD?x >+-D($4&%'v >%CBf& D?1H)2*,213xJ*#1'*,4%9E>IBf&1<2>V**#8&8-(%9=d%r; }?& >}A~ > 9}w # 9Le: 5 T C C \ T g*,+m%'* x*,2=( #1'% ;>+2>$m4&%9SD8&8-?DGW?+ 1'+?>V19 %3* )=D?%919?+ G C \ 1'2 #+7+C;*,&2#13Dm%'* )=(2 #%'21'#+ B? ;(+?2 #:(D)&+&a2@>>+&*#+f2s<;&+&a@A%3*,-Ds 5 T T { F}>~EL 9}?} d:+e!<;:=a >J} #}m }?& >}>~ > 9}a # 9Le: G*ž.#@)] n {p, r} q n #q nfqan.?q p q r q s% ;(+2>$4%9sa@>%'*#D n.#q p q n?q?+`?421'>+fv2@>@>(d1'.#($ (+f n?q n 3 q>n q p q n,q n #q>n fq p p n 3 q n?q nfqan #q r = e!9v2e>a:=atk *D2V*0=> #1'ws%r;>+-D($4&%'sD&88-?D)(Dv@A?$8&%9 >2w8 #&E%3* 2=>!*,21'#+C5?3:=ae:tl%'%9s8 >$Ava2g!F@(*,%'1'2>E2&)&+ 2 #D >+2>$m4&%9ws@A%3*,-D)B?1 Jx #&w&+:#%'w8*,2213@a&%'1'>5 S\>T/^sUWTAXaR ^ #&)$?+?2()B? ;&+w!#$&%' x)&+&wv*, &>V@V &>V*a+&s2=>!*,21'#+: 5 F C( F ) l+:aplaat V*, #%'#?19 *#+f21'%9? F F C( F ) C( F ) 4?1H F = (p (q r)) ((p q) (p r)) *,%'#V F ( p q r) ( p q) p r C( 8%9s=>21'=s>+ ) = { p q r, p q, p, r} 3 G?+ *m.f:bf& C( F ) #+-@ (DE&+&wV*, #%'#?19?5 F

9 M #$ %& & +& & / * ^*,d*#8&8-?dg*, *#%'@>&%W82?8-fD1921'#+&+>% G#+ Av&(! #+-@h21'#+->5 NPOIQSRSTVUWTAXRZYS[]\A^R`_a^_ab c +:* u %r;>+2>$m4&%'w&a@a#++&(@a2>di,,,, j {, } u &+`(+2>$4%9 Ž-+&1C #:=>+#$4*#4&%9wa@>#+D*#+?( C u &+ >+-D($4&%' Ž+&1I # =(+& #$m4&v*,4&%'s;f$4 #%'(m!?+@h19?+mam+&`*#8&8&%'1'@(*021'#+ GW*,88-(%9=( F *,1H=#G& &*#+ F N u &+ (+D($4&%' Ž+&1I #š=>+#$4*#4&%9s:;f$4 #%'( S2(%'*,21'#+mA&+&:*,88&%913@>*,21'#+CG *,88-(%9=( R *,1H=#G& &*,+ R N u &+`(+2>$4%9 =(+& #$m4&v*,4&%'g&s.ž*#213*,4%9>5 V E+:?œL7 x u #S4&19(+:&+:=>%'=>$>+fE C V u #S4&19(+ w%'*!?2$ f(t 1,..., t n ) xe<;*,192= > f F n t 1,..., t n D?+?E2(2$((5 { F}>~EL 9}h gt t = g(y, f(h(x, x))) t = c x)&+&a@> #+D*#+f2#G c &+ ;F$m4-?%9g!#+-@h21'#++&>%C+*,1'2?G f > );F$4 #%'(!?+@h19?+&+&(%'i4&1'+*#19((5 g h E+ 5 œl7 W r 37 rª< W 7(Dm %3*S!#$ R(t 1,..., t n ) GW<;*,1H= G > R R n t 1,..., t n 2 #+f( 2(2$>5 E+K5 -œl7 J UB Wœ0 V7 r?œ -œlb œ0 x u #S4&19(+:&+&g!#$&%'s*0#$1'Bf& u #S4&19(+ w%'*!?2$ A (DE&+&w! #$%9 A u #4&1'>+ )%3*g!#$ G G # A1 A2 A1 A2 A1 A2 A1 A2 > 2 #+fe(!#$&%'( A1 A2 u #S4&19(+ w%'*!?2$ xa xe&+w! #$&%' A u #S4&19(+ w%'*!?2$ xa xe&+w! #$&%'#5 A y(r(y, c) x(r(t, c) R(c, t))) x)&+&a@> #+D*#+f2#G c &+&s>%3*021' #+S4&1'+*,1' R> %'g2(2$s8&=(@>=((+?g t 2 #19 y(r(y, c) x(r(g(y, f(h(x, x))), c) R(c, g(y, f(h(x, x)))))) ˆ OEŠ ^armuwtaœ[sb E+a J L ( fœž œžÿf ( r - 2a@A?$8-fDw I

10 !#$&% ')( *+, - u &+`(+2>$4%9 D I G+& #+:.F1'&#G&*,88-(%9=KB 37 & J I u 8-?&E@V *#Bf&w@A?+D*,+f G&&+:=>%'=>$>+fE c D I G+&,= # G*,8&8 >%'=s1'+?>8&=a*,21' #+S I(c) c I c u 8-?&g@V -*#Bf&;F$4 #%'a!?+@h19?+&+&(% <;*#2192= GL&+&*,8&8&%'13@>*019?+ n f n D I &*,+- D I GL+&,=> I(f) # G&*#8&8-(%9=(s19+f>8&2=>*,21' #+`& f I f u 8-?&l@V *#Bf&) ;F$4 #%'I&)2(%'*,21'#+ <;*,1H= G,+&E*,8&8%913@>*,21' #+ n R n D I &*,+- Gž+& #2=> {v, f} I(R) # G&*,88-(%9=(s19+f2(282=>*019?+` 5 R I R E+a &«(«1 ( r - (DE&+&a*,88&%913@>*,21' #+:<;&+ D?x >+-D($4&%'vŽ-+&1 &*#+ σ V D I 5 kl; J L fœ œ0ÿf ( J B 9 ( fœl7 J* h> #+:D19.0*,+f t G 8 #m&+s*?2219?+*019?+ G +,2=(? IσG (D=>Ž+&1'S&%3* t σ I σ (t) t u D1 (D)&+&a@A?+D*,+f G t c t Iσ = c u D1 (D)&+&s.0*,1'*#4&%' G I t x t Iσ = σ(x) = x u D1 G σ t = f(t 1, t 2,...t n ) t Iσ = f I I (t σ I 1, t σ I 2,...t σ n ) kl; J L fœ œ0ÿf ( J B ' W 5 -œl7 J G-8-?&E&+&a*?221' F σ G+& #2=( # IσG&(Dv=>Ž+&1'w&s%'*mJ* h> F I σ #+:D19.0*,+f t (F ) F u D1 (D)&+&w! #$%9s*0#$13B? G F R(t 1, t 2,...t n ) F Iσ = R I (t Iσ 1, tiσ 2,...tIσ n ) u D1 F = Fy G F Iσ I = f (Fy σ ) u D1 F = Fy Fz 2D8C5 Fy Fz G Fy Fz G Fy FzA G F Iσ I = f (Fy σ I, Fz σ I (28C5 f (Fy σ I, Fz σ I ), f (Fy σ I, Fz σ I ), f ( y σ I, Fz σ u D1 F = xfy G 5 >;19% >KF13D2+ =>%'=>$(+? F Iσ = v a D I >%Bf& *ž.#@ F Iσa = v σ a => f*,%' 2*#l>+ σ x σ a (x) = a &?+ F Iσ = f u D1 F = xfy G 5 D1 8-?&I2?E=>%'=>$>+f & F Iσ = v a D I #+:* F Iσa = v &?+C5 F Iσ = f 4?1H %'*m!?2$m&%9w8&=(@>=((+f2 F *ž.?(@ 5 y(r(y, c) x(r(t, c) R(c, t))) t = g(y, f(h(x, x))) 4?1H % ;1'+f2(28&=AV*019?+Sa&#$ *,1'+&w%r;>+-D($4&%'s(I=>(%'(G I x)1'+?>8&=a2=(w8*, c 8*,)% ; #8&8?2=s&*,+I%'(E2=(>%3>G G 0 f > g 8*,)%3* 2(%'*,21' 8*#% ;*#&1921' #+:>E%'* $%H198&%'13@>*019?+:*,+I%'(I=>(%'(G h #+ G R *,%'#V t σ = y σ (x σ ) 2 R(t, c) Iσ = (y σ (x σ ) 2 0) (R(t, c) R(c, t)) Iσ = (y σ (x σ ) 2 = 0) R(y, c) Iσ = (y σ 0) (R(y, c) x(r(t, c) R(c, t))) Iσ = (D G19%C>K1'D2w&+:=>>% 2(% Bf& y σ 0 r y σr = r 2 ( y(r(y, c) x(r(t, c) R(c, t)))) Iσ = F Iσ = #&I#E=>(% s, (R(y, c) x(r(t, c) R(c, t))) Iσs = v 4 1<?+`8&>+-@A?$$ s?$ *,1'+&d% ;>+2>$m4&%9sI>+f21'>V#S(IV*019?+&+&>%3(Gf>)%9*#2(1'+f2>8&=AV*021'#+- 1'+@V *,+ #=>wnj&*#+)%'(i>+f19(i?`%'(iv*019?+&+&>%3vqhg { *,%'#V f D1 x AE+C;(DE8*#I+:@(*,2=g8*#DJ*#1H y (R(y, c) x(r(t, c) R(c, t))) Iσ = σ D1'+& #+ v ( y(r(y, c) x(r(t, c) R(c, t)))) Iσ = F Iσ = f E+b5 -œ7 J 6# J «nj@,;(dd 1'2m2*#+v.0*#213*,4&%'%'1'4&2žqv*`&+& 1'+?>8&=A*,21' #+71'+=(8-(+&*#+? &# *#219?+*019?+Av2>V*m+&,=> # 5 ;xe%'a@>*?&s%'*m!#$%9 8&2=@A=>+f2?5 I(F ) F I F

11 q!#$&% ')( *+, - c +:1HvBf& &! 21rG8-?&I2?2a*?2219?+*019?+ G 5 I «( J«5 F σ F Iσ = v 4 1%'(g.0*,1'*#4&%'(d%91'4&(w 2 #+f F x 1 G x 2 G5'595 x n G<%'*! #$%9 x 1 x 2... x n (Ds@A%'?2m>wxs*,88-(%9=( F 6# r œ0 7! - fœ0«f r J 5 F E+w!#$&%'w(DI & J B 21C>%'%'gxv*01'DJ*,192w8*#I2?2s1'+f2>8&=AV*021'#+ 5 x(a(x) B(x)) A(a) B(a) d>ks! #$&%'( > 2 #+fvÿfª<! & & r f L F«n q)21 >vd(&%9($>+fd21 (Dv.0*,%'1'&#G@#;x2 F G F G 1'w21 8 #&I# AI2?2 G I σ F Iσ = G Iσ x(a(x) B(x)) xa(x) xb(x) 9L?3:e2} F}A~v '}wt A +&sd?+fi8-*#i=(bf&1'.0*,%'>+f2 x(a(x) B(x)) xa(x) xb(x) D È XGFŠ [`\>bih F O)RSb bih E+d!?2$m&%' x) œžÿf C D1L>%'%'dx)w%'*a! #$ *ž.#@ F Q 1 x 1 Q 2 x 2...Q n x n G G Av@V -*#Bf& Q i =AV*,+fvD?1H 2 #19 5 ZK[ ŸF œl\v7 :t ]W#g!#$&%'w(DE=(Bf&1'.ž*#%9(+f2 &+&w! #$%9w8&=>+&>K#5 `baa~c9l? d:+e!<;:>9;?ƒ@> #+D2@A21'.#S8*, &+:2&1927S*#+x!?2$ *,21'#+* ;?*#+f 8 #& >p >:8-?D(m# %'()Bf*,+f1HŽ-@(*0>&V %r;akf2=(21'>>e21'%'1'*,+fe%'(i8&#8&1'=a2=s2&1'.ž*#+f2((5 ^ #&g#m!?2$m&%' G #.0*#213*,4&%' Aw2?2!#$%9 &*#+w%'*?bf&>%'%9 +C;*`8-*#w<;F@(@A&>+@> F x G x %'194&#G&21 (DE&+ Bf*,+f219Ž-@(*02(&)B?>%3@A #+-B?g> % ;*#2sBf*,+f219Ž-@(*02(&G#+:*`t Q Q QxF Q x F QxF G Qx(F G) G QxF Qx(G F ) QxF G Qx(F G) G QxF Qx(G F ) G QxF Qx(G F ) n!(+`>p > QxF G Q x(f G) QxF G QxF G Q x F G Q x( F G) Q x(f G) v}>~c<le }gt,9 %C8 >E+s8*# ;*ž.?#1'/+&1'@>1h=#5 { F}>~EL 9}h gt xp 0 (x) yq(y) x y(p (x) Q(y)) y x(p (x) Q(y)) D>+f21' #+ +C;(DE8*?I>+: #=(+&=>V*,%L=(Bf&1'.0*,%'>+f2 x y(p (x, y)) y x(p (x, y)) xp (x) yq(y) x y(p (x) Q(y)) È XGFŠ [`\>bih R`XKF Š ^a\>bih E+ 6? J & «s(d&+&)13-jx?+@h19?+až+&1')!#$&%'( *,2#$13Bf&( #+&=> f*019?+!#$&%'( *0#$13B?((5 4 1 G<2*#+gB?-*,+f219Ž-@>*,2(&(Gxw&+&@A?+LJx#+-@h21'#+ Ž+&1' &m@a%3*,-d>gl*,%'#vv%3*! F?2$m&%' x 1 x 2... x n F (Dv192a2 # 5 -œ7 W -œl7 & J U6? - VM - $6,! - N OIQYX0Tž5 ZK[ ŸF œl\v7 :t ]W#g!#$&%'w(DE=(Bf&1'.ž*#%9(+f2 &+&w! #$%9w8&=>+&>Kw2#- 5 `baa~c9l? d:+e!<;:>9;?st21'%9132>)%'s2=d&%9*,e*,+*#%9? #&w *,%3@A&% ^e2?8-fd1921'#+&+>%<ae%'g2 &=(# >$w8&=(@a=>+f5

12 !#$&% ')( *+, - # g*,+%3* 8*,213B?#G #+š=>%'1'$ 1'+&:%'( A %'( G #+ J*,19 &(@A>+-2S%'(+&=(?*019?+(G?+š2&8&821'$`%'(?&4&%'(I+&=(?*019?+(G#+`J*,19E$?+f2>)%'()@>#+LJx?+@h19?+IA)%'(EBf*#+f219Ž-@>*,2>(5 ˆ axa\aoeš TH ^wuwt>xar c + V*LJx?2m*, %3*,+?*,?a&+& 1'+Ž+&192=@A#+-xV*,+f2v>(GL8-?&w@V *#Bf& G +&m1'+ž+1h= ;f$m4-?%9d& n! #+-@h21'#+-)<;*#2192= G*#8&8 >%'=>(I6? - «( ( & L f«aa 5 - $6, J W«0B - J V7 5 n E+s >œ0 & «5 -œ7 ( r - B - r 7 *#D@>19 &+&w!?2$m&%9w8&=>+akg@>%9fdss%3*m!#$ x 1 x 2... x n xf %3*!?2$m&%' x 1 x 2... x n F (x f(x 1, x 2,..., x n )) F (x f(x 1, x 2,..., x n )) (D:%'*! #$&%'?42(+F& 8*#D19: >+ 2($ 8%'*ih>*,+f`2#&2(S%'(S@>@>&2(+@A`& 8*#`%' >$ F x f(x 1, x 2,..., x n ) (DE&+&s+ #&.?>%'%9d!#+@A21'#+ #%'>$ <;*,192= 5 f 4 n <ž <Le:+>f >J}ewt&%3*!?2$m&%' xf (D)2V*,+-x!?2$=>g>+ Bf&1 (De%'*a! F (x c)#$&%'e?42(+f 8*#D19/ (+ 2($8&%'*ih>*#+? #(/%9l@>@>&>+@>(e& 8-*,e%3*a@> #+D*#+? F x 4 #%'>$ 5 c c + *,8&8 >%'%9 «- JŸV7 r«(ÿf & &+&S! #$%9S&+&1'.#(2>%'%9S#4&2>+F&:(+š*#8&8&%'1'Bf*#+?192=>V*019.?>$>+f ( F 2V*,+-x!?2$ *019?+I 4?%9($ JxBfC; abfc;1'%c+c; ;`*,19)8&%'E 5 &%3*m!#$&%' (Dv x y z u vp (x, y, z, u, v) #%'=>$13D=(d(+ y up A D (c,?+f)&(! y, f(y), #+-@h21' u, g(y, #+-) u))?%9($<;*,1h=(i(28-@h19.?( A).mA &+&s@> #+D*#+?s f g 4 4?%9($:5 c v}>~c<le }h dt,9 %<+C; ;`* 8*#I+&1'@>1H=sAE%3* #%'=>$1'2=> <;&+&w!?+@h19?+ 67*#1' (DE@A #+-D=B?>+@>w2=>$ *#+?1'Bf&s n +&s%'&1cxe8*?i=(bf&1'.0*,%'>+f2?5 F qe>e #+ * F 5 F {F } {F } = F = F F xp (x) yq(y) 2w?%9=($1'2w>+ y(p (a) Q(y)) #S>+ Bf&1C+&w%'&1 2 y(p (f(y)) #+f)8*#i=bf&19.0*#%9(+?((5 Q(y)) 67*#1'I?+S* %' ZK[ ŸF œl\v7 ƒt 4?19(+? &+ >+2>$m4&%9!?2$m&%'(g8&=>+&>KFs@A%'?2(gA ;<?42(+f >+?%9=($ 13*,+fd%'( Σ Σ! #$%9) 5 0 %'#V (Dv*0213DJ*,132*#4&%9n!(28C5&@> #+f2v*#13@h#1'2žqe21c>)2>%9($ (+fvd1 %r;(d(5 Σ Σ Σ {/ >3 >} #} a>~09;?& :+e!<l:+>9l?7t 4 1C+&s19+f>8&2=>*,21' #+ 2*,213xJ*#1H G&>%'%9s*01'DJ*,19v*, I Σ Σ 1C+&s19+f>8&2=>*,21' #+ 2*,213xJ*#1H G& #+:@A?+D2&1H)&+s19+f2(282=>*019?+ Bf&1 *01'DJ*,19 5 I Σ I Σ

13 !#$&% ')( *+, -. PR`ThQ ^wuwtaxar E+`2&4D219219?+ #+ % ;>+2>$m4&%9`(.ž*#213*,4%9a&*,+-a% ;(+D($4&%'(m(2(2$ σ 2(%9%'dBf& (D/=(?*#%9 σ 5 %r;13>+f2192=w*,c8 #&&+&d+&#$m4&)ž+&1<d.ž*#213*,4%9 {x 1, x 2,...x n A2>V*s+& #2=( } (x 1 t 1, x 2 t 2,..., x n t n ) c +S8&2?%9?+& #g>+:&+a*,8&8&%'13@>*019?+`s% ;>+2>$m4&%9sI2(2$sn!8&&13E!#$&%'(Vq&*,+-)>KF $0/>$#5 *ž.?(@ G σ = (x a, y f(x)) R(f(x), f(y)) σ = R(f(a), f(f(x))) kcw8&&19 τ σ &w&>k:d&4-x1h21' #+ σ > τ xe%3*d&4-x1h21' #+: #4&2>+F&w(+:*#8&8&%'1'@(*,+f σ 8&1' τ 5 21 σ = (y f(x)) A τ = (x a) G R(f(x), f(y)) τ σ = R(f(a), f(f(a))) R(f(x), f(y)) σ τ = R(f(a), f(f(x))) 5 σ(l 1 ) = σ(l 2 ) =... = σ(l n ) E+:(+D($4&%'ss%'1H22=>V*,&K xe W,@ r >;1'%CAK13xg+&aD4x1H21' #+ 2>%'%'aB? (Dv*,8&8 >%'=w&+&19ž-@>*,2>e {l 1, l 2,...l n } 5 σ σ {l 1, l 2,..., l n } E+`&+&19Ž-@>*,2(& (D œ0 J $6# J & WD1 #)*,2d&+&19Ž-@>*,2(&/x)g%'*a!#$ σ τ 8 5-kCs=(2&%HV*0Ew% ;*#8&8&%'1'@(*021' σ #+:<;&+S2(%C&+&19Ž-@>*,2(&E(DE*,8&8 >%'=K5,6, F œf5 τ x)&+&19ž-*#4&%9s8-*, {R(x, g(x)), R(f(z), y), R(x, u)} G&&+1HŽ-@(*0>&)+&?+:8&21'+@>198*#% τ = (x f(f(c)), y g(f(f(c))), z f(c), u g(f(f(c)))) nwj*#@a2> q R(f(f(c)), g(f(f(c)))) AE8-*, σ nwj*#@a2> GF&+1HŽ-@(*0>&E821'+@A1'8*#%r5 = (x f(z), y g(f(z)), u g(f(z)))) c q R(f(z), g(f(z))) +:* 5 τ = (z f(c) σ = e!9v2e>a:=atkl;&+&19ž-@(*02(&)8&19+-@a1'8*,% G&>;1'%CAK13x#G(DE&+&13Bf& *,+&?>$>+fes.0*,13*,4&%'(8& ((5 1- -œ0! [ 7 B 9 W $6? ( r - a&>k%'19d2=(*#k l 1 > l 2 t c &(@V &?G/&*#+ + 8*,V@A?&V 8&=AŽ&K#G/%'* 8-fD1921' #+ & 8&>$19(` ;F$4 #%' Bf&1d+ ;xs8*#%9 $b/($ &*#+I%9)>&K`%'1H22=>V*,&KL5 u 4 1,%'(C(Ks;F$4 #%'(<@>#2D8 #+-&*,+fl2#+f C@A?+xV*,+f2L#sC;F$m4-?%9L!#+@A21'#+&+>%3lnJ19pL=>>+fVqhG 19% ;S*m=@V &(@#5 u #+S%r;&+ *,S$#1'+)(D)&+&s.0*,13*,4&%' 5 x 4 #19 %'g2>$a@a?$ $(+ h>*#+? *,8&8*#*#1H)&*#+ G&1'% ;`**,21<=(@V &@ n V(D)C;@>@>&>+@> >qag x t 5E219+ #+ >a%3*2&4d219219?+ Aw #+ 2=(1H >a%9m8&f@>(d-v*ž.#@ A σ = (x t) σ(l 1 ) σ(l 2 ) c +S #=>+=>V*,%'1'2vJ*?@A1'%9($>+f &+:>+2>$m4&%9ss8%9)&a>KS%919D=>V*,K<5 ZK[ ŸF œl\v7 t kl;*,%' ##192 &$ <;&+&19Ž-@>*,21'#+ D 2>$19+ Am@>#+1HGW21l#+ƒ% ;*#8&8&%'1'Bf& &>Kƒ%919D=>V*,K &+&19Ž-*#4&%9>G +&ad1haad4x1h21' #+- σ 1, σ 2,..., σ k #+f)%'s8&21h 5'595 σ k σ 1 8&19+-@A1'8*,%<>E%9a(2+19(%'19D2=(*#%C #4&2>+FS&+SJ*#@A2>E(I%'1H22=(*#K`1'+&19213*,K<5

14 !#$&% ')( *+, - %'19D2=(*#K 2&4D219219?+ > R(x, g(c, z)) R(f(y), g(y, k(x)) (x f(y)) A R(f(y), g(c, z)) R(f(y), g(y, k(f(y))) (y c) A R(f(c), g(c, z)) R(f(c), g(c, k(f(c))) (z k(f(c))) R(f(c), J*?@h2(&)8&19+-@A1'8*,% g(c, k(f(c)))) (x f(c)), &+1HŽ-@(*0>&E821'+@A1'8*#% (y c), (z k(f(c))) :+e2}a} f}>~v 9}wt %'19D2=(*#K 2&4D219219?+ > R(x, f(x)) R(f(y), y) (x f(y)) A =@V &(@ n!2xv<;@>@>&2(+@ažq R(f(y), f(f(y))) R(f(y), y) :+e2}a} f}>~v 9}wt %'19D2=(*#K > =@V &(@n xe&+a@a #+-xv*,+f2žq P (f(x)) P (c) c

15 ! % & & / % / & M,.-)_a\Ab YSb/, OIH X\A[ U TAXaR c +*.f`b?e2?2v!#$&%'e&+19.?>vd(%9%'e(d/=bf&19.0*#%9(+? *?B? x 1 x 2... x n (F 1... F k ) F i (DE&+&a&1'Jx #+@A21'#+:a@A%3*,-D>5 A22w!?2$m&%9wxE*,-221<=(Bf&1'.0*,%'>+f2 x 1 x 2... x n F 1... x 1 x 2... x n F n 5 c + 2($ 19%e(D 1'$ 8%913@A192aBf&s%'(I.0*,13*,4&%'(I2 #+feb?-*,+f219ž=>i&+&1'.#(2>%'%9($>+f(5 8&9;:=<;:>9;?& gt 4 1 (DE&+ D?D (+D($4&%'ws%r;>+-D($4&%'a(I%'19D2=(*#KS&s%'*@A%3*,-D G& #+:+&#2(* S C C \ S *, %9)=>%'=>$(+?V) 5 C S P (x) Q(y) R(x, y) T (z) \ {Q(y), T (z)} = P (x) R(x, y) 4 1 (DE&+:>+-D($4&%'ss%919D=>V*,K<G?+`+&#2(* % ;>+2>$m4&%9 5 S S { l l S} {Q(y), T (z)} = { Q(y), T (z)} k (D&+& œžÿf«- J & & L ( š@a%3*,-d C Cy A Cz (;1'%v>K1'D2 &(@>%'*#2( C y > C z G( >+-D($4&%'()s%'19D2=(*#K Sy > Sz AE&+aD&4-x1H21' #+ 2>%3EBf& σ u C y > C z 2 #+fl(.0*,13*,+f2 & Cy > Cz Bf&1+C;?+f 8-*# <;+&g! #$&%'w(de&+&w!#$%9sbf&1<%'&1c(de13>+f213bf& &+:2(+& #$$ *,?g&(i.0*,1'*#4&%'(8& (Vq u Sy A Sz 2 #+fe()2#-x >+2>$m4&%'(I(I(+2>$4%9)s%919D=>V*,KS C y > C z 2D8 (@A21'.#>$(+? u Sy Sz (DE&+&19Ž-*,4&%'wA >+:(D)&+:&+&19Ž-@>*,2(&E8&19+-@A1'8*,% σ u C = σ((c y \ Sy ) (C z \ Sz )) { F}>~EL 9}h gt P (x) Q(x) A P (a) x)&+a@a #+-xv*,+f2?g&#+&+&(+fi%3*=(2#%'.ž*#+f2 a Q(a) P (f(x), y) Q(x, y) A?+&+&>+f 2=(=(@A19I(+ P (x, f(y)) R(x, y) P (x, f(y )) R(x, y ) Q(x, f(y )) R(f(x), y ) *ž.#@ σ = (x f(x), y f(y )) 3

16 D!#$&% ')( *+, - P (f(x), y) Q(x, y) P (z, z) A =>=(@>219 P (x, f(y)) R(x, y) P (z, z) P (x, f(y )) R(x, y ) P (z, z )?+&+&>+f Q(x, f(x)) R(f(x), x) *ž.#@ σ = (x f(x), y f(x), z f(x), y x, z f(x)) c +?=>+&=(*#%9132m*, *,%3@A&%^l2=13@>*0Vg%9d+,21'#+-g& œ0 F Awœ0Ÿ5 W ( J.F&(w&*#+g%9 *,%3@A&% 8& #8?21H19?+&+&(%r5 ^a\at>ỳ TVU O bu /XŠ:\AOUW[SYSb c + *%9I$0/>$()=(2&%9*,EBf&a&*#+I%' *,%3@A&%<82?8-fD1921'#+&+>%t%3* $=A2 F& )=D?%919?+:(Dv«( & r W A 6? 37 \? 8-?&E%3*m=A!&*019?+S*,+I%' *,%3@A&%C&( ^l=(13@>*,(5 ^ #& a>~c9l? 8*,221' <;+ >+-D($4&%'S@> #+f2v*#&1'@a2#1':@a%3*,2( *,%3@A&%I( 8&=(13@>*,C&g82($ 1'> #V2?Gh#+s@A?+D2&1HC+g(+2>$4%9l@>#+f2V*#13@h#1'2l@>%'*#2(< *#%'@>&%;^l#8?21h19?+&+&(% Bf&1-*w+&E=A!V*021' #+ 8*#D19/E%3*#Bf&>%'%') E% ;>+2>$m4&%9v19+&19213*,%-v@A%3*,-D # #@> &+S &=>?2 ($ s@>#+&+f:d?i%9w+&?$ A%9($ $as>%' >.?>$>+f 05 ˆ U F ^wuwo)_at>bih c +=>Ž+&19(G@A?$$ g&*,+-l%' *,%3@A%-8&#8?21921'#+&+&(%rG?%9/D2V*0=>?19l8?21H19.?#Gf+&=> f*021'.#?g#%'19+=(*,1'#g#%'1'+&=(*#19 8*#)>+f2=>?G&w%r;>+2>$m4&%'aD&88-?D5 c +:* %'(I$0/>$)2=D%HV*0Ia@> #$8&%'=A#5 S\>T/^sUWTAXaR H ^ #&)$?+?2(EBfC;+&g! #$&%' F x).0*,%'1'&#g&?+s@v &(@V >V*&+g=A!&*019?+ C( F ) 5 l+:aplaat F.0*#%913 C( F #1'2 C( F ) 4?1H F = x(a(x) B(x)) x(b(x) C(x)) x(a(x) C(x)) $1'2a2 #/!#$w8&2=(+&akt #%'=>$132*,21'#+ t x)&+a@a #+-xv*,+f2s 4 #%'>$:5 a n q A(x) n.#q B(y) C(y) n?q A(a) n 3 q C(a) =A!V*021'#+S8?21H19.?g>E%91'+&=*,1'2w8*#)>+f2=>t F x(a(x) B(x)) x(b(x) C(x)) x(a(x) C(x)) F z x y((a(x) B(x)) (B(y) C(y)) A(z) C(z)) x y((a(x) B(x)) (B(y) C(y)) A(a) C(a)) =A!V*021' #+S+&=(?*019.?wA)%91'+&=*,1'2w8*#)>+f2=>t n?q n q>nfq n fq B(a) n.#q>n?q C(a) n #q n 3 qan fq n &;q n+.#qaan 3 q n ;q B(a) n qaan ; q A(a) n ;q n?q>an &; q

17 !#$&% ')( *+, - D2V*02=( #1' % ;(+D($4&%' 2&8&8 #2 (+2>$4%9 2&8&8 #2(Gg@,;(DD {A(a), C(a)} t x(a(x) C(x)) n >q nfqan q B(a) n >q n 3 q>n.?q B(a) n >q/an (q n >q ^ #&)$?+?2( G&?+`$#+f2(* Σ = F C(Σ { F }) Σ = F C(Σ { F }) C(Σ { F }) 4?1H A Σ = { x yp (x, y), x y( z(p (x, z) P (z, y)) Q(x, y))} F = x yq(x, y) F = x y Q(x, y) n q P (x, (DE&+&w!?+@h19?+S& 4 %9?%9($ f n.#q P (x, z) P (z, y)) Q(x, y) n?q Q(a, y) =A!V*021' n #+S8?21H19.?g>E%91'+&=*,1'2w8*#)>+f2=>t 3 q n qan+.#q P (f(x), y) Q(x, y) n?q n ;qgt n fq >n 3 q Q(x, f(f(x))) nfqan P #q (x, f(x )) =A!V*021'#+ 8*#S(+f2=>#G)>7@,;(D *#D1s%3* x*,2=>?19ƒƒ% ;(+D($4&%' D8&8-?D7*ž.?(@ { Q(a, y)} 1'2S%'*ƒ@>%'*#DS1322& %3* +=>?*,21' #+š : a9;e >~} &*#+I%'* F : n,ad9le>j} t 3 ;q Σ n.?qanfq P (a, z) P (z, y) n &;q n qan 3 ;q P (f(a), y) n ;q n q>n &;q

18 A 1... A n B 1... B p, ˆ ˆ N N, N 0 8*#D19E)@A?+&+*,132*#+@A/2&1'.ž*#+f2(wt 0 %3*,1'+`xE%'w8- (2a!>+*#<5 /(2+-*,V (D)%9s8 >s %3*,?5 kcw8- (2s:8 >a<;&+&s8 >VD?+&+&g(DvD?+`?*#+F 8- (2g8*,22(2+>% 5 %'*# Jx #&s*ž.?(@wd?+`?*#+f 8- (2d8*,22(2+>% 5 ^ >2?+S&=>$#+f2>G>+S19%'1'*,+fE%'G^l19+@>198 aw=(2#%'21'#+:bf& %'*# Jx #&s*ž.?(@ 0 %3*,1'+# M È XGFŠ ^a\at HW^wUWTAXaRZb)R]RSXwUW^wU TAXaR F O)Ỳ T^wUWT%$ab 0 8*#D19EI=>+&?+@A=)D19.0*,+fVwt (DD 8-(2> (a, b) (DD 8-(2> x2 8 >A (b, c) x( x2?*#+f 8-(2> (DD #V*,+-F 8 >A (y, z) Jx #& (x, z)) x( (x, c) (c, x)) G G D?+f)&()@A?+D*,+f(IA G G (I.0*,13*,4&%'((G #+S8-(v=($?+f2>IJ*#@>19%'>$(+?EBf& a b c x y z Jx #& (c, a) >+:$ n #+f*#+?)bf&s% ;(+D($4&%'sa@A%3*,-D'&vD19.0*,+fExE@A 5 q(dd 8-(2>?+f2V*#13@h#1'2?5 n (a, b) 5.?q(DD 8-(2> (b, c) n 5fq(DD 8-(2> x2?*#+f 8-(2> & (y, z) (x, z) n 5 3 q(dd #V*,+F 8 >A n Jx?& (x, c) (c, x) 5?q Jx #& (c, a) +&=> f*021' #+:`J*,19 =>$ #+f2( n 5 fqex2?*#+f 8-(2> & n 5 qagcn 5.?qe>n 5fq (a, c) )=>!*,21' #+ƒt n 5 #q Jx?& n 5?q/>n 5 3 q (c, a) n 5 fq n 5 #qe>n 5?q )($*#Bf&()B?g% ; #+:*&G&&*#+)@A>I>K>$8&%'#G+&?+:2>%9($ (+fi+&w2=>!*,21'#+cg$ *,13)*,-221L&+&a=@h19?+ 1'2@hw&s%'* 8& #8&1'=A2= Jx?& nj@>%'*#d n 5,qDqA5 (c, a) È XGFŠ ^a\at HW^wUWTAXaRZb)R]RSXwUW^wU TAXaR)(0XR euwtaxarsrsb)\a\ab 0 %3*:8&%'*?@A s8&=(13@>*,w B?s%r; #+S+&,>V* 1'@>1?+ &21'%913D`(a ;F$4 #%'(g!?+@h19?+&+&>%3s>a#+ *:4-D?19+ *#221l<;&+ 8&=(13@>*, $ *,13EBf&1<+&s8?D s*,@>&+&w8&#8&19=>2=w1'$ 8%913@A192as% ;=>?*#%9192=?5 egal = *,+.-0/213546/ :67;:6<5:646: =?>@8BA'C274697A'353546:6D,AE=;:6-0C579C#FG9H=I=JA'C%=9CG>HK2:6829,C2D,9L469,7M46: =;=;>1?A'/2NPO2Q0RJSUTVS WHRX9H=469746: =I=?>1;A'/2NZY\[^]%_%T`S WJRJa -0Cb>D1I: =L7;-0/2K%9Cc=4697dD4eA'/57;97859f4eAhg`-01IFG9 7?-0/274eAig`-01IFG9 A 1... A n B 1... B p

19 !#$&% ')( *+, - n+.5 q 8-(2 a = (b) n+.5#.#q 0 b = 8*#D19E)@A%3*,2()2&1'.0*,+f2 n+.5?q?*#+f 8 > 8-(2 (c) 8 >?nj8-(2 (x) = (x) n+.5 3 q Jx?& n+.5 #q Jx?& #V*,+ 8 > (c, (c)) (c, a) +&=(?*,21' #+`&`J*#1H =($ #+f2> #+ +&8 >a19(+ &=(&1'@(*,w #+ +&8 >a8*?g&+&19ž>s%'(s@>%'*#djx?& A Jx #& #V*,+-F 8 > (c, a) (c, (c)) >E%9g>$ s?*#+f 8-(2 +s8-(&.#(+?e8-*#)(;&+1hž((5 a (c)) ^ #&)=(2#-2w@Aw8&#4%9 ($?G#+ *>&K`D?%9&21'#+wt u *LJx#&2>g(c<( >9,~}A 0a <0 > : <L> e}h dbf&1 >KF821'$>+fd>KF8%913@A192($ (+fd #(g%9v82?8&21'=a=(d19$8&%'13@a192( w%r;=> f*,%'1h= u *LJx#&2>)s+#&.?>%'%9e(@V &+1'Bf&wt&%'*0B ŸV7 B W r ( r - AE%3* œ0,7 BW J & ( J D TAXŠ biho)_a^a\atvu ^atf bih ^ #&E@V -*#Bf&w8&=(13@>*, *#19#GF?+:*;Jx#w%r;*0K1'#$ P n 8 > 8 > x 1 x 2... x n y 1 y 2... y n (x 1 = y 1 x 2 = y 2... x n = y n ( (x 1, x 2,...x n ) (y 1,..., y n ))) ^ #&E@V -*#Bf&s;F$m4-?%9g!#+-@h21'#++&>% *,1'#G#+:*LJx?2w% ;*,K19?$ f n x 1 x 2... x n y 1 y 2... y n (x 1 = y 1 x 2 = y 2... x n = y n f(x 1, x 2,...x n ) = f(y 1,..., y n 1'2?GF8-?&)%r;AK($ 8%9s8&=(@>=((+f(G%'()@A%3*,-D n5 q x 1 = y 1 x 2 = y 2 Jx?& (x 1, x 2 Jx?& ) (y 1, y 2 ) n5.?q 8 > 8-(2 x = y (x) = (y) c +:*LJx #&2ss8&%')%'()*,KF1'#$ n5fq A x (x = x) n5 3 q x (x = y y = x) c +:* *,%' #Ve%3*2=>!*,21'#+:D19.0*,+f n5?q x 2 = y 2 Jx?& (x 1, x 2 Jx # ) (x 1, y 2 n5 q/an5fq n5 fq Jx #& 8 >?nj8-(2 ) n.5?qagcn.&5 3 qe>n5?q (c, (c))) n5 #q8-(2 8 >?nj8-(2 n.5#.#q/>n &5#.#q (b) = (c)) n5 fq8-(2fn!8 > 8 > n &5 3 q/>n &5,q n5fq Jx #& 8 > (c)) = (b) n &5?qAGCn5 fqe>n5?q (c, (b)) n5 q/8-(2 n.5 q/>n &5#.#q (b) = a n5? q Jx #& n5 qhgcn &5?q/An &5#q (c, a) n5.#q n.&5?q/an5? q NPOEŠ X Ỳ [S\A^wUWT>XaR A22s+&?&.#(%9%'g2@V &+&13B?gxE%3*D19.0*,+f 4 1C?+S* &+a@a%3*,2w&+&19*,1'g=(?*#%919*#19d8?21921'.# AE+&a@A%3*,2 r = s C = P (..., t,...) H G-&+:2(2$a*,88&8*,V*,132*#+?)&*#+)&+7v*#2?&$>+f) (DE&+&a1'+D*#+@Aa t P (...) r # s G-@,;(DD 1'2 8 >g>;&+&19ž>g*ž.#@ t r # s G-$ *,13ED(&%'(E%'(v.0*,1'*#4&%9) r # s 8 >&.?>+f /A22&4D2192=>((G 8*#)&+&a2&4D219219?+ G-D?1H σ t = σ(r) # t = σ(s) G *,%'#V#+Sœ0 7 r,6# 8*, t σ(s)? σ(t) &*#+ C -} eh~b>?29, 9 >J}t

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22 ,0 ˆ ^a\[s\ F X XKH TVU TAXaRSR`b)\ 5I^?&E@V *?@A&+&wI!#$&%'()2&1'.0*,+f2>G&1'w21C@,;(DE&+&s & W ( J 21- r #S&+&a & L J J 21 J #+CG&1'g8 #vbf&>%'%9i.0*,%'>&vi().0*#213*,4&%'(8&#8?21921'#+&+&(%9%'(%3*!#$&%'w(d).fv*,1'#5 p q p q p p p p p p p (p q) q p (p q) q (p r) (q r) p q.&5 219?+&1HŽ-wBf&s%'(!?2$m&%'( > 2 #+feÿfª< W J & & J F L ( F«?5 ; G?+S* F G p q p q p (q r) p q r (p q) p q (p q) p q (p q) p q p q p q p q q p (p q) p q 5 Σ = F 219?+&19ŽBf&a%3*! #$%9 š(d)6? - «(ŸFª< F 6? «ŸV7 g! #$&%'( Σ 6 #+f2>)bf& {p, p q} = q {p r, q r} = p q = p q 21CAv2>&%'>$(+?E21 {p} = q & L ( rª< W n!# %9? #13B?qIa% ;>+2>$m4&%9 3 5E6 A22%'(! #$&%'(md&1'.0*,+f2m2#- 5 -œ7 œl7 J 6# M 6# (! N OIQYX0T>2 #- 5 -œ7 W -œl7 & J UB J«=M - $6,! - ANPOIQSRUT p (q (r s)) p q p q p q p q r s p q r s p (p q) r p ((p q) r).

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