n!2 π(1) < π(3) < < π(2n 1). 1 n 1

Transcript

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2 <!" #!$ & &#! '()'&* '( '+,.- / &0 '(1!34& 5 6! '(879," : &;!<!" 4=,!+!>? (!6@-A!-B6>!<C"? 0!$D 5E!$#>=!,+,4 FC'(* GH+,'& " I J!K5! L!0 &!.- MN,GO '(64B &P "RQS, 9 N>=!,+,4 P " J(!$T N! "5,'5 S, 5 U K 0 S"G!$1 0=!'I & E!$V " K!!W= 0 S >, & BXY I! &Z,'&'([\= I] ' I!$P!! &H^$_G` Qa ''Bbc3*"RQL J( &H!''! #!! & &-df9v 'B Z6>Q9 e " > 9"fbg'+!Z!' <C \=*!'h\= e V&"5h!$;> 5! igs79," ^f6>qa eqa3hr=! P ] '&H!$.GO"'& 0jh*"G< -kl <"!&H 0Q $_G '+, 8$_N m& "&<"'&,!0 ),$a m'+,z!';g, njo5! "6pGO! C ]!'&'B(6Qa N qj"" 9 " Gr!#$_!0 D!$TC'B(f"GO&!'&?&m e > & a,$.gosb BjmC\Q9 -P79!, a! e&>=,!+!> # " Pst"5>,.6>< fä F4 GH,!! u" GH&, ),$ qj"" Q&''a<C $v!1$_gw<c 5K"-L *"6P $_G '+, Z!H<C =* Qa E,P" 0jfC S 0 #!$st"5>, m&>=,!&!> $_ 0 x!$ GO, & -x qj""'+! s &!)x,ẍ a*"go, ;! 8" GH& sb!6 5! [ GH '= F"fbg>=!,+,4 6"!y<C I! 5 >ë!$,e'&! $_ 0!$? B!<"'& N së! V\=,!+!> " Rt 5>, &.- igz79!" 8{f6pQa hq9!3l\= 1 m Z,'] '.6x! Q9 m R=! O }4,'& & N$_ CB =! I GOnbg 0] IGs -# 1 *!$_! Q y> 0 5h V5B= CB =! ~ GOnbg 0] B ~GO, BjE!HV!G GO, Bj6*@-A!-B6 hgs nje,$? *fb 0 6,Q & 8!3! <!31 NJ!5!'44=,!+!> -x f!$_x "<>()Q 5 " 0jf &! Qa Q9!> o \Q &) fbc 5>s =! E!o Gƒ!$ }4! - B O ' g([y] "& 5 qj"" ~ }4! &FZ! L<>([ 8$v, N, F &" =*&"!'; GHVZ!"! )< H z [<C OJ!5!'D>=!,+,4 6T " 5 "G &,[>=!,+,4 - IQ9\(E" 9 & Z5* <!3SGO! N! ˆ>Š8(! o o!j=*&u y! V &V "& 3L!$9 H 0jf &" 1 }4! O5! f &,!$h!< '& F " Q y \=,!&!>! 0$_ V Z,& &

3 " - )<!H ZS'& 34)79!" ){E o79!" 8^f-E79!" 8{ &I GO! m \Qfb@,bc Z!J!J m!$f & &6aBḧ "HQB!! '&, l L!< '& G!$.<C "& 5V$_G <C '\QU e G!$pV * 0 9,$T'& Z!$_" &,'&- N GH&!> Q&''a"'+Z(z,u&GHJ!> 8' O " 5! " - df! " hgo, Bj $_ 0 &!K &K,'&'p," I&e!$P n n G, nj C = (c i,j ) pf C = 1 n! n Q &P F(*GHGO 0 F5! '& 0 -k &9!> nbg(*ghgo 0 &!6 S n n C Q9 mgoz(k 0QB "&I!1EC'(*!GO&!'xQB L4 5 I* H > VK mgo, Bj > & Qa 1"H"! B=f I<>( 6 < " Qa V &0 I G V G*b c i,j n!, &" n! π S n ( 1) π c π(1),π() c π(n 1),π(n), π QB π(1) < π(),..., π(n 1) < π(n) π(1) < π(3) < < π(n 1). igi P!> nbgj(fghgh &xz!!6rq9 # Z=! - / (pf C) #!-A5 - Ik 6ZdF C nj = det C N- ^T$_! <,&V C &!$$v, O!"- "!,6! ' = J(J!!.6,Q&'&'4<C 0jf '&!&" 8 a 0j* - & > g( GO, BjoQ&''D< mqb 9t -E! 3!$ ygo, BjKQ'&'D<C 1 n 1 "! L<4([3C-# N 0 GO!> eq&''x<c 8"! K<>([= Z!'x<!F!N<>([ -Nig 79!" e{"6 I, C ygh '& qjs!st 5>, I!$;HGO, Bj Q&''Y<C 1"! A <>( A - $&')(+*#,.- /10+3/14 /1( 576 n n i.!g 5, 'B(I& " <" V eg8(i" =*&; 98, g(:8!; FGO!3!x! 1dN" <8! />= 6 Q " Q "G &; &D '&"@?A Z= <C mqb.-?i?,g 5, 0$_ 'f BA#''&! CT& <Y6"D0 a o\t6* 98s <De o, '& 6 />= &'E8,GF1D=D =! AD " />=,<IH~$_; 0$_ ' & 6! ) F!">(fGH? 0$_,$ G8(V!C?$_x $_ '*GHGH > i!'! 3EG)(E$v!GH&'(m$_ ",4ë C - &

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5 ! #! ( 5"* 0 5 * (! #"$'&()+*-,.( &)!" )&V<! u m gq9o!c +/9:6 +/D^ 6?Q u! ~t4 VQB 98, g(:8! NGH3!-10~=!![!'5 <"!Z!''(h'& <! ~] '!$x!! & K ^f6!qa "(8 & 5 V OC &!'P 5!'D5!!$&>=,!+!> 6!Q &x! "5,'f5 K[n m] G -[ 5 G, )<>(l' $ O n (K) SO n (K) G G 'B &"'&&, [ m,!$ G, V, 4 8, 1!' K n m n m 8*, 8 5!$ 6YQ & L#t J VC'(* sb O(K n m ) = K[n m] K GO&!'p 5E& n m sï H 0jf&J =,!&!< '& -Nk h R=! $_! K$_! n m O n (K) (n ) SO n (K) (n 3) m '&&" Z!~>=!,+,4 VC'B(f"GO&!'&Q!<"!'( m '+!5 F,ë! ~&" GOC!< '& 43v -,-6 "! 0jf &<"'& V! C'(*!GO&!'&#&S>=!,Bb!>!$p'\Q D &9S,S!4!J Q E ~" $_Gr "!$T " N 58,$p&>=,!&!>äC'B(f"GO&!'&=,!'&H&S,'&'! " 9!!0 &J -DigO$v! 6 Q9 I!''Y 0jf '& 'B(m" GH!!< ' m '&&" Z!a&>=,!&!> 9C'(*!GO&!'& $_P!'&'"< '& =,!' #!$ -Dig GH!!< '&B g(8!$y C 5V&>=,!&!> D!$ m VGO"'& 0j73v &!'85a! "5,'Y5 \= Q'&'&GHGO +, 'B(h$_'&'RQV-? Z " &"h 'B(!!'B( = &"5o D$vs,u!$ E3! 0Q bg(*ghgo 0 &SGO, Bj Q H > 1!<C\= h m &!5!'! h I * 0 V!$P =! V=!,Bb!< ' -1 h 0 V! 5=! & / 0 &! :!- ^f6y! &" GOC!< &' g(s \=! o / 0 &! :!- ˆ - 0Ih! D! Y6Q9 Q&''" \Q, #&>=,!&!>,!'*$_ &"D&h! b :

6 d!0 &J &9^e<C \=! <!&!'&'B(~ a!gh aq9z()!;!''*,?,!0 &J & 3_ 0b G :- {"- 506*! E 8 N>=!,+,4 PC'(*!GO&!'&B$ 3v " GHN:- {"-B: "6 m n :- {"-B: ~! :- {"-B: 50-xkl!'' $_!D!'&'!, e!'&5! < & n m K[n m] H] '([5, KGHf "'& 1R=! N 8 <!'5 <"5, L<>(y 8}4!", & G &>=,!+!> 3@79'&'&!(y:- {"- ^ 50- +& ".,+""+*'" CT <C SKCB =! 4 5 -! L$_I,!'&5 <,&Z,'&'(E'&! L] '&L!$a!<!J([!,!-3CT F N! S*, n!a<>( <>( $ F n x 1 y 1 x r y 6 6C $ - "! #; $&(' r n = r x 1 y 1 x r y r z n = r + 1 & =,!+!< ' n Q q = x 1 y x r y r n = r, Q *)+$&, $&(' k 1" ] q = x 1 y x r y r + z n = r + 1. β!$ q V(*GHGO 0 &I< '&&" Z!P$_G 5!= <4( β ( v (1), v ()) = q ( v (1) + v ()) q ( v (1)) q ( v ()). ker β = {v F n β(v, ) = 0}. D& " B=*+!'f <! F "'&! 6>&Q "& HZ! eb char F = n = r +1 & sjf&- e}4!, $_!G z, 6C D Z!&'B(h O )<C 3-$+./ ,-64$ q -MF! 1, FR=! I,4([,'&5 <!Z!''(O' v ker β q(v) = 0 v = 0 ] '&.6 & V"'(E"fbg" 5, V}4!, ~$_G m! 5! )!$?<!&kl 8Z(o, ~ h'& Z!! J$_GO,! +5 (! q +e$ q g = q - g : F n n, F q = 0 u F n! 1, SB$, -$+.! 11,, Q&!-Od q(u) = 0 '& Z, <"! )78$&,9,;:<! 1, B$?!'&'.B =! F! 1& 5! '+!dn>( *bg 5 '&!a=! 0 u 5B= e& ~ h =>0@?A06#B! $ " ] o<>( g u β(v, u) 3J:-B: 5 g u v = v q(u) u. dn''d C 0 &! )' ZZ= q &>=,!&!> Z-dF'&'D DC 1 Z= E" GH&,4 -S 1 &" > B g(s&fh DC &y$d, o"'(sb$ 6@-A!-B6 $! o! '(SB$ ker β 0 char F =! -)ig,'&'x! IZ! 6p m& > g(y&n! 1E f " V!$a! * n = r G<C!$? C 0 &! 3 ^

7 $&( $ 1$,&>$& ) &T" ]!T a5! V!$"'& Z!Y&!GO GH O n (F)!$, #' ZZ= " e}4,, & $_G 4=,!+!> -; 5 #.,&34 F n q O n (F) ' <>=!,& 0 g(!$e!!$f'&&" Z!8!"$_G, &"!$ - dn'&' M n (F) F ' GH > H!$ Z= y GO!> $v!0 Z6P 0(, y!'&' "* O,$ n O n (F) ±1 C 0 &! 3 Fe Z!'',Ö&u.,&34 C'&!5!(6D " y &< ' SGH! > H!$~!>(!'&5! <!&S5 SQ o U!GOC 4 -U " 3)+06#!, $&(. $ 1$,+6$& ) ] " K, VGHC >ë!$ >! &"5m SO n (F) O n (F) & > g(!6fq S&P ~,$p '& GO > a,, N"*!$x!S 0= [4 G8< 9!$ C 0 &! - "5 " ; ;!f J! ";$_,h!'&5! <!&!'&'B(~' ] '!$,Jb K! & &P^f-x P ' GH > x!$!'' Z= a GH&!> - / 6 O n (K) ±1 = 1 n = r #5 " = '?!. gqafgh" > 6, " # $v!0 pgo > &! O r (K)!<CR=! F, P F& 4 g(ma, 1 * 0!$p,* 4 G<C a!$t DC 6 V5! '( V!GOC 4 6" 4 -/! n = r + 1 O r+1 (K) O r+1 (K) = SO r+1 (K). &D5! h!& DI, "!'f fbc " O GH 0!-k J C ] &!$ 8 GH / & :- 6< Z!" EQa Q,4 8 LZ=!& O GO V " J(E$_ " V &GH ~<C &"5 - r+1,k,!( CT <C SK <"&"5L!$,'&5 <!Z!''(o'&! u] '& S 1 $_S l,'&5 <!$C'B(f"GO&!'&R=! & -Lkl OQ F =,!&!< '& - kl R = S[n m] S n m!! 5 h H=,!+!<"'& I [$_G " Gs nj -E '&"GO V!$, n m V V n "&GH &,'= 6T---6 -#d g(* Z!'T '& GH >ë!$![< IQB v (1) v (m) R! f = f(v ) = f ( v (1),..., v (m)). 9C'B(f"GO&!' &x!v N<C 7050 HB$"Bẍ&?F"G,$ GH5!?C'B(f"GO&!'& f!$d 0= o 5,-e " 8"<!'&5! < O!$ $_GH [<>([ = L!'(* GH+!' & Q R - R 0 dn G, nj, 9 GO, & a<>(o'& 0$ G 'B '&Z, & n n g O n (F) n m 3v }*"=,!'& > '(!6!0 S V'& GHS=!e<4(E'&&" Z! < J " 5 g : V = ( v (1),..., v (m)) gv = ( gv (1),..., gv (m)). {

8 k D \Qz 0] # 5T!$*&>=,!+!> TC'(*!GO&!'&-/ 6 G = O n (F) G = SO n (F) Qa 1" ] V " 1& 5h,$ G &>=,!+!> / =! S[n m] G = {f S[n m] f(gv ) = f(v ) (g G)}. /!'' 6 Q9 V 0] n S[n m] On,F = S[n m] On(F) 6 Qa V ]. n / * 6 Qa 1" ] n S[n m] SOn,F = S[n m] SOn(F). S[n m] O n,f = S[n m] 0 S[n m] SOn(F). k 8GH&5 > FGH &GH e $_G ) <"&" ~!$ " C &fï$db n F &!<4=* $_G V> 0j* - E 0] B &!!$ $_!)* GZ( G! ] +!' - F\Qa =! 6 S[n m] O n,f n < J= V, B$ 6" " char F S[(r + 1) m] O F = S[(r + 1) m] O(F). igu!! & &E^f6 8 K' 5 )!6D<"" Qa S =! ' HJ 3 y!<c\= ~ 0] B &!E!$ -D "&9 GH&"'&5,(Q&''<C ~!4=! > $_P S[n m] O n,f $_G '+s &!K!$aGH m!$ "~ GO6;!" BÏQ&'&';< Nt J ] & / 0 :!- Q [Qa )> * ~ VJ!5!'Y5"[ GH!- MF qjf Qa )" ] 1!GO I & 5"& " ' GH > & - / 0 R! (kk) = (kk) n = q ( v (k)) (1 k m) (kl) = (kl) n = β ( v (k), v (l)) (1 k m, 1 l m, k l). y V 0jf '&B '(!6*$_! n = r Qa V Z= Q Z,e$_ (kl) = x (k) 1 y(l) 1 + y (k) 1 x(l) n = r + 1 Q9 1 Z= x(k) r y (l) r + y r (k) x r (l) (k l), (kl) = x (k) 1 y(l) 1 + y (k) 1 x(l) x(k) r ˆ y (l) r + y r (k) x (l) r + z (k) z (l) (k l).

9 CT [i 1,..., i n ] = det [ v (i 1),..., v (in)] 3 :- ^ 5 < # " a 0 GH&!>ẍ!$f PGs nj~,ẍ! v 1)6*---6!x p' GO"-x (i v!'&'! H 5!!'D&>=,!+!> 6;, z!'' (in)! HC +!'#J!5!' (kl) [i 1,..., i n ] &>=,!&!> - z'&!&!' g] ÿ$_!gh >!'I G L$_![ 3_C &!'85EJ!5!' 5! u! J, 8Q " &)!$!! & & F = 6 " E!'5 < F[n m] &V5, z<>(l O& V * 0!$ O 0 GO, = 0 8 " O (kl) "&" s &!.64! m F!'5 < &D5!, <>(h F D * 0 F[n m]! S " I GH&!> -; &, < & o!' 58QB E V! SO!'5"9 qb 'B $_ I! '+!&!'Y5! "e<>(e G!"Ekl 0(f'T k ^@-# 1 '! m " GH )&GH!,4 V B] 64!"mQa '+! qj,go ' )!$ [5!'# " Gw!$ k (*' $! <>(m!>(5!!$'&&" Z!x, $_GRb char F = 0 O n SO n G!$ 6 " V 5 5, y<4(e '& GO > Z! y!$xq & F n F[n m] C " F!o, GHJ,$? =! =!,+,< '& - G n m N e79 &, D* 7?6 6! m'&, F& GO!+ O!" D! b! & &#$_ a Z, GO > x!$ " a " J(),$ =!;&>=,!+!> x!$ '+!Z!'45 &)! & '+!6\ 0()"\= 8 s? a]?$_!gh >!'> G $_!x 3_C +!' 5 5!!'#5! l GO!& V "! 5 u& f!! &!- igz!!0 & b 6 N 79! x! #f? \Q9 o 1,GO V$_ 3v"fbg " 55 GO J=*& 5 #< '& Z!$_G -;F& "G!~Qa3! G x (1) 1 x() 1 + +x(1) n x () n Q 8 9! &"5I " H5! Y6>! \Q9 H,? >=!,+,4 ;!'5 < &5, o& 5!" -F\Qa =! 6 " 5 5" " J= F 1 G }4 6?! 1 =! u o! 5 O!$ <!, N$_G 6"! 9 ~! "5,'Y5 S&!, x x n,& }4!", &h$_g 1 x x }4! ),$h'& Z,9$_G[6" " ) 5!,!- / m&!! & " 1}4 Jb n E!<C"ä=! &>=,!&!> P!$T " NJ 5!!'5 E GO!& PC.-k N!'' 8& / & :- {m, V] '&[!$,!'3_C +!' 59 5!'.&>=,!+!> 5 " s h<>()<>=*&?'&\q bg" 5 9>=!,+,4? 0= h&8!!0 &J -F\Qa = 6 1<C Z=*&"F!$'B(* GH+!'&>=,!&!> Q&''Y of N / & 1:- ^H, l:-aˆ V<C =! J(mG H X > Z- " F5!' " G!$F GO! mkl 0(*' GO 4 &"!<CR=! y$v!'&h! &!'&'B(- &mq,34 \Q!" $_m&>=,!&!> m!$f] 5! 9CB B= e!,!6f< " B PGO5 > 9<C N4 & 51 ~B P! C $_h'&!&z,'.gs nje5! y!0 &"5h&,![=! >,.-

10 * ( ( ( 0 /1 * 4 * 6 - / ( 3( 576 dnu '& GO 4!$ SC'(*!GO&!'P!'&5 < Z!''&,;! -0 ob$ R = S[n m] m 8&)G ' Bbc'&&" Z!Il E=! 0 =,!+!< ' dn '& GH >ḧ,$ v (1) v (m) S!'5 <"!$x&>=,!&!> C'(* GH+,'&DQ [f 0 H& > & &!&E H<C V! R G S! 106#6$&' )+$,40e ' GH > B$T 9&9"! >! S N < & 58!$ 5 " s R <>(E " 1 '& GO 4!$;'\Q9 e" 5!- G ic$ &)f l &) =.6a! 6Q9 S!''a 0Ḧ!z 0b n m m n 3!GOC,< '& m '& Z, SO n,k &>=,!&!> Z-;ic$ &9 = [! 6fQa I!'&' m m > n! 0 N,y GH!< ' m '& Z, O n,k 4=,!+!> - MF! e, P$_a!>( SO n,k &>=,!+!> D,$T 5! ) 0jf&J s!''@6>bä# Jb m,(h, <C F*! < F = H B a N " G :- {"- Yt ä! n m " V! &S!$x " r "&GH &,'C Q 1 '& GO 4 e! ~ I +,5,b SO (K),'Gs ~& -hic V$_'&'\QN s r SO r (K) O n,k &>=,!&!> ~! 0jf&J V '(o& =! y 5 dn''x,$d F& GHC!<"'& V&>=,!&!>ë'B(* GH+!'!''p< ) J! GHf "'&V^I&GO!5!$&>=,!&!> #'B(* GH+!'xQB H&> 5 Df 0 O& 4 -Dic #$_''&\Q, D '&, Q&''" Z=! e! &"51 GH!!< '&B g()"c J & #R=! - Z kl P5B= # P" &";&V "&p.-? 9 f,$_x,$ &" GOC!< &' g(! IJ C [ / & :-Aˆ - H!C I#>,& lsgh h Z, *!$a!$p m 0jf!$? &5! fbg 5 I& "!GOC,< '& ~>=!,+,4 &E Z!!- SO ic GH =! J( '&&3, 'B(6e< " SQa,!<"'& o "\=,6N 4,K s S&>=,!+!> 6!5 Q [ V!!o}4!", &I&>=,!+!> ] " o[ V 0=*& (kl) &.6"5, 1 V 5h!$?&>=,!&!>!'(* GH+!'- '&()&, " " $ / & k F!'&' F$_!'&'&\Q 5~ F'& GHGO!-x (OZ(m,ä e&"5)!gogh b G #I GHGH" &G Z S F Z[n m] S[n m] F[n m] Q O,3! a&>=,!&!> D 1&>=,!+!> - a(hv'&5 >ä!< N!$Y!,.64)C'B(f",b GH+!'RQ 1> 5!.f 0 O& 4 pgo5 > T< D"! 1<>(~ D!GH D'& p!t.&go!5,- CT < I " 1G,5 ~!$ & B$ 6 " - F p = Z/pZ Z S char F = 0 F p = F 0 = Z!#" f Z[r m] O r,c $! F p [r m] O r,f 0!9@<60 +#"! $ ' $ p!4 #6$+! -0

11 6$ $ 9 & 5H&>=,!&!> &e }4 B=!,'& > O< & 5m4=,!+!>ë " F!>( O C 0 &!.-Ukl [ \= y r,ö$ ' O\=! $_H!'' f(g u V ) = f(v ) C 6 6" [!'& \= $_!'' 6 - u C r q(u) 0 F u F r q(u) 0 79f 0 O& 4 N!$?<C! [ egz(s<c I=*& Qa o!, &!'Y$_ 0 Q yf 0$+b ] > I.-!$D 8= ~=,!&!< '& 6.!" bg4=,!+!" 1!$ <!&'& Z F p u O r f \Q $_GO!' }*!'&B g(!$i!h!$i Us &!!' $_ 0 - / $_G!' }4!' g(r=! &GH '& P s R=! 6 V' GHG8 \=!.- Z F p! " f Z[n m] SO n,c $ 0!9@<60 +#"! $ ' $ p!4 #6$+! -0 F p [n m] SO n,f 6$ $ *!$!,'&5! 1 o " E 0=*& ),- 9 5o>=!,+,4 ) " &; }4 =,!' > D I<C 5I&>=,!+!> " D * 0 #!$!>( gqa1 DC & - SO n kl h" \= h 6" yb ' \= s V$ f(g u g w V ) $_!!'&' "'& I\=! $_V,'&' 6 = f(v ) C u, w C 6 - n q(u)q(w) 0 F u F n q(u)q(w) 0 79f 0 O& 4 h!$ <C! l&" 8GOZ( < H=f Qa!), &!'D$_ &")Q l,b H > I.-!$# 8=! 0 ~=,!+!< '!" 6., bc&>=,!&! Z F p u w SO!$ <'& \Qo O$_G,'. }4!' g(!$!&f,$d o, &,'.$_ &"- n / &" f $_GO!' }4!' g(e\=! &GH '& P, \=! 6 V'& GOGO8 \=.- Z F p " f Z[(r + 1) m] O r+1,c $ #6$+! -0! F p [(r + 1) m] O r+1,f 6$ $ 0!99 >0 #"! $ ' $ a( V 0=*& ' GHGf6* V&GO!5 N,$ &>=,!&!> " - f SO r+1 (F) kl V m R=! I, I&GO!5 N!$ &"G!$? GH5 " C'B(f"GO&!'& f!$p =! 5 -icï" H 1 S \=, B '$#~".- N$_'&'\QF$_!G f V 0] B &![,$ 4=,!+!"!- O r+1,c & ( * *, "+( kl 8!''p 8J(fG8<' GO!K!>(y,$ ) gq9s'! -Fk x y ] 1 V5!$;! sgn w m '& Z, GO! GH+!' 3 :- { 5 Q w = z (i 01) z (i 0m 0 ) (i 11) 1 (i 1m 1 ) 1 (i r1) r (irmr ) r i j1 < < i jmj $_! j m<c V V5!$? V Gf, & i 01,..., i 0m0, i 11,..., i 1m1,......, i r1,..., i rmr!$x V&" & I:6T---6 m - p!4

12 $ $ igy! 0j*ë gq9h! &"9Q9 1!'&'.<C ~Qa34&"5O\= -Dk IQ Z Pf (i 1,..., i µ ) = Pf (i 1,..., i µ ) n = 0 (i 1 i ) (i 1 i 3 )... (i 1 i µ ) (i 1 i ) 0 (i i 3 )... (i i µ ) = pf (i 1 i 3 ) (i i 3 ) (i 3 i µ ) - - (i 1 i µ ) (i i µ ) (i 3 i µ )... 0 $_9 B$v, O!S!$p "!> BbgJ(*GOGH &ego, BjHQ 1"!'B$pfb µ µ e,$; "?A! &"5H H ) 6T---6 -adn'&'y > & e!$? (, ) i 1 i GO, Bj6*! $_ I!'& $v, O!Y6! ~&>=,!&!> 9" µ - $v,$+b O n (C) ]! 3 :- ˆ 5DDG 'B Bbc'& Z!pm =! v 1) $C " P"1 C (i v (i µ) &S I& "& -! I 0#6$ 0 #! 0 +! '<!, w &*! $&50!4&06$>+,4066 x j 0 3J:-Aˆ 5 Pf (1,..., µ) $,;! -0 ' $-$&. µ $& 0 1#!4 050 < 0 j$ m j = µ $ #6#7!,;30 B! 1 $&10 "!4<# & 0 j 0<#6$ 0 #! 0 +!4 y j µ {j>0:µ j 1} sgn w. >$ $ " ef 0 H& > #!$ &D5B= m<>(h < " 5~ " & 5I=! w!$ " [J!"! <,&h> -l m!gh >!'& '+s &"5y " Pf (1,..., µ) $v, O!!$) 3, Q bgj(*gogh & GO, Bj 5=! <4(W β? Gf &"5 RQy! ' GH S yb &> & [ G?!$) = 0 -!$GO, &! "RQ~, ~ $v, O!L& &GH I"* V,$ $v, O! 6. sgn w m j m $v!0 9$_! Z! 0j - / 6! GH&"5,!h!fb = hf 0 H& > Z6 j Z, j w G"ë< I 0=.- / 0 - m j µ j = m j / ic$ 6$_e!GO 6CQa Z=! gq9e* I!$ Q o &K<C gqa j > 0 x j y 3v! 5! #6050> 506 " W j j D$vs,!E gq9 &" > Z!'43!st, > 5S\Q6 Q EG,3! 9 = - / 8! G8 9f E!,' s &"5) P$_ x j y Z, -) " 0$_,6T j j > 0 j $v, O!.6Y$_ 6.&F, ~!$# j > 0 m j m 3! <! GO, BjLQB l > $_!)*!" = j b (kl) = sgn(l k) l k! JQ&,- A;j"! 5h<>( " ~] J \Q! & 5 0 E "RQ, µ "& & B$ -Dicë& B$ - j µ j 1 µ j 1 1 = µ j µ j = 0 / ~*! 6Q9 m Z= h G$v, O!!$# G, njyqb n j = 0 m 0 m > & - A;jf! "& 5S<>(o ] V\Q! & 5S 0 &! 0 b (kl) = sgn(l k) \Q s & - µ 0 µ 0 d 5m [ V 0jfC > N!$^8(f ' V "' Z-

13 $ $ $! $!" m n,;! -0 )+$&,;:-$&'<!, m!9 m n mod $ ( 1)π [π(1),..., π(n)]pf (π(n + 1),..., π(m)), 0 µ = m/ 0 1' 0! $&50 $ 0)+0('<1 "! $ "!4 (: π S m π(1) < < π(n) 0 #6$ 0 #! 0 +A! π(n + 1) < < π(m) $ ) m,;! -0 ' $-$&'<!, w &! $&50!4 &0>$7 +,4066 $& 0 1#!4 050 j > 0$ m ="! #",;:7)+$!9B! $ #6#7!,;80 "! 1 $&10 j " x j y!4<# & 0 0<#6$ 0 #! 0 +!4 j µ r ( 1) j j>0 : (i j1 ) j =y (i j1 ) j ff sgn w. 6$ $ df GO 5,!1 fb = of 0 H& > Z6xÏ$_''&\Q~ =*+!''(, w $_~! 6 G JÏ<C h = l, &0 '([CB B=!- / 0-1k j > 0 m j µ j = m j / Z!'' "& y!$n! $_G 1$ $ [$e [ G! &"5 x j y j w (µ $_!G!$D! * ~K!L,' s &"5 -NdN>(y!$#! j 1) x j y!''#< HZ!''& l 5*f!- Me\Q j #CB &! :!- ^f-aô \Q), z w GOI h"g 3J:- 5FKQ &,I fb = * H > V! C w S " H&G"', "~5*f 3 $_~! 5N!$a h!$a Z, l'& 0 N$_G 6 j w! Vf 0 O& 4ë!$ & K G w µ r {j>0:µ j } sgn w ±1, Q O j &I " O&5!$P, 1C G" s &!,$9 " O'!" ±1 x & 3_.-F 8' & 5 Q "& K f N 8! L! 3_ 5ey$_>Ï! o& j w z w!'&"!<c Z!'" 6 ' ZZ=*& 5 V ë&s " 1! (E![& - w icë$_!'&'&\q9, " 1f 0 H& > F,$ 3J:- 59& w µ r {j>0:µ j } sgn w ±1, Q y j GOGO, &uh\=! O [5f* & H!$I"!" x j y 3v Y-P! 59$_G - j z w ic$ Q9 H Z= H gq9yf 8!$ Q <C gqa 3v=5! #6050> 506 x j y j j G = 6p$_~Qa O!!&I!Xl m )<>( &>! 5 5 8' )!$; 8 gqa!st, >?A6T, o " ) gqa? ~o! L!eQ'&'pZ!" ' - x j ±1 j r 3 :- 5 y j

14 /! G Nf F&L,L!', 5H F$_F Z! -FigK & x j y j j > 0 Z,!6 j [ G & ( m j ) 1 ±1 if µ j = 1 and ± 1 + ( 1) l 1 = ± if µ j. V5 & $ GH ] F! [ & + x / f" 6 1Š j n G & m 0 l=1 B$ y j GH ] - d 5l y 0jfC > E!$V^! 'B Z- l=1 ( 1) l 1 = 1. ( 1) = '(!69Q9 K!= [,Ḧ " +& " (,)', "+( / 6* =* e $v, O!O&+D! & :- ^f-aˆ<>( 5 0 µ r 4=,!+!>äQ 4 5 * H > 6 Z!''. µ r O n (C) µ '&! (1,..., µ) = (1,..., µ) n Z[n µ] O n,c. dn" '( 5V GHGO! G V5 0ä Z F µ '&! * H& > e 6" Q&''Y<C V '&'Z!''& F p O n,f 4=,!+!> QB (1,..., µ) = (1,..., µ) n F p [n µ] O n,f. dn!'&5! 'B(6"$_! QB 6 =* I V"G 3 :- 59!$ Dsb m n m n mod B o:- ^f- <>( h5! ë! µ r m '&! SO n (C) &>=,!+!> 9QB S4 5 * hb > 6 Z!''. dn" '( Z,'&'.B Z F y5! 8! m [1,..., m] = [1,..., m] n Z[n m] SO n,c. '&&" Z! SO n (F) &>=,!&!>ÏQ * H& > ) [1,..., m] = [1,..., m] n F p [n m] SO n,f. MF, S s m&>=,!&! S\=! &!! $_'&'\Q8$_Gƒ sö\=! <>(u F C '& GOGO!N!<\=!!-hMe! m!'s, ~ " H}4! m<!3! 0 1! s &!L&> f " 5!'& = V V 0] y&e$_g '&3J:- ^ 5- :\Š F p 6

15 MF\Q! h " [! SQ :- ^*- ˆ 6f V&>=,!+!> F = K!m,!0 &J &[^*- P( DCnb (1,..., m) F [n m] O n,k, Q &h 0] Y6a S G!$e! m '&!GH GH+!' s Z6DQ w QB l m$_g 3 :- { 5q6? Z= H &0 '(K = H!"l =! $_1!'' m!, * V&!!'B, & 5E! - a( D! &l:- j ^*- 6T x j y!gh I ' $_ j [1,..., m] F [n m] SO n,k. j > 0 6 m n <! L =!.6 (1,..., m) = [1,..., m] 0 0 0! +5 (! + 0 µ r O r+1,k (1,..., µ)! +5 (! + 50 '<1,;"! ),;! #!9B: r+1 SO r+1,k [1,..., µ + 1] r+1 r! x$ y &N \QV&K! &"'+! s ~$_ &!! & &1^f- /!O*!!>( 6 qj, 'B( o,$!" & n m n (1,... m) [1,..., m] ].-1kl h!''? \= m& / &u:!- Ê,ÏB ~&~!L& "!GOC,< '& SO &>=,!&!> 9B$ - "&aq'&' '(H F$_!'&'&\Q 5)!< =,,.-;k N" ] ~ n,k n 3 '<,;"! ),;! #!9":!!$ [C'(*!GO&!'? y<c h H&"] G G!$a m,!'d 5 x$ y!$p 9GH GH+,'& 6 =,!&!< ' - 3 / a8 fb =!GO5! 9C'B(f",b x y GO&!'@6C I }4!'I h BXY m<c gqa h 5, L m 5 HK =,!&!< '& - 5 6$ $ OO!<4=* h$_g [ "!<\=! o!$f [GO! GH+!'),! C Z! '<,;"! ),;! #!9":! µ r!4 ' $ (1,..., µ) r r kl!''8,'& R=! z& / 0 :-Aˆ, [$_!" 6I n = r µ > r O r,k &>=,!&!> &I&" GOC!< '&,-h &IQ'&' '(o h$_''&\q b (1,..., µ) & 5"6"'&5 > '(HGO! ~ <" ' ~G 'B '& g(< =,, &.-k I! I Q '& Z! *"&,!" -pk ] " 7'<1,;B! ),;! #!9": t j = x j s j = x j +y j (j = 1,..., r)!!$?mc'b(f"go&!' m<c N " I&"] G8 G,$x ~,!' 5 e,$?b GH!GO&!'& s &S " =,!&!< '& - s 6$ $ µ '&&" Z!GH GH+!' Q [f 0 H& > ):!- s $ 0 O r,k s (1) 1 s (r) r t (r+1) 1 t (r) r t (r+1) 1 t (µ) 1 :: z! +5! + *"

16 igz O G! )!$ &).6' <C! < =.-Kk l m n " H J "!$, m '&! SO n (F) &>=,!&!> NQ li& GHC!<"'& $! - / <f,ï " h G 3 :- 5F!$ DCB z:- ^f- $_G char F = n 4 " $v, O! & D! & :- ^f-aˆ, "=*& S<>( 5 Ö! µ r+1 m '&! >=!,+,4äQ y4 5 * H > -a7!'&'.b SO n (C)! & 5, dn" '( Z F m5 0 N, m 1,..., m = 1,..., m n Z[n m] SO n(c), 1,..., m = ((1,..., m) [1,..., m]) /. '&&" Z! SO n (F) &>=,!&!> 1,..., m = 1,..., m n F p [n m] SOn(F). 0! +5 (! + >$ $ µ r "!9""! 1 1,..., µ r $ / '&'\Q& 51 V 0] &"6"Qa ) \=! z! ), 1#60 $! 0 x r y r SO r (C)! +5 (! + O r 1 (C) (1,..., µ) r 1 1,..., µ r xr=y r=z = 1 (1,..., µ) r xr=y r=z = = r µ 1 Pf (1,..., µ) r xr=y r=z = 0 5 (! +>10 = r µ 1 Pf (1,..., µ) r 1 =(1,..., µ) r 1, 0 0 µ r O r (F)! +5! +!4+-$&! +. SO r (F) 1,..., µ ig>=,!&!,$!5!&$_''&\q;$_g, # " 1,..., m SO n (F) SO n (C) <>( CT GHGE:!- ^f- ^f- S&" GOC!< &' g(l"*!$ $_ 6 K<C E5B= u / W:-Aˆ 6 1,..., m Q&''Y '(! '+!GO.- k 1!''T!'&h >$ $ H 5!'5 >,& I " O DC 6p " VQ & x 1 y 1 1,..., µ &x&>=,!&!> p &9"! 4=,!+!> -ig.6 GH G<C P s F[n m] (1,..., µ) Z[r µ] 6,Q9 "'() m, 6s $_!'e O n (C) &?! D>=!,+,4 6,Q m&;!<4=* -;ic$ x 1 y 1 char F [1,..., µ] F[r µ] $_!G DC :- ^f-!" O" ] B L!$ 6T&" & - [1,..., µ] 1 1 F / ) S!! & & ^o!,6dq9 u s &"! [1,..., µ] Z[r µ] &>=,!+!> GHfSˆ"6 Q & [&!5!&S<>=*&! - :R^

17 ! F& 5Hh$_G"'+8!$xk &3 V"6CD:6"k T$v, O! 6,'&'Y &>=,!&!> P'B(* GH+!'D J S&H a 0 &!EZ!< e & OQ O,& <!& 79'X!'&5 < " > s &! - k L z!ë K "&E!- / 5 * ( * * 576 GO &U & & 6x<"" 1GOJ V!$P G! hqa ''D3*"RQ $! K=!,'&&W\=! y!>(!'&5! <!&!'&'B( '&! ] '& -h " O4 & 5, V& F char F s ; " (h, a=,!'&8!'&~&!!0 &J ^f-? f!$_? k B? x G& I6! G *- ˆ 6R!!I$v! x "& 5 0 =! a5! "6s! 1<!&#!'5 <!& 5!GO 0 (!- & +* * " & Fe Z!''?,Ï " h*"'&'bbc" 8! C 5S S5!,'&5 < E!$PC'(*!GO&!'& &V 0] [< h m'&* V!$aGHGH = V!$ B I!GO5! V ' GH > V!$! =! I 5,- Cp 0ë &> f " I V,, & A = F [(kl) : 1 k l m]. &9& <!'5 <!$ -D! ~! &S5!'.!'5 <"!Z!''( F[n m] O F (kl) & " C > Z6 & [5!!'T,ëHC'(*!GO&!'Y,'&5 < "- A! 0B+1,9,;. #6$-06#6$&(>063)+$! 13$ 0 >060, 10 & >0 07 ' 0 6$ $ s ic$ char F F[n m] SO F[n m] O F A 64 " N " G $_'&'\Q#$_G F 'B 9 7 YZ(*& 5 F[n m] SO = A [[i 1,..., i n ] : 1 i 1 < < i n m], Q - / O' e! GH I s!!, 1^f- [i 1,..., i n ] A F = K / C, D " ec4 ( v (1),..., v (m)) < '&"5; I *"'&'Bbc"!$, A &6? " O= 6a---6, &J$ (L O }4, & - O <! v (1) v (m) (kl) = 0 0(L! N "!!'&'B(y& 5! '+! 3v@-A!- 0= J(4Q EL h <! 650- :\{ q = 0

18 ' CT <C O GOsjf&GO!'x!!'&'B(o& 5"'+!~ <! O>!& 5S Gy-mic V$_''&\Q W $_!G k B? G`, F "&GH &[!$ & 6Y! o, N W r = n/ 0jf ehgosjf&go!'!!''(h 5 '&! <! [ s W 1 K n = W W 1 ker β. / 6 '! y$_! VC +!'T! "5,'! J$_GOsb 0 t K g t SO n (K) <>(L:-DdF>(, FG 'B '& 9=! N& <4( 6 =!~ <>( 6C! o=! 0 N& W t W 1 1/t & 4=,!+!> " 6 ker β f K[n m] SO g t f ( tv (1),..., tv (m)) = f ( v (1),..., v (m)). I '&"N$_I!<!J( 6T ÏG8 1!'S!'&K$_ -8 I&GH '& t 0 t = 0,ä NC4 ( v (1),..., v (m)) &#!4!&" EH N4 ''Bbg e!$ 0, 10 & 0 80 & A $1,406 K[n m] SO - F[n m] SO F[n m] O 60 +!930,;:<10 +. F >$ $ / &" &1!$ ] " H g(* OR=! h ] '.6B )Mef 0 +,.- *" 1Q&'' A f O 1 m \= V " 879!''+!J(O$_ V'+!5!'&5 < - F[n m] MF\Q &a) " B= N!'&5 <,& 5.6*hMF!5>,?A#! G SO AM)6 G = SO n (F) G {"- ˆ TZ(*e, & ] " '(5, y!e![!'&5! < "- F[n m] 79 & Fh!GO5! ' GH > G - a(! G :- {f-:1! f F[n m] " KMF"'&'&J '&'& s = 6!S \Qa E K&" Z!'!$ G 5, W<>( f F[n m] " -8ic ~$_''&\Q~<>( AM)6@Cp GOGOE{"-Aˆ - ^;,!VS\Qa I&K h&" Z!'?!$ (kl) f 5, [<>(S - F[n m] G (kl) dn "'(*& 5m, F O! K '& GH >!$#h] )J(* G,$ GH5! F5! fb f,!p!$y ~!'5 <" \QP, P " N&" Z!'!$ 5 " s F[n m] G F[n m] <>( >!&!'&'C ' GH >,$,, ~!GO5!!$p "&5 G (kl) F[n m] " 5 1 5!,- / 6!!?! G F[n m] G A GH* '&,6\&p5, 8<>(V ' GH > x!$ " 0b 5 P'&\Q9 x!1gh #] #*"G< -? 9 ' GH > p$_g ] " qbg GO!' =!!6O] B I4 G<C!$x G Q&''.f O!- *8& +'" &&" &,) * &( k H!'&'DGO mqa '&'nbg34 \Q $v! 1$_G!'5 <!&85 GH (!- mq9! A=,!nb g(?4<c '&\Q! #$_#!H& " < '& s O" N!'5 <"!9=,! g(h\= D F!<"!(!'5 <!&!'&'B(m' [] '& -#k B $_! 1!'5 < h!$;c'b(f"go&!'$_" 0b F O(X) ) 6D, zqb $_) ] '&u!$,!'d$_ 0 8! - CT X K(X) X :Zˆ

19 ! 0 $ <C )hgh" &G!$x=,! - 1GO G!S m< 1$&'<!9. φ : X Y φ +9B$ & " 1 -DigS "&!,6 IGO! G & > ] φ(x) Y φ Q 1 <"] '&,$ -# VGH "&G φ K(Y ) K(X) φ $ Fm!!< '& I] '&y 0j*.-ak &,&S m<c < 03),40 1 y " 1$_''&\Q&"5 K(X) φ K(φ(X)).6s a$_? qj"!go"'& # 9"6 3J: f- { 5p G GH "&G &x GH&!> φ! L &V fbc 5 '&!~&!!< ' $P!"L '(y$d &~E fbc&"5 '+,FC&> 6?! " H"X > +,'! L, x X, φ(x) Y d x φ : T x X T φ(x) Y x t 0 =!!- K(Y ) 0 φ : X Y 0 1$&'<! + $ 03),40 ' $& )!4' $ 5 (!9. 0B! 06 ) )+$ 0 f!4 B! $,> (>-#"! $ $ # " $& $&' 0 -$+. X 0' )B: " (!4 &! $6)+0 0 U $ X $ 0 >06"! #"! $!4>0,!4<#6$+. f +#,4$1 0 0> $ U φ U 0!4< 0 )1,9,;. 1# $ G "! $, ( -#B! $ f $ Y $!4 $ f φ K(Y ) 6$ $ k NGO\(O! GH F, 91 >(fc bg $v! ~& - &P 'B( X \ U X f &,!< ' \= $_1GH m φ K(Y,!'#4 G<C ) <4( a 6 3J: f- -S 4 ^ 5q6.- 'B$9)>! &,ḧ6 &8>! u f ps φ K(Y ) s f φ K(Y ) < Z!" &!!< '& V<>(S"F,"GO" &Y- φ y V!y<C V!SQ 0 Y & G,'@- / I$_ 0j"!GH '& : "- { 506".- φ : X Y #B!950 ' $& )!4' $ 5 (! 0"! 06 $ &6' 0 Y!4 -$&', ) )+$ 0 h!4 )+$&,;: -$&'<!, (>-#"! $ $ X $ # h!4 0 )1,9,;. 1# $ G B! $, ( -#B! $ $ Y $ $!4#6$+! -0 h! φ 0 K(Y ) h!4 0),9,;. 1# $ )+$&,;:-$&'<!,: ( -#B! $ $ Y $!4 h φ O(Y ) 6$ $ / ~$_! 0j"!GH '& 9"63J: f- { 5q6".- 6!"S! ~, & ~Q9 1! I Z,'&&"5 Q o, H V=,! & 6 5"'+! $_ 0? [ 1 " 8,$ 9 3v -,-9 =! J(*Q " ] y, &,'$_ 0 5, I V!GO V, C'(* GH+,'C$_ "?-, "+* 8 1,.( CT - k 0] < G = SO n (F) G = O n (F) K = F(n m) $_!0 K] ',$a m'b(* GH+!'?!'&5! < -m H '& GO > V!$ R = F[n m] K! I!'&' S,!'$_ 0 &! & =,!+!< ' -ig>=!,+,4,!'$_ 0 n m! m 0] l&l m!gh mg!" V!V4=,!+!> VC'(*!GO&!'&-8ic$ 6? char F, &!'>$_ &8& &>=,!&!>?B$! 8 '(1B$ ;& &>=,!&!> O r+1 (F) : SO r+1 (F)

20 ! 050 EO N] '&E5, S<>(!GO5! 9C'B(f"GO&!'&!$N = U kl [! f "&H!h [ 0] &!$ O r+1,f &>=,!+! S&!<!J(,!0 &J & $_ * 6 Q9 V 0] n K O F = K 0 K SO(F), Q & I] 'y!$? =! y, &,'$_ 0 kl o \Q 0 K '&*3, y] '&!$V&>=,!+!>Ö, &,' $_ &"6 Q W& K G O Z!& # )" Z!' QB m!h e!'5 <" G!$Y&>=,!&!> #!'(* GH+!'-pMe! R, &D e$_! &h] '&H!$ -; &D$_''&\Q# Z!&'()$_G G $v! #, # K G R 5 &9C J$_ 0 Z6 -,-6fä&95!, S<>(EGOG8", P!$pB 9 '& GO 4 kl 8!''; h"!, &! -1 " G SO n (F) D = [1,..., n] n = 1,..., r! &>=,!+!> Qa V Z= 1" y& / 0 :- ^*-kl V 0] r SO 1,..., r 1, i = 1,..., r 1, i r = ( v (1),..., v (r 1), v (i)). 0 0,! 103) ! +5 (! , O! :7 0#, 10! #,9,;: F(r m) (kl) (1 k l m, k r). SO! : 0), 10! #,9,;:! 103)+0 +. F((r + 1) m) 10 +! +5 (! +@ # 0 0, 5 (! +9 (kl) (1 k l m, k r) [1,..., r, i] (r + 1 i m). SO!4A : 0, 10! #,9,;:! 103) ! +. F(r m) (kl) (1 k l m, k r 1) 1,..., r 1, i (r i m) , F((r+1) m) O! : 01, 10! #,9,;:! 103) F! +5 (! +9 (kl) (1 k l m, k r) D [1,..., r, i] (r + 1 i m). :

21 0 $& 0 50 m < n F(n m) SO = F(n m) O F $& 0 m n$ 0, F(n m) SO!4 B! #0 30! $$ F(n m) O F $ $& 0 ' ),40 : 07! +5! +! n!4<050 : D! n!4<$ &" &8& 3_ 5TQ&'&'4<C 9GO! agh '& 0 P& / < &E:!- {"-AˆNQ 9Qa GO F ~}4!, NC'(*!GO&!'&D\= s!, $ (!- F(n m) O F D /! V f!$ 6fQ9 V&> * I I$_'&'RQ 5, &!.- ;7 3v 5Ed GO, Bjm r m O 5 '&!#$ ] J!'& GH, N'&!'B(m& " C > Z- / h<c F G, njmq E r m :V VGO!&S &!5!'.!" = O '& 0Q!- 3v< 5Sd Gs nj & V F (r+1) m SO 5"'+! B$ " D&GO!5!$* ] ' GO"T! D'&!'B(e& 4 min(m, r) & 6! F n / ker β " [ 0jf O "fbg& 5! '+!)= 0 HQ & & β J 5!!'9 ] '& GH - min(m, r) / 0 N< 9 GO, BjVQB E: ) GO!8 &!5,'@6R 0jf " E (r+1) m $_P ~C H! &"58 8 f &,,6"!" = ' 0Q!- z 3v 5Sd Gs nj & V F r m SO 5"'+! B$ ] J!'& GH! V'&&" Z!'(H& " C > Z6C! min(m, r 1) " [ 0jf O "fbg& 5! '+!)= 0 HQ & & β J 5!!'9 ] '& GH 6! min(m, r 1) &u o! 6a y qj" = 0 Q & m r v F J 5!!'C h I] ' GO"!! r β k 1' r 1 ( v (1),..., v (r 1), v ) 0. E <C I V'& 0$ GH r m "< Gs njs!$x " r GO, Bj 1 r ! ",9, >060 # & 06 $ 0 +!9B! $ $ $&(' " (!4 &! $6)+0 0 #6$+! +! 1 0 )+$&! + E : min(m, r) >0 1, ' B(! #606

22 >$ $ 0~C e$_'&'rq!'(e$_g " )$v!0ë,ë!!'&'b(e& 5"'+! <! $_G h' y ë&s " 8I,GO! "+!.- Fe 5 '&!B g(z!$ &m<>=*& h$_ 3v 5! 3v< 5q- / 3_ 506a< =! y, E v = &!fbg& 5"'+!6&! "5,'a!''a'&"GO 8!$ 6 (0,..., 0, 1, 1) F r E!, = 1 ( pf ( ( ) (r 1) ( ) ) ( 0 0 det r ( e (1),..., e (r 1), v ) = ( )) ) 1 1 = kl V!'&'Y y " V$_!'&'&\Q 58 }4!$xk?A Gy-! I " V, V r m >0 F O >0, 3 = "!4 (: " (kl)(v ) = (kl)(v ) (1 k l m, k r), 0 060<0!49 " $&' "! $ V, V F (r+1) m >0 SO >0, g O r (F) # gv = V = "!4 (: (kl)(v ) = (kl)(v ) (1 k l m, k r), [1,, r, i](v ) = [1,, r, i](v ) (r + 1 i m), 0 0>0 0!4@ 3)+06#!,*$&( $ 1$," $&(' B! $ +# gv = V # " V, V r m >0 F SO "!4 (: >0, g SO r+1 (F) 0 $ (kl)(v ) = (kl)(v ) (1 k l m, k r 1), 1,, r 1, i (V ) = 1,, r 1, i (V ) (r i m), 060<0!49 3)+06#!, $& 1$, " $&' "! $ gv = V g SO r (F) +# :

23 6$ $ " H].-.- ' GH min(m, r) min(m, r) min(m, r 1)!$, o,'&m!$!'bq Z(*!oH <! QB -#k? V V W W ker β = 0 G R=* F, '& Z= 4=,!+!>, g M n (F) q gv (i) (i) 3 :- 5 = v $_ Y i min(m, r) 1 i min(m, r) 1 i min(m, r 1) ig Z! 3v< 59 Y- 3v 5q6" " V qj" F"fbg& 5! '+!a= 0 5!!' u v $_ Y- (i) i = 1,..., r i = 1,..., r 1 -[ " C 0 0] u 3 :-: 5 g ]"jf ~ u v (i) - / 6Y<>(y '+!& 5 Q B$D!J(6.Qa Z= m g gg u g SO [ s 3J:- 59 ' $_ " 1!GH I&,e!& <C $_!- i kl " L E \Q, 3 :- 5F!' "'& e$_ Y- r < i m r < i m.- -)Cp 0 1 1! GO m, &I'&!5! h "5 $_!1 r 1 < i m m i 0jf&J Z- 3 5;dF &D"fbg" 5, e! &#I<,&D,$ 6>B Df O β v (1) v (r) F m "RQ, r ( β v (k), gv (i) ) ( = β v (k), v (i) ) 3 :- 5 $_!" -D & }4 B=,!'& 4 1 i m 1 k r ( β gv (k), gv (i) ) ( = β v (k), v (i) ), - g O r (F) Q & $_''&\Q9$_!G 3v< 5 AD}4!'&B g( 3J:- 5a \= K!,<\=,6C! \Q së V X &~! "5,'? S =! G <C Z, 1!$ gv (i) (i) 3 :- 5 v $_ 6Y< " V ~&I!'S! <>( v (k) 1 k r [ det v (1),..., v (r), gv (i) ] [ = det gv (1),..., gv (r), gv (i) ] = [ = det v (1),..., v (r), v (i) ] [ = det v (1),..., v (r), v (i) ], S " h X 3J:- 5F = -8MF!, ~ }4!' g( SO r+1 (F) -,<\= $_!'&'&\Qe$_G 3v 5 k )!y"* K! 3_< 50-Pkl V \Q Z=! 8 }4,'& g( 3 :- 5a$_! - 4 I h"x 3 :- 5F&IJ!5!'x E =! 1 k r 1 : g 1 i m

24 $_ 6 " 0$_,6# hk!'+,8g 'B '& H!$ " E= 0 v (k) 1 k r 1 v R=* [<>(E V'&! B &!S& N ] &L:- {"- "- 3v 5q-Dkl 1!'h Z= ( v (1),..., v (r 1), gv (i) ) ( = gv (1),..., gv (r 1), gv (i) ) = ( = v (1),..., v (r 1), v (i) ) ( = v (1),..., v (r 1), v (i) ), S h X 3J:- 5F = -8MF! sï }4!' g( SO r (F) -!<CR=! h$_'&'rqe$_g >$ $ $ 06$&>0' 3v 506 3v< 5;, 3v65 k B $_? 5 '&!;GO! 0] φ,! V4=,!+!> &E I G! * & s -kl I \Q F, n m m GH&!>Ḧ!"!!< '!- k '+,&G,Ḧ BXY 4 +!',$, φ φ "!&> 5=! K N ] &z:!- {"- N> "-~, &!'p =,, = I!! E $_''&\Q- g (kl) x (k) j! [! 3v< 506 = y (l) j, (kl) y (k) j = x (l) j, (kl) x (l) j (kl) = z (k), z (l) dn''.!, &!'Y" B=,, B=!$ / Z! 3v< 5q6 Q9 V< J= ), / Z! 3v 5q6 Q9 1< J= V, ( x (i) r (kl), = - = y (k) j, (kl) z = (k) z(l). (kl) y (l) j = x (k) j, (kl)/ z (i) = 0, E $_!'&' k 6 l, i - y (i) r ) (kl) = 0 3J:- 5 3 :-B:\Š 5, $_,'&' 6! - E k l i!$ igk!''y " 1Z, 6, Gs nje$_go y<>(s )! +!'T B=!s =! e, n m QB!" * & s e&>! 5 L!NB E (kl) e (k) x y l K'&"GO!! Q!" f &, )4! 5 u!1 e (l) x y k ' GOY6;!''! V!'& GH V< & 5 = "-Ok O!'(K m s 8,'&'; GO, & 1! n m '& Z,'(K& 4-0~ 8'+!G $_''&\Q)l! 3v 50- / 3_< 506; )" H R=! N, 9 GO, & a$_!gh <>(H F!J +,'C" B=,, B= 9,!$p! 1 m E ^!Š

25 QB S C 0ë F=!,+,< '& [1,..., r, i] z (i )! ~'& Z,'(H C " > Z-# & <>=*&! 6 & )s 6 E [1,..., r, i] $_ z (i ) = δ (i) (i ) r + 1 i, i m -/ 3_ 5q6 Q9 1 I I$v!, F, E 6 ( x (i ) r y (i ) r ) 1,..., r 1, i = δ (i) (i ) $_ - r i, i m MF\QV6 ~"\= 3v 506 3v< 5p 3_ 506! C, 6!Q &? f F(n m) G G 5! 8) " 9, GO > ;Qa aq9!> F \=,-;k P & 3V!$!?N,!'$_ " f -x &p!!> #!'&! 5 a<" x!$ 6! D! &H:- {"- "RQ F n m f G s &1 J!>!'&"5[ O] < 8,$ 3v, )'& Z, )lgo H *bg GH" g( C f φ a(+cp GOGOE:- {"- {"6 & " 1" '&'nbg<,3o!$;m, &,'$_ 0.- f )J, GH > 3_ 5! 3v 59!'(O$_'&'RQ! V, $_ 6 &, n = r + 1 D O &>=,!&!> <C Z, 1!$?f [ 5 $_ 6 & "! n = r O &>=,!&!> <>( /! V:!- ^f- - & ()& m n igi T < 0 &!.6sQa # &<C #&>=,!+!> T,'&5 <!!" F[n m] $_ -xkl!'&'*], < Z=*& ;&,!0 &J & ^N&; > &!'&'B( SO m n O!GH E,8&l!'&'#! )!! &6; -,-6; E! qj", &?p&>=,!&!>!'(* GH+!'P$_! - m n &EQ'&'e< K=! (U Z!J(U$_ 6e< " S z,ë " LZ! 6Q9 m < n m = n,'&';" E$_ Q GO ' 8' GHG!-)igL!6TQ9 h & 3y!$# hj(*g<!'&!8 >ḧ=,!+!< ' 6;@-A!-B6;Qa GOC!'(K$_5 ) & $_ (kl) 1 k l m ] & 5B= / < & :-B:- {f-kl V 0] ) )+$ 0 (lk) = (kl) - F[n m] O i 1 $ $ 60!4"! -# i µ 0 0)+$&,;:-$&'<!,!4!9(>0 #!,40! Pf (i 1,..., i µ ) $ & 10-0 :3 $&'<1, 3 :-Aˆ 5! 06#"! $ $ 0 )+$&,;:-$&'<!, (! 1 F[(kl) : 1 k l m] ^f:

26 MF! F, 0] o!'&h\= Pf (i 1 -,..., i µ ) F!a&> 5 P* H& > 3v F D $v! 506fB P& >$ $ kl [ f <>( & " - k Z= 6PQ & µ Pf (i 1 i ) = (i 1 i ) &O& " < '&,- kl K!''F \= L o s GH >Ë$_!! GH& 5,Ë µ > 1 &m y$_! -Wkl ogz(! GH y, 6I---6-0~< =! µ 1 i 1 = 1 i µ = µ, &H C'B(f"GO&!'9,$I 5! :o 6PQB '& Z,fb Pf (1,..., µ) (µ 1, µ) &"5K* H > Pf B=*& 6Q & u& < '&,-ok! O S "'& (1,..., µ ) -?ig)$v! 6! & 5N &;!5! GO >?$_x! Pf (1,..., µ ) Pf (1,..., µ)!,$y & 6Q9 e! "'& B=*& $_ Pf (i 1,..., i µ ) Pf (1,..., µ)!'' - dne ) =*&F! ) > +,'&'( "& & 0 N 0b 1 i 1 < < i µ µ &< ' I!'(* GH+!'6>Qa 1Qa ' [5 0ë V 5! V }4!' g( Q y!< ".- kl V 0] V '<! ( ) µ (µ ) µ, (i 1 i 1 ) (i 1 i ) (i 1 i m ) (i i 1 ) (i i ) (i i m ) Gr(i 1,..., i m ) = (i m i 1 ) (i m i ) (i m i m ) / ev!gh+! a!$?f [ =,6 Q9 V Z=! Gr(i 1,..., i µ+1 ) Z[(kl) : 1 k l m]. >$ $ A#! o qjf! & G V GO!> B!em$v,! $_G &!5,'! O $_ N,!E 0= S* H > Z6fa91 * 0!$.!XCbc +,5!' > & 9, O!<C e!& mq H e!"c O G!,! 61' 0,9,!! # !9' 10$ 0 )+$&,;:-$&'<!, (kl) = (lk) - i 1 $ $ i µ+1 >0!4"! -# Gr(i 1,..., i µ+1 )/ Z[(kl) : 1 k l m]!4!9(>0 +#!,40! 0 )+$&,;:-$&'<!,-(! 1 ^^ F[(kl) : 1 k l m]

27 ' 0 char F,9,!! #606 i 1 $ $ >0!4"! -# i 0 < 0 )+$&,;:-$&'<!, µ Gr(i 1,..., i µ )!4!9(>0 +#!,40! 0 )+$&,;: -$&'<!, (! 1 F[(kl) : 1 k l m] 6$ $ k O"* l<>( & " & -S7! 3v 5N$_ V<>=f -+CT µ µ = 0 O! GO 6 3 5h$_! \=! 3v< 5h$_ - kl [GO\(u! GO [ s µ 1 µ i 1 = 1 i = 6 0,-u yc'b(f"go&!' Q K'&! &"5* H& >!m 5 : &z S=!,+,< '& Gr(1,..., µ) - / & 6C & (µ, µ) Gr(1,..., µ 1) char F '&! &"5yf 0 O& 4 8&V& < '& m<>( O& " & >(fc! &- k!! ' Gr(1,..., µ "=*& =* 1) Gr(1,. $_m!'&' -8igL$v!0 Z6.Qa O, h " h '&.., µ) 6 Gr(i 1,..., i µ 1 ) Gr(1,..., µ) 1 i 1 < < i µ 1 µ Q & y'&! 6 4! eo!'+,eg8 ' & ' µ(µ 1) µ µ = 1 Gr(1)!$ 6"Q & [&,<.- (11)() MF\Q 6 ' V 1! GH 0,-~ C'(*!GO&!' 3v< 5F$_!" "\= 3 5F$_! -k hgz(l,"go h, µ µ ^H!N 5 E:8&o i 1 = 1 i = Gr(1,..., µ + 1) =,!&!< '& 6"Q '& Z, & 5hf 0 H& > - (µ + 1, µ + 1) Gr(1,..., µ) k 6 F' Z!"& 5H* H >ÏF &< ' 1<>([ )& " & char F >(fc! &-Hkl O! H" m ' "=*& - Gr(1,..., µ) Gr(1,..., µ + 1)/ ig $v!0 Z6pQ9 O! H" H '& 6fQ "& S'&! 9 B=*& Gr(i 1,..., i µ ) Gr(1,..., µ 6f 4 $_1!'&' + 1) 6 1 i 1 < < i µ µ + 1 (µ + 1)µ µ + 1 µ = 0 &SQ y! & &" y &< '!- Gr(1)/ = (11) k &,$;,!0 &J &)^f6fq9 1 Z= F (11) (1) (1, µ) (1) () (, µ) Gr(1,..., µ) = = (µ, 1) (µ, ) (µ, µ) 0 (1) (1, µ) (1) 0 (, µ) = = Pf (1,..., µ). (µ, 1) (µ, ) 0 F& 5BCT GHGO):- $_P!'' {"- 6!Qa F,! ' =*&" Pf (i 1,..., i µ ) Gr(1,..., µ+ 1), 1 i 1 < < 91!'+!PG 'B '&!$ 6Q '&! D i µ µ + 1 (µ + 6Q E&a!< Y- 6> 4 1)µ µ + 1 µ = 1 Gr(1,, 3) (1)(3)(31) /! *.6 Qa )"RQ &> f " 1!K, &!'Y C " > N=,!+!<"'& m = n D! Q9 V 0] 3J:-B:: 5 P = ( 1) r D Gr(1,..., n)/ Z[D, (kl) : 1 k l n]. ^!{

28 $_''&\Q&"5["CB " Z!')Q l " >(* Jbg $v, = 8! F (n+1 )+1 Q " O*"&, VQ&''D< O, l<4( Q 1 k l n - {P = 0} " D, l<>( (kl)! 0 :()+0(. 1 1#60 {P = 0}! F (n+1 )+1!43-$&(', P (kk) = Gr(1,..., ˆk,..., n), Q L s & qj &S!GOB.- "&S =,, =! L f! &>=!'=! K =,!&!< ' -8k m'&!&g sï h'f I!$aGHGO! = V!$# h HC'B(f",b D GH+!' 6! P P/ (11) = Gr(,..., n) P/ (nn) = Gr(1,..., n 1)!$F* &GH "&! {! F (n+1 )+1- &,$Nf GO AD}*"=,!'& > '(!6D '&*"H,$NGOGH ^K&z 4(*C '+!" =!O!$ - P/ (11) P/ (nn) {D = 0} x 1 &e }4 B=,!'& 1! 1,ë Rt &[$_G 1 & 0 &!y!$?,b D & s sjf&m 0] EL] 0b@,bc SGO! $_G 4(*C bg"$v! {P = 0} > y E >(*C "'+! -L =,! 5K!$!" {D = 0} P/ (11) P/ (nn) yhghgo![ >(*C Jbc J$v! 1&S 1, H )! Qa '[GH Z,y Z=*&"5 {D = 0} " ] & 5'B(* GH+!'T!$#, ~ >(*C Jbc J$v! m,i!gogho$v!0-9" 6.<>( CT GHGK:!- {"-B::6. h gq9[!'(* GH+!'N, m > +!''(K"& & 0Ï& < '& I$ -uic$ &!$F!! & &[^f6# " (! S 5, B= S}4, H!$ char F F = K 4 +!''( &J 8& "&<"'& 3v!GH 'B(6 $v, O!" 58<>( CT GHGOl:- {"- - " ( Z= ) GHGO $v!0 Fy " ez!!- / O 8'f!$DGOGH = e!$ >$ $ d 4(*C bg"$v! 3_ = O'f!$;H& 5' I!'(* GH+!'C&[!y, H H! 65P&#!GO!' $.!! '(mb$y " F,$T 5 '&!D!&> P!P* &GH "&! & & $_''&\Q $_ 0j"!GH '& I$_G / & *<C 5?AN S$_ G!' g(9 /!6 H G 7!'& '+s! &,$p* &GH &{"-D I& 5"'+!P'&* a!$ P/ (11) P/ (nn) {P = 0} &I>!, V'f 6TyB V!I* GO & 6.Q {P = 0} " 0$_ V!GO!'@- kl O \Qr 0 l [ O5&,'&>, Q H!! (kl) D ' GH > 9!$ 6! $_ I!'!$ -D I&5"]! F!$T " Z[n m] F[n m] 0jf &! "& L,<\= V 5=! [<4( P! I! $& $, "! $ n $ 0 )+$&,;: -$&'<!, P = 0 $&50 Z, $ $&50 F ^,ˆ (kl) D P 6 "!4 (: 0=>0.

29 6$ $ k 34&"51\=! Q 64 ego, Bjh,$Y " FC'+,$_G β!$y F}4,, & $_G & /!<"!( M = V Q n n Q q = x 1 y x r y r + z ( ) ( ) 0 1 (). 1 0 i [!'& GH v (k) 6 Qa 1 \=! V T MV = ( β ( v (k), v (l))) n k,l=1. ;,3* 5m GO!> 5B= ( 1) r (det V ) = det ( β ( v (k), v (l))) n k,l=1. 1 C $_''&\Q6f β ( v (k), v (k)) = q ( v (k)) - kl oq'&'e!'l" W y$_''&\q&"5 CB &! - Fe Z!'',ËlGO, Bj B &1 ] z o<c,;30 B! 1 $ B 1&)!> Bbc(*GHGO 0 3v -,-6?&!, E^*6 (*GHGO 0 5q6>! H!''" &!5,'" ' GH > #, = "-ic #&Qa '' 34 \QH, 5! "& > B ( ) 0 1 J 0 = Q!$? J = V }4,'. h V! 3E!$ 3_Q y 4 e!'bq Z(*F 0= 50- B Cp 0 ω(v, v ) = r (x iy i y ix i ) < I )J!"!y!'B, & 5m< '&&" Z!P$_G! -aicë&!fbg 5 " s 1$;! F 'B(yB$ &F =!.-Vk 6 J(*GO ' 0 &)5! n &F ] " L n n = r Sp r (F) < I V5"[,$;'&!9! J$_GO,!$,ë'& Z= >=!,+,4 - g F r ω!! 0 n m 0 +: )+$!9B!950! > $,40 ( ) B = β (kl) 0 +:,;80 "! 1 m m ' "! $ 0 0>0 0!4 n 506#8$&> u (1),..., u (m) F n!9 ω ( u (k), u (l)) = β (kl) (k, l = 1,..., m). ^

30 >$ $ k 1GOZ(S! GH 3v<>(SC J$_GH& 5mh'&!!"5 V!$;<! 59 s ( ) ( ) β (kl) 0 1 = J 0 = 0 1 0! I!4*1 06#B!950! 0 m n 0 0' ) ((kl) : 1 k l m) : F n m F (m+1 ) >$ $ N&Qa '&'x34 \QK&y!! & 6E! GH ), &e!$ F = K!! & &1^f-D " - β = ω -Dkl VG J \=! 1 CT ( ) β (kl) <C 9!>(,' s &"5 G, nji\= 6!, )' m m, " 1 0jf = K q (1),..., q (m) Q K v (1),..., v (m) K n! β ( v (k), v (l)) = β (kl) (k, l = 1,..., m) q ( v (k)) = q (k) (k = 1,..., m). Q,$ J = -y E! & <>=*&"'(!'& )& &)Z!!6#!"l n 5!'TZ! I$_'&'RQ- dn)!'q9z(*6pq9 H 0 -E79 "f h= 0!V& r = n/ u (1),..., u (m) K " 1" =*! &.- r u!>( 79 & '& O!!''& '# ] J Z-DMe! 9, J!!m}4!, a$_g &!4 n = r+1 3_ q bgsjf& 50-K 0$_,6 E qj" )= 0 K ker β z, F! IG!" h <>(E V st 0 v (k) K r+1 u (k) K r+1 K r+1 / ker β = K r q ( v (k)) - = q (k)! Z=! MF\Q '& -/;J " ~ s! <!&!$ n = r m = n u (1),..., u - N 0] )m" Q }4,, &I$_G <>(E I$_G"'+ (m) K n q CT ( m ) q λ i u (k) = β, [$_! VC'+!a$_G!$ m λ i q(k) + q -D 1 i<l m λ i λ l β (kl). β ( u (k), u (l)) = q ( u (k) + u (l)) q ( u (k)) q ( u (l)) = β (kl) = β ( u (k), u (l)), ^

31 0$_,6 -Uic m$_''&\qh, &m!fbg 5 " s!- / &,'&5 qb β = β q K <,&Z,'&'(m'&.6!'&' fbc 5, I}4,, &N$_!GH! N }4 =,!' > Z- / &'&! &GH "&G, $_!'' - g : K n K n q(gu) = q (u) u K icë!$; I$_'&'\Q, n $_e!'&'! β (gu, gu ) = β (u, u ) = β (u, u ) -#,ë&6 - N 0] - u, u K n g Sp n (K) v (k) = gu (k) (k = 1,..., m) β ( v (k), v (l)) = β ( u (k), u (l)) = β (kl) q ( v (k)) = q ( u (k)) = q \=! 1 V & [ C J v (1) v / C H]!''(y (m), < " y! 1, -E79 f! n = r u (1),..., u (m) K GH!fb = O= 0 J!5!'p Z!!$ 6?---6 n -I79 f! 0 u (0) u (1) u ['&!N$_ " Q (m) f : K n K f ( u (0)) - F ] H m 0Q }4!", & 0 $_G <>(S I$_G '& q q = q + λf, Q!GO s 95B= λ K q ( u (0)) -D F}4!, e$_g!9š8!a 0 f!'+!;$_!g[6!!!db C'&!;$_Gy-xic D$_'&'\Q; s &; fbc 5,,- q β q kl a $_! Z=! N'& Z!x&GH "&G ", g : K n K $_!'' - 0N$?" -# I= n \=! u K n g Sp n (K) gu (k) β ( gu (k), gu (l)) = β ( u (k), u (l)) = β (kl). q(gu) = q (u) MF! [!'&, gu (0) β 5!!'9!,$ 6e---6!, gu (1) gu (m) q ( gu (0)) -l '&, 8 " m, >!,4(l'&,!''& '# 0 q K - / H 8, V= 0 Q Kgu (0) v (k) gu (k) + Kgu (0) q ( v (k)) -9 0( = q Z= )!'&' & [ C J - (k)! $& m n $ 0, 10 F[n m] O F ), 10! #,9,;:! 103) ! +5 (! + )+$&,;:-$&'<!, 0 50 F[n m] SO = F[n m] O F ( m+1! : 0 (kl) 0 m < n $ 6$ $ '&!&G $_''&\Q$_G G :- {"- 3v 650-Nk \= " )] '&!&G[-;CT 0 φ : F n m F (m+1 ) J! E$_!a " ~ 5 '+,ago!o,!a! (kl)?f &, -;79 f!!>(1 ' GH > -? G :- {"- Z(*;, f F[n m] O F ^

32 ! 0 0 m = n = r + 1 D G $& 0 [1,..., n] 0, 10 F[n m] SO! : 0 ( n+1 )! +5 (! + )+$&,;: -$&'<!, + 1 (kl) 0*!"10, $, 10! # 60, "! $ $&'! :< 0 D$! 1,400,40' 0 + >$ $ 5 P 10-0 :1 $&'<1, 3J:-B:: 79 & VGO! φ : F n n F (n+1 )+1,! 6" 6!! * & s -ic $_''&\Qa$_G&Dsb (kk) (kl) D B ~:- {"-B:\{m! :- {"-:, V&GO!5 N,$ & V >(fc bg $v! - φ {P = 0} 79 * e!>( -; G :- {"- f F[n n] SO Z(f#, &; ''Bbg<!3),$ f m, &,'.$_ &[ - 9f &Fm'B(* GH+!' 6""6C<>(CT GHGS:- {"-Aˆ {P = 0} f! D! &L:- {"-B:R^f6 V "'&'Bbc<!3O!$hC'(*!GO&!'@- f kl a \Q 8 N &f &!8!$!'5 <"F,$C &!'* 5!'4&>=,!nb!> e& " )! -9k 1Q m = n = r! 6 6 Pf = Pf (1,..., r) D = [1,..., r] = 1,..., r = (Pf D)/ = (Pf + D)/ -! 0 0 0, 10 m = n = r F[n m] SO! : 0 ( n+1 )! +5! (kl) 0<!"10,$, 10! #>0, "! $ $ $&'! : 07! 1,400,40' Π & Π = Pf + ( Pf ( 1) r Gr(1,..., n) ) / )>$ $ $& 0 ' 0 +! 1 $ 0>0 3J:-B:R^ 5 &m, &!'$_ " &S &S VZ! 6! I, f (kl) m = n = r + 1 IE!'(* GH+!'po <>( D! & :- {f-:\{f- / &N ''Bbc<!3 D (kl) f "!$YV,!'*$_ 0 &!.-9f DV'B(* GH+!' 6,, D! &:- {f-: φ f Z(*a, &P t =!!- a( CT GHGOH:- {"-Aˆ 6 D e ''Bbg<!3!$pVC'(* GH+,' φ f $_ 0.- >$ $ " F f,$.&d, " P&GH&'&!p V, P!$Y! G :!- {"-B: f-xkl $_! L BXY " ~E! > & 3 :-:\^ 5a!<C\=,-D & )'B(* GH+!' O " L (kl) Q &> 5!'Y* H > - /;& P&> " P ",!PC'(* GH+,'&\= H =!,+,< '& k (kl) Z x, y 1 \=! = ( Pf D ) /4 = L/4, ^

33 Q a # }4!'&B g(~x"\= &I " a!go #GO! x!7#cb &!H:!- {"-B:\{"- MF!, + - ic [$_'&'RQE, ", 3_"&"! = Pf (kl)!'(* GH+!'a\=! &S " =,!&!< ' 5P, &$ (O V '+s &! Z x, y 3J:-B:\{ 5 Pf + L/4 = 0. kl #'+!G, T " Pf 0 H& > p!$! D"=*&&< ' ;<>(I$_ 6s &p!'(* GH+!' L L/4 &K h=!,+,< '& QB L4 5 I* H > -hig.6xg 'B &"'(S " H '&, (kl) <>(! & ;GO* ' '& 0$ ;, h <C!GO = 6> = L/4 4 Q9 9<",&18 " 95 > ;! & 9!!'5 <"!P '&, )\=! ' &"5F<C gqa K 3_ ] \=! 5q-9f?! P!'5 <"!Z!''(e& " C > ' GH > (kl) K (kl)!$ <>( G :!- {"- K[n m] O 6*8 &a '+, &HG J <C N =*+!' -? "&9GH Z, s N,'&'T* H& > F!$ 3,Nm!'(* GH+!'&S " 5! V 0= Y- ;,3* 5H \Q L (kl) y '+, & GO* '! U C Z, & 5 y!gh K,5 GH >ḦQ9 = L/ <"!& '+!Gy- / 3 :-:Z{ 59&!y!'5 <!&N '&, EQ y> 5!!'.f 0 H& > ' & 5I<C gqa O!" 3v! O!#'B(* GH+!'?\= &h " (kl) Z x, y =,!&!< '& 5qicḧ$_'&'RQ8&GHGO +, 'B(L, 3J:-B:\{ 5)GO!3, m y! ' h!ml '+s &! \=,ë&6f V '+, & G,3! 8!"y!'& - F Π = 0 F[n m] 79 \Q VG, SO φ : F n n F (n+1 )+1 sä!p!"!#b #*"&, -;ic #$_'&'\Q$_G e '&, (kl) Π = 0! #CB &! :!- {"-B:,Ö [&GO!5 E!$ h [ >(fc bg $v! φ {Π = 0} & F (n+1 )+1-3 /!8"_t 0 B=fB g(6qa!'&k z, " E O r (F)\SO r (F) &>, 5! 6>1~!&> Q h*"&,! & G!5! (kl)!$ - 5y79 "f!>( B$#! L"'([$D C&> NQB Kf!,! &F&o h&go!5 V!$ φ (kl) - G :- {"- φ f F[n n] SO \(*N, e 8" '&'nbg<!3 f!$dms &!!'$_" &[ V >(* Jbg $v, - 9" &ehc'b(f"go&!'@6 {Π = 0} f 6P<>( CT GHGz:- {"-Aˆ! #CB &!U:- {"-: <C '\QV6 m ''Bbg<!3,$~ f!'(* GH+!' -! $!" #8$& 3) 1#60 :!43-$&(' F (n+1 )+1 $ (kl) (1 k l n 0 0 :()+0(. 1 1#60,!9 #6$ $&! $&30 {Π = 0}! F (n+1 )+1 6$ $ " 1!I Cp GOGOo:- {"-B:R^f6Ï" H I ["\=,Ï " 5 '&!N'f!* GO &S " 1 >(*C Jbc J$v!,- ^

34 7!'& '+s Π (kk) = ( 1)r+1 Gr(1,..., ˆk,..., n)/, - Q y *, &>='B= kl I'+!G, 9 I'f!$xGHGH =,$ 6! Π Π/ (11) Π/ (nn) &e!$#f GO K{O&, F (n+1!$;f"&gh &y^h& V >(* '&! )+1- AD}4 B=!,'& > '(6 " '&* F!$DGOGH = ~!$ -Pic Π/ (11) Π/ (nn) { = 0} f O K \Q s! Z= S"KGHGH $v!0! Π/ (11) Π/ (nn) C'(* GH+,'&P&S -D $_'&'RQa$_G CT GOGOE:- {"-:!:- (kl) / E " '&* F!$#!GOGH =!~!$ 6, &N,$#* Bb Π Π/ (11) Π/ (nn) GH & {"-H H&"5 '+,~'&* V!$ &I!4!&" &, V'f 6po {Π = 0}!f GO & 6"Q "& $_! V&!GO!'@- {Π = 0} "* * 6&* ( 0+/1 * 4 * 6-5 ig &m & 6 Q9 y,'&' R=! y!, m y &5! 5 yg8 ' Bbg'&!8>=!,+,4 0] S& / 0 &!K:- ^)! N&" GOC b (1,..., µ) [1,..., m] 1,..., µ!< ' I\= 1!'&5 <,&Z,'&'(m'& ] '!$?!, )^f- K /;& 1'& 0 - F!'&'$_G / 0 :- ^, VQa Z= ] " l n = r + 1 '<1,;B! ),;! #!9":<!,$x8!'(* GH+!' <C N N&"] G8 Gr!$T F!!'" 5,$ x$ y GO! GH+!'a 6 =,!+!<"'& - x y! 0 '<,;"! ),;! #!9":!!4,40 & min(m, r) x $ y $ +: m,;! -0 SO r+1 (K)! +5 (! + >$ $ / Q9 Z= G :- {f-: z, Z(*,!'&5 < m r &O5 " s W<>( - y 'B nbg'& Z,h&>=,!&!> K[(r + 1) m] SO r+1(k) (kl)!! =!,+,< '& h& <>(l * 0!$!! &[^f6a G 'B '& g( Q 6Q & u L"! 4=!'=! " (kl) k l 6,$I!>( z x y m '&! &>=,!+!> -P " 1! " y h &! $_''&\Qig 0 'B(,"GO N,! m m r m '&! SO r+1 (K) &>=,!&!> 9, 8GH GH+,'!$x I$_G (1) j 1 (d) j d z (d+1) z (m) Q -;Fe Z!''f, Rj"&, &; 3! '!$ 0 d < r z K (*GHGO 0 &m< &'&!I$_G 6?KB 1 1= &1!< '& r+1 β e = <>( - SO r+1 (K) icï$_''&\qf, V"< f &"5 $_N = I=!,+,< '& L e v (d+) v (m) {Š

35 &>=,!&!> 8(*& '& GO! GH+!'C!$x I$_G SO r+1 (K) (d + 1) (1) j 1 '&&" Z! (d) j d z (d+1). SO r+1 (K) 4=,!+!> 6 Z=*& 5 9" 6 HQa ) \=! 1H>, &0 &!.- d + 1 r / & h mg 'B '& g( 6!$ S f" 1!$a GH5 "1C'B(f"GO&!'& x y &I " O Gw!$9 mg ' & '& e!$ h$v! 6;Ï$_!'&'&\Q~$_, ) r 1 G 'B &"'&& g(& 6!$P!>(o"!GOC,< '& x y m '& Z,F4=,!+!>ëQ & m > r &0 'B(E5 Z,! 6"mQ9 V5 0 r 0 n 3 0$ 0 0,;! -0 m n m SO n (K)! +5 (! + $&! # 050 $-0! (1,..., m) n [1,..., m] n!4! 106#6$&'<. )+$,40 6$ $ 0 n 3 07$ G 0 0,;! -0 A! +5 (! + m n m (1,..., m) $& [1,..., m]! # 050<$-0!4=10-0!4! 106#6$&' )+$,40! 0 (! 1 Z[n m] SO n(c) 0 m!4 050 $!4 0>0 $&>0! 106#6$&' )+$,40 (1,..., m), $=! 0! 1 Z[n m] On(C) & $_''&\Q9$_G /!0 V:- ^*- "- dn! 6& 4 N *!$?!$ " G :-Aˆ - ^h$_ Q'&'.<C V5=! y& m > n G!3H$_!'&'&\Q 5) I & S!$p!,-#MF! ~, 9 I! O & GOGH &, e$_g! G :!- {"-B: hb [S V f!$;!<c\=!- r kl F \QW 1 FZ! -;k e< =! ~, P$_ 64 " n = r µ r O &>=,!&!>!9 ~GO!''& J 9C&<"'& NG8 ' & '&B g(8 -Dig.6"Qa r (K) (1,..., µ) r s Z= 0*'<1,;B! ),;! #!9":! f ( v (1),..., v (µ))!4 6$ $ / µ r s $ +:,;! -0 µ,40 & min(µ, r) O r (K)! +5 (! + m = n Qa E Z= S " G :- {f-: Q & z ''&1 8, 8 E!'5 <" &T5, )<>(~ 6\Q 8! D}4!, D&~ " P " & 5 K[r µ] O (kk) 6!"y<>(E v (k) (kl) = r j=1 ( ( t (k) j t (l) j = r ( j=1 ) ( + s (l) j + t (k) j t (k) j s (l) j {": + s (k) j ) t (l) j ) + s (k) j ) t (l) j = (k < l), f =

36 Q E! e<"&'& Z,& 6! H Z= FG 'B &"'&& g(m:e& 6*) G ' & '& g( v (k) v (l) s &!$,4( s µ '&!9>=!,+,4 - f µ MF\Q ' -9icë" H e H \=! ),ë$ 6 [ 1* H > µ r 0 d < r &!$x VGH GH+,' f 3 :-B:Zˆ 5 s (1) j 1 s (d) j d t (d+1) j d+1 t (µ) j & = -# &* H& > 6 0jf K & 5m " 1!&5!& µ!' 6 f!, 6 x y "G!$x V* H& > F,$? GO"GO&!'& f µ d y (1) j 1 y (d) j d (d+1) j d+1 (µ) j µ. ic$ 6\( 6 [ 8GH!GOnb {j d+1,..., j µ } {j 1,..., j d } j d+1 {j 1,..., j d }!' Z! <C 1!&!XK=*+ V DC & x jd+1 y jd+1,! m " e gqa)f 0 H& > 9m Z! O!&#, }4!' " e 1 O r &>=,!&!> D fb J g(s!$ - f MF\Q C H s - / & 6T {j d+1,..., j µ } {j 1,..., j d } d < µ d 4 G<C,$pf "!$xs '&! ~!$T ~ & 6---6,GO" r j & '![!GH 5 6T---6 -kl VGO\(S! GO 1 j d+1 j µ! 79 & O r (K) j 1 = = j a = 1 j a+1,..., j d &>=,!+!> j d+1 = = j d+a+1 = 1. f ( u (j 1),..., u (j d), v (d+1),..., v (d+a+1), w (j d+a+),..., w (j µ) ) &"5H!oh 0Q 0 F,$?=! =!,+,< '& 9Q 1!"&!' g(o& {j 1,..., j d } +a+1+ {j d+a+,..., j µ } d a+1+a+1+d = (d+1) r. a( G :- {"-B: l,5>!&y69 O>=!,+,4ÖG JË<C KlC'B(f"GO&!' " q!?an,$# e=!n=,!&!< '& -FdNNB ~F'&!FK Z, L!$ 6---6? 6 β v (d+1) v (d+a+1) "! < L4=!'=! 'B( =*+z β? EQB Z!! SOQ W?A u!? -H x 5 V m* H > 1!$a mgo! GH+!'13J:!-:Zˆ 5q6TQa O < " :h$_ w " f &, m!$9 Z! 6p$_I f &, m!$ 6B---6x$_I s j u (j) t jd+1 v * & s V!$ 6Y!"y$_ " *"&, V!$! (d+1) 6C! yq9 t jd+a+1 v (d+a+1) t j w "< f = V$_#!'&'f! "!" * & s -;de$ D &D < ".6> Z, (j) s t!$ 6B---6 Q'&'p< v (d+1) v (d+a+1) β 5!'T S!''p < J B " K= I qj" " 64< " P #C'(*!GO&!'f!a! '( =,!' Q9 F5 P = - u (1) a u (1) {>^ j d

37 / & ) )G8 ' & '&B g(!$o f" N!$C'(*!GO&!'& e 8 G!$ s G 'B &"'&& & a!$ )$v!0 6C $_''&\Q $_ s F 8G"' '&B g(o,$ r 1 s!>(o"!goc,< '& µ '&!e&>=,!&!> FQB &F &0 '([5 Z, N! - µ > r r kl 1!B= V, 0 m > n $& ! +5 (! +!4! 106#6$&' )+$,40 (1,..., m) n 6$ $ & $_''&\Q9$_G /!0 V:- ^*- f- m,;! -0 O n (K) 0 m > n $& ,;! -0 m O n (C)! +5 (! +!4! 106#6$&' )+$,40! 0! 1 (1,..., m) n Z[n m] On(C) G :!- ˆ - ^$_ $_!'&'&\Q~$_G G :!- ˆ - -Higfb m > n.6>&" > B$ ( QB 1 < 5 h!$!0 5N) 6 * nb O n 1 (K) O n (K) x y, 9 >(* '&! 91 9J!"!)Q Z()! 8]fj" 5 -; h!>( z O n (K) &>=,!&!>!'(* GH+!'xGO\(y<C m=* Qa! =,!+!<"'& +3v 5>, 51!V! O n 1 (K) &>=,!+!>ÏC'(*!GO&!'p&Ot J 1 x y, #Q Z(6f! H< Z!3B # > V DG ' Bbc GH5 "?GHC > - =,!+!<"'& P!a",4 5q- N& 0Q z (1,... m) [1,..., m] 0~ I!$x V& 6 Q &! Ghb (1,..., m) n 1 (1,..., m 1) n 1 z!!< ' (m) O n 1 (K) &>=,!&!>Ḧ<4( " G :-Aˆ - - icḧ$_'&'rq,!! e"! 9H e <,'&5 < V!$ 5 " s O<>( [1,..., µ + 1] n K[n m] I '& GO 4!$? 5 I'! - / & I SOn(K) µ SO n (K) 4=,!+!>!$? 5 h: 0jf&J Z6> GH!< '& g(i$_!'&'&\qx$_!q9 '&',;$_ - [1,..., µ+1] n (1,..., µ) n (1,..., µ) n 0 m n 4 $& ,;! -0 m SO n (K)! +5 (! + 1,..., m!4! 106#6$&' )+$,40 0! +5 (! + m n 4 $& 0< ,;! -0 m 1,..., m!4! 106#6$&' )+$,40*! 0 (! 1 Z[n m] SOn(C) SO n (C) 6$ $ / < " 5 $_y Z! * z!$!! z x r y r SO &>=,!&!> (*& ' e! r (K) O r 1,K &>=,!&!> 9 & $_''&\Q Z,&'B(E$_!G k?af qb G I6N " G *- ˆ - N 5! z! L Z!.- P( /! l:- ^f- f61 &GO!5!$ & 6 Q & W!$~ K!GO [ 5 L! U 1,..., m n (1,..., m) & "!GOC,< '& H<>(z G :-Aˆ - ^f-k EG,5 O,$eK fb@ B=*+!'D GHCnb n 1!$ Qa 'E< F8!fbc =*&!' GH &m!$ 6f 1,..., m n (1,..., m) G Jë!'&H<C V& "!GOC,< '&,- n 1 1,..., m n {{

38 / * 75 *33* ( B- ;* 6 /14 / igy &e &.6 ) Z!" Ne! GO o H<C )$v!gh&'+!#q o<!&1! & fb 5m5! GO -/ a!6 1!-A $& $ 1$,+>$& ) # 0' 0 O n = O n,z & 0] y H<C 1 )J!<"&'& =!$? 1,!o}4!, I$_G QB [ C q K [! &u,$f S5 ",' '& Z!85" GH u GL n,z Z!!$ }4!, ~$_GH-; *" &NC'(* GH+,'T&"5E& &a VC 0 G!$x V 5 O n =,!+!< ' -F Z[g]/(q g q) 6., K 8&!' Z[g] n n g i,j (q g q) 5, E<>(H e* H > a!$. " N}4!, $_G 64Q 6 q g q g = (g i,j ) 4 I h* H > V!$# h}4!, $_G! h" 5 O^E'B(* GH+!'e\=! &S - g dn<c $_!6 i,j '& 0 <C 1![!'&5 <,&Z,'&'(m'& ] '.-k 1" ] F O n,f = O n Spec F. h5 L!$ F C&> F!$# 8 GO F " h 5!'?5! O 0] & / "< :-:!- ^f-8 m GO n &I " ' O! -# & Q9 '&'T34 \QK, y!(!- n,f n = r + 1 CT $ äg ' & '&, 85B=!! &H!$. F5! O GO H F, H O! -dwc'(* GH+,' ; 0] H I<, n,f! +5! + Spec F[n m] f F[n m] $ 0 $& $ 1$,6$& ) # 0' 0aB$ ; "'&'Bbc<!3I " x 5! ) GO, GO! O n,f Spec F[n m] Spec F[n m]!& & NQ K ~ ''Bbg<!3 ~ "st K " 1!GH V!Z(*& 5h, Z O n (F) char F = Spec F[n m] -) ~& f g f (q g q)f[g][n m]. 79'&!'B(6 >=!,+,4!$; )! "5,'.5! y GH )$_!G C'(* GH+,'Y!'5 <" F[n m] - m <!'&5 < m!$?! 0!9 " $& f, s F[n m] s 0 $ 7 0<$&( $ 1$, >$& ) # 0' 0 O n,f $ 0 $=!4 f {!ˆ s sf >0! +5 (! +@