Francois Maurel- Un cadre quantitatif pour la Ludique
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tel-00152476, version 1 - 7 Jun 2007
Transcript of Francois Maurel- Un cadre quantitatif pour la Ludique
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1 H= M H mz L%lo1pErmSmKypÂshmûxr]mKvYo1mr xms l[pL #Ky]m%liyðshm+!:nxmKs7yg9ipÂmR@gy mKyxmyðz Lg9y]mKy>ig9n 9mKshs1my
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smmKvYshmxn /#e=;:Dai⊗ Fid
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JF,Dai⊗ FidK = Fid = Dai
mðxrg9xs$#vwmðlv #ixm32%ixo1rnxièixg9n m6ln xmyy]mKohi @%shl xo mr D mixzKmÂi D lMpEo mðixgp m Fid− @RmKp g9y]mKr
∀D,qD,Fid−
y= Fid
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m(pF(Fid) = Fid− @s1mr]g9xs$#vwmxm %lrpYy]m r yg9npYmKpsl g9ixzKp]ohg9i xm +ohmKip
JF,Dai⊗ FidK = JF(Dai),FidK = Fid =qFid−,Dai
y
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D := D+ | D−ξ
D+ := Fid | D+t
D+t := z | zξ | (+, ξ, i1, . . . , in).(D
−ξ.i1, . . . ,D−
ξ.in)
D−ξ := Fid− | ((−, ξ, I).D+
t )I∈N
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, H= M H <O#, %£ 3¼§%¸xñK®6¯º9¬®6¯°±
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X 6 H
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mvwg9ifgRzlsho1y zξ
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mKypÌr Kzro1p|mizξ
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4 myWlnpErmy|z6lyÌyg9ipÌohixz L%li D y 4 ohip Kr]XKp xnûr]mvwxshlzmKvYmKi:p xmKy KvYgixy Ìs l|rElzKohixm xmKy xmKy]ymohiy7%lr xmKy KvYg9iy fg+z6ls1ohy y
mypxmÚxo1mi xo Kr]mKixzo1mrbnixmÚmrr]mKnxrP!:nxoy]mr]l4xr]g%l D Km vwg9i fg+z6ls1ohy nixmmrr]mnrP!:nxoOl dmKn2sho1mn KvYgi2ixg9i fgRzlsho1y 4 m KvYg9i fgRzlsho1y zξ
Yshlwr]lzo1ixm nxi my]ymo1i2ixmSymr]ls lixz !:nxmy]oxs1m xmKy]y]mKohiwmypzg9nx |l 9mKzÌnimy]ymo1iwp]gp_ls xo Kr]mKi:pxm|s lxo! 9mKr D mKixzm|i D lpEo! 9m 4QlbixgpEohgixm KvYgi fgRzlsho1y r yg9nps1m xrg9xs$#vwm xmshlbixg9rvdlshohy]lpEo1g9iÂmi%ly]mvÚns1pEo1xshmmKixr Ky]mizm mxo mr D mixzKmûi D lp]o 9m4 o1ixy]o @shlÚixg9r]vYls1ohyElMpEohgiYynxr|nxixmÂ%lymðvÚnxs1p]ohxs1mbmKyp" 2xixohm%lr
g9o!pDshmxmKy]ymohiäy]nxr|shlÚ%ly]m
` ξ1, ξ2fgr]vxn vwg9ifgRzlsho1y
zξi
4 g9o!p
E1
nxi xmKy]y]mKohiy]nxr|shl[xly]mξ1 `
m(pE2
nxi xmyy]mKohiy]nxr|s l%ly]mξ2 `
4D i l JD,E1,E2K = z
y]oEi
i mypÌ%lyÌshlxo mr D mixzKmûi D lp]o 9m@ JD,E1,E2K = Fid
yohixg9i4
qWlpEnxrms1shmvwmip5@g9i D lr xmðs lÚixgpEohgi m>r ym6lnzg9i mr D mip è7#f %£,¡ ¡<§©¨µ:ñª:¬V¸Õº9°±¹+ª%½ D¤ª%±®
¾¿èß(çEÛ(ÀáÝRÀEÛ6Ü 69ÈÉOÑÃÁ GÃÉOÆà,ãßÛ(éÝ+À JRK = z
ì
nxmKs !:nxmy|mmKvYshmyxmÂr y]mlnzKg9i 9mr D mip]ybg9nixg9ièy]gi:pWnp]ohshmKy
QmÂr y]mln ynxo lipxo! 9mKr D m
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T H= M H
(+, ξ, 0)
(−, ξ.0, 1)
(+, ξ.0.1, 0)
Fid−
(−, ξ.0, 2)
(+, ξ.0.2, 0)
(−, ξ, 0)
(+, ξ.0, 1)
(−, ξ.0.1, 0)
zξ.0.1.0
mðr Ky]mln ynxo lip|zgi mr D m
(+, ξ, 0)
(−, ξ.0, 1)
(+, ξ.0.1, 0)
Fid−
(−, ξ.0, 2)
(+, ξ.0.2, 0)
(−, ξ, 0)
(+, ξ.0, 2)
(−, ξ.0.2, 0)
zξ.0.2.0
mðr Ky]mln ynxo lip, g9nxri ∈ 1; 2)
zg9i 9mKr D m
(+, ξ, 0)
(−, ξ.0, 1)
(+, ξ.0.1, 0)
Fid−
(−, ξ.0, 2)
(+, ξ.0.2, 0)
(−, ξ, 0)
(+, ξ.0, i)
(−, ξ.0.i, 0)
z
[C 8 LûGèG = % ! = ; my|zgixixmz(pEmKnxr]yW@mKn 9mKi:pWvYlohipEmKi%lipÌl D o1ry]nrÌs1myixgn 9mlnxmyy]mKohixy 4
<O#, %£,¡K%2§('%ª 2_%`a bòOÀ>p]mixymnxrðÀ_ÛÜóç *x¿xâQÛÝß|à,À_Ûbóç_æ<ã¿xÛ : ãRõáàhâ1Û(çEÛWÀÜà#áwóâ~ëÀEßôÀE¿@õÀöáß
D+ ~ Fid = Fid
zξ ~ D+ = zξ
Û]âD+ ¿ À_ÛÜöáÛ
Fid
Ýß|à,ÀEÛÌáÝxÜAßÀ_ÛÞóxÀEÛ]Û(ÀEâh¿xÛ ) âhàÀEÛ6Ü óxç *¿xâOõãæwæèÀ O[à áwóxç *x¿â~ÜAâfã¿ =9ìXAðöáMôÀ ARìD iøixm 32%ixo!p%lys1mYpEmKixy]mKnxrmip]r]m<shm vwg9i m(ps l o 9mKr D mKixzm<zlr5@zg9vwvwmdzmKs l¼l (p
vwg9ipEr <mi xnpxmèz L%lo1pErmA@o1si YþlÕ%ly l g9ixzKp]ohg9iK D lnxz Lmäm(pM xrg9o1p]m<i%lpEnr]ms1shm4 mK@mKi%lip@%o1sVmKypÌ@gy]y]o1xshmxm 2%iohrnxièpEmKixy]mKnxr D lnxz Lm ?_eca7K r]g9o!p g9nxrs1m+!:nxmKs z mKp
Fidly]g9r@mKi:p D lnxz Lxm ?_eca7K r]g9o!pEm 4 D ièr]m(pEr]gn 9mKrElo!p|nxixm>l g9izKpEo1g9i D lnxz Lm>gn xrg9o1p]m%lixyÂz L%lzni xmwzmKyûzly54 mwi mypû%ly Kzro1p>xshnyûl lip>vdlohyÞg9iÕr]m(pErg9n 9mwnxiÕ LKixg9v #ixmyohvwohs lohr]mmidshnxo$!:nxm|xr]g9xlxohs1ohypEm f%l D m>j !:nxo¤myp mnptcX(pEr]mÌxs1nxy%lr]shlip4:qWlpEnxrms1shmvwmip5@zKms lÞimÌ g9ymÌxly xm|xr]gxs #KvYm g9nxr 2%ixo1rshmp]mixymnxrmip]r]mxmnwzg9vw@grpEmKvYmKipEy 9s1m vwg9iym>rmKp]r]g9n mxlixy|z L%lzKnxi xmÂzKmy|zg9vw g9rp]mvwmipEy 4
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T=H
è7#f %£,¡H-2§('%ª ²@µ+º9¬V«¬ D¤ªòádóç *x¿xâÜAâfã¿ ó Ý4óxçõáà#áMôRÀÌÛ çÜ ÀE¿@óáRÝ/è¿@ã+ÝëÀ(áRÝ/¼óxÀ_ÛÛ(ÀEâh¿xÛì
; âDÀEÛ6Ü|Ý¿¼óxÀ_ÛÛ(À_â1¿döOãÛâ~ÜAâ : óxÀ9áÛ(À ` ξ.i áà,ãßÛ
D =
(−, ξ, i).DÛ]â
D 6= Fid
SkunkÛ]â1¿@ã¿
; âDÀEÛ6Ü|Ý¿¼óxÀ_ÛÛ(À_â1¿ä¿ çEô+áÜ,â : óxÀ9áÛÀ ξ.i ` áà,ã.ß]Û
D = (+, ξ, i).D
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R1ÀÜ
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R1,R2ÀEÛ6Üõã¿ ë.À_ßôRÀ_¿ Ü]ì
wC = ! #èP 9 L % 9 I !QI(LSG nxy !:n o1zo @RmKi<shnxo$!:nxmbyohvwxshm@+s1m",l5
Faxξ`ξ′y]mbzg9vw@grp_lo1pzg9vwvwmbnxixms1gRzlsho1yElp]ohg9ixm
ξ%lixy
ξ′f%l D m 4
m[i mypSxs1nxy +rElol mzws l xo! 9mr D mizmwi D lMpEo m[mKpÂshmKy vwg9ixyfg+z6ls1ohy y54UiÕm@mKp5@y]o DmypWy]nrÌshlÚ%lym` ξ
@xshl fg9r]vwmðixg9rvdlshmxnr Ky]mlnD,Faxξ`ξ′
mypÌshmðr Ky]nxs!p_lp nxixm s1gRzlsho1yElp]ohg9ixm
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SkunkmKpxmKyKvYg9iy
fgRzlsho1y Ky%lrshm vwg9i,ixg9ifg+z6ls1ohy 4
mKs ldg9n RrmÚs lY@g9rpEmYnxi¼ixg9n msOg mKpð g9nxrbs lYs1nxo$!+nmS g9nxrÞrmxr y]mKipEmrÂslRo1g9vYm4¤Uim@mKp@o1sOy]mKrElo!pÂy K%lr xn ,l³zlrðohs7ohvwxs KvYmKi:p]mr]lo1pWmlz(pEmvwmipûnxixm Kshg+z6ls1ohyElMpEohgi 4 D i³ixm@g9nr]y]no1p%lyxshnxyl lip|zKmy !:nxmypEohgixyvdlohy zKmshlxm +rElX(pEr]m(pEno A4Uixlrp]ohznshohmKr5@Rl 9mzÂnxipEmsg mKp5@%g9iä g9nxrrElo!pr]mKyp]r]mo1ixrm>s1myxmKy]ymohiyblnxmyy]mKohixy"xmÂxr]g fg9ixmKnxr 2%ixohm4
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ixg9n mshs1mÂzs ly]y]m xm xmKy]ymohiy xs1myxmKy]y]mKohixyaL?+_e=#.fe34è7#f %£,¡R¥2§©ïðª:ññKªx¯h± »%½µ:ñª%±®
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D i D lr mÕmlzKp]mvwmip2shlùvwXvwm2pEr]l xnxz(pEo1g9i m 2
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, H= M H [C # =
my32%ixo1p]ohg9iy"my|g9r r]myÌypElxs1mðmKpÌm+pEmKixy]o1g9ixixmKshy"xg9o! 9mipÌXKp]r]mûl %lRp mKyÌlnixg9n 9m6lnxmKy]ymohiy54<O#, %£,¡+í2§CB4½~²½ª ñK®¬ EV«ª
ò ÈRÁÊVÁ9Ãè÷Æ_Å 6¤ÄcÃYÀ_Û6Üà ãßó:ß(ÀðÀE¿ôÀ_¿ ó:ß(ç öáß7>; < áÛöã.Û]â~Ü,â : ì
Fid v D
; < áÛ|¿@çEô:áÜAâ : ìRÛ]â N ⊆ Máà,ã.ß]Û
((−, ξ, I).DI)I∈N v ((−, ξ, I).DI)I∈M
<O#, %£,¡Qä§CB4½~²½ª ª /:®ª¤±ñM¯°±V±xª%«ò ÈRÁÊVÁ9à ÃÆÃ9É÷6Ç~ÈÉÉOÃÄÀ_ÛÜà ã.ß(ó:ßÀðÀ_¿ôÀE¿@ó:ßçöáMß>
; < áÛöã.Û]â~Ü,â : ìFid 4 D+ D+ 4 z (+, ξ, I).R 4 zξ; < áÛ|¿@çEô:áÜAâ : ì
Fid− 4 D−
ÀÜ ) Û]â N ⊆ Máà,ãßÛ
((−, ξ, I).DI)I∈N 4 ((−, ξ, I).DI)I∈M
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[C <P L: |N = ûG 9 !QI ; =my>pLKg9rV#KvYmKyli%ls@:pEo !:nxmKyûymdzKg9ixy]mKr mip4[Qläymnxs1m !:nxmKypEo1g9i mKypÚ g9nxrûshm[pLKg9rV#KvYmxm
y %lr]lpEo1g9i 4 o mðmKyp xmÂr]mKvdlr !:nxmr !:nxm m"xmyy]mKohi
Fid− mypy K%lr]lxshm"xmKylnp]r]mKy xmKy]y]mKohixy i D lMpEo fyynxr shl>%ly]m ξ ` Λ D rzKmbln vwg9ifgRzlsho1y
zξ
4 m my]ymo1i
zξ
mypäy %lrElxs1m xn KvYg9i ynxr<s lkxly]m` ξ,Λ D rzmG s l o 9mKr D mKixzm
i D lp]o m Fid− 4 o1ixy]o @s l<z Lxrg9ixo$!+nm
c.(+, ξ, i).Fid−mypxo Kr]mip]o lxs1mxm
c.(+, ξ, i) D rzm nikg9Rt g9yEli:p !+no%lrzg9nxrp
cxnxo1y g9nxmðs1mKvYgi fgRzlsho1y
zξ
4 +v(pEr]o$!:nxmvwmip@Rs lÚz Lxrg9ixo$!+nmc.zξmKyp"xo rmipEohlxs1mxm
c.z D rzKm[niäg9@g9y]lip !+no%lr]zKg9nxrp cm(p|pEms1!:nxm
D−ξ = Fid− 4
m>pVLgrV#vwm xm>yp_lxohs1o1p ðymûzKg9ixymr mÚzm7!:nxo xg9iximûnximûixgpEohgi ohiz6lri%lpEo1g9i<pEr #yÌy]ohvwo!tshlohrmA4
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H R 1 = ,j 8 L N ' L: ! =|N = G"!
mKnxshmKy|shmyigpEo1g9ixy KpVLxo !:nxm>mKpxmÂxout (pLxo$!+nmxg9o mipÌXKp]r]m>l %lRp mKy54è7#f %£,¡2§©ÿ®@¯¶·¸ª
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ß(ÀEà#áÜ,â,ã¿äó çKéxÝâ~ë(áà,ÀE¿@õÀ öáMß6ÜAâfÀ_à1à,À≡ÛÝß
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âD ≡ D′ áMà,ãßÛ?>
; Û]âD = Fid
áàfãß]ÛD′ = Fid @; Û]â
D = záàfãß]Û
D′ = zÀÜÛâ
D = zξ
áàfãß]ÛD′ = zξ
ì;/< áÛAìxòá9áÛÀ β À_ÛÜ¿ çEô+áÜ,â~ë.Àì
Skunk ≡ SkunkÀ.Ü
Dai− ≡ Dai−ì
]D ûG %>= ! ! =Ql<ixgp]ohg9i xmxo mr D mixzKmYi D lp]o mw mnRpûX(pErmYvwohymmiÕshnxvwo #Kr]m <slo mxmyÂpEr]lixz LxmKyûg9n
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y]g9o!p∅
%z mypWs1mðz6lyxn KvYg9i y]g9o!pÌlnp]r]m xz mypbs1mðz6lyxmKyblnpErmyxmyy]mKohixyÌ@gy]o1p]o fy
mKp@gnxrnxixmyy]mo1idi D lp]o ¤pEgpEls @smixymvÚshmxmKyp]rElixz Lmyzg9ipEo1miplnwvYgohixyOshlbpEr]lixz Lxm +o xm,z6lr|p]g9npxmKy]ymohiäi D lpEo!yohvwxshmWpEgpElszgi mr D m ,lzm>ln vwg9i 4Ql xo! 9mKr D mKixzmèi D lpEo! 9m4mKyp xgixz4shm xmyy]mo1iùi D lMpEo !:nxoi l2%lyxmdp]rEliz LxmKppEmèxr]gxr]mlshg9ry !:nxm s1m \ "9 \I
SkunkmKypYshmxmKy]ymohi i D lpEo!P!:nxo|i lF!:n nxixmèpEr]lixz Lxm(p]p]m xrg9xr]m shl
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nr>nixmYxly]mwixg9iÕnxi%lo1r]m@%lrÂmRmvwxs1m` ξ1, ξ2
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ξ1mKpWnxixm
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@∅ xz mKypWshmðzlyxn vwg9ifgRzlsho1y ÞmKi ξ1 y]g9o!p
∅@∅ xz mKypWshmðzlyxn vwg9ifgRzlsho1y ÞmKi ξ2 y]g9o!pÌlnp]r]m xz mypbs1mðz6lyxmKyblnpErmyxmyy]mKohixyÌ@gy]o1p]o fy
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,F H= M H 01235476# %£ 38# 6^"(7e;: &-'/+#R:+`M+ "^# GI: D ^#a%^:9 "( 6^:/X<#D C:'?_eX<_e_e="(#e?1_e <_e_e="(#e <+^#.X?1_e &"^#eSaLE^_aL?_ede= -+ca< "he_e=#.A# E
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nY*uv;=.>F5o.f8A',35F.3T_.3+\5wIJAxN)H68+)2Q5?24$'+yN).WIJA>F$'TRT_.S2Q+68+)24.N).>V$'24N)>N).[>F.>zP)(BA+8,FK8.>v>F.G5F.3(?TR2Q+MA+\5eVMA(&)+8.Ggr.3&)2QIQI4.G@/24N).<Y
AC ) $8%," &&9$.%..% )+*$ **;:Ms &K*?%$T%<&l1) & )+*?%<&)&" *" $9* &0%9*)': )!$8M: &+)'&+""&q8q% $'&5*$*"X'r$4 *" 9*K * )%l&& +& )>$ *9*$&%"6/0)'0=
bP_`\CjkP9[QTcI[mc'\*^?a*^ecI`<\*V7^i_P'`Fa&Sw'cI`V6a&^ej?^e\TSrjeP]`7c'a.P_a*^ecI`c = c′.(c1, . . . , cn)
[mc'VQ j?Sfdhc6Sm(dh^eSh`Lac′
gcI`Fa jeSh\>f"SV^?jejeSh\Ii^gSh\ NP'V #c6dhdVQTQ*S`7dS\DgcI``hS\α1, . . . , αn
\*cI`Fa Q*ShU([jeP'dhhS\ [$P'Q j?S\dcS+d^?S`Fa&\
ci ;v;`[$P'QTa*^edhVje^?SQsdSa*a*S+`7c'a.P_a*^ecI` Sh\Ta P'V\T\*^;V7a*^eje^?\hhS[mc'VQ c′ P'|FP_`La<V``cIU!Q*S^e`,$`7^ZgSEf"SV^?jejeSh\9i^gSh\
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r
z<>?sq?sL:=<>:@?8 AC F #MZ H KNZO,IsQAZ VMK/AQ1 l>hp
c = (s1.c1, . . . , sn.cn)zJn~ld]9 _@]_np n u
W (c) = 1⇐⇒ ∀i,W (ci) = 1 ∧∑
i si = 1
¢n~ld]9 _@]_npyR]tql>@y[^1]_^pqvmrfzJU@]_va]op#l<«,nRh
]thzJ^ y[U Unzd]fvqzJn ^msuw=uwvm]qldzvm]ftY=u[]_vqzJn^mldzw^.l ] _E]fnpdtuwv 0 y uwnR^zJnSld]9 fE]_npyR]tql>@y[^1y[ nYz[]^LpEvEop.]f5]fnYpq^sl>nt¦l>@y[^L
0n,«Rn zJn l ]fE]fnYp c = c′.(c1, . . . , cn)]_^Lpay,]atql>@y[^
1^% ]>p ^m]>zwE]_]_np#^%
c′]opE]f^
(ci)^ml>npyR]tqloEy[^1
/t%l1 YG`a.GV)(F.3TR24.3(eV$'2Q+/5*.>?57&)+8.,$'+8>FHn/&8.3+8,.N2Q(F.,35F.N).IJA#N)H68+)2Q5?24$'+wN).SVd$'24N)>Y YG`a.LN).3&f24^3T_.UVd$'2Q+\5SN)H,$'&)I4.UN&wg:A2Q5vn/&a; &)+w,$.01,324.3+\5>BA+8>e+G&8NW,$hr68+)2 .>F5eN).UV$'24N)>+\&)I:YM`A#(FH3q<&)IJA(?2Q5FH[.>F57&)+8.G2Q+8N&8,35?24$'+w>?&)(7I4.SV)(F.3TR24.3(*V$'2Q+/5Y YG`a.G5?(F$'24>?24^3T_.[V$'2Q+/5*.>?57&)+8.,$'+8>BHn\&8.3+8,.N2Q(F.,35F.VMA(2Q+8N&8,35?24$'+wN& V)(F.3TR24.3(eV$'2Q+/5Y8YG"Z$'&)(I4.n/&MA5?(?24^3T_.SV$'2Q+/598$'+-V)(F$'&)@ .SI4.>*N).3&f >B.3+8>⇒ &)V)V$<>F$'+8> W (c) = 1
Y`a.S,$.01,324.3+\5c′.>?57&)+8.[AV)V)(F$f2QT1A5?24$'+p>?&)VdH3(?24.3&)(F.N).
cN)$'+8,
W (c′) = 1Y
$'2Q5 αi0
&)+8.$,,3&)(?(F.3+8,.[Nb; &)+8.Ggr.3&)2QIQI4.S@24N).SN).c98T_$'+\5?(F$'+8>vn\&8.
ci0
.>?5N).SVd$'24N)>1Y
$'2Q5 c′i0I4.e,$.01,324.3+\5$'P)5F.3+/&-.3+ (F.3TRV)IJAA+/5
αi0
VMA(ci0
T1A24>7.3+!q\A(FN8A+\57I4.>7A&)5?(F.>gE.3&)2QIQI4.>@/24N).>N).c′Y`a.S,$.01,324.3+\5
c′i0.>?57&)+8.AV)V)(F$f2QT1A5?24$'+p>?&)VdH3(?24.3&)(F.N).
cN)$'+8,.>?5N).SVd$'24N)>
1Y
$'2Q5 bIJA!P)(BA+8,BK8.1N).
c′>F.#5F.3(?TR2Q+MA+\5L.3+
αi0
Y e+8.#P)(BA+8,FK8.#68+)24._>B.U5F.3(?TR2Q+MA+/5[VMA(&)+8. gr.3&)2QIQI4.-@24N). N).c′i0
.>?5!>F$'2Q51&)+8. P)(BA+8,FK8.WN).c′>F.-5F.3(?TR2Q+MA+\5oVMA(1&)+8.-gE.3&)2QIQI4.@/24N).vN2bH3(F.3+\5F.SN).
b>F$'2Q5
b>?&)2Q@/24.GNb; &)+8.vP)(BA+8,FK8.SN).
ci0
>F.*5F.3(?TR2Q+MA+/5VMA(z&)+8.egE.3&)2QIQI4.@/24N).<YG$'+8,!IJAW>F$'TRT_. N).>#V$'24N)>#N).>#P)(BA+8,BK8.>1>F. 5F.3(?TR2Q+MA+\5_VMA(R&)+8. gr.3&)2QIQI4.o@/24N).N).c′i0
.>F5!H3q\AI4.]A& V$'24N)>!N).c′T_$'2Q+8>!I4. Vd$'24N)>oN).
bV)IQ&8>!I4. Vd$'24N)>oN).
bgE$'24>!IJA>F$'TRT_. N).> Vd$'24N)> N).> P)(BA+8,FK8.> >F.-5F.3(?TR2Q+MA+\5 VMA(1&)+8. gr.3&)2QIQI4.-@24N).wN).
ci0
YG$'+8,1 =W
(
c′i0)
=W(
c′i0)
−W (b)+W (b) ∗W (ci0)Ye(W (b) 6= 0
N)$'+8,W (ci0) = 1
Y⇐ &)V)V$<>F$'+8>en\&8. c′
.357I4.>(ci)
>F$'+/5*N).GV$'24N)>1Y
$'2Q5 ε ∈ ]0; 1]Y $'2Q5 c′ε
.35(ciε)
N).> AV)V)(F$f2QT1A5?24$'+8> 2Q+gEH3(?24.3&)(F.> 68+)24.> N).c′.35 N).>
(ci)N).GV$'24N)>*>?&)VH3(?24.3&)(F>vj1− ε
Y $'2Q5 cε = c′ε.(ciε)
I:;OAV)V)(F$f2QT1A5?24$'+-2Q+grH3(?24.3&)(F.e68+)24.GN).cn/&)2,$'+8>?24>F5F.Gj[(F.3TRV)IJA',.3(7I4.>gE.3&)2QIQI4.>@/24N).>UN).
c′εVMA(I4.1,$/.01,324.3+/5
ciε
,$'(?(F.>?Vd$'+8N8A+\5Ye+cAW (c) ≥ W (cε) ≥
(1− ε)2CVMA(68+)2Q5?&8N).[N&-,$.01,324.3+\5 XYu .,32a.>?5@(BA2bV$'&)(75F$'&)5
εV$<>F2Q5?2 g N)$'+8,
W (c) = 1Y
"!"! P1 1P+./21Gc'VQ6ghm$`^?QXjkP+`c'a*^ecI`DgS]gSh\*\*Sh^e` Hgh$`7^?a&^?cI` W W+[$P IS m^?j;fKP_V7a6g dZPscIQg [$P'\T\*SQX[$P'Q
V`7S@i'SQT\*^ec'` \*P'`\9dcI`g^?a*^ecI`>gSpdhckhQTS`dhS 1j?S\]P'Q Q*Sh\][7Q*c$P/^ej?^e\Ta*S\
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u
3547698;:=<>:@?8BAC~FBG"Z VhMPRYYKX KNQIsZ¢n ` E¦7g`*eEejJcm
D]_^p zJn uwvLwvm]«RnR l[z UnR«,nRx]fn,,]fnYy[vmr tuwv =u~7v_u| uwUvs]
^LzJ"muwnp.]
uwvLwvm]D := D+ | D−
ξ
uwvLwvm]t¦l>^"pUD+ := 0 | D+
tuwvLwvm]t¦l>^"pUPnl>n o@yR]D+
t := c.(D+s , . . . ,D
+s )uwvLwvm]t¦l>^"pUP^s]_ E^Ut@]
D+s := z | zξ | (+, ξ, i1, . . . , in).(D
−ξ.i1, . . . ,D−
ξ.in)uwvLwvm]anYrdu,p
D−ξ := c.(D−
sξ, . . . ,D−
sξ)uwvLwvm]anYrdu,p^m]_ E^t@]
D−sξ
:= ((−, ξ, I).D+t )I∈N
l N
]_^pazwn]fnR^m]f5wE]xyR]a^sldzJ^tPuwvpE]f^«RnR@]_^yR]N c.(D+
s , . . . ,D+s )vm] tvmrf^m]fnYp ]xzJn uwvo|vm]l>t¦l>^mryzSld]9 _@]_np
cy[Pr_vs]_np9yR]
0^LzJ">
tPuwv zJn5yR]_^%^m]fUnD+
s
t¦ldzJv#ª*u ,zd]"]ozJUE] o@yR]ay,]c ]2ld]9 fE]_np
c]_^p0y["poEg,jLecfeKOf
c.(D−sξ, . . . ,D−
sξ)vm] tvmrf^m]_np ]kzJn uwvLwvm]l>tqlo^mr#yzl ]fE]fnYp
c t¦l>^%^ |E]_]_npPrfduw
0^zJho*tPuwv zJnyR]_^%^m]fUn
D−sξ
t¦ldzJvª*u ,zd]2"]>zJU@] oEyR] yR]c ] ld]9 _@]_np
c]_^p
y["p¦i>:<;`cfeKOf¢n uwvLwvm]tvsl uRwU^Lp.] ]_^pXEg,jLecfeKO^d Ua]f^LpyR] =u "lov]
D+ ]>pki:<;`YcfeNO ^d Ua]f^LpyR] =u"l>v]D−
ξ
` P mSh\*cI^?` g7dV` dcS+d^?S`Fa jkP Q&P_d^e`7S>gS\(P'Q7Q*S\w`h FP_a*^f"\[scIVQQ*Sh`gQTS dhcIU+[7a&S>gSjkPDg7^iIShQ ISh`dS(`h IP_a&^QiIS~"^e`Fa&QTc\g7V^?a*S+\TSda&^ec'` Y [$P IS>ur [scIVQ<jxd Pg/eTcI`7dha&^?cI`LgV *+" *$ Sh`jeVg^jLVSf[7Q*c$P/^ej?^e\Ta*SK[7Q*cI[scI\*^1a&^?cI`8u Yu9[$P IS r iSh\ dhc6Sm(dh^eSh`La*\ \TSfQTSha&QTcIVi'S`Fa0 a&cIV7aYhha&P ISgS\gP'Q7Q*S\ dqP_V\*SCgS\3ghhdqP'jeP ISh\
P dhcI`g^1a&^?cI` \Ta&^?[VjeP'`Fa5jLV7dV`S P_dha&^?cI``h FPa&^i'S(Sh\Ta \*V^Qi6^?S,g d V`dcS+d^?S`Fa`c'``LVjrS\Ta`hdhS\*\*P'^eQTSR[scIVQ jeP\hh[$P'Q&Pa&^ec'` "[Q*c'[mcI\T^?a*^ecI` W Y p[$P IS t iS[sS`g$P'`FaL|dc'U(U+S S` jeVg^jLVS\*^eU+[j?SNP2i'SdjkP,g^i'SQ 'S`dhS
Fid iVQ r ^eQ mcI` \*S[sSQTU(SaLs[$P'Q P/V\EgS`7c'a.P_a*^ecI`Fg7S`c'a&ShQ9jxd P\*Sh`dS@g7dV`SRP'da&^ec'` `h FPa&^i'S
(−, ξ, I)[$P'Q
(−, ξ, I)
3547698;:=<>:@?8BACED FZ[QwAK dZ 6*ZOHZ[QFid
Z7IFid−
kuwnR^E]su[y[vs]ayR]=u"zdy[ ,zd];tYvml u,wUU^Lp ] s u|vowvs];tql>^%hpEU*vm]f^Lpvs]_nYp k=u"]>zJU@]0nzJU@] 0 ]_^LpnYldp rFid
]opm uwvLwvm]2nYrf[u,pqvm]_^pvs]_Unp =u "]ozJUE]anzJU@]0]_^pnYldp r
Fid− ] ,zJ]%tYU ,zd]m uRw^s]_nY] y9 zJn]ku7>p@l>n5^ht¦rfU« ,zd]
Fid−
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s
3476¦8:=<>:E?8 APCED*DF A"Z ACBKOH,X7RQ>K MPO\ eEi0b|=je g,i ]fnpvm]uwvLwvm]f^tvml JuRwUU^p.]_^ ]_^Lpqs l>vsy[vm]k]fn,,]fnYy[vmr#tPuwvx
quw^tqlo^hpEU.0 ⊆ D
quw^nYrf[u,p. ^N ⊆ M
uwElov^
((−, ξ, I).DI)I∈N ⊆ ((−, ξ, I).DI)I∈M
lg` dc'`La*Q*SyS 6ShU([j?SpS\Ta]V7a*^ej?Sb[mc'VQS^eSh` dhcIU+[Q*Sh`gQ*Spj d ^?`dj?V\*^?cI` S@gSh\*\TS^e`
D =
0.1
(+, ξ, 1; 2)
0.5
(−, ξ.1, 0)
z
(−, ξ.1, 1; 2; 3)
0.3
(+, ξ.1.1, 0)
0.6
0.3
(−, ξ.2, 0)
(+, ξ.2.0, 0)
0.2
(−, ξ.2, 0)
(+, ξ.2.0, 0)
`7dS\ a][$P'\9^e`dhjeV\AgP'`\
0.1
(+, ξ, 1; 2)
0.5
(−, ξ.1, 0)
z
(−, ξ.1, 1; 2; 3)
0.3
(+, ξ.1.1, 0)
0.6
0.123
0.3
(−, ξ.2, 0)
(+, ξ.2.0, 0)
0.2
(−, ξ.2, 0)
(+, ξ.2.0, 0)
dP'Q jkP]f"SV7^ejej?SX`LVj?jeS0QTSU+[jkP_dhS Sh\Tag`h IP_a&^QiIS R P'Q9dc'`La*Q*S
DS\ ag^?`dj?V\Ag$P'`7\
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0.1
(+, ξ, 3; 4)
0.9
(+, ξ, 1; 2)
0.5
(−, ξ.1, 0)
z
(−, ξ.1, 1; 2; 3)
0.3
(+, ξ.1.1, 0)
0.6
0.3
(−, ξ.2, 0)
(+, ξ.2.0, 0)
0.2
(−, ξ.2, 0)
(+, ξ.2.0, 0)
(−, ξ.2, 1)
z
dqP'QRjeSwd%k$P'` 'SU+S`FaS`Fa*Q*S dhS\8gSV wgSh\*\*Sh^e`\ S\Ta \*^?U([j?SU+S`Fapjxd P eTcIV6a5g7dV`S(`cIVi'Sj?jeS(P'dha*^ecI`
(−, ξ.2, 1)Sa@gSw\&P dcI`Fa*^e`LV$P_a*^ecI`Rg$P_`\pjeSKgSh\*\TS^e``h FP_a&^QfrjeS[j?V\x>gQ*c'^?a&SNP'VMgSh\*\*V7\5gV
\*dqP_jkP'^?Q*S0.2 P'^?`\*^ jV7S<j?S Q*ShU([jeP'dShU(Sh`Fa3g7d V7`SCf"ShV^ej?jeSg`V7jejeS [scI\T^?a&^QiIS [$P_Q V`Dg7S\*\TS^?`2[scI\*^1a&^f
3547698;:=<>:@?8BACED WFu#"M_ZdHJILKNMPOQ|KTWVXZ¢nY]rEqgJdbdcfehgRi(jeNJ Eqh¨y¦ zJnuwvo|vm]¦tvml JuRwUU^p.]]_^pzJn5u|vowvs]^tYE]l ,p ]_nzktPuwvvm]f t=u7]_]_np yR]a^s]_^kld]9 _@]_np^
cE]opyR]aE]>zwv^2lonYpEUnz7u,pEEl>n,^B tPuwvx
^ c ]_^LpyR]t¦l>Ey7^nYzw =u y7h>]fv%,]fnY] yR]a=ut¦l>uwv"p.r2y,l>nRnYr] ^nYl>n cuy,]_^ "]>zJUE]f^koEyR]f^B =u lonYpEUnz7u,pEEl>nWy¦ zJnY]a"]ozJUE] oEy,]kyR] c ldzr>>]fnpzd]fUE]f5]fnp u y[ho]_v% ]_nY] y,] =ut¦l>=uwv%hp rayRlonRnYr]a^% c ]_^p yR] t¦l>Ey[^ y[Pr'vm]_npyR]1
3547698;:=<>:@?8BACED AWFkZ[QQZRKUO V"MPRKUX K QmIsZ¢n xJjLj|Jei ^LzJv zJnY] uw^s]
β]_^LpzJn uwvo|vm] tvsl uR|UU^Lp ]¨^zJv
βp.]_ ,zd] ^m]f^xtvml]>pEEl>nR^
^t@]_^#^sl>npyR]_^kyR]_^%^m]fUnR^ ^t@]_^#^zJv=u5uw^s]β
bP_`\CjkP]\*V^1a&S*g7SjkP9akyh\*SJ'\&P'Vfs[Qhdh^e\*^?cI`jkP]`c'a&^?cI`5g7S*gS\T\*Sh^e`wV7a&^?je^?\hSrS\ a;dShjej?S9gS*g7S\*\TS^?`[Q*cP^ej?^e\ a&S
XShV S 6ShU([j?S\3gS@gSh\*\*Sh^e`\][QTc$P^?je^?\Ta&Sh\[sSVi'S`Fa6hhdjeP'^eQTSQ jkP]gh$`^1a&^?cI` 1
3lg`UgS\T\*Sh^e`2[scI\T^?a&^Qf
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t
0.4
(+, ξ, 1; 2; 3)
(−, ξ.1, 0)
0.3
z
0.4
(+, ξ.1.0, ∅)
0.3
0.9
(+, ξ, 4; 5)
0.4
0.3
(−, ξ.5, 0)
z
0.6
(−, ξ.5, 0)
zξ′
(−, ξ.5, 1)
(+, ξ.5.1, ∅)
`LiIcI^1a5jLVS(jeS\<dcS+dh^eS`Fa*\\*cI`Fa V7a&^?je^e\h\ jeP1f"c'^e\RP i_P'`FaRjeSh\P'da&^ec'`\p[mcI\T^?a*^iISh\ Sa j?S\P_dha&^?cI`\9`h FP_a&^QiISh\ S<dcS+dh^eS`Fa
0S\TabP'[Qy\]V`SP_dha&^?cI` [mcI\T^?a*^iISw|^ejndc'Q*Q*Sh\*[scI`g g7cI`d
jeP]g^i'SQ 'S`dhSR`h FP_a*^iIS ` Pw^edh^D−
ξ.4 = 0.4.0Sha
D−ξ.5 =
0.3
(−, ξ.5, 0)
z
0.6
(−, ξ.5, 0)
zξ′
(−, ξ.5, 1)
(+, ξ.5.1, ∅)
Y l]`1gS\T\*S^?`2`h FP_a*^f
0.2
(−, ξ, 1; 3)
0.2
(+, ξ.1, 5)
(−, ξ, 1; 4)
z
0.8
(−, ξ, 1; 3)
z
iS@gSh\*\TS^e` `h FP_a&^QfCPS P_dha&ShU(Sh`Fa6gShV [QTc/eTSda&^?cI`\9\*^?U([j?S\"1jeP[$P'QTa*^eS#g7S IP'Vd%kS
(−, ξ, 1; 3)
(+, ξ.1, 5)
(−, ξ, 1; 4)
z
Sa9jkP[$P'Q a&^?S#gS@gQ*c'^?a&S
(−, ξ, 1; 3)
z
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J
iS8jLV^nS 6[je^jV7S<jeP[Qhh\*Sh`dS8gS@gSV P'dha*^ecI`\(−, ξ, 1; 3)
W dZP'Q Q*S9[QTc$P7^eje^?\Ta*Si\*V7^i_P'`Far`7d Sh\Tar[$P'\lV`gSh\*\*Sh^e`(dqP'Q;^?j|wPEgSV(P'dha*^ecI`\(−, ξ, 1; 3)g$P'`\9jePwUqU+S [QTc/eTSda&^ec'` \T^eU+[jeS
0.2
(−, ξ, 1; 3)
0.2
(+, ξ.1, 5)
(−, ξ, 1; 3)
z
0.8
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0.s1s2 . . .KjxdV`^?d^1ah c'V `cI`XgSj d-hhdQT^?a&V7Q*S ` d ^?`La*SQi6^?S`Fa[$P_\ ` a*Q&Pg7V^?a V` dcS+d^?S`Fa s
^
[$P_Q
(+, ξc, 0; 1)
(−, ξc.0, s1)
(+, ξc.0.s1, s2)
(−, ξc.0.s1.s2, s3)
(+, ξ.0.s1.s2.s3, s4)
(−, ξc.1, 0)
P [$P'Q a&^eSAg7S IP'Vd%kS]dc?gS9jkP#i_P'jeShVQgSsSafjeP [$P'QTa*^eSAgS3gQTcI^?a*S je^ByhQ*S\T^eU+[jeShU(Sh`Lag7S]jePb[jkP_dS
[scIVQ V`1f"V7a*VQAgS\T\*Sh^e` lg` dcS+dh^eS`Fa
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]t'u
s1
^
. . . . . . . . . sn
^
c -si = 0.si
1si2 . . .
Sh\Ta]a&Q*PgV^1a [$P'Q
(+, ξc, n)
(−, ξc.n, 1)
(+, ξc.n.1, 0; 1)
(−, ξc.n.1.0, s11)
(+, ξc.n.1.0.s11, s
12)
(−, ξc.n.1.0.s11.s
12, s
13)
(+, ξc.n.1.0.s11.s
12.s
13, s
14)
(−, ξc.n.1.1, 0)
. . . . . . . . . (−, ξc.n, n)
(+, ξc.n.n, 0; 1)
(−, ξc.n.n.0, sn1)
(+, ξc.n.n.0.sn1 , s
n2)
(−, ξc.n.n.0.sn1 .s
n2 , s
n3)
(+, ξc.n.n.0.sn1 .s
n2 .s
n3 , s
n4)
(−, ξc.n.n.1, 0)
` Q*ShU P_QjLVS@jLVSpjeSRQ*P'`d%kShU(Sh`LagS\ a3fKP'^?awjxd P'^gS#g7dV`tjecd
(+, ξc, n)
(−, ξc.n, 1) . . . . . . . . . (−, ξc.n, n)
dV7a*^eje^?\&P_a*^ecI` g7d P'dha*^ecI`7\2\*V7Q V`UqU+Sje^?SV ξc.n
S\Ta V7a&^?jeS^ed^RdqP'QgdhS\ P'da&^?cI`\Jha.P'`Fa ^?` dckhhQ*Sh`La*S\R^?j`7dO| P P_VdV`7S dhcI`g^1a&^?cI` \*VQ jeS [P'QTa&P IS gSh\ j?^eSV P'V pgS\T\*V\LgSh\ P_dha&^?cI`\
(−, ξc.n.i.1, 0) 6dS\ a`hhdSh\*\&P_^eQ*S [mc'VQwdc?gSQ+jeSh\+dhc6Sm+d^eSh`Fa&\ dP'Q+^ej`7dO| P[P'\,g7S dhcI`g^
a&^ec'`(Sh`La*Q*SgjeS\rdc'`La*^e`LV$P_a*^ecI`7\0gSSgShV f"SV^?jejeSh\g7d V7` dcS+dh^eS`FakKQ*ShU P_QjLVSgWs<[$P IS]\*V7^i_P'`Fa&S `2V7a&^?je^e\TS jeS<je^?SVξc.n
Sab`c'`2jeS<je^?SVξc.0
[$P'Q]S 6SU+[jeSRdqP'Q]jeSn
gcI``SRjeSpdqP_Qg^?`$P'j;gVx7Q&P'` d%kSU+S`Fa
nSpdcS+d^?S`Fa0Sh\Ta9a&Q*PgV^1a9[$P'Q jeSEg7S\*\TS^?`2Qh gV^1a wj dZP_dha&^?cI`
(+, ξc, 0)
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ts
3476¦8:=<>:E?8 APC|>WFZ AZdQQZRKODs(ξc) l>hp
s ∈ ]0; 1[R\9] bgnx`+;P yz©^msuw=uwvm]
s]_^Lpq@]kyR]_^%^m]_n
Ds(ξc)
(−, ξc, s1)
(+, ξc.s1, s2)
(−, ξc.s1.s2, s3)
(+, ξ.s1.s2.s3, s4)
ls = 0.s1s2 . . .
nS\3ghm$`^?a*^ecI`7\ S\jLV^?\*\hS\eTV\jLV7d^edh^\*SpdcIU+[scI\*Sh`La ? ^?`\*^7jeS dhc6Sm+d^eSh`Fa 1
0.4
^
0.2
0.7
^
Sh\Ta9a&Q*PgV^1a [$P'Q
(+, ξc, 2)
(−, ξc.2, 1)
(+, ξc.2.1, 0; 1)
D0.4(ξc.2.1.0) (−, ξc.2.1.1, 0)
(−, ξc.2, 2)
(+, ξc.2.2, 0; 1)
D0.2(ξc.2.2.0) (−, ξc.2.2.1, 0)
(+, ξc.2.2.1.0, 0; 1)
D0.7(ξc.2.2.1.0.0) (−, ξc.2.2.1.0.1, 0)
yj8|P]`$P_a*VQ*ShjejeShU(Sh`Fa[gShV P'dha*^ecI`\G`h IP_a&^QiIS\ U P6^?U P'j?S\ZjLV^6dcIQTQ*S\T[mc'`gS`FarP'V8f"ShV^ej?jeSh\ i^g7S\gV dcS+dh^eS`Fa
7_o 5WAPC?> * )+*"" * &
0.5
(+, ξ, 0)
0.5
(+, ξ, 1)
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]t
*"9.)%*
(+, ξc, 2)
(−, ξc.2, 1)
(+, ξc.2.1, 0; 1)
(−, ξc.2.1.0, 1)
===
(−, ξc.2.1.1, )
(+, ξ, 0)
(−, ξc.2, 2)
(+, ξc.2.2, 0; 1)
(−, ξc.2.2.0, 1)
===
(−, ξc.2.2.1, )
(+, ξ, 1)
*"2)&+"(+, ξ, 0)
*(+, ξ, 1)
8C *)' &F" *$5"+$ * : * &5*7=U *"2)&+" o%5(F)'&7)#*")'6, )+*""" )+*+ *$()'o.:$*6
(+, ξc, 2))><&7)#*$ )>o 8C *)' &F" * $+)'H"" &G"+$w *+ : *
&5* %+$b *47)&" :$9*2)F)'+" 1)w : *b$T)'o*/ '=43R4 *"%+$*%<&I 6& &F" *)+*"p) &+""$X * &5*
ξc.2'(−, ξc.2, 1)
*(−, ξc.2, 2)
=
iSh\gS 7ShU([7jeS\9U+c'a&^QiISh`La9jeP]gh`^?a*^ecI`Vf"cIQTU(Shjej?Sb\*V^Qi_P'`Fa&S 1
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tt
3476¦8:=<>:E?8 APCmt£F "$A 7RHJILK MPO Z"Q5X X7$AK 7,Z5Q|KTWVXZ p"uwnpyRl>n,nYrzJn5U@]>z
ξc zJn y,]_^^s]_nt¦l>^"p">]_]_npYtvml JuRwUU^p.]^LzJvxzJn]uw^m] β y[^ l>np.]yR]ξc]_^Lpcw`sx =Re%c0tPuwvu"l>nYopElonvsrozJv^%h>]
φξc
yR]a=u¨ u|nRfvm]^zJhmuwnp.]
φξc: 0 7→ Fid φξc
: z 7→ z φξc: zξ 7→ zξ
φξc: (+, ξ, 1; . . . ;n)
D1
Dn
7−→ (+, ξc, 0)
(−, ξc.0, 1; . . . ;n)
(+, ξ, 1; . . . ;n)
φξc.0.1(D1)
φξc.0.n(Dn)
φξc: s
D
7−→ (+, ξc, 0; 1)
Ds(ξc.0) (−, ξc.1, 0)
φξc.1.0(D)
φξc:
s1
D1
. . . sn
Dn
7−→
(+, ξc, n)
(−, ξc.n, 1)
(+, ξc.n.1, 0; 1)
Ds1(ξc.n.1.0) (−, ξc.n.1.1, 0)
φξc.n.1.1.0(D1)
(−, ξc.n, n)
(+, ξc.n.n, 0; 1)
Dsn(ξc.n.n.0) (−, ξc.n.n.1, 0)
φξc.n.n.1.0(Dn)
φξc: ((−, ξ, I))I
DI
7−→ ((−, ξ, I))I
φξc(DI)
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]t
3547698;:=<>:@?8BACWFh$A 7HJILKNMPO ACB 7O 4[QZ[ 7\u cw`sx =bdce g,i x =Ri :j>7`8= yR]£yR]_^%^m]fUnR^St¦l>^%hp">]f5]fnYptvsl uR|UU^Lp ]_^
D1, . . . ,Dnl>n,^U^p.] ªYl>^vkzJnUE]oz ξc y[U^ l>Unpy,]_^ UE]oz| y,]_^uw^s]_^kyR]_^ (Di)
]>ppEv_u[y*zJUvs]kª*u ,zd] Di
u7>]ξc.i
lo ]U@]>zyR]pvfu[yzdop@l>nWyR]_^ld]9 fE]_npE^L
zR>?|q?sL:@<o:E?8BACkt£F "$A 7RHJILK MPO ACB 7O 54[QZ[ 7\9]pv_u7yzJhp y9 zJnvmrf^m]su,z5]f^LpzJnvmr_^s]mu,z,
! mIv>?&0_5oN). @ H3(?2 6M.3( n/&8.-IJAt(F.3IJA5?24$'+ N).W,$'&)V)&)(F.p.>?515F$'&?$'&)(F>o,$'+)+8.f.W.35A',,3IQ24n/&8.<Y
P a&Q&PgVdha*^ecI`w^?`+i'SQT\*SX` dZPp[$P'\ <\TSg[7QhcddhV[sSQ9g7S]jeP 'S\Ta*^ecI`GgSh\i\TcIV\ je^?SV Gg7SξcdhcIU(U+S
jkPa&Q*PgVda&^ec'`mgS:i_P'^?apj?S]fKP'^eQTS yj;S\ a5g7cI`dw[mc'\*\*^jeSKg7S,gc'``SQCgSh\RQy Ij?S\ a&Qy\p\*^?U([j?S\ [scIVQ^e`'iIShQ*\*ShQgjePa&Q&PgVdha*^ecI`
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3476¦8:=<>:E?8 APC WF "$A 7RHJILK MPO KUO Z"QZ A B 7O AZdQQZRKO\;uWc%J`|x =YbdcfehgRi ei Rw|j|y¦ zJn yR]_^%^m]_n ^%UtE]Spvfu[yzJ"p]_^p#m zJnR Rz[] "l>v]knYlovuwE]l #p.]_nzd]xtPuwvvsrr_v"pzJvs] UnR«RnR@]^LzJ"muwnpaE]f^v f7@]_^ l>nPzd]fnp.]_^S^zJhmuwnp ]_^ ^LzJv¨@]_^¨uwvowvs]_^tvml JuRwUU^p.]_^F n5nYl[p.]
ξ∗czJn^mldzw^E]oz y,]
ξc ,zd]fEl>n ,zd]>
Fid 0
(+, ξ∗c , 0)
(−, ξ∗c .0, 1; . . . ;n)
(+, ξ, 1; . . . ;n)
D1
Dn
(+, ξ, 1; . . . ;n)
D1
Dn
@]>p¦@]kmuw^^_u|nR^xl>npnYzu,p@l>n(+, ξ∗c , 0) 0
(+, ξ∗c , 0; 1)
Ds(ξ∗c .0) (−, ξ∗c .1, 0)
D
s
D
\u~pvfu[yzdop@l>n~Un>]fv^s] y¦ zJnnY§zdy l> ]_nmMsuwnp¦tPuwv(+, ξ∗c , n)
^s]Eu|hp ^zJv p.ldz,p.]_^E]_^wv_u|nYªY]_^ ]_n _]kp ]_t^ayR]#=ux uwn,_vs] ^LzJ"u|nYp ] E]_^
?UnYy[ ,zd]_np ,z U*Eu,z,p;Euwvm]=u
f5]klmt¦r_vfu,p@l>n^Lzwv @]_^ u,z,pvm]f^wv_uwnªY]_^F©
(+, ξ∗c , n)
? (−, ξ∗c .n, i)
(+, ξ∗c .n.i, 0; 1)
Ds(ξ∗c .n.i.0) (−, ξ∗c .n.i.1, 0)
Di
?
? s
Di
?
PVeTV7\Ta&^ $dqPa&^ec'` jLVS jkP a*Q&PgV7dha&^?cI` ^e`'iIShQ*\*SUg7dV` Qh\TSqP'V S\Ta(dc'Q*Q*Shdha*S2Q*Sh[mc'\*S \*VQwjkP QTS U(P'QjLVS \TV^i_P'`Fa*S 1
7_o 5WAPCkt W $&%" **)'$" &1s * P *$ 7)#$w$*))'$& )54 6 $.%" **)' )+* )+*"" * &+"%," & &,)#* *$E$P.)P.& &F" *"Ro4 *"@7)&" *es*D'l *"R9& 88"A 9<$&9**)' : )+*U$* )#*/ "+$ )+*"R &5*:ξ∗c
$*);BCy%"U7)#$p$T))'|& )%l&&TL=17) (+*M 2",$i/H" "0)P *" )' " *+" * " &l*4 ),& )'P.)&"5"+$<6 &5*
ξ∗c*"E$.%" *$ * )F)'+"<1)g8C<$3 *<<$3+)' * )54 $&%" **)'Ji)'F<$8"p *R9 & 8
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¨
* *$&5* $ *"*o<$*1$&%" *$ (# 7)#$3b%!)#*$K * * *$ *"w) &+" o%5(F)'&7)#*" )' 7)'Jg9*++&$9*/$&%" *$ *<*4 &+) (+*)54 *l8 *& *
0 '=
3547698;:=<>:@?8BACA FMY%TX K Q,ILK MPO~ZO(X7$APK 7RZ~VhMPRYYKX KNQIsZ\u iqg J(`,EejL`cfehgRi JRK y¦ zJn vmrf^m]mu z tvsl uR|UU^Lp ] R
yR]©yR]f^^m]fUnR^ t¦l>^"p">]_]_npqtvsl u#wUU^p.]_^^d ] P]>p%zd]kuwUnR^% 2^%^mld_@]_vkzJn5vsr_^m]su,z©^Ut@]
Rt
]2vsr_^s]mu,z yR]kyR]f^^m]fUnR^Rtuwv=upEv_u[y*zd>pEEl>ns@yRr_«
nR"pElond2tPu|,] o,xlovuwU^m]_v E] vsr_^m]su,z©^Ut@]
Rt
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d]_]>ppEvm]E]_^a^msuw=uwvm]f^WEy,r_«RnR"p@l>ndktu|,] tvsrryR]fnYp ]^Lzwv
Dt
n l ,p@]_np2zwnyR]f^^m]fUn
D = JRK
P a*Q&Pg7Vdha*^ecI`ng d V`wgS\T\*S^?` `cI` [scI\*^1a&^QiISU+S`FaR[Q*c$P^?je^e\ a&SwS\ a[j?V\ dhcIU([7je^jLVhhS(dP'Q ^?jfKP'V7a<[Q*Sh`gQTS(Sh`dc'U([7a*S\*Sh\ dcS+dh^eS`Fa*\R`h FP_a*^f"\ `V7a*^eje^?\*SV7`S+a*Sd%k`7^jLVSgS+a*Q&Pg7Vdha*^ecI`iISQT\ jeSh\6gSh\*\*Sh^e`\ [scI\T^?a&^QiIShU(Sh`Lag[QTc$P^?je^?\Ta&Sh\ P(a&Q*PgVda&^?cI`2d%k$P_` ISjeP(\Ta*Q*Vda&VQTSKgS\Xje^eShV vlj?jeS [sSV7ar\ dZP'[7[$P'Q*Sh`Fa&SQ pV`S # a&Q*PgVda&^?cI`wdqP'QlSj?jeS9P eTc'V7a&S3gSh\f^?`La*SQ*P'dha*^ecI`\;[scIVQ;^?U([scI\TSQV` c'QgQTSCg d-h:i'P_jeV$P_a*^ecI`
dPhha&VgS@g7d V7` S 7ShU([7jeSpS\ agV6a&^ej?S
S@gSh\*\TS^e` 1
(+, ξ, 1; 2)
0.5
(−, ξ.1, I)
σ
(−, ξ.1, J)
0.3
(−, ξ.1, I)
(−, ξ.1, J)
0.2
D−ξ2
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FY
Sh\Ta9a&Q*PgV^1a S`
(+, ξ, 0)
(−, ξ.0, 0)
(+, ξ.0.0, 1; 2)
(−, ξ.0.0.1, 0)
0.5
(+, ξ.0.0.1.0, 0)
(−, ξ.0.0.1.0.0, I)
σ
(−, ξ.0.0.1.0.0, J)
0.3
(+, ξ.0.0.1.0, 0)
(−, ξ.0.0.1.0.0, I)
(−, ξ.0.0.1.0.0, J)
(−, ξ.0.0.2, 0)
0.2
D−ξ2
5cIVQ9V`UgS\T\*S^?` `h FP_a*^f.^?jBfKP'V6aAfKP'^eQTS V`S a&Q*P'`\f"c'Q*U(P_a&^?cI`\*^?U(^?jkP'^?Q*S1
0.1
(−, ξ, 0)
(−, ξ, 1; 2)
D+1
(+, ξ.1, I)
0.3
(−, ξ, 0)
(−, ξ, 1; 2)
D+2
zξ.i
Sh\Ta9a&Q*PgV^1a S`
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¨IW
(−, ξ, 0)
(+, ξ.0, 0)
(−, ξ.0.0, 0)
0.1
0.3
(−, ξ.0.0, 1; 2)
0.1
D+1
(+, ξ.0.0.1, 0)
(−, ξ.0.0.1.0, 0)
(+, ξ.0.0.1.0.0, I)
0.3
D+2
(+, ξ.0.0.1, 0)
(−, ξ.0.0.1.0, 0)
z
P`cIQTU P'j?^e\*P_a&^?cI`IgSpdhS\3gSVUgS\T\*S^?`\SgcI``S
0.1
1
0.5
σ
0.3
0.3
2
0.5
0.3
z
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r
3476¦8:=<>:E?8 APCAPDF "$A 7RHJILK MPO Z*OAZ[QQZRKORQVM*Q|KILK ZTZO I V"MPRKUX K QmIsZ[Q\;u pv_u7yzd>pEEl>n l>nR^%U^p.] u ldzRp.]fvWzJn]£u[>pEEl>n nYrfdu pho]u7muwnp ]>pzJnY]£u[op@l>n t¦l>^"p">]EyzLtYU ,zdr]2utYv _^ª*u Rz[] l ] _E]fnptYvmr_«du|nYpyR]f^yR]_^%^m]_nR^ nYrfdu pU=^ qU@]au,zw7]_np ] =uEl>n,zd]>zJv2y,]_^U@]>z| hzJnU@]>z
ξ]_^ppv_u|nR^"l>v%5rk]fn
ξ>
zJnY]u[opElon (+, ξ, I)]_^papEv_u[y*zJhp ]k]_n
(+, ξ, 0)
(−, ξ.0, 0)
(+, ξ.0.0, I)
zJnY]u[opElon zξ
]_^p pEv_u[y*zJhp ]k]_n
(+, ξ, 0)
(−, ξ.0, 0)
z
zJn~yR]_^%^m]_n£nrfdu,pEU c.(D1, . . . ,Dn)^zJv zJn£E]oz
ξ]_^p pEv_u[y*zJhp]fn
(−, ξ, 0)
c
(+, ξ.0, 0)
D1
. . . (+, ξ.0, 0)
Dn
lc]_^pqrduw
cy uwnR^E], Rz[]_*E]f^"]ozJUE]f^¦nzJU@]_^#l>npqrop.r vm]ftY=u[r]_^;tPuwv#yR]f^#u7>p@l>nR^
(+, ξ.0, ∅)USuzJnxt¦]>pEhpuR,zJ^ y,]qnYldp"u,p@l>nt9zJU^' Rz[]#yR]_^l ] _E]fnp^ ]opyR]_^qu[>pEEl>nR^
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]f^Lppvfu[yzJ"p ]_n
( (−, ξ.0.0, I)
DI
)I
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c $T))'&7" )+* *"!# &5*$7"+o%5(F)' & 88"/ A")'+"]8 *+& *+$6 *=|+$ 9*"
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0 ⊆ D+
%/* $T)#& *4 <$ )#$*5"$p *"po%5(F)'& 88"* )>$ * $*)#&
0 ⊆ D−
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¨'u
9* &*" $9*"83" % T*)#$p*4 &o*)#$3s)' & )54 )+*"" * & ")'+"0
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IY $
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(α) . . . UDξ.in`Λ′in
(α)
?
c - j?S\(Λik)
Sa(Λ′
ik)hdQT^ ISh`FaXj?S\ [$P'Q a&^1a&^ec'`\ S
ΛSh`
n[$P'Q a&^?S\ \*^jePo$P'\*SpSh\Ta
` ξ,Λ
nS Sh\*\*Sh^e` V`^ 'SQT\*Sj\TVQ9jkPo$P'\TSξ ` Λ
S\ agj?S S\T\*Sh^e`
?
?
(−, ξ, j1, . . . , jn)
UD`ξ.j1,...,ξ.jn,Λ(α)
. . . (−, ξ, l1, . . . , lm)
UD`ξ.l1,...,ξ.lm,Λ(α)
yjiQTS\ a&S % hdQ*^?Q*S j?S\\TdqP'jeP'^eQTS\ LV^P'[[$P'Q*P'^e\T\*Sh`La $P'`\j?S Sh\*\*Sh^e` V`^ IShQ*\*Shj& +]`S V`^?cI`h`cIUo7Q&Pj?S Sh`\*ShUoj?S\ `cIU!Q&P7jeS\ a.P'`Fa `c'UoQ&P/jeS$ jeS\5[scI\*^1a&^ec'`\n[mc'VQjeS\5\*dP'jkP_^eQ*Sh\?\*c'`La `c'UoQ&P/jeSh\ ` \*S c'``SbV`7S `LVU Q*P_a&^?cI`W^ TSda&^ 'S
p = (pi)i∈N
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?dhcIU(U+S
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iSh\Tafj ^e`7^edS $P'`\
pS9jkP [scI\*^1a&^?cI`V+\*dqP_jkP'^?Q*S
?
3476¦8:=<>:E?8 APC% (kZ[QQZRKO 7YOYK Z"QZX l>hp
βzJn]auw^s]>|\¦]ixJjLj>eEi =Ri;e,J|j|J
UDβ(α)]_^pE]yR]_^%^m]fUn ,zd]0s lonoE]fnpyR] yRrfvUvs]>
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. ,
ACD+3 *" )+*"" * &+" &7)#* $8" *" ",$ P.&5* )+*" )+*"" * &+"= * s/"J< *"A" *)'1)#& $9*")54 6 )+*"" * & 6&7)#* $8" *E",$]/" )#& % $*$" *] *+$L", */0)' * *"P!<$o%9*7)#$
1= R&+" &FJ
( *"R *".", *"47)#$3 &5* *" )+*5" *)'1)#& $*"",$NP!<$3o%9*"N7)#$1=Ep4 ));B *. & 8 V`7^ IShQ*\TSj@$8&) &5*$
)+**4 P!" * $*))' &*/H" *" )+*"" * &+" " &1s *" )+*2P.)&" *β",$b^e`dhjeV\ )F)'+" * )+*"" * &&,
)#* $" * )+*5P.)&" *β)'9" *+"")+*"I$8(B *. &+"U" &1s *"L )%l& &KL= 57) (+* ; '*4 *+" *P *")+*"
",H", )+*"" * &+" )+*"L$8(B *.&+"9" &1s *" )' )+*"" * & &7)#* $" * )+*KP.)&" *β
*"p*4 *+" *P * )+*")+*"" * &+"." &1s *" )+*XP.)&" *
β=
zR>?|q?sL:@<o:E?8BACEDM ( HYXNYKhZ[QAORQ qD, UDβ⊥(α)
y l>"p
DzJn~yR]_^%^m]_n yR]¨u|^m]zJn*uwvm]
β qldzJv p.ldz,p
α l>n£u
Scal (D) ⊆ Scal(q
D,UDβ⊥(α)y)
⊆ Scal (D) ∪ 2−nα | n ∈ N
! #`A#V)(F.3&)@ .S(F.3Vd$<>F.S>?&)(7V)IQ&8>F24.3&)(F>(F.3T1A(Fn/&8.> V)(F^>U&)+8. A',35?24$'+x+8H3q\A5?2Q@ .<9%I4. N).>F>B.32Q+t&)+)2Q@ .3(F>F.3I
UDβ⊥(α)&)5?2QIQ24>F.WCE.3+\5?(F. A&)5?(F.>#A',h5?24$'+8> X&)+RN)H3T_$'+_.35Z&)+RN)H3T_$'+#gr$/,AIQ24>BH >?&)( 5F$'&8> I4.> IQ24.3&f CV$<>?2Q5?2 gr> XN). IJA*PMA'>B.GCI4$,AI4.XY
V)(F^>&)+8. A',35?24$'+LV$<>F2Q5?2Q@ .<9I4. N).>B>F.32Q+&)+)2Q@ .3(F>F.3I UDβ⊥(α).>F> A24. N).Dgr$/,AIQ24>B.3(%>?&)(a5F$'&)5F.>I4.>*A',35?24$'+8>Vd$<>F>?2QP)I4.>Y
`a.>eA',35?24$'+8>V$<>?2Q5?2Q@ .>e>F$'+\5>F&)2Q@/24.>N).5F$'&)5F.>7I4.>eA',35?24$'+8>7+8H3q\A5?2Q@ .>eVd$<>F>?2QP)I4.>Yu . n/&)2\2QTRV)IQ24n/&8.SCVMA(%(FH,3&)(?(F.3+8,.XZn/&8. 5F$'&)5 +G&8NLN).D.>?5Z@/24>?2Q5FH I4$'(F> N). IJA*+8$'(?T1AIQ24>BA5?24$'+RN&(FH>F.A&
D,UDβ⊥(α).35en\&8.GI4.,$.01,324.3+\5GA'>F>F$,324HSAV)VMA(BA45eN8A+8>7I4.S(FH>F&)IQ5BA5 q
D,UDβ⊥(α)y Y
ACD> ) $8%," && $&%..% )+*$ *2&1s &K*R*
Scal∗ (D) = Scal∗(q
D,UDβ⊥(1)y)
zR>?|q?sL:@<o:E?8BACED (('@KNQIsZOHZ2AYZαScal∗(D) l>"p
DzJn~yR]_^%^m]_n yR]¨u|^m]zJn*uwvm]
β7D 9]%,^Lp ]¨zwn£^smuwuwUvs]
αScal∗(D) ∈ ]0; 1[p.]f Rz[]
Scal∗ (D) ∩ αScal∗(D)Q = ∅
! #` ;=.3+8>F.3TUP)I4.-N).>R>F,AIJA2Q(F.>Scal∗ (D)
N).D
.>?5_A& V)IQ&8>RN)H3+8$'T[P)(BAP)I4.wN)$'+8,Scal∗ (D) ∗ Q
.>?5[A&]V)IQ&8>N)H3+8$'TUP)(BAP)I4.1$'(]0, 1[
+a;=.>?5SVMA'>N)H3+8$'TUP)(BAP)I4.1N)$'+8,R2QI .f24>?5F.L&)+>F,AIJA2Q(F.αScal∗(D) ∈ ]0; 1[
5F.3I%n\&8.Scal∗ (D) ∩ αScal∗(D)Q = ∅
Y3547698;:=<>:@?8BAC (Z£QLHXYKhZ
αD l>"pDzwnyR]f^^m]fUn£y,]xu|^m]zJn*uwvm]> l>hp
αD
zJnl ]fE]fnYpαScal∗(D)
or_v«*uwnp=utvmlmtYvErop.ryR]a=utvslmt¦l>^"pElon d xnY]y,r.t¦]_nYy u|nYp ,zd] yR]
Scal∗ (D)
3547698;:=<>:@?8BAC03 (VVM*QO I 7YOYK dZ*"QZX l>"p
DzJn~yR]_^%^m]_n ^zJv zJnY]¨uw^s]SzJnuwUvs]
β,\ gFEGEg,jo` i¦c =Rie ,w|j|JyR]
D]_^p
UOD = UDβ⊥(αJD,UDβ⊥(1)K)
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$
7_o 5WAPCED7t 3 *K * )%l&& )54 %,")'$$&7)#* $" *+*"GdhcI`\Ta*Q*Vda&^ IS2)F)'+"l *0" *+"rl4 * *)%/*) &K* *$)+*" $&%"/ )'7" )+*"+<$3+)' &F")' &+" )+*
D))#*. )54 )'$9*" )+*"" * &+" *g
)+*b*4 &$ $8,"/*.& )+*." *"." *)'1)#& $*"= *""$.%"/0)'" )+*" <$+)' &F")'&+"")+*D",$NP!" * $*))P *"
s+&F" * 9*5",$ *"" *$&5* * *$ )+*" *"!# &5*$"=P[Q*cI[scI\T^?a&^?cI` dj?S G[mc'VQ jkP\ [$P'Q*P_a&^?cI` S\ aXjeP\*V^ 'P_`La*S 1
z<>?sq?sL:=<>:@?8 AC ( *O _ZdHJILKKI'4 AZD 7→ JD, UODK
zJv zJnY]¨uw^s]zJn*uwvm]kyRl>n,nYr] s utRtU@mu,pEEl>n D 7→ JD,UODK ]_^p0n ]op">]> /t%l1 e+ V)(F$'&)@ . VMA( (FH,3&)(?(F.3+8,.Wn/&8.-I4.>
nV)(F.3TR24.3(F>!+)2Q@ .A&f N& N).>F>F.32Q+ >B$'+\5,AI4,3&)IJAP)I4.>GjLVMA(?5?2Q(*N). JD,UODK Y
z<>?sq?sL:=<>:@?8 ACPD ( 4VR",ILK MPO\;u"zdy[ Rz[]0tvsl uRwU^Lp.]a]_^p¦^mr.tuwvmr] $Jt¦ldzJvkp ldzJ^2yR]f^^m]fUnR^
D1]>p
D2^zJvkzJn] f5]Juw^m]
zJn*uwvm] l>n£u $
D1 = D2 ⇔ ∀E, JD1,EK = JD2,EK
/t%l1 #`a.#Vd$'2Q+\5S2QTRVd$'(?5BA+\59a,<;=.>F5Gn/&8.#Vd$'&)(v5F$'&)5N).>B>F.32Q+D9I:;=$'V)Vd$<>BA+/5&)+)2Q@ .3(F>F.3I
UOD
N)H3V.3+8N &)+)24n/&8.3T_.3+\5 N).>#2Q+\5F.3(BA',35?24$'+8>!N).DY $'2Q5 D1 6= D2
N).3&f N).>F>F.32Q+8>R>?&)(#&)+8.T3T_.SPMA'>F.G&)+MA2Q(F.<Y 2 Scal∗ (D1) 6= Scal∗ (D2)
AI4$'(F> qD1,UDβ⊥(1)
y6=
qD2,UDβ⊥(1)
y Nb;OAV)(F^>SIJA!V)(F$hVd$<>?2Q5?24$'+ Y #VMAq .SV)(FH,HN).3+/5F.<Y 2 Scal∗ (D1) = Scal∗ (D2)
AI4$'(F>UOD1 = UOD2
YD`ApV)(F$'Vd$<>?2Q5?24$'+ Y V.3(?T_.35_N).,$'+8,3IQ&)(F.<Y7_o 5WAPCEDM $ " %/7)#$9* $)+*: )+*"" * &+"J &E"3! )>o )+* )+*:%,")'$"D'rK$*, &5* $%+$(*)'e * $ *"L" *)'1)#& $*" *b%!)#*$N* * *$ )+*: & *U%+$L" %/7)#$9* $ *" )+*"" * &+"" &f9*1)+ *5"! 7)&"= *)'"" &; * $ ;4 o4 ) 7)&"p1) )#& &
D1 = D2 ⇔ (∀E,D1 ⊥ E⇐⇒ D2 ⊥ E)
R&+" &FJ 7)#$*;:*1s *J *" )+*"" * &+" & % )#&K)' * *$n(0)' : )%l& &9L=I7) (+* ! o4 $7)&" )54 <$3 % (>s)':L=
iShaTa&Sp[QTcI[mc'\*^?a*^ecI` [sSV6a(a&QTS<QTS`"c'Q*d hSz<>?sq?sL:=<>:@?8 AC* ( 4VR",ILK MPO V,]AYZ[QwAYZ[QQZRKUO,Q 7OKdZhQZ*XQ¡n@+4[Q
l>hpα]>pα′ y,]>z| ^msuw=uwvm]f^ky7U^LpEUnYop^xy uwnR^ ]0; 1]
l>hpD1
]opD2
yR]>z> yR]f^^s]_Un,^yR] Juw^m]zJn*uwvm]
β
\¦]f^yR]_^%^m]_nR^D1
]>pD2
^ml>nprf[u,z|5^%9]>pq^m]ozJE]f5]fnYpq^% $qD1,UDβ⊥(α)
y=
qD2,UDβ⊥(α)
y∧
qD1,UDβ⊥(α′)
y=
qD2,UDβ⊥(α′)
y
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. , "
! $'2Q5 D&)+ N).>F>F.32Q+-N).PMA'>F.
βY
`a.>G,$.01,324.3+\5F>c =
qD,UDβ⊥(α)
y .35c′ =
qD,UDβ⊥(α′)
y >F$'+/5G24N).3+\5?24n/&8.>j_VMA(?5GI4.>>F,AIJA2Q(F.>7V)(F$@ .3+MA+\5GN).>N).>F>F.32Q+8>7&)+)2Q@ .3(F>B.3I4>*n/&)2>F$'+\57TU&)IQ5?2QV)IQ24H>*VMA(α′/α
Ye(
α′/α.>?5DN2dH3(F.3+/5 N).
1N)$'+8,zN8A+8>
c92QI).>?5 V$<>B>?2QP)I4.zN).z(F.3VH3(F.3( I4.>D>F,AIJA2Q(F.>DV)(F$@ .3+MA+\5N).
DCI4.>>F,AIJA2Q(F.>724N).3+/5?24n\&8.>N8A+8>
c′X N).>7>B,AIJA2Q(F.>7V)(F$@ .3+MA+/5vN&-N).>F>B.32Q+!&)+)2Q@ .3(F>B.3I:Yuv;=.>?5I:; &)+)24n\&8.e2Q+)q<(FHN24.3+\5n/&)2d>?&0_5zjU,AI4,3&)I4.3(
DCEN8A+8> IJA[V)(F$'V$<>F2Q5?24$'+ Y VMAq .vV)(FH,HN).3+\5F.X $'+V.3&)5,AI4,3&)I4.3(eVMA(7(FH,3&)(?(F.3+8,.[,BKMA'n\&8.[+)2Q@ .A& N).
DY
ACD )U$8%," & & $.%..% )+*$ *]9$$9*X*] *")+*"" &+" )+*1P.)&" *β",$ " %/7)#$.%"
7)#$ a&cIV6a&S[$P_^eQ*S )+* )+*"" * &+"R6&7)#* $8" *")#&F"&o" "$ 1) P.)&" *β⊥ =V) $%," && L= 7) (+*
&,*$ $* 9$ $T)#&X" &1s * *$ ;4 &2*;: &F"0)#& * 7)#& $* +"$$& P * * 9* $30)#& " *+"UDβ⊥(1)
*UOD
(+& " %/7)#$T)#& =
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α*α′ * +" &)% $* $i *"] *"!# &5*$" c *
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)'$ )+*X1)+<$3+)' &F")'& )' $.%" **)'D,UDβ⊥(α+α′ ∗ i)
= *"" *)'1)#& $*"")' )+*"" * & &7)#* $8" * )+*<1)]8C<$3 *
2−n(α + α′ ∗ i) D",$ )#& % $*$7" )+*"
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f * $8 )#* )'H"" &g)' 7)&"") (+* ;4 & *"4%,"" & P * )+* $*"$* &)#$9*(*4 *"/7)9* )+*"" *)'1)#& $*"9#)+*" *+" *P *"1P **)'$Qs/"I*&7"p*b*4 &$ * $*))' *
]0; 1[ *S7)#$2*;:*1s *] *" $T)'&$ *"
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: * *C8h)' & *(an)n∈N
*"(an)
",$ )#& % $*$" *l+);BC<$.%"I7)#$ 12
%+$6 *&Cl&y% )54 *$$9* * :L=
%$#h! & ) 2+ .+ P\Ta&P^ej?^?aX\Ta*^e[V7jeS LVS<j ^e`Fa&ShQ*\TSdha*^ecI` V` S`7\*SU!jeS S Q \*SP'V 17)'& P *"p`cIQTU P_je^e\TS
S`j ^e`Fa*SQ*\TSda&^ec'` Sh\C`cIQTU P'j?^e\*P_a&^?cI`\ SdhS\CQ \TSqP'V P[mcI`7`S`c'a*^ecI` ^e`Fa&ShQ*\*Shdha*^ecI`Sh\Ta;dhSj?jeS ^?`Fa&SQT\*Shdha&^?cI` [jeSh^e`S 3547698;:=<>:@?8BAC03D ( MPTWVR ILKUKUX KNI'4~VYXNZRKORZ
]>z>WyR]_^%^m]fUnR^^mlonYp5EqheEi¦IJ i9cbg7J E`Ycfe *hjaEl>v%^ ,z UU^ ^ml>npt@]_Un]_]_np9Un_hzw^ay uwnR^zJnf5]y,]_^^s]_nP
zR>?|q?sL:@<o:E?8BAC*A ( #MPTWV,,ILKYKX KI'4Z7IORMTYX KNQ,ILKNMPO l>"p
R ⊆ R′ n£uJRK ⊆ JR′K
! e+ &)5?2QIQ24>F. I4. ,$N8Aq .<Y $'24.3+/5 R1 ⊆ R′1
N).3&f (FH>B.A&fdY $'24.3+\5 R2.35
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R2 ⊆ R′2
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R n u
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JR1, . . . ,RnK = JJR1K , . . . , JRnKK
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∀i, Di 4 D′i =⇒ JD1, . . . ,DnK 4 JD′
1, . . . ,D′nK
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Y4;!ADC;1YfF4>1 $B|+$,|F4SBpKY4C6AOC;Y b& LKJ!X> ¡G\$T4CVL`>HNO],>HAuF>¢*@;1Y4POC6F4SN?>N)£eT4Y4>[LKJ!N?ADC6>pKYC>FR:5T4YF4>P?PO>C6YqL4N?;<KJ1<CUC6P?AO>.@;],]/>.TYq>YPO>]q<4U>vF4>vF4>P?PO>C6Y4PP?C],L4U>HPQ]T4Y4CFR:5T4Y4>.¢*L4N?;<KJ1<CUCMADS£b¤ Y¥Y>s@H;Y4P?CF4^HNO>[\$T4>[F4>PF4>P?PO>HCY4P@;],L4U6>HADPu@J1NuT4Y~<4NDJ!Y4@I4>],>YA
s1 . . . . . . . . . sn
YR:5>P)A2LKJ1Pd¦CP?C6AOStFKJ1Y4PuTY4>CYAD>HNDJ1@BADC6;Yg@;Y¦1>NOX1>YAD>|U6>tL`;1CF4PdF>U0JqY4;1NO]/J1UC6PDJ!AOC;Y,>P)A1¡§P?CRU0J
P?;]/],>tF4>HPsi
>HP?AF4C6¨`SN?>YAD>[F4>1|LNO;L©;POCMADC6;Y~ªb L4J1X> ¬« ¡Bb
¤ Y¥@H;],]/>HY4@>LKJ!NuT4Y¥L©>HAOC6A2U6>],]/>tP?T4N2U>HPuPOT4CMAD>P2N?S>U6U>HPuF4>[PO;],]/>LNO;@I4>[F4>1b
®s¯°©±.°©²i³µ´³¶°©·¹¸Qº|»¼l½.¾©¿&ÀÁ$ÂfÃWÄÁKÅMÅ&Á$ ƩÁzÂHÇQÈzÈgÁÊÉKÃWÇVËÌ4Á¥Æ`Á1ÍÏÎÐÒÑ
(ai) Ó Ñ (bi) Ô4ÓÕÖ~×Õ ÐÒÑ Ó×2Ø4Ù Ð Ó×eÔ4ÓÚÛÓÜ6×tÔÝ!Ù4× [0; 1] Ô4ÓÞßDÞÓÜ Î ÙàÏÕÓ¬Õ1Ú n áÍÏÎÐÒÑε ∈ ]0; 1] á ÍÐâ ∑
aibi ≥ 1− ε2â ∑
ai ≤ 1Ý!Ü Î Ú?× Ð ÜZÓOÖ Ð × Ñ Ó I Ñ ÓDÜäãÕÓ
∑
i∈I
ai ≥ 1− ε Ó Ñ ∀i ∈ I, bi ≥ 1− ε
å ACIJ58Ed?B<T:G?HDFI/5sæ[ç |keRI = i | bi ≥ 1− ε
^ ç |keRA =
∑
i∈I ai
^è gOj
1− ε2 ≤∑
aibi =∑
i6∈I
aibi +∑
i∈I
aibi ≤ (1− ε)(1−A) + A = 1− ε + εA
=|kgW−ε2 ≤ −ε + εA
=|kgWA ≥
ε− ε2
ε= 1− ε
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c := ^ | (s.0, . . . , s.0, s.c, s.0, . . . , s.0)
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β Õ1Ù`Ó u Ý!×ÓqÕ1ÙÏÝ Ð ÚBÓá ÍÏÎÐÒÑ D Õ!Ù Ô4ÓD×O×Ó Ð Ù SwÚ Îvu Ý u!Ð Ü Ð × Ñ ÓeÔ4Ó u Ý!×BÓ β áa Ù`Ó ± r ?CM¸?OM C Ô4Ó D ÓD× Ñ Õ1Ù¹ÓDÙ4×BÓDÞ u Ü|ÓØ4Ù Ð Ô4Ó/Ô4Ó×?×BÓ Ð Ù4×,ãÕÝ× Ðji × Ð ÞySwܶÓ×Ø4Ù Ð × Ð Ùµ8DÜÒÕ1×ÔÝ!Ù4× D
Ñ ÓDÜ×ãÕÓ× Ð X ∈ C Ó Ñ X ′ ∈ C Ý U ÓG8 X 6= X ′ Ý!Ü Î ÚO× Ð Ü×w¶ Î Ù Ñ ÝÕÞ ÎÐ Ù4× Õ1Ù8g ÎÐ ÖÐ Ùµ8 Î ÞySRÝ ÑoÐNu Ü|Ó¬áÍÏÎÐÒÑ
E Õ1ÙzÔ4ÓD×O×BÓ Ð ÙzÔ4Ó u Ý!×BÓ β⊥ á a Ù`Ó (α, β) E± r ?4M¸?OM|At±OM E Ó× Ñ Õ1Ù`Ó"8 Î Õ U ÓÚ Ñ Õ1ÚBÓ CÔ4Ó D
Ñ ÓÜ6Ü|ÓãÕÓâ ∀X ∈ C, W(JOs(X),EK) ≥ αâ ∑
X∈CW (X) ≥ β
~<¯n¥¸Zº[Z§dK¨ne_;;¬XD¢@ K|vµFW9 w
(α, β)O1X|vD"
DY
E;BvvwyX+tDO1FW
D
(1− β)t; BKtd
E
(1− α)t;
' (+*·Z³µ´³|°©·¹¸Zº~¼ \Á$ÂÂHÁK¿[/ Æ©ÄËÃ?¿'À/ÉC3Ïþµ/4ÁzËÇQ¾ÁÏÃoÀ¬¾wÃWÁÍÏÎÐÒÑ
D Õ1Ù Ô4ÓD×O×BÓ Ð Ù Ó Ñ C Õ1Ù`Ó8 Î Õ U ÓDÚ Ñ Õ!ÚBÓyÔ4Ó D á 7 Ó§bMdBBMdDFe DC ± r ?CM¸J¨A C Ó× Ñ Ü|ÓSwÜÒÕ1×SÓ Ñ|ÐÒÑ Ô4ÓD×O×Ó Ð Ù Ð Ùµ8DÜÒÕ1×ÔÝ!Ù4× D 8 Î Ù Ñ ÓÙÏÝ!Ù Ñ C á' (+*·Z³µ´³|°©·¹¸Zºz¼flÃkÆZÃWÁz¾wÃÅ'ÁÂËÇQ¾ÁÏÃoÀ¬¾wÃWÁÂyÁÀ ËÇV¾ÁKÃoÀ¬¾wÃÒÁ È 3C°K¿6È 3wÅ'Á7 ÓD×"8 Î Õ U ÓÚ Ñ Õ1ÚÓ×e×Õ1ÚsÕ1Ù Ô4Ó×?×BÓ Ð Ù D SÓÕ U ÓÙ Ñ ß Ñ ÚÓÞgÕ1Ù Ð Ó×tÔ4ÓÜ q r bOM b DIeL±Kh?BD r e§áa Ù`Ó 8 Î Õ U ÓDÚ Ñ Õ1ÚBÓ C ÓD× Ñ HA³DIHAvK|M PS Î Õ1Ú DC R,× Ð S Î Õ1Ú Ñ*Î Õ Ñ Ó 8 Î Õ U ÓÚ Ñ Õ1ÚÓ C ′ 8 Î Ù Ñ ÓÙÏÝÙ Ñ× Ñ Ú Ð 8 Ñ ÓÞgÓÙ Ñ C !
DC ⊂ DC′
®s¯°©±.°©²i³µ´³¶°©·¹¸Qº[Z ¼#"c°K¿&ÂHÀÁ/KËÁzÆ`ÁzËÇQ¾ÁÏÃoÀ¬¾wÃWÁÊÈ 3C°K¿6È 3wÅ'Á$ Î Õ1Ú ÑkÎ Õ Ñ Óx8 Î Õ U ÓDÚ Ñ Õ1ÚBÓ C ! Ð ÜZÓ?Ö Ð × Ñ ÓÕ!Ù`Ó8 Î Õ U ÓÚ Ñ Õ1ÚBÓeÞÝÖ Ð ÞÝ!Ü|ÓpS Î Õ1Ú DC 8 Î Ù Ñ ÓDÙÏÝ!Ù Ñ C á
å ACIJ58Ed?B<T:G?HDFI/5sæ&% rg 'jZhCi i=g6afgWc-afp)(=rsavWg=e=a+*d|ki=~;aflli=-adc q/|ki=DC
*d|kg -afgSj>g C^
®s¯°©±.°©²i³µ´³¶°©·¹¸Qº[k ¼-,dÇV¾ÁKÃoÀ¬¾wÃÒÁÍÏÎÐÒÑC Õ1Ù`Óx8 Î Õ U ÓÚ Ñ Õ1ÚÓ¥S Î Õ1Ú[Õ1Ù Ô4ÓD×O×BÓ Ð Ù_8 Î ÞSwÜ¶Ó Ñ D PF8 q ÓD× ÑIi-,di Ô Ð ÚÓ DC = D R¬á/.uÙ¥Ý Vâ Ñ*Î Õ Ñ × Î Õ1× i ÓDÙ4×BÓDÞ u Ü|Ó[Ô4Ó C ÓD× Ñ Õ1Ù`Óx8 Î Õ U ÓÚ Ñ Õ1ÚÓ0â ∑
X∈CW (X) ≤ 1 0â × Ð C ÓD× Ñ ÞfÝ¬Ö Ð ÞfÝÜ|ÓuÝ!Ü Î Ú?× ∑
X∈CW (X) = 1 áå ACIJ58Ed?B<T:G?HDFI/5sæ&1 avq=-afpZesaf'q/|keRgC'adcl2*frj>eRB^
3j>4*f-|kesc-c j>g5*daw=aC 7→
∑
X∈C
W (X)
af+a6Gescl-afg5*daw i=gWa7*d|ki=~;aflli=-aup j6eRp j>rsa98SeRrcli;:'=abp|kg l-af rsavl-|kescle=<fpavq |keRg B^ç |keRCi=gWa4*d|ki=~;aflli=-a'p j6GeRp j>rsa@^>3j>cleRpZq=rRe=*feR-98Caf cj>gWc q afl-a'=a2?;fgWfj>rReR-A@*Bj>
Dadcl
*d|kpZq=rsafCB8S|kg6c-i=q=q/|@c-abrsadc2*EDW|keF6G(=eRgSj>eR-adc(pi, 1− pi)i∈1,...,n
^
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D"8qt8!0 *9!o (O¬ D"8 % 35" "7"8$%5 *9y
è ggW|k-apεi
i =
pi
c-eε = 1
1cle
ε = 01− pi
c-eε = −1è g6es=afgClenSaursadc4*EDW|keF6[=adc+~@j>lej (=rsadc q/|kr 'gW|kpZej>rsadcB^ ç |keR
Er afgWc-afp)(=rsau=adc+q |kr 'gkpadc
n~kj>lej (=rsadc
(pi)^ ç |keR
X ∈ C^ 1 avq=-|Gi=eR'=adc * DW|keF6=a
Xadcl i=g6q |kr 'Cgkpauafgrsadc
(pi)^
ç |keRFr afgWc-afp)(=rsau=adc q |kr 'gkpadc j>eRgWclejkc-c|*fesdc'j>i;6frsfpafg -c=abrj *d|ki=~;aflli=-a
C^
ç |keRP ∈ F
^ 1 avq/|kr 'gkpaPcH *fleR
P = pεP1
1 ∗ . . . ∗ pεPn
n
è g -d *fleRFafgq=rRiWclesafi=-cfj>q/adcur zj>es=ab=a'rjulj>gWc |klp j>les|kg6cli=eR~kj>gC-acle
εPi = 0
8|kg-afpZq=rj *da
PWj>gWc
FqSj> rsadc =afi;6Zq |kr 'Cgkpadc
P ∗pi
afP ∗ (1−pi)
^afl-a |kq/fj>les|kg -aflpZeRgWa*Bj>+afrRrsauj>i ?@pafg -aursaw=a?@- @3hCi=eadcl ( |klgW B ^ ç |keR
F ′ r afgWc-afp)(=rsau=abq |kr 'Cgkpadc | (=-afg iWcB^3j>w- *fi=l-afg5*da98rjc-|kpZpa[=adcvq |kr 'Cgkpadc
P=a
Fadc-b? j>rsa *dafrRrsaZ=adcbq |kr 'gkpadc
P=aF ′ ^ç e
P ∈ F ′ j>rs|k-c P cB *fleR P = pεP1
1 ∗ . . . ∗ pεPn
njB~;a *vq/|ki= -|ki=
i8εPi 6= 0^
3j> - *fi=l-afg5*da'cli=n8;jB~;a * rjbp j6GeRp j>rReR-+=a
C8@|kg| (=lesafg hCiWaq |ki= -|ki=
i ∈ 1, . . . , ncleP = p
εP1
1 ∗ . . . ∗ pεPn
n ∈ F ′ j>rs|k-c 8=q |ki= -|ki= i0 8 pε′P1
1 ∗ . . . ∗ pε′
Pn
n ∈ F ′
| ε′Pi =
−εPi
clei = i0
εPi
cleRgW|kgafl-a'- *fi=l-afg5*da-afq |@c-acli= rju-afp j>-h iWacli=eR~@j>g -a@^G`bafi;6 *EDW|keF6
(p, p)af
(q, q)q/afi=~;afgCfl-a
q/|@c-eRles|kg=gWdc c-|keR+c-dhCiWafg lesafrRrsafpafgCA@Fi=g * DW|keF6=fq afgW6 i=gj>i=l-a7* DW|keF6afg *dawcafgWc h i eRrg adcl q |@c mcle (=rsafpafg q=-dc-afg Wj>gWc+i=g
X ∈ ChCiWavclei=gOj>i=l-a7* DW|keF6adc-+Wj>gWc
XB
Wj>gWc i=gc-afgWcp
q q
p
|ki6Wj>gWc'r zj>i=l-aq
p p
q
c-|keR qSj>j>rRr=<frsafpafg p p q q
afl-a\gW|kles|kgcH f-afgWj>i;6 afgWc-afp (=rsadcv=a\q |kr 'CgkpadcB^ `vj>gWc *da)*BjkG-a98/rsa\qSjkc-cj ?;a\=aF
F ′-af~CesafgC\qSj>j>rRrsfrResc-af -|kiWc rsadc2* DW|keF6afgCl-awafi;6^è gafg *d|kg5*frRi= eRpZpdGej>-afpafg bh iWa
∑
P∈F ′
P = 1
=|kg5*∑
P∈F
P = 1
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/ !"#$#&%' () *,+.-/0"1"(2435"$6 "7"8%' *9af+afgGWg
∑
X∈C
W (X) = 1
~<¯n¥¸Zº[kdCtDB1w"tD;K XµIW2th; XKk;¢@ XC¬XG¨K;F ′wD;|KtxC w1XoD. F ^¬sv 4G ¨§BFÏB|2 dYGD
®s¯°©±.°©²i³µ´³¶°©·¹¸Qº ¸ ¼#"c°K¿&ÂHÀÁ/KËÁzÆ`ÁzËÇQ¾ÁÏÃoÀ¬¾wÃWÁÊÈ 3C°K¿6È 3wÅ'ÁÍÏÎÐÒÑ
D0 Õ!Ù Ô4Ó×?×BÓ Ð Ù Ø4Ù ÐäÐ Ùµ8DÜÒÕ1×[ÔÝ!Ù4× Õ!Ù Ô4Ó×?×BÓ Ð Ù D ákÜvÓ?Ö Ð × Ñ Ó,Õ1Ù`Ó28 Î Õ U ÓÚ Ñ Õ1ÚBÓsÞÝÖ Ð ÞÝ!ܶÓC Ô4Ó D
Ñ ÓÜ6Ü|ÓtãÕÓ DC = D0 áå ACIJ58Ed?B<T:G?HDFI/5sæ[ç |keR
Cr afgWc-afp (=rsa =adc+=adc-c-afeRgWc+hCiSjkclenmocleRpZq=rsadc'p j6eRp j>i;6eRg5*frRiWc+Wj>gWc
D0^
3j>+p j6GeRp j>rReR-v=a+*EDSjkhCiWav=adc-c-afeRgh iSjkc-enmocleRpZq=rsa98Gr afgWc-afp (=rsaCadcl (=esafgi=gWa+*d|ki=~;aflli=-a
@Frsadc+=adc-cafeRgWc+h iSjkc-enmocleRpZq=rsadc'c-|kgC+eRg5*d|kpZqSj>le (=rsadc=|kg5*vp j6eRp j>i;6 B ^1 j *d|ki=~;aflli=-a
Cadcl (=esafgp j6eRp j>rsa&*Bj>b-|ki=v=adc-c-afeRgh iSjkc-enmocleRpZq=rsa\eRg5*frRiWcWj>gWc
D0adcl
eRg5*frRiWc'Wj>gWc+i=g=adc-c-afeRg6hCiSjkclenmocleRpZq=rsaw=aC^
¤ ¨ +.! « . &#.¨® \.J.*" *$V>P§F4>P?PO>C6Y4PdF4SB¦>HU;L4L©SPrPO;YAdU6>Pv@;]q<4CYKJ!CPO;1Y4PGUCYSiJ1C6NO>PGF4>HPrF4>HPOP?>CYPdPOC6]/L4U6>PHbkU6PV=?;1T4>YA
T4Y~N1U>tC6]/L©;N?ADJ1YAuL©;T4NuU>HPu@;T¦>N)ADT4N?>P|L4N?;L`;1POC6AOC;Y b LKJ1X1> 1« ¡b' (+*·Z³µ´³|°©·¹¸Zºz¼ \Á$ÂÂHÁK¿[/ Æ©Ä ÁKÅ'ÇVÉwÉ©Äa ÙlÔ4Ó×?×BÓ Ð Ùb4MdK r JJ ÓD× Ñ Õ1ÙlÔ4Ó×?×BÓ Ð Ù ÔÓÜ0Ý9¶ Î ÚOÞgÓ c.(D1, . . . ,Dn)
Î Ü|ÓD× (Di) × Î Ù Ñ× Ð ÞSwܶÓ×iá~<¯n¥¸Zº ¸dC¦yµFWth; XKk;¢yK1XKFW/F ^ t;BXw¥ |©*X;Gtd^¬C" n dD; wµ;;¬XCxw;
c.(D1, . . . ,Dn) $QJzF4SBpKY4C6AOC;YÊPOT4CM¦!J1YAD>fF4;1Y4Y4>fT4Y ]/;¬j>HY F4>fLKJ!POPO>HN FR:5T4Y F4>P?PO>HCY pKY4CrmzT4Y F>POP?>C6Y p4Y4C
F4SB¦>HU;L4L©S1b#ä;1T4NUJ~],;1AOC6¦>HNhVT4YÊ>Bx>],L4U6>/F4>,F4>HPOPO>HCY >BAF4>,F4>HPOPO>HCY FSH¦>HU;LL`S/>P?AFR:'J1<©;NOFF4;1Y4Y4S
(+, ξ, 1; 2)
(−, ξ.1, 0)
0.1
z
0.9
zξ.1.0
(−, ξ.2, 0)
0.4
(+, ξ.2.0, 0)
0.5
z
0.1
0.45
z
0.55
(+, ξ.2.0, 0)
!
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D"8qt8!0 *9!o (O¬ D"8 % 35" "7"8$%5 *9y
0.1
0.9
0.4
(+, ξ, 1; 2)
(−, ξ.1, 0)
zξ.1.0
(−, ξ.2, 0)
(+, ξ.2.0,0)
0.5
(+, ξ, 1; 2)
(−, ξ.1, 0)
zξ.1.0
(−, ξ.2, 0)
z
0.1
0.45
(+, ξ, 1; 2)
(−, ξ.1, 0)
zξ.1.0
(−, ξ.2,0)
z
0.55
(+, ξ, 1; 2)
(−, ξ.1, 0)
zξ.1.0
(−, ξ.2, 0)
(+, ξ.2.0, 0)
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/ !"#$#&%' () *,+.-/0"1"(2435"$6 "7"8%' *9' (+*·Z³µ´³|°©·¹¸Zº|» ! ¼ \Ä ÁKÅ'ÇVÉwÉ©ÁKÈgÁ/ÀsÆpo¾µ/ Æ`Á$ÂÂHÁK¿[/L/©¿a Ùb4MdK r JJ¥MdHºMdep¸Ô q Õ!Ù Ô4Ó×?×BÓ Ð Ù Ø4Ù Ð 8 Î ÞSwÜ¶Ó Ñ D Ó× Ñ Õ1Ù©Óx¶ Î ÚOÞgÓeÙ Î ÚOÞÝ!Ü¶Ó , SVÝ!Ú ÑoÐ ÚÔ4ÓD S Î Õ!ÚuÜ|Ó×× Ñ ÞgÓsÔ4ÓÚBÛÛY8Ú ÐÒÑ Õ!ÚBÓ¶ Î ÚOÞgÛâ Ô4ÓܵÝqÚ àܶÓ
(+, ξ, I)
(−, ξ.i, J)
s1
E1
. . . . . . . . . sn
En
R
s1
(+, ξ, I)
(−, ξ.i, J)
E1
R
. . . . . . . . . sn
(+, ξ, I)
(−, ξ.i, J)
En
R
â Ô4ÓܵÝqÚ àܶÓ
(+, ξ, I)
E′1 . . .
s1
E1
. . . . . . . . . sn
En
. . . E′k
s1
(+, ξ, I)
E′1 . . . E1 . . . E′
k
sn
(+, ξ, I)
E′1 . . . En . . . E′
k
â Ý Ð Ù× Ð ãÕÓ!©× Ð D Ó× Ñ Ù©ÛDàÝ Ñ|Ð ¶Q×iÕ!ÚÜ0Ý u Ý!×BÓ ξ0 ` Λ !VÔ4ÓÜ0ÝqÚ àÜ|Ó
(−, ξ0, J)
s1
E1
. . . . . . . . . sn
En
R
s1
(−, ξ0.i, J)
E1
R
. . . . . . . . . sn
(−, ξ0.i, J)
En
R
~<¯n¥¸Zº[¯Rµh C1*B"1 ;;¬X
DBn¥wGDt K=2
D Q BEXxh;Xµ|vDsI ^ y+dCx µFWth; XKk[kXXw d C1kBv CKxh 1vµu;B¦Bn¦Cd h+ n
B CÏBXK;¢¦<Bvµk§1_wGD BIWC1 C1*Bh 1
¢2¢ ¢ h ¢ ¢ h + nXD µpBIWBkXXKBµdK dCOWdX K
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D"8qt8!0 *9!o (O¬ D"8 % 35" "7"8$%5 *9yt ;¢.v¬B1X;+KXXK¨ ¬;;¬Xr;BGWXn¬;;¬X_B1X;+
®¯°Ï±.°©²i³¶´¬³|°©· ¸Qº[¯n¼,rÇc/ÂÁKà 3Ài¿&Ç/ ÆZ¾ É©ÇR¿ Æ`ÂÉC3ÏÃÆ©Ä ÁKÅ'ÇVÉwÉ©ÁKÈgÁ/À$ Î Õ!Ú Ñ*Î Õ Ñ Ô4Û U ÓÜ Î S4SÓÞgÓÙ Ñ D′ Ô q Õ1Ù Ô4Ó×?×BÓ Ð Ù Ø4Ù Ð 8 Î ÞSwÜ¶Ó Ñ D ! Î Ù ÝyS Î Õ1Ú ÑkÎ Õ Ñ E Ô4Ó u Ý!×BÓÎ Ú Ñ g Î à Î ÙÏÝ!Ü|ÓxV
W (JD,EK) =W (JD′,EK)
å A%I/5Ed?B<>:G?HDFIJ5qæ è grsa ~;flenSa *EDSjkhCiWa fj>q a\GiO=f~;afrs|kq=q afpafg B^ 1 auq/|keRgC *fli5*fej>radclh iWaq/|ki= -|ki=2(=j>g5*EDWafpafg
s1 . . . . . . . . . sn
rj[c-|kpZpaw=adcsiadcl+? j>rsa
1^
~<4¯%cn¸Qº x ;K ;¬"[t^B|4"1);;¬Xz;B§XKXFd ntB¢§CXFXDCv¬;;¬X K XµF DBKtd ¬;;¬X
E 0 D ;B¦¤i
0.2
0.5
¬B1X;+KXXKD;B 0.2
0.5
Bn;BC Ktd E KFyskkK*XBFW¬
D11[¬1IDy;;¬XÊ;BBn¦Cd
0.2 + 0.5 = 0.7 d
TY4>[L4NO;=?>H@HAOC;Y~POC6]/L4U6>eFKJ!Y4PD@H;NON?>P?L`;YF¥L4U6T4POC6>T4N?P2L4NO;=?>H@HAOC;Y4PuP?C],L4U>HPh4U6>Pwh;XF1+Bb
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1, !"#$#&%' () *,+.-/0"1"(2435"$6 "7"8%' *9' (+*·Z³µ´³|°©·¹¸Zº|»Ï»g¼GÄ¿ ÆQ¾4ÂÉC3KÃÆ`Ä ÁÏÅ'ÇVÉwÉ©ÁKÈgÁ/ÀÍÏÎÐÒÑ
D Õ1ÙfÔ4Ó×?×BÓ Ð Ù/ØÙ Ð 8 Î ÞySwÜ¶Ó Ñ á a Ù)bCMdK r JJ¥MdHºMdep¸¤Mde B =@?tMde±OMqÔ4Ó D Ó× Ñ Õ!Ù`Ó§×iÕ ÐWÑ ÓD = D1 . . . Dn = D′ 6
7 ÓD×¥ÚÛ× Ð Ô©Õ1× ri(X) Ô q Õ1Ù ÔÓ×O×Ó Ð Ù ãÕÝ!× Ðji × Ð ÞSwÜ|Ó X Ô4Ó Di
, Ñ ÚÝ U ÓDÚO× Õ!Ù`ÓÊÛ Ñ ÝGSÓ Di
Di+1 × Î Ù Ñ Ô4ÛØÙ Ð ×Ý Ð Ù4× Ð VpP Î Ù~Ù`Ó8 Î Ù4× Ð Ô ÚBÓ[ãÕÓtÜ0ÝwSwÚBÓDÞ Ð ÚBÓÚ àÜ¶Ó !©Ü¶Ó×ÝÕ Ñ ÚBÓ×2Ú àܶÓ×2× Î Ù Ñ× Ð Þ Ð Ü0Ý Ð ÚÓ×RáÍÏÎÐÒÑX Õ1Ù Ô4ÓD×O×BÓ Ð Ù ãÕÝ!× Ð^i × Ð ÞSwÜ|ÓtÔÓ Di á
(+, ξ, I)
(−, ξ.i, J)
s1
E1
. . . . . . . . . sn
En
R
s1
(+, ξ, I)
(−, ξ.i, J)
E1
R
. . . . . . . . . sn
(+, ξ, I)
(−, ξ.i, J)
En
R
7 ÓD×ÚÛ× Ð Ô©Õ1×Ô4Ó X × Î Ù Ñ Ü|ÓD×tÔ4Ó×?×BÓ Ð Ù4×ã4ÕÝ!× Ð^i × Ð ÞyS`Ü|Ó× X ′ Ô4Ó Di+1
Ñ ÓDÜ×eãÕÓâ X Ó Ñ X ′ × Î Ù Ñ ÛDà$ÝÕÖ×Õ1Ú2ܵÝ"SVÝ!Ú ÑoÐ ÓÓÙ Ô4Ó;g Î ÚO×tÔ©ÕyÞ ÎÑ|Ð ¶*áâ × Ð X Ù q Ð Ù Ñ ÓDÚO×ÓG8 Ñ ÓSRÝ!×2Ü|Ów8g ÎÐ Ös1 . . . . . . . . . sn
ÞfÝ Ð × Ð Ù Ñ ÓÚO×ÓG8 Ñ Ó (+, ξ, I)
R
Ý!Ü Î ÚO× X ′Ð Ù Ñ ÓÚO×ÓG8 Ñ ÓrÔÓ.ܵÝuÞgßÞgÓÞÝ!Ù Ð ÚÓdÕ!Ù/Ô4Ó× (+, ξ, I)
R
×Ý!Ù× Ð Ù Ñ ÓÚ?×BÓY8 Ñ ÓDÚܶÓ×Ý8 ÑoжΠÙ4× (−, ξ.i, J) áâ × Ð X Ð Ù Ñ ÓÚ?×BÓG8 Ñ Ó Ej0 !Ý!Ü Î Ú?× X ′Ð Ù Ñ ÓÚ?×BÓG8 Ñ Ó2Ô4ÓvÜ0ÝeÞßDÞÓGÞÝ!Ù Ð ÚÓ Ej0 Ó Ñ Ô4Ó§ÞgßÞgÓS Î Õ1Ú
ܵÝ"SVÝÚ ÑoÐ Ó (+, ξ, I)
R
á7 ÓD×rÚÛ× Ð Ô©Õ1× Ri(X) Ô q Õ!Ù¥ÓÙ4×ÓÞ u Ü|ÓeÔÓeÔ4ÓD×O×BÓ Ð Ù4×ã4ÕÝ!× Ð^i × Ð ÞyS`Ü|Ó× X1, . . . , Xm × Î Ù Ñ Ô4ÛDØ4Ù Ð ×SVÝ!Ú[Õ1٠жΠ٬V ∪jri(Xj) á7 ÓD×ÚBÛD× Ð Ô©Õ1× R(X) Ô q Õ1Ù_Ô4ÓD×O×Ó Ð Ù_ãÕÝ× Ðji × Ð ÞySwÜ¶Ó X Ô4Ó D
, Ñ ÚÝ U ÓDÚO× ÑkÎ Õ Ñ ÓD×Ü|ÓD×/Û Ñ ÝGSÓ×/Ô4ÓÔ4Û U ÓÜ Î S4S.ÓDÞÓDÙ Ñ × Î Ù Ñ
Rn(Rn−1(. . . (R1(X)) . . .))
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D"8qt8!0 *9!o (O¬ D"8 % 35" "7"8$%5 *9y1«®¯°Ï±.°©²i³¶´¬³|°©· ¸Qº ¼GÄ¿ ÆQ¾4Â
â 7 Ó×2ÚBÛD× Ð Ô©Õ1×tÔ q Õ1Ù ÔÓ×O×Ó Ð Ù ãÕÝ!× Ðji × Ð ÞSwÜ|Ó Î Ù Ñ ÞßDÞÓ Î Õ u Ü Ð × Ð ÞSwܶÓáâ ´ Ó¦SwÜÒÕ!× !©ÜµÝ× Î Þ/ÞgÓÔ4ÓD×S ÎÐ Ô×ÔÓ×ÚBÛ× Ð Ô©Õ!×[Ô q Õ1Ù Ô4Ó×?×BÓ Ð Ù ãÕÝ!× Ðji × Ð ÞSwܶÓtÓ× Ñ ÛOàÝ!Ü|ÓeÝÕS ÎÐ Ô$×Ô©ÕgÔÓ×O×Ó Ð ÙVáâ $ Î Õ1Ú Ñ*Î Õ Ñ E Ó Ñ2Ñ*Î Õ Ñ D ! W (JD,EK) =
∑
D′∈R(D)W (JD′,EK) áå A%I/5Ed?B<>:G?HDFIJ5qæ è g6rsab~;flenSa * DSjkh iWaufj>q/av=aw=f~;afrs|kq=q afpafg B^
')(+*·ä³µ´³¶°©· ¸Qº|»vZn¼,rÇV¾ÁKÃWÀ¬¾`ÃÒÁ Æ©Ä5ÁKÅ'ÇQÉ`É©ÄÁa Ù`Óx8 Î Õ U ÓDÚ Ñ Õ1ÚBÓ C Ô q Õ1Ù ÔÓ×O×Ó Ð Ù D Ó× Ñ b CMdK r JJM × Ð Ü|ÓÔÓ×O×Ó Ð Ù DC Ó× Ñ ÔÛ U ÓDÜ Î SCS.Û¬á
®¯°Ï±.°©²i³¶´¬³|°©· ¸Qº¥¼,rÇQ¾ÁÏÃoÀ¬¾wÃWÁnÆpo¾µ/ Æ`Á$ÂÂHÁ4¿^/ Æ©Ä ÁKÅ'ÇVÉwÉ©ÄÍÏÎÐ ÓÙ Ñ D Õ1ÙÔ4Ó×?×BÓ Ð ÙgØ4Ù Ð 8 Î ÞySwÜ¶Ó Ñ Ô4Û U ÓÜ Î S4SÛeÓ Ñ E Õ1Ù¥ÔÓ×O×Ó Ð Ùf×iÕ1ÚeÕ1Ù`Ó u Ý!×BÓ Î Ú Ñ g Î à Î ÙÏÝ!Ü|Ó¬áÍÏÎÐWÑC Õ1Ù`Óx8 Î Õ U ÓDÚ Ñ Õ!ÚBÓsÔ4Û U ÓÜ Î S4S.ÛHÓS Î Õ1Ú D á/.dÙzÝ)Vâ W (JDC,EK) =
∑
X∈CW (JX,EK)â ∀X ∈ C, W (JX,EK) =W(JOs(X),EK)W (X)
å A%I/5Ed?B<>:G?HDFIJ5qæè gp|kg l-aursa7*Bjkc+|
Dadcl q |@cleRle l^
1 aw=adc-cafeRg DCadcl'j>iWc-cleq |@cleRle af+cB *fleR
DC = c.(Os(X1), . . . ,Os(Xn))^
è gOjJDC,EK = c.(JOs(X1),EK, . . . , JOs(Xn),EK)
adcl4*frj>eR&@ X ∈ C adcl'adc-cafg lesafrRrsafpafg bi=gWa (=j>g5*EDWa\=aucE*Bj>rj>eR-adccli=eR~esaw i=gO=adc-cafeRgcleRpZq=rsa B ^1 a *Bjkc |
Dadcl gW? j>le 8adcl i=g q/afleRq afi q=rRiWc *d|kpZq=rResh iW |kg q=-|* <d=a=a rjvpfpa p j>g=e=<f-a
p j>esc * DSjkh iWa afi=eRrRrsav~es=avGi6q=-afpZesaf2*d|a: *fesafg =aE^
®¯°Ï±.°©²i³¶´¬³|°©· ¸Qºn¼,rÇQ¾ÁÏÃoÀ¬¾wÃWÁnÆ`Ä ÁKÅ&ÇVÉ`É`Ä$ÁÍÏÎÐWÑ
D Õ1Ù Ô4ÓD×O×Ó Ð Ù Ø4Ù Ð 8 Î ÞSwÜ¶Ó Ñ á ÍÏÎÐWÑ C Õ!Ù`Ó (α, β)i 8 Î Õ U ÓÚ Ñ Õ1ÚÓs×iÕ1Ú D á kÜGÓOÖ Ð × Ñ Ó Cdev Ó Ñ
DdevC
Ñ ÓÜ×tãÕÓyVâ 7 ÓÔ4ÓD×O×BÓ Ð Ù DdevC Ó× Ñ ÜµÝ)8 Ð|u ܶÓqÔ q Õ1Ù Ô4Û U ÓÜ Î S4SÓÞgÓÙ Ñ ÓÙ ×ÛãÕÓÙt8ÓyÔ4Ó DC á ÍÏÎÐWÑ R ܵݶ Î Ùµ8 ÑoжΠÙzÔ4ÓÚBÛD× Ð ÔÏÕ1×Ý!×O× Î 8 Ð ÛÓ , 8ÓsÔ4Û U ÓÜ Î S4SÓÞgÓÙ Ñ ÓÙ~×BÛHã4Õ$ÓÙµ8HÓáâ Cdev ÓD× Ñ Õ1Ù`Ó (α, β)
i 8 Î Õ U ÓÚ Ñ Õ1ÚÓ×iÕ1Ú DdevC áâ ÍÐ X ∈ C !wÝ!Ü Î ÚO×dܶÓ×2ÚBÛD× Ð ÔÏÕ1×tÔ4Ó X × Î Ù Ñ ÔÝ!Ù4× Cdev áâ $ Î Õ1Ú Ñ*Î Õ Ñ E ! W (JDC,EK) =W
(qDdev
C ,Ey) áâ ∑
X∈CW (X) =∑
X∈Cdev W (X) áâ $ Î Õ1Ú Ñ*Î Õ Ñ E Ó Ñ2Ñ*Î Õ Ñ X ∈ C ! W (JX,EK) =∑
X′∈R(X)W (JX ′,EK) áâ ÍÐ C ÓD× Ñ ÞÝÖ Ð ÞÝ!Ü|Ó!ÏÝ!Ü Î ÚO× Cdev ÓD× Ñ ÝÕ1×?× Ð ÞÝÖ Ð ÞÝ!Ü|Ó¬á.dÙ~Ù ÎÑ Ó Dev(C) = Cdev á
å A%I/5Ed?B<>:G?HDFIJ5qæ è g *d|kgWcllli=eRwrj *d|ki=~;aflli=-aCdevqSj>wfj>q adcwGi=j>g rsaZ=f~;afrs|kq=q/afpafgC =a
Dafg q=-afgSj>gCZrsadc[-dclesGiWc=a rj *d|ki=~;aflli=-aq=- *dd=afg -a@^ adclafg5*d|k-ai=gWa *d|ki=~;aflli=-a @Frj
*d|kgWGeRles|kg\ eRg5*d|kpZqSj>le (=eRrReR- adc- (=esafg q=-dcafl~;da B ^T`ba q=rRiWc 8 *@ adcl8i=gWa *d|ki=~;aflli=-acli=rsa =adc-c-afeRg-dGi=eRB^
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!"#$#&%' () *,+.-/0"1"(2435"$6 "7"8%' *9
% r-adcl-a \q=-|ki=~;af Cdev
adc- i=gWa(α, β)m *d|ki=~;aflli=-a@^
adcl+~j>eqSj>+rj\q=-|kq |@cleRles|kg6]W^[qSj ?;awq=- *dd=afg -a@^ 3M|ki='-|ki= E 8 W (JDC,EK) =W
(qDdev
C ,Ey) ^
adcl+~j>eqSj>+rj\q=-|kq |@cleRles|kg6]W^[qSj ?;a _ByG_@^ ∑
X∈CW (X) =∑
X∈Cdev W (X)^
adcl+~j>eqSj>+rj\q=-|kq |@cleRles|kg6]W^[qSj ?;awq=- *dd=afg -a@^ 3M|ki='-|ki= E af+-|ki= X ∈ C 8 W (JX,EK) =
∑
X′∈R(X)W (JX ′,EK) ^ adcl+~j>eqSj>+rj\q=-|kq |@cleRles|kg6]W^[qSj ?;awq=- *dd=afg -a@^ ç e C adcl p j6eRp j>rsa98/j>rs|k-c Cdev adcl'j>iWc-clep j6GeRp j>rsa@^% r8cli;:=|kg5* =a p|kg l-afbhCiWaucle
Cadc-'p j6GeRp j>rsa98 j>rs|k-c
Cdevadclj>iWcclep j6GeRp j>rsa@^3M|ki=
*dafrj;8eRr/c-i;:=a p|kg l-af+hCi i=gWabfj>q/ab=a=f~;afrs|kq=q/afpafgCD D′ lj>gWc |klpa i=gWa *d|ki=~;af mli=-avp j6GeRp j>rsa
C=a
Dafgi=gWa7*d|ki=~;aflli=-awp j6GeRp j>rsa
C′=a
D′ ^ç i=q=q |@c-|kgWch iWaC′gWa c-|keRMqSjkcp j6GeRp j>rsa@^ ç |keR
X ′ν /∈ C′
-afrChCiWaC′∪X ′
ν
adcli=gWa *d|ki=~;aflli=-a=a
D′ ^è g q afi=-afp|kg -af Xν
afgj>lle=<f-abqSj>rsav=f~;afrs|kq=q/afpafgCB^X ′
ν /∈ C′=|kg5*
Xν /∈ C^ 3j>
p j6GeRp j>rReR- =aC8Xν
adcl *d|kpZqSj>le (=rsa jB~;a * i=g *daflj>eRgX0 ∈ C
^j>esc 8Hj>rs|k-c 8X ′
ν
adcl *d|kpZqSj>le (=rsajd~;a *\rsadcb-dclesGiWcb=a
X0=|kg5*
C′ ∪X ′ν
g adclqSjkcbi=gWa&*d|ki=~;aflli=-a@^ % r 'Oj *d|kg ljkGe=*fles|kg=|kg5*C′adc- p j6eRp j>rsa@^
¤ ¨ ª ) , )t® ,/*S)p®J* « kU>P?A]J!CYAD>HYKJ1YAL©;P?POC<U>vFR:'SHY4;Y4@H>N>BAF4>§F4S],;YADN?>NäU0J2L4NO;L©;P?C6ADC6;YsPOT4CM¦1J!Y$AO>r\$T4CP?AOCL4T4U6>
\$T4>F4>HTxgF>POP?>C6Y4PD>HA
EP?;YA2;N?AOI4;X;Y4J1TxPOCw>BAuPO>HT4U>H]/>HYA2POCw;1YgL`>HTA2@;T¦N?CN
DE|J1@>tm
EF4>[]/J1Y4C6^NO>tJ1T4POP?CQL4NO;@I4>[F4>1\$T4>FSPOC6NOSH>1b
®s¯°©±.°©²i³µ´³¶°©·¹¸Qº ¼-,¦3KÃ3KË1ÀÄKÃ?¿&ÂG3Ài¿&Ç/ Æ`Á Åo Ç`ÃoÀ¬Ì4ÇÏÇ/43wÅ ¿&ÀÄÍÏÎÐ ÓDÙ Ñ D Ó Ñ E Ô4Ó¬ÕÖ¹Ô4Ó×?×BÓ Ð Ù4×T8 Î ÞySwÜ¶Ó Ñ ×iá 7 Ó×/Ô4ÓD×O×Ó Ð Ù× D Ó Ñ E × Î Ù ÑÎ Ú Ñ g Î à Î ÙÏÝÕÖÊ× Ð Ó Ñ×BÓÕ!Ü|ÓÞgÓÙ Ñ × Ð
∀ε ∈ ]0; 1[, ∃C, Õ1Ù`Ó ((1− ε), (1− ε))− 8 Î Õ U ÓÚ Ñ Õ1ÚBÓsÔ4Ó D ¶|Ý8HÓ , E
å ACIJ58Ed?B<T:G?HDFI/5sæ è g6p|kgCl-a *EDSjkhCiWaveRpZq=rRe=*Bj>les|kg^⇒ç i=q=q |@c-|kgWc
D ⊥ E^
ç |keRε ∈ ]0; 1[
^ç |keR
D0i=g=adcc-afeRgWg=eq=rsafeRgWafpafg eRg5*frRiWc'Wj>gWc
D-afr8h iWa
W (JD0,EK) ≥ 1− ε2 ^ç |keRCi=gWa *d|ki=~;aflli=-a =a
Dp j6eRp j>rsa -afrRrsah iWa
DC = D0@Fr a6escl-afg5*da adcl jkc-cli=-da
qSj>+rj\q=-|kq |@cleRles|kg]W^ ]ZqSj ?;a _@_B B ^ç |keR
Cdev = Dev(C)^è g6gW|k-a
DdevC = Ddev
0 Cdev
^1 j[q=-|kq |@cleRles|kg6]W^zZqSj ?;avq=- *dd=afg -aueRpZq=rResh iWawhCiWa
Cdevadc- p j6eRp j>rsa@^
è gOj;8GqSj>+p j6GeRp j>rReR-u=aCdev
8
1− ε2 ≤ W (JD0,EK) =W(JDdevC ,EK) =
∑
X∈Cdev
W(JOs(X),EK)W (X)
è g%j ∑
X∈Cdev W (X) ≤ 1=|kg5*98 zj>q=E<dcbrj q=-|kq/|@cleRles|kg]W^R_\qSj ?;a_@_f]58/eRrMa6Gescl-a i=gWa
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D"8qV% "87!8¬ *(%%' (s% , ª*d|ki=~;aflli=-a
C0 ⊆ Cdev -afrhCiWa
∑
X∈C0
W (X) ≥ 1− ε
af
∀X ∈ C0,W(JOs(X),EK) ≥ 1− ε
è gafg *d|kg5*frRi= hCiWaC0adc- i=gWa
(1− ε, 1− ε)m *d|ki=~;aflli=-a =a
DdevC
^ç |keR
C′ = X | R(X) ∩ C0 6= ∅|
Radcl rj 3|kg5*fles|kg=a -dclesGiWcGi =f~;afrs|kq=q afpafg
=aCafCdev^
è g *d|kg5*frRi='afg6-afp j>-hCiSj>g h iWaC′adcl i=gWa
(1− ε, 1− ε)m *d|ki=~;aflli=-a =a
D*Bj>
∀X ∈ C′, ∀X ′ ∈ R(X), W(JOs(X), EK) =W(JOs(X ′), EK) ≥ 1− ε ∑
X∈C′W (X) ≥∑
X′∈C0W (X ′) ≥ 1− ε
⇐ç i=q=q/|@c|kgWc
∀ε ∈ ]0; 1[, ∃C,i=gWa
(1− ε, 1− ε)−*d|ki=~;aflli=-au=a
D j *da
E
ç |keRε ∈ ]0; 1[
^ ç |keRCi=gWa
(1− ε, 1− ε)m *d|ki=~;aflli=-au=a
D j *da
E^
è g *d|kgWcles <f-avi=gO=f~;afrs|kq=q/afpafgCCdev^è ggW|k-a
DdevC = Ddev
Cdev
^è gOj
W (JD,EK) ≥ W (JDC ,EK) =W(q
DdevC ,E
y)
≥∑
X′∈Cdev W (JX ′,EK) =∑
X′∈Cdev W(JOs(X ′),EK)W (X ′) =∑
X∈CW(JOs(X),EK)W (X) ≥ (1− ε)2
a *feadcl ~j>eq/|ki='-|ki=ε ∈ ]0; 1[
=|kg5*W (JD,EK) = 1
afD ⊥ E
^
Hg 2 Z g gg V B
>BAOAD>zPO>@BADC6;YaXSHY4SNOJ1UC6PO>~U>~L4U;YX>],>YA/@iJ1Y4;1Y4C\$T4>¥F4SHpKYCtm UJ P?>@HAOC;Y « b « b LKJ1X> ªb$G:5;<=?>H@HAOC6Eu>HP?AFR:'>xL4UC6\T>NsU0JP?ADN?T4@HAOT4NO>/>HAY4;YÊF4>/FSHpKY4C6NF>Ps@I4;PO>HPqAD>H@I4Y4C6\T>],>YATAOCU6>PLKJ1N2U0JP?T4C6AO>1b$V>t XXK0t +wXd ^GdKäF4SHp4Y4C`F4J1Y4Pv@>HA?AD>PO>@BADC6;Y/>HP?ArT4Y4>2E¶;Y4@HAOC;Y,F4>HP§F4>HPOP?>CYP
POC],L4U6>PZFKJ1Y4PäU>PäF4>P?PO>HCY4PLNO;<KJ!<4CU6CP?AO>PQF4>d]],>v<KJ1PO>§\$T4CN?>HAO;T4NOY>dT4YF4>HPOP?>CY \$T4CY4;N?]J!UCP?>J¬¦1>@sL4UTPF>sF>POP?>C6Y4P\$T4>[U6>F4>P?PO>C6YzF4>[F4SHLKJ1N)Aib
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!"#$#&%' () *,+.-/0"1"(2435"$6 "7"8%' *9' (+*·Z³µ´³|°©·¹¸Zº|»kn¼ zdÅ&Ç/ ©ÁKÈgÁ/À,É`Å6¾ÂT/4Ç©ÃOÈ 3`ÅM¿&ÂG3n/À7 Ý& r eL±¸hD r e φp Ô4Ó×rÔ4ÓD×O×Ó Ð Ù×v× Ð ÞSwÜ|ÓD×vÔ q Õ1Ù`Ó u Ý×BÓuÔ Î ÙÙ`ÛÓuÔÝÙ4×.Ü|ÓDקÔ4Ó×?×BÓ Ð Ù4×SwÚ Îvu Ý u!Ð Ü Ð × Ñ ÓD×Ô4ÓÞßDÞÓ u Ý×BÓ[Ó× Ñ ÔÛØ4Ù Ð ÓSVÝ!ÚxV
φ0 : K+
D1 . . . Dn
7→ K+
D1 . . . Dn
φ1 : K+
D1 . . . Dn
7→ z
φp : K+
D1 . . . Dn
7→
1− p
K+
φp(D1) . . . φp(Dn)
p
z
φp : (K−)
D+
7→ (K−)
φp(D+)
φp : Fid− 7→ Skunk
®s¯°©±.°©²i³µ´³¶°©·¹¸Qº|» ! ¼-lfÃWÀ¬ÌÇ©Ç/C3wÅM¿'ÀÄ~Á$À/ÉwÅ'Ç/ ÏÁKÈgÁ/ÀyÉwÅM¾4ÂT/Ç`Ã?È 3wÅ ¿ ÂY3n/ÀÍÏÎÐ ÓDÙ Ñ D Ó Ñ E Ô4Ó¬ÕÖ Ô4Ó×?×BÓ Ð Ù4××iÕ1ÚtÔ4ÓD× u Ý!×BÓD×[ÔÏÕÝ!Ü|ÓD×iá/.dÙ¥Ý V
D ⊥ E⇔ ∀p, W (Jφp(D),EK) = 1
D 6⊥ E⇔ ∀p < 1, W (Jφp(D),EK) < 1
$V>qL4U6;Y4X>H]/>HYAtL4U6T4PeY4;1NO]/J1UC6PDJ1YAφp
>HP?AeT4Y4>Y4;!ADC;1YzNDJ!CPO;1Y4YKJ1<4U6>FTnL`;1CYAeF>q¦T4>F4>U0JU6T4F4C\$T4>L4NO;1<KJ1<4C6UCP)AD>@iJ1Nt>HUU6>qL©>TA BADN?> CYAD>HNOYKJ1U6CP?S>q@H;],]/>U6>q],;YADN?>YA[U0JFSHpKY4CMADC6;Yn>HAtU0JL4N?;L©;POCMADC;1YfPOT4CM¦!J1YAD>1b' (+*·Z³µ´³|°©·¹¸Zº|»¸ ¼ Á|3v° ÉwÅM¾4 /4Ç`Ã?È 3wÅ ¿ ÂY3n/À7 ÓA|A³ JLK;?4B§e r HACKFDFB ACep¸ Fax
pξ`ξ′ ÓD× Ñ φp(Faxξ`ξ′) á
$QJL4NO;1L`;P?C6AOC;Y/POT4CM¦1J!Y$AO>tNO>H]J!NO\$T4>\$T4>U0JsE¶;Y4@HAOC;Yφp
>P)A(+*·ä³|²i²G>HYgUT4F4C6\$T4>tL4N?;&<KJ!<4CU6CP?AO>tJ1TP?>Y4P\$T4>1hÏ],;F4TU;T4Y>sY;NO]/J1UC6PDJ!AOC;YJi¦>@
Faxξ`ξ′Uo:'C6]J!X>eLKJ!N
φp
FR:'TY¥F4>HPOP?>CY>HP?APO;Y~C]/J1X>eLKJ1NuU0JY;NO]/J1UC6PDJ!AOC;YJi¦>@sU>tE|JxL4UTPuY4;NO]/J1U6CPDJ!Y$A
Faxpξ`ξ′
b
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D"8qV% "87!8¬ *(%%' (s% , ®¯°Ï±.°©²i³¶´¬³|°©· ¸Qº|»Ï»g¼ \Ä L/©¿&Ài¿&Ç/Ê¿[/ÀÁKà /ÁnÆ©Á
φp Å'Á N3C° ÉwÅM¾4Â /4Ç`Ã?È 3wÅ ¿ ÂY3n/À
$ Î Õ!Ú ÑkÎ Õ Ñ Ô4ÓD×O×Ó Ð Ù D ×iÕ1ÚsÕ1Ù©Ó u Ý!×BÓ ` ξ ! Î ÙzÝJD,Fax
pξ`ξ′K = φp(JD,Faxξ`ξ′K)
å A%I/5Ed?B<>:G?HDFIJ5qæ 3Mj>'- *fi=l-afg5*daucli=n8=|kg6q=-|ki=~;avqSj>4*Bjkc+cli=
DhCiWabr ? j>rReR-
JD,Faxpξ`ξ′K = φp(JD,Faxξ`ξ′K)
adcl ~j>esa iWc-hCi \rj&DSj>i=-afi=n^
$QJ L4NO;1L`;P?C6AOC;Y P?T4C6¦!J1YAO>~],;YADN?>g\$T4>1h.¦C0JnUJ E¶;1Y4@HAOC;Yφp
hU0JnU6T4F4C\$T4>gLNO;<KJ!<4CU6CP?AO>fPDJ!C6AF4C6¨`SN?>Y4@HC>N2UJF4C6¦1>NOX1>Y4@H>>HY~AO>],L4PupKY4CRF4>sU0JFC6¦>HNOX>HY4@>s>YAD>H]/LPuCYpKYCWb
®¯°Ï±.°©²i³¶´¬³|°©· ¸Qº|»Zn¼ \q¿ ÂÉw¾ÀÁ$Â/¿[/ L/©¿ Á$ÂÍÏÎÐ ÓÙ Ñ D Ó Ñ E Ô4Ó¬ÕÖ Ô4ÓD×O×BÓ Ð Ù4×× Ð ÞySwܶÓ×d×Õ1ÚtÔ4Ó× u Ý!×Ó×Ô©ÕÝ!ܶÓ× Ñ ÓDÜ×eãÕÓeܵÝ/Ô Ð ×NSÕ Ñ Ó[ÓÙ Ñ ÚÓ D
Ó Ñ E × ÎÐÒÑ ÔÓeÜ Î ÙàÏÕ$ÓÕ1Ú k ∈ N ∪ +∞ Ó Ñ JD,EK = Fid á/.dÙ¥Ý,ÝÜ Î Ú?×xVW (Jφp(D),EK) = (1− (1− p)k)
')(+*·ä³µ´³¶°©· ¸Qº|»v¯n¼ \Á$ÂÂHÁ4¿^/ ¿ÈzÉwÅ'ÁzËÇ/Ài¿[/w¾a ÙnÔ4ÓD×O×BÓ Ð Ù¥× Ð ÞSwÜ¶Ó D Ó× Ñ ± r ep¸hDFe¨?f× Ð S Î Õ1Ú Ñ*Î Õ Ñ Ô4Ó×?×BÓ Ð Ù¥× Ð ÞSwÜ|Ó E !©Ü0Ýy¶ Î Ùµ8 ÑoÐ|Î Ù
p 7→ W (Jφp(D),EK)
Ó× Ñ 8 Î Ù ÑoÐ Ù`Õ$ÓsÓDÙ 0 á 7 Ó[Ô4Ó×?×BÓ Ð Ù D ÓD× Ñ b%DFB± r ep¸hDFe¨?nÔÝ!Ù4×2Ü|Ów8Ý!×"8 Î Ù Ñ ÚDÝ Ð ÚÓá
')(+*·ä³µ´³¶°©· ¸Qº|» ¼,Ì4ÁKÈz¿[/a Ù_±G²LMdH DFe¹Ô q Õ1Ù Ô4Ó×?×BÓ Ð Ù D ÓD× Ñ ÜµÝ"SVÝÚ ÑoÐ ÓÔ4Ó D
UÐ × ÐWÑ ÛÓÜ Î ÚO×eÔ q Õ1Ù©Ó Ð Ù Ñ ÓÚDÝ8 Ñ|Ð|Î ÙzÔ Î Ù4Ù`ÛHÓá$QJ L4NO;1L`;P?C6AOC;YgP?T4C6¦!J1YAD>@J1NDJ!@HADSHNOC6PO>[U>HPF>POP?>C6Y4P@H;YADC6YTPb
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/ !"#$#&%' () *,+.-/0"1"(2435"$6 "7"8%' *9®s¯°©±.°©²i³µ´³¶°©·¹¸Qº|»k ¼]\[Á$ÂHÂÁ4¿^/¿6ÈnÉ`Å&Á$Ây33n/À/¾n/4Á ¿[/ÀÁKÃN3KË1Ài¿&Ç/ ¿[/ L/`¿&ÁÍÏÎÐÒÑ
D Õ!Ù Ô4ÓD×O×Ó Ð Ù~× Ð ÞyS`Ü|Óá 7 ÓרSwÚ Î SwÚ Ð Û Ñ ÛD×2×iÕ ÐU ÝÙ Ñ Ó×2× Î Ù Ñ ÛãÕ ÐU Ý!Ü|ÓDÙ Ñ Ó×V á 7 ÓsÔÓ×O×Ó Ð Ù D ÝÕ1Ù`Óx8g4Ú Î Ù Ð ãÕÓ Ð Ù4Ø4Ù Ð Óá á 7 ÓsÔÓ×O×Ó Ð Ù D ÝÕ1Ù_8g`ÓÞ Ð Ù Ð ÙØ4Ù Ð á á 7 ÓsÔÓ×O×Ó Ð Ù D ÝÕ1Ù`Ó Ñ ÚDÝ!Ùµ8g`Ó Ð Ù4Ø4Ù Ð Óá á 7 Ó,ÔÓ×O×Ó Ð Ù D Ó× Ñ Ô Ð ×X8 Î Ù ÑoÐ Ù©Õ¬V Ð ÜrÓ?Ö Ð × Ñ ÓgÕ1Ù Ô4ÓD×O×BÓ Ð ÙÊ× Ð ÞSwÜ¶Ó E
Ñ ÓDÜrã4Õ$ÓÜ0Ý ¶ Î Ùµ8 Ñ|Ð|Î Ùp 7→ W (Jφp(D),EK) Ó× Ñ Ô Ð ×X8 Î Ù ÑoÐ Ù©ÕÓÓDÙ 0 á
á kÜZÓ?Ö Ð × Ñ ÓÕ1ÙzÔ4Ó×?×BÓ Ð Ù× Ð ÞySwÜ¶Ó EÑ ÓDÜäãÕÓ W (Jφp(D),EK) Ó× Ñ ÛDàÝ!Ü¶Ó , 0 × Ð p Ó× Ñ Ù©Õ1ÜäÓ Ñ
1 × Ð Ù Î ÙVá á kÜZÓ?Ö Ð × Ñ ÓÕ1ÙzÔ4Ó×?×BÓ Ð Ù× Ð ÞySwÜ¶Ó E
Ñ ÓDÜäãÕÓ W (JD, φp(E)K) Ó× Ñ ÛDàÝ!Ü¶Ó , 0 × Ð p Ó× Ñ Ù©Õ1ÜäÓ Ñ1 × Ð Ù Î ÙVá
á kÜQÓ?Ö Ð × Ñ ÓsÕ1ÙzÔ4ÓD×O×Ó Ð Ùg× Ð ÞSwÜ|Ó EÑ ÓDÜZãÕÓ2Ü0ݶ Î Ùµ8 ÑoжΠ٠p 7→ W (JD, φp(E)K) ÓD× Ñ Ô Ð ×X8 Î Ù iÑoÐ Ù`Õ$ÓÓÙ 0 á
å ACIJ58Ed?B<T:G?HDFI/5sæ1⇒ 2
^u adcl ~j>e *Bj>+-|ki=-a7* D=-|kg=esh iWa adcl i=g * DWafpZeRg^2⇒ 3
^u adcl ~j>e *Bj>+-|ki=2* DWafpZeRgOadcl i=gWawlj>g5*EDWa@^3⇒ 1
^u DSjkhCiWa lj>g5*EDWaadcl (=j>g5*EDWafpafg Wg=e 8M=|kg5*98zj>q=E<dc[rsa rsafpZpa=a'|afg=e ?58-|ki=-avlj>g5* DWaweRgGWg=esa *d|kgClesafg i=gWa7*ED=-|kg=eshCiWaweRgGWg=esa@^
2⇒ 4af
5^è g cli=q=q/|@ca%h iWa
Dj i=g * DWafpZeRg eRgGWg=e ^ ç |keR
EqSj>E*d|ki=j>g *da *EDWafpZeRg
eRgGWg=e ^+`[zj>q=E<dcrj q=-|kq |@cleRles|kg ]W^R_By qSj ?;aq=- *dd=afgC-a98+q |ki=-|ki=pGe f-afgC=a
08
W (Jφp(D),EK) = 1p j>esc
W (Jφ0(D),EK) =W (JD,EK) = 0^
4⇒ 2^ ç |keR
E-afrJh iWa
p 7→ Jφp(D),EK c|keR GescE*d|kg leRgCi afg 0^>3i=esc-hCiWa -|ki= q |kr 'Cgkpa+adc-
*d|kg leRgCi 8;zj>q=E<dcrj q=-|kq/|@c-eRles|kg\]W^R_By qSj ?;aq=- *dd=afg -a'af8rj q=-|kq |@cleRles|kg[]W^R_ 'qSj ?;ab_By ;8j>rs|k-c+rj[Gesc-q=i=-awafg l-a
Daf
Eadcl eRgGWg=esauaf
Dj\i=g * DWafpZeRgeRgGWg=e ^
5⇒ 4^ 1 j |kg5*fles|kg *Bj>j *f-flesclleshCiWau=a
]0, 1]adcl (=esafgGescE*d|kgCleRg iWa@^
5⇔ 6^v`vj>gWc+rjGesclq=i=-auafgCl-a
Daf
E8/j>i5*fi=gO=fp|kgOg adcl j>l-afeRg *Bj>
W (JD,EK) = 0=|kg5* rsadc Gesclq=i=-adcq=-| (Sj (=eRrRescl-adcvafgCl-aφp(D)
afE i=gWauqSj>lbaf
Daf
φp(E)zj>i=l-a
qSj>l'c-|kgC'? j>rsadc j>i6= *Bj>rj ?;auq=E<dc+ i=g *d|ki=q6q |ki=+rsadc *d|a: *fesafg -cp^
6⇒ 7^ 1 j |kg5*fles|kg *Bj>j *f-flesclleshCiWau=a
]0, 1]adcl (=esafgGescE*d|kgCleRg iWa@^
7⇒ 6^è gq=-| * <d=a7*d|kpZpavq=- *dd=afpZpafgCB^
$QJL4N?;L`;1POC6AOC;Y b ],;YADN?>qT4Y J!L4L`;1N?AtF>U0J,UT4F4C6\$T4>qL4N?;<KJ1<CUC6P?AO>m/U0J/U6T4F4C6\T>qPOC6]/LU>1bkYAD>HNDJ1@BADCM¦>],>YAh©UJ UT4F4C6\$T4>[L4NO;<4J1<4CU6CP)AD>t>HP?A@iJ1L4J1<4U>[F4>s@iJ!U@TU>NuU6>[Y4;]<4N?>[FR:'SBAJ1L©>Pq|F4J1Y4PN∪+∞
¡QJi¦1J!Y$AF4>dF4CM¦>HNOX>HNU;1NOPO\$TR:5T4Y4>dJ1@HAOC;YK+ Y4>rAONO;T¦>dLKJ!PFR:'J1@HAOC;YF4TKJ1U6> K− beC6Y4POCohUJUT4FC\$T4>qLNO;<KJ!<4CU6CP?AO>FCP?AOCY4X1T4>U0J/F4C6¦1>NOX1>Y4@H> >Y AD>H]/L4PpKY4CF4>qU0JF4CM¦>HNOX>HY4@>y@iJ!T4POSH> LKJ1N
T4Y~@iJ!U@TURCYp4Y4CRPDJ!Y4PuE¶;NO],>Y4;N?]J1U6>tF4>A BAD>!b$QJ~LNO;L©;POCMADC6;Y b ¬« ]/;YAONO>1hwF4>/LUT4PHhV\T>,UJgUT4F4C6\$T4> L4NO;1<KJ1<4C6UCP)AD>POJ1C6A[@J1NDJ!@HADSHNOC6PO>N[C6Yv&
AO>NDJ!@HADCM¦>H]/>HY$AU6>PuF4>HPOPO>HCY4PJi¦>H@F4>HP@INO;Y4C6\$T4>P2CYpKY4C>HP K@>[PO;1Y$AU6>P2F4>P?PO>HCY4PF4CP?@;YAOCY$T4Pb
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" #""
« &(&#)+*-,/.132 !uI/% IJ<H?B7%7C5/?BE 1Z1\1[1Z1[1[1Z1[1\1[1Z1[1[1Z1[1\1Z1[1[1Z1[1[1\1Z1[1[12#"%$
G^R_@^R_ `b Wg=eRles|kgWc ^v^w^v^w^w^v^u^v^w^v^w^w^v^w^v^u^v^w^w^v^w^v^w^v^u^w^v^w^x_B G^R_@^zy |kp (=eRgSj>esc-|kgWc rReRgWBj>eR-adc^u^v^w^v^w^w^v^w^v^u^v^w^w^v^w^v^w^v^u^w^v^w^x_B G^R_@^z % g5*Bj>lgSj>les|kg ^v^w^v^w^w^v^u^v^w^v^w^w^v^w^v^u^v^w^w^v^w^v^w^v^u^w^v^w^x_B@yG^R_@^ ] ç i=q/aflq |@cleRles|kg ^w^v^w^w^v^u^v^w^v^w^w^v^w^v^u^v^w^w^v^w^v^w^v^u^w^v^w^x_B>]G^R_@^ & |kles|kgWc'q/|ki= rj *d|kpZq=rsfliW=aweRg -aflgWa =adc *d|kpZq |kl-afpafg -c ^w^v^w^x_f] y
1 !uI/58587 9@?B7<>E[1Z1[1[1Z1\1[1Z1[1[1Z1[1\1[1Z1[1[1Z1[1\1Z1[1[1Z1[1[1\1Z1[1[120/G^zyG^R_ `b *Bj>rj ?;adc ^w^v^w^v^w^w^v^u^v^w^v^w^w^v^w^v^u^v^w^w^v^w^v^w^v^u^w^v^w^x_f] G^zyG^zy ' =GeRle 3c ^u^w^v^w^v^w^w^v^u^v^w^v^w^w^v^w^v^u^v^w^w^v^w^v^w^v^u^w^v^w^x_f] G^zyG^z i=rRleRq=rRe=*Bj>le c ^w^v^w^w^v^u^v^w^v^w^w^v^w^v^u^v^w^w^v^w^v^w^v^u^w^v^w^x_f] G^zyG^ ] ( iSj>g len *Bj>-afi=-c ^v^w^w^v^u^v^w^v^w^w^v^w^v^u^v^w^w^v^w^v^w^v^u^w^v^w^x_f]
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@H;Y4Y4>H@HAD>HT4NOPzPOC],L4U6>Pb¤$V>PzY4;1ADC6;Y4P~F4> UJ U6T4F4C\$T4> POC6]/LU> @;],]/> @H>UU6>P¥FR:'C6Y4@iJ!NOYKJ!AOC;Y ;TFR:5SHAOI4C\$T4>¥@;],L4U6^HAD>P?;YAgJ1FKJ!LADSH>P>BAF>zY4;T¦>U6U>HP/Y4;!ADC;1Y4P/@H;],]/>@>HUU6>~F>z@;1]<4CY4J1CP?;YU6CY4SJ1CN?>tJ1L4L4J1NDJ1C6POP?>YAib
y [äQ 2 Z#J!NPOC6]/LUC@HC6AOS1h;Y_PO>nU6C],C6AD>J!Ta@iJ1P/F4>nUJÊ<KJ1P?>zT4Y4J1CN?>1b$Q>HPf@;1]/L©;N?AO>],>YADP/L4NO;1<KJ1<4C &
U6CP?AO>PP?;YAFSHpKY4C6PF>,]/J1Y4C6^N?> POC],CUJ1CN?>qmgUJU6T4F4C6\T> POC],L4U6>1hw]/J1C6PU>TNXS;],SHAONOC6> >P?AsL4U6T4PC6]/L©;N?ADJ1YAD>!b kU6PeL©;POP?^F4>HYALUT4P?C>TNOPeX1SY4SHNDJ!AO>T4N?PL`;TNtF4C6¨`SN?>YAD>HPt;L©SNOJ!ADC6;Y4P `U>HPtF4>P?PO>HCY4PP?>],C&kP?C],L4U>HP L`;1T4NU6>P,@;]<CYKJ1C6PO;YP/U6CY4SJ1CN?>P¥|P?>@HAOC;Ylªb b ¡ >HAfUW:5CY4@J1NOYKJADC;1Y¹@H;],]/>~>YU6T4F4C\$T4>P?C],L4U>eL`;TNuUW:5;NOF4N?>P?ADJ1<4U>,¶PO>H@HADC6;Yzªb b « LKJ1X1> ¬« ¡b
¨ " *|®J* « Y¹F4>HPOPO>HCY J TY ;N)ADI4;1X;YKJ1U§Y4;Y ¦CF4>POCu>BA,PO>TU>],>YA,POCuC6Ud>P)A,C],]/SHF4C0J!AO>],>YAqAO;1AJ!U
¶F4SHpKYC6ADC6;Y « b LKJ1X1> ¡Bbd$ZJY4;!ADC;1Y FR:'SBADI4C6\$T4>2L4NO;<4J1<4CU6CP)AD>r>HP?AvF4;Y@N?>P)ADNO>HCYAD>J1TxyF4>P?PO>HCY4PC6]/],SF4CJ!AD>H]/>HYA§AO;1AJ1Txwb' (+*·Z³µ´³|°©· ¯Vº|»f¼VÀ¬Ì`¿1Z¾Áa Ù`Ó C¸h²D|=@?tMgÓ× Ñ Õ1Ù ÓÙ4×ÓÞ u ܶÓtÙ Î Ù U¬Ð ÔÓÔ4ÓsÔ4ÓD×O×BÓ Ð Ù4× Ð Þ,ÞÛHÔ Ð Ý Ñ ÓÞgÓÙ ÑdÑkÎÑ ÝÕÖ Ô4ÓtÞgßÞgÓu Ý!×BÓ¬á
' (+*·Z³µ´³|°©· ¯Vº[Zz¼flÃWÀ¬Ì4ÇÏÇ/43wÅRÆpo¾n/Á¥ÄÀ¬Ì©¿ 1Q¾4Á7"q r ¸h² rr e@AvKvÔ q Õ1Ù`Ó[Û Ñ g Ð ãÕÓ[Ó× Ñ ÔÛØ4Ù Ð SRÝ!Ú
E⊥ = F | ∀D ∈ E, W (JF,DK) = 1
' (+*·Z³µ´³|°©· ¯Vº[kz¼f,rÇVÈnÉ©Ç`ÃWÀÁKÈgÁ/Àa Ù>± r H_J r ¸XMdHºMdep¸2Ó× Ñ Õ1Ù`Ó[Û Ñ g Ð ã4Õ$ÓsÛOàÝ!Ü¶Ó , × Î Ù u!ÐjikÎ Ú Ñ g Î à Î ÙKÝ!ÜÒá$QJq@U1ADTNO>tLKJ!Nr<4C &k;N)ADI4;X1;YKJ1UÏFR:5T4Yg@;1]/L©;N?AO>],>YAdC6]/LUC\$T4>2FR:'J1TAONO>HPu@U !ADT4N?>Pd@H;]/],>U0J
@HU 1AOT4NO>vLKJ1NZ@;]q<4CYKJ!CPO;1YU6CY4SJ1CN?>v>BALKJ!NäP?T4L`>HNOL©;POCMADC6;Yg¶F4SHpKYC6ADC6;Yqªb LKJ1X1> ¬« ª1¡Bb kU>HP?AJ1U;N?PYKJADT4N?>UÏF4>@;Y4P?CF4SHNO>HN§U6>PG>Y4P?>]<U>P§XSYSNDJAD>TNOPrL4J1NG@;]<CYKJ1C6PO;YyU6CY4SJ1CN?>u;T/P?T4L©>NOL©;P?C6ADC6;YFR:5T4Y~@;],L©;N?AO>],>YAib
¨© ª¬« &[*\)+* « º!-* )+*l, .$.:'TADCU6CPOJ!ADC6;Y¥F> @;>YW,@C6>YADPsJ1],^Y4>m@H;Y4P?CF4SHNO>NUJY4;1AOC;YzF4>y@;]<CYKJ1C6PO;Y¥UC6Y4SiJ1C6NO>!bn$V>
N?SPOTU6AJAgL4NOC6Y4@C6LKJ1Uq|L4N?;L©;POCMADC;1Ylªb « LKJ1X> ¬« ¡y>P?A\$T4> U6>Pf@H;]/L©;N)AD>],>YAOPfPO;YAg@HU;PLKJ1N@H;]<4C6YKJ1CP?;Y UCYSiJ1C6NO>y>HAq\T>fU0J~LKJ1N?AOC>,PO>H]/C &kPOC6]/LU>yFR:5T4Y @;1]/L©;N?AO>],>YA>HY4X>HY4F4NO>U>,@;]&L©;N?AO>],>YAib
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GB D "8 tuR" 1"$6 *!8¬ *(s¬«')(+*·ä³µ´³¶°©·Q¯Rº ¸ ¼]z3KÃWÀi¿&ÁnÂÁKÈz¿`5O¿6ÈnÉwÅ'Á7 Ý JA¸hD|M]BMdH DFEBGDIH>JLKNM_Ô q Õ1Ù 8 Î ÞyS Î Ú Ñ ÓÞgÓÙ Ñ G Ó× Ñ Ü q ÓDÙ4×BÓDÞ u Ü|Ó Ô4ÓD×ÊÔ4Ó×?×BÓ Ð Ù4×n×ÓÞ Ðji× Ð ÞySwܶÓ×ÔÓ G á.dÙ~ܵÝqÙ Î$Ñ Ó Pss(G) á
')(+*·ä³µ´³¶°©·Q¯Rº[¯ ¼-,dÇVÈ9m©¿^/C3`¿ ÂHÇ/ Å ¿^/Ä3©¿ÃWÁ7q MdeBMdH s0KNM Cl(E) Ó× Ñ Ü q ÓÙ4×ÓÞ u Ü|ÓÔ4Ó×8 Î Þ u!Ð ÙÏÝ Ð × Î Ù4×vÜ Ð Ù`ÛBÝ Ð ÚBÓD×uÔÓ"8 Î Ó;:<8 Ð ÓÙ Ñ ÔÓ¨S άРÔ$× 1Ô q Õ1Ù ÓDÙ4×BÓDÞ u ܶÓsÔÓsÔÓ×O×Ó Ð Ù4× E ×iÕ1ÚsÕ1Ù`Ó u Ý×BÓ[Ô Î Ù4Ù©ÛÓá
')(+*·ä³µ´³¶°©·Q¯Rº ¼]zuÃWÇ ÁË1Ài¿&Ç/ ÂÁKÈz¿`5D¿ÈzÉwÅ'Áa Ù Ô4Ó×?×BÓ Ð Ù ×BÓÞ Ðji × Ð ÞySwÜ¶Ó D0 ÓD× Ñ Õ1Ù`ÓwJ r M±¸hD r e B¼MOHDIEBDFH>JKNMÔ q Õ1Ù ÔÓ×O×Ó Ð Ù D × Ð D0ÝGS4SVÝÚ ÑoÐ ÓÙ Ñ , Ü0Ý ¶|Ý!Þ Ð ÜÜ¶Ó (Dα) Ô4Ó Ô4Ó×?×BÓ Ð Ù4×z×ÓÞ Ðji × Ð ÞSwܶÓ×¥Ô Î Ù Ñ D Ó× Ñ Ü0Ý 8 Î Þ u!Ð ÙÏÝ Ð × Î ÙÜ Ð Ù`ÛBÝ Ð ÚBÓ¬áctJ!Y4P§UJL4NO;L©;P?C6ADC6;Y POTC6¦!J1YAD>!h;1Y/TAOCU6CPO>uUJY4;1AJADC;1Y
D = c.(D1, . . . ,Dn)L`;1T4NGNO>LNOSP?>Yv&
AD>N2T4Y>s@H;]<4C6YKJ1C6PO;YgU6CY4SJ1CN?>t\$T4>U6@;Y4\$T4>|UJqE|J1],CUU6>(Di)i
YR:5>P)AeLKJ!P2E¶;1NO@SH]/>HYAupKY4C>i¡Bb®¯°Ï±.°©²i³¶´¬³|°©·Q¯Rº|»g¼ zdÃÒÇ ÁË1Ài¿ Ç/ ÂHÁÏÈ¥¿65O¿6ÈnÉwÅ'ÁÁÀqÇ©ÃWÀ¬Ì4ÇÏÇ/43wÅ ¿'ÀÄÍÏÎÐWÑ
D = c.(D1, . . . ,Dn) Õ1Ù Ô4Ó×?×BÓ Ð Ù Ô4Ó u Ý!×BÓ Õ1ÙÏÝ Ð ÚBÓ β Ó Ñ E Õ1ÙÊÔ4ÓD×O×BÓ Ð Ù Ô4Ó u Ý!×Ó Î Ú Ñ g Îià Î ÙÏÝÜ|ÓáeÜ Î ÚO× !
D ⊥ E⇔W (c) = 1 Ó Ñ ∀i, Di ⊥ E
å A%I/5Ed?B<>:G?HDFIJ5qæ&% r j>i= j>eR-aw=afi;6Bjkc'cli=eR~kj>gC+rj[q |krj>leR-w=aD^
jkcw_@^ D adcl q |@cleRle l^''rs|k-cJD,EK = c.(JD1,EK , . . . , JDn,EK)
3Mj>urjq=-|kq/|@c-eRles|kgG^ ]qSj ?;aT]58|kgd|kgfrRi=uhCiWaD ⊥ E
cle afwcafi=rsafpafg c-ecadc-w=a
q/|kes=c1af+h iWabrsadc JDi,EK c-|kg j>iWc-cle=avq |kes=c 1 ^
jkc'yG^ D adcl gW? j>le l^è gLfleRE = c′.(E1, . . . ,Ek)
| rsadcEi
c-|kgC cafpZenmocleRpZq=rsadc @3|kg i=leRrResc-aO-|ki |ki=-c rjd|kg ~;afgCles|kg hCiWabrj j>pZeRrRrsaw=adc
(Ei)q/afi= fl-aveRgGWg=esa B ^
è gOjJD,EK = c′.(JD,E1K , . . . , JD,EkK)`b|kg
JD,EK = c′.(c.(JD1,E1K , . . . , JDn,E1K), . . . , c.(JD1,EkK , . . . , JDn,EkK))3Mj> rjq=-|kq |@cleRles|kgG^ ]qSj ?;a T]58|kg=dGi=eR hCiWa
D ⊥ Ecle afucafi=rsafpafg c-e
c′adclw=a
q/|kes=c18cj>iWccleaf'hCiWabq/|ki= -|ki=
iaf
j8Di ⊥ Ej
^% r-adc--a q=-|ki=~;afuq |ki=v-|ki=
i8h iWa
Di ⊥ Ecle afwc-afi=rsafpafgC cle
c′adclw=aZq |kes=c
1af
q/|ki= -|ki=j8Di ⊥ Ej
^MeF6G|kgWc
i^è gj
JDi,EK = c′.(JDi,E1K , . . . , JDi,EkK)1 j q=-|kq/|@cleRles|kgG^ ] qSj ?;aT] eRpZq=rResh iWa\hCiWaDi ⊥ E
cle8afbc-afi=rsafpafgCbc-ec′adcl=auq |kes=c
1af q/|ki='-|ki=
j8Di ⊥ Ej
^Y¥@H;NO;U6U0J1C6NO>C],]/SHF4C0JAd>P)AU0JL4N?;L`;1POC6AOC;YgP?T4C6¦!J1YAO>1b
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¬«, !"#$#&%' () *,+LH" 1"$6 *!8¬ *(H ¦H"88GB *!(®s¯°©±.°©²i³µ´³¶°©· ¯Rº[Z ¼]zdÃÒÇ ÁË1Ài¿ Ç/4 ÂHÁKÈz¿65O¿6ÈnÉ`Å&Á$ÂqÁÀËÇQÈzÉ`Ç©ÃWÀÁKÈgÁ/ÀÂa Ù ÔÓ×O×Ó Ð Ù Ð Þ,ÞÛHÔ Ð Ý Ñ ÓÞgÓÙ ÑÑkÎÑ ÝÜeÓ× Ñ ÔÝ!Ù4×zÕ1Ù-8 Î ÞS Î Ú Ñ ÓDÞÓDÙ Ñ × Ð Ó Ñ ×ÓÕ1ܶÓÞgÓÙ Ñ × Ð ×BÓ×SwÚ Î ÓY8 Ñ|Ð|Î Ù4×2×ÓÞ Ðji × Ð ÞSwÜ|ÓD×r× Î Ù Ñ ÔÓÔÝ!Ù×iá$QJzL4N?;L`;1POC6AOC;YÊPOT4CM¦!J1YAD>f],;YADN?>\$T4>gU>/XSYSNDJAD>TN]/C6Y4C]/J1U.LKJ!Nq@;1]<4CY4J1CP?;Y U6CY4SJ1CN?>
>HP?AUW:5>Y4P?>]<U>F4>HPF4>HPOP?>CYPP?>],C&kP?C],L4U>HPb®s¯°©±.°©²i³µ´³¶°©· ¯Rº[k ¼-,dÇVÈ9m©¿^/C3`¿&ÂÇ/4ÂyÅM¿[/4Ä3`¿6ÃÒÁ$Â$ Î Õ1Ú ÑkÎ Õ Ñ 8 Î ÞyS Î Ú Ñ ÓÞgÓÙ Ñ G ! Î ÙzÝ
Cl(Pss(G)) = G
´ ÓSwÜÒÕ1× !vS Î Õ1Ú Ñ*Î Õ Ñ Ó[Û Ñ g Ð ãÕÓ E !Cl(E) = G⇔ Pss(G) ⊆ E ⊆ G
å ACIJ58Ed?B<T:G?HDFI/5sæ 3M|ki= -|ki=-a f D=esh iWaEaf-|ki= d|kpZq |kl-afpafg
G8@|kg j
E ⊆ Cl(E)8Pss(G) ⊆
Gaf
Pss(E) ⊆ E^
_@^ g=adcc-afeRgeRpZpdGej>-afpafg '-|kj>r @3c|kg d|Ca: fesafgC rj (Sjkcavadcl =a q |kes=c1BadcleRgfrRiWc
Wj>gWcZrsadc d|kp)(=eRgSj>esc-|kgWc rReRgWBj>eR-adc =a d|Ca: fesafgC =aq |kes=c1=ac-adcZq=-| lafles|kgWcc-afpZenm
cleRpZq=rsadcB^è rsadcq=-| afles|kgWc'c-afpZenmocleRpZq=rsadc+=adc =adc-cafeRgWc =a
Gc-|kg =adc+=adc-cafeRgWc c-afpZenmocleRpZq=rsadc =a
G@Fq=-|kq |@cleRles|kg G^zy B ^`b|kg
Cl(Pss(G)) ⊇ Gè 8GqSj>+p|kgW|k-|kg=esa =aCl8
Cl(Pss(G)) ⊆ G`b|kgCl(Pss(G)) = GyG^
⇒ç i=q=q/|@c-|kgWc
Cl(E) = G^è g j
E ⊆ G^W`babq=rRiWc 8
Pss(G) = Pss(Cl(E)) ⊆ E^
⇐ç i=q=q/|@c-|kgWc
Pss(G) ⊆ E ⊆ G^B`[zj>q=E<dc8rsaq=-afpZesaf Bjkc 8
Cl(Pss(G)) = G^Hè g d|kgfrRi=
afgeRg ~;|h iSj>gC'rj\p|kgW|k-|kg=esa =aCl^
¨ M) ,4 )p®/* « $.:'C6Y4@iJ!NOYKJ!AOC;Y[FR:5T4YF4>P?PO>C6Y
DFKJ1Y4PZT4Y@;],L`;1N?AD>H]/>HYA
G>P)AU0JuLKJ!N?ADC6>GF4>
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LKJ!Ne<4C &k;N)ADI4;X1;YKJ1UobKce>]/J1Y4C6^NO>L4UTPe@iJ!U@TU0J!AO;CN?>1hK@1:5>P)A[U>sL4UTPtL`>BADCMAeF4>HPOP?>CY F4Tz@;],L`;1N?AD>&],>YA
GwX +FKJ1YP
Dbv$ZJY;1ADC6;YFR:5CY@iJ1N?YKJ!ADC6;YfN?>L©;PO>tP?T4NuU>ADI4SH;NO^H]/>eF4>tP?ADJ1<4C6UC6AOSµADI4SH;&
N?^],> « b ª L4J1X> ¡b4$.:5CY4@HUT4P?C;YgTAOCU6>tC@HCR>P)AeUo:'C6Y4@U6T4POC6;YfL4U6>CY>,|F4SBpKY4C6AOC;Y « b LKJ!X> ¡b®s¯°©±.°©²i³µ´³¶°©· ¯Rº ¸ ¼2Ë1Ài¿ Ç/ K¿ ¿&ÀÄ$Á¥Á$ÀqËÇRÁ+.ÊË¿ Á/ÀÆ©ÁÊÉ©ÇR¿ Æ©Â
1ÍÏÎÐÒÑD Ó Ñ E Ô4ÓÕÖ Ô4Ó×?×BÓ Ð Ù4× Î Ú Ñ g Î à Î ÙÏÝÕÖqÓ Ñ σ Õ1Ù`Ó.Ý8 ÑoÐ|Î Ù ÔÓ D U¬Ð × ÐÒÑ ÛÓZÜ Î ÚO×Ô4ÓÜ q Ð Ù Ñ ÓÚÝ+8 ÑoжΠÙ
Ý U ÓG8 E á7 Óx8 Î Óh: 8 Ð ÓDÙ Ñ SwÚÛØÖ4Ý!Ù Ñ σ Ó× Ñ Ô4ÓS ÎÐ Ô×eÛDàÝ!Ü , 1 á
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07
GB D "8 tuR" 1"$6 *!8¬ *(s¬««å A%I/5Ed?B<>:G?HDFIJ5qæ[ç |keR
crsa d|Ca: fesafgC q=- 6Wj>g
σ^ 1 zj fles|kg
σadclZ~escleR-da6=|kg
cj>q=qSj>js
Wj>gWc JD,EK ^è JD,EK adcl+=avq |kes=c 1 =|kg @Fq=-|kq |@cleRles|kg G^ ]ZqSj ?;a T]B cadc-'=abq/|kes=c
1^
ce>HTxAD>HNO],>P2PO;1Y$AL4N?S@HCPOSHP2L`;TNUJL4NO;1L`;P?C6AOC;YgP?T4C6¦!J1YAD> eY4Y$T4U6>NT4Yz@H;>XW/@HC>HY$AeL`;1POC6AOC6E
cFKJ1Y4PT4YnF4>HPOP?>CYwh`@!:'>HP?AtU6>N?>],L4U0J!@>N2LKJ!N
0>BAe;T<4UC6>N
PDJy@;YAOCY$TKJ!AOC;YRbCV@>[@;>YW,@HC>YAPOTC6A2T4Y4>J1@BADC;1YgY4SXJADC6¦1>sJ1U;N?PuUW:'J1@HAOC;YY4SXJADC6¦1>[>P?AJ1T4P?POCQ>Y4U6>H¦1S>oJ1C6Y4POCR\$T4>[POJ,@H;YADCY$TKJADC;1YÏ¡Bb
Y~@;>YW,@C6>YAL`;1POC6AOC6EZ>P)AQ³o²³¶´ (,POC ;T<4C6>Y~U>@H;>XW/@HC>HY$A>HP?Am U0J NDJ1@HCY4>FT¥F4>HPOP?>CY ;T<4C6>Y~UW:'J1@HAOC;YY4SXJADC6¦1>[L4NOSH@SF>YAD>>P?A2¦CP?C6AOS>1b
®¯°Ï±.°©²i³¶´¬³|°©·Q¯Rº[¯n¼j/KËO3KÃ/43Ài¿&Ç/$ Î Õ!Ú ÑkÎ Õ Ñ 8 Î ÞyS Î Ú Ñ ÓÞgÓÙ Ñ G !tÓ Ñ~Ñ*Î Õ Ñ Ô4Ó×?×BÓ Ð Ù D Ô4Ó G ! Ð ÜÓ?Ö Ð × Ñ Ó Õ!Ù Õ1Ù Ð ãÕÓ Ô4Ó×?×BÓ Ð ÙD0 ⊆ D Ô4Ó G
Ñ ÓÜãÕÓ∀E ∈ G,E ⊆ D0 ⇒ E = D0´ Ó¨SwÜWÕ1× !t8ÓÔÓ×O×Ó Ð Ù D0 ×BÓ8BÝ!Ü8Õ1ܶÓeÓÙ~ÝÙ4Ù©Õ1ܵÝ!Ù Ñ Ü|Óפ8 Î Ó;:<8 Ð ÓÙ Ñ ×LS Î × ÐÒÑoÐ ¶µ×2ÔÓ D Ù Î Ù UÐ × ÐWÑ Û×
SVÝ!Ú[Õ1Ù Ô4ÓD×O×BÓ Ð Ù Ô4Ó G⊥ áå A%I/5Ed?B<>:G?HDFIJ5qæ è g6p|kg l-awr a6escl-afgdawq=i=esc+r i=g=e feR-@^
6Gescl-afgda@^ ç |keR D0rsa=adc-c-afeRg | (=-afg iafg j>g=gCi=rj>g rsadc d|Ca: fesafgC-cq/|@c-eRle 3c gW|kg ~escleR-dc
qSj>+i=g6=adc-c-afeRgO=aG⊥ ^
1 ab=adc-c-afeRg D0j>q=qSj>llesafgC (=esafg
GBj>eRrjwrsadcpfpadceRg -afj fles|kgWc'h iWa
Djd~;arsadc
=adc-c-afeRgWc+=aG⊥ ^
ç |keR E ⊆ D0i=g6=adc-cafeRg6=a
G^
ç i=q=q/|@c-|kgWcvhCiWa\rsa d|Ca: fesafgCvq/|@c-eRle c′Wj>gWc
Ec-|keRvGef-afg uGi d|a: fesafg
cd|k m
-adclq/|kgWWj>gCWj>gWcD0^53j>= Wg=eRles|kg=avr eRgfrRiWcles|kg 8
c′adcl'| (=-afg i [qSj>lleR=a
cafg
j>g=g i=rj>g [=adc d|a: fesafg -cugW|kg%g i=rsc =|kgc-|kgq |kes=cwadclwcllle f-afpafgC eRg flesafi=1^
è cadc-ZgW|kg g i=r'=|kg6~CescleR-=|kg
c′r adc- j>iWc-cle ^ afrj d|kgCl-adGeR rjq=-|kq |@cleRles|kg
q=-dd=afg -a@^ 'g=e feR-@^ ç |keR D1 ∈ G
i=g6=adc-cafeRgq/|@c-cdWj>g +rj[q=-|kq=lesf-
∀E ∈ G,E ⊆ D1 ⇒ E = D1
ç |keRci=g d|Ca: fesafgC'q |@cleRle =a
D1^
jkc _@^ 1 a d|a: fesafg d|kl-adc-q/|kgWWj>gC c′ adcl~escleR- Wj>gWc D ^ 3j>eRgfrRiWcles|kg\=adc8=adcc-afeRgWc 8rsa d|a: fesafg c=a
D1adcleRgfrRiWcWj>gWc
c′^ ç i=q=q/|@c-|kgWc hCiWa rsa d|a: fesafg
c=a
D1c-|keR
cllle f-afpafg eRgfrRiWc Wj>gWcc′^ ç |kgZq |kes=cadcl=|kg cllle f-afpafg eRg flesafi=
1af 8@zj>q=E<dc
rj\q=-|kq/|@cleRles|kgOq=-dd=afg -a98SeRr ' j d|kgCljkGe fles|kg^ jkcMyG^ 1 a d|Ca: fesafgC c′ d|kl-adclq |kgWWj>g Wj>gWc D g adclMqSjkc8~CescleR-@^iè g q=-|ki=~;aj>esc-fpafgCh iWa
cadcl gCi=r ^
è gafg6=dGi=eR'hCiWaD1 = D0
^
')(+*·ä³µ´³¶°©·Q¯Rºz¼ j/KËO3Kà /C3Ài¿ Ç/´ Ý!Ù4×Õ1Ù)8 Î ÞS Î Ú Ñ ÓDÞÓDÙ Ñ G !CK9 DFeL±Ae@An¸ D r e |D|G Ô q Õ1Ù¥Ô4ÓD×O×Ó Ð Ù D ÓD× Ñ Ü|ÓÔ4Ó×?×BÓ Ð Ù D0 ⊆ D
Ô4Ó GÑ ÓÜãÕÓ
∀E ∈ G,E ⊆ D0 ⇒ E = D0
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¬«! !"#$#&%' () *,+LH" 1"$6 *!8¬ *(H ¦H"88GB *!(®s¯°©±.°©²i³µ´³¶°©· ¯Rº ¼ j/KËO3Kà /C3Ài¿&Çc/nÁ$ÀÆ`Á$ÂÂHÁ4¿^/ É©ÇÏ¿'Ài¿ ÁKÈgÁ/ÀqËÇVÈnÉ`Å&ÁÀ7 ÓD× Ð Ùµ8Ý!ÚOÙKÝ ÑoжΠÙ4×vÔÓ×rÔ4ÓD×O×BÓ Ð Ù4×dÔ q Õ1Ù<8 Î ÞyS Î Ú Ñ ÓÞgÓÙ Ñ × Î Ù Ñ Ô4Ó×rÔ4ÓD×O×Ó Ð Ù×S Î × ÐÒÑoÐU ÓDÞÓDÙ Ñ 8 Î Þ iSwÜ|Ó Ñ ×iá
å ACIJ58Ed?B<T:G?HDFI/5sæ[ç |keRDi=g=adc-c-afeRgp j>-flesafr @3? j>r 6c-|kgeRgBj>lgSj>les|kgBuWj>gWc i=g d|kpZq |kl-a m
pafg G^`[zj>q=E<dc rjq=-|kq |@cleRles|kg G^qSj ?;a q=-dd=afg -a98Mrsadc d|a: fesafg -c\q |@cleRle cvgW|kg~escleR-dc
c-|kgC j>g=g i=rsdc @3afw=|kgafg=rsaf~;dcw=a[r eRgBj>lgSj>les|kgB ^M`|kg98 -|ki=-adcursadcuj fles|kgWcvq |@cleRleR~;adcw=aDc-|kgC~escleR-dadcqSj>i=g=adc-c-afeRg=a
G⊥ afrsadc d|Ca: fesafgC-cq=- 6=j>gC dadc j fles|kgWc c-|kg =aq/|kes=c 1@Fq=-|kq |@cleRles|kg G^ ]qSj ?;a_B@y B ^`b|kgvrsav=adc-c-afeRg
Dadc- q/|@cleRleR~;afpafgC d|kpZq=rsafB^
®s¯°©±.°©²i³µ´³¶°©· ¯Rºz¼ j/KËO3Kà /C3Ài¿&Çc/nÁ$À,¿^/ÀÁKÃÒÂÁË1Ài¿&Ç/´ Ý!Ù4× Õ!Ùf8 Î ÞS Î Ú Ñ ÓDÞÓDÙ Ñ G !dÜ q Ð Ùµ8BÝÚOÙÏÝ Ñ|Ð|Î Ù |D|G Ô q Õ1Ù Ô4ÓD×O×Ó Ð Ù D Ó× Ñ 8Ý!ÚÝ+8 Ñ ÛDÚ Ð ×BÛÓ§SVÝ!Ú|D|G =
⋂
E ∈ G | E ⊆ D
å ACIJ58Ed?B<T:G?HDFI/5sæ è g6i=leRrResc-abrsaw DWd|kE<fpau=avclj (=eRrReR-@^
~<¯n9¯Vº|»x KkX;¬BFW¹X Ï;BFW +dCt;K XµFW£th; XKk;B¥ K*Xh¬XFW tXw $QJÊY4;!ADC;1Y FR:5SHAOI4C\$T4>¥@;],L4U6^HAD>~L©>NO],>HA/F4>~F4SHp4Y4CN,U0J @;],L4USBADT4F>~C6YAD>N?Y4> µ¦;C6N/FSHpKY4C &AOC;Y b $ LKJ!X> « ¡ U6>P.@H;Y4Y4>H@HAD>HT4NOP§Y4>uNO>HY¦;C6>YArL4J1PE¶;NO@HS],>YAvYKJ!AOT4NO>HUU>H]/>HYA.F4>PG@H;]/L©;N&AO>],>YADP2]/J1CPdLUTA1A2F4>P2SBADI4C6\T>P@;],L4U^BAD>HPb».Y~UT4F4C6\$T4>[L4NO;<4J1<4CU6CP)AD>1h>U6U>>HP?A]J!CYAD>HYKJ1YAE|J1CMAD>],;F4T4U;@;]<CYKJ1C6PO;YPtUC6Y4SiJ1C6NO>HPe@>y\$T4C.P?CXY4CMpK> \$T4>yPO>HT4U> UJfLKJ1N?AOC> P?>],C&kP?C],L4U>sFR:5T4Y4>SBADI4C6\T>s@H;],LAD>!b' (+*·Z³µ´³|°©· ¯Vºz¼VÀ¬Ì`¿1Z¾ÁzËÇVÈnÉ`ÅÀÁa Ù`Ó[Û Ñ g Ð ãÕÓ E ÓD× Ñ ± r H>JLKC¸XMq× Ð
Pss(|E⊥⊥|) ⊆ E
$.:'TADCU6CPOJ!ADC6;Y F4> U0J L4J1N?AOC>zP?>],C&ÒPOC],L4U6>Pss( )
>HP?AgYS@>HPOPOJ1CN?> L`;TNfU>Ê*XC¬Bl¶F4SHpKYC&AOC;Yªb L4J1X> ¡LKJ1Nv>Bx>],L4U6>\$T4C©Y4S@H>P?POC6AO>tT4Y4>@U1ADT4N?>L4J1N§@H;]<4C6YKJ1C6PO;Y4PGU6CY4SJ1CN?>P.L©;T4N;<AD>YCNuT4Y¥@;],L`;1N?AD>H]/>HYAib :5>P)AeYKJADT4N?>UU6>],>YA2SXJ1U>H]/>HY$A2¦NDJ!CVL`;TNU6>P d9;|FSHpKY4C &AOC;Y~ªb LKJ1X1> ª¡BhU>t +bHbb®s¯°©±.°©²i³µ´³¶°©· ¯Rº ¼-,dÇVÈnÉ©Ç©ÃWÀÁKÈgÁ/À É43KÃWÀi¿6ÃÆpo¾n/Á¥ÄÀ¬Ì©¿ 1Q¾4Á
Î Õ Ñ 8 Î ÞS Î Ú Ñ ÓÞgÓÙ Ñ Ó× Ñ ÜµÝ)8DÜ Ñ Õ1ÚBÓSVÝ!ÚtÜ q Î ÚBÔ$ÚÓ[× Ñ Ý u Ü|ÓqÔ4ÓsÜ q ÓÙ4×ÓÞ u Ü|ÓqÔÓ×28 Î Þ u!Ð ÙÏÝ Ð × Î Ù4×Ü Ð Ù©ÛBÝ Ð ÚBÓ×Ô4Ó×BÓ×Û Ñ g Ð ãÕÓD×"8 Î ÞSwÜ ¬Ñ ÓD×iá
¨ ¤ .,v « S*N®/* « $V>P@;]<CYKJ1C6PO;YPU6CY4SJ1CN?>P2CYAD>HN?¦C>HY4Y4>HY$Am/U0J,NDJ1@HCY4>sF4>HPtF4>P?PO>HCY4Pt]/J1C6PTY4> J1TAONO>;L`S&
NOJ!ADC6;Y E|J1C6AyC6Y$AO>N)¦>YCN,U>P,@;>YW,@HC>YAOPL©;T4Ny@;YP?ADN?T4CN?>¥U6>P,F4>P?PO>HCY4P vUJ P?T4L©>NOL©;P?C6ADC6;YRbL$QJP?T4L`>HNOL©;POCMADC6;YgNO>HUC>eF4>PF4>P?PO>HCY4PPOC6]/L4U6>Pd>BAF4>PF4>P?PO>HCY4PL4NO;1<KJ1<4C6UCP)AD>HPb
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GB D "8 tuR" 1"$6 *!8¬ *(s¬« ªctJ!Y4P§@H>HA?AD>PO>@BADC6;Y/>BArL4J1NGU0JPOT4CMAD>!hUJY4;1AJADC;1Y
E⊥pF4SP?CXY4>UW:5;N)ADI4;X1;YKJ1UFKJ1Y4PvU0JUTF4C\$T4>
L4NO;<4J1<4CU6CP)AD>tFR:5T4Y4>[SBADI4C6\T>P?C],L4U>>HAE⊥s
PO;Y~;N?AOI4;X;1YKJ1URFKJ!Y4P2U0JUTF4C\$T4>[POC6]/LU>1b')(+*·ä³µ´³¶°©·Q¯Rº ¼ Á$ kÇ/4Ë1Ài¿ Ç/4Â
SimpÁÀ
Proba7 Î ÚO×ã4Õ$Ó G Ó× Ñ Õ1Ù 8 Î ÞyS Î Ú Ñ ÓÞgÓÙ Ñ SwÚ Îvu Ý u!Ð Ü Ð × Ñ Ó! Î Ù Ù Î$Ñ Ó Simp(G) Ü q ÓDÙ4×BÓDÞ u Ü¶Ó ÔÓz×Ó×Ô4Ó×?×BÓ Ð Ù4×× Ð ÞySwܶÓ×á7 Î ÚO×ã4Õ$Ó G Ó× Ñ Õ1Ù¹8 Î ÞyS Î Ú Ñ ÓÞgÓÙ Ñ × Ð ÞySwÜ¶Ó ! Î ÙzÙ ÎÑ Ó Proba(G) × Î Ù uÐji*Î Ú Ñ g Î à Î ÙÏÝÜ0SwÚ Îvu Ý iu!Ð Ü Ð × Ñ Óá
')(+*·ä³µ´³¶°©·Q¯Rº|» ! ¼ ½¾wÉ©ÁKÃOÉ©ÇÏ¿'Ài¿&Çc/ Æ`ÁnÆ©Á$ÂÂÁ4¿[/4¿ÈzÉwÅ'ÁÂ.dÙ Ô ÐÒÑ ãÕ q Õ1Ù ÔÓ×O×Ó Ð Ù<SwÚ Îvu Ý u!Ð Ü Ð × Ñ Ó D Ó× Ñ ÔÝ!Ù4×2ܵÝB?J¥MJ r BD¸hD r e Sp(E) Ô q Õ1Ù©ÓÛ Ñ g Ð ã4Õ$Ó× Ð ÞySwÜ¶Ó E × Ð
D ∈ E⊥p⊥p
~<4¯%c>¯Rº^Z % iwBFWSp( )
XnXd ^¬9 iwBIWProba( )
; K*XXKXt;1 £BK;"Xt; ®¯°Ï±.°©²i³¶´¬³|°©·Q¯Rºn¼ 3 kÇ/KË1Ài¿&Çc/
ProbaÁ$ÂHÀy¿[/ ÁË1Ài¿ Á
7 Ý ¶ Î Ùµ8 ÑoжΠ٠Proba Ó× Ñ Õ1Ù`Ó Ð Ù ÓG8 ÑoжΠٹÔ4ÓD×8 Î ÞyS Î Ú Ñ ÓÞgÓÙ Ñ ×q× Ð ÞyS`Ü|Ó× ÔÝ!Ù×qÜ|ÓD×8 Î ÞyS Î Ú Ñ Ó iÞÓDÙ Ñ ×¨SwÚ Îu Ý u!Ð Ü Ð × Ñ ÓD×iá ÍKÎ Ù Ð Ù U ÓDÚO×BÓ , àÝÕ8g`Ó[ÓD× Ñ Simp á
å A%I/5Ed?B<>:G?HDFIJ5qæ[ç |keRGi=g d|kpZq |kl-afpafg cleRpZq=rsa@^ |kg l-|kgWc'hCiWa
G = Simp(Proba(G))^
3Mj> p|kgW|k-|kg=esaw=a Simpaf
Proba8W|kgj
G ⊆ Simp(Proba(G))^
O|kgCl-|kgWc'hCiWaG ⊇ Simp(Proba(G))
^ç |keR
Di=g6=adc-c-afeRgOcleRpZq=rsavWj>gWc
Proba(G)^ ç |keR
E ∈ G⊥s^ |kg l-|kgWch iWa
D ⊥ E^
è gOjE ∈ G⊥s
=|kgE ∈ G⊥p
^=`b|kgD ⊥ E
^=`b|kgD ∈ G
^`b|kg
G ⊇ Simp(Proba(G))af
G = Simp(Proba(G))^
~<4¯%c>¯Rº^k %ºBiBIµ µ|¹£iwBIWProba
n;BwC XdD B;B d^; XiBoD X;¢ävwxB*Xt1FW£"wIW ¤B¼KXNK XµFW)GdC"GO KtdB dµBFXn Q w TB KXNK XµIWl;B[F ^ yK£B1 KXTXd );;¬XCXt; ;B¦;¬,t TCdIWB WXh¢¨TXd K C9§tdK _wKBL¬K*XKkXxD +d Xx;¦tDiBFWC"Xt; $QJfY4;1AOC;YzPOT4CM¦1J!Y$AO>L©>NO],>HAtF4> PO> LKJ!POPO>HN[F4> U0JP?T4N¶=?>H@HAOC6¦C6AOSF4> U0J,E¶;Y4@HAOC;Y
ProbaLKJ1NtUJ
POT4CMAD>1b')(+*·ä³µ´³¶°©·Q¯Rº|»©»g¼,rÇVÈnÉ©Ç`ÃoÀÁÏÈfÁ0/ÀqÂ5ÃWÄ Q¾`ÅM¿&ÁKÃa Ù98 Î ÞyS Î Ú Ñ ÓÞgÓÙ Ñ SwÚ Îvu Ý u!Ð Ü Ð × Ñ Ó G Ó× Ñ EE ?4KIDNMS Î Õ1Ú[Õ1Ù_8 Î ÞyS Î Ú Ñ ÓÞgÓÙ Ñ × Ð ÞyS`Ü|Ó H × Ðâ H = Simp(G)â H⊥s = Simp(G⊥p)
a Ù£8 Î ÞS Î Ú Ñ ÓDÞÓDÙ Ñ SwÚ Îu Ý u!Ð Ü Ð × Ñ Ó2Ó× Ñ EE ?4KIDNM× q Ð Ü©Ü q Ó× Ñ S Î Õ1ÚtÕ!Ù 8 Î ÞS Î Ú Ñ ÓDÞÓDÙ Ñ × Ð ÞyS`Ü|ÓÔ Î Ù4Ù©Ûá
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¬« !"#$#&%' () *,+LH" 1"$6 *!8¬ *(H ¦H"88GB *!(®s¯°©±.°©²i³µ´³¶°©· ¯Rº|» ! ¼-,¦3KÃN34Ë1ÀÄÏÃ)¿ ÂY3Ài¿&Çc/ Æ`Á$ÂyËÇVÈnÉ©Ç`ÃWÀÁKÈgÁ/ÀÂÂX5HÃÒÄ V¾wÅ ¿ ÁKÃWÂa Ù_8 Î ÞS Î Ú Ñ ÓDÞÓDÙ Ñ SwÚ Îu Ý u!Ð Ü Ð × Ñ ÓtÓD× Ñi ÚBÛOàÏÕ1Ü Ð ÓÚr× Ð Ó Ñ ×ÓÕ1ܶÓÞgÓÙ Ñ × Ð
Simp(G⊥p) = Simp(G)⊥s
®s¯°©±.°©²i³µ´³¶°©· ¯Rº|»Ï»¼#"c°K¿&ÂHÀÁ/KËÁzÆ`ÁnËÇVÈnÉ©Ç©ÃWÀÁKÈgÁ/À Â5ÃWÄ V¾`ÅM¿&ÁKÃÒÂÍÏÎÐÒÑ
H Õ!Ù>8 Î ÞyS Î Ú Ñ ÓÞgÓÙ Ñ × Ð ÞSwÜ|Ó¬ákÜÓOÖ Ð × Ñ ÓqÕ1Ù_8 Î ÞS Î Ú Ñ ÓDÞÓDÙ Ñ SwÚ Îu Ý u!Ð Ü Ð × Ñ Ó G
i ÚBÛOàÏÕ1Ü Ð ÓÚLS Î Õ1Ú H áå ACIJ58Ed?B<T:G?HDFI/5sæ[ç |keR
G = Proba(H)^ O|kgCl-|kgWc'hCiWa
H = Simp(G) H⊥s = Simp(G⊥p)1 avq=-afpZesaf'q/|keRgC'adcl eRpZq=rReshCiWwqSj> rj[q=-|kq |@cleRles|kg G^[qSj ?;abq=-dd=afgC-a@^è gOj
H⊥s ⊥ H=|kg
H⊥s ⊆ H⊥p=|kg
H⊥s ⊥ G^
`b|kgH⊥s ⊆ G⊥ 8 dawh i=e=|kg=gWa H⊥s ⊆ Simp(G⊥)
è H ⊆ G
=|kgH ⊥ Simp(G⊥p)
=|kgSimp(G⊥p) ⊆ (Simp(G))⊥s
af
Simp(G⊥p) = (Simp(G))⊥s = H⊥s
~<¯n9¯Vº ¸ x Ktd vÊKkXXK |h 1 WX ;B |; 1 WX ' (+*·Z³µ´³|°©· ¯Vº|»Zn¼,rÇc/µ/4ÁË1ÀÁK¾wÃ2Â5ÃWÄ V¾`ÅM¿&ÁKÃa Ù¬8 Î Ù4Ù`ÓY8 Ñ ÓÕ1Ú[Ó× Ñ EE ?4KFD|Mq× q Ð ÜÓÙ UάРÓqÔÓ×w8 Î ÞS Î Ú Ñ ÓDÞÓDÙ Ñ × i ÚBÛOàÏÕ1Ü Ð ÓÚ?××Õ1ÚÔ4Ó×x8 Î Þ iS Î Ú Ñ ÓDÞÓDÙ Ñ × i ÚBÛDàKÕ1Ü Ð ÓÚO×á
®s¯°©±.°©²i³µ´³¶°©· ¯Rº|»Z ¼]\ ¾C3wÅVÆ@o¾n/ ËÇ/n/ÁË1ÀÁϾ`ÃÂX5HÃÒÄ V¾wÅ ¿ ÁKÃ7 Ó[Ô©ÕÝ!ÜäÔ q Õ1Ù¬8 Î Ù4Ù`ÓY8 Ñ ÓÕ1Ú i ÚBÛOàÏÕ1Ü Ð ÓÚÓD× Ñi ÚBÛOàÏÕ1Ü Ð ÓÚá
®s¯°©±.°©²i³µ´³¶°©· ¯Rº|»k ¼-,dÇ/n/ÁË1ÀÁϾ`ÃÒÂÌC3nm`¿'À¬¾4ÁKÅ'ÂqÁÀÂ5ÃWÄ Q¾`Å 3KÃ)¿&ÀÄ7 ÓD×"8 Î Ù4Ù©ÓG8 Ñ ÓÕ1Ú?×gÏÝ u!ÐÒÑ ÕÓÜ6×d× Î Ù Ñi ÚBÛOàÏÕ1Ü Ð ÓÚ?×iá
å ACIJ58Ed?B<T:G?HDFI/5sæ è g6rsab~;flenSa cleRpZq=rsafpafg q/|ki= rsad^
Simp((G1&G2)⊥p)⊥s = Simp(G
⊥p
1 ⊕G⊥p
2 )⊥s
= Simp(G⊥p
1 )⊥s&Simp(G⊥p
2 )⊥s
= Simp(G1)&Simp(G2)
3j>'qSjkc-cj ?;awj>iOGiSj>r 8=rsa d|kg=gWaf-afi= uadcl+=|kguj>iWc-cle m -?@i=rResafB^ è g6q=-| <d=aw=abp j>g=e=<f-auc-eRpZeRrj>eR-avq/|ki='rsadc'j>i=l-adc d|kg=gWaf-afi=-cB^
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GB D "8 tuR" 1"$6 *!8¬ *(s¬« ®¯°Ï±.°©²i³¶´¬³|°©·Q¯Rº|»¸ ¼j/KËO3Kà /C3Ài¿&Çc/nÁ$ÀÆ`Á$ÂÂHÁ4¿^/Âq¿6ÈnÉ`Å&Á$Â$ Î Õ!Ú ÑkÎ Õ Ñ 8 Î ÞyS Î Ú Ñ ÓÞgÓÙ Ñ SwÚ Îvu Ý uÐ Ü Ð × Ñ Ó G !
Simp(|G|) = |Simp(G)|
å A%I/5Ed?B<>:G?HDFIJ5qæ è g6p|kg l-awrjZ=|ki (=rsaveRgfrRiWcles|kg^ ç |keR D ∈ Simp(|G|)
^è gOj
D = |D|G = ∩D′ ∈ G|D′ ⊆ D = ∩D′ ∈ Simp(G)|D′ ⊆ D = |D|Simp(G)`b|kgD ∈ |Simp(G)|
^ ç |keR D ∈ |Simp(G)|
^è gOj
D = |D|Simp(G) = ∩D′ ∈ Simp(G)|D′ ⊆ D = ∩D′ ∈ G|D′ ⊆ D = |D|G`b|kgD ∈ Simp(|G|)
^$QJ¥P?T4C6AO>/F>U0JPO>H@HAOC;Y >xL4UC6\T>/T4Y>fLKJ1N)ADC>yF4>/U0JF4C[W,@HT4U6AOS/F4>/U0J@iJ1NOJ1@HAOSN?CPDJADC;1YÊF4>/U0J
POT4L©>N?L`;P?C6AOC;Y>Y~XSHY4SNOJ1UV>HA>BxL4U6C@HC6AD>eU>@iJ!PF>PF4>P?PO>C6Y4P2pKY4CPHb$QJY4;1ADC6;Y POT4CM¦!J1YAD>,L4NO;¬¦C>HY$AF4>,U0JgNO>H]J!NO\$T4>y\$TR:'TY @;>YW,@C6>YAF4>/L©;C6F4P[F4C6¨`SN?>YAsF4>
1FKJ1Y4PT4YF4>P?PO>HCYzYw:'>HP?ALKJ1Pd¦CP?C6ADSL4J1N2T4Y¥F>POP?>C6Y¥;N)ADI4;1X;YKJ1U.|L4N?;L`;1POC6AOC;Ygªb LKJ!X> ¬« ¡b')(+*·ä³µ´³¶°©·Q¯Rº|»vkn¼ zdÃWÇ ÁË1Ài¿&Ç/ ¿6ÈnÉwÅ'ÁzËÇVÈnÉ`ÅÀÁa Ù`Ó_J r M±¸hD r e BGDIH>JLKNM ± r H_JK C¸XM Ô q Õ1Ù Ô4ÓD×O×BÓ Ð ÙfSwÚ Îvu Ý u!Ð Ü Ð × Ñ Ó ÓD× Ñ Õ1Ù`Ó9SwÚ Î ÓG8 Ñ|Ð|Î Ù× Ð ÞySwܶӥSwÚ ÐNU ÛÓyÔ4Ó×Ý8 ÑoжΠÙ4×¥SwÚ Î SwÚBÓD×)P|Ó Ñ Ô4ÓsܶÓÕ1Úy8 Î Ù ÑoÐ Ù`ÕÝ ÑoжΠÙvRÔ Î Ù Ñ Ü¶Ó×y8 Î Ó;:<8 Ð ÓÙ Ñ ×S`ÚBÛ iØÖÏÓ×× Î Ù Ñ ÔÓ¥S ÎÐ Ô$×tÔ Ð ÛDÚBÓDÙ Ñ ×sÔ4Ó 1 Ý Ð Ù4× Ð ãÕÓÔ4Ó×tÝ8 Ñ|Ð|Î Ù×Ù`ÛOàÝ Ñ|ÐNU ÓDצSwÚ Î SwÚÓ×eÞÝÖ Ð ÞÝ!ܶÓ×Õ1Ù`Ó¶ ÎÐ ×dܵÝwSwÚÓÞ Ð ÚÓ Î SÛÚÝ ÑoÐ|Î Ù~Ý8g`Ó U ÛÓ¬á
~<4¯%c>¯Rº^¯c ;;¬X¹n +1I c.(D1, . . . ,Dn)
*X KW (c) < 1
¥wyvKtDiBIW Xtt B* @ dµ1X Xw"n +1Iµ1
Fid− ;;¬XTK XµI
c.(D1, . . . ,Dn)*X K
W (c) < 1gwvK¥tDiBIW9Xt
t B* @<dµ1X Xw¦K XµFµ1Fid
CY4P?CWhU6>P2L4N?;=?>@BADC;1Y4PuPOC6]/L4U6>PuF4>
0.4
(+, ξ, 1)
(−, ξ.1, 1)
0.4
0.3
(+, ξ.1.1, 1; 2)
0.7
z
0.6
(+, ξ, 1; 2)
(−, ξ.1, 2)
0.2
(+, ξ.1.2, ∅)
0.8
(+, ξ.1.2, 42)
(−, ξ.1, 3)
z
(−, ξ.2, 4)
(+, ξ.2.4, ∅)
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¬« / !"#$#&%' () *,+LH" 1"$6 *!8¬ *(H ¦H"88GB *!(PO;1Y$A
(+, ξ, 1)
(−, ξ.1, 1)
(+, ξ.1.1, 1; 2)
(+, ξ, 1)
(−, ξ.1, 1)
z
(+, ξ, 1; 2)
(−, ξ.1, 2)
(+, ξ.1.2, ∅)
(−, ξ.1, 3)
z
(−, ξ.2, 4)
(+, ξ.2.4, ∅)
(+, ξ, 1; 2)
(−, ξ.1, 2)
(+, ξ.1.2, 42)
(−, ξ.1, 3)
z
(−, ξ.2, 4)
(+, ξ.2.4, ∅)
>BA2PO>PL4N?;=?>@BADC;1Y4P2POC],L4U6>Pd@H;]/LU^HAO>P2PO;1Y$A
(+, ξ, 1)
(+, ξ, 1; 2)
(−, ξ.1, 2)
(+, ξ.1.2, ∅)
(−, ξ.1, 3)
z
(−, ξ.2, 4)
(+, ξ.2.4, ∅)
(+, ξ, 1; 2)
(−, ξ.1, 2)
(+, ξ.1.2, 42)
(−, ξ.1, 3)
z
(−, ξ.2, 4)
(+, ξ.2.4, ∅)
$V>P2L4N?;=?>@BADC;1Y4P2POC],L4U6>Pd@H;]/LU^HAO>P2F4>
(+, ξ, 0)
0.3
Skunk
0.5
(−, ξ.0, 0)
z
P?;YAN?SF4TC6AD>HPm
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GB D "8 tuR" 1"$6 *!8¬ *(s¬«
(+, ξ, 0)
Fid−
J1U;N?Pu\$T4>[PO>PL4N?;=?>@BADC;1Y4P2POC],L4U6>PdP?;YA
(+, ξ, 0)
Fid−
(+, ξ, 0) (+, ξ, 0)
(−, ξ.0, 0)
z
®¯°Ï±.°©²i³¶´¬³|°©·Q¯Rº|»¯n¼ zdÃÒÇ ÁË1Ài¿ Ç/4 ¿ÈzÉwÅ'ÁÂqÆ`ÁzÆ`Á$ÂÂHÁK¿[/4ÂÆ`ÁÊÉKÃWÇ kÇ/KÆ`ÁK¾`à L/`¿&Á á ÍÏÎÐWÑ D ÓD× Ñ Õ1ÙfÔ4Ó×?×BÓ Ð ÙgÔ4Ó@SwÚ Î ¶ Î Ù`Ô4ÓÕ!Ú.Ø4Ù Ð Ó¬á 7 ÓuÔ4Ó×?×BÓ Ð Ù D ÝGS4SVÝ!Ú ÑoÐ ÓÙ Ñ/, Õ1Ù 8 Î ÞS Î Ú iÑ ÓDÞÓDÙ Ñ SwÚ Îvu Ý u!Ð Ü Ð × Ñ Ó G × Ð Ó Ñ ×BÓ¬Õ1Ü|ÓDÞÓDÙ Ñ × Ð`Ñ*Î Õ Ñ Ó××Ó×S`Ú Î ÓY8 ÑoжΠÙ4×× Ð ÞSwÜ|ÓD×8 Î ÞSwÜ ¬Ñ Ó×ÝGS4SRÝ!Ú Ñ|Ð ÓÙÙ`ÓÙ Ñ , G á
á ´ ÓSwÜWÕ1× !Ï× Ð D ∈ G !ÏÜ|ÓD×S`Ú Î ÓY8 ÑoжΠÙ4×u× Ð ÞSwÜ|ÓD×eÔ4Ó D × Î Ù Ñ ÔÝ!Ù4× G áå A%I/5Ed?B<>:G?HDFIJ5qæ
_@^2è g j>eR i=gWaweRgWGifles|kgOcli= rjADSj>i=-afi=h=a
D= Wg=esauj>eRgWc-e
h :
0 7−→ 0z 7−→ 0zξ 7−→ 0(s1.D1, . . . , sn.Dn) 7−→ 1 + max h(Di)(+, ξ, I).(D1, . . . ,Dn) 7−→ 1 + max h(Di)((−, ξ, I).DI ) 7−→ 1 + suph(DI)
ç e h = 08Gj>rs|k-c
Dadcl i=g =fp|kg|ki
0=|kg
Dadclc-|kg i=g=eshCiWa'q=-| lafles|kgcleRpZq=rsaA@3afg
es=afgClenJj>g 0
Fid|ki
Fid−cli=eR~@j>g +rj[q |krj>leR- B ^
ç e h = n + 1af q/|ki=+-|ki==adc-c-afeRg
D′ =a+DSj>i=-afi= n 8=af'-|ki= d|kpZq/|kl-afpafgC H 8 D′j>q=qSj>llesafgC Hcleaf'c-afi=rsafpafgC clecadcq=-| afles|kgWc cleRpZq=rsadc d|kpZq=r=<f-adc c-|kg 'Wj>gWc
H^
è gq=-| <d=avqSj> Bjkc+cli= rj |klpaw=aD^
jkc _@^ D = (s1.D1, . . . , sn.Dn)^
ç e ∑i si < 1
j>rs|k-cDgzj>q=qSj>llesafgC j>ifi=g d|kpZq |kl-afpafg B^©è 8 c-adc q=-| afles|kgWc
cleRpZq=rsadc d|kpZq=r=<f-adcc-|kg -dGi=eR-adc vrjwGeR~;af ?;afgda&@Fq |@cleRleR~;a |ki gW? j>leR~;a c-i=eR~kj>gC rjq |krj>leR- BhCi=egzj>q=qSj>llesafg=gWafg (=esafgZj>ifi=g d|kpZq |kl-afpafg B^è gc-i=q=q/|@c-av=|kgwqSj>+rj[cli=eR-awhCiWa ∑
i si = 1^
ç e D adcl q |@cleRle l^è gj[q |ki= -|ki='=adc-c-afeRg E ∈ G⊥
JD,EK = (s1. JD1,EK , . . . , sn. JDn,EK)ç i=q=q |@c-|kgWc
D ∈ G^è gj
W (JD,EK) = 1^`[zj>q=E<dc rjuq=-|kq |@cleRles|kgG^ ]\qSj ?;a T]58
|kg =dGi=eR\hCiWaq |ki=u-|ki=i8W (JDi,EK) = 1
^`b|kg988qSj>ueRgWGifles|kg 8Mrsadcuq=-|>mlafles|kgWcZcleRpZq=rsadc d|kpZq=r=<f-adc=a
Di
c|kg Wj>gWcG^Qè g d|kgfrRi=afg -afp j>-h iSj>gC
hCiWa+rsadc q=-| lafles|kgWccleRpZq=rsadc d|kpZq=r=<f-adc =adcDi
c-|kg a6Wj f-afpafg rsadc q=-| lafles|kgWccleRpZq=rsadc d|kpZq=r=<f-adc'=a
D^
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i / !"#$#&%' () *,+LH" 1"$6 *!8¬ *(H ¦H"88GB *!( feRq=-|ChCiWafpafg 8c-e
D /∈ G8JeRra6Gescl-a\i=g
E-afrh iWa
W (JD,EK) < 1^3M|ki=vi=g
-afrE8 eRra6esc--a i=g
i-afrhCiWa
W (JDi,EK) < 1^ 3j>veRgWGifles|kg 8|kg| (=lesafg ui=gWa
q=-| lafles|kgcleRpZq=rsa d|kpZq=r=<f-aw=aDhCi=eg adcl qSjkc'Wj>gWc
G^
ç e D adc- gW? j>le ^1 a =adc-cafeRgEadcl=a rj 3|klpa
c.(E1, . . . ,Ek)| rsadc
(Ei)c|kg c-afpZenmocleRpZq=rsadcB^è g
jJD,EK = c.(c1, . . . , cn)
| 8Gq |ki=+-|ki=i ∈ 1, . . . , k
8
ci = (s1. JD1,EiK , . . . , sn. JDn,EiK)è g d|kgfrRi=afgq=-|ddWj>gC d|kpZpavq |ki=+rsa Bjkc q=-dd=afgC&@
Dq |@cleRle B ^
jkc'yG^ D = (+, ξ, I).(D′1, . . . ,D
′n)^
ç i=q=q/|@c-|kgWcD ∈ G
^ ç |keRE ∈ G⊥ ^ 1 ab=adc-cafeRg E
adcl=a rj |klpac.(E1, . . . ,Ek)
| rsadc
(Ei)c|kg 'c-afpZenmoc-eRpZq=rsadcB^
DSjkh iWaEicB fleR
Ei = . . . (−, ξ, I).E(i)I . . .
q/|ki='-|ki= i 8 W(JE(i)I ,DiK) = 1
^1 eRgWGifles|kgcBzj>q=q=rResh iWa[j>i;6
Di
d|kpZpa\q=-dd=afpZpafg wWj>gWcrsadc d|kpZq/|kl-afpafgC-cjk=dh iSj>-cB^
'feRq=-|h iWafpafgC 8=rsa BjkcD /∈ G
cablj>eR-aw=avp j>g=e=<f-awcleRpZeRrj>eR-a@^ jkc'G^ D = ((−, ξ, I).D′
I)I^è g6q=-| <d=aw=abpfpa@^
yG^wa Bjkc+c-av=dGi=eRGi Bjkc1Bj>+rsadc d|kpZq |kl-afpafg -cc-|kgC frs|@c qSj>'|k-G-awa6G-afgWcles|kg=gWafr ^
®s¯°©±.°©²i³µ´³¶°©· ¯Rº|» ¼Ë1Ài¿&Ç/4 K¿&¿'ÀÄÁ$ÂÍÏÎÐÒÑ
G Õ1Ù8 Î ÞS Î Ú Ñ ÓDÞÓDÙ Ñ × Ð ÞyS`Ü|Óá 7 ÓD×~Ô4ÓD×O×BÓ Ð Ù4×~Ô4Ó G Ó Ñ 8ÓÕÖ Ô4Ó G⊥p⊥pU¬Ð × ÐÒÑ ÓÙ Ñ Ü¶Ó×
ÞßDÞÓD×Ý8 Ñ|Ð|Î Ù4×u×Õ1Ú2Ü|ÓD×ÔÓ×O×Ó Ð Ù4×[ÔÓ G⊥s áå ACIJ58Ed?B<T:G?HDFI/5sæ&1 a d|kpZq |kl-afpafg
GadclveRgfrRiWcwWj>gWc
G⊥p⊥p=|kgZrsadcwj fles|kgWcv~escleR-dadcwcli=
rsadcu=adcc-afeRgWcw=aG⊥sqSj>wrsadc =adc-c-afeRgWcw=a
Gc-|kg eRgfrRiWc-adcuWj>gWcursadc j fles|kgWcu~CescleR-dadcuqSj>wrsadc
=adc-c-afeRgWc[=aG⊥p⊥p
^ 1 jO-feRq=-|h iWac-a q=-|ki=~;aafgqSjkccj>g [qSj>[rsadc[=adc-c-afeRgWc[=a q=-| |kgW=afi=Wg=esa@^ç |keR
σ@Fr | fi=l-afgda% B i=gWa%j fles|kg i=g =adc-c-afeRg
E ∈ G⊥s~escleR-daqSj>i=g =adc-c-afeRg
D ∈ G⊥p⊥p^ afl-a6j fles|kg adc-\~escleR-daafg -afpZqWc[Wg=e 8 @ adcl m TmoGeR-a6hCi i=g c-|kiWc mo=adc-c-afeRg Wg=e
D0=a
D~escleR-aZj>iWc-cle dafl-aj fles|kg^ ç |keR
D′ i=g%=adc-c-afeRg d|kgC-afgSj>g D0cli=q flesafi=
Dq/|ki=
r |k-G-aa6-afgWc-es|kg=gWafr @Fi=gWa d|kgWc-llifles|kg q/|@c-c-e (=rsa adclZ=a j |ki=-afZ-|ki=-adc[rsadcZj fles|kgWc[gW? jTmleR~;adcbq |@c-cle (=rsadcvWj>gWc
D0afv=a[pafll-aZ=adcv=fp|kgWcuj>q=E<dcCB ^©è g%j| (=-afg ii=g=adcc-afeRg
D′ =aq=-| |kgW=afi= Wg=esa ~escleRj>g σ^3j>[rjq=-|kq |@cleRles|kg G^R_6qSj ?;a q=-dd=afgC-a98|kg =dGi=eR[h iWa rsadc
q=-| lafles|kgWc cleRpZq=rsadc d|kpZq=r=<f-adc=aD′ c-|kgC Wj>gWc G⊥p⊥p
^W`bavq=rRiWc 8Sq=i=esc-hCiWaD′ ~escleR-a σ
8=i=gWa=a+c-adc q=-| afles|kgWc cleRpZq=rsadc~escleR-a
σ^@`b|kg eRrSa6Gescl-ai=g =adc-c-afeRgc-eRpZq=rsa+=a
G⊥p⊥ph i=e=~escleR-a
σ^
è G⊥p⊥p
afG|kg rsadc+pfpadcb=adc-cafeRgWc'cleRpZq=rsadc&@Fq=-|kq/|@c-eRles|kg G^qSj ?;a _B B'=|kgweRr8a6Gescl-a
i=g=adc-cafeRg6cleRpZq=rsaw=aGhCi=e~Cesc-eR-a
σ^
~<¯n9¯Vº x ;K ;¬tDitBpW;B¨td^|KXCG+G tB¢Ly;;¬X
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GB D "8 tuR" 1"$6 *!8¬ *(si4
0.5
zξ
0.5
(+, ξ, 0)
(−, ξ.0, 0)
0.5
zξ.0.0
0.5
(+, ξ.0.0, 0)
;B¦+dCxB KXNK XµIW_x XC¬XG; DK;qpKYC>P2xi
c0 = zξ c1 (+, ξ, 0)
(−, ξ.0, 0)
zξ.0.0
cn (+, ξ, 0)
(−, ξ.0, 0)
(+, ξ.0.0, 0)
(−, ξ.0.0.0, 0)
(+, ξ.0.0.0.0, 0)
(−, . . . , 0)
zξ.0.0...
X¤ R;Bp1;X@+dC¤2B KXNK XµIW XC¬XGCBFµow;cn
>HA/ DÏW
c∞ (+, ξ, 0)
(−, ξ.0, 0)
(+, ξ.0.0, 0)
(−, ξ.0.0.0, 0)
(+, ξ.0.0.0.0, 0)
(−, . . . , 0)
¦dKID;¢y;;¬X C|th CXtC9
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i, !"#$#&%' () *,+LH" 1"$6 *!8¬ *(H ¦H"88GB *!(
12
zξ
1− 12
(+, ξ, 0)
(−, ξ.0, 0)
14
zξ.0.0
1− 14
(+, ξ.0.0, 0)
X+CdFWXK §B¼KXNK XµFW_ci | i ∈ N ∪ ∞
d^¦C Xxci | i ∈ N
dwKF K§Wf;BBFXWBkXXKLK XµI |KXt+dK¢ew Xw Xw2n;BC§¬Byd lBv |¦K ¦CdxB KXNK XµIW
¨ « ®J* « « u, !-) « & ! ®v\. *®J.,4 .\. « & « ,v®J. &#.®v
#ä;1T4NF4;YY4>NTY4>dSHAOI4C\$T4>r@;],L4U^BAD>rJ1Txq@;Y4Y>@HAO>T4N?P.]T4UMADCLUC@J!ADCME¶PhH;YNO>HL4NO>HY4FyU0JY4;1ADC6;YFR:5CYF4SL©>Y4F4J1Y4@>[F4>sU0JU6T4F4C6\T>POC],L4U6>y|P?>@HAOC;Y b&ªb « LKJ1X> 4 ¡b
¤ Y F4SHp4Y4C6AGUJeY4;1AOC;YF>uNOSL©>N)AD;C6NO>dFw:'T4Yy@H;]/L©;N)AD>],>YA.F>u]J1YC^N?>dPOC6]/C6U0J1C6NO>GmtUJtUT4F4C6\$T4>P?C],L4U>y¶F4SHp4Y4C6AOC;Y b LKJ1X1> ¡Bb
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GB D "8 tuR" 1"$6 *!8¬ *(si«')(+*·ä³µ´³¶°©·Q¯Rº|»$¸ ¼GÄKÉ©ÁÏÃoÀÇR¿ÃWÁ Æpo¾n/ ËÇQÈzÉ`Ç©ÃWÀÁKÈgÁ/À
â 7 Ó) J¥M¸ r D OM/Ô q Õ!Ù>8 Î ÞyS Î Ú Ñ ÓÞgÓÙ Ñ S Î × ÐWÑoÐ ¶ G Ó× Ñ Ü q ÓÙ×BÓÞ u Ü|Ó[Ô4ÓD×ÚDÝ!Þ Ð Øµ8BÝ ÑoÐ|Î Ù4× IÑ ÓDÜܶÓ×ãÕÓeÜ|ÓÔ4ÓD×O×Ó Ð Ù~×iÕ ÐNU Ý!Ù Ñ ÝGS4SVÝ!Ú ÑoÐ ÓÙ4Ù`Ó , G
(+, ξ, I)
(−, ξ.i1, J)
z
(−, ξ.i1, J′)
z
(−, ξ.in, J)
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(−, ξ.i1, J′)
z
Î
I = i1, . . . , inJ, J ′ SVÝ!Ú8 Î Õ1ÚÓÙ Ñ Pfin(N)â 7 Ó ¼J¥M¸ r D +MÊÔ q Õ1Ù 8 Î ÞS Î Ú Ñ ÓDÞÓDÙ Ñ Ù©ÛDàÝ Ñ|Ð ¶tÓ× Ñ Ü q ÓÙ4×ÓÞ u ܶÓÔ4Ó×yÚÝÞ Ð Øµ8BÝ Ñ|Ð|Î Ù× I
Ñ ÓDÜܶÓ×ãÕÓeÜ|ÓÔ4ÓD×O×Ó Ð Ù (−, ξ, I)
z
Ó× Ñ.Ð Ùµ8ÜWÕ1×eÔÝ!Ù4×2Ü q Ð Ùµ8BÝ!Ú?ÙÏÝ Ñ|Ð|Î Ù¥ÔÝ!Ù4× G Ô4Ó
Dai− = (−, ξ, J)
z
áHáá (−, ξ, J ′)
zÎ
J, J ′ SVÝ!ÚX8 Î Õ1ÚÓÙ Ñ Pfin(N) áâ Ü|Ó+ J¥M¸ r D OM/Ô q Õ!Ù`Ó[Û Ñ g Ð ãÕÓ[Ó× Ñ Ü¶Ó2ÚBÛ|S.ÓDÚ Ñ*άРÚBÓÔÓt× Î Ù u!ÐjikÎ Ú Ñ g Î à Î ÙKÝ!ÜÒᮯ°Ï±.°©²i³¶´¬³|°©·Q¯Rº|»¥¼,¦3KÃ3KË1ÀÄKÃ?¿&ÂG3Ài¿&Ç/ ÆZ¾ ÃWÄKÉ`ÁKÃoÀÇV¿6ÃÒÁzÆpo¾n/ ËÇVÈnÉ©Ç©ÃWÀÁKÈgÁ/À,ɩǩ¿&Ài¿ 7 ÓeÚÛSÓÚ Ñ*ÎÐ ÚBÓ[Ô q Õ1Ù¬8 Î ÞyS Î Ú Ñ ÓÞgÓÙ Ñ S Î × ÐWÑoÐ ¶ G ×Õ1ÚÜµÝ u Ý!×BÓ ` ξ Ó× Ñ Ü q ÓDÙ4×BÓDÞ u Ü|Ó[Ô4ÓD×eÚDÝ!Þ Ð Ø i8BÝ Ñ|Ð|Î Ù× I Ñ ÓDÜãÕ q Õ!Ù Ô4ÓD×O×BÓ Ð Ù~×BÓDÞ Ð^i × Ð ÞyS`Ü|ÓeÔÓ G 8 Î Þ,ÞgÓÙµ8HÓSVÝ!Ú (+, ξ, I) á
å A%I/5Ed?B<>:G?HDFIJ5qæ 3Mj>[p|kgW|k-|kg=esa @Fq=-|kq |@cleRles|kg G^zy qSj ?;a_@_@_ B88rsa -fq afl-|keR-a i=g d|kpZq |k m-afpafg q/|@cleRle +cli=urj (Sjkc-a
` ξd|kg lesafgC[r afgWc-afp (=rsa =adcwj>pZenBj>les|kgWc
I-afr hCi i=g=adc-cafeRg
c-afpZenmocleRpZq=rsaw=aGd|kpZpafgdawqSj>
(+, ξ, I)^
1 j[-feRq=-|hCiWawadcl+f~es=afg -a =c-ersav=adc-c-afeRg
(+, ξ, I)
(−, ξ.i1, J)
z
(−, ξ.i1, J′)
z
(−, ξ.in, J)
z
(−, ξ.i1, J′)
z
j>q=qSj>llesafg Gj>rs|k-ceRr 'jA(=esafgi=g=adc-c-afeRg6c-afpZenmocleRpZq=rsav=a
Gh i=e d|kpZpafgdavqSj>
(+, ξ, I)^
®¯°Ï±.°©²i³¶´¬³|°©·Q¯Rº|» n¼vÄKÉ©ÁKÃWÀÇR¿6ÃÒÁ Î Õ Ñ 8 Î ÞS Î Ú Ñ ÓÞgÓÙ Ñ ×iÕ1ÚsÕ1Ù©Ó u Ý!×BÓÕ1ÙÏÝ Ð ÚBÓeÝÜ|ÓÞgßÞgÓÚBÛ|S.ÓDÚ ÑkÎÐ ÚÓã4Õ$Óe× Î Ù Î Ú Ñ g Î à Î ÙKÝ!ÜÒá
å A%I/5Ed?B<>:G?HDFIJ5qæ[ç |keRGi=g d|kpZq |kl-afpafg q |@cleRle Scli=rj2(Sjkc-a
` ξ^ ç |keR
Ii=gWaj>pZenBj>les|kg=a
c-|kg6-fq afl-|keR-auafDi=gO=adc-c-afeRg d|kpZpafgBj>gCqSj>
(+, ξ, I)Wj>gWc
G^ 1 aw-dc-aBj>i
D, |Dai−|G⊥d|kg ~;af ?;a =|kgIj>q=qSj>llesafg vj>i-fq afl-|keR-aw=a
G⊥ ^
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i1 !"#$#&%' () *,+LH" 1"$6 *!8¬ *(H ¦H"88GB *!('feRq=-|hCiWafpafg 8Gcle
IadclWj>gWc rsa'-fq afl-|keR-ab=a
G⊥ j>rs|k-c (−, ξ, I)adcl~Cesc-eR-'Wj>gWc
Dai−qSj>
i=g=adc-c-afeRg=aGafb=|kg @FqSj>brj q=-|kq/|@c-eRles|kg G^zy qSj ?;a_B@y B eRr'j i=g=adc-c-afeRgc-afpZenmocleRpZq=rsa
Wj>gWcGhCi=e d|kpZpafgdawqSj>
(+, ξ, I)^è gafg6=dGi=eR hCiWa
Ij>q=qSj>llesafg vj>i-fq afl-|keR-aw=a
G^
' (+*·Z³µ´³|°©· ¯Vº|»¯n¼,rÇQÈzÉ`Ç©ÃWÀÁKÈgÁ/ÀÂÆQ¿& ÇR¿^/ÀÂÁÀ ËÇ/n/Á+°Á$Ââ ´ ÓÕÖ_8 Î ÞS Î Ú Ñ ÓDÞÓDÙ Ñ ×u× Î Ù Ñ bnDIB dr DIep¸ Bt× Ð Ü|Ó¬Õ1ÚO×uÚÛSÓÚ ÑkÎÐ ÚÓ×2× Î Ù Ñ Ô Ð × ÎÐ Ù Ñ ×iáâ a Ù`ÓÛ Ñ g Ð ãÕÓ E ÓD× Ñ ± r eeLM³@M/× Ð × Î ÙzÚBÛSÓÚ Ñ*ÎÐ ÚBÓÓ× Ñ Õ1Ù × Ð ÙàÜ¶Ó Ñ*Î Ù I á 7q ÓDÙ4×BÓDÞ u ܶÓI ÓD× Ñ ÝGS4SÓܶÛÜµÝ ACH D¨±An¸ D r e ÔÓ E á
®s¯°©±.°©²i³µ´³¶°©· ¯Rº|» ¼-,¦3KÃN34Ë1ÀÄÏÃ)¿ ÂY3Ài¿&Çc/ Æ`Á$ÂyËÇVÈnÉ©Ç`ÃWÀÁKÈgÁ/À ÆQ¿& ÇR¿[/ÀÂâ ´ ÓÕÖ8 Î ÞyS Î Ú Ñ ÓÞgÓÙ Ñ ×¦S Î × ÐÒÑ|Ð ¶µ×e× Î Ù Ñ Ô Ð × ÎÐ Ù Ñ ×e× Ð Ó Ñ ×BÓ¬Õ1Ü|ÓDÞÓDÙ Ñ × Ð Ü¶ÓÕ1Ú Ð Ù Ñ ÓDÚO×BÓY8 Ñ|Ð|Î ÙÓ× Ñ ÚBÛÔÏÕ ÐÒÑ ÓÝÕÖ£8 Î Þ uÐ ÙÏÝ Ð × Î Ù×rÜ Ð Ù`ÛBÝ Ð ÚBÓD×2ÔÓS ÎÐ Ô$× 1 ÔÓÔ4ÛÞ Î Ù4×2P[¶ Î 8Ý!Ü Ð ×BÛD× Î ÕÙ Î ÙvRáâ ´ ÓÕÖ_8 Î ÞS Î Ú Ñ ÓDÞÓDÙ Ñ ×uÙ`ÛOàÝ ÑoÐ ¶µ×u× Î Ù Ñ Ô Ð × ÎÐ Ù Ñ ×u× Ð Ó Ñ ×ÓÕ1ܶÓÞgÓÙ Ñ × Ð
∀D ∈ Pss(G), ∀E ∈ Pss(H), |D|G⋂
|E|H = ∅.
å ACIJ58Ed?B<T:G?HDFI/5sæ è g6p|kgCl-awrsadc+=afi;6eRpZq=rRe Bj>les|kgWc Wj>gWc+rsa Bjkc q |@cleRle ^⇒ç |kesafg
Gaf
H=afi;6 d|kpZq/|kl-afpafgC-c Gesc l|keRg -cH^ ç |keR
D ∈ G ∩ H^ ç |keR
D′ i=gWa[q=-|>mlafles|kg cafpZenmocleRpZq=rsa=a
D^.è g j
D′ ∈ G ∩ H^`[zj>q=E<dcrj%q=-|kq |@cleRles|kg G^R_ qSj ?;a
q=-dd=afg -a98D′ gWa d|kpZpafgdawqSjkc'qSj>+i=gWauj fles|kg6q=-|kq=-a@^ adcl+=|kgvi=g=fp|kg @ 3|>m
Bj>rResc-v|kigW|kgB ^è g6afg=dGi=eR+h iWab-|ki=-adcrsadcq=-| lafles|kgWc+c-afpZenmoc-eRpZq=rsadc+=aDc|kg +=adc
=fp|kgWc'af =|kgwh iWaDadc- i=gWa d|kp (=eRgSj>esc-|kg rReRgWBj>eR-aw=aw=fp|kgWcB^
⇐è g6i=leRrResc-abrjZq=-|kq/|@cleRles|kg G^R_ \qSj ?;avq=-dd=afg -a@^
;],]/>vL©;T4NZU>TNOPSH\$T4C6¦!J1U6>YADPä>Y UTF4C\$T4>vPOC6]/L4U6>1h¬U6>PLNO;L©;POCMADC6;Y4PQPOT4CM¦!J1YAD>PL`>HNO],>HAOAO>YAF4>[L4N?;T¦1>N2U0J @;1]/L4U6SHAOT4F4>CYAO>NOY>sL©;T4NuU6> *XC¬B¶L4NO;L©;P?C6ADC6;Y~ªb LKJ1X1> ¡Bb®s¯°©±.°©²i³µ´³¶°©· ¯Rº[Z ! ¼#"c°K¿&ÂHÀÁ/KËÁzÆpo¾n/Á ÉKÃWÇ ÁË1Ài¿&Ç/ÍÏÎÐÒÑ
M ⊆ N Õ!ÙzÚBÛD×BÓDÚ UάРÚiá Î Õ Ñ Ô4Ó×?×BÓ Ð Ù<S Î × ÐÒÑoÐ ¶ä×ÓÞ Ðji × Ð ÞSwÜ¶Ó D = (+, ξ, I).R Ó× Ñ ÛOàÝ!Ü ,DM~DN\M
ÎDM 8 Î Þ,ÞgÓÙµ8HÓSVÝ!Ú (+, ξ, I ∩M) Ó Ñ DN\M 8 Î Þ/ÞgÓÙt8ÓSRÝ!Ú (+, ξ, I \M) á7 Ó[Ô4ÓD×O×Ó Ð Ù DM Ó× Ñ Ü|Ó¦J r M4¸ fÔ4Ó D ×iÕ1Ú M á
.uÙnÛ Ñ ÓÙ©Ô 8Ó ÑOÑ ÓsÔÛØ4Ù ÐWÑoжΠÙnÓÙ S Î ×DÝ!Ù Ñ zξM= zξ á
å ACIJ58Ed?B<T:G?HDFI/5sæ è g6-afq=-afgWrjZq=-afi=~;aw=abrj[q=-|kq |@cleRles|kg_@^R_[qSj ?;av] G^
®s¯°©±.°©²i³µ´³¶°©· ¯Rº[ZV»¼]zuÃWÇ ÁË1Ài¿&Ç/ÍÏÎÐÒÑ
E Õ1Ù`Ó[Û Ñ g Ð ãÕÓx8 Î Ù4Ù`Ó?Ö©ÓÓ Ñ M Õ1Ù~ÚBÛD×BÓÚ UÎÐ Úiá/.dÙzÝ (E⊥⊥)M = (EM)⊥⊥ áå ACIJ58Ed?B<T:G?HDFI/5sæ&1 jq=-afi=~;a adcl cleRpZeRrj>eR-a dafrRrsauWj>gWc rsa BjkG-a\=awrjZrRiWGeshCiWauc-eRpZq=rsaeR G_q |ki=wrsadcu=adc-cafeRgWcuc-afpZenmocleRpZq=rsadcH^ 1 jq=-afi=~;a d|kpZq=r=<f-a adc- j>rs|k-cweRpZpdGej>-a afg%i=leRrRescj>gC rsadcd|kp (=eRgSj>esc|kgWc'rReRgWBj>eR-adcB^
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GB D "8 quR"88B *0(si ª')(+*·ä³µ´³¶°©·Q¯Rº|» ¼j/4Æ`ÄKÉ©Á/KƵ3µ/KËÁ´ ÓÕÖ~ÚBÛSÓÚ Ñ*ÎÐ ÚBÓD× R1 Ó Ñ R2 × Î Ù Ñ DIeLb J¥MdeLbACe@¸hB× Ð
∀I1, J1 ∈ R1, I2, J2 ∈ R2, I1 ∪ I2 = J1 ∪ J2 ⇒ I1 = J1, I2 = J2
´ ÓÕÖ_8 Î ÞS Î Ú Ñ ÓDÞÓDÙ Ñ ×u× Î Ù Ñ DFeb J¥Mdeb%ACep¸hBs× Ð Ü¶ÓÕ1Ú?×uÚBÛSÓÚ Ñ*ÎÐ ÚBÓD×× Î Ù Ñ.Ð Ù`Ô4Û|S.ÓDÙ`ÔÝ!Ù Ñ ×iá
, qg2§Qf$V>Pr@;Y4Y>@HAO>T4N?PuF4>eUJU6T4F4C\$T4>L4NO;1<KJ1<4C6UCP)AD>2XSHY4SNOJ1UC6PO>YAr@>TxgF4>U0JU6T4F4C6\T>ePOC6]/LU>|P?>@X&
ADC;1Y b&ª¥LKJ1X1> « ¡Bb$ZJ~Y4;1AOC;YÊF4>@H;]/LUSHAOT4F4>/CYAD>HNOY4>n|U>,NO>HPOL©>@BA F4>PqSHADIC\$T4>P@;],L4U6^HAD>HP¡F4;C6ABADN?>J!FKJ1LAOS>/|F4SBpKY4CMADC;1Y~ªb LKJ!X> ¬«1 ¡b
©¦ M)!-)+.$V>PF4SH@iJ1UJ1X>PF4>sF4>P?PO>HCY4P|F4SBpKY4C6AOC;Y¥ªb ¡dF4;1C6¦>HYA BADNO>sF4SBpKY4CPPOL©S@HC6pK\$T4>H]/>HYAtL©;T4NU>
@iJ1F4N?>LNO;<KJ!<4CU6CP?AO>d@iJ!N.CUP.C6YAD>NOJ1XC6POPO>HYArJi¦1>@U>PG@H;>XW/@HC>HY$AOPrmUJ[NOJ1@C6Y4>uF4>HP§F4>HPOP?>CYPbkUP.PO;1Y$Am,NOJ1L4L4N?;@I4>HN2F4>[U0JF4SBpKY4C6AOC;Y b ¬« LKJ1X1>sª b')(+*·ä³µ´³¶°©·Q¯Rº|»¥¼ \ÄËO3wÅ63 ÏÁzÆ@o¾n/ Æ©ÁÂHÂÁ4¿[/
â ÍÐ D ÓD× Ñ Õ1Ù Ô4Ó×?×BÓ Ð ÙTS Î × ÐÒÑoÐ ¶Ô4Ó u Ý!×BÓ ` ξ.i Ý!Ü Î Ú?×D =
(−, ξ, i).D × Ð D 6= Fid
Skunk × Ð Ù Î Ùâ ÍÐ D ÓD× Ñ Õ1Ù Ô4Ó×?×BÓ Ð Ù¥Ù©ÛDàÝ Ñ|Ð ¶.Ô4Ó u Ý!×Ó ξ.i ` Ý!Ü Î ÚO×
D = (+, ξ, i).D
')(+*·ä³µ´³¶°©·Q¯Rº|»n¼ \ÄËO3wÅ63 ÏÁ$ÂÍ Õ1ÚrÕ1Ù`Ó u Ý!×BÓÕ!ÙÏÝ Ð ÚÓ !1Ü|Ó F4S@J1U0J!X> Ô q Õ1Ù 8 Î ÞS Î Ú Ñ ÓDÞÓDÙ Ñ Ó× Ñ Ü|Ó u!ÐjikÎ Ú Ñ g Î à Î ÙÏÝ!ÜÔ4ÓDקÔ4ÛY8BÝ!Ü0ÝàÓ×Ô4Ó2×Ó×Ô4ÓD×O×Ó Ð Ù×iá 7 Î Ú?×BãÕÓ G Ó× Ñ S Î × ÐÒÑ|Ð ¶QÔ4Ó u Ý!×Ó ` ξ.i P NO>HPOLRb Ù`ÛDà$Ý ÑoÐ ¶Ô4Ó u Ý!×Ó ξ.i ` R ! Î ÙfܶÓÙ ÎÑ Ó G P NO>P?LRb G Rá
®¯°Ï±.°©²i³¶´¬³|°©·Q¯Rº[Z0Zn¼,dÇVÈnÉ`Å&ÄÀ¬¾4Æ`Án¿^/ÀÁKà /4Á É©ÇQ¾`ÃÅ&Á$ ƩÄËO3wÅ 3 ©Á$ÂÍÏÎÐWÑ
G Õ1Ù_8 Î ÞyS Î Ú Ñ ÓÞgÓÙ Ñ áâ ÍÐ G Ó× Ñ S Î × ÐWÑoÐ ¶ !ÏÝ!Ü Î ÚO× D | D ∈ G ÓD× Ñ Õ1Ù`Ó[Û Ñ g Ð ãÕÓx8 Î ÞyS`Ü Ñ ÓS Î Õ1Ú G áâ ÍÐ G Ó× Ñ Ù©ÛDàÝ Ñ|Ð ¶[Ô4Ó u Ý×BÓ ξ.i ` !dÝ!Ü Î ÚO× D | D ∈ G⋃
zξ ÓD× Ñ Õ!Ù`ÓzÛ Ñ g Ð ã4Õ$Ó8 Î ÞySwÜ Ñ Ó¨S Î Õ1Ú G áå qæ 38|ki=rsadc6=adc-c-afeRgWcc-afpZenmoc-eRpZq=rsadc 8+|kg j>eR rj pfpa%q=-afi=~;a%h i afg rRiWGeshCiWacleRpZq=rsaè g d|kgfrRi=bjd~;avrj[q=-|kq |@cleRles|kg z\qSj ?;aB@y
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i !"#$#&%' () *,+H" 1"$6 *!8¬ *(H H"88B *!( ©© N®;$V>P.@H;Y4Y4>H@HAD>HT4NOPvJ1FF6A6E¶PP?;YAGF4SHp4YP.>x4J!@HAD>H]/>HYA§F>uU0Je]],>r]/J1Y6^NO>r\T>uFKJ1Y4P.U6>d@iJ1FNO>
F4>[UJUT4F6\$T4>P6]/L4U6> b&ªb L4J1X> « b$QJg@;],L4USBADT4F> YAD>HNOY4> L©;T4NU6> 11>HP?AsP6]/L4U6>],>YAU6>]qjP?AO^NO>yF4>yUo:6Y4@iJ1N?YKJ!A;Y µ¦;6N
L4N?;L©;PMA;1Y b LKJ1X> 4 ¡b®s¯°©±.°©²i³µ´³¶°©· ¯Rº[Z0k ¼-,dÇVÈnÉ`Å&ÄÀ¬¾KÆ©Á ¿[/ÀÁKÃ/4Á É`ÇV¾`ÃÅ&Á$ Î Õ1Ú ÑkÎ Õ1×"8 Î ÞS Î Ú Ñ ÓÞgÓÙ Ñ ×eÔ Ð × ÎÐ Ù Ñ × G Ó Ñ H ×iÕ1ÚsÕ1Ù`ÓeÞgßÞgÓ u Ý!×ÓtÙ`ÛOàÝ Ñ|ÐNU Ó!
Pss(|G & H|) = Pss(|G|)× Pss(|H|)
å sæ&1 jbq=-afi=~;a'adcl cleRpZeRrj>eR-a dafrRrsa'afgZrRiWGesh iWa+c-eRpZq=rsaki=g =adc-c-afeRgZp j>-flesafr@3h i=eadclvcj q=-|kq=-a[eRgBj>lgSj>les|kgBbcafpZenmocleRpZq=rsa=a
G & HadclwcleRpZq=rsafpafg ur ! "$#&% '(#)
* ,+-(.' 0/ 132 34-55467298;:<=>?6@4A548B6DCE5?678BF3A@4HG34G4<GJI 132KG34-55467298;:<=>?6@4A548B6DCE5?678BF3A@4
G34H
$QJ @;],L4U6SHADTF4>KYAD>HNOY4>zL`;TNgU>£t + >P)A~;<AD>HY$T4> LKJ1NfUJ L4NO;LNSBADSnF> U0J F6P¶=?;Y@HA6;Y¶L4NO;L©;PL6A6;Y b ¬« LKJ1X1> ¡Bb®s¯°©±.°©²i³µ´³¶°©· ¯Rº[Z4¸ ¼-,dÇVÈnÉ`Å&ÄÀ¬¾KÆ©Á ¿[/ÀÁKÃ/4Á É`ÇV¾`ÃÅ&ÁNMPO,Q $ Î Õ1Ú ÑkÎ Õ1×"8 Î ÞS Î Ú Ñ ÓÞgÓÙ Ñ ×S Î × ÐWÑoÐ ¶µ×Ô Ð × ÎÐ Ù Ñ × !
G⊕ H = Cl(G ∪ H)
Õ Ñ ÚBÓÞgÓÙ Ñ Ô ÐWÑ !Pss(G⊕ H) = Pss(G) ∪ Pss(H)
å sæ[ç 678B67AR:67>4:1 S:G>4TG34UAR:$A71G6@V&14T5?678BF3A@4
~<¯n9¯VºXW¦¹wKBLC XnX d ^¬Xy;tD;K XµFWC k+§;;;¬XC/ dvwGtd^¬ n dD £;;¬XC
GB
HTXd ;Tn;BCiD;XXKvwn
Gtd^¬ n dDT<;;¬XCyÏXXK"GÊvwgGtd^¬ n dT<;;¬XC
ÏXXKH
© Y !N®|"! [Zp®;]\^3_`a `Lbcdb^3_afegih\^^bhj_be`kPafll _ ,m kone k\^3_plqjrJ^ kPaegB^ M¦ bafeslbkph\fc L \`L_bcdb^3_kut
v bkil M¨ qu`bu^_bk ¦ b`k \^kBle *XC¬Bwm \^3_ ^3_b`afx `lbuegylbukkLb ^k;bu^3_`b buegoz $ bkBl M¨ qu`bu^_bklqjrJ^ _ \f^kUle *XC¬B l\ M¦ bu^_Tl\f^h _`Lb;aflJa L _qbukae9hafl`bslbukh\.b W h b^3_kuz]b v a kb L afkLkb;k c L v bcdb^3_Ub^|e_ v kaf^3_ v a G n dµ le|_b^kLbe`uz
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')(+*·ä³µ´³¶°©·Q¯Rº|» n¼ Á Q7 Ó×tÔ Ð ÛÚÓÙ Ñ ÓD× U ÓÚ?× Ð¶Î Ù4×Ô©Õ Ñ ÓÙ4×ÓÕ1Ú× Î Ù Ñ Ô4ÛDØ4Ù Ð ÓרSVÝ!Ú u!Ð Ü Ð Ù©ÛBÝ!Ú ÐÒÑ Ûâ $ Î Õ1ÚyÔÓ×,ÔÓ×O×Ó Ð Ù4×q×ÓÞ Ðji × Ð ÞSwÜ|ÓD× !QÜ0Ý£8 Î Ù4× Ñ ÚÕ8 Ñ|Ð|Î Ù Ó× Ñ × Ð Þ Ð ÜµÝ Ð ÚBÓ , ÜµÝ 8 Î Ù4× Ñ ÚiÕ8 ÑoÐ|Î Ù
ÓÙ~ÜÒÕ$Ô Ð ãÕÓ× Ð ÞSwÜ|Ó§P¶Ô4ÛØÙ ÐÒÑ|Ð|Î Ù4× á SRÝàÓ Ó Ñ á SVÝàÓ Ráâ 7 ÓSwÚ Î Ô©Õ ÐÒÑqÑ ÓÙ4× Î Ú Ð ÓÜÔÓgÔ4ÓÕÖ 8 Î ÞS Î Ú Ñ ÓÞgÓÙ Ñ ×/ÓD× Ñ Ü q ÓDÙ4×BÓDÞ u Ü|ÓÔ4Ó×T8 Î Þ u!Ð ÙÏÝ Ð × Î Ù4×Ü Ð Ù`ÛBÝ Ð ÚBÓD×Ô4ÓD×S`Ú Î Ô©Õ ÐÒÑ × Ñ ÓDÙ4× Î Ú Ð ÓDÜ×eÔ4ÓD×SRÝ!Ú Ñ|Ð Ó×2×ÓÞ Ðji × Ð ÞSwÜ|ÓD×
Pss(
G⊗ H)
= Pss(G)⊗ Pss(H)
Ó Ñ Ô Î Ùµ8G⊗ H = Cl
(
Pss(G)⊗ Pss(H))
¤B^ L \e`L`a _KafekLk lqjrJ^ `e^ *XC¬B bu^3_`b lbukkLb ^k a`"! _`a `buk ^\^ m \`huqcdb^3_KkLbc &k c L v bk ¡ lb v a c a^ $# `bBke 6¦ af^3_b
c.(D1, . . . ,Dn)~ c′.(D′1, . . . ,D
′m)
= c.(
c′.(D1 ~D′1, . . . ,D1 ~D′
m), . . . , c′.(Dn ~D′1, . . . ,Dn ~D′
m))
\e
c′.(D1, . . . ,Dn)~ c.(D′1, . . . ,D
′m)
= c.(
c′.(D1 ~D′1, . . . ,D1 ~D′
m), . . . , c′.(Dn ~D′1, . . . ,Dn ~D′
m))
]bk lbe.g h&%\ g l\^^b`a bu^_ v bue('Ke^b ^\e ¦ b v,v b h v afkLk rha_ \^ lb~
b^3_`b lbkLkb ^k lJaf^kv*) bk L ` _lb v a h v afkk rJha_ \^ _bu^kbue`h\cdc e_a_ Rm,+ ^\^ hu\cdc e_-a&_ ,m z]b L b^lJaf^3_t v a l 6¨ q`bu^hbb^3_`LbBhbkUlbeg h&%\ g L \e` v b kXC¬Bv b^3_`LbBlbkLkb ^k^\^ kbc & k c L v bk ^ ) bkL_ L afk\-!kLb` ¦ a ! v bafe ^ 6¦ bafe lbk|h\c L \f`L_bucwbu^3_k bu_|\^ ^b kb L `qu\huhe L b L afk L v ek9a ¦ af^3_|l ) e^b q ¦ bu^3_eb v(v bh v akk rJha&_ \f^ z®¯°Ï±.°©²i³¶´¬³|°©·Q¯Rº[Z0¯n¼ 2Æ ÇR¿^/À´ Ó×Ý$Ô ÎÐ Ù Ñ ×SÓÕ U ÓÙ Ñ ÓÙµ8 Î ÚBÓ[ß Ñ ÚBÓ[Ô4ÛDØ4Ù Ð ×áâ 7 Ó _b^kbe`x3aeh&%b ~ Ñ ÓDÜãÕÓ
c.(D1, . . . ,Dn)~ c′.(D′1, . . . ,D
′m)
= c.(
c′.(D1 ~D′1, . . . ,D1 ~D′
m), . . . , c′.(Dn ~D′1, . . . ,Dn ~D′
m))
ÝgÕ1ÙzÝ$Ô ÎÐ Ù ÑJF,D1 ~D2K = JF(D2),D1K
â 7 Ó _b^kbe`l`L\ _ ~ Ñ ÓDÜã4Õ$Óc′.(D1, . . . ,Dn)~ c.(D′
1, . . . ,D′m)
= c.(
c′.(D1 ~D′1, . . . ,D1 ~D′
m), . . . , c′.(Dn ~D′1, . . . ,Dn ~D′
m))
ÝgÕ1ÙzÝ$Ô ÎÐ Ù ÑJF,D1 ~D2K = JF(D1),D2K
å/. qæ 3013>]G34-5 G34-55 46725 548B6DCE5?678BF3A@4-5 8A@4-5 :G 10672&<5 54G3="2236@55 42&< 46: <48s42&<G34AR:$88s4U8;:236=<>4TV1 I 42 A71G6@V145?678BF3A@4
33013>]G34-5 G34-5546725 2402 548B6DCE5?678BF3A@4-5 8 02 13<?67A76@54TA@4-5 :G 10672&<5]548B6DCE5?678BF3A@4-5]4< G34-5 &0f4:BC6@42&<5 ç 067<
F = cF(F1, . . . ,Fl)132 G34-554672d2=?&:<?6 è 2 F50 54
F(c.(D1, . . . ,Dn)) = c(cF.(F1(D1), . . . ,Fl(D1)), . . . , cF.(F1(Dn), . . . ,Fl(Dn)))
tel-0
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$ - $
0 Fi(Dj)
4-5?<A I :G 10672&< F013> G34-5 G34-5546725 5 48B6DCE5?678BF3A@4-5
$ bkl M¨ qu`b^3_bk ¦ b`Lk \^kle Cd k\^3_Tk c L v bcdb^3_UlqurJ^ bk L af`Ulea v _qfz®s¯°©±.°©²i³µ´³¶°©· ¯Rº[Z ¼-,dÇVÈnÉ`Å&ÄÀ¬¾KÆ©Á ¿[/ÀÁKÃ/4Á É`ÇV¾`ÃÅ&Á QÍÐ
G1 Ó Ñ G2 × Î Ù Ñ S Î × ÐÒÑoÐ ¶µ×Ó ÑÐ Ù`Ô4Û|S.ÓDÙ`ÔÝ!Ù Ñ × Ý!Ü Î Ú?× D1 ~ D2 | D1 ∈ G1,D2 ∈ G2 Ó ÑD1 ~D2 | D1 ∈ Pss(G1),D2 ∈ Pss(G2) × Î Ù Ñ Ô4Ó×tÛ Ñ g Ð ãÕÓD×"8 Î ÞSwÜ ¬Ñ ÓD×S Î Õ!Ú G1 ~G2 á
å . sæ 08B8s4HF013>AR: A71G6@V14H5?678BF3A@49802N13<?67A76@54 AR: F3> 0F50 5?67<?6 02GJI 46f6@5<42-4 G34F3> 0 ?4<?6 02 @(F3> 0F0 567<?6 02 iF.: ?4 B
© Z¨¨® [Zp®4$ bk n.eaf^3_ rha_be`Lk ¶¦ ek;h\fcwcdb lbk h\^^bhu_be`Lk afll _ Rm k ^r^ k ¡ kL\^3_ lqjrJ^ k lbwc a &
^ # `Lb af^Ja v \xeb kbhj_ \^ z gL afxfb « ¡ ' v a v el neb k c L v bfz $ bdneJa^_ rJha_bue`se^ M¦ b`Lkb v bukL_k c L v bucwbu^_Te^b ^3_b`kLbhj_ \f^Nlb h\c L \f`L_bucwbu^3_ka v \`Lkn.eb v b;neJaf^3_ rJha_be`Tbjg kL_bu^3_ b v bukL_e^b;e^ \^ h v \kLb L af` ! & \`L_"%\x\f^Ja v z
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! Z" #%$'& ( *),+-$.$.$/$.$.$/$0$.$/$.$.$/$.$0$.$/$.$.$/$.$0$/$.$.$/$.$.$0$/$.$.$&2143#%$65 7 689) .: ;+ . + <$0$.$/$.$.$/$.$0$.$/$.$.$/$.$0$/$.$.$/$.$.$0$/$.$.$&21=5#%$?> @ *+A);+CB.$/$.$.$/$0$.$/$.$.$/$.$0$.$/$.$.$/$.$0$/$.$.$/$.$.$0$/$.$.$&21ED
FHGJIHGLK M =A 0ONS:A76@5 :<?6 02 4< G3=PNS:AR:RQ4 GSGGSGSGGSGGTGGSGSGGSGGSGGTGSGGSGUK;V,WFHGJIHG XG3G67<?6ZY 5 GTGSGGSGGSGSGGTGGSGGSGSGGSGGTGGSGSGGSGGSGGTGSGGSGUK;V\[FHGJIHGJI ] 13A7<?67F3A76?NS:<?6ZY5 GSGGSGSGGTGGSGGSGSGGSGGTGGSGSGGSGGSGGTGSGGSGUK;V\^
#%$L_ @T` .:acbAd Bef+hgikjl89mC8ngikj.ef+ akoa Beikp%B+hjqi .:fa +$0$/$.$.$/$.$.$0$/$.$.$&H#=3FHG GLK Msr N;tRu6tRQ,vPw GSGGSGGSGSGGTGGSGGSGSGGSGGTGGSGSGGSGGSGGTGSGGSGUK;F FHG G Xyx2xHzL|zZY'w GTGSGGSGGSGSGGTGGSGGSGSGGSGGTGGSGSGGSGGSGGTGSGGSGUK;F\IFHG GJI ] uL|zL~2uLz?N;tR|zZYw GSGGSGSGGTGGSGGSGSGGSGGTGGSGSGGSGGSGGTGSGGSGUK;F FHG G %vPw tRA|z 2=N;tRnv 2 w GSGGTGGSGGSGSGGSGGTGGSGSGGSGGSGGTGSGGSGUK;F FHG GJV Cr vlOnwx2vs2zZN&0/~50 nvlvlAnw GSGGSGGTGGSGSGGSGGSGGTGSGGSGUK;F\V
#%$61 @T` .:acbAd Bef+`= d + $0$.$/$.$.$/$.$0$.$/$.$.$/$.$0$/$.$.$/$.$.$0$/$.$.$&H#1FHGJVHGLK 04xHzL|z 0x2vQAtRzL GSGSGGTGGSGGSGSGGSGGTGGSGSGGSGGSGGTGSGGSGUK;F\VFHGJVHG 0 | vPNl|z 0 GSGGSGGSGSGGTGGSGGSGSGGSGGTGGSGSGGSGGSGGTGSGGSGUK;F\FFHGJVHGJI 0/~2u r x2v GGSGGSGSGGTGGSGGSGSGGSGGTGGSGSGGSGGSGGTGSGGSGUK;F,W
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$ - - b L afkkafxb lb v a v el ,n.eb k(c L v b ' v a v el (neb L `L\-!Ja/! v ,kL_b(c L \kb lb _`af^k m \`cdb` v a
^\_(\f^9lb! h\c L \f`L_bucwbu^3_$afbh lqur^ ,_(\^ z L afxfb - b^e^bbu`k (\^"! L `\/!Ja/! #v (kL_b%$z a L `Lqkbu^hb$lb$hu\b%&wh'(bu^_k(,^h)R_b ';lqjrJ^ ,` e^b$^\f_(\^dlbl (k _-af^hub kbhj_,\^+*z L \e`L \ef\,(` L a` v bu`lbslbukkb'(^k L `L\h&%buk\fe9q v \,,x^qkuz
- ,^k0t v*) e^ m \`c.,_qUe_ v (kLbUe^b^\f_,\^,^3_b`afhu_/fblb0!l ,kL_af^hb1$Ekaf^k]hu\^3_`a2(^3_buk]kLe` v bklbukkLb)(^k 'Kl ,kL_-a^hb ^e v(v b z bk ! h\c L \f`L_bucwbu^3_k kL\^3_wlbk L a2(`Lbk h\fc L \kqubk l ) e^ h\cL \`L_bcdb^3_ L `L\-!Ja/! v ,kL_b bj_Bl ) e^b l ,kL_-a^hb kLe` v bukilbkLkb'(^k L a`L_,b v k$leyh\fc L \`L_bcdb^3_z bklbukkLb)(^k e^ m \f`cdbkHkL\^3_ v bukHlbukkb'(^k 'l ,kL_af^hb ^.e v(v bz43]bj__b ^\f_,\^ l ) e^ m \`c.,_q L bu`cdbu_lb lqurJ^ ,` v aw^\f_,\^lb xa2(^KlJa^k v b hafl`Lb L `\/!Ja/! v (k _b n.e l\^^b e^y`qkLe v _a_Blb hu\c L v q1_elb m \`L_b kbuhu_,\^5*z L afxfb * , 6(k71'896,kUlb v a v el ,nebsk(c L v bBbu_t L af`h\c L \fk,_(\^[t L \e`: - ;
2
bu_ : -<- 2
z
=afe m ,^l (ha_,\^ h\f^_`a2,`bt v buk!JafkbukkL\^3_Tke L4L \kLqbke^Ja2,`bukz
>@?)A BDC'EGFIHKJML(N a ^\f_(\^ lbBl(kL_af^hub v aPO v ekk (c.O v b ' l\^^bu`Ubk _hb v,v bikLe` v bukUh\.b'&dh),b^3_kuzRQ^bB^\f_,\^
lbsl ,kL_af^hbskLe` v bkUlbkkLb),^kbu^9lqh\fe v bzSUTWV4XIY[ZY]\^X`_Ga]b+cedfhg'i1jlk4Ë,m5g8nloqprm6gË,stmWu ËvfhmRki1g
wyxzl| ¸~,~^]%K 8'v+~]^ /q( ] x / 4δc(0, 0) = 0
δc((s1.c1, . . . , sn.cn), (s′1.c′1, . . . , s
′n.c
′n))
= 1n
∑
i
(
|si − s′i|+ min(si, s
′i).δc(ci, c
′i))
δc(c, c′) = 1 /wyx / x )^% 8'vU]]P ~]'6 c c′ 0 x /.9)21 x
2¡q ¢/ x )~^1~ x %1v x £)21~ <]v¤¥6 n c c′ ¦;§¨ ©7ª §2« ¬' ,®)¯ /° ¢±/ ©O ¯²2O ¯/®'ª®'©0©³ §2« ª ©K´
µK¶\^·\^¸Y[ZY\^X`_ta]bMc¹dfºg%i1jlkKË,m"gnoqphmgyË,stmWuÊËvfºmRkvi1gwyx / x )¢]%¢ 8)6~+~~^q~)() l/%q x %» ^¼ x [0; 1]
δc(c, c′) = 0⇐⇒ c = c′
½ (l921¾ δc ~)¿M6 ¦
À+Á6ÂÄöÅlÆIÁÇ_Ga]bÉÈtÊUË2ÌÍ%ΡÊ2ÏlÐ~ÑPÑ~Í%ΣÒ%Ó8Ô1ÏlÕ~Ñ0ÖÊ2Ô1 × Ô Ø6Ð~Ñ Ê2Ù
minËWÊ2ÏÍqÚÊUË6ÕÛÏRÌ[ÎÌ¡Ó8ÏÜË2ÙÜÐÊ8ÍÍ%Ù6Ì[Ý
Þ Ê2ÏRΡß
δc((s1.c1, . . . , sn.cn), (s′1.c′1, . . . , s
′n.c
′n)) =
1
n
∑
i
(
|si − s′i|+ min(si, s
′i).δc(ci, c
′i))
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P Þ Ñ~ÐKÙvÏ
max ÖÊ2ÔÑ! vÑ#" Ö^ÚÑ ÚÊË2ÌÍ%Î9Ê2ÏlÐ~Ñ£ÖRÓ8Ù6Ô1Ô Ê2Ì[Î$¡Ñ! 'Ö^ÚÓÍÑ%Ô%ßÈ'& Ì[ÏRÎÕ%Ô)(%Î.Ë*& Ê Þ Ó8Ì[Ô5ÙvÏlÑÜË2ÌÍ%Î9Ê2ÏlÐ~ÑÒ%Ó8Ô1ÏlÕ~Ñ Ñ~Í%Î+)Ù,& Ó8Ï ÖRÑ%ÙvÎ Ö^ÔÑ%Ï^Ë2ÔÑÇÙ6Ï
supÌ[ÏÛÏRÌËWÊ2ÏÍ ÚÊ
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- « © ?> ±§¨ ©7ª §2« ¬'q)« ª¯ ±/ © ¬ ² & ¬ '« ª©<© ³ / §2« ª©K´
0.1
0.9
^
0.7
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0.4
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^
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0.6
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0.4
^
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^
© ª0.1 = 1
3
∑
(0 + 0.1 ∗ 1; 0.1 + 0.6 ∗ 0; 0.1 + 0.1 ∗ 0) @A ± §@¨ © ª §2« ¬' ©³¯ ± © ¬ ² & ¬ )« ª©δc>^² « O ³ª ¨ ® ¨ ³¯ ³ « P¨ © ª §2« ¬'
δ© ³ ¯ ±/ © ¨ © © « © @3 )±±/ ¬ § ³ « ³ª ± ª®qª '¬)B « DC³ P§ ©© ;EP± DF./ª® F § ©© ¯ ª ¨HGÌ[ÏÍÖ^Ì[Ô Ê2ÎÌ¡Ó8Ï O^²,³ ¯ ± §¨ ©7ª §2« ¬' C³
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wIx W x ' δc (~¥6 x 8Ç % q < x x 0R~%02¡¼ x δ((s1.D1, . . . , sn.Dn), (s′1.D
′1, . . . , s
′n.D
′n))
= 1n
∑
i
(
|si − s′i|+ min(si, s
′i).δ(Di,D
′i))
δ(K+.(D1, . . . ,Dn),K+.(D′
1, . . . ,D′n)) = maxi δ(Di,D
′i)
δ(z,z) = 0δ(zξ,zξ) = 0δ(0, 0) = 0
δ(((−, ξ, I).DI)I∈Pfin(N), ((−, ξ, I).D′I)I∈Pfin(N)) = maxI δ(DI ,D
′I)
δ(Fid−,Fid−) = 0
δ(D,D′) = 1
µ ¶\¥·\^¸YZY]\^X _taJMÇcTSRjUP;fhg'i1jlk N,mδwIxWV ]] δ YX] 2K) x ' ¦
;§Ä« ²ª² « ¨ Ǩ /© ª §2« ¬) © ªM¯ )±DZ2¬)B ® )« ± § « ²2ª/² « ¨ O ¯®1 ¨ /© ª §« ¬) C³ ©³&ªMO^²,³ ¯ ±/ ©¨ ®'° « /ª/² « © Ì[ÏRÎÑ%Ô Ê6Ð%Î]Ì Þ Ñ~Í@¨ [I ®%ª B JC³ ¨ ®'° « /ª² « * @ *5O §\K, =LW @ I²,³ ¯ ±?G ³ « ]²,¯-F./ª®=>l² « « O²,³¯¯ § /ªKO § © KW§ ¯ ¨ ¯ ±?G ®;C³ / §2±/)« ¬'
d(x, y) = 0 ⇐⇒ x = yC³ ©7ª^] § ³©© « ª ¯ §2¬ ª F )« ª
O²,³¯ ±§P¨ ®'° « /ª² «M¨HG ²,¯ ª B ² K ² «R§2±l¬ ² « © ¨ ®'¯® ¨ ®'° « ª/² « * @`_ O §=K, \a @
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D^ D SUTWV4XIY[ZY]\^X`_Ga c£o "!P;fhg'i1jlk N,m
# %$'&() zt]| ¸~v, d Ü % X] X ~)2 V ]]U¿M6d : X ×X → [0; 1]
>@?+* , C.- L/10 23/54IF;N60 NJFIE;§ « ²ª² «.¨ ¨ ©© « O § ¯ ª )±R © ª ©JF@ ± § /¯ ¨ ®'° « /ª² « @ * O §=K, a a87 ¬')±±/¨R§2« © ± ¬§2¨ ¯
¨ ¢±§P± ³ ¨ JC³ ©JF@O ±/ @SUTWV4XIY[ZY]\^X`_Ga:9cedqmg)g'mRf/k<;jRo¡ifhm¥p
# z |)| =$>&¸ ? zA@:B 4C(C¸+D FEB E ~)+2 % l~/~U~^ /~¡2" x 2 ~ %~/ÜK' KHG]6 ¦w » ~%~ X]¢ q~1~/ ( x ]]~/ ~)^6 Ep ¦
À+Á6ÂÄöÅlÆIÁÇ_GaJM È;ÑË6Ñ~Í~ÍÑ%Ì[Ï ÏlÕ × Ê2Î]Ì I Fid− = 0Ï & Ñ~Í%Î Ù6Ï Ë6Ñ~Í~ÍÑ%Ì[Ï.ÖÊ2Ô1ÎÌ¡Ñ%ÚIË*& Ê2ÙRÐ%Ù6ÏlÑPÕ%Î0.6Ì1+)ÙRÑ)ß
;§Ç¨ ®'° « /ª² « ©³ §« ª O ¯-F ª ¨ O § ¯ ± ¯ ¨ +±§ O § ¯7ª @¨HG ³ « ¬ ² & ¬ )« ªQC³ (O ¯ ® ¬KJ)¨ ³ «'« © F I ±¢¨HGr§2¬ ª² « © @SUTWV4XIY[ZY]\^X`_GaL cNM(stmWu4NvfºmRkiKg's;ngO!QP~jON,mRki5R nlkmRk g'mTSVUlprm PWºjON2ifºsGkg
w @"XtY 2p¸ | X BR| )[Z8^,2p¸]\ 2 ~ 1~ X8]4» x ]] X x @2 'v+~]^ c ])6~+~~^ c′ X~ x x ^;]~ V ~ ¼] c ^y1» x » ~ x x X ¦^ D ~2` ~ % x X x % ` ξ ¾ ^6 Coefz,zξ
(D) ]@)6~+~~^ 162+_a` x )^ x ]¢l%@~)6~+]~^cb+)6~+]~^5\Ç x x ~^ed+ D @1» % X] z;zξ
¦½ cf~U)¾l+6 Coefpr (D) ]'v+~]^ %6+_g` x )~^ x q%~.] 'vU] D %» ~ 1~ X ~ x ]] Il1%l1~ ¦ w 4l%@~()6~+]~^¾'1~94 x ^\q%» ~ 1~ X8] ~x ]] 6 Coef fst(D) ¦ § ¯ .hv F.O ± >
Coefz,zξ(D)
²"i
D = 0.2
0.3
zξ
0.4
0.2
(+, ξ, 1; 2)
0.4
(+, ξ, 1; 3)
© ª0.2
0.3
^
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0.2
j0.4
j
4ªCoefz,zξ
(D′)²"i
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# ^ 5 J \ '=L
D′ = 0.2
0.3
zξ
0.4
0.2
(+, ξ, 1; 2)
0.4
zξ
© ª0.2
0.3
^
0.4
0.2
j0.4
^
G © ¬ ² & ¬ '« ª©<© ²,³ © §2¬')« ª©Coefpr (D
′) ª
Coefpr (D)©² « ª ¯ ©O )¬ ª/ F )« ª
0.2
0.3
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j
;§ ¨ ®%° « ª² «P¨ I #®'ª B DC³ ¨ £±§ ± ³ ¨ DC³ © DF.O ± ¨ ®%° « ª² « @ * a O §=K2 a, © ª !O ¯² IR§=I ± ©® $ '« ¯ F@O ± §)§2« ª ±/ © ®C³ / §2±/)« ¬' ©4O § ¯ ª )±/± ©;O § ¯ ¨ ©4O ¯ ®% ¨ © ª §2« ¬' © @ ,G ¨ ® K ® « ®'¯ §2±/ © ª,C³ ¨ ³ h+¨ © © « ©0© DF.O ± © ®C6³/ §2±'« ª© § ³© )« © ¨ ©£®C³ / §2±/)« ¬' ©£O § ¯ ª )±/± © ©² « ª 7 ¨ ©7ª §2«¬)« ³ ±/±¨HGr ³ h F;F ©> ¨ ³ h5¨ © © « ©<©DF.O ±/ © « ² « ®;C³ / §2±/)« ª© ©² « ª 7 ¨ © ª §2« ¬'
1 @ G © ¬ ² F I «R§ © ² « ©± « ® § /¯ © ¨ ¢¨ © © « © ®;C³ / §2±/)« ª© ª « ² « ®;C³ / §2±/)« ª©© ² « ª 7 ¨ © ¨ © ª §2« ¬' © « ª ¯-F.® ¨ § ¯ © @SMTWV XyY[ZY\^X _ta/_c fg!eWi fInm
# )+(C¸+D EB 2( x % (E, d) 6 E 2¢ G ]6 ¦ d ~) 0%._) x ^) <~ % x )~ E ]q6 x 9] V ¦ d(Fid,Fid) = 0 d(z,z) = 0 d(zξ,zξ) = 0 ∀D1,D2 ∈ Ep, d(D1,D2) ≥ δc(Coef fst(D1),Coef fst(D2)) ∀D1,D2 ∈ Ep, d(D1,D2) ≥ δc(Coefz,zξ
(D1),Coefz,zξ(D2)) x ¤ x V ¦
∀D1,D2 ∈ Ep, d(D1,D2) ≥ δc(Coef fst(D1),Coef fst(D2)) d(Dai−,Dai−) = 0 d(Skunk,Skunk) = 0
À+Á6ÂÄà ¶8ÅtÆ;ÁÜ_taQ£Ï Ô~Ñ#".Ê2Ô!+)ÙRÑ8+)ÙRÑ%Ú+)Ù¥Ñ~ÍÍ%Ì/"Ì[Ú Ì[Î]ÙË6Ñ~ÍKÊ Þ Ñ~ТÚÑ~ÍPÐ~Ó8Ï^Ë2Ì[Î]Ì¡Ó8ÏÍ Ë6Ñ¢ÚÊË6ÕÛÏRÌ[ÎÌ¡Ó8ÏÇË6ÑÒ)Ì[ÝÕ%Î0.6Ì1+)ÙRÑqÍ%Ì/" Ö^ÚÑQ2Ë6ÕÛÏRÌ[Î]Ì¡Ó8Ï 7Wß13#:@ÖÊ × Ñ : 9#<
Ê8Í^7Wß;ÈtÊUÒ~Ê8ÍÑβ
Ñ~Í%ÎGÖRÓÍ%Ì[Î]Ì Þ Ñ)ß Ì
D ≡ D′ Ê2ÚÓ8ÔÍ Í%Ì
D = FidÊ2ÚÓ8Ô~Í
D′ = Fid Í%Ì
D = zÊ2ÚÓ8ÔÍ
D′ = zÑ%ÎÍ%Ì
D = zξ
Ê2ÚÓ8ÔÍD′ = zξ
ß Ê8Í6ß;ÈtÊUÒ~Ê8ÍÑ
βÑ~Í%Î ÏlÕ × Ê2Î]Ì Þ Ñ)ß
Skunk ≡ SkunkÑ%Î
Dai− ≡ Dai−ß
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a D^ D Ê2ÏÍPÚÑ+ÐÊWË2ÔÑ Ö^ÔÓ2Ò~ÊWÒ)Ì[Ú ÌÍ%ÎÑ Ì[Ú5Ê +)ÙÊ2ÎÔÑUÐ~Ó8Ï^Ë2Ì[Î]Ì¡Ó8ÏÍ ÖRÓ8Ù6ÔÚÑ+ÐÊ8Í ÖRÓÍ%Ì[Î]Ì IÑ%ÎÎ]Ô~Ó8ÌÍÐ~Ó8Ï^Ë2Ì[ÎÌ¡Ó8ÏÍÖRÓ8Ù6Ô ÚÑKÐÊ8ÍÏlÕ × Ê2Î]Ì I'ßlÈGÊÐ~Ó8Ï^Ë2Ì[Î]Ì¡Ó8ÏUÍ%Ù6Ô
FidÑ~Í%ÎÊ2Ù6ÎÓ".Ê2ÎÌ1+)ÙRÑ#"Ñ%ÏRÎ Þ Õ%Ô1Ì Û£Õ~Ñ¢Ê2ÚÓ8ÔÍ +)Ù¥Ñ ÚÑ~ÍqÊ2ÙvÎ]ÔÑ~Í
Ð~Ó8Ï^Ë2Ì[Î]Ì¡Ó8ÏÍ<ÍÓ8ÏRÎ Ë2ÙÊ2ÚÑ~Í Ú-& ÙvÏlÑ¢Ë6ÑqÚ-& Ê2Ù6ÎÔÑ29ÚÊ.Ë6Ñ%Ù= ,Ì#"ÑKÐ~Ó8Ï^Ë2Ì[Î]Ì¡Ó8Ï 2Í%Ù6Ôzξ
< ÖRÓ8Ù6ÔÚÑÐÊ8Í ÖRÓÍ%Ì[ÎÌ IÑ~Í%Î(Ë2Ù Ê2ÚÑKË6ÑKÚÊ¢Ö^ÔÑ#"Ì%Ô~ÑÐ~Ó8Ï^Ë2Ì[ÎÌ¡Ó8Ï@ÖRÓ8Ù6Ô¢ÚÑPÐÊ8ÍKÏlÕ × Ê2Î]Ì I'ßßß<2ßÈ;Ñ~Í Ð~Ó8Ï^Ë2Ì[Î]Ì¡Ó8ÏÍ
d(Skunk,Skunk) = 0Ñ%Î
d(Dai−,Dai−) = 0ËWÊ2ÏÍUÚÑ5ÐÊ8ÍUÏlÕ × Ê2Î]Ì I@ÍÓ8ÏRÎ
Î]Ô~Í Í%Ì/"Ì[ÚÊ2Ì[ÔÑ~Í+Ð~Ñ%Ú[ÚÑ~Í ÙvÎ]Ì[Ú ÌÍÕ~Ñ~ÍÑ%Ï"Ú ÙË2Ì1+)Ù¥Ñ Í%Ì/" Ö^ÚÑ ;ÙvÏlÑKÐ~Ó8Ï^Ë2Ì[Î]Ì¡Ó8ÏÇË*& Õ-+)Ù6Ì Þ Ê2ÚÑ%ÏlÐ~Ñ¢Ñ%Ï5Ú ÙË2Ì1+)Ù¥ÑÍ%Ì/" Ö^ÚÑ 2
Skunk ≡ SkunkÑ%Î
Dai− ≡ Dai−<Ñ~Í%ÎÎ]ÔÊ2ÏÍ IÓ8Ô"Õ~Ñ@Ñ%ÏÙvÏlÑUÐ~Ó8Ï^Ë2Ì[Î]Ì¡Ó8Ï Ë6Ñ.Ë2ÌÍ%Î9Ê2ÏlÐ~Ñ
ÏRÙ6Ú[ÚÑ42d(Skunk,Skunk) = 0
Ñ%Îd(Dai−,Dai−) = 0
< Ñ%Ï Ú ÙË2Ì1+)Ù¥ÑÖ^ÔÓ2Ò~ÊWÒ)Ì[Ú ÌÍ%ÎÑ)ß È;Ñ~Í Ê2ÙvÎ]ÔÑ~ÍÐ~Ó8Ï^Ë2Ì[Î]Ì¡Ó8ÏÍ ËWÊ2ÏÍ¢ÚÑÐÊ8Í¢ÏlÕ × Ê2Î]Ì IÍÓ8ÏRÎ ÚÑ~ÍPÐ~Ó8Ï^Ë2Ì[ÎÌ¡Ó8ÏÍKË2Ù Ê2ÚÑ~ÍË6Ñ~ÍPÐ~Ó8Ï^Ë2Ì[Î]Ì¡Ó8ÏÍ¢ÔÑ~ÍÖRÑ~Ð%Î]Ì Þ Ñ~Í Ë2Ù ÐÊ8ÍÖRÓÍ%Ì[Î]Ì I'ß(Ú[ÚÑ~Í Ï & Ì/" Ö^Ú Ì1+)ÙRÑ%ÏRÎÔ1Ì¡Ñ%ÏÖRÓ8ÙvÔÇÚÊÚ ÙË2Ì1+)Ù¥Ñ"Í%Ì/" Ö^ÚÑÜÐÊ2Ô ÚÑ Ð~ÓÑ Ð%Ì¡Ñ%ÏRÎ ÚÊ Ô Ê6Ð%Ì[ÏlÑÇË2ÙÎ]Ô ÊWË2ÙvÌ[Î(Ë*& ÙvÏ Ë6Ñ~Í~ÍÑ%Ì[Ï5ÏlÕ × Ê2ÎÌ I Ë2Ì Õ%ÔÑ%ÏRÎ Ë6Ñ
Fid− Ñ~Í%Î1ß
È;Ñ~ÍÐ~Ó8Ï^Ë2Ì[ÎÌ¡Ó8ÏÍ Í%ÙvÔ¢ÚÊUË2Ì Þ Ñ%Ô × Ñ%ÏlÐ~Ñd(Fid,Fid) = 0
2Fid ≡ Fid
<UÑ%Î(ÚÑ Ë6Õ#"Ó8Ïd(z,z) = 02
z ≡ z<UÏlÑPÍÓ8ÏRÎ ÖÊ8Í@Ñ! 'Ö^Ú Ì¡Ð%Ì[ÎÑ#"Ñ%ÏRÎ Ë6Ó8ÏRÏlÕ~Ñ~Í ÖÊ2ÔRÑÊ2ÏRÝ Þ Ñ~Í£Ì[Ô Ê2ÔË ".Ê2ÌÍÍÓ8ÏRÎ Ë6Ñ~ÍÐ~Ó8ÏÍÕ%Ý
+)ÙRÑ%ÏlÐ~Ñ~Í5Ë6ÑMÚ-& Ó8Ô1Î/.RÓ × Ó8Ï^Ê2Ú Ì[ÎÕ)ߣÈIÑ~Í Ê2Ù6Î]Ô~Ñ~ÍÐ~Ó8Ï^Ë2Ì[Î]Ì¡Ó8ÏÍ"ËWÊ2ÏÍUÚÑÇÐÊ8ÍÖRÓÍ%Ì[ÎÌ I@ÍÓ8ÏRÎKÚÑ~Í Ð~Ó8Ï^Ë2Ì[ÎÌ¡Ó8ÏÍË2ÙÊ2ÚÑ~ÍË6Ñ~Í.Ð~Ó8Ï^Ë2Ì[Î]Ì¡Ó8ÏÍPÔÑ~ÍÖRÑ~Ð%Î]Ì Þ Ñ~ÍË2ÙÄÐÊ8ÍPÏlÕ × Ê2ÎÌ I'ßPÑ Ö^Ú ÙÍ Ñ%Ú[ÚÑ~ÍÌ/" ÖRÓÍÑ%ÏRÎ +)Ù¥Ñ
Fid ≡ D ⇒D = Fid
Ñ%Î(Ë6Ñ'" (#"Ñ£ÖRÓ8Ù6ÔKÚÑË6Õ#"Ó8Ï^ßSUTWV4XIY[ZY]\^X`_Ga"co¡i s¥sGk jlpHP'Wnlkm U^fg!eWi fInm
w » X>&+D8X X >?4» 2 XQ_ G ]6 (E, d) )2K2^X x %2 x /% x x 1 (E⊥, d⊥) xl%_/ x ) d⊥ ~)0~ x ∀E1,E2 ∈ E⊥p
, d⊥(E1,E2) = supD1,D2∈Simp(Ep)
(
δc(JD1,E1K , JD2,E2K)− d(D1,D2))
^ 2 2 X x 1 ~]'6 ξε11 , . . . , ξ
εnn ¾v2 XQ_ G ]6 (E, d) l ~ ]l x HG^¤ x 9
b] 9]] ¦"!$# x ¤ ! &%KdK∀D,D′ ∈ Ep, d(D,D′) = sup δc
(qD, (Eξi
)y,qD′, (E′
ξi)y)
−∑
i
di(Eξi,E′
ξi)
A§2« © ±§ ]²,¯-FP³ ± O^²,³ ¯ ¨ ®%° « ¯ ± G ²2¯ ª B ² K ² «R§±¨HG ³ « YI 9®'ª B DC³ > ±sup
« O § ¯ ¬ ²2³ ¯ ª O § ©ª²,³ ©±/ © ¨ © © « ©KO § ¯7ª '± © F § © ©DF.O ±/ F )« ª ± © ¨ ©© « © O § ¯ ª )± ©q©JF@O ±/ © @ 3 ª ª ¯ ©7ª¯ ¬ ª² « § ³ h¨ ©© « ©©JF@O ± © © ªP³vª ±/ O²,³¯ ¨ ®%° « /¯ © DF.O ± F '« ª ¨ © ¬ ² « « '¬ ª ³ ¯© ¬ ² FF ±/ © ¨ ® ¬)§2± §=K2 ©O § ¯ Ohv F.O ± ¨ ®'° « /ª/² « * @ O §=K, _ @ - « © ?> ±4«HG ©7ªO § © « ® ¬' ©© § ¯ ¨ ¨ ®'° « ¯¢³ « O ¯ ®%¨ /© ª §2« ¬) © ³ ¯ª²2³ © ± © ¨ © © « © F § /© ³ « DC³ F )« ªK©³¯ ± © ¨ ©© « ©¢©DF.O ±/ © @ ;§ O¯®% ¨ ©7ª §2«¬)© ³ ¯ ± © § ³ª¯ © ¨ © © « © ©7ª ¬)§2±¬ ³ ± ® O § ¯0²,¯7ª B ² K ² «R§2± /ª® @µK¶\^·\^¸Y[ZY\^X`_tac'+oi s¥s;kjlp PWnlk m Ufa!+Wi ^f ;n m
w » )HG¤v x y » 2^WX8 _HG 6q~) X _HG]6 ¦( b)( `+**j d oHd ik`$*-,. u%w &/ x2vs10A v v
2 d⊥ vPw|y2z?vl 4vS~ nr xHz?w|9tR4NPv G%v~30zLA zL/~$0 9tROvPw|x2v54 rl z76=v v
∀D1,D2, d⊥(D1,D2) ≥ 0
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# ^ 5 J \ ', 0 2 NPvlu6tHzLuw &/ x2v 0 4,v hx2vPwnwnvlzL.xHz?w|9tR4NPvS uLu?vSx2vu zZ lvTx4tR4wN=t v2zZ r 2z? v G 0~ vl4x.=tR 2 vluLu?vlvlA
Fid~30 2 u?vPwf2zZ r 2z? vPw~$0\wnzL|z 4,vPwfvl
Skunk~$0 2
u?vPwy2zZ r 2z? vPw r QAtR|z 4,vPw Gd⊥(E1,E2) = sup δc(JD1,E1K , JD2,E2K)− d(D1,D2)
≥ δc(JFid,E1K , JFid,E2K) ≥ 0vld⊥(E1,E2) = sup δc(JD1,E1K , JD2,E2K)− d(D1,D2)
≥ δc(JSkunk,E1K , JSkunk,E2K) ≥ 02 twy~30\w|zL|zZY G2 ] 0O 04w v d⊥(Fid,Fid) = 0
G t.x r vl t 4r
d⊥(Fid,Fid) ≥ 0G
. u%w &/ x 04NTx2v~ 0 4,v vd⊥(Fid,Fid) ≤ 0
Gd⊥(Fid,Fid) = sup δc(JE1,FidK , JE2,FidK)− d(E1,E2)
= sup δc(0,0)− d(E1,E2)= sup0− d(E1,E2)≤ 0M 04N
d⊥(Fid,Fid) ≤ 02 ] 0O 04w v d⊥(zξ,zξ) = 0
G t.x r vl t 4r
d⊥(zξ,zξ) ≥ 0G
. u%w &/ x 04N.~ 0 4,v vd⊥(zξ,zξ) ≤ 0
Gd⊥(zξ,zξ) = sup δc(JE1,zξK , JE2,zξK)− d(E1,E2)
= sup δc(Coef fst(E1),Coef fst (E2))− d(E1,E2) ~$0 2 0 E1vl
E2δc(Coef fst(E1),Coef fst(E2))− d(E1,E2) ≤ 0
GM 04N
d⊥(zξ,zξ) ≤ 0G
M vslv d⊥(z,z) = sup δc(JE1,zK , JE2,zK)− d(E1,E2)= sup δc(1,1)− d(E1,E2)≤ 0
2 ] 0O 04w d⊥(D1,D2) ≥ δc(Coef fst(D1),Coef fst(D2))G
d⊥(D1,D2) ≥ δc(qD1,Dai−
y,qD2,Dai−
y)− d(Dai−,Dai−)
= δc(Coef fst(D1),Coef fst(D2))− d(Dai−,Dai−)= δc(Coef fst(D1),Coef fst(D2))
2 ] 0O 04w d⊥(D1,D2) ≥ δc(Coefzξ(D1),Coefzξ
(D2))G
d⊥(D1,D2) ≥ δc(JD1,SkunkK , JD2,SkunkK)− d(Skunk,Skunk)= δc(Coefzξ
(D1),Coefzξ(D2))− d(Skunk,Skunk)
= δc(Coefzξ(D1),Coefzξ
(D2))2 twy r QAtR|zZY G2 ] 0O 04w d⊥(D1,D2) ≥ δc(Coef fst(D1),Coef fst(D2))
Gd⊥(D1,D2) ≥ δc(JD1,zξK , JD2,zξK)− d(zξ,zξ)
= δc(Coef fst(D1),Coef fst(D2))− d(zξ,zξ)= δc(Coef fst(D1),Coef fst(D2))
2 ] 0O 04w d⊥(Dai−,Dai−) = 0d⊥(Dai−,Dai−) = sup δc(
qD1,Dai−
y−
qD2,Dai−
y)− d(D1,D2)
= sup δc(Coef fst(D1),Coef fst(D2))− d(D1,D2)= 0
2 ] 0O 04w d⊥(Skunk,Skunk) = 0G
d⊥(Skunk,Skunk) = sup δc(JD1,SkunkK− JD2,SkunkK)− d(D1,D2)= sup δc(Coefzξ
(D1),Coefzξ(D2))− d(D1,D2)
= 0
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* D^ D SUTWV4XIY[ZY]\^X`_Ga c fa!N,s S ;s^o¡i1mTSUmRki
# )1"X $6X>&% 2 2WXQ_ G ]6 ¤ x \14X _ G¤ x ¦;§ O ¯²,O^²,© /ª/² « ©³/ §2« ª ¨ ² «« ³ « F.² '« ª¯ J ©;© DF.O ±4¨ ¬ ² « ©7ª¯³¯ ³ « I # ¬ ²=F@O^²,¯ ª F )« ª @µK¶\^·\^¸Y[ZY\^X`_ta:9 c'+oi s¥s;kjlp PWnlk m Ufa!+Wi ^f ;n mMmWi Ufa!N,s S ;s^o¡i1mTSUmRkiw » )HG¤v x y » 2^WX8 _HG 6q~) XQ_'4)~U~^ ¦½ (l921¾v2¢6 X _ G ]6 (E, d) 62 ~ % /l~ D1 D2 E ¾
d⊥⊥(D1,D2) ≤ d(D1,D2)
( b)( `+**j d oHd ik`$*-, 0zL(E,d)
4v 2zZ r 2z? v GE] 0A 04w v(E⊥,d⊥) = (E⊥⊥⊥,d⊥⊥⊥)
G t2z?vl
E⊥ = E⊥⊥⊥ G . u vPw|nv ~ 0 4,v v d⊥ = d⊥⊥⊥ G 0 2 NPvlu6t)0 N 04w|z?x vu 0 x v ≤dvlA vT~ nr xHz?w|9tR4NPvPw N 0/vSu vCnvl4w|z 0~$0zLO~=t ~30zLA x2vu 0 x vTw|9tR4x4t xhw 2[0; 1]
GK\G t
∀d1d2. d1 ≤d d2 =⇒ d⊥2 ≤d d⊥
1
G vlu6t 4Cz?vlAx2v0u6t N 0A tq4\t z6tR4NPv x 0zLnv.x2vu 0~ rl tRnv 2−x4tR4wyu?vPw nr vlu?w vlx2vu6t.N 0R4t z6tR4NPv x
supG
HG t∀d. d⊥⊥ ≤d d
G t ~30 2 0 w E1vl
E2wnzL/~2u?vPw
d⊥⊥(E1,E2)= supD1,D2
δc(JD1,E1K , JD2,E2K)− d⊥(D1,D2)
= supD1,D2δc(JD1,E1K , JD2,E2K)
−(supE′1,E′
2δc(JD1,E
′1K , JD2,E
′2K)− d(E′
1,E′2))
≤ supD1,D2δc(JD1,E1K , JD2,E2K)
−(δc(JD1,E1K , JD2,E2K)− d(E1,E2))= d(E1,E2)
t~=t d⊥⊥ ≤d d
x 04N ~=t K d⊥ ≤d d⊥⊥⊥ G t tR~2~2uLz? 4r
d⊥ 0 t t wnw|zd⊥⊥⊥ ≤d d⊥ G hN 04Nlu d⊥ = d⊥⊥⊥ G
SUTWV4XIY[ZY]\^X`_Ga cedqmg)g'mRf/kÄnkfso SUm# ~ %~/ D B X>& x XQ_'4)~U~^ (G, d)
d(D,D) = 0
À+Á6ÂÄöÅlÆIÁÇ_Ga:9 Ï Ú ÙË2Ì1+)Ù¥ÑÍ%Ì/" Ö^ÚÑ ÚÑ5Ò)Ì[ÝÓ8Ô1Î/.RÓ × Ó8Ï^Ê2Ú≡⊥⊥ Ë*& Ù6ÏlÑÇÕ-+)Ù6Ì Þ Ê2ÚÑ%ÏlÐ~Ñ ÖÊ2Ô1Î]Ì¡Ñ%Ú[ÚÑ
≡Ð~Ó8ÏRÎ]Ì¡Ñ%ÏRÎ≡
ß-PÓ8ÏlÐÚÑ+ÖÊ8Í~Í)Ê × ÑÜÊ2ÙÉÒ)Ì[ÝÓ8Ô%Î/.RÓ × Ó8Ï^Ê2Ú^$]Ê2Ù × "Ñ%ÏRÎÑ\%MÚ-& Ù6ÏRÌ IÓ8Ô"Ì[ÎÕ)ß Ì/"Ì[ÚÊ2Ì[ÔÑ#"Ñ%ÏRÎ Ñ%Ï Ú ÙË2Ì1+)٥ѢÖ^ÔÓ2Ò~ÊWÒ)Ì[Ú ÌÍ%ÎÑ ÚÑ¢ÖÊ8Í~Í)Ê × ÑÊ2ÙÒ)Ì[ÝÓ8Ô1Î/.RÓ × Ó8Ï^Ê2Ú Ò~Ê2ÌÍ~ÍÑ.ÚÊË2ÌÍ%Î9Ê2ÏlÐ~ÑUÐ~Ñ +)Ù6Ì $]Ê2Ù × "Ñ%ÏRÎÑ\%Ú-& Ù6ÏRÌ IÓ8Ô"Ì[ÎÕ)ßSUTWV4XIY[ZY]\^X`_Ga]b j c fa!'f/kON,jRo k jifhsGk
wyx ) >&8y X 4» 2 ~1~/Ä2 V M D x K2 X8 _' )M (G, d) ||D||G =
⋂
D′ | D′ ⊆ D, d(D,D′) = 0
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;# ^%H # _Q «5¨ ©© « ® K,§2± 7 © §QI 9 « ¬)§ ¯ «R§ ª² « ©7ª ¨ /ª YÂÄÃZT6¶8YÁ ²,³ YY]X,à ¶8X;T @
À+Á6ÂÄÃ ¶8ÅtÆ;ÁÜ_taQL È'& .~ÖRÓ8Î0. ~ÍÑ Ë*& Ù6ÏRÌ IÓ8Ô"Ì[ÎÕ Ë6ÑD
Ñ~Í%Î Ù6ÎÌ[ÚÑ PÚ-& Ñ! ,ÌÍ%ÎÑ%ÏlÐ~Ñ<Ë6Ñ ÚÊÒ)Ì[ÝÌ[ÏlÐÊ2Ô1Ï^Ê2ÎÌ¡Ó8Ï2Í%Ì[ÏlÓ8Ï Ú-& Ñ%ÏÍÑ#".Ò)ÚÑ
D′ | D′ ⊆ D, d(D,D′) = 0ÖRÑ%Ù6Î (%Î]ÔÑ Þ Ì]Ë6Ñ)ß
>? /JMJUNL F;N 44EA F F C³ Gr'« ± ³ ¨ DC³ © DF.O ± © )¬ ª/² « @`_@ O §=K, a * >I ± ] § ³ª « § ª³ ¯ )±/± F )« ª ¨ ®'° « ¯
±?Gr§2¬ ª² «U¨ © ¬ ² « « '¬ ª ³ ¯ ©P© )¬ ª/² « @ O §=K2 a, © ³ ¯ ± © I # ¬ ² F@O^²,¯7ª F )« ª© @ « © .¬ ² « ª '« ª ¨@¨ ®%° « /¯ ±/ © ¨ /© ª §« ¬) ©©³¯ ±/ © ¨ © © « © ©DF.O ±/ © @ I§M¨ ®%° « /ª² « © ³ ¯ ±/ ©§ ³ª¯ © ¨ ©© « © © ª,] § /ª O § ¯ I 9²,¯ ª B ² K ² « §2± @ ;§ « ²ª² « ³ª ± ©7ª ¬)'±± ¨ I #®'ª B DC³ ¬ ² F.O ±J ª @SMTWV XyY[ZY\^X _ta]b^bUc fg!eWi fIn m N,s SV;lpWi1m
# XQ_HG (E, d) 0"X $ ?$% Pss(|E⊥⊥|) ⊆ E
G462K2]~ ~1~/ < l]d(D1,D2) = d⊥⊥(D1,D2)
! #" n"%$ "'& " )(+*#&-, ." "0/0*SMTWV XyY[ZY\^X _ta]bMÇc d N,jlprj3¥mg
w z (6?GK4¡ V » 2 XQ_)4~U~^y^¤ x ] V X x 1 ξ.i ` ~)1(G, d) = (
1G,
1d) = (
1G,
1′d)⊥⊥
1′d(
1D1,
1D2) = d(D1,D2)
w z (6?G¤ x ] V 4» XQ_'4)~U~^ 9] V X x 1 ` ξ.i ~)2(G, d) = (
2G,
2d) = (
2G,
2′d)⊥⊥
2′d(. . .
2D1 . . . , . . .
2D2 . . .) = d(D1,D2)
. . . 2 Dk . . . ~ 2q^6 x ]]2 ~ %¤ x V ^ x '^/ x (−, ξ, i) ~) Dk
¦
À+Á6ÂÄà ¶8ÅtÆ;ÁÜ_ta_ ÈtÊKË6ÕÛÏRÌ[ÎÌ¡Ó8Ï@Ë2Ù.Ë6Õ~ÐÊ2ÚÊ × Ñ;ÖRÓÍ%Ì[Î]Ì I Ñ~Í%Î Þ Ê2Ú Ì]Ë6ÑÐÊ2Ô£Ú-& Ñ%ÏÍÑ#".Ò)ÚÑ(Ë6Ñ~Í0Ë6Õ~ÐÊ2ÚÊ × Ñ~Í£Ë6ÑË6Ñ~Í~ÍÑ%Ì[ÏÍKÏlÕ × Ê2ÎÌ IÍÐ~Ó8ÏRÎ]Ì¡Ñ%ÏRÏRÎ(ÚÑ~ÍPË6Ñ~Í~ÍÑ%Ì[ÏÍ0ÖRÓÍ%Ì[ÎÌ IÍqÍ%Ì/" Ö^ÚÑ~Í0Ö^ÔÓ~Ö^ÔÑ~ÍPË2Ù Ð~Ó" ÖRÓ8Ô1ÎÑ#"Ñ%ÏRÎ 1
GßIÈtÊ
Ë6ÕÛÏRÌ[ÎÌ¡Ó8Ï Ë2ÙÇË6Õ~ÐÊ2ÚÊ × Ñ¢ÏlÕ × Ê2ÎÌ I¢Ù6Î]Ì[Ú ÌÍÑKÚÑ~ÍË6Ñ~Í~ÍÑ%Ì[ÏÍÍÓ8ÙÍ ÚÊIÓ8Ô"Ñ. . .
2Dk . . .
ÐÊ2ÔKÚÑË6Õ~ÐÊ2ÚÊ × ÑÏlÕ × Ê2ÎÌ I¢Ï & Ñ~Í%ÎtÖÊ8ÍqÍ%Ù6Ô43Ñ~Ð%Î]Ì I Í%Ù6Ô¢ÚÑ~ÍË6Ñ~Í~ÍÑ%Ì[ÏÍqÏlÕ × Ê2Î]Ì IÍ)ß
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D^ D ;§ O ¯ ²,O^²,©ª² « © ³ / §2« ª ©JF@O ± /° ±§5¨ ®%° « ª² « * @ O §=K, O ¯® ¬ ® ¨)« ª U¬)§ ¯ '±±.Oh O ¯ DF
C³ ©³¯ ±/ © ¨ ©© « © ©JF@O ± ©> ±/ I 9²,¯ ª B ² K ² « §2±l © ª « ³ª ± ´ ² «5§ ³ « 'I 9®'ª B DC³ q¬ ²=F@O ±J ª @µK¶\^·\^¸Y[ZY\^X`_taLc¹d N,jphj3¥mg 62Ü6 X8 _' )M¢¤ x V (G, d) X x 1 ξ.i ` ¾ x 462Ç62 ~1~/ l~ 1 ′d(D1,D2) =
1d(D1,D2)½ . fM'¾;462U6 WX8 _' )M 9] V (G, d) X x % ` ξ.i ¾£ x 6U6
~ %~/ l~ 2 ′d(D1,D2) =2d(D1,D2)
( b)( `+**j d oHd ik`$*-,. u w &/ x2v10O v vu?vPw ~ nr xHz?wn9tR4NPvPw ′d vl ′d w 0Ox2vPw 0 0Q0=tRu?vPw ~ 0~30\w|zL|z 0 FHG ~=tRQ,v K;V\F G t ~ v 4,v/10O vx2v0~2u wS vluLu?vPwSw 0ASvl Y tRzL50 0Q0=tRu?vPwu 4vSx2vuJt v G2 0zL (G,d)
02zZN 0/~30 nvlvlA r QAtR|zZYx2v =twnvξ.i `
GR 0z?vlOD1
vlD2
x2v x2vPwnwnvlzL4ww|zL/~2u?vPwx2v
GG
t ′d(
D1,
D2) = d(D1,D2)
= supE1,E2δc(JD1,E1K , JD2,E2K)− d⊥(E1,E2)
= supE1,E2δc(J
D1,
E1K , J D2,
E2K)− (
′d)⊥(E1,
E2)
= supE′1,E′
2δc(J
D1,E
′1K , J D2,E
′2K)− (
′d)⊥(E′1,E
′2)
M 04N ′d(
D1,
D2) =
d(D1,
D2)
N 04Nlu ~=t u6t/w 2 vPNl|z 4OzL r x x r N;tRu6tRQ,vS~$0\w|zL|zZYw 2 u?vPwx2vPw9wnvlzL4ww|zL/~2u?vPw G2 h~ 0CN Px2vSx2vslvTx4tR4wu?vx r N;tRu6tRQ,vS r QAtR|zZY G 02|z?vlO
′d(. . .D1 . . . , . . .
D2 . . .) =
d(. . .
D1 . . . , . . .
D2 . . .)
, , " & " $SUTWV4XIY[ZY]\^X`_Ga]b Çc ki1mRo¡g)m N2ifºsGk
w » « ª ¯© )¬ ª² « XQ_'4)~U~^]£¤ x V < fM8X x %K~)∩k(Gk, dk) = (∩kGk,∩kdk)
x ¼)∩kdk = sup dk
µK¶\^·\^¸Y[ZY\^X`_ta/_c ki1mRo9g'm N2ifºsGkUPm Ufa!)N,s SV;^s^o¡i1mTSUmRki1gw » ^~1)]WX8 _' )M]< X x 1 β ~) XQ_'4)~U~^ X x % β ¦
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;# ^%H # 2SMTWV XyY[ZY\^X _ta]b9Üc sokm f/kFRo fhmRnlo¡m5g8nloYPmgQP^s S+jfkmg P;fhg'if/kON2i1g
^ ¡ B @]%/;) l] ¦ ^ 9 (fi) 2 V]x .y V ]] fi : Ai → B ¦ ^ ¡ A = ∪Ai¦wIx X>&84 (& B &,@ < xWV]x @/] (fi)
inf fi : A → Bx 7→ infx∈Ai
fi(x)
SMTWV XyY[ZY\^X _ta]b>LÇck^fºsGkw » ³ « /² « XQ_)4~U~^4 9] V < <cf~U^X x 1K
∪∗k(Gk, dk) = (∪∗kGk,∪∗kdk) = (∪∗kGk,∪kdk)
⊥⊥
x ¼)∪kdk = inf dk
3 © ¨ ®%° « /ª² « ©K Ì[ÏRÎÑ%Ô~ÍÑ~Ð%Î]Ì¡Ó8ÏÜ ª Ù6ÏRÌ¡Ó8Ï ¨ ®'° « © © )« ª ± © ¬ ² « « '¬ ª ³ ¯© Ê Þ Ñ~Т ª Ö^Ú ÙWÍ © ³ ¯ ±/ ©¬ ² F.O²2¯ ª F '« ª© ¨ © ², « ª© @ & " n" #"'& " $
;§P« ²2ª² «¨ FP³ ± ªO ± ¬)§ ª] « ® KW§ ªJ] ±<ÖÊ2Ô )«M± ² K DC³ ± « ® § ¯ ©7ª¯ )± ® 7 ¬)'±±<¨HG DF.O ± #¬§ ª² «M ª ¨ ¢« ® KW§ ª² « 7 ¬§ ³© ¢¨ ¢± § ]²,¯!FP³ ± ¨ ¢± §P± ² K DC³ ± « ® § ¯
A( B = A⊥ B
- « © ±§¨ ®'° « /ª² «+¨Ï(Ù ËÖÊ%Ö^Ì[Ú[ÚÓ8Ï5 © ªO ¯ ² ¬)B q¨q± §¨ ®%° « ª² « * @`_ O §=K, \a ¨HG ²,¯ ª B ² K ² « §2±¨HG ³ « 8I ®%ª B JC³ @SMTWV XyY[ZY\^X _ta]bv_ÇcTSRm
w ' '2 ./ ( 8 X _) M^;¤ x V cf~U X x % (G1, d1) (G2, d2)~) (G1, d1) ./ (G2, d2) = (G1 ./ G2, d1 ./ d2)x ¼)
(d1 ./ d2)(F1,F2) = sup d2((F1)D1, (F2)D2)− d⊥1 (D1,D2)
µ ¶\¥·\^¸YZY]\^X _ta5cTSRm ! 62 XQ_'4)~U~^] ^¤ x ] V £ £ fM X x 1 (G1, d1) (G2, d2) ¾ x X _ G ]6(G1 ./ G2, d1 ./ d2) ~)2 X8 _' )M ¦
( b ( `$*j d oHd ik`+* ,. uHw &/ ~ 0~$0\wnzL|z 0 FHG ~=tRQ,v K;V\F x2v 10O v v N vPw|u 0 0Q0=tRu4x 4v2zZ r 2z? v G vPw|.u 0 0Q0=tRux2vPw.2zZ r 2z? vPw (G⊥
1 ~ G⊥2 ,d′)
nvluLu?vPw v w 2 u?vPwx2vPw9wnvlzL4ww|zL/~2u?vPw
d′(D01,D
02) = min1; inf
∀i D0i =Di~D′
i
d1(D1,D2) + d2(D′1,D
′2)
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* D^ D v vl
(d1 ./ d2)(F1,F2)= supd2((F1)D1, (F2)D2)− d⊥
1 (D1,D2)= sup δc(J(F1)D1,D
′1K , J(F2)D2,D
′2K)− d⊥
2 (D′1,D
′2)− d⊥
1 (D1,D2)= sup δc(JF1,D1 ~D′
1K , JF2,D2 ~D′2K)− (d⊥
1 (D1,D2) + d⊥2 (D′
1,D′2))
= sup δc(JF1,D1 ~D′1K , JF2,D2 ~D′
2K)− d′(D1 ~D′1,D2 ~D′
2)
. u 4v vPw|nvS 0zL 40C v u6tw 2 vPNl|z 4CzL r x nvl4wnv 2 w 2 u?vPwx2vPwnwnvlzL4w w|zL/~2u?vPw G
À+Á6ÂÄöÅlÆIÁÇ_Ga £Ï ÊÙvÎ]Ì[Ú ÌÍÕ@ÚÑ I'Ê2Ì[Î'+)ÙRÑ@ÚÊÐ~Ó8ÏRÏ^Ê2ÌÍ~Í)Ê2ÏlÐ~Ñ.Ë6Ñ+ÚÊÜË2ÌÍ%ΡÊ2ÏlÐ~Ñ.Í%Ù6ÔÚÑ~ÍUË6Ñ~Í~ÍÑ%Ì[ÏÍÍ%Ì/" Ö^ÚÑ~Í<Í%Ù Î5Ë6ÕÛÏRÌ[ÔqÚ-& Ó8Ô%Î/.RÓ × Ó8Ï^Ê2ÚßµK¶\^·\^¸Y[ZY\^X`_tac SRm !Ümg'i^N,s SVSÇni1jvif
^ ]^ (G1, d1) (G2, d2) 8 XQ_)4~U~^(¤ x V cf~U^X x 1 ¦ x (d1 ./ d2)(F1,F2) = sup d2((F1)D1, (F2)D2)− d⊥
1 (D1,D2)= sup d1((F1)D1, (F2)D2)− d⊥
2 (D1,D2)
( b)( `+**j d oHd ik`$*-, M JtR~ Pwu6t0~ v 4,v x2vsu6t.~ 0~$0\wnzL|z 0 FHGJ[ 0t
(d1 ./ d2)(F1,F2)= sup δc(JF1,D1,D
′1K , JF2,D2,D
′2K)− (d⊥
1 (D1,D2) + d⊥2 (D′
1,D′2))
vl|nvSvH~ vPwnw|z 0vPwn 4Oz?w|zL2u?vlvlO wC r z? vTvl d1vl
d2G
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i ) ~(25~%~ X]Ξ,Λ XQ_)4~U~^ (Gi, di) 2 ] X x % ` ξi ¦w XQ_'4)~U~^ Ξ ` Λ x X x % Ξ ` Λ ~;%» )HG¤v x ¥ ~ V]x .~ <~1~/ (Eξi
) Eξi∈ Gi ξi ∈ Ξ Eξi
∈ G⊥i ^¾ x ¼) x l1._) x ' d
d(D,D′) = sup δc(q
D, (Eξi)y,qD′, (E′
ξi)y)
−∑
i
dεi⊥i (Eξi
,E′ξi)
dε⊥ =
d ε = −d⊥ ε = +
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% 1 % ] g )º5 * SMTWV XyY[ZY\^X _ta]b Çc fg!N,s SV;^so¡i1mTSUmRkviPgO!'o Gnlp#fhm¥o
# XQ_'4)~U~^ l%X x X/) (G, dG) ) &"( B ? &.2U2 XQ_)4~U~^/l (H,≡)
G _%¤¥2/]~46 H ¦ 462¢2 ~ %~/ l~ D1 D2 Hp
dG(D1,D2) = 0 ⇔ D1 ≡ D2
dG(D1,D2) = 1 ⇔ D1 6≡ D2
462¢2 ~ %~/ l~ E1 E2 H⊥sp
dG(E1,E2) = 0 ⇔ E1 ≡⊥ E2
dG(E1,E2) = 1 ⇔ E1 6≡⊥ E2
# X _) M^4l% X x X/) ) &( B ? &+6» 1» 6@ XQ_)4~U~^/l ¦
SMTWV XyY[ZY\^X _ta]b Çc M(sGklkmN2i1mRno<g.!)o ;npfhmRo# ' '2Ç~) ) &"() B ? &Ü6» K^¼ ~UXQ_'4)~U~^] _%¤¥2/]~ )25 X _) 4~U~^ _1¤¥] ¦
µ ¶\¥·\^¸YZY]\^X _taÇc dPn jlpHP'Wnlk<Ufa!)N,s SV;^soi1mTSUmRkiKgO!'o Gnlp#fºmRow » )HG¤ x ;4» 2 X8 _' )M _]%¤R2~< _]%¤¥2~ ¦
( b ( `$*j d oHd ik`+* , 0zL(G,d)
S2zZN 0/~30 nvlvlA nr Q uLz?v ~30 2 u?v2zZN 0/~30 nvlvlAw|zL/~2u?v(H,≡)
G 0z?vlA
Evl
E′ x2v x2vPw9wnvlzL4ww|zL/~2u?vPwx2v G⊥ G2 tw K\G E ≡⊥ E′ G t
D ≡ D′ ⇒qE,D
y=
qE′,D′
y
x 04ND ≡ D′ ⇒ δc(
qE,D
y,qE′,D′
y) = 0
x 04Nd(D,D′) = 0⇒ δc(
qE,D
y,qE′,D′
y) = 0
x 04Nd⊥(E,E′)
= sup δc(JE,DK , JE′,D′K)− d(D,D′)= 0
2 tw HG E 6≡⊥ E′ G Xu 0 w HzLuvHz?w|nv D ≡ D′ w|zL/~2u?vPwynvlu?w vqE,D
y6=
qE′,D′
y
M tR4wNPvSN;tw 0t
δc(qE,D
y,qE′,D′
y)− d(D,D′) = 1− 0 = 1
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* D^ D
vlhx r x zLd(E,E′)
= sup δc(JE,DK , JE′,D′K)− d(D,D′)= 1
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w ¢^ x y4» 2 ' '2 _]%¤¥2~< _%¤¥2/]~ ¦SUTWV4XIY[ZY]\^X`_GaJM j c fa!s^o¡i s ¥sGk jp;Ro9s Uj Uf/pfhg'i1m PW nk Ufa!)N,s SV;^s^o¡i1mTSUmRkiKg8fSV;phm
w )+X>&+D8X X >?8$'&X ^ ? | %@ » 2 XQ_)4~U~^ l] (G,≡) ~(G,≡)⊥p⊥p = (G⊥p⊥p, d⊥⊥
≡ )
x ¼)'¾ 462¢62 % /]~ D1 D2 ¾d≡(D1,D2) = 0 D1 ≡ D2
= 1
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Ñ~Í%Î(Ò)Ì¡Ñ%Ï ÙvÏlÑ£Ö^ÔÕ%Ý7Ë2ÌÍ%ΡÊ2ÏlÐ~ÑKË6Ó8ÏlТÚÊË6ÕÛÏRÌ[ÎÌ¡Ó8ÏÖ^Ô~Õ~Ð~ÕË6Ñ%ÏRÎÑKÊÒ)Ì¡Ñ%ÏÇÙ6Ï ÍÑ%ÏÍ)ßµK¶\^·\^¸Y[ZY\^X`_ta]b¥bMc fg!soi s¥sGk jlp';Ro¡s U j U^fp#fºg%i1m P'Wnlk U^fg!N,s SV;^soi1m S+m¥kvigfQS ;lprm
626£' )M/] (G,≡) ¾l1 XQ_)HG¤ x l%X x X/) (G⊥p⊥p, d⊥⊥≡ )
~ _1¤¥2/] ¦( b)( `+**j d oHd ik`$*-,. u%w &/ x2vs10A v vw 2 u?vPw x2vPwnwnvlzL4w w|zL/~2u?vPw
d(D1,D2) = d⊥⊥≡ (D1,D2)
vPw| 4 tRz N;t *w 2 u?vPwSx2vPw9wnvlzL4wTw|zL/~2u?vPw d≡vPw| 0 0Q0=tRu u6t ~ nr xHz?w|9tR4NPv
d≡⊥
0 d≡⊥
vPw|Tu6t~ nr xHz?w|9tR4NPv N 0 | vPw|~304x4tRA(G,≡)⊥
G N 04Nlu .vlzL)40O tRA.u6t ~ 0~30\w|zZ|z 0 FHG ~=tRQ,v K;V\FHG
#" "0/)* $
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( b)( `+**j d oHd ik`$*-, 0zL(G,dG)
/2zZN 0/~30 nvlvlA ~ 0=tR2zLuLz?w|nvx2v =twnv` ξ
nr Q uLz?v ~30 2 h2zZN 0/~30 nvlvlAsw|zL/~2u?v
(H,≡)G
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% 1 % ] g )º5 * L
2 G vlHw 0A x2vs=twnv
ξ.i `G
t(G,d) = (
G,d)
0 d(D1,
D2) = d(D1,D2)vl
(H,≡) = (
H,≡ )
0 D1 ≡
D2 ⇐⇒ D1 ≡ D2 vlhx r x zL
D1 ≡
D2 ⇔ D1 ≡ D2 ⇔ d(D1,D2) = 0⇔
d(D1,
D2) = 0
vlD1 6≡
D2 ⇔ D1 6≡ D2 ⇔ d(D1,D2) = 1⇔
d(D1,
D2) = 1
2 G vlHw 0A x2vs=twnv
` ξ.iG
|zLuLz?wnvsu6t.~ 0~30\w|zL|z 0 GLK ~=tRQ,vSNlzZN 0A v G
, , " & " $µ ¶\¥·\^¸YZY]\^X _ta]b ÇcTSRj g.!)o ;nphjRo7fhi. PIn
w q) ^)2 & _%¤¥2/]~ ¦( b ( `$*j d oHd ik`+* , 0z?vlA
(G1,dG1)vl
(G2,dG2)x2v 2zZN 0/~30 nvlvlAnw nr Q uLz?v w ~30 2
(H1,≡H1)vl
(H2,≡H2)G
0z?vlAD1
vlD2
x2v x2vPwnwnvlzL4ww|zL/~2u?vPw G2 O ~2~$0\w 04w D1 ≡H1&H2 D2
G t tRu 0 wD1 ≡H1 D2
vlD1 ≡H2 D2
x 04N~=t nr Q u6t zL r
dG1(D1,D2) = 0vl
dG2(D1,D2) = 0x 04N
dG1&G2(D1,D2) = sup0; 0 = 0
2 r NlzL~ 0C vlvlO =w ~2~30\w 04w dG1&G2(D1,D2) = 0G
tdG1(D1,D2) = 0
vldG2(D1,D2) = 0
x 04ND1 ≡H1 D2
vlD1 ≡H2 D2
x 04ND1 ≡H1&H2 D2
G2 O ~2~$0\w 04w D1 6≡H1&H2 D2
G t tRu 0 wD1 6≡H1 D2
0 D1 6≡H2 D2
x 04N~=t nr Q u6t zL r
dG1(D1,D2) = 10
dG2(D1,D2) = 1x 04N
dG1&G2(D1,D2) = sup1; 1 = 1
2 r NlzL~ 0C vlvlO =w ~2~30\w 04w dG1&G2(D1,D2) = 1G
tdG1(D1,D2) = 1
0 dG2(D1,D2) = 1
x 04ND1 6≡H1 D2
0 D1 6≡H2 D2
x 04ND1 6≡H1&H2 D2
G
µ ¶\¥·\^¸YZY]\^X _ta]b9ÜcTSRj g.!)o ;nphjRo7fhi. PIn ! w q) ^)2 ⊕ _]%¤R2~ ¦
( b ( `$*j d oHd ik`+* , |zLuLz?wnvsu6t.~ 0~30\w|zL|z 0 GLK ~=tRQ,v~ nr N r x2vlAnv G
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* a D^ D & " n" #"'& " $µK¶\^·\^¸Y[ZY\^X`_ta]b Lc S¥m
^ ]^ (G, dG) (H, dH) 8 X _) M^4 9 V ¦ x 2U2@ % l~q
dG~H(D1 ~D2,D′1 ~D′
2) ≤ min1; dG(D1,D′1) + dH(D2,D
′2)
( b)( `+**j d oHd ik`$*-, z?wn vu?vPwy~ nr xHz?wn9tR4NPvPw w 0O$0 r vPw ~=t 12zLu%w &/ x2v54 rl z76=v v
dG~H(D1 ~D2,D′1 ~D′
2) ≤ dG(D1,D′1) + dH(D2,D
′2)
tdG⊥./H⊥(E,E′) = supdH⊥((E)D1, (E
′)D′1)− dG(D1,D
′1)
= sup δc(J(E)D1,D2K , J(E)D′1,D
′2K)− dH(D2,D
′2)− dG(D1,D
′1)
x 04N~30 2 0 wD1
D2
D′
1
vlD′
2
δc(J(E)D1,D2K ,q(E)D′
1,D′2
y)− dG./H(E,E′) ≤ dG(D1,D
′1) + dH(D2,D
′2)
x 04NdG~H(D1 ~D2,D
′1 ~D′
2) = sup δc(JD1 ~D2,EK , JD′1 ~D′
2,E′K)− dG⊥./H⊥(E,E′)
= sup δc(JD1, (E)D2K , JD′1, (E
′)D′2K)− dG⊥./H⊥(E,E′)
≤ dG(D1,D′1) + dH(D2,D
′2)
µK¶\^·\^¸Y[ZY\^X`_ta]b6_c S¥jÜg.!)o GnphjRo fri. PIn w ¢' '2 ~) _]%¤R2~ ¦
( b)( `+**j d oHd ik`$*-, 0zLGvl
Hx2v hN 0/~$0 nvlvlOnw r QAtR|zZYw G t
(dG./H(F1,F2) = supdH⊥(JF1,D1K , JF2,D2K)− dG(D1,D2) = 0)⇔ (dH⊥(JF1,D1K , JF2,D2K) = 1⇒ dG(D1,D2) = 1)⇔ (JF1,D1K ≡H⊥ JF2,D2K⇒ JF1,D1K ≡G⊥ JF2,D2K)⇔ (JF1,D1K ≡G JF2,D2K⇒ JF1,D1K ≡H JF2,D2K)⇔ (F1 ≡G F2 ⇒ F1 ≡H F2)
µK¶\^·\^¸Y[ZY\^X`_ta]b$ c S¥jÜg.!)o GnphjRo fri. PIn
w ¢' '2 ⊗ ~) _1¤¥] ¦( b)( `+**j d oHd ik`$*-, |zLuLz?wnvsu6t/~ 0~$0\w|zL|z 0 GLK ~=tRQ,v K HG
* $ ")()& " #"'& * $;§ 9¯® K ³ ± § ¯ /ª® ¨ © C³ §2« ª/° ¬)§ ª ³ ¯© ©7ª< ¨ )« ªDC³ 7 ¬))±/±q¨ © §2¨ ¨ ªJ]© @
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% 1 D Q * *( & $ , * " f & * *( & $µ ¶\¥·\^¸YZY]\^X _ta]b ÇcTSRj g.!)o ;nphjRo7fhi. Pmgg ;n mRki1gWP^m U^fg!N,s SV;^so¡i1mTSUmRkvi1g
1~ l] D » 21)6<[X _) 4~U~^/]~ Ξ ` Λx x ]]x %' ~^ XQ_'4)~U~^]l% X x X/) Ξ⊥p⊥p ` Λ⊥p⊥p ¦
6 62 % D1 D2 x Ξ ` Λ ¾G x
dΞ⊥p⊥p`Λ⊥p⊥p (D1,D2) = 0 ⇔ D1 ≡ D2
dΞ⊥p⊥p`Λ⊥p⊥p (D1,D2) = 1 ⇔ D1 6≡ D2
( b ( `$*j d oHd ik`+* , |zLuLz?w9tROu?vPwy 09tR|z 04w x2vsu6t/x r 642zL|z 0 GLK W ~=tRQ,v K 0t
dΞ⊥p⊥p`Λ⊥p⊥p (D,D′) = sup δc
(qD, (Eξi
)y
,qD′, (E′
ξi)y)
−∑
i
dεi⊥≡i
(Eξi,E′
ξi)
2 0zL D ≡ D′ Gft nr Q u6t zL r x2vPw
Ξ⊥⊥,Λ⊥⊥
∀i dεi⊥≡i
(Eξi,E′
ξi) = 1
w|zEξi6≡εi⊥
i E′ξiM 04N
dΞ⊥p⊥p`Λ⊥p⊥p (D,D′) = sup δc
(
rD, (Eξi
)z
,rD′, (E′
ξi)z)
−∑
i dεi⊥≡i
(Eξi,E′
ξi)
= supd
εi⊥≡i
(Eξi,E′
ξi)=0
δc
(
rD, (Eξi
)z
,rD′, (E′
ξi)z)
− 0
= supd
εi⊥≡i
(Eξi,E′
ξi)=0
0− 0
= 0
2 0zL D 6≡ D′ G 0zL
(Eξi)vl
(E′ξi
)nvlu?w v
∀i, Eξi≡εi⊥
i E′ξi
vl rD, (Eξi
)z6≡
rD′, (E′
ξi)z G t
dΞ⊥p⊥p`Λ⊥p⊥p (D,D′) = sup δc
(
rD, (Eξi
)z
,rD′, (E′
ξi)z)
−∑
i dεi⊥≡i
(Eξi,E′
ξi)
≥ δc
(
rD, (Eξi
)z
,rD′, (E′
ξi)z)
− 0
= 1− 0= 1
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# ~ %~/Ü X | '(» /t » /1 x q ~U ¦SUTWV4XIY[ZY]\^X`_GaJM MÇc dqmg)g)m f/k<;Ro 6g'mRki
# ~ %~/Ü$8&"( | 20G ] ¢ G x 2Ü <%~q)6~+]~^] ^¤ x ] V 1 ¦SUTWV4XIY[ZY]\^X`_GaJM Çc
c! io9jlk N m m6jON2i1m
# c _ x G¢(I^% ~/]£ < x^V M z zξ (+, ξ, ∅) (+, ξ, i1, . . . , in).(Dξ.i1, . . . ,Dξ.in) ~ Dξ.i1 %^( x ¦ (−, ξ, i1, . . . , in).D D ~£ x (] ξ.ik x x x /%^£ x D ¦ (−, ξ, i1, . . . , in).D D ~£ x ( ¦ c.(D1, . . . ,Dn) ~ Di 1^£ x ] ¦
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Í%Ì4ÎÓ8Ù6ÎÑ~Í¢ÍÑ~Í£Ö^ÔÓ%3Ñ~Ð%Î]Ì¡Ó8ÏÍÍ%Ì/" Ö^ÚÑ~Í ÍÓ8ÏRÎ0Ñ! Ê6Ð%ÎÑ~Í)ßSUTWV4XIY[ZY]\^X`_GaJM 9Üc dqmg)g)m f/kÇm6jON2i
# U ~ %~/"4;^ I6 ~% c _] x ^ G~ x / x ]~t6 1» /~¡2P1^ x ¦SUTWV4XIY[ZY]\^X`_GaJM LÇc yjRo6NvfQSUsGk^fºm
# ~ %~) $ >&W X B4 x x G¼ 9'P^2 x ^q6¢ x / x ]] x ¼)
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³ « J]²,¯!F ª ² I © ª « ®.© §2« © ¬ ² « ¨ ª/² «U¨ O § ¯ ¬ DF.² « ²,³ ¨HG Oh§2¬ ª/ª³ ¨ @µK¶\^·\^¸Y[ZY\^X`_ta]b c g)mjln ¥j3Gkjlki# "%~1 x + ¢ % ¤ x ¤, x ^ x 1¢¼~ 2 ~ %5¤ x ¤W x ^ ¦
! * & " )( « « ª ¯O¯ J ª ³ «¨ ©© « © DF.O ±
DO § ¯©² « O ± ² « K, F )« ª )«± ³ ¨ DC³ O ¯² I §=I ± ©7ª « ²ª®
Dp
O^²,³ ¯0³ « F ±±/ ³ ¯ ± /© I ± /ª® @µK¶\^·\^¸Y[ZY\^X`_taJM j c¹d¢mg'g)m fk<Ufa! SUjvi.¥o7fºmRp 62 ( ~ %~/ D 4» %'6~^ WXQ_)4~U~^0 l] Ξ ` Λ ¾]¢ ~ % Dpb = D ¼6 ).U"~1~/l1X x X// d x x ~^ x Ü%)6^. X _) M^l%X x X/) Ξ⊥p⊥p ` Λ⊥p⊥p ¦ w . ~ % D ~) XQ_ x ~ 12~U~^ Dp X _] x ~ ]~ ¦
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% 1 D Q * _( b ( `$*j d oHd ik`+* , ts~ 0~30\w|zL|z 0 GLK;[ ~=tRQ,v K V zL/~2uLz? v v
DvPw| 2zZY 0 vw|z4vl wnv u?vlvlO
w|zDp
vPw| 2zZY 0 v G] 0A 04w v w|z
DvPw|2zZ tR rl z?vlu2tRu 0 w
Dp
vPwn%2zZ tR rl z?vlu G %v x2vPwnw9vlzLDvPw|*2zZ tR rl z?vlu
w|z%vlwnv u?vlvlOw|z∀D′ ⊆ D, D ≡ D′ ⇒ D = D′
%vSx2vPwnw9vlzLDp
vPw|y2zZ tR rl z?vlufwnz vlwnv u?vlvlA w|z
∀D′p ⊆ Dp, d(Dp,D
′p) = 0⇒ Dp = D′
p
2 O ~2~$0\w 04w D2zZ tR rl z?vlu G
0zLD′
p ⊆ Dpnvluf v
d(Dp,D′p) = 0
G %v0x2vPwnwnvlzLD′
p
vPw|wnzL/~2u?v N;t wnvPw N 0Ov / Nlz?vlOnww 0A.zL4Nlu w/x4tR4w0u?vPw.N 0Ov / Nlz?vlOnwx x2vPwnwnvlzL w|zL/~2u?v
DpG t x 04N
Dp ≡ D′p
vlDp = D′
p
G2M 04NDp
vPw|y2zZ tR rl z?vlu G2 t nr NlzL~ 0C vTwnv YtRzLx2vSu6t0lv tR2z v G
G ©0ª B ®)²,¯ J F ©* @ ª * @ =L © ² « ª0 §± ¨ ©£O^²,³ ¯ ±/ © ¬ ² « ¨ /ª² « © ¨ 'KW§ « ¯ ©³ §« ª ©¢´ Q «P¨ ©© « © ª à XIÃRX^Z © G ±6 © ª ©JF@O ±/ >³ « J]²,¯!F >8² I © ª « ® ª O § ¯ ¬ DF.² « ³ h ¬ ² « ¨ /ª/² «
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: - ;2
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2
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; 1~ b/ led D 4» 2%)6^4X _) 4~U~^I/l Ξ ` Λ ¾W] ~ %Dp b = D ¼'.M %%X x X/ed x x ]] x K%' ~^ X _) 4~U~^l% X x X/) Ξ⊥p⊥p ` Λ⊥p⊥p ¦ w ( ~ %~/ Dp ~l¤ x ¤W x ^b ¯ ©O @ X8 _ x ~~:d %]._M^4 D ~)¤ x ¤W x ^ b ¯ © O @*XQ_ x ] d ¦
( b ( `$*j d oHd ik`+* , t N 04wnv 4tR|z 0hx2vu6t N 04xHzL|z 0hx2v QAtRzL vl~30\wnv 2z? vlvlAw 2 u6t N 04wnv 4tR|z 0hx2vSu 2zZY0 /zL r vlx2vu6t02zZ tR rl z6tRuLzL r =u?vPwt vPw~ 0~ z r r w r 9tRO z?x2vlO|z? vPw G2 t/~ 0~30\w|zL|z 0 GLK;[ ~=tRQ,v K V w||zL~ u?v v D
vPw| 2zZY0 v0w|zvl wnv u?vlvlAsw|zDp
vPw| 2zZY0 v G
2 tT~ 0~$0\w|zL|z 0 G ~=tRQ,v~ nr N r x2vlAnv zL/~2uLz? v v DvPw| 2zZ tR rl z?vlu w|zEvl wnv u?vlvlO
w|zDp
vPw|2zZ tR rl z?vlu G=v²2/ª
ψ±§ ª¯ §2¨ ³ ¬ ª² «
φ¨ : - ;
2
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^ π l1 ¼P4» 2 V U]q~% A x ] ψ(π) ∈ ψ(A) ψ(π) ¤ x ¤W x ^£X _] x ~ ] ¦
% & , *
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* D^ D T\¥¶v Á_taJM cNM(s SV;lpgWinOP^m
^ 9 D ~ %~/UX _] x ~ ]lI¤ x ¤W x ^4» U%)6^ 8XQ_)4~U~^yl1X x XQ_) _1¤¥] Ξ ` Λ ¦ 1¾W]0~1~/ Ds b = D ¼P'.M£~1~/@ l] d x x ]]~^x @1)6~^ XQ_)4~U~^ l] Simp(Ξ) ` Simp(Λ) X _] x ~ ]~I¤ x _¤W x ^ x Simp(Ξ) ` Simp(Λ) ¦
( b)( `+**j d oHd ik`$*-, tN 04xHzL|z 0 x2v QAtRzLw 2DzL/~2uLz? v v
DvPw| w|zL/~2u?v vl x 04N v
DsvHz?w|nvS2z?vl GM vT~2u w Ds
vPw|=tR 2 vluLu?vlvlAQAtRQ\=tRO G %v0x2vPwnwnvlzLDs
vPw| 2zZ tR rl z?vluf~=t u6t~ 0~30\w|zL|z 0 G ~=tRQ,v K G
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_ D^ ; ` ¬)BR§ ª¯ ¨ ® ¬ ¯ ªq³ « .h ª )« © ² «Ü¨ ±§+± ³ ¨ C³ §,'¬¨ ©q¯ ® ®'ª ª ² « © '« ³ .¨ © Oh ²
« '« ª '±±/ © @ G © ¨ © © /« © ¨ ± §± ³ ¨ C³ © F ±/ © ² « ª ¨ © § ¯ I ¯ © ¨ B F I ! "#"$&% ! '(*) ,+ .-/%*)102$4326527598 &272:;326 0% 2<32$;#%* ! 2 4#% =2$% 2> 2 )2?32@8A ): & + 6! %B% ! 32;#%* ! 2 DC E%F1&2 ! G32H02 I 32 > ! 3J2$
K!L '*' M #2A62$N3242"2 ! 2$G32$?"#"$&% ! 32<5$% O ! > &2 3% !! " + P2G > ! 32G52?598 2G &Q52 > #3 598 &2;RS0 C -UT > 3"$V ! &B% ! 325% !! 25$2 %*W32$N2 I D% ! 2 ! &2 >&> 2N% ! ?3"$V ! &2 + 2 > 6! 2; X K O#% > 2=8"%*YA62;324"$ &% ! +[Z % G5$%* L 2$\523"$] 32O5%12 X^5$B2 ! % ! 0% " 598 &2 L 2=_'a` + .- ! &]b%*Ac" ! -/2 ! % ! > )#%*3"2K3 ! 5$24598 &2=5 > 2=A ! C 2<32<"$ B% ! A6% ! 2 ! A ! C 2=3d-U%D% ! +
e7fg hDi^jEkmlnjolnp;irq K! %B% ! 32\#%* ! 2 s2S5$2 >&> 2 C #2Ac2n.32\2 ! 32N5%*A6232 > 2$]b%15 > B &% ! +Ft ! > - )2G !
(+, ξ, 0; 1)
(−, ξ.0, 1)
(+, ξ, 0; 2)
(−, ξ.0, 1; 2)
(+, ξ.0.1, 42)
! -/2:. ! 32$2$ ! BA6 > 2S5 > 2 > &2 ξ2: > &"S32 I ]b%*& + P2%*) > YA62Q5$ 5$ > TG"$% 322$.32 *%*& C u0 &Vv2 > - 5$&% !
(+, ξ.0.1, 42)+Fw : 52 > 25% ! 32\% > C v BY$Ac2 5$B% !Ox 2$2 C 2:&% ! 2$?"$% > 2= > @! %&% ! 32<D% ! 2 + .- )24"5$"32 ! G3%&N3% ! 54$-U"$5B2<2 ! ] =y
(+, ξ, 0; 1)
(−, ξ.0, 1)
(+, ξ, 0; 2)
(−, ξ.0, 1; 2)
(+, 0 : ξ.0.1, 42)
% (+, ξ, 0; 1)
(−, ξ.0, 1)
(+, ξ, 0; 2)
(−, ξ.0, 1; 2)
(+, 1 : ξ.0.1, 42)
.- 5$B% !(+, 0 : ξ.0.1, 42)
L*! &Vv2 C 2 > 2Q0 V ! N2: > - 5$B% !(−, ξ.0, 1; 2)
! 3B C 2> - 5n&% !(+, 1 : ξ.0.1, 42)
L*! &V2 C 2 > 2S0 &V ! 2 > - 5$&% !(−, ξ.0, 1) z % ! 5$%*Ac2 > 2! %A@)2=3- 5n&% ! ! " La *2$? > 2 > &2
ξ.02 ! 2 > - 5nB% ! 0 &Vv"$242n > 2Q0 :&V ! | + P2 2$ 5$&% ! ! -/% ! O N32<#%* ! 2 ~
n : D% \32 1I &% ! \3B ! 5n2$=y
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@ "!$#"% R&('
) P2 5$B% ! ! " La &2S% ! d0 :Vv"2$H > 2$ 5$B% ! .#%*&*2$."$5"32 ! 2$ > 2 #%* ! 2 ~n : 2$O3% ! 5<% 0% OT+*"% z 2$ ! -/2:? ! %"| +
) P2 5$&% ! N > 2 > B2 ξ% ! ? ! > &2 32 > ) 2;2$ ! 24D%* ! 2 ! ?3% ! 54 5 ! 2 5$B% ! +
, 2 >&> 2A62 ! > C 2:&% ! 32 > ! %*A > & B% ! 245$ 5$ > 2ryP324D%* ! 2 <3.-D"2 ! ! -U2 ! / ! 2 ! c > 27A10A62 ! %*A > B &% ! + % ! &3"% ! > 2@%*&^322$ ! D1
D2
2$E ! <y
D1 = (+, ξ, 0; 1)
(−, ξ.0, 1)
(+, ξ, 0; 2)
(−, ξ.0, 1; 2)
(+, 0 : ξ.0.1, 42)
D2 = (+, ξ, 0; 1)
(−, ξ.0, 1)
(+, ξ, 0; 2)
(−, ξ.0, 1; 2)
(+, 1 : ξ.0.1, 42)
E = (−, ξ, 0; 1)
(+, 0 : ξ.0, 1)
(−, ξ.0.1, 42)
(+, ξ2, ∅)
(−, ξ, 0; 2)
(+, 0 : ξ.0, 1; 2)
(−, ξ.0.1, 42)
(+, ξ1, ∅)
> 2G) 2$=2D2$5$&*2$` ξ
` ξ
2$ξ ` ξ1, ξ2
+ P2@"$ > G32 > ! %*A > B &% ! 3 "$2 D1,E
2$(+, ξ1, ∅)
> % C 25$2 > 13 "$2 D2,E
2(+, ξ2, ∅)
+ o2$s3"n > o32$s ! 2 5$&% ! % ! ?3"5$&\ L 27R32( +e6f54 687Nqoq97\li q:7 j<;>=Nq?7A@4kCB
P2=322 ! <BA6 > 2K% ! L " ! "$ > B"K2 ! %*B ! =32<#%* ! 2 n : ξ
T > > 5$2@32$> B2 I #% > 2 5nB% ! \D%&&2 +DFEHGFEI JLK9MNMOK QP M
RTSVUXW>Y[ZY]\^WL_a`cbTdfe>gihkjOlmonqprXsutwvAxyz|^~t^wsuOsua$ut9A[u1ksFt(H^Qs
π := ξ | n : ξ
Nξ
sut".css3n
saccsO tN
(t s[$s" su^]sun
st 95¡H3ktAt5t5>"¢scFvAxyzX|£~ ¤ ¥^NQsu
πξ
¢NN¦Nsuξ
Nn : ξ
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R& ( ! ukN#O# ! "O3Q#V
R SVUW Y[ZYc\^W _q` d(lhg?jNm^g9h.jlmonap ( (as ts
Ξ ` Λ>(as "!N|5y xz stA
# tHFt5H 3s¥F(^sC%$ccqqs s1[ ¡HTs(+, ξ, I)
$(+, n : ξ, I)
N (as%$cc .aqs
zzξ
Nzn:ξ &# tH¢aNO 3s+o $Hss'$"s aNO 3s
Fid− ( $]Tkqqsa5NHs*)N(as%$cc aqs s[ ¡H5 s
(−, ξ, I)Nn : ξ
stA(¢3k^Qs3(I ⊂fin N
sut"as@ A6&Vv5 &% ! sξ
(.cs3 rst%$cc t
(+, ξ, I)" 9u.t5ts^ c¦NsTsa?N(
ξ ∈ Λ
Z % N3"$V ! B > 2$N322 ! > ] ! N3"nV ! B > 2 )2$ 254D%* ! 2 +
R SVUW Y[ZYc\^W _q`+ dp*,op m-Nm.^gihkjOlm£nap/ 0O1oV.32!vAxycz|£~
Dsut"( 3s54 N.64 ¢suOsuaV9[+Vu1Cks
t(Ha s
sD := D+ | D−
ξ
swt5Hk¡D+ := Fid | D+
tswt5Hk¡9^ 3] sD+
t := z | zπξ| (+, πξ, i1, . . . , in).(D
−ξ.i1, . . . ,D−
ξ.in)sa5N.¡
D−ξ := Fid− | D−
sξsa5N.¡?a ] sD−
sξ:= ((−, ξ, I).D+
t )I∈Nk^Qs3(πξ := ξ | n : ξ
NN
stF( s tsT]s sAtN(t87.9"cst94 ]sutA sN & πξ
s qtsa s( k^Qs3(n
stF(su^]sus
ξ(¥k]s
/ 3s tkqcs sut" vAxy|y;:At$ .Fsut s[ ¡H5 sD+ s3z=<>?a|y@:Q tN kFsut" s [¡H5 s
D−ξ
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@ "!$#"% R&*`RTSVUXW>Y[ZY]\^WL_a` d£n6lh Olh g9j
/ ast Qs sua"s %$cc tsut"~ |y <N1t"( as1tsβ
t5a¢N N q84£sc
s[t(HQsts su.£^99 (^s %$3]
Kct
# N¥]suK
st¢cs T3z
N+ 1$ 3suOsu'$s a5Nc sFid− &# N¥]su
KstA( u T¡H$k.t
zξ
sξ+ sut"wO tA[Tts
β &# N¥]suK
stA( u T¡H$k.tzn:ξ.i
s3c
stF s[¡HTs
c′.(−, ξ, J).cn.(−, ξ,Kn) . . . c1.(−, ξ,K1)c0.zn:ξ.i
Vs $i ∈ J
s3ξ
9 iit O tAcstAkcs3¥ sut%$cct sut(ck) ¤ $ ¦Ns
(−, ξ, J)O~o|5y¢
zn:ξ.i # N¥]suT %$ccK
tN $5H(ε, ξ, I)
sξε
stwO tA ¥(tsβ
# N¥]suT %$ccK
tN $5H(+, n : ξ.i, I)
c
sut" s ¡H5 s
c′.(−, ξ, J).cn.(−, ξ,Kn) . . . c1.(−, ξ,K1)c0.(+, n : ξ.i, I)
Vs $i ∈ J
s3ξ
9 iit O tAcstAkcs3¥ sut%$cct sut(ck) ¤ $ ¦Ns
(−, ξ, J)O~o|5y¢
(+, n : ξ.i, I) # NCcsC %$3c3
Kt$ $u5H
(−, ξ.i, I)s %$ccq $" su^QsN t
csut" sF[¡H5 s
(+, ξ, J) ( 2 + (+, n : ξ, J))Vs $
i ∈ J ¤ $ ¦Ns(+, ξ, J) ( 2 + (+, n : ξ, J)
)~ |y (−, ξ.i, I)
t ! > 2QD% ! 2 n : ξ.i
2"2 ! 2 ! ?32 > &2 I *2$5 ! 2K ! ]b%*A B% ! T > ~ ! 3B5$232G2O 0 ! + 2#2 ! 3 ! > 2 n ! 2\5%*A62 > 2$ 5nB% ! C 2 > 5% ! C 2H2 ! 2 ! 2 5$B% ! 2nS% !0 &V ! ?A BN&Ac > 2A62 ! > 2$ 5n&% ! O > 2 > &2 I
ξ+ 2:Y$Ac2;BA6 > &Vv2 > 2G3"$V ! &% ! 2$ > > 2$5$ 2=3 32$2$ ! +
RTSVUXW>Y[ZY]\^WL_a` dfe p ap g9jah nm-Nm 0^gih.jlmonqp / as+v < ! xz>y"!X~^ 2 ! vAxycz|£~ Tt((as ts
βsut"1ast"( sT $] tVs $
¢3k^Qs3(5ts¢ 5HQt HQs5asut$#(tc 4as"9t t T $Hsusu"$"sa$HsFid− %
k^Q5]s( / as q 7 $& c¦Ns stvAxy|y 2 ( 2 + z=<>?a|y*2 )+tsu.cs tsQs51kas9 asC%$cc¢3t5 c 3s ( 2 + ^Nc 3s )
RTSVUXW>Y[ZY]\^WL_a`' d)(*ap g9jah nm/ as ! xz y+!X~^Cst# (aswq 7 $& c¦Ns Vs $Fk^Qs3(5t t5HHs # (aswq 7 $& c¦Ns Vs $Fk^Qs3(5t ( aNO 3s ) 3a¢ 1 su ,s %$cc st
Fid− ¤ ¥^NQs
Chronβ
s tsTcssut5$& c¦NsutAt"( (as+tsβ
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R& ! ukN#O# ! "O3Q#V
R SVUW Y[ZYc\^W _q` _¥dm% moh.j6/ 3s$3s $9¢.^Qs5t 2 y st?( 3s$3s $i¢3k^Qs3(5t¢¦(Vas $"^]su^$9t¢ %$cc t / ?((ycz |xi|8t((as+ts
βst( sVs $Ak^Qs3(5t$?su^F s
Fid− ^a] st5
βsut?t5HHssw 3a]sut'$&astt^Fsut5$& c¦NsutAt"(
β rs st5tsuk9"]su
Fidst¢ 3s3c st"( (as tsFt Hs rs st5tsuk
Fid− sut"¢cs st5tskC9ccsia$.¡ rst $" tsSkunk
sut s 3] st( (as tsaNO 3s sut" st5tsuk QNH
DwEHG EHG 9MNKK! " > B2 ! 2=]b%*B32< > > @! %&% ! 32<"$2 32 > 25nB% ! R + _ + '^ L 2 '*' +
R SVUW Y[ZYc\^W _q` d eFp p %"m%qn/ v < <( ?~1sut ( su tsuT]s 4 os st5tsuktXt st tsut
βi
su ¦ ( k]s" iu H q (t O t s (tst9as¡Hkt t5HHsu su^s3 (as¡HktF^Nc 3su su^
R SVUW Y[ZYc\^W _q` d (wg9n qnap m/ as !x?~$v~ N t (a 7ctsO
Rsut"[ as ss suttsu. tC s
R ^]sut
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/ <9~ sut"(¥q 7ts O t]s"¦NsuX[Cs[cc s $"$(s+sut $ as ^s s3 %$67$uk]¦Ns
R SVUW Y[ZYc\^W _q`cb£b d 6m ^m p %m anr>1' ( uts
RstX tsA¡HTs s su tsuT]s st k]s ( $3s $]s(X¢ 5HQ*)
suttsut sut st5tsk t sR
¦(9 " 9u.t5ts^F¦ (as¡H.t R SVUW Y[ZYc\^W _q`cb d m%qn!g"/ ¥utssut"9!" xct¦Nstu¥ts sutA3cs
R SVUW Y[ZYc\^W _q`cb + d (lh g9j mVl#! h mon$ oh h ,%!m%
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t$ sut(Tt$(t87ckcs3C.css Tts sR
/ as %$3c3 N t ( suttsu.sR
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Fid− % u$uk^s^sut5tsu. q5.'$u9qN+q ,t(asA%$cc¢3t5 c 3s.t5 cs &# cs T3 &# ( u T¡H$k.t ^ ]sk]ssut"A.t5 cs &# (as %$3c3aqs ^¢cs.csst3kt5 cs
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7>V ! 5!3 R&&
e6f p ; @ $l$q?@<jolnpi G! %A > % !=! "$52$ 2 ! 2A 598 ! 2 ): 2 > s"*% > "2 C 2\3 ! > 25 32 A > 2% %*) ) B> 2 +
DFEAEI P P MP2 ! % ! 32 Y2 > ^! %*A > % ! % ! > 2\ ! 2$4y) P%*\3d- ! ! 2 5$ % ! > A10A62 5n % ! D2 A0$2Q0% "$2 > 2 ]b% 5$2 C ! A" > C 2C 2 > A0$Ac2 2=3d- ! 322 ! #2 ?2 # > 2 ]b% +) Z % 2 ! 325$2 > 2 ! 5%*A"242n?] 2 ! 2 L > 2 5n % ! N2 ! 22 >&> 2$ % !r &> 2 ! 2! % % ! 3d-/8 % C 2^32 ! 2$ 5n % ! C 3"$5 "5 "$Ac2 ! > 2$ 5$% K] Q0 C -UT! 52: ! D% ! +) 6! % % ! 3-U%*32 z 3"$V ! % ! R + _2" L 2 ''a| &> "2=2 ! > 3 C 2= A" > 2 ! -U2$ O ]&%Vv AcA62 ! O2 I 2 2 +'N2(2 ! % ! > 2N32$2 !
D1 = (+, ξ, 0; 1)
(−, ξ.0, 1)
(+, ξ, 0; 2)
(−, ξ.0, 1; 2)
(+, 0 : ξ.0.1, 42)
D2 = (+, ξ, 0; 1)
(−, ξ.0, 1)
(+, ξ, 0; 2)
(−, ξ.0, 1; 2)
(+, 1 : ξ.0.1, 42)
E = (−, ξ, 0; 1)
(+, 0 : ξ.0, 1)
(−, ξ.0.1, 42)
(+, ξ2, ∅)
(−, ξ, 0; 2)
(+, 0 : ξ.0, 1; 2)
(−, ξ.0.1, 42)
(+, ξ1, ∅)
^! %*A > % ! 3 "$2 D1,E
2 2 ! yR + > 2A Y2 5$ % ! 3 !
D1
2$(+, ξ, 0, 1)
2nE#%*Y32 ! 2 5$ % !
(−, ξ, 0, 1)3% ! 5 > 2A Y$2="n #2=2:(ξ, 0, 1) )_ + > 5% ! ! v % ! 32
(−, ξ, 0, 1)2:
(+, 0 : ξ.0, 1)C * ! 2 L 25 > - 5$ % ! ! " La % *2<3 v > 2
(−, ξ.0, 1)C > - 5$ % !
(+, ξ, 0, 1)0% "$2;T > 2A Y$2="n #2 )' + > - 5$ % ! #%* *2 C > - 5$ % !
(−, ξ.0, 1)2: > - 5$ % !
(+, ξ, 0; 2)C * ! 2 L *25 > - 5n % !
(−, ξ, 0; 2)32
E )( + > 5$% ! !1 % ! 32(−, ξ, 0; 2)
2$(+, 0 : ξ.0, 1; 2)
C + ! 2$ L 25 > - 5n % !(−, ξ.0, 1; 2)
C > - 5n % ! D%* *2=32 > -U"n #2 "5"$32 ! 2 )` + > - 5$ % ! #%* 2; ! 22:(+, 0 : ξ.0.1, 42)
C , ! 2 L *2$5 > - 5$ % !r! " La *2(−, ξ.0.1, 42)
C > 32$ ! Y2 5n % ! D% 2 z 5-/2: > - 5$ % ! (−, ξ.0.1, 42)32> ) ! 5982<32<3% 2=32
E| +
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R&2 ! ukN#O# ! "O3Q#V
+ > 2?" > 32 > 4! %*A > % ! 2:(+, ξ1, ∅)
C > - 5n % !c! " La 2"5"$32 ! 2=5 > 2 > 2 ! -/2: N5% #" + =! %*A > % ! 3 "$2
D2,E2$QTD2 ! 2\3 -"$2 ! 2<y > 2$H32 I 32 ! Y2S"$ #2H% ! 2A" > 5$"2$
` + > - 5$ % ! D% 2; ! 2;2:(+, 1 : ξ.0.1, 42)
C , ! 2$ L *25 > - 5$ % !r! " L* *2(−, ξ.0.1, 42)
C > - ! % 32$ ! Y2 5$ % ! D%* *2 C ]b%5 > 2m ξ.0 z 5-U2$> - 5$ % !
(−, ξ.0.1, 42)32 > ) ! 5982;32 L* 5982<32
E| ) + > 2?" > 32 > 4! %*A > % ! 2:
(+, ξ2, ∅)C > - 5n % !c! " La 2"5"$32 ! 2=5 > 2 > 2 ! -/2: N5% #" +
! 22 % ! > \]b%*A62 >&> 2?32<52$O2 I 2$A > 2=2:G3% !! "$2425n % ! & + ' + '$ L 2^R32(' +DwEEHG P KM K A 598 ! 2 ) 2 &> "2<2$ ! 2 3 % ! 32 > t z % !L 2 t ): 5$
598 ! 2| 3"5$ 2 Z 2$2 % P% 2 ! 2 + 0 Vv5 % ! 32 > > 3 "?32$N3"$V ! % ! 3% !! "$2 5 2: #%*"2 5%13 L 2=3 ! > 2 )2\32=Q*8A ) z 2$5$ % ! & + ( L 2@R32a`| +1t ! % !m! 2K-/%5$5 #2 32H0 : V2> 2\3"nV ! % ! \3 ! N52n2;25$ % ! + ! 2 ! 2 5$ % ! 2 ! 2O 2 32N5% C -U% !c D2 >B> 2 8 :%* C 2 2 ! "$]b"$2 ! 52 I6 )23248A ) +
R SVUW Y[ZYc\^W _q`cb$d h lg^ph nmrst >yc|xOy+!>~a( 4 .tAt^wsuOsua$ut9A[u1ks
& kt"Qc¦NstH := He | Hn& kt"Qc¦Nsts qQs t5 cstHe := ε | He.mov& kt"Qc¦Nst a s qQs t5 cstHn := He.Vis(Fid−) | He.Fid | He.z | He.Vis(zπ)T$ sTsactmov := Cutπ,I | Vis(+,π,I) | Vis(−,ξ,I)
Nπ := ξ | n : ξ
sut"A(C.a s( K 2 > C 2N5%AcA62 ! 2$\ > 2\8 % C 2$N% ! &> 2$=y) P2$ 8 :%* C 2$ 2 I 2 ! ) > 2$\5%*2$ #% ! 32 ! I ! 2 5$ % ! ! % ! ]b%5"$Ac2 ! 2$A ! "$2 > %* C 2 > 2\8 %* C 2 ! % ! 2 I 2 ! ) > 2N2 "$2 ! 2 ! =32$ ! 2 5$ % ! 2A ! "2$ +) .-/8 :%* C 2
ε2: > -U8 %* C 2 32^y > - ! 2 5$ % !m! - N2 ! 5%*2<5%*A6Ac2 ! 5"$2 +
K!6! -/"5 εC 2?3 ! Q52$Q8 % C 2 +at ! H2 I 2A" > 2 > -/8 :%* C 2
ε.Cutξ,I .Cutξ.i,J2 6! %"Cutξ,I .Cutξ.i,J
+) .-/8 :%* C 2
H.Vis(Fid−)
5%2$ #% ! 3JT ! 2 5$ % !Fid− ) > 2 +
) .-/8 :%* C 2H.Fid
5%*2$ #% ! 3JT > 3 2 L 2 ! 5$242 ! 2A"\V ! 2 +) .-/8 :%* C 2
H.z5%*2 D% ! 3 > ! 52$Ac2 ! G3d- ! 3"$Ac% ! z "*2 ! 2 >&> 2A62 ! K]b%15 > "; > 2K32$2 !J! " La ]E5$%*2 #% ! 3 ! G2$O3 -D"2 ! O32
Fid− | +) .-/8 :%* C 2
Cutπ,I
5%*2$ #% ! 3JT ! 5% > 2=3- 5n % ! \3 v > 2s0% "2N A > ! "A62 ! +) .-/8 :%* C 2
H.Vis(zπ)
5$%*2 #% ! 3 T ! 3"$Ac% ! ]b%15 > " ) > 2 z > 2 > 2 32\]b%15 > % !! -U2$ N5% D"| +) .-/8 :%* C 2
Vis(+,π,I)
5%*2$ #% ! 3 T ! 2 5$ % ! #%* *2 % 2 ) > 2 z > 2 > 2 32]b%15 > % ! ! -U2$ N5% D"| +
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>V ! 5!3 R&) .-U8 %* C 2 Vis(−,ξ,I)
5$%*2 D% ! 3 T ! 2 5n % ! ! " La *2 % 2 ) > 2 z > 2 > 2 32]b%5 > % ! ! -U2$ N5$% #"u| + @! % % ! 32K N2: ! 2 ! % % ! 2 >B> 2 +1w >&> 2D2$A62$\32=3"$V ! > -/"$ 32 > A6"A6% 2 A6%*A62 ! \% > 2Q0 V ! N3d- ! 2 5$ % ! "$"0% " + =!r! %2
⊥! 2#%* % ! A"D%* ) > 2 +
RTSVUXW>Y[ZY]\^WL_a`cb d qnOlr ¡H'$cc sC ?~i|
(n, ξ) 7→ SautH(n, ξ)OtT( & kt"Qc¦Nss i su t ]s
Hsut (^s
¡H'$3c3 9ccskcs 4 csw9
SautH.Cutk:ξ′,I(n, ξ)
=
H.Cutk:ξ′,Itn = 0
s∃i, ξ = ξ′.i
SautElague(SautH(k,ξ′))((n− 1), ξ)t5n > 0
s3∃i, ξ = ξ ′.i
SautElague(SautH(k,ξ′))(n, ξ)t5k^
SautH.Cutξ′,I(n, ξ)
=
H.Cutξ′,I
t5n = 0
s3∃i, ξ = ξ′.i
⊥t5.a
SautH.Vis(+,k:ξ′,J).Vis(−,ξ′ .j,I)(n, ξ)
=
H.Vis(+,k:ξ′,J).Vis(−,ξ′.j,I)
tn = 0
s∃i, ξ = ξ′.j.i
SautH((n− 1), ξ)t5n > 0
s3∃i, ξ = ξ ′.j.i
SautH(n, ξ)t5k^
SautH.Vis(+,ξ′,J).Vis(−,ξ′.j,I)(n, ξ)
=
H.Vis(+,ξ′,J).Vis(−,ξ′.j,I)
t5n = 0
s3∃i, ξ = ξ′.j.i
SautH((n− 1), ξ)t5n > 0
s∃i, ξ = ξ ′.j.i
SautH(n, ξ)tka
SautH(n, ξ)= ⊥
N tAcst sut $tN
Elague( )st¢[ ¡H'$3] ¦(Xsu ,3s ]s su csucTs^ % (^st"( s
2$2 ! % % ! 2# 2 >&> 2N5 % ! ! 2G #%*2= 2 ! > 2Q8 :%* C 2$ +w >&> 2O2% > 2? > 28 :%* C 2N3"nV ! I 3"$V ! % ! & + R ^2n & + R& z 5-/2 ! 245% ! " C 2 ! 5$2@3 5%13 L 2<3 ! > 2 )2O324*8A ): | +Z > C 2 32\3"nV ! ! 2 ! % % ! 32 2\5$%*A6Ac2Q5-/2S8 ) 2 > 2 ! "A ! C 2 32 02 1I > 2598 ! C 2 B> "$2 5 5$% ! 2<T %5 2$ ! 2 ! 2A) > 2=3d- 5n % ! \T ! 8 %* C 2 C 5%2$%
D% ! 32 ! I 2H0% 2 O2$G% #%* ! z > 2 5$ % ! %15 "$2?% ! % 32$ K32;598% ! C 2C 5$%*2 D% ! 32 ! GT > @! % % ! 32 2<2 ! > 3 C 2 'a_ | + ! 8 % C 2
H % !J %15 2 ! W ZY]\aW
AH32=2 >&> 2K%*2 C 2cy
) H2A ! 2
Cutπξ,I
> %*Q% ! %5 2 ! 2 5$ % ! % 2 ! " La *2(−, ξ, I)
2$ ! 2 5$ % ! % 2 #%* *2(+, πξ, I)
+)
H2A ! 2+
Vis(+,π,I)
> %* % !J %5 2 ! 2 5n % ! % 2D% 2(+, π, I)
+)
H2A ! 2+
Vis(−,ξ,I)
> %*% !r %5 2 ! 2 5n % !r! " La 2(−, ξ, I)
+
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R32' ! ukN#O# ! "O3Q#V
) H2A ! 2 > 2=3"A6% !r > %* %5 2 ! 3"$Ac% ! z ]b%5 > "K% ! % ! | +
) H2A ! 2
Vis(Fid−)
> %* %15 2 ! 2 5n % !Fid−
+) ! % !
AH2# 32 +
R SVUW Y[ZYc\^W _q`cb ' d (lh g9j 6 g OhNm% nqj ah lgaph ?n m¤ +t5t$cs(1z 1' !N|y xz
AH% & kt"Qc¦Ns
HX(+uts
R ¤ q$ ," s9 $ts ,usAt ^Qs H = H′.Fid
c35tAH = ∅ H stF scO£Ns k ¡H5cs3(sN 5N]s % 1
# H = ε
]5tAH = ∅
# H = Fid− ctAH = σ
Nσ
sut" %$3]Fid− % T%$u.as ^sut5tsu.
a5k"$u i # H = z
c35tAH = σ
Nσ
st¢cs u ( ¡H $.ktN+a) % u%$kas^sut5tsu.qk'$ 9
# H = Cutξ,I
]5tAH = σ, τ
Nσ
st %$3c3(+, ξ, I) % [ %$u.as ^
sut5tsu.Cqk'$ 9 sR
s3τ
st¢ ( ]¦Ns%$3c3 aNO 3s^V]s ( % [%$u.as ^sut5tsu. a5N.¡ s ts
ξ ` Λ)
# H = Vis(−,ξ,I)
c35tAH = σ
Nσ
sut $](−, ξ, I) % u$uk^s^
sut5tsu.qk'$ 9 # H = Vis(+,ξ,I)
ctAH = σ
$σ
st %$3](+, ξ, I) % u$uk^s^
sut5tsu.qk'$ 9 H stF scO£Ns tucss NCNcs % 2
# H = H′.Fid− ]5tAH = ∅
# H = H′.z]5t
AH = σ$
σst]su ( ¡H$k.t N^) O 7Q st5t"(t
s $] a5NHs sAH′
# H = H′.Cutξ,I
]5tAH = σ, τ
Nσ
sut" $](+, ξ, I)
7 sutt(t s $] a5Nc s+s
AH′
sτ
st ( ]¦ $s%$cc ^Nc 3s+^V]s ( % [C%$u.as£ st5tsuka$.¡¢s+(ts
ξ ` Λ)
# H = H′.Cutn:ξ,I
]5tAH = σ, τ
$σ
sto %$cc(+, n : ξ, I)
7 sutt(t¢ s $]C^Nc 3s s
AH′
sτ
sut" %$cc1a$Hs ^csA 7Qsut5t"(t sA $]t Hs s
ASautH′(n,ξ) # H = H′.Vis(+,π,I)
]5tAH = σ
Nσ
sutF $](+, π, I)
O 7Q st5t"(t s $] a5NHs s
AH′ # H = H′.Vis(zξ)
c35tAH = σ
Nσ
sut %$3c3zξ
7 sutt(t s $]^Nc 3s s
AH′ # H = H′.Vis(−,ξ,I)
c35tAH = σ
Nσ
sutA %$3c3(−, ξ, I)
7Qsut5t"(t+ s $] t5H 3s s
AH′
R SVUW Y[ZYc\^W _q`cb_d.3ah.j m ,l3p^hlm
R uts 9. s
H0 = εw 4 H(^s < ~'!N|y xzfv v ( a3
su51kkt s )Xt(>cst & .tQ35c¦Nst / &kt 5]¦NsH
st >yc|xOy"!X~aN~ <( ?~Rt5
H0 + H
¤ 1q$ ," sF9 $t
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>V ! 5!3 R32R H = ε sut" ]s¢.^ s Qi
( ) 9]s suttskq5.'$u9D
sR
st?¢3t5 ck¡>]5t1q$ ," sw95$tAt(D
# t5
D = Fid]5t
H Fid &# t5D = zξ
ct# t
ξsutA3kt ]sc35t
H Vis(zξ)# tka# t5
D−ξ = Fid− ct
H Fid# t5k^
H z# t5
D = (+, ξ, I).(D1, . . . ,Dn)ct
# tξ
sutA3kt ]sc35tH Vis(+,ξ,I)# tka
# t5(−, ξ, I)
"9u O tD−
ξ
]5tH Cutξ,I &# t5k^
H Fid &( ) 9]s suttsu.qk'$ 9
D s
Rsut a5N.¡>ct csk]s a5Nck¡
ξ s[Tts
sD
sut"A.t5 cs ¤ Ca $ ,"sFi $tAt"(D
# t5
D = Fid− ]5tH Fid−
# t5 (as %$3]TaNO 3s(−, ξ, I)
" 9]su^ % Dq]5t
H H.Vis(−,ξ,I) ¤ TsuC¦Nscs^37QQs51k .t5 s s $"s35 s ,V]s # t5k^
D = SkunksH
sutA(^s¡H35 sa5C]s H = H′.Cutπ,I
NH = H′.Vis(−,ξ,I) H
σ $] a5NHs t5t$cs % H H
K %$3c3¢t c st"( ^
σ ¤ q$ ," sw9 $tAt(H^
K ( ) K = (+, ξ, I)¤ q $ , s $"1Ts ¢N]s5$t
1 ( )# t5
ξsut"A.t5 csct
H H.Vis(+,ξ,I)# t5k^# t
(−, ξ, I)" 9 O t
D−ξ
c35tH H.Cutξ,I &# tka
H H.Fid &( ) K = (+, n : ξ, I)
# t5ξ
sut"A.t5 csctH H.Vis(+,n:ξ,I)# t5k^atH
σ %$3]t Hs s
ASautH′(n,ξ)
¤ q $ , s 9$twt(Ha[ $"3ack^Vcc s
σ t5
(−, ξ, I)t
σct
H H.Cutn:ξ,I &k t5k^H H.Fid
( $*) K = z ¤ H H.z
( ) K = zξ ¤ q $ , s $"1Ts ¢N]s5$t(+, ξ, I)
# t5
ξsut"A.t5 csct
H H.Vis(zξ) &# t5k^# t
D−ξ = Fid− ]5t
H H.Fid &# tkaH H.z &
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R32*_ ! ukN#O# ! "O3Q#V
( s ) K = zn:ξ ¤ Ca $ ,"s $1TsN( ]s5$t(+, n : ξ, I)
# t
ξsut"3.t5 cs]5t
H H.Vis(zn:ξ) &# tka3a t σ
%$3c3 t5H 3ssASaut
H′(n,ξ) ¤ +a $ ,"s 9$tFt"( ^
$"^ck^VccD− s
σ # t5
D− = Fid− c35tH H.Fid &# t5.a
H H.z & H = H′.Vis(+,π,I) Hσ
$] ( ¢t c s*)t5t$cs % H ( ) σ sutt"( 3]s k t (as %$3] a5Nc s
(−, ξ.i, J) ]5t
H H.Vis(−,ξ.i,J) ¤ suC¦Ns]sa37 su51kkt5 ss $sQs ,cs
( ) σ sut" t"( 3]s sFid− ct
H H.Vis(Fid−) ( $ ) σ stt 3cs X %$(as$]a5Nc sa]5t
Hst (^s¡H5 s a35Ccs
_q`cb !"$#%&'()+*,$*'-/.01*32!"-4$!"$'5'-!"1'-/67 28$!"*9:<; 1+ 2'=*!">&? '=*,@@A**=B0DC(EFG FH 2*,IA*,J4*,2*K4 L!"I 4M1 2'N*!"4$!A*,'=*,@ @A**=B0-OPRQSG 72!"-4$!"EG T4U*2V '=*!":/WX'(!":? G +)7 28*,ZY\[^]`_Nab Ac"dHefP
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>V ! 5!3 R32'
RTSVUXW>Y[ZY]\^WL_a`cb dCn ,%! hknqj ^h lgaph ?n mr x?~ 1'cy
cH ( & kt"Qc¦Ns
Hst(^sq 7 $& c¦Ns ( 3su^5$suk]su su^ 3c s*) 84 cs
ka£ $c sTs^ rXs5$t s
cH.Vis(+,n:ξ.i,J)
stA su t"( s
cε = ε ( q 7 $& c¦Ns3cs )cH.z = cH.zcH.Fid = c′HcH.Vis(η,ξ,I)
= cH.(η, ξ, I)
cH.Vis(zξ)= cH.zξ
cH.Vis(Fid−)= cH.Fid−
cH.Cutπ,I= cH
Nη
stA5N % +$−
s [a 7 $& 3 c¦Nsc′H
sut"Ncs % [ q 7 $& ]¦NscH
q5H"s s %$ccCaNO 3s ( qqs*) O.]s3saNsu.cs N]s5$t
cH.Vis(+,n:ξ.i,J)
9 3 s 4 kAcs ¢.^Qsn : ξ
cH.Vis(+,n:ξ.i,J)= cH.(+, m : ξ.i, J)
Nm
st¢csa3Ts %$3c3 t s[¡HTsVis(−,ξ,K)
O tcH
q sSautH(n, ξ.i)
sTsNN( cs5$tcH.Vis(zn:ξ.i)
9 3 s 4 k ]s¢.^Qsn : ξ
cH.Vis(zn:ξ.i)= cH.(zm:ξ.i)
Nm
st¢csa3Ts %$3c3 t s[¡HTsVis(−,ξ,K)
O tcH
q sSautH(n, ξ.i)
\¢\ Y]Z3Yc\^WL_a`cb dn', ! h mVl qpHg9jah n m%r N k9 ( & kt"Qc¦Ns uts t"( (as ts
βstFasq 7 $& ]¦ $s ( 3su^Nsu.cs 7
Ts^ ] s )t"(β
RTSVUXW>Y[ZY]\^WL_a`cb dAgap ! h lh g9jr TzxO cy a|y xz JRK uts
R sts
βsut" cs suttsk s (ts
β¡HT+s
( ] sut N k.t sut & .tQ35c¦Nst t(cs utsR
RTSVUXW>Y[ZY]\^WL_a`& dm% m hkj gapl g£g?jqn$ s suttsu. t
Ds
E s tsut^VcsutAt^FxON| x >?xz ?~ t JD,EK = z
¤ ¥^NQs $"s35 st V]D ⊥ E
DFEAE AKK9MK P QM P M'N2(2 ! % ! > 2 32 2 !
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R32( ! ukN#O# ! "O3Q#V
D1 = (+, ξ, 0; 1)
(−, ξ.0, 1)
(+, ξ, 0; 2)
(−, ξ.0, 1; 2)
(+, 0 : ξ.0.1, 42)
D2 = (+, ξ, 0; 1)
(−, ξ.0, 1)
(+, ξ, 0; 2)
(−, ξ.0, 1; 2)
(+, 1 : ξ.0.1, 42)
E = (−, ξ, 0; 1)
(+, 0 : ξ.0, 1)
(−, ξ.0.1, 42)
(+, ξ2, ∅)
(−, ξ, 0; 2)
(+, 0 : ξ.0, 1; 2)
(−, ξ.0.1, 42)
(+, ξ1, ∅)
P2 8 % C 2 32 > "$3 5 % ! T" 3 " 2 D1,E
% !
ε Cutξ,0;1
Cutξ,0;1.Cut0:ξ.0,1
Cutξ,0;1.Cut0:ξ.0,1.Cutξ,0;2
Cutξ,0;1.Cut0:ξ.0,1.Cutξ,0;2.Cut0:ξ.0,1;2
Cutξ,0;1.Cut0:ξ.0,1.Cutξ,0;2.Cut0:ξ.0,1;2.Cut0:ξ.0.1,42
Cutξ,0;1.Cut0:ξ.0,1.Cutξ,0;2.Cut0:ξ.0,1;2.Cut0:ξ.0.1,42.Vis(+,ξ1,∅)
5$2 C 3% !! 2JD1,EK = (+, ξ1, ∅)
P2 8 % C 2 32 > "$3 5 % ! T" 3 " 2 D2,E
% !
ε Cutξ,0;1
Cutξ,0;1.Cut0:ξ.0,1
Cutξ,0;1.Cut0:ξ.0,1.Cutξ,0;2
Cutξ,0;1.Cut0:ξ.0,1.Cutξ,0;2.Cut0:ξ.0,1;2
Cutξ,0;1.Cut0:ξ.0,1.Cutξ,0;2.Cut0:ξ.0,1;2.Cut1:ξ.0.1,42
Cutξ,0;1.Cut0:ξ.0,1.Cutξ,0;2.Cut0:ξ.0,1;2.Cut1:ξ.0.1,42.Vis(+,ξ2,∅)
5$2 C 3% !! 2JD2,EK = (+, ξ2, ∅)
! 52 32 I 2 I 2A" > 2 > 2r5 > 5 > 32 8 % C 2 2 3" 2 A ! 2 5 &> ! - 3d- 5 % !=! " L *2 % 2 ) > 2 +PZ % ! 2 I 2$A > 273d- B> % ! 32 ! % ! % 3" 2 A ! A623 ! > 2\5 > 5 > 32 8 % C 2 z 2 H! % ! 3 ! > ! % A > % ! | % ! #2 5% ! 3" 2 > 2\32 2 !
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$#O # ! "O3Q#Vk !k" !Qu[! Q R32a`
D′1 = (+, ξ, 0; 1)
(−, ξ.0, 1)
(+, ξ′, 0)
(−, ξ′.0, 1)
(+, ξ, 0; 2)
(−, ξ.0, 1; 2)
(+, 0 : ξ.0.1, 42)
(−, ξ′.0, 2)
z
P2 2 A I A > 2 3d-/8 % C 2 3 " 2 D′
1,E % ! > %
ε Cutξ,0;1
Cutξ,0;1.Cut0:ξ.0,1
Cutξ,0;1.Cut0:ξ.0,1.Vis(+,ξ′,0)
Cutξ,0;1.Cut0:ξ.0,1.Vis(+,ξ′,0).Vis(−,ξ′.0,1)
Cutξ,0;1.Cut0:ξ.0,1.Vis(+,ξ′,0).Vis(−,ξ′.0,1).Cutξ,0;2
Cutξ,0;1.Cut0:ξ.0,1.Vis(+,ξ′,0).Vis(−,ξ′.0,1).Cutξ,0;2.Cut0:ξ.0,1;2
Cutξ,0;1.Cut0:ξ.0,1.Vis(+,ξ′,0).Vis(−,ξ′.0,1).Cutξ,0;2.Cut0:ξ.0,1;2.Cut0:ξ.0.1,42
Cutξ,0;1.Cut0:ξ.0,1.Vis(+,ξ′,0).Vis(−,ξ′.0,1).Cutξ,0;2.Cut0:ξ.0,1;2.Cut0:ξ.0.1,42.Vis(+,ξ1,∅)2
ε Cutξ,0;1
Cutξ,0;1.Cut0:ξ.0,1
Cutξ,0;1.Cut0:ξ.0,1.Vis(+,ξ′,0)
Cutξ,0;1.Cut0:ξ.0,1.Vis(+,ξ′,0).Vis(−,ξ′.0,2)
Cutξ,0;1.Cut0:ξ.0,1.Vis(+,ξ′,0).Vis(−,ξ′.0,2).z52 C 3% !! 2
JD1,EK = (+, ξ′, 0)
(−, ξ′.0, 1)
(+, ξ1, ∅)
(−, ξ′.0, 2)
z
e6f k mlck 7 @ 7p4li^j?7Nk; q 7 j @;T;>7Nq 7 @rqdj9;@<lnjEq !#"%$'&5 ]3(&O5)& & &5 !#*& 3(& $ V+& > > 3 -, 3(& 3 , V(* !#* & 3.& > /! 0! !#*3(& > &$5 !#* , 5 , 3(&)* &21 * 0& >&> & > & 3 , V+* !#* & /! 3! !#* > & /"'& 3(&465*8A " -72 1Z ! 6] 5 &> & > > &5 '&8& > 5)!*A" , 8(&)* !9*: > & *(! !#* B> /, & 3! > & '"/& 3(&
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R32 ! ukN#O# ! "O3Q#V
4658A " !#* * & A , 3 '& &* /& > 5 > &?3(& Z & /&% ! &)* & > & *(! !9* 3(& > &5 !#* , 5 , 3(&* &91DwE EI K9M K M M
R SVUW Y[ZYc\^W _q`9b d e>g hlhg9j H^w a s3 sutsu csts $Nqt
AsX
# / !x?~Nv xVv vAx z|
cosut"A(cTsas
• ∪ A ∪X # / !x?~$v (x?~a£~
cpstwu]u su^ s
X ( aNQ[x])$C(as 9ks
[i : a]N
i ∈ Nsa ∈ A r su^]su
isut "¢sc(¢3k^Qs3(
# / as vAxy|y x z v OycCstA(ast Qs^ 3] s su£^Qs s[¡H35 s
co1cp2 . . . c
o2n−1c
p2n
Qskcs¦ $s# co1 ∈ • ∪X
s∀i > 1, coi ∈ A# cp2i = [k : a]
c35tk < i t $"s $t? $H¦ $s
cp2i
st¢k]9co2(i−k)−1
R SVUW Y[ZYc\^W _q` d p*,op m ^m ',6l3p ahl/ O1£ ? ?1£|5 y|st( su tsuT]s
Da3 3cs $ct 9a 4 ^s 9.s s
¢t cc t $"1Tsa9i•
su.csut¦Ns
qcp ∈ D ∧ qcp′ ∈ D⇒ cp = cp′
/ ¥V(zx >9+t"(Acst3sut s & t"u tstw 84 ^9
[x← φ] = x.r | •.r ∈ φ
/ ~ |y N1o ?1^(|5 y|Asut as( ]t csut(xi)
Vktck'$3tA sxi ← φi
* , * !#* ,), &)* &9: * '&)*(! ! & ! 0 /& !" " !& * , ! & * & & * 0! ]& *# % /"'& & 4 5$% " & ! & * & & * * , ] 1 &
φi * & #&)* * * ' ( , * !#* & & & & * * , ] ! /& 0!#* &* *) & & * 0! ] * " & !#* * , 9& * * & & * * , ] ! 1R SVUW Y[ZYc\^W _q` + d h lg^ph nm0^g9nqp !m% £p*,op m% ^m ,l3p ahl / >yc|xOy+!>~a1sut"cu su^ s u1ks
Γ := •ν | Γ.Visaν| Γ.x | Γ. 〈m : a〉 | Γ.Vis[ν:a] | Γ.Vis[x]
rsi +-,/. u]u su^ s
Γst¢aN
Γ(i) * $ ! 10 &
Γ.x ! /& 0!#* 32 !#* * , #& ! , & ! 0 &
Γ. 〈m : a〉 ! /& 3!9*
2 *(& !#* 3! #& z #& * 0! * & 9| ! , &21
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$#O # ! "O3Q#Vk !k" !Qu[! Q 2V&
RTSVUXW>Y[ZY]\^WL_a` dnmr 2?~a
VueΓ(n)ast(HQs s $"N"qt sutF 84 cs.a^ $3Hsu su^?9
VueΓ(1) = •Γ(2n) = x
VueΓ(2n) = x
Γ(n) = 〈m : a〉
VueΓ(n) = VueΓ(m− 1).a
Γ(n) = Visaν
VueΓ(n) = VueΓ(n− 2).a
DFE E G K9M P P M
1 -, & , * !#* & 10 & #& 3! * & & !#* , & &)* & 0! * y 1# & 3! * &
n : ξ 1 * &* 0 & & $ * *(& !#*
(+, n : ξ.i, I)&
n !#* & ] ! &
(−, ξ, J) * $/!#* 0 & 2 &
(+, n : ξ.i, I) z , * % !#*& 1 & &*`*| 1 & , 0 &)* & !! & ! & & !#* * , #& & *(!#* &&)* & !#* & ] ! &
(−, ξ, J)& ! & & , * !9* & z , * !#*& 1 ` & &97a| &
! " * $ ! 0 & z , * !#*¥&%1 2$ & 2'a| 1 & 0 &2: 0 & 2 ! & & , * !9* : & , 0 &)* & !" & & !#* &( & , " & $/!#* 0 & & $ ! 0 & z , $(! & & & & 4 $'*[| ! &( * z , $(! & & & & 4 $'* *#& &| 1
_ 1 * ! " * $ ! 0 & z , * !#* &%1 2 & 2'a| : * ! #& &* Vis(+,n:ξ.i,J)
& ! &)* *(& !#*
(+, m : ξ.i, J)!
m !! & & !#* & ] ! &
Vis(−,ξ,K)
1 & , 0 &)* &6] '& & ! " /& &&)* * 9& & $ ! 0 & z * $ *(& " &9:3!#* *&" &
Vis(+,n:ξ.i,J)
Vis(+,m:ξ.i,J)
| 1 6& $ * & &* !#* /, 0 &* & *(! !#* & z , * !#*& 1 `$ ! & &97*| 1
'.1# & $ ! 0 & &* 10 & /!#" " & &)* & ~ !#* ! " & Cutn:ξ,I
! 0 & & '"/& & 465$% " , !"0! &* &)* & *)
〈n : ξ, I〉&
[ξ, I] z * & ! /&u| 1(1# & $ ! 10 & & /"'& &465!$% " &* & " &
ν !& & ) %
0 , * & & Z & /&% ! &)* -72 0! 1 & & $ * * ! " & $ ! 10 & 9& *(&@] !#* !#* z '& ! " | 0& , & 2 * 1 6& " & *(& !9* * 3&* " & y X & /& !#* '& & ! & !#* &6] 2 , * !#*& 1 2$ ! & 2'.1
, * !#* * &"!#* /& *.& ! !#* & & & * * *(&\] 1 & /"/& &465$% " z & * ] 1 & ] $(! & &)* & " & & !
A& " &
X0!
' , * !#*¥& 1 _*_ ! & % !#* '&u| 1
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2(2 ! ukN#O# ! "O3Q#V
R SVUW Y[ZYc\^W _q` dp n6(lh g9j am% am% m hkj 'aj6 !m £p*,op m% am ,l3p^hl
A = (ξ, I); Ω;zξs3X = z rst1 st5tsukt1sAcst 3sut1s' & t"u t
a sutA3suto[cu$£ $cct$ $ $3s ( qsut¦ $s c¦NsTsa ) s cu$£(ks cst %$3] tsu.$"Nat # cs st5tsuk
Fidsut"AuV^( s
Ω &# ( suttsu. a5N.¡AV 9us^ sFid− t k]s
ξsut" cu$£( 9
[zξ.z]sA[
$^ $3c3 sut csut%$3] t a5NHst &# (as %$3c3z
stA$^(HQss[z] &# (as %$3c3
zξ
sut"cu$£( ssu[zξ] &# (as %$3c3
(+, ξ, I)sut"cu$£( ssu
[ξ, I] &# (as %$3c3(−, ξ, I)
sutAcu$^ Qss[ξ, I] &# (as%$cc
(+, n : ξ.i, I)sut"cu$^ Qss
[n : ξ.i, I]N
nsut csa3Ts %$3c3 t
a$Hsut su^s t #(tc 4£^ s]s s[ $& 3 c¦Ns s3a3 t5kq]su su^¢csaTs %$cct^Nc 3sut sk]s
ξsu^cs
(+, n : ξ.i, I)s¢t #t4£^ &# (asF%$3c3
zn:ξ.i
st $^(HQs s[n : zξ.i]
Nn
st9cs ^TsX %$3] t a$Hsutsu^s t3 #t4£^ s3cs O s $& 3 c¦Nss3a tkqcsTs^]s aTs %$cc t ^Nc 3sut sk]s
ξsu^cs
zn:ξ.i
s¢t3 #(t"4£^ r> #(t"4'$cc t"( $su.cs s[ HN$c¦NsV3s $F¢.^Qs5t
_q` Y\ '=H ? 2'=*!" <; A*, &? '=* * ^&$'7Fid− '=*,U*A A!"'(
<; 2'=*!" zξ
*,@A* <; &) !" 2 .01*Z4$) '+ @!"*,zξ
2!")+) 2'N*!"!")7 0P
& ! * & & * 0! ]& * /"'& & 465!$% " 1 & ! * & & ** , ] & * % /"'& & 4 5$% " 1 & ! * , & & *G~ , & /"'& & 465!$% " 1
! !#* *#& & !#*(*(& * ! $% &3! & , *(&6*.! !#* '"/& &4 5$% " 2 * & & *1
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$#O # ! "O3Q#Vk !k" !Qu[! Q 2#7
RTSVUXW>Y[ZY]\^WL_a`' dAp ?n3(lhg9j hkjNm£p "m¤ ck.kts[-$"^3su^] sa$ cctc [^¦Ns
(−, ξ, I).Fidt5 V 4^s+¦ $s
(−, ξ, I) " 9 q9t O tA]s suttsu. % $"s35 sFt ] rXsut%$cct qasutAt^cu$£( sutA¡c%$u.csTsaw# ($"$ #"$Ns(
[ξ, I]sutAcu$^ su
(+, ξ, I) &# ($"$ ¢3tu^[ξ, I]
stA$^(Hsu(−, ξ, I) &# ( $N
[n : ξ, I]st cu$£( Fsu
(+, n : ξ, I)N
nstw t¡H5C]¥k^s5ts
s[ a $" s^Qs N]sut%$3] t kqqst£.i¡c su . $" s sut u ts3w sut$ 3susu'$sut# cs suttsu.
•tas (tswt5H 3ssut"cu$£( s
Fid &# cs suttsu.Ω
t( (as tsF¢t c ssut$^(H suFid &# ($"$
[z]stA$^(Hsu
z &# ($"$[n : zξ]
sutAcu$^ szn:ξ
# ($"$[zξ]
¢3t5 ck¡XstuV^( suzξ &# (aswt5H]a5NHstu t $"Nat
[ξ, I]sutAcu$^ Qsi
# cst $ tsSkunk
t$ .' (ast Qs s $Nqt[zξ.z] &# [1$Hsusu"$"saNO 3s
Fid− t5k^ \¢\ Y]Z3Yc\^WL_a` dp n6(lh g9j m$lj g^p ! h lhg9j
r Tcu$£ $cc19 ]sutst s " & tcuHt $"3 QsVs $ +^5Ck.tucc * &)* , 1 -, & , * !#* 16& & ! !#* 3& & & !#* '& * & % * 3& & , 0 !9* , * , 0 & 1
\¢\ Y]Z3Yc\^WL_a`+ dm h oph.j3Oh^m am% nlh g9j £joph n m%a$ Qs ¦V] auc¦Ns
E3ucsTt"(cstTst s( & t" ct sut"+(tt5s
NVc¦NsVs $ k^ s(t ! #"%$&(' )*$&,+ -."&/$&' (0"1' )0&23465*782$&)*9!:127<;=>2 !;=)*$' )*:?:@"% ,BA782 !9>2*0"&'DC
!;=)*$FE
G )*;< E H 9>2;=I!"%9J7= K29!L%9, B0%NMPO*Q%:R2L&! !9>2;< ,E G )*;=/$.
Di1 , . . . ,Din
0% 0% >,/;<$&2 H 'S )*;<$. ,/"%9,T/ Rj1 , . . . ,Rjm
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Di1 , . . . , Din
/ Rj1 , . . . , Rjm
7=/"%9,B !9>2*0"&'/ !;=)*$&^0&2$&K7= 29!L%9, _0%`MPO*Q%:a2L&! !9>2;< ,EbF _0% >,/;<$&
Di1 , . . . , Din
/ W7= P9,+ ,2"VRj1 , . . . , Rjm
,)*$c P ,/7=-c"&
E(Di, Rj) , dL%;=/$[e)*9!:+ fE%gh29_Q.i S )* !Q&j ,fk E(Di, Rj)
! H 9>2;=fElB)*$&'fk S 29P;<$*mn '/ !; H ;< ,+0%B782N !9>2*0"&'/ !;=)*$FkE(Di,Rj)
! H 9>2;=60%)*$&' E ! H +/9!;poY+ @/$7<"&0;=-."&@2 H ' S )*;<$. ,/"%9,E
"\ _a` +rq M; L4 " \4*,2*K4$S &V.0 'N*!" &&*=.0$ C 4!4$!A*,'N*!"s Petc34 AuYvO2V M+ 0B0+xwIyA8) ?B'=H *,' M A!"$'54 "+A*, )7 *MG F) !"*,'=*& 1'=*,7MG 4!"#4$!A*,'N*!">AJ2!"A@z J&V.0 'N*!"M&&*=.0$@AH * ? 7 0B0 wNyA8) ?B'NH *,',A!"$'S@AH * U*=.0 @2Z4$!"*,$'(0P
6!& 3! 0 & /!#" " &9: & & /! 0! !#* 3& & & !9* /& & $ , ! D| & ! -, z '! 3! !#*¥&%1 $ ! & 7'a| 1
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7 ! ukN#O# ! "O3Q#V
$p;i7# 7Ni^j 7 "@ $k ml)6k 7 qPl( 7
10 & " & &$ !#* &* /& & &* &)* 10 & 9& 3! * & &* & * & 0! * & *) !9* 1 * /& 0 & 0 & !#*#&)* !#* & !" ! & & !#* 3& & & , * D| " &&)* & !#* !#* y & & /& &
0 *) "0!#* &)* /! 1
R SVUW Y[ZYc\^W _q`a_d3 g9j3(lhg9jPointr>C¡H3'$cc
Point s1kts sutk^Qs3(5tstF C¡H'$cc ¦(iqsa sut5tsu. tkqcs
t( (as tsβ
ssu^3cs ( suttsu. Vs $ .a s(t s[C ,sAt"( ^ s
z 7→ z
zξ 7→ zξ
t5ξ ∈ β
z0:ξ
t5.a
(+, ξ, I) 7→ (+, ξ, I)t5ξ ∈ β
(+, 0 : ξ, I)tka
(−, ξ, I) 7→ (−, ξ, I)
\ ¢\ Y[ZY]\^W _a` d)(Fg lh,ah! hl n %!g9j£m mojl
R(uts1t5kq]s ¤ T
JPoint(R)K = Point(JRK)
; ;>7Nq & ! '& " &6& & ) &* !#*(*(& & 10 & " & z & !#* 1 1 ! & ' | !#* , * , %
/, * /& 1 &&)* 1 ! '& & ) &)* !#*(*.& * & -, ', !! & &* 10 & " &z /! 0! !#* 1 &A'*| 0 & #& 0! * & * & ', , & z & !#*1& 1& 1 ! & * &| 1R SVUW Y[ZYc\^W _q`& dp pHm l ', !m
r xO qT(| ?1& sut $sssu^$w9 # t ¢3t5 ck¡
Fid v D
# tAa$.¡ t5N ⊆ M
c35t
((−, ξ, I).DI)I∈N v ((−, ξ, I).DI)I∈M
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O "V !V! 3 # 7
RTSVUXW>Y[ZY]\^WL_a`' dCp?p m m $$lmoj6h g9jqjm !r xO q ¢|(z>y xz>z sut"¢ $ssuOsua$ i# t t5Hk¡
Fid 4 D+ D+ 4 z (+, ξ, I).R 4 zξ# tAa5N.¡ Fid− 4 D−
s £t5N ⊆ M
c35t
((−, ξ, I).DI)I∈N 4 ((−, ξ, I).DI)I∈M
; 7 = ;# 7:@T@ ) 7 & $ , ! D| & * 0 & *(& !#* ! , +, y ', !#* & &91 6&(0&)* %
* : & '& $ , ! |& & !#* & #&* & & #&* 0! & !#*(*.& & * 0 & 0! & $ /& 2$ ! & 1
DFE DwEI P M P
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NbF (0% ,>/;<$&D1
/ D2
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D1 = (+, ξ, 0)
(−, ξ.0, 0)
z
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(−, ξ.0, 0)
(+, ξ, 0)
(−, ξ.0, 0)
z
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E"%$\0% ,,/;<$\0%6LY2*,
ξ `EbF B0% ,,/;<$&
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(+, ξ.0, 0)
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z
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7 ! ukN#O# ! "O3Q#V
. . . (−, ξ, 0)
(+, ξ.0, 0)
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b' '(!"WX!"1 !" '(!"WX!" OP
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(+, ξ, 0)
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e1
(−, ξ.0, 2)
e2
(−, ξ.0, 2)
(+, ξ, 0)
(−, ξ.0, 1)
e3
(−, ξ.0, 2)
e4
Y A*, 2!"-4$!"? $'SE &' $'
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Y A*, ' ^A!"$'b4 " @A**,'-& *,$'(H 2'=*,@) $' P
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V !! # 7'
DFE DwE G MNM * , & , * !9* 1' ! & ' & , & *) & , & *) * 3 & ! /& &
0 & #& 3! * & 1RTSVUXW>Y[ZY]\^WL_a`+& d %"m%qn ^m p %m an$
/ T <( ?~ <( ?~R1, . . . ,Rn
stA(asF9" cc¥utsTsuts \¢\ Y]Z3Yc\^WL_a`' d "g Oh lh £hl
Nuts suts R1, . . . ,Rn
i
JR1, . . . ,RnK = JJR1K , . . . , JRnKK
Ng )*"%9B78217<"&0;=-c"& S 9,)*LY2L%;<7<;=! ,fk )*$ 2"% !;<7<;=,+^7= S 9!;<$&'/; S I0% B+ -c"Y2 !;=)*$&34+/$&+DC9!;=-c"& E S 234 "!E '/;Xk )*$"% !;<7<;=,7=0%/:?;pC S 9!;<$&'/; S x0% @+ -."Y2 !;=)*$&634+/$&+/9!;=-."& $# S 9,) S )f!;pC !;=)*$&%E S 234'"()*DE
DFE DwE +
RTSVUXW>Y[ZY]\^WL_a`+?b d-, j6!.nhg9jr ycz !"~ y xz
⊆su^cs suttsu. tsut¢t ^ s s $stH cs
"\ _a` Y ; *,2U?*!" 28!"$*=.0$6; '\4 "^V; *,2U *!" 28!"$*=.0 '-)/4$.0 6*,'(M<; 2'=*!"2V 2G *,)/4U*=.0$H *,'
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⊆ (+, ξ, 0)
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Fid−
2M.01* 2!"$'=@A*V *,'SE V; !"H+H '-*!"$GP
RTSVUXW>Y[ZY]\^WL_a`+" d (wg 6Olh ,^h ! hl s suttsu. tt^9!x v a|y 1& ]5t¦ .ktFt^¢k'$ (tO t ( u s st5tsk
RTSVUXW>Y[ZY]\^WL_a`+"+ d-, jOlmop"m (lhg9jr ycz| !N|y x z ⋂
i Di
(as¡c1k.cs(Di)
s st5tsukt$"9c ]sut sut"Xcs suttsk ¡HT sut $& ]¦ $sutk'$ (tsut O t $&£V¦ $s
Di * , &)* * /& 1 &&)* " & &*(! !9* *) , & *) 1
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\ ¢\ Y[ZY]\^W _a` _ d l ',ah! hl 3
(Ri)(^s¡c.kcs sts¥k"$u tO t (¥tsO
R ¤
J∩ RiK = ∩ JRiK
. Z1`$ S 2*,> S 29(7= _Q%;=! ,)*9!;=-."& EfE `$ H +/9!;poYW-c"& 7= FQ%;=! ,)*9!;=-."& h0%T7X ;<$c ,/9,> '/ !;=)*$N0% Ri
! F7X ;<$. ,/9,, '/ !;=)*$?0% FQ%;=! ,)*9!;=-."& 0% _9,+ ,2"V
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Ri
! 7X ;<$. ,/9,, '/ !;=)*$0% ()*"%L%7<;=(0% JQ%;=! ,)*9!;=-."& (0% _9,+ ,2"V
Ri
EE `$ /$ ' )*$&'/7<"% J-."& J∩ RiK = ∩ JRiK
E
_q`' Y\ 4@ $V $# @ *,H HM_ *,Xct%e 4 "A 4 K41,U#%B 2A 2 .01*2!"-4$!" E 2 !"'=*!" <; 8*'-!"*=.0 0P
DwE DFE P P
\ ¢\ Y[ZY]\^W _a` d g9jg£lg9j^h Ohl r> a35Ck.tuccTsut $.t5tua s.t87 % 753kts 3$s s qQs t5] as
4
∀i, Di 4 D′i =⇒ JD1, . . . ,DnK 4 JD′
1, . . . ,D′nK
. Z1`$ 7= H +/9!;poY@2" $%; H 2" 0% _Q%;=! ,)*9!;=-."& E
#
! * & , & ! 10 & * , '&9: !#* , * & ) *(! !#* & !0! &&)* & !0! &&)* & & & !!0! &&)* * , /& 1 6!#* '&&)* 2 * !"3! & %&)* &9: * !!0! &&)* * , '& & !"3! ', & & & * *(& , , * & & *) & " &21
6& & *(!&)* /& & * , & /& 3! , * " &&)* & / !#*(*(& & 0! 1"! 1 & & $ & * /& $# , '&)* & & !9*(*(& & & ! 0 & * , '&$ " & &% ! & & ) 0!#*(&)* & & ' & & !#*.*(& &
[& ]% & !9*)( 1 7 1 & ' 1 6& & '* &
& /!9* * & , -, * & !"3! & &* & ! 0 & & !#*(*(& & $ " & & !#* * & " !"3! & &* " * , /& 1
& !#*.*(& & $ " & & ! 0 & * , /& &)*9! &* & !"0! &&)* * , /& & !!0! &&)* * , '& 1
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7
" -, : & " & !9* ! 3! ', & * /& 1
"!$#&%('*),+.-"/103254637"8:95;=<3>@?A"BC8EDGFHI7J8E75DK8ELNMPO8
E7"Q.7SR.TU58VUW8:UW8EDDK8ET$7WDXF@QYF[Z5H]\D^HI_`HI7"8aL=L8bMIZPDK8.c
"!$#de'gf[+.-5h ih,jWk"lnmo2pj54rq+.-/s0n2W4tu vxwY;=<yvxz,v5|5~Uyu HI7"8:FWT5HY8:8=DGF
E⊥ = D | ∀E ∈ E, D ⊥ E
"!$#e'h eh"f+44ijx+ 2pl+G/o"l4637Jv (vxwY;KBBIy; A&;=>o B8ED^FaHI78`.FWT5HY8:=YZPO8DKQ7SMPTQ_GF"Q.5Q.7iZPOc
"\ "! Y ; !"',8$!V!" b<; +&' 8*=.0$+'L1>2!")/4$!"'-) $' )+U'=*K40P
* '& 0 & % , * !#* &4' 0 * !"3! & &* & & ! ! /&
& ) &* !#*(*(& & ! /& " & G. "!$e'al&+.2pf[4
T D 4 D′ 8.F D ∈ GZPOQ_¡D
D′ ∈ Gc
¢ 7g£¤ZP_GFT¥HPOoT8=_K¦5D¡T D v D′ 8.F D ∈ GZPOQ_¡D
D′ ∈ Gc
QT[F (Dk)HP7"8V§ZPL¨ToO$O8U58bUW8=D¡DK8=To7WD`UxZP7WDbHI7©¥^Q.Lb£yQ_GF8ELN8E7FLNHIO[FT1£"O8
G¦"D¡T ⋃
k Dk8ED^FHI7(U58ED¡D8ET$7eZPOQ_¡D ⋂
k Dk
8=DGFHI7JUW8EDDK8=To7ª8.F~Z^£W£,ZP_^FT8E7F Gc
NbF\0%/9!$%;=/9 S )*;<$. , S 9,)*" H /$ ' )*$&!;=0%+/9>2$. "%$ 0% ,,/;<$E0%
G⊥ / 7= Q%;=! ,)*9!;=-."& (0% (9,+ ,2"V
Di,EE
"!$#pr'*«jW¬4 e®pl¯4m4°¬±m"4°¬^¬4&/$jW¬Cl²/$jWq°k"/$f4°¬b¬]2pfVl¯k³®WkW¬4637´UW8EDDK8=To7
DD^HI_gHP7"8MZPDK8
β8=DGFµ>9|5>wBDT~Z5HY¥.HI7SO$T8HUW8
β7~u 8=DGF`HWFT$OoToDbZ5HLQT$7WD
UW8H\§[QToDUYZP7WD`HI7"8bF_=ZP7"¥"8`UW8Dc
QT[F E HI78FWT5HY8DGHI_bHI7"8CMZ]DK8 β c3¶7e7"QYF8 Lin(E)Ou BI3·]B¹¸iBJº3BI·rºnBI·G·]BI>·ºnB
E>y9|x>wBI··YA5wβc
"!$#»e'*¼/¯½¡h"f+.-Whihnj5kplyl¾/oj5q°k"/of[4t8b¸>¿EvxwY;=<yvYznv5|5>y9|5>wB¨Uyu HI7"8:FWT5HY8
E8=DGF
Lin(E⊥⊥)
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7
Ny3 p!$#ie'h"hf[+4 N4&j5+±l¾/oj5q°k"/of[4637(v Jv5wY;KBBIy;>9|5>wBC8=DGFHI7"8:.FWTWH°8`¡YZPO8±DQ7©MPTQ_GF"Q.5Q7&ZPOpO$To7"ZPTo_8c u 8=DGF[[I@UTo_K8XHP7"8:F5T5HY8
GF8EOoO8V5HY8
G = Lin(G⊥⊥)
(p!$» Y A*,+&? '=* B "Aξ `
A!"$' U*,&V *,0PS ^'R4$!"1^4$!"@!"*, WX!"$/ 2'=*!"
(−, ξ, I)3*, f H *,'4$!"@!"*,/'N*,U*A/^U*M4& "!
ξ #%$ '&"(*)+-,.)(/&"!0 )1324)56+78198(4+:(*;.<=+>, ?8+@! 0 )1324)56+A8198(4+(*;.<=+-, "BC,(*;D=,5EF&G)(4+@!)( 0 ,-!8(4+>,-HJI4'& #$ =K(*)+-,.)(L!!'&'&G8,(MBC,(*;D=,5EN(*O '&8+P2/=&9;DHJI,Q=BR8(4+ATSKBU=V(*)+-,.)(W!T!'&'&G8,(M&8,132XBR # @( F8+I(M!'&'&G8,(YBC,(*;D=,5EZ248I +[5E;724;8+A85\!'&\BC,.8I]V8(M!'^4)5E&_!F&J=a`'=&G #bdc3egf'!he3!iDjlkDj !#*_iDmgjUn]c9*k-&@e0&Jf &8k & , *gjlf oj * cf'*4c]kDj !#*\j * , e0&)* c &)*k &Ji 0 m(&piqkEj !#*gi
& j * , cfDjUk , 1r\sXtXuG "!-vxw('zy:4°¬¬^4W/oj©/ojg5l$25¬m"k"jW¬2pjJm"4°¬^¬4&/$j³l¾/oj5q°k"/of[4
"QYH5FU58ED¡D8ET$7rT$7"¥EO[HIDUxZP7WD`HI7JUW8=D¡DK8=To7rOoTo7KZPT$_K88=DGFOoT$7"KZ]To_K8.c|Fc/i l~ c f ~ ~ ieXhjRk ~ mgfDip4j X cTkDhmqhm/fDimg ~ g]kEjRViqj1j cjRf ~ * jl cfq4c]kDjRh%
r\sXtXuG "!-vv' ,j4k&f¡j5k5+G/1h,j QYHI_`FQYHWF¥^QLX£yQ_GF8ELN8E7FyLNHIO[FT1£"O8Q°H¨O$To7"ZPTo_8
G¦¤8FF@Q°HWFUW8EDDK8ET$7
DUW8
G¦ToO8\5T$DGF8HP7
HI7WT5HY8:UW8=D¡D8ETo7D0 ⊆ D
UW8GF8EO5HY8
∀E ∈ G,E ⊆ D⇒ D0 ⊆ E
`8£pOHID¦,¥^8:UW8=D¡DK8=To7D0
8ED^F~Ou T$7"F8E_DK8^¥.FTQ7UW8EDVUW8=D¡DK8=To7WDVUW8GT$7"¥EO[HIDUxZP75D
Dc
Ny3 p!¾%XwJ'-,j4k&fjWk5+G/h,jtu > P|xw]|p;=>v5 Uyu HI7´UW8EDDK8ET$7
DUxZP7WD¨HI7³¥^Q.Lb£yQ_GF8ELN8E7FLNHIO[FT£pO8`QYHNOoT$7"KZPT$_K8
G8=DGF O8
UW8EDDK8ET$7(U58OZV£p_KQK£yQDTFTQ7.cUhc
|D|G =⋂
E ∈ G | E ⊆ D
Ny3 p!¾%vN'yV4°¬^¬^4W/$j©/$j4k&f¡j5q637SUW8=D¡DK8=To7´8ED^F>P|xw]y9rUxZ]7WD±HI7³¥^QLX£yQ_GF8ELN8E7F DTyToO8ED^Fa=YZPOygDKQ7(T$7"¥KZP_7iZ5FTQ7(UxZP7WD¥^8:¥^QLX£yQ_GF8ELN8E7F¡c,¶7eU°T[F Z5HID¡DTyWHnu HI7JUW8=D¡DK8=To7rTo7"¥ZP_¡7"V8=DGF´|p;9Iw]>BIc
Ny3 p!¾%e'-,j4k&fjWk5+G/h,j(mPo2pjhd"hf[+44N4&j5+tu > P|xw]|p;=>v5 ºsAW v/(vxwY;KBBIy;K¦¤7"QYF@8
|G|¦8ED^FµOu 8=7WDK8=LMPO8¨U58bD8ED±UW8EDDK8ET$7WDbT$7&
¥KZP_7"ED^cmgf g /jRf R~ i hg ~ k ~ mgfqi g ~ ~ m ) gkEjlhgi % k $gj 0 m ~ e |pk ~
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(
"!¾%p#e'*),+.-"/103254M2pl+G/o"l4Kh epl+4637"8:.FWTWH°8`8=DGFv/(5;KBbD¡T
|E⊥⊥| ⊆ Ec
"!¾%%('*),+.-"/103254ªl¾/$jWq°k"/$f4Khdpl°+4637"8:.FWTWH°8`8=DGFHI7"8:95;=<3>@?A"BN>9|5>wBv/(W;BDT
Lin(|E⊥⊥|) ⊆ Lin(E) = Ec
"!#$%&')($*'*+5$*
"!¾%pde'-, q&4if+h¤/of[4JmPo2pjhd"hf[+44N4&j5+ t8 w9.BPwx;vW>wBUyu HI7¥^QLX£yQ_GF8ELN8E7FLNHPOFT£pO8QYHXOoTo7KZPT$_K8£Q.D¡TFTo§ G 8=DGFyOu 8=7WDK8=LMPO8UW8=Dµ_=ZPL¨T/."¥Z5FTQ7WD
IF@8EO$O8=Da5HY8O8VUW8EDDK8=To7 DGHIT[RZ]7"F~Z^£5£,ZP_GFT8=7W7"8
G
(+, ξ, I)
(−, ξ.i1, J)
z
(−, ξ.i1, J′)
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(−, ξ.in, J)
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Q10
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` ξ.i Q f ~ iDe%i7"¡YZ5FTo§yUW8MZPD8 ξ.i ` R ¦pQ7rO8µ7"QYF8 O G Q f ~ iqe% P G R cr_stXuG. "!-vYde'h e"lq+.2Wm"4Jl²/$jWq°k"/$f4e/$j5+4&f¡j54©h,2pfVl¯4°¬±m"qkpl¯k87i4Y¬ QT[F G HI7/^Q.Lb£yQ_GF8ELN8E7FyOoT$7"KZ]To_K8.c T G 8ED^F£yQD¡T[FT$§[¦iZPOQ_¡D O D | D ∈ G⊥⊥
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aD | D ∈ G
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[~ k]
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[~ iqk ) ~ e4ciDiDc$9 ~ # mgJ"pE(ehf k ~ ( ~ xk,)RjR cjlf ~ eXhiDjUkEj+& 0 iqh Igj+: hfqk CgE9hhgc$) " ~5xmgj ~ 2/e%)Rj*5xm ~ ) ~ i19hmgj')*) ~ ( ~ kEi@cm/kDhmgf1# ~ ; "phg ~ "pk ~ mgfqi = 6
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⋃
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UWV UWYdX E ∈ ?GiZVeZ[Dnn[cYdVV3aDYuQHXYpoN
V UWYdX E′ ∈ G⊥ [cXD = (+, ξ, i).E′ NZ V|Q
E ⊥ DjFQHR
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X E | E ∈ GN]PUWV3j
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D′ ⊥ E′ N^ VovQHY\nQHVuXWQHRY\[cRE′ ]ZUWVfeZaDeIiZYdXzi3[ D′ ∈ [G
N]PUWV3j X D | D ∈ [G
[DnXiZV3[acX<[ZY\zui3[7jDUWSlbZ^`_cX[b'UWiZR X G NU Z V|QBaG ⊇
aD | D ∈ G ∪ zξ
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£,ZP_T6⋂
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M % 6687:9<;>=@?A=CBD;FE G$d2H 4Yh,jpj54I+4i2"f ⋃
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¶ 7eUWc.W7WT[F O8 ^QLX£yQ_GF8ELN8E7FyLNHIO[FT1£"O8 Q f ~ iDe%&O$To7"ZPTo_8 R ⋃⊥⊥k Gk
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⊥⊥ Q f ~ iDe% ⋃⊥⊥k Gk = Lin(D | ∃k, D ∈ Gk
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b ~ "Jhg ~ "pk ~ mgf&
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r\sABXt BXu =@?A=BD;FE G-vY»HA@CBD9E-B;GF>FIHKJGLKHMGN D9OgPD9OQ/)0R(/)* 5/6TS'/0(67(8*C,23: 1 230*US'/*V2(82W=*X4>23*,&5/237(.*ZY
G⊕ H = G ∪ H
/s 2E G-vxw $ a5E;'&8IBC+ =+3&Z[1\[BU=[5E&8I 5BR'& 0 )1324)56+A8198(4+&91NI BC+.[ 2XBR'&K'&8+<p=I] 0 =5I (!'&'&GZ[(!
(G⊕H)⊥⊥ 248I+I+.[B][R&G85!=(/&I(*@1_^819 `J5q=( 0 ^4!'& = 0 +.[.)(/& (+, ξ, I)8+
(+, ξ, J))I=822/=56+.[.8(4+=IN5';724856+7)`[5E!
G8+J=822/=56+8[.8(4+=IT5E;724856+A)`[5E !
H #$a 8BU=\19)(4+-5'ZBDO I +.[B][+7;! BU=9(*)+.[.)( ! 0 )1324)56+78198(4+B][(*;D=[5' #
! LK" cb d $& )
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" 67:9 ; =@?A=BD; E G E H DMOO- B & %
k ~ gi ~ mgf~
4 4) 4*V*2* S'/*;2<(82W=*D1
(D2
*(4:c.2.Y /6 6XS'/)0 , 1 ,<)>4>2)*V26 S ,c5Y S 1 0 1 **5)0", 1 S 06 2LS0 1 (.2./k6
2$) 4+*X4+*V*Z+23* +*(,. 4:6/ W/. 1 ,23*Z:zξ
Y,.Fk ~ gi ~ mgf*($,.4:6/ W/. 1 ,23*:zξ
? 2 D1 = (+, ξ, I1).R1
(D2 = (+, ξ, I2).R2
?, 1 4) / 1 *6 2 I1 ∩ I2 = ∅ 1 ,c/0V* D1 ~D2 = (+, ξ, I1 ∪ I2).R1R2
? 2 I1 ∩ I2 6= ∅ 1 ,c/0V*C2, 1 ) 1 (.0Z 1 *"*5)2 1 7(', 1 4:D.2(82c/ 47)Tk ~ gi ~ mgf6 ~ gi ~ mgf "JE( (?m/kEc]kEj+&(jlgj*(c$)_6
D1 ~D2 = Fid 8 ~ gi ~ mgf "JE( (?m/kEc]kEj+&(9cG2/j'(9c6) 6
D1 ~D2 = zξ 8 ~ gi ~ mgf?/h3:<"pE( (?m/k'ckEj'&89c]m" C ~ 6
D1 ~ D2 = (+, ξ, I1 ∪ I2).R1R′2
/10R′
2*5(C: 1 ,R2
S0V2 :T4*R4*V*2* 5/ 6 6 1 7(MS 1 0 4* 1 (.2./*"*5)0 4* ,2. ) 4 1 * ξ.i | i ∈ I1 8
~ gi ~ mgfg3:<"pE(4( m/k'c]kDj'& #gfDhjUk ~ 6 D1~D2 = (+, ξ, I1 ∪ I2).R2R′1
/10R′
1
*(:! 1 ,"
R1
S 0V2#:"4*C4+*V*Z+23*5/6 6 1 7(7S 1 0 4* 1 (82./ *Q*)0C4* ,32c) 4 1 *ξ.i | i ∈ I2
?
67:9 ; =@?A=BD; E G"$ H DMOO- B./
&%\m%#Ke4c]egj*)')Rh./
*5(U,. 47) 1 , 47)Tk ~ gi ~ mgf "J$(4(?m k'c]kDj'&(c2 j*(c$)<?r_sABt BXu =? =CBD; E G-v.5 H(' !-GN-O4 IHP3 OKE-BD GD9O
/2( M ⊆ N) I0Z:+*Z+0)/ 230? 7/)( 4+*V*23_SU/ *V2<(.23W
D = (+, ξ, I).R*( :! 1 ,*
DM ~DN\M/10DM
/6 6 +65 S 1 0 (+, ξ, I ∩M)(
DN\M
5/6 6 +65 S 1 0 (+, ξ, I \M)?
4*;*Z2DM
*5( ,c,+.-0/ $21&354 4D
*) 0M?
6 : ( +4 (V( 4:D.2<(82c/ SU/ * 1 7(zξM
= zξ
?:;=<?>'@BADCFEHGICKJ/>@ML~ g Y\eZaD[7eZ[ ^rQjDUWV3nXRi3jcXY\UWV|eZ[
DM
[DnXxnYdSlbZ^\[cST[cVuXqehgwaDjcRYdR[
D = (+, ξ, I).(Dξ.i1 , . . . ,Dξ.in)
[cXxeZ[ V3[ jDUWV3n[cR [cRqzui3[P^\[DnDξ.ik
QF [Djik ∈M
N
r_sABt BXu =? =CBD; E Gl4wH@ B<DPD9O /2( E
) : (8729)> /: T (M
) 0Z:+*Z+0)/ 230? 6 1 (E⊥⊥)M = (EM)⊥⊥ ?+ E( ( ~ eXhmgf8) c )Rm#/j*5xm ~ iDj*(e) ~ # ~ m324"JE(efqk ~ ( ~ kEi kEfDc9 ~ fDiZiqhkZjl# e ~ %#4ckEip/| ~
e)Rmgip #4cgi" ~ "JciJ ) ~ i5xm4c]kDf ~ +A8(/&G8I5E&_iDhk 9c]m32b ~ f iDm%)lk'ck ~ iqkg]k G1 ⊗G2
r_sABt BXu =? =CBD; E Gl v8H<; D EJGF H 3O -BVOKE7DMB JG= &` %"
2 G1 (
G2*Z/7(SU/*;2<(.23W=*ZY ,3237: 1 20Z*(-234: S'4 1 (.* 1 ,./0;* D1⊗D2 | D1 ∈ G1,D2 ∈ G2*5() :(872)X,32: 1 20Z /6TS ,LS ( S'/)0
G1 ⊗G2?
:;=<?>'@BADCFEHGICKJ/>@ML UWSlST[Mb'UWiZR^rQ|^di3eIY\zui3[`nYdSlbZ^\[`UWi ^rQf^di3eIY\zui3[`bZRUWs1QHsZYd^dY\nX[] UWV iZXYd^dY\n[M^rQbZRUWb'UnYdXY\UWVfehgw[=Y\nX[cV3jD[7ehg iZV3[ bZRU?>[DjcXY\UWVA@/bZRUWb'UnYdXY\UWVCB=NEDGFH N
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5 M te
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! ! x'*
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9 N NED ]P[Dnn[cYdV3n NN7N N7N N7N7N NN N7N N7N7N N7N NN N7N7N N7N N7N NN7N N7N ;:;99 N N < aDn[FQHi NN7N N7N N7N7N NN N7N N7N7N N7N NN N7N7N N7N N7N NN7N N7N ;: F
"!>= ? >EH< G5@ J2AKGICKJ2>'@ !(!(!)!*!(!)!(!(!)!(!*!(!)!(!(!)!(!*!)!(!(!)!(!(!*!)!(!(!,+A#B-9 NDCINED ERQWeIi3jcXY\UWV [cV^di3eIY\zi3[7QD [Dj7bmUWYdVuX[ciZRnN NN N7N7N N7N N7N NN7N N7N D :9 NDCIN O ^\UWVY [cST[cVuXPeZ[ ^rQ$^di3eIY\zi3[QD [Djb'UWYdVX[ciZRn N N7N7N N7N N7N NN7N N7N DGF
"!IH J ELK&EM1uA !*!(!(!)!(!(!)!*!(!)!(!(!)!(!*!(!)!(!(!)!(!*!)!(!(!)!(!(!*!)!(!(!,+A#ON "!0N PRQ ;>1EMS<T1ATG3@ G5@0UhCKJWV"&X1uA !(!(!)!(!*!(!)!(!(!)!(!*!)!(!(!)!(!(!*!)!(!(!,+A#B/
9 NDYINED Z"nnUjcYrQHXYd=YdXa N7N N7N7N NN N7N N7N7N N7N NN N7N7N N7N N7N NN7N N7N D3[9 NDYIN V acb1QHRQHXY\UWV N7N N7N N7N7N NN N7N N7N7N N7N NN N7N7N N7N N7N NN7N N7N D5B9 NDYINDC V XQHsZYd^dYdXa NN7N N7N N7N7N NN N7N N7N7N N7N NN N7N7N N7N N7N NN7N N7N Y9 NDYIN F b|UWV3UWXUWVZY\jcYdXa N7N N7N7N NN N7N N7N7N N7N NN N7N7N N7N N7N NN7N N7N [
"!>\ ] >'<T^ >EKC31=<T1@'CFA !)!*!(!)!(!(!)!(!*!(!)!(!(!)!(!*!)!(!(!)!(!(!*!)!(!(!,+_+`79 ND[INED ]Pa y3VZYdXY\UWV3n N N7N N7N7N NN N7N N7N7N N7N NN N7N7N N7N N7N NN7N N7N B9 ND[IN a UWXY\UWV3nb'UWiZR^rQTjDUWSlbZ^\acXi3eZ[7YdVuX[cRV3[`N7N NN N7N7N N7N N7N NN7N N7N F
"!W7 ] >'@ @X1cbC31B&BEHA !)!(!(!)!*!(!)!(!(!)!(!*!(!)!(!(!)!(!*!)!(!(!)!(!(!*!)!(!(!,+.=.-9 N B=NED ]PaDjFQH^rQ Y [Dn-N7N N7N N7N7N NN N7N N7N7N N7N NN N7N7N N7N N7N NN7N N7N C :9 N B=N ^ Ib'UWV3[cVXY\[c^d^\[Dn N N7N7N NN N7N N7N7N N7N NN N7N7N N7N N7N NN7N N7N C0D9 N B=NDC ZeZeIYdXYpo n-NN7N N7N N7N7N NN N7N N7N7N N7N NN N7N7N N7N N7N NN7N N7N C 9 N B=N F bfiZ^dXYdbZ^dY\jFQHXYpo n N7N N7N7N NN N7N N7N7N N7N NN N7N7N N7N N7N NN7N N7N C
"!> d @&JWe >1EW< J2CF; !(!)!(!(!)!*!(!)!(!(!)!(!*!(!)!(!(!)!(!*!)!(!(!)!(!(!*!)!(!(!,+.=.=9 N 9 NED f Yhg jDUWSlb'UWRX[cST[cVXn N7N NN N7N N7N7N N7N NN N7N7N N7N N7N NN7N N7N C;C9 N 9 N UWVZV3[DjcX[ciZRnlN N7N N7N7N NN N7N N7N7N N7N NN N7N7N N7N N7N NN7N N7N C;Y9 N 9 NDC ]PaDjFQH^rQ Y [Dn-N7N N7N N7N7N NN N7N N7N7N N7N NN N7N7N N7N N7N NN7N N7N C;Y9 N 9 N F ^ Ib'UWV3[cVXY\[c^d^\[Dn N N7N7N NN N7N N7N7N N7N NN N7N7N N7N N7N NN7N N7N C;[9 N 9 NDY ZeZeIYdXYpo n-NN7N N7N N7N7N NN N7N N7N7N N7N NN N7N7N N7N N7N NN7N N7N CB9 N 9 ND[ bfiZ^dXYdbZ^dY\jFQHXYpo n N7N N7N7N NN N7N N7N7N N7N NN N7N7N N7N N7N NN7N N7N CB
"!>i d @ <?>3CTAL&BEj@lk\J @'C31uEM^&E ;uCFGICKJ2>'@ !(!*!(!)!(!(!)!(!*!)!(!(!)!(!(!*!)!(!(!,+.=.i
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+ " C 3# ) ["!#$%$&(')*+$,*-.[.$,/$,0+ 12 3242 # (53#%/ ) )#'56 # ) )E92'56 )789 9:;<=23:>417# 912# ) )#'5? A@ "B C 14 4 ", : : $9 AD *(5 C24 #FEG`:)4 1# 4H2I@ 932J# 44 4J 32K47# ) 4 # 414 4L8@ " 32 4 ? " 42 ) 4 # 44 4
EI$I*)4 4M8@ "J 32 14 : ;MEG*#7 H 4# "#24N 42 # ; )24%9 # )#'5? 4 =O # #7 - ) )#'56 P 2I C )') "#$(4( 412N # )4H 1(QSR21# ) )#'56 EI$I*)4 J # ) )#'56 8@ "B 32 14 :
)#'5? T 2I C )')
)#*56 1EI6I*)4 )#*56 8@ "B C2 14
)%#*56 4*(5 )
U #V W YX.<X. ;<4)#*56 8@ " 32 14ZFE 4J 4H417( 17 12 32412 : F $9 AD : ;<Q @ #
3:[47# 192Y1*)4 ) 41# 32 # 414 4H41@94>?
D1 = (+, ξ, 0)
(−, ξ.0, 0)
z
D2 = (+, ξ, 0)
(−, ξ.0, 0)
(+, ξ, 0)
(−, ξ.0, 0)
z
56\2 ) (5] ( 2NDC$922$) :^ C2424 56 ).EG2_46" C 99E( )` )%#*56 8@ "a C2 14b8@ "J# 4 "# (c "# 4"(E(4( " )`KO7(17@&"< 32 ),)#'56 41'(Q ) 8@ "),)#'56 1$I$I')4N - N * *C $9 F :d4124
E = (−, ξ, 0)
0.5
(+, ξ.0, 0)
e f= )*) (
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e e #
JD1,EK = 0.5
z JD2,EK = 0.5
0.5
z
" 5?\417# D1
D2
: (5] ( 0%&"6" # 44
(+, ξ, 0)
(−, ξ.0, 1)
(+, ξ, 0)
(−, ξ.0, 1)
e1
(−, ξ.0, 2)
e2
(−, ξ.0, 2)
(+, ξ, 0)
(−, ξ.0, 1)
e3
(−, ξ.0, 2)
e4
2 IZI*)4 JI3I$)7 a 1EI$I*)4N =
(−, ξ, 0)
α
(+, ξ.0, 1)
β
(+, ξ.0, 2)
; b2 ( 6)4192 #2 $)214
α
α
e1
β
e2
β
α
e3
β
e4
" 5?\ ( #FEGEI4 @ :+ )`ZR41 - ) #7 - 2# 4 "( ,c "( 614"#E( ( # 4H I 4 Z2J# 4K17 )4 :
U OX , e a - ) 41#7 - 241# )` )#*56 1EI6I*)4 J # ) )#*56 8@ " 32 4 :
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*() )1* & )
687:9<;>=@?A=CBD; $(G%HTB-B< ! E7D9O-OG-J- -01(1('+ / 1 3 1
D*5() 1 0 0?.2T/>) 2m.2 & 4>0:_S 1 0, 1 0 1 6 6 1 230
*)2# 1 (M6
1 0 0D := D+ | D−
ξ
1 0 0 S'/*;2<(82WD+ := 0 | D+
t1 0 0 S'/*;2<(82WM/ 2.4D+
t := c.(D+s , . . . ,D
+s )1 0 0 S'/*;2<(82WM*Z6 2.*V26 S ,c
D+s := z | zπξ
| (+, πξ, i1, . . . , in).(D−ξ.i1, . . . ,D−
ξ.in)1 0 0 7:! 1 (.23W\, 1 1 *
D−ξ := c.(D−
sξ, . . . ,D−
sξ)1 0 0 7:! 1 (.23WM*Z+6 28*;236TS ,.
D−sξ
:= ((−, ξ, I).D+t )I∈NS'/2()0
πξ := ξ | n : ξ
/10 N
*(") *Z6 ,. 4X*Z/>)*S 1 0(.2.+* .2.+*4N
c.(D+s , . . . ,D
+s )
0ZS 0Z:+*Z7(R) 1 0 05/6TSU/ *Z:"47) / 2.+7(c4 29:0Z+7('4
0*5)2 2
S 1 0 ) 4+*V*Z+23D+
s
SU/>)0 7 1 ) W)2,3,c 2.4 4 c ? c.(D−
sξ, . . . ,D−
sξ)0 S 0:*7( ) 1 0 0Z/6TSU/*: 47)5/ +2.7(
c Q S'/*V*;2 ,c67(: 1 ,
0 R *5)2 2 S 1 0T) 4+*V*23D−
sξ
SU/>)0 7 1 ) W)2,3,. 2.4 4 c ? 1 0 0 *;236TS ,. *5(+ /! 3""*$# 2, *( 4 , 1 W/0V6
D+ (% 4'&(3)" Y**# 23, *5( 4 , 1 W/0V6D−
ξ
?
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1 N
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687:9<;>=@?A=CBD; $(G H ; DXSTOLGN5D9OKJ OF>LB< /2<( c )#5/ +2.7((
αi
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5/ 2c7( , 1 1 * +*( c (C,.+*TW )23,,.* 2c4*R*Z/7( 1 *V*/+2.:* 1 ) 4+*V*Z+23*Dαi
+*(, 1Y /ZV[ \)]!/^) 49- 1 4 5/ +2.7(
c4+* 4*V*2*
(Dαi)?
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e e # 2D67:9 ; =@?A=BD;$ G HA@CBDGD9O N5L ON D P-ON
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c Q ( 4X,. )0V* /7(82) 1 (82c/* R S 1 0 *V2
c*( 4 S'/2.4 *C) ,.Y7, 1 4 2+0:+65R4X, 1 S'/, 1 0V2( : 4/:
*V2/ Q c 1 4*W)2,3,.+* 2.4+* R Y , 1 5/ (.237) 1 (.2./ 4 # )XW )23,,. 2.4 4
c
/)_:+7(V)+,3,.+6 +7( Y , 1 4>2 0: 6R4, 1 S'/, 1 0;2<(:X4/ :5X*;2 c *5( 4 S'/2.4>*4>29:;0Z7( 41?
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β(, ) *Z+*RS 0Z/ &5-(.2./*
*V26 S ,c* */7( 4* 4*;*Z+23* S'/7(.2.,*"*)0", 1 1 *Z β ?e a #7(*)') 4)` #7 - 12# )ZR41 - "#9256 4NB#7 - 9 )') ( 6 :
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1 * S'/*V2(823W ?0 ⊆ D
1 *": 1 (82W ? *;2 N ⊆ M1 ,c/0V*
((−, ξ, I).DI)I∈N ⊆ ((−, ξ, I).DI)I∈M
e Y #)` #7 - 2# 4c:.1" C 41# )` )#'56 EI$I*)4 32,#7 - 7" )*) #
c: " C12*56 :67:9 ; =@?A=BD;$ G E H
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c BL O H3 O H>N5N-O
cV(.0 1 6 77
S+*(")
c53- Y 1 W"!#Z W*1!!1
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S+*(U236+,<)* 4 1 *
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67:9 ; =@?A=BD;$ G,5Hc B<D9O%$
c Y - /1'&(1 *()
cV(.0 1 2 7 ( ,,. )
7 1 ) 1 (.2./S'/*;2<(82#(+, π, i1, . . . , in)
1 - n ≥ 1*5( *) 2 2.CS 1 0 1 )\S ,<)* )
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R*5( , 1 4/: 4 4) 4*;*Z2*_4
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1 (82./ 4 1 *R) 4*;*Z+23K4R
+*(1Z! 91 *;2# *5(Q,c 4:6/Y$) 4:+6 / W/ 1 ,323*:/)) 1 (.2./ S 0/ZS 0 4/7(U,.,32c) 23*;2 ,c?
U "! # $ &%,Xe <X.+ E( ( 32 )Q2 ( 6)4192 )#'5? EI$I')4 4 ",2 : : $9 ' ,0 )` (Q7 :
C3# C #7 - !)`J2 (Q$)4912 M$" "(7 # 0 4 4M 12 17,7(4 4N@#FEG')4 1$#%"(2#94)`@ 142 44 "( ,c "( 614>? )#'56 8@ "B 32 14 :+ E( ( C2)` (Q$)4912 )%#*56 1EI6I*)4 08)` 2 (Q$)4912 4N #7 - #FE $I32 #a 321) 4# 44 4H C411@ ( K 1$I$I')4N 4 :
687:9<;>=@?A=CBD; $(G H Q >N5N5O E7D N` )(US-O E-B<DXL J GN 4*;*Z+23 *5( + /! 3+*&1V=1 3 + -0/X 9( \)!0351X*V2**,5/-+2.7(.* 7:! 1 (.23W=*'*/7($:! 1 ) 1
?
) " ./110"3243576& 0(&98'3057: 3"0 1<;40=2?>615 &@3023A/ )+ BCB T EDGFIH? 1JKJD4N 0I2aID4 1KFGL(12 EF MY BQ 4N1KFL(2
F 4@F 414 4 3241@ B 1KJKJID4N 4 -DGFH6 8@ LJ 32 4 4MFGL(12CF 4OF 4+P
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4 4 M 32 D4 F 414 4 MI C D4 C411@ B 1JJD4 4 : ; 4 D FC7# GL 4AH6FEGD"I B 1 F 4K 32 14
0 : ξc K2 4B5 GD B F 4 D M
ξc:
e aID4 D BQ] B F 414 32 L# F D 4K4 L#KD`9 4 :
67:9 ; =@?A=BD;$ G E H IH>N5N-ODs(ξc)
/2( s ∈ ]0; 1[?&' Y /(WX*& 147) * 1 , 1 230Z s *5( ,c 4*;*Z+23 Ds(ξc)
(−, ξc, s1)
(+, 0 : ξc.s1, s2)
(−, ξc.s1.s2, s3)
(+, 0 : ξ.s1.s2.s3, s4)
???
/s = 0.s1s2 . . .
;<B1KFGL(12 F 4@F 44 4OFE 4N 4 GD4OL#KBQ GDH67 H6 C2@D DFH6 1KJKJID4N 9:M NIL#GD (0 D 4P 32 14
n : ξ.iFEG2P 4Y](1 BCBQ7(4 L8 D E
nL#B5 D 4
KL(124J7@ 4 F D` 2 B (−, ξ, J)
1 (+, x : ξ.i, I)
T412QR4 T2f12 4D 4HKL,24O7!9@ 4 :
L94D 4 F M F7"12441@9 4#0 2_ID4 D.E KJ4F 9 912 L(244 (+, 0 : ξ, I)
KL(12(+, ξ, I)
41ξ 4Z D F D` J4 :
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$$BL-HPD9OE( -BN J L J -HM%$ L( E7D 3O B<N ( 1 7(R4/: ) ,2.)
ξcY ) 4+*V*Z+23 S'/+(.2.+,USU/ *V2<(.23W./): 1 (.23W *6 28*;236TS ,T*)0
) T 1 * β 4>23*"!5/2#7($_4%ξc
*( 3- 9W4U 3 S 1 0 , 1 W/ (.2./ 0:) 0V*V2# φξc
4&_, 1 6 1 2'0*)2# 1 ((
φξc: 0 7→ Fid φξc
: z 7→ z φξc: zπ 7→ zπ
φξc: (+, π, 1; . . . ;n)
D1
?5?5?Dn
7−→ (+, 0 : ξc, 0)
(−, ξc.0, 1; . . . ;n)
(+, π, 1; . . . ;n)
φξc.0.1(D1)??5?
φξc.0.n(Dn)
φξc: s
D
7−→ (+, 0 : ξc, 0; 1)
Ds(ξc.0) (−, ξc.1, 0)
φξc.1.0(D)
φξc:
s1
D1
. . . sn
Dn
7−→
(+, 0 : ξc, n)
(−, ξc.n, 1)
(+, 0 : ξc.n.1, 0; 1)
Ds1(ξc.n.1.0) (−, ξc.n.1.1, 0)
φξc.n.1.1.0(D1)
(−, ξc.n, n)
(+, 0 : ξc.n.n, 0; 1)
Dsn(ξc.n.n.0) (−, ξc.n.n.1, 0)
φξc.n.n.1.0(Dn)
),+*-/.0+*-,1+ $PO/. [<#!# $b10 )* '$,I-J ?)'67,*98:'8*- 67['U7,-[$56=$<b][.$,9&+2T)* ')6;68[/-HI-.[/][<A$56J'343<Z[.$,*6<658)`[<
ξc<D8*<58)F 8][<A$56 <,6B ?$< <A)$<73[][.$,9&% ?$J)*98$,6 86CB5[/-68[?6 $P ?$
ξc N a $,M8=*-0 8,'')'6- $'8'<:868*3 F<;)<A$b$,- F $,M8 $,*#=5:$568[/-M8 '96) '96Z[@7,- 7 0 [/*?>8$F,-8[.)* [?4G4 71 [8-[$ ?$M8Y ["!#$H'96)B83B5[/][<,-[$ $,* ["!# $ $&(')*+$,*-.[.$,/$
N
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6 $ "B<L-HGD9O3O( BN H OKH>N5N O 3:-\W Y 3/Z )* 1 -'!1 #+%# S < !" #$%& '!5# 9(9W)*#+ ,*#-+./!((<9 S01#2#4353#-&-#7 6#G%(1714+8!9#7':!A ;09 ,(<14+$ 1#9!<1#/#7 S < !6
6 ,(ξ∗c
=41>!6.&ξc
(9! 5 (9*?
Fid 0
(+, 0 : ξ∗c , 0)
(−, ξ∗c .0, 1; . . . ;n)
(+, π, 1; . . . ;n)
D1
? ?5?Dn
(+, π, 1; . . . ;n)
D1
? ?5?Dn
"! 41<+ >":(>@ 4(+, 0 : ξ∗c , 0) 0
(+, 0 : ξ∗c , 0; 1)
Ds(ξ∗c .0) (−, ξ∗c .1, 0)
D
s
D
AB>#: * B#" #C #+ ,DE ++< F+ SG1#(+, 0 : ξ∗c , n)
UW 1) #CH$$! 1#+ 7 <.+JI+K$+TSL<%<!MN+O+'#77#4+$ 6!
?#,:(9< (1# 6!W $W 16#72!M
+=I+ PZS'3#( = #<!Q># 1# 7 $
(+, 0 : ξ∗c , n)
? (−, ξ∗c .n, i)
(+, 0 : ξ∗c .n.i, 0; 1)
Ds(ξ∗c .n.i.0) (−, ξ∗c .n.i.1, 0)
Di
?
? s
Di
?
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$PO D7B SLJ GN5L(GD9O -O J -H%$(= E7D9O-O( -JJ A ./-'V )! 3/Z JRK # J#437F( S' <> !
R%K&"QS'& +< SL#F Z/
! $C*# *-9N1#- ?P-4 # #434(O S*#$#
Rt
SG1#<!M.>#$ >@ 0? ? *#-+./!#2!Q#437F(O "S*",(*#-
Rt
< %-7" S'*"$*1#Dt
? ? + #7N! K 41! 1# 3%)> ? S04"SL#F3 35%<(1B#
Dt
? 6 *(> <P%-7#
D = JRK ?
;<B1KFL(2 F+E F 414 MI 32 1 D %-F 44 MI 32 61 D+ 3241@ B 1J PJID4 4#E L, L#BCB = DGFH6 JKJID4 9: D "9P41IBQ GD B T1SR2 T2L D 4 L894H4@914&N F 414 f7!9
c.(D1, . . . ,Dn)41Z D
ξ 4K1KF
(−, ξ, 0)
c
(+, 0 : ξ.0, 0)
D1
. . . (+, 0 : ξ.0, 0)
Dn
c 4N<7D
cF4=D H6 D D 4 IDD 4+6GDID 4\2<7(7 B5 GD`L#7 4\ F 4 KL(124
(+, 0 :ξ.0, ∅)
: 1 7( 9 L 4 MI GDIH?7 T DGFIH? JKJID4 D`B B "H6 : ' D : KL(12
zn:ξ
4NKF T
(+, n : ξ, 0)
(−, ξ.0, 0)
z
KL(12(+, n : ξ, I)
4K1KF T
(+, n : ξ, 0)
(−, ξ.0, 0)
(+, 0 : ξ.0.0, I)
BCB T EDGFIH? 1JKJD4N 0 D`= 2 3241%41@9 P 4NH@7#7 :
0 ? $ "!#%$"! &%, ')( +*,(%-"! .!(/*10 2AJ #$*>@ =#3 #'+= ,H *?
)46574 * 8 876:9<;40>= 0"9@3 0 0 8'1 8 3A057: 3"0 1 ;40=2?>615?9@3023A/#@
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6 PO ')"! %USJ !( (!"!,%$( %, 0 %, 005.!#0 (( G.!F *,026 &V ] & P%"&"A 3 '#$#8 !8&" ' <> ! ( $! #$%#E! B> "'*-& ,3
σ< ! 9-<
113C&
σ* D $
,3$#$(>&%'G1#C! 54 ,1
,38% D $?
e aJ1 H91 DD B )(" 1KFGL,+*OD` 2 3241Y41@96 :
0 / ? PO '-"!,USJ ! % 0 %, 005 !0 0/PS-' 026,! ,#4+< 4+ /.(9<%<&"9 3 '*"$*1#, *+ +J(7 3 7!M:,#+O1!6( 0?
0H D24H6 D 4L7 H?92427(7#H6 41( F7!12 : D2 *Z416F7! 4T 9 D`2KBCBQ N
E ::= R = R
R ::= z | zξ | Fid | Di | Di,R | JRK | Rj | Rj,R
ED 4Di,Ri
412 F 4K B59M :; GL( F 4a7 H?924 27(7#H6 43( 12 3242 : 54 D2 * 4 27#7#D4 D`
DGFIH6 T MI 32 1 DID ( 2 "9 D,41654 $P @ * : 0 / ? &7 *- !&&% 0!8 $((-"!09+8 ! 8 * #$ 0
:,$$B3;.(:( <#3<,3#/.(9E
3#1 N #P! &" 3 =*",(*#-P- 77!+ ,6!! 3#/Q1# ! P%7# > , !?
?!@ACB,DEGFIH;J2FLKMBND9OQPRTSGUWVXZY[S]\G^_Y[Ua`GbcX[S$Y[\GS]dZSegfhY[^_iWj)f;elkmUf;VWkm^_Yonpf;iq\G^cd[f;rStsnpf;iudZ`>v[YZklVWkm^_Y+w2egfY[^_iWj)f;elkmUxf;VWkm^_Yy\G^_jtj=XZVaS=zhegfVWixf_d2X[\$VWkm^_Y+|⇐ ^_klV E ~ ixf;kmShUaXZiemSGU]dZSGUxUaS$klY[U]S>n ^_Y[S$YVWkmS$emUL|P^_jtjQStemSGU]dZSGUaUaS$klY[U]f ~ SG\Qn,^_klYcVaS$XZiaUUaSnZem^_YZrS$YVud[f;Y[UemSGUdZSGUaUaS$klY[US>n ^_Y[S$YVWkmS$emUIw
ESGUWV ~ ixf;kmSUWXZiemSGUdZSGUaUxS$klY[Uf ~ SG\n ^_klYVaS$XZiaUL|
⇒ ^_klV E ~ ixf;kmShUWXZiemSGUdZSGUaUaS$klY[U]f ~ SG\hn ^_klYVaS$XZiaUI| ^_kmS$YV Di1 , . . . ,Din
dZSGUdZSGUxUaS$klY[US> gn,^_Y[S$YcVWkmS$emUS$V
Rj1 , . . . ,Rjm
dZSGU!ia`GUxSIf;X2S>2n,^_Y[S$YcVWkmS$emUoVaS$emU!bcX[SemSGU!dZS$X2jQS$j=ZiaSGUdZSeRT`GbXpf;VWkm^_Y
EUa^_YVtZkmS$Y-^_iWjQ`GUI| ^_kmS$YV Di1 , . . . , Din
S$VRj1 , . . . , Rjm
emS$XZiaUtVWixfLgd2X[\$VWkm^_Y[Ud[f;Y[UemSGUdZSGUaUxS$klY[Uf ~ SG\"n ^_klYVaS$XZiaUI| +SGUdZSGUaUxS$klY[U Di1 , . . . , Din
S$VemSGUia`GUaSIf;X2Rj1 , . . . , Rjm
Ua^_YVhVaS$emU=bcX[SE(Di, Rj)
SGUWV"ZkmS$Y^_iWjQ`GS5|'f;in,^_VW[GUaS5wE(Di, Rj)SGUWV ~ ixf;kmS5|^_Y[\5wnpf;i=klY_WSG\$VWk ~ klVa`dZSegfyVWixf_d2X[\$VWkm^_Y+w E(Di,Rj)
SGUWV ~ ixf;kmS9dZ^_Y[\ ESGUWV
~ `$iWkvp`GSUWXZiuemSGUdZSGUaUaS$klY[UuS>2n,^_Y[S$YcVWkmS$emUI|
U1
; 4K2"F 4K4 JGD T M? ; 41 D<4 AF7! 4 L#BCB T¡ DGFIH? 1JKJD4N :
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8
PO\O *,% *)&05$( ' 2 # W2_!( \W "!5# *#F$#7 ,#&<$#4301# 1'& M%H?
0 v D
1 ,3$#$(>&%H?N ⊆ M
1! #
((−, ξ, I).DI)I∈N v ((−, ξ, I).DI)I∈M
PO+7 *,% *) $F/!#0/-"!! '2 # W2<uI1!] "!;# #4$# <,#& ,$#F3]0/# 1'& M%H?
0 4 D+ D+ 4 z (+, ξ, I).R 4 zξ 1 ,3$#$(>&%H?0 4 D−
*N ⊆ M
1! *#-
((−, ξ, I).DI)I∈N 4 ((−, ξ, I).DI)I∈M
Uy Q $ %"!$#&%('$)
+*-,.,*0/1* L ,1243657*(/8, L 3659/1: L(;.< * D := ; *->@?6* F *(/ H >A:B, ; *C, @< 3 ;.DB *(/E:F5A: DHG ,2 H >9*-/I*(5 D > F 2 H >9** M =J3659*-5,24* DD *FK
)4ML48N O @ @6 2 5'135 ; 5 3.
PQ5 ;.< >@,12 D 24/1*R>959* F * ; 592 D-; * 362S/ D : F9<" 592S,124365 K Z =A: * 6T F *(/ ;1< /1*(:F> M F * ;1< /1*(:F> M K SU9V u8 05 (L %, *8 0 (LWXYFZ6[&\^] 6YFZ6[&\
R1, . . . ,Rn + 01# & _ #437F( #F37F(L?
E` ? aV cb<00 ed-(()( I8f"# #437F(J&P#F34(
R1, . . . ,RnG
JR1, . . . ,RnK = JJR1K , . . . , JRnKK
?!@cA BND'EGFIH¡J2FLKMB,D)OIgY9XhWklelkmUaSemSnZiWklY[\$kln SdZSGU`GbcXpf(hWkm^_Y[Ur`$Y[`$iWkmbX[SGUiMnZia^_n ^5UWkjhWkm^_Y0k2|ml=npf;rSnZia`1n\G`GdZS$YhaS"o>|
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m
74ML 4 > 1/1 3576 9
PQ5c= ; 3 L-DF * L 3 BCB *X*(5 D > F 2 H >9* = ; 3 J : J 2 D 24/8,* N 365 F9<! 5@2S, F *(/ F *-/1/1*-2459/>@592S?F* ; /.* D / ( F *B :F5@2 D(; * /12 B 2 D :F2 ; * * =9>@24/ 365 B 3F5 , ; * D : / < =A: ; :B,124365 KPQ5 L 3 BCB *-5 L * =A: ; >95 * M * B = D * = 3F> ; B 365, ; * ; H >9* D *-/ F *(/./1*(2S59/ >@592S?F* ; /.* D /X/.365, = D >9/
L 3 B = D 2 H > < / *", H > 2 D 5@*Q/1>0, =A:B/ F * L 3F59/12 F9<(; * ; >95 F *-/1/.*(245 L 365,*(59:F5,,36>9/ D *(/ F *(/./1*(2S59/ : ?6* L= 362S5,*(> ; / ( *-5 >95 L * ; ,:F245 /1*-59/ * B :F24/ H > 2 D :F>@,I*(57= D >9/ D *-/ L 365,*-592 ; = D >9/12S*(> ; / 3624/ L :F>9/1*F *(/ ;1< = < ,2S,124365@/(K(K-KPQ5 L 365@/12 F@D(; * F *-> M F *-/1/1*-2459/
D0 = (+, ξ, 0)
(−, ξ.0, 0)
(+, 0 : ξ.0.0, 0)
(−, ξ.0.0.0, 0)
(+, ξ, 0)
(−, ξ.0, 0)
(+, 0 : ξ.0.0, 0)
(−, ξ.0.0.0, 0)
KKK
*",D1 = (+, ξ, 0)
(−, ξ.0, 0)
(+, 0 : ξ.0.0, 0)
(−, ξ.0.0.0, 0)
(+, ξ, 0)
(−, ξ.0, 0)
(+, 1 : ξ.0.0, 0)
(−, ξ.0.0.0, 0)
KKK
P 5 ?6*->@, / < =A: ; * ;D0
*-,D1
/:B59/ >@,2 D 24/.* ;,D : =A: ; ,24* 5@365?2S/12 JGD * F *(/ F *(/./1*-2459/ ( ; *(= ;.< /1*-5, < *=A: ;F *(/ KKK * K 36> ; L * D :9365 =J*(>@, = ; *(5 F9; *
E =
0.21
(−, ξ, 0)
0.22
(+, 0 : ξ.0, 0)
0.23
(−, ξ.0.0, 0)
0.24
(+, 0 : ξ.0.0.0, 0)
0.31
(−, ξ, 0)
0.32
(+, 0 : ξ.0, 0)
0.33
(−, ξ.0.0, 0)
0.34
(+, 0 : ξ.0.0.0, 0)
*(/ 593 ; B : D 2S/:B,1243659/ /13F5 , F * D : 3 ; B *
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'
JD0,EK =
0.21
0.22
0.23
0.24
0.21
∗
0.31
0.32
0.33
0.34
KKK
0.31
∗
*", JD1,EK =
0.21
0.22
0.23
0.24
0.21
∗
0.31
0.32
0.23
0.24
KKK
0.31
∗
3F5 LE/ < =A: ; *
D0
*",D1
K :F5@/ D *(/$593 ; B : D 24/:,243F59/ F *
D0,E*-,
D1,E D *(/ L 3* L 24*-5 ,1/ 593B, < /
∗; *(= ;1< /.*(5,*-5, F *(/
L 3* L 24*-5 ,1/ 245 >@,12 D *(/+=J36> ;@D : / < =A: ; :B,124365 K *Q= 9< 593 B$D 59*R*(/., F< 5 <-; : D( *-, D4<FD(; * B *-5, /12 B = D 2 9<F :B59/ L *-, * M * B = D * *N =J36> ; / < =A: ; * ;
D0
*-,D1
92 D /1>0, F *I= ;1< /1*-5 ,1* ; >95 F *(/1/.*(2S5 F * J :B/1*ξ `
H >92 J@; :F5 L */.>$/: BCB *-5, /1> ; D *-/ D 24*-> M; * 3 L : D 24/ < / ( 2 L 2365 5@* D >,2 D 24/.* H >9*I= 3F> ;ξ*
*", F 3 J /.* ; ?F* ; F :F5@/ D * L 3* L 24*-5, ;1< /.> D ,:B,C>95@* J9; :B5 L * H >92+59* ;.< = D ,1*=A:F/ F * L 36=92S*(/( L *(/8,=J36> ; L * D : H > 365 59* ; * : ; F * =9:F/ D * L 3 * L 24*(5,
∗:F> F *(/./1>9/ F > / L : D :F2 ; *
0.21* K
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d (')(,.*) 0 %! %, 005 !2Q_ Z "!#W ] FZZcB[$# [$% 2FZ
Scal (D) _ &"
D '& _ 7+)(&*5*+9*F3 &Q+*4,
&M16#7C; 01#1-+2 -&.*54/0* , K%D
?
2Q_ Z "!#W0] FZQZcB[$# [$% 26Z21eY YBB[$#3% ZYFZScal∗ (D)
_ O%7#D4& _ 7+)(& 5* *+6,
'*F3 %Q+*F&M16#7Q; 01#1-+7+(7& .* 98/9*< %D
*9%8-+:* ,3$#>,3 % %*L?
Scal∗ (D) = Scal (D) ∪ 2−n | n ∈ N
:F5@/ D *(/ F *-/1/1*-2459/ >9592H?6* ; /1* D / 2 D :F>@,X:F>@/1/12I= ; *(5 F@; *7*(5 L 3 B =@,* D *(/ ;1< = < ,2S,124365@/(K 36> ;*-5 *(5 F9; * ;D *-/ F *-/1/.*(245@/R>9592H?6* ; /1* D /Q*-5 = ; *(5A:F5,,36>@,1*(/ D *(/ ;1< = < ,2H,243F59/ *(5 L 3 B =,* J2 D *-/.,Q: D 3 ; />@,12 D * F * F9<! 592 ; D :05@3F,2S365 F * J :F/.*E2S5,* ; 59*FK
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$ ! m !
2459/.2 9:B= ;1D />95@*E= ;1< P L9; 3F592 H >9*
(+, ξ, 0)
(−, ξ.0, 1; 2)
(+, ξ, 0)
(−, ξ.0, 1; 2; 3)
D *(/E: L ,124365@/ = *->@?6*-5 ,-, ; * F * D : 3 ;"B *zzξ
z0:ξ.0.1
z0:ξ.0.2
z0:ξ.0.3
z1:ξ.0.1
z1:ξ.0.2
(+, ξ, I)(+, 0 : ξ.0.1, I)
(+, 0 : ξ.0.2, I)
(+, 0 : ξ.0.3, I)
(+, 1 : ξ.0.1, I)
36> *(5 5 F * D : 3 ; B *
(+, 1 : ξ.0.2, I)K
:C593F,124365 F * J :B/1*E245,* ; 59* F9< L(; 2S, L *-/ *-59/1* B JGD *(/(K
6 SU (0 !(F * !W 0!J[@Z % IF E <+)( & % &6* $"! *# +. _ t'&M1#-&H3E 3 * 4&&1 &6 ,3$#>M% *2_K+J&&> 4& *)H3
n ≥ 1# * # &1"P%7& *!'& M%M ?
W ( /7 #$#1 , 3
` ξ1, . . . , ξk; ξ′1 7→ m1, . . . , ξ
′n 7→ mn
ξ ` ξ1, . . . , ξk; ξ′1 7→ m1, . . . , ξ
′n 7→ mn
#7&M ( 17β4& ,$> % ( 1N#(#1
βc
_ ,#43, *#4 /.(9c <3 %K#,% ** ,
3 + , 0/#
βε =
` ξ1, . . . , ξn;-β = ` ξ1, . . . , ξn
ξ ` ξ1, . . . , ξn;-β = ξ ` ξ1, . . . , ξn βc.(+,ξ,I) = ξ.j ` Γ; ∆m
'1# * # &1"j ∈ I
βc = ` Γ; ∆m
? *-$.#7'$#& , , 3*(#+ #(( *F1# J1&"01 L#- *#-Q #.(9& & * 34%) *4&67# & _ *>@ J134+$*?
βc.(−,ξ,I) = ` ξ1, . . . , ξk;
(
ξ′i 7→
mi + 1∃i ∈ I, ξ′i = ξ.i
mi
-#, )
βc = ξ ` ξ1, . . . , ξk; (ξ
′i 7→ mi)
3 *& ,$&( ξ′i 7→ 0
ξ′i _ ;,01# , 01
&M0( /7 #$#1 ?
2 !J[@Z)% IF <B[q$ %\ E[@Z Z %Y[ = ( /7β
βε
.(9 &"_ = % , h β
?
: B > D ,2S= D 2 L 2S, < = 36> ; D *(/ D 24*(> M =J36/12H,2 / 245 F 2 H >9*(5, D * =J362S5 ,1*(> ; B : M 2 B : D >@,2 D 2S/: JGD *FK : ; * M * B = D * 9/1> ; D : J :F/1*
` ξ D : J :F/1*I2S5 ,1* ; 59* F * D : = ;.< P L@; 36592 H >9*
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4
(+, ξ, 0)
(−, ξ.0, 1; 2)
(+, ξ, 0)
(−, ξ.0, 1; 2; 3)
*-/.,` ξ; ξ.0.1 7→ 2, ξ.0.2 7→ 2, ξ.0.3 7→ 1
*", D *(/=J36245,1*(> ; / =J36/./12 JD *(//.365,
ξ; 0 : ξ.0.1; 1 : ξ.0.1; 0 : ξ.0.2; 1 : ξ.0.2; 0 : ξ.0.3
*(/ F *-/1/1*-2459/ >9592H?6* ; /1* D / /1365, F9<" 5924/ 2S5 F > L ,2S?F* B *(5, /1> ;AD *(/ J :F/.*(/ 245,* ; 59*-/(K+3 BCB * *-5 D > F 2 H >@*E= ; 3 J : J 2 D 24/.,1* = 36> ; D : F9<! 592S,124365 F *-/ F *-/1/.*(245@/>95@2S?6* ; /.* D / >95 / L : D :F2 ; *
*-/., 593F, <?*-,In
F9<L-; 2S, >95@* < 5 > B<-; :B,124365 2S5.* L ,2S?F* F *Pfin(N)
K SU G !(F *10 (0 0ed 8 0 ! (05 !(F * !(/0/ 2Q_ Z "!#W ] FZ9$% IJ\@Z$[@Z Z % YFZ) (17 #(#1*-&>13
` ξ1, . . . , ξk; ξ′1 7→ m1, . . . , ξ
′n 7→ mn
ξ1; . . . ; ξk ∪n
⋃
j=1
p : ξ′j | p < mj
55@3F, :F5,?>@5/ L : D :F2 ; * = 36> ;=D * B 3 B *-5,&59365 F9<! 5@2*-, =A: ;
(In)>@59* < 5 > B$<(; :,243F5E2S5.* L ,2H?6*
F *Pfin(N)
D * F *(/./1*(2S5 >95@2S?6* ; /.* D F * J :F/.*E2S5,* ; 59*
` ξ1, . . . , ξk; ξ′1 7→ m1, . . . , ξ
′n 7→ mn
*-/.,
?
Du
?
?
Du
?
?
Du
. . .
3 Du
*-/., D * F *-/1/1*-245X/1>92H?B:F5,
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$ ! m !
?
z
?
?
zπ0
. . .
. . . ?
?
zπl
?
?
(+, π0, I0)
KKK
. . .
. . . ?
?
(+, πl, I0)
KKK
?
?
(+, π0, I1)
KKK
. . .
. . . ?
?
(+, πl, I1)
KKK
?
KKK
3 π1, . . . , πl
*-/., D *(59/.* B JD * F *-/=J36245,1*(> ; / :F/1/.3 L 2 < *-/ CD : J :F/1*I2S5 ,1* ; 59*
` ξ1, . . . , ξk; ξ′1 7→ m1, . . . , ξ
′n 7→ mn
: ?F* L
(+, ξ, i1, . . . , in)
KKK
= (+, ξ, i1, . . . , in)
UDξ.i1`Γ;∆(α) . . . UDξ.in`Γ;∆(α)
*-,
(+, p : ξ, i1, . . . , in)
KKK
= (+, p : ξ, i1, . . . , in)
UDξ.i1`Γ;∆(α) . . . UDξ.in`Γ;∆(α)
+365, ; :F2 ; * B *(5, :F> L :B/ F * D : D > F 2 H >9*+= ; 3 J : J 2 D 24/.,1* ( =A: * 46T * 365C5 :R=A:F/ :F2 ; * :B,.,*(5,124365:F> M L 365,* M ,1*(/ L : ; 2 D 5 G :$= D >9/ F * L 3F5 , ; :F2S5,*(/ F * D 245 < : ; 2H, < K
* F *-/1/.*(245 >9592H?6* ; /1* D /.> ; D : J :B/1*ξ ` Γ; ∆
*-/., D * F *(/1/.*(2S5
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?
?
?
Du
?
?
Du
?
?
Du
. . .
3 Du
*(/8, D * F *(/./1*-245 /.>92S?B:F5, N
(−, ξ, j1, . . . , jn)
UD`Γ;∆(ξ,j1,...,jn)(α)
. . . (−, ξ, l1, . . . , lm)
UD`Γ;∆(ξ,l1,...,lm)(α)
3 ∆(ξ, I) =
(
ξ′i 7→
mi + 1/12∃i ∈ I, ξ′i = ξ.i
mi
/12S59365) *",
∆ = (ξ′i 7→ mi)*-5 >@,12 D 24/:B5 , D :
593B, :B,124365ξ′i 7→ 0
/12ξ′i5 :B=9=A: ; ,24*-5,=9:F/ CD : J :F/.*E2S5,* ; 59*FK
D+; *(/8,* F9< L(; 2 ; * D *(/ / L : D :F2 ; *(/ H >92 :F=9=A: ; :F24/./1*-5 , F :F59/ D * F *-/1/1*-245^>9592H?6* ; /1* D K 59*>9592S365F9< 593 B J@; : JGD * F *-59/1* B JGD *(/ F@< 593 BJ9; : JD *(/ < , :F5, F9< 593 B J9; : JGD * D *(/ =J36/12H,243F59/e= 3F> ; D *(/ / L : D :B2 ; *-/?/13F5 , F9< 593 B J9; : JGD *-/(K@P 5 /1* F 3F5959*Q>95@* < 5 > B$<(; :B,2S365 J 2 .* L ,2S?F*
p = (pi)i∈N
F * L *(/ =J36/.2S,2S3659/-K * / L : D :B2 ; *
?*(/8,
2−(i+1)α3
i*(/., D 2S5 F 2 L * F :F5@/
pF * D : = 36/.2S,124365 F >X/ L : D :B2 ; *
?K
SU+N 00 ! !, *10 ' *&
β ( 1*? 2+] FZ Z%3 \ %,FZ#
UDβ(α) 4& %-7#y.(9 & _ 3 %23 *##*?
)@`+*-, -&7 !"#$%&'()+*,-./,012+34$56<2$7&8:9!$;9!"<0 = !"#$4'()>)?'(6@! ACB"#)@)D$7"E'(FGFHC)I'()-9!$7G)J*'("KLB&39!"
10LME#2>)I'()?
$7N'(FGFHO39!"K)P-$$7QN8:9!$;9!"#='()G*'("LBR39!"10C 0 9S>T8)P 5 >9592H?6* ; /1* DC34"'( -)R
$10U'!*"K (9!)P-'(WV,XH)?'(6O$7H9! !8Y3'()I"#Z*9(β'() 2S5 LD >9/ S9!R$7
!"#$DZ*9(β9![H\34"#'$>T8)]-'(^,9!Z8'`_a8-)`cbIB`d>)P-'(We60%fg39 8Hhi!jkcl
$10UFR*,$7C/'(673C&34"#'$>T8)P-'(,9!m8'`_a8-)`Y!N. !"#$Lm*9(β)D$10UFR*,$7mnm9! !83'()I"#@n*9(
β0
E` ? ao d ( '-(, *& 0 %(!0 qD, UDβ⊥(α)
y *&
D+%# % ( 1E%16#7
β?Af"1#B
αG
Scal (D) ⊆ Scal(q
D,UDβ⊥(α)y)
⊆ Scal (D) ∪ 2−nα | n ∈ N
?!@cA BND'EGFIH¡J2FLKMB,D)OIgYonZia^\GGdZS\G^_jtjQSS$YoelX[d2kmbcX[SnZia^_pf;ZklelkmUhaS5|ftnZiaS$X ~ S]iaS$n,^5UxSUWXZiqnZelX[UWkmS$XZiaUiaS$j)f;iabX[SGUs
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+ $ ! m !
nZiaGU=XZY[Sf_\hWkm^_Y<Y[`$rf(hWk ~ S5w'emSdZSGUaUxS$klY XZYZk ~ S$iaUaS$e UDβ⊥(α)XhWklelkmUaS i S$YhWiaS!f;XhWiaSGUf_\1n
hWkm^_Y[U.oXZY9dZ`$jQ^_Y9ShXZYdZ`$jQ^_Y ^\If;elkmUa`uUaXZi ha^_X[UemSGU elkmS$X2iMn ^5UWkjhWkU.odZSuegfpf_UaSuklYBhaS$iWY[SiMem^c\If;emS"o>|
nZiaGU9XZY[S f_\hWkm^_Y Y[`$rf(hWk ~ S5wemS1dZSGUaUaS$klY XZYZk ~ S$iaUaS$e UDβ⊥(α)SGUaUxf;kmSdZSo^\If;elkmUaS$i!UWXZi
ha^_XhaSGUuemSGUf_\hWkm^_Y[Un,^5UxUWklZemSGUI| +SGUf_\hWkm^_Y[Uun,^5UWkjhWk ~ SGUUa^_YhuUaXZk ~ kmSGUudZS ha^_XhaSGUqemSGUf_\hWkm^_Y[UqY[`$rf(hWk ~ SGUn ^5UaUWklZemSGUI|PSbcXZkkljtnZelkmbcX[SiMnpf;i'ia`G\$XZiWiaS$Y[\GS"obcX[S&ha^_XhYX[dhdZS
DSGUh ~ kmUWkjha` em^_iaUdZS egfY[^_iWj)f;elkmUxf(hWkm^_Ytd2Xia`GUaSIf;X
D,UDβ⊥(α)ShbX[SemS\G^S)\$kmS$Yhf_UaUa^\$km`]f;nZnpf;ixfahd[f;Y[UqemS]ia`GUaXZejhxf(h q
D,UDβ⊥(α)y |
)9` * , HV 9\34"#'3')P-'(O34"#B8B:)IYFE34$5PV,XnV,
Scal∗ (D) = Scal∗(q
D,UDβ⊥(1)y)
` / ? %-05F !9d %,αScal∗(D)
*)D
%-7" % ( /7E%16#7β
? &6(6( + *4& 1#αScal∗(D) ∈ ]0; 1[
$&.
Scal∗ (D) ∩ αScal∗(D)Q = ∅
?!@ACB,DEGFIH;J2FLKMBND9OIgYonZia^\GGdZS]S>Zf_\haS$jQS$YBh\G^_jtjQSS$Y!elX[d2kmbcX[S]nZia^_pf;ZklelkmUhaS5|
6 SU 0 d (')(, *&αD
*)D
+ %7# &( /7K%16#7*? *&αD
*54/0* ,αScal∗(D)
33# %+ &ML#4L,# 3H3% &ML#FL*)> ? 3W ,+.(9 %
Scal∗ (D)?
6 &7 1/0(! !,/*10 ' *)
D %-7" # ( 1 1#
β?#2Q_ Z@Z [\ % ,BZ# %
D
UOD = UDβ⊥(αJD,UDβ⊥(1)K)
)9` * , J'(FGFH@ $5 !PV,<34"'!*9S*,$57)?<28)P)I2B`d>)P-'( 0U'3S3',9!)> !"#$ )L 3659/8, ; > L ,12S?6* S9!G$7@RV, 0U$$7^B?34NPV,XFH)/^"#B$5) 9!)O2L'("FR9!$57,9!)P-'(GD
9! !8@0 9!)P"#22)L'( m$10 )P"'?38)P-'( Y\8:9!$;9!"#,0@"#B$5) 9!)2mL'("3FR9!$57,9!)]-'(n
D'() '!*" (9S*,$7J347,V,G8C'() 5T'(4S9!FH) 9!$7FH)D@8'`_ 8-),0
6 &7 o&d (')(, *& ( ('"%(!0O !7d G d !(WJ*4& 1# 6 % , ,3" B4 * *##<:* 1 [ % [#+N*54/0* ,A _ & ,301#2& * 98/9* <
1?
:C= ; 3F= 36/.2S,124365 !4* =J36> ; ! : / < =A: ; :B,124365 *(/8,"! :C/1>92H?B:F5,*$#` / ? &% !' d I8&%,
D 7→ JD, UODK # ( /7E%16#7N %,3 -:&"_ ; 4&63*4>
D 7→ JD,UODK '#?! **>13 ?
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7 Ju
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07
?!@cA BND'EGFIH¡J2FLKMB,D)OIgYnZia^_X ~ Snpf;iia`G\$XZiWiaS$Y[\GS1bcX[SoemSGU nnZiaS$jtkmS$iaU9YZk ~ SIf;X2d2X dZSGUaUaS$klY3Ux^_Yh
\If;em\$XZegf;ZemSGUzhnpf;ihWklidZS JD,UODK |
^_XZiQnpf_UaUaS$itd2X YZk ~ SIf;X nf;X YZk ~ SIf;X n + 1
wegf1d2k )\$XZejha`oSGUhtdZS!dZ`haS$iWjtklY[S$iQegf ~ f;emS$XZidZSGU!f_\hWkm^_Y[U!f;X2 S$XZklelemSGUdZSGU\G^cS)\$kmS$YBhaU i bXZkUaS1\If;em\$XZemS$Yhof;kmUa`$jQS$YhIwq\G^_jtjQS1S$YelX[d2kmbX[SnZia^_pf;ZklelkmUhaS"o>|N+S]nZia^_Zem$jQSSGUh\GS$elXZkdZSegftdZ`haS$iWjtklYpf(hWkm^_YydZSGUun,^_klYBhaS$XZiaUI|
R kmdZ`GSu ^_Y[d[f;jQS$YBhxf;emSSGUhdZSq\G^_Y[UWkmdZ`$iaS$i XZYZkmbcX[S$jQS$YhemSGUZixf;Y[\a[SGUd[f;Y[UemSGUabcX[S$elemSGU \xpf_bX[SUa\If;egf;kliaS]j)f¡2klj)f;ed2XdZSGUxUaS$klY!XZYZk ~ S$iaUaS$e+klYBhaS$i ~ kmS$YBhf;X!nZelX[UqXZY[S] ^_kmUI| \GSGUZixf;Y[\a[SGUUaX )UaS$Yhn ^_XZidZ`G\$iWkliaSha^_XhaSGUemSGU
cn/\aZia^_YZkmbcX[SGU dZS
Di \5RTSGU hkm\$k2bcX[SUxS$i ~ S$YhemSGUQia`$n `hWkjhWkm^_Y[UQklY2v[YZkmSGUdZSGU)dZSGUaUxS$klY[U
Du
d[f;Y[UQegfdZ`>v[YZkjhWkm^_YdZSGU)dZSGUaUxS$klY[UtXZYZk ~ S$iaUaS$emUY[`$rf(hWkU.o d[f;Y[UqXZY[ShaS$elemSZixf;Y[\x[S5w[emS]\If;em\$XZed R XZY!n ^_klYhaS$XZiSGUhuf;kmUa`5|Z+S]n ^_klYhaS$XZidZSeR f_\hWkm^_Y
KndZS]egfcn/\aZia^_YZkmbcX[S
c1.K1 . . . cn.Kn
SGU hdZ^_YZY[`"f;klY[UWk s egfnZia`1n/\aZia^_YZkmbcX[S
c1.K1 . . . cn
dZ^_YZY[S]emS\G^S)\$kmS$Yh
|||c1
. . .
|||c2
. . .
|||
|||cn
. . .
. . .
egfcn/\aZia^_YZkmbcX[S
c1.K1 . . . cn.Kn
dZ^_YZY[S
|||c1
. . .
|||c2
. . .
|||
|||cn
c
. . .
. . .
k Kn = (+, k : ξ.i, I)f;em^_iaUemSGUUa\If;egf;kliaSGUdZS
c\G^_iWiaSGUWn ^_Y[dZS$Yh]zXZY[S"\GS$ihxf;klY[S=\G^_nZkmS
d[f;Y[UUOD
dZSeR f_\hWkm^_Y d2Xpf;emS(−, ξ.i, I)
|PShhaSof_\hWkm^_Y(−, ξ.i, I)
UWXZkjhtXZY[Sof_\hWkm^_Yn,^5UakjhWk ~ S (+, πξ, J)
|f;i)klY[d2X[\hWkm^_Y3UWXZiQegf pf;XhaS$XZi)dZS!egfcn/\aZia^_YZkmbcX[S\G^_Y[UakmdZ`$ia`GS5w
eR f_\hWkm^_Y9d2Xpf;emSdZS(+, πξ, J)
d[f;Y[UDSGUh ZkmS$YdZ`haS$iWjtklY[`GS\GSbcXZkpn S$iWjQShdZS\If;em\$XZemS$i
emSkdZS
Kn = (+, k : ξ.i, I)
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+ $ m 2
k Kn = (+, ξ, I)^_X
Kn = zξ
f;em^_iaUqkle+Y+R !ftf;X[\$XZYon ^_klYhaS$XZiz\If;em\$XZemS$iI| k Kn = zk:ξ.i
f;em^_iaU"emSGU"Ua\If;egf;kliaSGU=dZSc\G^_iWiaSGUWn ^_Y[dZS$YhtzXZY[SQ\GS$ihxf;klY[S)\G^_nZkmS)d[f;Y[U
UOD
d R XZY-dZSGUxUaS$klY Y[`$rf(hWkdZS9pf_UaSξ.i ` Λ
| PS9dZSGUxUaS$klY Y[`$rf(hWkUaXZkjhhXZY[Sof_\hWkm^_Yn,^5UWkjhWk ~ S (+, πξ, J)
| f;i)klY[d2X[\hWkm^_YUWXZiQegf1pf;XhaS$XZi9dZS!egfcn/\aZia^_YZkmbcX[S\G^_Y[UWkmdZ`$ia`GS5w
eR f_\hWkm^_Y9d2Xpf;emSdZS(+, πξ, J)
d[f;Y[UDSGUh ZkmS$Y9dZ`haS$iWjtklY[`GS\GSbXZkNn,S$iWjQShdZS\If;em\$XZemS$i
emSkdZS
Kn = zk:ξ.i
|
` / ? & o8+#( *,((-"!2 &&:/.(9L#F( (/&6( 43W01#F3 #N P%#(
D1*
D2#N+ Q+=I+ ( 17
%16#7-G
D1 = D2 ⇔ ∀E, JD1,EK = JD2,EK
?!@ACB,DEGFIH;J2FLKMBND9O+Sn ^_klYh]kljtn ^_ihxf;YhIw+\5RTSGUh]bcX[Shn,^_XZiha^_XhdZSGUaUxS$klYDw,eRT^_nZn ^5Uxf;Yh"XZYZk ~ S$iaUaS$e
UOD
Y[StdZ`$n,S$Y[d bcX[StdZSGU]klYBhaS$ixf_\hWkm^_Y[U=dZSD| ^_kjh D1 6= D2
dZS$X2dZSGUaUxS$klY[UUWXZiXZY[Stj$jQSpf_UaSXZYpf;kliaS5| k Scal∗ (D1) 6= Scal∗ (D2)
f;em^_iaU qD1,UDβ⊥(1)
y6=
qD2,UDβ⊥(1)
y d R f;nZiaGU]egf9nZia^(nn,^5UakjhWkm^_Yk2| hnpf;rS2|
k Scal∗ (D1) = Scal∗ (D2)f;em^_iaU
UOD1 = UOD2
|;fnZia^_n,^5UWkjhWkm^_YIk2|npf;rS ln S$iWjQShdZS\G^_Y[\$elXZiaS5|
)9` * , J'(FGFHEZ$5 !PV,>34"'!*9S*,$57)?<2 3'("<B?39!"#"^<2$ _ )'(L8nZ'3S3',9!)`\l+O34"#FG-"D3'("@8:9!$78$7"C$7E8:9!$;9!"m)B !)]$$7FH)Q =FH3'("CB?39!"#"Y$7@E&8$;9OLYK_ )39(,0
+3$*Q*(5 !4>@2 >9* = ; 3A:92 !42S/.,* -*-,1,1*I= ; 36=J36/.2S,2S365=J*(>@, -, ; * ; *-5 3 ; < *BK` / ? ! o8+#( *,((-"! ( *7%, 0 % 0 0 !0O !, *10 '-0#"p8 0
*)α*α′ %* *4& 1#N$ ># **>N+(
]0; 1]? *&
D1
*D2
%%-7"B% ( 17%16#7
β?
26K%-7#D1
*D2
F 93>#( -6**& +<'$
qD1,UDβ⊥(α)
y=
qD2,UDβ⊥(α)
y∧
qD1,UDβ⊥(α′)
y=
qD2,UDβ⊥(α′)
y
?!@ACB,DEGFIH;J2FLKMBND9O R f;iWr5XZjQS$YhSGUhkmdZS$YhWkmbcX[Sozy\GS$elXZkudZS9egfynZia^_n ^5UWkjhWkm^_Y l2| 1npf;rS&%(' k1dZS9egfelX[d2kmbX[SnZia^_pf;ZklelkmUhaS5|
74ML 4 5 *)+-,/.102.3)
46587:9<;>=?;A@B9DC&7JUFEHG -(.' I8 ' !I#J* J-KJL<4L, #%3 2$ ['% !#6Z & *#-L.( _ &24?L 4&AJLBJ+JML8L *8&&1 LL +=I+6JKJ(J!L0?
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E`?@-@ ;>=?;A@B9 DC E % !Ad#' 0/-"! : !# * (' 0 (( "! *&
R ⊆ R′ ? L
JRK ⊆ JR′K
?!@cA BND'EGFIH¡J2FLKMB,D)OIgYQXhWklelkmUaSemSu\G^d[f;rS5| ^_kmS$Yh R1 ⊆ R′1
dZS$X2tia`GUaSIf;X2 | ^_kmS$Yh R2Sh
R′2
emS$XZiaUhWixf_d2XZkjhaUI|5f;i jQ^_Y[^ ha^_YZkmSudZSegf hWixf_d2X[\hWkm^_Y+w^_YQf
R2 ⊆ R′2
|5f;iemS hW[`G^_ia$jQSudZSjQ^_Y[^ ha^_YZkm\$kjha`dZSegfelX[d2kmbcX[Suf ~ SG\n,^_klYBhaS$XZiaU iMnZia^_n ^5UWkjhWkm^_Y|mknpf;rS %Bo JR2K ⊆ JR′
2KSh dZ^_Y[\ JR1K ⊆ JR′
1Knpf;iujQ^_Y[^ ha^_YZkmS=dZSegfQhWixf_d2X[\hWkm^_YklY ~ S$iaUaS5| 2S5 !S>9/12S365= !S*(245@* = * ; $*-, 9* < 592 ; >@59*5@3F,2S365 245,* ; /1*-,124365 # ! 245,1* ; /.* ",243F5X=!4*(2S59*FK
E`?@-@ ;>=?;A@B9 DC 6E % ! *,0 d-"! '.!f2J!%! " J
(Di) # J # J%$&$KJ'L$6* ()* (+ J'$", "_ JML$KJ%0(+ J9%-(.0/ # J%$ c , *?L/. J%$1 J'L J'6J L23!L:*% -$KJ'$ # L$.*4Z. J Di # / *%&56L # J'$.$KJ'L)7?!@cA BND'EGFIH¡J2FLKMB,D)O ^_kjh X eRTS$Y[UaS$j=ZemS ^_iWjQ`dZSGU
cn/\xZia^_YZkmbX[SGUnZemS$klY[S$jQS$YBhklY[\$elX[UxSGU'd[f;Y[U\apf_bcX[S
Di
|+8 e UWX IhqdZSiaS$j)f;iabcX[S$iqbX[SdZS$X2cn/\aZia^_YZkmbcX[SGUud[f;Y[U
XUx^_Yhq\G^_[`$iaS$YhaSGUI| gY!\G^_Y[\$elXhnpf;i
egfhnZia^_n ^5UWkjhWkm^_Y l2| %(hnpf;rS9 k2|
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(Di) # J # J%$&$KJ'L$ * (9=* (+ J'$ ⋂
i Di
J'$1 AJ # J%$.$(J%!L%-(.0/ # J'$ c , *?L. J%$1 J'L J'6J L>3L:*' 5$(J%$ # L$.*4Z. J Di
7E`?@-@ ;>=?;A@B9 DC U Eao)$(4>.'?I8
@5(Ri)
LBJh%! " J # JA/'$KJBL *% 5+$ # L$CLB/'$KJDR7 LE
J∩ RiK = ∩ JRiK
?!@cA BND'EGFIH¡J2FLKMB,D)OIgYonZia^\GGdZS\G^_jtjQSS$YoelX[d2kmbcX[S]UWkljtnZemS5|8 epf;Xhd R f;,^_iad ~ `$iWkvpS$iqbX[S\GShhaSdZ`>v[YZkjhWkm^_YfZkmS$YXZYUaS$Y[U\5RTSGUh n z-n/d2kliaSbX[SemSGUdZSGUaUaS$klY[U
JRiKUa^_YBhuZkmS$Y\G^_jtnpf(hWklZemSGUI|,PRTSGUh ~ ixf;k npf;iuegfia^_n ^5UWkjhWkm^_Y k2| %('2|gYCXhWklelkmUxSQegfY[^_iWj)f;elkmUxf(hWkm^_Ynpf;i=\G^d[f;rS$i dZ`>v[YZkmS)UxSG\hWkm^_Y k2|ml!npf;rS#%(' o]\GSQbXZk n,S$iWjQSh
dZSiaS$emS ~ S$iemS hW[`G^_ia$jQSdZSU hxf;Zklelkjha`dZSegfelX[d2kmbcX[S]f ~ SG\n ^_klYhaS$XZiaUqf;X9\If_US>2n,^_Y[S$YBhWkmS$e| gYY[Sn S$Xhd2kliaSG\haS$jQS$YBh]XhWklelkmUaS$iegf)nZia^_n ^5UWkjhWkm^_Y k2|ml)npf;rS %()\If;i\GShhaS"nZia^_n ^5UWkjhWkm^_YY[SnZiaS$Y[d1npf_US$Y\G^_jtnhaS]eR klYBhaS$iaUaSG\hWkm^_Y+|
4ML4GF HJI 9 I)DI 9 .K .3) E`?@-@ ;>=?;A@B9 DC p7 E "!L$"!2 dDI82, L2@.+ "!$%?L6J%$ *'(!$.$%L2 J 3(!$, , 3@"$ # JM _ @ # (J J>12J L$.G?LL J' 4 7
∀i, Di 4 D′i =⇒ JD1, . . . ,DnK 4 JD′
1, . . . ,D′nK
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?!@ACB,DEGFIH;J2FLKMBND9OIgYemS ~ `$iWkvpSnpf;i hWixf_d2X[\hWkm^_Y!S$Y9elX[d2kmbX[Sf ~ SG\n ^_klYhaS$XZiaU]seRT^_iad2iaSS>6haS$Y[UWkm^_Y6nY[S$e+\G^_jtj"XhaSf ~ SG\]egfRhWixf_d2X[\hWkm^_Y+|
1 Q # #
*(/ (3=J3 ; ,1* $*(5,/ *", -365959*-,1*(> ; / 9*"! : !4>@2 >9* *=J3659*(5,124* ! !4* < 5 <(; : !42S/1*-5 , ! : 3624/ !S*(/(3$= 3 ; ,**-5,/ *-,"(3F5959* ",*-> ; / 9* ! : !4>@2 >9* = ; 3A:92 !42S/.,*R*-,9* ! : !4>92 >9*I: ?6* =J36245,1*(> ; /(K *-/5@3F,2S3659/ 245 (: ; 59:B,2S365 < ,92! >@* -3$= ! D ,* 36>69* (3 92459:F24/.365 !42S5 < :F2 ; *Q/.365,:A:F=@, < *-/(K
744.N 9 .3).I 9<@+3$**-5 !4>@2 >9*E= ; 39:92 !S24/8,* A>@5 9*(/./1*(2S5 : >95 3 ; ,9363659: !&59365 ? 2@* /12 *-,Q/1*(>!4* $*(5,
/12e2 ! *(/8,2! < @2 :B,1* $*(5, ,13F, : ! ( < 592H,2S365@K =A:6* 4 * K : 593B,243F5 < ,92! >9*I*= 3F59*(5,2S* ! !4**(/., 9365 ; *(/8, ; *-245,*E:F>69*(/./1*-2459/ 2 < 92 :,* $*(5, ,3F,:F> K46587:9<;>=?;A@B9DC&7AV E )"!2$#&%W L JIY,' % \' J%$ L JML$KJ%0(+ J # J # J'$.$KJ'L$9"<0/ # 2J'6J L2 - # JM)(%J ( $KJ(7
46587:9<;>=?;A@B9DC&7 FE+*-"!-,.,0/2143657%8/8 "!>9#6%:R_ p'1$ [$# # _ L J /( /. JJ'$ # / L+)+
E⊥ = F | ∀D ∈ E, W (JF,DK) = 1
46587:9<;>=?;A@B9DC&7FEHG;, -<,4** ./D6=%83 M43:W L 2$ pL \#x%M # J%$ALBJC/@ . J /$#+ AJh $?L (+ , ( 25#L4+ 57
46587:9<;>=?;A@B9DC&7 E>/2? 83=5>@?A52@?B? LC/2?D3 C/ 8@12C*5@??E%8*F3G1 212? W L # J%$&$KJ%!L D
$ L J ( +$(JβJ'$#3%3 Y6[$% 2 $. $(J%$2IH J *(*G?L$<$%L$ * J8/9*' JML>$$L2
"!L2/+!KJ'$7 (-
ELBJ /( /. J $ L J6( $KJ
β7 LFL2 J
Lin(E) _ Z !# ] FZ ] FZ Z%3 Z ]
E# %3 Y[% 2FZ Z \@β7
46587:9<;>=?;A@B9DC&7FEHG;, -<,4** ./DD3?C/ 8@12C*5W L 2$ pL # %3 Y[% 2J%$ LBJC/@ . J
G2J' " AJt. J
G = Lin(G⊥⊥)
245 (: ; 5A:,243F5*(/., < 592S*#@*#:F592 D(; *Q/12!2 ! :F2 ; *FK
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E`?@-@ ;>=?;A@B9DC V E % /Ad1 */-1$,0/fC *(3( J%JML>30 -*2 JM < !L2/+"(J
GDJ( # J%$&$KJ%!L D # J G
>" <J!$ J LL/. J # J'$.$(J%L D0 ⊆ D # J G
J% . J
∀E ∈ G,E ⊆ D⇒ D0 ⊆ E
I#J1 -$":* J # J'$.$(J%L D0
$KJ *+ 3*@ JJMLBLL2+ L> AJ%$7* J/9*% J L>*$u3($.-*M% $ # J DL2?L@"$&5 /%$
)+ L # J'$.$KJ'L # J G⊥ 7<:J # J'$.$KJ'L D0
J%$ _ !L2 J%&$KJ *@*G?L # J'$ # J'$.$(J%L$ # J GL:*' 5$ # L$
D7
?!@cA BND'EGFIH¡J2FLKMB,D)OIgY<nZia^c\GGdZS9\G^_jtjQSS$YelX[d2kmbcX[SUakljtnZemSS$YXhWklelkmUxf;YhegfyUhxf;Zklelkjha` iMnZia^_n ^5UWk nhWkm^_Yk2| %(npf;rS o>|
45879 ;>=?; @B9 C&7 FE&% /Ad1 * /21 ,0/:Q_ %3uB[ [%$ # _;L # J%$&$KJ%!L D # L$ L * ( (2J'6J L2
GJ%$ J # J'$.$KJ'L # J Q1L3 ,$.5?L 7 7
|D|G =⋂
E ∈ G | E ⊆ D
+3* *-5 !S>92! >@* /12$= !S* @3F5X>,2 !S24/.* 9*(>X593F,2S3659/ < ,92! >@* -3=! D ,1*FK45879 ;>=?; @B9 CaV E )"!2$#&% %43 83Xd, 83 W L J /( JJ'$R\ Y' % \0 c\ #% # "! # #$% $&
Pss(|E⊥⊥|) ⊆ E7
45879 ;>=?; @B9 CaV E )"!2$#&% 3?7/&1>7*- d, '43( W L J /( JJ'$ L JEY' % \0#3%3 Y6[$% ""!$ #)$% $.$
Pss(Lin(|E⊥⊥|)) ⊆ Lin(E) = E
*@`,+.- ' C 92L'()]-'(=0/UB)21PV,X/$5LB:9!"#\8'(FE34$43)?Y) L/'("K)?\65KS-'(=L'()]-'(8'("K"#?3'(4S9!)Ing$5!PV,X 34"#'!*9S*,$57)?2)<c$5 !PV,@9! !83'()?"#,07J`AC)98& $5 !PV,34"#'!*9S*,$57)I:8LGB)21PV,X@8'(FE34$43)I\ !B"K dnl
Pss(|E⊥⊥|) ⊆ E
) g$5!PV,X@9! !83'()?":8LGB)21PV,Xm$5LB:9!"#28'(FE34$43)?\ !B"K dnl
Lin(|E⊥⊥|) ⊆ Lin(E) = E
45879 ;>=?; @B9 CaV UFE9<;,<*-"!-,=.,0/2143 3?7/&1>7*-:J !%?>@!'A! 1B!$[# #3% Y6[$% " # _ L J /( J E
J'$
Lin(E⊥⊥)
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I-) .I I 01+ K2I/0 )H. )PQ5 /1* (365,1*(5,*#9*I=A:F/./1* ; *-5 ; *-? >9* !4*(/ 593B,243F59/ :B,1,: < *(/ :F> (365 -*(=@,9* ;1< = * ; ,1362 ; *FK
46587:9<;>=?;A@B9 DCaVFE & >.**I,=7*- 57%8/ d, 4,<*-0 / ::JY B%!$% " # _;L * (3(2J%JML>3($.-*
GJ%$ "_ JML$KJ%0(+ J # J'$A'+ :*D*L$
I2J' " AJ%$ JM J # J'$.$(J%LB$+L> !")+*AJMLL J#G
(+, ξ, I)
(−, ξ.i1, J)
z
(−, ξ.i1, J′)
z
(−, ξ.in, J)
z
(−, ξ.i1, J′)
z
%$
I = i1, . . . , inJ, J ′ )+*(JML>
Pfin(N)& ::J('6Y )'%!$%*' # _ L * (+ (2J'6J L2 L>/-, J%$ "_ JML$(J%)(+ AJ # J%$ % :*?L$
I
2J' " AJ%$ JM J # J'$.$(J%L (−, ξ, I)
z
J%$ !L:*% -$ # L$ _ !L:*+ L4?L # L$ G # J
Dai− = (−, ξ, J)
z
7 77(−, ξ, J ′)
z
%$J, J ′ )++* (JML>
Pfin(N)7
& J.'Y )'%!$%/'" # _;L J /( JJ%$ AJ /0-J'@"KJ # J $?L (+ , ( 21,?LL+ 57
`?@ -@ ;A=; @B9 DC FEHG 1 * 1Ad & *& ?B1 ,0/ 5&% * & 2 **I,)C*5=57%8/7d, '<,<*-0 / <,<?(:32::J9/4J%(!KJ # _ L *(+ @2J'6J L> 0 -*52 JM < !L2/+"(J3($.-*
G$ 0(+$KJ
` ξJ'$
76 "_ JML$KJ%0(+ J # J'$ %+<:**G?L$I2J' &_;L # J'$.$KJ'L # J G
*@JML:*J)+(+, ξ, I) 6
7 76 "_ JML$KJ%0(+ J # J'$,'+<:**G?L$I2J% _ L # J%$&$KJ%!L # J G
CLBJ *@*G?L(+, ξ, I)
@"$&5/J)+
G⊥ 7
8:9<;>=?A@CBED!FGBIH =J?KMLONQPCRhTSVUXWMY-Z\[!UV]^PM_VPCRT`Va4P-SVb^PCR.`"cdSVa.Z\[+ZeSf_GgihjSfPk`Va4cdl%[!lVgeZegiRh4PmPhOZ\[ZeSf_Ggih)SfP[Cb^PCn.`"cdgeUh4P-SVa4RIo
p :B,> ; * ! !4* $*(5, ! : -3659/1* ; ?B:B,124365 =A: ; 3 ; ,@36365A: !42S, < 9> ;1< = * ; ,1362 ; * *(/., *-5 (3 ; *E? <(; 2 A< * #`?@ -@ ;A=; @B9 DC FE & > *-I,=C*5
q2 *(+3( J%JML>3$CLBJ (+$KJ L4+!KJM J9)(%J9/0-J' ("(J JM$?L ( 21,?LL+ 57
46587:9<;>=?;A@B9 DCaVJV EHG;, '<,4** ./DI?A5 ? ' ,=C/I? d,6/4/@?& I#J@jr*@3( J%JML>*$ $?L>]4% Zts !$% Z $.) AJ(&$9/4J%(!KJ%$6$?L> # "$CH(L>*$7& W LBJ /( J E
J'$ "!$ Aju $& $L /0-J'@"KJ#J%$ L $&L, J( ?LI
74:R_ JML$(J%)(+ AJIJ'$ !"J% G/9 v'B[$ %*w B[%)!$ # J E
7
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E`?@-@ ;>=?;A@B9DC E G 1 * 19d &.* $? 1 ,6/ 52@?0d, '<,4*-0 /DI? 5 ? ' ,=C/I?& I#J(^r *@3( J%JML>*$T ($&5 $M$?L> # "$CH(L>$M$. J( $KJ@ J'6J L> $.3 AJ(ML>2J'.$KJ *(?LJ%$/ # +52JMDjr*@)(+LL+"$?L$6 "!L2/"KJ'$ # JO3( # $ 1 # J # /%0?L$-*+ "!$/'$7
& I#J(^r *(+ @2J'6J L>*$ L2/ ,* $6$?L> # !$7H(!L2$6$.:J@$(J( AJ%JML> $.
∀D ∈ Pss(G), ∀E ∈ Pss(H), |D|G⋂
|E|H = ∅.
8:9); =J? @CBED FGBIH =?mKIgUka4Y-Sh0geZegiR4P Z\[`Va4P-SVb^P P-U ZeSf_GgihjSfP `Va4cdl%[!lVgeZegiRh4Pi `Va4cd`c1R0gjh0gicdU#Go %`%[!]^P %'@Bo o
+3* =J36> ; ,36>@,1*(/ !4*(/ ?B: ; 24:F5,*(/ 9* !4:"!S>92! >@* ? >9*-/ .>9/ > 2 -2 !4*-/ = ; 36=J36/12H,243F59/&/1>92H?F:B5 ,1*(/=J* ; *",1,*-5, @*E= ; 36>@?F* ; ! : -3=! < ,1>9* 2S5 ,1* ; 59* = 3F> ; !4* )?" KE`?@-@ ;>=?;A@B9DC E L ? /Ad =57%8/ * , '% d ,0/
(5M ⊆ N
L /'$KJ' (@"7 q> # J%$&$KJ'L 3($&5* ,$(J%< ,$."+1 AJD = (+, ξ, I).R
J%$6/ ,+ #DM~DN\M
%$DM
*(<JML:*J)+(+, ξ, I ∩M)
J(DN\M
* (JML * J=+(+, ξ, I \M)
7:J # J'$.$(J%L DM
J%$ J '"!<s, Y # J D$
M7
L /@2JML # * J@. J # / L-*G?LFJML ($'L>zξM
= zξ
7
E`?@-@ ;>=?;A@B9DC E * , '%d ,0/ (5
EL J /( J* ?LL J rBJ J(
MLB/'$KJ% ((!7 L
(E⊥⊥)M = (EM)⊥⊥ 7
45879 ;>=?; @B9 CaV:FE&% / 5& 2 /25418/9d I#J(jrB/4J%(!KJ%$
R1
J(R2
$?L> %3 ]&Y ] [$Z $.
∀I1, J1 ∈ R1, I2, J2 ∈ R2, I1 ∪ I2 = J1 ∪ J2 ⇒ I1 = J1, I2 = J2
I#J(jr * (+ (2J'6J L2$6$?L> %3 ]&Y ] [$ZC$.) J@.$6/4J%(!KJ%$ $?L>!L # /0-J L # L>*$7
45879 ;>=?; @B9 CaV FEHG;, 4,<*-0 /I? & "* 18/ =4 * ?I#J(jr * (+ (2J'6J L2$6$?L>+Y 'B[$ 16)'Z $.) J@.$ /%$KJ' @("&$9$?L> # "$CH(L>*$7
# )
*(/$(3F5959* ",*-> ; //1365, < 5924/$(3 $*= 3F> ; ! : !4>92 >9*= ; 3A:@2 !42S/.,1* *-, !4: !S>92! >@* : ?F* =J36245,1*(> ; /(K 5X=A: ; ,12 -> !42S* ; !4*-/ (36595@* -,1*(> ; /R*I@=J3659*-5,24*!4/ /13F5 , < 592S/@*#:F592 D(; *Q/12!2 ! :F2 ; *FK
*(/ -365959*-,1*(> ; / 9:92S,1>9* !S/ =J3 ; ,1*(5,C/.> ; !4*-/ -3=J3 ; ,* $*(5,1/ !42S5 < :B2 ; *-/(K *[*", !4*
] 365,
=A:B/1/1* ; 9*(/"(3$=J3 ; ,1* $*(5,/ !S245 < :F2 ; *(/:F> (3$=J3 ; ,1* $*(5,/ >!S,2S= !4*-/(K
T8N )K +$02+!
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+ $ +
46587:9<;>=?;A@B9 DCaV E & d183G1 =4&
DJ'$ L # J%$&$KJ%!L 3($&5* # J ( +$KJ ` ξ.i + (&$
D =
(−, ξ, i).D$.
D 6= Fid
Skunk$.!L2?L
& D
J'$ L # J%$&$KJ%!L L>/-, # J ( +$(J ξ.i ` + @.$D = (+, ξ, i).D
& L J ( $KJ L4"KJ AJ ]&YB[$#M[ 16 # _ L *(+3( J%JML> !L2/+"(JG
J%$ J"(+ ,( 21,D?L4+ = !L2/+"(J # J'$ # / *+ j,J'$ # J $(J%$ # J%$.$(J%!L$7<: (.$ J G
J'$ @$.5 # J ( +$(J` ξ.i ; *-/1= K L2/ , # J ( +$(J ξ.i ` 1?LE AJL2 J
G ; *(/.= K G 7+3$* =J36> ; ! : !4>92! >9*: ?6* =J36245,1*(> ; / ! < ,92! >9* (3$= ! D ,*= 3F> ; !4* < : ! :6* 5 < :,2! >95-3$= 3 ; ,**-5 , = 36/.2S,12! :F2S, 245,* ; ?6*-592 ; >@5 923 ; ,936365A: ! K
`?@ -@ ;A=; @B9 DCSU EHG , '43(& "%252 3 C/ &@12C*5 C/ */ <,0%8*F3@?A5& d183G1 =4? (-
GL *@3( J%JML>3 "!L2/"KJ@7
& G
J%$ 3($.-* 4+ G(.$D | D ∈ G⊥⊥
J%$ L J /( J*@1 ( J + G7
& G
J%$ L2/ , > (&$ G =
D | D ∈ G
⋃
zξ7
8:9<;>=?A@CBED!FGBIH =J?KA[O`Va4P-SVb^PPCRhSVUfP SfR0gicdUM_VPCR`Va4P-SVb^PCR P-U ZeSf_Ggih)SfP `Va4cdl%[!lVgeZegiRh4PPhP-U ZeSf_Ggih)SfP[Cb^PCn.`"cdgeUh4P-SVa4RIo
O . I *) .4 020 PQ5 ; *-= ; *-5!S*(/ (3F59/., ; > ",243F59/ 9* ! : !4>@2 >9*I: ?6*=J36245,1*(> ; /(K
46587:9<;>=?;A@B9 DCaV.FE <,6/ /:.3 3?&
GJ%$ L *(+3( J%JML>, !L2/+"(J 3($&5* + (&$
?GJ%$ J(+ , ( 21,L4+ L "!L2/"KJ
# J'$ # / * j,J'$ L2/-,D* $ # J%$ # J%$&$KJ'L$ # (+ , ( 21,?LL+ # J G7
& G
J%$ L *@3( J%JML> "L>/+!KJ*L2/-, * L (&$!G
J%$, J ( ,@ >1,?L4 "!L2/+!KJ# J'$ # / * j,J'$ ($&5 $ # J'$ # J%$&$KJ%!L$ # J G
7
46587:9<;>=?;A@B9 DCaV FEL@?0d,0/8/d %4* ?9&),03[ 5(@?
]
& G
J%$AL* (3(2J%JML>3 !L2/+"(J @$.5 >?L L22J[G
$L (+,( 2d,?L4+ -7&
GJ%$ L * (+ (2J'6J L2 "!L2/"KJL>/-, ?L L22J
]G$L (+ , ( 21,L4+ /-,+ #
G 7`?@ -@ ;A=; @B9 DCSUEHG , '43(& "%252 3 C/ &@12C*5 C/ */ <,0%8*F3@? <,0/ /: 3C3@?
(-G
L *@3( J%JML>3 "!L2/"KJ@7&
GJ%$ 3($.-* 4+ G(.$
D | D ∈ [G
J%$ LBJC/@ 9 J*(+1 @2J3?G
7&
GJ%$ L2/ , > (&$
!G = D | D ∈ ]G
⋃
zξ = D | D ∈ G
⋃
zξ7
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: < 592H,243F5 9*-/+*I@=J3659*-5,24*! !S*(/+*-/., 5936>?6* :B> 29*-5,2 >9* -* ! !4*9*(/* < (: ! :F*(/(K-+* !4:E*-/.,-3 ;1; 27 < : ?F* ! >9592! 3 ; $2S, < ! : /1*-,124365 @KG@K0=A: 6* 6T K
T O .3).C
45879 ;>=?; @B9 C EL d,0/8/ d.%4* ⋂ (5
(Gk)L J +<" ! J # J * (3(2J%JML>$ 551 AJ%$ ; *(/.= K !L2/+"(J%$ $ L J ('6J2( +$(J(7
L # / L- AJ *(+3( J%JML>30 -*52 J ; *-/1= K !L2/+"(J ⋂
k Gk
)+⋂
k Gk = D | ∀k, D ∈ Gk
45879 ;>=?; @B9 C EL d,0/8/ d.%4* ⋃⊥⊥
(5(Gk)
L J +<" ! J # J * (3(2J%JML>$ 551 AJ%$ ; *(/.= K !L2/+"(J%$ $ L J ('6J2( +$(J(7 L # / L- AJ *(+3( J%JML>30 -*52 J ; *-/1= K !L2/+"(J ⋃⊥⊥
k Gk
=+#⋃⊥⊥
k Gk = D | ∃k, D ∈ Gk⊥⊥ ; *-/1= K ⋃⊥⊥
k Gk = Lin(D | ∃k, D ∈ Gk⊥⊥)
45879 ;>=?; @B9 C UFEL d,0/8/ d.%4* &
:J2* ?LL J *( J(&J%$) AJ5*?LL J *@2J(+ ⋂ !1 "9 /# # J%$ * (3(2J%JML>$-L2/-,D* $ # "$CH(L>*$7
* (36595@* -,1*(> ;&*(/., *-5 (3 ; *>@5 = ; 3 @>92S, (: ; , < /.24*(5X/1> ; !S*(/ 245 : ; 5A:B,2S3659/ #
E`?@-@ ;>=?;A@B9DCSU U E@? *- 52 3 !C/Ad1 */-1$,0/fC $.*(+3( J%JML>*$ # !$7H(!L>*$ G
J@H
$CL JM)(%J ( +$(J L2/ ,?JPss(|G & H|) = Pss(|G|)× Pss(|H|)
45879 ;>=?; @B9 C FEL d,0/8/ d.%4*⊕
:J * ?LL J *( J(⊕J'$L J* ?LL J *( J( ⋃⊥⊥ !"2 "/ # # J%$*(+3( J%JML>*$) !L2/+"(J%$"3($&5* $
# "$CH(L>*$7
E`?@-@ ;>=?;A@B9DCSU E * ,B *&(& &=52 3 1 5 ? ' ,0/9d$,0/fC $.*(+3( J%JML>*$6 !L2/+"(J%$ 3($.-* $ # "$CH(L>$
Pss(G⊕ H) = Pss(G) ∪ Pss(H)
TGF H /03) .0 02.K +) .C+3* *-5 !S>92! >@* = ; 39:92 !S24/8,* 365 < 592S, !S*Q,*(5@/1*(> ; =A: ; 92 !S245 < : ; 2H, < K
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46587:9<;>=?;A@B9 DC AV EL ::J%$ # )/%(JML>2J'$ ?J%&$.G?L$ # E JML$(J(9$L2 # / LAJ%$ )+ (+! "L>/+&5/
& ::J 1 # - 2JML$(& J% # J # J@jr0*@3( J%JML>*$ !L2/+"(J%$3($.-* $ J'$) "_ JML$KJ%0(+ J # J'$*@)(+LL+"$?L$ !L2/+"(J%$ # J'$ 1 # -*$ 2J L$@. J' "$ # J'$O=+ J%$6$(J%< ,$."+1 AJ%$
Pss(
G⊗ H)
= Pss(G)⊗ Pss(H)
J( # ?L:*G⊗ H = Cl
(
Pss(G)⊗ Pss(H))
& f # J%$ # J%$.$(J%!L$ $(J%< ,*$&"+1 J'$ *?L$* *(?L J%$9$.!! +!KJm#0 *L$ *@*L4 (5. J% " AJ 5 # 9 J9$.!1 AJ V1( (+" !"$2J < ?J *.3(!L2 J(&$ 7
46587:9<;>=?;A@B9 DC FEL d,6/4/ d %8*./
::J 5E> =A:B=92 ! !4365./
J%$ J # # ,1*(59/.*(> ; (3 >, :B,12! :E2: ! 7
`?@ -@ ;A=; @B9 DCSU9V EHG , '43(& "%252 7/D */ <,6%4*F3:
G1
J@G2
$L2 @$.5 $ J(9L # /0-J L # L>$ + @.$D1 ~ D2 | D1 ∈ G1,D2 ∈ G2
J@D1 ~D2 | D1 ∈ Pss(G1),D2 ∈ Pss(G2)
$?L> # J'$ /@ 9 J%$.*(+1 (2J'$ G1 ~G2
7
8:9<;>=?A@CBED!FGBIH =J?KML cdW WMPm`"cdSVaZiPCRk[!Sh0a4PCRb^P-a4R4gicdUfR_VPMZ\[ ZeSf_GgihjSfP cdU Sh0geZegiR4PmZ\[:`Va4cd`"c1R0gjh0gicdU_ NQP<giR h4P-UfnCP_VPT`Va4c PCnh0gicdU i `Va4cd`c1R4gjh0gicdUkGo % kk`%[!]^P l' o o
% % #
PQ5 >,2 !S24/.*E>@59*,1* 9592! >9* /12!2 ! :F2 ; * (*! !4*I*= !S3 G6< *= 3F> ; !4: !S>92 >9*E= ; 3A:92 !42S/.,* A: =92S, ; * T =A:6* "K
.N .(K1I I ) )
PQ5 ; *-= ; *-5!S*(/593F,124365@/ 9* !4>92! >9* = ; 3A:92 !424/8,*BK
46587:9<;>=?;A@B9 DC FE @?B? 7/ -1 *-:.3W L]66Z(Z% [J'%# ]j\ Y' % \ J%$L # J'$.$KJ'L+1 J'L J'6J L>=L:*' 5$ # L$9L # J%$&$KJ%!L# J* [email protected] /@ 9 J(7
: < 592H,243F5 /1>92H?B:F5,*=J* ; $*-,9*=A: ; !4* ; 9* !4: =A: ; ,2S*& >95 (3 * (2S*(5, >92 = ;1< D 9* >95*(59/.* !S* : -,2S3659/-K
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45879 ;>=?; @B9 C EHG;,= dD: / ?B,0%-?%; ' 1Ad / %4/ /2? 83=5 919d$,0/2?:J "! %$Z!&\AZ >es["v# L JML$(J%)( J # _ *(?L$ X # L$L"* J/9*% J L>
cJ'$ J
* J8/9*%AJML>c′ (2J L2 JML LL2+ L> J%$ J(+" " AJ%$ @ # J'$ # J c # ?L>, _ *(*G?L L _ J%$ )+$ # L$ X 7
D
J%$ L # J%$&$KJ'L $+ (+$KJ` ξ
?L L>2JCoefz,zξ
(D) J6* J8/9*%AJML>M$$,H * J L>
1KJ'AJ%-* J8/9*%AJML> AJ.* J8/9*%AJML># C% *'L J M# J D # J9 )"KJz;zξ
7 I#J9)(%J ?L L2 J
Coefpr (D) AJ * J/9*% J L>,$ $,H * J L23k1(J%< J%2* J8/9*' JML> # J D # J "_ JML$KJ%0(+ J
# J%$ *@*L$ 1I2KJ%$7 ::J 2KJ%< J' * J8/9*%AJML> * (.(J%$ ?L # L> # _ J L$KJ')(+ J # J%$ *(?L$ J'$L2/Coeffst(D)
7
45879 ;>=?; @B9 C FE9<;@& "!>9#6%W L J !%?>Y' % \&J'$ L JO)+!KJ
(E, d)2J' " J J
& EJ%$ L J /( J@7
& dJ%$9L JO1/, # "$5L:*JA$ J'$ # J%$&$KJ'L$)+* J' "$ # J E
2J% ! J J& $3($&5* 7
& d(Fid,Fid) = 0& d(zξ,zξ) = 0& ∀D1,D2 ∈ Ep, d(D1,D2) ≥ δc(Coeffst(D1),Coeffst(D2))& ∀D1,D2 ∈ Ep, d(D1,D2) ≥ δc(Coefz,zξ
(D1),Coefz,zξ(D2))& $L>/-, 7
& ∀D1,D2 ∈ Ep, d(D1,D2) ≥ δc(Coeffst(D1),Coeffst(D2))& d(Dai−,Dai−) = 0& d(Skunk,Skunk) = 0
:F5@/ ! : < 592S,124365 /1>92H?B:F5,* 365 593F,*Semi− Simp(E)
!4*(/ 9*(/./1*-2459/ /.* $2 /12$= !S*(/ >95*-59/1* !4*9*9*-/1/.*(245@/
EK
45879 ;>=?; @B9 C FE **"!2, =.,0/218357%8/ >; & "!2$#&%:Q_ !J''A! 1B!$[# # _ L J (+,/( J (E, d)
$CL J( $KJ L4+!KJ J'$ k=+"(J(E⊥, d⊥)
%$ 1/8, # !$5L:* J d⊥ J%$ # / LAJ)+
∀E1,E2 ∈ E⊥p, d⊥(E1,E2) = sup
D1,D2∈ Semi−Simp(Ep)
(
δc(JD1,E1K , JD2,E2K)− d(D1,D2))
*@`,+.- ' C 9GB`d>)P-'(./$ /U'("K)91''(49!$X0/ L*,:B)21PV,XYZ$5!PV,34"'!*9S*,$57)?mbIBd>)P-'(!0Y39 ,f!kZ)P$57@FE34$7FH)$7OmFE34$7R8:9!"G$7R*,)YB)-9!)E34"#'( !"" 934PFH) "B$5)-9!) 28'(FE34$7B)P 5T'(")?,0
92B`d>)P-'()P$57B<-8+8'(P3"#$7&FG-FE34$7/8:9!"E8$;9<39!" 9) 34$56<49!)P"$)<$;9N"#8:1"#8:1RB !)]$$7R0/ )21B'(":3FH^H8'(FE34$7B)P 2L@,9!" 9!)/)P$57"G$7G)?8:1PV,XR$;9Q 3FHY39!")P-[8:9!6H. 3'(L)P-$$7Ol$ / )P$57,9!)]-'( RL'( !:9!8'(L8)?"#FE34$5PV,:8/9! FGFGF 8<R"#95 9!"G$7C34"# !OG$;9c$5 !PV,2FE34$7)/L'(W^$7^)P$57" 5 98)I'("K7,9!)$7"#B$5) 9!)@3'("H$;98'(FE34$7B)] 5T'(")?N8'(FGFHm3'("H$;9 $5!PV,Xm34"#'!*9S*,$57)Ib]8)P-'(!0\39k!0
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`?@ -@ ;A=; @B9 DCSUFE **"!2, =.,6/-18305 7%8/ 2<; & "!>9#6%:R_ ( 2d,?L4+ # _ L J (+ , /( J J%$ L J (+ , /( J@7
8:9<;>=?A@CBED!FGBIH =J?KA[`Va4P-SVb^P PCRh R0geW geZ\[!gea4PnCP-ZeZiP _VP Z\[ `Va4cd`c1R0gjh0gicdUMnCcda0a4PCR4`cdUf_f[!UBh4P P-UZeSf_Ggih)SfP`Va4cdl%[!lVgeZegiRh4PEi `Va4cd`"c1R0gjh0gicdU Goml`%[!]^P#%(Bo o
46587:9<;>=?;A@B9 DC E ;d, '<,4*-0 /DW L !%?>:"!$ !J' J%$ LBJ ( ,/@ 9 J /-, J# $?L (+,( 21,D?L4+ -7
`?@ -@ ;A=; @B9 DCSU FE **"!2, =.,6/-18305 7%8/ 2<; & "!>9#6%B 2<;d, 4,<*-0 /:R_ ( 2d,?L4+ # _ L J (+ , /( J J%$ L ( , * (+ (2J'6J L2&7I#JO1 5+$ %3C 2J(+,/@ 9 J
(E, d)J@9 $ # J'$.$KJ'L$9$KJ',*$&"+1 J'$
D1
J(D2 # J E
d⊥⊥(D1,D2) ≤ d(D1,D2)
8:9<;>=?A@CBED!FGBIH =J?KIgU `Va4c<nC_VP.nCcdW WMP`"cdSVa Z\[k`Va4cd`c1R4gjh0gicdU Go `%[!]^P%(Go
46587:9<;>=?;A@B9 DC E @?B? 7/ %4/232 ,4* 0W L # J'$.$KJ'L D
J%$Q\ %!J' # L$CL (+ , *(+ @2J'6J L>(G, d)
$&
d(D,D) = 0
I)KB)J
>9592! 3 ; $2S, < /1> ; !4*-/ -365959*-,1*(> ; /9* ! : !43F2 >9* !42S5 < :F2 ; * *-/.,< < 5924* (3$* *(5 !S>92 >9* = ; 3 A:92 !424/8,*BK : 5@36>@?6*(:F>@, < ;1< /.29*/A:F59/ !4: 6*-/.,1243659*-/ < : !4:6*(/ >@2=J* ; $*-, 9* !S*(/ @2 <(; *-5 (2S* ;9*(/ *=J3659*(5,124* ! !4*-/(K
)K +$02+!
* < : !4:6* 5 < :B,12! >95 -3$= 3 ; ,**-5 , !4245 < :F2 ; * :F2S,I:B=9=A: ; :4, ; *@*(/9*(/./1*-2459/# >92 /13F5 ,9*(/ < (: ! :F*(/9* @*(/1/.*(2S59/ 593F5 !42S5 < :B2 ; *-/(K 36> ; *-5 !4*"?6* ; =A: ; >9592 3 ; $2S, < (*-/&9*-/1/1*-2459/ 9*(/92 (3$= 3 ; ,**-5,/ @365 >@,2 !424/.* ! : ,1* 95@2 >9*# < ; 2S,1*A:F59/ ! : < 592H,2S365 /1>92H?F:B5 ,1*FK
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FT
45879 ;>=?; @B9 C UFE & d183 1=>@?& ::J]&YB[$# [ 1B L2/ ,* # _;L (+, *@3( J%JML>, !L2/+"(J 3($.-* # J (+$KJ ` ξ.i J'$
(G, d) = (
G,
d) = (
G,
′d)⊥⊥
%$
′d(. . .
D1 . . . , . . .
D2 . . .) =
d(D1,D2)$.
D1
J(D2
$?L> !L2/+"(J%$1
$.!L2?LJ(. . .
Dk . . .
J'$ L J L2 5?L + L # J%$.$(J%!L L2/-,D* # ?L> 0* ?L>*L> D*L # J(−, ξ, i)
J%$Dk
7& ::J]&YB[$# [ 1BM3($.-* # _ L (+ , *(+3( J%JML>:L2/ ,* # J ( +$KJ ξ.i ` J%$
(G, d) = (
G,
d) = (
(G⊥, d⊥))⊥
*@`,+.- ' C! XD1
'(D2
./U) 39(\$5LB:9!"#n9!$7'("# ′d(. . .
D1 . . . , . . .
D2 . . .) = 1
0
J$;9gFE34$5PV,XZV,$7OL'( ?$5LB:9!"#R./ )I"K -L)D39(ZS9!O$7=8:9!$78$&H$;9!7)-9!L83'("
G⊥ )'(L83'(" G0
J?34S9!)28E$7.8'(B:V,L8R$7'!PV,XN8)])?ZB`d>)]-'(WL^'()J39(N8$;9!"cb 3'("O$;9"#8:1"#8:1m0/ B !)]$ )21B'(":3FHm28'(FE34$7B)]39!"2 FE34$7?k!0
GF . I *) .4 0104 +365, ; :F2 ; * $*(5, :F>& < : ! :6* 5 < :B,12! !S* 3'(" V,X'( 39( >95 -3=J3 ; ,* $*(5, !42S5 < :B2 ; *+= 36/.2S,12!
:F>@,3 ; 24/.* !S*(/# < (: ! : 6*(/ 9*&9*-/1/.*(245@/ 59365 !4245 < :F2 ; *(/ -*& >92 *= !42! >9* !S*(/#92 <(; *(5 -*(/ *-5 , ; * ! : < 592H,243F5/1>92H?F:B5 ,1*I*-,"! : < 592H,243F5 K T K45879 ;>=?; @B9 C FE<,0/ /D 3 3:@?
& ::J =J36> ; >9362e=A:F/ # _ L (+ , *(+3( J%JML>, !L2/+"(J @$.5 # J ( $KJ ` ξ.i J'$
?(G, d) = (?G, ?d) = (?G, ?′d)⊥⊥
%$?′d(. . .
D1 . . . , . . .
D2 . . .) = d(D1,D2)
J(. . .
Dk . . .
J'$ L J L2 5?L + L # J%$.$(J%!L L2/-,D* # ?L> 0* ?L>*L> D*L # J(−, ξ, i)
J%$Dk
7& ::J @24*(5 / ; # _;L (+, * (3(2J%JML>:L>/-, # J ( +$(J ξ.i ` J%$
!(G, d) = (!G, !d) = (!G, !′d)⊥⊥
+3* = 36> ; !4*(/ < : ! :6*-/I*(5 !4>@2 >9*C= ; 3A:92 !42S/.,* !S*(/ 92S/.,:F5 (*-/!′d*",
?′d/.365, (3
= ! D ,1*(/ :F> /1*(5@//.>92S?B:F5,9* ! : = ; 36=J36/12H,243F5/1>92H?F:B5 ,1*FK
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`?@ -@ ;A=; @B9 DCSU EHG , '43(& "%252 7/D */ <,6%4*F3:@? <,0/ /: 3C3@?f+ -(+, * (3(2J%JML>:L>/-,
(G, d)
(!G, !d) = (!G, !′d)⊥⊥ = (!G, !′d)
f+ -(+, * (3(2J%JML> !L2/+"(J3($.-* (G, d)
(?G, ?d) = (?G, ?′d)⊥⊥ = (?G, ?′d)
8:9<;>=?A@CBED!FGBIH =J?KIgUXa4Y-Sh0geZegiR4PkZ\[h4PCnVUVgih)SfP `%[!acdahfcd]^cdU%[!Zegjh4Y$i?′d
Ph!′d
R4cdUhcdahfcd]^cdU%[!ZiPCRZ*N SVUfP_VPTZ*N [!Sh0a4P"o o
L O/.3) . C
*(/ :@2S,2 / /13F5 ,< < 5924/ /1> ; !S*(/(3=J3 ; ,1* $*(5,/!S245 < :F2 ; *(/ -3 $* *(5 !4>@2 >9* = ; 3A:92 !424/8,*BK46587:9<;>=?;A@B9 DC AV E&% /D * ? d ,0/:R_ 245,1* ; /.* ",243F5 # J ( , * (+ (2J'6J L2$ L2/ ,* $ # JM('6J ( +$KJ#J%$
∩k(Gk, dk) = (∩kGk,∩kdk)
?J *∩kdk = sup dk
46587:9<;>=?;A@B9 DC FE />$,0/:R_ >9592S365 # J (+, *@3( J%JML>*$ ($&5 $ !L2/+"(J%$ # JA)(%J( +$(J#J'$
∪∗k(Gk, dk) = (∪∗kGk,∪∗kdk) = (∪∗kGk,∪kdk)
⊥⊥
?J *∪kdk = inf dk
+*-// < 592S,124365@/ )?"8)P-'( *-, -'( < 5924/./1*(5, !4*(/ -365959*-,*-> ; / 9! !8 *-, 34$5S /.> ; !S*(/(3$= 3 ; ,**-5,/ 924/ .36245,1/(K
H 0 ) .0 02.K +) .C *(/ > !S,124= !S2 :,2! / /.365, < 5@24/I/1> ; !S*(/ -3=J3 ; ,* $*(5,1/ !4245 < :F2 ; *(/ 9* :F592 D-; * /.2$2 !4:F2 ; *
! : !4>@2 >9* = ; 3A:92 !42S/.,*BK
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45879 ;>=?; @B9 C FEL :J'*LL J *@2J@
./ # J # J(^r0( , * (+ (2J'6J L2$ L2/-,D* $ # J3('6J2( +$(J(G1, d1)
J@(G2, d2)J%$
(G1, d1) ./ (G2, d2) = (G1 ./ G2, d1 ./ d2) J *(d1 ./ d2)(F1,F2) = sup d2((F1)D1, (F2)D2)− d⊥
1 (D1,D2)
E`?@-@ ;>=?;A@B9DC E&4 fM+$ (+, *@3( J%JML>*$$L2/ , $ # J6)(%J.( +$KJ
(G1, d1)J(
(G2, d2) (+ , /( J
(G1 ./ G2, d1 ./ d2)J%$9L (+, * (3(2J%JML>&7
8:9); =J? @CBED FGBIH =?mKML cdW WMPP-U ZeSf_Ggih)SfPM`Va4cdl%[!lVgeZegiRh4P geZ R0S Ih i `Va4cd`c1R0gjh0gicdU kGo :`%[!]^P l o._VPWMcdUh0a4P-a h)SfP n1NQPCRh Z*NQcdahfcd]^cdU%[!Z _ N SVUfP lVg n/YhVgih)SfP1oVLONQPCRh Z*NQcdahfcd]^cdU%[!Z _VPCR lVg n/YhVgihjSfPCR
(G⊥1 ~
G⊥2 ,d′)
h4P-ZeZiPCR h)SfP.R0SVa ZiPCR_VPCR4RP-geUfRR4P-W g n/R0geW `VZiPCR
d′(D01,D
02) = min1; inf
∀i D0i =Di~D′
i
d1(D1,D2) + d2(D′1,D
′2)
U(P Ph
(d1 ./ d2)(F1,F2)= supd2((F1)D1, (F2)D2)− d⊥
1 (D1,D2)= sup δc(J(F1)D1,D
′1K , J(F2)D2,D
′2K)− d⊥
2 (D′1,D
′2)− d⊥
1 (D1,D2)= sup δc(JF1,D1 ~D′
1K , JF2,D2 ~D′2K)− (d⊥
1 (D1,D2) + d⊥2 (D′
1,D′2))
= sup δc(JF1,D1 ~D′1K , JF2,D2 ~D′
2K)− d′(D1 ~D′1,D2 ~D′
2)
8tZ UfPTa4PCRh4Ph)S N geUjb^c<hjSfP-aZ\[MR0SVa PCnh0geb<gjh4Y._GS0h4P-UfR4P-SVaR0SVa ZiPCR_VPCR4R4P-geUfR R4P-W g n/R0geW `VZiPCREo
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! ! !"# !$"%& '()%&*'+% !,-$"%'.$"/0-'+% #
21?@-@ ;>=?;A@B9DC E ? I14/#3 B ( J L>
(G, dG)J(
(H, dH) # J(jr (+ , *(+ @2J'6J L>*$k3($&5* $7 L 0 $ # J'$.$KJ'L$-jr %1I1KJ'$J@ $."+1 AJ%$
dG⊗H(D1 ⊗D2,D′1 ⊗D′
2) ≤ min1; dG(D1,D′1) + dH(D2,D
′2)
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+ $ &
$ ! B ! $ -'$!#" !" $ !*$ '' *% "$ I '*'+%&!$ $-'!%"! $ $ /0-'+%&*$ #
-%'Γ, P ` ∆;
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Γ, !P⊥ ` ∆;
0 !$$" '+%Γ ` ∆;
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' '+%& ! ! $ $/ -% * !*$%& %&'$ $ $ !*$ /#" ! - $&' %(*),+ $ ! B ! $.- -! !$$" '+% *%# '+%& %&-' ' $"'+% #0$2$*"/0 ! $2 ' !! *% $'+%#' %& % $ #- !,- ! '+%%/" + *%"%& '-' $ /0 !%$" / *'!% !2" - ! 0- ' )% !$*'2' ' %/"/ $ $" %& % $$" ' '#" +0%& + %%& " !$& %&-' $$" ! $ #! '#" +0%& 1"*/ ! '#0%& ! !*'!% ! !! !'#" -"
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-%'Γ, P ` ∆;
Γ ` ∆; !P⊥ #"*$" ! %&-' Γ ` ∆;P
Γ, !P⊥ ` ∆;
0 !$$" '+%Γ ` ∆;
Γ, P ` ∆;*'+%&& *%&' Γ, P, P ` ∆;
Γ, P ` ∆;8 -'$" # !<5!*$ )-' '+%& ! ! $ $"'+%="B+ ! $ - #" ! $ !.- ! -)% ' ' !*/- +
$ 0"* ' $ '+%&" * $ '+%&*!"*% %&-'$ > */ -*'!%@?*%##"*% % ! $#-'$! $ #" '%&-'$ .- '! - %/" +
> ' *%"%& &-! $" %&-' $ !4- '+%&*!"*%&0%&' $ I '*'+%&! ! $;5' )% 0 !,- $ "2$ ! !! !'#" -" #
!(A&B) = AA⊗!B
$2CB# *%�'$ !4- $"2$ !'#" 0(A&B) =
A⊗
B
$ "% '+%&$ '#" +0%! $A
*%B
$ !<5$" !*$ -"%&*'+%&$ '#"B0%& $;5
!G =G
' !!<5 % $"2$ '.- $ % -$ 'D"B+ !%$" $ !$! $ *$$*'$2$ $" !*$ (A&B)
$'+% ! -
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(+, ξ, 0)
(−, ξ.0, I)
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(−, ξ.0, J)
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$J- #-"%*'!% 0 !" %&
B*$"% ' $"$ ' ' 0'#"(A⊗
B)⊥
% !" *" !*'!% +
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%%&(%% $ -"%($"*$ -'!% %&'$ !( $A "*'#" *'!%D $$"-'+%&!! " "#!$'#" 0&%
( +(' '")# "*# $"% / ' 2' -%&'+ ,-* '+%$ "7'# - % -'+%;5 / ,' " " %&.#- %& ".*$ ,- '+%&$C5/ ' /- $,% '! $-(* /*$ % * %%&-'+%(.-" %&*'0*$ !" $% %&$(,-**'+% " '%" &1$ $ +
2 +(' ' %#" -$ ! 3 * "*%& 4 #-"%/'#" - / $ / $5!6!!798 2
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2 + %#" ! Q9 * "*%&9 % $;- " ' ' *.#- &0-"%+ '9 *"% 0'"#" '%' $ )' '+%*$ +RE $"% -$$!K ( $ %&"*%' 9* % -' $0-$$" $& K*$$" ' $/ "* -'+%H $%#"*$ *$(-'#@? %!*$.*%H$" $ #- $" *%' N %%&*'+%H#" ' " " '+%&*$ )'*'!%*$ +S' ' $$"%$" $ %F (-$$"!*$T*$$" ' $ %&$5 ! '*& $"* " %&%' $>.- '9 ''#"#-' $B !&-' *$"%6"'#" +
%+(' ' - %!*$"%&' '#0%& *$"%- " $U "*%/ %$"HH Q= ' %&'Proba
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& 8ed 7 <) 1$"'wx "!:#&%('*) 51h Wiyjp!"m1!$2:9[9[%C!:0 5*% > -$M!#1k3z!$p!5!:|! +k~ $"%&*%'= > ~ + Y J 5 @8> \ VIO_@8JyVIJ;Fk`?@ \] UbO_VIO_@8J 5 (C % v 5 2 %+
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& c 2 ) '" -MwN9[l) % > " $6e7*|K9[0 +T '+%& < %Ai $ '; -' "*%&# % $"% %&" $ + M RDE@8>ADPO_BGVIX1`?@ \] UbOQDP> BP_DPJ BED 5 2 % 2I *2 ( 5 ( c 2 +& 8 h v ) > 0%&"Ex!$|Kbp.: 5 '* '+% &!$0$.:% 5 i -$1¡-¢**¢!$*25 *% -" '+%£&9[m¤0$|K9[ +
*/ % '0%65 N2- $ < 1*$F)$$ $ ) = -$$" I' -5 < + 8 -'$M¥ DPXtOK S JRJRUWVIX Y:7 7 7 [Z8\] @ L KU \ @8J^@ _B?KJ`?@ \] UpOQDP> BP_DPJ BED 5 # < $ c v +E !! * %&* ) % ? > *$$;5 ( I v +& 8 h v ) > 0%&"Ex!$|Kbp.: 5 '* '+% &!$0$.:% 5 i -$1¡-¢**¢!$*25 *% -" '+%£&9[m¤0$|K9[ +
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& > » <) > * $67**|K9[0 5.'N'-p.:'1|K0 5 %'@? '' |K0$%('19[ + $ %&"*%*$;5*-'$ -'Q' -> < + 8 -'$&=©>A@C@¦¤^VIJ UV D©VIJ;F Y JROQDP>GVpBPO_@8J LEL V Z(LKJ@8J @8Ub>@¦ @ r KJ ª KXtJ DP> + 7 EQ " $"$;5 2 +
& AIc ) > * $67*|K9;0 + 8 $ %&*%?±i %&* $ +ª VIO O>qURBPOUp>AD L `?@ \] UpO BP 5c % ( 5 ( Ic +
& + ) > * $N67**|K9[0 +4E '+%)*%'.%D' -/ < -'*$;5)#-"% + -'!$ " %;5 2 +& * ) > * $ 6e7*|K9[0 + E '!%) %&-' %C' -O < -'K $C5+#- % + -'!$ " %;5 2 +& 8 -'R ) ' *'+%e!:0*.1% + # # $ -' %$(? '# 2 < E % -$ " $%&$ +> *'!% v 5O!3 < !>7 0%Ai#" %&!<5 ' '/- $"% ? = > -"$: EME 5S= "*/
( %+& 8 -'R ) ' *'+% &!$0$.:% + ^V^@ URDk^$KJ «GVIK>AD S-]] XtUR«ED X «POUFpDfFpD/IDP> L =?>E@8¨
BED LEL U L FpD@8> \ VIXt L VIO_@8J µ =©>qKJ BP ] VIXD \ DPJRONFIUλ¨q` VIXBPUpX ¹ +¯ i# $".K)*%&-&0%;5
' ' /*$"% ? = > 0$ EME 5S "' ( %+& 8 *2 ) ' *'+%e!$0$.1% *%h $$"©,/!$*#&9[ + > #$ %& < 1*$$-'+%& $ + S ` ª
M >VIJ L VpBPO_@8J L @8J`?@ \] UpO_VIO_@8J;VIXg^@ _B 5 2 2 +& 8 h <) ' *'+%e!$0$.1% 5" < f,9[ 9[p|K05 % - '+%£&9;m¤0*|K9[ + '2 $$"-'+%& $
-' $ %&*%(Ei' $ + 8 -'$e^@ _B/KJa`?@ \] UbOQDP> BP_DPJ BED 5 ( I %+& 8 d ) ' *'+%se!:0*.1% 5$d- -' ¦ -)%&$ %&¯jb.1|K0*9R 5 *%x -"*;5"¢*9[bp|K0 +s i $ %&
%& ,= ! )' '+%&$#% ' ' /*' < %Gi#? '#1 $ 1=L' -B < 2 = $ + 8 -'$' +" .:"p.15 8 +# 9[|%85"¢ 5 % 8+ 347*0*2:|K5"|45 " %&*$C5=©>A@CBEDEDGFIKJ 8L @¦OKRD M pK>F¶ Up>PO%$'&FpDPXx`?@8XKX@UbKU \ @8J `?@ \] UbO_VIO_@8J;VIX^@ _BVIJ;F4=©>A@C@¦ M RDE@8> Z 5 # < $( ((v ( 5' #5 )( Eifh <5#- )% ( * +R ' < *,+¬ *M < -e */.c +
& 8 h ) ' *'+%e!$0$.1% *% -" '+%£/9[m¤0*|K9; + > =0+ ' *%&$ 0'4 *"%$"# + 8 -'$d + +21 +3" |K"!:2541 +3# !45l.10 5 *% + £/9[m¤0*|K9; 5 "% "$;5 S F/8VIJ BED L KJ ^:KJ DGVI>^@6I_B 5 # < $7*8 v * 2 c + # < ' ' /*$"% ? > *$$C5 ( :9 + > "I* ' < $ =%Gi» G!$Gi -' ' - < :5 E %Ai# %5 - ° 1 5;d-' ( * +
&<;¤iA:9 ) + ¡-¢$¢!:2 + B? N i**' *$3% 8 $"%' < ?$ % O)=I' - < +<8 -'$d + +21 +3" |K"!:2541 +3# !45l.10 5 *% + £/9[m¤0*|K9; 5 "% "$;5 S F/8VIJ BED L KJ ^:KJ DGVI>^@6I_B 5 # < $/c* ( 8c + # < ' ' /*$"% ? > " $$C5 ( :9 + > "I* ' < $1= %Gi» G!$Gi -' ' - < :5 E %Ai# %5 - ° 1 5;d-' ( * +
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&<;¤i8/. ) i*-$?¡-¢*¢*!$2 +R~ '% '*$$ $"#* $ +"ª VIORD \ VIO_BGVIX O>PURBPOUb>AD L KJf`?@ \] UpO¦DP> BP_DPJ BED 5 2 88/. + & < ' '/- $"% ? > *$$ +; - - +
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& ~ < 8 2 ) -[!"m¤m¤|!$0 + J OKRDxF Z J;V \ _B L @¦/XtUWFI_B L ?V L OUF Z @¦/KJROQDP>VpBPO_@8J +: i# $")*%& %65 E '$ %&%& %(C7 %Ai#" 0%!*$B ' ?:5 2 88 21+
& ~1~98 A 8 2 ) -[!m¤m¤|!:0 5#7 -C+¦h '%" ?p9[7$)Ex.10*0!:2$|K9;7 5 *% 7"? " x7*!|K0*| + 8 ' '+%)*%' %& '= -M % ? ' *$ +28 -'$ i*-$s¡-¢**¢!$*25d- -' +21 /- $ " |K"!$2 5 > -e£N7*9[ 5 *% > i ;5".$" 59" %&*$C5-^:KJ DGVI>§^@6I_B¯KJ`?@ \] UpO¦DP> BP_DPJ BED + 2 88 2 +
& ~ '8 2 ) -R!"m¤m¤|!:0 *%L7 -"%',e)1p!$0$2 +18 $" < '$C5$)%& $ -' $"%&0%& < *$ + 8 -'$`?@ \] UpO¦DP> BP_DPJ BED^*@6I_B 5 2 88 2 +& ~ A 8/. ) 7 0!+¦h '#"*?b9[7*l) %7H?)" 7!W|K0*| + ~ $ %(M* ' $ + ª VI¨
OKRD \ VIO_BGVIX[O>qU[BPOUb>AD L KJ `?@ \] UpO¦DP>BP_DPJ BED 5 ( . % ( c 2 ( *%5 2 88 . + & < ' '/- $"% ? > " $$ +
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8 0'$ + N.10*.:". 5¤h + ?9[ . 5 + !:p9[0*|K0*|45 *%-8 + !:0!$*2$. 59"% "$;5^@6I_Bs`?@8XKX@UpKU \ ¸¸ 5# < *$ 2*2 ( 2³ 8 + - "%Gi +¦ *-'.5 ( cI +
& 'Gc ) d- -' +21 /- $ " |K"!:2 +& ° 0M$2 < ** *%& ? 1= '+%& *%' +8 -'$d +;» + " "!b) %$8 + ;5"9[2:g. 5 " %&*$C5©` VIOQD26p@8>q_D L KJ´`?@ \] UpOQDP>BP_DPJ BEDVIJ;FT^@6I_B 5 < $ ( 8c + 8 " -' 7 0%Gi0% 1 ) *% ?:5 ( cI + > "I* ' < $.= %Ai8$7 +E 7 + IE 8$7 d-'+% K Nh $ -MEi 'I= *' <5d-' ( . 2 8 5 ( c v 5;*;5 *&1 '+% -N?H7 0%Ai 0%& $- *e 2 +
& 'G8 ) d- -' +21 /- $ " |K"!$2 + ' * *%N? ='+%& *%' ENE % 8 IE+ = " < %Ai $ +8 0'$ > + 3z!$b|K0 # 5 % ' + 34|K0*% 5" %&*$C5®` ^ $-¨ ¸¸ 5 # < $ vI /* + ' < +¬ N < - . ((v 5 ( I8 +& 'G ( ) d- -' +21 /- $ " |K"!$2 + 8 ' ° *'$ %&*%&/0 < #% -$"$ < +©ª VIOKRD \ VIO_BGVIX
[O>qU[BPOUb>AD L KJa`?@ \] UpO¦DP> BP_DPJ BED 5 ( * % 2 99 2 5 ( (:+& 'G ( ) d- -' +21 /- $ " |K!$2 + A #-'+%#*$ '' -T < EME + 8 -' $ ' + 6.1%(|*% ' + [!$# g|K05 "% "$;5 eUR@ C ] >A@ r XD \ FpDPXKXVkX@6I_BGVfD§FpDPXKXV! X@ L @"!?VFpDPXKXV L BP_DPJ$#CV:5/ +
EME 5[* < '# 5 E % 1? 5 ( ( + ' ; +& 'G:9- ) d- -' +21 /- $ " |K"!:2 + ' * *%N? = '+%& *%' EMENE % i < ' &1! 0$ + 8 -'$
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& 'G:91 ) d- -' +21 /- $ " |K"!$2 + = '" '0% 0%'#1 *% '*$$ + 7 -'!$N " %;5 ( :9 +& 'G ) d- -' +21 /- $ " |K!$2 + i " '+%9#-'#Ei $ *$;% *'+%&' $H ' -% 0%'
$"-'+%&*$ + ' $Gi " -'!$N " %;5 ( +& 'G ) d- -' +21 /- $ " |K"!:2 + ' $"%&"*%/ %$".% /*$ ' ##%$"0 '$"% + '$i
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