Capítulo de muestra

i
i
i
i
"! ' H {> 2LNHOPABG947ABNHAI>{2TH8-:<; ABGXa >@OPX<; CPAI4J;mCE8FW@OPAI4J; 258 OE87X>@GMC5; G ; :I>{2; 9>{2LCPOE87ABGMC5; TH87:
2EA7I{:I> 6 132EC5;V47AI87G947A<;V258.9x;TH8F25; OPOE>@:B:<;@TH> I{O5;@47A<;@2t;:<;4F>@G96987G947A<; TH84J; X*Nl>{2
87G ; Nx; OPAI87G947A<; X6f
E T\Ala 87OE87GŽCE8F2:<; ABGMRS8F2PCPA7ŽI ;@47AIW@G 2E>>NHOE8V:I>{2<a 6HG9T9; X87GŽCE>{2TH8V:<;@2
Z[; CE87X<; CPAI4J;@2 ?:<;LCE8F>@O <;VTH8Z ;>M= 6HABGx;@2 r:<; H ABGI I2PCPAI4J; 67676 K 8CPO5; C5;rNl>@OC5; GŽCE> vTH8
6HGx;Y47AI87G947A<;*X6H:BCPAIT\AI2E47ABNH:BABGx; O )M= 698$2E8; Nl>ZŽE ;x2E>>HN OE8(CE> TH> 987G87:998F4 99>YABG98F2PNl87O5;@TH>
TH8 M= 698:I>{2 X*AI2PX>{2Sa 87G9W@X87G9>{2$;@47C x; G E 2PABOPRS87G[TH8ja 6HG9T9; X87GMCE>87G ; OE8J;@2:=M698 #;
NHOPABX87O5;RMAI2EC5;\Nx; OE8F4F87GVCE>@C5; :BX87GŽCE8tTH8F2E4F>@G98F47C5;@T9;@2 87GMCPOE82 6

€‘ŽŽ’‡1–D$€‘‡1‚
!"#"
$&%('*)&+#,.-/'1032546)(-7"+#498:+;03-/,.<0>=@?+A4
“ Ž•‰‚–
’‡1Œq‚L–
H ;ABGMRS8F2PCPA7IŽ;@47AIW@G 2E>>NHOE8$:I>{2 a 6HG9T9; X87GMCE>{2(TH8$:<;@2 Z; CE87X<; CPAI4J;@24F>@X87GhJWL; Q9Gx; :I8F2
TH87: 2PA7{I :I>B _ 6 >@CPCP:I>>N A OE8,IS8 5
fPf k 5
U$B$L a 698c6HG9> TH83:I>{2I{O5; G9TH8F2?XY; CE87X<; CPAI4F>{2oTH8
Q9Gx; :I8F2oTH8a8F2E8a2PA7{I :I> 6A 6HG9T9;@TH>@O mJ 6HGŽCE>4F>@G 8F>@O IS8 8 > >@:I8 TH8":<; :IW>I{AI4J; 2EABX@NW@:BAI4J; 258
:I84F>@G92PAITH87O5;87:lNx;@T\OE8(TH87:#4
; :I476H:I>*TH8 NHOE>@Nl>{2PAI47AI>@G98F2 TH87:#4F>@G94F87NHCE>*TH8 NHOE8FT\AI4J;@TH> E
TH8:<; ABGMCPOE>MT\694F47AIW@GTH8:I>{23TH>{23476x; GMCPAlxQ 4J;@TH>@OE8F2 S6HGHABRS87OE25; : ∀ DCFEHGIEKJMLON>LE8 AI2PCE87GXk
47A<; : ∃ PCEHGQEREHSUTHVXWL 6 1cG2E692 H:BCPABX>{2; 9>{2TH8 ;@47CPABRMAIT9;@T A OE8,SI 8iABGMCE87GŽCEW4F>@G92PCPOP6HABO
6HGL2PAI2PCE87XY;$:IW>{I AI4F>4F>@X*NH:I87CE> 87:YCGQLZTHGQEH[RERS\LZTH]_^`]bacJME&Ž= 698OE8FT\6>PJ 87O5;$:<;t; OPABCPXD7CPAI4J;;
:<;:IW>{I AI4J;#TH8FT\6947AI87G9TH>CE> TH>{2(2P692 CE8F>@OE87XY;@2;Nx; OPCPABO$TH86HG 4F>@G>J 6HGMCE>TH8; AI>@XY;@2
>RS87OET9;@TH8F2$87R AITH87GŽCE8F2X8FT\A<; GŽCE8*:<;L; NH:BAI4J;@47AIW@G TH8t6HG[G
HX87OE>OE8F2ECPOPABG{I AITH>TH8OE8Rk
{I :<;@2aTH8†NHOE> T\694F47AIW@G M4F>@X>:<;ABG9T\694F47AIW@GNl87O a 8F47C5;?*
E 87:1[dLONHVea&CLHW;fcWYac47: @; 2EAI4F> 6 132PCE8
NHOE>>{I O5; XY;2E8iTH8F2 ; OPOE>@:B:<;$87G2E692"TH>{2">>HN O5;@2hgi]MfkjlGcVhWNFSmEDThfcWnNefcGoGp]qJsre[f`Jq]bt A 6HGXk
T9; X87GMCE>{2YTH8:<;[UOPABCPXD7CPAI4J; 5
f$fP iu
E jlGcVhWNOTefpavfvJxwOfyNefcGoGc]sJsre[f`Jq]bt H ;@2*:I8,SE 8F2
NT@; 2EAI4J;@2iTH8(:<;UOPABCPXD7CPAI4J; 5
f$U$G E 5
U$c$G 6
H ; >>HN O5;TH8 A OE8,SI 8†;HN OPAIW87:\4J; X*ABG9> x9 ;@47A<; :<; :IW>{I AI4J;iXY; CE87X<; CPAI4J;(;@47CP6x; :\8"ABG96SE W
N> TH87OE>{25; X87GŽCE8 87GXY; CE87X<; CPAI4F>{2]E@9Q :IW{25>a >{24F>@X> 8 87OPCPO5; G9T d 692E2E87:B: 5
fO>B k 5
UO>c i
i
i
i
i
i
i
B
U:la OE8FT >@OPC 9 9HABCE8,998J;@T 5
fX5 k 5
UP'O E H 69T †A7IŒABCPC IS87G92PCE87ABG 5
f$f$U k 5
U$LX5 {N87OE>
G9> a 698c4F>@X*NHOE87G9T\AIT9;iNl>@Or2E692/4F>@GŽCE87X*N>@Om; G98F>{2JOE8F47A7NHAIW47O BCPAI4J;@2/G98,IŽ; CPABR{;@2rTH8 8F>@O I
; GMCE>@O 5
fPL k 5
UX5
f ETH8 AB692587NHNl8†=v8J; G9> 5
f$L$f k 5
U$G$B ŽX*AI87GŽCPO5;@2:<;Q GHABRS87OE2PAIT9;@T
TH8 87Gx;2E8"G9,8 ISW(;iOE8F4F>@G9> 4F87O32P6tCPO5; Nx; J S> E$:I8G98,ISW:<; T\AI2PCPABG947AIW@G.=M69825>@: <; 4F>@G94F8FTH87O
;t2P692aNHOE> a 8F2E>@OE8F2 ; :#476HX*NH:BABO†:I>{2†2E8F2E87GMC5;; 9>{2 9;@T\6947AI87G9TH.
> =Ž698 acVEe^vJq]O ]qN>EeN^ZE3GQfQ^pf
Nhf ]sW/J"fcGp alCE3GIES\E1 WY]P fcGpap]_N>E>N 6
KMABGL87j
X Nx; O IS> \87GL87:#; 9> 5
U$c$c :I>{2CPO5; Nx; JP>{2†TH8 A OE,8 IS8 9x; N <; GLOE8F477A NHAITH>87:; N
> ES>
ABX*N>@OPC5; GŽCE8TH8 M ;JRMAIT AB7: N87OPC 5
f$B k 5
UPG =Ž6HAI87Gm87G 8F25; a 8Fm4 9x;V87GŽ6HG947AIW o87G 6HG
; OPC I476H:I>$NHOE8F2E87GŽC5;@TH>$87G*87: a>@G I{OE8F2E>(Z[; CE87<
X ; CPAI4F> _ GŽCE87OPGx;@47AI>@Gx; :xTH8†=o; O I2 AB7: N87OPC
5
U$c$c B$G NHOE>> NH:I87XY;@2"Nl87G9T\AI87GMCE8F23TH8aOE8F2E>@:B6947AIW@<G =M698 9x; G*2E87OPR AITH>(TH8aABG94F87GŽCPABRS>(Nx; O5;
:<;ABGMRS8F2PCP7A IŽ;@47AIW@G XY; CE87<
X ; CPAI4J; TH87:32P7A I{:I> 6 Q G9>LTH8*87:B:I>{2$87O5;L87:"l! W/Jsa`^crFfc]qN3VhWhTHa$#
C;GIL&`% S\fc[ 87:rNHOE>> NH:I87XY;TH8$:<;YTH8F47AI2PAIW@G T=M6984F>@G92PAI2ECE887G-87:?TH8F2E476 NHOPABX*AI87GŽCE>TH8(6HG
XD7CE> TH.
> IS87G987O5; :/Nx; O5;TH8F47AIT\ABO2PA#6HGxj
; a W@OPX6H:<;:I>W I{AI4J;*8F2†RS87OET9;@TH87O5;i
> ad; :I25; 6
=?> 4F>TH8F2ENH69DF2 :<;>> NHO5;TH8 A OE,8 IS8OE8F477A NHAIW 6H@
G IS>@:BN8TH8 I{O5;@47A<; 87G 5
U$c$B 8 87OPCPO5; G9T
d 692E2E87:B:\TH8F2E476 NHOPAI?W =M698:<; CE8F>@O <;TH84F>@>G J 6HGŽCE>{2 @C5; :H4F>@X>:<; a >@OPX6H:<; Nx; A OE,8 IS8 {T9; Nx;
:B6 IŽ; O;*6HGx;Nx; O5;@TH> JP; ‡87:?4F>@>G J 6HGŽCE>TH8$CE> TH>{2i:I>{2 4F>@>G J 6HGŽCE>{:
2 =M698(G9>N87OPCE87G98F4F87G ;
2 HX*AI2PX>{23G9>(NH698FTH8 Nl87OPCE87G98F4F87OGHAxTH8 JE; O"TH8†N87OPCE87G98F4F87O;(2 9X*AI2EX> 6A OE,8 IS8iABGMCE87GŽCEW
2EABGyD ABCE> 87:BABX*ABGx; OP:<; &Em;Nx; OPCPABOYTH8 5
U$c$G 4J;@2EAiTH8 J W TH8LNH6 NH:BAI4J; O 6 =v>@OY2P6‰Nx; OPCE8
d 692E2E87:B: 94F>@G; E 69T9;TH8 9HABCE,8 998J;@T xABGMCE87GŽCEW4F>@GŽCPABGM6x; O 87:/CPO5; Nx; JP>YTH8 A OE,8 IS825>> NHOE8
:<
; a >@OPXY; :B7A h ;@47AIW@G :I>W I{AI4J; TH8:<;m; OPABCPXD7CPAI4J; AITH8FWm:<; CE8F>@O <;mTH8CPABNl>{2Nx; O5; 87RMABC5; O
:<; Nx; O5;@TH> JP
; ENH6 NH:BAI4FW :<; >> NHO5;(' Gc]bW^v] C]_E [RE3Jsrfc[RE3Jq]_^ZE H >{2YNHOPABG947ABNHAI>{2TH8V:<;@2
XY; CE87<
X ; CPAI4J;@2 5
UX5
c k 5
UX5
G 6
KMABG 87@
X Nx; O IS> EL4F>@X>476HOPAI>{25;tAB:B692ECPO5;@47AIW@G TH87:/OER8 a mO ; GV4J;@2PCE87:B:<; G9>TH?
8 =M698*`) Vh]MfcW
Ere]MfcGpGQL6[RE3JME,+ E re]MfcGcGILn[KVFf`GQf /C5; @X NHAID7G :<;L>> NHO5;LTH8 d 692E2E87:B:c2E8tR ABG9>V; Nx; J > 6 1cG
5
U$GX5 6HOPC .- TH87: 5
U$c k 5
UO>f oNH6 NH:BAI4FW$2P6 ad; X>{2E>$; OPC I476H:I>l/ W.v0 LHGp[REHSsS2i1 VhWNhfQ^`]_N3#
E4c% S\fC;GILpCFLPap]qJq]qL3WFai]bW5' Gc]bW^v] C]_E*y
6 E3Jsrfc[RE3Jq]_^ZE EHWNGZfcSmE>J"fQN a$P1 a`J"fc[ka .- TH87: 5
U$GX57
=M698iNH698FTH84F>@G92PAITH87O5; OE2E8 4F>@X>$6HGx;TH8i:<;@2"OE8J; :B7A h ;@47AI>@G98F2XY; CE87<X ; CPAI4J;@2<X ;@2"ABX*Nl>@O k
C5; GMCE8F2TH87:a2P7A I{:I
> 6 1cG 2 BGMCE8F2PAI2 /87:CE8F>@OE87XY;VTH8 - TH87:aT\AI4F8*:I> 2P7A I{6HAI87GMCE8 .8 LDN>E
0vLHGc[KVhS\E>^`]:H9 W E<3; ]qLH[*3= Jq]_^ZE6^ZLHWFac]xa`J"fcW/J"fRNhfKS\EJ"fQLHG7q> E6NefKWh? [fcGQLPaR^ZLHW/Jq]MfcW;f CGILcCLPap]#
^v]qLHWAfpaB]bWNhfQ^`]_NH]:`% S\fpa 6 132cTH8F47ABO {476x; : =M6HAI87OvCE8F>@O <; XY; CE87<X ; CPAI4J?; 9x;TH82E87OoABG94F>@X*NH:I87C5; 6
KMAI87X*NHOE8 9x; NHmO ;-87Gm87:B:<;-; Q9OPXY;@47AI>@G98F2 CE8F>@OE87XY;@2 476 EŽ;VRS87OET9;@Tm> a ; :I2E8FT9;@T G9>-258
NH698FTH8TH87X>{2PCPO5; O 6 1c:@l! W/Jsa`^crFfc]qN3VhWhTHaMCGQL&c% S fc[ TH8 AB7: N87OPC†G9>*CPAI87G98$2E>@:B6947AIW@G 6
1cG 5
U$GO ‡87:vXY; CE87<
X ; CPAI4F>LABG I{:IDF2U :<; G[Z; C 9HAI25>@G `r6HOPABG I 5
UX5
B k 5
U$L$G NH6 NH:BAI4FW
>@CPOE>; OPC I476H:I.
> ad; X>{2E> 2E>> NHOE8:I>{i
2 WAX? [fcGILHak^ZEHS\^`VXSmE&`% S\fpa S=Ž698$TH8F2 ; OPOE>@:B:IWY87:rCE8F>@OE87XY;
TH8 - TH87: E =M698NH698FTH8*4F>@G92PAITH87O5; OE2E8Y87:o>@OP7A IS87G > Qx47A<; :3TH8:<;ABXG a >@OP<
X ; CPAI4J;LCE8FW@OPAI4J; 6
13GfT\AI4 99> ; OPC I476H:I> ABGŽCPOE> T\>6 J > :<
; [B=&`) Vh]sWENefY8 VXGp]sWhT o6HGx; 87GMCPAIT9;@TyXY; CE87<
X ; CPAI4J;
; N92ECPO5;@47C5j; =M698 a >@OPXY; :B7A hJWNl>@OaNHOPABX87O5;RS,8 h 87:4F>@G94F87NHCE>*TH8; 7: IS>@OPABCPXj> EYOE8F2P6H:BCEW2587O
NHOE8F476HOE25>@O5;*TH8:<;@2<
X ;>=Ž6HABGx;@2 TH8(4J; :I476H:<; O; 6HCE>@<
X ; CPAI4J;@2 V=M6984F>@X87G h ; OE>@G;t8 CE87XG k
TH87OE258i; Nx; OPCPABO"TH8:<;THDF4J;@T9;(2P7A I{6HAI87GMCE8 6 1c:HCE8F>@OE87XY;(TH8 `r6HOPABG I =M698†87G47AI87OPCE>X>MTH>
i
i
i
i
i
i
i
i
i
258OE8FT\694F8;Y6HGx;<a >@OPX6H:<;@47AIW@G 8=Ž6HABR{; :I87GŽCE8; :vCE8F>@OE87XY;LTH8 .- TH87: lTH87X698F2PCPO5; =M698
8 \AI2PCE87GmNHOE>> NH:I87XY;@\
2 =M698YG9>2E8NH698FTH87GmOE8F25>@:BRS87O 6 1"2tTH8F47ABO vGHABGI{6HGx;X<;>=M6HABGx;TH8
`r6HOPABG I E87G4F>@G92E8F476987G947A<; GHABG I HG>@OETH87Gx;@TH>@O "8F2"4J; Nx;hTH8 >>NHCE87G987Oa2P62E>@:B6947AIW@G 6
13G4F>@GMCPO5; Nl>{2EAI47AIW@G oCE>MTH>{2t:I>{2tNHOE>> NH:I87XY;@.
2 =Ž698NH698FTH87GOE8F2E>@:BRS87OE2E8X8FT\A<; GŽCE8L6HGx;
X<;>=M6HABGx;TH@
8 `/6HOPABG IL2E8$TH87G9>@X*ABGx; n
G CGIL4c% S\f`[dEHak^ZLH[CVJME&c% S fpa k
> CGQL&c% S fc[REPakGQfQ^vVhGpap]#
P EH[fcW/J"f fcWYVX[Rf`GIE&`% S\fpa 6 =v>@O$8F2PC5;O5; hJW@G /K
; `r6HOPABG I 2E8:I84F>@G92PAITH87O5;L87:cNx;@T\OE8TH8:<;
J"fZLHG$_> E Nef SmE ^ZLH[CVJME&c% ]sS ]qN>EeN 6
H ;<X ;>=M6HABGx;(TH8 `/6HOPABG I$NH698FTH8 4F>@G92PAITH87O5; OE2E8i4F>@X>$87:HX> TH87:I>(AITH8J; :9TH8†6HGx; <X ;k
=M6HABGx;TH>@C5;@T9;*TH86HGx;X87X>@OPA<;*ABXG Q9GHABC5; 94F>@G ;@4F4F8F2E>G9>8F2PCPOPAI47C5; X87GMCE82E8F476987G947A<; :
8F2*TH8F47ABO v:<;-<
X ;>=M6HABGx;-8F2*4J; Nx; hTH8L4J; @
X NHA<; OTH8T\ABOE8F4F47AIW@GyT\6HO5; GMCE8L:<;-:I8F47CP6HO5;[>
8F2547OPABCP6HO5;-TH8:<;-X87X>@OPA<; 6 132>> NŽR AID
> =Ž6982P6mOE8J; :B7A h ;@47AIW@G‰NHmO ;@47CPAI4J;-8F2EmC ;ba 6987O5;TH8
GM698F2PCPOE> ; :I4J; G94F8 G9>8F2(N>{2P7A NH:I84F>@G92PCPOP6HABO$<
X ;>=M6HABGx;@2$4F>@G X87X>@OPA<;ABXG Q9GHABC5; 6 KMABG
87j
X Nx; O IS> @:I>{2?>@OETH87Gx;@TH>@OE8F23;@47CP6x; :I8F2oCPAI87G987GtC5; GŽC5;@2oN>{2P7A NHAB:BAIT9;@TH8F2oTH&8 a 6HG947AI>@Gx; X*AI87XG k
CE>T\lA a 87OE87GŽCE8F2 E X87X>@OPA<;@2C5; G I{O5; G9TH8F2 =Ž698:<;V<
X ;>=M6HABGx;-THK
8 `r6HOPABG I OE8F2P6H:BC5; 2587O
6HGX> TH87:I>Y;@TH8F476x;@TH>TH8(2P6 ;@47CP6x;@47AIW@G ‡; 6HVG =M698 98F2PCPOPAI47C5; X87GMCE1
8 9x; NH:<; G9TH> x87:#RS87O k
T9;@TH87OE>X>MTH87:I>*TH86HGV>@OETH87Gx;@TH>@O 8F2 4F>@X>2E8 RS87mO ; 87:r; 6HCEW@XY; C5;:BABG98J; :r;@4F>@C5;@TH>
6HGx;*<
X ;>=Ž6HABGx;; 7: IS><
X ;@2†OE8F2PCPOPABG I{AIT9\
; =M698:<;TH1
8 `r6HOPABG I 6
!"#
8:<%(="'(+#464P-/?F%(-/'(?F="+A7"-/4
u+#%&01, +;03+#4
13:32E8,I{6HG9TH>8F2E4J; :IW@G TH88F2PC5; 9HAI2PCE>@OPA<;L258NHOE> T\6>JP>87G 6HG 4J; X*Nl>LCE>@C5; :BX87GMCE8YT\Ala 8Rk
OE87GMCE8 x:<; _ GIS87GHAI87O <;1c:IDF47CPOPAI4J; 6 1cG 5
U$G$f x87:?XY; CE87X<; CPAI4F>G9>@OPCE8J; X87OPAI4J; G9> :<; 69TH8
13: > >MTVK'9x; GHG9>@G 5
UX5 k B$c$cX5 NH6NH:BAI4FW87:; OPC I476H:I>o1H[ %QLHS ]q^ oWEHS21 ac]xaL 0 fcSmE 1
E3WN ]qJM^cre]sWhT ]bGQ^`VX]sJsa 'K 9x; GHG9>@G 5
U$G$f S87G87:)=M698; NH:BAI4FW:<; H W>I{AI4J;Z; CE87X<; CPAI4J;
Nx; O5;87:/; TG ; :BAI2PAI2†TH8 :I>{2†47ABOE476HABCE>{2i4F>@j
X NHABGx; CE>@OPAI>{F2 E2E8F476987G947A<; :I8F2 =M698N>@O87GŽCE>@G94F8F2
258*4F>@G92PCPOP6 <; G 4F>@G OE87:IDF?
2 E ;LNx; OPCPABOTH8:<;VTHDF4J;@T9;2P7A I{6HAI87GŽCE8YNx;@25; OE>@G ;LOE8J; :B7A h ; OE2E8
X8FT\A<; GMCE8T\AI2PN>{2PABCPABRS>{2i87:I8F47CPOEW@GHAI4F>{2 HNHOPABX87OE>4F>@GV$R ; :BRM6H:<;@2 TH8R@;@4 I> NHOPABX87O5; ISR8 k
G987O5;@47AIW@G ?TH8F2PNH69DF24F>@G CPO5; G92PAI2PCE>@OE8F2 25,8 I{6HG9T9; IS87G987O5;@47AIW@G Q9Gx; :BX87GMCE8Y4F>@G 47ABO k
476HABCE>{2"ABGŽCE,8 I{O5;@TH>{2 CE87OE4F87O5?
; IS87G987O5;@47AIW@G 6 U†TH87<
X ;@2TH8 8F2PC5;$; N>@OPC5;@47AIW@G 'K 9x; GHG9>@G8F2
C5; j
X NHAID7e
G ad; X>{2E>4F>@X>47OE8J;@TH>@O TH8:<;tCE8F>@O <;tXY; CE87<
X ; CPAI4J;YTH8:<;4F>@X6HGHAI4J;@47AIW@G lTH8
I{O5; G ABX*Nl>@OPC5; G947A<;Y87GV87:r4J; X*Nl>*TH8$:<; _ G IS87GHAI87O <;YTH?8 `r87:I8F4F>@X6HGHAI4J;@47AIW@G 6
Uy:I> :<; O IS>TH8:<;@2cTHDF4J;@T9;@2"2P7A I{6HAI87GŽCE8F2 @:<;@2oAITH8J;@2cTH8i'K 9x; GHG9>@G2E>> NHOE8a:I>{2c47ABOE476HABCE>{2
258F476987G947A<; :I8F2 2E8(TH8F25; OPOE>@:B:<; OE>@G4F>@G92PAITH87O5; NH:I87X87GŽCE8 lT9; G9TH>:B6 IŽ; O ;t6HGx;CE8F>@O <i
; a >@O k
XY; :B7A h ;@T9;fTH8[:<;@2<
X ;>=M6HABGx;@2 2E8F476987G947A<; :I8F2 NHOPABX87OE> FE…TH8 :I>{2 ; 6HCEW@XY; C5;@b
2 Q9GHABCE>{2
TH8F2ENH69DF2 6 132PC5;@2<
X ;>=M6HABGx;@2†2E>@GLX694 99><
X ;@2†2PABX*NH:I8F2 =Ž698:<;@2†TH:
8 `/6HOPABG I 6M 8 998F4 99>
25>@G:<;@2a<
X ;>=M6HABGx;@2<
X ;@2a2587G947AB:B:<;@2†4F>@G:<;@2 =M698 G9>{2aR{; X>{2i;87XG a OE87GŽC5; O ;:I>:<; O IS>*TH8
8F2ECE8:B7A NHOE> 6 KMABG 87j
X Nx; O IS> /;YN8F25; O(TH82P692 :BABX*ABC5;@47AI>@G98F2 NH698FTH87G ; NH:BAI4J; OE2E8*87G[XX6 k
m4 99>{2†4J; X*Nl>{2T\lA a 87OE87GŽCE8F2 6 ;2ES8 9x;$X87G947AI>@Gx;@TH>:<;CE8F>@O <;tTH8 :<;4F>@X6HGHAI4J;@47AIW@eG E87:
; TG ; :BAI2EAI2TH8 47ABOE476HABCE>{287:I8F47CPOEW@GHAI4F>{2 6 =c698FTH87GL; x;@T\ABOE258>@CPOE>{2 MC5; GT\lA a 87OE87GŽCE8F2†87GMCPOE82
4F>@X> :<;CE8F>@O <; TH87:H4F>@GŽCPOE>@: @:<; 4F>@G92PCPOP694F47AIW@GTH84F>@X*NHAB:<;@TH>@OE8F238ABGŽCED7OPNHOE87CE8F2 Ul: a >@XG k
i
i
i
i
i
i
i
P
i
258F4J;nf`JlEHS B$c$c 987:?T\AI2E89>TH8(>@OETH87Gx;@TH>@OE8F2 ‡:<;2PABX6H:<;@47AIW@G[TH87:r2PAI2ECE87XY;tG987OPRMAI>{2E>
>-:<; :BABG I I2PCPAI4J; 6gW N92ED7OPRS8F2E8 =Ž69887G CE> TH>{28F2PCE>{24J; X*Nl>{22E8YXY; G98 JP; G‰4F>@G94F87NHCE>{2
4F>@X> ^ZLHW/JqGQLHS E>^ ^`] 93W F[fc[RLHGc]qE Ei=M698†:I>{23>>NXJP87CE>{2"2E>@X87CPAITH>{2;4F>@GŽCPOE>@: 2E>>NHOE8†:I>{2
=M6982E8(;@47C x;.> =Ž6982E8 ABGMCPOE>MT\694F87G 87GL:<;tX87X>@OPA<; H2E>@GV8F2587G947A<; :BX87GŽCE8$2 BXjNl>@:I>{2 >
4J;@TH87Gx;@2TH8$2 Bj
X Nl>@:I>{2 6
a>@GV:<;@2†<
X ;>=M6HABGx;@2 TH1
8 `r6HOPABG I<E:I>{2 ; 6HCEW@XY; C5;@F2 Q9GHABCE>{22E8(T\AI2PN>@G <;*TH8(:I>{2iTH>{2
8 \CPOE87X>{2 @:<;@2cNHOPABX87O5;@2"2E>@G:<;@2c<
X ;>=Ž6HABGx;@23<
X ;@2c4F>@X*NH:I8 JP;@2 =Ž6982E8aNH698FTH8i4F>@G92PCPOP6HABO
:I>{22E,8 I{6HG9TH>{2:<;@2 <
X ;@22E87G947AB:B:<;@2 6YA ; :BC5; Nx;OE87:B:I87Gx; O(87:c8F2PNx;@47AI> =M698:<;@22E87Nx; O5; Nx; E
8F2ECE>*2E:
8 9H7A hJ>*N>@OiTH>{2 4J; X*ABG9>{2 5$6 U x;@T\AI87G9TH> N>@CE87G947A<; TH8"
4 ; :I476H:I>(;i:I>{2o; 6HCEW@XY; C5;@2 Q9GHABCE>{2 :IS> =M698"2584F>@G92P7A I{6HAIW
X8FT\A<; GŽCE887:#NHOE>M4F8FT\ABX*AI87GMCE>2E87G947AB:B:I>*TH8$; x;@T\ABOP:I8F2 6HGx;NHAB:<; \6HGx;*8F2PCPOP6947CP6HO5;
TH8T9; CE>{2 X6 E6HCPAB:B7A h ;@T9; l87GV:<\
; =M698$2E8(NH698FTH8ABGŽCPOE> T\6947ABO ABXG a >@OPXY;@47AIW@G xNx; O5;
OE8F476HNl87O5; OP:<;iNl>{2ECE87OPAI>@OPX87GŽCE8a87G$87:Ž>@OETH87G$ABGŽRS87OE25>;†>; =M6987:Ž87@
G =M698 a 6983; x;@T\AIT9; 6
1c:/OE8F2P6H:BC5;@TH>Y8F2†6HG EHVJ H9 [RE3JME EC;]bS\E 6
BX6 d 8FT\6947AI87G9TH>:<;N>@CE87G947A<;TH8:<; <X ;>=Ž6HABGx;-TH8 `/6HOPABG I 6 =v; O5;87:B:I> Nx;@2PCEW 4F>@G
2P692PCPABCP6HABO*:<;47ABGŽC5; ABXG Q9GHABC5;Nl>@O6HGx;[TH8C5; XY; 9> :BABX*ABC5;@TH> 6 1c:†OE8F2P6H:BC5;@TH> 258
TH87G9>@X*ABGxd
; EHVJ H9 [RE3JMEiS ]bWAfQEHSYE>^ZL3JMEeN>L 6 U6HVG =Ž698a8F2PC5;<
X ;>=Ž6HABGx; 2E8"Nx; OE8F4F8<
X ;@2c;
:I>{2c>@OETH87Gx;@TH>@OE8F2;@47CP6x; :I8F2 =M698a:<; NHOE>@NHA<; <
X ;>=Ž6HABGx; TH8 `/6HOPABG I >9x;2EAITH>X694 99>
X87G9>{2 8F2PCP69T\A<;@T9.
; E2EW@:I>*258CPO5; C5; mO ;Y>; =M6 /TH:
8 a >@OPXY;25>@X87O5; E2E6HNl87O Qx47A<; : 6
d 8F4J; NHABCP6H:<; G9TH> H87G8F2PCE8:B7A NHOE>; Nx; OE8F4F87GV476x; CPOE>CPABN>{2TH8 <X ;>=M6HABGx;@2i; N92PCPO5;@47C5;@2
> ; 6HCEW@XY; C5;@2 87Gf87:i2E87GMCPAITH> <
X ;@2Y; X*NH:BAI> TH87:iCED7OPX*ABG9> &=M698V; 4F>@GŽCPABGM6x;@47AIW@GŒ258
87GM6HX87O5; G-87G >@OETH87G-TH8F47OE8F47AI87GŽCE8tTH8(4F>@X*NH:I8 J AIT9;@T 6
Z ;>=M6HABGx;@2 TH?
8 `r6HOPABG I 6
•
U 6HCEW@XY; C5;@2 :BABG98J; :I8F2;@4F>@C5;@TH>{2 6
•
U 6HCEW@XY; C5;@2 ;NHAB:<; 6
•
U 6HCEW@XY; C5;@F2 Q9GHABCE>{F2 E<
X ;>=Ž6HABGx;@22E8F476987G947A<; :I8F2 6
•
`r> T9;@2Y87:B:<;@22E84J; O5;@47CE87OP7A h ; G…Nl>@O2587O2EAI2PCE87XY;@24J; Nx;@4F8F2YTH8LCPO5; G92EX*ABCPABOABXG a >@O k
XY;@47AIW@G =M69887G 4J;@T9;ABG92EC5; GŽCE82E887G9476987GŽCPO5; G87G 6HG 8F2PC5;@TH> TH87CE87OPX*ABGx;@TH>[87GŽCPOE8
R{; OPAI>{2N>{2P7A NH:I8F2 3;@4F87NHC5; Gy2E8 x; :I8F2TH87:†X8FT\AI> 8F2EC BX6H:I>{2 387GMCPO5;@T9;@2 B]sWHCVJ87GmABXG k
I{:IDF2 E*87\G a 6HG947AIW@GTH8iT\AIm4 9x;@22E8 x; :I8F2"4J; jX NHA<; GLTH8 8F2PC5;@TH1> ECPO5; G92PX*ABCE87GL2E8 x; :I8F2; :
X8FT\AI> OE8F2PNH698F2PC5;@2 x25; :BAIT9;@2 LHVJ CVJr87GABG I{:IDF2 6 132ECE88F287:?4J;@2E><
X ;@2 IS87G987O5; : =M698
N> TH87X>{2:B:<; XY; O EHVXJ 39 [dE>JMERJqGQEHWFa`N3V^vJML3G 6 `v; @
X NHAID7G 8F2N>{2P7A NH:I8 AITH8J; O6HGx;t<
X ;>=Ž6HABGx;
=M698 G9>tOE8F477A Nx;87GMCPO5;@T9;@2 HN87OE>*; 6HG;@2 l2E8J;t4J; Nx; h$TH8 IS87G987O5; O†25; :BAIT9;@2 CE87G9T\O <; X>{2
87G8F2PCE8t4J;@2E> 96HG EHVXJ 39 [dE>JMERTefcW;f`GIE>NeLHG 6 =v>@O H:BCPABX> ‡N>MT\O <;T9; OE2E887:v4J;@2E>YTHi
8 =M698
i
i
i
i
i
i
i
. i
6HGx; X<;>=M6HABGx;VOE8F47A7Nx; ABGXa >@OPXY;@47AIW@G‰TH87:aX8FT\AI> rN87OE>V:<;-4J; GŽCPAIT9;@TTH8YABGXa >@OPXY;@47AIW@G
=M698TH87RM6987:BR{;2E8J;X6E-OE8FT\6947AIT9; 6HGx;2EABX*NH:I8*2E8x; : NHABGx; OPA<; 4F>@X>87:c87G94F87G9T\AITH>
>; Nx; IŽ;@TH>TH8 6HGx;: ; X*Nx; O5; =M698 ABG9T\AI4J;2PA87:; 6HCEW@XY; C5;;@4F87NHC5;t>OE8F4m9x;h ;t:<;2587OPAI8
TH8*258 x; :I8F2(TH8*87GŽCPO5;@T9b
; =M69i
8 9x;OE8F477A NHAITH>VT\6HO5; GŽCE847AI87OPCE>LCPAI87X*N> 6 1cG 8F2PCE8*4J;@25>258
CPAI87G98$6HG EHVJ H9 [RE3JMEEe^ f"C;JME>NeLHG 6
¯Ž!Œ’‚DŽ!–
$€‘‚LG¥‡1}=‚–

`?>MT9;JR <;Sad; :BC5;Nx; 6HGCE87OE4F87O38F2E4J; :IW@G87G*87:H4J; X*ABG9>?=Ž698a:B:I87RSW$;:<; _ GXa >@OPX<; CPAI4J;:`r8FW@OPAlk
4J;M= 698NHOE>JR ABG9>YTH86HGV4J; X*N>\=M698CPO5;@T\AI47AI>@Gx; :BX87GŽCE8G9> 9x;N <;OE8F47A7NHAITH>*:<;4F>@G92PAlk
TH87O5;@47AIW@GLTH8i47AI87GMC lxQ 4F>M:<;(:BABGI I2PCPAI4J;Ž:<;(CE8F>@O <;$TH8i:I>{2":I87GI{6x;J 8F2&E:<;@2gI{O5; X<; CPAI4J;@2 6
K 8CPO5; C5;TH86HGV4J; X*Nl>X6L
E ; GMCPA7{I 69> TM= 698(2E8 OE87X>@GŽC5;Y; :#X87G9>{2 TH>{2†X*AB:=Ž6HABGHAI87GXk
CE>{2; 9>{2 x9 ;@47A<;Y87:vNx;@25;@TH> 6 1c:vNHOPABX87O:BABGI HAI2PC5;TH87: Ž= 698t2E8$CPAI87G98t4F>@G92PC5; G947A<;8F287:
I{O5; X<; CPAI4F>YABG9T\AI>*=v; GHABGHA )Ž= 698R ABR AIWNHOE>>xN ;HN :I87X87GŽCE8t87GV87:/2PA7{I :I> !; GMCE8F2 TH8OPAI2PCE>
EYOE8F2P6HX*AIW:<;j{I O5; X<; CPAI4J;TH87:2 ; G92E47OPABCE>*87G476x; CPOE>X*AB:#OE8,{I :<;@2+M= 698(; GŽCPAI47ABNx; OE>@G 25>@O k
NHOE87G9TH87GMCE87X87GŽCE8; :7{I 6HG9>{2 ;FR{; G94F8F2TH8(:<;*:BABGI I2PCPAI4J;*X> TH87OPGx;87G 87:?2PA7{I :I>d 6 1cG
OE8F47A<;c:<;@2*NHOPABX87O5;@2\{
U X6 M >@O 6 a >@o:I87>>O5I ; <;G xN 2P;@A7{I 2P:IC5> ; GM_ CE_ 8L;I ABGŽGXO5;CEa 878FX<OP2LAI; >@THCPOEAI8 8F4J2;@2; OP2PAI:<2EAI;2PCECE>87TH\X<8yE ; =oCPUAI; 4JGHN;@AB2>@GH":IA >@6g4FGH>@H AIX*> >{2iNHM AB{I :<\E ;@O52E;T94FX<;@>@2Y:I; > CPNAI4F>@ 2P>{O AB2*GMM C5OE; AI>@>@XYGAI2\E ; 2PG9AI2P>{> A72{I :I‡X<> O5; @; _ 2_
ad; X>{2E>{2 U†87:BAB692 M >@Gx; CP692 \2PA7{I :I> _ E =cOPAI2E47A<; GM692 ;@8F25; OPAI87G92PAI2M2PA7{I :I> _ "2E8i:BABX*Alk
C5; OE>@G;2E8,{I 6HABOi:<;NHAI2PC5;TH82P692i; GŽCE8F4F8FTH87GMCE8F2S{I OPAI8,SI >{2 6
M 6HO5; GMCE8 :<;‰8FT9;@T Z-8FT\A<;i:<; {I O5; X<; CPAI4J; a 698 6HGx;‰TH8 :<;@2VO5; XY;@2 NHOPABG947ABNx; :I8F2
TH87:c4F>@G9> 47ABX*AI87GŽCE> 6 =v87OPCE87G98F4 <;V; :BJqGc]D]sVX[ M= 698;xN ; OE4J;xN ;:<;@2 CPOE8F2O5; XY;@2:NT@; 2PAI4J;@2 {I O5; X<; CPAI4J;lOE87CEW@OPAI4J;KV
E T\A<; :IDF47CPAI4J;#87G[>@Nl>{2PAI47AIW@G ; : )`VEeNHGp]D]sVh[ ‡:<;@2476x; CPOE>YO5; XY;@2
; NH:BAI4J;@T9;@2 9; OPABCPXD7CPAI4J;XSI 8F>@X87CPO <;x;@2PCPOE>@G9>@X <;i
E X 92PAI4J; 6 1cG87: d 87Gx;@47ABX*AI87GŽCE> 9:<;
:BABGI I2ECPAI4J;-4F>@G9> 47AIW-6HG GM6987RS>TH8F2 ; OPOE>@:B:I> v; :†; NH:BAI4J; OE2E8L;V:<;@2GŽ6987R{;@2t:I87G{I 6x;@2OE>k
XY; G94F8F2iE SI 87OPX<; GHAI4J;@2 o4F>@X>8F287:4J;@2E>-TH8Y:<; NHOPABX87O5;e{I O5; X<; CPAI4J;TH8:<;V:I87G{I 6x;
4J;@2ECE87:B:<; Gx;Ž8F2E47OPABC5;N>@O31c:BAI>U GMCE>@GHAI>$TH8 8,HN OPA PJ ; 4 65,P$P$P k 5
L$B$B 6 =v87OE>$:I>{2";JR@; G94F8F2
X<@; 2L2PA7{I GHAlxQ 4J; CPABRS>{2LCP6HR AI87OE>@G :B6ŽI ; O87GŒ:<; THDF4J;@T9; TH8 5
U$L$c 476x; G9TH>87::BABGI HAI2PC5;
G9>@OPCE8J; X87OPAI4J; G9>[U?RMO5; X >S; X 99 >@X2'[
E ABGMCPOE>MT\6>PJ >V:<; J"fQLHG7>qE NefS\EPa THGQEH[*=3Jq]_^ZEPa
JqGQEHWFa0`L3Gp[RE>^`]_LHWE3S\fpa >6J"fQLHG7>qE6NefkS fcWhTHVFE Dfpa"0vLHGc[REHS\fca Ž= 698t4F>@GŽR ABOPCPAIW:<;:BABGI I2PCPAI4J;
87GY6HGx;(47AI87G947A<;?t
E NHOE>@Nl>@OE47AI>@G9W6HGx;?99 87OPO5; X*AI87GMC5;@M= 698G9>$2EW@:I>(N> T <;$; NH:BAI4J; OE2E8 ;:I>{2
:I87G{I 6x;PJ 8F2?Gx; CP6HO5; :I8F2 >>NXPJ 87CE> 8 H47:B692PABRS> x9 ;@2PC5;i87GMCE>@G94F8F2vTH8":I>{2r8F2PCP69T\AI>{2?:BABGI I2PCPAI4F>{2 2EABG9>KM= 698t;@TH87X<@; 2 da ;@47AB:BABC5;xN ;87:v8F2ECP69T\AI><V
E :<;\a >@OPXY; :BA7h ;@47AIW@G TH8:I>{2 :I87G{I 6x;PJ 8F2Nx; O5;
:<;(NHOE>>{I O5; XY;@47AIW@GLTH8 >@OETH87Gx;@TH>@OE8F2 'M= 698i4F>@X87Gh ;xN ; GV;$; Nx; OE8F4F87ONHOE8F47AI25; X87GŽCE8 N>@O
;>M= 6987:B:<;*D7N>M4J; 99 >@X2'E 5
U $G 6
1c:(8F2PCP69T\AI> TH8:I>{2:I87G{I 6x;PJ 8F2258-T\ABR AITH8 CPO5;@T\AI47AI>@Gx; :BX87GMCE8 87G TH>{2Nx; OPCE8F287:
; GT; :BAI2EAI2aTH8:<;8F2PCPOP6947CP6HO5;tTH8 :<;@2 a O5;@2E8F2 {I O5; X<; CPAI4J;&Y
2587X<; GŽCPAI4J; 6 U 2E6ŒRS8,h a:<; {I O5; X<; CPAI4J;mNH698FTH8 2P6N T\E ABR 87AI:T\8FABOE2P2ECP8-69T\87AIG >tTHX8>@2EO 6a >@2P:I>>A7{I I GH<;AlxQ 4J ;@M= TH69>8
i
i
i
i
i
i
i
i
`ABN> c
'$
)
h
P
PP
I
@i
P
@
&%
@
@
`ABN> 5
`ABN> B
`ABN> G
A A7I{6HO5; 5$675 78 O5; O =Ž6 <;TH8(:<;@2 I{O5; X<; CPAI4J;@2 TH8 99>@X2'E 6
8F2ECP69T\A<;$:<;@2ga >@OPXY;@2aT\ABRS87OE25;@2 =Ž698 CE>@XY; GY:<;@2Nx; :<;NHO5;@2 2E8,I HG2E6R@; :I>@Oa87G:<;?a O5;@258
2EABGŽC5; \AI2 =Ž698(; Gx; :B7A h ;t:<;@2†T\ABRS87OE25;@2]a >@OPXY;@2†87Gb=Ž698 2E84F>@X@NHABGx; GV:<;@2aNx; :<;NHO5;@2†Nx; O5;
a >@OPXY; KO a O5;@2E8F24F>@OPOE8F47C5;@2 \E a >@G9D7CPAI4J; 8F2PCP69T\AI> TH8 :<;@2YNHOE>@NHAI8FT9;@TH8F2TH87::I87GI{6x;J 8
9x; NH:<;@TH> 6 =v>@O 87:vX>@X87GŽCE> 2EW@:I>8F2PC5; H:BCPABXY;G9>8F2 ; NH:BAI4J; NH:I8;Y:I>{2 :I87G I{6x; JP8F2TH8
>@OETH87Gx;@TH>@O 6
=?87OE>:I><
X ;@2476HOPAI>{2Ej
> a 698:<;OE87:<;@47AIW@GABX*NHOE87R AI2PC5i
; =M6982E8 TH8F2E476 NHOPAIW H;Nx; OPCPABOiTH8
:I>{2t; 9>{247ABG9476987GMC5; TH87:2P7A I{:I>VNx;@25;@TH> ?87GŽCPOE8TH>{24J; X*N>{2C5; G T\lA a 87OE87GŽCE8F287G ; Nx; k
OPAI87G947A<;4F>@X>:<;CE8F>@O <;TH8a<
X ;>=M6HABGx;@23; N92PCPO5;@47C5;@2 E:<;CE8F>@O <;TH8a:I87G I{6x; JP8F2 a >@OPXY; :I8F2 6
99>@X2 'E 47:<;@2ElA Qx4FW :I>{2t:I87
G I{6x; JP8F2t87Gm6HGx\
; J 87O5; O =M6 <;VABG947:B692EABR@;-TH8476x; CPOE> I{O5;@TH>{2
NH698F2 4J;@T9; I{O5;@TH>4F>@GŽCPAI87G98;tCE> TH>{2†:I>{2i2P7A I{6HAI87GMCE8F2 RSDJ;@2E8(:<; A 7A I{6HO5; 5$675 6
1c:/<
X ;@2 IS87G987O5; : x; : =M698N87OPCE87G98F4F8(:<;*2E87<
X ; GMCPAI4J;*TH8:I>{2†:I87G I{6x; JP8F2iGx; CP6HO5; :I8F2
•
Et; OPCPlA Qx47A<; :I8F2 Ž2E8a:B:<; XYi; THGIE3[B>= Jq]q^ZEKNef`S/Jq] CFL Nef1 rXLH[ka t 1 6 Uf8F2ECE>{2o:I87G I{6x; J 8F2
G9>2E8:I8F2 ABX*Nl>@G98(OE8F2ECPOPAI4F47AIW@G-; 7: I{6HGx; 6 6HOPAI>{25; X87GMCE8 ‡TH8F2ENH69DF2 2E8(TH8F2E476 NHOPAIW
=Ž69887:H4F>@>G J 6HGŽCE>(TH8a:<;@2 I{O5; <X ; CPAI4J;@23TH87:\CPABNl> c 8F2PmC ; BGŽCPABXY; X87GMCE8iOE87:<;@47AI>@Gx;@TH>
4F>@Gf87: 4F>@>G J 6HGŽCE> TH8LCE>MTH>{2Y:I>{2*NHOE>> NH:I87XY;@2Y4F>@X*NH6HC5; NH:I8F2> OE8F476HOE2PABR{; X87GŽCE8
87GŽ6HX87O5; NH:I8F2 :I>{2 =Ž69832583NH698FTH83OE8F25>@:BRS87OrX8FT\A<; GŽCE86HGx;†<
X ;>=M6HABGx;iTH8 `/6HOPABG I 6
1c:i2E,8 I{6HG9TH> I{O5;@TH> 8F287:†TH8L:<;@
2 T3GIEH[*3= Jq]_^ZEPanNef`SlJq] CL =Ž6984F>@OPOE8F2ENl>@G9TH87G
•
;:I>{2 S\f`WhTHVFE vfca aDfcWFap] c% S fpakEHS^ZLHW/J"f3; JML 6 1cG[8F2PCE84J;@2E>2E8$ABGMCPOE>MT\694F87G ; 7: I{6HGx;@2
:BABX*ABC5;@47AI>@G98F287GV:<;8F2PCPOP6947CP6HO5\
; I{O5; XY; CPAI4J; : l; 6HVG =M698258(Nl87OPX*ABCEj
8 =Ž69887:rR{; :I>@O
2PABGŽmC ;@47CPAI4F> TH8:<;@2tNx; :<; NHO5;@2*TH87Nl87G9T9;TH82P6mNl>{2PAI47AIW@G87Gm:<b
; a O5;@2E8 v8F2tTH8F47ABO
TH82P6 4F>@GMCE8 CE> 6 ; E[TH>{2(:I87G I{6x; JP8F2$Gx; CP6HO5; :I8Fj
2 =Ž698N>{2E8F87G #2PABG T\69T9; #6HGx;
2PABGŽC5; \AI2TH8Y8F2PCE8CPABN> /6HGx; :I87G I{6x;-; a OPAI4J; Gx; 87]: Nx; @
X Nx; O5; @E >@CPO5;-876HOE>@Nl8J;
87:r; :I87<
X ; XG kz2P6H7A hJ> 6 13GV>@CPOE>{2:I87G I{6x; J 8F2†Gx; CP6HO5; :I8F2 x; 6HVG =Ž698G9>*25S8 9x;tN> T\AITH>
TH87X>{2PCPO5; O =Ž6982E8J;(G98F4F8F2 ; OPAI>OE8F476HOPOPABO†;8F2EC5;@&2 I{O5; <
X ; CPAI4J;@2 MC5; X*N>M4F>t2E8 8F2EmC ;
i
i
i
i
i
i
i
4
•
•
i
2E,8 I{6HOE>TH8:I> 4F>@GMCPO5; OPAI> 6 =c698F2jNHAI87G rOE8F2P6H:BC5; =Ž698YCE>MT9;@2:<;@2jI{O5; X<; CPAI4J;@2tTH87:
CPABNl> 5 NH698FTH87G-; Gx; :BA7h ; OE2E8$X8FT\A<; GMCE8; 6HCEW@XY; C5;@2 :BABG98J; :I8F2;@4F>@C5;@TH>{2 6
K 8"T\AI4F8]=Ž698":<;@2I{O5; X<; CPAI4J;@2oTH87:ŽCE87OE4F87OI{O5;@TH> N87OPCE87G98F4F87G*; :Jq] CL Nef1rLH[ka$#
t 1 E 4F>@OPOE8F2PNl>@G9TH87Gf; :I>{2 S\f`WhTHVFE vfca ]sWNef"CYfcWNH]MfcW/J"fpa9NefcSB^ZLHWJ"f ;3JML 6 U =M6 a258
OE8F2PCPOPABG IS8; HG <
X ;@2i:<;:BA7Nl87OPC5;@T-TH8$:<;.a >@OPXY;@47AIW@G[TH8(OE8,I{:<;@2SI{O5; XY; CPAI4J; :I8F287G
:I>{2:I87G I{6x; JP8F2TH8 8F2PCE8CPABN> 387:iR{; :I>@O2PABGMmC ;@47CPAI4F> TH8V6HGx; Nx; :<; NHO5; 8F2CE>@C5; :lk
X87GŽCE8ABG9TH87Nl87G9T\AI87GMCE8TH8Y2P6 N>{2PAI47AIW@G 87G :<; a O5;@2E8 6 `r> TH>{2$:I>{2$:I87G I{6x; JP8F2tTH8
>@OETH87Gx;@TH>@O x2PABGV8 H4F87N47AIW@G HCPAI87G987GV6HGx;*2PABGMC5; AI2 =Ž698(2E8NH698FTH88 NHOE8F25; O†XR8 k
T\A<; GŽCE8 I{O5; <
X ; CPAI4J;@2TH87:‡CPABN> B TH8 99>@X2 'E E2PAI87X*NHOE88F2N>{2P7A NH:I8; Gx; :B7A h ; OP:<;
X8FT\A<; GŽCE86HG ; 6HCEW@XY; C5;Y;tNHAB:<; 6
=?>@O H:BCPABX> ?:<;@@
2 I{O5; <
X ; CPAI4J;@2*TH87
: Jq] CFL Nef 1 rXLH[ka t 1 =Ž6984F>@OPOE8F2PN>@G9TH87G‰;
6HGCPABNl>*TH8$:I87G I{6x; J 8F2†X6 E2587G947AB:B:I> H:B:<; XY;@TH>{2 S fcWhTHVFE Dfpa GZf_THVXSmEHGZfpa ‡2E8(4F>@G92 k
CPOP6 ES87Gy4F>@G‰OE8F2PCPOPAI4F47AI>@G98F2*X6 Em8F2PCPOE8Fm4 9x;@2 6 1cG‰:<;-NHmO ;@47CPAI4J; 32EW@:I>[87: ; TG ; :BAI2PAI2
X>@O a >@:I>W I{AI4F>YTH8:I>{2:I87G I{6x; JP8F2†TH8>@OETH87Gx;@TH>@OiN87OPCE87G98F4F8$;8F2ECE8GHABRS87: 6TW 476HOPOE8
=Ž698i:<:; I{O5; <X ; CPAI4J;$TH8i476x; : =M6HAI87O3:I87G I{6x; JP8†OE,8 I{6H:<; O258NH698FTH8; Gx; :B7A h ; O"X8FT\A<; XG k
CE86HG ; 6HCEW@XY; C5; Q9GHABCE> 6 1"2PC5;@2(<
X ;>=M6HABGx;@2; N92PCPO5;@47C5;@2$CPAI87G987G C5; @
X NHAID7G >@CPO5;@2
; NH:BAI4J;@47AI>@G98F2 @N>@Ov8 JP87X*NH:I> @87Gt87:H; TG ; :BAI2PAI2 E$T\AI2E8 9>>@OPAI87GŽC5;@TH>$;>> NXJP87CE>{2 Nx; O5;
:<;(8F2PN8F47lA Qx4J;@47AIW@GYTH8†:I>{234J; j
X NHAI>{2TH8†8F2EC5;@TH>$TH8:I>{2">> NXJP87CE>{g2 =M698Nl87OPCE87G98F4F87GL;
6HGx;(47:<;@2E8iTH87CE87OPX*ABGx;@T9; 8 >M> 4 96fvJE3S 5
U$U$f 6$A ABGx; :BX87GŽCE8 8 AI2ECE86HGCPABNl>TH8
NHOE>> NH:I87XY;@1
2 =M698NH698FTH87G TH8F2E47OP7A NHABOE258X8FT\A<; GMCE8*6HGx;O5; XY;TH8:<;@2 XY; CE87<
X ; CPlA k
4J;@2 =Ž6982E8 :B:<; XY;H= SUTef<`% GIENefiSmEHa fp; CGZfpap]_LHW;fpa GZf_THVXS\EHGQfca T=M698 C5; @
X NHAID7GVNH698FTH8
OE8F2E>@:BRS87OE2E8(X8FT\A<; GMCE8; 6HCEW@XY; C5;@2 Q9GHABCE>{2 6
H ; A A7I{6HO5; 5$6CB OE8F2P6HX8(:<;tOE87:<;@47AIW@G TH8:<;@2 476x; CPOE> J 87O5; O =M6 <;@2iTH8:I{O5; X<; CPAI4J;@2 TH8
99>@X2'E l:I87GI{6x;J 8F2:a >@OPXY; :I8F2 X<;>=M6HABGx;@2;N92PCPO5;@47C5;@2?E :I>{2NHOE>>NH:I87XY;@2@=M698t4J;@T9;
6HGx;8F2i4J; Nx;
h TH8OE8F2E>@:BRS87O 6
 Ž!–[‚$€f€;L‰1‰Z
“ Ž!‰°‰1‡’€;
13:a:BA7NHOE>[2PA7I{69887:TH8F25; OPOE>@:B:I> 9HAI2PCEW@OPAI4F>[TH8:<; CE8F>@O <;TH8X<;>=Ž6HABGx;@2*;N92PCPO5;@47C5;@2 6 1c:
4J; N BCP6H:I>2E,8 I{6HG9TH>8F2EmC ;(TH8FT\AI4J;@TH>t;(:<;@23X<;>=M6HABGx;@2"TH8 `r6HOPABGI Ž87:‡CE87OE4F87OE>; Gx; :BA7h ;(:I>{2
; 6HCEW@XY; C5;@2 Q9GHABCE>{2 E :<;@2r<
X ;>=Ž6HABGx;@2?2E8F476987G947A<; :I8F2 E 87: 476x; OPCE> 8F2PCP69T\A<;i:I>{2?; 6HCEW@XY; C5;@2
; NHAB:<; 6 1c:a4J; N BCP6H:ID
> =M6HABGŽCE>[2PABOPRS8TH8YABGMCPOE>MT\694F47AIW@Gf; :<;CE8F>@O <;THK
8 I{O5; <
X ; CPAI4J;@2*TH8
99>@X2 'E 6
1cG02P62E,8 I{6HG9T9;ŒNx; OPCE8 (87::B7A NHOE> ; Nl>@OET9;…:<;@2 47698F2PCPAI>@G98F2 NH:<; GMCE8J;@T9;@2 N>@O[:<;
OE87:<;@47AIW@G X6HCP6x;87GŽCPOE8$:<;CE8F>@O <;TH8<
X ;>=M6HABGx;@2 ; N92PCPO5;@47C5;@2 E:<;tCE8F>@O <;TH8 :I87G I{6x; J 8F2
a >@OPXY; :I8F2 6 U 2 #87:32E8 \CE>4J; N BCP6H:I>-; N>@OET9;:<;8 =M6HABR@; :I87G947A<;V87GŽCPOE8:<;@2$<X ;>=Ž6HABGx;@2TH8
`r6HOPABG I\E:I>{2i; 6HCEW@XY; C5;@2i:BABG98J; :I8F2;@4F>@C5;@TH>{2 HNl>@O6HG:<;@TH> E:<;@2 I{O5; <X ; CPAI4J;@2 TH8:I>{2
i
i
i
i
i
i
i
f
!V4 XZ !"
&%'()!+*,%'"
!"
i
/1T2 3%'4 !"
>
4F>@X*NH6HC5;NH:I8F2
d 8F476HOE2PABR{;k
X87GŽCE8
87GŽ6HX87O5;NH:I8F2
`ABN> c
TH8 99 >@X2+E
`ABN> 5
TH8 99 >@X2+E
ZD;>=Ž6HABGx;@2 TH8
`r6HOPABG I
H 87GI{6x;J 8F2
2E87G92PA7HN :I8F2 ; :
U6HCEW@XY; C5;@2
:BABG98J; :I8F2
;@4F>@C5;@TH>{2
U6HCEW@XY; C5;@2
;tNHAB:<;
4F>@GŽCE8 \CE>
H 87G I{6x; J 8F2 ABGXk
TH87Nl87G9T\AI87GMCE8F2
TH87:/4F>@GŽCE8 \CE>
H 87G I{6x; J 8F2
OE,8 I{6H:<; OE8F2
`ABN> B
TH8 99 >@X2+E
`ABN> G
TH8 99 >@X2+E
a>@X*NH6HC5;NH:I8F2
U6HCEW@XY; C5;@2
1 \NHOE8F2PAI>@G98F2
OE8,I{6H:<; OE8F2
Q9GHABCE>{2
TH87CE87OPX*ABGHAI2PC5;@2
A 7A I{6HO5; 5$6CB d 87:<;@47AIW@G 87GŽCPOE8:<; J 87O5; O =M6 <; TH8 I{O5; <X ; CPAI4J;@2TH8 99>@X2+E a:I>{2T\AI2 k
CPABGMCE>{2*CPABN>{2tTH8L:I87GI{6x;JP8F2 v:<;-CE8F>@O <;[TH8X<;>=Ž6HABGx;@2Y;N92ECPO5;@47C5;@2 E <: ;-CE8F>@O <;TH8L:<;
4F>@X*NH6HC5; NHAB:BAIT9;@T 6
CPABN>{2 c E 5 TH8 99>@X2'E vNl>@O>@CPOE> c87:†2ED7NHCPABX> o:<;@2tOE87:<;@47AI>@G98F287GŽCPOE8L:I>{2; 6HCEWk
XY; C5;@2FQ9GHABCE>{2FE:<;@2 I{O5; X<; CPAI4J;@2 TH87:#CPABNl> G TH8 99>@X2'E H4F>@G 8F2PN8F47A<; :?; CE87G947AIW@G[;
:<;@28 \NHOE8F2PAI>@G98F2OE,8 I{6H:<; OE8F2 E;:<;@2aNHOE>@NHAI8FT9;@TH8F2†X<;@2aABX*Nl>@OPC5; GMCE8F2iTH8 :I>{2a:I87GI{6x;J 8F2
OE,8 I{6H:<; OE8F2 4F>@X*NH:I87C5; G9TH>;@2 l87:8F2PCP69T\AI>TH8 :<?
; Q9:<;ABXG a 87OPAI>@O†TH8:<; A A7I{6HO5; 5$6CB Q9Gx; :lk
X87GMCE8 M87:l>M47C5;JRS>$OE87R AI25;(:<;8 =Ž6HABR{; :I87G947A<;$87GMCPOE8 ; 6HCEW@XY; C5;@2†;NHAB:<1
; E.I{O5; <
X ; CPAI4J;@2TH87:
CPABN> B TH8 99>@X2 +E J 6HGŽCE>(4F>@G:<;@2vNHOE>@NHAI8FT9;@TH8F2c<
X ;@2vABX*Nl>@OPC5; GMCE8F23TH8a:I>{2v:I87G I{6x; J 8F2
ABG9TH87N87G9T\AI87GŽCE8F2 TH87:r4F>@GMCE8 CE> 6
H ;-CE87OE4F87O5; Nx; OPCE8LTH87::B7A NHOE>[2E8OER8 Qx87OE8L;-47698F2PCPAI>@G98F2<X ;@2;JR@; G h ;@T9;@2 c; 7: I{6HGx;@2
TH8 :<;@2476x; :I8F2aRMAI87G987G2PAI87G9TH>>> NXJP87CE>TH8 ABGMRS8F2PCP7A IŽ;@47AIW@G 87G; 9>{2X6 EOE8F47AI87GMCE8F2 6 U2
87:v4J; N BCP6H:I>G9> RS87G9>YNH698FTH8t4F>@G92PAITH87O5; OE2E86HGx;*ABGŽCPOE> T\694F47AIW@G ;*:<;CE8F>@O <;TH8(:<;4F>@X k
NH6HC5; NHAB:BAIT9;@T $=M6982P6987:I8a2E87OoNl>@Oo2 X*AI2PXY; >> NXJ 87CE>TH8aNH6 NH:BAI4J;@47AI>@G98F2cABG9TH87N87G9T\AI87GŽCE8F2 87:lTHDF47ABX>8F2PCP69T\A<;>@CPOE>{2"CPABN>{23X87G9>{2a4F>@G9> 47AITH>{2TH8†<
X ;>=M6HABGx;@2 ; 6HCEW@XY; C5;@2NHOE>> Nx; k
NHAB: I2ECPAI4F>{2 E4F87:B6H:<; OE8F2 947AI87OPCE>{2†CPABN>{2†2E87G947AB:B:I>{2 TH8OE8FTH8F2†G9876HOE>@Gx; :I8F2 ]ETH:8 I{O5; <X ; CPlA k
4J;@2 :<;@2(TH8 H ABG9TH87GHXY; ES87Ot>TH8*TH87OPABR@;@47AIW@G 87G Nx; O5; :I87:I> Q9Gx; :BX87GŽCE8 r87:o6HG9THDF47ABX>
i
i
i
i
i
i
i
i
ABGMCPOE>MT\694F8; :7I{6HG9>{2vNx; O5;@T\A7I{XY;@2vX>MTH87OPG9>{2oTH8"4F>@X*NH6HC5;@47AIW@G @4F>@X>:<;i4F>@X*NH6HC5;@47AIW@G
4F>@G X87@
X NHO5; Gx;@2 x:<;4F>@X*NH6HC5;@47AIW@G4F>@GVU M E:<;4F>@X*NH6HC5;@47AIW@G476T; GMCPAI4J; 6
Q G; ND7G9T\AI4F8cOE8F2P6HX8; 7: I{6HG9>{2?4F>@G94F87NHCE>{2rXY; CE87X<; CPAI4F>{2?G98F4F8F2 ; OPAI>{2/Nx; O5;:<;†4F>@X k
NHOE87G92EAIW@Gm4F>@X*NH:I87C5;TH87:":B7A NHOE> =M698NHOE> 4F8FTH84F>@:I> 4J; O; &: Q9Gx; :aTH8DF2PCE8Nx; O5; G9>VABGXk
CE87OPOP6HX*NHABO 87:/TH8F25; OPOE>@:B:I>YTH87:#OE8F2ECE>TH8:<;@2†XY; CE87OPA<;@2 6
1cG :Ie
> =M69.
8 =Ž698FT9;VTH8*8F2PCE84J; N BCP6H:I> 258ABGŽCPOE> T\694F87Gm; 7: I{6HG9>{24F>@G94F87NHCE>{1
2 a 6HG9T9; k
X87GMC5; :I8F2 9G98F4F8F2 ; OPAI>{2iNx; O5;:<;4F>@X*NHOE87G92PAIW@GTH8(8F2PCE8(CE8 \CE> 6

Ÿ‰
qŒ’‚–
“ Ž’Œ’‡1}=‡ZŒ’Ž!–
K–[‡1}=‚–
b% ) H :<; XY; OE87X>{2 !V3 !2g%
;t6HGV4F>@G>J 6HGMCE>.Q9GHABCE> HG9>*R{;@4 I> 6
H >{287:I87X87GŽCE>{2TH8L6HGf; :lad;Nl87CE> 2E8L:B:<; XY; G 3 %
!" > " 4 2&)3T" 6 U6HGV=Ž698L:I>{2
; :lda ;N 87CE>{2]M= 698; Nx; OE8,Jh 4J; GV87G8F2PCE8:BA7NHOE>8F2PC5; Om; G a >@OPXY;@TH>{2aN>@O:I87CPO5;@2a87GL87:#2E87GŽCPAITH>
692E6x; :vTH87:vCED7OPX*ABG9> 8F2PCE>YG9>2PAI87X*NHOE88F2G98F4F8F25; OPAI> 6 =?>@O(8 JP87X*NH:I> ‡:<;@2:I87CPO5;@2TH86HG
; :lda ;N 87CE>VN>MT\O <; G4F>@OPOE8F2PN>@G9TH87O; 8F2PCPOP6947CP6HO5;@2TH8T9; CE>{2t87G 6HG>@OETH87Gx;@TH>@O 6 U :I>
:<; O SI >TH87:9:BA7HN OE> S6HG; :la ;N 87CE>2E8OE87NHOE8F2587GŽC5; Om
; 692P6x; :BX87GŽCE8X8FT\A<; GŽCE8:<;:I87CPO5;?{I OPAI8,ŽI ;
a:<;@2:I87CPO5;@2b{I OPAI8,ŽI ;@2LX*ABG 92E476H:<;@2 †4F>@X> TH87G9>@C5; Om; G 2 BX@N >@:I>{2bSI 87G9D7OPAI4F>{2TH87:
Σ
; :lda ;N 87CE> X*AI87GMCPO5;@2&M= 698a:I>{2v2 BX@N >@:I>{234F>@G947OE87σCE>{2gM= 698":I> a >@OPX87G*2E87Om; GG9>@OPXY; :BX87GŽCE8
:<;@2iNHOPABX87O5;@2i:I87CPO5;@2iX*ABG 92E476H:<;@2 TH87:v; :la ;N 87CE>*692P6x; : a, b, c . . . 6
K 8(6HCPAB:BA7h ; Om
; 87:r2 BX@N >@:I>TH8(N87OPCE87G987G947A<; x9 ;HN ABCP6x; :?87GVCE8F>@O <;YTH8(4F>@G>J 6HGMCE>{2 Nx; O5;*ABG9T\AI4J; O Ž= 698$6HGx;t:I87CPO5;Nl87OPCE87G98F4F8t;t6HG ; :la ;N 87CE> N>@Oi8 J 87X*NH:I> a ∈ Σ 6 ∈
Q G; :la ;N 87CE> 2E8TH8R9Q G98<SI 87G987O5; :BX87GMCE8LNl>@Ot87GŽ6HX87O5;@47AIW@G‰TH8:I>{22 BXjlN >@:I>{2 M= 698
4F>@GMCPAI87G98 6 =v>@OV8 J 87X*NH:I> Σ = {a, b, c, d, e, . . . , z} 8F2L87:(; :lda ;N 87CE>mTH8[:<;@2:I87CPO5;@2
:<; CPABGx;@2X*ABG
92E476H:<;@2 Σ = {0, 1} 8F2$87:; :lda ;N 87CE>VTH8*:I>{2$G HX87OE>{2@HN ABGx; OPAI>{2 Σ =
8F2 87:c; :la ;N 87CE><M= 698N87OPX*ABCE8t47>@G92ECPOP6HABO 476x; : M= 6HAI87OG HX87OE>
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
Gx; CP6HO5; :o87G xN ;@2E8 5
c Σ = {/, \} 8F287:c; :la ;N 87CE>.a >@OPXY;@TH>Nl>@O:I>{2TH>{2 2 BXjlN >@:I>{2
E \6
1
2
3
4
! &"
K+A7"+ >+#4
b% ) [ K 8:B:<; XY; !V3!2! Za >@OPXY;@T9;4F>@G:I>{2v2 BX@N>@:I>{2cTH86HG; :lad;Nl87CE> ;tCE> T9;*2E8F476987G947A<; 9Q GHABC5;*TH8(:<;@2†:I87CPO5;@2iTH88F2E8$; :la ;N87CE> 6
1YJ 87X*NH:I>{2TH8VNx; :<;HN O5;@22E>>HN OE8-87:; :la ;N 87CE>
2E>@G cVEHW CFfQNHGQL E ^ZLD^crf 6 K >@G
Nx; :<;HN O5;@22E>>HN OE8*87:"; :la ;N 87CE> Σ 0 1 01010101Σ E 1111 6 =v>@O H:BCPABX> 0 1 125 E
12357384
HTH:B8 CPAB6HXYG ;@;2:lda :I;87CPlN 2EO5>@87;@GCE2$> Nx6 X*;U AB:<G 2;
9HN 2EO5Nl47;@6H>@2tO†:<;@2E8 2>>PJ HN 87THOEX*878:NH87:I; :a> :la ; ;:lN da ;87lN CE>b87CEx9> ;ΣHN ABCP8F66x21c; 6H:GmGxNx;:I; > NxO5;L; 2P69:<;OE4F87HN 8FNHO52EOE;*AB8FRS2E2E> 87>>?GŽHN 2EC5OE8; 8 O6H:<CP;@AB:B2$ A7h Nx; ;Om;:<;GmHN O5:<;@;@22
x = juan
Σ y = 1234
8F2i6HGx;Nx; :<;HN O5;*2E>>HN OE8 Σ 6
1
2
3
1
3
i
i
i
i
i
i
i
5
c
i
K 8?x9 ;6HCPAB:BA7h ;@TH>Y87:rCED7OPX*ABG9>\"
%'%+! Nx; O5;*TH8RQ9GHABOi6HGx;Nx; :<;NHO5;‡TH8 JP; G9TH>; :
J 6HAI47AI>TH87:‡:I8F47CE>@Oi2E6ABGMCE87OPNHOE87C5;@47AIW@G 6 =?>@O:I>{28 JP87X*NH:I>{2i;@T\6947AITH>{2 \2E8 NH698FTH8(TH8FT\6947ABO
M= 698:<;tNx; :<;HN O5;258F476987G947A<;*2E8 OE8RxQ 87OE8;t:<;*4F>@:I> 4J;@47AIW@Gba I2PAI4J;>@OETH87Gx;@T9;*TH8(:<;@2:I87CPO5;@2
6HGx;[;\J 6HGMCE> ;->@CPO5;o4F>@X>[2E8Kx9 ;@4F8 9x;NHABCP6x; :BX87GŽCE8V4F>@G:I>{2tCE8 CE>{2*8F2547OPABCE>{2 6 KMABG
87XjxN ; O SI > "8F2PC5; ;@4F87N47AIW@GyG9> ;@47:<; O5; :I> Ž= 698LN>MT\O <; 2E87O*6HGx;Nx; :<;HN O5; 476x; G9TH> :I>{2
2 BX@N >@:I>{2 TH87:c; :lda ;lN 87CE>G9>CE87GŽI ; G[6HGx;YOE87NHOE8F2E87GŽC5;@47AIW@GDa I2PAI4J;2587G947AB:B:<;l4F>@X>87G[87:
4J;@25>-TH8 Ž= 6982E8YCPO5; CE8TH88F2PCPOP6947CP6HO5;@2TH8T9; CE>{2 6 U6HGVM= 698; :I> :<; O SI >[TH87:a:BA7HN OE>-258
692 ; Om; GŒ2 BXjlN >@:I>{2ABX*NHOE8F2E>{2 
E :<;@2YNx; :<;HN O5;@2L4F>@OPOE8F2PN>@G9TH87Om; G ; :<; ABGŽCE87OPNHOE87C5;@47AIW@G
M= 698*2E8*;@4J;xN ;TH8*T9; O #4F>@GMRMAI87G98T9; O(6HGx;ABGŽCE87OPNHOE87C5;@47AIW@G a >@OPXY; :3TH87:34F>@G94F87NHCE>-TH8
Nx; :<;HN O5;\Ž= 698$258J;tR>; :BAIT9;*87GV87:?4F>@GŽCE8 \CE>.SI 87G987O5; :?TH8$; :lda ;lN 87CE>{2 476x; :I8F2 Ž= 6HAI87O5; 6
H ;VAITH8J;-8F2E87G947A<; :aNx; O5; TH8R9Q GHABOtNx; :<;HN O5;@2*8F2iŽ= 698DF2PC5;@2258NH698FTH87G4F>@G92PCPOP6HABO;
Nx; OPCPABOTH8*:I87CPO5;@2 X*AI87GŽCPO5;@2:<;@2(:I87CPO5;@22E8NH698FTH87G OE8F476HN87O5; O;Nx; OPCPABOTH8*:<;@2(Nx; :<;k
HN O5;@2 6 =v>@O:I>tC5; GŽCE> 6HGx;tTH8R9Q GHAI47AIW@G a >@OPXY; :#TH8 Nx; :<;HN O5;Nx; OPCPABOm
; TH8 :<;t8 \AI2PCE87G947A<;tTH8
4F>@G92ECPOP6947CE>@OE8F2 Y
E TH8F2PCPOP6947CE>@OE8F2†TH8 Nx; :<;HN O5;@2 Nx; O5;Nl> TH87OTH8R9Q GHABO†Nx; :<;HN O5;@2†2E>>HN OE8 6HG
; :lda ;N 87CE>2E>@GNHOE8F47AI25;@2†CPOE8F24F>@X*Nl>@G987GMCE8F2
Q Gx;Nx; :<;HN O5;8F2PN8F47A<; : {:B:<; XY;@T9;!V3!2! ! ! M= 698i2E8iTH87G9>@C5; Om$
; X8FT\A<; GŽCE8
•
:<;:I87CPO5;?{I OPAI8,ŽI ; λ 6_ GŽCP6HABCPABR{; X87GŽCE8 λ 8F2c:<;Nx; :<;HN O5;1M= 698G9>$4F>@GMCPAI87G98 GHABG{I 6HGx;
:I87CPO5; 6
Q Gm4F>@G92PCPOP6947CE>@OTH8YNx; :<;HN O5;@2 :<; >@Nl87O5;@47AIW@Gm4F>@G94J; CE87Gx;@47AIW@GTH86HG 2 BXjlN >@:I>
•
4F>@G 6HGx;mNx; :<;HN O5; Ž= 698 ;mNx; OPCPABOVTH86HG 2 BXjlN >@:I> σ f
E TH8[6HGx;mNx; :<;HN O5;
4F>@G92PCPOP6SE 86HGx;*Nx; :<;HN O5;YGŽ6987R{; σ.x )Ž= 6982E8>>HN CPAI87G98t4F>@:I>M4J; G9TH>Y:<;*:I87CPO5; σ x;
:<;A7h
M= 6HAI87OET9;*TH8(:<;@2†:I87CPO5;@2iTH8 x 6
M >{2(; Gx; :BA7h ;@TH>@OE8F2(TH8Nx; :<;HN O5;@2 YŽ= 698:B:<; XY; OE87X>{2 inicial E resto ‡TH8X> TH>
•
Ž= 6982PA x 8F2"6HGx;$Nx; :<;HN O5;$G9>R@;@4 <; inicial(x) 8F236HGx;$:I87CPO5; :<;$NHOPABX87O5;$:I87CPO5;
TH8 x E resto(x) 8F2i>@CPO5;Nx; :<;HN O5; 6
132†G98F4F8F25; OPAI>\M= 698:I>{2†4F>@G92PCPOP6947CE>@OE8F2S
E ; Gx; :BA7h ;@TH>@OE8F2TH8(Nx; :<;HN O5;@2†ABGŽCPOE> T\6947AITH>{2
RS87OPAlQTM= 6987GV:<;@22PA7{I 6HAI87GŽCE8F2 OE87:<;@47AI>@G98F2
5$6 KMA x 8F2†6HGx;*Nx; :<;HN O5;; O HN ABCPO5; OPA<;T\AI2PCPABGMC5;*TH8 λ H2E8(476HX*NH:I8@M= 698
BX6 KMA x 8F2i6HGx;YNx; :<;NHO5;476x; : =Ž6HAI87O5;KE σ 8F2 476x; : =M6HAI87O 2 BXjNl>@:I> l2E8$476HX*NH:I87G[:<;@2
TH>{2iNHOE>@NHAI8FT9;@TH8F22PA7{I 6HAI87GŽCE8F2
inicial(x).resto(x) = x
inicial(σ.x) = σ
resto(σ.x) = x
i
i
i
i
i
i
i
i
U 6HVG =M698;:I>:<; O IS>YTH87:r:BA7NHOE>Y2E8(6HCPAB:BA7h ; Om;4F>@G92PC5; GŽCE87X87GMCE8t87:?4F>@G94F87NHCE>TH8(Nx;k
:<;NHO5;4F>@X>Y2PA/2E8CPO5; C5; O5;YTH8(6HGx;*4J;@TH87Gx;YTH8(4J; O5;@47CE87OE8F2 9CE> TH>{2i:I>{2i4F>@G94F87NHCE>{2 =M698
6HCPAB:B7A h ; GNx; :<; NHO5;@2 4F>@X>87:9:I87GI{6x;JP8;@4F87NHC5;@TH>>(OE8F4F>@G9> 47AITH>Nl>@O6HG; 6HCEW@XY; C5; \87:
:I87G I{6x; JP8 IS87G987O5;@TH>>(OE87NHOE8F2E87GŽC5;@TH>N>@Oc6HGx:
; I{O5; <
X ; CPAI4J;>N>@O36HGx;(8 \NHOE8F2PAIW@GOE8,I{6Xk
:<; O 87CE4 6 /258cNH698FTH87Gt8 \CE87G9TH87OvABGHX8FT\A<; C5; X87GŽCE8i;Nx; :<; NHO5;@2 Nx;@25;@T9;@2v87G:I>{2o; AI>@XY;@2
; GMCE87OPAI>@OE8F2 9N>@O†:I.
> =M698CE>MT9;@2†:<;@2; Q9OPXY;@47AI>@G98F2 ECE8F>@OE87XY;@2iC5; j
X NHAID7G-2E87mO ; GV47AI87OPCE>{2
87G-8F2PCE8(4F>@GŽCE8 \CE><
X ;@2 ; X*NH:BAI> 6
; EmX694 9x;@2XY; G987O5;@2YTH8VABX*NH:I87X87GŽC5; OL6HGf2PAI2PCE87XY; TH8Nx; :<; NHO5;@K2 =M698V25; CPAI2 k
ad; IŽ;[:I>{2; AI>@XY;@2L; GŽCE87OPAI>@OE8F2X8FT\A<; GŽCE88F2PCPOP6947CP6HO5;@2TH8 T9; CE>{287Gf6HGf>@OETH87Gx;@TH>@O
2E6HNl>@GHAI87G9THj
> =Ž698CE87G IŽ;$X87X>@OPA<;(ABXG Q9GHABC5; Ž4F>@GYRS8F47CE>@OE8F2TH8i4J; O5;@47CE87OE8F2 Ž4F>@G:BAI2PC5;@2
87GH:<; h ;@T9;@2 #87CE4 6 132PCE>Nl87OPX*ABCE8*THR8 Q9GHABO$8F2ECPOP6947CP6HO5;@2$TH8T9; CE>{2<
X ;@2$4F>@X*NH:I8 JP;@2Nx; O5;
OE87NHOE8F2587GŽC5; OL:I>{24F>@G94F87NHCE>{2 =M698VABmO ; G ; Nx; OE8F47AI87G9TH> ;@2 4F>@X> NHOE>> I{O5; XY;@2 =M698 258
4F>@X*N>@OPC5; G-4F>@X>:I>{2 ; 7: IS>@OPABCPX>{2 TH8F2E47OPABCE>{287GV8F2PCE8(:B7A NHOE> 6
b% ) K 8$:B:<; XY;Y:I>@G I{ABCP69TTH8$6HGx;YNx; :<; NHO5;; :?G HX87OE>LTH8$:I87CPO5;@:2 =M698
:<;4F>@X*Nl>@G987G 6
H ;t:I>@G I{ABCP69TTH8 :<;Nx; :<; NHO5; x 258 OE87NHOE8F2E87GMC5;*4F>@GL:<;G9>@C5;@47AIW@G |x| 6 =?>@Oi8 JP87X*NH:I> |juan| = 4
|1| = 1
|12345678| = 8
|λ| = 0
1cGL6HGL2EAI2PCE87XY;jIS87G987O5; :TH8 Nx; :<;NHO5;@2 Nx;@2 ;@TH>t87GL:<;t; AI>@X<; CPAI4J;tABGMCPOE>MT\6947AIT9; 9:<;
HT 8RQ9GHAI47AIW@G 9x;NHO <;b=M698*8Ra 8F47CP6x; OP:<; TH8XY; G987O5; a >@OPXY; : 6 a>@X>:<;TH8RQ9GHAI47AIW@G TH8*:<;@2
Nx; :<;NHO5;@2;tNx; OPCPABOTH8$:I>{2 2 BXjNl>@:I>{2 E:I>{2>@Nl87O5;@TH>@OE8F2Fa 6HG9T9; X87GMC5; :I8F2 8F2iOE8F476HOE2PABR@;
476x; : =M6HAI87O>@CPO5;YTH8RQ9GHAI47AIW@G )EŽ;Y258J;*TH81a 6HG947AI>@G98F2 >YTH8t4F>@G94F87NHCE>{2 xOE8J; :BA7h ;@T9;;*Nx; O k
CPABO$TH8:<;@2$Nx; :<; NHO5;@2 YE476x; : =M6HAI87OTH87X>{2PCPO5;@47AIW@G TH8NHOE>@NHAI8FT9;@TH8F2TH8:<;@2(Nx; :<;NHO5;@2
9x; NHmO ; G TH8t2E87O(;@2 ?X*AI2EX>OE8F476HOE2EABR@;@2 6 =v>@O8 JP87X*NH:I> l:<;YTHR8 Q9GHAI47AIW@G OE8F476HOE2EABR@;TH8t:<;
:I>@G I{ABCP69T-TH86HGx;Nx; :<; NHO5;*2E:
8 9x; O <;Y;@2 ENx; |λ|O5;*=476x0; : =M6HAI87O 2 BXjNl>@:I> σ LE 476x; : =Ž6HAI87O Nx; :<;NHO5; x |σ.x| = 1 + |x|
1cG :I>y2E694F8F2PABRS> i2PAI87X*NHOE8=M698 2E8J;‰Nl>{2PA7NH:I8 ABGŽCPOE> T\6947ABO 6HG 4F>@G94F87NHCE>ŒTH8Da >@O k
XY;ABGŽCP6HABCPABR{;H2E887RMABC5; Om; GL:<;@2aTH8R9Q GHAI47AI>@G98F2 E TH87X>{2PCPO5;@47AI>@G98F2OE8F476HOE2PABR{;@2 a >@OPXY; :I8F2 ; 6HGVM= 698†87G; :7I HG4J;@2E>2E8aOE8F476HOPOPABOm(
; ;:<;(; O {I 6HX87GŽC5;@47AIW@G\a >@OPXY; : Nx; O5;87RMABC5; OcT\69T9;@2
>ABGMCE87OPNHOE87C5;@47AI>@G98F287OPOEW@G98J;@2 6
1c:v4F>@G>J 6HGŽCE>TH8$CE> T9;@2:<;@2 Nx; :<;HN O5;@2:M= 6982E8(NH698FTH8t4F>@G92PCPOP6HABO4F>@G :<;@2:I87CPO5;@2TH8
6HG ; :lda ;lN 87CE> Σ 2E8:B:<; XY; )%"
%+3F "
"
> 3 %+(!X*,% )%'"
!V33 TH8 Σ 6
K 8aOE87NHOE8F2E87GMC5; Om
; 4F>@X> 6 1"2v87R AITH87GŽCE8 M= 698 8F2v6HG*4F>@G>J 6HGŽCE>ABGX9Q GHABCE> 6 1cG*8Ra 8F47CE> 87G-87:#Nl8F>@Oi4J;@2E> 6HG-; :lΣa ;N 87CE>*TH8(6HGx;2E>@:<;*:I87ΣCPO5; Σ ; 92E8CE87G9T\Om;.M= 698 ∗
∗
i
i
i
i
i
i
i
5
B
i
=M698ΣCPAI87G9=8 9{λ,>>NMa,RMA<; aa,
X87aaa,
GMCE8 x.6H.GV.}G HX87OE>ABGXQ9GHABCE>TH887:I87X87GŽCE>{2 6
g%!)%"\) !V3!2!"
1"2PC5;2E8F4F47AIW@G[TH8R9Q G98; :7{I 6HGx;@2>@N87O5;@47AI>@G98F225>>HN OE887:?4F>@G>J 6HGŽCE>
TH8CE>MT9;@2:<;@2iNx;k
:<;HN O5;@24F>@G92PCPOP6HAIT9;@2 4F>@G :<;@2†:I87CPO5;@2iTH8(6HG ; :lda ;lN 87CE> Σ = {a , a Σ, a , . . . , a } 6
b% ) K 8J; GŒTH>{2YNx; :<;HN O5;@2 x, y ∈ Σ 6 K 8-2E6HNl>@G9T\Om; Ž= 698 |x| = i 6 K 8$:B:<; XY;\)!X
%'!) %D3 !" !V3!2!" x % y E 2E8$OE87NHOE8F2E87GŽC5;ABGXk
|y| = j
T\AI2ECPABGŽC5; X87GŽCE8 TH8:<;@2a >@OPXY;@2 > x.y ; >@CPO5;Nx; :<;HN O5; z Ž= 698a2E8>>HN CPAI87G98†; x;@T\AI87G9TH>
:<;@2i:I87CPO5;@2 >*2 BXjlN >@:I>{2TH8 y ;*xy
4F>@GMCPABGŽ6x;@47AIW@G[TH8:<;@2i:I87CPO5;@2i>2 BX@N >@:I>{2iTH8 x 6
=?>@O 8 J 87X*NH:I>92EA x = abbaa 8 y = bbbbb xy = abbaabbbbb 6
H ;G9>@C5;@47AIW@G x.y 2E86HCPAB:BA7h ; OmL; 476x; G9TH>LNH698FT9;NHOE>MT\6947ABOE2E8; :7{I 6HGx;L; XjHN A7I 98FT9;@T
2EAr2E8(6HCPAB:BA7h ;:<;*G9>@C5;@47AIW@G xy x>476x; G9TH>258$TH8F2E8F88F2ENl8F47AlxQ 4J; O:<;*>@Nl87O5;@47AIW@G[TH8$4F>@GXk
4J; CE87Gx;@47AIW@G-X8FT\A<; GŽCE86HG 2 BX@N >@:I> 6
W 9N 2ED7OPRS8F2E8\M= 698*:<;4F>@G94J; CE87Gx;@47AIW@G vTH8R9Q GHAIT9; ;@2 r8F2(6HGx;8 \CE87G92PAIW@G TH8*:<;4F>@GXk
4J; CE87Gx;@47AIW@GTH86HG*2 BXjlN >@:I>4F>@G*6HGx;Nx; :<;HN O5;@C5; :94F>@X>2E8TH8R9Q GHAIW(87G 5$6 PV675 NH698F2o6HG
2 BX@N >@:I>t>t:I87CPO5;t2E8 NH698FTH84F>@G92PAITH87O5; Oi4F>@X>t6HGx;tNx; :<;HN O5;tTH86HG2E>@:I>2 BXjlN >@:I> +
E :<;
4F>@G94J; CE87Gx;@47AIW@G TH86HGx;*Nx; :<;HN O5;KŽ= 698$4F>@GMCPAI87G986HGx;2E>@:<;:I87CPO5;Y4F>@G->@CPO5;476x; : Ž= 6HAI87O5;
4F>@ABG947AITH84F>@G:<;*4F>@G94J; CE87Gx;@47AIW@G-TH8$:<;t:I87CPO5;*4F>@GV:<;*2E8,{I 6HG9T9;*Nx; :<;HN O5; 6
?` ; GŽCE>8F2PC5;(TH8R9Q GHAI47AIW@GYTH8i:<;4F>@G94J; CE87Gx;@47AIW@GTH8Nx; :<;HN O5;@24F>@X>(:<;@2cNHOE>@NHAI8FT9;@TH8F2
M= 698L; Nx; OE8F4F87G‰; 4F>@GŽCPABGM6x;@47AIW@GyNl> T\O <; G x9 ;lN 87OE2E88F2PN8F47AlxQ 4J;@TH>-TH8<a >@OPXY;VOPA7{I 6HOE>{25;
TH8;@476987OETH>[4F>@G :I>{2; AI>@XY;@2tTH8Y:<; 4F>@G92PCPOP694F47AIW@GTH8YNx; :<;HN O5;@2 6H ;VG98F4F8F2PAIT9;@TTH8
OE8J; :BA7h ; O(;>M= 6 r8F2PCE88F2 a 6987O Jh >TH8?a >@OPXY; :BA7h ;@47AIW@G8F2T\AI2E476HCPA7HN :I8 6YW NHABGx; X>{2:M= 698$87:?8F2 k
CP69T\A<; GMCE8VTH8,N 87O <;[2E87O*4J; Nx;V
h TH8 a >@OPXY; :BA7h ; O8F2PCE>{2*4F>@G94F87NHCE>{2Nl>@O*2 X*AI2PX>Dm
E TH8
OE8J; :BA7h ; O:<;@24F>@G92PCPOP694F47AI>@G98F2OE8F476HOE2PABR{;@2 G98F4F8F25; OPA<;@2 ; 6HGVŽ= 698t:<;8 N>{2PAI47AIW@G[2E8$NH698Rk
T9;<x9 ;@4F87O 4F>@X>;>Ž= 6 87G a >@OPXY;ABGŽCP6HABCPABR{; 6 =?>@O(87:B:I>L2E8tTH8 PJ ; G:<;@2TH87X>{2PCPO5;@47AI>@G98F2
a >@OPXY; :I8F2(Nx; O5;L:I>{28 J 87OE47AI47AI>{2 6 1"2PCE>LG9>V2PA7{I GHAlxQ 4J; M= 698*87:cCPO5;xN ;J > TH8ia >@OPXY; :BA7h ;@47AIW@G
258J;[X87O5; X87GŽCE8OE87CEW@OPAI4F>"8F287RMAITH87GMCE8 M= 6986HGx; Nl87OE2E>@Gx; 4J; Nx;h TH8V8Ra 8F47CP6x; OTH8Rk
X>{2ECPO5;@47AI>@G98F2iOPA7{I 6HOE>{25;@2TH8NHOE>@NHAI8FT9;@TH8F2 2E87G947AB:B:<;@2 x4F>@X>*:<;@2FM= 698(2E8 ABG9T\AI4J; G;>M= 6 2587Om
; 4J; Nx;h TH8 ; Gx; :BA7h ; O†87GNHOE>a 6HG9T\AIT9;@TL>@CPOE>{2aNHOE>>HN :I87XY;@2X<@; 2TH87:BAI4J;@TH>{2 Ž= 6982E8:I8
NH:<; GMCE8F87G 6
H ;*4F>@G94J; CE87Gx;@47AIW@G[TH8Nx; :<;HN O5;@2iCPAI87G98(:<;@2i2EA7{I 6HAI87GŽCE8F2NHOE>@NHAI8FT9;@TH8F2
H ;4F>@G94J; CE87Gx;@47AIW@Gy8F26HGx;->@N87O5;@47AIW@G4F87OPO5;@T9;87G87:a4F>@G>J 6HGŽCE> TH8Nx; :<;HN O5;@2
•
6
Σ
=cOE>@NHAI8FT9;@T-;@2E> 47A<; CPABR{; x(yz) = (xy)z 6
•
=?>@O†476HX*NH:BABO:<;@2aTH>{2"NHOE>@NHAI8FT9;@TH8F2i; GMCE87OPAI>@OE8F2 \:<;$4F>@G94J; CE87Gx;@47AIW@GVTH8:<;@2"Nx; :<;k
HN O5;@2 TH86HG-; :lda ;N 87CE>*8F2†6HG "
%+4 (T> & 6
∗
∗
1
2
3
N
∗
∗
i
i
i
i
i
i
i
•
•
•
•
i
1 AI2ECE87G947A<;*TH887:I87X87GŽCE>*G9876HCPOE> 6VH ;tNx; :<;NHO5;tR@;@4 <; 8F2†87:/87:I87X87GŽCE>*G9876HCPOE>
TH8":<;4F>@G94J; CE87Gx;@47AIW@GTH8Nx; :<;NHO5;@2 C5; GMCE> Nl>@O?:<; TH87OE8F4mλ9x;4F>@X> N>@Or:<; A7h
=Ž6HAI87O k
T9; 6 1cGR8 a 8F47CE> S2E8J; x 6HGx; Nx; :<;NHO5;476x; : =Ž6HAI87O5; 6 K 8†476HX*NH:I8F=Ž698 λx = xλ = x 6
H ; H:BCPABXY;7A I{6x; :IT9;@T[2E8tTH87X698F2ECPO5\; a ;@47AB:BX87GŽCE8tN>@OOE8F476HOE2EAIW@G[87G x l; 6HGV=Ž698
8F2i>> NŽR A<;;GHABRS87:rABGŽCP6HABCPABRS> 6
=?>@Oa476HX*NH:BABO":<;@2cCPOE8F2"NHOE>@NHAI8FT9;@TH8F2a; GMCE87OPAI>@OE8F2 S:<;4F>@G94J; CE87Gx;@47AIW@GLTH8†Nx; :<; NHO5;@2
8F2†6HG 4 ))% 6
H ;t4F>@G94J; CE87Gx;@47AIW@G TH8 Nx; :<; NHO5;@2G9>CPAI87G98 :<;NHOE>@NHAI8FT9;@T4F>@GHX6HC5; CPABR{; 6 132aCPOPlA k
RMA<; : 9x; :B:<; OL4F>@GŽCPO5;@8 JP87X*NH:I>{2 ABG947:B692E>m4F>@GfNx; :<; NHO5;@2TH8V6HGx; 2E>@:<; :I87CPO5; ":<;@2
Nx; :<; NHO5;@2 a E b RS87OPlA Qx4J; G =M698 ab G9>8F2 7A I{6x; :c; ba ‡2PAI87X*NHOEi
8 =M698 a E b 2E8J; G
4J; O5;@47CE87OE8F2T\AI2ECPABGŽCE>{2 6
=cOE>@NHAI8FT9;@T TH8-4J; G94F87:<;@47AIW@G N>@O:<; 7A h
=M6HAI87OET9; 6 =v; O5;mCE> TH> a, b, c ∈ Σ∗ 258
476HX*NH:I@
8 =M698 ab = ac =⇒ b = c 6
`?; @X NHAID7G[2E8(476HX*NH:I8(:<;*:I,8 ETH8(4J; G94F87:<;@47AIW@GNl>@O†:<;TH87OE8Fm4 9x; ac = bc =⇒ a = b
6
b% ) K 8J;
6 K 8T\AI4F8?=Ž698 x 8F2 !2&%!TH8 z +E =Ž698 y 8F2 3!
HT 8 z 6 U†TH87X<;@2 x 8F2S!2gz%=! xy
!VTH8
2PA G9>Y8F2:<;*Nx; :<;NHO5;R@;@4 <; 6)M 8$A7I{6x; :
XY; 9G 87O5; y 8F2F)3! !LTH8 z 2PA x G9>*8Fz2i:<;yNx; :<;NHO5;R{;@4 <; 6
=?>@O8 J 87X*NH:I>?2E8J; x = abc 6H ;@22EA7I{6HAI87GŽCE8F2Nx; :<;NHO5;@22E>@G 4J;Nl8,h ; TH8 x λ a
6XH ;@22P7A I{6HAI87GMCE8F2†Nx; :<; NHO5;@2†2E>@G4F>@:<;tTH8 x λ c bc abc 6 `r> T9;@2†2E>@GNHOE>@NHA<;@2
ab abc
8 H4F87NHCE> abc 6
K 8V>> N92E87OPR{; mO ; =Ž698L:<D
; a 6HG947AIW@G S\LHWhT3]sJqVFN NhfVXWECEHS\E&c% GQEVCPAI87G98 cOE8F2ENl8F47CE> ; :<;
4F>@G94J; CE87Gx;@47AIW@G ?6HGx;NHOE>@NHAI8FT9;@T 2E87X8 JE; GŽCE8;:<; a 6HG947AIW@G SmLZTeEHGp]qJq[RLOE8F2ENl8F47CE> ;:<;
X6H:BCPABNH:BAI4J;@47AIW@GTH8$G HX87OE>{2iOE8J; :I8F2 |xy| = |x| + |y| 6
b% ) K 8:B:<; XY; &V
%'! "Z 4!TH8 6HGx;Nx; :<; NHO5; a :<;t>@Nl87O5;@47AIW@G
=M6984F>@G92PAI2PCE8L87G‰4F>@G94J; CE87Gx; OP:<;[4F>@G92P7A IS>X*AI2PXY; i RS8F4F8F2 6 a>@X>:<;-4F>@G94J; CE87Gx;@47AIW@G
CPAI87G98t:<;NHOE>@NHAI8FT9;@T ;@2E> 47A<; CPABR@; lG9>L8F2NHOE8F47AI2E>L8F2PN8F47lA Qx4J; O(87:o>@OETH87G 87G =M698CPAI87G987G
=M698(R8 a 8F47CP6x; OE2E8(:<;@2 i >@N87O5;@47AI>@G98F2 6
RS8F4F8F2
xi = xxx . . . x i
132PCPOPAI47C5; X87GŽCE<
8 9x; NH:<; G9TH> :<;Nl>@CE87G947A<;G9>L8F26HGx;>@Nl87O5;@47AIW@G GM6987R@; #2EABG9>6HGx;
G9>@C5;@47AIW@G-Nx; O5;OE8FT\6947ABO; 7: I{6HG9>{24J;@2E>{2iTH8:<;*>@N87O5;@47AIW@G 4F>@G94J; CE87Gx;@47AIW@G 6
K 8$THR8 Q9G987GVC5; @
X NHAID7G x1 = x XE x0 = λ 6
132i87R AITH87GŽCE@
8 =M698(2E8RS87OPlA Qx4J<
; =Ž698 xi xj = xi+j (i, j ≥ 0)
i
i
i
i
i
i
i
5,P
1YJ 87X*NH:I>‡2E8J;
i
6 13GŽCE>@G94F8F2
x = abc
2
x = xx = abcabc
x3 = xxx = abcabcabc
`?; @X NHAID7G87G*8F2PCE84J;@2E> @:<;@2oNHOE>@NHAI8FT9;@TH8F2"TH8a:<; a 6HG947AIW@GS\LHWhT3]sJqVFN†2E>@G*2E87X8 JE; GŽCE8F2
; :<;@2"TH87:‡:I>>IŽ; OPABCPX> |xi| = i × |x| ŽTH>@G9TH8 87:‡2PA7I{G9> × OE87NHOE8F2E87GMC5;$:<;$X6H:BCPABNH:BAI4J;@47AIW@G
$
TH8$G HX87OE>{2 87GMCE87OE>{2 6
b% ) K 8J; x = σ σ . . . σ 6 K 8:B:<; XY; !V3!2! %%*,! > !V3 !2!
%"
! TH8 x E 2E8*OE87NHOE8F2E87GMC5; x ; x = σ . . . σ σ /8F2TH8F47ABO v;:<;b=M698Y8F2ECm;
a >@OPXY;@T9;N>@O†:<;@2†X*AI2PXY;@2 :I87CPO5;@2i87G >@OETH87GVABGMRS87OE2E> 6
H ;ia 6HG947AIW@GV:I>@G{I ABCP69T 8F2ABGMR@; OPA<; GMCE84F>@GVOE8F2ENl8F47CE>;:<;tOE8cx8 \AIW@G TH8(Nx; :<;HN O5;@2 1 2
R
n
R
n
2 1
|xR | = |x|
! &
-/'5%&+P-/4
b% ) K 8 :B:<; XY; 3%'()!+*,%*25>>NHOE8 87:#; :lad;Nl87CE> Σ ;476x; : M= 6HAI87Oi2P6lN 4F>@GXk
J 6HGŽCE>YTH87:/:I87GI{6x;J 8$6HGHABRS87OE25; :rTH8 Σ 6
EŒNl>@OV87:B:I>
1cG Nx; OPCPAI476H:<; O 87:4F>@G>J 6HGMCE>‰R@;@4 I> 8F2V6HG 2P6N4F>@G>J 6HGMCE>yTH8
258*:B:<; XY;LC5; X@HN AID7G03%'()!+*,% ! 6 1"2P∅CE8:I87G{I 6x;JP8*G9> TH8,N84F>@GXa 6HΣG9T\ABOE2E84F>@Gm87:
M= 6984F>@GŽCPAI87G98L4F>@X> HGHAI4F>-87:I87X87GMCE>[:<;VNx; :<;HN O5; R@;@4 <; {λ} =M698YC5; XjNHAID7G8F26HG
2E6lN 4F>@G>J 6HGŽCE>TH8 Σ JT\Ala 87OE87GŽCE8TH87: ; GŽCE87OPAI>@O 6 =o; O5;iT\AI2PCPABG{I 6HABOP:I>{2 ZxN ;@2PC5;iT9; OE2583476987GŽC5;
TH8 M= 69887:lG
HX87OE>*TH8 87:I87X87GŽCE>{2iTH8:I>{2TH>{24F>@G>J 6HGMCE>{2†8F2aT\AI2ECPABGŽCE> 5 87GL87:4J;@2E>TH8
E c 87G-87:/TH8 ∅ 6
{λ}
W 9N 2ED7OPRS8F2E8@M= 698$C5; GMCE> 4F>@X> 2E>@GV:I87G{I 6x;PJ 8F2:Ž= 698$NH698FTH87GTH8R9Q GHABOE2E8t2E>>HN OE8
476x; : M= 6HAI87O ; :lda ;lN 87CE> 6 =v>@Oi>@CP∅O5;tNx; OPCE8 {λ}
H87:#NHOE>@NHAI>; :la ;N 87CE>NH698FTH8(4F>@G92PAITH87O5; OE2E8(4F>@X>
6HG9>-TH8*:I>{2$:I87G{I 6x;PJ 8F21SI 87G987O5;@TH>{2tN>@OD7: r;>M= 69D7:&M= 6984F>@GŽCPAI87G98CE>MT9;@2:<;@2$Nx; :<;HN O5;@2
N>{2PA7HN :I8F2iTH8$6HGx;2E>@:<;:I87CPO5; 6
K 8iABGŽCPOE> T\694F8;99 >@O5;(6HGx;(G9>@C5;@47AIW@G Ž= 6982PABX*NH:BAlxQ 4J;$87G; :7{I 6HG9>{24J;@2E>{2:<;(TH8F2E47OPABNXk
47AIW@GTH8:I>{23:I87G{I 6x;PJ 8F2 6 1cG:I>$2E694F8F2PABRS> {2PA987:B:I>(G9>$T9; :B6ŽI ; Oa;(8Ž= 6 BRS> 4F> {2E8†TH87G9>@C5; Om;
4F>@G 87:c2 BXjlN >@:I> x 87:v:I87G{I 6x;J 8 {x} a >@OPXY;@TH> HGHAI4J; X87GŽCE8*N>@O:<;YNx; :<;HN O5; x ∈ Σ 6
=v>@O8 J 87X*NH:I> l87:?:I87G{I 6x;PJ 8iŽ= 6982EW@:I>Y4F>@GMCPAI87G98:<;Nx; :<;HN O5;R@;@4 <;Y2E8TH87G9>@C5; Om; 2PA?G9>
T9;t:B6ŽI ; O ;*4F>@GXa 692EAIW@G a4F>@G 87:/2 BX@N >@:I> λ X
E 87:#:I87G{I 6x;PJ 81M= 698$4F>@GŽCPAI87G98 HGHAI4J; X87GMCE8
;:<;Nx; :<;HN O5;<a >@OPXY;@T9;N>@O :<; HGHAI4J;:I87CPO5; σ 2E8tTH87G9>@C5; Om
; A7{I 6x; :BX87GŽCE8*N>@O σ 6W NXk
25D7OPRS8F2E8 Ž= 6984F>@Gy8F2PCE>{24F>@GŽRS87GHAI>{2.x9 ;
E 47AI87OPCE>D{I O5;@TH>[TH8; X@HN A7I 98FT9;@Ty87Gy; :7{I 6HGx;@2
8 \NHOE8F2PAI>@G98F2 9NH698F2 σ NH698FTH8TH87G9>@C5; O6HGx;:I87CPO5;H6HGx;Nx; :<;HN O5;Y>6HGV:I87G{I 6x;J 8 6 >@O k
XY; :BX87GMCE8 87:94F>@GMCE8 CE>$87GKŽ= 698i; Nx; OE8,Jh 4J;:<;8 \NHOE8F2PAIW@GTH87CE87OPX*ABGx; Om
; ;:Ž= 69D†2E8OE8RxQ 87OE8
87:/2 BX@N >@:I> V
E 2EA#G9> a 698F258$;@2 /2E86HCPAB:BA7h ; Om*; :<;tG9>@C5;@47AIW@G-G9>; X@HN A7{I 6x; 6
∗
∗
∗
i
i
i
i
i
i
i
i
[ g%!)%"\) 3%+(!+*%"
b% ) ^ K 8J; GVTH>{2 :I87GI{6x;J 8F2 TH8RQ9GHAITH>{22E>>NHOE8(87:rX*AI2PX>; :la ;N87CE>
6
L ,L ⊂ Σ
K 8(:B:<; XY;\) TH8:I>{2iTH>{2i:I87GI{6x;JP8F2 L + L x; :#:I87GI{6x;JP8$TH8RQ9GHAITH>;@2 8F2"THL8F47ABO +Ž; L:x4F=
>@G>J {x|x
6HGŽCE>:∈a >@LOPXY∨;@THx>(∈N>@LO3:<};@2cNx; :<;HN O5;@2&=M698N87OPCE87G98,hJ4J; GYABG9T\AI2PCPABGMC5; X87GŽCE8
;t6HG9>6 >@CPOE>*TH8(:I>{2iTH>{2†:I87G{I 6x;PJ 8F2 6
=?>@OoO5;Jh >@G98F2H9 AI2PCEW@OPAI4J;@2 2E8"2E6987:I836HCPAB:BA7h ; Oc87:M2 BXjlN >@:I>yNx; O5;†OE8Ra 87OPABOE2E8a;i:<;i6HGHAIW@G
TH8$:I87G{I 6x;J 8F2 H87G RS8,
h TH87:/2 BX@N >@:I>TH8(6HGHAIW@G-TH8(4F>@G>J 6HGMCE>{2 6
M 8;@476987OETH>4F>@G8F2PC5;*TH8R9Q GHAI47AIW@G E :I>{2 4F>@GŽRS87GHAI>{2ABG9T\AI4J;@∪TH>{2 X<@; 2; OPOPA7xN ;x476x; :lk
M= 6HAI87O :I87G{I 6x;J 8ja >@OPXY;@TH>N>@O 6HG[G HX87OE><9Q GHABCE>LTH8Nx; :<;HN O5;@2 {x , x , . . . , x } 2E8
NH698FTH88F2E47OPA7HN ABO 87GV:<;ja >@OPXY;
6
x + x + ... + x
H ;6HGHAIW@G TH8(:I87G{I 6x;PJ 8F2†CPAI87G98(:<;@22PA7{I 6HAI87GŽCE8F2 NHOE>@NHAI8FT9;@TH8F2
W Nl87O5;@47AIW@G 4F87OPO5;@T9; 9:<;t6HGHAIW@GVTH8(TH>{2†:I87G{I 6x;J 8F2 2E>>HN OE8(87:#X*AI2PX>; :lda ;lN 87CE>*8F2
•
C5; X@HN AID7G-6HGV:I87G{I 6x;J 82E>>HN OE8$T\AI4 99 >; :la ;N 87CE> 6
=cOE>@NHAI8FT9;@T-;@2E> 47A<; CPABR{; (L + L ) + L = L + (L + L ) 6
•
=?>@O$476HX*NH:BABO(:<;@2TH>{2 NHOE>@NHAI8FT9;@TH8F2; GMCE87OPAI>@OE8F2 :<;6HGHAIW@G[TH8:I87G{I 6x;PJ 8F2(8F26HG
"
%'4 (T> g 6
1 AI2ECE87G947A<;†TH8387:I87X87GŽCE> G9876HCPOE>J476x; : M= 6HAI87O5;FM= 698c2E8J;87:S:I87G{I 6x;PJ 8 L F87:{:I87G{I 6x;k
•
J 8R{;@4 I>476HX*NH:I81M= 698 ∅ + L = L + ∅ = L 6
=?>@O(476HX*NH:BABO :<;@2 CPOE8F2NHOE>@NHAI8FT9;@TH8F2(; GŽCE87OPAI>@OE8F2 l:<;6HGHAIW@GTH8:I87G{I 6x;J 8F28F2i6HG
4 % 6
=cOE>@NHAI8FT9;@T4F>@GHX6HC5; CPABR{; v476x; :I8F2mŽ= 6HAI87O5; Ž= 6982E8J; G L E L ?2E8RS87OPAlxQ 4J; M= 698
•
6
L +L =L +L
=?>@O CE87G987O :<;@2 476x; CPOE>NHOE>@NHAI8FT9;@TH8F2; GMCE87OPAI>@OE8F2 ‡:<;*6HGHAIW@G-TH8:I87G{I 6x;J 8F2 8F2i6HG
4 %e!2g%+3 !V 6
=cOE>@NHAI8FT9;@TLAITH87X*Nl>@CE87GMCE8 H476x; : Ž= 6HAI87O5;iM= 698258J; L Ž2E8RS87OPAlxQ 4J;iŽ= 698 L + L = L 6
•
b% ) K 8J; G TH>{2 :I87G{I 6x;J 8F2-TH8R9Q GHAITH>{2-2E>>HN OE8 87:$X*AI2PX>f; :lda ;lN 87CE> TH8:I>{2 TH>{2†:I87G{I 6x;PJ 8F2 L L > x; :BCE87OPGx;k
6 K 8(:B:<; XY;.!X
%+!' 
L ,L ⊂ Σ
CPABR{; X87GŽCE8 L .L 9; :r:I87G{I 6x;J 8$TH8R9Q GHAITH>;@2 8F2(THL8F47ABLO :<=;@2 {xy|x
Nx; :<;HN ∈O5;@2$L TH,87y:c∈:I87GL{I 6x};PJ 84F>@G94J; CE87Gx;@47AIW@G TH8>@CPOE>{2(TH>{2(2E8ja >@OPXY; G[4F>@GXk
4J; CE87Gx; G9TH>Y6HGx;Nx; :<;HN O5;*TH87:/NHOPABX87O :I87G{I 6x;J 8$4F>@G->@CPO5;*TH87:/2E8,{I 6HG9TH> 6
1
∗
2
1
1
2
1
2
2
1
1
2
2
N
N
1
2
3
1
2
3
1
1
2
1
2
i
i
2
1
∗
1
1 2
2
1 2
2
1
2
i
i
i
i
i
5
i
H ;*4F>@G94J; CE87Gx;@47AIW@G[TH8:I87GI{6x;J 8F2 CPAI87G98(:<;@2i2PA7I{6HAI87GMCE8F2iNHOE>@NHAI8FT9;@TH8F2
W Nl87O5;@47AIW@G 4F87OPO5;@T9; :<; 4F>@G94J; CE87Gx;@47AIW@G TH8-TH>{2:I87GI{6x;JP8F22E>>NHOE8-87: X*AI2PX>
•
; :la ;N 87CE>8F2i>@CPOE>:I87G{I 6x;J 8$25>>HN OE8(87:/X*AI2PX>Y; :lad;Nl87CE> 6
=cOE>@NHAI8FT9;@T-;@2E> 47A<; CPABR{; (L L )L = L (L L ) 6
•
=?>@Oo476HX*NH:BABOo:<;@2rTH>{2?NHOE>@NHAI8FT9;@TH8F23; GŽCE87OPAI>@OE8F2 :<;i>@N87O5;@47AIW@GTH8"4F>@G94J; CE87Gx;@47AIW@G
TH8(:I87G I{6x; JP8F2 8F2†6He
G "
%+4 (T> g 6
1 AI2ECE87G947A<;†TH8387:I87X87GŽCE> G9876HCPOE> J476x; : =M6HAI87O5;F=M698c2E8J;87:S:I87GI{6x;JP8 F87:{:I87GI{6x;k
•
J 8 a >@OPXY;@TH>*N>@Oi:<;tNx; :<; NHO5;R{;@4 <;*476HX*NH:I18 =M698 λL = Lλ = L 6 L
=?>@Oo476HX*NH:BABOo:<;@2rTH>{2?NHOE>@NHAI8FT9;@TH8F23; GŽCE87OPAI>@OE8F2 :<;i>@N87O5;@47AIW@GTH8"4F>@G94J; CE87Gx;@47AIW@G
TH8(:I87G I{6x; JP8F2 8F2†6HG 4 ))% 6
1c:"4F>@>G J 6HGŽCE> TH8:I>{2(:I87G I{6x; J 8F22E>> NHOE8*6HG ; l: ad; N87CE> XJ 6HGMCE>V4F>@G :<;@2>@Nl87O5;@47AI>@G98F2
6HGHAIW@G E 4F>@G94J; CE87Gx;@47AIW@G o8F2$6HG 2 % NH698F2 r4F>@X>2E.
8 9x;VRMAI2PCE> r4J;@T9;6HGx;-TH8
8F2 ;@2 >@N87O5;@47AI>@G98F2(8F26HGX>@G9>@AITH8 6 U†TH87<
X ;@2 l;Nx; OPCPABO(TH8:I>{2 :I87G I{6x; JP8FS2 a >@OPXY;@TH>{2
HGHAI4J; X87GMCE8Nl>@O:<;Nx; :<; NHO5;YR{;@4 <;Y>YNl>@O 6HGx; HGHAI4J;Nx; :<; NHO5;TH86HGx;2E>@:<;*:I87CPO5; )E
X8FT\A<; GMCE8L:<;@2>@Nl87O5;@47AI>@G98F2TH86HGHAIW@G E 4F>@G94J; CE87Gx;@47AIW@G 38F2tN>{2P7A NH:I8 IS87G987O5; O476x; l: k
=M6HAI87O†:I87G I{6x; JP8(2E>> NHOE86HG ; l: a ; N87CE> Σ 8 H4F87NHCE> ∅ 6 =?>@Oi:I>C5; GMCE> 987:r; l: ad; Nl87CE>*8F26HG
4F>@>G J 6HGŽCE>*THS8 IS87G987O5;@TH>@OE8F2†Nx; O5;t87:#4F>@>G J 6HGMCE> L TH8 CE> TH>{2:I>{2a:I87G I{6x; JP8F2 =Ž698 NH698FTH87G
THR8 Q9GHABOE258$2E>> NHOE8 Σ \N>@O†:I.
> =Ž698 L 2E8(TH87G9>@X*ABGx\
; 2 % 3 2%.IS87G987O5;@TH>YNl>@O Σ 6
b% ) T K 8†:B:<; XY; gT
%'!e "Z 4!*TH8i6HG*:I87G I{6x; J 8 ; :<;(>@Nl87O5;@47AIW@G
=M6984F>@G92PAI2PCE8L87G‰4F>@G94J; CE87Gx; OP:I> 4F>@G92P7A IS>X*AI2PX> i RS8F4F8F2 6 a>@X>:<;-4F>@G94J; CE87Gx;@47AIW@G
CPAI87G98t:<;NHOE>@NHAI8FT9;@T ;@2E> 47A<; CPABR@; lG9>L8F2NHOE8F47AI2E>L8F2PN8F47lA Qx4J; O(87:o>@OETH87G 87G =M698CPAI87G987G
=M698(R8 a 8F47CP6x; OE2E8(:<;@2 i >@N87O5;@47AI>@G98F2 Li = LLL . . . L i RS8F4F8F2 6
132PCPOPAI47C5; X87GŽCES8 9x; NH:<; G9TH> S8F2PC5;(G9>(8F2c6HGx;>@Nl87O5;@47AIW@GGŽ6987R{; S2PABG9>6HGx; G9>@C5;@47AIW@G
=M698(2PABX*NH:BlA Qx4J;; 7: I{6HG9>{2 4J;@2E>{2iTH8:<;>@Nl87O5;@47AIW@G 4F>@G94J; CE87Gx;@47AIW@G 6
K 8(THR8 Q9G987GC5; @
X NHAID7G 1
E 0 6 132†87R AITH87GŽCE?8 =M698(2E8 RS87OPlA Qx4J; =Ž698 Nx; O5;
CE> TH>:I87G I{6x; JP8 L ENx; O5;LCE>MTH=> Li, j ≥L 0 = λ
1
2
3
2
3
1
b% ) [ K 8:B:<; XY;i %'!' '%%>j3!V"Z!YTH86HGY:I87GI{6x;JP8 L 8F2rTHL8F47ABO =J87∪:S:I87GI{L6x;JP8g=Ž698c258c>>NHCPAI87G98c6HGHAI87G9TH> 87:{:I87GI{6x;JP8 H 4F>@GCE>MT9;@2/2E692Nl>@CE87G947A<;@2
N>{2PA7HN :I8F2 \ABG947:B6925> L 6
W ŽN R A<; X87GŽCE8 ‡:<;47:<; 692P6HO5;TH8(476x; : M= 6HAI87Oi:I87G{I 6x;PJ 8$4F>@GŽCPAI87G98:<;Nx; :<;HN O5;R{;@4 <; 6
b% ) K 8$TH8R9Q G98(:<;.3!V"Z! g)"Z Z !TH8(6HG:I87G{I 6x;PJ 8 L ;@2 Li Lj = Li+j
∗
∞
i=0 i
0
L+ = ∪ ∞
i=1 Li
i
i
i
i
i
i
i
4
i
H ;@2 2PA7I{6HAI87GMCE8F2iAITH87GMCPAIT9;@TH8F22E>@GV87R AITH87GŽCE8F2 L∗ = L + ∪ λ
L+ = LL∗ = L∗ L
H ;Y>@N87O5;@47AIW@G TH847:<; 692E6HO5;?J 692PCPAlQx4J;Y:<;*G9>@C5;@47AIW@G l6HCPAB:BA7h ;@T9;87G-:<;LK 8F4F47AIW@G 5$6 P TH8 Σ Nx; O5;*87:3%'()!+*,% )%"
!V3vTH86HG-; :lad;Nl87CE> Σ 6
b% ) K 8J; L 6HG:I87GI{6x;JP8 476x; : =M6HAI87O5; 6 K 8 :B:<; XY;i3%+(!+*% %%*,
E 2E8OE87NHOE8F2587GŽC5; L 9;
>\ %"
TH8 L X
8F2 TH{x
8F47ABO |xx; ∈:M= L}
698(4F>@GŽCPAI87G98$:<;@2iNx; :<;HN O5;@2iABGŽRS87OE2 ;@2;:<;@2iTH8 L 6
=c698F2PCE>?M= 698:I>{2o:I87G{I 6x;PJ 8F2c2E>@G*4F>@G>J 6HGŽCE>{2 @C5; XjHN AID7GNH698FTH87GTH8R9Q GHABOE2E8:<;@2o>@Nl87O5;k
47AI>@G98F2tTH8K %'"
%') E)4 3%'4 %'
!) 4F>@GmOE8F2PN8F47CE>-; :":I87G{I 6x;PJ 86HGHAlk
RS87OE2 ; : 6 M ;@TH> M= 698DF2PC5;@2Y2E>@GŒ>@Nl87O5;@47AI>@G98F2L47: @; 2PAI4J;@2LTH8 CE8F>@O <; TH8-4F>@G>J 6HGMCE>{2 aG9>
8F2NHOE8F47AI2E>TH87C5; :B:<; OP:<;@2$;>Ž= 6 6)d 8F4769D7OETH8F2E8jŽ= 69887:v4F>@G>J 6HGŽCE> %"
%' 
TH8>@CPOE>{2
TH>{24F>@GMCPAI87G98L:I>{2t87:I87X87GMCE>{2\M= 6982E887G9476987GŽCPO5; G…87G; XjlN >{2 L ∪ L = {w|w ∈
E 87:4F>@X*NH:I87X87GMC5; OPAI>[TH86HG 4F>@G>J 6HGŽCE>4F>@G OE8F2ENl8F47CE>;V>@CPOE> M= 698:I>
L ,w ∈ L }
4F>@GMCPAI87G988F2PCm; a >@OPXY;@TH> N>@O/:I>{2v87:I87X87GŽCE>{2cTH87: 2E8,{I 6HG9TH>SM= 6983G9>8F2PCm; G87Gt87:ŽNHOPABX87OE>
∗
R
R
1
1
2
2
Lc = Σ ∗ −L = {w|w ∈ Σ∗, w ∈
/ L}
” D
Ž)€f}=‡1}=‡Z–

E
6+M 8F47ABOa476T; :I8F22E>@GY:<;@2"Nx; :<;NHO5;@2]=M698 258 >>NHCPAI87G987GL4F>@X>
OE8F2P6H:BC5;@TH>TH8(:<;@2 2PA7I{6HAI87GŽCE8F2>@Nl87O5;@47AI>@G98F2 xx xxx x x x 6 6T; :I8F2i25>@G
2P692†:I>@GI{ABCP69TH8F2 M 8RQ9GHABO Σ 6
K 8J; Σ = {a, b, c} x = aa y = b z = cba 6M 8R9Q GHABOi:<;@2i2PA7{I 6HAI87GŽCE8F2 Nx; :<;NHO5;@2
2E>@G 2P692 :I>@G I{ABCP69 TH8F2x 4J;N 8,h ;@ 2 E 4F>@:<;@ 2 6 6T; :I8F2
xx xy xz yx yy yz zx zy zz xyz x x z (xz) (yxx)
K 8J; Σ = {a, b, c} 6 132E47OPA7HN ABO†2E87AI2†TH8 :<;@2†4J;@TH87Gx;@2X<@; 2†4F>@OPC5;@2†TH8 Σ E TH8 Σ 6
M 8R9Q GHABO(:<;4F>@G94J; CE87Gx;@47AIW@G TH8Nx; :<;HN O5;@2(TH8ja >@OPXY;OPA7{I 6HOE>{25;#TH8t;@476987OETH>4F>@G
:I>{23; AI>@XY;@2cTH8:<; 4F>@G92PCPOP694F47AIW@GTH8aNx; :<;HN O5;@2 6 1 \NHOE8F25; OcTH8a:<; X*AI2EXY; XY; G987O5;
:<;@2†NHOE>@NHAI8FT9;@TH8F2 TH8$8F2PC5;>@N87O5;@47AIW@G 6
M 8R9Q GHABO :<;4F>@G94J; CE87Gx;@47AIW@G[TH8:I87G{I 6x;J 8F2TH8:a >@OPXY;*OPA7{I 6HOE>{25;9TH8;@476987OETH>4F>@G
:I>{23; AI>@XY;@2cTH8:<; 4F>@G92PCPOP694F47AIW@GTH8aNx; :<;HN O5;@2 6 1 \NHOE8F25; OcTH8a:<; X*AI2EXY; XY; G987O5;
:<;@2†NHOE>@NHAI8FT9;@TH8F2 TH8$8F2PC5;>@N87O5;@47AIW@G 6
5$6 K 8J;
Σ = {$}
x=$
3
BX6
5
0 ∗
3
2 2
2
3 XG 6
PV6
LX6
i
i
+
∗
i
i