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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
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3
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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
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:<;: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$476HX*NH:I87G[:<;@2
TH>{2iNHOE>@NHAI8FT9;@TH8F22PA7{I 6HAI87GŽCE8F2
inicial(x).resto(x) = x
inicial(σ.x) = σ
resto(σ.x) = x
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U 6HVG =M698;:I>:<; O IS>YTH87:r:BA7NHOE>Y2E8(6HCPAB:BA7h ; Om;4F>@G92PC5; GŽCE87X87GMCE8t87:?4F>@G94F87NHCE>TH8(Nx;k
:<;NHO5;4F>@X>Y2PA/2E8CPO5; C5; O5;YTH8(6HGx;*4J;@TH87Gx;YTH8(4J; O5;@47CE87OE8F2 9CE> TH>{2i:I>{2i4F>@G94F87NHCE>{2 =M698
6HCPAB:B7A h ; GNx; :<; NHO5;@2 4F>@X>87:9:I87GI{6x;JP8;@4F87NHC5;@TH>>(OE8F4F>@G9> 47AITH>Nl>@O6HG; 6HCEW@XY; C5; \87:
:I87G I{6x; JP8 IS87G987O5;@TH>>(OE87NHOE8F2E87GŽC5;@TH>N>@Oc6HGx:
; I{O5; <
X ; CPAI4J;>N>@O36HGx;(8 \NHOE8F2PAIW@GOE8,I{6Xk
:<; O 87CE4 6 /258cNH698FTH87Gt8 \CE87G9TH87OvABGHX8FT\A<; C5; X87GŽCE8i;Nx; :<; NHO5;@2 Nx;@25;@T9;@2v87G:I>{2o; AI>@XY;@2
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X ;@2 ; X*NH:BAI> 6
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ad; IŽ;[:I>{2; AI>@XY;@2L; GŽCE87OPAI>@OE8F2X8FT\A<; GŽCE88F2PCPOP6947CP6HO5;@2TH8 T9; CE>{287Gf6HGf>@OETH87Gx;@TH>@O
2E6HNl>@GHAI87G9THj
> =Ž698CE87G IŽ;$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; )%"
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!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 ∗
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1"2PC5;2E8F4F47AIW@G[TH8R9Q G98; :7{I 6HGx;@2>@N87O5;@47AI>@G98F225>>HN OE887:?4F>@G>J 6HGŽCE>
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:<;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
•
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=cOE>@NHAI8FT9;@T-;@2E> 47A<; CPABR{; x(yz) = (xy)z 6
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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
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=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
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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
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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$G HX87OE>{2iOE8J; :I8F2 |xy| = |x| + |y| 6
b% ) K 8:B:<; XY; &V
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=M6984F>@G92PAI2PCE8L87G‰4F>@G94J; CE87Gx; OP:<;[4F>@G92P7A IS>X*AI2PXY; i RS8F4F8F2 6 a>@X>:<;-4F>@G94J; CE87Gx;@47AIW@G
CPAI87G98t:<;NHOE>@NHAI8FT9;@T ;@2E> 47A<; CPABR@; lG9>L8F2NHOE8F47AI2E>L8F2PN8F47lA Qx4J; O(87:o>@OETH87G 87G =M698CPAI87G987G
=M698(R8 a 8F47CP6x; OE2E8(:<;@2 i >@N87O5;@47AI>@G98F2 6
RS8F4F8F2
xi = xxx . . . x i
132PCPOPAI47C5; X87GŽCE<
8 9x; NH:<; G9TH> :<;Nl>@CE87G947A<;G9>L8F26HGx;>@Nl87O5;@47AIW@G GM6987R@; #2EABG9>6HGx;
G9>@C5;@47AIW@G-Nx; O5;OE8FT\6947ABO; 7: I{6HG9>{24J;@2E>{2iTH8:<;*>@N87O5;@47AIW@G 4F>@G94J; CE87Gx;@47AIW@G 6
K 8$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
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i
6 13GŽCE>@G94F8F2
x = abc
2
x = xx = abcabc
x3 = xxx = abcabcabc
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; :<;@2"TH87:‡:I>>IŽ; OPABCPX> |xi| = i × |x| ŽTH>@G9TH8 87:‡2PA7I{G9> × OE87NHOE8F2E87GMC5;$:<;$X6H:BCPABNH:BAI4J;@47AIW@G
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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
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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
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6
L ,L ⊂ Σ
K 8(:B:<; XY;\) TH8:I>{2iTH>{2i:I87GI{6x;JP8F2 L + L x; :#:I87GI{6x;JP8$TH8RQ9GHAITH>;@2 8F2"THL8F47ABO +Ž; L:x4F=
>@G>J {x|x
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;t6HG9>6 >@CPOE>*TH8(:I>{2iTH>{2†:I87G{I 6x;PJ 8F2 6
=?>@OoO5;Jh >@G98F2H9 AI2PCEW@OPAI4J;@2 2E8"2E6987:I836HCPAB:BA7h ; Oc87:M2 BXjlN >@:I>yNx; O5;†OE8Ra 87OPABOE2E8a;i:<;i6HGHAIW@G
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M 8;@476987OETH>4F>@G8F2PC5;*TH8R9Q GHAI47AIW@G E :I>{2 4F>@GŽRS87GHAI>{2ABG9T\AI4J;@∪TH>{2 X<@; 2; OPOPA7xN ;x476x; :lk
M= 6HAI87O :I87G{I 6x;J 8ja >@OPXY;@TH>N>@O 6HG[G HX87OE><9Q GHABCE>LTH8Nx; :<;HN O5;@2 {x , x , . . . , x } 2E8
NH698FTH88F2E47OPA7HN ABO 87GV:<;ja >@OPXY;
6
x + x + ... + x
H ;6HGHAIW@G TH8(:I87G{I 6x;PJ 8F2†CPAI87G98(:<;@22PA7{I 6HAI87GŽCE8F2 NHOE>@NHAI8FT9;@TH8F2
W Nl87O5;@47AIW@G 4F87OPO5;@T9; 9:<;t6HGHAIW@GVTH8(TH>{2†:I87G{I 6x;J 8F2 2E>>HN OE8(87:#X*AI2PX>; :lda ;lN 87CE>*8F2
•
C5; X@HN AID7G-6HGV:I87G{I 6x;J 82E>>HN OE8$T\AI4 99 >; :la ;N 87CE> 6
=cOE>@NHAI8FT9;@T-;@2E> 47A<; CPABR{; (L + L ) + L = L + (L + L ) 6
•
=?>@O$476HX*NH:BABO(:<;@2TH>{2 NHOE>@NHAI8FT9;@TH8F2; GMCE87OPAI>@OE8F2 :<;6HGHAIW@G[TH8:I87G{I 6x;PJ 8F2(8F26HG
"
%'4 (T> g 6
1 AI2ECE87G947A<;†TH8387:I87X87GŽCE> G9876HCPOE>J476x; : M= 6HAI87O5;FM= 698c2E8J;87:S:I87G{I 6x;PJ 8 L F87:{:I87G{I 6x;k
•
J 8R{;@4 I>476HX*NH:I81M= 698 ∅ + L = L + ∅ = L 6
=?>@O(476HX*NH:BABO :<;@2 CPOE8F2NHOE>@NHAI8FT9;@TH8F2(; GŽCE87OPAI>@OE8F2 l:<;6HGHAIW@GTH8:I87G{I 6x;J 8F28F2i6HG
4 % 6
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•
6
L +L =L +L
=?>@O CE87G987O :<;@2 476x; CPOE>NHOE>@NHAI8FT9;@TH8F2; GMCE87OPAI>@OE8F2 ‡:<;*6HGHAIW@G-TH8:I87G{I 6x;J 8F2 8F2i6HG
4 %e!2g%+3 !V 6
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•
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6 K 8(:B:<; XY;.!X
%+!' 
L ,L ⊂ Σ
CPABR{; X87GŽCE8 L .L 9; :r:I87G{I 6x;J 8$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
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1
2
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1
1
2
2
N
N
1
2
3
1
2
3
1
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1 2
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W Nl87O5;@47AIW@G 4F87OPO5;@T9; :<; 4F>@G94J; CE87Gx;@47AIW@G TH8-TH>{2:I87GI{6x;JP8F22E>>NHOE8-87: X*AI2PX>
•
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=cOE>@NHAI8FT9;@T-;@2E> 47A<; CPABR{; (L L )L = L (L L ) 6
•
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TH8(:I87G I{6x; JP8F2 8F2†6He
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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
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=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
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i
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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}
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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
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