p D (1/(1 q)) = p/p 2 = 1/p

! " #$% & %' () *++, -./0123 2/1456726 8/9:3;. /05< 2/=3 /0 543 93053; <= 543 ;<<>? @AA B;/.<03;..4CD3 543/; 3E3. =;<> 543 12C;3 F 5430 =<97. <0 543 B;/.<0 GC;D30.5C0D/01 6E 543 2/145.G/594 F G/54 C.5C0DC;D 3H/2IB7JJ23;K. 12/05 /0 4/. 3E3? L3 631/0. 5<.B3C:M NO0 <03 4<7;F E<7 G/22 C22 63 5C:30 5< E<7; 9322. 5< 63 :3B5 /0.<2/5C;E 9<0P03> 305 F G/54 0< B<../6/2/5E <= 9<> >70/9C5/01 G/54 C0E <= E<7; =322<G /0> C53.? Q322F C2><.5 0< B<../6/2/5E??? 3H3;E 0/145 =;<> 0<G <0F O G/22 94<<.3 <03 <= E<7 C5 ;C0D<> F ;35;/3H3 E<7 =;<> E<7; 9322F C0D 5C:3 E<7 5< 54/. ;<<> F G43;3 E<7 >CE.33 /= 543 2/1456726 /. RS <; RT F C0D E<7 >CE 57;0 /5 RS <; RT C. E<7 G/.4?N - >7;>7; ;/BB23. C;<70D 543 ;<<> C. 543 B;/.<03;. 9<0./D3; 543 B;<.B395 <= 4CH/01.794 C0 3U395 <0 543/; 4/543;5< /> B<5305 C0D 3V53;0C22E 9<05;<223D 3V/.I 53093.? NO= C5.<>3 B</05F O 5C:3 E<7 5< 54/. ;<<> C0D E<7 632/3H3 54C5 C22 @AA B;/.<03;. 4CH3 6330 94<.30 C0D 5C:30 43;3 C5.<>3 5/>3F 5430 E<7 >CE 5322 >3 54/.? O= E<7 C;3 9<;;395F O G/22 =;33 E<7 C22? O= <= 9<7;.3 E<7 C;3 /09<;;395??? G322 235K..CE 0<03 <= E<7 G/22 2/H3 5< 8/B C0E ><;3 2/1456726.G/5943. /0 54/. G<;2D?N L3 3V/5. G/54 C 8<7;/.4 <= 4/. 92<C:F 54<7145=722E 23CH/01 543 2/1456726 <0? W43 B;/.<03;. C;3 /0 543 DC;: C. 5< 4<G 5< 135 =;33F 675 543E C;3 B3;=3952E 923C; C6<75 GC05/01 5< 63 C623 5< C5 23C.5 8/B 2/145.G/5943. /05< <2D C13 XC0D /5 2<<:. 2/:3 543E >/145 033D 5<YZ? [< 543E >7.5 9<>3 7B G/54 C.5;C531E 54C5 G/22 C00<7093 54C5 C22 @AA B;/.<03;. 4CH3 6330 94<.30 <02E /= 543E C957C22E 4CH3 F G/54 @AA\ 93;5C/05EF 4<B3=722E 63=<;3 543E C22 D/3 <= <2D C13? -5 P;.5 /5.33>. />B<../623 54C5 C0E <03 B;/.<03; 9<72D :0<G C6<75 G4C5 543 <543; ]] 4CH3 6330 7B 5<? ^<>/01 /05< 543 ;<<> C0D.33/01 543 2/1456726 /. RS D<3.0K5.33> 5< 1/H3 E<7 >794 /0=<;>C5/<0? _<7 D<0K5 :0<G G4<.35 /5F C0D /= E<7 8/B 543.G/594 E<7 4CH3 0< /D3C G4< G/22.33 54C5 E<7 8/BB3D /5? W43;3 `abc bc dae fggehdei jecblm ln f hfhe dafd ofc hpqrbcaei bm dae s tuvwxtuyzt{ }uw{{y~ w}zw ƒ mlˆ Š hh Œ Žˆ `ae hpqrbcabm cdf fie cl e bml glegdblmcˆ `ae eibdlc fhhemiei f je cald qbl fha ln efga ln dae fpdalcˆ @

.33>. 5< 63 0< GCE 5<.30D C >3..C13 5< C0E<03 /0 BC;5/972C;? O5.33>. 4<B323.. 54C5 543E G/22 135 <75 C5 C22? 75 /0 =C95M S L3;3 /. 4<G? X_<7 >CE G/.4 5< B<0D3; <0 E<7; <G0 63=<;3 ;3CD/01 <0?Z! " # $!%%&&" ''' ( ) * + ^75 543.3,73093 <= DCE. /05< 62<9:. <= 230154 @AA? W43 P;.5 B;/.<03; 5< 3053; 543 ;<<> /0 C 1/H30 62<9: 57;0. 543 2/1456726 RT? O= C B;/.<03; 3053;. 543 ;<<> C.39<0D 5/>3 /0 543.C>3 @AA DCE 62<9:F 5430 43 57;0. 543 2/1456726 RS? O= C B;/.<03; 3053;. 543 ;<<> <0 543 2C.5 DCE <= C @AA DCE 62<9:F C0D /5 /. 4/. P;.5 5/>3F C0D 543 2/1456726 /..5/22 RT F 5430 54C5 B;/.<03; :0<G. 54C5 3H3;E B;/.<03; 4C. 6330 94<.30 3VC952E <093 /0 54/. @AA DCE 62<9:? L3 5430 9<;;3952E D392C;3. 54C5 C22 B;/.<03;. 4CH3 6330 /0 543 9305;C2 ;<<> C5 23C.5 <093? O= 543 2/1456726 /. RS <0 543 2C.5 DCE /0 C 62<9: 5430 G3 4CH3 =C/23D 54/. 5/>3.< G3 5;E C1C/0 /0 543 03V5 62<9: <= @AA DCE.F C0D :33B 5;E/01 705/2.<>3<03 C00<7093.? -./ * *0 )*123 1 R) 1 ) * + W43 B;<6C6/2/5E <=.79933D/01 /0 C0E 1/H30 62<9: /. 543 07>63; <= <;D3;/01. <= 543 @AA B;/.<03;. D/H/D3D 6E 543 07>63; <= B<../623 GCE. 543 B;/.<03;. 9<72D 4CH3 3053;3D 543 ;<<>? Q /54 B;/.<03;.F 54C5 /. n W43 (n!/n 3VB3953D n )? 07>63; <= 62<9:. G4/94 >7.5 63 7.3D 63=<;3.79933D/01 /. 3,7C2 5< G43;3 /. 543 B;<6C6/2/5E <=.79933D/01 G/54 <03 62<9:? W<.33 54/.F.7BB<.3 1/p /. 543 p 94C093 <=.79933D/01 /0 C0E 1/H30 62<9:? W430 543 3VB3953D 07>63; <= 62<9:. p 705/2 G3.79933D /. 3,7C2 5< /. 3,7C2 5< G43;3 pq 1 q = 1 p? W4/. D p D (1/(1 q)) = p/p 2 = 1/p q W47. 543 3VB3953D 07>63; <= 62<9:. /..< 543 3VB3953D 07>63; <= DCE. 705/2 =;33D<> n n /. /n!? 4C94 62<9: 4C. 230154 n X7./01 [5/;2/01K. =<;>72CZ? 5<; B;/.<03;.F 543 3VB3953D n n+1 /n! F HC273 G4/94 /. /. O(n 1/2 e n ) <; CBB;<V/>C532E? 100 100 101 /100! 10 44 W43E 9C0 135 <75F C254<714 54/. /. C D/.CBB</05/012E 2C;13 07>63; =<; 543 B;/.<03;.M C6<75 E3C;.? [CD2E 543 70/H3;.3 >CE 4CH3 30D3D 2<01 63=<;3 10 41 543E C;3 =;33 678? S 9 :; < =><? @? A!BC% BB&%D#E ( ) * + 9 F03 B;/.<03;F G4< G/22 63 :0<G0 C. G??F G/22 63 ;3.B<0I./623 =<; C00<709/01 54C5 3H3;E <543; B;/.<03; 4C. 3053;3D 543 ;<<> C5.<>3 5/>3? W43 <543; (n 1) B;/.<03;. G/22 63 <@=<? H

4C94 <;D/0C;E B;/.<03;.5C;5. G/54 C./0123 5<:30 F 9C223D C? F G4/94 543E G/22 5;E 5< 23CH3 /0 543 9305;C2 ;<<>? W43 ^<7053.. G/22 9<22395.<72. =;<> 543 9305;C2 ;<<> 705/2.43 4C. C22 <= 543>? [43 >CE 5430 D392C;3.7993..? Q3 >CE C..7>3 54C5 543 2/1456726.5C;5. /0 543 RT B<./5/<0 XC. 543 B;/.<03; G4< 3053;. <0 543 P;.5 DCE >CE 57;0 /5 RT 63=<;3 D</01 C0E54/01 32.3Z? Q 430 C0 <;D/0C;E B3;.<0 3053;. 543 ;<<> C0D P0D. 543 2/1456726 RT F 543E >CE D;<B 543/;.<72 /0 543 ;<<> F /= 543E 4CH3 0<5 C2;3CDE D<03.<F 6E 57;0/01 543 2/1456726 RS? O= 543 2/1456726 /. C2;3CDE RS 5430 543E 23CH3 /5 C2<03? Q 430 543 ^<7053.. 3053;. 543 ;<<> F /=.43 P0D. 543 2/1456726 RS 5430.43 57;0. /5 RT C0D CDD. <03 5< 43;.<72 9<705? O= 43; 9<705 /. 0<G n 1 5430.43 :0<G. 54C5 3H3;E<03 >7.5 4CH3 3053;3D 543 ;<<>.<.43 9C0 D392C;3? O= 543 2/1456726 /. RT G430.43 3053;..43 23CH3. /5 RT? -./ * *0 )*123 1 R) 1 ) * + 9 5<; 543.5;C531E 5< 9<>B2353 G3 033D 5< 4CH3 C.3,73093 <= 3H305. 4CBB30? Q3 P;.5 033D C.<72 D;<BB3D /0 543 ;<<> F 5430 =<; 543 ^<7053.. 5< B/9: /5 7BF 5430 C0<543;.<72 D;<BB3DF 359? -. 543 07>63; <= 709<7053D.<72. 1<3. D<G0F 543 B;<6C6/2/5E <= C 03G <03 57;0/01 7B <0 543 03V5 DCE 1<3. D<G0 =;<> (n 1)/n =<; 543 P;.5.<72 5< 1/n =<; 543 2C.5.<72? 3C0G4/23F 543 B;<6C6/2/5E =<; 543 ^<7053.. 5<.4<G 7B <0 543 03V5 DCE /. 9<0.5C052E 1/n? [/093 543 3VB3953D 5/>3 033D3D =<; C0 3H305 <997;/01 G/54 B;<6C6/2/5E p <0 543 03V5 DCE /. 1/p F G3 />>3D/C532E 135 54C5 543 5/>3 /. n ( n 1 ) 1 + (n 1)(n), =1 G4/94 /. 635G330 C0D n 2 2n2? W43;3=<;3 543 3VB3953D 07>63; <= DCE. 705/2 543 B;/.<03;. 3.9CB3 /. W4/. /. >794 63553; O(n 54C0 2 )? <7; B;3H/<7. 3VB<0305/C2.<275/<0? W43 @AA B;/.<03;..4<72D 135 <75 /0 C;<70D @AF7AA DCE.F <; C6<75 H] E3C;.? W43E G/22 63 BC.5 543/; 63.5F 675 543E G/22 2/H3 5<.33 543 <75./D3 G<;2D? L<G3H3;F 543E 9C0 D< >794 63553; 54C0 54C5M S :; < < = ==: ; < = @? @ D" A% & & W4/. /. C1C/0 C >354<D =<; 9<22395/01.<72.? W4/. 5/>3 543;3 /. 0<./0123 9<7053;? C543; 3H3;E<03 /. /0H<2H3D /0 C B;<93.. <= 9<22395/01.<72. 5<13543;? W43 2/145I 6726 G/22 63 G<;54 D/U3;305 07>63;. <=.<72. <0 D/U3;305 0/145.? ( ) * + -.3,73093 /. 1/H30 G4/94 D3.9;/63. 4<G >C0E.<72. 543 2/1456726 /. G<;54 <0 3C94 0/145F G4/94 /. C2GCE. C B<G3; <= 5G<? 35 V (n) D30<53 543

07>63; <=.<72. 54C5 543 2/1456726 /. G<;54 /= /5 /. 23=5 /0 543 RS B<./5/<0 <0 0/145 F <; D/.9<H3;3D RS <0 0/145? -..7>3 n 54C5 543 07>63; <= B;/.<03;. n + 1 /. C B<G3; <= 5G<? W4/. G/22 57;0 <75 0<5 5< >C553; /0 543 30D? - B;/.<03; 3053;. 543 ;<<> <0 C 0/145 C0D 9<22395. 4<G3H3; >C0E.<72. 4CH3 6330 23=5 543;3 =;<> 2C.5 0/145 X.< /= /5 /. 0/145 C0D 543 2/1456726 /. RS 43 B/9:. 7B.<72.Z C0D 57;0. 543 2/1456726 RT n? L3 0<G 2<<:. C5 543 07>63; <=.<72. V (n 54C5 1) 43 4C. 9<223953DF 675 ;3B;3.3053D /0 6C.3? O= 543 9<>/01 0/145 /. G<;54 M.<72. 5430 43 2<<:. C5 543 6/0C;E 6/5 2 <= V (n) = 2 M G<;54.<72.? O= 54/. 6/5 /. 5430 43 D;<B..<72. 6E 57;0/01 543 2/1456726 RS C0D.765;C95/01 2 1 2 2 =;<> M F 4/. <G0 5<5C2 <=.<72. 9<223953D? O= 543 2I6/5 /. J3;< 5430 43 23CH3. 543 2/1456726 RT? <5/93 54C5 54/. 4C. 543 3U395 54C5.<72. C;3 1273D 5<13543;K /05< 27>B. <=./J3 G4/94 9C0 63 5;C0.=3;;3D <0 0/145. G4/94 C;3 G<;54 2 2? Q 4303H3; C B;/.<03; 4C. 5G< 27>B. <=./J3 43 1273. 543> 5<13543; /05< C 27>B <=./J3 2 2 +1? W4/. >CE <997; /= 43 4C. 7.5 B/9:3D 7B C 27>B <; 4C. 7.5 1273D 5G<.>C223; 27>B. 5<13543;? 2 2 1 Q 430 C22 543.<72. 4CH3 6330 1273D 5<13543; /05< <03 27>B <=./J3 n = 2 log 2 5430 543 B;/.<03; G4< 4<2D. 54/. 27>B D392C;3. n.7993..? Q3 4CH3 E35 5<.CE G4C5 G<72D 63 C0 3 9/305 94</93 <= HC273. =<; 543? [5C;5/01 G/54 C 62<9: <= 0/145. G<;54 @ /. C 1<<D /D3CF 5< 4<B3=722E 1273 V (n) C22 543./0123.<72. /05< BC/;.? W430 =<22<G G/54 C 62<9: <= 0/145. G<;54 H 5< 1273 /05< 62<9:. <= 7.<72.F C0D.< <0? Q3 GC05 543 230154. <= 543 62<9:. 5< 63 2<01 30<714 5< 1/H3 C 1<<D 94C093 <= 127/01 C22 543 27>B. <=./J3 /05< 27>B. <= 2 1./J3 F 675 0<5 5<< 2<01 C. G3 D<0 K5 GC05 5< GC.53 5/>3 <093 543E 4CH3 C22 6330 2 1273D? -./ * *0 )*123 1 R) 1 ) * + O0 <;D3; 5< C94/3H3 C 1<<D C.E>B5<5/9.F G3.5C;5 G/54 C 62<9: <= I0/145.F 5430 I 0/145.F 5430 (n log n+n loglog I0/145.F n) 1 C22 543 GCE 7B (n 5< log n+n loglog n) 2 X ZI0/145.? (n log O= n+n G3 4CH3 loglog =C/23D n) 4 C=53; 54/. 07>63; <= DCE. (n 5430 log G3 n+n 9C0 loglog./>b2e n) 54;<G log2 n7b <7; 4C0D. C0D.5C;5 <H3; C1C/0? O0 <543; G<;D.F 543.3,73093 ;3B3C5.? V (n) W43 B;<6C6/2/5E <= 127/01 C22 27>B. <= 2 /05< 27>B. <=./J3 1 2 G/54/0 n log(n) + cn 0/145. XG43;3 c /..<>3 9<0.5C05Z /. 6<70D3D 6E C.E>B5<5I e c /9C22E? W4/. /. :0<G0 C. 543 e <? <: 6H8? Q /54.<>3 9C;3=72 3.5/>C5/<0 54/. ;3.725 9C0 63 3V530D3D 6E 94C01/01 5< C =7095/<0? W4/. 1/H3. 7. C B;<6C6/2/5E <=.7993..=722E 9<>B235/01 c 3C94.5C13 c(n) <= C5 log(n) 23C.5 530D. 5< @?.5C13. /. C5 23C.5 e W47. 543 3VB3953D 07>63; <= 5/>3. G3 033D 5< 1< 54;<714 543 G4<23 9E923 /. 23.. 54C0 e log 2 (e) 1? W4/. 1/H3. 54C5 543 3VB3953D 07>63; <= DCE. /. <= ɛ(n) log 2 <;D3; (n) e log (e) 2 [n log(n)(log(n) + log(log(n)))] F G4/94 /. O(n log(n) 2 )? F G43;3 /. C0 3;;<; =C95<;.794 54C5 e 1 log(n) ɛ(n) ɛ(n) ɛ(n) log 2 (n) W43 94C093 <=.7993..=722E 9<>B235/01 C22 log2 (n) log 2? (e) ɛ(n) log 2 (n) O5 9C0 63 B;<H3D 54C5 0< 94C013. 5< 543 230154. <= 62<9:. <= 0/145. V (n) 9C0 7

/>B;<H3 543 C.E>B5<5/9.F 675 G4C5 /= G3 GC05 543 63.5.3,73093 =<; B;39/.32E @AA B;/.<03;. F7; C..7>B5/<0 54C5 543 07>63; <= B;/.<03;. /. C B<G3; <= 5G< 9C0 63 ;32CV3D? W< CBB2E <7;.5;C531EF G3 7.5 033D 54C5 3H3;E<03.5C;5. G/54 C5 23C.5 <03.<72? [<F G/54 @AA B;/.<03;. =<; 3VC>B23F <03 B;/.<03; 9<72D 63 1/H30 H].<72. C0D 543 <543; ]] B;/.<03;. 1/H30 @.<72 5<.5C;5 G/54? W43 B;/.<03; G4< P;.5 9<22395. @H.<72. D392C;3.? W< 5;E 5< 135 C 1<<D 7BB3; 6<70D <0 543 3VB3953D 07>63; <= DCE. 5< =;33D<> G3 7.3D C 9<>B753;./>72C5/<0 5<.3C;94 54;<714 543 94</93. =<; V (n)? F7; 63.5 1/H3 C;<70D 77AA DCE.F <; @H E3C;.? F03.3,73093 <= 62<9: 230154. G4/94 4C. C6<75 54/. CH3;C13 /. 6 AF AF @AF AF HAF 7 AF AF HAF 7]AF AF AF AF ]AF ]A8F 54C5 /. 5<.CEM A @IDCE.F 5430 A HIDCE.F??? 5430 A 7I DCE.F 5430 HA @IDCE.F??? 5430 ]A 7IDCE.F 5430 X6C9: 5< 543.5C;5Z A @IDCE.F??? W43 <B5/>/.C5/<0 C21<;/54> G<;:. 6E 5;E/01 5< <B5/>/.3 543 V (n) =<; 5G< BC..3. 54;<714 543 5EB3. <= DCE. X@IDCE. 5430 HIDCE.F 5430 7IDCE. 359?Z 5430 7.5 ;3B3C5. 54C5.3,73093 =<; 543 702/:32E 9C.3. /0 G4/94 543 B;/.<03;. 4CH3.5/22 0<5 P0/.43D C=53; H BC..3.? O5 /. 0<5 305/;32E 923C; G4E 543 C6<H3.3,73093 /. 1<<DF 54<714 /5 >C:3..30.3 54C5 543 P;.5./V 53;>. C;3 D39;3C./01./093 23.. B3<B23 4CH3 5< >335K /0 2C53;.5C13.? O5 C2.< >C:3..30.3 54C5 543.3H3054 53;> /. 2C;13;F./093 <03 G<72D GC05 5< 1/H3 C 2<5 <= 5/>3 =<; 543 2C.5 5G< 62<9:. <= 7.<72. 5< >335? /H/01 7B C5 54/..5C13 >3C0. 4CH/01 5< 9<05/073 54;<714 543 03V5.3H30.5C13. 5< P0/.4 7B? S *+ 0R * *) 1R * R *) 1 ) * + 5<; @AA B;/.<03;. G3.7.B395 54C5.<>3.<;5 <= 4E6;/D C21<;/54> /. B;<6C62E 543 63.5F 5< 7.3 1<<D B</05. <= ><;3 54C0 <03.5;C531E? ^<22395/01 5<13543;.<72. C. /0 543 BE;C>/D.943>3 /. 93;5C/02E C 1<<D /D3C 5<.5C;5 G/54F 675.<>354/01 32.3 >CE 63 63553; /0 543 30D1C>3? - 4E6;/D 1/H30 6E? 5321304C73; 6@8 7.3. 543 BE;C>/D.943>3 5<.5C;5 G/54F 675 4C. C ^<7053...5C;5 9<22395/01 >/DGCE 54;<714? L/..3,73093 <= 62<9: 230154. X94<.30 6E 4C0DZ 4C. 3VB3953D DCE. <= C;<70D ]7]F C0D ;700/01 <7; <B5/>/.C5/<0 B;<1;C> <0 <0 543 HC;/C623. =<; 4/..5;C531E 1/H3. C;<70D ]A? -.E>B5<5/9C22EF 543E 9C00<5 D< 63553; 54C0 3VB3953D 07>63; <= DCE. 5< =;33D<> F 639C7.3 54C5 /. 543 3VB3953D 07>63; O(n log <= n) DCE. =<; 3H3;E<03 5< 4CH3 H/./53D 543 ;<<> F /10<;/01 54C5 C22 B;/.<03;. C957C22E 4CH3 5< 9<> >70/9C53 Y L3;3 C;3.<>3 HC;/C5/<0. E<7 >/145 2/:3 5< 54/0: C6<75? 4C94 HC;/C5/<0 C..7>3. C22 543 9<0D/5/<0. /0 543 <;/1/0C2 B;<623> F 675 G/54.<>3 C.B395. C253;3D? O0 3C94 9C.3F E<7 >/145 2/:3 5< C.: E<7;.32= G43543; 543 B;/.<03;. 9C0 3.9CB3F C0D /=.< G4C5 /. C0 3 9/305 GCE 5< D< 54/.? Q3 C..7>3 54C5 543 2/1456726 C2GCE..5C;5. RT? @? 23 /3* 23 1 W43 9305;C2 ;<<> 9<05C/0. 5G< X<; ><;3Z 2/1456726. X543 9<> >70/9C5/<0 94C0032 /. G/D3;Z?

H? 23 /3* )RR 1 W43;3 C;3 5G< X<; ><;3Z /D305/9C2 ;<<>.? W43 B;/.<03;. C;3 5C:30 5< <03 C5 ;C0D<> 675 D<0K5 :0<G G4/94 543E C;3 /0?? (*/ ) * ) S1 *) )* * *) W43 GC;D30 57;0. 543 2/1456726 RT C5 @H- F 94<<.3. <03 B;/.<03; 5< H/./5 C5 @- F C0D 94<<.3. C1C/0 =<;.<>3<03 5< H/./5 C5 H-? W43 H/./5<;. <02E 5;C0.>/5 <; ;393/H3F 0<5 6<54? 7? 3 R21 )0*S W43 GC;D30 /. >C2/9/<7. C0D :0<G. 543.5;C531E 54C5 543 B;/.<03;. G/22 7.3 X43 2/.530. 5< 543> C1;33/01 <0 G4C5 5< D<Z? 4C94 DCE 43 G/22 94<<.3 G4/94 B;/.<03; 5< C22<G /05< 543 ;<<>? L/. 9<0I.9/3093 D3>C0D. 54C5 43 C22<G. 3H3;E B;/.<03; 5< H/./5 543 ;<<> /0P0/532E <=530?? 33 /) 1RS*)1 * R SSR2S * W43 9<0D/5/<0 =<; 3H3;E<03 5< 63 =;33D /. 54C5 3H3;E B;/.<03; >7.5 9<;;3952E C00<7093 XC5.<>3 5/>3Z? O0 <543; G<;D.M 3H3;E B;/.<03; >7.5 63.7;3 54C5 C22 B;/.<03;. 4CH3 6330 5< 543 ;<<>?? ( 23 S*R21 SSR2S S -0E<03 9C0 C00<7093 <0 C0E DCEF 0<5 7.5 543 B;/.<03; G4< GC..323953D 54C5 ><;0/01? W43 9<0D/5/<0 =<; 3H3;EI <03 5< 63 =;33D /. 54C5 3H3;E<03 >7.5 9<;;3952E C00<7093 <0 543? DCE? O= 543E C;3 /09<;;395 <; /=.<>3 C00<7093 G4/23.<>3 D< 0<5 5430 543E C22 D/3?? ) 1RS*)1 )* )**0 *S *+ SSR2S * 4H3;E<03 >7.5 9<;I ;3952E C00<7093 C5.<>3 5/>3? Q 430.<>3<03 C00<7093.F 543E C;3 =;33D X03H3; H/./5 543 ;<<> C1C/0ZF 675 543 <543;..5CE 705/2 543E 5<< C00<7093? /./5<;. C;3 94<.30 70/=<;>2E C><01 543 ;3>C/0/01 B;/.<03;.? <53 54C5 543 B;/.<03;. C;3.5/22 > <.5 /053;3.53D /0 3H3;E<03 3.9CB/01 F ;C543; 54C0 >/0/>/J/01 543/; <G0 5/>3 5< 3.9CB3?? *0 32* *331 RS* SSR2S *) W43 B;/.<03;. C;3 C22<9C53D ;3D <; 6273 B;/.<0 9322.? W43 C00<709/01 B;/.<03; >7.5 9<;;3952E.CE 4<G > C0E ;3D 9322. 543;3 C;3 /0 <;D3; =<; 543> C22 5< 63 ;323C.3D? ]? 2 *)1 S *331 RS* SSR2S *) 4C94 9322 4C. C 0C57;C2 07>63; G;/5530 <0 543 GC22? W43 C00<7093; >7.5 1/H3 C22 543 07>63;.? @A? 33 /) 1RS*)1 1*S0 *11 *1 R 33 /) 1RS*)1 X54/. /. C 9<>6/0C5/<0 <= C0D ]Z? @@? S0R 1 S *1 W43 B;/.<0 /..7653;;C03C0F G/54 0< 92<9:.F 9C230DC;.F <; C0E <543; /0=<;>C5/<0 C. 5< G4C5 543 5/>3 /.? W43 B;/.<03;. 2<.3 C22 5;C9: <= 5/>3F C0D 543 GC;D30 94<<.3. B;/.<03;. C5 ;C0D<> 5/>3.? O0 <543; G<;D.F 543 B;/.<03;. 4CH3 0< /D3C 4<G >C0E B3<B23 4CH3 H/./53D 543 ;<<>./093 543E G3;3 2C.5 543;3 F C0D 9C00<5 7.3.5;C531/3. G4/94 9<705 DCE.? W43E <02E :0<G 543 <;D3; /0 G4/94 3H305. <997;?

@H? S0R *1 33 /) 1RS*)1 21 SSR2S * ^<>6/03 HC;/C5/<0. C0D @@? 4 H3;E B;/.<03; >7.5 C00<7093 54C5 3H3;E<03 4C. H/./53D C5.<> 3 5/>3F C0D 543E 9C00<5 7.3 DCE 9<705/01.5;C531/3.? @? S0R *1 *11 * )R RS* R RS* W43;3 C;3 <02E 5G< B;/.<03;. C0D 543 5;C0.>/553; 4C. 5<.30D C >3..C13 XC 0C57;C2 07>63;Z 5< 543 ;393/H3;F 675 C. /0 @@ 543E 9C00<5 9<705 DCE.? @7? S0R *1 *11 *1 )R S+ R RS* ^<>6/03 HC;/C5/<0. ] C0D @@? F03 C00<7093; >7.5 1/H3 C22 07>63;. G;/5530 <0 543 GC22.F C0D 543 B;/.<03;. 2<.3 5;C9: <= 5/>3? @? S0R *1 *11 *1 )R S+ R S+ 9 3 23 1 @? S0R *1 *11 *1 )R S+ R S+ 3 23 Q3 0<G 1/H3 C.B</23; =<; ><.5 <= 543.3 B;<623>.? O5 57;0. <75 54C5 543.5;C531/3. 2/.53D C6<H3 X<;.2/145 ><D/P9C5/<0. <= 543> Z C;3.7/5C623 =<; ><.5 <= 543.3 HC;/C5/<0.? @? 23 /3* 23 1 ^<705/01.<72. X.5;C531E HZ G/22.5/22 G<;:F C0D 9C0 63 >CD3 3H30 =C.53; C..<72. 9C0 63 23=5 /0 C ;<<> G4/94 4C. 2 2/1456726.? 2/1456726. C22<G =<; 543 63.5 B<../623 5/>3 5< 3.9CB3 D/.5/095 C..<<0 C. 3H3;E<03 log2 n 4C. C957C22E 6330 /0 543 ;<<> 5430 543 B;/.<03; /0 543 ;<<> 9C0 D392C;3? [5;C531E 9C0 C2.< 63 />B;<H3D 6E C22<G/01 127/01. <=.<72. /05< 2C;13; 27>B.F.794 C. 27>B. <=./J3 2/1456726.? (2 ) /= 543;3 C;3 l D/.5/095 H? 23 /3* )RR 1 ^<705/01.<72. X.5;C531E HZ G/22.5/22 G<;:? O5 G/22 63.2<G3;F C254<714 543 3VB3953D 5/>3 705/2 3.9CB3 X=<; 07>63; <= ;<<>. /0D3B30D305 <= n Z /..5/22 O(n 2 ) DCE.?? (*/ *) * ) S1 *) )* * *) -.5;C531E./>/2C; 5<.<72I9<22395/01 X.5;C531E HZ G<;:.? W43 ^<7053.. C2GCE. B/9:. 7B C0D 03H3; D;<B..<72.? 4 H3;E<03 32.3 D;<B..<72. C5 3H3;E <BB<;570/5E X54<714 543E C;3 =<;93D 5< B/9: 543> 7B /= 543E P0D 543> Z? W4/..5;C531E 4C. 3VB3953D 5/>3 635G330 n 2 log 2 n C0D n 3? XO= 543;3 C;3.<72. <75.5C0D/01 5430 543 94C093 <= 543 9<7053.. B/9:/01 <03 7B 543 9<>/01 0/145 /. 635G330 n C0D 1 1 F D3B30D/01 <0 4<G 54<.3.<72. C;3 D/.5;/6753D? W4/. 1/H3. C 5<5C2 n3vb3953d n 2 5/>3 <= 635G330 n 2 log 2 n C0D n 3?Z 7? 3 R21 )0*S [5;C531E H G/22 G<;:F C254<714 543;3 /. 923C;2E 0< 6<70D <0 543 5/>3 705/2 3.9CB3 /5 D3B30D. <0 4<G >3C0 543 QC;D30 GC05. 5< 63?? 33 /) 1RS*)1 * R SSR2S * W;EI5;EIC1C/0 K X.5;C531E @Z G<;:.? O053;23CH/01 9E923. <=.5;C531E G/22 C2.< G<;: M 4 C94 B;/.<03; 4C. <03 5EB3 <=.<72 =<; 3C94 B;/.<03; G4< G/22 4CH3 5< C00<7093? F03 9E923 /. 1/H30 5< 3C94 B;/.<03;K. C553>B5. 5< 9<22395 543.<72. D3.5/03D =<; 43;F 5430 C=53; n

<= 543.3 C.39<0D 9E923 /. D3H<53D 5< 3C94 B;/.<03; C0D.< <0? W4/. 1/H3. C0 3VB3953D 5/>3 <= n 2 log 2 (n)?? ( 23 S*R21 SSR2S S W43 B;/.<03;. 9C00<5 63.7;3 <= 3.9CB/01? [7BB<.3 543E G/22 C00<7093 <0 DCE A? W43;3 /. C P;.5 DCEF D F <0 G4/94 543E C22 :0<G 54/.? W43 B;/.<03; G4< 3053;. <0 DCE D :0<G. 54C5 543E 4CH3 3053;3D C0D 543.5C53 <= 543 2/1456726? 4H3;E <543; B;/.<03; <02E :0<G. 54C5 543E D/D 0<5 3053; 543 ;<<> <0 54C5 DCE? O= C D/U3;305 B;/.<03; 3053;3D <0 DCE D 5430 C22 <= 543 <543; B;/.<03;. G4< D/D 0<5 3053; G<72D 4CH3 543.C>3 /0=<;>C5/<0F C0D.< G<72D 4CH3 5< 9<>3 5< 543.C>3 9<0927./<0M 54C5 543E.4<72D C00<7093 <0 DCE A XB;<H/D3D 543;3 C;3 C5 23C.5 B;/.<03;.Z? W43;3=<;3 /5 9C00<5 >C553; G4< 3053;. <0 DCE D F.< 543E >7.5 C22 :0<G <0 DCE D 1? Q 4/94 9<05;CD/95. 543 C..7>B5/<0 54C5 D GC. 543 P;.5 DCE 543E C22 :03G 543E G<72D C00<7093 <0 DCE A?? ) 1RS*)1 )* )**0 *S *+ SSR2S * [2<.4EK.<72I9<22395/01 XC. /0 543 C0.G3; 5< HC;/C5/<0 @H 632<GZ G/22 G<;:? Q 430 C B;/.<03; 4C. 9<223953D @AA.<72. C0D 5430 1/H30 543> C22 CGCE C1C/0 543E >CE D392C;3 C0D 63.35 =;33? F ]F @A? *0 32* *331 R) 2 *)1 S *331 RS* R) 33 SSR2S *) 1 W43 B;/.<03;. 9C0 3.9CB3? [33 543 63;I543<;3> X632<G Z =<; C.5;C531E C0D B;<<=? @@? S0R 1 S *1 ^<705/01.<72. X.5;C531E HZ G/22.5/22 G<;:? @H? S0R *1 33 /) 1RS*)1 21 SSR2S * [2<.4EK.<72 9<705I /01 G/22 G<;:? W43 2/1456726 /. C2GCE. G<;54 <03.<72? -0E B;/.<03; G4< 4C. 0<5 C00<7093D D<3. 543 =<22<G/01M O= 543 2/1456726 /. RS G430 543E 3053; 5430 543E 9<22395 543.<72 C0D 57;0 543 2/1456726 RT? O= 543 2/1456726 /. RT G430 543E 3053; C0D 543E 4CH3 <03 <; ><;3.<72. 5430 543E D;<B <03 C0D 57;0 543 2/1456726 RS? -0E B;/.<03; G4< 4C. C2;3CDE C00<7093D C2I GCE. D;<B. C0E.<72. G4/94 543E 4CH3 F C0D 23CH3. C0E 54C5 C;3 /0 543 ;<<>? W4/..5;C531E 4C. 3VB3953D 5/>3 <;D3; 23.. 54C0 <; 3,7C2 5<? e n W4/. 9C0 63.4<G0 6E 9<0.5;795/01 C0 CBB;<B;/C53 C;:<H 94C/0 C0D 1/H/01 2<G3; 6<70D. =<; 543 94C093 54C5 C 1/H30 B;/.<03; G/22 C00<7093 /0 543 03V5 HAA DCE.? <5/93 54C5 G430 543;3 /. <02E <03 B;/.<03; 23=5 5< C00<7093F 54/..5;C531E ;3D793. 5<.5;C531E H F.<72I9<22395/01 G/54 <03 ^<7053..? <53 54C5 54/..5;C531E G<72D C2.< G<;: X23.. 3 9/3052EZ /= B;/.<03;. G4< 4CH3 C2;3CDE C00<7093D 7.5 9<05/073 5<.2<.4.<72. C;<70D X1/H3 C0D 5C:3.<72. ;C543; 54C0 7.5 1/H3Z? W4/. /. 639C7.3 C ;C0D<> GC2: /0 C P0/53.BC93 G/22 3H3057C22E 135 3H3;EB2C93? Q3 G/22 7.3 54/. =C95 3V530./H32E 2C53; <0? -0<543;.5;C531E /. 54C5 3C94 B;/.<03; G4< /. 0<5 C.<72I9<22395<; 4C. C XH3;E.>C22Z 94C093 3C94 DCE 543E 3053; <= 639<>/01 <03? -=53; C 07>63; <= H/./5. 5< 543 ;<<> C. C.<72I9<22395<; 543E 1/H3 7B C0D 1< 6C9: 5< 63/01

C0 <;D/0C;E.<72I1/H3;? -0E B;/.<03; G4< 4C. C2;3CDE C00<7093D C2GCE. 1/H3..<72. C0D 03H3; 9<22395.? ^C0 E<7 54/0: <= C HC;/C5/<0 G43;3 543 63.5.5;C531E /. G<;.3 54C0 3VB<I 0305/C2 /0 543 07>63; <= B;/.<03;. @? S0R *1 *11 * )R RS* R RS* Q3 4CH3 5G< B;/.<03;.F <03 <= G4<> /. 5;E/01 5<.30D C >3..C13 5< 543 <543;? W43 5;C0.>/55/01 B;/.<03; 309<D3. 543 >3..C13 C. C 0C57;C2 07>63;F? L3 5;/3. 5< 1/H3 543 ;393/H/01 B;/.<03;.<72.? W43 B;<623> 0<G M /. 4<G 543 ;393/H3; :0<G. G430 543 >3..C13 M 4C. 6330.305 4<G D<3..43 :0<G G430.43 4C. ;393/H3D C22 <= 543.<72. W< D< 54/..43 <99C./<0C22E B75. C.<72 6C9: /05< 543 ;<<> G430.43 P0D. /5 3>B5E? L<B3=722E 543 5;C0.>/553;F 4CH/01 D;<BB3D C22 <= 4/..<72.F G/22 5C:3 543 2C.5.<72 6C9: 547. /0D/9C5/01 54C5 43 /. P0/.43D? W43 ;393/H3; G/22 5430.33 54C5 543.<72 4C. 6330 5C:30 C0D :0<G 54C5 C22 <= 543.<72. 4CH3 6330.305 F 639C7.3 543 5;C0.>/553; G/22 <02E B/9: 7B C.<72 G430 43 /. D<03? W< D< 54/. ;32/C62E 543 5G< B;/.<03;. 634CH3 C. =<22<G.M W43 5;C0.>/553; D;<B. C22 <= 4/..<72. 705/2 43 4C. 0<03 23=5? Q 430 43 4C. 0<.<72. 23=5 43 G/22 5C:3 <03.<72 =;<> 543 ;<<> /= 43 9C0? Q 430 43 4C. <03.<72 43 G/22 D;<B /5 /0 543 ;<<> /= 43 9C0? W43 ;393/H3; 5C:3. 3H3;E.<72 54C5.43 9C0 F C254<714.43 <99C./<0C22E D;<B. <03 6C9: /0 543 ;<<> X B/01.KZ? O= G430.43 03V5 3053;. 543 ;<<>.43 P0D. 54C5 543.<72.43 4C. D;<BB3D 4C. 6330 5C:30F 5430.43 :0<G. 54C5 543 5;C0.>/553; /. P0/.43D C0D.< :0<G. 543 5<5C2 07>63; <=.<72..305? @7? S0R *1 *11 *1 )R S+ R RS* 5<; B;/.<03;. 5;C0.>/55/01 C0D <03 ;393/H/01F 543 5;C0.>/553;. C22 634CH3 C. n /0 HC;/C5/<0 @? 5/;.5.7BB<.3 54C5 543 ;393/H3; GC05. 5< :0<G 543.7> <= 543 07>63;. <= 543 5;C0.>/553;.F? M W4/. 5/>3 543 ;393/H3; <99C./<0C22E 5;/3. 5< D;<B.<72. 6C9: C5 543.C>3 5/>3? W43 <02E GCE 54C5 C22 B/01. G/22 63 5C:30 n /. /= C22 <= 543 5;C0.>/5I 5/01 B;/.<03;. C;3 P0/.43D? n Q 430.43.79933D. 5430 43; n >CV/>7> HC273 GC. 543.7> <= 543 07>63;. <= 543 5;C0.>/553;.F? M Q 4C5 4CBB30. /. 54C5 543 ;393/H3;K. 5<5C2 9<223953D.<72. 7.7C22E /09;3C.3.F 675 03H3; =C22. 6C9: C. >794 C. =;<> 543 97;;305 C22I5/>3 >CV/>7> 70I 23.. 54C5 >CV/>7> /. 543 5<5C2 07>63; n <=.<72. 63/01 5;C0.>/553D? Q 430 C 5;C0.>/553; P0/.43.F 543 ;393/H3;K. 5<5C2 /. C22<G3D 5<.2<.4 6C9: 6E <03 ><;3 54C0 63=<;3? Q 430 C22 5;C0.>/553;. C;3 P0/.43D 5430 543 ;393/H3;K. 5<5C2 G/22.2<.4 635G330 C0D F C0D G430.43.33. 6<54 3V5;3>3. /0 54C5 <;D3; 5430.43 :0<G. M /5 M /. D<03? n 0<G/01 543 5<5C2F F /. 30<714 5< C22<G C22 543 5;C0.>/553;. 5<.30D C;6/5;C;E >3..C13.? M ^4<<./01 6C.3 HF 1/H3 543 I54 5;C0.>/553; n D/1/5. /0 G4/94 5< 309<D3 4/. >3..C13? i i, i+ n, i + 2n,... ]

@? S0R *1 *11 *1 )R S+ R S+ 9 3 23 1 Q3 9C0 7.3 543.<275/<0 5< HC;/C5/<0 @ 5<13543; G/54 C GCE 5< BC.. C;<70D G4< /. 5;C0.>/553; C0D G4< /. ;393/H3;? W< 63 B;39/.3F 543E 7.3 2/1456726 <03 7.5 C. /0 @? [<>3 B;/.<03; /. 94<.30 5< 63 543 P;.5 5;C0.>/553;? Q3 C..7>3 2/1456726 5G< /. RS 5<.5C;5 G/54? Q 4<3H3; 57;0. /5 RT XB/9:. 7B 543 2/.530/01.5/9:KZ /. 543 P;.5 ;393/H3;? W43 5;C0.>/553;.33. 54C5 543 2/.530/01.5/9: 4C. 6330 B/9:3D 7B F C0D.5C;5. 5;C0.>/55/01 <0 2/1456726 <03? Q 430 543 ;393/H3; :0<G. 543 >3..C13 /. D<03F 43 B75. D<G0 543 2/.530/01.5/9: C0D 639<> 3. 543 03G 5;C0.>/553;? W43 03G ;393/H3; /. G4<3H3; 03V5 B/9:. 7B 543.5/9:? W43 B;/.<03;. :33B.30D/01 > 3..C13. C;<70D XG/54<75 :0<G/01 G4< 543E C;3 5;C0.>/55/01 5<Z C0D 3H3057C22E 3C94 B;/.<03; 9<22395. C22 543 > 3..C13.? @? S0R *1 *11 *1 )R S+ R S+ 3 23 [33 543 63;I 63;I543<;3> 632<G? Q3 G/22 0<G 1/H3 <7; >354<D =<; HC;/C5/<0. F ] C0D @A? O5 57;0. <75 54C5 3C94 B;/.<03; 9C0 5;C0.>/5 C0 C;6/5;C;E >3..C13 5< C22 <= 543 <543; B;/.<03;.F 7./01 <02E 543 <03 2/145? Q3 G/22.5C;5 G/54 <03 B;/.<03; 5;C0.>/55/01 <03 6/5 5< 3H3;E <543; B;/.<03;? O= 543 5;C0.>/553; GC05. 5<.30D C AI6/5 5430 <0 C0E 3H30I07>63;3D DCE 43 23CH3. 543 2/1456726 RS C0D <0 C0E <DDI07>63;3D DCE 23CH3. 543 2/1456726 RT? O= 43 GC05. 5<.30D C @I6/5 5430 <0 C0E 3H30 DCE 43 23CH3. 543 2/1456726 RT C0D <0 C0E <DD DCE 43 23CH3. 543 2/1456726 RS? 4H3;E <543; B;/.<03; 23CH3. 543 2/1456726 RT? <G C0E B;/.<03; G4< P0D. 543 2/1456726 RS G430 543E 3053; 543 ;<<> G/22 :0<G =<;.7;3 54C5 543 5;C0.>/553D 6/5 /. C A <; C @F D3B30D/01 <0 G43543; 543 B;3H/<7. DCE GC. 3H30 <; <DD? 4 H3;E B;/.<03; G/22 P0D 543 2/1456726 RS C5.<>3 5/>3 XG/54 B;<6C6/2/5E @ZF C0D.< G/22 ;393/H3 543 >3..C13? F= 9<7;.3F 543;3 /. 0<54/01.B39/C2 C6<75 3H30 C0D <DD DCE.? -0E 6/ 395/<0 635G330 C0D N {0, 1} N G<72D D< 7.5 C. G322? 5<; 3VC>B23F G<72D >3C0 54C5 DCE j (0, ) j /. 543 AI6/5 DCE? W4<.3 DCE. G4<.3 07>63; 9<;;3.B<0D 5< C;3 3H30 DCE.K C0D 54<.3 G4/94 9<;;3.B<0D 5< C;3 <DD DCE.K? (0, n) W<.30D 5G< 6/5.F D/H/D3 543 DCE. (1, /05< m) =<7;.35.? O0 <543; G<;D.F B;<H/D3 C 6/ 395/<0 635G330 C0D? W43 P;.5 6/5 /. ;3B;3.3053D 6E 543 P;.5 5G< 5EB3. <= DCEF N A C0D {0, 1} @ ><D {0, 1} N 7.CEF C0D 543.39<0D 6/5 6E 543 <543; 5G< 5EB3. <= DCE.F H C0D ><D 7.CE? -0E B;/.<03; G4< P0D. 543 2/1456726 RS G/22 :0<G =<;.7;3 <03 <= 543 6/5. 63/01 5;C0.>/553D? W< 5;C0.>/5 C >3..C13 <= C;6/5;C;E 230154F B;<H/D3 C 6/ 395/<0 635G330 C0D? N W< {0, C22<G 1} 3H3;E N N B;/.<03; 5< 5;C0.>/5 C >3..C13 5< 3H3;E <543; B;/.<03;F P;.5 D/H/D3 543 DCE. C><01 543 B;/.<03;. X.< 54C5 3C94 /. C22<9C53D C0 /0P0/53 07>63;Z C0D 5430 ;70 543 C6<H3 C21<;/54> G/54 B;/.<03; 5;C0.>/55/01 <0 DCE. G4/94 C;3 C22<9C53D 5< 43;? 5<; B;/.<03;.F 54/. 9C0 63 54<7145 <= C. 1/H30 6E C M @A

6/ 395/<0 635G330 C0D? W<.B33D 7B 5;C0.>/../<0F N {1, /=, C0<543; M} {0, B;/.<03; 1} N :0<G. N C 1/H30 6/5 /0 <03 <= 543 >3..C13. 63/01 5;C0.>/553D 5430 543E 9C0 ;35;C0.>/5 54/. 6/5 6E C95/01 C. 543 5;C0.>/553; G<72D 394</01 K 543 > 3..C13? &C &C & & Q3 G/22 0<G D/.97.. C >354<D 54C5 C22<G. 3C94 <= 543 B;/.<03;. 5<.30D C.35 <= C;6/5;C;/2E 2<01 > 3..C13.F <03 5< 3C94 <543; B;/.<03;? Q3 C..7> 3 =7;543; 54C5 G3 C;3 /0 543.355/01 <= HC;/C5/<0 @@ X C0D<> H/./5/01 5/>3.KZF C0D 43093 54C5 543 B;/.<03;. 4CH3 0< 5/> 3 ;3=3;3093 <543; 54C0 543 <;D3; /0 G4/94 3H305. <997;? 02/:3 C22 543 HC;/C5/<0. G3 D/.97..3D 705/2 0<G F 54/. <03 9<72D 0<5 63.<2H3D 7./01 D/;395 ><D/P9C5/<0. <=.5;C531/3. @ <; H? F03 <= 543 C754<;. X?5?Z 9C>3 7B G/54 G4C5 G3 54/0: /. C0 <;/1/0C2.5;C531E? * 0* R * * R 0 W43 B;/.<03;. G/22 4CH3 C1;33D 7B<0 C0 <;D3;/01 C><01 543> C43CD <= 5/>3? n ;/.<03; @ G/22 63 543 :?<><F 2<<:/01 C5 543?? =<;>3D 6E C22 543 <543; B;/.<03;. XC0D 543 2/1456726Z? W4<.3 0<0I<6.3;H3;. G/22 63 9C223D <:? 639C7.3 543E G/22 =<22<G C./>B23 ;723? 3=<;3.5C;5/01 4/. ;723F 543 P;.5 5;C0.>/553;F.CE B;/.<03; F /05;<D793. A <; @.<72. /05< 543.E.53>? n W43 <6.3;H3; G/22 5;E 5< D3D793 4<G >C0E.<72. G3;3 <;/1/0C22E /05;<I D793D =;<> 543 634CH/<; <= 543 ;<6<5.? 5<; 54/.F B;/.<03; @ 4C. D/U3;305 B;<93D7;3. C5 4/. D/.B<.C2M WG<?= B;<93D7;3. P0, P 1 54C5 C22<G B;/.<03; @ 5< 9<0D795 3VB3I ;/> 305.? L3 /. 5;E/01 5< C0.G3; B<./5/H32E 5< <03 <= 543 5G<,73.5/<0. Q 0, Q 1 M =@ <=?< n =<@ i?? i = 0 < 1 =??? L<G3H3;F 6<54 P0 C0D P1 9C0 <02E B;<D793 B<./5/H3 ;3.725.F <; 63 /09<0927./H3? L3093 B;/.<03; @ G/22 <02E C0.G3; 031C5/H32E 5< Q 1 G430 P0 /. 9<0927./H3? - <?= B;<93D7;3 54C5 C22<G. B;/.<03; @ 5<.35 543.E.53> 6C9: 5< /5. <;/1/0C2 B<./5/<0 X543 07>63; <=.<72. /0 543.E.53> /. C. 543 5;C0.>/553; 23=5 /5Z? W4/. C22<G. 4/> 5< B;<933D G/54 CDD/5/<0C2 3VB3I ;/>305.? W43 5G< 53.5/01 B;<93D7;3. G/22 3H3057C22E 1/H3 C0 C0.G3; 5< 543 <6.3;H3;? <G B;/.<03; @ 5;/113;. B;/.<03; H /05< C0 <6.3;H/01 B4C.3? W4C5 /.F 543E X><;3 <; 23..Z 3V94C013 543/; ;<23.F C0D B;/.<03; H 639<>3. C0 <6.3;H3;F G4/23 B;/.<03; @.5C;5. =<22<G/01 C./>B23 ;723 C0D.< 639<>3. C ;<6<5? 4H3057C22EF =;<> 543 3VB3;/>305. 43 G/22 9<0D795 <0 543.E.53> =<;>3D @@

6E 543 <543; B;/.<03;.F B;/.<03; H G/22 P0D <75 G4/94 6/5 B;/.<03; @ 23=5 /0 543.E.53> C0D 5430 639<>3. C ;<6<5? W4/. 9<05/073.F 9E92/01 54;<714 C22 543 B;/.<03;.? Q3 4CH3 3C94 B;/.<03; i.30d/01 C P;.5 6/5 5< B;/.<03; F 5430 C22 <= 543>.30D/01 C.39<0D 6/5F 359??? i + 1 mod n./01 /053;>3D/C53.F C0E B;/.<03; 9C0.30D C >3..C13 5< C0E <543;F C0D 0<5 <02E 5< 4/. =<22<G3; /0 543 <;D3;/01? * 1 /3*) 1* n = 3 Q3 0<G D3.9;/63 3C94.53B /0 =722 =<; 543 9C.3 n = 3? ( /3* )23*1 W43 634CH/<; <= 543 B;/.<03;. G4< C;3 0<5 97;;3052E <6.3;H/01 G/22 63 1/H30 6E 543 D/;3953D 1;CB4. ϕ F G/54 C B<./5/H3 /05313; X.33 D/C1;C> @Z? W43.3 1;CB4. D3.9;/63 543 07>63; <=.<72. 3C94 B;/.<03; /. 3C13; 5< 4CH3 C5 C0E 5/>3F C0D 43093 D353;>/03 G43543; 43 GC05. 5< D;<B <; 1;C6 C.<72 3C94 5/>3 43 3053;. /0 543 ;<<>? W43 1;CB4. C;3 5< 63 ;3CD 23=5 5< ;/145F C0D 9<0./D3;3D 5< ;3B3C5 X543 DC.43D 2/03Z? -5 C0E 5/>3 G43;3 ><;3 54C0 <03 <B5/<0 /. <U3;3D F 543 B;/.<03; 94<<.3. G4/94 <B5/<0 5< 5;E G/54 3,7C2 B;<6C6/2/5E? -1-2 Trigger 10 9 8 7 6 5 4 3 2 1 0 slosh W<.5C;5 G/54F <03 <= 543 ;<6<5. G/22 =<22<G C0D <03 G/22 =<22<G -..7>3 54C5 /. 6/1F C0D /. 6/113;? W4/. G/22 ϕ0 63 >CD3 B;39/.3 2C53;? ϕ1 Q3 B2CE 543 ;<23 <= 0 B;/.<03; @F C0D 1 X=<; 0<G Z <02E <6.3;H3 B;/.<03; H XC0D Z ;700/01 543/; /0.5;795/<0. X;3.B? Z? <;3 B;39/.32EF G430 G3 135 C 94C093 5< 1< /0 543 ;<<> F G3 0<53 ϕ0 G43543; ϕ 1 543.5C53 <= 543 2/1456726 4C. 94C013D =;<> 543 2C.5 5/>3 G3 G3;3 543;3 XG4C5 G3 9C22 C = <=Z? O= 543 5<5C2 07>63; <=.<72. /0 543.E.53> /. X;3>3>63; G3 /0927D3 543 2/1456726 /0 543.E.53> YZF 0<54/01 9C0 4CBB30 639C7.3 0 6<54 B;/.<03;. C;3 6<54 3C13; 5< 135 ><;3.<72.F 675 0<03 C;3 CHC/2C623? O= 543 5<5C2 /. @F 543 2/1456726 >/145 63.G/5943D RS C0D RT.<>3.>C22 07>63; <= 5/>3. X/= 543 B;/.<03; G4<.5C;5. G/54 543.<72 /. /0/5/C22E 3C13; 5< 135 ;/D <= /5ZF 675 3H3057C22E <03 <= 543 @H

H B;/.<03;. G/22 4CH3 @.<72 C0D 63 3C13; 5< 4CH3 HF C0D 543 <543; G/22 4CH3 A C0D 63 3C13; 5< 4CH3 @? [< 543./57C5/<0 G/22.5C22 543;3 C=53; C P0/53 07>63; <= 8/9:3;/01.? [/>/2C;.5<B. G/22 <997; /= 543;3 C;3 7 <;.<72. /0 543.E.53>? F0 543 <543; 4C0DF /= H.<72. C;3 CHC/2C623 /0 543.E.53> F 543.E.53> >/145.5<B /0 C./57C5/<0 G43;3 3C94 4CH3 @.<72 C0D C;3 3C13; 5< 4CH3 HF 675 ><;3 />I B<;5C052EF 543 2/1456726 = 63 57;03D RS C0D RT C0 C;6/5;C;/2E 2C;13 C><705 <= 5/>3.F /= 543E 6<54 :33B 1</01 54;<714 C.3,73093 2, 1, 0, 1, 2, 1, 0,... XG/54 C D32CE /0 543/; B4C.3.Z? W43 2/1456726 /. 5430.C/D 5< = < =@ =? W43.C>3 54/01 9<72D 4CBB30 /= 543;3 C;3 F F???.<72. /0 543.E.53>? 5/0C22EF /0 543 9C.3 <=.<72.F 543.E.53> >/145 B;<D793 /0D3P0/53 8/9:3;/01 /0 543 2C>B /0 C ><;3 9<>B2/9C53D =C.4/<0? W4/. 634CH/<; /..7>>C;/J3D /0 543 =<22<G/01 94C;5? 7>63; <=.<72. O0D3P0/53 /0 543.E.53> 8/9:3;/01 B<../623 6, 7,..., 0 + 1 />B<../623 4, 5 B<../623 2, 3 />B<../623 0, 1 O5 /. C2.< G<;54 0<5/01 54C5 543;3 3V/.5. C0 /05313; M.794 54C5 /= 543;3 C;3 AF @F 7 <;.<72. /0 543.E.53> F 543.E.53> G/22.5C22 /0 23.. 54C0 M 8/9:3;/01.? L3093F <6.3;H/01 M + 1 8/9:3;/01. G/22 17C;C0533 54C5 G3 C;3 0<5 /0 C0E <= 543 9C.3. AF @F 7 <; F G4C5 G3 9C22 C?==> ;3.725? -./ *) *S RS -..7>3 543.E.53> 9<05C/0. 3/543; A <; @.<72F C0D 9<0D795 <03 <= 543 =<22<G/01 B;<93D7;3.M P 1 -DD @.<72 5< 543.E.53>? QC/5 =<; C B<./5/H3 ;3.725 =<;.<>3 5/>3? O= 54/. B<./5/H3 ;3.725 C;;/H3.F ;357;0 *1F <543;G/.3 ;357;0 2S SR S? P 0 -DD.<72. 5< 543.E.53>? QC/5 =<; C B<./5/H3 ;3.725? O= 54/. B<./5/H3 ;3.725 C;;/H3.F ;357;0 * 1 F <543;G/.3 ;357;0 2S SR S? W43 GC/5/01 5/>3..4<72D 63 5C:30.< 54C5 G3 9C0 B<5305/C22E <6.3;H3 C5 23C.5 8/9:3;/01. C0D 4<B3=722E 135 C B<./5/H3 ;3.725? M + Q3 1 4CH3 543 =<22<G/01 94C;5 <= <759<>3.M P0 P0!" P1 # P1!" L3093F C B<./5/H3 ;3.725 5< Pi 17C;C0533. C B<./5/H3 C0.G3; 5< Qi? -..7>/01 G3 D/D 0<5 135 C 9<0927./H3 ;3.725 G3 G<72D 93;5C/02E 2/:3 5< ;70 =7;543; 3VB3;/>305.F 675 543.E.53> 4C. B;<6C62E.5C223D? Q 4C5.4<72D G3 D< 0<G @

*1* S O= G3 9<72D ;357;0 543.E.53> 5< /5. <;/1/0C2.5C53 G/54 A <; @.<72 XC..35 7B 6E B;/.<03; ZF G3 9<72D 3VB3;/>305 =7;543;? W< D< 54/.F G3 G<72D 2/:3 5< 5C:3.<72. <75 <= 543.E.53>? O5.33>. 4<B323..F /= =<; /0.5C093 <03 ;<6<5 4C. 0<.<72.F 543 <543; 4C. C0D 543E C;3 6<54 3C13; 5< 4CH3 ><;3? 75 /= G3 C;3 ;3CDE 5< 1/H3 543>.<>3F 543E G/22 3H3057C22E 4CH3 C0D 63 G/22/01 5< D;<B 543.<72. C1C/0? Q3 9C0 5430 1;C6 54<.3 23=5<H3;.F 705/2 G3 C;3 6C9: 5< 543 /0/5/C2 07>63; XA <; @Z? W4/. C22<G. 7. 5< 9<0D795 <543; 3VB3;/>305.F C0D 43093 5< D353;>/03 3H3057C22E G43543; B;/.<03; 23=5 634/0D C.<72 <; 0<5? <53 54C5 G3 03H3; 4CH3 5< ;C/.3 543 07>63; <=.<72. CDD3D 5< 543.E.53> 5< ><;3 54C0 @H 5< 135 /5 ><H/01 C1C/0 F./093 G/54 @HF C5 23C.5 <03 ;<6<5 B;/.<03; /. C5 543.5C;5 <; /05< 4/..2<.4 K ;31/<0F C0D /. G/22/01 5< 3/543; 5C:3 <; 1/H3.<72.? ) *) S <G 54C5 G3 :0<G G4C5 543 6/5.305 6E B;/.<03; GC.F G3 B;3BC;3 <7; >3..C13 =<; B;/.<03; H 6E.355/01 543 5<5C2 07>63; <=.<72. 5< A <; @? -=53; 54C5F G3 G<72D 2/:3 5<./10C2 B;/.<03; H 5<.5C;5 4/. ;<23 <= <6.3;H3;? W4/. /. G43;3 543 07>63;. 9<>3 /05< B2CE? ;/.<03; i H 4C. C1;33D 63=<;34C0D 54C5 43 G/22 63 5;/113;3DK G430 43 4C. 3VC952E @.<72. X Z? <53 P;.5 54C5 G3 03H3; 033D3D 5< 1< 54C5 4/14 D7;/01 <7; 3VB3;/>305C5/<0 0 = 18 B4C.3 XG3 033D3D 5< 1< C5 ><.5 7B 5< @HZ? [< G3 9C0 63.7;3 54C5 G3 4CH3 0<5 5;/113;3D B;/.<03; H 63=<;3 0<G? Q3 D;<B 54<.3 @.<72. /0 543.E.53> F C0D 5430.5C;5 5< CBB2E 543 ;723 =<;.<>3 C1;33D <0 C43CD <= 5/>3F 6/113; 54C0? ϕ2 2 Q3 0<G 4CH3 1 @ <; @].<72. /0 543.E.53> F C0D 3C94 B;/.<03; /. ;700/01 C ;723? Q3 <02E 4CH3 ;<6<5. ;700/01 543 B2C93Y [< 543 G4<23.E.53> 3H<2H3. C99<;D/01 ϕ 5< C ;C0D<> GC2:? [/093 543;3 C;3 <02E @ <; @].<72. 543;3 C;3 P0/532E >C0E B<../623.5C53.? <;3<H3;F G3 :0<G 54C5 <03 <= 543 B;/.<03;. 4C. C5 23C.5.<72.F C0D 43093 543 <B5/<0 <= /09;3C./01 <; D39;3C./01 4/. C><705 <=.<72.? W4/. 17C;C0533. 54C5 <7; ;C0D<> GC2: 03H3;.5<B.F C0D 543;3 /. C 0<0IJ3;< B;<6C6/2/5E <= 1355/01 =;<> C0E.5C53 5< C0E <543;.5C53? L3093 B;/.<03; H G/22 3H3057C22E 30D 7B G/54 @.<72.? <G 54C5 43 4C. 4/. @ 5;/113;/01.<72.F B;/.<03; H 7.5 033D. 5< 3;C.3 543> /0 4/. >305C2 9<705 <=.<72.? L3 /. 6C9: 5< A.<72.F C0D 543;3 >/145 63 @.<72 23=5.<>3G43;3 32.3 /0 543.E.53>? L3 639<>3. C0 <6.3;H3; C0D 4/../57C5/<0 /../>/2C; 5< 543 <03 30 <E3D 6E B;/.<03; @ C5 543.5C;5? O0 543 9C.3 <= B;/.<03;.F G3 9C0 C957C22E 5C:3 0 = 18, 1 = 20, 2 = 22, 3 = 24, F C0D /0 1303;C2 543 K. G/22 /09;3>305 6E H 3C94 5/>3? W43 <02E ;3,7/;3>305. C;3 54C5 543E C;3 6/1 30<714 54C5 G/54 54C5 >C0E.<72. /0 543.E.53> X<; <03 ><;3 /= 543 >3..C13 /. C Z 543.E.53> 03H3; 135..579: XG430 C22 B;/.<03;. C;3 ;<6<5.ZF C0D 54C5 B;/.<03;. 1 C;3 0<5 5;/113;3D 5<< 3C;2E G430 <03 /. 5;E/01 5< 5;/113;.<>3<03 32.3? O09;3>305/01 6E H 1/H3. 7.5 30<714 233GCE.< 54C5 543 @ 6/5 >3..C13 D<3.0K5.35.<>3<03 32.3 <U 5<< 3C;2E? + 3 S <G 543 B;/.<03;. 7.5 4CH3 5< 9E923 54;<714 54C5 C21<;/54> F C0D 1/H3 =7;543; 6/5. 5< 543 B;/.<03; =<22<G/01 543> /0 543 <;D3;/01? W4/. G/22 3H3057C22E C22<G 543> 5< 3V94C013 C;6/5;C;/2E 2<01 > 3..C13. G/54 543 <543; B;/.<03;. 5<<? @7

* 1* R R)* /) 1RS*)1 Q3 G<72D 2/:3 P;.5 5< /D305/=E 543 />B<;5C05 B;<B3;5/3. 54C5 543 ;723. ϕ 4CH3 54C5 C22<G 543 C21<;/54> 5< G<;:? 3C22E C22 G3 9C;3 C6<75 /. 543 634CH/<7; <= 543.E.53> C. C G4<23? [B39/P9C22E G3 GC05 /5 5< 634CH3 /0 D/U3;305 GCE. D3B30D/01 <0 543 07>63; <=.<72. /0 543.E.53> F C..4<G0 /0 D/C1;C> H? O0 543 trigger upper bound for resetting procedure test for P 0 test for P 1 will run indefinitely can run indefinitely will stop after at most M flicers can run indefinitely will stop after at most M flicers number of souls 0 9C.3 <= B;/.<03;.F 543 53.5 =<; P1 /. D<03 C5 543 6<70DC;E 635G330 @ C0D H.<72. C0D 543 53.5 =<; P0 C5 543 6<70DC;E 635G330 C0D 7.<72.? W43 >CV/>7> 07>63; <=.<72. 54C5 543 <6.3;H3; 033D. 5< CDD 5< 543.E.53> 5< ;3.35 /5 /. @H? -2.<F /0.30D/01 543 P;.5 6/5F 543 5;/113; HC273 /. @? <53 54C5 /0 543 9C.3 <= n = 3 F 543 =C95 54C5 C22 543 ;723. 7.3D C;3 <= 543.C>3 5EB3 /. 0<5 ;3C22E />B<;5C05? O0 543 1303;C2 9C.3F G3 G/22 4CH3 n 1 5EB3. <= ;723F C22 G/54 D/U3;305 5;/113; HC273. F C0D G3 ;3,7/;3 543 5;/113;/01 B;/.<03; 5< CD<B5 543.C>3 5EB3 <= ;723 C. 543 <03 543 5;/113;3D B;/.<03; /. ;700/01? L3 :0<G. G4/94 ;723 54/. /. 7.5 6E 9<705/01 543 07>63; <= 9E923. C22 543 B;/.<03;. 4CH3 1<03 54;<714? * )23*1 -.35 <= ;723. 54C5 G/22 G<;: =<; 1303;C2 C;3.4<G0 /0 D/C1;C>? Q3.35 5< HF C0D 543 <543; C;3 D3P03D n ;397;./H32EF.< 54C5 h1 hi hi 2 i 1 j=1 h? W43 HC273 ;3=3;. 5< C 07>63; <=.<72 3V94C013. ;3,7/;3D 5< 9<H3; 54C5.395/<0 j <= 543 1;CB4F T ;C543; 54C0 543 07>63; <= B3C:.? WC:3 543 07>63; <= B3C:. 5< 63.794 54C5 543 5<5C2 230154K si 2si h i T? W43 HC273 <= G/22 63.B39/P3D 2C53;? T -22 3VB3;/>305C5/<0 4CBB30. =<; HC273. <=.<72. 23.. 54C0 F.< <093 C B;/.<03;.5C;5. 7B <0 543 2<01 <7;03E 5<GC;D. F 543E G/22 03H3; H 63 C623 5< 9<>3 6C9: D<G0 705/2 543 <6.3;H3; GC05. 5< ;3.35 H 543.E.53>? 4C. 5< 63.35 2C;13; 54C0 H hn 2 + n 2 i=1 h X543.7> <= 543 4/143.5 B3C: /0 3C94 Z.< 54C5 i ϕ,i @

Trigger Trigger Trigger Trigger slosh slosh slosh slosh H H H H h n-2 h n-2 h n-2 h n-2 h n-3 h n-3 h n-3 h n-3 h n-4 h n-4 h n-4 h n-4 h 2 h 2 h 2 h 2 h 1 h 1 h 1 h 1 0 0 0 0,0 T T T,1,2,n-2 543 0<;>C2 ;700/01 <= 543.E.53> F G/54 <543; B;/.<03;. <0 543/; J/1IJC1. D<G0 632<G G/22 0<5 6;/01 C0 3.9CB33 5< C0D C22<G 4/> 5< 1< 6C9: D<G0? -1C/0 G3 D3P03 543 3VC95 HC273 <= C 2/5523 H 2C53;F 675 C..7>3 =<; 0<G 54C5 /5 /. 6/1? W43 5;/113; HC273. C;3 H D/U3;305 =<; 3C94 B;/.<03;? W43 C21<;/54> G/22 G<;: G/54.< 54C5 G/54 54C5 >C0E.<72. /0 543.E.53> F C5 23C.5 <03 B;/.<03; /. (n /05< 1)H 4/..2<.4 ;C013 C0D 543;3=<;3 543.E.53> 9C00<5 135.579: G430 54C5 >C0E C;3 CDD3D? W43E 033D <02E /09;3>305 6E H 3C94 5/>3F C. /0 543 9C.3 =<; B;/.<03;.? W< 5;/113; 543 B;/.<03; ;700/01 ;723./>B2E CDD.<72.F C0D E<7; >3..C13 XA <; @ZF C0D 639<>3 C ;<6<5? -. /0 ϕ, 543 9C.3 <= B;/.<03;.F 543 ;C0D<> GC2: G/22 3H3057C22E 30D 7B G/54 543 B;/.<03; 63/01 5;/113;3D <0.<72.F C0D C22 <543; B;/.<03;. 4CH3 5;/113;. <= C5 23C.5 H ><;3 54C0.< G/22 0<5 63 5;/113;3D B;3> C57;32E? ^23C;2E 54/..E.53> G/22 0<5 ;70 /0D3P0/532E G/54 A.<72.? O5 /. C2.< 923C; 54C5 /5 = ;70 /0D3P0/532E G/54 n 2.<72.? L3;3 /. <03.3,73093 54C5F /= =<22<G3DF G/22 ;70 =<;3H3;M ^C22 543 B;/.<03; i=1 h i CBB2E/01 543 ;723 ;<6<5? W< ϕ,i i.5c;5f.35 ;<6<5 A 5< 63 C5 543 6<55<> <= C0E HC223E <0 4/. 9E923F 7.5 63=<;3 C B3C: <= C25/57D3 F C0D 1/H3 5< 3C94 ;<6<5 3VC952E.<72. X0393..C;/2E.5C;5/01 C5 543 B3C: <= 4/. hj 9E923Z? O= ;<6<5 1/H3. 4/. i.<72. hi 5< ;<6<5 A C0D 5430 5C:3. 543> 6C9:F G3 C;3 /0 C./>/2C; B<./5/<0 j 5< 543 hj <03 G3.5C;53D G/54? Q3 9C0 9<05/073 D</01 54/. /0D3P0/532E? Q3 033D 5<.4<G 54C5 543.E.53> 135..579: =<;.<>3 4/143; HC273 <=.<72.? W4/. G/22 ;3,7/;3 7. 5< B;<H3 54C5 0< B;<B3;.76.35 <= 543 ;<6<5. 9C0 ;70 /0D3=I /0/532EF /= 543;3 C;3 23.. 54C0.<72. /0 543.E.53> F G4/94 /. B;<H3D 2C53; <0? /H30 54/.F /5 <02E 5C:3. <03 ;<6<5 H <0 4/. GCE 7B 5< 5<.5C22 543.E.53>? Q3 H 9C0 30.7;3 54/. G/22 4CBB30 6E B755/01 /0 hn 2 + n 2 i=1 h.<72.? Q3 9C0 i + 1 0<G 5C:3 5< 63 C0E 07>63; 2C;13; 54C0 54/. 07>63;F.CE H hn 2 + n 2 i=1 h i +2? * / R *) R * R33* * Q3 0<G.4<G 54C5 G/54 23.. 54C0.<72.F 543.E.53> 9C00<5 ;70 /0D3P0/532E /= 0<5 C22 ;<6<5. C;3 /0H<2H3D? -..7>3 H 543.E.53> /. ;700/01 /0D3P0/532E G/54 C >/0/>C2 07>63; <=.<72. 94C01/01 4C0D.? @

F093 C ;<6<5.5C;5. 7B 5<GC;D. H F 43 9C0 <02E 5C:3.<72. C0D 03H3; ;357;0 543>? E >/0/>C2/5EF 43 03H3; 5C:3. C0E 03G.<72. C0D >/145 C. G322 0<5 63 543;3? [< G3 9C0 C..7>3 54C5 <7;.76.35 <= ;<6<5. >7.5 63 C623 5< ;70 /0D3P0/532E G/54<75 C0E<03 23CH/01 5<GC;D. H <; 1/H/01.<72. 5< C0E ;<6<5 1</01 5<GC;D. H? 5/;.5F G3 G/22 C..7>3 54C5 /. >/../01? -..7>3 C.76.35 0<5 /0927D/01 ;70.F 5430 543;3 /. C >/0/>C2 ϕ,0.76.35 0<5 /0927D/01 G4/94 ;70.? ϕ,0 <G F 235 63 543 2C;13.5 07>63;.794 54C5 /. ϕ,0 /0927D3D /0 54/..76.35? -. 543.76.35 m /. >/0/>C2 >7.5 9<>B2353 C ϕ,m =722 9E923F C. /= /5 D/D 0<5 5430 G3 9<72D./>B2E 23CH3 ;<6<5 ϕ,m <75? W47.F C5.<>3 5/>3 ;<6<5 >7.5 4CH3.<72.? L<G3H3;F 6E 543 94</93 m <= 543./J3. <= 543 B3C:. m h m hm > m 1 i=0 h F /5 /. 923C; 54C5 43 9C0 03H3; 135 ;/D <= 543> C22 G/54<75 B7.4/01 <03 <= 543 <543; i ;<6<5. <05< /5. BC54 5<GC;D.? -. 543;3 C;3 C P0/53 07>63; H <= /0/5/C2.5C53. X2<<:/01 <02E C5 543 ;<6<5. 632<G 543/; B3C:. ZF 543;3 /. C 12<6C2 6<70D <0 543 07>63; <= 3V94C013.F F G4/94 9C0 <997; 63=<;3 hi 543.E.53> 4C25.? /. 94<.30 5< 63 2C;13; 54C0? L [< G3 C;3 23=5 G/54 543 9C.3 G430 T /. /0927D3D /0 543.76.35F 675 L.<>3 ϕ,0 ϕ,m /. >/../01? -. ϕ,0 /. /0927D3D /5 >7.5 9<>B2353 C =722 9E923 XC. <543;G/.3 /5 9<72D 63 23=5 <75 C0D G3 G<72D 4CH3 543 B;3H/<7. 9C.3Z? F093 /5 4C. ;3C943D 543 631/00/01 <= 543.3;/3. <= B3C:. <= 43/145 hm /5 G/22F =<; C5 23C.5 543 03V5 T.<72 3V94C013.F 634CH3 3VC952E C. ϕ,m D/D /0 543 B;3H/<7. 9C.3? 75 54/..76.E.53> /. 17C;C0533D 5<.5<B 63=<;3 ϕ,0 P0/.43. /5. 43/145 hm B3C:.F C. =<; C5 23C.5 543 03V5 T 5;C0./5/<0. 54/..76.E.53> 634CH3. 3VC952E C. /0 543 B;3H/<7. 9C.3? W47. ϕ,0 G/22 03H3; 9<>B2353 /5. hm B3C:.F C0D.< 03H3; 9<>B2353 C =722 9E923? -/ 3R 2* Q3 4CH3 0<G B;<H3D 3V/.53093 <= C.5;C531E? W< C957C22E CBB2E 54/..5;C531EF G3 G<72D 033D 5< 9C2972C53 B;39/.32E 543 HC273 <=.3H3;C2 9<0.5C05. 7.3D /0 <7; C21<;/54>? 5<; /0.5C093F 543 9<0.5C05. T C0D M C0D 543 HC273. C5 G4/94 G3 53.5 =<; P0 F P1 C;3 4C;D 5< P0DF BC;5/972C;2E G/54/0 543 4<7; 54C5 543 GC;D30 4C. 1/H30 7.? W43 <;/1/0. <= 54/. B;<623> C;3 0<5 923C;? -99<;D/01 5< 23130D 6 F 8F./>/2C; B;<623>. 4CH3 6330 543 D32/145X6726Z <= L701C;/C0 >C543>C5/9/C0.? W43 P;.5 G;/5530 <997;3093 G3 9<72D P0D <= 543 B;<623> /0H<2H/01 @AA B;/.<03;. C0D C 70/,73 2/1456726 /. <0 C0 <02/03 =<;7> 4<.53D C5 3;:323E 6 8? -0<543; HC;/C05 /0H<2H3. H <; H7 B;/.<03;.F 5G< 2/1456726. C0D 543 <62/1C5/<0 =<; 543 B;/.<03;. 5< C2GCE. 94C013 3VC952E <03 <= 543 2/1456726.? W4/. <03 CBB3C;3D <0 543 @< =? G36./53 <= O 3.3C;94 6 8? W4<.3 5G< <02/03 <997;3093.F <0 543 3;:323E =<;7> /0 536;7C;E HAAH C0D 543 O G36./53 /0 72E HAAHF G3;3 =<22<G3D 6E.3H3;C2 <543;.F /0 3/543; <= 543 5G< H3;./<0.? - 54;3CD GC. /> > 3D/C532E.5C;53D <0 6 8F /5 GC. B762/.43D /0 543 5C22 /..73 <= 543 C543>C5/9C2 [9/3093. 3.3C;94 O0.5/5753 03G.23553; =??< 68F C0D 543 B;<623> GC. P0C22E C.:3D @

<0 543 B<B72C; ;CD/<.4<G G< 6 8Y Q3 G<72D 2/:3 5< 54C0: -0D;3G 300355. =<; /05;<D79/01 7. 5< 543 <;/1/0C2 B;<623>? 6@8 3;5;C> 5321304C73;? @AA B;/.<03;. C0D C 2/1456726? 3G.1;<7B F CHC/2C623 54;<714 F 72E H HAAH? 6H8 Q /22/C> 53223;? =<@ = <::= = < @ =? ==? F BC13. 7 F ]? [39<0D 3D/5/<0? <40 Q /23E [<0. O09?F @]? 68 C543> C5/9C2 [9/3093. 3.3C;94 O0.5/5753? 4 >/..C;E 03G.23553;F <H3>63; HAAH? -2.< CHC/2C623 C5? 678 30C5C C22<.4 C0D -0D;3/ /0D3? C;: 303;1E C0D 543 =C53 <= 543 70/H3;.3? HAA?? 6 8 NF231 N? @AA B;/.<03;. C0D C 2/1456726? 3G.1;<7B F CHC/2C623 54;<714 F 72E H7 HAAH? 6 8 C5/<0C2 762/9 CD/<? ^C;5C2: CD/< [4<G? W;C0.9;/B5/<0 CHC/2C623 C5!? 6 8 O 3.3C;94? <0D3; W4/. ^4C223013? " # $ % F 72E HAAH? 6 8 Q /22/C> Q7? LC;D ;/DD23.? $ & ' ( ) F 536;7C;E HAAH? @