Chaînes de Markov triplets et filtrage optimal dans les systemes à sauts

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https://tel.archives-ouvertes.fr/tel-00873630

Submitted on 16 Oct 2013

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Noufel Abbassi

To cite this version:

Noufel Abbassi. Chaînes de Markov triplets et filtrage optimal dans les systemes à sauts. Architec-ture, aménagement de l’espace. Institut National des Télécommunications, 2012. Français. �NNT : 2012TELE0018�. �tel-00873630�

Ecole Doctorale SMPC

Thèse présentée pour l’obtention du diplôme de

DOCTEUR DE TELECOM SUDPARIS

Doctorat conjoint

Télécom Sudparis et l’Université Pierre et Marie Curie – Paris VI

Spécialité

:

MATHEMATIQUES APPLIQUEES

Présentée par

Noufel ABBASSI

Sujet de la thèse

CHAÎNES DE MARKOV TRIPLETS ET FILTRAGE

OPTIMAL DANS LES SYSTEMES A SAUTS

Soutenue le 26 Avril 2012, devant le jury composé de

Président Cédric Richard Professeur à l’Université de Nice

Rapporteur Fabrice Heitz Professeur à l’Université de Strasbourg

Rapporteur Eric Moreau Professeur à l’Université de Toulon

Examinateur Michel Broniatowski Professeur à l’Université Paris VI

Examinateur Emmanuel Duflos Professeur à l’école Centrale de Lille

Examinateur Thierry Chonavel Professeur à Telecom Bretagne

Directeur de thèse Wojciech Pieczynski Professeur à Telecom Sudparis

Je tiens tout d’abord à exprimer ma reconnaissance à mon directeur de thèse Monsieur Wojciech Pieczynski qui m’a toujours encouragé et soutenu jusqu’au bout, ses conseils très instructifs m’ont permis de faire évoluer mes compétences et d’enrichir mes connaissances.

Un grand merci à mes rapporteurs Messieurs Eric Moreau et Fabrice Heitz pour le temps qu’ils ont consacré pour juger mon travail avec tant d’attention et pertinence par leurs remarques et conclusions objectives.

Je remercie les autres membres du jury les professeurs Michel Broniatowski, Cedric Richard, Emmanuel Duflos et Thierry Chonavel pour l’intérêt qu’ils ont porté à ce travail et leurs remarques et questions. Je remercie également tous les membres du département CITI.

Une pensée particulièrement amicale pour les personnes avec qui j’ai partagé des bons moments durant ma thèse : Lahouari, Mohamed, Soumeya, Boujemaa, Jerome, Marc, Abdelkader, Noémie, yahia, …

Ma gratitude sans limite est pour les deux personnes qui me sont les plus chères, mes parents ainsi que ma petite sœur, ma femme et mes enfants.

Mes derniers remerciements et qui ne sont pas les moindres sont pour mes amis : Brahim Mirisse, Driss Asrary, Driss Mounir, Hassan Rizzi, Youness Kaab, Rachid Elyasini…

Cette thèse est consacrée à la restauration et l'estimation des paramètres par filtrage, tout d'abord dans les modèles de chaîne de Markov cachée classique, puis couple et enfin triplet à sauts Markoviens. Nous proposons deux nouvelles méthodes d'approximation dans le cas des systèmes linéaires gaussiens à sauts Markoviens. La première est fondée sur l'utilisation des chaînes de Markov cachées par du bruit à mémoire longue, on obtient alors une méthode " partiellement non supervisée" dans la quelle certains paramètres, peuvent être estimés en utilisant une version adaptative de l'algorithme EM ou ICE. La deuxième technique est basée sur l'exploration de l'arbre des réalisations de la chaîne des sauts, l'idée consiste à garder à chaque instant les trajectoires les plus probables sous des contraintes prédéfinies; on obtient là un algorithme rapide et les résultats sont encourageants.

Nous proposons par la suite deux familles de modèles à sauts qui sont originaux. Dans la première, conditionnellement aux sauts le processus couple composé du processus d'intérêt et celui des observations, est de Markov, on montre alors qu'une extension du filtrage particulaire est possible. La deuxième, est une sous famille de la première le processus couple composé du processus d'observations et de la chaîne des sauts est Markovien, pour cette classe on peut faire du filtrage optimal exact avec une complexité linéaire dans le temps. L'utilisation de la deuxième famille en tant qu'approximation de la première est alors étudiée et les résultats exposés dans ce mémoire sont prometteurs.

Mots clés :

Chaînes de Markov cachées, Chaînes de Markov couples continue, System gaussien avec saut Markovien, Filtrage Kalman, Filtrage Particulaire, Algorithmes

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₁λ ₁₅λ _15@@@51λ_D

}

1 4 = Ω 1168165668114918481₇∈ℵ_{1681491848} 1 4 4 35 5@@@5 5 I H 1 + + ∈Ω 7 7 7 8 8 8 8 5141 A'1 1 I 1 H D I 15 1@ 1@ 1@ 15 15 1 H D 4_{7+4 =}8₇₊₄ 4_{7 =}8₇ 4₇₋₄₌8₇₋₄ 4_{3 =}8_{3 =} 4_{7+4 =}8₇₊₄ 4_{7 =}8₇ 1 H4@4I1 1 1 !+1615+69515161A1A1A9V851561++96861541A4361715A13AA61H4 I_{7 7}∈ℵ1 3461DH4_{3 =} 8₃54_{4 =}8₄5111@1@1@154_{7 =} 8₇I511DH4₇+₄ =8₇+₄14₇ =8₇IT1 69481 A9+41 4861AH8₃58₄5111@1@1@158₇I51AH8₇₊₄18₇IT11 1 BFBABF6!1 1 61A89361843A8F6168161A893613A99614813AF61568168119651349168961 31681416814815A1461615681613AF615961$A814@11 1

A1 541 61 3AA61 61 CA9<4$1 681 431 +61 A91 5A1 541 61 4 1 681 61 861 61 A89361₃

843A8F61 4A81 561 5411DH4₇₊₄₌ 8₇₊₄14_{7 =}8₇I@1 21 6++6851 481 61 +A5561 61 A89361

843A8F6 { }1 5@@@5 4 I 5 H II 5 H H 11 D F  F  7 7   ∈ = λ λ @1B1π316815A1541614 516811418699861561689613

61 3AF61 5961 3461 561 94A581 61 AA961 61 5B8A81λ_{1 $691 5B1 61 8A81 4561}

Ω ∈ F λ _{1715B8A81B 7 B1'1 H 5 I DH 1 I} 4 F 7  7 F  7 4 4  λ λ _{= =}λ ₌λ + 5141A1 1 I 5 H @@@ I 5 H I 5 H I H I 5@@@5 5 H8₃ 8₄ 8_{B 3} 8₃ ₄ 8₃ 8₄ ₁ 8₄ 8₁ _B 8_{B 4} 8_B A − × × × × = π 511 H4@1I1 1 E68861+495613456196386681615A1+495616194A58134461681615A194981 61CA9<4$@11 1 "#5$4561 1 A99616189A#6384961613AA6161CA9<4$1716:18A816817149841+:6 B1 16819681 715A1.99614@4@1 1

D1 1 1 .99614@416968A841615A99616189A#6384961613AA6161CA9<4$1491D =11681 ? = B @1 1 1

!1 2F58675692D6$CF586

6

BFBABF6%1 1

61 3AA61 61 CA9<4$1 H4₇I₇∈ℵ1 681 861 44961 HA1 561 86I1 1 ∀7∈ℵ51

I 1 H D 4_{7+4 =}8₇₊₄ 4_{7 =}8₇ 16161A16 71 @11 1 !15A1541613AA6161CA9<4$144961681461A916194A581π_{31681616561} A89361 843A8F6 _{{ }}1 5@@@5 4 I 5 H 5 I H 11 D F  F    ∈

= 51 861 J1A89361 61 89A841K1 41 J1A89361 61

CA9<4$K@1 1

"51 681 4$681 9A8F61 A43691 71 61 A89361 61 89A841 1 99A61 496851 4M1 561 4681 481 561 8A81 61 5A1 3AA61 681 5B4968A841 681 461 A91 561

94A58 _{{ }}1 5@@@5 4 I 5 H 5 I H 1 D F  F   ∈ @176:14681  F λ λ 115 1481A54919651A9161+5361A55A8161λ_1 $691λ 1168165668115A194A5816189A841_F 1_5F161λ_1$691 F λ 168141556@1 1 1 1 1

431 "#5$456!6 6 B46866D ₌151Ω=

{

λ 6₄5 λ₁

}

168 _      − − =      4 4 66 @1 1 6199A6149681869L8A81349964A81681561$A8'1 1 1 1 .99614@112:65616199A6149681A43171613AA61716:18A8@1 1 1

E4941 61 3AA61 61 CA9<4$1 44961A$63161A893616189A84 1 @1D4918481

ℵ ∈ 3 1 41 4861 { }1 5@@@5 4 I 5 H I H 5 I H _{1H I} D F  3 F  3 _  ∈

= 1 5A1 A89361 61 89A841 +61

A91 H I DH 1 I 5 7 3 F 7  3 F  4 4  _{= =}λ ₌λ + 51 681 41 4861 3

 1 5A1 61 A361 6 1 @1 1 A1 A5491

5498A81958A81$A851A6518496161J1EAAL-4549494$1K1'1 1 1DC$5666 1 A1A893616189A841  1B613AA6161CA9<4$144961$9+6'11 1 ℵ ∈ ∀_{H315I 1111111}H3+I _=H3I_×HI_{1 1 1 H4@?I1} 1 111∀₃∈ℵ_{11111111111111}H3I _=3_{@1 1 1 1 H4@EI1} 1

A1489A84161&%'134561615A1949816194A581848A565168136556161&('1681

+46191&%'5165561681N89618561491+A96119A46681A919399636@1 1 6156616815A1944841F1$6814681146619A8F614915691613AA6161 CA9<4$@1 1 )5$$561 6 B481 =H₇I₇∈ℵ16186161$A9A561A5A849616A86161N6154171$A5691A1

1 6A361 69A56 Λ1 @1 B481 _{4 1 61 $A9A561 A5A84961 71 $A5691 A1 Ω 1 681 61 5413} π₃₁

6A861 61 5A1 86 1 @1 B481  1 61 +43841 69A561 +61 91 _Ω×Λ1 71 $A5691

441

!5491 5A1 861 61 $A9A561 A5A84961 4 =H47I7∈ℵ1 +61 491 8481 17∈ℵ1 A91

I 5 H ₄ 4 + + = 7 7 7  4 

4 1 681 61 3AA61 61 CA9<4$1 44961 61 89841 8A561π_{31 681 61}

A893616189A8411 1 { }1 { }1 5@@@5 4 I 5 H 5 5@@@5 4 I 5 H 4I II H I 5 H H D H D F  F  D F  F      ∈ ∈ = = = λ λ @11 H4@GI1 1

A1 96$61 61 361 5661 681 A8651 491 4_{7 =}8₇1 5A1 $A9A561 4₇₊₄₌ H8₇5₇₊₄I1 61

61₇₊₄51 681 561 $638691 H ₃5@@@5 ₄I − 7 4 4 1 61 1 $638691H₄5@@@5₇₋₄I@1 BA3A81 F61₇₊₄1 681 I @@@5 5 H₄ ₇₋₄ 1 481 6A8651 4_{7+4 =} H8₇5₇₊₄I1 681 H ₃5@@@5 ₄I − 7 4 4 1 561 481 9A56681 H34846556681714_{7 =}8₇I514M15A1A9<4$A8@1 1 6189A8411_{5F =}DH4_{7 4=}λ_F14_{7 =}λ_I + 616681A161B 7 B64315A13AA61 4 168161

61 3AA61 61 CA9<4$1 44961 61 89841 8A561 π₃1 681 61 A89361 61

89A84 _{{ }}1 5@@@5 4 I 5 H 5 I H 11 D F  F    ∈ = @1 1 1A19A566815A19394F651F6141441A15A1+4961$A861'1 1 *D8BABF66 6

B481 4 =H4₇I₇∈ℵ1 681 61 3AA61 61 CA9<4$1 44961 71 $A5691 A Ω6 51 61 A89361 61

89A84 _{{ }}1 5@@@5 4 I 5 H 5 I H 11 D F  F    ∈ = @1"516:8616186161$A9A561A5A84961H@@I1 =H₇I₇∈ℵ171

$A5691 A1Q354R1 681 61 +43841  1 69A561 +61 91 _Ω×Q354R_{1 71 $A5691 A Ω}1 51

6A8616664₃1681865561F664_{7+4 =} H4₇5₇₊₄I@11

1

+5$2D,566 6

B1 41 61 461 61 A89361 61 89A84 6 51 41 348981 61 +43841  1 61 61 71 #491

A43651 41 561 61 861  =H₇I₇∈ℵ1 61 $A9A561 A5A84961 H@@I1 $A81 61 541

995961 461 681 41 561 4 1 6541 5A1 89841 8A56₃ π₃51 A5491 5A1 5A841 61 5A1

3AA61 61 CA9<4$1 4 =H4₇I₇∈ℵ1 61 +A81 A1 71 A1 $A81 5A1 965A841

939968664_{7+4 =} H4₇5₇₊₄I@1*41469415A13489384161865561+43841  1A15A1

638412@1 6

"51 681 4561 61 3A535691 61 69A361 41 61 541 348465561 491 61 3AA61 61 CA9<4$1 44961 61 +43841 61 A361 61 A1 A89361 61 89A84@1 *41 A$41 561 958A81$A81'1 6 *D8BABF6!6 6 B4814 =H4₇I₇∈ℵ1613AA6161CA9<4$14496161A893616189A84 6 @114861π₇15A1 54161 _{4 16814134961  161+438414961+6191 Ω 171$A5691A7} ℜ@11A1491 8481₇∈ℵW_'1 1

411 1 π7 =π36 1O11 1 !1 21QH4 I143R

(

1

)

H43I 7 7 = 1O1 1 %1 21Q H I1 ₄5@@@5 ₃R

(

1

)

H ₄I − − = 7 7 7 4 4   4 4  1O1 1 (1 21QH47IR=<π75 >1=<

(

π₃17

)

51 >1=<π₃5

(

7

)

>@1 6 *D55-6 6 •1 614814134561615A199561613A61848A561O141A1'1 1 W ℵ ∈ ∀_{7 51∀1 ∈F}

{

_145@@@5D

}

₅₁ 1

∑

∑

∑

= − = − − = − = = = = = × = = × = D  F   7 D   7  7 F 7 D  F 7  7 F 7 4 4 4 4 4  4 5 I H 4 4 4 4 4 4 I H I H D I 1 H D 1 I 15 H D λ λ λ λ λ π π ₁ 1 B41 +4961 A89365561 41 681 431 39961 _{7 7 4}6 − = π

π @1 DA91 89A841 61 368861 9A581 41 A'1

ℵ ∈ ∀_{7 51} 7 7 π36 π ₌ @1 1

•1 61 481 11 34561 61 5A1 +841 61 5B69A361 348465561 681 61 5BFA841 61

H2345B4CDEFBFF8FI5141A'1 1 ℵ ∈ ∀_{7 51∀∈}

{

_145@@@5D

}

₅₁

∑

= × = = = = D F F  F 7  7 4 4 4  4  4 3 3 R DH 1 I H I 1 I H Q 1 2 λ λ λ λ ₁ 1 7BA8961A98₆∀₇∈ℵ_{51∀H5FI∈}

{

_145@@@5D

}

1₁₁ 1

( )

H I 5 5 3 I 1 H D 4_{7 =}λ_F 4 ₌λ_{ =} 7 _{ F =}_7_F1 1 43∀₇∈ℵ_51∀∈

{

_145@@@5D

}

₁ 1

(

)

(

)

H I I H R 1 I H Q 1 2 HI 4 I H 5 3  7  7 D F F 7 F   7 4       4  ₌λ ₌

∑

_× λ _{= =} λ = 1 1 B4M1561958A8@1 1 •1 61481?1681N8961811958A81936851341A$6315A19498161CA9<4$@121 6++685₆∀₇∈ℵ_{51∀H5FI∈}

{

_145@@@5D

}

1₅₁₆₈₁

{

}

4 3 1 1 3 1 5@@@5 − − − Ω ∈ = ∀ 7 7 7 ω ω ω 5141A'1 1

∑

= − − − − − − = = = = = = × D F F 7 7  7 F 7 7 7  7 7 4 4 4 4 4  4  4 1 3 1 3 4 1 3 1 3 4 15 R DH 1 511 I H I 1 I H Q 1 2 λ ω λ λ ω λ ₁ 1

4?1 9'1 F   7 F 7 7 7  7 F 7 4 4 4 4  4 1 _{4 5} 3 1 3 4 15 I DH 1 I 1 H D _{= = = = = = =} − − − − λ λ ω λ λ ₅₁ 1 43'1

(

1

)

(

1

)

H I I H R 15 1 I H Q 1 2 4 5 1 3 1 3 4   D F F F  7 7  7 7 4 4       4  ₌λ ₌ω ₌

∑

_× λ _{= =} λ = − − − 51 1 B4M1561958A8@11 1 •1 6169691481E1614896171A98916148193685141A'1 1 ℵ ∈ ∀_{7 51 =}

∑

_{= × =< >} =   4 4  ₇ D F F F 7 7IR DH I H I 5 H Q 1 2 4 π λ λ ₁ 1 7148141418196386681F614918481₇∈ℵ_{'  4 =<}

(

_7

)

_{ >} 7IR 1 15 H Q 1 2 π₃ 1O1 41A1A54914918481₇∈ℵ_1'1 1

(

)

=<

(

)

> =         × = ×         = × =

∑

∑

∑

∑ ∑

∑

= = = = = =          4  7 D   7  D  F D F 7 F   D F F D  7 F   D F F F 7 7 5 1 I H I H I H IR H Q 1 2 3 4 I H I H 3 4 4 I H 5 I H 3 4 4 I H 5 I H 3 4 I H π π λ π λ π λ π 1 6 6 +5$2D,56!6 6 *4841561958A81$A8518918561619A8F651F134561196691481'1 1 ℵ ∈ ∀_{H 7E5 I} 6 51 _{7 E = ×}7+E _{= ×}E _×7 _{= E×}7 + π π π π H I 3 3 1 1 1

%1 2F58675692D6BDD7AB34586

1

B9938581 681 61 4841 F1 41 69681 61 BA9691 F61 841 561 8A81 4561 6$681N8961A88681A915A13AA6@1D49189A961368861484141A95619A56681618A81F1 J134F681K51 A3A81 F61 6:1 8A81 34F681 1 5A1 94A581 BA55691 61 5B1 71 5BA8961 681 89386681 48$6@1 1 9A1 FB61 3AA61 61 CA9<4$1 681 9938561 1 841 561 8A81 4561 34F681 68961 6:@1 D491 3693691 561 8A81 F1 34F681 51 681 4$681 51 34461 61 89A$A55691 91 561 99A61 61 5A1 3AA61 5881 F61 91 5A1 A89361 61 89A84@1613AA6161CA9<4$1681A5491993856116816566811516:8611361 +691A156199A61AA81A14161+41A918415618A81615A13AA6@11865199A61 68181J1+4986681346:61K@1!1199A61H4114L99A6I1681+4986681346:611 84161468134F681689616:@11

1 681 344691 A5491 1 99A61 F6534F61 61 8A81 B61 3AA61 61 CA9<4$1 61 61 A98841614L99A651F13486816:18661F1481+4986681346:61'1

4E1 1

1 E6:1F16134F681A1A$63156:89691'1A3136114L99A616169681

BA886961 1 4681 6:86961 71 361 6969@1 E61 481 61 35A61 861 +69651 41 561 A65561A1X35A61+A56X@1 1 !1 E6:1F134F681A$63156:89691'171A98916F6556141681A886961A141 146816:8696@11 1 BFBABF6(1 6 B481H47I7∈ℵ1613AA6161CA9<4$14496171$A5691A11Ω=

{

1λ415λ115@@@51λD

}

@1 11

HI1 1 8A81 λ_{F∈Ω1 681 81 J1A336561K51 J1A8869A561K51 71 A9891 61 5B8A81 ∈Ω}

 λ _{1 1 681} 6566811516:861₃∈ℵ_{18651F61DH 1 I 3} 3 = > = F  3 4 4 λ λ _1O1 1 HI1181F615618A81λ_1681 F λ 134F685168141486₁λ_1↔₁λ_F5116816566811 F λ 1681 A33656171A989161λ_1H41486 F  λ λ_{1 1} 1 → I1681  λ _{1681A33656171A989161} F λ 1Hλ_F6→6λ_I1O1 1

HI1 61 35A61 BF$A56361 349964A861 71 5A1 965A84₁λ_1↔₁λ_F1 481 861 J135A61

B8A81343A81K@11 1

21 894A81 A1 566561 61 35A61 B8A81 343A81 5A1 965A841 49961

$A861'1_{ ≤4} ₁1516:861λ∈E₄681λF∈E118651F611λ1→1λF5141681489691FA$6315A1 94A5819A5617115A13AA61+851A911861+51A1616135A61869A561491 368861965A84149961A9865@1 219A8F65141A94171+A961A:13AA61J19938561K514M18415618A8134F68@1 *41A$415A1+841$A861 1 BFBABF6.6 6

61 3AA61 61 CA9<4$1 481 841 561 5681 61 5B6A361 B8A81 34F681 681 861 J19938561K@11

1

*4841 F61 +841 F$A56861 34861 71 961 F61 56A36 Ω11 1 681 981 71 61 6561

35A61 B8A81 343A8@1 1 681 A++9691 F61 5A1 4841 9938581 61 F66816194A5816189A84168141615A15418A56@1

1

1

(1 /A2ABFF2DBA65A6$DA5$5FA6280$AAB,56

1

E461 931 936681 41 B89661 6:35$6681 A:1 3AA61 61 CA9<4$1 44961 71 6A361 B8A81 +@1 7A1 361 A9A99A61 41 $A1 8691 5A1 8A84A981 61 361

3AA61 A1 F61 5691 J1344986681 A6848F61K'1 561 81 681 61 39961 5A1 541 61 4₇1

549F6117→+1∞@11

6 6

4G1

(1 /A2ABFF2DBA68ADBA56

1 613AA61A5A849614 =H4₇I₇∈ℵ171$A5691A1166561+1Ω=

{

1λ415λ115@@@51λD

}

11681861 J18A84A961499613₊41K111 1 4 4 35 5@@@5 I H + Ω ∈ ∀ 8 8 83 3 5 4 4 35 5@@@5 I H 1_{∀ ∈ℵ}3+ 3 7 7 7 1168₁∀_1E∈ℵ_141A1 1 I 5@@@5 5 H D I 5@@@5 5 H D _{3 4 3 4} 4 3 4 3 E 7 E 7 E 3 7 7 7 3 7 8 4 8 4 8 4 8 4 8 4 8 4 3 3 = = = = = = = + + + 1 1 A1 541 34#4861 1 1 $638691 A5A84961H 5 5@@@5 I 4 3 E 7 E 7 E 7 4 43 4 _{+ + +} 1 681 A1 5A1 N61 F61 365561 1 $63869H 5 5@@@5 I 4 3 7 73 7 4 4 4 @1 1

61 3AA61 681 8A84A961 A1 61 89381 H41 +4986681 8A84A96I1 1 ∀₁₃∈ℵ_{1 65561 681}

8A84A9617154996_{1 +}3 4@1 1 14861F613AA6161CA9<4$18A84A9614996111681+49866818A84A96@1 1

(!1 B8ADB3ABF68A2ABFF2BD565A6$58D56BF2DB2FA56

6 B481 1 6A361 B8A81 +₁

{

₁λ ₁₅λ _15@@@51λ_D

}

1 4 =

Ω 1O1 61 541 61 94A581 91 Ω 1 69A1 3491

34618A811$6386915961_{π =}HπH4I5@@@5πHDII134416194A58161++96818A81 456@1 1 BFBABF6616 6 B481H4 I7 7∈ℵ1613AA6161CA9<4$14496171$A5691A1Ω=

{

1λ415λ1 15@@@51λD

}

5161A89361

61 89A84 1 @1 61 94A581 _{π =}HπH4I5@@@5πHDII1 681 $A9A861 A91  1 H41 681 61

J189841 8A84A961K1 6 1 I1 1 681 656681 1 65561 $9+61 5BFA841 A89365566π6₌π @1 1 61$6386915961π16814311$63869194961719A3614915A1$A5691949614@11 1$9+61A6815A194981$A861'1 1 *D8BABF6(6 6 613AA6161CA9<4$14496161A893616189A841  1681+49866818A84A9611681 65668115A15414 1681$A9A861A9 ₃ 1 @1 1 *41A$41A5491561958A81498A81$A81'1 6 1DC$56!6 6 613AA6161CA9<4$1449619938561A6816194A581$A9A86@1 1 1 1

421

D9341 5A1 4841 61 9$6958@1 B481π1 61 94A581 $A9A861 A9 ₁

I 5 H 5I H 1 Ω ∈ =    @1 D491848_{1 ∈}

{

1415@@@51D

}

5141A1'1 1

∑

= = D       4 5 1 1 1π π 1 1 A1 A89361 ₅ I_{H 5I} 1 HDH 1 ₄ II_{H 5I} 1 Y H Y 1 Ω ∈ + Ω ∈ = = = =   47  47  _

 λ λ _{1 681 A5491 9A56681 61 A89361}

843A8F6@1216++6851₁∀₇∈ℵ_{51681∀1H5I∈}

{

_1415@@@51D

}

1_{141A$451} 1 O 1 F 1 I H D I H D I H D I H D I 1 H D 111111 1111111111 1111111111 1111111111 1111111111 1111111111 I H D I 1 5 H D I 1 H D Y 111111111 5 5 4 4 4 4 4 4 5        7  7  7  7  7  7  7  7  7  7  7    4 4 4 4 4 4 4 4 4 4 4  π π λ λ λ λ λ λ λ λ λ λ λ = = = = = = = = = = = = = = = = = + + + + + + 1 1 1A16111 4 4 5 =

∑

= D     51431 Y168161A8936161CA9<4$@11

A1 A8936 6Y1 B8699861 3461 5A1 A89361 61 89A841 61 5A1 3AA664 1 A91

J19684966811861K@11 1

BFBABF626 6

1 81 FB61 3AA61 61 CA9<4$1 44961 61 A89361 61 89A841=H_5I_H5I∈Ω151 681

9$69561A919A4981715A194A586π11∀1H5I∈

{

1415@@@51D

}

111      1 5 15 1π ₌π @11 1 +5$2D,56%6 6 "513456196386681615A169961+841F611613AA6161CA9<4$16819$69561A91 9A4981716194A586π51A549136886194A581681$A9A86@1 1

(%1 1DC$565D7B,56D62F586BDD7AB3456BFB586

1

B481 H4 I_{7 7}∈ℵ1 61 3AA61 61 CA9<4$1 44961 681 99385651 71 $A5691

A₁

{

₁λ ₁₅λ _15@@@51λ_D

}

1 4 = Ω @1181F65561681619461_∈ℵW_{15161A98A811}λ_∈Ω51561 6561 4581 61 968491 71 λ_{1 481 A:1 8A81 58561 61@1 1 48961 F61 1 61}

61 A1 61 λ_{1 H51 99A566851 549F51 61 A1 5691 35A61 F$A56361 B8A81}

343A8514148961F613AF6135A61A681619461F16815A1N6149184161 568I@11

1

49F6_{1 =} 415A13AA61681861J1A94F61K@1

1

BA3A81F6143 = λ∈ΩO1414869A1τλ15A1$A9A561A5A84961J186119669196849171λ1K1

461A91'1 1

4C1

{

1 411Z1 1

}

1 1τ_{λ = ≥ =}λ 7 4 7 3 @11 1 69A36121

[

1τ_λ11_{43 =}λ1

]

_{1461561861466119669196849171}λ_@1 1

14869A11λ15A1$A9A561A5A84961J1496161$86161λK1461A91'1

1 { }

∑

+∞ = = = 4 4 1 7 47 1λ λ @1 1 BFBABF631 6 B481H4 I_{7 7}∈ℵ1613AA6161CA9<4$14496171$A5691A Ω6 +@1 1

&B'1 8A81λ∈Ω_{168181J1A49A81K1168165668115A194A58161561F886916815561O1}

1

&BB'118A81λ∈Ω_{168181J189A84961K116816566811D1}

[

_{1 11 1}

]

₄

3 = < +∞ < λ τ_{λ 4 O1} 1 &BBB'118A81λ∈Ω_{168181J19399681K116816566811D1}

[

_{1 11 1}

]

₄ 3 = = +∞ < λ τ_{λ 4 @11} 1 A1489A841618941958A81$A81681N8961345861A1Q?DR@1 1 1DC$56%6 6 B4864 1613AA6161CA9<4$144961993856@1!549111 11 &B'1 D491848₁λ∈Ω₅ { }

_[

_]

11111 @ @ 1 1 1 1 1 2 4 I H 11 1 4 4 1 3 4 E A 4 B B B 7 47 _{τ λ} λ π λ λ = =   →  +∞ → = =

∑

1O11 11

&BB'1 61$638696π1+1A1616815BF6194A581$A9A861615A13AA6@1761551

3 I Hλ > π _14918481λ_∈Ω1O1 1 &BBB'1 D49184861+43841  148$61+619 Ω6 511 =

∑

<∞ Ω ∈ λ λ λ π π _{15 H I1H I 51A549'1} 11 1 14 1H I1 11 15 H I1 H I11111 @ @ 4 E A   4  B B B 7 7

∑

∑

Ω ∈ +∞ → = =   →  λ λ λ π π ₁ 1 6 1DC$56(-63F8B9191EFBFF8F14198F696 6

B481 4 1 61 3AA61 61 CA9<4$1 6994F61 H9938561 61 9466 ₌4I1 61 94A581

$A9A861F66π@1!5495141A11 1

{

}

1 1 @@@5 15 4 1 I 5 H 1_∀   _∈ D 511 11 H 15 I 1 11 HI H I 5  7   7   7 7    λ π λ λ _{= =} +∞ → +∞ → 1 6

4>1 6

+5$2D,56(6 6

7A1 561 3A1 4M1 5A1 3AA61 681 9938561 71 6A361 B8A81 +1 H93996861 48$6I1 A1 61

9466 >451 5B6A361 B8A81 A681 61 9A98841

{

135@@@5−4

}

1 61 1 35A61 #4861

J135A613635F61K@17A13613A11 1× H 15 I≠3    7  λ λ _{1681656681}  λ 6 168₆λ_{1AA98668171} 5A1N6135A6@116814896919A56681F61'1 1   4 I H 3 1 =

∑

∈ λ π λ 51681∀H15I∈

{

1451@@@51D

}

11 11 7 H _15 _I H _I 7    × λ λ _{= ×}π λ +∞ → 1 1 1

.1 2F5675692D6675#6A2A86

6

7A1 561 3A1 B61 3AA61H4₇I₇∈ℵ1 71 6:1 8A81 HD =1I1 34961 A1 361 A9A99A61 41

A484156148A841$A861'1 1 L1 Ω=

{

1λ415λ1

}

51       − − =      4 4 51A$636H5I_∈

[ ] [ ]

3564 _× 3564O1 1 L1 4918481∈

{

14151

}

51   τ τ λ = 16811λ =1@1 1

.1 EDD7AB3B4BA6

6 A1 3AA61 H4 I_{7 7}∈ℵ1 681 9938561 491 8486H5I∈

]

635646

] ]

× 635646

]

@1 21 6++6851 13<11<411 A549 Ω6 16816135A61343A861+6961993856168141A'1 1 4 I 1 1 1 H D 11A₅ = B <∞ 4₃ =λ = 5DH1 =∞1143 =λI=41 6

61 6:1 8A81 61 5A1 3AA61 481 A5491 9399681 48+51 681 431 5A1 3AA61 681 9938561 9399686148$6@17A13613A15A13AA616819A56681A94F6514316994F6@11681 9A56681561348A86916148A81F615A1A893616189A84168199596@111

1

B∈

[ ]

31541 681=341 5B8A86λ11 681 1 8A81 A49A81 H431939968I5168151BA981B613AA61

A49A86@1 B8A8₆λ_{41A6816135A+3A8416541561$A5691961A91@1*41A$4'1}

1

]

[

]

[

[ ]

       ∈ = = = ∈ = ∈ = = = I 11 11 H 111111111 4 15 3 111 111 1 11111 111111111 1111111111 4 111111111 I 11 11 H 1111 1111111111 3 111 111 4 11111 111111111 1111111111 4 11111111 I 11 11 H 111111111 4 15 3 1 111 111 4 11111 1111111 4 15 3 1 11111111 I 11 11 H 1111 1111111111 4 8111 111 4 11111 111111111 1111111111 3 1111111 11 1 4 4 4 4 5 E37 E7  7  E E37 E7  7  E 73E7 E7  7  E 73E7 E7   E A_ λ λ λ λ l 6 1 7BA8961A986 1

4D1

[ [

   = = ∈ = = ∞ = 1 111 1111111 1111111111 4 11 4 111 111111 4 5 3 I 1 1 H D 11 ₃ 1  E  E 4 1_{ } l λ ₆ 6 61344A86616861#4681619561689F61A15A1A893616189A845143141 499A1343596161A961AA549611₌31681∈

[

3154

]

@11 6 +5$2D,56.6 6 B_{1 =} 41 681₌46 A5491 561 6:1 8A8₆λ_{41 68} 1 λ 6 1 481 9399681 48+1 681 5A1 3AA61 681 861 J14355A861K@125561681A13613A194F61619466 ₌151A$6311 ₌E ₌2 _=L=1@1 1

.!1 $DA5$5FA6280$AAB,565A67B8ADB3ABF6BF2DB2FA56

6 .!₁ *B882F586756426$2ADB56756AD2F8BABF6 6 148961A681HA916:6561A919399636I1F61491"+!≠3141A11 1 11 1 I 4 H 1 4 1111111111 1111111111 1111111111 1 I 4 H 1 I 4 H 1 I 4 H 1 I 4 H 1 1 4 5 _{H I} 1 5 1 I H 4 5 1 I H 1 5 4 I H 4 5 4       − − + − − +       + =         − − + − − − − − − − − + + =         = ℵ ∈ ∀                                      3 3 3 3 3 3 3 3 3 3 3 1 1

1 681 9A56681 489691 361 958A81 61 A94A5A81  @1 A1 A89361  1 A681 3461

$A5691 94961 41 686H4_{− −}I51 A$631 561 $638691 94961H1451141IB1 681H1_{ −}511 1IB@1 1 21 4A81

      − − =   # 4 3 3 4 1 68 _      − =   $ 4 4 6 51 41 A51 49₆H+I≠351 $−4₆#₆$ = 1 681 431 491 8481 ℵ ∈ 3 141A13 $−41#31$ = 51361F161A1958A8@11 1 B6486%"+!≠31681A1+498613A911"_{+ !=}351365A1349964171_{= =}3@1!5491       = = 4 3 3 4 1   16813 =₁3 =₁@1

1 A8961 3A1 A9835691 71 93691 681_=_=4_{51 361 F1 3499641 7 }

     = 3 4 4 3 1 51 681 F1 46    = 1111 11 11 11111 1 11 11 111111 1 1 E 3 A  A 3 E  3 @11 1 616:1696913A16:89N61FB41$681B6:469134996468171613AA6161CA9<4$1 7134498668186986149118A818A514@1 1 1 1 1

131 .!!₁ 24467564264B65A6$DA5$5FA664BFBFB111

(

7→+∞

)

1 6 6196691481615A1D9448414@?14169681613A5356916:53866815A1541π₇161 7 4 @1 216++685141A1 7 7 π31 π ₌ 149184817∈ℵ@174351491"+!≠3141A'1 1 1

(

)

(

)

_      − − + − − − − − − − − + + = = 7 7 7 7 7 7 7                   1H4 I 1H4 I I 4 H 1 I 4 H 1 11 4 H1I 3 I 4 H 3 I 1 H I 4 H π π π π π 51 1 361F1461 1

(

)

(

)

[

]

(

)

(

)

[

]

     − − − − + + = − − − + + + = 7 7 7 7               I 4 H 1 1 1 1 1 4 I 4 H 1 1 1 1 1 4 I 1 H 3 I 4 H 3 I 1 H 3 I 4 H 3 I 1 H I 1 H 3 I 4 H 3 I 1 H 3 I 4 H 3 I 4 H π π π π π π π π π π 1 1 BA3A81F611π₃H4I + π₃H1I =45151$68'1 1             + − − − + + =       + − − − + + =                 7 7 7 7 I 1 H 3 I 1 H I 4 H 3 I 4 H I 4 H I 4 H π π π π 1 1 *416141561344986681615A1541549F66

(

7 →+∞

)

1'1 1 ₁ B663_<₊_<1141A6664₋H₊I6 _<41A5491              + + =   →        + =   →  +∞ → +∞ →               7 7 7 7 15 11 1 4 11 11 1 π π 1 1 +5$2D,586 1 •1 A15861615A1541549F616

(

7 →+∞

)

16816A861615A15418A561O1 1 •1 A1A89361586168134461616:1596168F61O1 1 •1 1A15F$A56361$A861'1Q3<+<1R1⇔1Q3<1<4R@11 1 D491 561 6:1 3A1 6:89N61 11_{+ =}31 681 _{+ =}11 HF1 F$A56851 96638$66851 71 3 = =_  16811_=_=4_{I141A$41'1} 1 !₁ B1_{+ =}31A5491 ∀ ∈ℵ      = = = = 7    7 7 1 111111 1 11 11 3 1 π π π 1

141 1 %_{1 B}1_{+ =}151A549141BA1A16134$6996361H51BA981B613AA614355A86I'141A1491 8481₇∈ℵ_'1 1 1 1 11 4 1 1 1 1 1    = = + × ×     7 7 51 1 I 4 15 4 H 1 1111111111 1111111111 I 1 H 3 I 4 H 3 4 1 1 3 1 1    − − = = + × × π π π π π 7 7 1 1 196A9F61F61549F6π_{3 =}H3@G153@GI5141A 3 4 1 1 1 1 π π π _{= =} + × ×7 7 @1 1 .!%₁ )B68A2ABFF2BD56 1

7A1 561 3A1 4M63_<6_<41 5A1 3AA61 681 6994F61O1 41 681 A5491 A5F691 561 84961 61 -4549494$1 681 5A1 94A581 $A9A861 B681 A8961 F61 5A1 94A581 5861

      + + =       15

π @1 1 681 $9+691 361 958A81 61 85A81 5A1 +841 61 5A1 89841

$A9A86161945$A815616861$A8'1 1    = + × = 4 I 1 H I 4 H π π π π  51A$631_{π =}HπH4I5πH1II1 1 148961A681F615A145841F61613616861B681A8961F6 π6 @11 1 7A15616:1A89613A141A$4'1 1 •1 B6= =3684861561898416194A5814818A84A96@1 1 •1 B16= =415BF61898416194A5818A84A961686 =π H3@G563@GI@1 1 .!(₁ 2F56D5D8B3456 1

1 $A1 3693691 41 F65561 34841 5A1 3AA61 61 CA9<4$1 681 9$695651 3B68L7L961 41

F6556134841516:86161898416194A5816ν =HνH4I56νH1II1865561F6'1 1

{

}

1 1 5 4 I 5 H 6 66 6 6 6∀   ∈ 1111νHI_×1_{5 =}νHI_×1_51 1 B161865561898416194A5816:86165561681A54914584116861 1 1 3 11111 1111 A989A96 11 '1 4 1 3 11111 11 1 5 1 1 I H I H 11 11 1 11111 4 11 I 1 H I 4 H I 1 H I 4 H I 1 H I 4 H     = = < <       + + = ⇔    = × + = × + ⇔    = + × = ×   E   E               ν ν ν ν ν ν ν ν 1 1 74315A13AA616819$69561A919A498171A1898416194A5818A84A961_{ν =}π @1 1 1 1 1

111

11 54DBA$5867568B$42ABF6

6

A1 5A841 61 9A5A841 61 3AA61 61 CA9<4$1 681 1 A81 4851 61 3A535@1 *41 441 F61 5B41 461 B1 99A8691 61 $A9A561 A5A84961 6A861 61 541

+4961 91

[ ]

3514@1 *41 A5541 6:538691 5B61 61 8461 F1 69681 B49691 61

89A+49A841 91 361 $A9A561 A+1 61 94961 61 $A9A561 A5A84961 4A81 61 541 46@1 25561 69681 A5491 5A1 5A841 61 9A5A841 61 3AA61 61 CA9<4$1 +6@1 21 9A8F61 41 461 1 99A8691 61 $A9A561 J164LA5A84961K51 F41 681 961 A94:A8$6681A5A8496@11469A1A915A186116:6561BA594986149199691

61 4961 64LA5A84961 A1

[

3154

]

51 F1 69481 3491 3461 1 3A85541 1

B6189841+4961Q?>R@1 1

11 )B867B8DCA58668DA6BFB6

6 B481 4 161$A9A561A5A8496171$A5691A1

{

λ ₁₅λ _{15 15}λ_D

}

1 4 K = Ω 5161541 1

{

D

}

 4 A_ DH _I511111 1 415 15 1 = =λ ∀ ∈ K 1 1 1 461 E_{3 =}31 68 E A  D     =

∑

≤ ≤ = 4 4911 11 1 4 @1 1 A51 61 A98356951 4 4 = =

∑

= D   D A E @1 61

4811E₃ 15E₄15K15E_D1 681 61 A98841 61 5B869$A5561

[ ]

3514 @1 DA91 A556951 481  1 61

$A9A561A5A8496161541+496191

[ ]

3514 @11A1A5491 1

{

D

}

 1415 15 1 ∈ K ∀ 51D1

(

1∈

[

E_−415E_

[

1

)

=E_ −1E_−4 = A_56 1 5A1$A9A561

∑

_{[ [} = − = D  E E   4   4 11 5 1 H I 4 1 1 Y 4 λ _{168143161N61541F6 46 @1}

E461 41 A1 41 FB41 681 99691 61 $A9A561 A5A84961 61 541 +4961 91

[

35141

[

151

A5491A915A189A+49A841944613L664 16815BN8961A171A98916 @1*4841F61

5B361 61 5B8A81 FB41 349A1 3461 9A5A841 4964 1 B681 A8961 F61 5B361 61

5B869$A55613486A815A19A5A841615A1$A9A566@1 1 A59498616194561615A1A961$A86@1 1 1 F8673B91AA1/969161496164LA5A849611&Y1 _[1351141[111 6

•1 E491&3∈

[

135141

[

13461$A56918A561H9696I@11

•1 .:6916149616869%% 53 %% 41681O1 •1 113489815A1861614961H' I_{3 3}∈ℵ189A8$668'1 1 %%%% %%%%%%% %' %'₃+₄ = _{4 3} + ₃ 1C4541 1 111111111111111111111 6666666666666666666666!5491'6

1?1 1 1111111111111111111111111111111111111111'_3+4∈

{

14(%)))(%₋4

}

1 1 •1 1A++638617  ' & 3 3 4 4115A1$A569111 11 + + @1 1 111111111111111111111!549'1 11111111111 111111111111111111111111111111&3+₄∈

[

35141

[

1 111111111111111111111111 111111111111111111111!94:A841 1 1111111111111111111111111111111111111134961111111&3+₄Y1 [1351141[11 111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111.1 6 6 A5949861F181681146616++3A361615A84161541714981+1 6 1 F8673B91A!1C84616134A961B869$A5561491569181Y1 11 1[ 1 4 I H

∑

= D    λ π @1 6 •1 E489961561$638691E=

[

1E315E415K15ED

]

O11 111111111111111111111111111 E E  D     = ≤ ≤ =

∑

= 4 4911111 111111 1 11111111 5 3 11 4 I H 3 π 1 •1 /96911&1Y1 [1351141[1@1 •1 D4911 =41#FB711 =D 1+A961 1 11111111111111B111&

[

E_{ 4}15E_

[

− ∈ 11A549'1 1111111111 1111111111111111111₁8₌λ_1 1 111111111111111.LHI1 1111111 111111111111.LH49I1 1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111.1 6 6 6

1!1 6/B$42ABF67561D265ABD567F562F5675692D6BFB6

6 B4814 =H4₇I₇∈ℵ13AA6161CA9<4$14496171$A5691A1Ω=

{

1λ₄5λ₁5@@@5λ_D

}

5161A89361 6189A841 _{{ }}1 5@@@5 4 5 # 5 I H D F    ∈ = 168161898418A56

(

3H I

)

I 4 H 3 3 5@@@5 11π ₌ π π D @1 1 A18461FB41$A18569196461915A1D9448414@11O165561+496114661615691 613AA6171A9891B61861614961A5A84961349168961316814@1

1E1 1 1346361A9156915A1$A9A5618A5614₃@1146'1 1 D  E E     = ≤ ≤ =

∑

= 4 4911111 111111 1 11111111 5 3 11 4 I H 3 3 π 1 1 11111111

{

}

1 4 5 5 3 5 3511111111 1 1111114911111H 5 I 45@@@5 11* *    D        = =

∑

∈ = 1 1 1 +851 491 841H5I_∈

{

145@@@5D

}

11 6811&∈

[

35141

[

1 61 +43841 861 61 61 71 #491 A436171 518651F6'1 1 1 Q 5 Q 11 1111111111 I H ₄ 3 &  & E E  = λ ∀ ∈ − 11 1 1 H4@2I1 1 Q 5 Q 1111 1111111111 I 5 H _ & _ & *_{5H 4I} *_5  − ∈ ∀ = λ λ _{6 6 H4@CI1} 6 *D8BABF6.-6 6

B46811₃51₄51@1@1@1@51_B1 61 $A9A561 A5A84961 6A861 681 68F6681 8961

911

[

35141

[

@1 619436114 =

( )

47 ₇∈ℵ+61A9'1 1 I H ₃ 3 3   4 = 5168114_{7+4 =} H4₇5₇₊₄I14917₌45@@@51 ₋451 6 A$631  1681  1$9+A8196638$6681H4@2I1681H4@CI51681613AA6161CA9<4$171$A5691A1₃ Ω @11 6 +5$2D,5616 6 B481Y1 _[1351141[11958A81F1345619638668161361696918496'1 1

{

D

}

 145@@@5 1 1_{∀ ∈} 111₁₁Hλ_5I_1Y1

∑

= D      4 51[ 1 1 λ 1 1 1 1 F8673B91A"1B5A841B613AA6161CA9<4$1#FB71149841+:61 B @1 6 6

•1 1 561 H +B 4I1 $A9A561 A5A849611H&35&45@@@5&BI1 8961 +49681 91

5B869$A5561

[

35141

[

1O6 6

•1 13489815616:1+43841  1681  13461+81A1H4@2I1681H4@CI@13 6

6

1G1 1 •1 D4911 =7 41#FB7117 =B 14691 1 111111111111111118₇ H8_{7 4}5&₇I − = 1 1 111111111111.LH49I1 1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111.1 6 1 61$6941+461915BA5949861A!13499641715A5949861F18@1 1 1 1 F8673B91A#1B5A841B613AA6161CA9<4$1#FB71149841+:6 B1 @1 6 •1 15615A1$A9A5618A561831Y

∑

= D    4 I H 3 1[ 1 1 π λ @11 1 •1 D49117 =41#FB7117 =B 1+A961 1 11111111111111D4911 ₌41#FB711_{ =}D 1+A961 1 111111111111111111111B 8_{7 =}λ_ −4 11 111A549'1 111111111111111111111111 1111111111111111111111111111 1Y8₇

∑

= D      4 51[ 1 1 λ 1 111111111111111111111111 111111111111111111111.LHI1 1 11111111111111111111.LH49I1 1 111111111111.LH49I1 1 1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111.1 6 6 "#5$456(-6 6 B481Ω=

{

135114

}

1 681 _      − − = ρ ρ ρ ρ 4 4  @1 A1 8A5561 61 53A85541 681 +:61 71B ₌G3@1 *41 441F65F616:6561615A841613AA6161CA9<4$18A84A961H5A194A581 8A561681A184#4916194A5818A84A9614915A13AA6I@111 1

1 61 4961 61 89A#6384961 4561 491 3AF61 $A5691 961 A91 ρ 1 681 D @1 *41B

48941715A1.99614@11616:65616189A#6384961A1549961394A81491561$A5691 9681A91ρ_∈Ξ=

{

13@4153@1153@?153@E153@G153@2153@C153@>153@D

}

@11

121 2 34 32 54 52 64 62 74 72 24 4 482 3 9AB 2 34 32 54 52 64 62 74 72 24 4 482 3 9CB 2 34 32 54 52 64 62 74 72 24 4 482 3 9DB 1 1 2 34 32 54 52 64 62 74 72 24 4 482 3 9EB 2 34 32 54 52 64 62 74 72 24 4 482 3 9FB 2 34 32 54 52 64 62 74 72 24 4 482 3 9B 1 1

1C1 2 34 32 54 52 64 62 74 72 24 4 482 3 9B 2 34 32 54 52 64 62 74 72 24 4 482 3 9B 2 34 32 54 52 64 62 74 72 24 4 482 3 9B 1 6

.99614@11'1615A8416189A#6384961HAI149ρ ₌3@451HI1491ρ =3@151H3I1491ρ =3@?51

HI1491ρ =3@E51H6I1491ρ =3@G51H+I1491ρ =3@251H9I1491ρ =3@C51HI1491ρ =3@>1681

H<I149ρ ₌3@D@1

1

*414413L64161A961958A816189A+49A84141+4961A8936556161

$63869151618A5561B ₌21G₌1G_×1G1HA5A6A961615A96149848A5651$6983A55168161

898A9I1 491 61 A53A841 61 A96961 681 491

{

13@4513@1513@?513@E513@G513@2513@C153@>513@D

}

= ∈ Ξ ρ @1 21 6++6851 $6961 89A+49A841 61 5665616465161:65161618614464655616968868185691 561++968618461$65461A13688618616189A86681A96@1 1 9A₃B 2 34 32 54 52 2 34 32 54 52 9A₅B 2 34 32 54 52 2 34 32 54 52 9A₆B 2 34 32 54 52 2 34 32 54 52 9C₃B 2 34 32 54 52 2 34 32 54 52 9C₅B 2 34 32 54 52 2 34 32 54 52 9C₆B 2 34 32 54 52 2 34 32 54 52 9D₃B 2 34 32 54 52 2 34 32 54 52 9D₅B 2 34 32 54 52 2 34 32 54 52 9D₆B 2 34 32 54 52 2 34 32 54 52 1

1>1 1 9E₃B 2 34 32 54 52 2 34 32 54 52 9E₅B 2 34 32 54 52 2 34 32 54 52 9E₆B 2 34 32 54 52 2 34 32 54 52 9F₃B 2 34 32 54 52 2 34 32 54 52 9F₅B 2 34 32 54 52 2 34 32 54 52 9F₆B 2 34 32 54 52 2 34 32 54 52 9₃B 2 34 32 54 52 2 34 32 54 52 9₅B 2 34 32 54 52 2 34 32 54 52 9₆B 2 34 32 54 52 2 34 32 54 52 1 1 9₃B 2 34 32 54 52 2 34 32 54 52 9₅B 2 34 32 54 52 2 34 32 54 52 9₆B 2 34 32 54 52 2 34 32 54 52 9₃B 2 34 32 54 52 2 34 32 54 52 9₅B 2 34 32 54 52 2 34 32 54 52 9₆B 2 34 32 54 52 2 34 32 54 52 9₃B 2 34 32 54 52 2 34 32 54 52 9₅B 2 34 32 54 52 2 34 32 54 52 9₆B 2 34 32 54 52 2 34 32 54 52 1 6

7BD56 %6-1 F9A+49A841 61 89A#6384961 561 61 A961 61 8A5561 H1G:1GI1 76D=11

$6A:1 61 991 6541 8941 A93491 ++9681'1 61 A5A6A961 49848A561 461 561 +9961

HH4I51H4I51H-4I51H4I51H4I51H4I51H,4I51H+4I51H4II51 561 A5A6A961 $6983A561 461 561 +9961

HH1I56H1I56H-1I56H1I56H 1I56H1I56H,1I56H+1I56H1II1 681 491 561 A5A6A961 61 898A961 H_{H I51H I51H I51H I51H I51H I51H I51H I51H I} ? ? ? ? ? ? ? ? ?  -   , +   I@1 1

1D1 1 5 7   34 35 37 3 5 7   34 35 37 3 4 2 34 32 4 2 34 32 5 7   34 35 37 3 1 1

7BD56 (6-1 A1 34961 61 9A361 9696861 561 A93491 61 (5698LD6A41 61 641 11 AA81 A91 1G21 481 61 5A1 9+A361 1 3A991 61 8A5561 H42:42I@1 A1 34961 61 94861 681 1 A9349118661(5698LD6A4161641?1AA81A91E3D214811$4561136161 8A5561H42:42:42I@1 1 9AB 5 7   34 35 37 3 5 7   34 35 37 3 9CB 5 7   34 35 37 3 5 7   34 35 37 3 9DB 5 7   34 35 37 3 5 7   34 35 37 3 9EB 5 7   34 35 37 3 5 7   34 35 37 3 1 1

7BD56 .6-1 F9A+49A841 61 89A#6384961 561 61 A961 61 641 11 61 8A5561

H42:42I1 71 6:1 $6A:1 61 991 6541 561 A93491 61 (5698LD6A41'1 HAI1 49_{1 =}ρ 3@151 HI1

49_{1 =}ρ 3@E51H3I1491 =ρ 3@21681HI1491 =ρ 3@>@1

1 1

F9A+49A841 61 171 681 ?71 61 $638691 51 61 85A81 561 A93491 3481 1 8661

(5698LD6A41 491 61 A53A841 61 A96961 HB ₌B_{1× =}1G2₌42_×421 61 641 11

?31 1

1 9696861 71 5A1 .9961 4@2@1 5A1 89A+49A841 $A81 561 A93491 61 D6A41 61 FA8961

$63869151349964A817_{1 ∈}ρ

{

13@1153@E153@2153@>

}

@11

1

1 1

.9961 4@21'1 F9A+49A841 61 89A#6384961 561 61 A961 61 641 ?1 61 8A5561

H42:42:42I1716:1$6A:1619916541561A9349161(5698LD6A41'6H+I1491_{1 =}ρ 3@151H9I1

49_{1 =}ρ 3@E51HI1491 =ρ 3@21681H<I1491 =ρ 3@>@1 1 1 1 1 1