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Filtros de Kalman robustos para sistemas dinâmicos singulares em tempo discreto

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Aline Fernanda Bianco

Filtros de Kalman Robustos para Sistemas Dinâmicos

Singulares em Tempo Discreto

Tese apresentada à Escola de Engenharia de

São Carlos da Universidade de São Paulo, como

parte dos requisitos para obtenção do título de

Doutora em Engenharia Elétrica.

Área de Concentração: Sistemas Dinâmicos

Orientador: Prof. Dr. Marco Henrique Terra

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AUTORIZO A REPRODUÇÃO E DIVULGAÇÃO TOTAL OU PARCIAL DESTE

TRABALHO, POR QUALQUER MEIO CONVENCIONAL OU ELETRÔNICO,

PARA FINS DE ESTUDO E PESQUISA, DESDE QUE CITADA A FONTE.

Ficha catalográfica preparada pela Seção de Tratamento

da Informação do Serviço de Biblioteca – EESC/USP

Bianco, Aline Fernanda

B578f

Filtros de Kalman robustos para sistemas dinâmicos

singulares em tempo discreto / Aline Fernanda Bianco ;

orientador Marco Henrique Terra. –- São Carlos, 2009.

Tese (Doutorado-Programa de Pós-Graduação em

Engenharia Elétrica e Área de Concentração em Sistemas

Dinâmicos –- Escola de Engenharia de São Carlos da

Universidade de São Paulo, 2009).

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✒✔✓✖✕✘✗✍✙✛✚✑✜✢✓

✣✥✤✧✦✩★✫✪ ✒✭✬

✣✯✮✱✰

✮✱✲ ✒✭✳

✮✵✴

✜✷✶✛✕✘✗✍✕✖✓✡✸✍✜✢✹✡✶✻✺✽✼

✓✡✕✿✹✡✕✖✓✡✸❀✼✱✹❂❁✡✕❃✜✷✸❄✹✻❅✽❆❂✓❇✗❉❈✡✸❄✹✻✚✑✜✢❁❂❆❂✓❊✼

✓✡✸❍●■✗✍✙

✹✡✸✍✜✷✶❑❏✏✓❊✼▲✜✷✶

✗✍✶▼✚✑✜✷◆❊✓❇✗❖✹

✓✡✜✢✓☎P

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➌✌☞✎✍✂✏✑☞➌⑥➋✒✍✂✏✑✓✕✔✖✏❏➋✘✗✙✏✵➐✌✚❃➐➅➊✿➌✌☞➏➐✌✔☎➉✜✛✜✓❃➉➑➐☎➐☎➉✢✍✂✏✵➐✌✔➎➌✖✏✝✛✣✏✵➐✌✤✜✥➌✌☞✎✔☎➉✂✦✜✏✧➍✎✚✜✧★✏✪✩✌➉✫✓✣✦❃➋✫✦✜✏✝✬

✍✂✏✑✍✂✮⑩➌✌☞✇➋✘✦✜✏✯✍✂✏✑✚❃➐➄➋✱✰✙✮➑➐✲✏✯✔✌☞➏➉➑➐✴✳✕✚✣✏✯✵✱✏✝➌✌✔⑥➋✒✍✂✏✑✓✕✔✖✏★☞✎✓✕✔✖✏✝➌✖✵✱✏✱✦✜✏✑✍✶✤✌➉⑩➌✷✍✘☞✎✍✸✬

✹ ➉➑➐✴✤✩➌☎➉✒✺✻✏✵➐☎➐☎➉⑩➌✖✏✵➐✲✼P➋❲➌✖✵✵➉✂✽✾✏✑✓✩➌✌☞✿✳✕✚✣✏❁❀❂✏✝➌➎➌⑥➋❃✏❁✗ ➉ ➈⑩➉✘❄✿➉➑➐✌❅✜☞✎❆❇✚✜❈❇☞❂❉❇➐✌❅✜☞✎❅❿➋❲➌⑥➋✢✤❊✏✝➍✇➋✟➉⑩➌✌☞✿✏✑✓✕✔⑥➋✫❋ ➈⑩➉✜✛●✤❿➋✫✵✑☞✿❍✑✓✣✵✑☞✇➋✡✛

✦✜✏✱✦✡☞✿✵ ➋✫❋ ➈⑩➉✂✏✯☞✎✓✣✵✱✏✑✓✕✔✌☞✎✰ ➉✂✓✣✏✵➐✌✔✖✏✵➐➄➋✒✓❃➉➑➐✲✦✜✏✪✔➎➌⑥➋✒✩❿➋❲➍✎❅❃➉✜✬

✹ ➉➑➐■✍✂✏✑✚❃➐■✤❿➋✒☞➏➐✴✼❏☞✎❑✝✚✣✏✝➍▲✏✪▼✲✏✑❑✝☞✎✓❿➋✡✛✡☞➌✌✍✎➈ ✹ ➍✿✏✑✔✌❅✣◆ ➋✡✛✡✔✌☞✇➋✢❖✾✏✑☞✿✦✜✏✝✛✡✤✩➌✌☞✎✍✒➉➑➐✷P▲✚✣✵✑☞✇➋✒✓❿➋✡✛✣✼❘◗❲➌✖✵✑☞➏➉✘✏✪✼P➋❲➌✖✵✱✏✝➍➏➉✜✛

✵✑✚✜✓✜❅❿➋✫✦✩➉❙✗✝✚✜✓✜☞➏➉⑩➌❚✛●✤✩➌✌☞✎✍✘☞✎✓✜❅❃➉➑➐✯❯✾✚✜☞➍✎❅✣✏✝➌✌✍✂✏❱✏★▼➅➋❲✺✥➋✫✏✝➍❂✏❁✍✂✏✑✚✓➐☎➉✫✩✩➌✌☞✎✓✜❅❃➉❃✼❏✚✩➌✌☞➍➏➉❃✳✕✚✣✏★✏✵➐✌✔❳◗✒➋✂✵ ➋✒✍✘☞✎✓✜❅❃➉✜✛●✤✌➉⑩➌

➐✖✏✝➌✖✏✑✍ ➋⑩➐✷✤❊✏✵➐☎➐☎➉ ➋⑩➐✷✍✎➋✒☞➏➐✷☞✎✍✘✤✌➉⑩➌✌✔⑥➋✒✓✕✔✖✏✵➐✾✦❃➋❱✍✘☞✎✓✜❅❿➋✘✰❇☞✿✦❃➋

➉➑➐✂✵✵➉⑩➍✿✏✑❑ ➋⑩➐❙✦✩➉❨P

➇❇❉❬❩☎P ➋✒✩✌➉⑩➌⑥➋✒✔✖✮⑩➌✌☞➏➉❭✦✜✏✓➇✕☞➏➐✌✔✖✏✑✍✎➋⑩➐❃❉❪✓✕✔✖✏✝➍✎☞✎❑✙✏✑✓✕✔✖✏✵➐✟❫✟✛■✤✌➉⑩➌❃✏✵➐✌✔⑥➋❲➌✖✏✑✍❴✵✵➉✫✍✘☞✎❑ ➉❨✦✡✚✩➌⑥➋✒✓✕✔✖✏

✔☎➉❇✦✩➉✘✏✵➐☎➐✖✏✪✔✖✏✑✍✘✤✌➉✂✏✑✍❵❅❿➋❲➌✌✍✒➉✫✓✜☞✇➋✡✬

➉➑➐❘➋✒✍✘☞✎❑ ➉➑➐■✦✩➉★✵✵➉⑩➌⑥➋✫❋ ➈⑩➉✢❛✣✏✝➌✌✓❿➋✒✓✣✦❃➋✡✛✡❀■❅❿➋✒✥➏➐❚✛❃➊❘➋✒✍✘☞➍✇➋✡✛✡❜❝✚✜❑✙❍✑✓✜☞✇➋✡✛❃➊✿➍✿✏✑✩❊✏✝➌❚✛

✦❤➌✌☞✇➋✒✓❃➉✜✛✡▼✴☞✿✵ ➋❲➌✖✦✩➉✜✛❇❞✚➋✒✓✜☞✿✏✝➍❡✛

✗✝✚✩➍✎☞✇➋✡✛✢❛✣✏✝➌✌✓❿➋✒✓✣✦✩➉✜✛☛➊❘➋❲➌☎➉⑩➍✎☞✎✓❿➋✡✛✘✼P➋❲➌✖✵✵➉ ✹ ✚✩➌✖◆✝➍✎☞➏➉❢✏ ✹ ✓✕✔☎➉✫✓✜☞➏➉ ➊❘➋❲➌➎➍➏➉➑➐❣❩☎✼P➋❲➌⑥➋✒✓✜❅❿➈⑩➉✕❫❏✤❊✏✝➍✇➋❤❑✙✏✑✓✣✏✝➌☎➉➑➐✌☞✿✦❃➋✫✦✜✏✝✛

✤❿➋✫✵✑☞✿❍✑✓✣✵✑☞✇➋✘✏✚➍✿✏ ➋❲➍✿✦❃➋✫✦✜✏❁✵✵➉✫✍✶✳✕✚✣✏✚➐✖✏✑✍✘✤✩➌✖✏✯✍✂✏✯✔➎➌⑥➋✒✔⑥➋❲➌⑥➋✒✍✸✬

✹ ➉➑➐❏✤✩➌☎➉✒✺✻✏✵➐☎➐☎➉⑩➌✖✏✵➐❘✏✐✺☎✚✜✓✣✵✑☞➏➉✫✓●◗❲➌✌☞➏➉➑➐✐✦✩➉❥❞✪✏✑✤❿➋❲➌✌✔⑥➋✒✍✂✏✑✓✕✔☎➉❢✦✜✏❭❜❝✓✜❑✙✏✑✓✜❅❿➋❲➌✌☞✇➋❥❜ ➍✿◆✑✔➎➌✌☞✿✵ ➋✡✛❱✤✌➉⑩➌❏✍✂✏❭✦❃➋❲➌✖✏✑✍

➌✖✏✵➐✌✤❿➋❲➍✿✦✩➉✘✔☎➉❇✦❃➋⑩➐➄➋⑩➐✷✰✙✏✑✧❚✏✵➐✴✳✕✚✣✏✯✤✩➌✖✏✱✵✑☞➏➐✖✏✑☞❡✬

☛ ❜②➇ ☛ ❩☎❛✜✚✜✓✣✦❃➋✫❋ ➈⑩➉✾✦✜✏

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✞✝✟✡✠☞☛✍✌✏✎✒✑✓☛✔✠☞☛✕☛✔✖✗✎✘✑✚✙✛✠✒✜✢✑✚✙✤✣✦✥✚✧✒✟★✑✓☛✔✩✚✩✚✩✪✎✒✫✬✠✗✭

✖✗✎✰✯✪✱✘✎✒✜✢✑✚✧✒✎✳✲✴✠✒✫✬✠✵✙✛✖✗✎✷✶★✸✴✟★✫✬✹★✺✼✻✠☞☛✔✩

✽✾✸✴✟✕✜✢✫✿✑✓☛✿✜✼✟✿☛✍✎☞☛✍✌✏✠✒✱✡✑✚✙❁❀❂✎☞☛✏❃❄☛✔✟✪✙✛✖✗✎❆❅★✎✒✫✬✠✰✠

✲✛✫✬✟✿☛✔✟★✙❈❇☞✠✰❉✒✑✓☛✿✜✼✠✒✙✴✜✼✟✡❉✗✠☞☛✍✟✿☛✿✜✢✫✬✟★✻✠☞☛✔✭

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❉ ✹ ❖✚➊✌☞❱✛ ✹ ✬♠❛■✬♠❛ ☞➍✎✔➎➌☎➉➑➐✘✦✜✏✎✍s➋❲➍✎✍✎➋✒✓❣▼➄➉✫✩✜✚❃➐✌✔☎➉➑➐✢✤❿➋❲➌⑥➋✮➇✕☞➏➐✌✔✖✏✑✍✎➋⑩➐✘❞✾☞✎✓✑✏✒✍✘☞✿✵✵➉➑➐☛➇✕☞✎✓✜❑✝✚✩➍✇➋❲➌✖✏✵➐✂✏✑✍ ❀❂✏✑✍✘✤✌➉

❞✾☞➏➐✖✵✝➌✖✏✑✔☎➉✜✬ ➒➑➓➑➓➑➔ ✬✢❀❂✏✵➐✖✏✸❩☎❞✧➉✫✚✜✔☎➉⑩➌⑥➋✫✦✩➉✕❫✓✒✴❜ ➐✖✵✵➉⑩➍✇➋ ✦✜✏✘❜❝✓✜❑✙✏✑✓✜❅❿➋❲➌✌☞✇➋ ✦✜✏✎➇❤➈⑩➉✓➊❘➋❲➌➎➍➏➉➑➐❚✛✕✔✲✓✜☞✎✰✙✏✝➌☎➐✌☞✿✦❃➋✫✦✜✏✂✦✜✏✎➇❤➈⑩➉

➋✒✚✩➍➏➉✜✛✌➇❤➈⑩➉✎➊❘➋❲➌➎➍➏➉➑➐❚✛ ➒➑➓➑➓➑➔

❜ ➐✌✔⑥➋❘✔✖✏✵➐✖✏ ✔➎➌⑥➋✒✔⑥➋❨✦✩➉✐✤✩➌☎➉✫✩✩➍✿✏✑✍✎➋ ✦✜✏✸✏✵➐✌✔✌☞✎✍✎➋✒✔✌☞✎✰➑➋✳➌☎➉✫✩✜✚❃➐✌✔⑥➋❨✮✫✔✌☞✎✍✎➋✐✤❿➋❲➌⑥➋✳➐✌☞➏➐✌✔✖✏✑✍✎➋⑩➐❃✦✡☞✎✓✑✏✒✍✘☞✿✵✵➉➑➐☛➐✌☞✎✓✜❑✝✚✩➍✇➋❲➌✖✏✵➐

✦✡☞➏➐✖✵✝➌✖✏✑✔☎➉➑➐✢✓❃➉ ✔✖✏✑✍✘✤✌➉✜✬ ❖➅➉❚✰ ➉➑➐✒➋❲➍✎❑ ➉⑩➌✌☞✎✔✌✍✒➉➑➐✟➌✖✏✱✵✑✚✩➌☎➐✌☞✎✰ ➉➑➐✒➐⑥➈⑩➉❏✺➉⑩➌✌✍★✚✩➍✇➋✫✦✩➉➑➐✘✤❿➋❲➌⑥➋P➋⑩➐✘✏✵➐✌✔✌☞✎✍✎➋✒✔✌☞✎✰➑➋⑩➐❱❧❃➍✎✔➎➌⑥➋✫✦❃➋⑩➐

✏✸✤✩➌✖✏✱✦✡☞✎✔☎➉⑩➌⑥➋⑩➐❬✵✵➉✫✍▲➋⑩➐❙✵✵➉⑩➌➎➌✖✏✵➐✌✤✌➉✫✓✣✦✜✏✑✓✕✔✖✏✵➐ ✏✱✳✕✚❿➋✫❋✗✖❇✏✵➐❃✦✜✏ ▼✴☞✿✵✱✵ ➋✒✔✌☞❡✬✘☞ ❧❃➍✎✔➎➌☎➉✮➌☎➉✫✩✜✚❃➐✌✔☎➉ ✔✌☞✎✤✌➉✙✍s➋❲➍✎✍✎➋✒✓ ✏✓➋

✏✱✳✕✚❿➋✫❋ ➈⑩➉ ✦✜✏✂▼✴☞✿✵✱✵ ➋✒✔✌☞ ✵✵➉⑩➌➎➌✖✏✵➐✌✤✌➉✫✓✣✦✜✏✑✓✕✔✖✏✎➐⑥➈⑩➉❂➉✫✩✜✔✌☞✿✦✩➉➑➐❁✓✕✚✜✍✎➋❙✺➉⑩➌✌✍★✚✩➍✇➋✫❋ ➈⑩➉✸✍✎➋✒☞➏➐✯❑✙✏✝➌⑥➋❲➍❡✛ ✏✵➐✌✔✖✏✑✓✣✦✜✏✑✓✣✦✩➉✏➉➑➐✚➌✖✏✚✒

➐✌✚✩➍✎✔⑥➋✫✦✩➉➑➐❞➋✒✤✩➌✖✏✵➐✖✏✑✓✕✔⑥➋✫✦✩➉➑➐■✓❿➋s➍✎☞✎✔✖✏✝➌⑥➋✒✔✌✚✩➌⑥➋✡✬✛☞ ✺☎✚✜✓✣✵✑☞➏➉✫✓❿➋❲➍▲✳✕✚❿➋✫✦❤➌❳◗✒✔✌☞✿✵✵➉❱✤✩➌☎➉✫✤✌➉➑➐✌✔☎➉★✤❿➋❲➌⑥➋❱✦✜✏✱✦✡✚✜✧✱☞➌✷✏✵➐✌✔✖✏✴❧❃➍✎✔➎➌☎➉❁✺✥➋✒✧

➋✸✵✵➉✫✍★✩✜☞✎✓❿➋✫❋ ➈⑩➉✐✦❃➋⑩➐★✔✖◆✱✵✑✓✜☞✿✵ ➋⑩➐❱✍✘✥✎✓✜☞✎✍✒➉➑➐✢✳✕✚❿➋✫✦❤➌⑥➋✫✦✩➉➑➐❙➌✖✏✑❑✝✚✩➍✇➋❲➌✌☞✎✧❻➋✫✦✩➉➑➐✘✏✂✺☎✚✜✓✣❋✗✖❇✏✵➐★✤❊✏✑✓❿➋❲➍✎☞✿✦❃➋✫✦✜✏✝✬✜☞ ➐✌☞➏➐✌✔✖✏✑✍✎➋

✵✵➉✫✓❃➐✌☞✿✦✜✏✝➌⑥➋✫✦✩➉✯✤❿➋❲➌⑥➋✧➉✫✩✜✔✖✏✑✓✣❋ ➈⑩➉❁✦✜✏■✔⑥➋✒☞➏➐❝✏✵➐✌✔✌☞✎✍✎➋✒✔✌☞✎✰➑➋⑩➐ ◆❞➐✌☞✎✓✜❑✝✚✩➍✇➋❲➌❚✛✙✦✡☞➏➐✖✵✝➌✖✏✑✔☎➉✜✛✙✰➑➋❲➌✌☞✇➋✒✓✕✔✖✏■✓❃➉✪✔✖✏✑✍✘✤✌➉✜✛✕✵✵➉✫✍ ➌✌✚✜✥✿✦✩➉➑➐

✵✵➉⑩➌➎➌✖✏✝➍✇➋✫✵✑☞➏➉✫✓❿➋✫✦✩➉➑➐■✏✴✔☎➉❇✦✩➉➑➐✿➉➑➐ ✤❿➋❲➌✢✏✒✍✂✏✑✔➎➌☎➉➑➐■✦✩➉✯✍✒➉❇✦✜✏✝➍➏➉s➍✎☞✎✓✣✏ ➋❲➌■✏✵➐✌✔⑥➈⑩➉s➐✌✚✤✣✌✏✑☞✎✔☎➉➑➐❘➋✪☞✎✓✣✵✱✏✝➌✌✔✖✏✑✧❻➋⑩➐❚✬ ✹ ➐ ☞✎✓✣✵✱✏✝➌✌✔✖✏✑✧❻➋⑩➐

✤❿➋❲➌⑥➋✒✍✂◆✑✔➎➌✌☞✿✵ ➋⑩➐➅➐⑥➈⑩➉✟➍✎☞✎✍✘☞✎✔⑥➋✫✦❃➋⑩➐✾✤✌➉⑩➌✴✓❃➉⑩➌✌✍✎➋✡✬ ✹ ➐✷✤✩➌☎➉✫✤✩➌✌☞✿✏✱✦❃➋✫✦✜✏✵➐✪✦✜✏❁✏✵➐✌✔⑥➋✒✩✜☞➍✎☞✿✦❃➋✫✦✜✏❱✏❁✵✵➉✫✓✕✰✙✏✝➌✌❑✙❍✑✓✣✵✑☞✇➋❙✦✩➉✢❧❃➍✎✔➎➌☎➉

✦✜✏✥✍s➋❲➍✎✍✎➋✒✓✸✤❿➋❲➌⑥➋❏➐✌☞➏➐✌✔✖✏✑✍✎➋⑩➐✷✓❃➉✫✍✘☞✎✓❿➋✒☞➏➐✴✏✪☞✎✓✣✵✱✏✝➌✌✔☎➉➑➐ ➐⑥➈⑩➉✢✤✩➌☎➉❚✰➑➋✫✦❃➋⑩➐❚✛✡✍✒➉➑➐✌✔➎➌⑥➋✒✓✣✦✩➉✦✒➐✖✏❁✳✕✚✣✏✚➉★❧❃➍✎✔➎➌☎➉✂✏✑✍ ✏✵➐✌✔⑥➋✫✦✩➉

✤❊✏✝➌✌✍✎➋✒✓✣✏✑✓✕✔✖✏✸◆ ✏✵➐✌✔❳◗✱✰✙✏✝➍✲✏✏➋✓➌✖✏✱✵✑✚✩➌☎➐⑥➈⑩➉✐✦✜✏ ▼✴☞✿✵✱✵ ➋✒✔✌☞➄➋⑩➐☎➐☎➉❇✵✑☞✇➋✫✦❃➋✳➋❘✏✝➍✿✏ ◆❬✚✜✍✎➋P➐✖✏✱✳✕✚✣❍✑✓✣✵✑☞✇➋ ✍✒➉✫✓✣✮✫✔☎➉✫✓❿➋✐✓❿➈⑩➉

✦✜✏✱✵✝➌✖✏✵➐✖✵✱✏✑✓✕✔✖✏✝✛❃➍✎☞✎✍✘☞✎✔⑥➋✫✦❃➋✟➐✌✚✜✤❊✏✝➌✌☞➏➉⑩➌✌✍✂✏✑✓✕✔✖✏★✤❊✏✝➍✇➋❏➐☎➉⑩➍✎✚✣❋ ➈⑩➉✂✦❃➋✢❜ ✳✕✚❿➋✫❋ ➈⑩➉ ✹ ➍✎❑✙◆✑✩✩➌✌☞✿✵ ➋✘✦✜✏❁▼✴☞✿✵✱✵ ➋✒✔✌☞❡✬

✧✓★✪✩✫★✭✬✕✮✦★✰✯✚✱✳✲✵✴✶★✭✬✸✷✪✹ ➇✕☞➏➐✌✔✖✏✑✍✎➋⑩➐✒➐✌☞✎✓✜❑✝✚✩➍✇➋❲➌✖✏✵➐❚✛❝❧❃➍✎✔➎➌⑥➋✒❑✙✏✑✍❋➌☎➉✫✩✜✚❃➐✌✔⑥➋✡✛ ✏✵➐✌✔✌☞✎✍✎➋✒✔✌☞✎✰➑➋❘✦✜✏ ✏✵➐✌✔⑥➋✫✦✩➉✜✛ ✏✵➐✌✔⑥➋✒✩✜☞➍✎☞✿✦❃➋✫✦✜✏✝✛

(10)
(11)

☎☎✄✝✆✟✞✡✠☛✄

☞✱✦ ✒✭✳

✪✱✮

✼ ✒ P✍✌✯P▼✬ ✓✏✎✍✙☎● ✚✒✑❖✹✔✓✕✖✹✡✶☛✌▲✜✕✓✚✗✍✸❀●✒✖ ✓✡✸✘✗❍✜●

✸✩✗ ✚✗✚✙ ✚✑✜✷✕✘✗

✜✷✶ ❈✡✙✛✓✹✡✸ ✰✢✜

● ✚ ✗✍✕❖●■P✤✣✦✥✧✥✔★ P✪✩✝❏❑✗❀●✍✜●

✗ ✓

✚✟✓✡✸❄✹✔✓✺✟✙✬✫ ●

✓✔✓✹ ❁☎✗✭✫✱✶ ❈☎✗✍✶❑❏✏✹✡✸✍✜✢✹ ❁☎✗ ✰ ❆❂✓ ✪ ✹✡✸✚✓✓❊●❀✼ ✤ ✶ ✜✷◆✡✗✍✸❀●✍✜✢❁❂✹❂❁☎✗❖❁☎✗ ✰ ❆❂✓✪✮ ✹✡✙✛✓✓❊✼ ✰ ❆❂✓ ✪ ✹✡✸✚✓✓❊●❀✼✯✣✦✥✧✥✔★ P

✩✝❏☎✜● ✚✷❏❑✗❀●✍✜●

✓✡✶❑●✍✜✢❁☎✗✍✸❀● ✚✷❏❑✗ ✓

✚✑✜✷✕✖✹✔✓✯✸❄✓✏✎✍✙☎● ✚❃✗❀● ✚✑✜✷✕✖✹✻✚✗❀●

✸❄✓✏✎✚✓✗✍✕✰✖ ✓✡✸ ❁✡✜●

✸✩✗ ✚ ✗✚✙ ✚✑✜✷✕✘✗ ●✍✜✷✶ ❈✡✙✛✓✹✡✸✛❁ ✜ ✶ ✹✡✕❃✜

● ✚ ✗✍✕❖●■P❍✳ ✗✚✱✫✸✩✗

✙ ✸❀●✍✜✷◆✡✗ ✹✔✓❈❂✓✡✸✍✜✑✚✷❏☎✕❖●❃✹✡✸✩✗✖❁☎✗✍◆✡✗✚✓✓

✗❄❁✲✖ ✓✡✸❖✚✷❏❑✗☎✑❖✹✔✓✕✖✹✡✶✴✳✯✓✚ ✗✍✸✩✗❄❁✛✹✡✶ ❁

✸✩✗❄❁✡✜

✚✗❄❁ ✗❀● ✚✑✜✷✕✖✹✻✚ ✗

✸✩✗

✙ ✸❀●✍✜✢✓✡✶❑●✵✱ ✜✑✚✷❏ ✚✷❏❑✗

✓✡✸✍✸✩✗❀●

✓✡✶ ❁✡✜✷✶ ❈✘✬✔✜ ✾✩✾

✹✻✚✑✜ ✗✷✶ ✙❑✹✻✚✑✜✢✓✡✶❑●■P☛✩✝❏❑✗ ●✍✜✷✶ ❈✡✙✛✓✹✡✸ ✸❄✓✏✎✍✙☎● ✚✸✑❖✹✔✓✕✖✹✡✶✹✙✚✜

✗✬✳✯✓✚ ✗✍✸❖✹✡✶ ❁

✚✷❏❑✗

✓✡✸✍✸✩✗❀●

✓✡✶ ❁✡✜✷✶ ❈ ✸✩✗

✙ ✸❀●✍✜✷◆✡✗❖✬✔✜ ✾✩✾

✹✻✚✑✜ ✗✷✶ ✙❑✹✻✚✑✜✢✓✡✶ ✹✡✸✩✗ ✓✏✎ ✚✟✹✡✜✷✶☞✗❄❁ ✜✷✶ ✚✷❏❑✗✍✜✷✸ ✕✖✓❊● ✚ ❈☎✗✍✶☞✗✍✸❄✹✔✓✦✖ ✓✡✸✍✕❃✙✛✓✹✻✚✑✜✢✓✡✶❑✼✔✗✷✺✔✙

✚ ✗✍✶ ❁✡✜✷✶ ❈ ✚✷❏❑✗ ✸✩✗❀●✍✙✛✓✚✷●

✸✩✗❀●■✗✍✶▼✚ ✗❄❁ ✜✷✶ ✚✷❏❑✗✲✓✜✑✚ ✗✍✸❄✹✻✚✑✙ ✸✩✗■P✝✩✝❏❑✗✪✶ ✙❑✹❂❁✡✸❄✹✻✚✑✜

✖✍✙ ✶

✚✑✜✢✓✡✶ ✹✔✓ ❁☎✗✍◆✡✗✚✓✓

✗❄❁✛✚✟✓ ❁☎✗❄❁✡✙

✗ ✚✷❏☎✜●

✳✯✓✚ ✗✍✸

✓✡✕✻✎✍✜✷✶☞✗❀● ✸✩✗✢❈✡✙✛✓✹✡✸✍✜✽✼✽✗❄❁✾✓✗❄✹❊● ✚❉●✿✶ ✙❑✹✡✸✩✗❀●❇✹✡✶ ❁

✗✍✶ ✹✔✓✚

✖✍✙ ✶

✚✑✜✢✓✡✶❑● ✹ ❋❂❋

✸❄✓■✹

❏❑✗❀●■P❀✩✝❏❑✗ ●

● ✚ ✗✍✕

✓✡✶❑●✍✜✢❁✔✙

✗✍✸✩✗❄❁ ✚✟✓ ✓✏✎ ✚✟✹✡✜✷✶ ✚✷❏❑✗ ✗❀● ✚✑✜✷✕✖✹✻✚ ✗❀●✘✜● ●✍✜✷✶ ❈✡✙✛✓✹✡✸❀✼ ✚✑✜✷✕✘✗✚✙◆❊✹✡✸ ✜ ✜✷✶ ❈❁✱ ✜✑✚✷❏

✓✡✸✍✸✩✗✚✓✹✻✚✗❄❁ ✶ ✓✡✜●■✗❀● ✹✡✶ ❁ ✹✔✓✕✓

✹✡✸❄✹✡✕✘✗ ✚ ✗✍✸

✕✖✹✻✚✑✸✍✜

✗❀● ✓❂✖❍✚✷❏❑✗ ✙ ✶ ❁☎✗✍✸✚✓

✜✷✶ ❈✴✓✜✷✶☞✗❄✹✡✸❍✕✖✓■❁☎✗✚✓ ✹✡✸✩✗❍●✍✙✹✎✕❃■✗

✚✱✚✟✓✖✙ ✶

✗✍✸ ✚✟✹✡✜✷✶▼✚✑✜✟✗❀●■P✟✩✝❏❑✗

✹✡✸❄✹✡✕✘✗ ✚✑✸✍✜

✙ ✶

✗✍✸ ✚✟✹✡✜✷✶▼✚

✜●❉✹❊●❀●✍✙ ✕✘✗❄❁ ✚✟✓✻✎✩✗❉✶ ✓✡✸✍✕❄✎❄✓✡✙ ✶ ❁☎✗❄❁☎P✟✩✝❏❑✗

✸❄✓

✗✍✸ ✚✑✜✟✗❀● ✓❂✖✔● ✚✟✹✏✎✍✜✕✓✜✑✚✜ ✹✡✶ ❁

✓✡✶ ◆✡✗✍✸✢❈☎✗✍✶

✗❖✓❂✖❉✚✷❏❑✗✘✑❖✹✔✓✕✖✹✡✶✝✌▲✜✕✓✚✗✍✸

✖ ✓✡✸✖✶ ✓✡✕❃✜✷✶ ✹✔✓✱✹✡✶ ❁✂✙ ✶

✗✍✸ ✚✟✹✡✜✷✶ ●

● ✚ ✗✍✕ ✕✖✓■❁☎✗✚✓● ✹✡✸✩✗

✸❄✓✡◆✡✗❄❁❊✼☎✱ ❏❑✗✍✸✩✗✝✱✱✗✖●❀❏✏✓✔✱ ✚✷❏✏✹✻✚✭● ✚ ✗❄✹❂❁

✙✟● ✚✟✹✻✚ ✗❅✳✯✓✚ ✗✍✸✘✜●

● ✚✟✹✏✎✚✓✗❃✹✡✶ ❁ ✚✷❏❑✗ ✬✔✜ ✾✩✾

✹✻✚✑✜ ✸✩✗

✙ ✸❀●✍✜✢✓✡✶ ✹❊●❀● ✓

✜✢✹✻✚✗❄❁❆✱ ✜✑✚✷❏ ✚✷❏☎✜● ✜●❖✹ ✶ ✓✡✶ ❁☎✗

✸✩✗❄✹❊●✍✜✷✶ ❈❇✕✖✓✡✶ ✓✻✚✟✓✡✶☞✗❍●■✗✷✶ ✙ ✗✍✶

✗✴✱ ✜✑✚✷❏

❋❂❋

✗✍✸✪✎❄✓✡✙ ✶ ❁☎P

❇ ✷✧❈❊❉✵❋✑✮❍●✳✯ ✹✥✰

✜✷✶ ❈✡✙✛✓✹✡✸✥●

● ✚ ✗✍✕❖●❀✼▲✸❄✓✏✎✍✙☎● ✚✢✳✯✓✚ ✗✍✸✍✜✷✶ ❈❊✼▲● ✚✟✹✻✚ ✗❖✗❀● ✚✑✜✷✕✖✹✻✚✑✜✢✓✡✶❑✼ ● ✚✟✹✏✎✍✜✕✓✜✑✚

✓✡✶ ◆✡✗✍✸✢❈☎✗✍✶

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☎ ✷ ✯✝✆✟✞ ❋ ✠☛✡

☞✍✌ ✯✏✎ ✮✦★✒✑✓✎ ✡✔✠

✕✖✠✫✯✏✎ ★✾●✳✷✘✗✙✠✛✚✜✆✶✮✦★✰✯ ✡ ✬

✕✖✠✫✯✏✎ ★✾●✳✷ ☞✍✌ ✮✦✷✭✬✔✠✫★✢✎✝✆✶✮✦★✰✯✞✷✤✣✥✠✛✚✸✩✫★✰✯ ✡ ✬✔✠✦✠

✕✖✠✫✯✏✎ ★✾●✳✷✤✣✥✧✦✞ ✌ ❋✸✩✕❋✑✯ ✡✔✠☛✡

★✪✩✏✫

✎ ✮❍❋❊●✬✆✔✭✯✮ ❋

✼✓➉✫✔✌☞✎✰➑➋✫❋ ➈⑩➉ ✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬

✰ ✬➒ ❜ ➐✌✔✌☞✎✍✎➋✒✔✌☞✎✰➑➋⑩➐✲✱✲✔✌☞✎✍✎➋⑩➐✴✤❿➋❲➌⑥➋✎➇✕☞➏➐✌✔✖✏✑✍✎➋⑩➐✧➇✕☞✎✓✜❑✝✚✩➍✇➋❲➌✖✏✵➐ ✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬ ➒

✰ ✬✳ ❜ ➐✌✔✌☞✎✍✎➋✒✔✌☞✎✰➑➋⑩➐✲❛ ☞➍✎✔➎➌⑥➋✫✦❃➋✡✛ ☛ ➌✖✏✱✦✡☞✎✔☎➉⑩➌⑥➋❃✏✉➇✕✚❿➋✱✰❇☞✎✧❻➋✫✦❃➋ ✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬ ➒

✬✴ ☞✲✩ ✣✌✏✑✔✌☞✎✰ ➉➑➐❴✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬ ✳

✰ ✬✵ ❜ ➐✌✔➎➌✌✚✜✔✌✚✩➌⑥➋✂✦✩➉✘❀❂✏✷✶✡✔☎➉❢✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬ ✴

✬✸

✚✜✩✩➍✎☞✿✵ ➋✫❋✗✖❇✏✵➐✾❞✪✏✱✵✵➉⑩➌➎➌✖✏✑✓✕✔✖✏✵➐✲✦❃➋

✏✵➐✖✳✕✚✜☞➏➐⑥➋ ✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬ ✵

✹✻✺

✷✏❋✑✮✼✠✫★ ●✳✷✤✽✾✎✝✠✦✞✿✠✛❀✭★✒✭✯✮ ❋❂❁❃✮ ❋❂✕✖✠

✷ ★✰✮ ❄

➒ ✬✰ ☞ ☛ ➌☎➉✫✩✩➍✿✏✑✍✎➋✂✦✜✏✯✼❏✥✎✓✜☞✎✍✒➉➑➐✲❅✲✚❿➋✫✦❤➌⑥➋✫✦✩➉➑➐ ☛ ➉✫✓✣✦✜✏✝➌⑥➋✫✦✩➉➑➐✪✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬ ❆

➌☎➉✫✩✩➍✿✏✑✍✎➋✂✦✜✏✯✼❏✥✎✓✜☞✎✍✒➉➑➐✲❅✲✚❿➋✫✦❤➌⑥➋✫✦✩➉➑➐✲▼✲✏✑❑✝✚✩➍✇➋❲➌✌☞✎✧❻➋✫✦✩➉➑➐ ✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬ ✰✓✰

➒ ✬✳ ☞ ✼ ◆✑✔☎➉❇✦✩➉✘✦❃➋⑩➐✴❛✜✚✜✓✣❋✗✖❇✏✵➐ ☛ ✏✑✓❿➋❲➍✎☞✿✦❃➋✫✦✜✏ ✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬ ✰ ✴

❇✻❈ ✯✏✎✝✠✦✞ ★✢✎✝✠✬✰★✰✯✍☎ ✷❉✑❉✆✶✮✦✯✝✠ ✬✰★✰✯❋❊✾✎✝✠✦✞ ★✰✯❍●✳★✰✮✦★✾❋✿■ ❋❊●✳✷ ✩✕❋ ✫ ❋ ❈ ✯❏●✳★✒✭✧❋ ●✳✷ ❈ ✯✏✎ ★ ● ❋ ★✢❑

✳✡✬

❜ ➐✌✔✌☞✎✍✎➋✒✔✌☞✎✰➑➋✘❛ ☞➍✎✔➎➌⑥➋✫✦❃➋✘✤❿➋❲➌⑥➋✟➉✘✼✓➉❇✦✜✏✝➍➏➉✢✓❃➉✂❜ ➐✌✤❿➋✫❋✵➉✂✦✜✏✯❜ ➐✌✔⑥➋✫✦✩➉ ✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬ ✰ ➔

✳✡✬➒ ❜ ➐✌✔✌☞✎✍✎➋✒✔✌☞✎✰➑➋ ☛ ➌✖✏✱✦✡☞✎✔☎➉⑩➌⑥➋✘✤❿➋❲➌⑥➋✟➉✂✼✓➉❇✦✜✏✝➍➏➉✢✓❃➉✂❜ ➐✌✤❿➋✫❋✵➉✂✦✜✏✯❜ ➐✌✔⑥➋✫✦✩➉ ✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬ ➒ ✳

✳✡✬✳ ❜✬✶✡✤✩➌✖✏✵➐☎➐ ✖❇✏✵➐✴❜ ✳✕✚✜☞✎✰➑➋❲➍✿✏✑✓✕✔✖✏✵➐✴✤❿➋❲➌⑥➋✒➋⑩➐✴❜ ➐✌✔✌☞✎✍✎➋✒✔✌☞✎✰➑➋⑩➐✲▼✲✏✱✵✑✚✩➌☎➐✌☞✎✰➑➋⑩➐▲✱✲✔✌☞✎✍✎➋⑩➐❨✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬

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✴✣✬✰ ❜ ➐✌✔✌☞✎✍✎➋✒✔✌☞✎✰➑➋✘❛ ☞➍✎✔➎➌⑥➋✫✦❃➋✢✤❿➋❲➌⑥➋✎➇✕☞➏➐✌✔✖✏✑✍✎➋⑩➐✧➇✕☞✎✓✜❑✝✚✩➍✇➋❲➌✖✏✵➐✲❖➅➉✫✍✘☞✎✓❿➋✒☞➏➐✘✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬ ➒➑➔

✴✣✬

❜ ➐✌✔✌☞✎✍✎➋✒✔✌☞✎✰➑➋

➌✖✏✱✦✡☞✎✔☎➉⑩➌⑥➋✘✤❿➋❲➌⑥➋✎➇✕☞➏➐✌✔✖✏✑✍✎➋⑩➐✧➇✕☞✎✓✜❑✝✚✩➍✇➋❲➌✖✏✵➐✲❖➅➉✫✍✘☞✎✓❿➋✒☞➏➐❥✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬✻✳✓✳

✴✣✬✳ ➊❘➋⑩➐☎➉➑➐ ☛ ➋❲➌✌✔✌☞✿✵✑✚✩➍✇➋❲➌✖✏✵➐✪✦❃➋⑩➐✴❜ ➐✌✔✌☞✎✍✎➋✒✔✌☞✎✰➑➋⑩➐✲▼✲✏✱✵✑✚✩➌☎➐✌☞✎✰➑➋⑩➐ ✱✲✔✌☞✎✍✎➋⑩➐ ✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬✻✳✓✸

✁✻❈ ✯✏✎ ★ ✌ ✠ ✩✦✠✕●✳★ ●✳✷ ✷ ✲✲❋ ✫ ✬✸✷ ✮ ✚✄✂ ✫ ✑❉✠✫★✾● ❋✑✯✍✗ ✣✥❁ ✯ ▼☎✁

✵✡✬

▼✲✏✵➐✌✚✩➍✎✔⑥➋✫✦✩➉➑➐ ☛ ➌✖✏✝➍✎☞✎✍✘☞✎✓❿➋❲➌✖✏✵➐ ✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬ ✴❉✵

✵✡✬

❜ ➐✌✔⑥➋✒✩✜☞➍✎☞✿✦❃➋✫✦✜✏❘✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬✻✵

✵✡✬✳ ➊②➉✫✓✕✰✙✏✝➌✌❑✙❍✑✓✣✵✑☞✇➋ ✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬✻✵ ✴

✵✡✬✴ ❜✬✶✜✏✑✍✘✤✩➍➏➉✂❖✲✚✜✍✂◆✝➌✌☞✿✵✵➉❵✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬✻✸✯❆

✆◆❈ ✯✏✎✝✠✦✞ ★✢✎✝✠✬✰★ ☎✝❋ ✌ ✆✶✯✏✎ ★ ●✳★✰✮✦★❂✣✥✠✫✯✏✎ ✷ ✞ ★✰✯✞✝ ✠ ✫✠✟ ✞✿✠✛✑✧❋✑✯❋✣✥✠ ✫ ✚✜✆✳✩✫★✰✮✦✷ ✯ ❄ ★

✸✡✬✰ ❜ ➐✌✔⑥➋✒✩❊✏✝➍✿✏✱✵✑☞✎✍✂✏✑✓✕✔☎➉❙✦✩➉ ☛ ➌☎➉✫✩✩➍✿✏✑✍✎➋✂✦✜✏❁❛ ☞➍✎✔➎➌⑥➋✒❑✙✏✑✍✶▼➄➉✫✩✜✚❃➐✌✔⑥➋ ✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬◆❆ ➒

✸✡✬

❜ ➐✌✔✌☞✎✍✎➋✒✔✌☞✎✰➑➋⑩➐✲❛ ☞➍✎✔➎➌⑥➋✫✦❃➋⑩➐✲▼➄➉✫✩✜✚❃➐✌✔⑥➋⑩➐❢✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬◆❆✼✳

✸✡✬➒ ❜ ➐✌✔⑥➋✒✩❊✏✝➍✿✏✱✵✑☞✎✍✂✏✑✓✕✔☎➉❙✦✩➉ ☛ ➌☎➉✫✩✩➍✿✏✑✍✎➋✂✦✜✏ ☛ ➌✖✏✱✦✡☞✿❋ ➈⑩➉❃▼➄➉✫✩✜✚❃➐✌✔⑥➋ ✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬◆❆✓❆

✸✡✬

❜ ➐✌✔✌☞✎✍✎➋✒✔✌☞✎✰➑➋⑩➐

➌✖✏✱✦✡☞✎✔☎➉⑩➌⑥➋⑩➐✾▼➄➉✫✩✜✚❃➐✌✔⑥➋⑩➐ ✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬☛✡

✸✡✬✳ ❛❃➉⑩➌✌✍✎➋⑩➐✲✼P➋✒✔➎➌✌☞✿✵✑☞✇➋✒☞➏➐✲❜ ✳✕✚✜☞✎✰➑➋❲➍✿✏✑✓✕✔✖✏✵➐✲✤❿➋❲➌⑥➋✟➋⑩➐✴❜ ➐✌✔✌☞✎✍✎➋✒✔✌☞✎✰➑➋⑩➐✲▼➄➉✫✩✜✚❃➐✌✔⑥➋⑩➐ ✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬☛✡ ✴

❄ ❈ ✯✏✎ ★ ✌ ✠ ✩✦✠✕●✳★ ●✳✷ ✷ ✲✲❋ ✫ ✬✸✷ ✮ ✚✄✂ ✫ ✑❉✠✫★✾● ❋✑✯✍✗ ✣ ☎ ✯ ❑ ★

❆❇✬

▼✲✏✵➐✌✚✩➍✎✔⑥➋✫✦✩➉➑➐

➌✖✏✝➍✎☞✎✍✘☞✎✓❿➋❲➌✖✏✵➐ ✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬ ➔✢✰

❆❇✬➒ ❜ ➐✌✔⑥➋✒✩✜☞➍✎☞✿✦❃➋✫✦✜✏❘✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬ ➔ ✵

❆❇✬✳ ➊②➉✫✓✕✰✙✏✝➌✌❑✙❍✑✓✣✵✑☞✇➋ ✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬

☎ ✷ ✯✝✆✳✩☛✎ ★ ● ❋✑✯✍❁ ✆✟✞✍✌ ✮✼✠✛✑✧❋✑✯ ✷ ✲✲❋

✑ ✩✦✆✶✯✏✎ ✷ ✯

❑ ❑

✡✡✬✰ ➊②➉✫✍✘✤❿➋❲➌⑥➋✫❋ ➈⑩➉❃✵✵➉✫✍ ➉✘❛ ☞➍✎✔➎➌☎➉ ☛ ❞✥✔ ✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬ ➔➑➔

✡✡✬➒ ➊②➉✫✓✣✵✝➍✎✚❃➐ ✖❇✏✵➐✴✏❙➊②➉✫✓✕✔✌☞✎✓✕✚✜☞✿✦❃➋✫✦✜✏❱✦❃➋ ☛ ✏✵➐✖✳✕✚✜☞➏➐⑥➋❃✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬ ✰ ➓➑➒

(15)

✂✪☎☎✄❆✞ ✁ ✂ ✂ ✂☎✄✘✝ ✆✟✞ ☎

✵✡✬

❜❝✰ ➉⑩➍✎✚✣❋ ➈⑩➉❃✦✜✏

P

i

+1

|

i

+1

✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬✻✸ ✡

✵✡✬➒ ❜❝✰ ➉⑩➍✎✚✣❋ ➈⑩➉❃✦✜✏

P

i

+1

|

i

✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬✻✸ ➔

✡✡✬

❛ ➇❇▼❱❩✝✆❢❫■✏✯❛ ☞➍✎✔➎➌☎➉

❞✥✔ ❩✒ ✒ ✒ ❫ ✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬❁✬✯✬❁✬✯✬❁✬❁✬✯✬ ✰ ➓✢✰

✡✡✬

❜❝✰ ➉⑩➍✎✚✣❋ ➈⑩➉❃✦✜✏

P

i

+1

|

i

+1

(16)
(17)

✂✪☎☎✄❆✞ ✁ ✂

✂✁

✆ ✂✁ ✂❊✞ ✄ ✝ ✆✟✞ ☎ ✂ ✂☎✄✄✂❊✞ ☎

❜ ✹ ▼

✳✕✚❿➋✫❋ ➈⑩➉ ☞ ➍✎❑✙◆✑✩✩➌✌☞✿✵ ➋✘✦✜✏ ☎ ☞✿✵✱✵ ➋✒✔✌☞

❛ ➇❇❖➅➐

✗ ☞➍✎✔➎➌☎➉➑➐ ✣ ☞✎✓✜❑✝✚✩➍✇➋❲➌✖✏✵➐ ❁ ➉✫✍✘☞✎✓❿➋✒☞➏➐

❛ ➇❇▼➄➐

✗ ☞➍✎✔➎➌☎➉➑➐ ✣ ☞✎✓✜❑✝✚✩➍✇➋❲➌✖✏✵➐ ☎ ➉✫✩✜✚❃➐✌✔☎➉➑➐

❛ ✍✚➐

☞➍✎✔➎➌☎➉➑➐✲✦✜✏

➋❲➍✎✍✎➋✒✓

❛♠▼ ✍✚➐

✗ ☞➍✎✔➎➌☎➉➑➐ ☎ ➉✫✩✜✚❃➐✌✔☎➉➑➐✴✦✜✏ ❇ ➋❲➍✎✍✎➋✒✓

P ❞✥✔

❉❪✓✡✺✻✏✝➌✌☞➏➉⑩➌✲❞✾☞✇➋✒❑ ➉✫✓❿➋❲➍ ➇✕✚✜✤❊✏✝➌✌☞➏➉⑩➌❱❩✆☎ ✓✔✱✱✗✍✸✞✝ ✜✢✹■❈❂✓✡✶ ✹✔✓✠✟ ❋❂❋

✗✍✸ ❫

☛ ❞✥✔

❞✚➋✫✦✩➉➑➐✴✵✵➉✫✍✶❉❪✓✣✵✱✏✝➌✌✔✖✏✑✧❻➋⑩➐✲P▲☞✎✍✘☞✎✔⑥➋✫✦❃➋⑩➐❱❩✆✡ ✓✡✙ ✶ ❁☎✗❄❁☛✝ ✹✻✚✟✹☞✟✂✶

(18)
(19)

✂✪☎☎✄❆✞ ✁ ✂ ✟ ✡ ✂✑✡ ☎

R

✵✵➉⑩➌✌✤✌➉✂✦✩➉➑➐✷✓✄✂✜✍✂✏✝➌☎➉➑➐➄➌✖✏ ➋✒☞➏➐

R

n

✵✵➉✫✓✤✣ ✚✜✓✕✔☎➉❃✦✩➉➑➐✷✰✙✏✑✔☎➉⑩➌✖✏✵➐✲✦✜✏✯✦✡☞✎✍✂✏✑✓❃➐⑥➈⑩➉

n

R

n

×

m

✵✵➉✫✓✤✣ ✚✜✓✕✔☎➉❃✦❃➋⑩➐✷✍✎➋✒✔➎➌✌☞✎✧❚✏✵➐✲✦✜✏✯✦✡☞✎✍✂✏✑✓❃➐⑥➈⑩➉

n

×

m

A

T

✔➎➌⑥➋✒✓❃➐✌✤✌➉➑➐✌✔⑥➋✢✦❃➋✢✍✎➋✒✔➎➌✌☞✎✧

A

A

1

☞✎✓✕✰✙✏✝➌☎➐⑥➋✘✦❃➋✢✍✎➋✒✔➎➌✌☞✎✧

A

A

1

2

➌⑥➋✒☞✎✧★✳✕✚❿➋✫✦❤➌⑥➋✫✦❃➋✘✦✜✏✯✚✜✍✎➋✢✍✎➋✒✔➎➌✌☞✎✧

A

➐✖✏✑✍✘☞✿✦✜✏✟❧✣✓✜☞✿✦❃➋✘✤✌➉➑➐✌☞✎✔✌☞✎✰➑➋

A

✤❃➐✖✏✑✚✣✦✩➉✘☞✎✓✕✰✙✏✝➌☎➐⑥➋✂✦❃➋❱✍✎➋✒✔➎➌✌☞✎✧

A

P >

0

✦✜✏✑✓❃➉✫✔⑥➋✘✳✕✚✣✏

P

◆❁✦✜✏✟❧✣✓✜☞✿✦❃➋✢✤✌➉➑➐✌☞✎✔✌☞✎✰➑➋

P

0

✦✜✏✑✓❃➉✫✔⑥➋✘✳✕✚✣✏

P

◆s➐✖✏✑✍✘☞✿✦✜✏✟❧✣✓✜☞✿✦❃➋✘✤✌➉➑➐✌☞✎✔✌☞✎✰➑➋

k · k

✓❃➉⑩➌✌✍✎➋✘✏✑✚✣✵✝➍✎☞✿✦✡☞✇➋✒✓❿➋✂✦✜✏✯✚✜✍❵✰✙✏✑✔☎➉⑩➌❱❩

k

x

k

2

=

x

T

x

✤❿➋❲➌⑥➋

x

R

n

k · kP

✓❃➉⑩➌✌✍✎➋✢✤✌➉✫✓✣✦✜✏✝➌⑥➋✫✦❃➋✘✦✜✏✯✚✜✍❵✰✙✏✑✔☎➉⑩➌❱❩

k

x

k

2

P

=

x

T

P x

✤❿➋❲➌⑥➋

x

R

n

x

T

P

(

)

✏✷✶✡✤✩➌✖✏✵➐☎➐⑥➈⑩➉✟➐✌☞✎✍✘✤✩➍✎☞❧●✵ ➋✫✦❃➋✘✤❿➋❲➌⑥➋

x

T

P x

σ

(

A

)

✵✵➉✫✓✤✣ ✚✜✓✕✔☎➉❃✦✩➉➑➐➄➋✒✚✜✔☎➉❚✰➑➋❲➍➏➉⑩➌✖✏✵➐✾✦✡☞➏➐✌✔✌☞✎✓✕✔☎➉➑➐✲✦❃➋✢✍✎➋✒✔➎➌✌☞✎✧

A

E

(20)
(21)

Cap´ıtulo

1

✁✄✂❀✄✝✆ ✡ ✁✠✝✆☎✞✝ ✡

✟✡✠☛✟ ☞✍✌✏✎✒✑✔✓✖✕✘✗✚✙✛✌

❞✪✏✵➐✖✦✜✏✲✳✕✚✣✏✾▼✴✚✣✦✩➉⑩➍✺❂❜✴✬ ✍s➋❲➍✎✍✎➋✒✓❂➋✒✤✩➌✖✏✵➐✖✏✑✓✕✔☎➉✫✚ ✏✑✍ ✰ ➔ ✸ ➓ ➉❱✵✵➉✫✓✣✵✱✏✑☞✎✔☎➉❁✺☎✚✜✓✣✦❃➋✒✍✂✏✑✓✕✔⑥➋❲➍ ✦✜✏✷❧❃➍✎✔➎➌⑥➋✒❑✙✏✑✍❵✮✫✔✌☞✎✍✎➋

➌✖✏✱✵✑✚✩➌☎➐✌☞✎✰➑➋✢✜✳ ➓✤✣ ✛✜✰✝◗❲➌✌☞➏➉➑➐✴✤✩➌☎➉✫✩✩➍✿✏✑✍✎➋⑩➐✲✔✖❍✑✍ ➐✌☞✿✦✩➉✟➍✿✏✑✰➑➋✒✓✕✔⑥➋✫✦✩➉➑➐✾✏✧➌✖✏✵➐☎➉⑩➍✎✰❇☞✿✦✩➉➑➐✴✤❿➋❲➌⑥➋✒➋✒✤❊✏✝➌ ✺✻✏✑☞✿❋✵➉ ➋❲➌✲✚✜✍✎➋✘✵✝➍✇➋⑩➐☎➐✖✏❁✦✜✏

❧❃➍✎✔➎➌☎➉➑➐ ➉⑩➌✌☞✎❑✝☞✎✓❿➋✫✦❃➋✂✦✜✏✵➐☎➐✖✏✯✵✵➉✫✓✣✵✱✏✑☞✎✔☎➉✜✛●✳✕✚✣✏✚➐⑥➈⑩➉✘✦✜✏✑✓❃➉✫✍✘☞✎✓❿➋✫✦✩➉➑➐✾❛ ☞➍✎✔➎➌☎➉➑➐✲✦✜✏ ✍s➋❲➍✎✍✎➋✒✓ ❩☎❛ ✍✚➐✟❫✟✬

✔✲✍ ✦✩➉➑➐✾✤✩➌☎➉✫✩✩➍✿✏✑✍✎➋⑩➐✾☞✿✦✜✏✑✓✕✔✌☞❧●✵ ➋✫✦✩➉➑➐❁◆★✳✕✚✣✏❙➉➑➐✴❧❃➍✎✔➎➌☎➉➑➐✪✳✕✚✣✏★✏✵➐✌✔⑥➈⑩➉❙✦✜✏✑✓✕✔➎➌☎➉❬✦✜✏✵➐✌✔⑥➋❃✵ ➋✒✔✖✏✑❑ ➉⑩➌✌☞✇➋❃✤✌➉❇✦✜✏✑✍ ➐✖✏✝➌

➐✖✏✑✓❃➐✌✥✎✰✙✏✑☞➏➐❬❦⑩➐❃☞✎✓✣✵✱✏✝➌✌✔✖✏✑✧❻➋⑩➐❬✦✩➉❥➐✌☞➏➐✌✔✖✏✑✍✎➋✡✛✴✏❏✏✵➐☎➐⑥➋✮➐✖✏✑✓❃➐✌☞✎✩✜☞➍✎☞✿✦❃➋✫✦✜✏❏✤✌➉❇✦✜✏❏✵✵➉✫✍✘✤✩➌☎➉✫✍✂✏✑✔✖✏✝➌✏➉ ✦✜✏✵➐✖✏✑✍✘✤❊✏✑✓✜❅❃➉ ✏✓➋

✏✵➐✌✔⑥➋✒✩✜☞➍✎☞✿✦❃➋✫✦✜✏❱✦✩➉➑➐✾✍✂✏✵➐✌✍✒➉➑➐❚✬ ✔✲✍ ✵ ➋⑩➐☎➉❃✔✌✥✎✤✜☞✿✵✵➉☛➉❇✵✵➉⑩➌➎➌✖✏★✳✕✚❿➋✒✓✣✦✩➉☛➉❙✦✜✏✵➐✖✏✑✍✘✤❊✏✑✓✜❅❃➉❙✦✩➉✘❧❃➍✎✔➎➌☎➉✜✛❊✏✑✍★✩✌➉⑩➌⑥➋❃✮✫✔✌☞✎✍✒➉

✤❿➋❲➌⑥➋ ➉➅➐✌☞➏➐✌✔✖✏✑✍✎➋✷✓❃➉✫✍✘☞✎✓❿➋❲➍❡✛✫✤✌➉❇✦✜✏✿➐✖✏ ✦✜✏✑✔✖✏✝➌✌☞➏➉⑩➌⑥➋❲➌♠✍★✚✜☞✎✔☎➉➄➌⑥➋✒✤✜☞✿✦❃➋✒✍✂✏✑✓✕✔✖✏✷✳✕✚❿➋✒✓✣✦✩➉➅➉➑➐ ✦❃➋✫✦✩➉➑➐ ✦✩➉➅➐✌☞➏➐✌✔✖✏✑✍✎➋ ➐☎➉✒✺✐➌✖✏✑✍

✤❊✏✝➌✌✔✌✚✩➌✌✩❿➋✫❋✗✖❇✏✵➐❚✬ ❉❇➐✌✔☎➉❱✓❿➈⑩➉✂◆✉➋✫✵✱✏✑☞✎✔❳◗✱✰✙✏✝➍❡✛✣✚✜✍✎➋✢✰✙✏✑✧❁✳✕✚✣✏✯✚✜✍❵✍✒➉❇✦✜✏✝➍➏➉✢❧●✏✝➍❂✦✩➉✟➐✌☞➏➐✌✔✖✏✑✍✎➋✘◆✯✦✡☞✺☎✥✿✵✑☞➍❂✦✜✏✚➐✖✏✚➉✫✩✜✔✖✏✝➌❚✬

✼✓➉✫✔✌☞✎✰➑➋✫✦✩➉ ✤✌➉⑩➌✘✏✵➐✌✔✖✏❙✤✩➌☎➉✫✩✩➍✿✏✑✍✎➋✡✛ ✚✜✍ ✓✄✂✜✍✂✏✝➌☎➉✳➌✖✏✝➍✿✏✑✰➑➋✒✓✕✔✖✏✸✦✜✏❂➋❲➌✌✔✌☞✎❑ ➉➑➐✢✔✖✏✑✍ ➋✒✤✩➌✖✏✵➐✖✏✑✓✕✔⑥➋✫✦✩➉❸➋❲➍✎❑ ➉⑩➌✌☞✎✔✌✍✒➉➑➐

➋❲➍✎✔✖✏✝➌✌✓❿➋✒✔✌☞✎✰ ➉➑➐♠✤❿➋❲➌⑥➋➄➉✲❛ ✍❤✚❃➐✌✚❿➋❲➍❇✤❿➋❲➌⑥➋➄➐✌☞➏➐✌✔✖✏✑✍✎➋⑩➐ ➐✌✚✤✣✌✏✑☞✎✔☎➉➑➐ ➋✷☞✎✓✣✵✱✏✝➌✌✔✖✏✑✧❻➋⑩➐♠✓❃➉➑➐ ✤❿➋❲➌✢✏✒✍✂✏✑✔➎➌☎➉➑➐❚✛✫✰✙✏ ✣☎➋✴✤✌➉⑩➌ ✏✷✶✜✏✑✍✘✤✩➍➏➉

✜➒➑➓✤✣✛✛✜✴ ➒✤✣✛✛✜✴❉✳ ✣✛✛✜✴❉✵ ✣ ✏✚➋⑩➐②➌✖✏✟✺✻✏✝➌✖❍✑✓✣✵✑☞✇➋⑩➐✷✵✵➉✫✓✕✔✌☞✿✦❃➋⑩➐■✓✣✏✝➍✇➋⑩➐❚✬❝❀ ➋✒☞➏➐ ❧❃➍✎✔➎➌☎➉➑➐❞➐⑥➈⑩➉❱✦✜✏✑✓❃➉✫✍✘☞✎✓❿➋✫✦✩➉➑➐✴❛ ☞➍✎✔➎➌☎➉➑➐■▼➄➉✫✩✜✚❃➐✌✔☎➉➑➐

✦✜✏ ✍s➋❲➍✎✍✎➋✒✓ ❩☎❛♠▼ ✍✚➐✟❫■✏❙➋❲➍✎❑✝✚✜✍✎➋⑩➐✚➋✒✤✩➍✎☞✿✵ ➋✫❋✗✖❇✏✵➐✪✦✜✏✵➐☎➐⑥➋✘✵✝➍✇➋⑩➐☎➐✖✏★✦✜✏✯❧❃➍✎✔➎➌☎➉➑➐✾✤✌➉❇✦✜✏✑✍ ➐✖✏✝➌✲✏✑✓✣✵✵➉✫✓✕✔➎➌⑥➋✫✦❃➋⑩➐✯✏✑✍✥✜ ✰ ➔✦✣

✜✳ ➒✤✣ ✏✧✜✳✓✳ ✣ ✬

☞ ❛ ✍ ◆✾✚✜✍ ✦✩➉➑➐✷✦✜✏✵➐✖✏✑✓✕✰ ➉⑩➍✎✰❇☞✎✍✂✏✑✓✕✔☎➉➑➐✲✍✎➋✒☞➏➐❞➌✖✏✝➍✿✏✑✰➑➋✒✓✕✔✖✏✵➐✴✏✑✍ ✵✑☞✿❍✑✓✣✵✑☞✇➋✒➋✒✤✩➍✎☞✿✵ ➋✫✦❃➋❱✓❃➉ ✂✩➍✎✔✌☞✎✍✒➉✟➐✖◆✱✵✑✚✩➍➏➉✜✛✣✵✵➉✫✍

✺➉⑩➌✌✔✖✏❙✦✜✏✵➐✌✔⑥➋✫✳✕✚✣✏❃✤❿➋❲➌⑥➋✓➋⑩➐✟➋✒✤✩➍✎☞✿✵ ➋✫❋✗✖❇✏✵➐✢✏✑✍ ✏✑✓✜❑✙✏✑✓✜❅❿➋❲➌✌☞✇➋✡✬ ✹ ✔✌✚❿➋❲➍✎✍✂✏✑✓✕✔✖✏✝✛ ✍★✚✜☞✎✔⑥➋⑩➐✘✏✷✶✡✤✌➉➑➐✌☞✿❋✗✖❇✏✵➐❱✦✜✏✵➐✌✔⑥➋✸✔✖✏✵➉⑩➌✌☞✇➋

✏✓➐✌✚❿➋⑩➐❂➋✒✤✩➍✎☞✿✵ ➋✫❋✗✖❇✏✵➐ ✏✵➐✌✔⑥➈⑩➉ ✦✡☞➏➐✌✤✌➉✫✓✜✥✎✰✙✏✑☞➏➐ ✏ ✍★✚✜☞✎✔⑥➋⑩➐✏➋✒☞✎✓✣✦❃➋❭✏✵➐✌✔⑥➈⑩➉❝➐✖✏✑✓✣✦✩➉ ✦✜✏✵➐✖✏✑✓✕✰ ➉⑩➍✎✰❇☞✿✦❃➋⑩➐❚✬ ☞ ❛ ✍ ✤✌➉➑➐☎➐✌✚✜☞

✰✝◗❲➌✌☞➏➉➑➐➄➋⑩➐✌✤❊✏✱✵✑✔☎➉➑➐✷☞✎✍✘✤✌➉⑩➌✌✔⑥➋✒✓✕✔✖✏✵➐❚✛✣✔⑥➋✒☞➏➐✲✵✵➉✫✍✒➉✘✤✩➌☎➉❇✦✡✚✜✧✱☞➌✲✏✵➐✌✔✌☞✎✍✎➋✒✔✌☞✎✰➑➋⑩➐✴✩❿➋⑩➐✖✏ ➋✫✦❃➋⑩➐✴✏✑✍ ☞✎✓✡✺ ➉⑩➌✌✍✎➋✫❋✗✖❇✏✵➐✴✤❿➋⑩➐☎➐⑥➋✫✦❃➋⑩➐❚✛

(22)

➊②➉✫✓❃➐✌☞✿✦✜✏✝➌✖✏✯✚✜✍ ➐✌☞➏➐✌✔✖✏✑✍✎➋✢✦✡☞✎✓✑✏✒✍✘☞✿✵✵➉✒➍✎☞✎✓✣✏ ➋❲➌✴✰➑➋❲➌✌☞✇➋✒✓✕✔✖✏✯✓❃➉✘✔✖✏✑✍✘✤✌➉✂✦✜✏✵➐✖✵✝➌✌☞✎✔☎➉✘✤✌➉⑩➌

E

i

+1

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+1

=

F

i

x

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+

G

w,i

w

i

+

G

v,i

v

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=

H

i

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w,i

w

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+

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.

❜ ➐✌✔✖✏✏➐✌☞➏➐✌✔✖✏✑✍✎➋✐◆✸✦✜✏✑✓❃➉✫✍✘☞✎✓❿➋✫✦✩➉❥➐✌☞✎✓✜❑✝✚✩➍✇➋❲➌❚✛❘➐✖✏✑✓✣✦✩➉

x

i

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n

➋❘✰➑➋❲➌✌☞ ◗✱✰✙✏✝➍➅➐✌☞✎✓✜❑✝✚✩➍✇➋❲➌ ❩✐➉✫✚ ➐✖✏✑✍✘☞✒❪✏✵➐✌✔⑥➋✫✦✩➉✕❫✟✛

z

i

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➋ ✍✂✏✱✦✡☞✿✦❃➋✐✦❃➋P➐⑥➋✒✥✿✦❃➋✡✛

w

i

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m

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➉➑➐❏➌✌✚✜✥✿✦✩➉➑➐✂✤✩➌✖✏✵➐✖✏✑✓✕✔✖✏✵➐✂✓❃➉✐✏✵➐✌✔⑥➋✫✦✩➉✐✏❬✓❿➋❘✍✂✏✱✦✡☞✿✦❃➋✡✛

E

i

+1

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m

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➋⑩➐ ✍✎➋✒✔➎➌✌☞✎✧❚✏✵➐✷✦✩➉❙➐✌☞➏➐✌✔✖✏✑✍✎➋❁✓❃➉✫✍✘☞✎✓❿➋❲➍❡✬ ❅✲✚❿➋✒✓✣✦✩➉

E

i

+1

=

I

➉❙➐✌☞➏➐✌✔✖✏✑✍✎➋❙❩ ✰ ✬✰ ❫ ➌✖✏✱✦✡✚✜✧ ✒➐✖✏✧➋⑩➉❱✍✒➉❇✦✜✏✝➍➏➉★✓❃➉❱✏✵➐✌✤❿➋✫❋✵➉

✦✜✏❱✏✵➐✌✔⑥➋✫✦✩➉✂✚❃➐✌✚❿➋❲➍❡✬ ✹ ✔✖✏✵➉⑩➌✌☞✇➋❃✦✜✏✵➐☎➐⑥➋✂✵✝➍✇➋⑩➐☎➐✖✏★✦✜✏✉➐✌☞➏➐✌✔✖✏✑✍✎➋⑩➐✷✺➉✫☞ ✦✜✏✵➐✖✏✑✓✕✰ ➉⑩➍✎✰❇☞✿✦❃➋❃✤❿➋❲➌⑥➋✂✦✜✏✵➐✖✵✝➌✖✏✑✰✙✏✝➌✚➋⑩➐✾✦✡☞✎✓✑✏✒✍✘☞✿✵ ➋⑩➐

✦✜✏❁✵✱✏✝➌✌✔☎➉➑➐➄➐✌☞➏➐✌✔✖✏✑✍✎➋⑩➐❞➍✎☞✎✓✣✏ ➋❲➌✖✏✵➐✲✏✑✍✶✳✕✚✣✏✉➋❙➌✖✏✑✤✩➌✖✏✵➐✖✏✑✓✕✔⑥➋✫❋ ➈⑩➉❃✓❃➉✂✏✵➐✌✤❿➋✫❋✵➉✘✦✜✏❁✏✵➐✌✔⑥➋✫✦✩➉✘✓❿➈⑩➉✒➐✖✏s➋✒✤✩➍✎☞✿✵ ➋✡✛✣✔⑥➋✒☞➏➐✴✵✵➉✫✍✒➉

➐✌☞➏➐✌✔✖✏✑✍✎➋⑩➐✴✍✎➋✒✔➎➌✌☞✿✵✑☞✇➋✒☞➏➐✲✳✕✚❿➋✫✦❤➌⑥➋✫✦✩➉➑➐✴✓❿➈⑩➉✘☞✎✓✕✰✙✏✝➌✌✔✌✥✎✰✙✏✑☞➏➐✾✏✚➐✌☞➏➐✌✔✖✏✑✍✎➋⑩➐✷✍✎➋✒✔➎➌✌☞✿✵✑☞✇➋✒☞➏➐ ➌✖✏✑✔⑥➋✒✓✜❑✝✚✩➍✇➋❲➌✖✏✵➐❚✬

☞➅➐✪✏✵➐✌✔✌✚✣✦✩➉➑➐✧➐☎➉✫✩✩➌✖✏❙➐✌☞➏➐✌✔✖✏✑✍✎➋⑩➐✚➐✌☞✎✓✜❑✝✚✩➍✇➋❲➌✖✏✵➐✘❩✔⑥➋✒✍★✩❊◆✑✍ ✦✜✏✑✓❃➉✫✍✘☞✎✓❿➋✫✦✩➉➑➐s➐✌☞➏➐✌✔✖✏✑✍✎➋⑩➐✪✦✜✏✵➐✖✵✝➌✌☞✎✔☎➉⑩➌✖✏✵➐✟❫➄➐⑥➈⑩➉❃✍✒➉✫✔✌☞✒

✰➑➋✫✦✩➉➑➐✯✤❊✏✝➍➏➉❙✺✥➋✒✔☎➉ ✦✜✏✘✳✕✚✣✏✢✍✒➉❇✦✜✏✝➍➏➉➑➐❁✵✵➉✫✍ ✏✵➐☎➐⑥➋⑩➐❁✵ ➋❲➌⑥➋✫✵✑✔✖✏✝➌✌✥➏➐✌✔✌☞✿✵ ➋⑩➐❙➋✒✤❿➋❲➌✖✏✱✵✱✏✝➌✖✏✑✍ ✦✜✏✘✍✎➋✒✓✣✏✑☞➌⑥➋❙✺✐➌✖✏✱✳✕✚✣✏✑✓✕✔✖✏✘✏✑✍

✦✡☞✎✰✙✏✝➌☎➐⑥➋⑩➐➅➋✒✤✩➍✎☞✿✵ ➋✫❋✗✖❇✏✵➐❚✛✜✵✵➉✫✍✒➉✂✏✑✍ ➐✌☞➏➐✌✔✖✏✑✍✎➋⑩➐✴✏✱✵✵➉✫✓✤✣✫✍✘☞✿✵✵➉➑➐ ✜✳✓✸

✛✡✍✒➉❇✦✜✏✝➍✇➋✒❑✙✏✑✍✶✦✜✏✪☞✎✍✎➋✒❑✙✏✑✓❃➐ ✜ ➒➑➒✤✣

✛✜✍✒➉❇✦✜✏✝➍✇➋✒❑✙✏✑✍

✦✜✏✚➋✫✏✝➌☎➉✫✓❿➋✱✰✙✏✵➐ ✜✴✓✴

✛✕✍✒➉✝✦✜✏✝➍✇➋✒❑✙✏✑✍ ✦✜✏✲❅✣✏✝➍✎☞✿✵✱✮✫✤✜✔✖✏✝➌☎➉➑➐ ✜✳✯❆

✛❬➌☎➉✫✩❊✮✫✔✌☞✿✵ ➋ ✜✳ ✡

✛❤➐✌☞➏➐✌✔✖✏✑✍✎➋⑩➐■✦✜✏✲✤✌➉✫✔✖❍✑✓✣✵✑☞✇➋❱✏✾✵✑☞➌✖✵✑✚✜☞✎✔☎➉➑➐

✏✝➍✿◆✑✔➎➌✌☞✿✵✵➉➑➐❚✬

☛ ➋❲➌⑥➋❂➐✌☞➏➐✌✔✖✏✑✍✎➋⑩➐✚➐✌☞✎✓✜❑✝✚✩➍✇➋❲➌✖✏✵➐❁✦✡☞➏➐✖✵✝➌✖✏✑✔☎➉➑➐✯✓❃➉❬✔✖✏✑✍✘✤✌➉✜✛ ✏✵➐✌✔✌☞✎✍✎➋✒✔✌☞✎✰➑➋⑩➐❁✦✜✏✘✏✵➐✌✔⑥➋✫✦✩➉❬✔✖❍✑✍ ➐✌☞✿✦✩➉❙✺ ➉⑩➌✌✍★✚✩➍✇➋✫✦❃➋⑩➐★✦✜✏

✍✎➋✒✓✣✏✑☞➌⑥➋✎➌✖✏✱✵✑✚✩➌☎➐✌☞✎✰➑➋✂✏✉➉➑➐✴❧❃➍✎✔➎➌☎➉➑➐✧➐✌☞✎✓✜❑✝✚✩➍✇➋❲➌✖✏✵➐✲✓❃➉✫✍✘☞✎✓❿➋✒☞➏➐✘❩☎❛ ➇❇❖➅➐✟❫❘➌✖✏✵➐✌✚✩➍✎✔⑥➋✒✓✕✔✖✏✵➐✪✔✖❍✑✍ ➐✌☞✿✦✩➉✎➉✫✩ ✣✌✏✑✔☎➉❙✦✜✏★✏✵➐✌✔✌✚✣✦✩➉

✦✜✏★✦✡☞✎✰✙✏✝➌☎➐☎➉➑➐✴✔➎➌⑥➋✒✩❿➋❲➍✎❅❃➉➑➐✢❩ ✰✙✏ ✣☎➋✡✛✜✤✌➉⑩➌✲✏✷✶✜✏✑✍✘✤✩➍➏➉✜✛ ✜✸ ✣ ✛✏✜✰ ✳ ✣ ✛ ✜✰ ✴ ✣✛ ✜✰ ✡ ✣ ✛✏✜➒✢✰ ✣✛✏✜➒ ✡ ✣ ✛✏✜✳ ✰ ✣ ✛ ✜✳ ➔✤✣ ✛✏✜✴ ➓✦✣✛✏✜✴✒✡ ✣ ✛✏✜✴ ➔✦✣❫✟✬

✔✲✍ ☞✎✓✣✦✡☞✿✵ ➋✫✦✩➉⑩➌❱✦✩➉✸✵✝➌✖✏✵➐✖✵✑☞✎✍✂✏✑✓✕✔☎➉ ✦✜✏✵➐✌✔✖✏✂✵ ➋✒✍✘✤✌➉✸✦✜✏✘✤❊✏✵➐✖✳✕✚✜☞➏➐⑥➋❬◆✒➉ ✦✜✏✵➐✖✏✑✓✕✰ ➉⑩➍✎✰❇☞✎✍✂✏✑✓✕✔☎➉❘✦✜✏✘✚✜✍ ✔☎➉❬➉⑩➍✎✩✌➉✏✶❨✦✜✏

➐✌☞➏➐✌✔✖✏✑✍✎➋⑩➐➄➐✌☞✎✓✜❑✝✚✩➍✇➋❲➌✖✏✵➐✷✤❿➋❲➌⑥➋✟➉✒➐☎➉✒✺☎✔✦✥❘➋❲➌✖✏❁✼P➋✒✔➎➍✇➋✒✩ ✜➒ ✴ ✣✬

✟✡✠✂✧ ✄✆☎✦✎✒✑✂✝✞✕ ✎✒✑✔✓✖✕✞☎✩★ ✑✂✚✎✪☞ ✕✓✫ ✕✙✬✮✭✯☞✜✑✌✫ ✑✎✤✌✗☞ ✕✰✑✰✍✱✘✘✕✛✓ ✑✏✲✚✕✓✫ ✕

❖➅➉✓➐✌☞➏➐✌✔✖✏✑✍✎➋ ❩

z

i

◆❬✵✵➉✫✓❃➐✌☞✿✦✜✏✝➌⑥➋✫✦✩➉✳➉❘✵✵➉✫✓✤✣ ✚✜✓✕✔☎➉❨✦✜✏❙✍✂✏✱✦✡☞✿✦❃➋⑩➐✒➉✫✚✼➉✫✩❃➐✖✏✝➌✌✰➑➋✫❋✗✖❇✏✵➐❚✬ ✹ ➐☎➐✌☞✎✍✸✛❝✤✌➉❇✦✜✏✚✒➐✖✏

✦✜✏✑✔✖✏✝➌✌✍✘☞✎✓❿➋❲➌

ˆ

x

i

✳✕✚✣✏❙◆☛➉✸✰➑➋❲➍➏➉⑩➌✘✏✵➐✌✔✌☞✎✍✎➋✫✦✩➉ ✦✜✏

x

i

✛ ➋✸✤❿➋❲➌✌✔✌☞➌✘✦❃➋⑩➐❏➉✫✩❃➐✖✏✝➌✌✰➑➋✫❋✗✖❇✏✵➐

z

i

✬ ☛ ➋❲➌⑥➋❏✔⑥➋✒✓✕✔☎➉✜✛❝✤✌➉❇✦✜✏✚✒ ➐✖✏

✏✵➐✌✔✌☞✎✍✎➋❲➌

x

i

➋✒✔➎➌⑥➋✱✰✙◆✵➐✾✦✜✏❁✦✡☞✎✰✙✏✝➌☎➐☎➉➑➐✴✍✂◆✑✔☎➉❇✦✩➉➑➐❚✬②➊②➉✫✓✡✺➉⑩➌✌✍✂✏❙➋✘✦✡☞➏➐✌✤✌➉✫✓✜☞✎✩✜☞➍✎☞✿✦❃➋✫✦✜✏★✦❃➋✘✳✕✚❿➋✒✓✕✔✌☞✿✦❃➋✫✦✜✏★✦✜✏✯✍✂✏✱✦✡☞✿✦❃➋⑩➐❚✛

➉✂✵✵➉✫✓✣✵✱✏✑☞✎✔☎➉✂✦✜✏❁✏✵➐✌✔✌☞✎✍✎➋✒✔✌☞✎✰➑➋❏➌✖✏✱✵✱✏✑✩❊✏✯✚✜✍✎➋✢✦✜✏✑✓❃➉✫✍✘☞✎✓❿➋✫❋ ➈⑩➉❙✦✡☞✺✻✏✝➌✖✏✑✓✕✔✖✏✝✬

❅✲✚❿➋✒✓✣✦✩➉❙➐✖✏✷✔✖✏✑✍

i

✍✂✏✱✦✡☞✿✦❃➋⑩➐❚✛❬➉✫✚☛➐✖✏✣☎➋

z

0

, z

1

, . . . z

i

✛❇✏✑✓✕✔⑥➈⑩➉❏➋❁✏✵➐✌✔✌☞✎✍✎➋✒✔✌☞✎✰➑➋❁✦✜✏

x

i

(23)

◆✸✵✖❅❿➋✒✍✎➋✫✦❃➋ ✦✜✏ ❧❃➍✎✔➎➌⑥➋✫✦❃➋❨✏✸◆✸❑✙✏✝➌⑥➋❲➍✎✍✂✏✑✓✕✔✖✏❈➌✖✏✑✤✩➌✖✏✵➐✖✏✑✓✕✔⑥➋✫✦❃➋ ✤✌➉⑩➌

ˆ

x

i

|

i

✬ ✹ ➐✖✏✑❑✝✚✜✓✣✦❃➋❨✦✜✏✑✓❃➉✫✍✘☞✎✓❿➋✫❋ ➈⑩➉❭◆❈➋❨✦✜✏

✏✵➐✌✔✌☞✎✍✎➋✒✔✌☞✎✰➑➋❁✤✩➌✖✏✱✦✡☞✎✔☎➉⑩➌⑥➋✡✬ ❖✾✏✵➐✌✔✖✏✾✵ ➋⑩➐☎➉✜✛❇✦✜✏✵➐✖✏ ✣☎➋ ✒ ➐✖✏✾✏✵➐✌✔✌☞✎✍✎➋❲➌❘➉❙➐✌☞✎✓❿➋❲➍

x

i

+1

✦✡☞➏➐✌✤✌➉✫✓✣✦✩➉✦✒ ➐✖✏✧➋✒✤❊✏✑✓❿➋⑩➐ ✦✜✏

i

✍✂✏✱✦✡☞✿✦❃➋⑩➐ ➉✫✚✏➉✫✩❃➐✖✏✝➌✌✰➑➋✫❋✗✖❇✏✵➐❚✛✩➉✫✚❈➐✖✏ ✣☎➋✡✛

z

0

, z

1

, . . . z

i

✹ ✔✖✏✝➌✖✵✱✏✑☞➌⑥➋ ✦✜✏✑✓❃➉✫✍✘☞✎✓❿➋✫❋ ➈⑩➉❏◆✒➋❙✦✜✏✘✏✵➐✌✔✌☞✎✍✎➋✒✔✌☞✎✰➑➋❂➐✌✚❿➋✱✰❇☞✎✧❻➋✫✦❃➋✡✬✘❖✾✏✵➐✌✔✖✏✢✵ ➋⑩➐☎➉✜✛ ➋❬✳✕✚❿➋✒✓✕✔✌☞✿✦❃➋✫✦✜✏✂✦✜✏✢✍✂✏✱✦✡☞✿✦❃➋⑩➐

✦✡☞➏➐✌✤✌➉✫✓✜✥✎✰✙✏✝➍♠◆❙➐✌✚✜✤❊✏✝➌✌☞➏➉⑩➌✚➋⑩➉❃✓✄✂✜✍✂✏✝➌☎➉❬✏✵➐✌✔⑥➋✫✦✩➉➑➐✪✳✕✚✣✏❁✤✩➌✖✏✑✔✖✏✑✓✣✦✜✏✚✒ ➐✖✏✢✏✵➐✌✔✌☞✎✍✎➋❲➌❚✛✌➉✫✚P➐✖✏ ✣☎➋✡✛✌➋⑩➐✾✍✂✏✱✦✡☞✿✦❃➋⑩➐✧➐✖✏✑❑✝✚✜☞✎✓✕✔✖✏✵➐

✤✌➉❇✦✜✏✑✍ ➐✖✏✝➌♠✚❃➐⑥➋✫✦❃➋⑩➐♠✤❿➋❲➌⑥➋✧➉✫✩✜✔✖✏✑✓✣❋ ➈⑩➉❁✦❃➋✪✏✵➐✌✔✌☞✎✍✎➋✒✔✌☞✎✰➑➋✡✬ ✹ ➐☎➐✌☞✎✍✸✛✫✤✌➉❇✦✜✏✚✒ ➐✖✏❘➉✫✩✜✔✖✏✝➌✿➋✾✏✵➐✌✔✌☞✎✍✎➋✒✔✌☞✎✰➑➋

ˆ

x

i

✦✡☞➏➐✌✤✌➉✫✓✣✦✩➉✦✒ ➐✖✏

✦❃➋⑩➐ ✍✂✏✱✦✡☞✿✦❃➋⑩➐

z

0

, z

1

, . . . z

i

+1

✬♠❖➅➉⑩➌✌✍✎➋❲➍✎✍✂✏✑✓✕✔✖✏✧➉✯✤✩➌☎➉❇✵✱✏✵➐☎➐☎➉★✦✜✏➄➐✌✚❿➋✱✰❇☞✎✧❻➋✫❋ ➈⑩➉★✤✌➉❇✦✜✏ ➐✖✏✷✔☎➉⑩➌✌✓❿➋❲➌ ✍✎➋✒☞➏➐ ✏✷✶❃➋✒✔☎➉★✦✩➉

✳✕✚✣✏✚➉✢✤✩➌☎➉❇✵✱✏✵➐☎➐☎➉✘✦✜✏✪❧❃➍✎✔➎➌⑥➋✒❑✙✏✑✍✸✛✣✤✌➉⑩➌✴✏✑✓✕✰ ➉⑩➍✎✰✙✏✝➌✲✚✜✍✎➋✘✳✕✚❿➋✒✓✕✔✌☞✿✦❃➋✫✦✜✏❁✍✎➋✒☞➏➉⑩➌✲✦✜✏✯✍✂✏✱✦✡☞✿✦❃➋⑩➐❚✬

❜ ➐✌✔✖✏✵➐✷✔➎➌✖❍✵➐✷✔✌☞✎✤✌➉➑➐✴✦✜✏❁✏✵➐✌✔✌☞✎✍✎➋✒✔✌☞✎✰➑➋⑩➐➄➐⑥➈⑩➉❏➌✖✏✑✤✩➌✖✏✵➐✖✏✑✓✕✔⑥➋✫✦❃➋⑩➐✲✍✎➋✒✔✖✏✑✍✎➋✒✔✌☞✿✵ ➋✒✍✂✏✑✓✕✔✖✏★✤✌➉⑩➌

x

ˆ

i

|

i

➋✢✏✵➐✌✔✌☞✎✍✎➋✒✔✌☞✎✰➑➋❏➍✎☞✎✓✣✏ ➋❲➌✷❧❃➍✎✔➎➌⑥➋✫✦❃➋✂✁

x

ˆ

i

+1

|

i

➋✢✏✵➐✌✔✌☞✎✍✎➋✒✔✌☞✎✰➑➋❏➍✎☞✎✓✣✏ ➋❲➌✴✤✩➌✖✏✱✦✡☞✎✔☎➉⑩➌⑥➋✂✁

x

ˆ

i

|

i

+1

➋✢✏✵➐✌✔✌☞✎✍✎➋✒✔✌☞✎✰➑➋❏➍✎☞✎✓✣✏ ➋❲➌➄➐✌✚❿➋✱✰❇☞✎✧❻➋✫✦❃➋✂✁

✏✏➐⑥➈⑩➉❸➋✒✤✩➌✖✏✵➐✖✏✑✓✕✔⑥➋✫✦❃➋⑩➐❙✓✣✏✵➐✌✔⑥➋✐✔✖✏✵➐✖✏✝✛ ✔⑥➋✒✓✕✔☎➉ ✤❿➋❲➌⑥➋✮➉✐✍✒➉❇✦✜✏✝➍➏➉❸➐✌☞✎✓✜❑✝✚✩➍✇➋❲➌❃✓❃➉✫✍✘☞✎✓❿➋❲➍❡✛✴✳✕✚❿➋✒✓✕✔☎➉❨✤❿➋❲➌⑥➋✮➉✐✍✒➉❇✦✜✏✝➍➏➉

➐✌☞✎✓✜❑✝✚✩➍✇➋❲➌➄➐✌✚✤✣✌✏✑☞✎✔☎➉✒➋❱☞✎✓✣✵✱✏✝➌✌✔✖✏✑✧❻➋⑩➐✲✤❿➋❲➌⑥➋✒✍✂◆✑✔➎➌✌☞✿✵ ➋⑩➐❚✬

✟✡✠☎✄ ✆✞✝✠✟ ✑✖✎✒✑✔✓✖✌ ☎

✹ ✔✖◆❏➉❃✍✒➉✫✍✂✏✑✓✕✔☎➉✜✛ ➉❃✤✩➌☎➉✫✩✩➍✿✏✑✍✎➋ ✦✜✏❱✏✵➐✌✔✌☞✎✍✎➋✒✔✌☞✎✰➑➋☛➌☎➉✫✩✜✚❃➐✌✔⑥➋❙✮✫✔✌☞✎✍✎➋❃✤❿➋❲➌⑥➋☛➐✌☞➏➐✌✔✖✏✑✍✎➋⑩➐✚➐✌☞✎✓✜❑✝✚✩➍✇➋❲➌✖✏✵➐✪✔✖✏✑✍ ➐✌☞✿✦✩➉

➋⑩➐☎➐✌✚✜✓✕✔☎➉❃✦✜✏✯✤✌➉✫✚✣✵ ➋⑩➐✴✤❊✏✵➐✖✳✕✚✜☞➏➐⑥➋⑩➐ ✜

✛ ✜

✛ ✜

✛ ✜ ➒➑➔✤✣

✏✚➍✎☞✎✰❬➌☎➉➑➐ ✜✴❉✸

✬■❖✾✏✵➐✌✔⑥➋✢✔✖✏✵➐✖✏✝✛✣✵✵➉✫✓❃➐✌☞✿✦✜✏✝➌⑥➋ ✒ ➐✖✏❬❩

❫✷✵✵➉✫✍

☞✎✓✣✵✱✏✝➌✌✔✖✏✑✧❻➋⑩➐✲✏✑✍ ✔☎➉❇✦❃➋⑩➐➄➋⑩➐✷✍✎➋✒✔➎➌✌☞✎✧❚✏✵➐✲✦✩➉✟➐✌☞➏➐✌✔✖✏✑✍✎➋

(

E

i

+1

+

δE

i

+1

)

x

i

+1

= (

F

i

+

δF

i

)

x

i

+ (

G

w,i

+

δG

w,i

)

w

i

+ (

G

v,i

+1

+

δG

v,i

+1

)

v

i

+1

,

z

i

+1

= (

H

i

+1

+

δH

i

+1

)

x

i

+1

+ (

K

w,i

+

δK

w,i

)

w

i

+ (

K

v,i

+1

+

δK

v,i

+1

)

v

i

+1

.

❩ ✰ ✬➒ ❫

➐✖✏✑✓✣✦✩➉❏✳✕✚✣✏

δE

i

+1

, δF

i

, δG

w,i

, δG

v,i

, δH

i

, δK

v,i

δK

v,i

➐⑥➈⑩➉✸✤❊✏✝➌✌✔✌✚✩➌✌✩❿➋✫❋✗✖❇✏✵➐❱✰➑➋❲➌✌☞✇➋✒✓✕✔✖✏✵➐★✓❃➉❏✔✖✏✑✍✘✤✌➉

➍✎☞✎✍✘☞✎✔⑥➋✫✦❃➋⑩➐✴✤✌➉⑩➌✷✓❃➉⑩➌✌✍✎➋⑩➐❚✬

✹ ✤❿➋❲➌✌✔✌☞➌ ✦✜✏❁❩ ✰ ✬➒ ❫✟✛ ➐✖✏✝➌⑥➈⑩➉✯✦✜✏✵➐✖✏✑✓✕✰ ➉⑩➍✎✰❇☞✿✦✩➉➑➐❝❧❃➍✎✔➎➌☎➉➑➐ ➐✌☞✎✓✜❑✝✚✩➍✇➋❲➌✖✏✵➐ ➌☎➉✫✩✜✚❃➐✌✔☎➉➑➐✲❩☎❛ ➇❇▼➄➐✟❫❂✰❇☞✇➋✧➋✒✩✌➉⑩➌✖✦❃➋✒❑✙✏✑✍ ✦✜✏✑✔✖✏✝➌✒

Referências

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