Uma construção do conceito de área para figuras planas

FRANCISCO CARLOS JACOB

UMA CONSTRUÇÃO DO CONCEITO DE ÁREA PARA FIGURAS PLANAS

Dissertação apresentada à Universidade Federal de Viçosa, como parte das exigências do PROFMAT (Mestrado Profissional em Matemática em Rede Nacional), para obtenção do título de Magister Scientiae.

VIÇOSA

Ficha catalográfica preparada pela Biblioteca Central da Universidade Federal de Viçosa - Câmpus Viçosa

T

Jacob, Francisco Carlos,

1968-J15c

2015

Uma construção do conceito de área para figuras planas /

Francisco Carlos Jacob. – Viçosa, MG, 2015.

vi, 45f. : il. (algumas color.) ; 29 cm.

Orientador: Kennedy Martins Pedroso.

Dissertação (mestrado) - Universidade Federal de Viçosa.

Referências bibliográficas: f.45.

1. Geometria plana. 2. Geometria Sólida. 3. Teoria das

medidas . I. Universidade Federal de Viçosa. Departamento de

Matemática. Programa de Pós-graduação em Matemática.

II. Título.

II

III

IV

Sumário

Resumo V

Abstract VI

1 Um pouco de história 3

2 Área de figuras planas elementares 7

2.1 Área do quadrado e do retângulo 7

2.2 Área do paralelogramo e do triângulo 11

2.3 Definição geral de área 14

3 O Método de Exaustão de Eudoxo-Arquimedes 18

4 Área de Lebesgue 23

4.1 Conceitos Preliminares 23

4.2 Figuras discretas 25

4.3 Figuras planas abertas 27

4.4 Área exterior 34

4.5 Área interior 37

4.6 Figuras mensuráveis 38

4.7 Experimento: cálculo da área de figuras planas não elementares 39

V RESUMO

JACOB, Francisco Carlos, M.Sc., Universidade Federal de Viçosa, novembro de 2015. Uma Construção do conceito de Área para Figuras Planas. Orientador: Kennedy Martins Pedroso.

VI ABSTRACT

JACOB, Francisco Carlos, M.Sc., Universidade Federal de Viçosa, November, 2015. An Area Concept to Construction Figures Planas. Advisor: Kennedy Martins Pedroso.

■♥tr♦❞✉çã♦

❆ ✐♠❡♥s❛ ❞✐✈❡rs✐❞❛❞❡ ♥❛ ❢♦r♠❛ q✉❡ ✉♠❛ ✜❣✉r❛ ♣❧❛♥❛ ♣♦❞❡ ❛ss✉♠✐r ❞❡✈❡r✐❛ s❡r s✉✜❝✐❡♥t❡ ♣❛r❛ ♥♦s ✐♥❝❡♥t✐✈❛r ❛ ❛♣r❡♥❞❡r ♠❛✐s s♦❜r❡ ♦ ❝♦♥❝❡✐t♦ ❞❛ ♠❡❞✐❞❛ ❞❡ s✉❛ ár❡❛✳ ◆♦ ❡♥t❛♥t♦✱ ❞♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧ ❡ ♠é❞✐♦✱ ❡①tr❛✐✲s❡ ♠❛✐s ❢ór♠✉❧❛s ♣❛r❛ ✜❣✉r❛s ♣❧❛♥❛s ❡❧❡♠❡♥t❛r❡s ❞♦ q✉❡ ✉♠❛ ♠❡t♦❞♦❧♦❣✐❛ ❞❡ ❝♦♠♦ ❛ ár❡❛ ❞❡ss❛s ✜❣✉r❛s ♣♦❞❡ s❡r ✉t✐❧✐③❛❞❛ ♣❛r❛ ♦❜t❡r r❡s♣♦st❛s ♣❛r❛ ✉♠ ❣r✉♣♦ ♠✉✐t♦ ♠❛✐s ❛♠♣❧♦ ❡ ♥✉♠❡r♦s♦ ❞❡ ❢♦r♠❛s ♣❧❛♥❛s✳

◆❡st❡ tr❛❜❛❧❤♦✱ ♣r♦♣♦♠♦s ✉♠❛ ✏tr❛♥s✐çã♦ s✉❛✈❡✑ ❡♥tr❡ ❛ ár❡❛ ❞❡ r❡tâ♥❣✉❧♦s ❡ tr✐â♥❣✉❧♦s ❡ ♦s ♠ét♦❞♦s q✉❡ ♥♦s ❧❡✈❛♠ à ♠❡❞✐❞❛ ❞❛ ár❡❛ ❞❡ ✉♠❛ ✜❣✉r❛ ♣❧❛♥❛ ❞✐s❢♦r♠❡✱ ❛♣r❡s❡♥t❛♥❞♦ ✉♠ ❝❛♠✐♥❤♦ r❡♣❧❡t♦ ❞❡ ♣♦ss✐❜✐❧✐❞❛❞❡s ♣❛r❛ ♦r✐❡♥t❛r ❡ ♠♦t✐✈❛r ♠❡❧❤♦r ♦ ❡♥s✐♥♦ ❞❡ ❣❡♦♠❡tr✐❛✱ t❛♥t♦ ❡♠ ❛✉❧❛s ♣rát✐❝❛s q✉❛♥t♦ ❡♠ ❛✉❧❛s t❡ór✐❝❛s✳

P❛r❛ t❛❧✱ ✈❛♠♦s ✐♥tr♦❞✉③✐r ❛ ♠❡❞✐❞❛ ❞❛ ár❡❛ ❞❡ ✜❣✉r❛s ♣❧❛♥❛s ❞❡ ❢♦r♠❛ s✐st❡✲ ♠❛t✐③❛❞❛✱ ❝♦♠❡ç❛♥❞♦ ♣❡❧❛s ✜❣✉r❛s ♠❛✐s s✐♠♣❧❡s✱ ♠♦str❛♥❞♦ ❝♦♠♦ ♦ ▼ét♦❞♦ ❞❡ ❊①❛✉stã♦✱ ♣r♦♣♦st♦ ♣♦r ❊✉❞♦①♦ ❞❡ ❈♥✐❞♦ ✭✹✵✽ ❛ ✸✺✺ ❛✳❈✳✮ ❡ ❞❡s❡♥✈♦❧✈✐❞♦ ♠❛✐s ❛♠♣❧❛♠❡♥t❡ ♣♦r ❆rq✉✐♠❡❞❡s ❞❡ ❙✐r❛❝✉s❛ ✭✷✽✼ ❛ ✷✶✷ ❛✳❈✳✮✱ ✉t✐❧✐③❛✲s❡ ❞❛ ár❡❛ ❞❡ ❛❧❣✉♠❛s ✜❣✉r❛s ♣❧❛♥❛s ❡❧❡♠❡♥t❛r❡s ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞❛ ár❡❛ ❞❡ ♦✉tr❛s ✜❣✉r❛s ♣❧❛♥❛s✳

❈♦♠♦ ✉♠ ❝♦♠♣❧❡♠❡♥t♦ ❞❡ ❣r❛♥❞❡ ✐♠♣♦rtâ♥❝✐❛✱ ❛♣r❡s❡♥t❛r❡♠♦s ♥♦çõ❡s ❡❧❡✲ ♠❡♥t❛r❡s ❞❛ ❚❡♦r✐❛ ❞❡ ▼❡❞✐❞❛✱ ♣r♦♣♦st❛ ♣♦r ▲❡❜❡s❣✉❡ ♥♦ ✐♥í❝✐♦ ❞♦ sé❝✉❧♦ ❳❳✱ ♠❛✐s ♣r❡❝✐s❛♠❡♥t❡ ❡♠ ✶✾✵✹✱ ❝♦♠ ✉♠❛ ❧✐♥❣✉❛❣❡♠ ❡ ❡①♣♦s✐çã♦ q✉❡✱ ❡♠ s✉❛ ♠❛✐♦r ♣❛rt❡✱ ❜❡♥❡✜❝✐❛♠ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞❡ ♣r♦❢❡ss♦r❡s ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✱ ❞❛♥❞♦ ♦♣♦rt✉✲ ♥✐❞❛❞❡ ♣❛r❛ q✉❡ ❝r✐❡♠ s❡✉ ♣ró♣r✐♦ ♠❛t❡r✐❛❧ ❞❡ ❛✉❧❛s s♦❜r❡ ✉♠ t❡①t♦ r✐❣♦r♦s❛♠❡♥t❡ ❝♦♥str✉í❞♦✳

❖ ❈❛♣ít✉❧♦ ✶ tr❛③ ✉♠❛ ❜r❡✈❡ ❤✐stór✐❛ s♦❜r❡ ♦ ❝á❧❝✉❧♦ ❞❛ ár❡❛s ❞❡ ✜❣✉r❛s ♣❧❛♥❛s ❡♠ s❡✉s ♣r✐♠ór❞✐♦s ❣r❡❣♦s ✐♥❞✐❝❛♥❞♦ ❛s ❝♦♥tr✐❜✉✐çõ❡s ✐♠♣♦rt❛♥t❡s ❞♦s ♠❛t❡♠át✐❝♦s ❊✉❞♦①♦ ❡ ❆rq✉✐♠❡❞❡s✳ ❆ t❡♦r✐❛ ♠❛✐s r❡❣✉❧❛r♠❡♥t❡ ✉t✐❧✐③❛❞❛ ♥♦ ❝á❧❝✉❧♦ ❞❛ ♠❡❞✐❞❛ ❞❡ ✜❣✉r❛s✱ q✉❡ ♥♦ ♥♦ss♦ ❝❛s♦ s❡rá ❛ ár❡❛✱ é ❛tr✐❜✉í❞❛ ❛ ❍❡♥r✐ ▲❡❜❡s❣✉❡ ❡ ❛♦ s❡✉ tr❛❜❛❧❤♦✱ t❛♠❜é♠ ♥❡ss❡ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦✱ ❢❛r❡♠♦s ❛❧❣✉♠❛s ♦❜s❡r✈❛çõ❡s ❡ r❡❢❡rê♥❝✐❛s✳

❖ ❈❛♣ít✉❧♦ ✷ ❞á ✉♠❛ ✐♥tr♦❞✉çã♦ ❛♦ ❝á❧❝✉❧♦ ❞❛ ár❡❛ ❞❡ ✜❣✉r❛s ♣❧❛♥❛s ❡❧❡♠❡♥✲ t❛r❡s✳ ❖ ❡❧❡♠❡♥t♦ ❞❡ ár❡❛ ✉♥✐tár✐❛ é ❞❡✜♥✐❞♦✿ q✉❛❧q✉❡r q✉❛❞r❛❞♦ ❝✉❥♦ ❧❛❞♦ ♠❡ç❛

✉♠❛ ✉♥✐❞❛❞❡ ❞❡ ❝♦♠♣r✐♠❡♥t♦ t❡rá ár❡❛ ✐❣✉❛❧ ❛ ✉♠❛ ✉♥✐❞❛❞❡✳ ❊♠ s❡❣✉✐❞❛ sã♦ ❝❛❧❝✉❧❛❞❛s ❛s ár❡❛s ❞❡ ✉♠ q✉❛❞r❛❞♦ q✉❛❧q✉❡r✱ r❡tâ♥❣✉❧♦s✱ ♣❛r❛❧❡❧♦❣r❛♠♦s✱ tr✐â♥✲ ❣✉❧♦s ❡ ♣♦❧í❣♦♥♦s✳ ❚♦❞❛ ❛❜♦r❞❛❣❡♠ é ❝♦♥str✉t✐✈❛✳ ❆ ú❧t✐♠❛ s❡çã♦ ❞♦ ❝❛♣ít✉❧♦ tr❛t❛ ❞❛ ❞❡✜♥✐çã♦ ❣❡r❛❧ ❞❡ ár❡❛ ❝♦♠ ✐❞é✐❛s q✉❡ s❡r✈✐rã♦ ❞❡ ❜❛s❡ ♣❛r❛ ♦ ❝♦♥❝❡✐t♦ ❞❡ ár❡❛ ❝♦♥str✉í❞♦ ♣❡❧❛ t❡♦r✐❛ ❞❡ ▲❡❜❡s❣✉❡✳

❖ ❈❛♣ít✉❧♦ ✸ ❛♣r❡s❡♥t❛ ♦ ▼ét♦❞♦ ❞❡ ❊①❛✉stã♦ ♣r♦♣♦st♦ ♣♦r ❊✉❞♦①♦ ❡ ❛♣r✐✲ ♠♦r❛❞♦ ♣♦r ❆rq✉✐♠❡❞❡s✳ ❆t❡♥t❛♠♦s ♣❛r❛ ❛ ❝♦♥str✉çã♦ ♦r✐❣✐♥❛❧ ♣r♦♣♦st❛ ♥♦s ❊❧❡♠❡♥t♦s ❞❡ ❊✉❝❧✐❞❡s✱ ✉♠ ❝❧áss✐❝♦ ❞❛ ❧✐t❡r❛t✉r❛ ♠✉♥❞✐❛❧✳ ❊ss❡ ❝❛♣ít✉❧♦ ✐❧✉str❛ ❛ ❞❡✜♥✐çã♦ ❣❡r❛❧ ❞❡ ár❡❛ ❞❛❞❛ ♥♦ ❝❛♣ít✉❧♦ ❛♥t❡r✐♦r ❡ ✐♥❞✐❝❛ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ✉♠❛ t❡♦r✐❛ ♦r❣❛♥✐③❛❞❛ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❛ ár❡❛ ❞❡ ✜❣✉r❛s ♠❛✐s ❝♦♠♣❧❡①❛s t❛❧ ❝♦♠♦ é ❛♣r❡s❡♥t❛❞❛ ♥♦ ❝❛♣ít✉❧♦ s❡❣✉✐♥t❡✳

❖ ❈❛♣ít✉❧♦ ✹ ✐♥✐❝✐❛ ❛ ❝♦♥str✉çã♦ ❞♦ ❝♦♥❝❡✐t♦ ❞❡ ár❡❛ s❡ ✈❛❧❡♥❞♦ ❞❛ ár❡❛ ❞❡ q✉❛❞r❛❞♦s✳ ❆s ♣r✐♠❡✐r❛s ✜❣✉r❛s tr❛t❛❞❛s sã♦ ❛q✉❡❧❛s ♦❜t✐❞❛s ❝♦♠♦ ✉♥✐ã♦ ✭❞✐s✲ ❥✉♥t❛✮ ❞❡ q✉❛❞r❛❞♦s❀ ❡ss❛s ✜❣✉r❛s s❡rã♦ ❝❤❛♠❛❞❛s ❞❡ ❞✐s❝r❡t❛s✳ ❖ ♣ró①✐♠♦ ♣❛ss♦ é tr❛❜❛❧❤❛r ♦ ❝♦♥❝❡✐t♦ ❞❡ ár❡❛ ♣❛r❛ ✜❣✉r❛s ♣❧❛♥❛s q✉❡ s❡♠♣r❡♠ ♣♦ss❛♠ ❝♦♥t❡r ✜❣✉r❛s ❞✐s❝r❡t❛s✱ ♥❡ss❡ ♣♦♥t♦ ✐♥tr♦❞✉③✐♠♦s ♦ ✐♠♣♦rt❛♥t❡ ❝♦♥❝❡✐t♦ ❞❡ ✜❣✉r❛ ♣❧❛♥❛ ❛❜❡rt❛✳ ◆❛ s❡q✉ê♥❝✐❛✱ ❛s ✜❣✉r❛s ♣❧❛♥❛s ❛❜❡rt❛s s❡rã♦ ❛♣r❡s❡♥t❛❞❛s ❝♦♠♦ ♦s ❜❧♦❝♦s ❞❡ ❝♦♥str✉çã♦ ♣❛r❛ ❛ ár❡❛ ❡①t❡r✐♦r ❞❡ ▲❡❜❡s❣✉❡✱ ❝♦♥❝❡✐t♦ q✉❡ ♥♦rt❡❛rá t♦❞❛s ❛s ❝♦♥str✉çõ❡s ❞♦ ❝❛♣ít✉❧♦✳ ❋✐♥❛❧✐③❛♥❞♦ ♦ ♥♦ss♦ tr❛❜❛❧❤♦✱ ♦ ❝❛♣ít✉❧♦ t❛♠❜é♠ ♠♦s✲ tr❛ ✉♠ ❡①♣❡r✐♠❡♥t♦ ♣rát✐❝♦ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❛ ár❡❛ ❞❡ ✉♠❛ ✜❣✉r❛ ♣❧❛♥❛ ❞✐s❢♦r♠❡ ✉s❛♥❞♦ ✜❣✉r❛s ❡❧❡♠❡♥t❛r❡s✱ ✉♠ ✏♣r✐♥❝í♣✐♦ ❞❡ ❡①❛✉stã♦✑ ❛ss♦❝✐❛❞♦ ❛ r✉❞✐♠❡♥t♦s ❞❛ t❡♦r✐❛ ❞❡ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ q✉❡ ❡①❡♠♣❧✐✜❝❛ ❛ ✐❞é✐❛ ❞❡ t♦❞♦ ♦ t❡①t♦✳

❈❛♣ít✉❧♦ ✶

❯♠ ♣♦✉❝♦ ❞❡ ❤✐stór✐❛

❆s ❝♦♥tr✐❜✉✐çõ❡s ❞❛ ❣❡♦♠❡tr✐❛ ❣r❡❣❛ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ❝á❧❝✉❧♦ ❡stã♦ ❣r❛✈❛❞❛s✱ s♦❜r❡t✉❞♦✱ ♥♦s tr❛❜❛❧❤♦s ❞❡ ❊✉❞♦①♦ ❞❡ ❈♥✐❞♦ ✭✹✵✽ ❛ ✸✺✺ ❛✳❈✳✮ ❡ ❆r✲ q✉✐♠❡❞❡s ✭✷✽✼ ❛ ✷✶✷ ❛✳❈✳✮ ❡♥✈♦❧✈❡♥❞♦ ♦ ▼ét♦❞♦ ❞❡ ❊①❛✉stã♦✳ ❊♥tr❡t❛♥t♦✱ é ✐♠♣♦ssí✈❡❧ ♥♦s r❡♠❡t❡r♠♦s ❞✐r❡t❛♠❡♥t❡ ❛♦s tr❛❜❛❧❤♦s ❞❡ss❡s ❞♦✐s ♠❛t❡♠át✐❝♦s ❞❛ ❛♥t✐❣✉✐❞❛❞❡ s❡♠ ❝♦♠♣r❡❡♥❞❡r♠♦s ♦ ♣ró♣r✐♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ●❡♦♠❡tr✐❛ ❡ ❞❛ ▼❛t❡♠át✐❝❛ ❣r❡❣❛ ❛♥t❡r✐♦r ❛ ❡st❡s✱ ✐♥❝❧✉✐♥❞♦ ❛s ❝♦♥tr✐❜✉✐çõ❡s ❞♦s ❡❣í♣❝✐♦s ❡ ❜❛❜✐❧ô♥✐♦s✳

❙❡❣✉♥❞♦ ❊✈❡s ❬✶❪✱ ❛ ❣❡♦♠❡tr✐❛ ❜❛❜✐❧ô♥✐❝❛ s❡ r❡❧❛❝✐♦♥❛ ✐♥t✐♠❛♠❡♥t❡ ❝♦♠ ❛ ♠❡♥s✉r❛çã♦ ♣rát✐❝❛✳ ❉❡ ♥✉♠❡r♦s♦s ❡①❡♠♣❧♦s ❝♦♥❝r❡t♦s ✐♥❢❡r❡✲s❡ q✉❡ ♦s ❜❛❜✐❧ô♥✐♦s ❞♦ ♣❡rí♦❞♦ ✷✵✵✵ ❛✳❈ ❛ ✶✻✵✵ ❛✳❈ ❞❡✈✐❛♠ ❡st❛r ❢❛♠✐❧✐❛r✐③❛❞♦s ❝♦♠ ❛s r❡❣r❛s ❣❡r❛✐s ❞❛ ár❡❛ ❞♦ r❡tâ♥❣✉❧♦✱ ❞❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ❡ ❞♦ tr✐â♥❣✉❧♦ ✐sós❝❡❧❡s ✭❡ t❛❧✈❡③ ❞❛ ár❡❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❣❡♥ér✐❝♦✮✱ ❞❛ ár❡❛ ❞❡ ✉♠ tr❛♣é③✐♦ r❡tâ♥❣✉❧♦✱ ❞♦ ✈♦❧✉♠❡ ❞❡ ✉♠ ♣❛r❛❧❡❧❡♣í♣❡❞♦ r❡t♦✲r❡tâ♥❣✉❧♦ ❡✱ ♠❛✐s ❣❡r❛❧♠❡♥t❡✱ ❞♦ ✈♦❧✉♠❡ ❞❡ ✉♠ ♣r✐s♠❛ r❡t♦ tr❛♣❡③♦✐❞❛❧✳ ❈♦♥s✐❞❡r❛✈❛✲s❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❝♦♠♦ ♦ tr✐♣❧♦ ❞❡ s❡✉ ❞✐â♠❡tr♦ ❡ ❛ ár❡❛ ❞♦ ❝ír❝✉❧♦ ❝♦♠♦ ✉♠ ❞✉♦❞é❝✐♠♦ ❞❛ ár❡❛ ❞♦ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ ✐❣✉❛❧ à ❝✐r❝✉♥❢❡rê♥❝✐❛ r❡s♣❡❝t✐✈❛ ✭r❡❣r❛s ❝♦rr❡t❛s ♣❛r❛π = 3✮✳

Pr❡❧❡❝✐♦♥❛ ❛✐♥❞❛ ❊✈❡s ❬✶❪ q✉❡ ✷✻ ❞♦s ✶✶✵ ♣r♦❜❧❡♠❛s ❞♦s ♣❛♣✐r♦s ❞❡ ▼♦s❝♦✉ ❡ ❘❤✐♥❞ sã♦ ❣❡♦♠étr✐❝♦s✳ ▼✉✐t♦s ❞❡❧❡s ❞❡❝♦rr❡♠ ❞❡ ❢ór♠✉❧❛s ❞❡ ♠❡♥s✉r❛çã♦ ♥❡❝❡ssár✐❛s ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❡ ár❡❛s ❞❡ t❡rr❛s ❡ ✈♦❧✉♠❡s ❞❡ ❣rã♦s✳ ❆ss✉♠❡✲s❡ q✉❡ ❛ ár❡❛ ❞❡ ✉♠ ❝ír❝✉❧♦ é ✐❣✉❛❧ à ❞❡ ✉♠ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ ✐❣✉❛❧ ❛8/9❞♦ ❞✐â♠❡tr♦ ❡ q✉❡ ♦ ✈♦❧✉♠❡ ❞❡ ✉♠ ❝✐❧✐♥❞r♦ r❡t♦ é ♦ ♣r♦❞✉t♦ ❞❛ ár❡❛ ❞❛ ❜❛s❡ ♣❡❧♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ❛❧t✉r❛✳ ■♥✈❡st✐❣❛çõ❡s r❡❝❡♥t❡s ♣❛r❡❝❡♠ ♠♦str❛r q✉❡ ♦s ❡❣í♣❝✐♦s s❛❜✐❛♠ q✉❡ ❛ ár❡❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ q✉❛❧q✉❡r é ♦ s❡♠✐✲♣r♦❞✉t♦ ❞❛ ❜❛s❡ ♣❡❧❛ ❛❧t✉r❛✳ ❆❧❣✉♥s ♣r♦❜❧❡♠❛s ♣❛r❡❝❡♠ ❡♥✈♦❧✈❡r ❝♦t❛♥❣❡♥t❡ ❞♦ â♥❣✉❧♦ ❞✐❡❞r♦ ❡♥tr❡ ❛ ❜❛s❡ ❡ ❛ ❢❛❝❡ ❞❛ ♣✐râ♠✐❞❡✱ ❡ ♦✉tr♦s ♠♦str❛♠ ❛❧❣✉♠ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❛ t❡♦r✐❛ ❞❛s ♣r♦♣♦rçõ❡s✳ ❈♦♥tr❛r✐❛♥❞♦ ❤✐stór✐❛s ♠✉✐t♦ r❡♣❡t✐❞❛s ❡ ❛♣❛r❡♥t❡♠❡♥t❡ ✐♥❢✉♥❞❛❞❛s✱ ♥ã♦ s❡ ❡♥❝♦♥tr♦✉ ♥❡♥❤✉♠❛ ❡✈✐❞ê♥❝✐❛ ❞♦❝✉♠❡♥t❛❧ ❞❡ q✉❡ ♦s ❡❣í♣❝✐♦s t✐♥❤❛♠ ❝✐ê♥❝✐❛✱ ♠❡s♠♦ q✉❡ ♥✉♠ ❝❛s♦

♣❛rt✐❝✉❧❛r✱ ❞♦ t❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s✳

❖s ú❧t✐♠♦s sé❝✉❧♦s ❞♦ s❡❣✉♥❞♦ ♠✐❧ê♥✐♦ ❛✳❈✳ t❡st❡♠✉♥❤❛r❛♠ ♠✉✐t❛s ♠✉❞❛♥ç❛s ❡❝♦♥ô♠✐❝❛s ❡ ♣♦❧ít✐❝❛s✳ ❆❧❣✉♠❛s ❝✐✈✐❧✐③❛çõ❡s ❞❡s❛♣❛r❡❝❡r❛♠✱ ♦ ♣♦❞❡r ❞♦ ❊❣✐t♦ ❡ ❞❛ ❇❛❜✐❧ô♥✐❛ ❞❡❝❧✐♥♦✉✱ ❡ ♦✉tr♦s ♣♦✈♦s✱ ❡s♣❡❝✐❛❧♠❡♥t❡ ♦s ❤❡❜r❡✉s✱ ♦s ❛ssír✐♦s✱ ♦s ❢❡♥í❝✐♦s ❡ ♦s ❣r❡❣♦s✱ ♣❛ss❛r❛♠ ❛♦ ♣r✐♠❡✐r♦ ♣❧❛♥♦✳ ❖ ❛♣❛r❡❝✐♠❡♥t♦ ❞❡ss❛ ♥♦✈❛ ❝✐✈✐❧✐③❛çã♦ s❡ ❞❡✉ ♥❛s ❝✐❞❛❞❡s ❝♦♠❡r❝✐❛✐s ❡s♣❛❧❤❛❞❛s ❛♦ ❧♦♥❣♦ ❞❛s ❝♦st❛s ❞❛ ➪s✐❛ ▼❡♥♦r ❡✱ ♠❛✐s t❛r❞❡✱ ♥❛ ♣❛rt❡ ❝♦♥t✐♥❡♥t❛❧ ❞❛ ●ré❝✐❛✱ ♥❛ ❙✐❝í❧✐❛ ❡ ♥♦ ❧✐t♦r❛❧ ❞❛ ■tá❧✐❛✳ ❆ ✈✐sã♦ ❞❡ r❛❝✐♦♥❛❧✐s♠♦ ❝r❡s❝❡♥t❡✱ ♦ ❤♦♠❡♠ ❝♦♠❡ç♦✉ ❛ ✐♥❞❛❣❛r ❝♦♠♦ ❡ ♣♦r q✉ê✳

◆❡ss❛ ❛t♠♦s❢❡r❛✱ ❛ ●ré❝✐❛ ❛♥t✐❣❛ ❞❡s♣♦♥t❛ ❝♦♠♦ ♣r✐♥❝✐♣❛❧ ♣♦❧♦ ❝✐❡♥tí✜❝♦ ❞♦ ♠✉♥❞♦ ❡ ♠❛r❝❛ ♦ ✐♥í❝✐♦ ❞❛ ▼❛t❡♠át✐❝❛ ❣r❡❣❛ ❡ ❞♦ ♥❛s❝✐♠❡♥t♦ ❞❡ ✉♠❛ ❧❡❣✐ã♦ ❞❡ ♠❛t❡♠át✐❝♦s ❡ ✜❧ós♦❢♦s q✉❡ ✈ã♦ ✐♥✢✉❡♥❝✐❛r ♣r♦❢✉♥❞❛♠❡♥t❡ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♠❛t❡♠át✐❝♦ ❡ ❝✐❡♥tí✜❝♦✱ ✈❡❥❛ ❬✷❪ ❡ ❬✻❪✳

❈♦♠❡ç❡♠♦s ♣♦r P✐tá❣♦r❛s✳

❆❞♠✐t❡✲s❡ ❣❡r❛❧♠❡♥t❡ q✉❡ ♦s ♣r✐♠❡✐r♦s ♣❛ss♦s ♥♦ s❡♥t✐❞♦ ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s ❡✱ ❛♦ ♠❡s♠♦ t❡♠♣♦✱ ❞♦ ❧❛♥ç❛♠❡♥t♦ ❞❛s ❜❛s❡s ❞♦ ❢✉✲ t✉r♦ ♠✐st✐❝✐s♠♦ ♥✉♠ér✐❝♦✱ ❢♦r❛♠ ❞❛❞♦s ♣♦r P✐tá❣♦r❛s ❡ s❡✉s s❡❣✉✐❞♦r❡s ♠♦✈✐❞♦s ♣❡❧❛ ✜❧♦s♦✜❛ ❞❛ ❢r❛t❡r♥✐❞❛❞❡✳ ➱ ❛tr✐❜✉í❞♦ ❛ P✐tá❣♦r❛s ❛ ❞❡s❝♦❜❡rt❛ ✐♥❞❡♣❡♥❞❡♥t❡ ❞♦ t❡♦r❡♠❛ s♦❜r❡s tr✐â♥❣✉❧♦s r❡tâ♥❣✉❧♦s ❤♦❥❡ ✉♥✐✈❡rs❛❧♠❡♥t❡ ❝♦♥❤❡❝✐❞♦ ♣❡❧♦ s❡✉ ♥♦♠❡ ✲ q✉❡ ♦ q✉❛❞r❛❞♦ s♦❜r❡ ❛ ❤✐♣♦t❡♥✉s❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ é ✐❣✉❛❧ à s♦♠❛ ❞♦s q✉❛❞r❛❞♦s s♦❜r❡ ♦s ❝❛t❡t♦s✳

P✐tá❣♦r❛s ❛❝r❡❞✐t❛✈❛ q✉❡ ♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ❡r❛♠ ❛ ♠❛tér✐❛ ❡ss❡♥❝✐❛❧ ♣❛r❛ ❛ ❝♦♥str✉çã♦ ❞♦ ✉♥✐✈❡rs♦✳ P♦rt❛♥t♦✱ ❛ ❞❡s❝♦❜❡rt❛ ❞❛ ✐rr❛❝✐♦♥❛❧✐❞❛❞❡ ❞❡√2♣r♦✈♦❝♦✉ ❛❧❣✉♠❛ ❝♦♥st❡r♥❛çã♦ ♥♦s ♠❡✐♦s ♣✐t❛❣ór✐❝♦s✳ ❚ã♦ ❣r❛♥❞❡ ❢♦✐ ♦ ✏❡s❝â♥❞❛❧♦ ❧ó❣✐❝♦✑ q✉❡ ♣♦r ❛❧❣✉♠ t❡♠♣♦ s❡ ✜③❡r❛♠ ❡s❢♦rç♦s ♣❛r❛ ♠❛♥t❡r ❛ q✉❡stã♦ ❡♠ s✐❣✐❧♦✳

P♦r ✈♦❧t❛ ❞❡ ✸✼✵ ❛✳❈✳✱ ♦ ♣r♦❜❧❡♠❛ ❢♦✐ r❡s♦❧✈✐❞♦ ♣♦r ❊✉❞♦①♦✱ ✉♠ ❜r✐❧❤❛♥t❡

❞✐s❝í♣✉❧♦ ❞❡ P❧❛tã♦ ❡ ❞♦ ♣✐t❛❣ór✐❝♦ ❆rq✉✐t❛s✱ ❛tr❛✈és ❞❡ ✉♠❛ ♥♦✈❛ ❞❡✜♥✐çã♦ ❞❡ ♣r♦♣♦rçã♦✳

❊✉❞♦①♦ ❛♣r❡s❡♥t♦✉ ❛ s✉❛ t❡♦r✐❛ ❞❛s ♣r♦♣♦rçõ❡s ❝♦♠♦ ♠♦❞♦ ❞❡ ✉❧tr❛♣❛ss❛r ❛ ✏❝r✐s❡✑ s✉r❣✐❞❛ ♥❛ ▼❛t❡♠át✐❝❛ ❣r❡❣❛ ♥♦ ♠♦♠❡♥t♦ ❞❛ ❞❡s❝♦❜❡rt❛ ❞♦s ✐♥❝♦♠❡♥s✉✲ rá✈❡✐s✱ q✉❡ ❞❡✐t❛✈❛ ♣♦r t❡rr❛ ❛ t❡♦r✐❛ ❞❛s ♣r♦♣♦rçõ❡s ❞♦s ♣✐t❛❣ór✐❝♦s✳

❖ ♠❛❣✐str❛❧ tr❛t❛♠❡♥t♦ ❞♦s ✐♥❝♦♠❡♥s✉rá✈❡✐s ❢♦r♠✉❧❛❞♦ ♣♦r ❊✉❞♦①♦ ❛♣❛r❡❝❡ ♥♦ q✉✐♥t♦ ❧✐✈r♦ ❞♦s ❊❧❡♠❡♥t♦s ❞❡ ❊✉❝❧✐❞❡s✱ ❡ ❡ss❡♥❝✐❛❧♠❡♥t❡ ❝♦✐♥❝✐❞❡ ❝♦♠ ❛ ❡①✲ ♣♦s✐çã♦ ♠♦❞❡r♥❛ ❞♦s ♥ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s ❞❛❞❛ ❡♠ ✶✽✼✷ ♣❡❧♦ ❛❧❡♠ã♦ ❘✐❝❤❛r❞ ❉❡❞❡❦✐♥❞ ✭✶✽✸✶ ❛ ✶✾✶✻✮✳

❈♦♠ ❆rq✉✐♠❡❞❡s ✭sé❝✉❧♦ ■■ ❛✳❈✮✱ ❛ ▼❛t❡♠át✐❝❛ ❞❛ ❛♥t✐❣✉✐❞❛❞❡ ❛❧❝❛♥ç♦✉ s✉❛ ♠❛✐♦r ♣r♦❥❡çã♦✱ ✉♠❛ ✈❡③ q✉❡ s✉❛ r✐q✉❡③❛ ❞❡ ♣❡♥s❛♠❡♥t♦s ❡♠ ✈ár✐❛s ár❡❛s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♦ t♦r♥♦✉ ♦ ♠❛✐s ❝é❧❡❜r❡ ❞♦s ♠❛t❡♠át✐❝♦s ❣r❡❣♦s ❡ ♦ ♠❛✐s ❡st✉❞❛❞♦ ❛ ♣❛rt✐r ❞♦ sé❝✉❧♦ ❳■❱ ❞❛ ❡r❛ ❝r✐stã✳

❆rq✉✐♠❡❞❡s ❛♣❧✐❝♦✉ ♦ ▼ét♦❞♦ ❞❡ ❊①❛✉stã♦ ♣❛r❛ ♣r♦✈❛r ♦s ✐♥ú♠❡r♦s r❡s✉❧t❛✲ ❞♦s r❡❧❛t✐✈♦s ❛ ❝♦♠♣r✐♠❡♥t♦s✱ ár❡❛s ❡ ✈♦❧✉♠❡s ❞❡ ❞✐✈❡rs❛s ✜❣✉r❛s ❣❡♦♠étr✐❝❛s ❡ t❛♠❜é♠ ❛♦ ❝á❧❝✉❧♦ ❞❡ ❝❡♥tr♦s ❞❡ ❣r❛✈✐❞❛❞❡❀ ❛❧❣✉♥s ❞❡st❡s r❡s✉❧t❛❞♦s ❥á ❡r❛♠ ❝♦♥❤❡❝✐❞♦s ♠❛s ♦✉tr♦s ❡r❛♠ ✐♥t❡✐r❛♠❡♥t❡ ♥♦✈♦s✳

❊♠ r❡❧❛çã♦ ❛♦ ❡st✉❞♦ ❞♦ ❝á❧❝✉❧♦✱ s♦❜r❡t✉❞♦ ♦ ✐♥t❡❣r❛❧✱ ❆rq✉✐♠❡❞❡s ❢♦✐ s❡♠ ❞ú✈✐❞❛ ❞❡t❡r♠✐♥❛♥t❡ ♣❛r❛ ❛ ❛❧❛✈❛♥❝❛❞❛ ❞♦s ❡st✉❞♦s ♠❛✐s ❛✈❛♥ç❛❞♦s ♥❡ss❡ ❝❛♠♣♦ ❛té ❛❧❝❛♥ç❛r ✉♠ ❢♦r♠❛t♦ ♠❛✐s ♣ró①✐♠♦ ❞♦ ❛t✉❛❧ ♣r♦♣♦st♦ ♣♦r ◆❡✇t♦♥ ❡ ▲❡✐❜♥✐③✱ ♥♦ sé❝✉❧♦ ❳❱■■✳

❍❡♥r✐ ▲é♦♥ ▲❡❜❡s❣✉❡ ✭❇❡❛✉✈❛✐s✱ ✷✽ ❞❡ ❥✉♥❤♦ ❞❡ ✶✽✼✺ ✖ P❛r✐s✱ ✷✻ ❞❡ ❥✉❧❤♦ ❞❡ ✶✾✹✶✮ ❢♦✐ ✉♠ ♠❛t❡♠át✐❝♦ ❢r❛♥❝ês✳ ❊st✉❞♦✉ ❞❡ ✶✽✾✹ ❛ ✶✽✾✼ ♥❛ ❊s❝♦❧❛ ◆♦r♠❛❧ ❙✉♣❡r✐♦r ❞❡ P❛r✐s ❡ ❢♦✐ ♣r♦❢❡ss♦r ♥♦ ▲②❝é❡ ❍❡♥r✐✲P♦✐♥❝❛ré ❞❡ ◆❛♥❝②✳ ▲á ❡❧❡ ❞❡s❝♦✲ ❜r✐✉ ❛ ✐♥t❡❣r❛❧ q✉❡ ❧❡✈❛ s❡✉ ♥♦♠❡ ✭❙✉r ✉♥❡ ❣é♥ér❛❧✐s❛t✐♦♥ ❞❡ ❧✬✐♥té❣r❛❧❡ ❞é✜♥✐❡✱ ❈♦♠♣t❡s ❘❡♥❞✉s ✶✾✵✶✮✳ ❆♣ós ♦ ❞♦✉t♦r❛❞♦ ❡♠ ✶✾✵✷ ✭■♥té❣r❛❧❡✱ ▲♦♥❣✉❡✉r✱ ❆✐r❡✱ ❆♥♥❛❧✐ ❞✐ ▼❛t❤❡♠❛t✐❝❛✮✱ ❢♦✐ ♣r♦❢❡ss♦r ❡♠ ❘❡♥♥❡s✳ ❊♠ ✶✾✵✻ ♦❜t❡✈❡ ✉♠❛ ❝át❡❞r❛ ❡♠ ♠❡❝â♥✐❝❛ ❡♠ P♦✐t✐❡rs✳ ❊♠ r❡❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ s❡✉ tr❛❜❛❧❤♦ ♠✐♥✐str♦✉ ♥❡st❡ ♠❡✐♦ t❡♠♣♦ ❝✉rs♦s ♥♦ ❈♦❧❧è❣❡ ❞❡ ❋r❛♥❝❡✱ ❞♦s q✉❛✐s r❡s✉❧t❛r❛♠ ♦s ❧✐✈r♦s ▲❡ç♦♥s s✉r ❧✬✐♥té❣r❛t✐♦♥ ❡t ❧❛ r❡❝❤❡r❝❤❡ ❞❡s ❢♦♥❝t✐♦♥s ♣r✐♠✐t✐✈❡s ✭✶✾✵✹✮ ❡ ▲❡ç♦♥s s✉r ❧❡s sér✐❡s tr✐❣♦♥♦♠étr✐q✉❡s ✭✶✾✵✻✮✳ ❊♠ ✶✾✶✵ ❢♦✐ ♣r♦❢❡ss♦r ❛ss✐st❡♥t❡ ♥❛ ❙♦r❜♦♥♥❡✱ ♦♥❞❡ ♦❜t❡✈❡ ❛ ❝át❡❞r❛ ❡♠ ✶✾✶✽✳ ❆ ♣❛rt✐r ❞❡ ✶✾✷✶ ❢♦✐ ♣r♦❢❡ss♦r ♥♦ ❈♦❧❧è❣❡ ❞❡ ❋r❛♥❝❡✳

▲❡❜❡s❣✉❡ ❣❡♥❡r❛❧✐③♦✉ ♦ ❝♦♥❝❡✐t♦ ❞❡ ✐♥t❡❣r❛❧✱ ✐♥tr♦❞✉③✐♥❞♦ ❛ss✐♠ ♦ ❝♦♥❝❡✐t♦ ❞❡ ♠❡❞✐❞❛✳ ▲❡✈❛♠ s❡✉ ♥♦♠❡ ❛ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ ❡ ❛ ✐♥t❡❣r❛❧ ❞❡ ▲❡❜❡s❣✉❡✳ ❆ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ ❣❡♥❡r❛❧✐③❛ ❛s ♠❡❞✐❞❛s ❛♥t❡r✐♦r♠❡♥t❡ ✉s❛❞❛s✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦ ❛ ♠❡❞✐❞❛ ❞❡ ❏♦r❞❛♥✱ ❡ t♦r♥♦✉✲s❡ ❧♦❣♦ ❡♠ s❡❣✉✐❞❛✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ❛ ✐♥t❡❣r❛❧ ❞❡ ▲❡❜❡s❣✉❡✱ ✉♠❛ ❢❡rr❛♠❡♥t❛ ♣❛❞rã♦ ❞❛ ❆♥á❧✐s❡ ❘❡❛❧✳ ❆ ✐♠♣♦rtâ♥❝✐❛ ❞❛s ✐❞é✐❛s ❞❡ ▲❡❜❡s❣✉❡ r❡s✐❞❡♠ ♥♦ ❢❛t♦ ❞❡ q✉❡ s✉❛ t❡♦r✐❛ ❞❛ ✐♥t❡❣r❛çã♦ ✭✐♥t❡❣r❛❧ ❞❡ ▲❡❜❡s❣✉❡✮ ♣♦ss✉✐ ✉♠❛ sér✐❡ ❞❡ ❝❛r❛❝t❡ríst✐❝❛s ♣rát✐❝❛s q✉❡ ❢❛❧t❛♠ à ✐♥t❡❣r❛❧ ❞❡ ❘✐❡♠❛♥♥✳

❈❛♣ít✉❧♦ ✷

➪r❡❛ ❞❡ ✜❣✉r❛s ♣❧❛♥❛s ❡❧❡♠❡♥t❛r❡s

◆❡ss❡ ❝❛♣ít✉❧♦ ✐r❡♠♦s tr❛t❛r ❞❡ ♠❡❞✐r ❛ ♣♦rçã♦ ❞♦ ♣❧❛♥♦ ♦❝✉♣❛❞❛ ♣♦r ✉♠❛ ✜❣✉r❛ ♣❧❛♥❛ F✳ P❛r❛ ✐ss♦✱ ❝♦♠♣❛r❛r❡♠♦s F ❝♦♠ ✉♠❛ ✉♥✐❞❛❞❡ ❞❡ ár❡❛✳ ❖ r❡s✉❧t❛❞♦ ❞❡ss❛ ❝♦♠♣❛r❛çã♦ s❡rá ✉♠ ♥ú♠❡r♦ q✉❡ ✐rá ❡①♣r✐♠✐r q✉❛♥t❛s ✈❡③❡s ❛ ✜❣✉r❛ F ❝♦♥té♠ ❛ ✉♥✐❞❛❞❡ ❞❡ ár❡❛✳ ❆q✉✐ ❞❛r❡♠♦s ✉♠ s✐❣♥✐✜❝❛❞♦ ♣r❡❝✐s♦ ❛ ❡st❛ ✐❞é✐❛ ❡ ❡st❛❜❡❧❡❝❡r❡♠♦s ❛s ❢ór♠✉❧❛s ♣❛r❛ ❛ ár❡❛ ❞❛s ✜❣✉r❛s ❣❡♦♠étr✐❝❛s ♠❛✐s ❝♦♥❤❡❝✐❞❛s✳ Pr✐♥❝✐♣❛✐s r❡❢❡rê♥❝✐❛s ❜✐❜❧✐♦❣rá✜❝❛s ✉t✐❧✐③❛❞❛s✿ ❬✸❪ ❡ ❬✹❪✳

✷✳✶ ➪r❡❛ ❞♦ q✉❛❞r❛❞♦ ❡ ❞♦ r❡tâ♥❣✉❧♦

❖ q✉❛❞r❛❞♦ é ♦ q✉❛❞r✐❧át❡r♦ q✉❡ ♣♦ss✉✐ ♦s ✹ ❧❛❞♦s ✐❣✉❛✐s ❡ ♦s ✹ â♥❣✉❧♦s r❡t♦s✳ ❈♦♥✲ ✈❡♥❝✐♦♥❛r❡♠♦s ❝♦♠♦ ✉♥✐❞❛❞❡ ❞❡ ár❡❛ ✉♠ q✉❛❞r❛❞♦ ❝✉❥♦ ❧❛❞♦ ♠❡❞❡ ✉♠❛ ✉♥✐❞❛❞❡ ❞❡ ❝♦♠♣r✐♠❡♥t♦✳ ❱❛♠♦s ❝❤❛♠á✲❧♦ ♦ q✉❛❞r❛❞♦ ✉♥✐tár✐♦✳

◗✉❛❧q✉❡r q✉❛❞r❛❞♦ ❝✉❥♦ ❧❛❞♦ ♠❡ç❛ ✉♠❛ ✉♥✐❞❛❞❡ ❞❡ ❝♦♠♣r✐♠❡♥t♦ t❡rá✱ ♣♦r ❞❡✜♥✐çã♦✱ ár❡❛ ✐❣✉❛❧ ❛ ✉♠❛ ✉♥✐❞❛❞❡✳

P♦r ♠❡✐♦ ❞❡ ♣❛r❛❧❡❧❛s ❛♦s s❡✉s ❧❛❞♦s✱ ♣♦❞❡♠♦s ❞❡❝♦♠♣♦r ✉♠ q✉❛❞r❛❞♦Q✱ ❝✉❥♦ ❧❛❞♦ t❡♠ ❝♦♠♦ ♠❡❞✐❞❛ ♦ ♥ú♠❡r♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦n✱ ❡♠ n2 q✉❛❞r❛❞♦s ❥✉st❛♣♦st♦s✱ ❝❛❞❛ ✉♠ ❞❡❧❡s ❞❡ ❧❛❞♦ ✉♥✐tár✐♦ ❡ ♣♦rt❛♥t♦ ❝♦♠ ár❡❛ ✶✳ ❆ss✐♠ ♦ q✉❛❞r❛❞♦Q t❡rá ár❡❛ n2_✳

❉❡ ♠♦❞♦ ❛♥á❧♦❣♦✱ s❡ ♦ ❧❛❞♦ ❞❡ ✉♠ q✉❛❞r❛❞♦ Q t❡♠ ♣♦r ♠❡❞✐❞❛ 1/n✱ ♦♥❞❡ n é ✐♥t❡✐r♦ ♣♦s✐t✐✈♦✱ ❡♥tã♦ ♣♦❞❡♠♦s ❞❡❝♦♠♣♦r ♦ q✉❛❞r❛❞♦ ✉♥✐tár✐♦✱ ♠❡❞✐❛♥t❡ ♣❛r❛❧❡❧❛s ❛♦s s❡✉s ❧❛❞♦s✱ ❡♠ n2 _{q✉❛❞r❛❞♦s ❥✉st❛♣♦st♦s✱ t♦❞♦s ❝♦♥❣r✉❡♥t❡s ❛ Q✳} ❊st❡s n2 q✉❛❞r❛❞♦s ❝♦♥❣r✉❡♥t❡s ❛ _Q ❝♦♠♣õ❡♠ ✉♠ q✉❛❞r❛❞♦ ❞❡ ár❡❛₁✳ ❙❡❣✉❡✲s❡ q✉❡ ❛ ár❡❛ ❞❡ Q ❞❡✈❡ s❛t✐s❢❛③❡r à ❝♦♥❞✐çã♦ n2 _×[ár❡❛ ❞❡ _{Q] = 1} ❡✱ ♣♦rt❛♥t♦✱ [ár❡❛ ❞❡Q] = 1/n2✳

P♦rt❛♥t♦✱ s❡ ♦ ❧❛❞♦ ❞❡ ✉♠ q✉❛❞r❛❞♦ Q t❡♠ ♣♦r ♠❡❞✐❞❛ ♦ ♥ú♠❡r♦ r❛❝✐♦♥❛❧ m/n ♣♦s✐t✐✈♦✱ ❡♥tã♦ ♣♦❞❡♠♦s ❞❡❝♦♠♣♦r ❝❛❞❛ ❧❛❞♦ ❞❡ Q ❡♠ m s❡❣♠❡♥t♦s✱ ❝❛❞❛ ✉♠ ❞♦s q✉❛✐s ❞❡ ❝♦♠♣r✐♠❡♥t♦ 1/n✳ ❚r❛ç❛♥❞♦ ♣❛r❛❧❡❧❛s ❛♦s ❧❛❞♦s ❞❡ Q ❛ ♣❛rt✐r ❞♦s ♣♦♥t♦s ❞❡ ❞✐✈✐sã♦✱ ♦❜t❡♠♦s ✉♠❛ ❞❡❝♦♠♣♦s✐çã♦ ❞❡ Q❡♠ m2 _{q✉❛❞r❛❞♦s✱ ❝❛❞❛} ✉♠ ❞♦s q✉❛✐s ❝♦♠ ❧❛❞♦1/n✳ ❆ss✐♠✱ ❛ ár❡❛ ❞❡ ❝❛❞❛ ✉♠ ❞❡ss❡s q✉❛❞r❛❞♦s ♠❡♥♦r❡s é 1/n2_{✳ ❙❡❣✉❡✲s❡ q✉❡ ❛ ár❡❛ ❞❡ Q❞❡✈❡ s❡r}

m2_· 1 n2 =

m2 n2.

♦✉ s❡❥❛✱

[ár❡❛ ❞❡ Q] =m n

2

.

P♦❞❡♠♦s ❡♥tã♦ ❝♦♥❝❧✉✐r q✉❡ ❛ ár❡❛ ❞❡ ✉♠ q✉❛❞r❛❞♦ Q ❝✉❥♦ ❧❛❞♦ t❡♠ ♣❛r❛ ♠❡❞✐❞❛ ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧ a=m/né ❞❛❞❛ ♣❡❧❛ ❡①♣r❡ssã♦✿

[ár❡❛ ❞❡ Q] =a2_.

▼❛s ❡①✐st❡♠ q✉❛❞r❛❞♦s ❝✉❥♦s ❧❛❞♦s sã♦ ✐♥❝♦♠❡♥s✉rá✈❡✐s ❝♦♠ ♦ s❡❣♠❡♥t♦ ✉♥✐tár✐♦✳ ❙❡❥❛ Q ✉♠ ❞❡ss❡s✿ ♦ ❧❛❞♦ ❞❡ Q t❡♠ ❝♦♠♦ ♠❡❞✐❞❛ ♦ ♥ú♠❡r♦ ✐rr❛❝✐♦♥❛❧ a✳ ▼♦str❛r❡♠♦s ❛❣♦r❛ q✉❡✱ ❛✐♥❞❛ ♥❡st❡ ❝❛s♦✱ ❞❡✈❡✲s❡ t❡r [ár❡❛ ❞❡Q] =a2_.

❱❛♠♦s r❛❝✐♦❝✐♥❛r ❞❡ ♠♦❞♦ ✐♥❞✐r❡t♦✳ ❉❛❞♦ q✉❛❧q✉❡r ♥ú♠❡r♦b < a2✱ ♠♦str❛r❡♠♦s q✉❡ ❞❡✈❡ s❡r b < [ár❡❛ ❞❡Q]✳ ❊♠ s❡❣✉✐❞❛✱ ♣r♦✈❛r❡♠♦s q✉❡ a2 _{< c} ✐♠♣❧✐❝❛ ❡♠ [ár❡❛ ❞❡Q] < c✳ ■st♦ ♠♦str❛rá q✉❡ ❛ ár❡❛ ❞❡ Q ♥ã♦ ♣♦❞❡ s❡r ✉♠ ♥ú♠❡r♦ b ♠❡♥♦r ♥❡♠ ✉♠ ♥ú♠❡r♦ c ♠❛✐♦r ❞♦ q✉❡ a2_. P♦rt❛♥t♦✱ ❝♦♥❝❧✉✐r❡♠♦s q✉❡ ❛ [ár❡❛ ❞❡Q] = a2_{✳ ❱❛♠♦s ❞❡♠♦♥str❛r ❛ ♣r✐♠❡✐r❛ ♣❛rt❡ ❞❡st❛ ❛✜r♠❛çã♦✳ ❆ s❡✲} ❣✉♥❞❛ ♣❛rt❡ é ✐♥t❡✐r❛♠❡♥t❡ ❛♥á❧♦❣❛ ❡ ♣♦r ✐ss♦ s❡rá ♦♠✐t✐❞❛✳

❙❡❥❛✱ ♣♦✐s✱ b ✉♠ ♥ú♠❡r♦ t❛❧ q✉❡ b < a2_{✳ ❚♦♠❛♠♦s ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧ r✱} ✐♥❢❡r✐♦r ❛ a✱ ♣♦ré♠✱ tã♦ ♣ró①✐♠♦ ❞❡ a q✉❡ s❡ t❡♥❤❛ b < r2 _{< a}2 _{✭❜❛st❛ t♦♠❛r r✱} ✉♠❛ ❛♣r♦①✐♠❛çã♦ ♣♦r ❢❛❧t❛ ❞❡ a✱ ❝♦♠ ❡rr♦ ✐♥❢❡r✐♦r ❛a₋√b✳ ❊♥tã♦ √b < r < a ❡ ♣♦rt❛♥t♦ b < r2 _{< a}2✮✳

◆♦ ✐♥t❡r✐♦r ❞❡ Q✱ t♦♠❛♠♦s ✉♠ q✉❛❞r❛❞♦ Q′ ❞❡ ❧❛❞♦ _r✳ ❙❡♥❞♦ _r r❛❝✐♦♥❛❧✱ ❛ ár❡❛ ❞❡st❡ q✉❛❞r❛❞♦ é r2_{✳ ❝♦♠♦ Q}′ ❡stá ❝♦♥t✐❞♦ ♥♦ ✐♥t❡r✐♦r ❞❡ _Q✱ ❞❡✈❡♠♦s t❡r [ár❡❛ ❞❡Q′_{] < [}ár❡❛ ❞❡ _Q]✱ ♦✉ s❡❥❛ _r2 _{< [ár❡❛ ❞❡Q]✳ ▼❛s s❛❜❡♠♦s q✉❡ b < r}2_✳ ❈♦♥❝❧✉sã♦✿ b <[ár❡❛ ❞❡ Q]✳ ❆ss✐♠✱ t♦❞♦ ♥ú♠❡r♦ r❡❛❧ b✱ ✐♥❢❡r✐♦r ❛ a2_{✱ é t❛♠❜é♠} ♠❡♥♦r ❞♦ q✉❡ ❛ ár❡❛ ❞❡ Q✳ ❉❛ ♠❡s♠❛ ♠❛♥❡✐r❛ s❡ ♣r♦✈❛ q✉❡ t♦❞♦ ♥ú♠❡r♦ r❡❛❧c✱ ♠❛✐♦r ❞♦ q✉❡ a2✱ é ♠❛✐♦r ❞♦ q✉❡ ❛ ár❡❛ ❞❡ _Q✳ ▲♦❣♦✱ ❛ ár❡❛ ❞❡ _Q ♥ã♦ ♣♦❞❡ s❡r ♠❡♥♦r ♥❡♠ ♠❛✐♦r ❞♦ q✉❡ a2✳

❈♦♥❝❧✉í♠♦s✱ ❞❡st❛ ♠❛♥❡✐r❛✱ q✉❡ ❛ ár❡❛ ❞❡ ✉♠ q✉❛❞r❛❞♦ Q✱ ❝✉❥♦ ❧❛❞♦ ♠❡❞❡ a✱ ❞❡✈❡ s❡r ❡①♣r❡ss❛ ♣❡❧❛ ❢ór♠✉❧❛

[ár❡❛ ❞❡ Q] =a2_.

◆❛ ❢ór♠✉❧❛ ❛❝✐♠❛✱ a é ✉♠ ♥ú♠❡r♦ r❡❛❧ ♣♦s✐t✐✈♦ q✉❛❧q✉❡r✿ ✐♥t❡✐r♦✱ ❢r❛❝✐♦♥ár✐♦ ♦✉ ✐rr❛❝✐♦♥❛❧✳

❈♦♥s✐❞❡r❡♠♦s ❛❣♦r❛ ❛ ár❡❛ ❞♦ r❡tâ♥❣✉❧♦✳ ❖ r❡tâ♥❣✉❧♦ é ♦ q✉❛❞r✐❧át❡r♦ q✉❡ t❡♠ ♦s q✉❛tr♦ â♥❣✉❧♦s r❡t♦s✳

❙❡ ♦s ❧❛❞♦s ❞❡ ✉♠ r❡tâ♥❣✉❧♦ R tê♠ ♣❛r❛ ♠❡❞✐❞❛s ♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s m ❡ n✱ ❡♥tã♦✱ ♠❡❞✐❛♥t❡ ♣❛r❛❧❡❧❛s ❛♦s ❧❛❞♦s✱ ♣♦❞❡♠♦s ❞❡❝♦♠♣♦rR ❡♠ mn q✉❛❞r❛❞♦s ✉♥✐tár✐♦s✱ ❞❡ ♠♦❞♦ q✉❡ s❡ ❞❡✈❡ t❡r[ár❡❛ ❞❡ R] =mn✳

▼❛✐s ❣❡r❛❧♠❡♥t❡✱ s❡ ♦s ❧❛❞♦s ❞♦ r❡tâ♥❣✉❧♦ R tê♠ ❝♦♠♦ ♠❡❞✐❞❛s ❞♦✐s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s a ❡ b✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❡st❡s ♥ú♠❡r♦s ❝♦♠♦ ❞✉❛s ❢r❛çõ❡s a = p/q ❡ b =r/q✱ ❝♦♠ ♦ ♠❡s♠♦ ❞❡♥♦♠✐♥❛❞♦r q✳ ❉✐✈✐❞✐♠♦s ❝❛❞❛ ❧❛❞♦ ❞❡ R ❡♠ s❡❣♠❡♥t♦s ❞❡ ❝♦♠♣r✐♠❡♥t♦ ✶✴q✳ ❖ ❧❛❞♦ q✉❡ ♠❡❞❡ a ✜❝❛rá ❞❡❝♦♠♣♦st♦ ❡♠ p s❡❣♠❡♥t♦s ❥✉st❛♣♦st♦s✱ ❝❛❞❛ ✉♠ ❞❡❧❡s ♠❡❞✐♥❞♦ 1/q✳ ❖ ❧❛❞♦ q✉❡ ♠❡❞❡ b ✜❝❛rá s✉❜❞✐✈✐❞✐❞♦ ❡♠rs❡❣♠❡♥t♦s ✐❣✉❛✐s✱ ❞❡ ❝♦♠♣r✐♠❡♥t♦1/q✳ ❚r❛ç❛♥❞♦ ♣❛r❛❧❡❧❛s ❛♦s ❧❛❞♦s ❛ ♣❛rt✐r ❞♦s ♣♦♥t♦s ❞❡ s✉❜❞✐✈✐sã♦✱ ♦ r❡tâ♥❣✉❧♦R ✜❝❛rá s✉❜❞✐✈✐❞✐❞♦ ❡♠p·rq✉❛❞r❛❞♦s✱ ❝❛❞❛ ✉♠ ❞❡❧❡s ❞❡ ❧❛❞♦ 1/q✳ ❆ ár❡❛ ❞❡ ❝❛❞❛ ✉♠ ❞❡ss❡s q✉❛❞r❛❞✐♥❤♦s é (1/q)2 = 1/q2✳ ▲♦❣♦ ❛ ár❡❛ ❞❡ R ❞❡✈❡rá s❡r ✐❣✉❛❧ ❛

(pr)· 1

q2 = pr q2 =

p q ·

r q ♦✉ s❡❥❛✱ [ár❡❛ ❞❡ R] =ab✳

❱❡♠♦s ❛ss✐♠ q✉❡✱ q✉❛♥❞♦ ♦s ❧❛❞♦s ❞❡ ✉♠ r❡tâ♥❣✉❧♦ R tê♠ ♣♦r ♠❡❞✐❞❛s ♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s a ❡ b✱ ❛ ár❡❛ ❞❡ R é ❡①♣r❡ss❛ ♣❡❧❛ ❢ór♠✉❧❛✿

[ár❡❛ ❞❡ R] =ab✳

❉✐③✲s❡✱ ❡♥tã♦✱ q✉❡ ❛ ár❡❛ ❞♦ r❡tâ♥❣✉❧♦ é ♦ ♣r♦❞✉t♦ ❞❛ ❜❛s❡ ♣❡❧❛ ❛❧t✉r❛✳ ■st♦ ❢♦✐ ♠♦str❛❞♦ ❛❝✐♠❛ ❛♣❡♥❛s q✉❛♥❞♦ a ❡ b sã♦ ♥ú♠❡r♦s r❛❝✐♦♥❛✐s✱ ♠❛s é ✉♠❛ ❢ór♠✉❧❛ ❣❡r❛❧✱ ✈á❧✐❞❛ ♠❡s♠♦ q✉❡ ♦s ♥ú♠❡r♦s a ❡ b s❡❥❛♠ ✐rr❛❝✐♦♥❛✐s ✭♦✉ ✉♠ ❞❡❧❡s é r❛❝✐♦♥❛❧ ❡ ♦ ♦✉tr♦ ✐rr❛❝✐♦♥❛❧✮✳

❱❛♠♦s ♣r♦✈❛r ✐ss♦✱ ✉s❛♥❞♦ ✉♠ ❛rt✐❢í❝✐♦ s✐♠♣❧❡s ❡ ❡❧❡❣❛♥t❡✱ ❢❛③❡♥❞♦ r❡❝❛✐r ❛ ár❡❛ ❞♦ r❡tâ♥❣✉❧♦ ♥❛ ár❡❛ ❞♦ q✉❛❞r❛❞♦✳ Pr♦❝❡❞❡♥❞♦ ❛ss✐♠✱ ✜❝❛♠♦s ✐♥❝❧✉s✐✈❡ ❞✐s♣❡♥s❛❞♦s ❞❡ ❝♦♥s✐❞❡r❛r s❡♣❛r❛❞❛♠❡♥t❡ ♦ ❝❛s♦ ❡♠ q✉❡ ❛ ❜❛s❡ ❡ ❛ ❛❧t✉r❛ tê♠ ♠❡❞✐❞❛s r❛❝✐♦♥❛✐s✳

❉❛❞♦ ♦ r❡tâ♥❣✉❧♦ R✱ ❞❡ ❜❛s❡ b ❡ ❛❧t✉r❛ a✱ ❝♦♥str✉ír❡♠♦s ♦ q✉❛❞r❛❞♦ Q✱ ❞❡ ❧❛❞♦ a+b✱ ♦ q✉❛❧ ❝♦♥té♠ ✷ ❝ó♣✐❛s ❞❡ R ❡ ♠❛✐s ❞♦✐s q✉❛❞r❛❞♦s✱ ✉♠ ❞❡ ❧❛❞♦ a ❡ ♦✉tr♦ ❞❡ ❧❛❞♦ b✳ ❈♦♠♦ s❛❜❡♠♦s✱

[ár❡❛ ❞❡ Q] = (a+b)2 =a2_{+ 2ab+b}2_.

P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ ♦s q✉❛❞r❛❞♦s ♠❡♥♦r❡s tê♠ ár❡❛s ✐❣✉❛✐s ❛ a2 ❡_b2 r❡s♣❡❝✲ t✐✈❛♠❡♥t❡✱ t❡♠♦s

[ár❡❛ ❞❡Q] =a2_+b2_{+ 2×[ár❡❛ ❞❡ R].} ❙❡❣✉❡✲s❡ q✉❡ [ár❡❛ ❞❡ R] =ab✳

✷✳✷ ➪r❡❛ ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ❡ ❞♦ tr✐â♥❣✉❧♦

❉❛ ár❡❛ ❞♦ r❡tâ♥❣✉❧♦✱ ♣❛ss❛✲s❡ ❢❛❝✐❧♠❡♥t❡ ♣❛r❛ ❛ ár❡❛ ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦✳ ❯♠ ♣❛r❛❧❡❧♦❣r❛♠♦ é ✉♠ q✉❛❞r✐❧át❡r♦ ♥♦ q✉❛❧ ♦s ❧❛❞♦s ♦♣♦st♦s sã♦ ♣❛r❛❧❡❧♦s✳

◗✉❛♥❞♦ s❡ t♦♠❛ ✉♠ ❧❛❞♦ ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ❝♦♠♦ ❜❛s❡✱ ❝❤❛♠❛✲s❡ ❛❧t✉r❛ ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ❛ ✉♠ s❡❣♠❡♥t♦ ♣❡r♣❡♥❞✐❝✉❧❛r q✉❡ ❧✐❣❛ ❛ ❜❛s❡ ❛♦ ❧❛❞♦ ♦♣♦st♦ ✭♦✉ ❛♦ s❡✉ ♣r♦❧♦♥❣❛♠❡♥t♦✮✳

❙❡❥❛ABDC ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦ ❝✉❥❛ ár❡❛ S q✉❡r❡♠♦s ❝❛❧❝✉❧❛r✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ❛ s✉❛ ❜❛s❡ AB t❡♠ ❝♦♠♣r✐♠❡♥t♦ b ❡ s✉❛ ❛❧t✉r❛DE t❡♠ ❝♦♠♣r✐♠❡♥t♦ a✳

❖ ♣❛r❛❧❡❧♦❣r❛♠♦ ABDC ❡stá ❝♦♥t✐❞♦ ♥✉♠ r❡tâ♥❣✉❧♦ ❞❡ ❜❛s❡ b+c ❡ ❛❧t✉r❛ a✳ ❈♦♠♦ ✈✐♠♦s✱ ❛ ár❡❛ ❞❡ss❡ r❡tâ♥❣✉❧♦ é (b+c)·a = ba+ca✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ♦ r❡tâ♥❣✉❧♦ é ❢♦r♠❛❞♦ ♣❡❧♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ❞❛❞♦ ♠❛✐s ❞♦✐s tr✐â♥❣✉❧♦s q✉❡✱ ❥✉♥t♦s✱ ❢♦r♠❛♠ ✉♠ r❡tâ♥❣✉❧♦ ❞❡ ár❡❛ca✳ P♦rt❛♥t♦ba+ca=S+ca✱ ❞❡ ♦♥❞❡ ❝♦♥❝❧✉✐♠♦s q✉❡ S =ba✳

❆ss✐♠✱ ❛ ár❡❛ ❞❡ ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦ é ✐❣✉❛❧ ❛♦ ♣r♦❞✉t♦ ❞♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ q✉❛❧q✉❡r ✉♠❛ ❞❡ s✉❛s ❜❛s❡s ♣❡❧♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ❛❧t✉r❛ ❝♦rr❡s♣♦♥❞❡♥t❡✳

❊♠ ♣❛rt✐❝✉❧❛r✱ ✈❡♠♦s q✉❡ ♦ ♣r♦❞✉t♦ ❞♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ q✉❛❧q✉❡r ❜❛s❡ ❞❡ ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦ ♣❡❧♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ❛❧t✉r❛ ❝♦rr❡s♣♦♥❞❡♥t❡ é ❝♦♥st❛♥t❡ ✭♥ã♦ ❞❡♣❡♥❞❡ ❞❛ ❜❛s❡ ❡s❝♦❧❤✐❞❛✮✳

❱❡♠♦s t❛♠❜é♠ q✉❡✱ ❞❛❞❛s ❛s r❡t❛s ♣❛r❛❧❡❧❛s r✱ s ❡ ♦ s❡❣♠❡♥t♦ AB s♦❜r❡ r✱ t♦❞♦s ♦s ♣❛r❛❧❡❧♦❣r❛♠♦s ABDC✱ ❝♦♠ C ❡ Ds♦❜r❡ ❛ r❡t❛ s✱ tê♠ ❛ ♠❡s♠❛ ár❡❛✳

❉❛ ár❡❛ ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦✱ ♣❛ss❛✲s❡ ✐♠❡❞✐❛t❛♠❡♥t❡ ♣❛r❛ ❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦✱ ♣♦✐s t♦❞♦ tr✐â♥❣✉❧♦ é ❛ ♠❡t❛❞❡ ❞❡ ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦✳

▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ ❞❛❞♦ ✉♠ tr✐â♥❣✉❧♦ ABC✱ ❝✉❥❛ ár❡❛ ❞❡s❡❥❛♠♦s ❝❛❧❝✉❧❛r✱ tr❛ç❛♠♦s✱ ♣❡❧♦s ✈ért✐❝❡s C ❡ B✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♣❛r❛❧❡❧❛s ❛♦s ❧❛❞♦s AB ❡ AC✳ ❊st❛s r❡t❛s s❡ ❡♥❝♦♥tr❛♠ ♥♦ ♣♦♥t♦ D ❡ ❢♦r♥❡❝❡♠ ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦ ABDC✳ ❚♦♠❡♠♦s ❛ ❛❧t✉r❛ CE ❞❡st❡ ♣❛r❛❧❡❧♦❣r❛♠♦✳ ❙❡ AB =b ❡ CE =a✱ s❛❜❡♠♦s q✉❡ ❛[ár❡❛ ❞❡ABDC] =ba✳ ❙❡♥❞♦ ❛ss✐♠✱ ♦s tr✐â♥❣✉❧♦sABC❡BCDsã♦ ❝♦♥❣r✉❡♥t❡s ✭tê♠ ✉♠ ❧❛❞♦ ❝♦♠✉♠ ❝♦♠♣r❡❡♥❞✐❞♦ ❡♥tr❡ ❞♦✐s â♥❣✉❧♦s ✐❣✉❛✐s✮✱ ❧♦❣♦ tê♠ ❛ ♠❡s♠❛ ár❡❛✳

P♦rt❛♥t♦✱ [ár❡❛ ❞❡ABDC] = 2_×[ár❡❛ ❞❡ ABC] ❡ ♣♦r ❝♦♥s❡❣✉✐♥t❡✿ [ár❡❛ ❞❡ABC] = 1

2ba .

■st♦ s❡ ❡①♣r✐♠❡ ❞✐③❡♥❞♦ q✉❡ ❛ ár❡❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ é ❛ ♠❡t❛❞❡ ❞♦ ♣r♦❞✉t♦ ❞❡ ✉♠❛ ❜❛s❡ ♣❡❧❛ ❛❧t✉r❛ ❝♦rr❡s♣♦♥❞❡♥t❡✳

◆✉♠ tr✐â♥❣✉❧♦✱ t❡♠♦s três ❡s❝♦❧❤❛s ♣❛r❛ ❛ ❜❛s❡ b ❡✱ ♣♦rt❛♥t♦✱ três ❡s❝♦❧❤❛s ♣❛r❛ ❛ ❛❧t✉r❛ a✳ ❙❡❥❛ q✉❛❧ ❢♦r ❛ ❡s❝♦❧❤❛✱ ♦ ♣r♦❞✉t♦ ba s❡rá ♦ ♠❡s♠♦✱ ♣♦✐s✱ ❡♠ ❝❛❞❛ ❝❛s♦ ❡❧❡ ❢♦r♥❡❝❡ ♦ ❞♦❜r♦ ❞❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦✳

❙❡❥❛♠ r ❡s r❡t❛s ♣❛r❛❧❡❧❛s ❡ b ✉♠ ♥ú♠❡r♦ r❡❛❧ ♣♦s✐t✐✈♦✳ ❙❡❣✉❡✲s❡ ❞❛ ❢ór♠✉❧❛ ❛❝✐♠❛ q✉❡ t♦❞♦s ♦s tr✐â♥❣✉❧♦s ABC ❝♦♠ ✈ért✐❝❡ A s♦❜r❡ r✱ ❜❛s❡ BC s♦❜r❡ s ❡ BC =b✱ tê♠ ❛ ♠❡s♠❛ ár❡❛✳

P❛r❛ ✉♠ ♣♦❧í❣♦♥♦ q✉❛❧q✉❡r✱ ♦ ♣r♦❝❡ss♦ ❞❡ ❝❛❧❝✉❧❛r s✉❛ ár❡❛ ✭❡s♣❛ç♦ q✉❡ ❛ ✜❣✉r❛ ♦❝✉♣❛ ♥♦ ♣❧❛♥♦✮ ❝♦♥s✐st❡ ❡♠ s✉❜❞✐✈✐❞✐✲❧♦ ❡♠ tr✐â♥❣✉❧♦s✱ ♣❛r❛❧❡❧♦❣r❛♠♦s ♦✉ q✉❛✐sq✉❡r ♦✉tr❛s ✜❣✉r❛s ❝✉❥❛s ár❡❛s s❛❜❡♠♦s ❝❛❧❝✉❧❛r✳ ❆ ár❡❛ ❞♦ ♣♦❧í❣♦♥♦ ♣r♦❝✉r❛❞❛ s❡rá ❛ s♦♠❛ ❞❛s ár❡❛s ❞❛s ✜❣✉r❛s ❡♠ q✉❡ ♦ ❞❡❝♦♠♣✉s❡♠♦s✳

P♦r ❡①❡♠♣❧♦✱ ❝♦♥s✐❞❡r❡ ❛ ♠❛❧❤❛ q✉❛❞r✐❝✉❧❛❞❛ ♦♥❞❡ ❝❛❞❛ q✉❛❞r❛❞♦ t❡♠ ❧❛✲ ❞♦s ♠❡❞✐♥❞♦ ✉♠❛ ✉♥✐❞❛❞❡ ❞❡ ❝♦♠♣r✐♠❡♥t♦ ✭❡ ♣♦rt❛♥t♦ ár❡❛ ✉♥✐tár✐❛✮ ❡ s❡❥❛ ♦ ❤❡①á❣♦♥♦ ABCDEF ♣❛r❛ ♦ q✉❛❧ q✉❡r❡♠♦s ❝❛❧❝✉❧❛r s✉❛ ár❡❛✳

❙❡❥❛ S ♦ ✈❛❧♦r ❛ s❡r ❝❛❧❝✉❧❛❞♦✳ ❆ss✐♠✱

S = [ár❡❛ ❞❡ ABDE] + [ár❡❛ ❞❡ AF E] + [ár❡❛ ❞❡ BCD]

= 2_·6 + 3·6 2 +

3_·6

2 = 12 + 9 + 9 = 30.

P♦❞❡rí❛♠♦s t❛♠❜é♠ t❡r ❝❛❧❝✉❧❛❞♦S ♣♦r ♠❡✐♦ ❞❛s ár❡❛s ❞♦s q✉❛❞r❛❞♦s ✉♥✐tár✐♦s ❡ tr✐â♥❣✉❧♦s ❞❡ ❜❛s❡s ❡ ❛❧t✉r❛s ✐❣✉❛✐s ❛ ✶✳ ❱❡❥❛ q✉❡ ♦ ❤❡①á❣♦♥♦ABCDEF ❝♦♥té♠ ✷✹ q✉❛❞r❛❞♦s ✉♥✐tár✐♦s ❡ ✶✷ tr✐â♥❣✉❧♦s ❞❡ ❜❛s❡ ❡ ❛❧t✉r❛ ❝♦♠ ♠❡❞✐❞❛s ✐❣✉❛✐s ❛ ✶✳ ❆ss✐♠✱

S= 12·1·1

2 + 24·1 = 6 + 24 = 30.

✷✳✸ ❉❡✜♥✐çã♦ ❣❡r❛❧ ❞❡ ár❡❛

◆♦s ♣❛rá❣r❛❢♦s ❛♥t❡r✐♦r❡s✱ ♠♦str❛♠♦s q✉❡ s❡ ♣♦❞❡ ❛ss♦❝✐❛r ❛ ❝❛❞❛ ♣♦❧í❣♦♥♦ P ✉♠ ♥ú♠❡r♦ r❡❛❧ ♥ã♦✲♥❡❣❛t✐✈♦✱ ❝❤❛♠❛❞♦ ❛ ár❡❛ ❞❡ P✱ ❝♦♠ ❛s s❡❣✉✐♥t❡s ♣r♦✲ ♣r✐❡❞❛❞❡s✿

✶✳ P♦❧í❣♦♥♦s ❝♦♥❣r✉❡♥t❡s tê♠ ár❡❛s ✐❣✉❛✐s❀

✷✳ ❙❡P é ✉♠ q✉❛❞r❛❞♦ ❝♦♠ ❧❛❞♦ ✉♥✐tár✐♦✱ ❡♥tã♦ [ár❡❛ ❞❡P] = 1❀

✸✳ ❙❡P ♣♦❞❡ s❡r ❞❡❝♦♠♣♦st♦ ❝♦♠♦ r❡✉♥✐ã♦ ❞❡ n♣♦❧í❣♦♥♦s✱P1, ..., Pn✱ t❛✐s q✉❡ ❞♦✐s q✉❛✐sq✉❡r ❞❡❧❡s tê♠ ❡♠ ❝♦♠✉♠ ♥♦ ♠á①✐♠♦ ❛❧❣✉♥s ❧❛❞♦s✱ ❡♥tã♦ ❛ ár❡❛ ❞❡P é ❛ s♦♠❛ ❞❛s ár❡❛s ❞♦s Pi✳

❙❡❣✉❡✲s❡ ❞❛ t❡r❝❡✐r❛ ♣r♦♣r✐❡❞❛❞❡ ❛❝✐♠❛ q✉❡ s❡ ♦ ♣♦❧í❣♦♥♦P ❡stá ❝♦♥t✐❞♦ ♥♦ ♣♦❧í✲ ❣♦♥♦ Q ❡♥tã♦ ❛ ár❡❛ ❞❡P é ♠❡♥♦r ❞♦ q✉❡ ❛ ár❡❛ ❞❡Q✳

P❡r❝❡❜❛ q✉❡ ❛s ❢ór♠✉❧❛s ♣❛r❛ ❛s ár❡❛s ❞♦ q✉❛❞r❛❞♦✱ ❞♦ r❡tâ♥❣✉❧♦✱ ❞♦ tr✐â♥❣✉❧♦ ❡ ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦✱ q✉❡ ♦❜t✐✈❡♠♦s ❛❝✐♠❛ ❢♦r❛♠ t♦❞❛s ❞❡❞✉③✐❞❛s ❛ ♣❛rt✐r ❞❡st❛s três ♣r♦♣r✐❡❞❛❞❡s✳

❱❛♠♦s ❞❛r ❛❣♦r❛ ✉♠❛ ❞❡✜♥✐çã♦ ♣❛r❛ ❛ ár❡❛ ❞❡ ✉♠❛ ✜❣✉r❛ ♣❧❛♥❛ F✳

❆ ár❡❛ ❞❡ ✉♠❛ ✜❣✉r❛ ♣❧❛♥❛ F ❞❡✈❡ s❡r ✉♠ ♥ú♠❡r♦ r❡❛❧ ♥ã♦ ♥❡❣❛t✐✈♦✱ q✉❡ ✐♥❞✐❝❛r❡♠♦s ♣♦r a(F). ❊❧❡ s❡rá ❜❡♠ ❞❡t❡r♠✐♥❛❞♦ s❡ ❝♦♥❤❡❝❡r♠♦s s❡✉s ✈❛❧♦r❡s ❛♣r♦①✐♠❛❞♦s✱ ♣♦r ❢❛❧t❛ ♦✉ ♣♦r ❡①❝❡ss♦✳

❖s ✈❛❧♦r❡s ❞❡ a(F) ❛♣r♦①✐♠❛❞♦s ♣♦r ❢❛❧t❛ sã♦✱ ♣♦r ❞❡✜♥✐çã♦✱ ❛s ár❡❛s ❞♦s ♣♦❧í❣♦♥♦s P ❝♦♥t✐❞♦s ❡♠F✳ ❖s ✈❛❧♦r❡s ❞❡ a(F) ❛♣r♦①✐♠❛❞♦s ♣♦r ❡①❝❡ss♦ sã♦ ❛s ár❡❛s ❞♦s ♣♦❧í❣♦♥♦s P′ q✉❡ ❝♦♥té♠ _F✳ ❆ss✐♠✱ q✉❛✐sq✉❡r q✉❡ s❡❥❛♠ ♦s ♣♦❧í❣♦♥♦s P ✭❝♦♥t✐❞♦s ❡♠ F✮ ❡ P′ ✭❝♦♥t❡♥❞♦_F✮✱ ♦ ♥ú♠❡r♦ r❡❛❧ _a(F) s❛t✐s❢❛③

a(P)_≤a(F)_≤a(P′_).

P❛r❛ s✐♠♣❧✐✜❝❛r✱ ❡♠ ✈❡③ ❞❡ ✉s❛r♠♦s ♣♦❧í❣♦♥♦s q✉❛✐sq✉❡r✱ ✈❛♠♦s ✉s❛r ♦s ♣♦❧í✲ ❣♦♥♦s r❡t❛♥❣✉❧❛r❡s✱ ♣❛r❛ ♦s q✉❛✐s é ♠❛✐s ❢á❝✐❧ ❝❛❧❝✉❧❛r ❛ ár❡❛✳

❯♠ ♣♦❧í❣♦♥♦ r❡t❛♥❣✉❧❛r é ❛ r❡✉♥✐ã♦ ❞❡ ✈ár✐♦s r❡tâ♥❣✉❧♦s ❥✉st❛♣♦st♦s ✭✐st♦ é✱ ❞♦✐s ❞❡ss❡s r❡tâ♥❣✉❧♦s tê♠ ❡♠ ❝♦♠✉♠ ♥♦ ♠á①✐♠♦ ✉♠ ❧❛❞♦✮✳ ❆ ár❡❛ ❞❡ ✉♠ ♣♦❧í❣♦♥♦ r❡t❛♥❣✉❧❛r é ❛ s♦♠❛ ❞❛s ár❡❛s ❞♦s r❡tâ♥❣✉❧♦s q✉❡ ♦ ❝♦♠♣õ❡♠✳

P♦❞❡♠♦s t❛♠❜é♠ ❝♦❧♦❝❛r ❛ ✜❣✉r❛ ♣❧❛♥❛ F ❡♠ ✉♠ ♣❛♣❡❧ q✉❛❞r✐❝✉❧❛❞♦ ♦♥❞❡ t♦❞♦s ♦s q✉❛❞r❛❞♦s sã♦ ✉♥✐tár✐♦s ✭ár❡❛ ✐❣✉❛♠ ❛ ✉♠✮✳ ❆ss✐♠✱ ♦ ✈❛❧♦r ❞❡ a(F) ❛♣r♦①✐♠❛❞♦ ♣♦r ❢❛❧t❛ s❡rá ❛ s♦♠❛ ❞❛s ár❡❛s ❞❡ t♦❞♦s ♦s q✉❛❞r❛❞♦s ✉♥✐tár✐♦s ❝♦♥t✐❞♦s ❡♠ F ❡ ♦ ✈❛❧♦r ❞❡ a(F) ❛♣r♦①✐♠❛❞♦ ♣♦r ❡①❝❡ss♦ s❡rá ❛ s♦♠❛ ❞❛s ár❡❛s ❞❡ t♦❞♦s ♦s q✉❛❞r❛❞♦s ✉♥✐tár✐♦s q✉❡ ❝♦♥té♠ F✳

❱❡❥❛♠♦s✳ ❙❡❥❛ F ✉♠❛ ✜❣✉r❛ ♣❧❛♥❛ ❡♠ ✉♠ ♣❛♣❡❧ q✉❛❞r✐❝✉❧❛❞♦ ❝♦♠ q✉❛❞r❛❞♦s ✉♥✐tár✐♦s✳

◆❛ ♠❛❧❤❛ q✉❛❞r✐❝✉❧❛❞❛ ❞❛ ✜❣✉r❛ ❛❝✐♠❛✱ F ❝♦♥té♠ ✶✾ q✉❛❞r❛❞♦s ✉♥✐tár✐♦s ❡ ❡stá ❝♦♥t✐❞❛ ❡♠ ✸✾ q✉❛❞r❛❞♦s ✉♥✐tár✐♦s✳ ❆ss✐♠✱ ♣♦❞❡♠♦s ❞✐③❡r q✉❡ ♦ ✈❛❧♦r ❞❡ a(F) ♣♦r ❢❛❧t❛ é ✶✾ ❡ ♣♦r ❡①❝❡ss♦ é ✸✾ ❡ ♣♦rt❛♥t♦19_≤a(F)_≤39✳

❱❛♠♦s ❛❣♦r❛✱ ❛tr❛✈és ❞❡ ♣❛r❛❧❡❧❛s ❛♦s ❧❛❞♦s✱ s✉❜❞✐✈✐❞✐r ❝❛❞❛ q✉❛❞r❛❞♦ ✉♥✐tár✐♦ ❛♠❛r❡❧♦ ❞❛ ✜❣✉r❛ ❛♥t❡r✐♦r ❡♠ ✶✻ q✉❛❞r❛❞♦s ❝♦♥❣r✉❡♥t❡s✳

❉❛ ár❡❛ ❡①t❡r♥❛ t❡♠♦s q✉❡ s✉❜tr❛✐r146 q✉❛❞r❛❞♦s ❞❡ ár❡❛ 1/16❡ ♥❛ ár❡❛ ✐♥✲ t❡r♥❛ s♦♠❛r112 q✉❛❞r❛❞♦s ❞❡ ár❡❛1/16✳ ❆ss✐♠✱ ✈❛♠♦s ❛✉♠❡♥t❛r ❛ ❛♣r♦①✐♠❛çã♦ ♣♦r ❢❛❧t❛ ❞❡ a(F)❡ ❞✐♠✐♥✉✐r ❛ ❛♣r♦①✐♠❛çã♦ ♣♦r ❡①❝❡ss♦ ❞❡ a(F)✳ P♦rt❛♥t♦

19 + 112

16 < a(F)<39− 146

16 ⇔26< a(F)<29,875.

Pr♦❝❡❞❡♥❞♦ ❞❛ ♠❡s♠❛ ❢♦r♠❛ q✉❡ ❛ ❛♥t❡r✐♦r✱✈❛♠♦s ❛❣♦r❛✱ ❛tr❛✈és ❞❡ ♣❛r❛❧❡❧❛s ❛♦s ❧❛❞♦s✱ ❞✐✈✐❞✐r ❝❛❞❛ q✉❛❞r❛❞♦ ✉♥✐tár✐♦ ❛♠❛r❡❧♦ ❡♠100q✉❛❞r❛❞♦s ❝♦♥❣r✉❡♥t❡s✳

❉❛ ár❡❛ ❡①t❡r♥❛ t❡♠♦s q✉❡ s✉❜tr❛✐r 1010 q✉❛❞r❛❞♦s ❞❡ ár❡❛ 1/100 ❡ ♥❛ ár❡❛ ✐♥t❡r♥❛ s♦♠❛r 791 q✉❛❞r❛❞♦s ❞❡ ár❡❛ 1/100✳ ❆ss✐♠✱ ✈❛♠♦s ❛✉♠❡♥t❛r ❛ ❛♣r♦①✐✲ ♠❛çã♦ ♣♦r ❢❛❧t❛ ❞❡a(F)❡ ❞✐♠✐♥✉✐r ❛ ❛♣r♦①✐♠❛çã♦ ♣♦r ❡①❝❡ss♦ ❞❡a(F)✳ P♦rt❛♥t♦

19 + 791

100 < a(F)<39− 1010

100 ⇔26,9< a(F)<28,9.

Pr♦❝❡❞❡♥❞♦ ❞❡st❛ ❢♦r♠❛ ✈❛♠♦s ❝❛❞❛ ✈❡③ ♠❛✐s ❛✉♠❡♥t❛r ❛ ❛♣r♦①✐♠❛çã♦ ♣♦r ❢❛❧t❛ ❡ ❞✐♠✐♥✉✐r ❛ ❛♣r♦①✐♠❛çã♦ ♣♦r ❡①❝❡ss♦ ❛té ♦❜t❡r♠♦s ♦ ✈❛❧♦r ❞❡ a(F)✳

◆♦ ♣ró①✐♠♦ ❝❛♣ít✉❧♦ ❞❛r❡♠♦s ✉♠❛ ❞❡♠♦♥str❛çã♦ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❛ ár❡❛ ❞♦ ❝ír❝✉❧♦ ✉t✐❧✐③❛♥❞♦ ♦ ▼ét♦❞♦ ❞❡ ❊①❛✉stã♦ ❞❡ ❊✉❞♦①♦✳

❈❛♣ít✉❧♦ ✸

❖ ▼ét♦❞♦ ❞❡ ❊①❛✉stã♦ ❞❡

❊✉❞♦①♦✲❆rq✉✐♠❡❞❡s

❖ ▼ét♦❞♦ ❞❡ ❊①❛✉stã♦ é t❛♠❜é♠ ❝♦♥❤❡❝✐❞♦ ♣♦r ♣r✐♥❝í♣✐♦ ❞❡ ❊✉❞♦①♦✲❆rq✉✐♠❡❞❡s ♣♦r t❡r ♥❛ s✉❛ ❜❛s❡ ❛ t❡♦r✐❛ ❞❛s ♣r♦♣♦rçõ❡s ❛♣r❡s❡♥t❛❞❛ ♣♦r ❊✉❞♦①♦ ❞❡ ❈♥✐❞♦ ✭✹✵✽ ❛ ✸✺✺ ❛✳❈✳✮ ❡ ♣♦r ❆rq✉✐♠❡❞❡s ❞❡ ❙✐r❛❝✉s❛ ✭✷✽✼ ❛ ✷✶✷ ❛✳❈✳✮ t❡r s✐❞♦ ♦ ♠❛t❡♠át✐❝♦ q✉❡ ♠❛✐♦r ✈✐s✐❜✐❧✐❞❛❞❡ ❧❤❡ ❞❡✉✳ ❚❛❧ ❡st✉❞♦ ❣✐r❛✈❛ ❡♠ t♦r♥♦ ❞❛ q✉❛❞r❛t✉r❛ ❞♦ ❝ír❝✉❧♦✳

❖ ♠ét♦❞♦ ❞❡ ❊✉❞♦①♦ ❝♦♥s✐st✐❛ ❡♠ ✐♥s❝r❡✈❡r ♣♦❧í❣♦♥♦s r❡❣✉❧❛r❡s ❡♠ ✉♠❛ ✜❣✉r❛ ❝✉r✈✐❧í♥❡❛✱ ❡ ✐r ❞♦❜r❛♥❞♦ ♦ ♥ú♠❡r♦ ❞❡ ❧❛❞♦s ❛té q✉❡ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❛ ár❡❛ ❞❛ ✜❣✉r❛ ❡ ❛ ❞♦ ♣♦❧í❣♦♥♦ ✐♥s❝r✐t♦ s❡ t♦r♥❛ss❡ ♠❡♥♦r ❞♦ q✉❡ q✉❛❧q✉❡r q✉❛♥t✐❞❛❞❡ ❞❛❞❛✳

❆rq✉✐♠❡❞❡s ♣r♦♣ôs ✉♠ r❡✜♥❛♠❡♥t♦ ❞❡ss❡ ♠ét♦❞♦✱ ❝♦♠♣r✐♠✐♥❞♦ ❛ ✜❣✉r❛ ❡♥tr❡ ❞✉❛s ♦✉tr❛s ❝✉❥❛s ár❡❛s ♠✉❞❛♠ ❡ t❡♥❞❡♠ ♣❛r❛ ❛ ❞❛ ✜❣✉r❛ ✐♥✐❝✐❛❧✱ ✉♠❛ ❝r❡s❝❡♥❞♦ ❡ ♦✉tr❛ ❞❡❝r❡s❝❡♥❞♦✳

❯♠ ❡①❡♠♣❧♦ ✐♥t❡r❡ss❛♥t❡ ❞❡ss❡ ♠ét♦❞♦ é ❛ ár❡❛ ❞♦ ❝ír❝✉❧♦✱ q✉❡ ❡r❛ ❡♥✈♦❧✈✐❞❛ ♣♦r ♣♦❧í❣♦♥♦s ✐♥s❝r✐t♦s ❡ ❝✐r❝✉♥s❝r✐t♦s✱ ❞❡ ♠♦❞♦ q✉❡✱ ❛✉♠❡♥t❛♥❞♦✲s❡ ♦ ♥ú♠❡r♦ ❞❡ ❧❛❞♦s✱ s✉❛s ár❡❛s s❡ ❛♣r♦①✐♠❛♠ ❞❛ ár❡❛ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ♦✉ s❡❥❛✱ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❛s ár❡❛s ❞♦s ❞♦✐s ♣♦❧í❣♦♥♦s ❞❡✈❡ s❡r t♦r♥❛❞❛ ♠❡♥♦r ❞♦ q✉❡ q✉❛❧q✉❡r q✉❛♥t✐❞❛❞❡ ♣♦s✐t✐✈❛ ❞❛❞❛ q✉❛♥❞♦ ♦ ♥ú♠❡r♦ ❞❡ ❧❛❞♦s ❛✉♠❡♥t❛✳ P♦r ❡ss❛ r❛③ã♦ ❛✜r♠❛✲s❡ q✉❡ ❆rq✉✐♠❡❞❡s ✉s❛✈❛ ✉♠ ♠ét♦❞♦ ✐♥❞✐r❡t♦ ♣❛r❛ ❛ ♠❡❞✐❞❛ ❞❛ ár❡❛ ❞❡ ✜❣✉r❛s ❝✉r✈✐❧í♥❡❛s✳

❆♥❛❧✐s❛r❡♠♦s✱ ❡♠ s❡❣✉✐❞❛✱ ♦ ♠♦❞♦ ❝♦♠♦ ❆rq✉✐♠❡❞❡s ❝❛❧❝✉❧❛✈❛ ❛ ár❡❛ ❞❡ ✉♠ ❝ír❝✉❧♦ ♥❛ ♣r✐♠❡✐r❛ ♣r♦♣♦s✐çã♦ ❞❡ ✉♠ ❞❡ s❡✉s ❧✐✈r♦s ♠❛✐s ❛♥t✐❣♦s✿ ▼❡❞✐❞❛ ❞♦ ❈ír❝✉❧♦✳

❊ss❛ ♣r♦♣♦s✐çã♦ é ✉♠❛ ♠❛♥❡✐r❛ ❞❡ ❞❡t❡r♠✐♥❛r ❛ ár❡❛ ❞♦ ❝ír❝✉❧♦ ❡♥❝♦♥tr❛♥❞♦ ✉♠❛ ✜❣✉r❛ r❡t✐❧í♥❡❛✱ ✉♠ tr✐â♥❣✉❧♦✱ ♥♦ ❝❛s♦✱ ❝✉❥❛ ár❡❛ s❡❥❛ ✐❣✉❛❧ à ár❡❛ ❞♦ ❝ír❝✉❧♦✳ ❊ss❡ r❡s✉❧t❛❞♦ ❢♦✐✱ s❡♠ ❞ú✈✐❞❛✱ ✉♠ ❞♦s ♠❛✐s ❢❛♠♦s♦s ❡♠ s✉❛ é♣♦❝❛✱ ❡ ♦ ♣r♦❝❡❞✲

✐♠❡♥t♦ é ❛♥á❧♦❣♦ ❛♦ ❡♠♣r❡❣❛❞♦ ♥❛ Pr♦♣♦s✐çã♦ ❳■■ ✲ ✷ ❞♦s ❊❧❡♠❡♥t♦s ❞❡ ❊✉❝❧✐❞❡s✱ ❛tr✐❜✉í❞♦s ❛ ❊✉❞♦①♦✳

❆ ❞❡♠♦♥str❛çã♦ ✉s❛ ✉♠ ♣r✐♥❝í♣✐♦ ❢✉♥❞❛♠❡♥t❛❧ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ▲❡♠❛ ❞❡ ❊✉✲ ❝❧✐❞❡s✱ ❡♥✉♥❝✐❛❞♦ ♥❛ Pr♦♣♦s✐çã♦ ✶ ❞♦ ▲✐✈r♦ ❳ ❞♦s ❊❧❡♠❡♥t♦s ❞❡ ❊✉❝❧✐❞❡s✳

Pr♦♣♦s✐çã♦ ❳ ✲ ✶ ✭▲❡♠❛ ❞❡ ❊✉❝❧✐❞❡s✮ ❙❡♥❞♦ ❡①♣♦st❛s ❞✉❛s ♠❛❣♥✐t✉❞❡s ❞❡s✐❣✉❛✐s✱ ❝❛s♦ ❞❛ ♠❛✐♦r s❡❥❛ s✉❜tr❛í❞❛ ✉♠❛ ♠❛✐♦r ❞♦ q✉❡ ❛ ♠❡t❛❞❡ ❡✱ ❞❛ q✉❡ é ❞❡✐①❛❞❛✱ ✉♠❛ ♠❛✐♦r ❞♦ q✉❡ ❛ ♠❡t❛❞❡✱ ❡ ✐ss♦ ❛❝♦♥t❡ç❛ s❡♠♣r❡✱ ❛❧❣✉♠❛ ♠❛❣♥✐t✉❞❡ s❡rá ❞❡✐①❛❞❛✱ ❛ q✉❛❧ s❡rá ♠❡♥♦r ❞♦ q✉❡ ❛ ♠❡♥♦r ♠❛❣♥✐t✉❞❡ ❡①♣♦st❛✳

❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ❞❛❞❛s ❞✉❛s ❣r❛♥❞❡③❛sA❡B ✭✈❛♠♦s s✉♣♦r q✉❡A >B✮✱ s❡ s✉❜tr❛ír♠♦s ✉♠❛ t❡r❝❡✐r❛ ❣r❛♥❞❡③❛ C₁❞❡A✱ s❡♥❞♦ C₁ ♠❛✐♦r q✉❡ ❛ ♠❡t❛❞❡ ❞❡ A✱ ♦❜t❡r❡♠♦sR1✳ ❈♦♥t✐♥✉❛♥❞♦ ♦ ♣r♦❝❡ss♦✱ s❡ s✉❜tr❛ír♠♦s ✉♠❛ ♦✉tr❛ ❣r❛♥❞❡③❛C2❞❡ R1✱ s❡♥❞♦ C2♠❛✐♦r q✉❡ ❛ ♠❡t❛❞❡ ❞❡ R1✱ ♦❜t❡r❡♠♦s R2✳ Pr♦❝❡❞❡♥❞♦ ❛ss✐♠✱ ♣❛r❛ n s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✱ ♦❜t❡r❡♠♦s ✉♠❛ ❣r❛♥❞❡③❛ Rn ♠❡♥♦r q✉❡ ❛ ❣r❛♥❞❡③❛ B ❞❛❞❛ ✐♥✐❝✐❛❧♠❡♥t❡✳ ❆ ♣r♦♣♦s✐çã♦ ❣❛r❛♥t❡✱ ❡♥tã♦✱ q✉❡ ♣♦❞❡♠♦s t♦r♥❛r ❛ ❞✐❢❡r❡♥ç❛ Rn ♠❡♥♦r ❞♦ q✉❡ q✉❛❧q✉❡r ❣r❛♥❞❡③❛ ❞❛❞❛✳ ❆ ✜❣✉r❛ ✶ r❡♣r❡s❡♥t❛ ❡ss❡ ♣r♦❝❡ss♦✱ ❝♦♥s✐❞❡r❛♥❞♦ s❡❣♠❡♥t♦s ❞❡ r❡t❛s ❝♦♠♦ ❢r❛♥❞❡③❛s ♣❛r❛ ✉♠❛ s✐t✉❛çã♦ ❡♠ q✉❡ ♦ r❡s✉❧t❛❞♦ é ❛t✐♥❣✐❞♦ ❡♠ ❞✉❛s ❡t❛♣❛s✳ ❱❡❥❛ ❬✻❪✳

❱❡r❡♠♦s ❝♦♠♦ ❡ss❡ ❧❡♠❛ é ✉s❛❞♦ ♣❛r❛ ❞❡♠♦♥str❛r ❛ ár❡❛ ❞♦ ❝ír❝✉❧♦✳

Pr♦♣♦s✐çã♦ ✶ ✭❆rq✉✐♠❡❞❡s✮ ❞♦s ❡❧❡♠❡♥t♦s ❞❡ ❊✉❝❧✐❞❡s ❆ ár❡❛ ❞❡ ✉♠ ❝ír❝✉❧♦ é ✐❣✉❛❧ à ❞♦ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ♥♦ q✉❛❧ ✉♠ ❞♦s ❧❛❞♦s q✉❡ ❢♦r♠❛♠ ♦ â♥❣✉❧♦ r❡t♦ é ✐❣✉❛❧ ❛♦ r❛✐♦ ❡ ♦ ♦✉tr♦ ❧❛❞♦ q✉❡ ❢♦r♠❛ ♦ â♥❣✉❧♦ r❡t♦ é ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡st❡ ❝ír❝✉❧♦✳

❉❡♠♦♥str❛çã♦✳ ❆ ✐❞é✐❛ ♣r✐♥❝✐♣❛❧ ❞❛ ❞❡♠♦♥str❛çã♦ é ❛♣r♦①✐♠❛r ❛ ár❡❛ ❞♦ ❝ír❝✉❧♦ ♣❡❧❛s ár❡❛s ❞❡ ♣♦❧í❣♦♥♦s r❡❣✉❧❛r❡s ✐♥s❝r✐t♦s ❡ ❝✐r❝✉♥s❝r✐t♦s✱ ❝✉❥♦s ❧❛❞♦s sã♦ s✉❝❡ss✐✈❛♠❡♥t❡ ❞✉♣❧✐❝❛❞♦s✳ ❈❛❞❛ ♣♦❧í❣♦♥♦ é ✉♠❛ ✉♥✐ã♦ ❞❡ tr✐â♥❣✉❧♦s✱ ❧♦❣♦✱ ❛

ár❡❛ ❞♦ ♣♦❧í❣♦♥♦ é ✐❣✉❛❧ à ár❡❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❝✉❥❛ ❛❧t✉r❛ é ♦ ❛♣ót❡♠❛ ❡ ❝✉❥❛ ❜❛s❡ é ♦ ♣❡rí♠❡tr♦✳ ❆ss✐♠✱ s❡ ♦ ❛♣ót❡♠❛ é ♦ r❛✐♦ ❞♦ ❝ír❝✉❧♦ ❡ s❡ ♦ ♣❡rí♠❡tr♦ ❞♦ ♣♦❧í❣♦♥♦ é ♦ ♣❡rí♠❡tr♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ t❡♠♦s ♦ t❡♦r❡♠❛✳

❙❡❥❛♠ C ❡T ❛s ár❡❛s ❞♦ ❝ír❝✉❧♦ ❡ ❞♦ tr✐â♥❣✉❧♦ ❡In ❡Cn♣♦❧í❣♦♥♦s ❞❡n ❧❛❞♦s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡ ✐♥s❝r✐t♦s ❡ ❝✐r❝✉♥s❝r✐t♦s ♥❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ❝♦♠♦ ♥❛ ✜❣✉r❛ ✸✳

❱❛♠♦s s✉♣♦r C > T ❡ C < T ❡ ♦❜t❡r ❝♦♥tr❛❞✐çõ❡s✱ ♦ q✉❡ ♠♦str❛ q✉❡ C = T✳ ❙✉♣♦♠♦s ✐♥✐❝✐❛❧♠❡♥t❡ C > T✳ ◆❡ss❡ ❝❛s♦✱ ♣♦❞❡♠♦s ♦❜t❡r ✉♠❛ q✉❛♥t✐✲ ❞❛❞❡ d = C −T > 0✳ ❙❛❜❡♠♦s✱ ❛✐♥❞❛✱ q✉❡ In t❡♠ ❛ ♠❡s♠❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ♥♦ q✉❛❧ ♦s ❧❛❞♦s q✉❡ ❢♦r♠❛♠ ♦ â♥❣✉❧♦ r❡t♦ sã♦ ✐❣✉❛✐s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❛♦ ❛♣ót❡♠❛ ❡ ❛♦ ♣❡rí♠❡tr♦ ❞♦ ♣♦❧í❣♦♥♦ r❡❣✉❧❛r ❞❡ n ❧❛❞♦s ✐♥s❝r✐t♦ ♥♦ ❝ír❝✉❧♦ ✭ár❡❛ ❂ ♣❡rí♠❡tr♦×❛♣ót❡♠❛✮✳ ❈♦♠♦ ♦s ❛♣ót❡♠❛s ❡ ♦s ♣❡rí♠❡tr♦s ❞♦s ♣♦❧í❣♦♥♦s

✐♥s❝r✐t♦s sã♦ s✉❝❡ss✐✈❛♠❡♥t❡ ♠❡♥♦r❡s q✉❡ ♦ r❛✐♦ ❡ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞♦ ❝ír❝✉❧♦✱ ✐st♦ é✱ ♠❡♥♦r❡s ❞♦ q✉❡ ♦s ❧❛❞♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ❞♦ tr✐â♥❣✉❧♦ ❞❡ ár❡❛ T✱ é ♣♦ssí✈❡❧ ❝♦♥❝❧✉✐r q✉❡ [ár❡❛ ❞❡In]< T ♣❛r❛ t♦❞♦n✳ ▲♦❣♦✱[ár❡❛ ❞❡ In]< T < C✳

❈♦♠♦[ár❡❛ ❞❡ In]< C✱ ❡①✐st❡ ✉♠❛ q✉❛♥t✐❞❛❞❡kn =C−[ár❡❛ ❞❡In]✳ ❱❡r❡♠♦s ❛❞✐❛♥t❡✱ ✉s❛♥❞♦ ♦ ▲❡♠❛ ❞❡ ❊✉❝❧✐❞❡s✱ q✉❡ q✉❛♥❞♦ ❛✉♠❡♥t❛♠♦s ♦ ♥ú♠❡r♦ ❞❡ ❧❛❞♦s ❞♦ ♣♦❧í❣♦♥♦ ❡ss❛ q✉❛♥t✐❞❛❞❡ ♣♦❞❡ s❡r t♦r♥❛❞❛ ♠❡♥♦r ❞♦ q✉❡ q✉❛❧q✉❡r q✉❛♥t✐❞❛❞❡ ❞❛❞❛✳ ▲♦❣♦✱ ♣❛r❛ n s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✱ é ♣♦ssí✈❡❧ ♦❜t❡r kn < d✳ ▼❛s ❛ [ár❡❛ ❞❡In]< T < C✱ ❧♦❣♦✱ d=C−T < C−[ár❡❛ ❞❡ In] =kn✱ ♦ q✉❡ ❧❡✈❛ ❛ ✉♠❛ ❝♦♥tr❛❞✐çã♦✳

❘❡st❛ ♠♦str❛r q✉❡ ❛s ❝♦♥❞✐çõ❡s ❞❛ Pr♦♣♦s✐çã♦ ❳ ✲ ✶ ❞❡ ❡✉❝❧✐❞❡s sã♦ s❛t✲ ✐s❢❡✐t❛s✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ♣❛r❛ ❝♦♥❝❧✉✐r q✉❡ kn ♣♦❞❡ s❡r t♦r♥❛❞❛ ♠❡♥♦r q✉❡ q✉❛❧q✉❡r q✉❛♥t✐❞❛❞❡ ❞❛❞❛✱ t❡♠♦s ❞❡ ♠♦str❛r q✉❡✱ ❛♦ ❞✉♣❧✐❝❛r ♦ ♥ú♠❡r♦ ❞❡ ❧❛❞♦s ❞♦ ♣♦❧í❣♦♥♦✱ ❡st❛♠♦s r❡t✐r❛♥❞♦ ❞❡ss❛ q✉❛♥t✐❞❛❞❡ ♠❛✐s q✉❡ ❛ s✉❛ ♠❡t❛❞❡✳

■ss♦ s✐❣♥✐✜❝❛ ♠♦str❛r q✉❡ ♦ ❡①❝❡ss♦ ❡♥tr❡ ❛ ár❡❛ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡ ❞♦ ♣♦❧í❣♦♥♦ ❞❡ 2n ❧❛❞♦s é ♠❡♥♦r ❞♦ q✉❡ ❛ ♠❡t❛❞❡ ❞♦ ❡①❝❡ss♦ ❡♥tr❡ ❛ ár❡❛ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡ ❞♦ ♣♦❧í❣♦♥♦ ❞❡ n ❧❛❞♦s✱ ♦✉ s❡❥❛✱ k₂n <

kn

2 ✳ ▼❛s q✉❛♥❞♦ ✉♠ ❛r❝♦ ❞❡ ❝ír❝✉❧♦ é s✉❜❞✐✈✐❞✐❞♦✱ ♦ ❡①❝❡ss♦ é ❞✐♠✐♥✉í❞♦ ❞❡ ✉♠ ❢❛t♦r ♠❛✐♦r q✉❡ ✷✳ ■ss♦ é ❞❡♠♦♥str❛❞♦ ♣♦r ❊✉❝❧✐❞❡s ♥❛ Pr♦♣♦s✐çã♦ ❳■■ ✲ ✷✱ ❞♦ ♠♦❞♦ ❝♦♠♦ s❡ s❡❣✉❡✿

❙❡❥❛M ♦ ♣♦♥t♦ ♠é❞✐♦ ❞♦ ❛r❝♦ ❞❡ ❝✐r❝✉♥❢❡rê♥❝✐❛AM B ✭✜❣✉r❛ ❛❝✐♠❛✮ ❡ s❡❥❛ ♦ tr✐â♥❣✉❧♦AM B ❢♦r♠❛❞♦ ♣♦r ❞♦✐s ❧❛❞♦s ❞♦ ♣♦❧í❣♦♥♦ ✐♥s❝r✐t♦ ♥❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✳ ❙❡ RS é ♦ ❧❛❞♦ ❞♦ ♣♦❧í❣♦♥♦ ❝✐r❝✉♥s❝r✐t♦✱ ❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦AM B é ♠❡t❛❞❡ ❞❛ ár❡❛ ❞♦ r❡tâ♥❣✉❧♦ARSB✱ ❧♦❣♦✱ é ♠❛✐♦r ❞♦ q✉❡ ❛ ♠❡t❛❞❡ ❞❛ ár❡❛ ❞♦ s❡❣♠❡♥t♦ ❝✐r❝✉❧❛r AM B✱ ✉♠❛ ✈❡③ q✉❡ ♦ r❡tâ♥❣✉❧♦ é ❢♦r♠❛❞♦ ♣♦r ✉♠ ♣❡❞❛ç♦ ❞♦ ❧❛❞♦ ❞♦ ♣♦❧í❣♦♥♦ ❝✐r❝✉♥s❝r✐t♦ á ❝✐r❝✉♥❢❡rê♥❝✐❛✳ ❙❡♥❞♦ ❛ss✐♠✱ s✉❜tr❛í♥❞♦ ❞♦ s❡❣♠❡♥t♦ ❝✐r❝✉❧❛rAM B ♦ tr✐â♥❣✉❧♦AM B✱ r❡t✐r❛♠♦s ✉♠❛ ✜❣✉r❛ ❝♦♠ ár❡❛ ♠❛✐♦r ❞♦ q✉❡ ❛ ♠❡t❛❞❡ ❞❛ ár❡❛ ❞♦ s❡❣♠❡♥t♦ ❝✐r❝✉❧❛r✳

❘❡♣❡t✐♥❞♦ ♦ ♣r♦❝❡❞✐♠❡♥t♦✱ ♣♦r ❡①❡♠♣❧♦✱ ♣❛r❛ ✉♠ tr✐â♥❣✉❧♦ AN M✱ ❢♦r♠❛❞♦ ♣♦r ❞♦✐s ❧❛❞♦s ❞❡ ✉♠ ♣♦❧í❣♦♥♦ ✐♥s❝r✐t♦ ❝♦♠ ♦ ❞♦❜r♦ ❞♦ ♥ú♠❡r♦ ❞❡ ❧❛❞♦s ❞♦ ♣♦✲ ❧í❣♦♥♦ ♣r♦❝❡❞❡♥t❡✱ ♣♦❞❡♠♦s s❡♠♣r❡ r❡t✐r❛r ❞❛ ár❡❛ q✉❡ r❡st❛ ✉♠❛ q✉❛♥t✐❞❛❞❡ ♠❛✐♦r ❞♦ q✉❡ ❛ ♠❡t❛❞❡ ❞❛ ár❡❛ ❞♦ s❡❣♠❡♥t♦ ❝✐r❝✉❧❛r ♦r✐❣✐♥❛❧✳ ❙❡♥❞♦ ❛ss✐♠✱ ❛ ❞✐❢❡r❡♥ç❛ kn ❡♥tr❡ ❛ ár❡❛ ❞♦ ❝ír❝✉❧♦ ❡ ❛ ❞♦ ♣♦❧í❣♦♥♦ ♣♦❞❡ s❡r t♦r♥❛❞❛ ♠❡♥♦r ❞♦ q✉❡ q✉❛❧q✉❡r q✉❛♥t✐❞❛❞❡ ❞❛❞❛✳ ■ss♦ ♠♦str❛ q✉❡ q✉❛♥❞♦ ❞♦❜r❛♠♦s ♦ ♥ú♠❡r♦ ❞❡ ❧❛❞♦s ❞♦ ♣♦❧í❣♦♥♦ ♦ ❡①❝❡ss♦ ❡♥tr❡ ❛ ár❡❛ ❞♦ ❝ír❝✉❧♦ ❡ ❛ ❞♦ ♣♦❧í❣♦♥♦ é ❞✐♠✐♥✉í❞♦ ♣♦r ✉♠ ❢❛t♦r ♠❛✐♦r q✉❡ ✷✳

❱♦❧t❛♥❞♦ à ❞❡♠♦♥str❛çã♦ ❞❛ Pr♦♣♦s✐çã♦ ✶ ❞❡ ❆rq✉✐♠❡❞❡s✱ ✐ss♦ ✐♠♣❧✐❝❛ q✉❡ ♣♦❞❡♠♦s t♦♠❛r kn < d ♥♦ ❛r❣✉♠❡♥t♦ ❛♥t❡r✐♦r✳ P❛r❛ ✜♥❛❧✐③❛r ❛ ❞❡♠♦♥str❛çã♦✱ s✉♣♦♠♦s ❛❣♦r❛ q✉❡ C < T ❡ ✈❛♠♦s ❡♥❝♦♥tr❛r ♥♦✈❛♠❡♥t❡ ✉♠❛ ❝♦♥tr❛❞✐çã♦✳ ❙❡ C < T✱ t❡♠♦s d = T ₋C > 0✳ ❖ ❛r❣✉♠❡♥t♦ é ❛♥á❧♦❣♦✱ ✉s❛♥❞♦ ♣♦❧í❣♦♥♦s ❝✐r❝✉♥s❝r✐t♦s✱ ♦ q✉❡ ❞❡♠♦♥str❛ ❛ ♣r♦♣♦s✐çã♦

◆❛ ♦❜r❛ ❞❡ ❆rq✉✐♠❡❞❡s✱ ✉♠ ♣r♦❝❡ss♦ ✐♥✜♥✐t♦ ❛♥á❧♦❣♦ ❛ ❡ss❡ é ✉t✐❧✐③❛❞♦ ♣❛r❛ ❡st❛❜❡❧❡❝❡r ❧✐♠✐t❡s ♣❛r❛ ❛ r❛③ã♦ ❡♥tr❡ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡ ♦ r❛✐♦ ❞♦ ❝ír❝✉❧♦✱ ♦✉ s❡❥❛✱ ♣❛r❛ ❛ q✉❛♥t✐❞❛❞❡ q✉❡ ❝❤❛♠❛♠♦s ❤♦❥❡ ❞❡π✳

❈❛♣ít✉❧♦ ✹

➪r❡❛ ❞❡ ▲❡❜❡s❣✉❡

❆❝r❡❞✐t❛♠♦s q✉❡ ♦ ❝♦♥t❡ú❞♦ ❞❡ss❡ ❝❛♣ít✉❧♦ ♣♦ss❛ s❡r ❡♥t❡♥❞✐❞♦ ♣♦r q✉❛❧q✉❡r ♣r♦❢❡ss♦r q✉❡ t❡♥❤❛ r❡❛❧✐③❛❞♦ ✉♠ ❝✉rs♦ ✐♥tr♦❞✉tór✐♦ ❞❡ ❆♥á❧✐s❡ ❘❡❛❧ t❛❧ ❝♦♠♦ ❛♣r❡s❡♥t❛❞♦ ❡♠ ❬✼❪ ❡ ♥♦r♠❛❧♠❡♥t❡ ♦❢❡r❡❝✐❞♦ ❛♦s ❛❧✉♥♦s ❞♦ P❘❖❋▼❆❚ ❡♠ ♣❡❧♦ ♠❡♥♦s ❞✉❛s ❞❛s ❞✐s❝✐♣❧✐♥❛s ✐♥tr♦❞✉tór✐❛s ❞♦ ❝✉rs♦✱ ▼❆ ✶✶ ✕ ◆ú♠❡r♦s ❡ ❋✉♥çõ❡s ❘❡❛✐s ❡ ▼❆ ✷✷ ✕ ❋✉♥❞❛♠❡♥t♦s ❞❡ ❈á❧❝✉❧♦✳

✹✳✶ ❈♦♥❝❡✐t♦s Pr❡❧✐♠✐♥❛r❡s

❖ sí♠❜♦❧♦ R s❡rá ✉t✐❧✐③❛❞♦ ♣❛r❛ r❡♣r❡s❡♥t❛r ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s✳ ▲❡♠❛ ✶✳ ❈♦♥s✐❞❡r❡ ♦s ♥ú♠❡r♦s r❡❛✐s x ❡ y✳ ❙❡✱ ♣❛r❛ t♦❞♦ ε > 0✱ t❡♠♦s

|x−y|< ε✱ ❡♥tã♦ x=y✳

❉❡♠♦♥str❛çã♦✳ ❙❡ ❢♦ss❡ x 6= y✱ t❡rí❛♠♦s |x−y| > 0 ❡✱ ❛♦ ❡s❝♦❧❤❡r♠♦s ε=|x−y|✱ ❝❤❡❣❛rí❛♠♦s ❛ ✉♠❛ ❝♦♥tr❛❞✐çã♦

❙❡❥❛A ⊂R✳ ❙✉♣♦♥❤❛ q✉❡A♣♦ss✉✐ ✉♠ ❧✐♠✐t❛♥t❡ s✉♣❡r✐♦r✱ ♦✉ s❡❥❛✱ ✉♠ ♥ú♠❡r♦ a _∈R t❛❧ q✉❡ x_≤ a ♣❛r❛ q✉❛❧q✉❡rx_∈ A✳ P♦r ❡①❡♠♣❧♦✱ a= 1/3 é ✉♠ ❧✐♠✐t❛♥t❡ s✉♣❡r✐♦r ♣❛r❛

A=_{0.3, 0.33, 0.333, ..._}.

◆♦t❡ q✉❡ q✉❛❧q✉❡r ♥ú♠❡r♦ ♠❛✐♦r q✉❡ 1/3 t❛♠❜é♠ é ✉♠ ❧✐♠✐t❛♥t❡ s✉♣❡r✐♦r ♣❛r❛ A✳ ❯♠❛ ❜♦❛ ♣❡r❣✉♥t❛ ❛ s❡ ❢❛③❡r é✿ ❡①✐st❡ ✉♠ ❧✐♠✐t❛♥t❡ s✉♣❡r✐♦r ♣❛r❛ A q✉❡ s❡❥❛ ♠❡♥♦r q✉❡ 1/3 ❄ ❆ r❡s♣♦st❛ é ♥ã♦✳ ❉❡ ❢❛t♦✱ s✉♣♦♥❤❛ q✉❡ ❡①✐st❡ c∈ R t❛❧ q✉❡

x_≤c < 1 3