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 MINAS GERAIS – BRASIL

2015

Ficha catalográfica preparada pela Biblioteca Central da UniversidadeFederal de Viçosa - Câmpus Viçosa

T

Jacob, Francisco Carlos, 1968-

J15c2015

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 dasmedidas . I. Universidade Federal de Viçosa. Departamento deMatemática. Programa de Pós-graduação em Matemática.II. Título.

CDD 22. ed. 516.15

II

A Deus e às minhas amadas esposa e filha, que me apoiaram em todos os momentos.

III

Agradeço a Deus, que me deu sabedoria para vencer mais uma etapa de minha vida, e a meu orientador Kennedy Pedroso, pelas orientações precisas em todos os momentos solicitados, pela dedicação e carinho que me foram dispensados.

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

4.8 Conclusão 44

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.

A presente dissertação tem o objetivo de oferecer ao estudante uma visão global do conceito

da área de figuras planas. No desenvolvimento, procuramos seguir uma ordem lógica na

apresentação de definições e propriedades. Procuramos também, sempre que possível, expor

os conceitos logo após as figuras, pois, com elas, o leitor poderá ter uma melhor compressão

dos mesmos. Sempre que possível, apresentamos as demonstrações das fórmulas de cálculos

de áreas das figuras planas elementares, pois estas demonstrações ajudam na compreensão da

generalização do conceito de área. No final, temos um experimento prático usando triângulos

e quadrados na aplicação do Princípio da Exaustão. Aconselhamos o leitor a buscar

experimentos usando outras figuras planas, pois a prática constante aprimora o aprendizado,

aguça a crítica e amplia a visão. Finalmente, como há sempre certa distância entre o anseio

dos autores e o valor de sua obra, gostaria de receber dos colegas professores uma apreciação

sobre este trabalho, notadamente os comentários críticos, pelos quais agradeço.

VI

ABSTRACT

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

The present dissertation has the aim of offering students the global view of the concept of the

area of flat figures. Throughout the development, we tried to follow a logical order in the

presentation of definitions and properties. We also tried whenever possible to expose the

concepts right after the figures, since with them, the reader will also have a better

understanding. Whenever possible, we presented the demonstrations of the formulas for the

calculation of the areas of elementary flat figures, for these demonstrations help the

understanding of the generalization of the concept of area. At the end, we present a practical

experiment using triangles and squares in the application of the Exhaustion Principle.

■♥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/n é ❞❛❞❛ ♣❡❧❛ ❡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 < a2 ✭❜❛st❛ t♦♠❛r r✱

✉♠❛ ❛♣r♦①✐♠❛çã♦ ♣♦r ❢❛❧t❛ ❞❡ a✱ ❝♦♠ ❡rr♦ ✐♥❢❡r✐♦r ❛ a−√b✳ ❊♥tã♦

√b < r < a

❡ ♣♦rt❛♥t♦ b < r2 < a2✮✳

◆♦ ✐♥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 < r2✳

❈♦♥❝❧✉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 ❞❡❝♦♠♣♦r R ❡♠ 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✉❜❞✐✈✐❞✐❞♦

❡♠ r s❡❣♠❡♥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·r q✉❛❞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· rq

♦✉ 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+ b2.

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 ❡ D s♦❜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 ❡BCD sã♦ ❝♦♥❣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❡❛ ❞❡ AFE] + [ár❡❛ ❞❡ BCD]

= 2 · 6 + 3 · 62

+3 · 62

= 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 · 12

+ 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❛✐r 146 q✉❛❞r❛❞♦s ❞❡ ár❡❛ 1/16 ❡ ♥❛ ár❡❛ ✐♥✲

t❡r♥❛ s♦♠❛r 112 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❡❧♦ ❡♠ 100 q✉❛❞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❛s A ❡ 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❡♠♦s R1✳ ❈♦♥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 <kn2✳ ▼❛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♥❝✐❛ AMB ✭✜❣✉r❛ ❛❝✐♠❛✮ ❡ s❡❥❛ ♦

tr✐â♥❣✉❧♦ AMB ❢♦r♠❛❞♦ ♣♦r ❞♦✐s ❧❛❞♦s ❞♦ ♣♦❧í❣♦♥♦ ✐♥s❝r✐t♦ ♥❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✳ ❙❡

RS é ♦ ❧❛❞♦ ❞♦ ♣♦❧í❣♦♥♦ ❝✐r❝✉♥s❝r✐t♦✱ ❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ AMB é ♠❡t❛❞❡ ❞❛ ár❡❛

❞♦ r❡tâ♥❣✉❧♦ ARSB✱ ❧♦❣♦✱ é ♠❛✐♦r ❞♦ q✉❡ ❛ ♠❡t❛❞❡ ❞❛ ár❡❛ ❞♦ s❡❣♠❡♥t♦ ❝✐r❝✉❧❛r

AMB✱ ✉♠❛ ✈❡③ q✉❡ ♦ r❡tâ♥❣✉❧♦ é ❢♦r♠❛❞♦ ♣♦r ✉♠ ♣❡❞❛ç♦ ❞♦ ❧❛❞♦ ❞♦ ♣♦❧í❣♦♥♦

❝✐r❝✉♥s❝r✐t♦ á ❝✐r❝✉♥❢❡rê♥❝✐❛✳ ❙❡♥❞♦ ❛ss✐♠✱ s✉❜tr❛í♥❞♦ ❞♦ s❡❣♠❡♥t♦ ❝✐r❝✉❧❛rAMB

♦ tr✐â♥❣✉❧♦ AMB✱ r❡t✐r❛♠♦s ✉♠❛ ✜❣✉r❛ ❝♦♠ ár❡❛ ♠❛✐♦r ❞♦ q✉❡ ❛ ♠❡t❛❞❡ ❞❛ ár❡❛

❞♦ s❡❣♠❡♥t♦ ❝✐r❝✉❧❛r✳

❘❡♣❡t✐♥❞♦ ♦ ♣r♦❝❡❞✐♠❡♥t♦✱ ♣♦r ❡①❡♠♣❧♦✱ ♣❛r❛ ✉♠ tr✐â♥❣✉❧♦ ANM ✱ ❢♦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✉❡r x ∈ 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

✷✸

♣❛r❛ q✉❛❧q✉❡r x ∈ A✳ ❊ss❛ ú❧t✐♠❛ ❛✜r♠❛çã♦ ✐♠♣❧✐❝❛ ❡♠

0.9 < 3c < 1

0.99 < 3c < 1

0.999 < 3c < 1✳✳✳

♦ q✉❡ ♥♦s ❧❡✈❛ ❛ ❝♦♥❝❧✉✐r q✉❡ 1 − 3c < 10−m✱ ♣❛r❛ q✉❛❧q✉❡r m = 1, 2, 3, ..., ❡

❛ss✐♠✱ ❞❛❞♦ ε > 0✱ t♦♠❡ m > log(1/ε) ❡✱ ♣❡❧♦ ▲❡♠❛ ✶✱ ♦❜t❡♠♦s c = 1/3✱ ✉♠❛

❝♦♥tr❛❞✐çã♦✳ ◆♦ ❡①❡♠♣❧♦ ❛❝✐♠❛✱ sup(A) /∈ A ♠❛s t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❞❡ A ❡ ♦

♣ró♣r✐♦ sup(A) sã♦ ♥ú♠❡r♦s r❛❝✐♦♥❛✐s✳

❙✉♣♦♥❤❛ q✉❡ A ⊂ R t❡♠ ✉♠ ❧✐♠✐t❛♥t❡ s✉♣❡r✐♦r✳ ❊♥tã♦ A t❡♠ ✉♠❛ ✐♥✜♥✐❞❛❞❡

❞❡ ❧✐♠✐t❛♥t❡s s✉♣❡r✐♦r❡s ❡ ❛♦ ♠❡♥♦r ❡♥tr❡ ❡ss❡s ❧✐♠✐t❛♥t❡s s✉♣❡r✐♦r❡s ❞❛♠♦s ♦

♥♦♠❡ ❞❡ s✉♣r❡♠♦ ❞❡ A✱ ❞❡♥♦t❛❞♦ ♣♦r sup(A)✳ ❆ ❡①✐stê♥❝✐❛ ❞❡ sup(A) ∈ R é

❣❛r❛♥t✐❞❛ ♣❡❧♦ ❆①✐♦♠❛ ❞♦ ❙✉♣r❡♠♦ ♣❛r❛ q✉❛❧q✉❡r ❝♦♥❥✉♥t♦ A ⊂ R q✉❡ ♣♦ss✉❛

❧✐♠✐t❛♥t❡ s✉♣❡r✐♦r✳ ❈♦♥❝❧✉✐♠♦s q✉❡ ♦ ✐♥t❡r✈❛❧♦ J = (−∞, sup(A)] é ♦ ✐♥t❡r✈❛❧♦

❞❡ss❡ t✐♣♦ q✉❡ ✏♠❡❧❤♦r s❡ ❛❥✉st❛✑ ❛♦ ❝♦♥❥✉♥t♦ A ♥♦ s❡♥t✐❞♦ ❞❡ q✉❡ A ⊂ J ❡

♣❛r❛ q✉❛❧q✉❡r ♦✉tr♦ ✐♥t❡r✈❛❧♦ I = (−∞, a] t❛❧ q✉❡ A ⊂ I t❡♠♦s J ⊂ I✳ ❯♠❛

♦✉tr❛ ♠❛♥❡✐r❛ ❞❡ ❝♦♥✜r♠❛r ❡ss❡ ❛❥✉st❡ é✿ ❞❛❞♦ ε > 0✱ ❡①✐st❡ x ∈ A t❛❧ q✉❡

x > sup(A)− ε✳

❆♥❛❧♦❣❛♠❡♥t❡✱ ❞❛❞♦ ✉♠ ❝♦♥❥✉♥t♦ B ⊂ R q✉❡ ♣♦ss✉✐ ❧✐♠✐t❛♥t❡ ✐♥❢❡r✐♦r✱ ♦✉

s❡❥❛✱ ✉♠ ♥ú♠❡r♦ b ∈ R t❛❧ q✉❡ b ≤ x ♣❛r❛ q✉❛❧q✉❡r x ∈ B✱ ❡①✐st❡ inf(B)✱ ♦ í♥✜♠♦

❞❡ B✱ ♦ ♠❛✐♦r ❡♥tr❡ s❡✉s ❧✐♠✐t❛♥t❡s ✐♥❢❡r✐♦r❡s✳ ❙❡❣✉❡ ❞❛ ❞❡✜♥✐çã♦ ❞❡ í♥✜♠♦ ❛

s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡✿ ❞❛❞♦ ε > 0✱ ❡①✐st❡ x ∈ B t❛❧ q✉❡ x < inf(A) + ε✳

P♦r ❡①❡♠♣❧♦✱ ♦ í♥✜♠♦ ❞♦ ❝♦♥❥✉♥t♦ B = {x r❛❝✐♦♥❛❧ ♣♦s✐t✐✈♦ : x2 > 2}, ❧✐♠✐✲

t❛❞♦ ✐♥❢❡r✐♦r♠❡♥t❡ ♣♦r b = 1✱ é ♦ ♥ú♠❡r♦ ✐rr❛❝✐♦♥❛❧√2✳ ❊ss❡ ❡①❡♠♣❧♦ ❞❡✐①❛ ❝❧❛r♦

q✉❡✱ s❡ ❝♦♥s✐❞❡r❛♠♦s ❛♣❡♥❛s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s✱ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ í♥✜♠♦ ❡♥tr❡ ♦s

♥ú♠❡r♦s r❛❝✐♦♥❛✐s ♥ã♦ ♣♦❞❡ s❡r ❛ss❡❣✉r❛❞❛✱ ♠♦str❛♥❞♦ q✉❡✱ ❛ ♣r✐♦r✐✱ inf(B) é ✉♠

♥ú♠❡r♦ r❡❛❧✱ ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ r❛❝✐♦♥❛❧✳ ❖ r❡s✉❧t❛❞♦ ❛ s❡❣✉✐r ♠♦str❛✱ ❡♠ ♣❛r✲

t✐❝✉❧❛r✱ q✉❡ ♦ s✉♣r❡♠♦ ♣♦❞❡ ❛✉♠❡♥t❛r s❡✉ ✈❛❧♦r q✉❛♥❞♦ ❛❞✐❝✐♦♥❛♠♦s ❡❧❡♠❡♥t♦s

❛♦ ❝♦♥❥✉♥t♦✳

▲❡♠❛ ✷✳ ❙❡ A ❡ B ♣♦ss✉❡♠ ❧✐♠✐t❛♥t❡ s✉♣❡r✐♦r ❡ A ⊂ B ⊂ R✱ ❡♥tã♦ sup(A) ≤sup(B)✳

❉❡♠♦♥str❛çã♦✳ ❇❛st❛ ♥♦t❛r q✉❡ sup(B) é ✉♠ ❧✐♠✐t❛♥t❡ s✉♣❡r✐♦r ♣❛r❛ A �

❯♠ r❡s✉❧t❛❞♦ ❛♥á❧♦❣♦ ❛♦ ▲❡♠❛ ✷ ❛✜r♠❛ q✉❡ s❡ A ❡ B sã♦ s✉❜❝♦♥❥✉♥t♦s ❞❡ R✱

♣♦ss✉❡♠ ❧✐♠✐t❛♥t❡ ✐♥❢❡r✐♦r ❡ A ⊂ B✱ ❡♥tã♦ inf(B) ≤ inf(A)✳

✷✹

❈♦♥✈❡♥❝✐♦♥❛♠♦s✿ sup(∅) = −∞✱ inf(∅) = +∞ ❡ −∞ < x < +∞ ♣❛r❛ t♦❞♦

x ∈ R✳ ❯♠ ❝♦♥❥✉♥t♦ q✉❡ ♣♦ss✉✐ ❧✐♠✐t❛♥t❡s ✐♥❢❡r✐♦r❡s ❡ ❧✐♠✐t❛♥t❡s s✉♣❡r✐♦r❡s é

❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❧✐♠✐t❛❞♦✳ ➱ út✐❧ ♦❜s❡r✈❛r q✉❡ s❡ A ⊂ R é ✉♠ ❝♦♥❥✉♥t♦ ❧✐♠✐t❛❞♦

✜♥✐t♦✱ ❡♥tã♦ sup(A) é ♦ ♠❛✐♦r ❡❧❡♠❡♥t♦ ❞❡ A ❡ inf(A) é ♦ ♠❡♥♦r ❡❧❡♠❡♥t♦ ❞❡ A✳

✹✳✷ ❋✐❣✉r❛s ❞✐s❝r❡t❛s

❱❛♠♦s ❛❞♠✐t✐r q✉❡ q✉❛❧q✉❡r ✜❣✉r❛ ♣❧❛♥❛ ❡stá ❝♦♥t✐❞❛ ❡♠ ✉♠ ♣❛♣❡❧ q✉❛❞r✐❝✉❧❛❞♦✳

❆❞♠✐t✐r❡♠♦s t❛♠❜é♠ q✉❡ ❝❛❞❛ q✉❛❞r❛❞✐♥❤♦ t❡♠ ❛ ♠❡s♠❛ ár❡❛ α = 2−2k✱

♦♥❞❡ k é ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ❡s❝♦❧❤✐❞♦ ❛r❜✐tr❛r✐❛♠❡♥t❡✱ ♦✉ s❡❥❛✱ ♣♦❞❡♠♦s ❛✉♠❡♥✲

t❛r ♦✉ ❞✐♠✐♥✉✐r ❛ ár❡❛ ❞❡ ❝❛❞❛ q✉❛❞r❛❞✐♥❤♦ ❞❛ ❢♦r♠❛ q✉❡ ♥♦s ❢♦r ♠❛✐s ❝♦♥✈❡♥✐❡♥t❡

♠✉❧t✐♣❧✐❝❛♥❞♦ ♦✉ ❞✐✈✐❞✐♥❞♦ ♣♦r ❞♦✐s ♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦s ❧❛❞♦s ❞❡ ❝❛❞❛ q✉❛❞r❛❞✐✲

♥❤♦✳ ❉❛q✉✐ ♣❛r❛ ❢r❡♥t❡✱ ✈❛♠♦s ❝❤❛♠❛r ❝❛❞❛ q✉❛❞r❛❞✐♥❤♦ ❞♦ ♣❛♣❡❧ q✉❛❞r✐❝✉❧❛❞♦

❞❡ ❡❧❡♠❡♥t♦ ❞❡ ár❡❛ α✳

❙❡❥❛ Qα ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ✜❣✉r❛s ♣❧❛♥❛s ❢♦r♠❛❞❛s ♣♦r ✉♥✐õ❡s ✭✜♥✐t❛s ♦✉

✐♥✜♥✐t❛s✮ ❞❡ ❡❧❡♠❡♥t♦s ❞❡ ár❡❛ α✳ ❙❡♥❞♦ ❛ss✐♠✱ ❞❛❞♦ X ∈ Qα✱ ❡①✐st❡ n ✐♥t❡✐r♦

♣♦s✐t✐✈♦ t❛❧ q✉❡

X = Q1 ∪Q2 ∪ . . . ∪Qn ,

♦♥❞❡ Qj é ✉♠ ❡❧❡♠❡♥t♦ ❞❡ ár❡❛ α✱ ♣❛r❛ ❝❛❞❛ j = 1, ..., n✱ ✭❝❛s♦ ❝♦♥trár✐♦✱ X

é ✉♠❛ ✜❣✉r❛ ❢♦r♠❛❞❛ ♣♦r ✐♥✜♥✐t♦s ❡❧❡♠❡♥t♦s ❞❡ ár❡❛ α✮✳ ❆s ✜❣✉r❛s ❡♠ Qα

s❡rã♦ ❝❤❛♠❛❞❛s ❞❡ ✜❣✉r❛s ❞✐s❝r❡t❛s✳ ❆ ✜❣✉r❛ ❛❜❛✐①♦✱ ♣♦r ❡①❡♠♣❧♦✱ t❡♠ n = 334

❡❧❡♠❡♥t♦s ❞❡ ár❡❛ α = 1♠♠2✳

✷✺

❆ ❝❛❞❛ X ∈ Qα✱ ♣♦❞❡♠♦s ❛ss♦❝✐❛r ♦ ♥ú♠❡r♦ ♣♦s✐t✐✈♦ λ1(X) ❞❡✜♥✐❞♦ ♣♦r

λ1(X) =

nα, s❡ X = Q1 ∪Q2 ∪ . . . ∪Qn,

+∞, s❡ X é ❢♦r♠❛❞❛ ♣♦r ✐♥✜♥✐t♦s ❡❧❡♠❡♥t♦s ❞❡ ár❡❛ α .

❆ ❢✉♥çã♦ λ1 : Qα → (0,+∞] ❛ss✐♠ ♦❜t✐❞❛ é ❛ ár❡❛ ❞❡ ▲❡❜❡s❣✉❡ ❞❛s ✜❣✉r❛s

❞✐s❝r❡t❛s✳

➱ ✐♠♣♦rt❛♥t❡ ♦❜s❡r✈❛r q✉❡✿

✶✳ ❆ ár❡❛ ❞❡ ▲❡❜❡s❣✉❡ ❞❡ ✜❣✉r❛s ❞✐s❝r❡t❛s ♥❛❞❛ ♠❛✐s é q✉❡ ❛ ár❡❛ ❝♦♠ ❛ q✉❛❧

❥á ❡st❛♠♦s ❤❛❜✐t✉❛❞♦s✱ ❧❡✈❛♥❞♦ ❡♠ ❝♦♥s✐❞❡r❛çã♦ q✉❡ ♦s ❡❧❡♠❡♥t♦s ❞❡ ár❡❛

α q✉❡ ❝♦♠♣õ❡♠ ❛ ✜❣✉r❛ ♣♦ss✉❡♠✱ ♥♦ ♠á①✐♠♦✱ ✈ért✐❝❡s ❡✴♦✉ ❛r❡st❛s ❡♠

❝♦♠✉♠❀

✷✳ ❆ ✉♥✐ã♦ ❞❡ ✜❣✉r❛s ❞✐s❝r❡t❛s ❡♠Qα é ✉♠❛ ✜❣✉r❛ ❞✐s❝r❡t❛ ❡♠Qα✳ ❘❡❝✐♣r♦❝❛✲

♠❡♥t❡✱ q✉❛❧q✉❡r ✜❣✉r❛ ❞✐s❝r❡t❛ ❡♠ Qα✱ ❢♦r♠❛❞❛ ♣♦r ❞♦✐s ♦✉ ♠❛✐s ❡❧❡♠❡♥t♦s

❞❡ ár❡❛ α✱ ♣♦❞❡ s❡r ❞❡❝♦♠♣♦st❛ ❡♠ ✉♠❛ ✉♥✐ã♦ ❞❡ ❞✉❛s ✜❣✉r❛s ❞✐s❝r❡t❛s ❡♠

Qα❀

✸✳ ❙❡ X, Y ∈ Qα ❡ X ⊂ Y ✱ ❡♥tã♦ λ1(X) ≤ λ1(Y )✳ ■ss♦ s✐❣♥✐✜❝❛ q✉❡ λ1

é ✏❝r❡s❝❡♥t❡✑ ❡ ❞❛♠♦s ♦ ♥♦♠❡ ❞❡ ♠♦♥♦t♦♥✐❝✐❞❛❞❡ ❛ ❡ss❛ ❝❛r❛❝t❡ríst✐❝❛ ❞❛

ár❡❛ ❞❡ ▲❡❜❡s❣✉❡ ❞❛s ✜❣✉r❛s ❞✐s❝r❡t❛s❀

✹✳ ❙❡ X, Y ∈ Qα ❡ X∩Y é ✉♠❛ ❛r❡st❛✱ ✉♠ ✈ért✐❝❡ ♦✉ ✈❛③✐❛✱ ❡♥tã♦ λ1(X∪Y ) =

λ1(X) + λ1(Y )✳ ❊ss❛ ♣r♦♣r✐❡❞❛❞❡ ❝❤❛♠❛✲s❡ ❛❞✐t✐✈✐❞❛❞❡❀

✺✳ ❊♠ ❣❡r❛❧✱ ✈❛❧❡ ❛ s✉❜❛❞✐t✐✈✐❞❛❞❡✱ ♦✉ s❡❥❛✱ λ1(X ∪ Y ) ≤ λ1(X) + λ1(Y ) ♣❛r❛

q✉❛✐sq✉❡r X, Y ∈ Qα❀

✻✳ ❙❡♥❞♦√α = 2−k ❡ ✜①❛♥❞♦ X ∈ Qα✱ t❡♠♦s q✉❡✱ ❞❛❞♦ β ❞✐st✐♥t♦ ❞❡ α✱ ❝♦♠√

β = 2−ℓ✱ ℓ > k ✐♥t❡✐r♦✱ ♦ ✈❛❧♦r ❞❡ λ1(X) é ♦ ♠❡s♠♦ ♣❛r❛ X ✈✐st♦ ❝♦♠♦

✜❣✉r❛ ❞✐s❝r❡t❛ ❡♠ Qα ♦✉ ❡♠ Qβ✳ ◗✉❛♥❞♦ β < α✱ ❞✐③❡♠♦s q✉❡ Qβ t❡♠ ✉♠❛

✷✻

r❡s♦❧✉çã♦ ♠❛✐♦r q✉❡ ❛ ❞❡ Qα✱ k ❡ ℓ sã♦ ❛s ♦r❞❡♥s ❞❛s r❡s♦❧✉çõ❡s ❞❡ Qα ❡

Qβ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❘❡s✉♠✐♥❞♦✱ ❛ ár❡❛ ❞❡ ✉♠❛ ✜❣✉r❛ ❞✐s❝r❡t❛ ♥ã♦ ✈❛r✐❛

❝♦♠ ♦ ❛✉♠❡♥t♦ ❞❛ r❡s♦❧✉çã♦ ✉t✐❧✐③❛❞❛❀

✼✳ ➱ s❡♠♣r❡ ♣♦ssí✈❡❧ ❛✉♠❡♥t❛r ❛ r❡s♦❧✉çã♦ ❞❡ ✉♠❛ ✜❣✉r❛ ❞✐s❝r❡t❛ ❞❡✈✐❞♦ à

❡s❝♦❧❤❛ q✉❡ ✜③❡♠♦s ♣❛r❛ ❛ ❢♦r♠❛ ❞♦ ♥ú♠❡r♦ q✉❡ r❡♣r❡s❡♥t❛ ❛ ár❡❛ ❞❡ ❝❛❞❛

q✉❛❞r❛❞✐♥❤♦✳ ◗✉❛♥❞♦ ❛ ♦r❞❡♠ ❞❛ r❡s♦❧✉çã♦ ❛✉♠❡♥t❛ ❞❡ k ♣❛r❛ k + 1✱ ♦

♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ❞❡ ár❡❛ α = 2−2k ❛✉♠❡♥t❛ ❞❡ n ♣❛r❛ 4n ❡❧❡♠❡♥t♦s ❞❡

ár❡❛ β = α/4❀

✽✳ P♦❞❡rí❛♠♦s ✉s❛r ✉♠ ♣❛♣❡❧ tr✐❛♥❣✉❧❛❞♦ ♥❛ ❝♦♥str✉çã♦ ❞❡ ✜❣✉r❛s ❞✐s❝r❡t❛s✳

◆❡ss❡ ❝❛s♦✱ ♦s ❡❧❡♠❡♥t♦s ❞❡ ár❡❛ s❡r✐❛♠ tr✐â♥❣✉❧♦s ❡q✉✐❧át❡r♦s ❡ ❝♦♥s✐❞❡r❛✲

rí❛♠♦s ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ✜❣✉r❛s ❞✐s❝r❡t❛s ❢♦r♠❛❞❛s ♣♦r ✉♥✐õ❡s ✭✜♥✐t❛s

♦✉ ✐♥✜♥✐t❛s✮ ❞❡ tr✐â♥❣✉❧♦s ❞♦ ♣❛♣❡❧ tr✐❛♥❣✉❧❛❞♦✳

✹✳✸ ❋✐❣✉r❛s ♣❧❛♥❛s ❛❜❡rt❛s

❱❛♠♦s ❝❤❛♠❛r ❞❡ q✉❛❞r❛❞♦ ❛❜❡rt♦ ❛ ✜❣✉r❛ ♣❧❛♥❛ ♦❜t✐❞❛ q✉❛♥❞♦ r❡t✐r❛♠♦s ♦s ❧❛❞♦s

❞♦ q✉❛❞r❛❞♦✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ❛s ♦✉tr❛s ✜❣✉r❛s ♣❧❛♥❛s ❛❜❡rt❛s

❡❧❡♠❡♥t❛r❡s✳

❊♠ ❣❡r❛❧✱ ❞✐③❡♠♦s q✉❡ ✉♠❛ ✜❣✉r❛ ♣❧❛♥❛ G é ❛❜❡rt❛ q✉❛♥❞♦ ❝❛❞❛ ✉♠ ❞❡ s❡✉s

♣♦♥t♦s✱ ❞✐❣❛♠♦s x ∈ G✱ é ♦ ❝❡♥tr♦ ❞❡ ❛❧❣✉♠ ❝ír❝✉❧♦ ❛❜❡rt♦ ♣r♦♣r✐❛♠❡♥t❡ ❝♦♥t✐❞♦

❡♠ G✳ P♦♥t♦s ❝♦♠ t❛❧ ♣r♦♣r✐❡❞❛❞❡ s❡rã♦ ❝❤❛♠❛❞♦s ❞❡ ♣♦♥t♦s ✐♥t❡r✐♦r❡s ❡ ♦

❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ♣♦♥t♦s ✐♥t❡r✐♦r❡s ❞❡ G s❡rá ❝❤❛♠❛❞♦ ❞❡ ✐♥t❡r✐♦r ❞❡ G✳

✷✼

❆ ✉♥✐ã♦ ❞❡ ❞✉❛s ✜❣✉r❛s ♣❧❛♥❛s ❛❜❡rt❛s é ❛❜❡rt❛ ♣♦rq✉❡ q✉❛❧q✉❡r ♣♦♥t♦ ❞❛

✉♥✐ã♦ é ♣♦♥t♦ ✐♥t❡r✐♦r ❞❡ ♣❡❧♦ ♠❡♥♦s ✉♠❛ ❞❛s ✜❣✉r❛s✳ ❆ ❛✜r♠❛çã♦ ❛ s❡❣✉✐r ❝✉✐❞❛

❞♦ ❝❛s♦ ❞❛ ✐♥t❡rs❡çã♦✳

▲❡♠❛ ✸✳ ❆ ✐♥t❡rs❡çã♦ ❞❡ ❞✉❛s ✜❣✉r❛s ♣❧❛♥❛s ❛❜❡rt❛s é ✉♠❛ ✜❣✉r❛ ♣❧❛♥❛

❛❜❡rt❛✳

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ G1 ❡ G2 ❞✉❛s ✜❣✉r❛s ♣❧❛♥❛s ❛❜❡rt❛s ❡ x ∈ G1 ∩ G2✳

❊♥tã♦ ❡①✐st❡♠ ❝ír❝✉❧♦s ❛❜❡rt♦s✱ ❞✐❣❛♠♦s C1 ❡ C2✱ ❛♠❜♦s ❝❡♥tr❛❞♦s ❡♠ x✱ t❛✐s q✉❡

C1 ⊂ G1 ❡ C2 ⊂ G2✳ ❇❛st❛ ♦❜s❡r✈❛r ❛❣♦r❛ q✉❡ ♦ ♠❡♥♦r ❞♦s ❞♦✐s ❝ír❝✉❧♦s ❡stá

❝♦♥t✐❞♦ ❡♠ G1 ∩G2 �

❈❛s♦ ✉♠❛ ✜❣✉r❛ ♣❧❛♥❛ S ♥ã♦ s❡❥❛ ❛❜❡rt❛✱ ❡①✐st❡♠ ♣♦♥t♦s x ∈ S t❛✐s q✉❡✱

♣❛r❛ q✉❛❧q✉❡r ❝ír❝✉❧♦ ❛❜❡rt♦ C ❝❡♥tr❛❞♦ ❡♠ x✱ C ∩ S ❡ C ∩ Sc sã♦ ♥ã♦ ✈❛③✐♦s✱

♦♥❞❡ Sc ✐♥❞✐❝❛ ♦ ❝♦♥❥✉♥t♦ ❝♦♠♣❧❡♠❡♥t❛r ❞❡ S ❡♠ r❡❧❛çã♦ ❛♦ ♣❧❛♥♦✳ P♦♥t♦s ❛ss✐♠

❞❡✜♥✐❞♦s s❡rã♦ ❝❤❛♠❛❞♦s ❞❡ ♣♦♥t♦s ❞❡ ❢r♦♥t❡✐r❛ ❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ♣♦♥t♦s

❞❡ ❢r♦♥t❡✐r❛ ❞❡ G s❡rá ❝❤❛♠❛❞♦ ❞❡ ❢r♦♥t❡✐r❛ ❞❡ G✳

❙❡♥❞♦ ❛ss✐♠✱ ✉♠❛ ✜❣✉r❛ ♣❧❛♥❛ ❛❜❡rt❛ G é ❛q✉❡❧❛ ❡♠ q✉❡ s❡✉s ♣♦♥t♦s ❞❡

❢r♦♥t❡✐r❛ ♥ã♦ ♣❡rt❡♥❝❡♠ ❛ G ✭t♦❞♦s ♦s s❡✉s ♣♦♥t♦s sã♦ ✐♥t❡r✐♦r❡s✮✳ ❖❜✈✐❛♠❡♥t❡✱

✉♠ ♣♦♥t♦ ♥ã♦ ♣♦❞❡ s❡r ✐♥t❡r✐♦r ❡ ❞❡ ❢r♦♥t❡✐r❛ ❛♦ ♠❡s♠♦ t❡♠♣♦✳ ❖ ❝♦♥❥✉♥t♦

✈❛③✐♦ ❡ ♦ ♣ró♣r✐♦ ♣❧❛♥♦ sã♦ ❡①❡♠♣❧♦s ❞❡ ❝♦♥❥✉♥t♦s ❛❜❡rt♦s ♣♦rq✉❡ ❛♠❜♦s tê♠

❢r♦♥t❡✐r❛ ✈❛③✐❛ ✭♦ ❝♦♥❥✉♥t♦ ✈❛③✐♦ t❛♠❜é♠ ♥ã♦ t❡♠ ♣♦♥t♦s ✐♥t❡r✐♦r❡s✱ ♠❛s ♥ã♦

✈❛♠♦s ♣r♦❧♦♥❣❛r ❡ss❛ ❛r❣✉♠❡♥t❛çã♦✮✳ ❙❡❣✉❡ ❞✐r❡t❛♠❡♥t❡ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ♣♦♥t♦

❞❡ ❢r♦♥t❡✐r❛ ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✳

▲❡♠❛ ✹✳ ❆s ✜❣✉r❛s ♣❧❛♥❛s S ❡ Sc tê♠ ❛ ♠❡s♠❛ ❢r♦♥t❡✐r❛✳

❯♠❛ ✜❣✉r❛ ♣❧❛♥❛ S q✉❡ ❝♦♥té♠ s✉❛ ♣ró♣r✐❛ ❢r♦♥t❡✐r❛ é ❝❤❛♠❛❞❛ ❞❡ ✜❣✉r❛

♣❧❛♥❛ ❢❡❝❤❛❞❛✳ ❋✐❣✉r❛s ❞✐s❝r❡t❛s sã♦ ❡①❡♠♣❧♦s ❞❡ ✜❣✉r❛s ♣❧❛♥❛s ❢❡❝❤❛❞❛s✳ ❆

✉♥✐ã♦ ❞❡ ✉♠❛ ✜❣✉r❛ ♣❧❛♥❛ q✉❛❧q✉❡r S ❝♦♠ t♦❞♦s ♦s s❡✉s ♣♦♥t♦s ❞❡ ❢r♦♥t❡✐r❛ é

✷✽

❞❡♥♦t❛❞❛ ♣♦r S ❡ r❡❝❡❜❡ ♦ ♥♦♠❡ ❞❡ ❢❡❝❤♦ ❞❡ S✳ ❉❛ ❞❡✜♥✐çã♦ q✉❡ ❛❝❛❜❛♠♦s ❞❡

❛♣r❡s❡♥t❛r ❡ ❞♦ ▲❡♠❛ ✹ ❝♦♥❝❧✉✐♠♦s ♦

▲❡♠❛ ✺✳ ❙❡ S é ✉♠❛ ✜❣✉r❛ ♣❧❛♥❛ ❢❡❝❤❛❞❛✱ ❡♥tã♦ Sc é ✉♠❛ ✜❣✉r❛ ♣❧❛♥❛

❛❜❡rt❛✳

❋✐❣✉r❛s ♣❧❛♥❛s q✉❡ ❝♦♥s✐st❡♠ ❛♣❡♥❛s ❞❡ ♣♦♥t♦s ❞❡ ❢r♦♥t❡✐r❛ ✭♣♦r ❡①❡♠♣❧♦✱

❛♣❡♥❛s ❛s ❛r❡st❛s ❞❡ ✉♠ q✉❛❞r❛❞♦✮ sã♦ ❡①❡♠♣❧♦s ❞❡ ✜❣✉r❛s ♣❧❛♥❛s ❢❡❝❤❛❞❛s ❝✉❥♦

✐♥t❡r✐♦r é ✈❛③✐♦✳ ❉❛❞❛ ✉♠❛ ✜❣✉r❛ ♣❧❛♥❛ S q✉❛❧q✉❡r✱ ✈❛♠♦s ❞❡♥♦t❛r ♦ ❝♦♥❥✉♥t♦

❞❡ t♦❞♦s ♦s s❡✉s ♣♦♥t♦s ✐♥t❡r✐♦r❡s ♣♦r ✐♥t(S) ❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s s❡✉s ♣♦♥t♦s

❞❡ ❢r♦♥t❡✐r❛ ♣♦r ❢r(S)✳ ◆ã♦ é ❞✐❢í❝✐❧ ✈❡r✐✜❝❛r q✉❡ ✐♥t(S) é✱ ♥❡❝❡ss❛r✐❛♠❡♥t❡✱ ✉♠❛

✜❣✉r❛ ♣❧❛♥❛ ❛❜❡rt❛ ❡ q✉❡ ❢r(S) é ❢❡❝❤❛❞❛✳

P♦❞❡ ♦❝♦rr❡r ♦ ❝❛s♦ ❡♠ q✉❡ ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ❛❜❡rt♦ ♣♦ss✉✐ ❛♣❡♥❛s ❛❧❣✉♥s ❞♦s

s❡✉s ♣♦♥t♦s ❞❡ ❢r♦♥t❡✐r❛❀

s✐❣♥✐✜❝❛♥❞♦ q✉❡ ❡①✐st❡♠ ✜❣✉r❛s ♣❧❛♥❛s q✉❡ ♥ã♦ sã♦ ♥❡♠ ❛❜❡rt❛s ❡ ♥❡♠ ❢❡❝❤❛❞❛s✳

❆❧é♠ ❞✐ss♦✱ ❛♣❡s❛r ❞❛ ❛♣❛rê♥❝✐❛ s✐♠♣❧❡s ❞❛s ✜❣✉r❛s ❛❝✐♠❛✱ ✜❣✉r❛s ♣❧❛♥❛s ❡♠ ❣❡r❛❧

♣♦❞❡♠ t❡r ❢♦r♠❛t♦s ❜❡♠ ❞✐✈❡rs✐✜❝❛❞♦s ❡♠ r❡❧❛çã♦ às ✜❣✉r❛s ♣❧❛♥❛s ❡❧❡♠❡♥t❛r❡s✱

❝♦♠♦ ♠♦str❛ ❛ s❡❣✉✐♥t❡ ✜❣✉r❛ ♣❧❛♥❛ ❝❤❛♠❛❞❛ ❞❡ ❈♦♥❥✉♥t♦ ❞❡ ❏✉❧✐❛✱ ✉♠ ❡①❡♠♣❧♦

❜❡♠ ❝♦♥❤❡❝✐❞♦ q✉❡ ❛♣r❡s❡♥t❛ ✉♠ ♣❛❞rã♦ ❞❡ ❢♦r♠❛çã♦ ❡♠ q✉❛❧q✉❡r ❢r❛❣♠❡♥t♦ ❞❡

s✉❛ ❢r♦♥t❡✐r❛ ✭♠♦t✐✈♦ ♣❡❧♦ q✉❛❧ é ❝❤❛♠❛❞♦ ❞❡ ❢r❛❝t❛❧✮✳

❯♠ ♦✉tr♦ ❡①❡♠♣❧♦ ❜❡♠ ❞✐st✐♥t♦ ❞♦s ❛♥t❡r✐♦r❡s é ♦ ❝♦♥❥✉♥t♦ P ❞❡ ♣❛r❡s ❞❡

♥ú♠❡r♦s r❛❝✐♦♥❛✐s (r, s) ♣❡rt❡♥❝❡♥t❡s ❛♦ q✉❛❞r❛❞♦ ❢❡❝❤❛❞♦ [0, 1]× [0, 1]✳ ▲❡✈❛♥❞♦

❡♠ ❝♦♥s✐❞❡r❛çã♦ q✉❡ q✉❛❧q✉❡r ♥ú♠❡r♦ r❡❛❧ ♣♦❞❡ s❡r ❛♣r♦①✐♠❛❞♦ ♣♦r ♥ú♠❡r♦s

✷✾

r❛❝✐♦♥❛✐s✱ ♥ã♦ é ❞✐❢í❝✐❧ ✈❡r✐✜❝❛r q✉❡ P ♥ã♦ ♣♦ss✉✐ ♣♦♥t♦s ✐♥t❡r✐♦r❡s ❡ t♦❞♦s ♦s

♣♦♥t♦s ❞❡ [0, 1]× [0, 1] sã♦ ♣♦♥t♦s ❞❛ ❢r♦♥t❡✐r❛ ❞❡ P ✳

❈♦♥s✐❞❡r❡ ❛❣♦r❛ ♦ ❝♦♥❥✉♥t♦ Q ❞❡ t♦❞❛s ❛s ✜❣✉r❛s ❞✐s❝r❡t❛s q✉❡ ♣♦ss✉❡♠ r❡✲

s♦❧✉çõ❡s ❛r❜✐trár✐❛s✿

Q =⋃

Qα ,

♦✉ s❡❥❛✱ Q é ♦ ❝♦♥❥✉♥t♦ ❞❛s ✜❣✉r❛s ❞✐s❝r❡t❛s ❢♦r♠❛❞❛s ♣♦r ❡❧❡♠❡♥t♦s ❞❡ ár❡❛ α

❝♦♠ q✉❛❧q✉❡r ✉♠ ❞♦s ✈❛❧♦r❡s ♣♦ssí✈❡✐s ♣❛r❛ α = 2−2k✱ k ✐♥t❡✐r♦✳ ❱❛♠♦s ❡st❡♥❞❡r

❛ ár❡❛ ❞❡ ▲❡❜❡s❣✉❡ ♣❛r❛ ♦ ❝♦♥❥✉♥t♦ Q ∪A✱ ♦♥❞❡ A é ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s

✜❣✉r❛s ♣❧❛♥❛s ❛❜❡rt❛s✱ ❞❡✜♥✐♥❞♦ ♣❛r❛ ✐ss♦ ✉♠❛ ❢✉♥çã♦

λ2 : Q ∪A → [0,+∞]

q✉❡ ❝♦✐♥❝✐❞❡ ❝♦♠ λ1 ❡♠ Q✱ ✐st♦ é✱ λ2(W ) = λ1(W ) ♣❛r❛ q✉❛❧q✉❡r W ∈ Q✳

❯♠❛ ✜❣✉r❛ ♣❧❛♥❛ é ❝❤❛♠❛❞❛ ❞❡ ❧✐♠✐t❛❞❛ s❡ ❡①✐st❡ ✉♠ r❡tâ♥❣✉❧♦ R ∈ Qq✉❡ ❛ ❝♦♥t❡♥❤❛ ♣r♦♣r✐❛♠❡♥t❡✳ ❙❡ Y ∈ Q ∪A ❢♦r ✉♠❛ ✜❣✉r❛ ♥ã♦ ❧✐♠✐t❛❞❛ ♣♦❞❡

♦❝♦rr❡r ♦ ❝❛s♦ ❞❡ s❡r s✉❛ ár❡❛ ❞❡ ▲❡❜❡s❣✉❡ ✜♥✐t❛ ♦✉ ✐♥✜♥✐t❛ ♦ q✉❡✱ s✐♠❜♦❧✐❝❛♠❡♥t❡✱

r❡♣r❡s❡♥t❛♠♦s ♣♦r λ2(Y ) < ∞ ♦✉ λ2(Y ) = ∞ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❊♠ ♣❛rt✐❝✉❧❛r✱

♦ ♣❧❛♥♦ é ✉♠ ❛❜❡rt♦ ❝♦♠ ár❡❛ ❞❡ ▲❡❜❡s❣✉❡ ✐♥✜♥✐t❛✳ ❋♦✐ ✈✐st♦ q✉❡ ∅ ∈ A ❡

✈❛♠♦s ❝♦♥✈❡♥❝✐♦♥❛r q✉❡ λ2(∅) = 0✳ ❖✉tr♦ ❡①❡♠♣❧♦ é ❛ ✜❣✉r❛ E ∈ Q∪A ❛❜❛✐①♦✱

❝♦♠❡ç❛♥❞♦ ❞❛ ❡sq✉❡r❞❛ ♣❛r❛ ❛ ❞✐r❡✐t❛ ❝♦♠ ✉♠ q✉❛❞r❛❞♦ ❞❡ ár❡❛ ✐❣✉❛❧ ❛ 1✱ é ✉♠❛

✏❡s❝❛❞❛ ✐♥✜♥✐t❛✑ ♦♥❞❡ ❝❛❞❛ r❡tâ♥❣✉❧♦ ✭❞❡❣r❛✉✮ t❡♠ ❜❛s❡ ✐❣✉❛❧ ❛ 1 ❡ ❛❧t✉r❛ ✐❣✉❛❧

à ♠❡t❛❞❡ ❞❛ ❛❧t✉r❛ ❞♦ ❞❡❣r❛✉ ✐♠❡❞✐❛t❛♠❡♥t❡ ❛♥t❡r✐♦r✳ ◆ã♦ é ❞✐❢í❝✐❧ ✈❡r✐✜❝❛r q✉❡

❛ ár❡❛ ❞❡ ▲❡❜❡s❣✉❡ ❞❡ E é ✜♥✐t❛ ✭✐❣✉❛❧ ❛ 2✮ ❛♣❡s❛r ❞❡ E ♥ã♦ s❡r ❧✐♠✐t❛❞❛✳

❙✉♣♦♥❞♦ ❡♥tã♦ q✉❡ Y ∈ Q ∪A é ✉♠❛ ✜❣✉r❛ ❧✐♠✐t❛❞❛✱ ❢❛③❡♠♦s

λ2(Y ) = sup{λ1(X) : X ⊂ Y, X ∈ Q} .

❉❡ss❛ ❢♦r♠❛✱ ❛ ár❡❛ ❞❡ ▲❡❜❡s❣✉❡ ❞❡ ✉♠❛ ✜❣✉r❛ ♣❧❛♥❛ ❛❜❡rt❛ é ❛♣r♦①✐♠❛❞❛✱

♣♦r ❢❛❧t❛✱ ♣❡❧❛ ár❡❛ ❞❡ ✜❣✉r❛s ❞✐s❝r❡t❛s✳

✸✵

❖ ♠❛♣❛D ❛❝✐♠❛ é ✉♠❛ ✜❣✉r❛ ❞✐s❝r❡t❛ ♦♥❞❡ ❛ r❡s♦❧✉çã♦ ♠❛✐♦r ❢♦✐ ✉t✐❧✐③❛❞❛ ♥❛s

♣r♦①✐♠✐❞❛❞❡s ❞❛ ❢r♦♥t❡✐r❛ ❞♦ ♠❛♣❛ r❡❛❧ M ✭✈❡❥❛ r❡❣✐ã♦ ❞❡st❛❝❛❞❛ ♥❛ ❝♦r ❧❛r❛♥❥❛

♥❛ ✜❣✉r❛ ❛❜❛✐①♦✮✳ ◆♦t❡ q✉❡M é ✉♠❛ ✜❣✉r❛ ♣❧❛♥❛ ❛❜❡rt❛ ♣♦rq✉❡ s❡✉ ✐♥t❡r✐♦r é ♥ã♦

✈❛③✐♦ ❡ ❛ ❞✐✈✐s❛ ✭❢r♦♥t❡✐r❛✮ ❞❡ ✉♠❛ ✉♥✐❞❛❞❡ ❞❛ ❢❡❞❡r❛çã♦ é ✉♠❛ ❧✐♥❤❛ ✐♠❛❣✐♥ár✐❛

q✉❡ ♥ã♦ ♣❡rt❡♥❝❡ ❛ t❡rr✐tór✐♦ ❛❧❣✉♠✳ ❖ ✉s♦ ❞♦ s✉♣r❡♠♦ ♥❛ ❞❡✜♥✐çã♦ ❞❡t❡r♠✐♥❛

q✉❡ λ2(M) é ✉♠ ✈❛❧♦r ❡①❛t♦✱ ♠❛✐♦r ❡ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ✐❣✉❛❧ ❛ λ1(D)✳

❖ r❡s✉❧t❛❞♦ ❛ s❡❣✉✐r ❛✜r♠❛ q✉❡ ❛✉♠❡♥t❛r ❛ r❡s♦❧✉çã♦ ✐♠♣❧✐❝❛ ❡♠ ♠❡❧❤♦r❛r

❛ ❛♣r♦①✐♠❛çã♦ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❛ ár❡❛ ❞❡ ▲❡❜❡s❣✉❡ ❞❡ ✉♠❛ ✜❣✉r❛ ♣❧❛♥❛ ❛❜❡rt❛

❧✐♠✐t❛❞❛✳

Pr♦♣♦s✐çã♦ ✶✳ P❛r❛ q✉❛✐sq✉❡r G ∈ A ❡ β < α✱ G ❧✐♠✐t❛❞❛ ❡ β s✉✜❝✐❡♥t❡✲

♠❡♥t❡ ♣❡q✉❡♥♦✱ t❡♠♦s

sup{λ1(X) : X ⊂ G, X ∈ Qα} < sup{λ1(X) : X ⊂ G, X ∈ Qβ} ≤ λ2(G) .

✸✶

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ A = {λ1(X) : X ⊂ G, X ∈ Qα} ❡ B = {λ1(X) :

X ⊂ G, X ∈ Qβ}✳ ◆♦t❡ q✉❡ A ❡ B sã♦ ❝♦♥❥✉♥t♦s ✜♥✐t♦s ♣♦rq✉❡ G é ❧✐♠✐✲

t❛❞❛✳ ❯♠❛ ✈❡③ q✉❡ Qα ❡ Qβ sã♦ s✉❜❝♦♥❥✉♥t♦s ❞❡ Q✱ ♦❜t❡♠♦s q✉❡ A ❡ B sã♦

s✉❜❝♦♥❥✉♥t♦s ❞❡ {λ1(X) : X ⊂ G, X ∈ Q}✳ ▲♦❣♦✱ ♣❡❧♦ ▲❡♠❛ ✷✱ ❝♦♥❝❧✉✐♠♦s

q✉❡ sup(A) ❡ sup(B) ♥ã♦ s✉♣❡r❛♠ ♦ ✈❛❧♦r ❞❡ λ2(G)✳ ❱❛♠♦s ♠♦str❛r ❛❣♦r❛ q✉❡

sup(A) < sup(B)✳ ❈❛s♦ ♥ã♦ ❡①✐st❛ X ∈ Qα t❛❧ q✉❡ X ⊂ G✱ ❡♥tã♦ A = ∅✱ ✐♠✲

♣❧✐❝❛♥❞♦ ❡♠ sup(A) = −∞ ❡ ♦ r❡s✉❧t❛❞♦ s❡❣✉❡ ❛♦ ❡s❝♦❧❤❡r♠♦s β s✉✜❝✐❡♥t❡♠❡♥t❡

♣❡q✉❡♥♦ ❛ ♣♦♥t♦ ❞❡ ❡①✐st✐r X ∈ Qβ t❛❧ q✉❡ X ⊂ G✳ ❈❛s♦ ❝♦♥trár✐♦✱ s❡❥❛ W ❛

♠❛✐♦r ✜❣✉r❛ ❞✐s❝r❡t❛ ❞❡ Qα ❝♦♥t✐❞❛ ❡♠ G✳ P♦❞❡♠♦s ❡s❝r❡✈❡r

G = (G ∩W c) ∪W .

P❡❧♦ ▲❡♠❛ ✺✱ W c é ❛❜❡rt❛ ❡✱ ♣❡❧♦ ▲❡♠❛ ✸✱ G∩W c t❛♠❜é♠ é ❛❜❡rt❛✳ ▲♦❣♦✱ ♣❛r❛

β s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦✱ ❡①✐st❡ ✉♠ ❡❧❡♠❡♥t♦ ❞❡ ár❡❛ β✱ ❞✐❣❛♠♦s Q✱ ❝♦♥t✐❞♦ ❡♠

G ∩W c✳ ❈♦♥s✐❞❡r❡ Y = W ∪Q ❝♦♠♦ ✉♠❛ ✜❣✉r❛ ❞✐s❝r❡t❛ ❡♠ Qβ✳ ❙❡♥❞♦ ❛ss✐♠✱

sup(B) ≥ λ1(Y ) = λ1(W ) + β > λ1(W ) = sup(A) �

❖❜s❡r✈❡ q✉❡✿

✶✳ ❖ ❢❛t♦ ❞❡ G s❡r ✉♠❛ ✜❣✉r❛ ♣❧❛♥❛ ❛❜❡rt❛ é ❡①tr❡♠❛♠❡♥t❡ ✐♠♣♦rt❛♥t❡ ♣♦rq✉❡

♣♦ss✐❜✐❧✐t❛ ❛ ❡①✐stê♥❝✐❛ ❞❡ ♣❡❧♦ ♠❡♥♦s ✉♠ ❡❧❡♠❡♥t♦ ❞❡ ár❡❛ α s✉✜❝✐❡♥t❡♠❡♥t❡

♣❡q✉❡♥♦✱ ❞✐❣❛♠♦s Q ∈ Q✱ t❛❧ q✉❡ Q ⊂ G✳ ❙❡♥❞♦ ❛ss✐♠✱ ❛ ár❡❛ ❞❡ ▲❡❜❡s❣✉❡

❞❡ ✜❣✉r❛s ❡♠ Q ∪A é s❡♠♣r❡ ♣♦s✐t✐✈❛ ♦✉ ✐♥✜♥✐t❛❀

✷✳ ❉❛❞♦ G ∈ A ❧✐♠✐t❛❞❛✱ λ1(R) é ✉♠ ❧✐♠✐t❛♥t❡ s✉♣❡r✐♦r ♣❛r❛ {λ1(X) : X ⊂G, X ∈ Q}✱ ♦♥❞❡ R é ✉♠ r❡tâ♥❣✉❧♦ ❡♠ Q ❝♦♥t❡♥❞♦ G✱ ❣❛r❛♥t✐♥❞♦ ❛ss✐♠

q✉❡ λ(G) t❡♠ ✉♠ ✈❛❧♦r ✜♥✐t♦❀

✸✳ ◆ã♦ é ❞✐❢í❝✐❧ ✈❡r✐✜❝❛r q✉❡ λ2(W ) = λ1(W ) ♣❛r❛ q✉❛❧q✉❡r W ∈ Q✳ ❉❡

❢❛t♦✱ ✉♠❛ ✈❡③ q✉❡ W ⊂ W ❡ W ∈ Q✱ t❡♠♦s λ1(W ) ≤ λ2(W ) ♣♦r ❝❛✉s❛ ❞❛

❞❡✜♥✐çã♦ ❞❡ λ2✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❞✐❛♥t❡ ❞❛ ♠♦♥♦t♦♥✐❝✐❞❛❞❡ ❞❡ λ1✱ ✈❡r✐✜❝❛♠♦s

q✉❡ λ1(W ) é ❧✐♠✐t❛♥t❡ s✉♣❡r✐♦r ♣❛r❛ {λ1(X) : X ⊂ W, X ∈ Q}✱ ❞❡ ♦♥❞❡

❝♦♥❝❧✉✐♠♦s λ1(W ) ≥ λ2(W )❀

✹✳ ❙❡❥❛♠ U ❡ V ❞✉❛s ✜❣✉r❛s ♣❧❛♥❛s ❛❜❡rt❛s ♥ã♦ ✈❛③✐❛s ❡ W ⊂ U ∪ V ✉♠❛

✜❣✉r❛ ❞✐s❝r❡t❛✳ ❱❛♠♦s ❛❞♠✐t✐r s❡♠ ❞❡♠♦♥str❛çã♦ q✉❡ ❡①✐st❡♠ D,E ∈ Qt❛✐s q✉❡ D ⊂ U ✱ E ⊂ V ❡ W = D ∪ E✳ ❊ss❡ ❢❛t♦✱ ❞❡ ❡♥✉♥❝✐❛❞♦ s✐♠♣❧❡s ❡

❞❡♠♦♥str❛çã♦ ❛❧é♠ ❞♦ ❛❧❝❛♥❝❡ ❞❛ t❡♦r✐❛ q✉❡ ♥♦s ♣r♦♣♦♠♦s ❛ ❛♣r❡s❡♥t❛r✱ é

✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ❞♦ ▲❡♠❛ ❞♦ ◆ú♠❡r♦ ❞❡ ▲❡❜❡s❣✉❡ q✉❡ ❣❛r❛♥t❡ ❛ ❡①✐stê♥❝✐❛

❞❡ α s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦ ✭♦ ♥ú♠❡r♦ ❞❡ ▲❡❜❡s❣✉❡✮ ❞❡ ❢♦r♠❛ q✉❡ ❝❛❞❛

✸✷

❡❧❡♠❡♥t♦ ❞❡ ár❡❛ α q✉❡ ❝♦♥st✐t✉✐ W ❡st❡❥❛ ♣r♦♣r✐❛♠❡♥t❡ ❝♦♥t✐❞♦ ❡♠ ✉♠ ❞♦s

❛❜❡rt♦s U ♦✉ V ❀

✺✳ ❆ ❢✉♥çã♦ λ2 : Q ∪A → [0,+∞] t❛♠❜é♠ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ár❡❛ ❞❡ P❡❛♥♦✲

❏♦r❞❛♥✳

❆ ♣ró①✐♠❛ ♣r♦♣♦s✐çã♦ ♠♦str❛ ❝♦♠♦ λ2 s❡ ❝♦♠♣♦rt❛ ❞✐❛♥t❡ ❞❡ ✐♥❝❧✉sõ❡s ❡

✉♥✐õ❡s✳ ❈♦♠♦ ❥á ❢♦✐ ♦❜s❡r✈❛❞♦✱ ♦ ❝❛s♦ ❞❛s ✜❣✉r❛s ❞✐s❝r❡t❛s é ❜❛st❛♥t❡ s✐♠♣❧❡s

❡ ♥♦s ❝♦♥❝❡♥tr❛r❡♠♦s ❛♣❡♥❛s ♥❛s ✜❣✉r❛s ♣❧❛♥❛s ❛❜❡rt❛s✳ P❛r❛ ❞❡♠♦♥strá✲❧❛✱

♣r❡❝✐s❛♠♦s ❞♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦ s♦❜r❡ ♦ s✉♣r❡♠♦ ❞❛ s♦♠❛ ❞❡ ❞♦✐s ❝♦♥❥✉♥t♦s✳

▲❡♠❛ ✻✳ ❙✉♣♦♥❤❛ q✉❡ A,B ⊂ R ♣♦ss✉❡♠ ❧✐♠✐t❛♥t❡s s✉♣❡r✐♦r❡s ❡ ❝♦♥s✐❞❡r❡ ♦

❝♦♥❥✉♥t♦

A+B = {x+ y ∈ R : x ∈ A ❡ y ∈ B}.

❊♥tã♦ sup(A + B) = sup(A) + sup(B)✳ ❈❛s♦ A,B ⊂ R ♣♦ss✉❛♠ ❧✐♠✐t❛♥t❡s

✐♥❢❡r✐♦r❡s✱ t❡r❡♠♦s inf(A+B) = inf(A) + inf(B)✳

❉❡♠♦♥str❛çã♦✳ ❉❡ x ≤ sup(A) ❡ y ≤ sup(B)✱ ♣❛r❛ q✉❛✐sq✉❡r x ∈ A ❡

y ∈ B✱ ❝♦♥❝❧✉✐♠♦s q✉❡ x + y ≤ sup(A) + sup(B) ♣❛r❛ q✉❛❧q✉❡r x + y ∈ A + B✳

▲♦❣♦✱ sup(A+B) ≤ sup(A) + sup(B)✳

❉❡ x+ y ≤ sup(A+B) ♣❛r❛ q✉❛✐sq✉❡r x ∈ A ❡ y ∈ B✱ ♦❜t❡♠♦s q✉❡✱ ❞❛❞♦ y ∈B✱ sup(A+B)−y é ✉♠ ❧✐♠✐t❛♥t❡ s✉♣❡r✐♦r ♣❛r❛ A✱ ❡ ❛ss✐♠ sup(A) ≤ sup(A+B)−y

♣❛r❛ q✉❛❧q✉❡r y ∈ B✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ y ≤ sup(A + B) − sup(A) ♣❛r❛ t♦❞♦

y ∈ B✱ ❞❡ ♦♥❞❡ ❝♦♥❝❧✉✐♠♦s q✉❡ sup(B) ≤ sup(A+B)− sup(A)✳ ❆ ❞❡♠♦♥str❛çã♦

♣❛r❛ ♦ ❝❛s♦ ❞♦ í♥✜♠♦ é ❛♥á❧♦❣❛ �

Pr♦♣♦s✐çã♦ ✷✳ ❙❡❥❛♠ U ❡ V ❞✉❛s ✜❣✉r❛s ♣❧❛♥❛s ❛❜❡rt❛s ❧✐♠✐t❛❞❛s✳

❛✮ ✭▼♦♥♦t♦♥✐❝✐❞❛❞❡✮ ❙❡ U ⊂ V ✱ ❡♥tã♦ λ2(U) ≤ λ2(V )❀

❜✮ ✭❙✉❜❛❞✐t✐✈✐❞❛❞❡✮ ❊♠ ❣❡r❛❧✱ λ2(U ∪ V ) ≤ λ2(U) + λ2(V )❀

❝✮ ✭❆❞✐t✐✈✐❞❛❞❡✮ ❙❡ U ∩ V = ∅✱ ❡♥tã♦ λ2(U ∪ V ) = λ2(U) + λ2(V )✳

❉❡♠♦♥str❛çã♦✳

❛✮ ❇❛st❛ ♦❜s❡r✈❛r q✉❡ λ1(X) ≤ λ2(V ) ♣❛r❛ q✉❛❧q✉❡r X ∈ Q t❛❧ q✉❡ X ⊂ U ❀

❜✮ ❙❡ U ♦✉ V é ✈❛③✐♦✱ ♦ r❡s✉❧t❛❞♦ s❡❣✉❡ ✈❛❧❡♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡✳ ❙❡❥❛ W ∈ Qt❛❧ q✉❡ W ⊂ U ∪ V ♦♥❞❡ U ❡ V sã♦ ♥ã♦ ✈❛③✐♦s✳ ❊♥tã♦ ❡①✐st❡♠ D,E ∈ Q t❛✐s

q✉❡ D ⊂ U ✱ E ⊂ V ❡ W = D ∪ E✳ ❉❛ ❞❡✜♥✐çã♦ ❞❡ λ2 ❡ ❞❛ s✉❜❛❞✐t✐✈✐❞❛❞❡

❞❡ λ1✱ ❝♦♥❝❧✉✐♠♦s q✉❡ λ2(U) + λ2(V ) ≥ λ1(D) + λ1(E) ≥ λ1(D ∪ E)✱ ♦✉ s❡❥❛✱

λ2(U) + λ2(V ) é ✉♠ ❧✐♠✐t❛♥t❡ s✉♣❡r✐♦r ♣❛r❛ {λ1(W ) : W ⊂ U ∪ V, W ∈ Q}❀

❝✮ ❙❡❥❛ W ∈ Q ❝♦♠♦ ♥♦ ✐t❡♠ ❜ ❛❝✐♠❛ ♦♥❞❡ D,E ∈ Q t❛✐s q✉❡ D ⊂ U ✱

E ⊂ V ✳ ❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❞❡ U ∩ V = ∅✱ t❡♠♦s t❛♠❜é♠ D ∩ E = ∅✳ ▲♦❣♦✱

✸✸

♣❡❧❛ ❛❞✐t✐✈✐❞❛❞❡ ❞❡ λ1✱ t❡♠♦s

λ1(D) + λ1(E) = λ1(D ∪ E),

❡ ❛ss✐♠ λ2(U ∪V ) = sup(S)✱ ♦♥❞❡ S = {λ1(D)+λ1(E) : D ⊂ U,E ⊂ V ❡ D,E ∈Q}, ♦ q✉❡ ✐♠♣❧✐❝❛✱ ♣❡❧♦ ▲❡♠❛ ✻✱ ❡♠ λ2(U) + λ2(V ) = sup(S) = λ2(U ∪ V ) �

✹✳✹ ➪r❡❛ ❡①t❡r✐♦r

❙❡❥❛ X ✉♠❛ ✜❣✉r❛ ♣❧❛♥❛ q✉❛❧q✉❡r✳ ❉❡✜♥✐♠♦s ❛ ár❡❛ ❡①t❡r✐♦r ❞❡ ▲❡❜❡s❣✉❡ ❞❡ X

♣♦r

λ∗(X) = inf{λ2(G) : G ⊃ X, G ∈ A}.

❚r❛t❛✲s❡ ❞❡ ✉♠❛ ❢✉♥çã♦ λ∗ : F → [0,+∞]✱ ♦♥❞❡ F é ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s

✜❣✉r❛s ♣❧❛♥❛s✳ ❙❡♥❞♦ ❛ss✐♠✱ ❛ ár❡❛ ❡①t❡r✐♦r ❞❡ ▲❡❜❡s❣✉❡ ❞❡ ✉♠❛ ✜❣✉r❛ ♣❧❛♥❛

q✉❛❧q✉❡r X é ❛♣r♦①✐♠❛❞❛✱ ♣♦r ❡①❝❡ss♦✱ ♣❡❧❛ ár❡❛ ❞❡ ✉♠❛ ✜❣✉r❛ ♣❧❛♥❛ ❛❜❡rt❛

G q✉❡ ✏❝♦❜r❡✑ X✳ ❉✐r❡♠♦s ❡♥tã♦ q✉❡ G ❝♦❜r❡ X q✉❛♥❞♦ X ⊂ G ✭❡q✉✐✈❛❧❡♥t❡

❛ G ⊃ X✮✳ ◆♦t❡ q✉❡ ♦ í♥✜♠♦ ❞♦ ❝♦♥❥✉♥t♦ {λ2(G) : G ⊃ X, G ∈ A} s❡♠♣r❡

❡①✐st❡ ♣♦rq✉❡ λ2(G) é ✉♠ ♥ú♠❡r♦ r❡❛❧ ♥ã♦ ♥❡❣❛t✐✈♦ ♣❛r❛ q✉❛❧q✉❡r G ∈ A ♦✉

λ2(G) = +∞✳ ❱❛♠♦s s✐♠❜♦❧✐③❛r ❝♦♠ Γ(X) ❛ ❢❛♠í❧✐❛ ❞❡ t♦❞❛s ❛s ✜❣✉r❛s ♣❧❛♥❛s

❛❜❡rt❛s q✉❡ ❝♦❜r❡♠ ❛ ✜❣✉r❛ ♣❧❛♥❛ X✳

❖ r❡s✉❧t❛❞♦ ❛ s❡❣✉✐r é ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ✐♠❡❞✐❛t❛ ❞♦ ▲❡♠❛ ✶ ❡ ✉♠❛ ❢❡rr❛♠❡♥t❛

út✐❧ ♣❛r❛ ❡♥❝♦♥tr❛r ✜❣✉r❛s ♣❧❛♥❛s ❝♦♠ ár❡❛ ❡①t❡r✐♦r ❞❡ ▲❡❜❡s❣✉❡ ♥✉❧❛✳

▲❡♠❛ ✼✳ ❙❡❥❛ X ∈ F ✳ ❙❡✱ ♣❛r❛ q✉❛❧q✉❡r ε > 0✱ ❡①✐st❡ Gε ∈ Γ(X) t❛❧ q✉❡

λ2(Gε) < ε✱ ❡♥tã♦ λ∗(X) = 0.

P♦r ❡①❡♠♣❧♦✱ ❛ ✜❣✉r❛ ♣❧❛♥❛ ❛❜❡rt❛ ❛♠❛r❡❧❛ ✭✈❡❥❛ ✜❣✉r❛ ❛❜❛✐①♦✮ ♣♦❞❡ t❡r ár❡❛

❛r❜✐tr❛r✐❛♠❡♥t❡ ♣❡q✉❡♥❛✱ ♠❡♥♦r q✉❡ ε✱ ❛✐♥❞❛ ❝♦♥t❡♥❞♦ ❛s ❛r❡st❛s ♣r❡t❛s✳ ❋✐❣✉r❛s

❛❜❡rt❛s s❡♠❡❧❤❛♥t❡s à ✜❣✉r❛ ❛♠❛r❡❧❛ ♣♦❞❡♠ s❡r ❢❡✐t❛s ❡♥✈♦❧✈❡♥❞♦ q✉❛❧q✉❡r ♦✉tr❛

✜❣✉r❛ ♣❧❛♥❛ ❡❧❡♠❡♥t❛r ✭tr✐â♥❣✉❧♦✱ ♣❛r❛❧❡❧♦❣r❛♠♦✱ q✉❛❞r❛❞♦✱ r❡tâ♥❣✉❧♦ ❡ ❝ír❝✉❧♦✮✳

❆♣❧✐❝❛♥❞♦ ♦ ▲❡♠❛ ✼✱ ❝♦♥❝❧✉✐♠♦s q✉❡ ❛ ✜❣✉r❛ ❢♦r♠❛❞❛ ❛♣❡♥❛s ♣❡❧❛s ❛r❡st❛s

❞❡ ✉♠❛ ✜❣✉r❛ ♣❧❛♥❛ ❡❧❡♠❡♥t❛r t❡♠ ár❡❛ ❡①t❡r✐♦r ❞❡ ▲❡❜❡s❣✉❡ ♥✉❧❛✳

✸✹

❆ Pr♦♣♦s✐çã♦ ✸ ❛ s❡❣✉✐r ❢♦r♥❡❝❡ ♣r♦♣r✐❡❞❛❞❡s ♠✉✐t♦ ✐♠♣♦rt❛♥t❡s ♣❛r❛ ❛ ár❡❛

❡①t❡r✐♦r ❞❡ ▲❡❜❡s❣✉❡ q✉❡ ❛ t♦r♥❛♠ ❧❡❣ít✐♠❛ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❛ ár❡❛ ❞❡ ✜❣✉r❛s

♣❧❛♥❛s✳ ❆♥t❡s ❞❡ ♣r♦✈á✲❧❛✱ ♣r❡❝✐s❛♠♦s ❞♦ s❡❣✉✐♥t❡ ❧❡♠❛✳

▲❡♠❛ ✽✳ ❙✉♣♦♥❤❛ q✉❡ A ⊂ R t❡♠ ❧✐♠✐t❛♥t❡s ✐♥❢❡r✐♦r❡s✳ ❙❡✱ ♣❛r❛ t♦❞♦ ε > 0✱

❡①✐st❡ x ∈ A✱ ♣♦ss✐✈❡❧♠❡♥t❡ ❞❡♣❡♥❞❡♥t❡ ❞❡ ε✱ t❛❧ q✉❡ x < c + ε ♣❛r❛ ✉♠ ❝❡rt♦

c ∈ R✱ ❡♥tã♦ inf(A) ≤ c✳

❉❡♠♦♥str❛çã♦✳ ❆❞♠✐t❛ q✉❡ inf(A) > c ❡ ❢❛ç❛ ε = inf(A) − c✳ ❊①✐st✐r✐❛

❡♥tã♦ x ∈ A t❛❧ q✉❡ x < c+ inf(A)− c = inf(A), ✉♠ ❛❜s✉r❞♦ �

Pr♦♣♦s✐çã♦ ✸✳ ❙❡❥❛♠ X ❡ Y ❡♠ F ❧✐♠✐t❛❞❛s✳

❛✮ ✭▼♦♥♦t♦♥✐❝✐❞❛❞❡✮ ❙❡ X ⊂ Y ✱ ❡♥tã♦ λ∗(X) ≤ λ∗(Y )❀

❜✮ ✭❙✉❜❛❞✐t✐✈✐❞❛❞❡✮ ❊♠ ❣❡r❛❧✱ λ∗(X ∪ Y ) ≤ λ∗(X) + λ∗(Y )❀

❝✮ ✭❆❞✐t✐✈✐❞❛❞❡ ❘❡str✐t❛✮ ❙❡ ❡①✐st❡♠ U ∈ Γ(X) ❡ V ∈ Γ(Y ) t❛✐s q✉❡ U∩V = ∅✱❡♥tã♦ λ∗(X ∪ Y ) = λ∗(X) + λ∗(X)✳

❉❡♠♦♥str❛çã♦✳

❛✮ ❙❡ G ⊃ Y ❡ Y ⊃ X✱ ❡♥tã♦ G ⊃ X✱ ✐st♦ é✱ s♦❜ ❛ ❤✐♣ót❡s❡ X ⊂ Y ✱

t♦❞❛ ✜❣✉r❛ ♣❧❛♥❛ ❛❜❡rt❛ q✉❡ ❝♦❜r❡ Y t❛♠❜é♠ ❝♦❜r✐rá X✳ ▲♦❣♦✱ {λ2(G) : G ∈Γ(Y )} ⊂ {λ2(G) : G ∈ Γ(X)}, ♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠ λ∗(Y ) = inf{λ2(G) : G ∈Γ(Y )} ≥ inf{λ2(G) : G ∈ Γ(X)} = λ∗(X)❀

❜✮ ❉❛❞♦ ε > 0✱ ❡①✐st❡♠ U ∈ Γ(X) ❡ V ∈ Γ(Y )✱ ♣♦ss✐✈❡❧♠❡♥t❡ ❞❡♣❡♥❞❡♥t❡s

❞❡ ε✱ t❛✐s q✉❡

λ2(U) < λ∗(X) +ε

2❡ λ2(V ) < λ∗(Y ) +

ε

2.

▲♦❣♦✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✷✱ ✐t❡♠ ❜✱ ♦❜t❡♠♦s λ2(U ∪V ) ≤ λ2(U)+λ2(V ) < λ∗(X)+

λ∗(Y ) + ε✱ ♣❛r❛ q✉❛❧q✉❡r ε > 0✳ ❈♦♥❝❧✉✐♠♦s q✉❡ λ∗(X ∪ Y ) ≤ λ∗(X) + λ∗(Y )

♦❜s❡r✈❛♥❞♦ q✉❡ U ∪ V ∈ Γ(X ∪ Y ) ❡ ✉s❛♥❞♦ ♦ ▲❡♠❛ ✽❀

❝✮ ❈♦♥s✐❞❡r❡ ♦s s❡❣✉✐♥t❡s ❝♦♥❥✉♥t♦s ❞❡ ✜❣✉r❛s ♣❧❛♥❛s ❛❜❡rt❛s✿

M = Γ(X ∪ Y ) ❡ N = {U ∪ V : U ∈ Γ(X), V ∈ Γ(Y ) ❡ U ∩ V = ∅}.

❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ✐♠❡❞✐❛t❛ ❞❛ ❤✐♣ót❡s❡✱ t❡♠♦s N 6= ∅✳ ❆❧é♠ ❞✐ss♦✱ ❞❛❞♦ G ∈M✱ ❡①✐st❡ H = U ∪ V ∈ N t❛❧ q✉❡ H ⊂ G✳ ❆ ❞❡✜♥✐çã♦ ❞❡ í♥✜♠♦ ✐♠♣❧✐❝❛ q✉❡✱

❞❛❞♦ ε > 0✱ ❡①✐st❡ G ∈ M t❛❧ q✉❡ λ2(G) ≤ λ∗(X ∪ Y ) + ε✳ ❙❡♥❞♦ ❛ss✐♠✱ ♣❡❧❛

Pr♦♣♦s✐çã♦ ✷✱ ✐t❡♥s ❛ ❡ ❝✱ t❡♠♦s

λ2(U) + λ2(V ) = λ2(H) ≤ λ2(G) < λ∗(X ∪ Y ) + ε,

❞❡ ♦♥❞❡ ❝♦♥❝❧✉✐♠♦s q✉❡ λ∗(X) + λ∗(Y ) < λ∗(X ∪ Y ) + ε ♣❛r❛ t♦❞♦ ε > 0✳ P❡❧♦

✸✺

✐t❡♠ ❜ ❛❝✐♠❛✱

λ∗(X) + λ∗(Y )− λ∗(X ∪ Y ) ≥ 0

❡ ❛ss✐♠✱ ♣❡❧♦ ▲❡♠❛ ✶✱ λ∗(X) + λ∗(Y ) = λ∗(X ∪ Y ) �

◆♦ ✐t❡♠ ❝ ❞❛ Pr♦♣♦s✐çã♦ ✸✱ ♥ã♦ é s✉✜❝✐❡♥t❡ s✉♣♦r ❛♣❡♥❛s X ∩ Y = ∅ ♣❛r❛

♦❜t❡r♠♦s λ∗(X ∪ Y ) = λ∗(X) + λ∗(Y ) ❝♦♠♦ ♦❝♦rr❡✉ ❝♦♠ ❛ ❛❞✐t✐✈✐❞❛❞❡ ❞❡ λ1 ❡

λ2 ✭♣♦r ✐ss♦ ❛ ♣r♦♣r✐❡❞❛❞❡ ❢♦✐ ❝❤❛♠❛❞❛ ❞❡ ✏❛❞✐t✐✈✐❞❛❞❡ r❡str✐t❛✑✮✳ ❊①❡♠♣❧♦s ❞❡

❝♦♥❥✉♥t♦s X, Y ∈ F t❛✐s q✉❡ X ∩ Y = ∅ ♣❛r❛ ♦s q✉❛✐s ♥ã♦ ✈❛❧❡ ❛ ❛❞✐t✐✈✐❞❛❞❡ ❞❛

ár❡❛ ❡①t❡r✐♦r ❞❡ ▲❡❜❡s❣✉❡ r❡q✉❡r❡♠ ❛r❣✉♠❡♥t♦s t❡ór✐❝♦s ❛❧é♠ ❞❛q✉❡❧❡s q✉❡ ♥♦s

♣r♦♣♦♠♦s ❛ ❛♣r❡s❡♥t❛r ❛q✉✐✳ ■♥❢♦r♠❛♠♦s ❛♣❡♥❛s q✉❡✱ ♦❜✈✐❛♠❡♥t❡ ❡ ♣❛r❛ ❝♦♠❡ç❛r✱

❞❡✈❡♠♦s ❞❡✐①❛r ❞❡ s❛t✐s❢❛③❡r ❛ ❤✐♣ót❡s❡ ❛❞✐❝✐♦♥❛❧ ✭r❡str✐çã♦ ✏❡①✐st❡♠ U ∈ Γ(X) ❡

V ∈ Γ(Y ) t❛✐s q✉❡ U ∩V = ∅✑✮✳ ❖❝♦rr❡✱ ♣♦r ❡①❡♠♣❧♦✱ ❝♦♠ ♦ ❝♦♥❥✉♥t♦ P ❞❡ ♣❛r❡s

❞❡ ♥ú♠❡r♦s r❛❝✐♦♥❛✐s (r, s) ♣❡rt❡♥❝❡♥t❡s ❛♦ q✉❛❞r❛❞♦ ❢❡❝❤❛❞♦ [0, 1]× [0, 1] ❡ ♦ s❡✉

❝♦♠♣❧❡♠❡♥t❛r ❡♠ r❡❧❛çã♦ ❛♦ q✉❛❞r❛❞♦ ❢❡❝❤❛❞♦✱ ♦✉ s❡❥❛✱ I = P c∩ [0, 1]× [0, 1]✿ P

❡ I sã♦ ❞✐s❥✉♥t♦s✱ ✐st♦ é✱ P ∩ I = ∅✱ ♠❛s ♥ã♦ ❡①✐st❡♠ ❛❜❡rt♦s ❞✐s❥✉♥t♦s ❝♦♥t❡♥❞♦

P ❡ I✳

❈♦r♦❧ár✐♦ ✶✳ ❋✐❣✉r❛s ♣❧❛♥❛s ❡❧❡♠❡♥t❛r❡s ✭tr✐â♥❣✉❧♦✱ ♣❛r❛❧❡❧♦❣r❛♠♦✱ q✉❛✲

❞r❛❞♦✱ r❡tâ♥❣✉❧♦ ❡ ❝ír❝✉❧♦✮ ❢❡❝❤❛❞❛s ♥ã♦ tê♠ ❛ s✉❛ ár❡❛ ❡①t❡r✐♦r ❞❡ ▲❡❜❡s❣✉❡

❛❧t❡r❛❞❛ q✉❛♥❞♦ ❞❡❧❛s r❡t✐r❛♠♦s ❛s ❛r❡st❛s✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❞❛❞❛ X ∈ Q ❧✐♠✐t❛❞❛✱

t❡♠♦s λ∗(X) = λ∗(✐♥t(X))✳

❉❡♠♦♥str❛çã♦✳ ◆♦t❡ q✉❡ q✉❛❧q✉❡r ✜❣✉r❛ ♣❧❛♥❛ ❡❧❡♠❡♥t❛r ❢❡❝❤❛❞❛ S é ❛

✉♥✐ã♦ ❞❛ ✜❣✉r❛ ♣❧❛♥❛ ❡❧❡♠❡♥t❛r ❛❜❡rt❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ✐♥t(S) ❝♦♠ s✉❛ ❢r♦♥t❡✐r❛

❢r(S)✱ ✐st♦ é✱ S = ✐♥t(S) ∪ ❢r(S)✳ P❡❧❛ s✉❜❛❞✐t✐✈✐❞❛❞❡ ❞❡ λ∗ ❡ ♣❡❧♦ ▲❡♠❛ ✼ t❡♠♦s

λ∗(S) ≤ λ∗ ( ✐♥t(S)) + λ∗( ❢r(S)) = λ∗ ( ✐♥t(S)) .

❉❛ ♠♦♥♦t♦♥✐❝✐❞❛❞❡ ❞❡ λ∗ ❝♦♥❝❧✉✐♠♦s q✉❡ λ∗ ( ✐♥t(S)) ≤ λ∗(S)✳ ◆♦ ❝❛s♦ X ∈ Q✱

❜❛st❛ s✉❜st✐t✉✐r ♦s q✉❛❞r❛❞✐♥❤♦s ❢❡❝❤❛❞♦s q✉❡ ❝♦♥tê♠ ❛ ❢r♦♥t❡✐r❛ ♣♦r q✉❛❞r❛❞✐✲

♥❤♦s ❛❜❡rt♦s ❡ ❛♣❧✐❝❛r ❛ s✉❜❛❞✐t✐✈✐❞❛❞❡ r❡str✐t❛ ❞❡ λ∗�

❖ ♣ró①✐♠♦ r❡s✉❧t❛❞♦ ❞❡ss❡ ❝❛♣ít✉❧♦✱ ♠♦str❛ q✉❡ ❛ ♠❡❞✐❞❛ ❡①t❡r✐♦r ❞❡ ▲❡❜❡s❣✉❡

❡st❡♥❞❡ ❛ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ ❞❡ Q ∪ A ♣❛r❛ F ✳ ◆❛ ♣rát✐❝❛✱ ✐ss♦ s✐❣♥✐✜❝❛ q✉❡

❛q✉✐❧♦ q✉❡ ❥á s❛❜í❛♠♦s ❝❛❧❝✉❧❛r ✉s❛♥❞♦ λ2 ♥ã♦ ♣r❡❝✐s❛ s❡r r❡❝❛❧❝✉❧❛❞♦ ❛♦ ✉s❛r♠♦s

λ∗✳

▲❡♠❛ ✾✳ P❛r❛ q✉❛❧q✉❡r ✜❣✉r❛ ♣❧❛♥❛ ❛❜❡rt❛ W ✱ t❡♠♦s λ2(W ) = λ∗(W )✳

❉❡♠♦♥str❛çã♦✳ ❯♠❛ ✈❡③ q✉❡W é ❛❜❡rt❛✱ t❡♠♦sW ∈ Γ(W ) ❡✱ ♣❡❧❛ ❞❡✜♥✐çã♦

❞❡ λ∗✱ ❝♦♥❝❧✉✐♠♦s q✉❡ λ∗(W ) ≤ λ2(W )✱ ♦✉ s❡❥❛✱ λ2(W ) − λ∗(W ) ≥ 0✳ P♦r

♦✉tr♦ ❧❛❞♦✱ ❞❛❞♦ ε > 0✱ ❡①✐st❡ G ∈ Γ(W ) t❛❧ q✉❡ λ2(G) < λ∗(W ) + ε✳ ❉❛

✸✻

♠♦♥♦t♦♥✐❝✐❞❛❞❡ ❞❡ λ2 ♦❜t❡♠♦s

λ2(W ) ≤ λ2(G) < λ∗(W ) + ε,

♦✉ s❡❥❛✱ λ2(W )−λ∗(W ) < ε ♣❛r❛ q✉❛❧q✉❡r ε > 0✱ ✐♠♣❧✐❝❛♥❞♦ ❡♠ λ∗(W ) = λ2(W )

♣❡❧♦ ▲❡♠❛ ✶ �

▲❡♠❛ ✶✵✳ ❙❡ Q é ✉♠ ❡❧❡♠❡♥t♦ ❞❡ ár❡❛ α✱ ❡♥tã♦ λ1(Q) = λ∗(Q)✳

❉❡♠♦♥str❛çã♦✳ ❉❛❞♦ G ∈ Γ(Q)✱ t❡♠♦s λ1(Q) ≤ λ2(G) ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡

λ2✳ ▲♦❣♦✱ λ1(Q) é ✉♠ ❧✐♠✐t❛♥t❡ ✐♥❢❡r✐♦r ♣❛r❛ {λ2(G) : G ∈ Γ(Q)} ❡ ❛ss✐♠✱

♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ λ∗✱ ❝♦♥❝❧✉✐♠♦s q✉❡ λ1(Q) ≤ λ∗(Q)✳ P♦r ♦✉tr♦ ❧❛❞♦✱ λ∗(Q) =

λ∗( ✐♥t(Q)) ❝♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❞♦ ❈♦r♦❧ár✐♦ ✶✳ ❉♦ ▲❡♠❛ ✾✱ ❞❛ ❞❡✜♥✐çã♦ ❞❡ λ2 ❡

❞❛ ♠♦♥♦t♦♥✐❝✐❞❛❞❡ ❞❡ λ1 ❝♦♥❝❧✉✐♠♦s q✉❡ λ∗( ✐♥t(Q)) = λ2( ✐♥t(Q)) ≤ λ2(Q) �

Pr♦♣♦s✐çã♦ ✹✳ P❛r❛ q✉❛❧q✉❡r Y ∈ Q ∪A✱ t❡♠♦s λ∗(Y ) = λ2(Y )✳

❉❡♠♦♥str❛çã♦✳ ❙❡ Y ∈ A✱ ❡♥tã♦ ❛ ♣r♦♣♦s✐çã♦ s❡ r❡s✉♠❡ ❛♦ ▲❡♠❛ ✾✳ ❙✉✲

♣♦♥❤❛♠♦s ❡♥tã♦ Y ∈ Q✳ ❙❡♥❞♦ ❛ss✐♠✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ λ2✱ λ1(Y ) ≤ λ2(G) ♣❛r❛

q✉❛❧q✉❡r G ∈ Γ(Y )✳ ▲♦❣♦✱ λ1(Y ) é ✉♠ ❧✐♠✐t❛♥t❡ ✐♥❢❡r✐♦r ♣❛r❛ {λ2(G) : G ∈ Γ(Y )}❡ ❛ss✐♠✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ λ∗✱ ❝♦♥❝❧✉✐♠♦s q✉❡ λ1(Y ) ≤ λ∗(Y )✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❞❛

s✉❜❛❞✐t✐✈✐❞❛❞❡ ❞❡ λ∗✱ ❞♦ ▲❡♠❛ ✶✵ ❡ ❞❛ ❛❞✐t✐✈✐❞❛❞❡ ❞❡ λ1✱ ♦❜t❡♠♦s

λ∗(Y ) ≤n∑

j=1

λ∗(Qj) =n∑

j=1

λ1(Qj) = λ1

(

n⋃

j=1

Qj

)

= λ1(Y ),

♦♥❞❡ Y = Q1 ∪Q2 ∪ ... ∪Qn✱ Qj ∈ Qα ✱ ♣❛r❛ ❝❛❞❛ j = 1, ..., n✱ α s✉✜❝✐❡♥t❡♠❡♥t❡

♣❡q✉❡♥♦ ❞❡ ❢♦r♠❛ ❛ ❢❛③❡r ❝♦♠ q✉❡ t♦❞♦s ♦s q✉❛❞r❛❞✐♥❤♦s q✉❡ ❝♦♥st✐t✉❡♠ Y

t❡♥❤❛♠ ❛ ♠❡s♠❛ ár❡❛ α �

❙❡❣✉❡ ❞❛ ú❧t✐♠❛ ♣❛rt❡ ❞❛ ❞❡♠♦♥str❛çã♦ ❛❝✐♠❛ ♦

❈♦r♦❧ár✐♦ ✷✳ λ1(X) = λ∗(X) ♣❛r❛ q✉❛❧q✉❡r X ∈ Q✳

✹✳✺ ➪r❡❛ ✐♥t❡r✐♦r

❯♠❛ ✜❣✉r❛ ♣❧❛♥❛ K é ❝❤❛♠❛❞❛ ❞❡ ❝♦♠♣❛❝t❛ q✉❛♥❞♦ ❢♦r ❢❡❝❤❛❞❛ ❡ ❧✐♠✐t❛❞❛✳

❱❛♠♦s ❞❡♥♦t❛r ♣♦r K ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ✜❣✉r❛s ♣❧❛♥❛s ❝♦♠♣❛❝t❛s✳ ❆s ✜❣✉r❛s

♣❧❛♥❛s ❡❧❡♠❡♥t❛r❡s ❢❡❝❤❛❞❛s sã♦ ❡①❡♠♣❧♦s ❞❡ ✜❣✉r❛s ❝♦♠♣❛❝t❛s✳ ❖ ❝♦♥❥✉♥t♦ ✈❛③✐♦

é ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠♣❛❝t♦✳ ❙❡ P é ♦ s✉❜❝♦♥❥✉♥t♦ ❞❡ Q ❞❛s ✜❣✉r❛s ♣❧❛♥❛s ❢♦r♠❛❞❛s

♣♦r ✉♥✐õ❡s ✜♥✐t❛s ❞❡ q✉❛❞r❛❞♦s ❞♦ ♣❛♣❡❧ q✉❛❞r✐❝✉❧❛❞♦✱ ❡♥tã♦ P ⊂ K✱ ♦✉ s❡❥❛✱

q✉❛❧q✉❡r ❡❧❡♠❡♥t♦ ❞❡ P é ✉♠❛ ✜❣✉r❛ ♣❧❛♥❛ ❝♦♠♣❛❝t❛✳

❆ ár❡❛ ✐♥t❡r✐♦r ❞❡ ▲❡❜❡s❣✉❡ ❞❡ ✉♠❛ ✜❣✉r❛ ♣❧❛♥❛ q✉❛❧q✉❡r X é ❞❡✜♥✐❞❛ ♣♦r

✸✼

λ∗(X) = sup{λ∗(K) : K ⊂ X, K ∈ K}.

❆ ♣ró①✐♠❛ ♣r♦♣♦s✐çã♦ ♠♦str❛ ❛ r❡❧❛çã♦ ♠❛✐s ❣❡r❛❧ ❡♥tr❡ ❛ ár❡❛ ✐♥t❡r✐♦r ❡ ❛

ár❡❛ ❡①t❡r✐♦r ❞❡ ✉♠❛ ✜❣✉r❛ ♣❧❛♥❛✳

Pr♦♣♦s✐çã♦ ✺✳ P❛r❛ q✉❛❧q✉❡r X ∈ F ✱ λ∗(X) ≤ λ∗(X)✳

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ K ⊂ X ❝♦♠♣❛❝t♦✳ ❊♥tã♦✱ ❞❛ ♠♦♥♦t♦♥✐❝✐❞❛❞❡ ❞❡ λ∗✱

♦❜t❡♠♦s λ∗(K) ≤ λ∗(X)✳ ▲♦❣♦✱ λ∗(X) é ✉♠ ❧✐♠✐t❛♥t❡ s✉♣❡r✐♦r ♣❛r❛ {λ∗(K) :

K ⊂ X, K ∈ K} �

❖s ♣ró①✐♠♦s ❞♦✐s r❡s✉❧t❛❞♦s✱ ❢♦r♥❡❝❡♠ ❡①❡♠♣❧♦s ❞❡ ✜❣✉r❛s ♣❧❛♥❛s ♣❛r❛ ❛s

q✉❛✐s ✈❛❧❡ ❛ ✐❣✉❛❧❞❛❞❡ ♥❛ Pr♦♣♦s✐çã♦ ✺✳

Pr♦♣♦s✐çã♦ ✻✳ ❙❡ G ∈ A✱ ❡♥tã♦ λ∗(G) = λ∗(G)✳

❉❡♠♦♥str❛çã♦✳ ❙❛❜❡♥❞♦ q✉❡ t♦❞♦ W ∈ Q é ❝♦♠♣❛❝t♦ ❡ q✉❡ λ1(W ) =

λ∗(W )✱ ❝♦♥❝❧✉✐♠♦s q✉❡

{λ1(W ) : W ⊂ G, W ∈ Q} ⊂ {λ∗(K) : K ⊂ G, K ∈ K}.

▲♦❣♦✱ ♣❡❧♦ ▲❡♠❛ ✷✱ λ2(G) ≤ λ∗(G) ❡✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✹✱ ❝♦♥❝❧✉✐♠♦s q✉❡ λ∗(G) ≤λ∗(G) ❡ ♦❜t❡♠♦s ❛ ✐❣✉❛❧❞❛❞❡ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✺ �

Pr♦♣♦s✐çã♦ ✼✳ ❙❡ W ∈ K✱ ❡♥tã♦ λ∗(W ) = λ∗(W )✳

❉❡♠♦♥str❛çã♦✳ ❇❛st❛ ♦❜s❡r✈❛r q✉❡ λ∗(W ) ∈ {λ∗(K) : K ⊂ W, K ∈ K} �

✹✳✻ ❋✐❣✉r❛s ♠❡♥s✉rá✈❡✐s

❯♠❛ ✜❣✉r❛ ♣❧❛♥❛ X é ❝❤❛♠❛❞❛ ❞❡ ♠❡♥s✉rá✈❡❧ q✉❛♥❞♦ λ∗(X) = λ∗(X)✳ ❊ss❡

✈❛❧♦r ❝♦♠✉♠ é ❞❡♥♦t❛❞♦ ♣♦r λ(X) ❡ ❛ ❢✉♥çã♦

λ : M → [0,+∞]

é ❛ ár❡❛ ❞❡ ▲❡❜❡s❣✉❡✳ ❖ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ✜❣✉r❛s ♣❧❛♥❛s ♠❡♥s✉rá✈❡✐s é ❞❡♥♦✲

t❛❞♦ ♣♦r M✳ ❆ ♣ró①✐♠❛ ♣r♦♣♦s✐çã♦ ❛✜r♠❛ q✉❡ ♥❡♠ t♦❞❛ ✜❣✉r❛ ♣❧❛♥❛ é ▲❡❜❡s❣✉❡

♠❡♥s✉rá✈❡❧✳

Pr♦♣♦s✐çã♦ ✽✳ ❆ ✐♥❝❧✉sã♦ M ⊂ F é ♣ró♣r✐❛✳

❆ ❞❡♠♦♥str❛çã♦ ❞❛ Pr♦♣♦s✐çã♦ ✽ r❡q✉❡r ♦ ❢❛♠♦s♦ ❆①✐♦♠❛ ❞❛ ❊s❝♦❧❤❛✱ q✉❡ é

✉♠❛ ♣❡ç❛ s✐♥❣✉❧❛r ♥♦s ♠♦❞❡❧♦s ❧ó❣✐❝♦s ❞♦s ❢✉♥❞❛♠❡♥t♦s ❞❛ ❚❡♦r✐❛ ❞♦s ❈♦♥❥✉♥t♦s

✸✽

❡ ❡stá ❛❧é♠ ❞♦s ♥♦ss♦s ♣r♦♣ós✐t♦s ❛q✉✐✳ ❖ ❧❡✐t♦r ✐♥t❡r❡ss❛❞♦ ♣♦❞❡rá ❝♦♥s✉❧t❛r

q✉❛❧q✉❡r ❧✐✈r♦ ❞❡ ❚❡♦r✐❛ ❞❛ ▼❡❞✐❞❛✱ ♣♦r ❡①❡♠♣❧♦ ❬✺❪✳

✹✳✼ ❊①♣❡r✐♠❡♥t♦✿ ❝á❧❝✉❧♦ ❞❛ ár❡❛ ❞❡ ✜❣✉r❛s ♣❧❛♥❛s

♥ã♦ ❡❧❡♠❡♥t❛r❡s

◆❡ss❡ ❡①♣❡r✐♠❡♥t♦ ✉s❛♠♦s t♦❞♦s ♦s ❝♦♥❝❡✐t♦s ❛♥t❡r✐♦r♠❡♥t❡ ♠❡♥❝✐♦♥❛❞♦s✱ ❛ss♦✲

❝✐❛❞♦s ❛♦ ▼ét♦❞♦ ❞❡ ❊①❛✉stã♦ ❡ à ár❡❛ ❞❡ ▲❡❜❡s❣✉❡✳ ❱❛♠♦s ❡♥❝♦♥tr❛r ❛ ár❡❛ ❞❡

✉♠❛ ✜❣✉r❛ ♥ã♦ ❡❧❡♠❡♥t❛r ✉s❛♥❞♦ ❛s ✜❣✉r❛s ❡❧❡♠❡♥t❛r❡s tr✐â♥❣✉❧♦s ❡ q✉❛❞r❛❞♦s✳

P❛r❛ t❛❧ ✈❛♠♦s ✉s❛r ✉♠❛ ✜❣✉r❛ ♥ã♦ ❡❧❡♠❡♥t❛r ❛❜❡rt❛ ❡ ❝❛❧❝✉❧❛r s✉❛ ár❡❛ ♣♦r

❞❡♥tr♦ ❡ ♣♦r ❢♦r❛ ❝♦♠ ❛s ✜❣✉r❛s ❡❧❡♠❡♥t❛r❡s ♠❡♥❝✐♦♥❛❞❛s✳

❱❛♠♦s ✐♥s❡r✐r ❛ ✜❣✉r❛ ❡❧❡♠❡♥t❛r ♥♦ ✐♥t❡r✐♦r ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❡ ❞❡ ✉♠ q✉❛❞r❛❞♦

❡ ❞❡♣♦✐s ❞✐✈✐❞✐r❡♠♦s ♦s ♠❡s♠♦s ❡♠ ✹✱ ✶✻✱ ✻✹✱ ✷✺✻ ❡ ✶✵✷✹ ♣❛rt❡s✱ r❡t✐r❛♥❞♦ ❛s ár❡❛s

❡①t❡r✐♦r❡s ❡ s♦♠❛♥❞♦ ❛s ár❡❛s ✐♥t❡r✐♦r❡s✳

✸✾

✹✵

✹✶

✹✷

✹✸

✹✳✽ ❈♦♥❝❧✉sã♦

❈♦♥❝❧✉✐♠♦s q✉❡ é ♣♦ssí✈❡❧ ❝❛❧❝✉❧❛r ❛ ár❡❛ ❞❡ ✉♠❛ ✜❣✉r❛ ♥ã♦ ❡❧❡♠❡♥t❛r ✉t✐❧✐③❛♥❞♦

✜❣✉r❛s ❡❧❡♠❡♥t❛r❡s ❛tr❛✈és ❞♦ ▼ét♦❞♦ ❞❡ ❊①❛✉stã♦ ❛ss♦❝✐❛❞♦ à ár❡❛ ❞❡ ▲❡❜❡s❣✉❡✳

❈♦♥❥❡❝t✉r❛♠♦s q✉❡ t❛♥t♦ ❢❛③ ✉s❛r♠♦s tr✐â♥❣✉❧♦s ♦✉ q✉❛❞r❛❞♦s ♣♦✐s ♦ ✈❛❧♦r ❞❛

ár❡❛ s❡ ♠♦❞✐✜❝❛ ❝♦♠ ✉♠❛ ❞✐❢❡r❡♥ç❛ ♠✉✐t♦ ✐♥s✐❣♥✐✜❝❛♥t❡✳ P♦rt❛♥t♦ é ♣♦ssí✈❡❧

❛♣❧✐❝❛r♠♦s ❡ss❡ ♠ét♦❞♦ ♥♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧✱ ❡♥s✐♥♦ ♠é❞✐♦ ❡ t❛♠❜é♠ ❝♦♠♦

✉♠❛ ✐♥tr♦❞✉çã♦ ♣❛r❛ ✉♠ ❝✉rs♦ ❞❡ ♠❡❞✐❞❛ ♥♦ ❡♥s✐♥♦ s✉♣❡r✐♦r✳

✹✹

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s

❬✶❪ ❊❱❊❙✱ ❍✳ ❲✳ ■♥tr♦❞✉çã♦ à ❤✐stór✐❛ ❞❛ ♠❛t❡♠át✐❝❛✳ ❚r❛❞✳ ❍②❣✐♥♦ ❍✳

❉♦♠✐♥❣✉❡s✳ ❈❛♠♣✐♥❛s✿ ❊❞✐t♦r❛ ❞❛ ❯♥✐❝❛♠♣✱ ✷✵✵✹✳

❬✷❪ ❇❖❨❊❘✱ ❈❛r❧ ❇✳ ❍✐stór✐❛ ❞❛ ▼❛t❡♠át✐❝❛✳ ✷ ❡❞✳ ❙ã♦ P❛✉❧♦✿ ❊❞❣❛r❞ ❇❧ü✲

❝❤❡r✱ ✶✾✾✻✳

❬✸❪ ▲■▼❆✱ ❊❧♦♥ ▲✳ ▼❡❞✐❞❛ ❡ ❢♦r♠❛ ❡♠ ❣❡♦♠❡tr✐❛✳ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✿ ❙❇▼✱

✶✾✾✼✳

❬✹❪ ❇❆❘❇❖❙❆✱ ❏♦ã♦ ▲✳ ▼✳ ●❡♦♠❡tr✐❛ ❡✉❝❧✐❞✐❛♥❛ ♣❧❛♥❛✳ ✾ ❡❞✳ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✿

❙❇▼✱ ✷✵✵✻✳

❬✺❪ ❋❊❘◆❆◆❉❊❩✱ P✳ ❏✳ ▼❡❞✐❞❛ ❡ ✐♥t❡❣r❛çã♦✳ ✷ ❡❞✳ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✿ ❙❇▼✱

✶✾✾✻✳

❬✻❪ ❘❖◗❯❊✱ ❚❛t✐❛♥❛✳ ❍✐stór✐❛ ❞❛ ▼❛t❡♠át✐❝❛✿ ✉♠❛ ✈✐sã♦ ❝rít✐❝❛✱ ❞❡s❢❛✲

③❡♥❞♦ ♠✐t♦s ❡ ❧❡♥❞❛s✳ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✿ ❩❛❤❛r✳

❬✼❪ ▲■▼❆✱ ❊❧♦♥ ▲✳ ❆♥á❧✐s❡ ❘❡❛❧✱ ❱♦❧✉♠❡ ✶✳ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✿ ■▼P❆✱ ✶✾✽✾✳

✹✺